LMFDB - Embedded newform 343.2.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [343,2,Mod(1,343)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(343, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("343.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 343 = 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	343.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:	$2.73886878933$
Analytic rank:	$0$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-3.09901$ of defining polynomial
Character	$\chi$	$=$	343.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.554958 q^{2} +3.09901 q^{3} -1.69202 q^{4} +1.37919 q^{5} +1.71982 q^{6} -2.04892 q^{8} +6.60388 q^{9} +O(q^{10})$	\(q+0.554958 q^{2} +3.09901 q^{3} -1.69202 q^{4} +1.37919 q^{5} +1.71982 q^{6} -2.04892 q^{8} +6.60388 q^{9} +0.765393 q^{10} +1.80194 q^{11} -5.24359 q^{12} -4.20504 q^{13} +4.27413 q^{15} +2.24698 q^{16} -4.47820 q^{17} +3.66487 q^{18} +5.58423 q^{19} -2.33362 q^{20} +1.00000 q^{22} -0.0489173 q^{23} -6.34962 q^{24} -3.09783 q^{25} -2.33362 q^{26} +11.1685 q^{27} -3.15883 q^{29} +2.37196 q^{30} -6.34962 q^{31} +5.34481 q^{32} +5.58423 q^{33} -2.48521 q^{34} -11.1739 q^{36} +3.18598 q^{37} +3.09901 q^{38} -13.0315 q^{39} -2.82585 q^{40} -9.44863 q^{41} +2.45473 q^{43} -3.04892 q^{44} +9.10800 q^{45} -0.0271471 q^{46} -9.10800 q^{47} +6.96342 q^{48} -1.71917 q^{50} -13.8780 q^{51} +7.11501 q^{52} +9.74094 q^{53} +6.19802 q^{54} +2.48521 q^{55} +17.3056 q^{57} -1.75302 q^{58} +7.30405 q^{59} -7.23191 q^{60} -0.954429 q^{61} -3.52377 q^{62} -1.52781 q^{64} -5.79954 q^{65} +3.09901 q^{66} -1.43296 q^{67} +7.57721 q^{68} -0.151595 q^{69} -2.13706 q^{71} -13.5308 q^{72} -0.273166 q^{73} +1.76809 q^{74} -9.60023 q^{75} -9.44863 q^{76} -7.23191 q^{78} +2.18060 q^{79} +3.09901 q^{80} +14.7995 q^{81} -5.24359 q^{82} +4.20504 q^{83} -6.17629 q^{85} +1.36227 q^{86} -9.78926 q^{87} -3.69202 q^{88} +7.45564 q^{89} +5.05456 q^{90} +0.0827692 q^{92} -19.6775 q^{93} -5.05456 q^{94} +7.70171 q^{95} +16.5636 q^{96} -5.70580 q^{97} +11.8998 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{2} + 6 q^{8} + 22 q^{9}+O(q^{10}) $ 6 * q + 4 * q^2 + 6 * q^8 + 22 * q^9	$ 6 q + 4 q^{2} + 6 q^{8} + 22 q^{9} + 2 q^{11} + 4 q^{15} + 4 q^{16} + 24 q^{18} + 6 q^{22} + 18 q^{23} + 18 q^{25} - 2 q^{29} - 44 q^{30} - 14 q^{32} - 10 q^{37} - 28 q^{39} - 30 q^{43} + 12 q^{46} + 12 q^{50} - 44 q^{51} + 30 q^{53} + 32 q^{57} - 20 q^{58} - 84 q^{60} - 22 q^{64} + 56 q^{65} + 30 q^{67} - 2 q^{71} - 6 q^{72} - 30 q^{74} - 84 q^{78} - 10 q^{79} - 2 q^{81} - 52 q^{85} - 6 q^{86} - 12 q^{88} + 14 q^{92} + 12 q^{93} - 8 q^{95} + 26 q^{99}+O(q^{100}) $ 6 * q + 4 * q^2 + 6 * q^8 + 22 * q^9 + 2 * q^11 + 4 * q^15 + 4 * q^16 + 24 * q^18 + 6 * q^22 + 18 * q^23 + 18 * q^25 - 2 * q^29 - 44 * q^30 - 14 * q^32 - 10 * q^37 - 28 * q^39 - 30 * q^43 + 12 * q^46 + 12 * q^50 - 44 * q^51 + 30 * q^53 + 32 * q^57 - 20 * q^58 - 84 * q^60 - 22 * q^64 + 56 * q^65 + 30 * q^67 - 2 * q^71 - 6 * q^72 - 30 * q^74 - 84 * q^78 - 10 * q^79 - 2 * q^81 - 52 * q^85 - 6 * q^86 - 12 * q^88 + 14 * q^92 + 12 * q^93 - 8 * q^95 + 26 * q^99

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	$-0.437137\pi$
$2$	0.554958	0.392415	0.196207	+	0.980562i	$0.437137\pi$
$3$	3.09901	1.78922	0.894608	−	0.446852i	$-0.147455\pi$
$3$	3.09901	1.78922	0.894608	+	0.446852i	$0.147455\pi$
$4$	−1.69202	−0.846011
$5$	1.37919	0.616793	0.308396	−	0.951258i	$-0.400208\pi$
$5$	1.37919	0.616793	0.308396	+	0.951258i	$0.400208\pi$
$6$	1.71982	0.702114
$7$	0	0
$7$	0	0
$8$	−2.04892	−0.724402
$9$	6.60388	2.20129
$10$	0.765393	0.242038
$11$	1.80194	0.543305	0.271652	−	0.962395i	$-0.412430\pi$
$11$	1.80194	0.543305	0.271652	+	0.962395i	$0.412430\pi$
$12$	−5.24359	−1.51370
$13$	−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
$15$	4.27413	1.10357
$16$	2.24698	0.561745
$17$	−4.47820	−1.08612	−0.543062	−	0.839693i	$-0.682735\pi$
$17$	−4.47820	−1.08612	−0.543062	+	0.839693i	$0.682735\pi$
$18$	3.66487	0.863819
$19$	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	$-0.501623\pi$
$23$	−0.0489173	−0.0102000	−0.00509999	+	0.999987i	$0.501623\pi$
$24$	−6.34962	−1.29611
$25$	−3.09783	−0.619567
$26$	−2.33362	−0.457660
$27$	11.1685	2.14937
$28$	0	0
$29$	−3.15883	−0.586581	−0.293290	−	0.956023i	$-0.594750\pi$
$29$	−3.15883	−0.586581	−0.293290	+	0.956023i	$0.594750\pi$
$30$	2.37196	0.433059
$31$	−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
$33$	5.58423	0.972089
$34$	−2.48521	−0.426211
$35$	0	0
$36$	−11.1739	−1.86232
$37$	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
