LMFDB - Embedded newform 1127.4.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1127,4,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, 4, names="a")

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

chi := DirichletCharacter("1127.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1127 = 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
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:	$66.4951525765$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.334189.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 16x^{2} - 5x + 4 $ x^4 - 2x^3 - 16x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 23)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.83969$ 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)

$f(q)$	$=$	\(q+2.86845 q^{2} -3.43737 q^{3} +0.228032 q^{4} +17.9704 q^{5} -9.85995 q^{6} -22.2935 q^{8} -15.1845 q^{9} +51.5473 q^{10} +26.7049 q^{11} -0.783832 q^{12} +14.4956 q^{13} -61.7710 q^{15} -65.7723 q^{16} -24.7016 q^{17} -43.5560 q^{18} -94.6224 q^{19} +4.09784 q^{20} +76.6018 q^{22} -23.0000 q^{23} +76.6312 q^{24} +197.935 q^{25} +41.5801 q^{26} +145.004 q^{27} -57.5965 q^{29} -177.187 q^{30} -88.8691 q^{31} -10.3165 q^{32} -91.7948 q^{33} -70.8553 q^{34} -3.46255 q^{36} -305.467 q^{37} -271.420 q^{38} -49.8269 q^{39} -400.624 q^{40} +179.205 q^{41} -96.5826 q^{43} +6.08959 q^{44} -272.871 q^{45} -65.9745 q^{46} -218.484 q^{47} +226.084 q^{48} +567.769 q^{50} +84.9085 q^{51} +3.30548 q^{52} -519.174 q^{53} +415.937 q^{54} +479.898 q^{55} +325.252 q^{57} -165.213 q^{58} +37.2884 q^{59} -14.0858 q^{60} -96.3052 q^{61} -254.917 q^{62} +496.586 q^{64} +260.493 q^{65} -263.309 q^{66} +497.552 q^{67} -5.63276 q^{68} +79.0596 q^{69} -19.6235 q^{71} +338.515 q^{72} +208.235 q^{73} -876.219 q^{74} -680.378 q^{75} -21.5770 q^{76} -142.926 q^{78} -446.200 q^{79} -1181.95 q^{80} -88.4513 q^{81} +514.042 q^{82} -501.151 q^{83} -443.897 q^{85} -277.043 q^{86} +197.981 q^{87} -595.347 q^{88} -1102.82 q^{89} -782.718 q^{90} -5.24475 q^{92} +305.476 q^{93} -626.711 q^{94} -1700.40 q^{95} +35.4615 q^{96} -1814.37 q^{97} -405.500 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 7 q^{3} + 20 q^{4} - 14 q^{5} + 17 q^{6} - 63 q^{8} - 33 q^{9} + 70 q^{10} + 8 q^{11} + 67 q^{12} - 111 q^{13} + 10 q^{15} + 64 q^{16} - 98 q^{17} + 49 q^{18} - 96 q^{19} - 140 q^{20} + 220 q^{22}+ \cdots - 1498 q^{99}+O(q^{100}) $ 4 * q + 2 * q^2 - 7 * q^3 + 20 * q^4 - 14 * q^5 + 17 * q^6 - 63 * q^8 - 33 * q^9 + 70 * q^10 + 8 * q^11 + 67 * q^12 - 111 * q^13 + 10 * q^15 + 64 * q^16 - 98 * q^17 + 49 * q^18 - 96 * q^19 - 140 * q^20 + 220 * q^22 - 92 * q^23 + 188 * q^24 + 184 * q^25 + 229 * q^26 + 155 * q^27 + 21 * q^29 - 406 * q^30 + 193 * q^31 - 432 * q^32 + 418 * q^33 - 666 * q^34 - 629 * q^36 + 170 * q^37 - 748 * q^38 - 291 * q^39 + 26 * q^40 + 125 * q^41 + 2 * q^43 + 830 * q^44 - 168 * q^45 - 46 * q^46 + 677 * q^47 - 551 * q^48 + 414 * q^50 - 340 * q^51 - 2247 * q^52 - 230 * q^53 - 641 * q^54 + 972 * q^55 + 1322 * q^57 - 1835 * q^58 + 1140 * q^59 - 804 * q^60 - 754 * q^61 - 443 * q^62 - 805 * q^64 + 1318 * q^65 + 398 * q^66 + 488 * q^67 - 284 * q^68 + 161 * q^69 - 401 * q^71 + 1503 * q^72 - 1509 * q^73 + 1366 * q^74 - 1401 * q^75 + 3832 * q^76 - 1907 * q^78 - 838 * q^79 - 2846 * q^80 - 932 * q^81 + 949 * q^82 - 142 * q^83 + 112 * q^85 + 918 * q^86 - 2223 * q^87 - 404 * q^88 - 2342 * q^89 - 1784 * q^90 - 460 * q^92 - 509 * q^93 - 1567 * q^94 - 956 * q^95 - 799 * q^96 - 1062 * q^97 - 1498 * 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 $

$2$	2.86845	1.01415	0.507076	−	0.861901i	$-0.330726\pi$
$2$	2.86845	1.01415	0.507076	+	0.861901i	$0.330726\pi$
$3$	−3.43737	−0.661523	−0.330761	−	0.943714i	$-0.607306\pi$
$3$	−3.43737	−0.661523	−0.330761	+	0.943714i	$0.607306\pi$
$4$	0.228032	0.0285041
$5$	17.9704	1.60732	0.803661	−	0.595087i	$-0.202883\pi$
$5$	17.9704	1.60732	0.803661	+	0.595087i	$0.202883\pi$
$6$	−9.85995	−0.670885
$7$	0	0
$7$	0	0
$8$	−22.2935	−0.985244
$9$	−15.1845	−0.562388
$10$	51.5473	1.63007
$11$	26.7049	0.731985	0.365993	−	0.930618i	$-0.380730\pi$
$11$	26.7049	0.731985	0.365993	+	0.930618i	$0.380730\pi$
$12$	−0.783832	−0.0188561
$13$	14.4956	0.309259	0.154630	−	0.987973i	$-0.450582\pi$
$13$	14.4956	0.309259	0.154630	+	0.987973i	$0.450582\pi$
$14$	0	0
$15$	−61.7710	−1.06328
$16$	−65.7723	−1.02769
$17$	−24.7016	−0.352412	−0.176206	−	0.984353i	$-0.556383\pi$
$17$	−24.7016	−0.352412	−0.176206	+	0.984353i	$0.556383\pi$
$18$	−43.5560	−0.570347
$19$	−94.6224	−1.14252	−0.571259	−	0.820770i	$-0.693545\pi$
$19$	−94.6224	−1.14252	−0.571259	+	0.820770i	$0.693545\pi$
$20$	4.09784	0.0458152
$21$	0	0
$22$	76.6018	0.742344
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	76.6312	0.651762
$25$	197.935	1.58348
$26$	41.5801	0.313636
$27$	145.004	1.03355
$28$	0	0
$29$	−57.5965	−0.368807	−0.184403	−	0.982851i	$-0.559035\pi$
$29$	−57.5965	−0.368807	−0.184403	+	0.982851i	$0.559035\pi$
$30$	−177.187	−1.07833
$31$	−88.8691	−0.514882	−0.257441	−	0.966294i	$-0.582879\pi$
$31$	−88.8691	−0.514882	−0.257441	+	0.966294i	$0.582879\pi$
$32$	−10.3165	−0.0569909
$33$	−91.7948	−0.484225
$34$	−70.8553	−0.357400
$35$	0	0
$36$	−3.46255	−0.0160303
$37$	−305.467	−1.35726	−0.678629	−	0.734482i	$-0.737425\pi$
$37$	−305.467	−1.35726	−0.678629	+	0.734482i	$0.737425\pi$
$38$	−271.420	−1.15869
$39$	−49.8269	−0.204582
$40$	−400.624	−1.58360
$41$	179.205	0.682613	0.341306	−	0.939952i	$-0.389131\pi$
$41$	179.205	0.682613	0.341306	+	0.939952i	$0.389131\pi$
$42$	0	0
$43$	−96.5826	−0.342528	−0.171264	−	0.985225i	$-0.554785\pi$
$43$	−96.5826	−0.342528	−0.171264	+	0.985225i	$0.554785\pi$
$44$	6.08959	0.0208645
$45$	−272.871	−0.903938
