LMFDB - Embedded newform 1160.4.a.d.1.7

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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 153x^{7} + 229x^{6} + 7393x^{5} - 8331x^{4} - 115371x^{3} + 125775x^{2} + 306882x + 29241 $ x^9 - 2x^8 - 153x^7 + 229x^6 + 7393x^5 - 8331x^4 - 115371x^3 + 125775x^2 + 306882x + 29241
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$4.19773$ 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)

$f(q)$	$=$	$q+3.19773 q^{3} +5.00000 q^{5} +17.9152 q^{7} -16.7745 q^{9} -52.6158 q^{11} +6.67297 q^{13} +15.9886 q^{15} +62.1701 q^{17} -94.9258 q^{19} +57.2878 q^{21} -61.4391 q^{23} +25.0000 q^{25} -139.979 q^{27} +29.0000 q^{29} +6.81367 q^{31} -168.251 q^{33} +89.5759 q^{35} -296.251 q^{37} +21.3383 q^{39} -184.550 q^{41} +215.352 q^{43} -83.8727 q^{45} -428.438 q^{47} -22.0463 q^{49} +198.803 q^{51} -472.395 q^{53} -263.079 q^{55} -303.547 q^{57} +335.523 q^{59} -507.339 q^{61} -300.519 q^{63} +33.3649 q^{65} -424.873 q^{67} -196.465 q^{69} +1013.16 q^{71} +666.292 q^{73} +79.9431 q^{75} -942.621 q^{77} -82.7376 q^{79} +5.29834 q^{81} -1021.81 q^{83} +310.850 q^{85} +92.7340 q^{87} +1610.87 q^{89} +119.547 q^{91} +21.7883 q^{93} -474.629 q^{95} +503.200 q^{97} +882.606 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 7 q^{3} + 45 q^{5} - 39 q^{7} + 72 q^{9} - 108 q^{11} - 19 q^{13} - 35 q^{15} + 91 q^{17} - 36 q^{19} - 200 q^{21} - 209 q^{23} + 225 q^{25} - 316 q^{27} + 261 q^{29} - 599 q^{31} - 192 q^{33} - 195 q^{35}+ \cdots - 2946 q^{99}+O(q^{100}) $ 9 * q - 7 * q^3 + 45 * q^5 - 39 * q^7 + 72 * q^9 - 108 * q^11 - 19 * q^13 - 35 * q^15 + 91 * q^17 - 36 * q^19 - 200 * q^21 - 209 * q^23 + 225 * q^25 - 316 * q^27 + 261 * q^29 - 599 * q^31 - 192 * q^33 - 195 * q^35 + 204 * q^37 - 221 * q^39 - 328 * q^41 - 717 * q^43 + 360 * q^45 - 646 * q^47 + 200 * q^49 - 1038 * q^51 + 71 * q^53 - 540 * q^55 + 494 * q^57 - 1591 * q^59 - 439 * q^61 - 354 * q^63 - 95 * q^65 - 1222 * q^67 + 873 * q^69 - 1240 * q^71 - 69 * q^73 - 175 * q^75 + 102 * q^77 - 1013 * q^79 - 171 * q^81 - 2728 * q^83 + 455 * q^85 - 203 * q^87 - 312 * q^89 - 1652 * q^91 - 702 * q^93 - 180 * q^95 + 1145 * q^97 - 2946 * 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
$3$	3.19773	0.615403	0.307701	−	0.951483i	$-0.400440\pi$
$3$	3.19773	0.615403	0.307701	+	0.951483i	$0.400440\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	17.9152	0.967329	0.483664	−	0.875254i	$-0.339306\pi$
$7$	17.9152	0.967329	0.483664	+	0.875254i	$0.339306\pi$
$8$	0	0
$9$	−16.7745	−0.621280
$10$	0	0
$11$	−52.6158	−1.44221	−0.721103	−	0.692828i	$-0.756364\pi$
$11$	−52.6158	−1.44221	−0.721103	+	0.692828i	$0.756364\pi$
$12$	0	0
$13$	6.67297	0.142365	0.0711827	−	0.997463i	$-0.477323\pi$
$13$	6.67297	0.142365	0.0711827	+	0.997463i	$0.477323\pi$
$14$	0	0
$15$	15.9886	0.275216
$16$	0	0
$17$	62.1701	0.886968	0.443484	−	0.896282i	$-0.353742\pi$
$17$	62.1701	0.886968	0.443484	+	0.896282i	$0.353742\pi$
$18$	0	0
$19$	−94.9258	−1.14618	−0.573091	−	0.819492i	$-0.694256\pi$
$19$	−94.9258	−1.14618	−0.573091	+	0.819492i	$0.694256\pi$
$20$	0	0
$21$	57.2878	0.595297
$22$	0	0
$23$	−61.4391	−0.556997	−0.278499	−	0.960437i	$-0.589837\pi$
$23$	−61.4391	−0.556997	−0.278499	+	0.960437i	$0.589837\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−139.979	−0.997740
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	6.81367	0.0394765	0.0197383	−	0.999805i	$-0.493717\pi$
$31$	6.81367	0.0394765	0.0197383	+	0.999805i	$0.493717\pi$
$32$	0	0
$33$	−168.251	−0.887537
$34$	0	0
$35$	89.5759	0.432603
$36$	0	0
$37$	−296.251	−1.31631	−0.658153	−	0.752884i	$-0.728662\pi$
$37$	−296.251	−1.31631	−0.658153	+	0.752884i	$0.728662\pi$
$38$	0	0
$39$	21.3383	0.0876120
$40$	0	0
$41$	−184.550	−0.702974	−0.351487	−	0.936193i	$-0.614324\pi$
$41$	−184.550	−0.702974	−0.351487	+	0.936193i	$0.614324\pi$
$42$	0	0
$43$	215.352	0.763742	0.381871	−	0.924216i	$-0.375280\pi$
$43$	215.352	0.763742	0.381871	+	0.924216i	$0.375280\pi$
$44$	0	0
$45$	−83.8727	−0.277845
$46$	0	0
$47$	−428.438	−1.32966	−0.664831	−	0.746994i	$-0.731496\pi$
$47$	−428.438	−1.32966	−0.664831	+	0.746994i	$0.731496\pi$
$48$	0	0
$49$	−22.0463	−0.0642751
$50$	0	0
$51$	198.803	0.545842
$52$	0	0
$53$	−472.395	−1.22431	−0.612155	−	0.790738i	$-0.709697\pi$
$53$	−472.395	−1.22431	−0.612155	+	0.790738i	$0.709697\pi$
$54$	0	0
$55$	−263.079	−0.644974
$56$	0	0
$57$	−303.547	−0.705364
$58$	0	0
$59$	335.523	0.740361	0.370181	−	0.928960i	$-0.379296\pi$
$59$	335.523	0.740361	0.370181	+	0.928960i	$0.379296\pi$
$60$	0	0
$61$	−507.339	−1.06489	−0.532444	−	0.846465i	$-0.678726\pi$
$61$	−507.339	−1.06489	−0.532444	+	0.846465i	$0.678726\pi$
$62$	0	0
