LMFDB - Embedded newform 7105.2.a.p.1.2

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 = [3,3,2,5,3] 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			1.2
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)

$f(q)$	$=$	\(q+1.53919 q^{2} -1.70928 q^{3} +0.369102 q^{4} +1.00000 q^{5} -2.63090 q^{6} -2.51026 q^{8} -0.0783777 q^{9} +1.53919 q^{10} +0.290725 q^{11} -0.630898 q^{12} +0.921622 q^{13} -1.70928 q^{15} -4.60197 q^{16} -4.97107 q^{17} -0.120638 q^{18} +6.04945 q^{19} +0.369102 q^{20} +0.447480 q^{22} +2.29072 q^{23} +4.29072 q^{24} +1.00000 q^{25} +1.41855 q^{26} +5.26180 q^{27} +1.00000 q^{29} -2.63090 q^{30} -10.0494 q^{31} -2.06278 q^{32} -0.496928 q^{33} -7.65142 q^{34} -0.0289294 q^{36} +1.55252 q^{37} +9.31124 q^{38} -1.57531 q^{39} -2.51026 q^{40} -0.340173 q^{41} -5.70928 q^{43} +0.107307 q^{44} -0.0783777 q^{45} +3.52586 q^{46} +1.12783 q^{47} +7.86603 q^{48} +1.53919 q^{50} +8.49693 q^{51} +0.340173 q^{52} -0.340173 q^{53} +8.09890 q^{54} +0.290725 q^{55} -10.3402 q^{57} +1.53919 q^{58} -9.75872 q^{59} -0.630898 q^{60} -3.07838 q^{61} -15.4680 q^{62} +6.02893 q^{64} +0.921622 q^{65} -0.764867 q^{66} -5.70928 q^{67} -1.83483 q^{68} -3.91548 q^{69} +9.07838 q^{71} +0.196748 q^{72} +6.94441 q^{73} +2.38962 q^{74} -1.70928 q^{75} +2.23287 q^{76} -2.42469 q^{78} +12.3896 q^{79} -4.60197 q^{80} -8.75872 q^{81} -0.523590 q^{82} -2.78765 q^{83} -4.97107 q^{85} -8.78765 q^{86} -1.70928 q^{87} -0.729794 q^{88} -4.73820 q^{89} -0.120638 q^{90} +0.845512 q^{92} +17.1773 q^{93} +1.73594 q^{94} +6.04945 q^{95} +3.52586 q^{96} +15.8927 q^{97} -0.0227863 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 2 q^{3} + 5 q^{4} + 3 q^{5} - 4 q^{6} + 9 q^{8} + 3 q^{9} + 3 q^{10} + 8 q^{11} + 2 q^{12} + 6 q^{13} + 2 q^{15} + 5 q^{16} - 13 q^{18} + 5 q^{20} + 2 q^{22} + 14 q^{23} + 20 q^{24} + 3 q^{25}+ \cdots + 20 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + 2 * q^3 + 5 * q^4 + 3 * q^5 - 4 * q^6 + 9 * q^8 + 3 * q^9 + 3 * q^10 + 8 * q^11 + 2 * q^12 + 6 * q^13 + 2 * q^15 + 5 * q^16 - 13 * q^18 + 5 * q^20 + 2 * q^22 + 14 * q^23 + 20 * q^24 + 3 * q^25 - 10 * q^26 + 8 * q^27 + 3 * q^29 - 4 * q^30 - 12 * q^31 + 11 * q^32 + 16 * q^33 + 14 * q^34 - 15 * q^36 + 4 * q^37 + 2 * q^38 + 16 * q^39 + 9 * q^40 + 10 * q^41 - 10 * q^43 + 12 * q^44 + 3 * q^45 + 8 * q^46 - 18 * q^47 + 10 * q^48 + 3 * q^50 + 8 * q^51 - 10 * q^52 + 10 * q^53 - 12 * q^54 + 8 * q^55 - 20 * q^57 + 3 * q^58 - 4 * q^59 + 2 * q^60 - 6 * q^61 - 14 * q^62 + 33 * q^64 + 6 * q^65 - 12 * q^66 - 10 * q^67 + 36 * q^68 + 20 * q^69 + 24 * q^71 - 3 * q^72 + 4 * q^73 - 22 * q^74 + 2 * q^75 - 16 * q^76 - 28 * q^78 + 8 * q^79 + 5 * q^80 - q^81 + 14 * q^82 + 2 * q^83 - 16 * q^86 + 2 * q^87 + 38 * q^88 - 22 * q^89 - 13 * q^90 + 22 * q^92 + 12 * q^93 + 8 * q^96 + 36 * q^97 + 20 * 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 $

$2$	1.53919	1.08837	0.544185	−	0.838965i	$-0.316839\pi$
$2$	1.53919	1.08837	0.544185	+	0.838965i	$0.316839\pi$
$3$	−1.70928	−0.986851	−0.493425	−	0.869788i	$-0.664255\pi$
$3$	−1.70928	−0.986851	−0.493425	+	0.869788i	$0.664255\pi$
$4$	0.369102	0.184551
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.63090	−1.07406
$7$	0	0
$7$	0	0
$8$	−2.51026	−0.887511
$9$	−0.0783777	−0.0261259
$10$	1.53919	0.486734
$11$	0.290725	0.0876568	0.0438284	−	0.999039i	$-0.486045\pi$
$11$	0.290725	0.0876568	0.0438284	+	0.999039i	$0.486045\pi$
$12$	−0.630898	−0.182124
$13$	0.921622	0.255612	0.127806	−	0.991799i	$-0.459207\pi$
$13$	0.921622	0.255612	0.127806	+	0.991799i	$0.459207\pi$
$14$	0	0
$15$	−1.70928	−0.441333
$16$	−4.60197	−1.15049
$17$	−4.97107	−1.20566	−0.602831	−	0.797869i	$-0.705961\pi$
$17$	−4.97107	−1.20566	−0.602831	+	0.797869i	$0.705961\pi$
$18$	−0.120638	−0.0284347
$19$	6.04945	1.38784	0.693919	−	0.720053i	$-0.255882\pi$
$19$	6.04945	1.38784	0.693919	+	0.720053i	$0.255882\pi$
$20$	0.369102	0.0825338
$21$	0	0
$22$	0.447480	0.0954031
$23$	2.29072	0.477649	0.238825	−	0.971063i	$-0.423238\pi$
$23$	2.29072	0.477649	0.238825	+	0.971063i	$0.423238\pi$
$24$	4.29072	0.875840
$25$	1.00000	0.200000
$26$	1.41855	0.278201
$27$	5.26180	1.01263
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−2.63090	−0.480334
$31$	−10.0494	−1.80493	−0.902467	−	0.430759i	$-0.858246\pi$
$31$	−10.0494	−1.80493	−0.902467	+	0.430759i	$0.858246\pi$
$32$	−2.06278	−0.364651
$33$	−0.496928	−0.0865041
$34$	−7.65142	−1.31221
$35$	0	0
$36$	−0.0289294	−0.00482157
$37$	1.55252	0.255233	0.127616	−	0.991824i	$-0.459267\pi$
$37$	1.55252	0.255233	0.127616	+	0.991824i	$0.459267\pi$
$38$	9.31124	1.51048
$39$	−1.57531	−0.252251
$40$	−2.51026	−0.396907
$41$	−0.340173	−0.0531261	−0.0265630	−	0.999647i	$-0.508456\pi$
$41$	−0.340173	−0.0531261	−0.0265630	+	0.999647i	$0.508456\pi$
$42$	0	0
$43$	−5.70928	−0.870656	−0.435328	−	0.900272i	$-0.643368\pi$
$43$	−5.70928	−0.870656	−0.435328	+	0.900272i	$0.643368\pi$
$44$	0.107307	0.0161772
$45$	−0.0783777	−0.0116839
$46$	3.52586	0.519859
$47$	1.12783	0.164510	0.0822552	−	0.996611i	$-0.473788\pi$
$47$	1.12783	0.164510	0.0822552	+	0.996611i	$0.473788\pi$
$48$	7.86603	1.13536
$49$	0	0
$50$	1.53919	0.217674
$51$	8.49693	1.18981
$52$	0.340173	0.0471735
$53$	−0.340173	−0.0467264	−0.0233632	−	0.999727i	$-0.507437\pi$
$53$	−0.340173	−0.0467264	−0.0233632	+	0.999727i	$0.507437\pi$
$54$	8.09890	1.10212
$55$	0.290725	0.0392013
$56$	0	0
$57$	−10.3402	−1.36959
$58$	1.53919	0.202105
$59$	−9.75872	−1.27048	−0.635239	−	0.772316i	$-0.719098\pi$
$59$	−9.75872	−1.27048	−0.635239	+	0.772316i	$0.719098\pi$
$60$	−0.630898	−0.0814485
$61$	−3.07838	−0.394146	−0.197073	−	0.980389i	$-0.563144\pi$
