LMFDB - Embedded newform 2175.2.c.j.349.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2175,2,Mod(349,2175)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2175.349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2175.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,2,0,-2,0,0,-4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$17.3674624396$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	2175.349
Dual form			2175.2.c.j.349.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.61803i q^{2} -1.00000i q^{3} -0.618034 q^{4} -1.61803 q^{6} -3.00000i q^{7} -2.23607i q^{8} -1.00000 q^{9} +3.47214 q^{11} +0.618034i q^{12} +1.76393i q^{13} -4.85410 q^{14} -4.85410 q^{16} -5.47214i q^{17} +1.61803i q^{18} +5.70820 q^{19} -3.00000 q^{21} -5.61803i q^{22} -2.23607 q^{24} +2.85410 q^{26} +1.00000i q^{27} +1.85410i q^{28} -1.00000 q^{29} -8.00000 q^{31} +3.38197i q^{32} -3.47214i q^{33} -8.85410 q^{34} +0.618034 q^{36} -8.00000i q^{37} -9.23607i q^{38} +1.76393 q^{39} +4.47214 q^{41} +4.85410i q^{42} +1.23607i q^{43} -2.14590 q^{44} -6.70820i q^{47} +4.85410i q^{48} -2.00000 q^{49} -5.47214 q^{51} -1.09017i q^{52} +11.2361i q^{53} +1.61803 q^{54} -6.70820 q^{56} -5.70820i q^{57} +1.61803i q^{58} -0.763932 q^{59} +7.70820 q^{61} +12.9443i q^{62} +3.00000i q^{63} -4.23607 q^{64} -5.61803 q^{66} -2.52786i q^{67} +3.38197i q^{68} +2.76393 q^{71} +2.23607i q^{72} +8.00000i q^{73} -12.9443 q^{74} -3.52786 q^{76} -10.4164i q^{77} -2.85410i q^{78} -16.1803 q^{79} +1.00000 q^{81} -7.23607i q^{82} -9.70820i q^{83} +1.85410 q^{84} +2.00000 q^{86} +1.00000i q^{87} -7.76393i q^{88} +11.1803 q^{89} +5.29180 q^{91} +8.00000i q^{93} -10.8541 q^{94} +3.38197 q^{96} +7.23607i q^{97} +3.23607i q^{98} -3.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 2 q^{6} - 4 q^{9} - 4 q^{11} - 6 q^{14} - 6 q^{16} - 4 q^{19} - 12 q^{21} - 2 q^{26} - 4 q^{29} - 32 q^{31} - 22 q^{34} - 2 q^{36} + 16 q^{39} - 22 q^{44} - 8 q^{49} - 4 q^{51} + 2 q^{54}+ \cdots + 4 q^{99}+O(q^{100}) $ 4 * q + 2 * q^4 - 2 * q^6 - 4 * q^9 - 4 * q^11 - 6 * q^14 - 6 * q^16 - 4 * q^19 - 12 * q^21 - 2 * q^26 - 4 * q^29 - 32 * q^31 - 22 * q^34 - 2 * q^36 + 16 * q^39 - 22 * q^44 - 8 * q^49 - 4 * q^51 + 2 * q^54 - 12 * q^59 + 4 * q^61 - 8 * q^64 - 18 * q^66 + 20 * q^71 - 16 * q^74 - 32 * q^76 - 20 * q^79 + 4 * q^81 - 6 * q^84 + 8 * q^86 + 48 * q^91 - 30 * q^94 + 18 * q^96 + 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times$.

$n$	$901$	$1451$	$2002$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 1.61803i	− 1.14412i	−0.820211	−	0.572061i	$-0.806144\pi$
$2$	− 1.61803i	− 1.14412i	0.820211	−	0.572061i	$-0.193856\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−0.618034	−0.309017
$5$	0	0
$5$	0	0
$6$	−1.61803	−0.660560
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	− 2.23607i	− 0.790569i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.47214	1.04689	0.523444	−	0.852060i	$-0.324647\pi$
$11$	3.47214	1.04689	0.523444	+	0.852060i	$0.324647\pi$
$12$	0.618034i	0.178411i
$13$	1.76393i	0.489227i	0.969621	+	0.244613i	$0.0786610\pi$
$13$	1.76393i	0.489227i	−0.969621	+	0.244613i	$0.921339\pi$
$14$	−4.85410	−1.29731
$15$	0	0
$16$	−4.85410	−1.21353
$17$	− 5.47214i	− 1.32719i	−0.748093	−	0.663594i	$-0.769030\pi$
$17$	− 5.47214i	− 1.32719i	0.748093	−	0.663594i	$-0.230970\pi$
$18$	1.61803i	0.381374i
$19$	5.70820	1.30955	0.654776	−	0.755823i	$-0.272763\pi$
$19$	5.70820	1.30955	0.654776	+	0.755823i	$0.272763\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	− 5.61803i	− 1.19777i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−2.23607	−0.456435
$25$	0	0
$26$	2.85410	0.559735
$27$	1.00000i	0.192450i
$28$	1.85410i	0.350392i
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	3.38197i	0.597853i
$33$	− 3.47214i	− 0.604421i
$34$	−8.85410	−1.51847
$35$	0	0
$36$	0.618034	0.103006
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	− 9.23607i	− 1.49829i
$39$	1.76393	0.282455
$40$	0	0
$41$	4.47214	0.698430	0.349215	−	0.937043i	$-0.386448\pi$
$41$	4.47214	0.698430	0.349215	+	0.937043i	$0.386448\pi$
$42$	4.85410i	0.749004i
$43$	1.23607i	0.188499i	0.995549	+	0.0942493i	$0.0300451\pi$
$43$	1.23607i	0.188499i	−0.995549	+	0.0942493i	$0.969955\pi$
$44$	−2.14590	−0.323506
$45$	0	0
$46$	0	0
$47$	− 6.70820i	− 0.978492i	−0.872146	−	0.489246i	$-0.837272\pi$
$47$	− 6.70820i	− 0.978492i	0.872146	−	0.489246i	$-0.162728\pi$
$48$	4.85410i	0.700629i
$49$	−2.00000	−0.285714
$50$	0	0
$51$	−5.47214	−0.766252
$52$	− 1.09017i	− 0.151179i
$53$	11.2361i	1.54339i	0.635991	+	0.771696i	$0.280591\pi$
$53$	11.2361i	1.54339i	−0.635991	+	0.771696i	$0.719409\pi$
$54$	1.61803	0.220187
$55$	0	0
$56$	−6.70820	−0.896421
$57$	− 5.70820i	− 0.756070i
$58$	1.61803i	0.212458i
$59$	−0.763932	−0.0994555	−0.0497277	−	0.998763i	$-0.515835\pi$
$59$	−0.763932	−0.0994555	−0.0497277	+	0.998763i	$0.515835\pi$
$60$	0	0
$61$	7.70820	0.986934	0.493467	−	0.869764i	$-0.335729\pi$
$61$	7.70820	0.986934	0.493467	+	0.869764i	$0.335729\pi$
$62$	12.9443i	1.64392i
$63$	3.00000i	0.377964i
$64$	−4.23607	−0.529508
$65$	0	0
$66$	−5.61803	−0.691532
$67$	− 2.52786i	− 0.308828i	−0.988006	−	0.154414i	$-0.950651\pi$
$67$	− 2.52786i	− 0.308828i	0.988006	−	0.154414i	$-0.0493489\pi$
$68$	3.38197i	0.410124i
$69$	0	0
$70$	0	0
$71$	2.76393	0.328018	0.164009	−	0.986459i	$-0.447557\pi$
