LMFDB - Embedded newform 1127.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1127,2,Mod(1,1127)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1127, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1127.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1127 = 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1127.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$8.99914030780$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 10x^{5} - x^{4} + 29x^{3} + 9x^{2} - 24x - 11 $ x^7 - 10x^5 - x^4 + 29x^3 + 9x^2 - 24x - 11
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 161)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.04500$ of defining polynomial
Character	$\chi$	$=$	1127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.04500 q^{2} +0.886215 q^{3} +2.18203 q^{4} +1.51297 q^{5} -1.81231 q^{6} -0.372248 q^{8} -2.21462 q^{9} -3.09402 q^{10} -3.71824 q^{11} +1.93375 q^{12} +3.46012 q^{13} +1.34081 q^{15} -3.60281 q^{16} +1.88268 q^{17} +4.52890 q^{18} -7.83967 q^{19} +3.30134 q^{20} +7.60380 q^{22} +1.00000 q^{23} -0.329892 q^{24} -2.71093 q^{25} -7.07595 q^{26} -4.62128 q^{27} -8.98292 q^{29} -2.74197 q^{30} -0.0978463 q^{31} +8.11224 q^{32} -3.29516 q^{33} -3.85009 q^{34} -4.83237 q^{36} -7.98903 q^{37} +16.0321 q^{38} +3.06641 q^{39} -0.563199 q^{40} +1.24879 q^{41} +3.32571 q^{43} -8.11330 q^{44} -3.35065 q^{45} -2.04500 q^{46} +6.09232 q^{47} -3.19286 q^{48} +5.54386 q^{50} +1.66846 q^{51} +7.55008 q^{52} -1.23169 q^{53} +9.45052 q^{54} -5.62557 q^{55} -6.94764 q^{57} +18.3701 q^{58} -0.0313522 q^{59} +2.92569 q^{60} -1.87734 q^{61} +0.200096 q^{62} -9.38393 q^{64} +5.23505 q^{65} +6.73860 q^{66} +11.9802 q^{67} +4.10807 q^{68} +0.886215 q^{69} +0.158944 q^{71} +0.824389 q^{72} -13.9659 q^{73} +16.3376 q^{74} -2.40247 q^{75} -17.1064 q^{76} -6.27082 q^{78} -3.07077 q^{79} -5.45093 q^{80} +2.54842 q^{81} -2.55377 q^{82} +12.2989 q^{83} +2.84844 q^{85} -6.80107 q^{86} -7.96080 q^{87} +1.38411 q^{88} -14.2496 q^{89} +6.85208 q^{90} +2.18203 q^{92} -0.0867129 q^{93} -12.4588 q^{94} -11.8612 q^{95} +7.18919 q^{96} -12.9734 q^{97} +8.23449 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 5 q^{3} + 6 q^{4} - 4 q^{5} - 6 q^{6} - 3 q^{8} + 4 q^{9} + 2 q^{10} + 4 q^{11} - 9 q^{12} - 14 q^{13} - 3 q^{15} - 8 q^{16} - 4 q^{17} + 19 q^{18} - 9 q^{19} - 12 q^{20} - 10 q^{22} + 7 q^{23} + 4 q^{24}+ \cdots + 18 q^{99}+O(q^{100}) $ 7 * q - 5 * q^3 + 6 * q^4 - 4 * q^5 - 6 * q^6 - 3 * q^8 + 4 * q^9 + 2 * q^10 + 4 * q^11 - 9 * q^12 - 14 * q^13 - 3 * q^15 - 8 * q^16 - 4 * q^17 + 19 * q^18 - 9 * q^19 - 12 * q^20 - 10 * q^22 + 7 * q^23 + 4 * q^24 + 3 * q^25 - 14 * q^26 - 20 * q^27 - 5 * q^29 - 17 * q^30 - 10 * q^31 + 19 * q^32 - 3 * q^33 - 20 * q^34 + 4 * q^36 - 5 * q^37 + 17 * q^38 + 18 * q^39 - 4 * q^40 - 23 * q^41 - 11 * q^43 - 34 * q^44 - 6 * q^45 - 4 * q^47 - 11 * q^48 - 9 * q^50 + 7 * q^51 - 12 * q^52 + 2 * q^53 - 15 * q^54 - 17 * q^55 - 24 * q^57 - 7 * q^58 - 7 * q^59 + 32 * q^60 - 12 * q^61 + 4 * q^62 - 9 * q^64 + 9 * q^65 + 27 * q^66 + 2 * q^67 + 12 * q^68 - 5 * q^69 - 28 * q^71 - 20 * q^72 - 59 * q^73 + 2 * q^74 + 22 * q^75 - q^76 - 17 * q^78 - 4 * q^79 + 22 * q^80 + 11 * q^81 - 54 * q^82 + 9 * q^83 + 9 * q^85 - 15 * q^86 - 11 * q^87 - 33 * q^88 - q^89 - 2 * q^90 + 6 * q^92 + 15 * q^93 + 14 * q^94 - 16 * q^95 - 20 * q^96 - 35 * q^97 + 18 * 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 $

$2$	−2.04500	−1.44603	−0.723017	−	0.690830i	$-0.757245\pi$
$2$	−2.04500	−1.44603	−0.723017	+	0.690830i	$0.757245\pi$
$3$	0.886215	0.511657	0.255828	−	0.966722i	$-0.417652\pi$
$3$	0.886215	0.511657	0.255828	+	0.966722i	$0.417652\pi$
$4$	2.18203	1.09101
$5$	1.51297	0.676619	0.338310	−	0.941035i	$-0.390145\pi$
$5$	1.51297	0.676619	0.338310	+	0.941035i	$0.390145\pi$
$6$	−1.81231	−0.739873
$7$	0	0
$7$	0	0
$8$	−0.372248	−0.131610
$9$	−2.21462	−0.738207
$10$	−3.09402	−0.978414
$11$	−3.71824	−1.12109	−0.560545	−	0.828124i	$-0.689408\pi$
$11$	−3.71824	−1.12109	−0.560545	+	0.828124i	$0.689408\pi$
$12$	1.93375	0.558225
$13$	3.46012	0.959665	0.479832	−	0.877360i	$-0.340697\pi$
$13$	3.46012	0.959665	0.479832	+	0.877360i	$0.340697\pi$
$14$	0	0
$15$	1.34081	0.346197
$16$	−3.60281	−0.900702
$17$	1.88268	0.456618	0.228309	−	0.973589i	$-0.426680\pi$
$17$	1.88268	0.456618	0.228309	+	0.973589i	$0.426680\pi$
$18$	4.52890	1.06747
$19$	−7.83967	−1.79854	−0.899272	−	0.437389i	$-0.855903\pi$
$19$	−7.83967	−1.79854	−0.899272	+	0.437389i	$0.855903\pi$
$20$	3.30134	0.738201
$21$	0	0
$22$	7.60380	1.62114
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−0.329892	−0.0673389
$25$	−2.71093	−0.542187
$26$	−7.07595	−1.38771
$27$	−4.62128	−0.889365
$28$	0	0
$29$	−8.98292	−1.66809	−0.834043	−	0.551699i	$-0.813980\pi$
$29$	−8.98292	−1.66809	−0.834043	+	0.551699i	$0.813980\pi$
$30$	−2.74197	−0.500612
$31$	−0.0978463	−0.0175737	−0.00878685	−	0.999961i	$-0.502797\pi$
$31$	−0.0978463	−0.0175737	−0.00878685	+	0.999961i	$0.502797\pi$
$32$	8.11224	1.43406
$33$	−3.29516	−0.573614
$34$	−3.85009	−0.660285
$35$	0	0
$36$	−4.83237	−0.805395
$37$	−7.98903	−1.31339	−0.656694	−	0.754157i	$-0.728046\pi$
$37$	−7.98903	−1.31339	−0.656694	+	0.754157i	$0.728046\pi$
$38$	16.0321	2.60076
$39$	3.06641	0.491019
$40$	−0.563199	−0.0890496
$41$	1.24879	0.195028	0.0975139	−	0.995234i	$-0.468911\pi$
$41$	1.24879	0.195028	0.0975139	+	0.995234i	$0.468911\pi$
$42$	0	0
$43$	3.32571	0.507165	0.253583	−	0.967314i	$-0.418391\pi$
$43$	3.32571	0.507165	0.253583	+	0.967314i	$0.418391\pi$
$44$	−8.11330	−1.22313
$45$	−3.35065	−0.499485
$46$	−2.04500	−0.301519
$47$	6.09232	0.888656	0.444328	−	0.895864i	$-0.353443\pi$
