LMFDB - Embedded newform 931.2.a.e.1.1

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 = [2,-2,0,2,-2,4] 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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 133)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	\(q-2.41421 q^{2} -1.41421 q^{3} +3.82843 q^{4} -1.00000 q^{5} +3.41421 q^{6} -4.41421 q^{8} -1.00000 q^{9} +2.41421 q^{10} -0.414214 q^{11} -5.41421 q^{12} +2.24264 q^{13} +1.41421 q^{15} +3.00000 q^{16} +4.00000 q^{17} +2.41421 q^{18} +1.00000 q^{19} -3.82843 q^{20} +1.00000 q^{22} -5.58579 q^{23} +6.24264 q^{24} -4.00000 q^{25} -5.41421 q^{26} +5.65685 q^{27} -6.58579 q^{29} -3.41421 q^{30} +6.24264 q^{31} +1.58579 q^{32} +0.585786 q^{33} -9.65685 q^{34} -3.82843 q^{36} +9.07107 q^{37} -2.41421 q^{38} -3.17157 q^{39} +4.41421 q^{40} +3.17157 q^{41} +8.07107 q^{43} -1.58579 q^{44} +1.00000 q^{45} +13.4853 q^{46} +4.41421 q^{47} -4.24264 q^{48} +9.65685 q^{50} -5.65685 q^{51} +8.58579 q^{52} -4.24264 q^{53} -13.6569 q^{54} +0.414214 q^{55} -1.41421 q^{57} +15.8995 q^{58} -6.82843 q^{59} +5.41421 q^{60} -11.8284 q^{61} -15.0711 q^{62} -9.82843 q^{64} -2.24264 q^{65} -1.41421 q^{66} +1.17157 q^{67} +15.3137 q^{68} +7.89949 q^{69} -14.2426 q^{71} +4.41421 q^{72} -15.1421 q^{73} -21.8995 q^{74} +5.65685 q^{75} +3.82843 q^{76} +7.65685 q^{78} +1.41421 q^{79} -3.00000 q^{80} -5.00000 q^{81} -7.65685 q^{82} -9.24264 q^{83} -4.00000 q^{85} -19.4853 q^{86} +9.31371 q^{87} +1.82843 q^{88} +11.4142 q^{89} -2.41421 q^{90} -21.3848 q^{92} -8.82843 q^{93} -10.6569 q^{94} -1.00000 q^{95} -2.24264 q^{96} +3.17157 q^{97} +0.414214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{6} - 6 q^{8} - 2 q^{9} + 2 q^{10} + 2 q^{11} - 8 q^{12} - 4 q^{13} + 6 q^{16} + 8 q^{17} + 2 q^{18} + 2 q^{19} - 2 q^{20} + 2 q^{22} - 14 q^{23} + 4 q^{24} - 8 q^{25}+ \cdots - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^6 - 6 * q^8 - 2 * q^9 + 2 * q^10 + 2 * q^11 - 8 * q^12 - 4 * q^13 + 6 * q^16 + 8 * q^17 + 2 * q^18 + 2 * q^19 - 2 * q^20 + 2 * q^22 - 14 * q^23 + 4 * q^24 - 8 * q^25 - 8 * q^26 - 16 * q^29 - 4 * q^30 + 4 * q^31 + 6 * q^32 + 4 * q^33 - 8 * q^34 - 2 * q^36 + 4 * q^37 - 2 * q^38 - 12 * q^39 + 6 * q^40 + 12 * q^41 + 2 * q^43 - 6 * q^44 + 2 * q^45 + 10 * q^46 + 6 * q^47 + 8 * q^50 + 20 * q^52 - 16 * q^54 - 2 * q^55 + 12 * q^58 - 8 * q^59 + 8 * q^60 - 18 * q^61 - 16 * q^62 - 14 * q^64 + 4 * q^65 + 8 * q^67 + 8 * q^68 - 4 * q^69 - 20 * q^71 + 6 * q^72 - 2 * q^73 - 24 * q^74 + 2 * q^76 + 4 * q^78 - 6 * q^80 - 10 * q^81 - 4 * q^82 - 10 * q^83 - 8 * q^85 - 22 * q^86 - 4 * q^87 - 2 * q^88 + 20 * q^89 - 2 * q^90 - 6 * q^92 - 12 * q^93 - 10 * q^94 - 2 * q^95 + 4 * q^96 + 12 * q^97 - 2 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	3.82843	1.91421
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	3.41421	1.39385
$7$	0	0
$7$	0	0
$8$	−4.41421	−1.56066
$9$	−1.00000	−0.333333
$10$	2.41421	0.763441
$11$	−0.414214	−0.124890	−0.0624450	−	0.998048i	$-0.519890\pi$
$11$	−0.414214	−0.124890	−0.0624450	+	0.998048i	$0.519890\pi$
$12$	−5.41421	−1.56295
$13$	2.24264	0.621997	0.310998	−	0.950410i	$-0.399337\pi$
$13$	2.24264	0.621997	0.310998	+	0.950410i	$0.399337\pi$
$14$	0	0
$15$	1.41421	0.365148
$16$	3.00000	0.750000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	2.41421	0.569036
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−3.82843	−0.856062
$21$	0	0
$22$	1.00000	0.213201
$23$	−5.58579	−1.16472	−0.582358	−	0.812932i	$-0.697870\pi$
$23$	−5.58579	−1.16472	−0.582358	+	0.812932i	$0.697870\pi$
$24$	6.24264	1.27427
$25$	−4.00000	−0.800000
$26$	−5.41421	−1.06181
$27$	5.65685	1.08866
$28$	0	0
$29$	−6.58579	−1.22295	−0.611475	−	0.791264i	$-0.709423\pi$
$29$	−6.58579	−1.22295	−0.611475	+	0.791264i	$0.709423\pi$
$30$	−3.41421	−0.623347
$31$	6.24264	1.12121	0.560606	−	0.828083i	$-0.310568\pi$
$31$	6.24264	1.12121	0.560606	+	0.828083i	$0.310568\pi$
$32$	1.58579	0.280330
$33$	0.585786	0.101972
$34$	−9.65685	−1.65614
$35$	0	0
$36$	−3.82843	−0.638071
$37$	9.07107	1.49127	0.745637	−	0.666352i	$-0.232145\pi$
$37$	9.07107	1.49127	0.745637	+	0.666352i	$0.232145\pi$
$38$	−2.41421	−0.391637
$39$	−3.17157	−0.507858
$40$	4.41421	0.697948
$41$	3.17157	0.495316	0.247658	−	0.968847i	$-0.420339\pi$
$41$	3.17157	0.495316	0.247658	+	0.968847i	$0.420339\pi$
$42$	0	0
$43$	8.07107	1.23083	0.615413	−	0.788205i	$-0.288989\pi$
$43$	8.07107	1.23083	0.615413	+	0.788205i	$0.288989\pi$
$44$	−1.58579	−0.239066
$45$	1.00000	0.149071
$46$	13.4853	1.98830
$47$	4.41421	0.643879	0.321940	−	0.946760i	$-0.395665\pi$
$47$	4.41421	0.643879	0.321940	+	0.946760i	$0.395665\pi$
$48$	−4.24264	−0.612372
$49$	0	0
$50$	9.65685	1.36569
$51$	−5.65685	−0.792118
$52$	8.58579	1.19063
$53$	−4.24264	−0.582772	−0.291386	−	0.956606i	$-0.594116\pi$
$53$	−4.24264	−0.582772	−0.291386	+	0.956606i	$0.594116\pi$
$54$	−13.6569	−1.85846
$55$	0.414214	0.0558525
$56$	0	0
$57$	−1.41421	−0.187317
$58$	15.8995	2.08771
$59$	−6.82843	−0.888985	−0.444493	−	0.895782i	$-0.646616\pi$
$59$	−6.82843	−0.888985	−0.444493	+	0.895782i	$0.646616\pi$
$60$	5.41421	0.698972
$61$	−11.8284	−1.51447	−0.757237	−	0.653140i	$-0.773451\pi$
$61$	−11.8284	−1.51447	−0.757237	+	0.653140i	$0.773451\pi$
$62$	−15.0711	−1.91403
$63$	0	0
$64$	−9.82843	−1.22855
$65$	−2.24264	−0.278165
$66$	−1.41421	−0.174078
$67$	1.17157	0.143130	0.0715652	−	0.997436i	$-0.477201\pi$
$67$	1.17157	0.143130	0.0715652	+	0.997436i	$0.477201\pi$
$68$	15.3137	1.85706
