LMFDB - Embedded newform 2790.2.a.bk.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2790,2,Mod(1,2790)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2790, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2790.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2790.a (trivial)

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:	yes
Analytic conductor:	$22.2782621639$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.17428.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 4x + 6 $ x^4 - x^3 - 6x^2 + 4x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.10710$ of defining polynomial
Character	$\chi$	$=$	2790.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.92682 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.92682 q^{7} +1.00000 q^{8} +1.00000 q^{10} -0.926817 q^{11} +6.21419 q^{13} +2.92682 q^{14} +1.00000 q^{16} +7.09390 q^{17} -1.28738 q^{19} +1.00000 q^{20} -0.926817 q^{22} -3.80653 q^{23} +1.00000 q^{25} +6.21419 q^{26} +2.92682 q^{28} -8.42839 q^{29} +1.00000 q^{31} +1.00000 q^{32} +7.09390 q^{34} +2.92682 q^{35} -7.97361 q^{37} -1.28738 q^{38} +1.00000 q^{40} -8.42839 q^{41} +11.0468 q^{43} -0.926817 q^{44} -3.80653 q^{46} -5.75942 q^{47} +1.56626 q^{49} +1.00000 q^{50} +6.21419 q^{52} +5.14101 q^{53} -0.926817 q^{55} +2.92682 q^{56} -8.42839 q^{58} +8.11998 q^{59} -14.8012 q^{61} +1.00000 q^{62} +1.00000 q^{64} +6.21419 q^{65} +5.39886 q^{67} +7.09390 q^{68} +2.92682 q^{70} +7.50157 q^{71} +3.43374 q^{73} -7.97361 q^{74} -1.28738 q^{76} -2.71262 q^{77} -0.472045 q^{79} +1.00000 q^{80} -8.42839 q^{82} +7.61305 q^{83} +7.09390 q^{85} +11.0468 q^{86} -0.926817 q^{88} -11.4074 q^{89} +18.1878 q^{91} -3.80653 q^{92} -5.75942 q^{94} -1.28738 q^{95} +8.57475 q^{97} +1.56626 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 5 * q^7 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8} + 4 q^{10} + 3 q^{11} + 6 q^{13} + 5 q^{14} + 4 q^{16} + 7 q^{19} + 4 q^{20} + 3 q^{22} + q^{23} + 4 q^{25} + 6 q^{26} + 5 q^{28} + 4 q^{29} + 4 q^{31} + 4 q^{32} + 5 q^{35} + 6 q^{37} + 7 q^{38} + 4 q^{40} + 4 q^{41} + 13 q^{43} + 3 q^{44} + q^{46} - 4 q^{47} + 5 q^{49} + 4 q^{50} + 6 q^{52} - 5 q^{53} + 3 q^{55} + 5 q^{56} + 4 q^{58} + 8 q^{59} - 4 q^{61} + 4 q^{62} + 4 q^{64} + 6 q^{65} + 8 q^{67} + 5 q^{70} - q^{71} + 15 q^{73} + 6 q^{74} + 7 q^{76} - 23 q^{77} + 5 q^{79} + 4 q^{80} + 4 q^{82} - 2 q^{83} + 13 q^{86} + 3 q^{88} - 9 q^{89} + 16 q^{91} + q^{92} - 4 q^{94} + 7 q^{95} + 10 q^{97} + 5 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 5 * q^7 + 4 * q^8 + 4 * q^10 + 3 * q^11 + 6 * q^13 + 5 * q^14 + 4 * q^16 + 7 * q^19 + 4 * q^20 + 3 * q^22 + q^23 + 4 * q^25 + 6 * q^26 + 5 * q^28 + 4 * q^29 + 4 * q^31 + 4 * q^32 + 5 * q^35 + 6 * q^37 + 7 * q^38 + 4 * q^40 + 4 * q^41 + 13 * q^43 + 3 * q^44 + q^46 - 4 * q^47 + 5 * q^49 + 4 * q^50 + 6 * q^52 - 5 * q^53 + 3 * q^55 + 5 * q^56 + 4 * q^58 + 8 * q^59 - 4 * q^61 + 4 * q^62 + 4 * q^64 + 6 * q^65 + 8 * q^67 + 5 * q^70 - q^71 + 15 * q^73 + 6 * q^74 + 7 * q^76 - 23 * q^77 + 5 * q^79 + 4 * q^80 + 4 * q^82 - 2 * q^83 + 13 * q^86 + 3 * q^88 - 9 * q^89 + 16 * q^91 + q^92 - 4 * q^94 + 7 * q^95 + 10 * q^97 + 5 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.92682	1.10623	0.553116	−	0.833104i	$-0.313438\pi$
$7$	2.92682	1.10623	0.553116	+	0.833104i	$0.313438\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−0.926817	−0.279446	−0.139723	−	0.990191i	$-0.544621\pi$
$11$	−0.926817	−0.279446	−0.139723	+	0.990191i	$0.544621\pi$
$12$	0	0
$13$	6.21419	1.72351	0.861753	−	0.507327i	$-0.169366\pi$
$13$	6.21419	1.72351	0.861753	+	0.507327i	$0.169366\pi$
$14$	2.92682	0.782225
$15$	0	0
$16$	1.00000	0.250000
$17$	7.09390	1.72052	0.860262	−	0.509852i	$-0.170300\pi$
$17$	7.09390	1.72052	0.860262	+	0.509852i	$0.170300\pi$
$18$	0	0
$19$	−1.28738	−0.295344	−0.147672	−	0.989036i	$-0.547178\pi$
$19$	−1.28738	−0.295344	−0.147672	+	0.989036i	$0.547178\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−0.926817	−0.197598
$23$	−3.80653	−0.793716	−0.396858	−	0.917880i	$-0.629899\pi$
$23$	−3.80653	−0.793716	−0.396858	+	0.917880i	$0.629899\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.21419	1.21870
$27$	0	0
$28$	2.92682	0.553116
$29$	−8.42839	−1.56511	−0.782556	−	0.622580i	$-0.786084\pi$
$29$	−8.42839	−1.56511	−0.782556	+	0.622580i	$0.786084\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000	0.176777
$33$	0	0
$34$	7.09390	1.21659
$35$	2.92682	0.494722
$36$	0	0
$37$	−7.97361	−1.31085	−0.655427	−	0.755259i	$-0.727511\pi$
$37$	−7.97361	−1.31085	−0.655427	+	0.755259i	$0.727511\pi$
$38$	−1.28738	−0.208840
$39$	0	0
$40$	1.00000	0.158114
$41$	−8.42839	−1.31629	−0.658146	−	0.752890i	$-0.728659\pi$
$41$	−8.42839	−1.31629	−0.658146	+	0.752890i	$0.728659\pi$
$42$	0	0
$43$	11.0468	1.68462	0.842310	−	0.538993i	$-0.181195\pi$
$43$	11.0468	1.68462	0.842310	+	0.538993i	$0.181195\pi$
$44$	−0.926817	−0.139723
$45$	0	0
$46$	−3.80653	−0.561242
$47$	−5.75942	−0.840098	−0.420049	−	0.907501i	$-0.637987\pi$
$47$	−5.75942	−0.840098	−0.420049	+	0.907501i	$0.637987\pi$
$48$	0	0
$49$	1.56626	0.223751
$50$	1.00000	0.141421
$51$	0	0
$52$	6.21419	0.861753
