LMFDB - Embedded newform 1160.4.a.h.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1160,4,Mod(1,1160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1160, 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("1160.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1160 = 2^{3} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1160.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:	$68.4422156067$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 3 x^{10} - 200 x^{9} + 418 x^{8} + 13211 x^{7} - 14353 x^{6} - 314463 x^{5} + 64817 x^{4} + \cdots + 712233 $ x^11 - 3x^10 - 200x^9 + 418x^8 + 13211x^7 - 14353x^6 - 314463x^5 + 64817x^4 + 1676824x^3 - 292536x^2 - 2502333x + 712233
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{3}\cdot 43 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-7.27289$ of defining polynomial
Character	$\chi$	$=$	1160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.27289 q^{3} -5.00000 q^{5} -31.8915 q^{7} +41.4407 q^{9} -63.2476 q^{11} +12.2067 q^{13} -41.3644 q^{15} +46.6824 q^{17} +85.1852 q^{19} -263.835 q^{21} +114.108 q^{23} +25.0000 q^{25} +119.466 q^{27} +29.0000 q^{29} +249.909 q^{31} -523.240 q^{33} +159.457 q^{35} +60.5079 q^{37} +100.985 q^{39} -46.7949 q^{41} +399.796 q^{43} -207.203 q^{45} +28.9658 q^{47} +674.067 q^{49} +386.198 q^{51} -311.057 q^{53} +316.238 q^{55} +704.728 q^{57} +533.147 q^{59} +137.649 q^{61} -1321.60 q^{63} -61.0334 q^{65} +20.3883 q^{67} +944.004 q^{69} -875.918 q^{71} -549.269 q^{73} +206.822 q^{75} +2017.06 q^{77} +1142.94 q^{79} -130.570 q^{81} +1031.80 q^{83} -233.412 q^{85} +239.914 q^{87} +708.780 q^{89} -389.289 q^{91} +2067.47 q^{93} -425.926 q^{95} +1765.18 q^{97} -2621.02 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 8 q^{3} - 55 q^{5} + 16 q^{7} + 117 q^{9} + 26 q^{11} + 52 q^{13} - 40 q^{15} + 14 q^{17} + 146 q^{19} - 288 q^{21} - 80 q^{23} + 275 q^{25} + 224 q^{27} + 319 q^{29} - 38 q^{31} - 152 q^{33} - 80 q^{35}+ \cdots + 4690 q^{99}+O(q^{100}) $ 11 * q + 8 * q^3 - 55 * q^5 + 16 * q^7 + 117 * q^9 + 26 * q^11 + 52 * q^13 - 40 * q^15 + 14 * q^17 + 146 * q^19 - 288 * q^21 - 80 * q^23 + 275 * q^25 + 224 * q^27 + 319 * q^29 - 38 * q^31 - 152 * q^33 - 80 * q^35 + 376 * q^37 + 108 * q^39 - 146 * q^41 + 644 * q^43 - 585 * q^45 - 464 * q^47 + 393 * q^49 - 576 * q^51 - 188 * q^53 - 130 * q^55 + 1380 * q^57 + 672 * q^59 + 472 * q^61 + 628 * q^63 - 260 * q^65 + 1688 * q^67 + 138 * q^69 + 92 * q^71 + 1514 * q^73 + 200 * q^75 + 2884 * q^77 + 2130 * q^79 + 4067 * q^81 + 2584 * q^83 - 70 * q^85 + 232 * q^87 + 1606 * q^89 + 3208 * q^91 + 3540 * q^93 - 730 * q^95 + 3322 * q^97 + 4690 * 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
$3$	8.27289	1.59212	0.796059	−	0.605219i	$-0.206915\pi$
$3$	8.27289	1.59212	0.796059	+	0.605219i	$0.206915\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−31.8915	−1.72198	−0.860989	−	0.508623i	$-0.830155\pi$
$7$	−31.8915	−1.72198	−0.860989	+	0.508623i	$0.830155\pi$
$8$	0	0
$9$	41.4407	1.53484
$10$	0	0
$11$	−63.2476	−1.73362	−0.866812	−	0.498635i	$-0.833835\pi$
$11$	−63.2476	−1.73362	−0.866812	+	0.498635i	$0.833835\pi$
$12$	0	0
$13$	12.2067	0.260425	0.130213	−	0.991486i	$-0.458434\pi$
$13$	12.2067	0.260425	0.130213	+	0.991486i	$0.458434\pi$
$14$	0	0
$15$	−41.3644	−0.712017
$16$	0	0
$17$	46.6824	0.666009	0.333004	−	0.942925i	$-0.391938\pi$
$17$	46.6824	0.666009	0.333004	+	0.942925i	$0.391938\pi$
$18$	0	0
$19$	85.1852	1.02857	0.514285	−	0.857619i	$-0.328057\pi$
$19$	85.1852	1.02857	0.514285	+	0.857619i	$0.328057\pi$
$20$	0	0
$21$	−263.835	−2.74159
$22$	0	0
$23$	114.108	1.03449	0.517243	−	0.855838i	$-0.326958\pi$
$23$	114.108	1.03449	0.517243	+	0.855838i	$0.326958\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	119.466	0.851527
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	249.909	1.44790	0.723951	−	0.689852i	$-0.242324\pi$
$31$	249.909	1.44790	0.723951	+	0.689852i	$0.242324\pi$
$32$	0	0
$33$	−523.240	−2.76013
$34$	0	0
$35$	159.457	0.770092
$36$	0	0
$37$	60.5079	0.268850	0.134425	−	0.990924i	$-0.457081\pi$
$37$	60.5079	0.268850	0.134425	+	0.990924i	$0.457081\pi$
$38$	0	0
$39$	100.985	0.414627
$40$	0	0
$41$	−46.7949	−0.178247	−0.0891235	−	0.996021i	$-0.528407\pi$
$41$	−46.7949	−0.178247	−0.0891235	+	0.996021i	$0.528407\pi$
$42$	0	0
$43$	399.796	1.41787	0.708935	−	0.705274i	$-0.249176\pi$
$43$	399.796	1.41787	0.708935	+	0.705274i	$0.249176\pi$
$44$	0	0
$45$	−207.203	−0.686401
$46$	0	0
$47$	28.9658	0.0898958	0.0449479	−	0.998989i	$-0.485688\pi$
$47$	28.9658	0.0898958	0.0449479	+	0.998989i	$0.485688\pi$
$48$	0	0
$49$	674.067	1.96521
$50$	0	0
$51$	386.198	1.06036
$52$	0	0
$53$	−311.057	−0.806170	−0.403085	−	0.915163i	$-0.632062\pi$
