LMFDB - Embedded newform 2475.2.c.q.199.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(199,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("2475.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5161984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 4x^{3} + 25x^{2} - 20x + 8 $ x^6 - 4x^3 + 25x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 825)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.5
Root			$1.32001 + 1.32001i$ of defining polynomial
Character	$\chi$	$=$	2475.199
Dual form			2475.2.c.q.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.12489i q^{2} -2.51514 q^{4} -3.64002i q^{7} -1.09461i q^{8} +O(q^{10})$	$q+2.12489i q^{2} -2.51514 q^{4} -3.64002i q^{7} -1.09461i q^{8} -1.00000 q^{11} +1.51514i q^{13} +7.73463 q^{14} -2.70436 q^{16} +1.15516i q^{17} -2.60975 q^{19} -2.12489i q^{22} +5.73463i q^{23} -3.21949 q^{26} +9.15516i q^{28} +6.24977 q^{29} +5.51514 q^{31} -7.93567i q^{32} -2.45459 q^{34} -0.454586i q^{37} -5.54541i q^{38} -4.12489 q^{41} +11.7044i q^{43} +2.51514 q^{44} -12.1854 q^{46} -3.48486i q^{47} -6.24977 q^{49} -3.81078i q^{52} +12.5601i q^{53} -3.98440 q^{56} +13.2800i q^{58} -7.73463 q^{59} -12.0147 q^{61} +11.7190i q^{62} +11.4537 q^{64} +14.2645i q^{67} -2.90539i q^{68} -8.51514 q^{71} -9.21949i q^{73} +0.965943 q^{74} +6.56387 q^{76} +3.64002i q^{77} -5.09461 q^{79} -8.76491i q^{82} +14.7493i q^{83} -24.8704 q^{86} +1.09461i q^{88} -10.4995 q^{89} +5.51514 q^{91} -14.4234i q^{92} +7.40493 q^{94} +6.77959i q^{97} -13.2800i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 16 q^{4}+O(q^{10}) $ 6 * q - 16 * q^4	$ 6 q - 16 q^{4} - 6 q^{11} + 12 q^{14} + 20 q^{16} + 2 q^{19} + 16 q^{26} + 4 q^{29} + 34 q^{31} - 12 q^{34} - 8 q^{41} + 16 q^{44} - 60 q^{46} - 4 q^{49} + 44 q^{56} - 12 q^{59} - 6 q^{61} - 68 q^{64} - 52 q^{71} + 28 q^{74} - 48 q^{76} - 12 q^{79} - 56 q^{86} + 4 q^{89} + 34 q^{91} - 4 q^{94}+O(q^{100}) $ 6 * q - 16 * q^4 - 6 * q^11 + 12 * q^14 + 20 * q^16 + 2 * q^19 + 16 * q^26 + 4 * q^29 + 34 * q^31 - 12 * q^34 - 8 * q^41 + 16 * q^44 - 60 * q^46 - 4 * q^49 + 44 * q^56 - 12 * q^59 - 6 * q^61 - 68 * q^64 - 52 * q^71 + 28 * q^74 - 48 * q^76 - 12 * q^79 - 56 * q^86 + 4 * q^89 + 34 * q^91 - 4 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$.

$n$	$551$	$2026$	$2377$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	2.12489i	1.50252i	0.660006	+	0.751260i	$0.270554\pi$
$2$	2.12489i	1.50252i	−0.660006	+	0.751260i	$0.729446\pi$
$3$	0	0
$3$	0	0
$4$	−2.51514	−1.25757
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.64002i	− 1.37580i	−0.725806	−	0.687900i	$-0.758533\pi$
$7$	− 3.64002i	− 1.37580i	0.725806	−	0.687900i	$-0.241467\pi$
$8$	− 1.09461i	− 0.387003i
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.51514i	0.420224i	0.977677	+	0.210112i	$0.0673828\pi$
$13$	1.51514i	0.420224i	−0.977677	+	0.210112i	$0.932617\pi$
$14$	7.73463	2.06717
$15$	0	0
$16$	−2.70436	−0.676089
$17$	1.15516i	0.280168i	0.990140	+	0.140084i	$0.0447372\pi$
$17$	1.15516i	0.280168i	−0.990140	+	0.140084i	$0.955263\pi$
$18$	0	0
$19$	−2.60975	−0.598717	−0.299359	−	0.954141i	$-0.596773\pi$
$19$	−2.60975	−0.598717	−0.299359	+	0.954141i	$0.596773\pi$
$20$	0	0
$21$	0	0
$22$	− 2.12489i	− 0.453027i
$23$	5.73463i	1.19575i	0.801588	+	0.597877i	$0.203989\pi$
$23$	5.73463i	1.19575i	−0.801588	+	0.597877i	$0.796011\pi$
$24$	0	0
$25$	0	0
$26$	−3.21949	−0.631395
$27$	0	0
$28$	9.15516i	1.73016i
$29$	6.24977	1.16055	0.580277	−	0.814419i	$-0.302944\pi$
$29$	6.24977	1.16055	0.580277	+	0.814419i	$0.302944\pi$
$30$	0	0
$31$	5.51514	0.990548	0.495274	−	0.868737i	$-0.335068\pi$
$31$	5.51514	0.990548	0.495274	+	0.868737i	$0.335068\pi$
$32$	− 7.93567i	− 1.40284i
$33$	0	0
$34$	−2.45459	−0.420958
$35$	0	0
$36$	0	0
$37$	− 0.454586i	− 0.0747335i	−0.999302	−	0.0373667i	$-0.988103\pi$
$37$	− 0.454586i	− 0.0747335i	0.999302	−	0.0373667i	$-0.0118970\pi$
$38$	− 5.54541i	− 0.899585i
$39$	0	0
$40$	0	0
$41$	−4.12489	−0.644199	−0.322099	−	0.946706i	$-0.604389\pi$
$41$	−4.12489	−0.644199	−0.322099	+	0.946706i	$0.604389\pi$
$42$	0	0
$43$	11.7044i	1.78490i	0.451149	+	0.892449i	$0.351014\pi$
$43$	11.7044i	1.78490i	−0.451149	+	0.892449i	$0.648986\pi$
$44$	2.51514	0.379171
$45$	0	0
$46$	−12.1854	−1.79664
$47$	− 3.48486i	− 0.508319i	−0.967162	−	0.254160i	$-0.918201\pi$
$47$	− 3.48486i	− 0.508319i	0.967162	−	0.254160i	$-0.0817989\pi$
$48$	0	0
$49$	−6.24977	−0.892824
$50$	0	0
$51$	0	0
$52$	− 3.81078i	− 0.528460i
$53$	12.5601i	1.72526i	0.505834	+	0.862631i	$0.331185\pi$
$53$	12.5601i	1.72526i	−0.505834	+	0.862631i	$0.668815\pi$
$54$	0	0
$55$	0	0
$56$	−3.98440	−0.532438
$57$	0	0
$58$	13.2800i	1.74376i
$59$	−7.73463	−1.00696	−0.503482	−	0.864006i	$-0.667948\pi$
$59$	−7.73463	−1.00696	−0.503482	+	0.864006i	$0.667948\pi$
$60$	0	0
$61$	−12.0147	−1.53832	−0.769161	−	0.639055i	$-0.779326\pi$
$61$	−12.0147	−1.53832	−0.769161	+	0.639055i	$0.779326\pi$
$62$	11.7190i	1.48832i
