LMFDB - Embedded newform 925.2.b.f.149.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [925,2,Mod(149,925)] 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("925.149"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 925 = 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	925.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,-20,0,-12,0,0,-12,0,-10] 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:	$7.38616218697$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.60703296077824.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 20x^{8} + 142x^{6} + 420x^{4} + 457x^{2} + 144 $ x^10 + 20x^8 + 142x^6 + 420x^4 + 457x^2 + 144
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 185)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.6
Root			$-0.728950i$ of defining polynomial
Character	$\chi$	$=$	925.149
Dual form			925.2.b.f.149.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.728950i q^{2} -2.62871i q^{3} +1.46863 q^{4} +1.91620 q^{6} +2.55244i q^{7} +2.52846i q^{8} -3.91009 q^{9} +2.46863 q^{11} -3.86060i q^{12} -1.55854i q^{13} -1.86060 q^{14} +1.09414 q^{16} -6.83662i q^{17} -2.85026i q^{18} +7.66011 q^{19} +6.70960 q^{21} +1.79951i q^{22} +7.50003i q^{23} +6.64658 q^{24} +1.13610 q^{26} +2.39236i q^{27} +3.74859i q^{28} -3.25741 q^{29} +0.658785 q^{31} +5.85449i q^{32} -6.48930i q^{33} +4.98356 q^{34} -5.74248 q^{36} +1.00000i q^{37} +5.58384i q^{38} -4.09694 q^{39} +2.46863 q^{41} +4.89097i q^{42} -10.9579i q^{43} +3.62551 q^{44} -5.46715 q^{46} +3.11521i q^{47} -2.87617i q^{48} +0.485072 q^{49} -17.9715 q^{51} -2.28892i q^{52} -8.64184i q^{53} -1.74391 q^{54} -6.45373 q^{56} -20.1362i q^{57} -2.37449i q^{58} +6.23634 q^{59} +3.27808 q^{61} +0.480222i q^{62} -9.98026i q^{63} -2.07935 q^{64} +4.73038 q^{66} +1.47764i q^{67} -10.0405i q^{68} +19.7154 q^{69} -8.06686 q^{71} -9.88651i q^{72} +4.96199i q^{73} -0.728950 q^{74} +11.2499 q^{76} +6.30102i q^{77} -2.98647i q^{78} -12.8206 q^{79} -5.44146 q^{81} +1.79951i q^{82} -1.14934i q^{83} +9.85393 q^{84} +7.98779 q^{86} +8.56277i q^{87} +6.24184i q^{88} -11.5207 q^{89} +3.97807 q^{91} +11.0148i q^{92} -1.73175i q^{93} -2.27083 q^{94} +15.3897 q^{96} +17.2929i q^{97} +0.353594i q^{98} -9.65257 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 20 q^{4} - 12 q^{6} - 12 q^{9} - 10 q^{11} + 16 q^{14} + 32 q^{16} + 8 q^{19} + 6 q^{21} + 84 q^{24} - 8 q^{26} + 8 q^{29} + 16 q^{31} + 64 q^{34} + 32 q^{36} - 4 q^{39} - 10 q^{41} + 92 q^{44} - 44 q^{49}+ \cdots + 20 q^{99}+O(q^{100}) $ 10 * q - 20 * q^4 - 12 * q^6 - 12 * q^9 - 10 * q^11 + 16 * q^14 + 32 * q^16 + 8 * q^19 + 6 * q^21 + 84 * q^24 - 8 * q^26 + 8 * q^29 + 16 * q^31 + 64 * q^34 + 32 * q^36 - 4 * q^39 - 10 * q^41 + 92 * q^44 - 44 * q^49 - 4 * q^51 + 20 * q^54 + 16 * q^56 + 60 * q^59 - 28 * q^61 - 40 * q^64 + 96 * q^66 - 16 * q^69 - 14 * q^71 - 4 * q^74 + 24 * q^76 - 56 * q^79 - 62 * q^81 + 36 * q^84 + 88 * q^86 - 12 * q^89 - 28 * q^91 - 8 * q^94 - 84 * q^96 + 20 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/925\mathbb{Z}\right)^\times$.

$n$	$76$	$852$
$\chi(n)$	$1$	$-1$

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 $

$2$	0.728950i	0.515446i	0.966219	+	0.257723i	$0.0829722\pi$
$2$	0.728950i	0.515446i	−0.966219	+	0.257723i	$0.917028\pi$
$3$	− 2.62871i	− 1.51768i	−0.651275	−	0.758842i	$-0.725766\pi$
$3$	− 2.62871i	− 1.51768i	0.651275	−	0.758842i	$-0.274234\pi$
$4$	1.46863	0.734316
$5$	0	0
$5$	0	0
$6$	1.91620	0.782284
$7$	2.55244i	0.964730i	0.875970	+	0.482365i	$0.160222\pi$
$7$	2.55244i	0.964730i	−0.875970	+	0.482365i	$0.839778\pi$
$8$	2.52846i	0.893946i
$9$	−3.91009	−1.30336
$10$	0	0
$11$	2.46863	0.744320	0.372160	−	0.928169i	$-0.378617\pi$
$11$	2.46863	0.744320	0.372160	+	0.928169i	$0.378617\pi$
$12$	− 3.86060i	− 1.11446i
$13$	− 1.55854i	− 0.432261i	−0.976364	−	0.216131i	$-0.930656\pi$
$13$	− 1.55854i	− 0.432261i	0.976364	−	0.216131i	$-0.0693437\pi$
$14$	−1.86060	−0.497266
$15$	0	0
$16$	1.09414	0.273535
$17$	− 6.83662i	− 1.65812i	−0.559156	−	0.829062i	$-0.688875\pi$
$17$	− 6.83662i	− 1.65812i	0.559156	−	0.829062i	$-0.311125\pi$
$18$	− 2.85026i	− 0.671813i
$19$	7.66011	1.75735	0.878675	−	0.477421i	$-0.158428\pi$
$19$	7.66011	1.75735	0.878675	+	0.477421i	$0.158428\pi$
$20$	0	0
$21$	6.70960	1.46415
$22$	1.79951i	0.383657i
$23$	7.50003i	1.56387i	0.623363	+	0.781933i	$0.285766\pi$
$23$	7.50003i	1.56387i	−0.623363	+	0.781933i	$0.714234\pi$
$24$	6.64658	1.35673
$25$	0	0
$26$	1.13610	0.222807
$27$	2.39236i	0.460410i
$28$	3.74859i	0.708416i
$29$	−3.25741	−0.604886	−0.302443	−	0.953167i	$-0.597802\pi$
$29$	−3.25741	−0.604886	−0.302443	+	0.953167i	$0.597802\pi$
$30$	0	0
$31$	0.658785	0.118321	0.0591607	−	0.998248i	$-0.481158\pi$
$31$	0.658785	0.118321	0.0591607	+	0.998248i	$0.481158\pi$
$32$	5.85449i	1.03494i
$33$	− 6.48930i	− 1.12964i
$34$	4.98356	0.854673
$35$	0	0
$36$	−5.74248	−0.957080
$37$	1.00000i	0.164399i
$37$	1.00000i	0.164399i
$38$	5.58384i	0.905818i
$39$	−4.09694	−0.656036
$40$	0	0
$41$	2.46863	0.385535	0.192768	−	0.981244i	$-0.438254\pi$
$41$	2.46863	0.385535	0.192768	+	0.981244i	$0.438254\pi$
$42$	4.89097i	0.754692i
$43$	− 10.9579i	− 1.67107i	−0.549438	−	0.835535i	$-0.685158\pi$
$43$	− 10.9579i	− 1.67107i	0.549438	−	0.835535i	$-0.314842\pi$
$44$	3.62551	0.546566
$45$	0	0
$46$	−5.46715	−0.806088
$47$	3.11521i	0.454400i	0.973848	+	0.227200i	$0.0729571\pi$
$47$	3.11521i	0.454400i	−0.973848	+	0.227200i	$0.927043\pi$
$48$	− 2.87617i	− 0.415140i
$49$	0.485072	0.0692960
$50$	0	0
$51$	−17.9715	−2.51651
$52$	− 2.28892i	− 0.317416i
$53$	− 8.64184i	− 1.18705i	−0.804816	−	0.593524i	$-0.797736\pi$