$63$	−300.519	−0.600982
$64$	0	0
$65$	33.3649	0.0636677
$66$	0	0
$67$	−424.873	−0.774724	−0.387362	−	0.921928i	$-0.626614\pi$
$67$	−424.873	−0.774724	−0.387362	+	0.921928i	$0.626614\pi$
$68$	0	0
$69$	−196.465	−0.342778
$70$	0	0
$71$	1013.16	1.69353	0.846763	−	0.531971i	$-0.178548\pi$
$71$	1013.16	1.69353	0.846763	+	0.531971i	$0.178548\pi$
$72$	0	0
$73$	666.292	1.06827	0.534134	−	0.845400i	$-0.320638\pi$
$73$	666.292	1.06827	0.534134	+	0.845400i	$0.320638\pi$
$74$	0	0
$75$	79.9431	0.123081
$76$	0	0
$77$	−942.621	−1.39509
$78$	0	0
$79$	−82.7376	−0.117832	−0.0589159	−	0.998263i	$-0.518764\pi$
$79$	−82.7376	−0.117832	−0.0589159	+	0.998263i	$0.518764\pi$
$80$	0	0
$81$	5.29834	0.00726796
$82$	0	0
$83$	−1021.81	−1.35131	−0.675653	−	0.737220i	$-0.736138\pi$
$83$	−1021.81	−1.35131	−0.675653	+	0.737220i	$0.736138\pi$
$84$	0	0
$85$	310.850	0.396664
$86$	0	0
$87$	92.7340	0.114277
$88$	0	0
$89$	1610.87	1.91856	0.959281	−	0.282452i	$-0.0911478\pi$
$89$	1610.87	1.91856	0.959281	+	0.282452i	$0.0911478\pi$
$90$	0	0
$91$	119.547	0.137714
$92$	0	0
$93$	21.7883	0.0242939
$94$	0	0
$95$	−474.629	−0.512588
$96$	0	0
$97$	503.200	0.526724	0.263362	−	0.964697i	$-0.415169\pi$
$97$	503.200	0.526724	0.263362	+	0.964697i	$0.415169\pi$
$98$	0	0
$99$	882.606	0.896013
$100$	0	0
$101$	−135.869	−0.133856	−0.0669282	−	0.997758i	$-0.521320\pi$
$101$	−135.869	−0.133856	−0.0669282	+	0.997758i	$0.521320\pi$
$102$	0	0
$103$	−759.802	−0.726850	−0.363425	−	0.931624i	$-0.618393\pi$
$103$	−759.802	−0.726850	−0.363425	+	0.931624i	$0.618393\pi$
$104$	0	0
$105$	286.439	0.266225
$106$	0	0
$107$	−278.066	−0.251230	−0.125615	−	0.992079i	$-0.540090\pi$
$107$	−278.066	−0.251230	−0.125615	+	0.992079i	$0.540090\pi$
$108$	0	0
$109$	−1314.25	−1.15488	−0.577440	−	0.816433i	$-0.695948\pi$
$109$	−1314.25	−1.15488	−0.577440	+	0.816433i	$0.695948\pi$
$110$	0	0
$111$	−947.329	−0.810058
$112$	0	0
$113$	345.367	0.287517	0.143758	−	0.989613i	$-0.454081\pi$
$113$	345.367	0.287517	0.143758	+	0.989613i	$0.454081\pi$
$114$	0	0
$115$	−307.196	−0.249097
$116$	0	0
$117$	−111.936	−0.0884487
$118$	0	0
$119$	1113.79	0.857989
$120$	0	0
$121$	1437.42	1.07996
$122$	0	0
$123$	−590.142	−0.432612
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	586.061	0.409484	0.204742	−	0.978816i	$-0.434364\pi$
$127$	586.061	0.409484	0.204742	+	0.978816i	$0.434364\pi$
$128$	0	0
$129$	688.637	0.470009
$130$	0	0
$131$	403.077	0.268832	0.134416	−	0.990925i	$-0.457084\pi$
$131$	403.077	0.268832	0.134416	+	0.990925i	$0.457084\pi$
$132$	0	0
$133$	−1700.61	−1.10874
$134$	0	0
$135$	−699.895	−0.446203
$136$	0	0
$137$	−2772.78	−1.72916	−0.864580	−	0.502496i	$-0.832415\pi$
$137$	−2772.78	−1.72916	−0.864580	+	0.502496i	$0.832415\pi$
$138$	0	0
$139$	−872.749	−0.532558	−0.266279	−	0.963896i	$-0.585794\pi$
$139$	−872.749	−0.532558	−0.266279	+	0.963896i	$0.585794\pi$
$140$	0	0
$141$	−1370.03	−0.818278
$142$	0	0
$143$	−351.104	−0.205320
$144$	0	0
$145$	145.000	0.0830455
$146$	0	0
$147$	−70.4982	−0.0395550
$148$	0	0
$149$	−634.182	−0.348686	−0.174343	−	0.984685i	$-0.555780\pi$
$149$	−634.182	−0.348686	−0.174343	+	0.984685i	$0.555780\pi$
$150$	0	0
$151$	−3537.42	−1.90643	−0.953215	−	0.302292i	$-0.902248\pi$
$151$	−3537.42	−1.90643	−0.953215	+	0.302292i	$0.902248\pi$
$152$	0	0
$153$	−1042.87	−0.551055
$154$	0	0
$155$	34.0684	0.0176544
$156$	0	0
$157$	1718.64	0.873646	0.436823	−	0.899548i	$-0.356104\pi$
$157$	1718.64	0.873646	0.436823	+	0.899548i	$0.356104\pi$
$158$	0	0
$159$	−1510.59	−0.753444
$160$	0	0
$161$	−1100.69	−0.538800
$162$	0	0
$163$	−2706.14	−1.30037	−0.650187	−	0.759774i	$-0.725309\pi$
$163$	−2706.14	−1.30037	−0.650187	+	0.759774i	$0.725309\pi$
$164$	0	0
$165$	−841.254	−0.396919
$166$	0	0
$167$	725.949	0.336381	0.168190	−	0.985755i	$-0.446208\pi$
$167$	725.949	0.336381	0.168190	+	0.985755i	$0.446208\pi$
$168$	0	0
$169$	−2152.47	−0.979732
$170$	0	0
$171$	1592.34	0.712100
$172$	0	0
$173$	4296.47	1.88817	0.944087	−	0.329696i	$-0.106946\pi$
$173$	4296.47	1.88817	0.944087	+	0.329696i	$0.106946\pi$
$174$	0	0
$175$	447.879	0.193466
$176$	0	0
$177$	1072.91	0.455620
$178$	0	0
$179$	−2209.49	−0.922600	−0.461300	−	0.887244i	$-0.652617\pi$
$179$	−2209.49	−0.922600	−0.461300	+	0.887244i	$0.652617\pi$
$180$	0	0
$181$	1819.49	0.747191	0.373595	−	0.927592i	$-0.378125\pi$
$181$	1819.49	0.747191	0.373595	+	0.927592i	$0.378125\pi$
$182$	0	0
$183$	−1622.33	−0.655335
$184$	0	0
$185$	−1481.25	−0.588670
$186$	0	0
$187$	−3271.13	−1.27919
$188$	0	0
$189$	−2507.75	−0.965142
$190$	0	0
$191$	716.891	0.271583	0.135792	−	0.990737i	$-0.456642\pi$
$191$	716.891	0.271583	0.135792	+	0.990737i	$0.456642\pi$
$192$	0	0
$193$	1460.72	0.544793	0.272396	−	0.962185i	$-0.412184\pi$
