LMFDB - Embedded newform 1232.4.a.x.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,4,Mod(1,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1232.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1232.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:	$72.6903531271$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 116x^{3} - 22x^{2} + 2859x - 2034 $ x^5 - 116x^3 - 22x^2 + 2859*x - 2034
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 308)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$5.11231$ of defining polynomial
Character	$\chi$	$=$	1232.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.11231 q^{3} -17.6741 q^{5} -7.00000 q^{7} -10.0889 q^{9} +11.0000 q^{11} -38.6100 q^{13} -72.6815 q^{15} -68.4884 q^{17} -5.12965 q^{19} -28.7862 q^{21} -61.6628 q^{23} +187.375 q^{25} -152.521 q^{27} +103.792 q^{29} +15.5970 q^{31} +45.2354 q^{33} +123.719 q^{35} +201.963 q^{37} -158.776 q^{39} +335.950 q^{41} -267.787 q^{43} +178.313 q^{45} +17.5452 q^{47} +49.0000 q^{49} -281.645 q^{51} -505.495 q^{53} -194.416 q^{55} -21.0947 q^{57} +770.119 q^{59} -160.322 q^{61} +70.6225 q^{63} +682.400 q^{65} +33.8051 q^{67} -253.577 q^{69} -613.808 q^{71} +1137.42 q^{73} +770.546 q^{75} -77.0000 q^{77} -259.075 q^{79} -354.813 q^{81} +81.0856 q^{83} +1210.47 q^{85} +426.823 q^{87} -112.157 q^{89} +270.270 q^{91} +64.1398 q^{93} +90.6621 q^{95} +522.596 q^{97} -110.978 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + 15 q^{5} - 35 q^{7} + 102 q^{9} + 55 q^{11} + 70 q^{13} - 17 q^{15} + 104 q^{17} + 94 q^{19} + 35 q^{21} - 73 q^{23} + 576 q^{25} - 365 q^{27} + 14 q^{29} - 457 q^{31} - 55 q^{33} - 105 q^{35}+ \cdots + 1122 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + 15 * q^5 - 35 * q^7 + 102 * q^9 + 55 * q^11 + 70 * q^13 - 17 * q^15 + 104 * q^17 + 94 * q^19 + 35 * q^21 - 73 * q^23 + 576 * q^25 - 365 * q^27 + 14 * q^29 - 457 * q^31 - 55 * q^33 - 105 * q^35 + 139 * q^37 - 1020 * q^39 + 724 * q^41 - 64 * q^43 + 978 * q^45 - 378 * q^47 + 245 * q^49 - 246 * q^51 + 126 * q^53 + 165 * q^55 + 290 * q^57 + 1121 * q^59 + 740 * q^61 - 714 * q^63 + 400 * q^65 - 907 * q^67 + 47 * q^69 - 1435 * q^71 + 1380 * q^73 - 842 * q^75 - 385 * q^77 + 326 * q^79 + 1057 * q^81 + 416 * q^83 + 858 * q^85 + 3798 * q^87 + 1709 * q^89 - 490 * q^91 - 1235 * q^93 + 2590 * q^95 + 723 * q^97 + 1122 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	4.11231	0.791414	0.395707	−	0.918377i	$-0.370500\pi$
$3$	4.11231	0.791414	0.395707	+	0.918377i	$0.370500\pi$
$4$	0	0
$5$	−17.6741	−1.58082	−0.790412	−	0.612576i	$-0.790133\pi$
$5$	−17.6741	−1.58082	−0.790412	+	0.612576i	$0.790133\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−10.0889	−0.373664
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−38.6100	−0.823731	−0.411865	−	0.911245i	$-0.635123\pi$
$13$	−38.6100	−0.823731	−0.411865	+	0.911245i	$0.635123\pi$
$14$	0	0
$15$	−72.6815	−1.25109
$16$	0	0
$17$	−68.4884	−0.977110	−0.488555	−	0.872533i	$-0.662476\pi$
$17$	−68.4884	−0.977110	−0.488555	+	0.872533i	$0.662476\pi$
$18$	0	0
$19$	−5.12965	−0.0619380	−0.0309690	−	0.999520i	$-0.509859\pi$
$19$	−5.12965	−0.0619380	−0.0309690	+	0.999520i	$0.509859\pi$
$20$	0	0
$21$	−28.7862	−0.299126
$22$	0	0
$23$	−61.6628	−0.559026	−0.279513	−	0.960142i	$-0.590173\pi$
$23$	−61.6628	−0.559026	−0.279513	+	0.960142i	$0.590173\pi$
$24$	0	0
$25$	187.375	1.49900
$26$	0	0
$27$	−152.521	−1.08714
$28$	0	0
$29$	103.792	0.664607	0.332304	−	0.943172i	$-0.392174\pi$
$29$	103.792	0.664607	0.332304	+	0.943172i	$0.392174\pi$
$30$	0	0
$31$	15.5970	0.0903648	0.0451824	−	0.998979i	$-0.485613\pi$
$31$	15.5970	0.0903648	0.0451824	+	0.998979i	$0.485613\pi$
$32$	0	0
$33$	45.2354	0.238620
$34$	0	0
$35$	123.719	0.597495
$36$	0	0
$37$	201.963	0.897367	0.448683	−	0.893691i	$-0.351893\pi$
$37$	201.963	0.897367	0.448683	+	0.893691i	$0.351893\pi$
$38$	0	0
$39$	−158.776	−0.651912
$40$	0	0
$41$	335.950	1.27967	0.639836	−	0.768512i	$-0.279002\pi$
$41$	335.950	1.27967	0.639836	+	0.768512i	$0.279002\pi$
$42$	0	0
$43$	−267.787	−0.949702	−0.474851	−	0.880066i	$-0.657498\pi$
$43$	−267.787	−0.949702	−0.474851	+	0.880066i	$0.657498\pi$
$44$	0	0
$45$	178.313	0.590697
$46$	0	0
$47$	17.5452	0.0544516	0.0272258	−	0.999629i	$-0.491333\pi$
$47$	17.5452	0.0544516	0.0272258	+	0.999629i	$0.491333\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−281.645	−0.773299
$52$	0	0
$53$	−505.495	−1.31010	−0.655048	−	0.755587i	$-0.727352\pi$
$53$	−505.495	−1.31010	−0.655048	+	0.755587i	$0.727352\pi$
$54$	0	0
$55$	−194.416	−0.476636
$56$	0	0
$57$	−21.0947	−0.0490186
$58$	0	0
$59$	770.119	1.69934	0.849669	−	0.527317i	$-0.176802\pi$
$59$	770.119	1.69934	0.849669	+	0.527317i	$0.176802\pi$
$60$	0	0