$71$	2.76393	0.328018	0.164009	+	0.986459i	$0.447557\pi$
$72$	2.23607i	0.263523i
$73$	8.00000i	0.936329i	0.883641	+	0.468165i	$0.155085\pi$
$73$	8.00000i	0.936329i	−0.883641	+	0.468165i	$0.844915\pi$
$74$	−12.9443	−1.50474
$75$	0	0
$76$	−3.52786	−0.404674
$77$	− 10.4164i	− 1.18706i
$78$	− 2.85410i	− 0.323163i
$79$	−16.1803	−1.82043	−0.910215	−	0.414136i	$-0.864084\pi$
$79$	−16.1803	−1.82043	−0.910215	+	0.414136i	$0.864084\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 7.23607i	− 0.799090i
$83$	− 9.70820i	− 1.06561i	−0.846237	−	0.532807i	$-0.821137\pi$
$83$	− 9.70820i	− 1.06561i	0.846237	−	0.532807i	$-0.178863\pi$
$84$	1.85410	0.202299
$85$	0	0
$86$	2.00000	0.215666
$87$	1.00000i	0.107211i
$88$	− 7.76393i	− 0.827638i
$89$	11.1803	1.18511	0.592557	−	0.805529i	$-0.298119\pi$
$89$	11.1803	1.18511	0.592557	+	0.805529i	$0.298119\pi$
$90$	0	0
$91$	5.29180	0.554731
$92$	0	0
$93$	8.00000i	0.829561i
$94$	−10.8541	−1.11952
$95$	0	0
$96$	3.38197	0.345170
$97$	7.23607i	0.734711i	0.930081	+	0.367356i	$0.119737\pi$
$97$	7.23607i	0.734711i	−0.930081	+	0.367356i	$0.880263\pi$
$98$	3.23607i	0.326892i
$99$	−3.47214	−0.348963
$100$	0	0
$101$	−0.236068	−0.0234896	−0.0117448	−	0.999931i	$-0.503739\pi$
$101$	−0.236068	−0.0234896	−0.0117448	+	0.999931i	$0.503739\pi$
$102$	8.85410i	0.876687i
$103$	19.4164i	1.91316i	0.291477	+	0.956578i	$0.405853\pi$
$103$	19.4164i	1.91316i	−0.291477	+	0.956578i	$0.594147\pi$
$104$	3.94427	0.386768
$105$	0	0
$106$	18.1803	1.76583
$107$	16.4721i	1.59242i	0.605019	+	0.796211i	$0.293165\pi$
$107$	16.4721i	1.59242i	−0.605019	+	0.796211i	$0.706835\pi$
$108$	− 0.618034i	− 0.0594703i
$109$	−10.4164	−0.997711	−0.498855	−	0.866685i	$-0.666246\pi$
$109$	−10.4164	−0.997711	−0.498855	+	0.866685i	$0.666246\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	14.5623i	1.37601i
$113$	− 9.94427i	− 0.935478i	−0.883867	−	0.467739i	$-0.845069\pi$
$113$	− 9.94427i	− 0.935478i	0.883867	−	0.467739i	$-0.154931\pi$
$114$	−9.23607	−0.865037
$115$	0	0
$116$	0.618034	0.0573830
$117$	− 1.76393i	− 0.163076i
$118$	1.23607i	0.113789i
$119$	−16.4164	−1.50489
$120$	0	0
$121$	1.05573	0.0959753
$122$	− 12.4721i	− 1.12917i
$123$	− 4.47214i	− 0.403239i
$124$	4.94427	0.444009
$125$	0	0
$126$	4.85410	0.432438
$127$	− 6.00000i	− 0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$	− 6.00000i	− 0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$	13.6180i	1.20368i
$129$	1.23607	0.108830
$130$	0	0
$131$	−5.47214	−0.478103	−0.239051	−	0.971007i	$-0.576836\pi$
$131$	−5.47214	−0.478103	−0.239051	+	0.971007i	$0.576836\pi$
$132$	2.14590i	0.186776i
$133$	− 17.1246i	− 1.48489i
$134$	−4.09017	−0.353337
$135$	0	0
$136$	−12.2361	−1.04923
$137$	6.94427i	0.593289i	0.954988	+	0.296645i	$0.0958677\pi$
$137$	6.94427i	0.593289i	−0.954988	+	0.296645i	$0.904132\pi$
$138$	0	0
$139$	−12.7082	−1.07790	−0.538948	−	0.842339i	$-0.681178\pi$
$139$	−12.7082	−1.07790	−0.538948	+	0.842339i	$0.681178\pi$
$140$	0	0
$141$	−6.70820	−0.564933
$142$	− 4.47214i	− 0.375293i
$143$	6.12461i	0.512166i
$144$	4.85410	0.404508
$145$	0	0
$146$	12.9443	1.07128
$147$	2.00000i	0.164957i
$148$	4.94427i	0.406417i
$149$	2.18034	0.178620	0.0893102	−	0.996004i	$-0.471534\pi$
$149$	2.18034	0.178620	0.0893102	+	0.996004i	$0.471534\pi$
$150$	0	0
$151$	−6.47214	−0.526695	−0.263347	−	0.964701i	$-0.584827\pi$
$151$	−6.47214	−0.526695	−0.263347	+	0.964701i	$0.584827\pi$
$152$	− 12.7639i	− 1.03529i
$153$	5.47214i	0.442396i
$154$	−16.8541	−1.35814
$155$	0	0
$156$	−1.09017	−0.0872835
$157$	1.70820i	0.136330i	0.997674	+	0.0681648i	$0.0217144\pi$
$157$	1.70820i	0.136330i	−0.997674	+	0.0681648i	$0.978286\pi$
$158$	26.1803i	2.08280i
$159$	11.2361	0.891078
$160$	0	0
$161$	0	0
$162$	− 1.61803i	− 0.127125i
$163$	− 25.1246i	− 1.96791i	−0.178413	−	0.983956i	$-0.557096\pi$
$163$	− 25.1246i	− 1.96791i	0.178413	−	0.983956i	$-0.442904\pi$
$164$	−2.76393	−0.215827
$165$	0	0
$166$	−15.7082	−1.21919
$167$	17.8885i	1.38426i	0.721774	+	0.692129i	$0.243327\pi$
$167$	17.8885i	1.38426i	−0.721774	+	0.692129i	$0.756673\pi$
$168$	6.70820i	0.517549i
$169$	9.88854	0.760657
$170$	0	0
$171$	−5.70820	−0.436517
$172$	− 0.763932i	− 0.0582493i
$173$	− 7.41641i	− 0.563859i	−0.959435	−	0.281930i	$-0.909026\pi$
$173$	− 7.41641i	− 0.563859i	0.959435	−	0.281930i	$-0.0909744\pi$
$174$	1.61803	0.122663
$175$	0	0
$176$	−16.8541	−1.27043
$177$	0.763932i	0.0574206i
$178$	− 18.0902i	− 1.35592i
$179$	18.1803	1.35886	0.679431	−	0.733739i	$-0.262227\pi$
$179$	18.1803	1.35886	0.679431	+	0.733739i	$0.262227\pi$
$180$	0	0
$181$	−18.4164	−1.36888	−0.684440	−	0.729069i	$-0.739953\pi$
$181$	−18.4164	−1.36888	−0.684440	+	0.729069i	$0.739953\pi$
$182$	− 8.56231i	− 0.634680i
$183$	− 7.70820i	− 0.569807i
$184$	0	0
$185$	0	0
$186$	12.9443	0.949120
$187$	− 19.0000i	− 1.38942i
$188$	4.14590i	0.302371i
$189$	3.00000	0.218218
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	4.23607i	0.305712i
$193$	6.00000i	0.431889i	0.976406	+	0.215945i	$0.0692831\pi$
$193$	6.00000i	0.431889i	−0.976406	+	0.215945i	$0.930717\pi$
$194$	11.7082	0.840600
$195$	0	0
$196$	1.23607	0.0882906
$197$	− 2.94427i	− 0.209771i	−0.994484	−	0.104885i	$-0.966552\pi$
$197$	− 2.94427i	− 0.209771i	0.994484	−	0.104885i	$-0.0334476\pi$
$198$	5.61803i	0.399256i
$199$	−7.29180	−0.516902	−0.258451	−	0.966024i	$-0.583212\pi$
$199$	−7.29180	−0.516902	−0.258451	+	0.966024i	$0.583212\pi$
