LMFDB - Embedded newform 2475.2.a.bd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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, 0, 0]))

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

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

chi := DirichletCharacter("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 825)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.12489$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.12489 q^{2} +2.51514 q^{4} +3.64002 q^{7} -1.09461 q^{8} +O(q^{10})$	$q-2.12489 q^{2} +2.51514 q^{4} +3.64002 q^{7} -1.09461 q^{8} -1.00000 q^{11} +1.51514 q^{13} -7.73463 q^{14} -2.70436 q^{16} -1.15516 q^{17} +2.60975 q^{19} +2.12489 q^{22} +5.73463 q^{23} -3.21949 q^{26} +9.15516 q^{28} -6.24977 q^{29} +5.51514 q^{31} +7.93567 q^{32} +2.45459 q^{34} +0.454586 q^{37} -5.54541 q^{38} -4.12489 q^{41} +11.7044 q^{43} -2.51514 q^{44} -12.1854 q^{46} +3.48486 q^{47} +6.24977 q^{49} +3.81078 q^{52} +12.5601 q^{53} -3.98440 q^{56} +13.2800 q^{58} +7.73463 q^{59} -12.0147 q^{61} -11.7190 q^{62} -11.4537 q^{64} -14.2645 q^{67} -2.90539 q^{68} -8.51514 q^{71} -9.21949 q^{73} -0.965943 q^{74} +6.56387 q^{76} -3.64002 q^{77} +5.09461 q^{79} +8.76491 q^{82} +14.7493 q^{83} -24.8704 q^{86} +1.09461 q^{88} +10.4995 q^{89} +5.51514 q^{91} +14.4234 q^{92} -7.40493 q^{94} -6.77959 q^{97} -13.2800 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 8 q^{4} + 3 q^{7} + 6 q^{8}+O(q^{10}) $ 3 * q + 2 * q^2 + 8 * q^4 + 3 * q^7 + 6 * q^8	$ 3 q + 2 q^{2} + 8 q^{4} + 3 q^{7} + 6 q^{8} - 3 q^{11} + 5 q^{13} - 6 q^{14} + 10 q^{16} + 4 q^{17} - q^{19} - 2 q^{22} + 8 q^{26} + 20 q^{28} - 2 q^{29} + 17 q^{31} + 34 q^{32} + 6 q^{34} - 18 q^{38} - 4 q^{41} + 17 q^{43} - 8 q^{44} - 30 q^{46} + 10 q^{47} + 2 q^{49} + 30 q^{52} + 6 q^{53} + 22 q^{56} + 24 q^{58} + 6 q^{59} - 3 q^{61} + 16 q^{62} + 34 q^{64} + 7 q^{67} - 18 q^{68} - 26 q^{71} - 10 q^{73} - 14 q^{74} - 24 q^{76} - 3 q^{77} + 6 q^{79} + 10 q^{82} - 6 q^{83} - 28 q^{86} - 6 q^{88} - 2 q^{89} + 17 q^{91} - 26 q^{92} + 2 q^{94} + 29 q^{97} - 24 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 8 * q^4 + 3 * q^7 + 6 * q^8 - 3 * q^11 + 5 * q^13 - 6 * q^14 + 10 * q^16 + 4 * q^17 - q^19 - 2 * q^22 + 8 * q^26 + 20 * q^28 - 2 * q^29 + 17 * q^31 + 34 * q^32 + 6 * q^34 - 18 * q^38 - 4 * q^41 + 17 * q^43 - 8 * q^44 - 30 * q^46 + 10 * q^47 + 2 * q^49 + 30 * q^52 + 6 * q^53 + 22 * q^56 + 24 * q^58 + 6 * q^59 - 3 * q^61 + 16 * q^62 + 34 * q^64 + 7 * q^67 - 18 * q^68 - 26 * q^71 - 10 * q^73 - 14 * q^74 - 24 * q^76 - 3 * q^77 + 6 * q^79 + 10 * q^82 - 6 * q^83 - 28 * q^86 - 6 * q^88 - 2 * q^89 + 17 * q^91 - 26 * q^92 + 2 * q^94 + 29 * q^97 - 24 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−2.12489	−1.50252	−0.751260	−	0.660006i	$-0.770554\pi$
$2$	−2.12489	−1.50252	−0.751260	+	0.660006i	$0.770554\pi$
$3$	0	0
$3$	0	0
$4$	2.51514	1.25757
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.64002	1.37580	0.687900	−	0.725806i	$-0.258533\pi$
$7$	3.64002	1.37580	0.687900	+	0.725806i	$0.258533\pi$
$8$	−1.09461	−0.387003
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.51514	0.420224	0.210112	−	0.977677i	$-0.432617\pi$
$13$	1.51514	0.420224	0.210112	+	0.977677i	$0.432617\pi$
$14$	−7.73463	−2.06717
$15$	0	0
$16$	−2.70436	−0.676089
$17$	−1.15516	−0.280168	−0.140084	−	0.990140i	$-0.544737\pi$
$17$	−1.15516	−0.280168	−0.140084	+	0.990140i	$0.544737\pi$
$18$	0	0
$19$	2.60975	0.598717	0.299359	−	0.954141i	$-0.403227\pi$
$19$	2.60975	0.598717	0.299359	+	0.954141i	$0.403227\pi$
$20$	0	0
$21$	0	0
$22$	2.12489	0.453027
$23$	5.73463	1.19575	0.597877	−	0.801588i	$-0.296011\pi$
$23$	5.73463	1.19575	0.597877	+	0.801588i	$0.296011\pi$
$24$	0	0
$25$	0	0
$26$	−3.21949	−0.631395
$27$	0	0
$28$	9.15516	1.73016
$29$	−6.24977	−1.16055	−0.580277	−	0.814419i	$-0.697056\pi$
$29$	−6.24977	−1.16055	−0.580277	+	0.814419i	$0.697056\pi$
$30$	0	0
$31$	5.51514	0.990548	0.495274	−	0.868737i	$-0.335068\pi$
$31$	5.51514	0.990548	0.495274	+	0.868737i	$0.335068\pi$
$32$	7.93567	1.40284
$33$	0	0
$34$	2.45459	0.420958
$35$	0	0
$36$	0	0
$37$	0.454586	0.0747335	0.0373667	−	0.999302i	$-0.488103\pi$
$37$	0.454586	0.0747335	0.0373667	+	0.999302i	$0.488103\pi$
$38$	−5.54541	−0.899585
$39$	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
$43$	11.7044	1.78490	0.892449	−	0.451149i	$-0.148986\pi$
$43$	11.7044	1.78490	0.892449	+	0.451149i	$0.148986\pi$
$44$	−2.51514	−0.379171
$45$	0	0
$46$	−12.1854	−1.79664
$47$	3.48486	0.508319	0.254160	−	0.967162i	$-0.418201\pi$
