LMFDB - Embedded newform 3087.2.a.g.1.3

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			1.3
Root			$-3.09901$ 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)

$f(q)$	$=$	$q-0.554958 q^{2} -1.69202 q^{4} -1.37919 q^{5} +2.04892 q^{8} +O(q^{10})$	$q-0.554958 q^{2} -1.69202 q^{4} -1.37919 q^{5} +2.04892 q^{8} +0.765393 q^{10} -1.80194 q^{11} -4.20504 q^{13} +2.24698 q^{16} +4.47820 q^{17} +5.58423 q^{19} +2.33362 q^{20} +1.00000 q^{22} +0.0489173 q^{23} -3.09783 q^{25} +2.33362 q^{26} +3.15883 q^{29} -6.34962 q^{31} -5.34481 q^{32} -2.48521 q^{34} +3.18598 q^{37} -3.09901 q^{38} -2.82585 q^{40} +9.44863 q^{41} +2.45473 q^{43} +3.04892 q^{44} -0.0271471 q^{46} +9.10800 q^{47} +1.71917 q^{50} +7.11501 q^{52} -9.74094 q^{53} +2.48521 q^{55} -1.75302 q^{58} -7.30405 q^{59} -0.954429 q^{61} +3.52377 q^{62} -1.52781 q^{64} +5.79954 q^{65} -1.43296 q^{67} -7.57721 q^{68} +2.13706 q^{71} -0.273166 q^{73} -1.76809 q^{74} -9.44863 q^{76} +2.18060 q^{79} -3.09901 q^{80} -5.24359 q^{82} -4.20504 q^{83} -6.17629 q^{85} -1.36227 q^{86} -3.69202 q^{88} -7.45564 q^{89} -0.0827692 q^{92} -5.05456 q^{94} -7.70171 q^{95} -5.70580 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.554958	−0.392415	−0.196207	−	0.980562i	$-0.562863\pi$
$2$	−0.554958	−0.392415	−0.196207	+	0.980562i	$0.562863\pi$
$3$	0	0
$3$	0	0
$4$	−1.69202	−0.846011
$5$	−1.37919	−0.616793	−0.308396	−	0.951258i	$-0.599792\pi$
$5$	−1.37919	−0.616793	−0.308396	+	0.951258i	$0.599792\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.04892	0.724402
$9$	0	0
$10$	0.765393	0.242038
$11$	−1.80194	−0.543305	−0.271652	−	0.962395i	$-0.587570\pi$
$11$	−1.80194	−0.543305	−0.271652	+	0.962395i	$0.587570\pi$
$12$	0	0
$13$	−4.20504	−1.16627	−0.583134	−	0.812376i	$-0.698174\pi$
$13$	−4.20504	−1.16627	−0.583134	+	0.812376i	$0.698174\pi$
$14$	0	0
$15$	0	0
$16$	2.24698	0.561745
$17$	4.47820	1.08612	0.543062	−	0.839693i	$-0.317265\pi$
$17$	4.47820	1.08612	0.543062	+	0.839693i	$0.317265\pi$
$18$	0	0
$19$	5.58423	1.28111	0.640555	−	0.767913i	$-0.278704\pi$
$19$	5.58423	1.28111	0.640555	+	0.767913i	$0.278704\pi$
$20$	2.33362	0.521813
$21$	0	0
$22$	1.00000	0.213201
$23$	0.0489173	0.0102000	0.00509999	−	0.999987i	$-0.498377\pi$
$23$	0.0489173	0.0102000	0.00509999	+	0.999987i	$0.498377\pi$
$24$	0	0
$25$	−3.09783	−0.619567
$26$	2.33362	0.457660
$27$	0	0
$28$	0	0
$29$	3.15883	0.586581	0.293290	−	0.956023i	$-0.405250\pi$
$29$	3.15883	0.586581	0.293290	+	0.956023i	$0.405250\pi$
$30$	0	0
$31$	−6.34962	−1.14043	−0.570213	−	0.821497i	$-0.693139\pi$
$31$	−6.34962	−1.14043	−0.570213	+	0.821497i	$0.693139\pi$
$32$	−5.34481	−0.944839
$33$	0	0
$34$	−2.48521	−0.426211
$35$	0	0
$36$	0	0
$37$	3.18598	0.523772	0.261886	−	0.965099i	$-0.415656\pi$
$37$	3.18598	0.523772	0.261886	+	0.965099i	$0.415656\pi$
$38$	−3.09901	−0.502726
$39$	0	0
$40$	−2.82585	−0.446806
$41$	9.44863	1.47563	0.737814	−	0.675004i	$-0.235858\pi$
$41$	9.44863	1.47563	0.737814	+	0.675004i	$0.235858\pi$
$42$	0	0
$43$	2.45473	0.374343	0.187171	−	0.982327i	$-0.440068\pi$
$43$	2.45473	0.374343	0.187171	+	0.982327i	$0.440068\pi$
$44$	3.04892	0.459642
$45$	0	0
$46$	−0.0271471	−0.00400262
$47$	9.10800	1.32854	0.664269	−	0.747493i	$-0.268743\pi$
$47$	9.10800	1.32854	0.664269	+	0.747493i	$0.268743\pi$
$48$	0	0
$49$	0	0
$50$	1.71917	0.243127
$51$	0	0
$52$	7.11501	0.986675
$53$	−9.74094	−1.33802	−0.669010	−	0.743253i	$-0.733282\pi$
$53$	−9.74094	−1.33802	−0.669010	+	0.743253i	$0.733282\pi$
$54$	0	0
$55$	2.48521	0.335106
$56$	0	0
$57$	0	0
$58$	−1.75302	−0.230183
$59$	−7.30405	−0.950906	−0.475453	−	0.879741i	$-0.657716\pi$
$59$	−7.30405	−0.950906	−0.475453	+	0.879741i	$0.657716\pi$
$60$	0	0
$61$	−0.954429	−0.122202	−0.0611011	−	0.998132i	$-0.519461\pi$
$61$	−0.954429	−0.122202	−0.0611011	+	0.998132i	$0.519461\pi$
$62$	3.52377	0.447520
$63$	0	0
$64$	−1.52781	−0.190976
$65$	5.79954	0.719345
$66$	0	0
$67$	−1.43296	−0.175064	−0.0875320	−	0.996162i	$-0.527898\pi$
$67$	−1.43296	−0.175064	−0.0875320	+	0.996162i	$0.527898\pi$
$68$	−7.57721	−0.918872
$69$	0	0
$70$	0	0
$71$	2.13706	0.253623	0.126811	−	0.991927i	$-0.459526\pi$
$71$	2.13706	0.253623	0.126811	+	0.991927i	$0.459526\pi$
$72$	0	0
$73$	−0.273166	−0.0319716	−0.0159858	−	0.999872i	$-0.505089\pi$
$73$	−0.273166	−0.0319716	−0.0159858	+	0.999872i	$0.505089\pi$
$74$	−1.76809	−0.205536
$75$	0	0
$76$	−9.44863	−1.08383
$77$	0	0
$78$	0	0
$79$	2.18060	0.245337	0.122669	−	0.992448i	$-0.460855\pi$
$79$	2.18060	0.245337	0.122669	+	0.992448i	$0.460855\pi$
$80$	−3.09901	−0.346480
$81$	0	0
$82$	−5.24359	−0.579058
$83$	−4.20504	−0.461563	−0.230781	−	0.973006i	$-0.574128\pi$
$83$	−4.20504	−0.461563	−0.230781	+	0.973006i	$0.574128\pi$
$84$	0	0
$85$	−6.17629	−0.669913
