LMFDB - Embedded newform 1472.4.a.bg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1472,4,Mod(1,1472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1472, 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("1472.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1472.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:	$86.8508115285$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 137x^{6} + 344x^{5} + 6175x^{4} - 7924x^{3} - 89643x^{2} + 45072x + 51084 $ x^8 - 4x^7 - 137x^6 + 344x^5 + 6175x^4 - 7924x^3 - 89643x^2 + 45072*x + 51084
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 736)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-5.87424$ of defining polynomial
Character	$\chi$	$=$	1472.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.87424 q^{3} -13.0818 q^{5} -35.1038 q^{7} +35.0036 q^{9} -66.4666 q^{11} +40.5066 q^{13} +103.009 q^{15} +39.8530 q^{17} -93.1529 q^{19} +276.415 q^{21} -23.0000 q^{23} +46.1334 q^{25} -63.0226 q^{27} -234.363 q^{29} +226.367 q^{31} +523.374 q^{33} +459.220 q^{35} +275.095 q^{37} -318.959 q^{39} +59.0584 q^{41} -184.026 q^{43} -457.911 q^{45} +506.104 q^{47} +889.275 q^{49} -313.812 q^{51} +8.25645 q^{53} +869.503 q^{55} +733.509 q^{57} +483.221 q^{59} -411.848 q^{61} -1228.76 q^{63} -529.899 q^{65} -324.945 q^{67} +181.108 q^{69} +674.173 q^{71} -752.991 q^{73} -363.265 q^{75} +2333.23 q^{77} -1125.07 q^{79} -448.843 q^{81} +136.505 q^{83} -521.349 q^{85} +1845.43 q^{87} -944.763 q^{89} -1421.94 q^{91} -1782.47 q^{93} +1218.61 q^{95} +271.386 q^{97} -2326.57 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{3} + 12 q^{5} - 14 q^{7} + 90 q^{9} - 88 q^{11} + 30 q^{13} + 30 q^{15} + 58 q^{17} - 190 q^{19} + 66 q^{21} - 184 q^{23} + 28 q^{25} - 432 q^{27} - 190 q^{29} - 60 q^{31} + 346 q^{33} - 192 q^{35}+ \cdots - 5986 q^{99}+O(q^{100}) $ 8 * q - 12 * q^3 + 12 * q^5 - 14 * q^7 + 90 * q^9 - 88 * q^11 + 30 * q^13 + 30 * q^15 + 58 * q^17 - 190 * q^19 + 66 * q^21 - 184 * q^23 + 28 * q^25 - 432 * q^27 - 190 * q^29 - 60 * q^31 + 346 * q^33 - 192 * q^35 + 156 * q^37 + 156 * q^39 + 282 * q^41 - 810 * q^43 - 358 * q^45 + 564 * q^47 + 752 * q^49 - 1614 * q^51 + 230 * q^53 + 1292 * q^55 + 796 * q^57 - 1916 * q^59 - 22 * q^61 + 1036 * q^63 + 222 * q^65 - 2292 * q^67 + 276 * q^69 + 2376 * q^71 - 630 * q^73 - 3952 * q^75 + 948 * q^77 + 892 * q^79 - 1044 * q^81 - 2800 * q^83 - 288 * q^85 + 2756 * q^87 - 308 * q^89 - 4610 * q^91 + 982 * q^93 + 1300 * q^95 + 854 * q^97 - 5986 * 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
$3$	−7.87424	−1.51540	−0.757699	−	0.652604i	$-0.773676\pi$
$3$	−7.87424	−1.51540	−0.757699	+	0.652604i	$0.773676\pi$
$4$	0	0
$5$	−13.0818	−1.17007	−0.585036	−	0.811008i	$-0.698920\pi$
$5$	−13.0818	−1.17007	−0.585036	+	0.811008i	$0.698920\pi$
$6$	0	0
$7$	−35.1038	−1.89543	−0.947713	−	0.319125i	$-0.896611\pi$
$7$	−35.1038	−1.89543	−0.947713	+	0.319125i	$0.896611\pi$
$8$	0	0
$9$	35.0036	1.29643
$10$	0	0
$11$	−66.4666	−1.82186	−0.910929	−	0.412563i	$-0.864634\pi$