$53$	− 8.64184i	− 1.18705i	0.804816	−	0.593524i	$-0.202264\pi$
$54$	−1.74391	−0.237317
$55$	0	0
$56$	−6.45373	−0.862416
$57$	− 20.1362i	− 2.66710i
$58$	− 2.37449i	− 0.311786i
$59$	6.23634	0.811903	0.405951	−	0.913895i	$-0.366940\pi$
$59$	6.23634	0.811903	0.405951	+	0.913895i	$0.366940\pi$
$60$	0	0
$61$	3.27808	0.419716	0.209858	−	0.977732i	$-0.432700\pi$
$61$	3.27808	0.419716	0.209858	+	0.977732i	$0.432700\pi$
$62$	0.480222i	0.0609882i
$63$	− 9.98026i	− 1.25739i
$64$	−2.07935	−0.259919
$65$	0	0
$66$	4.73038	0.582270
$67$	1.47764i	0.180523i	0.995918	+	0.0902615i	$0.0287703\pi$
$67$	1.47764i	0.180523i	−0.995918	+	0.0902615i	$0.971230\pi$
$68$	− 10.0405i	− 1.21759i
$69$	19.7154	2.37345
$70$	0	0
$71$	−8.06686	−0.957360	−0.478680	−	0.877989i	$-0.658885\pi$
$71$	−8.06686	−0.957360	−0.478680	+	0.877989i	$0.658885\pi$
$72$	− 9.88651i	− 1.16514i
$73$	4.96199i	0.580757i	0.956912	+	0.290379i	$0.0937812\pi$
$73$	4.96199i	0.580757i	−0.956912	+	0.290379i	$0.906219\pi$
$74$	−0.728950	−0.0847388
$75$	0	0
$76$	11.2499	1.29045
$77$	6.30102i	0.718068i
$78$	− 2.98647i	− 0.338151i
$79$	−12.8206	−1.44243	−0.721214	−	0.692713i	$-0.756415\pi$
$79$	−12.8206	−1.44243	−0.721214	+	0.692713i	$0.756415\pi$
$80$	0	0
$81$	−5.44146	−0.604607
$82$	1.79951i	0.198723i
$83$	− 1.14934i	− 0.126157i	−0.998009	−	0.0630784i	$-0.979908\pi$
$83$	− 1.14934i	− 0.126157i	0.998009	−	0.0630784i	$-0.0200918\pi$
$84$	9.85393	1.07515
$85$	0	0
$86$	7.98779	0.861346
$87$	8.56277i	0.918026i
$88$	6.24184i	0.665382i
$89$	−11.5207	−1.22119	−0.610596	−	0.791942i	$-0.709070\pi$
$89$	−11.5207	−1.22119	−0.610596	+	0.791942i	$0.709070\pi$
$90$	0	0
$91$	3.97807	0.417015
$92$	11.0148i	1.14837i
$93$	− 1.73175i	− 0.179574i
$94$	−2.27083	−0.234218
$95$	0	0
$96$	15.3897	1.57071
$97$	17.2929i	1.75583i	0.478815	+	0.877916i	$0.341067\pi$
$97$	17.2929i	1.75583i	−0.478815	+	0.877916i	$0.658933\pi$
$98$	0.353594i	0.0357183i
$99$	−9.65257	−0.970120
$100$	0	0
$101$	−7.33253	−0.729614	−0.364807	−	0.931083i	$-0.618865\pi$
$101$	−7.33253	−0.729614	−0.364807	+	0.931083i	$0.618865\pi$
$102$	− 13.1003i	− 1.29712i
$103$	− 9.26962i	− 0.913363i	−0.889630	−	0.456681i	$-0.849038\pi$
$103$	− 9.26962i	− 0.913363i	0.889630	−	0.456681i	$-0.150962\pi$
$104$	3.94071	0.386418
$105$	0	0
$106$	6.29947	0.611859
$107$	− 12.0987i	− 1.16962i	−0.811170	−	0.584811i	$-0.801169\pi$
$107$	− 12.0987i	− 1.16962i	0.811170	−	0.584811i	$-0.198831\pi$
$108$	3.51350i	0.338086i
$109$	2.24262	0.214804	0.107402	−	0.994216i	$-0.465747\pi$
$109$	2.24262	0.214804	0.107402	+	0.994216i	$0.465747\pi$
$110$	0	0
$111$	2.62871	0.249506
$112$	2.79272i	0.263888i
$113$	11.3787i	1.07042i	0.844719	+	0.535210i	$0.179768\pi$
$113$	11.3787i	1.07042i	−0.844719	+	0.535210i	$0.820232\pi$
$114$	14.6783	1.37475
$115$	0	0
$116$	−4.78394	−0.444177
$117$	6.09403i	0.563394i
$118$	4.54598i	0.418492i
$119$	17.4500	1.59964
$120$	0	0
$121$	−4.90586	−0.445987
$122$	2.38956i	0.216341i
$123$	− 6.48930i	− 0.585121i
$124$	0.967513	0.0868852
$125$	0	0
$126$	7.27511	0.648118
$127$	15.6642i	1.38997i	0.719024	+	0.694985i	$0.244589\pi$
$127$	15.6642i	1.38997i	−0.719024	+	0.694985i	$0.755411\pi$
$128$	10.1932i	0.900964i
$129$	−28.8052	−2.53615
$130$	0	0
$131$	−13.6723	−1.19456	−0.597278	−	0.802034i	$-0.703751\pi$
$131$	−13.6723	−1.19456	−0.597278	+	0.802034i	$0.703751\pi$
$132$	− 9.53040i	− 0.829514i
$133$	19.5519i	1.69537i
$134$	−1.07713	−0.0930498
$135$	0	0
$136$	17.2861	1.48227
$137$	− 14.2031i	− 1.21346i	−0.794910	−	0.606728i	$-0.792482\pi$
$137$	− 14.2031i	− 1.21346i	0.794910	−	0.606728i	$-0.207518\pi$
$138$	14.3715i	1.22339i
$139$	11.8416	1.00440	0.502198	−	0.864753i	$-0.332525\pi$
$139$	11.8416	1.00440	0.502198	+	0.864753i	$0.332525\pi$
$140$	0	0
$141$	8.18896	0.689635
$142$	− 5.88034i	− 0.493467i
$143$	− 3.84746i	− 0.321741i
$144$	−4.27819	−0.356516
$145$	0	0
$146$	−3.61705	−0.299349
$147$	− 1.27511i	− 0.105169i
$148$	1.46863i	0.120721i
$149$	−4.11726	−0.337299	−0.168649	−	0.985676i	$-0.553941\pi$
$149$	−4.11726	−0.337299	−0.168649	+	0.985676i	$0.553941\pi$
$150$	0	0
$151$	16.9092	1.37605	0.688025	−	0.725687i	$-0.258478\pi$
$151$	16.9092	1.37605	0.688025	+	0.725687i	$0.258478\pi$
$152$	19.3683i	1.57097i
$153$	26.7318i	2.16114i
$154$	−4.59313	−0.370125
$155$	0	0
$156$	−6.01690	−0.481737
$157$	4.02473i	0.321208i	0.987019	+	0.160604i	$0.0513442\pi$
$157$	4.02473i	0.321208i	−0.987019	+	0.160604i	$0.948656\pi$
$158$	− 9.34556i	− 0.743493i
$159$	−22.7169	−1.80156
$160$	0	0
$161$	−19.1434	−1.50871
$162$	− 3.96655i	− 0.311642i
$163$	− 3.57756i	− 0.280216i	−0.990136	−	0.140108i	$-0.955255\pi$
$163$	− 3.57756i	− 0.280216i	0.990136	−	0.140108i	$-0.0447450\pi$
$164$	3.62551	0.283105
$165$	0	0
$166$	0.837814	0.0650270
$167$	23.6674i	1.83144i	0.401816	+	0.915720i	$0.368379\pi$
$167$	23.6674i	1.83144i	−0.401816	+	0.915720i	$0.631621\pi$
$168$	16.9650i	1.30887i
$169$	10.5710	0.813150
$170$	0	0
$171$	−29.9517	−2.29047
$172$	− 16.0932i	− 1.22709i
$173$	15.7383i	1.19656i	0.801288	+	0.598279i	$0.204148\pi$
$173$	15.7383i	1.19656i	−0.801288	+	0.598279i	$0.795852\pi$
$174$	−6.24184	−0.473192
$175$	0	0
$176$	2.70103	0.203598
$177$	− 16.3935i	− 1.23221i
$178$	− 8.39802i	− 0.629459i
$179$	−10.1749	−0.760510	−0.380255	−	0.924882i	$-0.624164\pi$
$179$	−10.1749	−0.760510	−0.380255	+	0.924882i	$0.624164\pi$
$180$	0	0
$181$	−2.54781	−0.189377	−0.0946886	−	0.995507i	$-0.530186\pi$
$181$	−2.54781	−0.189377	−0.0946886	+	0.995507i	$0.530186\pi$
$182$	2.89982i	0.214949i
$183$	− 8.61711i	− 0.636995i
$184$	−18.9635	−1.39801
$185$	0	0
$186$	1.26236	0.0925608
$187$	− 16.8771i	− 1.23418i
$188$	4.57509i	0.333673i
$189$	−6.10635	−0.444172
$190$	0	0
