LMFDB - Embedded newform 112.12.i.c.65.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,12,Mod(65,112)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 12, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("112.65");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	112.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$86.0544362227$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + 1846 x^{10} + 9475 x^{9} + 2735534 x^{8} + 11305015 x^{7} + 1247863105 x^{6} + \cdots + 4089842896896 $ x^12 - x^11 + 1846x^10 + 9475x^9 + 2735534x^8 + 11305015x^7 + 1247863105x^6 + 8299724444x^5 + 456730004116x^4 + 1967575859568x^3 + 7281700309008x^2 + 5939576563968x + 4089842896896
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{3}\cdot 7^{6} $
Twist minimal:	no (minimal twist has level 7)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			65.4
Root			$-17.5066 + 30.3223i$ of defining polynomial
Character	$\chi$	$=$	112.65
Dual form			112.12.i.c.81.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(82.0189 + 142.061i) q^{3} +(1670.28 - 2893.01i) q^{5} +(26385.1 - 35793.2i) q^{7} +(75119.3 - 130110. i) q^{9} +O(q^{10})$	\(q+(82.0189 + 142.061i) q^{3} +(1670.28 - 2893.01i) q^{5} +(26385.1 - 35793.2i) q^{7} +(75119.3 - 130110. i) q^{9} +(178457. + 309097. i) q^{11} +2.06258e6 q^{13} +547978. q^{15} +(1.53495e6 + 2.65861e6i) q^{17} +(-6.51852e6 + 1.12904e7i) q^{19} +(7.24889e6 + 812572. i) q^{21} +(2.24585e7 - 3.88993e7i) q^{23} +(1.88344e7 + 3.26221e7i) q^{25} +5.37036e7 q^{27} -1.16423e8 q^{29} +(-1.09920e8 - 1.90386e8i) q^{31} +(-2.92737e7 + 5.07036e7i) q^{33} +(-5.94796e7 - 1.36117e8i) q^{35} +(-3.99465e7 + 6.91894e7i) q^{37} +(1.69171e8 + 2.93012e8i) q^{39} +9.01434e8 q^{41} -5.79780e8 q^{43} +(-2.50940e8 - 4.34642e8i) q^{45} +(5.34035e8 - 9.24976e8i) q^{47} +(-5.84980e8 - 1.88881e9i) q^{49} +(-2.51790e8 + 4.36113e8i) q^{51} +(2.00050e8 + 3.46497e8i) q^{53} +1.19229e9 q^{55} -2.13857e9 q^{57} +(2.26396e9 + 3.92130e9i) q^{59} +(-8.45655e8 + 1.46472e9i) q^{61} +(-2.67504e9 - 6.12174e9i) q^{63} +(3.44509e9 - 5.96707e9i) q^{65} +(4.36812e9 + 7.56581e9i) q^{67} +7.36810e9 q^{69} +5.61370e8 q^{71} +(4.25598e9 + 7.37158e9i) q^{73} +(-3.08955e9 + 5.35126e9i) q^{75} +(1.57722e10 + 1.76800e9i) q^{77} +(1.02217e10 - 1.77045e10i) q^{79} +(-8.90244e9 - 1.54195e10i) q^{81} +2.59465e8 q^{83} +1.02552e10 q^{85} +(-9.54890e9 - 1.65392e10i) q^{87} +(-4.21641e10 + 7.30304e10i) q^{89} +(5.44214e10 - 7.38264e10i) q^{91} +(1.80310e10 - 3.12305e10i) q^{93} +(2.17755e10 + 3.77163e10i) q^{95} +1.58145e11 q^{97} +5.36223e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 244 q^{3} - 8782 q^{5} + 504 q^{7} - 172348 q^{9}+O(q^{10}) $ 12 * q + 244 * q^3 - 8782 * q^5 + 504 * q^7 - 172348 * q^9	$ 12 q + 244 q^{3} - 8782 q^{5} + 504 q^{7} - 172348 q^{9} + 1001572 q^{11} + 3864504 q^{13} + 1286512 q^{15} - 6704802 q^{17} - 4192212 q^{19} + 44745358 q^{21} + 33871872 q^{23} + 13695456 q^{25} - 73859384 q^{27} - 255125224 q^{29} + 331783920 q^{31} - 80899438 q^{33} - 1407354844 q^{35} - 833082774 q^{37} - 737605904 q^{39} + 3104076808 q^{41} + 1722177552 q^{43} - 7406493484 q^{45} + 1327587552 q^{47} + 11976558636 q^{49} + 13921261140 q^{51} + 6725755626 q^{53} - 26323921200 q^{55} - 16884487756 q^{57} + 26237179548 q^{59} - 14411013726 q^{61} - 45955779184 q^{63} - 16224702172 q^{65} + 4241860068 q^{67} + 46750854252 q^{69} + 37335334656 q^{71} + 6005568990 q^{73} - 17116276792 q^{75} - 51928077698 q^{77} - 11712395640 q^{79} - 12455008366 q^{81} + 100821781200 q^{83} + 138884613396 q^{85} - 119455310144 q^{87} - 48633519778 q^{89} + 160908361488 q^{91} + 266530114134 q^{93} + 72161225128 q^{95} - 401308415928 q^{97} - 367357472240 q^{99}+O(q^{100}) $ 12 * q + 244 * q^3 - 8782 * q^5 + 504 * q^7 - 172348 * q^9 + 1001572 * q^11 + 3864504 * q^13 + 1286512 * q^15 - 6704802 * q^17 - 4192212 * q^19 + 44745358 * q^21 + 33871872 * q^23 + 13695456 * q^25 - 73859384 * q^27 - 255125224 * q^29 + 331783920 * q^31 - 80899438 * q^33 - 1407354844 * q^35 - 833082774 * q^37 - 737605904 * q^39 + 3104076808 * q^41 + 1722177552 * q^43 - 7406493484 * q^45 + 1327587552 * q^47 + 11976558636 * q^49 + 13921261140 * q^51 + 6725755626 * q^53 - 26323921200 * q^55 - 16884487756 * q^57 + 26237179548 * q^59 - 14411013726 * q^61 - 45955779184 * q^63 - 16224702172 * q^65 + 4241860068 * q^67 + 46750854252 * q^69 + 37335334656 * q^71 + 6005568990 * q^73 - 17116276792 * q^75 - 51928077698 * q^77 - 11712395640 * q^79 - 12455008366 * q^81 + 100821781200 * q^83 + 138884613396 * q^85 - 119455310144 * q^87 - 48633519778 * q^89 + 160908361488 * q^91 + 266530114134 * q^93 + 72161225128 * q^95 - 401308415928 * q^97 - 367357472240 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	82.0189	+	142.061i	0.194871	+	0.337526i	0.946858	−	0.321651i	$-0.104238\pi$
$3$	82.0189	+	142.061i	0.194871	+	0.337526i	−0.751987	+	0.659178i	$0.770905\pi$
$4$		0			0
$5$	1670.28	−	2893.01i	0.239031	−	0.414014i	−0.721406	−	0.692513i	$-0.756504\pi$
$5$	1670.28	−	2893.01i	0.239031	−	0.414014i	0.960436	+	0.278499i	$0.0898369\pi$
$6$		0			0
$7$	26385.1	−	35793.2i	0.593362	−	0.804936i
$7$	26385.1	−	35793.2i	0.593362	−	0.804936i
$8$		0			0
$9$	75119.3	−	130110.i	0.424051	−	0.734477i
$10$		0			0
$11$	178457.	+	309097.i	0.334098	+	0.578675i	0.983311	−	0.181931i	$-0.0582349\pi$
$11$	178457.	+	309097.i	0.334098	+	0.578675i	−0.649213	+	0.760607i	$0.724902\pi$
$12$		0			0
$13$	2.06258e6			1.54072			0.770359	−	0.637611i	$-0.220077\pi$
$13$	2.06258e6			1.54072			0.770359	+	0.637611i	$0.220077\pi$
$14$		0			0
$15$	547978.			0.186321
$16$		0			0
$17$	1.53495e6	+	2.65861e6i	0.262196	+	0.454136i	0.966825	−	0.255439i	$-0.0822201\pi$
$17$	1.53495e6	+	2.65861e6i	0.262196	+	0.454136i	−0.704630	+	0.709575i	$0.748887\pi$
$18$		0			0
$19$	−6.51852e6	+	1.12904e7i	−0.603955	+	1.04608i	0.388261	+	0.921549i	$0.373076\pi$
$19$	−6.51852e6	+	1.12904e7i	−0.603955	+	1.04608i	−0.992216	+	0.124531i	$0.960258\pi$
$20$		0			0
$21$	7.24889e6	+	812572.i	0.387316	+	0.0434166i
$22$		0			0
$23$	2.24585e7	−	3.88993e7i	0.727576	−	1.26020i	−0.230329	−	0.973113i	$-0.573980\pi$
$23$	2.24585e7	−	3.88993e7i	0.727576	−	1.26020i	0.957905	−	0.287086i	$-0.0926866\pi$
$24$		0			0
$25$	1.88344e7	+	3.26221e7i	0.385728	+	0.668101i
$26$		0			0
$27$	5.37036e7			0.720283
$28$		0			0
$29$	−1.16423e8			−1.05402			−0.527012	−	0.849858i	$-0.676688\pi$
$29$	−1.16423e8			−1.05402			−0.527012	+	0.849858i	$0.676688\pi$
$30$		0			0
$31$	−1.09920e8	−	1.90386e8i	−0.689581	−	1.19439i	−0.971973	−	0.235091i	$-0.924461\pi$
$31$	−1.09920e8	−	1.90386e8i	−0.689581	−	1.19439i	0.282392	−	0.959299i	$-0.408872\pi$
$32$		0			0
$33$	−2.92737e7	+	5.07036e7i	−0.130212	+	0.225534i
$34$		0			0
$35$	−5.94796e7	−	1.36117e8i	−0.191423	−	0.438064i
$36$		0			0
$37$	−3.99465e7	+	6.91894e7i	−0.0947042	+	0.164032i	−0.909485	−	0.415737i	$-0.863524\pi$
$37$	−3.99465e7	+	6.91894e7i	−0.0947042	+	0.164032i	0.814781	+	0.579769i	$0.196857\pi$
$38$		0			0
$39$	1.69171e8	+	2.93012e8i	0.300241	+	0.520033i
$40$		0			0
$41$	9.01434e8			1.21513			0.607565	−	0.794270i	$-0.292146\pi$
$41$	9.01434e8			1.21513			0.607565	+	0.794270i	$0.292146\pi$
$42$		0			0
$43$	−5.79780e8			−0.601432			−0.300716	−	0.953714i	$-0.597226\pi$
$43$	−5.79780e8			−0.601432			−0.300716	+	0.953714i	$0.597226\pi$
$44$		0			0
$45$	−2.50940e8	−	4.34642e8i	−0.202722	−	0.351126i
