LMFDB - Embedded newform 7105.2.a.o.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7105,2,Mod(1,7105)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7105.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7105, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7105 = 5 \cdot 7^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7105.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,1,-2,1,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$56.7337106361$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	7105.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.17009 q^{2} +1.70928 q^{3} +2.70928 q^{4} -1.00000 q^{5} +3.70928 q^{6} +1.53919 q^{8} -0.0783777 q^{9} -2.17009 q^{10} -0.630898 q^{11} +4.63090 q^{12} +4.34017 q^{13} -1.70928 q^{15} -2.07838 q^{16} +1.55252 q^{17} -0.170086 q^{18} +5.70928 q^{19} -2.70928 q^{20} -1.36910 q^{22} +6.63090 q^{23} +2.63090 q^{24} +1.00000 q^{25} +9.41855 q^{26} -5.26180 q^{27} -1.00000 q^{29} -3.70928 q^{30} +2.29072 q^{31} -7.58864 q^{32} -1.07838 q^{33} +3.36910 q^{34} -0.212347 q^{36} -2.44748 q^{37} +12.3896 q^{38} +7.41855 q^{39} -1.53919 q^{40} -5.60197 q^{41} +12.5464 q^{43} -1.70928 q^{44} +0.0783777 q^{45} +14.3896 q^{46} -2.29072 q^{47} -3.55252 q^{48} +2.17009 q^{50} +2.65368 q^{51} +11.7587 q^{52} +0.921622 q^{53} -11.4186 q^{54} +0.630898 q^{55} +9.75872 q^{57} -2.17009 q^{58} +3.60197 q^{59} -4.63090 q^{60} +13.0205 q^{61} +4.97107 q^{62} -12.3112 q^{64} -4.34017 q^{65} -2.34017 q^{66} +10.6309 q^{67} +4.20620 q^{68} +11.3340 q^{69} +15.6020 q^{71} -0.120638 q^{72} +10.9444 q^{73} -5.31124 q^{74} +1.70928 q^{75} +15.4680 q^{76} +16.0989 q^{78} -10.2062 q^{79} +2.07838 q^{80} -8.75872 q^{81} -12.1568 q^{82} +3.12783 q^{83} -1.55252 q^{85} +27.2267 q^{86} -1.70928 q^{87} -0.971071 q^{88} -1.41855 q^{89} +0.170086 q^{90} +17.9649 q^{92} +3.91548 q^{93} -4.97107 q^{94} -5.70928 q^{95} -12.9711 q^{96} -13.4680 q^{97} +0.0494483 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - 2 q^{3} + q^{4} - 3 q^{5} + 4 q^{6} + 3 q^{8} + 3 q^{9} - q^{10} + 2 q^{11} + 10 q^{12} + 2 q^{13} + 2 q^{15} - 3 q^{16} + 4 q^{17} + 5 q^{18} + 10 q^{19} - q^{20} - 8 q^{22} + 16 q^{23}+ \cdots - 18 q^{99}+O(q^{100}) $ 3 * q + q^2 - 2 * q^3 + q^4 - 3 * q^5 + 4 * q^6 + 3 * q^8 + 3 * q^9 - q^10 + 2 * q^11 + 10 * q^12 + 2 * q^13 + 2 * q^15 - 3 * q^16 + 4 * q^17 + 5 * q^18 + 10 * q^19 - q^20 - 8 * q^22 + 16 * q^23 + 4 * q^24 + 3 * q^25 + 14 * q^26 - 8 * q^27 - 3 * q^29 - 4 * q^30 + 14 * q^31 - 3 * q^32 + 14 * q^34 - 11 * q^36 - 8 * q^37 + 8 * q^38 + 8 * q^39 - 3 * q^40 + 2 * q^41 + 2 * q^43 + 2 * q^44 - 3 * q^45 + 14 * q^46 - 14 * q^47 - 10 * q^48 + q^50 - 16 * q^51 + 10 * q^52 + 6 * q^53 - 20 * q^54 - 2 * q^55 + 4 * q^57 - q^58 - 8 * q^59 - 10 * q^60 + 6 * q^61 - 11 * q^64 - 2 * q^65 + 4 * q^66 + 28 * q^67 - 12 * q^68 - 12 * q^69 + 28 * q^71 - 13 * q^72 + 16 * q^73 + 10 * q^74 - 2 * q^75 + 14 * q^76 + 12 * q^78 - 6 * q^79 + 3 * q^80 - q^81 - 30 * q^82 - 12 * q^83 - 4 * q^85 + 24 * q^86 + 2 * q^87 + 12 * q^88 + 10 * q^89 - 5 * q^90 + 4 * q^92 - 20 * q^93 - 10 * q^95 - 24 * q^96 - 8 * q^97 - 18 * q^99

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 $

$2$	2.17009	1.53448	0.767241	−	0.641358i	$-0.221629\pi$
$2$	2.17009	1.53448	0.767241	+	0.641358i	$0.221629\pi$
$3$	1.70928	0.986851	0.493425	−	0.869788i	$-0.335745\pi$
$3$	1.70928	0.986851	0.493425	+	0.869788i	$0.335745\pi$
$4$	2.70928	1.35464
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	3.70928	1.51431
$7$	0	0
$7$	0	0
$8$	1.53919	0.544185
$9$	−0.0783777	−0.0261259
$10$	−2.17009	−0.686242
$11$	−0.630898	−0.190223	−0.0951114	−	0.995467i	$-0.530321\pi$
$11$	−0.630898	−0.190223	−0.0951114	+	0.995467i	$0.530321\pi$
$12$	4.63090	1.33682
$13$	4.34017	1.20375	0.601874	−	0.798591i	$-0.294421\pi$
$13$	4.34017	1.20375	0.601874	+	0.798591i	$0.294421\pi$
$14$	0	0
$15$	−1.70928	−0.441333
$16$	−2.07838	−0.519594
$17$	1.55252	0.376541	0.188271	−	0.982117i	$-0.439712\pi$
$17$	1.55252	0.376541	0.188271	+	0.982117i	$0.439712\pi$
$18$	−0.170086	−0.0400898
$19$	5.70928	1.30980	0.654899	−	0.755717i	$-0.272711\pi$
$19$	5.70928	1.30980	0.654899	+	0.755717i	$0.272711\pi$
$20$	−2.70928	−0.605812
$21$	0	0
$22$	−1.36910	−0.291894
$23$	6.63090	1.38264	0.691319	−	0.722550i	$-0.257030\pi$
$23$	6.63090	1.38264	0.691319	+	0.722550i	$0.257030\pi$
$24$	2.63090	0.537030
$25$	1.00000	0.200000
$26$	9.41855	1.84713
$27$	−5.26180	−1.01263
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−3.70928	−0.677218
$31$	2.29072	0.411426	0.205713	−	0.978612i	$-0.434049\pi$
$31$	2.29072	0.411426	0.205713	+	0.978612i	$0.434049\pi$
$32$	−7.58864	−1.34149
$33$	−1.07838	−0.187721
$34$	3.36910	0.577796
$35$	0	0
$36$	−0.212347	−0.0353911
$37$	−2.44748	−0.402363	−0.201182	−	0.979554i	$-0.564478\pi$
$37$	−2.44748	−0.402363	−0.201182	+	0.979554i	$0.564478\pi$
$38$	12.3896	2.00986
$39$	7.41855	1.18792
$40$	−1.53919	−0.243367
$41$	−5.60197	−0.874880	−0.437440	−	0.899247i	$-0.644115\pi$
$41$	−5.60197	−0.874880	−0.437440	+	0.899247i	$0.644115\pi$
$42$	0	0
$43$	12.5464	1.91330	0.956652	−	0.291233i	$-0.0940654\pi$
$43$	12.5464	1.91330	0.956652	+	0.291233i	$0.0940654\pi$
$44$	−1.70928	−0.257683
$45$	0.0783777	0.0116839
$46$	14.3896	2.12163
$47$	−2.29072	−0.334137	−0.167068	−	0.985945i	$-0.553430\pi$
$47$	−2.29072	−0.334137	−0.167068	+	0.985945i	$0.553430\pi$
$48$	−3.55252	−0.512762
$49$	0	0
$50$	2.17009	0.306897
$51$	2.65368	0.371590
$52$	11.7587	1.63064
$53$	0.921622	0.126595	0.0632973	−	0.997995i	$-0.479838\pi$
$53$	0.921622	0.126595	0.0632973	+	0.997995i	$0.479838\pi$
$54$	−11.4186	−1.55387
$55$	0.630898	0.0850702