$61$	−3.07838	−0.394146	−0.197073	+	0.980389i	$0.563144\pi$
$62$	−15.4680	−1.96444
$63$	0	0
$64$	6.02893	0.753616
$65$	0.921622	0.114313
$66$	−0.764867	−0.0941486
$67$	−5.70928	−0.697499	−0.348749	−	0.937216i	$-0.613394\pi$
$67$	−5.70928	−0.697499	−0.348749	+	0.937216i	$0.613394\pi$
$68$	−1.83483	−0.222506
$69$	−3.91548	−0.471368
$70$	0	0
$71$	9.07838	1.07741	0.538703	−	0.842496i	$-0.318915\pi$
$71$	9.07838	1.07741	0.538703	+	0.842496i	$0.318915\pi$
$72$	0.196748	0.0231870
$73$	6.94441	0.812782	0.406391	−	0.913699i	$-0.366787\pi$
$73$	6.94441	0.812782	0.406391	+	0.913699i	$0.366787\pi$
$74$	2.38962	0.277788
$75$	−1.70928	−0.197370
$76$	2.23287	0.256127
$77$	0	0
$78$	−2.42469	−0.274543
$79$	12.3896	1.39394	0.696971	−	0.717100i	$-0.254531\pi$
$79$	12.3896	1.39394	0.696971	+	0.717100i	$0.254531\pi$
$80$	−4.60197	−0.514516
$81$	−8.75872	−0.973192
$82$	−0.523590	−0.0578209
$83$	−2.78765	−0.305985	−0.152992	−	0.988227i	$-0.548891\pi$
$83$	−2.78765	−0.305985	−0.152992	+	0.988227i	$0.548891\pi$
$84$	0	0
$85$	−4.97107	−0.539188
$86$	−8.78765	−0.947597
$87$	−1.70928	−0.183254
$88$	−0.729794	−0.0777963
$89$	−4.73820	−0.502249	−0.251124	−	0.967955i	$-0.580800\pi$
$89$	−4.73820	−0.502249	−0.251124	+	0.967955i	$0.580800\pi$
$90$	−0.120638	−0.0127164
$91$	0	0
$92$	0.845512	0.0881507
$93$	17.1773	1.78120
$94$	1.73594	0.179048
$95$	6.04945	0.620660
$96$	3.52586	0.359856
$97$	15.8927	1.61366	0.806829	−	0.590785i	$-0.201182\pi$
$97$	15.8927	1.61366	0.806829	+	0.590785i	$0.201182\pi$
$98$	0	0
$99$	−0.0227863	−0.00229011
$100$	0.369102	0.0369102
$101$	12.2557	1.21948	0.609741	−	0.792600i	$-0.291273\pi$
$101$	12.2557	1.21948	0.609741	+	0.792600i	$0.291273\pi$
$102$	13.0784	1.29495
$103$	7.86603	0.775063	0.387532	−	0.921856i	$-0.373328\pi$
$103$	7.86603	0.775063	0.387532	+	0.921856i	$0.373328\pi$
$104$	−2.31351	−0.226858
$105$	0	0
$106$	−0.523590	−0.0508556
$107$	12.7298	1.23064	0.615318	−	0.788279i	$-0.289028\pi$
$107$	12.7298	1.23064	0.615318	+	0.788279i	$0.289028\pi$
$108$	1.94214	0.186883
$109$	12.4391	1.19145	0.595723	−	0.803190i	$-0.296865\pi$
$109$	12.4391	1.19145	0.595723	+	0.803190i	$0.296865\pi$
$110$	0.447480	0.0426656
$111$	−2.65368	−0.251877
$112$	0	0
$113$	12.5730	1.18277	0.591386	−	0.806389i	$-0.298581\pi$
$113$	12.5730	1.18277	0.591386	+	0.806389i	$0.298581\pi$
$114$	−15.9155	−1.49062
$115$	2.29072	0.213611
$116$	0.369102	0.0342703
$117$	−0.0722347	−0.00667810
$118$	−15.0205	−1.38275
$119$	0	0
$120$	4.29072	0.391688
$121$	−10.9155	−0.992316
$122$	−4.73820	−0.428977
$123$	0.581449	0.0524275
$124$	−3.70928	−0.333103
$125$	1.00000	0.0894427
$126$	0	0
$127$	−20.9132	−1.85575	−0.927874	−	0.372895i	$-0.878365\pi$
$127$	−20.9132	−1.85575	−0.927874	+	0.372895i	$0.878365\pi$
$128$	13.4052	1.18487
$129$	9.75872	0.859208
$130$	1.41855	0.124415
$131$	13.4680	1.17670	0.588352	−	0.808605i	$-0.299777\pi$
$131$	13.4680	1.17670	0.588352	+	0.808605i	$0.299777\pi$
$132$	−0.183417	−0.0159644
$133$	0	0
$134$	−8.78765	−0.759138
$135$	5.26180	0.452863
$136$	12.4787	1.07004
$137$	−13.5525	−1.15787	−0.578935	−	0.815374i	$-0.696531\pi$
$137$	−13.5525	−1.15787	−0.578935	+	0.815374i	$0.696531\pi$
$138$	−6.02666	−0.513024
$139$	4.89496	0.415185	0.207593	−	0.978215i	$-0.433437\pi$
$139$	4.89496	0.415185	0.207593	+	0.978215i	$0.433437\pi$
$140$	0	0
$141$	−1.92777	−0.162347
$142$	13.9733	1.17262
$143$	0.267938	0.0224061
$144$	0.360692	0.0300577
$145$	1.00000	0.0830455
$146$	10.6888	0.884608
$147$	0	0
$148$	0.573039	0.0471035
$149$	−12.5236	−1.02597	−0.512986	−	0.858397i	$-0.671461\pi$
$149$	−12.5236	−1.02597	−0.512986	+	0.858397i	$0.671461\pi$
$150$	−2.63090	−0.214812
$151$	7.60197	0.618639	0.309320	−	0.950958i	$-0.399899\pi$
$151$	7.60197	0.618639	0.309320	+	0.950958i	$0.399899\pi$
$152$	−15.1857	−1.23172
$153$	0.389621	0.0314990
$154$	0	0
$155$	−10.0494	−0.807191
$156$	−0.581449	−0.0465532
$157$	24.8865	1.98616	0.993081	−	0.117428i	$-0.0374648\pi$
$157$	24.8865	1.98616	0.993081	+	0.117428i	$0.0374648\pi$
$158$	19.0700	1.51713
$159$	0.581449	0.0461119
$160$	−2.06278	−0.163077
$161$	0	0
$162$	−13.4813	−1.05919
$163$	0.447480	0.0350493	0.0175247	−	0.999846i	$-0.494421\pi$
$163$	0.447480	0.0350493	0.0175247	+	0.999846i	$0.494421\pi$
$164$	−0.125559	−0.00980448
$165$	−0.496928	−0.0386858
$166$	−4.29072	−0.333025
$167$	−19.8660	−1.53728	−0.768640	−	0.639682i	$-0.779066\pi$
$167$	−19.8660	−1.53728	−0.768640	+	0.639682i	$0.779066\pi$
$168$	0	0
$169$	−12.1506	−0.934662
$170$	−7.65142	−0.586837
$171$	−0.474142	−0.0362586
$172$	−2.10731	−0.160681
$173$	25.4329	1.93363	0.966815	−	0.255478i	$-0.0822329\pi$
$173$	25.4329	1.93363	0.966815	+	0.255478i	$0.0822329\pi$
$174$	−2.63090	−0.199448
$175$	0	0
$176$	−1.33791	−0.100848
$177$	16.6803	1.25377
$178$	−7.29299	−0.546633
$179$	14.8371	1.10898	0.554489	−	0.832191i	$-0.312914\pi$
$179$	14.8371	1.10898	0.554489	+	0.832191i	$0.312914\pi$
$180$	−0.0289294	−0.00215627
$181$	5.91548	0.439694	0.219847	−	0.975534i	$-0.429444\pi$
$181$	5.91548	0.439694	0.219847	+	0.975534i	$0.429444\pi$
$182$	0	0
$183$	5.26180	0.388963
$184$	−5.75031	−0.423919
$185$	1.55252	0.114144
$186$	26.4391	1.93861
$187$	−1.44521	−0.105684
$188$	0.416283	0.0303606
$189$	0	0
$190$	9.31124	0.675509
$191$	7.02893	0.508595	0.254298	−	0.967126i	$-0.418156\pi$
$191$	7.02893	0.508595	0.254298	+	0.967126i	$0.418156\pi$
$192$	−10.3051	−0.743707
$193$	17.8660	1.28603	0.643013	−	0.765856i	$-0.277684\pi$
$193$	17.8660	1.28603	0.643013	+	0.765856i	$0.277684\pi$
$194$	24.4619	1.75626
$195$	−1.57531	−0.112810
$196$	0	0
$197$	6.09890	0.434528	0.217264	−	0.976113i	$-0.430287\pi$
