LMFDB - Embedded newform 2175.2.c.f.349.3

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,-14,0,-2,0,0,-4,0,20] 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:	$\Q(i, \sqrt{21})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 11x^{2} + 25 $ x^4 + 11*x^2 + 25
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.3
Root			$1.79129i$ of defining polynomial
Character	$\chi$	$=$	2175.349
Dual form			2175.2.c.f.349.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.79129i q^{2} -1.00000i q^{3} -1.20871 q^{4} +1.79129 q^{6} +1.00000i q^{7} +1.41742i q^{8} -1.00000 q^{9} +5.00000 q^{11} +1.20871i q^{12} +4.58258i q^{13} -1.79129 q^{14} -4.95644 q^{16} -3.00000i q^{17} -1.79129i q^{18} -3.58258 q^{19} +1.00000 q^{21} +8.95644i q^{22} +4.00000i q^{23} +1.41742 q^{24} -8.20871 q^{26} +1.00000i q^{27} -1.20871i q^{28} -1.00000 q^{29} +4.00000 q^{31} -6.04356i q^{32} -5.00000i q^{33} +5.37386 q^{34} +1.20871 q^{36} -4.00000i q^{37} -6.41742i q^{38} +4.58258 q^{39} -9.16515 q^{41} +1.79129i q^{42} +9.58258i q^{43} -6.04356 q^{44} -7.16515 q^{46} +10.5826i q^{47} +4.95644i q^{48} +6.00000 q^{49} -3.00000 q^{51} -5.53901i q^{52} -0.417424i q^{53} -1.79129 q^{54} -1.41742 q^{56} +3.58258i q^{57} -1.79129i q^{58} +7.58258 q^{59} +12.7477 q^{61} +7.16515i q^{62} -1.00000i q^{63} +0.912878 q^{64} +8.95644 q^{66} -4.16515i q^{67} +3.62614i q^{68} +4.00000 q^{69} -9.58258 q^{71} -1.41742i q^{72} -4.00000i q^{73} +7.16515 q^{74} +4.33030 q^{76} +5.00000i q^{77} +8.20871i q^{78} -7.58258 q^{79} +1.00000 q^{81} -16.4174i q^{82} +11.5826i q^{83} -1.20871 q^{84} -17.1652 q^{86} +1.00000i q^{87} +7.08712i q^{88} -1.41742 q^{89} -4.58258 q^{91} -4.83485i q^{92} -4.00000i q^{93} -18.9564 q^{94} -6.04356 q^{96} +11.5826i q^{97} +10.7477i q^{98} -5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 14 q^{4} - 2 q^{6} - 4 q^{9} + 20 q^{11} + 2 q^{14} + 26 q^{16} + 4 q^{19} + 4 q^{21} + 24 q^{24} - 42 q^{26} - 4 q^{29} + 16 q^{31} - 6 q^{34} + 14 q^{36} - 70 q^{44} + 8 q^{46} + 24 q^{49} - 12 q^{51}+ \cdots - 20 q^{99}+O(q^{100}) $ 4 * q - 14 * q^4 - 2 * q^6 - 4 * q^9 + 20 * q^11 + 2 * q^14 + 26 * q^16 + 4 * q^19 + 4 * q^21 + 24 * q^24 - 42 * q^26 - 4 * q^29 + 16 * q^31 - 6 * q^34 + 14 * q^36 - 70 * q^44 + 8 * q^46 + 24 * q^49 - 12 * q^51 + 2 * q^54 - 24 * q^56 + 12 * q^59 - 4 * q^61 - 88 * q^64 - 10 * q^66 + 16 * q^69 - 20 * q^71 - 8 * q^74 - 56 * q^76 - 12 * q^79 + 4 * q^81 - 14 * q^84 - 32 * q^86 - 24 * q^89 - 30 * q^94 - 70 * q^96 - 20 * 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 $

$2$	1.79129i	1.26663i	0.773893	+	0.633316i	$0.218307\pi$
$2$	1.79129i	1.26663i	−0.773893	+	0.633316i	$0.781693\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−1.20871	−0.604356
$5$	0	0
$5$	0	0
$6$	1.79129	0.731290
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	1.41742i	0.501135i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	1.20871i	0.348925i
$13$	4.58258i	1.27098i	0.772110	+	0.635489i	$0.219201\pi$
$13$	4.58258i	1.27098i	−0.772110	+	0.635489i	$0.780799\pi$
$14$	−1.79129	−0.478742
$15$	0	0
$16$	−4.95644	−1.23911
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	− 1.79129i	− 0.422211i
$19$	−3.58258	−0.821899	−0.410950	−	0.911658i	$-0.634803\pi$
$19$	−3.58258	−0.821899	−0.410950	+	0.911658i	$0.634803\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	8.95644i	1.90952i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	1.41742	0.289331
$25$	0	0
$26$	−8.20871	−1.60986
$27$	1.00000i	0.192450i
$28$	− 1.20871i	− 0.228425i
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	− 6.04356i	− 1.06836i
$33$	− 5.00000i	− 0.870388i
$34$	5.37386	0.921610
$35$	0	0
$36$	1.20871	0.201452
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	− 6.41742i	− 1.04104i
$39$	4.58258	0.733799
$40$	0	0
$41$	−9.16515	−1.43136	−0.715678	−	0.698430i	$-0.753882\pi$
$41$	−9.16515	−1.43136	−0.715678	+	0.698430i	$0.753882\pi$
$42$	1.79129i	0.276402i
$43$	9.58258i	1.46133i	0.682737	+	0.730665i	$0.260790\pi$
$43$	9.58258i	1.46133i	−0.682737	+	0.730665i	$0.739210\pi$
$44$	−6.04356	−0.911101
$45$	0	0
$46$	−7.16515	−1.05644
$47$	10.5826i	1.54363i	0.635849	+	0.771814i	$0.280650\pi$
$47$	10.5826i	1.54363i	−0.635849	+	0.771814i	$0.719350\pi$
$48$	4.95644i	0.715400i
$49$	6.00000	0.857143
$50$	0	0
$51$	−3.00000	−0.420084
$52$	− 5.53901i	− 0.768123i
$53$	− 0.417424i	− 0.0573376i	−0.999589	−	0.0286688i	$-0.990873\pi$
$53$	− 0.417424i	− 0.0573376i	0.999589	−	0.0286688i	$-0.00912682\pi$
$54$	−1.79129	−0.243763
$55$	0	0
$56$	−1.41742	−0.189411
$57$	3.58258i	0.474524i
$58$	− 1.79129i	− 0.235208i
$59$	7.58258	0.987167	0.493584	−	0.869698i	$-0.335687\pi$
$59$	7.58258	0.987167	0.493584	+	0.869698i	$0.335687\pi$
$60$	0	0
$61$	12.7477	1.63218	0.816090	−	0.577925i	$-0.196138\pi$
$61$	12.7477	1.63218	0.816090	+	0.577925i	$0.196138\pi$
$62$	7.16515i	0.909975i
$63$	− 1.00000i	− 0.125988i
$64$	0.912878	0.114110
$65$	0	0
$66$	8.95644	1.10246
$67$	− 4.16515i	− 0.508854i	−0.967092	−	0.254427i	$-0.918113\pi$
$67$	− 4.16515i	− 0.508854i	0.967092	−	0.254427i	$-0.0818869\pi$
$68$	3.62614i	0.439734i
$69$	4.00000	0.481543
$70$	0	0
$71$	−9.58258	−1.13724	−0.568621	−	0.822599i	$-0.692523\pi$
$71$	−9.58258	−1.13724	−0.568621	+	0.822599i	$0.692523\pi$
$72$	− 1.41742i	− 0.167045i
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	7.16515	0.832932