$53$	5.14101	0.706172	0.353086	−	0.935591i	$-0.385132\pi$
$53$	5.14101	0.706172	0.353086	+	0.935591i	$0.385132\pi$
$54$	0	0
$55$	−0.926817	−0.124972
$56$	2.92682	0.391112
$57$	0	0
$58$	−8.42839	−1.10670
$59$	8.11998	1.05713	0.528566	−	0.848892i	$-0.322730\pi$
$59$	8.11998	1.05713	0.528566	+	0.848892i	$0.322730\pi$
$60$	0	0
$61$	−14.8012	−1.89510	−0.947548	−	0.319614i	$-0.896447\pi$
$61$	−14.8012	−1.89510	−0.947548	+	0.319614i	$0.896447\pi$
$62$	1.00000	0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	6.21419	0.770776
$66$	0	0
$67$	5.39886	0.659576	0.329788	−	0.944055i	$-0.393023\pi$
$67$	5.39886	0.659576	0.329788	+	0.944055i	$0.393023\pi$
$68$	7.09390	0.860262
$69$	0	0
$70$	2.92682	0.349822
$71$	7.50157	0.890272	0.445136	−	0.895463i	$-0.353155\pi$
$71$	7.50157	0.890272	0.445136	+	0.895463i	$0.353155\pi$
$72$	0	0
$73$	3.43374	0.401889	0.200945	−	0.979603i	$-0.435599\pi$
$73$	3.43374	0.401889	0.200945	+	0.979603i	$0.435599\pi$
$74$	−7.97361	−0.926914
$75$	0	0
$76$	−1.28738	−0.147672
$77$	−2.71262	−0.309132
$78$	0	0
$79$	−0.472045	−0.0531092	−0.0265546	−	0.999647i	$-0.508454\pi$
$79$	−0.472045	−0.0531092	−0.0265546	+	0.999647i	$0.508454\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−8.42839	−0.930759
$83$	7.61305	0.835641	0.417821	−	0.908530i	$-0.362794\pi$
$83$	7.61305	0.835641	0.417821	+	0.908530i	$0.362794\pi$
$84$	0	0
$85$	7.09390	0.769442
$86$	11.0468	1.19121
$87$	0	0
$88$	−0.926817	−0.0987990
$89$	−11.4074	−1.20918	−0.604589	−	0.796538i	$-0.706662\pi$
$89$	−11.4074	−1.20918	−0.604589	+	0.796538i	$0.706662\pi$
$90$	0	0
$91$	18.1878	1.90660
$92$	−3.80653	−0.396858
$93$	0	0
$94$	−5.75942	−0.594039
$95$	−1.28738	−0.132082
$96$	0	0
$97$	8.57475	0.870634	0.435317	−	0.900277i	$-0.356636\pi$
$97$	8.57475	0.870634	0.435317	+	0.900277i	$0.356636\pi$
$98$	1.56626	0.158216
$99$	0	0
$100$	1.00000	0.100000
$101$	−10.9004	−1.08463	−0.542317	−	0.840174i	$-0.682453\pi$
$101$	−10.9004	−1.08463	−0.542317	+	0.840174i	$0.682453\pi$
$102$	0	0
$103$	−0.119979	−0.0118219	−0.00591094	−	0.999983i	$-0.501882\pi$
$103$	−0.119979	−0.0118219	−0.00591094	+	0.999983i	$0.501882\pi$
$104$	6.21419	0.609352
$105$	0	0
$106$	5.14101	0.499339
$107$	3.04680	0.294545	0.147272	−	0.989096i	$-0.452951\pi$
$107$	3.04680	0.294545	0.147272	+	0.989096i	$0.452951\pi$
$108$	0	0
$109$	5.90579	0.565672	0.282836	−	0.959168i	$-0.408725\pi$
$109$	5.90579	0.565672	0.282836	+	0.959168i	$0.408725\pi$
$110$	−0.926817	−0.0883686
$111$	0	0
$112$	2.92682	0.276558
$113$	4.32568	0.406926	0.203463	−	0.979083i	$-0.434780\pi$
$113$	4.32568	0.406926	0.203463	+	0.979083i	$0.434780\pi$
$114$	0	0
$115$	−3.80653	−0.354961
$116$	−8.42839	−0.782556
$117$	0	0
$118$	8.11998	0.747505
$119$	20.7626	1.90330
$120$	0	0
$121$	−10.1410	−0.921910
$122$	−14.8012	−1.34004
$123$	0	0
$124$	1.00000	0.0898027
$125$	1.00000	0.0894427
$126$	0	0
$127$	14.6426	1.29932	0.649659	−	0.760225i	$-0.274912\pi$
$127$	14.6426	1.29932	0.649659	+	0.760225i	$0.274912\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	6.21419	0.545021
$131$	−13.6809	−1.19530	−0.597652	−	0.801756i	$-0.703900\pi$
$131$	−13.6809	−1.19530	−0.597652	+	0.801756i	$0.703900\pi$
$132$	0	0
$133$	−3.76791	−0.326719
$134$	5.39886	0.466391
$135$	0	0
$136$	7.09390	0.608297
$137$	4.22642	0.361087	0.180544	−	0.983567i	$-0.442214\pi$
$137$	4.22642	0.361087	0.180544	+	0.983567i	$0.442214\pi$
$138$	0	0
$139$	7.24027	0.614112	0.307056	−	0.951692i	$-0.400656\pi$
$139$	7.24027	0.614112	0.307056	+	0.951692i	$0.400656\pi$
$140$	2.92682	0.247361
$141$	0	0
$142$	7.50157	0.629518
$143$	−5.75942	−0.481627
$144$	0	0
$145$	−8.42839	−0.699939
$146$	3.43374	0.284178
$147$	0	0
$148$	−7.97361	−0.655427
$149$	−0.618411	−0.0506622	−0.0253311	−	0.999679i	$-0.508064\pi$
$149$	−0.618411	−0.0506622	−0.0253311	+	0.999679i	$0.508064\pi$
$150$	0	0
$151$	−18.1878	−1.48010	−0.740051	−	0.672550i	$-0.765199\pi$
$151$	−18.1878	−1.48010	−0.740051	+	0.672550i	$0.765199\pi$
$152$	−1.28738	−0.104420
$153$	0	0
$154$	−2.71262	−0.218590
$155$	1.00000	0.0803219
$156$	0	0
$157$	6.32568	0.504844	0.252422	−	0.967617i	$-0.418773\pi$
$157$	6.32568	0.504844	0.252422	+	0.967617i	$0.418773\pi$
$158$	−0.472045	−0.0375539
$159$	0	0
$160$	1.00000	0.0790569
$161$	−11.1410	−0.878035
$162$	0	0
$163$	12.4548	0.975533	0.487767	−	0.872974i	$-0.337812\pi$
$163$	12.4548	0.975533	0.487767	+	0.872974i	$0.337812\pi$
$164$	−8.42839	−0.658146
$165$	0	0
$166$	7.61305	0.590888
$167$	−22.1828	−1.71655	−0.858277	−	0.513187i	$-0.828465\pi$
$167$	−22.1828	−1.71655	−0.858277	+	0.513187i	$0.828465\pi$
$168$	0	0
$169$	25.6162	1.97048
$170$	7.09390	0.544078
$171$	0	0
$172$	11.0468	0.842310
$173$	−5.18467	−0.394183	−0.197092	−	0.980385i	$-0.563150\pi$
$173$	−5.18467	−0.394183	−0.197092	+	0.980385i	$0.563150\pi$
$174$	0	0
$175$	2.92682	0.221247
$176$	−0.926817	−0.0698615
$177$	0	0
$178$	−11.4074	−0.855017
$179$	−11.2525	−0.841051	−0.420526	−	0.907281i	$-0.638154\pi$
$179$	−11.2525	−0.841051	−0.420526	+	0.907281i	$0.638154\pi$
$180$	0	0
$181$	−2.62186	−0.194881	−0.0974406	−	0.995241i	$-0.531066\pi$
$181$	−2.62186	−0.194881	−0.0974406	+	0.995241i	$0.531066\pi$
$182$	18.1878	1.34817
$183$	0	0
$184$	−3.80653	−0.280621