$69$	7.89949	0.950987
$70$	0	0
$71$	−14.2426	−1.69029	−0.845145	−	0.534537i	$-0.820486\pi$
$71$	−14.2426	−1.69029	−0.845145	+	0.534537i	$0.820486\pi$
$72$	4.41421	0.520220
$73$	−15.1421	−1.77225	−0.886126	−	0.463444i	$-0.846614\pi$
$73$	−15.1421	−1.77225	−0.886126	+	0.463444i	$0.846614\pi$
$74$	−21.8995	−2.54576
$75$	5.65685	0.653197
$76$	3.82843	0.439151
$77$	0	0
$78$	7.65685	0.866968
$79$	1.41421	0.159111	0.0795557	−	0.996830i	$-0.474650\pi$
$79$	1.41421	0.159111	0.0795557	+	0.996830i	$0.474650\pi$
$80$	−3.00000	−0.335410
$81$	−5.00000	−0.555556
$82$	−7.65685	−0.845558
$83$	−9.24264	−1.01451	−0.507256	−	0.861796i	$-0.669340\pi$
$83$	−9.24264	−1.01451	−0.507256	+	0.861796i	$0.669340\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	−19.4853	−2.10115
$87$	9.31371	0.998534
$88$	1.82843	0.194911
$89$	11.4142	1.20990	0.604952	−	0.796262i	$-0.293192\pi$
$89$	11.4142	1.20990	0.604952	+	0.796262i	$0.293192\pi$
$90$	−2.41421	−0.254480
$91$	0	0
$92$	−21.3848	−2.22952
$93$	−8.82843	−0.915465
$94$	−10.6569	−1.09917
$95$	−1.00000	−0.102598
$96$	−2.24264	−0.228889
$97$	3.17157	0.322024	0.161012	−	0.986952i	$-0.448524\pi$
$97$	3.17157	0.322024	0.161012	+	0.986952i	$0.448524\pi$
$98$	0	0
$99$	0.414214	0.0416300
$100$	−15.3137	−1.53137
$101$	−18.3137	−1.82228	−0.911141	−	0.412095i	$-0.864797\pi$
$101$	−18.3137	−1.82228	−0.911141	+	0.412095i	$0.864797\pi$
$102$	13.6569	1.35223
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	−9.89949	−0.970725
$105$	0	0
$106$	10.2426	0.994853
$107$	−13.8995	−1.34371	−0.671857	−	0.740680i	$-0.734503\pi$
$107$	−13.8995	−1.34371	−0.671857	+	0.740680i	$0.734503\pi$
$108$	21.6569	2.08393
$109$	−9.65685	−0.924959	−0.462479	−	0.886630i	$-0.653040\pi$
$109$	−9.65685	−0.924959	−0.462479	+	0.886630i	$0.653040\pi$
$110$	−1.00000	−0.0953463
$111$	−12.8284	−1.21762
$112$	0	0
$113$	−0.242641	−0.0228257	−0.0114129	−	0.999935i	$-0.503633\pi$
$113$	−0.242641	−0.0228257	−0.0114129	+	0.999935i	$0.503633\pi$
$114$	3.41421	0.319770
$115$	5.58579	0.520877
$116$	−25.2132	−2.34099
$117$	−2.24264	−0.207332
$118$	16.4853	1.51759
$119$	0	0
$120$	−6.24264	−0.569873
$121$	−10.8284	−0.984402
$122$	28.5563	2.58537
$123$	−4.48528	−0.404424
$124$	23.8995	2.14624
$125$	9.00000	0.804984
$126$	0	0
$127$	4.24264	0.376473	0.188237	−	0.982124i	$-0.439723\pi$
$127$	4.24264	0.376473	0.188237	+	0.982124i	$0.439723\pi$
$128$	20.5563	1.81694
$129$	−11.4142	−1.00497
$130$	5.41421	0.474858
$131$	−3.51472	−0.307082	−0.153541	−	0.988142i	$-0.549068\pi$
$131$	−3.51472	−0.307082	−0.153541	+	0.988142i	$0.549068\pi$
$132$	2.24264	0.195197
$133$	0	0
$134$	−2.82843	−0.244339
$135$	−5.65685	−0.486864
$136$	−17.6569	−1.51406
$137$	−19.8284	−1.69406	−0.847028	−	0.531548i	$-0.821611\pi$
$137$	−19.8284	−1.69406	−0.847028	+	0.531548i	$0.821611\pi$
$138$	−19.0711	−1.62344
$139$	6.07107	0.514941	0.257471	−	0.966286i	$-0.417111\pi$
$139$	6.07107	0.514941	0.257471	+	0.966286i	$0.417111\pi$
$140$	0	0
$141$	−6.24264	−0.525725
$142$	34.3848	2.88551
$143$	−0.928932	−0.0776812
$144$	−3.00000	−0.250000
$145$	6.58579	0.546920
$146$	36.5563	3.02542
$147$	0	0
$148$	34.7279	2.85462
$149$	−0.171573	−0.0140558	−0.00702790	−	0.999975i	$-0.502237\pi$
$149$	−0.171573	−0.0140558	−0.00702790	+	0.999975i	$0.502237\pi$
$150$	−13.6569	−1.11508
$151$	−17.3137	−1.40897	−0.704485	−	0.709719i	$-0.748822\pi$
$151$	−17.3137	−1.40897	−0.704485	+	0.709719i	$0.748822\pi$
$152$	−4.41421	−0.358040
$153$	−4.00000	−0.323381
$154$	0	0
$155$	−6.24264	−0.501421
$156$	−12.1421	−0.972149
$157$	−6.17157	−0.492545	−0.246273	−	0.969201i	$-0.579206\pi$
$157$	−6.17157	−0.492545	−0.246273	+	0.969201i	$0.579206\pi$
$158$	−3.41421	−0.271620
$159$	6.00000	0.475831
$160$	−1.58579	−0.125367
$161$	0	0
$162$	12.0711	0.948393
$163$	3.92893	0.307738	0.153869	−	0.988091i	$-0.450827\pi$
$163$	3.92893	0.307738	0.153869	+	0.988091i	$0.450827\pi$
$164$	12.1421	0.948141
$165$	−0.585786	−0.0456034
$166$	22.3137	1.73188
$167$	−13.0711	−1.01147	−0.505735	−	0.862689i	$-0.668779\pi$
$167$	−13.0711	−1.01147	−0.505735	+	0.862689i	$0.668779\pi$
$168$	0	0
$169$	−7.97056	−0.613120
$170$	9.65685	0.740647
$171$	−1.00000	−0.0764719
$172$	30.8995	2.35606
$173$	7.31371	0.556051	0.278025	−	0.960574i	$-0.410320\pi$
$173$	7.31371	0.556051	0.278025	+	0.960574i	$0.410320\pi$
$174$	−22.4853	−1.70460
$175$	0	0
$176$	−1.24264	−0.0936676
$177$	9.65685	0.725854
$178$	−27.5563	−2.06544
$179$	17.7990	1.33036	0.665179	−	0.746684i	$-0.268355\pi$
$179$	17.7990	1.33036	0.665179	+	0.746684i	$0.268355\pi$
$180$	3.82843	0.285354
$181$	−9.89949	−0.735824	−0.367912	−	0.929861i	$-0.619927\pi$
$181$	−9.89949	−0.735824	−0.367912	+	0.929861i	$0.619927\pi$
$182$	0	0
$183$	16.7279	1.23656
$184$	24.6569	1.81773
$185$	−9.07107	−0.666918
$186$	21.3137	1.56280
$187$	−1.65685	−0.121161
$188$	16.8995	1.23252
$189$	0	0
$190$	2.41421	0.175145
$191$	8.41421	0.608831	0.304416	−	0.952539i	$-0.401539\pi$
$191$	8.41421	0.608831	0.304416	+	0.952539i	$0.401539\pi$
$192$	13.8995	1.00311
$193$	7.89949	0.568618	0.284309	−	0.958733i	$-0.408236\pi$
$193$	7.89949	0.568618	0.284309	+	0.958733i	$0.408236\pi$
$194$	−7.65685	−0.549730
$195$	3.17157	0.227121
$196$	0	0
$197$	4.51472	0.321660	0.160830	−	0.986982i	$-0.448583\pi$
$197$	4.51472	0.321660	0.160830	+	0.986982i	$0.448583\pi$
$198$	−1.00000	−0.0710669
$199$	−16.0711	−1.13925	−0.569624	−	0.821905i	$-0.692911\pi$
$199$	−16.0711	−1.13925	−0.569624	+	0.821905i	$0.692911\pi$
$200$	17.6569	1.24853