$191$	−10.4639	−0.757141	−0.378571	−	0.925572i	$-0.623584\pi$
$191$	−10.4639	−0.757141	−0.378571	+	0.925572i	$0.623584\pi$
$192$	5.46601i	0.394475i
$193$	9.81172i	0.706263i	0.935574	+	0.353131i	$0.114883\pi$
$193$	9.81172i	0.706263i	−0.935574	+	0.353131i	$0.885117\pi$
$194$	−12.6057	−0.905036
$195$	0	0
$196$	0.712392	0.0508852
$197$	13.3945i	0.954317i	0.878817	+	0.477158i	$0.158333\pi$
$197$	13.3945i	0.954317i	−0.878817	+	0.477158i	$0.841667\pi$
$198$	− 7.03625i	− 0.500044i
$199$	−4.20221	−0.297887	−0.148943	−	0.988846i	$-0.547587\pi$
$199$	−4.20221	−0.297887	−0.148943	+	0.988846i	$0.547587\pi$
$200$	0	0
$201$	3.88429	0.273977
$202$	− 5.34505i	− 0.376077i
$203$	− 8.31433i	− 0.583552i
$204$	−26.3935	−1.84791
$205$	0	0
$206$	6.75709	0.470789
$207$	− 29.3258i	− 2.03829i
$208$	− 1.70526i	− 0.118239i
$209$	18.9100	1.30803
$210$	0	0
$211$	20.0610	1.38106	0.690530	−	0.723304i	$-0.257377\pi$
$211$	20.0610	1.38106	0.690530	+	0.723304i	$0.257377\pi$
$212$	− 12.6917i	− 0.871668i
$213$	21.2054i	1.45297i
$214$	8.81932	0.602876
$215$	0	0
$216$	−6.04899	−0.411582
$217$	1.68151i	0.114148i
$218$	1.63476i	0.110720i
$219$	13.0436	0.881406
$220$	0	0
$221$	−10.6552	−0.716743
$222$	1.91620i	0.128607i
$223$	− 1.72511i	− 0.115522i	−0.998330	−	0.0577610i	$-0.981604\pi$
$223$	− 1.72511i	− 0.115522i	0.998330	−	0.0577610i	$-0.0183961\pi$
$224$	−14.9432	−0.998436
$225$	0	0
$226$	−8.29452	−0.551744
$227$	4.59930i	0.305266i	0.988283	+	0.152633i	$0.0487753\pi$
$227$	4.59930i	0.305266i	−0.988283	+	0.152633i	$0.951225\pi$
$228$	− 29.5726i	− 1.95849i
$229$	23.4902	1.55227	0.776137	−	0.630565i	$-0.217177\pi$
$229$	23.4902	1.55227	0.776137	+	0.630565i	$0.217177\pi$
$230$	0	0
$231$	16.5635	1.08980
$232$	− 8.23623i	− 0.540735i
$233$	− 2.43465i	− 0.159499i	−0.996815	−	0.0797496i	$-0.974588\pi$
$233$	− 2.43465i	− 0.159499i	0.996815	−	0.0797496i	$-0.0254121\pi$
$234$	−4.44225	−0.290399
$235$	0	0
$236$	9.15889	0.596193
$237$	33.7015i	2.18915i
$238$	12.7202i	0.824529i
$239$	−2.03195	−0.131436	−0.0657180	−	0.997838i	$-0.520934\pi$
$239$	−2.03195	−0.131436	−0.0657180	+	0.997838i	$0.520934\pi$
$240$	0	0
$241$	−20.2246	−1.30278	−0.651390	−	0.758743i	$-0.725814\pi$
$241$	−20.2246	−1.30278	−0.651390	+	0.758743i	$0.725814\pi$
$242$	− 3.57613i	− 0.229882i
$243$	21.4811i	1.37801i
$244$	4.81430	0.308204
$245$	0	0
$246$	4.73038	0.301598
$247$	− 11.9386i	− 0.759634i
$248$	1.66571i	0.105773i
$249$	−3.02128	−0.191466
$250$	0	0
$251$	0.695568	0.0439038	0.0219519	−	0.999759i	$-0.493012\pi$
$251$	0.695568	0.0439038	0.0219519	+	0.999759i	$0.493012\pi$
$252$	− 14.6573i	− 0.923324i
$253$	18.5148i	1.16402i
$254$	−11.4184	−0.716454
$255$	0	0
$256$	−11.5891	−0.724317
$257$	7.13610i	0.445138i	0.974917	+	0.222569i	$0.0714442\pi$
$257$	7.13610i	0.445138i	−0.974917	+	0.222569i	$0.928556\pi$
$258$	− 20.9975i	− 1.30725i
$259$	−2.55244	−0.158601
$260$	0	0
$261$	12.7368	0.788386
$262$	− 9.96644i	− 0.615729i
$263$	− 21.2754i	− 1.31190i	−0.754806	−	0.655948i	$-0.772269\pi$
$263$	− 21.2754i	− 1.31190i	0.754806	−	0.655948i	$-0.227731\pi$
$264$	16.4079	1.00984
$265$	0	0
$266$	−14.2524	−0.873870
$267$	30.2845i	1.85338i
$268$	2.17011i	0.132561i
$269$	−18.3525	−1.11897	−0.559486	−	0.828840i	$-0.689002\pi$
$269$	−18.3525	−1.11897	−0.559486	+	0.828840i	$0.689002\pi$
$270$	0	0
$271$	−23.2195	−1.41048	−0.705241	−	0.708967i	$-0.749161\pi$
$271$	−23.2195	−1.41048	−0.705241	+	0.708967i	$0.749161\pi$
$272$	− 7.48023i	− 0.453555i
$273$	− 10.4572i	− 0.632897i
$274$	10.3534	0.625471
$275$	0	0
$276$	28.9546	1.74286
$277$	5.65674i	0.339880i	0.985454	+	0.169940i	$0.0543574\pi$
$277$	5.65674i	0.339880i	−0.985454	+	0.169940i	$0.945643\pi$
$278$	8.63197i	0.517711i
$279$	−2.57591	−0.154216
$280$	0	0
$281$	−24.4373	−1.45781	−0.728903	−	0.684616i	$-0.759970\pi$
$281$	−24.4373	−1.45781	−0.728903	+	0.684616i	$0.759970\pi$
$282$	5.96935i	0.355469i
$283$	− 20.8498i	− 1.23939i	−0.784841	−	0.619697i	$-0.787256\pi$
$283$	− 20.8498i	− 1.23939i	0.784841	−	0.619697i	$-0.212744\pi$
$284$	−11.8472	−0.703005
$285$	0	0
$286$	2.80461	0.165840
$287$	6.30102i	0.371938i
$288$	− 22.8916i	− 1.34890i
$289$	−29.7394	−1.74938
$290$	0	0
$291$	45.4580	2.66480
$292$	7.28734i	0.426459i
$293$	− 7.72199i	− 0.451123i	−0.974229	−	0.225562i	$-0.927578\pi$
$293$	− 7.72199i	− 0.451123i	0.974229	−	0.225562i	$-0.0724217\pi$
$294$	0.929493	0.0542091
$295$	0	0
$296$	−2.52846	−0.146964
$297$	5.90586i	0.342693i
$298$	− 3.00128i	− 0.173859i
$299$	11.6891	0.675998
$300$	0	0
$301$	27.9694	1.61213
$302$	12.3260i	0.709280i
$303$	19.2751i	1.10732i
$304$	8.38124	0.480697
$305$	0	0
$306$	−19.4862	−1.11395
$307$	− 9.01041i	− 0.514251i	−0.966378	−	0.257126i	$-0.917225\pi$
$307$	− 9.01041i	− 0.514251i	0.966378	−	0.257126i	$-0.0827754\pi$
$308$	9.25388i	0.527289i
$309$	−24.3671	−1.38620
$310$	0	0
$311$	−32.1220	−1.82147	−0.910735	−	0.412991i	$-0.864484\pi$
$311$	−32.1220	−1.82147	−0.910735	+	0.412991i	$0.864484\pi$
$312$	− 10.3590i	− 0.586460i
$313$	− 19.8623i	− 1.12268i	−0.827584	−	0.561342i	$-0.810285\pi$
$313$	− 19.8623i	− 1.12268i	0.827584	−	0.561342i	$-0.189715\pi$
$314$	−2.93383	−0.165565
$315$	0	0
$316$	−18.8287	−1.05920
$317$	− 8.58984i	− 0.482453i	−0.970469	−	0.241227i	$-0.922450\pi$
$317$	− 8.58984i	− 0.482453i	0.970469	−	0.241227i	$-0.0775497\pi$
$318$	− 16.5595i	− 0.928609i
$319$	−8.04135	−0.450229
$320$	0	0
$321$	−31.8038	−1.77512
$322$	− 13.9546i	− 0.777657i
$323$	− 52.3693i	− 2.91390i
$324$	−7.99150	−0.443972
$325$	0	0
$326$	2.60786	0.144436
$327$	− 5.89520i	− 0.326005i
$328$	6.24184i	0.344648i
$329$	−7.95137	−0.438373
$330$	0	0
$331$	−22.9711	−1.26260	−0.631302	−	0.775537i	$-0.717479\pi$
