LMFDB - Embedded newform 1160.4.a.g.1.8

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} - 2 x^{10} - 179 x^{9} + 370 x^{8} + 10353 x^{7} - 19394 x^{6} - 210392 x^{5} + 267796 x^{4} + \cdots + 567808 $ x^11 - 2x^10 - 179x^9 + 370x^8 + 10353x^7 - 19394x^6 - 210392x^5 + 267796x^4 + 1107520x^3 - 894176x^2 - 757696x + 567808
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.81857$ 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)

$f(q)$	$=$	$q+2.81857 q^{3} -5.00000 q^{5} -26.0684 q^{7} -19.0556 q^{9} -23.3506 q^{11} +1.24079 q^{13} -14.0929 q^{15} +85.2728 q^{17} -111.018 q^{19} -73.4757 q^{21} +35.9888 q^{23} +25.0000 q^{25} -129.811 q^{27} -29.0000 q^{29} +58.5275 q^{31} -65.8153 q^{33} +130.342 q^{35} -226.421 q^{37} +3.49725 q^{39} +447.304 q^{41} -490.884 q^{43} +95.2782 q^{45} +223.382 q^{47} +336.562 q^{49} +240.347 q^{51} +137.117 q^{53} +116.753 q^{55} -312.912 q^{57} +352.244 q^{59} +133.867 q^{61} +496.751 q^{63} -6.20394 q^{65} +450.653 q^{67} +101.437 q^{69} +462.170 q^{71} -17.1569 q^{73} +70.4643 q^{75} +608.712 q^{77} +1221.19 q^{79} +148.620 q^{81} -481.608 q^{83} -426.364 q^{85} -81.7386 q^{87} -179.628 q^{89} -32.3454 q^{91} +164.964 q^{93} +555.089 q^{95} +956.201 q^{97} +444.960 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 2 q^{3} - 55 q^{5} + 56 q^{7} + 65 q^{9} - 8 q^{11} + 16 q^{13} - 10 q^{15} + 64 q^{17} + 160 q^{19} - 48 q^{21} + 140 q^{23} + 275 q^{25} - 136 q^{27} - 319 q^{29} - 144 q^{31} - 424 q^{33} - 280 q^{35}+ \cdots + 6884 q^{99}+O(q^{100}) $ 11 * q + 2 * q^3 - 55 * q^5 + 56 * q^7 + 65 * q^9 - 8 * q^11 + 16 * q^13 - 10 * q^15 + 64 * q^17 + 160 * q^19 - 48 * q^21 + 140 * q^23 + 275 * q^25 - 136 * q^27 - 319 * q^29 - 144 * q^31 - 424 * q^33 - 280 * q^35 - 50 * q^37 + 572 * q^39 + 62 * q^41 + 326 * q^43 - 325 * q^45 + 1074 * q^47 + 1513 * q^49 - 144 * q^51 + 576 * q^53 + 40 * q^55 - 896 * q^57 - 468 * q^59 + 660 * q^61 + 2552 * q^63 - 80 * q^65 + 892 * q^67 - 510 * q^69 + 920 * q^71 - 676 * q^73 + 50 * q^75 + 1708 * q^77 + 2308 * q^79 + 1287 * q^81 - 1548 * q^83 - 320 * q^85 - 58 * q^87 + 1854 * q^89 + 3100 * q^91 + 5424 * q^93 - 800 * q^95 + 4724 * q^97 + 6884 * 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
$3$	2.81857	0.542435	0.271217	−	0.962518i	$-0.412574\pi$
$3$	2.81857	0.542435	0.271217	+	0.962518i	$0.412574\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−26.0684	−1.40756	−0.703781	−	0.710417i	$-0.748506\pi$
$7$	−26.0684	−1.40756	−0.703781	+	0.710417i	$0.748506\pi$
$8$	0	0
$9$	−19.0556	−0.705765
$10$	0	0
$11$	−23.3506	−0.640042	−0.320021	−	0.947410i	$-0.603690\pi$
$11$	−23.3506	−0.640042	−0.320021	+	0.947410i	$0.603690\pi$
$12$	0	0
$13$	1.24079	0.0264717	0.0132359	−	0.999912i	$-0.495787\pi$
$13$	1.24079	0.0264717	0.0132359	+	0.999912i	$0.495787\pi$
$14$	0	0
$15$	−14.0929	−0.242584
$16$	0	0
$17$	85.2728	1.21657	0.608285	−	0.793719i	$-0.291858\pi$
$17$	85.2728	1.21657	0.608285	+	0.793719i	$0.291858\pi$
$18$	0	0
$19$	−111.018	−1.34049	−0.670243	−	0.742142i	$-0.733810\pi$
$19$	−111.018	−1.34049	−0.670243	+	0.742142i	$0.733810\pi$
$20$	0	0
$21$	−73.4757	−0.763510
$22$	0	0
$23$	35.9888	0.326268	0.163134	−	0.986604i	$-0.447840\pi$
$23$	35.9888	0.326268	0.163134	+	0.986604i	$0.447840\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−129.811	−0.925266
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	58.5275	0.339092	0.169546	−	0.985522i	$-0.445770\pi$
$31$	58.5275	0.339092	0.169546	+	0.985522i	$0.445770\pi$
$32$	0	0
$33$	−65.8153	−0.347181
$34$	0	0
$35$	130.342	0.629481
$36$	0	0
$37$	−226.421	−1.00604	−0.503018	−	0.864276i	$-0.667777\pi$
$37$	−226.421	−1.00604	−0.503018	+	0.864276i	$0.667777\pi$
$38$	0	0
$39$	3.49725	0.0143592
$40$	0	0
$41$	447.304	1.70383	0.851917	−	0.523677i	$-0.175440\pi$
$41$	447.304	1.70383	0.851917	+	0.523677i	$0.175440\pi$
$42$	0	0
$43$	−490.884	−1.74091	−0.870454	−	0.492249i	$-0.836175\pi$
$43$	−490.884	−1.74091	−0.870454	+	0.492249i	$0.836175\pi$
$44$	0	0
$45$	95.2782	0.315628
$46$	0	0
$47$	223.382	0.693267	0.346634	−	0.938001i	$-0.387325\pi$
$47$	223.382	0.693267	0.346634	+	0.938001i	$0.387325\pi$
$48$	0	0
$49$	336.562	0.981232
$50$	0	0
$51$	240.347	0.659909
$52$	0	0
$53$	137.117	0.355367	0.177683	−	0.984088i	$-0.443140\pi$
$53$	137.117	0.355367	0.177683	+	0.984088i	$0.443140\pi$
$54$	0	0
$55$	116.753	0.286235
$56$	0	0
$57$	−312.912	−0.727126
$58$	0	0
$59$	352.244	0.777260	0.388630	−	0.921394i	$-0.372949\pi$
$59$	352.244	0.777260	0.388630	+	0.921394i	$0.372949\pi$
$60$	0	0
$61$	133.867	0.280982	0.140491	−	0.990082i	$-0.455132\pi$
$61$	133.867	0.280982	0.140491	+	0.990082i	$0.455132\pi$
$62$	0	0
$63$	496.751	0.993408
$64$	0	0