$75$	0	0
$76$	4.33030	0.496720
$77$	5.00000i	0.569803i
$78$	8.20871i	0.929454i
$79$	−7.58258	−0.853106	−0.426553	−	0.904462i	$-0.640272\pi$
$79$	−7.58258	−0.853106	−0.426553	+	0.904462i	$0.640272\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 16.4174i	− 1.81300i
$83$	11.5826i	1.27135i	0.771956	+	0.635676i	$0.219279\pi$
$83$	11.5826i	1.27135i	−0.771956	+	0.635676i	$0.780721\pi$
$84$	−1.20871	−0.131881
$85$	0	0
$86$	−17.1652	−1.85097
$87$	1.00000i	0.107211i
$88$	7.08712i	0.755490i
$89$	−1.41742	−0.150247	−0.0751233	−	0.997174i	$-0.523935\pi$
$89$	−1.41742	−0.150247	−0.0751233	+	0.997174i	$0.523935\pi$
$90$	0	0
$91$	−4.58258	−0.480384
$92$	− 4.83485i	− 0.504068i
$93$	− 4.00000i	− 0.414781i
$94$	−18.9564	−1.95521
$95$	0	0
$96$	−6.04356	−0.616818
$97$	11.5826i	1.17603i	0.808849	+	0.588016i	$0.200091\pi$
$97$	11.5826i	1.17603i	−0.808849	+	0.588016i	$0.799909\pi$
$98$	10.7477i	1.08568i
$99$	−5.00000	−0.502519
$100$	0	0
$101$	−0.582576	−0.0579684	−0.0289842	−	0.999580i	$-0.509227\pi$
$101$	−0.582576	−0.0579684	−0.0289842	+	0.999580i	$0.509227\pi$
$102$	− 5.37386i	− 0.532092i
$103$	− 15.1652i	− 1.49427i	−0.664674	−	0.747133i	$-0.731430\pi$
$103$	− 15.1652i	− 1.49427i	0.664674	−	0.747133i	$-0.268570\pi$
$104$	−6.49545	−0.636932
$105$	0	0
$106$	0.747727	0.0726257
$107$	5.16515i	0.499334i	0.968332	+	0.249667i	$0.0803212\pi$
$107$	5.16515i	0.499334i	−0.968332	+	0.249667i	$0.919679\pi$
$108$	− 1.20871i	− 0.116308i
$109$	−14.1652	−1.35678	−0.678388	−	0.734704i	$-0.737321\pi$
$109$	−14.1652	−1.35678	−0.678388	+	0.734704i	$0.737321\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	− 4.95644i	− 0.468339i
$113$	14.1652i	1.33255i	0.745708	+	0.666273i	$0.232111\pi$
$113$	14.1652i	1.33255i	−0.745708	+	0.666273i	$0.767889\pi$
$114$	−6.41742	−0.601047
$115$	0	0
$116$	1.20871	0.112226
$117$	− 4.58258i	− 0.423659i
$118$	13.5826i	1.25038i
$119$	3.00000	0.275010
$120$	0	0
$121$	14.0000	1.27273
$122$	22.8348i	2.06737i
$123$	9.16515i	0.826394i
$124$	−4.83485	−0.434182
$125$	0	0
$126$	1.79129	0.159581
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	− 10.4519i	− 0.923826i
$129$	9.58258	0.843699
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	6.04356i	0.526024i
$133$	− 3.58258i	− 0.310649i
$134$	7.46099	0.644531
$135$	0	0
$136$	4.25227	0.364629
$137$	16.3303i	1.39519i	0.716491	+	0.697596i	$0.245747\pi$
$137$	16.3303i	1.39519i	−0.716491	+	0.697596i	$0.754253\pi$
$138$	7.16515i	0.609938i
$139$	9.41742	0.798776	0.399388	−	0.916782i	$-0.369223\pi$
$139$	9.41742	0.798776	0.399388	+	0.916782i	$0.369223\pi$
$140$	0	0
$141$	10.5826	0.891214
$142$	− 17.1652i	− 1.44047i
$143$	22.9129i	1.91607i
$144$	4.95644	0.413037
$145$	0	0
$146$	7.16515	0.592992
$147$	− 6.00000i	− 0.494872i
$148$	4.83485i	0.397422i
$149$	−16.7477	−1.37203	−0.686014	−	0.727589i	$-0.740641\pi$
$149$	−16.7477	−1.37203	−0.686014	+	0.727589i	$0.740641\pi$
$150$	0	0
$151$	7.16515	0.583092	0.291546	−	0.956557i	$-0.405830\pi$
$151$	7.16515	0.583092	0.291546	+	0.956557i	$0.405830\pi$
$152$	− 5.07803i	− 0.411883i
$153$	3.00000i	0.242536i
$154$	−8.95644	−0.721730
$155$	0	0
$156$	−5.53901	−0.443476
$157$	10.7477i	0.857762i	0.903361	+	0.428881i	$0.141092\pi$
$157$	10.7477i	0.857762i	−0.903361	+	0.428881i	$0.858908\pi$
$158$	− 13.5826i	− 1.08057i
$159$	−0.417424	−0.0331039
$160$	0	0
$161$	−4.00000	−0.315244
$162$	1.79129i	0.140737i
$163$	− 7.58258i	− 0.593913i	−0.954891	−	0.296957i	$-0.904028\pi$
$163$	− 7.58258i	− 0.593913i	0.954891	−	0.296957i	$-0.0959717\pi$
$164$	11.0780	0.865049
$165$	0	0
$166$	−20.7477	−1.61034
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	1.41742i	0.109357i
$169$	−8.00000	−0.615385
$170$	0	0
$171$	3.58258	0.273966
$172$	− 11.5826i	− 0.883163i
$173$	3.16515i	0.240642i	0.992735	+	0.120321i	$0.0383924\pi$
$173$	3.16515i	0.240642i	−0.992735	+	0.120321i	$0.961608\pi$
$174$	−1.79129	−0.135797
$175$	0	0
$176$	−24.7822	−1.86803
$177$	− 7.58258i	− 0.569941i
$178$	− 2.53901i	− 0.190307i
$179$	−4.74773	−0.354862	−0.177431	−	0.984133i	$-0.556779\pi$
$179$	−4.74773	−0.354862	−0.177431	+	0.984133i	$0.556779\pi$
$180$	0	0
$181$	16.1652	1.20155	0.600773	−	0.799420i	$-0.294860\pi$
$181$	16.1652	1.20155	0.600773	+	0.799420i	$0.294860\pi$
$182$	− 8.20871i	− 0.608470i
$183$	− 12.7477i	− 0.942339i
$184$	−5.66970	−0.417976
$185$	0	0
$186$	7.16515	0.525374
$187$	− 15.0000i	− 1.09691i
$188$	− 12.7913i	− 0.932901i
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	− 0.912878i	− 0.0658813i
$193$	20.3303i	1.46341i	0.681623	+	0.731704i	$0.261274\pi$
$193$	20.3303i	1.46341i	−0.681623	+	0.731704i	$0.738726\pi$
$194$	−20.7477	−1.48960
$195$	0	0
$196$	−7.25227	−0.518019
$197$	− 16.3303i	− 1.16349i	−0.813373	−	0.581743i	$-0.802371\pi$
$197$	− 16.3303i	− 1.16349i	0.813373	−	0.581743i	$-0.197629\pi$
$198$	− 8.95644i	− 0.636506i
$199$	−13.4174	−0.951136	−0.475568	−	0.879679i	$-0.657757\pi$
$199$	−13.4174	−0.951136	−0.475568	+	0.879679i	$0.657757\pi$
$200$	0	0
$201$	−4.16515	−0.293787
$202$	− 1.04356i	− 0.0734247i
$203$	− 1.00000i	− 0.0701862i
$204$	3.62614	0.253880
$205$	0	0
$206$	27.1652	1.89269
$207$	− 4.00000i	− 0.278019i
$208$	− 22.7133i	− 1.57488i
$209$	−17.9129	−1.23906
$210$	0	0
$211$	20.3303	1.39960	0.699798	−	0.714341i	$-0.253273\pi$
$211$	20.3303	1.39960	0.699798	+	0.714341i	$0.253273\pi$
$212$	0.504546i	0.0346523i
$213$	9.58258i	0.656587i