$193$	1460.72	0.544793	0.272396	+	0.962185i	$0.412184\pi$
$194$	0	0
$195$	106.692	0.0391813
$196$	0	0
$197$	−1340.32	−0.484740	−0.242370	−	0.970184i	$-0.577925\pi$
$197$	−1340.32	−0.484740	−0.242370	+	0.970184i	$0.577925\pi$
$198$	0	0
$199$	−4141.10	−1.47515	−0.737574	−	0.675266i	$-0.764029\pi$
$199$	−4141.10	−1.47515	−0.737574	+	0.675266i	$0.764029\pi$
$200$	0	0
$201$	−1358.63	−0.476767
$202$	0	0
$203$	519.540	0.179628
$204$	0	0
$205$	−922.752	−0.314380
$206$	0	0
$207$	1030.61	0.346051
$208$	0	0
$209$	4994.59	1.65303
$210$	0	0
$211$	3672.55	1.19824	0.599121	−	0.800659i	$-0.295517\pi$
$211$	3672.55	1.19824	0.599121	+	0.800659i	$0.295517\pi$
$212$	0	0
$213$	3239.82	1.04220
$214$	0	0
$215$	1076.76	0.341556
$216$	0	0
$217$	122.068	0.0381868
$218$	0	0
$219$	2130.62	0.657415
$220$	0	0
$221$	414.859	0.126273
$222$	0	0
$223$	−2683.80	−0.805922	−0.402961	−	0.915217i	$-0.632019\pi$
$223$	−2683.80	−0.805922	−0.402961	+	0.915217i	$0.632019\pi$
$224$	0	0
$225$	−419.364	−0.124256
$226$	0	0
$227$	284.860	0.0832901	0.0416451	−	0.999132i	$-0.486740\pi$
$227$	284.860	0.0832901	0.0416451	+	0.999132i	$0.486740\pi$
$228$	0	0
$229$	801.992	0.231428	0.115714	−	0.993283i	$-0.463084\pi$
$229$	801.992	0.231428	0.115714	+	0.993283i	$0.463084\pi$
$230$	0	0
$231$	−3014.24	−0.858540
$232$	0	0
$233$	1519.67	0.427283	0.213642	−	0.976912i	$-0.431468\pi$
$233$	1519.67	0.427283	0.213642	+	0.976912i	$0.431468\pi$
$234$	0	0
$235$	−2142.19	−0.594643
$236$	0	0
$237$	−264.572	−0.0725140
$238$	0	0
$239$	623.331	0.168703	0.0843513	−	0.996436i	$-0.473118\pi$
$239$	623.331	0.168703	0.0843513	+	0.996436i	$0.473118\pi$
$240$	0	0
$241$	−1902.72	−0.508569	−0.254284	−	0.967129i	$-0.581840\pi$
$241$	−1902.72	−0.508569	−0.254284	+	0.967129i	$0.581840\pi$
$242$	0	0
$243$	3796.38	1.00221
$244$	0	0
$245$	−110.232	−0.0287447
$246$	0	0
$247$	−633.437	−0.163177
$248$	0	0
$249$	−3267.47	−0.831597
$250$	0	0
$251$	1063.32	0.267394	0.133697	−	0.991022i	$-0.457315\pi$
$251$	1063.32	0.267394	0.133697	+	0.991022i	$0.457315\pi$
$252$	0	0
$253$	3232.67	0.803304
$254$	0	0
$255$	994.014	0.244108
$256$	0	0
$257$	2456.41	0.596214	0.298107	−	0.954533i	$-0.403645\pi$
$257$	2456.41	0.596214	0.298107	+	0.954533i	$0.403645\pi$
$258$	0	0
$259$	−5307.39	−1.27330
$260$	0	0
$261$	−486.462	−0.115369
$262$	0	0
$263$	−928.726	−0.217748	−0.108874	−	0.994056i	$-0.534725\pi$
$263$	−928.726	−0.217748	−0.108874	+	0.994056i	$0.534725\pi$
$264$	0	0
$265$	−2361.97	−0.547528
$266$	0	0
$267$	5151.13	1.18069
$268$	0	0
$269$	−6970.48	−1.57992	−0.789959	−	0.613160i	$-0.789898\pi$
$269$	−6970.48	−1.57992	−0.789959	+	0.613160i	$0.789898\pi$
$270$	0	0
$271$	6555.07	1.46934	0.734671	−	0.678423i	$-0.237336\pi$
$271$	6555.07	1.46934	0.734671	+	0.678423i	$0.237336\pi$
$272$	0	0
$273$	382.280	0.0847496
$274$	0	0
$275$	−1315.39	−0.288441
$276$	0	0
$277$	−4361.45	−0.946043	−0.473022	−	0.881051i	$-0.656837\pi$
$277$	−4361.45	−0.946043	−0.473022	+	0.881051i	$0.656837\pi$
$278$	0	0
$279$	−114.296	−0.0245260
$280$	0	0
$281$	8137.78	1.72761	0.863806	−	0.503824i	$-0.168074\pi$
$281$	8137.78	1.72761	0.863806	+	0.503824i	$0.168074\pi$
$282$	0	0
$283$	−7638.55	−1.60447	−0.802234	−	0.597009i	$-0.796356\pi$
$283$	−7638.55	−1.60447	−0.802234	+	0.597009i	$0.796356\pi$
$284$	0	0
$285$	−1517.73	−0.315448
$286$	0	0
$287$	−3306.25	−0.680007
$288$	0	0
$289$	−1047.88	−0.213288
$290$	0	0
$291$	1609.09	0.324147
$292$	0	0
$293$	4.38130	0.000873578 0	0.000436789	−	1.00000i	$-0.499861\pi$
$293$	4.38130	0.000873578 0	0.000436789		1.00000i	$0.499861\pi$
$294$	0	0
$295$	1677.61	0.331100
$296$	0	0
$297$	7365.11	1.43895
$298$	0	0
$299$	−409.981	−0.0792971
$300$	0	0
$301$	3858.07	0.738790
$302$	0	0
$303$	−434.473	−0.0823756
$304$	0	0
$305$	−2536.70	−0.476232
$306$	0	0
$307$	−7951.94	−1.47831	−0.739155	−	0.673535i	$-0.764775\pi$
$307$	−7951.94	−1.47831	−0.739155	+	0.673535i	$0.764775\pi$
$308$	0	0
$309$	−2429.64	−0.447305
$310$	0	0
$311$	−9691.14	−1.76699	−0.883495	−	0.468440i	$-0.844816\pi$
$311$	−9691.14	−1.76699	−0.883495	+	0.468440i	$0.844816\pi$
$312$	0	0
$313$	−1947.15	−0.351627	−0.175814	−	0.984423i	$-0.556256\pi$
$313$	−1947.15	−0.351627	−0.175814	+	0.984423i	$0.556256\pi$
$314$	0	0
$315$	−1502.60	−0.268767
$316$	0	0
$317$	3617.14	0.640880	0.320440	−	0.947269i	$-0.396169\pi$
$317$	3617.14	0.640880	0.320440	+	0.947269i	$0.396169\pi$
$318$	0	0
$319$	−1525.86	−0.267811
$320$	0	0
$321$	−889.177	−0.154608
$322$	0	0
$323$	−5901.54	−1.01663
$324$	0	0
$325$	166.824	0.0284731
$326$	0	0
$327$	−4202.60	−0.710717
$328$	0	0
$329$	−7675.55	−1.28622
$330$	0	0
$331$	5949.33	0.987931	0.493965	−	0.869482i	$-0.335547\pi$
$331$	5949.33	0.987931	0.493965	+	0.869482i	$0.335547\pi$