$61$	−160.322	−0.336511	−0.168255	−	0.985743i	$-0.553813\pi$
$61$	−160.322	−0.336511	−0.168255	+	0.985743i	$0.553813\pi$
$62$	0	0
$63$	70.6225	0.141232
$64$	0	0
$65$	682.400	1.30217
$66$	0	0
$67$	33.8051	0.0616411	0.0308205	−	0.999525i	$-0.490188\pi$
$67$	33.8051	0.0616411	0.0308205	+	0.999525i	$0.490188\pi$
$68$	0	0
$69$	−253.577	−0.442421
$70$	0	0
$71$	−613.808	−1.02599	−0.512997	−	0.858390i	$-0.671465\pi$
$71$	−613.808	−1.02599	−0.512997	+	0.858390i	$0.671465\pi$
$72$	0	0
$73$	1137.42	1.82363	0.911816	−	0.410600i	$-0.134681\pi$
$73$	1137.42	1.82363	0.911816	+	0.410600i	$0.134681\pi$
$74$	0	0
$75$	770.546	1.18633
$76$	0	0
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−259.075	−0.368965	−0.184483	−	0.982836i	$-0.559061\pi$
$79$	−259.075	−0.368965	−0.184483	+	0.982836i	$0.559061\pi$
$80$	0	0
$81$	−354.813	−0.486711
$82$	0	0
$83$	81.0856	0.107233	0.0536163	−	0.998562i	$-0.482925\pi$
$83$	81.0856	0.107233	0.0536163	+	0.998562i	$0.482925\pi$
$84$	0	0
$85$	1210.47	1.54464
$86$	0	0
$87$	426.823	0.525980
$88$	0	0
$89$	−112.157	−0.133580	−0.0667898	−	0.997767i	$-0.521276\pi$
$89$	−112.157	−0.133580	−0.0667898	+	0.997767i	$0.521276\pi$
$90$	0	0
$91$	270.270	0.311341
$92$	0	0
$93$	64.1398	0.0715159
$94$	0	0
$95$	90.6621	0.0979130
$96$	0	0
$97$	522.596	0.547027	0.273513	−	0.961868i	$-0.411814\pi$
$97$	522.596	0.547027	0.273513	+	0.961868i	$0.411814\pi$
$98$	0	0
$99$	−110.978	−0.112664
$100$	0	0
$101$	−1013.32	−0.998305	−0.499153	−	0.866514i	$-0.666355\pi$
$101$	−1013.32	−0.998305	−0.499153	+	0.866514i	$0.666355\pi$
$102$	0	0
$103$	1764.71	1.68818	0.844089	−	0.536203i	$-0.180142\pi$
$103$	1764.71	1.68818	0.844089	+	0.536203i	$0.180142\pi$
$104$	0	0
$105$	508.771	0.472866
$106$	0	0
$107$	1226.42	1.10806	0.554029	−	0.832497i	$-0.313090\pi$
$107$	1226.42	1.10806	0.554029	+	0.832497i	$0.313090\pi$
$108$	0	0
$109$	1870.86	1.64400	0.821998	−	0.569490i	$-0.192859\pi$
$109$	1870.86	1.64400	0.821998	+	0.569490i	$0.192859\pi$
$110$	0	0
$111$	830.536	0.710189
$112$	0	0
$113$	−414.741	−0.345270	−0.172635	−	0.984986i	$-0.555228\pi$
$113$	−414.741	−0.345270	−0.172635	+	0.984986i	$0.555228\pi$
$114$	0	0
$115$	1089.84	0.883721
$116$	0	0
$117$	389.534	0.307798
$118$	0	0
$119$	479.419	0.369313
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	1381.53	1.01275
$124$	0	0
$125$	−1102.43	−0.788837
$126$	0	0
$127$	−1859.92	−1.29954	−0.649770	−	0.760131i	$-0.725135\pi$
$127$	−1859.92	−1.29954	−0.649770	+	0.760131i	$0.725135\pi$
$128$	0	0
$129$	−1101.22	−0.751607
$130$	0	0
$131$	1550.55	1.03414	0.517069	−	0.855944i	$-0.327023\pi$
$131$	1550.55	1.03414	0.517069	+	0.855944i	$0.327023\pi$
$132$	0	0
$133$	35.9075	0.0234104
$134$	0	0
$135$	2695.68	1.71857
$136$	0	0
$137$	2480.88	1.54713	0.773563	−	0.633719i	$-0.218473\pi$
$137$	2480.88	1.54713	0.773563	+	0.633719i	$0.218473\pi$
$138$	0	0
$139$	892.408	0.544554	0.272277	−	0.962219i	$-0.412223\pi$
$139$	892.408	0.544554	0.272277	+	0.962219i	$0.412223\pi$
$140$	0	0
$141$	72.1512	0.0430938
$142$	0	0
$143$	−424.710	−0.248364
$144$	0	0
$145$	−1834.43	−1.05063
$146$	0	0
$147$	201.503	0.113059
$148$	0	0
$149$	−2886.72	−1.58718	−0.793588	−	0.608456i	$-0.791789\pi$
$149$	−2886.72	−1.58718	−0.793588	+	0.608456i	$0.791789\pi$
$150$	0	0
$151$	−2088.91	−1.12578	−0.562890	−	0.826532i	$-0.690311\pi$
$151$	−2088.91	−1.12578	−0.562890	+	0.826532i	$0.690311\pi$
$152$	0	0
$153$	690.974	0.365111
$154$	0	0
$155$	−275.664	−0.142851
$156$	0	0
$157$	−807.585	−0.410524	−0.205262	−	0.978707i	$-0.565805\pi$
$157$	−807.585	−0.410524	−0.205262	+	0.978707i	$0.565805\pi$
$158$	0	0
$159$	−2078.75	−1.03683
$160$	0	0
$161$	431.640	0.211292
$162$	0	0
$163$	2274.59	1.09300	0.546501	−	0.837458i	$-0.315959\pi$
$163$	2274.59	1.09300	0.546501	+	0.837458i	$0.315959\pi$
$164$	0	0
$165$	−799.497	−0.377217
$166$	0	0
$167$	−638.140	−0.295693	−0.147847	−	0.989010i	$-0.547234\pi$
$167$	−638.140	−0.295693	−0.147847	+	0.989010i	$0.547234\pi$
$168$	0	0
$169$	−706.265	−0.321468
$170$	0	0
$171$	51.7526	0.0231440
$172$	0	0
$173$	3426.83	1.50599	0.752996	−	0.658025i	$-0.228608\pi$
$173$	3426.83	1.50599	0.752996	+	0.658025i	$0.228608\pi$
$174$	0	0
$175$	−1311.63	−0.566570
$176$	0	0
$177$	3166.97	1.34488
$178$	0	0
$179$	−467.573	−0.195241	−0.0976203	−	0.995224i	$-0.531123\pi$
$179$	−467.573	−0.195241	−0.0976203	+	0.995224i	$0.531123\pi$
$180$	0	0
$181$	1525.94	0.626641	0.313320	−	0.949647i	$-0.398559\pi$
$181$	1525.94	0.626641	0.313320	+	0.949647i	$0.398559\pi$
$182$	0	0
$183$	−659.294	−0.266319
$184$	0	0
$185$	−3569.53	−1.41858
$186$	0	0
$187$	−753.372	−0.294610
$188$	0	0
$189$	1067.65	0.410899
$190$	0	0
$191$	−420.374	−0.159253	−0.0796263	−	0.996825i	$-0.525373\pi$
$191$	−420.374	−0.159253	−0.0796263	+	0.996825i	$0.525373\pi$