$86$	−1.36227	−0.146898
$87$	0	0
$88$	−3.69202	−0.393571
$89$	−7.45564	−0.790297	−0.395148	−	0.918617i	$-0.629307\pi$
$89$	−7.45564	−0.790297	−0.395148	+	0.918617i	$0.629307\pi$
$90$	0	0
$91$	0	0
$92$	−0.0827692	−0.00862928
$93$	0	0
$94$	−5.05456	−0.521338
$95$	−7.70171	−0.790179
$96$	0	0
$97$	−5.70580	−0.579336	−0.289668	−	0.957127i	$-0.593545\pi$
$97$	−5.70580	−0.579336	−0.289668	+	0.957127i	$0.593545\pi$
$98$	0	0
$99$	0	0
$100$	5.24160	0.524160
$101$	12.8883	1.28243	0.641216	−	0.767361i	$-0.278430\pi$
$101$	12.8883	1.28243	0.641216	+	0.767361i	$0.278430\pi$
$102$	0	0
$103$	−4.32661	−0.426313	−0.213157	−	0.977018i	$-0.568374\pi$
$103$	−4.32661	−0.426313	−0.213157	+	0.977018i	$0.568374\pi$
$104$	−8.61577	−0.844846
$105$	0	0
$106$	5.40581	0.525059
$107$	−8.76271	−0.847123	−0.423562	−	0.905867i	$-0.639220\pi$
$107$	−8.76271	−0.847123	−0.423562	+	0.905867i	$0.639220\pi$
$108$	0	0
$109$	−2.08277	−0.199493	−0.0997466	−	0.995013i	$-0.531803\pi$
$109$	−2.08277	−0.199493	−0.0997466	+	0.995013i	$0.531803\pi$
$110$	−1.37919	−0.131501
$111$	0	0
$112$	0	0
$113$	−18.2959	−1.72113	−0.860567	−	0.509338i	$-0.829890\pi$
$113$	−18.2959	−1.72113	−0.860567	+	0.509338i	$0.829890\pi$
$114$	0	0
$115$	−0.0674663	−0.00629127
$116$	−5.34481	−0.496254
$117$	0	0
$118$	4.05344	0.373150
$119$	0	0
$120$	0	0
$121$	−7.75302	−0.704820
$122$	0.529668	0.0479539
$123$	0	0
$124$	10.7437	0.964812
$125$	11.1685	0.998937
$126$	0	0
$127$	−0.295897	−0.0262566	−0.0131283	−	0.999914i	$-0.504179\pi$
$127$	−0.295897	−0.0262566	−0.0131283	+	0.999914i	$0.504179\pi$
$128$	11.5375	1.01978
$129$	0	0
$130$	−3.21850	−0.282282
$131$	0.492227	0.0430061	0.0215030	−	0.999769i	$-0.493155\pi$
$131$	0.492227	0.0430061	0.0215030	+	0.999769i	$0.493155\pi$
$132$	0	0
$133$	0	0
$134$	0.795233	0.0686977
$135$	0	0
$136$	9.17547	0.786790
$137$	−11.7313	−1.00227	−0.501134	−	0.865370i	$-0.667084\pi$
$137$	−11.7313	−1.00227	−0.501134	+	0.865370i	$0.667084\pi$
$138$	0	0
$139$	6.53866	0.554602	0.277301	−	0.960783i	$-0.410560\pi$
$139$	6.53866	0.554602	0.277301	+	0.960783i	$0.410560\pi$
$140$	0	0
$141$	0	0
$142$	−1.18598	−0.0995253
$143$	7.57721	0.633638
$144$	0	0
$145$	−4.35663	−0.361799
$146$	0.151595	0.0125461
$147$	0	0
$148$	−5.39075	−0.443117
$149$	−7.24027	−0.593146	−0.296573	−	0.955010i	$-0.595844\pi$
$149$	−7.24027	−0.593146	−0.296573	+	0.955010i	$0.595844\pi$
$150$	0	0
$151$	−15.2838	−1.24378	−0.621890	−	0.783105i	$-0.713635\pi$
$151$	−15.2838	−1.24378	−0.621890	+	0.783105i	$0.713635\pi$
$152$	11.4416	0.928038
$153$	0	0
$154$	0	0
$155$	8.75733	0.703406
$156$	0	0
$157$	13.0098	1.03830	0.519149	−	0.854684i	$-0.326249\pi$
$157$	13.0098	1.03830	0.519149	+	0.854684i	$0.326249\pi$
$158$	−1.21014	−0.0962739
$159$	0	0
$160$	7.37151	0.582769
$161$	0	0
$162$	0	0
$163$	17.2228	1.34900	0.674498	−	0.738277i	$-0.264360\pi$
$163$	17.2228	1.34900	0.674498	+	0.738277i	$0.264360\pi$
$164$	−15.9873	−1.24840
$165$	0	0
$166$	2.33362	0.181124
$167$	−12.3586	−0.956338	−0.478169	−	0.878268i	$-0.658699\pi$
$167$	−12.3586	−0.956338	−0.478169	+	0.878268i	$0.658699\pi$
$168$	0	0
$169$	4.68233	0.360179
$170$	3.42758	0.262884
$171$	0	0
$172$	−4.15346	−0.316698
$173$	−16.0714	−1.22189	−0.610944	−	0.791674i	$-0.709210\pi$
$173$	−16.0714	−1.22189	−0.610944	+	0.791674i	$0.709210\pi$
$174$	0	0
$175$	0	0
$176$	−4.04892	−0.305199
$177$	0	0
$178$	4.13757	0.310124
$179$	−23.4426	−1.75219	−0.876093	−	0.482142i	$-0.839859\pi$
$179$	−23.4426	−1.75219	−0.876093	+	0.482142i	$0.839859\pi$
$180$	0	0
$181$	24.1409	1.79438	0.897188	−	0.441649i	$-0.145606\pi$
$181$	24.1409	1.79438	0.897188	+	0.441649i	$0.145606\pi$
$182$	0	0
$183$	0	0
$184$	0.100228	0.00738888
$185$	−4.39407	−0.323059
$186$	0	0
$187$	−8.06944	−0.590096
$188$	−15.4109	−1.12396
$189$	0	0
$190$	4.27413	0.310078
$191$	−8.07606	−0.584364	−0.292182	−	0.956363i	$-0.594381\pi$
$191$	−8.07606	−0.584364	−0.292182	+	0.956363i	$0.594381\pi$
$192$	0	0
$193$	−1.73125	−0.124618	−0.0623091	−	0.998057i	$-0.519846\pi$
$193$	−1.73125	−0.124618	−0.0623091	+	0.998057i	$0.519846\pi$
$194$	3.16648	0.227340
$195$	0	0
$196$	0	0
$197$	−12.8847	−0.917997	−0.458999	−	0.888437i	$-0.651792\pi$
$197$	−12.8847	−0.917997	−0.458999	+	0.888437i	$0.651792\pi$
$198$	0	0
$199$	−20.0033	−1.41800	−0.708998	−	0.705211i	$-0.750852\pi$
$199$	−20.0033	−1.41800	−0.708998	+	0.705211i	$0.750852\pi$
$200$	−6.34721	−0.448815
$201$	0	0
$202$	−7.15245	−0.503245
$203$	0	0
$204$	0	0
$205$	−13.0315	−0.910157
$206$	2.40109	0.167292
$207$	0	0
$208$	−9.44863	−0.655145
$209$	−10.0624	−0.696033
$210$	0	0
$211$	−26.3424	−1.81349	−0.906744	−	0.421683i	$-0.861440\pi$
$211$	−26.3424	−1.81349	−0.906744	+	0.421683i	$0.861440\pi$
$212$	16.4819	1.13198
$213$	0	0
$214$	4.86294	0.332424