$11$	−66.4666	−1.82186	−0.910929	+	0.412563i	$0.864634\pi$
$12$	0	0
$13$	40.5066	0.864194	0.432097	−	0.901827i	$-0.357774\pi$
$13$	40.5066	0.864194	0.432097	+	0.901827i	$0.357774\pi$
$14$	0	0
$15$	103.009	1.77312
$16$	0	0
$17$	39.8530	0.568575	0.284288	−	0.958739i	$-0.408243\pi$
$17$	39.8530	0.568575	0.284288	+	0.958739i	$0.408243\pi$
$18$	0	0
$19$	−93.1529	−1.12478	−0.562388	−	0.826873i	$-0.690117\pi$
$19$	−93.1529	−1.12478	−0.562388	+	0.826873i	$0.690117\pi$
$20$	0	0
$21$	276.415	2.87232
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	46.1334	0.369067
$26$	0	0
$27$	−63.0226	−0.449211
$28$	0	0
$29$	−234.363	−1.50069	−0.750345	−	0.661046i	$-0.770113\pi$
$29$	−234.363	−1.50069	−0.750345	+	0.661046i	$0.770113\pi$
$30$	0	0
$31$	226.367	1.31151	0.655755	−	0.754974i	$-0.272351\pi$
$31$	226.367	1.31151	0.655755	+	0.754974i	$0.272351\pi$
$32$	0	0
$33$	523.374	2.76084
$34$	0	0
$35$	459.220	2.21778
$36$	0	0
$37$	275.095	1.22231	0.611154	−	0.791512i	$-0.290706\pi$
$37$	275.095	1.22231	0.611154	+	0.791512i	$0.290706\pi$
$38$	0	0
$39$	−318.959	−1.30960
$40$	0	0
$41$	59.0584	0.224960	0.112480	−	0.993654i	$-0.464120\pi$
$41$	59.0584	0.224960	0.112480	+	0.993654i	$0.464120\pi$
$42$	0	0
$43$	−184.026	−0.652644	−0.326322	−	0.945259i	$-0.605809\pi$
$43$	−184.026	−0.652644	−0.326322	+	0.945259i	$0.605809\pi$
$44$	0	0
$45$	−457.911	−1.51692
$46$	0	0
$47$	506.104	1.57070	0.785349	−	0.619053i	$-0.212483\pi$
$47$	506.104	1.57070	0.785349	+	0.619053i	$0.212483\pi$
$48$	0	0
$49$	889.275	2.59264
$50$	0	0
$51$	−313.812	−0.861618
$52$	0	0
$53$	8.25645	0.0213983	0.0106992	−	0.999943i	$-0.496594\pi$
$53$	8.25645	0.0213983	0.0106992	+	0.999943i	$0.496594\pi$
$54$	0	0
$55$	869.503	2.13170
$56$	0	0
$57$	733.509	1.70448
$58$	0	0
$59$	483.221	1.06627	0.533136	−	0.846029i	$-0.321013\pi$
$59$	483.221	1.06627	0.533136	+	0.846029i	$0.321013\pi$
$60$	0	0
$61$	−411.848	−0.864454	−0.432227	−	0.901765i	$-0.642272\pi$
$61$	−411.848	−0.864454	−0.432227	+	0.901765i	$0.642272\pi$
$62$	0	0
$63$	−1228.76	−2.45729
$64$	0	0
$65$	−529.899	−1.01117
$66$	0	0
$67$	−324.945	−0.592513	−0.296256	−	0.955108i	$-0.595738\pi$
$67$	−324.945	−0.592513	−0.296256	+	0.955108i	$0.595738\pi$
$68$	0	0
$69$	181.108	0.315982
$70$	0	0
$71$	674.173	1.12690	0.563448	−	0.826151i	$-0.309474\pi$
$71$	674.173	1.12690	0.563448	+	0.826151i	$0.309474\pi$
$72$	0	0
$73$	−752.991	−1.20727	−0.603637	−	0.797260i	$-0.706282\pi$
$73$	−752.991	−1.20727	−0.603637	+	0.797260i	$0.706282\pi$
$74$	0	0
$75$	−363.265	−0.559284
$76$	0	0
$77$	2333.23	3.45320
$78$	0	0
$79$	−1125.07	−1.60228	−0.801140	−	0.598478i	$-0.795773\pi$
$79$	−1125.07	−1.60228	−0.801140	+	0.598478i	$0.795773\pi$
$80$	0	0
$81$	−448.843	−0.615697
$82$	0	0
$83$	136.505	0.180522	0.0902610	−	0.995918i	$-0.471230\pi$
$83$	136.505	0.180522	0.0902610	+	0.995918i	$0.471230\pi$
$84$	0	0
$85$	−521.349	−0.665273
$86$	0	0
$87$	1845.43	2.27414
$88$	0	0
$89$	−944.763	−1.12522	−0.562611	−	0.826722i	$-0.690203\pi$
$89$	−944.763	−1.12522	−0.562611	+	0.826722i	$0.690203\pi$
$90$	0	0
$91$	−1421.94	−1.63801
$92$	0	0
$93$	−1782.47	−1.98746