$39$	−13.0315	−2.08670
$40$	−2.82585	−0.446806
$41$	−9.44863	−1.47563	−0.737814	−	0.675004i	$-0.764142\pi$
$41$	−9.44863	−1.47563	−0.737814	+	0.675004i	$0.764142\pi$
$42$	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
$45$	9.10800	1.35774
$46$	−0.0271471	−0.00400262
$47$	−9.10800	−1.32854	−0.664269	−	0.747493i	$-0.731257\pi$
$47$	−9.10800	−1.32854	−0.664269	+	0.747493i	$0.731257\pi$
$48$	6.96342	1.00508
$49$	0	0
$50$	−1.71917	−0.243127
$51$	−13.8780	−1.94331
$52$	7.11501	0.986675
$53$	9.74094	1.33802	0.669010	−	0.743253i	$-0.266718\pi$
$53$	9.74094	1.33802	0.669010	+	0.743253i	$0.266718\pi$
$54$	6.19802	0.843444
$55$	2.48521	0.335106
$56$	0	0
$57$	17.3056	2.29218
$58$	−1.75302	−0.230183
$59$	7.30405	0.950906	0.475453	−	0.879741i	$-0.342284\pi$
$59$	7.30405	0.950906	0.475453	+	0.879741i	$0.342284\pi$
$60$	−7.23191	−0.933636
$61$	−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
$66$	3.09901	0.381462
$67$	−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
$69$	−0.151595	−0.0182499
$70$	0	0
$71$	−2.13706	−0.253623	−0.126811	−	0.991927i	$-0.540474\pi$
$71$	−2.13706	−0.253623	−0.126811	+	0.991927i	$0.540474\pi$
$72$	−13.5308	−1.59462
$73$	−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
$75$	−9.60023	−1.10854
$76$	−9.44863	−1.08383
$77$	0	0
$78$	−7.23191	−0.818853
$79$	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
$81$	14.7995	1.64439
$82$	−5.24359	−0.579058
$83$	4.20504	0.461563	0.230781	−	0.973006i	$-0.425872\pi$
$83$	4.20504	0.461563	0.230781	+	0.973006i	$0.425872\pi$
$84$	0	0
$85$	−6.17629	−0.669913
$86$	1.36227	0.146898
$87$	−9.78926	−1.04952
$88$	−3.69202	−0.393571
$89$	7.45564	0.790297	0.395148	−	0.918617i	$-0.370693\pi$
$89$	7.45564	0.790297	0.395148	+	0.918617i	$0.370693\pi$
$90$	5.05456	0.532797
$91$	0	0
$92$	0.0827692	0.00862928
$93$	−19.6775	−2.04047
$94$	−5.05456	−0.521338
$95$	7.70171	0.790179
$96$	16.5636	1.69052
$97$	−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
$99$	11.8998	1.19597
$100$	5.24160	0.524160
$101$	−12.8883	−1.28243	−0.641216	−	0.767361i	$-0.721570\pi$
$101$	−12.8883	−1.28243	−0.641216	+	0.767361i	$0.721570\pi$
$102$	−7.70171	−0.762583
$103$	−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	$-0.360780\pi$
$107$	8.76271	0.847123	0.423562	+	0.905867i	$0.360780\pi$
$108$	−18.8973	−1.81839
$109$	−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
$111$	9.87339	0.937141
$112$	0	0
$113$	18.2959	1.72113	0.860567	−	0.509338i	$-0.170110\pi$
$113$	18.2959	1.72113	0.860567	+	0.509338i	$0.170110\pi$
$114$	9.60388	0.899485
$115$	−0.0674663	−0.00629127
$116$	5.34481	0.496254
$117$	−27.7695	−2.56729
$118$	4.05344	0.373150
$119$	0	0
$120$	−8.75733	−0.799431
$121$	−7.75302	−0.704820
$122$	−0.529668	−0.0479539
$123$	−29.2814	−2.64022
$124$	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
$129$	7.60724	0.669780
$130$	−3.21850	−0.282282
$131$	−0.492227	−0.0430061	−0.0215030	−	0.999769i	$-0.506845\pi$
$131$	−0.492227	−0.0430061	−0.0215030	+	0.999769i	$0.506845\pi$
$132$	−9.44863	−0.822398
$133$	0	0
$134$	−0.795233	−0.0686977
$135$	15.4034	1.32572
$136$	9.17547	0.786790
$137$	11.7313	1.00227	0.501134	−	0.865370i	$-0.332916\pi$
$137$	11.7313	1.00227	0.501134	+	0.865370i	$0.332916\pi$
$138$	−0.0841291	−0.00716155
$139$	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
$141$	−28.2258	−2.37704
$142$	−1.18598	−0.0995253
$143$	−7.57721	−0.633638
$144$	14.8388	1.23656
$145$	−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	$-0.404156\pi$
$149$	7.24027	0.593146	0.296573	+	0.955010i	$0.404156\pi$
$150$	−5.32772	−0.435007
$151$	−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
$153$	−29.5735	−2.39087
$154$	0	0
$155$	−8.75733	−0.703406
$156$	22.0495	1.76537
$157$	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
$159$	30.1873	2.39401
$160$	7.37151	0.582769
$161$	0	0
$162$	8.21313	0.645284
$163$	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
$165$	7.70171	0.599577
$166$	2.33362	0.181124
$167$	12.3586	0.956338	0.478169	−	0.878268i	$-0.341301\pi$
$167$	12.3586	0.956338	0.478169	+	0.878268i	$0.341301\pi$
$168$	0	0
$169$	4.68233	0.360179
$170$	−3.42758	−0.262884
$171$	36.8775	2.82010
$172$	−4.15346	−0.316698
$173$	16.0714	1.22189	0.610944	−	0.791674i	$-0.290790\pi$
$173$	16.0714	1.22189	0.610944	+	0.791674i	$0.290790\pi$
$174$	−5.43263	−0.411847
$175$	0	0
$176$	4.04892	0.305199
$177$	22.6353	1.70138
$178$	4.13757	0.310124
$179$	23.4426	1.75219	0.876093	−	0.482142i	$-0.160141\pi$
$179$	23.4426	1.75219	0.876093	+	0.482142i	$0.160141\pi$
$180$	−15.4109	−1.14866