$331$	−22.9711	−1.26260	−0.631302	+	0.775537i	$0.717479\pi$
$332$	− 1.68796i	− 0.0926389i
$333$	− 3.91009i	− 0.214272i
$334$	−17.2524	−0.944008
$335$	0	0
$336$	7.34125	0.400498
$337$	26.2320i	1.42895i	0.699663	+	0.714473i	$0.253334\pi$
$337$	26.2320i	1.42895i	−0.699663	+	0.714473i	$0.746666\pi$
$338$	7.70570i	0.419135i
$339$	29.9113	1.62456
$340$	0	0
$341$	1.62630	0.0880690
$342$	− 21.8333i	− 1.18061i
$343$	19.1052i	1.03158i
$344$	27.7067	1.49385
$345$	0	0
$346$	−11.4724	−0.616760
$347$	− 0.390348i	− 0.0209550i	−0.999945	−	0.0104775i	$-0.996665\pi$
$347$	− 0.390348i	− 0.0209550i	0.999945	−	0.0104775i	$-0.00333515\pi$
$348$	12.5756i	0.674121i
$349$	6.37205	0.341088	0.170544	−	0.985350i	$-0.445448\pi$
$349$	6.37205	0.341088	0.170544	+	0.985350i	$0.445448\pi$
$350$	0	0
$351$	3.72859	0.199017
$352$	14.4526i	0.770326i
$353$	9.35970i	0.498167i	0.968482	+	0.249083i	$0.0801293\pi$
$353$	9.35970i	0.498167i	−0.968482	+	0.249083i	$0.919871\pi$
$354$	11.9501	0.635138
$355$	0	0
$356$	−16.9197	−0.896741
$357$	− 45.8710i	− 2.42775i
$358$	− 7.41702i	− 0.392002i
$359$	5.38443	0.284179	0.142090	−	0.989854i	$-0.454618\pi$
$359$	5.38443	0.284179	0.142090	+	0.989854i	$0.454618\pi$
$360$	0	0
$361$	39.6773	2.08828
$362$	− 1.85723i	− 0.0976137i
$363$	12.8961i	0.676868i
$364$	5.84232	0.306221
$365$	0	0
$366$	6.28145	0.328337
$367$	− 22.1964i	− 1.15864i	−0.815099	−	0.579321i	$-0.803318\pi$
$367$	− 22.1964i	− 1.15864i	0.815099	−	0.579321i	$-0.196682\pi$
$368$	8.20609i	0.427772i
$369$	−9.65257	−0.502493
$370$	0	0
$371$	22.0577	1.14518
$372$	− 2.54331i	− 0.131864i
$373$	31.6626i	1.63943i	0.572774	+	0.819713i	$0.305867\pi$
$373$	31.6626i	1.63943i	−0.572774	+	0.819713i	$0.694133\pi$
$374$	12.3026	0.636151
$375$	0	0
$376$	−7.87668	−0.406209
$377$	5.07680i	0.261469i
$378$	− 4.45123i	− 0.228946i
$379$	15.6622	0.804515	0.402257	−	0.915527i	$-0.368226\pi$
$379$	15.6622	0.804515	0.402257	+	0.915527i	$0.368226\pi$
$380$	0	0
$381$	41.1765	2.10953
$382$	− 7.62766i	− 0.390265i
$383$	− 7.84825i	− 0.401027i	−0.979691	−	0.200513i	$-0.935739\pi$
$383$	− 7.84825i	− 0.401027i	0.979691	−	0.200513i	$-0.0642610\pi$
$384$	26.7950	1.36738
$385$	0	0
$386$	−7.15226	−0.364040
$387$	42.8465i	2.17801i
$388$	25.3970i	1.28933i
$389$	−17.6518	−0.894981	−0.447491	−	0.894289i	$-0.647682\pi$
$389$	−17.6518	−0.894981	−0.447491	+	0.894289i	$0.647682\pi$
$390$	0	0
$391$	51.2749	2.59308
$392$	1.22649i	0.0619469i
$393$	35.9405i	1.81296i
$394$	−9.76391	−0.491899
$395$	0	0
$396$	−14.1761	−0.712374
$397$	− 22.7648i	− 1.14253i	−0.820765	−	0.571267i	$-0.806452\pi$
$397$	− 22.7648i	− 1.14253i	0.820765	−	0.571267i	$-0.193548\pi$
$398$	− 3.06320i	− 0.153544i
$399$	51.3963	2.57303
$400$	0	0
$401$	−0.168397	−0.00840934	−0.00420467	−	0.999991i	$-0.501338\pi$
$401$	−0.168397	−0.00840934	−0.00420467	+	0.999991i	$0.501338\pi$
$402$	2.83146i	0.141220i
$403$	− 1.02674i	− 0.0511457i
$404$	−10.7688	−0.535767
$405$	0	0
$406$	6.06073	0.300789
$407$	2.46863i	0.122366i
$408$	− 45.4401i	− 2.24962i
$409$	21.1075	1.04370	0.521850	−	0.853037i	$-0.325242\pi$
$409$	21.1075	1.04370	0.521850	+	0.853037i	$0.325242\pi$
$410$	0	0
$411$	−37.3359	−1.84164
$412$	− 13.6137i	− 0.670697i
$413$	15.9179i	0.783267i
$414$	21.3771	1.05063
$415$	0	0
$416$	9.12446	0.447364
$417$	− 31.1282i	− 1.52435i
$418$	13.7844i	0.674219i
$419$	22.3217	1.09049	0.545243	−	0.838278i	$-0.316437\pi$
$419$	22.3217	1.09049	0.545243	+	0.838278i	$0.316437\pi$
$420$	0	0
$421$	−27.4605	−1.33835	−0.669173	−	0.743107i	$-0.733351\pi$
$421$	−27.4605	−1.33835	−0.669173	+	0.743107i	$0.733351\pi$
$422$	14.6235i	0.711861i
$423$	− 12.1807i	− 0.592248i
$424$	21.8506	1.06116
$425$	0	0
$426$	−15.4577	−0.748927
$427$	8.36710i	0.404912i
$428$	− 17.7685i	− 0.858871i
$429$	−10.1138	−0.488301
$430$	0	0
$431$	36.7380	1.76960	0.884802	−	0.465966i	$-0.154293\pi$
$431$	36.7380	1.76960	0.884802	+	0.465966i	$0.154293\pi$
$432$	2.61758i	0.125938i
$433$	− 0.370435i	− 0.0178020i	−0.999960	−	0.00890098i	$-0.997167\pi$
$433$	− 0.370435i	− 0.0178020i	0.999960	−	0.00890098i	$-0.00283331\pi$
$434$	−1.22574	−0.0588372
$435$	0	0
$436$	3.29359	0.157734
$437$	57.4511i	2.74826i
$438$	9.50815i	0.454317i
$439$	−30.1768	−1.44026	−0.720130	−	0.693839i	$-0.755918\pi$
$439$	−30.1768	−1.44026	−0.720130	+	0.693839i	$0.755918\pi$
$440$	0	0
$441$	−1.89668	−0.0903179
$442$	− 7.76708i	− 0.369442i
$443$	− 38.1789i	− 1.81393i	−0.421203	−	0.906966i	$-0.638392\pi$
$443$	− 38.1789i	− 1.81393i	0.421203	−	0.906966i	$-0.361608\pi$
$444$	3.86060	0.183216
$445$	0	0
$446$	1.25752	0.0595454
$447$	10.8231i	0.511913i
$448$	− 5.30742i	− 0.250752i
$449$	3.77840	0.178314	0.0891569	−	0.996018i	$-0.471583\pi$
$449$	3.77840	0.178314	0.0891569	+	0.996018i	$0.471583\pi$
$450$	0	0
$451$	6.09414	0.286962
$452$	16.7111i	0.786026i
$453$	− 44.4493i	− 2.08841i
$454$	−3.35266	−0.157348
$455$	0	0
$456$	50.9135	2.38424
$457$	25.3343i	1.18509i	0.805538	+	0.592544i	$0.201876\pi$
$457$	25.3343i	1.18509i	−0.805538	+	0.592544i	$0.798124\pi$
$458$	17.1232i	0.800113i
$459$	16.3557	0.763418
$460$	0	0
$461$	−14.7582	−0.687359	−0.343680	−	0.939087i	$-0.611673\pi$
$461$	−14.7582	−0.687359	−0.343680	+	0.939087i	$0.611673\pi$
$462$	12.0740i	0.561733i
$463$	− 8.63700i	− 0.401395i	−0.979653	−	0.200698i	$-0.935679\pi$
$463$	− 8.63700i	− 0.401395i	0.979653	−	0.200698i	$-0.0643209\pi$
$464$	−3.56407	−0.165458
$465$	0	0
$466$	1.77474	0.0822132
$467$	− 1.64618i	− 0.0761762i	−0.999274	−	0.0380881i	$-0.987873\pi$
$467$	− 1.64618i	− 0.0761762i	0.999274	−	0.0380881i	$-0.0121268\pi$
$468$	8.94989i	0.413709i
$469$	−3.77159	−0.174156
$470$	0	0
$471$	10.5798	0.487493
$472$	15.7683i	0.725797i