$214$	−9.25227	−0.632472
$215$	0	0
$216$	−1.41742	−0.0964435
$217$	4.00000i	0.271538i
$218$	− 25.3739i	− 1.71853i
$219$	−4.00000	−0.270295
$220$	0	0
$221$	13.7477	0.924772
$222$	− 7.16515i	− 0.480893i
$223$	− 7.00000i	− 0.468755i	−0.972146	−	0.234377i	$-0.924695\pi$
$223$	− 7.00000i	− 0.468755i	0.972146	−	0.234377i	$-0.0753051\pi$
$224$	6.04356	0.403802
$225$	0	0
$226$	−25.3739	−1.68784
$227$	− 7.58258i	− 0.503273i	−0.967822	−	0.251637i	$-0.919031\pi$
$227$	− 7.58258i	− 0.503273i	0.967822	−	0.251637i	$-0.0809688\pi$
$228$	− 4.33030i	− 0.286781i
$229$	17.1652	1.13431	0.567153	−	0.823613i	$-0.308045\pi$
$229$	17.1652	1.13431	0.567153	+	0.823613i	$0.308045\pi$
$230$	0	0
$231$	5.00000	0.328976
$232$	− 1.41742i	− 0.0930585i
$233$	13.1652i	0.862478i	0.902238	+	0.431239i	$0.141923\pi$
$233$	13.1652i	0.862478i	−0.902238	+	0.431239i	$0.858077\pi$
$234$	8.20871	0.536620
$235$	0	0
$236$	−9.16515	−0.596601
$237$	7.58258i	0.492541i
$238$	5.37386i	0.348336i
$239$	26.0000	1.68180	0.840900	−	0.541190i	$-0.182026\pi$
$239$	26.0000	1.68180	0.840900	+	0.541190i	$0.182026\pi$
$240$	0	0
$241$	29.3303	1.88933	0.944665	−	0.328035i	$-0.106387\pi$
$241$	29.3303	1.88933	0.944665	+	0.328035i	$0.106387\pi$
$242$	25.0780i	1.61208i
$243$	− 1.00000i	− 0.0641500i
$244$	−15.4083	−0.986417
$245$	0	0
$246$	−16.4174	−1.04674
$247$	− 16.4174i	− 1.04462i
$248$	5.66970i	0.360026i
$249$	11.5826	0.734016
$250$	0	0
$251$	16.1652	1.02034	0.510168	−	0.860075i	$-0.329583\pi$
$251$	16.1652	1.02034	0.510168	+	0.860075i	$0.329583\pi$
$252$	1.20871i	0.0761417i
$253$	20.0000i	1.25739i
$254$	−3.58258	−0.224791
$255$	0	0
$256$	20.5481	1.28426
$257$	12.7477i	0.795181i	0.917563	+	0.397591i	$0.130154\pi$
$257$	12.7477i	0.795181i	−0.917563	+	0.397591i	$0.869846\pi$
$258$	17.1652i	1.06866i
$259$	4.00000	0.248548
$260$	0	0
$261$	1.00000	0.0618984
$262$	− 26.8693i	− 1.65999i
$263$	− 30.3303i	− 1.87025i	−0.354322	−	0.935123i	$-0.615288\pi$
$263$	− 30.3303i	− 1.87025i	0.354322	−	0.935123i	$-0.384712\pi$
$264$	7.08712	0.436182
$265$	0	0
$266$	6.41742	0.393478
$267$	1.41742i	0.0867450i
$268$	5.03447i	0.307529i
$269$	−22.5826	−1.37688	−0.688442	−	0.725291i	$-0.741705\pi$
$269$	−22.5826	−1.37688	−0.688442	+	0.725291i	$0.741705\pi$
$270$	0	0
$271$	1.16515	0.0707779	0.0353890	−	0.999374i	$-0.488733\pi$
$271$	1.16515	0.0707779	0.0353890	+	0.999374i	$0.488733\pi$
$272$	14.8693i	0.901585i
$273$	4.58258i	0.277350i
$274$	−29.2523	−1.76719
$275$	0	0
$276$	−4.83485	−0.291024
$277$	− 24.9129i	− 1.49687i	−0.663208	−	0.748435i	$-0.730806\pi$
$277$	− 24.9129i	− 1.49687i	0.663208	−	0.748435i	$-0.269194\pi$
$278$	16.8693i	1.01175i
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−0.417424	−0.0249014	−0.0124507	−	0.999922i	$-0.503963\pi$
$281$	−0.417424	−0.0249014	−0.0124507	+	0.999922i	$0.503963\pi$
$282$	18.9564i	1.12884i
$283$	0.834849i	0.0496266i	0.999692	+	0.0248133i	$0.00789913\pi$
$283$	0.834849i	0.0496266i	−0.999692	+	0.0248133i	$0.992101\pi$
$284$	11.5826	0.687299
$285$	0	0
$286$	−41.0436	−2.42696
$287$	− 9.16515i	− 0.541002i
$288$	6.04356i	0.356120i
$289$	8.00000	0.470588
$290$	0	0
$291$	11.5826	0.678983
$292$	4.83485i	0.282938i
$293$	− 30.1652i	− 1.76227i	−0.472868	−	0.881133i	$-0.656781\pi$
$293$	− 30.1652i	− 1.76227i	0.472868	−	0.881133i	$-0.343219\pi$
$294$	10.7477	0.626820
$295$	0	0
$296$	5.66970	0.329544
$297$	5.00000i	0.290129i
$298$	− 30.0000i	− 1.73785i
$299$	−18.3303	−1.06007
$300$	0	0
$301$	−9.58258	−0.552330
$302$	12.8348i	0.738563i
$303$	0.582576i	0.0334681i
$304$	17.7568	1.01842
$305$	0	0
$306$	−5.37386	−0.307203
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	− 6.04356i	− 0.344364i
$309$	−15.1652	−0.862715
$310$	0	0
$311$	−3.00000	−0.170114	−0.0850572	−	0.996376i	$-0.527107\pi$
$311$	−3.00000	−0.170114	−0.0850572	+	0.996376i	$0.527107\pi$
$312$	6.49545i	0.367733i
$313$	− 10.5826i	− 0.598163i	−0.954228	−	0.299081i	$-0.903320\pi$
$313$	− 10.5826i	− 0.598163i	0.954228	−	0.299081i	$-0.0966802\pi$
$314$	−19.2523	−1.08647
$315$	0	0
$316$	9.16515	0.515580
$317$	− 25.0000i	− 1.40414i	−0.712108	−	0.702070i	$-0.752259\pi$
$317$	− 25.0000i	− 1.40414i	0.712108	−	0.702070i	$-0.247741\pi$
$318$	− 0.747727i	− 0.0419305i
$319$	−5.00000	−0.279946
$320$	0	0
$321$	5.16515	0.288291
$322$	− 7.16515i	− 0.399298i
$323$	10.7477i	0.598020i
$324$	−1.20871	−0.0671507
$325$	0	0
$326$	13.5826	0.752269
$327$	14.1652i	0.783335i
$328$	− 12.9909i	− 0.717303i
$329$	−10.5826	−0.583436
$330$	0	0
$331$	−8.33030	−0.457875	−0.228937	−	0.973441i	$-0.573525\pi$
$331$	−8.33030	−0.457875	−0.228937	+	0.973441i	$0.573525\pi$
$332$	− 14.0000i	− 0.768350i
$333$	4.00000i	0.219199i
$334$	0	0
$335$	0	0
$336$	−4.95644	−0.270396
$337$	− 21.1652i	− 1.15294i	−0.817119	−	0.576470i	$-0.804430\pi$
$337$	− 21.1652i	− 1.15294i	0.817119	−	0.576470i	$-0.195570\pi$
$338$	− 14.3303i	− 0.779466i
$339$	14.1652	0.769345
$340$	0	0
$341$	20.0000	1.08306
$342$	6.41742i	0.347015i
$343$	13.0000i	0.701934i
$344$	−13.5826	−0.732323
$345$	0	0
$346$	−5.66970	−0.304805
$347$	7.16515i	0.384645i	0.981332	+	0.192323i	$0.0616020\pi$
$347$	7.16515i	0.384645i	−0.981332	+	0.192323i	$0.938398\pi$
$348$	− 1.20871i	− 0.0647938i
$349$	4.33030	0.231796	0.115898	−	0.993261i	$-0.463025\pi$
$349$	4.33030	0.231796	0.115898	+	0.993261i	$0.463025\pi$
$350$	0	0
$351$	−4.58258	−0.244600
$352$	− 30.2178i	− 1.61061i