$181$	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
$183$	−2.95779	−0.218646
$184$	0.100228	0.00738888
$185$	4.39407	0.323059
$186$	−10.9202	−0.800709
$187$	−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	$-0.405619\pi$
$191$	8.07606	0.584364	0.292182	+	0.956363i	$0.405619\pi$
$192$	−4.73470	−0.341698
$193$	−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
$195$	−17.9729	−1.28706
$196$	0	0
$197$	12.8847	0.917997	0.458999	−	0.888437i	$-0.348208\pi$
$197$	12.8847	0.917997	0.458999	+	0.888437i	$0.348208\pi$
$198$	6.60388	0.469317
$199$	−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
$201$	−4.44076	−0.313227
$202$	−7.15245	−0.503245
$203$	0	0
$204$	23.4819	1.64406
$205$	−13.0315	−0.910157
$206$	−2.40109	−0.167292
$207$	−0.323044	−0.0224531
$208$	−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
$213$	−6.62278	−0.453786
$214$	4.86294	0.332424
$215$	3.38554	0.230892
$216$	−22.8832	−1.55701
$217$	0	0
$218$	−1.15585	−0.0782840
$219$	−0.846543	−0.0572041
$220$	−4.20504	−0.283504
$221$	18.8310	1.26671
$222$	5.47932	0.367748
$223$	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
$225$	−20.4577	−1.36385
$226$	10.1535	0.675398
$227$	0.765393	0.0508009	0.0254005	−	0.999677i	$-0.491914\pi$
$227$	0.765393	0.0508009	0.0254005	+	0.999677i	$0.491914\pi$
$228$	−29.2814	−1.93921
$229$	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	$-0.218422\pi$
$233$	23.6189	1.54733	0.773664	+	0.633596i	$0.218422\pi$
$234$	−15.4109	−1.00744
$235$	−12.5617	−0.819433
$236$	−12.3586	−0.804477
$237$	6.75772	0.438961
$238$	0	0
$239$	7.15883	0.463066	0.231533	−	0.972827i	$-0.425626\pi$
$239$	7.15883	0.463066	0.231533	+	0.972827i	$0.425626\pi$
$240$	9.60388	0.619927
$241$	−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
$243$	12.3586	0.792805
$244$	1.61491	0.103384
$245$	0	0
$246$	−16.2500	−1.03606
$247$	−23.4819	−1.49412
$248$	13.0098	0.826126
$249$	13.0315	0.825835
$250$	−6.19802	−0.391997
$251$	9.44863	0.596392	0.298196	−	0.954505i	$-0.403615\pi$
$251$	9.44863	0.596392	0.298196	+	0.954505i	$0.403615\pi$
$252$	0	0
$253$	−0.0881460	−0.00554169
$254$	−0.164210	−0.0103035
$255$	−19.1404	−1.19862
$256$	−3.34721	−0.209200
$257$	−4.90296	−0.305838	−0.152919	−	0.988239i	$-0.548867\pi$
$257$	−4.90296	−0.305838	−0.152919	+	0.988239i	$0.548867\pi$
$258$	4.22170	0.262832
$259$	0	0
$260$	9.81295	0.608574
$261$	−20.8605	−1.29124
$262$	−0.273166	−0.0168762
$263$	−20.8442	−1.28531	−0.642653	−	0.766158i	$-0.722166\pi$
$263$	−20.8442	−1.28531	−0.642653	+	0.766158i	$0.722166\pi$
$264$	−11.4416	−0.704183
$265$	13.4346	0.825281
$266$	0	0
$267$	23.1051	1.41401
$268$	2.42460	0.148106
$269$	−24.7847	−1.51115	−0.755574	−	0.655063i	$-0.772642\pi$
$269$	−24.7847	−1.51115	−0.755574	+	0.655063i	$0.772642\pi$
$270$	8.54825	0.520230
$271$	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
$276$	0.256503	0.0154396
$277$	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
$279$	−41.9321	−2.51041
$280$	0	0
$281$	−6.02177	−0.359229	−0.179614	−	0.983737i	$-0.557485\pi$
$281$	−6.02177	−0.359229	−0.179614	+	0.983737i	$0.557485\pi$
$282$	−15.6641	−0.932786
$283$	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
$285$	23.8677	1.41380
$286$	−4.20504	−0.248649
$287$	0	0
$288$	35.2965	2.07987
$289$	3.05429	0.179664
$290$	−2.41775	−0.141975
$291$	−17.6823	−1.03656
$292$	0.462202	0.0270483
$293$	1.40922	0.0823272	0.0411636	−	0.999152i	$-0.486894\pi$
$293$	1.40922	0.0823272	0.0411636	+	0.999152i	$0.486894\pi$
$294$	0	0
$295$	10.0737	0.586512
$296$	−6.52781	−0.379421
$297$	20.1249	1.16776
$298$	4.01805	0.232759
$299$	0.205699	0.0118959
$300$	16.2438	0.937836
$301$	0	0
$302$	−8.48188	−0.488077
$303$	−39.9409	−2.29455
$304$	12.5476	0.719657
$305$	−1.31634	−0.0753734
$306$	−16.4120	−0.938214
$307$	−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
$309$	−13.4082	−0.762766
$310$	−4.85995	−0.276027
$311$	−7.64468	−0.433490	−0.216745	−	0.976228i	$-0.569544\pi$
$311$	−7.64468	−0.433490	−0.216745	+	0.976228i	$0.569544\pi$
$312$	26.7004	1.51161
$313$	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	$-0.343466\pi$
$317$	16.8140	0.944369	0.472185	+	0.881500i	$0.343466\pi$
$318$	16.7527	0.939444
$319$	−5.69202	−0.318692
$320$	−2.10714	−0.117793
$321$	27.1557	1.51569
$322$	0	0
$323$	−25.0073	−1.39144
$324$	−25.0411	−1.39117
$325$	13.0265	0.722581
$326$	9.55794	0.529365
$327$	−6.45453	−0.356936
$328$	19.3595	1.06895
$329$	0	0
$330$	4.27413	0.235283
$331$	−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
$333$	21.0398	1.15298
$334$	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
