LMFDB - Embedded newform 112.12.i.b.81.3

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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 149344 x^{6} + 5578711 x^{5} + 20557200983 x^{4} + 408905884576 x^{3} + \cdots + 30\!\cdots\!24 $ x^8 - x^7 + 149344x^6 + 5578711x^5 + 20557200983x^4 + 408905884576x^3 + 267736482219682x^2 - 4732823471679688x + 3039499286293024324
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12}\cdot 3\cdot 7^{3} $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			81.3
Root			$-62.4133 + 108.103i$ of defining polynomial
Character	$\chi$	$=$	112.81
Dual form			112.12.i.b.65.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(158.327 - 274.230i) q^{3} +(-6009.47 - 10408.7i) q^{5} +(-34168.3 + 28457.9i) q^{7} +(38438.9 + 66578.1i) q^{9} +O(q^{10})$	\(q+(158.327 - 274.230i) q^{3} +(-6009.47 - 10408.7i) q^{5} +(-34168.3 + 28457.9i) q^{7} +(38438.9 + 66578.1i) q^{9} +(-187066. + 324007. i) q^{11} -625709. q^{13} -3.80583e6 q^{15} +(3.56150e6 - 6.16869e6i) q^{17} +(9.63596e6 + 1.66900e7i) q^{19} +(2.39426e6 + 1.38756e7i) q^{21} +(1.19610e7 + 2.07171e7i) q^{23} +(-4.78134e7 + 8.28152e7i) q^{25} +8.04377e7 q^{27} +7.06535e7 q^{29} +(7.36573e7 - 1.27578e8i) q^{31} +(5.92349e7 + 1.02598e8i) q^{33} +(5.01544e8 + 1.84631e8i) q^{35} +(-2.68355e8 - 4.64805e8i) q^{37} +(-9.90663e7 + 1.71588e8i) q^{39} +4.82553e8 q^{41} -5.66805e8 q^{43} +(4.61994e8 - 8.00198e8i) q^{45} +(-5.35695e8 - 9.27851e8i) q^{47} +(3.57619e8 - 1.94472e9i) q^{49} +(-1.12776e9 - 1.95334e9i) q^{51} +(-2.88664e9 + 4.99981e9i) q^{53} +4.49666e9 q^{55} +6.10251e9 q^{57} +(-1.69150e9 + 2.92976e9i) q^{59} +(-2.55605e9 - 4.42721e9i) q^{61} +(-3.20807e9 - 1.18097e9i) q^{63} +(3.76018e9 + 6.51282e9i) q^{65} +(1.24117e9 - 2.14977e9i) q^{67} +7.57500e9 q^{69} +2.40523e9 q^{71} +(9.20676e8 - 1.59466e9i) q^{73} +(1.51403e10 + 2.62237e10i) q^{75} +(-2.82886e9 - 1.63943e10i) q^{77} +(-1.37771e10 - 2.38626e10i) q^{79} +(5.92610e9 - 1.02643e10i) q^{81} -3.20498e10 q^{83} -8.56108e10 q^{85} +(1.11863e10 - 1.93753e10i) q^{87} +(-1.01067e10 - 1.75053e10i) q^{89} +(2.13794e10 - 1.78064e10i) q^{91} +(-2.33238e10 - 4.03980e10i) q^{93} +(1.15814e11 - 2.00596e11i) q^{95} +1.43007e11 q^{97} -2.87624e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 266 q^{3} + 3808 q^{5} - 110328 q^{7} - 503848 q^{9}+O(q^{10}) $ 8 * q + 266 * q^3 + 3808 * q^5 - 110328 * q^7 - 503848 * q^9	$ 8 q + 266 q^{3} + 3808 q^{5} - 110328 q^{7} - 503848 q^{9} - 920150 q^{11} + 997472 q^{13} + 500444 q^{15} + 1333724 q^{17} + 21551726 q^{19} - 90442480 q^{21} - 72510158 q^{23} - 77154744 q^{25} - 156683212 q^{27} + 213575104 q^{29} + 194359774 q^{31} - 164547124 q^{33} + 981719074 q^{35} + 171517048 q^{37} + 653125236 q^{39} - 3312095136 q^{41} - 850279648 q^{43} + 6784014272 q^{45} - 2223880974 q^{47} - 4232044312 q^{49} - 6968362082 q^{51} - 7185483360 q^{53} - 624915620 q^{55} + 4959469112 q^{57} - 6997401502 q^{59} - 6476463280 q^{61} + 17252507684 q^{63} + 30625528248 q^{65} + 18660972186 q^{67} + 31319762096 q^{69} - 23224449248 q^{71} + 3731641452 q^{73} + 91204646176 q^{75} - 29345595440 q^{77} - 12221157926 q^{79} - 97333443028 q^{81} - 316739523968 q^{83} - 147794862544 q^{85} + 218122807364 q^{87} - 71204406084 q^{89} - 189816116528 q^{91} + 9838293624 q^{93} + 215091896614 q^{95} + 688251797184 q^{97} + 545487662552 q^{99}+O(q^{100}) $ 8 * q + 266 * q^3 + 3808 * q^5 - 110328 * q^7 - 503848 * q^9 - 920150 * q^11 + 997472 * q^13 + 500444 * q^15 + 1333724 * q^17 + 21551726 * q^19 - 90442480 * q^21 - 72510158 * q^23 - 77154744 * q^25 - 156683212 * q^27 + 213575104 * q^29 + 194359774 * q^31 - 164547124 * q^33 + 981719074 * q^35 + 171517048 * q^37 + 653125236 * q^39 - 3312095136 * q^41 - 850279648 * q^43 + 6784014272 * q^45 - 2223880974 * q^47 - 4232044312 * q^49 - 6968362082 * q^51 - 7185483360 * q^53 - 624915620 * q^55 + 4959469112 * q^57 - 6997401502 * q^59 - 6476463280 * q^61 + 17252507684 * q^63 + 30625528248 * q^65 + 18660972186 * q^67 + 31319762096 * q^69 - 23224449248 * q^71 + 3731641452 * q^73 + 91204646176 * q^75 - 29345595440 * q^77 - 12221157926 * q^79 - 97333443028 * q^81 - 316739523968 * q^83 - 147794862544 * q^85 + 218122807364 * q^87 - 71204406084 * q^89 - 189816116528 * q^91 + 9838293624 * q^93 + 215091896614 * q^95 + 688251797184 * q^97 + 545487662552 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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
$3$	158.327	−	274.230i	0.376172	−	0.651550i	−0.614329	−	0.789050i	$-0.710573\pi$
$3$	158.327	−	274.230i	0.376172	−	0.651550i	0.990502	+	0.137500i	$0.0439067\pi$
$4$		0			0
$5$	−6009.47	−	10408.7i	−0.860005	−	1.48957i	−0.871923	−	0.489644i	$-0.837127\pi$
$5$	−6009.47	−	10408.7i	−0.860005	−	1.48957i	0.0119176	−	0.999929i	$-0.496206\pi$
$6$		0			0
$7$	−34168.3	+	28457.9i	−0.768394	+	0.639977i
$7$	−34168.3	+	28457.9i	−0.768394	+	0.639977i
$8$		0			0
$9$	38438.9	+	66578.1i	0.216989	+	0.375835i
$10$		0			0
$11$	−187066.	+	324007.i	−0.350215	+	0.606590i	−0.986287	−	0.165040i	$-0.947225\pi$
$11$	−187066.	+	324007.i	−0.350215	+	0.606590i	0.636072	+	0.771630i	$0.280558\pi$
$12$		0			0
$13$	−625709.			−0.467395			−0.233697	−	0.972309i	$-0.575082\pi$
$13$	−625709.			−0.467395			−0.233697	+	0.972309i	$0.575082\pi$
$14$		0			0
$15$	−3.80583e6			−1.29404
$16$		0			0
$17$	3.56150e6	−	6.16869e6i	0.608364	−	1.05372i	−0.383146	−	0.923688i	$-0.625159\pi$
$17$	3.56150e6	−	6.16869e6i	0.608364	−	1.05372i	0.991510	−	0.130030i	$-0.0415072\pi$
$18$		0			0
$19$	9.63596e6	+	1.66900e7i	0.892792	+	1.54636i	0.836514	+	0.547945i	$0.184590\pi$
$19$	9.63596e6	+	1.66900e7i	0.892792	+	1.54636i	0.0562775	+	0.998415i	$0.482077\pi$
$20$		0			0
$21$	2.39426e6	+	1.38756e7i	0.127928	+	0.741389i
$22$		0			0
$23$	1.19610e7	+	2.07171e7i	0.387494	+	0.671160i	0.992112	−	0.125356i	$-0.0400073\pi$
$23$	1.19610e7	+	2.07171e7i	0.387494	+	0.671160i	−0.604617	+	0.796516i	$0.706674\pi$
$24$		0			0
$25$	−4.78134e7	+	8.28152e7i	−0.979218	+	1.69605i
$26$		0			0
$27$	8.04377e7			1.07885
$28$		0			0
$29$	7.06535e7			0.639654			0.319827	−	0.947476i	$-0.396375\pi$
$29$	7.06535e7			0.639654			0.319827	+	0.947476i	$0.396375\pi$
$30$		0			0
$31$	7.36573e7	−	1.27578e8i	0.462090	−	0.800363i	−0.536975	−	0.843598i	$-0.680433\pi$
$31$	7.36573e7	−	1.27578e8i	0.462090	−	0.800363i	0.999065	+	0.0432348i	$0.0137664\pi$
$32$		0			0
$33$	5.92349e7	+	1.02598e8i	0.263482	+	0.456365i
$34$		0			0
$35$	5.01544e8	+	1.84631e8i	1.61411	+	0.594196i
$36$		0			0
$37$	−2.68355e8	−	4.64805e8i	−0.636210	−	1.10195i	−0.986257	−	0.165216i	$-0.947168\pi$
$37$	−2.68355e8	−	4.64805e8i	−0.636210	−	1.10195i	0.350048	−	0.936732i	$-0.386165\pi$
$38$		0			0
$39$	−9.90663e7	+	1.71588e8i	−0.175821	+	0.304531i
$40$		0			0
$41$	4.82553e8			0.650480			0.325240	−	0.945632i	$-0.394555\pi$
$41$	4.82553e8			0.650480			0.325240	+	0.945632i	$0.394555\pi$
$42$		0			0
$43$	−5.66805e8			−0.587973			−0.293986	−	0.955810i	$-0.594982\pi$
$43$	−5.66805e8			−0.587973			−0.293986	+	0.955810i	$0.594982\pi$
$44$		0			0
$45$	4.61994e8	−	8.00198e8i	0.373223	−	0.646441i
$46$		0			0
$47$	−5.35695e8	−	9.27851e8i	−0.340706	−	0.590119i	0.643858	−	0.765145i	$-0.277333\pi$
$47$	−5.35695e8	−	9.27851e8i	−0.340706	−	0.590119i	−0.984564	+	0.175025i	$0.943999\pi$
$48$		0			0