$63$	0	0
$64$	11.4537	1.43171
$65$	0	0
$66$	0	0
$67$	14.2645i	1.74268i	0.490680	+	0.871340i	$0.336749\pi$
$67$	14.2645i	1.74268i	−0.490680	+	0.871340i	$0.663251\pi$
$68$	− 2.90539i	− 0.352330i
$69$	0	0
$70$	0	0
$71$	−8.51514	−1.01056	−0.505280	−	0.862955i	$-0.668611\pi$
$71$	−8.51514	−1.01056	−0.505280	+	0.862955i	$0.668611\pi$
$72$	0	0
$73$	− 9.21949i	− 1.07906i	−0.841966	−	0.539530i	$-0.818602\pi$
$73$	− 9.21949i	− 1.07906i	0.841966	−	0.539530i	$-0.181398\pi$
$74$	0.965943	0.112289
$75$	0	0
$76$	6.56387	0.752928
$77$	3.64002i	0.414819i
$78$	0	0
$79$	−5.09461	−0.573188	−0.286594	−	0.958052i	$-0.592523\pi$
$79$	−5.09461	−0.573188	−0.286594	+	0.958052i	$0.592523\pi$
$80$	0	0
$81$	0	0
$82$	− 8.76491i	− 0.967922i
$83$	14.7493i	1.61895i	0.587156	+	0.809474i	$0.300247\pi$
$83$	14.7493i	1.61895i	−0.587156	+	0.809474i	$0.699753\pi$
$84$	0	0
$85$	0	0
$86$	−24.8704	−2.68185
$87$	0	0
$88$	1.09461i	0.116686i
$89$	−10.4995	−1.11295	−0.556475	−	0.830865i	$-0.687846\pi$
$89$	−10.4995	−1.11295	−0.556475	+	0.830865i	$0.687846\pi$
$90$	0	0
$91$	5.51514	0.578144
$92$	− 14.4234i	− 1.50374i
$93$	0	0
$94$	7.40493	0.763760
$95$	0	0
$96$	0	0
$97$	6.77959i	0.688363i	0.938903	+	0.344181i	$0.111844\pi$
$97$	6.77959i	0.688363i	−0.938903	+	0.344181i	$0.888156\pi$
$98$	− 13.2800i	− 1.34149i
$99$	0	0
$100$	0	0
$101$	−7.40493	−0.736818	−0.368409	−	0.929664i	$-0.620097\pi$
$101$	−7.40493	−0.736818	−0.368409	+	0.929664i	$0.620097\pi$
$102$	0	0
$103$	− 16.4995i	− 1.62575i	−0.582439	−	0.812874i	$-0.697902\pi$
$103$	− 16.4995i	− 1.62575i	0.582439	−	0.812874i	$-0.302098\pi$
$104$	1.65848	0.162628
$105$	0	0
$106$	−26.6888	−2.59224
$107$	3.93945i	0.380841i	0.981703	+	0.190420i	$0.0609851\pi$
$107$	3.93945i	0.380841i	−0.981703	+	0.190420i	$0.939015\pi$
$108$	0	0
$109$	−6.73463	−0.645061	−0.322530	−	0.946559i	$-0.604533\pi$
$109$	−6.73463	−0.645061	−0.322530	+	0.946559i	$0.604533\pi$
$110$	0	0
$111$	0	0
$112$	9.84392i	0.930163i
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	0	0
$115$	0	0
$116$	−15.7190	−1.45948
$117$	0	0
$118$	− 16.4352i	− 1.51298i
$119$	4.20482	0.385455
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 25.5298i	− 2.31136i
$123$	0	0
$124$	−13.8713	−1.24568
$125$	0	0
$126$	0	0
$127$	− 8.06433i	− 0.715594i	−0.933799	−	0.357797i	$-0.883528\pi$
$127$	− 8.06433i	− 0.715594i	0.933799	−	0.357797i	$-0.116472\pi$
$128$	8.46640i	0.748331i
$129$	0	0
$130$	0	0
$131$	12.8099	1.11920	0.559602	−	0.828762i	$-0.310954\pi$
$131$	12.8099	1.11920	0.559602	+	0.828762i	$0.310954\pi$
$132$	0	0
$133$	9.49954i	0.823715i
$134$	−30.3103	−2.61841
$135$	0	0
$136$	1.26445	0.108426
$137$	22.8099i	1.94878i	0.224868	+	0.974389i	$0.427805\pi$
$137$	22.8099i	1.94878i	−0.224868	+	0.974389i	$0.572195\pi$
$138$	0	0
$139$	−7.59037	−0.643807	−0.321903	−	0.946773i	$-0.604323\pi$
$139$	−7.59037	−0.643807	−0.321903	+	0.946773i	$0.604323\pi$
$140$	0	0
$141$	0	0
$142$	− 18.0937i	− 1.51839i
$143$	− 1.51514i	− 0.126702i
$144$	0	0
$145$	0	0
$146$	19.5904	1.62131
$147$	0	0
$148$	1.14335i	0.0939825i
$149$	−1.81456	−0.148655	−0.0743274	−	0.997234i	$-0.523681\pi$
$149$	−1.81456	−0.148655	−0.0743274	+	0.997234i	$0.523681\pi$
$150$	0	0
$151$	24.3250	1.97954	0.989770	−	0.142670i	$-0.0455687\pi$
$151$	24.3250	1.97954	0.989770	+	0.142670i	$0.0455687\pi$
$152$	2.85665i	0.231705i
$153$	0	0
$154$	−7.73463	−0.623274
$155$	0	0
$156$	0	0
$157$	− 9.76491i	− 0.779325i	−0.920958	−	0.389662i	$-0.872592\pi$
$157$	− 9.76491i	− 0.779325i	0.920958	−	0.389662i	$-0.127408\pi$
$158$	− 10.8255i	− 0.861227i
$159$	0	0
$160$	0	0
$161$	20.8742	1.64512
$162$	0	0
$163$	− 6.98440i	− 0.547061i	−0.961863	−	0.273530i	$-0.911809\pi$
$163$	− 6.98440i	− 0.547061i	0.961863	−	0.273530i	$-0.0881914\pi$
$164$	10.3747	0.810125
$165$	0	0
$166$	−31.3406	−2.43250
$167$	− 6.31032i	− 0.488307i	−0.969737	−	0.244154i	$-0.921490\pi$
$167$	− 6.31032i	− 0.488307i	0.969737	−	0.244154i	$-0.0785102\pi$
$168$	0	0
$169$	10.7044	0.823412
$170$	0	0
$171$	0	0
$172$	− 29.4381i	− 2.24463i
$173$	12.8448i	0.976575i	0.872683	+	0.488287i	$0.162378\pi$
$173$	12.8448i	0.976575i	−0.872683	+	0.488287i	$0.837622\pi$
$174$	0	0
$175$	0	0
$176$	2.70436	0.203849
$177$	0	0
$178$	− 22.3103i	− 1.67223i
$179$	13.4849	1.00791	0.503953	−	0.863731i	$-0.331878\pi$
$179$	13.4849	1.00791	0.503953	+	0.863731i	$0.331878\pi$
$180$	0	0
$181$	−23.0899	−1.71626	−0.858130	−	0.513433i	$-0.828374\pi$
$181$	−23.0899	−1.71626	−0.858130	+	0.513433i	$0.828374\pi$
$182$	11.7190i	0.868673i
$183$	0	0
$184$	6.27718	0.462760
$185$	0	0
$186$	0	0
$187$	− 1.15516i	− 0.0844738i
$188$	8.76491i	0.639247i
$189$	0	0
$190$	0	0
$191$	7.98440	0.577731	0.288866	−	0.957370i	$-0.406722\pi$
$191$	7.98440	0.577731	0.288866	+	0.957370i	$0.406722\pi$
$192$	0	0
$193$	11.7649i	0.846857i	0.905930	+	0.423428i	$0.139173\pi$
$193$	11.7649i	0.846857i	−0.905930	+	0.423428i	$0.860827\pi$
$194$	−14.4058	−1.03428
$195$	0	0
$196$	15.7190	1.12279