$332$	0	0
$333$	4969.47	0.817794
$334$	0	0
$335$	−2124.36	−0.346467
$336$	0	0
$337$	−3873.87	−0.626181	−0.313091	−	0.949723i	$-0.601364\pi$
$337$	−3873.87	−0.626181	−0.313091	+	0.949723i	$0.601364\pi$
$338$	0	0
$339$	1104.39	0.176939
$340$	0	0
$341$	−358.507	−0.0569332
$342$	0	0
$343$	−6539.87	−1.02950
$344$	0	0
$345$	−982.327	−0.153295
$346$	0	0
$347$	4859.89	0.751852	0.375926	−	0.926650i	$-0.377325\pi$
$347$	4859.89	0.751852	0.375926	+	0.926650i	$0.377325\pi$
$348$	0	0
$349$	7470.46	1.14580	0.572901	−	0.819625i	$-0.305818\pi$
$349$	7470.46	1.14580	0.572901	+	0.819625i	$0.305818\pi$
$350$	0	0
$351$	−934.076	−0.142044
$352$	0	0
$353$	−8770.79	−1.32244	−0.661221	−	0.750191i	$-0.729961\pi$
$353$	−8770.79	−1.32244	−0.661221	+	0.750191i	$0.729961\pi$
$354$	0	0
$355$	5065.81	0.757368
$356$	0	0
$357$	3561.59	0.528009
$358$	0	0
$359$	6966.77	1.02421	0.512106	−	0.858923i	$-0.328866\pi$
$359$	6966.77	1.02421	0.512106	+	0.858923i	$0.328866\pi$
$360$	0	0
$361$	2151.90	0.313734
$362$	0	0
$363$	4596.48	0.664607
$364$	0	0
$365$	3331.46	0.477744
$366$	0	0
$367$	6601.00	0.938881	0.469440	−	0.882964i	$-0.344456\pi$
$367$	6601.00	0.938881	0.469440	+	0.882964i	$0.344456\pi$
$368$	0	0
$369$	3095.75	0.436744
$370$	0	0
$371$	−8463.04	−1.18431
$372$	0	0
$373$	8555.25	1.18760	0.593799	−	0.804614i	$-0.297628\pi$
$373$	8555.25	1.18760	0.593799	+	0.804614i	$0.297628\pi$
$374$	0	0
$375$	399.716	0.0550433
$376$	0	0
$377$	193.516	0.0264366
$378$	0	0
$379$	−7739.90	−1.04900	−0.524501	−	0.851410i	$-0.675748\pi$
$379$	−7739.90	−1.04900	−0.524501	+	0.851410i	$0.675748\pi$
$380$	0	0
$381$	1874.06	0.251998
$382$	0	0
$383$	269.048	0.0358949	0.0179474	−	0.999839i	$-0.494287\pi$
$383$	269.048	0.0358949	0.0179474	+	0.999839i	$0.494287\pi$
$384$	0	0
$385$	−4713.11	−0.623902
$386$	0	0
$387$	−3612.44	−0.474497
$388$	0	0
$389$	1006.78	0.131223	0.0656113	−	0.997845i	$-0.479100\pi$
$389$	1006.78	0.131223	0.0656113	+	0.997845i	$0.479100\pi$
$390$	0	0
$391$	−3819.67	−0.494039
$392$	0	0
$393$	1288.93	0.165440
$394$	0	0
$395$	−413.688	−0.0526960
$396$	0	0
$397$	13629.4	1.72302	0.861511	−	0.507739i	$-0.169518\pi$
$397$	13629.4	1.72302	0.861511	+	0.507739i	$0.169518\pi$
$398$	0	0
$399$	−5438.09	−0.682318
$400$	0	0
$401$	−8894.57	−1.10767	−0.553833	−	0.832628i	$-0.686835\pi$
$401$	−8894.57	−1.10767	−0.553833	+	0.832628i	$0.686835\pi$
$402$	0	0
$403$	45.4674	0.00562009
$404$	0	0
$405$	26.4917	0.00325033
$406$	0	0
$407$	15587.5	1.89838
$408$	0	0
$409$	11566.2	1.39832	0.699158	−	0.714968i	$-0.253559\pi$
$409$	11566.2	1.39832	0.699158	+	0.714968i	$0.253559\pi$
$410$	0	0
$411$	−8866.60	−1.06413
$412$	0	0
$413$	6010.95	0.716173
$414$	0	0
$415$	−5109.06	−0.604322
$416$	0	0
$417$	−2790.81	−0.327738
$418$	0	0
$419$	−5313.67	−0.619546	−0.309773	−	0.950811i	$-0.600253\pi$
$419$	−5313.67	−0.619546	−0.309773	+	0.950811i	$0.600253\pi$
$420$	0	0
$421$	8441.45	0.977223	0.488612	−	0.872501i	$-0.337504\pi$
$421$	8441.45	0.977223	0.488612	+	0.872501i	$0.337504\pi$
$422$	0	0
$423$	7186.86	0.826092
$424$	0	0
$425$	1554.25	0.177394
$426$	0	0
$427$	−9089.08	−1.03010
$428$	0	0
$429$	−1122.73	−0.126354
$430$	0	0
$431$	3013.08	0.336740	0.168370	−	0.985724i	$-0.446150\pi$
$431$	3013.08	0.336740	0.168370	+	0.985724i	$0.446150\pi$
$432$	0	0
$433$	13117.3	1.45583	0.727916	−	0.685666i	$-0.240489\pi$
$433$	13117.3	1.45583	0.727916	+	0.685666i	$0.240489\pi$
$434$	0	0
$435$	463.670	0.0511064
$436$	0	0
$437$	5832.15	0.638421
$438$	0	0
$439$	−12694.2	−1.38010	−0.690048	−	0.723764i	$-0.742411\pi$
$439$	−12694.2	−1.38010	−0.690048	+	0.723764i	$0.742411\pi$
$440$	0	0
$441$	369.818	0.0399328
$442$	0	0
$443$	−1465.19	−0.157141	−0.0785704	−	0.996909i	$-0.525036\pi$
$443$	−1465.19	−0.157141	−0.0785704	+	0.996909i	$0.525036\pi$
$444$	0	0
$445$	8054.36	0.858007
$446$	0	0
$447$	−2027.94	−0.214582
$448$	0	0
$449$	18063.8	1.89862	0.949312	−	0.314336i	$-0.101782\pi$
$449$	18063.8	1.89862	0.949312	+	0.314336i	$0.101782\pi$
$450$	0	0
$451$	9710.27	1.01383
$452$	0	0
$453$	−11311.7	−1.17322
$454$	0	0
$455$	597.737	0.0615876
$456$	0	0
$457$	12527.9	1.28234	0.641172	−	0.767397i	$-0.278448\pi$
$457$	12527.9	1.28234	0.641172	+	0.767397i	$0.278448\pi$
$458$	0	0
$459$	−8702.50	−0.884963
$460$	0	0
$461$	9493.78	0.959153	0.479576	−	0.877500i	$-0.340790\pi$
$461$	9493.78	0.959153	0.479576	+	0.877500i	$0.340790\pi$
$462$	0	0
$463$	−12422.3	−1.24690	−0.623448	−	0.781865i	$-0.714269\pi$
$463$	−12422.3	−1.24690	−0.623448	+	0.781865i	$0.714269\pi$
$464$	0	0
$465$	108.941	0.0108646
$466$	0	0
$467$	−2687.23	−0.266275	−0.133137	−	0.991098i	$-0.542505\pi$
$467$	−2687.23	−0.266275	−0.133137	+	0.991098i	$0.542505\pi$
$468$	0	0
$469$	−7611.68	−0.749413