$53$	−311.057	−0.806170	−0.403085	+	0.915163i	$0.632062\pi$
$54$	0	0
$55$	316.238	0.775300
$56$	0	0
$57$	704.728	1.63760
$58$	0	0
$59$	533.147	1.17644	0.588219	−	0.808702i	$-0.299829\pi$
$59$	533.147	1.17644	0.588219	+	0.808702i	$0.299829\pi$
$60$	0	0
$61$	137.649	0.288921	0.144461	−	0.989511i	$-0.453855\pi$
$61$	137.649	0.288921	0.144461	+	0.989511i	$0.453855\pi$
$62$	0	0
$63$	−1321.60	−2.64296
$64$	0	0
$65$	−61.0334	−0.116466
$66$	0	0
$67$	20.3883	0.0371766	0.0185883	−	0.999827i	$-0.494083\pi$
$67$	20.3883	0.0371766	0.0185883	+	0.999827i	$0.494083\pi$
$68$	0	0
$69$	944.004	1.64703
$70$	0	0
$71$	−875.918	−1.46412	−0.732059	−	0.681241i	$-0.761440\pi$
$71$	−875.918	−1.46412	−0.732059	+	0.681241i	$0.761440\pi$
$72$	0	0
$73$	−549.269	−0.880645	−0.440323	−	0.897840i	$-0.645136\pi$
$73$	−549.269	−0.880645	−0.440323	+	0.897840i	$0.645136\pi$
$74$	0	0
$75$	206.822	0.318424
$76$	0	0
$77$	2017.06	2.98526
$78$	0	0
$79$	1142.94	1.62773	0.813867	−	0.581051i	$-0.197358\pi$
$79$	1142.94	1.62773	0.813867	+	0.581051i	$0.197358\pi$
$80$	0	0
$81$	−130.570	−0.179108
$82$	0	0
$83$	1031.80	1.36451	0.682257	−	0.731112i	$-0.260998\pi$
$83$	1031.80	1.36451	0.682257	+	0.731112i	$0.260998\pi$
$84$	0	0
$85$	−233.412	−0.297848
$86$	0	0
$87$	239.914	0.295649
$88$	0	0
$89$	708.780	0.844164	0.422082	−	0.906558i	$-0.361299\pi$
$89$	708.780	0.844164	0.422082	+	0.906558i	$0.361299\pi$
$90$	0	0
$91$	−389.289	−0.448446
$92$	0	0
$93$	2067.47	2.30523
$94$	0	0
$95$	−425.926	−0.459991
$96$	0	0
$97$	1765.18	1.84770	0.923851	−	0.382753i	$-0.125024\pi$
$97$	1765.18	1.84770	0.923851	+	0.382753i	$0.125024\pi$
$98$	0	0
$99$	−2621.02	−2.66083
$100$	0	0
$101$	156.369	0.154053	0.0770263	−	0.997029i	$-0.475457\pi$
$101$	156.369	0.154053	0.0770263	+	0.997029i	$0.475457\pi$
$102$	0	0
$103$	−1137.89	−1.08854	−0.544270	−	0.838910i	$-0.683193\pi$
$103$	−1137.89	−1.08854	−0.544270	+	0.838910i	$0.683193\pi$
$104$	0	0
$105$	1319.17	1.22608
$106$	0	0
$107$	1477.41	1.33482	0.667412	−	0.744688i	$-0.267402\pi$
$107$	1477.41	1.33482	0.667412	+	0.744688i	$0.267402\pi$
$108$	0	0
$109$	−1753.81	−1.54114	−0.770570	−	0.637355i	$-0.780028\pi$
$109$	−1753.81	−1.54114	−0.770570	+	0.637355i	$0.780028\pi$
$110$	0	0
$111$	500.575	0.428040
$112$	0	0
$113$	−434.992	−0.362129	−0.181065	−	0.983471i	$-0.557954\pi$
$113$	−434.992	−0.362129	−0.181065	+	0.983471i	$0.557954\pi$
$114$	0	0
$115$	−570.541	−0.462637
$116$	0	0
$117$	505.853	0.399710
$118$	0	0
$119$	−1488.77	−1.14685
$120$	0	0
$121$	2669.26	2.00545
$122$	0	0
$123$	−387.129	−0.283790
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−100.068	−0.0699180	−0.0349590	−	0.999389i	$-0.511130\pi$
$127$	−100.068	−0.0699180	−0.0349590	+	0.999389i	$0.511130\pi$
$128$	0	0
$129$	3307.47	2.25742
$130$	0	0
$131$	1466.15	0.977850	0.488925	−	0.872326i	$-0.337389\pi$
$131$	1466.15	0.977850	0.488925	+	0.872326i	$0.337389\pi$
$132$	0	0
$133$	−2716.68	−1.77118
$134$	0	0
$135$	−597.330	−0.380814
$136$	0	0
$137$	−1382.82	−0.862350	−0.431175	−	0.902268i	$-0.641901\pi$
$137$	−1382.82	−0.862350	−0.431175	+	0.902268i	$0.641901\pi$
$138$	0	0
$139$	−1125.98	−0.687083	−0.343541	−	0.939138i	$-0.611627\pi$
$139$	−1125.98	−0.687083	−0.343541	+	0.939138i	$0.611627\pi$
$140$	0	0
$141$	239.631	0.143125
$142$	0	0
$143$	−772.043	−0.451479
$144$	0	0
$145$	−145.000	−0.0830455
$146$	0	0
$147$	5576.48	3.12885
$148$	0	0
$149$	−2063.74	−1.13469	−0.567344	−	0.823481i	$-0.692029\pi$
$149$	−2063.74	−1.13469	−0.567344	+	0.823481i	$0.692029\pi$
$150$	0	0
$151$	−271.065	−0.146086	−0.0730430	−	0.997329i	$-0.523271\pi$
$151$	−271.065	−0.146086	−0.0730430	+	0.997329i	$0.523271\pi$
$152$	0	0
$153$	1934.55	1.02222
$154$	0	0
$155$	−1249.54	−0.647521
$156$	0	0
$157$	−231.948	−0.117907	−0.0589536	−	0.998261i	$-0.518776\pi$
$157$	−231.948	−0.117907	−0.0589536	+	0.998261i	$0.518776\pi$
$158$	0	0
$159$	−2573.34	−1.28352
$160$	0	0
$161$	−3639.08	−1.78136
$162$	0	0
$163$	1875.15	0.901064	0.450532	−	0.892760i	$-0.351234\pi$
$163$	1875.15	0.901064	0.450532	+	0.892760i	$0.351234\pi$
$164$	0	0
$165$	2616.20	1.23437
$166$	0	0
$167$	−2893.71	−1.34085	−0.670426	−	0.741977i	$-0.733888\pi$
$167$	−2893.71	−1.34085	−0.670426	+	0.741977i	$0.733888\pi$
$168$	0	0
$169$	−2048.00	−0.932179
$170$	0	0
$171$	3530.13	1.57869
$172$	0	0
$173$	1232.03	0.541444	0.270722	−	0.962658i	$-0.412738\pi$
$173$	1232.03	0.541444	0.270722	+	0.962658i	$0.412738\pi$
$174$	0	0
$175$	−797.287	−0.344396
$176$	0	0
$177$	4410.66	1.87303
$178$	0	0
$179$	1246.13	0.520338	0.260169	−	0.965563i	$-0.416222\pi$
$179$	1246.13	0.520338	0.260169	+	0.965563i	$0.416222\pi$
$180$	0	0
$181$	−2785.61	−1.14394	−0.571968	−	0.820276i	$-0.693820\pi$
$181$	−2785.61	−1.14394	−0.571968	+	0.820276i	$0.693820\pi$
$182$	0	0