$192$	0	0
$193$	−830.253	−0.309653	−0.154826	−	0.987942i	$-0.549482\pi$
$193$	−830.253	−0.309653	−0.154826	+	0.987942i	$0.549482\pi$
$194$	0	0
$195$	2806.24	1.03056
$196$	0	0
$197$	461.652	0.166961	0.0834806	−	0.996509i	$-0.473396\pi$
$197$	461.652	0.166961	0.0834806	+	0.996509i	$0.473396\pi$
$198$	0	0
$199$	−4811.55	−1.71398	−0.856990	−	0.515333i	$-0.827668\pi$
$199$	−4811.55	−1.71398	−0.856990	+	0.515333i	$0.827668\pi$
$200$	0	0
$201$	139.017	0.0487836
$202$	0	0
$203$	−726.541	−0.251198
$204$	0	0
$205$	−5937.63	−2.02294
$206$	0	0
$207$	622.112	0.208888
$208$	0	0
$209$	−56.4261	−0.0186750
$210$	0	0
$211$	689.791	0.225058	0.112529	−	0.993648i	$-0.464105\pi$
$211$	689.791	0.225058	0.112529	+	0.993648i	$0.464105\pi$
$212$	0	0
$213$	−2524.17	−0.811986
$214$	0	0
$215$	4732.91	1.50131
$216$	0	0
$217$	−109.179	−0.0341547
$218$	0	0
$219$	4677.43	1.44325
$220$	0	0
$221$	2644.34	0.804876
$222$	0	0
$223$	2263.55	0.679725	0.339863	−	0.940475i	$-0.389619\pi$
$223$	2263.55	0.679725	0.339863	+	0.940475i	$0.389619\pi$
$224$	0	0
$225$	−1890.42	−0.560124
$226$	0	0
$227$	−671.396	−0.196309	−0.0981545	−	0.995171i	$-0.531294\pi$
$227$	−671.396	−0.196309	−0.0981545	+	0.995171i	$0.531294\pi$
$228$	0	0
$229$	4019.93	1.16002	0.580010	−	0.814610i	$-0.303049\pi$
$229$	4019.93	1.16002	0.580010	+	0.814610i	$0.303049\pi$
$230$	0	0
$231$	−316.648	−0.0901900
$232$	0	0
$233$	1292.78	0.363488	0.181744	−	0.983346i	$-0.441826\pi$
$233$	1292.78	0.363488	0.181744	+	0.983346i	$0.441826\pi$
$234$	0	0
$235$	−310.096	−0.0860785
$236$	0	0
$237$	−1065.40	−0.292004
$238$	0	0
$239$	−4517.42	−1.22263	−0.611313	−	0.791389i	$-0.709359\pi$
$239$	−4517.42	−1.22263	−0.611313	+	0.791389i	$0.709359\pi$
$240$	0	0
$241$	2297.08	0.613975	0.306988	−	0.951714i	$-0.400679\pi$
$241$	2297.08	0.613975	0.306988	+	0.951714i	$0.400679\pi$
$242$	0	0
$243$	2658.97	0.701947
$244$	0	0
$245$	−866.033	−0.225832
$246$	0	0
$247$	198.056	0.0510202
$248$	0	0
$249$	333.449	0.0848653
$250$	0	0
$251$	−7628.70	−1.91840	−0.959202	−	0.282723i	$-0.908762\pi$
$251$	−7628.70	−1.91840	−0.959202	+	0.282723i	$0.908762\pi$
$252$	0	0
$253$	−678.291	−0.168553
$254$	0	0
$255$	4977.84	1.22245
$256$	0	0
$257$	6925.57	1.68095	0.840477	−	0.541848i	$-0.182275\pi$
$257$	6925.57	1.68095	0.840477	+	0.541848i	$0.182275\pi$
$258$	0	0
$259$	−1413.74	−0.339173
$260$	0	0
$261$	−1047.15	−0.248340
$262$	0	0
$263$	−5692.15	−1.33457	−0.667287	−	0.744801i	$-0.732544\pi$
$263$	−5692.15	−1.33457	−0.667287	+	0.744801i	$0.732544\pi$
$264$	0	0
$265$	8934.20	2.07103
$266$	0	0
$267$	−461.223	−0.105717
$268$	0	0
$269$	−5606.13	−1.27068	−0.635338	−	0.772234i	$-0.719139\pi$
$269$	−5606.13	−1.27068	−0.635338	+	0.772234i	$0.719139\pi$
$270$	0	0
$271$	1984.56	0.444847	0.222424	−	0.974950i	$-0.428603\pi$
$271$	1984.56	0.444847	0.222424	+	0.974950i	$0.428603\pi$
$272$	0	0
$273$	1111.43	0.246400
$274$	0	0
$275$	2061.13	0.451967
$276$	0	0
$277$	5968.02	1.29453	0.647263	−	0.762267i	$-0.275914\pi$
$277$	5968.02	1.29453	0.647263	+	0.762267i	$0.275914\pi$
$278$	0	0
$279$	−157.357	−0.0337660
$280$	0	0
$281$	−581.065	−0.123357	−0.0616787	−	0.998096i	$-0.519645\pi$
$281$	−581.065	−0.123357	−0.0616787	+	0.998096i	$0.519645\pi$
$282$	0	0
$283$	1844.26	0.387385	0.193693	−	0.981062i	$-0.437954\pi$
$283$	1844.26	0.387385	0.193693	+	0.981062i	$0.437954\pi$
$284$	0	0
$285$	372.831	0.0774897
$286$	0	0
$287$	−2351.65	−0.483670
$288$	0	0
$289$	−222.340	−0.0452553
$290$	0	0
$291$	2149.08	0.432925
$292$	0	0
$293$	−1233.40	−0.245926	−0.122963	−	0.992411i	$-0.539240\pi$
$293$	−1233.40	−0.245926	−0.122963	+	0.992411i	$0.539240\pi$
$294$	0	0
$295$	−13611.2	−2.68635
$296$	0	0
$297$	−1677.73	−0.327784
$298$	0	0
$299$	2380.80	0.460487
$300$	0	0
$301$	1874.51	0.358954
$302$	0	0
$303$	−4167.07	−0.790073
$304$	0	0
$305$	2833.56	0.531964
$306$	0	0
$307$	−3818.52	−0.709884	−0.354942	−	0.934888i	$-0.615499\pi$
$307$	−3818.52	−0.709884	−0.354942	+	0.934888i	$0.615499\pi$
$308$	0	0
$309$	7257.04	1.33605
$310$	0	0
$311$	7872.68	1.43543	0.717715	−	0.696337i	$-0.245188\pi$
$311$	7872.68	1.43543	0.717715	+	0.696337i	$0.245188\pi$
$312$	0	0
$313$	8834.84	1.59545	0.797723	−	0.603024i	$-0.206037\pi$
$313$	8834.84	1.59545	0.797723	+	0.603024i	$0.206037\pi$
$314$	0	0
$315$	−1248.19	−0.223262
$316$	0	0
$317$	6393.43	1.13278	0.566389	−	0.824138i	$-0.308340\pi$
$317$	6393.43	1.13278	0.566389	+	0.824138i	$0.308340\pi$
$318$	0	0
$319$	1141.71	0.200387
$320$	0	0
$321$	5043.40	0.876932
$322$	0	0
$323$	351.321	0.0605202
$324$	0	0
$325$	−7234.57	−1.23478
$326$	0	0
$327$	7693.54	1.30108
$328$	0	0
$329$	−122.816	−0.0205808
$330$	0	0
$331$	−3793.39	−0.629920	−0.314960	−	0.949105i	$-0.601991\pi$