$56$	0	0
$57$	9.75872	1.29257
$58$	−2.17009	−0.284946
$59$	3.60197	0.468936	0.234468	−	0.972124i	$-0.424665\pi$
$59$	3.60197	0.468936	0.234468	+	0.972124i	$0.424665\pi$
$60$	−4.63090	−0.597846
$61$	13.0205	1.66711	0.833553	−	0.552439i	$-0.186303\pi$
$61$	13.0205	1.66711	0.833553	+	0.552439i	$0.186303\pi$
$62$	4.97107	0.631327
$63$	0	0
$64$	−12.3112	−1.53891
$65$	−4.34017	−0.538332
$66$	−2.34017	−0.288055
$67$	10.6309	1.29877	0.649385	−	0.760459i	$-0.275026\pi$
$67$	10.6309	1.29877	0.649385	+	0.760459i	$0.275026\pi$
$68$	4.20620	0.510077
$69$	11.3340	1.36446
$70$	0	0
$71$	15.6020	1.85161	0.925806	−	0.377998i	$-0.123387\pi$
$71$	15.6020	1.85161	0.925806	+	0.377998i	$0.123387\pi$
$72$	−0.120638	−0.0142173
$73$	10.9444	1.28095	0.640473	−	0.767981i	$-0.278738\pi$
$73$	10.9444	1.28095	0.640473	+	0.767981i	$0.278738\pi$
$74$	−5.31124	−0.617420
$75$	1.70928	0.197370
$76$	15.4680	1.77430
$77$	0	0
$78$	16.0989	1.82284
$79$	−10.2062	−1.14829	−0.574144	−	0.818754i	$-0.694665\pi$
$79$	−10.2062	−1.14829	−0.574144	+	0.818754i	$0.694665\pi$
$80$	2.07838	0.232370
$81$	−8.75872	−0.973192
$82$	−12.1568	−1.34249
$83$	3.12783	0.343324	0.171662	−	0.985156i	$-0.445086\pi$
$83$	3.12783	0.343324	0.171662	+	0.985156i	$0.445086\pi$
$84$	0	0
$85$	−1.55252	−0.168394
$86$	27.2267	2.93593
$87$	−1.70928	−0.183254
$88$	−0.971071	−0.103516
$89$	−1.41855	−0.150366	−0.0751830	−	0.997170i	$-0.523954\pi$
$89$	−1.41855	−0.150366	−0.0751830	+	0.997170i	$0.523954\pi$
$90$	0.170086	0.0179287
$91$	0	0
$92$	17.9649	1.87297
$93$	3.91548	0.406016
$94$	−4.97107	−0.512727
$95$	−5.70928	−0.585759
$96$	−12.9711	−1.32385
$97$	−13.4680	−1.36747	−0.683734	−	0.729731i	$-0.739645\pi$
$97$	−13.4680	−1.36747	−0.683734	+	0.729731i	$0.739645\pi$
$98$	0	0
$99$	0.0494483	0.00496974
$100$	2.70928	0.270928
$101$	−1.10504	−0.109956	−0.0549778	−	0.998488i	$-0.517509\pi$
$101$	−1.10504	−0.109956	−0.0549778	+	0.998488i	$0.517509\pi$
$102$	5.75872	0.570199
$103$	−15.6248	−1.53955	−0.769776	−	0.638314i	$-0.779632\pi$
$103$	−15.6248	−1.53955	−0.769776	+	0.638314i	$0.779632\pi$
$104$	6.68035	0.655062
$105$	0	0
$106$	2.00000	0.194257
$107$	2.81432	0.272070	0.136035	−	0.990704i	$-0.456564\pi$
$107$	2.81432	0.272070	0.136035	+	0.990704i	$0.456564\pi$
$108$	−14.2557	−1.37175
$109$	5.91548	0.566600	0.283300	−	0.959031i	$-0.408571\pi$
$109$	5.91548	0.566600	0.283300	+	0.959031i	$0.408571\pi$
$110$	1.36910	0.130539
$111$	−4.18342	−0.397072
$112$	0	0
$113$	−1.95055	−0.183492	−0.0917462	−	0.995782i	$-0.529245\pi$
$113$	−1.95055	−0.183492	−0.0917462	+	0.995782i	$0.529245\pi$
$114$	21.1773	1.98343
$115$	−6.63090	−0.618334
$116$	−2.70928	−0.251550
$117$	−0.340173	−0.0314490
$118$	7.81658	0.719575
$119$	0	0
$120$	−2.63090	−0.240167
$121$	−10.6020	−0.963815
$122$	28.2557	2.55815
$123$	−9.57531	−0.863376
$124$	6.20620	0.557334
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−22.4885	−1.99553	−0.997767	−	0.0667962i	$-0.978722\pi$
$127$	−22.4885	−1.99553	−0.997767	+	0.0667962i	$0.978722\pi$
$128$	−11.5392	−1.01993
$129$	21.4452	1.88815
$130$	−9.41855	−0.826062
$131$	−3.86603	−0.337777	−0.168888	−	0.985635i	$-0.554018\pi$
$131$	−3.86603	−0.337777	−0.168888	+	0.985635i	$0.554018\pi$
$132$	−2.92162	−0.254295
$133$	0	0
$134$	23.0700	1.99294
$135$	5.26180	0.452863
$136$	2.38962	0.204908
$137$	21.2846	1.81846	0.909232	−	0.416289i	$-0.136670\pi$
$137$	21.2846	1.81846	0.909232	+	0.416289i	$0.136670\pi$
$138$	24.5958	2.09374
$139$	−8.09890	−0.686939	−0.343470	−	0.939164i	$-0.611602\pi$
$139$	−8.09890	−0.686939	−0.343470	+	0.939164i	$0.611602\pi$
$140$	0	0
$141$	−3.91548	−0.329743
$142$	33.8576	2.84127
$143$	−2.73820	−0.228980
$144$	0.162899	0.0135749
$145$	1.00000	0.0830455
$146$	23.7503	1.96559
$147$	0	0
$148$	−6.63090	−0.545056
$149$	12.8371	1.05166	0.525828	−	0.850591i	$-0.323755\pi$
$149$	12.8371	1.05166	0.525828	+	0.850591i	$0.323755\pi$
$150$	3.70928	0.302861
$151$	−20.5958	−1.67606	−0.838032	−	0.545621i	$-0.816294\pi$
$151$	−20.5958	−1.67606	−0.838032	+	0.545621i	$0.816294\pi$
$152$	8.78765	0.712773
$153$	−0.121683	−0.00983749
$154$	0	0
$155$	−2.29072	−0.183995
$156$	20.0989	1.60920
$157$	6.04945	0.482799	0.241399	−	0.970426i	$-0.422394\pi$
$157$	6.04945	0.482799	0.241399	+	0.970426i	$0.422394\pi$
$158$	−22.1483	−1.76203
$159$	1.57531	0.124930
$160$	7.58864	0.599934
$161$	0	0
$162$	−19.0072	−1.49335
$163$	−15.9649	−1.25047	−0.625235	−	0.780437i	$-0.714997\pi$
$163$	−15.9649	−1.25047	−0.625235	+	0.780437i	$0.714997\pi$
$164$	−15.1773	−1.18515
$165$	1.07838	0.0839516
$166$	6.78765	0.526824
$167$	11.3112	0.875290	0.437645	−	0.899148i	$-0.355813\pi$
$167$	11.3112	0.875290	0.437645	+	0.899148i	$0.355813\pi$
$168$	0	0
$169$	5.83710	0.449008
$170$	−3.36910	−0.258398
$171$	−0.447480	−0.0342197
$172$	33.9916	2.59183
$173$	10.4969	0.798067	0.399033	−	0.916936i	$-0.369346\pi$
$173$	10.4969	0.798067	0.399033	+	0.916936i	$0.369346\pi$
$174$	−3.70928	−0.281199
$175$	0	0
$176$	1.31124	0.0988387
$177$	6.15676	0.462770
$178$	−3.07838	−0.230734
$179$	−12.3135	−0.920355	−0.460178	−	0.887827i	$-0.652214\pi$
$179$	−12.3135	−0.920355	−0.460178	+	0.887827i	$0.652214\pi$
$180$	0.212347	0.0158274
$181$	1.60197	0.119073	0.0595367	−	0.998226i	$-0.481038\pi$
$181$	1.60197	0.119073	0.0595367	+	0.998226i	$0.481038\pi$
$182$	0	0
$183$	22.2557	1.64519
$184$	10.2062	0.752411
$185$	2.44748	0.179942
$186$	8.49693	0.623025
$187$	−0.979481	−0.0716268
$188$	−6.20620	−0.452634
$189$	0	0
$190$	−12.3896	−0.898838
$191$	13.6248	0.985853	0.492926	−	0.870071i	$-0.335927\pi$
$191$	13.6248	0.985853	0.492926	+	0.870071i	$0.335927\pi$
$192$	−21.0433	−1.51867
$193$	−16.9711	−1.22160	−0.610802	−	0.791783i	$-0.709153\pi$
