LMFDB - Embedded newform 2156.4.a.m.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2156,4,Mod(1,2156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2156, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2156.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2156 = 2^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2156.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:	$127.208117972$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4 x^{9} - 178 x^{8} + 572 x^{7} + 9767 x^{6} - 23476 x^{5} - 169595 x^{4} + 254142 x^{3} + \cdots + 89424 $ x^10 - 4x^9 - 178x^8 + 572x^7 + 9767x^6 - 23476x^5 - 169595x^4 + 254142x^3 + 725472x^2 - 539352*x + 89424
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 308)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.28467$ of defining polynomial
Character	$\chi$	$=$	2156.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.28467 q^{3} +20.1685 q^{5} -16.2110 q^{9} +11.0000 q^{11} +18.0138 q^{13} +66.2468 q^{15} +6.74170 q^{17} +158.204 q^{19} +85.0708 q^{23} +281.769 q^{25} -141.934 q^{27} -49.3654 q^{29} -149.492 q^{31} +36.1313 q^{33} +284.122 q^{37} +59.1693 q^{39} -223.913 q^{41} -38.0640 q^{43} -326.951 q^{45} +411.068 q^{47} +22.1442 q^{51} -357.458 q^{53} +221.854 q^{55} +519.648 q^{57} -748.006 q^{59} -125.145 q^{61} +363.312 q^{65} +637.971 q^{67} +279.429 q^{69} -176.322 q^{71} +1053.44 q^{73} +925.516 q^{75} +419.986 q^{79} -28.5081 q^{81} -1472.87 q^{83} +135.970 q^{85} -162.149 q^{87} +1342.12 q^{89} -491.033 q^{93} +3190.75 q^{95} +1289.25 q^{97} -178.321 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 6 q^{3} + 10 q^{5} + 104 q^{9} + 110 q^{11} + 8 q^{13} + 54 q^{15} + 166 q^{17} + 342 q^{19} - 54 q^{23} + 198 q^{25} + 306 q^{27} - 80 q^{29} + 492 q^{31} + 66 q^{33} + 258 q^{37} - 418 q^{39} + 634 q^{41}+ \cdots + 1144 q^{99}+O(q^{100}) $ 10 * q + 6 * q^3 + 10 * q^5 + 104 * q^9 + 110 * q^11 + 8 * q^13 + 54 * q^15 + 166 * q^17 + 342 * q^19 - 54 * q^23 + 198 * q^25 + 306 * q^27 - 80 * q^29 + 492 * q^31 + 66 * q^33 + 258 * q^37 - 418 * q^39 + 634 * q^41 - 274 * q^43 + 138 * q^45 + 312 * q^47 - 196 * q^51 + 656 * q^53 + 110 * q^55 - 738 * q^57 + 934 * q^59 - 228 * q^61 + 524 * q^65 + 478 * q^67 + 1572 * q^69 + 138 * q^71 - 364 * q^73 + 234 * q^75 - 616 * q^79 + 4082 * q^81 - 132 * q^83 - 496 * q^85 - 2120 * q^87 + 1906 * q^89 - 2984 * q^93 + 2322 * q^95 + 1350 * q^97 + 1144 * 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
$3$	3.28467	0.632134	0.316067	−	0.948737i	$-0.397638\pi$
$3$	3.28467	0.632134	0.316067	+	0.948737i	$0.397638\pi$
$4$	0	0
$5$	20.1685	1.80393	0.901963	−	0.431813i	$-0.142126\pi$
$5$	20.1685	1.80393	0.901963	+	0.431813i	$0.142126\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−16.2110	−0.600406
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	18.0138	0.384318	0.192159	−	0.981364i	$-0.438451\pi$
$13$	18.0138	0.384318	0.192159	+	0.981364i	$0.438451\pi$
$14$	0	0
$15$	66.2468	1.14032
$16$	0	0
$17$	6.74170	0.0961825	0.0480912	−	0.998843i	$-0.484686\pi$
$17$	6.74170	0.0961825	0.0480912	+	0.998843i	$0.484686\pi$
$18$	0	0
$19$	158.204	1.91024	0.955120	−	0.296219i	$-0.0957259\pi$
$19$	158.204	1.91024	0.955120	+	0.296219i	$0.0957259\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	85.0708	0.771239	0.385619	−	0.922658i	$-0.373988\pi$
$23$	85.0708	0.771239	0.385619	+	0.922658i	$0.373988\pi$
$24$	0	0
$25$	281.769	2.25415
$26$	0	0
$27$	−141.934	−1.01167
$28$	0	0
$29$	−49.3654	−0.316101	−0.158050	−	0.987431i	$-0.550521\pi$
$29$	−49.3654	−0.316101	−0.158050	+	0.987431i	$0.550521\pi$
$30$	0	0
$31$	−149.492	−0.866117	−0.433059	−	0.901366i	$-0.642566\pi$
$31$	−149.492	−0.866117	−0.433059	+	0.901366i	$0.642566\pi$
$32$	0	0
$33$	36.1313	0.190596
$34$	0	0
$35$	0	0
$36$	0	0
$37$	284.122	1.26241	0.631207	−	0.775614i	$-0.282560\pi$
$37$	284.122	1.26241	0.631207	+	0.775614i	$0.282560\pi$
$38$	0	0
$39$	59.1693	0.242940
$40$	0	0
$41$	−223.913	−0.852911	−0.426456	−	0.904509i	$-0.640238\pi$
$41$	−223.913	−0.852911	−0.426456	+	0.904509i	$0.640238\pi$
$42$	0	0
$43$	−38.0640	−0.134993	−0.0674965	−	0.997720i	$-0.521501\pi$
$43$	−38.0640	−0.134993	−0.0674965	+	0.997720i	$0.521501\pi$
$44$	0	0
$45$	−326.951	−1.08309
$46$	0	0
$47$	411.068	1.27576	0.637878	−	0.770138i	$-0.279813\pi$
$47$	411.068	1.27576	0.637878	+	0.770138i	$0.279813\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	22.1442	0.0608002
$52$	0	0
$53$	−357.458	−0.926427	−0.463213	−	0.886247i	$-0.653304\pi$
$53$	−357.458	−0.926427	−0.463213	+	0.886247i	$0.653304\pi$
$54$	0	0
$55$	221.854	0.543904
$56$	0	0
$57$	519.648	1.20753
$58$	0	0
$59$	−748.006	−1.65054	−0.825272	−	0.564735i	$-0.808978\pi$