$49$	3.57619e8	−	1.94472e9i	0.180860	−	0.983509i
$50$		0			0
$51$	−1.12776e9	−	1.95334e9i	−0.457699	−	0.792759i
$52$		0			0
$53$	−2.88664e9	+	4.99981e9i	−0.948147	+	1.64224i	−0.198824	+	0.980035i	$0.563712\pi$
$53$	−2.88664e9	+	4.99981e9i	−0.948147	+	1.64224i	−0.749323	+	0.662204i	$0.769621\pi$
$54$		0			0
$55$	4.49666e9			1.20475
$56$		0			0
$57$	6.10251e9			1.34337
$58$		0			0
$59$	−1.69150e9	+	2.92976e9i	−0.308024	+	0.533514i	−0.977930	−	0.208932i	$-0.933001\pi$
$59$	−1.69150e9	+	2.92976e9i	−0.308024	+	0.533514i	0.669906	+	0.742446i	$0.266335\pi$
$60$		0			0
$61$	−2.55605e9	−	4.42721e9i	−0.387485	−	0.671144i	0.604626	−	0.796510i	$-0.293323\pi$
$61$	−2.55605e9	−	4.42721e9i	−0.387485	−	0.671144i	−0.992111	+	0.125366i	$0.959989\pi$
$62$		0			0
$63$	−3.20807e9	−	1.18097e9i	−0.407259	−	0.149922i
$64$		0			0
$65$	3.76018e9	+	6.51282e9i	0.401962	+	0.696218i
$66$		0			0
$67$	1.24117e9	−	2.14977e9i	0.112310	−	0.194527i	−0.804391	−	0.594100i	$-0.797508\pi$
$67$	1.24117e9	−	2.14977e9i	0.112310	−	0.194527i	0.916701	+	0.399573i	$0.130842\pi$
$68$		0			0
$69$	7.57500e9			0.583059
$70$		0			0
$71$	2.40523e9			0.158210			0.0791052	−	0.996866i	$-0.474794\pi$
$71$	2.40523e9			0.158210			0.0791052	+	0.996866i	$0.474794\pi$
$72$		0			0
$73$	9.20676e8	−	1.59466e9i	0.0519794	−	0.0900309i	−0.838865	−	0.544340i	$-0.816780\pi$
$73$	9.20676e8	−	1.59466e9i	0.0519794	−	0.0900309i	0.890844	+	0.454309i	$0.150114\pi$
$74$		0			0
$75$	1.51403e10	+	2.62237e10i	0.736709	+	1.27602i
$76$		0			0
$77$	−2.82886e9	−	1.63943e10i	−0.119100	−	0.690229i
$78$		0			0
$79$	−1.37771e10	−	2.38626e10i	−0.503742	−	0.872506i	−0.999991	−	0.00432604i	$-0.998623\pi$
$79$	−1.37771e10	−	2.38626e10i	−0.503742	−	0.872506i	0.496249	−	0.868180i	$-0.334710\pi$
$80$		0			0
$81$	5.92610e9	−	1.02643e10i	0.188843	−	0.327086i
$82$		0			0
$83$	−3.20498e10			−0.893090			−0.446545	−	0.894761i	$-0.647346\pi$
$83$	−3.20498e10			−0.893090			−0.446545	+	0.894761i	$0.647346\pi$
$84$		0			0
$85$	−8.56108e10			−2.09278
$86$		0			0
$87$	1.11863e10	−	1.93753e10i	0.240620	−	0.416766i
$88$		0			0
$89$	−1.01067e10	−	1.75053e10i	−0.191851	−	0.332295i	0.754013	−	0.656860i	$-0.228116\pi$
$89$	−1.01067e10	−	1.75053e10i	−0.191851	−	0.332295i	−0.945864	+	0.324565i	$0.894782\pi$
$90$		0			0
$91$	2.13794e10	−	1.78064e10i	0.359143	−	0.299122i
$92$		0			0
$93$	−2.33238e10	−	4.03980e10i	−0.347651	−	0.602149i
$94$		0			0
$95$	1.15814e11	−	2.00596e11i	1.53561	−	2.65976i
$96$		0			0
$97$	1.43007e11			1.69088			0.845439	−	0.534072i	$-0.179339\pi$
$97$	1.43007e11			1.69088			0.845439	+	0.534072i	$0.179339\pi$
$98$		0			0
$99$	−2.87624e10			−0.303970
$100$		0			0
$101$	−7.13461e8	+	1.23575e9i	−0.00675465	+	0.0116994i	−0.869383	−	0.494139i	$-0.835483\pi$
$101$	−7.13461e8	+	1.23575e9i	−0.00675465	+	0.0116994i	0.862628	+	0.505838i	$0.168817\pi$
$102$		0			0
$103$	7.14015e10	+	1.23671e11i	0.606880	+	1.05115i	0.991751	+	0.128177i	$0.0409125\pi$
$103$	7.14015e10	+	1.23671e11i	0.606880	+	1.05115i	−0.384871	+	0.922970i	$0.625754\pi$
$104$		0			0
$105$	1.30039e11	−	1.08306e11i	0.994334	−	0.828156i
$106$		0			0
$107$	1.84715e10	+	3.19935e10i	0.127318	+	0.220522i	0.922637	−	0.385670i	$-0.126030\pi$
$107$	1.84715e10	+	3.19935e10i	0.127318	+	0.220522i	−0.795319	+	0.606192i	$0.792696\pi$
$108$		0			0
$109$	4.86415e10	−	8.42495e10i	0.302803	−	0.524471i	−0.673966	−	0.738762i	$-0.735411\pi$
$109$	4.86415e10	−	8.42495e10i	0.302803	−	0.524471i	0.976770	+	0.214291i	$0.0687441\pi$
$110$		0			0
$111$	−1.69951e11			−0.957298
$112$		0			0
$113$	2.51526e11			1.28425			0.642127	−	0.766598i	$-0.278052\pi$
$113$	2.51526e11			1.28425			0.642127	+	0.766598i	$0.278052\pi$
$114$		0			0
$115$	1.43759e11	−	2.48998e11i	0.666494	−	1.15440i
$116$		0			0
$117$	−2.40515e10	−	4.16585e10i	−0.101419	−	0.175663i
$118$		0			0
$119$	5.38579e10	+	3.12127e11i	0.206891	+	1.19901i
$120$		0			0
$121$	7.26687e10	+	1.25866e11i	0.254699	+	0.441152i
$122$		0			0
$123$	7.64009e10	−	1.32330e11i	0.244692	−	0.423820i
$124$		0			0
$125$	5.62469e11			1.64852
$126$		0			0
$127$	4.50975e11			1.21124			0.605622	−	0.795752i	$-0.292924\pi$
$127$	4.50975e11			1.21124			0.605622	+	0.795752i	$0.292924\pi$
$128$		0			0
$129$	−8.97403e10	+	1.55435e11i	−0.221179	+	0.383094i
$130$		0			0
$131$	−5.58869e10	−	9.67990e10i	−0.126566	−	0.219219i	0.795778	−	0.605589i	$-0.207062\pi$
$131$	−5.58869e10	−	9.67990e10i	−0.126566	−	0.219219i	−0.922344	+	0.386369i	$0.873729\pi$
$132$		0			0
$133$	−8.04206e11	−	2.96049e11i	−1.67565	−	0.616849i
$134$		0			0
$135$	−4.83388e11	−	8.37253e11i	−0.927813	−	1.60702i
$136$		0			0
$137$	3.08019e11	−	5.33504e11i	0.545273	−	0.944441i	−0.453316	−	0.891350i	$-0.649759\pi$
$137$	3.08019e11	−	5.33504e11i	0.545273	−	0.944441i	0.998590	−	0.0530914i	$-0.0169075\pi$
$138$		0			0
$139$	7.99626e11			1.30709			0.653545	−	0.756888i	$-0.273281\pi$
$139$	7.99626e11			1.30709			0.653545	+	0.756888i	$0.273281\pi$
$140$		0			0
$141$	−3.39259e11			−0.512656
$142$		0			0
$143$	1.17049e11	−	2.02734e11i	0.163688	−	0.283517i
$144$		0			0
$145$	−4.24590e11	−	7.35412e11i	−0.550105	−	0.952810i
$146$		0			0
$147$	−4.76679e11	−	4.05970e11i	−0.572770	−	0.487808i
$148$		0			0
$149$	7.62790e10	+	1.32119e11i	0.0850904	+	0.147381i	0.905430	−	0.424496i	$-0.139549\pi$
$149$	7.62790e10	+	1.32119e11i	0.0850904	+	0.147381i	−0.820339	+	0.571877i	$0.806215\pi$
$150$		0			0
$151$	7.92887e11	−	1.37332e12i	0.821936	−	1.42363i	−0.0823028	−	0.996607i	$-0.526227\pi$
$151$	7.92887e11	−	1.37332e12i	0.821936	−	1.42363i	0.904239	−	0.427027i	$-0.140439\pi$
$152$		0			0
$153$	5.47600e11			0.528032
$154$		0			0
$155$	−1.77057e12			−1.58960
$156$		0			0
$157$	8.01268e11	−	1.38784e12i	0.670393	−	1.16115i	−0.307400	−	0.951581i	$-0.599459\pi$
$157$	8.01268e11	−	1.38784e12i	0.670393	−	1.16115i	0.977793	−	0.209574i	$-0.0672079\pi$
$158$		0			0
$159$	9.14065e11	+	1.58321e12i	0.713334	+	1.23553i
$160$		0			0
$161$	−9.98254e11	−	3.67482e11i	−0.727275	−	0.267728i
$162$		0			0
$163$	6.37573e11	+	1.10431e12i	0.434008	+	0.751724i	0.997214	−	0.0745923i	$-0.0237655\pi$
$163$	6.37573e11	+	1.10431e12i	0.434008	+	0.751724i	−0.563206	+	0.826317i	$0.690432\pi$
$164$		0			0
$165$	7.11941e11	−	1.23312e12i	0.453192	−	0.784952i
$166$		0			0
$167$	8.99273e11			0.535736			0.267868	−	0.963456i	$-0.413681\pi$
$167$	8.99273e11			0.535736			0.267868	+	0.963456i	$0.413681\pi$
$168$		0			0
$169$	−1.40065e12			−0.781542
$170$		0			0
$171$	−7.40791e11	+	1.28309e12i	−0.387451	+	0.671085i
$172$		0			0
$173$	2.01703e12	+	3.49359e12i	0.989596	+	1.71403i	0.619395	+	0.785080i	$0.287378\pi$
$173$	2.01703e12	+	3.49359e12i	0.989596	+	1.71403i	0.370202	+	0.928951i	$0.379289\pi$
$174$		0			0
$175$	−7.23047e11	−	4.19032e12i	−0.333010	−	1.92992i
$176$		0			0
$177$	5.35618e11	+	9.27717e11i	0.231741	+	0.401386i
$178$		0			0
$179$	−5.94728e11	+	1.03010e12i	−0.241895	+	0.418975i	−0.961254	−	0.275664i	$-0.911102\pi$
$179$	−5.94728e11	+	1.03010e12i	−0.241895	+	0.418975i	0.719359	+	0.694638i	$0.244436\pi$
$180$		0			0
$181$	1.14571e12			0.438372			0.219186	−	0.975683i	$-0.429660\pi$
$181$	1.14571e12			0.438372			0.219186	+	0.975683i	$0.429660\pi$
$182$		0			0
$183$	−1.61876e12			−0.583045
$184$		0			0
$185$	−3.22534e12	+	5.58646e12i	−1.09429	+	1.89536i
$186$		0			0
$187$	1.33247e12	+	2.30790e12i	0.426116	+	0.738054i
$188$		0			0
