LMFDB - Embedded newform 1386.2.ba.a.1187.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(989,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 2, 3]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1386.989");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.ba (of order $6$, degree $2$, 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:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1187.1
Character	$\chi$	$=$	1386.1187
Dual form			1386.2.ba.a.989.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-3.19684 + 1.84570i) q^{5} +(-1.52985 + 2.15860i) q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-3.19684 + 1.84570i) q^{5} +(-1.52985 + 2.15860i) q^{7} +1.00000 q^{8} +(3.19684 + 1.84570i) q^{10} +(0.412474 - 3.29088i) q^{11} +5.60520i q^{13} +(2.63433 + 0.245585i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-1.39461 + 2.41554i) q^{17} +(-2.14631 + 1.23917i) q^{19} -3.69140i q^{20} +(-3.05622 + 1.28823i) q^{22} +(3.26215 - 1.88340i) q^{23} +(4.31321 - 7.47069i) q^{25} +(4.85424 - 2.80260i) q^{26} +(-1.10448 - 2.40419i) q^{28} -1.51011 q^{29} +(-1.70296 + 2.94961i) q^{31} +(-0.500000 + 0.866025i) q^{32} +2.78923 q^{34} +(0.906550 - 9.72435i) q^{35} +(2.85865 + 4.95132i) q^{37} +(2.14631 + 1.23917i) q^{38} +(-3.19684 + 1.84570i) q^{40} -6.24731 q^{41} -7.72006i q^{43} +(2.64375 + 2.00265i) q^{44} +(-3.26215 - 1.88340i) q^{46} +(8.98162 - 5.18554i) q^{47} +(-2.31914 - 6.60467i) q^{49} -8.62641 q^{50} +(-4.85424 - 2.80260i) q^{52} +(-9.27677 - 5.35595i) q^{53} +(4.75535 + 11.2817i) q^{55} +(-1.52985 + 2.15860i) q^{56} +(0.755057 + 1.30780i) q^{58} +(-11.6232 - 6.71066i) q^{59} +(-2.13230 + 1.23108i) q^{61} +3.40592 q^{62} +1.00000 q^{64} +(-10.3455 - 17.9189i) q^{65} +(5.66594 - 9.81369i) q^{67} +(-1.39461 - 2.41554i) q^{68} +(-8.87481 + 4.07708i) q^{70} -5.15087i q^{71} +(-4.53946 - 2.62086i) q^{73} +(2.85865 - 4.95132i) q^{74} -2.47835i q^{76} +(6.47267 + 5.92490i) q^{77} +(-0.250209 + 0.144458i) q^{79} +(3.19684 + 1.84570i) q^{80} +(3.12366 + 5.41033i) q^{82} -10.2522 q^{83} -10.2961i q^{85} +(-6.68577 + 3.86003i) q^{86} +(0.412474 - 3.29088i) q^{88} +(13.7391 - 7.93225i) q^{89} +(-12.0994 - 8.57509i) q^{91} +3.76681i q^{92} +(-8.98162 - 5.18554i) q^{94} +(4.57428 - 7.92289i) q^{95} +4.58628 q^{97} +(-4.56024 + 5.31076i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 16 q^{2} - 16 q^{4} + 32 q^{8}+O(q^{10}) $ 32 * q - 16 * q^2 - 16 * q^4 + 32 * q^8	$ 32 q - 16 q^{2} - 16 q^{4} + 32 q^{8} - 2 q^{11} - 16 q^{16} - 4 q^{17} + 4 q^{22} + 4 q^{25} - 16 q^{29} + 4 q^{31} - 16 q^{32} + 8 q^{34} - 16 q^{35} + 4 q^{37} + 32 q^{41} - 2 q^{44} + 20 q^{49} - 8 q^{50} - 12 q^{55} + 8 q^{58} - 8 q^{62} + 32 q^{64} - 8 q^{67} - 4 q^{68} - 4 q^{70} + 4 q^{74} - 14 q^{77} - 16 q^{82} - 88 q^{83} - 2 q^{88} + 24 q^{95} - 32 q^{97} + 8 q^{98}+O(q^{100}) $ 32 * q - 16 * q^2 - 16 * q^4 + 32 * q^8 - 2 * q^11 - 16 * q^16 - 4 * q^17 + 4 * q^22 + 4 * q^25 - 16 * q^29 + 4 * q^31 - 16 * q^32 + 8 * q^34 - 16 * q^35 + 4 * q^37 + 32 * q^41 - 2 * q^44 + 20 * q^49 - 8 * q^50 - 12 * q^55 + 8 * q^58 - 8 * q^62 + 32 * q^64 - 8 * q^67 - 4 * q^68 - 4 * q^70 + 4 * q^74 - 14 * q^77 - 16 * q^82 - 88 * q^83 - 2 * q^88 + 24 * q^95 - 32 * q^97 + 8 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$1135$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$-1$

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 $

$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$	−3.19684	+	1.84570i	−1.42967	+	0.825421i	−0.997094	−	0.0761754i	$-0.975729\pi$
$5$	−3.19684	+	1.84570i	−1.42967	+	0.825421i	−0.432577	+	0.901597i	$0.642396\pi$
$6$		0			0
$7$	−1.52985	+	2.15860i	−0.578228	+	0.815875i
$7$	−1.52985	+	2.15860i	−0.578228	+	0.815875i
$8$	1.00000			0.353553
$9$		0			0
$10$	3.19684	+	1.84570i	1.01093	+	0.583661i
$11$	0.412474	−	3.29088i	0.124365	−	0.992236i
$11$	0.412474	−	3.29088i	0.124365	−	0.992236i
$12$		0			0
$13$			5.60520i			1.55460i	0.629129	+	0.777301i	$0.283412\pi$
$13$			5.60520i			1.55460i	−0.629129	+	0.777301i	$0.716588\pi$
$14$	2.63433	+	0.245585i	0.704054	+	0.0656352i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	−1.39461	+	2.41554i	−0.338243	+	0.585855i	−0.984102	−	0.177602i	$-0.943166\pi$
$17$	−1.39461	+	2.41554i	−0.338243	+	0.585855i	0.645859	+	0.763457i	$0.276499\pi$
$18$		0			0
$19$	−2.14631	+	1.23917i	−0.492398	+	0.284286i	−0.725569	−	0.688150i	$-0.758423\pi$
$19$	−2.14631	+	1.23917i	−0.492398	+	0.284286i	0.233171	+	0.972436i	$0.425090\pi$
$20$		−	3.69140i		−	0.825421i
$21$		0			0
$22$	−3.05622	+	1.28823i	−0.651588	+	0.274651i
$23$	3.26215	−	1.88340i	0.680206	−	0.392717i	−0.119727	−	0.992807i	$-0.538202\pi$
$23$	3.26215	−	1.88340i	0.680206	−	0.392717i	0.799933	+	0.600090i	$0.204869\pi$
$24$		0			0
$25$	4.31321	−	7.47069i	0.862641	−	1.49414i
$26$	4.85424	−	2.80260i	0.951995	−	0.549635i
$27$		0			0
$28$	−1.10448	−	2.40419i	−0.208727	−	0.454349i
$29$	−1.51011			−0.280421			−0.140210	−	0.990122i	$-0.544778\pi$
$29$	−1.51011			−0.280421			−0.140210	+	0.990122i	$0.544778\pi$
$30$		0			0
$31$	−1.70296	+	2.94961i	−0.305860	+	0.529766i	−0.977453	−	0.211155i	$-0.932277\pi$
$31$	−1.70296	+	2.94961i	−0.305860	+	0.529766i	0.671592	+	0.740921i	$0.265611\pi$
$32$	−0.500000	+	0.866025i	−0.0883883	+	0.153093i
$33$		0			0
$34$	2.78923			0.478348
$35$	0.906550	−	9.72435i	0.153235	−	1.64372i
$36$		0			0
$37$	2.85865	+	4.95132i	0.469959	+	0.813993i	0.999410	−	0.0343479i	$-0.0109354\pi$
$37$	2.85865	+	4.95132i	0.469959	+	0.813993i	−0.529451	+	0.848340i	$0.677602\pi$
$38$	2.14631	+	1.23917i	0.348178	+	0.201021i
$39$		0			0
$40$	−3.19684	+	1.84570i	−0.505465	+	0.291831i
$41$	−6.24731			−0.975666			−0.487833	−	0.872937i	$-0.662213\pi$
$41$	−6.24731			−0.975666			−0.487833	+	0.872937i	$0.662213\pi$
$42$		0			0
$43$		−	7.72006i		−	1.17730i	−0.808389	−	0.588649i	$-0.799660\pi$
$43$		−	7.72006i		−	1.17730i	0.808389	−	0.588649i	$-0.200340\pi$
$44$	2.64375	+	2.00265i	0.398560	+	0.301911i
$45$		0			0
$46$	−3.26215	−	1.88340i	−0.480978	−	0.277693i
$47$	8.98162	−	5.18554i	1.31010	−	0.756389i	0.327991	−	0.944681i	$-0.393629\pi$
$47$	8.98162	−	5.18554i	1.31010	−	0.756389i	0.982113	+	0.188292i	$0.0602953\pi$
$48$		0			0
$49$	−2.31914	−	6.60467i	−0.331305	−	0.943524i
$50$	−8.62641			−1.21996
$51$		0			0
$52$	−4.85424	−	2.80260i	−0.673162	−	0.388650i
$53$	−9.27677	−	5.35595i	−1.27426	−	0.735696i	−0.298475	−	0.954417i	$-0.596478\pi$
$53$	−9.27677	−	5.35595i	−1.27426	−	0.735696i	−0.975787	+	0.218721i	$0.929811\pi$
$54$		0			0
$55$	4.75535	+	11.2817i	0.641211	+	1.52123i
$56$	−1.52985	+	2.15860i	−0.204434	+	0.288456i
$57$		0			0
$58$	0.755057	+	1.30780i	0.0991438	+	0.171722i
$59$	−11.6232	−	6.71066i	−1.51321	−	0.873654i	−0.999880	−	0.0154644i	$-0.995077\pi$
$59$	−11.6232	−	6.71066i	−1.51321	−	0.873654i	−0.513333	−	0.858190i	$-0.671589\pi$
$60$		0			0
$61$	−2.13230	+	1.23108i	−0.273013	+	0.157624i	−0.630256	−	0.776387i	$-0.717050\pi$
$61$	−2.13230	+	1.23108i	−0.273013	+	0.157624i	0.357243	+	0.934011i	$0.383717\pi$
$62$	3.40592			0.432552
$63$		0			0
$64$	1.00000			0.125000
$65$	−10.3455	−	17.9189i	−1.28320	−	2.22257i
$66$		0			0
$67$	5.66594	−	9.81369i	0.692204	−	1.19893i	−0.278910	−	0.960317i	$-0.589973\pi$
$67$	5.66594	−	9.81369i	0.692204	−	1.19893i	0.971114	−	0.238616i	$-0.0766936\pi$