$59$	−748.006	−1.65054	−0.825272	+	0.564735i	$0.808978\pi$
$60$	0	0
$61$	−125.145	−0.262674	−0.131337	−	0.991338i	$-0.541927\pi$
$61$	−125.145	−0.262674	−0.131337	+	0.991338i	$0.541927\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	363.312	0.693281
$66$	0	0
$67$	637.971	1.16329	0.581646	−	0.813442i	$-0.302409\pi$
$67$	637.971	1.16329	0.581646	+	0.813442i	$0.302409\pi$
$68$	0	0
$69$	279.429	0.487526
$70$	0	0
$71$	−176.322	−0.294727	−0.147363	−	0.989082i	$-0.547079\pi$
$71$	−176.322	−0.294727	−0.147363	+	0.989082i	$0.547079\pi$
$72$	0	0
$73$	1053.44	1.68899	0.844494	−	0.535564i	$-0.179901\pi$
$73$	1053.44	1.68899	0.844494	+	0.535564i	$0.179901\pi$
$74$	0	0
$75$	925.516	1.42493
$76$	0	0
$77$	0	0
$78$	0	0
$79$	419.986	0.598128	0.299064	−	0.954233i	$-0.403326\pi$
$79$	419.986	0.598128	0.299064	+	0.954233i	$0.403326\pi$
$80$	0	0
$81$	−28.5081	−0.0391057
$82$	0	0
$83$	−1472.87	−1.94781	−0.973907	−	0.226949i	$-0.927125\pi$
$83$	−1472.87	−1.94781	−0.973907	+	0.226949i	$0.927125\pi$
$84$	0	0
$85$	135.970	0.173506
$86$	0	0
$87$	−162.149	−0.199818
$88$	0	0
$89$	1342.12	1.59847	0.799237	−	0.601016i	$-0.205237\pi$
$89$	1342.12	1.59847	0.799237	+	0.601016i	$0.205237\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−491.033	−0.547502
$94$	0	0
$95$	3190.75	3.44593
$96$	0	0
$97$	1289.25	1.34952	0.674759	−	0.738038i	$-0.264248\pi$
$97$	1289.25	1.34952	0.674759	+	0.738038i	$0.264248\pi$
$98$	0	0
$99$	−178.321	−0.181029
$100$	0	0
$101$	−1358.24	−1.33811	−0.669057	−	0.743211i	$-0.733302\pi$
$101$	−1358.24	−1.33811	−0.669057	+	0.743211i	$0.733302\pi$
$102$	0	0
$103$	−1149.72	−1.09986	−0.549929	−	0.835212i	$-0.685345\pi$
$103$	−1149.72	−1.09986	−0.549929	+	0.835212i	$0.685345\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1431.53	1.29337	0.646687	−	0.762755i	$-0.276154\pi$
$107$	1431.53	1.29337	0.646687	+	0.762755i	$0.276154\pi$
$108$	0	0
$109$	1118.06	0.982486	0.491243	−	0.871022i	$-0.336543\pi$
$109$	1118.06	0.982486	0.491243	+	0.871022i	$0.336543\pi$
$110$	0	0
$111$	933.245	0.798015
$112$	0	0
$113$	584.060	0.486228	0.243114	−	0.969998i	$-0.421831\pi$
$113$	584.060	0.486228	0.243114	+	0.969998i	$0.421831\pi$
$114$	0	0
$115$	1715.75	1.39126
$116$	0	0
$117$	−292.021	−0.230747
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−735.480	−0.539154
$124$	0	0
$125$	3161.79	2.26239
$126$	0	0
$127$	137.723	0.0962277	0.0481139	−	0.998842i	$-0.484679\pi$
$127$	137.723	0.0962277	0.0481139	+	0.998842i	$0.484679\pi$
$128$	0	0
$129$	−125.027	−0.0853337
$130$	0	0
$131$	−1547.45	−1.03207	−0.516034	−	0.856568i	$-0.672592\pi$
$131$	−1547.45	−1.03207	−0.516034	+	0.856568i	$0.672592\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2862.59	−1.82498
$136$	0	0
$137$	−599.214	−0.373681	−0.186841	−	0.982390i	$-0.559825\pi$
$137$	−599.214	−0.373681	−0.186841	+	0.982390i	$0.559825\pi$
$138$	0	0
$139$	1760.91	1.07452	0.537260	−	0.843417i	$-0.319459\pi$
$139$	1760.91	1.07452	0.537260	+	0.843417i	$0.319459\pi$
$140$	0	0
$141$	1350.22	0.806448
$142$	0	0
$143$	198.152	0.115876
$144$	0	0
$145$	−995.627	−0.570223
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2474.91	−1.36076	−0.680379	−	0.732861i	$-0.738185\pi$
$149$	−2474.91	−1.36076	−0.680379	+	0.732861i	$0.738185\pi$
$150$	0	0
$151$	−419.776	−0.226231	−0.113116	−	0.993582i	$-0.536083\pi$
$151$	−419.776	−0.226231	−0.113116	+	0.993582i	$0.536083\pi$
$152$	0	0
$153$	−109.290	−0.0577486
$154$	0	0
$155$	−3015.04	−1.56241
$156$	0	0
$157$	1277.60	0.649449	0.324724	−	0.945809i	$-0.394728\pi$
$157$	1277.60	0.649449	0.324724	+	0.945809i	$0.394728\pi$
$158$	0	0
$159$	−1174.13	−0.585626
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2653.83	1.27524	0.637620	−	0.770351i	$-0.279919\pi$
$163$	2653.83	1.27524	0.637620	+	0.770351i	$0.279919\pi$
$164$	0	0
$165$	728.715	0.343820
$166$	0	0
$167$	3219.63	1.49187	0.745935	−	0.666019i	$-0.232003\pi$
$167$	3219.63	1.49187	0.745935	+	0.666019i	$0.232003\pi$
$168$	0	0
$169$	−1872.50	−0.852300
$170$	0	0
$171$	−2564.65	−1.14692
$172$	0	0
$173$	911.626	0.400634	0.200317	−	0.979731i	$-0.435803\pi$
$173$	911.626	0.400634	0.200317	+	0.979731i	$0.435803\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2456.95	−1.04337
$178$	0	0
$179$	−1493.62	−0.623677	−0.311838	−	0.950135i	$-0.600945\pi$
$179$	−1493.62	−0.623677	−0.311838	+	0.950135i	$0.600945\pi$
$180$	0	0
$181$	−1967.65	−0.808035	−0.404018	−	0.914751i	$-0.632387\pi$
$181$	−1967.65	−0.808035	−0.404018	+	0.914751i	$0.632387\pi$
$182$	0	0
$183$	−411.058	−0.166045
$184$	0	0
$185$	5730.31	2.27730
$186$	0	0
$187$	74.1587	0.0290001
$188$	0	0
$189$	0	0
$190$	0	0
$191$	336.180	0.127357	0.0636784	−	0.997970i	$-0.479717\pi$