$197$	6.09890	0.434528	0.217264	+	0.976113i	$0.430287\pi$
$198$	−0.0350725	−0.00249249
$199$	9.75872	0.691778	0.345889	−	0.938276i	$-0.387577\pi$
$199$	9.75872	0.691778	0.345889	+	0.938276i	$0.387577\pi$
$200$	−2.51026	−0.177502
$201$	9.75872	0.688327
$202$	18.8638	1.32725
$203$	0	0
$204$	3.13624	0.219580
$205$	−0.340173	−0.0237587
$206$	12.1073	0.843556
$207$	−0.179542	−0.0124790
$208$	−4.24128	−0.294080
$209$	1.75872	0.121653
$210$	0	0
$211$	9.86603	0.679206	0.339603	−	0.940569i	$-0.389707\pi$
$211$	9.86603	0.679206	0.339603	+	0.940569i	$0.389707\pi$
$212$	−0.125559	−0.00862340
$213$	−15.5174	−1.06324
$214$	19.5936	1.33939
$215$	−5.70928	−0.389369
$216$	−13.2085	−0.898723
$217$	0	0
$218$	19.1461	1.29674
$219$	−11.8699	−0.802094
$220$	0.107307	0.00723465
$221$	−4.58145	−0.308182
$222$	−4.08452	−0.274135
$223$	−10.9711	−0.734677	−0.367339	−	0.930087i	$-0.619731\pi$
$223$	−10.9711	−0.734677	−0.367339	+	0.930087i	$0.619731\pi$
$224$	0	0
$225$	−0.0783777	−0.00522518
$226$	19.3523	1.28729
$227$	12.5464	0.832732	0.416366	−	0.909197i	$-0.363303\pi$
$227$	12.5464	0.832732	0.416366	+	0.909197i	$0.363303\pi$
$228$	−3.81658	−0.252759
$229$	−23.3607	−1.54372	−0.771859	−	0.635794i	$-0.780673\pi$
$229$	−23.3607	−1.54372	−0.771859	+	0.635794i	$0.780673\pi$
$230$	3.52586	0.232488
$231$	0	0
$232$	−2.51026	−0.164807
$233$	12.4703	0.816954	0.408477	−	0.912769i	$-0.366060\pi$
$233$	12.4703	0.816954	0.408477	+	0.912769i	$0.366060\pi$
$234$	−0.111183	−0.00726825
$235$	1.12783	0.0735713
$236$	−3.60197	−0.234468
$237$	−21.1773	−1.37561
$238$	0	0
$239$	13.7587	0.889978	0.444989	−	0.895536i	$-0.353208\pi$
$239$	13.7587	0.889978	0.444989	+	0.895536i	$0.353208\pi$
$240$	7.86603	0.507750
$241$	14.6803	0.945644	0.472822	−	0.881158i	$-0.343235\pi$
$241$	14.6803	0.945644	0.472822	+	0.881158i	$0.343235\pi$
$242$	−16.8010	−1.08001
$243$	−0.814315	−0.0522383
$244$	−1.13624	−0.0727401
$245$	0	0
$246$	0.894960	0.0570606
$247$	5.57531	0.354748
$248$	25.2267	1.60190
$249$	4.76487	0.301961
$250$	1.53919	0.0973469
$251$	−15.4413	−0.974649	−0.487324	−	0.873221i	$-0.662027\pi$
$251$	−15.4413	−0.974649	−0.487324	+	0.873221i	$0.662027\pi$
$252$	0	0
$253$	0.665970	0.0418692
$254$	−32.1894	−2.01974
$255$	8.49693	0.532098
$256$	8.57531	0.535957
$257$	6.28231	0.391880	0.195940	−	0.980616i	$-0.437224\pi$
$257$	6.28231	0.391880	0.195940	+	0.980616i	$0.437224\pi$
$258$	15.0205	0.935137
$259$	0	0
$260$	0.340173	0.0210966
$261$	−0.0783777	−0.00485146
$262$	20.7298	1.28069
$263$	10.0761	0.621320	0.310660	−	0.950521i	$-0.399450\pi$
$263$	10.0761	0.621320	0.310660	+	0.950521i	$0.399450\pi$
$264$	1.24742	0.0767734
$265$	−0.340173	−0.0208967
$266$	0	0
$267$	8.09890	0.495644
$268$	−2.10731	−0.128724
$269$	−28.1711	−1.71762	−0.858812	−	0.512291i	$-0.828797\pi$
$269$	−28.1711	−1.71762	−0.858812	+	0.512291i	$0.828797\pi$
$270$	8.09890	0.492883
$271$	−28.8020	−1.74960	−0.874799	−	0.484485i	$-0.839007\pi$
$271$	−28.8020	−1.74960	−0.874799	+	0.484485i	$0.839007\pi$
$272$	22.8767	1.38710
$273$	0	0
$274$	−20.8599	−1.26019
$275$	0.290725	0.0175314
$276$	−1.44521	−0.0869916
$277$	0.0266620	0.00160196	0.000800982	−	1.00000i	$-0.499745\pi$
$277$	0.0266620	0.00160196	0.000800982		1.00000i	$0.499745\pi$
$278$	7.53427	0.451875
$279$	0.787653	0.0471556
$280$	0	0
$281$	−28.0722	−1.67465	−0.837325	−	0.546706i	$-0.815881\pi$
$281$	−28.0722	−1.67465	−0.837325	+	0.546706i	$0.815881\pi$
$282$	−2.96719	−0.176694
$283$	20.8143	1.23728	0.618641	−	0.785674i	$-0.287683\pi$
$283$	20.8143	1.23728	0.618641	+	0.785674i	$0.287683\pi$
$284$	3.35085	0.198836
$285$	−10.3402	−0.612499
$286$	0.412408	0.0243862
$287$	0	0
$288$	0.161676	0.00952685
$289$	7.71154	0.453620
$290$	1.53919	0.0903843
$291$	−27.1650	−1.59244
$292$	2.56320	0.150000
$293$	15.4101	0.900270	0.450135	−	0.892961i	$-0.351376\pi$
$293$	15.4101	0.900270	0.450135	+	0.892961i	$0.351376\pi$
$294$	0	0
$295$	−9.75872	−0.568175
$296$	−3.89723	−0.226522
$297$	1.52973	0.0887641
$298$	−19.2762	−1.11664
$299$	2.11118	0.122093
$300$	−0.630898	−0.0364249
$301$	0	0
$302$	11.7009	0.673309
$303$	−20.9483	−1.20345
$304$	−27.8394	−1.59670
$305$	−3.07838	−0.176267
$306$	0.599701	0.0342826
$307$	28.4307	1.62262	0.811312	−	0.584614i	$-0.198754\pi$
$307$	28.4307	1.62262	0.811312	+	0.584614i	$0.198754\pi$
$308$	0	0
$309$	−13.4452	−0.764871
$310$	−15.4680	−0.878523
$311$	19.6248	1.11282	0.556409	−	0.830909i	$-0.312179\pi$
$311$	19.6248	1.11282	0.556409	+	0.830909i	$0.312179\pi$
$312$	3.95443	0.223875
$313$	−22.9093	−1.29491	−0.647456	−	0.762103i	$-0.724167\pi$
$313$	−22.9093	−1.29491	−0.647456	+	0.762103i	$0.724167\pi$
$314$	38.3051	2.16168
$315$	0	0
$316$	4.57304	0.257254
$317$	22.8599	1.28394	0.641970	−	0.766730i	$-0.278118\pi$
$317$	22.8599	1.28394	0.641970	+	0.766730i	$0.278118\pi$
$318$	0.894960	0.0501869
$319$	0.290725	0.0162775
$320$	6.02893	0.337027
$321$	−21.7587	−1.21445
$322$	0	0
$323$	−30.0722	−1.67326
$324$	−3.23287	−0.179604
$325$	0.921622	0.0511224
$326$	0.688756	0.0381467
$327$	−21.2618	−1.17578
$328$	0.853922	0.0471500
$329$	0	0
$330$	−0.764867	−0.0421045
$331$	24.0905	1.32413	0.662066	−	0.749445i	$-0.269680\pi$
$331$	24.0905	1.32413	0.662066	+	0.749445i	$0.269680\pi$
$332$	−1.02893	−0.0564698
$333$	−0.121683	−0.00666819
$334$	−30.5776	−1.67313
$335$	−5.70928	−0.311931
$336$	0	0
$337$	12.7877	0.696588	0.348294	−	0.937385i	$-0.386761\pi$
$337$	12.7877	0.696588	0.348294	+	0.937385i	$0.386761\pi$
$338$	−18.7021	−1.01726
$339$	−21.4908	−1.16722
$340$	−1.83483	−0.0995078
$341$	−2.92162	−0.158215
$342$	−0.729794	−0.0394628
$343$	0	0
$344$	14.3318	0.772717
$345$	−3.91548	−0.210802