$94$	0	0
$95$	1218.61	1.31607
$96$	0	0
$97$	271.386	0.284073	0.142036	−	0.989861i	$-0.454635\pi$
$97$	271.386	0.284073	0.142036	+	0.989861i	$0.454635\pi$
$98$	0	0
$99$	−2326.57	−2.36191
$100$	0	0
$101$	1776.73	1.75041	0.875204	−	0.483754i	$-0.160727\pi$
$101$	1776.73	1.75041	0.875204	+	0.483754i	$0.160727\pi$
$102$	0	0
$103$	240.423	0.229996	0.114998	−	0.993366i	$-0.463314\pi$
$103$	240.423	0.229996	0.114998	+	0.993366i	$0.463314\pi$
$104$	0	0
$105$	−3616.01	−3.36082
$106$	0	0
$107$	308.001	0.278276	0.139138	−	0.990273i	$-0.455567\pi$
$107$	308.001	0.278276	0.139138	+	0.990273i	$0.455567\pi$
$108$	0	0
$109$	353.112	0.310294	0.155147	−	0.987891i	$-0.450415\pi$
$109$	353.112	0.310294	0.155147	+	0.987891i	$0.450415\pi$
$110$	0	0
$111$	−2166.16	−1.85228
$112$	0	0
$113$	−449.483	−0.374193	−0.187096	−	0.982342i	$-0.559908\pi$
$113$	−449.483	−0.374193	−0.187096	+	0.982342i	$0.559908\pi$
$114$	0	0
$115$	300.881	0.243977
$116$	0	0
$117$	1417.88	1.12037
$118$	0	0
$119$	−1398.99	−1.07769
$120$	0	0
$121$	3086.81	2.31917
$122$	0	0
$123$	−465.040	−0.340905
$124$	0	0
$125$	1031.72	0.738237
$126$	0	0
$127$	1259.21	0.879816	0.439908	−	0.898043i	$-0.355011\pi$
$127$	1259.21	0.879816	0.439908	+	0.898043i	$0.355011\pi$
$128$	0	0
$129$	1449.07	0.989016
$130$	0	0
$131$	−1674.98	−1.11713	−0.558564	−	0.829461i	$-0.688648\pi$
$131$	−1674.98	−1.11713	−0.558564	+	0.829461i	$0.688648\pi$
$132$	0	0
$133$	3270.02	2.13193
$134$	0	0
$135$	824.449	0.525610
$136$	0	0
$137$	−133.744	−0.0834051	−0.0417025	−	0.999130i	$-0.513278\pi$
$137$	−133.744	−0.0834051	−0.0417025	+	0.999130i	$0.513278\pi$
$138$	0	0
$139$	870.562	0.531224	0.265612	−	0.964080i	$-0.414426\pi$
$139$	870.562	0.531224	0.265612	+	0.964080i	$0.414426\pi$
$140$	0	0
$141$	−3985.18	−2.38023
$142$	0	0
$143$	−2692.34	−1.57444
$144$	0	0
$145$	3065.88	1.75592
$146$	0	0
$147$	−7002.36	−3.92888
$148$	0	0
$149$	1796.43	0.987714	0.493857	−	0.869543i	$-0.335587\pi$
$149$	1796.43	0.987714	0.493857	+	0.869543i	$0.335587\pi$
$150$	0	0
$151$	262.176	0.141295	0.0706477	−	0.997501i	$-0.477493\pi$
$151$	262.176	0.141295	0.0706477	+	0.997501i	$0.477493\pi$
$152$	0	0
$153$	1395.00	0.737119
$154$	0	0
$155$	−2961.29	−1.53456
$156$	0	0
$157$	−953.475	−0.484685	−0.242343	−	0.970191i	$-0.577916\pi$
$157$	−953.475	−0.484685	−0.242343	+	0.970191i	$0.577916\pi$
$158$	0	0
$159$	−65.0132	−0.0324270
$160$	0	0
$161$	807.387	0.395224
$162$	0	0
$163$	−2314.74	−1.11230	−0.556148	−	0.831083i	$-0.687721\pi$
$163$	−2314.74	−1.11230	−0.556148	+	0.831083i	$0.687721\pi$
$164$	0	0
$165$	−6846.67	−3.23038
$166$	0	0
$167$	−3085.96	−1.42993	−0.714966	−	0.699159i	$-0.753558\pi$
$167$	−3085.96	−1.42993	−0.714966	+	0.699159i	$0.753558\pi$
$168$	0	0
$169$	−556.213	−0.253169
$170$	0	0
$171$	−3260.69	−1.45820
$172$	0	0
$173$	2372.47	1.04263	0.521316	−	0.853364i	$-0.325441\pi$
$173$	2372.47	1.04263	0.521316	+	0.853364i	$0.325441\pi$
$174$	0	0
$175$	−1619.46	−0.699539
$176$	0	0
$177$	−3805.00	−1.61583
$178$	0	0
$179$	831.748	0.347306	0.173653	−	0.984807i	$-0.444443\pi$