$46$	−65.9745	−0.211465
$47$	−218.484	−0.678067	−0.339034	−	0.940774i	$-0.610100\pi$
$47$	−218.484	−0.678067	−0.339034	+	0.940774i	$0.610100\pi$
$48$	226.084	0.679841
$49$	0	0
$50$	567.769	1.60589
$51$	84.9085	0.233129
$52$	3.30548	0.00881514
$53$	−519.174	−1.34555	−0.672774	−	0.739848i	$-0.734897\pi$
$53$	−519.174	−1.34555	−0.672774	+	0.739848i	$0.734897\pi$
$54$	415.937	1.04818
$55$	479.898	1.17654
$56$	0	0
$57$	325.252	0.755802
$58$	−165.213	−0.374026
$59$	37.2884	0.0822803	0.0411401	−	0.999153i	$-0.486901\pi$
$59$	37.2884	0.0822803	0.0411401	+	0.999153i	$0.486901\pi$
$60$	−14.0858	−0.0303078
$61$	−96.3052	−0.202141	−0.101071	−	0.994879i	$-0.532227\pi$
$61$	−96.3052	−0.202141	−0.101071	+	0.994879i	$0.532227\pi$
$62$	−254.917	−0.522169
$63$	0	0
$64$	496.586	0.969894
$65$	260.493	0.497079
$66$	−263.309	−0.491077
$67$	497.552	0.907248	0.453624	−	0.891193i	$-0.350131\pi$
$67$	497.552	0.907248	0.453624	+	0.891193i	$0.350131\pi$
$68$	−5.63276	−0.0100452
$69$	79.0596	0.137937
$70$	0	0
$71$	−19.6235	−0.0328011	−0.0164005	−	0.999866i	$-0.505221\pi$
$71$	−19.6235	−0.0328011	−0.0164005	+	0.999866i	$0.505221\pi$
$72$	338.515	0.554089
$73$	208.235	0.333865	0.166932	−	0.985968i	$-0.446614\pi$
$73$	208.235	0.333865	0.166932	+	0.985968i	$0.446614\pi$
$74$	−876.219	−1.37647
$75$	−680.378	−1.04751
$76$	−21.5770	−0.0325664
$77$	0	0
$78$	−142.926	−0.207477
$79$	−446.200	−0.635461	−0.317730	−	0.948181i	$-0.602921\pi$
$79$	−446.200	−0.635461	−0.317730	+	0.948181i	$0.602921\pi$
$80$	−1181.95	−1.65183
$81$	−88.4513	−0.121332
$82$	514.042	0.692273
$83$	−501.151	−0.662752	−0.331376	−	0.943499i	$-0.607513\pi$
$83$	−501.151	−0.662752	−0.331376	+	0.943499i	$0.607513\pi$
$84$	0	0
$85$	−443.897	−0.566440
$86$	−277.043	−0.347375
$87$	197.981	0.243974
$88$	−595.347	−0.721184
$89$	−1102.82	−1.31347	−0.656733	−	0.754124i	$-0.728062\pi$
$89$	−1102.82	−1.31347	−0.656733	+	0.754124i	$0.728062\pi$
$90$	−782.718	−0.916731
$91$	0	0
$92$	−5.24475	−0.00594351
$93$	305.476	0.340606
$94$	−626.711	−0.687663
$95$	−1700.40	−1.83640
$96$	35.4615	0.0377008
$97$	−1814.37	−1.89919	−0.949595	−	0.313478i	$-0.898506\pi$
$97$	−1814.37	−1.89919	−0.949595	+	0.313478i	$0.898506\pi$
$98$	0	0
$99$	−405.500	−0.411659
$100$	45.1357	0.0451357
$101$	1386.24	1.36570	0.682852	−	0.730557i	$-0.260739\pi$
$101$	1386.24	1.36570	0.682852	+	0.730557i	$0.260739\pi$
$102$	243.556	0.236428
$103$	−1372.33	−1.31281	−0.656407	−	0.754407i	$-0.727925\pi$
$103$	−1372.33	−1.31281	−0.656407	+	0.754407i	$0.727925\pi$
$104$	−323.159	−0.304696
$105$	0	0
$106$	−1489.23	−1.36459
$107$	1657.83	1.49784	0.748919	−	0.662662i	$-0.230573\pi$
$107$	1657.83	1.49784	0.748919	+	0.662662i	$0.230573\pi$
$108$	33.0656	0.0294605
$109$	−821.369	−0.721770	−0.360885	−	0.932610i	$-0.617525\pi$
$109$	−821.369	−0.721770	−0.360885	+	0.932610i	$0.617525\pi$
$110$	1376.57	1.19319
$111$	1050.01	0.897857
$112$	0	0
$113$	−267.142	−0.222395	−0.111197	−	0.993798i	$-0.535469\pi$
$113$	−267.142	−0.222395	−0.111197	+	0.993798i	$0.535469\pi$
$114$	932.972	0.766498
$115$	−413.319	−0.335150
$116$	−13.1339	−0.0105125
$117$	−220.109	−0.173924
$118$	106.960	0.0834447
$119$	0	0
$120$	1377.09	1.04759
$121$	−617.847	−0.464198
$122$	−276.247	−0.205002
$123$	−615.995	−0.451564
$124$	−20.2650	−0.0146762
$125$	1310.68	0.937846
$126$	0	0
$127$	−1446.21	−1.01048	−0.505239	−	0.862979i	$-0.668596\pi$
$127$	−1446.21	−1.01048	−0.505239	+	0.862979i	$0.668596\pi$
$128$	1506.97	1.04061
$129$	331.990	0.226590
$130$	747.211	0.504114
$131$	−1459.33	−0.973297	−0.486649	−	0.873598i	$-0.661781\pi$
$131$	−1459.33	−0.973297	−0.486649	+	0.873598i	$0.661781\pi$
$132$	−20.9322	−0.0138024
$133$	0	0
$134$	1427.20	0.920088
$135$	2605.78	1.66126
$136$	550.685	0.347212
$137$	−1889.31	−1.17821	−0.589105	−	0.808056i	$-0.700520\pi$
$137$	−1889.31	−1.17821	−0.589105	+	0.808056i	$0.700520\pi$
$138$	226.779	0.139889
$139$	1506.31	0.919159	0.459580	−	0.888137i	$-0.348000\pi$
$139$	1506.31	0.919159	0.459580	+	0.888137i	$0.348000\pi$
$140$	0	0
$141$	751.011	0.448557
$142$	−56.2890	−0.0332653
$143$	387.105	0.226373
$144$	998.717	0.577961
$145$	−1035.03	−0.592791
$146$	597.314	0.338589
$147$	0	0
$148$	−69.6565	−0.0386873
$149$	−2946.38	−1.61998	−0.809989	−	0.586444i	$-0.800527\pi$
$149$	−2946.38	−1.61998	−0.809989	+	0.586444i	$0.800527\pi$
$150$	−1951.63	−1.06233
$151$	1979.37	1.06675	0.533374	−	0.845880i	$-0.320924\pi$
$151$	1979.37	1.06675	0.533374	+	0.845880i	$0.320924\pi$
$152$	2109.47	1.12566
$153$	375.080	0.198192
$154$	0	0
$155$	−1597.01	−0.827582
$156$	−11.3622	−0.00583142
$157$	−505.403	−0.256915	−0.128457	−	0.991715i	$-0.541003\pi$
$157$	−505.403	−0.256915	−0.128457	+	0.991715i	$0.541003\pi$
$158$	−1279.90	−0.644454
$159$	1784.59	0.890110
$160$	−185.391	−0.0916027
$161$	0	0
$162$	−253.719	−0.123049
$163$	−2604.26	−1.25142	−0.625711	−	0.780055i	$-0.715191\pi$
$163$	−2604.26	−1.25142	−0.625711	+	0.780055i	$0.715191\pi$
$164$	40.8646	0.0194572
$165$	−1649.59	−0.778305
$166$	−1437.53	−0.672131
$167$	845.304	0.391686	0.195843	−	0.980635i	$-0.437256\pi$
$167$	845.304	0.391686	0.195843	+	0.980635i	$0.437256\pi$
$168$	0	0
$169$	−1986.88	−0.904359
$170$	−1273.30	−0.574456
$171$	1436.79	0.642539
$172$	−22.0240	−0.00976343
$173$	−886.843	−0.389742	−0.194871	−	0.980829i	$-0.562429\pi$
$173$	−886.843	−0.389742	−0.194871	+	0.980829i	$0.562429\pi$
$174$	567.898	0.247427
$175$	0	0
$176$	−1756.44	−0.752255
$177$	−128.174	−0.0544303
$178$	−3163.38	−1.33205
$179$	1103.01	0.460575	0.230288	−	0.973123i	$-0.426033\pi$
$179$	1103.01	0.460575	0.230288	+	0.973123i	$0.426033\pi$
$180$	−62.2234	−0.0257659
$181$	1200.43	0.492967	0.246483	−	0.969147i	$-0.420725\pi$
$181$	1200.43	0.492967	0.246483	+	0.969147i	$0.420725\pi$
