LMFDB - Embedded newform 961.2.a.j.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [961,2,Mod(1,961)] 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("961.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(961, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 961 = 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	961.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,2,3,8,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.67362363425$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.2051578125.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 19x^{5} + 14x^{4} - 28x^{3} - 11x^{2} + 6x + 1 $ x^8 - 2x^7 - 9x^6 + 19x^5 + 14x^4 - 28x^3 - 11x^2 + 6*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 31)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.143490$ of defining polynomial
Character	$\chi$	$=$	961.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.23217 q^{2} -2.07497 q^{3} -0.481752 q^{4} -1.54562 q^{5} -2.55673 q^{6} -3.80376 q^{7} -3.05795 q^{8} +1.30552 q^{9} -1.90447 q^{10} +3.76152 q^{11} +0.999622 q^{12} +2.63506 q^{13} -4.68689 q^{14} +3.20713 q^{15} -2.80441 q^{16} -3.77434 q^{17} +1.60863 q^{18} +6.09831 q^{19} +0.744606 q^{20} +7.89270 q^{21} +4.63485 q^{22} -0.909847 q^{23} +6.34516 q^{24} -2.61105 q^{25} +3.24685 q^{26} +3.51600 q^{27} +1.83247 q^{28} +6.80859 q^{29} +3.95173 q^{30} +2.66037 q^{32} -7.80507 q^{33} -4.65064 q^{34} +5.87917 q^{35} -0.628936 q^{36} +1.81406 q^{37} +7.51417 q^{38} -5.46769 q^{39} +4.72643 q^{40} +0.337145 q^{41} +9.72517 q^{42} -3.88400 q^{43} -1.81212 q^{44} -2.01784 q^{45} -1.12109 q^{46} -1.18915 q^{47} +5.81908 q^{48} +7.46858 q^{49} -3.21726 q^{50} +7.83166 q^{51} -1.26945 q^{52} +2.34413 q^{53} +4.33232 q^{54} -5.81390 q^{55} +11.6317 q^{56} -12.6538 q^{57} +8.38936 q^{58} +7.77883 q^{59} -1.54504 q^{60} +2.72343 q^{61} -4.96588 q^{63} +8.88686 q^{64} -4.07281 q^{65} -9.61719 q^{66} -7.42118 q^{67} +1.81829 q^{68} +1.88791 q^{69} +7.24416 q^{70} -5.09818 q^{71} -3.99221 q^{72} -5.39174 q^{73} +2.23524 q^{74} +5.41786 q^{75} -2.93787 q^{76} -14.3079 q^{77} -6.73714 q^{78} +9.73723 q^{79} +4.33456 q^{80} -11.2122 q^{81} +0.415420 q^{82} +8.39515 q^{83} -3.80232 q^{84} +5.83370 q^{85} -4.78575 q^{86} -14.1277 q^{87} -11.5025 q^{88} -5.09560 q^{89} -2.48633 q^{90} -10.0231 q^{91} +0.438320 q^{92} -1.46523 q^{94} -9.42569 q^{95} -5.52020 q^{96} +10.9142 q^{97} +9.20257 q^{98} +4.91075 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} + 3 q^{3} + 8 q^{4} + 3 q^{5} + 11 q^{6} - 2 q^{7} - 9 q^{8} + 5 q^{9} - 13 q^{10} + 18 q^{11} + 8 q^{13} - 9 q^{14} + 18 q^{15} + 4 q^{16} + 14 q^{17} + 23 q^{18} - 6 q^{19} - 7 q^{20} - q^{21}+ \cdots + 6 q^{99}+O(q^{100}) $ 8 * q + 2 * q^2 + 3 * q^3 + 8 * q^4 + 3 * q^5 + 11 * q^6 - 2 * q^7 - 9 * q^8 + 5 * q^9 - 13 * q^10 + 18 * q^11 + 8 * q^13 - 9 * q^14 + 18 * q^15 + 4 * q^16 + 14 * q^17 + 23 * q^18 - 6 * q^19 - 7 * q^20 - q^21 + 4 * q^22 + 22 * q^23 + 30 * q^24 + 13 * q^25 + 9 * q^26 + 18 * q^27 - 5 * q^28 + 12 * q^29 + 11 * q^30 - 21 * q^32 + 6 * q^33 - 7 * q^34 - 12 * q^35 - q^36 - 8 * q^37 + 7 * q^38 + q^39 + 11 * q^40 - 22 * q^41 - 6 * q^42 - 2 * q^43 + 4 * q^44 + 18 * q^46 - 18 * q^47 - q^48 - 2 * q^49 + 27 * q^50 + 2 * q^51 - q^52 + 6 * q^53 - 7 * q^54 + 28 * q^55 + 30 * q^56 - 17 * q^57 - 5 * q^58 - 4 * q^59 - 40 * q^60 + 30 * q^61 - 23 * q^63 + 9 * q^64 + 3 * q^65 + 10 * q^66 - 13 * q^67 + 30 * q^68 + 22 * q^69 - 39 * q^70 - q^71 + 3 * q^72 + 2 * q^73 + 8 * q^74 + 23 * q^75 - 33 * q^76 - 24 * q^77 + 8 * q^79 + 24 * q^80 - 28 * q^81 - 19 * q^82 + 39 * q^83 + 8 * q^84 - 26 * q^85 - 16 * q^86 - 15 * q^87 - 17 * q^88 + 27 * q^89 + 8 * q^90 + 16 * q^91 + 32 * q^92 + 22 * q^94 - 6 * q^95 - 12 * q^96 + 34 * q^97 + 10 * q^98 + 6 * 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 $

$2$	1.23217	0.871277	0.435639	−	0.900122i	$-0.356523\pi$
$2$	1.23217	0.871277	0.435639	+	0.900122i	$0.356523\pi$
$3$	−2.07497	−1.19799	−0.598994	−	0.800754i	$-0.704433\pi$
$3$	−2.07497	−1.19799	−0.598994	+	0.800754i	$0.704433\pi$
$4$	−0.481752	−0.240876
$5$	−1.54562	−0.691223	−0.345612	−	0.938378i	$-0.612329\pi$
$5$	−1.54562	−0.691223	−0.345612	+	0.938378i	$0.612329\pi$
$6$	−2.55673	−1.04378
$7$	−3.80376	−1.43769	−0.718843	−	0.695173i	$-0.755328\pi$
$7$	−3.80376	−1.43769	−0.718843	+	0.695173i	$0.755328\pi$
$8$	−3.05795	−1.08115
$9$	1.30552	0.435173
$10$	−1.90447	−0.602247
$11$	3.76152	1.13414	0.567071	−	0.823669i	$-0.308076\pi$
$11$	3.76152	1.13414	0.567071	+	0.823669i	$0.308076\pi$
$12$	0.999622	0.288566
$13$	2.63506	0.730835	0.365418	−	0.930844i	$-0.380926\pi$
$13$	2.63506	0.730835	0.365418	+	0.930844i	$0.380926\pi$
$14$	−4.68689	−1.25262
$15$	3.20713	0.828077
$16$	−2.80441	−0.701103
$17$	−3.77434	−0.915412	−0.457706	−	0.889104i	$-0.651329\pi$
$17$	−3.77434	−0.915412	−0.457706	+	0.889104i	$0.651329\pi$
$18$	1.60863	0.379157
$19$	6.09831	1.39905	0.699525	−	0.714608i	$-0.253395\pi$
$19$	6.09831	1.39905	0.699525	+	0.714608i	$0.253395\pi$
$20$	0.744606	0.166499
$21$	7.89270	1.72233
$22$	4.63485	0.988152
$23$	−0.909847	−0.189716	−0.0948581	−	0.995491i	$-0.530240\pi$
$23$	−0.909847	−0.189716	−0.0948581	+	0.995491i	$0.530240\pi$
$24$	6.34516	1.29520
$25$	−2.61105	−0.522210
$26$	3.24685	0.636760
$27$	3.51600	0.676655
$28$	1.83247	0.346304
$29$	6.80859	1.26432	0.632162	−	0.774836i	$-0.282168\pi$
$29$	6.80859	1.26432	0.632162	+	0.774836i	$0.282168\pi$
$30$	3.95173	0.721485
$31$	0	0
$31$	0	0
$32$	2.66037	0.470292
$33$	−7.80507	−1.35869
$34$	−4.65064	−0.797578
$35$	5.87917	0.993762
$36$	−0.628936	−0.104823
$37$	1.81406	0.298230	0.149115	−	0.988820i	$-0.452357\pi$
$37$	1.81406	0.298230	0.149115	+	0.988820i	$0.452357\pi$
$38$	7.51417	1.21896
$39$	−5.46769	−0.875531
$40$	4.72643	0.747314
$41$	0.337145	0.0526531	0.0263266	−	0.999653i	$-0.491619\pi$
$41$	0.337145	0.0526531	0.0263266	+	0.999653i	$0.491619\pi$
$42$	9.72517	1.50063
$43$	−3.88400	−0.592304	−0.296152	−	0.955141i	$-0.595703\pi$
$43$	−3.88400	−0.592304	−0.296152	+	0.955141i	$0.595703\pi$
$44$	−1.81212	−0.273187
$45$	−2.01784	−0.300802
$46$	−1.12109	−0.165295
$47$	−1.18915	−0.173455	−0.0867275	−	0.996232i	$-0.527641\pi$
$47$	−1.18915	−0.173455	−0.0867275	+	0.996232i	$0.527641\pi$