$179$	831.748	0.347306	0.173653	+	0.984807i	$0.444443\pi$
$180$	0	0
$181$	−1530.78	−0.628628	−0.314314	−	0.949319i	$-0.601774\pi$
$181$	−1530.78	−0.628628	−0.314314	+	0.949319i	$0.601774\pi$
$182$	0	0
$183$	3242.99	1.30999
$184$	0	0
$185$	−3598.74	−1.43019
$186$	0	0
$187$	−2648.90	−1.03586
$188$	0	0
$189$	2212.33	0.851447
$190$	0	0
$191$	1277.79	0.484070	0.242035	−	0.970268i	$-0.422185\pi$
$191$	1277.79	0.484070	0.242035	+	0.970268i	$0.422185\pi$
$192$	0	0
$193$	−1630.36	−0.608061	−0.304030	−	0.952662i	$-0.598332\pi$
$193$	−1630.36	−0.608061	−0.304030	+	0.952662i	$0.598332\pi$
$194$	0	0
$195$	4172.55	1.53232
$196$	0	0
$197$	−2256.96	−0.816254	−0.408127	−	0.912925i	$-0.633818\pi$
$197$	−2256.96	−0.816254	−0.408127	+	0.912925i	$0.633818\pi$
$198$	0	0
$199$	1839.55	0.655288	0.327644	−	0.944801i	$-0.393745\pi$
$199$	1839.55	0.655288	0.327644	+	0.944801i	$0.393745\pi$
$200$	0	0
$201$	2558.69	0.897893
$202$	0	0
$203$	8227.01	2.84445
$204$	0	0
$205$	−772.591	−0.263220
$206$	0	0
$207$	−805.084	−0.270325
$208$	0	0
$209$	6191.56	2.04918
$210$	0	0
$211$	4418.46	1.44161	0.720804	−	0.693139i	$-0.243773\pi$
$211$	4418.46	1.44161	0.720804	+	0.693139i	$0.243773\pi$
$212$	0	0
$213$	−5308.60	−1.70770
$214$	0	0
$215$	2407.39	0.763640
$216$	0	0
$217$	−7946.35	−2.48587
$218$	0	0
$219$	5929.23	1.82950
$220$	0	0
$221$	1614.31	0.491359
$222$	0	0
$223$	−3252.68	−0.976752	−0.488376	−	0.872633i	$-0.662411\pi$
$223$	−3252.68	−0.976752	−0.488376	+	0.872633i	$0.662411\pi$
$224$	0	0
$225$	1614.84	0.478470
$226$	0	0
$227$	−704.737	−0.206057	−0.103029	−	0.994678i	$-0.532853\pi$
$227$	−704.737	−0.206057	−0.103029	+	0.994678i	$0.532853\pi$
$228$	0	0
$229$	−1803.87	−0.520537	−0.260269	−	0.965536i	$-0.583811\pi$
$229$	−1803.87	−0.520537	−0.260269	+	0.965536i	$0.583811\pi$
$230$	0	0
$231$	−18372.4	−5.23297
$232$	0	0
$233$	−5406.49	−1.52013	−0.760067	−	0.649845i	$-0.774834\pi$
$233$	−5406.49	−1.52013	−0.760067	+	0.649845i	$0.774834\pi$
$234$	0	0
$235$	−6620.75	−1.83783
$236$	0	0
$237$	8859.05	2.42809
$238$	0	0
$239$	5261.38	1.42398	0.711988	−	0.702192i	$-0.247795\pi$
$239$	5261.38	1.42398	0.711988	+	0.702192i	$0.247795\pi$
$240$	0	0
$241$	1503.05	0.401743	0.200872	−	0.979618i	$-0.435623\pi$
$241$	1503.05	0.401743	0.200872	+	0.979618i	$0.435623\pi$
$242$	0	0
$243$	5235.91	1.38224
$244$	0	0
$245$	−11633.3	−3.03357
$246$	0	0
$247$	−3773.31	−0.972024
$248$	0	0
$249$	−1074.87	−0.273563
$250$	0	0
$251$	−7296.14	−1.83477	−0.917387	−	0.397996i	$-0.869706\pi$
$251$	−7296.14	−1.83477	−0.917387	+	0.397996i	$0.869706\pi$
$252$	0	0
$253$	1528.73	0.379884
$254$	0	0
$255$	4105.23	1.00815
$256$	0	0
$257$	−1061.47	−0.257638	−0.128819	−	0.991668i	$-0.541119\pi$
$257$	−1061.47	−0.257638	−0.128819	+	0.991668i	$0.541119\pi$
$258$	0	0
$259$	−9656.88	−2.31679
$260$	0	0
$261$	−8203.54	−1.94554
$262$	0	0
$263$	748.926	0.175592	0.0877961	−	0.996138i	$-0.472018\pi$
$263$	748.926	0.175592	0.0877961	+	0.996138i	$0.472018\pi$
$264$	0	0
$265$	−108.009	−0.0250375
$266$	0	0
$267$	7439.29	1.70516
$268$	0	0
$269$	−324.497	−0.0735500	−0.0367750	−	0.999324i	$-0.511708\pi$