$189$	−2.74842e12	+	2.28909e12i	−0.828979	+	0.690436i
$190$		0			0
$191$	5.67271e11	+	9.82541e11i	0.161476	+	0.279684i	0.935398	−	0.353596i	$-0.115041\pi$
$191$	5.67271e11	+	9.82541e11i	0.161476	+	0.279684i	−0.773923	+	0.633280i	$0.781708\pi$
$192$		0			0
$193$	3.38313e12	−	5.85975e12i	0.909397	−	1.57512i	0.0944924	−	0.995526i	$-0.469877\pi$
$193$	3.38313e12	−	5.85975e12i	0.909397	−	1.57512i	0.814904	−	0.579596i	$-0.196789\pi$
$194$		0			0
$195$	2.38134e12			0.604828
$196$		0			0
$197$	7.83137e12			1.88050			0.940251	−	0.340483i	$-0.110591\pi$
$197$	7.83137e12			1.88050			0.940251	+	0.340483i	$0.110591\pi$
$198$		0			0
$199$	8.07265e11	−	1.39822e12i	0.183368	−	0.317603i	−0.759657	−	0.650324i	$-0.774633\pi$
$199$	8.07265e11	−	1.39822e12i	0.183368	−	0.317603i	0.943025	+	0.332721i	$0.107967\pi$
$200$		0			0
$201$	−3.93020e11	−	6.80731e11i	−0.0844960	−	0.146351i
$202$		0			0
$203$	−2.41411e12	+	2.01065e12i	−0.491506	+	0.409363i
$204$		0			0
$205$	−2.89989e12	−	5.02275e12i	−0.559416	−	0.968937i
$206$		0			0
$207$	−9.19537e11	+	1.59269e12i	−0.168164	+	0.291268i
$208$		0			0
$209$	−7.21023e12			−1.25067
$210$		0			0
$211$	−8.54136e12			−1.40596			−0.702981	−	0.711209i	$-0.748148\pi$
$211$	−8.54136e12			−1.40596			−0.702981	+	0.711209i	$0.748148\pi$
$212$		0			0
$213$	3.80811e11	−	6.59585e11i	0.0595144	−	0.103082i
$214$		0			0
$215$	3.40620e12	+	5.89971e12i	0.505660	+	0.875828i
$216$		0			0
$217$	1.11387e12	+	6.45527e12i	0.157146	+	0.910722i
$218$		0			0
$219$	−2.91535e11	−	5.04953e11i	−0.0391064	−	0.0677343i
$220$		0			0
$221$	−2.22846e12	+	3.85981e12i	−0.284346	+	0.492502i
$222$		0			0
$223$	4.63010e12			0.562229			0.281115	−	0.959674i	$-0.409296\pi$
$223$	4.63010e12			0.562229			0.281115	+	0.959674i	$0.409296\pi$
$224$		0			0
$225$	−7.35157e12			−0.849916
$226$		0			0
$227$	4.35358e12	−	7.54061e12i	0.479406	−	0.830356i	−0.520315	−	0.853975i	$-0.674185\pi$
$227$	4.35358e12	−	7.54061e12i	0.479406	−	0.830356i	0.999721	+	0.0236183i	$0.00751865\pi$
$228$		0			0
$229$	1.11477e12	+	1.93085e12i	0.116975	+	0.202606i	0.918567	−	0.395264i	$-0.129347\pi$
$229$	1.11477e12	+	1.93085e12i	0.116975	+	0.202606i	−0.801593	+	0.597870i	$0.796014\pi$
$230$		0			0
$231$	−4.94368e12	−	1.81989e12i	−0.494521	−	0.182046i
$232$		0			0
$233$	1.22410e12	+	2.12021e12i	0.116778	+	0.202265i	0.918489	−	0.395447i	$-0.129410\pi$
$233$	1.22410e12	+	2.12021e12i	0.116778	+	0.202265i	−0.801711	+	0.597712i	$0.796077\pi$
$234$		0			0
$235$	−6.43848e12	+	1.11518e13i	−0.586017	+	1.01501i
$236$		0			0
$237$	−8.72511e12			−0.757975
$238$		0			0
$239$	−5.62092e12			−0.466250			−0.233125	−	0.972447i	$-0.574895\pi$
$239$	−5.62092e12			−0.466250			−0.233125	+	0.972447i	$0.574895\pi$
$240$		0			0
$241$	−2.21149e12	+	3.83041e12i	−0.175223	+	0.303495i	−0.940238	−	0.340517i	$-0.889398\pi$
$241$	−2.21149e12	+	3.83041e12i	−0.175223	+	0.303495i	0.765015	+	0.644012i	$0.222731\pi$
$242$		0			0
$243$	5.24813e12	+	9.09003e12i	0.397347	+	0.688226i
$244$		0			0
$245$	−2.23911e13	+	7.96437e12i	−1.62055	+	0.576419i
$246$		0			0
$247$	−6.02930e12	−	1.04431e13i	−0.417286	−	0.722761i
$248$		0			0
$249$	−5.07433e12	+	8.78900e12i	−0.335956	+	0.581893i
$250$		0			0
$251$	−9.24639e12			−0.585823			−0.292912	−	0.956140i	$-0.594624\pi$
$251$	−9.24639e12			−0.585823			−0.292912	+	0.956140i	$0.594624\pi$
$252$		0			0
$253$	−8.94999e12			−0.542825
$254$		0			0
$255$	−1.35545e13	+	2.34770e13i	−0.787248	+	1.36355i
$256$		0			0
$257$	5.95716e12	+	1.03181e13i	0.331441	+	0.574073i	0.982795	−	0.184701i	$-0.0591317\pi$
$257$	5.95716e12	+	1.03181e13i	0.331441	+	0.574073i	−0.651353	+	0.758775i	$0.725798\pi$
$258$		0			0
$259$	2.23966e13	+	8.24475e12i	1.19408	+	0.439571i
$260$		0			0
$261$	2.71584e12	+	4.70398e12i	0.138798	+	0.240404i
$262$		0			0
$263$	−1.14495e13	+	1.98311e13i	−0.561086	+	0.971830i	0.436316	+	0.899794i	$0.356283\pi$
$263$	−1.14495e13	+	1.98311e13i	−0.561086	+	0.971830i	−0.997402	+	0.0720364i	$0.977050\pi$
$264$		0			0
$265$	6.93888e13			3.26165
$266$		0			0
$267$	−6.40062e12			−0.288676
$268$		0			0
$269$	3.05703e12	−	5.29494e12i	0.132331	−	0.229205i	−0.792243	−	0.610205i	$-0.791087\pi$
$269$	3.05703e12	−	5.29494e12i	0.132331	−	0.229205i	0.924575	+	0.381000i	$0.124420\pi$
$270$		0			0
$271$	5.40943e11	+	9.36941e11i	0.0224813	+	0.0389387i	0.877047	−	0.480404i	$-0.159510\pi$
$271$	5.40943e11	+	9.36941e11i	0.0224813	+	0.0389387i	−0.854566	+	0.519343i	$0.826177\pi$
$272$		0			0
$273$	−1.49811e12	−	8.68209e12i	−0.0597928	−	0.346521i
$274$		0			0
$275$	−1.78885e13	−	3.09837e13i	−0.685873	−	1.18797i
$276$		0			0
$277$	−5.14069e12	+	8.90394e12i	−0.189401	+	0.328053i	−0.945051	−	0.326924i	$-0.893988\pi$
$277$	−5.14069e12	+	8.90394e12i	−0.189401	+	0.328053i	0.755649	+	0.654976i	$0.227321\pi$
$278$		0			0
$279$	1.13252e13			0.401073
$280$		0			0
$281$	2.64698e13			0.901292			0.450646	−	0.892703i	$-0.351194\pi$
$281$	2.64698e13			0.901292			0.450646	+	0.892703i	$0.351194\pi$
$282$		0			0
$283$	1.20657e13	−	2.08984e13i	0.395118	−	0.684364i	−0.597999	−	0.801497i	$-0.704037\pi$
$283$	1.20657e13	−	2.08984e13i	0.395118	−	0.684364i	0.993116	+	0.117133i	$0.0373705\pi$
$284$		0			0
$285$	−3.66729e13	−	6.35193e13i	−1.15531	−	2.00105i
$286$		0			0
$287$	−1.64880e13	+	1.37325e13i	−0.499825	+	0.416292i
$288$		0			0
$289$	−8.23257e12	−	1.42592e13i	−0.240213	−	0.416062i
$290$		0			0
$291$	2.26418e13	−	3.92167e13i	0.636062	−	1.10169i
$292$		0			0
$293$	−2.33914e13			−0.632825			−0.316413	−	0.948622i	$-0.602478\pi$
$293$	−2.33914e13			−0.632825			−0.316413	+	0.948622i	$0.602478\pi$
$294$		0			0
$295$	4.06600e13			1.05961
$296$		0			0
$297$	−1.50471e13	+	2.60624e13i	−0.377827	+	0.654416i
$298$		0			0
$299$	−7.48412e12	−	1.29629e13i	−0.181113	−	0.313697i
$300$		0			0
$301$	1.93668e13	−	1.61301e13i	0.451795	−	0.376289i
$302$		0			0
$303$	2.25920e11	+	3.91305e11i	0.00508183	+	0.00880198i
$304$		0			0
$305$	−3.07210e13	+	5.32103e13i	−0.666478	+	1.15437i
$306$		0			0
$307$	−3.45847e13			−0.723807			−0.361903	−	0.932216i	$-0.617873\pi$
$307$	−3.45847e13			−0.723807			−0.361903	+	0.932216i	$0.617873\pi$
$308$		0			0
$309$	4.52191e13			0.913166
$310$		0			0
$311$	3.75500e13	−	6.50385e13i	0.731860	−	1.26762i	−0.224228	−	0.974537i	$-0.571986\pi$
$311$	3.75500e13	−	6.50385e13i	0.731860	−	1.26762i	0.956087	−	0.293082i	$-0.0946807\pi$
$312$		0			0
$313$	−1.89963e13	−	3.29025e13i	−0.357417	−	0.619064i	0.630112	−	0.776505i	$-0.283009\pi$
$313$	−1.89963e13	−	3.29025e13i	−0.357417	−	0.619064i	−0.987528	+	0.157440i	$0.949676\pi$
$314$		0			0
$315$	6.98641e12	+	4.04888e13i	0.126925	+	0.735575i
$316$		0			0
$317$	−3.61930e12	−	6.26882e12i	−0.0635037	−	0.109992i	0.832526	−	0.553987i	$-0.186894\pi$
$317$	−3.61930e12	−	6.26882e12i	−0.0635037	−	0.109992i	−0.896029	+	0.443995i	$0.853561\pi$
$318$		0			0
$319$	−1.32168e13	+	2.28923e13i	−0.224016	+	0.388007i
$320$		0			0
$321$	1.16981e13			0.191574
$322$		0			0
$323$	1.37274e14			2.17257
$324$		0			0
$325$	2.99172e13	−	5.18182e13i	0.457681	−	0.792727i
$326$		0			0
$327$	−1.54025e13	−	2.66779e13i	−0.227813	−	0.394583i
$328$		0			0
$329$	4.47085e13	+	1.64583e13i	0.639459	+	0.235401i
$330$		0			0
$331$	8.29737e12	+	1.43715e13i	0.114785	+	0.198814i	0.917694	−	0.397288i	$-0.130049\pi$
$331$	8.29737e12	+	1.43715e13i	0.114785	+	0.198814i	−0.802909	+	0.596102i	$0.796715\pi$
$332$		0			0