$65$	−6.20394	−0.0118385
$66$	0	0
$67$	450.653	0.821733	0.410866	−	0.911696i	$-0.365226\pi$
$67$	450.653	0.821733	0.410866	+	0.911696i	$0.365226\pi$
$68$	0	0
$69$	101.437	0.176979
$70$	0	0
$71$	462.170	0.772528	0.386264	−	0.922388i	$-0.373765\pi$
$71$	462.170	0.772528	0.386264	+	0.922388i	$0.373765\pi$
$72$	0	0
$73$	−17.1569	−0.0275078	−0.0137539	−	0.999905i	$-0.504378\pi$
$73$	−17.1569	−0.0275078	−0.0137539	+	0.999905i	$0.504378\pi$
$74$	0	0
$75$	70.4643	0.108487
$76$	0	0
$77$	608.712	0.900899
$78$	0	0
$79$	1221.19	1.73918	0.869590	−	0.493775i	$-0.164383\pi$
$79$	1221.19	1.73918	0.869590	+	0.493775i	$0.164383\pi$
$80$	0	0
$81$	148.620	0.203869
$82$	0	0
$83$	−481.608	−0.636908	−0.318454	−	0.947938i	$-0.603164\pi$
$83$	−481.608	−0.636908	−0.318454	+	0.947938i	$0.603164\pi$
$84$	0	0
$85$	−426.364	−0.544066
$86$	0	0
$87$	−81.7386	−0.100728
$88$	0	0
$89$	−179.628	−0.213938	−0.106969	−	0.994262i	$-0.534115\pi$
$89$	−179.628	−0.213938	−0.106969	+	0.994262i	$0.534115\pi$
$90$	0	0
$91$	−32.3454	−0.0372606
$92$	0	0
$93$	164.964	0.183935
$94$	0	0
$95$	555.089	0.599483
$96$	0	0
$97$	956.201	1.00090	0.500451	−	0.865765i	$-0.333168\pi$
$97$	956.201	1.00090	0.500451	+	0.865765i	$0.333168\pi$
$98$	0	0
$99$	444.960	0.451719
$100$	0	0
$101$	1798.35	1.77171	0.885854	−	0.463965i	$-0.153574\pi$
$101$	1798.35	1.77171	0.885854	+	0.463965i	$0.153574\pi$
$102$	0	0
$103$	−1427.36	−1.36546	−0.682728	−	0.730673i	$-0.739207\pi$
$103$	−1427.36	−1.36546	−0.682728	+	0.730673i	$0.739207\pi$
$104$	0	0
$105$	367.379	0.341452
$106$	0	0
$107$	470.074	0.424708	0.212354	−	0.977193i	$-0.431887\pi$
$107$	470.074	0.424708	0.212354	+	0.977193i	$0.431887\pi$
$108$	0	0
$109$	−1965.70	−1.72734	−0.863668	−	0.504060i	$-0.831839\pi$
$109$	−1965.70	−1.72734	−0.863668	+	0.504060i	$0.831839\pi$
$110$	0	0
$111$	−638.184	−0.545709
$112$	0	0
$113$	1183.13	0.984953	0.492477	−	0.870326i	$-0.336092\pi$
$113$	1183.13	0.984953	0.492477	+	0.870326i	$0.336092\pi$
$114$	0	0
$115$	−179.944	−0.145912
$116$	0	0
$117$	−23.6440	−0.0186828
$118$	0	0
$119$	−2222.93	−1.71240
$120$	0	0
$121$	−785.751	−0.590346
$122$	0	0
$123$	1260.76	0.924219
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1059.72	0.740435	0.370218	−	0.928945i	$-0.379283\pi$
$127$	1059.72	0.740435	0.370218	+	0.928945i	$0.379283\pi$
$128$	0	0
$129$	−1383.59	−0.944329
$130$	0	0
$131$	−1533.41	−1.02271	−0.511354	−	0.859370i	$-0.670856\pi$
$131$	−1533.41	−1.02271	−0.511354	+	0.859370i	$0.670856\pi$
$132$	0	0
$133$	2894.06	1.88682
$134$	0	0
$135$	649.056	0.413791
$136$	0	0
$137$	15.4018	0.00960484	0.00480242	−	0.999988i	$-0.498471\pi$
$137$	15.4018	0.00960484	0.00480242	+	0.999988i	$0.498471\pi$
$138$	0	0
$139$	2534.21	1.54640	0.773198	−	0.634165i	$-0.218656\pi$
$139$	2534.21	1.54640	0.773198	+	0.634165i	$0.218656\pi$
$140$	0	0
$141$	629.617	0.376052
$142$	0	0
$143$	−28.9731	−0.0169430
$144$	0	0
$145$	145.000	0.0830455
$146$	0	0
$147$	948.626	0.532254
$148$	0	0
$149$	−2113.08	−1.16181	−0.580907	−	0.813970i	$-0.697302\pi$
$149$	−2113.08	−1.16181	−0.580907	+	0.813970i	$0.697302\pi$
$150$	0	0
$151$	−212.741	−0.114653	−0.0573265	−	0.998355i	$-0.518258\pi$
$151$	−212.741	−0.114653	−0.0573265	+	0.998355i	$0.518258\pi$
$152$	0	0
$153$	−1624.93	−0.858612
$154$	0	0
$155$	−292.637	−0.151646
$156$	0	0
$157$	3239.00	1.64650	0.823251	−	0.567678i	$-0.192158\pi$
$157$	3239.00	1.64650	0.823251	+	0.567678i	$0.192158\pi$
$158$	0	0
$159$	386.473	0.192763
$160$	0	0
$161$	−938.170	−0.459243
$162$	0	0
$163$	−2007.15	−0.964493	−0.482247	−	0.876036i	$-0.660179\pi$
$163$	−2007.15	−0.964493	−0.482247	+	0.876036i	$0.660179\pi$
$164$	0	0
$165$	329.076	0.155264
$166$	0	0
$167$	3510.22	1.62652	0.813261	−	0.581899i	$-0.197690\pi$
$167$	3510.22	1.62652	0.813261	+	0.581899i	$0.197690\pi$
$168$	0	0
$169$	−2195.46	−0.999299
$170$	0	0
$171$	2115.52	0.946067
$172$	0	0
$173$	−1855.20	−0.815310	−0.407655	−	0.913136i	$-0.633653\pi$
$173$	−1855.20	−0.815310	−0.407655	+	0.913136i	$0.633653\pi$
$174$	0	0
$175$	−651.710	−0.281512
$176$	0	0
$177$	992.826	0.421612
$178$	0	0
$179$	−761.071	−0.317794	−0.158897	−	0.987295i	$-0.550794\pi$
$179$	−761.071	−0.317794	−0.158897	+	0.987295i	$0.550794\pi$
$180$	0	0
$181$	3988.76	1.63802	0.819011	−	0.573778i	$-0.194523\pi$
$181$	3988.76	1.63802	0.819011	+	0.573778i	$0.194523\pi$
$182$	0	0
$183$	377.313	0.152414
$184$	0	0
$185$	1132.10	0.449913
$186$	0	0
$187$	−1991.17	−0.778656
$188$	0	0
$189$	3383.97	1.30237
$190$	0	0
$191$	598.925	0.226894	0.113447	−	0.993544i	$-0.463811\pi$
$191$	598.925	0.226894	0.113447	+	0.993544i	$0.463811\pi$
$192$	0	0
$193$	−3120.67	−1.16389	−0.581945	−	0.813228i	$-0.697708\pi$
$193$	−3120.67	−1.16389	−0.581945	+	0.813228i	$0.697708\pi$