$346$	39.1461	2.10451
$347$	8.41628	0.451810	0.225905	−	0.974149i	$-0.427466\pi$
$347$	8.41628	0.451810	0.225905	+	0.974149i	$0.427466\pi$
$348$	−0.630898	−0.0338197
$349$	−22.1978	−1.18822	−0.594110	−	0.804384i	$-0.702496\pi$
$349$	−22.1978	−1.18822	−0.594110	+	0.804384i	$0.702496\pi$
$350$	0	0
$351$	4.84939	0.258841
$352$	−0.599701	−0.0319642
$353$	6.18342	0.329110	0.164555	−	0.986368i	$-0.447381\pi$
$353$	6.18342	0.329110	0.164555	+	0.986368i	$0.447381\pi$
$354$	25.6742	1.36457
$355$	9.07838	0.481830
$356$	−1.74888	−0.0926906
$357$	0	0
$358$	22.8371	1.20698
$359$	5.05559	0.266824	0.133412	−	0.991061i	$-0.457407\pi$
$359$	5.05559	0.266824	0.133412	+	0.991061i	$0.457407\pi$
$360$	0.196748	0.0103696
$361$	17.5958	0.926096
$362$	9.10504	0.478550
$363$	18.6576	0.979268
$364$	0	0
$365$	6.94441	0.363487
$366$	8.09890	0.423336
$367$	−29.5402	−1.54199	−0.770994	−	0.636843i	$-0.780240\pi$
$367$	−29.5402	−1.54199	−0.770994	+	0.636843i	$0.780240\pi$
$368$	−10.5418	−0.549531
$369$	0.0266620	0.00138797
$370$	2.38962	0.124230
$371$	0	0
$372$	6.34017	0.328723
$373$	14.4124	0.746246	0.373123	−	0.927782i	$-0.378287\pi$
$373$	14.4124	0.746246	0.373123	+	0.927782i	$0.378287\pi$
$374$	−2.22446	−0.115024
$375$	−1.70928	−0.0882666
$376$	−2.83114	−0.146005
$377$	0.921622	0.0474660
$378$	0	0
$379$	14.1340	0.726013	0.363007	−	0.931787i	$-0.381750\pi$
$379$	14.1340	0.726013	0.363007	+	0.931787i	$0.381750\pi$
$380$	2.23287	0.114544
$381$	35.7464	1.83135
$382$	10.8188	0.553541
$383$	15.7815	0.806397	0.403199	−	0.915112i	$-0.367898\pi$
$383$	15.7815	0.806397	0.403199	+	0.915112i	$0.367898\pi$
$384$	−22.9132	−1.16928
$385$	0	0
$386$	27.4992	1.39967
$387$	0.447480	0.0227467
$388$	5.86603	0.297803
$389$	−13.8166	−0.700529	−0.350264	−	0.936651i	$-0.613908\pi$
$389$	−13.8166	−0.700529	−0.350264	+	0.936651i	$0.613908\pi$
$390$	−2.42469	−0.122779
$391$	−11.3874	−0.575883
$392$	0	0
$393$	−23.0205	−1.16123
$394$	9.38735	0.472928
$395$	12.3896	0.623390
$396$	−0.00841049	−0.000422643 0
$397$	−9.05172	−0.454293	−0.227146	−	0.973861i	$-0.572940\pi$
$397$	−9.05172	−0.454293	−0.227146	+	0.973861i	$0.572940\pi$
$398$	15.0205	0.752911
$399$	0	0
$400$	−4.60197	−0.230098
$401$	19.7587	0.986704	0.493352	−	0.869830i	$-0.335772\pi$
$401$	19.7587	0.986704	0.493352	+	0.869830i	$0.335772\pi$
$402$	15.0205	0.749155
$403$	−9.26180	−0.461363
$404$	4.52359	0.225057
$405$	−8.75872	−0.435224
$406$	0	0
$407$	0.451356	0.0223729
$408$	−21.3295	−1.05597
$409$	1.71769	0.0849341	0.0424670	−	0.999098i	$-0.486478\pi$
$409$	1.71769	0.0849341	0.0424670	+	0.999098i	$0.486478\pi$
$410$	−0.523590	−0.0258583
$411$	23.1650	1.14264
$412$	2.90337	0.143039
$413$	0	0
$414$	−0.276349	−0.0135818
$415$	−2.78765	−0.136841
$416$	−1.90110	−0.0932093
$417$	−8.36683	−0.409726
$418$	2.70701	0.132404
$419$	35.5318	1.73584	0.867922	−	0.496701i	$-0.165456\pi$
$419$	35.5318	1.73584	0.867922	+	0.496701i	$0.165456\pi$
$420$	0	0
$421$	−12.0722	−0.588365	−0.294182	−	0.955749i	$-0.595047\pi$
$421$	−12.0722	−0.588365	−0.294182	+	0.955749i	$0.595047\pi$
$422$	15.1857	0.739228
$423$	−0.0883965	−0.00429798
$424$	0.853922	0.0414701
$425$	−4.97107	−0.241132
$426$	−23.8843	−1.15720
$427$	0	0
$428$	4.69860	0.227115
$429$	−0.457980	−0.0221115
$430$	−8.78765	−0.423778
$431$	19.8310	0.955224	0.477612	−	0.878571i	$-0.341503\pi$
$431$	19.8310	0.955224	0.477612	+	0.878571i	$0.341503\pi$
$432$	−24.2146	−1.16503
$433$	−14.8143	−0.711931	−0.355965	−	0.934499i	$-0.615848\pi$
$433$	−14.8143	−0.711931	−0.355965	+	0.934499i	$0.615848\pi$
$434$	0	0
$435$	−1.70928	−0.0819535
$436$	4.59129	0.219883
$437$	13.8576	0.662900
$438$	−18.2700	−0.872976
$439$	17.8576	0.852298	0.426149	−	0.904653i	$-0.359870\pi$
$439$	17.8576	0.852298	0.426149	+	0.904653i	$0.359870\pi$
$440$	−0.729794	−0.0347916
$441$	0	0
$442$	−7.05172	−0.335416
$443$	33.5936	1.59608	0.798039	−	0.602606i	$-0.205871\pi$
$443$	33.5936	1.59608	0.798039	+	0.602606i	$0.205871\pi$
$444$	−0.979481	−0.0464841
$445$	−4.73820	−0.224612
$446$	−16.8865	−0.799601
$447$	21.4063	1.01248
$448$	0	0
$449$	7.07838	0.334049	0.167025	−	0.985953i	$-0.446584\pi$
$449$	7.07838	0.334049	0.167025	+	0.985953i	$0.446584\pi$
$450$	−0.120638	−0.00568694
$451$	−0.0988967	−0.00465686
$452$	4.64074	0.218282
$453$	−12.9939	−0.610505
$454$	19.3112	0.906322
$455$	0	0
$456$	25.9565	1.21553
$457$	−5.81658	−0.272088	−0.136044	−	0.990703i	$-0.543439\pi$
$457$	−5.81658	−0.272088	−0.136044	+	0.990703i	$0.543439\pi$
$458$	−35.9565	−1.68014
$459$	−26.1568	−1.22089
$460$	0.845512	0.0394222
$461$	32.3090	1.50478	0.752390	−	0.658718i	$-0.228901\pi$
$461$	32.3090	1.50478	0.752390	+	0.658718i	$0.228901\pi$
$462$	0	0
$463$	1.44134	0.0669846	0.0334923	−	0.999439i	$-0.489337\pi$
$463$	1.44134	0.0669846	0.0334923	+	0.999439i	$0.489337\pi$
$464$	−4.60197	−0.213641
$465$	17.1773	0.796577
$466$	19.1941	0.889149
$467$	11.7503	0.543740	0.271870	−	0.962334i	$-0.412358\pi$
$467$	11.7503	0.543740	0.271870	+	0.962334i	$0.412358\pi$
$468$	−0.0266620	−0.00123245
$469$	0	0
$470$	1.73594	0.0800728
$471$	−42.5380	−1.96005
$472$	24.4969	1.12756
$473$	−1.65983	−0.0763189
$474$	−32.5958	−1.49718
$475$	6.04945	0.277568
$476$	0	0
$477$	0.0266620	0.00122077
$478$	21.1773	0.968626
$479$	−17.1689	−0.784465	−0.392233	−	0.919866i	$-0.628297\pi$
$479$	−17.1689	−0.784465	−0.392233	+	0.919866i	$0.628297\pi$
$480$	3.52586	0.160933
$481$	1.43084	0.0652405
$482$	22.5958	1.02921
$483$	0	0
$484$	−4.02893	−0.183133
$485$	15.8927	0.721650
$486$	−1.25338	−0.0568547
$487$	−4.10277	−0.185914	−0.0929572	−	0.995670i	$-0.529632\pi$
$487$	−4.10277	−0.185914	−0.0929572	+	0.995670i	$0.529632\pi$
