LMFDB - Embedded newform 6825.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6825,2,Mod(1,6825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6825.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6825.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:	$54.4978993795$
Analytic rank:	$1$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.10710$ of defining polynomial
Character	$\chi$	$=$	6825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.43986 q^{2} -1.00000 q^{3} +0.0731828 q^{4} +1.43986 q^{6} -1.00000 q^{7} +2.77434 q^{8} +1.00000 q^{9} +4.21419 q^{11} -0.0731828 q^{12} -1.00000 q^{13} +1.43986 q^{14} -4.14101 q^{16} +2.87971 q^{17} -1.43986 q^{18} -1.28738 q^{19} +1.00000 q^{21} -6.06783 q^{22} +8.02072 q^{23} -2.77434 q^{24} +1.43986 q^{26} -1.00000 q^{27} -0.0731828 q^{28} +3.28738 q^{29} -7.04680 q^{31} +0.413779 q^{32} -4.21419 q^{33} -4.14637 q^{34} +0.0731828 q^{36} -8.57475 q^{37} +1.85363 q^{38} +1.00000 q^{39} -12.0678 q^{41} -1.43986 q^{42} -7.14101 q^{43} +0.308407 q^{44} -11.5487 q^{46} -1.95289 q^{47} +4.14101 q^{48} +1.00000 q^{49} -2.87971 q^{51} -0.0731828 q^{52} +5.14101 q^{53} +1.43986 q^{54} -2.77434 q^{56} +1.28738 q^{57} -4.73334 q^{58} -7.33448 q^{59} +7.75942 q^{61} +10.1464 q^{62} -1.00000 q^{63} +7.68624 q^{64} +6.06783 q^{66} -12.0414 q^{67} +0.210745 q^{68} -8.02072 q^{69} +10.7889 q^{71} +2.77434 q^{72} +8.32568 q^{73} +12.3464 q^{74} -0.0942138 q^{76} -4.21419 q^{77} -1.43986 q^{78} +4.47204 q^{79} +1.00000 q^{81} +17.3759 q^{82} +3.80653 q^{83} +0.0731828 q^{84} +10.2820 q^{86} -3.28738 q^{87} +11.6916 q^{88} -5.64793 q^{89} +1.00000 q^{91} +0.586979 q^{92} +7.04680 q^{93} +2.81188 q^{94} -0.413779 q^{96} -6.90043 q^{97} -1.43986 q^{98} +4.21419 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9} - 2 q^{11} - 7 q^{12} - 4 q^{13} + q^{14} + 9 q^{16} + 2 q^{17} - q^{18} + 7 q^{19} + 4 q^{21} + 8 q^{22} - 3 q^{23} + 3 q^{24}+ \cdots - 2 q^{99}+O(q^{100}) $ 4 * q - q^2 - 4 * q^3 + 7 * q^4 + q^6 - 4 * q^7 - 3 * q^8 + 4 * q^9 - 2 * q^11 - 7 * q^12 - 4 * q^13 + q^14 + 9 * q^16 + 2 * q^17 - q^18 + 7 * q^19 + 4 * q^21 + 8 * q^22 - 3 * q^23 + 3 * q^24 + q^26 - 4 * q^27 - 7 * q^28 + q^29 + 3 * q^31 - 7 * q^32 + 2 * q^33 - 30 * q^34 + 7 * q^36 - 10 * q^37 - 6 * q^38 + 4 * q^39 - 16 * q^41 - q^42 - 3 * q^43 - 12 * q^44 - 18 * q^46 - 5 * q^47 - 9 * q^48 + 4 * q^49 - 2 * q^51 - 7 * q^52 - 5 * q^53 + q^54 + 3 * q^56 - 7 * q^57 + 4 * q^58 - 20 * q^59 + 12 * q^61 + 54 * q^62 - 4 * q^63 + 5 * q^64 - 8 * q^66 + 22 * q^67 + 10 * q^68 + 3 * q^69 - 3 * q^72 + 13 * q^73 - 6 * q^74 - 6 * q^76 + 2 * q^77 - q^78 + 11 * q^79 + 4 * q^81 - 10 * q^82 - q^83 + 7 * q^84 - 10 * q^86 - q^87 + 60 * q^88 - 5 * q^89 + 4 * q^91 - 34 * q^92 - 3 * q^93 + 34 * q^94 + 7 * q^96 + 17 * q^97 - q^98 - 2 * q^99

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 $

$2$	−1.43986	−1.01813	−0.509066	−	0.860728i	$-0.670009\pi$
$2$	−1.43986	−1.01813	−0.509066	+	0.860728i	$0.670009\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.0731828	0.0365914
$5$	0	0
$5$	0	0
$6$	1.43986	0.587818
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.77434	0.980876
$9$	1.00000	0.333333
$10$	0	0
$11$	4.21419	1.27063	0.635313	−	0.772254i	$-0.280871\pi$
$11$	4.21419	1.27063	0.635313	+	0.772254i	$0.280871\pi$
$12$	−0.0731828	−0.0211261
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.43986	0.384817
$15$	0	0
$16$	−4.14101	−1.03525
$17$	2.87971	0.698432	0.349216	−	0.937042i	$-0.386448\pi$
$17$	2.87971	0.698432	0.349216	+	0.937042i	$0.386448\pi$
$18$	−1.43986	−0.339377
$19$	−1.28738	−0.295344	−0.147672	−	0.989036i	$-0.547178\pi$
$19$	−1.28738	−0.295344	−0.147672	+	0.989036i	$0.547178\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	−6.06783	−1.29367
$23$	8.02072	1.67244	0.836218	−	0.548397i	$-0.184762\pi$
$23$	8.02072	1.67244	0.836218	+	0.548397i	$0.184762\pi$
$24$	−2.77434	−0.566309
$25$	0	0
$26$	1.43986	0.282379
$27$	−1.00000	−0.192450
$28$	−0.0731828	−0.0138303
$29$	3.28738	0.610450	0.305225	−	0.952280i	$-0.401268\pi$
$29$	3.28738	0.610450	0.305225	+	0.952280i	$0.401268\pi$
$30$	0	0
$31$	−7.04680	−1.26564	−0.632821	−	0.774298i	$-0.718103\pi$
$31$	−7.04680	−1.26564	−0.632821	+	0.774298i	$0.718103\pi$
$32$	0.413779	0.0731465
$33$	−4.21419	−0.733597
$34$	−4.14637	−0.711096
$35$	0	0
$36$	0.0731828	0.0121971
$37$	−8.57475	−1.40968	−0.704840	−	0.709366i	$-0.748981\pi$
$37$	−8.57475	−1.40968	−0.704840	+	0.709366i	$0.748981\pi$
$38$	1.85363	0.300699
$39$	1.00000	0.160128
$40$	0	0
$41$	−12.0678	−1.88468	−0.942339	−	0.334660i	$-0.891379\pi$
$41$	−12.0678	−1.88468	−0.942339	+	0.334660i	$0.891379\pi$
$42$	−1.43986	−0.222174
$43$	−7.14101	−1.08899	−0.544497	−	0.838763i	$-0.683279\pi$
$43$	−7.14101	−1.08899	−0.544497	+	0.838763i	$0.683279\pi$
$44$	0.308407	0.0464940
$45$	0	0
$46$	−11.5487	−1.70276
$47$	−1.95289	−0.284859	−0.142429	−	0.989805i	$-0.545491\pi$
$47$	−1.95289	−0.284859	−0.142429	+	0.989805i	$0.545491\pi$
$48$	4.14101	0.597703
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.87971	−0.403240
$52$	−0.0731828	−0.0101486
$53$	5.14101	0.706172	0.353086	−	0.935591i	$-0.385132\pi$
$53$	5.14101	0.706172	0.353086	+	0.935591i	$0.385132\pi$
$54$	1.43986	0.195939
$55$	0	0
$56$	−2.77434	−0.370736
$57$	1.28738	0.170517
$58$	−4.73334	−0.621519
$59$	−7.33448	−0.954868	−0.477434	−	0.878668i	$-0.658433\pi$
$59$	−7.33448	−0.954868	−0.477434	+	0.878668i	$0.658433\pi$
$60$	0	0
