LMFDB - Embedded newform 931.2.a.m.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [931,2,Mod(1,931)] 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("931.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(931, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 931 = 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	931.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,2,2,8,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.43407242818$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.5744.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} - 2x + 1 $ x^4 - 5x^2 - 2x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.291367$ of defining polynomial
Character	$\chi$	$=$	931.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.291367 q^{2} -2.43210 q^{3} -1.91511 q^{4} +2.06574 q^{5} -0.708633 q^{6} -1.14073 q^{8} +2.91511 q^{9} +0.601888 q^{10} -4.94909 q^{11} +4.65773 q^{12} -1.62374 q^{13} -5.02408 q^{15} +3.49784 q^{16} +6.99010 q^{17} +0.849365 q^{18} -1.00000 q^{19} -3.95611 q^{20} -1.44200 q^{22} -7.41294 q^{23} +2.77437 q^{24} -0.732718 q^{25} -0.473104 q^{26} +0.206472 q^{27} +3.88335 q^{29} -1.46385 q^{30} -0.808361 q^{31} +3.30062 q^{32} +12.0367 q^{33} +2.03668 q^{34} -5.58273 q^{36} +4.10674 q^{37} -0.291367 q^{38} +3.94909 q^{39} -2.35646 q^{40} +8.50709 q^{41} +7.31545 q^{43} +9.47803 q^{44} +6.02185 q^{45} -2.15989 q^{46} +5.41950 q^{47} -8.50709 q^{48} -0.213490 q^{50} -17.0006 q^{51} +3.10963 q^{52} -1.99010 q^{53} +0.0601592 q^{54} -10.2235 q^{55} +2.43210 q^{57} +1.13148 q^{58} +5.75522 q^{59} +9.62165 q^{60} +11.3812 q^{61} -0.235529 q^{62} -6.03399 q^{64} -3.35422 q^{65} +3.50709 q^{66} +13.9950 q^{67} -13.3868 q^{68} +18.0290 q^{69} +6.51969 q^{71} -3.32535 q^{72} +2.08982 q^{73} +1.19657 q^{74} +1.78204 q^{75} +1.91511 q^{76} +1.15063 q^{78} -2.31977 q^{79} +7.22563 q^{80} -9.24748 q^{81} +2.47868 q^{82} -5.01753 q^{83} +14.4397 q^{85} +2.13148 q^{86} -9.44470 q^{87} +5.64559 q^{88} -6.69441 q^{89} +1.75457 q^{90} +14.1966 q^{92} +1.96601 q^{93} +1.57906 q^{94} -2.06574 q^{95} -8.02743 q^{96} -14.0663 q^{97} -14.4271 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 2 q^{4} + 8 q^{5} - 4 q^{6} + 6 q^{8} + 2 q^{9} + 10 q^{10} - 6 q^{11} + 6 q^{12} + 2 q^{13} + 4 q^{15} + 2 q^{16} + 8 q^{17} - 6 q^{18} - 4 q^{19} - 14 q^{22} - 8 q^{23} + 12 q^{24} + 20 q^{25}+ \cdots + 8 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 2 * q^4 + 8 * q^5 - 4 * q^6 + 6 * q^8 + 2 * q^9 + 10 * q^10 - 6 * q^11 + 6 * q^12 + 2 * q^13 + 4 * q^15 + 2 * q^16 + 8 * q^17 - 6 * q^18 - 4 * q^19 - 14 * q^22 - 8 * q^23 + 12 * q^24 + 20 * q^25 + 16 * q^26 - 10 * q^27 + 2 * q^29 + 2 * q^30 + 2 * q^32 + 18 * q^33 - 22 * q^34 - 20 * q^36 + 10 * q^37 + 2 * q^39 + 22 * q^40 + 12 * q^41 + 4 * q^43 - 14 * q^44 + 8 * q^45 - 8 * q^46 + 16 * q^47 - 12 * q^48 + 12 * q^50 - 10 * q^51 + 28 * q^52 + 12 * q^53 + 4 * q^54 - 8 * q^55 - 2 * q^57 + 4 * q^58 + 14 * q^59 - 2 * q^60 + 20 * q^61 - 20 * q^62 - 20 * q^64 + 10 * q^65 - 8 * q^66 + 2 * q^67 - 24 * q^68 + 14 * q^69 - 2 * q^71 - 8 * q^72 - 16 * q^73 - 26 * q^74 + 36 * q^75 - 2 * q^76 + 14 * q^78 - 8 * q^79 + 28 * q^80 - 20 * q^81 - 12 * q^82 + 20 * q^83 - 14 * q^85 + 8 * q^86 - 20 * q^87 - 2 * q^88 + 16 * q^89 - 32 * q^90 + 26 * q^92 + 12 * q^93 + 10 * q^94 - 8 * q^95 - 12 * q^96 + 2 * q^97 + 8 * 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 $

$2$	0.291367	0.206028	0.103014	−	0.994680i	$-0.467151\pi$
$2$	0.291367	0.206028	0.103014	+	0.994680i	$0.467151\pi$
$3$	−2.43210	−1.40417	−0.702086	−	0.712092i	$-0.747748\pi$
$3$	−2.43210	−1.40417	−0.702086	+	0.712092i	$0.747748\pi$
$4$	−1.91511	−0.957553
$5$	2.06574	0.923827	0.461914	−	0.886925i	$-0.347163\pi$
$5$	2.06574	0.923827	0.461914	+	0.886925i	$0.347163\pi$
$6$	−0.708633	−0.289298
$7$	0	0
$7$	0	0
$8$	−1.14073	−0.403310
$9$	2.91511	0.971702
$10$	0.601888	0.190334
$11$	−4.94909	−1.49221	−0.746104	−	0.665830i	$-0.768078\pi$
$11$	−4.94909	−1.49221	−0.746104	+	0.665830i	$0.768078\pi$
$12$	4.65773	1.34457
$13$	−1.62374	−0.450344	−0.225172	−	0.974319i	$-0.572294\pi$
$13$	−1.62374	−0.450344	−0.225172	+	0.974319i	$0.572294\pi$
$14$	0	0
$15$	−5.02408	−1.29721
$16$	3.49784	0.874460
$17$	6.99010	1.69535	0.847674	−	0.530518i	$-0.178002\pi$
$17$	6.99010	1.69535	0.847674	+	0.530518i	$0.178002\pi$
$18$	0.849365	0.200197
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−3.95611	−0.884613
$21$	0	0
$22$	−1.44200	−0.307436
$23$	−7.41294	−1.54571	−0.772853	−	0.634585i	$-0.781171\pi$
$23$	−7.41294	−1.54571	−0.772853	+	0.634585i	$0.781171\pi$
$24$	2.77437	0.566317
$25$	−0.732718	−0.146544
$26$	−0.473104	−0.0927833
$27$	0.206472	0.0397356
$28$	0	0
$29$	3.88335	0.721120	0.360560	−	0.932736i	$-0.382586\pi$
$29$	3.88335	0.721120	0.360560	+	0.932736i	$0.382586\pi$
$30$	−1.46385	−0.267262
$31$	−0.808361	−0.145186	−0.0725929	−	0.997362i	$-0.523127\pi$
$31$	−0.808361	−0.145186	−0.0725929	+	0.997362i	$0.523127\pi$
$32$	3.30062	0.583472
$33$	12.0367	2.09532
$34$	2.03668	0.349288
$35$	0	0
$36$	−5.58273	−0.930456
$37$	4.10674	0.675145	0.337572	−	0.941300i	$-0.390394\pi$
$37$	4.10674	0.675145	0.337572	+	0.941300i	$0.390394\pi$
$38$	−0.291367	−0.0472660
$39$	3.94909	0.632361
$40$	−2.35646	−0.372588
$41$	8.50709	1.32858	0.664292	−	0.747473i	$-0.268733\pi$
$41$	8.50709	1.32858	0.664292	+	0.747473i	$0.268733\pi$
$42$	0	0
$43$	7.31545	1.11560	0.557798	−	0.829977i	$-0.311646\pi$
$43$	7.31545	1.11560	0.557798	+	0.829977i	$0.311646\pi$
$44$	9.47803	1.42887
$45$	6.02185	0.897684
$46$	−2.15989	−0.318458
$47$	5.41950	0.790515	0.395258	−	0.918570i	$-0.370655\pi$