$473$	− 27.0511i	− 1.24381i
$474$	−24.5667	−1.12839
$475$	0	0
$476$	25.6277	1.17464
$477$	33.7904i	1.54716i
$478$	− 1.48119i	− 0.0677482i
$479$	−16.0571	−0.733669	−0.366835	−	0.930286i	$-0.619558\pi$
$479$	−16.0571	−0.733669	−0.366835	+	0.930286i	$0.619558\pi$
$480$	0	0
$481$	1.55854	0.0710633
$482$	− 14.7427i	− 0.671513i
$483$	50.3222i	2.28974i
$484$	−7.20490	−0.327495
$485$	0	0
$486$	−15.6586	−0.710290
$487$	− 6.60786i	− 0.299431i	−0.988729	−	0.149715i	$-0.952164\pi$
$487$	− 6.60786i	− 0.299431i	0.988729	−	0.149715i	$-0.0478357\pi$
$488$	8.28850i	0.375203i
$489$	−9.40435	−0.425279
$490$	0	0
$491$	−18.2039	−0.821532	−0.410766	−	0.911741i	$-0.634739\pi$
$491$	−18.2039	−0.821532	−0.410766	+	0.911741i	$0.634739\pi$
$492$	− 9.53040i	− 0.429663i
$493$	22.2697i	1.00298i
$494$	8.70264	0.391550
$495$	0	0
$496$	0.720804	0.0323650
$497$	− 20.5901i	− 0.923594i
$498$	− 2.20237i	− 0.0986904i
$499$	26.1795	1.17196	0.585978	−	0.810327i	$-0.300711\pi$
$499$	26.1795	1.17196	0.585978	+	0.810327i	$0.300711\pi$
$500$	0	0
$501$	62.2147	2.77955
$502$	0.507034i	0.0226301i
$503$	− 0.546197i	− 0.0243537i	−0.999926	−	0.0121769i	$-0.996124\pi$
$503$	− 0.546197i	− 0.0243537i	0.999926	−	0.0121769i	$-0.00387611\pi$
$504$	25.2347	1.12404
$505$	0	0
$506$	−13.4964	−0.599988
$507$	− 27.7879i	− 1.23410i
$508$	23.0049i	1.02068i
$509$	27.4237	1.21553	0.607767	−	0.794115i	$-0.292065\pi$
$509$	27.4237	1.21553	0.607767	+	0.794115i	$0.292065\pi$
$510$	0	0
$511$	−12.6652	−0.560274
$512$	11.9386i	0.527618i
$513$	18.3258i	0.809102i
$514$	−5.20186	−0.229444
$515$	0	0
$516$	−42.3042	−1.86234
$517$	7.69030i	0.338219i
$518$	− 1.86060i	− 0.0817500i
$519$	41.3712	1.81600
$520$	0	0
$521$	39.2211	1.71831	0.859154	−	0.511717i	$-0.170990\pi$
$521$	39.2211	1.71831	0.859154	+	0.511717i	$0.170990\pi$
$522$	9.28448i	0.406370i
$523$	2.89217i	0.126466i	0.997999	+	0.0632329i	$0.0201411\pi$
$523$	2.89217i	0.126466i	−0.997999	+	0.0632329i	$0.979859\pi$
$524$	−20.0796	−0.877181
$525$	0	0
$526$	15.5087	0.676212
$527$	− 4.50387i	− 0.196191i
$528$	− 7.10021i	− 0.308997i
$529$	−33.2505	−1.44567
$530$	0	0
$531$	−24.3847	−1.05820
$532$	28.7146i	1.24494i
$533$	− 3.84746i	− 0.166652i
$534$	−22.0759	−0.955319
$535$	0	0
$536$	−3.73616	−0.161378
$537$	26.7469i	1.15421i
$538$	− 13.3781i	− 0.576770i
$539$	1.19746	0.0515784
$540$	0	0
$541$	−30.3727	−1.30582	−0.652912	−	0.757434i	$-0.726453\pi$
$541$	−30.3727	−1.30582	−0.652912	+	0.757434i	$0.726453\pi$
$542$	− 16.9258i	− 0.727027i
$543$	6.69744i	0.287415i
$544$	40.0250	1.71606
$545$	0	0
$546$	7.62277	0.326224
$547$	18.8597i	0.806384i	0.915115	+	0.403192i	$0.132099\pi$
$547$	18.8597i	0.806384i	−0.915115	+	0.403192i	$0.867901\pi$
$548$	− 20.8592i	− 0.891060i
$549$	−12.8176	−0.547042
$550$	0	0
$551$	−24.9521	−1.06300
$552$	49.8496i	2.12174i
$553$	− 32.7237i	− 1.39155i
$554$	−4.12348	−0.175190
$555$	0	0
$556$	17.3910	0.737543
$557$	24.2064i	1.02566i	0.858491	+	0.512828i	$0.171402\pi$
$557$	24.2064i	1.02566i	−0.858491	+	0.512828i	$0.828598\pi$
$558$	− 1.87771i	− 0.0794898i
$559$	−17.0784	−0.722339
$560$	0	0
$561$	−44.3649	−1.87309
$562$	− 17.8136i	− 0.751420i
$563$	− 5.13444i	− 0.216391i	−0.994130	−	0.108196i	$-0.965493\pi$
$563$	− 5.13444i	− 0.216391i	0.994130	−	0.108196i	$-0.0345072\pi$
$564$	12.0266	0.506410
$565$	0	0
$566$	15.1985	0.638840
$567$	− 13.8890i	− 0.583282i
$568$	− 20.3967i	− 0.855828i
$569$	11.9797	0.502214	0.251107	−	0.967959i	$-0.419205\pi$
$569$	11.9797	0.502214	0.251107	+	0.967959i	$0.419205\pi$
$570$	0	0
$571$	−25.5874	−1.07080	−0.535399	−	0.844599i	$-0.679839\pi$
$571$	−25.5874	−1.07080	−0.535399	+	0.844599i	$0.679839\pi$
$572$	− 5.65050i	− 0.236259i
$573$	27.5065i	1.14910i
$574$	−4.59313	−0.191714
$575$	0	0
$576$	8.13047	0.338769
$577$	− 40.3272i	− 1.67884i	−0.543481	−	0.839421i	$-0.682894\pi$
$577$	− 40.3272i	− 1.67884i	0.543481	−	0.839421i	$-0.317106\pi$
$578$	− 21.6786i	− 0.901709i
$579$	25.7921	1.07188
$580$	0	0
$581$	2.93362	0.121707
$582$	33.1367i	1.37356i
$583$	− 21.3335i	− 0.883544i
$584$	−12.5462	−0.519165
$585$	0	0
$586$	5.62894	0.232530
$587$	− 31.6751i	− 1.30737i	−0.756766	−	0.653686i	$-0.773222\pi$
$587$	− 31.6751i	− 1.30737i	0.756766	−	0.653686i	$-0.226778\pi$
$588$	− 1.87267i	− 0.0772276i
$589$	5.04637	0.207932
$590$	0	0
$591$	35.2101	1.44835
$592$	1.09414i	0.0449689i
$593$	− 6.26075i	− 0.257098i	−0.991703	−	0.128549i	$-0.958968\pi$
$593$	− 6.26075i	− 0.257098i	0.991703	−	0.128549i	$-0.0410320\pi$
$594$	−4.30508	−0.176640
$595$	0	0
$596$	−6.04673	−0.247684
$597$	11.0464i	0.452098i
$598$	8.52078i	0.348440i
$599$	−34.4197	−1.40635	−0.703175	−	0.711017i	$-0.748235\pi$
$599$	−34.4197	−1.40635	−0.703175	+	0.711017i	$0.748235\pi$
$600$	0	0
$601$	12.2491	0.499650	0.249825	−	0.968291i	$-0.419627\pi$
$601$	12.2491	0.499650	0.249825	+	0.968291i	$0.419627\pi$
$602$	20.3883i	0.830966i
$603$	− 5.77772i	− 0.235287i
$604$	24.8334	1.01046
$605$	0	0
$606$	−14.0506	−0.570765
$607$	− 31.0932i	− 1.26204i	−0.775768	−	0.631018i	$-0.782637\pi$
$607$	− 31.0932i	− 1.26204i	0.775768	−	0.631018i	$-0.217363\pi$
$608$	44.8461i	1.81875i
$609$	−21.8559	−0.885647
$610$	0	0
$611$	4.85518	0.196419
$612$	39.2592i	1.58696i
$613$	− 15.0998i	− 0.609876i	−0.952372	−	0.304938i	$-0.901364\pi$
$613$	− 15.0998i	− 0.609876i	0.952372	−	0.304938i	$-0.0986357\pi$
$614$	6.56814	0.265069
$615$	0	0
$616$	−15.9319	−0.641914
$617$	− 27.3635i	− 1.10161i	−0.834632	−	0.550807i	$-0.814320\pi$
$617$	− 27.3635i	− 1.10161i	0.834632	−	0.550807i	$-0.185680\pi$
$618$	− 17.7624i	− 0.714509i
$619$	11.1466	0.448021	0.224010	−	0.974587i	$-0.428085\pi$
$619$	11.1466	0.448021	0.224010	+	0.974587i	$0.428085\pi$
$620$	0	0