$339$	56.6992	3.07948
$340$	10.4504	0.566754
$341$	−11.4416	−0.619598
$342$	20.4655	1.10665
$343$	0	0
$344$	−5.02954	−0.271175
$345$	−0.209079	−0.0112564
$346$	8.91896	0.479486
$347$	−17.5797	−0.943728	−0.471864	−	0.881671i	$-0.656419\pi$
$347$	−17.5797	−0.943728	−0.471864	+	0.881671i	$0.656419\pi$
$348$	16.5636	0.887905
$349$	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
$351$	−46.9638	−2.50674
$352$	9.63102	0.513335
$353$	24.0734	1.28130	0.640649	−	0.767834i	$-0.278666\pi$
$353$	24.0734	1.28130	0.640649	+	0.767834i	$0.278666\pi$
$354$	12.5617	0.667645
$355$	−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	$-0.641800\pi$
$359$	−16.3284	−0.861781	−0.430891	+	0.902404i	$0.641800\pi$
$360$	−18.6615	−0.983549
$361$	12.1836	0.641241
$362$	13.3972	0.704139
$363$	−24.0267	−1.26107
$364$	0	0
$365$	−0.376747	−0.0197198
$366$	−1.64145	−0.0857999
$367$	−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
$369$	−62.3976	−3.24829
$370$	2.43853	0.126773
$371$	0	0
$372$	33.2948	1.72626
$373$	−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
$375$	−34.6112	−1.78731
$376$	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
$381$	−0.916988	−0.0469787
$382$	4.48188	0.229313
$383$	−29.0946	−1.48667	−0.743333	−	0.668922i	$-0.766756\pi$
$383$	−29.0946	−1.48667	−0.743333	+	0.668922i	$0.766756\pi$
$384$	−35.7549	−1.82461
$385$	0	0
$386$	−0.960771	−0.0489020
$387$	16.2107	0.824038
$388$	9.65433	0.490124
$389$	−27.9801	−1.41865	−0.709325	−	0.704882i	$-0.751000\pi$
$389$	−27.9801	−1.41865	−0.709325	+	0.704882i	$0.751000\pi$
$390$	−9.97418	−0.505062
$391$	0.219062	0.0110784
$392$	0	0
$393$	−1.52542	−0.0769472
$394$	7.15047	0.360236
$395$	3.00747	0.151322
$396$	−20.1347	−1.01181
$397$	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	$-0.297599\pi$
$401$	23.7845	1.18774	0.593870	+	0.804561i	$0.297599\pi$
$402$	−2.46444	−0.122915
$403$	26.7004	1.33004
$404$	21.8072	1.08495
$405$	20.4114	1.01425
$406$	0	0
$407$	5.74094	0.284568
$408$	28.4349	1.40774
$409$	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
$411$	36.3553	1.79327
$412$	7.32071	0.360666
$413$	0	0
$414$	−0.179276	−0.00881093
$415$	5.79954	0.284688
$416$	−22.4751	−1.10193
$417$	20.2634	0.992302
$418$	5.58423	0.273133
$419$	−7.78291	−0.380220	−0.190110	−	0.981763i	$-0.560884\pi$
$419$	−7.78291	−0.380220	−0.190110	+	0.981763i	$0.560884\pi$
$420$	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
$423$	−60.1481	−2.92450
$424$	−19.9584	−0.969265
$425$	13.8727	0.672926
$426$	−3.67537	−0.178072
$427$	0	0
$428$	−14.8267	−0.716675
$429$	−23.4819	−1.13372
$430$	1.87883	0.0906054
$431$	21.4319	1.03234	0.516169	−	0.856487i	$-0.327358\pi$
$431$	21.4319	1.03234	0.516169	+	0.856487i	$0.327358\pi$
$432$	25.0953	1.20740
$433$	−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
$435$	−13.5013	−0.647336
$436$	3.52409	0.168773
$437$	−0.273166	−0.0130673
$438$	−0.469796	−0.0224477
$439$	−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	$-0.311140\pi$
$443$	23.5362	1.11824	0.559119	+	0.829088i	$0.311140\pi$
$444$	−16.7060	−0.792831
$445$	10.2828	0.487449
$446$	2.44777	0.115905
$447$	22.4377	1.06127
$448$	0	0
$449$	−21.6383	−1.02118	−0.510588	−	0.859826i	$-0.670572\pi$
$449$	−21.6383	−1.02118	−0.510588	+	0.859826i	$0.670572\pi$
$450$	−11.3532	−0.535194
$451$	−17.0258	−0.801716
$452$	−30.9571	−1.45610
$453$	−47.3647	−2.22539
$454$	0.424761	0.0199350
$455$	0	0
$456$	−35.4577	−1.66046
$457$	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
$459$	−50.0146	−2.33448
$460$	0.114154	0.00532248
$461$	3.83438	0.178585	0.0892924	−	0.996005i	$-0.471539\pi$
$461$	3.83438	0.178585	0.0892924	+	0.996005i	$0.471539\pi$
$462$	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
$465$	−27.1391	−1.25854
$466$	13.1075	0.607194
$467$	3.25061	0.150420	0.0752101	−	0.997168i	$-0.476037\pi$
$467$	3.25061	0.150420	0.0752101	+	0.997168i	$0.476037\pi$
$468$	46.9867	2.17196
$469$	0	0
$470$	−6.97120	−0.321557
$471$	40.3177	1.85774
$472$	−14.9654	−0.688838
$473$	4.42327	0.203382
$474$	3.75025	0.172255
$475$	−17.2990	−0.793733
$476$	0	0
$477$	64.3279	2.94537
$478$	3.97285	0.181714
$479$	−1.36253	−0.0622555	−0.0311277	−	0.999515i	$-0.509910\pi$
$479$	−1.36253	−0.0622555	−0.0311277	+	0.999515i	$0.509910\pi$
$480$	22.8444	1.04270
$481$	−13.3972	−0.610858
$482$	−14.8438	−0.676118
$483$	0	0
$484$	13.1183	0.596285
$485$	−7.86938	−0.357330
$486$	6.85851	0.311108
$487$	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
$489$	53.3737	2.41364
$490$	0	0
$491$	−16.5418	−0.746522	−0.373261	−	0.927726i	$-0.621760\pi$