$470$	0	0
$471$	5495.74	0.537644
$472$	0	0
$473$	−11330.9	−1.10147
$474$	0	0
$475$	−2373.14	−0.229236
$476$	0	0
$477$	7924.21	0.760639
$478$	0	0
$479$	6265.59	0.597666	0.298833	−	0.954305i	$-0.403403\pi$
$479$	6265.59	0.597666	0.298833	+	0.954305i	$0.403403\pi$
$480$	0	0
$481$	−1976.87	−0.187396
$482$	0	0
$483$	−3519.71	−0.331579
$484$	0	0
$485$	2516.00	0.235558
$486$	0	0
$487$	−2182.45	−0.203073	−0.101536	−	0.994832i	$-0.532376\pi$
$487$	−2182.45	−0.203073	−0.101536	+	0.994832i	$0.532376\pi$
$488$	0	0
$489$	−8653.48	−0.800254
$490$	0	0
$491$	12012.1	1.10407	0.552033	−	0.833822i	$-0.313852\pi$
$491$	12012.1	1.10407	0.552033	+	0.833822i	$0.313852\pi$
$492$	0	0
$493$	1802.93	0.164706
$494$	0	0
$495$	4413.03	0.400709
$496$	0	0
$497$	18151.0	1.63820
$498$	0	0
$499$	−5124.94	−0.459767	−0.229883	−	0.973218i	$-0.573834\pi$
$499$	−5124.94	−0.459767	−0.229883	+	0.973218i	$0.573834\pi$
$500$	0	0
$501$	2321.39	0.207010
$502$	0	0
$503$	11674.0	1.03482	0.517412	−	0.855736i	$-0.326896\pi$
$503$	11674.0	1.03482	0.517412	+	0.855736i	$0.326896\pi$
$504$	0	0
$505$	−679.346	−0.0598624
$506$	0	0
$507$	−6883.01	−0.602930
$508$	0	0
$509$	−6986.81	−0.608418	−0.304209	−	0.952605i	$-0.598392\pi$
$509$	−6986.81	−0.608418	−0.304209	+	0.952605i	$0.598392\pi$
$510$	0	0
$511$	11936.7	1.03337
$512$	0	0
$513$	13287.6	1.14359
$514$	0	0
$515$	−3799.01	−0.325057
$516$	0	0
$517$	22542.6	1.91765
$518$	0	0
$519$	13738.9	1.16199
$520$	0	0
$521$	3951.22	0.332257	0.166129	−	0.986104i	$-0.446873\pi$
$521$	3951.22	0.332257	0.166129	+	0.986104i	$0.446873\pi$
$522$	0	0
$523$	12384.7	1.03546	0.517730	−	0.855544i	$-0.326777\pi$
$523$	12384.7	1.03546	0.517730	+	0.855544i	$0.326777\pi$
$524$	0	0
$525$	1432.20	0.119059
$526$	0	0
$527$	423.606	0.0350144
$528$	0	0
$529$	−8392.24	−0.689754
$530$	0	0
$531$	−5628.24	−0.459971
$532$	0	0
$533$	−1231.50	−0.100079
$534$	0	0
$535$	−1390.33	−0.112353
$536$	0	0
$537$	−7065.36	−0.567770
$538$	0	0
$539$	1159.99	0.0926978
$540$	0	0
$541$	18209.6	1.44712	0.723561	−	0.690261i	$-0.242504\pi$
$541$	18209.6	1.44712	0.723561	+	0.690261i	$0.242504\pi$
$542$	0	0
$543$	5818.22	0.459823
$544$	0	0
$545$	−6571.23	−0.516478
$546$	0	0
$547$	−6706.17	−0.524196	−0.262098	−	0.965041i	$-0.584414\pi$
$547$	−6706.17	−0.524196	−0.262098	+	0.965041i	$0.584414\pi$
$548$	0	0
$549$	8510.39	0.661593
$550$	0	0
$551$	−2752.85	−0.212841
$552$	0	0
$553$	−1482.26	−0.113982
$554$	0	0
$555$	−4736.64	−0.362269
$556$	0	0
$557$	−2253.63	−0.171435	−0.0857177	−	0.996319i	$-0.527318\pi$
$557$	−2253.63	−0.171435	−0.0857177	+	0.996319i	$0.527318\pi$
$558$	0	0
$559$	1437.04	0.108730
$560$	0	0
$561$	−10460.2	−0.787217
$562$	0	0
$563$	−11102.2	−0.831091	−0.415545	−	0.909572i	$-0.636409\pi$
$563$	−11102.2	−0.831091	−0.415545	+	0.909572i	$0.636409\pi$
$564$	0	0
$565$	1726.84	0.128581
$566$	0	0
$567$	94.9208	0.00703051
$568$	0	0
$569$	5931.20	0.436992	0.218496	−	0.975838i	$-0.429885\pi$
$569$	5931.20	0.436992	0.218496	+	0.975838i	$0.429885\pi$
$570$	0	0
$571$	20111.2	1.47395	0.736976	−	0.675919i	$-0.236253\pi$
$571$	20111.2	1.47395	0.736976	+	0.675919i	$0.236253\pi$
$572$	0	0
$573$	2292.42	0.167133
$574$	0	0
$575$	−1535.98	−0.111399
$576$	0	0
$577$	−8488.16	−0.612421	−0.306210	−	0.951964i	$-0.599061\pi$
$577$	−8488.16	−0.612421	−0.306210	+	0.951964i	$0.599061\pi$
$578$	0	0
$579$	4670.98	0.335267
$580$	0	0
$581$	−18305.9	−1.30716
$582$	0	0
$583$	24855.4	1.76571
$584$	0	0
$585$	−559.680	−0.0395554
$586$	0	0
$587$	1833.65	0.128932	0.0644659	−	0.997920i	$-0.479466\pi$
$587$	1833.65	0.128932	0.0644659	+	0.997920i	$0.479466\pi$
$588$	0	0
$589$	−646.793	−0.0452473
$590$	0	0
$591$	−4285.97	−0.298310
$592$	0	0
$593$	2105.52	0.145806	0.0729032	−	0.997339i	$-0.476774\pi$
$593$	2105.52	0.145806	0.0729032	+	0.997339i	$0.476774\pi$
$594$	0	0
$595$	5568.94	0.383705
$596$	0	0
$597$	−13242.1	−0.907810
$598$	0	0
$599$	908.499	0.0619704	0.0309852	−	0.999520i	$-0.490136\pi$
$599$	908.499	0.0619704	0.0309852	+	0.999520i	$0.490136\pi$
$600$	0	0
$601$	−9802.80	−0.665332	−0.332666	−	0.943045i	$-0.607948\pi$
$601$	−9802.80	−0.665332	−0.332666	+	0.943045i	$0.607948\pi$
$602$	0	0
$603$	7127.05	0.481320
$604$	0	0
$605$	7187.10	0.482971
$606$	0	0
$607$	9792.26	0.654787	0.327394	−	0.944888i	$-0.393830\pi$
$607$	9792.26	0.654787	0.327394	+	0.944888i	$0.393830\pi$
$608$	0	0
$609$	1661.35	0.110544
$610$	0	0
$611$	−2858.95	−0.189298
$612$	0	0
$613$	4093.99	0.269747	0.134873	−	0.990863i	$-0.456937\pi$
$613$	4093.99	0.269747	0.134873	+	0.990863i	$0.456937\pi$
$614$	0	0
$615$	−2950.71	−0.193470
$616$	0	0
$617$	−23664.7	−1.54409	−0.772045	−	0.635568i	$-0.780766\pi$
$617$	−23664.7	−1.54409	−0.772045	+	0.635568i	$0.780766\pi$