$47$	5.41950	0.790515	0.395258	+	0.918570i	$0.370655\pi$
$48$	−8.50709	−1.22789
$49$	0	0
$50$	−0.213490	−0.0301920
$51$	−17.0006	−2.38056
$52$	3.10963	0.431228
$53$	−1.99010	−0.273361	−0.136680	−	0.990615i	$-0.543643\pi$
$53$	−1.99010	−0.273361	−0.136680	+	0.990615i	$0.543643\pi$
$54$	0.0601592	0.00818663
$55$	−10.2235	−1.37854
$56$	0	0
$57$	2.43210	0.322139
$58$	1.13148	0.148571
$59$	5.75522	0.749266	0.374633	−	0.927173i	$-0.377769\pi$
$59$	5.75522	0.749266	0.374633	+	0.927173i	$0.377769\pi$
$60$	9.62165	1.24215
$61$	11.3812	1.45721	0.728606	−	0.684933i	$-0.240169\pi$
$61$	11.3812	1.45721	0.728606	+	0.684933i	$0.240169\pi$
$62$	−0.235529	−0.0299123
$63$	0	0
$64$	−6.03399	−0.754248
$65$	−3.35422	−0.416040
$66$	3.50709	0.431693
$67$	13.9950	1.70976	0.854882	−	0.518822i	$-0.173629\pi$
$67$	13.9950	1.70976	0.854882	+	0.518822i	$0.173629\pi$
$68$	−13.3868	−1.62338
$69$	18.0290	2.17044
$70$	0	0
$71$	6.51969	0.773745	0.386872	−	0.922133i	$-0.373555\pi$
$71$	6.51969	0.773745	0.386872	+	0.922133i	$0.373555\pi$
$72$	−3.32535	−0.391897
$73$	2.08982	0.244595	0.122298	−	0.992493i	$-0.460974\pi$
$73$	2.08982	0.244595	0.122298	+	0.992493i	$0.460974\pi$
$74$	1.19657	0.139098
$75$	1.78204	0.205772
$76$	1.91511	0.219678
$77$	0	0
$78$	1.15063	0.130284
$79$	−2.31977	−0.260995	−0.130497	−	0.991449i	$-0.541657\pi$
$79$	−2.31977	−0.260995	−0.130497	+	0.991449i	$0.541657\pi$
$80$	7.22563	0.807850
$81$	−9.24748	−1.02750
$82$	2.47868	0.273725
$83$	−5.01753	−0.550745	−0.275373	−	0.961338i	$-0.588801\pi$
$83$	−5.01753	−0.550745	−0.275373	+	0.961338i	$0.588801\pi$
$84$	0	0
$85$	14.4397	1.56621
$86$	2.13148	0.229843
$87$	−9.44470	−1.01258
$88$	5.64559	0.601822
$89$	−6.69441	−0.709606	−0.354803	−	0.934941i	$-0.615452\pi$
$89$	−6.69441	−0.709606	−0.354803	+	0.934941i	$0.615452\pi$
$90$	1.75457	0.184948
$91$	0	0
$92$	14.1966	1.48009
$93$	1.96601	0.203866
$94$	1.57906	0.162868
$95$	−2.06574	−0.211940
$96$	−8.02743	−0.819296
$97$	−14.0663	−1.42822	−0.714111	−	0.700033i	$-0.753169\pi$
$97$	−14.0663	−1.42822	−0.714111	+	0.700033i	$0.753169\pi$
$98$	0	0
$99$	−14.4271	−1.44998
$100$	1.40323	0.140323
$101$	−1.79414	−0.178523	−0.0892616	−	0.996008i	$-0.528451\pi$
$101$	−1.79414	−0.178523	−0.0892616	+	0.996008i	$0.528451\pi$
$102$	−4.95341	−0.490461
$103$	16.9009	1.66529	0.832647	−	0.553805i	$-0.186825\pi$
$103$	16.9009	1.66529	0.832647	+	0.553805i	$0.186825\pi$
$104$	1.85225	0.181628
$105$	0	0
$106$	−0.579848	−0.0563199
$107$	9.00925	0.870957	0.435479	−	0.900199i	$-0.356579\pi$
$107$	9.00925	0.870957	0.435479	+	0.900199i	$0.356579\pi$
$108$	−0.395416	−0.0380489
$109$	7.00925	0.671365	0.335682	−	0.941975i	$-0.391033\pi$
$109$	7.00925	0.671365	0.335682	+	0.941975i	$0.391033\pi$
$110$	−2.97880	−0.284018
$111$	−9.98801	−0.948020
$112$	0	0
$113$	−0.261701	−0.0246188	−0.0123094	−	0.999924i	$-0.503918\pi$
$113$	−0.261701	−0.0246188	−0.0123094	+	0.999924i	$0.503918\pi$
$114$	0.708633	0.0663696
$115$	−15.3132	−1.42796
$116$	−7.43703	−0.690511
$117$	−4.73337	−0.437600
$118$	1.67688	0.154369
$119$	0	0
$120$	5.73113	0.523179
$121$	13.4935	1.22668
$122$	3.31610	0.300226
$123$	−20.6901	−1.86556
$124$	1.54810	0.139023
$125$	−11.8423	−1.05921
$126$	0	0
$127$	−8.07337	−0.716395	−0.358198	−	0.933646i	$-0.616609\pi$
$127$	−8.07337	−0.716395	−0.358198	+	0.933646i	$0.616609\pi$
$128$	−8.35934	−0.738868
$129$	−17.7919	−1.56649
$130$	−0.977309	−0.0857157
$131$	−19.4327	−1.69784	−0.848922	−	0.528519i	$-0.822748\pi$
$131$	−19.4327	−1.69784	−0.848922	+	0.528519i	$0.822748\pi$
$132$	−23.0515	−2.00638
$133$	0	0
$134$	4.07769	0.352259
$135$	0.426518	0.0367088
$136$	−7.97383	−0.683750
$137$	−6.28146	−0.536662	−0.268331	−	0.963327i	$-0.586472\pi$
$137$	−6.28146	−0.536662	−0.268331	+	0.963327i	$0.586472\pi$
$138$	5.25306	0.447170
$139$	18.1959	1.54336	0.771679	−	0.636012i	$-0.219417\pi$
$139$	18.1959	1.54336	0.771679	+	0.636012i	$0.219417\pi$
$140$	0	0
$141$	−13.1808	−1.11002
$142$	1.89962	0.159413
$143$	8.03603	0.672007
$144$	10.1966	0.849714
$145$	8.02200	0.666191
$146$	0.608906	0.0503934
$147$	0	0
$148$	−7.86485	−0.646487
$149$	8.75806	0.717488	0.358744	−	0.933436i	$-0.383205\pi$
$149$	8.75806	0.717488	0.358744	+	0.933436i	$0.383205\pi$
$150$	0.519228	0.0423948
$151$	−12.0445	−0.980167	−0.490084	−	0.871675i	$-0.663034\pi$
$151$	−12.0445	−0.980167	−0.490084	+	0.871675i	$0.663034\pi$
$152$	1.14073	0.0925256
$153$	20.3769	1.64737
$154$	0	0
$155$	−1.66986	−0.134127
$156$	−7.56293	−0.605519
$157$	−4.27588	−0.341253	−0.170626	−	0.985336i	$-0.554579\pi$
$157$	−4.27588	−0.341253	−0.170626	+	0.985336i	$0.554579\pi$
$158$	−0.675905	−0.0537721
$159$	4.84011	0.383846
$160$	6.81822	0.539028
$161$	0	0
$162$	−2.69441	−0.211693
$163$	−9.02678	−0.707032	−0.353516	−	0.935428i	$-0.615014\pi$
$163$	−9.02678	−0.707032	−0.353516	+	0.935428i	$0.615014\pi$
$164$	−16.2920	−1.27219
$165$	24.8647	1.93571
$166$	−1.46194	−0.113469
$167$	21.2318	1.64297	0.821484	−	0.570232i	$-0.193147\pi$
$167$	21.2318	1.64297	0.821484	+	0.570232i	$0.193147\pi$
$168$	0	0
$169$	−10.3635	−0.797190
$170$	4.20726	0.322682
$171$	−2.91511	−0.222924
$172$	−14.0099	−1.06824
$173$	−6.80069	−0.517047	−0.258524	−	0.966005i	$-0.583236\pi$
$173$	−6.80069	−0.517047	−0.258524	+	0.966005i	$0.583236\pi$
$174$	−2.75187	−0.208619
$175$	0	0
$176$	−17.3111	−1.30488
$177$	−13.9973	−1.05210
$178$	−1.95053	−0.146198
$179$	19.2722	1.44047	0.720235	−	0.693730i	$-0.244034\pi$
$179$	19.2722	1.44047	0.720235	+	0.693730i	$0.244034\pi$
$180$	−11.5325	−0.859580
$181$	20.5508	1.52753	0.763764	−	0.645495i	$-0.223349\pi$
$181$	20.5508	1.52753	0.763764	+	0.645495i	$0.223349\pi$
$182$	0	0
$183$	−27.6802	−2.04618
$184$	8.45618	0.623398