$191$	336.180	0.127357	0.0636784	+	0.997970i	$0.479717\pi$
$192$	0	0
$193$	−2281.88	−0.851052	−0.425526	−	0.904946i	$-0.639911\pi$
$193$	−2281.88	−0.851052	−0.425526	+	0.904946i	$0.639911\pi$
$194$	0	0
$195$	1193.36	0.438247
$196$	0	0
$197$	1810.04	0.654619	0.327309	−	0.944917i	$-0.393858\pi$
$197$	1810.04	0.654619	0.327309	+	0.944917i	$0.393858\pi$
$198$	0	0
$199$	1793.38	0.638840	0.319420	−	0.947613i	$-0.396512\pi$
$199$	1793.38	0.638840	0.319420	+	0.947613i	$0.396512\pi$
$200$	0	0
$201$	2095.52	0.735357
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−4515.99	−1.53859
$206$	0	0
$207$	−1379.08	−0.463057
$208$	0	0
$209$	1740.25	0.575959
$210$	0	0
$211$	−701.349	−0.228829	−0.114414	−	0.993433i	$-0.536499\pi$
$211$	−701.349	−0.228829	−0.114414	+	0.993433i	$0.536499\pi$
$212$	0	0
$213$	−579.160	−0.186307
$214$	0	0
$215$	−767.694	−0.243518
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3460.21	1.06767
$220$	0	0
$221$	121.444	0.0369646
$222$	0	0
$223$	3527.11	1.05916	0.529580	−	0.848260i	$-0.322350\pi$
$223$	3527.11	1.05916	0.529580	+	0.848260i	$0.322350\pi$
$224$	0	0
$225$	−4567.75	−1.35341
$226$	0	0
$227$	−1594.82	−0.466307	−0.233153	−	0.972440i	$-0.574904\pi$
$227$	−1594.82	−0.466307	−0.233153	+	0.972440i	$0.574904\pi$
$228$	0	0
$229$	−2965.48	−0.855741	−0.427870	−	0.903840i	$-0.640736\pi$
$229$	−2965.48	−0.855741	−0.427870	+	0.903840i	$0.640736\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3675.63	−1.03347	−0.516734	−	0.856146i	$-0.672853\pi$
$233$	−3675.63	−1.03347	−0.516734	+	0.856146i	$0.672853\pi$
$234$	0	0
$235$	8290.64	2.30137
$236$	0	0
$237$	1379.51	0.378097
$238$	0	0
$239$	−3671.87	−0.993780	−0.496890	−	0.867814i	$-0.665525\pi$
$239$	−3671.87	−0.993780	−0.496890	+	0.867814i	$0.665525\pi$
$240$	0	0
$241$	−3990.86	−1.06670	−0.533348	−	0.845896i	$-0.679066\pi$
$241$	−3990.86	−1.06670	−0.533348	+	0.845896i	$0.679066\pi$
$242$	0	0
$243$	3738.57	0.986951
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2849.86	0.734139
$248$	0	0
$249$	−4837.89	−1.23128
$250$	0	0
$251$	777.368	0.195486	0.0977430	−	0.995212i	$-0.468838\pi$
$251$	777.368	0.195486	0.0977430	+	0.995212i	$0.468838\pi$
$252$	0	0
$253$	935.779	0.232537
$254$	0	0
$255$	446.616	0.109679
$256$	0	0
$257$	−3297.90	−0.800457	−0.400229	−	0.916415i	$-0.631069\pi$
$257$	−3297.90	−0.800457	−0.400229	+	0.916415i	$0.631069\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	800.261	0.189789
$262$	0	0
$263$	4018.93	0.942273	0.471136	−	0.882060i	$-0.343844\pi$
$263$	4018.93	0.942273	0.471136	+	0.882060i	$0.343844\pi$
$264$	0	0
$265$	−7209.39	−1.67121
$266$	0	0
$267$	4408.41	1.01045
$268$	0	0
$269$	−2867.91	−0.650036	−0.325018	−	0.945708i	$-0.605370\pi$
$269$	−2867.91	−0.650036	−0.325018	+	0.945708i	$0.605370\pi$
$270$	0	0
$271$	7583.57	1.69989	0.849943	−	0.526874i	$-0.176636\pi$
$271$	7583.57	1.69989	0.849943	+	0.526874i	$0.176636\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3099.46	0.679652
$276$	0	0
$277$	1856.66	0.402730	0.201365	−	0.979516i	$-0.435462\pi$
$277$	1856.66	0.402730	0.201365	+	0.979516i	$0.435462\pi$
$278$	0	0
$279$	2423.42	0.520023
$280$	0	0
$281$	146.754	0.0311551	0.0155776	−	0.999879i	$-0.495041\pi$
$281$	146.754	0.0311551	0.0155776	+	0.999879i	$0.495041\pi$
$282$	0	0
$283$	1176.87	0.247200	0.123600	−	0.992332i	$-0.460556\pi$
$283$	1176.87	0.247200	0.123600	+	0.992332i	$0.460556\pi$
$284$	0	0
$285$	10480.5	2.17829
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4867.55	−0.990749
$290$	0	0
$291$	4234.75	0.853076
$292$	0	0
$293$	−2767.43	−0.551792	−0.275896	−	0.961187i	$-0.588975\pi$
$293$	−2767.43	−0.551792	−0.275896	+	0.961187i	$0.588975\pi$
$294$	0	0
$295$	−15086.2	−2.97746
$296$	0	0
$297$	−1561.27	−0.305030
$298$	0	0
$299$	1532.45	0.296401
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4461.35	−0.845867
$304$	0	0
$305$	−2523.98	−0.473845
$306$	0	0
$307$	2937.35	0.546069	0.273035	−	0.962004i	$-0.411973\pi$
$307$	2937.35	0.546069	0.273035	+	0.962004i	$0.411973\pi$
$308$	0	0
$309$	−3776.45	−0.695258
$310$	0	0
$311$	3235.74	0.589975	0.294987	−	0.955501i	$-0.404685\pi$
$311$	3235.74	0.589975	0.294987	+	0.955501i	$0.404685\pi$
$312$	0	0
$313$	9509.69	1.71731	0.858657	−	0.512550i	$-0.171299\pi$
$313$	9509.69	1.71731	0.858657	+	0.512550i	$0.171299\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4435.54	−0.785883	−0.392941	−	0.919563i	$-0.628543\pi$
$317$	−4435.54	−0.785883	−0.392941	+	0.919563i	$0.628543\pi$
$318$	0	0
$319$	−543.019	−0.0953080
$320$	0	0
$321$	4702.09	0.817586
$322$	0	0
$323$	1066.57	0.183732
$324$	0	0
$325$	5075.73	0.866310
$326$	0	0
$327$	3672.46	0.621063