$185$	−7.97361	−0.586232
$186$	0	0
$187$	−6.57475	−0.480793
$188$	−5.75942	−0.420049
$189$	0	0
$190$	−1.28738	−0.0933960
$191$	19.6809	1.42406	0.712029	−	0.702150i	$-0.247776\pi$
$191$	19.6809	1.42406	0.712029	+	0.702150i	$0.247776\pi$
$192$	0	0
$193$	−1.70727	−0.122892	−0.0614459	−	0.998110i	$-0.519571\pi$
$193$	−1.70727	−0.122892	−0.0614459	+	0.998110i	$0.519571\pi$
$194$	8.57475	0.615631
$195$	0	0
$196$	1.56626	0.111876
$197$	−26.4284	−1.88294	−0.941472	−	0.337090i	$-0.890557\pi$
$197$	−26.4284	−1.88294	−0.941472	+	0.337090i	$0.890557\pi$
$198$	0	0
$199$	−11.5694	−0.820133	−0.410066	−	0.912056i	$-0.634494\pi$
$199$	−11.5694	−0.820133	−0.410066	+	0.912056i	$0.634494\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−10.9004	−0.766952
$203$	−24.6683	−1.73138
$204$	0	0
$205$	−8.42839	−0.588664
$206$	−0.119979	−0.00835933
$207$	0	0
$208$	6.21419	0.430877
$209$	1.19316	0.0825327
$210$	0	0
$211$	9.80997	0.675347	0.337673	−	0.941263i	$-0.390360\pi$
$211$	9.80997	0.675347	0.337673	+	0.941263i	$0.390360\pi$
$212$	5.14101	0.353086
$213$	0	0
$214$	3.04680	0.208275
$215$	11.0468	0.753385
$216$	0	0
$217$	2.92682	0.198685
$218$	5.90579	0.399990
$219$	0	0
$220$	−0.926817	−0.0624860
$221$	44.0829	2.96534
$222$	0	0
$223$	25.0119	1.67492	0.837461	−	0.546497i	$-0.184039\pi$
$223$	25.0119	1.67492	0.837461	+	0.546497i	$0.184039\pi$
$224$	2.92682	0.195556
$225$	0	0
$226$	4.32568	0.287740
$227$	−21.6215	−1.43507	−0.717536	−	0.696521i	$-0.754730\pi$
$227$	−21.6215	−1.43507	−0.717536	+	0.696521i	$0.754730\pi$
$228$	0	0
$229$	−28.3285	−1.87200	−0.936000	−	0.352000i	$-0.885502\pi$
$229$	−28.3285	−1.87200	−0.936000	+	0.352000i	$0.885502\pi$
$230$	−3.80653	−0.250995
$231$	0	0
$232$	−8.42839	−0.553351
$233$	−7.47580	−0.489756	−0.244878	−	0.969554i	$-0.578748\pi$
$233$	−7.47580	−0.489756	−0.244878	+	0.969554i	$0.578748\pi$
$234$	0	0
$235$	−5.75942	−0.375703
$236$	8.11998	0.528566
$237$	0	0
$238$	20.7626	1.34584
$239$	11.3203	0.732251	0.366125	−	0.930566i	$-0.380684\pi$
$239$	11.3203	0.732251	0.366125	+	0.930566i	$0.380684\pi$
$240$	0	0
$241$	4.18781	0.269760	0.134880	−	0.990862i	$-0.456935\pi$
$241$	4.18781	0.269760	0.134880	+	0.990862i	$0.456935\pi$
$242$	−10.1410	−0.651889
$243$	0	0
$244$	−14.8012	−0.947548
$245$	1.56626	0.100065
$246$	0	0
$247$	−8.00000	−0.509028
$248$	1.00000	0.0635001
$249$	0	0
$250$	1.00000	0.0632456
$251$	11.2525	0.710251	0.355126	−	0.934819i	$-0.384438\pi$
$251$	11.2525	0.710251	0.355126	+	0.934819i	$0.384438\pi$
$252$	0	0
$253$	3.52796	0.221801
$254$	14.6426	0.918757
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.14415	0.508018	0.254009	−	0.967202i	$-0.418251\pi$
$257$	8.14415	0.508018	0.254009	+	0.967202i	$0.418251\pi$
$258$	0	0
$259$	−23.3373	−1.45011
$260$	6.21419	0.385388
$261$	0	0
$262$	−13.6809	−0.845208
$263$	−12.0556	−0.743380	−0.371690	−	0.928357i	$-0.621222\pi$
$263$	−12.0556	−0.743380	−0.371690	+	0.928357i	$0.621222\pi$
$264$	0	0
$265$	5.14101	0.315810
$266$	−3.76791	−0.231026
$267$	0	0
$268$	5.39886	0.329788
$269$	−5.42525	−0.330783	−0.165392	−	0.986228i	$-0.552889\pi$
$269$	−5.42525	−0.330783	−0.165392	+	0.986228i	$0.552889\pi$
$270$	0	0
$271$	23.9978	1.45776	0.728881	−	0.684641i	$-0.240041\pi$
$271$	23.9978	1.45776	0.728881	+	0.684641i	$0.240041\pi$
$272$	7.09390	0.430131
$273$	0	0
$274$	4.22642	0.255327
$275$	−0.926817	−0.0558892
$276$	0	0
$277$	9.92146	0.596123	0.298061	−	0.954547i	$-0.403660\pi$
$277$	9.92146	0.596123	0.298061	+	0.954547i	$0.403660\pi$
$278$	7.24027	0.434242
$279$	0	0
$280$	2.92682	0.174911
$281$	16.8568	1.00559	0.502795	−	0.864405i	$-0.332305\pi$
$281$	16.8568	1.00559	0.502795	+	0.864405i	$0.332305\pi$
$282$	0	0
$283$	23.9736	1.42508	0.712542	−	0.701630i	$-0.247544\pi$
$283$	23.9736	1.42508	0.712542	+	0.701630i	$0.247544\pi$
$284$	7.50157	0.445136
$285$	0	0
$286$	−5.75942	−0.340562
$287$	−24.6683	−1.45613
$288$	0	0
$289$	33.3235	1.96020
$290$	−8.42839	−0.494932
$291$	0	0
$292$	3.43374	0.200945
$293$	10.0414	0.586627	0.293314	−	0.956016i	$-0.405242\pi$
$293$	10.0414	0.586627	0.293314	+	0.956016i	$0.405242\pi$
$294$	0	0
$295$	8.11998	0.472763
$296$	−7.97361	−0.463457
$297$	0	0
$298$	−0.618411	−0.0358236
$299$	−23.6545	−1.36797
$300$	0	0
$301$	32.3320	1.86358
$302$	−18.1878	−1.04659
$303$	0	0
$304$	−1.28738	−0.0738361
$305$	−14.8012	−0.847513
$306$	0	0
$307$	−27.3882	−1.56312	−0.781562	−	0.623827i	$-0.785577\pi$
$307$	−27.3882	−1.56312	−0.781562	+	0.623827i	$0.785577\pi$
$308$	−2.71262	−0.154566
$309$	0	0
$310$	1.00000	0.0567962
$311$	−16.1093	−0.913473	−0.456736	−	0.889602i	$-0.650982\pi$
$311$	−16.1093	−0.913473	−0.456736	+	0.889602i	$0.650982\pi$
$312$	0	0
$313$	20.4698	1.15702	0.578511	−	0.815674i	$-0.303634\pi$
$313$	20.4698	1.15702	0.578511	+	0.815674i	$0.303634\pi$
$314$	6.32568	0.356979
$315$	0	0
$316$	−0.472045	−0.0265546
$317$	−2.29273	−0.128773	−0.0643863	−	0.997925i	$-0.520509\pi$
$317$	−2.29273	−0.128773	−0.0643863	+	0.997925i	$0.520509\pi$
$318$	0	0
$319$	7.81157	0.437364
$320$	1.00000	0.0559017
$321$	0	0
$322$	−11.1410	−0.620864
$323$	−9.13252	−0.508147
$324$	0	0
$325$	6.21419	0.344701
$326$	12.4548	0.689806
$327$	0	0
$328$	−8.42839	−0.465380
$329$	−16.8568	−0.929344