$47$	3.48486	0.508319	0.254160	+	0.967162i	$0.418201\pi$
$48$	0	0
$49$	6.24977	0.892824
$50$	0	0
$51$	0	0
$52$	3.81078	0.528460
$53$	12.5601	1.72526	0.862631	−	0.505834i	$-0.168815\pi$
$53$	12.5601	1.72526	0.862631	+	0.505834i	$0.168815\pi$
$54$	0	0
$55$	0	0
$56$	−3.98440	−0.532438
$57$	0	0
$58$	13.2800	1.74376
$59$	7.73463	1.00696	0.503482	−	0.864006i	$-0.332052\pi$
$59$	7.73463	1.00696	0.503482	+	0.864006i	$0.332052\pi$
$60$	0	0
$61$	−12.0147	−1.53832	−0.769161	−	0.639055i	$-0.779326\pi$
$61$	−12.0147	−1.53832	−0.769161	+	0.639055i	$0.779326\pi$
$62$	−11.7190	−1.48832
$63$	0	0
$64$	−11.4537	−1.43171
$65$	0	0
$66$	0	0
$67$	−14.2645	−1.74268	−0.871340	−	0.490680i	$-0.836749\pi$
$67$	−14.2645	−1.74268	−0.871340	+	0.490680i	$0.836749\pi$
$68$	−2.90539	−0.352330
$69$	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
$73$	−9.21949	−1.07906	−0.539530	−	0.841966i	$-0.681398\pi$
$73$	−9.21949	−1.07906	−0.539530	+	0.841966i	$0.681398\pi$
$74$	−0.965943	−0.112289
$75$	0	0
$76$	6.56387	0.752928
$77$	−3.64002	−0.414819
$78$	0	0
$79$	5.09461	0.573188	0.286594	−	0.958052i	$-0.407477\pi$
$79$	5.09461	0.573188	0.286594	+	0.958052i	$0.407477\pi$
$80$	0	0
$81$	0	0
$82$	8.76491	0.967922
$83$	14.7493	1.61895	0.809474	−	0.587156i	$-0.199753\pi$
$83$	14.7493	1.61895	0.809474	+	0.587156i	$0.199753\pi$
$84$	0	0
$85$	0	0
$86$	−24.8704	−2.68185
$87$	0	0
$88$	1.09461	0.116686
$89$	10.4995	1.11295	0.556475	−	0.830865i	$-0.312154\pi$
$89$	10.4995	1.11295	0.556475	+	0.830865i	$0.312154\pi$
$90$	0	0
$91$	5.51514	0.578144
$92$	14.4234	1.50374
$93$	0	0
$94$	−7.40493	−0.763760
$95$	0	0
$96$	0	0
$97$	−6.77959	−0.688363	−0.344181	−	0.938903i	$-0.611844\pi$
$97$	−6.77959	−0.688363	−0.344181	+	0.938903i	$0.611844\pi$
$98$	−13.2800	−1.34149
$99$	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
$103$	−16.4995	−1.62575	−0.812874	−	0.582439i	$-0.802098\pi$
$103$	−16.4995	−1.62575	−0.812874	+	0.582439i	$0.802098\pi$
$104$	−1.65848	−0.162628
$105$	0	0
$106$	−26.6888	−2.59224
$107$	−3.93945	−0.380841	−0.190420	−	0.981703i	$-0.560985\pi$
$107$	−3.93945	−0.380841	−0.190420	+	0.981703i	$0.560985\pi$
$108$	0	0
$109$	6.73463	0.645061	0.322530	−	0.946559i	$-0.395467\pi$
$109$	6.73463	0.645061	0.322530	+	0.946559i	$0.395467\pi$
$110$	0	0
$111$	0	0
$112$	−9.84392	−0.930163
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	−15.7190	−1.45948
$117$	0	0
$118$	−16.4352	−1.51298
$119$	−4.20482	−0.385455
$120$	0	0
$121$	1.00000	0.0909091
$122$	25.5298	2.31136
$123$	0	0
$124$	13.8713	1.24568
$125$	0	0
$126$	0	0
$127$	8.06433	0.715594	0.357797	−	0.933799i	$-0.383528\pi$
$127$	8.06433	0.715594	0.357797	+	0.933799i	$0.383528\pi$
$128$	8.46640	0.748331
$129$	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
$133$	9.49954	0.823715
$134$	30.3103	2.61841
$135$	0	0
$136$	1.26445	0.108426
$137$	−22.8099	−1.94878	−0.974389	−	0.224868i	$-0.927805\pi$
$137$	−22.8099	−1.94878	−0.974389	+	0.224868i	$0.927805\pi$
$138$	0	0
$139$	7.59037	0.643807	0.321903	−	0.946773i	$-0.395677\pi$
$139$	7.59037	0.643807	0.321903	+	0.946773i	$0.395677\pi$
$140$	0	0
$141$	0	0
$142$	18.0937	1.51839
$143$	−1.51514	−0.126702
$144$	0	0
$145$	0	0
$146$	19.5904	1.62131
$147$	0	0
$148$	1.14335	0.0939825
$149$	1.81456	0.148655	0.0743274	−	0.997234i	$-0.476319\pi$
$149$	1.81456	0.148655	0.0743274	+	0.997234i	$0.476319\pi$
$150$	0	0
$151$	24.3250	1.97954	0.989770	−	0.142670i	$-0.0455687\pi$
$151$	24.3250	1.97954	0.989770	+	0.142670i	$0.0455687\pi$
$152$	−2.85665	−0.231705
$153$	0	0
$154$	7.73463	0.623274
$155$	0	0
$156$	0	0
$157$	9.76491	0.779325	0.389662	−	0.920958i	$-0.372592\pi$
$157$	9.76491	0.779325	0.389662	+	0.920958i	$0.372592\pi$
$158$	−10.8255	−0.861227
$159$	0	0
$160$	0	0
$161$	20.8742	1.64512
$162$	0	0
$163$	−6.98440	−0.547061	−0.273530	−	0.961863i	$-0.588191\pi$
$163$	−6.98440	−0.547061	−0.273530	+	0.961863i	$0.588191\pi$
$164$	−10.3747	−0.810125
$165$	0	0
$166$	−31.3406	−2.43250
$167$	6.31032	0.488307	0.244154	−	0.969737i	$-0.421490\pi$
$167$	6.31032	0.488307	0.244154	+	0.969737i	$0.421490\pi$
$168$	0	0
$169$	−10.7044	−0.823412
$170$	0	0
$171$	0	0
$172$	29.4381	2.24463
$173$	12.8448	0.976575	0.488287	−	0.872683i	$-0.337622\pi$
$173$	12.8448	0.976575	0.488287	+	0.872683i	$0.337622\pi$
$174$	0	0
$175$	0	0
$176$	2.70436	0.203849
$177$	0	0
$178$	−22.3103	−1.67223
$179$	−13.4849	−1.00791	−0.503953	−	0.863731i	$-0.668122\pi$
$179$	−13.4849	−1.00791	−0.503953	+	0.863731i	$0.668122\pi$
$180$	0	0
$181$	−23.0899	−1.71626	−0.858130	−	0.513433i	$-0.828374\pi$
$181$	−23.0899	−1.71626	−0.858130	+	0.513433i	$0.828374\pi$