$491$	−16.5418	−0.746522	−0.373261	+	0.927726i	$0.621760\pi$
$492$	49.5448	2.23365
$493$	14.1459	0.637099
$494$	−13.0315	−0.586313
$495$	16.4120	0.737667
$496$	−14.2675	−0.640628
$497$	0	0
$498$	7.23191	0.324070
$499$	−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
$501$	38.2995	1.71109
$502$	5.24359	0.234033
$503$	−36.9617	−1.64804	−0.824020	−	0.566561i	$-0.808273\pi$
$503$	−36.9617	−1.64804	−0.824020	+	0.566561i	$0.808273\pi$
$504$	0	0
$505$	−17.7754	−0.790994
$506$	−0.0489173	−0.00217464
$507$	14.5106	0.644439
$508$	0.500664	0.0222134
$509$	−1.53079	−0.0678509	−0.0339254	−	0.999424i	$-0.510801\pi$
$509$	−1.53079	−0.0678509	−0.0339254	+	0.999424i	$0.510801\pi$
$510$	−10.6221	−0.470355
$511$	0	0
$512$	21.2174	0.937687
$513$	62.3672	2.75358
$514$	−2.72094	−0.120015
$515$	−5.96721	−0.262947
$516$	−12.8716	−0.566641
$517$	−16.4120	−0.721801
$518$	0	0
$519$	49.8055	2.18622
$520$	11.8828	0.521095
$521$	26.4070	1.15691	0.578456	−	0.815714i	$-0.303655\pi$
$521$	26.4070	1.15691	0.578456	+	0.815714i	$0.303655\pi$
$522$	−11.5767	−0.506700
$523$	−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
$528$	12.5476	0.546066
$529$	−22.9976	−0.999896
$530$	7.45564	0.323852
$531$	48.2350	2.09322
$532$	0	0
$533$	39.7318	1.72098
$534$	12.8224	0.554879
$535$	12.0854	0.522499
$536$	2.93602	0.126817
$537$	72.6491	3.13504
$538$	−13.7545	−0.592997
$539$	0	0
$540$	−26.0629	−1.12157
$541$	−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
$543$	74.8128	3.21052
$544$	−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
$549$	−6.30293	−0.269003
$550$	−3.09783	−0.132092
$551$	−17.6396	−0.751474
$552$	0.310606	0.0132203
$553$	0	0
$554$	0.0314589	0.00133656
$555$	13.6173	0.578022
$556$	−11.0635	−0.469199
$557$	−35.9560	−1.52350	−0.761752	−	0.647869i	$-0.775661\pi$
$557$	−35.9560	−1.52350	−0.761752	+	0.647869i	$0.775661\pi$
$558$	−23.2706	−0.985121
$559$	−10.3222	−0.436584
$560$	0	0
$561$	−25.0073	−1.05581
$562$	−3.34183	−0.140967
$563$	28.5349	1.20260	0.601302	−	0.799022i	$-0.294649\pi$
$563$	28.5349	1.20260	0.601302	+	0.799022i	$0.294649\pi$
$564$	47.7587	2.01100
$565$	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	$-0.572673\pi$
$569$	−10.7976	−0.452660	−0.226330	+	0.974051i	$0.572673\pi$
$570$	13.2456	0.554796
$571$	−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
$573$	25.0278	1.04555
$574$	0	0
$575$	0.151538	0.00631956
$576$	−10.0895	−0.420395
$577$	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
$579$	−5.36516	−0.222969
$580$	7.37151	0.306085
$581$	0	0
$582$	−9.81295	−0.406760
$583$	17.5526	0.726953
$584$	0.559694	0.0231603
$585$	−38.2995	−1.58349
$586$	0.782056	0.0323064
$587$	−17.3965	−0.718031	−0.359015	−	0.933332i	$-0.616887\pi$
$587$	−17.3965	−0.718031	−0.359015	+	0.933332i	$0.616887\pi$
$588$	0	0
$589$	−35.4577	−1.46101
$590$	5.59047	0.230156
$591$	39.9299	1.64250
$592$	7.15883	0.294226
$593$	−7.49309	−0.307704	−0.153852	−	0.988094i	$-0.549168\pi$
$593$	−7.49309	−0.307704	−0.153852	+	0.988094i	$0.549168\pi$
$594$	11.1685	0.458247
$595$	0	0
$596$	−12.2507	−0.501808
$597$	−61.9904	−2.53710
$598$	0.114154	0.00466812
$599$	−46.3564	−1.89407	−0.947036	−	0.321127i	$-0.895938\pi$
$599$	−46.3564	−1.89407	−0.947036	+	0.321127i	$0.895938\pi$
$600$	19.6701	0.803027
$601$	−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
$603$	−9.46309	−0.385367
$604$	25.8605	1.05225
$605$	−10.6929	−0.434728
$606$	−22.1655	−0.900413
$607$	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
$612$	50.0390	2.02271
$613$	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
$615$	−40.3846	−1.62847
$616$	0	0
$617$	0.635808	0.0255967	0.0127983	−	0.999918i	$-0.495926\pi$
$617$	0.635808	0.0255967	0.0127983	+	0.999918i	$0.495926\pi$
$618$	−7.44099	−0.299321
$619$	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
$621$	−0.546331	−0.0219235
$622$	−4.24248	−0.170108
$623$	0	0
$624$	−29.2814	−1.17220
$625$	0.0857531	0.00343012
$626$	4.50823	0.180185
$627$	31.1836	1.24535
$628$	−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
$633$	−81.6355	−3.24472
$634$	9.33108	0.370584
$635$	−0.408098	−0.0161949
$636$	−51.0775	−2.02536
$637$	0	0
$638$	−3.15883	−0.125059
$639$	−14.1129	−0.558298
$640$	−15.9124	−0.628993
$641$	−29.3739	−1.16020	−0.580099	−	0.814546i	$-0.696986\pi$
$641$	−29.3739	−1.16020	−0.580099	+	0.814546i	$0.696986\pi$
$642$	15.0703	0.594777
$643$	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
$645$	10.4918	0.413115
$646$	−13.8780	−0.546023
$647$	−38.9006	−1.52934	−0.764669	−	0.644423i	$-0.777098\pi$
$647$	−38.9006	−1.52934	−0.764669	+	0.644423i	$0.777098\pi$
