LMFDB - Embedded newform 637.2.a.l.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [637,2,Mod(1,637)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("637.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 637 = 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	637.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$5.08647060876$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.746052.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 7x^{3} + 8x + 2 $ x^5 - x^4 - 7x^3 + 8x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.265608$ of defining polynomial
Character	$\chi$	$=$	637.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.26561 q^{2} -2.62728 q^{3} -0.398235 q^{4} -2.90260 q^{5} -3.32511 q^{6} -3.03523 q^{8} +3.90260 q^{9} -3.67356 q^{10} +2.03656 q^{11} +1.04628 q^{12} +1.00000 q^{13} +7.62594 q^{15} -3.04494 q^{16} +3.99866 q^{17} +4.93916 q^{18} +6.96210 q^{19} +1.15592 q^{20} +2.57749 q^{22} -0.627280 q^{23} +7.97439 q^{24} +3.42509 q^{25} +1.26561 q^{26} -2.37138 q^{27} +1.09606 q^{29} +9.65146 q^{30} -10.4325 q^{31} +2.21675 q^{32} -5.35062 q^{33} +5.06074 q^{34} -1.55415 q^{36} -3.08537 q^{37} +8.81129 q^{38} -2.62728 q^{39} +8.81005 q^{40} -0.521150 q^{41} +0.329024 q^{43} -0.811031 q^{44} -11.3277 q^{45} -0.793891 q^{46} +10.5457 q^{47} +7.99991 q^{48} +4.33482 q^{50} -10.5056 q^{51} -0.398235 q^{52} +7.11900 q^{53} -3.00124 q^{54} -5.91133 q^{55} -18.2914 q^{57} +1.38719 q^{58} +2.03656 q^{59} -3.03692 q^{60} +2.40081 q^{61} -13.2034 q^{62} +8.89542 q^{64} -2.90260 q^{65} -6.77179 q^{66} +14.6942 q^{67} -1.59241 q^{68} +1.64804 q^{69} +3.60141 q^{71} -11.8453 q^{72} +2.97573 q^{73} -3.90487 q^{74} -8.99866 q^{75} -2.77255 q^{76} -3.32511 q^{78} -8.76150 q^{79} +8.83824 q^{80} -5.47751 q^{81} -0.659572 q^{82} +12.8039 q^{83} -11.6065 q^{85} +0.416416 q^{86} -2.87966 q^{87} -6.18143 q^{88} -2.68098 q^{89} -14.3364 q^{90} +0.249805 q^{92} +27.4090 q^{93} +13.3467 q^{94} -20.2082 q^{95} -5.82403 q^{96} -2.32902 q^{97} +7.94789 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 8 q^{4} + 2 q^{5} - 5 q^{6} + 9 q^{8} + 3 q^{9} - 5 q^{10} + 11 q^{11} + 5 q^{12} + 5 q^{13} + 10 q^{16} - 5 q^{17} + 9 q^{18} + 9 q^{19} + q^{20} + 8 q^{22} + 10 q^{23} + 9 q^{25} + 4 q^{26}+ \cdots + 11 q^{99}+O(q^{100}) $ 5 * q + 4 * q^2 + 8 * q^4 + 2 * q^5 - 5 * q^6 + 9 * q^8 + 3 * q^9 - 5 * q^10 + 11 * q^11 + 5 * q^12 + 5 * q^13 + 10 * q^16 - 5 * q^17 + 9 * q^18 + 9 * q^19 + q^20 + 8 * q^22 + 10 * q^23 + 9 * q^25 + 4 * q^26 - 3 * q^29 - 13 * q^30 - 6 * q^31 + 22 * q^32 + 8 * q^33 - 22 * q^34 + 7 * q^36 + 4 * q^37 - 10 * q^38 + 28 * q^40 + 14 * q^41 + 2 * q^43 - 32 * q^45 + 3 * q^46 + q^47 - 23 * q^48 + 9 * q^50 - 8 * q^51 + 8 * q^52 + 17 * q^53 + 23 * q^54 - 16 * q^57 - 27 * q^58 + 11 * q^59 - 29 * q^60 - 11 * q^61 - 23 * q^62 + 9 * q^64 + 2 * q^65 + 21 * q^66 + 13 * q^67 - 32 * q^68 + 18 * q^69 + 15 * q^71 - 19 * q^72 - 33 * q^74 - 20 * q^75 + 8 * q^76 - 5 * q^78 + 2 * q^79 + 55 * q^80 - 19 * q^81 + 34 * q^82 + 6 * q^83 - 22 * q^85 + 28 * q^86 - 8 * q^87 - 3 * q^88 - 4 * q^89 - 34 * q^90 + 21 * q^92 + 18 * q^93 + 20 * q^94 - 12 * q^95 - 37 * q^96 - 12 * q^97 + 11 * 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 $

$2$	1.26561	0.894920	0.447460	−	0.894304i	$-0.352329\pi$
$2$	1.26561	0.894920	0.447460	+	0.894304i	$0.352329\pi$
$3$	−2.62728	−1.51686	−0.758430	−	0.651754i	$-0.774033\pi$
$3$	−2.62728	−1.51686	−0.758430	+	0.651754i	$0.774033\pi$
$4$	−0.398235	−0.199118
$5$	−2.90260	−1.29808	−0.649041	−	0.760753i	$-0.724830\pi$
$5$	−2.90260	−1.29808	−0.649041	+	0.760753i	$0.724830\pi$
$6$	−3.32511	−1.35747
$7$	0	0
$7$	0	0
$8$	−3.03523	−1.07311
$9$	3.90260	1.30087
$10$	−3.67356	−1.16168
$11$	2.03656	0.614047	0.307024	−	0.951702i	$-0.400667\pi$
$11$	2.03656	0.614047	0.307024	+	0.951702i	$0.400667\pi$
$12$	1.04628	0.302034
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	7.62594	1.96901
$16$	−3.04494	−0.761235
$17$	3.99866	0.969818	0.484909	−	0.874565i	$-0.338853\pi$
$17$	3.99866	0.969818	0.484909	+	0.874565i	$0.338853\pi$
$18$	4.93916	1.16417
$19$	6.96210	1.59722	0.798608	−	0.601852i	$-0.205570\pi$
$19$	6.96210	1.59722	0.798608	+	0.601852i	$0.205570\pi$
$20$	1.15592	0.258471
$21$	0	0
$22$	2.57749	0.549523
$23$	−0.627280	−0.130797	−0.0653985	−	0.997859i	$-0.520832\pi$
$23$	−0.627280	−0.130797	−0.0653985	+	0.997859i	$0.520832\pi$
$24$	7.97439	1.62777
$25$	3.42509	0.685017
$26$	1.26561	0.248206
$27$	−2.37138	−0.456373
$28$	0	0
$29$	1.09606	0.203534	0.101767	−	0.994808i	$-0.467550\pi$
$29$	1.09606	0.203534	0.101767	+	0.994808i	$0.467550\pi$
$30$	9.65146	1.76211
$31$	−10.4325	−1.87373	−0.936864	−	0.349693i	$-0.886286\pi$
$31$	−10.4325	−1.87373	−0.936864	+	0.349693i	$0.886286\pi$
$32$	2.21675	0.391870
$33$	−5.35062	−0.931424
$34$	5.06074	0.867910
$35$	0	0
$36$	−1.55415	−0.259025
$37$	−3.08537	−0.507232	−0.253616	−	0.967305i	$-0.581620\pi$
$37$	−3.08537	−0.507232	−0.253616	+	0.967305i	$0.581620\pi$
$38$	8.81129	1.42938
$39$	−2.62728	−0.420701
$40$	8.81005	1.39299
$41$	−0.521150	−0.0813900	−0.0406950	−	0.999172i	$-0.512957\pi$
$41$	−0.521150	−0.0813900	−0.0406950	+	0.999172i	$0.512957\pi$
$42$	0	0
$43$	0.329024	0.0501757	0.0250879	−	0.999685i	$-0.492013\pi$
$43$	0.329024	0.0501757	0.0250879	+	0.999685i	$0.492013\pi$
$44$	−0.811031	−0.122268
$45$	−11.3277	−1.68863
$46$	−0.793891	−0.117053
$47$	10.5457	1.53825	0.769123	−	0.639101i	$-0.220693\pi$
$47$	10.5457	1.53825	0.769123	+	0.639101i	$0.220693\pi$
$48$	7.99991	1.15469
$49$	0	0
$50$	4.33482	0.613036
$51$	−10.5056	−1.47108
$52$	−0.398235	−0.0552253
$53$	7.11900	0.977870	0.488935	−	0.872320i	$-0.337386\pi$