$61$	7.75942	0.993492	0.496746	−	0.867896i	$-0.334528\pi$
$61$	7.75942	0.993492	0.496746	+	0.867896i	$0.334528\pi$
$62$	10.1464	1.28859
$63$	−1.00000	−0.125988
$64$	7.68624	0.960780
$65$	0	0
$66$	6.06783	0.746898
$67$	−12.0414	−1.47110	−0.735548	−	0.677473i	$-0.763075\pi$
$67$	−12.0414	−1.47110	−0.735548	+	0.677473i	$0.763075\pi$
$68$	0.210745	0.0255566
$69$	−8.02072	−0.965581
$70$	0	0
$71$	10.7889	1.28041	0.640206	−	0.768203i	$-0.278849\pi$
$71$	10.7889	1.28041	0.640206	+	0.768203i	$0.278849\pi$
$72$	2.77434	0.326959
$73$	8.32568	0.974447	0.487224	−	0.873277i	$-0.338010\pi$
$73$	8.32568	0.974447	0.487224	+	0.873277i	$0.338010\pi$
$74$	12.3464	1.43524
$75$	0	0
$76$	−0.0942138	−0.0108071
$77$	−4.21419	−0.480252
$78$	−1.43986	−0.163031
$79$	4.47204	0.503144	0.251572	−	0.967839i	$-0.419052\pi$
$79$	4.47204	0.503144	0.251572	+	0.967839i	$0.419052\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	17.3759	1.91885
$83$	3.80653	0.417821	0.208910	−	0.977935i	$-0.433008\pi$
$83$	3.80653	0.417821	0.208910	+	0.977935i	$0.433008\pi$
$84$	0.0731828	0.00798490
$85$	0	0
$86$	10.2820	1.10874
$87$	−3.28738	−0.352444
$88$	11.6916	1.24633
$89$	−5.64793	−0.598680	−0.299340	−	0.954147i	$-0.596766\pi$
$89$	−5.64793	−0.598680	−0.299340	+	0.954147i	$0.596766\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0.586979	0.0611968
$93$	7.04680	0.730719
$94$	2.81188	0.290024
$95$	0	0
$96$	−0.413779	−0.0422312
$97$	−6.90043	−0.700633	−0.350316	−	0.936631i	$-0.613926\pi$
$97$	−6.90043	−0.700633	−0.350316	+	0.936631i	$0.613926\pi$
$98$	−1.43986	−0.145447
$99$	4.21419	0.423542
$100$	0	0
$101$	−10.9739	−1.09195	−0.545973	−	0.837803i	$-0.683840\pi$
$101$	−10.9739	−1.09195	−0.545973	+	0.837803i	$0.683840\pi$
$102$	4.14637	0.410551
$103$	−16.4284	−1.61874	−0.809368	−	0.587301i	$-0.800190\pi$
$103$	−16.4284	−1.61874	−0.809368	+	0.587301i	$0.800190\pi$
$104$	−2.77434	−0.272046
$105$	0	0
$106$	−7.40231	−0.718976
$107$	3.45446	0.333955	0.166978	−	0.985961i	$-0.446599\pi$
$107$	3.45446	0.333955	0.166978	+	0.985961i	$0.446599\pi$
$108$	−0.0731828	−0.00704202
$109$	4.66896	0.447206	0.223603	−	0.974680i	$-0.428218\pi$
$109$	4.66896	0.447206	0.223603	+	0.974680i	$0.428218\pi$
$110$	0	0
$111$	8.57475	0.813879
$112$	4.14101	0.391289
$113$	3.28738	0.309250	0.154625	−	0.987973i	$-0.450583\pi$
$113$	3.28738	0.309250	0.154625	+	0.987973i	$0.450583\pi$
$114$	−1.85363	−0.173609
$115$	0	0
$116$	0.240579	0.0223372
$117$	−1.00000	−0.0924500
$118$	10.5606	0.972181
$119$	−2.87971	−0.263983
$120$	0	0
$121$	6.75942	0.614493
$122$	−11.1724	−1.01151
$123$	12.0678	1.08812
$124$	−0.515704	−0.0463116
$125$	0	0
$126$	1.43986	0.128272
$127$	0.428386	0.0380131	0.0190065	−	0.999819i	$-0.493950\pi$
$127$	0.428386	0.0380131	0.0190065	+	0.999819i	$0.493950\pi$
$128$	−11.8946	−1.05135
$129$	7.14101	0.628731
$130$	0	0
$131$	14.1878	1.23959	0.619797	−	0.784762i	$-0.287215\pi$
$131$	14.1878	1.23959	0.619797	+	0.784762i	$0.287215\pi$
$132$	−0.308407	−0.0268433
$133$	1.28738	0.111630
$134$	17.3379	1.49777
$135$	0	0
$136$	7.98929	0.685076
$137$	−14.7070	−1.25650	−0.628250	−	0.778011i	$-0.716229\pi$
$137$	−14.7070	−1.25650	−0.628250	+	0.778011i	$0.716229\pi$
$138$	11.5487	0.983089
$139$	9.85363	0.835774	0.417887	−	0.908499i	$-0.362771\pi$
$139$	9.85363	0.835774	0.417887	+	0.908499i	$0.362771\pi$
$140$	0	0
$141$	1.95289	0.164463
$142$	−15.5345	−1.30363
$143$	−4.21419	−0.352409
$144$	−4.14101	−0.345084
$145$	0	0
$146$	−11.9878	−0.992115
$147$	−1.00000	−0.0824786
$148$	−0.627525	−0.0515822
$149$	−3.38663	−0.277444	−0.138722	−	0.990331i	$-0.544299\pi$
$149$	−3.38663	−0.277444	−0.138722	+	0.990331i	$0.544299\pi$
$150$	0	0
$151$	−21.8951	−1.78180	−0.890898	−	0.454204i	$-0.849924\pi$
$151$	−21.8951	−1.78180	−0.890898	+	0.454204i	$0.849924\pi$
$152$	−3.57161	−0.289696
$153$	2.87971	0.232811
$154$	6.06783	0.488959
$155$	0	0
$156$	0.0731828	0.00585932
$157$	0.867482	0.0692326	0.0346163	−	0.999401i	$-0.488979\pi$
$157$	0.867482	0.0692326	0.0346163	+	0.999401i	$0.488979\pi$
$158$	−6.43910	−0.512267
$159$	−5.14101	−0.407709
$160$	0	0
$161$	−8.02072	−0.632121
$162$	−1.43986	−0.113126
$163$	−4.42839	−0.346858	−0.173429	−	0.984846i	$-0.555485\pi$
$163$	−4.42839	−0.346858	−0.173429	+	0.984846i	$0.555485\pi$
$164$	−0.883158	−0.0689630
$165$	0	0
$166$	−5.48085	−0.425396
$167$	−20.5276	−1.58848	−0.794238	−	0.607606i	$-0.792130\pi$
$167$	−20.5276	−1.58848	−0.794238	+	0.607606i	$0.792130\pi$
$168$	2.77434	0.214045
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.28738	−0.0984481
$172$	−0.522599	−0.0398478
$173$	−15.0154	−1.14160	−0.570799	−	0.821090i	$-0.693366\pi$
$173$	−15.0154	−1.14160	−0.570799	+	0.821090i	$0.693366\pi$
$174$	4.73334	0.358834
$175$	0	0
$176$	−17.4510	−1.31542
$177$	7.33448	0.551293
$178$	8.13221	0.609535
$179$	−5.35176	−0.400009	−0.200004	−	0.979795i	$-0.564096\pi$
$179$	−5.35176	−0.400009	−0.200004	+	0.979795i	$0.564096\pi$
$180$	0	0
$181$	11.4667	0.852312	0.426156	−	0.904650i	$-0.359867\pi$
$181$	11.4667	0.852312	0.426156	+	0.904650i	$0.359867\pi$
$182$	−1.43986	−0.106729
$183$	−7.75942	−0.573593
$184$	22.2522	1.64045
$185$	0	0
$186$	−10.1464	−0.743968
$187$	12.1357	0.887447
$188$	−0.142918	−0.0104234
$189$	1.00000	0.0727393
$190$	0	0
$191$	20.1756	1.45985	0.729927	−	0.683525i	$-0.239554\pi$
$191$	20.1756	1.45985	0.729927	+	0.683525i	$0.239554\pi$
$192$	−7.68624	−0.554706
$193$	26.3756	1.89856	0.949279	−	0.314435i	$-0.101815\pi$