$201$	−1.65685	−0.116865
$202$	44.2132	3.11083
$203$	0	0
$204$	−21.6569	−1.51628
$205$	−3.17157	−0.221512
$206$	−14.4853	−1.00924
$207$	5.58579	0.388239
$208$	6.72792	0.466497
$209$	−0.414214	−0.0286518
$210$	0	0
$211$	−1.17157	−0.0806544	−0.0403272	−	0.999187i	$-0.512840\pi$
$211$	−1.17157	−0.0806544	−0.0403272	+	0.999187i	$0.512840\pi$
$212$	−16.2426	−1.11555
$213$	20.1421	1.38012
$214$	33.5563	2.29386
$215$	−8.07107	−0.550442
$216$	−24.9706	−1.69903
$217$	0	0
$218$	23.3137	1.57900
$219$	21.4142	1.44704
$220$	1.58579	0.106914
$221$	8.97056	0.603425
$222$	30.9706	2.07861
$223$	−2.24264	−0.150178	−0.0750892	−	0.997177i	$-0.523924\pi$
$223$	−2.24264	−0.150178	−0.0750892	+	0.997177i	$0.523924\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0.585786	0.0389659
$227$	20.2426	1.34355	0.671776	−	0.740755i	$-0.265532\pi$
$227$	20.2426	1.34355	0.671776	+	0.740755i	$0.265532\pi$
$228$	−5.41421	−0.358565
$229$	−12.4853	−0.825051	−0.412525	−	0.910946i	$-0.635353\pi$
$229$	−12.4853	−0.825051	−0.412525	+	0.910946i	$0.635353\pi$
$230$	−13.4853	−0.889193
$231$	0	0
$232$	29.0711	1.90861
$233$	0.485281	0.0317918	0.0158959	−	0.999874i	$-0.494940\pi$
$233$	0.485281	0.0317918	0.0158959	+	0.999874i	$0.494940\pi$
$234$	5.41421	0.353938
$235$	−4.41421	−0.287952
$236$	−26.1421	−1.70171
$237$	−2.00000	−0.129914
$238$	0	0
$239$	2.00000	0.129369	0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000	0.129369	0.0646846	+	0.997906i	$0.479396\pi$
$240$	4.24264	0.273861
$241$	26.0416	1.67749	0.838744	−	0.544525i	$-0.183290\pi$
$241$	26.0416	1.67749	0.838744	+	0.544525i	$0.183290\pi$
$242$	26.1421	1.68048
$243$	−9.89949	−0.635053
$244$	−45.2843	−2.89903
$245$	0	0
$246$	10.8284	0.690395
$247$	2.24264	0.142696
$248$	−27.5563	−1.74983
$249$	13.0711	0.828345
$250$	−21.7279	−1.37419
$251$	13.7279	0.866499	0.433249	−	0.901274i	$-0.357367\pi$
$251$	13.7279	0.866499	0.433249	+	0.901274i	$0.357367\pi$
$252$	0	0
$253$	2.31371	0.145462
$254$	−10.2426	−0.642680
$255$	5.65685	0.354246
$256$	−29.9706	−1.87316
$257$	−9.31371	−0.580973	−0.290487	−	0.956879i	$-0.593817\pi$
$257$	−9.31371	−0.580973	−0.290487	+	0.956879i	$0.593817\pi$
$258$	27.5563	1.71558
$259$	0	0
$260$	−8.58579	−0.532468
$261$	6.58579	0.407650
$262$	8.48528	0.524222
$263$	16.4853	1.01653	0.508263	−	0.861202i	$-0.330288\pi$
$263$	16.4853	1.01653	0.508263	+	0.861202i	$0.330288\pi$
$264$	−2.58579	−0.159144
$265$	4.24264	0.260623
$266$	0	0
$267$	−16.1421	−0.987883
$268$	4.48528	0.273982
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	13.6569	0.831130
$271$	15.2426	0.925924	0.462962	−	0.886378i	$-0.346787\pi$
$271$	15.2426	0.925924	0.462962	+	0.886378i	$0.346787\pi$
$272$	12.0000	0.727607
$273$	0	0
$274$	47.8701	2.89194
$275$	1.65685	0.0999121
$276$	30.2426	1.82039
$277$	−22.4558	−1.34924	−0.674620	−	0.738165i	$-0.735693\pi$
$277$	−22.4558	−1.34924	−0.674620	+	0.738165i	$0.735693\pi$
$278$	−14.6569	−0.879060
$279$	−6.24264	−0.373737
$280$	0	0
$281$	−28.2426	−1.68481	−0.842407	−	0.538841i	$-0.818862\pi$
$281$	−28.2426	−1.68481	−0.842407	+	0.538841i	$0.818862\pi$
$282$	15.0711	0.897469
$283$	−30.0711	−1.78754	−0.893770	−	0.448526i	$-0.851949\pi$
$283$	−30.0711	−1.78754	−0.893770	+	0.448526i	$0.851949\pi$
$284$	−54.5269	−3.23558
$285$	1.41421	0.0837708
$286$	2.24264	0.132610
$287$	0	0
$288$	−1.58579	−0.0934434
$289$	−1.00000	−0.0588235
$290$	−15.8995	−0.933650
$291$	−4.48528	−0.262932
$292$	−57.9706	−3.39247
$293$	−3.17157	−0.185285	−0.0926426	−	0.995699i	$-0.529531\pi$
$293$	−3.17157	−0.185285	−0.0926426	+	0.995699i	$0.529531\pi$
$294$	0	0
$295$	6.82843	0.397566
$296$	−40.0416	−2.32737
$297$	−2.34315	−0.135963
$298$	0.414214	0.0239947
$299$	−12.5269	−0.724450
$300$	21.6569	1.25036
$301$	0	0
$302$	41.7990	2.40526
$303$	25.8995	1.48789
$304$	3.00000	0.172062
$305$	11.8284	0.677294
$306$	9.65685	0.552046
$307$	−29.3137	−1.67302	−0.836511	−	0.547950i	$-0.815408\pi$
$307$	−29.3137	−1.67302	−0.836511	+	0.547950i	$0.815408\pi$
$308$	0	0
$309$	−8.48528	−0.482711
$310$	15.0711	0.855979
$311$	23.7990	1.34952	0.674758	−	0.738039i	$-0.264248\pi$
$311$	23.7990	1.34952	0.674758	+	0.738039i	$0.264248\pi$
$312$	14.0000	0.792594
$313$	6.65685	0.376268	0.188134	−	0.982143i	$-0.439756\pi$
$313$	6.65685	0.376268	0.188134	+	0.982143i	$0.439756\pi$
$314$	14.8995	0.840827
$315$	0	0
$316$	5.41421	0.304573
$317$	−20.2426	−1.13694	−0.568470	−	0.822704i	$-0.692464\pi$
$317$	−20.2426	−1.13694	−0.568470	+	0.822704i	$0.692464\pi$
$318$	−14.4853	−0.812294
$319$	2.72792	0.152734
$320$	9.82843	0.549426
$321$	19.6569	1.09714
$322$	0	0
$323$	4.00000	0.222566
$324$	−19.1421	−1.06345
$325$	−8.97056	−0.497597
$326$	−9.48528	−0.525341
$327$	13.6569	0.755226
$328$	−14.0000	−0.773021
$329$	0	0
$330$	1.41421	0.0778499
$331$	−31.4558	−1.72897	−0.864485	−	0.502659i	$-0.832355\pi$
$331$	−31.4558	−1.72897	−0.864485	+	0.502659i	$0.832355\pi$
$332$	−35.3848	−1.94199
$333$	−9.07107	−0.497091
$334$	31.5563	1.72669
$335$	−1.17157	−0.0640099
$336$	0	0
$337$	19.5563	1.06530	0.532651	−	0.846335i	$-0.321196\pi$
$337$	19.5563	1.06530	0.532651	+	0.846335i	$0.321196\pi$
$338$	19.2426	1.04666
$339$	0.343146	0.0186371
$340$	−15.3137	−0.830502
$341$	−2.58579	−0.140028
$342$	2.41421	0.130546
$343$	0	0
$344$	−35.6274	−1.92090
$345$	−7.89949	−0.425295
$346$	−17.6569	−0.949238
$347$	27.0416	1.45167	0.725835	−	0.687868i	$-0.241453\pi$
$347$	27.0416	1.45167	0.725835	+	0.687868i	$0.241453\pi$
$348$	35.6569	1.91141