$46$		0			0
$47$	5.34035e8	−	9.24976e8i	0.339650	−	0.588291i	−0.644717	−	0.764422i	$-0.723025\pi$
$47$	5.34035e8	−	9.24976e8i	0.339650	−	0.588291i	0.984367	+	0.176130i	$0.0563580\pi$
$48$		0			0
$49$	−5.84980e8	−	1.88881e9i	−0.295844	−	0.955236i
$50$		0			0
$51$	−2.51790e8	+	4.36113e8i	−0.102189	+	0.176996i
$52$		0			0
$53$	2.00050e8	+	3.46497e8i	0.0657086	+	0.113811i	0.897008	−	0.442014i	$-0.145736\pi$
$53$	2.00050e8	+	3.46497e8i	0.0657086	+	0.113811i	−0.831300	+	0.555825i	$0.812403\pi$
$54$		0			0
$55$	1.19229e9			0.319439
$56$		0			0
$57$	−2.13857e9			−0.470773
$58$		0			0
$59$	2.26396e9	+	3.92130e9i	0.412271	+	0.714075i	0.995138	−	0.0984933i	$-0.0314023\pi$
$59$	2.26396e9	+	3.92130e9i	0.412271	+	0.714075i	−0.582867	+	0.812568i	$0.698069\pi$
$60$		0			0
$61$	−8.45655e8	+	1.46472e9i	−0.128197	+	0.222044i	−0.922978	−	0.384852i	$-0.874252\pi$
$61$	−8.45655e8	+	1.46472e9i	−0.128197	+	0.222044i	0.794781	+	0.606897i	$0.207586\pi$
$62$		0			0
$63$	−2.67504e9	−	6.12174e9i	−0.339592	−	0.777144i
$64$		0			0
$65$	3.44509e9	−	5.96707e9i	0.368279	−	0.637878i
$66$		0			0
$67$	4.36812e9	+	7.56581e9i	0.395260	+	0.684611i	0.993134	−	0.116979i	$-0.0373211\pi$
$67$	4.36812e9	+	7.56581e9i	0.395260	+	0.684611i	−0.597874	+	0.801590i	$0.703988\pi$
$68$		0			0
$69$	7.36810e9			0.567134
$70$		0			0
$71$	5.61370e8			0.0369256			0.0184628	−	0.999830i	$-0.494123\pi$
$71$	5.61370e8			0.0369256			0.0184628	+	0.999830i	$0.494123\pi$
$72$		0			0
$73$	4.25598e9	+	7.37158e9i	0.240284	+	0.416183i	0.960795	−	0.277260i	$-0.0894262\pi$
$73$	4.25598e9	+	7.37158e9i	0.240284	+	0.416183i	−0.720511	+	0.693443i	$0.756093\pi$
$74$		0			0
$75$	−3.08955e9	+	5.35126e9i	−0.150335	+	0.260387i
$76$		0			0
$77$	1.57722e10	+	1.76800e9i	0.664038	+	0.0744360i
$78$		0			0
$79$	1.02217e10	−	1.77045e10i	0.373743	−	0.647342i	−0.616395	−	0.787437i	$-0.711407\pi$
$79$	1.02217e10	−	1.77045e10i	0.373743	−	0.647342i	0.990138	+	0.140095i	$0.0447408\pi$
$80$		0			0
$81$	−8.90244e9	−	1.54195e10i	−0.283688	−	0.491363i
$82$		0			0
$83$	2.59465e8			0.00723018			0.00361509	−	0.999993i	$-0.498849\pi$
$83$	2.59465e8			0.00723018			0.00361509	+	0.999993i	$0.498849\pi$
$84$		0			0
$85$	1.02552e10			0.250691
$86$		0			0
$87$	−9.54890e9	−	1.65392e10i	−0.205398	−	0.355761i
$88$		0			0
$89$	−4.21641e10	+	7.30304e10i	−0.800384	+	1.38631i	0.118980	+	0.992897i	$0.462038\pi$
$89$	−4.21641e10	+	7.30304e10i	−0.800384	+	1.38631i	−0.919364	+	0.393408i	$0.871296\pi$
$90$		0			0
$91$	5.44214e10	−	7.38264e10i	0.914202	−	1.24018i
$92$		0			0
$93$	1.80310e10	−	3.12305e10i	0.268759	−	0.465504i
$94$		0			0
$95$	2.17755e10	+	3.77163e10i	0.288728	+	0.500091i
$96$		0			0
$97$	1.58145e11			1.86987			0.934933	−	0.354823i	$-0.115459\pi$
$97$	1.58145e11			1.86987			0.934933	+	0.354823i	$0.115459\pi$
$98$		0			0
$99$	5.36223e10			0.566699
$100$		0			0
$101$	−7.61940e10	−	1.31972e11i	−0.721362	−	1.24944i	−0.960454	−	0.278439i	$-0.910183\pi$
$101$	−7.61940e10	−	1.31972e11i	−0.721362	−	1.24944i	0.239092	−	0.970997i	$-0.423150\pi$
$102$		0			0
$103$	6.31803e10	−	1.09432e11i	0.537004	−	0.930117i	−0.462060	−	0.886849i	$-0.652890\pi$
$103$	6.31803e10	−	1.09432e11i	0.537004	−	0.930117i	0.999063	−	0.0432687i	$-0.0137772\pi$
$104$		0			0
$105$	1.44585e10	−	1.96139e10i	0.110556	−	0.149976i
$106$		0			0
$107$	−6.89604e10	+	1.19443e11i	−0.475323	+	0.823284i	−0.999601	−	0.0282634i	$-0.991002\pi$
$107$	−6.89604e10	+	1.19443e11i	−0.475323	+	0.823284i	0.524277	+	0.851548i	$0.324336\pi$
$108$		0			0
$109$	−4.33107e10	−	7.50164e10i	−0.269619	−	0.466993i	0.699145	−	0.714980i	$-0.253564\pi$
$109$	−4.33107e10	−	7.50164e10i	−0.269619	−	0.466993i	−0.968763	+	0.247987i	$0.920231\pi$
$110$		0			0
$111$	−1.31055e10			−0.0738204
$112$		0			0
$113$	1.53430e11			0.783389			0.391695	−	0.920095i	$-0.371889\pi$
$113$	1.53430e11			0.783389			0.391695	+	0.920095i	$0.371889\pi$
$114$		0			0
$115$	−7.50241e10	−	1.29946e11i	−0.347826	−	0.602453i
$116$		0			0
$117$	1.54940e11	−	2.68364e11i	0.653342	−	1.13162i
$118$		0			0
$119$	1.35660e11	+	1.52070e10i	0.521127	+	0.0584163i
$120$		0			0
$121$	7.89618e10	−	1.36766e11i	0.276756	−	0.479356i
$122$		0			0
$123$	7.39346e10	+	1.28059e11i	0.236793	+	0.410138i
$124$		0			0
$125$	2.88948e11			0.846866
$126$		0			0
$127$	5.58699e11			1.50057			0.750287	−	0.661112i	$-0.229915\pi$
$127$	5.58699e11			1.50057			0.750287	+	0.661112i	$0.229915\pi$
$128$		0			0
$129$	−4.75529e10	−	8.23640e10i	−0.117202	−	0.202999i
$130$		0			0
$131$	3.74086e11	−	6.47936e11i	0.847187	−	1.46737i	−0.0365215	−	0.999333i	$-0.511628\pi$
$131$	3.74086e11	−	6.47936e11i	0.847187	−	1.46737i	0.883708	−	0.468038i	$-0.155039\pi$
$132$		0			0
$133$	2.32128e11	+	5.31217e11i	0.483664	+	1.10685i
$134$		0			0
$135$	8.97001e10	−	1.55365e11i	0.172170	−	0.298207i
$136$		0			0
$137$	−1.38619e11	−	2.40095e11i	−0.245391	−	0.425030i	0.716850	−	0.697227i	$-0.245583\pi$
$137$	−1.38619e11	−	2.40095e11i	−0.245391	−	0.425030i	−0.962241	+	0.272197i	$0.912250\pi$
$138$		0			0
$139$	−1.59443e11			−0.260630			−0.130315	−	0.991473i	$-0.541599\pi$
$139$	−1.59443e11			−0.260630			−0.130315	+	0.991473i	$0.541599\pi$
$140$		0			0
$141$	1.75204e11			0.264752
$142$		0			0
$143$	3.68083e11	+	6.37538e11i	0.514751	+	0.891575i
$144$		0			0
$145$	−1.94459e11	+	3.36813e11i	−0.251944	+	0.436380i
$146$		0			0
$147$	2.20347e11	−	2.38021e11i	0.264766	−	0.286003i
$148$		0			0
$149$	2.23936e11	−	3.87868e11i	0.249804	−	0.432673i	−0.713668	−	0.700485i	$-0.752967\pi$
$149$	2.23936e11	−	3.87868e11i	0.249804	−	0.432673i	0.963471	+	0.267812i	$0.0863006\pi$
$150$		0			0
$151$	−7.28136e11	−	1.26117e12i	−0.754813	−	1.30737i	−0.945467	−	0.325718i	$-0.894394\pi$
$151$	−7.28136e11	−	1.26117e12i	−0.754813	−	1.30737i	0.190654	−	0.981657i	$-0.438939\pi$
$152$		0			0
$153$	4.61218e11			0.444737
$154$		0			0
$155$	−7.34385e11			−0.659325
$156$		0			0
$157$	3.16025e11	+	5.47372e11i	0.264407	+	0.457967i	0.967408	−	0.253222i	$-0.0814903\pi$
$157$	3.16025e11	+	5.47372e11i	0.264407	+	0.457967i	−0.703001	+	0.711189i	$0.748157\pi$
$158$		0			0
$159$	−3.28158e10	+	5.68387e10i	−0.0256094	+	0.0443568i
$160$		0			0
$161$	−7.99761e11	−	1.83023e12i	−0.582664	−	1.33341i
$162$		0			0
$163$	1.44029e11	−	2.49466e11i	0.0980436	−	0.169817i	−0.812831	−	0.582499i	$-0.802075\pi$
$163$	1.44029e11	−	2.49466e11i	0.0980436	−	0.169817i	0.910875	+	0.412683i	$0.135408\pi$
$164$		0			0
$165$	9.77907e10	+	1.69378e11i	0.0622495	+	0.107819i
$166$		0			0
$167$	3.87053e11			0.230584			0.115292	−	0.993332i	$-0.463220\pi$
$167$	3.87053e11			0.230584			0.115292	+	0.993332i	$0.463220\pi$
$168$		0			0
$169$	2.46209e12			1.37381
$170$		0			0
$171$	9.79333e11	+	1.69626e12i	0.512215	+	0.887182i
$172$		0			0
$173$	−4.92857e11	+	8.53654e11i	−0.241806	+	0.418821i	−0.961229	−	0.275752i	$-0.911073\pi$
$173$	−4.92857e11	+	8.53654e11i	−0.241806	+	0.418821i	0.719423	+	0.694573i	$0.244407\pi$
$174$		0			0
$175$	1.66460e12	+	1.86595e11i	0.766655	+	0.0859390i
$176$		0			0
$177$	−3.71375e11	+	6.43241e11i	−0.160679	+	0.278305i
$178$		0			0
$179$	1.14907e11	+	1.99024e11i	0.0467362	+	0.0809495i	0.888447	−	0.458979i	$-0.151785\pi$
$179$	1.14907e11	+	1.99024e11i	0.0467362	+	0.0809495i	−0.841711	+	0.539928i	$0.818451\pi$
$180$		0			0
$181$	−2.49612e12			−0.955066			−0.477533	−	0.878614i	$-0.658469\pi$
$181$	−2.49612e12			−0.955066			−0.477533	+	0.878614i	$0.658469\pi$