$353$	14.8348i	0.789579i	0.918772	+	0.394790i	$0.129183\pi$
$353$	14.8348i	0.789579i	−0.918772	+	0.394790i	$0.870817\pi$
$354$	13.5826	0.721906
$355$	0	0
$356$	1.71326	0.0908025
$357$	− 3.00000i	− 0.158777i
$358$	− 8.50455i	− 0.449479i
$359$	27.1652	1.43372	0.716861	−	0.697216i	$-0.245578\pi$
$359$	27.1652	1.43372	0.716861	+	0.697216i	$0.245578\pi$
$360$	0	0
$361$	−6.16515	−0.324482
$362$	28.9564i	1.52192i
$363$	− 14.0000i	− 0.734809i
$364$	5.53901	0.290323
$365$	0	0
$366$	22.8348	1.19360
$367$	− 37.4955i	− 1.95725i	−0.205661	−	0.978623i	$-0.565934\pi$
$367$	− 37.4955i	− 1.95725i	0.205661	−	0.978623i	$-0.434066\pi$
$368$	− 19.8258i	− 1.03349i
$369$	9.16515	0.477119
$370$	0	0
$371$	0.417424	0.0216716
$372$	4.83485i	0.250675i
$373$	− 28.3303i	− 1.46689i	−0.679750	−	0.733444i	$-0.737912\pi$
$373$	− 28.3303i	− 1.46689i	0.679750	−	0.733444i	$-0.262088\pi$
$374$	26.8693	1.38938
$375$	0	0
$376$	−15.0000	−0.773566
$377$	− 4.58258i	− 0.236015i
$378$	− 1.79129i	− 0.0921339i
$379$	26.0000	1.33553	0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000	1.33553	0.667765	+	0.744372i	$0.267251\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	− 7.16515i	− 0.366601i
$383$	− 3.58258i	− 0.183061i	−0.995802	−	0.0915305i	$-0.970824\pi$
$383$	− 3.58258i	− 0.183061i	0.995802	−	0.0915305i	$-0.0291759\pi$
$384$	−10.4519	−0.533371
$385$	0	0
$386$	−36.4174	−1.85360
$387$	− 9.58258i	− 0.487110i
$388$	− 14.0000i	− 0.710742i
$389$	6.58258	0.333750	0.166875	−	0.985978i	$-0.446632\pi$
$389$	6.58258	0.333750	0.166875	+	0.985978i	$0.446632\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	8.50455i	0.429544i
$393$	15.0000i	0.756650i
$394$	29.2523	1.47371
$395$	0	0
$396$	6.04356	0.303700
$397$	10.8348i	0.543785i	0.962328	+	0.271893i	$0.0876496\pi$
$397$	10.8348i	0.543785i	−0.962328	+	0.271893i	$0.912350\pi$
$398$	− 24.0345i	− 1.20474i
$399$	−3.58258	−0.179353
$400$	0	0
$401$	12.4174	0.620097	0.310048	−	0.950721i	$-0.399655\pi$
$401$	12.4174	0.620097	0.310048	+	0.950721i	$0.399655\pi$
$402$	− 7.46099i	− 0.372120i
$403$	18.3303i	0.913097i
$404$	0.704166	0.0350336
$405$	0	0
$406$	1.79129	0.0889001
$407$	− 20.0000i	− 0.991363i
$408$	− 4.25227i	− 0.210519i
$409$	2.74773	0.135866	0.0679332	−	0.997690i	$-0.478360\pi$
$409$	2.74773	0.135866	0.0679332	+	0.997690i	$0.478360\pi$
$410$	0	0
$411$	16.3303	0.805514
$412$	18.3303i	0.903069i
$413$	7.58258i	0.373114i
$414$	7.16515	0.352148
$415$	0	0
$416$	27.6951	1.35786
$417$	− 9.41742i	− 0.461173i
$418$	− 32.0871i	− 1.56943i
$419$	37.1652	1.81564	0.907818	−	0.419364i	$-0.137747\pi$
$419$	37.1652	1.81564	0.907818	+	0.419364i	$0.137747\pi$
$420$	0	0
$421$	4.41742	0.215292	0.107646	−	0.994189i	$-0.465669\pi$
$421$	4.41742	0.215292	0.107646	+	0.994189i	$0.465669\pi$
$422$	36.4174i	1.77277i
$423$	− 10.5826i	− 0.514542i
$424$	0.591667	0.0287339
$425$	0	0
$426$	−17.1652	−0.831654
$427$	12.7477i	0.616906i
$428$	− 6.24318i	− 0.301776i
$429$	22.9129	1.10624
$430$	0	0
$431$	−39.1652	−1.88652	−0.943259	−	0.332057i	$-0.892258\pi$
$431$	−39.1652	−1.88652	−0.943259	+	0.332057i	$0.892258\pi$
$432$	− 4.95644i	− 0.238467i
$433$	− 25.0780i	− 1.20517i	−0.798053	−	0.602587i	$-0.794137\pi$
$433$	− 25.0780i	− 1.20517i	0.798053	−	0.602587i	$-0.205863\pi$
$434$	−7.16515	−0.343938
$435$	0	0
$436$	17.1216	0.819975
$437$	− 14.3303i	− 0.685511i
$438$	− 7.16515i	− 0.342364i
$439$	16.9129	0.807208	0.403604	−	0.914934i	$-0.367757\pi$
$439$	16.9129	0.807208	0.403604	+	0.914934i	$0.367757\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	24.6261i	1.17135i
$443$	− 0.582576i	− 0.0276790i	−0.999904	−	0.0138395i	$-0.995595\pi$
$443$	− 0.582576i	− 0.0276790i	0.999904	−	0.0138395i	$-0.00440539\pi$
$444$	4.83485	0.229452
$445$	0	0
$446$	12.5390	0.593740
$447$	16.7477i	0.792140i
$448$	0.912878i	0.0431295i
$449$	30.0780	1.41947	0.709735	−	0.704469i	$-0.248815\pi$
$449$	30.0780	1.41947	0.709735	+	0.704469i	$0.248815\pi$
$450$	0	0
$451$	−45.8258	−2.15785
$452$	− 17.1216i	− 0.805332i
$453$	− 7.16515i	− 0.336648i
$454$	13.5826	0.637462
$455$	0	0
$456$	−5.07803	−0.237801
$457$	− 3.74773i	− 0.175311i	−0.996151	−	0.0876556i	$-0.972062\pi$
$457$	− 3.74773i	− 0.175311i	0.996151	−	0.0876556i	$-0.0279375\pi$
$458$	30.7477i	1.43675i
$459$	3.00000	0.140028
$460$	0	0
$461$	−9.16515	−0.426864	−0.213432	−	0.976958i	$-0.568464\pi$
$461$	−9.16515	−0.426864	−0.213432	+	0.976958i	$0.568464\pi$
$462$	8.95644i	0.416691i
$463$	0.165151i	0.00767524i	0.999993	+	0.00383762i	$0.00122155\pi$
$463$	0.165151i	0.00767524i	−0.999993	+	0.00383762i	$0.998778\pi$
$464$	4.95644	0.230097
$465$	0	0
$466$	−23.5826	−1.09244
$467$	− 8.00000i	− 0.370196i	−0.982720	−	0.185098i	$-0.940740\pi$
$467$	− 8.00000i	− 0.370196i	0.982720	−	0.185098i	$-0.0592602\pi$
$468$	5.53901i	0.256041i
$469$	4.16515	0.192329
$470$	0	0
$471$	10.7477	0.495229
$472$	10.7477i	0.494704i
$473$	47.9129i	2.20304i
$474$	−13.5826	−0.623868
$475$	0	0
$476$	−3.62614	−0.166204
$477$	0.417424i	0.0191125i
$478$	46.5735i	2.13022i
$479$	19.1652	0.875678	0.437839	−	0.899053i	$-0.355744\pi$
$479$	19.1652	0.875678	0.437839	+	0.899053i	$0.355744\pi$
$480$	0	0
$481$	18.3303	0.835790
$482$	52.5390i	2.39309i
$483$	4.00000i	0.182006i
$484$	−16.9220	−0.769180
$485$	0	0
$486$	1.79129	0.0812545
$487$	2.33030i	0.105596i	0.998605	+	0.0527980i	$0.0168140\pi$
$487$	2.33030i	0.105596i	−0.998605	+	0.0527980i	$0.983186\pi$