$68$	−1.39461	−	2.41554i	−0.169122	−	0.292927i
$69$		0			0
$70$	−8.87481	+	4.07708i	−1.06074	+	0.487304i
$71$		−	5.15087i		−	0.611296i	−0.952145	−	0.305648i	$-0.901127\pi$
$71$		−	5.15087i		−	0.611296i	0.952145	−	0.305648i	$-0.0988730\pi$
$72$		0			0
$73$	−4.53946	−	2.62086i	−0.531303	−	0.306748i	0.210244	−	0.977649i	$-0.432574\pi$
$73$	−4.53946	−	2.62086i	−0.531303	−	0.306748i	−0.741547	+	0.670901i	$0.765908\pi$
$74$	2.85865	−	4.95132i	0.332311	−	0.575580i
$75$		0			0
$76$		−	2.47835i		−	0.284286i
$77$	6.47267	+	5.92490i	0.737630	+	0.675205i
$78$		0			0
$79$	−0.250209	+	0.144458i	−0.0281507	+	0.0162528i	−0.514009	−	0.857785i	$-0.671840\pi$
$79$	−0.250209	+	0.144458i	−0.0281507	+	0.0162528i	0.485859	+	0.874037i	$0.338507\pi$
$80$	3.19684	+	1.84570i	0.357418	+	0.206355i
$81$		0			0
$82$	3.12366	+	5.41033i	0.344950	+	0.597471i
$83$	−10.2522			−1.12533			−0.562664	−	0.826685i	$-0.690224\pi$
$83$	−10.2522			−1.12533			−0.562664	+	0.826685i	$0.690224\pi$
$84$		0			0
$85$		−	10.2961i		−	1.11677i
$86$	−6.68577	+	3.86003i	−0.720945	+	0.416238i
$87$		0			0
$88$	0.412474	−	3.29088i	0.0439698	−	0.350809i
$89$	13.7391	−	7.93225i	1.45634	−	0.840816i	0.457508	−	0.889205i	$-0.348742\pi$
$89$	13.7391	−	7.93225i	1.45634	−	0.840816i	0.998829	+	0.0483889i	$0.0154087\pi$
$90$		0			0
$91$	−12.0994	−	8.57509i	−1.26836	−	0.898914i
$92$			3.76681i			0.392717i
$93$		0			0
$94$	−8.98162	−	5.18554i	−0.926383	−	0.534848i
$95$	4.57428	−	7.92289i	0.469311	−	0.812871i
$96$		0			0
$97$	4.58628			0.465666			0.232833	−	0.972517i	$-0.425200\pi$
$97$	4.58628			0.465666			0.232833	+	0.972517i	$0.425200\pi$
$98$	−4.56024	+	5.31076i	−0.460654	+	0.536468i
$99$		0			0
$100$	4.31321	+	7.47069i	0.431321	+	0.747069i
$101$	−1.45801	+	2.52536i	−0.145078	+	0.251282i	−0.929402	−	0.369069i	$-0.879677\pi$
$101$	−1.45801	+	2.52536i	−0.145078	+	0.251282i	0.784324	+	0.620351i	$0.213010\pi$
$102$		0			0
$103$	−4.61616	−	7.99543i	−0.454844	−	0.787813i	0.543835	−	0.839192i	$-0.316972\pi$
$103$	−4.61616	−	7.99543i	−0.454844	−	0.787813i	−0.998679	+	0.0513790i	$0.983638\pi$
$104$			5.60520i			0.549635i
$105$		0			0
$106$			10.7119i			1.04043i
$107$	1.81237	+	3.13911i	0.175208	+	0.303469i	0.940233	−	0.340531i	$-0.110607\pi$
$107$	1.81237	+	3.13911i	0.175208	+	0.303469i	−0.765025	+	0.644000i	$0.777274\pi$
$108$		0			0
$109$	13.9355	+	8.04565i	1.33478	+	0.770634i	0.986028	−	0.166583i	$-0.0532732\pi$
$109$	13.9355	+	8.04565i	1.33478	+	0.770634i	0.348749	+	0.937216i	$0.386607\pi$
$110$	7.39258	−	9.75911i	0.704855	−	0.930495i
$111$		0			0
$112$	2.63433	+	0.245585i	0.248921	+	0.0232056i
$113$			18.8371i			1.77204i	0.463643	+	0.886022i	$0.346542\pi$
$113$			18.8371i			1.77204i	−0.463643	+	0.886022i	$0.653458\pi$
$114$		0			0
$115$	−6.95239	+	12.0419i	−0.648314	+	1.12291i
$116$	0.755057	−	1.30780i	0.0701052	−	0.121426i
$117$		0			0
$118$			13.4213i			1.23553i
$119$	−3.08065	−	6.70582i	−0.282403	−	0.614722i
$120$		0			0
$121$	−10.6597	−	2.71480i	−0.969066	−	0.246800i
$122$	2.13230	+	1.23108i	0.193049	+	0.111457i
$123$		0			0
$124$	−1.70296	−	2.94961i	−0.152930	−	0.264883i
$125$			13.3865i			1.19733i
$126$		0			0
$127$			8.08283i			0.717235i	0.933485	+	0.358618i	$0.116752\pi$
$127$			8.08283i			0.717235i	−0.933485	+	0.358618i	$0.883248\pi$
$128$	−0.500000	−	0.866025i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	−10.3455	+	17.9189i	−0.907361	+	1.57159i
$131$	7.83557	+	13.5716i	0.684597	+	1.18576i	0.973563	+	0.228417i	$0.0733549\pi$
$131$	7.83557	+	13.5716i	0.684597	+	1.18576i	−0.288967	+	0.957339i	$0.593312\pi$
$132$		0			0
$133$	0.608644	−	6.52878i	0.0527761	−	0.566117i
$134$	−11.3319			−0.978925
$135$		0			0
$136$	−1.39461	+	2.41554i	−0.119587	+	0.207131i
$137$	−2.55426	−	1.47471i	−0.218226	−	0.125993i	0.386903	−	0.922120i	$-0.373545\pi$
$137$	−2.55426	−	1.47471i	−0.218226	−	0.125993i	−0.605128	+	0.796128i	$0.706878\pi$
$138$		0			0
$139$		−	4.00952i		−	0.340083i	−0.985437	−	0.170042i	$-0.945610\pi$
$139$		−	4.00952i		−	0.340083i	0.985437	−	0.170042i	$-0.0543902\pi$
$140$	7.96826	+	5.64727i	0.673441	+	0.477282i
$141$		0			0
$142$	−4.46078	+	2.57543i	−0.374341	+	0.216126i
$143$	18.4460	+	2.31200i	1.54253	+	0.193339i
$144$		0			0
$145$	4.82759	−	2.78721i	0.400910	−	0.231465i
$146$			5.24171i			0.433807i
$147$		0			0
$148$	−5.71730			−0.469959
$149$	3.23457	+	5.60244i	0.264986	+	0.458969i	0.967560	−	0.252641i	$-0.0812993\pi$
$149$	3.23457	+	5.60244i	0.264986	+	0.458969i	−0.702574	+	0.711611i	$0.747966\pi$
$150$		0			0
$151$	−13.3957	−	7.73400i	−1.09012	−	0.629384i	−0.156515	−	0.987676i	$-0.550026\pi$
$151$	−13.3957	−	7.73400i	−1.09012	−	0.629384i	−0.933610	+	0.358292i	$0.883359\pi$
$152$	−2.14631	+	1.23917i	−0.174089	+	0.100510i
$153$		0			0
$154$	1.89478	−	8.56795i	0.152686	−	0.690425i
$155$		−	12.5726i		−	1.00985i
$156$		0			0
$157$	9.34257	−	16.1818i	0.745618	−	1.29145i	−0.204287	−	0.978911i	$-0.565488\pi$
$157$	9.34257	−	16.1818i	0.745618	−	1.29145i	0.949905	−	0.312538i	$-0.101179\pi$
$158$	0.250209	+	0.144458i	0.0199056	+	0.0114925i
$159$		0			0
$160$		−	3.69140i		−	0.291831i
$161$	−0.925070	+	9.92301i	−0.0729057	+	0.782043i
$162$		0			0
$163$	4.62801	+	8.01595i	0.362494	+	0.627858i	0.988371	−	0.152064i	$-0.0485920\pi$
$163$	4.62801	+	8.01595i	0.362494	+	0.627858i	−0.625877	+	0.779922i	$0.715259\pi$
$164$	3.12366	−	5.41033i	0.243917	−	0.422476i
$165$		0			0
$166$	5.12612	+	8.87869i	0.397864	+	0.689120i
$167$	16.3875			1.26811			0.634053	−	0.773289i	$-0.281390\pi$
$167$	16.3875			1.26811			0.634053	+	0.773289i	$0.281390\pi$
$168$		0			0
$169$	−18.4182			−1.41679
$170$	−8.91672	+	5.14807i	−0.683881	+	0.394839i
$171$		0			0
$172$	6.68577	+	3.86003i	0.509785	+	0.294325i
$173$	−10.4622	−	18.1210i	−0.795424	−	1.37771i	−0.922570	−	0.385831i	$-0.873915\pi$
$173$	−10.4622	−	18.1210i	−0.795424	−	1.37771i	0.127146	−	0.991884i	$-0.459418\pi$
$174$		0			0
$175$	9.52771	+	20.7395i	0.720228	+	1.56776i
$176$	−3.05622	+	1.28823i	−0.230371	+	0.0971036i
$177$		0			0
$178$	−13.7391	−	7.93225i	−1.02979	−	0.594547i
$179$	−12.9584	−	7.48153i	−0.968555	−	0.559196i	−0.0697596	−	0.997564i	$-0.522223\pi$
$179$	−12.9584	−	7.48153i	−0.968555	−	0.559196i	−0.898795	+	0.438368i	$0.855557\pi$
$180$		0			0
$181$	−10.1125			−0.751657			−0.375828	−	0.926689i	$-0.622642\pi$
$181$	−10.1125			−0.751657			−0.375828	+	0.926689i	$0.622642\pi$
$182$	−1.37655	+	14.7659i	−0.102037	+	1.09452i
$183$		0			0
$184$	3.26215	−	1.88340i	0.240489	−	0.138846i
$185$	−18.2773	−	10.5524i	−1.34377	−	0.775828i
$186$		0			0
$187$	7.37400	+	5.58584i	0.539240	+	0.408477i
$188$			10.3711i			0.756389i
$189$		0			0
$190$	−9.14856			−0.663707
$191$	−4.57254	+	2.63995i	−0.330857	+	0.191020i	−0.656222	−	0.754568i	$-0.727846\pi$
$191$	−4.57254	+	2.63995i	−0.330857	+	0.191020i	0.325364	+	0.945589i	$0.394513\pi$
$192$		0			0
$193$	19.8361	+	11.4524i	1.42784	+	0.824362i	0.996950	−	0.0780447i	$-0.0248677\pi$
$193$	19.8361	+	11.4524i	1.42784	+	0.824362i	0.430886	+	0.902406i	$0.358201\pi$
$194$	−2.29314	−	3.97184i	−0.164638	−	0.285161i
$195$		0			0
$196$	6.87938	+	1.29390i	0.491384	+	0.0924215i
$197$	8.56673			0.610354			0.305177	−	0.952296i	$-0.401284\pi$
$197$	8.56673			0.610354			0.305177	+	0.952296i	$0.401284\pi$
$198$		0			0
$199$	2.28887	−	3.96444i	0.162254	−	0.281032i	−0.773423	−	0.633890i	$-0.781457\pi$