$215$	−3.38554	−0.230892
$216$	0	0
$217$	0	0
$218$	1.15585	0.0782840
$219$	0	0
$220$	−4.20504	−0.283504
$221$	−18.8310	−1.26671
$222$	0	0
$223$	4.41074	0.295365	0.147682	−	0.989035i	$-0.452819\pi$
$223$	4.41074	0.295365	0.147682	+	0.989035i	$0.452819\pi$
$224$	0	0
$225$	0	0
$226$	10.1535	0.675398
$227$	−0.765393	−0.0508009	−0.0254005	−	0.999677i	$-0.508086\pi$
$227$	−0.765393	−0.0508009	−0.0254005	+	0.999677i	$0.508086\pi$
$228$	0	0
$229$	9.78926	0.646893	0.323446	−	0.946247i	$-0.395158\pi$
$229$	9.78926	0.646893	0.323446	+	0.946247i	$0.395158\pi$
$230$	0.0374410	0.00246878
$231$	0	0
$232$	6.47219	0.424920
$233$	−23.6189	−1.54733	−0.773664	−	0.633596i	$-0.781578\pi$
$233$	−23.6189	−1.54733	−0.773664	+	0.633596i	$0.781578\pi$
$234$	0	0
$235$	−12.5617	−0.819433
$236$	12.3586	0.804477
$237$	0	0
$238$	0	0
$239$	−7.15883	−0.463066	−0.231533	−	0.972827i	$-0.574374\pi$
$239$	−7.15883	−0.463066	−0.231533	+	0.972827i	$0.574374\pi$
$240$	0	0
$241$	−26.7476	−1.72297	−0.861484	−	0.507785i	$-0.830464\pi$
$241$	−26.7476	−1.72297	−0.861484	+	0.507785i	$0.830464\pi$
$242$	4.30260	0.276582
$243$	0	0
$244$	1.61491	0.103384
$245$	0	0
$246$	0	0
$247$	−23.4819	−1.49412
$248$	−13.0098	−0.826126
$249$	0	0
$250$	−6.19802	−0.391997
$251$	−9.44863	−0.596392	−0.298196	−	0.954505i	$-0.596385\pi$
$251$	−9.44863	−0.596392	−0.298196	+	0.954505i	$0.596385\pi$
$252$	0	0
$253$	−0.0881460	−0.00554169
$254$	0.164210	0.0103035
$255$	0	0
$256$	−3.34721	−0.209200
$257$	4.90296	0.305838	0.152919	−	0.988239i	$-0.451133\pi$
$257$	4.90296	0.305838	0.152919	+	0.988239i	$0.451133\pi$
$258$	0	0
$259$	0	0
$260$	−9.81295	−0.608574
$261$	0	0
$262$	−0.273166	−0.0168762
$263$	20.8442	1.28531	0.642653	−	0.766158i	$-0.277834\pi$
$263$	20.8442	1.28531	0.642653	+	0.766158i	$0.277834\pi$
$264$	0	0
$265$	13.4346	0.825281
$266$	0	0
$267$	0	0
$268$	2.42460	0.148106
$269$	24.7847	1.51115	0.755574	−	0.655063i	$-0.227358\pi$
$269$	24.7847	1.51115	0.755574	+	0.655063i	$0.227358\pi$
$270$	0	0
$271$	6.41709	0.389810	0.194905	−	0.980822i	$-0.437560\pi$
$271$	6.41709	0.389810	0.194905	+	0.980822i	$0.437560\pi$
$272$	10.0624	0.610124
$273$	0	0
$274$	6.51035	0.393305
$275$	5.58211	0.336614
$276$	0	0
$277$	0.0566871	0.00340600	0.00170300	−	0.999999i	$-0.499458\pi$
$277$	0.0566871	0.00340600	0.00170300	+	0.999999i	$0.499458\pi$
$278$	−3.62868	−0.217634
$279$	0	0
$280$	0	0
$281$	6.02177	0.359229	0.179614	−	0.983737i	$-0.442515\pi$
$281$	6.02177	0.359229	0.179614	+	0.983737i	$0.442515\pi$
$282$	0	0
$283$	12.0854	0.718405	0.359202	−	0.933260i	$-0.383049\pi$
$283$	12.0854	0.718405	0.359202	+	0.933260i	$0.383049\pi$
$284$	−3.61596	−0.214568
$285$	0	0
$286$	−4.20504	−0.248649
$287$	0	0
$288$	0	0
$289$	3.05429	0.179664
$290$	2.41775	0.141975
$291$	0	0
$292$	0.462202	0.0270483
$293$	−1.40922	−0.0823272	−0.0411636	−	0.999152i	$-0.513106\pi$
$293$	−1.40922	−0.0823272	−0.0411636	+	0.999152i	$0.513106\pi$
$294$	0	0
$295$	10.0737	0.586512
$296$	6.52781	0.379421
$297$	0	0
$298$	4.01805	0.232759
$299$	−0.205699	−0.0118959
$300$	0	0
$301$	0	0
$302$	8.48188	0.488077
$303$	0	0
$304$	12.5476	0.719657
$305$	1.31634	0.0753734
$306$	0	0
$307$	−31.2559	−1.78387	−0.891933	−	0.452167i	$-0.850651\pi$
$307$	−31.2559	−1.78387	−0.891933	+	0.452167i	$0.850651\pi$
$308$	0	0
$309$	0	0
$310$	−4.85995	−0.276027
$311$	7.64468	0.433490	0.216745	−	0.976228i	$-0.430456\pi$
$311$	7.64468	0.433490	0.216745	+	0.976228i	$0.430456\pi$
$312$	0	0
$313$	8.12355	0.459170	0.229585	−	0.973289i	$-0.426263\pi$
$313$	8.12355	0.459170	0.229585	+	0.973289i	$0.426263\pi$
$314$	−7.21992	−0.407444
$315$	0	0
$316$	−3.68963	−0.207558
$317$	−16.8140	−0.944369	−0.472185	−	0.881500i	$-0.656534\pi$
$317$	−16.8140	−0.944369	−0.472185	+	0.881500i	$0.656534\pi$
$318$	0	0
$319$	−5.69202	−0.318692
$320$	2.10714	0.117793
$321$	0	0
$322$	0	0
$323$	25.0073	1.39144
$324$	0	0
$325$	13.0265	0.722581
$326$	−9.55794	−0.529365
$327$	0	0
$328$	19.3595	1.06895
$329$	0	0
$330$	0	0
$331$	−26.5429	−1.45893	−0.729464	−	0.684019i	$-0.760231\pi$
$331$	−26.5429	−1.45893	−0.729464	+	0.684019i	$0.760231\pi$
$332$	7.11501	0.390487
$333$	0	0
$334$	6.85851	0.375281
$335$	1.97632	0.107978
$336$	0	0
$337$	−24.6136	−1.34079	−0.670393	−	0.742006i	$-0.733875\pi$
$337$	−24.6136	−1.34079	−0.670393	+	0.742006i	$0.733875\pi$
$338$	−2.59850	−0.141340
$339$	0	0
$340$	10.4504	0.566754
$341$	11.4416	0.619598
$342$	0	0
$343$	0	0
$344$	5.02954	0.271175
$345$	0	0
$346$	8.91896	0.479486
$347$	17.5797	0.943728	0.471864	−	0.881671i	$-0.343581\pi$
$347$	17.5797	0.943728	0.471864	+	0.881671i	$0.343581\pi$
$348$	0	0
$349$	15.1544	0.811198	0.405599	−	0.914051i	$-0.367063\pi$
$349$	15.1544	0.811198	0.405599	+	0.914051i	$0.367063\pi$
$350$	0	0
$351$	0	0