$349$	−4.48528	−0.240092	−0.120046	−	0.992768i	$-0.538304\pi$
$349$	−4.48528	−0.240092	−0.120046	+	0.992768i	$0.538304\pi$
$350$	0	0
$351$	12.6863	0.677144
$352$	−0.656854	−0.0350104
$353$	−0.485281	−0.0258289	−0.0129145	−	0.999917i	$-0.504111\pi$
$353$	−0.485281	−0.0258289	−0.0129145	+	0.999917i	$0.504111\pi$
$354$	−23.3137	−1.23911
$355$	14.2426	0.755921
$356$	43.6985	2.31602
$357$	0	0
$358$	−42.9706	−2.27106
$359$	5.24264	0.276696	0.138348	−	0.990384i	$-0.455821\pi$
$359$	5.24264	0.276696	0.138348	+	0.990384i	$0.455821\pi$
$360$	−4.41421	−0.232649
$361$	1.00000	0.0526316
$362$	23.8995	1.25613
$363$	15.3137	0.803761
$364$	0	0
$365$	15.1421	0.792576
$366$	−40.3848	−2.11095
$367$	24.3431	1.27070	0.635351	−	0.772224i	$-0.280855\pi$
$367$	24.3431	1.27070	0.635351	+	0.772224i	$0.280855\pi$
$368$	−16.7574	−0.873538
$369$	−3.17157	−0.165105
$370$	21.8995	1.13850
$371$	0	0
$372$	−33.7990	−1.75240
$373$	11.3137	0.585802	0.292901	−	0.956143i	$-0.405379\pi$
$373$	11.3137	0.585802	0.292901	+	0.956143i	$0.405379\pi$
$374$	4.00000	0.206835
$375$	−12.7279	−0.657267
$376$	−19.4853	−1.00488
$377$	−14.7696	−0.760671
$378$	0	0
$379$	4.24264	0.217930	0.108965	−	0.994046i	$-0.465246\pi$
$379$	4.24264	0.217930	0.108965	+	0.994046i	$0.465246\pi$
$380$	−3.82843	−0.196394
$381$	−6.00000	−0.307389
$382$	−20.3137	−1.03934
$383$	−19.7574	−1.00955	−0.504777	−	0.863250i	$-0.668425\pi$
$383$	−19.7574	−1.00955	−0.504777	+	0.863250i	$0.668425\pi$
$384$	−29.0711	−1.48353
$385$	0	0
$386$	−19.0711	−0.970692
$387$	−8.07107	−0.410275
$388$	12.1421	0.616424
$389$	−11.5147	−0.583819	−0.291910	−	0.956446i	$-0.594291\pi$
$389$	−11.5147	−0.583819	−0.291910	+	0.956446i	$0.594291\pi$
$390$	−7.65685	−0.387720
$391$	−22.3431	−1.12994
$392$	0	0
$393$	4.97056	0.250732
$394$	−10.8995	−0.549109
$395$	−1.41421	−0.0711568
$396$	1.58579	0.0796888
$397$	8.62742	0.432998	0.216499	−	0.976283i	$-0.430536\pi$
$397$	8.62742	0.432998	0.216499	+	0.976283i	$0.430536\pi$
$398$	38.7990	1.94482
$399$	0	0
$400$	−12.0000	−0.600000
$401$	0.485281	0.0242338	0.0121169	−	0.999927i	$-0.496143\pi$
$401$	0.485281	0.0242338	0.0121169	+	0.999927i	$0.496143\pi$
$402$	4.00000	0.199502
$403$	14.0000	0.697390
$404$	−70.1127	−3.48824
$405$	5.00000	0.248452
$406$	0	0
$407$	−3.75736	−0.186245
$408$	24.9706	1.23623
$409$	2.68629	0.132829	0.0664143	−	0.997792i	$-0.478844\pi$
$409$	2.68629	0.132829	0.0664143	+	0.997792i	$0.478844\pi$
$410$	7.65685	0.378145
$411$	28.0416	1.38319
$412$	22.9706	1.13168
$413$	0	0
$414$	−13.4853	−0.662765
$415$	9.24264	0.453703
$416$	3.55635	0.174364
$417$	−8.58579	−0.420448
$418$	1.00000	0.0489116
$419$	−6.07107	−0.296591	−0.148296	−	0.988943i	$-0.547379\pi$
$419$	−6.07107	−0.296591	−0.148296	+	0.988943i	$0.547379\pi$
$420$	0	0
$421$	−37.2132	−1.81366	−0.906830	−	0.421496i	$-0.861505\pi$
$421$	−37.2132	−1.81366	−0.906830	+	0.421496i	$0.861505\pi$
$422$	2.82843	0.137686
$423$	−4.41421	−0.214626
$424$	18.7279	0.909508
$425$	−16.0000	−0.776114
$426$	−48.6274	−2.35601
$427$	0	0
$428$	−53.2132	−2.57216
$429$	1.31371	0.0634264
$430$	19.4853	0.939664
$431$	−10.0416	−0.483688	−0.241844	−	0.970315i	$-0.577752\pi$
$431$	−10.0416	−0.483688	−0.241844	+	0.970315i	$0.577752\pi$
$432$	16.9706	0.816497
$433$	−28.0000	−1.34559	−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000	−1.34559	−0.672797	+	0.739827i	$0.734907\pi$
$434$	0	0
$435$	−9.31371	−0.446558
$436$	−36.9706	−1.77057
$437$	−5.58579	−0.267204
$438$	−51.6985	−2.47025
$439$	24.4853	1.16862	0.584309	−	0.811531i	$-0.301365\pi$
$439$	24.4853	1.16862	0.584309	+	0.811531i	$0.301365\pi$
$440$	−1.82843	−0.0871668
$441$	0	0
$442$	−21.6569	−1.03011
$443$	17.6569	0.838902	0.419451	−	0.907778i	$-0.362223\pi$
$443$	17.6569	0.838902	0.419451	+	0.907778i	$0.362223\pi$
$444$	−49.1127	−2.33079
$445$	−11.4142	−0.541086
$446$	5.41421	0.256370
$447$	0.242641	0.0114765
$448$	0	0
$449$	9.65685	0.455735	0.227868	−	0.973692i	$-0.426825\pi$
$449$	9.65685	0.455735	0.227868	+	0.973692i	$0.426825\pi$
$450$	−9.65685	−0.455228
$451$	−1.31371	−0.0618601
$452$	−0.928932	−0.0436933
$453$	24.4853	1.15042
$454$	−48.8701	−2.29359
$455$	0	0
$456$	6.24264	0.292338
$457$	−7.82843	−0.366198	−0.183099	−	0.983094i	$-0.558613\pi$
$457$	−7.82843	−0.366198	−0.183099	+	0.983094i	$0.558613\pi$
$458$	30.1421	1.40845
$459$	22.6274	1.05616
$460$	21.3848	0.997070
$461$	−29.9706	−1.39587	−0.697934	−	0.716162i	$-0.745897\pi$
$461$	−29.9706	−1.39587	−0.697934	+	0.716162i	$0.745897\pi$
$462$	0	0
$463$	−4.07107	−0.189199	−0.0945993	−	0.995515i	$-0.530157\pi$
$463$	−4.07107	−0.189199	−0.0945993	+	0.995515i	$0.530157\pi$
$464$	−19.7574	−0.917212
$465$	8.82843	0.409409
$466$	−1.17157	−0.0542721
$467$	−19.1005	−0.883866	−0.441933	−	0.897048i	$-0.645707\pi$
$467$	−19.1005	−0.883866	−0.441933	+	0.897048i	$0.645707\pi$
$468$	−8.58579	−0.396878
$469$	0	0
$470$	10.6569	0.491564
$471$	8.72792	0.402161
$472$	30.1421	1.38740
$473$	−3.34315	−0.153718
$474$	4.82843	0.221777
$475$	−4.00000	−0.183533
$476$	0	0
$477$	4.24264	0.194257
$478$	−4.82843	−0.220847
$479$	25.3848	1.15986	0.579930	−	0.814666i	$-0.303080\pi$
$479$	25.3848	1.15986	0.579930	+	0.814666i	$0.303080\pi$
$480$	2.24264	0.102362
$481$	20.3431	0.927568
$482$	−62.8701	−2.86365
$483$	0	0
$484$	−41.4558	−1.88436
$485$	−3.17157	−0.144014
$486$	23.8995	1.08410
$487$	15.1716	0.687490	0.343745	−	0.939063i	$-0.388304\pi$
$487$	15.1716	0.687490	0.343745	+	0.939063i	$0.388304\pi$
$488$	52.2132	2.36358
$489$	−5.55635	−0.251267
$490$	0	0