$194$	0	0
$195$	−17.4862	−0.00642162
$196$	0	0
$197$	3767.68	1.36262	0.681309	−	0.731996i	$-0.261411\pi$
$197$	3767.68	1.36262	0.681309	+	0.731996i	$0.261411\pi$
$198$	0	0
$199$	−3884.76	−1.38384	−0.691918	−	0.721976i	$-0.743234\pi$
$199$	−3884.76	−1.38384	−0.691918	+	0.721976i	$0.743234\pi$
$200$	0	0
$201$	1270.20	0.445736
$202$	0	0
$203$	755.984	0.261378
$204$	0	0
$205$	−2236.52	−0.761978
$206$	0	0
$207$	−685.789	−0.230269
$208$	0	0
$209$	2592.33	0.857967
$210$	0	0
$211$	98.8592	0.0322547	0.0161274	−	0.999870i	$-0.494866\pi$
$211$	98.8592	0.0322547	0.0161274	+	0.999870i	$0.494866\pi$
$212$	0	0
$213$	1302.66	0.419046
$214$	0	0
$215$	2454.42	0.778558
$216$	0	0
$217$	−1525.72	−0.477293
$218$	0	0
$219$	−48.3581	−0.0149212
$220$	0	0
$221$	105.805	0.0322047
$222$	0	0
$223$	−962.114	−0.288915	−0.144457	−	0.989511i	$-0.546144\pi$
$223$	−962.114	−0.288915	−0.144457	+	0.989511i	$0.546144\pi$
$224$	0	0
$225$	−476.391	−0.141153
$226$	0	0
$227$	−2092.95	−0.611955	−0.305977	−	0.952039i	$-0.598983\pi$
$227$	−2092.95	−0.611955	−0.305977	+	0.952039i	$0.598983\pi$
$228$	0	0
$229$	−1525.39	−0.440178	−0.220089	−	0.975480i	$-0.570635\pi$
$229$	−1525.39	−0.440178	−0.220089	+	0.975480i	$0.570635\pi$
$230$	0	0
$231$	1715.70	0.488679
$232$	0	0
$233$	1682.88	0.473171	0.236586	−	0.971611i	$-0.423972\pi$
$233$	1682.88	0.473171	0.236586	+	0.971611i	$0.423972\pi$
$234$	0	0
$235$	−1116.91	−0.310039
$236$	0	0
$237$	3442.03	0.943391
$238$	0	0
$239$	886.758	0.239998	0.119999	−	0.992774i	$-0.461711\pi$
$239$	886.758	0.239998	0.119999	+	0.992774i	$0.461711\pi$
$240$	0	0
$241$	4111.82	1.09903	0.549513	−	0.835485i	$-0.314813\pi$
$241$	4111.82	1.09903	0.549513	+	0.835485i	$0.314813\pi$
$242$	0	0
$243$	3923.80	1.03585
$244$	0	0
$245$	−1682.81	−0.438820
$246$	0	0
$247$	−137.749	−0.0354850
$248$	0	0
$249$	−1357.45	−0.345481
$250$	0	0
$251$	−7052.67	−1.77355	−0.886775	−	0.462202i	$-0.847059\pi$
$251$	−7052.67	−1.77355	−0.886775	+	0.462202i	$0.847059\pi$
$252$	0	0
$253$	−840.358	−0.208825
$254$	0	0
$255$	−1201.74	−0.295120
$256$	0	0
$257$	824.415	0.200100	0.100050	−	0.994982i	$-0.468100\pi$
$257$	824.415	0.200100	0.100050	+	0.994982i	$0.468100\pi$
$258$	0	0
$259$	5902.43	1.41606
$260$	0	0
$261$	552.614	0.131057
$262$	0	0
$263$	−1553.82	−0.364307	−0.182154	−	0.983270i	$-0.558307\pi$
$263$	−1553.82	−0.364307	−0.182154	+	0.983270i	$0.558307\pi$
$264$	0	0
$265$	−685.584	−0.158925
$266$	0	0
$267$	−506.294	−0.116047
$268$	0	0
$269$	5304.92	1.20240	0.601202	−	0.799097i	$-0.294689\pi$
$269$	5304.92	1.20240	0.601202	+	0.799097i	$0.294689\pi$
$270$	0	0
$271$	1988.76	0.445789	0.222894	−	0.974843i	$-0.428449\pi$
$271$	1988.76	0.445789	0.222894	+	0.974843i	$0.428449\pi$
$272$	0	0
$273$	−91.1677	−0.0202114
$274$	0	0
$275$	−583.764	−0.128008
$276$	0	0
$277$	4779.07	1.03663	0.518315	−	0.855190i	$-0.326560\pi$
$277$	4779.07	1.03663	0.518315	+	0.855190i	$0.326560\pi$
$278$	0	0
$279$	−1115.28	−0.239319
$280$	0	0
$281$	362.267	0.0769075	0.0384538	−	0.999260i	$-0.487757\pi$
$281$	362.267	0.0769075	0.0384538	+	0.999260i	$0.487757\pi$
$282$	0	0
$283$	4932.90	1.03615	0.518075	−	0.855335i	$-0.326649\pi$
$283$	4932.90	1.03615	0.518075	+	0.855335i	$0.326649\pi$
$284$	0	0
$285$	1564.56	0.325180
$286$	0	0
$287$	−11660.5	−2.39825
$288$	0	0
$289$	2358.44	0.480042
$290$	0	0
$291$	2695.12	0.542924
$292$	0	0
$293$	9504.43	1.89507	0.947534	−	0.319655i	$-0.103567\pi$
$293$	9504.43	1.89507	0.947534	+	0.319655i	$0.103567\pi$
$294$	0	0
$295$	−1761.22	−0.347601
$296$	0	0
$297$	3031.17	0.592209
$298$	0	0
$299$	44.6544	0.00863689
$300$	0	0
$301$	12796.6	2.45044
$302$	0	0
$303$	5068.78	0.961035
$304$	0	0
$305$	−669.334	−0.125659
$306$	0	0
$307$	−7000.28	−1.30139	−0.650696	−	0.759339i	$-0.725523\pi$
$307$	−7000.28	−1.30139	−0.650696	+	0.759339i	$0.725523\pi$
$308$	0	0
$309$	−4023.12	−0.740670
$310$	0	0
$311$	866.348	0.157962	0.0789809	−	0.996876i	$-0.474833\pi$
$311$	866.348	0.157962	0.0789809	+	0.996876i	$0.474833\pi$
$312$	0	0
$313$	388.214	0.0701060	0.0350530	−	0.999385i	$-0.488840\pi$
$313$	388.214	0.0701060	0.0350530	+	0.999385i	$0.488840\pi$
$314$	0	0
$315$	−2483.75	−0.444266
$316$	0	0
$317$	−1832.34	−0.324652	−0.162326	−	0.986737i	$-0.551900\pi$
$317$	−1832.34	−0.324652	−0.162326	+	0.986737i	$0.551900\pi$
$318$	0	0
$319$	677.167	0.118853
$320$	0	0
$321$	1324.94	0.230376
$322$	0	0
$323$	−9466.79	−1.63079
$324$	0	0
$325$	31.0197	0.00529435
$326$	0	0
$327$	−5540.46	−0.936967
$328$	0	0
$329$	−5823.21	−0.975817
$330$	0	0
$331$	−11334.6	−1.88220	−0.941098	−	0.338134i	$-0.890204\pi$
$331$	−11334.6	−1.88220	−0.941098	+	0.338134i	$0.890204\pi$
$332$	0	0
$333$	4314.60	0.710025
$334$	0	0