$47$	6.09232	0.888656	0.444328	+	0.895864i	$0.353443\pi$
$48$	−3.19286	−0.460850
$49$	0	0
$50$	5.54386	0.784020
$51$	1.66846	0.233632
$52$	7.55008	1.04701
$53$	−1.23169	−0.169185	−0.0845927	−	0.996416i	$-0.526959\pi$
$53$	−1.23169	−0.169185	−0.0845927	+	0.996416i	$0.526959\pi$
$54$	9.45052	1.28605
$55$	−5.62557	−0.758551
$56$	0	0
$57$	−6.94764	−0.920237
$58$	18.3701	2.41211
$59$	−0.0313522	−0.00408170	−0.00204085	−	0.999998i	$-0.500650\pi$
$59$	−0.0313522	−0.00408170	−0.00204085	+	0.999998i	$0.500650\pi$
$60$	2.92569	0.377705
$61$	−1.87734	−0.240369	−0.120184	−	0.992752i	$-0.538349\pi$
$61$	−1.87734	−0.240369	−0.120184	+	0.992752i	$0.538349\pi$
$62$	0.200096	0.0254122
$63$	0	0
$64$	−9.38393	−1.17299
$65$	5.23505	0.649328
$66$	6.73860	0.829465
$67$	11.9802	1.46361	0.731806	−	0.681513i	$-0.238678\pi$
$67$	11.9802	1.46361	0.731806	+	0.681513i	$0.238678\pi$
$68$	4.10807	0.498177
$69$	0.886215	0.106688
$70$	0	0
$71$	0.158944	0.0188632	0.00943162	−	0.999956i	$-0.496998\pi$
$71$	0.158944	0.0188632	0.00943162	+	0.999956i	$0.496998\pi$
$72$	0.824389	0.0971552
$73$	−13.9659	−1.63458	−0.817290	−	0.576226i	$-0.804525\pi$
$73$	−13.9659	−1.63458	−0.817290	+	0.576226i	$0.804525\pi$
$74$	16.3376	1.89920
$75$	−2.40247	−0.277413
$76$	−17.1064	−1.96224
$77$	0	0
$78$	−6.27082	−0.710030
$79$	−3.07077	−0.345489	−0.172745	−	0.984967i	$-0.555264\pi$
$79$	−3.07077	−0.345489	−0.172745	+	0.984967i	$0.555264\pi$
$80$	−5.45093	−0.609432
$81$	2.54842	0.283158
$82$	−2.55377	−0.282017
$83$	12.2989	1.34998	0.674989	−	0.737828i	$-0.264148\pi$
$83$	12.2989	1.34998	0.674989	+	0.737828i	$0.264148\pi$
$84$	0	0
$85$	2.84844	0.308956
$86$	−6.80107	−0.733378
$87$	−7.96080	−0.853488
$88$	1.38411	0.147546
$89$	−14.2496	−1.51045	−0.755225	−	0.655465i	$-0.772473\pi$
$89$	−14.2496	−1.51045	−0.755225	+	0.655465i	$0.772473\pi$
$90$	6.85208	0.722273
$91$	0	0
$92$	2.18203	0.227492
$93$	−0.0867129	−0.00899171
$94$	−12.4588	−1.28503
$95$	−11.8612	−1.21693
$96$	7.18919	0.733744
$97$	−12.9734	−1.31725	−0.658625	−	0.752471i	$-0.728862\pi$
$97$	−12.9734	−1.31725	−0.658625	+	0.752471i	$0.728862\pi$
$98$	0	0
$99$	8.23449	0.827598
$100$	−5.91533	−0.591533
$101$	−5.68802	−0.565979	−0.282990	−	0.959123i	$-0.591326\pi$
$101$	−5.68802	−0.565979	−0.282990	+	0.959123i	$0.591326\pi$
$102$	−3.41201	−0.337839
$103$	−5.18929	−0.511316	−0.255658	−	0.966767i	$-0.582292\pi$
$103$	−5.18929	−0.511316	−0.255658	+	0.966767i	$0.582292\pi$
$104$	−1.28802	−0.126301
$105$	0	0
$106$	2.51880	0.244648
$107$	5.77421	0.558214	0.279107	−	0.960260i	$-0.409962\pi$
$107$	5.77421	0.558214	0.279107	+	0.960260i	$0.409962\pi$
$108$	−10.0838	−0.970310
$109$	−6.02633	−0.577218	−0.288609	−	0.957447i	$-0.593193\pi$
$109$	−6.02633	−0.577218	−0.288609	+	0.957447i	$0.593193\pi$
$110$	11.5043	1.09689
$111$	−7.08000	−0.672004
$112$	0	0
$113$	8.44589	0.794522	0.397261	−	0.917706i	$-0.369961\pi$
$113$	8.44589	0.794522	0.397261	+	0.917706i	$0.369961\pi$
$114$	14.2079	1.33069
$115$	1.51297	0.141085
$116$	−19.6010	−1.81991
$117$	−7.66286	−0.708432
$118$	0.0641152	0.00590228
$119$	0	0
$120$	−0.499115	−0.0455628
$121$	2.82529	0.256845
$122$	3.83916	0.347581
$123$	1.10669	0.0997873
$124$	−0.213503	−0.0191732
$125$	−11.6664	−1.04347
$126$	0	0
$127$	0.416632	0.0369701	0.0184851	−	0.999829i	$-0.494116\pi$
$127$	0.416632	0.0369701	0.0184851	+	0.999829i	$0.494116\pi$
$128$	2.96565	0.262129
$129$	2.94729	0.259495
$130$	−10.7057	−0.938950
$131$	15.0765	1.31724	0.658618	−	0.752477i	$-0.271141\pi$
$131$	15.0765	1.31724	0.658618	+	0.752477i	$0.271141\pi$
$132$	−7.19013	−0.625821
$133$	0	0
$134$	−24.4995	−2.11643
$135$	−6.99184	−0.601762
$136$	−0.700825	−0.0600953
$137$	−0.149166	−0.0127441	−0.00637204	−	0.999980i	$-0.502028\pi$
$137$	−0.149166	−0.0127441	−0.00637204	+	0.999980i	$0.502028\pi$
$138$	−1.81231	−0.154274
$139$	−19.6184	−1.66401	−0.832005	−	0.554768i	$-0.812807\pi$
$139$	−19.6184	−1.66401	−0.832005	+	0.554768i	$0.812807\pi$
$140$	0	0
$141$	5.39910	0.454687
$142$	−0.325042	−0.0272769
$143$	−12.8656	−1.07587
$144$	7.97886	0.664905
$145$	−13.5909	−1.12866
$146$	28.5602	2.36366
$147$	0	0
$148$	−17.4323	−1.43292
$149$	9.63713	0.789504	0.394752	−	0.918788i	$-0.370830\pi$
$149$	9.63713	0.789504	0.394752	+	0.918788i	$0.370830\pi$
$150$	4.91305	0.401149
$151$	−17.5582	−1.42887	−0.714434	−	0.699703i	$-0.753316\pi$
$151$	−17.5582	−1.42887	−0.714434	+	0.699703i	$0.753316\pi$
$152$	2.91830	0.236706
$153$	−4.16943	−0.337079
$154$	0	0
$155$	−0.148038	−0.0118907
$156$	6.69100	0.535709
$157$	−22.0414	−1.75909	−0.879546	−	0.475813i	$-0.842154\pi$
$157$	−22.0414	−1.75909	−0.879546	+	0.475813i	$0.842154\pi$
$158$	6.27974	0.499589
$159$	−1.09154	−0.0865649
$160$	12.2736	0.970309
$161$	0	0
$162$	−5.21152	−0.409456
$163$	16.2174	1.27025	0.635123	−	0.772411i	$-0.280949\pi$
$163$	16.2174	1.27025	0.635123	+	0.772411i	$0.280949\pi$
$164$	2.72489	0.212778
$165$	−4.98546	−0.388118
$166$	−25.1512	−1.95211
$167$	−16.6608	−1.28925	−0.644624	−	0.764499i	$-0.722986\pi$
$167$	−16.6608	−1.28925	−0.644624	+	0.764499i	$0.722986\pi$
$168$	0	0
$169$	−1.02756	−0.0790434
$170$	−5.82506	−0.446761
$171$	17.3619	1.32770
$172$	7.25679	0.553325
$173$	−3.72382	−0.283117	−0.141559	−	0.989930i	$-0.545211\pi$
$173$	−3.72382	−0.283117	−0.141559	+	0.989930i	$0.545211\pi$
$174$	16.2798	1.23417
$175$	0	0
$176$	13.3961	1.00977
$177$	−0.0277848	−0.00208843
$178$	29.1404	2.18416
$179$	−18.1757	−1.35852	−0.679259	−	0.733899i	$-0.737699\pi$
$179$	−18.1757	−1.35852	−0.679259	+	0.733899i	$0.737699\pi$
$180$	−7.31121	−0.544946
$181$	9.63548	0.716200	0.358100	−	0.933683i	$-0.383425\pi$
