LMFDB - Embedded newform 3087.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3087,2,Mod(1,3087)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3087, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3087.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3087 = 3^{2} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3087.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:	$24.6498191040$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.35650048.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 20x^{4} + 124x^{2} - 232 $ x^6 - 20x^4 + 124x^2 - 232
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 343)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.62490$ of defining polynomial
Character	$\chi$	$=$	3087.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.24698 q^{2} +3.04892 q^{4} -3.27319 q^{5} -2.35690 q^{8} +O(q^{10})$	$q-2.24698 q^{2} +3.04892 q^{4} -3.27319 q^{5} -2.35690 q^{8} +7.35480 q^{10} -0.445042 q^{11} +4.44138 q^{13} -0.801938 q^{16} +0.648296 q^{17} -1.16819 q^{19} -9.97969 q^{20} +1.00000 q^{22} -4.35690 q^{23} +5.71379 q^{25} -9.97969 q^{26} +2.13706 q^{29} -6.18661 q^{31} +6.51573 q^{32} -1.45671 q^{34} -7.65279 q^{37} +2.62490 q^{38} +7.71457 q^{40} +3.56171 q^{41} -6.02177 q^{43} -1.35690 q^{44} +9.78986 q^{46} +12.7330 q^{47} -12.8388 q^{50} +13.5414 q^{52} -0.594187 q^{53} +1.45671 q^{55} -4.80194 q^{58} +7.06628 q^{59} +13.2529 q^{61} +13.9012 q^{62} -13.0368 q^{64} -14.5375 q^{65} +12.4547 q^{67} +1.97660 q^{68} -4.29590 q^{71} -5.08968 q^{73} +17.1957 q^{74} -3.56171 q^{76} +6.57002 q^{79} +2.62490 q^{80} -8.00309 q^{82} +4.44138 q^{83} -2.12200 q^{85} +13.5308 q^{86} +1.04892 q^{88} -4.37012 q^{89} -13.2838 q^{92} -28.6108 q^{94} +3.82371 q^{95} +7.51491 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{2} - 6 q^{8}+O(q^{10}) $ 6 * q - 4 * q^2 - 6 * q^8	$ 6 q - 4 q^{2} - 6 q^{8} - 2 q^{11} + 4 q^{16} + 6 q^{22} - 18 q^{23} + 18 q^{25} + 2 q^{29} + 14 q^{32} - 10 q^{37} - 30 q^{43} + 12 q^{46} - 12 q^{50} - 30 q^{53} - 20 q^{58} - 22 q^{64} - 56 q^{65} + 30 q^{67} + 2 q^{71} + 30 q^{74} - 10 q^{79} - 52 q^{85} + 6 q^{86} - 12 q^{88} - 14 q^{92} + 8 q^{95}+O(q^{100}) $ 6 * q - 4 * q^2 - 6 * q^8 - 2 * q^11 + 4 * q^16 + 6 * q^22 - 18 * q^23 + 18 * q^25 + 2 * q^29 + 14 * q^32 - 10 * q^37 - 30 * q^43 + 12 * q^46 - 12 * q^50 - 30 * q^53 - 20 * q^58 - 22 * q^64 - 56 * q^65 + 30 * q^67 + 2 * q^71 + 30 * q^74 - 10 * q^79 - 52 * q^85 + 6 * q^86 - 12 * q^88 - 14 * q^92 + 8 * q^95

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 $

$2$	−2.24698	−1.58885	−0.794427	−	0.607359i	$-0.792229\pi$
$2$	−2.24698	−1.58885	−0.794427	+	0.607359i	$0.792229\pi$
$3$	0	0
$3$	0	0
$4$	3.04892	1.52446
$5$	−3.27319	−1.46382	−0.731908	−	0.681403i	$-0.761370\pi$
$5$	−3.27319	−1.46382	−0.731908	+	0.681403i	$0.761370\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.35690	−0.833289
$9$	0	0
$10$	7.35480	2.32579
$11$	−0.445042	−0.134185	−0.0670926	−	0.997747i	$-0.521372\pi$
$11$	−0.445042	−0.134185	−0.0670926	+	0.997747i	$0.521372\pi$
$12$	0	0
$13$	4.44138	1.23182	0.615909	−	0.787817i	$-0.288789\pi$
$13$	4.44138	1.23182	0.615909	+	0.787817i	$0.288789\pi$
$14$	0	0
$15$	0	0
$16$	−0.801938	−0.200484
$17$	0.648296	0.157235	0.0786174	−	0.996905i	$-0.474949\pi$
$17$	0.648296	0.157235	0.0786174	+	0.996905i	$0.474949\pi$
$18$	0	0
$19$	−1.16819	−0.268001	−0.134000	−	0.990981i	$-0.542782\pi$
$19$	−1.16819	−0.268001	−0.134000	+	0.990981i	$0.542782\pi$
$20$	−9.97969	−2.23153
$21$	0	0
$22$	1.00000	0.213201
$23$	−4.35690	−0.908476	−0.454238	−	0.890880i	$-0.650088\pi$
$23$	−4.35690	−0.908476	−0.454238	+	0.890880i	$0.650088\pi$
$24$	0	0
$25$	5.71379	1.14276
$26$	−9.97969	−1.95718
$27$	0	0
$28$	0	0
$29$	2.13706	0.396843	0.198421	−	0.980117i	$-0.436419\pi$
$29$	2.13706	0.396843	0.198421	+	0.980117i	$0.436419\pi$
$30$	0	0
$31$	−6.18661	−1.11115	−0.555574	−	0.831467i	$-0.687501\pi$
$31$	−6.18661	−1.11115	−0.555574	+	0.831467i	$0.687501\pi$
$32$	6.51573	1.15183
$33$	0	0
$34$	−1.45671	−0.249823
$35$	0	0
$36$	0	0
$37$	−7.65279	−1.25811	−0.629056	−	0.777360i	$-0.716558\pi$
$37$	−7.65279	−1.25811	−0.629056	+	0.777360i	$0.716558\pi$
$38$	2.62490	0.425815
$39$	0	0
$40$	7.71457	1.21978
$41$	3.56171	0.556246	0.278123	−	0.960545i	$-0.410288\pi$
$41$	3.56171	0.556246	0.278123	+	0.960545i	$0.410288\pi$
$42$	0	0
$43$	−6.02177	−0.918311	−0.459156	−	0.888356i	$-0.651848\pi$
$43$	−6.02177	−0.918311	−0.459156	+	0.888356i	$0.651848\pi$
$44$	−1.35690	−0.204560
$45$	0	0
$46$	9.78986	1.44344
$47$	12.7330	1.85730	0.928649	−	0.370959i	$-0.120971\pi$
$47$	12.7330	1.85730	0.928649	+	0.370959i	$0.120971\pi$
$48$	0	0
$49$	0	0
$50$	−12.8388	−1.81568
$51$	0	0
$52$	13.5414	1.87786
$53$	−0.594187	−0.0816178	−0.0408089	−	0.999167i	$-0.512993\pi$
$53$	−0.594187	−0.0816178	−0.0408089	+	0.999167i	$0.512993\pi$
$54$	0	0
$55$	1.45671	0.196422
$56$	0	0
$57$	0	0
$58$	−4.80194	−0.630525