$53$	7.11900	0.977870	0.488935	+	0.872320i	$0.337386\pi$
$54$	−3.00124	−0.408417
$55$	−5.91133	−0.797084
$56$	0	0
$57$	−18.2914	−2.42275
$58$	1.38719	0.182147
$59$	2.03656	0.265138	0.132569	−	0.991174i	$-0.457677\pi$
$59$	2.03656	0.265138	0.132569	+	0.991174i	$0.457677\pi$
$60$	−3.03692	−0.392065
$61$	2.40081	0.307393	0.153696	−	0.988118i	$-0.450882\pi$
$61$	2.40081	0.307393	0.153696	+	0.988118i	$0.450882\pi$
$62$	−13.2034	−1.67684
$63$	0	0
$64$	8.89542	1.11193
$65$	−2.90260	−0.360023
$66$	−6.77179	−0.833550
$67$	14.6942	1.79518	0.897589	−	0.440832i	$-0.145317\pi$
$67$	14.6942	1.79518	0.897589	+	0.440832i	$0.145317\pi$
$68$	−1.59241	−0.193108
$69$	1.64804	0.198401
$70$	0	0
$71$	3.60141	0.427409	0.213704	−	0.976898i	$-0.431447\pi$
$71$	3.60141	0.427409	0.213704	+	0.976898i	$0.431447\pi$
$72$	−11.8453	−1.39598
$73$	2.97573	0.348283	0.174141	−	0.984721i	$-0.444285\pi$
$73$	2.97573	0.348283	0.174141	+	0.984721i	$0.444285\pi$
$74$	−3.90487	−0.453932
$75$	−8.99866	−1.03908
$76$	−2.77255	−0.318034
$77$	0	0
$78$	−3.32511	−0.376494
$79$	−8.76150	−0.985746	−0.492873	−	0.870101i	$-0.664053\pi$
$79$	−8.76150	−0.985746	−0.492873	+	0.870101i	$0.664053\pi$
$80$	8.83824	0.988145
$81$	−5.47751	−0.608613
$82$	−0.659572	−0.0728376
$83$	12.8039	1.40541	0.702703	−	0.711483i	$-0.251976\pi$
$83$	12.8039	1.40541	0.702703	+	0.711483i	$0.251976\pi$
$84$	0	0
$85$	−11.6065	−1.25890
$86$	0.416416	0.0449033
$87$	−2.87966	−0.308732
$88$	−6.18143	−0.658943
$89$	−2.68098	−0.284184	−0.142092	−	0.989853i	$-0.545383\pi$
$89$	−2.68098	−0.284184	−0.142092	+	0.989853i	$0.545383\pi$
$90$	−14.3364	−1.51119
$91$	0	0
$92$	0.249805	0.0260440
$93$	27.4090	2.84219
$94$	13.3467	1.37661
$95$	−20.2082	−2.07332
$96$	−5.82403	−0.594413
$97$	−2.32902	−0.236477	−0.118238	−	0.992985i	$-0.537725\pi$
$97$	−2.32902	−0.236477	−0.118238	+	0.992985i	$0.537725\pi$
$98$	0	0
$99$	7.94789	0.798793
$100$	−1.36399	−0.136399
$101$	1.45324	0.144603	0.0723014	−	0.997383i	$-0.476966\pi$
$101$	1.45324	0.144603	0.0723014	+	0.997383i	$0.476966\pi$
$102$	−13.2960	−1.31650
$103$	−11.6353	−1.14646	−0.573230	−	0.819394i	$-0.694310\pi$
$103$	−11.6353	−1.14646	−0.573230	+	0.819394i	$0.694310\pi$
$104$	−3.03523	−0.297628
$105$	0	0
$106$	9.00987	0.875115
$107$	19.6259	1.89731	0.948656	−	0.316310i	$-0.102444\pi$
$107$	19.6259	1.89731	0.948656	+	0.316310i	$0.102444\pi$
$108$	0.944368	0.0908719
$109$	−1.10676	−0.106008	−0.0530040	−	0.998594i	$-0.516880\pi$
$109$	−1.10676	−0.106008	−0.0530040	+	0.998594i	$0.516880\pi$
$110$	−7.48143	−0.713326
$111$	8.10613	0.769400
$112$	0	0
$113$	−1.09606	−0.103109	−0.0515545	−	0.998670i	$-0.516418\pi$
$113$	−1.09606	−0.103109	−0.0515545	+	0.998670i	$0.516418\pi$
$114$	−23.1497	−2.16817
$115$	1.82074	0.169785
$116$	−0.436491	−0.0405272
$117$	3.90260	0.360796
$118$	2.57749	0.237277
$119$	0	0
$120$	−23.1465	−2.11297
$121$	−6.85241	−0.622946
$122$	3.03849	0.275092
$123$	1.36921	0.123457
$124$	4.15458	0.373092
$125$	4.57134	0.408873
$126$	0	0
$127$	5.18143	0.459778	0.229889	−	0.973217i	$-0.426164\pi$
$127$	5.18143	0.459778	0.229889	+	0.973217i	$0.426164\pi$
$128$	6.82461	0.603216
$129$	−0.864439	−0.0761096
$130$	−3.67356	−0.322192
$131$	10.5667	0.923217	0.461609	−	0.887084i	$-0.347272\pi$
$131$	10.5667	0.923217	0.461609	+	0.887084i	$0.347272\pi$
$132$	2.13081	0.185463
$133$	0	0
$134$	18.5971	1.60654
$135$	6.88318	0.592410
$136$	−12.1368	−1.04073
$137$	−5.87177	−0.501659	−0.250830	−	0.968031i	$-0.580703\pi$
$137$	−5.87177	−0.501659	−0.250830	+	0.968031i	$0.580703\pi$
$138$	2.08577	0.177553
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−27.7065	−2.33331
$142$	4.55797	0.382497
$143$	2.03656	0.170306
$144$	−11.8832	−0.990265
$145$	−3.18143	−0.264204
$146$	3.76611	0.311685
$147$	0	0
$148$	1.22870	0.100999
$149$	−10.1054	−0.827868	−0.413934	−	0.910307i	$-0.635846\pi$
$149$	−10.1054	−0.827868	−0.413934	+	0.910307i	$0.635846\pi$
$150$	−11.3888	−0.929890
$151$	−0.187726	−0.0152769	−0.00763847	−	0.999971i	$-0.502431\pi$
$151$	−0.187726	−0.0152769	−0.00763847	+	0.999971i	$0.502431\pi$
$152$	−21.1316	−1.71400
$153$	15.6052	1.26160
$154$	0	0
$155$	30.2813	2.43225
$156$	1.04628	0.0837691
$157$	−12.0718	−0.963434	−0.481717	−	0.876327i	$-0.659987\pi$
$157$	−12.0718	−0.963434	−0.481717	+	0.876327i	$0.659987\pi$
$158$	−11.0886	−0.882164
$159$	−18.7036	−1.48329
$160$	−6.43435	−0.508680
$161$	0	0
$162$	−6.93239	−0.544660
$163$	14.9136	1.16812	0.584060	−	0.811711i	$-0.301463\pi$
$163$	14.9136	1.16812	0.584060	+	0.811711i	$0.301463\pi$
$164$	0.207540	0.0162062
$165$	15.5307	1.20906
$166$	16.2047	1.25773
$167$	5.05664	0.391294	0.195647	−	0.980674i	$-0.437319\pi$
$167$	5.05664	0.391294	0.195647	+	0.980674i	$0.437319\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−14.6893	−1.12662
$171$	27.1703	2.07776
$172$	−0.131029	−0.00999087
$173$	−0.595615	−0.0452837	−0.0226419	−	0.999744i	$-0.507208\pi$
$173$	−0.595615	−0.0452837	−0.0226419	+	0.999744i	$0.507208\pi$
$174$	−3.64453	−0.276291
$175$	0	0
$176$	−6.20121	−0.467434
$177$	−5.35062	−0.402177
$178$	−3.39308	−0.254322
$179$	8.07664	0.603676	0.301838	−	0.953359i	$-0.402400\pi$
$179$	8.07664	0.603676	0.301838	+	0.953359i	$0.402400\pi$
$180$	4.51108	0.336236
$181$	1.89324	0.140724	0.0703618	−	0.997522i	$-0.477585\pi$
$181$	1.89324	0.140724	0.0703618	+	0.997522i	$0.477585\pi$
$182$	0	0
$183$	−6.30761	−0.466272
$184$	1.90394	0.140360
$185$	8.95559	0.658428
$186$	34.6891	2.54353
$187$	8.14353	0.595514
$188$	−4.19966	−0.306292
$189$	0	0
$190$	−25.5757	−1.85545
$191$	3.70174	0.267849	0.133924	−	0.990992i	$-0.457242\pi$
$191$	3.70174	0.267849	0.133924	+	0.990992i	$0.457242\pi$