$193$	26.3756	1.89856	0.949279	+	0.314435i	$0.101815\pi$
$194$	9.93562	0.713336
$195$	0	0
$196$	0.0731828	0.00522734
$197$	20.1322	1.43436	0.717180	−	0.696888i	$-0.245432\pi$
$197$	20.1322	1.43436	0.717180	+	0.696888i	$0.245432\pi$
$198$	−6.06783	−0.431222
$199$	1.75942	0.124722	0.0623610	−	0.998054i	$-0.480137\pi$
$199$	1.75942	0.124722	0.0623610	+	0.998054i	$0.480137\pi$
$200$	0	0
$201$	12.0414	0.849338
$202$	15.8009	1.11174
$203$	−3.28738	−0.230729
$204$	−0.210745	−0.0147551
$205$	0	0
$206$	23.6545	1.64809
$207$	8.02072	0.557479
$208$	4.14101	0.287127
$209$	−5.42525	−0.375272
$210$	0	0
$211$	15.5694	1.07184	0.535921	−	0.844268i	$-0.319965\pi$
$211$	15.5694	1.07184	0.535921	+	0.844268i	$0.319965\pi$
$212$	0.376234	0.0258398
$213$	−10.7889	−0.739246
$214$	−4.97392	−0.340010
$215$	0	0
$216$	−2.77434	−0.188770
$217$	7.04680	0.478368
$218$	−6.72263	−0.455314
$219$	−8.32568	−0.562597
$220$	0	0
$221$	−2.87971	−0.193710
$222$	−12.3464	−0.828636
$223$	11.5694	0.774744	0.387372	−	0.921923i	$-0.373383\pi$
$223$	11.5694	0.774744	0.387372	+	0.921923i	$0.373383\pi$
$224$	−0.413779	−0.0276468
$225$	0	0
$226$	−4.73334	−0.314857
$227$	11.0418	0.732867	0.366433	−	0.930444i	$-0.380579\pi$
$227$	11.0418	0.732867	0.366433	+	0.930444i	$0.380579\pi$
$228$	0.0942138	0.00623946
$229$	−13.7073	−0.905802	−0.452901	−	0.891561i	$-0.649611\pi$
$229$	−13.7073	−0.905802	−0.452901	+	0.891561i	$0.649611\pi$
$230$	0	0
$231$	4.21419	0.277274
$232$	9.12029	0.598776
$233$	−1.23522	−0.0809222	−0.0404611	−	0.999181i	$-0.512883\pi$
$233$	−1.23522	−0.0809222	−0.0404611	+	0.999181i	$0.512883\pi$
$234$	1.43986	0.0941263
$235$	0	0
$236$	−0.536758	−0.0349400
$237$	−4.47204	−0.290491
$238$	4.14637	0.268769
$239$	19.1061	1.23587	0.617936	−	0.786228i	$-0.287969\pi$
$239$	19.1061	1.23587	0.617936	+	0.786228i	$0.287969\pi$
$240$	0	0
$241$	−12.0329	−0.775110	−0.387555	−	0.921847i	$-0.626680\pi$
$241$	−12.0329	−0.775110	−0.387555	+	0.921847i	$0.626680\pi$
$242$	−9.73259	−0.625634
$243$	−1.00000	−0.0641500
$244$	0.567856	0.0363533
$245$	0	0
$246$	−17.3759	−1.10785
$247$	1.28738	0.0819137
$248$	−19.5502	−1.24144
$249$	−3.80653	−0.241229
$250$	0	0
$251$	−15.0031	−0.946990	−0.473495	−	0.880797i	$-0.657008\pi$
$251$	−15.0031	−0.946990	−0.473495	+	0.880797i	$0.657008\pi$
$252$	−0.0731828	−0.00461008
$253$	33.8009	2.12504
$254$	−0.616813	−0.0387023
$255$	0	0
$256$	1.75406	0.109629
$257$	13.0261	0.812544	0.406272	−	0.913752i	$-0.366828\pi$
$257$	13.0261	0.812544	0.406272	+	0.913752i	$0.366828\pi$
$258$	−10.2820	−0.640131
$259$	8.57475	0.532809
$260$	0	0
$261$	3.28738	0.203483
$262$	−20.4284	−1.26207
$263$	2.59547	0.160044	0.0800218	−	0.996793i	$-0.474501\pi$
$263$	2.59547	0.160044	0.0800218	+	0.996793i	$0.474501\pi$
$264$	−11.6916	−0.719568
$265$	0	0
$266$	−1.85363	−0.113654
$267$	5.64793	0.345648
$268$	−0.881227	−0.0538295
$269$	−23.3081	−1.42112	−0.710560	−	0.703637i	$-0.751558\pi$
$269$	−23.3081	−1.42112	−0.710560	+	0.703637i	$0.751558\pi$
$270$	0	0
$271$	−6.05215	−0.367642	−0.183821	−	0.982960i	$-0.558847\pi$
$271$	−6.05215	−0.367642	−0.183821	+	0.982960i	$0.558847\pi$
$272$	−11.9249	−0.723054
$273$	−1.00000	−0.0605228
$274$	21.1759	1.27928
$275$	0	0
$276$	−0.586979	−0.0353320
$277$	−15.1932	−0.912869	−0.456434	−	0.889757i	$-0.650874\pi$
$277$	−15.1932	−0.912869	−0.456434	+	0.889757i	$0.650874\pi$
$278$	−14.1878	−0.850928
$279$	−7.04680	−0.421881
$280$	0	0
$281$	−7.57506	−0.451890	−0.225945	−	0.974140i	$-0.572547\pi$
$281$	−7.57506	−0.451890	−0.225945	+	0.974140i	$0.572547\pi$
$282$	−2.81188	−0.167445
$283$	−19.0974	−1.13522	−0.567610	−	0.823298i	$-0.692132\pi$
$283$	−19.0974	−1.13522	−0.567610	+	0.823298i	$0.692132\pi$
$284$	0.789565	0.0468521
$285$	0	0
$286$	6.06783	0.358798
$287$	12.0678	0.712341
$288$	0.413779	0.0243822
$289$	−8.70727	−0.512192
$290$	0	0
$291$	6.90043	0.404510
$292$	0.609297	0.0356564
$293$	3.79430	0.221665	0.110833	−	0.993839i	$-0.464648\pi$
$293$	3.79430	0.221665	0.110833	+	0.993839i	$0.464648\pi$
$294$	1.43986	0.0839741
$295$	0	0
$296$	−23.7893	−1.38272
$297$	−4.21419	−0.244532
$298$	4.87626	0.282474
$299$	−8.02072	−0.463850
$300$	0	0
$301$	7.14101	0.411601
$302$	31.5257	1.81410
$303$	10.9739	0.630435
$304$	5.33104	0.305756
$305$	0	0
$306$	−4.14637	−0.237032
$307$	5.19316	0.296389	0.148195	−	0.988958i	$-0.452654\pi$
$307$	5.19316	0.296389	0.148195	+	0.988958i	$0.452654\pi$
$308$	−0.308407	−0.0175731
$309$	16.4284	0.934578
$310$	0	0
$311$	−13.8536	−0.785568	−0.392784	−	0.919631i	$-0.628488\pi$
$311$	−13.8536	−0.785568	−0.392784	+	0.919631i	$0.628488\pi$
$312$	2.77434	0.157066
$313$	32.6162	1.84358	0.921788	−	0.387694i	$-0.126728\pi$
$313$	32.6162	1.84358	0.921788	+	0.387694i	$0.126728\pi$
$314$	−1.24905	−0.0704879
$315$	0	0
$316$	0.327277	0.0184108
$317$	−12.0380	−0.676121	−0.338061	−	0.941124i	$-0.609771\pi$
$317$	−12.0380	−0.676121	−0.338061	+	0.941124i	$0.609771\pi$
$318$	7.40231	0.415101
$319$	13.8536	0.775655
$320$	0	0
$321$	−3.45446	−0.192809
$322$	11.5487	0.643583
$323$	−3.70727	−0.206278
$324$	0.0731828	0.00406571
$325$	0	0
$326$	6.37623	0.353147
$327$	−4.66896	−0.258194
$328$	−33.4802	−1.84864
$329$	1.95289	0.107666
$330$	0	0
$331$	−26.2820	−1.44459	−0.722295	−	0.691585i	$-0.756913\pi$
$331$	−26.2820	−1.44459	−0.722295	+	0.691585i	$0.756913\pi$
$332$	0.278572	0.0152886
$333$	−8.57475	−0.469893
$334$	29.5568	1.61728
$335$	0	0
$336$	−4.14101	−0.225911
$337$	−26.2905	−1.43214	−0.716068	−	0.698031i	$-0.754060\pi$