$618$	0	0
$619$	9143.61	0.593720	0.296860	−	0.954921i	$-0.404060\pi$
$619$	9143.61	0.593720	0.296860	+	0.954921i	$0.404060\pi$
$620$	0	0
$621$	8600.19	0.555738
$622$	0	0
$623$	28859.1	1.85588
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	15971.3	1.01728
$628$	0	0
$629$	−18417.9	−1.16752
$630$	0	0
$631$	27746.1	1.75049	0.875243	−	0.483684i	$-0.160701\pi$
$631$	27746.1	1.75049	0.875243	+	0.483684i	$0.160701\pi$
$632$	0	0
$633$	11743.8	0.737401
$634$	0	0
$635$	2930.30	0.183127
$636$	0	0
$637$	−147.115	−0.00915054
$638$	0	0
$639$	−16995.3	−1.05215
$640$	0	0
$641$	18111.0	1.11598	0.557990	−	0.829848i	$-0.311573\pi$
$641$	18111.0	1.11598	0.557990	+	0.829848i	$0.311573\pi$
$642$	0	0
$643$	−5147.09	−0.315679	−0.157839	−	0.987465i	$-0.550453\pi$
$643$	−5147.09	−0.315679	−0.157839	+	0.987465i	$0.550453\pi$
$644$	0	0
$645$	3443.19	0.210194
$646$	0	0
$647$	−25849.3	−1.57070	−0.785348	−	0.619055i	$-0.787516\pi$
$647$	−25849.3	−1.57070	−0.785348	+	0.619055i	$0.787516\pi$
$648$	0	0
$649$	−17653.8	−1.06775
$650$	0	0
$651$	390.341	0.0235002
$652$	0	0
$653$	−3407.37	−0.204197	−0.102099	−	0.994774i	$-0.532556\pi$
$653$	−3407.37	−0.204197	−0.102099	+	0.994774i	$0.532556\pi$
$654$	0	0
$655$	2015.38	0.120225
$656$	0	0
$657$	−11176.7	−0.663693
$658$	0	0
$659$	−20168.4	−1.19218	−0.596092	−	0.802916i	$-0.703281\pi$
$659$	−20168.4	−1.19218	−0.596092	+	0.802916i	$0.703281\pi$
$660$	0	0
$661$	−20849.4	−1.22685	−0.613425	−	0.789753i	$-0.710209\pi$
$661$	−20849.4	−1.22685	−0.613425	+	0.789753i	$0.710209\pi$
$662$	0	0
$663$	1326.61	0.0777090
$664$	0	0
$665$	−8503.06	−0.495841
$666$	0	0
$667$	−1781.73	−0.103432
$668$	0	0
$669$	−8582.06	−0.495967
$670$	0	0
$671$	26694.1	1.53579
$672$	0	0
$673$	22770.2	1.30420	0.652099	−	0.758133i	$-0.273888\pi$
$673$	22770.2	1.30420	0.652099	+	0.758133i	$0.273888\pi$
$674$	0	0
$675$	−3499.48	−0.199548
$676$	0	0
$677$	109.708	0.00622812	0.00311406	−	0.999995i	$-0.499009\pi$
$677$	109.708	0.00622812	0.00311406	+	0.999995i	$0.499009\pi$
$678$	0	0
$679$	9014.91	0.509515
$680$	0	0
$681$	910.906	0.0512570
$682$	0	0
$683$	24062.9	1.34809	0.674043	−	0.738692i	$-0.264556\pi$
$683$	24062.9	1.34809	0.674043	+	0.738692i	$0.264556\pi$
$684$	0	0
$685$	−13863.9	−0.773304
$686$	0	0
$687$	2564.55	0.142422
$688$	0	0
$689$	−3152.28	−0.174299
$690$	0	0
$691$	−2469.01	−0.135927	−0.0679635	−	0.997688i	$-0.521650\pi$
$691$	−2469.01	−0.135927	−0.0679635	+	0.997688i	$0.521650\pi$
$692$	0	0
$693$	15812.0	0.866739
$694$	0	0
$695$	−4363.74	−0.238167
$696$	0	0
$697$	−11473.5	−0.623515
$698$	0	0
$699$	4859.49	0.262951
$700$	0	0
$701$	21976.8	1.18410	0.592050	−	0.805901i	$-0.298319\pi$
$701$	21976.8	1.18410	0.592050	+	0.805901i	$0.298319\pi$
$702$	0	0
$703$	28121.8	1.50873
$704$	0	0
$705$	−6850.14	−0.365945
$706$	0	0
$707$	−2434.12	−0.129483
$708$	0	0
$709$	−12148.5	−0.643509	−0.321755	−	0.946823i	$-0.604273\pi$
$709$	−12148.5	−0.643509	−0.321755	+	0.946823i	$0.604273\pi$
$710$	0	0
$711$	1387.89	0.0732065
$712$	0	0
$713$	−418.626	−0.0219883
$714$	0	0
$715$	−1755.52	−0.0918219
$716$	0	0
$717$	1993.24	0.103820
$718$	0	0
$719$	−30341.3	−1.57377	−0.786885	−	0.617100i	$-0.788307\pi$
$719$	−30341.3	−1.57377	−0.786885	+	0.617100i	$0.788307\pi$
$720$	0	0
$721$	−13612.0	−0.703103
$722$	0	0
$723$	−6084.38	−0.312975
$724$	0	0
$725$	725.000	0.0371391
$726$	0	0
$727$	−34508.1	−1.76043	−0.880217	−	0.474572i	$-0.842603\pi$
$727$	−34508.1	−1.76043	−0.880217	+	0.474572i	$0.842603\pi$
$728$	0	0
$729$	11996.7	0.609496
$730$	0	0
$731$	13388.5	0.677415
$732$	0	0
$733$	22674.2	1.14255	0.571275	−	0.820759i	$-0.306449\pi$
$733$	22674.2	1.14255	0.571275	+	0.820759i	$0.306449\pi$
$734$	0	0
$735$	−352.491	−0.0176896
$736$	0	0
$737$	22355.0	1.11731
$738$	0	0
$739$	−23215.0	−1.15559	−0.577793	−	0.816183i	$-0.696086\pi$
$739$	−23215.0	−1.15559	−0.577793	+	0.816183i	$0.696086\pi$
$740$	0	0
$741$	−2025.56	−0.100419
$742$	0	0
$743$	32946.8	1.62679	0.813393	−	0.581714i	$-0.197618\pi$
$743$	32946.8	1.62679	0.813393	+	0.581714i	$0.197618\pi$
$744$	0	0
$745$	−3170.91	−0.155937
$746$	0	0
$747$	17140.4	0.839539
$748$	0	0
$749$	−4981.59	−0.243022
$750$	0	0
$751$	31354.7	1.52350	0.761750	−	0.647872i	$-0.224341\pi$
$751$	31354.7	1.52350	0.761750	+	0.647872i	$0.224341\pi$
$752$	0	0
$753$	3400.20	0.164555
$754$	0	0
$755$	−17687.1	−0.852582
$756$	0	0
$757$	5133.20	0.246459	0.123229	−	0.992378i	$-0.460675\pi$
$757$	5133.20	0.246459	0.123229	+	0.992378i	$0.460675\pi$
$758$	0	0
$759$	10337.2	0.494356
$760$	0	0
$761$	1128.72	0.0537664	0.0268832	−	0.999639i	$-0.491442\pi$
$761$	1128.72	0.0537664	0.0268832	+	0.999639i	$0.491442\pi$
$762$	0	0
$763$	−23545.0	−1.11715
$764$	0	0
$765$	−5214.37	−0.246439