$182$	−11.7190	−0.868673
$183$	0	0
$184$	−6.27718	−0.462760
$185$	0	0
$186$	0	0
$187$	1.15516	0.0844738
$188$	8.76491	0.639247
$189$	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
$193$	11.7649	0.846857	0.423428	−	0.905930i	$-0.360827\pi$
$193$	11.7649	0.846857	0.423428	+	0.905930i	$0.360827\pi$
$194$	14.4058	1.03428
$195$	0	0
$196$	15.7190	1.12279
$197$	3.81456	0.271776	0.135888	−	0.990724i	$-0.456611\pi$
$197$	3.81456	0.271776	0.135888	+	0.990724i	$0.456611\pi$
$198$	0	0
$199$	−12.0752	−0.855990	−0.427995	−	0.903781i	$-0.640780\pi$
$199$	−12.0752	−0.855990	−0.427995	+	0.903781i	$0.640780\pi$
$200$	0	0
$201$	0	0
$202$	15.7346	1.10708
$203$	−22.7493	−1.59669
$204$	0	0
$205$	0	0
$206$	35.0596	2.44272
$207$	0	0
$208$	−4.09747	−0.284109
$209$	−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	$0.385054\pi$
$212$	31.5904	2.16964
$213$	0	0
$214$	8.37088	0.572221
$215$	0	0
$216$	0	0
$217$	20.0752	1.36280
$218$	−14.3103	−0.969217
$219$	0	0
$220$	0	0
$221$	−1.75023	−0.117733
$222$	0	0
$223$	12.9239	0.865445	0.432723	−	0.901527i	$-0.357553\pi$
$223$	12.9239	0.865445	0.432723	+	0.901527i	$0.357553\pi$
$224$	28.8860	1.93003
$225$	0	0
$226$	−12.7493	−0.848072
$227$	−22.8099	−1.51394	−0.756972	−	0.653447i	$-0.773322\pi$
$227$	−22.8099	−1.51394	−0.756972	+	0.653447i	$0.773322\pi$
$228$	0	0
$229$	14.7796	0.976663	0.488331	−	0.872658i	$-0.337606\pi$
$229$	14.7796	0.976663	0.488331	+	0.872658i	$0.337606\pi$
$230$	0	0
$231$	0	0
$232$	6.84106	0.449137
$233$	−4.96594	−0.325330	−0.162665	−	0.986681i	$-0.552009\pi$
$233$	−4.96594	−0.325330	−0.162665	+	0.986681i	$0.552009\pi$
$234$	0	0
$235$	0	0
$236$	19.4537	1.26633
$237$	0	0
$238$	8.93475	0.579154
$239$	14.9991	0.970210	0.485105	−	0.874456i	$-0.338781\pi$
$239$	14.9991	0.970210	0.485105	+	0.874456i	$0.338781\pi$
$240$	0	0
$241$	5.04496	0.324974	0.162487	−	0.986711i	$-0.448048\pi$
$241$	5.04496	0.324974	0.162487	+	0.986711i	$0.448048\pi$
$242$	−2.12489	−0.136593
$243$	0	0
$244$	−30.2186	−1.93455
$245$	0	0
$246$	0	0
$247$	3.95413	0.251595
$248$	−6.03692	−0.383345
$249$	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
$253$	−5.73463	−0.360533
$254$	−17.1358	−1.07519
$255$	0	0
$256$	4.91721	0.307325
$257$	13.6509	0.851521	0.425761	−	0.904836i	$-0.360007\pi$
$257$	13.6509	0.851521	0.425761	+	0.904836i	$0.360007\pi$
$258$	0	0
$259$	1.65470	0.102818
$260$	0	0
$261$	0	0
$262$	−27.2195	−1.68163
$263$	−12.5601	−0.774489	−0.387244	−	0.921977i	$-0.626573\pi$
$263$	−12.5601	−0.774489	−0.387244	+	0.921977i	$0.626573\pi$
$264$	0	0
$265$	0	0
$266$	−20.1854	−1.23765
$267$	0	0
$268$	−35.8771	−2.19154
$269$	24.6888	1.50530	0.752650	−	0.658421i	$-0.228775\pi$
$269$	24.6888	1.50530	0.752650	+	0.658421i	$0.228775\pi$
$270$	0	0
$271$	7.56479	0.459528	0.229764	−	0.973246i	$-0.426205\pi$
$271$	7.56479	0.459528	0.229764	+	0.973246i	$0.426205\pi$
$272$	3.12397	0.189418
$273$	0	0
$274$	48.4683	2.92808
$275$	0	0
$276$	0	0
$277$	1.92477	0.115648	0.0578241	−	0.998327i	$-0.481584\pi$
$277$	1.92477	0.115648	0.0578241	+	0.998327i	$0.481584\pi$
$278$	−16.1287	−0.967333
$279$	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
$283$	30.1396	1.79161	0.895806	−	0.444446i	$-0.146599\pi$
$283$	30.1396	1.79161	0.895806	+	0.444446i	$0.146599\pi$
$284$	−21.4167	−1.27085
$285$	0	0
$286$	3.21949	0.190373
$287$	−15.0147	−0.886289
$288$	0	0
$289$	−15.6656	−0.921506
$290$	0	0
$291$	0	0
$292$	−23.1883	−1.35699
$293$	29.1552	1.70326	0.851631	−	0.524141i	$-0.175614\pi$
$293$	29.1552	1.70326	0.851631	+	0.524141i	$0.175614\pi$
$294$	0	0
$295$	0	0
$296$	−0.497594	−0.0289221
$297$	0	0
$298$	−3.85574	−0.223357
$299$	8.68876	0.502484
$300$	0	0
$301$	42.6041	2.45566
$302$	−51.6878	−2.97430
$303$	0	0
$304$	−7.05769	−0.404786
$305$	0	0
$306$	0	0
$307$	27.8548	1.58976	0.794879	−	0.606768i	$-0.207534\pi$
$307$	27.8548	1.58976	0.794879	+	0.606768i	$0.207534\pi$
$308$	−9.15516	−0.521664
$309$	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
$313$	28.3094	1.60014	0.800071	−	0.599905i	$-0.204795\pi$
$313$	28.3094	1.60014	0.800071	+	0.599905i	$0.204795\pi$
$314$	−20.7493	−1.17095
$315$	0	0
$316$	12.8136	0.720824
$317$	8.80986	0.494811	0.247406	−	0.968912i	$-0.420422\pi$
$317$	8.80986	0.494811	0.247406	+	0.968912i	$0.420422\pi$
$318$	0	0
$319$	6.24977	0.349920
$320$	0	0
$321$	0	0
$322$	−44.3553	−2.47182
$323$	−3.01468	−0.167741
$324$	0	0
$325$	0	0
$326$	14.8411	0.821970
$327$	0	0
$328$	4.51514	0.249307
$329$	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	$0.153264\pi$
$332$	37.0966	2.03594