$333$	2.06305e13	−	3.57331e13i	0.276100	−	0.478220i
$334$		0			0
$335$	−2.98351e13			−0.386350
$336$		0			0
$337$	4.49127e13			0.562865			0.281433	−	0.959581i	$-0.409190\pi$
$337$	4.49127e13			0.562865			0.281433	+	0.959581i	$0.409190\pi$
$338$		0			0
$339$	3.98232e13	−	6.89758e13i	0.483101	−	0.836756i
$340$		0			0
$341$	2.75575e13	+	4.77310e13i	0.323661	+	0.560598i
$342$		0			0
$343$	4.31234e13	+	7.66248e13i	0.490451	+	0.871469i
$344$		0			0
$345$	−4.55217e13	−	7.88459e13i	−0.501434	−	0.868509i
$346$		0			0
$347$	−6.08265e13	+	1.05355e14i	−0.649054	+	1.12419i	0.334295	+	0.942468i	$0.391502\pi$
$347$	−6.08265e13	+	1.05355e14i	−0.649054	+	1.12419i	−0.983349	+	0.181726i	$0.941832\pi$
$348$		0			0
$349$	−2.22919e13			−0.230466			−0.115233	−	0.993338i	$-0.536761\pi$
$349$	−2.22919e13			−0.230466			−0.115233	+	0.993338i	$0.536761\pi$
$350$		0			0
$351$	−5.03306e13			−0.504247
$352$		0			0
$353$	8.78147e13	−	1.52099e14i	0.852719	−	1.47695i	−0.0260253	−	0.999661i	$-0.508285\pi$
$353$	8.78147e13	−	1.52099e14i	0.852719	−	1.47695i	0.878745	−	0.477292i	$-0.158382\pi$
$354$		0			0
$355$	−1.44541e13	−	2.50353e13i	−0.136062	−	0.235666i
$356$		0			0
$357$	9.41215e13	+	3.46485e13i	0.859041	+	0.316234i
$358$		0			0
$359$	−8.08719e13	−	1.40074e14i	−0.715778	−	1.23976i	−0.962659	−	0.270717i	$-0.912739\pi$
$359$	−8.08719e13	−	1.40074e14i	−0.715778	−	1.23976i	0.246881	−	0.969046i	$-0.420594\pi$
$360$		0			0
$361$	−1.27458e14	+	2.20764e14i	−1.09415	+	1.89513i
$362$		0			0
$363$	4.60216e13			0.383244
$364$		0			0
$365$	−2.21311e13			−0.178810
$366$		0			0
$367$	−3.77419e12	+	6.53709e12i	−0.0295911	+	0.0512532i	−0.880442	−	0.474154i	$-0.842754\pi$
$367$	−3.77419e12	+	6.53709e12i	−0.0295911	+	0.0512532i	0.850851	+	0.525408i	$0.176087\pi$
$368$		0			0
$369$	1.85488e13	+	3.21275e13i	0.141147	+	0.244473i
$370$		0			0
$371$	−4.36526e13	−	2.52983e14i	−0.322444	−	1.86868i
$372$		0			0
$373$	−1.17613e14	−	2.03712e14i	−0.843445	−	1.46089i	−0.886965	−	0.461837i	$-0.847191\pi$
$373$	−1.17613e14	−	2.03712e14i	−0.843445	−	1.46089i	0.0435200	−	0.999053i	$-0.486143\pi$
$374$		0			0
$375$	8.90538e13	−	1.54246e14i	0.620127	−	1.07409i
$376$		0			0
$377$	−4.42085e13			−0.298971
$378$		0			0
$379$	−5.99930e13			−0.394080			−0.197040	−	0.980395i	$-0.563133\pi$
$379$	−5.99930e13			−0.394080			−0.197040	+	0.980395i	$0.563133\pi$
$380$		0			0
$381$	7.14013e13	−	1.23671e14i	0.455637	−	0.789186i
$382$		0			0
$383$	−7.45203e12	−	1.29073e13i	−0.0462042	−	0.0800280i	0.841998	−	0.539480i	$-0.181379\pi$
$383$	−7.45203e12	−	1.29073e13i	−0.0462042	−	0.0800280i	−0.888203	+	0.459452i	$0.848046\pi$
$384$		0			0
$385$	−1.53643e14	+	1.27966e14i	−0.925720	+	0.771009i
$386$		0			0
$387$	−2.17874e13	−	3.77368e13i	−0.127583	−	0.220981i
$388$		0			0
$389$	1.66207e14	−	2.87879e14i	0.946078	−	1.63866i	0.192501	−	0.981297i	$-0.438340\pi$
$389$	1.66207e14	−	2.87879e14i	0.946078	−	1.63866i	0.753578	−	0.657359i	$-0.228326\pi$
$390$		0			0
$391$	1.70397e14			0.942951
$392$		0			0
$393$	−3.53936e13			−0.190443
$394$		0			0
$395$	−1.65586e14	+	2.86803e14i	−0.866441	+	1.50072i
$396$		0			0
$397$	7.83509e13	+	1.35708e14i	0.398746	+	0.690648i	0.993571	−	0.113207i	$-0.0361122\pi$
$397$	7.83509e13	+	1.35708e14i	0.398746	+	0.690648i	−0.594826	+	0.803855i	$0.702779\pi$
$398$		0			0
$399$	−2.08513e14	+	1.73665e14i	−1.03224	+	0.859728i
$400$		0			0
$401$	1.42128e14	+	2.46172e14i	0.684518	+	1.18562i	0.973588	+	0.228311i	$0.0733204\pi$
$401$	1.42128e14	+	2.46172e14i	0.684518	+	1.18562i	−0.289071	+	0.957308i	$0.593346\pi$
$402$		0			0
$403$	−4.60880e13	+	7.98268e13i	−0.215978	+	0.374086i
$404$		0			0
$405$	−1.42451e14			−0.649625
$406$		0			0
$407$	2.00800e14			0.891240
$408$		0			0
$409$	−1.39877e13	+	2.42275e13i	−0.0604323	+	0.104672i	−0.894659	−	0.446750i	$-0.852581\pi$
$409$	−1.39877e13	+	2.42275e13i	−0.0604323	+	0.104672i	0.834226	+	0.551422i	$0.185915\pi$
$410$		0			0
$411$	−9.75352e13	−	1.68936e14i	−0.410234	−	0.710545i
$412$		0			0
$413$	−2.55793e13	−	1.48241e14i	−0.104752	−	0.607078i
$414$		0			0
$415$	1.92602e14	+	3.33597e14i	0.768062	+	1.33032i
$416$		0			0
$417$	1.26602e14	−	2.19281e14i	0.491691	−	0.851634i
$418$		0			0
$419$	3.30060e14			1.24858			0.624288	−	0.781194i	$-0.285389\pi$
$419$	3.30060e14			1.24858			0.624288	+	0.781194i	$0.285389\pi$
$420$		0			0
$421$	−2.84800e14			−1.04951			−0.524756	−	0.851252i	$-0.675844\pi$
$421$	−2.84800e14			−1.04951			−0.524756	+	0.851252i	$0.675844\pi$
$422$		0			0
$423$	4.11830e13	−	7.13311e13i	0.147858	−	0.256098i
$424$		0			0
$425$	3.40574e14	+	5.89892e14i	1.19144	+	2.06364i
$426$		0			0
$427$	2.13325e14	+	7.85302e13i	0.727258	+	0.267722i
$428$		0			0
$429$	−3.70638e13	−	6.41964e13i	−0.123150	−	0.213302i
$430$		0			0
$431$	−1.10769e14	+	1.91858e14i	−0.358753	+	0.621378i	−0.987753	−	0.156028i	$-0.950131\pi$
$431$	−1.10769e14	+	1.91858e14i	−0.358753	+	0.621378i	0.629000	+	0.777405i	$0.283464\pi$
$432$		0			0
$433$	−9.21086e13			−0.290815			−0.145407	−	0.989372i	$-0.546449\pi$
$433$	−9.21086e13			−0.290815			−0.145407	+	0.989372i	$0.546449\pi$
$434$		0			0
$435$	−2.68896e14			−0.827738
$436$		0			0
$437$	−2.30512e14	+	3.99259e14i	−0.691904	+	1.19841i
$438$		0			0
$439$	−1.10254e14	−	1.90966e14i	−0.322731	−	0.558986i	0.658320	−	0.752738i	$-0.271268\pi$
$439$	−1.10254e14	−	1.90966e14i	−0.322731	−	0.558986i	−0.981050	+	0.193752i	$0.937934\pi$
$440$		0			0
$441$	1.43222e14	−	5.09432e13i	0.408882	−	0.145437i
$442$		0			0
$443$	2.47563e14	+	4.28791e14i	0.689389	+	1.19406i	0.972036	+	0.234833i	$0.0754543\pi$
$443$	2.47563e14	+	4.28791e14i	0.689389	+	1.19406i	−0.282647	+	0.959224i	$0.591212\pi$
$444$		0			0
$445$	−1.21471e14	+	2.10395e14i	−0.329985	+	0.571551i
$446$		0			0
$447$	4.83080e13			0.128035
$448$		0			0
$449$	2.17377e13			0.0562158			0.0281079	−	0.999605i	$-0.491052\pi$
$449$	2.17377e13			0.0562158			0.0281079	+	0.999605i	$0.491052\pi$
$450$		0			0
$451$	−9.02691e13	+	1.56351e14i	−0.227807	+	0.394574i
$452$		0			0
$453$	−2.51070e14	−	4.34866e14i	−0.618379	−	1.07106i
$454$		0			0
$455$	−3.13820e14	−	1.15525e14i	−0.754429	−	0.277724i
$456$		0			0
$457$	3.48611e14	+	6.03813e14i	0.818093	+	1.41698i	0.907086	+	0.420946i	$0.138302\pi$
$457$	3.48611e14	+	6.03813e14i	0.818093	+	1.41698i	−0.0889930	+	0.996032i	$0.528365\pi$
$458$		0			0
$459$	2.86479e14	−	4.96196e14i	0.656331	−	1.13680i
$460$		0			0
$461$	−2.66359e14			−0.595817			−0.297908	−	0.954594i	$-0.596289\pi$
$461$	−2.66359e14			−0.595817			−0.297908	+	0.954594i	$0.596289\pi$
$462$		0			0
$463$	−1.80648e14			−0.394583			−0.197291	−	0.980345i	$-0.563214\pi$
$463$	−1.80648e14			−0.394583			−0.197291	+	0.980345i	$0.563214\pi$
$464$		0			0
$465$	−2.80328e14	+	4.85542e14i	−0.597963	+	1.03570i
$466$		0			0
$467$	3.56563e14	+	6.17585e14i	0.742837	+	1.28663i	0.951199	+	0.308579i	$0.0998534\pi$
$467$	3.56563e14	+	6.17585e14i	0.742837	+	1.28663i	−0.208362	+	0.978052i	$0.566813\pi$
$468$		0			0
$469$	1.87693e13	+	1.08775e14i	0.0381942	+	0.221349i
$470$		0			0
$471$	−2.53724e14	−	4.39463e14i	−0.504367	−	0.873589i
$472$		0			0
$473$	1.06030e14	−	1.83649e14i	0.205917	−	0.356658i
$474$		0			0
$475$	−1.84291e15			−3.49695
$476$		0			0
$477$	−4.43837e14			−0.822949
$478$		0			0
$479$	−1.92124e14	+	3.32769e14i	−0.348126	+	0.602972i	−0.985917	−	0.167238i	$-0.946515\pi$