$185$	8.48347	0.623717
$186$	0.572831	0.0420020
$187$	−34.5946	−2.52981
$188$	−10.3789	−0.756960
$189$	0	0
$190$	−0.601888	−0.0436656
$191$	−4.48366	−0.324426	−0.162213	−	0.986756i	$-0.551863\pi$
$191$	−4.48366	−0.324426	−0.162213	+	0.986756i	$0.551863\pi$
$192$	14.6753	1.05910
$193$	−25.0290	−1.80163	−0.900814	−	0.434205i	$-0.857029\pi$
$193$	−25.0290	−1.80163	−0.900814	+	0.434205i	$0.857029\pi$
$194$	−4.09847	−0.294253
$195$	8.15780	0.584192
$196$	0	0
$197$	−0.100378	−0.00715167	−0.00357583	−	0.999994i	$-0.501138\pi$
$197$	−0.100378	−0.00715167	−0.00357583	+	0.999994i	$0.501138\pi$
$198$	−4.20359	−0.298736
$199$	−26.9299	−1.90901	−0.954506	−	0.298191i	$-0.903617\pi$
$199$	−26.9299	−1.90901	−0.954506	+	0.298191i	$0.903617\pi$
$200$	0.835834	0.0591024
$201$	−34.0373	−2.40081
$202$	−0.522752	−0.0367807
$203$	0	0
$204$	32.5580	2.27951
$205$	17.5734	1.22738
$206$	4.92436	0.343096
$207$	−21.6095	−1.50197
$208$	−5.67958	−0.393808
$209$	4.94909	0.342336
$210$	0	0
$211$	14.5345	1.00060	0.500299	−	0.865853i	$-0.333223\pi$
$211$	14.5345	1.00060	0.500299	+	0.865853i	$0.333223\pi$
$212$	3.81125	0.261757
$213$	−15.8565	−1.08647
$214$	2.62500	0.179441
$215$	15.1118	1.03062
$216$	−0.235529	−0.0160258
$217$	0	0
$218$	2.04226	0.138320
$219$	−5.08266	−0.343454
$220$	19.5792	1.32003
$221$	−11.3501	−0.763490
$222$	−2.91018	−0.195318
$223$	−2.30829	−0.154574	−0.0772872	−	0.997009i	$-0.524626\pi$
$223$	−2.30829	−0.154574	−0.0772872	+	0.997009i	$0.524626\pi$
$224$	0	0
$225$	−2.13595	−0.142397
$226$	−0.0762511	−0.00507215
$227$	19.8522	1.31764	0.658819	−	0.752302i	$-0.271056\pi$
$227$	19.8522	1.31764	0.658819	+	0.752302i	$0.271056\pi$
$228$	−4.65773	−0.308465
$229$	17.4271	1.15162	0.575808	−	0.817585i	$-0.304687\pi$
$229$	17.4271	1.15162	0.575808	+	0.817585i	$0.304687\pi$
$230$	−4.46176	−0.294200
$231$	0	0
$232$	−4.42986	−0.290835
$233$	8.89228	0.582553	0.291276	−	0.956639i	$-0.405920\pi$
$233$	8.89228	0.582553	0.291276	+	0.956639i	$0.405920\pi$
$234$	−1.37915	−0.0901577
$235$	11.1953	0.730300
$236$	−11.0219	−0.717461
$237$	5.64192	0.366482
$238$	0	0
$239$	−10.9234	−0.706575	−0.353287	−	0.935515i	$-0.614936\pi$
$239$	−10.9234	−0.706575	−0.353287	+	0.935515i	$0.614936\pi$
$240$	−17.5734	−1.13436
$241$	5.18397	0.333929	0.166964	−	0.985963i	$-0.446603\pi$
$241$	5.18397	0.333929	0.166964	+	0.985963i	$0.446603\pi$
$242$	3.93156	0.252731
$243$	21.8714	1.40305
$244$	−21.7962	−1.39536
$245$	0	0
$246$	−6.02841	−0.384357
$247$	1.62374	0.103316
$248$	0.922123	0.0585549
$249$	12.2031	0.773342
$250$	−3.45046	−0.218226
$251$	12.2672	0.774301	0.387151	−	0.922016i	$-0.373459\pi$
$251$	12.2672	0.774301	0.387151	+	0.922016i	$0.373459\pi$
$252$	0	0
$253$	36.6873	2.30651
$254$	−2.35231	−0.147597
$255$	−35.1188	−2.19923
$256$	9.63234	0.602021
$257$	−22.3740	−1.39565	−0.697825	−	0.716268i	$-0.745849\pi$
$257$	−22.3740	−1.39565	−0.697825	+	0.716268i	$0.745849\pi$
$258$	−5.18397	−0.322740
$259$	0	0
$260$	6.42369	0.398380
$261$	11.3204	0.700714
$262$	−5.66205	−0.349802
$263$	9.07625	0.559666	0.279833	−	0.960049i	$-0.409721\pi$
$263$	9.07625	0.559666	0.279833	+	0.960049i	$0.409721\pi$
$264$	−13.7306	−0.845062
$265$	−4.11102	−0.252538
$266$	0	0
$267$	16.2815	0.996409
$268$	−26.8020	−1.63719
$269$	18.8198	1.14747	0.573733	−	0.819042i	$-0.305495\pi$
$269$	18.8198	1.14747	0.573733	+	0.819042i	$0.305495\pi$
$270$	0.124273	0.00756303
$271$	24.9815	1.51752	0.758758	−	0.651373i	$-0.225807\pi$
$271$	24.9815	1.51752	0.758758	+	0.651373i	$0.225807\pi$
$272$	24.4502	1.48251
$273$	0	0
$274$	−1.83021	−0.110567
$275$	3.62629	0.218673
$276$	−34.5275	−2.07831
$277$	20.8741	1.25420	0.627100	−	0.778939i	$-0.284242\pi$
$277$	20.8741	1.25420	0.627100	+	0.778939i	$0.284242\pi$
$278$	5.30169	0.317974
$279$	−2.35646	−0.141077
$280$	0	0
$281$	29.3117	1.74859	0.874296	−	0.485393i	$-0.161324\pi$
$281$	29.3117	1.74859	0.874296	+	0.485393i	$0.161324\pi$
$282$	−3.84044	−0.228695
$283$	0.566275	0.0336616	0.0168308	−	0.999858i	$-0.494642\pi$
$283$	0.566275	0.0336616	0.0168308	+	0.999858i	$0.494642\pi$
$284$	−12.4859	−0.740901
$285$	5.02408	0.297601
$286$	2.34143	0.138452
$287$	0	0
$288$	9.62165	0.566961
$289$	31.8615	1.87420
$290$	2.33734	0.137254
$291$	34.2108	2.00547
$292$	−4.00223	−0.234213
$293$	25.2701	1.47630	0.738148	−	0.674639i	$-0.235700\pi$
$293$	25.2701	1.47630	0.738148	+	0.674639i	$0.235700\pi$
$294$	0	0
$295$	11.8888	0.692192
$296$	−4.68470	−0.272292
$297$	−1.02185	−0.0592938
$298$	2.55181	0.147822
$299$	12.0367	0.696099
$300$	−3.41280	−0.197038
$301$	0	0
$302$	−3.50937	−0.201941
$303$	4.36352	0.250677
$304$	−3.49784	−0.200615
$305$	23.5106	1.34621
$306$	5.93715	0.339404
$307$	−6.37114	−0.363620	−0.181810	−	0.983334i	$-0.558196\pi$
$307$	−6.37114	−0.363620	−0.181810	+	0.983334i	$0.558196\pi$
$308$	0	0
$309$	−41.1046	−2.33836
$310$	−0.486543	−0.0276338
$311$	0.295689	0.0167670	0.00838348	−	0.999965i	$-0.497331\pi$
$311$	0.295689	0.0167670	0.00838348	+	0.999965i	$0.497331\pi$
$312$	−4.50486	−0.255037
$313$	−20.3679	−1.15126	−0.575632	−	0.817709i	$-0.695244\pi$
$313$	−20.3679	−1.15126	−0.575632	+	0.817709i	$0.695244\pi$
$314$	−1.24585	−0.0703074
$315$	0	0
$316$	4.44261	0.249916
$317$	−30.5160	−1.71395	−0.856974	−	0.515360i	$-0.827658\pi$
$317$	−30.5160	−1.71395	−0.856974	+	0.515360i	$0.827658\pi$
$318$	1.41025	0.0790828
$319$	−19.2191	−1.07606
$320$	−12.4646	−0.696795
$321$	−21.9114	−1.22297
$322$	0	0
$323$	−6.99010	−0.388939
$324$	17.7099	0.983883
$325$	1.18974	0.0659950
$326$	−2.63010	−0.145668
$327$	−17.0472	−0.942712
$328$	−9.70431	−0.535831
$329$	0	0
$330$	7.24474	0.398810
$331$	−28.2814	−1.55449	−0.777244	−	0.629200i	$-0.783383\pi$