$337$	−26.2905	−1.43214	−0.716068	+	0.698031i	$0.754060\pi$
$338$	−1.43986	−0.0783178
$339$	−3.28738	−0.178546
$340$	0	0
$341$	−29.6966	−1.60816
$342$	1.85363	0.100233
$343$	−1.00000	−0.0539949
$344$	−19.8116	−1.06817
$345$	0	0
$346$	21.6199	1.16230
$347$	−16.4928	−0.885378	−0.442689	−	0.896675i	$-0.645975\pi$
$347$	−16.4928	−0.885378	−0.442689	+	0.896675i	$0.645975\pi$
$348$	−0.240579	−0.0128964
$349$	14.1793	0.759001	0.379501	−	0.925191i	$-0.376096\pi$
$349$	14.1793	0.759001	0.379501	+	0.925191i	$0.376096\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	1.74375	0.0929419
$353$	29.8272	1.58754	0.793772	−	0.608215i	$-0.208114\pi$
$353$	29.8272	1.58754	0.793772	+	0.608215i	$0.208114\pi$
$354$	−10.5606	−0.561289
$355$	0	0
$356$	−0.413332	−0.0219065
$357$	2.87971	0.152410
$358$	7.70575	0.407262
$359$	1.34671	0.0710767	0.0355383	−	0.999368i	$-0.488685\pi$
$359$	1.34671	0.0710767	0.0355383	+	0.999368i	$0.488685\pi$
$360$	0	0
$361$	−17.3427	−0.912772
$362$	−16.5104	−0.867766
$363$	−6.75942	−0.354778
$364$	0.0731828	0.00383582
$365$	0	0
$366$	11.1724	0.583993
$367$	−36.1771	−1.88843	−0.944214	−	0.329331i	$-0.893177\pi$
$367$	−36.1771	−1.88843	−0.944214	+	0.329331i	$0.893177\pi$
$368$	−33.2139	−1.73139
$369$	−12.0678	−0.628226
$370$	0	0
$371$	−5.14101	−0.266908
$372$	0.515704	0.0267380
$373$	16.9089	0.875511	0.437755	−	0.899094i	$-0.355774\pi$
$373$	16.9089	0.875511	0.437755	+	0.899094i	$0.355774\pi$
$374$	−17.4736	−0.903538
$375$	0	0
$376$	−5.41798	−0.279411
$377$	−3.28738	−0.169308
$378$	−1.43986	−0.0740582
$379$	−11.6131	−0.596523	−0.298261	−	0.954484i	$-0.596407\pi$
$379$	−11.6131	−0.596523	−0.298261	+	0.954484i	$0.596407\pi$
$380$	0	0
$381$	−0.428386	−0.0219469
$382$	−29.0499	−1.48632
$383$	−14.6134	−0.746708	−0.373354	−	0.927689i	$-0.621792\pi$
$383$	−14.6134	−0.746708	−0.373354	+	0.927689i	$0.621792\pi$
$384$	11.8946	0.606995
$385$	0	0
$386$	−37.9771	−1.93298
$387$	−7.14101	−0.362998
$388$	−0.504993	−0.0256371
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	23.0974	1.16808
$392$	2.77434	0.140125
$393$	−14.1878	−0.715680
$394$	−28.9875	−1.46037
$395$	0	0
$396$	0.308407	0.0154980
$397$	−25.6982	−1.28975	−0.644877	−	0.764287i	$-0.723091\pi$
$397$	−25.6982	−1.28975	−0.644877	+	0.764287i	$0.723091\pi$
$398$	−2.53331	−0.126983
$399$	−1.28738	−0.0644494
$400$	0	0
$401$	−19.5637	−0.976966	−0.488483	−	0.872573i	$-0.662450\pi$
$401$	−19.5637	−0.976966	−0.488483	+	0.872573i	$0.662450\pi$
$402$	−17.3379	−0.864737
$403$	7.04680	0.351026
$404$	−0.803103	−0.0399559
$405$	0	0
$406$	4.73334	0.234912
$407$	−36.1357	−1.79118
$408$	−7.98929	−0.395529
$409$	3.19316	0.157892	0.0789458	−	0.996879i	$-0.474845\pi$
$409$	3.19316	0.157892	0.0789458	+	0.996879i	$0.474845\pi$
$410$	0	0
$411$	14.7070	0.725441
$412$	−1.20228	−0.0592319
$413$	7.33448	0.360906
$414$	−11.5487	−0.567586
$415$	0	0
$416$	−0.413779	−0.0202872
$417$	−9.85363	−0.482535
$418$	7.81157	0.382076
$419$	30.2292	1.47680	0.738398	−	0.674366i	$-0.235583\pi$
$419$	30.2292	1.47680	0.738398	+	0.674366i	$0.235583\pi$
$420$	0	0
$421$	26.9510	1.31351	0.656755	−	0.754104i	$-0.271928\pi$
$421$	26.9510	1.31351	0.656755	+	0.754104i	$0.271928\pi$
$422$	−22.4177	−1.09128
$423$	−1.95289	−0.0949529
$424$	14.2629	0.692668
$425$	0	0
$426$	15.5345	0.752650
$427$	−7.75942	−0.375505
$428$	0.252807	0.0122199
$429$	4.21419	0.203463
$430$	0	0
$431$	16.7713	0.807847	0.403923	−	0.914793i	$-0.367646\pi$
$431$	16.7713	0.807847	0.403923	+	0.914793i	$0.367646\pi$
$432$	4.14101	0.199234
$433$	−25.8530	−1.24242	−0.621208	−	0.783646i	$-0.713358\pi$
$433$	−25.8530	−1.24242	−0.621208	+	0.783646i	$0.713358\pi$
$434$	−10.1464	−0.487041
$435$	0	0
$436$	0.341688	0.0163639
$437$	−10.3257	−0.493944
$438$	11.9878	0.572798
$439$	25.0553	1.19582	0.597912	−	0.801562i	$-0.295997\pi$
$439$	25.0553	1.19582	0.597912	+	0.801562i	$0.295997\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	4.14637	0.197223
$443$	−4.64762	−0.220815	−0.110408	−	0.993886i	$-0.535216\pi$
$443$	−4.64762	−0.220815	−0.110408	+	0.993886i	$0.535216\pi$
$444$	0.627525	0.0297810
$445$	0	0
$446$	−16.6583	−0.788791
$447$	3.38663	0.160182
$448$	−7.68624	−0.363141
$449$	0.494696	0.0233462	0.0116731	−	0.999932i	$-0.496284\pi$
$449$	0.494696	0.0233462	0.0116731	+	0.999932i	$0.496284\pi$
$450$	0	0
$451$	−50.8561	−2.39472
$452$	0.240579	0.0113159
$453$	21.8951	1.02872
$454$	−15.8985	−0.746155
$455$	0	0
$456$	3.57161	0.167256
$457$	−26.4698	−1.23821	−0.619103	−	0.785310i	$-0.712504\pi$
$457$	−26.4698	−1.23821	−0.619103	+	0.785310i	$0.712504\pi$
$458$	19.7365	0.922225
$459$	−2.87971	−0.134413
$460$	0	0
$461$	−1.08168	−0.0503786	−0.0251893	−	0.999683i	$-0.508019\pi$
$461$	−1.08168	−0.0503786	−0.0251893	+	0.999683i	$0.508019\pi$
$462$	−6.06783	−0.282301
$463$	32.7098	1.52015	0.760076	−	0.649834i	$-0.225162\pi$
$463$	32.7098	1.52015	0.760076	+	0.649834i	$0.225162\pi$
$464$	−13.6131	−0.631970
$465$	0	0
$466$	1.77854	0.0823894
$467$	17.0797	0.790356	0.395178	−	0.918605i	$-0.370683\pi$
$467$	17.0797	0.790356	0.395178	+	0.918605i	$0.370683\pi$
$468$	−0.0731828	−0.00338288
$469$	12.0414	0.556022
$470$	0	0
$471$	−0.867482	−0.0399715
$472$	−20.3483	−0.936608
$473$	−30.0936	−1.38370
$474$	6.43910	0.295758
$475$	0	0
$476$	−0.210745	−0.00965950
$477$	5.14101	0.235391
$478$	−27.5101	−1.25828
$479$	−17.1790	−0.784929	−0.392464	−	0.919767i	$-0.628377\pi$
$479$	−17.1790	−0.784929	−0.392464	+	0.919767i	$0.628377\pi$