$183$	1138.76	0.459996
$184$	0	0
$185$	−302.539	−0.120233
$186$	0	0
$187$	−2952.55	−1.15461
$188$	0	0
$189$	−3809.95	−1.46631
$190$	0	0
$191$	−2977.04	−1.12781	−0.563903	−	0.825841i	$-0.690701\pi$
$191$	−2977.04	−1.12781	−0.563903	+	0.825841i	$0.690701\pi$
$192$	0	0
$193$	−340.400	−0.126956	−0.0634781	−	0.997983i	$-0.520219\pi$
$193$	−340.400	−0.126956	−0.0634781	+	0.997983i	$0.520219\pi$
$194$	0	0
$195$	−504.923	−0.185427
$196$	0	0
$197$	−3015.59	−1.09062	−0.545309	−	0.838235i	$-0.683588\pi$
$197$	−3015.59	−1.09062	−0.545309	+	0.838235i	$0.683588\pi$
$198$	0	0
$199$	3444.68	1.22707	0.613535	−	0.789668i	$-0.289747\pi$
$199$	3444.68	1.22707	0.613535	+	0.789668i	$0.289747\pi$
$200$	0	0
$201$	168.670	0.0591895
$202$	0	0
$203$	−924.853	−0.319763
$204$	0	0
$205$	233.974	0.0797145
$206$	0	0
$207$	4728.72	1.58777
$208$	0	0
$209$	−5387.76	−1.78315
$210$	0	0
$211$	5107.30	1.66635	0.833177	−	0.553006i	$-0.186519\pi$
$211$	5107.30	1.66635	0.833177	+	0.553006i	$0.186519\pi$
$212$	0	0
$213$	−7246.37	−2.33105
$214$	0	0
$215$	−1998.98	−0.634091
$216$	0	0
$217$	−7969.96	−2.49325
$218$	0	0
$219$	−4544.04	−1.40209
$220$	0	0
$221$	569.838	0.173445
$222$	0	0
$223$	−2965.53	−0.890523	−0.445261	−	0.895401i	$-0.646889\pi$
$223$	−2965.53	−0.890523	−0.445261	+	0.895401i	$0.646889\pi$
$224$	0	0
$225$	1036.02	0.306968
$226$	0	0
$227$	−3638.58	−1.06388	−0.531940	−	0.846782i	$-0.678537\pi$
$227$	−3638.58	−1.06388	−0.531940	+	0.846782i	$0.678537\pi$
$228$	0	0
$229$	2597.17	0.749458	0.374729	−	0.927134i	$-0.377736\pi$
$229$	2597.17	0.749458	0.374729	+	0.927134i	$0.377736\pi$
$230$	0	0
$231$	16686.9	4.75289
$232$	0	0
$233$	5145.83	1.44684	0.723422	−	0.690407i	$-0.242568\pi$
$233$	5145.83	1.44684	0.723422	+	0.690407i	$0.242568\pi$
$234$	0	0
$235$	−144.829	−0.0402026
$236$	0	0
$237$	9455.43	2.59154
$238$	0	0
$239$	5260.87	1.42384	0.711919	−	0.702262i	$-0.247826\pi$
$239$	5260.87	1.42384	0.711919	+	0.702262i	$0.247826\pi$
$240$	0	0
$241$	1295.77	0.346339	0.173170	−	0.984892i	$-0.444599\pi$
$241$	1295.77	0.346339	0.173170	+	0.984892i	$0.444599\pi$
$242$	0	0
$243$	−4305.77	−1.13669
$244$	0	0
$245$	−3370.34	−0.878869
$246$	0	0
$247$	1039.83	0.267865
$248$	0	0
$249$	8535.96	2.17247
$250$	0	0
$251$	585.397	0.147211	0.0736055	−	0.997287i	$-0.476549\pi$
$251$	585.397	0.147211	0.0736055	+	0.997287i	$0.476549\pi$
$252$	0	0
$253$	−7217.07	−1.79341
$254$	0	0
$255$	−1930.99	−0.474210
$256$	0	0
$257$	6985.74	1.69556	0.847779	−	0.530350i	$-0.177940\pi$
$257$	6985.74	1.69556	0.847779	+	0.530350i	$0.177940\pi$
$258$	0	0
$259$	−1929.69	−0.462953
$260$	0	0
$261$	1201.78	0.285012
$262$	0	0
$263$	3903.02	0.915098	0.457549	−	0.889184i	$-0.348727\pi$
$263$	3903.02	0.915098	0.457549	+	0.889184i	$0.348727\pi$
$264$	0	0
$265$	1555.29	0.360530
$266$	0	0
$267$	5863.66	1.34401
$268$	0	0
$269$	5133.61	1.16358	0.581788	−	0.813341i	$-0.302353\pi$
$269$	5133.61	1.16358	0.581788	+	0.813341i	$0.302353\pi$
$270$	0	0
$271$	4957.05	1.11114	0.555571	−	0.831469i	$-0.312500\pi$
$271$	4957.05	1.11114	0.555571	+	0.831469i	$0.312500\pi$
$272$	0	0
$273$	−3220.55	−0.713979
$274$	0	0
$275$	−1581.19	−0.346725
$276$	0	0
$277$	6232.33	1.35186	0.675928	−	0.736968i	$-0.263743\pi$
$277$	6232.33	1.35186	0.675928	+	0.736968i	$0.263743\pi$
$278$	0	0
$279$	10356.4	2.22230
$280$	0	0
$281$	1108.22	0.235269	0.117635	−	0.993057i	$-0.462469\pi$
$281$	1108.22	0.235269	0.117635	+	0.993057i	$0.462469\pi$
$282$	0	0
$283$	3473.84	0.729677	0.364838	−	0.931071i	$-0.381124\pi$
$283$	3473.84	0.729677	0.364838	+	0.931071i	$0.381124\pi$
$284$	0	0
$285$	−3523.64	−0.732359
$286$	0	0
$287$	1492.36	0.306938
$288$	0	0
$289$	−2733.75	−0.556432
$290$	0	0
$291$	14603.1	2.94176
$292$	0	0
$293$	−9239.71	−1.84229	−0.921143	−	0.389224i	$-0.872743\pi$
$293$	−9239.71	−1.84229	−0.921143	+	0.389224i	$0.872743\pi$
$294$	0	0
$295$	−2665.74	−0.526119
$296$	0	0
$297$	−7555.93	−1.47623
$298$	0	0
$299$	1392.88	0.269406
$300$	0	0
$301$	−12750.1	−2.44154
$302$	0	0
$303$	1293.62	0.245270
$304$	0	0
$305$	−688.246	−0.129209
$306$	0	0
$307$	78.6313	0.0146180	0.00730900	−	0.999973i	$-0.497673\pi$
$307$	78.6313	0.0146180	0.00730900	+	0.999973i	$0.497673\pi$
$308$	0	0
$309$	−9413.63	−1.73308
$310$	0	0
$311$	8512.12	1.55202	0.776010	−	0.630721i	$-0.217241\pi$
$311$	8512.12	1.55202	0.776010	+	0.630721i	$0.217241\pi$
$312$	0	0
$313$	7716.09	1.39342	0.696708	−	0.717355i	$-0.254647\pi$
$313$	7716.09	1.39342	0.696708	+	0.717355i	$0.254647\pi$
$314$	0	0
$315$	6608.02	1.18197
$316$	0	0
$317$	1661.19	0.294328	0.147164	−	0.989112i	$-0.452986\pi$
$317$	1661.19	0.294328	0.147164	+	0.989112i	$0.452986\pi$
$318$	0	0
$319$	−1834.18	−0.321926
$320$	0	0
$321$	12222.4	2.12520
$322$	0	0
$323$	3976.65	0.685037
$324$	0	0