$333$	0	0
$334$	−13.4087	−0.733692
$335$	0	0
$336$	0	0
$337$	28.9844	1.57888	0.789441	−	0.613827i	$-0.210371\pi$
$337$	28.9844	1.57888	0.789441	+	0.613827i	$0.210371\pi$
$338$	22.7455	1.23719
$339$	0	0
$340$	0	0
$341$	−5.51514	−0.298661
$342$	0	0
$343$	−2.73085	−0.147452
$344$	−12.8117	−0.690760
$345$	0	0
$346$	−27.2938	−1.46732
$347$	35.7190	1.91750	0.958749	−	0.284253i	$-0.0917457\pi$
$347$	35.7190	1.91750	0.958749	+	0.284253i	$0.0917457\pi$
$348$	0	0
$349$	−23.2800	−1.24615	−0.623076	−	0.782161i	$-0.714117\pi$
$349$	−23.2800	−1.24615	−0.623076	+	0.782161i	$0.714117\pi$
$350$	0	0
$351$	0	0
$352$	−7.93567	−0.422972
$353$	9.75023	0.518952	0.259476	−	0.965750i	$-0.416450\pi$
$353$	9.75023	0.518952	0.259476	+	0.965750i	$0.416450\pi$
$354$	0	0
$355$	0	0
$356$	26.4078	1.39961
$357$	0	0
$358$	28.6538	1.51440
$359$	−33.9007	−1.78921	−0.894605	−	0.446858i	$-0.852543\pi$
$359$	−33.9007	−1.78921	−0.894605	+	0.446858i	$0.852543\pi$
$360$	0	0
$361$	−12.1892	−0.641538
$362$	49.0634	2.57872
$363$	0	0
$364$	13.8713	0.727055
$365$	0	0
$366$	0	0
$367$	1.88601	0.0984491	0.0492245	−	0.998788i	$-0.484325\pi$
$367$	1.88601	0.0984491	0.0492245	+	0.998788i	$0.484325\pi$
$368$	−15.5085	−0.808436
$369$	0	0
$370$	0	0
$371$	45.7190	2.37361
$372$	0	0
$373$	−16.3250	−0.845277	−0.422638	−	0.906298i	$-0.638896\pi$
$373$	−16.3250	−0.845277	−0.422638	+	0.906298i	$0.638896\pi$
$374$	−2.45459	−0.126924
$375$	0	0
$376$	−3.81456	−0.196721
$377$	−9.46927	−0.487692
$378$	0	0
$379$	−26.0440	−1.33779	−0.668896	−	0.743356i	$-0.733233\pi$
$379$	−26.0440	−1.33779	−0.668896	+	0.743356i	$0.733233\pi$
$380$	0	0
$381$	0	0
$382$	−16.9659	−0.868053
$383$	−12.4702	−0.637197	−0.318598	−	0.947890i	$-0.603212\pi$
$383$	−12.4702	−0.637197	−0.318598	+	0.947890i	$0.603212\pi$
$384$	0	0
$385$	0	0
$386$	−24.9991	−1.27242
$387$	0	0
$388$	−17.0516	−0.865664
$389$	18.0899	0.917195	0.458597	−	0.888644i	$-0.348352\pi$
$389$	18.0899	0.917195	0.458597	+	0.888644i	$0.348352\pi$
$390$	0	0
$391$	−6.62443	−0.335012
$392$	−6.84106	−0.345526
$393$	0	0
$394$	−8.10551	−0.408350
$395$	0	0
$396$	0	0
$397$	−15.2342	−0.764581	−0.382291	−	0.924042i	$-0.624865\pi$
$397$	−15.2342	−0.764581	−0.382291	+	0.924042i	$0.624865\pi$
$398$	25.6585	1.28614
$399$	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
$403$	8.35620	0.416252
$404$	−18.6244	−0.926600
$405$	0	0
$406$	48.3397	2.39906
$407$	−0.454586	−0.0225330
$408$	0	0
$409$	−3.98532	−0.197061	−0.0985307	−	0.995134i	$-0.531414\pi$
$409$	−3.98532	−0.197061	−0.0985307	+	0.995134i	$0.531414\pi$
$410$	0	0
$411$	0	0
$412$	−41.4986	−2.04449
$413$	28.1542	1.38538
$414$	0	0
$415$	0	0
$416$	12.0236	0.589507
$417$	0	0
$418$	5.54541	0.271235
$419$	5.13578	0.250899	0.125450	−	0.992100i	$-0.459963\pi$
$419$	5.13578	0.250899	0.125450	+	0.992100i	$0.459963\pi$
$420$	0	0
$421$	−8.94657	−0.436029	−0.218014	−	0.975946i	$-0.569958\pi$
$421$	−8.94657	−0.436029	−0.218014	+	0.975946i	$0.569958\pi$
$422$	−21.8108	−1.06173
$423$	0	0
$424$	−13.7484	−0.667681
$425$	0	0
$426$	0	0
$427$	−43.7337	−2.11642
$428$	−9.90826	−0.478934
$429$	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
$433$	7.58325	0.364428	0.182214	−	0.983259i	$-0.441674\pi$
$433$	7.58325	0.364428	0.182214	+	0.983259i	$0.441674\pi$
$434$	−42.6576	−2.04763
$435$	0	0
$436$	16.9385	0.811209
$437$	14.9659	0.715918
$438$	0	0
$439$	17.3903	0.829991	0.414996	−	0.909823i	$-0.363783\pi$
$439$	17.3903	0.829991	0.414996	+	0.909823i	$0.363783\pi$
$440$	0	0
$441$	0	0
$442$	3.71904	0.176897
$443$	−10.6438	−0.505702	−0.252851	−	0.967505i	$-0.581368\pi$
$443$	−10.6438	−0.505702	−0.252851	+	0.967505i	$0.581368\pi$
$444$	0	0
$445$	0	0
$446$	−27.4617	−1.30035
$447$	0	0
$448$	−41.6916	−1.96974
$449$	26.9310	1.27095	0.635476	−	0.772121i	$-0.280804\pi$
$449$	26.9310	1.27095	0.635476	+	0.772121i	$0.280804\pi$
$450$	0	0
$451$	4.12489	0.194233
$452$	15.0908	0.709813
$453$	0	0
$454$	48.4683	2.27473
$455$	0	0
$456$	0	0
$457$	−15.7796	−0.738138	−0.369069	−	0.929402i	$-0.620323\pi$
$457$	−15.7796	−0.738138	−0.369069	+	0.929402i	$0.620323\pi$
$458$	−31.4049	−1.46746
$459$	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
$463$	16.0899	0.747762	0.373881	−	0.927477i	$-0.378027\pi$
$463$	16.0899	0.747762	0.373881	+	0.927477i	$0.378027\pi$
$464$	16.9016	0.784638
$465$	0	0
$466$	10.5521	0.488815
$467$	29.4693	1.36367	0.681837	−	0.731504i	$-0.261181\pi$
$467$	29.4693	1.36367	0.681837	+	0.731504i	$0.261181\pi$
$468$	0	0
$469$	−51.9229	−2.39758
$470$	0	0
$471$	0	0
$472$	−8.46640	−0.389698
$473$	−11.7044	−0.538167