$59$	7.06628	0.919951	0.459976	−	0.887932i	$-0.347858\pi$
$59$	7.06628	0.919951	0.459976	+	0.887932i	$0.347858\pi$
$60$	0	0
$61$	13.2529	1.69686	0.848429	−	0.529309i	$-0.177549\pi$
$61$	13.2529	1.69686	0.848429	+	0.529309i	$0.177549\pi$
$62$	13.9012	1.76545
$63$	0	0
$64$	−13.0368	−1.62960
$65$	−14.5375	−1.80315
$66$	0	0
$67$	12.4547	1.52159	0.760794	−	0.648994i	$-0.224810\pi$
$67$	12.4547	1.52159	0.760794	+	0.648994i	$0.224810\pi$
$68$	1.97660	0.239698
$69$	0	0
$70$	0	0
$71$	−4.29590	−0.509829	−0.254915	−	0.966964i	$-0.582047\pi$
$71$	−4.29590	−0.509829	−0.254915	+	0.966964i	$0.582047\pi$
$72$	0	0
$73$	−5.08968	−0.595702	−0.297851	−	0.954612i	$-0.596270\pi$
$73$	−5.08968	−0.595702	−0.297851	+	0.954612i	$0.596270\pi$
$74$	17.1957	1.99896
$75$	0	0
$76$	−3.56171	−0.408556
$77$	0	0
$78$	0	0
$79$	6.57002	0.739185	0.369593	−	0.929194i	$-0.379497\pi$
$79$	6.57002	0.739185	0.369593	+	0.929194i	$0.379497\pi$
$80$	2.62490	0.293472
$81$	0	0
$82$	−8.00309	−0.883794
$83$	4.44138	0.487505	0.243752	−	0.969837i	$-0.421622\pi$
$83$	4.44138	0.487505	0.243752	+	0.969837i	$0.421622\pi$
$84$	0	0
$85$	−2.12200	−0.230163
$86$	13.5308	1.45906
$87$	0	0
$88$	1.04892	0.111815
$89$	−4.37012	−0.463232	−0.231616	−	0.972807i	$-0.574401\pi$
$89$	−4.37012	−0.463232	−0.231616	+	0.972807i	$0.574401\pi$
$90$	0	0
$91$	0	0
$92$	−13.2838	−1.38493
$93$	0	0
$94$	−28.6108	−2.95098
$95$	3.82371	0.392304
$96$	0	0
$97$	7.51491	0.763024	0.381512	−	0.924364i	$-0.375404\pi$
$97$	7.51491	0.763024	0.381512	+	0.924364i	$0.375404\pi$
$98$	0	0
$99$	0	0
$100$	17.4209	1.74209
$101$	−8.23447	−0.819360	−0.409680	−	0.912229i	$-0.634360\pi$
$101$	−8.23447	−0.819360	−0.409680	+	0.912229i	$0.634360\pi$
$102$	0	0
$103$	10.7881	1.06298	0.531492	−	0.847063i	$-0.321632\pi$
$103$	10.7881	1.06298	0.531492	+	0.847063i	$0.321632\pi$
$104$	−10.4679	−1.02646
$105$	0	0
$106$	1.33513	0.129679
$107$	−5.02715	−0.485993	−0.242996	−	0.970027i	$-0.578130\pi$
$107$	−5.02715	−0.485993	−0.242996	+	0.970027i	$0.578130\pi$
$108$	0	0
$109$	−15.2838	−1.46392	−0.731962	−	0.681345i	$-0.761395\pi$
$109$	−15.2838	−1.46392	−0.731962	+	0.681345i	$0.761395\pi$
$110$	−3.27319	−0.312087
$111$	0	0
$112$	0	0
$113$	−10.8412	−1.01985	−0.509926	−	0.860219i	$-0.670327\pi$
$113$	−10.8412	−1.01985	−0.509926	+	0.860219i	$0.670327\pi$
$114$	0	0
$115$	14.2610	1.32984
$116$	6.51573	0.604970
$117$	0	0
$118$	−15.8778	−1.46167
$119$	0	0
$120$	0	0
$121$	−10.8019	−0.981994
$122$	−29.7790	−2.69606
$123$	0	0
$124$	−18.8625	−1.69390
$125$	−2.33638	−0.208972
$126$	0	0
$127$	7.15883	0.635244	0.317622	−	0.948217i	$-0.397116\pi$
$127$	7.15883	0.635244	0.317622	+	0.948217i	$0.397116\pi$
$128$	16.2620	1.43738
$129$	0	0
$130$	32.6655	2.86495
$131$	2.26512	0.197904	0.0989522	−	0.995092i	$-0.468451\pi$
$131$	2.26512	0.197904	0.0989522	+	0.995092i	$0.468451\pi$
$132$	0	0
$133$	0	0
$134$	−27.9855	−2.41758
$135$	0	0
$136$	−1.52797	−0.131022
$137$	−9.36898	−0.800446	−0.400223	−	0.916418i	$-0.631067\pi$
$137$	−9.36898	−0.800446	−0.400223	+	0.916418i	$0.631067\pi$
$138$	0	0
$139$	−14.4211	−1.22318	−0.611590	−	0.791175i	$-0.709470\pi$
$139$	−14.4211	−1.22318	−0.611590	+	0.791175i	$0.709470\pi$
$140$	0	0
$141$	0	0
$142$	9.65279	0.810044
$143$	−1.97660	−0.165292
$144$	0	0
$145$	−6.99502	−0.580905
$146$	11.4364	0.946483
$147$	0	0
$148$	−23.3327	−1.91794
$149$	23.2325	1.90328	0.951640	−	0.307214i	$-0.0993967\pi$
$149$	23.2325	1.90328	0.951640	+	0.307214i	$0.0993967\pi$
$150$	0	0
$151$	4.36658	0.355348	0.177674	−	0.984089i	$-0.443143\pi$
$151$	4.36658	0.355348	0.177674	+	0.984089i	$0.443143\pi$
$152$	2.75330	0.223322
$153$	0	0
$154$	0	0
$155$	20.2500	1.62652
$156$	0	0
$157$	−14.5812	−1.16371	−0.581853	−	0.813294i	$-0.697672\pi$
$157$	−14.5812	−1.16371	−0.581853	+	0.813294i	$0.697672\pi$
$158$	−14.7627	−1.17446
$159$	0	0
$160$	−21.3272	−1.68607
$161$	0	0
$162$	0	0
$163$	−10.2174	−0.800292	−0.400146	−	0.916451i	$-0.631041\pi$
$163$	−10.2174	−0.800292	−0.400146	+	0.916451i	$0.631041\pi$
$164$	10.8594	0.847974
$165$	0	0
$166$	−9.97969	−0.774575
$167$	−21.5445	−1.66716	−0.833582	−	0.552396i	$-0.813714\pi$
$167$	−21.5445	−1.66716	−0.833582	+	0.552396i	$0.813714\pi$
$168$	0	0
$169$	6.72587	0.517375
$170$	4.76809	0.365695
$171$	0	0
$172$	−18.3599	−1.39993
$173$	−14.8380	−1.12811	−0.564056	−	0.825736i	$-0.690760\pi$
$173$	−14.8380	−1.12811	−0.564056	+	0.825736i	$0.690760\pi$
$174$	0	0
$175$	0	0
$176$	0.356896	0.0269020
$177$	0	0
$178$	9.81958	0.736009
$179$	−2.77048	−0.207075	−0.103538	−	0.994626i	$-0.533016\pi$
$179$	−2.77048	−0.207075	−0.103538	+	0.994626i	$0.533016\pi$
$180$	0	0
$181$	15.1265	1.12435	0.562173	−	0.827020i	$-0.309966\pi$
$181$	15.1265	1.12435	0.562173	+	0.827020i	$0.309966\pi$
$182$	0	0
$183$	0	0
$184$	10.2687	0.757022
$185$	25.0491	1.84164
$186$	0	0
$187$	−0.288519	−0.0210986
$188$	38.8218	2.83137
$189$	0	0
$190$	−8.59179	−0.623314
$191$	6.14675	0.444763	0.222382	−	0.974960i	$-0.428617\pi$
$191$	6.14675	0.444763	0.222382	+	0.974960i	$0.428617\pi$