$193$	−16.9711	−1.22160	−0.610802	+	0.791783i	$0.709153\pi$
$194$	−29.2267	−2.09836
$195$	−7.41855	−0.531253
$196$	0	0
$197$	−0.0578588	−0.00412227	−0.00206114	−	0.999998i	$-0.500656\pi$
$197$	−0.0578588	−0.00412227	−0.00206114	+	0.999998i	$0.500656\pi$
$198$	0.107307	0.00762599
$199$	−5.39189	−0.382221	−0.191110	−	0.981569i	$-0.561209\pi$
$199$	−5.39189	−0.382221	−0.191110	+	0.981569i	$0.561209\pi$
$200$	1.53919	0.108837
$201$	18.1711	1.28169
$202$	−2.39803	−0.168725
$203$	0	0
$204$	7.18956	0.503370
$205$	5.60197	0.391258
$206$	−33.9071	−2.36242
$207$	−0.519715	−0.0361227
$208$	−9.02052	−0.625460
$209$	−3.60197	−0.249153
$210$	0	0
$211$	4.14834	0.285584	0.142792	−	0.989753i	$-0.454392\pi$
$211$	4.14834	0.285584	0.142792	+	0.989753i	$0.454392\pi$
$212$	2.49693	0.171490
$213$	26.6681	1.82727
$214$	6.10731	0.417487
$215$	−12.5464	−0.855656
$216$	−8.09890	−0.551060
$217$	0	0
$218$	12.8371	0.869438
$219$	18.7070	1.26410
$220$	1.70928	0.115239
$221$	6.73820	0.453261
$222$	−9.07838	−0.609301
$223$	6.72979	0.450660	0.225330	−	0.974282i	$-0.427654\pi$
$223$	6.72979	0.450660	0.225330	+	0.974282i	$0.427654\pi$
$224$	0	0
$225$	−0.0783777	−0.00522518
$226$	−4.23287	−0.281566
$227$	22.2472	1.47660	0.738301	−	0.674472i	$-0.235629\pi$
$227$	22.2472	1.47660	0.738301	+	0.674472i	$0.235629\pi$
$228$	26.4391	1.75097
$229$	7.16290	0.473338	0.236669	−	0.971590i	$-0.423944\pi$
$229$	7.16290	0.473338	0.236669	+	0.971590i	$0.423944\pi$
$230$	−14.3896	−0.948824
$231$	0	0
$232$	−1.53919	−0.101053
$233$	30.1978	1.97832	0.989162	−	0.146831i	$-0.0469073\pi$
$233$	30.1978	1.97832	0.989162	+	0.146831i	$0.0469073\pi$
$234$	−0.738205	−0.0482580
$235$	2.29072	0.149430
$236$	9.75872	0.635239
$237$	−17.4452	−1.13319
$238$	0	0
$239$	−6.43907	−0.416509	−0.208254	−	0.978075i	$-0.566778\pi$
$239$	−6.43907	−0.416509	−0.208254	+	0.978075i	$0.566778\pi$
$240$	3.55252	0.229314
$241$	−10.9939	−0.708177	−0.354088	−	0.935212i	$-0.615209\pi$
$241$	−10.9939	−0.708177	−0.354088	+	0.935212i	$0.615209\pi$
$242$	−23.0072	−1.47896
$243$	0.814315	0.0522383
$244$	35.2762	2.25833
$245$	0	0
$246$	−20.7792	−1.32484
$247$	24.7792	1.57667
$248$	3.52586	0.223892
$249$	5.34632	0.338809
$250$	−2.17009	−0.137248
$251$	−9.41014	−0.593963	−0.296981	−	0.954883i	$-0.595980\pi$
$251$	−9.41014	−0.593963	−0.296981	+	0.954883i	$0.595980\pi$
$252$	0	0
$253$	−4.18342	−0.263009
$254$	−48.8020	−3.06211
$255$	−2.65368	−0.166180
$256$	−0.418551	−0.0261594
$257$	−5.81658	−0.362828	−0.181414	−	0.983407i	$-0.558067\pi$
$257$	−5.81658	−0.362828	−0.181414	+	0.983407i	$0.558067\pi$
$258$	46.5380	2.89733
$259$	0	0
$260$	−11.7587	−0.729245
$261$	0.0783777	0.00485146
$262$	−8.38962	−0.518313
$263$	−8.91321	−0.549612	−0.274806	−	0.961500i	$-0.588614\pi$
$263$	−8.91321	−0.549612	−0.274806	+	0.961500i	$0.588614\pi$
$264$	−1.65983	−0.102155
$265$	−0.921622	−0.0566148
$266$	0	0
$267$	−2.42469	−0.148389
$268$	28.8020	1.75936
$269$	16.4391	1.00231	0.501154	−	0.865358i	$-0.332909\pi$
$269$	16.4391	1.00231	0.501154	+	0.865358i	$0.332909\pi$
$270$	11.4186	0.694911
$271$	−29.4101	−1.78654	−0.893269	−	0.449522i	$-0.851594\pi$
$271$	−29.4101	−1.78654	−0.893269	+	0.449522i	$0.851594\pi$
$272$	−3.22672	−0.195649
$273$	0	0
$274$	46.1894	2.79040
$275$	−0.630898	−0.0380446
$276$	30.7070	1.84834
$277$	11.0784	0.665635	0.332818	−	0.942991i	$-0.392001\pi$
$277$	11.0784	0.665635	0.332818	+	0.942991i	$0.392001\pi$
$278$	−17.5753	−1.05410
$279$	−0.179542	−0.0107489
$280$	0	0
$281$	−21.1194	−1.25988	−0.629939	−	0.776644i	$-0.716920\pi$
$281$	−21.1194	−1.25988	−0.629939	+	0.776644i	$0.716920\pi$
$282$	−8.49693	−0.505985
$283$	13.7815	0.819226	0.409613	−	0.912259i	$-0.365664\pi$
$283$	13.7815	0.819226	0.409613	+	0.912259i	$0.365664\pi$
$284$	42.2700	2.50826
$285$	−9.75872	−0.578057
$286$	−5.94214	−0.351366
$287$	0	0
$288$	0.594780	0.0350478
$289$	−14.5897	−0.858217
$290$	2.17009	0.127432
$291$	−23.0205	−1.34949
$292$	29.6514	1.73522
$293$	−6.14834	−0.359190	−0.179595	−	0.983741i	$-0.557479\pi$
$293$	−6.14834	−0.359190	−0.179595	+	0.983741i	$0.557479\pi$
$294$	0	0
$295$	−3.60197	−0.209715
$296$	−3.76713	−0.218960
$297$	3.31965	0.192626
$298$	27.8576	1.61375
$299$	28.7792	1.66435
$300$	4.63090	0.267365
$301$	0	0
$302$	−44.6947	−2.57189
$303$	−1.88882	−0.108510
$304$	−11.8660	−0.680564
$305$	−13.0205	−0.745553
$306$	−0.264063	−0.0150955
$307$	−10.3896	−0.592967	−0.296484	−	0.955038i	$-0.595814\pi$
$307$	−10.3896	−0.592967	−0.296484	+	0.955038i	$0.595814\pi$
$308$	0	0
$309$	−26.7070	−1.51931
$310$	−4.97107	−0.282338
$311$	−18.7565	−1.06358	−0.531791	−	0.846876i	$-0.678481\pi$
$311$	−18.7565	−1.06358	−0.531791	+	0.846876i	$0.678481\pi$
$312$	11.4186	0.646448
$313$	−12.3402	−0.697508	−0.348754	−	0.937214i	$-0.613395\pi$
$313$	−12.3402	−0.697508	−0.348754	+	0.937214i	$0.613395\pi$
$314$	13.1278	0.740846
$315$	0	0
$316$	−27.6514	−1.55551
$317$	−30.6986	−1.72421	−0.862103	−	0.506734i	$-0.830853\pi$
$317$	−30.6986	−1.72421	−0.862103	+	0.506734i	$0.830853\pi$
$318$	3.41855	0.191703
$319$	0.630898	0.0353235
$320$	12.3112	0.688219
$321$	4.81044	0.268493
$322$	0	0
$323$	8.86376	0.493193
$324$	−23.7298	−1.31832
$325$	4.34017	0.240749
$326$	−34.6453	−1.91882
$327$	10.1112	0.559150
$328$	−8.62249	−0.476097
$329$	0	0
$330$	2.34017	0.128822
$331$	−4.08065	−0.224293	−0.112146	−	0.993692i	$-0.535773\pi$
$331$	−4.08065	−0.224293	−0.112146	+	0.993692i	$0.535773\pi$
$332$	8.47414	0.465079
$333$	0.191828	0.0105121
$334$	24.5464	1.34312
$335$	−10.6309	−0.580828
$336$	0	0
$337$	−18.3630	−1.00029	−0.500147	−	0.865940i	$-0.666721\pi$
$337$	−18.3630	−1.00029	−0.500147	+	0.865940i	$0.666721\pi$