$200$	0	0
$201$	−2.52786	−0.178302
$202$	0.381966i	0.0268750i
$203$	3.00000i	0.210559i
$204$	3.38197	0.236785
$205$	0	0
$206$	31.4164	2.18888
$207$	0	0
$208$	− 8.56231i	− 0.593689i
$209$	19.8197	1.37095
$210$	0	0
$211$	7.88854	0.543070	0.271535	−	0.962429i	$-0.412469\pi$
$211$	7.88854	0.543070	0.271535	+	0.962429i	$0.412469\pi$
$212$	− 6.94427i	− 0.476935i
$213$	− 2.76393i	− 0.189382i
$214$	26.6525	1.82193
$215$	0	0
$216$	2.23607	0.152145
$217$	24.0000i	1.62923i
$218$	16.8541i	1.14150i
$219$	8.00000	0.540590
$220$	0	0
$221$	9.65248	0.649296
$222$	12.9443i	0.868763i
$223$	13.0000i	0.870544i	0.900299	+	0.435272i	$0.143348\pi$
$223$	13.0000i	0.870544i	−0.900299	+	0.435272i	$0.856652\pi$
$224$	10.1459	0.677901
$225$	0	0
$226$	−16.0902	−1.07030
$227$	− 28.1803i	− 1.87039i	−0.354127	−	0.935197i	$-0.615222\pi$
$227$	− 28.1803i	− 1.87039i	0.354127	−	0.935197i	$-0.384778\pi$
$228$	3.52786i	0.233639i
$229$	17.4164	1.15091	0.575454	−	0.817834i	$-0.304825\pi$
$229$	17.4164	1.15091	0.575454	+	0.817834i	$0.304825\pi$
$230$	0	0
$231$	−10.4164	−0.685349
$232$	2.23607i	0.146805i
$233$	25.4164i	1.66508i	0.553962	+	0.832542i	$0.313115\pi$
$233$	25.4164i	1.66508i	−0.553962	+	0.832542i	$0.686885\pi$
$234$	−2.85410	−0.186578
$235$	0	0
$236$	0.472136	0.0307334
$237$	16.1803i	1.05103i
$238$	26.5623i	1.72178i
$239$	23.8885	1.54522	0.772611	−	0.634880i	$-0.218950\pi$
$239$	23.8885	1.54522	0.772611	+	0.634880i	$0.218950\pi$
$240$	0	0
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	− 1.70820i	− 0.109808i
$243$	− 1.00000i	− 0.0641500i
$244$	−4.76393	−0.304979
$245$	0	0
$246$	−7.23607	−0.461355
$247$	10.0689i	0.640668i
$248$	17.8885i	1.13592i
$249$	−9.70820	−0.615232
$250$	0	0
$251$	10.8885	0.687279	0.343639	−	0.939102i	$-0.388340\pi$
$251$	10.8885	0.687279	0.343639	+	0.939102i	$0.388340\pi$
$252$	− 1.85410i	− 0.116797i
$253$	0	0
$254$	−9.70820	−0.609147
$255$	0	0
$256$	13.5623	0.847644
$257$	3.70820i	0.231311i	0.993289	+	0.115656i	$0.0368969\pi$
$257$	3.70820i	0.231311i	−0.993289	+	0.115656i	$0.963103\pi$
$258$	− 2.00000i	− 0.124515i
$259$	−24.0000	−1.49129
$260$	0	0
$261$	1.00000	0.0618984
$262$	8.85410i	0.547008i
$263$	− 24.0000i	− 1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$	− 24.0000i	− 1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$	−7.76393	−0.477837
$265$	0	0
$266$	−27.7082	−1.69890
$267$	− 11.1803i	− 0.684226i
$268$	1.56231i	0.0954330i
$269$	25.7639	1.57085	0.785427	−	0.618954i	$-0.212443\pi$
$269$	25.7639	1.57085	0.785427	+	0.618954i	$0.212443\pi$
$270$	0	0
$271$	−30.3607	−1.84428	−0.922140	−	0.386856i	$-0.873561\pi$
$271$	−30.3607	−1.84428	−0.922140	+	0.386856i	$0.873561\pi$
$272$	26.5623i	1.61058i
$273$	− 5.29180i	− 0.320274i
$274$	11.2361	0.678796
$275$	0	0
$276$	0	0
$277$	− 4.70820i	− 0.282889i	−0.989946	−	0.141444i	$-0.954825\pi$
$277$	− 4.70820i	− 0.282889i	0.989946	−	0.141444i	$-0.0451746\pi$
$278$	20.5623i	1.23325i
$279$	8.00000	0.478947
$280$	0	0
$281$	17.1246	1.02157	0.510784	−	0.859709i	$-0.329355\pi$
$281$	17.1246	1.02157	0.510784	+	0.859709i	$0.329355\pi$
$282$	10.8541i	0.646352i
$283$	− 9.52786i	− 0.566373i	−0.959065	−	0.283186i	$-0.908609\pi$
$283$	− 9.52786i	− 0.566373i	0.959065	−	0.283186i	$-0.0913915\pi$
$284$	−1.70820	−0.101363
$285$	0	0
$286$	9.90983	0.585981
$287$	− 13.4164i	− 0.791946i
$288$	− 3.38197i	− 0.199284i
$289$	−12.9443	−0.761428
$290$	0	0
$291$	7.23607	0.424186
$292$	− 4.94427i	− 0.289342i
$293$	0.0557281i	0.00325567i	0.999999	+	0.00162783i	$0.000518156\pi$
$293$	0.0557281i	0.00325567i	−0.999999	+	0.00162783i	$0.999482\pi$
$294$	3.23607	0.188731
$295$	0	0
$296$	−17.8885	−1.03975
$297$	3.47214i	0.201474i
$298$	− 3.52786i	− 0.204364i
$299$	0	0
$300$	0	0
$301$	3.70820	0.213737
$302$	10.4721i	0.602604i
$303$	0.236068i	0.0135618i
$304$	−27.7082	−1.58917
$305$	0	0
$306$	8.85410	0.506155
$307$	7.05573i	0.402692i	0.979520	+	0.201346i	$0.0645315\pi$
$307$	7.05573i	0.402692i	−0.979520	+	0.201346i	$0.935468\pi$
$308$	6.43769i	0.366822i
$309$	19.4164	1.10456
$310$	0	0
$311$	14.5279	0.823800	0.411900	−	0.911229i	$-0.364865\pi$
$311$	14.5279	0.823800	0.411900	+	0.911229i	$0.364865\pi$
$312$	− 3.94427i	− 0.223300i
$313$	21.1803i	1.19718i	0.801054	+	0.598592i	$0.204273\pi$
$313$	21.1803i	1.19718i	−0.801054	+	0.598592i	$0.795727\pi$
$314$	2.76393	0.155978
$315$	0	0
$316$	10.0000	0.562544
$317$	− 1.58359i	− 0.0889434i	−0.999011	−	0.0444717i	$-0.985840\pi$
$317$	− 1.58359i	− 0.0889434i	0.999011	−	0.0444717i	$-0.0141605\pi$
$318$	− 18.1803i	− 1.01950i
$319$	−3.47214	−0.194402
$320$	0	0
$321$	16.4721	0.919385
$322$	0	0
$323$	− 31.2361i	− 1.73802i
$324$	−0.618034	−0.0343352
$325$	0	0
$326$	−40.6525	−2.25153
$327$	10.4164i	0.576029i
$328$	− 10.0000i	− 0.552158i
$329$	−20.1246	−1.10951
$330$	0	0
$331$	19.8885	1.09317	0.546587	−	0.837403i	$-0.315927\pi$
$331$	19.8885	1.09317	0.546587	+	0.837403i	$0.315927\pi$
$332$	6.00000i	0.329293i
$333$	8.00000i	0.438397i
$334$	28.9443	1.58376
$335$	0	0
$336$	14.5623	0.794439
$337$	− 11.5279i	− 0.627963i	−0.949429	−	0.313981i	$-0.898337\pi$
$337$	− 11.5279i	− 0.627963i	0.949429	−	0.313981i	$-0.101663\pi$
$338$	− 16.0000i	− 0.870285i
$339$	−9.94427	−0.540099
$340$	0	0
$341$	−27.7771	−1.50421
$342$	9.23607i	0.499429i
$343$	− 15.0000i	− 0.809924i
$344$	2.76393	0.149021
$345$	0	0
$346$	−12.0000	−0.645124