$48$	5.81908	0.839913
$49$	7.46858	1.06694
$50$	−3.21726	−0.454990
$51$	7.83166	1.09665
$52$	−1.26945	−0.176040
$53$	2.34413	0.321990	0.160995	−	0.986955i	$-0.448530\pi$
$53$	2.34413	0.321990	0.160995	+	0.986955i	$0.448530\pi$
$54$	4.33232	0.589554
$55$	−5.81390	−0.783946
$56$	11.6317	1.55435
$57$	−12.6538	−1.67604
$58$	8.38936	1.10158
$59$	7.77883	1.01272	0.506359	−	0.862323i	$-0.330991\pi$
$59$	7.77883	1.01272	0.506359	+	0.862323i	$0.330991\pi$
$60$	−1.54504	−0.199464
$61$	2.72343	0.348700	0.174350	−	0.984684i	$-0.444218\pi$
$61$	2.72343	0.348700	0.174350	+	0.984684i	$0.444218\pi$
$62$	0	0
$63$	−4.96588	−0.625642
$64$	8.88686	1.11086
$65$	−4.07281	−0.505170
$66$	−9.61719	−1.18379
$67$	−7.42118	−0.906642	−0.453321	−	0.891347i	$-0.649761\pi$
$67$	−7.42118	−0.906642	−0.453321	+	0.891347i	$0.649761\pi$
$68$	1.81829	0.220501
$69$	1.88791	0.227278
$70$	7.24416	0.865842
$71$	−5.09818	−0.605043	−0.302521	−	0.953143i	$-0.597828\pi$
$71$	−5.09818	−0.605043	−0.302521	+	0.953143i	$0.597828\pi$
$72$	−3.99221	−0.470486
$73$	−5.39174	−0.631056	−0.315528	−	0.948916i	$-0.602182\pi$
$73$	−5.39174	−0.631056	−0.315528	+	0.948916i	$0.602182\pi$
$74$	2.23524	0.259841
$75$	5.41786	0.625601
$76$	−2.93787	−0.336997
$77$	−14.3079	−1.63054
$78$	−6.73714	−0.762830
$79$	9.73723	1.09552	0.547762	−	0.836634i	$-0.315480\pi$
$79$	9.73723	1.09552	0.547762	+	0.836634i	$0.315480\pi$
$80$	4.33456	0.484619
$81$	−11.2122	−1.24580
$82$	0.415420	0.0458755
$83$	8.39515	0.921487	0.460743	−	0.887533i	$-0.347583\pi$
$83$	8.39515	0.921487	0.460743	+	0.887533i	$0.347583\pi$
$84$	−3.80232	−0.414867
$85$	5.83370	0.632754
$86$	−4.78575	−0.516061
$87$	−14.1277	−1.51464
$88$	−11.5025	−1.22617
$89$	−5.09560	−0.540132	−0.270066	−	0.962842i	$-0.587046\pi$
$89$	−5.09560	−0.540132	−0.270066	+	0.962842i	$0.587046\pi$
$90$	−2.48633	−0.262082
$91$	−10.0231	−1.05071
$92$	0.438320	0.0456980
$93$	0	0
$94$	−1.46523	−0.151127
$95$	−9.42569	−0.967056
$96$	−5.52020	−0.563403
$97$	10.9142	1.10817	0.554086	−	0.832460i	$-0.313068\pi$
$97$	10.9142	1.10817	0.554086	+	0.832460i	$0.313068\pi$
$98$	9.20257	0.929600
$99$	4.91075	0.493549
$100$	1.25788	0.125788
$101$	−0.398727	−0.0396748	−0.0198374	−	0.999803i	$-0.506315\pi$
$101$	−0.398727	−0.0396748	−0.0198374	+	0.999803i	$0.506315\pi$
$102$	9.64995	0.955488
$103$	3.27680	0.322873	0.161436	−	0.986883i	$-0.448387\pi$
$103$	3.27680	0.322873	0.161436	+	0.986883i	$0.448387\pi$
$104$	−8.05788	−0.790140
$105$	−12.1991	−1.19051
$106$	2.88837	0.280543
$107$	3.15685	0.305184	0.152592	−	0.988289i	$-0.451238\pi$
$107$	3.15685	0.305184	0.152592	+	0.988289i	$0.451238\pi$
$108$	−1.69384	−0.162990
$109$	6.95622	0.666285	0.333142	−	0.942877i	$-0.391891\pi$
$109$	6.95622	0.666285	0.333142	+	0.942877i	$0.391891\pi$
$110$	−7.16372	−0.683034
$111$	−3.76414	−0.357276
$112$	10.6673	1.00797
$113$	−15.2468	−1.43430	−0.717148	−	0.696920i	$-0.754553\pi$
$113$	−15.2468	−1.43430	−0.717148	+	0.696920i	$0.754553\pi$
$114$	−15.5917	−1.46030
$115$	1.40628	0.131136
$116$	−3.28005	−0.304545
$117$	3.44013	0.318040
$118$	9.58486	0.882358
$119$	14.3567	1.31607
$120$	−9.80722	−0.895273
$121$	3.14907	0.286279
$122$	3.35574	0.303814
$123$	−0.699567	−0.0630778
$124$	0	0
$125$	11.7638	1.05219
$126$	−6.11882	−0.545108
$127$	20.2264	1.79480	0.897400	−	0.441218i	$-0.145453\pi$
$127$	20.2264	1.79480	0.897400	+	0.441218i	$0.145453\pi$
$128$	5.62940	0.497573
$129$	8.05919	0.709572
$130$	−5.01841	−0.440143
$131$	12.6513	1.10535	0.552674	−	0.833398i	$-0.313608\pi$
$131$	12.6513	1.10535	0.552674	+	0.833398i	$0.313608\pi$
$132$	3.76010	0.327275
$133$	−23.1965	−2.01139
$134$	−9.14417	−0.789936
$135$	−5.43441	−0.467720
$136$	11.5417	0.989695
$137$	9.87801	0.843936	0.421968	−	0.906611i	$-0.361340\pi$
$137$	9.87801	0.843936	0.421968	+	0.906611i	$0.361340\pi$
$138$	2.32623	0.198022
$139$	15.4990	1.31461	0.657303	−	0.753626i	$-0.271697\pi$
$139$	15.4990	1.31461	0.657303	+	0.753626i	$0.271697\pi$
$140$	−2.83230	−0.239373
$141$	2.46745	0.207797
$142$	−6.28184	−0.527160
$143$	9.91185	0.828871
$144$	−3.66122	−0.305101
$145$	−10.5235	−0.873930
$146$	−6.64356	−0.549825
$147$	−15.4971	−1.27818
$148$	−0.873928	−0.0718364
$149$	−15.2432	−1.24878	−0.624388	−	0.781115i	$-0.714651\pi$
$149$	−15.2432	−1.24878	−0.624388	+	0.781115i	$0.714651\pi$
$150$	6.67574	0.545072
$151$	0.261828	0.0213073	0.0106536	−	0.999943i	$-0.496609\pi$
$151$	0.261828	0.0213073	0.0106536	+	0.999943i	$0.496609\pi$
$152$	−18.6483	−1.51258
$153$	−4.92748	−0.398363
$154$	−17.6298	−1.42065
$155$	0	0
$156$	2.63407	0.210894
$157$	15.9265	1.27108	0.635538	−	0.772070i	$-0.280778\pi$
$157$	15.9265	1.27108	0.635538	+	0.772070i	$0.280778\pi$
$158$	11.9979	0.954505
$159$	−4.86400	−0.385740
$160$	−4.11193	−0.325077
$161$	3.46084	0.272752
$162$	−13.8153	−1.08544
$163$	−0.133175	−0.0104311	−0.00521555	−	0.999986i	$-0.501660\pi$
$163$	−0.133175	−0.0104311	−0.00521555	+	0.999986i	$0.501660\pi$
$164$	−0.162420	−0.0126829
$165$	12.0637	0.939157
$166$	10.3443	0.802871
$167$	3.94878	0.305566	0.152783	−	0.988260i	$-0.451176\pi$
$167$	3.94878	0.305566	0.152783	+	0.988260i	$0.451176\pi$
$168$	−24.1355	−1.86209
$169$	−6.05644	−0.465880
$170$	7.18813	0.551304
$171$	7.96147	0.608829
$172$	1.87112	0.142672
$173$	−14.4569	−1.09914	−0.549570	−	0.835447i	$-0.685209\pi$
$173$	−14.4569	−1.09914	−0.549570	+	0.835447i	$0.685209\pi$
$174$	−17.4077	−1.31967
$175$	9.93180	0.750774
$176$	−10.5489	−0.795151
$177$	−16.1409	−1.21322
$178$	−6.27865	−0.470605
$179$	17.7199	1.32445	0.662223	−	0.749307i	$-0.269613\pi$
$179$	17.7199	1.32445	0.662223	+	0.749307i	$0.269613\pi$
$180$	0.972098	0.0724559
$181$	12.0592	0.896350	0.448175	−	0.893946i	$-0.352074\pi$
$181$	12.0592	0.896350	0.448175	+	0.893946i	$0.352074\pi$
$182$	−12.3502	−0.915461
$183$	−5.65105	−0.417738
$184$	2.78226	0.205111
$185$	−2.80386	−0.206144
$186$	0	0
$187$	−14.1973	−1.03821
$188$	0.572874	0.0417811
$189$	−13.3740	−0.972817
$190$	−11.6141	−0.842574
$191$	5.07152	0.366962	0.183481	−	0.983023i	$-0.441263\pi$
$191$	5.07152	0.366962	0.183481	+	0.983023i	$0.441263\pi$
$192$	−18.4400	−1.33079
$193$	−20.0417	−1.44264	−0.721318	−	0.692604i	$-0.756463\pi$
$193$	−20.0417	−1.44264	−0.721318	+	0.692604i	$0.756463\pi$