$648$	−30.3230	−1.19120
$649$	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	$-0.618730\pi$
$653$	−18.6243	−0.728826	−0.364413	+	0.931237i	$0.618730\pi$
$654$	−3.58199	−0.140067
$655$	−0.678875	−0.0265258
$656$	−21.2309	−0.828927
$657$	−1.80395	−0.0703788
$658$	0	0
$659$	22.9778	0.895086	0.447543	−	0.894262i	$-0.352299\pi$
$659$	22.9778	0.895086	0.447543	+	0.894262i	$0.352299\pi$
$660$	−13.0315	−0.507249
$661$	−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
$663$	58.3575	2.26642
$664$	−8.61577	−0.334357
$665$	0	0
$666$	11.6762	0.452444
$667$	0.154522	0.00598311
$668$	−20.9110	−0.809072
$669$	13.6689	0.528471
$670$	−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
$675$	−34.5980	−1.33168
$676$	−7.92261	−0.304716
$677$	27.7395	1.06612	0.533058	−	0.846079i	$-0.321043\pi$
$677$	27.7395	1.06612	0.533058	+	0.846079i	$0.321043\pi$
$678$	31.4657	1.20843
$679$	0	0
$680$	12.6547	0.485286
$681$	2.37196	0.0908938
$682$	−6.34962	−0.243139
$683$	−14.6872	−0.561991	−0.280996	−	0.959709i	$-0.590665\pi$
$683$	−14.6872	−0.561991	−0.280996	+	0.959709i	$0.590665\pi$
$684$	−62.3976	−2.38583
$685$	16.1796	0.618192
$686$	0	0
$687$	30.3370	1.15743
$688$	5.51573	0.210285
$689$	−40.9610	−1.56049
$690$	−0.116030	−0.00441719
$691$	−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
$696$	20.0574	0.760273
$697$	42.3129	1.60271
$698$	8.41007	0.318326
$699$	73.1954	2.76850
$700$	0	0
$701$	28.3889	1.07224	0.536118	−	0.844143i	$-0.319890\pi$
$701$	28.3889	1.07224	0.536118	+	0.844143i	$0.319890\pi$
$702$	−26.0629	−0.983681
$703$	17.7912	0.671009
$704$	−2.75302	−0.103758
$705$	−38.9287	−1.46614
$706$	13.3597	0.502800
$707$	0	0
$708$	−38.2995	−1.43938
$709$	−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
$711$	14.4004	0.540059
$712$	−15.2760	−0.572492
$713$	0.310606	0.0116323
$714$	0	0
$715$	−10.4504	−0.390823
$716$	−39.6655	−1.48237
$717$	22.1853	0.828526
$718$	−9.06159	−0.338176
$719$	−10.7737	−0.401792	−0.200896	−	0.979613i	$-0.564385\pi$
$719$	−10.7737	−0.401792	−0.200896	+	0.979613i	$0.564385\pi$
$720$	20.4655	0.762704
$721$	0	0
$722$	6.76138	0.251633
$723$	−82.8913	−3.08276
$724$	−40.8468	−1.51806
$725$	9.78554	0.363426
$726$	−13.3338	−0.494864
$727$	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
$729$	−6.09916	−0.225895
$730$	−0.209079	−0.00773836
$731$	−10.9928	−0.406583
$732$	5.00464	0.184977
$733$	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
$738$	−34.6280	−1.27468
$739$	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
$741$	−72.7706	−2.67330
$742$	0	0
$743$	−26.6848	−0.978972	−0.489486	−	0.872011i	$-0.662816\pi$
$743$	−26.6848	−0.978972	−0.489486	+	0.872011i	$0.662816\pi$
$744$	40.3177	1.47812
$745$	9.98572	0.365848
$746$	−8.76510	−0.320913
$747$	27.7695	1.01603
$748$	13.6537	0.499228
$749$	0	0
$750$	−19.2078	−0.701368
$751$	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
$753$	29.2814	1.06707
$754$	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
$759$	−0.273166	−0.00991528
$760$	−15.7802	−0.572407
$761$	49.4552	1.79275	0.896375	−	0.443297i	$-0.146191\pi$
$761$	49.4552	1.79275	0.896375	+	0.443297i	$0.146191\pi$
$762$	−0.508890	−0.0184351
$763$	0	0
$764$	−13.6649	−0.494378
$765$	−40.7875	−1.47467
$766$	−16.1463	−0.583389
$767$	−30.7138	−1.10901
$768$	−10.3730	−0.374305
$769$	−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
$771$	−15.1943	−0.547211
$772$	2.92931	0.105428
$773$	45.4692	1.63541	0.817707	−	0.575634i	$-0.195245\pi$
$773$	45.4692	1.63541	0.817707	+	0.575634i	$0.195245\pi$
$774$	8.99628	0.323365
$775$	19.6701	0.706570
$776$	11.6907	0.419672
$777$	0	0
$778$	−15.5278	−0.556699
$779$	−52.7633	−1.89044
$780$	30.4105	1.08887
$781$	−3.85086	−0.137794
$782$	0.121570	0.00434734
$783$	−35.2793	−1.26078
$784$	0	0
$785$	17.9430	0.640415
$786$	−0.846543	−0.0301952
$787$	−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
$789$	−64.5963	−2.29969
$790$	1.66902	0.0593810
$791$	0	0
$792$	−24.3817	−0.866364
$793$	4.01341	0.142520
$794$	14.5614	0.516765
$795$	41.6340	1.47661
$796$	33.8460	1.19964
$797$	22.0637	0.781538	0.390769	−	0.920489i	$-0.372209\pi$
$797$	22.0637	0.781538	0.390769	+	0.920489i	$0.372209\pi$
$798$	0	0
$799$	40.7875	1.44296
$800$	−16.5574	−0.585391
$801$	49.2361	1.73967
$802$	13.1994	0.466087
$803$	−0.492227	−0.0173703
$804$	7.51386	0.264994
$805$	0	0
$806$	14.8176	0.521927
$807$	−76.8080	−2.70377
$808$	26.4070	0.928995
$809$	9.10752	0.320203	0.160102	−	0.987101i	$-0.448818\pi$
$809$	9.10752	0.320203	0.160102	+	0.987101i	$0.448818\pi$
$810$	11.3275	0.398007