$352$	9.63102	0.513335
$353$	−24.0734	−1.28130	−0.640649	−	0.767834i	$-0.721334\pi$
$353$	−24.0734	−1.28130	−0.640649	+	0.767834i	$0.721334\pi$
$354$	0	0
$355$	−2.94742	−0.156433
$356$	12.6151	0.668599
$357$	0	0
$358$	13.0097	0.687583
$359$	16.3284	0.861781	0.430891	−	0.902404i	$-0.358200\pi$
$359$	16.3284	0.861781	0.430891	+	0.902404i	$0.358200\pi$
$360$	0	0
$361$	12.1836	0.641241
$362$	−13.3972	−0.704139
$363$	0	0
$364$	0	0
$365$	0.376747	0.0197198
$366$	0	0
$367$	−21.9662	−1.14663	−0.573314	−	0.819335i	$-0.694343\pi$
$367$	−21.9662	−1.14663	−0.573314	+	0.819335i	$0.694343\pi$
$368$	0.109916	0.00572978
$369$	0	0
$370$	2.43853	0.126773
$371$	0	0
$372$	0	0
$373$	−15.7942	−0.817791	−0.408896	−	0.912581i	$-0.634086\pi$
$373$	−15.7942	−0.817791	−0.408896	+	0.912581i	$0.634086\pi$
$374$	4.47820	0.231562
$375$	0	0
$376$	18.6615	0.962395
$377$	−13.2830	−0.684110
$378$	0	0
$379$	27.7318	1.42449	0.712244	−	0.701931i	$-0.247679\pi$
$379$	27.7318	1.42449	0.712244	+	0.701931i	$0.247679\pi$
$380$	13.0315	0.668500
$381$	0	0
$382$	4.48188	0.229313
$383$	29.0946	1.48667	0.743333	−	0.668922i	$-0.233244\pi$
$383$	29.0946	1.48667	0.743333	+	0.668922i	$0.233244\pi$
$384$	0	0
$385$	0	0
$386$	0.960771	0.0489020
$387$	0	0
$388$	9.65433	0.490124
$389$	27.9801	1.41865	0.709325	−	0.704882i	$-0.249000\pi$
$389$	27.9801	1.41865	0.709325	+	0.704882i	$0.249000\pi$
$390$	0	0
$391$	0.219062	0.0110784
$392$	0	0
$393$	0	0
$394$	7.15047	0.360236
$395$	−3.00747	−0.151322
$396$	0	0
$397$	26.2388	1.31689	0.658443	−	0.752631i	$-0.271216\pi$
$397$	26.2388	1.31689	0.658443	+	0.752631i	$0.271216\pi$
$398$	11.1010	0.556442
$399$	0	0
$400$	−6.96077	−0.348039
$401$	−23.7845	−1.18774	−0.593870	−	0.804561i	$-0.702401\pi$
$401$	−23.7845	−1.18774	−0.593870	+	0.804561i	$0.702401\pi$
$402$	0	0
$403$	26.7004	1.33004
$404$	−21.8072	−1.08495
$405$	0	0
$406$	0	0
$407$	−5.74094	−0.284568
$408$	0	0
$409$	5.80329	0.286954	0.143477	−	0.989654i	$-0.454172\pi$
$409$	5.80329	0.286954	0.143477	+	0.989654i	$0.454172\pi$
$410$	7.23191	0.357159
$411$	0	0
$412$	7.32071	0.360666
$413$	0	0
$414$	0	0
$415$	5.79954	0.284688
$416$	22.4751	1.10193
$417$	0	0
$418$	5.58423	0.273133
$419$	7.78291	0.380220	0.190110	−	0.981763i	$-0.439116\pi$
$419$	7.78291	0.380220	0.190110	+	0.981763i	$0.439116\pi$
$420$	0	0
$421$	22.1806	1.08102	0.540508	−	0.841339i	$-0.318232\pi$
$421$	22.1806	1.08102	0.540508	+	0.841339i	$0.318232\pi$
$422$	14.6189	0.711639
$423$	0	0
$424$	−19.9584	−0.969265
$425$	−13.8727	−0.672926
$426$	0	0
$427$	0	0
$428$	14.8267	0.716675
$429$	0	0
$430$	1.87883	0.0906054
$431$	−21.4319	−1.03234	−0.516169	−	0.856487i	$-0.672642\pi$
$431$	−21.4319	−1.03234	−0.516169	+	0.856487i	$0.672642\pi$
$432$	0	0
$433$	−34.3082	−1.64875	−0.824373	−	0.566047i	$-0.808472\pi$
$433$	−34.3082	−1.64875	−0.824373	+	0.566047i	$0.808472\pi$
$434$	0	0
$435$	0	0
$436$	3.52409	0.168773
$437$	0.273166	0.0130673
$438$	0	0
$439$	−36.5069	−1.74238	−0.871189	−	0.490947i	$-0.836651\pi$
$439$	−36.5069	−1.74238	−0.871189	+	0.490947i	$0.836651\pi$
$440$	5.09200	0.242752
$441$	0	0
$442$	10.4504	0.497076
$443$	−23.5362	−1.11824	−0.559119	−	0.829088i	$-0.688860\pi$
$443$	−23.5362	−1.11824	−0.559119	+	0.829088i	$0.688860\pi$
$444$	0	0
$445$	10.2828	0.487449
$446$	−2.44777	−0.115905
$447$	0	0
$448$	0	0
$449$	21.6383	1.02118	0.510588	−	0.859826i	$-0.329428\pi$
$449$	21.6383	1.02118	0.510588	+	0.859826i	$0.329428\pi$
$450$	0	0
$451$	−17.0258	−0.801716
$452$	30.9571	1.45610
$453$	0	0
$454$	0.424761	0.0199350
$455$	0	0
$456$	0	0
$457$	5.88338	0.275213	0.137606	−	0.990487i	$-0.456059\pi$
$457$	5.88338	0.275213	0.137606	+	0.990487i	$0.456059\pi$
$458$	−5.43263	−0.253850
$459$	0	0
$460$	0.114154	0.00532248
$461$	−3.83438	−0.178585	−0.0892924	−	0.996005i	$-0.528461\pi$
$461$	−3.83438	−0.178585	−0.0892924	+	0.996005i	$0.528461\pi$
$462$	0	0
$463$	−21.4359	−0.996213	−0.498106	−	0.867116i	$-0.665971\pi$
$463$	−21.4359	−0.996213	−0.498106	+	0.867116i	$0.665971\pi$
$464$	7.09783	0.329509
$465$	0	0
$466$	13.1075	0.607194
$467$	−3.25061	−0.150420	−0.0752101	−	0.997168i	$-0.523963\pi$
$467$	−3.25061	−0.150420	−0.0752101	+	0.997168i	$0.523963\pi$
$468$	0	0
$469$	0	0
$470$	6.97120	0.321557
$471$	0	0
$472$	−14.9654	−0.688838
$473$	−4.42327	−0.203382
$474$	0	0
$475$	−17.2990	−0.793733
$476$	0	0
$477$	0	0
$478$	3.97285	0.181714
$479$	1.36253	0.0622555	0.0311277	−	0.999515i	$-0.490090\pi$
$479$	1.36253	0.0622555	0.0311277	+	0.999515i	$0.490090\pi$
$480$	0	0
$481$	−13.3972	−0.610858
$482$	14.8438	0.676118
$483$	0	0
$484$	13.1183	0.596285
$485$	7.86938	0.357330
$486$	0	0
$487$	19.0664	0.863980	0.431990	−	0.901878i	$-0.357812\pi$
$487$	19.0664	0.863980	0.431990	+	0.901878i	$0.357812\pi$
$488$	−1.95555	−0.0885234