$480$	0	0
$481$	8.57475	0.390975
$482$	17.3257	0.789164
$483$	8.02072	0.364955
$484$	0.494674	0.0224852
$485$	0	0
$486$	1.43986	0.0653132
$487$	4.28264	0.194065	0.0970325	−	0.995281i	$-0.469065\pi$
$487$	4.28264	0.194065	0.0970325	+	0.995281i	$0.469065\pi$
$488$	21.5273	0.974493
$489$	4.42839	0.200259
$490$	0	0
$491$	27.4545	1.23900	0.619501	−	0.784996i	$-0.287335\pi$
$491$	27.4545	1.23900	0.619501	+	0.784996i	$0.287335\pi$
$492$	0.883158	0.0398158
$493$	9.46669	0.426358
$494$	−1.85363	−0.0833990
$495$	0	0
$496$	29.1809	1.31026
$497$	−10.7889	−0.483950
$498$	5.48085	0.245603
$499$	−37.0452	−1.65837	−0.829185	−	0.558974i	$-0.811195\pi$
$499$	−37.0452	−1.65837	−0.829185	+	0.558974i	$0.811195\pi$
$500$	0	0
$501$	20.5276	0.917108
$502$	21.6023	0.964160
$503$	−3.23744	−0.144350	−0.0721752	−	0.997392i	$-0.522994\pi$
$503$	−3.23744	−0.144350	−0.0721752	+	0.997392i	$0.522994\pi$
$504$	−2.77434	−0.123579
$505$	0	0
$506$	−48.6683	−2.16357
$507$	−1.00000	−0.0444116
$508$	0.0313505	0.00139095
$509$	−30.4877	−1.35134	−0.675672	−	0.737202i	$-0.736147\pi$
$509$	−30.4877	−1.35134	−0.675672	+	0.737202i	$0.736147\pi$
$510$	0	0
$511$	−8.32568	−0.368306
$512$	21.2637	0.939730
$513$	1.28738	0.0568390
$514$	−18.7557	−0.827277
$515$	0	0
$516$	0.522599	0.0230062
$517$	−8.22987	−0.361949
$518$	−12.3464	−0.542470
$519$	15.0154	0.659101
$520$	0	0
$521$	−25.3602	−1.11105	−0.555526	−	0.831499i	$-0.687483\pi$
$521$	−25.3602	−1.11105	−0.555526	+	0.831499i	$0.687483\pi$
$522$	−4.73334	−0.207173
$523$	7.00314	0.306226	0.153113	−	0.988209i	$-0.451070\pi$
$523$	7.00314	0.306226	0.153113	+	0.988209i	$0.451070\pi$
$524$	1.03830	0.0453585
$525$	0	0
$526$	−3.73710	−0.162945
$527$	−20.2927	−0.883965
$528$	17.4510	0.759458
$529$	41.3320	1.79704
$530$	0	0
$531$	−7.33448	−0.318289
$532$	0.0942138	0.00408469
$533$	12.0678	0.522716
$534$	−8.13221	−0.351915
$535$	0	0
$536$	−33.4070	−1.44296
$537$	5.35176	0.230945
$538$	33.5603	1.44689
$539$	4.21419	0.181518
$540$	0	0
$541$	−1.42463	−0.0612495	−0.0306248	−	0.999531i	$-0.509750\pi$
$541$	−1.42463	−0.0612495	−0.0306248	+	0.999531i	$0.509750\pi$
$542$	8.71422	0.374308
$543$	−11.4667	−0.492083
$544$	1.19156	0.0510879
$545$	0	0
$546$	1.43986	0.0616201
$547$	22.0499	0.942787	0.471394	−	0.881923i	$-0.343751\pi$
$547$	22.0499	0.942787	0.471394	+	0.881923i	$0.343751\pi$
$548$	−1.07630	−0.0459771
$549$	7.75942	0.331164
$550$	0	0
$551$	−4.23209	−0.180293
$552$	−22.2522	−0.947116
$553$	−4.47204	−0.190171
$554$	21.8760	0.929420
$555$	0	0
$556$	0.721117	0.0305822
$557$	9.08319	0.384867	0.192434	−	0.981310i	$-0.438362\pi$
$557$	9.08319	0.384867	0.192434	+	0.981310i	$0.438362\pi$
$558$	10.1464	0.429530
$559$	7.14101	0.302033
$560$	0	0
$561$	−12.1357	−0.512368
$562$	10.9070	0.460084
$563$	−34.1350	−1.43862	−0.719310	−	0.694689i	$-0.755542\pi$
$563$	−34.1350	−1.43862	−0.719310	+	0.694689i	$0.755542\pi$
$564$	0.142918	0.00601794
$565$	0	0
$566$	27.4974	1.15580
$567$	−1.00000	−0.0419961
$568$	29.9322	1.25593
$569$	1.23522	0.0517833	0.0258916	−	0.999665i	$-0.491758\pi$
$569$	1.23522	0.0517833	0.0258916	+	0.999665i	$0.491758\pi$
$570$	0	0
$571$	−26.4613	−1.10737	−0.553686	−	0.832725i	$-0.686779\pi$
$571$	−26.4613	−1.10737	−0.553686	+	0.832725i	$0.686779\pi$
$572$	−0.308407	−0.0128951
$573$	−20.1756	−0.842847
$574$	−17.3759	−0.725257
$575$	0	0
$576$	7.68624	0.320260
$577$	23.2852	0.969374	0.484687	−	0.874688i	$-0.338934\pi$
$577$	23.2852	0.969374	0.484687	+	0.874688i	$0.338934\pi$
$578$	12.5372	0.521479
$579$	−26.3756	−1.09613
$580$	0	0
$581$	−3.80653	−0.157921
$582$	−9.93562	−0.411845
$583$	21.6652	0.897281
$584$	23.0982	0.955812
$585$	0	0
$586$	−5.46324	−0.225684
$587$	10.7155	0.442274	0.221137	−	0.975243i	$-0.429023\pi$
$587$	10.7155	0.442274	0.221137	+	0.975243i	$0.429023\pi$
$588$	−0.0731828	−0.00301801
$589$	9.07187	0.373800
$590$	0	0
$591$	−20.1322	−0.828128
$592$	35.5081	1.45938
$593$	−14.1008	−0.579049	−0.289525	−	0.957171i	$-0.593497\pi$
$593$	−14.1008	−0.579049	−0.289525	+	0.957171i	$0.593497\pi$
$594$	6.06783	0.248966
$595$	0	0
$596$	−0.247843	−0.0101521
$597$	−1.75942	−0.0720083
$598$	11.5487	0.472260
$599$	−38.9296	−1.59062	−0.795311	−	0.606202i	$-0.792692\pi$
$599$	−38.9296	−1.59062	−0.795311	+	0.606202i	$0.792692\pi$
$600$	0	0
$601$	−32.6162	−1.33044	−0.665221	−	0.746646i	$-0.731663\pi$
$601$	−32.6162	−1.33044	−0.665221	+	0.746646i	$0.731663\pi$
$602$	−10.2820	−0.419064
$603$	−12.0414	−0.490365
$604$	−1.60234	−0.0651984
$605$	0	0
$606$	−15.8009	−0.641866
$607$	−19.3203	−0.784188	−0.392094	−	0.919925i	$-0.628249\pi$
$607$	−19.3203	−0.784188	−0.392094	+	0.919925i	$0.628249\pi$
$608$	−0.532689	−0.0216034
$609$	3.28738	0.133211
$610$	0	0
$611$	1.95289	0.0790056
$612$	0.210745	0.00851888
$613$	−35.3197	−1.42655	−0.713275	−	0.700885i	$-0.752789\pi$
$613$	−35.3197	−1.42655	−0.713275	+	0.700885i	$0.752789\pi$
$614$	−7.47740	−0.301763
$615$	0	0
$616$	−11.6916	−0.471068
$617$	24.4318	0.983589	0.491794	−	0.870711i	$-0.336341\pi$
$617$	24.4318	0.983589	0.491794	+	0.870711i	$0.336341\pi$
$618$	−23.6545	−0.951523
$619$	2.90892	0.116919	0.0584597	−	0.998290i	$-0.481381\pi$
$619$	2.90892	0.116919	0.0584597	+	0.998290i	$0.481381\pi$
$620$	0	0
$621$	−8.02072	−0.321860
$622$	19.9472	0.799811
$623$	5.64793	0.226280
$624$	−4.14101	−0.165773
$625$	0	0
$626$	−46.9626	−1.87700
$627$	5.42525	0.216664
$628$	0.0634848	0.00253332
$629$	−24.6928	−0.984566
$630$	0	0