$488$	7.72753	0.349809
$489$	−0.764867	−0.0345885
$490$	0	0
$491$	40.7708	1.83996	0.919981	−	0.391963i	$-0.128204\pi$
$491$	40.7708	1.83996	0.919981	+	0.391963i	$0.128204\pi$
$492$	0.214614	0.00967556
$493$	−4.97107	−0.223886
$494$	8.58145	0.386098
$495$	−0.0227863	−0.00102417
$496$	46.2472	2.07656
$497$	0	0
$498$	7.33403	0.328646
$499$	−18.4703	−0.826843	−0.413421	−	0.910540i	$-0.635666\pi$
$499$	−18.4703	−0.826843	−0.413421	+	0.910540i	$0.635666\pi$
$500$	0.369102	0.0165068
$501$	33.9565	1.51707
$502$	−23.7671	−1.06078
$503$	21.4947	0.958400	0.479200	−	0.877706i	$-0.340927\pi$
$503$	21.4947	0.958400	0.479200	+	0.877706i	$0.340927\pi$
$504$	0	0
$505$	12.2557	0.545369
$506$	1.02505	0.0455692
$507$	20.7687	0.922372
$508$	−7.71912	−0.342480
$509$	−3.75872	−0.166602	−0.0833012	−	0.996524i	$-0.526546\pi$
$509$	−3.75872	−0.166602	−0.0833012	+	0.996524i	$0.526546\pi$
$510$	13.0784	0.579120
$511$	0	0
$512$	−13.6114	−0.601546
$513$	31.8310	1.40537
$514$	9.66967	0.426511
$515$	7.86603	0.346619
$516$	3.60197	0.158568
$517$	0.327887	0.0144204
$518$	0	0
$519$	−43.4719	−1.90820
$520$	−2.31351	−0.101454
$521$	−12.8059	−0.561037	−0.280518	−	0.959849i	$-0.590506\pi$
$521$	−12.8059	−0.561037	−0.280518	+	0.959849i	$0.590506\pi$
$522$	−0.120638	−0.00528019
$523$	21.1278	0.923855	0.461928	−	0.886918i	$-0.347158\pi$
$523$	21.1278	0.923855	0.461928	+	0.886918i	$0.347158\pi$
$524$	4.97107	0.217162
$525$	0	0
$526$	15.5090	0.676226
$527$	49.9565	2.17614
$528$	2.28685	0.0995223
$529$	−17.7526	−0.771851
$530$	−0.523590	−0.0227433
$531$	0.764867	0.0331924
$532$	0	0
$533$	−0.313511	−0.0135797
$534$	12.4657	0.539445
$535$	12.7298	0.550357
$536$	14.3318	0.619038
$537$	−25.3607	−1.09439
$538$	−43.3607	−1.86941
$539$	0	0
$540$	1.94214	0.0835764
$541$	32.7382	1.40753	0.703763	−	0.710435i	$-0.251502\pi$
$541$	32.7382	1.40753	0.703763	+	0.710435i	$0.251502\pi$
$542$	−44.3318	−1.90421
$543$	−10.1112	−0.433912
$544$	10.2542	0.439646
$545$	12.4391	0.532831
$546$	0	0
$547$	−22.1073	−0.945240	−0.472620	−	0.881266i	$-0.656692\pi$
$547$	−22.1073	−0.945240	−0.472620	+	0.881266i	$0.656692\pi$
$548$	−5.00227	−0.213686
$549$	0.241276	0.0102974
$550$	0.447480	0.0190806
$551$	6.04945	0.257715
$552$	9.82887	0.418344
$553$	0	0
$554$	0.0410378	0.00174353
$555$	−2.65368	−0.112643
$556$	1.80674	0.0766229
$557$	−39.8720	−1.68943	−0.844715	−	0.535216i	$-0.820230\pi$
$557$	−39.8720	−1.68943	−0.844715	+	0.535216i	$0.820230\pi$
$558$	1.21235	0.0513227
$559$	−5.26180	−0.222550
$560$	0	0
$561$	2.47027	0.104295
$562$	−43.2085	−1.82264
$563$	−10.1217	−0.426578	−0.213289	−	0.976989i	$-0.568418\pi$
$563$	−10.1217	−0.426578	−0.213289	+	0.976989i	$0.568418\pi$
$564$	−0.711543	−0.0299614
$565$	12.5730	0.528952
$566$	32.0372	1.34662
$567$	0	0
$568$	−22.7891	−0.956209
$569$	−24.4391	−1.02454	−0.512270	−	0.858825i	$-0.671195\pi$
$569$	−24.4391	−1.02454	−0.512270	+	0.858825i	$0.671195\pi$
$570$	−15.9155	−0.666626
$571$	−28.2511	−1.18227	−0.591136	−	0.806572i	$-0.701320\pi$
$571$	−28.2511	−1.18227	−0.591136	+	0.806572i	$0.701320\pi$
$572$	0.0988967	0.00413508
$573$	−12.0144	−0.501908
$574$	0	0
$575$	2.29072	0.0955298
$576$	−0.472534	−0.0196889
$577$	−46.1171	−1.91988	−0.959941	−	0.280202i	$-0.909598\pi$
$577$	−46.1171	−1.91988	−0.959941	+	0.280202i	$0.909598\pi$
$578$	11.8695	0.493707
$579$	−30.5380	−1.26911
$580$	0.369102	0.0153261
$581$	0	0
$582$	−41.8120	−1.73317
$583$	−0.0988967	−0.00409588
$584$	−17.4323	−0.721352
$585$	−0.0722347	−0.00298654
$586$	23.7191	0.979828
$587$	0.715418	0.0295285	0.0147642	−	0.999891i	$-0.495300\pi$
$587$	0.715418	0.0295285	0.0147642	+	0.999891i	$0.495300\pi$
$588$	0	0
$589$	−60.7936	−2.50496
$590$	−15.0205	−0.618385
$591$	−10.4247	−0.428815
$592$	−7.14465	−0.293643
$593$	−15.5441	−0.638320	−0.319160	−	0.947701i	$-0.603401\pi$
$593$	−15.5441	−0.638320	−0.319160	+	0.947701i	$0.603401\pi$
$594$	2.35455	0.0966083
$595$	0	0
$596$	−4.62249	−0.189344
$597$	−16.6803	−0.682681
$598$	3.24951	0.132882
$599$	−9.59809	−0.392167	−0.196084	−	0.980587i	$-0.562822\pi$
$599$	−9.59809	−0.392167	−0.196084	+	0.980587i	$0.562822\pi$
$600$	4.29072	0.175168
$601$	−6.81044	−0.277804	−0.138902	−	0.990306i	$-0.544357\pi$
$601$	−6.81044	−0.277804	−0.138902	+	0.990306i	$0.544357\pi$
$602$	0	0
$603$	0.447480	0.0182228
$604$	2.80590	0.114171
$605$	−10.9155	−0.443777
$606$	−32.2434	−1.30980
$607$	−31.6970	−1.28654	−0.643271	−	0.765639i	$-0.722423\pi$
$607$	−31.6970	−1.28654	−0.643271	+	0.765639i	$0.722423\pi$
$608$	−12.4787	−0.506077
$609$	0	0
$610$	−4.73820	−0.191844
$611$	1.03943	0.0420508
$612$	0.143810	0.00581318
$613$	1.20394	0.0486265	0.0243133	−	0.999704i	$-0.492260\pi$
$613$	1.20394	0.0486265	0.0243133	+	0.999704i	$0.492260\pi$
$614$	43.7602	1.76602
$615$	0.581449	0.0234463
$616$	0	0
$617$	−37.9337	−1.52715	−0.763577	−	0.645716i	$-0.776559\pi$
$617$	−37.9337	−1.52715	−0.763577	+	0.645716i	$0.776559\pi$
$618$	−20.6947	−0.832464
$619$	4.60424	0.185060	0.0925299	−	0.995710i	$-0.470505\pi$
$619$	4.60424	0.185060	0.0925299	+	0.995710i	$0.470505\pi$
$620$	−3.70928	−0.148968
$621$	12.0533	0.483683
$622$	30.2062	1.21116
$623$	0	0
$624$	7.24951	0.290213
$625$	1.00000	0.0400000
$626$	−35.2618	−1.40934
$627$	−3.00614	−0.120054
$628$	9.18568	0.366549
$629$	−7.71769	−0.307724
$630$	0	0
$631$	−8.41241	−0.334893	−0.167446	−	0.985881i	$-0.553552\pi$
$631$	−8.41241	−0.334893	−0.167446	+	0.985881i	$0.553552\pi$
$632$	−31.1012	−1.23714
$633$	−16.8638	−0.670274
$634$	35.1857	1.39740
$635$	−20.9132	−0.829915
$636$	0.214614	0.00851001
$637$	0	0
$638$	0.447480	0.0177159
$639$	−0.711543	−0.0281482
$640$	13.4052	0.529888