$192$	−23.3708	−1.68664
$193$	13.5875	0.978047	0.489024	−	0.872271i	$-0.337353\pi$
$193$	13.5875	0.978047	0.489024	+	0.872271i	$0.337353\pi$
$194$	−2.94763	−0.211628
$195$	7.62594	0.546105
$196$	0	0
$197$	−9.70258	−0.691280	−0.345640	−	0.938367i	$-0.612338\pi$
$197$	−9.70258	−0.691280	−0.345640	+	0.938367i	$0.612338\pi$
$198$	10.0589	0.714856
$199$	26.2720	1.86237	0.931185	−	0.364547i	$-0.118776\pi$
$199$	26.2720	1.86237	0.931185	+	0.364547i	$0.118776\pi$
$200$	−10.3959	−0.735102
$201$	−38.6057	−2.72304
$202$	1.83923	0.129408
$203$	0	0
$204$	4.18370	0.292918
$205$	1.51269	0.105651
$206$	−14.7257	−1.02599
$207$	−2.44802	−0.170149
$208$	−3.04494	−0.211128
$209$	14.1788	0.980765
$210$	0	0
$211$	10.0338	0.690758	0.345379	−	0.938463i	$-0.387750\pi$
$211$	10.0338	0.690758	0.345379	+	0.938463i	$0.387750\pi$
$212$	−2.83504	−0.194711
$213$	−9.46191	−0.648320
$214$	24.8388	1.69794
$215$	−0.955026	−0.0651322
$216$	7.19769	0.489740
$217$	0	0
$218$	−1.40072	−0.0948688
$219$	−7.81807	−0.528296
$220$	2.35410	0.158713
$221$	3.99866	0.268979
$222$	10.2592	0.688552
$223$	17.4961	1.17163	0.585813	−	0.810446i	$-0.300775\pi$
$223$	17.4961	1.17163	0.585813	+	0.810446i	$0.300775\pi$
$224$	0	0
$225$	13.3667	0.891116
$226$	−1.38719	−0.0922743
$227$	−9.51630	−0.631619	−0.315810	−	0.948823i	$-0.602276\pi$
$227$	−9.51630	−0.631619	−0.315810	+	0.948823i	$0.602276\pi$
$228$	7.28427	0.482413
$229$	21.1170	1.39545	0.697725	−	0.716366i	$-0.254196\pi$
$229$	21.1170	1.39545	0.697725	+	0.716366i	$0.254196\pi$
$230$	2.30435	0.151944
$231$	0	0
$232$	−3.32680	−0.218415
$233$	14.1788	0.928881	0.464441	−	0.885604i	$-0.346255\pi$
$233$	14.1788	0.928881	0.464441	+	0.885604i	$0.346255\pi$
$234$	4.93916	0.322883
$235$	−30.6099	−1.99677
$236$	−0.811031	−0.0527936
$237$	23.0189	1.49524
$238$	0	0
$239$	−16.5275	−1.06907	−0.534536	−	0.845145i	$-0.679514\pi$
$239$	−16.5275	−1.06907	−0.534536	+	0.845145i	$0.679514\pi$
$240$	−23.2205	−1.49888
$241$	−13.6890	−0.881786	−0.440893	−	0.897560i	$-0.645338\pi$
$241$	−13.6890	−0.881786	−0.440893	+	0.897560i	$0.645338\pi$
$242$	−8.67247	−0.557487
$243$	21.5051	1.37955
$244$	−0.956089	−0.0612073
$245$	0	0
$246$	1.73288	0.110484
$247$	6.96210	0.442988
$248$	31.6649	2.01073
$249$	−33.6393	−2.13181
$250$	5.78553	0.365909
$251$	−14.6603	−0.925349	−0.462674	−	0.886528i	$-0.653110\pi$
$251$	−14.6603	−0.925349	−0.462674	+	0.886528i	$0.653110\pi$
$252$	0	0
$253$	−1.27750	−0.0803155
$254$	6.55767	0.411464
$255$	30.4936	1.90958
$256$	−9.15355	−0.572097
$257$	−1.75277	−0.109335	−0.0546675	−	0.998505i	$-0.517410\pi$
$257$	−1.75277	−0.109335	−0.0546675	+	0.998505i	$0.517410\pi$
$258$	−1.09404	−0.0681120
$259$	0	0
$260$	1.15592	0.0716870
$261$	4.27750	0.264770
$262$	13.3733	0.826206
$263$	26.9416	1.66129	0.830645	−	0.556802i	$-0.187972\pi$
$263$	26.9416	1.66129	0.830645	+	0.556802i	$0.187972\pi$
$264$	16.2404	0.999525
$265$	−20.6636	−1.26936
$266$	0	0
$267$	7.04370	0.431067
$268$	−5.85174	−0.357452
$269$	−22.0691	−1.34558	−0.672789	−	0.739835i	$-0.734904\pi$
$269$	−22.0691	−1.34558	−0.672789	+	0.739835i	$0.734904\pi$
$270$	8.71141	0.530159
$271$	−8.96210	−0.544409	−0.272204	−	0.962239i	$-0.587753\pi$
$271$	−8.96210	−0.544409	−0.272204	+	0.962239i	$0.587753\pi$
$272$	−12.1757	−0.738259
$273$	0	0
$274$	−7.43137	−0.448945
$275$	6.97541	0.420633
$276$	−0.656308	−0.0395051
$277$	−7.52925	−0.452389	−0.226194	−	0.974082i	$-0.572628\pi$
$277$	−7.52925	−0.452389	−0.226194	+	0.974082i	$0.572628\pi$
$278$	−5.06243	−0.303625
$279$	−40.7138	−2.43747
$280$	0	0
$281$	29.7762	1.77630	0.888151	−	0.459553i	$-0.151990\pi$
$281$	29.7762	1.77630	0.888151	+	0.459553i	$0.151990\pi$
$282$	−35.0655	−2.08812
$283$	0.301451	0.0179194	0.00895970	−	0.999960i	$-0.497148\pi$
$283$	0.301451	0.0179194	0.00895970	+	0.999960i	$0.497148\pi$
$284$	−1.43421	−0.0851046
$285$	53.0926	3.14493
$286$	2.57749	0.152410
$287$	0	0
$288$	8.65110	0.509771
$289$	−1.01069	−0.0594526
$290$	−4.02645	−0.236441
$291$	6.11900	0.358702
$292$	−1.18504	−0.0693492
$293$	−19.2471	−1.12443	−0.562214	−	0.826992i	$-0.690050\pi$
$293$	−19.2471	−1.12443	−0.562214	+	0.826992i	$0.690050\pi$
$294$	0	0
$295$	−5.91133	−0.344171
$296$	9.36480	0.544318
$297$	−4.82947	−0.280234
$298$	−12.7895	−0.740876
$299$	−0.627280	−0.0362765
$300$	3.58358	0.206898
$301$	0	0
$302$	−0.237588	−0.0136716
$303$	−3.81807	−0.219342
$304$	−21.1992	−1.21586
$305$	−6.96860	−0.399021
$306$	19.7501	1.12904
$307$	−3.57779	−0.204195	−0.102098	−	0.994774i	$-0.532555\pi$
$307$	−3.57779	−0.204195	−0.102098	+	0.994774i	$0.532555\pi$
$308$	0	0
$309$	30.5692	1.73902
$310$	38.3243	2.17667
$311$	−23.8306	−1.35131	−0.675655	−	0.737218i	$-0.736139\pi$
$311$	−23.8306	−1.35131	−0.675655	+	0.737218i	$0.736139\pi$
$312$	7.97439	0.451461
$313$	−18.0814	−1.02202	−0.511009	−	0.859575i	$-0.670728\pi$
$313$	−18.0814	−1.02202	−0.511009	+	0.859575i	$0.670728\pi$
$314$	−15.2782	−0.862196
$315$	0	0
$316$	3.48914	0.196279
$317$	−27.5482	−1.54726	−0.773630	−	0.633638i	$-0.781561\pi$
$317$	−27.5482	−1.54726	−0.773630	+	0.633638i	$0.781561\pi$
$318$	−23.6714	−1.32743
$319$	2.23220	0.124979
$320$	−25.8198	−1.44337
$321$	−51.5628	−2.87796
$322$	0	0
$323$	27.8391	1.54901
$324$	2.18134	0.121185
$325$	3.42509	0.189990
$326$	18.8747	1.04537
$327$	2.90776	0.160799
$328$	1.58181	0.0873408
$329$	0	0
$330$	19.6558	1.08202
$331$	−18.1814	−0.999339	−0.499669	−	0.866216i	$-0.666545\pi$
$331$	−18.1814	−0.999339	−0.499669	+	0.866216i	$0.666545\pi$
$332$	−5.09895	−0.279841
$333$	−12.0410	−0.659841
$334$	6.39972	0.350177
$335$	−42.6513	−2.33029
$336$	0	0
$337$	−17.1381	−0.933572	−0.466786	−	0.884370i	$-0.654588\pi$