$330$	0	0
$331$	−35.8043	−1.96798	−0.983991	−	0.178216i	$-0.942967\pi$
$331$	−35.8043	−1.96798	−0.983991	+	0.178216i	$0.942967\pi$
$332$	7.61305	0.417821
$333$	0	0
$334$	−22.1828	−1.21379
$335$	5.39886	0.294971
$336$	0	0
$337$	−12.0942	−0.658814	−0.329407	−	0.944188i	$-0.606849\pi$
$337$	−12.0942	−0.658814	−0.329407	+	0.944188i	$0.606849\pi$
$338$	25.6162	1.39334
$339$	0	0
$340$	7.09390	0.384721
$341$	−0.926817	−0.0501900
$342$	0	0
$343$	−15.9036	−0.858712
$344$	11.0468	0.595603
$345$	0	0
$346$	−5.18467	−0.278730
$347$	−1.51946	−0.0815690	−0.0407845	−	0.999168i	$-0.512986\pi$
$347$	−1.51946	−0.0815690	−0.0407845	+	0.999168i	$0.512986\pi$
$348$	0	0
$349$	11.4422	0.612489	0.306244	−	0.951953i	$-0.400928\pi$
$349$	11.4422	0.612489	0.306244	+	0.951953i	$0.400928\pi$
$350$	2.92682	0.156445
$351$	0	0
$352$	−0.926817	−0.0493995
$353$	−32.2679	−1.71744	−0.858722	−	0.512441i	$-0.828741\pi$
$353$	−32.2679	−1.71744	−0.858722	+	0.512441i	$0.828741\pi$
$354$	0	0
$355$	7.50157	0.398142
$356$	−11.4074	−0.604589
$357$	0	0
$358$	−11.2525	−0.594713
$359$	6.48772	0.342409	0.171204	−	0.985236i	$-0.445234\pi$
$359$	6.48772	0.342409	0.171204	+	0.985236i	$0.445234\pi$
$360$	0	0
$361$	−17.3427	−0.912772
$362$	−2.62186	−0.137802
$363$	0	0
$364$	18.1878	0.953300
$365$	3.43374	0.179730
$366$	0	0
$367$	−12.5835	−0.656855	−0.328428	−	0.944529i	$-0.606519\pi$
$367$	−12.5835	−0.656855	−0.328428	+	0.944529i	$0.606519\pi$
$368$	−3.80653	−0.198429
$369$	0	0
$370$	−7.97361	−0.414528
$371$	15.0468	0.781191
$372$	0	0
$373$	−5.80997	−0.300829	−0.150415	−	0.988623i	$-0.548061\pi$
$373$	−5.80997	−0.300829	−0.150415	+	0.988623i	$0.548061\pi$
$374$	−6.57475	−0.339972
$375$	0	0
$376$	−5.75942	−0.297019
$377$	−52.3756	−2.69748
$378$	0	0
$379$	10.0085	0.514102	0.257051	−	0.966398i	$-0.417249\pi$
$379$	10.0085	0.514102	0.257051	+	0.966398i	$0.417249\pi$
$380$	−1.28738	−0.0660410
$381$	0	0
$382$	19.6809	1.00696
$383$	14.2434	0.727804	0.363902	−	0.931437i	$-0.381444\pi$
$383$	14.2434	0.727804	0.363902	+	0.931437i	$0.381444\pi$
$384$	0	0
$385$	−2.71262	−0.138248
$386$	−1.70727	−0.0868977
$387$	0	0
$388$	8.57475	0.435317
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	−27.0031	−1.36561
$392$	1.56626	0.0791080
$393$	0	0
$394$	−26.4284	−1.33144
$395$	−0.472045	−0.0237512
$396$	0	0
$397$	−17.3288	−0.869708	−0.434854	−	0.900501i	$-0.643200\pi$
$397$	−17.3288	−0.869708	−0.434854	+	0.900501i	$0.643200\pi$
$398$	−11.5694	−0.579921
$399$	0	0
$400$	1.00000	0.0500000
$401$	−2.49781	−0.124735	−0.0623673	−	0.998053i	$-0.519865\pi$
$401$	−2.49781	−0.124735	−0.0623673	+	0.998053i	$0.519865\pi$
$402$	0	0
$403$	6.21419	0.309551
$404$	−10.9004	−0.542317
$405$	0	0
$406$	−24.6683	−1.22427
$407$	7.39008	0.366313
$408$	0	0
$409$	38.2713	1.89239	0.946197	−	0.323591i	$-0.104890\pi$
$409$	38.2713	1.89239	0.946197	+	0.323591i	$0.104890\pi$
$410$	−8.42839	−0.416248
$411$	0	0
$412$	−0.119979	−0.00591094
$413$	23.7657	1.16943
$414$	0	0
$415$	7.61305	0.373710
$416$	6.21419	0.304676
$417$	0	0
$418$	1.19316	0.0583595
$419$	−24.9767	−1.22019	−0.610097	−	0.792327i	$-0.708870\pi$
$419$	−24.9767	−1.22019	−0.610097	+	0.792327i	$0.708870\pi$
$420$	0	0
$421$	−20.7632	−1.01194	−0.505968	−	0.862552i	$-0.668865\pi$
$421$	−20.7632	−1.01194	−0.505968	+	0.862552i	$0.668865\pi$
$422$	9.80997	0.477542
$423$	0	0
$424$	5.14101	0.249670
$425$	7.09390	0.344105
$426$	0	0
$427$	−43.3203	−2.09642
$428$	3.04680	0.147272
$429$	0	0
$430$	11.0468	0.532724
$431$	27.9736	1.34744	0.673721	−	0.738986i	$-0.264695\pi$
$431$	27.9736	1.34744	0.673721	+	0.738986i	$0.264695\pi$
$432$	0	0
$433$	−7.32944	−0.352230	−0.176115	−	0.984370i	$-0.556353\pi$
$433$	−7.32944	−0.352230	−0.176115	+	0.984370i	$0.556353\pi$
$434$	2.92682	0.140492
$435$	0	0
$436$	5.90579	0.282836
$437$	4.90043	0.234419
$438$	0	0
$439$	−19.0031	−0.906970	−0.453485	−	0.891264i	$-0.649819\pi$
$439$	−19.0031	−0.906970	−0.453485	+	0.891264i	$0.649819\pi$
$440$	−0.926817	−0.0441843
$441$	0	0
$442$	44.0829	2.09681
$443$	−22.9425	−1.09003	−0.545015	−	0.838426i	$-0.683476\pi$
$443$	−22.9425	−1.09003	−0.545015	+	0.838426i	$0.683476\pi$
$444$	0	0
$445$	−11.4074	−0.540760
$446$	25.0119	1.18435
$447$	0	0
$448$	2.92682	0.138279
$449$	−36.0678	−1.70215	−0.851073	−	0.525047i	$-0.824048\pi$
$449$	−36.0678	−1.70215	−0.851073	+	0.525047i	$0.824048\pi$
$450$	0	0
$451$	7.81157	0.367833
$452$	4.32568	0.203463
$453$	0	0
$454$	−21.6215	−1.01475
$455$	18.1878	0.852658
$456$	0	0
$457$	−4.67968	−0.218906	−0.109453	−	0.993992i	$-0.534910\pi$
$457$	−4.67968	−0.218906	−0.109453	+	0.993992i	$0.534910\pi$
$458$	−28.3285	−1.32370
$459$	0	0
$460$	−3.80653	−0.177480
$461$	15.2261	0.709151	0.354575	−	0.935027i	$-0.384625\pi$
$461$	15.2261	0.709151	0.354575	+	0.935027i	$0.384625\pi$
$462$	0	0
$463$	10.4026	0.483451	0.241725	−	0.970345i	$-0.422287\pi$
$463$	10.4026	0.483451	0.241725	+	0.970345i	$0.422287\pi$
$464$	−8.42839	−0.391278
$465$	0	0
$466$	−7.47580	−0.346310
$467$	−12.6168	−0.583836	−0.291918	−	0.956443i	$-0.594294\pi$
$467$	−12.6168	−0.583836	−0.291918	+	0.956443i	$0.594294\pi$
$468$	0	0
$469$	15.8015	0.729645
$470$	−5.75942	−0.265662
$471$	0	0
$472$	8.11998	0.373752