$197$	− 3.81456i	− 0.271776i	−0.990724	−	0.135888i	$-0.956611\pi$
$197$	− 3.81456i	− 0.271776i	0.990724	−	0.135888i	$-0.0433888\pi$
$198$	0	0
$199$	12.0752	0.855990	0.427995	−	0.903781i	$-0.359220\pi$
$199$	12.0752	0.855990	0.427995	+	0.903781i	$0.359220\pi$
$200$	0	0
$201$	0	0
$202$	− 15.7346i	− 1.10708i
$203$	− 22.7493i	− 1.59669i
$204$	0	0
$205$	0	0
$206$	35.0596	2.44272
$207$	0	0
$208$	− 4.09747i	− 0.284109i
$209$	2.60975	0.180520
$210$	0	0
$211$	10.2645	0.706634	0.353317	−	0.935504i	$-0.385054\pi$
$211$	10.2645	0.706634	0.353317	+	0.935504i	$0.385054\pi$
$212$	− 31.5904i	− 2.16964i
$213$	0	0
$214$	−8.37088	−0.572221
$215$	0	0
$216$	0	0
$217$	− 20.0752i	− 1.36280i
$218$	− 14.3103i	− 0.969217i
$219$	0	0
$220$	0	0
$221$	−1.75023	−0.117733
$222$	0	0
$223$	12.9239i	0.865445i	0.901527	+	0.432723i	$0.142447\pi$
$223$	12.9239i	0.865445i	−0.901527	+	0.432723i	$0.857553\pi$
$224$	−28.8860	−1.93003
$225$	0	0
$226$	−12.7493	−0.848072
$227$	22.8099i	1.51394i	0.653447	+	0.756972i	$0.273322\pi$
$227$	22.8099i	1.51394i	−0.653447	+	0.756972i	$0.726678\pi$
$228$	0	0
$229$	−14.7796	−0.976663	−0.488331	−	0.872658i	$-0.662394\pi$
$229$	−14.7796	−0.976663	−0.488331	+	0.872658i	$0.662394\pi$
$230$	0	0
$231$	0	0
$232$	− 6.84106i	− 0.449137i
$233$	− 4.96594i	− 0.325330i	−0.986681	−	0.162665i	$-0.947991\pi$
$233$	− 4.96594i	− 0.325330i	0.986681	−	0.162665i	$-0.0520089\pi$
$234$	0	0
$235$	0	0
$236$	19.4537	1.26633
$237$	0	0
$238$	8.93475i	0.579154i
$239$	−14.9991	−0.970210	−0.485105	−	0.874456i	$-0.661219\pi$
$239$	−14.9991	−0.970210	−0.485105	+	0.874456i	$0.661219\pi$
$240$	0	0
$241$	5.04496	0.324974	0.162487	−	0.986711i	$-0.448048\pi$
$241$	5.04496	0.324974	0.162487	+	0.986711i	$0.448048\pi$
$242$	2.12489i	0.136593i
$243$	0	0
$244$	30.2186	1.93455
$245$	0	0
$246$	0	0
$247$	− 3.95413i	− 0.251595i
$248$	− 6.03692i	− 0.383345i
$249$	0	0
$250$	0	0
$251$	3.03028	0.191269	0.0956347	−	0.995417i	$-0.469512\pi$
$251$	3.03028	0.191269	0.0956347	+	0.995417i	$0.469512\pi$
$252$	0	0
$253$	− 5.73463i	− 0.360533i
$254$	17.1358	1.07519
$255$	0	0
$256$	4.91721	0.307325
$257$	− 13.6509i	− 0.851521i	−0.904836	−	0.425761i	$-0.860007\pi$
$257$	− 13.6509i	− 0.851521i	0.904836	−	0.425761i	$-0.139993\pi$
$258$	0	0
$259$	−1.65470	−0.102818
$260$	0	0
$261$	0	0
$262$	27.2195i	1.68163i
$263$	− 12.5601i	− 0.774489i	−0.921977	−	0.387244i	$-0.873427\pi$
$263$	− 12.5601i	− 0.774489i	0.921977	−	0.387244i	$-0.126573\pi$
$264$	0	0
$265$	0	0
$266$	−20.1854	−1.23765
$267$	0	0
$268$	− 35.8771i	− 2.19154i
$269$	−24.6888	−1.50530	−0.752650	−	0.658421i	$-0.771225\pi$
$269$	−24.6888	−1.50530	−0.752650	+	0.658421i	$0.771225\pi$
$270$	0	0
$271$	7.56479	0.459528	0.229764	−	0.973246i	$-0.426205\pi$
$271$	7.56479	0.459528	0.229764	+	0.973246i	$0.426205\pi$
$272$	− 3.12397i	− 0.189418i
$273$	0	0
$274$	−48.4683	−2.92808
$275$	0	0
$276$	0	0
$277$	− 1.92477i	− 0.115648i	−0.998327	−	0.0578241i	$-0.981584\pi$
$277$	− 1.92477i	− 0.115648i	0.998327	−	0.0578241i	$-0.0184162\pi$
$278$	− 16.1287i	− 0.967333i
$279$	0	0
$280$	0	0
$281$	1.87511	0.111860	0.0559300	−	0.998435i	$-0.482188\pi$
$281$	1.87511	0.111860	0.0559300	+	0.998435i	$0.482188\pi$
$282$	0	0
$283$	30.1396i	1.79161i	0.444446	+	0.895806i	$0.353401\pi$
$283$	30.1396i	1.79161i	−0.444446	+	0.895806i	$0.646599\pi$
$284$	21.4167	1.27085
$285$	0	0
$286$	3.21949	0.190373
$287$	15.0147i	0.886289i
$288$	0	0
$289$	15.6656	0.921506
$290$	0	0
$291$	0	0
$292$	23.1883i	1.35699i
$293$	29.1552i	1.70326i	0.524141	+	0.851631i	$0.324386\pi$
$293$	29.1552i	1.70326i	−0.524141	+	0.851631i	$0.675614\pi$
$294$	0	0
$295$	0	0
$296$	−0.497594	−0.0289221
$297$	0	0
$298$	− 3.85574i	− 0.223357i
$299$	−8.68876	−0.502484
$300$	0	0
$301$	42.6041	2.45566
$302$	51.6878i	2.97430i
$303$	0	0
$304$	7.05769	0.404786
$305$	0	0
$306$	0	0
$307$	− 27.8548i	− 1.58976i	−0.606768	−	0.794879i	$-0.707534\pi$
$307$	− 27.8548i	− 1.58976i	0.606768	−	0.794879i	$-0.292466\pi$
$308$	− 9.15516i	− 0.521664i
$309$	0	0
$310$	0	0
$311$	−23.9083	−1.35571	−0.677856	−	0.735194i	$-0.737091\pi$
$311$	−23.9083	−1.35571	−0.677856	+	0.735194i	$0.737091\pi$
$312$	0	0
$313$	28.3094i	1.60014i	0.599905	+	0.800071i	$0.295205\pi$
$313$	28.3094i	1.60014i	−0.599905	+	0.800071i	$0.704795\pi$
$314$	20.7493	1.17095
$315$	0	0
$316$	12.8136	0.720824
$317$	− 8.80986i	− 0.494811i	−0.968912	−	0.247406i	$-0.920422\pi$
$317$	− 8.80986i	− 0.494811i	0.968912	−	0.247406i	$-0.0795780\pi$
$318$	0	0
$319$	−6.24977	−0.349920
$320$	0	0
$321$	0	0
$322$	44.3553i	2.47182i
$323$	− 3.01468i	− 0.167741i
$324$	0	0
$325$	0	0
$326$	14.8411	0.821970
$327$	0	0
$328$	4.51514i	0.249307i
$329$	−12.6850	−0.699346
$330$	0	0
$331$	32.2498	1.77261	0.886304	−	0.463104i	$-0.153264\pi$
$331$	32.2498	1.77261	0.886304	+	0.463104i	$0.153264\pi$
$332$	− 37.0966i	− 2.03594i
$333$	0	0
$334$	13.4087	0.733692
$335$	0	0
$336$	0	0
$337$	− 28.9844i	− 1.57888i	−0.613827	−	0.789441i	$-0.710371\pi$