$811$	−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
$813$	19.8866	0.697454
$814$	3.18598	0.111669
$815$	23.7535	0.832050
$816$	−31.1836	−1.09164
$817$	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	$-0.318172\pi$
$821$	30.9836	1.08134	0.540668	+	0.841236i	$0.318172\pi$
$822$	20.1757	0.703707
$823$	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
$825$	−17.2990	−0.602274
$826$	0	0
$827$	18.4483	0.641510	0.320755	−	0.947162i	$-0.396063\pi$
$827$	18.4483	0.641510	0.320755	+	0.947162i	$0.396063\pi$
$828$	0.546597	0.0189956
$829$	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
$831$	0.175674	0.00609406
$832$	6.42450	0.222730
$833$	0	0
$834$	11.2453	0.389394
$835$	17.0449	0.589862
$836$	−17.0258	−0.588851
$837$	−70.9154	−2.45120
$838$	−4.31919	−0.149204
$839$	48.5382	1.67573	0.837863	−	0.545881i	$-0.183805\pi$
$839$	48.5382	1.67573	0.837863	+	0.545881i	$0.183805\pi$
$840$	0	0
$841$	−19.0218	−0.655923
$842$	12.3093	0.424207
$843$	−18.6615	−0.642738
$844$	44.5719	1.53423
$845$	6.45783	0.222156
$846$	−33.3797	−1.14762
$847$	0	0
$848$	21.8877	0.751626
$849$	37.4529	1.28538
$850$	7.69878	0.264066
$851$	−0.155850	−0.00534246
$852$	11.2059	0.383908
$853$	−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
$855$	50.8611	1.73941
$856$	−17.9541	−0.613657
$857$	33.1106	1.13104	0.565519	−	0.824735i	$-0.308676\pi$
$857$	33.1106	1.13104	0.565519	+	0.824735i	$0.308676\pi$
$858$	−13.0315	−0.444887
$859$	−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	$-0.299912\pi$
$863$	34.5478	1.17602	0.588010	+	0.808854i	$0.299912\pi$
$864$	59.6933	2.03081
$865$	22.1655	0.753651
$866$	−19.0396	−0.646992
$867$	9.46529	0.321458
$868$	0	0
$869$	3.92931	0.133293
$870$	−7.49263	−0.254024
$871$	6.02565	0.204171
$872$	4.26742	0.144513
$873$	−37.6804	−1.27529
$874$	−0.151595	−0.00512779
$875$	0	0
$876$	1.43237	0.0483953
$877$	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
$879$	4.36718	0.147301
$880$	5.58423	0.188244
$881$	56.4861	1.90306	0.951532	−	0.307549i	$-0.0995087\pi$
$881$	56.4861	1.90306	0.951532	+	0.307549i	$0.0995087\pi$
$882$	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
$885$	31.2184	1.04940
$886$	13.0616	0.438813
$887$	−5.10866	−0.171532	−0.0857660	−	0.996315i	$-0.527334\pi$
$887$	−5.10866	−0.171532	−0.0857660	+	0.996315i	$0.527334\pi$
$888$	−20.2298	−0.678866
$889$	0	0
$890$	5.70650	0.191282
$891$	26.6679	0.893407
$892$	−7.46306	−0.249882
$893$	−50.8611	−1.70200
$894$	12.4520	0.416457
$895$	32.3319	1.08074
$896$	0	0
$897$	0.637464	0.0212843
$898$	−12.0084	−0.400724
$899$	20.0574	0.668951
$900$	34.6149	1.15383
$901$	−43.6219	−1.45326
$902$	−9.44863	−0.314605
$903$	0	0
$904$	−37.4868	−1.24679
$905$	33.2948	1.10676
$906$	−26.2854	−0.873275
$907$	−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
$909$	−85.1126	−2.82301
$910$	0	0
$911$	−55.5666	−1.84100	−0.920501	−	0.390740i	$-0.872219\pi$
$911$	−55.5666	−1.84100	−0.920501	+	0.390740i	$0.872219\pi$
$912$	38.8853	1.28762
$913$	7.57721	0.250769
$914$	3.26503	0.107998
$915$	−4.07935	−0.134859
$916$	−16.5636	−0.547278
$917$	0	0
$918$	−27.7560	−0.916085
$919$	−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
$921$	−96.8623	−3.19172
$922$	2.12792	0.0700793
$923$	8.98643	0.295792
$924$	0	0
$925$	−9.86964	−0.324512
$926$	−11.8961	−0.390929
$927$	−28.5724	−0.938440
$928$	−16.8834	−0.554224
$929$	−4.53231	−0.148700	−0.0743501	−	0.997232i	$-0.523688\pi$
$929$	−4.53231	−0.148700	−0.0743501	+	0.997232i	$0.523688\pi$
$930$	−15.0611	−0.493871
$931$	0	0
$932$	−39.9638	−1.30906
$933$	−23.6910	−0.775607
$934$	1.80395	0.0590271
$935$	−11.1293	−0.363967
$936$	56.8975	1.85975
$937$	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
$939$	25.1750	0.821554
$940$	21.2546	0.693249
$941$	−35.1069	−1.14445	−0.572226	−	0.820096i	$-0.693920\pi$
$941$	−35.1069	−1.14445	−0.572226	+	0.820096i	$0.693920\pi$
$942$	22.3746	0.729004
$943$	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	$-0.616378\pi$
$947$	−22.0043	−0.715044	−0.357522	+	0.933905i	$0.616378\pi$
$948$	−11.4342	−0.371366
$949$	1.14867	0.0372874
$950$	−9.60023	−0.311472
$951$	52.1068	1.68968
$952$	0	0
$953$	17.3864	0.563202	0.281601	−	0.959532i	$-0.409135\pi$
$953$	17.3864	0.563202	0.281601	+	0.959532i	$0.409135\pi$
$954$	35.6993	1.15581
$955$	11.1384	0.360431
$956$	−12.1129	−0.391759
$957$	−17.6396	−0.570209
$958$	−0.756146	−0.0244300
$959$	0	0
$960$	−6.53006	−0.210757
$961$	9.31767	0.300570
$962$	−7.43487	−0.239710
$963$	57.8678	1.86477
$964$	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
$969$	−77.4979	−2.48959
$970$	−4.36718	−0.140222