$473$	−10.2384	−0.470760
$474$	0	0
$475$	−1.28738	−0.0590688
$476$	20.7626	0.951650
$477$	0	0
$478$	11.3203	0.517780
$479$	−5.64793	−0.258061	−0.129030	−	0.991641i	$-0.541186\pi$
$479$	−5.64793	−0.258061	−0.129030	+	0.991641i	$0.541186\pi$
$480$	0	0
$481$	−49.5496	−2.25927
$482$	4.18781	0.190749
$483$	0	0
$484$	−10.1410	−0.460955
$485$	8.57475	0.389359
$486$	0	0
$487$	−39.4993	−1.78989	−0.894943	−	0.446180i	$-0.852784\pi$
$487$	−39.4993	−1.78989	−0.894943	+	0.446180i	$0.852784\pi$
$488$	−14.8012	−0.670018
$489$	0	0
$490$	1.56626	0.0707564
$491$	18.3521	0.828217	0.414109	−	0.910227i	$-0.364093\pi$
$491$	18.3521	0.828217	0.414109	+	0.910227i	$0.364093\pi$
$492$	0	0
$493$	−59.7901	−2.69281
$494$	−8.00000	−0.359937
$495$	0	0
$496$	1.00000	0.0449013
$497$	21.9557	0.984849
$498$	0	0
$499$	−22.8533	−1.02306	−0.511528	−	0.859267i	$-0.670920\pi$
$499$	−22.8533	−1.02306	−0.511528	+	0.859267i	$0.670920\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	11.2525	0.502223
$503$	37.2852	1.66246	0.831231	−	0.555926i	$-0.187636\pi$
$503$	37.2852	1.66246	0.831231	+	0.555926i	$0.187636\pi$
$504$	0	0
$505$	−10.9004	−0.485063
$506$	3.52796	0.156837
$507$	0	0
$508$	14.6426	0.649659
$509$	−2.86748	−0.127099	−0.0635495	−	0.997979i	$-0.520242\pi$
$509$	−2.86748	−0.127099	−0.0635495	+	0.997979i	$0.520242\pi$
$510$	0	0
$511$	10.0499	0.444583
$512$	1.00000	0.0441942
$513$	0	0
$514$	8.14415	0.359223
$515$	−0.119979	−0.00528691
$516$	0	0
$517$	5.33793	0.234762
$518$	−23.3373	−1.02538
$519$	0	0
$520$	6.21419	0.272510
$521$	24.1357	1.05740	0.528701	−	0.848808i	$-0.322679\pi$
$521$	24.1357	1.05740	0.528701	+	0.848808i	$0.322679\pi$
$522$	0	0
$523$	20.5135	0.896992	0.448496	−	0.893785i	$-0.351960\pi$
$523$	20.5135	0.896992	0.448496	+	0.893785i	$0.351960\pi$
$524$	−13.6809	−0.597652
$525$	0	0
$526$	−12.0556	−0.525649
$527$	7.09390	0.309015
$528$	0	0
$529$	−8.51035	−0.370015
$530$	5.14101	0.223311
$531$	0	0
$532$	−3.76791	−0.163360
$533$	−52.3756	−2.26864
$534$	0	0
$535$	3.04680	0.131724
$536$	5.39886	0.233195
$537$	0	0
$538$	−5.42525	−0.233899
$539$	−1.45164	−0.0625264
$540$	0	0
$541$	−22.4698	−0.966053	−0.483027	−	0.875606i	$-0.660463\pi$
$541$	−22.4698	−0.966053	−0.483027	+	0.875606i	$0.660463\pi$
$542$	23.9978	1.03079
$543$	0	0
$544$	7.09390	0.304149
$545$	5.90579	0.252976
$546$	0	0
$547$	−6.60114	−0.282244	−0.141122	−	0.989992i	$-0.545071\pi$
$547$	−6.60114	−0.282244	−0.141122	+	0.989992i	$0.545071\pi$
$548$	4.22642	0.180544
$549$	0	0
$550$	−0.926817	−0.0395196
$551$	10.8505	0.462247
$552$	0	0
$553$	−1.38159	−0.0587512
$554$	9.92146	0.421523
$555$	0	0
$556$	7.24027	0.307056
$557$	−21.0468	−0.891781	−0.445891	−	0.895087i	$-0.647113\pi$
$557$	−21.0468	−0.891781	−0.445891	+	0.895087i	$0.647113\pi$
$558$	0	0
$559$	68.6469	2.90346
$560$	2.92682	0.123681
$561$	0	0
$562$	16.8568	0.711060
$563$	19.4145	0.818225	0.409113	−	0.912484i	$-0.365838\pi$
$563$	19.4145	0.818225	0.409113	+	0.912484i	$0.365838\pi$
$564$	0	0
$565$	4.32568	0.181983
$566$	23.9736	1.00769
$567$	0	0
$568$	7.50157	0.314759
$569$	6.48772	0.271979	0.135990	−	0.990710i	$-0.456579\pi$
$569$	6.48772	0.271979	0.135990	+	0.990710i	$0.456579\pi$
$570$	0	0
$571$	−10.0135	−0.419053	−0.209527	−	0.977803i	$-0.567192\pi$
$571$	−10.0135	−0.419053	−0.209527	+	0.977803i	$0.567192\pi$
$572$	−5.75942	−0.240813
$573$	0	0
$574$	−24.6683	−1.02964
$575$	−3.80653	−0.158743
$576$	0	0
$577$	15.5081	0.645612	0.322806	−	0.946465i	$-0.395374\pi$
$577$	15.5081	0.645612	0.322806	+	0.946465i	$0.395374\pi$
$578$	33.3235	1.38607
$579$	0	0
$580$	−8.42839	−0.349970
$581$	22.2820	0.924414
$582$	0	0
$583$	−4.76478	−0.197337
$584$	3.43374	0.142089
$585$	0	0
$586$	10.0414	0.414808
$587$	37.2852	1.53892	0.769462	−	0.638693i	$-0.220525\pi$
$587$	37.2852	1.53892	0.769462	+	0.638693i	$0.220525\pi$
$588$	0	0
$589$	−1.28738	−0.0530454
$590$	8.11998	0.334294
$591$	0	0
$592$	−7.97361	−0.327713
$593$	−36.5634	−1.50148	−0.750740	−	0.660598i	$-0.770303\pi$
$593$	−36.5634	−1.50148	−0.750740	+	0.660598i	$0.770303\pi$
$594$	0	0
$595$	20.7626	0.851182
$596$	−0.618411	−0.0253311
$597$	0	0
$598$	−23.6545	−0.967304
$599$	34.5047	1.40982	0.704912	−	0.709295i	$-0.250986\pi$
$599$	34.5047	1.40982	0.704912	+	0.709295i	$0.250986\pi$
$600$	0	0
$601$	−47.7066	−1.94599	−0.972997	−	0.230816i	$-0.925860\pi$
$601$	−47.7066	−1.94599	−0.972997	+	0.230816i	$0.925860\pi$
$602$	32.3320	1.31775
$603$	0	0
$604$	−18.1878	−0.740051
$605$	−10.1410	−0.412291
$606$	0	0
$607$	11.0097	0.446870	0.223435	−	0.974719i	$-0.428273\pi$
$607$	11.0097	0.446870	0.223435	+	0.974719i	$0.428273\pi$
$608$	−1.28738	−0.0522100
$609$	0	0
$610$	−14.8012	−0.599282
$611$	−35.7901	−1.44791
$612$	0	0
$613$	32.8718	1.32768	0.663840	−	0.747874i	$-0.268925\pi$
$613$	32.8718	1.32768	0.663840	+	0.747874i	$0.268925\pi$
$614$	−27.3882	−1.10530
$615$	0	0
$616$	−2.71262	−0.109295
$617$	−8.23147	−0.331386	−0.165693	−	0.986177i	$-0.552986\pi$
$617$	−8.23147	−0.331386	−0.165693	+	0.986177i	$0.552986\pi$
$618$	0	0
$619$	−31.9858	−1.28562	−0.642810	−	0.766026i	$-0.722231\pi$
$619$	−31.9858	−1.28562	−0.642810	+	0.766026i	$0.722231\pi$
$620$	1.00000	0.0401610
$621$	0	0
$622$	−16.1093	−0.645923