$192$	0	0
$193$	0.631023	0.0454220	0.0227110	−	0.999742i	$-0.492770\pi$
$193$	0.631023	0.0454220	0.0227110	+	0.999742i	$0.492770\pi$
$194$	−16.8859	−1.21233
$195$	0	0
$196$	0	0
$197$	−24.7289	−1.76186	−0.880929	−	0.473248i	$-0.843081\pi$
$197$	−24.7289	−1.76186	−0.880929	+	0.473248i	$0.843081\pi$
$198$	0	0
$199$	−5.30694	−0.376199	−0.188099	−	0.982150i	$-0.560233\pi$
$199$	−5.30694	−0.376199	−0.188099	+	0.982150i	$0.560233\pi$
$200$	−13.4668	−0.952247
$201$	0	0
$202$	18.5027	1.30184
$203$	0	0
$204$	0	0
$205$	−11.6582	−0.814242
$206$	−24.2407	−1.68893
$207$	0	0
$208$	−3.56171	−0.246960
$209$	0.519893	0.0359618
$210$	0	0
$211$	4.49827	0.309674	0.154837	−	0.987940i	$-0.450515\pi$
$211$	4.49827	0.309674	0.154837	+	0.987940i	$0.450515\pi$
$212$	−1.81163	−0.124423
$213$	0	0
$214$	11.2959	0.772172
$215$	19.7104	1.34424
$216$	0	0
$217$	0	0
$218$	34.3424	2.32596
$219$	0	0
$220$	4.44138	0.299438
$221$	2.87933	0.193685
$222$	0	0
$223$	14.9093	0.998398	0.499199	−	0.866487i	$-0.333628\pi$
$223$	14.9093	0.998398	0.499199	+	0.866487i	$0.333628\pi$
$224$	0	0
$225$	0	0
$226$	24.3599	1.62040
$227$	−7.35480	−0.488155	−0.244078	−	0.969756i	$-0.578485\pi$
$227$	−7.35480	−0.488155	−0.244078	+	0.969756i	$0.578485\pi$
$228$	0	0
$229$	−5.60957	−0.370691	−0.185345	−	0.982673i	$-0.559340\pi$
$229$	−5.60957	−0.370691	−0.185345	+	0.982673i	$0.559340\pi$
$230$	−32.0441	−2.11292
$231$	0	0
$232$	−5.03684	−0.330684
$233$	1.10752	0.0725563	0.0362781	−	0.999342i	$-0.488450\pi$
$233$	1.10752	0.0725563	0.0362781	+	0.999342i	$0.488450\pi$
$234$	0	0
$235$	−41.6775	−2.71874
$236$	21.5445	1.40243
$237$	0	0
$238$	0	0
$239$	−6.13706	−0.396974	−0.198487	−	0.980104i	$-0.563603\pi$
$239$	−6.13706	−0.396974	−0.198487	+	0.980104i	$0.563603\pi$
$240$	0	0
$241$	−10.2365	−0.659391	−0.329695	−	0.944087i	$-0.606946\pi$
$241$	−10.2365	−0.659391	−0.329695	+	0.944087i	$0.606946\pi$
$242$	24.2717	1.56025
$243$	0	0
$244$	40.4070	2.58679
$245$	0	0
$246$	0	0
$247$	−5.18837	−0.330128
$248$	14.5812	0.925907
$249$	0	0
$250$	5.24979	0.332026
$251$	−3.56171	−0.224813	−0.112407	−	0.993662i	$-0.535856\pi$
$251$	−3.56171	−0.224813	−0.112407	+	0.993662i	$0.535856\pi$
$252$	0	0
$253$	1.93900	0.121904
$254$	−16.0858	−1.00931
$255$	0	0
$256$	−10.4668	−0.654176
$257$	17.1744	1.07131	0.535654	−	0.844438i	$-0.320065\pi$
$257$	17.1744	1.07131	0.535654	+	0.844438i	$0.320065\pi$
$258$	0	0
$259$	0	0
$260$	−44.3236	−2.74884
$261$	0	0
$262$	−5.08968	−0.314441
$263$	−12.3424	−0.761066	−0.380533	−	0.924767i	$-0.624260\pi$
$263$	−12.3424	−0.761066	−0.380533	+	0.924767i	$0.624260\pi$
$264$	0	0
$265$	1.94489	0.119474
$266$	0	0
$267$	0	0
$268$	37.9734	2.31960
$269$	28.8280	1.75768	0.878838	−	0.477120i	$-0.158319\pi$
$269$	28.8280	1.75768	0.878838	+	0.477120i	$0.158319\pi$
$270$	0	0
$271$	−8.07435	−0.490482	−0.245241	−	0.969462i	$-0.578867\pi$
$271$	−8.07435	−0.490482	−0.245241	+	0.969462i	$0.578867\pi$
$272$	−0.519893	−0.0315231
$273$	0	0
$274$	21.0519	1.27179
$275$	−2.54288	−0.153341
$276$	0	0
$277$	−0.597171	−0.0358805	−0.0179403	−	0.999839i	$-0.505711\pi$
$277$	−0.597171	−0.0358805	−0.0179403	+	0.999839i	$0.505711\pi$
$278$	32.4039	1.94345
$279$	0	0
$280$	0	0
$281$	11.4330	0.682033	0.341017	−	0.940057i	$-0.389229\pi$
$281$	11.4330	0.682033	0.341017	+	0.940057i	$0.389229\pi$
$282$	0	0
$283$	16.4548	0.978138	0.489069	−	0.872245i	$-0.337337\pi$
$283$	16.4548	0.978138	0.489069	+	0.872245i	$0.337337\pi$
$284$	−13.0978	−0.777213
$285$	0	0
$286$	4.44138	0.262624
$287$	0	0
$288$	0	0
$289$	−16.5797	−0.975277
$290$	15.7177	0.922973
$291$	0	0
$292$	−15.5180	−0.908123
$293$	−21.0563	−1.23012	−0.615062	−	0.788479i	$-0.710869\pi$
$293$	−21.0563	−1.23012	−0.615062	+	0.788479i	$0.710869\pi$
$294$	0	0
$295$	−23.1293	−1.34664
$296$	18.0368	1.04837
$297$	0	0
$298$	−52.2030	−3.02404
$299$	−19.3506	−1.11908
$300$	0	0
$301$	0	0
$302$	−9.81163	−0.564596
$303$	0	0
$304$	0.936815	0.0537300
$305$	−43.3793	−2.48389
$306$	0	0
$307$	−28.6679	−1.63616	−0.818082	−	0.575101i	$-0.804963\pi$
$307$	−28.6679	−1.63616	−0.818082	+	0.575101i	$0.804963\pi$
$308$	0	0
$309$	0	0
$310$	−45.5013	−2.58430
$311$	−16.2376	−0.920748	−0.460374	−	0.887725i	$-0.652285\pi$
$311$	−16.2376	−0.920748	−0.460374	+	0.887725i	$0.652285\pi$
$312$	0	0
$313$	8.20275	0.463647	0.231824	−	0.972758i	$-0.425531\pi$
$313$	8.20275	0.463647	0.231824	+	0.972758i	$0.425531\pi$
$314$	32.7636	1.84896
$315$	0	0
$316$	20.0315	1.12686
$317$	−27.6528	−1.55314	−0.776568	−	0.630034i	$-0.783041\pi$
$317$	−27.6528	−1.55314	−0.776568	+	0.630034i	$0.783041\pi$
$318$	0	0
$319$	−0.951083	−0.0532504
$320$	42.6721	2.38544
$321$	0	0
$322$	0	0
$323$	−0.757332	−0.0421391
$324$	0	0
$325$	25.3771	1.40767
$326$	22.9584	1.27155
$327$	0	0
$328$	−8.39458	−0.463513
$329$	0	0
$330$	0	0
$331$	−16.0392	−0.881596	−0.440798	−	0.897606i	$-0.645304\pi$
$331$	−16.0392	−0.881596	−0.440798	+	0.897606i	$0.645304\pi$
$332$	13.5414	0.743181
$333$	0	0