$194$	13.4482	0.965525
$195$	8.45099	0.605188
$196$	−3.59800	−0.257000
$197$	−21.7084	−1.54666	−0.773328	−	0.634006i	$-0.781410\pi$
$197$	−21.7084	−1.54666	−0.773328	+	0.634006i	$0.781410\pi$
$198$	6.05089	0.430018
$199$	−15.1655	−1.07505	−0.537527	−	0.843246i	$-0.680641\pi$
$199$	−15.1655	−1.07505	−0.537527	+	0.843246i	$0.680641\pi$
$200$	7.98445	0.564586
$201$	15.3988	1.08615
$202$	−0.491300	−0.0345678
$203$	−25.8982	−1.81770
$204$	−3.77291	−0.264157
$205$	−0.521098	−0.0363951
$206$	4.03759	0.281312
$207$	−1.18782	−0.0825595
$208$	−7.38980	−0.512391
$209$	22.9390	1.58672
$210$	−15.0314	−1.03727
$211$	6.30441	0.434013	0.217007	−	0.976170i	$-0.430371\pi$
$211$	6.30441	0.434013	0.217007	+	0.976170i	$0.430371\pi$
$212$	−1.12929	−0.0775597
$213$	10.5786	0.724833
$214$	3.88978	0.265900
$215$	6.00319	0.409414
$216$	−10.7517	−0.731564
$217$	0	0
$218$	8.57126	0.580519
$219$	11.1877	0.755997
$220$	2.80085	0.188834
$221$	−9.94562	−0.669015
$222$	−4.63806	−0.311287
$223$	−7.38911	−0.494811	−0.247406	−	0.968912i	$-0.579578\pi$
$223$	−7.38911	−0.494811	−0.247406	+	0.968912i	$0.579578\pi$
$224$	−10.1194	−0.676132
$225$	−3.40878	−0.227252
$226$	−18.7867	−1.24967
$227$	19.2398	1.27699	0.638495	−	0.769626i	$-0.279557\pi$
$227$	19.2398	1.27699	0.638495	+	0.769626i	$0.279557\pi$
$228$	6.09601	0.403718
$229$	6.08710	0.402247	0.201124	−	0.979566i	$-0.435541\pi$
$229$	6.08710	0.402247	0.201124	+	0.979566i	$0.435541\pi$
$230$	1.73278	0.114256
$231$	29.6886	1.95337
$232$	−20.8203	−1.36692
$233$	17.5280	1.14830	0.574150	−	0.818750i	$-0.305333\pi$
$233$	17.5280	1.14830	0.574150	+	0.818750i	$0.305333\pi$
$234$	4.23883	0.277101
$235$	1.83797	0.119896
$236$	−3.74746	−0.243939
$237$	−20.2045	−1.31242
$238$	17.6899	1.14667
$239$	27.9120	1.80548	0.902740	−	0.430187i	$-0.141552\pi$
$239$	27.9120	1.80548	0.902740	+	0.430187i	$0.141552\pi$
$240$	−8.99411	−0.580567
$241$	−22.7497	−1.46544	−0.732720	−	0.680531i	$-0.761749\pi$
$241$	−22.7497	−1.46544	−0.732720	+	0.680531i	$0.761749\pi$
$242$	3.88019	0.249428
$243$	12.7170	0.815794
$244$	−1.31202	−0.0839933
$245$	−11.5436	−0.737494
$246$	−0.861987	−0.0549583
$247$	16.0694	1.02247
$248$	0	0
$249$	−17.4197	−1.10393
$250$	14.4950	0.916747
$251$	16.2345	1.02471	0.512356	−	0.858773i	$-0.328773\pi$
$251$	16.2345	1.02471	0.512356	+	0.858773i	$0.328773\pi$
$252$	2.39232	0.150702
$253$	−3.42241	−0.215165
$254$	24.9224	1.56377
$255$	−12.1048	−0.758031
$256$	−10.8373	−0.677333
$257$	23.3521	1.45666	0.728332	−	0.685224i	$-0.240296\pi$
$257$	23.3521	1.45666	0.728332	+	0.685224i	$0.240296\pi$
$258$	9.93031	0.618234
$259$	−6.90026	−0.428761
$260$	1.96208	0.121683
$261$	8.88875	0.550200
$262$	15.5886	0.963064
$263$	10.3944	0.640944	0.320472	−	0.947258i	$-0.396159\pi$
$263$	10.3944	0.640944	0.320472	+	0.947258i	$0.396159\pi$
$264$	23.8675	1.46894
$265$	−3.62313	−0.222567
$266$	−28.5821	−1.75248
$267$	10.5732	0.647072
$268$	3.57517	0.218388
$269$	4.41805	0.269373	0.134686	−	0.990888i	$-0.456997\pi$
$269$	4.41805	0.269373	0.134686	+	0.990888i	$0.456997\pi$
$270$	−6.69613	−0.407514
$271$	−6.15286	−0.373760	−0.186880	−	0.982383i	$-0.559838\pi$
$271$	−6.15286	−0.373760	−0.186880	+	0.982383i	$0.559838\pi$
$272$	10.5848	0.641798
$273$	20.7978	1.25874
$274$	12.1714	0.735302
$275$	−9.82153	−0.592261
$276$	−0.909503	−0.0547457
$277$	−22.0474	−1.32470	−0.662350	−	0.749194i	$-0.730441\pi$
$277$	−22.0474	−1.32470	−0.662350	+	0.749194i	$0.730441\pi$
$278$	19.0974	1.14539
$279$	0	0
$280$	−17.9782	−1.07440
$281$	−30.7736	−1.83580	−0.917899	−	0.396815i	$-0.870115\pi$
$281$	−30.7736	−1.83580	−0.917899	+	0.396815i	$0.870115\pi$
$282$	3.04032	0.181049
$283$	−21.3453	−1.26884	−0.634422	−	0.772987i	$-0.718762\pi$
$283$	−21.3453	−1.26884	−0.634422	+	0.772987i	$0.718762\pi$
$284$	2.45606	0.145740
$285$	19.5581	1.15852
$286$	12.2131	0.722176
$287$	−1.28242	−0.0756987
$288$	3.47317	0.204658
$289$	−2.75436	−0.162021
$290$	−12.9668	−0.761436
$291$	−22.6467	−1.32758
$292$	2.59748	0.152006
$293$	−1.79903	−0.105100	−0.0525501	−	0.998618i	$-0.516735\pi$
$293$	−1.79903	−0.105100	−0.0525501	+	0.998618i	$0.516735\pi$
$294$	−19.0951	−1.11365
$295$	−12.0231	−0.700014
$296$	−5.54731	−0.322431
$297$	13.2255	0.767423
$298$	−18.7823	−1.08803
$299$	−2.39750	−0.138651
$300$	−2.61006	−0.150692
$301$	14.7738	0.851546
$302$	0.322617	0.0185645
$303$	0.827349	0.0475299
$304$	−17.1022	−0.980878
$305$	−4.20940	−0.241030
$306$	−6.07150	−0.347085
$307$	26.7694	1.52781	0.763905	−	0.645328i	$-0.223279\pi$
$307$	26.7694	1.52781	0.763905	+	0.645328i	$0.223279\pi$
$308$	6.89287	0.392758
$309$	−6.79928	−0.386798
$310$	0	0
$311$	4.18114	0.237090	0.118545	−	0.992949i	$-0.462177\pi$
$311$	4.18114	0.237090	0.118545	+	0.992949i	$0.462177\pi$
$312$	16.7199	0.946578
$313$	11.3428	0.641135	0.320567	−	0.947226i	$-0.396126\pi$
$313$	11.3428	0.641135	0.320567	+	0.947226i	$0.396126\pi$
$314$	19.6242	1.10746
$315$	7.67538	0.432459
$316$	−4.69092	−0.263885
$317$	25.8813	1.45364	0.726820	−	0.686828i	$-0.240997\pi$
$317$	25.8813	1.45364	0.726820	+	0.686828i	$0.240997\pi$
$318$	−5.99329	−0.336087
$319$	25.6107	1.43392
$320$	−13.7357	−0.767851
$321$	−6.55038	−0.365606
$322$	4.26435	0.237643
$323$	−23.0171	−1.28071
$324$	5.40148	0.300082
$325$	−6.88028	−0.381649
$326$	−0.164095	−0.00908838
$327$	−14.4340	−0.798201
$328$	−1.03097	−0.0569258
$329$	4.52323	0.249374
$330$	14.8645	0.818266
$331$	−0.765788	−0.0420915	−0.0210457	−	0.999779i	$-0.506700\pi$
$331$	−0.765788	−0.0420915	−0.0210457	+	0.999779i	$0.506700\pi$
$332$	−4.04437	−0.221964
$333$	2.36830	0.129782
$334$	4.86558	0.266232
$335$	11.4703	0.626692
$336$	−22.1344	−1.20753
$337$	−2.39434	−0.130428	−0.0652140	−	0.997871i	$-0.520773\pi$
$337$	−2.39434	−0.130428	−0.0652140	+	0.997871i	$0.520773\pi$
$338$	−7.46258	−0.405911
$339$	31.6367	1.71827
$340$	−2.81040	−0.152415
$341$	0	0
$342$	9.80991	0.530459
$343$	−1.78235	−0.0962378
$344$	11.8770	0.640367
$345$	−2.91800	−0.157100
$346$	−17.8134	−0.957656
$347$	−5.65780	−0.303727	−0.151863	−	0.988402i	$-0.548527\pi$
$347$	−5.65780	−0.303727	−0.151863	+	0.988402i	$0.548527\pi$
$348$	6.80602	0.364841
$349$	29.0676	1.55595	0.777977	−	0.628293i	$-0.216246\pi$
$349$	29.0676	1.55595	0.777977	+	0.628293i	$0.216246\pi$