$623$	−33.3872	−1.33763
$624$	0	0
$625$	1.00000	0.0400000
$626$	20.4698	0.818139
$627$	0	0
$628$	6.32568	0.252422
$629$	−56.5640	−2.25536
$630$	0	0
$631$	10.8238	0.430890	0.215445	−	0.976516i	$-0.430880\pi$
$631$	10.8238	0.430890	0.215445	+	0.976516i	$0.430880\pi$
$632$	−0.472045	−0.0187769
$633$	0	0
$634$	−2.29273	−0.0910560
$635$	14.6426	0.581073
$636$	0	0
$637$	9.73303	0.385637
$638$	7.81157	0.309263
$639$	0	0
$640$	1.00000	0.0395285
$641$	−13.7745	−0.544059	−0.272030	−	0.962289i	$-0.587695\pi$
$641$	−13.7745	−0.544059	−0.272030	+	0.962289i	$0.587695\pi$
$642$	0	0
$643$	−49.0775	−1.93543	−0.967714	−	0.252050i	$-0.918895\pi$
$643$	−49.0775	−1.93543	−0.967714	+	0.252050i	$0.918895\pi$
$644$	−11.1410	−0.439017
$645$	0	0
$646$	−9.13252	−0.359314
$647$	9.03326	0.355134	0.177567	−	0.984109i	$-0.443177\pi$
$647$	9.03326	0.355134	0.177567	+	0.984109i	$0.443177\pi$
$648$	0	0
$649$	−7.52574	−0.295411
$650$	6.21419	0.243741
$651$	0	0
$652$	12.4548	0.487767
$653$	0.240579	0.00941460	0.00470730	−	0.999989i	$-0.498502\pi$
$653$	0.240579	0.00941460	0.00470730	+	0.999989i	$0.498502\pi$
$654$	0	0
$655$	−13.6809	−0.534556
$656$	−8.42839	−0.329073
$657$	0	0
$658$	−16.8568	−0.657145
$659$	44.9240	1.74999	0.874995	−	0.484132i	$-0.160865\pi$
$659$	44.9240	1.74999	0.874995	+	0.484132i	$0.160865\pi$
$660$	0	0
$661$	24.9504	0.970457	0.485228	−	0.874387i	$-0.338736\pi$
$661$	24.9504	0.970457	0.485228	+	0.874387i	$0.338736\pi$
$662$	−35.8043	−1.39157
$663$	0	0
$664$	7.61305	0.295444
$665$	−3.76791	−0.146113
$666$	0	0
$667$	32.0829	1.24225
$668$	−22.1828	−0.858277
$669$	0	0
$670$	5.39886	0.208576
$671$	13.7180	0.529577
$672$	0	0
$673$	−15.1847	−0.585326	−0.292663	−	0.956216i	$-0.594541\pi$
$673$	−15.1847	−0.585326	−0.292663	+	0.956216i	$0.594541\pi$
$674$	−12.0942	−0.465852
$675$	0	0
$676$	25.6162	0.985238
$677$	−45.4293	−1.74599	−0.872995	−	0.487729i	$-0.837825\pi$
$677$	−45.4293	−1.74599	−0.872995	+	0.487729i	$0.837825\pi$
$678$	0	0
$679$	25.0967	0.963124
$680$	7.09390	0.272039
$681$	0	0
$682$	−0.926817	−0.0354897
$683$	38.1686	1.46048	0.730240	−	0.683190i	$-0.239408\pi$
$683$	38.1686	1.46048	0.730240	+	0.683190i	$0.239408\pi$
$684$	0	0
$685$	4.22642	0.161483
$686$	−15.9036	−0.607201
$687$	0	0
$688$	11.0468	0.421155
$689$	31.9472	1.21709
$690$	0	0
$691$	41.8929	1.59368	0.796840	−	0.604191i	$-0.206504\pi$
$691$	41.8929	1.59368	0.796840	+	0.604191i	$0.206504\pi$
$692$	−5.18467	−0.197092
$693$	0	0
$694$	−1.51946	−0.0576780
$695$	7.24027	0.274639
$696$	0	0
$697$	−59.7901	−2.26471
$698$	11.4422	0.433095
$699$	0	0
$700$	2.92682	0.110623
$701$	−18.7961	−0.709920	−0.354960	−	0.934882i	$-0.615506\pi$
$701$	−18.7961	−0.709920	−0.354960	+	0.934882i	$0.615506\pi$
$702$	0	0
$703$	10.2650	0.387153
$704$	−0.926817	−0.0349307
$705$	0	0
$706$	−32.2679	−1.21442
$707$	−31.9036	−1.19986
$708$	0	0
$709$	−27.4372	−1.03043	−0.515213	−	0.857062i	$-0.672287\pi$
$709$	−27.4372	−1.03043	−0.515213	+	0.857062i	$0.672287\pi$
$710$	7.50157	0.281529
$711$	0	0
$712$	−11.4074	−0.427509
$713$	−3.80653	−0.142556
$714$	0	0
$715$	−5.75942	−0.215390
$716$	−11.2525	−0.420526
$717$	0	0
$718$	6.48772	0.242120
$719$	−31.5257	−1.17571	−0.587856	−	0.808966i	$-0.700028\pi$
$719$	−31.5257	−1.17571	−0.587856	+	0.808966i	$0.700028\pi$
$720$	0	0
$721$	−0.351157	−0.0130778
$722$	−17.3427	−0.645427
$723$	0	0
$724$	−2.62186	−0.0974406
$725$	−8.42839	−0.313022
$726$	0	0
$727$	−26.9438	−0.999290	−0.499645	−	0.866230i	$-0.666536\pi$
$727$	−26.9438	−0.999290	−0.499645	+	0.866230i	$0.666536\pi$
$728$	18.1878	0.674085
$729$	0	0
$730$	3.43374	0.127088
$731$	78.3649	2.89843
$732$	0	0
$733$	23.0031	0.849640	0.424820	−	0.905278i	$-0.360337\pi$
$733$	23.0031	0.849640	0.424820	+	0.905278i	$0.360337\pi$
$734$	−12.5835	−0.464467
$735$	0	0
$736$	−3.80653	−0.140310
$737$	−5.00376	−0.184316
$738$	0	0
$739$	−2.00345	−0.0736980	−0.0368490	−	0.999321i	$-0.511732\pi$
$739$	−2.00345	−0.0736980	−0.0368490	+	0.999321i	$0.511732\pi$
$740$	−7.97361	−0.293116
$741$	0	0
$742$	15.0468	0.552385
$743$	48.3706	1.77454	0.887272	−	0.461247i	$-0.152598\pi$
$743$	48.3706	1.77454	0.887272	+	0.461247i	$0.152598\pi$
$744$	0	0
$745$	−0.618411	−0.0226568
$746$	−5.80997	−0.212718
$747$	0	0
$748$	−6.57475	−0.240397
$749$	8.91742	0.325835
$750$	0	0
$751$	0.239958	0.00875619	0.00437809	−	0.999990i	$-0.498606\pi$
$751$	0.239958	0.00875619	0.00437809	+	0.999990i	$0.498606\pi$
$752$	−5.75942	−0.210024
$753$	0	0
$754$	−52.3756	−1.90741
$755$	−18.1878	−0.661922
$756$	0	0
$757$	−3.45101	−0.125429	−0.0627146	−	0.998032i	$-0.519976\pi$
$757$	−3.45101	−0.125429	−0.0627146	+	0.998032i	$0.519976\pi$
$758$	10.0085	0.363525
$759$	0	0
$760$	−1.28738	−0.0466980
$761$	28.3514	1.02774	0.513870	−	0.857868i	$-0.328212\pi$
$761$	28.3514	1.02774	0.513870	+	0.857868i	$0.328212\pi$
$762$	0	0
$763$	17.2852	0.625765
$764$	19.6809	0.712029
$765$	0	0
$766$	14.2434	0.514635
$767$	50.4591	1.82197
$768$	0	0
$769$	0.894157	0.0322441	0.0161221	−	0.999870i	$-0.494868\pi$
$769$	0.894157	0.0322441	0.0161221	+	0.999870i	$0.494868\pi$
$770$	−2.71262	−0.0977562
$771$	0	0
$772$	−1.70727	−0.0614459
$773$	45.2340	1.62695	0.813477	−	0.581598i	$-0.197572\pi$