$491$	−1.44365	−0.0651510	−0.0325755	−	0.999469i	$-0.510371\pi$
$491$	−1.44365	−0.0651510	−0.0325755	+	0.999469i	$0.510371\pi$
$492$	−17.1716	−0.774154
$493$	−26.3431	−1.18644
$494$	−5.41421	−0.243597
$495$	−0.414214	−0.0186175
$496$	18.7279	0.840909
$497$	0	0
$498$	−31.5563	−1.41407
$499$	34.5563	1.54695	0.773477	−	0.633824i	$-0.218516\pi$
$499$	34.5563	1.54695	0.773477	+	0.633824i	$0.218516\pi$
$500$	34.4558	1.54091
$501$	18.4853	0.825861
$502$	−33.1421	−1.47921
$503$	29.0416	1.29490	0.647451	−	0.762107i	$-0.275835\pi$
$503$	29.0416	1.29490	0.647451	+	0.762107i	$0.275835\pi$
$504$	0	0
$505$	18.3137	0.814949
$506$	−5.58579	−0.248318
$507$	11.2721	0.500611
$508$	16.2426	0.720651
$509$	14.9706	0.663559	0.331779	−	0.943357i	$-0.392351\pi$
$509$	14.9706	0.663559	0.331779	+	0.943357i	$0.392351\pi$
$510$	−13.6569	−0.604736
$511$	0	0
$512$	31.2426	1.38074
$513$	5.65685	0.249756
$514$	22.4853	0.991783
$515$	−6.00000	−0.264392
$516$	−43.6985	−1.92372
$517$	−1.82843	−0.0804141
$518$	0	0
$519$	−10.3431	−0.454014
$520$	9.89949	0.434122
$521$	−41.5563	−1.82062	−0.910308	−	0.413931i	$-0.864156\pi$
$521$	−41.5563	−1.82062	−0.910308	+	0.413931i	$0.864156\pi$
$522$	−15.8995	−0.695902
$523$	12.3431	0.539728	0.269864	−	0.962898i	$-0.413021\pi$
$523$	12.3431	0.539728	0.269864	+	0.962898i	$0.413021\pi$
$524$	−13.4558	−0.587821
$525$	0	0
$526$	−39.7990	−1.73532
$527$	24.9706	1.08773
$528$	1.75736	0.0764792
$529$	8.20101	0.356566
$530$	−10.2426	−0.444912
$531$	6.82843	0.296328
$532$	0	0
$533$	7.11270	0.308085
$534$	38.9706	1.68642
$535$	13.8995	0.600928
$536$	−5.17157	−0.223378
$537$	−25.1716	−1.08623
$538$	−9.65685	−0.416337
$539$	0	0
$540$	−21.6569	−0.931963
$541$	−23.1421	−0.994958	−0.497479	−	0.867476i	$-0.665741\pi$
$541$	−23.1421	−0.994958	−0.497479	+	0.867476i	$0.665741\pi$
$542$	−36.7990	−1.58065
$543$	14.0000	0.600798
$544$	6.34315	0.271960
$545$	9.65685	0.413654
$546$	0	0
$547$	−1.55635	−0.0665447	−0.0332723	−	0.999446i	$-0.510593\pi$
$547$	−1.55635	−0.0665447	−0.0332723	+	0.999446i	$0.510593\pi$
$548$	−75.9117	−3.24279
$549$	11.8284	0.504825
$550$	−4.00000	−0.170561
$551$	−6.58579	−0.280564
$552$	−34.8701	−1.48417
$553$	0	0
$554$	54.2132	2.30330
$555$	12.8284	0.544536
$556$	23.2426	0.985708
$557$	15.8284	0.670672	0.335336	−	0.942099i	$-0.391150\pi$
$557$	15.8284	0.670672	0.335336	+	0.942099i	$0.391150\pi$
$558$	15.0711	0.638009
$559$	18.1005	0.765570
$560$	0	0
$561$	2.34315	0.0989277
$562$	68.1838	2.87616
$563$	−5.89949	−0.248634	−0.124317	−	0.992243i	$-0.539674\pi$
$563$	−5.89949	−0.248634	−0.124317	+	0.992243i	$0.539674\pi$
$564$	−23.8995	−1.00635
$565$	0.242641	0.0102080
$566$	72.5980	3.05152
$567$	0	0
$568$	62.8701	2.63797
$569$	−9.02944	−0.378534	−0.189267	−	0.981926i	$-0.560611\pi$
$569$	−9.02944	−0.378534	−0.189267	+	0.981926i	$0.560611\pi$
$570$	−3.41421	−0.143006
$571$	0.899495	0.0376427	0.0188213	−	0.999823i	$-0.494009\pi$
$571$	0.899495	0.0376427	0.0188213	+	0.999823i	$0.494009\pi$
$572$	−3.55635	−0.148698
$573$	−11.8995	−0.497109
$574$	0	0
$575$	22.3431	0.931774
$576$	9.82843	0.409518
$577$	5.97056	0.248558	0.124279	−	0.992247i	$-0.460338\pi$
$577$	5.97056	0.248558	0.124279	+	0.992247i	$0.460338\pi$
$578$	2.41421	0.100418
$579$	−11.1716	−0.464275
$580$	25.2132	1.04692
$581$	0	0
$582$	10.8284	0.448853
$583$	1.75736	0.0727824
$584$	66.8406	2.76588
$585$	2.24264	0.0927218
$586$	7.65685	0.316302
$587$	−14.0000	−0.577842	−0.288921	−	0.957353i	$-0.593296\pi$
$587$	−14.0000	−0.577842	−0.288921	+	0.957353i	$0.593296\pi$
$588$	0	0
$589$	6.24264	0.257224
$590$	−16.4853	−0.678688
$591$	−6.38478	−0.262635
$592$	27.2132	1.11846
$593$	−11.8284	−0.485735	−0.242868	−	0.970059i	$-0.578088\pi$
$593$	−11.8284	−0.485735	−0.242868	+	0.970059i	$0.578088\pi$
$594$	5.65685	0.232104
$595$	0	0
$596$	−0.656854	−0.0269058
$597$	22.7279	0.930192
$598$	30.2426	1.23671
$599$	27.6985	1.13173	0.565865	−	0.824498i	$-0.308542\pi$
$599$	27.6985	1.13173	0.565865	+	0.824498i	$0.308542\pi$
$600$	−24.9706	−1.01942
$601$	34.7279	1.41658	0.708291	−	0.705921i	$-0.249467\pi$
$601$	34.7279	1.41658	0.708291	+	0.705921i	$0.249467\pi$
$602$	0	0
$603$	−1.17157	−0.0477101
$604$	−66.2843	−2.69707
$605$	10.8284	0.440238
$606$	−62.5269	−2.53998
$607$	32.5269	1.32023	0.660113	−	0.751166i	$-0.270508\pi$
$607$	32.5269	1.32023	0.660113	+	0.751166i	$0.270508\pi$
$608$	1.58579	0.0643121
$609$	0	0
$610$	−28.5563	−1.15621
$611$	9.89949	0.400491
$612$	−15.3137	−0.619020
$613$	−31.3137	−1.26475	−0.632374	−	0.774663i	$-0.717920\pi$
$613$	−31.3137	−1.26475	−0.632374	+	0.774663i	$0.717920\pi$
$614$	70.7696	2.85603
$615$	4.48528	0.180864
$616$	0	0
$617$	8.45584	0.340419	0.170210	−	0.985408i	$-0.445555\pi$
$617$	8.45584	0.340419	0.170210	+	0.985408i	$0.445555\pi$
$618$	20.4853	0.824039
$619$	−34.5563	−1.38894	−0.694468	−	0.719523i	$-0.744360\pi$
$619$	−34.5563	−1.38894	−0.694468	+	0.719523i	$0.744360\pi$
$620$	−23.8995	−0.959827
$621$	−31.5980	−1.26798
$622$	−57.4558	−2.30377
$623$	0	0
$624$	−9.51472	−0.380894
$625$	11.0000	0.440000
$626$	−16.0711	−0.642329
$627$	0.585786	0.0233941
$628$	−23.6274	−0.942837
$629$	36.2843	1.44675
$630$	0	0
$631$	−36.5563	−1.45529	−0.727643	−	0.685956i	$-0.759384\pi$
$631$	−36.5563	−1.45529	−0.727643	+	0.685956i	$0.759384\pi$
$632$	−6.24264	−0.248319
$633$	1.65685	0.0658540
$634$	48.8701	1.94088
$635$	−4.24264	−0.168364
$636$	22.9706	0.910842
$637$	0	0
$638$	−6.58579	−0.260734
$639$	14.2426	0.563430
$640$	−20.5563	−0.812561