$337$	−17.1381	−0.933572	−0.466786	+	0.884370i	$0.654588\pi$
$338$	1.26561	0.0688400
$339$	2.87966	0.156402
$340$	4.62212	0.250670
$341$	−21.2464	−1.15056
$342$	34.3869	1.85943
$343$	0	0
$344$	−0.998663	−0.0538443
$345$	−4.78360	−0.257540
$346$	−0.753815	−0.0405253
$347$	22.2688	1.19545	0.597725	−	0.801701i	$-0.296071\pi$
$347$	22.2688	1.19545	0.597725	+	0.801701i	$0.296071\pi$
$348$	1.14678	0.0614741
$349$	19.9368	1.06719	0.533595	−	0.845740i	$-0.320841\pi$
$349$	19.9368	1.06719	0.533595	+	0.845740i	$0.320841\pi$
$350$	0	0
$351$	−2.37138	−0.126575
$352$	4.51456	0.240627
$353$	22.9152	1.21965	0.609825	−	0.792536i	$-0.291240\pi$
$353$	22.9152	1.21965	0.609825	+	0.792536i	$0.291240\pi$
$354$	−6.77179	−0.359917
$355$	−10.4535	−0.554812
$356$	1.06766	0.0565860
$357$	0	0
$358$	10.2219	0.540242
$359$	27.2314	1.43722	0.718610	−	0.695413i	$-0.244779\pi$
$359$	27.2314	1.43722	0.718610	+	0.695413i	$0.244779\pi$
$360$	34.3821	1.81210
$361$	29.4708	1.55110
$362$	2.39611	0.125936
$363$	18.0032	0.944923
$364$	0	0
$365$	−8.63735	−0.452099
$366$	−7.98297	−0.417276
$367$	−10.8564	−0.566702	−0.283351	−	0.959016i	$-0.591446\pi$
$367$	−10.8564	−0.566702	−0.283351	+	0.959016i	$0.591446\pi$
$368$	1.91003	0.0995671
$369$	−2.03384	−0.105878
$370$	11.3343	0.589241
$371$	0	0
$372$	−10.9152	−0.565929
$373$	−2.37144	−0.122789	−0.0613943	−	0.998114i	$-0.519555\pi$
$373$	−2.37144	−0.122789	−0.0613943	+	0.998114i	$0.519555\pi$
$374$	10.3065	0.532938
$375$	−12.0102	−0.620204
$376$	−32.0085	−1.65071
$377$	1.09606	0.0564501
$378$	0	0
$379$	−29.2197	−1.50092	−0.750458	−	0.660918i	$-0.770167\pi$
$379$	−29.2197	−1.50092	−0.750458	+	0.660918i	$0.770167\pi$
$380$	8.04761	0.412834
$381$	−13.6131	−0.697419
$382$	4.68496	0.239703
$383$	−3.06595	−0.156663	−0.0783313	−	0.996927i	$-0.524959\pi$
$383$	−3.06595	−0.156663	−0.0783313	+	0.996927i	$0.524959\pi$
$384$	−17.9302	−0.914995
$385$	0	0
$386$	17.1964	0.875274
$387$	1.28405	0.0652719
$388$	0.927499	0.0470866
$389$	27.7410	1.40652	0.703261	−	0.710932i	$-0.251726\pi$
$389$	27.7410	1.40652	0.703261	+	0.710932i	$0.251726\pi$
$390$	9.65146	0.488721
$391$	−2.50828	−0.126849
$392$	0	0
$393$	−27.7617	−1.40039
$394$	−12.2797	−0.618641
$395$	25.4311	1.27958
$396$	−3.16513	−0.159054
$397$	17.2312	0.864808	0.432404	−	0.901680i	$-0.357665\pi$
$397$	17.2312	0.864808	0.432404	+	0.901680i	$0.357665\pi$
$398$	33.2500	1.66667
$399$	0	0
$400$	−10.4292	−0.521459
$401$	16.6440	0.831163	0.415582	−	0.909556i	$-0.363578\pi$
$401$	16.6440	0.831163	0.415582	+	0.909556i	$0.363578\pi$
$402$	−48.8597	−2.43690
$403$	−10.4325	−0.519679
$404$	−0.578731	−0.0287930
$405$	15.8990	0.790029
$406$	0	0
$407$	−6.28355	−0.311464
$408$	31.8869	1.57864
$409$	−13.6338	−0.674147	−0.337073	−	0.941478i	$-0.609437\pi$
$409$	−13.6338	−0.674147	−0.337073	+	0.941478i	$0.609437\pi$
$410$	1.91447	0.0945491
$411$	15.4268	0.760948
$412$	4.63359	0.228280
$413$	0	0
$414$	−3.09824	−0.152270
$415$	−37.1645	−1.82433
$416$	2.21675	0.108685
$417$	10.5091	0.514634
$418$	17.9448	0.877707
$419$	−10.8502	−0.530066	−0.265033	−	0.964239i	$-0.585383\pi$
$419$	−10.8502	−0.530066	−0.265033	+	0.964239i	$0.585383\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	12.6989	0.618173
$423$	41.1556	2.00105
$424$	−21.6078	−1.04937
$425$	13.6958	0.664342
$426$	−11.9751	−0.580194
$427$	0	0
$428$	−7.81574	−0.377788
$429$	−5.35062	−0.258331
$430$	−1.20869	−0.0582881
$431$	1.20953	0.0582609	0.0291304	−	0.999576i	$-0.490726\pi$
$431$	1.20953	0.0582609	0.0291304	+	0.999576i	$0.490726\pi$
$432$	7.22072	0.347407
$433$	−5.56422	−0.267399	−0.133700	−	0.991022i	$-0.542686\pi$
$433$	−5.56422	−0.267399	−0.133700	+	0.991022i	$0.542686\pi$
$434$	0	0
$435$	8.35851	0.400760
$436$	0.440749	0.0211081
$437$	−4.36719	−0.208911
$438$	−9.89461	−0.472783
$439$	−19.7192	−0.941146	−0.470573	−	0.882361i	$-0.655953\pi$
$439$	−19.7192	−0.941146	−0.470573	+	0.882361i	$0.655953\pi$
$440$	17.9422	0.855362
$441$	0	0
$442$	5.06074	0.240715
$443$	22.2310	1.05623	0.528113	−	0.849174i	$-0.322900\pi$
$443$	22.2310	1.05623	0.528113	+	0.849174i	$0.322900\pi$
$444$	−3.22815	−0.153201
$445$	7.78182	0.368894
$446$	22.1432	1.04851
$447$	26.5498	1.25576
$448$	0	0
$449$	18.4579	0.871082	0.435541	−	0.900169i	$-0.356557\pi$
$449$	18.4579	0.871082	0.435541	+	0.900169i	$0.356557\pi$
$450$	16.9171	0.797478
$451$	−1.06136	−0.0499773
$452$	0.436491	0.0205308
$453$	0.493209	0.0231730
$454$	−12.0439	−0.565249
$455$	0	0
$456$	55.5185	2.59989
$457$	−29.9819	−1.40250	−0.701248	−	0.712917i	$-0.747373\pi$
$457$	−29.9819	−1.40250	−0.701248	+	0.712917i	$0.747373\pi$
$458$	26.7258	1.24882
$459$	−9.48236	−0.442599
$460$	−0.725084	−0.0338072
$461$	29.1498	1.35764	0.678821	−	0.734304i	$-0.262491\pi$
$461$	29.1498	1.35764	0.678821	+	0.734304i	$0.262491\pi$
$462$	0	0
$463$	1.55900	0.0724530	0.0362265	−	0.999344i	$-0.488466\pi$
$463$	1.55900	0.0724530	0.0362265	+	0.999344i	$0.488466\pi$
$464$	−3.33744	−0.154937
$465$	−79.5575	−3.68939
$466$	17.9448	0.831275
$467$	12.4231	0.574874	0.287437	−	0.957800i	$-0.407197\pi$
$467$	12.4231	0.574874	0.287437	+	0.957800i	$0.407197\pi$
$468$	−1.55415	−0.0718407
$469$	0	0
$470$	−38.7402	−1.78695
$471$	31.7160	1.46139
$472$	−6.18143	−0.284523
$473$	0.670079	0.0308103
$474$	29.1329	1.33812
$475$	23.8458	1.09412
$476$	0	0
$477$	27.7826	1.27208
$478$	−20.9173	−0.956735
$479$	36.0558	1.64743	0.823716	−	0.567003i	$-0.191897\pi$
$479$	36.0558	1.64743	0.823716	+	0.567003i	$0.191897\pi$
$480$	16.9048	0.771597
$481$	−3.08537	−0.140681
$482$	−17.3249	−0.789128
$483$	0	0
$484$	2.72887	0.124040
$485$	6.76023	0.306966
$486$	27.2170	1.23459
$487$	7.30004	0.330796	0.165398	−	0.986227i	$-0.447109\pi$