$182$	0	0
$183$	331.037	0.133721
$184$	512.751	0.205438
$185$	−5489.37	−2.18155
$186$	876.244	0.345427
$187$	−659.653	−0.257961
$188$	−49.8214	−0.0193277
$189$	0	0
$190$	−4877.53	−1.86238
$191$	2415.93	0.915237	0.457618	−	0.889149i	$-0.348703\pi$
$191$	2415.93	0.915237	0.457618	+	0.889149i	$0.348703\pi$
$192$	−1706.95	−0.641607
$193$	232.884	0.0868568	0.0434284	−	0.999057i	$-0.486172\pi$
$193$	232.884	0.0868568	0.0434284	+	0.999057i	$0.486172\pi$
$194$	−5204.44	−1.92607
$195$	−895.410	−0.328829
$196$	0	0
$197$	−1418.48	−0.513008	−0.256504	−	0.966543i	$-0.582571\pi$
$197$	−1418.48	−0.513008	−0.256504	+	0.966543i	$0.582571\pi$
$198$	−1163.16	−0.417485
$199$	−1068.21	−0.380520	−0.190260	−	0.981734i	$-0.560933\pi$
$199$	−1068.21	−0.380520	−0.190260	+	0.981734i	$0.560933\pi$
$200$	−4412.68	−1.56012
$201$	−1710.27	−0.600165
$202$	3976.37	1.38503
$203$	0	0
$204$	19.3619	0.00664511
$205$	3220.39	1.09718
$206$	−3936.47	−1.33139
$207$	349.243	0.117266
$208$	−953.412	−0.317823
$209$	−2526.88	−0.836307
$210$	0	0
$211$	−1537.74	−0.501717	−0.250859	−	0.968024i	$-0.580713\pi$
$211$	−1537.74	−0.501717	−0.250859	+	0.968024i	$0.580713\pi$
$212$	−118.388	−0.0383535
$213$	67.4532	0.0216987
$214$	4755.41	1.51903
$215$	−1735.63	−0.550553
$216$	−3232.65	−1.01830
$217$	0	0
$218$	−2356.06	−0.731984
$219$	−715.783	−0.220859
$220$	109.432	0.0335360
$221$	−358.065	−0.108987
$222$	3011.89	0.910563
$223$	5359.68	1.60946	0.804732	−	0.593638i	$-0.202309\pi$
$223$	5359.68	1.60946	0.804732	+	0.593638i	$0.202309\pi$
$224$	0	0
$225$	−3005.54	−0.890532
$226$	−766.285	−0.225542
$227$	−1108.36	−0.324072	−0.162036	−	0.986785i	$-0.551806\pi$
$227$	−1108.36	−0.324072	−0.162036	+	0.986785i	$0.551806\pi$
$228$	74.1681	0.0215434
$229$	5046.34	1.45621	0.728104	−	0.685467i	$-0.240402\pi$
$229$	5046.34	1.45621	0.728104	+	0.685467i	$0.240402\pi$
$230$	−1185.59	−0.339893
$231$	0	0
$232$	1284.03	0.363365
$233$	7102.11	1.99689	0.998444	−	0.0557635i	$-0.0177593\pi$
$233$	7102.11	1.99689	0.998444	+	0.0557635i	$0.0177593\pi$
$234$	−631.372	−0.176385
$235$	−3926.25	−1.08987
$236$	8.50296	0.00234532
$237$	1533.75	0.420372
$238$	0	0
$239$	1556.22	0.421187	0.210594	−	0.977574i	$-0.432460\pi$
$239$	1556.22	0.421187	0.210594	+	0.977574i	$0.432460\pi$
$240$	4062.82	1.09272
$241$	−3028.88	−0.809573	−0.404787	−	0.914411i	$-0.632654\pi$
$241$	−3028.88	−0.809573	−0.404787	+	0.914411i	$0.632654\pi$
$242$	−1772.27	−0.470767
$243$	−3611.06	−0.953291
$244$	−21.9607	−0.00576184
$245$	0	0
$246$	−1766.95	−0.457954
$247$	−1371.61	−0.353334
$248$	1981.21	0.507285
$249$	1722.64	0.438426
$250$	3759.63	0.951118
$251$	−1449.39	−0.364480	−0.182240	−	0.983254i	$-0.558335\pi$
$251$	−1449.39	−0.364480	−0.182240	+	0.983254i	$0.558335\pi$
$252$	0	0
$253$	−614.213	−0.152629
$254$	−4148.40	−1.02478
$255$	1525.84	0.374713
$256$	349.976	0.0854433
$257$	3099.62	0.752332	0.376166	−	0.926552i	$-0.377242\pi$
$257$	3099.62	0.752332	0.376166	+	0.926552i	$0.377242\pi$
$258$	952.299	0.229797
$259$	0	0
$260$	59.4008	0.0141688
$261$	874.572	0.207412
$262$	−4186.01	−0.987071
$263$	−1087.92	−0.255073	−0.127537	−	0.991834i	$-0.540707\pi$
$263$	−1087.92	−0.255073	−0.127537	+	0.991834i	$0.540707\pi$
$264$	2046.43	0.477080
$265$	−9329.76	−2.16273
$266$	0	0
$267$	3790.79	0.868887
$268$	113.458	0.0258603
$269$	−4721.01	−1.07006	−0.535028	−	0.844834i	$-0.679699\pi$
$269$	−4721.01	−1.07006	−0.535028	+	0.844834i	$0.679699\pi$
$270$	7474.55	1.68477
$271$	−1401.15	−0.314074	−0.157037	−	0.987593i	$-0.550194\pi$
$271$	−1401.15	−0.314074	−0.157037	+	0.987593i	$0.550194\pi$
$272$	1624.68	0.362171
$273$	0	0
$274$	−5419.41	−1.19488
$275$	5285.85	1.15909
$276$	18.0281	0.00393176
$277$	4122.82	0.894283	0.447142	−	0.894463i	$-0.352442\pi$
$277$	4122.82	0.894283	0.447142	+	0.894463i	$0.352442\pi$
$278$	4320.77	0.932167
$279$	1349.43	0.289564
$280$	0	0
$281$	803.897	0.170664	0.0853318	−	0.996353i	$-0.472805\pi$
$281$	803.897	0.170664	0.0853318	+	0.996353i	$0.472805\pi$
$282$	2154.24	0.454905
$283$	3147.53	0.661134	0.330567	−	0.943782i	$-0.392760\pi$
$283$	3147.53	0.661134	0.330567	+	0.943782i	$0.392760\pi$
$284$	−4.47479	−0.000934964 0
$285$	5844.92	1.21482
$286$	1110.39	0.229577
$287$	0	0
$288$	156.650	0.0320510
$289$	−4302.83	−0.875806
$290$	−2968.94	−0.601181
$291$	6236.67	1.25636
$292$	47.4844	0.00951649
$293$	2812.88	0.560854	0.280427	−	0.959875i	$-0.409524\pi$
$293$	2812.88	0.560854	0.280427	+	0.959875i	$0.409524\pi$
$294$	0	0
$295$	670.088	0.132251
$296$	6809.95	1.33723
$297$	3872.31	0.756547
$298$	−8451.56	−1.64290
$299$	−333.400	−0.0644850
$300$	−155.148	−0.0298583
$301$	0	0
$302$	5677.73	1.08184
$303$	−4765.02	−0.903444
$304$	6223.53	1.17416
$305$	−1730.64	−0.324906
$306$	1075.90	0.200997
$307$	7885.52	1.46596	0.732981	−	0.680249i	$-0.238128\pi$
$307$	7885.52	1.46596	0.732981	+	0.680249i	$0.238128\pi$
$308$	0	0
$309$	4717.22	0.868457
$310$	−4580.96	−0.839294
$311$	1904.46	0.347240	0.173620	−	0.984813i	$-0.444453\pi$
$311$	1904.46	0.347240	0.173620	+	0.984813i	$0.444453\pi$
$312$	1110.82	0.201563
$313$	3854.89	0.696137	0.348069	−	0.937469i	$-0.386838\pi$
$313$	3854.89	0.696137	0.348069	+	0.937469i	$0.386838\pi$
$314$	−1449.73	−0.260550
$315$	0	0
$316$	−101.748	−0.0181132
$317$	1878.70	0.332865	0.166432	−	0.986053i	$-0.446775\pi$
$317$	1878.70	0.332865	0.166432	+	0.986053i	$0.446775\pi$
$318$	5119.03	0.902707
$319$	−1538.11	−0.269961
$320$	8923.85	1.55893
$321$	−5698.58	−0.990853
$322$	0	0
$323$	2337.32	0.402638
$324$	−20.1698	−0.00345846
$325$	2869.20	0.489707
$326$	−7470.21	−1.26913
$327$	2823.35	0.477467
$328$	−3995.11	−0.672541
$329$	0	0
$330$	−4731.77	−0.789320
$331$	6829.96	1.13417	0.567083	−	0.823661i	$-0.308072\pi$