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2711.75	−0.450307	−0.225153	−	0.974323i	$-0.572288\pi$
$331$	−2711.75	−0.450307	−0.225153	+	0.974323i	$0.572288\pi$
$332$	0	0
$333$	−4605.89	−0.757962
$334$	0	0
$335$	12866.9	2.09849
$336$	0	0
$337$	3897.67	0.630029	0.315015	−	0.949087i	$-0.397991\pi$
$337$	3897.67	0.630029	0.315015	+	0.949087i	$0.397991\pi$
$338$	0	0
$339$	1918.44	0.307361
$340$	0	0
$341$	−1644.42	−0.261144
$342$	0	0
$343$	0	0
$344$	0	0
$345$	5635.67	0.879462
$346$	0	0
$347$	208.013	0.0321808	0.0160904	−	0.999871i	$-0.494878\pi$
$347$	208.013	0.0321808	0.0160904	+	0.999871i	$0.494878\pi$
$348$	0	0
$349$	2513.14	0.385459	0.192729	−	0.981252i	$-0.438266\pi$
$349$	2513.14	0.385459	0.192729	+	0.981252i	$0.438266\pi$
$350$	0	0
$351$	−2556.76	−0.388803
$352$	0	0
$353$	8788.17	1.32506	0.662531	−	0.749034i	$-0.269482\pi$
$353$	8788.17	1.32506	0.662531	+	0.749034i	$0.269482\pi$
$354$	0	0
$355$	−3556.16	−0.531666
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7764.82	1.14154	0.570768	−	0.821111i	$-0.306645\pi$
$359$	7764.82	1.14154	0.570768	+	0.821111i	$0.306645\pi$
$360$	0	0
$361$	18169.6	2.64902
$362$	0	0
$363$	397.445	0.0574667
$364$	0	0
$365$	21246.4	3.04681
$366$	0	0
$367$	−3695.33	−0.525599	−0.262800	−	0.964850i	$-0.584646\pi$
$367$	−3695.33	−0.525599	−0.262800	+	0.964850i	$0.584646\pi$
$368$	0	0
$369$	3629.85	0.512093
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−10739.0	−1.49073	−0.745366	−	0.666656i	$-0.767725\pi$
$373$	−10739.0	−1.49073	−0.745366	+	0.666656i	$0.767725\pi$
$374$	0	0
$375$	10385.4	1.43014
$376$	0	0
$377$	−889.259	−0.121483
$378$	0	0
$379$	3267.42	0.442839	0.221419	−	0.975179i	$-0.428931\pi$
$379$	3267.42	0.442839	0.221419	+	0.975179i	$0.428931\pi$
$380$	0	0
$381$	452.373	0.0608288
$382$	0	0
$383$	14519.2	1.93707	0.968536	−	0.248873i	$-0.0800600\pi$
$383$	14519.2	1.93707	0.968536	+	0.248873i	$0.0800600\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	617.054	0.0810507
$388$	0	0
$389$	−115.798	−0.0150930	−0.00754652	−	0.999972i	$-0.502402\pi$
$389$	−115.798	−0.0150930	−0.00754652	+	0.999972i	$0.502402\pi$
$390$	0	0
$391$	573.522	0.0741797
$392$	0	0
$393$	−5082.84	−0.652406
$394$	0	0
$395$	8470.50	1.07898
$396$	0	0
$397$	−10165.3	−1.28509	−0.642545	−	0.766248i	$-0.722122\pi$
$397$	−10165.3	−1.28509	−0.642545	+	0.766248i	$0.722122\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13876.0	1.72802	0.864011	−	0.503473i	$-0.167945\pi$
$401$	13876.0	1.72802	0.864011	+	0.503473i	$0.167945\pi$
$402$	0	0
$403$	−2692.93	−0.332864
$404$	0	0
$405$	−574.965	−0.0705438
$406$	0	0
$407$	3125.34	0.380632
$408$	0	0
$409$	−11520.6	−1.39281	−0.696403	−	0.717651i	$-0.745217\pi$
$409$	−11520.6	−1.39281	−0.696403	+	0.717651i	$0.745217\pi$
$410$	0	0
$411$	−1968.22	−0.236217
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−29705.6	−3.51371
$416$	0	0
$417$	5783.99	0.679241
$418$	0	0
$419$	−14017.0	−1.63431	−0.817155	−	0.576419i	$-0.804450\pi$
$419$	−14017.0	−1.63431	−0.817155	+	0.576419i	$0.804450\pi$
$420$	0	0
$421$	−7481.04	−0.866042	−0.433021	−	0.901384i	$-0.642552\pi$
$421$	−7481.04	−0.866042	−0.433021	+	0.901384i	$0.642552\pi$
$422$	0	0
$423$	−6663.82	−0.765972
$424$	0	0
$425$	1899.60	0.216810
$426$	0	0
$427$	0	0
$428$	0	0
$429$	650.863	0.0732493
$430$	0	0
$431$	6857.34	0.766372	0.383186	−	0.923671i	$-0.374827\pi$
$431$	6857.34	0.766372	0.383186	+	0.923671i	$0.374827\pi$
$432$	0	0
$433$	−6537.32	−0.725551	−0.362775	−	0.931877i	$-0.618171\pi$
$433$	−6537.32	−0.725551	−0.362775	+	0.931877i	$0.618171\pi$
$434$	0	0
$435$	−3270.30	−0.360457
$436$	0	0
$437$	13458.6	1.47325
$438$	0	0
$439$	5429.91	0.590331	0.295166	−	0.955446i	$-0.404625\pi$
$439$	5429.91	0.590331	0.295166	+	0.955446i	$0.404625\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3919.73	−0.420388	−0.210194	−	0.977660i	$-0.567410\pi$
$443$	−3919.73	−0.420388	−0.210194	+	0.977660i	$0.567410\pi$
$444$	0	0
$445$	27068.5	2.88353
$446$	0	0
$447$	−8129.27	−0.860181
$448$	0	0
$449$	9796.86	1.02972	0.514858	−	0.857275i	$-0.327845\pi$
$449$	9796.86	1.02972	0.514858	+	0.857275i	$0.327845\pi$
$450$	0	0
$451$	−2463.04	−0.257162
$452$	0	0
$453$	−1378.82	−0.143008
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−18507.0	−1.89435	−0.947177	−	0.320711i	$-0.896078\pi$
$457$	−18507.0	−1.89435	−0.947177	+	0.320711i	$0.896078\pi$
$458$	0	0
$459$	−956.874	−0.0973051
$460$	0	0
$461$	−4871.70	−0.492185	−0.246093	−	0.969246i	$-0.579147\pi$
$461$	−4871.70	−0.492185	−0.246093	+	0.969246i	$0.579147\pi$
$462$	0	0
$463$	6158.63	0.618177	0.309088	−	0.951033i	$-0.399976\pi$
$463$	6158.63	0.618177	0.309088	+	0.951033i	$0.399976\pi$
$464$	0	0
$465$	−9903.40	−0.987654