$337$	− 28.9844i	− 1.57888i	0.613827	−	0.789441i	$-0.289629\pi$
$338$	22.7455i	1.23719i
$339$	0	0
$340$	0	0
$341$	−5.51514	−0.298661
$342$	0	0
$343$	− 2.73085i	− 0.147452i
$344$	12.8117	0.690760
$345$	0	0
$346$	−27.2938	−1.46732
$347$	− 35.7190i	− 1.91750i	−0.284253	−	0.958749i	$-0.591746\pi$
$347$	− 35.7190i	− 1.91750i	0.284253	−	0.958749i	$-0.408254\pi$
$348$	0	0
$349$	23.2800	1.24615	0.623076	−	0.782161i	$-0.285883\pi$
$349$	23.2800	1.24615	0.623076	+	0.782161i	$0.285883\pi$
$350$	0	0
$351$	0	0
$352$	7.93567i	0.422972i
$353$	9.75023i	0.518952i	0.965750	+	0.259476i	$0.0835499\pi$
$353$	9.75023i	0.518952i	−0.965750	+	0.259476i	$0.916450\pi$
$354$	0	0
$355$	0	0
$356$	26.4078	1.39961
$357$	0	0
$358$	28.6538i	1.51440i
$359$	33.9007	1.78921	0.894605	−	0.446858i	$-0.147457\pi$
$359$	33.9007	1.78921	0.894605	+	0.446858i	$0.147457\pi$
$360$	0	0
$361$	−12.1892	−0.641538
$362$	− 49.0634i	− 2.57872i
$363$	0	0
$364$	−13.8713	−0.727055
$365$	0	0
$366$	0	0
$367$	− 1.88601i	− 0.0984491i	−0.998788	−	0.0492245i	$-0.984325\pi$
$367$	− 1.88601i	− 0.0984491i	0.998788	−	0.0492245i	$-0.0156750\pi$
$368$	− 15.5085i	− 0.808436i
$369$	0	0
$370$	0	0
$371$	45.7190	2.37361
$372$	0	0
$373$	− 16.3250i	− 0.845277i	−0.906298	−	0.422638i	$-0.861104\pi$
$373$	− 16.3250i	− 0.845277i	0.906298	−	0.422638i	$-0.138896\pi$
$374$	2.45459	0.126924
$375$	0	0
$376$	−3.81456	−0.196721
$377$	9.46927i	0.487692i
$378$	0	0
$379$	26.0440	1.33779	0.668896	−	0.743356i	$-0.266767\pi$
$379$	26.0440	1.33779	0.668896	+	0.743356i	$0.266767\pi$
$380$	0	0
$381$	0	0
$382$	16.9659i	0.868053i
$383$	− 12.4702i	− 0.637197i	−0.947890	−	0.318598i	$-0.896788\pi$
$383$	− 12.4702i	− 0.637197i	0.947890	−	0.318598i	$-0.103212\pi$
$384$	0	0
$385$	0	0
$386$	−24.9991	−1.27242
$387$	0	0
$388$	− 17.0516i	− 0.865664i
$389$	−18.0899	−0.917195	−0.458597	−	0.888644i	$-0.651648\pi$
$389$	−18.0899	−0.917195	−0.458597	+	0.888644i	$0.651648\pi$
$390$	0	0
$391$	−6.62443	−0.335012
$392$	6.84106i	0.345526i
$393$	0	0
$394$	8.10551	0.408350
$395$	0	0
$396$	0	0
$397$	15.2342i	0.764581i	0.924042	+	0.382291i	$0.124865\pi$
$397$	15.2342i	0.764581i	−0.924042	+	0.382291i	$0.875135\pi$
$398$	25.6585i	1.28614i
$399$	0	0
$400$	0	0
$401$	−2.74931	−0.137294	−0.0686471	−	0.997641i	$-0.521868\pi$
$401$	−2.74931	−0.137294	−0.0686471	+	0.997641i	$0.521868\pi$
$402$	0	0
$403$	8.35620i	0.416252i
$404$	18.6244	0.926600
$405$	0	0
$406$	48.3397	2.39906
$407$	0.454586i	0.0225330i
$408$	0	0
$409$	3.98532	0.197061	0.0985307	−	0.995134i	$-0.468586\pi$
$409$	3.98532	0.197061	0.0985307	+	0.995134i	$0.468586\pi$
$410$	0	0
$411$	0	0
$412$	41.4986i	2.04449i
$413$	28.1542i	1.38538i
$414$	0	0
$415$	0	0
$416$	12.0236	0.589507
$417$	0	0
$418$	5.54541i	0.271235i
$419$	−5.13578	−0.250899	−0.125450	−	0.992100i	$-0.540037\pi$
$419$	−5.13578	−0.250899	−0.125450	+	0.992100i	$0.540037\pi$
$420$	0	0
$421$	−8.94657	−0.436029	−0.218014	−	0.975946i	$-0.569958\pi$
$421$	−8.94657	−0.436029	−0.218014	+	0.975946i	$0.569958\pi$
$422$	21.8108i	1.06173i
$423$	0	0
$424$	13.7484	0.667681
$425$	0	0
$426$	0	0
$427$	43.7337i	2.11642i
$428$	− 9.90826i	− 0.478934i
$429$	0	0
$430$	0	0
$431$	−22.7493	−1.09580	−0.547898	−	0.836545i	$-0.684572\pi$
$431$	−22.7493	−1.09580	−0.547898	+	0.836545i	$0.684572\pi$
$432$	0	0
$433$	7.58325i	0.364428i	0.983259	+	0.182214i	$0.0583263\pi$
$433$	7.58325i	0.364428i	−0.983259	+	0.182214i	$0.941674\pi$
$434$	42.6576	2.04763
$435$	0	0
$436$	16.9385	0.811209
$437$	− 14.9659i	− 0.715918i
$438$	0	0
$439$	−17.3903	−0.829991	−0.414996	−	0.909823i	$-0.636217\pi$
$439$	−17.3903	−0.829991	−0.414996	+	0.909823i	$0.636217\pi$
$440$	0	0
$441$	0	0
$442$	− 3.71904i	− 0.176897i
$443$	− 10.6438i	− 0.505702i	−0.967505	−	0.252851i	$-0.918632\pi$
$443$	− 10.6438i	− 0.505702i	0.967505	−	0.252851i	$-0.0813683\pi$
$444$	0	0
$445$	0	0
$446$	−27.4617	−1.30035
$447$	0	0
$448$	− 41.6916i	− 1.96974i
$449$	−26.9310	−1.27095	−0.635476	−	0.772121i	$-0.719196\pi$
$449$	−26.9310	−1.27095	−0.635476	+	0.772121i	$0.719196\pi$
$450$	0	0
$451$	4.12489	0.194233
$452$	− 15.0908i	− 0.709813i
$453$	0	0
$454$	−48.4683	−2.27473
$455$	0	0
$456$	0	0
$457$	15.7796i	0.738138i	0.929402	+	0.369069i	$0.120323\pi$
$457$	15.7796i	0.738138i	−0.929402	+	0.369069i	$0.879677\pi$
$458$	− 31.4049i	− 1.46746i
$459$	0	0
$460$	0	0
$461$	−8.18922	−0.381410	−0.190705	−	0.981647i	$-0.561077\pi$
$461$	−8.18922	−0.381410	−0.190705	+	0.981647i	$0.561077\pi$
$462$	0	0
$463$	16.0899i	0.747762i	0.927477	+	0.373881i	$0.121973\pi$
$463$	16.0899i	0.747762i	−0.927477	+	0.373881i	$0.878027\pi$
$464$	−16.9016	−0.784638
$465$	0	0
$466$	10.5521	0.488815
$467$	− 29.4693i	− 1.36367i	−0.731504	−	0.681837i	$-0.761181\pi$
$467$	− 29.4693i	− 1.36367i	0.731504	−	0.681837i	$-0.238819\pi$
$468$	0	0
$469$	51.9229	2.39758
$470$	0	0
$471$	0	0
$472$	8.46640i	0.389698i
$473$	− 11.7044i	− 0.538167i
$474$	0	0
$475$	0	0
$476$	−10.5757	−0.484736
$477$	0	0
$478$	− 31.8713i	− 1.45776i