$331$	6829.96	1.13417	0.567083	+	0.823661i	$0.308072\pi$
$332$	−114.279	−0.0188911
$333$	4638.36	0.763305
$334$	2424.72	0.397229
$335$	8941.21	1.45824
$336$	0	0
$337$	11048.9	1.78597	0.892985	−	0.450087i	$-0.148607\pi$
$337$	11048.9	1.78597	0.892985	+	0.450087i	$0.148607\pi$
$338$	−5699.26	−0.917157
$339$	918.267	0.147119
$340$	−101.223	−0.0161458
$341$	−2373.24	−0.376886
$342$	4121.37	0.651632
$343$	0	0
$344$	2153.17	0.337474
$345$	1420.73	0.221709
$346$	−2543.87	−0.395258
$347$	−7093.14	−1.09735	−0.548674	−	0.836037i	$-0.684867\pi$
$347$	−7093.14	−1.09735	−0.548674	+	0.836037i	$0.684867\pi$
$348$	45.1460	0.00695425
$349$	5363.16	0.822588	0.411294	−	0.911503i	$-0.365077\pi$
$349$	5363.16	0.822588	0.411294	+	0.911503i	$0.365077\pi$
$350$	0	0
$351$	2101.92	0.319636
$352$	−275.500	−0.0417165
$353$	−11789.3	−1.77757	−0.888787	−	0.458320i	$-0.848451\pi$
$353$	−11789.3	−1.77757	−0.888787	+	0.458320i	$0.848451\pi$
$354$	−367.662	−0.0552006
$355$	−352.642	−0.0527219
$356$	−251.478	−0.0374391
$357$	0	0
$358$	3163.94	0.467093
$359$	−2645.66	−0.388949	−0.194474	−	0.980908i	$-0.562300\pi$
$359$	−2645.66	−0.388949	−0.194474	+	0.980908i	$0.562300\pi$
$360$	6083.26	0.890600
$361$	2094.39	0.305349
$362$	3443.37	0.499943
$363$	2123.77	0.307077
$364$	0	0
$365$	3742.07	0.536628
$366$	949.564	0.135613
$367$	9775.68	1.39043	0.695213	−	0.718804i	$-0.255310\pi$
$367$	9775.68	1.39043	0.695213	+	0.718804i	$0.255310\pi$
$368$	1512.76	0.214289
$369$	−2721.13	−0.383893
$370$	−15746.0	−2.21242
$371$	0	0
$372$	69.6585	0.00970866
$373$	10981.1	1.52435	0.762174	−	0.647373i	$-0.224132\pi$
$373$	10981.1	1.52435	0.762174	+	0.647373i	$0.224132\pi$
$374$	−1892.19	−0.261611
$375$	−4505.30	−0.620407
$376$	4870.78	0.668062
$377$	−834.899	−0.114057
$378$	0	0
$379$	−2313.77	−0.313589	−0.156795	−	0.987631i	$-0.550116\pi$
$379$	−2313.77	−0.313589	−0.156795	+	0.987631i	$0.550116\pi$
$380$	−387.747	−0.0523447
$381$	4971.18	0.668455
$382$	6929.97	0.928189
$383$	−9219.25	−1.22998	−0.614989	−	0.788535i	$-0.710840\pi$
$383$	−9219.25	−1.22998	−0.614989	+	0.788535i	$0.710840\pi$
$384$	−5180.00	−0.688388
$385$	0	0
$386$	668.017	0.0880860
$387$	1466.55	0.192633
$388$	−413.736	−0.0541346
$389$	5876.90	0.765991	0.382996	−	0.923750i	$-0.374892\pi$
$389$	5876.90	0.765991	0.382996	+	0.923750i	$0.374892\pi$
$390$	−2568.44	−0.333483
$391$	568.136	0.0734830
$392$	0	0
$393$	5016.25	0.643858
$394$	−4068.85	−0.520268
$395$	−8018.39	−1.02139
$396$	−92.4671	−0.0117340
$397$	−14268.6	−1.80383	−0.901916	−	0.431911i	$-0.857840\pi$
$397$	−14268.6	−1.80383	−0.901916	+	0.431911i	$0.857840\pi$
$398$	−3064.12	−0.385905
$399$	0	0
$400$	−13018.7	−1.62733
$401$	−11556.1	−1.43911	−0.719557	−	0.694434i	$-0.755655\pi$
$401$	−11556.1	−1.43911	−0.719557	+	0.694434i	$0.755655\pi$
$402$	−4905.84	−0.608659
$403$	−1288.21	−0.159232
$404$	316.108	0.0389281
$405$	−1589.51	−0.195020
$406$	0	0
$407$	−8157.48	−0.993492
$408$	−1892.91	−0.229689
$409$	14148.3	1.71049	0.855245	−	0.518223i	$-0.173406\pi$
$409$	14148.3	1.71049	0.855245	+	0.518223i	$0.173406\pi$
$410$	9237.54	1.11271
$411$	6494.27	0.779413
$412$	−312.936	−0.0374205
$413$	0	0
$414$	1001.79	0.118925
$415$	−9005.88	−1.06526
$416$	−149.544	−0.0176250
$417$	−5177.73	−0.608045
$418$	−7248.25	−0.848142
$419$	2222.16	0.259092	0.129546	−	0.991573i	$-0.458648\pi$
$419$	2222.16	0.259092	0.129546	+	0.991573i	$0.458648\pi$
$420$	0	0
$421$	2518.51	0.291555	0.145777	−	0.989317i	$-0.453432\pi$
$421$	2518.51	0.291555	0.145777	+	0.989317i	$0.453432\pi$
$422$	−4410.94	−0.508818
$423$	3317.56	0.381337
$424$	11574.2	1.32569
$425$	−4889.31	−0.558039
$426$	193.486	0.0220057
$427$	0	0
$428$	378.039	0.0426944
$429$	−1330.62	−0.149751
$430$	−4978.57	−0.558344
$431$	9935.17	1.11035	0.555174	−	0.831734i	$-0.312652\pi$
$431$	9935.17	1.11035	0.555174	+	0.831734i	$0.312652\pi$
$432$	−9537.22	−1.06218
$433$	8427.47	0.935331	0.467665	−	0.883906i	$-0.345095\pi$
$433$	8427.47	0.935331	0.467665	+	0.883906i	$0.345095\pi$
$434$	0	0
$435$	3557.79	0.392145
$436$	−187.299	−0.0205734
$437$	2176.31	0.238232
$438$	−2053.19	−0.223985
$439$	−4886.04	−0.531203	−0.265601	−	0.964083i	$-0.585570\pi$
$439$	−4886.04	−0.531203	−0.265601	+	0.964083i	$0.585570\pi$
$440$	−10698.6	−1.15918
$441$	0	0
$442$	−1027.09	−0.110529
$443$	9724.04	1.04290	0.521448	−	0.853283i	$-0.325392\pi$
$443$	9724.04	1.04290	0.521448	+	0.853283i	$0.325392\pi$
$444$	239.435	0.0255926
$445$	−19818.1	−2.11116
$446$	15374.0	1.63224
$447$	10127.8	1.07165
$448$	0	0
$449$	−10093.7	−1.06091	−0.530457	−	0.847712i	$-0.677980\pi$
$449$	−10093.7	−1.06091	−0.530457	+	0.847712i	$0.677980\pi$
$450$	−8621.27	−0.903134
$451$	4785.66	0.499662
$452$	−60.9171	−0.00633915
$453$	−6803.83	−0.705678
$454$	−3179.28	−0.328658
$455$	0	0
$456$	−7251.02	−0.744650
$457$	3561.77	0.364578	0.182289	−	0.983245i	$-0.441649\pi$
$457$	3561.77	0.364578	0.182289	+	0.983245i	$0.441649\pi$
$458$	14475.2	1.47682
$459$	−3581.82	−0.364237
$460$	−94.2502	−0.00955313
$461$	7492.00	0.756914	0.378457	−	0.925619i	$-0.376455\pi$
$461$	7492.00	0.756914	0.378457	+	0.925619i	$0.376455\pi$
$462$	0	0
$463$	15427.0	1.54849	0.774247	−	0.632884i	$-0.218129\pi$
$463$	15427.0	1.54849	0.774247	+	0.632884i	$0.218129\pi$
$464$	3788.25	0.379020
$465$	5489.53	0.547464
$466$	20372.1	2.02515
$467$	−11868.7	−1.17606	−0.588028	−	0.808841i	$-0.700096\pi$
$467$	−11868.7	−1.17606	−0.588028	+	0.808841i	$0.700096\pi$
$468$	−50.1919	−0.00495753
$469$	0	0
$470$	−11262.3	−1.10530
$471$	1737.26	0.169955
$472$	−831.290	−0.0810662
$473$	−2579.23	−0.250725
$474$	4399.51	0.426321
$475$	−18729.1	−1.80916
$476$	0	0
$477$	7883.38	0.756719
$478$	4463.96	0.427148
$479$	−697.153	−0.0665005	−0.0332503	−	0.999447i	$-0.510586\pi$