$474$	0	0
$475$	0	0
$476$	−10.5757	−0.484736
$477$	0	0
$478$	−31.8713	−1.45776
$479$	−32.2186	−1.47210	−0.736052	−	0.676925i	$-0.763312\pi$
$479$	−32.2186	−1.47210	−0.736052	+	0.676925i	$0.763312\pi$
$480$	0	0
$481$	0.688760	0.0314048
$482$	−10.7200	−0.488280
$483$	0	0
$484$	2.51514	0.114324
$485$	0	0
$486$	0	0
$487$	−35.8936	−1.62649	−0.813247	−	0.581919i	$-0.802302\pi$
$487$	−35.8936	−1.62649	−0.813247	+	0.581919i	$0.802302\pi$
$488$	13.1514	0.595335
$489$	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
$493$	7.21949	0.325150
$494$	−8.40207	−0.378027
$495$	0	0
$496$	−14.9149	−0.669699
$497$	−30.9953	−1.39033
$498$	0	0
$499$	27.0743	1.21201	0.606006	−	0.795460i	$-0.292771\pi$
$499$	27.0743	1.21201	0.606006	+	0.795460i	$0.292771\pi$
$500$	0	0
$501$	0	0
$502$	−6.43899	−0.287386
$503$	−26.9991	−1.20383	−0.601915	−	0.798560i	$-0.705595\pi$
$503$	−26.9991	−1.20383	−0.601915	+	0.798560i	$0.705595\pi$
$504$	0	0
$505$	0	0
$506$	12.1854	0.541709
$507$	0	0
$508$	20.2829	0.899909
$509$	−15.5904	−0.691031	−0.345515	−	0.938413i	$-0.612296\pi$
$509$	−15.5904	−0.691031	−0.345515	+	0.938413i	$0.612296\pi$
$510$	0	0
$511$	−33.5592	−1.48457
$512$	−27.3813	−1.21009
$513$	0	0
$514$	−29.0066	−1.27943
$515$	0	0
$516$	0	0
$517$	−3.48486	−0.153264
$518$	−3.51605	−0.154487
$519$	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
$523$	−10.5786	−0.462568	−0.231284	−	0.972886i	$-0.574293\pi$
$523$	−10.5786	−0.462568	−0.231284	+	0.972886i	$0.574293\pi$
$524$	32.2186	1.40748
$525$	0	0
$526$	26.6888	1.16369
$527$	−6.37088	−0.277520
$528$	0	0
$529$	9.88601	0.429827
$530$	0	0
$531$	0	0
$532$	23.8927	1.03588
$533$	−6.24977	−0.270708
$534$	0	0
$535$	0	0
$536$	15.6140	0.674422
$537$	0	0
$538$	−52.4608	−2.26175
$539$	−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	$0.578066\pi$
$542$	−16.0743	−0.690451
$543$	0	0
$544$	−9.16698	−0.393031
$545$	0	0
$546$	0	0
$547$	6.09369	0.260547	0.130274	−	0.991478i	$-0.458414\pi$
$547$	6.09369	0.260547	0.130274	+	0.991478i	$0.458414\pi$
$548$	−57.3700	−2.45072
$549$	0	0
$550$	0	0
$551$	−16.3103	−0.694843
$552$	0	0
$553$	18.5445	0.788592
$554$	−4.08991	−0.173764
$555$	0	0
$556$	19.0908	0.809631
$557$	5.90069	0.250020	0.125010	−	0.992155i	$-0.460104\pi$
$557$	5.90069	0.250020	0.125010	+	0.992155i	$0.460104\pi$
$558$	0	0
$559$	17.7337	0.750056
$560$	0	0
$561$	0	0
$562$	−3.98440	−0.168072
$563$	−3.03028	−0.127711	−0.0638555	−	0.997959i	$-0.520340\pi$
$563$	−3.03028	−0.127711	−0.0638555	+	0.997959i	$0.520340\pi$
$564$	0	0
$565$	0	0
$566$	−64.0431	−2.69193
$567$	0	0
$568$	9.32075	0.391090
$569$	−13.4049	−0.561964	−0.280982	−	0.959713i	$-0.590660\pi$
$569$	−13.4049	−0.561964	−0.280982	+	0.959713i	$0.590660\pi$
$570$	0	0
$571$	26.8851	1.12511	0.562553	−	0.826761i	$-0.309819\pi$
$571$	26.8851	1.12511	0.562553	+	0.826761i	$0.309819\pi$
$572$	−3.81078	−0.159337
$573$	0	0
$574$	31.9045	1.33167
$575$	0	0
$576$	0	0
$577$	2.03028	0.0845215	0.0422607	−	0.999107i	$-0.486544\pi$
$577$	2.03028	0.0845215	0.0422607	+	0.999107i	$0.486544\pi$
$578$	33.2876	1.38458
$579$	0	0
$580$	0	0
$581$	53.6878	2.22735
$582$	0	0
$583$	−12.5601	−0.520186
$584$	10.0917	0.417599
$585$	0	0
$586$	−61.9514	−2.55919
$587$	−21.8245	−0.900795	−0.450398	−	0.892828i	$-0.648718\pi$
$587$	−21.8245	−0.900795	−0.450398	+	0.892828i	$0.648718\pi$
$588$	0	0
$589$	14.3931	0.593058
$590$	0	0
$591$	0	0
$592$	−1.22936	−0.0505265
$593$	−8.06811	−0.331318	−0.165659	−	0.986183i	$-0.552975\pi$
$593$	−8.06811	−0.331318	−0.165659	+	0.986183i	$0.552975\pi$
$594$	0	0
$595$	0	0
$596$	4.56387	0.186944
$597$	0	0
$598$	−18.4626	−0.754993
$599$	−7.61353	−0.311080	−0.155540	−	0.987830i	$-0.549712\pi$
$599$	−7.61353	−0.311080	−0.155540	+	0.987830i	$0.549712\pi$
$600$	0	0
$601$	3.57569	0.145855	0.0729277	−	0.997337i	$-0.476766\pi$
$601$	3.57569	0.145855	0.0729277	+	0.997337i	$0.476766\pi$
$602$	−90.5289	−3.68968
$603$	0	0
$604$	61.1807	2.48941
$605$	0	0
$606$	0	0
$607$	17.5298	0.711513	0.355757	−	0.934579i	$-0.384223\pi$
$607$	17.5298	0.711513	0.355757	+	0.934579i	$0.384223\pi$
$608$	20.7101	0.839905
$609$	0	0
$610$	0	0
$611$	5.28005	0.213608
$612$	0	0
$613$	12.5601	0.507297	0.253649	−	0.967296i	$-0.418369\pi$
$613$	12.5601	0.507297	0.253649	+	0.967296i	$0.418369\pi$
$614$	−59.1883	−2.38865
$615$	0	0
$616$	3.98440	0.160536
$617$	−15.9612	−0.642576	−0.321288	−	0.946982i	$-0.604116\pi$
$617$	−15.9612	−0.642576	−0.321288	+	0.946982i	$0.604116\pi$
$618$	0	0
$619$	−9.23417	−0.371153	−0.185576	−	0.982630i	$-0.559415\pi$
$619$	−9.23417	−0.371153	−0.185576	+	0.982630i	$0.559415\pi$