$199$	2.28887	−	3.96444i	0.162254	−	0.281032i	0.935677	+	0.352859i	$0.114790\pi$
$200$	4.31321	−	7.47069i	0.304990	−	0.528258i
$201$		0			0
$202$	2.91603			0.205171
$203$	2.31024	−	3.25974i	0.162147	−	0.228789i
$204$		0			0
$205$	19.9717	−	11.5307i	1.39488	−	0.805336i
$206$	−4.61616	+	7.99543i	−0.321623	+	0.557068i
$207$		0			0
$208$	4.85424	−	2.80260i	0.336581	−	0.194325i
$209$	3.19267	+	7.57437i	0.220842	+	0.523930i
$210$		0			0
$211$		−	10.0586i		−	0.692463i	−0.938149	−	0.346231i	$-0.887461\pi$
$211$		−	10.0586i		−	0.692463i	0.938149	−	0.346231i	$-0.112539\pi$
$212$	9.27677	−	5.35595i	0.637131	−	0.367848i
$213$		0			0
$214$	1.81237	−	3.13911i	0.123891	−	0.214585i
$215$	14.2489	+	24.6798i	0.971767	+	1.68315i
$216$		0			0
$217$	−3.76177	−	8.18846i	−0.255366	−	0.555869i
$218$		−	16.0913i		−	1.08984i
$219$		0			0
$220$	−12.1479	−	1.52260i	−0.819013	−	0.102654i
$221$	−13.5396	−	7.81708i	−0.910770	−	0.525834i
$222$		0			0
$223$	14.7430			0.987266			0.493633	−	0.869670i	$-0.335669\pi$
$223$	14.7430			0.987266			0.493633	+	0.869670i	$0.335669\pi$
$224$	−1.10448	−	2.40419i	−0.0737963	−	0.160637i
$225$		0			0
$226$	16.3134	−	9.41854i	1.08515	−	0.626512i
$227$	−9.41469	+	16.3067i	−0.624875	+	1.08232i	0.363690	+	0.931520i	$0.381517\pi$
$227$	−9.41469	+	16.3067i	−0.624875	+	1.08232i	−0.988565	+	0.150795i	$0.951817\pi$
$228$		0			0
$229$	−6.82233	−	11.8166i	−0.450832	−	0.780865i	0.547606	−	0.836737i	$-0.315539\pi$
$229$	−6.82233	−	11.8166i	−0.450832	−	0.780865i	−0.998438	+	0.0558720i	$0.982206\pi$
$230$	13.9048			0.916854
$231$		0			0
$232$	−1.51011			−0.0991438
$233$	−3.51518	−	6.08847i	−0.230287	−	0.398869i	0.727606	−	0.685996i	$-0.240633\pi$
$233$	−3.51518	−	6.08847i	−0.230287	−	0.398869i	−0.957893	+	0.287127i	$0.907300\pi$
$234$		0			0
$235$	−19.1419	+	33.1547i	−1.24868	+	2.16278i
$236$	11.6232	−	6.71066i	0.756607	−	0.436827i
$237$		0			0
$238$	−4.26709	+	6.02083i	−0.276594	+	0.390273i
$239$	−10.3908			−0.672126			−0.336063	−	0.941840i	$-0.609095\pi$
$239$	−10.3908			−0.672126			−0.336063	+	0.941840i	$0.609095\pi$
$240$		0			0
$241$	−9.24972	−	5.34033i	−0.595827	−	0.344001i	0.171571	−	0.985172i	$-0.445116\pi$
$241$	−9.24972	−	5.34033i	−0.595827	−	0.344001i	−0.767398	+	0.641171i	$0.778449\pi$
$242$	2.97878	+	10.5890i	0.191483	+	0.680687i
$243$		0			0
$244$		−	2.46216i		−	0.157624i
$245$	19.6041	+	16.8337i	1.25246	+	1.07546i
$246$		0			0
$247$	−6.94581	−	12.0305i	−0.441951	−	0.765482i
$248$	−1.70296	+	2.94961i	−0.108138	+	0.187300i
$249$		0			0
$250$	11.5931	−	6.69326i	0.733210	−	0.423319i
$251$			9.19949i			0.580667i	0.956926	+	0.290333i	$0.0937662\pi$
$251$			9.19949i			0.580667i	−0.956926	+	0.290333i	$0.906234\pi$
$252$		0			0
$253$	−4.85250	−	11.5122i	−0.305074	−	0.723765i
$254$	6.99993	−	4.04141i	0.439215	−	0.253581i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	−17.8743	+	10.3197i	−1.11497	+	0.643728i	−0.940112	−	0.340865i	$-0.889280\pi$
$257$	−17.8743	+	10.3197i	−1.11497	+	0.643728i	−0.174858	+	0.984594i	$0.555947\pi$
$258$		0			0
$259$	−15.0612	−	1.40408i	−0.935860	−	0.0872453i
$260$	20.6910			1.28320
$261$		0			0
$262$	7.83557	−	13.5716i	0.484083	−	0.838456i
$263$	−13.1613	+	22.7960i	−0.811561	+	1.40566i	0.100211	+	0.994966i	$0.468048\pi$
$263$	−13.1613	+	22.7960i	−0.811561	+	1.40566i	−0.911771	+	0.410698i	$0.865285\pi$
$264$		0			0
$265$	39.5419			2.42904
$266$	−5.95841	+	2.73729i	−0.365334	+	0.167834i
$267$		0			0
$268$	5.66594	+	9.81369i	0.346102	+	0.599467i
$269$	−7.04033	−	4.06473i	−0.429256	−	0.247831i	0.269773	−	0.962924i	$-0.413051\pi$
$269$	−7.04033	−	4.06473i	−0.429256	−	0.247831i	−0.699030	+	0.715093i	$0.746385\pi$
$270$		0			0
$271$	12.6065	−	7.27836i	0.765790	−	0.442129i	−0.0655809	−	0.997847i	$-0.520890\pi$
$271$	12.6065	−	7.27836i	0.765790	−	0.442129i	0.831371	+	0.555718i	$0.187557\pi$
$272$	2.78923			0.169122
$273$		0			0
$274$			2.94941i			0.178180i
$275$	−22.8060	−	17.2757i	−1.37526	−	1.04176i
$276$		0			0
$277$	−25.9763	−	14.9974i	−1.56077	−	0.901109i	−0.997180	−	0.0750526i	$-0.976088\pi$
$277$	−25.9763	−	14.9974i	−1.56077	−	0.901109i	−0.563587	−	0.826057i	$-0.690579\pi$
$278$	−3.47235	+	2.00476i	−0.208258	+	0.120238i
$279$		0			0
$280$	0.906550	−	9.72435i	0.0541767	−	0.581141i
$281$	12.8423			0.766110			0.383055	−	0.923726i	$-0.374872\pi$
$281$	12.8423			0.766110			0.383055	+	0.923726i	$0.374872\pi$
$282$		0			0
$283$	−6.35099	−	3.66675i	−0.377527	−	0.217965i	0.299215	−	0.954186i	$-0.403275\pi$
$283$	−6.35099	−	3.66675i	−0.377527	−	0.217965i	−0.676742	+	0.736220i	$0.736609\pi$
$284$	4.46078	+	2.57543i	0.264699	+	0.152824i
$285$		0			0
$286$	−7.22076	−	17.1307i	−0.426972	−	1.01296i
$287$	9.55743	−	13.4855i	0.564157	−	0.796022i
$288$		0			0
$289$	4.61011	+	7.98495i	0.271183	+	0.469703i
$290$	−4.82759	−	2.78721i	−0.283486	−	0.163671i
$291$		0			0
$292$	4.53946	−	2.62086i	0.265652	−	0.153374i
$293$	−6.05343			−0.353645			−0.176823	−	0.984243i	$-0.556582\pi$
$293$	−6.05343			−0.353645			−0.176823	+	0.984243i	$0.556582\pi$
$294$		0			0
$295$	49.5434			2.88453
$296$	2.85865	+	4.95132i	0.166156	+	0.287790i
$297$		0			0
$298$	3.23457	−	5.60244i	0.187373	−	0.324540i
$299$	10.5569	+	18.2850i	0.610518	+	1.05745i
$300$		0			0
$301$	16.6646	+	11.8105i	0.960529	+	0.680747i
$302$			15.4680i			0.890083i
$303$		0			0
$304$	2.14631	+	1.23917i	0.123099	+	0.0710715i
$305$	4.54441	−	7.87116i	0.260212	−	0.450701i
$306$		0			0
$307$		−	6.24654i		−	0.356509i	−0.983984	−	0.178255i	$-0.942955\pi$
$307$		−	6.24654i		−	0.356509i	0.983984	−	0.178255i	$-0.0570450\pi$
$308$	−8.36745	+	2.64305i	−0.476780	+	0.150602i
$309$		0			0
$310$	−10.8882	+	6.28629i	−0.618407	+	0.357038i
$311$	−8.45677	−	4.88252i	−0.479539	−	0.276862i	0.240685	−	0.970603i	$-0.422628\pi$
$311$	−8.45677	−	4.88252i	−0.479539	−	0.276862i	−0.720225	+	0.693741i	$0.755961\pi$
$312$		0			0
$313$	−16.0158	−	27.7402i	−0.905266	−	1.56797i	−0.820559	−	0.571561i	$-0.806338\pi$
$313$	−16.0158	−	27.7402i	−0.905266	−	1.56797i	−0.0847068	−	0.996406i	$-0.526995\pi$
$314$	−18.6851			−1.05446
$315$		0			0
$316$		−	0.288917i		−	0.0162528i
$317$	−18.5774	+	10.7256i	−1.04341	+	0.602412i	−0.920797	−	0.390042i	$-0.872460\pi$
$317$	−18.5774	+	10.7256i	−1.04341	+	0.602412i	−0.122612	+	0.992455i	$0.539127\pi$
$318$		0			0
$319$	−0.622882	+	4.96959i	−0.0348747	+	0.278244i
$320$	−3.19684	+	1.84570i	−0.178709	+	0.103178i
$321$		0			0
$322$	9.05612	−	4.16037i	0.504678	−	0.231848i
$323$		−	6.91267i		−	0.384631i
$324$		0			0
$325$	41.8747	+	24.1764i	2.32279	+	1.34106i
$326$	4.62801	−	8.01595i	0.256322	−	0.443963i
$327$		0			0
$328$	−6.24731			−0.344950
$329$	−2.54698	+	27.3208i	−0.140419	+	1.50625i
$330$		0			0
$331$	15.3327	+	26.5569i	0.842759	+	1.45970i	0.887553	+	0.460705i	$0.152403\pi$
$331$	15.3327	+	26.5569i	0.842759	+	1.45970i	−0.0447947	+	0.998996i	$0.514263\pi$
$332$	5.12612	−	8.87869i	0.281332	−	0.487282i
$333$		0			0
$334$	−8.19377	−	14.1920i	−0.448343	−	0.776554i
$335$			41.8304i			2.28544i
$336$		0			0
$337$		−	33.3731i		−	1.81795i	−0.416853	−	0.908974i	$-0.636867\pi$
$337$		−	33.3731i		−	1.81795i	0.416853	−	0.908974i	$-0.363133\pi$
$338$	9.20911	+	15.9507i	0.500910	+	0.867601i
$339$		0			0
$340$	8.91672	+	5.14807i	0.483577	+	0.279193i
$341$	9.00438	+	6.82086i	0.487614	+	0.369370i
$342$		0			0
$343$	17.8048	+	5.09803i	0.961368	+	0.275268i
$344$		−	7.72006i		−	0.416238i
$345$		0			0