$479$	−697.153	−0.0665005	−0.0332503	+	0.999447i	$0.510586\pi$
$480$	637.258	0.0605973
$481$	−4427.95	−0.419744
$482$	−8688.20	−0.821030
$483$	0	0
$484$	−140.889	−0.0132315
$485$	−32605.0	−3.05261
$486$	−10358.2	−0.966782
$487$	14414.0	1.34119	0.670597	−	0.741822i	$-0.266038\pi$
$487$	14414.0	1.34119	0.670597	+	0.741822i	$0.266038\pi$
$488$	2146.98	0.199159
$489$	8951.83	0.827844
$490$	0	0
$491$	17538.9	1.61206	0.806029	−	0.591876i	$-0.201612\pi$
$491$	17538.9	1.61206	0.806029	+	0.591876i	$0.201612\pi$
$492$	−140.467	−0.0128714
$493$	1422.72	0.129972
$494$	−3934.41	−0.358335
$495$	−7287.00	−0.661669
$496$	5845.12	0.529140
$497$	0	0
$498$	4941.32	0.444630
$499$	−18798.6	−1.68645	−0.843226	−	0.537559i	$-0.819347\pi$
$499$	−18798.6	−1.68645	−0.843226	+	0.537559i	$0.819347\pi$
$500$	298.878	0.0267324
$501$	−2905.62	−0.259109
$502$	−4157.50	−0.369638
$503$	−4634.11	−0.410785	−0.205392	−	0.978680i	$-0.565847\pi$
$503$	−4634.11	−0.410785	−0.205392	+	0.978680i	$0.565847\pi$
$504$	0	0
$505$	24911.3	2.19513
$506$	−1761.84	−0.154789
$507$	6829.63	0.598254
$508$	−329.784	−0.0288027
$509$	2193.57	0.191018	0.0955092	−	0.995429i	$-0.469552\pi$
$509$	2193.57	0.191018	0.0955092	+	0.995429i	$0.469552\pi$
$510$	4376.80	0.380016
$511$	0	0
$512$	−11051.8	−0.953958
$513$	−13720.6	−1.18086
$514$	8891.13	0.762979
$515$	−24661.4	−2.11012
$516$	75.7045	0.00645873
$517$	−5834.60	−0.496335
$518$	0	0
$519$	3048.41	0.257823
$520$	−5807.30	−0.489744
$521$	4401.81	0.370147	0.185074	−	0.982725i	$-0.440748\pi$
$521$	4401.81	0.370147	0.185074	+	0.982725i	$0.440748\pi$
$522$	2508.67	0.210348
$523$	9974.09	0.833913	0.416957	−	0.908926i	$-0.363097\pi$
$523$	9974.09	0.833913	0.416957	+	0.908926i	$0.363097\pi$
$524$	−332.774	−0.0277429
$525$	0	0
$526$	−3120.66	−0.258683
$527$	2195.20	0.181451
$528$	6037.55	0.497634
$529$	529.000	0.0434783
$530$	−26762.0	−2.19333
$531$	−566.204	−0.0462734
$532$	0	0
$533$	2597.69	0.211104
$534$	10873.7	0.881183
$535$	29791.9	2.40751
$536$	−11092.2	−0.893861
$537$	−3791.46	−0.304681
$538$	−13542.0	−1.08520
$539$	0	0
$540$	594.201	0.0473525
$541$	−9161.72	−0.728083	−0.364042	−	0.931383i	$-0.618603\pi$
$541$	−9161.72	−0.728083	−0.364042	+	0.931383i	$0.618603\pi$
$542$	−4019.15	−0.318519
$543$	−4126.31	−0.326109
$544$	254.833	0.0200843
$545$	−14760.3	−1.16012
$546$	0	0
$547$	1113.51	0.0870390	0.0435195	−	0.999053i	$-0.486143\pi$
$547$	1113.51	0.0870390	0.0435195	+	0.999053i	$0.486143\pi$
$548$	−430.825	−0.0335838
$549$	1462.34	0.113682
$550$	15162.2	1.17549
$551$	5449.92	0.421369
$552$	−1762.52	−0.135902
$553$	0	0
$554$	11826.1	0.906939
$555$	18869.0	1.44314
$556$	343.486	0.0261998
$557$	−7660.96	−0.582775	−0.291387	−	0.956605i	$-0.594117\pi$
$557$	−7660.96	−0.582775	−0.291387	+	0.956605i	$0.594117\pi$
$558$	3870.78	0.293661
$559$	−1400.03	−0.105930
$560$	0	0
$561$	2267.47	0.170647
$562$	2305.94	0.173079
$563$	−17217.7	−1.28888	−0.644441	−	0.764654i	$-0.722910\pi$
$563$	−17217.7	−1.28888	−0.644441	+	0.764654i	$0.722910\pi$
$564$	171.255	0.0127857
$565$	−4800.65	−0.357460
$566$	9028.54	0.670491
$567$	0	0
$568$	437.476	0.0323171
$569$	−2385.63	−0.175766	−0.0878830	−	0.996131i	$-0.528010\pi$
$569$	−2385.63	−0.175766	−0.0878830	+	0.996131i	$0.528010\pi$
$570$	16765.9	1.23201
$571$	16927.9	1.24065	0.620323	−	0.784347i	$-0.287002\pi$
$571$	16927.9	1.24065	0.620323	+	0.784347i	$0.287002\pi$
$572$	88.2725	0.00645255
$573$	−8304.44	−0.605450
$574$	0	0
$575$	−4552.52	−0.330179
$576$	−7540.39	−0.545457
$577$	−2886.69	−0.208275	−0.104137	−	0.994563i	$-0.533208\pi$
$577$	−2886.69	−0.208275	−0.104137	+	0.994563i	$0.533208\pi$
$578$	−12342.5	−0.888200
$579$	−800.509	−0.0574578
$580$	−236.021	−0.0168970
$581$	0	0
$582$	17889.6	1.27414
$583$	−13864.5	−0.984920
$584$	−4642.30	−0.328938
$585$	−3955.44	−0.279551
$586$	8068.62	0.568791
$587$	−503.810	−0.0354250	−0.0177125	−	0.999843i	$-0.505638\pi$
$587$	−503.810	−0.0354250	−0.0177125	+	0.999843i	$0.505638\pi$
$588$	0	0
$589$	8409.00	0.588263
$590$	1922.12	0.134122
$591$	4875.85	0.339366
$592$	20091.3	1.39484
$593$	7654.92	0.530101	0.265050	−	0.964235i	$-0.414611\pi$
$593$	7654.92	0.530101	0.265050	+	0.964235i	$0.414611\pi$
$594$	11107.6	0.767253
$595$	0	0
$596$	−671.870	−0.0461760
$597$	3671.84	0.251723
$598$	−956.343	−0.0653976
$599$	17366.5	1.18460	0.592301	−	0.805717i	$-0.298220\pi$
$599$	17366.5	1.18460	0.592301	+	0.805717i	$0.298220\pi$
$600$	15168.0	1.03205
$601$	−3872.81	−0.262854	−0.131427	−	0.991326i	$-0.541956\pi$
$601$	−3872.81	−0.262854	−0.131427	+	0.991326i	$0.541956\pi$
$602$	0	0
$603$	−7555.06	−0.510225
$604$	451.361	0.0304066
$605$	−11103.0	−0.746115
$606$	−13668.3	−0.916229
$607$	12823.3	0.857464	0.428732	−	0.903432i	$-0.358961\pi$
$607$	12823.3	0.857464	0.428732	+	0.903432i	$0.358961\pi$
$608$	976.167	0.0651132
$609$	0	0
$610$	−4964.27	−0.329504
$611$	−3167.07	−0.209699
$612$	85.5304	0.00564928
$613$	−17226.4	−1.13502	−0.567509	−	0.823367i	$-0.692093\pi$
$613$	−17226.4	−1.13502	−0.567509	+	0.823367i	$0.692093\pi$
$614$	22619.3	1.48671
$615$	−11069.7	−0.725809
$616$	0	0
$617$	−5892.12	−0.384454	−0.192227	−	0.981351i	$-0.561571\pi$
$617$	−5892.12	−0.384454	−0.192227	+	0.981351i	$0.561571\pi$
$618$	13531.1	0.880747
$619$	13036.6	0.846503	0.423251	−	0.906012i	$-0.360889\pi$
$619$	13036.6	0.846503	0.423251	+	0.906012i	$0.360889\pi$
$620$	−364.171	−0.0235894
$621$	−3335.09	−0.215511
$622$	5462.84	0.352154
$623$	0	0
$624$	3277.23	0.210247
$625$	−1188.48	−0.0760630
$626$	11057.6	0.705989
$627$	8685.84	0.553236
$628$	−115.248	−0.00732310
$629$	7545.52	0.478314
$630$	0	0
$631$	−10044.6	−0.633707	−0.316853	−	0.948475i	$-0.602626\pi$
$631$	−10044.6	−0.633707	−0.316853	+	0.948475i	$0.602626\pi$