$487$	7.30004	0.330796	0.165398	+	0.986227i	$0.447109\pi$
$488$	−7.28702	−0.329868
$489$	−39.1821	−1.77188
$490$	0	0
$491$	4.49178	0.202711	0.101356	−	0.994850i	$-0.467682\pi$
$491$	4.49178	0.202711	0.101356	+	0.994850i	$0.467682\pi$
$492$	−0.545267	−0.0245825
$493$	4.38279	0.197391
$494$	8.81129	0.396439
$495$	−23.0696	−1.03690
$496$	31.7663	1.42635
$497$	0	0
$498$	−42.5742	−1.90780
$499$	11.3674	0.508873	0.254437	−	0.967089i	$-0.418110\pi$
$499$	11.3674	0.508873	0.254437	+	0.967089i	$0.418110\pi$
$500$	−1.82047	−0.0814139
$501$	−13.2852	−0.593539
$502$	−18.5542	−0.828113
$503$	17.1080	0.762806	0.381403	−	0.924409i	$-0.375441\pi$
$503$	17.1080	0.762806	0.381403	+	0.924409i	$0.375441\pi$
$504$	0	0
$505$	−4.21818	−0.187706
$506$	−1.61681	−0.0718759
$507$	−2.62728	−0.116682
$508$	−2.06343	−0.0915498
$509$	3.28284	0.145509	0.0727547	−	0.997350i	$-0.476821\pi$
$509$	3.28284	0.145509	0.0727547	+	0.997350i	$0.476821\pi$
$510$	38.5929	1.70892
$511$	0	0
$512$	−25.2340	−1.11520
$513$	−16.5098	−0.728926
$514$	−2.21833	−0.0978462
$515$	33.7726	1.48820
$516$	0.344250	0.0151548
$517$	21.4770	0.944555
$518$	0	0
$519$	1.56485	0.0686891
$520$	8.81005	0.386346
$521$	−4.77061	−0.209004	−0.104502	−	0.994525i	$-0.533325\pi$
$521$	−4.77061	−0.209004	−0.104502	+	0.994525i	$0.533325\pi$
$522$	5.41363	0.236948
$523$	25.5124	1.11558	0.557789	−	0.829983i	$-0.311650\pi$
$523$	25.5124	1.11558	0.557789	+	0.829983i	$0.311650\pi$
$524$	−4.20803	−0.183829
$525$	0	0
$526$	34.0975	1.48672
$527$	−41.7160	−1.81718
$528$	16.2923	0.709032
$529$	−22.6065	−0.982892
$530$	−26.1520	−1.13597
$531$	7.94789	0.344909
$532$	0	0
$533$	−0.521150	−0.0225735
$534$	8.91456	0.385771
$535$	−56.9663	−2.46287
$536$	−44.6001	−1.92643
$537$	−21.2196	−0.915693
$538$	−27.9309	−1.20418
$539$	0	0
$540$	−2.74112	−0.117959
$541$	−16.5157	−0.710064	−0.355032	−	0.934854i	$-0.615530\pi$
$541$	−16.5157	−0.710064	−0.355032	+	0.934854i	$0.615530\pi$
$542$	−11.3425	−0.487202
$543$	−4.97408	−0.213458
$544$	8.86405	0.380043
$545$	3.21247	0.137607
$546$	0	0
$547$	23.3317	0.997591	0.498796	−	0.866720i	$-0.333776\pi$
$547$	23.3317	0.997591	0.498796	+	0.866720i	$0.333776\pi$
$548$	2.33835	0.0998892
$549$	9.36942	0.399877
$550$	8.82814	0.376433
$551$	7.63090	0.325087
$552$	−5.00218	−0.212907
$553$	0	0
$554$	−9.52909	−0.404852
$555$	−23.5289	−0.998744
$556$	1.59294	0.0675557
$557$	−20.0471	−0.849422	−0.424711	−	0.905329i	$-0.639624\pi$
$557$	−20.0471	−0.849422	−0.424711	+	0.905329i	$0.639624\pi$
$558$	−51.5277	−2.18134
$559$	0.329024	0.0139162
$560$	0	0
$561$	−21.3953	−0.903312
$562$	37.6851	1.58965
$563$	−40.5284	−1.70807	−0.854034	−	0.520218i	$-0.825851\pi$
$563$	−40.5284	−1.70807	−0.854034	+	0.520218i	$0.825851\pi$
$564$	11.0337	0.464602
$565$	3.18143	0.133844
$566$	0.381519	0.0160364
$567$	0	0
$568$	−10.9311	−0.458659
$569$	−21.4504	−0.899246	−0.449623	−	0.893219i	$-0.648442\pi$
$569$	−21.4504	−0.899246	−0.449623	+	0.893219i	$0.648442\pi$
$570$	67.1944	2.81446
$571$	−10.9559	−0.458489	−0.229244	−	0.973369i	$-0.573625\pi$
$571$	−10.9559	−0.458489	−0.229244	+	0.973369i	$0.573625\pi$
$572$	−0.811031	−0.0339109
$573$	−9.72552	−0.406289
$574$	0	0
$575$	−2.14849	−0.0895982
$576$	34.7153	1.44647
$577$	34.7415	1.44631	0.723154	−	0.690687i	$-0.242692\pi$
$577$	34.7415	1.44631	0.723154	+	0.690687i	$0.242692\pi$
$578$	−1.27914	−0.0532053
$579$	−35.6981	−1.48356
$580$	1.26696	0.0526076
$581$	0	0
$582$	7.74426	0.321010
$583$	14.4983	0.600458
$584$	−9.03201	−0.373747
$585$	−11.3277	−0.468342
$586$	−24.3593	−1.00627
$587$	−22.8463	−0.942967	−0.471483	−	0.881875i	$-0.656281\pi$
$587$	−22.8463	−0.942967	−0.471483	+	0.881875i	$0.656281\pi$
$588$	0	0
$589$	−72.6320	−2.99275
$590$	−7.48143	−0.308006
$591$	25.4914	1.04858
$592$	9.39476	0.386122
$593$	17.5935	0.722480	0.361240	−	0.932473i	$-0.382354\pi$
$593$	17.5935	0.722480	0.361240	+	0.932473i	$0.382354\pi$
$594$	−6.11222	−0.250787
$595$	0	0
$596$	4.02433	0.164843
$597$	−69.0238	−2.82496
$598$	−0.793891	−0.0324646
$599$	31.0073	1.26692	0.633461	−	0.773774i	$-0.281634\pi$
$599$	31.0073	1.26692	0.633461	+	0.773774i	$0.281634\pi$
$600$	27.3130	1.11505
$601$	−1.43754	−0.0586385	−0.0293193	−	0.999570i	$-0.509334\pi$
$601$	−1.43754	−0.0586385	−0.0293193	+	0.999570i	$0.509334\pi$
$602$	0	0
$603$	57.3455	2.33529
$604$	0.0747591	0.00304191
$605$	19.8898	0.808635
$606$	−4.83218	−0.196294
$607$	−33.0171	−1.34012	−0.670061	−	0.742306i	$-0.733732\pi$
$607$	−33.0171	−1.34012	−0.670061	+	0.742306i	$0.733732\pi$
$608$	15.4333	0.625901
$609$	0	0
$610$	−8.81953	−0.357092
$611$	10.5457	0.426633
$612$	−6.21453	−0.251208
$613$	−43.1657	−1.74345	−0.871723	−	0.489999i	$-0.836997\pi$
$613$	−43.1657	−1.74345	−0.871723	+	0.489999i	$0.836997\pi$
$614$	−4.52808	−0.182738
$615$	−3.97426	−0.160258
$616$	0	0
$617$	2.45772	0.0989441	0.0494721	−	0.998776i	$-0.484246\pi$
$617$	2.45772	0.0989441	0.0494721	+	0.998776i	$0.484246\pi$
$618$	38.6886	1.55628
$619$	37.7789	1.51846	0.759231	−	0.650822i	$-0.225575\pi$
$619$	37.7789	1.51846	0.759231	+	0.650822i	$0.225575\pi$
$620$	−12.0591	−0.484305
$621$	1.48752	0.0596922
$622$	−30.1602	−1.20932
$623$	0	0
$624$	7.99991	0.320253
$625$	−30.3942	−1.21577
$626$	−22.8839	−0.914625
$627$	−37.2516	−1.48768
$628$	4.80741	0.191837
$629$	−12.3374	−0.491922
$630$	0	0
$631$	−28.4828	−1.13388	−0.566942	−	0.823758i	$-0.691874\pi$
$631$	−28.4828	−1.13388	−0.566942	+	0.823758i	$0.691874\pi$
$632$	26.5932	1.05782
$633$	−26.3617	−1.04778
$634$	−34.8652	−1.38467
$635$	−15.0396	−0.596829
$636$	7.44843	0.295350
$637$	0	0
$638$	2.82509	0.111847
$639$	14.0549	0.556002
$640$	−19.8091	−0.783024