$489$	0	0
$490$	0	0
$491$	16.5418	0.746522	0.373261	−	0.927726i	$-0.378240\pi$
$491$	16.5418	0.746522	0.373261	+	0.927726i	$0.378240\pi$
$492$	0	0
$493$	14.1459	0.637099
$494$	13.0315	0.586313
$495$	0	0
$496$	−14.2675	−0.640628
$497$	0	0
$498$	0	0
$499$	−9.91915	−0.444042	−0.222021	−	0.975042i	$-0.571265\pi$
$499$	−9.91915	−0.444042	−0.222021	+	0.975042i	$0.571265\pi$
$500$	−18.8973	−0.845111
$501$	0	0
$502$	5.24359	0.234033
$503$	36.9617	1.64804	0.824020	−	0.566561i	$-0.191727\pi$
$503$	36.9617	1.64804	0.824020	+	0.566561i	$0.191727\pi$
$504$	0	0
$505$	−17.7754	−0.790994
$506$	0.0489173	0.00217464
$507$	0	0
$508$	0.500664	0.0222134
$509$	1.53079	0.0678509	0.0339254	−	0.999424i	$-0.489199\pi$
$509$	1.53079	0.0678509	0.0339254	+	0.999424i	$0.489199\pi$
$510$	0	0
$511$	0	0
$512$	−21.2174	−0.937687
$513$	0	0
$514$	−2.72094	−0.120015
$515$	5.96721	0.262947
$516$	0	0
$517$	−16.4120	−0.721801
$518$	0	0
$519$	0	0
$520$	11.8828	0.521095
$521$	−26.4070	−1.15691	−0.578456	−	0.815714i	$-0.696345\pi$
$521$	−26.4070	−1.15691	−0.578456	+	0.815714i	$0.696345\pi$
$522$	0	0
$523$	−36.4153	−1.59233	−0.796166	−	0.605079i	$-0.793142\pi$
$523$	−36.4153	−1.59233	−0.796166	+	0.605079i	$0.793142\pi$
$524$	−0.832859	−0.0363836
$525$	0	0
$526$	−11.5676	−0.504373
$527$	−28.4349	−1.23864
$528$	0	0
$529$	−22.9976	−0.999896
$530$	−7.45564	−0.323852
$531$	0	0
$532$	0	0
$533$	−39.7318	−1.72098
$534$	0	0
$535$	12.0854	0.522499
$536$	−2.93602	−0.126817
$537$	0	0
$538$	−13.7545	−0.592997
$539$	0	0
$540$	0	0
$541$	−20.2204	−0.869344	−0.434672	−	0.900589i	$-0.643136\pi$
$541$	−20.2204	−0.869344	−0.434672	+	0.900589i	$0.643136\pi$
$542$	−3.56121	−0.152967
$543$	0	0
$544$	−23.9352	−1.02621
$545$	2.87253	0.123046
$546$	0	0
$547$	28.3889	1.21382	0.606912	−	0.794769i	$-0.292408\pi$
$547$	28.3889	1.21382	0.606912	+	0.794769i	$0.292408\pi$
$548$	19.8495	0.847930
$549$	0	0
$550$	−3.09783	−0.132092
$551$	17.6396	0.751474
$552$	0	0
$553$	0	0
$554$	−0.0314589	−0.00133656
$555$	0	0
$556$	−11.0635	−0.469199
$557$	35.9560	1.52350	0.761752	−	0.647869i	$-0.224339\pi$
$557$	35.9560	1.52350	0.761752	+	0.647869i	$0.224339\pi$
$558$	0	0
$559$	−10.3222	−0.436584
$560$	0	0
$561$	0	0
$562$	−3.34183	−0.140967
$563$	−28.5349	−1.20260	−0.601302	−	0.799022i	$-0.705351\pi$
$563$	−28.5349	−1.20260	−0.601302	+	0.799022i	$0.705351\pi$
$564$	0	0
$565$	25.2335	1.06158
$566$	−6.70691	−0.281913
$567$	0	0
$568$	4.37867	0.183725
$569$	10.7976	0.452660	0.226330	−	0.974051i	$-0.427327\pi$
$569$	10.7976	0.452660	0.226330	+	0.974051i	$0.427327\pi$
$570$	0	0
$571$	−5.97152	−0.249901	−0.124950	−	0.992163i	$-0.539877\pi$
$571$	−5.97152	−0.249901	−0.124950	+	0.992163i	$0.539877\pi$
$572$	−12.8208	−0.536065
$573$	0	0
$574$	0	0
$575$	−0.151538	−0.00631956
$576$	0	0
$577$	13.8427	0.576279	0.288140	−	0.957588i	$-0.406963\pi$
$577$	13.8427	0.576279	0.288140	+	0.957588i	$0.406963\pi$
$578$	−1.69501	−0.0705029
$579$	0	0
$580$	7.37151	0.306085
$581$	0	0
$582$	0	0
$583$	17.5526	0.726953
$584$	−0.559694	−0.0231603
$585$	0	0
$586$	0.782056	0.0323064
$587$	17.3965	0.718031	0.359015	−	0.933332i	$-0.383113\pi$
$587$	17.3965	0.718031	0.359015	+	0.933332i	$0.383113\pi$
$588$	0	0
$589$	−35.4577	−1.46101
$590$	−5.59047	−0.230156
$591$	0	0
$592$	7.15883	0.294226
$593$	7.49309	0.307704	0.153852	−	0.988094i	$-0.450832\pi$
$593$	7.49309	0.307704	0.153852	+	0.988094i	$0.450832\pi$
$594$	0	0
$595$	0	0
$596$	12.2507	0.501808
$597$	0	0
$598$	0.114154	0.00466812
$599$	46.3564	1.89407	0.947036	−	0.321127i	$-0.104062\pi$
$599$	46.3564	1.89407	0.947036	+	0.321127i	$0.104062\pi$
$600$	0	0
$601$	−13.0773	−0.533435	−0.266717	−	0.963775i	$-0.585939\pi$
$601$	−13.0773	−0.533435	−0.266717	+	0.963775i	$0.585939\pi$
$602$	0	0
$603$	0	0
$604$	25.8605	1.05225
$605$	10.6929	0.434728
$606$	0	0
$607$	11.9172	0.483704	0.241852	−	0.970313i	$-0.422245\pi$
$607$	11.9172	0.483704	0.241852	+	0.970313i	$0.422245\pi$
$608$	−29.8467	−1.21044
$609$	0	0
$610$	−0.730513	−0.0295776
$611$	−38.2995	−1.54943
$612$	0	0
$613$	5.02044	0.202774	0.101387	−	0.994847i	$-0.467672\pi$
$613$	5.02044	0.202774	0.101387	+	0.994847i	$0.467672\pi$
$614$	17.3457	0.700015
$615$	0	0
$616$	0	0
$617$	−0.635808	−0.0255967	−0.0127983	−	0.999918i	$-0.504074\pi$
$617$	−0.635808	−0.0255967	−0.0127983	+	0.999918i	$0.504074\pi$
$618$	0	0
$619$	30.2006	1.21387	0.606933	−	0.794753i	$-0.292400\pi$
$619$	30.2006	1.21387	0.606933	+	0.794753i	$0.292400\pi$
$620$	−14.8176	−0.595089
$621$	0	0
$622$	−4.24248	−0.170108
$623$	0	0
$624$	0	0
$625$	0.0857531	0.00343012
$626$	−4.50823	−0.180185
$627$	0	0
$628$	−22.0129	−0.878412
$629$	14.2675	0.568881
$630$	0	0
$631$	−2.02475	−0.0806042	−0.0403021	−	0.999188i	$-0.512832\pi$