$334$	48.4101	2.64888
$335$	−40.7667	−2.22732
$336$	0	0
$337$	−15.1153	−0.823382	−0.411691	−	0.911323i	$-0.635062\pi$
$337$	−15.1153	−0.823382	−0.411691	+	0.911323i	$0.635062\pi$
$338$	−15.1129	−0.822033
$339$	0	0
$340$	−6.46980	−0.350874
$341$	2.75330	0.149100
$342$	0	0
$343$	0	0
$344$	14.1927	0.765218
$345$	0	0
$346$	33.3407	1.79241
$347$	−9.52542	−0.511351	−0.255676	−	0.966763i	$-0.582298\pi$
$347$	−9.52542	−0.511351	−0.255676	+	0.966763i	$0.582298\pi$
$348$	0	0
$349$	−3.95320	−0.211610	−0.105805	−	0.994387i	$-0.533742\pi$
$349$	−3.95320	−0.211610	−0.105805	+	0.994387i	$0.533742\pi$
$350$	0	0
$351$	0	0
$352$	−2.89977	−0.154558
$353$	−29.3875	−1.56414	−0.782069	−	0.623192i	$-0.785835\pi$
$353$	−29.3875	−1.56414	−0.782069	+	0.623192i	$0.785835\pi$
$354$	0	0
$355$	14.0613	0.746296
$356$	−13.3241	−0.706178
$357$	0	0
$358$	6.22521	0.329013
$359$	−16.1715	−0.853499	−0.426750	−	0.904370i	$-0.640341\pi$
$359$	−16.1715	−0.853499	−0.426750	+	0.904370i	$0.640341\pi$
$360$	0	0
$361$	−17.6353	−0.928175
$362$	−33.9890	−1.78642
$363$	0	0
$364$	0	0
$365$	16.6595	0.871998
$366$	0	0
$367$	13.2846	0.693450	0.346725	−	0.937967i	$-0.387294\pi$
$367$	13.2846	0.693450	0.346725	+	0.937967i	$0.387294\pi$
$368$	3.49396	0.182135
$369$	0	0
$370$	−56.2847	−2.92610
$371$	0	0
$372$	0	0
$373$	−10.6853	−0.553265	−0.276632	−	0.960976i	$-0.589218\pi$
$373$	−10.6853	−0.553265	−0.276632	+	0.960976i	$0.589218\pi$
$374$	0.648296	0.0335226
$375$	0	0
$376$	−30.0103	−1.54767
$377$	9.49151	0.488838
$378$	0	0
$379$	−27.8189	−1.42896	−0.714481	−	0.699655i	$-0.753337\pi$
$379$	−27.8189	−1.42896	−0.714481	+	0.699655i	$0.753337\pi$
$380$	11.6582	0.598051
$381$	0	0
$382$	−13.8116	−0.706664
$383$	−21.9184	−1.11998	−0.559989	−	0.828500i	$-0.689195\pi$
$383$	−21.9184	−1.11998	−0.559989	+	0.828500i	$0.689195\pi$
$384$	0	0
$385$	0	0
$386$	−1.41789	−0.0721689
$387$	0	0
$388$	22.9124	1.16320
$389$	12.0325	0.610073	0.305037	−	0.952341i	$-0.401331\pi$
$389$	12.0325	0.610073	0.305037	+	0.952341i	$0.401331\pi$
$390$	0	0
$391$	−2.82456	−0.142844
$392$	0	0
$393$	0	0
$394$	55.5652	2.79934
$395$	−21.5050	−1.08203
$396$	0	0
$397$	−31.9869	−1.60538	−0.802689	−	0.596397i	$-0.796598\pi$
$397$	−31.9869	−1.60538	−0.802689	+	0.596397i	$0.796598\pi$
$398$	11.9246	0.597725
$399$	0	0
$400$	−4.58211	−0.229105
$401$	−25.4601	−1.27142	−0.635709	−	0.771929i	$-0.719292\pi$
$401$	−25.4601	−1.27142	−0.635709	+	0.771929i	$0.719292\pi$
$402$	0	0
$403$	−27.4771	−1.36873
$404$	−25.1062	−1.24908
$405$	0	0
$406$	0	0
$407$	3.40581	0.168820
$408$	0	0
$409$	−3.99275	−0.197429	−0.0987143	−	0.995116i	$-0.531473\pi$
$409$	−3.99275	−0.197429	−0.0987143	+	0.995116i	$0.531473\pi$
$410$	26.1957	1.29371
$411$	0	0
$412$	32.8920	1.62047
$413$	0	0
$414$	0	0
$415$	−14.5375	−0.713618
$416$	28.9388	1.41884
$417$	0	0
$418$	−1.16819	−0.0571380
$419$	17.3740	0.848777	0.424389	−	0.905480i	$-0.360489\pi$
$419$	17.3740	0.848777	0.424389	+	0.905480i	$0.360489\pi$
$420$	0	0
$421$	26.5700	1.29494	0.647472	−	0.762089i	$-0.275826\pi$
$421$	26.5700	1.29494	0.647472	+	0.762089i	$0.275826\pi$
$422$	−10.1075	−0.492027
$423$	0	0
$424$	1.40044	0.0680112
$425$	3.70423	0.179681
$426$	0	0
$427$	0	0
$428$	−15.3274	−0.740876
$429$	0	0
$430$	−44.2889	−2.13580
$431$	−31.2161	−1.50363	−0.751814	−	0.659376i	$-0.770821\pi$
$431$	−31.2161	−1.50363	−0.751814	+	0.659376i	$0.770821\pi$
$432$	0	0
$433$	31.6984	1.52333	0.761665	−	0.647972i	$-0.224382\pi$
$433$	31.6984	1.52333	0.761665	+	0.647972i	$0.224382\pi$
$434$	0	0
$435$	0	0
$436$	−46.5991	−2.23169
$437$	5.08968	0.243472
$438$	0	0
$439$	13.1562	0.627911	0.313956	−	0.949438i	$-0.398346\pi$
$439$	13.1562	0.627911	0.313956	+	0.949438i	$0.398346\pi$
$440$	−3.43331	−0.163677
$441$	0	0
$442$	−6.46980	−0.307737
$443$	14.3913	0.683753	0.341877	−	0.939745i	$-0.388938\pi$
$443$	14.3913	0.683753	0.341877	+	0.939745i	$0.388938\pi$
$444$	0	0
$445$	14.3043	0.678087
$446$	−33.5008	−1.58631
$447$	0	0
$448$	0	0
$449$	−16.6571	−0.786097	−0.393049	−	0.919518i	$-0.628580\pi$
$449$	−16.6571	−0.786097	−0.393049	+	0.919518i	$0.628580\pi$
$450$	0	0
$451$	−1.58511	−0.0746400
$452$	−33.0538	−1.55472
$453$	0	0
$454$	16.5261	0.775607
$455$	0	0
$456$	0	0
$457$	−24.9245	−1.16592	−0.582960	−	0.812501i	$-0.698106\pi$
$457$	−24.9245	−1.16592	−0.582960	+	0.812501i	$0.698106\pi$
$458$	12.6046	0.588974
$459$	0	0
$460$	43.4805	2.02729
$461$	13.0532	0.607949	0.303975	−	0.952680i	$-0.401686\pi$
$461$	13.0532	0.607949	0.303975	+	0.952680i	$0.401686\pi$
$462$	0	0
$463$	26.6601	1.23900	0.619500	−	0.784997i	$-0.287336\pi$
$463$	26.6601	1.23900	0.619500	+	0.784997i	$0.287336\pi$
$464$	−1.71379	−0.0795608
$465$	0	0
$466$	−2.48858	−0.115281
$467$	−8.81151	−0.407748	−0.203874	−	0.978997i	$-0.565353\pi$
$467$	−8.81151	−0.407748	−0.203874	+	0.978997i	$0.565353\pi$
$468$	0	0
$469$	0	0
$470$	93.6486	4.31969
$471$	0	0
$472$	−16.6545	−0.766585
$473$	2.67994	0.123224
$474$	0	0
$475$	−6.67479	−0.306260
$476$	0	0