$338$	12.6670	0.688995
$339$	−3.33403	−0.181080
$340$	−4.20620	−0.228113
$341$	−1.44521	−0.0782627
$342$	−0.971071	−0.0525095
$343$	0	0
$344$	19.3112	1.04119
$345$	−11.3340	−0.610204
$346$	22.7792	1.22462
$347$	8.97107	0.481592	0.240796	−	0.970576i	$-0.422591\pi$
$347$	8.97107	0.481592	0.240796	+	0.970576i	$0.422591\pi$
$348$	−4.63090	−0.248242
$349$	−26.1978	−1.40234	−0.701168	−	0.712996i	$-0.747338\pi$
$349$	−26.1978	−1.40234	−0.701168	+	0.712996i	$0.747338\pi$
$350$	0	0
$351$	−22.8371	−1.21895
$352$	4.78765	0.255183
$353$	26.2823	1.39887	0.699433	−	0.714698i	$-0.253436\pi$
$353$	26.2823	1.39887	0.699433	+	0.714698i	$0.253436\pi$
$354$	13.3607	0.710113
$355$	−15.6020	−0.828066
$356$	−3.84324	−0.203692
$357$	0	0
$358$	−26.7214	−1.41227
$359$	22.8722	1.20715	0.603574	−	0.797307i	$-0.293743\pi$
$359$	22.8722	1.20715	0.603574	+	0.797307i	$0.293743\pi$
$360$	0.120638	0.00635819
$361$	13.5958	0.715570
$362$	3.47641	0.182716
$363$	−18.1217	−0.951142
$364$	0	0
$365$	−10.9444	−0.572857
$366$	48.2967	2.52451
$367$	11.0700	0.577848	0.288924	−	0.957352i	$-0.406703\pi$
$367$	11.0700	0.577848	0.288924	+	0.957352i	$0.406703\pi$
$368$	−13.7815	−0.718411
$369$	0.439070	0.0228571
$370$	5.31124	0.276118
$371$	0	0
$372$	10.6081	0.550005
$373$	11.5753	0.599347	0.299673	−	0.954042i	$-0.403122\pi$
$373$	11.5753	0.599347	0.299673	+	0.954042i	$0.403122\pi$
$374$	−2.12556	−0.109910
$375$	−1.70928	−0.0882666
$376$	−3.52586	−0.181832
$377$	−4.34017	−0.223530
$378$	0	0
$379$	−9.31124	−0.478286	−0.239143	−	0.970984i	$-0.576867\pi$
$379$	−9.31124	−0.478286	−0.239143	+	0.970984i	$0.576867\pi$
$380$	−15.4680	−0.793492
$381$	−38.4391	−1.96929
$382$	29.5669	1.51277
$383$	−33.9649	−1.73553	−0.867763	−	0.496978i	$-0.834443\pi$
$383$	−33.9649	−1.73553	−0.867763	+	0.496978i	$0.834443\pi$
$384$	−19.7237	−1.00652
$385$	0	0
$386$	−36.8287	−1.87453
$387$	−0.983357	−0.0499868
$388$	−36.4885	−1.85242
$389$	−4.12556	−0.209174	−0.104587	−	0.994516i	$-0.533352\pi$
$389$	−4.12556	−0.209174	−0.104587	+	0.994516i	$0.533352\pi$
$390$	−16.0989	−0.815199
$391$	10.2946	0.520620
$392$	0	0
$393$	−6.60811	−0.333335
$394$	−0.125559	−0.00632555
$395$	10.2062	0.513530
$396$	0.133969	0.00673220
$397$	−17.1050	−0.858477	−0.429239	−	0.903191i	$-0.641218\pi$
$397$	−17.1050	−0.858477	−0.429239	+	0.903191i	$0.641218\pi$
$398$	−11.7009	−0.586511
$399$	0	0
$400$	−2.07838	−0.103919
$401$	−0.554787	−0.0277048	−0.0138524	−	0.999904i	$-0.504409\pi$
$401$	−0.554787	−0.0277048	−0.0138524	+	0.999904i	$0.504409\pi$
$402$	39.4329	1.96674
$403$	9.94214	0.495253
$404$	−2.99386	−0.148950
$405$	8.75872	0.435224
$406$	0	0
$407$	1.54411	0.0765387
$408$	4.08452	0.202214
$409$	20.6537	1.02126	0.510629	−	0.859801i	$-0.329412\pi$
$409$	20.6537	1.02126	0.510629	+	0.859801i	$0.329412\pi$
$410$	12.1568	0.600379
$411$	36.3812	1.79455
$412$	−42.3318	−2.08554
$413$	0	0
$414$	−1.12783	−0.0554296
$415$	−3.12783	−0.153539
$416$	−32.9360	−1.61482
$417$	−13.8432	−0.677907
$418$	−7.81658	−0.382322
$419$	6.02666	0.294422	0.147211	−	0.989105i	$-0.452970\pi$
$419$	6.02666	0.294422	0.147211	+	0.989105i	$0.452970\pi$
$420$	0	0
$421$	−12.5380	−0.611063	−0.305532	−	0.952182i	$-0.598834\pi$
$421$	−12.5380	−0.611063	−0.305532	+	0.952182i	$0.598834\pi$
$422$	9.00227	0.438224
$423$	0.179542	0.00872962
$424$	1.41855	0.0688909
$425$	1.55252	0.0753083
$426$	57.8720	2.80391
$427$	0	0
$428$	7.62475	0.368556
$429$	−4.68035	−0.225969
$430$	−27.2267	−1.31299
$431$	−18.0410	−0.869006	−0.434503	−	0.900670i	$-0.643076\pi$
$431$	−18.0410	−0.869006	−0.434503	+	0.900670i	$0.643076\pi$
$432$	10.9360	0.526158
$433$	−18.8143	−0.904158	−0.452079	−	0.891978i	$-0.649318\pi$
$433$	−18.8143	−0.904158	−0.452079	+	0.891978i	$0.649318\pi$
$434$	0	0
$435$	1.70928	0.0819535
$436$	16.0267	0.767538
$437$	37.8576	1.81098
$438$	40.5958	1.93974
$439$	−5.54411	−0.264606	−0.132303	−	0.991209i	$-0.542237\pi$
$439$	−5.54411	−0.264606	−0.132303	+	0.991209i	$0.542237\pi$
$440$	0.971071	0.0462940
$441$	0	0
$442$	14.6225	0.695521
$443$	17.8082	0.846092	0.423046	−	0.906108i	$-0.360961\pi$
$443$	17.8082	0.846092	0.423046	+	0.906108i	$0.360961\pi$
$444$	−11.3340	−0.537889
$445$	1.41855	0.0672458
$446$	14.6042	0.691531
$447$	21.9421	1.03783
$448$	0	0
$449$	−10.6947	−0.504715	−0.252358	−	0.967634i	$-0.581206\pi$
$449$	−10.6947	−0.504715	−0.252358	+	0.967634i	$0.581206\pi$
$450$	−0.170086	−0.00801795
$451$	3.53427	0.166422
$452$	−5.28458	−0.248566
$453$	−35.2039	−1.65403
$454$	48.2784	2.26582
$455$	0	0
$456$	15.0205	0.703400
$457$	−21.7998	−1.01975	−0.509875	−	0.860249i	$-0.670308\pi$
$457$	−21.7998	−1.01975	−0.509875	+	0.860249i	$0.670308\pi$
$458$	15.5441	0.726329
$459$	−8.16904	−0.381298
$460$	−17.9649	−0.837619
$461$	−22.4124	−1.04385	−0.521925	−	0.852991i	$-0.674786\pi$
$461$	−22.4124	−1.04385	−0.521925	+	0.852991i	$0.674786\pi$
$462$	0	0
$463$	−2.10277	−0.0977241	−0.0488621	−	0.998806i	$-0.515559\pi$
$463$	−2.10277	−0.0977241	−0.0488621	+	0.998806i	$0.515559\pi$
$464$	2.07838	0.0964863
$465$	−3.91548	−0.181576
$466$	65.5318	3.03570
$467$	18.6042	0.860901	0.430451	−	0.902614i	$-0.358355\pi$
$467$	18.6042	0.860901	0.430451	+	0.902614i	$0.358355\pi$
$468$	−0.921622	−0.0426020
$469$	0	0
$470$	4.97107	0.229298
$471$	10.3402	0.476450
$472$	5.54411	0.255188
$473$	−7.91548	−0.363954
$474$	−37.8576	−1.73886
$475$	5.70928	0.261960
$476$	0	0
$477$	−0.0722347	−0.00330740
$478$	−13.9733	−0.639126
$479$	−8.89884	−0.406598	−0.203299	−	0.979117i	$-0.565166\pi$
$479$	−8.89884	−0.406598	−0.203299	+	0.979117i	$0.565166\pi$
$480$	12.9711	0.592046
$481$	−10.6225	−0.484344
$482$	−23.8576	−1.08668
$483$	0	0