$773$	45.2340	1.62695	0.813477	+	0.581598i	$0.197572\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	8.57475	0.307816
$777$	0	0
$778$	−12.0000	−0.430221
$779$	10.8505	0.388759
$780$	0	0
$781$	−6.95258	−0.248783
$782$	−27.0031	−0.965630
$783$	0	0
$784$	1.56626	0.0559378
$785$	6.32568	0.225773
$786$	0	0
$787$	−9.24594	−0.329582	−0.164791	−	0.986329i	$-0.552695\pi$
$787$	−9.24594	−0.329582	−0.164791	+	0.986329i	$0.552695\pi$
$788$	−26.4284	−0.941472
$789$	0	0
$790$	−0.472045	−0.0167946
$791$	12.6605	0.450155
$792$	0	0
$793$	−91.9773	−3.26621
$794$	−17.3288	−0.614977
$795$	0	0
$796$	−11.5694	−0.410066
$797$	−28.5050	−1.00970	−0.504849	−	0.863207i	$-0.668452\pi$
$797$	−28.5050	−1.00970	−0.504849	+	0.863207i	$0.668452\pi$
$798$	0	0
$799$	−40.8568	−1.44541
$800$	1.00000	0.0353553
$801$	0	0
$802$	−2.49781	−0.0882007
$803$	−3.18245	−0.112306
$804$	0	0
$805$	−11.1410	−0.392669
$806$	6.21419	0.218886
$807$	0	0
$808$	−10.9004	−0.383476
$809$	18.5989	0.653903	0.326952	−	0.945041i	$-0.393979\pi$
$809$	18.5989	0.653903	0.326952	+	0.945041i	$0.393979\pi$
$810$	0	0
$811$	26.9526	0.946433	0.473217	−	0.880946i	$-0.343093\pi$
$811$	26.9526	0.946433	0.473217	+	0.880946i	$0.343093\pi$
$812$	−24.6683	−0.865689
$813$	0	0
$814$	7.39008	0.259022
$815$	12.4548	0.436272
$816$	0	0
$817$	−14.2214	−0.497543
$818$	38.2713	1.33812
$819$	0	0
$820$	−8.42839	−0.294332
$821$	−22.2820	−0.777648	−0.388824	−	0.921312i	$-0.627119\pi$
$821$	−22.2820	−0.777648	−0.388824	+	0.921312i	$0.627119\pi$
$822$	0	0
$823$	−49.3002	−1.71850	−0.859249	−	0.511558i	$-0.829068\pi$
$823$	−49.3002	−1.71850	−0.859249	+	0.511558i	$0.829068\pi$
$824$	−0.119979	−0.00417967
$825$	0	0
$826$	23.7657	0.826914
$827$	15.0031	0.521710	0.260855	−	0.965378i	$-0.415996\pi$
$827$	15.0031	0.521710	0.260855	+	0.965378i	$0.415996\pi$
$828$	0	0
$829$	−33.0496	−1.14786	−0.573930	−	0.818904i	$-0.694582\pi$
$829$	−33.0496	−1.14786	−0.573930	+	0.818904i	$0.694582\pi$
$830$	7.61305	0.264253
$831$	0	0
$832$	6.21419	0.215438
$833$	11.1109	0.384969
$834$	0	0
$835$	−22.1828	−0.767666
$836$	1.19316	0.0412664
$837$	0	0
$838$	−24.9767	−0.862808
$839$	−20.2572	−0.699357	−0.349679	−	0.936870i	$-0.613709\pi$
$839$	−20.2572	−0.699357	−0.349679	+	0.936870i	$0.613709\pi$
$840$	0	0
$841$	42.0377	1.44958
$842$	−20.7632	−0.715546
$843$	0	0
$844$	9.80997	0.337673
$845$	25.6162	0.881224
$846$	0	0
$847$	−29.6809	−1.01985
$848$	5.14101	0.176543
$849$	0	0
$850$	7.09390	0.243319
$851$	30.3518	1.04045
$852$	0	0
$853$	47.8508	1.63838	0.819190	−	0.573522i	$-0.194423\pi$
$853$	47.8508	1.63838	0.819190	+	0.573522i	$0.194423\pi$
$854$	−43.3203	−1.48239
$855$	0	0
$856$	3.04680	0.104137
$857$	32.2990	1.10331	0.551656	−	0.834071i	$-0.313996\pi$
$857$	32.2990	1.10331	0.551656	+	0.834071i	$0.313996\pi$
$858$	0	0
$859$	−52.3514	−1.78621	−0.893103	−	0.449853i	$-0.851476\pi$
$859$	−52.3514	−1.78621	−0.893103	+	0.449853i	$0.851476\pi$
$860$	11.0468	0.376693
$861$	0	0
$862$	27.9736	0.952785
$863$	24.0993	0.820348	0.410174	−	0.912007i	$-0.365468\pi$
$863$	24.0993	0.820348	0.410174	+	0.912007i	$0.365468\pi$
$864$	0	0
$865$	−5.18467	−0.176284
$866$	−7.32944	−0.249064
$867$	0	0
$868$	2.92682	0.0993426
$869$	0.437499	0.0148412
$870$	0	0
$871$	33.5496	1.13678
$872$	5.90579	0.199995
$873$	0	0
$874$	4.90043	0.165760
$875$	2.92682	0.0989445
$876$	0	0
$877$	15.1074	0.510142	0.255071	−	0.966922i	$-0.417901\pi$
$877$	15.1074	0.510142	0.255071	+	0.966922i	$0.417901\pi$
$878$	−19.0031	−0.641325
$879$	0	0
$880$	−0.926817	−0.0312430
$881$	8.86617	0.298709	0.149354	−	0.988784i	$-0.452280\pi$
$881$	8.86617	0.298709	0.149354	+	0.988784i	$0.452280\pi$
$882$	0	0
$883$	12.8760	0.433311	0.216656	−	0.976248i	$-0.430485\pi$
$883$	12.8760	0.433311	0.216656	+	0.976248i	$0.430485\pi$
$884$	44.0829	1.48267
$885$	0	0
$886$	−22.9425	−0.770768
$887$	32.1457	1.07935	0.539674	−	0.841874i	$-0.318547\pi$
$887$	32.1457	1.07935	0.539674	+	0.841874i	$0.318547\pi$
$888$	0	0
$889$	42.8561	1.43735
$890$	−11.4074	−0.382375
$891$	0	0
$892$	25.0119	0.837461
$893$	7.41454	0.248118
$894$	0	0
$895$	−11.2525	−0.376129
$896$	2.92682	0.0977781
$897$	0	0
$898$	−36.0678	−1.20360
$899$	−8.42839	−0.281102
$900$	0	0
$901$	36.4698	1.21499
$902$	7.81157	0.260097
$903$	0	0
$904$	4.32568	0.143870
$905$	−2.62186	−0.0871535
$906$	0	0
$907$	−21.9629	−0.729266	−0.364633	−	0.931151i	$-0.618806\pi$
$907$	−21.9629	−0.729266	−0.364633	+	0.931151i	$0.618806\pi$
$908$	−21.6215	−0.717536
$909$	0	0
$910$	18.1878	0.602920
$911$	−32.1702	−1.06585	−0.532923	−	0.846164i	$-0.678907\pi$
$911$	−32.1702	−1.06585	−0.532923	+	0.846164i	$0.678907\pi$
$912$	0	0
$913$	−7.05591	−0.233517
$914$	−4.67968	−0.154790
$915$	0	0
$916$	−28.3285	−0.936000
$917$	−40.0414	−1.32228
$918$	0	0
$919$	5.92339	0.195395	0.0976973	−	0.995216i	$-0.468852\pi$
$919$	5.92339	0.195395	0.0976973	+	0.995216i	$0.468852\pi$
$920$	−3.80653	−0.125497
$921$	0	0
$922$	15.2261	0.501445
$923$	46.6162	1.53439
$924$	0	0
$925$	−7.97361	−0.262171
$926$	10.4026	0.341851
$927$	0	0
$928$	−8.42839	−0.276675
$929$	−13.8533	−0.454514	−0.227257	−	0.973835i	$-0.572976\pi$
$929$	−13.8533	−0.454514	−0.227257	+	0.973835i	$0.572976\pi$
$930$	0	0
$931$	−2.01636	−0.0660836