$631$	−2.02475	−0.0806042	−0.0403021	+	0.999188i	$0.512832\pi$
$632$	4.46788	0.177723
$633$	0	0
$634$	9.33108	0.370584
$635$	0.408098	0.0161949
$636$	0	0
$637$	0	0
$638$	3.15883	0.125059
$639$	0	0
$640$	−15.9124	−0.628993
$641$	29.3739	1.16020	0.580099	−	0.814546i	$-0.303014\pi$
$641$	29.3739	1.16020	0.580099	+	0.814546i	$0.303014\pi$
$642$	0	0
$643$	8.66658	0.341776	0.170888	−	0.985290i	$-0.445336\pi$
$643$	8.66658	0.341776	0.170888	+	0.985290i	$0.445336\pi$
$644$	0	0
$645$	0	0
$646$	−13.8780	−0.546023
$647$	38.9006	1.52934	0.764669	−	0.644423i	$-0.222902\pi$
$647$	38.9006	1.52934	0.764669	+	0.644423i	$0.222902\pi$
$648$	0	0
$649$	13.1614	0.516632
$650$	−7.22917	−0.283551
$651$	0	0
$652$	−29.1414	−1.14126
$653$	18.6243	0.728826	0.364413	−	0.931237i	$-0.381270\pi$
$653$	18.6243	0.728826	0.364413	+	0.931237i	$0.381270\pi$
$654$	0	0
$655$	−0.678875	−0.0265258
$656$	21.2309	0.828927
$657$	0	0
$658$	0	0
$659$	−22.9778	−0.895086	−0.447543	−	0.894262i	$-0.647701\pi$
$659$	−22.9778	−0.895086	−0.447543	+	0.894262i	$0.647701\pi$
$660$	0	0
$661$	−6.60612	−0.256948	−0.128474	−	0.991713i	$-0.541008\pi$
$661$	−6.60612	−0.256948	−0.128474	+	0.991713i	$0.541008\pi$
$662$	14.7302	0.572505
$663$	0	0
$664$	−8.61577	−0.334357
$665$	0	0
$666$	0	0
$667$	0.154522	0.00598311
$668$	20.9110	0.809072
$669$	0	0
$670$	−1.09678	−0.0423722
$671$	1.71982	0.0663930
$672$	0	0
$673$	27.6819	1.06706	0.533529	−	0.845782i	$-0.320866\pi$
$673$	27.6819	1.06706	0.533529	+	0.845782i	$0.320866\pi$
$674$	13.6595	0.526144
$675$	0	0
$676$	−7.92261	−0.304716
$677$	−27.7395	−1.06612	−0.533058	−	0.846079i	$-0.678957\pi$
$677$	−27.7395	−1.06612	−0.533058	+	0.846079i	$0.678957\pi$
$678$	0	0
$679$	0	0
$680$	−12.6547	−0.485286
$681$	0	0
$682$	−6.34962	−0.243139
$683$	14.6872	0.561991	0.280996	−	0.959709i	$-0.409335\pi$
$683$	14.6872	0.561991	0.280996	+	0.959709i	$0.409335\pi$
$684$	0	0
$685$	16.1796	0.618192
$686$	0	0
$687$	0	0
$688$	5.51573	0.210285
$689$	40.9610	1.56049
$690$	0	0
$691$	−26.0197	−0.989836	−0.494918	−	0.868940i	$-0.664802\pi$
$691$	−26.0197	−0.989836	−0.494918	+	0.868940i	$0.664802\pi$
$692$	27.1932	1.03373
$693$	0	0
$694$	−9.75600	−0.370333
$695$	−9.01805	−0.342074
$696$	0	0
$697$	42.3129	1.60271
$698$	−8.41007	−0.318326
$699$	0	0
$700$	0	0
$701$	−28.3889	−1.07224	−0.536118	−	0.844143i	$-0.680110\pi$
$701$	−28.3889	−1.07224	−0.536118	+	0.844143i	$0.680110\pi$
$702$	0	0
$703$	17.7912	0.671009
$704$	2.75302	0.103758
$705$	0	0
$706$	13.3597	0.502800
$707$	0	0
$708$	0	0
$709$	−1.57540	−0.0591654	−0.0295827	−	0.999562i	$-0.509418\pi$
$709$	−1.57540	−0.0591654	−0.0295827	+	0.999562i	$0.509418\pi$
$710$	1.63569	0.0613865
$711$	0	0
$712$	−15.2760	−0.572492
$713$	−0.310606	−0.0116323
$714$	0	0
$715$	−10.4504	−0.390823
$716$	39.6655	1.48237
$717$	0	0
$718$	−9.06159	−0.338176
$719$	10.7737	0.401792	0.200896	−	0.979613i	$-0.435615\pi$
$719$	10.7737	0.401792	0.200896	+	0.979613i	$0.435615\pi$
$720$	0	0
$721$	0	0
$722$	−6.76138	−0.251633
$723$	0	0
$724$	−40.8468	−1.51806
$725$	−9.78554	−0.363426
$726$	0	0
$727$	45.0011	1.66900	0.834499	−	0.551010i	$-0.185757\pi$
$727$	45.0011	1.66900	0.834499	+	0.551010i	$0.185757\pi$
$728$	0	0
$729$	0	0
$730$	−0.209079	−0.00773836
$731$	10.9928	0.406583
$732$	0	0
$733$	32.5209	1.20119	0.600594	−	0.799555i	$-0.294931\pi$
$733$	32.5209	1.20119	0.600594	+	0.799555i	$0.294931\pi$
$734$	12.1903	0.449954
$735$	0	0
$736$	−0.261454	−0.00963733
$737$	2.58211	0.0951131
$738$	0	0
$739$	39.6722	1.45936	0.729682	−	0.683786i	$-0.239668\pi$
$739$	39.6722	1.45936	0.729682	+	0.683786i	$0.239668\pi$
$740$	7.43487	0.273311
$741$	0	0
$742$	0	0
$743$	26.6848	0.978972	0.489486	−	0.872011i	$-0.337184\pi$
$743$	26.6848	0.978972	0.489486	+	0.872011i	$0.337184\pi$
$744$	0	0
$745$	9.98572	0.365848
$746$	8.76510	0.320913
$747$	0	0
$748$	13.6537	0.499228
$749$	0	0
$750$	0	0
$751$	7.83685	0.285971	0.142985	−	0.989725i	$-0.454330\pi$
$751$	7.83685	0.285971	0.142985	+	0.989725i	$0.454330\pi$
$752$	20.4655	0.746300
$753$	0	0
$754$	7.37151	0.268455
$755$	21.0793	0.767154
$756$	0	0
$757$	−16.2494	−0.590593	−0.295297	−	0.955406i	$-0.595419\pi$
$757$	−16.2494	−0.590593	−0.295297	+	0.955406i	$0.595419\pi$
$758$	−15.3900	−0.558990
$759$	0	0
$760$	−15.7802	−0.572407
$761$	−49.4552	−1.79275	−0.896375	−	0.443297i	$-0.853809\pi$
$761$	−49.4552	−1.79275	−0.896375	+	0.443297i	$0.853809\pi$
$762$	0	0
$763$	0	0
$764$	13.6649	0.494378
$765$	0	0
$766$	−16.1463	−0.583389
$767$	30.7138	1.10901
$768$	0	0
$769$	−39.1811	−1.41291	−0.706454	−	0.707759i	$-0.749706\pi$
$769$	−39.1811	−1.41291	−0.706454	+	0.707759i	$0.749706\pi$
$770$	0	0
$771$	0	0
$772$	2.92931	0.105428
$773$	−45.4692	−1.63541	−0.817707	−	0.575634i	$-0.804755\pi$
$773$	−45.4692	−1.63541	−0.817707	+	0.575634i	$0.804755\pi$