$484$	−28.7237	−1.30562
$485$	13.4680	0.611550
$486$	1.76713	0.0801588
$487$	−12.9711	−0.587775	−0.293888	−	0.955840i	$-0.594949\pi$
$487$	−12.9711	−0.587775	−0.293888	+	0.955840i	$0.594949\pi$
$488$	20.0410	0.907215
$489$	−27.2885	−1.23403
$490$	0	0
$491$	−13.8615	−0.625561	−0.312780	−	0.949826i	$-0.601260\pi$
$491$	−13.8615	−0.625561	−0.312780	+	0.949826i	$0.601260\pi$
$492$	−25.9421	−1.16956
$493$	−1.55252	−0.0699220
$494$	53.7731	2.41937
$495$	−0.0494483	−0.00222254
$496$	−4.76099	−0.213775
$497$	0	0
$498$	11.6020	0.519897
$499$	−22.3545	−1.00073	−0.500364	−	0.865815i	$-0.666800\pi$
$499$	−22.3545	−1.00073	−0.500364	+	0.865815i	$0.666800\pi$
$500$	−2.70928	−0.121162
$501$	19.3340	0.863781
$502$	−20.4208	−0.911426
$503$	−34.4885	−1.53777	−0.768884	−	0.639389i	$-0.779187\pi$
$503$	−34.4885	−1.53777	−0.768884	+	0.639389i	$0.779187\pi$
$504$	0	0
$505$	1.10504	0.0491736
$506$	−9.07838	−0.403583
$507$	9.97721	0.443104
$508$	−60.9276	−2.70322
$509$	28.7526	1.27444	0.637218	−	0.770684i	$-0.280085\pi$
$509$	28.7526	1.27444	0.637218	+	0.770684i	$0.280085\pi$
$510$	−5.75872	−0.255001
$511$	0	0
$512$	22.1701	0.979789
$513$	−30.0410	−1.32634
$514$	−12.6225	−0.556754
$515$	15.6248	0.688509
$516$	58.1010	2.55775
$517$	1.44521	0.0635604
$518$	0	0
$519$	17.9421	0.787573
$520$	−6.68035	−0.292953
$521$	21.6020	0.946399	0.473200	−	0.880955i	$-0.343099\pi$
$521$	21.6020	0.946399	0.473200	+	0.880955i	$0.343099\pi$
$522$	0.170086	0.00744448
$523$	−4.20620	−0.183924	−0.0919622	−	0.995762i	$-0.529314\pi$
$523$	−4.20620	−0.183924	−0.0919622	+	0.995762i	$0.529314\pi$
$524$	−10.4741	−0.457565
$525$	0	0
$526$	−19.3424	−0.843370
$527$	3.55640	0.154919
$528$	2.24128	0.0975390
$529$	20.9688	0.911687
$530$	−2.00000	−0.0868744
$531$	−0.282314	−0.0122514
$532$	0	0
$533$	−24.3135	−1.05314
$534$	−5.26180	−0.227700
$535$	−2.81432	−0.121673
$536$	16.3630	0.706772
$537$	−21.0472	−0.908253
$538$	35.6742	1.53802
$539$	0	0
$540$	14.2557	0.613466
$541$	26.7792	1.15133	0.575665	−	0.817686i	$-0.304743\pi$
$541$	26.7792	1.15133	0.575665	+	0.817686i	$0.304743\pi$
$542$	−63.8225	−2.74141
$543$	2.73820	0.117508
$544$	−11.7815	−0.505128
$545$	−5.91548	−0.253391
$546$	0	0
$547$	2.33176	0.0996990	0.0498495	−	0.998757i	$-0.484126\pi$
$547$	2.33176	0.0996990	0.0498495	+	0.998757i	$0.484126\pi$
$548$	57.6658	2.46336
$549$	−1.02052	−0.0435547
$550$	−1.36910	−0.0583787
$551$	−5.70928	−0.243223
$552$	17.4452	0.742518
$553$	0	0
$554$	24.0410	1.02141
$555$	4.18342	0.177576
$556$	−21.9421	−0.930554
$557$	30.8781	1.30835	0.654174	−	0.756344i	$-0.273016\pi$
$557$	30.8781	1.30835	0.654174	+	0.756344i	$0.273016\pi$
$558$	−0.389621	−0.0164940
$559$	54.4534	2.30314
$560$	0	0
$561$	−1.67420	−0.0706849
$562$	−45.8310	−1.93326
$563$	34.1750	1.44030	0.720152	−	0.693816i	$-0.244072\pi$
$563$	34.1750	1.44030	0.720152	+	0.693816i	$0.244072\pi$
$564$	−10.6081	−0.446682
$565$	1.95055	0.0820603
$566$	29.9071	1.25709
$567$	0	0
$568$	24.0144	1.00762
$569$	−30.6947	−1.28679	−0.643395	−	0.765535i	$-0.722475\pi$
$569$	−30.6947	−1.28679	−0.643395	+	0.765535i	$0.722475\pi$
$570$	−21.1773	−0.887018
$571$	−25.7275	−1.07666	−0.538332	−	0.842733i	$-0.680945\pi$
$571$	−25.7275	−1.07666	−0.538332	+	0.842733i	$0.680945\pi$
$572$	−7.41855	−0.310185
$573$	23.2885	0.972889
$574$	0	0
$575$	6.63090	0.276528
$576$	0.964928	0.0402053
$577$	−16.0228	−0.667037	−0.333519	−	0.942743i	$-0.608236\pi$
$577$	−16.0228	−0.667037	−0.333519	+	0.942743i	$0.608236\pi$
$578$	−31.6609	−1.31692
$579$	−29.0082	−1.20554
$580$	2.70928	0.112497
$581$	0	0
$582$	−49.9565	−2.07076
$583$	−0.581449	−0.0240812
$584$	16.8455	0.697072
$585$	0.340173	0.0140644
$586$	−13.3424	−0.551171
$587$	−19.6248	−0.810000	−0.405000	−	0.914317i	$-0.632729\pi$
$587$	−19.6248	−0.810000	−0.405000	+	0.914317i	$0.632729\pi$
$588$	0	0
$589$	13.0784	0.538885
$590$	−7.81658	−0.321804
$591$	−0.0988967	−0.00406807
$592$	5.08679	0.209066
$593$	−30.9627	−1.27148	−0.635742	−	0.771902i	$-0.719306\pi$
$593$	−30.9627	−1.27148	−0.635742	+	0.771902i	$0.719306\pi$
$594$	7.20394	0.295581
$595$	0	0
$596$	34.7792	1.42461
$597$	−9.21622	−0.377195
$598$	62.4534	2.55391
$599$	24.4619	0.999484	0.499742	−	0.866174i	$-0.333428\pi$
$599$	24.4619	0.999484	0.499742	+	0.866174i	$0.333428\pi$
$600$	2.63090	0.107406
$601$	−16.3857	−0.668389	−0.334194	−	0.942504i	$-0.608464\pi$
$601$	−16.3857	−0.668389	−0.334194	+	0.942504i	$0.608464\pi$
$602$	0	0
$603$	−0.833226	−0.0339316
$604$	−55.7998	−2.27046
$605$	10.6020	0.431031
$606$	−4.09890	−0.166506
$607$	−16.6986	−0.677775	−0.338888	−	0.940827i	$-0.610051\pi$
$607$	−16.6986	−0.677775	−0.338888	+	0.940827i	$0.610051\pi$
$608$	−43.3256	−1.75709
$609$	0	0
$610$	−28.2557	−1.14404
$611$	−9.94214	−0.402216
$612$	−0.329673	−0.0133262
$613$	5.83096	0.235510	0.117755	−	0.993043i	$-0.462430\pi$
$613$	5.83096	0.235510	0.117755	+	0.993043i	$0.462430\pi$
$614$	−22.5464	−0.909898
$615$	9.57531	0.386114
$616$	0	0
$617$	−11.7237	−0.471976	−0.235988	−	0.971756i	$-0.575833\pi$
$617$	−11.7237	−0.471976	−0.235988	+	0.971756i	$0.575833\pi$
$618$	−57.9565	−2.33135
$619$	8.41628	0.338279	0.169139	−	0.985592i	$-0.445901\pi$
$619$	8.41628	0.338279	0.169139	+	0.985592i	$0.445901\pi$
$620$	−6.20620	−0.249247
$621$	−34.8904	−1.40010
$622$	−40.7031	−1.63205
$623$	0	0
$624$	−15.4186	−0.617236
$625$	1.00000	0.0400000
$626$	−26.7792	−1.07031
$627$	−6.15676	−0.245877
$628$	16.3896	0.654017
$629$	−3.79976	−0.151506
$630$	0	0
$631$	−12.7792	−0.508734	−0.254367	−	0.967108i	$-0.581867\pi$
$631$	−12.7792	−0.508734	−0.254367	+	0.967108i	$0.581867\pi$
$632$	−15.7093	−0.624881
$633$	7.09066	0.281829
$634$	−66.6186	−2.64576
$635$	22.4885	0.892430