$350$	12.2377	0.654132
$351$	9.26489	0.494523
$352$	10.0071	0.533378
$353$	19.2752	1.02592	0.512958	−	0.858414i	$-0.328550\pi$
$353$	19.2752	1.02592	0.512958	+	0.858414i	$0.328550\pi$
$354$	−19.8883	−1.05705
$355$	7.87986	0.418220
$356$	2.45481	0.130105
$357$	−29.7897	−1.57664
$358$	21.8340	1.15396
$359$	10.3443	0.545950	0.272975	−	0.962021i	$-0.411992\pi$
$359$	10.3443	0.545950	0.272975	+	0.962021i	$0.411992\pi$
$360$	6.17045	0.325211
$361$	18.1894	0.957339
$362$	14.8590	0.780970
$363$	−6.53423	−0.342958
$364$	4.82867	0.253091
$365$	8.33360	0.436201
$366$	−6.96307	−0.363966
$367$	0.136564	0.00712857	0.00356428	−	0.999994i	$-0.498865\pi$
$367$	0.136564	0.00712857	0.00356428	+	0.999994i	$0.498865\pi$
$368$	2.55159	0.133011
$369$	0.440149	0.0229133
$370$	−3.45484	−0.179608
$371$	−8.91649	−0.462921
$372$	0	0
$373$	−7.36393	−0.381290	−0.190645	−	0.981659i	$-0.561058\pi$
$373$	−7.36393	−0.381290	−0.190645	+	0.981659i	$0.561058\pi$
$374$	−17.4935	−0.904566
$375$	−24.4096	−1.26051
$376$	3.63635	0.187530
$377$	17.9411	0.924012
$378$	−16.4791	−0.847593
$379$	−4.60813	−0.236704	−0.118352	−	0.992972i	$-0.537761\pi$
$379$	−4.60813	−0.236704	−0.118352	+	0.992972i	$0.537761\pi$
$380$	4.54084	0.232940
$381$	−41.9692	−2.15015
$382$	6.24899	0.319726
$383$	32.2948	1.65019	0.825093	−	0.564997i	$-0.191123\pi$
$383$	32.2948	1.65019	0.825093	+	0.564997i	$0.191123\pi$
$384$	−11.6809	−0.596087
$385$	22.1147	1.12707
$386$	−24.6949	−1.25694
$387$	−5.07063	−0.257755
$388$	−5.25794	−0.266932
$389$	35.2713	1.78832	0.894162	−	0.447743i	$-0.147772\pi$
$389$	35.2713	1.78832	0.894162	+	0.447743i	$0.147772\pi$
$390$	10.4131	0.527286
$391$	3.43407	0.173668
$392$	−22.8385	−1.15352
$393$	−26.2511	−1.32419
$394$	−26.7484	−1.34757
$395$	−15.0501	−0.757252
$396$	−2.36576	−0.118884
$397$	−4.03402	−0.202462	−0.101231	−	0.994863i	$-0.532278\pi$
$397$	−4.03402	−0.202462	−0.101231	+	0.994863i	$0.532278\pi$
$398$	−18.6865	−0.936671
$399$	48.1322	2.40962
$400$	7.32246	0.366123
$401$	−24.8832	−1.24261	−0.621304	−	0.783570i	$-0.713397\pi$
$401$	−24.8832	−1.24261	−0.621304	+	0.783570i	$0.713397\pi$
$402$	18.9739	0.946334
$403$	0	0
$404$	0.192087	0.00955671
$405$	17.3298	0.861124
$406$	−31.9111	−1.58372
$407$	6.82365	0.338236
$408$	−23.9488	−1.18564
$409$	−3.50722	−0.173421	−0.0867105	−	0.996234i	$-0.527636\pi$
$409$	−3.50722	−0.173421	−0.0867105	+	0.996234i	$0.527636\pi$
$410$	−0.642083	−0.0317102
$411$	−20.4966	−1.01102
$412$	−1.57860	−0.0777723
$413$	−29.5888	−1.45597
$414$	−1.46360	−0.0719322
$415$	−12.9757	−0.636953
$416$	7.01025	0.343706
$417$	−32.1600	−1.57488
$418$	28.2647	1.38247
$419$	4.12031	0.201290	0.100645	−	0.994922i	$-0.467909\pi$
$419$	4.12031	0.201290	0.100645	+	0.994922i	$0.467909\pi$
$420$	5.87695	0.286766
$421$	27.6014	1.34521	0.672605	−	0.740002i	$-0.265175\pi$
$421$	27.6014	1.34521	0.672605	+	0.740002i	$0.265175\pi$
$422$	7.76812	0.378146
$423$	−1.55246	−0.0754830
$424$	−7.16821	−0.348119
$425$	9.85499	0.478037
$426$	13.0347	0.631531
$427$	−10.3593	−0.501321
$428$	−1.52082	−0.0735114
$429$	−20.5668	−0.992977
$430$	7.39697	0.356713
$431$	−14.8739	−0.716452	−0.358226	−	0.933635i	$-0.616618\pi$
$431$	−14.8739	−0.716452	−0.358226	+	0.933635i	$0.616618\pi$
$432$	−9.86032	−0.474405
$433$	−9.26195	−0.445101	−0.222550	−	0.974921i	$-0.571438\pi$
$433$	−9.26195	−0.445101	−0.222550	+	0.974921i	$0.571438\pi$
$434$	0	0
$435$	21.8360	1.04696
$436$	−3.35117	−0.160492
$437$	−5.54853	−0.265422
$438$	13.7852	0.658683
$439$	17.7182	0.845643	0.422821	−	0.906213i	$-0.361040\pi$
$439$	17.7182	0.845643	0.422821	+	0.906213i	$0.361040\pi$
$440$	17.7786	0.847561
$441$	9.75038	0.464304
$442$	−12.2547	−0.582898
$443$	−18.4502	−0.876597	−0.438299	−	0.898829i	$-0.644419\pi$
$443$	−18.4502	−0.876597	−0.438299	+	0.898829i	$0.644419\pi$
$444$	1.81338	0.0860591
$445$	7.87587	0.373352
$446$	−9.10465	−0.431118
$447$	31.6294	1.49602
$448$	−33.8035	−1.59706
$449$	−22.7102	−1.07176	−0.535881	−	0.844294i	$-0.680020\pi$
$449$	−22.7102	−1.07176	−0.535881	+	0.844294i	$0.680020\pi$
$450$	−4.20020	−0.197999
$451$	1.26818	0.0597162
$452$	7.34516	0.345487
$453$	−0.543286	−0.0255258
$454$	23.7067	1.11261
$455$	15.4920	0.726276
$456$	38.6948	1.81205
$457$	16.1619	0.756020	0.378010	−	0.925802i	$-0.376608\pi$
$457$	16.1619	0.756020	0.378010	+	0.925802i	$0.376608\pi$
$458$	7.50036	0.350469
$459$	−13.2706	−0.619418
$460$	−0.677478	−0.0315876
$461$	−21.5388	−1.00316	−0.501580	−	0.865111i	$-0.667248\pi$
$461$	−21.5388	−1.00316	−0.501580	+	0.865111i	$0.667248\pi$
$462$	36.5815	1.70192
$463$	17.5098	0.813750	0.406875	−	0.913484i	$-0.366619\pi$
$463$	17.5098	0.813750	0.406875	+	0.913484i	$0.366619\pi$
$464$	−19.0941	−0.886421
$465$	0	0
$466$	21.5976	1.00049
$467$	−11.8066	−0.546346	−0.273173	−	0.961965i	$-0.588073\pi$
$467$	−11.8066	−0.546346	−0.273173	+	0.961965i	$0.588073\pi$
$468$	−1.65729	−0.0766081
$469$	28.2284	1.30347
$470$	2.26470	0.104463
$471$	−33.0472	−1.52273
$472$	−23.7872	−1.09490
$473$	−14.6097	−0.671757
$474$	−24.8954	−1.14348
$475$	−15.9230	−0.730598
$476$	−6.91635	−0.317010
$477$	3.06030	0.140122
$478$	34.3924	1.57307
$479$	−15.8087	−0.722319	−0.361159	−	0.932504i	$-0.617619\pi$
$479$	−15.8087	−0.722319	−0.361159	+	0.932504i	$0.617619\pi$
$480$	8.53215	0.389438
$481$	4.78017	0.217957
$482$	−28.0316	−1.27680
$483$	−7.18115	−0.326754
$484$	−1.51707	−0.0689576
$485$	−16.8693	−0.765994
$486$	15.6695	0.710783
$487$	−12.7916	−0.579645	−0.289822	−	0.957080i	$-0.593596\pi$
$487$	−12.7916	−0.579645	−0.289822	+	0.957080i	$0.593596\pi$
$488$	−8.32811	−0.376996
$489$	0.276335	0.0124963
$490$	−14.2237	−0.642561
$491$	18.5282	0.836166	0.418083	−	0.908409i	$-0.362702\pi$
$491$	18.5282	0.836166	0.418083	+	0.908409i	$0.362702\pi$
$492$	0.337017	0.0151939
$493$	−25.6979	−1.15738
$494$	19.8003	0.890859
$495$	−7.59016	−0.341152
$496$	0	0
$497$	19.3922	0.869861
$498$	−21.4641	−0.961829
$499$	−40.8873	−1.83036	−0.915182	−	0.403040i	$-0.867954\pi$
$499$	−40.8873	−1.83036	−0.915182	+	0.403040i	$0.867954\pi$
$500$	−5.66723	−0.253446
$501$	−8.19362	−0.366064
$502$	20.0037	0.892809
$503$	20.0771	0.895194	0.447597	−	0.894235i	$-0.352280\pi$
$503$	20.0771	0.895194	0.447597	+	0.894235i	$0.352280\pi$