$479$	32.2186	1.47210	0.736052	−	0.676925i	$-0.236688\pi$
$479$	32.2186	1.47210	0.736052	+	0.676925i	$0.236688\pi$
$480$	0	0
$481$	0.688760	0.0314048
$482$	10.7200i	0.488280i
$483$	0	0
$484$	−2.51514	−0.114324
$485$	0	0
$486$	0	0
$487$	35.8936i	1.62649i	0.581919	+	0.813247i	$0.302302\pi$
$487$	35.8936i	1.62649i	−0.581919	+	0.813247i	$0.697698\pi$
$488$	13.1514i	0.595335i
$489$	0	0
$490$	0	0
$491$	7.15894	0.323079	0.161539	−	0.986866i	$-0.448354\pi$
$491$	7.15894	0.323079	0.161539	+	0.986866i	$0.448354\pi$
$492$	0	0
$493$	7.21949i	0.325150i
$494$	8.40207	0.378027
$495$	0	0
$496$	−14.9149	−0.669699
$497$	30.9953i	1.39033i
$498$	0	0
$499$	−27.0743	−1.21201	−0.606006	−	0.795460i	$-0.707229\pi$
$499$	−27.0743	−1.21201	−0.606006	+	0.795460i	$0.707229\pi$
$500$	0	0
$501$	0	0
$502$	6.43899i	0.287386i
$503$	− 26.9991i	− 1.20383i	−0.798560	−	0.601915i	$-0.794405\pi$
$503$	− 26.9991i	− 1.20383i	0.798560	−	0.601915i	$-0.205595\pi$
$504$	0	0
$505$	0	0
$506$	12.1854	0.541709
$507$	0	0
$508$	20.2829i	0.899909i
$509$	15.5904	0.691031	0.345515	−	0.938413i	$-0.387704\pi$
$509$	15.5904	0.691031	0.345515	+	0.938413i	$0.387704\pi$
$510$	0	0
$511$	−33.5592	−1.48457
$512$	27.3813i	1.21009i
$513$	0	0
$514$	29.0066	1.27943
$515$	0	0
$516$	0	0
$517$	3.48486i	0.153264i
$518$	− 3.51605i	− 0.154487i
$519$	0	0
$520$	0	0
$521$	−11.1589	−0.488882	−0.244441	−	0.969664i	$-0.578604\pi$
$521$	−11.1589	−0.488882	−0.244441	+	0.969664i	$0.578604\pi$
$522$	0	0
$523$	− 10.5786i	− 0.462568i	−0.972886	−	0.231284i	$-0.925707\pi$
$523$	− 10.5786i	− 0.462568i	0.972886	−	0.231284i	$-0.0742926\pi$
$524$	−32.2186	−1.40748
$525$	0	0
$526$	26.6888	1.16369
$527$	6.37088i	0.277520i
$528$	0	0
$529$	−9.88601	−0.429827
$530$	0	0
$531$	0	0
$532$	− 23.8927i	− 1.03588i
$533$	− 6.24977i	− 0.270708i
$534$	0	0
$535$	0	0
$536$	15.6140	0.674422
$537$	0	0
$538$	− 52.4608i	− 2.26175i
$539$	6.24977	0.269197
$540$	0	0
$541$	−11.2947	−0.485598	−0.242799	−	0.970077i	$-0.578066\pi$
$541$	−11.2947	−0.485598	−0.242799	+	0.970077i	$0.578066\pi$
$542$	16.0743i	0.690451i
$543$	0	0
$544$	9.16698	0.393031
$545$	0	0
$546$	0	0
$547$	− 6.09369i	− 0.260547i	−0.991478	−	0.130274i	$-0.958414\pi$
$547$	− 6.09369i	− 0.260547i	0.991478	−	0.130274i	$-0.0415856\pi$
$548$	− 57.3700i	− 2.45072i
$549$	0	0
$550$	0	0
$551$	−16.3103	−0.694843
$552$	0	0
$553$	18.5445i	0.788592i
$554$	4.08991	0.173764
$555$	0	0
$556$	19.0908	0.809631
$557$	− 5.90069i	− 0.250020i	−0.992155	−	0.125010i	$-0.960104\pi$
$557$	− 5.90069i	− 0.250020i	0.992155	−	0.125010i	$-0.0398964\pi$
$558$	0	0
$559$	−17.7337	−0.750056
$560$	0	0
$561$	0	0
$562$	3.98440i	0.168072i
$563$	− 3.03028i	− 0.127711i	−0.997959	−	0.0638555i	$-0.979660\pi$
$563$	− 3.03028i	− 0.127711i	0.997959	−	0.0638555i	$-0.0203397\pi$
$564$	0	0
$565$	0	0
$566$	−64.0431	−2.69193
$567$	0	0
$568$	9.32075i	0.391090i
$569$	13.4049	0.561964	0.280982	−	0.959713i	$-0.409340\pi$
$569$	13.4049	0.561964	0.280982	+	0.959713i	$0.409340\pi$
$570$	0	0
$571$	26.8851	1.12511	0.562553	−	0.826761i	$-0.309819\pi$
$571$	26.8851	1.12511	0.562553	+	0.826761i	$0.309819\pi$
$572$	3.81078i	0.159337i
$573$	0	0
$574$	−31.9045	−1.33167
$575$	0	0
$576$	0	0
$577$	− 2.03028i	− 0.0845215i	−0.999107	−	0.0422607i	$-0.986544\pi$
$577$	− 2.03028i	− 0.0845215i	0.999107	−	0.0422607i	$-0.0134560\pi$
$578$	33.2876i	1.38458i
$579$	0	0
$580$	0	0
$581$	53.6878	2.22735
$582$	0	0
$583$	− 12.5601i	− 0.520186i
$584$	−10.0917	−0.417599
$585$	0	0
$586$	−61.9514	−2.55919
$587$	21.8245i	0.900795i	0.892828	+	0.450398i	$0.148718\pi$
$587$	21.8245i	0.900795i	−0.892828	+	0.450398i	$0.851282\pi$
$588$	0	0
$589$	−14.3931	−0.593058
$590$	0	0
$591$	0	0
$592$	1.22936i	0.0505265i
$593$	− 8.06811i	− 0.331318i	−0.986183	−	0.165659i	$-0.947025\pi$
$593$	− 8.06811i	− 0.331318i	0.986183	−	0.165659i	$-0.0529751\pi$
$594$	0	0
$595$	0	0
$596$	4.56387	0.186944
$597$	0	0
$598$	− 18.4626i	− 0.754993i
$599$	7.61353	0.311080	0.155540	−	0.987830i	$-0.450288\pi$
$599$	7.61353	0.311080	0.155540	+	0.987830i	$0.450288\pi$
$600$	0	0
$601$	3.57569	0.145855	0.0729277	−	0.997337i	$-0.476766\pi$
$601$	3.57569	0.145855	0.0729277	+	0.997337i	$0.476766\pi$
$602$	90.5289i	3.68968i
$603$	0	0
$604$	−61.1807	−2.48941
$605$	0	0
$606$	0	0
$607$	− 17.5298i	− 0.711513i	−0.934579	−	0.355757i	$-0.884223\pi$
$607$	− 17.5298i	− 0.711513i	0.934579	−	0.355757i	$-0.115777\pi$
$608$	20.7101i	0.839905i
$609$	0	0
$610$	0	0
$611$	5.28005	0.213608
$612$	0	0
$613$	12.5601i	0.507297i	0.967296	+	0.253649i	$0.0816307\pi$
$613$	12.5601i	0.507297i	−0.967296	+	0.253649i	$0.918369\pi$
$614$	59.1883	2.38865
$615$	0	0
$616$	3.98440	0.160536
$617$	15.9612i	0.642576i	0.946982	+	0.321288i	$0.104116\pi$
$617$	15.9612i	0.642576i	−0.946982	+	0.321288i	$0.895884\pi$
$618$	0	0
$619$	9.23417	0.371153	0.185576	−	0.982630i	$-0.440585\pi$
$619$	9.23417	0.371153	0.185576	+	0.982630i	$0.440585\pi$
$620$	0	0
$621$	0	0
$622$	− 50.8023i	− 2.03699i