$331$	−28.2814	−1.55449	−0.777244	+	0.629200i	$0.783383\pi$
$332$	9.60910	0.527368
$333$	11.9716	0.656039
$334$	6.18625	0.338496
$335$	28.9101	1.57953
$336$	0	0
$337$	0.0981441	0.00534625	0.00267312	−	0.999996i	$-0.499149\pi$
$337$	0.0981441	0.00534625	0.00267312	+	0.999996i	$0.499149\pi$
$338$	−3.01957	−0.164243
$339$	0.636484	0.0345691
$340$	−27.6536	−1.49973
$341$	4.00065	0.216647
$342$	−0.849365	−0.0459284
$343$	0	0
$344$	−8.34497	−0.449931
$345$	37.2433	2.00511
$346$	−1.98150	−0.106526
$347$	−1.07373	−0.0576410	−0.0288205	−	0.999585i	$-0.509175\pi$
$347$	−1.07373	−0.0576410	−0.0288205	+	0.999585i	$0.509175\pi$
$348$	18.0876	0.969597
$349$	−7.94431	−0.425249	−0.212625	−	0.977134i	$-0.568201\pi$
$349$	−7.94431	−0.425249	−0.212625	+	0.977134i	$0.568201\pi$
$350$	0	0
$351$	−0.335257	−0.0178947
$352$	−16.3351	−0.870662
$353$	8.75810	0.466147	0.233073	−	0.972459i	$-0.425122\pi$
$353$	8.75810	0.466147	0.233073	+	0.972459i	$0.425122\pi$
$354$	−4.07834	−0.216761
$355$	13.4680	0.714806
$356$	12.8205	0.679485
$357$	0	0
$358$	5.61527	0.296776
$359$	8.88559	0.468963	0.234482	−	0.972121i	$-0.424661\pi$
$359$	8.88559	0.468963	0.234482	+	0.972121i	$0.424661\pi$
$360$	−6.86932	−0.362045
$361$	1.00000	0.0526316
$362$	5.98782	0.314713
$363$	−32.8176	−1.72248
$364$	0	0
$365$	4.31703	0.225964
$366$	−8.06509	−0.421569
$367$	−14.0644	−0.734158	−0.367079	−	0.930190i	$-0.619642\pi$
$367$	−14.0644	−0.734158	−0.367079	+	0.930190i	$0.619642\pi$
$368$	−25.9293	−1.35166
$369$	24.7991	1.29099
$370$	2.47180	0.128503
$371$	0	0
$372$	−3.76512	−0.195212
$373$	−36.8568	−1.90837	−0.954187	−	0.299212i	$-0.903276\pi$
$373$	−36.8568	−1.90837	−0.954187	+	0.299212i	$0.903276\pi$
$374$	−10.0797	−0.521211
$375$	28.8017	1.48731
$376$	−6.18220	−0.318823
$377$	−6.30555	−0.324752
$378$	0	0
$379$	−17.2907	−0.888164	−0.444082	−	0.895986i	$-0.646470\pi$
$379$	−17.2907	−0.888164	−0.444082	+	0.895986i	$0.646470\pi$
$380$	3.95611	0.202944
$381$	19.6352	1.00594
$382$	−1.30639	−0.0668407
$383$	23.6224	1.20705	0.603525	−	0.797344i	$-0.293762\pi$
$383$	23.6224	1.20705	0.603525	+	0.797344i	$0.293762\pi$
$384$	20.3307	1.03750
$385$	0	0
$386$	−7.29263	−0.371185
$387$	21.3253	1.08403
$388$	26.9385	1.36760
$389$	−2.81761	−0.142859	−0.0714293	−	0.997446i	$-0.522756\pi$
$389$	−2.81761	−0.142859	−0.0714293	+	0.997446i	$0.522756\pi$
$390$	2.37691	0.120360
$391$	−51.8172	−2.62051
$392$	0	0
$393$	47.2623	2.38407
$394$	−0.0292469	−0.00147344
$395$	−4.79205	−0.241114
$396$	27.6295	1.38843
$397$	−24.0012	−1.20459	−0.602293	−	0.798275i	$-0.705746\pi$
$397$	−24.0012	−1.20459	−0.602293	+	0.798275i	$0.705746\pi$
$398$	−7.84649	−0.393309
$399$	0	0
$400$	−2.56293	−0.128146
$401$	33.3740	1.66662	0.833309	−	0.552808i	$-0.186444\pi$
$401$	33.3740	1.66662	0.833309	+	0.552808i	$0.186444\pi$
$402$	−9.91734	−0.494632
$403$	1.31257	0.0653836
$404$	3.43596	0.170945
$405$	−19.1029	−0.949230
$406$	0	0
$407$	−20.3247	−1.00746
$408$	19.3931	0.960103
$409$	28.4148	1.40502	0.702511	−	0.711673i	$-0.252062\pi$
$409$	28.4148	1.40502	0.702511	+	0.711673i	$0.252062\pi$
$410$	5.12032	0.252874
$411$	15.2771	0.753566
$412$	−32.3670	−1.59461
$413$	0	0
$414$	−6.29630	−0.309446
$415$	−10.3649	−0.508793
$416$	−5.35934	−0.262763
$417$	−44.2543	−2.16714
$418$	1.44200	0.0705306
$419$	−5.95276	−0.290812	−0.145406	−	0.989372i	$-0.546449\pi$
$419$	−5.95276	−0.290812	−0.145406	+	0.989372i	$0.546449\pi$
$420$	0	0
$421$	−2.95244	−0.143893	−0.0719465	−	0.997408i	$-0.522921\pi$
$421$	−2.95244	−0.143893	−0.0719465	+	0.997408i	$0.522921\pi$
$422$	4.23488	0.206151
$423$	15.7984	0.768145
$424$	2.27017	0.110249
$425$	−5.12177	−0.248442
$426$	−4.62007	−0.223843
$427$	0	0
$428$	−17.2537	−0.833987
$429$	−19.5444	−0.943614
$430$	4.40308	0.212336
$431$	9.97033	0.480254	0.240127	−	0.970741i	$-0.422811\pi$
$431$	9.97033	0.480254	0.240127	+	0.970741i	$0.422811\pi$
$432$	0.722207	0.0347472
$433$	19.1090	0.918319	0.459159	−	0.888354i	$-0.348151\pi$
$433$	19.1090	0.918319	0.459159	+	0.888354i	$0.348151\pi$
$434$	0	0
$435$	−19.5103	−0.935447
$436$	−13.4235	−0.642867
$437$	7.41294	0.354609
$438$	−1.48092	−0.0707610
$439$	−7.30681	−0.348735	−0.174367	−	0.984681i	$-0.555788\pi$
$439$	−7.30681	−0.348735	−0.174367	+	0.984681i	$0.555788\pi$
$440$	11.6623	0.555979
$441$	0	0
$442$	−3.30704	−0.157300
$443$	−16.4639	−0.782221	−0.391111	−	0.920344i	$-0.627909\pi$
$443$	−16.4639	−0.782221	−0.391111	+	0.920344i	$0.627909\pi$
$444$	19.1281	0.907779
$445$	−13.8289	−0.655553
$446$	−0.672558	−0.0318466
$447$	−21.3005	−1.00748
$448$	0	0
$449$	−9.71849	−0.458644	−0.229322	−	0.973351i	$-0.573651\pi$
$449$	−9.71849	−0.458644	−0.229322	+	0.973351i	$0.573651\pi$
$450$	−0.622345	−0.0293376
$451$	−42.1024	−1.98252
$452$	0.501186	0.0235738
$453$	29.2934	1.37632
$454$	5.78428	0.271470
$455$	0	0
$456$	−2.77437	−0.129922
$457$	−11.0016	−0.514632	−0.257316	−	0.966327i	$-0.582838\pi$
$457$	−11.0016	−0.514632	−0.257316	+	0.966327i	$0.582838\pi$
$458$	5.07769	0.237265
$459$	1.44326	0.0673657
$460$	29.3264	1.36735
$461$	28.6779	1.33567	0.667833	−	0.744311i	$-0.267222\pi$
$461$	28.6779	1.33567	0.667833	+	0.744311i	$0.267222\pi$
$462$	0	0
$463$	−33.7722	−1.56953	−0.784765	−	0.619794i	$-0.787216\pi$
$463$	−33.7722	−1.56953	−0.784765	+	0.619794i	$0.787216\pi$
$464$	13.5833	0.630591
$465$	4.06127	0.188337
$466$	2.59092	0.120022
$467$	16.8477	0.779620	0.389810	−	0.920895i	$-0.372541\pi$
$467$	16.8477	0.779620	0.389810	+	0.920895i	$0.372541\pi$
$468$	9.06490	0.419025
$469$	0	0
$470$	3.26193	0.150462
$471$	10.3994	0.479178
$472$	−6.56516	−0.302186
$473$	−36.2048	−1.66470
$474$	1.64387	0.0755054
$475$	0.732718	0.0336194
$476$	0	0
$477$	−5.80134	−0.265625