$641$	32.5380	1.28517	0.642586	−	0.766213i	$-0.277861\pi$
$641$	32.5380	1.28517	0.642586	+	0.766213i	$0.277861\pi$
$642$	−33.4908	−1.32178
$643$	2.09293	0.0825372	0.0412686	−	0.999148i	$-0.486860\pi$
$643$	2.09293	0.0825372	0.0412686	+	0.999148i	$0.486860\pi$
$644$	0	0
$645$	9.75872	0.384249
$646$	−46.2868	−1.82113
$647$	−45.1955	−1.77682	−0.888410	−	0.459051i	$-0.848189\pi$
$647$	−45.1955	−1.77682	−0.888410	+	0.459051i	$0.848189\pi$
$648$	21.9867	0.863718
$649$	−2.83710	−0.111366
$650$	1.41855	0.0556401
$651$	0	0
$652$	0.165166	0.00646840
$653$	−2.14834	−0.0840712	−0.0420356	−	0.999116i	$-0.513384\pi$
$653$	−2.14834	−0.0840712	−0.0420356	+	0.999116i	$0.513384\pi$
$654$	−32.7259	−1.27968
$655$	13.4680	0.526238
$656$	1.56547	0.0611211
$657$	−0.544287	−0.0212347
$658$	0	0
$659$	−45.0843	−1.75624	−0.878118	−	0.478444i	$-0.841201\pi$
$659$	−45.0843	−1.75624	−0.878118	+	0.478444i	$0.841201\pi$
$660$	−0.183417	−0.00713952
$661$	36.3234	1.41281	0.706407	−	0.707806i	$-0.250315\pi$
$661$	36.3234	1.41281	0.706407	+	0.707806i	$0.250315\pi$
$662$	37.0798	1.44115
$663$	7.83096	0.304129
$664$	6.99773	0.271565
$665$	0	0
$666$	−0.187293	−0.00725746
$667$	2.29072	0.0886972
$668$	−7.33260	−0.283707
$669$	18.7526	0.725017
$670$	−8.78765	−0.339497
$671$	−0.894960	−0.0345496
$672$	0	0
$673$	−17.4719	−0.673491	−0.336746	−	0.941596i	$-0.609326\pi$
$673$	−17.4719	−0.673491	−0.336746	+	0.941596i	$0.609326\pi$
$674$	19.6826	0.758146
$675$	5.26180	0.202527
$676$	−4.48482	−0.172493
$677$	40.0372	1.53875	0.769377	−	0.638796i	$-0.220567\pi$
$677$	40.0372	1.53875	0.769377	+	0.638796i	$0.220567\pi$
$678$	−33.0784	−1.27037
$679$	0	0
$680$	12.4787	0.478535
$681$	−21.4452	−0.821782
$682$	−4.49693	−0.172196
$683$	2.07611	0.0794402	0.0397201	−	0.999211i	$-0.487353\pi$
$683$	2.07611	0.0794402	0.0397201	+	0.999211i	$0.487353\pi$
$684$	−0.175007	−0.00669156
$685$	−13.5525	−0.517815
$686$	0	0
$687$	39.9299	1.52342
$688$	26.2739	1.00168
$689$	−0.313511	−0.0119438
$690$	−6.02666	−0.229431
$691$	−26.7070	−1.01598	−0.507991	−	0.861362i	$-0.669612\pi$
$691$	−26.7070	−1.01598	−0.507991	+	0.861362i	$0.669612\pi$
$692$	9.38735	0.356854
$693$	0	0
$694$	12.9542	0.491737
$695$	4.89496	0.185676
$696$	4.29072	0.162639
$697$	1.69102	0.0640521
$698$	−34.1666	−1.29322
$699$	−21.3151	−0.806212
$700$	0	0
$701$	−21.9155	−0.827736	−0.413868	−	0.910337i	$-0.635823\pi$
$701$	−21.9155	−0.827736	−0.413868	+	0.910337i	$0.635823\pi$
$702$	7.46412	0.281715
$703$	9.39189	0.354222
$704$	1.75276	0.0660596
$705$	−1.92777	−0.0726038
$706$	9.51745	0.358194
$707$	0	0
$708$	6.15676	0.231385
$709$	−4.60811	−0.173061	−0.0865306	−	0.996249i	$-0.527578\pi$
$709$	−4.60811	−0.173061	−0.0865306	+	0.996249i	$0.527578\pi$
$710$	13.9733	0.524410
$711$	−0.971071	−0.0364180
$712$	11.8941	0.445751
$713$	−23.0205	−0.862125
$714$	0	0
$715$	0.267938	0.0100203
$716$	5.47641	0.204663
$717$	−23.5174	−0.878275
$718$	7.78151	0.290403
$719$	−6.80590	−0.253817	−0.126909	−	0.991914i	$-0.540506\pi$
$719$	−6.80590	−0.253817	−0.126909	+	0.991914i	$0.540506\pi$
$720$	0.360692	0.0134422
$721$	0	0
$722$	27.0833	1.00794
$723$	−25.0928	−0.933210
$724$	2.18342	0.0811461
$725$	1.00000	0.0371391
$726$	28.7175	1.06581
$727$	26.9711	1.00030	0.500151	−	0.865938i	$-0.333278\pi$
$727$	26.9711	1.00030	0.500151	+	0.865938i	$0.333278\pi$
$728$	0	0
$729$	27.6681	1.02474
$730$	10.6888	0.395609
$731$	28.3812	1.04972
$732$	1.94214	0.0717836
$733$	30.0638	1.11043	0.555216	−	0.831706i	$-0.312635\pi$
$733$	30.0638	1.11043	0.555216	+	0.831706i	$0.312635\pi$
$734$	−45.4680	−1.67825
$735$	0	0
$736$	−4.72526	−0.174175
$737$	−1.65983	−0.0611405
$738$	0.0410378	0.00151062
$739$	51.1422	1.88130	0.940648	−	0.339383i	$-0.110218\pi$
$739$	51.1422	1.88130	0.940648	+	0.339383i	$0.110218\pi$
$740$	0.573039	0.0210653
$741$	−9.52973	−0.350084
$742$	0	0
$743$	11.1857	0.410363	0.205181	−	0.978724i	$-0.434222\pi$
$743$	11.1857	0.410363	0.205181	+	0.978724i	$0.434222\pi$
$744$	−43.1194	−1.58083
$745$	−12.5236	−0.458829
$746$	22.1834	0.812193
$747$	0.218490	0.00799413
$748$	−0.533431	−0.0195042
$749$	0	0
$750$	−2.63090	−0.0960668
$751$	−18.3630	−0.670074	−0.335037	−	0.942205i	$-0.608749\pi$
$751$	−18.3630	−0.670074	−0.335037	+	0.942205i	$0.608749\pi$
$752$	−5.19022	−0.189268
$753$	26.3935	0.961832
$754$	1.41855	0.0516606
$755$	7.60197	0.276664
$756$	0	0
$757$	15.8927	0.577630	0.288815	−	0.957385i	$-0.406739\pi$
$757$	15.8927	0.577630	0.288815	+	0.957385i	$0.406739\pi$
$758$	21.7548	0.790172
$759$	−1.13833	−0.0413186
$760$	−15.1857	−0.550843
$761$	−13.8843	−0.503305	−0.251652	−	0.967818i	$-0.580974\pi$
$761$	−13.8843	−0.503305	−0.251652	+	0.967818i	$0.580974\pi$
$762$	55.0205	1.99318
$763$	0	0
$764$	2.59439	0.0938619
$765$	0.389621	0.0140868
$766$	24.2907	0.877660
$767$	−8.99386	−0.324749
$768$	−14.6576	−0.528909
$769$	35.4063	1.27678	0.638391	−	0.769712i	$-0.279600\pi$
$769$	35.4063	1.27678	0.638391	+	0.769712i	$0.279600\pi$
$770$	0	0
$771$	−10.7382	−0.386727
$772$	6.59439	0.237337
$773$	0.488518	0.0175708	0.00878539	−	0.999961i	$-0.497203\pi$
$773$	0.488518	0.0175708	0.00878539	+	0.999961i	$0.497203\pi$
$774$	0.688756	0.0247568
$775$	−10.0494	−0.360987
$776$	−39.8948	−1.43214
$777$	0	0
$778$	−21.2663	−0.762435
$779$	−2.05786	−0.0737304
$780$	−0.581449	−0.0208192
$781$	2.63931	0.0944419
$782$	−17.5273	−0.626775
$783$	5.26180	0.188041
$784$	0	0
$785$	24.8865	0.888239
$786$	−35.4329	−1.26385
$787$	1.99159	0.0709925	0.0354962	−	0.999370i	$-0.488699\pi$
$787$	1.99159	0.0709925	0.0354962	+	0.999370i	$0.488699\pi$
$788$	2.25112	0.0801927
$789$	−17.2228	−0.613150
$790$	19.0700	0.678479