$182$		0			0
$183$	−2.77439e11			−0.0999278
$184$		0			0
$185$	1.33444e11	+	2.31131e11i	0.0452745	+	0.0784177i
$186$		0			0
$187$	−5.47846e11	+	9.48897e11i	−0.175198	+	0.303452i
$188$		0			0
$189$	1.41698e12	−	1.92222e12i	0.427388	−	0.579781i
$190$		0			0
$191$	−4.57871e10	+	7.93056e10i	−0.0130335	+	0.0225746i	−0.872469	−	0.488670i	$-0.837482\pi$
$191$	−4.57871e10	+	7.93056e10i	−0.0130335	+	0.0225746i	0.859435	+	0.511245i	$0.170815\pi$
$192$		0			0
$193$	8.49236e10	+	1.47092e11i	0.0228277	+	0.0395388i	0.877214	−	0.480100i	$-0.159400\pi$
$193$	8.49236e10	+	1.47092e11i	0.0228277	+	0.0395388i	−0.854386	+	0.519639i	$0.826066\pi$
$194$		0			0
$195$	1.13025e12			0.287068
$196$		0			0
$197$	−1.89808e12			−0.455774			−0.227887	−	0.973688i	$-0.573182\pi$
$197$	−1.89808e12			−0.455774			−0.227887	+	0.973688i	$0.573182\pi$
$198$		0			0
$199$	−5.35507e11	−	9.27526e11i	−0.121639	−	0.210685i	0.798775	−	0.601630i	$-0.205482\pi$
$199$	−5.35507e11	−	9.27526e11i	−0.121639	−	0.210685i	−0.920414	+	0.390945i	$0.872148\pi$
$200$		0			0
$201$	−7.16537e11	+	1.24108e12i	−0.154050	+	0.266822i
$202$		0			0
$203$	−3.07183e12	+	4.16716e12i	−0.625417	+	0.848421i
$204$		0			0
$205$	1.50565e12	−	2.60786e12i	0.290454	−	0.503080i
$206$		0			0
$207$	−3.37414e12	−	5.84418e12i	−0.617058	−	1.06878i
$208$		0			0
$209$	−4.65311e12			−0.807121
$210$		0			0
$211$	−9.82737e12			−1.61765			−0.808823	−	0.588052i	$-0.799895\pi$
$211$	−9.82737e12			−1.61765			−0.808823	+	0.588052i	$0.799895\pi$
$212$		0			0
$213$	4.60429e10	+	7.97487e10i	0.00719573	+	0.0124634i
$214$		0			0
$215$	−9.68394e11	+	1.67731e12i	−0.143761	+	0.249001i
$216$		0			0
$217$	−9.71477e12	−	1.08899e12i	−1.37058	−	0.153636i
$218$		0			0
$219$	−6.98142e11	+	1.20922e12i	−0.0936486	+	0.162204i
$220$		0			0
$221$	3.16596e12	+	5.48361e12i	0.403969	+	0.699695i
$222$		0			0
$223$	−1.43843e12			−0.174667			−0.0873337	−	0.996179i	$-0.527835\pi$
$223$	−1.43843e12			−0.174667			−0.0873337	+	0.996179i	$0.527835\pi$
$224$		0			0
$225$	5.65931e12			0.654274
$226$		0			0
$227$	2.24180e12	+	3.88291e12i	0.246862	+	0.427577i	0.962653	−	0.270737i	$-0.0872672\pi$
$227$	2.24180e12	+	3.88291e12i	0.246862	+	0.427577i	−0.715791	+	0.698314i	$0.753934\pi$
$228$		0			0
$229$	−7.53005e12	+	1.30424e13i	−0.790138	+	1.36856i	0.135743	+	0.990744i	$0.456658\pi$
$229$	−7.53005e12	+	1.30424e13i	−0.790138	+	1.36856i	−0.925881	+	0.377815i	$0.876676\pi$
$230$		0			0
$231$	1.04245e12	+	2.38562e12i	0.104278	+	0.238636i
$232$		0			0
$233$	−7.45866e12	+	1.29188e13i	−0.711546	+	1.23243i	0.252730	+	0.967537i	$0.418671\pi$
$233$	−7.45866e12	+	1.29188e13i	−0.711546	+	1.23243i	−0.964277	+	0.264898i	$0.914662\pi$
$234$		0			0
$235$	−1.78398e12	−	3.08994e12i	−0.162374	−	0.281240i
$236$		0			0
$237$	3.35348e12			0.291327
$238$		0			0
$239$	6.45438e12			0.535385			0.267693	−	0.963504i	$-0.413739\pi$
$239$	6.45438e12			0.535385			0.267693	+	0.963504i	$0.413739\pi$
$240$		0			0
$241$	−3.10091e12	−	5.37093e12i	−0.245694	−	0.425555i	0.716632	−	0.697451i	$-0.245683\pi$
$241$	−3.10091e12	−	5.37093e12i	−0.245694	−	0.425555i	−0.962327	+	0.271896i	$0.912349\pi$
$242$		0			0
$243$	6.21706e12	−	1.07683e13i	0.470707	−	0.815288i
$244$		0			0
$245$	−6.44144e12	−	1.46250e12i	−0.466197	−	0.105848i
$246$		0			0
$247$	−1.34450e13	+	2.32874e13i	−0.930523	+	1.61171i
$248$		0			0
$249$	2.12810e10	+	3.68598e10i	0.00140895	+	0.00244038i
$250$		0			0
$251$	1.07963e12			0.0684023			0.0342012	−	0.999415i	$-0.489111\pi$
$251$	1.07963e12			0.0684023			0.0342012	+	0.999415i	$0.489111\pi$
$252$		0			0
$253$	1.60316e13			0.972328
$254$		0			0
$255$	8.41119e11	+	1.45686e12i	0.0488525	+	0.0846150i
$256$		0			0
$257$	3.23723e12	−	5.60705e12i	0.180112	−	0.311962i	−0.761807	−	0.647804i	$-0.775687\pi$
$257$	3.23723e12	−	5.60705e12i	0.180112	−	0.311962i	0.941918	+	0.335842i	$0.109021\pi$
$258$		0			0
$259$	1.42252e12	+	3.25538e12i	0.0758418	+	0.173561i
$260$		0			0
$261$	−8.74562e12	+	1.51479e13i	−0.446959	+	0.774156i
$262$		0			0
$263$	1.54944e12	+	2.68370e12i	0.0759306	+	0.131516i	0.901491	−	0.432799i	$-0.142474\pi$
$263$	1.54944e12	+	2.68370e12i	0.0759306	+	0.131516i	−0.825560	+	0.564314i	$0.809141\pi$
$264$		0			0
$265$	1.33656e12			0.0628255
$266$		0			0
$267$	−1.38330e13			−0.623886
$268$		0			0
$269$	−1.35603e13	−	2.34872e13i	−0.586993	−	1.01670i	−0.994624	−	0.103555i	$-0.966978\pi$
$269$	−1.35603e13	−	2.34872e13i	−0.586993	−	1.01670i	0.407631	−	0.913147i	$-0.366355\pi$
$270$		0			0
$271$	−1.05204e13	+	1.82219e13i	−0.437221	+	0.757290i	−0.997474	−	0.0710322i	$-0.977371\pi$
$271$	−1.05204e13	+	1.82219e13i	−0.437221	+	0.757290i	0.560253	+	0.828322i	$0.310704\pi$
$272$		0			0
$273$	1.49514e13	+	1.67600e12i	0.596745	+	0.0668927i
$274$		0			0
$275$	−6.72227e12	+	1.16433e13i	−0.257743	+	0.446423i
$276$		0			0
$277$	−1.02743e13	−	1.77955e13i	−0.378540	−	0.655651i	0.612310	−	0.790618i	$-0.290240\pi$
$277$	−1.02743e13	−	1.77955e13i	−0.378540	−	0.655651i	−0.990850	+	0.134967i	$0.956907\pi$
$278$		0			0
$279$	−3.30283e13			−1.16967
$280$		0			0
$281$	−4.62210e13			−1.57382			−0.786910	−	0.617068i	$-0.788320\pi$
$281$	−4.62210e13			−1.57382			−0.786910	+	0.617068i	$0.788320\pi$
$282$		0			0
$283$	−4.43844e12	−	7.68760e12i	−0.145347	−	0.251748i	0.784156	−	0.620564i	$-0.213096\pi$
$283$	−4.43844e12	−	7.68760e12i	−0.145347	−	0.251748i	−0.929502	+	0.368817i	$0.879763\pi$
$284$		0			0
$285$	−3.57201e12	+	6.18690e12i	−0.112529	+	0.194906i
$286$		0			0
$287$	2.37844e13	−	3.22652e13i	0.721011	−	0.978102i
$288$		0			0
$289$	1.24238e13	−	2.15187e13i	0.362507	−	0.627881i
$290$		0			0
$291$	1.29709e13	+	2.24662e13i	0.364383	+	0.631129i
$292$		0			0
$293$	6.30880e13			1.70677			0.853385	−	0.521281i	$-0.174546\pi$
$293$	6.30880e13			1.70677			0.853385	+	0.521281i	$0.174546\pi$
$294$		0			0
$295$	1.51258e13			0.394182
$296$		0			0
$297$	9.58380e12	+	1.65996e13i	0.240645	+	0.416810i
$298$		0			0
$299$	4.63226e13	−	8.02331e13i	1.12099	−	1.94161i
$300$		0			0
$301$	−1.52975e13	+	2.07522e13i	−0.356867	+	0.484114i
$302$		0			0
$303$	1.24987e13	−	2.16484e13i	0.281145	−	0.486958i
$304$		0			0
$305$	2.82496e12	+	4.89298e12i	0.0612863	+	0.106151i
$306$		0			0
$307$	4.58598e13			0.959778			0.479889	−	0.877329i	$-0.340677\pi$
$307$	4.58598e13			0.959778			0.479889	+	0.877329i	$0.340677\pi$
$308$		0			0
$309$	2.07279e13			0.418586
$310$		0			0
$311$	2.02229e13	+	3.50271e13i	0.394150	+	0.682687i	0.992992	−	0.118179i	$-0.0377058\pi$
$311$	2.02229e13	+	3.50271e13i	0.394150	+	0.682687i	−0.598843	+	0.800867i	$0.704372\pi$
$312$		0			0
$313$	2.74355e12	−	4.75197e12i	0.0516201	−	0.0894087i	−0.839061	−	0.544038i	$-0.816895\pi$
$313$	2.74355e12	−	4.75197e12i	0.0516201	−	0.0894087i	0.890681	+	0.454629i	$0.150228\pi$
$314$		0			0
$315$	−2.21783e13	−	2.48610e12i	−0.402921	−	0.0451659i
$316$		0			0
$317$	2.71013e13	−	4.69408e13i	0.475514	−	0.823615i	−0.524092	−	0.851662i	$-0.675595\pi$
$317$	2.71013e13	−	4.69408e13i	0.475514	−	0.823615i	0.999607	+	0.0280464i	$0.00892862\pi$
$318$		0			0
$319$	−2.07765e13	−	3.59860e13i	−0.352147	−	0.609937i
$320$		0			0
$321$	−2.26242e13			−0.370507
$322$		0			0
$323$	−4.00224e13			−0.633417
$324$		0			0
$325$	3.88475e13	+	6.72858e13i	0.594298	+	1.02936i
$326$		0			0
$327$	7.10460e12	−	1.23055e13i	0.105082	−	0.182007i
$328$		0			0
$329$	−1.90173e13	−	4.35204e13i	−0.272001	−	0.622466i
$330$		0			0