$632$	9947.37	0.626084
$633$	5285.79	0.331898
$634$	5388.95	0.337575
$635$	−25989.1	−1.62416
$636$	406.945	0.0253717
$637$	0	0
$638$	−4412.00	−0.273782
$639$	297.972	0.0184469
$640$	27080.8	1.67260
$641$	4583.33	0.282419	0.141209	−	0.989980i	$-0.454901\pi$
$641$	4583.33	0.282419	0.141209	+	0.989980i	$0.454901\pi$
$642$	−16346.1	−1.00488
$643$	−19260.0	−1.18125	−0.590623	−	0.806948i	$-0.701118\pi$
$643$	−19260.0	−1.18125	−0.590623	+	0.806948i	$0.701118\pi$
$644$	0	0
$645$	5966.00	0.364203
$646$	6704.50	0.408336
$647$	1771.91	0.107667	0.0538337	−	0.998550i	$-0.482856\pi$
$647$	1771.91	0.107667	0.0538337	+	0.998550i	$0.482856\pi$
$648$	1971.89	0.119542
$649$	995.784	0.0602279
$650$	8230.18	0.496637
$651$	0	0
$652$	−593.857	−0.0356706
$653$	−28000.8	−1.67803	−0.839017	−	0.544106i	$-0.816869\pi$
$653$	−28000.8	−1.67803	−0.839017	+	0.544106i	$0.816869\pi$
$654$	8098.66	0.484224
$655$	−26224.7	−1.56440
$656$	−11786.7	−0.701515
$657$	−3161.94	−0.187761
$658$	0	0
$659$	−27664.0	−1.63526	−0.817629	−	0.575745i	$-0.804712\pi$
$659$	−27664.0	−1.63526	−0.817629	+	0.575745i	$0.804712\pi$
$660$	−376.160	−0.0221848
$661$	−23392.1	−1.37647	−0.688234	−	0.725489i	$-0.741614\pi$
$661$	−23392.1	−1.37647	−0.688234	+	0.725489i	$0.741614\pi$
$662$	19591.4	1.15022
$663$	1230.80	0.0720972
$664$	11172.4	0.652973
$665$	0	0
$666$	13304.9	0.774107
$667$	1324.72	0.0769016
$668$	192.757	0.0111646
$669$	−18423.2	−1.06470
$670$	25647.5	1.47888
$671$	−2571.82	−0.147964
$672$	0	0
$673$	4318.41	0.247344	0.123672	−	0.992323i	$-0.460533\pi$
$673$	4318.41	0.247344	0.123672	+	0.992323i	$0.460533\pi$
$674$	31693.3	1.81124
$675$	28701.4	1.63662
$676$	−453.072	−0.0257779
$677$	−18272.8	−1.03734	−0.518671	−	0.854974i	$-0.673573\pi$
$677$	−18272.8	−1.03734	−0.518671	+	0.854974i	$0.673573\pi$
$678$	2634.01	0.149201
$679$	0	0
$680$	9896.04	0.558082
$681$	3809.84	0.214381
$682$	−6807.53	−0.382220
$683$	−7245.38	−0.405910	−0.202955	−	0.979188i	$-0.565055\pi$
$683$	−7245.38	−0.405910	−0.202955	+	0.979188i	$0.565055\pi$
$684$	327.635	0.0183150
$685$	−33951.7	−1.89376
$686$	0	0
$687$	−17346.2	−0.963314
$688$	6352.45	0.352013
$689$	−7525.76	−0.416123
$690$	4075.31	0.224847
$691$	12055.2	0.663679	0.331839	−	0.943336i	$-0.392331\pi$
$691$	12055.2	0.663679	0.331839	+	0.943336i	$0.392331\pi$
$692$	−202.229	−0.0111092
$693$	0	0
$694$	−20346.3	−1.11288
$695$	27068.9	1.47738
$696$	−4413.69	−0.240374
$697$	−4426.64	−0.240561
$698$	15384.0	0.834229
$699$	−24412.6	−1.32099
$700$	0	0
$701$	−15301.9	−0.824455	−0.412228	−	0.911081i	$-0.635249\pi$
$701$	−15301.9	−0.824455	−0.412228	+	0.911081i	$0.635249\pi$
$702$	6029.27	0.324160
$703$	28904.0	1.55069
$704$	13261.3	0.709948
$705$	13496.0	0.720975
$706$	−33817.2	−1.80273
$707$	0	0
$708$	−29.2279	−0.00155148
$709$	−12173.6	−0.644835	−0.322418	−	0.946597i	$-0.604496\pi$
$709$	−12173.6	−0.644835	−0.322418	+	0.946597i	$0.604496\pi$
$710$	−1011.54	−0.0534680
$711$	6775.30	0.357375
$712$	24585.7	1.29408
$713$	2043.99	0.107360
$714$	0	0
$715$	6956.44	0.363854
$716$	251.522	0.0131283
$717$	−5349.32	−0.278625
$718$	−7588.95	−0.394453
$719$	−2639.63	−0.136914	−0.0684572	−	0.997654i	$-0.521808\pi$
$719$	−2639.63	−0.136914	−0.0684572	+	0.997654i	$0.521808\pi$
$720$	17947.3	0.928970
$721$	0	0
$722$	6007.67	0.309671
$723$	10411.4	0.535551
$724$	273.736	0.0140515
$725$	−11400.4	−0.584000
$726$	6091.94	0.311423
$727$	32759.5	1.67123	0.835613	−	0.549319i	$-0.185113\pi$
$727$	32759.5	1.67123	0.835613	+	0.549319i	$0.185113\pi$
$728$	0	0
$729$	14800.7	0.751956
$730$	10734.0	0.544222
$731$	2385.74	0.120711
$732$	75.4871	0.00381159
$733$	−29178.0	−1.47028	−0.735139	−	0.677917i	$-0.762883\pi$
$733$	−29178.0	−1.47028	−0.735139	+	0.677917i	$0.762883\pi$
$734$	28041.1	1.41010
$735$	0	0
$736$	237.278	0.0118834
$737$	13287.1	0.664092
$738$	−7805.45	−0.389326
$739$	−13704.9	−0.682194	−0.341097	−	0.940028i	$-0.610798\pi$
$739$	−13704.9	−0.682194	−0.341097	+	0.940028i	$0.610798\pi$
$740$	−1251.75	−0.0621830
$741$	4714.74	0.233739
$742$	0	0
$743$	−991.593	−0.0489610	−0.0244805	−	0.999700i	$-0.507793\pi$
$743$	−991.593	−0.0489610	−0.0244805	+	0.999700i	$0.507793\pi$
$744$	−6810.14	−0.335581
$745$	−52947.6	−2.60383
$746$	31498.9	1.54592
$747$	7609.71	0.372724
$748$	−150.422	−0.00735292
$749$	0	0
$750$	−12923.2	−0.629186
$751$	9440.73	0.458718	0.229359	−	0.973342i	$-0.426337\pi$
$751$	9440.73	0.458718	0.229359	+	0.973342i	$0.426337\pi$
$752$	14370.2	0.696844
$753$	4982.09	0.241112
$754$	−2394.87	−0.115671
$755$	35570.1	1.71461
$756$	0	0
$757$	10480.1	0.503177	0.251589	−	0.967834i	$-0.419047\pi$
$757$	10480.1	0.503177	0.251589	+	0.967834i	$0.419047\pi$
$758$	−6636.94	−0.318027
$759$	2111.28	0.100968
$760$	37908.0	1.80930
$761$	31314.9	1.49168	0.745838	−	0.666128i	$-0.232049\pi$
$761$	31314.9	1.49168	0.745838	+	0.666128i	$0.232049\pi$
$762$	14259.6	0.677915
$763$	0	0
$764$	550.909	0.0260880
$765$	6740.34	0.318559
$766$	−26445.0	−1.24739
$767$	540.519	0.0254459
$768$	−1203.00	−0.0565227
$769$	20862.1	0.978292	0.489146	−	0.872202i	$-0.337309\pi$
$769$	20862.1	0.978292	0.489146	+	0.872202i	$0.337309\pi$
$770$	0	0
$771$	−10654.6	−0.497684
$772$	53.1051	0.00247577
$773$	20340.2	0.946426	0.473213	−	0.880948i	$-0.343094\pi$
$773$	20340.2	0.946426	0.473213	+	0.880948i	$0.343094\pi$
$774$	4206.75	0.195360
$775$	−17590.3	−0.815308
$776$	40448.8	1.87117
$777$	0	0
$778$	16857.6	0.776832
$779$	−16956.8	−0.779898
$780$	−204.183	−0.00937296
$781$	−524.043	−0.0240099
$782$	1629.67	0.0745230
$783$	−8351.71	−0.381182
$784$	0	0
$785$	−9082.30	−0.412944
$786$	14388.9	0.652970
$787$	−31293.8	−1.41741	−0.708707	−	0.705503i	$-0.750721\pi$
$787$	−31293.8	−1.41741	−0.708707	+	0.705503i	$0.750721\pi$