$641$	−27.1922	−1.07403	−0.537014	−	0.843573i	$-0.680448\pi$
$641$	−27.1922	−1.07403	−0.537014	+	0.843573i	$0.680448\pi$
$642$	−65.2584	−2.57554
$643$	−37.1664	−1.46570	−0.732849	−	0.680391i	$-0.761810\pi$
$643$	−37.1664	−1.46570	−0.732849	+	0.680391i	$0.761810\pi$
$644$	0	0
$645$	2.50912	0.0987965
$646$	35.2334	1.38624
$647$	18.8319	0.740357	0.370178	−	0.928961i	$-0.379297\pi$
$647$	18.8319	0.740357	0.370178	+	0.928961i	$0.379297\pi$
$648$	16.6255	0.653111
$649$	4.14759	0.162807
$650$	4.33482	0.170026
$651$	0	0
$652$	−5.93910	−0.232593
$653$	−26.0185	−1.01818	−0.509090	−	0.860713i	$-0.670018\pi$
$653$	−26.0185	−1.01818	−0.509090	+	0.860713i	$0.670018\pi$
$654$	3.68008	0.143903
$655$	−30.6709	−1.19841
$656$	1.58687	0.0619569
$657$	11.6131	0.453069
$658$	0	0
$659$	−33.3339	−1.29851	−0.649253	−	0.760573i	$-0.724918\pi$
$659$	−33.3339	−1.29851	−0.649253	+	0.760573i	$0.724918\pi$
$660$	−6.18488	−0.240746
$661$	6.29841	0.244980	0.122490	−	0.992470i	$-0.460912\pi$
$661$	6.29841	0.244980	0.122490	+	0.992470i	$0.460912\pi$
$662$	−23.0105	−0.894329
$663$	−10.5056	−0.408004
$664$	−38.8626	−1.50816
$665$	0	0
$666$	−15.2391	−0.590505
$667$	−0.687538	−0.0266216
$668$	−2.01373	−0.0779136
$669$	−45.9672	−1.77719
$670$	−53.9799	−2.08542
$671$	4.88941	0.188754
$672$	0	0
$673$	18.3188	0.706137	0.353068	−	0.935598i	$-0.385138\pi$
$673$	18.3188	0.706137	0.353068	+	0.935598i	$0.385138\pi$
$674$	−21.6901	−0.835473
$675$	−8.12219	−0.312623
$676$	−0.398235	−0.0153167
$677$	24.3392	0.935430	0.467715	−	0.883879i	$-0.345077\pi$
$677$	24.3392	0.935430	0.467715	+	0.883879i	$0.345077\pi$
$678$	3.64453	0.139967
$679$	0	0
$680$	35.2284	1.35095
$681$	25.0020	0.958078
$682$	−26.8896	−1.02966
$683$	11.7682	0.450297	0.225149	−	0.974324i	$-0.427713\pi$
$683$	11.7682	0.450297	0.225149	+	0.974324i	$0.427713\pi$
$684$	−10.8202	−0.413719
$685$	17.0434	0.651195
$686$	0	0
$687$	−55.4802	−2.11670
$688$	−1.00186	−0.0381955
$689$	7.11900	0.271212
$690$	−6.05417	−0.230478
$691$	1.17785	0.0448074	0.0224037	−	0.999749i	$-0.492868\pi$
$691$	1.17785	0.0448074	0.0224037	+	0.999749i	$0.492868\pi$
$692$	0.237195	0.00901679
$693$	0	0
$694$	28.1835	1.06983
$695$	11.6104	0.440408
$696$	8.74043	0.331305
$697$	−2.08390	−0.0789335
$698$	25.2321	0.955050
$699$	−37.2516	−1.40898
$700$	0	0
$701$	−31.2867	−1.18168	−0.590841	−	0.806788i	$-0.701204\pi$
$701$	−31.2867	−1.18168	−0.590841	+	0.806788i	$0.701204\pi$
$702$	−3.00124	−0.113275
$703$	−21.4806	−0.810158
$704$	18.1161	0.682776
$705$	80.4208	3.02882
$706$	29.0016	1.09149
$707$	0	0
$708$	2.13081	0.0800806
$709$	15.3748	0.577411	0.288706	−	0.957418i	$-0.406775\pi$
$709$	15.3748	0.577411	0.288706	+	0.957418i	$0.406775\pi$
$710$	−13.2300	−0.496512
$711$	−34.1926	−1.28232
$712$	8.13739	0.304962
$713$	6.54409	0.245078
$714$	0	0
$715$	−5.91133	−0.221071
$716$	−3.21640	−0.120203
$717$	43.4223	1.62163
$718$	34.4643	1.28620
$719$	−11.1417	−0.415517	−0.207759	−	0.978180i	$-0.566617\pi$
$719$	−11.1417	−0.415517	−0.207759	+	0.978180i	$0.566617\pi$
$720$	34.4921	1.28545
$721$	0	0
$722$	37.2985	1.38811
$723$	35.9648	1.33755
$724$	−0.753956	−0.0280206
$725$	3.75411	0.139424
$726$	22.7850	0.845631
$727$	−6.24735	−0.231702	−0.115851	−	0.993267i	$-0.536959\pi$
$727$	−6.24735	−0.231702	−0.115851	+	0.993267i	$0.536959\pi$
$728$	0	0
$729$	−40.0674	−1.48398
$730$	−10.9315	−0.404593
$731$	1.31566	0.0486613
$732$	2.51191	0.0928430
$733$	30.9669	1.14379	0.571894	−	0.820327i	$-0.306209\pi$
$733$	30.9669	1.14379	0.571894	+	0.820327i	$0.306209\pi$
$734$	−13.7400	−0.507153
$735$	0	0
$736$	−1.39053	−0.0512554
$737$	29.9256	1.10232
$738$	−2.57405	−0.0947520
$739$	2.33744	0.0859843	0.0429921	−	0.999075i	$-0.486311\pi$
$739$	2.33744	0.0859843	0.0429921	+	0.999075i	$0.486311\pi$
$740$	−3.56643	−0.131105
$741$	−18.2914	−0.671951
$742$	0	0
$743$	24.3612	0.893726	0.446863	−	0.894603i	$-0.352541\pi$
$743$	24.3612	0.893726	0.446863	+	0.894603i	$0.352541\pi$
$744$	−83.1927	−3.04999
$745$	29.3320	1.07464
$746$	−3.00132	−0.109886
$747$	49.9684	1.82825
$748$	−3.24304	−0.118577
$749$	0	0
$750$	−15.2002	−0.555033
$751$	−12.0253	−0.438810	−0.219405	−	0.975634i	$-0.570412\pi$
$751$	−12.0253	−0.438810	−0.219405	+	0.975634i	$0.570412\pi$
$752$	−32.1110	−1.17097
$753$	38.5167	1.40363
$754$	1.38719	0.0505184
$755$	0.544894	0.0198307
$756$	0	0
$757$	25.9905	0.944641	0.472321	−	0.881427i	$-0.343416\pi$
$757$	25.9905	0.944641	0.472321	+	0.881427i	$0.343416\pi$
$758$	−36.9807	−1.34320
$759$	3.35634	0.121827
$760$	61.3364	2.22491
$761$	13.3270	0.483103	0.241552	−	0.970388i	$-0.422344\pi$
$761$	13.3270	0.483103	0.241552	+	0.970388i	$0.422344\pi$
$762$	−17.2288	−0.624134
$763$	0	0
$764$	−1.47416	−0.0533334
$765$	−45.2956	−1.63767
$766$	−3.88029	−0.140200
$767$	2.03656	0.0735361
$768$	24.0489	0.867792
$769$	9.24486	0.333378	0.166689	−	0.986010i	$-0.446692\pi$
$769$	9.24486	0.333378	0.166689	+	0.986010i	$0.446692\pi$
$770$	0	0
$771$	4.60503	0.165846
$772$	−5.41101	−0.194746
$773$	−10.1419	−0.364780	−0.182390	−	0.983226i	$-0.558383\pi$
$773$	−10.1419	−0.364780	−0.182390	+	0.983226i	$0.558383\pi$
$774$	1.62510	0.0584132
$775$	−35.7322	−1.28354
$776$	7.06912	0.253766
$777$	0	0
$778$	35.1092	1.25873
$779$	−3.62830	−0.129997
$780$	−3.03692	−0.108739
$781$	7.33450	0.262449
$782$	−3.17450	−0.113520
$783$	−2.59919	−0.0928873
$784$	0	0
$785$	35.0396	1.25062
$786$	−35.1354	−1.25324
$787$	45.2823	1.61414	0.807070	−	0.590456i	$-0.201052\pi$
$787$	45.2823	1.61414	0.807070	+	0.590456i	$0.201052\pi$
$788$	3.86391	0.137646
$789$	−70.7831	−2.51995
$790$	32.1859	1.14512
$791$	0	0
$792$	−24.1237	−0.857197
$793$	2.40081	0.0852554
$794$	21.8079	0.773935
$795$	54.2891	1.92544