$504$	15.1854	0.676411
$505$	0.616282	0.0274242
$506$	−4.21700	−0.187469
$507$	12.5670	0.558118
$508$	−9.74408	−0.432324
$509$	31.9112	1.41444	0.707220	−	0.706994i	$-0.249949\pi$
$509$	31.9112	1.41444	0.707220	+	0.706994i	$0.249949\pi$
$510$	−14.9152	−0.660456
$511$	20.5089	0.907260
$512$	−24.6123	−1.08772
$513$	21.4417	0.946674
$514$	28.7738	1.26916
$515$	−5.06470	−0.223177
$516$	−3.88253	−0.170919
$517$	−4.47301	−0.196723
$518$	−8.50231	−0.373570
$519$	29.9978	1.31676
$520$	12.4544	0.546163
$521$	2.10756	0.0923339	0.0461670	−	0.998934i	$-0.485299\pi$
$521$	2.10756	0.0923339	0.0461670	+	0.998934i	$0.485299\pi$
$522$	10.9525	0.479377
$523$	−4.90777	−0.214602	−0.107301	−	0.994227i	$-0.534221\pi$
$523$	−4.90777	−0.214602	−0.107301	+	0.994227i	$0.534221\pi$
$524$	−6.09477	−0.266251
$525$	−20.6082	−0.899418
$526$	12.8076	0.558440
$527$	0	0
$528$	21.8886	0.952580
$529$	−22.1722	−0.964008
$530$	−4.46432	−0.193918
$531$	10.1554	0.440708
$532$	11.1750	0.484496
$533$	0.888398	0.0384808
$534$	13.0280	0.563779
$535$	−4.87929	−0.210950
$536$	22.6936	0.980213
$537$	−36.7683	−1.58667
$538$	5.44379	0.234699
$539$	28.0932	1.21006
$540$	2.61804	0.112662
$541$	−12.4370	−0.534709	−0.267355	−	0.963598i	$-0.586150\pi$
$541$	−12.4370	−0.534709	−0.267355	+	0.963598i	$0.586150\pi$
$542$	−7.58139	−0.325648
$543$	−25.0225	−1.07382
$544$	−10.0411	−0.430511
$545$	−10.7517	−0.460552
$546$	25.6264	1.09671
$547$	−13.9650	−0.597101	−0.298550	−	0.954394i	$-0.596503\pi$
$547$	−13.9650	−0.597101	−0.298550	+	0.954394i	$0.596503\pi$
$548$	−4.75875	−0.203284
$549$	3.55550	0.151745
$550$	−12.1018	−0.516023
$551$	41.5209	1.76885
$552$	−5.77312	−0.245721
$553$	−37.0381	−1.57502
$554$	−27.1662	−1.15418
$555$	5.81793	0.246958
$556$	−7.46666	−0.316657
$557$	37.2207	1.57709	0.788546	−	0.614976i	$-0.210834\pi$
$557$	37.2207	1.57709	0.788546	+	0.614976i	$0.210834\pi$
$558$	0	0
$559$	−10.2346	−0.432876
$560$	−16.4876	−0.696730
$561$	29.4590	1.24376
$562$	−37.9183	−1.59949
$563$	−30.0267	−1.26548	−0.632738	−	0.774366i	$-0.718069\pi$
$563$	−30.0267	−1.26548	−0.632738	+	0.774366i	$0.718069\pi$
$564$	−1.18870	−0.0500532
$565$	23.5658	0.991420
$566$	−26.3010	−1.10551
$567$	42.6484	1.79106
$568$	15.5900	0.654140
$569$	40.3713	1.69245	0.846227	−	0.532823i	$-0.178869\pi$
$569$	40.3713	1.69245	0.846227	+	0.532823i	$0.178869\pi$
$570$	24.0989	1.00939
$571$	−42.6015	−1.78282	−0.891410	−	0.453199i	$-0.850283\pi$
$571$	−42.6015	−1.78282	−0.891410	+	0.453199i	$0.850283\pi$
$572$	−4.77505	−0.199655
$573$	−10.5233	−0.439616
$574$	−1.58016	−0.0659545
$575$	2.37566	0.0990717
$576$	11.6020	0.483416
$577$	−6.53777	−0.272171	−0.136085	−	0.990697i	$-0.543452\pi$
$577$	−6.53777	−0.272171	−0.136085	+	0.990697i	$0.543452\pi$
$578$	−3.39385	−0.141165
$579$	41.5861	1.72826
$580$	5.06972	0.210509
$581$	−31.9331	−1.32481
$582$	−27.9047	−1.15669
$583$	8.81749	0.365183
$584$	16.4877	0.682264
$585$	−5.31714	−0.219837
$586$	−2.21671	−0.0915715
$587$	21.3777	0.882352	0.441176	−	0.897421i	$-0.354561\pi$
$587$	21.3777	0.882352	0.441176	+	0.897421i	$0.354561\pi$
$588$	7.46576	0.307883
$589$	0	0
$590$	−14.8146	−0.609907
$591$	45.0443	1.85287
$592$	−5.08738	−0.209090
$593$	25.9906	1.06731	0.533653	−	0.845704i	$-0.320819\pi$
$593$	25.9906	1.06731	0.533653	+	0.845704i	$0.320819\pi$
$594$	16.2961	0.668638
$595$	−22.1900	−0.909701
$596$	7.34346	0.300800
$597$	31.4680	1.28790
$598$	−2.95414	−0.120804
$599$	−9.18796	−0.375410	−0.187705	−	0.982225i	$-0.560105\pi$
$599$	−9.18796	−0.375410	−0.187705	+	0.982225i	$0.560105\pi$
$600$	−16.5675	−0.676367
$601$	−33.3190	−1.35911	−0.679556	−	0.733624i	$-0.737828\pi$
$601$	−33.3190	−1.35911	−0.679556	+	0.733624i	$0.737828\pi$
$602$	18.2038	0.741933
$603$	−9.68850	−0.394546
$604$	−0.126136	−0.00513240
$605$	−4.86727	−0.197883
$606$	1.01944	0.0414118
$607$	−48.7008	−1.97670	−0.988352	−	0.152188i	$-0.951368\pi$
$607$	−48.7008	−1.97670	−0.988352	+	0.152188i	$0.951368\pi$
$608$	16.2238	0.657961
$609$	53.7382	2.17758
$610$	−5.18671	−0.210004
$611$	−3.13348	−0.126767
$612$	2.37382	0.0959560
$613$	−20.9140	−0.844709	−0.422354	−	0.906431i	$-0.638796\pi$
$613$	−20.9140	−0.844709	−0.422354	+	0.906431i	$0.638796\pi$
$614$	32.9845	1.33115
$615$	1.08127	0.0436009
$616$	43.7529	1.76285
$617$	45.3840	1.82709	0.913545	−	0.406738i	$-0.133334\pi$
$617$	45.3840	1.82709	0.913545	+	0.406738i	$0.133334\pi$
$618$	−8.37789	−0.337008
$619$	10.2462	0.411832	0.205916	−	0.978570i	$-0.433983\pi$
$619$	10.2462	0.411832	0.205916	+	0.978570i	$0.433983\pi$
$620$	0	0
$621$	−3.19902	−0.128372
$622$	5.15188	0.206572
$623$	19.3824	0.776540
$624$	15.3337	0.613838
$625$	−5.12716	−0.205086
$626$	13.9763	0.558606
$627$	−47.5978	−1.90087
$628$	−7.67263	−0.306171
$629$	−6.84689	−0.273003
$630$	9.45739	0.376792
$631$	10.6770	0.425043	0.212522	−	0.977156i	$-0.431832\pi$
$631$	10.6770	0.425043	0.212522	+	0.977156i	$0.431832\pi$
$632$	−29.7759	−1.18442
$633$	−13.0815	−0.519942
$634$	31.8903	1.26652
$635$	−31.2623	−1.24061
$636$	2.34324	0.0929155
$637$	19.6802	0.779757
$638$	31.5568	1.24934
$639$	−6.65578	−0.263298
$640$	−8.70093	−0.343934
$641$	−23.8621	−0.942496	−0.471248	−	0.882001i	$-0.656196\pi$
$641$	−23.8621	−0.942496	−0.471248	+	0.882001i	$0.656196\pi$
$642$	−8.07119	−0.318545
$643$	7.93926	0.313094	0.156547	−	0.987671i	$-0.449964\pi$
$643$	7.93926	0.313094	0.156547	+	0.987671i	$0.449964\pi$
$644$	−1.66726	−0.0656994
$645$	−12.4565	−0.490473
$646$	−28.3610	−1.11585
$647$	−2.47761	−0.0974050	−0.0487025	−	0.998813i	$-0.515509\pi$
$647$	−2.47761	−0.0974050	−0.0487025	+	0.998813i	$0.515509\pi$
$648$	34.2862	1.34689
$649$	29.2603	1.14857
$650$	−8.47769	−0.332523
$651$	0	0
$652$	0.0641574	0.00251260
$653$	22.4940	0.880260	0.440130	−	0.897934i	$-0.354932\pi$
$653$	22.4940	0.880260	0.440130	+	0.897934i	$0.354932\pi$
$654$	−17.7851	−0.695454
$655$	−19.5541	−0.764042
$656$	−0.945493	−0.0369153
$657$	−7.03903	−0.274619
$658$	5.57340	0.217274
$659$	−11.9596	−0.465881	−0.232941	−	0.972491i	$-0.574835\pi$
$659$	−11.9596	−0.465881	−0.232941	+	0.972491i	$0.574835\pi$
$660$	−5.81170	−0.226220
$661$	23.8360	0.927112	0.463556	−	0.886068i	$-0.346573\pi$
$661$	23.8360	0.927112	0.463556	+	0.886068i	$0.346573\pi$
$662$	−0.943582	−0.0366734