$346$	−10.4622	+	18.1210i	−0.562450	+	0.974191i
$347$	10.0767	−	17.4533i	0.540943	−	0.936941i	−0.457907	−	0.889000i	$-0.651401\pi$
$347$	10.0767	−	17.4533i	0.540943	−	0.936941i	0.998850	−	0.0479408i	$-0.0152659\pi$
$348$		0			0
$349$			25.8276i			1.38252i	0.722607	+	0.691259i	$0.242944\pi$
$349$			25.8276i			1.38252i	−0.722607	+	0.691259i	$0.757056\pi$
$350$	13.1971	−	18.6210i	0.705414	−	0.995334i
$351$		0			0
$352$	2.64375	+	2.00265i	0.140912	+	0.106742i
$353$	−8.21248	−	4.74148i	−0.437106	−	0.252363i	0.265263	−	0.964176i	$-0.414541\pi$
$353$	−8.21248	−	4.74148i	−0.437106	−	0.252363i	−0.702369	+	0.711813i	$0.747875\pi$
$354$		0			0
$355$	9.50695	+	16.4665i	0.504577	+	0.873952i
$356$			15.8645i			0.840816i
$357$		0			0
$358$			14.9631i			0.790822i
$359$	−5.42112	−	9.38966i	−0.286116	−	0.495567i	0.686763	−	0.726881i	$-0.259031\pi$
$359$	−5.42112	−	9.38966i	−0.286116	−	0.495567i	−0.972879	+	0.231314i	$0.925698\pi$
$360$		0			0
$361$	−6.42890	+	11.1352i	−0.338363	+	0.586062i
$362$	5.05625	+	8.75769i	0.265751	+	0.460294i
$363$		0			0
$364$	13.4759	−	6.19084i	0.706331	−	0.324488i
$365$	19.3492			1.01279
$366$		0			0
$367$	4.46446	−	7.73268i	0.233043	−	0.403642i	−0.725659	−	0.688054i	$-0.758465\pi$
$367$	4.46446	−	7.73268i	0.233043	−	0.403642i	0.958702	+	0.284412i	$0.0917983\pi$
$368$	−3.26215	−	1.88340i	−0.170051	−	0.0981792i
$369$		0			0
$370$			21.1048i			1.09719i
$371$	25.7534	−	11.8311i	1.33705	−	0.614240i
$372$		0			0
$373$	−12.8877	+	7.44073i	−0.667301	+	0.385266i	−0.795053	−	0.606540i	$-0.792557\pi$
$373$	−12.8877	+	7.44073i	−0.667301	+	0.385266i	0.127752	+	0.991806i	$0.459224\pi$
$374$	1.15048	−	9.17900i	0.0594900	−	0.474635i
$375$		0			0
$376$	8.98162	−	5.18554i	0.463192	−	0.267424i
$377$		−	8.46448i		−	0.435943i
$378$		0			0
$379$	30.2827			1.55552			0.777759	−	0.628563i	$-0.216356\pi$
$379$	30.2827			1.55552			0.777759	+	0.628563i	$0.216356\pi$
$380$	4.57428	+	7.92289i	0.234656	+	0.406436i
$381$		0			0
$382$	4.57254	+	2.63995i	0.233951	+	0.135072i
$383$	18.4934	−	10.6772i	0.944969	−	0.545578i	0.0534545	−	0.998570i	$-0.482977\pi$
$383$	18.4934	−	10.6772i	0.944969	−	0.545578i	0.891514	+	0.452992i	$0.149643\pi$
$384$		0			0
$385$	−31.6277	−	6.99438i	−1.61190	−	0.356467i
$386$		−	22.9048i		−	1.16582i
$387$		0			0
$388$	−2.29314	+	3.97184i	−0.116417	+	0.201639i
$389$	6.05094	+	3.49351i	0.306795	+	0.177128i	0.645491	−	0.763768i	$-0.276653\pi$
$389$	6.05094	+	3.49351i	0.306795	+	0.177128i	−0.338696	+	0.940896i	$0.609986\pi$
$390$		0			0
$391$			10.5065i			0.531335i
$392$	−2.31914	−	6.60467i	−0.117134	−	0.333586i
$393$		0			0
$394$	−4.28337	−	7.41901i	−0.215793	−	0.373764i
$395$	0.533253	−	0.923622i	0.0268309	−	0.0464725i
$396$		0			0
$397$	−5.27929	−	9.14400i	−0.264960	−	0.458924i	0.702593	−	0.711592i	$-0.252025\pi$
$397$	−5.27929	−	9.14400i	−0.264960	−	0.458924i	−0.967553	+	0.252668i	$0.918692\pi$
$398$	−4.57774			−0.229462
$399$		0			0
$400$	−8.62641			−0.431321
$401$	8.21248	−	4.74148i	0.410112	−	0.236778i	−0.280726	−	0.959788i	$-0.590575\pi$
$401$	8.21248	−	4.74148i	0.410112	−	0.236778i	0.690838	+	0.723010i	$0.257242\pi$
$402$		0			0
$403$	−16.5331	−	9.54542i	−0.823575	−	0.475491i
$404$	−1.45801	−	2.52536i	−0.0725389	−	0.125641i
$405$		0			0
$406$	−3.97813	−	0.370860i	−0.197431	−	0.0184055i
$407$	17.4733	−	7.36517i	0.866120	−	0.365078i
$408$		0			0
$409$	−1.48918	−	0.859778i	−0.0736351	−	0.0425133i	0.462730	−	0.886499i	$-0.346870\pi$
$409$	−1.48918	−	0.859778i	−0.0736351	−	0.0425133i	−0.536366	+	0.843986i	$0.680203\pi$
$410$	−19.9717	−	11.5307i	−0.986331	−	0.569459i
$411$		0			0
$412$	9.23233			0.454844
$413$	32.2674	−	14.8236i	1.58777	−	0.729422i
$414$		0			0
$415$	32.7748	−	18.9225i	1.60885	−	0.928870i
$416$	−4.85424	−	2.80260i	−0.237999	−	0.137409i
$417$		0			0
$418$	4.96326	−	6.55212i	0.242761	−	0.320475i
$419$		−	5.34038i		−	0.260895i	−0.991455	−	0.130447i	$-0.958359\pi$
$419$		−	5.34038i		−	0.260895i	0.991455	−	0.130447i	$-0.0416414\pi$
$420$		0			0
$421$	−27.1036			−1.32095			−0.660475	−	0.750848i	$-0.729645\pi$
$421$	−27.1036			−1.32095			−0.660475	+	0.750848i	$0.729645\pi$
$422$	−8.71101	+	5.02930i	−0.424045	+	0.244823i
$423$		0			0
$424$	−9.27677	−	5.35595i	−0.450520	−	0.260108i
$425$	12.0305	+	20.8374i	0.583565	+	1.01076i
$426$		0			0
$427$	0.604670	−	6.48615i	0.0292620	−	0.313887i
$428$	−3.62473			−0.175208
$429$		0			0
$430$	14.2489	−	24.6798i	0.687143	−	1.19017i
$431$	3.80062	−	6.58286i	0.183069	−	0.317085i	−0.759855	−	0.650093i	$-0.774730\pi$
$431$	3.80062	−	6.58286i	0.183069	−	0.317085i	0.942924	+	0.333007i	$0.108063\pi$
$432$		0			0
$433$	−31.9514			−1.53549			−0.767744	−	0.640757i	$-0.778621\pi$
$433$	−31.9514			−1.53549			−0.767744	+	0.640757i	$0.778621\pi$
$434$	−5.21053	+	7.35202i	−0.250113	+	0.352908i
$435$		0			0
$436$	−13.9355	+	8.04565i	−0.667388	+	0.385317i
$437$	−4.66773	+	8.08475i	−0.223288	+	0.386746i
$438$		0			0
$439$	−3.37246	+	1.94709i	−0.160959	+	0.0929294i	−0.578316	−	0.815813i	$-0.696290\pi$
$439$	−3.37246	+	1.94709i	−0.160959	+	0.0929294i	0.417357	+	0.908743i	$0.362956\pi$
$440$	4.75535	+	11.2817i	0.226702	+	0.537835i
$441$		0			0
$442$			15.6342i			0.743641i
$443$	4.84958	−	2.79991i	0.230410	−	0.133028i	−0.380351	−	0.924842i	$-0.624197\pi$
$443$	4.84958	−	2.79991i	0.230410	−	0.133028i	0.610761	+	0.791815i	$0.290863\pi$
$444$		0			0
$445$	−29.2811	+	50.7163i	−1.38806	+	2.40418i
$446$	−7.37151	−	12.7678i	−0.349051	−	0.604575i
$447$		0			0
$448$	−1.52985	+	2.15860i	−0.0722785	+	0.101984i
$449$		−	19.7608i		−	0.932570i	−0.884635	−	0.466285i	$-0.845592\pi$
$449$		−	19.7608i		−	0.932570i	0.884635	−	0.466285i	$-0.154408\pi$
$450$		0			0
$451$	−2.57685	+	20.5591i	−0.121339	+	0.968092i
$452$	−16.3134	−	9.41854i	−0.767318	−	0.443011i
$453$		0			0
$454$	18.8294			0.883707
$455$	54.5069	+	5.08139i	2.55532	+	0.238219i
$456$		0			0
$457$	16.7231	−	9.65507i	0.782272	−	0.451645i	−0.0549627	−	0.998488i	$-0.517504\pi$
$457$	16.7231	−	9.65507i	0.782272	−	0.451645i	0.837235	+	0.546843i	$0.184171\pi$
$458$	−6.82233	+	11.8166i	−0.318787	+	0.552155i
$459$		0			0
$460$	−6.95239	−	12.0419i	−0.324157	−	0.561456i
$461$	6.95967			0.324144			0.162072	−	0.986779i	$-0.448182\pi$
$461$	6.95967			0.324144			0.162072	+	0.986779i	$0.448182\pi$
$462$		0			0
$463$	16.2724			0.756243			0.378121	−	0.925756i	$-0.376570\pi$
$463$	16.2724			0.756243			0.378121	+	0.925756i	$0.376570\pi$
$464$	0.755057	+	1.30780i	0.0350526	+	0.0607129i
$465$		0			0
$466$	−3.51518	+	6.08847i	−0.162837	+	0.282043i
$467$	−1.39696	+	0.806535i	−0.0646436	+	0.0373220i	−0.531973	−	0.846761i	$-0.678549\pi$
$467$	−1.39696	+	0.806535i	−0.0646436	+	0.0373220i	0.467330	+	0.884083i	$0.345216\pi$
$468$		0			0
$469$	12.5158	+	27.2440i	0.577928	+	1.25801i
$470$	38.2838			1.76590
$471$		0			0
$472$	−11.6232	−	6.71066i	−0.535002	−	0.308883i
$473$	−25.4058	−	3.18432i	−1.16816	−	0.146415i
$474$		0			0
$475$			21.3792i			0.980947i
$476$	7.34774	+	0.684991i	0.336783	+	0.0313965i
$477$		0			0
$478$	5.19541	+	8.99871i	0.237632	+	0.411591i
$479$	−16.9520	+	29.3616i	−0.774555	+	1.34157i	0.160490	+	0.987037i	$0.448693\pi$
$479$	−16.9520	+	29.3616i	−0.774555	+	1.34157i	−0.935045	+	0.354530i	$0.884641\pi$
$480$		0			0
$481$	−27.7531	+	16.0233i	−1.26543	+	0.730599i
$482$			10.6807i			0.486490i
$483$		0			0
$484$	7.68095	−	7.87420i	0.349134	−	0.357918i