$488$	18.0689i	0.817942i
$489$	−7.58258	−0.342896
$490$	0	0
$491$	16.0000	0.722070	0.361035	−	0.932552i	$-0.382424\pi$
$491$	16.0000	0.722070	0.361035	+	0.932552i	$0.382424\pi$
$492$	− 11.0780i	− 0.499436i
$493$	3.00000i	0.135113i
$494$	29.4083	1.32314
$495$	0	0
$496$	−19.8258	−0.890203
$497$	− 9.58258i	− 0.429837i
$498$	20.7477i	0.929728i
$499$	13.4174	0.600646	0.300323	−	0.953837i	$-0.402905\pi$
$499$	13.4174	0.600646	0.300323	+	0.953837i	$0.402905\pi$
$500$	0	0
$501$	0	0
$502$	28.9564i	1.29239i
$503$	22.9129i	1.02163i	0.859689	+	0.510817i	$0.170657\pi$
$503$	22.9129i	1.02163i	−0.859689	+	0.510817i	$0.829343\pi$
$504$	1.41742	0.0631371
$505$	0	0
$506$	−35.8258	−1.59265
$507$	8.00000i	0.355292i
$508$	− 2.41742i	− 0.107256i
$509$	−26.7477	−1.18557	−0.592786	−	0.805360i	$-0.701972\pi$
$509$	−26.7477	−1.18557	−0.592786	+	0.805360i	$0.701972\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	15.9038i	0.702855i
$513$	− 3.58258i	− 0.158175i
$514$	−22.8348	−1.00720
$515$	0	0
$516$	−11.5826	−0.509894
$517$	52.9129i	2.32711i
$518$	7.16515i	0.314819i
$519$	3.16515	0.138935
$520$	0	0
$521$	43.0780	1.88728	0.943641	−	0.330970i	$-0.107376\pi$
$521$	43.0780	1.88728	0.943641	+	0.330970i	$0.107376\pi$
$522$	1.79129i	0.0784025i
$523$	33.3303i	1.45743i	0.684816	+	0.728716i	$0.259883\pi$
$523$	33.3303i	1.45743i	−0.684816	+	0.728716i	$0.740117\pi$
$524$	18.1307	0.792043
$525$	0	0
$526$	54.3303	2.36891
$527$	− 12.0000i	− 0.522728i
$528$	24.7822i	1.07851i
$529$	7.00000	0.304348
$530$	0	0
$531$	−7.58258	−0.329056
$532$	4.33030i	0.187742i
$533$	− 42.0000i	− 1.81922i
$534$	−2.53901	−0.109874
$535$	0	0
$536$	5.90379	0.255005
$537$	4.74773i	0.204880i
$538$	− 40.4519i	− 1.74400i
$539$	30.0000	1.29219
$540$	0	0
$541$	−31.4955	−1.35410	−0.677048	−	0.735939i	$-0.736741\pi$
$541$	−31.4955	−1.35410	−0.677048	+	0.735939i	$0.736741\pi$
$542$	2.08712i	0.0896495i
$543$	− 16.1652i	− 0.693713i
$544$	−18.1307	−0.777347
$545$	0	0
$546$	−8.20871	−0.351300
$547$	4.16515i	0.178089i	0.996028	+	0.0890445i	$0.0283813\pi$
$547$	4.16515i	0.178089i	−0.996028	+	0.0890445i	$0.971619\pi$
$548$	− 19.7386i	− 0.843193i
$549$	−12.7477	−0.544060
$550$	0	0
$551$	3.58258	0.152623
$552$	5.66970i	0.241318i
$553$	− 7.58258i	− 0.322444i
$554$	44.6261	1.89598
$555$	0	0
$556$	−11.3830	−0.482745
$557$	− 22.7477i	− 0.963852i	−0.876212	−	0.481926i	$-0.839937\pi$
$557$	− 22.7477i	− 0.963852i	0.876212	−	0.481926i	$-0.160063\pi$
$558$	− 7.16515i	− 0.303325i
$559$	−43.9129	−1.85732
$560$	0	0
$561$	−15.0000	−0.633300
$562$	− 0.747727i	− 0.0315410i
$563$	14.5826i	0.614582i	0.951616	+	0.307291i	$0.0994225\pi$
$563$	14.5826i	0.614582i	−0.951616	+	0.307291i	$0.900577\pi$
$564$	−12.7913	−0.538610
$565$	0	0
$566$	−1.49545	−0.0628586
$567$	1.00000i	0.0419961i
$568$	− 13.5826i	− 0.569912i
$569$	−28.5826	−1.19824	−0.599122	−	0.800658i	$-0.704484\pi$
$569$	−28.5826	−1.19824	−0.599122	+	0.800658i	$0.704484\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	− 27.6951i	− 1.15799i
$573$	4.00000i	0.167102i
$574$	16.4174	0.685250
$575$	0	0
$576$	−0.912878	−0.0380366
$577$	− 19.1652i	− 0.797856i	−0.916982	−	0.398928i	$-0.869382\pi$
$577$	− 19.1652i	− 0.797856i	0.916982	−	0.398928i	$-0.130618\pi$
$578$	14.3303i	0.596062i
$579$	20.3303	0.844899
$580$	0	0
$581$	−11.5826	−0.480526
$582$	20.7477i	0.860021i
$583$	− 2.08712i	− 0.0864397i
$584$	5.66970	0.234614
$585$	0	0
$586$	54.0345	2.23214
$587$	− 11.5826i	− 0.478064i	−0.971012	−	0.239032i	$-0.923170\pi$
$587$	− 11.5826i	− 0.478064i	0.971012	−	0.239032i	$-0.0768301\pi$
$588$	7.25227i	0.299079i
$589$	−14.3303	−0.590470
$590$	0	0
$591$	−16.3303	−0.671739
$592$	19.8258i	0.814834i
$593$	− 8.41742i	− 0.345662i	−0.984951	−	0.172831i	$-0.944709\pi$
$593$	− 8.41742i	− 0.345662i	0.984951	−	0.172831i	$-0.0552915\pi$
$594$	−8.95644	−0.367487
$595$	0	0
$596$	20.2432	0.829193
$597$	13.4174i	0.549139i
$598$	− 32.8348i	− 1.34272i
$599$	−0.165151	−0.00674790	−0.00337395	−	0.999994i	$-0.501074\pi$
$599$	−0.165151	−0.00674790	−0.00337395	+	0.999994i	$0.501074\pi$
$600$	0	0
$601$	−32.3303	−1.31878	−0.659390	−	0.751801i	$-0.729185\pi$
$601$	−32.3303	−1.31878	−0.659390	+	0.751801i	$0.729185\pi$
$602$	− 17.1652i	− 0.699599i
$603$	4.16515i	0.169618i
$604$	−8.66061	−0.352395
$605$	0	0
$606$	−1.04356	−0.0423918
$607$	3.58258i	0.145412i	0.997353	+	0.0727061i	$0.0231635\pi$
$607$	3.58258i	0.145412i	−0.997353	+	0.0727061i	$0.976836\pi$
$608$	21.6515i	0.878085i
$609$	−1.00000	−0.0405220
$610$	0	0
$611$	−48.4955	−1.96192
$612$	− 3.62614i	− 0.146578i
$613$	43.7477i	1.76695i	0.468474	+	0.883477i	$0.344804\pi$
$613$	43.7477i	1.76695i	−0.468474	+	0.883477i	$0.655196\pi$
$614$	0	0
$615$	0	0
$616$	−7.08712	−0.285548
$617$	15.4955i	0.623823i	0.950111	+	0.311912i	$0.100969\pi$
$617$	15.4955i	0.623823i	−0.950111	+	0.311912i	$0.899031\pi$
$618$	− 27.1652i	− 1.09274i
$619$	−13.1652	−0.529152	−0.264576	−	0.964365i	$-0.585232\pi$
$619$	−13.1652	−0.529152	−0.264576	+	0.964365i	$0.585232\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	− 5.37386i	− 0.215472i
$623$	− 1.41742i	− 0.0567879i
$624$	−22.7133	−0.909258
$625$	0	0
$626$	18.9564	0.757652
$627$	17.9129i	0.715371i
$628$	− 12.9909i	− 0.518394i
$629$	−12.0000	−0.478471
$630$	0	0
$631$	10.9129	0.434435	0.217217	−	0.976123i	$-0.430302\pi$
$631$	10.9129	0.434435	0.217217	+	0.976123i	$0.430302\pi$