$774$	0	0
$775$	19.6701	0.706570
$776$	−11.6907	−0.419672
$777$	0	0
$778$	−15.5278	−0.556699
$779$	52.7633	1.89044
$780$	0	0
$781$	−3.85086	−0.137794
$782$	−0.121570	−0.00434734
$783$	0	0
$784$	0	0
$785$	−17.9430	−0.640415
$786$	0	0
$787$	−28.8756	−1.02930	−0.514651	−	0.857400i	$-0.672079\pi$
$787$	−28.8756	−1.02930	−0.514651	+	0.857400i	$0.672079\pi$
$788$	21.8012	0.776636
$789$	0	0
$790$	1.66902	0.0593810
$791$	0	0
$792$	0	0
$793$	4.01341	0.142520
$794$	−14.5614	−0.516765
$795$	0	0
$796$	33.8460	1.19964
$797$	−22.0637	−0.781538	−0.390769	−	0.920489i	$-0.627791\pi$
$797$	−22.0637	−0.781538	−0.390769	+	0.920489i	$0.627791\pi$
$798$	0	0
$799$	40.7875	1.44296
$800$	16.5574	0.585391
$801$	0	0
$802$	13.1994	0.466087
$803$	0.492227	0.0173703
$804$	0	0
$805$	0	0
$806$	−14.8176	−0.521927
$807$	0	0
$808$	26.4070	0.928995
$809$	−9.10752	−0.320203	−0.160102	−	0.987101i	$-0.551182\pi$
$809$	−9.10752	−0.320203	−0.160102	+	0.987101i	$0.551182\pi$
$810$	0	0
$811$	−1.29506	−0.0454757	−0.0227379	−	0.999741i	$-0.507238\pi$
$811$	−1.29506	−0.0454757	−0.0227379	+	0.999741i	$0.507238\pi$
$812$	0	0
$813$	0	0
$814$	3.18598	0.111669
$815$	−23.7535	−0.832050
$816$	0	0
$817$	13.7078	0.479574
$818$	−3.22058	−0.112605
$819$	0	0
$820$	22.0495	0.770002
$821$	−30.9836	−1.08134	−0.540668	−	0.841236i	$-0.681828\pi$
$821$	−30.9836	−1.08134	−0.540668	+	0.841236i	$0.681828\pi$
$822$	0	0
$823$	4.25428	0.148295	0.0741474	−	0.997247i	$-0.476376\pi$
$823$	4.25428	0.148295	0.0741474	+	0.997247i	$0.476376\pi$
$824$	−8.86486	−0.308822
$825$	0	0
$826$	0	0
$827$	−18.4483	−0.641510	−0.320755	−	0.947162i	$-0.603937\pi$
$827$	−18.4483	−0.641510	−0.320755	+	0.947162i	$0.603937\pi$
$828$	0	0
$829$	10.5847	0.367621	0.183811	−	0.982962i	$-0.441157\pi$
$829$	10.5847	0.367621	0.183811	+	0.982962i	$0.441157\pi$
$830$	−3.21850	−0.111716
$831$	0	0
$832$	6.42450	0.222730
$833$	0	0
$834$	0	0
$835$	17.0449	0.589862
$836$	17.0258	0.588851
$837$	0	0
$838$	−4.31919	−0.149204
$839$	−48.5382	−1.67573	−0.837863	−	0.545881i	$-0.816195\pi$
$839$	−48.5382	−1.67573	−0.837863	+	0.545881i	$0.816195\pi$
$840$	0	0
$841$	−19.0218	−0.655923
$842$	−12.3093	−0.424207
$843$	0	0
$844$	44.5719	1.53423
$845$	−6.45783	−0.222156
$846$	0	0
$847$	0	0
$848$	−21.8877	−0.751626
$849$	0	0
$850$	7.69878	0.264066
$851$	0.155850	0.00534246
$852$	0	0
$853$	−16.1014	−0.551303	−0.275651	−	0.961258i	$-0.588894\pi$
$853$	−16.1014	−0.551303	−0.275651	+	0.961258i	$0.588894\pi$
$854$	0	0
$855$	0	0
$856$	−17.9541	−0.613657
$857$	−33.1106	−1.13104	−0.565519	−	0.824735i	$-0.691324\pi$
$857$	−33.1106	−1.13104	−0.565519	+	0.824735i	$0.691324\pi$
$858$	0	0
$859$	−22.6701	−0.773495	−0.386747	−	0.922186i	$-0.626401\pi$
$859$	−22.6701	−0.773495	−0.386747	+	0.922186i	$0.626401\pi$
$860$	5.72841	0.195337
$861$	0	0
$862$	11.8938	0.405105
$863$	−34.5478	−1.17602	−0.588010	−	0.808854i	$-0.700088\pi$
$863$	−34.5478	−1.17602	−0.588010	+	0.808854i	$0.700088\pi$
$864$	0	0
$865$	22.1655	0.753651
$866$	19.0396	0.646992
$867$	0	0
$868$	0	0
$869$	−3.92931	−0.133293
$870$	0	0
$871$	6.02565	0.204171
$872$	−4.26742	−0.144513
$873$	0	0
$874$	−0.151595	−0.00512779
$875$	0	0
$876$	0	0
$877$	29.3062	0.989599	0.494800	−	0.869007i	$-0.335241\pi$
$877$	29.3062	0.989599	0.494800	+	0.869007i	$0.335241\pi$
$878$	20.2598	0.683735
$879$	0	0
$880$	5.58423	0.188244
$881$	−56.4861	−1.90306	−0.951532	−	0.307549i	$-0.900491\pi$
$881$	−56.4861	−1.90306	−0.951532	+	0.307549i	$0.900491\pi$
$882$	0	0
$883$	29.0183	0.976544	0.488272	−	0.872692i	$-0.337627\pi$
$883$	29.0183	0.976544	0.488272	+	0.872692i	$0.337627\pi$
$884$	31.8625	1.07165
$885$	0	0
$886$	13.0616	0.438813
$887$	5.10866	0.171532	0.0857660	−	0.996315i	$-0.472666\pi$
$887$	5.10866	0.171532	0.0857660	+	0.996315i	$0.472666\pi$
$888$	0	0
$889$	0	0
$890$	−5.70650	−0.191282
$891$	0	0
$892$	−7.46306	−0.249882
$893$	50.8611	1.70200
$894$	0	0
$895$	32.3319	1.08074
$896$	0	0
$897$	0	0
$898$	−12.0084	−0.400724
$899$	−20.0574	−0.668951
$900$	0	0
$901$	−43.6219	−1.45326
$902$	9.44863	0.314605
$903$	0	0
$904$	−37.4868	−1.24679
$905$	−33.2948	−1.10676
$906$	0	0
$907$	−17.9385	−0.595639	−0.297820	−	0.954622i	$-0.596259\pi$
$907$	−17.9385	−0.595639	−0.297820	+	0.954622i	$0.596259\pi$
$908$	1.29506	0.0429781
$909$	0	0
$910$	0	0
$911$	55.5666	1.84100	0.920501	−	0.390740i	$-0.127781\pi$
$911$	55.5666	1.84100	0.920501	+	0.390740i	$0.127781\pi$
$912$	0	0
$913$	7.57721	0.250769
$914$	−3.26503	−0.107998
$915$	0	0
$916$	−16.5636	−0.547278
$917$	0	0
$918$	0	0
$919$	−6.24996	−0.206167	−0.103084	−	0.994673i	$-0.532871\pi$
$919$	−6.24996	−0.206167	−0.103084	+	0.994673i	$0.532871\pi$
$920$	−0.138233	−0.00455740
$921$	0	0
$922$	2.12792	0.0700793
$923$	−8.98643	−0.295792