$663$	20.6369	0.801471
$664$	−25.6719	−0.996263
$665$	35.8531	1.39032
$666$	2.91815	0.113076
$667$	−6.19478	−0.239863
$668$	−1.90233	−0.0736034
$669$	15.3322	0.592777
$670$	14.1334	0.546022
$671$	10.2443	0.395475
$672$	20.9975	0.809997
$673$	−3.49412	−0.134688	−0.0673442	−	0.997730i	$-0.521453\pi$
$673$	−3.49412	−0.134688	−0.0673442	+	0.997730i	$0.521453\pi$
$674$	−2.95024	−0.113639
$675$	−9.18046	−0.353356
$676$	2.91770	0.112219
$677$	19.2457	0.739674	0.369837	−	0.929097i	$-0.379414\pi$
$677$	19.2457	0.739674	0.369837	+	0.929097i	$0.379414\pi$
$678$	38.9819	1.49709
$679$	−41.5151	−1.59320
$680$	−17.8391	−0.684100
$681$	−39.9221	−1.52982
$682$	0	0
$683$	−39.8738	−1.52573	−0.762865	−	0.646558i	$-0.776208\pi$
$683$	−39.8738	−1.52573	−0.762865	+	0.646558i	$0.776208\pi$
$684$	−3.83545	−0.146652
$685$	−15.2677	−0.583348
$686$	−2.19616	−0.0838498
$687$	−12.6306	−0.481887
$688$	10.8923	0.415266
$689$	6.17692	0.235322
$690$	−3.59547	−0.136877
$691$	38.4104	1.46120	0.730601	−	0.682804i	$-0.239240\pi$
$691$	38.4104	1.46120	0.730601	+	0.682804i	$0.239240\pi$
$692$	6.96465	0.264756
$693$	−18.6793	−0.709568
$694$	−6.97139	−0.264630
$695$	−23.9556	−0.908687
$696$	43.2016	1.63755
$697$	−1.27250	−0.0481993
$698$	35.8163	1.35567
$699$	−36.3703	−1.37565
$700$	−4.78466	−0.180843
$701$	−35.4255	−1.33800	−0.669001	−	0.743262i	$-0.733278\pi$
$701$	−35.4255	−1.33800	−0.669001	+	0.743262i	$0.733278\pi$
$702$	11.4159	0.430867
$703$	11.0627	0.417239
$704$	33.4281	1.25987
$705$	−3.81375	−0.143634
$706$	23.7504	0.893857
$707$	1.51666	0.0570399
$708$	7.77589	0.292236
$709$	25.8093	0.969288	0.484644	−	0.874711i	$-0.338949\pi$
$709$	25.8093	0.969288	0.484644	+	0.874711i	$0.338949\pi$
$710$	9.70935	0.364385
$711$	12.7121	0.476743
$712$	15.5821	0.583962
$713$	0	0
$714$	−36.7061	−1.37369
$715$	−15.3200	−0.572935
$716$	−8.53658	−0.319027
$717$	−57.9168	−2.16294
$718$	12.7459	0.475674
$719$	39.0467	1.45620	0.728099	−	0.685472i	$-0.240404\pi$
$719$	39.0467	1.45620	0.728099	+	0.685472i	$0.240404\pi$
$720$	5.65886	0.210893
$721$	−12.4642	−0.464190
$722$	22.4125	0.834108
$723$	47.2051	1.75558
$724$	−5.80952	−0.215909
$725$	−17.7776	−0.660243
$726$	−8.05130	−0.298812
$727$	24.9836	0.926592	0.463296	−	0.886204i	$-0.346667\pi$
$727$	24.9836	0.926592	0.463296	+	0.886204i	$0.346667\pi$
$728$	30.6502	1.13597
$729$	7.24913	0.268486
$730$	10.2684	0.380052
$731$	14.6595	0.542202
$732$	2.72240	0.100623
$733$	−16.9490	−0.626027	−0.313014	−	0.949749i	$-0.601339\pi$
$733$	−16.9490	−0.626027	−0.313014	+	0.949749i	$0.601339\pi$
$734$	0.168270	0.00621096
$735$	23.9527	0.883508
$736$	−2.42053	−0.0892220
$737$	−27.9150	−1.02826
$738$	0.542340	0.0199638
$739$	−14.3821	−0.529055	−0.264528	−	0.964378i	$-0.585216\pi$
$739$	−14.3821	−0.529055	−0.264528	+	0.964378i	$0.585216\pi$
$740$	1.35076	0.0496550
$741$	−33.3437	−1.22491
$742$	−10.9866	−0.403333
$743$	27.7705	1.01880	0.509400	−	0.860530i	$-0.329867\pi$
$743$	27.7705	1.01880	0.509400	+	0.860530i	$0.329867\pi$
$744$	0	0
$745$	23.5603	0.863183
$746$	−9.07363	−0.332209
$747$	10.9600	0.401007
$748$	6.83956	0.250079
$749$	−12.0079	−0.438758
$750$	−30.0768	−1.09825
$751$	−43.5702	−1.58990	−0.794950	−	0.606676i	$-0.792503\pi$
$751$	−43.5702	−1.58990	−0.794950	+	0.606676i	$0.792503\pi$
$752$	3.33486	0.121610
$753$	−33.6862	−1.22759
$754$	22.1065	0.805071
$755$	−0.404687	−0.0147281
$756$	6.44296	0.234328
$757$	−25.2171	−0.916531	−0.458266	−	0.888815i	$-0.651529\pi$
$757$	−25.2171	−0.916531	−0.458266	+	0.888815i	$0.651529\pi$
$758$	−5.67802	−0.206235
$759$	7.10142	0.257765
$760$	28.8233	1.04553
$761$	2.61593	0.0948274	0.0474137	−	0.998875i	$-0.484902\pi$
$761$	2.61593	0.0948274	0.0474137	+	0.998875i	$0.484902\pi$
$762$	−51.7133	−1.87337
$763$	−26.4598	−0.957908
$764$	−2.44321	−0.0883923
$765$	7.61602	0.275358
$766$	39.7927	1.43777
$767$	20.4977	0.740130
$768$	22.4872	0.811437
$769$	11.3507	0.409316	0.204658	−	0.978834i	$-0.434392\pi$
$769$	11.3507	0.409316	0.204658	+	0.978834i	$0.434392\pi$
$770$	27.2491	0.981988
$771$	−48.4550	−1.74507
$772$	9.65514	0.347496
$773$	11.4174	0.410655	0.205327	−	0.978693i	$-0.434174\pi$
$773$	11.4174	0.410655	0.205327	+	0.978693i	$0.434174\pi$
$774$	−6.24789	−0.224576
$775$	0	0
$776$	−33.3751	−1.19810
$777$	14.3179	0.513651
$778$	43.4603	1.55813
$779$	2.05601	0.0736644
$780$	−4.07128	−0.145775
$781$	−19.1769	−0.686204
$782$	4.23137	0.151313
$783$	23.9390	0.855511
$784$	−20.9450	−0.748035
$785$	−24.6164	−0.878597
$786$	−32.3459	−1.15374
$787$	27.0892	0.965626	0.482813	−	0.875723i	$-0.339615\pi$
$787$	27.0892	0.965626	0.482813	+	0.875723i	$0.339615\pi$
$788$	10.4580	0.372552
$789$	−21.5680	−0.767842
$790$	−18.5443	−0.659776
$791$	57.9951	2.06207
$792$	−15.0168	−0.533599
$793$	7.17642	0.254842
$794$	−4.97060	−0.176400
$795$	7.51791	0.266633
$796$	7.30601	0.258955
$797$	−48.6558	−1.72348	−0.861739	−	0.507352i	$-0.830624\pi$
$797$	−48.6558	−1.72348	−0.861739	+	0.507352i	$0.830624\pi$
$798$	59.3071	2.09945
$799$	4.48825	0.158783
$800$	−6.94637	−0.245591
$801$	−6.65241	−0.235051
$802$	−30.6604	−1.08266
$803$	−20.2812	−0.715707
$804$	−7.41838	−0.261626
$805$	−5.34915	−0.188533
$806$	0	0
$807$	−9.16733	−0.322705
$808$	1.21929	0.0428943
$809$	18.8091	0.661294	0.330647	−	0.943755i	$-0.392733\pi$
$809$	18.8091	0.661294	0.330647	+	0.943755i	$0.392733\pi$
$810$	21.3533	0.750278
$811$	−39.5385	−1.38839	−0.694193	−	0.719789i	$-0.744239\pi$
$811$	−39.5385	−1.38839	−0.694193	+	0.719789i	$0.744239\pi$
$812$	12.4765	0.437840
$813$	12.7670	0.447760
$814$	8.40791	0.294697
$815$	0.205839	0.00721022
$816$	−21.9632	−0.768866
$817$	−23.6858	−0.828662
$818$	−4.32150	−0.151098
$819$	−13.0854	−0.457241
$820$	0.251040	0.00876670
$821$	14.6367	0.510825	0.255412	−	0.966832i	$-0.417789\pi$
$821$	14.6367	0.510825	0.255412	+	0.966832i	$0.417789\pi$
$822$	−25.2554	−0.880882
$823$	−41.7867	−1.45659	−0.728296	−	0.685262i	$-0.759688\pi$
$823$	−41.7867	−1.45659	−0.728296	+	0.685262i	$0.759688\pi$
$824$	−10.0203	−0.349073
$825$	20.3794	0.709521
$826$	−36.4585	−1.26855
$827$	9.40613	0.327083	0.163542	−	0.986536i	$-0.447708\pi$
$827$	9.40613	0.327083	0.163542	+	0.986536i	$0.447708\pi$
$828$	0.572236	0.0198866