$636$	4.26794	0.169235
$637$	0	0
$638$	1.36910	0.0542033
$639$	−1.22285	−0.0483751
$640$	11.5392	0.456126
$641$	−0.0722347	−0.00285310	−0.00142655	−	0.999999i	$-0.500454\pi$
$641$	−0.0722347	−0.00285310	−0.00142655	+	0.999999i	$0.500454\pi$
$642$	10.4391	0.411997
$643$	−32.7175	−1.29025	−0.645126	−	0.764076i	$-0.723195\pi$
$643$	−32.7175	−1.29025	−0.645126	+	0.764076i	$0.723195\pi$
$644$	0	0
$645$	−21.4452	−0.844404
$646$	19.2351	0.756796
$647$	15.8082	0.621483	0.310742	−	0.950494i	$-0.399423\pi$
$647$	15.8082	0.621483	0.310742	+	0.950494i	$0.399423\pi$
$648$	−13.4813	−0.529597
$649$	−2.27247	−0.0892024
$650$	9.41855	0.369426
$651$	0	0
$652$	−43.2534	−1.69393
$653$	−15.3112	−0.599175	−0.299588	−	0.954069i	$-0.596849\pi$
$653$	−15.3112	−0.599175	−0.299588	+	0.954069i	$0.596849\pi$
$654$	21.9421	0.858006
$655$	3.86603	0.151058
$656$	11.6430	0.454583
$657$	−0.857798	−0.0334659
$658$	0	0
$659$	17.1278	0.667205	0.333603	−	0.942714i	$-0.391736\pi$
$659$	17.1278	0.667205	0.333603	+	0.942714i	$0.391736\pi$
$660$	2.92162	0.113724
$661$	26.2290	1.02019	0.510095	−	0.860118i	$-0.329610\pi$
$661$	26.2290	1.02019	0.510095	+	0.860118i	$0.329610\pi$
$662$	−8.85535	−0.344173
$663$	11.5174	0.447301
$664$	4.81432	0.186832
$665$	0	0
$666$	0.416283	0.0161307
$667$	−6.63090	−0.256749
$668$	30.6453	1.18570
$669$	11.5031	0.444734
$670$	−23.0700	−0.891271
$671$	−8.21461	−0.317122
$672$	0	0
$673$	46.4657	1.79112	0.895561	−	0.444938i	$-0.146774\pi$
$673$	46.4657	1.79112	0.895561	+	0.444938i	$0.146774\pi$
$674$	−39.8492	−1.53493
$675$	−5.26180	−0.202527
$676$	15.8143	0.608243
$677$	27.8394	1.06995	0.534977	−	0.844867i	$-0.320320\pi$
$677$	27.8394	1.06995	0.534977	+	0.844867i	$0.320320\pi$
$678$	−7.23513	−0.277864
$679$	0	0
$680$	−2.38962	−0.0916378
$681$	38.0267	1.45718
$682$	−3.13624	−0.120093
$683$	39.0966	1.49599	0.747995	−	0.663704i	$-0.231016\pi$
$683$	39.0966	1.49599	0.747995	+	0.663704i	$0.231016\pi$
$684$	−1.21235	−0.0463552
$685$	−21.2846	−0.813242
$686$	0	0
$687$	12.2434	0.467114
$688$	−26.0761	−0.994142
$689$	4.00000	0.152388
$690$	−24.5958	−0.936347
$691$	−24.7480	−0.941460	−0.470730	−	0.882277i	$-0.656009\pi$
$691$	−24.7480	−0.941460	−0.470730	+	0.882277i	$0.656009\pi$
$692$	28.4391	1.08109
$693$	0	0
$694$	19.4680	0.738995
$695$	8.09890	0.307209
$696$	−2.63090	−0.0997239
$697$	−8.69717	−0.329429
$698$	−56.8515	−2.15186
$699$	51.6163	1.95231
$700$	0	0
$701$	0.187952	0.00709886	0.00354943	−	0.999994i	$-0.498870\pi$
$701$	0.187952	0.00709886	0.00354943	+	0.999994i	$0.498870\pi$
$702$	−49.5585	−1.87046
$703$	−13.9733	−0.527014
$704$	7.76713	0.292735
$705$	3.91548	0.147465
$706$	57.0349	2.14654
$707$	0	0
$708$	16.6803	0.626886
$709$	13.6020	0.510833	0.255416	−	0.966831i	$-0.417787\pi$
$709$	13.6020	0.510833	0.255416	+	0.966831i	$0.417787\pi$
$710$	−33.8576	−1.27065
$711$	0.799939	0.0300001
$712$	−2.18342	−0.0818270
$713$	15.1896	0.568854
$714$	0	0
$715$	2.73820	0.102403
$716$	−33.3607	−1.24675
$717$	−11.0061	−0.411032
$718$	49.6346	1.85235
$719$	9.27617	0.345943	0.172971	−	0.984927i	$-0.444663\pi$
$719$	9.27617	0.345943	0.172971	+	0.984927i	$0.444663\pi$
$720$	−0.162899	−0.00607087
$721$	0	0
$722$	29.5041	1.09803
$723$	−18.7915	−0.698864
$724$	4.34017	0.161301
$725$	−1.00000	−0.0371391
$726$	−39.3256	−1.45951
$727$	−29.0121	−1.07600	−0.538000	−	0.842945i	$-0.680820\pi$
$727$	−29.0121	−1.07600	−0.538000	+	0.842945i	$0.680820\pi$
$728$	0	0
$729$	27.6681	1.02474
$730$	−23.7503	−0.879039
$731$	19.4785	0.720438
$732$	60.2967	2.22863
$733$	34.0638	1.25818	0.629088	−	0.777334i	$-0.283428\pi$
$733$	34.0638	1.25818	0.629088	+	0.777334i	$0.283428\pi$
$734$	24.0228	0.886697
$735$	0	0
$736$	−50.3195	−1.85480
$737$	−6.70701	−0.247056
$738$	0.952819	0.0350738
$739$	1.49466	0.0549820	0.0274910	−	0.999622i	$-0.491248\pi$
$739$	1.49466	0.0549820	0.0274910	+	0.999622i	$0.491248\pi$
$740$	6.63090	0.243757
$741$	42.3545	1.55593
$742$	0	0
$743$	17.8082	0.653318	0.326659	−	0.945142i	$-0.394077\pi$
$743$	17.8082	0.653318	0.326659	+	0.945142i	$0.394077\pi$
$744$	6.02666	0.220948
$745$	−12.8371	−0.470315
$746$	25.1194	0.919687
$747$	−0.245152	−0.00896964
$748$	−2.65368	−0.0970283
$749$	0	0
$750$	−3.70928	−0.135444
$751$	−23.7503	−0.866661	−0.433331	−	0.901235i	$-0.642662\pi$
$751$	−23.7503	−0.866661	−0.433331	+	0.901235i	$0.642662\pi$
$752$	4.76099	0.173615
$753$	−16.0845	−0.586153
$754$	−9.41855	−0.343003
$755$	20.5958	0.749559
$756$	0	0
$757$	26.1939	0.952034	0.476017	−	0.879436i	$-0.342080\pi$
$757$	26.1939	0.952034	0.476017	+	0.879436i	$0.342080\pi$
$758$	−20.2062	−0.733922
$759$	−7.15061	−0.259551
$760$	−8.78765	−0.318762
$761$	44.7214	1.62115	0.810574	−	0.585636i	$-0.199155\pi$
$761$	44.7214	1.62115	0.810574	+	0.585636i	$0.199155\pi$
$762$	−83.4161	−3.02185
$763$	0	0
$764$	36.9132	1.33547
$765$	0.121683	0.00439946
$766$	−73.7068	−2.66314
$767$	15.6332	0.564481
$768$	−0.715418	−0.0258154
$769$	−10.8950	−0.392882	−0.196441	−	0.980516i	$-0.562938\pi$
$769$	−10.8950	−0.392882	−0.196441	+	0.980516i	$0.562938\pi$
$770$	0	0
$771$	−9.94214	−0.358057
$772$	−45.9793	−1.65483
$773$	−34.1171	−1.22711	−0.613554	−	0.789653i	$-0.710261\pi$
$773$	−34.1171	−1.22711	−0.613554	+	0.789653i	$0.710261\pi$
$774$	−2.13397	−0.0767039
$775$	2.29072	0.0822853
$776$	−20.7298	−0.744156
$777$	0	0
$778$	−8.95282	−0.320974
$779$	−31.9832	−1.14592
$780$	−20.0989	−0.719656
$781$	−9.84324	−0.352219
$782$	22.3402	0.798883
$783$	5.26180	0.188041
$784$	0	0
$785$	−6.04945	−0.215914
$786$	−14.3402	−0.511497
$787$	−39.7548	−1.41711	−0.708554	−	0.705657i	$-0.750652\pi$
$787$	−39.7548	−1.41711	−0.708554	+	0.705657i	$0.750652\pi$