$335$	−2253.27	−0.367490
$336$	0	0
$337$	−1607.74	−0.259880	−0.129940	−	0.991522i	$-0.541478\pi$
$337$	−1607.74	−0.259880	−0.129940	+	0.991522i	$0.541478\pi$
$338$	0	0
$339$	3334.74	0.534273
$340$	0	0
$341$	−1366.65	−0.217033
$342$	0	0
$343$	167.817	0.0264176
$344$	0	0
$345$	−507.185	−0.0791475
$346$	0	0
$347$	3317.76	0.513276	0.256638	−	0.966508i	$-0.417385\pi$
$347$	3317.76	0.513276	0.256638	+	0.966508i	$0.417385\pi$
$348$	0	0
$349$	8700.51	1.33446	0.667231	−	0.744851i	$-0.267479\pi$
$349$	8700.51	1.33446	0.667231	+	0.744851i	$0.267479\pi$
$350$	0	0
$351$	−161.068	−0.0244934
$352$	0	0
$353$	11916.2	1.79669	0.898347	−	0.439286i	$-0.144768\pi$
$353$	11916.2	1.79669	0.898347	+	0.439286i	$0.144768\pi$
$354$	0	0
$355$	−2310.85	−0.345485
$356$	0	0
$357$	−6265.48	−0.928864
$358$	0	0
$359$	−4750.30	−0.698360	−0.349180	−	0.937056i	$-0.613540\pi$
$359$	−4750.30	−0.698360	−0.349180	+	0.937056i	$0.613540\pi$
$360$	0	0
$361$	5465.95	0.796901
$362$	0	0
$363$	−2214.70	−0.320224
$364$	0	0
$365$	85.7847	0.0123018
$366$	0	0
$367$	8332.78	1.18520	0.592599	−	0.805498i	$-0.298102\pi$
$367$	8332.78	1.18520	0.592599	+	0.805498i	$0.298102\pi$
$368$	0	0
$369$	−8523.68	−1.20251
$370$	0	0
$371$	−3574.42	−0.500201
$372$	0	0
$373$	−8718.56	−1.21027	−0.605134	−	0.796123i	$-0.706881\pi$
$373$	−8718.56	−1.21027	−0.605134	+	0.796123i	$0.706881\pi$
$374$	0	0
$375$	−352.322	−0.0485168
$376$	0	0
$377$	−35.9828	−0.00491568
$378$	0	0
$379$	−9266.54	−1.25591	−0.627955	−	0.778249i	$-0.716108\pi$
$379$	−9266.54	−1.25591	−0.627955	+	0.778249i	$0.716108\pi$
$380$	0	0
$381$	2986.91	0.401638
$382$	0	0
$383$	−4157.46	−0.554664	−0.277332	−	0.960774i	$-0.589450\pi$
$383$	−4157.46	−0.554664	−0.277332	+	0.960774i	$0.589450\pi$
$384$	0	0
$385$	−3043.56	−0.402894
$386$	0	0
$387$	9354.11	1.22867
$388$	0	0
$389$	5776.27	0.752876	0.376438	−	0.926442i	$-0.377149\pi$
$389$	5776.27	0.752876	0.376438	+	0.926442i	$0.377149\pi$
$390$	0	0
$391$	3068.86	0.396928
$392$	0	0
$393$	−4322.03	−0.554753
$394$	0	0
$395$	−6105.97	−0.777785
$396$	0	0
$397$	−8760.87	−1.10755	−0.553773	−	0.832668i	$-0.686812\pi$
$397$	−8760.87	−1.10755	−0.553773	+	0.832668i	$0.686812\pi$
$398$	0	0
$399$	8157.11	1.02347
$400$	0	0
$401$	−2850.16	−0.354939	−0.177469	−	0.984126i	$-0.556791\pi$
$401$	−2850.16	−0.354939	−0.177469	+	0.984126i	$0.556791\pi$
$402$	0	0
$403$	72.6201	0.00897635
$404$	0	0
$405$	−743.101	−0.0911728
$406$	0	0
$407$	5287.06	0.643906
$408$	0	0
$409$	−11829.9	−1.43020	−0.715098	−	0.699025i	$-0.753618\pi$
$409$	−11829.9	−1.43020	−0.715098	+	0.699025i	$0.753618\pi$
$410$	0	0
$411$	43.4110	0.00521000
$412$	0	0
$413$	−9182.45	−1.09404
$414$	0	0
$415$	2408.04	0.284834
$416$	0	0
$417$	7142.86	0.838818
$418$	0	0
$419$	2861.50	0.333636	0.166818	−	0.985988i	$-0.446651\pi$
$419$	2861.50	0.333636	0.166818	+	0.985988i	$0.446651\pi$
$420$	0	0
$421$	2642.71	0.305933	0.152966	−	0.988231i	$-0.451117\pi$
$421$	2642.71	0.305933	0.152966	+	0.988231i	$0.451117\pi$
$422$	0	0
$423$	−4256.68	−0.489284
$424$	0	0
$425$	2131.82	0.243314
$426$	0	0
$427$	−3489.70	−0.395499
$428$	0	0
$429$	−81.6627	−0.00919048
$430$	0	0
$431$	1147.71	0.128267	0.0641335	−	0.997941i	$-0.479572\pi$
$431$	1147.71	0.128267	0.0641335	+	0.997941i	$0.479572\pi$
$432$	0	0
$433$	15563.7	1.72735	0.863676	−	0.504048i	$-0.168156\pi$
$433$	15563.7	1.72735	0.863676	+	0.504048i	$0.168156\pi$
$434$	0	0
$435$	408.693	0.0450467
$436$	0	0
$437$	−3995.39	−0.437358
$438$	0	0
$439$	14046.7	1.52713	0.763567	−	0.645729i	$-0.223446\pi$
$439$	14046.7	1.52713	0.763567	+	0.645729i	$0.223446\pi$
$440$	0	0
$441$	−6413.42	−0.692519
$442$	0	0
$443$	2538.61	0.272264	0.136132	−	0.990691i	$-0.456533\pi$
$443$	2538.61	0.272264	0.136132	+	0.990691i	$0.456533\pi$
$444$	0	0
$445$	898.138	0.0956760
$446$	0	0
$447$	−5955.87	−0.630208
$448$	0	0
$449$	13802.5	1.45074	0.725370	−	0.688359i	$-0.241669\pi$
$449$	13802.5	1.45074	0.725370	+	0.688359i	$0.241669\pi$
$450$	0	0
$451$	−10444.8	−1.09053
$452$	0	0
$453$	−599.626	−0.0621918
$454$	0	0
$455$	161.727	0.0166634
$456$	0	0
$457$	4047.50	0.414298	0.207149	−	0.978309i	$-0.433581\pi$
$457$	4047.50	0.414298	0.207149	+	0.978309i	$0.433581\pi$
$458$	0	0
$459$	−11069.4	−1.12565
$460$	0	0
$461$	705.096	0.0712356	0.0356178	−	0.999365i	$-0.488660\pi$
$461$	705.096	0.0712356	0.0356178	+	0.999365i	$0.488660\pi$
$462$	0	0
$463$	3440.10	0.345302	0.172651	−	0.984983i	$-0.444767\pi$
$463$	3440.10	0.345302	0.172651	+	0.984983i	$0.444767\pi$
$464$	0	0
$465$	−824.820	−0.0822583
$466$	0	0
$467$	−5931.86	−0.587781	−0.293891	−	0.955839i	$-0.594950\pi$
$467$	−5931.86	−0.587781	−0.293891	+	0.955839i	$0.594950\pi$
$468$	0	0
$469$	−11747.8	−1.15664
$470$	0	0
$471$	9129.37	0.893119