$829$	−40.3307	−1.40074	−0.700372	−	0.713778i	$-0.746983\pi$
$829$	−40.3307	−1.40074	−0.700372	+	0.713778i	$0.746983\pi$
$830$	−15.9883	−0.554963
$831$	45.7478	1.58697
$832$	23.4174	0.811854
$833$	−28.1889	−0.976689
$834$	−39.6266	−1.37216
$835$	−6.10332	−0.211214
$836$	−11.0509	−0.382203
$837$	0	0
$838$	5.07693	0.175380
$839$	7.14965	0.246833	0.123417	−	0.992355i	$-0.460615\pi$
$839$	7.14965	0.246833	0.123417	+	0.992355i	$0.460615\pi$
$840$	37.3043	1.28712
$841$	17.3569	0.598514
$842$	34.0097	1.17205
$843$	63.8544	2.19926
$844$	−3.03716	−0.104543
$845$	9.36097	0.322027
$846$	−1.91289	−0.0657666
$847$	−11.9783	−0.411579
$848$	−6.57389	−0.225748
$849$	44.2909	1.52006
$850$	12.1430	0.416503
$851$	−1.65052	−0.0565791
$852$	−5.09625	−0.174595
$853$	1.12667	0.0385765	0.0192882	−	0.999814i	$-0.493860\pi$
$853$	1.12667	0.0385765	0.0192882	+	0.999814i	$0.493860\pi$
$854$	−12.7644	−0.436789
$855$	−12.3054	−0.420837
$856$	−9.65346	−0.329949
$857$	−23.8540	−0.814838	−0.407419	−	0.913241i	$-0.633571\pi$
$857$	−23.8540	−0.814838	−0.407419	+	0.913241i	$0.633571\pi$
$858$	−25.3419	−0.865158
$859$	−32.4657	−1.10771	−0.553857	−	0.832612i	$-0.686845\pi$
$859$	−32.4657	−1.10771	−0.553857	+	0.832612i	$0.686845\pi$
$860$	−2.89205	−0.0986180
$861$	2.66098	0.0906860
$862$	−18.3273	−0.624229
$863$	23.8418	0.811585	0.405793	−	0.913965i	$-0.366996\pi$
$863$	23.8418	0.811585	0.405793	+	0.913965i	$0.366996\pi$
$864$	9.35387	0.318225
$865$	22.3450	0.759752
$866$	−11.4123	−0.387806
$867$	5.71523	0.194099
$868$	0	0
$869$	36.6268	1.24248
$870$	26.9057	0.912190
$871$	−19.5553	−0.662605
$872$	−21.2717	−0.720352
$873$	14.2487	0.482247
$874$	−6.83675	−0.231256
$875$	−44.7467	−1.51271
$876$	−5.38971	−0.182101
$877$	29.2746	0.988534	0.494267	−	0.869310i	$-0.335436\pi$
$877$	29.2746	0.988534	0.494267	+	0.869310i	$0.335436\pi$
$878$	21.8319	0.736789
$879$	3.73294	0.125909
$880$	16.3046	0.549627
$881$	−9.03008	−0.304231	−0.152116	−	0.988363i	$-0.548609\pi$
$881$	−9.03008	−0.304231	−0.152116	+	0.988363i	$0.548609\pi$
$882$	12.0141	0.404537
$883$	−21.4076	−0.720422	−0.360211	−	0.932871i	$-0.617295\pi$
$883$	−21.4076	−0.720422	−0.360211	+	0.932871i	$0.617295\pi$
$884$	4.79132	0.161150
$885$	24.9477	0.838608
$886$	−22.7339	−0.763759
$887$	24.5950	0.825820	0.412910	−	0.910772i	$-0.364512\pi$
$887$	24.5950	0.825820	0.412910	+	0.910772i	$0.364512\pi$
$888$	11.5105	0.386268
$889$	−76.9362	−2.58036
$890$	9.70443	0.325293
$891$	−42.1749	−1.41291
$892$	3.55971	0.119188
$893$	−7.25179	−0.242672
$894$	38.9728	1.30345
$895$	−27.3883	−0.915488
$896$	−21.4129	−0.715354
$897$	4.97476	0.166102
$898$	−27.9829	−0.933801
$899$	0	0
$900$	1.64218	0.0547395
$901$	−8.84753	−0.294754
$902$	1.56261	0.0520293
$903$	−30.6552	−1.02014
$904$	46.6238	1.55069
$905$	−18.6389	−0.619578
$906$	−0.669422	−0.0222401
$907$	−19.7663	−0.656329	−0.328165	−	0.944621i	$-0.606430\pi$
$907$	−19.7663	−0.656329	−0.328165	+	0.944621i	$0.606430\pi$
$908$	−9.26880	−0.307596
$909$	−0.520546	−0.0172654
$910$	19.0888	0.632788
$911$	35.6216	1.18020	0.590098	−	0.807332i	$-0.299089\pi$
$911$	35.6216	1.18020	0.590098	+	0.807332i	$0.299089\pi$
$912$	35.4866	1.17508
$913$	31.5785	1.04510
$914$	19.9142	0.658703
$915$	8.73440	0.288750
$916$	−2.93247	−0.0968916
$917$	−48.1224	−1.58914
$918$	−16.3516	−0.539685
$919$	−17.1563	−0.565935	−0.282968	−	0.959129i	$-0.591319\pi$
$919$	−17.1563	−0.565935	−0.282968	+	0.959129i	$0.591319\pi$
$920$	−4.30033	−0.141778
$921$	−55.5458	−1.83030
$922$	−26.5395	−0.874030
$923$	−13.4340	−0.442186
$924$	−14.3025	−0.470519
$925$	−4.73661	−0.155739
$926$	21.5751	0.709002
$927$	4.27793	0.140506
$928$	18.1134	0.594601
$929$	−45.5222	−1.49353	−0.746767	−	0.665086i	$-0.768395\pi$
$929$	−45.5222	−1.49353	−0.746767	+	0.665086i	$0.768395\pi$
$930$	0	0
$931$	45.5457	1.49270
$932$	−8.44416	−0.276598
$933$	−8.67575	−0.284031
$934$	−14.5478	−0.476018
$935$	21.9436	0.717633
$936$	−10.5197	−0.343848
$937$	25.6175	0.836887	0.418443	−	0.908243i	$-0.362576\pi$
$937$	25.6175	0.836887	0.418443	+	0.908243i	$0.362576\pi$
$938$	34.7822	1.13568
$939$	−23.5361	−0.768071
$940$	−0.885446	−0.0288801
$941$	−12.1238	−0.395226	−0.197613	−	0.980280i	$-0.563319\pi$
$941$	−12.1238	−0.395226	−0.197613	+	0.980280i	$0.563319\pi$
$942$	−40.7198	−1.32672
$943$	−0.306750	−0.00998916
$944$	−21.8151	−0.710019
$945$	20.6712	0.672434
$946$	−18.0017	−0.585286
$947$	27.2011	0.883916	0.441958	−	0.897036i	$-0.354284\pi$
$947$	27.2011	0.883916	0.441958	+	0.897036i	$0.354284\pi$
$948$	9.73355	0.316131
$949$	−14.2076	−0.461198
$950$	−19.6199	−0.636553
$951$	−53.7031	−1.74144
$952$	−43.9019	−1.42287
$953$	−24.8927	−0.806353	−0.403177	−	0.915122i	$-0.632094\pi$
$953$	−24.8927	−0.806353	−0.403177	+	0.915122i	$0.632094\pi$
$954$	3.77082	0.122085
$955$	−7.83866	−0.253653
$956$	−13.4467	−0.434896
$957$	−53.1415	−1.71782
$958$	−19.4791	−0.629340
$959$	−37.5736	−1.21331
$960$	28.5013	0.919876
$961$	0	0
$962$	5.89000	0.189901
$963$	4.12133	0.132808
$964$	10.9597	0.352989
$965$	30.9770	0.997184
$966$	−8.84842	−0.284693
$967$	−52.1998	−1.67863	−0.839316	−	0.543643i	$-0.817044\pi$
$967$	−52.1998	−1.67863	−0.839316	+	0.543643i	$0.817044\pi$
$968$	−9.62967	−0.309509
$969$	47.7599	1.53427
$970$	−20.7858	−0.667393
$971$	−19.9608	−0.640574	−0.320287	−	0.947321i	$-0.603779\pi$
$971$	−19.9608	−0.640574	−0.320287	+	0.947321i	$0.603779\pi$
$972$	−6.12642	−0.196505
$973$	−58.9544	−1.88999
$974$	−15.7615	−0.505031
$975$	14.2764	0.457211
$976$	−7.63763	−0.244475
$977$	−60.2506	−1.92759	−0.963794	−	0.266648i	$-0.914084\pi$
$977$	−60.2506	−1.92759	−0.963794	+	0.266648i	$0.914084\pi$
$978$	0.340493	0.0108878
$979$	−19.1672	−0.612587
$980$	5.56115	0.177644
$981$	9.08148	0.289949
$982$	22.8299	0.728532
$983$	11.5387	0.368028	0.184014	−	0.982924i	$-0.441091\pi$
$983$	11.5387	0.368028	0.184014	+	0.982924i	$0.441091\pi$
$984$	2.13924	0.0681964
$985$	33.5529	1.06909
$986$	−31.6643	−1.00840
$987$	−9.38559	−0.298746
$988$	−7.74148	−0.246289
$989$	3.53384	0.112370
$990$	−9.35239	−0.297238
$991$	−37.3423	−1.18622	−0.593109	−	0.805122i	$-0.702100\pi$
$991$	−37.3423	−1.18622	−0.593109	+	0.805122i	$0.702100\pi$
$992$	0	0
$993$	1.58899	0.0504251
$994$	23.8946	0.757890
$995$	23.4402	0.743103
$996$	8.39197	0.265910