$620$	0	0
$621$	0	0
$622$	50.8023	2.03699
$623$	38.2186	1.53119
$624$	0	0
$625$	0	0
$626$	−60.1542	−2.40425
$627$	0	0
$628$	24.5601	0.980054
$629$	−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	$0.302311\pi$
$632$	−5.57661	−0.221826
$633$	0	0
$634$	−18.7200	−0.743464
$635$	0	0
$636$	0	0
$637$	9.46927	0.375186
$638$	−13.2800	−0.525762
$639$	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
$643$	12.6206	0.497710	0.248855	−	0.968541i	$-0.419946\pi$
$643$	12.6206	0.497710	0.248855	+	0.968541i	$0.419946\pi$
$644$	52.5015	2.06885
$645$	0	0
$646$	6.40585	0.252035
$647$	−29.6429	−1.16538	−0.582691	−	0.812694i	$-0.698000\pi$
$647$	−29.6429	−1.16538	−0.582691	+	0.812694i	$0.698000\pi$
$648$	0	0
$649$	−7.73463	−0.303611
$650$	0	0
$651$	0	0
$652$	−17.5667	−0.687967
$653$	−9.90069	−0.387444	−0.193722	−	0.981056i	$-0.562056\pi$
$653$	−9.90069	−0.387444	−0.193722	+	0.981056i	$0.562056\pi$
$654$	0	0
$655$	0	0
$656$	11.1552	0.435536
$657$	0	0
$658$	−26.9541	−1.05078
$659$	5.28005	0.205681	0.102841	−	0.994698i	$-0.467207\pi$
$659$	5.28005	0.205681	0.102841	+	0.994698i	$0.467207\pi$
$660$	0	0
$661$	−26.8548	−1.04453	−0.522266	−	0.852783i	$-0.674913\pi$
$661$	−26.8548	−1.04453	−0.522266	+	0.852783i	$0.674913\pi$
$662$	−68.5271	−2.66338
$663$	0	0
$664$	−16.1447	−0.626537
$665$	0	0
$666$	0	0
$667$	−35.8401	−1.38774
$668$	15.8713	0.614080
$669$	0	0
$670$	0	0
$671$	12.0147	0.463822
$672$	0	0
$673$	−3.81834	−0.147186	−0.0735932	−	0.997288i	$-0.523447\pi$
$673$	−3.81834	−0.147186	−0.0735932	+	0.997288i	$0.523447\pi$
$674$	−61.5885	−2.37230
$675$	0	0
$676$	−26.9229	−1.03550
$677$	15.6897	0.603003	0.301502	−	0.953466i	$-0.402512\pi$
$677$	15.6897	0.603003	0.301502	+	0.953466i	$0.402512\pi$
$678$	0	0
$679$	−24.6779	−0.947049
$680$	0	0
$681$	0	0
$682$	11.7190	0.448745
$683$	15.6353	0.598269	0.299135	−	0.954211i	$-0.403302\pi$
$683$	15.6353	0.598269	0.299135	+	0.954211i	$0.403302\pi$
$684$	0	0
$685$	0	0
$686$	5.80275	0.221550
$687$	0	0
$688$	−31.6528	−1.20675
$689$	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	$0.703973\pi$
$692$	32.3065	1.22811
$693$	0	0
$694$	−75.8989	−2.88108
$695$	0	0
$696$	0	0
$697$	4.76491	0.180484
$698$	49.4674	1.87237
$699$	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
$703$	1.18635	0.0447442
$704$	11.4537	0.431676
$705$	0	0
$706$	−20.7181	−0.779737
$707$	−26.9541	−1.01371
$708$	0	0
$709$	−26.7190	−1.00345	−0.501727	−	0.865026i	$-0.667302\pi$
$709$	−26.7190	−1.00345	−0.501727	+	0.865026i	$0.667302\pi$
$710$	0	0
$711$	0	0
$712$	−11.4929	−0.430714
$713$	31.6273	1.18445
$714$	0	0
$715$	0	0
$716$	−33.9163	−1.26751
$717$	0	0
$718$	72.0351	2.68833
$719$	6.78807	0.253152	0.126576	−	0.991957i	$-0.459601\pi$
$719$	6.78807	0.253152	0.126576	+	0.991957i	$0.459601\pi$
$720$	0	0
$721$	−60.0587	−2.23670
$722$	25.9007	0.963924
$723$	0	0
$724$	−58.0743	−2.15831
$725$	0	0
$726$	0	0
$727$	−19.9154	−0.738620	−0.369310	−	0.929306i	$-0.620406\pi$
$727$	−19.9154	−0.738620	−0.369310	+	0.929306i	$0.620406\pi$
$728$	−6.03692	−0.223743
$729$	0	0
$730$	0	0
$731$	−13.5204	−0.500071
$732$	0	0
$733$	−31.3388	−1.15752	−0.578762	−	0.815497i	$-0.696464\pi$
$733$	−31.3388	−1.15752	−0.578762	+	0.815497i	$0.696464\pi$
$734$	−4.00756	−0.147922
$735$	0	0
$736$	45.5081	1.67745
$737$	14.2645	0.525438
$738$	0	0
$739$	−2.25355	−0.0828982	−0.0414491	−	0.999141i	$-0.513197\pi$
$739$	−2.25355	−0.0828982	−0.0414491	+	0.999141i	$0.513197\pi$
$740$	0	0
$741$	0	0
$742$	−97.1477	−3.56640
$743$	6.74931	0.247608	0.123804	−	0.992307i	$-0.460491\pi$
$743$	6.74931	0.247608	0.123804	+	0.992307i	$0.460491\pi$
$744$	0	0
$745$	0	0
$746$	34.6888	1.27005
$747$	0	0
$748$	2.90539	0.106232
$749$	−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	$0.365736\pi$
$752$	−9.42431	−0.343669
$753$	0	0
$754$	20.1211	0.732767
$755$	0	0
$756$	0	0
$757$	25.4158	0.923754	0.461877	−	0.886944i	$-0.347176\pi$
$757$	25.4158	0.923754	0.461877	+	0.886944i	$0.347176\pi$
$758$	55.3406	2.01006
$759$	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
$763$	24.5142	0.887474
$764$	20.0819	0.726537
$765$	0	0
$766$	26.4977	0.957401
$767$	11.7190	0.423150
$768$	0	0
$769$	−16.2956	−0.587636	−0.293818	−	0.955861i	$-0.594926\pi$
$769$	−16.2956	−0.587636	−0.293818	+	0.955861i	$0.594926\pi$
$770$	0	0
$771$	0	0
$772$	29.5904	1.06498
$773$	−48.7787	−1.75445	−0.877223	−	0.480082i	$-0.840607\pi$
$773$	−48.7787	−1.75445	−0.877223	+	0.480082i	$0.840607\pi$
$774$	0	0
$775$	0	0
$776$	7.42100	0.266398
$777$	0	0
$778$	−38.4390	−1.37810
$779$	−10.7649	−0.385693
$780$	0	0