$269$	−324.497	−0.0735500	−0.0367750	+	0.999324i	$0.511708\pi$
$270$	0	0
$271$	−2872.18	−0.643809	−0.321905	−	0.946772i	$-0.604323\pi$
$271$	−2872.18	−0.643809	−0.321905	+	0.946772i	$0.604323\pi$
$272$	0	0
$273$	11196.7	2.48224
$274$	0	0
$275$	−3066.33	−0.672388
$276$	0	0
$277$	5076.51	1.10115	0.550574	−	0.834787i	$-0.314409\pi$
$277$	5076.51	1.10115	0.550574	+	0.834787i	$0.314409\pi$
$278$	0	0
$279$	7923.68	1.70028
$280$	0	0
$281$	−4594.57	−0.975406	−0.487703	−	0.873010i	$-0.662165\pi$
$281$	−4594.57	−0.975406	−0.487703	+	0.873010i	$0.662165\pi$
$282$	0	0
$283$	2410.70	0.506365	0.253183	−	0.967418i	$-0.418523\pi$
$283$	2410.70	0.506365	0.253183	+	0.967418i	$0.418523\pi$
$284$	0	0
$285$	−9595.61	−1.99437
$286$	0	0
$287$	−2073.17	−0.426396
$288$	0	0
$289$	−3324.74	−0.676722
$290$	0	0
$291$	−2136.96	−0.430483
$292$	0	0
$293$	6951.63	1.38607	0.693035	−	0.720904i	$-0.256273\pi$
$293$	6951.63	1.38607	0.693035	+	0.720904i	$0.256273\pi$
$294$	0	0
$295$	−6321.40	−1.24761
$296$	0	0
$297$	4188.90	0.818400
$298$	0	0
$299$	−931.652	−0.180197
$300$	0	0
$301$	6460.01	1.23704
$302$	0	0
$303$	−13990.4	−2.65256
$304$	0	0
$305$	5387.71	1.01147
$306$	0	0
$307$	−1003.23	−0.186506	−0.0932529	−	0.995642i	$-0.529727\pi$
$307$	−1003.23	−0.186506	−0.0932529	+	0.995642i	$0.529727\pi$
$308$	0	0
$309$	−1893.15	−0.348535
$310$	0	0
$311$	6903.16	1.25866	0.629328	−	0.777140i	$-0.283330\pi$
$311$	6903.16	1.25866	0.629328	+	0.777140i	$0.283330\pi$
$312$	0	0
$313$	6660.12	1.20272	0.601362	−	0.798977i	$-0.294625\pi$
$313$	6660.12	1.20272	0.601362	+	0.798977i	$0.294625\pi$
$314$	0	0
$315$	16074.4	2.87520
$316$	0	0
$317$	−1824.97	−0.323345	−0.161672	−	0.986844i	$-0.551689\pi$
$317$	−1824.97	−0.323345	−0.161672	+	0.986844i	$0.551689\pi$
$318$	0	0
$319$	15577.3	2.73405
$320$	0	0
$321$	−2425.27	−0.421699
$322$	0	0
$323$	−3712.43	−0.639520
$324$	0	0
$325$	1868.71	0.318945
$326$	0	0
$327$	−2780.49	−0.470219
$328$	0	0
$329$	−17766.2	−2.97714
$330$	0	0
$331$	1894.02	0.314515	0.157258	−	0.987558i	$-0.449735\pi$
$331$	1894.02	0.314515	0.157258	+	0.987558i	$0.449735\pi$
$332$	0	0
$333$	9629.33	1.58464
$334$	0	0
$335$	4250.86	0.693282
$336$	0	0
$337$	−5973.37	−0.965549	−0.482774	−	0.875745i	$-0.660371\pi$
$337$	−5973.37	−0.965549	−0.482774	+	0.875745i	$0.660371\pi$
$338$	0	0
$339$	3539.33	0.567051
$340$	0	0
$341$	−15045.9	−2.38938
$342$	0	0
$343$	−19176.3	−3.01873
$344$	0	0
$345$	−2369.21	−0.369722
$346$	0	0
$347$	7361.21	1.13882	0.569410	−	0.822054i	$-0.307172\pi$
$347$	7361.21	1.13882	0.569410	+	0.822054i	$0.307172\pi$
$348$	0	0
$349$	9820.75	1.50628	0.753141	−	0.657859i	$-0.228538\pi$
$349$	9820.75	1.50628	0.753141	+	0.657859i	$0.228538\pi$
$350$	0	0
$351$	−2552.83	−0.388206
$352$	0	0
$353$	2395.23	0.361148	0.180574	−	0.983561i	$-0.442204\pi$
$353$	2395.23	0.361148	0.180574	+	0.983561i	$0.442204\pi$
$354$	0	0
$355$	−8819.40	−1.31855
$356$	0	0
$357$	11016.0	1.63313
$358$	0	0
$359$	−1119.80	−0.164626	−0.0823131	−	0.996607i	$-0.526231\pi$
$359$	−1119.80	−0.164626	−0.0823131	+	0.996607i	$0.526231\pi$