$466$	0	0
$467$	−3685.92	−0.365234	−0.182617	−	0.983184i	$-0.558457\pi$
$467$	−3685.92	−0.365234	−0.182617	+	0.983184i	$0.558457\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	4196.48	0.410539
$472$	0	0
$473$	−418.704	−0.0407019
$474$	0	0
$475$	44577.0	4.30597
$476$	0	0
$477$	5794.74	0.556232
$478$	0	0
$479$	−10342.0	−0.986506	−0.493253	−	0.869886i	$-0.664192\pi$
$479$	−10342.0	−0.986506	−0.493253	+	0.869886i	$0.664192\pi$
$480$	0	0
$481$	5118.11	0.485168
$482$	0	0
$483$	0	0
$484$	0	0
$485$	26002.2	2.43443
$486$	0	0
$487$	−10203.9	−0.949454	−0.474727	−	0.880133i	$-0.657453\pi$
$487$	−10203.9	−0.949454	−0.474727	+	0.880133i	$0.657453\pi$
$488$	0	0
$489$	8716.95	0.806123
$490$	0	0
$491$	2590.24	0.238078	0.119039	−	0.992890i	$-0.462019\pi$
$491$	2590.24	0.238078	0.119039	+	0.992890i	$0.462019\pi$
$492$	0	0
$493$	−332.807	−0.0304034
$494$	0	0
$495$	−3596.46	−0.326564
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−3706.06	−0.332477	−0.166238	−	0.986086i	$-0.553162\pi$
$499$	−3706.06	−0.332477	−0.166238	+	0.986086i	$0.553162\pi$
$500$	0	0
$501$	10575.4	0.943061
$502$	0	0
$503$	−10724.5	−0.950661	−0.475330	−	0.879807i	$-0.657671\pi$
$503$	−10724.5	−0.950661	−0.475330	+	0.879807i	$0.657671\pi$
$504$	0	0
$505$	−27393.6	−2.41386
$506$	0	0
$507$	−6150.54	−0.538768
$508$	0	0
$509$	−1932.44	−0.168279	−0.0841395	−	0.996454i	$-0.526814\pi$
$509$	−1932.44	−0.168279	−0.0841395	+	0.996454i	$0.526814\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−22454.5	−1.93254
$514$	0	0
$515$	−23188.1	−1.98406
$516$	0	0
$517$	4521.75	0.384655
$518$	0	0
$519$	2994.39	0.253254
$520$	0	0
$521$	7116.60	0.598434	0.299217	−	0.954185i	$-0.403275\pi$
$521$	7116.60	0.598434	0.299217	+	0.954185i	$0.403275\pi$
$522$	0	0
$523$	6219.68	0.520015	0.260007	−	0.965607i	$-0.416275\pi$
$523$	6219.68	0.520015	0.260007	+	0.965607i	$0.416275\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1007.83	−0.0833053
$528$	0	0
$529$	−4929.96	−0.405191
$530$	0	0
$531$	12125.9	0.990997
$532$	0	0
$533$	−4033.53	−0.327789
$534$	0	0
$535$	28871.8	2.33315
$536$	0	0
$537$	−4906.03	−0.394247
$538$	0	0
$539$	0	0
$540$	0	0
$541$	11251.8	0.894186	0.447093	−	0.894488i	$-0.352459\pi$
$541$	11251.8	0.894186	0.447093	+	0.894488i	$0.352459\pi$
$542$	0	0
$543$	−6463.08	−0.510787
$544$	0	0
$545$	22549.7	1.77233
$546$	0	0
$547$	4334.19	0.338787	0.169393	−	0.985548i	$-0.445819\pi$
$547$	4334.19	0.338787	0.169393	+	0.985548i	$0.445819\pi$
$548$	0	0
$549$	2028.71	0.157711
$550$	0	0
$551$	−7809.82	−0.603829
$552$	0	0
$553$	0	0
$554$	0	0
$555$	18822.2	1.43956
$556$	0	0
$557$	−936.401	−0.0712327	−0.0356163	−	0.999366i	$-0.511339\pi$
$557$	−936.401	−0.0712327	−0.0356163	+	0.999366i	$0.511339\pi$
$558$	0	0
$559$	−685.677	−0.0518802
$560$	0	0
$561$	243.586	0.0183320
$562$	0	0
$563$	−11738.5	−0.878719	−0.439360	−	0.898311i	$-0.644795\pi$
$563$	−11738.5	−0.878719	−0.439360	+	0.898311i	$0.644795\pi$
$564$	0	0
$565$	11779.6	0.877119
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−23924.6	−1.76269	−0.881345	−	0.472474i	$-0.843361\pi$
$569$	−23924.6	−1.76269	−0.881345	+	0.472474i	$0.843361\pi$
$570$	0	0
$571$	−6784.46	−0.497235	−0.248617	−	0.968602i	$-0.579976\pi$
$571$	−6784.46	−0.497235	−0.248617	+	0.968602i	$0.579976\pi$
$572$	0	0
$573$	1104.24	0.0805066
$574$	0	0
$575$	23970.3	1.73849
$576$	0	0
$577$	−7887.28	−0.569067	−0.284534	−	0.958666i	$-0.591839\pi$
$577$	−7887.28	−0.569067	−0.284534	+	0.958666i	$0.591839\pi$
$578$	0	0
$579$	−7495.20	−0.537979
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−3932.04	−0.279328
$584$	0	0
$585$	−5889.63	−0.416250
$586$	0	0
$587$	11414.7	0.802613	0.401306	−	0.915944i	$-0.368556\pi$
$587$	11414.7	0.802613	0.401306	+	0.915944i	$0.368556\pi$
$588$	0	0
$589$	−23650.4	−1.65449
$590$	0	0
$591$	5945.37	0.413807
$592$	0	0
$593$	−13577.7	−0.940252	−0.470126	−	0.882599i	$-0.655792\pi$
$593$	−13577.7	−0.940252	−0.470126	+	0.882599i	$0.655792\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5890.65	0.403833
$598$	0	0
$599$	24789.7	1.69095	0.845476	−	0.534014i	$-0.179317\pi$
$599$	24789.7	1.69095	0.845476	+	0.534014i	$0.179317\pi$
$600$	0	0
$601$	−27467.5	−1.86426	−0.932132	−	0.362119i	$-0.882054\pi$
$601$	−27467.5	−1.86426	−0.932132	+	0.362119i	$0.882054\pi$
$602$	0	0
$603$	−10342.1	−0.698448
$604$	0	0
$605$	2440.39	0.163993
$606$	0	0
$607$	−16648.6	−1.11325	−0.556627	−	0.830763i	$-0.687905\pi$
$607$	−16648.6	−1.11325	−0.556627	+	0.830763i	$0.687905\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7404.91	0.490295
$612$	0	0
$613$	−25766.7	−1.69773	−0.848864	−	0.528611i	$-0.822713\pi$
$613$	−25766.7	−1.69773	−0.848864	+	0.528611i	$0.822713\pi$
$614$	0	0