$478$	−3.18271	−0.145574
$479$	23.0450	1.05295	0.526476	−	0.850190i	$-0.323513\pi$
$479$	23.0450	1.05295	0.526476	+	0.850190i	$0.323513\pi$
$480$	−16.5826	−0.756888
$481$	−6.66828	−0.304047
$482$	1.51044	0.0687985
$483$	0	0
$484$	−25.8415	−1.17461
$485$	−29.0574	−1.31943
$486$	6.37259	0.289067
$487$	−21.9468	−0.994505	−0.497253	−	0.867606i	$-0.665658\pi$
$487$	−21.9468	−0.994505	−0.497253	+	0.867606i	$0.665658\pi$
$488$	−12.9829	−0.587708
$489$	21.9540	0.992795
$490$	0	0
$491$	10.1883	0.459791	0.229896	−	0.973215i	$-0.426162\pi$
$491$	10.1883	0.459791	0.229896	+	0.973215i	$0.426162\pi$
$492$	39.6237	1.78637
$493$	27.1450	1.22255
$494$	0.473104	0.0212859
$495$	−29.8027	−1.33953
$496$	−2.82752	−0.126959
$497$	0	0
$498$	3.55559	0.159330
$499$	16.4300	0.735508	0.367754	−	0.929923i	$-0.380127\pi$
$499$	16.4300	0.735508	0.367754	+	0.929923i	$0.380127\pi$
$500$	22.6793	1.01425
$501$	−51.6379	−2.30701
$502$	3.57427	0.159527
$503$	−2.13595	−0.0952373	−0.0476186	−	0.998866i	$-0.515163\pi$
$503$	−2.13595	−0.0952373	−0.0476186	+	0.998866i	$0.515163\pi$
$504$	0	0
$505$	−3.70622	−0.164925
$506$	10.6895	0.475205
$507$	25.2050	1.11939
$508$	15.4613	0.685986
$509$	24.2247	1.07374	0.536869	−	0.843665i	$-0.319607\pi$
$509$	24.2247	1.07374	0.536869	+	0.843665i	$0.319607\pi$
$510$	−10.2325	−0.453101
$511$	0	0
$512$	19.5252	0.862901
$513$	−0.206472	−0.00911597
$514$	−6.51904	−0.287542
$515$	34.9128	1.53844
$516$	34.0734	1.50000
$517$	−26.8216	−1.17961
$518$	0	0
$519$	16.5400	0.726023
$520$	3.82627	0.167793
$521$	7.56441	0.331403	0.165701	−	0.986176i	$-0.447011\pi$
$521$	7.56441	0.331403	0.165701	+	0.986176i	$0.447011\pi$
$522$	3.29838	0.144366
$523$	11.7441	0.513532	0.256766	−	0.966474i	$-0.417343\pi$
$523$	11.7441	0.513532	0.256766	+	0.966474i	$0.417343\pi$
$524$	37.2157	1.62577
$525$	0	0
$526$	2.64452	0.115307
$527$	−5.65052	−0.246140
$528$	42.1024	1.83227
$529$	31.9517	1.38921
$530$	−1.19782	−0.0520298
$531$	16.7771	0.728063
$532$	0	0
$533$	−13.8133	−0.598320
$534$	4.74388	0.205288
$535$	18.6108	0.804614
$536$	−15.9646	−0.689565
$537$	−46.8718	−2.02267
$538$	5.48348	0.236410
$539$	0	0
$540$	−0.816827	−0.0351506
$541$	−4.44391	−0.191059	−0.0955293	−	0.995427i	$-0.530454\pi$
$541$	−4.44391	−0.191059	−0.0955293	+	0.995427i	$0.530454\pi$
$542$	7.27877	0.312650
$543$	−49.9816	−2.14491
$544$	23.0716	0.989189
$545$	14.4793	0.620225
$546$	0	0
$547$	−8.15542	−0.348700	−0.174350	−	0.984684i	$-0.555782\pi$
$547$	−8.15542	−0.348700	−0.174350	+	0.984684i	$0.555782\pi$
$548$	12.0297	0.513882
$549$	33.1774	1.41598
$550$	1.05658	0.0450527
$551$	−3.88335	−0.165436
$552$	−20.5663	−0.875359
$553$	0	0
$554$	6.08201	0.258400
$555$	−20.6326	−0.875806
$556$	−34.8471	−1.47785
$557$	5.23473	0.221803	0.110901	−	0.993831i	$-0.464626\pi$
$557$	5.23473	0.221803	0.110901	+	0.993831i	$0.464626\pi$
$558$	−0.686593	−0.0290658
$559$	−11.8784	−0.502402
$560$	0	0
$561$	84.1376	3.55229
$562$	8.54047	0.360258
$563$	−25.3904	−1.07008	−0.535040	−	0.844827i	$-0.679703\pi$
$563$	−25.3904	−1.07008	−0.535040	+	0.844827i	$0.679703\pi$
$564$	25.2425	1.06290
$565$	−0.540607	−0.0227435
$566$	0.164994	0.00693521
$567$	0	0
$568$	−7.43722	−0.312059
$569$	−7.18156	−0.301067	−0.150533	−	0.988605i	$-0.548099\pi$
$569$	−7.18156	−0.301067	−0.150533	+	0.988605i	$0.548099\pi$
$570$	1.46385	0.0613140
$571$	8.94189	0.374206	0.187103	−	0.982340i	$-0.440090\pi$
$571$	8.94189	0.374206	0.187103	+	0.982340i	$0.440090\pi$
$572$	−15.3898	−0.643482
$573$	10.9047	0.455550
$574$	0	0
$575$	5.43159	0.226513
$576$	−17.5897	−0.732905
$577$	−12.7644	−0.531390	−0.265695	−	0.964057i	$-0.585601\pi$
$577$	−12.7644	−0.531390	−0.265695	+	0.964057i	$0.585601\pi$
$578$	9.28337	0.386137
$579$	60.8730	2.52980
$580$	−15.3630	−0.637913
$581$	0	0
$582$	9.96788	0.413182
$583$	9.84918	0.407911
$584$	−2.38393	−0.0986477
$585$	−9.77791	−0.404267
$586$	7.36288	0.304158
$587$	−24.2518	−1.00098	−0.500489	−	0.865743i	$-0.666847\pi$
$587$	−24.2518	−1.00098	−0.500489	+	0.865743i	$0.666847\pi$
$588$	0	0
$589$	0.808361	0.0333079
$590$	3.46400	0.142611
$591$	0.244130	0.0100422
$592$	14.3647	0.590387
$593$	3.85095	0.158140	0.0790698	−	0.996869i	$-0.474805\pi$
$593$	3.85095	0.158140	0.0790698	+	0.996869i	$0.474805\pi$
$594$	−0.297733	−0.0122161
$595$	0	0
$596$	−16.7726	−0.687033
$597$	65.4963	2.68058
$598$	3.50709	0.143416
$599$	37.2717	1.52288	0.761440	−	0.648235i	$-0.224493\pi$
$599$	37.2717	1.52288	0.761440	+	0.648235i	$0.224493\pi$
$600$	−2.03283	−0.0829900
$601$	2.99633	0.122223	0.0611114	−	0.998131i	$-0.480536\pi$
$601$	2.99633	0.122223	0.0611114	+	0.998131i	$0.480536\pi$
$602$	0	0
$603$	40.7970	1.66138
$604$	23.0665	0.938562
$605$	27.8741	1.13324
$606$	1.27138	0.0516465
$607$	−25.3423	−1.02861	−0.514306	−	0.857607i	$-0.671950\pi$
$607$	−25.3423	−1.02861	−0.514306	+	0.857607i	$0.671950\pi$
$608$	−3.30062	−0.133858
$609$	0	0
$610$	6.85021	0.277357
$611$	−8.79985	−0.356004
$612$	−39.0239	−1.57745
$613$	11.2393	0.453953	0.226976	−	0.973900i	$-0.427116\pi$
$613$	11.2393	0.453953	0.226976	+	0.973900i	$0.427116\pi$
$614$	−1.85634	−0.0749158
$615$	−42.7403	−1.72346
$616$	0	0
$617$	−16.1879	−0.651701	−0.325851	−	0.945421i	$-0.605651\pi$
$617$	−16.1879	−0.651701	−0.325851	+	0.945421i	$0.605651\pi$
$618$	−11.9765	−0.481766
$619$	−14.6528	−0.588944	−0.294472	−	0.955660i	$-0.595144\pi$
$619$	−14.6528	−0.588944	−0.294472	+	0.955660i	$0.595144\pi$
$620$	3.19796	0.128433
$621$	−1.53057	−0.0614195
$622$	0.0861539	0.00345446
$623$	0	0
$624$	13.8133	0.552974
$625$	−20.7995	−0.831981
$626$	−5.93454	−0.237192
$627$	−12.0367	−0.480699
$628$	8.18877	0.326767
$629$	28.7065	1.14460
$630$	0	0