$997$	20.8024	0.658820	0.329410	−	0.944187i	$-0.393150\pi$
$997$	20.8024	0.658820	0.329410	+	0.944187i	$0.393150\pi$
$998$	−50.3801	−1.59476
$999$	6.37825	0.201799

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	961.2.a.j.1.5		8
3.2	odd	2		8649.2.a.be.1.4		8
31.2	even	5		961.2.d.n.531.3		16
31.3	odd	30		31.2.g.a.9.2	yes	16
31.4	even	5		961.2.d.q.388.2		16
31.5	even	3		961.2.c.i.521.5		16
31.6	odd	6		961.2.c.j.439.5		16
31.7	even	15		961.2.g.n.235.1		16
31.8	even	5		961.2.d.q.374.2		16
31.9	even	15		961.2.g.n.732.1		16
31.10	even	15		961.2.g.l.844.2		16
31.11	odd	30		961.2.g.s.338.1		16
31.12	odd	30		961.2.g.k.547.2		16
31.13	odd	30		961.2.g.k.448.2		16
31.14	even	15		961.2.g.m.816.1		16
31.15	odd	10		961.2.d.o.628.3		16
31.16	even	5		961.2.d.n.628.3		16
31.17	odd	30		961.2.g.s.816.1		16
31.18	even	15		961.2.g.j.448.2		16
31.19	even	15		961.2.g.j.547.2		16
31.20	even	15		961.2.g.m.338.1		16
31.21	odd	30		31.2.g.a.7.2	✓	16
31.22	odd	30		961.2.g.t.732.1		16
31.23	odd	10		961.2.d.p.374.2		16
31.24	odd	30		961.2.g.t.235.1		16
31.25	even	3		961.2.c.i.439.5		16
31.26	odd	6		961.2.c.j.521.5		16
31.27	odd	10		961.2.d.p.388.2		16
31.28	even	15		961.2.g.l.846.2		16
31.29	odd	10		961.2.d.o.531.3		16
31.30	odd	2		961.2.a.i.1.5		8
93.65	even	30		279.2.y.c.226.1		16
93.83	even	30		279.2.y.c.100.1		16
93.92	even	2		8649.2.a.bf.1.4		8
124.3	even	30		496.2.bg.c.257.2		16
124.83	even	30		496.2.bg.c.193.2		16
155.3	even	60		775.2.ck.a.474.2		32
155.34	odd	30		775.2.bl.a.226.1		16
155.52	even	60		775.2.ck.a.224.2		32
155.83	even	60		775.2.ck.a.224.3		32
155.114	odd	30		775.2.bl.a.751.1		16
155.127	even	60		775.2.ck.a.474.3		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.2.g.a.7.2	✓	16	31.21	odd	30
31.2.g.a.9.2	yes	16	31.3	odd	30
279.2.y.c.100.1		16	93.83	even	30
279.2.y.c.226.1		16	93.65	even	30
496.2.bg.c.193.2		16	124.83	even	30
496.2.bg.c.257.2		16	124.3	even	30
775.2.bl.a.226.1		16	155.34	odd	30
775.2.bl.a.751.1		16	155.114	odd	30
775.2.ck.a.224.2		32	155.52	even	60
775.2.ck.a.224.3		32	155.83	even	60
775.2.ck.a.474.2		32	155.3	even	60
775.2.ck.a.474.3		32	155.127	even	60
961.2.a.i.1.5		8	31.30	odd	2
961.2.a.j.1.5		8	1.1	even	1	trivial
961.2.c.i.439.5		16	31.25	even	3
961.2.c.i.521.5		16	31.5	even	3
961.2.c.j.439.5		16	31.6	odd	6
961.2.c.j.521.5		16	31.26	odd	6
961.2.d.n.531.3		16	31.2	even	5
961.2.d.n.628.3		16	31.16	even	5
961.2.d.o.531.3		16	31.29	odd	10
961.2.d.o.628.3		16	31.15	odd	10
961.2.d.p.374.2		16	31.23	odd	10
961.2.d.p.388.2		16	31.27	odd	10
961.2.d.q.374.2		16	31.8	even	5
961.2.d.q.388.2		16	31.4	even	5
961.2.g.j.448.2		16	31.18	even	15
961.2.g.j.547.2		16	31.19	even	15
961.2.g.k.448.2		16	31.13	odd	30
961.2.g.k.547.2		16	31.12	odd	30
961.2.g.l.844.2		16	31.10	even	15
961.2.g.l.846.2		16	31.28	even	15
961.2.g.m.338.1		16	31.20	even	15
961.2.g.m.816.1		16	31.14	even	15
961.2.g.n.235.1		16	31.7	even	15
961.2.g.n.732.1		16	31.9	even	15
961.2.g.s.338.1		16	31.11	odd	30
961.2.g.s.816.1		16	31.17	odd	30
961.2.g.t.235.1		16	31.24	odd	30
961.2.g.t.732.1		16	31.22	odd	30
8649.2.a.be.1.4		8	3.2	odd	2
8649.2.a.bf.1.4		8	93.92	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:	\( N \)	\(=\)	\( 961 = 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	961.a (trivial)

\(f(q)\)	\(=\)	\(q+1.23217 q^{2} -2.07497 q^{3} -0.481752 q^{4} -1.54562 q^{5} -2.55673 q^{6} -3.80376 q^{7} -3.05795 q^{8} +1.30552 q^{9} -1.90447 q^{10} +3.76152 q^{11} +0.999622 q^{12} +2.63506 q^{13} -4.68689 q^{14} +3.20713 q^{15} -2.80441 q^{16} -3.77434 q^{17} +1.60863 q^{18} +6.09831 q^{19} +0.744606 q^{20} +7.89270 q^{21} +4.63485 q^{22} -0.909847 q^{23} +6.34516 q^{24} -2.61105 q^{25} +3.24685 q^{26} +3.51600 q^{27} +1.83247 q^{28} +6.80859 q^{29} +3.95173 q^{30} +2.66037 q^{32} -7.80507 q^{33} -4.65064 q^{34} +5.87917 q^{35} -0.628936 q^{36} +1.81406 q^{37} +7.51417 q^{38} -5.46769 q^{39} +4.72643 q^{40} +0.337145 q^{41} +9.72517 q^{42} -3.88400 q^{43} -1.81212 q^{44} -2.01784 q^{45} -1.12109 q^{46} -1.18915 q^{47} +5.81908 q^{48} +7.46858 q^{49} -3.21726 q^{50} +7.83166 q^{51} -1.26945 q^{52} +2.34413 q^{53} +4.33232 q^{54} -5.81390 q^{55} +11.6317 q^{56} -12.6538 q^{57} +8.38936 q^{58} +7.77883 q^{59} -1.54504 q^{60} +2.72343 q^{61} -4.96588 q^{63} +8.88686 q^{64} -4.07281 q^{65} -9.61719 q^{66} -7.42118 q^{67} +1.81829 q^{68} +1.88791 q^{69} +7.24416 q^{70} -5.09818 q^{71} -3.99221 q^{72} -5.39174 q^{73} +2.23524 q^{74} +5.41786 q^{75} -2.93787 q^{76} -14.3079 q^{77} -6.73714 q^{78} +9.73723 q^{79} +4.33456 q^{80} -11.2122 q^{81} +0.415420 q^{82} +8.39515 q^{83} -3.80232 q^{84} +5.83370 q^{85} -4.78575 q^{86} -14.1277 q^{87} -11.5025 q^{88} -5.09560 q^{89} -2.48633 q^{90} -10.0231 q^{91} +0.438320 q^{92} -1.46523 q^{94} -9.42569 q^{95} -5.52020 q^{96} +10.9142 q^{97} +9.20257 q^{98} +4.91075 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} + 3 q^{3} + 8 q^{4} + 3 q^{5} + 11 q^{6} - 2 q^{7} - 9 q^{8} + 5 q^{9} - 13 q^{10} + 18 q^{11} + 8 q^{13} - 9 q^{14} + 18 q^{15} + 4 q^{16} + 14 q^{17} + 23 q^{18} - 6 q^{19} - 7 q^{20} - q^{21}+ \cdots + 6 q^{99}+O(q^{100}) \) 8 * q + 2 * q^2 + 3 * q^3 + 8 * q^4 + 3 * q^5 + 11 * q^6 - 2 * q^7 - 9 * q^8 + 5 * q^9 - 13 * q^10 + 18 * q^11 + 8 * q^13 - 9 * q^14 + 18 * q^15 + 4 * q^16 + 14 * q^17 + 23 * q^18 - 6 * q^19 - 7 * q^20 - q^21 + 4 * q^22 + 22 * q^23 + 30 * q^24 + 13 * q^25 + 9 * q^26 + 18 * q^27 - 5 * q^28 + 12 * q^29 + 11 * q^30 - 21 * q^32 + 6 * q^33 - 7 * q^34 - 12 * q^35 - q^36 - 8 * q^37 + 7 * q^38 + q^39 + 11 * q^40 - 22 * q^41 - 6 * q^42 - 2 * q^43 + 4 * q^44 + 18 * q^46 - 18 * q^47 - q^48 - 2 * q^49 + 27 * q^50 + 2 * q^51 - q^52 + 6 * q^53 - 7 * q^54 + 28 * q^55 + 30 * q^56 - 17 * q^57 - 5 * q^58 - 4 * q^59 - 40 * q^60 + 30 * q^61 - 23 * q^63 + 9 * q^64 + 3 * q^65 + 10 * q^66 - 13 * q^67 + 30 * q^68 + 22 * q^69 - 39 * q^70 - q^71 + 3 * q^72 + 2 * q^73 + 8 * q^74 + 23 * q^75 - 33 * q^76 - 24 * q^77 + 8 * q^79 + 24 * q^80 - 28 * q^81 - 19 * q^82 + 39 * q^83 + 8 * q^84 - 26 * q^85 - 16 * q^86 - 15 * q^87 - 17 * q^88 + 27 * q^89 + 8 * q^90 + 16 * q^91 + 32 * q^92 + 22 * q^94 - 6 * q^95 - 12 * q^96 + 34 * q^97 + 10 * q^98 + 6 * q^99

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