$477$	0	0
$478$	13.7899	0.630733
$479$	−36.6851	−1.67619	−0.838093	−	0.545527i	$-0.816330\pi$
$479$	−36.6851	−1.67619	−0.838093	+	0.545527i	$0.816330\pi$
$480$	0	0
$481$	−33.9890	−1.54976
$482$	23.0012	1.04768
$483$	0	0
$484$	−32.9342	−1.49701
$485$	−24.5978	−1.11693
$486$	0	0
$487$	11.6280	0.526917	0.263458	−	0.964671i	$-0.415137\pi$
$487$	11.6280	0.526917	0.263458	+	0.964671i	$0.415137\pi$
$488$	−31.2357	−1.41397
$489$	0	0
$490$	0	0
$491$	29.7101	1.34080	0.670398	−	0.742001i	$-0.266123\pi$
$491$	29.7101	1.34080	0.670398	+	0.742001i	$0.266123\pi$
$492$	0	0
$493$	1.38545	0.0623975
$494$	11.6582	0.524526
$495$	0	0
$496$	4.96127	0.222768
$497$	0	0
$498$	0	0
$499$	13.8183	0.618593	0.309297	−	0.950966i	$-0.399906\pi$
$499$	13.8183	0.618593	0.309297	+	0.950966i	$0.399906\pi$
$500$	−7.12342	−0.318569
$501$	0	0
$502$	8.00309	0.357196
$503$	21.1530	0.943166	0.471583	−	0.881822i	$-0.343683\pi$
$503$	21.1530	0.943166	0.471583	+	0.881822i	$0.343683\pi$
$504$	0	0
$505$	26.9530	1.19939
$506$	−4.35690	−0.193688
$507$	0	0
$508$	21.8267	0.968403
$509$	14.7096	0.651991	0.325996	−	0.945371i	$-0.394301\pi$
$509$	14.7096	0.651991	0.325996	+	0.945371i	$0.394301\pi$
$510$	0	0
$511$	0	0
$512$	−9.00538	−0.397985
$513$	0	0
$514$	−38.5905	−1.70215
$515$	−35.3116	−1.55601
$516$	0	0
$517$	−5.66672	−0.249222
$518$	0	0
$519$	0	0
$520$	34.2634	1.50255
$521$	−19.4078	−0.850270	−0.425135	−	0.905130i	$-0.639773\pi$
$521$	−19.4078	−0.850270	−0.425135	+	0.905130i	$0.639773\pi$
$522$	0	0
$523$	−10.9737	−0.479844	−0.239922	−	0.970792i	$-0.577122\pi$
$523$	−10.9737	−0.479844	−0.239922	+	0.970792i	$0.577122\pi$
$524$	6.90616	0.301697
$525$	0	0
$526$	27.7332	1.20922
$527$	−4.01075	−0.174711
$528$	0	0
$529$	−4.01746	−0.174672
$530$	−4.37012	−0.189826
$531$	0	0
$532$	0	0
$533$	15.8189	0.685194
$534$	0	0
$535$	16.4548	0.711404
$536$	−29.3545	−1.26792
$537$	0	0
$538$	−64.7760	−2.79269
$539$	0	0
$540$	0	0
$541$	26.2000	1.12643	0.563213	−	0.826312i	$-0.309565\pi$
$541$	26.2000	1.12643	0.563213	+	0.826312i	$0.309565\pi$
$542$	18.1429	0.779304
$543$	0	0
$544$	4.22412	0.181108
$545$	50.0269	2.14292
$546$	0	0
$547$	−25.8377	−1.10474	−0.552370	−	0.833599i	$-0.686277\pi$
$547$	−25.8377	−1.10474	−0.552370	+	0.833599i	$0.686277\pi$
$548$	−28.5652	−1.22025
$549$	0	0
$550$	5.71379	0.243637
$551$	−2.49649	−0.106354
$552$	0	0
$553$	0	0
$554$	1.34183	0.0570089
$555$	0	0
$556$	−43.9687	−1.86469
$557$	−4.38298	−0.185713	−0.0928564	−	0.995680i	$-0.529600\pi$
$557$	−4.38298	−0.185713	−0.0928564	+	0.995680i	$0.529600\pi$
$558$	0	0
$559$	−26.7450	−1.13119
$560$	0	0
$561$	0	0
$562$	−25.6896	−1.08365
$563$	9.92255	0.418186	0.209093	−	0.977896i	$-0.432949\pi$
$563$	9.92255	0.418186	0.209093	+	0.977896i	$0.432949\pi$
$564$	0	0
$565$	35.4852	1.49287
$566$	−36.9736	−1.55412
$567$	0	0
$568$	10.1250	0.424835
$569$	0.997016	0.0417971	0.0208985	−	0.999782i	$-0.493347\pi$
$569$	0.997016	0.0417971	0.0208985	+	0.999782i	$0.493347\pi$
$570$	0	0
$571$	26.8635	1.12420	0.562102	−	0.827068i	$-0.309993\pi$
$571$	26.8635	1.12420	0.562102	+	0.827068i	$0.309993\pi$
$572$	−6.02649	−0.251980
$573$	0	0
$574$	0	0
$575$	−24.8944	−1.03817
$576$	0	0
$577$	−21.4874	−0.894530	−0.447265	−	0.894401i	$-0.647602\pi$
$577$	−21.4874	−0.894530	−0.447265	+	0.894401i	$0.647602\pi$
$578$	37.2543	1.54957
$579$	0	0
$580$	−21.3272	−0.885565
$581$	0	0
$582$	0	0
$583$	0.264438	0.0109519
$584$	11.9958	0.496391
$585$	0	0
$586$	47.3131	1.95449
$587$	10.1970	0.420873	0.210437	−	0.977607i	$-0.432511\pi$
$587$	10.1970	0.420873	0.210437	+	0.977607i	$0.432511\pi$
$588$	0	0
$589$	7.22713	0.297789
$590$	51.9711	2.13961
$591$	0	0
$592$	6.13706	0.252232
$593$	−27.6740	−1.13643	−0.568217	−	0.822879i	$-0.692366\pi$
$593$	−27.6740	−1.13643	−0.568217	+	0.822879i	$0.692366\pi$
$594$	0	0
$595$	0	0
$596$	70.8340	2.90147
$597$	0	0
$598$	43.4805	1.77805
$599$	17.1750	0.701750	0.350875	−	0.936422i	$-0.385884\pi$
$599$	17.1750	0.701750	0.350875	+	0.936422i	$0.385884\pi$
$600$	0	0
$601$	28.8422	1.17650	0.588248	−	0.808681i	$-0.299818\pi$
$601$	28.8422	1.17650	0.588248	+	0.808681i	$0.299818\pi$
$602$	0	0
$603$	0	0
$604$	13.3134	0.541713
$605$	35.3568	1.43746
$606$	0	0
$607$	−34.9399	−1.41817	−0.709083	−	0.705125i	$-0.750891\pi$
$607$	−34.9399	−1.41817	−0.709083	+	0.705125i	$0.750891\pi$
$608$	−7.61160	−0.308691
$609$	0	0
$610$	97.4723	3.94654
$611$	56.5521	2.28785
$612$	0	0
$613$	−32.2204	−1.30137	−0.650685	−	0.759347i	$-0.725518\pi$
$613$	−32.2204	−1.30137	−0.650685	+	0.759347i	$0.725518\pi$
$614$	64.4162	2.59963
$615$	0	0
$616$	0	0
$617$	−26.0653	−1.04935	−0.524675	−	0.851303i	$-0.675813\pi$
$617$	−26.0653	−1.04935	−0.524675	+	0.851303i	$0.675813\pi$
$618$	0	0
$619$	−23.7349	−0.953985	−0.476993	−	0.878907i	$-0.658273\pi$
$619$	−23.7349	−0.953985	−0.476993	+	0.878907i	$0.658273\pi$
$620$	61.7405	2.47956
$621$	0	0
$622$	36.4855	1.46293
$623$	0	0
$624$	0	0
$625$	−20.9215	−0.836862
$626$	−18.4314	−0.736668
$627$	0	0