$347$	26.4721i	1.42110i	0.703647	+	0.710549i	$0.251554\pi$
$347$	26.4721i	1.42110i	−0.703647	+	0.710549i	$0.748446\pi$
$348$	− 0.618034i	− 0.0331301i
$349$	1.05573	0.0565118	0.0282559	−	0.999601i	$-0.491005\pi$
$349$	1.05573	0.0565118	0.0282559	+	0.999601i	$0.491005\pi$
$350$	0	0
$351$	−1.76393	−0.0941517
$352$	11.7426i	0.625885i
$353$	− 12.4721i	− 0.663825i	−0.943310	−	0.331912i	$-0.892306\pi$
$353$	− 12.4721i	− 0.663825i	0.943310	−	0.331912i	$-0.107694\pi$
$354$	1.23607	0.0656963
$355$	0	0
$356$	−6.90983	−0.366220
$357$	16.4164i	0.868848i
$358$	− 29.4164i	− 1.55471i
$359$	31.4164	1.65809	0.829047	−	0.559178i	$-0.188883\pi$
$359$	31.4164	1.65809	0.829047	+	0.559178i	$0.188883\pi$
$360$	0	0
$361$	13.5836	0.714926
$362$	29.7984i	1.56617i
$363$	− 1.05573i	− 0.0554114i
$364$	−3.27051	−0.171421
$365$	0	0
$366$	−12.4721	−0.651929
$367$	− 11.4164i	− 0.595932i	−0.954577	−	0.297966i	$-0.903692\pi$
$367$	− 11.4164i	− 0.595932i	0.954577	−	0.297966i	$-0.0963081\pi$
$368$	0	0
$369$	−4.47214	−0.232810
$370$	0	0
$371$	33.7082	1.75004
$372$	− 4.94427i	− 0.256349i
$373$	19.8885i	1.02979i	0.857253	+	0.514895i	$0.172169\pi$
$373$	19.8885i	1.02979i	−0.857253	+	0.514895i	$0.827831\pi$
$374$	−30.7426	−1.58966
$375$	0	0
$376$	−15.0000	−0.773566
$377$	− 1.76393i	− 0.0908471i
$378$	− 4.85410i	− 0.249668i
$379$	10.9443	0.562169	0.281085	−	0.959683i	$-0.409306\pi$
$379$	10.9443	0.562169	0.281085	+	0.959683i	$0.409306\pi$
$380$	0	0
$381$	−6.00000	−0.307389
$382$	− 19.4164i	− 0.993430i
$383$	− 38.0689i	− 1.94523i	−0.232424	−	0.972615i	$-0.574666\pi$
$383$	− 38.0689i	− 1.94523i	0.232424	−	0.972615i	$-0.425334\pi$
$384$	13.6180	0.694942
$385$	0	0
$386$	9.70820	0.494135
$387$	− 1.23607i	− 0.0628329i
$388$	− 4.47214i	− 0.227038i
$389$	−17.7639	−0.900667	−0.450334	−	0.892860i	$-0.648695\pi$
$389$	−17.7639	−0.900667	−0.450334	+	0.892860i	$0.648695\pi$
$390$	0	0
$391$	0	0
$392$	4.47214i	0.225877i
$393$	5.47214i	0.276033i
$394$	−4.76393	−0.240003
$395$	0	0
$396$	2.14590	0.107835
$397$	− 33.4164i	− 1.67712i	−0.544808	−	0.838561i	$-0.683398\pi$
$397$	− 33.4164i	− 1.67712i	0.544808	−	0.838561i	$-0.316602\pi$
$398$	11.7984i	0.591399i
$399$	−17.1246	−0.857303
$400$	0	0
$401$	−5.34752	−0.267043	−0.133521	−	0.991046i	$-0.542628\pi$
$401$	−5.34752	−0.267043	−0.133521	+	0.991046i	$0.542628\pi$
$402$	4.09017i	0.203999i
$403$	− 14.1115i	− 0.702942i
$404$	0.145898	0.00725870
$405$	0	0
$406$	4.85410	0.240905
$407$	− 27.7771i	− 1.37686i
$408$	12.2361i	0.605776i
$409$	22.6525	1.12009	0.560046	−	0.828461i	$-0.310783\pi$
$409$	22.6525	1.12009	0.560046	+	0.828461i	$0.310783\pi$
$410$	0	0
$411$	6.94427	0.342536
$412$	− 12.0000i	− 0.591198i
$413$	2.29180i	0.112772i
$414$	0	0
$415$	0	0
$416$	−5.96556	−0.292486
$417$	12.7082i	0.622323i
$418$	− 32.0689i	− 1.56854i
$419$	10.3607	0.506152	0.253076	−	0.967446i	$-0.418558\pi$
$419$	10.3607	0.506152	0.253076	+	0.967446i	$0.418558\pi$
$420$	0	0
$421$	−24.1803	−1.17848	−0.589239	−	0.807959i	$-0.700572\pi$
$421$	−24.1803	−1.17848	−0.589239	+	0.807959i	$0.700572\pi$
$422$	− 12.7639i	− 0.621338i
$423$	6.70820i	0.326164i
$424$	25.1246	1.22016
$425$	0	0
$426$	−4.47214	−0.216676
$427$	− 23.1246i	− 1.11908i
$428$	− 10.1803i	− 0.492085i
$429$	6.12461	0.295699
$430$	0	0
$431$	25.5279	1.22963	0.614817	−	0.788670i	$-0.289230\pi$
$431$	25.5279	1.22963	0.614817	+	0.788670i	$0.289230\pi$
$432$	− 4.85410i	− 0.233543i
$433$	− 26.6525i	− 1.28084i	−0.768026	−	0.640418i	$-0.778761\pi$
$433$	− 26.6525i	− 1.28084i	0.768026	−	0.640418i	$-0.221239\pi$
$434$	38.8328	1.86403
$435$	0	0
$436$	6.43769	0.308310
$437$	0	0
$438$	− 12.9443i	− 0.618501i
$439$	−30.1246	−1.43777	−0.718885	−	0.695129i	$-0.755347\pi$
$439$	−30.1246	−1.43777	−0.718885	+	0.695129i	$0.755347\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	− 15.6180i	− 0.742874i
$443$	− 28.2361i	− 1.34154i	−0.741667	−	0.670768i	$-0.765965\pi$
$443$	− 28.2361i	− 1.34154i	0.741667	−	0.670768i	$-0.234035\pi$
$444$	4.94427	0.234645
$445$	0	0
$446$	21.0344	0.996010
$447$	− 2.18034i	− 0.103127i
$448$	12.7082i	0.600406i
$449$	−6.23607	−0.294298	−0.147149	−	0.989114i	$-0.547010\pi$
$449$	−6.23607	−0.294298	−0.147149	+	0.989114i	$0.547010\pi$
$450$	0	0
$451$	15.5279	0.731179
$452$	6.14590i	0.289079i
$453$	6.47214i	0.304087i
$454$	−45.5967	−2.13996
$455$	0	0
$456$	−12.7639	−0.597726
$457$	− 34.1246i	− 1.59628i	−0.602471	−	0.798141i	$-0.705817\pi$
$457$	− 34.1246i	− 1.59628i	0.602471	−	0.798141i	$-0.294183\pi$
$458$	− 28.1803i	− 1.31678i
$459$	5.47214	0.255417
$460$	0	0
$461$	−42.3607	−1.97293	−0.986467	−	0.163961i	$-0.947573\pi$
$461$	−42.3607	−1.97293	−0.986467	+	0.163961i	$0.947573\pi$
$462$	16.8541i	0.784124i
$463$	11.4721i	0.533155i	0.963813	+	0.266578i	$0.0858929\pi$
$463$	11.4721i	0.533155i	−0.963813	+	0.266578i	$0.914107\pi$
$464$	4.85410	0.225346
$465$	0	0
$466$	41.1246	1.90506
$467$	4.94427i	0.228794i	0.993435	+	0.114397i	$0.0364935\pi$
$467$	4.94427i	0.228794i	−0.993435	+	0.114397i	$0.963506\pi$
$468$	1.09017i	0.0503931i
$469$	−7.58359	−0.350178
$470$	0	0
$471$	1.70820	0.0787099
$472$	1.70820i	0.0786265i
$473$	4.29180i	0.197337i
$474$	26.1803	1.20250
$475$	0	0
$476$	10.1459	0.465036
$477$	− 11.2361i	− 0.514464i
$478$	− 38.6525i	− 1.76792i
$479$	−2.47214	−0.112955	−0.0564774	−	0.998404i	$-0.517987\pi$
$479$	−2.47214	−0.112955	−0.0564774	+	0.998404i	$0.517987\pi$
$480$	0	0