$331$	−3793.39	−0.629920	−0.314960	+	0.949105i	$0.601991\pi$
$332$	0	0
$333$	−2037.59	−0.335314
$334$	0	0
$335$	−597.477	−0.0974437
$336$	0	0
$337$	3321.51	0.536896	0.268448	−	0.963294i	$-0.413489\pi$
$337$	3321.51	0.536896	0.268448	+	0.963294i	$0.413489\pi$
$338$	0	0
$339$	−1705.54	−0.273251
$340$	0	0
$341$	171.567	0.0272460
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	4481.75	0.699389
$346$	0	0
$347$	−4850.73	−0.750434	−0.375217	−	0.926937i	$-0.622432\pi$
$347$	−4850.73	−0.750434	−0.375217	+	0.926937i	$0.622432\pi$
$348$	0	0
$349$	3178.58	0.487522	0.243761	−	0.969835i	$-0.421619\pi$
$349$	3178.58	0.487522	0.243761	+	0.969835i	$0.421619\pi$
$350$	0	0
$351$	5888.84	0.895508
$352$	0	0
$353$	−12138.9	−1.83029	−0.915143	−	0.403130i	$-0.867922\pi$
$353$	−12138.9	−1.83029	−0.915143	+	0.403130i	$0.867922\pi$
$354$	0	0
$355$	10848.5	1.62192
$356$	0	0
$357$	1971.52	0.292279
$358$	0	0
$359$	4039.89	0.593920	0.296960	−	0.954890i	$-0.404027\pi$
$359$	4039.89	0.593920	0.296960	+	0.954890i	$0.404027\pi$
$360$	0	0
$361$	−6832.69	−0.996164
$362$	0	0
$363$	497.589	0.0719467
$364$	0	0
$365$	−20102.9	−2.88284
$366$	0	0
$367$	5765.75	0.820081	0.410041	−	0.912067i	$-0.365515\pi$
$367$	5765.75	0.820081	0.410041	+	0.912067i	$0.365515\pi$
$368$	0	0
$369$	−3389.37	−0.478167
$370$	0	0
$371$	3538.47	0.495170
$372$	0	0
$373$	9444.30	1.31101	0.655506	−	0.755190i	$-0.272456\pi$
$373$	9444.30	1.31101	0.655506	+	0.755190i	$0.272456\pi$
$374$	0	0
$375$	−4533.55	−0.624297
$376$	0	0
$377$	−4007.40	−0.547457
$378$	0	0
$379$	10942.6	1.48308	0.741538	−	0.670911i	$-0.234097\pi$
$379$	10942.6	1.48308	0.741538	+	0.670911i	$0.234097\pi$
$380$	0	0
$381$	−7648.58	−1.02847
$382$	0	0
$383$	−6614.65	−0.882488	−0.441244	−	0.897387i	$-0.645463\pi$
$383$	−6614.65	−0.882488	−0.441244	+	0.897387i	$0.645463\pi$
$384$	0	0
$385$	1360.91	0.180152
$386$	0	0
$387$	2701.69	0.354869
$388$	0	0
$389$	4425.60	0.576830	0.288415	−	0.957505i	$-0.406872\pi$
$389$	4425.60	0.576830	0.288415	+	0.957505i	$0.406872\pi$
$390$	0	0
$391$	4223.19	0.546230
$392$	0	0
$393$	6376.33	0.818431
$394$	0	0
$395$	4578.94	0.583269
$396$	0	0
$397$	3940.19	0.498116	0.249058	−	0.968489i	$-0.419879\pi$
$397$	3940.19	0.498116	0.249058	+	0.968489i	$0.419879\pi$
$398$	0	0
$399$	147.663	0.0185273
$400$	0	0
$401$	−2686.28	−0.334530	−0.167265	−	0.985912i	$-0.553493\pi$
$401$	−2686.28	−0.334530	−0.167265	+	0.985912i	$0.553493\pi$
$402$	0	0
$403$	−602.202	−0.0744362
$404$	0	0
$405$	6271.01	0.769405
$406$	0	0
$407$	2221.60	0.270566
$408$	0	0
$409$	−7641.24	−0.923801	−0.461901	−	0.886932i	$-0.652832\pi$
$409$	−7641.24	−0.923801	−0.461901	+	0.886932i	$0.652832\pi$
$410$	0	0
$411$	10202.2	1.22442
$412$	0	0
$413$	−5390.83	−0.642289
$414$	0	0
$415$	−1433.12	−0.169516
$416$	0	0
$417$	3669.86	0.430968
$418$	0	0
$419$	−1814.36	−0.211545	−0.105773	−	0.994390i	$-0.533732\pi$
$419$	−1814.36	−0.211545	−0.105773	+	0.994390i	$0.533732\pi$
$420$	0	0
$421$	10705.0	1.23927	0.619634	−	0.784891i	$-0.287281\pi$
$421$	10705.0	1.23927	0.619634	+	0.784891i	$0.287281\pi$
$422$	0	0
$423$	−177.012	−0.0203466
$424$	0	0
$425$	−12833.0	−1.46469
$426$	0	0
$427$	1122.26	0.127189
$428$	0	0
$429$	−1746.54	−0.196559
$430$	0	0
$431$	15741.5	1.75926	0.879631	−	0.475656i	$-0.157789\pi$
$431$	15741.5	1.75926	0.879631	+	0.475656i	$0.157789\pi$
$432$	0	0
$433$	10532.0	1.16890	0.584450	−	0.811429i	$-0.301310\pi$
$433$	10532.0	1.16890	0.584450	+	0.811429i	$0.301310\pi$
$434$	0	0
$435$	−7543.73	−0.831481
$436$	0	0
$437$	316.308	0.0346249
$438$	0	0
$439$	−6780.12	−0.737124	−0.368562	−	0.929603i	$-0.620150\pi$
$439$	−6780.12	−0.737124	−0.368562	+	0.929603i	$0.620150\pi$
$440$	0	0
$441$	−494.357	−0.0533805
$442$	0	0
$443$	8625.58	0.925087	0.462544	−	0.886597i	$-0.346937\pi$
$443$	8625.58	0.925087	0.462544	+	0.886597i	$0.346937\pi$
$444$	0	0
$445$	1982.27	0.211166
$446$	0	0
$447$	−11871.1	−1.25611
$448$	0	0
$449$	−15570.0	−1.63651	−0.818257	−	0.574853i	$-0.805059\pi$
$449$	−15570.0	−1.63651	−0.818257	+	0.574853i	$0.805059\pi$
$450$	0	0
$451$	3695.45	0.385836
$452$	0	0
$453$	−8590.23	−0.890958
$454$	0	0
$455$	−4776.80	−0.492175
$456$	0	0
$457$	−6885.93	−0.704837	−0.352418	−	0.935843i	$-0.614641\pi$
$457$	−6885.93	−0.704837	−0.352418	+	0.935843i	$0.614641\pi$
$458$	0	0
$459$	10445.9	1.06225
$460$	0	0
$461$	−9806.71	−0.990768	−0.495384	−	0.868674i	$-0.664973\pi$
$461$	−9806.71	−0.990768	−0.495384	+	0.868674i	$0.664973\pi$
$462$	0	0
$463$	−16156.6	−1.62173	−0.810863	−	0.585236i	$-0.801002\pi$
$463$	−16156.6	−1.62173	−0.810863	+	0.585236i	$0.801002\pi$
$464$	0	0
$465$	−1133.62	−0.113054
$466$	0	0
$467$	−14714.2	−1.45801	−0.729006	−	0.684507i	$-0.760018\pi$
$467$	−14714.2	−1.45801	−0.729006	+	0.684507i	$0.760018\pi$
$468$	0	0