$631$	5.66521	0.225528	0.112764	−	0.993622i	$-0.464030\pi$
$631$	5.66521	0.225528	0.112764	+	0.993622i	$0.464030\pi$
$632$	12.4070	0.493522
$633$	−15.5694	−0.618828
$634$	17.3330	0.688380
$635$	0	0
$636$	−0.376234	−0.0149186
$637$	−1.00000	−0.0396214
$638$	−19.9472	−0.789718
$639$	10.7889	0.426804
$640$	0	0
$641$	12.2842	0.485198	0.242599	−	0.970127i	$-0.422000\pi$
$641$	12.2842	0.485198	0.242599	+	0.970127i	$0.422000\pi$
$642$	4.97392	0.196305
$643$	−9.14950	−0.360821	−0.180411	−	0.983591i	$-0.557743\pi$
$643$	−9.14950	−0.360821	−0.180411	+	0.983591i	$0.557743\pi$
$644$	−0.586979	−0.0231302
$645$	0	0
$646$	5.33793	0.210018
$647$	−38.4108	−1.51008	−0.755042	−	0.655677i	$-0.772383\pi$
$647$	−38.4108	−1.51008	−0.755042	+	0.655677i	$0.772383\pi$
$648$	2.77434	0.108986
$649$	−30.9089	−1.21328
$650$	0	0
$651$	−7.04680	−0.276186
$652$	−0.324082	−0.0126920
$653$	−28.4805	−1.11453	−0.557265	−	0.830335i	$-0.688149\pi$
$653$	−28.4805	−1.11453	−0.557265	+	0.830335i	$0.688149\pi$
$654$	6.72263	0.262876
$655$	0	0
$656$	49.9730	1.95112
$657$	8.32568	0.324816
$658$	−2.81188	−0.109619
$659$	−30.4384	−1.18571	−0.592856	−	0.805309i	$-0.702000\pi$
$659$	−30.4384	−1.18571	−0.592856	+	0.805309i	$0.702000\pi$
$660$	0	0
$661$	36.7710	1.43023	0.715114	−	0.699008i	$-0.246375\pi$
$661$	36.7710	1.43023	0.715114	+	0.699008i	$0.246375\pi$
$662$	37.8423	1.47078
$663$	2.87971	0.111839
$664$	10.5606	0.409830
$665$	0	0
$666$	12.3464	0.478413
$667$	26.3671	1.02094
$668$	−1.50227	−0.0581246
$669$	−11.5694	−0.447299
$670$	0	0
$671$	32.6997	1.26236
$672$	0.413779	0.0159619
$673$	−26.7961	−1.03291	−0.516457	−	0.856313i	$-0.672750\pi$
$673$	−26.7961	−1.03291	−0.516457	+	0.856313i	$0.672750\pi$
$674$	37.8545	1.45810
$675$	0	0
$676$	0.0731828	0.00281472
$677$	20.3050	0.780383	0.390191	−	0.920734i	$-0.372409\pi$
$677$	20.3050	0.780383	0.390191	+	0.920734i	$0.372409\pi$
$678$	4.73334	0.181783
$679$	6.90043	0.264814
$680$	0	0
$681$	−11.0418	−0.423121
$682$	42.7587	1.63732
$683$	−44.9240	−1.71897	−0.859484	−	0.511162i	$-0.829215\pi$
$683$	−44.9240	−1.71897	−0.859484	+	0.511162i	$0.829215\pi$
$684$	−0.0942138	−0.00360235
$685$	0	0
$686$	1.43986	0.0549739
$687$	13.7073	0.522965
$688$	29.5710	1.12738
$689$	−5.14101	−0.195857
$690$	0	0
$691$	22.0085	0.837243	0.418621	−	0.908161i	$-0.362513\pi$
$691$	22.0085	0.837243	0.418621	+	0.908161i	$0.362513\pi$
$692$	−1.09887	−0.0417726
$693$	−4.21419	−0.160084
$694$	23.7472	0.901431
$695$	0	0
$696$	−9.12029	−0.345704
$697$	−34.7518	−1.31632
$698$	−20.4162	−0.772763
$699$	1.23522	0.0467205
$700$	0	0
$701$	31.4645	1.18840	0.594198	−	0.804319i	$-0.297469\pi$
$701$	31.4645	1.18840	0.594198	+	0.804319i	$0.297469\pi$
$702$	−1.43986	−0.0543438
$703$	11.0389	0.416341
$704$	32.3913	1.22079
$705$	0	0
$706$	−42.9469	−1.61633
$707$	10.9739	0.412717
$708$	0.536758	0.0201726
$709$	−25.3373	−0.951563	−0.475781	−	0.879564i	$-0.657835\pi$
$709$	−25.3373	−0.951563	−0.475781	+	0.879564i	$0.657835\pi$
$710$	0	0
$711$	4.47204	0.167715
$712$	−15.6693	−0.587231
$713$	−56.5204	−2.11670
$714$	−4.14637	−0.155174
$715$	0	0
$716$	−0.391657	−0.0146369
$717$	−19.1061	−0.713532
$718$	−1.93907	−0.0723654
$719$	−43.8002	−1.63347	−0.816737	−	0.577011i	$-0.804219\pi$
$719$	−43.8002	−1.63347	−0.816737	+	0.577011i	$0.804219\pi$
$720$	0	0
$721$	16.4284	0.611825
$722$	24.9709	0.929322
$723$	12.0329	0.447510
$724$	0.839165	0.0311873
$725$	0	0
$726$	9.73259	0.361210
$727$	−0.944090	−0.0350144	−0.0175072	−	0.999847i	$-0.505573\pi$
$727$	−0.944090	−0.0350144	−0.0175072	+	0.999847i	$0.505573\pi$
$728$	2.77434	0.102824
$729$	1.00000	0.0370370
$730$	0	0
$731$	−20.5640	−0.760588
$732$	−0.567856	−0.0209886
$733$	5.09957	0.188357	0.0941784	−	0.995555i	$-0.469978\pi$
$733$	5.09957	0.188357	0.0941784	+	0.995555i	$0.469978\pi$
$734$	52.0898	1.92267
$735$	0	0
$736$	3.31881	0.122333
$737$	−50.7450	−1.86921
$738$	17.3759	0.639617
$739$	−8.25129	−0.303529	−0.151764	−	0.988417i	$-0.548495\pi$
$739$	−8.25129	−0.303529	−0.151764	+	0.988417i	$0.548495\pi$
$740$	0	0
$741$	−1.28738	−0.0472929
$742$	7.40231	0.271747
$743$	−38.4962	−1.41229	−0.706145	−	0.708068i	$-0.749567\pi$
$743$	−38.4962	−1.41229	−0.706145	+	0.708068i	$0.749567\pi$
$744$	19.5502	0.716745
$745$	0	0
$746$	−24.3464	−0.891385
$747$	3.80653	0.139274
$748$	0.888121	0.0324729
$749$	−3.45446	−0.126223
$750$	0	0
$751$	−41.3175	−1.50770	−0.753848	−	0.657049i	$-0.771805\pi$
$751$	−41.3175	−1.50770	−0.753848	+	0.657049i	$0.771805\pi$
$752$	8.08695	0.294901
$753$	15.0031	0.546745
$754$	4.73334	0.172378
$755$	0	0
$756$	0.0731828	0.00266163
$757$	39.7158	1.44349	0.721747	−	0.692157i	$-0.243339\pi$
$757$	39.7158	1.44349	0.721747	+	0.692157i	$0.243339\pi$
$758$	16.7211	0.607338
$759$	−33.8009	−1.22689
$760$	0	0
$761$	10.7390	0.389289	0.194644	−	0.980874i	$-0.437645\pi$
$761$	10.7390	0.389289	0.194644	+	0.980874i	$0.437645\pi$
$762$	0.616813	0.0223448
$763$	−4.66896	−0.169028
$764$	1.47651	0.0534181
$765$	0	0
$766$	21.0411	0.760247
$767$	7.33448	0.264833
$768$	−1.75406	−0.0632944
$769$	−50.5549	−1.82306	−0.911529	−	0.411237i	$-0.865097\pi$
$769$	−50.5549	−1.82306	−0.911529	+	0.411237i	$0.865097\pi$
$770$	0	0
$771$	−13.0261	−0.469123
$772$	1.93024	0.0694709
$773$	−8.91770	−0.320747	−0.160374	−	0.987056i	$-0.551270\pi$
$773$	−8.91770	−0.320747	−0.160374	+	0.987056i	$0.551270\pi$
$774$	10.2820	0.369580
$775$	0	0
$776$	−19.1441	−0.687234
$777$	−8.57475	−0.307617
$778$	−8.63913	−0.309728
$779$	15.5358	0.556629