$641$	−35.7990	−1.41398	−0.706988	−	0.707226i	$-0.749946\pi$
$641$	−35.7990	−1.41398	−0.706988	+	0.707226i	$0.749946\pi$
$642$	−47.4558	−1.87293
$643$	−1.85786	−0.0732670	−0.0366335	−	0.999329i	$-0.511663\pi$
$643$	−1.85786	−0.0732670	−0.0366335	+	0.999329i	$0.511663\pi$
$644$	0	0
$645$	11.4142	0.449434
$646$	−9.65685	−0.379944
$647$	36.2132	1.42369	0.711844	−	0.702338i	$-0.247860\pi$
$647$	36.2132	1.42369	0.711844	+	0.702338i	$0.247860\pi$
$648$	22.0711	0.867033
$649$	2.82843	0.111025
$650$	21.6569	0.849452
$651$	0	0
$652$	15.0416	0.589076
$653$	−30.1421	−1.17955	−0.589776	−	0.807567i	$-0.700784\pi$
$653$	−30.1421	−1.17955	−0.589776	+	0.807567i	$0.700784\pi$
$654$	−32.9706	−1.28925
$655$	3.51472	0.137331
$656$	9.51472	0.371487
$657$	15.1421	0.590751
$658$	0	0
$659$	10.0416	0.391166	0.195583	−	0.980687i	$-0.437340\pi$
$659$	10.0416	0.391166	0.195583	+	0.980687i	$0.437340\pi$
$660$	−2.24264	−0.0872947
$661$	−2.72792	−0.106104	−0.0530519	−	0.998592i	$-0.516895\pi$
$661$	−2.72792	−0.106104	−0.0530519	+	0.998592i	$0.516895\pi$
$662$	75.9411	2.95154
$663$	−12.6863	−0.492695
$664$	40.7990	1.58331
$665$	0	0
$666$	21.8995	0.848588
$667$	36.7868	1.42439
$668$	−50.0416	−1.93617
$669$	3.17157	0.122620
$670$	2.82843	0.109272
$671$	4.89949	0.189143
$672$	0	0
$673$	−3.79899	−0.146440	−0.0732201	−	0.997316i	$-0.523328\pi$
$673$	−3.79899	−0.146440	−0.0732201	+	0.997316i	$0.523328\pi$
$674$	−47.2132	−1.81858
$675$	−22.6274	−0.870930
$676$	−30.5147	−1.17364
$677$	−23.3137	−0.896019	−0.448009	−	0.894029i	$-0.647867\pi$
$677$	−23.3137	−0.896019	−0.448009	+	0.894029i	$0.647867\pi$
$678$	−0.828427	−0.0318156
$679$	0	0
$680$	17.6569	0.677109
$681$	−28.6274	−1.09701
$682$	6.24264	0.239043
$683$	−6.48528	−0.248152	−0.124076	−	0.992273i	$-0.539597\pi$
$683$	−6.48528	−0.248152	−0.124076	+	0.992273i	$0.539597\pi$
$684$	−3.82843	−0.146384
$685$	19.8284	0.757605
$686$	0	0
$687$	17.6569	0.673651
$688$	24.2132	0.923120
$689$	−9.51472	−0.362482
$690$	19.0711	0.726023
$691$	−30.9706	−1.17818	−0.589088	−	0.808069i	$-0.700513\pi$
$691$	−30.9706	−1.17818	−0.589088	+	0.808069i	$0.700513\pi$
$692$	28.0000	1.06440
$693$	0	0
$694$	−65.2843	−2.47816
$695$	−6.07107	−0.230289
$696$	−41.1127	−1.55837
$697$	12.6863	0.480528
$698$	10.8284	0.409862
$699$	−0.686292	−0.0259579
$700$	0	0
$701$	−1.82843	−0.0690587	−0.0345294	−	0.999404i	$-0.510993\pi$
$701$	−1.82843	−0.0690587	−0.0345294	+	0.999404i	$0.510993\pi$
$702$	−30.6274	−1.15596
$703$	9.07107	0.342122
$704$	4.07107	0.153434
$705$	6.24264	0.235111
$706$	1.17157	0.0440927
$707$	0	0
$708$	36.9706	1.38944
$709$	−42.9411	−1.61269	−0.806344	−	0.591447i	$-0.798557\pi$
$709$	−42.9411	−1.61269	−0.806344	+	0.591447i	$0.798557\pi$
$710$	−34.3848	−1.29044
$711$	−1.41421	−0.0530372
$712$	−50.3848	−1.88825
$713$	−34.8701	−1.30589
$714$	0	0
$715$	0.928932	0.0347401
$716$	68.1421	2.54659
$717$	−2.82843	−0.105630
$718$	−12.6569	−0.472350
$719$	−52.0833	−1.94238	−0.971189	−	0.238311i	$-0.923406\pi$
$719$	−52.0833	−1.94238	−0.971189	+	0.238311i	$0.923406\pi$
$720$	3.00000	0.111803
$721$	0	0
$722$	−2.41421	−0.0898477
$723$	−36.8284	−1.36966
$724$	−37.8995	−1.40852
$725$	26.3431	0.978360
$726$	−36.9706	−1.37211
$727$	−20.0711	−0.744395	−0.372197	−	0.928154i	$-0.621396\pi$
$727$	−20.0711	−0.744395	−0.372197	+	0.928154i	$0.621396\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	−36.5563	−1.35301
$731$	32.2843	1.19408
$732$	64.0416	2.36705
$733$	−21.4558	−0.792490	−0.396245	−	0.918145i	$-0.629687\pi$
$733$	−21.4558	−0.792490	−0.396245	+	0.918145i	$0.629687\pi$
$734$	−58.7696	−2.16922
$735$	0	0
$736$	−8.85786	−0.326505
$737$	−0.485281	−0.0178756
$738$	7.65685	0.281853
$739$	−8.34315	−0.306908	−0.153454	−	0.988156i	$-0.549040\pi$
$739$	−8.34315	−0.306908	−0.153454	+	0.988156i	$0.549040\pi$
$740$	−34.7279	−1.27662
$741$	−3.17157	−0.116511
$742$	0	0
$743$	−2.87006	−0.105292	−0.0526461	−	0.998613i	$-0.516766\pi$
$743$	−2.87006	−0.105292	−0.0526461	+	0.998613i	$0.516766\pi$
$744$	38.9706	1.42873
$745$	0.171573	0.00628594
$746$	−27.3137	−1.00003
$747$	9.24264	0.338171
$748$	−6.34315	−0.231928
$749$	0	0
$750$	30.7279	1.12203
$751$	9.45584	0.345049	0.172524	−	0.985005i	$-0.444808\pi$
$751$	9.45584	0.345049	0.172524	+	0.985005i	$0.444808\pi$
$752$	13.2426	0.482909
$753$	−19.4142	−0.707493
$754$	35.6569	1.29855
$755$	17.3137	0.630110
$756$	0	0
$757$	−5.00000	−0.181728	−0.0908640	−	0.995863i	$-0.528963\pi$
$757$	−5.00000	−0.181728	−0.0908640	+	0.995863i	$0.528963\pi$
$758$	−10.2426	−0.372029
$759$	−3.27208	−0.118769
$760$	4.41421	0.160120
$761$	34.1127	1.23658	0.618292	−	0.785948i	$-0.287825\pi$
$761$	34.1127	1.23658	0.618292	+	0.785948i	$0.287825\pi$
$762$	14.4853	0.524746
$763$	0	0
$764$	32.2132	1.16543
$765$	4.00000	0.144620
$766$	47.6985	1.72342
$767$	−15.3137	−0.552946
$768$	42.3848	1.52943
$769$	43.4264	1.56600	0.782998	−	0.622024i	$-0.213689\pi$
$769$	43.4264	1.56600	0.782998	+	0.622024i	$0.213689\pi$
$770$	0	0
$771$	13.1716	0.474363
$772$	30.2426	1.08846
$773$	34.2426	1.23162	0.615811	−	0.787894i	$-0.288828\pi$
$773$	34.2426	1.23162	0.615811	+	0.787894i	$0.288828\pi$
$774$	19.4853	0.700384
$775$	−24.9706	−0.896969
$776$	−14.0000	−0.502571
$777$	0	0
$778$	27.7990	0.996642
$779$	3.17157	0.113633
$780$	12.1421	0.434758
$781$	5.89949	0.211101
$782$	53.9411	1.92893
$783$	−37.2548	−1.33138
$784$	0	0
$785$	6.17157	0.220273
$786$	−12.0000	−0.428026
$787$	−41.0122	−1.46193	−0.730963	−	0.682417i	$-0.760929\pi$