$932$	−7.47580	−0.244878
$933$	0	0
$934$	−12.6168	−0.412835
$935$	−6.57475	−0.215017
$936$	0	0
$937$	27.1495	0.886936	0.443468	−	0.896290i	$-0.353748\pi$
$937$	27.1495	0.886936	0.443468	+	0.896290i	$0.353748\pi$
$938$	15.8015	0.515937
$939$	0	0
$940$	−5.75942	−0.187852
$941$	−2.86748	−0.0934773	−0.0467386	−	0.998907i	$-0.514883\pi$
$941$	−2.86748	−0.0934773	−0.0467386	+	0.998907i	$0.514883\pi$
$942$	0	0
$943$	32.0829	1.04476
$944$	8.11998	0.264283
$945$	0	0
$946$	−10.2384	−0.332878
$947$	31.7487	1.03169	0.515847	−	0.856681i	$-0.327477\pi$
$947$	31.7487	1.03169	0.515847	+	0.856681i	$0.327477\pi$
$948$	0	0
$949$	21.3379	0.692659
$950$	−1.28738	−0.0417680
$951$	0	0
$952$	20.7626	0.672918
$953$	−35.2465	−1.14175	−0.570874	−	0.821038i	$-0.693395\pi$
$953$	−35.2465	−1.14175	−0.570874	+	0.821038i	$0.693395\pi$
$954$	0	0
$955$	19.6809	0.636858
$956$	11.3203	0.366125
$957$	0	0
$958$	−5.64793	−0.182476
$959$	12.3700	0.399447
$960$	0	0
$961$	1.00000	0.0322581
$962$	−49.5496	−1.59754
$963$	0	0
$964$	4.18781	0.134880
$965$	−1.70727	−0.0549589
$966$	0	0
$967$	−44.9761	−1.44633	−0.723167	−	0.690673i	$-0.757314\pi$
$967$	−44.9761	−1.44633	−0.723167	+	0.690673i	$0.757314\pi$
$968$	−10.1410	−0.325944
$969$	0	0
$970$	8.57475	0.275319
$971$	11.4686	0.368046	0.184023	−	0.982922i	$-0.441088\pi$
$971$	11.4686	0.368046	0.184023	+	0.982922i	$0.441088\pi$
$972$	0	0
$973$	21.1909	0.679350
$974$	−39.4993	−1.26564
$975$	0	0
$976$	−14.8012	−0.473774
$977$	11.8951	0.380557	0.190279	−	0.981730i	$-0.439061\pi$
$977$	11.8951	0.380557	0.190279	+	0.981730i	$0.439061\pi$
$978$	0	0
$979$	10.5725	0.337900
$980$	1.56626	0.0500323
$981$	0	0
$982$	18.3521	0.585638
$983$	34.6962	1.10664	0.553319	−	0.832969i	$-0.313361\pi$
$983$	34.6962	1.10664	0.553319	+	0.832969i	$0.313361\pi$
$984$	0	0
$985$	−26.4284	−0.842078
$986$	−59.7901	−1.90411
$987$	0	0
$988$	−8.00000	−0.254514
$989$	−42.0499	−1.33711
$990$	0	0
$991$	−32.5549	−1.03414	−0.517071	−	0.855943i	$-0.672978\pi$
$991$	−32.5549	−1.03414	−0.517071	+	0.855943i	$0.672978\pi$
$992$	1.00000	0.0317500
$993$	0	0
$994$	21.9557	0.696393
$995$	−11.5694	−0.366774
$996$	0	0
$997$	−5.37248	−0.170148	−0.0850740	−	0.996375i	$-0.527113\pi$
$997$	−5.37248	−0.170148	−0.0850740	+	0.996375i	$0.527113\pi$
$998$	−22.8533	−0.723409
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2790.2.a.bk.1.3	yes	4
3.2	odd	2		2790.2.a.bj.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2790.2.a.bj.1.3	✓	4	3.2	odd	2
2790.2.a.bk.1.3	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2790.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.92682 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.92682 q^{7} +1.00000 q^{8} +1.00000 q^{10} -0.926817 q^{11} +6.21419 q^{13} +2.92682 q^{14} +1.00000 q^{16} +7.09390 q^{17} -1.28738 q^{19} +1.00000 q^{20} -0.926817 q^{22} -3.80653 q^{23} +1.00000 q^{25} +6.21419 q^{26} +2.92682 q^{28} -8.42839 q^{29} +1.00000 q^{31} +1.00000 q^{32} +7.09390 q^{34} +2.92682 q^{35} -7.97361 q^{37} -1.28738 q^{38} +1.00000 q^{40} -8.42839 q^{41} +11.0468 q^{43} -0.926817 q^{44} -3.80653 q^{46} -5.75942 q^{47} +1.56626 q^{49} +1.00000 q^{50} +6.21419 q^{52} +5.14101 q^{53} -0.926817 q^{55} +2.92682 q^{56} -8.42839 q^{58} +8.11998 q^{59} -14.8012 q^{61} +1.00000 q^{62} +1.00000 q^{64} +6.21419 q^{65} +5.39886 q^{67} +7.09390 q^{68} +2.92682 q^{70} +7.50157 q^{71} +3.43374 q^{73} -7.97361 q^{74} -1.28738 q^{76} -2.71262 q^{77} -0.472045 q^{79} +1.00000 q^{80} -8.42839 q^{82} +7.61305 q^{83} +7.09390 q^{85} +11.0468 q^{86} -0.926817 q^{88} -11.4074 q^{89} +18.1878 q^{91} -3.80653 q^{92} -5.75942 q^{94} -1.28738 q^{95} +8.57475 q^{97} +1.56626 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 5 * q^7 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8} + 4 q^{10} + 3 q^{11} + 6 q^{13} + 5 q^{14} + 4 q^{16} + 7 q^{19} + 4 q^{20} + 3 q^{22} + q^{23} + 4 q^{25} + 6 q^{26} + 5 q^{28} + 4 q^{29} + 4 q^{31} + 4 q^{32} + 5 q^{35} + 6 q^{37} + 7 q^{38} + 4 q^{40} + 4 q^{41} + 13 q^{43} + 3 q^{44} + q^{46} - 4 q^{47} + 5 q^{49} + 4 q^{50} + 6 q^{52} - 5 q^{53} + 3 q^{55} + 5 q^{56} + 4 q^{58} + 8 q^{59} - 4 q^{61} + 4 q^{62} + 4 q^{64} + 6 q^{65} + 8 q^{67} + 5 q^{70} - q^{71} + 15 q^{73} + 6 q^{74} + 7 q^{76} - 23 q^{77} + 5 q^{79} + 4 q^{80} + 4 q^{82} - 2 q^{83} + 13 q^{86} + 3 q^{88} - 9 q^{89} + 16 q^{91} + q^{92} - 4 q^{94} + 7 q^{95} + 10 q^{97} + 5 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 5 * q^7 + 4 * q^8 + 4 * q^10 + 3 * q^11 + 6 * q^13 + 5 * q^14 + 4 * q^16 + 7 * q^19 + 4 * q^20 + 3 * q^22 + q^23 + 4 * q^25 + 6 * q^26 + 5 * q^28 + 4 * q^29 + 4 * q^31 + 4 * q^32 + 5 * q^35 + 6 * q^37 + 7 * q^38 + 4 * q^40 + 4 * q^41 + 13 * q^43 + 3 * q^44 + q^46 - 4 * q^47 + 5 * q^49 + 4 * q^50 + 6 * q^52 - 5 * q^53 + 3 * q^55 + 5 * q^56 + 4 * q^58 + 8 * q^59 - 4 * q^61 + 4 * q^62 + 4 * q^64 + 6 * q^65 + 8 * q^67 + 5 * q^70 - q^71 + 15 * q^73 + 6 * q^74 + 7 * q^76 - 23 * q^77 + 5 * q^79 + 4 * q^80 + 4 * q^82 - 2 * q^83 + 13 * q^86 + 3 * q^88 - 9 * q^89 + 16 * q^91 + q^92 - 4 * q^94 + 7 * q^95 + 10 * q^97 + 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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