$481$	14.1115	0.643427
$482$	− 11.3262i	− 0.515896i
$483$	0	0
$484$	−0.652476	−0.0296580
$485$	0	0
$486$	−1.61803	−0.0733955
$487$	− 12.0000i	− 0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$	− 12.0000i	− 0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$	− 17.2361i	− 0.780240i
$489$	−25.1246	−1.13617
$490$	0	0
$491$	−9.88854	−0.446264	−0.223132	−	0.974788i	$-0.571628\pi$
$491$	−9.88854	−0.446264	−0.223132	+	0.974788i	$0.571628\pi$
$492$	2.76393i	0.124608i
$493$	5.47214i	0.246453i
$494$	16.2918	0.733003
$495$	0	0
$496$	38.8328	1.74364
$497$	− 8.29180i	− 0.371938i
$498$	15.7082i	0.703901i
$499$	28.2361	1.26402	0.632010	−	0.774960i	$-0.282230\pi$
$499$	28.2361	1.26402	0.632010	+	0.774960i	$0.282230\pi$
$500$	0	0
$501$	17.8885	0.799201
$502$	− 17.6180i	− 0.786331i
$503$	− 18.5967i	− 0.829188i	−0.910006	−	0.414594i	$-0.863924\pi$
$503$	− 18.5967i	− 0.829188i	0.910006	−	0.414594i	$-0.136076\pi$
$504$	6.70820	0.298807
$505$	0	0
$506$	0	0
$507$	− 9.88854i	− 0.439166i
$508$	3.70820i	0.164525i
$509$	16.1803	0.717181	0.358590	−	0.933495i	$-0.383257\pi$
$509$	16.1803	0.717181	0.358590	+	0.933495i	$0.383257\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	5.29180i	0.233867i
$513$	5.70820i	0.252023i
$514$	6.00000	0.264649
$515$	0	0
$516$	−0.763932	−0.0336302
$517$	− 23.2918i	− 1.02437i
$518$	38.8328i	1.70622i
$519$	−7.41641	−0.325544
$520$	0	0
$521$	−25.2361	−1.10561	−0.552806	−	0.833310i	$-0.686443\pi$
$521$	−25.2361	−1.10561	−0.552806	+	0.833310i	$0.686443\pi$
$522$	− 1.61803i	− 0.0708194i
$523$	− 23.8328i	− 1.04214i	−0.853515	−	0.521068i	$-0.825534\pi$
$523$	− 23.8328i	− 1.04214i	0.853515	−	0.521068i	$-0.174466\pi$
$524$	3.38197	0.147742
$525$	0	0
$526$	−38.8328	−1.69319
$527$	43.7771i	1.90696i
$528$	16.8541i	0.733481i
$529$	23.0000	1.00000
$530$	0	0
$531$	0.763932	0.0331518
$532$	10.5836i	0.458857i
$533$	7.88854i	0.341691i
$534$	−18.0902	−0.782838
$535$	0	0
$536$	−5.65248	−0.244150
$537$	− 18.1803i	− 0.784540i
$538$	− 41.6869i	− 1.79725i
$539$	−6.94427	−0.299111
$540$	0	0
$541$	−6.36068	−0.273467	−0.136733	−	0.990608i	$-0.543660\pi$
$541$	−6.36068	−0.273467	−0.136733	+	0.990608i	$0.543660\pi$
$542$	49.1246i	2.11008i
$543$	18.4164i	0.790324i
$544$	18.5066	0.793463
$545$	0	0
$546$	−8.56231	−0.366433
$547$	18.5279i	0.792194i	0.918209	+	0.396097i	$0.129636\pi$
$547$	18.5279i	0.792194i	−0.918209	+	0.396097i	$0.870364\pi$
$548$	− 4.29180i	− 0.183336i
$549$	−7.70820	−0.328978
$550$	0	0
$551$	−5.70820	−0.243178
$552$	0	0
$553$	48.5410i	2.06417i
$554$	−7.61803	−0.323659
$555$	0	0
$556$	7.85410	0.333088
$557$	7.23607i	0.306602i	0.988180	+	0.153301i	$0.0489904\pi$
$557$	7.23607i	0.306602i	−0.988180	+	0.153301i	$0.951010\pi$
$558$	− 12.9443i	− 0.547975i
$559$	−2.18034	−0.0922186
$560$	0	0
$561$	−19.0000	−0.802181
$562$	− 27.7082i	− 1.16880i
$563$	− 3.87539i	− 0.163328i	−0.996660	−	0.0816641i	$-0.973977\pi$
$563$	− 3.87539i	− 0.163328i	0.996660	−	0.0816641i	$-0.0260235\pi$
$564$	4.14590	0.174574
$565$	0	0
$566$	−15.4164	−0.648000
$567$	− 3.00000i	− 0.125988i
$568$	− 6.18034i	− 0.259321i
$569$	−17.1803	−0.720237	−0.360119	−	0.932907i	$-0.617264\pi$
$569$	−17.1803	−0.720237	−0.360119	+	0.932907i	$0.617264\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	− 3.78522i	− 0.158268i
$573$	− 12.0000i	− 0.501307i
$574$	−21.7082	−0.906083
$575$	0	0
$576$	4.23607	0.176503
$577$	33.3050i	1.38650i	0.720696	+	0.693252i	$0.243823\pi$
$577$	33.3050i	1.38650i	−0.720696	+	0.693252i	$0.756177\pi$
$578$	20.9443i	0.871167i
$579$	6.00000	0.249351
$580$	0	0
$581$	−29.1246	−1.20829
$582$	− 11.7082i	− 0.485321i
$583$	39.0132i	1.61576i
$584$	17.8885	0.740233
$585$	0	0
$586$	0.0901699	0.00372489
$587$	− 24.1803i	− 0.998029i	−0.866594	−	0.499015i	$-0.833695\pi$
$587$	− 24.1803i	− 0.998029i	0.866594	−	0.499015i	$-0.166305\pi$
$588$	− 1.23607i	− 0.0509746i
$589$	−45.6656	−1.88162
$590$	0	0
$591$	−2.94427	−0.121111
$592$	38.8328i	1.59602i
$593$	− 6.65248i	− 0.273184i	−0.990627	−	0.136592i	$-0.956385\pi$
$593$	− 6.65248i	− 0.273184i	0.990627	−	0.136592i	$-0.0436150\pi$
$594$	5.61803	0.230511
$595$	0	0
$596$	−1.34752	−0.0551967
$597$	7.29180i	0.298433i
$598$	0	0
$599$	−26.8885	−1.09864	−0.549318	−	0.835613i	$-0.685112\pi$
$599$	−26.8885	−1.09864	−0.549318	+	0.835613i	$0.685112\pi$
$600$	0	0
$601$	43.8885	1.79025	0.895126	−	0.445814i	$-0.147086\pi$
$601$	43.8885	1.79025	0.895126	+	0.445814i	$0.147086\pi$
$602$	− 6.00000i	− 0.244542i
$603$	2.52786i	0.102943i
$604$	4.00000	0.162758
$605$	0	0
$606$	0.381966	0.0155163
$607$	9.34752i	0.379404i	0.981842	+	0.189702i	$0.0607522\pi$
$607$	9.34752i	0.379404i	−0.981842	+	0.189702i	$0.939248\pi$
$608$	19.3050i	0.782919i
$609$	3.00000	0.121566
$610$	0	0
$611$	11.8328	0.478704
$612$	− 3.38197i	− 0.136708i
$613$	− 20.7082i	− 0.836396i	−0.908356	−	0.418198i	$-0.862662\pi$
$613$	− 20.7082i	− 0.836396i	0.908356	−	0.418198i	$-0.137338\pi$
$614$	11.4164	0.460729
$615$	0	0
$616$	−23.2918	−0.938453
$617$	− 19.3050i	− 0.777188i	−0.921409	−	0.388594i	$-0.872961\pi$
$617$	− 19.3050i	− 0.777188i	0.921409	−	0.388594i	$-0.127039\pi$
$618$	− 31.4164i	− 1.26375i
$619$	12.4721	0.501297	0.250649	−	0.968078i	$-0.419356\pi$
$619$	12.4721	0.501297	0.250649	+	0.968078i	$0.419356\pi$
$620$	0	0
$621$	0	0
$622$	− 23.5066i	− 0.942528i
$623$	− 33.5410i	− 1.34379i
$624$	−8.56231	−0.342767
$625$	0	0
$626$	34.2705	1.36973