$469$	−236.636	−0.0232981
$470$	0	0
$471$	−3321.04	−0.324894
$472$	0	0
$473$	−2945.66	−0.286346
$474$	0	0
$475$	−961.170	−0.0928453
$476$	0	0
$477$	5099.90	0.489536
$478$	0	0
$479$	−7513.08	−0.716663	−0.358331	−	0.933594i	$-0.616654\pi$
$479$	−7513.08	−0.716663	−0.358331	+	0.933594i	$0.616654\pi$
$480$	0	0
$481$	−7797.81	−0.739189
$482$	0	0
$483$	1775.04	0.167219
$484$	0	0
$485$	−9236.44	−0.864753
$486$	0	0
$487$	2434.28	0.226505	0.113253	−	0.993566i	$-0.463873\pi$
$487$	2434.28	0.226505	0.113253	+	0.993566i	$0.463873\pi$
$488$	0	0
$489$	9353.80	0.865017
$490$	0	0
$491$	−16178.6	−1.48703	−0.743513	−	0.668721i	$-0.766842\pi$
$491$	−16178.6	−1.48703	−0.743513	+	0.668721i	$0.766842\pi$
$492$	0	0
$493$	−7108.52	−0.649395
$494$	0	0
$495$	1961.44	0.178102
$496$	0	0
$497$	4296.65	0.387789
$498$	0	0
$499$	−3060.57	−0.274569	−0.137284	−	0.990532i	$-0.543837\pi$
$499$	−3060.57	−0.274569	−0.137284	+	0.990532i	$0.543837\pi$
$500$	0	0
$501$	−2624.23	−0.234016
$502$	0	0
$503$	17466.1	1.54826	0.774129	−	0.633028i	$-0.218188\pi$
$503$	17466.1	1.54826	0.774129	+	0.633028i	$0.218188\pi$
$504$	0	0
$505$	17909.5	1.57814
$506$	0	0
$507$	−2904.38	−0.254414
$508$	0	0
$509$	−6519.58	−0.567731	−0.283866	−	0.958864i	$-0.591617\pi$
$509$	−6519.58	−0.567731	−0.283866	+	0.958864i	$0.591617\pi$
$510$	0	0
$511$	−7961.95	−0.689268
$512$	0	0
$513$	782.379	0.0673350
$514$	0	0
$515$	−31189.8	−2.66871
$516$	0	0
$517$	192.997	0.0164178
$518$	0	0
$519$	14092.2	1.19186
$520$	0	0
$521$	−7476.59	−0.628705	−0.314352	−	0.949306i	$-0.601787\pi$
$521$	−7476.59	−0.628705	−0.314352	+	0.949306i	$0.601787\pi$
$522$	0	0
$523$	2830.39	0.236643	0.118321	−	0.992975i	$-0.462249\pi$
$523$	2830.39	0.236643	0.118321	+	0.992975i	$0.462249\pi$
$524$	0	0
$525$	−5393.82	−0.448392
$526$	0	0
$527$	−1068.21	−0.0882963
$528$	0	0
$529$	−8364.70	−0.687490
$530$	0	0
$531$	−7769.67	−0.634981
$532$	0	0
$533$	−12971.0	−1.05410
$534$	0	0
$535$	−21675.9	−1.75164
$536$	0	0
$537$	−1922.80	−0.154516
$538$	0	0
$539$	539.000	0.0430730
$540$	0	0
$541$	−21924.3	−1.74233	−0.871165	−	0.490990i	$-0.836635\pi$
$541$	−21924.3	−1.74233	−0.871165	+	0.490990i	$0.836635\pi$
$542$	0	0
$543$	6275.12	0.495932
$544$	0	0
$545$	−33065.8	−2.59887
$546$	0	0
$547$	−10489.7	−0.819937	−0.409969	−	0.912100i	$-0.634460\pi$
$547$	−10489.7	−0.819937	−0.409969	+	0.912100i	$0.634460\pi$
$548$	0	0
$549$	1617.48	0.125742
$550$	0	0
$551$	−532.414	−0.0411644
$552$	0	0
$553$	1813.53	0.139456
$554$	0	0
$555$	−14679.0	−1.12268
$556$	0	0
$557$	19390.1	1.47502	0.737508	−	0.675338i	$-0.236002\pi$
$557$	19390.1	1.47502	0.737508	+	0.675338i	$0.236002\pi$
$558$	0	0
$559$	10339.3	0.782298
$560$	0	0
$561$	−3098.10	−0.233158
$562$	0	0
$563$	18053.0	1.35141	0.675703	−	0.737174i	$-0.263840\pi$
$563$	18053.0	1.35141	0.675703	+	0.737174i	$0.263840\pi$
$564$	0	0
$565$	7330.18	0.545811
$566$	0	0
$567$	2483.69	0.183960
$568$	0	0
$569$	−19010.3	−1.40062	−0.700311	−	0.713838i	$-0.746955\pi$
$569$	−19010.3	−1.40062	−0.700311	+	0.713838i	$0.746955\pi$
$570$	0	0
$571$	10933.2	0.801296	0.400648	−	0.916232i	$-0.368785\pi$
$571$	10933.2	0.801296	0.400648	+	0.916232i	$0.368785\pi$
$572$	0	0
$573$	−1728.71	−0.126035
$574$	0	0
$575$	−11554.1	−0.837981
$576$	0	0
$577$	5529.42	0.398948	0.199474	−	0.979903i	$-0.436077\pi$
$577$	5529.42	0.398948	0.199474	+	0.979903i	$0.436077\pi$
$578$	0	0
$579$	−3414.26	−0.245063
$580$	0	0
$581$	−567.599	−0.0405301
$582$	0	0
$583$	−5560.45	−0.395009
$584$	0	0
$585$	−6884.68	−0.486575
$586$	0	0
$587$	1775.79	0.124863	0.0624316	−	0.998049i	$-0.480114\pi$
$587$	1775.79	0.124863	0.0624316	+	0.998049i	$0.480114\pi$
$588$	0	0
$589$	−80.0072	−0.00559701
$590$	0	0
$591$	1898.46	0.132135
$592$	0	0
$593$	21576.6	1.49418	0.747088	−	0.664725i	$-0.231451\pi$
$593$	21576.6	1.49418	0.747088	+	0.664725i	$0.231451\pi$
$594$	0	0
$595$	−8473.32	−0.583819
$596$	0	0
$597$	−19786.6	−1.35647
$598$	0	0
$599$	−24669.6	−1.68276	−0.841379	−	0.540446i	$-0.818255\pi$
$599$	−24669.6	−1.68276	−0.841379	+	0.540446i	$0.818255\pi$
$600$	0	0
$601$	7791.77	0.528840	0.264420	−	0.964408i	$-0.414819\pi$
$601$	7791.77	0.528840	0.264420	+	0.964408i	$0.414819\pi$
$602$	0	0
$603$	−341.057	−0.0230330
$604$	0	0
$605$	−2138.57	−0.143711
$606$	0	0
$607$	−15801.6	−1.05662	−0.528309	−	0.849052i	$-0.677174\pi$
$607$	−15801.6	−1.05662	−0.528309	+	0.849052i	$0.677174\pi$
$608$	0	0
$609$	−2987.76	−0.198802
$610$	0	0
$611$	−677.420	−0.0448535
$612$	0	0
$613$	26965.8	1.77674	0.888368	−	0.459132i	$-0.151839\pi$
$613$	26965.8	1.77674	0.888368	+	0.459132i	$0.151839\pi$
$614$	0	0
$615$	−24417.3	−1.60098
$616$	0	0
$617$	12991.7	0.847690	0.423845	−	0.905735i	$-0.360680\pi$
$617$	12991.7	0.847690	0.423845	+	0.905735i	$0.360680\pi$