$181$	9.63548	0.716200	0.358100	+	0.933683i	$0.383425\pi$
$182$	0	0
$183$	−1.66373	−0.122986
$184$	−0.372248	−0.0274425
$185$	−12.0871	−0.888663
$186$	0.177328	0.0130023
$187$	−7.00026	−0.511910
$188$	13.2936	0.969536
$189$	0	0
$190$	24.2561	1.75972
$191$	14.3720	1.03992	0.519961	−	0.854190i	$-0.325947\pi$
$191$	14.3720	1.03992	0.519961	+	0.854190i	$0.325947\pi$
$192$	−8.31618	−0.600169
$193$	10.9713	0.789730	0.394865	−	0.918739i	$-0.370791\pi$
$193$	10.9713	0.789730	0.394865	+	0.918739i	$0.370791\pi$
$194$	26.5306	1.90479
$195$	4.63938	0.332233
$196$	0	0
$197$	5.10847	0.363964	0.181982	−	0.983302i	$-0.441749\pi$
$197$	5.10847	0.363964	0.181982	+	0.983302i	$0.441749\pi$
$198$	−16.8395	−1.19673
$199$	4.49584	0.318702	0.159351	−	0.987222i	$-0.449060\pi$
$199$	4.49584	0.318702	0.159351	+	0.987222i	$0.449060\pi$
$200$	1.00914	0.0713570
$201$	10.6170	0.748867
$202$	11.6320	0.818425
$203$	0	0
$204$	3.64063	0.254895
$205$	1.88937	0.131960
$206$	10.6121	0.739380
$207$	−2.21462	−0.153927
$208$	−12.4662	−0.864372
$209$	29.1498	2.01633
$210$	0	0
$211$	2.58550	0.177993	0.0889965	−	0.996032i	$-0.471634\pi$
$211$	2.58550	0.177993	0.0889965	+	0.996032i	$0.471634\pi$
$212$	−2.68758	−0.184584
$213$	0.140859	0.00965150
$214$	−11.8083	−0.807196
$215$	5.03168	0.343158
$216$	1.72026	0.117049
$217$	0	0
$218$	12.3239	0.834677
$219$	−12.3768	−0.836344
$220$	−12.2751	−0.827590
$221$	6.51431	0.438200
$222$	14.4786	0.971740
$223$	−5.65862	−0.378929	−0.189465	−	0.981888i	$-0.560675\pi$
$223$	−5.65862	−0.378929	−0.189465	+	0.981888i	$0.560675\pi$
$224$	0	0
$225$	6.00369	0.400246
$226$	−17.2718	−1.14891
$227$	10.5592	0.700841	0.350421	−	0.936592i	$-0.386039\pi$
$227$	10.5592	0.700841	0.350421	+	0.936592i	$0.386039\pi$
$228$	−15.1599	−1.00399
$229$	11.5726	0.764741	0.382371	−	0.924009i	$-0.375108\pi$
$229$	11.5726	0.764741	0.382371	+	0.924009i	$0.375108\pi$
$230$	−3.09402	−0.204013
$231$	0	0
$232$	3.34388	0.219536
$233$	−10.9110	−0.714805	−0.357403	−	0.933950i	$-0.616338\pi$
$233$	−10.9110	−0.714805	−0.357403	+	0.933950i	$0.616338\pi$
$234$	15.6706	1.02442
$235$	9.21747	0.601281
$236$	−0.0684113	−0.00445320
$237$	−2.72137	−0.176772
$238$	0	0
$239$	−17.8912	−1.15729	−0.578644	−	0.815580i	$-0.696418\pi$
$239$	−17.8912	−1.15729	−0.578644	+	0.815580i	$0.696418\pi$
$240$	−4.83070	−0.311820
$241$	−3.34537	−0.215494	−0.107747	−	0.994178i	$-0.534364\pi$
$241$	−3.34537	−0.215494	−0.107747	+	0.994178i	$0.534364\pi$
$242$	−5.77772	−0.371406
$243$	16.1223	1.03424
$244$	−4.09641	−0.262246
$245$	0	0
$246$	−2.26319	−0.144296
$247$	−27.1262	−1.72600
$248$	0.0364231	0.00231287
$249$	10.8995	0.690725
$250$	23.8578	1.50890
$251$	19.2286	1.21370	0.606851	−	0.794816i	$-0.292433\pi$
$251$	19.2286	1.21370	0.606851	+	0.794816i	$0.292433\pi$
$252$	0	0
$253$	−3.71824	−0.233764
$254$	−0.852013	−0.0534601
$255$	2.52433	0.158080
$256$	12.7031	0.793943
$257$	−8.79403	−0.548556	−0.274278	−	0.961650i	$-0.588439\pi$
$257$	−8.79403	−0.548556	−0.274278	+	0.961650i	$0.588439\pi$
$258$	−6.02721	−0.375238
$259$	0	0
$260$	11.4230	0.708426
$261$	19.8938	1.23139
$262$	−30.8314	−1.90477
$263$	10.5709	0.651832	0.325916	−	0.945399i	$-0.394327\pi$
$263$	10.5709	0.651832	0.325916	+	0.945399i	$0.394327\pi$
$264$	1.22662	0.0754930
$265$	−1.86350	−0.114474
$266$	0	0
$267$	−12.6282	−0.772832
$268$	26.1411	1.59682
$269$	30.8612	1.88164	0.940821	−	0.338904i	$-0.110056\pi$
$269$	30.8612	1.88164	0.940821	+	0.338904i	$0.110056\pi$
$270$	14.2983	0.870168
$271$	10.3192	0.626844	0.313422	−	0.949614i	$-0.398525\pi$
$271$	10.3192	0.626844	0.313422	+	0.949614i	$0.398525\pi$
$272$	−6.78295	−0.411277
$273$	0	0
$274$	0.305044	0.0184284
$275$	10.0799	0.607840
$276$	1.93375	0.116398
$277$	−21.3471	−1.28262	−0.641310	−	0.767282i	$-0.721609\pi$
$277$	−21.3471	−1.28262	−0.641310	+	0.767282i	$0.721609\pi$
$278$	40.1196	2.40622
$279$	0.216693	0.0129730
$280$	0	0
$281$	5.64761	0.336908	0.168454	−	0.985709i	$-0.446122\pi$
$281$	5.64761	0.336908	0.168454	+	0.985709i	$0.446122\pi$
$282$	−11.0412	−0.657492
$283$	−16.5416	−0.983294	−0.491647	−	0.870795i	$-0.663605\pi$
$283$	−16.5416	−0.983294	−0.491647	+	0.870795i	$0.663605\pi$
$284$	0.346821	0.0205801
$285$	−10.5115	−0.622650
$286$	26.3101	1.55575
$287$	0	0
$288$	−17.9656	−1.05863
$289$	−13.4555	−0.791500
$290$	27.7933	1.63208
$291$	−11.4972	−0.673980
$292$	−30.4739	−1.78335
$293$	15.8804	0.927744	0.463872	−	0.885902i	$-0.346460\pi$
$293$	15.8804	0.927744	0.463872	+	0.885902i	$0.346460\pi$
$294$	0	0
$295$	−0.0474348	−0.00276176
$296$	2.97390	0.172854
$297$	17.1830	0.997059
$298$	−19.7079	−1.14165
$299$	3.46012	0.200104
$300$	−5.24226	−0.302662
$301$	0	0
$302$	35.9066	2.06619
$303$	−5.04081	−0.289587
$304$	28.2448	1.61995
$305$	−2.84035	−0.162638
$306$	8.52649	0.487427
$307$	14.4841	0.826650	0.413325	−	0.910584i	$-0.364367\pi$
$307$	14.4841	0.826650	0.413325	+	0.910584i	$0.364367\pi$
$308$	0	0
$309$	−4.59883	−0.261618
$310$	0.302738	0.0171944
$311$	−13.1047	−0.743101	−0.371550	−	0.928413i	$-0.621174\pi$
$311$	−13.1047	−0.743101	−0.371550	+	0.928413i	$0.621174\pi$
$312$	−1.14147	−0.0646228
$313$	15.3284	0.866414	0.433207	−	0.901294i	$-0.357382\pi$
$313$	15.3284	0.866414	0.433207	+	0.901294i	$0.357382\pi$
$314$	45.0746	2.54371
$315$	0	0
$316$	−6.70052	−0.376934
$317$	32.7405	1.83889	0.919446	−	0.393217i	$-0.128638\pi$
$317$	32.7405	1.83889	0.919446	+	0.393217i	$0.128638\pi$
$318$	2.23220	0.125176
$319$	33.4006	1.87008
$320$	−14.1976	−0.793668
$321$	5.11719	0.285614
$322$	0	0
$323$	−14.7596	−0.821247
$324$	5.56072	0.308929
$325$	−9.38016	−0.520317
$326$	−33.1646	−1.83682
$327$	−5.34063	−0.295338