$632$	− 10.7477i	− 0.427522i
$633$	− 20.3303i	− 0.808057i
$634$	44.7822	1.77853
$635$	0	0
$636$	0.504546	0.0200065
$637$	27.4955i	1.08941i
$638$	− 8.95644i	− 0.354589i
$639$	9.58258	0.379081
$640$	0	0
$641$	−6.58258	−0.259996	−0.129998	−	0.991514i	$-0.541497\pi$
$641$	−6.58258	−0.259996	−0.129998	+	0.991514i	$0.541497\pi$
$642$	9.25227i	0.365158i
$643$	− 43.6606i	− 1.72181i	−0.508769	−	0.860903i	$-0.669899\pi$
$643$	− 43.6606i	− 1.72181i	0.508769	−	0.860903i	$-0.330101\pi$
$644$	4.83485	0.190520
$645$	0	0
$646$	−19.2523	−0.757471
$647$	24.7477i	0.972934i	0.873699	+	0.486467i	$0.161715\pi$
$647$	24.7477i	0.972934i	−0.873699	+	0.486467i	$0.838285\pi$
$648$	1.41742i	0.0556817i
$649$	37.9129	1.48821
$650$	0	0
$651$	4.00000	0.156772
$652$	9.16515i	0.358935i
$653$	− 26.1652i	− 1.02392i	−0.859009	−	0.511961i	$-0.828919\pi$
$653$	− 26.1652i	− 1.02392i	0.859009	−	0.511961i	$-0.171081\pi$
$654$	−25.3739	−0.992197
$655$	0	0
$656$	45.4265	1.77361
$657$	4.00000i	0.156055i
$658$	− 18.9564i	− 0.738999i
$659$	−18.1652	−0.707614	−0.353807	−	0.935318i	$-0.615113\pi$
$659$	−18.1652	−0.707614	−0.353807	+	0.935318i	$0.615113\pi$
$660$	0	0
$661$	25.0000	0.972387	0.486194	−	0.873851i	$-0.338385\pi$
$661$	25.0000	0.972387	0.486194	+	0.873851i	$0.338385\pi$
$662$	− 14.9220i	− 0.579959i
$663$	− 13.7477i	− 0.533917i
$664$	−16.4174	−0.637120
$665$	0	0
$666$	−7.16515	−0.277644
$667$	− 4.00000i	− 0.154881i
$668$	0	0
$669$	−7.00000	−0.270636
$670$	0	0
$671$	63.7386	2.46060
$672$	− 6.04356i	− 0.233135i
$673$	− 1.74773i	− 0.0673699i	−0.999433	−	0.0336850i	$-0.989276\pi$
$673$	− 1.74773i	− 0.0673699i	0.999433	−	0.0336850i	$-0.0107243\pi$
$674$	37.9129	1.46035
$675$	0	0
$676$	9.66970	0.371911
$677$	44.8258i	1.72279i	0.507932	+	0.861397i	$0.330410\pi$
$677$	44.8258i	1.72279i	−0.507932	+	0.861397i	$0.669590\pi$
$678$	25.3739i	0.974477i
$679$	−11.5826	−0.444498
$680$	0	0
$681$	−7.58258	−0.290565
$682$	35.8258i	1.37184i
$683$	− 7.16515i	− 0.274167i	−0.990560	−	0.137083i	$-0.956227\pi$
$683$	− 7.16515i	− 0.274167i	0.990560	−	0.137083i	$-0.0437728\pi$
$684$	−4.33030	−0.165573
$685$	0	0
$686$	−23.2867	−0.889092
$687$	− 17.1652i	− 0.654891i
$688$	− 47.4955i	− 1.81075i
$689$	1.91288	0.0728749
$690$	0	0
$691$	42.9129	1.63248	0.816241	−	0.577711i	$-0.196054\pi$
$691$	42.9129	1.63248	0.816241	+	0.577711i	$0.196054\pi$
$692$	− 3.82576i	− 0.145433i
$693$	− 5.00000i	− 0.189934i
$694$	−12.8348	−0.487204
$695$	0	0
$696$	−1.41742	−0.0537273
$697$	27.4955i	1.04146i
$698$	7.75682i	0.293600i
$699$	13.1652	0.497952
$700$	0	0
$701$	−23.0780	−0.871645	−0.435823	−	0.900033i	$-0.643542\pi$
$701$	−23.0780	−0.871645	−0.435823	+	0.900033i	$0.643542\pi$
$702$	− 8.20871i	− 0.309818i
$703$	14.3303i	0.540478i
$704$	4.56439	0.172027
$705$	0	0
$706$	−26.5735	−1.00011
$707$	− 0.582576i	− 0.0219100i
$708$	9.16515i	0.344447i
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	7.58258	0.284369
$712$	− 2.00909i	− 0.0752939i
$713$	16.0000i	0.599205i
$714$	5.37386	0.201112
$715$	0	0
$716$	5.73864	0.214463
$717$	− 26.0000i	− 0.970988i
$718$	48.6606i	1.81600i
$719$	−7.91288	−0.295101	−0.147550	−	0.989055i	$-0.547139\pi$
$719$	−7.91288	−0.295101	−0.147550	+	0.989055i	$0.547139\pi$
$720$	0	0
$721$	15.1652	0.564780
$722$	− 11.0436i	− 0.410999i
$723$	− 29.3303i	− 1.09081i
$724$	−19.5390	−0.726162
$725$	0	0
$726$	25.0780	0.930733
$727$	9.66970i	0.358629i	0.983792	+	0.179315i	$0.0573880\pi$
$727$	9.66970i	0.358629i	−0.983792	+	0.179315i	$0.942612\pi$
$728$	− 6.49545i	− 0.240738i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	28.7477	1.06327
$732$	15.4083i	0.569508i
$733$	38.4174i	1.41898i	0.704716	+	0.709490i	$0.251075\pi$
$733$	38.4174i	1.41898i	−0.704716	+	0.709490i	$0.748925\pi$
$734$	67.1652	2.47911
$735$	0	0
$736$	24.1742	0.891074
$737$	− 20.8258i	− 0.767127i
$738$	16.4174i	0.604334i
$739$	−19.2523	−0.708206	−0.354103	−	0.935206i	$-0.615214\pi$
$739$	−19.2523	−0.708206	−0.354103	+	0.935206i	$0.615214\pi$
$740$	0	0
$741$	−16.4174	−0.603109
$742$	0.747727i	0.0274499i
$743$	− 29.7477i	− 1.09134i	−0.838001	−	0.545669i	$-0.816276\pi$
$743$	− 29.7477i	− 1.09134i	0.838001	−	0.545669i	$-0.183724\pi$
$744$	5.66970	0.207861
$745$	0	0
$746$	50.7477	1.85801
$747$	− 11.5826i	− 0.423784i
$748$	18.1307i	0.662923i
$749$	−5.16515	−0.188731
$750$	0	0
$751$	−17.4955	−0.638418	−0.319209	−	0.947684i	$-0.603417\pi$
$751$	−17.4955	−0.638418	−0.319209	+	0.947684i	$0.603417\pi$
$752$	− 52.4519i	− 1.91272i
$753$	− 16.1652i	− 0.589091i
$754$	8.20871	0.298944
$755$	0	0
$756$	1.20871	0.0439604
$757$	− 2.33030i	− 0.0846963i	−0.999103	−	0.0423481i	$-0.986516\pi$
$757$	− 2.33030i	− 0.0846963i	0.999103	−	0.0423481i	$-0.0134839\pi$
$758$	46.5735i	1.69163i
$759$	20.0000	0.725954
$760$	0	0
$761$	36.4174	1.32013	0.660065	−	0.751208i	$-0.270529\pi$
$761$	36.4174	1.32013	0.660065	+	0.751208i	$0.270529\pi$
$762$	3.58258i	0.129783i
$763$	− 14.1652i	− 0.512813i
$764$	4.83485	0.174919
$765$	0	0
$766$	6.41742	0.231871
$767$	34.7477i	1.25467i
$768$	− 20.5481i	− 0.741466i
$769$	−25.5826	−0.922531	−0.461266	−	0.887262i	$-0.652604\pi$
$769$	−25.5826	−0.922531	−0.461266	+	0.887262i	$0.652604\pi$
$770$	0	0
$771$	12.7477	0.459098
$772$	− 24.5735i	− 0.884419i
$773$	5.16515i	0.185778i	0.995676	+	0.0928888i	$0.0296101\pi$
$773$	5.16515i	0.185778i	−0.995676	+	0.0928888i	$0.970390\pi$