$325$	305.167	0.0520850
$326$	0	0
$327$	−14509.0	−2.45368
$328$	0	0
$329$	−923.764	−0.154799
$330$	0	0
$331$	4244.71	0.704866	0.352433	−	0.935837i	$-0.385355\pi$
$331$	4244.71	0.704866	0.352433	+	0.935837i	$0.385355\pi$
$332$	0	0
$333$	2507.49	0.412641
$334$	0	0
$335$	−101.942	−0.0166259
$336$	0	0
$337$	−7770.04	−1.25597	−0.627984	−	0.778226i	$-0.716120\pi$
$337$	−7770.04	−1.25597	−0.627984	+	0.778226i	$0.716120\pi$
$338$	0	0
$339$	−3598.64	−0.576552
$340$	0	0
$341$	−15806.1	−2.51012
$342$	0	0
$343$	−10558.2	−1.66207
$344$	0	0
$345$	−4720.02	−0.736572
$346$	0	0
$347$	−1004.11	−0.155341	−0.0776707	−	0.996979i	$-0.524748\pi$
$347$	−1004.11	−0.155341	−0.0776707	+	0.996979i	$0.524748\pi$
$348$	0	0
$349$	4944.98	0.758449	0.379225	−	0.925305i	$-0.376191\pi$
$349$	4944.98	0.758449	0.379225	+	0.925305i	$0.376191\pi$
$350$	0	0
$351$	1458.28	0.221759
$352$	0	0
$353$	4651.17	0.701294	0.350647	−	0.936508i	$-0.385962\pi$
$353$	4651.17	0.701294	0.350647	+	0.936508i	$0.385962\pi$
$354$	0	0
$355$	4379.59	0.654773
$356$	0	0
$357$	−12316.4	−1.82593
$358$	0	0
$359$	7607.30	1.11838	0.559190	−	0.829040i	$-0.311112\pi$
$359$	7607.30	1.11838	0.559190	+	0.829040i	$0.311112\pi$
$360$	0	0
$361$	397.523	0.0579565
$362$	0	0
$363$	22082.5	3.19292
$364$	0	0
$365$	2746.35	0.393837
$366$	0	0
$367$	−11709.5	−1.66548	−0.832739	−	0.553666i	$-0.813229\pi$
$367$	−11709.5	−1.66548	−0.832739	+	0.553666i	$0.813229\pi$
$368$	0	0
$369$	−1939.21	−0.273581
$370$	0	0
$371$	9920.08	1.38821
$372$	0	0
$373$	−2294.53	−0.318515	−0.159258	−	0.987237i	$-0.550910\pi$
$373$	−2294.53	−0.318515	−0.159258	+	0.987237i	$0.550910\pi$
$374$	0	0
$375$	−1034.11	−0.142403
$376$	0	0
$377$	353.994	0.0483597
$378$	0	0
$379$	−10194.9	−1.38173	−0.690866	−	0.722983i	$-0.742771\pi$
$379$	−10194.9	−1.38173	−0.690866	+	0.722983i	$0.742771\pi$
$380$	0	0
$381$	−827.850	−0.111318
$382$	0	0
$383$	3389.41	0.452196	0.226098	−	0.974105i	$-0.427403\pi$
$383$	3389.41	0.452196	0.226098	+	0.974105i	$0.427403\pi$
$384$	0	0
$385$	−10085.3	−1.33505
$386$	0	0
$387$	16567.8	2.17620
$388$	0	0
$389$	−13231.5	−1.72459	−0.862293	−	0.506410i	$-0.830972\pi$
$389$	−13231.5	−1.72459	−0.862293	+	0.506410i	$0.830972\pi$
$390$	0	0
$391$	5326.85	0.688978
$392$	0	0
$393$	12129.3	1.55685
$394$	0	0
$395$	−5714.71	−0.727945
$396$	0	0
$397$	10910.3	1.37927	0.689635	−	0.724157i	$-0.257771\pi$
$397$	10910.3	1.37927	0.689635	+	0.724157i	$0.257771\pi$
$398$	0	0
$399$	−22474.8	−2.81992
$400$	0	0
$401$	−2012.31	−0.250599	−0.125299	−	0.992119i	$-0.539989\pi$
$401$	−2012.31	−0.250599	−0.125299	+	0.992119i	$0.539989\pi$
$402$	0	0
$403$	3050.56	0.377070
$404$	0	0
$405$	652.848	0.0800995
$406$	0	0
$407$	−3826.98	−0.466084
$408$	0	0
$409$	14586.3	1.76344	0.881718	−	0.471777i	$-0.156387\pi$
$409$	14586.3	1.76344	0.881718	+	0.471777i	$0.156387\pi$
$410$	0	0
$411$	−11439.9	−1.37296
$412$	0	0
$413$	−17002.9	−2.02580
$414$	0	0
$415$	−5159.00	−0.610229
$416$	0	0
$417$	−9315.12	−1.09392
$418$	0	0
$419$	925.500	0.107908	0.0539542	−	0.998543i	$-0.482817\pi$
$419$	925.500	0.107908	0.0539542	+	0.998543i	$0.482817\pi$
$420$	0	0
$421$	6443.92	0.745979	0.372990	−	0.927835i	$-0.378333\pi$
$421$	6443.92	0.745979	0.372990	+	0.927835i	$0.378333\pi$
$422$	0	0
$423$	1200.36	0.137976
$424$	0	0
$425$	1167.06	0.133202
$426$	0	0
$427$	−4389.84	−0.497516
$428$	0	0
$429$	−6387.03	−0.718808
$430$	0	0
$431$	−15826.4	−1.76875	−0.884373	−	0.466782i	$-0.845413\pi$
$431$	−15826.4	−1.76875	−0.884373	+	0.466782i	$0.845413\pi$
$432$	0	0
$433$	−860.751	−0.0955313	−0.0477656	−	0.998859i	$-0.515210\pi$
$433$	−860.751	−0.0955313	−0.0477656	+	0.998859i	$0.515210\pi$
$434$	0	0
$435$	−1199.57	−0.132218
$436$	0	0
$437$	9720.33	1.06404
$438$	0	0
$439$	−122.829	−0.0133537	−0.00667687	−	0.999978i	$-0.502125\pi$
$439$	−122.829	−0.0133537	−0.00667687	+	0.999978i	$0.502125\pi$
$440$	0	0
$441$	27933.8	3.01628
$442$	0	0
$443$	11937.0	1.28024	0.640118	−	0.768277i	$-0.278886\pi$
$443$	11937.0	1.28024	0.640118	+	0.768277i	$0.278886\pi$
$444$	0	0
$445$	−3543.90	−0.377522
$446$	0	0
$447$	−17073.1	−1.80656
$448$	0	0
$449$	−664.050	−0.0697961	−0.0348981	−	0.999391i	$-0.511111\pi$
$449$	−664.050	−0.0697961	−0.0348981	+	0.999391i	$0.511111\pi$
$450$	0	0
$451$	2959.66	0.309013
$452$	0	0
$453$	−2242.49	−0.232586
$454$	0	0
$455$	1946.45	0.200551
$456$	0	0
$457$	−8173.19	−0.836599	−0.418300	−	0.908309i	$-0.637374\pi$
$457$	−8173.19	−0.836599	−0.418300	+	0.908309i	$0.637374\pi$
$458$	0	0
$459$	5576.96	0.567125
$460$	0	0
$461$	5415.29	0.547104	0.273552	−	0.961857i	$-0.411801\pi$
$461$	5415.29	0.547104	0.273552	+	0.961857i	$0.411801\pi$
$462$	0	0
$463$	−4748.17	−0.476601	−0.238301	−	0.971191i	$-0.576590\pi$
$463$	−4748.17	−0.476601	−0.238301	+	0.971191i	$0.576590\pi$