$627$	− 19.8197i	− 0.791521i
$628$	− 1.05573i	− 0.0421281i
$629$	−43.7771	−1.74551
$630$	0	0
$631$	46.4853	1.85055	0.925275	−	0.379297i	$-0.123834\pi$
$631$	46.4853	1.85055	0.925275	+	0.379297i	$0.123834\pi$
$632$	36.1803i	1.43918i
$633$	− 7.88854i	− 0.313541i
$634$	−2.56231	−0.101762
$635$	0	0
$636$	−6.94427	−0.275358
$637$	− 3.52786i	− 0.139779i
$638$	5.61803i	0.222420i
$639$	−2.76393	−0.109339
$640$	0	0
$641$	41.7639	1.64958	0.824788	−	0.565442i	$-0.191294\pi$
$641$	41.7639	1.64958	0.824788	+	0.565442i	$0.191294\pi$
$642$	− 26.6525i	− 1.05189i
$643$	− 17.8328i	− 0.703258i	−0.936140	−	0.351629i	$-0.885628\pi$
$643$	− 17.8328i	− 0.703258i	0.936140	−	0.351629i	$-0.114372\pi$
$644$	0	0
$645$	0	0
$646$	−50.5410	−1.98851
$647$	6.76393i	0.265918i	0.991122	+	0.132959i	$0.0424478\pi$
$647$	6.76393i	0.265918i	−0.991122	+	0.132959i	$0.957552\pi$
$648$	− 2.23607i	− 0.0878410i
$649$	−2.65248	−0.104119
$650$	0	0
$651$	24.0000	0.940634
$652$	15.5279i	0.608118i
$653$	26.8885i	1.05223i	0.850413	+	0.526115i	$0.176352\pi$
$653$	26.8885i	1.05223i	−0.850413	+	0.526115i	$0.823648\pi$
$654$	16.8541	0.659048
$655$	0	0
$656$	−21.7082	−0.847563
$657$	− 8.00000i	− 0.312110i
$658$	32.5623i	1.26941i
$659$	21.0000	0.818044	0.409022	−	0.912525i	$-0.365870\pi$
$659$	21.0000	0.818044	0.409022	+	0.912525i	$0.365870\pi$
$660$	0	0
$661$	15.8328	0.615825	0.307913	−	0.951415i	$-0.400370\pi$
$661$	15.8328	0.615825	0.307913	+	0.951415i	$0.400370\pi$
$662$	− 32.1803i	− 1.25072i
$663$	− 9.65248i	− 0.374871i
$664$	−21.7082	−0.842442
$665$	0	0
$666$	12.9443	0.501580
$667$	0	0
$668$	− 11.0557i	− 0.427759i
$669$	13.0000	0.502609
$670$	0	0
$671$	26.7639	1.03321
$672$	− 10.1459i	− 0.391387i
$673$	46.7082i	1.80047i	0.435405	+	0.900234i	$0.356605\pi$
$673$	46.7082i	1.80047i	−0.435405	+	0.900234i	$0.643395\pi$
$674$	−18.6525	−0.718467
$675$	0	0
$676$	−6.11146	−0.235056
$677$	14.8885i	0.572213i	0.958198	+	0.286107i	$0.0923612\pi$
$677$	14.8885i	0.572213i	−0.958198	+	0.286107i	$0.907639\pi$
$678$	16.0902i	0.617939i
$679$	21.7082	0.833084
$680$	0	0
$681$	−28.1803	−1.07987
$682$	44.9443i	1.72101i
$683$	19.4164i	0.742948i	0.928443	+	0.371474i	$0.121148\pi$
$683$	19.4164i	0.742948i	−0.928443	+	0.371474i	$0.878852\pi$
$684$	3.52786	0.134891
$685$	0	0
$686$	−24.2705	−0.926652
$687$	− 17.4164i	− 0.664477i
$688$	− 6.00000i	− 0.228748i
$689$	−19.8197	−0.755069
$690$	0	0
$691$	−21.2918	−0.809978	−0.404989	−	0.914322i	$-0.632725\pi$
$691$	−21.2918	−0.809978	−0.404989	+	0.914322i	$0.632725\pi$
$692$	4.58359i	0.174242i
$693$	10.4164i	0.395687i
$694$	42.8328	1.62591
$695$	0	0
$696$	2.23607	0.0847579
$697$	− 24.4721i	− 0.926948i
$698$	− 1.70820i	− 0.0646565i
$699$	25.4164	0.961337
$700$	0	0
$701$	35.1246	1.32664	0.663319	−	0.748337i	$-0.269147\pi$
$701$	35.1246	1.32664	0.663319	+	0.748337i	$0.269147\pi$
$702$	2.85410i	0.107721i
$703$	− 45.6656i	− 1.72231i
$704$	−14.7082	−0.554336
$705$	0	0
$706$	−20.1803	−0.759497
$707$	0.708204i	0.0266348i
$708$	− 0.472136i	− 0.0177440i
$709$	15.8885	0.596707	0.298353	−	0.954455i	$-0.403563\pi$
$709$	15.8885	0.596707	0.298353	+	0.954455i	$0.403563\pi$
$710$	0	0
$711$	16.1803	0.606810
$712$	− 25.0000i	− 0.936915i
$713$	0	0
$714$	26.5623	0.994069
$715$	0	0
$716$	−11.2361	−0.419912
$717$	− 23.8885i	− 0.892134i
$718$	− 50.8328i	− 1.89706i
$719$	6.76393	0.252252	0.126126	−	0.992014i	$-0.459746\pi$
$719$	6.76393	0.252252	0.126126	+	0.992014i	$0.459746\pi$
$720$	0	0
$721$	58.2492	2.16931
$722$	− 21.9787i	− 0.817963i
$723$	− 7.00000i	− 0.260333i
$724$	11.3820	0.423007
$725$	0	0
$726$	−1.70820	−0.0633974
$727$	37.8885i	1.40521i	0.711581	+	0.702604i	$0.247980\pi$
$727$	37.8885i	1.40521i	−0.711581	+	0.702604i	$0.752020\pi$
$728$	− 11.8328i	− 0.438553i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	6.76393	0.250173
$732$	4.76393i	0.176080i
$733$	27.7082i	1.02343i	0.859156	+	0.511713i	$0.170989\pi$
$733$	27.7082i	1.02343i	−0.859156	+	0.511713i	$0.829011\pi$
$734$	−18.4721	−0.681819
$735$	0	0
$736$	0	0
$737$	− 8.77709i	− 0.323308i
$738$	7.23607i	0.266363i
$739$	43.4853	1.59963	0.799816	−	0.600245i	$-0.204930\pi$
$739$	43.4853	1.59963	0.799816	+	0.600245i	$0.204930\pi$
$740$	0	0
$741$	10.0689	0.369890
$742$	− 54.5410i	− 2.00226i
$743$	33.1803i	1.21727i	0.793451	+	0.608634i	$0.208282\pi$
$743$	33.1803i	1.21727i	−0.793451	+	0.608634i	$0.791718\pi$
$744$	17.8885	0.655826
$745$	0	0
$746$	32.1803	1.17821
$747$	9.70820i	0.355205i
$748$	11.7426i	0.429354i
$749$	49.4164	1.80564
$750$	0	0
$751$	16.5836	0.605144	0.302572	−	0.953127i	$-0.402155\pi$
$751$	16.5836	0.605144	0.302572	+	0.953127i	$0.402155\pi$
$752$	32.5623i	1.18743i
$753$	− 10.8885i	− 0.396801i
$754$	−2.85410	−0.103940
$755$	0	0
$756$	−1.85410	−0.0674330
$757$	18.8328i	0.684490i	0.939611	+	0.342245i	$0.111187\pi$
$757$	18.8328i	0.684490i	−0.939611	+	0.342245i	$0.888813\pi$
$758$	− 17.7082i	− 0.643191i
$759$	0	0
$760$	0	0
$761$	−43.0132	−1.55923	−0.779613	−	0.626262i	$-0.784584\pi$
$761$	−43.0132	−1.55923	−0.779613	+	0.626262i	$0.784584\pi$
$762$	9.70820i	0.351691i
$763$	31.2492i	1.13130i
$764$	−7.41641	−0.268316
$765$	0	0
$766$	−61.5967	−2.22558
$767$	− 1.34752i	− 0.0486563i
$768$	− 13.5623i	− 0.489388i
$769$	−3.34752	−0.120715	−0.0603574	−	0.998177i	$-0.519224\pi$
$769$	−3.34752	−0.120715	−0.0603574	+	0.998177i	$0.519224\pi$
$770$	0	0
$771$	3.70820	0.133548