$331$	8.92055e12	−	1.54509e13i	0.123406	−	0.213746i	−0.797702	−	0.603051i	$-0.793951\pi$
$331$	8.92055e12	−	1.54509e13i	0.123406	−	0.213746i	0.921109	+	0.389305i	$0.127285\pi$
$332$		0			0
$333$	6.00150e12	+	1.03949e13i	0.0803187	+	0.139116i
$334$		0			0
$335$	2.91839e13			0.377918
$336$		0			0
$337$	1.60124e12			0.0200674			0.0100337	−	0.999950i	$-0.496806\pi$
$337$	1.60124e12			0.0200674			0.0100337	+	0.999950i	$0.496806\pi$
$338$		0			0
$339$	1.25841e13	+	2.17964e13i	0.152660	+	0.264415i
$340$		0			0
$341$	3.92319e13	−	6.79516e13i	0.460776	−	0.798088i
$342$		0			0
$343$	−8.30415e13	−	2.88982e13i	−0.944446	−	0.328665i
$344$		0			0
$345$	1.23068e13	−	2.13160e13i	0.135563	−	0.234801i
$346$		0			0
$347$	5.89343e13	+	1.02077e14i	0.628863	+	1.08922i	0.987780	+	0.155853i	$0.0498126\pi$
$347$	5.89343e13	+	1.02077e14i	0.628863	+	1.08922i	−0.358918	+	0.933369i	$0.616854\pi$
$348$		0			0
$349$	1.24860e14			1.29087			0.645436	−	0.763815i	$-0.276676\pi$
$349$	1.24860e14			1.29087			0.645436	+	0.763815i	$0.276676\pi$
$350$		0			0
$351$	1.10768e14			1.10975
$352$		0			0
$353$	−1.78120e13	−	3.08514e13i	−0.172963	−	0.299581i	0.766491	−	0.642254i	$-0.222001\pi$
$353$	−1.78120e13	−	3.08514e13i	−0.172963	−	0.299581i	−0.939454	+	0.342674i	$0.888667\pi$
$354$		0			0
$355$	9.37644e11	−	1.62405e12i	0.00882637	−	0.0152877i
$356$		0			0
$357$	8.96638e12	+	2.05193e13i	0.0818355	+	0.187278i
$358$		0			0
$359$	5.26831e13	−	9.12498e13i	0.466285	−	0.807630i	−0.532973	−	0.846132i	$-0.678925\pi$
$359$	5.26831e13	−	9.12498e13i	0.466285	−	0.807630i	0.999259	+	0.0385022i	$0.0122587\pi$
$360$		0			0
$361$	−2.67371e13	−	4.63100e13i	−0.229522	−	0.397544i
$362$		0			0
$363$	2.59055e13			0.215727
$364$		0			0
$365$	2.84347e13			0.229741
$366$		0			0
$367$	−5.06480e13	−	8.77250e13i	−0.397099	−	0.687796i	0.596267	−	0.802786i	$-0.296650\pi$
$367$	−5.06480e13	−	8.77250e13i	−0.397099	−	0.687796i	−0.993367	+	0.114990i	$0.963317\pi$
$368$		0			0
$369$	6.77151e13	−	1.17286e14i	0.515276	−	0.892485i
$370$		0			0
$371$	1.76806e13	+	1.98192e12i	0.130599	+	0.0146396i
$372$		0			0
$373$	−6.47411e13	+	1.12135e14i	−0.464282	+	0.804160i	−0.999169	−	0.0407640i	$-0.987021\pi$
$373$	−6.47411e13	+	1.12135e14i	−0.464282	+	0.804160i	0.534887	+	0.844924i	$0.320354\pi$
$374$		0			0
$375$	2.36992e13	+	4.10482e13i	0.165030	+	0.285840i
$376$		0			0
$377$	−2.40132e14			−1.62395
$378$		0			0
$379$	1.53338e14			1.00724			0.503620	−	0.863925i	$-0.332001\pi$
$379$	1.53338e14			1.00724			0.503620	+	0.863925i	$0.332001\pi$
$380$		0			0
$381$	4.58239e13	+	7.93693e13i	0.292418	+	0.506483i
$382$		0			0
$383$	−1.32219e14	+	2.29010e14i	−0.819785	+	1.41991i	0.0860559	+	0.996290i	$0.472574\pi$
$383$	−1.32219e14	+	2.29010e14i	−0.819785	+	1.41991i	−0.905841	+	0.423619i	$0.860760\pi$
$384$		0			0
$385$	3.14588e13	−	4.26760e13i	0.189543	−	0.257128i
$386$		0			0
$387$	−4.35526e13	+	7.54354e13i	−0.255038	+	0.441738i
$388$		0			0
$389$	3.65688e13	+	6.33390e13i	0.208155	+	0.360536i	0.951133	−	0.308780i	$-0.0999207\pi$
$389$	3.65688e13	+	6.33390e13i	0.208155	+	0.360536i	−0.742978	+	0.669316i	$0.766587\pi$
$390$		0			0
$391$	1.37891e14			0.763069
$392$		0			0
$393$	1.22728e14			0.660369
$394$		0			0
$395$	−3.41461e13	−	5.91428e13i	−0.178672	−	0.309470i
$396$		0			0
$397$	4.45211e13	−	7.71128e13i	0.226578	−	0.392445i	−0.730213	−	0.683219i	$-0.760579\pi$
$397$	4.45211e13	−	7.71128e13i	0.226578	−	0.392445i	0.956792	+	0.290774i	$0.0939128\pi$
$398$		0			0
$399$	−5.64263e13	+	7.65462e13i	−0.279339	+	0.378942i
$400$		0			0
$401$	−6.38630e13	+	1.10614e14i	−0.307578	+	0.532740i	−0.977832	−	0.209391i	$-0.932852\pi$
$401$	−6.38630e13	+	1.10614e14i	−0.307578	+	0.532740i	0.670254	+	0.742132i	$0.266185\pi$
$402$		0			0
$403$	−2.26718e14	−	3.92687e14i	−1.06245	−	1.84022i
$404$		0			0
$405$	−5.94783e13			−0.271241
$406$		0			0
$407$	−2.85150e13			−0.126562
$408$		0			0
$409$	−1.34921e14	−	2.33691e14i	−0.582912	−	1.00963i	−0.995132	−	0.0985487i	$-0.968580\pi$
$409$	−1.34921e14	−	2.33691e14i	−0.582912	−	1.00963i	0.412220	−	0.911084i	$-0.364753\pi$
$410$		0			0
$411$	2.27387e13	−	3.93846e13i	0.0956392	−	0.165652i
$412$		0			0
$413$	2.00091e14	+	2.24294e13i	0.819410	+	0.0918526i
$414$		0			0
$415$	4.33379e11	−	7.50635e11i	0.00172824	−	0.00299339i
$416$		0			0
$417$	−1.30774e13	−	2.26507e13i	−0.0507893	−	0.0879696i
$418$		0			0
$419$	−2.34628e14			−0.887571			−0.443785	−	0.896133i	$-0.646365\pi$
$419$	−2.34628e14			−0.887571			−0.443785	+	0.896133i	$0.646365\pi$
$420$		0			0
$421$	−8.33234e12			−0.0307055			−0.0153527	−	0.999882i	$-0.504887\pi$
$421$	−8.33234e12			−0.0307055			−0.0153527	+	0.999882i	$0.504887\pi$
$422$		0			0
$423$	−8.02327e13	−	1.38967e14i	−0.288058	−	0.498931i
$424$		0			0
$425$	−5.78197e13	+	1.00147e14i	−0.202273	+	0.350346i
$426$		0			0
$427$	3.01143e13	+	6.89154e13i	0.102664	+	0.234943i
$428$		0			0
$429$	−6.03795e13	+	1.04580e14i	−0.200620	+	0.347484i
$430$		0			0
$431$	1.31729e14	+	2.28162e14i	0.426636	+	0.738955i	0.996572	−	0.0827346i	$-0.0263654\pi$
$431$	1.31729e14	+	2.28162e14i	0.426636	+	0.738955i	−0.569936	+	0.821689i	$0.693032\pi$
$432$		0			0
$433$	−2.82761e14			−0.892762			−0.446381	−	0.894843i	$-0.647287\pi$
$433$	−2.82761e14			−0.892762			−0.446381	+	0.894843i	$0.647287\pi$
$434$		0			0
$435$	−6.37973e13			−0.196386
$436$		0			0
$437$	2.92793e14	+	5.07132e14i	0.878846	+	1.52221i
$438$		0			0
$439$	−9.69318e12	+	1.67891e13i	−0.0283734	+	0.0491442i	−0.879863	−	0.475227i	$-0.842366\pi$
$439$	−9.69318e12	+	1.67891e13i	−0.0283734	+	0.0491442i	0.851490	+	0.524371i	$0.175699\pi$
$440$		0			0
$441$	−2.89698e14	−	6.57744e13i	−0.827052	−	0.187778i
$442$		0			0
$443$	−3.49344e13	+	6.05082e13i	−0.0972821	+	0.168498i	−0.910559	−	0.413380i	$-0.864348\pi$
$443$	−3.49344e13	+	6.05082e13i	−0.0972821	+	0.168498i	0.813277	+	0.581877i	$0.197682\pi$
$444$		0			0
$445$	1.40852e14	+	2.43962e14i	0.382633	+	0.662740i
$446$		0			0
$447$	7.34678e13			0.194718
$448$		0			0
$449$	−7.31084e14			−1.89065			−0.945327	−	0.326124i	$-0.894257\pi$
$449$	−7.31084e14			−1.89065			−0.945327	+	0.326124i	$0.894257\pi$
$450$		0			0
$451$	1.60867e14	+	2.78631e14i	0.405973	+	0.703166i
$452$		0			0
$453$	1.19442e14	−	2.06879e14i	0.294182	−	0.509539i
$454$		0			0
$455$	−1.22682e14	−	2.80752e14i	−0.294928	−	0.674933i
$456$		0			0
$457$	−9.23160e13	+	1.59896e14i	−0.216640	+	0.375231i	−0.953779	−	0.300510i	$-0.902843\pi$
$457$	−9.23160e13	+	1.59896e14i	−0.216640	+	0.375231i	0.737139	+	0.675741i	$0.236176\pi$
$458$		0			0
$459$	8.24324e13	+	1.42777e14i	0.188855	+	0.327106i
$460$		0			0
$461$	−1.57609e14			−0.352554			−0.176277	−	0.984341i	$-0.556405\pi$
$461$	−1.57609e14			−0.352554			−0.176277	+	0.984341i	$0.556405\pi$
$462$		0			0
$463$	−1.07731e14			−0.235312			−0.117656	−	0.993054i	$-0.537538\pi$
$463$	−1.07731e14			−0.235312			−0.117656	+	0.993054i	$0.537538\pi$
$464$		0			0
$465$	−6.02335e13	−	1.04327e14i	−0.128483	−	0.222540i
$466$		0			0
$467$	−3.52748e14	+	6.10977e14i	−0.734888	+	1.27286i	0.219884	+	0.975526i	$0.429432\pi$
$467$	−3.52748e14	+	6.10977e14i	−0.734888	+	1.27286i	−0.954772	+	0.297338i	$0.903901\pi$
$468$		0			0
$469$	3.86058e14	+	4.32755e13i	0.785600	+	0.0880626i
$470$		0			0
$471$	−5.18401e13	+	8.97897e13i	−0.103051	+	0.178489i
$472$		0			0
$473$	−1.03466e14	−	1.79208e14i	−0.200937	−	0.348034i
$474$		0			0
$475$	−4.91090e14			−0.931850
$476$		0			0
$477$	6.01106e13			0.111455