$774$	17.1652	0.616989
$775$	0	0
$776$	−16.4174	−0.589351
$777$	− 4.00000i	− 0.143499i
$778$	11.7913i	0.422738i
$779$	32.8348	1.17643
$780$	0	0
$781$	−47.9129	−1.71446
$782$	21.4955i	0.768676i
$783$	− 1.00000i	− 0.0357371i
$784$	−29.7386	−1.06209
$785$	0	0
$786$	−26.8693	−0.958397
$787$	− 24.0000i	− 0.855508i	−0.903895	−	0.427754i	$-0.859305\pi$
$787$	− 24.0000i	− 0.855508i	0.903895	−	0.427754i	$-0.140695\pi$
$788$	19.7386i	0.703160i
$789$	−30.3303	−1.07979
$790$	0	0
$791$	−14.1652	−0.503655
$792$	− 7.08712i	− 0.251830i
$793$	58.4174i	2.07446i
$794$	−19.4083	−0.688776
$795$	0	0
$796$	16.2178	0.574825
$797$	32.3303i	1.14520i	0.819836	+	0.572599i	$0.194065\pi$
$797$	32.3303i	1.14520i	−0.819836	+	0.572599i	$0.805935\pi$
$798$	− 6.41742i	− 0.227174i
$799$	31.7477	1.12315
$800$	0	0
$801$	1.41742	0.0500822
$802$	22.2432i	0.785434i
$803$	− 20.0000i	− 0.705785i
$804$	5.03447	0.177552
$805$	0	0
$806$	−32.8348	−1.15656
$807$	22.5826i	0.794944i
$808$	− 0.825757i	− 0.0290500i
$809$	−44.0780	−1.54970	−0.774851	−	0.632145i	$-0.782175\pi$
$809$	−44.0780	−1.54970	−0.774851	+	0.632145i	$0.782175\pi$
$810$	0	0
$811$	32.5826	1.14413	0.572064	−	0.820209i	$-0.306143\pi$
$811$	32.5826	1.14413	0.572064	+	0.820209i	$0.306143\pi$
$812$	1.20871i	0.0424175i
$813$	− 1.16515i	− 0.0408636i
$814$	35.8258	1.25569
$815$	0	0
$816$	14.8693	0.520530
$817$	− 34.3303i	− 1.20107i
$818$	4.92197i	0.172093i
$819$	4.58258	0.160128
$820$	0	0
$821$	−31.4955	−1.09920	−0.549599	−	0.835428i	$-0.685220\pi$
$821$	−31.4955	−1.09920	−0.549599	+	0.835428i	$0.685220\pi$
$822$	29.2523i	1.02029i
$823$	− 51.0780i	− 1.78047i	−0.455503	−	0.890234i	$-0.650541\pi$
$823$	− 51.0780i	− 1.78047i	0.455503	−	0.890234i	$-0.349459\pi$
$824$	21.4955	0.748830
$825$	0	0
$826$	−13.5826	−0.472598
$827$	− 43.8258i	− 1.52397i	−0.647594	−	0.761985i	$-0.724225\pi$
$827$	− 43.8258i	− 1.52397i	0.647594	−	0.761985i	$-0.275775\pi$
$828$	4.83485i	0.168023i
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	0	0
$831$	−24.9129	−0.864218
$832$	4.18333i	0.145031i
$833$	− 18.0000i	− 0.623663i
$834$	16.8693	0.584137
$835$	0	0
$836$	21.6515	0.748833
$837$	4.00000i	0.138260i
$838$	66.5735i	2.29974i
$839$	48.4955	1.67425	0.837125	−	0.547012i	$-0.184235\pi$
$839$	48.4955	1.67425	0.837125	+	0.547012i	$0.184235\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	7.91288i	0.272696i
$843$	0.417424i	0.0143769i
$844$	−24.5735	−0.845854
$845$	0	0
$846$	18.9564	0.651736
$847$	14.0000i	0.481046i
$848$	2.06894i	0.0710476i
$849$	0.834849	0.0286519
$850$	0	0
$851$	16.0000	0.548473
$852$	− 11.5826i	− 0.396813i
$853$	− 20.7477i	− 0.710389i	−0.934792	−	0.355194i	$-0.884415\pi$
$853$	− 20.7477i	− 0.710389i	0.934792	−	0.355194i	$-0.115585\pi$
$854$	−22.8348	−0.781392
$855$	0	0
$856$	−7.32121	−0.250234
$857$	− 46.6606i	− 1.59390i	−0.604048	−	0.796948i	$-0.706446\pi$
$857$	− 46.6606i	− 1.59390i	0.604048	−	0.796948i	$-0.293554\pi$
$858$	41.0436i	1.40120i
$859$	47.5826	1.62350	0.811748	−	0.584007i	$-0.198516\pi$
$859$	47.5826	1.62350	0.811748	+	0.584007i	$0.198516\pi$
$860$	0	0
$861$	−9.16515	−0.312348
$862$	− 70.1561i	− 2.38952i
$863$	6.41742i	0.218452i	0.994017	+	0.109226i	$0.0348372\pi$
$863$	6.41742i	0.218452i	−0.994017	+	0.109226i	$0.965163\pi$
$864$	6.04356	0.205606
$865$	0	0
$866$	44.9220	1.52651
$867$	− 8.00000i	− 0.271694i
$868$	− 4.83485i	− 0.164105i
$869$	−37.9129	−1.28611
$870$	0	0
$871$	19.0871	0.646742
$872$	− 20.0780i	− 0.679928i
$873$	− 11.5826i	− 0.392011i
$874$	25.6697	0.868290
$875$	0	0
$876$	4.83485	0.163354
$877$	19.6697i	0.664198i	0.943244	+	0.332099i	$0.107757\pi$
$877$	19.6697i	0.664198i	−0.943244	+	0.332099i	$0.892243\pi$
$878$	30.2958i	1.02243i
$879$	−30.1652	−1.01745
$880$	0	0
$881$	−32.0780	−1.08074	−0.540368	−	0.841429i	$-0.681715\pi$
$881$	−32.0780	−1.08074	−0.540368	+	0.841429i	$0.681715\pi$
$882$	− 10.7477i	− 0.361895i
$883$	− 16.0000i	− 0.538443i	−0.963078	−	0.269221i	$-0.913234\pi$
$883$	− 16.0000i	− 0.538443i	0.963078	−	0.269221i	$-0.0867663\pi$
$884$	−16.6170	−0.558892
$885$	0	0
$886$	1.04356	0.0350591
$887$	− 44.9129i	− 1.50803i	−0.656859	−	0.754013i	$-0.728115\pi$
$887$	− 44.9129i	− 1.50803i	0.656859	−	0.754013i	$-0.271885\pi$
$888$	− 5.66970i	− 0.190263i
$889$	−2.00000	−0.0670778
$890$	0	0
$891$	5.00000	0.167506
$892$	8.46099i	0.283295i
$893$	− 37.9129i	− 1.26871i
$894$	−30.0000	−1.00335
$895$	0	0
$896$	10.4519	0.349173
$897$	18.3303i	0.612031i
$898$	53.8784i	1.79795i
$899$	−4.00000	−0.133407
$900$	0	0
$901$	−1.25227	−0.0417193
$902$	− 82.0871i	− 2.73320i
$903$	9.58258i	0.318888i
$904$	−20.0780	−0.667785
$905$	0	0
$906$	12.8348	0.426409
$907$	23.5826i	0.783047i	0.920168	+	0.391523i	$0.128052\pi$
$907$	23.5826i	0.783047i	−0.920168	+	0.391523i	$0.871948\pi$
$908$	9.16515i	0.304156i
$909$	0.582576	0.0193228
$910$	0	0
$911$	−50.8258	−1.68393	−0.841966	−	0.539530i	$-0.818602\pi$
$911$	−50.8258	−1.68393	−0.841966	+	0.539530i	$0.818602\pi$
$912$	− 17.7568i	− 0.587987i
$913$	57.9129i	1.91664i
$914$	6.71326	0.222055
$915$	0	0
$916$	−20.7477	−0.685524
$917$	− 15.0000i	− 0.495344i
$918$	5.37386i	0.177364i
$919$	−18.9129	−0.623878	−0.311939	−	0.950102i	$-0.600979\pi$
$919$	−18.9129	−0.623878	−0.311939	+	0.950102i	$0.600979\pi$
$920$	0	0
$921$	0	0
$922$	− 16.4174i	− 0.540679i
$923$	− 43.9129i	− 1.44541i
$924$	−6.04356	−0.198819
$925$	0	0
$926$	−0.295834	−0.00972170