$485$	−14.6616	+	8.46489i	−0.665750	+	0.384371i
$486$		0			0
$487$	−1.52594	+	2.64300i	−0.0691468	+	0.119766i	−0.898526	−	0.438920i	$-0.855361\pi$
$487$	−1.52594	+	2.64300i	−0.0691468	+	0.119766i	0.829379	+	0.558686i	$0.188694\pi$
$488$	−2.13230	+	1.23108i	−0.0965246	+	0.0557285i
$489$		0			0
$490$	4.77630	−	25.3945i	0.215771	−	1.14721i
$491$	−35.7449			−1.61315			−0.806573	−	0.591134i	$-0.798680\pi$
$491$	−35.7449			−1.61315			−0.806573	+	0.591134i	$0.798680\pi$
$492$		0			0
$493$	2.10602	−	3.64774i	0.0948505	−	0.164286i
$494$	−6.94581	+	12.0305i	−0.312507	+	0.541278i
$495$		0			0
$496$	3.40592			0.152930
$497$	11.1187	+	7.88004i	0.498741	+	0.353468i
$498$		0			0
$499$	−3.75597	−	6.50553i	−0.168140	−	0.291227i	0.769626	−	0.638495i	$-0.220443\pi$
$499$	−3.75597	−	6.50553i	−0.168140	−	0.291227i	−0.937766	+	0.347268i	$0.887109\pi$
$500$	−11.5931	−	6.69326i	−0.518458	−	0.299332i
$501$		0			0
$502$	7.96699	−	4.59974i	0.355584	−	0.205297i
$503$	−21.5419			−0.960507			−0.480253	−	0.877130i	$-0.659455\pi$
$503$	−21.5419			−0.960507			−0.480253	+	0.877130i	$0.659455\pi$
$504$		0			0
$505$		−	10.7642i		−	0.479002i
$506$	−7.54360	+	9.95848i	−0.335354	+	0.442709i
$507$		0			0
$508$	−6.99993	−	4.04141i	−0.310572	−	0.179309i
$509$	−26.1028	+	15.0705i	−1.15699	+	0.667986i	−0.950580	−	0.310480i	$-0.899510\pi$
$509$	−26.1028	+	15.0705i	−1.15699	+	0.667986i	−0.206406	+	0.978466i	$0.566177\pi$
$510$		0			0
$511$	12.6021	−	5.78938i	0.557483	−	0.256107i
$512$	1.00000			0.0441942
$513$		0			0
$514$	17.8743	+	10.3197i	0.788403	+	0.455185i
$515$	29.5143	+	17.0401i	1.30056	+	0.750876i
$516$		0			0
$517$	−13.3603	−	31.6963i	−0.587585	−	1.39400i
$518$	6.31465	+	13.7455i	0.277450	+	0.603941i
$519$		0			0
$520$	−10.3455	−	17.9189i	−0.453680	−	0.785797i
$521$	27.8685	+	16.0899i	1.22094	+	0.704909i	0.965118	−	0.261815i	$-0.0843209\pi$
$521$	27.8685	+	16.0899i	1.22094	+	0.704909i	0.255821	+	0.966724i	$0.417654\pi$
$522$		0			0
$523$	3.03873	−	1.75441i	0.132874	−	0.0767149i	−0.432089	−	0.901831i	$-0.642224\pi$
$523$	3.03873	−	1.75441i	0.132874	−	0.0767149i	0.564964	+	0.825116i	$0.308890\pi$
$524$	−15.6711			−0.684597
$525$		0			0
$526$	26.3226			1.14772
$527$	−4.74993	−	8.22713i	−0.206910	−	0.358379i
$528$		0			0
$529$	−4.40558	+	7.63068i	−0.191547	+	0.331769i
$530$	−19.7709	−	34.2443i	−0.858794	−	1.48748i
$531$		0			0
$532$	5.34977	+	3.79149i	0.231942	+	0.164382i
$533$		−	35.0174i		−	1.51677i
$534$		0			0
$535$	−11.5877	−	6.69016i	−0.500980	−	0.289241i
$536$	5.66594	−	9.81369i	0.244731	−	0.423887i
$537$		0			0
$538$			8.12947i			0.350486i
$539$	−22.6917	+	4.90774i	−0.977401	+	0.211391i
$540$		0			0
$541$	−3.92948	+	2.26869i	−0.168942	+	0.0975385i	−0.582087	−	0.813127i	$-0.697764\pi$
$541$	−3.92948	+	2.26869i	−0.168942	+	0.0975385i	0.413145	+	0.910665i	$0.364430\pi$
$542$	−12.6065	−	7.27836i	−0.541495	−	0.312632i
$543$		0			0
$544$	−1.39461	−	2.41554i	−0.0597935	−	0.103565i
$545$	−59.3994			−2.54439
$546$		0			0
$547$			16.8271i			0.719477i	0.933053	+	0.359738i	$0.117134\pi$
$547$			16.8271i			0.719477i	−0.933053	+	0.359738i	$0.882866\pi$
$548$	2.55426	−	1.47471i	0.109113	−	0.0629963i
$549$		0			0
$550$	−3.55817	+	28.3884i	−0.151721	+	1.21049i
$551$	3.24117	−	1.87129i	0.138079	−	0.0797197i
$552$		0			0
$553$	0.0709535	−	0.761102i	0.00301725	−	0.0323653i
$554$			29.9949i			1.27436i
$555$		0			0
$556$	3.47235	+	2.00476i	0.147260	+	0.0850208i
$557$	−6.67157	+	11.5555i	−0.282684	+	0.489622i	−0.972045	−	0.234795i	$-0.924558\pi$
$557$	−6.67157	+	11.5555i	−0.282684	+	0.489622i	0.689361	+	0.724418i	$0.257891\pi$
$558$		0			0
$559$	43.2725			1.83023
$560$	−8.87481	+	4.07708i	−0.375029	+	0.172288i
$561$		0			0
$562$	−6.42117	−	11.1218i	−0.270861	−	0.469144i
$563$	16.4049	−	28.4141i	0.691384	−	1.19751i	−0.280001	−	0.960000i	$-0.590335\pi$
$563$	16.4049	−	28.4141i	0.691384	−	1.19751i	0.971385	−	0.237512i	$-0.0763318\pi$
$564$		0			0
$565$	−34.7676	−	60.2192i	−1.46268	−	2.53344i
$566$			7.33349i			0.308250i
$567$		0			0
$568$		−	5.15087i		−	0.216126i
$569$	2.97076	+	5.14551i	0.124541	+	0.215711i	0.921553	−	0.388252i	$-0.126921\pi$
$569$	2.97076	+	5.14551i	0.124541	+	0.215711i	−0.797012	+	0.603963i	$0.793588\pi$
$570$		0			0
$571$	−31.8007	−	18.3601i	−1.33082	−	0.768348i	−0.345393	−	0.938458i	$-0.612254\pi$
$571$	−31.8007	−	18.3601i	−1.33082	−	0.768348i	−0.985425	+	0.170110i	$0.945588\pi$
$572$	−11.2253	+	14.8187i	−0.469351	+	0.619602i
$573$		0			0
$574$	−16.4575	−	1.53424i	−0.686922	−	0.0640381i
$575$		−	32.4940i		−	1.35510i
$576$		0			0
$577$	0.409992	−	0.710127i	0.0170682	−	0.0295630i	−0.857365	−	0.514709i	$-0.827900\pi$
$577$	0.409992	−	0.710127i	0.0170682	−	0.0295630i	0.874433	+	0.485146i	$0.161233\pi$
$578$	4.61011	−	7.98495i	0.191755	−	0.332130i
$579$		0			0
$580$			5.57443i			0.231465i
$581$	15.6843	−	22.1305i	0.650696	−	0.918128i
$582$		0			0
$583$	−21.4522	+	28.3195i	−0.888459	+	1.17287i
$584$	−4.53946	−	2.62086i	−0.187844	−	0.108452i
$585$		0			0
$586$	3.02672	+	5.24243i	0.125032	+	0.216563i
$587$			42.0341i			1.73493i	0.497494	+	0.867467i	$0.334254\pi$
$587$			42.0341i			1.73493i	−0.497494	+	0.867467i	$0.665746\pi$
$588$		0			0
$589$		−	8.44105i		−	0.347807i
$590$	−24.7717	−	42.9059i	−1.01984	−	1.76641i
$591$		0			0
$592$	2.85865	−	4.95132i	0.117490	−	0.203498i
$593$	−13.1169	−	22.7191i	−0.538645	−	0.932960i	−0.998977	−	0.0452135i	$-0.985603\pi$
$593$	−13.1169	−	22.7191i	−0.538645	−	0.932960i	0.460333	−	0.887747i	$-0.347730\pi$
$594$		0			0
$595$	22.2253	+	15.7515i	0.911148	+	0.645749i
$596$	−6.46913			−0.264986
$597$		0			0
$598$	10.5569	−	18.2850i	0.431702	−	0.747729i
$599$	22.7563	+	13.1384i	0.929798	+	0.536819i	0.886748	−	0.462254i	$-0.152959\pi$
$599$	22.7563	+	13.1384i	0.929798	+	0.536819i	0.0430501	+	0.999073i	$0.486292\pi$
$600$		0			0
$601$		−	37.0989i		−	1.51329i	−0.653824	−	0.756647i	$-0.726836\pi$
$601$		−	37.0989i		−	1.51329i	0.653824	−	0.756647i	$-0.273164\pi$
$602$	1.89593	−	20.3372i	0.0772722	−	0.828882i
$603$		0			0
$604$	13.3957	−	7.73400i	0.545062	−	0.314692i
$605$	39.0882	−	10.9959i	1.58916	−	0.447045i
$606$		0			0
$607$	1.72911	−	0.998300i	0.0701822	−	0.0405197i	−0.464498	−	0.885574i	$-0.653765\pi$
$607$	1.72911	−	0.998300i	0.0701822	−	0.0405197i	0.534680	+	0.845054i	$0.320432\pi$
$608$		−	2.47835i		−	0.100510i
$609$		0			0
$610$	−9.08883			−0.367996
$611$	29.0660	+	50.3437i	1.17588	+	2.03669i
$612$		0			0
$613$	−6.55742	−	3.78593i	−0.264852	−	0.152912i	0.361694	−	0.932297i	$-0.382199\pi$
$613$	−6.55742	−	3.78593i	−0.264852	−	0.152912i	−0.626546	+	0.779385i	$0.715532\pi$
$614$	−5.40966	+	3.12327i	−0.218316	+	0.126045i
$615$		0			0
$616$	6.47267	+	5.92490i	0.260792	+	0.238721i
$617$		−	17.2159i		−	0.693084i	−0.938034	−	0.346542i	$-0.887356\pi$
$617$		−	17.2159i		−	0.693084i	0.938034	−	0.346542i	$-0.112644\pi$
$618$		0			0
$619$	−18.3426	+	31.7703i	−0.737250	+	1.27695i	0.216479	+	0.976287i	$0.430543\pi$
$619$	−18.3426	+	31.7703i	−0.737250	+	1.27695i	−0.953729	+	0.300667i	$0.902791\pi$
$620$	10.8882	+	6.28629i	0.437280	+	0.252464i
$621$		0			0
$622$			9.76504i			0.391542i
$623$	−3.89607	+	41.7923i	−0.156093	+	1.67437i
$624$		0			0
$625$	−3.14145	−	5.44116i	−0.125658	−	0.217646i
$626$	−16.0158	+	27.7402i	−0.640120	+	1.10872i
$627$		0			0
$628$	9.34257	+	16.1818i	0.372809	+	0.645725i