$787$	−41.0122	−1.46193	−0.730963	+	0.682417i	$0.760929\pi$
$788$	17.2843	0.615727
$789$	−23.3137	−0.829990
$790$	3.41421	0.121472
$791$	0	0
$792$	−1.82843	−0.0649703
$793$	−26.5269	−0.941998
$794$	−20.8284	−0.739173
$795$	−6.00000	−0.212798
$796$	−61.5269	−2.18076
$797$	30.2426	1.07125	0.535625	−	0.844456i	$-0.320076\pi$
$797$	30.2426	1.07125	0.535625	+	0.844456i	$0.320076\pi$
$798$	0	0
$799$	17.6569	0.624655
$800$	−6.34315	−0.224264
$801$	−11.4142	−0.403301
$802$	−1.17157	−0.0413697
$803$	6.27208	0.221337
$804$	−6.34315	−0.223706
$805$	0	0
$806$	−33.7990	−1.19052
$807$	−5.65685	−0.199131
$808$	80.8406	2.84396
$809$	31.1421	1.09490	0.547450	−	0.836839i	$-0.315599\pi$
$809$	31.1421	1.09490	0.547450	+	0.836839i	$0.315599\pi$
$810$	−12.0711	−0.424134
$811$	0.544156	0.0191079	0.00955395	−	0.999954i	$-0.496959\pi$
$811$	0.544156	0.0191079	0.00955395	+	0.999954i	$0.496959\pi$
$812$	0	0
$813$	−21.5563	−0.756014
$814$	9.07107	0.317941
$815$	−3.92893	−0.137624
$816$	−16.9706	−0.594089
$817$	8.07107	0.282371
$818$	−6.48528	−0.226753
$819$	0	0
$820$	−12.1421	−0.424022
$821$	10.1127	0.352936	0.176468	−	0.984306i	$-0.443533\pi$
$821$	10.1127	0.352936	0.176468	+	0.984306i	$0.443533\pi$
$822$	−67.6985	−2.36126
$823$	−6.21320	−0.216579	−0.108289	−	0.994119i	$-0.534537\pi$
$823$	−6.21320	−0.216579	−0.108289	+	0.994119i	$0.534537\pi$
$824$	−26.4853	−0.922658
$825$	−2.34315	−0.0815779
$826$	0	0
$827$	12.2843	0.427166	0.213583	−	0.976925i	$-0.431487\pi$
$827$	12.2843	0.427166	0.213583	+	0.976925i	$0.431487\pi$
$828$	21.3848	0.743172
$829$	28.2843	0.982353	0.491177	−	0.871060i	$-0.336567\pi$
$829$	28.2843	0.982353	0.491177	+	0.871060i	$0.336567\pi$
$830$	−22.3137	−0.774520
$831$	31.7574	1.10165
$832$	−22.0416	−0.764156
$833$	0	0
$834$	20.7279	0.717749
$835$	13.0711	0.452343
$836$	−1.58579	−0.0548456
$837$	35.3137	1.22062
$838$	14.6569	0.506313
$839$	19.7990	0.683537	0.341769	−	0.939784i	$-0.388974\pi$
$839$	19.7990	0.683537	0.341769	+	0.939784i	$0.388974\pi$
$840$	0	0
$841$	14.3726	0.495606
$842$	89.8406	3.09611
$843$	39.9411	1.37565
$844$	−4.48528	−0.154390
$845$	7.97056	0.274196
$846$	10.6569	0.366390
$847$	0	0
$848$	−12.7279	−0.437079
$849$	42.5269	1.45952
$850$	38.6274	1.32491
$851$	−50.6690	−1.73691
$852$	77.1127	2.64184
$853$	−43.9706	−1.50552	−0.752762	−	0.658293i	$-0.771279\pi$
$853$	−43.9706	−1.50552	−0.752762	+	0.658293i	$0.771279\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	61.3553	2.09708
$857$	12.9706	0.443066	0.221533	−	0.975153i	$-0.428894\pi$
$857$	12.9706	0.443066	0.221533	+	0.975153i	$0.428894\pi$
$858$	−3.17157	−0.108276
$859$	12.0711	0.411860	0.205930	−	0.978567i	$-0.433978\pi$
$859$	12.0711	0.411860	0.205930	+	0.978567i	$0.433978\pi$
$860$	−30.8995	−1.05366
$861$	0	0
$862$	24.2426	0.825708
$863$	4.58579	0.156102	0.0780510	−	0.996949i	$-0.475130\pi$
$863$	4.58579	0.156102	0.0780510	+	0.996949i	$0.475130\pi$
$864$	8.97056	0.305185
$865$	−7.31371	−0.248674
$866$	67.5980	2.29707
$867$	1.41421	0.0480292
$868$	0	0
$869$	−0.585786	−0.0198714
$870$	22.4853	0.762322
$871$	2.62742	0.0890266
$872$	42.6274	1.44355
$873$	−3.17157	−0.107341
$874$	13.4853	0.456146
$875$	0	0
$876$	81.9828	2.76994
$877$	−27.8995	−0.942099	−0.471050	−	0.882107i	$-0.656125\pi$
$877$	−27.8995	−0.942099	−0.471050	+	0.882107i	$0.656125\pi$
$878$	−59.1127	−1.99496
$879$	4.48528	0.151285
$880$	1.24264	0.0418894
$881$	52.2843	1.76150	0.880751	−	0.473580i	$-0.157038\pi$
$881$	52.2843	1.76150	0.880751	+	0.473580i	$0.157038\pi$
$882$	0	0
$883$	−5.65685	−0.190368	−0.0951842	−	0.995460i	$-0.530344\pi$
$883$	−5.65685	−0.190368	−0.0951842	+	0.995460i	$0.530344\pi$
$884$	34.3431	1.15508
$885$	−9.65685	−0.324612
$886$	−42.6274	−1.43210
$887$	21.3137	0.715644	0.357822	−	0.933790i	$-0.383519\pi$
$887$	21.3137	0.715644	0.357822	+	0.933790i	$0.383519\pi$
$888$	56.6274	1.90029
$889$	0	0
$890$	27.5563	0.923691
$891$	2.07107	0.0693834
$892$	−8.58579	−0.287473
$893$	4.41421	0.147716
$894$	−0.585786	−0.0195916
$895$	−17.7990	−0.594955
$896$	0	0
$897$	17.7157	0.591511
$898$	−23.3137	−0.777989
$899$	−41.1127	−1.37119
$900$	15.3137	0.510457
$901$	−16.9706	−0.565371
$902$	3.17157	0.105602
$903$	0	0
$904$	1.07107	0.0356232
$905$	9.89949	0.329070
$906$	−59.1127	−1.96389
$907$	31.4142	1.04309	0.521546	−	0.853223i	$-0.325356\pi$
$907$	31.4142	1.04309	0.521546	+	0.853223i	$0.325356\pi$
$908$	77.4975	2.57184
$909$	18.3137	0.607427
$910$	0	0
$911$	9.27208	0.307198	0.153599	−	0.988133i	$-0.450914\pi$
$911$	9.27208	0.307198	0.153599	+	0.988133i	$0.450914\pi$
$912$	−4.24264	−0.140488
$913$	3.82843	0.126702
$914$	18.8995	0.625140
$915$	−16.7279	−0.553008
$916$	−47.7990	−1.57932
$917$	0	0
$918$	−54.6274	−1.80297
$919$	−19.0416	−0.628125	−0.314063	−	0.949402i	$-0.601690\pi$
$919$	−19.0416	−0.628125	−0.314063	+	0.949402i	$0.601690\pi$
$920$	−24.6569	−0.812912
$921$	41.4558	1.36602
$922$	72.3553	2.38290
$923$	−31.9411	−1.05135
$924$	0	0
$925$	−36.2843	−1.19302
$926$	9.82843	0.322982
$927$	−6.00000	−0.197066
$928$	−10.4437	−0.342830
$929$	−15.0000	−0.492134	−0.246067	−	0.969253i	$-0.579138\pi$
$929$	−15.0000	−0.492134	−0.246067	+	0.969253i	$0.579138\pi$
$930$	−21.3137	−0.698904
$931$	0	0
$932$	1.85786	0.0608564
$933$	−33.6569	−1.10188
$934$	46.1127	1.50885
$935$	1.65685	0.0541849
$936$	9.89949	0.323575
$937$	33.6863	1.10048	0.550242	−	0.835006i	$-0.314536\pi$
$937$	33.6863	1.10048	0.550242	+	0.835006i	$0.314536\pi$
$938$	0	0
$939$	−9.41421	−0.307221
$940$	−16.8995	−0.551201