$478$		0			0
$479$	−1.92273e14	−	3.33027e14i	−0.348396	−	0.603440i	0.637569	−	0.770394i	$-0.279940\pi$
$479$	−1.92273e14	−	3.33027e14i	−0.348396	−	0.603440i	−0.985965	+	0.166954i	$0.946607\pi$
$480$		0			0
$481$	−8.23929e13	+	1.42709e14i	−0.145912	+	0.252728i
$482$		0			0
$483$	1.94408e14	−	2.63728e14i	0.336516	−	0.456507i
$484$		0			0
$485$	2.64146e14	−	4.57515e14i	0.446956	−	0.774151i
$486$		0			0
$487$	4.71105e14	+	8.15978e14i	0.779307	+	1.34980i	0.932341	+	0.361579i	$0.117762\pi$
$487$	4.71105e14	+	8.15978e14i	0.779307	+	1.34980i	−0.153034	+	0.988221i	$0.548904\pi$
$488$		0			0
$489$	4.72526e13			0.0764234
$490$		0			0
$491$	−3.10111e14			−0.490422			−0.245211	−	0.969470i	$-0.578857\pi$
$491$	−3.10111e14			−0.490422			−0.245211	+	0.969470i	$0.578857\pi$
$492$		0			0
$493$	−1.78704e14	−	3.09524e14i	−0.276360	−	0.478670i
$494$		0			0
$495$	8.95643e13	−	1.55130e14i	0.135458	−	0.234621i
$496$		0			0
$497$	1.48118e13	−	2.00932e13i	0.0219103	−	0.0297228i
$498$		0			0
$499$	−3.48001e14	+	6.02756e14i	−0.503533	+	0.872145i	0.496459	+	0.868060i	$0.334633\pi$
$499$	−3.48001e14	+	6.02756e14i	−0.503533	+	0.872145i	−0.999992	+	0.00408425i	$0.998700\pi$
$500$		0			0
$501$	3.17456e13	+	5.49850e13i	0.0449341	+	0.0778282i
$502$		0			0
$503$	−9.62084e14			−1.33226			−0.666130	−	0.745836i	$-0.732050\pi$
$503$	−9.62084e14			−1.33226			−0.666130	+	0.745836i	$0.732050\pi$
$504$		0			0
$505$	−5.09061e14			−0.689711
$506$		0			0
$507$	2.01938e14	+	3.49766e14i	0.267716	+	0.463697i
$508$		0			0
$509$	−5.59945e14	+	9.69852e14i	−0.726436	+	1.25822i	0.231945	+	0.972729i	$0.425491\pi$
$509$	−5.59945e14	+	9.69852e14i	−0.726436	+	1.25822i	−0.958380	+	0.285495i	$0.907842\pi$
$510$		0			0
$511$	3.76147e14	+	4.21646e13i	0.477576	+	0.0535344i
$512$		0			0
$513$	−3.50068e14	+	6.06336e14i	−0.435018	+	0.753473i
$514$		0			0
$515$	−2.11058e14	−	3.65562e14i	−0.256721	−	0.444654i
$516$		0			0
$517$	3.81210e14			0.453906
$518$		0			0
$519$	−1.61695e14			−0.188484
$520$		0			0
$521$	1.67014e13	+	2.89278e13i	0.0190610	+	0.0330147i	0.875399	−	0.483402i	$-0.160599\pi$
$521$	1.67014e13	+	2.89278e13i	0.0190610	+	0.0330147i	−0.856338	+	0.516417i	$0.827266\pi$
$522$		0			0
$523$	5.39399e14	−	9.34267e14i	0.602769	−	1.04403i	−0.389630	−	0.920971i	$-0.627397\pi$
$523$	5.39399e14	−	9.34267e14i	0.602769	−	1.04403i	0.992400	−	0.123056i	$-0.0392695\pi$
$524$		0			0
$525$	1.10021e14	+	2.51779e14i	0.120392	+	0.275513i
$526$		0			0
$527$	3.37442e14	−	5.84467e14i	0.361610	−	0.626327i
$528$		0			0
$529$	−5.32368e14	−	9.22088e14i	−0.558734	−	0.967756i
$530$		0			0
$531$	6.80269e14			0.699295
$532$		0			0
$533$	1.85928e15			1.87217
$534$		0			0
$535$	2.30366e14	+	3.99006e14i	0.227234	+	0.393581i
$536$		0			0
$537$	−1.88490e13	+	3.26475e13i	−0.0182151	+	0.0315494i
$538$		0			0
$539$	4.79433e14	−	5.17888e14i	0.453931	−	0.490341i
$540$		0			0
$541$	2.30958e13	−	4.00031e13i	0.0214263	−	0.0371115i	−0.855113	−	0.518441i	$-0.826513\pi$
$541$	2.30958e13	−	4.00031e13i	0.0214263	−	0.0371115i	0.876540	+	0.481329i	$0.159846\pi$
$542$		0			0
$543$	−2.04729e14	−	3.54601e14i	−0.186115	−	0.322360i
$544$		0			0
$545$	−2.89364e14			−0.257789
$546$		0			0
$547$	6.67534e14			0.582832			0.291416	−	0.956596i	$-0.405874\pi$
$547$	6.67534e14			0.582832			0.291416	+	0.956596i	$0.405874\pi$
$548$		0			0
$549$	1.27050e14	+	2.20057e14i	0.108724	+	0.188316i
$550$		0			0
$551$	7.58906e14	−	1.31446e15i	0.636582	−	1.10259i
$552$		0			0
$553$	−3.64000e14	−	8.33001e14i	−0.299304	−	0.684947i
$554$		0			0
$555$	−2.18898e13	+	3.79142e13i	−0.0176454	+	0.0305626i
$556$		0			0
$557$	1.11745e15	+	1.93547e15i	0.883126	+	1.52962i	0.847846	+	0.530242i	$0.177899\pi$
$557$	1.11745e15	+	1.93547e15i	0.883126	+	1.52962i	0.0352802	+	0.999377i	$0.488768\pi$
$558$		0			0
$559$	−1.19584e15			−0.926636
$560$		0			0
$561$	−1.79735e14			−0.136564
$562$		0			0
$563$	2.19006e14	+	3.79330e14i	0.163177	+	0.282632i	0.936007	−	0.351983i	$-0.114492\pi$
$563$	2.19006e14	+	3.79330e14i	0.163177	+	0.282632i	−0.772829	+	0.634614i	$0.781159\pi$
$564$		0			0
$565$	2.56270e14	−	4.43873e14i	0.187254	−	0.324334i
$566$		0			0
$567$	−7.86805e14	−	8.81976e13i	−0.563845	−	0.0632048i
$568$		0			0
$569$	1.33884e15	−	2.31894e15i	0.941048	−	1.62994i	0.177571	−	0.984108i	$-0.443176\pi$
$569$	1.33884e15	−	2.31894e15i	0.941048	−	1.62994i	0.763477	−	0.645835i	$-0.223491\pi$
$570$		0			0
$571$	1.96949e14	+	3.41126e14i	0.135786	+	0.235189i	0.925898	−	0.377775i	$-0.123311\pi$
$571$	1.96949e14	+	3.41126e14i	0.135786	+	0.235189i	−0.790111	+	0.612963i	$0.789977\pi$
$572$		0			0
$573$	−1.50216e13			−0.0101594
$574$		0			0
$575$	1.69197e15			1.12259
$576$		0			0
$577$	−4.15132e14	−	7.19029e14i	−0.270221	−	0.468036i	0.698697	−	0.715417i	$-0.253763\pi$
$577$	−4.15132e14	−	7.19029e14i	−0.270221	−	0.468036i	−0.968918	+	0.247381i	$0.920430\pi$
$578$		0			0
$579$	−1.39307e13	+	2.41286e13i	−0.00889693	+	0.0154099i
$580$		0			0
$581$	6.84601e12	−	9.28708e12i	0.00429011	−	0.00581983i
$582$		0			0
$583$	−7.14009e13	+	1.23670e14i	−0.0439063	+	0.0760479i
$584$		0			0
$585$	−5.17585e14	−	8.96484e14i	−0.312338	−	0.540985i
$586$		0			0
$587$	−1.82230e15			−1.07922			−0.539612	−	0.841914i	$-0.681429\pi$
$587$	−1.82230e15			−1.07922			−0.539612	+	0.841914i	$0.681429\pi$
$588$		0			0
$589$	2.86605e15			1.66590
$590$		0			0
$591$	−1.55678e14	−	2.69643e14i	−0.0888172	−	0.153836i
$592$		0			0
$593$	6.98238e14	−	1.20938e15i	0.391023	−	0.677272i	−0.601562	−	0.798826i	$-0.705455\pi$
$593$	6.98238e14	−	1.20938e15i	0.391023	−	0.677272i	0.992585	+	0.121554i	$0.0387879\pi$
$594$		0			0
$595$	2.70584e14	−	3.67066e14i	0.148751	−	0.201791i
$596$		0			0
$597$	8.78434e13	−	1.52149e14i	0.0474079	−	0.0821129i
$598$		0			0
$599$	1.25388e15	+	2.17178e15i	0.664366	+	1.15072i	0.979457	+	0.201654i	$0.0646316\pi$
$599$	1.25388e15	+	2.17178e15i	0.664366	+	1.15072i	−0.315091	+	0.949061i	$0.602035\pi$
$600$		0			0
$601$	−2.84773e15			−1.48146			−0.740729	−	0.671804i	$-0.765520\pi$
$601$	−2.84773e15			−1.48146			−0.740729	+	0.671804i	$0.765520\pi$
$602$		0			0
$603$	1.31252e15			0.670441
$604$		0			0
$605$	−2.63777e14	−	4.56875e14i	−0.132307	−	0.229162i
$606$		0			0
$607$	6.90245e14	−	1.19554e15i	0.339990	−	0.588879i	−0.644441	−	0.764654i	$-0.722910\pi$
$607$	6.90245e14	−	1.19554e15i	0.339990	−	0.588879i	0.984430	+	0.175775i	$0.0562431\pi$
$608$		0			0
$609$	−8.43938e14	−	9.46021e13i	−0.408240	−	0.0457621i
$610$		0			0
$611$	1.10149e15	−	1.90784e15i	0.523305	−	0.906390i
$612$		0			0
$613$	1.87127e15	+	3.24114e15i	0.873181	+	1.51239i	0.858687	+	0.512500i	$0.171280\pi$
$613$	1.87127e15	+	3.24114e15i	0.873181	+	1.51239i	0.0144940	+	0.999895i	$0.495386\pi$
$614$		0			0
$615$	4.93966e14			0.226404
$616$		0			0
$617$	3.03502e14			0.136645			0.0683223	−	0.997663i	$-0.478235\pi$
$617$	3.03502e14			0.136645			0.0683223	+	0.997663i	$0.478235\pi$
$618$		0			0
$619$	−6.72106e14	−	1.16412e15i	−0.297262	−	0.514873i	0.678246	−	0.734834i	$-0.262740\pi$
$619$	−6.72106e14	−	1.16412e15i	−0.297262	−	0.514873i	−0.975509	+	0.219961i	$0.929407\pi$
$620$		0			0
$621$	1.20611e15	−	2.08904e15i	0.524060	−	0.907699i
$622$		0			0
$623$	1.50149e15	+	3.43610e15i	0.640970	+	1.46684i
$624$		0			0
$625$	−4.37024e14	+	7.56948e14i	−0.183301	+	0.317487i
$626$		0			0
$627$	−3.81643e14	−	6.61025e14i	−0.157284	−	0.272425i
$628$		0			0
$629$	−2.45264e14			−0.0993241