$629$	−15.9468			−0.635842
$630$		0			0
$631$	−47.4286			−1.88810			−0.944051	−	0.329801i	$-0.893019\pi$
$631$	−47.4286			−1.88810			−0.944051	+	0.329801i	$0.893019\pi$
$632$	−0.250209	+	0.144458i	−0.00995279	+	0.00574625i
$633$		0			0
$634$	18.5774	+	10.7256i	0.737802	+	0.425970i
$635$	−14.9185	−	25.8395i	−0.592021	−	1.02541i
$636$		0			0
$637$	37.0204	−	12.9992i	1.46680	−	0.515048i
$638$	4.61524	−	1.94537i	0.182719	−	0.0770178i
$639$		0			0
$640$	3.19684	+	1.84570i	0.126366	+	0.0729576i
$641$	−28.5801	−	16.5007i	−1.12884	−	0.651739i	−0.185200	−	0.982701i	$-0.559293\pi$
$641$	−28.5801	−	16.5007i	−1.12884	−	0.651739i	−0.943644	+	0.330962i	$0.892627\pi$
$642$		0			0
$643$	−41.8847			−1.65177			−0.825885	−	0.563838i	$-0.809324\pi$
$643$	−41.8847			−1.65177			−0.825885	+	0.563838i	$0.809324\pi$
$644$	−8.13105	−	5.76264i	−0.320408	−	0.227080i
$645$		0			0
$646$	−5.98655	+	3.45634i	−0.235538	+	0.135988i
$647$	17.9616	+	10.3701i	0.706142	+	0.407691i	0.809631	−	0.586939i	$-0.199667\pi$
$647$	17.9616	+	10.3701i	0.706142	+	0.407691i	−0.103489	+	0.994631i	$0.533001\pi$
$648$		0			0
$649$	−26.8782	+	35.4826i	−1.05506	+	1.39281i
$650$		−	48.3527i		−	1.89655i
$651$		0			0
$652$	−9.25602			−0.362494
$653$	6.73045	−	3.88583i	0.263383	−	0.152064i	−0.362494	−	0.931986i	$-0.618075\pi$
$653$	6.73045	−	3.88583i	0.263383	−	0.152064i	0.625877	+	0.779922i	$0.284741\pi$
$654$		0			0
$655$	−50.0982	−	28.9242i	−1.95750	−	1.13016i
$656$	3.12366	+	5.41033i	0.121958	+	0.211238i
$657$		0			0
$658$	24.9340	−	11.4547i	0.972029	−	0.446549i
$659$	−44.6638			−1.73986			−0.869928	−	0.493180i	$-0.835835\pi$
$659$	−44.6638			−1.73986			−0.869928	+	0.493180i	$0.835835\pi$
$660$		0			0
$661$	3.14521	−	5.44767i	0.122335	−	0.211890i	−0.798353	−	0.602189i	$-0.794295\pi$
$661$	3.14521	−	5.44767i	0.122335	−	0.211890i	0.920688	+	0.390300i	$0.127629\pi$
$662$	15.3327	−	26.5569i	0.595920	−	1.03216i
$663$		0			0
$664$	−10.2522			−0.397864
$665$	10.1044	+	21.9949i	0.391833	+	0.852924i
$666$		0			0
$667$	−4.92622	+	2.84415i	−0.190744	+	0.110126i
$668$	−8.19377	+	14.1920i	−0.317027	+	0.549106i
$669$		0			0
$670$	36.2262	−	20.9152i	1.39954	−	0.808025i
$671$	3.17182	+	7.52491i	0.122447	+	0.290496i
$672$		0			0
$673$		−	4.96695i		−	0.191462i	−0.995407	−	0.0957309i	$-0.969481\pi$
$673$		−	4.96695i		−	0.191462i	0.995407	−	0.0957309i	$-0.0305188\pi$
$674$	−28.9019	+	16.6865i	−1.11326	+	0.642742i
$675$		0			0
$676$	9.20911	−	15.9507i	0.354197	−	0.613487i
$677$	−12.5168	−	21.6798i	−0.481061	−	0.833222i	0.518703	−	0.854955i	$-0.326415\pi$
$677$	−12.5168	−	21.6798i	−0.481061	−	0.833222i	−0.999764	+	0.0217327i	$0.993082\pi$
$678$		0			0
$679$	−7.01631	+	9.89996i	−0.269261	+	0.379926i
$680$		−	10.2961i		−	0.394839i
$681$		0			0
$682$	1.40485	−	11.2084i	0.0537945	−	0.429194i
$683$	−37.1556	−	21.4518i	−1.42172	−	0.820831i	−0.425275	−	0.905064i	$-0.639823\pi$
$683$	−37.1556	−	21.4518i	−1.42172	−	0.820831i	−0.996446	+	0.0842334i	$0.973156\pi$
$684$		0			0
$685$	10.8874			0.415988
$686$	−4.48737	−	17.9684i	−0.171328	−	0.686037i
$687$		0			0
$688$	−6.68577	+	3.86003i	−0.254893	+	0.147162i
$689$	30.0211	−	51.9981i	1.14371	−	1.98097i
$690$		0			0
$691$	−3.04518	−	5.27440i	−0.115844	−	0.200648i	0.802273	−	0.596958i	$-0.203624\pi$
$691$	−3.04518	−	5.27440i	−0.115844	−	0.200648i	−0.918117	+	0.396310i	$0.870291\pi$
$692$	20.9243			0.795424
$693$		0			0
$694$	−20.1533			−0.765009
$695$	7.40037	+	12.8178i	0.280712	+	0.486207i
$696$		0			0
$697$	8.71258	−	15.0906i	0.330013	−	0.571599i
$698$	22.3673	−	12.9138i	0.846616	−	0.488794i
$699$		0			0
$700$	−22.7248	−	2.11851i	−0.858917	−	0.0800723i
$701$	−49.6670			−1.87590			−0.937948	−	0.346777i	$-0.887276\pi$
$701$	−49.6670			−1.87590			−0.937948	+	0.346777i	$0.887276\pi$
$702$		0			0
$703$	−12.2711	−	7.08472i	−0.462813	−	0.267205i
$704$	0.412474	−	3.29088i	0.0155457	−	0.124030i
$705$		0			0
$706$			9.48296i			0.356896i
$707$	−3.22070	−	7.01068i	−0.121127	−	0.263664i
$708$		0			0
$709$	−1.37833	−	2.38734i	−0.0517643	−	0.0896584i	0.838982	−	0.544159i	$-0.183151\pi$
$709$	−1.37833	−	2.38734i	−0.0517643	−	0.0896584i	−0.890746	+	0.454500i	$0.849818\pi$
$710$	9.50695	−	16.4665i	0.356789	−	0.617978i
$711$		0			0
$712$	13.7391	−	7.93225i	0.514893	−	0.297274i
$713$			12.8294i			0.480466i
$714$		0			0
$715$	−63.2362	+	26.6547i	−2.36490	+	0.996828i
$716$	12.9584	−	7.48153i	0.484278	−	0.279598i
$717$		0			0
$718$	−5.42112	+	9.38966i	−0.202314	+	0.350419i
$719$	34.4448	−	19.8867i	1.28457	−	0.741650i	0.306894	−	0.951744i	$-0.400710\pi$
$719$	34.4448	−	19.8867i	1.28457	−	0.741650i	0.977681	+	0.210094i	$0.0673771\pi$
$720$		0			0
$721$	24.3210	+	2.26732i	0.905761	+	0.0844393i
$722$	12.8578			0.478518
$723$		0			0
$724$	5.05625	−	8.75769i	0.187914	−	0.325477i
$725$	−6.51343	+	11.2816i	−0.241903	+	0.418988i
$726$		0			0
$727$	25.4901			0.945376			0.472688	−	0.881230i	$-0.343284\pi$
$727$	25.4901			0.945376			0.472688	+	0.881230i	$0.343284\pi$
$728$	−12.0994	−	8.57509i	−0.448433	−	0.317814i
$729$		0			0
$730$	−9.67462	−	16.7569i	−0.358074	−	0.620202i
$731$	18.6481	+	10.7665i	0.689726	+	0.398213i
$732$		0			0
$733$	−42.1323	+	24.3251i	−1.55619	+	0.898467i	−0.558575	+	0.829454i	$0.688652\pi$
$733$	−42.1323	+	24.3251i	−1.55619	+	0.898467i	−0.997616	+	0.0690128i	$0.978015\pi$
$734$	−8.92893			−0.329573
$735$		0			0
$736$			3.76681i			0.138846i
$737$	−29.9586	−	22.6938i	−1.10354	−	0.835936i
$738$		0			0
$739$	10.9783	+	6.33835i	0.403845	+	0.233160i	0.688142	−	0.725576i	$-0.258427\pi$
$739$	10.9783	+	6.33835i	0.403845	+	0.233160i	−0.284297	+	0.958736i	$0.591760\pi$
$740$	18.2773	−	10.5524i	0.671887	−	0.387914i
$741$		0			0
$742$	−23.1227	−	16.3876i	−0.848862	−	0.601606i
$743$	−18.4512			−0.676910			−0.338455	−	0.940983i	$-0.609904\pi$
$743$	−18.4512			−0.676910			−0.338455	+	0.940983i	$0.609904\pi$
$744$		0			0
$745$	−20.6808	−	11.9401i	−0.757686	−	0.437450i
$746$	12.8877	+	7.44073i	0.471853	+	0.272424i
$747$		0			0
$748$	−8.52448	+	3.59315i	−0.311686	+	0.131379i
$749$	−9.54873	−	0.890178i	−0.348903	−	0.0325264i
$750$		0			0
$751$	25.5428	+	44.2414i	0.932070	+	1.61439i	0.779777	+	0.626057i	$0.215332\pi$
$751$	25.5428	+	44.2414i	0.932070	+	1.61439i	0.152293	+	0.988335i	$0.451334\pi$
$752$	−8.98162	−	5.18554i	−0.327526	−	0.189097i
$753$		0			0
$754$	−7.33046	+	4.23224i	−0.266959	+	0.154129i
$755$	57.0985			2.07803
$756$		0			0
$757$	13.3898			0.486659			0.243329	−	0.969944i	$-0.421760\pi$
$757$	13.3898			0.486659			0.243329	+	0.969944i	$0.421760\pi$
$758$	−15.1414	−	26.2256i	−0.549959	−	0.952556i
$759$		0			0
$760$	4.57428	−	7.92289i	0.165927	−	0.287393i
$761$	3.48214	+	6.03125i	0.126228	+	0.218633i	0.922212	−	0.386684i	$-0.126380\pi$
$761$	3.48214	+	6.03125i	0.126228	+	0.218633i	−0.795985	+	0.605317i	$0.793046\pi$
$762$		0			0
$763$	−38.6865	+	17.7726i	−1.40055	+	0.643410i
$764$		−	5.27991i		−	0.191020i
$765$		0			0
$766$	−18.4934	−	10.6772i	−0.668194	−	0.385782i
$767$	37.6146	−	65.1504i	1.35818	−	2.35244i
$768$		0			0
$769$			2.40816i			0.0868404i	0.999057	+	0.0434202i	$0.0138254\pi$
$769$			2.40816i			0.0868404i	−0.999057	+	0.0434202i	$0.986175\pi$
$770$	9.75654	+	30.8876i	0.351601	+	1.11311i
$771$		0			0
$772$	−19.8361	+	11.4524i	−0.713918	+	0.412181i
$773$	11.9339	+	6.89002i	0.429231	+	0.247817i	0.699019	−	0.715103i	$-0.253620\pi$