$618$	0	0
$619$	−10646.4	−0.691302	−0.345651	−	0.938363i	$-0.612342\pi$
$619$	−10646.4	−0.691302	−0.345651	+	0.938363i	$0.612342\pi$
$620$	0	0
$621$	9404.88	0.607737
$622$	0	0
$623$	785.097	0.0504883
$624$	0	0
$625$	−3937.37	−0.251991
$626$	0	0
$627$	−232.042	−0.0147797
$628$	0	0
$629$	−13832.1	−0.876826
$630$	0	0
$631$	30473.4	1.92255	0.961273	−	0.275597i	$-0.0888757\pi$
$631$	30473.4	1.92255	0.961273	+	0.275597i	$0.0888757\pi$
$632$	0	0
$633$	2836.63	0.178114
$634$	0	0
$635$	32872.6	2.05434
$636$	0	0
$637$	−1891.89	−0.117676
$638$	0	0
$639$	6192.66	0.383377
$640$	0	0
$641$	−11802.0	−0.727225	−0.363613	−	0.931550i	$-0.618457\pi$
$641$	−11802.0	−0.727225	−0.363613	+	0.931550i	$0.618457\pi$
$642$	0	0
$643$	−21675.7	−1.32940	−0.664700	−	0.747110i	$-0.731441\pi$
$643$	−21675.7	−1.32940	−0.664700	+	0.747110i	$0.731441\pi$
$644$	0	0
$645$	19463.2	1.18816
$646$	0	0
$647$	−20491.7	−1.24515	−0.622575	−	0.782560i	$-0.713913\pi$
$647$	−20491.7	−1.24515	−0.622575	+	0.782560i	$0.713913\pi$
$648$	0	0
$649$	8471.31	0.512370
$650$	0	0
$651$	−448.978	−0.0270305
$652$	0	0
$653$	−7670.49	−0.459677	−0.229839	−	0.973229i	$-0.573820\pi$
$653$	−7670.49	−0.459677	−0.229839	+	0.973229i	$0.573820\pi$
$654$	0	0
$655$	−27404.6	−1.63479
$656$	0	0
$657$	−11475.4	−0.681425
$658$	0	0
$659$	−4241.73	−0.250735	−0.125367	−	0.992110i	$-0.540011\pi$
$659$	−4241.73	−0.250735	−0.125367	+	0.992110i	$0.540011\pi$
$660$	0	0
$661$	33241.0	1.95602	0.978008	−	0.208567i	$-0.0668800\pi$
$661$	33241.0	1.95602	0.978008	+	0.208567i	$0.0668800\pi$
$662$	0	0
$663$	10874.3	0.636990
$664$	0	0
$665$	−634.635	−0.0370076
$666$	0	0
$667$	−6400.08	−0.371532
$668$	0	0
$669$	9308.43	0.537944
$670$	0	0
$671$	−1763.54	−0.101462
$672$	0	0
$673$	−14244.4	−0.815869	−0.407934	−	0.913011i	$-0.633751\pi$
$673$	−14244.4	−0.815869	−0.407934	+	0.913011i	$0.633751\pi$
$674$	0	0
$675$	−28578.7	−1.62962
$676$	0	0
$677$	26689.5	1.51516	0.757578	−	0.652745i	$-0.226383\pi$
$677$	26689.5	1.51516	0.757578	+	0.652745i	$0.226383\pi$
$678$	0	0
$679$	−3658.17	−0.206757
$680$	0	0
$681$	−2760.99	−0.155362
$682$	0	0
$683$	21550.1	1.20731	0.603653	−	0.797247i	$-0.293711\pi$
$683$	21550.1	1.20731	0.603653	+	0.797247i	$0.293711\pi$
$684$	0	0
$685$	−43847.5	−2.44573
$686$	0	0
$687$	16531.2	0.918055
$688$	0	0
$689$	19517.2	1.07917
$690$	0	0
$691$	8736.96	0.480998	0.240499	−	0.970649i	$-0.422689\pi$
$691$	8736.96	0.480998	0.240499	+	0.970649i	$0.422689\pi$
$692$	0	0
$693$	776.847	0.0425829
$694$	0	0
$695$	−15772.5	−0.860844
$696$	0	0
$697$	−23008.7	−1.25038
$698$	0	0
$699$	5316.30	0.287669
$700$	0	0
$701$	7377.01	0.397469	0.198735	−	0.980053i	$-0.436317\pi$
$701$	7377.01	0.397469	0.198735	+	0.980053i	$0.436317\pi$
$702$	0	0
$703$	−1036.00	−0.0555811
$704$	0	0
$705$	−1275.21	−0.0681237
$706$	0	0
$707$	7093.22	0.377324
$708$	0	0
$709$	36663.7	1.94208	0.971041	−	0.238914i	$-0.0767915\pi$
$709$	36663.7	1.94208	0.971041	+	0.238914i	$0.0767915\pi$
$710$	0	0
$711$	2613.79	0.137869
$712$	0	0
$713$	−961.756	−0.0505162
$714$	0	0
$715$	7506.40	0.392620
$716$	0	0
$717$	−18577.0	−0.967604
$718$	0	0
$719$	−17416.8	−0.903390	−0.451695	−	0.892173i	$-0.649180\pi$
$719$	−17416.8	−0.903390	−0.451695	+	0.892173i	$0.649180\pi$
$720$	0	0
$721$	−12353.0	−0.638071
$722$	0	0
$723$	9446.31	0.485909
$724$	0	0
$725$	19448.0	0.996249
$726$	0	0
$727$	−15872.3	−0.809725	−0.404863	−	0.914378i	$-0.632681\pi$
$727$	−15872.3	−0.809725	−0.404863	+	0.914378i	$0.632681\pi$
$728$	0	0
$729$	20514.4	1.04224
$730$	0	0
$731$	18340.3	0.927963
$732$	0	0
$733$	−21946.8	−1.10590	−0.552950	−	0.833215i	$-0.686498\pi$
$733$	−21946.8	−1.10590	−0.552950	+	0.833215i	$0.686498\pi$
$734$	0	0
$735$	−3561.40	−0.178727
$736$	0	0
$737$	371.856	0.0185855
$738$	0	0
$739$	15934.2	0.793165	0.396583	−	0.917999i	$-0.370196\pi$
$739$	15934.2	0.793165	0.396583	+	0.917999i	$0.370196\pi$
$740$	0	0
$741$	814.467	0.0403781
$742$	0	0
$743$	−17728.9	−0.875384	−0.437692	−	0.899125i	$-0.644204\pi$
$743$	−17728.9	−0.875384	−0.437692	+	0.899125i	$0.644204\pi$
$744$	0	0
$745$	51020.3	2.50905
$746$	0	0
$747$	−818.067	−0.0400689
$748$	0	0
$749$	−8584.92	−0.418806
$750$	0	0
$751$	6806.02	0.330699	0.165350	−	0.986235i	$-0.447125\pi$
$751$	6806.02	0.330699	0.165350	+	0.986235i	$0.447125\pi$
$752$	0	0
$753$	−31371.6	−1.51825
$754$	0	0
$755$	36919.7	1.77966
$756$	0	0
$757$	−1527.50	−0.0733395	−0.0366697	−	0.999327i	$-0.511675\pi$
$757$	−1527.50	−0.0733395	−0.0366697	+	0.999327i	$0.511675\pi$
$758$	0	0
$759$	−2789.34	−0.133395
$760$	0	0
$761$	23250.9	1.10755	0.553774	−	0.832667i	$-0.313187\pi$
$761$	23250.9	1.10755	0.553774	+	0.832667i	$0.313187\pi$
$762$	0	0
$763$	−13096.0	−0.621372
$764$	0	0
$765$	−12212.4	−0.577176
$766$	0	0
$767$	−29734.3	−1.39980