$630$		0			0
$631$	−3.46330e15			−1.37825			−0.689126	−	0.724641i	$-0.742005\pi$
$631$	−3.46330e15			−1.37825			−0.689126	+	0.724641i	$0.742005\pi$
$632$		0			0
$633$	−8.06030e14	−	1.39609e15i	−0.315232	−	0.545999i
$634$		0			0
$635$	9.33183e14	−	1.61632e15i	0.358684	−	0.621258i
$636$		0			0
$637$	−1.20657e15	−	3.89584e15i	−0.455812	−	1.47175i
$638$		0			0
$639$	4.21697e13	−	7.30400e13i	0.0156583	−	0.0271210i
$640$		0			0
$641$	1.79597e15	+	3.11071e15i	0.655511	+	1.13538i	0.981765	+	0.190097i	$0.0608801\pi$
$641$	1.79597e15	+	3.11071e15i	0.655511	+	1.13538i	−0.326254	+	0.945282i	$0.605787\pi$
$642$		0			0
$643$	3.61033e15			1.29535			0.647674	−	0.761918i	$-0.275742\pi$
$643$	3.61033e15			1.29535			0.647674	+	0.761918i	$0.275742\pi$
$644$		0			0
$645$	−3.17706e14			−0.112059
$646$		0			0
$647$	9.97032e14	+	1.72691e15i	0.345729	+	0.598820i	0.985486	−	0.169757i	$-0.0542984\pi$
$647$	9.97032e14	+	1.72691e15i	0.345729	+	0.598820i	−0.639757	+	0.768577i	$0.720965\pi$
$648$		0			0
$649$	−8.08041e14	+	1.39957e15i	−0.275478	+	0.477142i
$650$		0			0
$651$	−6.42092e14	−	1.46941e15i	−0.215230	−	0.492546i
$652$		0			0
$653$	−1.19001e15	+	2.06116e15i	−0.392218	+	0.679342i	−0.992742	−	0.120265i	$-0.961626\pi$
$653$	−1.19001e15	+	2.06116e15i	−0.392218	+	0.679342i	0.600524	+	0.799607i	$0.294959\pi$
$654$		0			0
$655$	−1.24966e15	−	2.16447e15i	−0.405008	−	0.701494i
$656$		0			0
$657$	1.27883e15			0.407570
$658$		0			0
$659$	−2.26075e15			−0.708570			−0.354285	−	0.935138i	$-0.615276\pi$
$659$	−2.26075e15			−0.708570			−0.354285	+	0.935138i	$0.615276\pi$
$660$		0			0
$661$	1.86779e15	+	3.23511e15i	0.575731	+	0.997196i	0.995962	+	0.0897784i	$0.0286159\pi$
$661$	1.86779e15	+	3.23511e15i	0.575731	+	0.997196i	−0.420231	+	0.907417i	$0.638051\pi$
$662$		0			0
$663$	−5.19338e14	+	8.99519e14i	−0.157444	+	0.272701i
$664$		0			0
$665$	1.92454e15	+	2.15733e14i	0.573861	+	0.0643275i
$666$		0			0
$667$	−2.61469e15	+	4.52878e15i	−0.766882	+	1.32828i
$668$		0			0
$669$	−1.17978e14	−	2.04345e14i	−0.0340376	−	0.0589549i
$670$		0			0
$671$	−6.03653e14			−0.171322
$672$		0			0
$673$	5.58217e14			0.155855			0.0779274	−	0.996959i	$-0.475170\pi$
$673$	5.58217e14			0.155855			0.0779274	+	0.996959i	$0.475170\pi$
$674$		0			0
$675$	1.01148e15	+	1.75193e15i	0.277833	+	0.481222i
$676$		0			0
$677$	1.14479e15	−	1.98283e15i	0.309376	−	0.535855i	−0.668850	−	0.743397i	$-0.733213\pi$
$677$	1.14479e15	−	1.98283e15i	0.309376	−	0.535855i	0.978226	+	0.207543i	$0.0665465\pi$
$678$		0			0
$679$	4.17267e15	−	5.66051e15i	1.10951	−	1.50512i
$680$		0			0
$681$	−3.67740e14	+	6.36944e14i	−0.0962125	+	0.166645i
$682$		0			0
$683$	2.44076e15	+	4.22751e15i	0.628363	+	1.08836i	0.987880	+	0.155218i	$0.0496080\pi$
$683$	2.44076e15	+	4.22751e15i	0.628363	+	1.08836i	−0.359518	+	0.933138i	$0.617059\pi$
$684$		0			0
$685$	−9.26128e14			−0.234624
$686$		0			0
$687$	−2.47043e15			−0.615900
$688$		0			0
$689$	4.12620e14	+	7.14679e14i	0.101238	+	0.175350i
$690$		0			0
$691$	1.29715e15	−	2.24672e15i	0.313227	−	0.542526i	−0.665832	−	0.746102i	$-0.731923\pi$
$691$	1.29715e15	−	2.24672e15i	0.313227	−	0.542526i	0.979059	+	0.203576i	$0.0652565\pi$
$692$		0			0
$693$	1.41483e15	−	1.91932e15i	0.336257	−	0.456156i
$694$		0			0
$695$	−2.66315e14	+	4.61271e14i	−0.0622987	+	0.107905i
$696$		0			0
$697$	1.38366e15	+	2.39656e15i	0.318602	+	0.551834i
$698$		0			0
$699$	−2.44700e15			−0.554639
$700$		0			0
$701$	−6.12895e15			−1.36753			−0.683765	−	0.729702i	$-0.739659\pi$
$701$	−6.12895e15			−1.36753			−0.683765	+	0.729702i	$0.739659\pi$
$702$		0			0
$703$	−5.20784e14	−	9.02024e14i	−0.114394	−	0.198136i
$704$		0			0
$705$	2.92640e14	−	5.06867e14i	0.0632839	−	0.109611i
$706$		0			0
$707$	−6.73408e15	−	7.54864e14i	−1.43374	−	0.160717i
$708$		0			0
$709$	−2.50958e15	+	4.34671e15i	−0.526073	+	0.911185i	0.473466	+	0.880812i	$0.343003\pi$
$709$	−2.50958e15	+	4.34671e15i	−0.526073	+	0.911185i	−0.999539	+	0.0303729i	$0.990331\pi$
$710$		0			0
$711$	−1.53569e15	−	2.65989e15i	−0.316972	−	0.549012i
$712$		0			0
$713$	−9.87453e15			−2.00689
$714$		0			0
$715$	2.45920e15			0.492166
$716$		0			0
$717$	5.29381e14	+	9.16916e14i	0.104331	+	0.180707i
$718$		0			0
$719$	−2.24731e15	+	3.89246e15i	−0.436169	+	0.755467i	−0.997390	−	0.0721987i	$-0.976998\pi$
$719$	−2.24731e15	+	3.89246e15i	−0.436169	+	0.755467i	0.561221	+	0.827666i	$0.310332\pi$
$720$		0			0
$721$	−2.24989e15	−	5.14879e15i	−0.430048	−	0.984150i
$722$		0			0
$723$	5.08666e14	−	8.81035e14i	0.0957573	−	0.165857i
$724$		0			0
$725$	−2.19276e15	−	3.79797e15i	−0.406567	−	0.704194i
$726$		0			0
$727$	−9.03428e15			−1.64989			−0.824943	−	0.565215i	$-0.808793\pi$
$727$	−9.03428e15			−1.64989			−0.824943	+	0.565215i	$0.808793\pi$
$728$		0			0
$729$	−1.11442e15			−0.200469
$730$		0			0
$731$	−8.89933e14	−	1.54141e15i	−0.157693	−	0.273132i
$732$		0			0
$733$	9.01549e14	−	1.56153e15i	0.157368	−	0.272570i	−0.776551	−	0.630055i	$-0.783032\pi$
$733$	9.01549e14	−	1.56153e15i	0.157368	−	0.272570i	0.933919	+	0.357485i	$0.116366\pi$
$734$		0			0
$735$	−3.20556e14	−	1.03503e15i	−0.0551218	−	0.177980i
$736$		0			0
$737$	−1.55905e15	+	2.70035e15i	−0.264112	+	0.457455i
$738$		0			0
$739$	4.09552e15	+	7.09365e15i	0.683541	+	1.18393i	0.973893	+	0.227008i	$0.0728945\pi$
$739$	4.09552e15	+	7.09365e15i	0.683541	+	1.18393i	−0.290351	+	0.956920i	$0.593772\pi$
$740$		0			0
$741$	−4.41097e15			−0.725328
$742$		0			0
$743$	7.52703e15			1.21951			0.609754	−	0.792590i	$-0.291268\pi$
$743$	7.52703e15			1.21951			0.609754	+	0.792590i	$0.291268\pi$
$744$		0			0
$745$	−7.48070e14	−	1.29570e15i	−0.119422	−	0.206844i
$746$		0			0
$747$	1.94908e13	−	3.37591e13i	0.00306596	−	0.00531040i
$748$		0			0
$749$	2.45572e15	+	5.61983e15i	0.380652	+	0.871110i
$750$		0			0
$751$	−3.18329e15	+	5.51362e15i	−0.486246	+	0.842203i	−0.999875	−	0.0158092i	$-0.994968\pi$
$751$	−3.18329e15	+	5.51362e15i	−0.486246	+	0.842203i	0.513629	+	0.858013i	$0.328301\pi$
$752$		0			0
$753$	8.85504e13	+	1.53374e14i	0.0133296	+	0.0230876i
$754$		0			0
$755$	−4.86476e15			−0.721695
$756$		0			0
$757$	6.09474e15			0.891103			0.445552	−	0.895256i	$-0.353008\pi$
$757$	6.09474e15			0.891103			0.445552	+	0.895256i	$0.353008\pi$
$758$		0			0
$759$	1.31489e15	+	2.27746e15i	0.189479	+	0.328187i
$760$		0			0
$761$	−1.63264e14	+	2.82782e14i	−0.0231886	+	0.0401639i	−0.877387	−	0.479784i	$-0.840715\pi$
$761$	−1.63264e14	+	2.82782e14i	−0.0231886	+	0.0401639i	0.854198	+	0.519948i	$0.174049\pi$
$762$		0			0
$763$	−3.82784e15	−	4.29085e14i	−0.535881	−	0.0600701i
$764$		0			0
$765$	7.70362e14	−	1.33431e15i	0.106306	−	0.184127i
$766$		0			0
$767$	4.66961e15	+	8.08800e15i	0.635193	+	1.10019i
$768$		0			0
$769$	9.11977e15			1.22290			0.611448	−	0.791285i	$-0.290588\pi$
$769$	9.11977e15			1.22290			0.611448	+	0.791285i	$0.290588\pi$
$770$		0			0
$771$	1.06206e15			0.140394
$772$		0			0
$773$	−8.31302e14	−	1.43986e15i	−0.108336	−	0.187643i	0.806760	−	0.590879i	$-0.201219\pi$
$773$	−8.31302e14	−	1.43986e15i	−0.108336	−	0.187643i	−0.915096	+	0.403236i	$0.867885\pi$
$774$		0			0
$775$	4.14054e15	−	7.17162e15i	0.531982	−	0.921420i
$776$		0			0
$777$	−3.45789e14	+	4.69087e14i	−0.0438022	+	0.0594207i
$778$		0			0
$779$	−5.87602e15	+	1.01776e16i	−0.733883	+	1.27112i
$780$		0			0
$781$	1.00180e14	+	1.73518e14i	0.0123368	+	0.0213680i
$782$		0			0
$783$	−6.25234e15			−0.759194
$784$		0			0