$773$	11.9339	+	6.89002i	0.429231	+	0.247817i	−0.269788	+	0.962920i	$0.586954\pi$
$774$		0			0
$775$	14.6904	+	25.4445i	0.527695	+	0.913995i
$776$	4.58628			0.164638
$777$		0			0
$778$		−	6.98703i		−	0.250497i
$779$	13.4087	−	7.74151i	0.480416	−	0.277368i
$780$		0			0
$781$	−16.9509	−	2.12460i	−0.606550	−	0.0760241i
$782$	9.09888	−	5.25324i	0.325375	−	0.187855i
$783$		0			0
$784$	−4.56024	+	5.31076i	−0.162866	+	0.189670i
$785$			68.9743i			2.46180i
$786$		0			0
$787$	4.15993	+	2.40174i	0.148285	+	0.0856127i	0.572307	−	0.820039i	$-0.306049\pi$
$787$	4.15993	+	2.40174i	0.148285	+	0.0856127i	−0.424022	+	0.905652i	$0.639382\pi$
$788$	−4.28337	+	7.41901i	−0.152589	+	0.264291i
$789$		0			0
$790$	−1.06651			−0.0379446
$791$	−40.6618	−	28.8179i	−1.44577	−	1.02465i
$792$		0			0
$793$	−6.90046	−	11.9519i	−0.245042	−	0.424426i
$794$	−5.27929	+	9.14400i	−0.187355	+	0.324508i
$795$		0			0
$796$	2.28887	+	3.96444i	0.0811269	+	0.140516i
$797$			17.4411i			0.617794i	0.951096	+	0.308897i	$0.0999598\pi$
$797$			17.4411i			0.617794i	−0.951096	+	0.308897i	$0.900040\pi$
$798$		0			0
$799$			28.9273i			1.02337i
$800$	4.31321	+	7.47069i	0.152495	+	0.264129i
$801$		0			0
$802$	−8.21248	−	4.74148i	−0.289993	−	0.167427i
$803$	−10.4973	+	13.8578i	−0.370442	+	0.489030i
$804$		0			0
$805$	−15.3576	−	33.4297i	−0.541284	−	1.17824i
$806$			19.0908i			0.672446i
$807$		0			0
$808$	−1.45801	+	2.52536i	−0.0512928	+	0.0888417i
$809$	−9.83186	+	17.0293i	−0.345670	+	0.598717i	−0.985475	−	0.169819i	$-0.945682\pi$
$809$	−9.83186	+	17.0293i	−0.345670	+	0.598717i	0.639806	+	0.768537i	$0.279015\pi$
$810$		0			0
$811$			41.2535i			1.44861i	0.689481	+	0.724304i	$0.257839\pi$
$811$			41.2535i			1.44861i	−0.689481	+	0.724304i	$0.742161\pi$
$812$	1.66789	+	3.63060i	0.0585316	+	0.127409i
$813$		0			0
$814$	−15.1151	−	11.4497i	−0.529783	−	0.401313i
$815$	−29.5901	−	17.0838i	−1.03649	−	0.598420i
$816$		0			0
$817$	9.56650	+	16.5697i	0.334689	+	0.579699i
$818$			1.71956i			0.0601228i
$819$		0			0
$820$			23.0613i			0.805336i
$821$	−6.89065	−	11.9350i	−0.240485	−	0.416533i	0.720367	−	0.693593i	$-0.243973\pi$
$821$	−6.89065	−	11.9350i	−0.240485	−	0.416533i	−0.960853	+	0.277060i	$0.910640\pi$
$822$		0			0
$823$	23.8569	−	41.3214i	0.831600	−	1.44037i	−0.0651682	−	0.997874i	$-0.520758\pi$
$823$	23.8569	−	41.3214i	0.831600	−	1.44037i	0.896769	−	0.442500i	$-0.145908\pi$
$824$	−4.61616	−	7.99543i	−0.160812	−	0.278534i
$825$		0			0
$826$	−28.9713	−	20.5326i	−1.00804	−	0.714420i
$827$	12.5107			0.435040			0.217520	−	0.976056i	$-0.430203\pi$
$827$	12.5107			0.435040			0.217520	+	0.976056i	$0.430203\pi$
$828$		0			0
$829$	−20.7935	+	36.0153i	−0.722187	+	1.25086i	0.237935	+	0.971281i	$0.423530\pi$
$829$	−20.7935	+	36.0153i	−0.722187	+	1.25086i	−0.960121	+	0.279583i	$0.909804\pi$
$830$	−32.7748	−	18.9225i	−1.13763	−	0.656811i
$831$		0			0
$832$			5.60520i			0.194325i
$833$	19.1881	+	3.60898i	0.664829	+	0.125044i
$834$		0			0
$835$	−52.3884	+	30.2465i	−1.81298	+	1.04672i
$836$	−8.15594	−	1.02225i	−0.282079	−	0.0353554i
$837$		0			0
$838$	−4.62491	+	2.67019i	−0.159765	+	0.0922402i
$839$		−	20.8388i		−	0.719436i	−0.933061	−	0.359718i	$-0.882873\pi$
$839$		−	20.8388i		−	0.719436i	0.933061	−	0.359718i	$-0.117127\pi$
$840$		0			0
$841$	−26.7196			−0.921364
$842$	13.5518	+	23.4724i	0.467026	+	0.808914i
$843$		0			0
$844$	8.71101	+	5.02930i	0.299845	+	0.173116i
$845$	58.8802	−	33.9945i	2.02554	−	1.16945i
$846$		0			0
$847$	22.1679	−	18.8569i	0.761699	−	0.647931i
$848$			10.7119i			0.367848i
$849$		0			0
$850$	12.0305	−	20.8374i	0.412643	−	0.714718i
$851$	18.6507	+	10.7680i	0.639337	+	0.369122i
$852$		0			0
$853$			21.8753i			0.748994i	0.927228	+	0.374497i	$0.122185\pi$
$853$			21.8753i			0.748994i	−0.927228	+	0.374497i	$0.877815\pi$
$854$	−5.91951	+	2.71942i	−0.202561	+	0.0930565i
$855$		0			0
$856$	1.81237	+	3.13911i	0.0619454	+	0.107293i
$857$	25.9670	−	44.9762i	0.887016	−	1.53636i	0.0436301	−	0.999048i	$-0.486108\pi$
$857$	25.9670	−	44.9762i	0.887016	−	1.53636i	0.843386	−	0.537309i	$-0.180559\pi$
$858$		0			0
$859$	−3.70523	−	6.41764i	−0.126421	−	0.218967i	0.795867	−	0.605472i	$-0.207016\pi$
$859$	−3.70523	−	6.41764i	−0.126421	−	0.218967i	−0.922287	+	0.386505i	$0.873682\pi$
$860$	−28.4978			−0.971767
$861$		0			0
$862$	−7.60124			−0.258899
$863$	−8.57220	+	4.94916i	−0.291801	+	0.168471i	−0.638754	−	0.769411i	$-0.720550\pi$
$863$	−8.57220	+	4.94916i	−0.291801	+	0.168471i	0.346953	+	0.937883i	$0.387216\pi$
$864$		0			0
$865$	66.8918	+	38.6200i	2.27439	+	1.31312i
$866$	15.9757	+	27.6707i	0.542877	+	0.940290i
$867$		0			0
$868$	8.97230	+	0.836440i	0.304540	+	0.0283906i
$869$	0.372190	+	0.882993i	0.0126257	+	0.0299535i
$870$		0			0
$871$	55.0077	+	31.7587i	1.86386	+	1.07610i
$872$	13.9355	+	8.04565i	0.471915	+	0.272460i
$873$		0			0
$874$	9.33546			0.315777
$875$	−28.8962	−	20.4793i	−0.976869	−	0.692328i
$876$		0			0
$877$	30.3992	−	17.5510i	1.02651	−	0.592655i	0.110526	−	0.993873i	$-0.464746\pi$
$877$	30.3992	−	17.5510i	1.02651	−	0.592655i	0.915983	+	0.401218i	$0.131413\pi$
$878$	3.37246	+	1.94709i	0.113815	+	0.0657110i
$879$		0			0
$880$	7.39258	−	9.75911i	0.249204	−	0.328980i
$881$			23.3064i			0.785212i	0.919707	+	0.392606i	$0.128426\pi$
$881$			23.3064i			0.785212i	−0.919707	+	0.392606i	$0.871574\pi$
$882$		0			0
$883$	−24.2116			−0.814786			−0.407393	−	0.913253i	$-0.633562\pi$
$883$	−24.2116			−0.814786			−0.407393	+	0.913253i	$0.633562\pi$
$884$	13.5396	−	7.81708i	0.455385	−	0.262917i
$885$		0			0
$886$	−4.84958	−	2.79991i	−0.162925	−	0.0940647i
$887$	19.5859	+	33.9237i	0.657629	+	1.13905i	0.981228	+	0.192852i	$0.0617737\pi$
$887$	19.5859	+	33.9237i	0.657629	+	1.13905i	−0.323599	+	0.946194i	$0.604893\pi$
$888$		0			0
$889$	−17.4476	−	12.3655i	−0.585174	−	0.414725i
$890$	58.5621			1.96301
$891$		0			0
$892$	−7.37151	+	12.7678i	−0.246817	+	0.427499i
$893$	−12.8516	+	22.2596i	−0.430061	+	0.744888i
$894$		0			0
$895$	55.2346			1.84629
$896$	2.63433	+	0.245585i	0.0880067	+	0.00820440i
$897$		0			0
$898$	−17.1134	+	9.88040i	−0.571080	+	0.329713i
$899$	2.57166	−	4.45425i	0.0857696	−	0.148557i
$900$		0			0
$901$	25.8750	−	14.9389i	0.862022	−	0.497688i
$902$	19.0932	−	8.04795i	0.635733	−	0.267967i
$903$		0			0
$904$			18.8371i			0.626512i
$905$	32.3281	−	18.6646i	1.07462	−	0.620433i
$906$		0			0
$907$	4.18723	−	7.25250i	0.139035	−	0.240815i	−0.788097	−	0.615551i	$-0.788933\pi$
$907$	4.18723	−	7.25250i	0.139035	−	0.240815i	0.927132	+	0.374736i	$0.122267\pi$
$908$	−9.41469	−	16.3067i	−0.312437	−	0.541158i
$909$		0			0
$910$	−22.8528	−	49.7451i	−0.757564	−	1.64903i
$911$			11.1411i			0.369121i	0.982821	+	0.184560i	$0.0590861\pi$
$911$			11.1411i			0.369121i	−0.982821	+	0.184560i	$0.940914\pi$
$912$		0			0
$913$	−4.22877	+	33.7388i	−0.139952	+	1.11659i
$914$	−16.7231	−	9.65507i	−0.553150	−	0.319361i
$915$		0			0
$916$	13.6447			0.450832
$917$	−41.2829	−	3.84859i	−1.36328	−	0.127092i
$918$		0			0
$919$	−38.7054	+	22.3466i	−1.27677	+	0.737146i	−0.976254	−	0.216630i	$-0.930494\pi$
$919$	−38.7054	+	22.3466i	−1.27677	+	0.737146i	−0.300520	+	0.953776i	$0.597160\pi$
$920$	−6.95239	+	12.0419i	−0.229214	+	0.397010i
$921$		0			0
$922$	−3.47983	−	6.02725i	−0.114602	−	0.198497i
$923$	28.8716			0.950321
$924$		0			0
$925$	49.3197			1.62162
$926$	−8.13620	−	14.0923i	−0.267372	−	0.463102i
$927$		0			0