$328$	−0.464859	−0.0256675
$329$	0	0
$330$	10.1953	0.561232
$331$	31.8302	1.74954	0.874772	−	0.484534i	$-0.161011\pi$
$331$	31.8302	1.74954	0.874772	+	0.484534i	$0.161011\pi$
$332$	26.8365	1.47285
$333$	17.6927	0.969553
$334$	34.0713	1.86430
$335$	18.1256	0.990307
$336$	0	0
$337$	30.6355	1.66882	0.834410	−	0.551144i	$-0.185809\pi$
$337$	30.6355	1.66882	0.834410	+	0.551144i	$0.185809\pi$
$338$	2.10137	0.114299
$339$	7.48487	0.406523
$340$	6.21537	0.337076
$341$	0.363816	0.0197017
$342$	−35.5051	−1.91990
$343$	0	0
$344$	−1.23799	−0.0667478
$345$	1.34081	0.0721870
$346$	7.61522	0.409397
$347$	−13.8571	−0.743888	−0.371944	−	0.928255i	$-0.621309\pi$
$347$	−13.8571	−0.743888	−0.371944	+	0.928255i	$0.621309\pi$
$348$	−17.3707	−0.931167
$349$	−22.8620	−1.22377	−0.611887	−	0.790945i	$-0.709589\pi$
$349$	−22.8620	−1.22377	−0.611887	+	0.790945i	$0.709589\pi$
$350$	0	0
$351$	−15.9902	−0.853493
$352$	−30.1632	−1.60771
$353$	−8.29661	−0.441584	−0.220792	−	0.975321i	$-0.570864\pi$
$353$	−8.29661	−0.441584	−0.220792	+	0.975321i	$0.570864\pi$
$354$	0.0568199	0.00301994
$355$	0.240478	0.0127632
$356$	−31.0929	−1.64792
$357$	0	0
$358$	37.1694	1.96446
$359$	25.7354	1.35826	0.679131	−	0.734017i	$-0.262357\pi$
$359$	25.7354	1.35826	0.679131	+	0.734017i	$0.262357\pi$
$360$	1.24727	0.0657370
$361$	42.4605	2.23476
$362$	−19.7046	−1.03565
$363$	2.50382	0.131416
$364$	0	0
$365$	−21.1299	−1.10599
$366$	3.40232	0.177842
$367$	−1.23382	−0.0644049	−0.0322024	−	0.999481i	$-0.510252\pi$
$367$	−1.23382	−0.0644049	−0.0322024	+	0.999481i	$0.510252\pi$
$368$	−3.60281	−0.187809
$369$	−2.76559	−0.143971
$370$	24.7182	1.28504
$371$	0	0
$372$	−0.189210	−0.00981008
$373$	−33.9548	−1.75811	−0.879055	−	0.476720i	$-0.841826\pi$
$373$	−33.9548	−1.75811	−0.879055	+	0.476720i	$0.841826\pi$
$374$	14.3155	0.740239
$375$	−10.3389	−0.533900
$376$	−2.26785	−0.116956
$377$	−31.0820	−1.60080
$378$	0	0
$379$	−25.2824	−1.29867	−0.649335	−	0.760502i	$-0.724953\pi$
$379$	−25.2824	−1.29867	−0.649335	+	0.760502i	$0.724953\pi$
$380$	−25.8814	−1.32769
$381$	0.369226	0.0189160
$382$	−29.3907	−1.50376
$383$	29.9889	1.53236	0.766180	−	0.642626i	$-0.222155\pi$
$383$	29.9889	1.53236	0.766180	+	0.642626i	$0.222155\pi$
$384$	2.62821	0.134120
$385$	0	0
$386$	−22.4363	−1.14198
$387$	−7.36518	−0.374393
$388$	−28.3084	−1.43714
$389$	−22.0442	−1.11768	−0.558842	−	0.829274i	$-0.688754\pi$
$389$	−22.0442	−1.11768	−0.558842	+	0.829274i	$0.688754\pi$
$390$	−9.48753	−0.480420
$391$	1.88268	0.0952114
$392$	0	0
$393$	13.3610	0.673973
$394$	−10.4468	−0.526304
$395$	−4.64598	−0.233765
$396$	17.9679	0.902921
$397$	−13.7288	−0.689031	−0.344515	−	0.938781i	$-0.611957\pi$
$397$	−13.7288	−0.689031	−0.344515	+	0.938781i	$0.611957\pi$
$398$	−9.19401	−0.460854
$399$	0	0
$400$	9.76697	0.488349
$401$	−5.81298	−0.290286	−0.145143	−	0.989411i	$-0.546364\pi$
$401$	−5.81298	−0.290286	−0.145143	+	0.989411i	$0.546364\pi$
$402$	−21.7118	−1.08289
$403$	−0.338560	−0.0168649
$404$	−12.4114	−0.617491
$405$	3.85567	0.191590
$406$	0	0
$407$	29.7051	1.47243
$408$	−0.621082	−0.0307482
$409$	29.5294	1.46014	0.730068	−	0.683375i	$-0.239488\pi$
$409$	29.5294	1.46014	0.730068	+	0.683375i	$0.239488\pi$
$410$	−3.86377	−0.190818
$411$	−0.132193	−0.00652059
$412$	−11.3232	−0.557853
$413$	0	0
$414$	4.52890	0.222584
$415$	18.6078	0.913421
$416$	28.0693	1.37621
$417$	−17.3861	−0.851402
$418$	−59.6113	−2.91568
$419$	−14.6409	−0.715255	−0.357628	−	0.933864i	$-0.616414\pi$
$419$	−14.6409	−0.715255	−0.357628	+	0.933864i	$0.616414\pi$
$420$	0	0
$421$	−4.25571	−0.207411	−0.103705	−	0.994608i	$-0.533070\pi$
$421$	−4.25571	−0.207411	−0.103705	+	0.994608i	$0.533070\pi$
$422$	−5.28734	−0.257384
$423$	−13.4922	−0.656012
$424$	0.458494	0.0222664
$425$	−5.10383	−0.247572
$426$	−0.288057	−0.0139564
$427$	0	0
$428$	12.5995	0.609019
$429$	−11.4016	−0.550477
$430$	−10.2898	−0.496218
$431$	−0.652698	−0.0314393	−0.0157197	−	0.999876i	$-0.505004\pi$
$431$	−0.652698	−0.0314393	−0.0157197	+	0.999876i	$0.505004\pi$
$432$	16.6496	0.801053
$433$	6.56790	0.315633	0.157817	−	0.987468i	$-0.449555\pi$
$433$	6.56790	0.315633	0.157817	+	0.987468i	$0.449555\pi$
$434$	0	0
$435$	−12.0444	−0.577486
$436$	−13.1496	−0.629753
$437$	−7.83967	−0.375022
$438$	25.3105	1.20938
$439$	14.0130	0.668805	0.334403	−	0.942430i	$-0.391465\pi$
$439$	14.0130	0.668805	0.334403	+	0.942430i	$0.391465\pi$
$440$	2.09411	0.0998326
$441$	0	0
$442$	−13.3218	−0.633652
$443$	37.1004	1.76269	0.881347	−	0.472469i	$-0.156637\pi$
$443$	37.1004	1.76269	0.881347	+	0.472469i	$0.156637\pi$
$444$	−15.4488	−0.733165
$445$	−21.5591	−1.02200
$446$	11.5719	0.547944
$447$	8.54057	0.403955
$448$	0	0
$449$	4.57195	0.215764	0.107882	−	0.994164i	$-0.465593\pi$
$449$	4.57195	0.215764	0.107882	+	0.994164i	$0.465593\pi$
$450$	−12.2776	−0.578770
$451$	−4.64329	−0.218644
$452$	18.4292	0.866835
$453$	−15.5604	−0.731089
$454$	−21.5937	−1.01344
$455$	0	0
$456$	2.58625	0.121112
$457$	−23.1292	−1.08194	−0.540969	−	0.841043i	$-0.681942\pi$
$457$	−23.1292	−1.08194	−0.540969	+	0.841043i	$0.681942\pi$
$458$	−23.6660	−1.10584
$459$	−8.70040	−0.406100
$460$	3.30134	0.153926
$461$	−21.5238	−1.00246	−0.501232	−	0.865313i	$-0.667120\pi$
$461$	−21.5238	−1.00246	−0.501232	+	0.865313i	$0.667120\pi$
$462$	0	0
$463$	3.69564	0.171751	0.0858754	−	0.996306i	$-0.472631\pi$
$463$	3.69564	0.171751	0.0858754	+	0.996306i	$0.472631\pi$
$464$	32.3638	1.50245
$465$	−0.131194	−0.00608396
$466$	22.3131	1.03363
$467$	−19.7442	−0.913654	−0.456827	−	0.889556i	$-0.651014\pi$
$467$	−19.7442	−0.913654	−0.456827	+	0.889556i	$0.651014\pi$
$468$	−16.7206	−0.772909
$469$	0	0
$470$	−18.8497	−0.869473