$785$	2.11140e15			0.252806
$786$		0			0
$787$	−1.14988e15	−	1.99164e15i	−0.135766	−	0.235153i	0.790124	−	0.612947i	$-0.210016\pi$
$787$	−1.14988e15	−	1.99164e15i	−0.135766	−	0.235153i	−0.925890	+	0.377794i	$0.876683\pi$
$788$		0			0
$789$	−2.54166e14	+	4.40228e14i	−0.0295933	+	0.0512572i
$790$		0			0
$791$	4.04825e15	−	5.49174e15i	0.464833	−	0.630578i
$792$		0			0
$793$	−1.74423e15	+	3.02110e15i	−0.197516	+	0.342108i
$794$		0			0
$795$	1.09623e14	+	1.89873e14i	0.0122429	+	0.0212053i
$796$		0			0
$797$	−9.27622e15			−1.02176			−0.510881	−	0.859651i	$-0.670681\pi$
$797$	−9.27622e15			−1.02176			−0.510881	+	0.859651i	$0.670681\pi$
$798$		0			0
$799$	3.27887e15			0.356219
$800$		0			0
$801$	6.33468e15	+	1.09720e16i	0.678806	+	1.17573i
$802$		0			0
$803$	−1.51902e15	+	2.63102e15i	−0.160557	+	0.278092i
$804$		0			0
$805$	−6.63068e15	−	7.43273e14i	−0.691323	−	0.0774945i
$806$		0			0
$807$	2.22441e15	−	3.85279e15i	0.228776	−	0.396251i
$808$		0			0
$809$	5.18871e15	+	8.98711e15i	0.526433	+	0.911808i	0.999526	+	0.0307955i	$0.00980406\pi$
$809$	5.18871e15	+	8.98711e15i	0.526433	+	0.911808i	−0.473093	+	0.881012i	$0.656863\pi$
$810$		0			0
$811$	−4.03954e15			−0.404313			−0.202156	−	0.979353i	$-0.564795\pi$
$811$	−4.03954e15			−0.404313			−0.202156	+	0.979353i	$0.564795\pi$
$812$		0			0
$813$	−3.45149e15			−0.340807
$814$		0			0
$815$	−4.81139e14	−	8.33357e14i	−0.0468709	−	0.0811828i
$816$		0			0
$817$	3.77931e15	−	6.54595e15i	0.363237	−	0.629146i
$818$		0			0
$819$	−5.51749e15	−	1.26266e16i	−0.523215	−	1.19736i
$820$		0			0
$821$	−8.41429e15	+	1.45740e16i	−0.787281	+	1.36361i	0.140345	+	0.990103i	$0.455179\pi$
$821$	−8.41429e15	+	1.45740e16i	−0.787281	+	1.36361i	−0.927627	+	0.373509i	$0.878155\pi$
$822$		0			0
$823$	1.67049e15	+	2.89338e15i	0.154222	+	0.267120i	0.932775	−	0.360458i	$-0.117380\pi$
$823$	1.67049e15	+	2.89338e15i	0.154222	+	0.267120i	−0.778554	+	0.627578i	$0.784046\pi$
$824$		0			0
$825$	−2.20541e15			−0.200906
$826$		0			0
$827$	1.63931e16			1.47360			0.736801	−	0.676110i	$-0.236336\pi$
$827$	1.63931e16			1.47360			0.736801	+	0.676110i	$0.236336\pi$
$828$		0			0
$829$	−7.39357e15	−	1.28060e16i	−0.655850	−	1.13597i	−0.981680	−	0.190537i	$-0.938977\pi$
$829$	−7.39357e15	−	1.28060e16i	−0.655850	−	1.13597i	0.325830	−	0.945428i	$-0.394356\pi$
$830$		0			0
$831$	1.68537e15	−	2.91914e15i	0.147533	−	0.255535i
$832$		0			0
$833$	4.12371e15	−	4.45447e15i	0.356238	−	0.384812i
$834$		0			0
$835$	6.46486e14	−	1.11975e15i	0.0551167	−	0.0954650i
$836$		0			0
$837$	−5.90308e15	−	1.02244e16i	−0.496693	−	0.860298i
$838$		0			0
$839$	−1.08076e16			−0.897507			−0.448754	−	0.893655i	$-0.648132\pi$
$839$	−1.08076e16			−0.897507			−0.448754	+	0.893655i	$0.648132\pi$
$840$		0			0
$841$	1.35382e15			0.110965
$842$		0			0
$843$	−3.79100e15	−	6.56620e15i	−0.306692	−	0.531206i
$844$		0			0
$845$	4.11237e15	−	7.12284e15i	0.328383	−	0.568776i
$846$		0			0
$847$	−2.81187e15	−	6.43488e15i	−0.221634	−	0.507203i
$848$		0			0
$849$	7.28072e14	−	1.26106e15i	0.0566477	−	0.0981166i
$850$		0			0
$851$	1.79428e15	+	3.10778e15i	0.137809	+	0.238692i
$852$		0			0
$853$	−8.19653e15			−0.621456			−0.310728	−	0.950499i	$-0.600573\pi$
$853$	−8.19653e15			−0.621456			−0.310728	+	0.950499i	$0.600573\pi$
$854$		0			0
$855$	6.54304e15			0.489741
$856$		0			0
$857$	1.33653e16	+	2.31495e16i	0.987611	+	1.71059i	0.629705	+	0.776834i	$0.283176\pi$
$857$	1.33653e16	+	2.31495e16i	0.987611	+	1.71059i	0.357906	+	0.933758i	$0.383491\pi$
$858$		0			0
$859$	3.90104e15	−	6.75680e15i	0.284589	−	0.492923i	−0.687920	−	0.725786i	$-0.741476\pi$
$859$	3.90104e15	−	6.75680e15i	0.284589	−	0.492923i	0.972509	+	0.232863i	$0.0748095\pi$
$860$		0			0
$861$	6.53440e15	+	7.32480e14i	0.470639	+	0.0527568i
$862$		0			0
$863$	−1.69766e15	+	2.94043e15i	−0.120723	+	0.209099i	−0.920053	−	0.391794i	$-0.871855\pi$
$863$	−1.69766e15	+	2.94043e15i	−0.120723	+	0.209099i	0.799330	+	0.600892i	$0.205188\pi$
$864$		0			0
$865$	1.64642e15	+	2.85168e15i	0.115598	+	0.200222i
$866$		0			0
$867$	4.07595e15			0.282568
$868$		0			0
$869$	7.29653e15			0.499468
$870$		0			0
$871$	9.00961e15	+	1.56051e16i	0.608984	+	1.05479i
$872$		0			0
$873$	1.18797e16	−	2.05763e16i	0.792918	−	1.37337i
$874$		0			0
$875$	7.62392e15	−	1.03424e16i	0.502498	−	0.681673i
$876$		0			0
$877$	1.40708e16	−	2.43713e16i	0.915842	−	1.58629i	0.110179	−	0.993912i	$-0.464857\pi$
$877$	1.40708e16	−	2.43713e16i	0.915842	−	1.58629i	0.805663	−	0.592374i	$-0.201809\pi$
$878$		0			0
$879$	5.17441e15	+	8.96234e15i	0.332600	+	0.576080i
$880$		0			0
$881$	−4.91831e15			−0.312211			−0.156105	−	0.987740i	$-0.549894\pi$
$881$	−4.91831e15			−0.312211			−0.156105	+	0.987740i	$0.549894\pi$
$882$		0			0
$883$	−4.50743e15			−0.282583			−0.141291	−	0.989968i	$-0.545125\pi$
$883$	−4.50743e15			−0.282583			−0.141291	+	0.989968i	$0.545125\pi$
$884$		0			0
$885$	1.24060e15	+	2.14878e15i	0.0768147	+	0.133047i
$886$		0			0
$887$	−3.31579e15	+	5.74312e15i	−0.202772	+	0.351211i	−0.949420	−	0.314008i	$-0.898328\pi$
$887$	−3.31579e15	+	5.74312e15i	−0.202772	+	0.351211i	0.746649	+	0.665218i	$0.231662\pi$
$888$		0			0
$889$	1.47413e16	−	1.99976e16i	0.890383	−	1.20787i
$890$		0			0
$891$	3.17741e15	−	5.50344e15i	0.189560	−	0.328327i
$892$		0			0
$893$	6.96224e15	+	1.20590e16i	0.410266	+	0.710602i
$894$		0			0
$895$	7.67704e14			0.0446856
$896$		0			0
$897$	1.51973e16			0.873793
$898$		0			0
$899$	1.27972e16	+	2.21653e16i	0.726835	+	1.25891i
$900$		0			0
$901$	−6.14135e14	+	1.06371e15i	−0.0344570	+	0.0596813i
$902$		0			0
$903$	−4.20276e15	−	4.71113e14i	−0.232944	−	0.0261121i
$904$		0			0
$905$	−4.16922e15	+	7.22130e15i	−0.228290	+	0.395410i
$906$		0			0
$907$	−1.14616e16	−	1.98521e16i	−0.620019	−	1.07390i	−0.989482	−	0.144658i	$-0.953792\pi$
$907$	−1.14616e16	−	1.98521e16i	−0.620019	−	1.07390i	0.369463	−	0.929245i	$-0.379542\pi$
$908$		0			0
$909$	−2.28946e16			−1.22358
$910$		0			0
$911$	7.39337e15			0.390383			0.195192	−	0.980765i	$-0.437467\pi$
$911$	7.39337e15			0.390383			0.195192	+	0.980765i	$0.437467\pi$
$912$		0			0
$913$	4.63034e13	+	8.01999e13i	0.00241559	+	0.00418393i
$914$		0			0
$915$	−4.63401e14	+	8.02633e14i	−0.0238858	+	0.0413715i
$916$		0			0
$917$	−1.33214e16	−	3.04856e16i	−0.678451	−	1.55261i
$918$		0			0
$919$	9.54983e15	−	1.65408e16i	0.480574	−	0.832379i	−0.519177	−	0.854667i	$-0.673762\pi$
$919$	9.54983e15	−	1.65408e16i	0.480574	−	0.832379i	0.999752	+	0.0222875i	$0.00709492\pi$
$920$		0			0
$921$	3.76137e15	+	6.51488e15i	0.187033	+	0.323951i
$922$		0			0
$923$	1.15787e15			0.0568920
$924$		0			0
$925$	−3.00947e15			−0.146120
$926$		0			0
$927$	−9.49212e15	−	1.64408e16i	−0.455433	−	0.788834i
$928$		0			0
$929$	4.71388e15	−	8.16468e15i	0.223508	−	0.387126i	−0.732363	−	0.680914i	$-0.761583\pi$
$929$	4.71388e15	−	8.16468e15i	0.223508	−	0.387126i	0.955871	+	0.293788i	$0.0949159\pi$
$930$		0			0
$931$	2.51387e16	+	5.70761e15i	1.17793	+	0.267443i
$932$		0			0
$933$	−3.31732e15	+	5.74577e15i	−0.153617	+	0.266072i
$934$		0			0
$935$	1.83011e15	+	3.16985e15i	0.0837556	+	0.145069i
$936$		0			0
$937$	9.59725e14			0.0434089			0.0217045	−	0.999764i	$-0.493091\pi$
$937$	9.59725e14			0.0434089			0.0217045	+	0.999764i	$0.493091\pi$
$938$		0			0
$939$	9.00092e14			0.0402371
$940$		0			0
$941$	1.12618e15	+	1.95061e15i	0.0497584	+	0.0861840i	0.889832	−	0.456289i	$-0.150822\pi$