$464$	0	0
$465$	−10337.3	−1.03093
$466$	0	0
$467$	1607.01	0.159236	0.0796182	−	0.996825i	$-0.474630\pi$
$467$	1607.01	0.159236	0.0796182	+	0.996825i	$0.474630\pi$
$468$	0	0
$469$	−650.214	−0.0640173
$470$	0	0
$471$	−1918.88	−0.187722
$472$	0	0
$473$	−25286.2	−2.45805
$474$	0	0
$475$	2129.63	0.205714
$476$	0	0
$477$	−12890.4	−1.23734
$478$	0	0
$479$	14479.4	1.38117	0.690586	−	0.723250i	$-0.257353\pi$
$479$	14479.4	1.38117	0.690586	+	0.723250i	$0.257353\pi$
$480$	0	0
$481$	738.601	0.0700152
$482$	0	0
$483$	−30105.7	−2.83614
$484$	0	0
$485$	−8825.91	−0.826317
$486$	0	0
$487$	13683.1	1.27318	0.636591	−	0.771201i	$-0.280344\pi$
$487$	13683.1	1.27318	0.636591	+	0.771201i	$0.280344\pi$
$488$	0	0
$489$	15512.9	1.43460
$490$	0	0
$491$	−11077.8	−1.01820	−0.509099	−	0.860708i	$-0.670021\pi$
$491$	−11077.8	−1.01820	−0.509099	+	0.860708i	$0.670021\pi$
$492$	0	0
$493$	1353.79	0.123675
$494$	0	0
$495$	13105.1	1.18996
$496$	0	0
$497$	27934.3	2.52118
$498$	0	0
$499$	1972.33	0.176941	0.0884704	−	0.996079i	$-0.471802\pi$
$499$	1972.33	0.176941	0.0884704	+	0.996079i	$0.471802\pi$
$500$	0	0
$501$	−23939.4	−2.13479
$502$	0	0
$503$	19211.7	1.70300	0.851500	−	0.524355i	$-0.175694\pi$
$503$	19211.7	1.70300	0.851500	+	0.524355i	$0.175694\pi$
$504$	0	0
$505$	−781.845	−0.0688944
$506$	0	0
$507$	−16942.8	−1.48414
$508$	0	0
$509$	−12040.6	−1.04851	−0.524254	−	0.851562i	$-0.675656\pi$
$509$	−12040.6	−1.04851	−0.524254	+	0.851562i	$0.675656\pi$
$510$	0	0
$511$	17517.0	1.51645
$512$	0	0
$513$	10176.7	0.875855
$514$	0	0
$515$	5689.45	0.486810
$516$	0	0
$517$	−1832.02	−0.155846
$518$	0	0
$519$	10192.5	0.862043
$520$	0	0
$521$	−16908.5	−1.42183	−0.710917	−	0.703276i	$-0.751720\pi$
$521$	−16908.5	−1.42183	−0.710917	+	0.703276i	$0.751720\pi$
$522$	0	0
$523$	−19177.5	−1.60339	−0.801695	−	0.597733i	$-0.796068\pi$
$523$	−19177.5	−1.60339	−0.801695	+	0.597733i	$0.796068\pi$
$524$	0	0
$525$	−6595.87	−0.548319
$526$	0	0
$527$	11666.3	0.964315
$528$	0	0
$529$	853.675	0.0701631
$530$	0	0
$531$	22094.0	1.80564
$532$	0	0
$533$	−571.210	−0.0464200
$534$	0	0
$535$	−7387.03	−0.596952
$536$	0	0
$537$	10309.1	0.828439
$538$	0	0
$539$	−42633.1	−3.40694
$540$	0	0
$541$	1919.78	0.152565	0.0762826	−	0.997086i	$-0.475695\pi$
$541$	1919.78	0.152565	0.0762826	+	0.997086i	$0.475695\pi$
$542$	0	0
$543$	−23045.0	−1.82128
$544$	0	0
$545$	8769.04	0.689219
$546$	0	0
$547$	9321.37	0.728616	0.364308	−	0.931279i	$-0.381306\pi$
$547$	9321.37	0.728616	0.364308	+	0.931279i	$0.381306\pi$
$548$	0	0
$549$	5704.28	0.443447
$550$	0	0
$551$	2470.37	0.191001
$552$	0	0
$553$	−36450.1	−2.80292
$554$	0	0
$555$	−2502.87	−0.191425
$556$	0	0
$557$	−4970.29	−0.378093	−0.189047	−	0.981968i	$-0.560540\pi$
$557$	−4970.29	−0.378093	−0.189047	+	0.981968i	$0.560540\pi$
$558$	0	0
$559$	4880.19	0.369249
$560$	0	0
$561$	−24426.1	−1.83827
$562$	0	0
$563$	9771.90	0.731503	0.365752	−	0.930712i	$-0.380812\pi$
$563$	9771.90	0.731503	0.365752	+	0.930712i	$0.380812\pi$
$564$	0	0
$565$	2174.96	0.161949
$566$	0	0
$567$	4164.06	0.308420
$568$	0	0
$569$	6481.43	0.477532	0.238766	−	0.971077i	$-0.423257\pi$
$569$	6481.43	0.477532	0.238766	+	0.971077i	$0.423257\pi$
$570$	0	0
$571$	537.118	0.0393655	0.0196827	−	0.999806i	$-0.493734\pi$
$571$	537.118	0.0393655	0.0196827	+	0.999806i	$0.493734\pi$
$572$	0	0
$573$	−24628.7	−1.79560
$574$	0	0
$575$	2852.70	0.206897
$576$	0	0
$577$	−9078.66	−0.655025	−0.327513	−	0.944847i	$-0.606210\pi$
$577$	−9078.66	−0.655025	−0.327513	+	0.944847i	$0.606210\pi$
$578$	0	0
$579$	−2816.09	−0.202129
$580$	0	0
$581$	−32905.6	−2.34966
$582$	0	0
$583$	19673.6	1.39760
$584$	0	0
$585$	−2529.27	−0.178756
$586$	0	0
$587$	13313.2	0.936107	0.468054	−	0.883700i	$-0.344955\pi$
$587$	13313.2	0.936107	0.468054	+	0.883700i	$0.344955\pi$
$588$	0	0
$589$	21288.5	1.48927
$590$	0	0
$591$	−24947.6	−1.73639
$592$	0	0
$593$	12396.4	0.858447	0.429223	−	0.903198i	$-0.358787\pi$
$593$	12396.4	0.858447	0.429223	+	0.903198i	$0.358787\pi$
$594$	0	0
$595$	7443.86	0.512888
$596$	0	0
$597$	28497.4	1.95364
$598$	0	0
$599$	−23782.7	−1.62226	−0.811130	−	0.584865i	$-0.801147\pi$
$599$	−23782.7	−1.62226	−0.811130	+	0.584865i	$0.801147\pi$
$600$	0	0
$601$	1671.44	0.113444	0.0567218	−	0.998390i	$-0.481935\pi$
$601$	1671.44	0.113444	0.0567218	+	0.998390i	$0.481935\pi$
$602$	0	0
$603$	844.906	0.0570601
$604$	0	0
$605$	−13346.3	−0.896866
$606$	0	0
$607$	1471.65	0.0984063	0.0492031	−	0.998789i	$-0.484332\pi$
$607$	1471.65	0.0984063	0.0492031	+	0.998789i	$0.484332\pi$
$608$	0	0
$609$	−7651.21	−0.509101
$610$	0	0
$611$	353.577	0.0234111
$612$	0	0
$613$	15383.8	1.01362	0.506808	−	0.862059i	$-0.330825\pi$
$613$	15383.8	1.01362	0.506808	+	0.862059i	$0.330825\pi$
$614$	0	0
$615$	1935.64	0.126915