$621$	−17.9428	−0.720020
$622$	− 23.4153i	− 0.938869i
$623$	− 29.4059i	− 1.17812i
$624$	−4.48263	−0.179449
$625$	0	0
$626$	14.4786	0.578683
$627$	− 49.7088i	− 1.98518i
$628$	5.91084i	0.235868i
$629$	6.83662	0.272594
$630$	0	0
$631$	11.9459	0.475558	0.237779	−	0.971319i	$-0.423581\pi$
$631$	11.9459	0.475558	0.237779	+	0.971319i	$0.423581\pi$
$632$	− 32.4163i	− 1.28945i
$633$	− 52.7346i	− 2.09601i
$634$	6.26156	0.248679
$635$	0	0
$636$	−33.3627	−1.32292
$637$	− 0.756004i	− 0.0299540i
$638$	− 5.86174i	− 0.232069i
$639$	31.5422	1.24779
$640$	0	0
$641$	27.9313	1.10322	0.551609	−	0.834103i	$-0.314014\pi$
$641$	27.9313	1.10322	0.551609	+	0.834103i	$0.314014\pi$
$642$	− 23.1834i	− 0.914976i
$643$	− 39.0497i	− 1.53997i	−0.638063	−	0.769984i	$-0.720264\pi$
$643$	− 39.0497i	− 1.53997i	0.638063	−	0.769984i	$-0.279736\pi$
$644$	−28.1145	−1.10787
$645$	0	0
$646$	38.1746	1.50196
$647$	− 3.60562i	− 0.141752i	−0.997485	−	0.0708759i	$-0.977421\pi$
$647$	− 3.60562i	− 0.141752i	0.997485	−	0.0708759i	$-0.0225794\pi$
$648$	− 13.7585i	− 0.540485i
$649$	15.3952	0.604316
$650$	0	0
$651$	4.42019	0.173241
$652$	− 5.25411i	− 0.205767i
$653$	16.9343i	0.662691i	0.943510	+	0.331345i	$0.107503\pi$
$653$	16.9343i	0.662691i	−0.943510	+	0.331345i	$0.892497\pi$
$654$	4.29731	0.168038
$655$	0	0
$656$	2.70103	0.105458
$657$	− 19.4018i	− 0.756938i
$658$	− 5.79615i	− 0.225958i
$659$	−31.2664	−1.21797	−0.608983	−	0.793183i	$-0.708422\pi$
$659$	−31.2664	−1.21797	−0.608983	+	0.793183i	$0.708422\pi$
$660$	0	0
$661$	−39.2941	−1.52836	−0.764181	−	0.645002i	$-0.776857\pi$
$661$	−39.2941	−1.52836	−0.764181	+	0.645002i	$0.776857\pi$
$662$	− 16.7448i	− 0.650804i
$663$	28.0093i	1.08779i
$664$	2.90607	0.112777
$665$	0	0
$666$	2.85026	0.110445
$667$	− 24.4307i	− 0.945960i
$668$	34.7587i	1.34486i
$669$	−4.53481	−0.175326
$670$	0	0
$671$	8.09238	0.312403
$672$	39.2813i	1.51531i
$673$	− 33.7258i	− 1.30004i	−0.759919	−	0.650018i	$-0.774762\pi$
$673$	− 33.7258i	− 1.30004i	0.759919	−	0.650018i	$-0.225238\pi$
$674$	−19.1218	−0.736544
$675$	0	0
$676$	15.5248	0.597109
$677$	− 20.4385i	− 0.785515i	−0.919642	−	0.392758i	$-0.871521\pi$
$677$	− 20.4385i	− 0.785515i	0.919642	−	0.392758i	$-0.128479\pi$
$678$	21.8039i	0.837372i
$679$	−44.1391	−1.69390
$680$	0	0
$681$	12.0902	0.463298
$682$	1.18549i	0.0453948i
$683$	31.0965i	1.18988i	0.803771	+	0.594938i	$0.202823\pi$
$683$	31.0965i	1.18988i	−0.803771	+	0.594938i	$0.797177\pi$
$684$	−43.9880	−1.68192
$685$	0	0
$686$	−13.9267	−0.531725
$687$	− 61.7487i	− 2.35586i
$688$	− 11.9895i	− 0.457096i
$689$	−13.4687	−0.513115
$690$	0	0
$691$	10.8292	0.411960	0.205980	−	0.978556i	$-0.433962\pi$
$691$	10.8292	0.411960	0.205980	+	0.978556i	$0.433962\pi$
$692$	23.1137i	0.878651i
$693$	− 24.6376i	− 0.935904i
$694$	0.284545	0.0108012
$695$	0	0
$696$	−21.6506	−0.820665
$697$	− 16.8771i	− 0.639266i
$698$	4.64491i	0.175812i
$699$	−6.39998	−0.242069
$700$	0	0
$701$	3.32122	0.125441	0.0627204	−	0.998031i	$-0.480022\pi$
$701$	3.32122	0.125441	0.0627204	+	0.998031i	$0.480022\pi$
$702$	2.71796i	0.102583i
$703$	7.66011i	0.288906i
$704$	−5.13316	−0.193463
$705$	0	0
$706$	−6.82276	−0.256778
$707$	− 18.7158i	− 0.703881i
$708$	− 24.0760i	− 0.904832i
$709$	−17.6659	−0.663458	−0.331729	−	0.943375i	$-0.607632\pi$
$709$	−17.6659	−0.663458	−0.331729	+	0.943375i	$0.607632\pi$
$710$	0	0
$711$	50.1296	1.88001
$712$	− 29.1296i	− 1.09168i
$713$	4.94091i	0.185039i
$714$	33.4377	1.25137
$715$	0	0
$716$	−14.9432	−0.558454
$717$	5.34140i	0.199478i
$718$	3.92498i	0.146479i
$719$	17.4938	0.652410	0.326205	−	0.945299i	$-0.394230\pi$
$719$	17.4938	0.652410	0.326205	+	0.945299i	$0.394230\pi$
$720$	0	0
$721$	23.6601	0.881148
$722$	28.9228i	1.07639i
$723$	53.1645i	1.97721i
$724$	−3.74179	−0.139063
$725$	0	0
$726$	−9.40059	−0.348888
$727$	19.2435i	0.713701i	0.934161	+	0.356851i	$0.116149\pi$
$727$	19.2435i	0.713701i	−0.934161	+	0.356851i	$0.883851\pi$
$728$	10.0584i	0.372789i
$729$	40.1430	1.48678
$730$	0	0
$731$	−74.9153	−2.77084
$732$	− 12.6554i	− 0.467756i
$733$	17.2213i	0.636082i	0.948077	+	0.318041i	$0.103025\pi$
$733$	17.2213i	0.636082i	−0.948077	+	0.318041i	$0.896975\pi$
$734$	16.1801	0.597217
$735$	0	0
$736$	−43.9089	−1.61850
$737$	3.64776i	0.134367i
$738$	− 7.03625i	− 0.259008i
$739$	−44.1590	−1.62442	−0.812208	−	0.583368i	$-0.801735\pi$
$739$	−44.1590	−1.62442	−0.812208	+	0.583368i	$0.801735\pi$
$740$	0	0
$741$	−31.3830	−1.15288
$742$	16.0790i	0.590279i
$743$	15.8768i	0.582461i	0.956653	+	0.291231i	$0.0940647\pi$
$743$	15.8768i	0.582461i	−0.956653	+	0.291231i	$0.905935\pi$
$744$	4.37867	0.160530
$745$	0	0
$746$	−23.0805	−0.845035
$747$	4.49404i	0.164428i
$748$	− 24.7862i	− 0.906275i
$749$	30.8811	1.12837
$750$	0	0
$751$	20.6030	0.751815	0.375907	−	0.926657i	$-0.377331\pi$
$751$	20.6030	0.751815	0.375907	+	0.926657i	$0.377331\pi$
$752$	3.40848i	0.124294i
$753$	− 1.82844i	− 0.0666322i
$754$	−3.70074	−0.134773
$755$	0	0
$756$	−8.96798	−0.326162
$757$	− 21.6914i	− 0.788387i	−0.919027	−	0.394194i	$-0.871024\pi$
$757$	− 21.6914i	− 0.788387i	0.919027	−	0.394194i	$-0.128976\pi$
$758$	11.4170i	0.414684i
$759$	48.6700	1.76661
$760$	0	0
$761$	22.5166	0.816225	0.408113	−	0.912932i	$-0.366187\pi$
$761$	22.5166	0.816225	0.408113	+	0.912932i	$0.366187\pi$
$762$	30.0156i	1.08735i
$763$	5.72415i	0.207228i
$764$	−15.3676	−0.555981
$765$	0	0
$766$	5.72098	0.206708
$767$	− 9.71959i	− 0.350954i
$768$	30.4643i	1.09928i
$769$	0.975961	0.0351941	0.0175970	−	0.999845i	$-0.494398\pi$
$769$	0.975961	0.0351941	0.0175970	+	0.999845i	$0.494398\pi$
$770$	0	0
$771$	18.7587	0.675578
$772$	14.4098i	0.518620i
$773$	− 15.6271i	− 0.562066i	−0.959698	−	0.281033i	$-0.909323\pi$