$628$	−44.4569	−1.77402
$629$	−4.96127	−0.197819
$630$	0	0
$631$	26.7724	1.06579	0.532896	−	0.846181i	$-0.321104\pi$
$631$	26.7724	1.06579	0.532896	+	0.846181i	$0.321104\pi$
$632$	−15.4849	−0.615955
$633$	0	0
$634$	62.1353	2.46771
$635$	−23.4322	−0.929880
$636$	0	0
$637$	0	0
$638$	2.13706	0.0846071
$639$	0	0
$640$	−53.2288	−2.10405
$641$	−2.84010	−0.112177	−0.0560886	−	0.998426i	$-0.517863\pi$
$641$	−2.84010	−0.112177	−0.0560886	+	0.998426i	$0.517863\pi$
$642$	0	0
$643$	−43.7514	−1.72539	−0.862694	−	0.505727i	$-0.831224\pi$
$643$	−43.7514	−1.72539	−0.862694	+	0.505727i	$0.831224\pi$
$644$	0	0
$645$	0	0
$646$	1.70171	0.0669529
$647$	12.4304	0.488688	0.244344	−	0.969689i	$-0.421427\pi$
$647$	12.4304	0.488688	0.244344	+	0.969689i	$0.421427\pi$
$648$	0	0
$649$	−3.14479	−0.123444
$650$	−57.0219	−2.23658
$651$	0	0
$652$	−31.1521	−1.22001
$653$	−21.3303	−0.834721	−0.417360	−	0.908741i	$-0.637045\pi$
$653$	−21.3303	−0.834721	−0.417360	+	0.908741i	$0.637045\pi$
$654$	0	0
$655$	−7.41417	−0.289696
$656$	−2.85627	−0.111519
$657$	0	0
$658$	0	0
$659$	11.9500	0.465507	0.232753	−	0.972536i	$-0.425227\pi$
$659$	11.9500	0.465507	0.232753	+	0.972536i	$0.425227\pi$
$660$	0	0
$661$	28.6820	1.11560	0.557801	−	0.829975i	$-0.311645\pi$
$661$	28.6820	1.11560	0.557801	+	0.829975i	$0.311645\pi$
$662$	36.0398	1.40073
$663$	0	0
$664$	−10.4679	−0.406232
$665$	0	0
$666$	0	0
$667$	−9.31096	−0.360522
$668$	−65.6874	−2.54152
$669$	0	0
$670$	91.6020	3.53889
$671$	−5.89809	−0.227693
$672$	0	0
$673$	0.208816	0.00804926	0.00402463	−	0.999992i	$-0.498719\pi$
$673$	0.208816	0.00804926	0.00402463	+	0.999992i	$0.498719\pi$
$674$	33.9638	1.30823
$675$	0	0
$676$	20.5066	0.788717
$677$	35.0605	1.34748	0.673742	−	0.738967i	$-0.264686\pi$
$677$	35.0605	1.34748	0.673742	+	0.738967i	$0.264686\pi$
$678$	0	0
$679$	0	0
$680$	5.00133	0.191792
$681$	0	0
$682$	−6.18661	−0.236897
$683$	−28.0140	−1.07193	−0.535963	−	0.844241i	$-0.680051\pi$
$683$	−28.0140	−1.07193	−0.535963	+	0.844241i	$0.680051\pi$
$684$	0	0
$685$	30.6665	1.17171
$686$	0	0
$687$	0	0
$688$	4.82908	0.184107
$689$	−2.63901	−0.100538
$690$	0	0
$691$	29.1624	1.10939	0.554695	−	0.832054i	$-0.312835\pi$
$691$	29.1624	1.10939	0.554695	+	0.832054i	$0.312835\pi$
$692$	−45.2398	−1.71976
$693$	0	0
$694$	21.4034	0.812463
$695$	47.2030	1.79051
$696$	0	0
$697$	2.30904	0.0874613
$698$	8.88276	0.336218
$699$	0	0
$700$	0	0
$701$	25.8377	0.975877	0.487938	−	0.872878i	$-0.337749\pi$
$701$	25.8377	0.975877	0.487938	+	0.872878i	$0.337749\pi$
$702$	0	0
$703$	8.93991	0.337175
$704$	5.80194	0.218669
$705$	0	0
$706$	66.0331	2.48519
$707$	0	0
$708$	0	0
$709$	33.9734	1.27590	0.637950	−	0.770078i	$-0.279783\pi$
$709$	33.9734	1.27590	0.637950	+	0.770078i	$0.279783\pi$
$710$	−31.5955	−1.18576
$711$	0	0
$712$	10.2999	0.386006
$713$	26.9544	1.00945
$714$	0	0
$715$	6.46980	0.241957
$716$	−8.44696	−0.315678
$717$	0	0
$718$	36.3370	1.35609
$719$	−1.07933	−0.0402523	−0.0201261	−	0.999797i	$-0.506407\pi$
$719$	−1.07933	−0.0402523	−0.0201261	+	0.999797i	$0.506407\pi$
$720$	0	0
$721$	0	0
$722$	39.6262	1.47474
$723$	0	0
$724$	46.1195	1.71402
$725$	12.2107	0.453495
$726$	0	0
$727$	3.65840	0.135683	0.0678413	−	0.997696i	$-0.478389\pi$
$727$	3.65840	0.135683	0.0678413	+	0.997696i	$0.478389\pi$
$728$	0	0
$729$	0	0
$730$	−37.4336	−1.38548
$731$	−3.90389	−0.144391
$732$	0	0
$733$	−11.5394	−0.426216	−0.213108	−	0.977029i	$-0.568359\pi$
$733$	−11.5394	−0.426216	−0.213108	+	0.977029i	$0.568359\pi$
$734$	−29.8502	−1.10179
$735$	0	0
$736$	−28.3884	−1.04641
$737$	−5.54288	−0.204174
$738$	0	0
$739$	18.9836	0.698323	0.349162	−	0.937063i	$-0.386466\pi$
$739$	18.9836	0.698323	0.349162	+	0.937063i	$0.386466\pi$
$740$	76.3725	2.80751
$741$	0	0
$742$	0	0
$743$	−34.9965	−1.28390	−0.641949	−	0.766747i	$-0.721874\pi$
$743$	−34.9965	−1.28390	−0.641949	+	0.766747i	$0.721874\pi$
$744$	0	0
$745$	−76.0445	−2.78605
$746$	24.0097	0.879057
$747$	0	0
$748$	−0.879670	−0.0321639
$749$	0	0
$750$	0	0
$751$	0.414911	0.0151403	0.00757016	−	0.999971i	$-0.497590\pi$
$751$	0.414911	0.0151403	0.00757016	+	0.999971i	$0.497590\pi$
$752$	−10.2111	−0.372359
$753$	0	0
$754$	−21.3272	−0.776692
$755$	−14.2927	−0.520164
$756$	0	0
$757$	−32.1806	−1.16962	−0.584812	−	0.811169i	$-0.698832\pi$
$757$	−32.1806	−1.16962	−0.584812	+	0.811169i	$0.698832\pi$
$758$	62.5086	2.27041
$759$	0	0
$760$	−9.01208	−0.326903
$761$	−14.1756	−0.513865	−0.256932	−	0.966429i	$-0.582712\pi$
$761$	−14.1756	−0.513865	−0.256932	+	0.966429i	$0.582712\pi$
$762$	0	0
$763$	0	0
$764$	18.7409	0.678023
$765$	0	0
$766$	49.2502	1.77948
$767$	31.3840	1.13321
$768$	0	0
$769$	32.3072	1.16503	0.582513	−	0.812821i	$-0.302069\pi$
$769$	32.3072	1.16503	0.582513	+	0.812821i	$0.302069\pi$
$770$	0	0
$771$	0	0
$772$	1.92394	0.0692440
$773$	−15.7924	−0.568014	−0.284007	−	0.958822i	$-0.591664\pi$
$773$	−15.7924	−0.568014	−0.284007	+	0.958822i	$0.591664\pi$
$774$	0	0
$775$	−35.3490	−1.26977
$776$	−17.7119	−0.635819