$471$	−19.5334	−0.900052
$472$	0.0116708	0.000537191 0
$473$	−12.3658	−0.568578
$474$	5.56520	0.255618
$475$	21.2528	0.975147
$476$	0	0
$477$	2.72773	0.124894
$478$	36.5876	1.67348
$479$	−36.8915	−1.68561	−0.842807	−	0.538216i	$-0.819098\pi$
$479$	−36.8915	−1.68561	−0.842807	+	0.538216i	$0.819098\pi$
$480$	10.8770	0.496465
$481$	−27.6430	−1.26041
$482$	6.84128	0.311612
$483$	0	0
$484$	6.16486	0.280221
$485$	−19.6283	−0.891277
$486$	−32.9701	−1.49555
$487$	21.5693	0.977399	0.488699	−	0.872452i	$-0.337471\pi$
$487$	21.5693	0.977399	0.488699	+	0.872452i	$0.337471\pi$
$488$	0.698836	0.0316348
$489$	14.3721	0.649930
$490$	0	0
$491$	−3.74105	−0.168831	−0.0844156	−	0.996431i	$-0.526902\pi$
$491$	−3.74105	−0.168831	−0.0844156	+	0.996431i	$0.526902\pi$
$492$	2.41484	0.108869
$493$	−16.9120	−0.761678
$494$	55.4731	2.49585
$495$	12.4585	0.559968
$496$	0.352521	0.0158287
$497$	0	0
$498$	−22.2894	−0.998812
$499$	−27.6275	−1.23678	−0.618388	−	0.785873i	$-0.712214\pi$
$499$	−27.6275	−1.23678	−0.618388	+	0.785873i	$0.712214\pi$
$500$	−25.4564	−1.13844
$501$	−14.7650	−0.659653
$502$	−39.3226	−1.75505
$503$	−35.6054	−1.58757	−0.793783	−	0.608201i	$-0.791891\pi$
$503$	−35.6054	−1.58757	−0.793783	+	0.608201i	$0.791891\pi$
$504$	0	0
$505$	−8.60578	−0.382952
$506$	7.60380	0.338030
$507$	−0.910643	−0.0404431
$508$	0.909103	0.0403349
$509$	7.65827	0.339447	0.169723	−	0.985492i	$-0.445713\pi$
$509$	7.65827	0.339447	0.169723	+	0.985492i	$0.445713\pi$
$510$	−5.16225	−0.228588
$511$	0	0
$512$	−31.9091	−1.41020
$513$	36.2293	1.59956
$514$	17.9838	0.793231
$515$	−7.85122	−0.345966
$516$	6.43107	0.283112
$517$	−22.6527	−0.996264
$518$	0	0
$519$	−3.30011	−0.144859
$520$	−1.94874	−0.0854577
$521$	−11.2253	−0.491788	−0.245894	−	0.969297i	$-0.579082\pi$
$521$	−11.2253	−0.491788	−0.245894	+	0.969297i	$0.579082\pi$
$522$	−40.6828	−1.78064
$523$	0.135161	0.00591018	0.00295509	−	0.999996i	$-0.499059\pi$
$523$	0.135161	0.00591018	0.00295509	+	0.999996i	$0.499059\pi$
$524$	32.8973	1.43712
$525$	0	0
$526$	−21.6176	−0.942572
$527$	−0.184214	−0.00802447
$528$	11.8718	0.516655
$529$	1.00000	0.0434783
$530$	3.81087	0.165533
$531$	0.0694332	0.00301315
$532$	0	0
$533$	4.32095	0.187161
$534$	25.8246	1.11754
$535$	8.73618	0.377698
$536$	−4.45960	−0.192625
$537$	−16.1076	−0.695094
$538$	−63.1112	−2.72092
$539$	0	0
$540$	−15.2564	−0.656530
$541$	−3.81157	−0.163872	−0.0819360	−	0.996638i	$-0.526110\pi$
$541$	−3.81157	−0.163872	−0.0819360	+	0.996638i	$0.526110\pi$
$542$	−21.1027	−0.906437
$543$	8.53911	0.366448
$544$	15.2728	0.654815
$545$	−9.11764	−0.390557
$546$	0	0
$547$	−10.7277	−0.458683	−0.229341	−	0.973346i	$-0.573657\pi$
$547$	−10.7277	−0.458683	−0.229341	+	0.973346i	$0.573657\pi$
$548$	−0.325483	−0.0139040
$549$	4.15760	0.177442
$550$	−20.6134	−0.878958
$551$	70.4232	3.00013
$552$	−0.329892	−0.0140411
$553$	0	0
$554$	43.6547	1.85471
$555$	−10.7118	−0.454690
$556$	−42.8079	−1.81546
$557$	31.3859	1.32986	0.664931	−	0.746904i	$-0.268461\pi$
$557$	31.3859	1.32986	0.664931	+	0.746904i	$0.268461\pi$
$558$	−0.443136	−0.0187595
$559$	11.5073	0.486709
$560$	0	0
$561$	−6.20374	−0.261922
$562$	−11.5494	−0.487181
$563$	29.6495	1.24958	0.624789	−	0.780793i	$-0.285185\pi$
$563$	29.6495	1.24958	0.624789	+	0.780793i	$0.285185\pi$
$564$	11.7810	0.496070
$565$	12.7783	0.537589
$566$	33.8275	1.42188
$567$	0	0
$568$	−0.0591668	−0.00248258
$569$	15.6224	0.654927	0.327464	−	0.944864i	$-0.393806\pi$
$569$	15.6224	0.654927	0.327464	+	0.944864i	$0.393806\pi$
$570$	21.4961	0.900373
$571$	8.37962	0.350676	0.175338	−	0.984508i	$-0.443898\pi$
$571$	8.37962	0.350676	0.175338	+	0.984508i	$0.443898\pi$
$572$	−28.0730	−1.17379
$573$	12.7367	0.532083
$574$	0	0
$575$	−2.71093	−0.113054
$576$	20.7819	0.865911
$577$	4.32594	0.180091	0.0900457	−	0.995938i	$-0.471299\pi$
$577$	4.32594	0.180091	0.0900457	+	0.995938i	$0.471299\pi$
$578$	27.5165	1.14454
$579$	9.72291	0.404070
$580$	−29.6556	−1.23138
$581$	0	0
$582$	23.5119	0.974598
$583$	4.57971	0.189672
$584$	5.19877	0.215127
$585$	−11.5937	−0.479338
$586$	−32.4755	−1.34155
$587$	8.87385	0.366263	0.183132	−	0.983088i	$-0.441377\pi$
$587$	8.87385	0.366263	0.183132	+	0.983088i	$0.441377\pi$
$588$	0	0
$589$	0.767083	0.0316071
$590$	0.0970042	0.00399360
$591$	4.52721	0.186224
$592$	28.7829	1.18297
$593$	16.4941	0.677332	0.338666	−	0.940907i	$-0.390024\pi$
$593$	16.4941	0.677332	0.338666	+	0.940907i	$0.390024\pi$
$594$	−35.1393	−1.44178
$595$	0	0
$596$	21.0285	0.861360
$597$	3.98429	0.163066
$598$	−7.07595	−0.289357
$599$	29.4201	1.20207	0.601036	−	0.799222i	$-0.294755\pi$
$599$	29.4201	1.20207	0.601036	+	0.799222i	$0.294755\pi$
$600$	0.894315	0.0365103
$601$	−41.9446	−1.71095	−0.855477	−	0.517841i	$-0.826736\pi$
$601$	−41.9446	−1.71095	−0.855477	+	0.517841i	$0.826736\pi$
$602$	0	0
$603$	−26.5316	−1.08045
$604$	−38.3125	−1.55891
$605$	4.27457	0.173786
$606$	10.3085	0.418753
$607$	−17.1819	−0.697393	−0.348696	−	0.937236i	$-0.613376\pi$
$607$	−17.1819	−0.697393	−0.348696	+	0.937236i	$0.613376\pi$
$608$	−63.5973	−2.57921
$609$	0	0
$610$	5.80852	0.235180
$611$	21.0802	0.852812
$612$	−9.09782	−0.367758
$613$	−27.2893	−1.10221	−0.551103	−	0.834437i	$-0.685793\pi$
$613$	−27.2893	−1.10221	−0.551103	+	0.834437i	$0.685793\pi$
$614$	−29.6200	−1.19536
$615$	1.67439	0.0675180
$616$	0	0
$617$	28.5612	1.14983	0.574915	−	0.818213i	$-0.305035\pi$
$617$	28.5612	1.14983	0.574915	+	0.818213i	$0.305035\pi$
$618$	9.40461	0.378309
$619$	23.3187	0.937258	0.468629	−	0.883395i	$-0.344748\pi$
$619$	23.3187	0.937258	0.468629	+	0.883395i	$0.344748\pi$
$620$	−0.323023	−0.0129729
$621$	−4.62128	−0.185446