$766$	0	0
$767$	2238.93	0.105402
$768$	0	0
$769$	7523.51	0.352802	0.176401	−	0.984318i	$-0.443554\pi$
$769$	7523.51	0.352802	0.176401	+	0.984318i	$0.443554\pi$
$770$	0	0
$771$	7854.94	0.366911
$772$	0	0
$773$	16606.5	0.772696	0.386348	−	0.922353i	$-0.373736\pi$
$773$	16606.5	0.772696	0.386348	+	0.922353i	$0.373736\pi$
$774$	0	0
$775$	170.342	0.00789530
$776$	0	0
$777$	−16971.6	−0.783593
$778$	0	0
$779$	17518.6	0.805736
$780$	0	0
$781$	−53308.3	−2.44241
$782$	0	0
$783$	−4059.39	−0.185276
$784$	0	0
$785$	8593.20	0.390706
$786$	0	0
$787$	−8302.34	−0.376044	−0.188022	−	0.982165i	$-0.560208\pi$
$787$	−8302.34	−0.376044	−0.188022	+	0.982165i	$0.560208\pi$
$788$	0	0
$789$	−2969.81	−0.134003
$790$	0	0
$791$	6187.32	0.278123
$792$	0	0
$793$	−3385.46	−0.151603
$794$	0	0
$795$	−7552.95	−0.336950
$796$	0	0
$797$	3798.67	0.168828	0.0844139	−	0.996431i	$-0.473098\pi$
$797$	3798.67	0.168828	0.0844139	+	0.996431i	$0.473098\pi$
$798$	0	0
$799$	−26636.0	−1.17937
$800$	0	0
$801$	−27021.7	−1.19196
$802$	0	0
$803$	−35057.5	−1.54066
$804$	0	0
$805$	−5503.46	−0.240959
$806$	0	0
$807$	−22289.7	−0.972285
$808$	0	0
$809$	14597.8	0.634401	0.317201	−	0.948358i	$-0.397257\pi$
$809$	14597.8	0.634401	0.317201	+	0.948358i	$0.397257\pi$
$810$	0	0
$811$	−30975.4	−1.34118	−0.670588	−	0.741830i	$-0.733958\pi$
$811$	−30975.4	−1.34118	−0.670588	+	0.741830i	$0.733958\pi$
$812$	0	0
$813$	20961.3	0.904238
$814$	0	0
$815$	−13530.7	−0.581545
$816$	0	0
$817$	−20442.5	−0.875388
$818$	0	0
$819$	−2005.35	−0.0855589
$820$	0	0
$821$	28057.6	1.19271	0.596355	−	0.802721i	$-0.296615\pi$
$821$	28057.6	1.19271	0.596355	+	0.802721i	$0.296615\pi$
$822$	0	0
$823$	−22257.8	−0.942718	−0.471359	−	0.881941i	$-0.656236\pi$
$823$	−22257.8	−0.942718	−0.471359	+	0.881941i	$0.656236\pi$
$824$	0	0
$825$	−4206.27	−0.177507
$826$	0	0
$827$	26900.2	1.13109	0.565546	−	0.824717i	$-0.308666\pi$
$827$	26900.2	1.13109	0.565546	+	0.824717i	$0.308666\pi$
$828$	0	0
$829$	−30057.2	−1.25926	−0.629631	−	0.776894i	$-0.716794\pi$
$829$	−30057.2	−1.25926	−0.629631	+	0.776894i	$0.716794\pi$
$830$	0	0
$831$	−13946.7	−0.582197
$832$	0	0
$833$	−1370.62	−0.0570099
$834$	0	0
$835$	3629.75	0.150434
$836$	0	0
$837$	−953.771	−0.0393873
$838$	0	0
$839$	20278.8	0.834450	0.417225	−	0.908803i	$-0.363003\pi$
$839$	20278.8	0.834450	0.417225	+	0.908803i	$0.363003\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	26022.4	1.06318
$844$	0	0
$845$	−10762.4	−0.438150
$846$	0	0
$847$	25751.6	1.04467
$848$	0	0
$849$	−24426.0	−0.987394
$850$	0	0
$851$	18201.4	0.733179
$852$	0	0
$853$	−28158.6	−1.13028	−0.565141	−	0.824994i	$-0.691178\pi$
$853$	−28158.6	−1.13028	−0.565141	+	0.824994i	$0.691178\pi$
$854$	0	0
$855$	7961.68	0.318461
$856$	0	0
$857$	20284.0	0.808506	0.404253	−	0.914647i	$-0.367532\pi$
$857$	20284.0	0.808506	0.404253	+	0.914647i	$0.367532\pi$
$858$	0	0
$859$	−33964.6	−1.34908	−0.674538	−	0.738240i	$-0.735657\pi$
$859$	−33964.6	−1.34908	−0.674538	+	0.738240i	$0.735657\pi$
$860$	0	0
$861$	−10572.5	−0.418478
$862$	0	0
$863$	−753.223	−0.0297103	−0.0148552	−	0.999890i	$-0.504729\pi$
$863$	−753.223	−0.0297103	−0.0148552	+	0.999890i	$0.504729\pi$
$864$	0	0
$865$	21482.3	0.844417
$866$	0	0
$867$	−3350.85	−0.131258
$868$	0	0
$869$	4353.31	0.169938
$870$	0	0
$871$	−2835.16	−0.110294
$872$	0	0
$873$	−8440.95	−0.327243
$874$	0	0
$875$	2239.40	0.0865205
$876$	0	0
$877$	−34014.3	−1.30967	−0.654836	−	0.755771i	$-0.727262\pi$
$877$	−34014.3	−1.30967	−0.654836	+	0.755771i	$0.727262\pi$
$878$	0	0
$879$	14.0102	0.000537602 0
$880$	0	0
$881$	−39180.0	−1.49830	−0.749152	−	0.662398i	$-0.769539\pi$
$881$	−39180.0	−1.49830	−0.749152	+	0.662398i	$0.769539\pi$
$882$	0	0
$883$	−17470.1	−0.665818	−0.332909	−	0.942959i	$-0.608030\pi$
$883$	−17470.1	−0.665818	−0.332909	+	0.942959i	$0.608030\pi$
$884$	0	0
$885$	5364.55	0.203760
$886$	0	0
$887$	23954.8	0.906792	0.453396	−	0.891309i	$-0.350212\pi$
$887$	23954.8	0.906792	0.453396	+	0.891309i	$0.350212\pi$
$888$	0	0
$889$	10499.4	0.396106
$890$	0	0
$891$	−278.777	−0.0104819
$892$	0	0
$893$	40669.8	1.52404
$894$	0	0
$895$	−11047.5	−0.412599
$896$	0	0
$897$	−1311.01	−0.0487996
$898$	0	0
$899$	197.597	0.00733060
$900$	0	0
$901$	−29368.8	−1.08592
$902$	0	0
$903$	12337.1	0.454653
$904$	0	0
$905$	9097.44	0.334154
$906$	0	0
$907$	−14323.5	−0.524371	−0.262186	−	0.965017i	$-0.584443\pi$
$907$	−14323.5	−0.524371	−0.262186	+	0.965017i	$0.584443\pi$
$908$	0	0
$909$	2279.15	0.0831623
$910$	0	0
$911$	12582.2	0.457593	0.228796	−	0.973474i	$-0.426521\pi$
$911$	12582.2	0.457593	0.228796	+	0.973474i	$0.426521\pi$
$912$	0	0
$913$	53763.4	1.94886
$914$	0	0
$915$	−8111.66	−0.293075
$916$	0	0
$917$	7221.20	0.260049
$918$	0	0