$777$	0	0
$778$	−27.0368	−0.969318
$779$	−4.16075	−0.149074
$780$	0	0
$781$	1.91185	0.0684115
$782$	6.34672	0.226958
$783$	0	0
$784$	0	0
$785$	47.7271	1.70345
$786$	0	0
$787$	19.0938	0.680622	0.340311	−	0.940313i	$-0.389468\pi$
$787$	19.0938	0.680622	0.340311	+	0.940313i	$0.389468\pi$
$788$	−75.3962	−2.68588
$789$	0	0
$790$	48.3212	1.71919
$791$	0	0
$792$	0	0
$793$	58.8611	2.09022
$794$	71.8740	2.55071
$795$	0	0
$796$	−16.1804	−0.573500
$797$	9.76243	0.345803	0.172902	−	0.984939i	$-0.444686\pi$
$797$	9.76243	0.345803	0.172902	+	0.984939i	$0.444686\pi$
$798$	0	0
$799$	8.25475	0.292032
$800$	37.2295	1.31626
$801$	0	0
$802$	57.2083	2.02010
$803$	2.26512	0.0799343
$804$	0	0
$805$	0	0
$806$	61.7405	2.17472
$807$	0	0
$808$	19.4078	0.682763
$809$	6.48858	0.228126	0.114063	−	0.993473i	$-0.463613\pi$
$809$	6.48858	0.228126	0.114063	+	0.993473i	$0.463613\pi$
$810$	0	0
$811$	22.4242	0.787419	0.393710	−	0.919235i	$-0.371192\pi$
$811$	22.4242	0.787419	0.393710	+	0.919235i	$0.371192\pi$
$812$	0	0
$813$	0	0
$814$	−7.65279	−0.268230
$815$	33.4437	1.17148
$816$	0	0
$817$	7.03457	0.246108
$818$	8.97162	0.313685
$819$	0	0
$820$	−35.5448	−1.24128
$821$	−10.3442	−0.361016	−0.180508	−	0.983574i	$-0.557774\pi$
$821$	−10.3442	−0.361016	−0.180508	+	0.983574i	$0.557774\pi$
$822$	0	0
$823$	−24.5593	−0.856082	−0.428041	−	0.903759i	$-0.640796\pi$
$823$	−24.5593	−0.856082	−0.428041	+	0.903759i	$0.640796\pi$
$824$	−25.4264	−0.885772
$825$	0	0
$826$	0	0
$827$	−48.8719	−1.69944	−0.849721	−	0.527233i	$-0.823230\pi$
$827$	−48.8719	−1.69944	−0.849721	+	0.527233i	$0.823230\pi$
$828$	0	0
$829$	19.5284	0.678248	0.339124	−	0.940742i	$-0.389869\pi$
$829$	19.5284	0.678248	0.339124	+	0.940742i	$0.389869\pi$
$830$	32.6655	1.13383
$831$	0	0
$832$	−57.9016	−2.00738
$833$	0	0
$834$	0	0
$835$	70.5193	2.44042
$836$	1.58511	0.0548222
$837$	0	0
$838$	−39.0391	−1.34858
$839$	4.61561	0.159349	0.0796743	−	0.996821i	$-0.474612\pi$
$839$	4.61561	0.159349	0.0796743	+	0.996821i	$0.474612\pi$
$840$	0	0
$841$	−24.4330	−0.842516
$842$	−59.7023	−2.05748
$843$	0	0
$844$	13.7149	0.472085
$845$	−22.0151	−0.757342
$846$	0	0
$847$	0	0
$848$	0.476501	0.0163631
$849$	0	0
$850$	−8.32333	−0.285488
$851$	33.3424	1.14296
$852$	0	0
$853$	−32.6211	−1.11693	−0.558463	−	0.829529i	$-0.688609\pi$
$853$	−32.6211	−1.11693	−0.558463	+	0.829529i	$0.688609\pi$
$854$	0	0
$855$	0	0
$856$	11.8485	0.404972
$857$	5.75209	0.196488	0.0982438	−	0.995162i	$-0.468678\pi$
$857$	5.75209	0.196488	0.0982438	+	0.995162i	$0.468678\pi$
$858$	0	0
$859$	−35.9832	−1.22773	−0.613865	−	0.789411i	$-0.710386\pi$
$859$	−35.9832	−1.22773	−0.613865	+	0.789411i	$0.710386\pi$
$860$	60.0954	2.04924
$861$	0	0
$862$	70.1420	2.38904
$863$	20.7006	0.704658	0.352329	−	0.935876i	$-0.385390\pi$
$863$	20.7006	0.704658	0.352329	+	0.935876i	$0.385390\pi$
$864$	0	0
$865$	48.5676	1.65135
$866$	−71.2257	−2.42035
$867$	0	0
$868$	0	0
$869$	−2.92394	−0.0991877
$870$	0	0
$871$	55.3162	1.87432
$872$	36.0224	1.21987
$873$	0	0
$874$	−11.4364	−0.386842
$875$	0	0
$876$	0	0
$877$	−38.1215	−1.28727	−0.643636	−	0.765332i	$-0.722575\pi$
$877$	−38.1215	−1.28727	−0.643636	+	0.765332i	$0.722575\pi$
$878$	−29.5617	−0.997659
$879$	0	0
$880$	−1.16819	−0.0393796
$881$	−2.01963	−0.0680431	−0.0340216	−	0.999421i	$-0.510831\pi$
$881$	−2.01963	−0.0680431	−0.0340216	+	0.999421i	$0.510831\pi$
$882$	0	0
$883$	39.1213	1.31654	0.658268	−	0.752784i	$-0.271290\pi$
$883$	39.1213	1.31654	0.658268	+	0.752784i	$0.271290\pi$
$884$	8.77884	0.295264
$885$	0	0
$886$	−32.3370	−1.08638
$887$	36.5250	1.22639	0.613195	−	0.789932i	$-0.289884\pi$
$887$	36.5250	1.22639	0.613195	+	0.789932i	$0.289884\pi$
$888$	0	0
$889$	0	0
$890$	−32.1414	−1.07738
$891$	0	0
$892$	45.4571	1.52202
$893$	−14.8745	−0.497758
$894$	0	0
$895$	9.06831	0.303120
$896$	0	0
$897$	0	0
$898$	37.4282	1.24899
$899$	−13.2212	−0.440951
$900$	0	0
$901$	−0.385209	−0.0128332
$902$	3.56171	0.118592
$903$	0	0
$904$	25.5515	0.849830
$905$	−49.5120	−1.64583
$906$	0	0
$907$	19.3679	0.643101	0.321550	−	0.946892i	$-0.395796\pi$
$907$	19.3679	0.643101	0.321550	+	0.946892i	$0.395796\pi$
$908$	−22.4242	−0.744172
$909$	0	0
$910$	0	0
$911$	39.9377	1.32319	0.661597	−	0.749860i	$-0.269879\pi$
$911$	39.9377	1.32319	0.661597	+	0.749860i	$0.269879\pi$
$912$	0	0
$913$	−1.97660	−0.0654159
$914$	56.0049	1.85248
$915$	0	0
$916$	−17.1031	−0.565103
$917$	0	0
$918$	0	0
$919$	31.0073	1.02284	0.511418	−	0.859332i	$-0.329120\pi$
$919$	31.0073	1.02284	0.511418	+	0.859332i	$0.329120\pi$
$920$	−33.6116	−1.10814
$921$	0	0
$922$	−29.3303	−0.965943
$923$	−19.0797	−0.628017
$924$	0	0
$925$	−43.7265	−1.43772
$926$	−59.9047	−1.96859
$927$	0	0
$928$	13.9245	0.457095
$929$	8.56253	0.280928	0.140464	−	0.990086i	$-0.455141\pi$
$929$	8.56253	0.280928	0.140464	+	0.990086i	$0.455141\pi$
$930$	0	0
$931$	0	0
$932$	3.37675	0.110609
$933$	0	0
$934$	19.7993	0.647852
$935$	0.944378	0.0308845
$936$	0	0
$937$	−56.0992	−1.83268	−0.916340	−	0.400400i	$-0.868871\pi$