$924$	0	0
$925$	−9.86964	−0.324512
$926$	11.8961	0.390929
$927$	0	0
$928$	−16.8834	−0.554224
$929$	4.53231	0.148700	0.0743501	−	0.997232i	$-0.476312\pi$
$929$	4.53231	0.148700	0.0743501	+	0.997232i	$0.476312\pi$
$930$	0	0
$931$	0	0
$932$	39.9638	1.30906
$933$	0	0
$934$	1.80395	0.0590271
$935$	11.1293	0.363967
$936$	0	0
$937$	43.1805	1.41064	0.705322	−	0.708887i	$-0.250802\pi$
$937$	43.1805	1.41064	0.705322	+	0.708887i	$0.250802\pi$
$938$	0	0
$939$	0	0
$940$	21.2546	0.693249
$941$	35.1069	1.14445	0.572226	−	0.820096i	$-0.306080\pi$
$941$	35.1069	1.14445	0.572226	+	0.820096i	$0.306080\pi$
$942$	0	0
$943$	0.462202	0.0150514
$944$	−16.4120	−0.534167
$945$	0	0
$946$	2.45473	0.0798102
$947$	22.0043	0.715044	0.357522	−	0.933905i	$-0.383622\pi$
$947$	22.0043	0.715044	0.357522	+	0.933905i	$0.383622\pi$
$948$	0	0
$949$	1.14867	0.0372874
$950$	9.60023	0.311472
$951$	0	0
$952$	0	0
$953$	−17.3864	−0.563202	−0.281601	−	0.959532i	$-0.590865\pi$
$953$	−17.3864	−0.563202	−0.281601	+	0.959532i	$0.590865\pi$
$954$	0	0
$955$	11.1384	0.360431
$956$	12.1129	0.391759
$957$	0	0
$958$	−0.756146	−0.0244300
$959$	0	0
$960$	0	0
$961$	9.31767	0.300570
$962$	7.43487	0.239710
$963$	0	0
$964$	45.2576	1.45765
$965$	2.38772	0.0768635
$966$	0	0
$967$	−46.9778	−1.51070	−0.755351	−	0.655320i	$-0.772534\pi$
$967$	−46.9778	−1.51070	−0.755351	+	0.655320i	$0.772534\pi$
$968$	−15.8853	−0.510573
$969$	0	0
$970$	−4.36718	−0.140222
$971$	5.36516	0.172176	0.0860882	−	0.996288i	$-0.472563\pi$
$971$	5.36516	0.172176	0.0860882	+	0.996288i	$0.472563\pi$
$972$	0	0
$973$	0	0
$974$	−10.5810	−0.339038
$975$	0	0
$976$	−2.14458	−0.0686464
$977$	−16.2658	−0.520388	−0.260194	−	0.965556i	$-0.583787\pi$
$977$	−16.2658	−0.520388	−0.260194	+	0.965556i	$0.583787\pi$
$978$	0	0
$979$	13.4346	0.429372
$980$	0	0
$981$	0	0
$982$	−9.18001	−0.292946
$983$	−44.7279	−1.42660	−0.713300	−	0.700859i	$-0.752800\pi$
$983$	−44.7279	−1.42660	−0.713300	+	0.700859i	$0.752800\pi$
$984$	0	0
$985$	17.7705	0.566214
$986$	−7.85038	−0.250007
$987$	0	0
$988$	39.7318	1.26404
$989$	0.120079	0.00381829
$990$	0	0
$991$	−14.8364	−0.471293	−0.235647	−	0.971839i	$-0.575721\pi$
$991$	−14.8364	−0.471293	−0.235647	+	0.971839i	$0.575721\pi$
$992$	33.9375	1.07752
$993$	0	0
$994$	0	0
$995$	27.5883	0.874609
$996$	0	0
$997$	−40.4013	−1.27952	−0.639761	−	0.768574i	$-0.720967\pi$
$997$	−40.4013	−1.27952	−0.639761	+	0.768574i	$0.720967\pi$
$998$	5.50471	0.174249
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3087.2.a.g.1.3		6
3.2	odd	2		343.2.a.e.1.4	yes	6
7.6	odd	2	inner	3087.2.a.g.1.4		6
12.11	even	2		5488.2.a.o.1.1		6
15.14	odd	2		8575.2.a.g.1.3		6
21.2	odd	6		343.2.c.c.18.3		12
21.5	even	6		343.2.c.c.18.4		12
21.11	odd	6		343.2.c.c.324.3		12
21.17	even	6		343.2.c.c.324.4		12
21.20	even	2		343.2.a.e.1.3	✓	6
84.83	odd	2		5488.2.a.o.1.6		6
105.104	even	2		8575.2.a.g.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
343.2.a.e.1.3	✓	6	21.20	even	2
343.2.a.e.1.4	yes	6	3.2	odd	2
343.2.c.c.18.3		12	21.2	odd	6
343.2.c.c.18.4		12	21.5	even	6
343.2.c.c.324.3		12	21.11	odd	6
343.2.c.c.324.4		12	21.17	even	6
3087.2.a.g.1.3		6	1.1	even	1	trivial
3087.2.a.g.1.4		6	7.6	odd	2	inner
5488.2.a.o.1.1		6	12.11	even	2
5488.2.a.o.1.6		6	84.83	odd	2
8575.2.a.g.1.3		6	15.14	odd	2
8575.2.a.g.1.4		6	105.104	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:	\( N \)	\(=\)	\( 3087 = 3^{2} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3087.a (trivial)

\(f(q)\)	\(=\)	\(q-0.554958 q^{2} -1.69202 q^{4} -1.37919 q^{5} +2.04892 q^{8} +O(q^{10})\)	\(q-0.554958 q^{2} -1.69202 q^{4} -1.37919 q^{5} +2.04892 q^{8} +0.765393 q^{10} -1.80194 q^{11} -4.20504 q^{13} +2.24698 q^{16} +4.47820 q^{17} +5.58423 q^{19} +2.33362 q^{20} +1.00000 q^{22} +0.0489173 q^{23} -3.09783 q^{25} +2.33362 q^{26} +3.15883 q^{29} -6.34962 q^{31} -5.34481 q^{32} -2.48521 q^{34} +3.18598 q^{37} -3.09901 q^{38} -2.82585 q^{40} +9.44863 q^{41} +2.45473 q^{43} +3.04892 q^{44} -0.0271471 q^{46} +9.10800 q^{47} +1.71917 q^{50} +7.11501 q^{52} -9.74094 q^{53} +2.48521 q^{55} -1.75302 q^{58} -7.30405 q^{59} -0.954429 q^{61} +3.52377 q^{62} -1.52781 q^{64} +5.79954 q^{65} -1.43296 q^{67} -7.57721 q^{68} +2.13706 q^{71} -0.273166 q^{73} -1.76809 q^{74} -9.44863 q^{76} +2.18060 q^{79} -3.09901 q^{80} -5.24359 q^{82} -4.20504 q^{83} -6.17629 q^{85} -1.36227 q^{86} -3.69202 q^{88} -7.45564 q^{89} -0.0827692 q^{92} -5.05456 q^{94} -7.70171 q^{95} -5.70580 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} - 6 q^{8}+O(q^{10}) \) 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

Modular forms

Varieties

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Database

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