$623$	38.2186i	1.53119i
$624$	0	0
$625$	0	0
$626$	−60.1542	−2.40425
$627$	0	0
$628$	24.5601i	0.980054i
$629$	0.525120	0.0209379
$630$	0	0
$631$	29.2342	1.16379	0.581897	−	0.813262i	$-0.302311\pi$
$631$	29.2342	1.16379	0.581897	+	0.813262i	$0.302311\pi$
$632$	5.57661i	0.221826i
$633$	0	0
$634$	18.7200	0.743464
$635$	0	0
$636$	0	0
$637$	− 9.46927i	− 0.375186i
$638$	− 13.2800i	− 0.525762i
$639$	0	0
$640$	0	0
$641$	−13.9612	−0.551436	−0.275718	−	0.961239i	$-0.588916\pi$
$641$	−13.9612	−0.551436	−0.275718	+	0.961239i	$0.588916\pi$
$642$	0	0
$643$	12.6206i	0.497710i	0.968541	+	0.248855i	$0.0800542\pi$
$643$	12.6206i	0.497710i	−0.968541	+	0.248855i	$0.919946\pi$
$644$	−52.5015	−2.06885
$645$	0	0
$646$	6.40585	0.252035
$647$	29.6429i	1.16538i	0.812694	+	0.582691i	$0.198000\pi$
$647$	29.6429i	1.16538i	−0.812694	+	0.582691i	$0.802000\pi$
$648$	0	0
$649$	7.73463	0.303611
$650$	0	0
$651$	0	0
$652$	17.5667i	0.687967i
$653$	− 9.90069i	− 0.387444i	−0.981056	−	0.193722i	$-0.937944\pi$
$653$	− 9.90069i	− 0.387444i	0.981056	−	0.193722i	$-0.0620560\pi$
$654$	0	0
$655$	0	0
$656$	11.1552	0.435536
$657$	0	0
$658$	− 26.9541i	− 1.05078i
$659$	−5.28005	−0.205681	−0.102841	−	0.994698i	$-0.532793\pi$
$659$	−5.28005	−0.205681	−0.102841	+	0.994698i	$0.532793\pi$
$660$	0	0
$661$	−26.8548	−1.04453	−0.522266	−	0.852783i	$-0.674913\pi$
$661$	−26.8548	−1.04453	−0.522266	+	0.852783i	$0.674913\pi$
$662$	68.5271i	2.66338i
$663$	0	0
$664$	16.1447	0.626537
$665$	0	0
$666$	0	0
$667$	35.8401i	1.38774i
$668$	15.8713i	0.614080i
$669$	0	0
$670$	0	0
$671$	12.0147	0.463822
$672$	0	0
$673$	− 3.81834i	− 0.147186i	−0.997288	−	0.0735932i	$-0.976553\pi$
$673$	− 3.81834i	− 0.147186i	0.997288	−	0.0735932i	$-0.0234466\pi$
$674$	61.5885	2.37230
$675$	0	0
$676$	−26.9229	−1.03550
$677$	− 15.6897i	− 0.603003i	−0.953466	−	0.301502i	$-0.902512\pi$
$677$	− 15.6897i	− 0.603003i	0.953466	−	0.301502i	$-0.0974879\pi$
$678$	0	0
$679$	24.6779	0.947049
$680$	0	0
$681$	0	0
$682$	− 11.7190i	− 0.448745i
$683$	15.6353i	0.598269i	0.954211	+	0.299135i	$0.0966979\pi$
$683$	15.6353i	0.598269i	−0.954211	+	0.299135i	$0.903302\pi$
$684$	0	0
$685$	0	0
$686$	5.80275	0.221550
$687$	0	0
$688$	− 31.6528i	− 1.20675i
$689$	−19.0303	−0.724996
$690$	0	0
$691$	−31.4305	−1.19567	−0.597836	−	0.801618i	$-0.703973\pi$
$691$	−31.4305	−1.19567	−0.597836	+	0.801618i	$0.703973\pi$
$692$	− 32.3065i	− 1.22811i
$693$	0	0
$694$	75.8989	2.88108
$695$	0	0
$696$	0	0
$697$	− 4.76491i	− 0.180484i
$698$	49.4674i	1.87237i
$699$	0	0
$700$	0	0
$701$	−3.24507	−0.122565	−0.0612824	−	0.998120i	$-0.519519\pi$
$701$	−3.24507	−0.122565	−0.0612824	+	0.998120i	$0.519519\pi$
$702$	0	0
$703$	1.18635i	0.0447442i
$704$	−11.4537	−0.431676
$705$	0	0
$706$	−20.7181	−0.779737
$707$	26.9541i	1.01371i
$708$	0	0
$709$	26.7190	1.00345	0.501727	−	0.865026i	$-0.332698\pi$
$709$	26.7190	1.00345	0.501727	+	0.865026i	$0.332698\pi$
$710$	0	0
$711$	0	0
$712$	11.4929i	0.430714i
$713$	31.6273i	1.18445i
$714$	0	0
$715$	0	0
$716$	−33.9163	−1.26751
$717$	0	0
$718$	72.0351i	2.68833i
$719$	−6.78807	−0.253152	−0.126576	−	0.991957i	$-0.540399\pi$
$719$	−6.78807	−0.253152	−0.126576	+	0.991957i	$0.540399\pi$
$720$	0	0
$721$	−60.0587	−2.23670
$722$	− 25.9007i	− 0.963924i
$723$	0	0
$724$	58.0743	2.15831
$725$	0	0
$726$	0	0
$727$	19.9154i	0.738620i	0.929306	+	0.369310i	$0.120406\pi$
$727$	19.9154i	0.738620i	−0.929306	+	0.369310i	$0.879594\pi$
$728$	− 6.03692i	− 0.223743i
$729$	0	0
$730$	0	0
$731$	−13.5204	−0.500071
$732$	0	0
$733$	− 31.3388i	− 1.15752i	−0.815497	−	0.578762i	$-0.803536\pi$
$733$	− 31.3388i	− 1.15752i	0.815497	−	0.578762i	$-0.196464\pi$
$734$	4.00756	0.147922
$735$	0	0
$736$	45.5081	1.67745
$737$	− 14.2645i	− 0.525438i
$738$	0	0
$739$	2.25355	0.0828982	0.0414491	−	0.999141i	$-0.486803\pi$
$739$	2.25355	0.0828982	0.0414491	+	0.999141i	$0.486803\pi$
$740$	0	0
$741$	0	0
$742$	97.1477i	3.56640i
$743$	6.74931i	0.247608i	0.992307	+	0.123804i	$0.0395095\pi$
$743$	6.74931i	0.247608i	−0.992307	+	0.123804i	$0.960491\pi$
$744$	0	0
$745$	0	0
$746$	34.6888	1.27005
$747$	0	0
$748$	2.90539i	0.106232i
$749$	14.3397	0.523961
$750$	0	0
$751$	22.4390	0.818810	0.409405	−	0.912353i	$-0.365736\pi$
$751$	22.4390	0.818810	0.409405	+	0.912353i	$0.365736\pi$
$752$	9.42431i	0.343669i
$753$	0	0
$754$	−20.1211	−0.732767
$755$	0	0
$756$	0	0
$757$	− 25.4158i	− 0.923754i	−0.886944	−	0.461877i	$-0.847176\pi$
$757$	− 25.4158i	− 0.923754i	0.886944	−	0.461877i	$-0.152824\pi$
$758$	55.3406i	2.01006i
$759$	0	0
$760$	0	0
$761$	30.7493	1.11466	0.557331	−	0.830291i	$-0.311826\pi$
$761$	30.7493	1.11466	0.557331	+	0.830291i	$0.311826\pi$
$762$	0	0
$763$	24.5142i	0.887474i
$764$	−20.0819	−0.726537
$765$	0	0
$766$	26.4977	0.957401
$767$	− 11.7190i	− 0.423150i
$768$	0	0
$769$	16.2956	0.587636	0.293818	−	0.955861i	$-0.405074\pi$
$769$	16.2956	0.587636	0.293818	+	0.955861i	$0.405074\pi$
$770$	0	0
$771$	0	0
$772$	− 29.5904i	− 1.06498i