$772$	− 3.70820i	− 0.133461i
$773$	− 28.2492i	− 1.01605i	−0.861341	−	0.508027i	$-0.830375\pi$
$773$	− 28.2492i	− 1.01605i	0.861341	−	0.508027i	$-0.169625\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	16.1803	0.580840
$777$	24.0000i	0.860995i
$778$	28.7426i	1.03047i
$779$	25.5279	0.914631
$780$	0	0
$781$	9.59675	0.343399
$782$	0	0
$783$	− 1.00000i	− 0.0357371i
$784$	9.70820	0.346722
$785$	0	0
$786$	8.85410	0.315815
$787$	− 33.8885i	− 1.20800i	−0.796986	−	0.603998i	$-0.793573\pi$
$787$	− 33.8885i	− 1.20800i	0.796986	−	0.603998i	$-0.206427\pi$
$788$	1.81966i	0.0648227i
$789$	−24.0000	−0.854423
$790$	0	0
$791$	−29.8328	−1.06073
$792$	7.76393i	0.275879i
$793$	13.5967i	0.482835i
$794$	−54.0689	−1.91883
$795$	0	0
$796$	4.50658	0.159731
$797$	− 10.9443i	− 0.387666i	−0.981035	−	0.193833i	$-0.937908\pi$
$797$	− 10.9443i	− 0.387666i	0.981035	−	0.193833i	$-0.0620920\pi$
$798$	27.7082i	0.980860i
$799$	−36.7082	−1.29864
$800$	0	0
$801$	−11.1803	−0.395038
$802$	8.65248i	0.305530i
$803$	27.7771i	0.980232i
$804$	1.56231	0.0550983
$805$	0	0
$806$	−22.8328	−0.804252
$807$	− 25.7639i	− 0.906933i
$808$	0.527864i	0.0185702i
$809$	−39.7639	−1.39803	−0.699013	−	0.715109i	$-0.746377\pi$
$809$	−39.7639	−1.39803	−0.699013	+	0.715109i	$0.746377\pi$
$810$	0	0
$811$	−9.29180	−0.326279	−0.163140	−	0.986603i	$-0.552162\pi$
$811$	−9.29180	−0.326279	−0.163140	+	0.986603i	$0.552162\pi$
$812$	− 1.85410i	− 0.0650662i
$813$	30.3607i	1.06480i
$814$	−44.9443	−1.57530
$815$	0	0
$816$	26.5623	0.929867
$817$	7.05573i	0.246849i
$818$	− 36.6525i	− 1.28152i
$819$	−5.29180	−0.184910
$820$	0	0
$821$	−37.4164	−1.30584	−0.652921	−	0.757426i	$-0.726457\pi$
$821$	−37.4164	−1.30584	−0.652921	+	0.757426i	$0.726457\pi$
$822$	− 11.2361i	− 0.391903i
$823$	33.2361i	1.15854i	0.815137	+	0.579268i	$0.196662\pi$
$823$	33.2361i	1.15854i	−0.815137	+	0.579268i	$0.803338\pi$
$824$	43.4164	1.51248
$825$	0	0
$826$	3.70820	0.129025
$827$	19.4164i	0.675175i	0.941294	+	0.337587i	$0.109611\pi$
$827$	19.4164i	0.675175i	−0.941294	+	0.337587i	$0.890389\pi$
$828$	0	0
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	0	0
$831$	−4.70820	−0.163326
$832$	− 7.47214i	− 0.259050i
$833$	10.9443i	0.379197i
$834$	20.5623	0.712014
$835$	0	0
$836$	−12.2492	−0.423648
$837$	− 8.00000i	− 0.276520i
$838$	− 16.7639i	− 0.579100i
$839$	20.8885	0.721153	0.360576	−	0.932730i	$-0.382580\pi$
$839$	20.8885	0.721153	0.360576	+	0.932730i	$0.382580\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	39.1246i	1.34832i
$843$	− 17.1246i	− 0.589803i
$844$	−4.87539	−0.167818
$845$	0	0
$846$	10.8541	0.373172
$847$	− 3.16718i	− 0.108826i
$848$	− 54.5410i	− 1.87295i
$849$	−9.52786	−0.326995
$850$	0	0
$851$	0	0
$852$	1.70820i	0.0585221i
$853$	33.2361i	1.13798i	0.822344	+	0.568991i	$0.192666\pi$
$853$	33.2361i	1.13798i	−0.822344	+	0.568991i	$0.807334\pi$
$854$	−37.4164	−1.28036
$855$	0	0
$856$	36.8328	1.25892
$857$	11.8885i	0.406105i	0.979168	+	0.203052i	$0.0650862\pi$
$857$	11.8885i	0.406105i	−0.979168	+	0.203052i	$0.934914\pi$
$858$	− 9.90983i	− 0.338316i
$859$	2.29180	0.0781951	0.0390975	−	0.999235i	$-0.487552\pi$
$859$	2.29180	0.0781951	0.0390975	+	0.999235i	$0.487552\pi$
$860$	0	0
$861$	−13.4164	−0.457230
$862$	− 41.3050i	− 1.40685i
$863$	33.5967i	1.14365i	0.820377	+	0.571823i	$0.193764\pi$
$863$	33.5967i	1.14365i	−0.820377	+	0.571823i	$0.806236\pi$
$864$	−3.38197	−0.115057
$865$	0	0
$866$	−43.1246	−1.46543
$867$	12.9443i	0.439611i
$868$	− 14.8328i	− 0.503459i
$869$	−56.1803	−1.90579
$870$	0	0
$871$	4.45898	0.151087
$872$	23.2918i	0.788760i
$873$	− 7.23607i	− 0.244904i
$874$	0	0
$875$	0	0
$876$	−4.94427	−0.167051
$877$	− 4.83282i	− 0.163193i	−0.996665	−	0.0815963i	$-0.973998\pi$
$877$	− 4.83282i	− 0.163193i	0.996665	−	0.0815963i	$-0.0260018\pi$
$878$	48.7426i	1.64498i
$879$	0.0557281	0.00187966
$880$	0	0
$881$	−0.708204	−0.0238600	−0.0119300	−	0.999929i	$-0.503798\pi$
$881$	−0.708204	−0.0238600	−0.0119300	+	0.999929i	$0.503798\pi$
$882$	− 3.23607i	− 0.108964i
$883$	32.0000i	1.07689i	0.842662	+	0.538443i	$0.180987\pi$
$883$	32.0000i	1.07689i	−0.842662	+	0.538443i	$0.819013\pi$
$884$	−5.96556	−0.200643
$885$	0	0
$886$	−45.6869	−1.53488
$887$	4.59675i	0.154344i	0.997018	+	0.0771718i	$0.0245890\pi$
$887$	4.59675i	0.154344i	−0.997018	+	0.0771718i	$0.975411\pi$
$888$	17.8885i	0.600300i
$889$	−18.0000	−0.603701
$890$	0	0
$891$	3.47214	0.116321
$892$	− 8.03444i	− 0.269013i
$893$	− 38.2918i	− 1.28139i
$894$	−3.52786	−0.117989
$895$	0	0
$896$	40.8541	1.36484
$897$	0	0
$898$	10.0902i	0.336713i
$899$	8.00000	0.266815
$900$	0	0
$901$	61.4853	2.04837
$902$	− 25.1246i	− 0.836558i
$903$	− 3.70820i	− 0.123401i
$904$	−22.2361	−0.739561
$905$	0	0
$906$	10.4721	0.347913
$907$	31.2361i	1.03718i	0.855024	+	0.518588i	$0.173542\pi$
$907$	31.2361i	1.03718i	−0.855024	+	0.518588i	$0.826458\pi$
$908$	17.4164i	0.577984i
$909$	0.236068	0.00782988
$910$	0	0
$911$	−35.9443	−1.19089	−0.595443	−	0.803397i	$-0.703024\pi$
$911$	−35.9443	−1.19089	−0.595443	+	0.803397i	$0.703024\pi$
$912$	27.7082i	0.917510i
$913$	− 33.7082i	− 1.11558i
$914$	−55.2148	−1.82634
$915$	0	0
$916$	−10.7639	−0.355650
$917$	16.4164i	0.542118i
$918$	− 8.85410i	− 0.292229i
$919$	20.1246	0.663850	0.331925	−	0.943306i	$-0.392302\pi$
$919$	20.1246	0.663850	0.331925	+	0.943306i	$0.392302\pi$
$920$	0	0
$921$	7.05573	0.232494
$922$	68.5410i	2.25728i