$479$	−1.92124e14	+	3.32769e14i	−0.348126	+	0.602972i	0.637790	+	0.770210i	$0.279849\pi$
$480$		0			0
$481$	1.67912e14	+	2.90832e14i	0.297361	+	0.515044i
$482$		0			0
$483$	−2.58825e14	+	2.15569e14i	−0.448019	+	0.373144i
$484$		0			0
$485$	−8.59395e14	−	1.48852e15i	−1.45416	−	2.51869i
$486$		0			0
$487$	9.35583e13	−	1.62048e14i	0.154765	−	0.268061i	−0.778208	−	0.628006i	$-0.783871\pi$
$487$	9.35583e13	−	1.62048e14i	0.154765	−	0.268061i	0.932973	+	0.359945i	$0.117205\pi$
$488$		0			0
$489$	4.03779e14			0.653048
$490$		0			0
$491$	−3.62679e14			−0.573553			−0.286777	−	0.957997i	$-0.592584\pi$
$491$	−3.62679e14			−0.573553			−0.286777	+	0.957997i	$0.592584\pi$
$492$		0			0
$493$	2.51632e14	−	4.35840e14i	0.389142	−	0.674014i
$494$		0			0
$495$	1.72847e14	+	2.99379e14i	0.261416	+	0.452786i
$496$		0			0
$497$	−8.21825e13	+	6.84478e13i	−0.121568	+	0.101251i
$498$		0			0
$499$	1.98331e13	+	3.43519e13i	0.0286970	+	0.0497047i	0.880017	−	0.474942i	$-0.157531\pi$
$499$	1.98331e13	+	3.43519e13i	0.0286970	+	0.0497047i	−0.851320	+	0.524646i	$0.824198\pi$
$500$		0			0
$501$	1.42379e14	−	2.46607e14i	0.201529	−	0.349059i
$502$		0			0
$503$	2.17027e14			0.300531			0.150265	−	0.988646i	$-0.451987\pi$
$503$	2.17027e14			0.300531			0.150265	+	0.988646i	$0.451987\pi$
$504$		0			0
$505$	1.71501e13			0.0232361
$506$		0			0
$507$	−2.21760e14	+	3.84100e14i	−0.293995	+	0.509214i
$508$		0			0
$509$	4.08850e13	+	7.08148e13i	0.0530415	+	0.0918706i	0.891327	−	0.453361i	$-0.149775\pi$
$509$	4.08850e13	+	7.08148e13i	0.0530415	+	0.0918706i	−0.838286	+	0.545231i	$0.816442\pi$
$510$		0			0
$511$	1.39227e13	+	8.06872e13i	0.0176770	+	0.102445i
$512$		0			0
$513$	7.75095e14	+	1.34250e15i	0.963184	+	1.66828i
$514$		0			0
$515$	8.58171e14	−	1.48640e15i	1.04384	−	1.80798i
$516$		0			0
$517$	4.00840e14			0.477280
$518$		0			0
$519$	1.27740e15			1.48904
$520$		0			0
$521$	−4.50760e14	+	7.80740e14i	−0.514444	+	0.891044i	0.485415	+	0.874284i	$0.338668\pi$
$521$	−4.50760e14	+	7.80740e14i	−0.514444	+	0.891044i	−0.999860	+	0.0167599i	$0.994665\pi$
$522$		0			0
$523$	5.25079e14	+	9.09464e14i	0.586767	+	1.01631i	0.994653	+	0.103277i	$0.0329327\pi$
$523$	5.25079e14	+	9.09464e14i	0.586767	+	1.01631i	−0.407886	+	0.913033i	$0.633734\pi$
$524$		0			0
$525$	−1.26359e15	−	4.65159e14i	−1.38270	−	0.509008i
$526$		0			0
$527$	−5.24661e14	−	9.08739e14i	−0.562238	−	0.973824i
$528$		0			0
$529$	1.90272e14	−	3.29561e14i	0.199696	−	0.345884i
$530$		0			0
$531$	−2.60077e14			−0.267351
$532$		0			0
$533$	−3.01938e14			−0.304031
$534$		0			0
$535$	2.22007e14	−	3.84528e14i	0.218989	−	0.379299i
$536$		0			0
$537$	1.88323e14	+	3.26184e14i	0.181989	+	0.315213i
$538$		0			0
$539$	5.63205e14	+	4.79661e14i	0.533246	+	0.454147i
$540$		0			0
$541$	8.68979e14	+	1.50512e15i	0.806166	+	1.39632i	0.915501	+	0.402315i	$0.131794\pi$
$541$	8.68979e14	+	1.50512e15i	0.806166	+	1.39632i	−0.109336	+	0.994005i	$0.534872\pi$
$542$		0			0
$543$	1.81397e14	−	3.14188e14i	0.164904	−	0.285621i
$544$		0			0
$545$	−1.16924e15			−1.04165
$546$		0			0
$547$	−3.12142e14			−0.272535			−0.136268	−	0.990672i	$-0.543511\pi$
$547$	−3.12142e14			−0.272535			−0.136268	+	0.990672i	$0.543511\pi$
$548$		0			0
$549$	1.96503e14	−	3.40354e14i	0.168160	−	0.291261i
$550$		0			0
$551$	6.80814e14	+	1.17921e15i	0.571077	+	0.989135i
$552$		0			0
$553$	1.14982e15	+	4.23277e14i	0.945456	+	0.348046i
$554$		0			0
$555$	1.02131e15	+	1.76897e15i	0.823281	+	1.42596i
$556$		0			0
$557$	5.34209e14	−	9.25277e14i	0.422190	−	0.731254i	−0.573964	−	0.818881i	$-0.694595\pi$
$557$	5.34209e14	−	9.25277e14i	0.422190	−	0.731254i	0.996153	+	0.0876266i	$0.0279282\pi$
$558$		0			0
$559$	3.54655e14			0.274815
$560$		0			0
$561$	8.43860e14			0.641172
$562$		0			0
$563$	3.17399e14	−	5.49751e14i	0.236488	−	0.409609i	−0.723216	−	0.690622i	$-0.757337\pi$
$563$	3.17399e14	−	5.49751e14i	0.236488	−	0.409609i	0.959704	+	0.281013i	$0.0906702\pi$
$564$		0			0
$565$	−1.51154e15	−	2.61806e15i	−1.10447	−	1.91299i
$566$		0			0
$567$	8.96162e13	+	5.19359e14i	0.0642214	+	0.372186i
$568$		0			0
$569$	−3.27952e14	−	5.68030e14i	−0.230512	−	0.399258i	0.727447	−	0.686164i	$-0.240707\pi$
$569$	−3.27952e14	−	5.68030e14i	−0.230512	−	0.399258i	−0.957959	+	0.286906i	$0.907373\pi$
$570$		0			0
$571$	8.48277e14	−	1.46926e15i	0.584843	−	1.01298i	−0.410052	−	0.912062i	$-0.634489\pi$
$571$	8.48277e14	−	1.46926e15i	0.584843	−	1.01298i	0.994895	−	0.100916i	$-0.0321772\pi$
$572$		0			0
$573$	3.59256e14			0.242971
$574$		0			0
$575$	−2.28759e15			−1.51777
$576$		0			0
$577$	7.34164e14	−	1.27161e15i	0.477888	−	0.827726i	−0.521791	−	0.853074i	$-0.674736\pi$
$577$	7.34164e14	−	1.27161e15i	0.477888	−	0.827726i	0.999679	+	0.0253472i	$0.00806914\pi$
$578$		0			0
$579$	−1.07128e15	−	1.85551e15i	−0.684180	−	1.18503i
$580$		0			0
$581$	1.09509e15	−	9.12070e14i	0.686246	−	0.571557i
$582$		0			0
$583$	−1.07998e15	−	1.87059e15i	−0.664110	−	1.15027i
$584$		0			0
$585$	−2.89074e14	+	5.00691e14i	−0.174442	+	0.302143i
$586$		0			0
$587$	−9.32618e14			−0.552324			−0.276162	−	0.961111i	$-0.589063\pi$
$587$	−9.32618e14			−0.552324			−0.276162	+	0.961111i	$0.589063\pi$
$588$		0			0
$589$	2.83904e15			1.65020
$590$		0			0
$591$	1.23991e15	−	2.14759e15i	0.707393	−	1.22524i
$592$		0			0
$593$	2.42040e14	+	4.19225e14i	0.135546	+	0.234772i	0.925806	−	0.378000i	$-0.123388\pi$
$593$	2.42040e14	+	4.19225e14i	0.135546	+	0.234772i	−0.790260	+	0.612772i	$0.790055\pi$
$594$		0			0
$595$	2.92518e15	−	2.43631e15i	1.60808	−	1.33933i
$596$		0			0
$597$	−2.55623e14	−	4.42752e14i	−0.137956	−	0.238947i
$598$		0			0
$599$	−1.72211e15	+	2.98279e15i	−0.912461	+	1.58043i	−0.101885	+	0.994796i	$0.532487\pi$
$599$	−1.72211e15	+	2.98279e15i	−0.912461	+	1.58043i	−0.810576	+	0.585633i	$0.800846\pi$
$600$		0			0
$601$	−1.43451e15			−0.746268			−0.373134	−	0.927778i	$-0.621717\pi$
$601$	−1.43451e15			−0.746268			−0.373134	+	0.927778i	$0.621717\pi$
$602$		0			0
$603$	1.90837e14			0.0974802
$604$		0			0
$605$	8.73401e14	−	1.51277e15i	0.438086	−	0.758786i
$606$		0			0
$607$	5.74851e14	+	9.95671e14i	0.283151	+	0.490431i	0.972159	−	0.234322i	$-0.0752871\pi$
$607$	5.74851e14	+	9.95671e14i	0.283151	+	0.490431i	−0.689008	+	0.724753i	$0.741954\pi$
$608$		0			0
$609$	1.69163e14	+	9.80361e14i	0.0818295	+	0.474232i
$610$		0			0
$611$	3.35189e14	+	5.80564e14i	0.159244	+	0.275819i
$612$		0			0
$613$	1.03786e15	−	1.79763e15i	0.484293	−	0.838820i	−0.515544	−	0.856863i	$-0.672410\pi$
$613$	1.03786e15	−	1.79763e15i	0.484293	−	0.838820i	0.999837	+	0.0180430i	$0.00574358\pi$
$614$		0			0
$615$	−1.83652e15			−0.841747
$616$		0			0
$617$	1.93418e14			0.0870822			0.0435411	−	0.999052i	$-0.486136\pi$
$617$	1.93418e14			0.0870822			0.0435411	+	0.999052i	$0.486136\pi$
$618$		0			0
$619$	2.02998e15	−	3.51604e15i	0.897830	−	1.55509i	0.0675677	−	0.997715i	$-0.478476\pi$
$619$	2.02998e15	−	3.51604e15i	0.897830	−	1.55509i	0.830262	−	0.557373i	$-0.188191\pi$
$620$		0			0
$621$	9.62118e14	+	1.66644e15i	0.418047	+	0.724078i
$622$		0			0
$623$	8.43492e14	+	3.10510e14i	0.360078	+	0.132554i
$624$		0			0
$625$	−1.04550e15	−	1.81087e15i	−0.438516	−	0.759533i
$626$		0			0
$627$	−1.14157e15	+	1.97726e15i	−0.470469	+	0.814877i
$628$		0			0
$629$	−3.82298e15			−1.54819
$630$		0			0
$631$	3.98410e15			1.58551			0.792754	−	0.609541i	$-0.208646\pi$