$615$	−14833.5	−0.972594
$616$	0	0
$617$	−21818.5	−1.42363	−0.711815	−	0.702367i	$-0.752126\pi$
$617$	−21818.5	−1.42363	−0.711815	+	0.702367i	$0.752126\pi$
$618$	0	0
$619$	5313.45	0.345017	0.172509	−	0.985008i	$-0.444813\pi$
$619$	5313.45	0.345017	0.172509	+	0.985008i	$0.444813\pi$
$620$	0	0
$621$	−12074.4	−0.780240
$622$	0	0
$623$	0	0
$624$	0	0
$625$	28547.5	1.82704
$626$	0	0
$627$	5716.13	0.364083
$628$	0	0
$629$	1915.46	0.121422
$630$	0	0
$631$	−26807.8	−1.69129	−0.845645	−	0.533746i	$-0.820784\pi$
$631$	−26807.8	−1.69129	−0.845645	+	0.533746i	$0.820784\pi$
$632$	0	0
$633$	−2303.70	−0.144650
$634$	0	0
$635$	2777.66	0.173588
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2858.36	0.176956
$640$	0	0
$641$	−4669.35	−0.287720	−0.143860	−	0.989598i	$-0.545951\pi$
$641$	−4669.35	−0.287720	−0.143860	+	0.989598i	$0.545951\pi$
$642$	0	0
$643$	15653.7	0.960067	0.480033	−	0.877250i	$-0.340625\pi$
$643$	15653.7	0.960067	0.480033	+	0.877250i	$0.340625\pi$
$644$	0	0
$645$	−2521.62	−0.153936
$646$	0	0
$647$	−4680.90	−0.284429	−0.142214	−	0.989836i	$-0.545422\pi$
$647$	−4680.90	−0.284429	−0.142214	+	0.989836i	$0.545422\pi$
$648$	0	0
$649$	−8228.07	−0.497658
$650$	0	0
$651$	0	0
$652$	0	0
$653$	21476.7	1.28706	0.643528	−	0.765423i	$-0.277470\pi$
$653$	21476.7	1.28706	0.643528	+	0.765423i	$0.277470\pi$
$654$	0	0
$655$	−31209.7	−1.86178
$656$	0	0
$657$	−17077.3	−1.01408
$658$	0	0
$659$	−17974.1	−1.06248	−0.531239	−	0.847222i	$-0.678273\pi$
$659$	−17974.1	−1.06248	−0.531239	+	0.847222i	$0.678273\pi$
$660$	0	0
$661$	13280.8	0.781487	0.390743	−	0.920500i	$-0.372218\pi$
$661$	13280.8	0.781487	0.390743	+	0.920500i	$0.372218\pi$
$662$	0	0
$663$	398.902	0.0233666
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4199.56	−0.243789
$668$	0	0
$669$	11585.4	0.669531
$670$	0	0
$671$	−1376.59	−0.0791992
$672$	0	0
$673$	28060.2	1.60719	0.803596	−	0.595176i	$-0.202917\pi$
$673$	28060.2	1.60719	0.803596	+	0.595176i	$0.202917\pi$
$674$	0	0
$675$	−39992.4	−2.28046
$676$	0	0
$677$	−14484.9	−0.822305	−0.411152	−	0.911567i	$-0.634874\pi$
$677$	−14484.9	−0.822305	−0.411152	+	0.911567i	$0.634874\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−5238.44	−0.294768
$682$	0	0
$683$	8480.37	0.475098	0.237549	−	0.971376i	$-0.423656\pi$
$683$	8480.37	0.475098	0.237549	+	0.971376i	$0.423656\pi$
$684$	0	0
$685$	−12085.3	−0.674093
$686$	0	0
$687$	−9740.62	−0.540943
$688$	0	0
$689$	−6439.18	−0.356042
$690$	0	0
$691$	24567.9	1.35254	0.676270	−	0.736654i	$-0.263595\pi$
$691$	24567.9	1.35254	0.676270	+	0.736654i	$0.263595\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	35514.9	1.93835
$696$	0	0
$697$	−1509.55	−0.0820351
$698$	0	0
$699$	−12073.2	−0.653291
$700$	0	0
$701$	23325.9	1.25679	0.628394	−	0.777895i	$-0.283712\pi$
$701$	23325.9	1.25679	0.628394	+	0.777895i	$0.283712\pi$
$702$	0	0
$703$	44949.3	2.41151
$704$	0	0
$705$	27232.0	1.45477
$706$	0	0
$707$	0	0
$708$	0	0
$709$	29148.6	1.54400	0.772001	−	0.635621i	$-0.219256\pi$
$709$	29148.6	1.54400	0.772001	+	0.635621i	$0.219256\pi$
$710$	0	0
$711$	−6808.39	−0.359120
$712$	0	0
$713$	−12717.4	−0.667983
$714$	0	0
$715$	3996.43	0.209032
$716$	0	0
$717$	−12060.9	−0.628202
$718$	0	0
$719$	−13331.2	−0.691472	−0.345736	−	0.938332i	$-0.612371\pi$
$719$	−13331.2	−0.691472	−0.345736	+	0.938332i	$0.612371\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−13108.6	−0.674295
$724$	0	0
$725$	−13909.6	−0.712539
$726$	0	0
$727$	18836.0	0.960920	0.480460	−	0.877017i	$-0.340470\pi$
$727$	18836.0	0.960920	0.480460	+	0.877017i	$0.340470\pi$
$728$	0	0
$729$	13049.7	0.662991
$730$	0	0
$731$	−256.616	−0.0129840
$732$	0	0
$733$	11574.3	0.583229	0.291614	−	0.956536i	$-0.405808\pi$
$733$	11574.3	0.583229	0.291614	+	0.956536i	$0.405808\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7017.69	0.350746
$738$	0	0
$739$	−16287.5	−0.810750	−0.405375	−	0.914150i	$-0.632859\pi$
$739$	−16287.5	−0.810750	−0.405375	+	0.914150i	$0.632859\pi$
$740$	0	0
$741$	9360.84	0.464074
$742$	0	0
$743$	18975.7	0.936947	0.468473	−	0.883478i	$-0.344804\pi$
$743$	18975.7	0.936947	0.468473	+	0.883478i	$0.344804\pi$
$744$	0	0
$745$	−49915.3	−2.45471
$746$	0	0
$747$	23876.7	1.16948
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−13068.3	−0.634978	−0.317489	−	0.948262i	$-0.602840\pi$
$751$	−13068.3	−0.634978	−0.317489	+	0.948262i	$0.602840\pi$
$752$	0	0
$753$	2553.39	0.123573
$754$	0	0
$755$	−8466.26	−0.408104
$756$	0	0
$757$	12393.2	0.595030	0.297515	−	0.954717i	$-0.403842\pi$
$757$	12393.2	0.595030	0.297515	+	0.954717i	$0.403842\pi$
$758$	0	0
$759$	3073.72	0.146995
$760$	0	0
$761$	−24454.7	−1.16489	−0.582445	−	0.812870i	$-0.697904\pi$