$616$	0	0
$617$	−26786.3	−1.74777	−0.873887	−	0.486129i	$-0.838408\pi$
$617$	−26786.3	−1.74777	−0.873887	+	0.486129i	$0.838408\pi$
$618$	0	0
$619$	−2144.28	−0.139234	−0.0696170	−	0.997574i	$-0.522178\pi$
$619$	−2144.28	−0.139234	−0.0696170	+	0.997574i	$0.522178\pi$
$620$	0	0
$621$	13632.0	0.880893
$622$	0	0
$623$	−22604.1	−1.45363
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−44572.3	−2.83899
$628$	0	0
$629$	2824.66	0.179056
$630$	0	0
$631$	9241.81	0.583059	0.291530	−	0.956562i	$-0.405836\pi$
$631$	9241.81	0.583059	0.291530	+	0.956562i	$0.405836\pi$
$632$	0	0
$633$	42252.1	2.65303
$634$	0	0
$635$	500.339	0.0312683
$636$	0	0
$637$	8228.12	0.511790
$638$	0	0
$639$	−36298.6	−2.24719
$640$	0	0
$641$	−26606.6	−1.63947	−0.819733	−	0.572746i	$-0.805878\pi$
$641$	−26606.6	−1.63947	−0.819733	+	0.572746i	$0.805878\pi$
$642$	0	0
$643$	−23626.5	−1.44905	−0.724525	−	0.689249i	$-0.757941\pi$
$643$	−23626.5	−1.44905	−0.724525	+	0.689249i	$0.757941\pi$
$644$	0	0
$645$	−16537.4	−1.00955
$646$	0	0
$647$	−14404.0	−0.875241	−0.437621	−	0.899160i	$-0.644179\pi$
$647$	−14404.0	−0.875241	−0.437621	+	0.899160i	$0.644179\pi$
$648$	0	0
$649$	−33720.3	−2.03950
$650$	0	0
$651$	−65934.6	−3.96956
$652$	0	0
$653$	20501.0	1.22858	0.614292	−	0.789079i	$-0.289442\pi$
$653$	20501.0	1.22858	0.614292	+	0.789079i	$0.289442\pi$
$654$	0	0
$655$	−7330.76	−0.437308
$656$	0	0
$657$	−22762.1	−1.35165
$658$	0	0
$659$	−11621.0	−0.686932	−0.343466	−	0.939165i	$-0.611601\pi$
$659$	−11621.0	−0.686932	−0.343466	+	0.939165i	$0.611601\pi$
$660$	0	0
$661$	956.471	0.0562820	0.0281410	−	0.999604i	$-0.491041\pi$
$661$	956.471	0.0562820	0.0281410	+	0.999604i	$0.491041\pi$
$662$	0	0
$663$	4714.20	0.276145
$664$	0	0
$665$	13583.4	0.792094
$666$	0	0
$667$	3309.14	0.192099
$668$	0	0
$669$	−24533.5	−1.41782
$670$	0	0
$671$	−8705.99	−0.500881
$672$	0	0
$673$	19146.6	1.09666	0.548328	−	0.836264i	$-0.315265\pi$
$673$	19146.6	1.09666	0.548328	+	0.836264i	$0.315265\pi$
$674$	0	0
$675$	2986.65	0.170305
$676$	0	0
$677$	−19087.3	−1.08358	−0.541791	−	0.840513i	$-0.682254\pi$
$677$	−19087.3	−1.08358	−0.541791	+	0.840513i	$0.682254\pi$
$678$	0	0
$679$	−56294.3	−3.18170
$680$	0	0
$681$	−30101.5	−1.69382
$682$	0	0
$683$	−2378.73	−0.133264	−0.0666322	−	0.997778i	$-0.521225\pi$
$683$	−2378.73	−0.133264	−0.0666322	+	0.997778i	$0.521225\pi$
$684$	0	0
$685$	6914.08	0.385655
$686$	0	0
$687$	21486.1	1.19323
$688$	0	0
$689$	−3796.98	−0.209947
$690$	0	0
$691$	−31752.8	−1.74810	−0.874049	−	0.485838i	$-0.838514\pi$
$691$	−31752.8	−1.74810	−0.874049	+	0.485838i	$0.838514\pi$
$692$	0	0
$693$	83588.3	4.58190
$694$	0	0
$695$	5629.91	0.307273
$696$	0	0
$697$	−2184.50	−0.118714
$698$	0	0
$699$	42570.9	2.30354
$700$	0	0
$701$	−29863.3	−1.60902	−0.804508	−	0.593942i	$-0.797571\pi$
$701$	−29863.3	−1.60902	−0.804508	+	0.593942i	$0.797571\pi$
$702$	0	0
$703$	5154.38	0.276531
$704$	0	0
$705$	−1198.16	−0.0640073
$706$	0	0
$707$	−4986.84	−0.265275
$708$	0	0
$709$	−18195.9	−0.963841	−0.481920	−	0.876215i	$-0.660061\pi$
$709$	−18195.9	−0.963841	−0.481920	+	0.876215i	$0.660061\pi$
$710$	0	0
$711$	47364.3	2.49831
$712$	0	0
$713$	28516.6	1.49783
$714$	0	0
$715$	3860.22	0.201908
$716$	0	0
$717$	43522.6	2.26692
$718$	0	0
$719$	12156.8	0.630561	0.315280	−	0.948999i	$-0.397901\pi$
$719$	12156.8	0.630561	0.315280	+	0.948999i	$0.397901\pi$
$720$	0	0
$721$	36289.0	1.87444
$722$	0	0
$723$	10719.7	0.551413
$724$	0	0
$725$	725.000	0.0371391
$726$	0	0
$727$	33816.2	1.72514	0.862569	−	0.505940i	$-0.168854\pi$
$727$	33816.2	1.72514	0.862569	+	0.505940i	$0.168854\pi$
$728$	0	0
$729$	−32095.8	−1.63063
$730$	0	0
$731$	18663.5	0.944314
$732$	0	0
$733$	19607.4	0.988019	0.494009	−	0.869457i	$-0.335531\pi$
$733$	19607.4	0.988019	0.494009	+	0.869457i	$0.335531\pi$
$734$	0	0
$735$	−27882.4	−1.39926
$736$	0	0
$737$	−1289.51	−0.0644502
$738$	0	0
$739$	13056.1	0.649902	0.324951	−	0.945731i	$-0.394652\pi$
$739$	13056.1	0.649902	0.324951	+	0.945731i	$0.394652\pi$
$740$	0	0
$741$	8602.39	0.426473
$742$	0	0
$743$	4228.03	0.208764	0.104382	−	0.994537i	$-0.466714\pi$
$743$	4228.03	0.208764	0.104382	+	0.994537i	$0.466714\pi$
$744$	0	0
$745$	10318.7	0.507448
$746$	0	0
$747$	42758.4	2.09431
$748$	0	0
$749$	−47116.7	−2.29854
$750$	0	0
$751$	566.117	0.0275072	0.0137536	−	0.999905i	$-0.495622\pi$
$751$	566.117	0.0275072	0.0137536	+	0.999905i	$0.495622\pi$
$752$	0	0
$753$	4842.93	0.234377
$754$	0	0
$755$	1355.33	0.0653317
$756$	0	0
$757$	38158.6	1.83210	0.916049	−	0.401066i	$-0.131360\pi$
$757$	38158.6	1.83210	0.916049	+	0.401066i	$0.131360\pi$
$758$	0	0
$759$	−59706.0	−2.85532
$760$	0	0
$761$	−15151.0	−0.721711	−0.360856	−	0.932622i	$-0.617515\pi$
$761$	−15151.0	−0.721711	−0.360856	+	0.932622i	$0.617515\pi$