$631$	3.98410e15			1.58551			0.792754	+	0.609541i	$0.208646\pi$
$632$		0			0
$633$	−1.35232e15	+	2.34229e15i	−0.528884	+	0.916054i
$634$		0			0
$635$	−2.71012e15	−	4.69406e15i	−1.04168	−	1.80424i
$636$		0			0
$637$	−2.23765e14	+	1.21683e15i	−0.0845330	+	0.459687i
$638$		0			0
$639$	9.24542e13	+	1.60135e14i	0.0343299	+	0.0594611i
$640$		0			0
$641$	4.16479e14	−	7.21364e14i	0.152011	−	0.263290i	−0.779956	−	0.625835i	$-0.784758\pi$
$641$	4.16479e14	−	7.21364e14i	0.152011	−	0.263290i	0.931967	+	0.362544i	$0.118092\pi$
$642$		0			0
$643$	−3.12211e15			−1.12018			−0.560090	−	0.828432i	$-0.689233\pi$
$643$	−3.12211e15			−1.12018			−0.560090	+	0.828432i	$0.689233\pi$
$644$		0			0
$645$	2.15717e15			0.760861
$646$		0			0
$647$	−1.63841e15	+	2.83782e15i	−0.568133	+	0.984036i	0.428617	+	0.903486i	$0.359001\pi$
$647$	−1.63841e15	+	2.83782e15i	−0.568133	+	0.984036i	−0.996751	+	0.0805495i	$0.974332\pi$
$648$		0			0
$649$	−6.32842e14	−	1.09611e15i	−0.215749	−	0.373689i
$650$		0			0
$651$	1.94658e15	+	7.16585e14i	0.652495	+	0.240200i
$652$		0			0
$653$	−8.72175e14	−	1.51065e15i	−0.287463	−	0.497900i	0.685741	−	0.727846i	$-0.259478\pi$
$653$	−8.72175e14	−	1.51065e15i	−0.287463	−	0.497900i	−0.973203	+	0.229946i	$0.926145\pi$
$654$		0			0
$655$	−6.71702e14	+	1.16342e15i	−0.217695	+	0.377060i
$656$		0			0
$657$	1.41559e14			0.0451157
$658$		0			0
$659$	2.29700e15			0.719931			0.359965	−	0.932966i	$-0.382788\pi$
$659$	2.29700e15			0.719931			0.359965	+	0.932966i	$0.382788\pi$
$660$		0			0
$661$	−5.25673e14	+	9.10492e14i	−0.162034	+	0.280652i	−0.935598	−	0.353067i	$-0.885139\pi$
$661$	−5.25673e14	+	9.10492e14i	−0.162034	+	0.280652i	0.773564	+	0.633718i	$0.218472\pi$
$662$		0			0
$663$	7.05649e14	+	1.22222e15i	0.213926	+	0.370531i
$664$		0			0
$665$	1.75137e15	+	1.01498e16i	0.522227	+	3.02650i
$666$		0			0
$667$	8.45089e14	+	1.46374e15i	0.247862	+	0.429310i
$668$		0			0
$669$	7.33068e14	−	1.26971e15i	0.211495	−	0.366320i
$670$		0			0
$671$	1.91260e15			0.542812
$672$		0			0
$673$	1.81381e15			0.506417			0.253208	−	0.967412i	$-0.418514\pi$
$673$	1.81381e15			0.506417			0.253208	+	0.967412i	$0.418514\pi$
$674$		0			0
$675$	−3.84600e15	+	6.66147e15i	−1.05642	+	1.82978i
$676$		0			0
$677$	−1.34262e15	−	2.32549e15i	−0.362840	−	0.628458i	0.625587	−	0.780155i	$-0.284860\pi$
$677$	−1.34262e15	−	2.32549e15i	−0.362840	−	0.628458i	−0.988427	+	0.151697i	$0.951526\pi$
$678$		0			0
$679$	−4.88630e15	+	4.06968e15i	−1.29926	+	1.08212i
$680$		0			0
$681$	−1.37857e15	−	2.38776e15i	−0.360679	−	0.624714i
$682$		0			0
$683$	2.14946e15	−	3.72298e15i	0.553371	−	0.958466i	−0.444658	−	0.895701i	$-0.646675\pi$
$683$	2.14946e15	−	3.72298e15i	0.553371	−	0.958466i	0.998028	−	0.0627655i	$-0.0199920\pi$
$684$		0			0
$685$	−7.40412e15			−1.87575
$686$		0			0
$687$	7.05994e14			0.176011
$688$		0			0
$689$	1.80620e15	−	3.12843e15i	0.443159	−	0.767574i
$690$		0			0
$691$	3.40151e15	+	5.89158e15i	0.821376	+	1.42267i	0.904658	+	0.426139i	$0.140127\pi$
$691$	3.40151e15	+	5.89158e15i	0.821376	+	1.42267i	−0.0832817	+	0.996526i	$0.526540\pi$
$692$		0			0
$693$	9.82762e14	−	8.18518e14i	0.233569	−	0.194534i
$694$		0			0
$695$	−4.80533e15	−	8.32307e15i	−1.12410	−	1.94700i
$696$		0			0
$697$	1.71861e15	−	2.97672e15i	0.395728	−	0.685422i
$698$		0			0
$699$	7.75232e14			0.175714
$700$		0			0
$701$	−3.11376e15			−0.694763			−0.347381	−	0.937724i	$-0.612929\pi$
$701$	−3.11376e15			−0.694763			−0.347381	+	0.937724i	$0.612929\pi$
$702$		0			0
$703$	5.17172e15	−	8.95767e15i	1.13601	−	1.96762i
$704$		0			0
$705$	2.03877e15	+	3.53125e15i	0.440887	+	0.763638i
$706$		0			0
$707$	−1.07892e13	−	6.25272e13i	−0.00229711	−	0.0133126i
$708$		0			0
$709$	7.68726e14	+	1.33147e15i	0.161145	+	0.279111i	0.935280	−	0.353910i	$-0.115148\pi$
$709$	7.68726e14	+	1.33147e15i	0.161145	+	0.279111i	−0.774135	+	0.633021i	$0.781815\pi$
$710$		0			0
$711$	1.05915e15	−	1.83450e15i	0.218612	−	0.378648i
$712$		0			0
$713$	3.52407e15			0.716229
$714$		0			0
$715$	−2.81360e15			−0.563092
$716$		0			0
$717$	−8.89940e14	+	1.54142e15i	−0.175390	+	0.303785i
$718$		0			0
$719$	4.99680e14	+	8.65472e14i	0.0969803	+	0.167975i	0.910433	−	0.413656i	$-0.135748\pi$
$719$	4.99680e14	+	8.65472e14i	0.0969803	+	0.167975i	−0.813453	+	0.581631i	$0.802415\pi$
$720$		0			0
$721$	−5.95909e15	−	2.19369e15i	−1.13903	−	0.419307i
$722$		0			0
$723$	7.00275e14	+	1.21291e15i	0.131828	+	0.228333i
$724$		0			0
$725$	−3.37818e15	+	5.85118e15i	−0.626360	+	1.08489i
$726$		0			0
$727$	2.08683e15			0.381108			0.190554	−	0.981677i	$-0.438972\pi$
$727$	2.08683e15			0.381108			0.190554	+	0.981677i	$0.438972\pi$
$728$		0			0
$729$	5.42326e15			0.975571
$730$		0			0
$731$	−2.01868e15	+	3.49645e15i	−0.357702	+	0.619557i
$732$		0			0
$733$	4.40887e15	+	7.63638e15i	0.769583	+	1.33296i	0.937790	+	0.347204i	$0.112869\pi$
$733$	4.40887e15	+	7.63638e15i	0.769583	+	1.33296i	−0.168207	+	0.985752i	$0.553798\pi$
$734$		0			0
$735$	−1.36104e15	+	7.40128e15i	−0.234040	+	1.27270i
$736$		0			0
$737$	4.64360e14	+	8.04295e14i	0.0786654	+	0.136252i
$738$		0			0
$739$	−1.07263e15	+	1.85784e15i	−0.179021	+	0.310074i	−0.941545	−	0.336886i	$-0.890626\pi$
$739$	−1.07263e15	+	1.85784e15i	−0.179021	+	0.310074i	0.762524	+	0.646959i	$0.223960\pi$
$740$		0			0
$741$	−3.81840e15			−0.627886
$742$		0			0
$743$	−1.27352e15			−0.206332			−0.103166	−	0.994664i	$-0.532897\pi$
$743$	−1.27352e15			−0.206332			−0.103166	+	0.994664i	$0.532897\pi$
$744$		0			0
$745$	9.16792e14	−	1.58793e15i	0.146356	−	0.253497i
$746$		0			0
$747$	−1.23196e15	−	2.13381e15i	−0.193790	−	0.335655i
$748$		0			0
$749$	−1.54161e15	−	5.67504e14i	−0.238959	−	0.0879669i
$750$		0			0
$751$	−2.64201e15	−	4.57610e15i	−0.403566	−	0.698997i	0.590587	−	0.806974i	$-0.298896\pi$
$751$	−2.64201e15	−	4.57610e15i	−0.403566	−	0.698997i	−0.994153	+	0.107977i	$0.965563\pi$
$752$		0			0
$753$	−1.46395e15	+	2.53563e15i	−0.220371	+	0.381693i
$754$		0			0
$755$	−1.90593e16			−2.82748
$756$		0			0
$757$	1.16812e15			0.170789			0.0853943	−	0.996347i	$-0.472785\pi$
$757$	1.16812e15			0.170789			0.0853943	+	0.996347i	$0.472785\pi$
$758$		0			0
$759$	−1.41702e15	+	2.45435e15i	−0.204196	+	0.353677i
$760$		0			0
$761$	2.88187e15	+	4.99155e15i	0.409316	+	0.708957i	0.994813	−	0.101718i	$-0.0324339\pi$
$761$	2.88187e15	+	4.99155e15i	0.409316	+	0.708957i	−0.585497	+	0.810675i	$0.699101\pi$
$762$		0			0
$763$	7.35570e14	+	4.26290e15i	0.102977	+	0.596788i
$764$		0			0
$765$	−3.29078e15	−	5.69980e15i	−0.454110	−	0.786542i
$766$		0			0
$767$	1.05838e15	−	1.83318e15i	0.143969	−	0.249362i
$768$		0			0
$769$	−6.97867e14			−0.0935788			−0.0467894	−	0.998905i	$-0.514899\pi$
$769$	−6.97867e14			−0.0935788			−0.0467894	+	0.998905i	$0.514899\pi$
$770$		0			0
$771$	3.77271e15			0.498717
$772$		0			0
$773$	7.91519e14	−	1.37095e15i	0.103151	−	0.178663i	−0.809830	−	0.586664i	$-0.800441\pi$
$773$	7.91519e14	−	1.37095e15i	0.103151	−	0.178663i	0.912981	+	0.408001i	$0.133774\pi$
$774$		0			0
$775$	7.04361e15	+	1.21999e16i	0.904973	+	1.56746i
$776$		0			0
$777$	5.80693e15	−	4.83645e15i	0.735582	−	0.612648i
$778$		0			0
$779$	4.64986e15	+	8.05379e15i	0.580743	+	1.00588i
$780$		0			0
$781$	−4.49935e14	+	7.79311e14i	−0.0554076	+	0.0959688i
$782$		0			0
$783$	5.68321e15			0.690087
$784$		0			0
$785$	−1.92608e16			−2.30617
$786$		0			0