$791$	0	0
$792$	0.0571996	0.00203250
$793$	−2.83710	−0.100748
$794$	−13.9323	−0.494439
$795$	0.581449	0.0206219
$796$	3.60197	0.127668
$797$	17.2702	0.611742	0.305871	−	0.952073i	$-0.401052\pi$
$797$	17.2702	0.611742	0.305871	+	0.952073i	$0.401052\pi$
$798$	0	0
$799$	−5.60650	−0.198344
$800$	−2.06278	−0.0729303
$801$	0.371370	0.0131217
$802$	30.4124	1.07390
$803$	2.01891	0.0712458
$804$	3.60197	0.127032
$805$	0	0
$806$	−14.2557	−0.502134
$807$	48.1522	1.69504
$808$	−30.7649	−1.08230
$809$	−56.5068	−1.98667	−0.993336	−	0.115254i	$-0.963232\pi$
$809$	−56.5068	−1.98667	−0.993336	+	0.115254i	$0.963232\pi$
$810$	−13.4813	−0.473686
$811$	8.77924	0.308281	0.154140	−	0.988049i	$-0.450739\pi$
$811$	8.77924	0.308281	0.154140	+	0.988049i	$0.450739\pi$
$812$	0	0
$813$	49.2306	1.72659
$814$	0.694722	0.0243500
$815$	0.447480	0.0156745
$816$	−39.1026	−1.36886
$817$	−34.5380	−1.20833
$818$	2.64384	0.0924398
$819$	0	0
$820$	−0.125559	−0.00438470
$821$	−28.1568	−0.982678	−0.491339	−	0.870969i	$-0.663492\pi$
$821$	−28.1568	−0.982678	−0.491339	+	0.870969i	$0.663492\pi$
$822$	35.6553	1.24362
$823$	20.7442	0.723096	0.361548	−	0.932353i	$-0.382248\pi$
$823$	20.7442	0.723096	0.361548	+	0.932353i	$0.382248\pi$
$824$	−19.7458	−0.687877
$825$	−0.496928	−0.0173008
$826$	0	0
$827$	−9.12783	−0.317406	−0.158703	−	0.987326i	$-0.550731\pi$
$827$	−9.12783	−0.317406	−0.158703	+	0.987326i	$0.550731\pi$
$828$	−0.0662693	−0.00230302
$829$	−31.8576	−1.10646	−0.553230	−	0.833028i	$-0.686605\pi$
$829$	−31.8576	−1.10646	−0.553230	+	0.833028i	$0.686605\pi$
$830$	−4.29072	−0.148933
$831$	−0.0455727	−0.00158090
$832$	5.55640	0.192633
$833$	0	0
$834$	−12.8781	−0.445933
$835$	−19.8660	−0.687492
$836$	0.649149	0.0224513
$837$	−52.8781	−1.82774
$838$	54.6902	1.88924
$839$	−27.4413	−0.947380	−0.473690	−	0.880692i	$-0.657078\pi$
$839$	−27.4413	−0.947380	−0.473690	+	0.880692i	$0.657078\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−18.5814	−0.640359
$843$	47.9832	1.65263
$844$	3.64158	0.125348
$845$	−12.1506	−0.417994
$846$	−0.136059	−0.00467780
$847$	0	0
$848$	1.56547	0.0537583
$849$	−35.5774	−1.22101
$850$	−7.65142	−0.262441
$851$	3.55640	0.121912
$852$	−5.72753	−0.196222
$853$	56.0515	1.91917	0.959584	−	0.281422i	$-0.0908061\pi$
$853$	56.0515	1.91917	0.959584	+	0.281422i	$0.0908061\pi$
$854$	0	0
$855$	−0.474142	−0.0162153
$856$	−31.9551	−1.09220
$857$	6.08452	0.207843	0.103922	−	0.994585i	$-0.466861\pi$
$857$	6.08452	0.207843	0.103922	+	0.994585i	$0.466861\pi$
$858$	−0.704918	−0.0240655
$859$	−35.5936	−1.21444	−0.607218	−	0.794535i	$-0.707715\pi$
$859$	−35.5936	−1.21444	−0.607218	+	0.794535i	$0.707715\pi$
$860$	−2.10731	−0.0718586
$861$	0	0
$862$	30.5236	1.03964
$863$	−12.8287	−0.436694	−0.218347	−	0.975871i	$-0.570066\pi$
$863$	−12.8287	−0.436694	−0.218347	+	0.975871i	$0.570066\pi$
$864$	−10.8539	−0.369258
$865$	25.4329	0.864745
$866$	−22.8020	−0.774844
$867$	−13.1812	−0.447655
$868$	0	0
$869$	3.60197	0.122188
$870$	−2.63090	−0.0891958
$871$	−5.26180	−0.178289
$872$	−31.2253	−1.05742
$873$	−1.24563	−0.0421583
$874$	21.3295	0.721481
$875$	0	0
$876$	−4.38121	−0.148027
$877$	−1.50307	−0.0507551	−0.0253776	−	0.999678i	$-0.508079\pi$
$877$	−1.50307	−0.0507551	−0.0253776	+	0.999678i	$0.508079\pi$
$878$	27.4863	0.927616
$879$	−26.3402	−0.888432
$880$	−1.33791	−0.0451008
$881$	−23.4908	−0.791425	−0.395712	−	0.918375i	$-0.629502\pi$
$881$	−23.4908	−0.791425	−0.395712	+	0.918375i	$0.629502\pi$
$882$	0	0
$883$	−29.0433	−0.977385	−0.488693	−	0.872456i	$-0.662526\pi$
$883$	−29.0433	−0.977385	−0.488693	+	0.872456i	$0.662526\pi$
$884$	−1.69102	−0.0568753
$885$	16.6803	0.560704
$886$	51.7068	1.73712
$887$	19.0700	0.640307	0.320153	−	0.947366i	$-0.396266\pi$
$887$	19.0700	0.640307	0.320153	+	0.947366i	$0.396266\pi$
$888$	6.66144	0.223543
$889$	0	0
$890$	−7.29299	−0.244462
$891$	−2.54638	−0.0853068
$892$	−4.04945	−0.135586
$893$	6.82273	0.228314
$894$	32.9483	1.10196
$895$	14.8371	0.495950
$896$	0	0
$897$	−3.60859	−0.120487
$898$	10.8950	0.363570
$899$	−10.0494	−0.335168
$900$	−0.0289294	−0.000964314 0
$901$	1.69102	0.0563362
$902$	−0.152221	−0.00506839
$903$	0	0
$904$	−31.5616	−1.04972
$905$	5.91548	0.196637
$906$	−20.0000	−0.664455
$907$	5.54023	0.183960	0.0919802	−	0.995761i	$-0.470680\pi$
$907$	5.54023	0.183960	0.0919802	+	0.995761i	$0.470680\pi$
$908$	4.63090	0.153682
$909$	−0.960570	−0.0318601
$910$	0	0
$911$	53.2990	1.76587	0.882937	−	0.469492i	$-0.155563\pi$
$911$	53.2990	1.76587	0.882937	+	0.469492i	$0.155563\pi$
$912$	47.5851	1.57570
$913$	−0.810439	−0.0268216
$914$	−8.95282	−0.296133
$915$	5.26180	0.173950
$916$	−8.62249	−0.284895
$917$	0	0
$918$	−40.2602	−1.32878
$919$	37.5897	1.23997	0.619985	−	0.784614i	$-0.287139\pi$
$919$	37.5897	1.23997	0.619985	+	0.784614i	$0.287139\pi$
$920$	−5.75031	−0.189582
$921$	−48.5958	−1.60129
$922$	49.7296	1.63776
$923$	8.36683	0.275398
$924$	0	0
$925$	1.55252	0.0510465
$926$	2.21849	0.0729041
$927$	−0.616522	−0.0202492
$928$	−2.06278	−0.0677140
$929$	−37.3197	−1.22442	−0.612209	−	0.790696i	$-0.709719\pi$
$929$	−37.3197	−1.22442	−0.612209	+	0.790696i	$0.709719\pi$
$930$	26.4391	0.866971
$931$	0	0
$932$	4.60281	0.150770
$933$	−33.5441	−1.09818
$934$	18.0860	0.591790
$935$	−1.44521	−0.0472635
$936$	0.181328	0.00592688
$937$	22.8638	0.746927	0.373463	−	0.927645i	$-0.378170\pi$
$937$	22.8638	0.746927	0.373463	+	0.927645i	$0.378170\pi$
$938$	0	0
$939$	39.1584	1.27788
$940$	0.416283	0.0135777
$941$	−0.523590	−0.0170686	−0.00853428	−	0.999964i	$-0.502717\pi$
$941$	−0.523590	−0.0170686	−0.00853428	+	0.999964i	$0.502717\pi$
$942$	−65.4740	−2.13326