$622$	26.7992	1.07455
$623$	0	0
$624$	−11.0477	−0.442262
$625$	−4.09618	−0.163847
$626$	−31.3467	−1.25286
$627$	25.8330	1.03167
$628$	−48.0949	−1.91920
$629$	−15.0408	−0.599716
$630$	0	0
$631$	−8.32633	−0.331466	−0.165733	−	0.986171i	$-0.552999\pi$
$631$	−8.32633	−0.331466	−0.165733	+	0.986171i	$0.552999\pi$
$632$	1.14309	0.0454697
$633$	2.29131	0.0910713
$634$	−66.9544	−2.65910
$635$	0.630350	0.0250147
$636$	−2.38177	−0.0944435
$637$	0	0
$638$	−68.3043	−2.70419
$639$	−0.352002	−0.0139250
$640$	4.48693	0.177361
$641$	−12.7773	−0.504675	−0.252337	−	0.967639i	$-0.581199\pi$
$641$	−12.7773	−0.504675	−0.252337	+	0.967639i	$0.581199\pi$
$642$	−10.4647	−0.413007
$643$	44.8214	1.76759	0.883793	−	0.467879i	$-0.154982\pi$
$643$	44.8214	1.76759	0.883793	+	0.467879i	$0.154982\pi$
$644$	0	0
$645$	4.45915	0.175579
$646$	30.1834	1.18755
$647$	9.13291	0.359052	0.179526	−	0.983753i	$-0.442544\pi$
$647$	9.13291	0.359052	0.179526	+	0.983753i	$0.442544\pi$
$648$	−0.948645	−0.0372663
$649$	0.116575	0.00457596
$650$	19.1824	0.752397
$651$	0	0
$652$	35.3869	1.38586
$653$	−11.4982	−0.449961	−0.224980	−	0.974363i	$-0.572232\pi$
$653$	−11.4982	−0.449961	−0.224980	+	0.974363i	$0.572232\pi$
$654$	10.9216	0.427068
$655$	22.8102	0.891268
$656$	−4.49914	−0.175662
$657$	30.9291	1.20666
$658$	0	0
$659$	−32.6074	−1.27021	−0.635103	−	0.772428i	$-0.719042\pi$
$659$	−32.6074	−1.27021	−0.635103	+	0.772428i	$0.719042\pi$
$660$	−10.8784	−0.423442
$661$	2.19097	0.0852189	0.0426095	−	0.999092i	$-0.486433\pi$
$661$	2.19097	0.0852189	0.0426095	+	0.999092i	$0.486433\pi$
$662$	−65.0927	−2.52990
$663$	5.77308	0.224208
$664$	−4.57824	−0.177670
$665$	0	0
$666$	−36.1815	−1.40201
$667$	−8.98292	−0.347820
$668$	−36.3543	−1.40659
$669$	−5.01475	−0.193882
$670$	−37.0669	−1.43202
$671$	6.98040	0.269475
$672$	0	0
$673$	−3.08634	−0.118969	−0.0594847	−	0.998229i	$-0.518946\pi$
$673$	−3.08634	−0.118969	−0.0594847	+	0.998229i	$0.518946\pi$
$674$	−62.6495	−2.41317
$675$	12.5280	0.482202
$676$	−2.24217	−0.0862374
$677$	11.8218	0.454348	0.227174	−	0.973854i	$-0.427051\pi$
$677$	11.8218	0.454348	0.227174	+	0.973854i	$0.427051\pi$
$678$	−15.3066	−0.587845
$679$	0	0
$680$	−1.06033	−0.0406616
$681$	9.35776	0.358590
$682$	−0.744003	−0.0284894
$683$	−15.7473	−0.602554	−0.301277	−	0.953537i	$-0.597413\pi$
$683$	−15.7473	−0.602554	−0.301277	+	0.953537i	$0.597413\pi$
$684$	37.8842	1.44854
$685$	−0.225682	−0.00862288
$686$	0	0
$687$	10.2558	0.391285
$688$	−11.9819	−0.456805
$689$	−4.26179	−0.162361
$690$	−2.74197	−0.104385
$691$	22.0929	0.840455	0.420227	−	0.907419i	$-0.361950\pi$
$691$	22.0929	0.840455	0.420227	+	0.907419i	$0.361950\pi$
$692$	−8.12549	−0.308885
$693$	0	0
$694$	28.3378	1.07569
$695$	−29.6820	−1.12590
$696$	2.96339	0.112327
$697$	2.35107	0.0890532
$698$	46.7528	1.76962
$699$	−9.66952	−0.365735
$700$	0	0
$701$	2.71631	0.102594	0.0512969	−	0.998683i	$-0.483665\pi$
$701$	2.71631	0.102594	0.0512969	+	0.998683i	$0.483665\pi$
$702$	32.6999	1.23418
$703$	62.6314	2.36219
$704$	34.8917	1.31503
$705$	8.16866	0.307650
$706$	16.9666	0.638545
$707$	0	0
$708$	−0.0606272	−0.00227851
$709$	19.2639	0.723471	0.361735	−	0.932281i	$-0.382184\pi$
$709$	19.2639	0.723471	0.361735	+	0.932281i	$0.382184\pi$
$710$	−0.491777	−0.0184561
$711$	6.80061	0.255043
$712$	5.30437	0.198790
$713$	−0.0978463	−0.00366437
$714$	0	0
$715$	−19.4651	−0.727955
$716$	−39.6599	−1.48216
$717$	−15.8555	−0.592134
$718$	−52.6289	−1.96409
$719$	−5.01738	−0.187117	−0.0935583	−	0.995614i	$-0.529824\pi$
$719$	−5.01738	−0.187117	−0.0935583	+	0.995614i	$0.529824\pi$
$720$	12.0717	0.449888
$721$	0	0
$722$	−86.8317	−3.23154
$723$	−2.96472	−0.110259
$724$	21.0249	0.781384
$725$	24.3521	0.904414
$726$	−5.12030	−0.190032
$727$	−16.1209	−0.597892	−0.298946	−	0.954270i	$-0.596635\pi$
$727$	−16.1209	−0.597892	−0.298946	+	0.954270i	$0.596635\pi$
$728$	0	0
$729$	6.64255	0.246021
$730$	43.2106	1.59930
$731$	6.26125	0.231581
$732$	−3.63030	−0.134180
$733$	−7.74265	−0.285981	−0.142991	−	0.989724i	$-0.545672\pi$
$733$	−7.74265	−0.285981	−0.142991	+	0.989724i	$0.545672\pi$
$734$	2.52316	0.0931317
$735$	0	0
$736$	8.11224	0.299021
$737$	−44.5451	−1.64084
$738$	5.65564	0.208187
$739$	8.40107	0.309038	0.154519	−	0.987990i	$-0.450617\pi$
$739$	8.40107	0.309038	0.154519	+	0.987990i	$0.450617\pi$
$740$	−26.3745	−0.969544
$741$	−24.0397	−0.883119
$742$	0	0
$743$	−32.4828	−1.19168	−0.595839	−	0.803104i	$-0.703180\pi$
$743$	−32.4828	−1.19168	−0.595839	+	0.803104i	$0.703180\pi$
$744$	0.0322787	0.00118339
$745$	14.5807	0.534194
$746$	69.4375	2.54229
$747$	−27.2374	−0.996564
$748$	−15.2748	−0.558501
$749$	0	0
$750$	21.1431	0.772037
$751$	−3.43943	−0.125506	−0.0627532	−	0.998029i	$-0.519988\pi$
$751$	−3.43943	−0.125506	−0.0627532	+	0.998029i	$0.519988\pi$
$752$	−21.9495	−0.800414
$753$	17.0407	0.620998
$754$	63.5627	2.31482
$755$	−26.5650	−0.966799
$756$	0	0
$757$	49.6866	1.80589	0.902946	−	0.429755i	$-0.141400\pi$
$757$	49.6866	1.80589	0.902946	+	0.429755i	$0.141400\pi$
$758$	51.7026	1.87792
$759$	−3.29516	−0.119607
$760$	4.41530	0.160160
$761$	−48.9256	−1.77355	−0.886776	−	0.462199i	$-0.847061\pi$
$761$	−48.9256	−1.77355	−0.886776	+	0.462199i	$0.847061\pi$
$762$	−0.755067	−0.0273532
$763$	0	0
$764$	31.3601	1.13457
$765$	−6.30821	−0.228074
$766$	−61.3273	−2.21584
$767$	−0.108482	−0.00391707
$768$	11.2577	0.406226
$769$	34.2752	1.23600	0.617998	−	0.786179i	$-0.287944\pi$
$769$	34.2752	1.23600	0.617998	+	0.786179i	$0.287944\pi$
$770$	0	0
$771$	−7.79340	−0.280673
$772$	23.9396	0.861606
$773$	−13.8297	−0.497421	−0.248710	−	0.968578i	$-0.580007\pi$
$773$	−13.8297	−0.497421	−0.248710	+	0.968578i	$0.580007\pi$