$788$	−323.460	−0.0146228
$789$	3739.60	0.168737
$790$	−23000.4	−1.03584
$791$	0	0
$792$	9040.03	0.405585
$793$	−1396.01	−0.0625140
$794$	−40928.9	−1.82936
$795$	32069.9	1.43069
$796$	−243.587	−0.0108464
$797$	−32015.4	−1.42289	−0.711446	−	0.702741i	$-0.751959\pi$
$797$	−32015.4	−1.42289	−0.711446	+	0.702741i	$0.751959\pi$
$798$	0	0
$799$	5396.89	0.238959
$800$	−2041.99	−0.0902442
$801$	16745.7	0.738677
$802$	−33148.2	−1.45948
$803$	5560.91	0.244384
$804$	−389.997	−0.0171071
$805$	0	0
$806$	−3695.19	−0.161486
$807$	16227.9	0.707867
$808$	−30904.2	−1.34555
$809$	5234.16	0.227470	0.113735	−	0.993511i	$-0.463719\pi$
$809$	5234.16	0.227470	0.113735	+	0.993511i	$0.463719\pi$
$810$	−4559.43	−0.197780
$811$	2377.40	0.102937	0.0514685	−	0.998675i	$-0.483610\pi$
$811$	2377.40	0.102937	0.0514685	+	0.998675i	$0.483610\pi$
$812$	0	0
$813$	4816.29	0.207767
$814$	−23399.4	−1.00755
$815$	−46799.7	−2.01144
$816$	−5584.62	−0.239584
$817$	9138.87	0.391345
$818$	40583.9	1.73470
$819$	0	0
$820$	734.353	0.0312740
$821$	5174.70	0.219974	0.109987	−	0.993933i	$-0.464919\pi$
$821$	5174.70	0.219974	0.109987	+	0.993933i	$0.464919\pi$
$822$	18628.5	0.790443
$823$	44178.3	1.87115	0.935576	−	0.353126i	$-0.114881\pi$
$823$	44178.3	1.87115	0.935576	+	0.353126i	$0.114881\pi$
$824$	30594.1	1.29344
$825$	−18169.4	−0.766762
$826$	0	0
$827$	5766.56	0.242470	0.121235	−	0.992624i	$-0.461314\pi$
$827$	5766.56	0.242470	0.121235	+	0.992624i	$0.461314\pi$
$828$	79.6387	0.00334255
$829$	−38465.6	−1.61154	−0.805770	−	0.592229i	$-0.798248\pi$
$829$	−38465.6	−1.61154	−0.805770	+	0.592229i	$0.798248\pi$
$830$	−25833.0	−1.08033
$831$	−14171.7	−0.591589
$832$	7198.33	0.299949
$833$	0	0
$834$	−14852.1	−0.616650
$835$	15190.5	0.629566
$836$	−576.211	−0.0238381
$837$	−12886.3	−0.532159
$838$	6374.17	0.262759
$839$	17261.8	0.710301	0.355151	−	0.934809i	$-0.384430\pi$
$839$	17261.8	0.710301	0.355151	+	0.934809i	$0.384430\pi$
$840$	0	0
$841$	−21071.6	−0.863981
$842$	7224.22	0.295681
$843$	−2763.29	−0.112898
$844$	−350.655	−0.0143010
$845$	−35705.0	−1.45360
$846$	9516.28	0.386733
$847$	0	0
$848$	34147.2	1.38281
$849$	−10819.2	−0.437355
$850$	−14024.8	−0.565936
$851$	7025.75	0.283008
$852$	15.3815	0.000618500 0
$853$	−29038.7	−1.16561	−0.582805	−	0.812612i	$-0.698045\pi$
$853$	−29038.7	−1.16561	−0.582805	+	0.812612i	$0.698045\pi$
$854$	0	0
$855$	25819.7	1.03277
$856$	−36958.9	−1.47574
$857$	9865.16	0.393217	0.196609	−	0.980482i	$-0.437007\pi$
$857$	9865.16	0.393217	0.196609	+	0.980482i	$0.437007\pi$
$858$	−3816.84	−0.151870
$859$	24476.6	0.972214	0.486107	−	0.873899i	$-0.338416\pi$
$859$	24476.6	0.972214	0.486107	+	0.873899i	$0.338416\pi$
$860$	−395.779	−0.0156930
$861$	0	0
$862$	28498.6	1.12606
$863$	−37353.6	−1.47339	−0.736693	−	0.676227i	$-0.763614\pi$
$863$	−37353.6	−1.47339	−0.736693	+	0.676227i	$0.763614\pi$
$864$	−1495.92	−0.0589032
$865$	−15936.9	−0.626442
$866$	24173.8	0.948567
$867$	14790.4	0.579365
$868$	0	0
$869$	−11915.7	−0.465148
$870$	10205.4	0.397695
$871$	7212.34	0.280575
$872$	18311.2	0.711120
$873$	27550.3	1.06808
$874$	6242.66	0.241603
$875$	0	0
$876$	−163.222	−0.00629538
$877$	−25920.7	−0.998040	−0.499020	−	0.866590i	$-0.666307\pi$
$877$	−25920.7	−0.998040	−0.499020	+	0.866590i	$0.666307\pi$
$878$	−14015.4	−0.538720
$879$	−9668.92	−0.371018
$880$	−31564.0	−1.20912
$881$	9337.06	0.357064	0.178532	−	0.983934i	$-0.442865\pi$
$881$	9337.06	0.357064	0.178532	+	0.983934i	$0.442865\pi$
$882$	0	0
$883$	−70.8656	−0.00270081	−0.00135041	−	0.999999i	$-0.500430\pi$
$883$	−70.8656	−0.00270081	−0.00135041	+	0.999999i	$0.500430\pi$
$884$	−81.6505	−0.00310656
$885$	−2303.34	−0.0874870
$886$	27893.0	1.05765
$887$	1058.85	0.0400819	0.0200409	−	0.999799i	$-0.493620\pi$
$887$	1058.85	0.0400819	0.0200409	+	0.999799i	$0.493620\pi$
$888$	−23408.3	−0.884608
$889$	0	0
$890$	−56847.2	−2.14104
$891$	−2362.08	−0.0888135
$892$	1222.18	0.0458762
$893$	20673.5	0.774705
$894$	29051.2	1.08682
$895$	19821.6	0.740293
$896$	0	0
$897$	1146.02	0.0426583
$898$	−28953.3	−1.07593
$899$	5118.55	0.189892
$900$	−685.362	−0.0253838
$901$	12824.4	0.474187
$902$	13727.4	0.506734
$903$	0	0
$904$	5955.54	0.219113
$905$	21572.1	0.792356
$906$	−19516.5	−0.715664
$907$	29050.2	1.06350	0.531751	−	0.846900i	$-0.321534\pi$
$907$	29050.2	1.06350	0.531751	+	0.846900i	$0.321534\pi$
$908$	−252.742	−0.00923736
$909$	−21049.3	−0.768055
$910$	0	0
$911$	16041.6	0.583404	0.291702	−	0.956509i	$-0.405778\pi$
$911$	16041.6	0.583404	0.291702	+	0.956509i	$0.405778\pi$
$912$	−21392.6	−0.776732
$913$	−13383.2	−0.485125
$914$	10216.8	0.369738
$915$	5948.87	0.214933
$916$	1150.73	0.0415078
$917$	0	0
$918$	−10274.3	−0.369392
$919$	−30151.2	−1.08226	−0.541130	−	0.840939i	$-0.682003\pi$
$919$	−30151.2	−1.08226	−0.541130	+	0.840939i	$0.682003\pi$
$920$	9214.35	0.330204
$921$	−27105.5	−0.969767
$922$	21490.5	0.767626
$923$	−284.455	−0.0101440
$924$	0	0
$925$	−60462.8	−2.14919
$926$	44251.6	1.57041
$927$	20838.1	0.738311
$928$	594.192	0.0210186
$929$	−43339.8	−1.53061	−0.765303	−	0.643670i	$-0.777411\pi$
$929$	−43339.8	−1.53061	−0.765303	+	0.643670i	$0.777411\pi$
$930$	15746.5	0.555212
$931$	0	0
$932$	1619.51	0.0569194
$933$	−6546.32	−0.229707
$934$	−34044.8	−1.19270
$935$	−11854.2	−0.414626
$936$	4907.00	0.171357
$937$	−27291.9	−0.951533	−0.475766	−	0.879572i	$-0.657829\pi$
$937$	−27291.9	−0.951533	−0.475766	+	0.879572i	$0.657829\pi$
$938$	0	0
$939$	−13250.7	−0.460511
$940$	−895.311	−0.0310658
$941$	4358.15	0.150980	0.0754898	−	0.997147i	$-0.475948\pi$
$941$	4358.15	0.150980	0.0754898	+	0.997147i	$0.475948\pi$
$942$	4983.25	0.172360
$943$	−4121.72	−0.142335
$944$	−2452.54	−0.0845587
$945$	0	0
$946$	−7398.40	−0.254274