$773$	− 15.6271i	− 0.562066i	0.959698	−	0.281033i	$-0.0906771\pi$
$774$	−31.2330	−1.12265
$775$	0	0
$776$	−43.7245	−1.56962
$777$	6.70960i	0.240706i
$778$	− 12.8673i	− 0.461314i
$779$	18.9100	0.677521
$780$	0	0
$781$	−19.9141	−0.712583
$782$	37.3769i	1.33659i
$783$	− 7.79290i	− 0.278496i
$784$	0.530737	0.0189549
$785$	0	0
$786$	−26.1988	−0.934482
$787$	44.8503i	1.59874i	0.600840	+	0.799370i	$0.294833\pi$
$787$	44.8503i	1.59874i	−0.600840	+	0.799370i	$0.705167\pi$
$788$	19.6715i	0.700770i
$789$	−55.9267	−1.99104
$790$	0	0
$791$	−29.0435	−1.03267
$792$	− 24.4061i	− 0.867235i
$793$	− 5.10902i	− 0.181427i
$794$	16.5944	0.588914
$795$	0	0
$796$	−6.17149	−0.218743
$797$	12.4685i	0.441658i	0.975313	+	0.220829i	$0.0708762\pi$
$797$	12.4685i	0.441658i	−0.975313	+	0.220829i	$0.929124\pi$
$798$	37.4653i	1.32626i
$799$	21.2975	0.753451
$800$	0	0
$801$	45.0470	1.59166
$802$	− 0.122753i	− 0.00433456i
$803$	12.2493i	0.432269i
$804$	5.70459	0.201185
$805$	0	0
$806$	0.748445	0.0263628
$807$	48.2434i	1.69825i
$808$	− 18.5400i	− 0.652236i
$809$	−18.8505	−0.662748	−0.331374	−	0.943500i	$-0.607512\pi$
$809$	−18.8505	−0.662748	−0.331374	+	0.943500i	$0.607512\pi$
$810$	0	0
$811$	29.8244	1.04728	0.523638	−	0.851941i	$-0.324575\pi$
$811$	29.8244	1.04728	0.523638	+	0.851941i	$0.324575\pi$
$812$	− 12.2107i	− 0.428511i
$813$	61.0371i	2.14067i
$814$	−1.79951	−0.0630728
$815$	0	0
$816$	−19.6633	−0.688354
$817$	− 83.9390i	− 2.93665i
$818$	15.3863i	0.537971i
$819$	−15.5546	−0.543523
$820$	0	0
$821$	11.9059	0.415517	0.207759	−	0.978180i	$-0.433383\pi$
$821$	11.9059	0.415517	0.207759	+	0.978180i	$0.433383\pi$
$822$	− 27.2160i	− 0.949267i
$823$	− 41.2544i	− 1.43804i	−0.694990	−	0.719020i	$-0.744591\pi$
$823$	− 41.2544i	− 1.43804i	0.694990	−	0.719020i	$-0.255409\pi$
$824$	23.4379	0.816497
$825$	0	0
$826$	−11.6033	−0.403732
$827$	27.0596i	0.940953i	0.882412	+	0.470477i	$0.155918\pi$
$827$	27.0596i	0.940953i	−0.882412	+	0.470477i	$0.844082\pi$
$828$	− 43.0688i	− 1.49674i
$829$	20.6787	0.718200	0.359100	−	0.933299i	$-0.383084\pi$
$829$	20.6787	0.718200	0.359100	+	0.933299i	$0.383084\pi$
$830$	0	0
$831$	14.8699	0.515831
$832$	3.24076i	0.112353i
$833$	− 3.31626i	− 0.114901i
$834$	22.6909	0.785722
$835$	0	0
$836$	27.7718	0.960508
$837$	1.57605i	0.0544763i
$838$	16.2714i	0.562086i
$839$	0.0974789	0.00336534	0.00168267	−	0.999999i	$-0.499464\pi$
$839$	0.0974789	0.00336534	0.00168267	+	0.999999i	$0.499464\pi$
$840$	0	0
$841$	−18.3893	−0.634113
$842$	− 20.0174i	− 0.689844i
$843$	64.2385i	2.21249i
$844$	29.4623	1.01413
$845$	0	0
$846$	8.87916	0.305272
$847$	− 12.5219i	− 0.430257i
$848$	− 9.45539i	− 0.324700i
$849$	−54.8080	−1.88101
$850$	0	0
$851$	−7.50003	−0.257098
$852$	31.1429i	1.06694i
$853$	− 0.919899i	− 0.0314967i	−0.999876	−	0.0157484i	$-0.994987\pi$
$853$	− 0.919899i	− 0.0314967i	0.999876	−	0.0157484i	$-0.00501307\pi$
$854$	−6.09920	−0.208710
$855$	0	0
$856$	30.5910	1.04558
$857$	20.3344i	0.694608i	0.937753	+	0.347304i	$0.112903\pi$
$857$	20.3344i	0.694608i	−0.937753	+	0.347304i	$0.887097\pi$
$858$	− 7.37249i	− 0.251693i
$859$	−15.8959	−0.542360	−0.271180	−	0.962529i	$-0.587414\pi$
$859$	−15.8959	−0.542360	−0.271180	+	0.962529i	$0.587414\pi$
$860$	0	0
$861$	16.5635	0.564484
$862$	26.7801i	0.912135i
$863$	37.0558i	1.26139i	0.776029	+	0.630697i	$0.217231\pi$
$863$	37.0558i	1.26139i	−0.776029	+	0.630697i	$0.782769\pi$
$864$	−14.0061	−0.476496
$865$	0	0
$866$	0.270029	0.00917595
$867$	78.1762i	2.65500i
$868$	2.46951i	0.0838208i
$869$	−31.6493	−1.07363
$870$	0	0
$871$	2.30297	0.0780331
$872$	5.67039i	0.192024i
$873$	− 67.6170i	− 2.28849i
$874$	−41.8790	−1.41658
$875$	0	0
$876$	19.1563	0.647230
$877$	− 25.4103i	− 0.858045i	−0.903294	−	0.429022i	$-0.858858\pi$
$877$	− 25.4103i	− 0.858045i	0.903294	−	0.429022i	$-0.141142\pi$
$878$	− 21.9974i	− 0.742376i
$879$	−20.2988	−0.684662
$880$	0	0
$881$	1.37793	0.0464238	0.0232119	−	0.999731i	$-0.492611\pi$
$881$	1.37793	0.0464238	0.0232119	+	0.999731i	$0.492611\pi$
$882$	− 1.38258i	− 0.0465540i
$883$	54.5280i	1.83501i	0.397722	+	0.917506i	$0.369801\pi$
$883$	54.5280i	1.83501i	−0.397722	+	0.917506i	$0.630199\pi$
$884$	−15.6485	−0.526316
$885$	0	0
$886$	27.8305	0.934984
$887$	− 6.43212i	− 0.215969i	−0.994153	−	0.107985i	$-0.965560\pi$
$887$	− 6.43212i	− 0.215969i	0.994153	−	0.107985i	$-0.0344397\pi$
$888$	6.64658i	0.223045i
$889$	−39.9818	−1.34095
$890$	0	0
$891$	−13.4330	−0.450021
$892$	− 2.53355i	− 0.0848297i
$893$	23.8628i	0.798539i
$894$	−7.88947	−0.263863
$895$	0	0
$896$	−26.0176	−0.869187
$897$	− 30.7272i	− 1.02595i
$898$	2.75427i	0.0919111i
$899$	−2.14593	−0.0715709
$900$	0	0
$901$	−59.0810	−1.96827
$902$	4.44233i	0.147913i
$903$	− 73.5234i	− 2.44670i
$904$	−28.7706	−0.956897
$905$	0	0
$906$	32.4013	1.07646
$907$	− 33.4197i	− 1.10968i	−0.831957	−	0.554841i	$-0.812779\pi$
$907$	− 33.4197i	− 1.10968i	0.831957	−	0.554841i	$-0.187221\pi$
$908$	6.75468i	0.224162i
$909$	28.6709	0.950953
$910$	0	0
$911$	52.4122	1.73649	0.868247	−	0.496132i	$-0.165247\pi$
$911$	52.4122	1.73649	0.868247	+	0.496132i	$0.165247\pi$
$912$	− 22.0318i	− 0.729546i
$913$	− 2.83730i	− 0.0939011i
$914$	−18.4674	−0.610849
$915$	0	0
$916$	34.4984	1.13986
$917$	− 34.8977i	− 1.15242i
$918$	11.9225i	0.393500i
$919$	11.6967	0.385838	0.192919	−	0.981215i	$-0.438205\pi$
$919$	11.6967	0.385838	0.192919	+	0.981215i	$0.438205\pi$
$920$	0	0
$921$	−23.6857	−0.780470
$922$	− 10.7580i	− 0.354297i
$923$	12.5725i	0.413830i
$924$	24.3257	0.800257
$925$	0	0
$926$	6.29594	0.206898
$927$	36.2451i	1.19044i
$928$	− 19.0705i	− 0.626020i
$929$	10.4868	0.344059	0.172030	−	0.985092i	$-0.444968\pi$