$774$	15.0618	0.541385
$775$	0.265255	0.00952823
$776$	4.82933	0.173363
$777$	0	0
$778$	45.0803	1.61621
$779$	−9.79008	−0.350766
$780$	10.1233	0.362471
$781$	−0.590993	−0.0211474
$782$	−3.85009	−0.137679
$783$	41.5126	1.48354
$784$	0	0
$785$	−33.3479	−1.19024
$786$	−27.3232	−0.974588
$787$	−17.6159	−0.627939	−0.313969	−	0.949433i	$-0.601659\pi$
$787$	−17.6159	−0.627939	−0.313969	+	0.949433i	$0.601659\pi$
$788$	11.1468	0.397089
$789$	9.36813	0.333514
$790$	9.50103	0.338031
$791$	0	0
$792$	−3.06527	−0.108920
$793$	−6.49582	−0.230673
$794$	28.0755	0.996362
$795$	−1.65147	−0.0585714
$796$	9.81006	0.347708
$797$	32.3522	1.14598	0.572988	−	0.819564i	$-0.305784\pi$
$797$	32.3522	1.14598	0.572988	+	0.819564i	$0.305784\pi$
$798$	0	0
$799$	11.4699	0.405776
$800$	−21.9917	−0.777526
$801$	31.5574	1.11503
$802$	11.8875	0.419764
$803$	51.9284	1.83251
$804$	23.1666	0.817024
$805$	0	0
$806$	0.692355	0.0243872
$807$	27.3497	0.962755
$808$	2.11735	0.0744883
$809$	15.0748	0.530003	0.265002	−	0.964248i	$-0.414627\pi$
$809$	15.0748	0.530003	0.265002	+	0.964248i	$0.414627\pi$
$810$	−7.88486	−0.277046
$811$	−36.2016	−1.27121	−0.635605	−	0.772014i	$-0.719249\pi$
$811$	−36.2016	−1.27121	−0.635605	+	0.772014i	$0.719249\pi$
$812$	0	0
$813$	9.14499	0.320729
$814$	−60.7469	−2.12918
$815$	24.5364	0.859473
$816$	−6.01115	−0.210432
$817$	−26.0725	−0.912160
$818$	−60.3877	−2.11141
$819$	0	0
$820$	4.12267	0.143970
$821$	−8.73365	−0.304807	−0.152403	−	0.988318i	$-0.548701\pi$
$821$	−8.73365	−0.304807	−0.152403	+	0.988318i	$0.548701\pi$
$822$	0.270334	0.00942899
$823$	49.4933	1.72523	0.862613	−	0.505864i	$-0.168826\pi$
$823$	49.4933	1.72523	0.862613	+	0.505864i	$0.168826\pi$
$824$	1.93170	0.0672941
$825$	8.93296	0.311006
$826$	0	0
$827$	26.4116	0.918423	0.459211	−	0.888327i	$-0.348132\pi$
$827$	26.4116	0.918423	0.459211	+	0.888327i	$0.348132\pi$
$828$	−4.83237	−0.167936
$829$	39.2206	1.36219	0.681093	−	0.732197i	$-0.261505\pi$
$829$	39.2206	1.36219	0.681093	+	0.732197i	$0.261505\pi$
$830$	−38.0530	−1.32084
$831$	−18.9181	−0.656261
$832$	−32.4695	−1.12568
$833$	0	0
$834$	35.5546	1.23116
$835$	−25.2072	−0.872330
$836$	63.6056	2.19985
$837$	0.452175	0.0156294
$838$	29.9407	1.03428
$839$	14.8912	0.514101	0.257050	−	0.966398i	$-0.417249\pi$
$839$	14.8912	0.514101	0.257050	+	0.966398i	$0.417249\pi$
$840$	0	0
$841$	51.6929	1.78251
$842$	8.70293	0.299923
$843$	5.00500	0.172381
$844$	5.64163	0.194193
$845$	−1.55467	−0.0534823
$846$	27.5915	0.948616
$847$	0	0
$848$	4.43754	0.152386
$849$	−14.6594	−0.503109
$850$	10.4373	0.357998
$851$	−7.98903	−0.273860
$852$	0.307358	0.0105299
$853$	42.1641	1.44367	0.721836	−	0.692064i	$-0.243298\pi$
$853$	42.1641	1.44367	0.721836	+	0.692064i	$0.243298\pi$
$854$	0	0
$855$	26.2680	0.898346
$856$	−2.14944	−0.0734663
$857$	−45.4482	−1.55248	−0.776240	−	0.630438i	$-0.782875\pi$
$857$	−45.4482	−1.55248	−0.776240	+	0.630438i	$0.782875\pi$
$858$	23.3164	0.796008
$859$	0.718209	0.0245050	0.0122525	−	0.999925i	$-0.496100\pi$
$859$	0.718209	0.0245050	0.0122525	+	0.999925i	$0.496100\pi$
$860$	10.9793	0.374390
$861$	0	0
$862$	1.33477	0.0454623
$863$	−24.3255	−0.828050	−0.414025	−	0.910265i	$-0.635877\pi$
$863$	−24.3255	−0.828050	−0.414025	+	0.910265i	$0.635877\pi$
$864$	−37.4889	−1.27540
$865$	−5.63402	−0.191562
$866$	−13.4314	−0.456416
$867$	−11.9245	−0.404976
$868$	0	0
$869$	11.4179	0.387325
$870$	24.6309	0.835064
$871$	41.4529	1.40458
$872$	2.24329	0.0759674
$873$	28.7312	0.972404
$874$	16.0321	0.542295
$875$	0	0
$876$	−27.0064	−0.912463
$877$	40.0789	1.35337	0.676684	−	0.736274i	$-0.263416\pi$
$877$	40.0789	1.35337	0.676684	+	0.736274i	$0.263416\pi$
$878$	−28.6566	−0.967115
$879$	14.0735	0.474686
$880$	20.2678	0.683229
$881$	−35.8620	−1.20822	−0.604111	−	0.796901i	$-0.706471\pi$
$881$	−35.8620	−1.20822	−0.604111	+	0.796901i	$0.706471\pi$
$882$	0	0
$883$	27.0684	0.910925	0.455463	−	0.890255i	$-0.349474\pi$
$883$	27.0684	0.910925	0.455463	+	0.890255i	$0.349474\pi$
$884$	14.2144	0.478083
$885$	−0.0420374	−0.00141307
$886$	−75.8704	−2.54892
$887$	−57.1807	−1.91994	−0.959970	−	0.280102i	$-0.909632\pi$
$887$	−57.1807	−1.91994	−0.959970	+	0.280102i	$0.909632\pi$
$888$	2.63552	0.0884421
$889$	0	0
$890$	44.0884	1.47785
$891$	−9.47563	−0.317446
$892$	−12.3473	−0.413417
$893$	−47.7618	−1.59829
$894$	−17.4655	−0.584133
$895$	−27.4993	−0.919199
$896$	0	0
$897$	3.06641	0.102385
$898$	−9.34964	−0.312002
$899$	0.878945	0.0293145
$900$	13.1002	0.436674
$901$	−2.31888	−0.0772531
$902$	9.49553	0.316166
$903$	0	0
$904$	−3.14397	−0.104567
$905$	14.5782	0.484595
$906$	31.8209	1.05718
$907$	15.9753	0.530451	0.265225	−	0.964186i	$-0.414554\pi$
$907$	15.9753	0.530451	0.265225	+	0.964186i	$0.414554\pi$
$908$	23.0406	0.764628
$909$	12.5968	0.417810
$910$	0	0
$911$	−12.2417	−0.405587	−0.202793	−	0.979222i	$-0.565002\pi$
$911$	−12.2417	−0.405587	−0.202793	+	0.979222i	$0.565002\pi$
$912$	25.0310	0.828860
$913$	−45.7302	−1.51345
$914$	47.2992	1.56452
$915$	−2.51716	−0.0832149
$916$	25.2518	0.834344
$917$	0	0
$918$	17.7923	0.587235
$919$	13.0071	0.429064	0.214532	−	0.976717i	$-0.431177\pi$
$919$	13.0071	0.429064	0.214532	+	0.976717i	$0.431177\pi$
$920$	−0.563199	−0.0185681
$921$	12.8360	0.422961
$922$	44.0162	1.44960
$923$	0.549967	0.0181024
$924$	0	0
$925$	21.6577	0.712101
$926$	−7.55758	−0.248358
$927$	11.4923	0.377457
$928$	−72.8717	−2.39213
$929$	−44.7855	−1.46936	−0.734682	−	0.678412i	$-0.762668\pi$
$929$	−44.7855	−1.46936	−0.734682	+	0.678412i	$0.762668\pi$
$930$	0.268291	0.00879761
$931$	0	0
$932$	−23.8082	−0.779863
$933$	−11.6136	−0.380212
$934$	40.3770	1.32117