$472$	0	0
$473$	11462.4	1.11425
$474$	0	0
$475$	−2775.44	−0.268097
$476$	0	0
$477$	−2612.85	−0.250805
$478$	0	0
$479$	−13018.9	−1.24186	−0.620929	−	0.783867i	$-0.713244\pi$
$479$	−13018.9	−1.24186	−0.620929	+	0.783867i	$0.713244\pi$
$480$	0	0
$481$	−280.940	−0.0266315
$482$	0	0
$483$	−2644.30	−0.249109
$484$	0	0
$485$	−4781.01	−0.447617
$486$	0	0
$487$	12450.0	1.15844	0.579222	−	0.815170i	$-0.303356\pi$
$487$	12450.0	1.15844	0.579222	+	0.815170i	$0.303356\pi$
$488$	0	0
$489$	−5657.31	−0.523174
$490$	0	0
$491$	−16627.1	−1.52825	−0.764126	−	0.645067i	$-0.776829\pi$
$491$	−16627.1	−1.52825	−0.764126	+	0.645067i	$0.776829\pi$
$492$	0	0
$493$	−2472.91	−0.225911
$494$	0	0
$495$	−2224.80	−0.202015
$496$	0	0
$497$	−12048.0	−1.08738
$498$	0	0
$499$	−21405.0	−1.92028	−0.960140	−	0.279518i	$-0.909825\pi$
$499$	−21405.0	−1.92028	−0.960140	+	0.279518i	$0.909825\pi$
$500$	0	0
$501$	9893.82	0.882282
$502$	0	0
$503$	−13585.1	−1.20423	−0.602117	−	0.798408i	$-0.705676\pi$
$503$	−13585.1	−1.20423	−0.602117	+	0.798408i	$0.705676\pi$
$504$	0	0
$505$	−8991.75	−0.792332
$506$	0	0
$507$	−6188.06	−0.542054
$508$	0	0
$509$	3332.50	0.290197	0.145099	−	0.989417i	$-0.453650\pi$
$509$	3332.50	0.290197	0.145099	+	0.989417i	$0.453650\pi$
$510$	0	0
$511$	447.254	0.0387189
$512$	0	0
$513$	14411.3	1.24031
$514$	0	0
$515$	7136.80	0.610650
$516$	0	0
$517$	−5216.09	−0.443720
$518$	0	0
$519$	−5229.03	−0.442252
$520$	0	0
$521$	−18894.0	−1.58879	−0.794396	−	0.607400i	$-0.792213\pi$
$521$	−18894.0	−1.58879	−0.794396	+	0.607400i	$0.792213\pi$
$522$	0	0
$523$	−9603.20	−0.802903	−0.401452	−	0.915880i	$-0.631494\pi$
$523$	−9603.20	−0.802903	−0.401452	+	0.915880i	$0.631494\pi$
$524$	0	0
$525$	−1836.89	−0.152702
$526$	0	0
$527$	4990.80	0.412529
$528$	0	0
$529$	−10871.8	−0.893549
$530$	0	0
$531$	−6712.25	−0.548562
$532$	0	0
$533$	555.010	0.0451034
$534$	0	0
$535$	−2350.37	−0.189935
$536$	0	0
$537$	−2145.13	−0.172382
$538$	0	0
$539$	−7858.92	−0.628029
$540$	0	0
$541$	14655.5	1.16468	0.582339	−	0.812946i	$-0.302137\pi$
$541$	14655.5	1.16468	0.582339	+	0.812946i	$0.302137\pi$
$542$	0	0
$543$	11242.6	0.888519
$544$	0	0
$545$	9828.49	0.772489
$546$	0	0
$547$	−10880.5	−0.850489	−0.425244	−	0.905079i	$-0.639812\pi$
$547$	−10880.5	−0.850489	−0.425244	+	0.905079i	$0.639812\pi$
$548$	0	0
$549$	−2550.92	−0.198307
$550$	0	0
$551$	3219.52	0.248922
$552$	0	0
$553$	−31834.6	−2.44800
$554$	0	0
$555$	3190.92	0.244049
$556$	0	0
$557$	−3863.82	−0.293923	−0.146962	−	0.989142i	$-0.546949\pi$
$557$	−3863.82	−0.293923	−0.146962	+	0.989142i	$0.546949\pi$
$558$	0	0
$559$	−609.082	−0.0460849
$560$	0	0
$561$	−5612.25	−0.422370
$562$	0	0
$563$	15893.3	1.18974	0.594870	−	0.803822i	$-0.297203\pi$
$563$	15893.3	1.18974	0.594870	+	0.803822i	$0.297203\pi$
$564$	0	0
$565$	−5915.66	−0.440485
$566$	0	0
$567$	−3874.29	−0.286958
$568$	0	0
$569$	6923.24	0.510083	0.255042	−	0.966930i	$-0.417911\pi$
$569$	6923.24	0.510083	0.255042	+	0.966930i	$0.417911\pi$
$570$	0	0
$571$	18659.2	1.36754	0.683768	−	0.729699i	$-0.260340\pi$
$571$	18659.2	1.36754	0.683768	+	0.729699i	$0.260340\pi$
$572$	0	0
$573$	1688.11	0.123075
$574$	0	0
$575$	899.719	0.0652537
$576$	0	0
$577$	−3522.67	−0.254161	−0.127080	−	0.991892i	$-0.540561\pi$
$577$	−3522.67	−0.254161	−0.127080	+	0.991892i	$0.540561\pi$
$578$	0	0
$579$	−8795.83	−0.631334
$580$	0	0
$581$	12554.8	0.896487
$582$	0	0
$583$	−3201.75	−0.227450
$584$	0	0
$585$	118.220	0.00835521
$586$	0	0
$587$	20375.0	1.43265	0.716324	−	0.697767i	$-0.245823\pi$
$587$	20375.0	1.43265	0.716324	+	0.697767i	$0.245823\pi$
$588$	0	0
$589$	−6497.59	−0.454548
$590$	0	0
$591$	10619.5	0.739131
$592$	0	0
$593$	−4174.07	−0.289053	−0.144526	−	0.989501i	$-0.546166\pi$
$593$	−4174.07	−0.289053	−0.144526	+	0.989501i	$0.546166\pi$
$594$	0	0
$595$	11114.6	0.765807
$596$	0	0
$597$	−10949.5	−0.750641
$598$	0	0
$599$	−9914.79	−0.676306	−0.338153	−	0.941091i	$-0.609802\pi$
$599$	−9914.79	−0.676306	−0.338153	+	0.941091i	$0.609802\pi$
$600$	0	0
$601$	9322.05	0.632703	0.316351	−	0.948642i	$-0.397542\pi$
$601$	9322.05	0.632703	0.316351	+	0.948642i	$0.397542\pi$
$602$	0	0
$603$	−8587.49	−0.579950
$604$	0	0
$605$	3928.75	0.264011
$606$	0	0
$607$	2971.29	0.198684	0.0993419	−	0.995053i	$-0.468326\pi$
$607$	2971.29	0.198684	0.0993419	+	0.995053i	$0.468326\pi$
$608$	0	0
$609$	2130.80	0.141780
$610$	0	0
$611$	277.169	0.0183520
$612$	0	0
$613$	−6394.07	−0.421295	−0.210648	−	0.977562i	$-0.567557\pi$
$613$	−6394.07	−0.421295	−0.210648	+	0.977562i	$0.567557\pi$
$614$	0	0
$615$	−6303.80	−0.413323
$616$	0	0
$617$	3743.82	0.244280	0.122140	−	0.992513i	$-0.461024\pi$
$617$	3743.82	0.244280	0.122140	+	0.992513i	$0.461024\pi$
$618$	0	0