$943$	−0.779243	−0.0253756
$944$	44.9093	1.46167
$945$	0	0
$946$	−2.55479	−0.0830633
$947$	−10.0228	−0.325697	−0.162848	−	0.986651i	$-0.552068\pi$
$947$	−10.0228	−0.325697	−0.162848	+	0.986651i	$0.552068\pi$
$948$	−7.81658	−0.253871
$949$	6.40012	0.207757
$950$	9.31124	0.302097
$951$	−39.0738	−1.26706
$952$	0	0
$953$	−8.15676	−0.264223	−0.132112	−	0.991235i	$-0.542176\pi$
$953$	−8.15676	−0.264223	−0.132112	+	0.991235i	$0.542176\pi$
$954$	0.0410378	0.00132865
$955$	7.02893	0.227451
$956$	5.07838	0.164246
$957$	−0.496928	−0.0160634
$958$	−26.4261	−0.853789
$959$	0	0
$960$	−10.3051	−0.332596
$961$	69.9914	2.25779
$962$	2.20233	0.0710059
$963$	−0.997733	−0.0321515
$964$	5.41855	0.174520
$965$	17.8660	0.575128
$966$	0	0
$967$	15.7671	0.507037	0.253518	−	0.967331i	$-0.418412\pi$
$967$	15.7671	0.507037	0.253518	+	0.967331i	$0.418412\pi$
$968$	27.4007	0.880691
$969$	51.4017	1.65126
$970$	24.4619	0.785423
$971$	17.8804	0.573810	0.286905	−	0.957959i	$-0.407374\pi$
$971$	17.8804	0.573810	0.286905	+	0.957959i	$0.407374\pi$
$972$	−0.300566	−0.00964065
$973$	0	0
$974$	−6.31494	−0.202344
$975$	−1.57531	−0.0504502
$976$	14.1666	0.453462
$977$	−55.1071	−1.76303	−0.881517	−	0.472153i	$-0.843477\pi$
$977$	−55.1071	−1.76303	−0.881517	+	0.472153i	$0.843477\pi$
$978$	−1.17727	−0.0376451
$979$	−1.37751	−0.0440255
$980$	0	0
$981$	−0.974946	−0.0311276
$982$	62.7540	2.00256
$983$	1.29687	0.0413637	0.0206818	−	0.999786i	$-0.493416\pi$
$983$	1.29687	0.0413637	0.0206818	+	0.999786i	$0.493416\pi$
$984$	−1.45959	−0.0465300
$985$	6.09890	0.194327
$986$	−7.65142	−0.243671
$987$	0	0
$988$	2.05786	0.0654692
$989$	−13.0784	−0.415868
$990$	−0.0350725	−0.00111468
$991$	−3.11942	−0.0990915	−0.0495458	−	0.998772i	$-0.515777\pi$
$991$	−3.11942	−0.0990915	−0.0495458	+	0.998772i	$0.515777\pi$
$992$	20.7298	0.658172
$993$	−41.1773	−1.30672
$994$	0	0
$995$	9.75872	0.309372
$996$	1.75872	0.0557273
$997$	−30.2472	−0.957940	−0.478970	−	0.877831i	$-0.658990\pi$
$997$	−30.2472	−0.957940	−0.478970	+	0.877831i	$0.658990\pi$
$998$	−28.4292	−0.899912
$999$	8.16904	0.258457

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7105.2.a.p.1.2		3
7.6	odd	2		145.2.a.d.1.2	✓	3
21.20	even	2		1305.2.a.o.1.2		3
28.27	even	2		2320.2.a.s.1.1		3
35.13	even	4		725.2.b.d.349.2		6
35.27	even	4		725.2.b.d.349.5		6
35.34	odd	2		725.2.a.d.1.2		3
56.13	odd	2		9280.2.a.bu.1.1		3
56.27	even	2		9280.2.a.bm.1.3		3
105.104	even	2		6525.2.a.bh.1.2		3
203.202	odd	2		4205.2.a.e.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.d.1.2	✓	3	7.6	odd	2
725.2.a.d.1.2		3	35.34	odd	2
725.2.b.d.349.2		6	35.13	even	4
725.2.b.d.349.5		6	35.27	even	4
1305.2.a.o.1.2		3	21.20	even	2
2320.2.a.s.1.1		3	28.27	even	2
4205.2.a.e.1.2		3	203.202	odd	2
6525.2.a.bh.1.2		3	105.104	even	2
7105.2.a.p.1.2		3	1.1	even	1	trivial
9280.2.a.bm.1.3		3	56.27	even	2
9280.2.a.bu.1.1		3	56.13	odd	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)

\(f(q)\)	\(=\)	\(q+1.53919 q^{2} -1.70928 q^{3} +0.369102 q^{4} +1.00000 q^{5} -2.63090 q^{6} -2.51026 q^{8} -0.0783777 q^{9} +1.53919 q^{10} +0.290725 q^{11} -0.630898 q^{12} +0.921622 q^{13} -1.70928 q^{15} -4.60197 q^{16} -4.97107 q^{17} -0.120638 q^{18} +6.04945 q^{19} +0.369102 q^{20} +0.447480 q^{22} +2.29072 q^{23} +4.29072 q^{24} +1.00000 q^{25} +1.41855 q^{26} +5.26180 q^{27} +1.00000 q^{29} -2.63090 q^{30} -10.0494 q^{31} -2.06278 q^{32} -0.496928 q^{33} -7.65142 q^{34} -0.0289294 q^{36} +1.55252 q^{37} +9.31124 q^{38} -1.57531 q^{39} -2.51026 q^{40} -0.340173 q^{41} -5.70928 q^{43} +0.107307 q^{44} -0.0783777 q^{45} +3.52586 q^{46} +1.12783 q^{47} +7.86603 q^{48} +1.53919 q^{50} +8.49693 q^{51} +0.340173 q^{52} -0.340173 q^{53} +8.09890 q^{54} +0.290725 q^{55} -10.3402 q^{57} +1.53919 q^{58} -9.75872 q^{59} -0.630898 q^{60} -3.07838 q^{61} -15.4680 q^{62} +6.02893 q^{64} +0.921622 q^{65} -0.764867 q^{66} -5.70928 q^{67} -1.83483 q^{68} -3.91548 q^{69} +9.07838 q^{71} +0.196748 q^{72} +6.94441 q^{73} +2.38962 q^{74} -1.70928 q^{75} +2.23287 q^{76} -2.42469 q^{78} +12.3896 q^{79} -4.60197 q^{80} -8.75872 q^{81} -0.523590 q^{82} -2.78765 q^{83} -4.97107 q^{85} -8.78765 q^{86} -1.70928 q^{87} -0.729794 q^{88} -4.73820 q^{89} -0.120638 q^{90} +0.845512 q^{92} +17.1773 q^{93} +1.73594 q^{94} +6.04945 q^{95} +3.52586 q^{96} +15.8927 q^{97} -0.0227863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 2 q^{3} + 5 q^{4} + 3 q^{5} - 4 q^{6} + 9 q^{8} + 3 q^{9} + 3 q^{10} + 8 q^{11} + 2 q^{12} + 6 q^{13} + 2 q^{15} + 5 q^{16} - 13 q^{18} + 5 q^{20} + 2 q^{22} + 14 q^{23} + 20 q^{24} + 3 q^{25}+ \cdots + 20 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 2 * q^3 + 5 * q^4 + 3 * q^5 - 4 * q^6 + 9 * q^8 + 3 * q^9 + 3 * q^10 + 8 * q^11 + 2 * q^12 + 6 * q^13 + 2 * q^15 + 5 * q^16 - 13 * q^18 + 5 * q^20 + 2 * q^22 + 14 * q^23 + 20 * q^24 + 3 * q^25 - 10 * q^26 + 8 * q^27 + 3 * q^29 - 4 * q^30 - 12 * q^31 + 11 * q^32 + 16 * q^33 + 14 * q^34 - 15 * q^36 + 4 * q^37 + 2 * q^38 + 16 * q^39 + 9 * q^40 + 10 * q^41 - 10 * q^43 + 12 * q^44 + 3 * q^45 + 8 * q^46 - 18 * q^47 + 10 * q^48 + 3 * q^50 + 8 * q^51 - 10 * q^52 + 10 * q^53 - 12 * q^54 + 8 * q^55 - 20 * q^57 + 3 * q^58 - 4 * q^59 + 2 * q^60 - 6 * q^61 - 14 * q^62 + 33 * q^64 + 6 * q^65 - 12 * q^66 - 10 * q^67 + 36 * q^68 + 20 * q^69 + 24 * q^71 - 3 * q^72 + 4 * q^73 - 22 * q^74 + 2 * q^75 - 16 * q^76 - 28 * q^78 + 8 * q^79 + 5 * q^80 - q^81 + 14 * q^82 + 2 * q^83 - 16 * q^86 + 2 * q^87 + 38 * q^88 - 22 * q^89 - 13 * q^90 + 22 * q^92 + 12 * q^93 + 8 * q^96 + 36 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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