$631$	−23.3309	−0.928790	−0.464395	−	0.885628i	$-0.653728\pi$
$631$	−23.3309	−0.928790	−0.464395	+	0.885628i	$0.653728\pi$
$632$	2.64624	0.105262
$633$	−35.3494	−1.40501
$634$	−8.89134	−0.353120
$635$	−16.6775	−0.661825
$636$	−9.26933	−0.367553
$637$	0	0
$638$	−5.59980	−0.221698
$639$	19.0056	0.751849
$640$	−17.2682	−0.682587
$641$	15.1832	0.599699	0.299850	−	0.953986i	$-0.403063\pi$
$641$	15.1832	0.599699	0.299850	+	0.953986i	$0.403063\pi$
$642$	−6.38425	−0.251966
$643$	31.7842	1.25345	0.626723	−	0.779242i	$-0.284396\pi$
$643$	31.7842	1.25345	0.626723	+	0.779242i	$0.284396\pi$
$644$	0	0
$645$	−36.7534	−1.44717
$646$	−2.03668	−0.0801322
$647$	13.3767	0.525893	0.262947	−	0.964810i	$-0.415306\pi$
$647$	13.3767	0.525893	0.262947	+	0.964810i	$0.415306\pi$
$648$	10.5489	0.414400
$649$	−28.4831	−1.11806
$650$	0.346651	0.0135968
$651$	0	0
$652$	17.2872	0.677020
$653$	−27.3705	−1.07109	−0.535544	−	0.844507i	$-0.679894\pi$
$653$	−27.3705	−1.07109	−0.535544	+	0.844507i	$0.679894\pi$
$654$	−4.96699	−0.194225
$655$	−40.1429	−1.56851
$656$	29.7564	1.16179
$657$	6.09206	0.237674
$658$	0	0
$659$	47.6905	1.85776	0.928879	−	0.370383i	$-0.120774\pi$
$659$	47.6905	1.85776	0.928879	+	0.370383i	$0.120774\pi$
$660$	−47.6184	−1.85355
$661$	−16.8217	−0.654289	−0.327144	−	0.944974i	$-0.606086\pi$
$661$	−16.8217	−0.654289	−0.327144	+	0.944974i	$0.606086\pi$
$662$	−8.24027	−0.320267
$663$	27.6045	1.07207
$664$	5.72366	0.222121
$665$	0	0
$666$	3.48813	0.135162
$667$	−28.7871	−1.11464
$668$	−40.6612	−1.57323
$669$	5.61398	0.217049
$670$	8.42344	0.325426
$671$	−56.3266	−2.17446
$672$	0	0
$673$	14.1611	0.545870	0.272935	−	0.962032i	$-0.412006\pi$
$673$	14.1611	0.545870	0.272935	+	0.962032i	$0.412006\pi$
$674$	0.0285959	0.00110147
$675$	−0.151286	−0.00582299
$676$	19.8471	0.763352
$677$	−5.05314	−0.194208	−0.0971040	−	0.995274i	$-0.530958\pi$
$677$	−5.05314	−0.194208	−0.0971040	+	0.995274i	$0.530958\pi$
$678$	0.185450	0.00712218
$679$	0	0
$680$	−16.4719	−0.631667
$681$	−48.2825	−1.85019
$682$	1.16566	0.0446353
$683$	20.3359	0.778130	0.389065	−	0.921210i	$-0.372798\pi$
$683$	20.3359	0.778130	0.389065	+	0.921210i	$0.372798\pi$
$684$	5.58273	0.213461
$685$	−12.9759	−0.495783
$686$	0	0
$687$	−42.3845	−1.61707
$688$	25.5883	0.975544
$689$	3.23140	0.123106
$690$	10.8515	0.413108
$691$	−23.1140	−0.879300	−0.439650	−	0.898169i	$-0.644897\pi$
$691$	−23.1140	−0.879300	−0.439650	+	0.898169i	$0.644897\pi$
$692$	13.0240	0.495100
$693$	0	0
$694$	−0.312850	−0.0118756
$695$	37.5880	1.42580
$696$	10.7739	0.408382
$697$	59.4654	2.25241
$698$	−2.31471	−0.0876130
$699$	−21.6269	−0.818005
$700$	0	0
$701$	−50.1400	−1.89376	−0.946882	−	0.321583i	$-0.895785\pi$
$701$	−50.1400	−1.89376	−0.946882	+	0.321583i	$0.895785\pi$
$702$	−0.0976828	−0.00368680
$703$	−4.10674	−0.154889
$704$	29.8628	1.12550
$705$	−27.2280	−1.02547
$706$	2.55182	0.0960391
$707$	0	0
$708$	26.8062	1.00744
$709$	−3.80739	−0.142989	−0.0714947	−	0.997441i	$-0.522777\pi$
$709$	−3.80739	−0.142989	−0.0714947	+	0.997441i	$0.522777\pi$
$710$	3.92412	0.147270
$711$	−6.76238	−0.253609
$712$	7.63653	0.286191
$713$	5.99233	0.224415
$714$	0	0
$715$	16.6004	0.620818
$716$	−36.9082	−1.37933
$717$	26.5667	0.992153
$718$	2.58897	0.0966193
$719$	−10.7018	−0.399108	−0.199554	−	0.979887i	$-0.563949\pi$
$719$	−10.7018	−0.399108	−0.199554	+	0.979887i	$0.563949\pi$
$720$	21.0635	0.784989
$721$	0	0
$722$	0.291367	0.0108436
$723$	−12.6079	−0.468894
$724$	−39.3569	−1.46269
$725$	−2.84540	−0.105676
$726$	−9.56195	−0.354877
$727$	−22.1589	−0.821829	−0.410914	−	0.911674i	$-0.634790\pi$
$727$	−22.1589	−0.821829	−0.410914	+	0.911674i	$0.634790\pi$
$728$	0	0
$729$	−25.4509	−0.942625
$730$	1.25784	0.0465548
$731$	51.1357	1.89132
$732$	53.0105	1.95932
$733$	37.3419	1.37925	0.689627	−	0.724165i	$-0.257775\pi$
$733$	37.3419	1.37925	0.689627	+	0.724165i	$0.257775\pi$
$734$	−4.09791	−0.151257
$735$	0	0
$736$	−24.4673	−0.901877
$737$	−69.2627	−2.55132
$738$	7.22563	0.265979
$739$	−16.0538	−0.590548	−0.295274	−	0.955413i	$-0.595411\pi$
$739$	−16.0538	−0.590548	−0.295274	+	0.955413i	$0.595411\pi$
$740$	−16.2467	−0.597242
$741$	−3.94909	−0.145074
$742$	0	0
$743$	24.6993	0.906131	0.453065	−	0.891477i	$-0.350330\pi$
$743$	24.6993	0.906131	0.453065	+	0.891477i	$0.350330\pi$
$744$	−2.24269	−0.0822211
$745$	18.0919	0.662835
$746$	−10.7389	−0.393177
$747$	−14.6266	−0.535160
$748$	66.2524	2.42243
$749$	0	0
$750$	8.39185	0.306427
$751$	26.1921	0.955762	0.477881	−	0.878425i	$-0.341405\pi$
$751$	26.1921	0.955762	0.477881	+	0.878425i	$0.341405\pi$
$752$	18.9565	0.691274
$753$	−29.8351	−1.08725
$754$	−1.83723	−0.0669079
$755$	−24.8808	−0.905505
$756$	0	0
$757$	23.2718	0.845826	0.422913	−	0.906170i	$-0.361008\pi$
$757$	23.2718	0.845826	0.422913	+	0.906170i	$0.361008\pi$
$758$	−5.03794	−0.182986
$759$	−89.2273	−3.23874
$760$	2.35646	0.0854776
$761$	34.5434	1.25220	0.626099	−	0.779744i	$-0.284651\pi$
$761$	34.5434	1.25220	0.626099	+	0.779744i	$0.284651\pi$
$762$	5.72105	0.207252
$763$	0	0
$764$	8.58668	0.310655
$765$	42.0933	1.52189
$766$	6.88280	0.248686
$767$	−9.34497	−0.337427
$768$	−23.4268	−0.845342
$769$	11.1106	0.400659	0.200329	−	0.979729i	$-0.435799\pi$
$769$	11.1106	0.400659	0.200329	+	0.979729i	$0.435799\pi$
$770$	0	0
$771$	54.4157	1.95974
$772$	47.9332	1.72515
$773$	8.30414	0.298679	0.149340	−	0.988786i	$-0.452285\pi$
$773$	8.30414	0.298679	0.149340	+	0.988786i	$0.452285\pi$
$774$	6.21349	0.223339
$775$	0.592300	0.0212760
$776$	16.0459	0.576015
$777$	0	0
$778$	−0.820959	−0.0294328
$779$	−8.50709	−0.304798
$780$	−15.6230	−0.559395
$781$	−32.2665	−1.15459
$782$	−15.0978	−0.539897
$783$	0.801804	0.0286542