$947$	39067.8	1.34059	0.670293	−	0.742097i	$-0.266169\pi$
$947$	39067.8	1.34059	0.670293	+	0.742097i	$0.266169\pi$
$948$	349.746	0.0119823
$949$	3018.51	0.103251
$950$	−53723.6	−1.83476
$951$	−6457.78	−0.220197
$952$	0	0
$953$	−52563.8	−1.78668	−0.893341	−	0.449379i	$-0.851645\pi$
$953$	−52563.8	−1.78668	−0.893341	+	0.449379i	$0.851645\pi$
$954$	22613.1	0.767428
$955$	43415.2	1.47108
$956$	354.869	0.0120055
$957$	5287.06	0.178585
$958$	−1999.75	−0.0674416
$959$	0	0
$960$	−30674.6	−1.03127
$961$	−21893.3	−0.734896
$962$	−12701.4	−0.425684
$963$	−25173.3	−0.842365
$964$	−690.682	−0.0230761
$965$	4185.02	0.139607
$966$	0	0
$967$	−2440.46	−0.0811580	−0.0405790	−	0.999176i	$-0.512920\pi$
$967$	−2440.46	−0.0811580	−0.0405790	+	0.999176i	$0.512920\pi$
$968$	13774.0	0.457348
$969$	−8034.24	−0.266354
$970$	−93525.9	−3.09581
$971$	−45490.7	−1.50347	−0.751733	−	0.659468i	$-0.770782\pi$
$971$	−45490.7	−1.50347	−0.751733	+	0.659468i	$0.770782\pi$
$972$	−823.439	−0.0271727
$973$	0	0
$974$	41345.9	1.36017
$975$	−9862.52	−0.323952
$976$	6334.21	0.207739
$977$	−33244.4	−1.08862	−0.544311	−	0.838884i	$-0.683209\pi$
$977$	−33244.4	−1.08862	−0.544311	+	0.838884i	$0.683209\pi$
$978$	25677.9	0.839559
$979$	−29450.6	−0.961437
$980$	0	0
$981$	12472.1	0.405914
$982$	50309.6	1.63487
$983$	−1171.32	−0.0380053	−0.0190027	−	0.999819i	$-0.506049\pi$
$983$	−1171.32	−0.0380053	−0.0190027	+	0.999819i	$0.506049\pi$
$984$	13732.7	0.444901
$985$	−25490.7	−0.824569
$986$	4081.02	0.131811
$987$	0	0
$988$	−312.772	−0.0100715
$989$	2221.40	0.0714220
$990$	−20902.4	−0.671033
$991$	−3714.32	−0.119061	−0.0595304	−	0.998226i	$-0.518960\pi$
$991$	−3714.32	−0.119061	−0.0595304	+	0.998226i	$0.518960\pi$
$992$	916.814	0.0293436
$993$	−23477.1	−0.750276
$994$	0	0
$995$	−19196.2	−0.611618
$996$	392.818	0.0124969
$997$	−19521.0	−0.620098	−0.310049	−	0.950721i	$-0.600345\pi$
$997$	−19521.0	−0.620098	−0.310049	+	0.950721i	$0.600345\pi$
$998$	−53922.9	−1.71032
$999$	−44293.9	−1.40280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1127.4.a.c.1.3		4
7.6	odd	2		23.4.a.b.1.3	✓	4
21.20	even	2		207.4.a.e.1.2		4
28.27	even	2		368.4.a.l.1.2		4
35.13	even	4		575.4.b.g.24.3		8
35.27	even	4		575.4.b.g.24.6		8
35.34	odd	2		575.4.a.i.1.2		4
56.13	odd	2		1472.4.a.y.1.2		4
56.27	even	2		1472.4.a.bf.1.3		4
161.160	even	2		529.4.a.g.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.4.a.b.1.3	✓	4	7.6	odd	2
207.4.a.e.1.2		4	21.20	even	2
368.4.a.l.1.2		4	28.27	even	2
529.4.a.g.1.3		4	161.160	even	2
575.4.a.i.1.2		4	35.34	odd	2
575.4.b.g.24.3		8	35.13	even	4
575.4.b.g.24.6		8	35.27	even	4
1127.4.a.c.1.3		4	1.1	even	1	trivial
1472.4.a.y.1.2		4	56.13	odd	2
1472.4.a.bf.1.3		4	56.27	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1127 = 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1127.a (trivial)

\(f(q)\)	\(=\)	\(q+2.86845 q^{2} -3.43737 q^{3} +0.228032 q^{4} +17.9704 q^{5} -9.85995 q^{6} -22.2935 q^{8} -15.1845 q^{9} +51.5473 q^{10} +26.7049 q^{11} -0.783832 q^{12} +14.4956 q^{13} -61.7710 q^{15} -65.7723 q^{16} -24.7016 q^{17} -43.5560 q^{18} -94.6224 q^{19} +4.09784 q^{20} +76.6018 q^{22} -23.0000 q^{23} +76.6312 q^{24} +197.935 q^{25} +41.5801 q^{26} +145.004 q^{27} -57.5965 q^{29} -177.187 q^{30} -88.8691 q^{31} -10.3165 q^{32} -91.7948 q^{33} -70.8553 q^{34} -3.46255 q^{36} -305.467 q^{37} -271.420 q^{38} -49.8269 q^{39} -400.624 q^{40} +179.205 q^{41} -96.5826 q^{43} +6.08959 q^{44} -272.871 q^{45} -65.9745 q^{46} -218.484 q^{47} +226.084 q^{48} +567.769 q^{50} +84.9085 q^{51} +3.30548 q^{52} -519.174 q^{53} +415.937 q^{54} +479.898 q^{55} +325.252 q^{57} -165.213 q^{58} +37.2884 q^{59} -14.0858 q^{60} -96.3052 q^{61} -254.917 q^{62} +496.586 q^{64} +260.493 q^{65} -263.309 q^{66} +497.552 q^{67} -5.63276 q^{68} +79.0596 q^{69} -19.6235 q^{71} +338.515 q^{72} +208.235 q^{73} -876.219 q^{74} -680.378 q^{75} -21.5770 q^{76} -142.926 q^{78} -446.200 q^{79} -1181.95 q^{80} -88.4513 q^{81} +514.042 q^{82} -501.151 q^{83} -443.897 q^{85} -277.043 q^{86} +197.981 q^{87} -595.347 q^{88} -1102.82 q^{89} -782.718 q^{90} -5.24475 q^{92} +305.476 q^{93} -626.711 q^{94} -1700.40 q^{95} +35.4615 q^{96} -1814.37 q^{97} -405.500 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 7 q^{3} + 20 q^{4} - 14 q^{5} + 17 q^{6} - 63 q^{8} - 33 q^{9} + 70 q^{10} + 8 q^{11} + 67 q^{12} - 111 q^{13} + 10 q^{15} + 64 q^{16} - 98 q^{17} + 49 q^{18} - 96 q^{19} - 140 q^{20} + 220 q^{22}+ \cdots - 1498 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 7 * q^3 + 20 * q^4 - 14 * q^5 + 17 * q^6 - 63 * q^8 - 33 * q^9 + 70 * q^10 + 8 * q^11 + 67 * q^12 - 111 * q^13 + 10 * q^15 + 64 * q^16 - 98 * q^17 + 49 * q^18 - 96 * q^19 - 140 * q^20 + 220 * q^22 - 92 * q^23 + 188 * q^24 + 184 * q^25 + 229 * q^26 + 155 * q^27 + 21 * q^29 - 406 * q^30 + 193 * q^31 - 432 * q^32 + 418 * q^33 - 666 * q^34 - 629 * q^36 + 170 * q^37 - 748 * q^38 - 291 * q^39 + 26 * q^40 + 125 * q^41 + 2 * q^43 + 830 * q^44 - 168 * q^45 - 46 * q^46 + 677 * q^47 - 551 * q^48 + 414 * q^50 - 340 * q^51 - 2247 * q^52 - 230 * q^53 - 641 * q^54 + 972 * q^55 + 1322 * q^57 - 1835 * q^58 + 1140 * q^59 - 804 * q^60 - 754 * q^61 - 443 * q^62 - 805 * q^64 + 1318 * q^65 + 398 * q^66 + 488 * q^67 - 284 * q^68 + 161 * q^69 - 401 * q^71 + 1503 * q^72 - 1509 * q^73 + 1366 * q^74 - 1401 * q^75 + 3832 * q^76 - 1907 * q^78 - 838 * q^79 - 2846 * q^80 - 932 * q^81 + 949 * q^82 - 142 * q^83 + 112 * q^85 + 918 * q^86 - 2223 * q^87 - 404 * q^88 - 2342 * q^89 - 1784 * q^90 - 460 * q^92 - 509 * q^93 - 1567 * q^94 - 956 * q^95 - 799 * q^96 - 1062 * q^97 - 1498 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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