$927$	15.1652i	0.498089i
$928$	6.04356i	0.198390i
$929$	−15.6697	−0.514106	−0.257053	−	0.966397i	$-0.582752\pi$
$929$	−15.6697	−0.514106	−0.257053	+	0.966397i	$0.582752\pi$
$930$	0	0
$931$	−21.4955	−0.704485
$932$	− 15.9129i	− 0.521244i
$933$	3.00000i	0.0982156i
$934$	14.3303	0.468902
$935$	0	0
$936$	6.49545	0.212311
$937$	− 18.5826i	− 0.607066i	−0.952821	−	0.303533i	$-0.901834\pi$
$937$	− 18.5826i	− 0.607066i	0.952821	−	0.303533i	$-0.0981663\pi$
$938$	7.46099i	0.243610i
$939$	−10.5826	−0.345349
$940$	0	0
$941$	19.9129	0.649141	0.324571	−	0.945861i	$-0.394780\pi$
$941$	19.9129	0.649141	0.324571	+	0.945861i	$0.394780\pi$
$942$	19.2523i	0.627273i
$943$	− 36.6606i	− 1.19383i
$944$	−37.5826	−1.22321
$945$	0	0
$946$	−85.8258	−2.79044
$947$	− 54.9129i	− 1.78443i	−0.451612	−	0.892214i	$-0.649151\pi$
$947$	− 54.9129i	− 1.78443i	0.451612	−	0.892214i	$-0.350849\pi$
$948$	− 9.16515i	− 0.297670i
$949$	18.3303	0.595027
$950$	0	0
$951$	−25.0000	−0.810681
$952$	4.25227i	0.137817i
$953$	37.5826i	1.21742i	0.793393	+	0.608710i	$0.208313\pi$
$953$	37.5826i	1.21742i	−0.793393	+	0.608710i	$0.791687\pi$
$954$	−0.747727	−0.0242086
$955$	0	0
$956$	−31.4265	−1.01641
$957$	5.00000i	0.161627i
$958$	34.3303i	1.10916i
$959$	−16.3303	−0.527333
$960$	0	0
$961$	−15.0000	−0.483871
$962$	32.8348i	1.05864i
$963$	− 5.16515i	− 0.166445i
$964$	−35.4519	−1.14183
$965$	0	0
$966$	−7.16515	−0.230535
$967$	9.25227i	0.297533i	0.988872	+	0.148767i	$0.0475303\pi$
$967$	9.25227i	0.297533i	−0.988872	+	0.148767i	$0.952470\pi$
$968$	19.8439i	0.637808i
$969$	10.7477	0.345267
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	1.20871i	0.0387695i
$973$	9.41742i	0.301909i
$974$	−4.17424	−0.133751
$975$	0	0
$976$	−63.1833	−2.02245
$977$	− 2.50455i	− 0.0801275i	−0.999197	−	0.0400638i	$-0.987244\pi$
$977$	− 2.50455i	− 0.0801275i	0.999197	−	0.0400638i	$-0.0127561\pi$
$978$	− 13.5826i	− 0.434323i
$979$	−7.08712	−0.226505
$980$	0	0
$981$	14.1652	0.452258
$982$	28.6606i	0.914597i
$983$	− 55.1652i	− 1.75950i	−0.475441	−	0.879748i	$-0.657712\pi$
$983$	− 55.1652i	− 1.75950i	0.475441	−	0.879748i	$-0.342288\pi$
$984$	−12.9909	−0.414135
$985$	0	0
$986$	−5.37386	−0.171139
$987$	10.5826i	0.336847i
$988$	19.8439i	0.631320i
$989$	−38.3303	−1.21883
$990$	0	0
$991$	16.0780	0.510735	0.255368	−	0.966844i	$-0.417803\pi$
$991$	16.0780	0.510735	0.255368	+	0.966844i	$0.417803\pi$
$992$	− 24.1742i	− 0.767533i
$993$	8.33030i	0.264354i
$994$	17.1652	0.544446
$995$	0	0
$996$	−14.0000	−0.443607
$997$	− 0.834849i	− 0.0264399i	−0.999913	−	0.0132200i	$-0.995792\pi$
$997$	− 0.834849i	− 0.0264399i	0.999913	−	0.0132200i	$-0.00420817\pi$
$998$	24.0345i	0.760798i
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.c.f.349.3		4
5.2	odd	4		2175.2.a.r.1.1		2
5.3	odd	4		435.2.a.f.1.2	✓	2
5.4	even	2	inner	2175.2.c.f.349.2		4
15.2	even	4		6525.2.a.t.1.2		2
15.8	even	4		1305.2.a.m.1.1		2
20.3	even	4		6960.2.a.bw.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.a.f.1.2	✓	2	5.3	odd	4
1305.2.a.m.1.1		2	15.8	even	4
2175.2.a.r.1.1		2	5.2	odd	4
2175.2.c.f.349.2		4	5.4	even	2	inner
2175.2.c.f.349.3		4	1.1	even	1	trivial
6525.2.a.t.1.2		2	15.2	even	4
6960.2.a.bw.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)

\(f(q)\)	\(=\)	\(q+1.79129i q^{2} -1.00000i q^{3} -1.20871 q^{4} +1.79129 q^{6} +1.00000i q^{7} +1.41742i q^{8} -1.00000 q^{9} +5.00000 q^{11} +1.20871i q^{12} +4.58258i q^{13} -1.79129 q^{14} -4.95644 q^{16} -3.00000i q^{17} -1.79129i q^{18} -3.58258 q^{19} +1.00000 q^{21} +8.95644i q^{22} +4.00000i q^{23} +1.41742 q^{24} -8.20871 q^{26} +1.00000i q^{27} -1.20871i q^{28} -1.00000 q^{29} +4.00000 q^{31} -6.04356i q^{32} -5.00000i q^{33} +5.37386 q^{34} +1.20871 q^{36} -4.00000i q^{37} -6.41742i q^{38} +4.58258 q^{39} -9.16515 q^{41} +1.79129i q^{42} +9.58258i q^{43} -6.04356 q^{44} -7.16515 q^{46} +10.5826i q^{47} +4.95644i q^{48} +6.00000 q^{49} -3.00000 q^{51} -5.53901i q^{52} -0.417424i q^{53} -1.79129 q^{54} -1.41742 q^{56} +3.58258i q^{57} -1.79129i q^{58} +7.58258 q^{59} +12.7477 q^{61} +7.16515i q^{62} -1.00000i q^{63} +0.912878 q^{64} +8.95644 q^{66} -4.16515i q^{67} +3.62614i q^{68} +4.00000 q^{69} -9.58258 q^{71} -1.41742i q^{72} -4.00000i q^{73} +7.16515 q^{74} +4.33030 q^{76} +5.00000i q^{77} +8.20871i q^{78} -7.58258 q^{79} +1.00000 q^{81} -16.4174i q^{82} +11.5826i q^{83} -1.20871 q^{84} -17.1652 q^{86} +1.00000i q^{87} +7.08712i q^{88} -1.41742 q^{89} -4.58258 q^{91} -4.83485i q^{92} -4.00000i q^{93} -18.9564 q^{94} -6.04356 q^{96} +11.5826i q^{97} +10.7477i q^{98} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 14 q^{4} - 2 q^{6} - 4 q^{9} + 20 q^{11} + 2 q^{14} + 26 q^{16} + 4 q^{19} + 4 q^{21} + 24 q^{24} - 42 q^{26} - 4 q^{29} + 16 q^{31} - 6 q^{34} + 14 q^{36} - 70 q^{44} + 8 q^{46} + 24 q^{49} - 12 q^{51}+ \cdots - 20 q^{99}+O(q^{100}) \) 4 * q - 14 * q^4 - 2 * q^6 - 4 * q^9 + 20 * q^11 + 2 * q^14 + 26 * q^16 + 4 * q^19 + 4 * q^21 + 24 * q^24 - 42 * q^26 - 4 * q^29 + 16 * q^31 - 6 * q^34 + 14 * q^36 - 70 * q^44 + 8 * q^46 + 24 * q^49 - 12 * q^51 + 2 * q^54 - 24 * q^56 + 12 * q^59 - 4 * q^61 - 88 * q^64 - 10 * q^66 + 16 * q^69 - 20 * q^71 - 8 * q^74 - 56 * q^76 - 12 * q^79 + 4 * q^81 - 14 * q^84 - 32 * q^86 - 24 * q^89 - 30 * q^94 - 70 * q^96 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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