$781$	8.51514	0.304696
$782$	14.0761	0.503362
$783$	0	0
$784$	−16.9016	−0.603629
$785$	0	0
$786$	0	0
$787$	−46.2001	−1.64686	−0.823428	−	0.567421i	$-0.807941\pi$
$787$	−46.2001	−1.64686	−0.823428	+	0.567421i	$0.807941\pi$
$788$	9.59415	0.341777
$789$	0	0
$790$	0	0
$791$	21.8401	0.776546
$792$	0	0
$793$	−18.2039	−0.646439
$794$	32.3709	1.14880
$795$	0	0
$796$	−30.3709	−1.07647
$797$	−36.3784	−1.28859	−0.644295	−	0.764777i	$-0.722849\pi$
$797$	−36.3784	−1.28859	−0.644295	+	0.764777i	$0.722849\pi$
$798$	0	0
$799$	−4.02558	−0.142415
$800$	0	0
$801$	0	0
$802$	5.84197	0.206287
$803$	9.21949	0.325349
$804$	0	0
$805$	0	0
$806$	−17.7560	−0.625427
$807$	0	0
$808$	8.10551	0.285151
$809$	−11.3737	−0.399879	−0.199940	−	0.979808i	$-0.564075\pi$
$809$	−11.3737	−0.399879	−0.199940	+	0.979808i	$0.564075\pi$
$810$	0	0
$811$	−13.3903	−0.470195	−0.235098	−	0.971972i	$-0.575541\pi$
$811$	−13.3903	−0.470195	−0.235098	+	0.971972i	$0.575541\pi$
$812$	−57.2177	−2.00795
$813$	0	0
$814$	0.965943	0.0338563
$815$	0	0
$816$	0	0
$817$	30.5454	1.06865
$818$	8.46835	0.296089
$819$	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
$823$	16.7952	0.585443	0.292722	−	0.956198i	$-0.405439\pi$
$823$	16.7952	0.585443	0.292722	+	0.956198i	$0.405439\pi$
$824$	18.0606	0.629169
$825$	0	0
$826$	−59.8245	−2.08156
$827$	45.5904	1.58533	0.792666	−	0.609656i	$-0.208692\pi$
$827$	45.5904	1.58533	0.792666	+	0.609656i	$0.208692\pi$
$828$	0	0
$829$	12.9385	0.449374	0.224687	−	0.974431i	$-0.427864\pi$
$829$	12.9385	0.449374	0.224687	+	0.974431i	$0.427864\pi$
$830$	0	0
$831$	0	0
$832$	−17.3539	−0.601638
$833$	−7.21949	−0.250141
$834$	0	0
$835$	0	0
$836$	−6.56387	−0.227016
$837$	0	0
$838$	−10.9130	−0.376982
$839$	1.59037	0.0549057	0.0274528	−	0.999623i	$-0.491260\pi$
$839$	1.59037	0.0549057	0.0274528	+	0.999623i	$0.491260\pi$
$840$	0	0
$841$	10.0596	0.346884
$842$	19.0104	0.655143
$843$	0	0
$844$	25.8165	0.888641
$845$	0	0
$846$	0	0
$847$	3.64002	0.125073
$848$	−33.9670	−1.16643
$849$	0	0
$850$	0	0
$851$	2.60688	0.0893628
$852$	0	0
$853$	10.5161	0.360063	0.180031	−	0.983661i	$-0.442380\pi$
$853$	10.5161	0.360063	0.180031	+	0.983661i	$0.442380\pi$
$854$	92.9291	3.17997
$855$	0	0
$856$	4.31216	0.147386
$857$	57.4637	1.96292	0.981460	−	0.191665i	$-0.0613886\pi$
$857$	57.4637	1.96292	0.981460	+	0.191665i	$0.0613886\pi$
$858$	0	0
$859$	32.7181	1.11633	0.558164	−	0.829731i	$-0.311506\pi$
$859$	32.7181	1.11633	0.558164	+	0.829731i	$0.311506\pi$
$860$	0	0
$861$	0	0
$862$	48.3397	1.64646
$863$	43.1807	1.46989	0.734945	−	0.678127i	$-0.237208\pi$
$863$	43.1807	1.46989	0.734945	+	0.678127i	$0.237208\pi$
$864$	0	0
$865$	0	0
$866$	−16.1135	−0.547560
$867$	0	0
$868$	50.4920	1.71381
$869$	−5.09461	−0.172823
$870$	0	0
$871$	−21.6126	−0.732315
$872$	−7.37179	−0.249640
$873$	0	0
$874$	−31.8009	−1.07568
$875$	0	0
$876$	0	0
$877$	−16.0752	−0.542822	−0.271411	−	0.962464i	$-0.587490\pi$
$877$	−16.0752	−0.542822	−0.271411	+	0.962464i	$0.587490\pi$
$878$	−36.9523	−1.24708
$879$	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
$883$	24.7640	0.833375	0.416687	−	0.909050i	$-0.363191\pi$
$883$	24.7640	0.833375	0.416687	+	0.909050i	$0.363191\pi$
$884$	−4.40207	−0.148058
$885$	0	0
$886$	22.6169	0.759828
$887$	26.3085	0.883353	0.441676	−	0.897174i	$-0.354384\pi$
$887$	26.3085	0.883353	0.441676	+	0.897174i	$0.354384\pi$
$888$	0	0
$889$	29.3544	0.984514
$890$	0	0
$891$	0	0
$892$	32.5053	1.08836
$893$	9.09461	0.304339
$894$	0	0
$895$	0	0
$896$	30.8179	1.02955
$897$	0	0
$898$	−57.2252	−1.90963
$899$	−34.4683	−1.14958
$900$	0	0
$901$	−14.5089	−0.483363
$902$	−8.76491	−0.291840
$903$	0	0
$904$	−6.56766	−0.218437
$905$	0	0
$906$	0	0
$907$	−55.9301	−1.85713	−0.928563	−	0.371174i	$-0.878955\pi$
$907$	−55.9301	−1.85713	−0.928563	+	0.371174i	$0.878955\pi$
$908$	−57.3700	−1.90389
$909$	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
$913$	−14.7493	−0.488131
$914$	33.5298	1.10907
$915$	0	0
$916$	37.1727	1.22822
$917$	46.6282	1.53980
$918$	0	0
$919$	−51.2598	−1.69090	−0.845452	−	0.534052i	$-0.820669\pi$
$919$	−51.2598	−1.69090	−0.845452	+	0.534052i	$0.820669\pi$
$920$	0	0
$921$	0	0
$922$	17.4012	0.573076
$923$	−12.9016	−0.424662
$924$	0	0
$925$	0	0
$926$	−34.1892	−1.12353
$927$	0	0
$928$	−49.5961	−1.62807
$929$	−14.8099	−0.485896	−0.242948	−	0.970039i	$-0.578114\pi$
$929$	−14.8099	−0.485896	−0.242948	+	0.970039i	$0.578114\pi$
$930$	0	0
$931$	16.3103	0.534549
$932$	−12.4900	−0.409125
$933$	0	0
$934$	−62.6188	−2.04895
$935$	0	0
$936$	0	0
$937$	27.2654	0.890721	0.445360	−	0.895351i	$-0.353076\pi$