$928$	0.755057	−	1.30780i	0.0247859	−	0.0429305i
$929$	−5.89281	+	3.40222i	−0.193337	+	0.111623i	−0.593544	−	0.804802i	$-0.702272\pi$
$929$	−5.89281	+	3.40222i	−0.193337	+	0.111623i	0.400207	+	0.916425i	$0.368938\pi$
$930$		0			0
$931$	13.1619	+	11.3019i	0.431365	+	0.370403i
$932$	7.03035			0.230287
$933$		0			0
$934$	1.39696	+	0.806535i	0.0457099	+	0.0263906i
$935$	−33.8833	−	4.24689i	−1.10810	−	0.138888i
$936$		0			0
$937$		−	56.2645i		−	1.83808i	−0.394165	−	0.919040i	$-0.628966\pi$
$937$		−	56.2645i		−	1.83808i	0.394165	−	0.919040i	$-0.371034\pi$
$938$	17.3360	−	24.4610i	0.566041	−	0.798681i
$939$		0			0
$940$	−19.1419	−	33.1547i	−0.624339	−	1.08139i
$941$	20.2704	−	35.1093i	0.660796	−	1.14453i	−0.319611	−	0.947549i	$-0.603552\pi$
$941$	20.2704	−	35.1093i	0.660796	−	1.14453i	0.980407	−	0.196983i	$-0.0631144\pi$
$942$		0			0
$943$	−20.3797	+	11.7662i	−0.663654	+	0.383161i
$944$			13.4213i			0.436827i
$945$		0			0
$946$	9.94518	+	23.5942i	0.323346	+	0.767114i
$947$	10.9708	−	6.33401i	0.356504	−	0.205828i	−0.311042	−	0.950396i	$-0.600678\pi$
$947$	10.9708	−	6.33401i	0.356504	−	0.205828i	0.667546	+	0.744568i	$0.267345\pi$
$948$		0			0
$949$	14.6904	−	25.4445i	0.476871	−	0.825965i
$950$	18.5150	−	10.6896i	0.600705	−	0.346817i
$951$		0			0
$952$	−3.08065	−	6.70582i	−0.0998444	−	0.217337i
$953$	−4.11699			−0.133362			−0.0666812	−	0.997774i	$-0.521241\pi$
$953$	−4.11699			−0.133362			−0.0666812	+	0.997774i	$0.521241\pi$
$954$		0			0
$955$	9.74512	−	16.8790i	0.315345	−	0.546193i
$956$	5.19541	−	8.99871i	0.168031	−	0.291039i
$957$		0			0
$958$	33.9039			1.09539
$959$	7.09094	−	3.25757i	0.228978	−	0.105192i
$960$		0			0
$961$	9.69987	+	16.8007i	0.312899	+	0.541957i
$962$	27.7531	+	16.0233i	0.894797	+	0.516611i
$963$		0			0
$964$	9.24972	−	5.34033i	0.297913	−	0.172000i
$965$	−84.5507			−2.72178
$966$		0			0
$967$		−	2.53643i		−	0.0815661i	−0.999168	−	0.0407831i	$-0.987015\pi$
$967$		−	2.53643i		−	0.0815661i	0.999168	−	0.0407831i	$-0.0129853\pi$
$968$	−10.6597	−	2.71480i	−0.342617	−	0.0872569i
$969$		0			0
$970$	14.6616	+	8.46489i	0.470756	+	0.271791i
$971$	4.36761	−	2.52164i	0.140163	−	0.0809234i	−0.428278	−	0.903647i	$-0.640880\pi$
$971$	4.36761	−	2.52164i	0.140163	−	0.0809234i	0.568442	+	0.822723i	$0.307546\pi$
$972$		0			0
$973$	8.65497	+	6.13395i	0.277466	+	0.196646i
$974$	3.05187			0.0977883
$975$		0			0
$976$	2.13230	+	1.23108i	0.0682532	+	0.0394060i
$977$	−51.9446	−	29.9902i	−1.66185	−	0.959472i	−0.971829	−	0.235686i	$-0.924266\pi$
$977$	−51.9446	−	29.9902i	−1.66185	−	0.959472i	−0.690025	−	0.723786i	$-0.742400\pi$
$978$		0			0
$979$	−20.4370	−	48.4854i	−0.653171	−	1.54960i
$980$	−24.3804	+	8.56086i	−0.778805	+	0.273467i
$981$		0			0
$982$	17.8725	+	30.9560i	0.570334	+	0.987847i
$983$	−19.0455	−	10.9959i	−0.607458	−	0.350716i	0.164512	−	0.986375i	$-0.447395\pi$
$983$	−19.0455	−	10.9959i	−0.607458	−	0.350716i	−0.771970	+	0.635659i	$0.780728\pi$
$984$		0			0
$985$	−27.3865	+	15.8116i	−0.872607	+	0.503800i
$986$	−4.21205			−0.134139
$987$		0			0
$988$	13.8916			0.441951
$989$	−14.5400	−	25.1840i	−0.462345	−	0.800805i
$990$		0			0
$991$	6.59341	−	11.4201i	0.209446	−	0.362772i	−0.742094	−	0.670296i	$-0.766167\pi$
$991$	6.59341	−	11.4201i	0.209446	−	0.362772i	0.951540	+	0.307524i	$0.0995005\pi$
$992$	−1.70296	−	2.94961i	−0.0540690	−	0.0936502i
$993$		0			0
$994$	1.26497	−	13.5691i	0.0401225	−	0.430385i
$995$			16.8983i			0.535711i
$996$		0			0
$997$	−41.2987	−	23.8438i	−1.30794	−	0.755141i	−0.326191	−	0.945304i	$-0.605765\pi$
$997$	−41.2987	−	23.8438i	−1.30794	−	0.755141i	−0.981753	+	0.190162i	$0.939099\pi$
$998$	−3.75597	+	6.50553i	−0.118893	+	0.205929i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.ba.a.1187.1	yes	32
3.2	odd	2		1386.2.ba.b.1187.16	yes	32
7.2	even	3	inner	1386.2.ba.a.989.16	yes	32
11.10	odd	2		1386.2.ba.b.1187.1	yes	32
21.2	odd	6		1386.2.ba.b.989.1	yes	32
33.32	even	2	inner	1386.2.ba.a.1187.16	yes	32
77.65	odd	6		1386.2.ba.b.989.16	yes	32
231.65	even	6	inner	1386.2.ba.a.989.1	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.ba.a.989.1	✓	32	231.65	even	6	inner
1386.2.ba.a.989.16	yes	32	7.2	even	3	inner
1386.2.ba.a.1187.1	yes	32	1.1	even	1	trivial
1386.2.ba.a.1187.16	yes	32	33.32	even	2	inner
1386.2.ba.b.989.1	yes	32	21.2	odd	6
1386.2.ba.b.989.16	yes	32	77.65	odd	6
1386.2.ba.b.1187.1	yes	32	11.10	odd	2
1386.2.ba.b.1187.16	yes	32	3.2	odd	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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.ba (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-3.19684 + 1.84570i) q^{5} +(-1.52985 + 2.15860i) q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-3.19684 + 1.84570i) q^{5} +(-1.52985 + 2.15860i) q^{7} +1.00000 q^{8} +(3.19684 + 1.84570i) q^{10} +(0.412474 - 3.29088i) q^{11} +5.60520i q^{13} +(2.63433 + 0.245585i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-1.39461 + 2.41554i) q^{17} +(-2.14631 + 1.23917i) q^{19} -3.69140i q^{20} +(-3.05622 + 1.28823i) q^{22} +(3.26215 - 1.88340i) q^{23} +(4.31321 - 7.47069i) q^{25} +(4.85424 - 2.80260i) q^{26} +(-1.10448 - 2.40419i) q^{28} -1.51011 q^{29} +(-1.70296 + 2.94961i) q^{31} +(-0.500000 + 0.866025i) q^{32} +2.78923 q^{34} +(0.906550 - 9.72435i) q^{35} +(2.85865 + 4.95132i) q^{37} +(2.14631 + 1.23917i) q^{38} +(-3.19684 + 1.84570i) q^{40} -6.24731 q^{41} -7.72006i q^{43} +(2.64375 + 2.00265i) q^{44} +(-3.26215 - 1.88340i) q^{46} +(8.98162 - 5.18554i) q^{47} +(-2.31914 - 6.60467i) q^{49} -8.62641 q^{50} +(-4.85424 - 2.80260i) q^{52} +(-9.27677 - 5.35595i) q^{53} +(4.75535 + 11.2817i) q^{55} +(-1.52985 + 2.15860i) q^{56} +(0.755057 + 1.30780i) q^{58} +(-11.6232 - 6.71066i) q^{59} +(-2.13230 + 1.23108i) q^{61} +3.40592 q^{62} +1.00000 q^{64} +(-10.3455 - 17.9189i) q^{65} +(5.66594 - 9.81369i) q^{67} +(-1.39461 - 2.41554i) q^{68} +(-8.87481 + 4.07708i) q^{70} -5.15087i q^{71} +(-4.53946 - 2.62086i) q^{73} +(2.85865 - 4.95132i) q^{74} -2.47835i q^{76} +(6.47267 + 5.92490i) q^{77} +(-0.250209 + 0.144458i) q^{79} +(3.19684 + 1.84570i) q^{80} +(3.12366 + 5.41033i) q^{82} -10.2522 q^{83} -10.2961i q^{85} +(-6.68577 + 3.86003i) q^{86} +(0.412474 - 3.29088i) q^{88} +(13.7391 - 7.93225i) q^{89} +(-12.0994 - 8.57509i) q^{91} +3.76681i q^{92} +(-8.98162 - 5.18554i) q^{94} +(4.57428 - 7.92289i) q^{95} +4.58628 q^{97} +(-4.56024 + 5.31076i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 16 q^{2} - 16 q^{4} + 32 q^{8}+O(q^{10}) \) 32 * q - 16 * q^2 - 16 * q^4 + 32 * q^8	\( 32 q - 16 q^{2} - 16 q^{4} + 32 q^{8} - 2 q^{11} - 16 q^{16} - 4 q^{17} + 4 q^{22} + 4 q^{25} - 16 q^{29} + 4 q^{31} - 16 q^{32} + 8 q^{34} - 16 q^{35} + 4 q^{37} + 32 q^{41} - 2 q^{44} + 20 q^{49} - 8 q^{50} - 12 q^{55} + 8 q^{58} - 8 q^{62} + 32 q^{64} - 8 q^{67} - 4 q^{68} - 4 q^{70} + 4 q^{74} - 14 q^{77} - 16 q^{82} - 88 q^{83} - 2 q^{88} + 24 q^{95} - 32 q^{97} + 8 q^{98}+O(q^{100}) \) 32 * q - 16 * q^2 - 16 * q^4 + 32 * q^8 - 2 * q^11 - 16 * q^16 - 4 * q^17 + 4 * q^22 + 4 * q^25 - 16 * q^29 + 4 * q^31 - 16 * q^32 + 8 * q^34 - 16 * q^35 + 4 * q^37 + 32 * q^41 - 2 * q^44 + 20 * q^49 - 8 * q^50 - 12 * q^55 + 8 * q^58 - 8 * q^62 + 32 * q^64 - 8 * q^67 - 4 * q^68 - 4 * q^70 + 4 * q^74 - 14 * q^77 - 16 * q^82 - 88 * q^83 - 2 * q^88 + 24 * q^95 - 32 * q^97 + 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more