$773$	− 48.7787i	− 1.75445i	−0.480082	−	0.877223i	$-0.659393\pi$
$773$	− 48.7787i	− 1.75445i	0.480082	−	0.877223i	$-0.340607\pi$
$774$	0	0
$775$	0	0
$776$	7.42100	0.266398
$777$	0	0
$778$	− 38.4390i	− 1.37810i
$779$	10.7649	0.385693
$780$	0	0
$781$	8.51514	0.304696
$782$	− 14.0761i	− 0.503362i
$783$	0	0
$784$	16.9016	0.603629
$785$	0	0
$786$	0	0
$787$	46.2001i	1.64686i	0.567421	+	0.823428i	$0.307941\pi$
$787$	46.2001i	1.64686i	−0.567421	+	0.823428i	$0.692059\pi$
$788$	9.59415i	0.341777i
$789$	0	0
$790$	0	0
$791$	21.8401	0.776546
$792$	0	0
$793$	− 18.2039i	− 0.646439i
$794$	−32.3709	−1.14880
$795$	0	0
$796$	−30.3709	−1.07647
$797$	36.3784i	1.28859i	0.764777	+	0.644295i	$0.222849\pi$
$797$	36.3784i	1.28859i	−0.764777	+	0.644295i	$0.777151\pi$
$798$	0	0
$799$	4.02558	0.142415
$800$	0	0
$801$	0	0
$802$	− 5.84197i	− 0.206287i
$803$	9.21949i	0.325349i
$804$	0	0
$805$	0	0
$806$	−17.7560	−0.625427
$807$	0	0
$808$	8.10551i	0.285151i
$809$	11.3737	0.399879	0.199940	−	0.979808i	$-0.435925\pi$
$809$	11.3737	0.399879	0.199940	+	0.979808i	$0.435925\pi$
$810$	0	0
$811$	−13.3903	−0.470195	−0.235098	−	0.971972i	$-0.575541\pi$
$811$	−13.3903	−0.470195	−0.235098	+	0.971972i	$0.575541\pi$
$812$	57.2177i	2.00795i
$813$	0	0
$814$	−0.965943	−0.0338563
$815$	0	0
$816$	0	0
$817$	− 30.5454i	− 1.06865i
$818$	8.46835i	0.296089i
$819$	0	0
$820$	0	0
$821$	−32.0975	−1.12021	−0.560105	−	0.828422i	$-0.689239\pi$
$821$	−32.0975	−1.12021	−0.560105	+	0.828422i	$0.689239\pi$
$822$	0	0
$823$	16.7952i	0.585443i	0.956198	+	0.292722i	$0.0945609\pi$
$823$	16.7952i	0.585443i	−0.956198	+	0.292722i	$0.905439\pi$
$824$	−18.0606	−0.629169
$825$	0	0
$826$	−59.8245	−2.08156
$827$	− 45.5904i	− 1.58533i	−0.609656	−	0.792666i	$-0.708692\pi$
$827$	− 45.5904i	− 1.58533i	0.609656	−	0.792666i	$-0.291308\pi$
$828$	0	0
$829$	−12.9385	−0.449374	−0.224687	−	0.974431i	$-0.572136\pi$
$829$	−12.9385	−0.449374	−0.224687	+	0.974431i	$0.572136\pi$
$830$	0	0
$831$	0	0
$832$	17.3539i	0.601638i
$833$	− 7.21949i	− 0.250141i
$834$	0	0
$835$	0	0
$836$	−6.56387	−0.227016
$837$	0	0
$838$	− 10.9130i	− 0.376982i
$839$	−1.59037	−0.0549057	−0.0274528	−	0.999623i	$-0.508740\pi$
$839$	−1.59037	−0.0549057	−0.0274528	+	0.999623i	$0.508740\pi$
$840$	0	0
$841$	10.0596	0.346884
$842$	− 19.0104i	− 0.655143i
$843$	0	0
$844$	−25.8165	−0.888641
$845$	0	0
$846$	0	0
$847$	− 3.64002i	− 0.125073i
$848$	− 33.9670i	− 1.16643i
$849$	0	0
$850$	0	0
$851$	2.60688	0.0893628
$852$	0	0
$853$	10.5161i	0.360063i	0.983661	+	0.180031i	$0.0576199\pi$
$853$	10.5161i	0.360063i	−0.983661	+	0.180031i	$0.942380\pi$
$854$	−92.9291	−3.17997
$855$	0	0
$856$	4.31216	0.147386
$857$	− 57.4637i	− 1.96292i	−0.191665	−	0.981460i	$-0.561389\pi$
$857$	− 57.4637i	− 1.96292i	0.191665	−	0.981460i	$-0.438611\pi$
$858$	0	0
$859$	−32.7181	−1.11633	−0.558164	−	0.829731i	$-0.688494\pi$
$859$	−32.7181	−1.11633	−0.558164	+	0.829731i	$0.688494\pi$
$860$	0	0
$861$	0	0
$862$	− 48.3397i	− 1.64646i
$863$	43.1807i	1.46989i	0.678127	+	0.734945i	$0.262792\pi$
$863$	43.1807i	1.46989i	−0.678127	+	0.734945i	$0.737208\pi$
$864$	0	0
$865$	0	0
$866$	−16.1135	−0.547560
$867$	0	0
$868$	50.4920i	1.71381i
$869$	5.09461	0.172823
$870$	0	0
$871$	−21.6126	−0.732315
$872$	7.37179i	0.249640i
$873$	0	0
$874$	31.8009	1.07568
$875$	0	0
$876$	0	0
$877$	16.0752i	0.542822i	0.962464	+	0.271411i	$0.0874903\pi$
$877$	16.0752i	0.542822i	−0.962464	+	0.271411i	$0.912510\pi$
$878$	− 36.9523i	− 1.24708i
$879$	0	0
$880$	0	0
$881$	−31.2876	−1.05411	−0.527053	−	0.849832i	$-0.676703\pi$
$881$	−31.2876	−1.05411	−0.527053	+	0.849832i	$0.676703\pi$
$882$	0	0
$883$	24.7640i	0.833375i	0.909050	+	0.416687i	$0.136809\pi$
$883$	24.7640i	0.833375i	−0.909050	+	0.416687i	$0.863191\pi$
$884$	4.40207	0.148058
$885$	0	0
$886$	22.6169	0.759828
$887$	− 26.3085i	− 0.883353i	−0.897174	−	0.441676i	$-0.854384\pi$
$887$	− 26.3085i	− 0.883353i	0.897174	−	0.441676i	$-0.145616\pi$
$888$	0	0
$889$	−29.3544	−0.984514
$890$	0	0
$891$	0	0
$892$	− 32.5053i	− 1.08836i
$893$	9.09461i	0.304339i
$894$	0	0
$895$	0	0
$896$	30.8179	1.02955
$897$	0	0
$898$	− 57.2252i	− 1.90963i
$899$	34.4683	1.14958
$900$	0	0
$901$	−14.5089	−0.483363
$902$	8.76491i	0.291840i
$903$	0	0
$904$	6.56766	0.218437
$905$	0	0
$906$	0	0
$907$	55.9301i	1.85713i	0.371174	+	0.928563i	$0.378955\pi$
$907$	55.9301i	1.85713i	−0.371174	+	0.928563i	$0.621045\pi$
$908$	− 57.3700i	− 1.90389i
$909$	0	0
$910$	0	0
$911$	15.3931	0.509997	0.254998	−	0.966941i	$-0.417925\pi$
$911$	15.3931	0.509997	0.254998	+	0.966941i	$0.417925\pi$
$912$	0	0
$913$	− 14.7493i	− 0.488131i
$914$	−33.5298	−1.10907
$915$	0	0
$916$	37.1727	1.22822
$917$	− 46.6282i	− 1.53980i
$918$	0	0
$919$	51.2598	1.69090	0.845452	−	0.534052i	$-0.179331\pi$
$919$	51.2598	1.69090	0.845452	+	0.534052i	$0.179331\pi$
$920$	0	0
$921$	0	0
$922$	− 17.4012i	− 0.573076i
$923$	− 12.9016i	− 0.424662i
$924$	0	0
$925$	0	0
$926$	−34.1892	−1.12353
$927$	0	0
$928$	− 49.5961i	− 1.62807i