$761$	−24454.7	−1.16489	−0.582445	+	0.812870i	$0.697904\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2204.21	−0.104174
$766$	0	0
$767$	−13474.4	−0.634334
$768$	0	0
$769$	9229.19	0.432787	0.216393	−	0.976306i	$-0.430571\pi$
$769$	9229.19	0.432787	0.216393	+	0.976306i	$0.430571\pi$
$770$	0	0
$771$	−10832.5	−0.505996
$772$	0	0
$773$	24943.5	1.16062	0.580308	−	0.814397i	$-0.302932\pi$
$773$	24943.5	1.16062	0.580308	+	0.814397i	$0.302932\pi$
$774$	0	0
$775$	−42122.3	−1.95236
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−35424.0	−1.62926
$780$	0	0
$781$	−1939.55	−0.0888635
$782$	0	0
$783$	7006.61	0.319790
$784$	0	0
$785$	25767.3	1.17156
$786$	0	0
$787$	−8253.16	−0.373816	−0.186908	−	0.982377i	$-0.559847\pi$
$787$	−8253.16	−0.373816	−0.186908	+	0.982377i	$0.559847\pi$
$788$	0	0
$789$	13200.8	0.595643
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2254.33	−0.100950
$794$	0	0
$795$	−23680.4	−1.05643
$796$	0	0
$797$	12314.7	0.547315	0.273657	−	0.961827i	$-0.411767\pi$
$797$	12314.7	0.547315	0.273657	+	0.961827i	$0.411767\pi$
$798$	0	0
$799$	2771.30	0.122705
$800$	0	0
$801$	−21757.0	−0.959734
$802$	0	0
$803$	11587.9	0.509249
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−9420.13	−0.410910
$808$	0	0
$809$	−21840.8	−0.949175	−0.474588	−	0.880208i	$-0.657403\pi$
$809$	−21840.8	−0.949175	−0.474588	+	0.880208i	$0.657403\pi$
$810$	0	0
$811$	−42180.2	−1.82632	−0.913161	−	0.407600i	$-0.866366\pi$
$811$	−42180.2	−1.82632	−0.913161	+	0.407600i	$0.866366\pi$
$812$	0	0
$813$	24909.5	1.07456
$814$	0	0
$815$	53523.8	2.30044
$816$	0	0
$817$	−6021.89	−0.257869
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−19618.3	−0.833961	−0.416980	−	0.908915i	$-0.636912\pi$
$821$	−19618.3	−0.833961	−0.416980	+	0.908915i	$0.636912\pi$
$822$	0	0
$823$	23634.6	1.00103	0.500516	−	0.865727i	$-0.333144\pi$
$823$	23634.6	1.00103	0.500516	+	0.865727i	$0.333144\pi$
$824$	0	0
$825$	10180.7	0.429631
$826$	0	0
$827$	1479.01	0.0621888	0.0310944	−	0.999516i	$-0.490101\pi$
$827$	1479.01	0.0621888	0.0310944	+	0.999516i	$0.490101\pi$
$828$	0	0
$829$	26614.5	1.11503	0.557515	−	0.830167i	$-0.311755\pi$
$829$	26614.5	1.11503	0.557515	+	0.830167i	$0.311755\pi$
$830$	0	0
$831$	6098.52	0.254579
$832$	0	0
$833$	0	0
$834$	0	0
$835$	64935.1	2.69122
$836$	0	0
$837$	21218.0	0.876226
$838$	0	0
$839$	−56.5620	−0.00232746	−0.00116373	−	0.999999i	$-0.500370\pi$
$839$	−56.5620	−0.00232746	−0.00116373	+	0.999999i	$0.500370\pi$
$840$	0	0
$841$	−21952.1	−0.900080
$842$	0	0
$843$	482.037	0.0196942
$844$	0	0
$845$	−37765.6	−1.53749
$846$	0	0
$847$	0	0
$848$	0	0
$849$	3865.62	0.156263
$850$	0	0
$851$	24170.5	0.973623
$852$	0	0
$853$	−40634.3	−1.63106	−0.815529	−	0.578716i	$-0.803554\pi$
$853$	−40634.3	−1.63106	−0.815529	+	0.578716i	$0.803554\pi$
$854$	0	0
$855$	−51725.1	−2.06896
$856$	0	0
$857$	−35344.6	−1.40881	−0.704403	−	0.709800i	$-0.748785\pi$
$857$	−35344.6	−1.40881	−0.704403	+	0.709800i	$0.748785\pi$
$858$	0	0
$859$	17995.8	0.714796	0.357398	−	0.933952i	$-0.383664\pi$
$859$	17995.8	0.714796	0.357398	+	0.933952i	$0.383664\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	21516.1	0.848688	0.424344	−	0.905501i	$-0.360505\pi$
$863$	21516.1	0.848688	0.424344	+	0.905501i	$0.360505\pi$
$864$	0	0
$865$	18386.1	0.722714
$866$	0	0
$867$	−15988.3	−0.626286
$868$	0	0
$869$	4619.85	0.180343
$870$	0	0
$871$	11492.3	0.447074
$872$	0	0
$873$	−20900.0	−0.810259
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−29888.4	−1.15081	−0.575405	−	0.817869i	$-0.695155\pi$
$877$	−29888.4	−1.15081	−0.575405	+	0.817869i	$0.695155\pi$
$878$	0	0
$879$	−9090.09	−0.348807
$880$	0	0
$881$	−42643.4	−1.63075	−0.815376	−	0.578932i	$-0.803470\pi$
$881$	−42643.4	−1.63075	−0.815376	+	0.578932i	$0.803470\pi$
$882$	0	0
$883$	−9486.36	−0.361542	−0.180771	−	0.983525i	$-0.557859\pi$
$883$	−9486.36	−0.361542	−0.180771	+	0.983525i	$0.557859\pi$
$884$	0	0
$885$	−49553.0	−1.88215
$886$	0	0
$887$	−15062.8	−0.570191	−0.285096	−	0.958499i	$-0.592025\pi$
$887$	−15062.8	−0.570191	−0.285096	+	0.958499i	$0.592025\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−313.589	−0.0117908
$892$	0	0
$893$	65032.8	2.43700
$894$	0	0
$895$	−30124.0	−1.12507
$896$	0	0
$897$	5033.58	0.187365
$898$	0	0
$899$	7379.76	0.273781
$900$	0	0
$901$	−2409.87	−0.0891060
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−39684.6	−1.45764
$906$	0	0
$907$	−43136.7	−1.57920	−0.789598	−	0.613625i	$-0.789711\pi$
$907$	−43136.7	−1.57920	−0.789598	+	0.613625i	$0.789711\pi$
$908$	0	0
$909$	22018.3	0.803412
$910$	0	0
$911$	−3708.67	−0.134878	−0.0674390	−	0.997723i	$-0.521483\pi$
$911$	−3708.67	−0.134878	−0.0674390	+	0.997723i	$0.521483\pi$