$971$	−5.36516	−0.172176	−0.0860882	−	0.996288i	$-0.527437\pi$
$971$	−5.36516	−0.172176	−0.0860882	+	0.996288i	$0.527437\pi$
$972$	−20.9110	−0.670722
$973$	0	0
$974$	10.5810	0.339038
$975$	40.3693	1.29285
$976$	−2.14458	−0.0686464
$977$	16.2658	0.520388	0.260194	−	0.965556i	$-0.416213\pi$
$977$	16.2658	0.520388	0.260194	+	0.965556i	$0.416213\pi$
$978$	29.6202	0.947149
$979$	13.4346	0.429372
$980$	0	0
$981$	−13.7543	−0.439143
$982$	−9.18001	−0.292946
$983$	44.7279	1.42660	0.713300	−	0.700859i	$-0.247200\pi$
$983$	44.7279	1.42660	0.713300	+	0.700859i	$0.247200\pi$
$984$	59.9952	1.91258
$985$	17.7705	0.566214
$986$	7.85038	0.250007
$987$	0	0
$988$	39.7318	1.26404
$989$	−0.120079	−0.00381829
$990$	9.10800	0.289471
$991$	−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
$993$	−82.2567	−2.61034
$994$	0	0
$995$	−27.5883	−0.874609
$996$	−22.0495	−0.698665
$997$	−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
$999$	35.5825	1.12578

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	343.2.a.e.1.4	yes	6
3.2	odd	2		3087.2.a.g.1.3		6
4.3	odd	2		5488.2.a.o.1.1		6
5.4	even	2		8575.2.a.g.1.3		6
7.2	even	3		343.2.c.c.18.3		12
7.3	odd	6		343.2.c.c.324.4		12
7.4	even	3		343.2.c.c.324.3		12
7.5	odd	6		343.2.c.c.18.4		12
7.6	odd	2	inner	343.2.a.e.1.3	✓	6
21.20	even	2		3087.2.a.g.1.4		6
28.27	even	2		5488.2.a.o.1.6		6
35.34	odd	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	7.6	odd	2	inner
343.2.a.e.1.4	yes	6	1.1	even	1	trivial
343.2.c.c.18.3		12	7.2	even	3
343.2.c.c.18.4		12	7.5	odd	6
343.2.c.c.324.3		12	7.4	even	3
343.2.c.c.324.4		12	7.3	odd	6
3087.2.a.g.1.3		6	3.2	odd	2
3087.2.a.g.1.4		6	21.20	even	2
5488.2.a.o.1.1		6	4.3	odd	2
5488.2.a.o.1.6		6	28.27	even	2
8575.2.a.g.1.3		6	5.4	even	2
8575.2.a.g.1.4		6	35.34	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:	\( N \)	\(=\)	\( 343 = 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	343.a (trivial)

\(f(q)\)	\(=\)	\(q+0.554958 q^{2} +3.09901 q^{3} -1.69202 q^{4} +1.37919 q^{5} +1.71982 q^{6} -2.04892 q^{8} +6.60388 q^{9} +O(q^{10})\)	\(q+0.554958 q^{2} +3.09901 q^{3} -1.69202 q^{4} +1.37919 q^{5} +1.71982 q^{6} -2.04892 q^{8} +6.60388 q^{9} +0.765393 q^{10} +1.80194 q^{11} -5.24359 q^{12} -4.20504 q^{13} +4.27413 q^{15} +2.24698 q^{16} -4.47820 q^{17} +3.66487 q^{18} +5.58423 q^{19} -2.33362 q^{20} +1.00000 q^{22} -0.0489173 q^{23} -6.34962 q^{24} -3.09783 q^{25} -2.33362 q^{26} +11.1685 q^{27} -3.15883 q^{29} +2.37196 q^{30} -6.34962 q^{31} +5.34481 q^{32} +5.58423 q^{33} -2.48521 q^{34} -11.1739 q^{36} +3.18598 q^{37} +3.09901 q^{38} -13.0315 q^{39} -2.82585 q^{40} -9.44863 q^{41} +2.45473 q^{43} -3.04892 q^{44} +9.10800 q^{45} -0.0271471 q^{46} -9.10800 q^{47} +6.96342 q^{48} -1.71917 q^{50} -13.8780 q^{51} +7.11501 q^{52} +9.74094 q^{53} +6.19802 q^{54} +2.48521 q^{55} +17.3056 q^{57} -1.75302 q^{58} +7.30405 q^{59} -7.23191 q^{60} -0.954429 q^{61} -3.52377 q^{62} -1.52781 q^{64} -5.79954 q^{65} +3.09901 q^{66} -1.43296 q^{67} +7.57721 q^{68} -0.151595 q^{69} -2.13706 q^{71} -13.5308 q^{72} -0.273166 q^{73} +1.76809 q^{74} -9.60023 q^{75} -9.44863 q^{76} -7.23191 q^{78} +2.18060 q^{79} +3.09901 q^{80} +14.7995 q^{81} -5.24359 q^{82} +4.20504 q^{83} -6.17629 q^{85} +1.36227 q^{86} -9.78926 q^{87} -3.69202 q^{88} +7.45564 q^{89} +5.05456 q^{90} +0.0827692 q^{92} -19.6775 q^{93} -5.05456 q^{94} +7.70171 q^{95} +16.5636 q^{96} -5.70580 q^{97} +11.8998 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} + 6 q^{8} + 22 q^{9}+O(q^{10}) \) 6 * q + 4 * q^2 + 6 * q^8 + 22 * q^9	\( 6 q + 4 q^{2} + 6 q^{8} + 22 q^{9} + 2 q^{11} + 4 q^{15} + 4 q^{16} + 24 q^{18} + 6 q^{22} + 18 q^{23} + 18 q^{25} - 2 q^{29} - 44 q^{30} - 14 q^{32} - 10 q^{37} - 28 q^{39} - 30 q^{43} + 12 q^{46} + 12 q^{50} - 44 q^{51} + 30 q^{53} + 32 q^{57} - 20 q^{58} - 84 q^{60} - 22 q^{64} + 56 q^{65} + 30 q^{67} - 2 q^{71} - 6 q^{72} - 30 q^{74} - 84 q^{78} - 10 q^{79} - 2 q^{81} - 52 q^{85} - 6 q^{86} - 12 q^{88} + 14 q^{92} + 12 q^{93} - 8 q^{95} + 26 q^{99}+O(q^{100}) \) 6 * q + 4 * q^2 + 6 * q^8 + 22 * q^9 + 2 * q^11 + 4 * q^15 + 4 * q^16 + 24 * q^18 + 6 * q^22 + 18 * q^23 + 18 * q^25 - 2 * q^29 - 44 * q^30 - 14 * q^32 - 10 * q^37 - 28 * q^39 - 30 * q^43 + 12 * q^46 + 12 * q^50 - 44 * q^51 + 30 * q^53 + 32 * q^57 - 20 * q^58 - 84 * q^60 - 22 * q^64 + 56 * q^65 + 30 * q^67 - 2 * q^71 - 6 * q^72 - 30 * q^74 - 84 * q^78 - 10 * q^79 - 2 * q^81 - 52 * q^85 - 6 * q^86 - 12 * q^88 + 14 * q^92 + 12 * q^93 - 8 * q^95 + 26 * q^99

Introduction

L-functions

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