$787$	−5.71998e15	+	9.90730e15i	−0.675357	+	1.16975i	0.301007	+	0.953622i	$0.402677\pi$
$787$	−5.71998e15	+	9.90730e15i	−0.675357	+	1.16975i	−0.976364	+	0.216131i	$0.930656\pi$
$788$		0			0
$789$	3.62552e15	+	6.27958e15i	0.422130	+	0.731151i
$790$		0			0
$791$	−8.59421e15	+	7.15790e15i	−0.986814	+	0.821893i
$792$		0			0
$793$	1.59934e15	+	2.77014e15i	0.181108	+	0.313689i
$794$		0			0
$795$	1.09861e16	−	1.90285e16i	1.22694	−	2.12512i
$796$		0			0
$797$	8.30315e15			0.914581			0.457291	−	0.889317i	$-0.348820\pi$
$797$	8.30315e15			0.914581			0.457291	+	0.889317i	$0.348820\pi$
$798$		0			0
$799$	−7.63150e15			−0.829092
$800$		0			0
$801$	7.76979e14	−	1.34577e15i	0.0832588	−	0.144209i
$802$		0			0
$803$	3.44454e14	+	5.96611e14i	0.0364079	+	0.0630603i
$804$		0			0
$805$	2.17396e15	+	1.25989e16i	0.226660	+	1.31358i
$806$		0			0
$807$	−9.68020e14	−	1.67666e15i	−0.0995588	−	0.172441i
$808$		0			0
$809$	−2.71897e15	+	4.70939e15i	−0.275859	+	0.477802i	−0.970351	−	0.241698i	$-0.922296\pi$
$809$	−2.71897e15	+	4.70939e15i	−0.275859	+	0.477802i	0.694492	+	0.719500i	$0.255629\pi$
$810$		0			0
$811$	−1.07407e16			−1.07502			−0.537511	−	0.843257i	$-0.680635\pi$
$811$	−1.07407e16			−1.07502			−0.537511	+	0.843257i	$0.680635\pi$
$812$		0			0
$813$	3.42583e14			0.0338273
$814$		0			0
$815$	7.66295e15	−	1.32726e16i	0.746499	−	1.29297i
$816$		0			0
$817$	−5.46171e15	−	9.45996e15i	−0.524937	−	0.909218i
$818$		0			0
$819$	2.00732e15	+	7.38943e14i	0.190351	+	0.0700728i
$820$		0			0
$821$	−4.49814e15	−	7.79100e15i	−0.420868	−	0.728964i	0.575157	−	0.818043i	$-0.304941\pi$
$821$	−4.49814e15	−	7.79100e15i	−0.420868	−	0.728964i	−0.996025	+	0.0890790i	$0.971608\pi$
$822$		0			0
$823$	1.51454e15	−	2.62327e15i	0.139824	−	0.242183i	−0.787606	−	0.616180i	$-0.788680\pi$
$823$	1.51454e15	−	2.62327e15i	0.139824	−	0.242183i	0.927430	+	0.373997i	$0.122013\pi$
$824$		0			0
$825$	−1.13289e16			−1.03203
$826$		0			0
$827$	1.73797e16			1.56229			0.781146	−	0.624349i	$-0.214636\pi$
$827$	1.73797e16			1.56229			0.781146	+	0.624349i	$0.214636\pi$
$828$		0			0
$829$	−5.68518e15	+	9.84703e15i	−0.504307	+	0.873485i	0.495681	+	0.868505i	$0.334919\pi$
$829$	−5.68518e15	+	9.84703e15i	−0.504307	+	0.873485i	−0.999988	+	0.00497994i	$0.998415\pi$
$830$		0			0
$831$	1.62782e15	+	2.81946e15i	0.142495	+	0.246809i
$832$		0			0
$833$	−1.07227e16	−	9.13215e15i	−0.926312	−	0.788907i
$834$		0			0
$835$	−5.40415e15	−	9.36027e15i	−0.460736	−	0.798018i
$836$		0			0
$837$	5.92483e15	−	1.02621e16i	0.498524	−	0.863468i
$838$		0			0
$839$	−7.75755e14			−0.0644219			−0.0322109	−	0.999481i	$-0.510255\pi$
$839$	−7.75755e14			−0.0644219			−0.0322109	+	0.999481i	$0.510255\pi$
$840$		0			0
$841$	−7.20859e15			−0.590843
$842$		0			0
$843$	4.19087e15	−	7.25880e15i	0.339041	−	0.587236i
$844$		0			0
$845$	8.41716e15	+	1.45789e16i	0.672130	+	1.16416i
$846$		0			0
$847$	−6.06485e15	−	2.23262e15i	−0.478037	−	0.175977i
$848$		0			0
$849$	−3.82064e15	−	6.61754e15i	−0.297265	−	0.514878i
$850$		0			0
$851$	6.41961e15	−	1.11191e16i	0.493055	−	0.853997i
$852$		0			0
$853$	1.32476e16			1.00443			0.502213	−	0.864744i	$-0.332519\pi$
$853$	1.32476e16			1.00443			0.502213	+	0.864744i	$0.332519\pi$
$854$		0			0
$855$	1.78070e16			1.33284
$856$		0			0
$857$	1.01543e16	−	1.75878e16i	0.750339	−	1.29963i	−0.197320	−	0.980339i	$-0.563224\pi$
$857$	1.01543e16	−	1.75878e16i	0.750339	−	1.29963i	0.947658	−	0.319286i	$-0.103443\pi$
$858$		0			0
$859$	4.32609e15	+	7.49302e15i	0.315598	+	0.546631i	0.979564	−	0.201131i	$-0.0644617\pi$
$859$	4.32609e15	+	7.49302e15i	0.315598	+	0.546631i	−0.663967	+	0.747762i	$0.731128\pi$
$860$		0			0
$861$	1.15536e15	+	6.69572e15i	0.0832144	+	0.482258i
$862$		0			0
$863$	−8.96775e15	−	1.55326e16i	−0.637712	−	1.10455i	−0.985934	−	0.167137i	$-0.946548\pi$
$863$	−8.96775e15	−	1.55326e16i	−0.637712	−	1.10455i	0.348222	−	0.937412i	$-0.386786\pi$
$864$		0			0
$865$	2.42425e16	−	4.19893e16i	1.70212	−	2.94815i
$866$		0			0
$867$	−5.21374e15			−0.361447
$868$		0			0
$869$	1.03089e16			0.705671
$870$		0			0
$871$	−7.76610e14	+	1.34513e15i	−0.0524932	+	0.0909209i
$872$		0			0
$873$	5.49703e15	+	9.52113e15i	0.366901	+	0.635492i
$874$		0			0
$875$	−1.92186e16	+	1.60067e16i	−1.26671	+	1.05501i
$876$		0			0
$877$	2.96085e15	+	5.12835e15i	0.192717	+	0.333795i	0.946150	−	0.323730i	$-0.104937\pi$
$877$	2.96085e15	+	5.12835e15i	0.192717	+	0.333795i	−0.753433	+	0.657525i	$0.771604\pi$
$878$		0			0
$879$	−3.70348e15	+	6.41461e15i	−0.238051	+	0.412317i
$880$		0			0
$881$	−1.75489e16			−1.11399			−0.556996	−	0.830515i	$-0.688046\pi$
$881$	−1.75489e16			−1.11399			−0.556996	+	0.830515i	$0.688046\pi$
$882$		0			0
$883$	−1.15068e16			−0.721392			−0.360696	−	0.932684i	$-0.617461\pi$
$883$	−1.15068e16			−0.721392			−0.360696	+	0.932684i	$0.617461\pi$
$884$		0			0
$885$	6.43756e15	−	1.11502e16i	0.398596	−	0.690389i
$886$		0			0
$887$	−1.40939e15	−	2.44113e15i	−0.0861886	−	0.149283i	0.819708	−	0.572781i	$-0.194135\pi$
$887$	−1.40939e15	−	2.44113e15i	−0.0861886	−	0.149283i	−0.905897	+	0.423498i	$0.860802\pi$
$888$		0			0
$889$	−1.54090e16	+	1.28338e16i	−0.930713	+	0.775168i
$890$		0			0
$891$	2.21714e15	+	3.84020e15i	0.132271	+	0.229101i
$892$		0			0
$893$	1.03239e16	−	1.78815e16i	0.608358	−	1.05371i
$894$		0			0
$895$	1.42960e16			0.832124
$896$		0			0
$897$	−4.73974e15			−0.272519
$898$		0			0
$899$	5.20415e15	−	9.01385e15i	0.295578	−	0.511955i
$900$		0			0
$901$	2.05615e16	+	3.56136e16i	1.15364	+	1.99816i
$902$		0			0
$903$	−1.35708e15	−	7.86477e15i	−0.0752181	−	0.435917i
$904$		0			0
$905$	−6.88512e15	−	1.19254e16i	−0.377002	−	0.652987i
$906$		0			0
$907$	8.05870e15	−	1.39581e16i	0.435938	−	0.755067i	−0.561433	−	0.827522i	$-0.689750\pi$
$907$	8.05870e15	−	1.39581e16i	0.435938	−	0.755067i	0.997372	+	0.0724547i	$0.0230833\pi$
$908$		0			0
$909$	−1.09699e14			−0.00586273
$910$		0			0
$911$	3.29746e15			0.174112			0.0870560	−	0.996203i	$-0.472254\pi$
$911$	3.29746e15			0.174112			0.0870560	+	0.996203i	$0.472254\pi$
$912$		0			0
$913$	5.99541e15	−	1.03844e16i	0.312773	−	0.541739i
$914$		0			0
$915$	9.72790e15	+	1.68492e16i	0.501421	+	0.868487i
$916$		0			0
$917$	4.66426e15	+	1.71703e15i	0.237548	+	0.0874474i
$918$		0			0
$919$	3.65486e14	+	6.33041e14i	0.0183923	+	0.0318564i	0.875075	−	0.483987i	$-0.160812\pi$
$919$	3.65486e14	+	6.33041e14i	0.0183923	+	0.0318564i	−0.856683	+	0.515844i	$0.827479\pi$
$920$		0			0
$921$	−5.47567e15	+	9.48415e15i	−0.272276	+	0.471596i
$922$		0			0
$923$	−1.50497e15			−0.0739467
$924$		0			0
$925$	5.13238e16			2.49195
$926$		0			0
$927$	−5.48919e15	+	9.50756e15i	−0.263372	+	0.456174i
$928$		0			0
$929$	−4.35522e13	−	7.54347e13i	−0.00206502	−	0.00357672i	0.864991	−	0.501787i	$-0.167324\pi$
$929$	−4.35522e13	−	7.54347e13i	−0.00206502	−	0.00357672i	−0.867056	+	0.498211i	$0.833991\pi$
$930$		0			0
$931$	3.59033e16	−	1.27706e16i	1.68233	−	0.598394i
$932$		0			0
$933$	−1.18903e16	−	2.05947e16i	−0.550611	−	0.953686i
$934$		0			0
$935$	1.60148e16	−	2.77385e16i	0.732924	−	1.26946i
$936$		0			0
$937$	−2.38126e16			−1.07706			−0.538530	−	0.842607i	$-0.681020\pi$
$937$	−2.38126e16			−1.07706			−0.538530	+	0.842607i	$0.681020\pi$
$938$		0			0
$939$	−1.20305e16			−0.537801
$940$		0			0
$941$	−1.76098e16	+	3.05011e16i	−0.778059	+	1.34764i	0.155000	+	0.987914i	$0.450462\pi$