$941$	−4.04163	−0.131753	−0.0658767	−	0.997828i	$-0.520984\pi$
$941$	−4.04163	−0.131753	−0.0658767	+	0.997828i	$0.520984\pi$
$942$	−21.0711	−0.686532
$943$	−17.7157	−0.576904
$944$	−20.4853	−0.666739
$945$	0	0
$946$	8.07107	0.262413
$947$	3.65685	0.118832	0.0594159	−	0.998233i	$-0.481076\pi$
$947$	3.65685	0.118832	0.0594159	+	0.998233i	$0.481076\pi$
$948$	−7.65685	−0.248683
$949$	−33.9584	−1.10234
$950$	9.65685	0.313310
$951$	28.6274	0.928308
$952$	0	0
$953$	−3.41421	−0.110597	−0.0552986	−	0.998470i	$-0.517611\pi$
$953$	−3.41421	−0.110597	−0.0552986	+	0.998470i	$0.517611\pi$
$954$	−10.2426	−0.331618
$955$	−8.41421	−0.272278
$956$	7.65685	0.247640
$957$	−3.85786	−0.124707
$958$	−61.2843	−1.98000
$959$	0	0
$960$	−13.8995	−0.448604
$961$	7.97056	0.257115
$962$	−49.1127	−1.58346
$963$	13.8995	0.447905
$964$	99.6985	3.21107
$965$	−7.89949	−0.254294
$966$	0	0
$967$	−36.2843	−1.16682	−0.583412	−	0.812177i	$-0.698283\pi$
$967$	−36.2843	−1.16682	−0.583412	+	0.812177i	$0.698283\pi$
$968$	47.7990	1.53632
$969$	−5.65685	−0.181724
$970$	7.65685	0.245847
$971$	−26.8701	−0.862301	−0.431151	−	0.902280i	$-0.641892\pi$
$971$	−26.8701	−0.862301	−0.431151	+	0.902280i	$0.641892\pi$
$972$	−37.8995	−1.21563
$973$	0	0
$974$	−36.6274	−1.17362
$975$	12.6863	0.406286
$976$	−35.4853	−1.13586
$977$	−44.7696	−1.43230	−0.716152	−	0.697944i	$-0.754098\pi$
$977$	−44.7696	−1.43230	−0.716152	+	0.697944i	$0.754098\pi$
$978$	13.4142	0.428939
$979$	−4.72792	−0.151105
$980$	0	0
$981$	9.65685	0.308320
$982$	3.48528	0.111220
$983$	46.0000	1.46717	0.733586	−	0.679597i	$-0.237845\pi$
$983$	46.0000	1.46717	0.733586	+	0.679597i	$0.237845\pi$
$984$	19.7990	0.631169
$985$	−4.51472	−0.143851
$986$	63.5980	2.02537
$987$	0	0
$988$	8.58579	0.273150
$989$	−45.0833	−1.43356
$990$	1.00000	0.0317821
$991$	3.79899	0.120679	0.0603394	−	0.998178i	$-0.480782\pi$
$991$	3.79899	0.120679	0.0603394	+	0.998178i	$0.480782\pi$
$992$	9.89949	0.314309
$993$	44.4853	1.41170
$994$	0	0
$995$	16.0711	0.509487
$996$	50.0416	1.58563
$997$	28.2843	0.895772	0.447886	−	0.894091i	$-0.352177\pi$
$997$	28.2843	0.895772	0.447886	+	0.894091i	$0.352177\pi$
$998$	−83.4264	−2.64082
$999$	51.3137	1.62349

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	931.2.a.e.1.1		2
3.2	odd	2		8379.2.a.bl.1.2		2
7.2	even	3		931.2.f.i.704.2		4
7.3	odd	6		133.2.f.c.58.2	yes	4
7.4	even	3		931.2.f.i.324.2		4
7.5	odd	6		133.2.f.c.39.2	✓	4
7.6	odd	2		931.2.a.f.1.1		2
21.5	even	6		1197.2.j.e.172.1		4
21.17	even	6		1197.2.j.e.856.1		4
21.20	even	2		8379.2.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
133.2.f.c.39.2	✓	4	7.5	odd	6
133.2.f.c.58.2	yes	4	7.3	odd	6
931.2.a.e.1.1		2	1.1	even	1	trivial
931.2.a.f.1.1		2	7.6	odd	2
931.2.f.i.324.2		4	7.4	even	3
931.2.f.i.704.2		4	7.2	even	3
1197.2.j.e.172.1		4	21.5	even	6
1197.2.j.e.856.1		4	21.17	even	6
8379.2.a.bi.1.2		2	21.20	even	2
8379.2.a.bl.1.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} -1.41421 q^{3} +3.82843 q^{4} -1.00000 q^{5} +3.41421 q^{6} -4.41421 q^{8} -1.00000 q^{9} +2.41421 q^{10} -0.414214 q^{11} -5.41421 q^{12} +2.24264 q^{13} +1.41421 q^{15} +3.00000 q^{16} +4.00000 q^{17} +2.41421 q^{18} +1.00000 q^{19} -3.82843 q^{20} +1.00000 q^{22} -5.58579 q^{23} +6.24264 q^{24} -4.00000 q^{25} -5.41421 q^{26} +5.65685 q^{27} -6.58579 q^{29} -3.41421 q^{30} +6.24264 q^{31} +1.58579 q^{32} +0.585786 q^{33} -9.65685 q^{34} -3.82843 q^{36} +9.07107 q^{37} -2.41421 q^{38} -3.17157 q^{39} +4.41421 q^{40} +3.17157 q^{41} +8.07107 q^{43} -1.58579 q^{44} +1.00000 q^{45} +13.4853 q^{46} +4.41421 q^{47} -4.24264 q^{48} +9.65685 q^{50} -5.65685 q^{51} +8.58579 q^{52} -4.24264 q^{53} -13.6569 q^{54} +0.414214 q^{55} -1.41421 q^{57} +15.8995 q^{58} -6.82843 q^{59} +5.41421 q^{60} -11.8284 q^{61} -15.0711 q^{62} -9.82843 q^{64} -2.24264 q^{65} -1.41421 q^{66} +1.17157 q^{67} +15.3137 q^{68} +7.89949 q^{69} -14.2426 q^{71} +4.41421 q^{72} -15.1421 q^{73} -21.8995 q^{74} +5.65685 q^{75} +3.82843 q^{76} +7.65685 q^{78} +1.41421 q^{79} -3.00000 q^{80} -5.00000 q^{81} -7.65685 q^{82} -9.24264 q^{83} -4.00000 q^{85} -19.4853 q^{86} +9.31371 q^{87} +1.82843 q^{88} +11.4142 q^{89} -2.41421 q^{90} -21.3848 q^{92} -8.82843 q^{93} -10.6569 q^{94} -1.00000 q^{95} -2.24264 q^{96} +3.17157 q^{97} +0.414214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{6} - 6 q^{8} - 2 q^{9} + 2 q^{10} + 2 q^{11} - 8 q^{12} - 4 q^{13} + 6 q^{16} + 8 q^{17} + 2 q^{18} + 2 q^{19} - 2 q^{20} + 2 q^{22} - 14 q^{23} + 4 q^{24} - 8 q^{25}+ \cdots - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^6 - 6 * q^8 - 2 * q^9 + 2 * q^10 + 2 * q^11 - 8 * q^12 - 4 * q^13 + 6 * q^16 + 8 * q^17 + 2 * q^18 + 2 * q^19 - 2 * q^20 + 2 * q^22 - 14 * q^23 + 4 * q^24 - 8 * q^25 - 8 * q^26 - 16 * q^29 - 4 * q^30 + 4 * q^31 + 6 * q^32 + 4 * q^33 - 8 * q^34 - 2 * q^36 + 4 * q^37 - 2 * q^38 - 12 * q^39 + 6 * q^40 + 12 * q^41 + 2 * q^43 - 6 * q^44 + 2 * q^45 + 10 * q^46 + 6 * q^47 + 8 * q^50 + 20 * q^52 - 16 * q^54 - 2 * q^55 + 12 * q^58 - 8 * q^59 + 8 * q^60 - 18 * q^61 - 16 * q^62 - 14 * q^64 + 4 * q^65 + 8 * q^67 + 8 * q^68 - 4 * q^69 - 20 * q^71 + 6 * q^72 - 2 * q^73 - 24 * q^74 + 2 * q^76 + 4 * q^78 - 6 * q^80 - 10 * q^81 - 4 * q^82 - 10 * q^83 - 8 * q^85 - 22 * q^86 - 4 * q^87 - 2 * q^88 + 20 * q^89 - 2 * q^90 - 6 * q^92 - 12 * q^93 - 10 * q^94 - 2 * q^95 + 4 * q^96 + 12 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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