$923$	4.87539i	0.160475i
$924$	6.43769	0.211785
$925$	0	0
$926$	18.5623	0.609995
$927$	− 19.4164i	− 0.637719i
$928$	− 3.38197i	− 0.111018i
$929$	3.16718	0.103912	0.0519560	−	0.998649i	$-0.483454\pi$
$929$	3.16718	0.103912	0.0519560	+	0.998649i	$0.483454\pi$
$930$	0	0
$931$	−11.4164	−0.374158
$932$	− 15.7082i	− 0.514539i
$933$	− 14.5279i	− 0.475621i
$934$	8.00000	0.261768
$935$	0	0
$936$	−3.94427	−0.128923
$937$	24.2361i	0.791758i	0.918303	+	0.395879i	$0.129560\pi$
$937$	24.2361i	0.791758i	−0.918303	+	0.395879i	$0.870440\pi$
$938$	12.2705i	0.400646i
$939$	21.1803	0.691194
$940$	0	0
$941$	12.0689	0.393434	0.196717	−	0.980460i	$-0.436972\pi$
$941$	12.0689	0.393434	0.196717	+	0.980460i	$0.436972\pi$
$942$	− 2.76393i	− 0.0900538i
$943$	0	0
$944$	3.70820	0.120692
$945$	0	0
$946$	6.94427	0.225778
$947$	0.708204i	0.0230135i	0.999934	+	0.0115068i	$0.00366280\pi$
$947$	0.708204i	0.0230135i	−0.999934	+	0.0115068i	$0.996337\pi$
$948$	− 10.0000i	− 0.324785i
$949$	−14.1115	−0.458077
$950$	0	0
$951$	−1.58359	−0.0513515
$952$	36.7082i	1.18972i
$953$	57.0132i	1.84684i	0.383794	+	0.923419i	$0.374617\pi$
$953$	57.0132i	1.84684i	−0.383794	+	0.923419i	$0.625383\pi$
$954$	−18.1803	−0.588610
$955$	0	0
$956$	−14.7639	−0.477500
$957$	3.47214i	0.112238i
$958$	4.00000i	0.129234i
$959$	20.8328	0.672727
$960$	0	0
$961$	33.0000	1.06452
$962$	− 22.8328i	− 0.736160i
$963$	− 16.4721i	− 0.530807i
$964$	−4.32624	−0.139339
$965$	0	0
$966$	0	0
$967$	5.12461i	0.164796i	0.996599	+	0.0823982i	$0.0262579\pi$
$967$	5.12461i	0.164796i	−0.996599	+	0.0823982i	$0.973742\pi$
$968$	− 2.36068i	− 0.0758751i
$969$	−31.2361	−1.00345
$970$	0	0
$971$	−40.9443	−1.31396	−0.656982	−	0.753906i	$-0.728167\pi$
$971$	−40.9443	−1.31396	−0.656982	+	0.753906i	$0.728167\pi$
$972$	0.618034i	0.0198234i
$973$	38.1246i	1.22222i
$974$	−19.4164	−0.622142
$975$	0	0
$976$	−37.4164	−1.19767
$977$	− 13.5279i	− 0.432795i	−0.976305	−	0.216397i	$-0.930569\pi$
$977$	− 13.5279i	− 0.432795i	0.976305	−	0.216397i	$-0.0694307\pi$
$978$	40.6525i	1.29992i
$979$	38.8197	1.24068
$980$	0	0
$981$	10.4164	0.332570
$982$	16.0000i	0.510581i
$983$	− 46.4721i	− 1.48223i	−0.671378	−	0.741115i	$-0.734297\pi$
$983$	− 46.4721i	− 1.48223i	0.671378	−	0.741115i	$-0.265703\pi$
$984$	−10.0000	−0.318788
$985$	0	0
$986$	8.85410	0.281972
$987$	20.1246i	0.640573i
$988$	− 6.22291i	− 0.197977i
$989$	0	0
$990$	0	0
$991$	−62.4853	−1.98491	−0.992455	−	0.122607i	$-0.960875\pi$
$991$	−62.4853	−1.98491	−0.992455	+	0.122607i	$0.960875\pi$
$992$	− 27.0557i	− 0.859020i
$993$	− 19.8885i	− 0.631144i
$994$	−13.4164	−0.425543
$995$	0	0
$996$	6.00000	0.190117
$997$	− 19.4164i	− 0.614924i	−0.951560	−	0.307462i	$-0.900520\pi$
$997$	− 19.4164i	− 0.614924i	0.951560	−	0.307462i	$-0.0994797\pi$
$998$	− 45.6869i	− 1.44619i
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.c.j.349.1		4
5.2	odd	4		2175.2.a.q.1.2		2
5.3	odd	4		435.2.a.e.1.1	✓	2
5.4	even	2	inner	2175.2.c.j.349.4		4
15.2	even	4		6525.2.a.s.1.1		2
15.8	even	4		1305.2.a.k.1.2		2
20.3	even	4		6960.2.a.bu.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.a.e.1.1	✓	2	5.3	odd	4
1305.2.a.k.1.2		2	15.8	even	4
2175.2.a.q.1.2		2	5.2	odd	4
2175.2.c.j.349.1		4	1.1	even	1	trivial
2175.2.c.j.349.4		4	5.4	even	2	inner
6525.2.a.s.1.1		2	15.2	even	4
6960.2.a.bu.1.1		2	20.3	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.61803i q^{2} -1.00000i q^{3} -0.618034 q^{4} -1.61803 q^{6} -3.00000i q^{7} -2.23607i q^{8} -1.00000 q^{9} +3.47214 q^{11} +0.618034i q^{12} +1.76393i q^{13} -4.85410 q^{14} -4.85410 q^{16} -5.47214i q^{17} +1.61803i q^{18} +5.70820 q^{19} -3.00000 q^{21} -5.61803i q^{22} -2.23607 q^{24} +2.85410 q^{26} +1.00000i q^{27} +1.85410i q^{28} -1.00000 q^{29} -8.00000 q^{31} +3.38197i q^{32} -3.47214i q^{33} -8.85410 q^{34} +0.618034 q^{36} -8.00000i q^{37} -9.23607i q^{38} +1.76393 q^{39} +4.47214 q^{41} +4.85410i q^{42} +1.23607i q^{43} -2.14590 q^{44} -6.70820i q^{47} +4.85410i q^{48} -2.00000 q^{49} -5.47214 q^{51} -1.09017i q^{52} +11.2361i q^{53} +1.61803 q^{54} -6.70820 q^{56} -5.70820i q^{57} +1.61803i q^{58} -0.763932 q^{59} +7.70820 q^{61} +12.9443i q^{62} +3.00000i q^{63} -4.23607 q^{64} -5.61803 q^{66} -2.52786i q^{67} +3.38197i q^{68} +2.76393 q^{71} +2.23607i q^{72} +8.00000i q^{73} -12.9443 q^{74} -3.52786 q^{76} -10.4164i q^{77} -2.85410i q^{78} -16.1803 q^{79} +1.00000 q^{81} -7.23607i q^{82} -9.70820i q^{83} +1.85410 q^{84} +2.00000 q^{86} +1.00000i q^{87} -7.76393i q^{88} +11.1803 q^{89} +5.29180 q^{91} +8.00000i q^{93} -10.8541 q^{94} +3.38197 q^{96} +7.23607i q^{97} +3.23607i q^{98} -3.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 2 q^{6} - 4 q^{9} - 4 q^{11} - 6 q^{14} - 6 q^{16} - 4 q^{19} - 12 q^{21} - 2 q^{26} - 4 q^{29} - 32 q^{31} - 22 q^{34} - 2 q^{36} + 16 q^{39} - 22 q^{44} - 8 q^{49} - 4 q^{51} + 2 q^{54}+ \cdots + 4 q^{99}+O(q^{100}) \) 4 * q + 2 * q^4 - 2 * q^6 - 4 * q^9 - 4 * q^11 - 6 * q^14 - 6 * q^16 - 4 * q^19 - 12 * q^21 - 2 * q^26 - 4 * q^29 - 32 * q^31 - 22 * q^34 - 2 * q^36 + 16 * q^39 - 22 * q^44 - 8 * q^49 - 4 * q^51 + 2 * q^54 - 12 * q^59 + 4 * q^61 - 8 * q^64 - 18 * q^66 + 20 * q^71 - 16 * q^74 - 32 * q^76 - 20 * q^79 + 4 * q^81 - 6 * q^84 + 8 * q^86 + 48 * q^91 - 30 * q^94 + 18 * q^96 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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