$762$	0	0
$763$	55931.5	2.65381
$764$	0	0
$765$	−9672.75	−0.457149
$766$	0	0
$767$	6507.96	0.306374
$768$	0	0
$769$	39802.6	1.86647	0.933236	−	0.359264i	$-0.116972\pi$
$769$	39802.6	1.86647	0.933236	+	0.359264i	$0.116972\pi$
$770$	0	0
$771$	57792.2	2.69953
$772$	0	0
$773$	21828.3	1.01566	0.507832	−	0.861456i	$-0.330447\pi$
$773$	21828.3	1.01566	0.507832	+	0.861456i	$0.330447\pi$
$774$	0	0
$775$	6247.72	0.289580
$776$	0	0
$777$	−15964.1	−0.737076
$778$	0	0
$779$	−3986.23	−0.183340
$780$	0	0
$781$	55399.7	2.53823
$782$	0	0
$783$	3464.51	0.158125
$784$	0	0
$785$	1159.74	0.0527297
$786$	0	0
$787$	−11785.6	−0.533815	−0.266908	−	0.963722i	$-0.586002\pi$
$787$	−11785.6	−0.533815	−0.266908	+	0.963722i	$0.586002\pi$
$788$	0	0
$789$	32289.3	1.45694
$790$	0	0
$791$	13872.5	0.623579
$792$	0	0
$793$	1680.24	0.0752423
$794$	0	0
$795$	12866.7	0.574006
$796$	0	0
$797$	14512.6	0.644996	0.322498	−	0.946570i	$-0.395477\pi$
$797$	14512.6	0.644996	0.322498	+	0.946570i	$0.395477\pi$
$798$	0	0
$799$	1352.20	0.0598714
$800$	0	0
$801$	29372.3	1.29566
$802$	0	0
$803$	34740.0	1.52671
$804$	0	0
$805$	18195.4	0.796650
$806$	0	0
$807$	42469.8	1.85255
$808$	0	0
$809$	−33761.5	−1.46723	−0.733617	−	0.679564i	$-0.762169\pi$
$809$	−33761.5	−1.46723	−0.733617	+	0.679564i	$0.762169\pi$
$810$	0	0
$811$	19758.2	0.855491	0.427745	−	0.903899i	$-0.359308\pi$
$811$	19758.2	0.855491	0.427745	+	0.903899i	$0.359308\pi$
$812$	0	0
$813$	41009.1	1.76907
$814$	0	0
$815$	−9375.77	−0.402968
$816$	0	0
$817$	34056.8	1.45838
$818$	0	0
$819$	−16132.4	−0.688293
$820$	0	0
$821$	−26419.7	−1.12308	−0.561542	−	0.827448i	$-0.689792\pi$
$821$	−26419.7	−1.12308	−0.561542	+	0.827448i	$0.689792\pi$
$822$	0	0
$823$	7250.84	0.307106	0.153553	−	0.988140i	$-0.450928\pi$
$823$	7250.84	0.307106	0.153553	+	0.988140i	$0.450928\pi$
$824$	0	0
$825$	−13081.0	−0.552027
$826$	0	0
$827$	−24562.2	−1.03278	−0.516392	−	0.856353i	$-0.672725\pi$
$827$	−24562.2	−1.03278	−0.516392	+	0.856353i	$0.672725\pi$
$828$	0	0
$829$	−27935.0	−1.17035	−0.585177	−	0.810906i	$-0.698975\pi$
$829$	−27935.0	−1.17035	−0.585177	+	0.810906i	$0.698975\pi$
$830$	0	0
$831$	51559.3	2.15231
$832$	0	0
$833$	31467.1	1.30885
$834$	0	0
$835$	14468.6	0.599647
$836$	0	0
$837$	29855.6	1.23293
$838$	0	0
$839$	45395.0	1.86795	0.933975	−	0.357338i	$-0.116316\pi$
$839$	45395.0	1.86795	0.933975	+	0.357338i	$0.116316\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	9168.15	0.374576
$844$	0	0
$845$	10240.0	0.416883
$846$	0	0
$847$	−85126.6	−3.45335
$848$	0	0
$849$	28738.7	1.16173
$850$	0	0
$851$	6904.45	0.278121
$852$	0	0
$853$	17380.4	0.697647	0.348824	−	0.937188i	$-0.386581\pi$
$853$	17380.4	0.697647	0.348824	+	0.937188i	$0.386581\pi$
$854$	0	0
$855$	−17650.7	−0.706011
$856$	0	0
$857$	−617.644	−0.0246188	−0.0123094	−	0.999924i	$-0.503918\pi$
$857$	−617.644	−0.0246188	−0.0123094	+	0.999924i	$0.503918\pi$
$858$	0	0
$859$	−9097.11	−0.361338	−0.180669	−	0.983544i	$-0.557826\pi$
$859$	−9097.11	−0.361338	−0.180669	+	0.983544i	$0.557826\pi$
$860$	0	0
$861$	12346.1	0.488681
$862$	0	0
$863$	−31046.4	−1.22460	−0.612302	−	0.790624i	$-0.709756\pi$
$863$	−31046.4	−1.22460	−0.612302	+	0.790624i	$0.709756\pi$
$864$	0	0
$865$	−6160.17	−0.242141
$866$	0	0
$867$	−22616.0	−0.885905
$868$	0	0
$869$	−72288.3	−2.82188
$870$	0	0
$871$	248.874	0.00968171
$872$	0	0
$873$	73150.3	2.83592
$874$	0	0
$875$	3986.44	0.154018
$876$	0	0
$877$	−11910.1	−0.458581	−0.229290	−	0.973358i	$-0.573641\pi$
$877$	−11910.1	−0.458581	−0.229290	+	0.973358i	$0.573641\pi$
$878$	0	0
$879$	−76439.1	−2.93314
$880$	0	0
$881$	−38773.3	−1.48275	−0.741377	−	0.671088i	$-0.765827\pi$
$881$	−38773.3	−1.48275	−0.741377	+	0.671088i	$0.765827\pi$
$882$	0	0
$883$	−27665.8	−1.05439	−0.527197	−	0.849743i	$-0.676757\pi$
$883$	−27665.8	−1.05439	−0.527197	+	0.849743i	$0.676757\pi$
$884$	0	0
$885$	−22053.3	−0.837643
$886$	0	0
$887$	1907.89	0.0722219	0.0361109	−	0.999348i	$-0.488503\pi$
$887$	1907.89	0.0722219	0.0361109	+	0.999348i	$0.488503\pi$
$888$	0	0
$889$	3191.31	0.120397
$890$	0	0
$891$	8258.22	0.310506
$892$	0	0
$893$	2467.46	0.0924641
$894$	0	0
$895$	−6230.67	−0.232702
$896$	0	0
$897$	11523.2	0.428926
$898$	0	0
$899$	7247.35	0.268868
$900$	0	0
$901$	−14520.9	−0.536916
$902$	0	0
$903$	−105480.	−3.88722
$904$	0	0
$905$	13928.0	0.511584
$906$	0	0
$907$	30501.0	1.11661	0.558307	−	0.829634i	$-0.311451\pi$
$907$	30501.0	1.11661	0.558307	+	0.829634i	$0.311451\pi$
$908$	0	0
$909$	6480.04	0.236446
$910$	0	0
$911$	−11978.4	−0.435635	−0.217818	−	0.975989i	$-0.569894\pi$
$911$	−11978.4	−0.435635	−0.217818	+	0.975989i	$0.569894\pi$
$912$	0	0
$913$	−65258.8	−2.36556
$914$	0	0
$915$	−5693.78	−0.205717
$916$	0	0
$917$	−46757.8	−1.68384
$918$	0	0