$619$	−14398.8	−0.934954	−0.467477	−	0.884005i	$-0.654837\pi$
$619$	−14398.8	−0.934954	−0.467477	+	0.884005i	$0.654837\pi$
$620$	0	0
$621$	−4671.74	−0.301885
$622$	0	0
$623$	4682.61	0.301131
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	7306.67	0.465391
$628$	0	0
$629$	−19307.5	−1.22391
$630$	0	0
$631$	−16775.9	−1.05838	−0.529190	−	0.848503i	$-0.677504\pi$
$631$	−16775.9	−1.05838	−0.529190	+	0.848503i	$0.677504\pi$
$632$	0	0
$633$	278.642	0.0174961
$634$	0	0
$635$	−5298.62	−0.331133
$636$	0	0
$637$	417.602	0.0259749
$638$	0	0
$639$	−8806.95	−0.545223
$640$	0	0
$641$	−20797.5	−1.28151	−0.640757	−	0.767744i	$-0.721379\pi$
$641$	−20797.5	−1.28151	−0.640757	+	0.767744i	$0.721379\pi$
$642$	0	0
$643$	21041.5	1.29050	0.645252	−	0.763970i	$-0.276752\pi$
$643$	21041.5	1.29050	0.645252	+	0.763970i	$0.276752\pi$
$644$	0	0
$645$	6917.96	0.422317
$646$	0	0
$647$	−1186.85	−0.0721171	−0.0360586	−	0.999350i	$-0.511480\pi$
$647$	−1186.85	−0.0721171	−0.0360586	+	0.999350i	$0.511480\pi$
$648$	0	0
$649$	−8225.11	−0.497479
$650$	0	0
$651$	−4300.35	−0.258900
$652$	0	0
$653$	−7217.10	−0.432507	−0.216253	−	0.976337i	$-0.569384\pi$
$653$	−7217.10	−0.432507	−0.216253	+	0.976337i	$0.569384\pi$
$654$	0	0
$655$	7667.06	0.457369
$656$	0	0
$657$	326.936	0.0194140
$658$	0	0
$659$	−12372.4	−0.731353	−0.365676	−	0.930742i	$-0.619162\pi$
$659$	−12372.4	−0.731353	−0.365676	+	0.930742i	$0.619162\pi$
$660$	0	0
$661$	13670.8	0.804434	0.402217	−	0.915544i	$-0.368240\pi$
$661$	13670.8	0.804434	0.402217	+	0.915544i	$0.368240\pi$
$662$	0	0
$663$	298.220	0.0174689
$664$	0	0
$665$	−14470.3	−0.843810
$666$	0	0
$667$	−1043.67	−0.0605865
$668$	0	0
$669$	−2711.79	−0.156717
$670$	0	0
$671$	−3125.87	−0.179840
$672$	0	0
$673$	11415.3	0.653830	0.326915	−	0.945054i	$-0.393991\pi$
$673$	11415.3	0.653830	0.326915	+	0.945054i	$0.393991\pi$
$674$	0	0
$675$	−3245.28	−0.185053
$676$	0	0
$677$	−28110.6	−1.59583	−0.797917	−	0.602767i	$-0.794065\pi$
$677$	−28110.6	−1.59583	−0.797917	+	0.602767i	$0.794065\pi$
$678$	0	0
$679$	−24926.6	−1.40883
$680$	0	0
$681$	−5899.12	−0.331945
$682$	0	0
$683$	6207.75	0.347779	0.173889	−	0.984765i	$-0.444366\pi$
$683$	6207.75	0.347779	0.173889	+	0.984765i	$0.444366\pi$
$684$	0	0
$685$	−77.0089	−0.00429541
$686$	0	0
$687$	−4299.43	−0.238768
$688$	0	0
$689$	170.133	0.00940717
$690$	0	0
$691$	13192.1	0.726266	0.363133	−	0.931737i	$-0.381707\pi$
$691$	13192.1	0.726266	0.363133	+	0.931737i	$0.381707\pi$
$692$	0	0
$693$	−11599.4	−0.635823
$694$	0	0
$695$	−12671.1	−0.691569
$696$	0	0
$697$	38142.9	2.07283
$698$	0	0
$699$	4743.31	0.256664
$700$	0	0
$701$	21056.4	1.13450	0.567252	−	0.823544i	$-0.308006\pi$
$701$	21056.4	1.13450	0.567252	+	0.823544i	$0.308006\pi$
$702$	0	0
$703$	25136.7	1.34858
$704$	0	0
$705$	−3148.09	−0.168176
$706$	0	0
$707$	−46880.1	−2.49379
$708$	0	0
$709$	11512.8	0.609833	0.304917	−	0.952379i	$-0.401371\pi$
$709$	11512.8	0.609833	0.304917	+	0.952379i	$0.401371\pi$
$710$	0	0
$711$	−23270.7	−1.22745
$712$	0	0
$713$	2106.33	0.110635
$714$	0	0
$715$	144.865	0.00757715
$716$	0	0
$717$	2499.39	0.130183
$718$	0	0
$719$	−9823.82	−0.509550	−0.254775	−	0.967000i	$-0.582001\pi$
$719$	−9823.82	−0.509550	−0.254775	+	0.967000i	$0.582001\pi$
$720$	0	0
$721$	37209.0	1.92196
$722$	0	0
$723$	11589.5	0.596150
$724$	0	0
$725$	−725.000	−0.0371391
$726$	0	0
$727$	18755.7	0.956824	0.478412	−	0.878136i	$-0.341213\pi$
$727$	18755.7	0.956824	0.478412	+	0.878136i	$0.341213\pi$
$728$	0	0
$729$	7046.77	0.358013
$730$	0	0
$731$	−41859.0	−2.11794
$732$	0	0
$733$	2594.67	0.130745	0.0653725	−	0.997861i	$-0.479176\pi$
$733$	2594.67	0.130745	0.0653725	+	0.997861i	$0.479176\pi$
$734$	0	0
$735$	−4743.13	−0.238031
$736$	0	0
$737$	−10523.0	−0.525943
$738$	0	0
$739$	23906.0	1.18998	0.594992	−	0.803732i	$-0.297155\pi$
$739$	23906.0	1.18998	0.594992	+	0.803732i	$0.297155\pi$
$740$	0	0
$741$	−388.257	−0.0192483
$742$	0	0
$743$	21038.5	1.03880	0.519400	−	0.854531i	$-0.326156\pi$
$743$	21038.5	1.03880	0.519400	+	0.854531i	$0.326156\pi$
$744$	0	0
$745$	10565.4	0.519579
$746$	0	0
$747$	9177.35	0.449507
$748$	0	0
$749$	−12254.1	−0.597803
$750$	0	0
$751$	34793.9	1.69061	0.845304	−	0.534285i	$-0.179419\pi$
$751$	34793.9	1.69061	0.845304	+	0.534285i	$0.179419\pi$
$752$	0	0
$753$	−19878.5	−0.962034
$754$	0	0
$755$	1063.71	0.0512744
$756$	0	0
$757$	−33348.7	−1.60116	−0.800581	−	0.599224i	$-0.795476\pi$
$757$	−33348.7	−1.60116	−0.800581	+	0.599224i	$0.795476\pi$
$758$	0	0
$759$	−2368.61	−0.113274
$760$	0	0
$761$	25093.6	1.19532	0.597661	−	0.801749i	$-0.296097\pi$
$761$	25093.6	1.19532	0.597661	+	0.801749i	$0.296097\pi$
$762$	0	0
$763$	51242.6	2.43133
$764$	0	0
$765$	8124.64	0.383983
$766$	0	0