$768$	0	0
$769$	−3388.31	−0.158889	−0.0794444	−	0.996839i	$-0.525315\pi$
$769$	−3388.31	−0.158889	−0.0794444	+	0.996839i	$0.525315\pi$
$770$	0	0
$771$	28480.1	1.33033
$772$	0	0
$773$	21704.4	1.00990	0.504951	−	0.863148i	$-0.331511\pi$
$773$	21704.4	1.00990	0.504951	+	0.863148i	$0.331511\pi$
$774$	0	0
$775$	2922.50	0.135457
$776$	0	0
$777$	−5813.75	−0.268426
$778$	0	0
$779$	−1723.30	−0.0792603
$780$	0	0
$781$	−6751.89	−0.309349
$782$	0	0
$783$	−15830.4	−0.722519
$784$	0	0
$785$	14273.4	0.648966
$786$	0	0
$787$	31201.3	1.41322	0.706612	−	0.707602i	$-0.250223\pi$
$787$	31201.3	1.41322	0.706612	+	0.707602i	$0.250223\pi$
$788$	0	0
$789$	−23407.9	−1.05620
$790$	0	0
$791$	2903.18	0.130500
$792$	0	0
$793$	6190.05	0.277194
$794$	0	0
$795$	36740.2	1.63904
$796$	0	0
$797$	28099.4	1.24885	0.624424	−	0.781085i	$-0.285334\pi$
$797$	28099.4	1.24885	0.624424	+	0.781085i	$0.285334\pi$
$798$	0	0
$799$	−1201.64	−0.0532053
$800$	0	0
$801$	1131.54	0.0499139
$802$	0	0
$803$	12511.6	0.549845
$804$	0	0
$805$	−7628.87	−0.334015
$806$	0	0
$807$	−23054.1	−1.00563
$808$	0	0
$809$	10856.4	0.471804	0.235902	−	0.971777i	$-0.424196\pi$
$809$	10856.4	0.471804	0.235902	+	0.971777i	$0.424196\pi$
$810$	0	0
$811$	42422.7	1.83682	0.918412	−	0.395626i	$-0.129472\pi$
$811$	42422.7	1.83682	0.918412	+	0.395626i	$0.129472\pi$
$812$	0	0
$813$	8161.13	0.352058
$814$	0	0
$815$	−40201.4	−1.72784
$816$	0	0
$817$	1373.65	0.0588226
$818$	0	0
$819$	−2726.74	−0.116337
$820$	0	0
$821$	7592.23	0.322741	0.161371	−	0.986894i	$-0.448409\pi$
$821$	7592.23	0.322741	0.161371	+	0.986894i	$0.448409\pi$
$822$	0	0
$823$	10393.5	0.440214	0.220107	−	0.975476i	$-0.429359\pi$
$823$	10393.5	0.440214	0.220107	+	0.975476i	$0.429359\pi$
$824$	0	0
$825$	8476.00	0.357693
$826$	0	0
$827$	35799.1	1.50527	0.752634	−	0.658439i	$-0.228783\pi$
$827$	35799.1	1.50527	0.752634	+	0.658439i	$0.228783\pi$
$828$	0	0
$829$	−16267.6	−0.681543	−0.340771	−	0.940146i	$-0.610688\pi$
$829$	−16267.6	−0.681543	−0.340771	+	0.940146i	$0.610688\pi$
$830$	0	0
$831$	24542.3	1.02451
$832$	0	0
$833$	−3355.93	−0.139587
$834$	0	0
$835$	11278.6	0.467439
$836$	0	0
$837$	−2378.87	−0.0982389
$838$	0	0
$839$	−45586.0	−1.87581	−0.937905	−	0.346894i	$-0.887237\pi$
$839$	−45586.0	−1.87581	−0.937905	+	0.346894i	$0.887237\pi$
$840$	0	0
$841$	−13616.3	−0.558297
$842$	0	0
$843$	−2389.52	−0.0976267
$844$	0	0
$845$	12482.6	0.508184
$846$	0	0
$847$	−847.000	−0.0343604
$848$	0	0
$849$	7584.18	0.306582
$850$	0	0
$851$	−12453.6	−0.501651
$852$	0	0
$853$	−10013.8	−0.401952	−0.200976	−	0.979596i	$-0.564411\pi$
$853$	−10013.8	−0.401952	−0.200976	+	0.979596i	$0.564411\pi$
$854$	0	0
$855$	−914.683	−0.0365866
$856$	0	0
$857$	−15420.1	−0.614633	−0.307316	−	0.951607i	$-0.599431\pi$
$857$	−15420.1	−0.614633	−0.307316	+	0.951607i	$0.599431\pi$
$858$	0	0
$859$	1411.56	0.0560673	0.0280336	−	0.999607i	$-0.491075\pi$
$859$	1411.56	0.0560673	0.0280336	+	0.999607i	$0.491075\pi$
$860$	0	0
$861$	−9670.70	−0.382784
$862$	0	0
$863$	−42553.1	−1.67848	−0.839238	−	0.543765i	$-0.816998\pi$
$863$	−42553.1	−1.67848	−0.839238	+	0.543765i	$0.816998\pi$
$864$	0	0
$865$	−60566.2	−2.38071
$866$	0	0
$867$	−914.329	−0.0358157
$868$	0	0
$869$	−2849.83	−0.111247
$870$	0	0
$871$	−1305.22	−0.0507757
$872$	0	0
$873$	−5272.43	−0.204404
$874$	0	0
$875$	7717.03	0.298152
$876$	0	0
$877$	35063.1	1.35005	0.675026	−	0.737794i	$-0.264132\pi$
$877$	35063.1	1.35005	0.675026	+	0.737794i	$0.264132\pi$
$878$	0	0
$879$	−5072.14	−0.194629
$880$	0	0
$881$	8276.97	0.316525	0.158262	−	0.987397i	$-0.449411\pi$
$881$	8276.97	0.316525	0.158262	+	0.987397i	$0.449411\pi$
$882$	0	0
$883$	10791.2	0.411273	0.205636	−	0.978628i	$-0.434074\pi$
$883$	10791.2	0.411273	0.205636	+	0.978628i	$0.434074\pi$
$884$	0	0
$885$	−55973.4	−2.12602
$886$	0	0
$887$	31684.7	1.19940	0.599701	−	0.800224i	$-0.295286\pi$
$887$	31684.7	1.19940	0.599701	+	0.800224i	$0.295286\pi$
$888$	0	0
$889$	13019.5	0.491180
$890$	0	0
$891$	−3902.94	−0.146749
$892$	0	0
$893$	−90.0006	−0.00337262
$894$	0	0
$895$	8263.96	0.308641
$896$	0	0
$897$	9790.60	0.364435
$898$	0	0
$899$	1618.84	0.0600571
$900$	0	0
$901$	34620.6	1.28011
$902$	0	0
$903$	7708.57	0.284081
$904$	0	0
$905$	−26969.6	−0.990608
$906$	0	0
$907$	7562.45	0.276854	0.138427	−	0.990373i	$-0.455795\pi$
$907$	7562.45	0.276854	0.138427	+	0.990373i	$0.455795\pi$
$908$	0	0
$909$	10223.3	0.373031
$910$	0	0
$911$	37906.2	1.37858	0.689292	−	0.724484i	$-0.257922\pi$
$911$	37906.2	1.37858	0.689292	+	0.724484i	$0.257922\pi$
$912$	0	0
$913$	891.942	0.0323318
$914$	0	0
$915$	11652.5	0.421004
$916$	0	0
$917$	−10853.8	−0.390867
$918$	0	0
$919$	17758.3	0.637422	0.318711	−	0.947852i	$-0.396750\pi$
$919$	17758.3	0.637422	0.318711	+	0.947852i	$0.396750\pi$