$919$	14637.5	0.525404	0.262702	−	0.964877i	$-0.415386\pi$
$919$	14637.5	0.525404	0.262702	+	0.964877i	$0.415386\pi$
$920$	0	0
$921$	650.508	0.0232736
$922$	0	0
$923$	−10692.1	−0.381293
$924$	0	0
$925$	1512.70	0.0537699
$926$	0	0
$927$	−47154.9	−1.67073
$928$	0	0
$929$	−42459.2	−1.49951	−0.749754	−	0.661717i	$-0.769828\pi$
$929$	−42459.2	−1.49951	−0.749754	+	0.661717i	$0.769828\pi$
$930$	0	0
$931$	57420.6	2.02136
$932$	0	0
$933$	70419.8	2.47100
$934$	0	0
$935$	14762.8	0.516357
$936$	0	0
$937$	−47339.8	−1.65051	−0.825253	−	0.564764i	$-0.808967\pi$
$937$	−47339.8	−1.65051	−0.825253	+	0.564764i	$0.808967\pi$
$938$	0	0
$939$	63834.4	2.21848
$940$	0	0
$941$	17460.4	0.604881	0.302441	−	0.953168i	$-0.402199\pi$
$941$	17460.4	0.604881	0.302441	+	0.953168i	$0.402199\pi$
$942$	0	0
$943$	−5339.68	−0.184394
$944$	0	0
$945$	19049.7	0.655754
$946$	0	0
$947$	−54969.2	−1.88623	−0.943115	−	0.332467i	$-0.892119\pi$
$947$	−54969.2	−1.88623	−0.943115	+	0.332467i	$0.892119\pi$
$948$	0	0
$949$	−6704.76	−0.229342
$950$	0	0
$951$	13742.9	0.468604
$952$	0	0
$953$	4867.82	0.165461	0.0827305	−	0.996572i	$-0.473636\pi$
$953$	4867.82	0.165461	0.0827305	+	0.996572i	$0.473636\pi$
$954$	0	0
$955$	14885.2	0.504370
$956$	0	0
$957$	−15174.0	−0.512544
$958$	0	0
$959$	44100.1	1.48495
$960$	0	0
$961$	32663.4	1.09642
$962$	0	0
$963$	61224.7	2.04874
$964$	0	0
$965$	1702.00	0.0567766
$966$	0	0
$967$	26258.4	0.873231	0.436616	−	0.899648i	$-0.356177\pi$
$967$	26258.4	0.873231	0.436616	+	0.899648i	$0.356177\pi$
$968$	0	0
$969$	32898.4	1.09066
$970$	0	0
$971$	20712.4	0.684543	0.342272	−	0.939601i	$-0.388804\pi$
$971$	20712.4	0.684543	0.342272	+	0.939601i	$0.388804\pi$
$972$	0	0
$973$	35909.2	1.18314
$974$	0	0
$975$	2524.61	0.0829255
$976$	0	0
$977$	36164.9	1.18426	0.592128	−	0.805844i	$-0.298288\pi$
$977$	36164.9	1.18426	0.592128	+	0.805844i	$0.298288\pi$
$978$	0	0
$979$	−44828.7	−1.46346
$980$	0	0
$981$	−72678.9	−2.36540
$982$	0	0
$983$	−6013.98	−0.195134	−0.0975668	−	0.995229i	$-0.531106\pi$
$983$	−6013.98	−0.195134	−0.0975668	+	0.995229i	$0.531106\pi$
$984$	0	0
$985$	15077.9	0.487739
$986$	0	0
$987$	−7642.19	−0.246458
$988$	0	0
$989$	45620.0	1.46677
$990$	0	0
$991$	−24940.8	−0.799467	−0.399734	−	0.916631i	$-0.630897\pi$
$991$	−24940.8	−0.799467	−0.399734	+	0.916631i	$0.630897\pi$
$992$	0	0
$993$	35116.0	1.12223
$994$	0	0
$995$	−17223.4	−0.548762
$996$	0	0
$997$	51877.1	1.64791	0.823954	−	0.566657i	$-0.191763\pi$
$997$	51877.1	1.64791	0.823954	+	0.566657i	$0.191763\pi$
$998$	0	0
$999$	7228.63	0.228933

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1160.4.a.h.1.10	✓	11
4.3	odd	2		2320.4.a.z.1.2		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.4.a.h.1.10	✓	11	1.1	even	1	trivial
2320.4.a.z.1.2		11	4.3	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:	\( N \)	\(=\)	\( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1160.a (trivial)

\(f(q)\)	\(=\)	\(q+8.27289 q^{3} -5.00000 q^{5} -31.8915 q^{7} +41.4407 q^{9} -63.2476 q^{11} +12.2067 q^{13} -41.3644 q^{15} +46.6824 q^{17} +85.1852 q^{19} -263.835 q^{21} +114.108 q^{23} +25.0000 q^{25} +119.466 q^{27} +29.0000 q^{29} +249.909 q^{31} -523.240 q^{33} +159.457 q^{35} +60.5079 q^{37} +100.985 q^{39} -46.7949 q^{41} +399.796 q^{43} -207.203 q^{45} +28.9658 q^{47} +674.067 q^{49} +386.198 q^{51} -311.057 q^{53} +316.238 q^{55} +704.728 q^{57} +533.147 q^{59} +137.649 q^{61} -1321.60 q^{63} -61.0334 q^{65} +20.3883 q^{67} +944.004 q^{69} -875.918 q^{71} -549.269 q^{73} +206.822 q^{75} +2017.06 q^{77} +1142.94 q^{79} -130.570 q^{81} +1031.80 q^{83} -233.412 q^{85} +239.914 q^{87} +708.780 q^{89} -389.289 q^{91} +2067.47 q^{93} -425.926 q^{95} +1765.18 q^{97} -2621.02 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 8 q^{3} - 55 q^{5} + 16 q^{7} + 117 q^{9} + 26 q^{11} + 52 q^{13} - 40 q^{15} + 14 q^{17} + 146 q^{19} - 288 q^{21} - 80 q^{23} + 275 q^{25} + 224 q^{27} + 319 q^{29} - 38 q^{31} - 152 q^{33} - 80 q^{35}+ \cdots + 4690 q^{99}+O(q^{100}) \) 11 * q + 8 * q^3 - 55 * q^5 + 16 * q^7 + 117 * q^9 + 26 * q^11 + 52 * q^13 - 40 * q^15 + 14 * q^17 + 146 * q^19 - 288 * q^21 - 80 * q^23 + 275 * q^25 + 224 * q^27 + 319 * q^29 - 38 * q^31 - 152 * q^33 - 80 * q^35 + 376 * q^37 + 108 * q^39 - 146 * q^41 + 644 * q^43 - 585 * q^45 - 464 * q^47 + 393 * q^49 - 576 * q^51 - 188 * q^53 - 130 * q^55 + 1380 * q^57 + 672 * q^59 + 472 * q^61 + 628 * q^63 - 260 * q^65 + 1688 * q^67 + 138 * q^69 + 92 * q^71 + 1514 * q^73 + 200 * q^75 + 2884 * q^77 + 2130 * q^79 + 4067 * q^81 + 2584 * q^83 - 70 * q^85 + 232 * q^87 + 1606 * q^89 + 3208 * q^91 + 3540 * q^93 - 730 * q^95 + 3322 * q^97 + 4690 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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