$941$	−1.76098e16	+	3.05011e16i	−0.778059	+	1.34764i	−0.933059	+	0.359723i	$0.882871\pi$
$942$		0			0
$943$	5.77183e15	+	9.99710e15i	0.252057	+	0.436576i
$944$		0			0
$945$	4.03430e16	+	1.48513e16i	1.74138	+	0.641046i
$946$		0			0
$947$	3.37685e15	+	5.84888e15i	0.144075	+	0.249544i	0.929027	−	0.370011i	$-0.120646\pi$
$947$	3.37685e15	+	5.84888e15i	0.144075	+	0.249544i	−0.784953	+	0.619556i	$0.787313\pi$
$948$		0			0
$949$	−5.76075e14	+	9.97791e14i	−0.0242949	+	0.0420800i
$950$		0			0
$951$	−2.29213e15			−0.0955533
$952$		0			0
$953$	2.08782e16			0.860363			0.430181	−	0.902742i	$-0.358450\pi$
$953$	2.08782e16			0.860363			0.430181	+	0.902742i	$0.358450\pi$
$954$		0			0
$955$	6.81799e15	−	1.18091e16i	0.277740	−	0.481059i
$956$		0			0
$957$	4.18516e15	+	7.24890e15i	0.168537	+	0.291915i
$958$		0			0
$959$	4.65795e15	+	2.69945e16i	0.185435	+	1.07467i
$960$		0			0
$961$	1.85344e15	+	3.21025e15i	0.0729456	+	0.126346i
$962$		0			0
$963$	−1.42004e15	+	2.45959e15i	−0.0552532	+	0.0957013i
$964$		0			0
$965$	−8.13232e16			−3.12834
$966$		0			0
$967$	1.65586e16			0.629766			0.314883	−	0.949130i	$-0.398035\pi$
$967$	1.65586e16			0.629766			0.314883	+	0.949130i	$0.398035\pi$
$968$		0			0
$969$	2.17341e16	−	3.76445e16i	0.817261	−	1.41554i
$970$		0			0
$971$	−2.20252e16	−	3.81487e16i	−0.818867	−	1.41832i	−0.906518	−	0.422167i	$-0.861269\pi$
$971$	−2.20252e16	−	3.81487e16i	−0.818867	−	1.41832i	0.0876514	−	0.996151i	$-0.472064\pi$
$972$		0			0
$973$	−2.73219e16	+	2.27557e16i	−1.00436	+	0.836507i
$974$		0			0
$975$	−9.47339e15	−	1.64084e16i	−0.344334	−	0.596404i
$976$		0			0
$977$	8.24600e15	−	1.42825e16i	0.296362	−	0.513315i	−0.678938	−	0.734195i	$-0.737560\pi$
$977$	8.24600e15	−	1.42825e16i	0.296362	−	0.513315i	0.975301	+	0.220880i	$0.0708930\pi$
$978$		0			0
$979$	7.56245e15			0.268756
$980$		0			0
$981$	7.47889e15			0.262820
$982$		0			0
$983$	−1.78403e16	+	3.09004e16i	−0.619953	+	1.07379i	0.369541	+	0.929214i	$0.379515\pi$
$983$	−1.78403e16	+	3.09004e16i	−0.619953	+	1.07379i	−0.989494	+	0.144575i	$0.953818\pi$
$984$		0			0
$985$	−4.70624e16	−	8.15144e16i	−1.61724	−	2.80114i
$986$		0			0
$987$	1.15919e16	−	9.65461e15i	0.393922	−	0.328088i
$988$		0			0
$989$	−6.77958e15	−	1.17426e16i	−0.227836	−	0.394624i
$990$		0			0
$991$	−1.46282e16	+	2.53368e16i	−0.486168	+	0.842067i	−0.999874	−	0.0158991i	$-0.994939\pi$
$991$	−1.46282e16	+	2.53368e16i	−0.486168	+	0.842067i	0.513706	+	0.857966i	$0.328272\pi$
$992$		0			0
$993$	5.25478e15			0.172716
$994$		0			0
$995$	−1.94049e16			−0.630791
$996$		0			0
$997$	−9.38310e15	+	1.62520e16i	−0.301664	+	0.522497i	−0.976513	−	0.215459i	$-0.930875\pi$
$997$	−9.38310e15	+	1.62520e16i	−0.301664	+	0.522497i	0.674849	+	0.737956i	$0.264209\pi$
$998$		0			0
$999$	−2.15859e16	−	3.73878e16i	−0.686372	−	1.18883i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.12.i.b.81.3		8
4.3	odd	2		14.12.c.b.11.2	yes	8
7.2	even	3	inner	112.12.i.b.65.3		8
12.11	even	2		126.12.g.c.109.4		8
28.3	even	6		98.12.a.h.1.2		4
28.11	odd	6		98.12.a.i.1.3		4
28.19	even	6		98.12.c.n.79.3		8
28.23	odd	6		14.12.c.b.9.2	✓	8
28.27	even	2		98.12.c.n.67.3		8
84.23	even	6		126.12.g.c.37.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.12.c.b.9.2	✓	8	28.23	odd	6
14.12.c.b.11.2	yes	8	4.3	odd	2
98.12.a.h.1.2		4	28.3	even	6
98.12.a.i.1.3		4	28.11	odd	6
98.12.c.n.67.3		8	28.27	even	2
98.12.c.n.79.3		8	28.19	even	6
112.12.i.b.65.3		8	7.2	even	3	inner
112.12.i.b.81.3		8	1.1	even	1	trivial
126.12.g.c.37.4		8	84.23	even	6
126.12.g.c.109.4		8	12.11	even	2

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)

\(f(q)\)	\(=\)	\(q+(158.327 - 274.230i) q^{3} +(-6009.47 - 10408.7i) q^{5} +(-34168.3 + 28457.9i) q^{7} +(38438.9 + 66578.1i) q^{9} +O(q^{10})\)	\(q+(158.327 - 274.230i) q^{3} +(-6009.47 - 10408.7i) q^{5} +(-34168.3 + 28457.9i) q^{7} +(38438.9 + 66578.1i) q^{9} +(-187066. + 324007. i) q^{11} -625709. q^{13} -3.80583e6 q^{15} +(3.56150e6 - 6.16869e6i) q^{17} +(9.63596e6 + 1.66900e7i) q^{19} +(2.39426e6 + 1.38756e7i) q^{21} +(1.19610e7 + 2.07171e7i) q^{23} +(-4.78134e7 + 8.28152e7i) q^{25} +8.04377e7 q^{27} +7.06535e7 q^{29} +(7.36573e7 - 1.27578e8i) q^{31} +(5.92349e7 + 1.02598e8i) q^{33} +(5.01544e8 + 1.84631e8i) q^{35} +(-2.68355e8 - 4.64805e8i) q^{37} +(-9.90663e7 + 1.71588e8i) q^{39} +4.82553e8 q^{41} -5.66805e8 q^{43} +(4.61994e8 - 8.00198e8i) q^{45} +(-5.35695e8 - 9.27851e8i) q^{47} +(3.57619e8 - 1.94472e9i) q^{49} +(-1.12776e9 - 1.95334e9i) q^{51} +(-2.88664e9 + 4.99981e9i) q^{53} +4.49666e9 q^{55} +6.10251e9 q^{57} +(-1.69150e9 + 2.92976e9i) q^{59} +(-2.55605e9 - 4.42721e9i) q^{61} +(-3.20807e9 - 1.18097e9i) q^{63} +(3.76018e9 + 6.51282e9i) q^{65} +(1.24117e9 - 2.14977e9i) q^{67} +7.57500e9 q^{69} +2.40523e9 q^{71} +(9.20676e8 - 1.59466e9i) q^{73} +(1.51403e10 + 2.62237e10i) q^{75} +(-2.82886e9 - 1.63943e10i) q^{77} +(-1.37771e10 - 2.38626e10i) q^{79} +(5.92610e9 - 1.02643e10i) q^{81} -3.20498e10 q^{83} -8.56108e10 q^{85} +(1.11863e10 - 1.93753e10i) q^{87} +(-1.01067e10 - 1.75053e10i) q^{89} +(2.13794e10 - 1.78064e10i) q^{91} +(-2.33238e10 - 4.03980e10i) q^{93} +(1.15814e11 - 2.00596e11i) q^{95} +1.43007e11 q^{97} -2.87624e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 266 q^{3} + 3808 q^{5} - 110328 q^{7} - 503848 q^{9}+O(q^{10}) \) 8 * q + 266 * q^3 + 3808 * q^5 - 110328 * q^7 - 503848 * q^9	\( 8 q + 266 q^{3} + 3808 q^{5} - 110328 q^{7} - 503848 q^{9} - 920150 q^{11} + 997472 q^{13} + 500444 q^{15} + 1333724 q^{17} + 21551726 q^{19} - 90442480 q^{21} - 72510158 q^{23} - 77154744 q^{25} - 156683212 q^{27} + 213575104 q^{29} + 194359774 q^{31} - 164547124 q^{33} + 981719074 q^{35} + 171517048 q^{37} + 653125236 q^{39} - 3312095136 q^{41} - 850279648 q^{43} + 6784014272 q^{45} - 2223880974 q^{47} - 4232044312 q^{49} - 6968362082 q^{51} - 7185483360 q^{53} - 624915620 q^{55} + 4959469112 q^{57} - 6997401502 q^{59} - 6476463280 q^{61} + 17252507684 q^{63} + 30625528248 q^{65} + 18660972186 q^{67} + 31319762096 q^{69} - 23224449248 q^{71} + 3731641452 q^{73} + 91204646176 q^{75} - 29345595440 q^{77} - 12221157926 q^{79} - 97333443028 q^{81} - 316739523968 q^{83} - 147794862544 q^{85} + 218122807364 q^{87} - 71204406084 q^{89} - 189816116528 q^{91} + 9838293624 q^{93} + 215091896614 q^{95} + 688251797184 q^{97} + 545487662552 q^{99}+O(q^{100}) \) 8 * q + 266 * q^3 + 3808 * q^5 - 110328 * q^7 - 503848 * q^9 - 920150 * q^11 + 997472 * q^13 + 500444 * q^15 + 1333724 * q^17 + 21551726 * q^19 - 90442480 * q^21 - 72510158 * q^23 - 77154744 * q^25 - 156683212 * q^27 + 213575104 * q^29 + 194359774 * q^31 - 164547124 * q^33 + 981719074 * q^35 + 171517048 * q^37 + 653125236 * q^39 - 3312095136 * q^41 - 850279648 * q^43 + 6784014272 * q^45 - 2223880974 * q^47 - 4232044312 * q^49 - 6968362082 * q^51 - 7185483360 * q^53 - 624915620 * q^55 + 4959469112 * q^57 - 6997401502 * q^59 - 6476463280 * q^61 + 17252507684 * q^63 + 30625528248 * q^65 + 18660972186 * q^67 + 31319762096 * q^69 - 23224449248 * q^71 + 3731641452 * q^73 + 91204646176 * q^75 - 29345595440 * q^77 - 12221157926 * q^79 - 97333443028 * q^81 - 316739523968 * q^83 - 147794862544 * q^85 + 218122807364 * q^87 - 71204406084 * q^89 - 189816116528 * q^91 + 9838293624 * q^93 + 215091896614 * q^95 + 688251797184 * q^97 + 545487662552 * q^99

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