$937$	27.2654	0.890721	0.445360	+	0.895351i	$0.353076\pi$
$938$	110.330	3.60241
$939$	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
$943$	−23.6547	−0.770303
$944$	−20.9172	−0.680797
$945$	0	0
$946$	24.8704	0.808607
$947$	13.9844	0.454432	0.227216	−	0.973844i	$-0.427038\pi$
$947$	13.9844	0.454432	0.227216	+	0.973844i	$0.427038\pi$
$948$	0	0
$949$	−13.9688	−0.453447
$950$	0	0
$951$	0	0
$952$	4.60263	0.149172
$953$	−17.1240	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$953$	−17.1240	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$954$	0	0
$955$	0	0
$956$	37.7248	1.22011
$957$	0	0
$958$	68.4608	2.21187
$959$	−83.0284	−2.68113
$960$	0	0
$961$	−0.583252	−0.0188146
$962$	−1.46354	−0.0471863
$963$	0	0
$964$	12.6888	0.408677
$965$	0	0
$966$	0	0
$967$	1.90826	0.0613653	0.0306827	−	0.999529i	$-0.490232\pi$
$967$	1.90826	0.0613653	0.0306827	+	0.999529i	$0.490232\pi$
$968$	−1.09461	−0.0351821
$969$	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
$973$	27.6291	0.885749
$974$	76.2697	2.44384
$975$	0	0
$976$	32.4920	1.04004
$977$	−41.4693	−1.32672	−0.663360	−	0.748301i	$-0.730870\pi$
$977$	−41.4693	−1.32672	−0.663360	+	0.748301i	$0.730870\pi$
$978$	0	0
$979$	−10.4995	−0.335567
$980$	0	0
$981$	0	0
$982$	−15.2119	−0.485432
$983$	6.23601	0.198898	0.0994489	−	0.995043i	$-0.468292\pi$
$983$	6.23601	0.198898	0.0994489	+	0.995043i	$0.468292\pi$
$984$	0	0
$985$	0	0
$986$	−15.3406	−0.488544
$987$	0	0
$988$	9.94518	0.316398
$989$	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	$0.357775\pi$
$992$	43.7663	1.38958
$993$	0	0
$994$	65.8615	2.08900
$995$	0	0
$996$	0	0
$997$	28.0606	0.888687	0.444343	−	0.895857i	$-0.353437\pi$
$997$	28.0606	0.888687	0.444343	+	0.895857i	$0.353437\pi$
$998$	−57.5298	−1.82107
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.a.i.1.3	✓	3	3.2	odd	2
825.2.a.m.1.1	yes	3	15.14	odd	2
825.2.c.f.199.2		6	15.8	even	4
825.2.c.f.199.5		6	15.2	even	4
2475.2.a.z.1.3		3	5.4	even	2
2475.2.a.bd.1.1		3	1.1	even	1	trivial
2475.2.c.q.199.2		6	5.2	odd	4
2475.2.c.q.199.5		6	5.3	odd	4
9075.2.a.cd.1.3		3	165.164	even	2
9075.2.a.cj.1.1		3	33.32	even	2

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.a (trivial)

\(f(q)\)	\(=\)	\(q-2.12489 q^{2} +2.51514 q^{4} +3.64002 q^{7} -1.09461 q^{8} +O(q^{10})\)	\(q-2.12489 q^{2} +2.51514 q^{4} +3.64002 q^{7} -1.09461 q^{8} -1.00000 q^{11} +1.51514 q^{13} -7.73463 q^{14} -2.70436 q^{16} -1.15516 q^{17} +2.60975 q^{19} +2.12489 q^{22} +5.73463 q^{23} -3.21949 q^{26} +9.15516 q^{28} -6.24977 q^{29} +5.51514 q^{31} +7.93567 q^{32} +2.45459 q^{34} +0.454586 q^{37} -5.54541 q^{38} -4.12489 q^{41} +11.7044 q^{43} -2.51514 q^{44} -12.1854 q^{46} +3.48486 q^{47} +6.24977 q^{49} +3.81078 q^{52} +12.5601 q^{53} -3.98440 q^{56} +13.2800 q^{58} +7.73463 q^{59} -12.0147 q^{61} -11.7190 q^{62} -11.4537 q^{64} -14.2645 q^{67} -2.90539 q^{68} -8.51514 q^{71} -9.21949 q^{73} -0.965943 q^{74} +6.56387 q^{76} -3.64002 q^{77} +5.09461 q^{79} +8.76491 q^{82} +14.7493 q^{83} -24.8704 q^{86} +1.09461 q^{88} +10.4995 q^{89} +5.51514 q^{91} +14.4234 q^{92} -7.40493 q^{94} -6.77959 q^{97} -13.2800 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 8 q^{4} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) 3 * q + 2 * q^2 + 8 * q^4 + 3 * q^7 + 6 * q^8	\( 3 q + 2 q^{2} + 8 q^{4} + 3 q^{7} + 6 q^{8} - 3 q^{11} + 5 q^{13} - 6 q^{14} + 10 q^{16} + 4 q^{17} - q^{19} - 2 q^{22} + 8 q^{26} + 20 q^{28} - 2 q^{29} + 17 q^{31} + 34 q^{32} + 6 q^{34} - 18 q^{38} - 4 q^{41} + 17 q^{43} - 8 q^{44} - 30 q^{46} + 10 q^{47} + 2 q^{49} + 30 q^{52} + 6 q^{53} + 22 q^{56} + 24 q^{58} + 6 q^{59} - 3 q^{61} + 16 q^{62} + 34 q^{64} + 7 q^{67} - 18 q^{68} - 26 q^{71} - 10 q^{73} - 14 q^{74} - 24 q^{76} - 3 q^{77} + 6 q^{79} + 10 q^{82} - 6 q^{83} - 28 q^{86} - 6 q^{88} - 2 q^{89} + 17 q^{91} - 26 q^{92} + 2 q^{94} + 29 q^{97} - 24 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 8 * q^4 + 3 * q^7 + 6 * q^8 - 3 * q^11 + 5 * q^13 - 6 * q^14 + 10 * q^16 + 4 * q^17 - q^19 - 2 * q^22 + 8 * q^26 + 20 * q^28 - 2 * q^29 + 17 * q^31 + 34 * q^32 + 6 * q^34 - 18 * q^38 - 4 * q^41 + 17 * q^43 - 8 * q^44 - 30 * q^46 + 10 * q^47 + 2 * q^49 + 30 * q^52 + 6 * q^53 + 22 * q^56 + 24 * q^58 + 6 * q^59 - 3 * q^61 + 16 * q^62 + 34 * q^64 + 7 * q^67 - 18 * q^68 - 26 * q^71 - 10 * q^73 - 14 * q^74 - 24 * q^76 - 3 * q^77 + 6 * q^79 + 10 * q^82 - 6 * q^83 - 28 * q^86 - 6 * q^88 - 2 * q^89 + 17 * q^91 - 26 * q^92 + 2 * q^94 + 29 * q^97 - 24 * q^98

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