$360$	0	0
$361$	1818.47	0.265122
$362$	0	0
$363$	−24306.3	−3.51446
$364$	0	0
$365$	9850.48	1.41260
$366$	0	0
$367$	8452.70	1.20225	0.601127	−	0.799153i	$-0.294718\pi$
$367$	8452.70	1.20225	0.601127	+	0.799153i	$0.294718\pi$
$368$	0	0
$369$	2067.26	0.291646
$370$	0	0
$371$	−289.832	−0.0405589
$372$	0	0
$373$	−2951.38	−0.409696	−0.204848	−	0.978794i	$-0.565670\pi$
$373$	−2951.38	−0.409696	−0.204848	+	0.978794i	$0.565670\pi$
$374$	0	0
$375$	−8123.99	−1.11872
$376$	0	0
$377$	−9493.24	−1.29689
$378$	0	0
$379$	5185.22	0.702762	0.351381	−	0.936233i	$-0.385712\pi$
$379$	5185.22	0.702762	0.351381	+	0.936233i	$0.385712\pi$
$380$	0	0
$381$	−9915.30	−1.33327
$382$	0	0
$383$	−7892.20	−1.05293	−0.526465	−	0.850197i	$-0.676483\pi$
$383$	−7892.20	−1.05293	−0.526465	+	0.850197i	$0.676483\pi$
$384$	0	0
$385$	−30522.8	−4.04049
$386$	0	0
$387$	−6441.58	−0.846109
$388$	0	0
$389$	−1228.55	−0.160128	−0.0800641	−	0.996790i	$-0.525513\pi$
$389$	−1228.55	−0.160128	−0.0800641	+	0.996790i	$0.525513\pi$
$390$	0	0
$391$	−916.619	−0.118556
$392$	0	0
$393$	13189.2	1.69289
$394$	0	0
$395$	14717.9	1.87478
$396$	0	0
$397$	1175.83	0.148648	0.0743240	−	0.997234i	$-0.476320\pi$
$397$	1175.83	0.148648	0.0743240	+	0.997234i	$0.476320\pi$
$398$	0	0
$399$	−25748.9	−3.23072
$400$	0	0
$401$	10907.6	1.35836	0.679178	−	0.733974i	$-0.262337\pi$
$401$	10907.6	1.35836	0.679178	+	0.733974i	$0.262337\pi$
$402$	0	0
$403$	9169.38	1.13340
$404$	0	0
$405$	5871.68	0.720410
$406$	0	0
$407$	−18284.6	−2.22687
$408$	0	0
$409$	−5589.01	−0.675694	−0.337847	−	0.941201i	$-0.609699\pi$
$409$	−5589.01	−0.675694	−0.337847	+	0.941201i	$0.609699\pi$
$410$	0	0
$411$	1053.13	0.126392
$412$	0	0
$413$	−16962.9	−2.02104
$414$	0	0
$415$	−1785.72	−0.211224
$416$	0	0
$417$	−6855.01	−0.805016
$418$	0	0
$419$	8169.39	0.952508	0.476254	−	0.879308i	$-0.341994\pi$
$419$	8169.39	0.952508	0.476254	+	0.879308i	$0.341994\pi$
$420$	0	0
$421$	7188.37	0.832161	0.416081	−	0.909328i	$-0.363403\pi$
$421$	7188.37	0.832161	0.416081	+	0.909328i	$0.363403\pi$
$422$	0	0
$423$	17715.5	2.03630
$424$	0	0
$425$	1838.55	0.209842
$426$	0	0
$427$	14457.4	1.63851
$428$	0	0
$429$	21200.1	2.38590
$430$	0	0
$431$	12185.9	1.36189	0.680947	−	0.732333i	$-0.261568\pi$
$431$	12185.9	1.36189	0.680947	+	0.732333i	$0.261568\pi$
$432$	0	0
$433$	6461.87	0.717177	0.358589	−	0.933496i	$-0.383258\pi$
$433$	6461.87	0.717177	0.358589	+	0.933496i	$0.383258\pi$
$434$	0	0
$435$	−24141.5	−2.66091
$436$	0	0
$437$	2142.52	0.234532
$438$	0	0
$439$	−12095.8	−1.31503	−0.657517	−	0.753439i	$-0.728393\pi$
$439$	−12095.8	−1.31503	−0.657517	+	0.753439i	$0.728393\pi$
$440$	0	0
$441$	31127.9	3.36118
$442$	0	0
$443$	8255.21	0.885366	0.442683	−	0.896678i	$-0.354027\pi$
$443$	8255.21	0.885366	0.442683	+	0.896678i	$0.354027\pi$
$444$	0	0
$445$	12359.2	1.31659
$446$	0	0
$447$	−14145.5	−1.49678
$448$	0	0
$449$	10637.0	1.11802	0.559009	−	0.829162i	$-0.311182\pi$
$449$	10637.0	1.11802	0.559009	+	0.829162i	$0.311182\pi$
$450$	0	0
$451$	−3925.41	−0.409846
$452$	0	0
$453$	−2064.44	−0.214119
$454$	0	0
$455$	18601.5	1.91659