$788$	−0.156755	−0.00558418
$789$	−15.2351	−0.542385
$790$	22.1483	0.788003
$791$	0	0
$792$	0.0761103	0.00270446
$793$	56.5113	2.00678
$794$	−37.1194	−1.31732
$795$	−1.57531	−0.0558704
$796$	−14.6081	−0.517771
$797$	−18.7298	−0.663443	−0.331722	−	0.943377i	$-0.607629\pi$
$797$	−18.7298	−0.663443	−0.331722	+	0.943377i	$0.607629\pi$
$798$	0	0
$799$	−3.55640	−0.125816
$800$	−7.58864	−0.268299
$801$	0.111183	0.00392845
$802$	−1.20394	−0.0425125
$803$	−6.90480	−0.243665
$804$	49.2306	1.73623
$805$	0	0
$806$	21.5753	0.759958
$807$	28.0989	0.989128
$808$	−1.70086	−0.0598362
$809$	−31.9421	−1.12303	−0.561513	−	0.827468i	$-0.689781\pi$
$809$	−31.9421	−1.12303	−0.561513	+	0.827468i	$0.689781\pi$
$810$	19.0072	0.667845
$811$	17.8888	0.628161	0.314081	−	0.949396i	$-0.398304\pi$
$811$	17.8888	0.628161	0.314081	+	0.949396i	$0.398304\pi$
$812$	0	0
$813$	−50.2700	−1.76305
$814$	3.35085	0.117447
$815$	15.9649	0.559227
$816$	−5.51536	−0.193076
$817$	71.6307	2.50604
$818$	44.8203	1.56710
$819$	0	0
$820$	15.1773	0.530013
$821$	30.9939	1.08169	0.540847	−	0.841121i	$-0.318104\pi$
$821$	30.9939	1.08169	0.540847	+	0.841121i	$0.318104\pi$
$822$	78.9504	2.75371
$823$	16.5008	0.575182	0.287591	−	0.957753i	$-0.407146\pi$
$823$	16.5008	0.575182	0.287591	+	0.957753i	$0.407146\pi$
$824$	−24.0494	−0.837802
$825$	−1.07838	−0.0375443
$826$	0	0
$827$	43.1155	1.49927	0.749637	−	0.661849i	$-0.230228\pi$
$827$	43.1155	1.49927	0.749637	+	0.661849i	$0.230228\pi$
$828$	−1.40805	−0.0489331
$829$	−22.5958	−0.784785	−0.392393	−	0.919798i	$-0.628353\pi$
$829$	−22.5958	−0.784785	−0.392393	+	0.919798i	$0.628353\pi$
$830$	−6.78765	−0.235603
$831$	18.9360	0.656882
$832$	−53.4329	−1.85245
$833$	0	0
$834$	−30.0410	−1.04024
$835$	−11.3112	−0.391442
$836$	−9.75872	−0.337513
$837$	−12.0533	−0.416624
$838$	13.0784	0.451785
$839$	−1.21235	−0.0418549	−0.0209274	−	0.999781i	$-0.506662\pi$
$839$	−1.21235	−0.0418549	−0.0209274	+	0.999781i	$0.506662\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−27.2085	−0.937666
$843$	−36.0989	−1.24331
$844$	11.2390	0.386863
$845$	−5.83710	−0.200802
$846$	0.389621	0.0133955
$847$	0	0
$848$	−1.91548	−0.0657778
$849$	23.5564	0.808453
$850$	3.36910	0.115559
$851$	−16.2290	−0.556323
$852$	72.2511	2.47528
$853$	−5.93618	−0.203251	−0.101625	−	0.994823i	$-0.532404\pi$
$853$	−5.93618	−0.203251	−0.101625	+	0.994823i	$0.532404\pi$
$854$	0	0
$855$	0.447480	0.0153035
$856$	4.33176	0.148057
$857$	−14.5503	−0.497027	−0.248514	−	0.968628i	$-0.579942\pi$
$857$	−14.5503	−0.497027	−0.248514	+	0.968628i	$0.579942\pi$
$858$	−10.1568	−0.346746
$859$	6.32192	0.215701	0.107851	−	0.994167i	$-0.465603\pi$
$859$	6.32192	0.215701	0.107851	+	0.994167i	$0.465603\pi$
$860$	−33.9916	−1.15910
$861$	0	0
$862$	−39.1506	−1.33348
$863$	−3.28005	−0.111654	−0.0558270	−	0.998440i	$-0.517780\pi$
$863$	−3.28005	−0.111654	−0.0558270	+	0.998440i	$0.517780\pi$
$864$	39.9299	1.35844
$865$	−10.4969	−0.356906
$866$	−40.8287	−1.38742
$867$	−24.9378	−0.846932
$868$	0	0
$869$	6.43907	0.218430
$870$	3.70928	0.125756
$871$	46.1399	1.56339
$872$	9.10504	0.308336
$873$	1.05559	0.0357264
$874$	82.1543	2.77891
$875$	0	0
$876$	50.6824	1.71240
$877$	18.2823	0.617350	0.308675	−	0.951168i	$-0.400114\pi$
$877$	18.2823	0.617350	0.308675	+	0.951168i	$0.400114\pi$
$878$	−12.0312	−0.406033
$879$	−10.5092	−0.354467
$880$	−1.31124	−0.0442020
$881$	−29.7464	−1.00218	−0.501091	−	0.865394i	$-0.667068\pi$
$881$	−29.7464	−1.00218	−0.501091	+	0.865394i	$0.667068\pi$
$882$	0	0
$883$	13.8127	0.464835	0.232417	−	0.972616i	$-0.425336\pi$
$883$	13.8127	0.464835	0.232417	+	0.972616i	$0.425336\pi$
$884$	18.2557	0.614004
$885$	−6.15676	−0.206957
$886$	38.6453	1.29831
$887$	56.3318	1.89144	0.945718	−	0.324989i	$-0.105361\pi$
$887$	56.3318	1.89144	0.945718	+	0.324989i	$0.105361\pi$
$888$	−6.43907	−0.216081
$889$	0	0
$890$	3.07838	0.103187
$891$	5.52586	0.185123
$892$	18.2329	0.610482
$893$	−13.0784	−0.437651
$894$	47.6163	1.59253
$895$	12.3135	0.411595
$896$	0	0
$897$	49.1917	1.64246
$898$	−23.2085	−0.774477
$899$	−2.29072	−0.0763999
$900$	−0.212347	−0.00707823
$901$	1.43084	0.0476681
$902$	7.66967	0.255372
$903$	0	0
$904$	−3.00227	−0.0998539
$905$	−1.60197	−0.0532512
$906$	−76.3956	−2.53807
$907$	21.8082	0.724128	0.362064	−	0.932153i	$-0.382072\pi$
$907$	21.8082	0.724128	0.362064	+	0.932153i	$0.382072\pi$
$908$	60.2739	2.00026
$909$	0.0866105	0.00287269
$910$	0	0
$911$	4.76099	0.157739	0.0788693	−	0.996885i	$-0.474869\pi$
$911$	4.76099	0.157739	0.0788693	+	0.996885i	$0.474869\pi$
$912$	−20.2823	−0.671615
$913$	−1.97334	−0.0653080
$914$	−47.3074	−1.56479
$915$	−22.2557	−0.735749
$916$	19.4063	0.641201
$917$	0	0
$918$	−17.7275	−0.585096
$919$	34.1256	1.12570	0.562849	−	0.826560i	$-0.309705\pi$
$919$	34.1256	1.12570	0.562849	+	0.826560i	$0.309705\pi$
$920$	−10.2062	−0.336489
$921$	−17.7587	−0.585170
$922$	−48.6369	−1.60177
$923$	67.7152	2.22887
$924$	0	0
$925$	−2.44748	−0.0804727
$926$	−4.56320	−0.149956
$927$	1.22463	0.0402222
$928$	7.58864	0.249109
$929$	12.5769	0.412635	0.206318	−	0.978485i	$-0.433852\pi$
$929$	12.5769	0.412635	0.206318	+	0.978485i	$0.433852\pi$
$930$	−8.49693	−0.278625
$931$	0	0
$932$	81.8141	2.67991
$933$	−32.0599	−1.04960
$934$	40.3728	1.32104
$935$	0.979481	0.0320325
$936$	−0.523590	−0.0171141
$937$	29.7464	0.971774	0.485887	−	0.874022i	$-0.338497\pi$
$937$	29.7464	0.971774	0.485887	+	0.874022i	$0.338497\pi$
$938$	0	0
$939$	−21.0928	−0.688336
$940$	6.20620	0.202424
$941$	7.47641	0.243724	0.121862	−	0.992547i	$-0.461113\pi$
$941$	7.47641	0.243724	0.121862	+	0.992547i	$0.461113\pi$
$942$	22.4391	0.731104
$943$	−37.1461	−1.20964
$944$	−7.48625	−0.243657