$767$	437.060	0.0205754
$768$	0	0
$769$	−33556.5	−1.57357	−0.786787	−	0.617225i	$-0.788257\pi$
$769$	−33556.5	−1.57357	−0.786787	+	0.617225i	$0.788257\pi$
$770$	0	0
$771$	2323.67	0.108541
$772$	0	0
$773$	35489.1	1.65130	0.825650	−	0.564182i	$-0.190809\pi$
$773$	35489.1	1.65130	0.825650	+	0.564182i	$0.190809\pi$
$774$	0	0
$775$	1463.19	0.0678184
$776$	0	0
$777$	16636.4	0.768120
$778$	0	0
$779$	−49658.7	−2.28397
$780$	0	0
$781$	−10791.9	−0.494450
$782$	0	0
$783$	3764.52	0.171818
$784$	0	0
$785$	−16195.0	−0.736338
$786$	0	0
$787$	−1449.49	−0.0656528	−0.0328264	−	0.999461i	$-0.510451\pi$
$787$	−1449.49	−0.0656528	−0.0328264	+	0.999461i	$0.510451\pi$
$788$	0	0
$789$	−4379.56	−0.197613
$790$	0	0
$791$	−30842.4	−1.38638
$792$	0	0
$793$	166.100	0.00743807
$794$	0	0
$795$	−1932.37	−0.0862063
$796$	0	0
$797$	34576.8	1.53673	0.768364	−	0.640013i	$-0.221071\pi$
$797$	34576.8	1.53673	0.768364	+	0.640013i	$0.221071\pi$
$798$	0	0
$799$	19048.4	0.843408
$800$	0	0
$801$	3422.92	0.150990
$802$	0	0
$803$	400.624	0.0176061
$804$	0	0
$805$	4690.85	0.205380
$806$	0	0
$807$	14952.3	0.652225
$808$	0	0
$809$	19613.4	0.852372	0.426186	−	0.904636i	$-0.359857\pi$
$809$	19613.4	0.852372	0.426186	+	0.904636i	$0.359857\pi$
$810$	0	0
$811$	37056.7	1.60448	0.802242	−	0.597000i	$-0.203641\pi$
$811$	37056.7	1.60448	0.802242	+	0.597000i	$0.203641\pi$
$812$	0	0
$813$	5605.47	0.241811
$814$	0	0
$815$	10035.8	0.431334
$816$	0	0
$817$	54496.8	2.33366
$818$	0	0
$819$	616.362	0.0262972
$820$	0	0
$821$	−42.7009	−0.00181519	−0.000907595	−	1.00000i	$-0.500289\pi$
$821$	−42.7009	−0.00181519	−0.000907595		1.00000i	$0.500289\pi$
$822$	0	0
$823$	2146.40	0.0909099	0.0454550	−	0.998966i	$-0.485526\pi$
$823$	2146.40	0.0909099	0.0454550	+	0.998966i	$0.485526\pi$
$824$	0	0
$825$	−1645.38	−0.0694362
$826$	0	0
$827$	36455.5	1.53287	0.766434	−	0.642323i	$-0.222029\pi$
$827$	36455.5	1.53287	0.766434	+	0.642323i	$0.222029\pi$
$828$	0	0
$829$	−3326.70	−0.139374	−0.0696871	−	0.997569i	$-0.522200\pi$
$829$	−3326.70	−0.139374	−0.0696871	+	0.997569i	$0.522200\pi$
$830$	0	0
$831$	13470.1	0.562303
$832$	0	0
$833$	28699.6	1.19374
$834$	0	0
$835$	−17551.1	−0.727403
$836$	0	0
$837$	−7597.52	−0.313750
$838$	0	0
$839$	−40324.7	−1.65931	−0.829656	−	0.558275i	$-0.811464\pi$
$839$	−40324.7	−1.65931	−0.829656	+	0.558275i	$0.811464\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	1021.07	0.0417173
$844$	0	0
$845$	10977.3	0.446900
$846$	0	0
$847$	20483.3	0.830949
$848$	0	0
$849$	13903.7	0.562043
$850$	0	0
$851$	−8148.61	−0.328238
$852$	0	0
$853$	27937.6	1.12141	0.560707	−	0.828014i	$-0.310529\pi$
$853$	27937.6	1.12141	0.560707	+	0.828014i	$0.310529\pi$
$854$	0	0
$855$	−10577.6	−0.423094
$856$	0	0
$857$	3858.91	0.153813	0.0769065	−	0.997038i	$-0.475496\pi$
$857$	3858.91	0.153813	0.0769065	+	0.997038i	$0.475496\pi$
$858$	0	0
$859$	−114.561	−0.00455038	−0.00227519	−	0.999997i	$-0.500724\pi$
$859$	−114.561	−0.00455038	−0.00227519	+	0.999997i	$0.500724\pi$
$860$	0	0
$861$	−32866.0	−1.30090
$862$	0	0
$863$	43027.0	1.69717	0.848585	−	0.529059i	$-0.177455\pi$
$863$	43027.0	1.69717	0.848585	+	0.529059i	$0.177455\pi$
$864$	0	0
$865$	9276.02	0.364618
$866$	0	0
$867$	6647.45	0.260391
$868$	0	0
$869$	−28515.6	−1.11315
$870$	0	0
$871$	559.165	0.0217527
$872$	0	0
$873$	−18221.0	−0.706401
$874$	0	0
$875$	3258.55	0.125896
$876$	0	0
$877$	−24387.2	−0.938995	−0.469498	−	0.882934i	$-0.655565\pi$
$877$	−24387.2	−0.938995	−0.469498	+	0.882934i	$0.655565\pi$
$878$	0	0
$879$	26788.9	1.02795
$880$	0	0
$881$	−9834.37	−0.376082	−0.188041	−	0.982161i	$-0.560214\pi$
$881$	−9834.37	−0.376082	−0.188041	+	0.982161i	$0.560214\pi$
$882$	0	0
$883$	21793.4	0.830585	0.415292	−	0.909688i	$-0.363679\pi$
$883$	21793.4	0.830585	0.415292	+	0.909688i	$0.363679\pi$
$884$	0	0
$885$	−4964.13	−0.188551
$886$	0	0
$887$	−12803.2	−0.484657	−0.242329	−	0.970194i	$-0.577911\pi$
$887$	−12803.2	−0.484657	−0.242329	+	0.970194i	$0.577911\pi$
$888$	0	0
$889$	−27625.3	−1.04221
$890$	0	0
$891$	−3470.37	−0.130484
$892$	0	0
$893$	−24799.3	−0.929315
$894$	0	0
$895$	3805.36	0.142122
$896$	0	0
$897$	125.862	0.00468495
$898$	0	0
$899$	−1697.30	−0.0629678
$900$	0	0
$901$	11692.3	0.432328
$902$	0	0
$903$	36068.0	1.32920
$904$	0	0
$905$	−19943.8	−0.732545
$906$	0	0
$907$	20331.0	0.744299	0.372149	−	0.928173i	$-0.378621\pi$
$907$	20331.0	0.744299	0.372149	+	0.928173i	$0.378621\pi$
$908$	0	0
$909$	−34268.7	−1.25041
$910$	0	0
$911$	−2439.59	−0.0887236	−0.0443618	−	0.999016i	$-0.514125\pi$
$911$	−2439.59	−0.0887236	−0.0443618	+	0.999016i	$0.514125\pi$
$912$	0	0
$913$	11245.8	0.407648
$914$	0	0
$915$	−1886.57	−0.0681617
$916$	0	0
$917$	39973.6	1.43953
$918$	0	0