$784$	0	0
$785$	−8.83286	−0.315258
$786$	13.7707	0.491183
$787$	−16.6191	−0.592407	−0.296203	−	0.955125i	$-0.595721\pi$
$787$	−16.6191	−0.592407	−0.296203	+	0.955125i	$0.595721\pi$
$788$	0.192235	0.00684810
$789$	−22.0743	−0.785867
$790$	−1.39624	−0.0496761
$791$	0	0
$792$	16.4575	0.584791
$793$	−18.4801	−0.656247
$794$	−6.99316	−0.248178
$795$	9.99842	0.354607
$796$	51.5737	1.82798
$797$	24.5479	0.869532	0.434766	−	0.900543i	$-0.356831\pi$
$797$	24.5479	0.869532	0.434766	+	0.900543i	$0.356831\pi$
$798$	0	0
$799$	37.8828	1.34020
$800$	−2.41842	−0.0855041
$801$	−19.5149	−0.689525
$802$	9.72407	0.343369
$803$	−10.3427	−0.364987
$804$	65.1850	2.29890
$805$	0	0
$806$	0.382438	0.0134708
$807$	−45.7717	−1.61124
$808$	2.04663	0.0720001
$809$	−3.99029	−0.140291	−0.0701455	−	0.997537i	$-0.522346\pi$
$809$	−3.99029	−0.140291	−0.0701455	+	0.997537i	$0.522346\pi$
$810$	−5.56595	−0.195567
$811$	41.1357	1.44447	0.722235	−	0.691648i	$-0.243115\pi$
$811$	41.1357	1.44447	0.722235	+	0.691648i	$0.243115\pi$
$812$	0	0
$813$	−60.7574	−2.13085
$814$	−5.92193	−0.207564
$815$	−18.6470	−0.653175
$816$	−59.4654	−2.08171
$817$	−7.31545	−0.255935
$818$	8.27914	0.289473
$819$	0	0
$820$	−33.6550	−1.17528
$821$	−2.43814	−0.0850917	−0.0425459	−	0.999095i	$-0.513547\pi$
$821$	−2.43814	−0.0850917	−0.0425459	+	0.999095i	$0.513547\pi$
$822$	4.45125	0.155255
$823$	35.8018	1.24797	0.623985	−	0.781436i	$-0.285512\pi$
$823$	35.8018	1.24797	0.623985	+	0.781436i	$0.285512\pi$
$824$	−19.2794	−0.671629
$825$	−8.81949	−0.307055
$826$	0	0
$827$	3.66665	0.127502	0.0637510	−	0.997966i	$-0.479694\pi$
$827$	3.66665	0.127502	0.0637510	+	0.997966i	$0.479694\pi$
$828$	41.3845	1.43821
$829$	48.5906	1.68762	0.843811	−	0.536640i	$-0.180307\pi$
$829$	48.5906	1.68762	0.843811	+	0.536640i	$0.180307\pi$
$830$	−3.01999	−0.104825
$831$	−50.7678	−1.76111
$832$	9.79762	0.339671
$833$	0	0
$834$	−12.8942	−0.446491
$835$	43.8594	1.51782
$836$	−9.47803	−0.327805
$837$	−0.166904	−0.00576905
$838$	−1.73444	−0.0599152
$839$	−9.11600	−0.314719	−0.157360	−	0.987541i	$-0.550298\pi$
$839$	−9.11600	−0.314719	−0.157360	+	0.987541i	$0.550298\pi$
$840$	0	0
$841$	−13.9196	−0.479985
$842$	−0.860243	−0.0296459
$843$	−71.2890	−2.45533
$844$	−27.8351	−0.958125
$845$	−21.4082	−0.736466
$846$	4.60314	0.158259
$847$	0	0
$848$	−6.96104	−0.239043
$849$	−1.37724	−0.0472666
$850$	−1.49231	−0.0511859
$851$	−30.4431	−1.04358
$852$	30.3669	1.04035
$853$	−1.01631	−0.0347979	−0.0173989	−	0.999849i	$-0.505539\pi$
$853$	−1.01631	−0.0347979	−0.0173989	+	0.999849i	$0.505539\pi$
$854$	0	0
$855$	−6.02185	−0.205943
$856$	−10.2771	−0.351265
$857$	−15.8851	−0.542626	−0.271313	−	0.962491i	$-0.587458\pi$
$857$	−15.8851	−0.542626	−0.271313	+	0.962491i	$0.587458\pi$
$858$	−5.69460	−0.194410
$859$	−5.72412	−0.195304	−0.0976522	−	0.995221i	$-0.531133\pi$
$859$	−5.72412	−0.195304	−0.0976522	+	0.995221i	$0.531133\pi$
$860$	−28.9407	−0.986871
$861$	0	0
$862$	2.90503	0.0989456
$863$	13.8950	0.472990	0.236495	−	0.971633i	$-0.424001\pi$
$863$	13.8950	0.472990	0.236495	+	0.971633i	$0.424001\pi$
$864$	0.681486	0.0231846
$865$	−14.0485	−0.477662
$866$	5.56772	0.189199
$867$	−77.4902	−2.63171
$868$	0	0
$869$	11.4808	0.389459
$870$	−5.68465	−0.192728
$871$	−22.7243	−0.769982
$872$	−7.99568	−0.270768
$873$	−41.0049	−1.38781
$874$	2.15989	0.0730593
$875$	0	0
$876$	9.73383	0.328876
$877$	−40.0210	−1.35141	−0.675707	−	0.737170i	$-0.736162\pi$
$877$	−40.0210	−1.35141	−0.675707	+	0.737170i	$0.736162\pi$
$878$	−2.12896	−0.0718490
$879$	−61.4594	−2.07298
$880$	−35.7603	−1.20548
$881$	8.38440	0.282478	0.141239	−	0.989976i	$-0.454891\pi$
$881$	8.38440	0.282478	0.141239	+	0.989976i	$0.454891\pi$
$882$	0	0
$883$	−20.0109	−0.673421	−0.336711	−	0.941608i	$-0.609314\pi$
$883$	−20.0109	−0.673421	−0.336711	+	0.941608i	$0.609314\pi$
$884$	21.7366	0.731082
$885$	−28.9147	−0.971957
$886$	−4.79702	−0.161159
$887$	−29.1581	−0.979032	−0.489516	−	0.871994i	$-0.662827\pi$
$887$	−29.1581	−0.979032	−0.489516	+	0.871994i	$0.662827\pi$
$888$	11.3936	0.382346
$889$	0	0
$890$	−4.02929	−0.135062
$891$	45.7666	1.53324
$892$	4.42061	0.148013
$893$	−5.41950	−0.181357
$894$	−6.20625	−0.207568
$895$	39.8113	1.33074
$896$	0	0
$897$	−29.2744	−0.977444
$898$	−2.83165	−0.0944933
$899$	−3.13915	−0.104696
$900$	4.09057	0.136352
$901$	−13.9110	−0.463442
$902$	−12.2672	−0.408454
$903$	0	0
$904$	0.298531	0.00992900
$905$	42.4526	1.41117
$906$	8.53513	0.283561
$907$	50.6610	1.68217	0.841086	−	0.540902i	$-0.181917\pi$
$907$	50.6610	1.68217	0.841086	+	0.540902i	$0.181917\pi$
$908$	−38.0191	−1.26171
$909$	−5.23010	−0.173471
$910$	0	0
$911$	−28.9746	−0.959970	−0.479985	−	0.877277i	$-0.659358\pi$
$911$	−28.9746	−0.959970	−0.479985	+	0.877277i	$0.659358\pi$
$912$	8.50709	0.281698
$913$	24.8322	0.821826
$914$	−3.20550	−0.106028
$915$	−57.1801	−1.89031
$916$	−33.3748	−1.10273
$917$	0	0
$918$	0.420518	0.0138792
$919$	50.1894	1.65559	0.827797	−	0.561028i	$-0.189594\pi$
$919$	50.1894	1.65559	0.827797	+	0.561028i	$0.189594\pi$
$920$	17.4683	0.575912
$921$	15.4952	0.510586
$922$	8.35581	0.275184
$923$	−10.5863	−0.348451
$924$	0	0
$925$	−3.00908	−0.0989381
$926$	−9.84011	−0.323366
$927$	49.2678	1.61817
$928$	12.8175	0.420754
$929$	33.9101	1.11255	0.556277	−	0.830997i	$-0.312229\pi$
$929$	33.9101	1.11255	0.556277	+	0.830997i	$0.312229\pi$
$930$	1.18332	0.0388026
$931$	0	0
$932$	−17.0297	−0.557825
$933$	−0.719144	−0.0235437
$934$	4.90887	0.160623
$935$	−71.4635	−2.33711
$936$	5.39951	0.176488
$937$	−17.4372	−0.569648	−0.284824	−	0.958580i	$-0.591935\pi$
$937$	−17.4372	−0.569648	−0.284824	+	0.958580i	$0.591935\pi$
$938$	0	0
$939$	49.5369	1.61657