$796$	−10.4624	−0.370831
$797$	27.0784	0.959165	0.479583	−	0.877497i	$-0.340788\pi$
$797$	27.0784	0.959165	0.479583	+	0.877497i	$0.340788\pi$
$798$	0	0
$799$	42.1686	1.49182
$800$	7.59257	0.268438
$801$	−10.4628	−0.369685
$802$	21.0648	0.743825
$803$	6.06026	0.213862
$804$	15.3741	0.542204
$805$	0	0
$806$	−13.2034	−0.465071
$807$	57.9817	2.04105
$808$	−4.41091	−0.155175
$809$	−25.7798	−0.906370	−0.453185	−	0.891417i	$-0.649712\pi$
$809$	−25.7798	−0.906370	−0.453185	+	0.891417i	$0.649712\pi$
$810$	20.1219	0.707013
$811$	−25.7829	−0.905362	−0.452681	−	0.891673i	$-0.649532\pi$
$811$	−25.7829	−0.905362	−0.452681	+	0.891673i	$0.649532\pi$
$812$	0	0
$813$	23.5459	0.825792
$814$	−7.95252	−0.278736
$815$	−43.2881	−1.51632
$816$	31.9889	1.11984
$817$	2.29070	0.0801414
$818$	−17.2550	−0.603308
$819$	0	0
$820$	−0.602407	−0.0210370
$821$	−1.71073	−0.0597050	−0.0298525	−	0.999554i	$-0.509504\pi$
$821$	−1.71073	−0.0597050	−0.0298525	+	0.999554i	$0.509504\pi$
$822$	19.5243	0.680987
$823$	−40.3773	−1.40747	−0.703733	−	0.710465i	$-0.748485\pi$
$823$	−40.3773	−1.40747	−0.703733	+	0.710465i	$0.748485\pi$
$824$	35.3158	1.23028
$825$	−18.3264	−0.638042
$826$	0	0
$827$	19.5698	0.680509	0.340254	−	0.940333i	$-0.389487\pi$
$827$	19.5698	0.680509	0.340254	+	0.940333i	$0.389487\pi$
$828$	0.974889	0.0338797
$829$	41.5742	1.44393	0.721966	−	0.691928i	$-0.243239\pi$
$829$	41.5742	1.44393	0.721966	+	0.691928i	$0.243239\pi$
$830$	−47.0357	−1.63263
$831$	19.7815	0.686211
$832$	8.89542	0.308393
$833$	0	0
$834$	13.3004	0.460556
$835$	−14.6774	−0.507932
$836$	−5.64648	−0.195288
$837$	24.7394	0.855119
$838$	−13.7321	−0.474367
$839$	45.8480	1.58285	0.791425	−	0.611266i	$-0.209340\pi$
$839$	45.8480	1.58285	0.791425	+	0.611266i	$0.209340\pi$
$840$	0	0
$841$	−27.7986	−0.958574
$842$	−12.6561	−0.436157
$843$	−78.2305	−2.69440
$844$	−3.99583	−0.137542
$845$	−2.90260	−0.0998525
$846$	52.0869	1.79078
$847$	0	0
$848$	−21.6769	−0.744388
$849$	−0.791997	−0.0271812
$850$	17.3335	0.594534
$851$	1.93539	0.0663443
$852$	3.76807	0.129092
$853$	40.0236	1.37038	0.685191	−	0.728364i	$-0.259719\pi$
$853$	40.0236	1.37038	0.685191	+	0.728364i	$0.259719\pi$
$854$	0	0
$855$	−78.8645	−2.69711
$856$	−59.5692	−2.03603
$857$	32.8702	1.12282	0.561412	−	0.827536i	$-0.310258\pi$
$857$	32.8702	1.12282	0.561412	+	0.827536i	$0.310258\pi$
$858$	−6.77179	−0.231185
$859$	−34.0503	−1.16178	−0.580891	−	0.813981i	$-0.697296\pi$
$859$	−34.0503	−1.16178	−0.580891	+	0.813981i	$0.697296\pi$
$860$	0.380325	0.0129690
$861$	0	0
$862$	1.53079	0.0521388
$863$	−14.0642	−0.478749	−0.239375	−	0.970927i	$-0.576942\pi$
$863$	−14.0642	−0.478749	−0.239375	+	0.970927i	$0.576942\pi$
$864$	−5.25677	−0.178839
$865$	1.72883	0.0587820
$866$	−7.04212	−0.239301
$867$	2.65537	0.0901813
$868$	0	0
$869$	−17.8434	−0.605295
$870$	10.5786	0.358648
$871$	14.6942	0.497893
$872$	3.35926	0.113759
$873$	−9.08925	−0.307625
$874$	−5.52715	−0.186959
$875$	0	0
$876$	3.11343	0.105193
$877$	51.0669	1.72441	0.862204	−	0.506562i	$-0.169084\pi$
$877$	51.0669	1.72441	0.862204	+	0.506562i	$0.169084\pi$
$878$	−24.9568	−0.842251
$879$	50.5675	1.70560
$880$	17.9996	0.606768
$881$	−18.4203	−0.620597	−0.310298	−	0.950639i	$-0.600429\pi$
$881$	−18.4203	−0.620597	−0.310298	+	0.950639i	$0.600429\pi$
$882$	0	0
$883$	0.126678	0.00426305	0.00213153	−	0.999998i	$-0.499322\pi$
$883$	0.126678	0.00426305	0.00213153	+	0.999998i	$0.499322\pi$
$884$	−1.59241	−0.0535585
$885$	15.5307	0.522059
$886$	28.1357	0.945239
$887$	3.87470	0.130100	0.0650498	−	0.997882i	$-0.479279\pi$
$887$	3.87470	0.130100	0.0650498	+	0.997882i	$0.479279\pi$
$888$	−24.6039	−0.825654
$889$	0	0
$890$	9.84874	0.330131
$891$	−11.1553	−0.373717
$892$	−6.96757	−0.233291
$893$	73.4201	2.45691
$894$	33.6016	1.12381
$895$	−23.4433	−0.783622
$896$	0	0
$897$	1.64804	0.0550265
$898$	23.3605	0.779549
$899$	−11.4347	−0.381367
$900$	−5.32311	−0.177437
$901$	28.4665	0.948356
$902$	−1.34326	−0.0447257
$903$	0	0
$904$	3.32680	0.110648
$905$	−5.49533	−0.182671
$906$	0.624210	0.0207380
$907$	−46.7741	−1.55311	−0.776555	−	0.630050i	$-0.783035\pi$
$907$	−46.7741	−1.55311	−0.776555	+	0.630050i	$0.783035\pi$
$908$	3.78973	0.125766
$909$	5.67142	0.188109
$910$	0	0
$911$	5.93675	0.196693	0.0983467	−	0.995152i	$-0.468645\pi$
$911$	5.93675	0.196693	0.0983467	+	0.995152i	$0.468645\pi$
$912$	55.6961	1.84428
$913$	26.0759	0.862986
$914$	−37.9454	−1.25512
$915$	18.3085	0.605260
$916$	−8.40952	−0.277858
$917$	0	0
$918$	−12.0010	−0.396091
$919$	−8.58701	−0.283259	−0.141630	−	0.989920i	$-0.545234\pi$
$919$	−8.58701	−0.283259	−0.141630	+	0.989920i	$0.545234\pi$
$920$	−5.52637	−0.182199
$921$	9.39985	0.309736
$922$	36.8922	1.21498
$923$	3.60141	0.118542
$924$	0	0
$925$	−10.5677	−0.347463
$926$	1.97309	0.0648396
$927$	−45.4079	−1.49139
$928$	2.42970	0.0797589
$929$	−8.45945	−0.277546	−0.138773	−	0.990324i	$-0.544316\pi$
$929$	−8.45945	−0.277546	−0.138773	+	0.990324i	$0.544316\pi$
$930$	−100.689	−3.30171
$931$	0	0
$932$	−5.64648	−0.184957
$933$	62.6097	2.04975
$934$	15.7228	0.514466
$935$	−23.6374	−0.773026
$936$	−11.8453	−0.387175
$937$	−33.3596	−1.08981	−0.544905	−	0.838498i	$-0.683434\pi$
$937$	−33.3596	−1.08981	−0.544905	+	0.838498i	$0.683434\pi$
$938$	0	0
$939$	47.5048	1.55026
$940$	12.1899	0.397592
$941$	−13.4037	−0.436949	−0.218475	−	0.975843i	$-0.570108\pi$
$941$	−13.4037	−0.436949	−0.218475	+	0.975843i	$0.570108\pi$
$942$	40.1400	1.30783
$943$	0.326907	0.0106456
$944$	−6.20121	−0.201832
$945$	0	0
$946$	0.848057	0.0275727
$947$	43.0794	1.39989	0.699946	−	0.714196i	$-0.253207\pi$
$947$	43.0794	1.39989	0.699946	+	0.714196i	$0.253207\pi$
$948$	−9.16695	−0.297729