$456$	0	0
$457$	16881.0	1.72792	0.863962	−	0.503557i	$-0.167976\pi$
$457$	16881.0	1.72792	0.863962	+	0.503557i	$0.167976\pi$
$458$	0	0
$459$	−2511.64	−0.255410
$460$	0	0
$461$	12873.6	1.30061	0.650306	−	0.759672i	$-0.274641\pi$
$461$	12873.6	1.30061	0.650306	+	0.759672i	$0.274641\pi$
$462$	0	0
$463$	−3681.29	−0.369512	−0.184756	−	0.982784i	$-0.559149\pi$
$463$	−3681.29	−0.369512	−0.184756	+	0.982784i	$0.559149\pi$
$464$	0	0
$465$	23317.9	2.32547
$466$	0	0
$467$	10621.5	1.05247	0.526236	−	0.850338i	$-0.323603\pi$
$467$	10621.5	1.05247	0.526236	+	0.850338i	$0.323603\pi$
$468$	0	0
$469$	11406.8	1.12306
$470$	0	0
$471$	7507.89	0.734491
$472$	0	0
$473$	12231.6	1.18903
$474$	0	0
$475$	−4297.46	−0.415118
$476$	0	0
$477$	289.006	0.0277414
$478$	0	0
$479$	8148.47	0.777271	0.388636	−	0.921392i	$-0.372946\pi$
$479$	8148.47	0.777271	0.388636	+	0.921392i	$0.372946\pi$
$480$	0	0
$481$	11143.2	1.05631
$482$	0	0
$483$	−6357.56	−0.598921
$484$	0	0
$485$	−3550.21	−0.332385
$486$	0	0
$487$	−15874.3	−1.47707	−0.738535	−	0.674215i	$-0.764482\pi$
$487$	−15874.3	−1.47707	−0.738535	+	0.674215i	$0.764482\pi$
$488$	0	0
$489$	18226.8	1.68557
$490$	0	0
$491$	−1478.97	−0.135937	−0.0679684	−	0.997687i	$-0.521652\pi$
$491$	−1478.97	−0.135937	−0.0679684	+	0.997687i	$0.521652\pi$
$492$	0	0
$493$	−9340.06	−0.853255
$494$	0	0
$495$	30435.8	2.76361
$496$	0	0
$497$	−23666.0	−2.13595
$498$	0	0
$499$	16863.5	1.51286	0.756429	−	0.654076i	$-0.226942\pi$
$499$	16863.5	1.51286	0.756429	+	0.654076i	$0.226942\pi$
$500$	0	0
$501$	24299.6	2.16692
$502$	0	0
$503$	13888.2	1.23110	0.615551	−	0.788097i	$-0.288933\pi$
$503$	13888.2	1.23110	0.615551	+	0.788097i	$0.288933\pi$
$504$	0	0
$505$	−23242.8	−2.04810
$506$	0	0
$507$	4379.76	0.383652
$508$	0	0
$509$	−7687.68	−0.669450	−0.334725	−	0.942316i	$-0.608644\pi$
$509$	−7687.68	−0.669450	−0.334725	+	0.942316i	$0.608644\pi$
$510$	0	0
$511$	26432.8	2.28830
$512$	0	0
$513$	5870.74	0.505262
$514$	0	0
$515$	−3145.16	−0.269111
$516$	0	0
$517$	−33639.0	−2.86159
$518$	0	0
$519$	−18681.4	−1.58000
$520$	0	0
$521$	−6311.16	−0.530704	−0.265352	−	0.964152i	$-0.585488\pi$
$521$	−6311.16	−0.530704	−0.265352	+	0.964152i	$0.585488\pi$
$522$	0	0
$523$	−805.074	−0.0673105	−0.0336553	−	0.999434i	$-0.510715\pi$
$523$	−805.074	−0.0673105	−0.0336553	+	0.999434i	$0.510715\pi$
$524$	0	0
$525$	12752.0	1.06008
$526$	0	0
$527$	9021.42	0.745691
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	16914.5	1.38235
$532$	0	0
$533$	2392.26	0.194409
$534$	0	0
$535$	−4029.20	−0.325603
$536$	0	0
$537$	−6549.38	−0.526306
$538$	0	0
$539$	−59107.1	−4.72342
$540$	0	0
$541$	−20484.3	−1.62789	−0.813946	−	0.580940i	$-0.802685\pi$
$541$	−20484.3	−1.62789	−0.813946	+	0.580940i	$0.802685\pi$
$542$	0	0
$543$	12053.7	0.952622
$544$	0	0
$545$	−4619.34	−0.363066
$546$	0	0
$547$	5933.69	0.463814	0.231907	−	0.972738i	$-0.425503\pi$
$547$	5933.69	0.463814	0.231907	+	0.972738i	$0.425503\pi$
$548$	0	0
$549$	−14416.2	−1.12071
$550$	0	0
$551$	21831.6	1.68794
$552$	0	0
$553$	39494.1	3.03700
$554$	0	0
$555$	28337.3	2.16730