$780$	0	0
$781$	45.4667	1.62693
$782$	−33.2568	−1.18926
$783$	−3.28738	−0.117481
$784$	−4.14101	−0.147893
$785$	0	0
$786$	20.4284	0.728656
$787$	−22.3671	−0.797302	−0.398651	−	0.917103i	$-0.630521\pi$
$787$	−22.3671	−0.797302	−0.398651	+	0.917103i	$0.630521\pi$
$788$	1.47333	0.0524853
$789$	−2.59547	−0.0924012
$790$	0	0
$791$	−3.28738	−0.116886
$792$	11.6916	0.415443
$793$	−7.75942	−0.275545
$794$	37.0016	1.31314
$795$	0	0
$796$	0.128759	0.00456375
$797$	−20.5518	−0.727983	−0.363991	−	0.931402i	$-0.618586\pi$
$797$	−20.5518	−0.727983	−0.363991	+	0.931402i	$0.618586\pi$
$798$	1.85363	0.0656179
$799$	−5.62377	−0.198955
$800$	0	0
$801$	−5.64793	−0.199560
$802$	28.1689	0.994680
$803$	35.0860	1.23816
$804$	0.881227	0.0310785
$805$	0	0
$806$	−10.1464	−0.357390
$807$	23.3081	0.820484
$808$	−30.4454	−1.07106
$809$	7.83443	0.275444	0.137722	−	0.990471i	$-0.456022\pi$
$809$	7.83443	0.275444	0.137722	+	0.990471i	$0.456022\pi$
$810$	0	0
$811$	−47.5502	−1.66971	−0.834857	−	0.550468i	$-0.814449\pi$
$811$	−47.5502	−1.66971	−0.834857	+	0.550468i	$0.814449\pi$
$812$	−0.240579	−0.00844268
$813$	6.05215	0.212258
$814$	52.0301	1.82365
$815$	0	0
$816$	11.9249	0.417455
$817$	9.19316	0.321628
$818$	−4.59769	−0.160754
$819$	1.00000	0.0349428
$820$	0	0
$821$	4.42494	0.154431	0.0772157	−	0.997014i	$-0.475397\pi$
$821$	4.42494	0.154431	0.0772157	+	0.997014i	$0.475397\pi$
$822$	−21.1759	−0.738594
$823$	28.7525	1.00225	0.501124	−	0.865375i	$-0.332920\pi$
$823$	28.7525	1.00225	0.501124	+	0.865375i	$0.332920\pi$
$824$	−45.5779	−1.58778
$825$	0	0
$826$	−10.5606	−0.367450
$827$	5.58667	0.194267	0.0971337	−	0.995271i	$-0.469033\pi$
$827$	5.58667	0.194267	0.0971337	+	0.995271i	$0.469033\pi$
$828$	0.586979	0.0203989
$829$	53.4974	1.85804	0.929021	−	0.370027i	$-0.120652\pi$
$829$	53.4974	1.85804	0.929021	+	0.370027i	$0.120652\pi$
$830$	0	0
$831$	15.1932	0.527045
$832$	−7.68624	−0.266472
$833$	2.87971	0.0997760
$834$	14.1878	0.491284
$835$	0	0
$836$	−0.397035	−0.0137317
$837$	7.04680	0.243573
$838$	−43.5257	−1.50357
$839$	−43.0405	−1.48592	−0.742962	−	0.669334i	$-0.766580\pi$
$839$	−43.0405	−1.48592	−0.742962	+	0.669334i	$0.766580\pi$
$840$	0	0
$841$	−18.1932	−0.627350
$842$	−38.8055	−1.33733
$843$	7.57506	0.260899
$844$	1.13941	0.0392202
$845$	0	0
$846$	2.81188	0.0966745
$847$	−6.75942	−0.232256
$848$	−21.2890	−0.731066
$849$	19.0974	0.655419
$850$	0	0
$851$	−68.7757	−2.35760
$852$	−0.789565	−0.0270501
$853$	30.7434	1.05263	0.526316	−	0.850289i	$-0.323573\pi$
$853$	30.7434	1.05263	0.526316	+	0.850289i	$0.323573\pi$
$854$	11.1724	0.382313
$855$	0	0
$856$	9.58384	0.327569
$857$	24.5449	0.838438	0.419219	−	0.907885i	$-0.362304\pi$
$857$	24.5449	0.838438	0.419219	+	0.907885i	$0.362304\pi$
$858$	−6.06783	−0.207152
$859$	−52.1243	−1.77846	−0.889229	−	0.457461i	$-0.848759\pi$
$859$	−52.1243	−1.77846	−0.889229	+	0.457461i	$0.848759\pi$
$860$	0	0
$861$	−12.0678	−0.411270
$862$	−24.1483	−0.822494
$863$	−22.2563	−0.757612	−0.378806	−	0.925476i	$-0.623665\pi$
$863$	−22.2563	−0.757612	−0.378806	+	0.925476i	$0.623665\pi$
$864$	−0.413779	−0.0140771
$865$	0	0
$866$	37.2246	1.26494
$867$	8.70727	0.295714
$868$	0.515704	0.0175041
$869$	18.8461	0.639309
$870$	0	0
$871$	12.0414	0.408009
$872$	12.9533	0.438654
$873$	−6.90043	−0.233544
$874$	14.8675	0.502900
$875$	0	0
$876$	−0.609297	−0.0205862
$877$	−41.0547	−1.38632	−0.693159	−	0.720785i	$-0.743782\pi$
$877$	−41.0547	−1.38632	−0.693159	+	0.720785i	$0.743782\pi$
$878$	−36.0760	−1.21751
$879$	−3.79430	−0.127979
$880$	0	0
$881$	−31.0568	−1.04633	−0.523165	−	0.852231i	$-0.675249\pi$
$881$	−31.0568	−1.04633	−0.523165	+	0.852231i	$0.675249\pi$
$882$	−1.43986	−0.0484824
$883$	56.5357	1.90258	0.951289	−	0.308300i	$-0.0997600\pi$
$883$	56.5357	1.90258	0.951289	+	0.308300i	$0.0997600\pi$
$884$	−0.210745	−0.00708813
$885$	0	0
$886$	6.69190	0.224819
$887$	−9.94785	−0.334016	−0.167008	−	0.985956i	$-0.553411\pi$
$887$	−9.94785	−0.334016	−0.167008	+	0.985956i	$0.553411\pi$
$888$	23.7893	0.798315
$889$	−0.428386	−0.0143676
$890$	0	0
$891$	4.21419	0.141181
$892$	0.846681	0.0283490
$893$	2.51411	0.0841314
$894$	−4.87626	−0.163087
$895$	0	0
$896$	11.8946	0.397372
$897$	8.02072	0.267804
$898$	−0.712291	−0.0237695
$899$	−23.1655	−0.772612
$900$	0	0
$901$	14.8046	0.493213
$902$	73.2255	2.43814
$903$	−7.14101	−0.237638
$904$	9.12029	0.303336
$905$	0	0
$906$	−31.5257	−1.04737
$907$	31.2936	1.03909	0.519544	−	0.854444i	$-0.326102\pi$
$907$	31.2936	1.03909	0.519544	+	0.854444i	$0.326102\pi$
$908$	0.808067	0.0268166
$909$	−10.9739	−0.363982
$910$	0	0
$911$	−9.39320	−0.311210	−0.155605	−	0.987819i	$-0.549733\pi$
$911$	−9.39320	−0.311210	−0.155605	+	0.987819i	$0.549733\pi$
$912$	−5.33104	−0.176528
$913$	16.0414	0.530894
$914$	38.1127	1.26066
$915$	0	0
$916$	−1.00314	−0.0331446
$917$	−14.1878	−0.468523
$918$	4.14637	0.136850
$919$	21.7066	0.716036	0.358018	−	0.933715i	$-0.383453\pi$
$919$	21.7066	0.716036	0.358018	+	0.933715i	$0.383453\pi$
$920$	0	0
$921$	−5.19316	−0.171120
$922$	1.55746	0.0512921
$923$	−10.7889	−0.355122
$924$	0.308407	0.0101458
$925$	0	0
$926$	−47.0974	−1.54771
$927$	−16.4284	−0.539579
$928$	1.36025	0.0446523
$929$	35.6787	1.17058	0.585289	−	0.810824i	$-0.300981\pi$
$929$	35.6787	1.17058	0.585289	+	0.810824i	$0.300981\pi$
$930$	0	0
$931$	−1.28738	−0.0421920
$932$	−0.0903972	−0.00296106
$933$	13.8536	0.453548
$934$	−24.5924	−0.804687
$935$	0	0
$936$	−2.77434	−0.0906821