$929$	14.8099	0.485896	0.242948	−	0.970039i	$-0.421886\pi$
$929$	14.8099	0.485896	0.242948	+	0.970039i	$0.421886\pi$
$930$	0	0
$931$	16.3103	0.534549
$932$	12.4900i	0.409125i
$933$	0	0
$934$	62.6188	2.04895
$935$	0	0
$936$	0	0
$937$	− 27.2654i	− 0.890721i	−0.895351	−	0.445360i	$-0.853076\pi$
$937$	− 27.2654i	− 0.890721i	0.895351	−	0.445360i	$-0.146924\pi$
$938$	110.330i	3.60241i
$939$	0	0
$940$	0	0
$941$	49.5630	1.61571	0.807853	−	0.589384i	$-0.200629\pi$
$941$	49.5630	1.61571	0.807853	+	0.589384i	$0.200629\pi$
$942$	0	0
$943$	− 23.6547i	− 0.770303i
$944$	20.9172	0.680797
$945$	0	0
$946$	24.8704	0.808607
$947$	− 13.9844i	− 0.454432i	−0.973844	−	0.227216i	$-0.927038\pi$
$947$	− 13.9844i	− 0.454432i	0.973844	−	0.227216i	$-0.0729623\pi$
$948$	0	0
$949$	13.9688	0.453447
$950$	0	0
$951$	0	0
$952$	− 4.60263i	− 0.149172i
$953$	− 17.1240i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$953$	− 17.1240i	− 0.554700i	0.960769	−	0.277350i	$-0.0894561\pi$
$954$	0	0
$955$	0	0
$956$	37.7248	1.22011
$957$	0	0
$958$	68.4608i	2.21187i
$959$	83.0284	2.68113
$960$	0	0
$961$	−0.583252	−0.0188146
$962$	1.46354i	0.0471863i
$963$	0	0
$964$	−12.6888	−0.408677
$965$	0	0
$966$	0	0
$967$	− 1.90826i	− 0.0613653i	−0.999529	−	0.0306827i	$-0.990232\pi$
$967$	− 1.90826i	− 0.0613653i	0.999529	−	0.0306827i	$-0.00976813\pi$
$968$	− 1.09461i	− 0.0351821i
$969$	0	0
$970$	0	0
$971$	−31.3856	−1.00721	−0.503605	−	0.863934i	$-0.667993\pi$
$971$	−31.3856	−1.00721	−0.503605	+	0.863934i	$0.667993\pi$
$972$	0	0
$973$	27.6291i	0.885749i
$974$	−76.2697	−2.44384
$975$	0	0
$976$	32.4920	1.04004
$977$	41.4693i	1.32672i	0.748301	+	0.663360i	$0.230870\pi$
$977$	41.4693i	1.32672i	−0.748301	+	0.663360i	$0.769130\pi$
$978$	0	0
$979$	10.4995	0.335567
$980$	0	0
$981$	0	0
$982$	15.2119i	0.485432i
$983$	6.23601i	0.198898i	0.995043	+	0.0994489i	$0.0317080\pi$
$983$	6.23601i	0.198898i	−0.995043	+	0.0994489i	$0.968292\pi$
$984$	0	0
$985$	0	0
$986$	−15.3406	−0.488544
$987$	0	0
$988$	9.94518i	0.316398i
$989$	−67.1202	−2.13430
$990$	0	0
$991$	27.2048	0.864189	0.432095	−	0.901828i	$-0.357775\pi$
$991$	27.2048	0.864189	0.432095	+	0.901828i	$0.357775\pi$
$992$	− 43.7663i	− 1.38958i
$993$	0	0
$994$	−65.8615	−2.08900
$995$	0	0
$996$	0	0
$997$	− 28.0606i	− 0.888687i	−0.895857	−	0.444343i	$-0.853437\pi$
$997$	− 28.0606i	− 0.888687i	0.895857	−	0.444343i	$-0.146563\pi$
$998$	− 57.5298i	− 1.82107i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.c.q.199.5		6
3.2	odd	2		825.2.c.f.199.2		6
5.2	odd	4		2475.2.a.bd.1.1		3
5.3	odd	4		2475.2.a.z.1.3		3
5.4	even	2	inner	2475.2.c.q.199.2		6
15.2	even	4		825.2.a.i.1.3	✓	3
15.8	even	4		825.2.a.m.1.1	yes	3
15.14	odd	2		825.2.c.f.199.5		6
165.32	odd	4		9075.2.a.cj.1.1		3
165.98	odd	4		9075.2.a.cd.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.a.i.1.3	✓	3	15.2	even	4
825.2.a.m.1.1	yes	3	15.8	even	4
825.2.c.f.199.2		6	3.2	odd	2
825.2.c.f.199.5		6	15.14	odd	2
2475.2.a.z.1.3		3	5.3	odd	4
2475.2.a.bd.1.1		3	5.2	odd	4
2475.2.c.q.199.2		6	5.4	even	2	inner
2475.2.c.q.199.5		6	1.1	even	1	trivial
9075.2.a.cd.1.3		3	165.98	odd	4
9075.2.a.cj.1.1		3	165.32	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.12489i q^{2} -2.51514 q^{4} -3.64002i q^{7} -1.09461i q^{8} +O(q^{10})\)	\(q+2.12489i q^{2} -2.51514 q^{4} -3.64002i q^{7} -1.09461i q^{8} -1.00000 q^{11} +1.51514i q^{13} +7.73463 q^{14} -2.70436 q^{16} +1.15516i q^{17} -2.60975 q^{19} -2.12489i q^{22} +5.73463i q^{23} -3.21949 q^{26} +9.15516i q^{28} +6.24977 q^{29} +5.51514 q^{31} -7.93567i q^{32} -2.45459 q^{34} -0.454586i q^{37} -5.54541i q^{38} -4.12489 q^{41} +11.7044i q^{43} +2.51514 q^{44} -12.1854 q^{46} -3.48486i q^{47} -6.24977 q^{49} -3.81078i q^{52} +12.5601i q^{53} -3.98440 q^{56} +13.2800i q^{58} -7.73463 q^{59} -12.0147 q^{61} +11.7190i q^{62} +11.4537 q^{64} +14.2645i q^{67} -2.90539i q^{68} -8.51514 q^{71} -9.21949i q^{73} +0.965943 q^{74} +6.56387 q^{76} +3.64002i q^{77} -5.09461 q^{79} -8.76491i q^{82} +14.7493i q^{83} -24.8704 q^{86} +1.09461i q^{88} -10.4995 q^{89} +5.51514 q^{91} -14.4234i q^{92} +7.40493 q^{94} +6.77959i q^{97} -13.2800i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 16 q^{4}+O(q^{10}) \) 6 * q - 16 * q^4	\( 6 q - 16 q^{4} - 6 q^{11} + 12 q^{14} + 20 q^{16} + 2 q^{19} + 16 q^{26} + 4 q^{29} + 34 q^{31} - 12 q^{34} - 8 q^{41} + 16 q^{44} - 60 q^{46} - 4 q^{49} + 44 q^{56} - 12 q^{59} - 6 q^{61} - 68 q^{64} - 52 q^{71} + 28 q^{74} - 48 q^{76} - 12 q^{79} - 56 q^{86} + 4 q^{89} + 34 q^{91} - 4 q^{94}+O(q^{100}) \) 6 * q - 16 * q^4 - 6 * q^11 + 12 * q^14 + 20 * q^16 + 2 * q^19 + 16 * q^26 + 4 * q^29 + 34 * q^31 - 12 * q^34 - 8 * q^41 + 16 * q^44 - 60 * q^46 - 4 * q^49 + 44 * q^56 - 12 * q^59 - 6 * q^61 - 68 * q^64 - 52 * q^71 + 28 * q^74 - 48 * q^76 - 12 * q^79 - 56 * q^86 + 4 * q^89 + 34 * q^91 - 4 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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