$912$	0	0
$913$	−16201.6	−0.587288
$914$	0	0
$915$	−8290.42	−0.299533
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−48744.9	−1.74967	−0.874834	−	0.484423i	$-0.839030\pi$
$919$	−48744.9	−1.74967	−0.874834	+	0.484423i	$0.839030\pi$
$920$	0	0
$921$	9648.20	0.345189
$922$	0	0
$923$	−3176.24	−0.113269
$924$	0	0
$925$	80056.6	2.84567
$926$	0	0
$927$	18638.1	0.660362
$928$	0	0
$929$	−5003.60	−0.176709	−0.0883547	−	0.996089i	$-0.528161\pi$
$929$	−5003.60	−0.176709	−0.0883547	+	0.996089i	$0.528161\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	10628.3	0.372943
$934$	0	0
$935$	1495.67	0.0523141
$936$	0	0
$937$	34591.3	1.20603	0.603014	−	0.797731i	$-0.293966\pi$
$937$	34591.3	1.20603	0.603014	+	0.797731i	$0.293966\pi$
$938$	0	0
$939$	31236.1	1.08557
$940$	0	0
$941$	2704.42	0.0936892	0.0468446	−	0.998902i	$-0.485083\pi$
$941$	2704.42	0.0936892	0.0468446	+	0.998902i	$0.485083\pi$
$942$	0	0
$943$	−19048.5	−0.657798
$944$	0	0
$945$	0	0
$946$	0	0
$947$	11042.5	0.378915	0.189457	−	0.981889i	$-0.439327\pi$
$947$	11042.5	0.378915	0.189457	+	0.981889i	$0.439327\pi$
$948$	0	0
$949$	18976.5	0.649108
$950$	0	0
$951$	−14569.3	−0.496783
$952$	0	0
$953$	14088.1	0.478864	0.239432	−	0.970913i	$-0.423039\pi$
$953$	14088.1	0.478864	0.239432	+	0.970913i	$0.423039\pi$
$954$	0	0
$955$	6780.26	0.229742
$956$	0	0
$957$	−1783.64	−0.0602474
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−7443.00	−0.249840
$962$	0	0
$963$	−23206.5	−0.776550
$964$	0	0
$965$	−46022.0	−1.53524
$966$	0	0
$967$	42880.4	1.42600	0.712999	−	0.701165i	$-0.247336\pi$
$967$	42880.4	1.42600	0.712999	+	0.701165i	$0.247336\pi$
$968$	0	0
$969$	3503.31	0.116143
$970$	0	0
$971$	51805.1	1.71216	0.856079	−	0.516845i	$-0.172894\pi$
$971$	51805.1	1.71216	0.856079	+	0.516845i	$0.172894\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	16672.1	0.547624
$976$	0	0
$977$	2321.57	0.0760223	0.0380111	−	0.999277i	$-0.487898\pi$
$977$	2321.57	0.0760223	0.0380111	+	0.999277i	$0.487898\pi$
$978$	0	0
$979$	14763.3	0.481958
$980$	0	0
$981$	−18124.9	−0.589891
$982$	0	0
$983$	−15591.4	−0.505887	−0.252943	−	0.967481i	$-0.581399\pi$
$983$	−15591.4	−0.505887	−0.252943	+	0.967481i	$0.581399\pi$
$984$	0	0
$985$	36505.8	1.18088
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3238.13	−0.104112
$990$	0	0
$991$	−11692.2	−0.374787	−0.187394	−	0.982285i	$-0.560004\pi$
$991$	−11692.2	−0.374787	−0.187394	+	0.982285i	$0.560004\pi$
$992$	0	0
$993$	−8907.20	−0.284654
$994$	0	0
$995$	36169.8	1.15242
$996$	0	0
$997$	−30785.4	−0.977917	−0.488959	−	0.872307i	$-0.662623\pi$
$997$	−30785.4	−0.977917	−0.488959	+	0.872307i	$0.662623\pi$
$998$	0	0
$999$	−40326.4	−1.27715

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2156.4.a.m.1.7		10
7.2	even	3		308.4.i.a.221.4	yes	20
7.4	even	3		308.4.i.a.177.4	✓	20
7.6	odd	2		2156.4.a.j.1.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.4.i.a.177.4	✓	20	7.4	even	3
308.4.i.a.221.4	yes	20	7.2	even	3
2156.4.a.j.1.4		10	7.6	odd	2
2156.4.a.m.1.7		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2156.a (trivial)

\(f(q)\)	\(=\)	\(q+3.28467 q^{3} +20.1685 q^{5} -16.2110 q^{9} +11.0000 q^{11} +18.0138 q^{13} +66.2468 q^{15} +6.74170 q^{17} +158.204 q^{19} +85.0708 q^{23} +281.769 q^{25} -141.934 q^{27} -49.3654 q^{29} -149.492 q^{31} +36.1313 q^{33} +284.122 q^{37} +59.1693 q^{39} -223.913 q^{41} -38.0640 q^{43} -326.951 q^{45} +411.068 q^{47} +22.1442 q^{51} -357.458 q^{53} +221.854 q^{55} +519.648 q^{57} -748.006 q^{59} -125.145 q^{61} +363.312 q^{65} +637.971 q^{67} +279.429 q^{69} -176.322 q^{71} +1053.44 q^{73} +925.516 q^{75} +419.986 q^{79} -28.5081 q^{81} -1472.87 q^{83} +135.970 q^{85} -162.149 q^{87} +1342.12 q^{89} -491.033 q^{93} +3190.75 q^{95} +1289.25 q^{97} -178.321 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 6 q^{3} + 10 q^{5} + 104 q^{9} + 110 q^{11} + 8 q^{13} + 54 q^{15} + 166 q^{17} + 342 q^{19} - 54 q^{23} + 198 q^{25} + 306 q^{27} - 80 q^{29} + 492 q^{31} + 66 q^{33} + 258 q^{37} - 418 q^{39} + 634 q^{41}+ \cdots + 1144 q^{99}+O(q^{100}) \) 10 * q + 6 * q^3 + 10 * q^5 + 104 * q^9 + 110 * q^11 + 8 * q^13 + 54 * q^15 + 166 * q^17 + 342 * q^19 - 54 * q^23 + 198 * q^25 + 306 * q^27 - 80 * q^29 + 492 * q^31 + 66 * q^33 + 258 * q^37 - 418 * q^39 + 634 * q^41 - 274 * q^43 + 138 * q^45 + 312 * q^47 - 196 * q^51 + 656 * q^53 + 110 * q^55 - 738 * q^57 + 934 * q^59 - 228 * q^61 + 524 * q^65 + 478 * q^67 + 1572 * q^69 + 138 * q^71 - 364 * q^73 + 234 * q^75 - 616 * q^79 + 4082 * q^81 - 132 * q^83 - 496 * q^85 - 2120 * q^87 + 1906 * q^89 - 2984 * q^93 + 2322 * q^95 + 1350 * q^97 + 1144 * q^99

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