$940$	−21.4401	−0.699300
$941$	0.598364	0.0195061	0.00975306	−	0.999952i	$-0.496895\pi$
$941$	0.598364	0.0195061	0.00975306	+	0.999952i	$0.496895\pi$
$942$	3.03003	0.0987238
$943$	−63.0626	−2.05360
$944$	20.1308	0.655203
$945$	0	0
$946$	−10.5489	−0.342974
$947$	33.3136	1.08255	0.541274	−	0.840846i	$-0.317942\pi$
$947$	33.3136	1.08255	0.541274	+	0.840846i	$0.317942\pi$
$948$	−10.8049	−0.350926
$949$	−3.39333	−0.110152
$950$	0.213490	0.00692652
$951$	74.2179	2.40668
$952$	0	0
$953$	8.41792	0.272683	0.136342	−	0.990662i	$-0.456466\pi$
$953$	8.41792	0.272683	0.136342	+	0.990662i	$0.456466\pi$
$954$	−1.69032	−0.0547261
$955$	−9.26207	−0.299714
$956$	20.9194	0.676583
$957$	46.7427	1.51098
$958$	6.71454	0.216937
$959$	0	0
$960$	30.3153	0.978421
$961$	−30.3466	−0.978921
$962$	−1.94292	−0.0626421
$963$	26.2629	0.846311
$964$	−9.92785	−0.319755
$965$	−51.7034	−1.66439
$966$	0	0
$967$	−17.5459	−0.564237	−0.282118	−	0.959380i	$-0.591037\pi$
$967$	−17.5459	−0.564237	−0.282118	+	0.959380i	$0.591037\pi$
$968$	−15.3925	−0.494733
$969$	17.0006	0.546138
$970$	−8.46637	−0.271839
$971$	−9.87266	−0.316829	−0.158414	−	0.987373i	$-0.550638\pi$
$971$	−9.87266	−0.316829	−0.158414	+	0.987373i	$0.550638\pi$
$972$	−41.8860	−1.34349
$973$	0	0
$974$	−6.39458	−0.204895
$975$	−2.89357	−0.0926684
$976$	39.8096	1.27427
$977$	−14.9641	−0.478743	−0.239371	−	0.970928i	$-0.576941\pi$
$977$	−14.9641	−0.478743	−0.239371	+	0.970928i	$0.576941\pi$
$978$	6.39668	0.204543
$979$	33.1312	1.05888
$980$	0	0
$981$	20.4327	0.652366
$982$	2.96853	0.0947296
$983$	−6.58111	−0.209905	−0.104952	−	0.994477i	$-0.533469\pi$
$983$	−6.58111	−0.209905	−0.104952	+	0.994477i	$0.533469\pi$
$984$	23.6018	0.752399
$985$	−0.207356	−0.00660690
$986$	7.90916	0.251879
$987$	0	0
$988$	−3.10963	−0.0989305
$989$	−54.2290	−1.72438
$990$	−8.68352	−0.275980
$991$	15.1300	0.480621	0.240311	−	0.970696i	$-0.422751\pi$
$991$	15.1300	0.480621	0.240311	+	0.970696i	$0.422751\pi$
$992$	−2.66809	−0.0847119
$993$	68.7832	2.18277
$994$	0	0
$995$	−55.6303	−1.76360
$996$	−23.3703	−0.740515
$997$	−28.4545	−0.901164	−0.450582	−	0.892735i	$-0.648783\pi$
$997$	−28.4545	−0.901164	−0.450582	+	0.892735i	$0.648783\pi$
$998$	4.78716	0.151535
$999$	0.847929	0.0268273

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	931.2.a.m.1.3	yes	4
3.2	odd	2		8379.2.a.bu.1.2		4
7.2	even	3		931.2.f.n.704.2		8
7.3	odd	6		931.2.f.o.324.2		8
7.4	even	3		931.2.f.n.324.2		8
7.5	odd	6		931.2.f.o.704.2		8
7.6	odd	2		931.2.a.l.1.3	✓	4
21.20	even	2		8379.2.a.bv.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
931.2.a.l.1.3	✓	4	7.6	odd	2
931.2.a.m.1.3	yes	4	1.1	even	1	trivial
931.2.f.n.324.2		8	7.4	even	3
931.2.f.n.704.2		8	7.2	even	3
931.2.f.o.324.2		8	7.3	odd	6
931.2.f.o.704.2		8	7.5	odd	6
8379.2.a.bu.1.2		4	3.2	odd	2
8379.2.a.bv.1.2		4	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 931 = 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	931.a (trivial)

\(f(q)\)	\(=\)	\(q+0.291367 q^{2} -2.43210 q^{3} -1.91511 q^{4} +2.06574 q^{5} -0.708633 q^{6} -1.14073 q^{8} +2.91511 q^{9} +0.601888 q^{10} -4.94909 q^{11} +4.65773 q^{12} -1.62374 q^{13} -5.02408 q^{15} +3.49784 q^{16} +6.99010 q^{17} +0.849365 q^{18} -1.00000 q^{19} -3.95611 q^{20} -1.44200 q^{22} -7.41294 q^{23} +2.77437 q^{24} -0.732718 q^{25} -0.473104 q^{26} +0.206472 q^{27} +3.88335 q^{29} -1.46385 q^{30} -0.808361 q^{31} +3.30062 q^{32} +12.0367 q^{33} +2.03668 q^{34} -5.58273 q^{36} +4.10674 q^{37} -0.291367 q^{38} +3.94909 q^{39} -2.35646 q^{40} +8.50709 q^{41} +7.31545 q^{43} +9.47803 q^{44} +6.02185 q^{45} -2.15989 q^{46} +5.41950 q^{47} -8.50709 q^{48} -0.213490 q^{50} -17.0006 q^{51} +3.10963 q^{52} -1.99010 q^{53} +0.0601592 q^{54} -10.2235 q^{55} +2.43210 q^{57} +1.13148 q^{58} +5.75522 q^{59} +9.62165 q^{60} +11.3812 q^{61} -0.235529 q^{62} -6.03399 q^{64} -3.35422 q^{65} +3.50709 q^{66} +13.9950 q^{67} -13.3868 q^{68} +18.0290 q^{69} +6.51969 q^{71} -3.32535 q^{72} +2.08982 q^{73} +1.19657 q^{74} +1.78204 q^{75} +1.91511 q^{76} +1.15063 q^{78} -2.31977 q^{79} +7.22563 q^{80} -9.24748 q^{81} +2.47868 q^{82} -5.01753 q^{83} +14.4397 q^{85} +2.13148 q^{86} -9.44470 q^{87} +5.64559 q^{88} -6.69441 q^{89} +1.75457 q^{90} +14.1966 q^{92} +1.96601 q^{93} +1.57906 q^{94} -2.06574 q^{95} -8.02743 q^{96} -14.0663 q^{97} -14.4271 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 2 q^{4} + 8 q^{5} - 4 q^{6} + 6 q^{8} + 2 q^{9} + 10 q^{10} - 6 q^{11} + 6 q^{12} + 2 q^{13} + 4 q^{15} + 2 q^{16} + 8 q^{17} - 6 q^{18} - 4 q^{19} - 14 q^{22} - 8 q^{23} + 12 q^{24} + 20 q^{25}+ \cdots + 8 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 2 * q^4 + 8 * q^5 - 4 * q^6 + 6 * q^8 + 2 * q^9 + 10 * q^10 - 6 * q^11 + 6 * q^12 + 2 * q^13 + 4 * q^15 + 2 * q^16 + 8 * q^17 - 6 * q^18 - 4 * q^19 - 14 * q^22 - 8 * q^23 + 12 * q^24 + 20 * q^25 + 16 * q^26 - 10 * q^27 + 2 * q^29 + 2 * q^30 + 2 * q^32 + 18 * q^33 - 22 * q^34 - 20 * q^36 + 10 * q^37 + 2 * q^39 + 22 * q^40 + 12 * q^41 + 4 * q^43 - 14 * q^44 + 8 * q^45 - 8 * q^46 + 16 * q^47 - 12 * q^48 + 12 * q^50 - 10 * q^51 + 28 * q^52 + 12 * q^53 + 4 * q^54 - 8 * q^55 - 2 * q^57 + 4 * q^58 + 14 * q^59 - 2 * q^60 + 20 * q^61 - 20 * q^62 - 20 * q^64 + 10 * q^65 - 8 * q^66 + 2 * q^67 - 24 * q^68 + 14 * q^69 - 2 * q^71 - 8 * q^72 - 16 * q^73 - 26 * q^74 + 36 * q^75 - 2 * q^76 + 14 * q^78 - 8 * q^79 + 28 * q^80 - 20 * q^81 - 12 * q^82 + 20 * q^83 - 14 * q^85 + 8 * q^86 - 20 * q^87 - 2 * q^88 + 16 * q^89 - 32 * q^90 + 26 * q^92 + 12 * q^93 + 10 * q^94 - 8 * q^95 - 12 * q^96 + 2 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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