$929$	10.4868	0.344059	0.172030	+	0.985092i	$0.444968\pi$
$930$	0	0
$931$	3.71571	0.121777
$932$	− 3.57560i	− 0.117123i
$933$	84.4392i	2.76442i
$934$	1.19999	0.0392647
$935$	0	0
$936$	−15.4085	−0.503643
$937$	38.3244i	1.25200i	0.779821	+	0.626002i	$0.215310\pi$
$937$	38.3244i	1.25200i	−0.779821	+	0.626002i	$0.784690\pi$
$938$	− 2.74930i	− 0.0897679i
$939$	−52.2122	−1.70388
$940$	0	0
$941$	33.4866	1.09163	0.545816	−	0.837905i	$-0.316220\pi$
$941$	33.4866	1.09163	0.545816	+	0.837905i	$0.316220\pi$
$942$	7.71217i	0.251276i
$943$	18.5148i	0.602926i
$944$	6.82344	0.222084
$945$	0	0
$946$	19.7189	0.641117
$947$	− 40.2997i	− 1.30956i	−0.755818	−	0.654782i	$-0.772760\pi$
$947$	− 40.2997i	− 1.30956i	0.755818	−	0.654782i	$-0.227240\pi$
$948$	49.4951i	1.60753i
$949$	7.73346	0.251039
$950$	0	0
$951$	−22.5801	−0.732211
$952$	44.1217i	1.42999i
$953$	− 10.8028i	− 0.349939i	−0.984574	−	0.174969i	$-0.944017\pi$
$953$	− 10.8028i	− 0.349939i	0.984574	−	0.174969i	$-0.0559826\pi$
$954$	−24.6315	−0.797475
$955$	0	0
$956$	−2.98419	−0.0965156
$957$	21.1383i	0.683305i
$958$	− 11.7049i	− 0.378167i
$959$	36.2526	1.17066
$960$	0	0
$961$	−30.5660	−0.986000
$962$	1.13610i	0.0366293i
$963$	47.3069i	1.52444i
$964$	−29.7025	−0.956652
$965$	0	0
$966$	−36.6824	−1.18024
$967$	46.0540i	1.48100i	0.672058	+	0.740499i	$0.265411\pi$
$967$	46.0540i	1.48100i	−0.672058	+	0.740499i	$0.734589\pi$
$968$	− 12.4043i	− 0.398688i
$969$	−137.663	−4.42239
$970$	0	0
$971$	−0.231513	−0.00742961	−0.00371481	−	0.999993i	$-0.501182\pi$
$971$	−0.231513	−0.00742961	−0.00371481	+	0.999993i	$0.501182\pi$
$972$	31.5478i	1.01190i
$973$	30.2250i	0.968970i
$974$	4.81680	0.154340
$975$	0	0
$976$	3.58668	0.114807
$977$	8.81121i	0.281896i	0.990017	+	0.140948i	$0.0450150\pi$
$977$	8.81121i	0.281896i	−0.990017	+	0.140948i	$0.954985\pi$
$978$	− 6.85530i	− 0.219208i
$979$	−28.4404	−0.908958
$980$	0	0
$981$	−8.76887	−0.279968
$982$	− 13.2698i	− 0.423455i
$983$	− 18.8943i	− 0.602634i	−0.953524	−	0.301317i	$-0.902574\pi$
$983$	− 18.8943i	− 0.602634i	0.953524	−	0.301317i	$-0.0974263\pi$
$984$	16.4079	0.523066
$985$	0	0
$986$	−16.2335	−0.516980
$987$	20.9018i	0.665312i
$988$	− 17.5334i	− 0.557811i
$989$	82.1849	2.61333
$990$	0	0
$991$	30.3969	0.965589	0.482794	−	0.875734i	$-0.339622\pi$
$991$	30.3969	0.965589	0.482794	+	0.875734i	$0.339622\pi$
$992$	3.85685i	0.122455i
$993$	60.3842i	1.91623i
$994$	15.0092	0.476063
$995$	0	0
$996$	−4.43715	−0.140597
$997$	35.2291i	1.11572i	0.829936	+	0.557859i	$0.188377\pi$
$997$	35.2291i	1.11572i	−0.829936	+	0.557859i	$0.811623\pi$
$998$	19.0836i	0.604079i
$999$	−2.39236	−0.0756910

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	925.2.b.f.149.6		10
5.2	odd	4		925.2.a.f.1.3		5
5.3	odd	4		185.2.a.e.1.3	✓	5
5.4	even	2	inner	925.2.b.f.149.5		10
15.2	even	4		8325.2.a.ch.1.3		5
15.8	even	4		1665.2.a.p.1.3		5
20.3	even	4		2960.2.a.w.1.1		5
35.13	even	4		9065.2.a.k.1.3		5
185.73	odd	4		6845.2.a.f.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.a.e.1.3	✓	5	5.3	odd	4
925.2.a.f.1.3		5	5.2	odd	4
925.2.b.f.149.5		10	5.4	even	2	inner
925.2.b.f.149.6		10	1.1	even	1	trivial
1665.2.a.p.1.3		5	15.8	even	4
2960.2.a.w.1.1		5	20.3	even	4
6845.2.a.f.1.3		5	185.73	odd	4
8325.2.a.ch.1.3		5	15.2	even	4
9065.2.a.k.1.3		5	35.13	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:	\( N \)	\(=\)	\( 925 = 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	925.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.728950i q^{2} -2.62871i q^{3} +1.46863 q^{4} +1.91620 q^{6} +2.55244i q^{7} +2.52846i q^{8} -3.91009 q^{9} +2.46863 q^{11} -3.86060i q^{12} -1.55854i q^{13} -1.86060 q^{14} +1.09414 q^{16} -6.83662i q^{17} -2.85026i q^{18} +7.66011 q^{19} +6.70960 q^{21} +1.79951i q^{22} +7.50003i q^{23} +6.64658 q^{24} +1.13610 q^{26} +2.39236i q^{27} +3.74859i q^{28} -3.25741 q^{29} +0.658785 q^{31} +5.85449i q^{32} -6.48930i q^{33} +4.98356 q^{34} -5.74248 q^{36} +1.00000i q^{37} +5.58384i q^{38} -4.09694 q^{39} +2.46863 q^{41} +4.89097i q^{42} -10.9579i q^{43} +3.62551 q^{44} -5.46715 q^{46} +3.11521i q^{47} -2.87617i q^{48} +0.485072 q^{49} -17.9715 q^{51} -2.28892i q^{52} -8.64184i q^{53} -1.74391 q^{54} -6.45373 q^{56} -20.1362i q^{57} -2.37449i q^{58} +6.23634 q^{59} +3.27808 q^{61} +0.480222i q^{62} -9.98026i q^{63} -2.07935 q^{64} +4.73038 q^{66} +1.47764i q^{67} -10.0405i q^{68} +19.7154 q^{69} -8.06686 q^{71} -9.88651i q^{72} +4.96199i q^{73} -0.728950 q^{74} +11.2499 q^{76} +6.30102i q^{77} -2.98647i q^{78} -12.8206 q^{79} -5.44146 q^{81} +1.79951i q^{82} -1.14934i q^{83} +9.85393 q^{84} +7.98779 q^{86} +8.56277i q^{87} +6.24184i q^{88} -11.5207 q^{89} +3.97807 q^{91} +11.0148i q^{92} -1.73175i q^{93} -2.27083 q^{94} +15.3897 q^{96} +17.2929i q^{97} +0.353594i q^{98} -9.65257 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 20 q^{4} - 12 q^{6} - 12 q^{9} - 10 q^{11} + 16 q^{14} + 32 q^{16} + 8 q^{19} + 6 q^{21} + 84 q^{24} - 8 q^{26} + 8 q^{29} + 16 q^{31} + 64 q^{34} + 32 q^{36} - 4 q^{39} - 10 q^{41} + 92 q^{44} - 44 q^{49}+ \cdots + 20 q^{99}+O(q^{100}) \) 10 * q - 20 * q^4 - 12 * q^6 - 12 * q^9 - 10 * q^11 + 16 * q^14 + 32 * q^16 + 8 * q^19 + 6 * q^21 + 84 * q^24 - 8 * q^26 + 8 * q^29 + 16 * q^31 + 64 * q^34 + 32 * q^36 - 4 * q^39 - 10 * q^41 + 92 * q^44 - 44 * q^49 - 4 * q^51 + 20 * q^54 + 16 * q^56 + 60 * q^59 - 28 * q^61 - 40 * q^64 + 96 * q^66 - 16 * q^69 - 14 * q^71 - 4 * q^74 + 24 * q^76 - 56 * q^79 - 62 * q^81 + 36 * q^84 + 88 * q^86 - 12 * q^89 - 28 * q^91 - 8 * q^94 - 84 * q^96 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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