$919$	−1187.35	−0.0426193	−0.0213097	−	0.999773i	$-0.506784\pi$
$919$	−1187.35	−0.0426193	−0.0213097	+	0.999773i	$0.506784\pi$
$920$	0	0
$921$	−19730.8	−0.705920
$922$	0	0
$923$	573.455	0.0204502
$924$	0	0
$925$	−5660.52	−0.201207
$926$	0	0
$927$	27199.3	0.963691
$928$	0	0
$929$	−8135.41	−0.287313	−0.143657	−	0.989628i	$-0.545886\pi$
$929$	−8135.41	−0.287313	−0.143657	+	0.989628i	$0.545886\pi$
$930$	0	0
$931$	−37364.4	−1.31533
$932$	0	0
$933$	2441.87	0.0856839
$934$	0	0
$935$	9955.84	0.348225
$936$	0	0
$937$	24483.0	0.853601	0.426800	−	0.904346i	$-0.359641\pi$
$937$	24483.0	0.853601	0.426800	+	0.904346i	$0.359641\pi$
$938$	0	0
$939$	1094.21	0.0380279
$940$	0	0
$941$	−11602.1	−0.401931	−0.200966	−	0.979598i	$-0.564408\pi$
$941$	−11602.1	−0.401931	−0.200966	+	0.979598i	$0.564408\pi$
$942$	0	0
$943$	16097.9	0.555907
$944$	0	0
$945$	−16919.9	−0.582437
$946$	0	0
$947$	13467.6	0.462130	0.231065	−	0.972938i	$-0.425779\pi$
$947$	13467.6	0.462130	0.231065	+	0.972938i	$0.425779\pi$
$948$	0	0
$949$	−21.2881	−0.000728178 0
$950$	0	0
$951$	−5164.59	−0.176102
$952$	0	0
$953$	−35164.0	−1.19525	−0.597625	−	0.801776i	$-0.703889\pi$
$953$	−35164.0	−1.19525	−0.597625	+	0.801776i	$0.703889\pi$
$954$	0	0
$955$	−2994.63	−0.101470
$956$	0	0
$957$	1908.64	0.0644699
$958$	0	0
$959$	−401.500	−0.0135194
$960$	0	0
$961$	−26365.5	−0.885017
$962$	0	0
$963$	−8957.56	−0.299744
$964$	0	0
$965$	15603.3	0.520507
$966$	0	0
$967$	−11414.1	−0.379580	−0.189790	−	0.981825i	$-0.560781\pi$
$967$	−11414.1	−0.379580	−0.189790	+	0.981825i	$0.560781\pi$
$968$	0	0
$969$	−26682.8	−0.884599
$970$	0	0
$971$	28522.6	0.942670	0.471335	−	0.881954i	$-0.343772\pi$
$971$	28522.6	0.942670	0.471335	+	0.881954i	$0.343772\pi$
$972$	0	0
$973$	−66062.9	−2.17665
$974$	0	0
$975$	87.4312	0.00287184
$976$	0	0
$977$	−22494.0	−0.736589	−0.368294	−	0.929709i	$-0.620058\pi$
$977$	−22494.0	−0.736589	−0.368294	+	0.929709i	$0.620058\pi$
$978$	0	0
$979$	4194.41	0.136929
$980$	0	0
$981$	37457.7	1.21909
$982$	0	0
$983$	44514.9	1.44436	0.722180	−	0.691706i	$-0.243140\pi$
$983$	44514.9	1.44436	0.722180	+	0.691706i	$0.243140\pi$
$984$	0	0
$985$	−18838.4	−0.609381
$986$	0	0
$987$	−16413.1	−0.529317
$988$	0	0
$989$	−17666.3	−0.568004
$990$	0	0
$991$	58738.5	1.88284	0.941418	−	0.337242i	$-0.109494\pi$
$991$	58738.5	1.88284	0.941418	+	0.337242i	$0.109494\pi$
$992$	0	0
$993$	−31947.4	−1.02097
$994$	0	0
$995$	19423.8	0.618871
$996$	0	0
$997$	16881.9	0.536262	0.268131	−	0.963382i	$-0.413594\pi$
$997$	16881.9	0.536262	0.268131	+	0.963382i	$0.413594\pi$
$998$	0	0
$999$	29392.0	0.930852

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1160.4.a.g.1.8	✓	11
4.3	odd	2		2320.4.a.bb.1.4		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.4.a.g.1.8	✓	11	1.1	even	1	trivial
2320.4.a.bb.1.4		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)

\(f(q)\)	\(=\)	\(q+2.81857 q^{3} -5.00000 q^{5} -26.0684 q^{7} -19.0556 q^{9} -23.3506 q^{11} +1.24079 q^{13} -14.0929 q^{15} +85.2728 q^{17} -111.018 q^{19} -73.4757 q^{21} +35.9888 q^{23} +25.0000 q^{25} -129.811 q^{27} -29.0000 q^{29} +58.5275 q^{31} -65.8153 q^{33} +130.342 q^{35} -226.421 q^{37} +3.49725 q^{39} +447.304 q^{41} -490.884 q^{43} +95.2782 q^{45} +223.382 q^{47} +336.562 q^{49} +240.347 q^{51} +137.117 q^{53} +116.753 q^{55} -312.912 q^{57} +352.244 q^{59} +133.867 q^{61} +496.751 q^{63} -6.20394 q^{65} +450.653 q^{67} +101.437 q^{69} +462.170 q^{71} -17.1569 q^{73} +70.4643 q^{75} +608.712 q^{77} +1221.19 q^{79} +148.620 q^{81} -481.608 q^{83} -426.364 q^{85} -81.7386 q^{87} -179.628 q^{89} -32.3454 q^{91} +164.964 q^{93} +555.089 q^{95} +956.201 q^{97} +444.960 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 2 q^{3} - 55 q^{5} + 56 q^{7} + 65 q^{9} - 8 q^{11} + 16 q^{13} - 10 q^{15} + 64 q^{17} + 160 q^{19} - 48 q^{21} + 140 q^{23} + 275 q^{25} - 136 q^{27} - 319 q^{29} - 144 q^{31} - 424 q^{33} - 280 q^{35}+ \cdots + 6884 q^{99}+O(q^{100}) \) 11 * q + 2 * q^3 - 55 * q^5 + 56 * q^7 + 65 * q^9 - 8 * q^11 + 16 * q^13 - 10 * q^15 + 64 * q^17 + 160 * q^19 - 48 * q^21 + 140 * q^23 + 275 * q^25 - 136 * q^27 - 319 * q^29 - 144 * q^31 - 424 * q^33 - 280 * q^35 - 50 * q^37 + 572 * q^39 + 62 * q^41 + 326 * q^43 - 325 * q^45 + 1074 * q^47 + 1513 * q^49 - 144 * q^51 + 576 * q^53 + 40 * q^55 - 896 * q^57 - 468 * q^59 + 660 * q^61 + 2552 * q^63 - 80 * q^65 + 892 * q^67 - 510 * q^69 + 920 * q^71 - 676 * q^73 + 50 * q^75 + 1708 * q^77 + 2308 * q^79 + 1287 * q^81 - 1548 * q^83 - 320 * q^85 - 58 * q^87 + 1854 * q^89 + 3100 * q^91 + 5424 * q^93 - 800 * q^95 + 4724 * q^97 + 6884 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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