$941$	1.12618e15	+	1.95061e15i	0.0497584	+	0.0861840i	−0.840073	+	0.542473i	$0.817488\pi$
$942$		0			0
$943$	2.02449e16	−	3.50652e16i	0.884100	−	1.53131i
$944$		0			0
$945$	−3.19427e15	−	7.30997e15i	−0.137878	−	0.315530i
$946$		0			0
$947$	−8.00021e15	+	1.38568e16i	−0.341332	+	0.591204i	−0.984680	−	0.174370i	$-0.944211\pi$
$947$	−8.00021e15	+	1.38568e16i	−0.341332	+	0.591204i	0.643349	+	0.765573i	$0.277545\pi$
$948$		0			0
$949$	8.77831e15	+	1.52045e16i	0.370209	+	0.641221i
$950$		0			0
$951$	8.89127e15			0.370656
$952$		0			0
$953$	1.42900e16			0.588872			0.294436	−	0.955671i	$-0.404868\pi$
$953$	1.42900e16			0.588872			0.294436	+	0.955671i	$0.404868\pi$
$954$		0			0
$955$	1.52954e14	+	2.64925e14i	0.00623080	+	0.0107921i
$956$		0			0
$957$	3.40814e15	−	5.90307e15i	0.137247	−	0.237718i
$958$		0			0
$959$	−1.22512e16	−	1.37331e15i	−0.487727	−	0.0546723i
$960$		0			0
$961$	−1.14604e16	+	1.98499e16i	−0.451045	+	0.781232i
$962$		0			0
$963$	1.03605e16	+	1.79449e16i	0.403122	+	0.698228i
$964$		0			0
$965$	5.67384e14			0.0218261
$966$		0			0
$967$	−3.03448e16			−1.15409			−0.577045	−	0.816713i	$-0.695794\pi$
$967$	−3.03448e16			−1.15409			−0.577045	+	0.816713i	$0.695794\pi$
$968$		0			0
$969$	−3.28260e15	−	5.68562e15i	−0.123435	−	0.213795i
$970$		0			0
$971$	5.78248e15	−	1.00155e16i	0.214985	−	0.372365i	−0.738283	−	0.674491i	$-0.764363\pi$
$971$	5.78248e15	−	1.00155e16i	0.214985	−	0.372365i	0.953268	+	0.302126i	$0.0976964\pi$
$972$		0			0
$973$	−4.20693e15	+	5.70699e15i	−0.154648	+	0.209791i
$974$		0			0
$975$	−6.37246e15	+	1.10374e16i	−0.231623	+	0.401183i
$976$		0			0
$977$	2.02859e16	+	3.51362e16i	0.729078	+	1.26280i	0.957273	+	0.289185i	$0.0933844\pi$
$977$	2.02859e16	+	3.51362e16i	0.729078	+	1.26280i	−0.228195	+	0.973616i	$0.573282\pi$
$978$		0			0
$979$	−3.00980e16			−1.06963
$980$		0			0
$981$	−1.30139e16			−0.457328
$982$		0			0
$983$	−1.15801e16	−	2.00574e16i	−0.402410	−	0.696995i	0.591606	−	0.806227i	$-0.298494\pi$
$983$	−1.15801e16	−	2.00574e16i	−0.402410	−	0.696995i	−0.994016	+	0.109233i	$0.965161\pi$
$984$		0			0
$985$	−3.17032e15	+	5.49115e15i	−0.108944	+	0.188697i
$986$		0			0
$987$	4.62277e15	−	6.27111e15i	0.157094	−	0.213108i
$988$		0			0
$989$	−1.30210e16	+	2.25530e16i	−0.437588	+	0.757924i
$990$		0			0
$991$	1.36159e16	+	2.35834e16i	0.452524	+	0.783794i	0.998542	−	0.0539794i	$-0.0171905\pi$
$991$	1.36159e16	+	2.35834e16i	0.452524	+	0.783794i	−0.546019	+	0.837773i	$0.683857\pi$
$992$		0			0
$993$	2.92662e15			0.0961933
$994$		0			0
$995$	−3.57779e15			−0.116302
$996$		0			0
$997$	−2.28819e16	−	3.96327e16i	−0.735647	−	1.27418i	−0.954439	−	0.298406i	$-0.903545\pi$
$997$	−2.28819e16	−	3.96327e16i	−0.735647	−	1.27418i	0.218792	−	0.975772i	$-0.429788\pi$
$998$		0			0
$999$	−2.14527e15	+	3.71572e15i	−0.0682138	+	0.118150i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.12.i.c.65.4		12
4.3	odd	2		7.12.c.a.2.6	✓	12
7.4	even	3	inner	112.12.i.c.81.4		12
12.11	even	2		63.12.e.b.37.1		12
28.3	even	6		49.12.c.i.18.6		12
28.11	odd	6		7.12.c.a.4.6	yes	12
28.19	even	6		49.12.a.f.1.1		6
28.23	odd	6		49.12.a.g.1.1		6
28.27	even	2		49.12.c.i.30.6		12
84.11	even	6		63.12.e.b.46.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.12.c.a.2.6	✓	12	4.3	odd	2
7.12.c.a.4.6	yes	12	28.11	odd	6
49.12.a.f.1.1		6	28.19	even	6
49.12.a.g.1.1		6	28.23	odd	6
49.12.c.i.18.6		12	28.3	even	6
49.12.c.i.30.6		12	28.27	even	2
63.12.e.b.37.1		12	12.11	even	2
63.12.e.b.46.1		12	84.11	even	6
112.12.i.c.65.4		12	1.1	even	1	trivial
112.12.i.c.81.4		12	7.4	even	3	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	112.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(82.0189 + 142.061i) q^{3} +(1670.28 - 2893.01i) q^{5} +(26385.1 - 35793.2i) q^{7} +(75119.3 - 130110. i) q^{9} +O(q^{10})\)	\(q+(82.0189 + 142.061i) q^{3} +(1670.28 - 2893.01i) q^{5} +(26385.1 - 35793.2i) q^{7} +(75119.3 - 130110. i) q^{9} +(178457. + 309097. i) q^{11} +2.06258e6 q^{13} +547978. q^{15} +(1.53495e6 + 2.65861e6i) q^{17} +(-6.51852e6 + 1.12904e7i) q^{19} +(7.24889e6 + 812572. i) q^{21} +(2.24585e7 - 3.88993e7i) q^{23} +(1.88344e7 + 3.26221e7i) q^{25} +5.37036e7 q^{27} -1.16423e8 q^{29} +(-1.09920e8 - 1.90386e8i) q^{31} +(-2.92737e7 + 5.07036e7i) q^{33} +(-5.94796e7 - 1.36117e8i) q^{35} +(-3.99465e7 + 6.91894e7i) q^{37} +(1.69171e8 + 2.93012e8i) q^{39} +9.01434e8 q^{41} -5.79780e8 q^{43} +(-2.50940e8 - 4.34642e8i) q^{45} +(5.34035e8 - 9.24976e8i) q^{47} +(-5.84980e8 - 1.88881e9i) q^{49} +(-2.51790e8 + 4.36113e8i) q^{51} +(2.00050e8 + 3.46497e8i) q^{53} +1.19229e9 q^{55} -2.13857e9 q^{57} +(2.26396e9 + 3.92130e9i) q^{59} +(-8.45655e8 + 1.46472e9i) q^{61} +(-2.67504e9 - 6.12174e9i) q^{63} +(3.44509e9 - 5.96707e9i) q^{65} +(4.36812e9 + 7.56581e9i) q^{67} +7.36810e9 q^{69} +5.61370e8 q^{71} +(4.25598e9 + 7.37158e9i) q^{73} +(-3.08955e9 + 5.35126e9i) q^{75} +(1.57722e10 + 1.76800e9i) q^{77} +(1.02217e10 - 1.77045e10i) q^{79} +(-8.90244e9 - 1.54195e10i) q^{81} +2.59465e8 q^{83} +1.02552e10 q^{85} +(-9.54890e9 - 1.65392e10i) q^{87} +(-4.21641e10 + 7.30304e10i) q^{89} +(5.44214e10 - 7.38264e10i) q^{91} +(1.80310e10 - 3.12305e10i) q^{93} +(2.17755e10 + 3.77163e10i) q^{95} +1.58145e11 q^{97} +5.36223e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 244 q^{3} - 8782 q^{5} + 504 q^{7} - 172348 q^{9}+O(q^{10}) \) 12 * q + 244 * q^3 - 8782 * q^5 + 504 * q^7 - 172348 * q^9	\( 12 q + 244 q^{3} - 8782 q^{5} + 504 q^{7} - 172348 q^{9} + 1001572 q^{11} + 3864504 q^{13} + 1286512 q^{15} - 6704802 q^{17} - 4192212 q^{19} + 44745358 q^{21} + 33871872 q^{23} + 13695456 q^{25} - 73859384 q^{27} - 255125224 q^{29} + 331783920 q^{31} - 80899438 q^{33} - 1407354844 q^{35} - 833082774 q^{37} - 737605904 q^{39} + 3104076808 q^{41} + 1722177552 q^{43} - 7406493484 q^{45} + 1327587552 q^{47} + 11976558636 q^{49} + 13921261140 q^{51} + 6725755626 q^{53} - 26323921200 q^{55} - 16884487756 q^{57} + 26237179548 q^{59} - 14411013726 q^{61} - 45955779184 q^{63} - 16224702172 q^{65} + 4241860068 q^{67} + 46750854252 q^{69} + 37335334656 q^{71} + 6005568990 q^{73} - 17116276792 q^{75} - 51928077698 q^{77} - 11712395640 q^{79} - 12455008366 q^{81} + 100821781200 q^{83} + 138884613396 q^{85} - 119455310144 q^{87} - 48633519778 q^{89} + 160908361488 q^{91} + 266530114134 q^{93} + 72161225128 q^{95} - 401308415928 q^{97} - 367357472240 q^{99}+O(q^{100}) \) 12 * q + 244 * q^3 - 8782 * q^5 + 504 * q^7 - 172348 * q^9 + 1001572 * q^11 + 3864504 * q^13 + 1286512 * q^15 - 6704802 * q^17 - 4192212 * q^19 + 44745358 * q^21 + 33871872 * q^23 + 13695456 * q^25 - 73859384 * q^27 - 255125224 * q^29 + 331783920 * q^31 - 80899438 * q^33 - 1407354844 * q^35 - 833082774 * q^37 - 737605904 * q^39 + 3104076808 * q^41 + 1722177552 * q^43 - 7406493484 * q^45 + 1327587552 * q^47 + 11976558636 * q^49 + 13921261140 * q^51 + 6725755626 * q^53 - 26323921200 * q^55 - 16884487756 * q^57 + 26237179548 * q^59 - 14411013726 * q^61 - 45955779184 * q^63 - 16224702172 * q^65 + 4241860068 * q^67 + 46750854252 * q^69 + 37335334656 * q^71 + 6005568990 * q^73 - 17116276792 * q^75 - 51928077698 * q^77 - 11712395640 * q^79 - 12455008366 * q^81 + 100821781200 * q^83 + 138884613396 * q^85 - 119455310144 * q^87 - 48633519778 * q^89 + 160908361488 * q^91 + 266530114134 * q^93 + 72161225128 * q^95 - 401308415928 * q^97 - 367357472240 * q^99

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