$945$	0	0
$946$	−17.1773	−0.558481
$947$	−15.2846	−0.496682	−0.248341	−	0.968673i	$-0.579885\pi$
$947$	−15.2846	−0.496682	−0.248341	+	0.968673i	$0.579885\pi$
$948$	−47.2639	−1.53506
$949$	47.5006	1.54194
$950$	12.3896	0.401972
$951$	−52.4724	−1.70153
$952$	0	0
$953$	−54.0288	−1.75016	−0.875081	−	0.483976i	$-0.839192\pi$
$953$	−54.0288	−1.75016	−0.875081	+	0.483976i	$0.839192\pi$
$954$	−0.156755	−0.00507515
$955$	−13.6248	−0.440887
$956$	−17.4452	−0.564219
$957$	1.07838	0.0348590
$958$	−19.3112	−0.623918
$959$	0	0
$960$	21.0433	0.679170
$961$	−25.7526	−0.830728
$962$	−23.0517	−0.743217
$963$	−0.220580	−0.00710808
$964$	−29.7854	−0.959323
$965$	16.9711	0.546318
$966$	0	0
$967$	−15.7671	−0.507037	−0.253518	−	0.967331i	$-0.581588\pi$
$967$	−15.7671	−0.507037	−0.253518	+	0.967331i	$0.581588\pi$
$968$	−16.3184	−0.524494
$969$	15.1506	0.486708
$970$	29.2267	0.938414
$971$	−48.1627	−1.54562	−0.772808	−	0.634640i	$-0.781148\pi$
$971$	−48.1627	−1.54562	−0.772808	+	0.634640i	$0.781148\pi$
$972$	2.20620	0.0707640
$973$	0	0
$974$	−28.1483	−0.901931
$975$	7.41855	0.237584
$976$	−27.0616	−0.866219
$977$	−8.28685	−0.265120	−0.132560	−	0.991175i	$-0.542320\pi$
$977$	−8.28685	−0.265120	−0.132560	+	0.991175i	$0.542320\pi$
$978$	−59.2183	−1.89359
$979$	0.894960	0.0286031
$980$	0	0
$981$	−0.463642	−0.0148029
$982$	−30.0806	−0.959912
$983$	−22.9177	−0.730963	−0.365481	−	0.930819i	$-0.619096\pi$
$983$	−22.9177	−0.730963	−0.365481	+	0.930819i	$0.619096\pi$
$984$	−14.7382	−0.469837
$985$	0.0578588	0.00184354
$986$	−3.36910	−0.107294
$987$	0	0
$988$	67.1338	2.13581
$989$	83.1937	2.64541
$990$	−0.107307	−0.00341045
$991$	22.3234	0.709125	0.354562	−	0.935032i	$-0.384630\pi$
$991$	22.3234	0.709125	0.354562	+	0.935032i	$0.384630\pi$
$992$	−17.3835	−0.551926
$993$	−6.97495	−0.221343
$994$	0	0
$995$	5.39189	0.170934
$996$	14.4846	0.458963
$997$	−14.3630	−0.454879	−0.227440	−	0.973792i	$-0.573035\pi$
$997$	−14.3630	−0.454879	−0.227440	+	0.973792i	$0.573035\pi$
$998$	−48.5113	−1.53560
$999$	12.8781	0.407446

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7105.2.a.o.1.3		3
7.6	odd	2		145.2.a.c.1.3	✓	3
21.20	even	2		1305.2.a.p.1.1		3
28.27	even	2		2320.2.a.n.1.3		3
35.13	even	4		725.2.b.e.349.1		6
35.27	even	4		725.2.b.e.349.6		6
35.34	odd	2		725.2.a.e.1.1		3
56.13	odd	2		9280.2.a.bj.1.3		3
56.27	even	2		9280.2.a.br.1.1		3
105.104	even	2		6525.2.a.be.1.3		3
203.202	odd	2		4205.2.a.f.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.c.1.3	✓	3	7.6	odd	2
725.2.a.e.1.1		3	35.34	odd	2
725.2.b.e.349.1		6	35.13	even	4
725.2.b.e.349.6		6	35.27	even	4
1305.2.a.p.1.1		3	21.20	even	2
2320.2.a.n.1.3		3	28.27	even	2
4205.2.a.f.1.1		3	203.202	odd	2
6525.2.a.be.1.3		3	105.104	even	2
7105.2.a.o.1.3		3	1.1	even	1	trivial
9280.2.a.bj.1.3		3	56.13	odd	2
9280.2.a.br.1.1		3	56.27	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7105 = 5 \cdot 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7105.a (trivial)

\(f(q)\)	\(=\)	\(q+2.17009 q^{2} +1.70928 q^{3} +2.70928 q^{4} -1.00000 q^{5} +3.70928 q^{6} +1.53919 q^{8} -0.0783777 q^{9} -2.17009 q^{10} -0.630898 q^{11} +4.63090 q^{12} +4.34017 q^{13} -1.70928 q^{15} -2.07838 q^{16} +1.55252 q^{17} -0.170086 q^{18} +5.70928 q^{19} -2.70928 q^{20} -1.36910 q^{22} +6.63090 q^{23} +2.63090 q^{24} +1.00000 q^{25} +9.41855 q^{26} -5.26180 q^{27} -1.00000 q^{29} -3.70928 q^{30} +2.29072 q^{31} -7.58864 q^{32} -1.07838 q^{33} +3.36910 q^{34} -0.212347 q^{36} -2.44748 q^{37} +12.3896 q^{38} +7.41855 q^{39} -1.53919 q^{40} -5.60197 q^{41} +12.5464 q^{43} -1.70928 q^{44} +0.0783777 q^{45} +14.3896 q^{46} -2.29072 q^{47} -3.55252 q^{48} +2.17009 q^{50} +2.65368 q^{51} +11.7587 q^{52} +0.921622 q^{53} -11.4186 q^{54} +0.630898 q^{55} +9.75872 q^{57} -2.17009 q^{58} +3.60197 q^{59} -4.63090 q^{60} +13.0205 q^{61} +4.97107 q^{62} -12.3112 q^{64} -4.34017 q^{65} -2.34017 q^{66} +10.6309 q^{67} +4.20620 q^{68} +11.3340 q^{69} +15.6020 q^{71} -0.120638 q^{72} +10.9444 q^{73} -5.31124 q^{74} +1.70928 q^{75} +15.4680 q^{76} +16.0989 q^{78} -10.2062 q^{79} +2.07838 q^{80} -8.75872 q^{81} -12.1568 q^{82} +3.12783 q^{83} -1.55252 q^{85} +27.2267 q^{86} -1.70928 q^{87} -0.971071 q^{88} -1.41855 q^{89} +0.170086 q^{90} +17.9649 q^{92} +3.91548 q^{93} -4.97107 q^{94} -5.70928 q^{95} -12.9711 q^{96} -13.4680 q^{97} +0.0494483 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - 2 q^{3} + q^{4} - 3 q^{5} + 4 q^{6} + 3 q^{8} + 3 q^{9} - q^{10} + 2 q^{11} + 10 q^{12} + 2 q^{13} + 2 q^{15} - 3 q^{16} + 4 q^{17} + 5 q^{18} + 10 q^{19} - q^{20} - 8 q^{22} + 16 q^{23}+ \cdots - 18 q^{99}+O(q^{100}) \) 3 * q + q^2 - 2 * q^3 + q^4 - 3 * q^5 + 4 * q^6 + 3 * q^8 + 3 * q^9 - q^10 + 2 * q^11 + 10 * q^12 + 2 * q^13 + 2 * q^15 - 3 * q^16 + 4 * q^17 + 5 * q^18 + 10 * q^19 - q^20 - 8 * q^22 + 16 * q^23 + 4 * q^24 + 3 * q^25 + 14 * q^26 - 8 * q^27 - 3 * q^29 - 4 * q^30 + 14 * q^31 - 3 * q^32 + 14 * q^34 - 11 * q^36 - 8 * q^37 + 8 * q^38 + 8 * q^39 - 3 * q^40 + 2 * q^41 + 2 * q^43 + 2 * q^44 - 3 * q^45 + 14 * q^46 - 14 * q^47 - 10 * q^48 + q^50 - 16 * q^51 + 10 * q^52 + 6 * q^53 - 20 * q^54 - 2 * q^55 + 4 * q^57 - q^58 - 8 * q^59 - 10 * q^60 + 6 * q^61 - 11 * q^64 - 2 * q^65 + 4 * q^66 + 28 * q^67 - 12 * q^68 - 12 * q^69 + 28 * q^71 - 13 * q^72 + 16 * q^73 + 10 * q^74 - 2 * q^75 + 14 * q^76 + 12 * q^78 - 6 * q^79 + 3 * q^80 - q^81 - 30 * q^82 - 12 * q^83 - 4 * q^85 + 24 * q^86 + 2 * q^87 + 12 * q^88 + 10 * q^89 - 5 * q^90 + 4 * q^92 - 20 * q^93 - 10 * q^95 - 24 * q^96 - 8 * q^97 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

Properties

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