$937$	−56.0992	−1.83268	−0.916340	+	0.400400i	$0.868871\pi$
$938$	0	0
$939$	0	0
$940$	−127.071	−4.14461
$941$	55.5730	1.81163	0.905814	−	0.423675i	$-0.139260\pi$
$941$	55.5730	1.81163	0.905814	+	0.423675i	$0.139260\pi$
$942$	0	0
$943$	−15.5180	−0.505336
$944$	−5.66672	−0.184436
$945$	0	0
$946$	−6.02177	−0.195785
$947$	30.4480	0.989428	0.494714	−	0.869056i	$-0.335273\pi$
$947$	30.4480	0.989428	0.494714	+	0.869056i	$0.335273\pi$
$948$	0	0
$949$	−22.6052	−0.733796
$950$	14.9981	0.486603
$951$	0	0
$952$	0	0
$953$	−26.8847	−0.870881	−0.435441	−	0.900217i	$-0.643407\pi$
$953$	−26.8847	−0.870881	−0.435441	+	0.900217i	$0.643407\pi$
$954$	0	0
$955$	−20.1195	−0.651052
$956$	−18.7114	−0.605170
$957$	0	0
$958$	82.4307	2.66322
$959$	0	0
$960$	0	0
$961$	7.27413	0.234649
$962$	76.3725	2.46235
$963$	0	0
$964$	−31.2102	−1.00521
$965$	−2.06546	−0.0664895
$966$	0	0
$967$	−12.0500	−0.387501	−0.193751	−	0.981051i	$-0.562065\pi$
$967$	−12.0500	−0.387501	−0.193751	+	0.981051i	$0.562065\pi$
$968$	25.4590	0.818285
$969$	0	0
$970$	55.2707	1.77463
$971$	1.65637	0.0531554	0.0265777	−	0.999647i	$-0.491539\pi$
$971$	1.65637	0.0531554	0.0265777	+	0.999647i	$0.491539\pi$
$972$	0	0
$973$	0	0
$974$	−26.1280	−0.837194
$975$	0	0
$976$	−10.6280	−0.340194
$977$	−52.8364	−1.69039	−0.845193	−	0.534462i	$-0.820514\pi$
$977$	−52.8364	−1.69039	−0.845193	+	0.534462i	$0.820514\pi$
$978$	0	0
$979$	1.94489	0.0621589
$980$	0	0
$981$	0	0
$982$	−66.7579	−2.13033
$983$	1.43128	0.0456506	0.0228253	−	0.999739i	$-0.492734\pi$
$983$	1.43128	0.0456506	0.0228253	+	0.999739i	$0.492734\pi$
$984$	0	0
$985$	80.9423	2.57904
$986$	−3.11308	−0.0991406
$987$	0	0
$988$	−15.8189	−0.503267
$989$	26.2362	0.834263
$990$	0	0
$991$	22.1021	0.702098	0.351049	−	0.936357i	$-0.385825\pi$
$991$	22.1021	0.702098	0.351049	+	0.936357i	$0.385825\pi$
$992$	−40.3103	−1.27985
$993$	0	0
$994$	0	0
$995$	17.3706	0.550686
$996$	0	0
$997$	−9.35683	−0.296334	−0.148167	−	0.988962i	$-0.547337\pi$
$997$	−9.35683	−0.296334	−0.148167	+	0.988962i	$0.547337\pi$
$998$	−31.0495	−0.982855
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3087.2.a.g.1.1		6
3.2	odd	2		343.2.a.e.1.5	✓	6
7.6	odd	2	inner	3087.2.a.g.1.2		6
12.11	even	2		5488.2.a.o.1.5		6
15.14	odd	2		8575.2.a.g.1.2		6
21.2	odd	6		343.2.c.c.18.2		12
21.5	even	6		343.2.c.c.18.1		12
21.11	odd	6		343.2.c.c.324.2		12
21.17	even	6		343.2.c.c.324.1		12
21.20	even	2		343.2.a.e.1.6	yes	6
84.83	odd	2		5488.2.a.o.1.2		6
105.104	even	2		8575.2.a.g.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
343.2.a.e.1.5	✓	6	3.2	odd	2
343.2.a.e.1.6	yes	6	21.20	even	2
343.2.c.c.18.1		12	21.5	even	6
343.2.c.c.18.2		12	21.2	odd	6
343.2.c.c.324.1		12	21.17	even	6
343.2.c.c.324.2		12	21.11	odd	6
3087.2.a.g.1.1		6	1.1	even	1	trivial
3087.2.a.g.1.2		6	7.6	odd	2	inner
5488.2.a.o.1.2		6	84.83	odd	2
5488.2.a.o.1.5		6	12.11	even	2
8575.2.a.g.1.1		6	105.104	even	2
8575.2.a.g.1.2		6	15.14	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 3087 = 3^{2} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3087.a (trivial)

\(f(q)\)	\(=\)	\(q-2.24698 q^{2} +3.04892 q^{4} -3.27319 q^{5} -2.35690 q^{8} +O(q^{10})\)	\(q-2.24698 q^{2} +3.04892 q^{4} -3.27319 q^{5} -2.35690 q^{8} +7.35480 q^{10} -0.445042 q^{11} +4.44138 q^{13} -0.801938 q^{16} +0.648296 q^{17} -1.16819 q^{19} -9.97969 q^{20} +1.00000 q^{22} -4.35690 q^{23} +5.71379 q^{25} -9.97969 q^{26} +2.13706 q^{29} -6.18661 q^{31} +6.51573 q^{32} -1.45671 q^{34} -7.65279 q^{37} +2.62490 q^{38} +7.71457 q^{40} +3.56171 q^{41} -6.02177 q^{43} -1.35690 q^{44} +9.78986 q^{46} +12.7330 q^{47} -12.8388 q^{50} +13.5414 q^{52} -0.594187 q^{53} +1.45671 q^{55} -4.80194 q^{58} +7.06628 q^{59} +13.2529 q^{61} +13.9012 q^{62} -13.0368 q^{64} -14.5375 q^{65} +12.4547 q^{67} +1.97660 q^{68} -4.29590 q^{71} -5.08968 q^{73} +17.1957 q^{74} -3.56171 q^{76} +6.57002 q^{79} +2.62490 q^{80} -8.00309 q^{82} +4.44138 q^{83} -2.12200 q^{85} +13.5308 q^{86} +1.04892 q^{88} -4.37012 q^{89} -13.2838 q^{92} -28.6108 q^{94} +3.82371 q^{95} +7.51491 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} - 6 q^{8}+O(q^{10}) \) 6 * q - 4 * q^2 - 6 * q^8	\( 6 q - 4 q^{2} - 6 q^{8} - 2 q^{11} + 4 q^{16} + 6 q^{22} - 18 q^{23} + 18 q^{25} + 2 q^{29} + 14 q^{32} - 10 q^{37} - 30 q^{43} + 12 q^{46} - 12 q^{50} - 30 q^{53} - 20 q^{58} - 22 q^{64} - 56 q^{65} + 30 q^{67} + 2 q^{71} + 30 q^{74} - 10 q^{79} - 52 q^{85} + 6 q^{86} - 12 q^{88} - 14 q^{92} + 8 q^{95}+O(q^{100}) \) 6 * q - 4 * q^2 - 6 * q^8 - 2 * q^11 + 4 * q^16 + 6 * q^22 - 18 * q^23 + 18 * q^25 + 2 * q^29 + 14 * q^32 - 10 * q^37 - 30 * q^43 + 12 * q^46 - 12 * q^50 - 30 * q^53 - 20 * q^58 - 22 * q^64 - 56 * q^65 + 30 * q^67 + 2 * q^71 + 30 * q^74 - 10 * q^79 - 52 * q^85 + 6 * q^86 - 12 * q^88 - 14 * q^92 + 8 * q^95

Introduction

L-functions

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Database

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