$920$	0	0
$921$	−15702.9	−0.561812
$922$	0	0
$923$	23699.1	0.845143
$924$	0	0
$925$	37843.0	1.34516
$926$	0	0
$927$	−17804.1	−0.630811
$928$	0	0
$929$	−6018.33	−0.212546	−0.106273	−	0.994337i	$-0.533892\pi$
$929$	−6018.33	−0.212546	−0.106273	+	0.994337i	$0.533892\pi$
$930$	0	0
$931$	−251.353	−0.00884828
$932$	0	0
$933$	32374.9	1.13602
$934$	0	0
$935$	13315.2	0.465726
$936$	0	0
$937$	24122.9	0.841046	0.420523	−	0.907282i	$-0.361847\pi$
$937$	24122.9	0.841046	0.420523	+	0.907282i	$0.361847\pi$
$938$	0	0
$939$	36331.6	1.26266
$940$	0	0
$941$	−34974.2	−1.21161	−0.605806	−	0.795612i	$-0.707149\pi$
$941$	−34974.2	−1.21161	−0.605806	+	0.795612i	$0.707149\pi$
$942$	0	0
$943$	−20715.6	−0.715369
$944$	0	0
$945$	−18869.8	−0.649559
$946$	0	0
$947$	−12841.9	−0.440662	−0.220331	−	0.975425i	$-0.570714\pi$
$947$	−12841.9	−0.440662	−0.220331	+	0.975425i	$0.570714\pi$
$948$	0	0
$949$	−43915.9	−1.50218
$950$	0	0
$951$	26291.7	0.896497
$952$	0	0
$953$	−41627.2	−1.41494	−0.707469	−	0.706744i	$-0.750163\pi$
$953$	−41627.2	−1.41494	−0.707469	+	0.706744i	$0.750163\pi$
$954$	0	0
$955$	7429.76	0.251750
$956$	0	0
$957$	4695.05	0.158589
$958$	0	0
$959$	−17366.2	−0.584759
$960$	0	0
$961$	−29547.7	−0.991834
$962$	0	0
$963$	−12373.2	−0.414041
$964$	0	0
$965$	14674.0	0.489506
$966$	0	0
$967$	47798.8	1.58956	0.794780	−	0.606897i	$-0.207586\pi$
$967$	47798.8	1.58956	0.794780	+	0.606897i	$0.207586\pi$
$968$	0	0
$969$	1444.74	0.0478966
$970$	0	0
$971$	−3328.28	−0.110000	−0.0549998	−	0.998486i	$-0.517516\pi$
$971$	−3328.28	−0.110000	−0.0549998	+	0.998486i	$0.517516\pi$
$972$	0	0
$973$	−6246.86	−0.205822
$974$	0	0
$975$	−29750.8	−0.977219
$976$	0	0
$977$	−16672.8	−0.545967	−0.272984	−	0.962019i	$-0.588010\pi$
$977$	−16672.8	−0.545967	−0.272984	+	0.962019i	$0.588010\pi$
$978$	0	0
$979$	−1233.72	−0.0402758
$980$	0	0
$981$	−18874.9	−0.614302
$982$	0	0
$983$	−3509.00	−0.113855	−0.0569276	−	0.998378i	$-0.518130\pi$
$983$	−3509.00	−0.113855	−0.0569276	+	0.998378i	$0.518130\pi$
$984$	0	0
$985$	−8159.31	−0.263936
$986$	0	0
$987$	−505.058	−0.0162879
$988$	0	0
$989$	16512.5	0.530908
$990$	0	0
$991$	17767.4	0.569524	0.284762	−	0.958598i	$-0.408085\pi$
$991$	17767.4	0.569524	0.284762	+	0.958598i	$0.408085\pi$
$992$	0	0
$993$	−15599.6	−0.498528
$994$	0	0
$995$	85040.1	2.70950
$996$	0	0
$997$	29983.0	0.952428	0.476214	−	0.879329i	$-0.342009\pi$
$997$	29983.0	0.952428	0.476214	+	0.879329i	$0.342009\pi$
$998$	0	0
$999$	−30803.7	−0.975561

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.x.1.4		5
4.3	odd	2		308.4.a.e.1.2	✓	5
28.27	even	2		2156.4.a.g.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.4.a.e.1.2	✓	5	4.3	odd	2
1232.4.a.x.1.4		5	1.1	even	1	trivial
2156.4.a.g.1.4		5	28.27	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:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1232.a (trivial)

\(f(q)\)	\(=\)	\(q+4.11231 q^{3} -17.6741 q^{5} -7.00000 q^{7} -10.0889 q^{9} +11.0000 q^{11} -38.6100 q^{13} -72.6815 q^{15} -68.4884 q^{17} -5.12965 q^{19} -28.7862 q^{21} -61.6628 q^{23} +187.375 q^{25} -152.521 q^{27} +103.792 q^{29} +15.5970 q^{31} +45.2354 q^{33} +123.719 q^{35} +201.963 q^{37} -158.776 q^{39} +335.950 q^{41} -267.787 q^{43} +178.313 q^{45} +17.5452 q^{47} +49.0000 q^{49} -281.645 q^{51} -505.495 q^{53} -194.416 q^{55} -21.0947 q^{57} +770.119 q^{59} -160.322 q^{61} +70.6225 q^{63} +682.400 q^{65} +33.8051 q^{67} -253.577 q^{69} -613.808 q^{71} +1137.42 q^{73} +770.546 q^{75} -77.0000 q^{77} -259.075 q^{79} -354.813 q^{81} +81.0856 q^{83} +1210.47 q^{85} +426.823 q^{87} -112.157 q^{89} +270.270 q^{91} +64.1398 q^{93} +90.6621 q^{95} +522.596 q^{97} -110.978 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + 15 q^{5} - 35 q^{7} + 102 q^{9} + 55 q^{11} + 70 q^{13} - 17 q^{15} + 104 q^{17} + 94 q^{19} + 35 q^{21} - 73 q^{23} + 576 q^{25} - 365 q^{27} + 14 q^{29} - 457 q^{31} - 55 q^{33} - 105 q^{35}+ \cdots + 1122 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + 15 * q^5 - 35 * q^7 + 102 * q^9 + 55 * q^11 + 70 * q^13 - 17 * q^15 + 104 * q^17 + 94 * q^19 + 35 * q^21 - 73 * q^23 + 576 * q^25 - 365 * q^27 + 14 * q^29 - 457 * q^31 - 55 * q^33 - 105 * q^35 + 139 * q^37 - 1020 * q^39 + 724 * q^41 - 64 * q^43 + 978 * q^45 - 378 * q^47 + 245 * q^49 - 246 * q^51 + 126 * q^53 + 165 * q^55 + 290 * q^57 + 1121 * q^59 + 740 * q^61 - 714 * q^63 + 400 * q^65 - 907 * q^67 + 47 * q^69 - 1435 * q^71 + 1380 * q^73 - 842 * q^75 - 385 * q^77 + 326 * q^79 + 1057 * q^81 + 416 * q^83 + 858 * q^85 + 3798 * q^87 + 1709 * q^89 - 490 * q^91 - 1235 * q^93 + 2590 * q^95 + 723 * q^97 + 1122 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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