$919$	−24674.7	−0.885683	−0.442842	−	0.896600i	$-0.646030\pi$
$919$	−24674.7	−0.885683	−0.442842	+	0.896600i	$0.646030\pi$
$920$	0	0
$921$	−25428.1	−0.909756
$922$	0	0
$923$	6760.80	0.241099
$924$	0	0
$925$	−7406.27	−0.263261
$926$	0	0
$927$	12745.3	0.451577
$928$	0	0
$929$	−4197.11	−0.148227	−0.0741135	−	0.997250i	$-0.523613\pi$
$929$	−4197.11	−0.148227	−0.0741135	+	0.997250i	$0.523613\pi$
$930$	0	0
$931$	2092.77	0.0736709
$932$	0	0
$933$	−30989.6	−1.08741
$934$	0	0
$935$	−16355.6	−0.572071
$936$	0	0
$937$	−39945.1	−1.39269	−0.696345	−	0.717707i	$-0.745192\pi$
$937$	−39945.1	−1.39269	−0.696345	+	0.717707i	$0.745192\pi$
$938$	0	0
$939$	−6226.45	−0.216392
$940$	0	0
$941$	−24254.1	−0.840235	−0.420117	−	0.907470i	$-0.638011\pi$
$941$	−24254.1	−0.840235	−0.420117	+	0.907470i	$0.638011\pi$
$942$	0	0
$943$	11338.6	0.391555
$944$	0	0
$945$	−12538.7	−0.431625
$946$	0	0
$947$	22187.2	0.761338	0.380669	−	0.924711i	$-0.375694\pi$
$947$	22187.2	0.761338	0.380669	+	0.924711i	$0.375694\pi$
$948$	0	0
$949$	4446.15	0.152084
$950$	0	0
$951$	11566.6	0.394399
$952$	0	0
$953$	16582.0	0.563634	0.281817	−	0.959468i	$-0.409063\pi$
$953$	16582.0	0.563634	0.281817	+	0.959468i	$0.409063\pi$
$954$	0	0
$955$	3584.46	0.121456
$956$	0	0
$957$	−4879.27	−0.164811
$958$	0	0
$959$	−49674.9	−1.67267
$960$	0	0
$961$	−29744.6	−0.998442
$962$	0	0
$963$	4664.42	0.156084
$964$	0	0
$965$	7303.60	0.243639
$966$	0	0
$967$	39863.3	1.32567	0.662833	−	0.748768i	$-0.269354\pi$
$967$	39863.3	1.32567	0.662833	+	0.748768i	$0.269354\pi$
$968$	0	0
$969$	−18871.5	−0.625635
$970$	0	0
$971$	7854.55	0.259593	0.129796	−	0.991541i	$-0.458568\pi$
$971$	7854.55	0.259593	0.129796	+	0.991541i	$0.458568\pi$
$972$	0	0
$973$	−15635.5	−0.515159
$974$	0	0
$975$	533.458	0.0175224
$976$	0	0
$977$	26872.1	0.879954	0.439977	−	0.898009i	$-0.354987\pi$
$977$	26872.1	0.879954	0.439977	+	0.898009i	$0.354987\pi$
$978$	0	0
$979$	−84757.3	−2.76696
$980$	0	0
$981$	22045.9	0.717504
$982$	0	0
$983$	−29487.8	−0.956781	−0.478390	−	0.878147i	$-0.658780\pi$
$983$	−29487.8	−0.956781	−0.478390	+	0.878147i	$0.658780\pi$
$984$	0	0
$985$	−6701.59	−0.216782
$986$	0	0
$987$	−24544.3	−0.791543
$988$	0	0
$989$	−13231.0	−0.425402
$990$	0	0
$991$	23743.6	0.761092	0.380546	−	0.924762i	$-0.375736\pi$
$991$	23743.6	0.761092	0.380546	+	0.924762i	$0.375736\pi$
$992$	0	0
$993$	19024.3	0.607975
$994$	0	0
$995$	−20705.5	−0.659706
$996$	0	0
$997$	−16962.5	−0.538824	−0.269412	−	0.963025i	$-0.586829\pi$
$997$	−16962.5	−0.538824	−0.269412	+	0.963025i	$0.586829\pi$
$998$	0	0
$999$	41468.9	1.31333

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1160.4.a.d.1.7	✓	9
4.3	odd	2		2320.4.a.y.1.3		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.4.a.d.1.7	✓	9	1.1	even	1	trivial
2320.4.a.y.1.3		9	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)

\(f(q)\)	\(=\)	\(q+3.19773 q^{3} +5.00000 q^{5} +17.9152 q^{7} -16.7745 q^{9} -52.6158 q^{11} +6.67297 q^{13} +15.9886 q^{15} +62.1701 q^{17} -94.9258 q^{19} +57.2878 q^{21} -61.4391 q^{23} +25.0000 q^{25} -139.979 q^{27} +29.0000 q^{29} +6.81367 q^{31} -168.251 q^{33} +89.5759 q^{35} -296.251 q^{37} +21.3383 q^{39} -184.550 q^{41} +215.352 q^{43} -83.8727 q^{45} -428.438 q^{47} -22.0463 q^{49} +198.803 q^{51} -472.395 q^{53} -263.079 q^{55} -303.547 q^{57} +335.523 q^{59} -507.339 q^{61} -300.519 q^{63} +33.3649 q^{65} -424.873 q^{67} -196.465 q^{69} +1013.16 q^{71} +666.292 q^{73} +79.9431 q^{75} -942.621 q^{77} -82.7376 q^{79} +5.29834 q^{81} -1021.81 q^{83} +310.850 q^{85} +92.7340 q^{87} +1610.87 q^{89} +119.547 q^{91} +21.7883 q^{93} -474.629 q^{95} +503.200 q^{97} +882.606 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 7 q^{3} + 45 q^{5} - 39 q^{7} + 72 q^{9} - 108 q^{11} - 19 q^{13} - 35 q^{15} + 91 q^{17} - 36 q^{19} - 200 q^{21} - 209 q^{23} + 225 q^{25} - 316 q^{27} + 261 q^{29} - 599 q^{31} - 192 q^{33} - 195 q^{35}+ \cdots - 2946 q^{99}+O(q^{100}) \) 9 * q - 7 * q^3 + 45 * q^5 - 39 * q^7 + 72 * q^9 - 108 * q^11 - 19 * q^13 - 35 * q^15 + 91 * q^17 - 36 * q^19 - 200 * q^21 - 209 * q^23 + 225 * q^25 - 316 * q^27 + 261 * q^29 - 599 * q^31 - 192 * q^33 - 195 * q^35 + 204 * q^37 - 221 * q^39 - 328 * q^41 - 717 * q^43 + 360 * q^45 - 646 * q^47 + 200 * q^49 - 1038 * q^51 + 71 * q^53 - 540 * q^55 + 494 * q^57 - 1591 * q^59 - 439 * q^61 - 354 * q^63 - 95 * q^65 - 1222 * q^67 + 873 * q^69 - 1240 * q^71 - 69 * q^73 - 175 * q^75 + 102 * q^77 - 1013 * q^79 - 171 * q^81 - 2728 * q^83 + 455 * q^85 - 203 * q^87 - 312 * q^89 - 1652 * q^91 - 702 * q^93 - 180 * q^95 + 1145 * q^97 - 2946 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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