$949$	2.97573	0.0965962
$950$	30.1794	0.979150
$951$	72.3768	2.34698
$952$	0	0
$953$	16.7332	0.542040	0.271020	−	0.962574i	$-0.412639\pi$
$953$	16.7332	0.542040	0.271020	+	0.962574i	$0.412639\pi$
$954$	35.1619	1.13841
$955$	−10.7447	−0.347690
$956$	6.58182	0.212871
$957$	−5.86462	−0.189576
$958$	45.6325	1.47432
$959$	0	0
$960$	67.8360	2.18940
$961$	77.8366	2.51086
$962$	−3.90487	−0.125898
$963$	76.5922	2.46815
$964$	5.45144	0.175579
$965$	−39.4390	−1.26959
$966$	0	0
$967$	44.7594	1.43937	0.719683	−	0.694303i	$-0.244287\pi$
$967$	44.7594	1.43937	0.719683	+	0.694303i	$0.244287\pi$
$968$	20.7986	0.668493
$969$	−73.1411	−2.34963
$970$	8.55580	0.274710
$971$	−4.20259	−0.134867	−0.0674337	−	0.997724i	$-0.521481\pi$
$971$	−4.20259	−0.134867	−0.0674337	+	0.997724i	$0.521481\pi$
$972$	−8.56409	−0.274693
$973$	0	0
$974$	9.23899	0.296036
$975$	−8.99866	−0.288188
$976$	−7.31033	−0.233998
$977$	−25.6899	−0.821892	−0.410946	−	0.911660i	$-0.634801\pi$
$977$	−25.6899	−0.821892	−0.410946	+	0.911660i	$0.634801\pi$
$978$	−49.5892	−1.58569
$979$	−5.45999	−0.174502
$980$	0	0
$981$	−4.31923	−0.137902
$982$	5.68483	0.181410
$983$	−31.8244	−1.01504	−0.507520	−	0.861640i	$-0.669438\pi$
$983$	−31.8244	−1.01504	−0.507520	+	0.861640i	$0.669438\pi$
$984$	−4.15586	−0.132484
$985$	28.1627	0.897339
$986$	5.54689	0.176649
$987$	0	0
$988$	−2.77255	−0.0882067
$989$	−0.206390	−0.00656283
$990$	−29.1970	−0.927942
$991$	9.47478	0.300976	0.150488	−	0.988612i	$-0.451915\pi$
$991$	9.47478	0.300976	0.150488	+	0.988612i	$0.451915\pi$
$992$	−23.1262	−0.734259
$993$	47.7676	1.51586
$994$	0	0
$995$	−76.2570	−2.41751
$996$	13.3964	0.424480
$997$	21.9511	0.695198	0.347599	−	0.937643i	$-0.386997\pi$
$997$	21.9511	0.695198	0.347599	+	0.937643i	$0.386997\pi$
$998$	14.3866	0.455401
$999$	7.31659	0.231487

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	637.2.a.l.1.3		5
3.2	odd	2		5733.2.a.bl.1.3		5
7.2	even	3		91.2.e.c.53.3	✓	10
7.3	odd	6		637.2.e.m.79.3		10
7.4	even	3		91.2.e.c.79.3	yes	10
7.5	odd	6		637.2.e.m.508.3		10
7.6	odd	2		637.2.a.k.1.3		5
13.12	even	2		8281.2.a.bw.1.3		5
21.2	odd	6		819.2.j.h.235.3		10
21.11	odd	6		819.2.j.h.352.3		10
21.20	even	2		5733.2.a.bm.1.3		5
28.11	odd	6		1456.2.r.p.625.1		10
28.23	odd	6		1456.2.r.p.417.1		10
91.25	even	6		1183.2.e.f.170.3		10
91.51	even	6		1183.2.e.f.508.3		10
91.90	odd	2		8281.2.a.bx.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.e.c.53.3	✓	10	7.2	even	3
91.2.e.c.79.3	yes	10	7.4	even	3
637.2.a.k.1.3		5	7.6	odd	2
637.2.a.l.1.3		5	1.1	even	1	trivial
637.2.e.m.79.3		10	7.3	odd	6
637.2.e.m.508.3		10	7.5	odd	6
819.2.j.h.235.3		10	21.2	odd	6
819.2.j.h.352.3		10	21.11	odd	6
1183.2.e.f.170.3		10	91.25	even	6
1183.2.e.f.508.3		10	91.51	even	6
1456.2.r.p.417.1		10	28.23	odd	6
1456.2.r.p.625.1		10	28.11	odd	6
5733.2.a.bl.1.3		5	3.2	odd	2
5733.2.a.bm.1.3		5	21.20	even	2
8281.2.a.bw.1.3		5	13.12	even	2
8281.2.a.bx.1.3		5	91.90	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	637.a (trivial)

\(f(q)\)	\(=\)	\(q+1.26561 q^{2} -2.62728 q^{3} -0.398235 q^{4} -2.90260 q^{5} -3.32511 q^{6} -3.03523 q^{8} +3.90260 q^{9} -3.67356 q^{10} +2.03656 q^{11} +1.04628 q^{12} +1.00000 q^{13} +7.62594 q^{15} -3.04494 q^{16} +3.99866 q^{17} +4.93916 q^{18} +6.96210 q^{19} +1.15592 q^{20} +2.57749 q^{22} -0.627280 q^{23} +7.97439 q^{24} +3.42509 q^{25} +1.26561 q^{26} -2.37138 q^{27} +1.09606 q^{29} +9.65146 q^{30} -10.4325 q^{31} +2.21675 q^{32} -5.35062 q^{33} +5.06074 q^{34} -1.55415 q^{36} -3.08537 q^{37} +8.81129 q^{38} -2.62728 q^{39} +8.81005 q^{40} -0.521150 q^{41} +0.329024 q^{43} -0.811031 q^{44} -11.3277 q^{45} -0.793891 q^{46} +10.5457 q^{47} +7.99991 q^{48} +4.33482 q^{50} -10.5056 q^{51} -0.398235 q^{52} +7.11900 q^{53} -3.00124 q^{54} -5.91133 q^{55} -18.2914 q^{57} +1.38719 q^{58} +2.03656 q^{59} -3.03692 q^{60} +2.40081 q^{61} -13.2034 q^{62} +8.89542 q^{64} -2.90260 q^{65} -6.77179 q^{66} +14.6942 q^{67} -1.59241 q^{68} +1.64804 q^{69} +3.60141 q^{71} -11.8453 q^{72} +2.97573 q^{73} -3.90487 q^{74} -8.99866 q^{75} -2.77255 q^{76} -3.32511 q^{78} -8.76150 q^{79} +8.83824 q^{80} -5.47751 q^{81} -0.659572 q^{82} +12.8039 q^{83} -11.6065 q^{85} +0.416416 q^{86} -2.87966 q^{87} -6.18143 q^{88} -2.68098 q^{89} -14.3364 q^{90} +0.249805 q^{92} +27.4090 q^{93} +13.3467 q^{94} -20.2082 q^{95} -5.82403 q^{96} -2.32902 q^{97} +7.94789 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 8 q^{4} + 2 q^{5} - 5 q^{6} + 9 q^{8} + 3 q^{9} - 5 q^{10} + 11 q^{11} + 5 q^{12} + 5 q^{13} + 10 q^{16} - 5 q^{17} + 9 q^{18} + 9 q^{19} + q^{20} + 8 q^{22} + 10 q^{23} + 9 q^{25} + 4 q^{26}+ \cdots + 11 q^{99}+O(q^{100}) \) 5 * q + 4 * q^2 + 8 * q^4 + 2 * q^5 - 5 * q^6 + 9 * q^8 + 3 * q^9 - 5 * q^10 + 11 * q^11 + 5 * q^12 + 5 * q^13 + 10 * q^16 - 5 * q^17 + 9 * q^18 + 9 * q^19 + q^20 + 8 * q^22 + 10 * q^23 + 9 * q^25 + 4 * q^26 - 3 * q^29 - 13 * q^30 - 6 * q^31 + 22 * q^32 + 8 * q^33 - 22 * q^34 + 7 * q^36 + 4 * q^37 - 10 * q^38 + 28 * q^40 + 14 * q^41 + 2 * q^43 - 32 * q^45 + 3 * q^46 + q^47 - 23 * q^48 + 9 * q^50 - 8 * q^51 + 8 * q^52 + 17 * q^53 + 23 * q^54 - 16 * q^57 - 27 * q^58 + 11 * q^59 - 29 * q^60 - 11 * q^61 - 23 * q^62 + 9 * q^64 + 2 * q^65 + 21 * q^66 + 13 * q^67 - 32 * q^68 + 18 * q^69 + 15 * q^71 - 19 * q^72 - 33 * q^74 - 20 * q^75 + 8 * q^76 - 5 * q^78 + 2 * q^79 + 55 * q^80 - 19 * q^81 + 34 * q^82 + 6 * q^83 - 22 * q^85 + 28 * q^86 - 8 * q^87 - 3 * q^88 - 4 * q^89 - 34 * q^90 + 21 * q^92 + 18 * q^93 + 20 * q^94 - 12 * q^95 - 37 * q^96 - 12 * q^97 + 11 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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