$937$	47.0866	1.53825	0.769127	−	0.639096i	$-0.220691\pi$
$937$	47.0866	1.53825	0.769127	+	0.639096i	$0.220691\pi$
$938$	−17.3379	−0.566103
$939$	−32.6162	−1.06439
$940$	0	0
$941$	45.2923	1.47649	0.738244	−	0.674534i	$-0.235655\pi$
$941$	45.2923	1.47649	0.738244	+	0.674534i	$0.235655\pi$
$942$	1.24905	0.0406962
$943$	−96.7927	−3.15200
$944$	30.3722	0.988530
$945$	0	0
$946$	43.3304	1.40879
$947$	−4.77134	−0.155048	−0.0775238	−	0.996991i	$-0.524701\pi$
$947$	−4.77134	−0.155048	−0.0775238	+	0.996991i	$0.524701\pi$
$948$	−0.327277	−0.0106295
$949$	−8.32568	−0.270263
$950$	0	0
$951$	12.0380	0.390359
$952$	−7.98929	−0.258934
$953$	3.95572	0.128138	0.0640692	−	0.997945i	$-0.479592\pi$
$953$	3.95572	0.128138	0.0640692	+	0.997945i	$0.479592\pi$
$954$	−7.40231	−0.239659
$955$	0	0
$956$	1.39824	0.0452223
$957$	−13.8536	−0.447824
$958$	24.7353	0.799160
$959$	14.7070	0.474912
$960$	0	0
$961$	18.6573	0.601850
$962$	−12.3464	−0.398064
$963$	3.45446	0.111318
$964$	−0.880605	−0.0283624
$965$	0	0
$966$	−11.5487	−0.371573
$967$	−50.2292	−1.61526	−0.807632	−	0.589687i	$-0.799251\pi$
$967$	−50.2292	−1.61526	−0.807632	+	0.589687i	$0.799251\pi$
$968$	18.7529	0.602742
$969$	3.70727	0.119095
$970$	0	0
$971$	38.6576	1.24058	0.620291	−	0.784372i	$-0.287014\pi$
$971$	38.6576	1.24058	0.620291	+	0.784372i	$0.287014\pi$
$972$	−0.0731828	−0.00234734
$973$	−9.85363	−0.315893
$974$	−6.16638	−0.197584
$975$	0	0
$976$	−32.1318	−1.02852
$977$	20.6134	0.659480	0.329740	−	0.944072i	$-0.393039\pi$
$977$	20.6134	0.659480	0.329740	+	0.944072i	$0.393039\pi$
$978$	−6.37623	−0.203889
$979$	−23.8015	−0.760699
$980$	0	0
$981$	4.66896	0.149069
$982$	−39.5304	−1.26147
$983$	4.48620	0.143088	0.0715438	−	0.997437i	$-0.477207\pi$
$983$	4.48620	0.143088	0.0715438	+	0.997437i	$0.477207\pi$
$984$	33.4802	1.06731
$985$	0	0
$986$	−13.6307	−0.434089
$987$	−1.95289	−0.0621613
$988$	0.0942138	0.00299734
$989$	−57.2760	−1.82127
$990$	0	0
$991$	−32.7971	−1.04183	−0.520917	−	0.853607i	$-0.674410\pi$
$991$	−32.7971	−1.04183	−0.520917	+	0.853607i	$0.674410\pi$
$992$	−2.91582	−0.0925773
$993$	26.2820	0.834035
$994$	15.5345	0.492725
$995$	0	0
$996$	−0.278572	−0.00882691
$997$	−0.491870	−0.0155777	−0.00778884	−	0.999970i	$-0.502479\pi$
$997$	−0.491870	−0.0155777	−0.00778884	+	0.999970i	$0.502479\pi$
$998$	53.3397	1.68844
$999$	8.57475	0.271293

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6825.2.a.bg.1.2		4
5.4	even	2		273.2.a.e.1.3	✓	4
15.14	odd	2		819.2.a.k.1.2		4
20.19	odd	2		4368.2.a.br.1.3		4
35.34	odd	2		1911.2.a.s.1.3		4
65.64	even	2		3549.2.a.w.1.2		4
105.104	even	2		5733.2.a.bf.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.a.e.1.3	✓	4	5.4	even	2
819.2.a.k.1.2		4	15.14	odd	2
1911.2.a.s.1.3		4	35.34	odd	2
3549.2.a.w.1.2		4	65.64	even	2
4368.2.a.br.1.3		4	20.19	odd	2
5733.2.a.bf.1.2		4	105.104	even	2
6825.2.a.bg.1.2		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:	\( N \)	\(=\)	\( 6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6825.a (trivial)

\(f(q)\)	\(=\)	\(q-1.43986 q^{2} -1.00000 q^{3} +0.0731828 q^{4} +1.43986 q^{6} -1.00000 q^{7} +2.77434 q^{8} +1.00000 q^{9} +4.21419 q^{11} -0.0731828 q^{12} -1.00000 q^{13} +1.43986 q^{14} -4.14101 q^{16} +2.87971 q^{17} -1.43986 q^{18} -1.28738 q^{19} +1.00000 q^{21} -6.06783 q^{22} +8.02072 q^{23} -2.77434 q^{24} +1.43986 q^{26} -1.00000 q^{27} -0.0731828 q^{28} +3.28738 q^{29} -7.04680 q^{31} +0.413779 q^{32} -4.21419 q^{33} -4.14637 q^{34} +0.0731828 q^{36} -8.57475 q^{37} +1.85363 q^{38} +1.00000 q^{39} -12.0678 q^{41} -1.43986 q^{42} -7.14101 q^{43} +0.308407 q^{44} -11.5487 q^{46} -1.95289 q^{47} +4.14101 q^{48} +1.00000 q^{49} -2.87971 q^{51} -0.0731828 q^{52} +5.14101 q^{53} +1.43986 q^{54} -2.77434 q^{56} +1.28738 q^{57} -4.73334 q^{58} -7.33448 q^{59} +7.75942 q^{61} +10.1464 q^{62} -1.00000 q^{63} +7.68624 q^{64} +6.06783 q^{66} -12.0414 q^{67} +0.210745 q^{68} -8.02072 q^{69} +10.7889 q^{71} +2.77434 q^{72} +8.32568 q^{73} +12.3464 q^{74} -0.0942138 q^{76} -4.21419 q^{77} -1.43986 q^{78} +4.47204 q^{79} +1.00000 q^{81} +17.3759 q^{82} +3.80653 q^{83} +0.0731828 q^{84} +10.2820 q^{86} -3.28738 q^{87} +11.6916 q^{88} -5.64793 q^{89} +1.00000 q^{91} +0.586979 q^{92} +7.04680 q^{93} +2.81188 q^{94} -0.413779 q^{96} -6.90043 q^{97} -1.43986 q^{98} +4.21419 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9} - 2 q^{11} - 7 q^{12} - 4 q^{13} + q^{14} + 9 q^{16} + 2 q^{17} - q^{18} + 7 q^{19} + 4 q^{21} + 8 q^{22} - 3 q^{23} + 3 q^{24}+ \cdots - 2 q^{99}+O(q^{100}) \) 4 * q - q^2 - 4 * q^3 + 7 * q^4 + q^6 - 4 * q^7 - 3 * q^8 + 4 * q^9 - 2 * q^11 - 7 * q^12 - 4 * q^13 + q^14 + 9 * q^16 + 2 * q^17 - q^18 + 7 * q^19 + 4 * q^21 + 8 * q^22 - 3 * q^23 + 3 * q^24 + q^26 - 4 * q^27 - 7 * q^28 + q^29 + 3 * q^31 - 7 * q^32 + 2 * q^33 - 30 * q^34 + 7 * q^36 - 10 * q^37 - 6 * q^38 + 4 * q^39 - 16 * q^41 - q^42 - 3 * q^43 - 12 * q^44 - 18 * q^46 - 5 * q^47 - 9 * q^48 + 4 * q^49 - 2 * q^51 - 7 * q^52 - 5 * q^53 + q^54 + 3 * q^56 - 7 * q^57 + 4 * q^58 - 20 * q^59 + 12 * q^61 + 54 * q^62 - 4 * q^63 + 5 * q^64 - 8 * q^66 + 22 * q^67 + 10 * q^68 + 3 * q^69 - 3 * q^72 + 13 * q^73 - 6 * q^74 - 6 * q^76 + 2 * q^77 - q^78 + 11 * q^79 + 4 * q^81 - 10 * q^82 - q^83 + 7 * q^84 - 10 * q^86 - q^87 + 60 * q^88 - 5 * q^89 + 4 * q^91 - 34 * q^92 - 3 * q^93 + 34 * q^94 + 7 * q^96 + 17 * q^97 - q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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