$935$	−10.5912	−0.346368
$936$	2.85249	0.0932364
$937$	−31.7090	−1.03589	−0.517943	−	0.855415i	$-0.673302\pi$
$937$	−31.7090	−1.03589	−0.517943	+	0.855415i	$0.673302\pi$
$938$	0	0
$939$	13.5843	0.443307
$940$	20.1128	0.656007
$941$	−15.0224	−0.489717	−0.244858	−	0.969559i	$-0.578741\pi$
$941$	−15.0224	−0.489717	−0.244858	+	0.969559i	$0.578741\pi$
$942$	39.9458	1.30151
$943$	1.24879	0.0406661
$944$	0.112956	0.00367640
$945$	0	0
$946$	25.2880	0.822184
$947$	−28.2856	−0.919159	−0.459580	−	0.888137i	$-0.652000\pi$
$947$	−28.2856	−0.919159	−0.459580	+	0.888137i	$0.652000\pi$
$948$	−5.93810	−0.192861
$949$	−48.3236	−1.56865
$950$	−43.4621	−1.41010
$951$	29.0152	0.940881
$952$	0	0
$953$	−21.1837	−0.686209	−0.343104	−	0.939297i	$-0.611478\pi$
$953$	−21.1837	−0.686209	−0.343104	+	0.939297i	$0.611478\pi$
$954$	−5.57820	−0.180601
$955$	21.7443	0.703631
$956$	−39.0392	−1.26262
$957$	29.6002	0.956837
$958$	75.4431	2.43746
$959$	0	0
$960$	−12.5821	−0.406086
$961$	−30.9904	−0.999691
$962$	56.5299	1.82260
$963$	−12.7877	−0.412078
$964$	−7.29969	−0.235107
$965$	16.5992	0.534346
$966$	0	0
$967$	−30.9508	−0.995310	−0.497655	−	0.867375i	$-0.665805\pi$
$967$	−30.9508	−0.995310	−0.497655	+	0.867375i	$0.665805\pi$
$968$	−1.05171	−0.0338032
$969$	−13.0802	−0.420197
$970$	40.1400	1.28882
$971$	−50.9647	−1.63554	−0.817768	−	0.575548i	$-0.804789\pi$
$971$	−50.9647	−1.63554	−0.817768	+	0.575548i	$0.804789\pi$
$972$	35.1793	1.12838
$973$	0	0
$974$	−44.1093	−1.41335
$975$	−8.31284	−0.266224
$976$	6.76370	0.216501
$977$	0.373632	0.0119535	0.00597677	−	0.999982i	$-0.498098\pi$
$977$	0.373632	0.0119535	0.00597677	+	0.999982i	$0.498098\pi$
$978$	−29.3910	−0.939821
$979$	52.9833	1.69335
$980$	0	0
$981$	13.3461	0.426107
$982$	7.65045	0.244136
$983$	−16.7820	−0.535262	−0.267631	−	0.963521i	$-0.586241\pi$
$983$	−16.7820	−0.535262	−0.267631	+	0.963521i	$0.586241\pi$
$984$	−0.411965	−0.0131330
$985$	7.72895	0.246265
$986$	34.5851	1.10141
$987$	0	0
$988$	−59.1902	−1.88309
$989$	3.32571	0.105751
$990$	−25.4777	−0.809733
$991$	−22.5556	−0.716504	−0.358252	−	0.933625i	$-0.616627\pi$
$991$	−22.5556	−0.716504	−0.358252	+	0.933625i	$0.616627\pi$
$992$	−0.793753	−0.0252017
$993$	28.2084	0.895166
$994$	0	0
$995$	6.80206	0.215640
$996$	23.7829	0.753591
$997$	−27.3439	−0.865990	−0.432995	−	0.901396i	$-0.642543\pi$
$997$	−27.3439	−0.865990	−0.432995	+	0.901396i	$0.642543\pi$
$998$	56.4982	1.78842
$999$	36.9195	1.16808

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1127.2.a.k.1.2		7
7.3	odd	6		161.2.e.a.93.6	✓	14
7.5	odd	6		161.2.e.a.116.6	yes	14
7.6	odd	2		1127.2.a.n.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.e.a.93.6	✓	14	7.3	odd	6
161.2.e.a.116.6	yes	14	7.5	odd	6
1127.2.a.k.1.2		7	1.1	even	1	trivial
1127.2.a.n.1.2		7	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1127 = 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1127.a (trivial)

\(f(q)\)	\(=\)	\(q-2.04500 q^{2} +0.886215 q^{3} +2.18203 q^{4} +1.51297 q^{5} -1.81231 q^{6} -0.372248 q^{8} -2.21462 q^{9} -3.09402 q^{10} -3.71824 q^{11} +1.93375 q^{12} +3.46012 q^{13} +1.34081 q^{15} -3.60281 q^{16} +1.88268 q^{17} +4.52890 q^{18} -7.83967 q^{19} +3.30134 q^{20} +7.60380 q^{22} +1.00000 q^{23} -0.329892 q^{24} -2.71093 q^{25} -7.07595 q^{26} -4.62128 q^{27} -8.98292 q^{29} -2.74197 q^{30} -0.0978463 q^{31} +8.11224 q^{32} -3.29516 q^{33} -3.85009 q^{34} -4.83237 q^{36} -7.98903 q^{37} +16.0321 q^{38} +3.06641 q^{39} -0.563199 q^{40} +1.24879 q^{41} +3.32571 q^{43} -8.11330 q^{44} -3.35065 q^{45} -2.04500 q^{46} +6.09232 q^{47} -3.19286 q^{48} +5.54386 q^{50} +1.66846 q^{51} +7.55008 q^{52} -1.23169 q^{53} +9.45052 q^{54} -5.62557 q^{55} -6.94764 q^{57} +18.3701 q^{58} -0.0313522 q^{59} +2.92569 q^{60} -1.87734 q^{61} +0.200096 q^{62} -9.38393 q^{64} +5.23505 q^{65} +6.73860 q^{66} +11.9802 q^{67} +4.10807 q^{68} +0.886215 q^{69} +0.158944 q^{71} +0.824389 q^{72} -13.9659 q^{73} +16.3376 q^{74} -2.40247 q^{75} -17.1064 q^{76} -6.27082 q^{78} -3.07077 q^{79} -5.45093 q^{80} +2.54842 q^{81} -2.55377 q^{82} +12.2989 q^{83} +2.84844 q^{85} -6.80107 q^{86} -7.96080 q^{87} +1.38411 q^{88} -14.2496 q^{89} +6.85208 q^{90} +2.18203 q^{92} -0.0867129 q^{93} -12.4588 q^{94} -11.8612 q^{95} +7.18919 q^{96} -12.9734 q^{97} +8.23449 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 5 q^{3} + 6 q^{4} - 4 q^{5} - 6 q^{6} - 3 q^{8} + 4 q^{9} + 2 q^{10} + 4 q^{11} - 9 q^{12} - 14 q^{13} - 3 q^{15} - 8 q^{16} - 4 q^{17} + 19 q^{18} - 9 q^{19} - 12 q^{20} - 10 q^{22} + 7 q^{23} + 4 q^{24}+ \cdots + 18 q^{99}+O(q^{100}) \) 7 * q - 5 * q^3 + 6 * q^4 - 4 * q^5 - 6 * q^6 - 3 * q^8 + 4 * q^9 + 2 * q^10 + 4 * q^11 - 9 * q^12 - 14 * q^13 - 3 * q^15 - 8 * q^16 - 4 * q^17 + 19 * q^18 - 9 * q^19 - 12 * q^20 - 10 * q^22 + 7 * q^23 + 4 * q^24 + 3 * q^25 - 14 * q^26 - 20 * q^27 - 5 * q^29 - 17 * q^30 - 10 * q^31 + 19 * q^32 - 3 * q^33 - 20 * q^34 + 4 * q^36 - 5 * q^37 + 17 * q^38 + 18 * q^39 - 4 * q^40 - 23 * q^41 - 11 * q^43 - 34 * q^44 - 6 * q^45 - 4 * q^47 - 11 * q^48 - 9 * q^50 + 7 * q^51 - 12 * q^52 + 2 * q^53 - 15 * q^54 - 17 * q^55 - 24 * q^57 - 7 * q^58 - 7 * q^59 + 32 * q^60 - 12 * q^61 + 4 * q^62 - 9 * q^64 + 9 * q^65 + 27 * q^66 + 2 * q^67 + 12 * q^68 - 5 * q^69 - 28 * q^71 - 20 * q^72 - 59 * q^73 + 2 * q^74 + 22 * q^75 - q^76 - 17 * q^78 - 4 * q^79 + 22 * q^80 + 11 * q^81 - 54 * q^82 + 9 * q^83 + 9 * q^85 - 15 * q^86 - 11 * q^87 - 33 * q^88 - q^89 - 2 * q^90 + 6 * q^92 + 15 * q^93 + 14 * q^94 - 16 * q^95 - 20 * q^96 - 35 * q^97 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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