$556$	0	0
$557$	4701.32	0.357633	0.178816	−	0.983882i	$-0.442773\pi$
$557$	4701.32	0.357633	0.178816	+	0.983882i	$0.442773\pi$
$558$	0	0
$559$	−7454.27	−0.564011
$560$	0	0
$561$	20858.0	1.56974
$562$	0	0
$563$	16968.6	1.27024	0.635118	−	0.772415i	$-0.280952\pi$
$563$	16968.6	1.27024	0.635118	+	0.772415i	$0.280952\pi$
$564$	0	0
$565$	5880.04	0.437832
$566$	0	0
$567$	15756.1	1.16701
$568$	0	0
$569$	−3628.85	−0.267363	−0.133681	−	0.991024i	$-0.542680\pi$
$569$	−3628.85	−0.267363	−0.133681	+	0.991024i	$0.542680\pi$
$570$	0	0
$571$	−18920.7	−1.38670	−0.693352	−	0.720599i	$-0.743867\pi$
$571$	−18920.7	−1.38670	−0.693352	+	0.720599i	$0.743867\pi$
$572$	0	0
$573$	−10061.6	−0.733559
$574$	0	0
$575$	−1061.07	−0.0769558
$576$	0	0
$577$	9551.63	0.689150	0.344575	−	0.938759i	$-0.388023\pi$
$577$	9551.63	0.689150	0.344575	+	0.938759i	$0.388023\pi$
$578$	0	0
$579$	12837.8	0.921454
$580$	0	0
$581$	−4791.82	−0.342166
$582$	0	0
$583$	−548.778	−0.0389847
$584$	0	0
$585$	−18548.4	−1.31091
$586$	0	0
$587$	−8874.67	−0.624015	−0.312007	−	0.950080i	$-0.601001\pi$
$587$	−8874.67	−0.624015	−0.312007	+	0.950080i	$0.601001\pi$
$588$	0	0
$589$	−21086.8	−1.47515
$590$	0	0
$591$	17771.9	1.23695
$592$	0	0
$593$	−17852.0	−1.23625	−0.618124	−	0.786080i	$-0.712107\pi$
$593$	−17852.0	−1.23625	−0.618124	+	0.786080i	$0.712107\pi$
$594$	0	0
$595$	18301.3	1.26098
$596$	0	0
$597$	−14485.1	−0.993022
$598$	0	0
$599$	6049.93	0.412677	0.206339	−	0.978481i	$-0.433845\pi$
$599$	6049.93	0.412677	0.206339	+	0.978481i	$0.433845\pi$
$600$	0	0
$601$	28911.1	1.96224	0.981121	−	0.193397i	$-0.0619506\pi$
$601$	28911.1	1.96224	0.981121	+	0.193397i	$0.0619506\pi$
$602$	0	0
$603$	−11374.3	−0.768152
$604$	0	0
$605$	−40381.0	−2.71359
$606$	0	0
$607$	−24725.2	−1.65332	−0.826660	−	0.562702i	$-0.809762\pi$
$607$	−24725.2	−1.65332	−0.826660	+	0.562702i	$0.809762\pi$
$608$	0	0
$609$	−64781.4	−4.31047
$610$	0	0
$611$	20500.6	1.35739
$612$	0	0
$613$	−16600.4	−1.09378	−0.546889	−	0.837205i	$-0.684188\pi$
$613$	−16600.4	−1.09378	−0.546889	+	0.837205i	$0.684188\pi$
$614$	0	0
$615$	6083.56	0.398883
$616$	0	0
$617$	−8358.55	−0.545385	−0.272692	−	0.962101i	$-0.587914\pi$
$617$	−8358.55	−0.545385	−0.272692	+	0.962101i	$0.587914\pi$
$618$	0	0
$619$	−12333.3	−0.800833	−0.400417	−	0.916333i	$-0.631135\pi$
$619$	−12333.3	−0.800833	−0.400417	+	0.916333i	$0.631135\pi$
$620$	0	0
$621$	1449.52	0.0936671
$622$	0	0
$623$	33164.7	2.13277
$624$	0	0
$625$	−19263.4	−1.23286
$626$	0	0
$627$	−48753.8	−3.10533
$628$	0	0
$629$	10963.4	0.694973
$630$	0	0
$631$	2383.28	0.150360	0.0751799	−	0.997170i	$-0.476047\pi$
$631$	2383.28	0.150360	0.0751799	+	0.997170i	$0.476047\pi$
$632$	0	0
$633$	−34792.0	−2.18461
$634$	0	0
$635$	−16472.7	−1.02945
$636$	0	0
$637$	36021.5	2.24054
$638$	0	0
$639$	23598.5	1.46094
$640$	0	0
$641$	18028.2	1.11087	0.555437	−	0.831559i	$-0.312551\pi$
$641$	18028.2	1.11087	0.555437	+	0.831559i	$0.312551\pi$
$642$	0	0
$643$	−7755.80	−0.475675	−0.237837	−	0.971305i	$-0.576439\pi$
$643$	−7755.80	−0.475675	−0.237837	+	0.971305i	$0.576439\pi$
$644$	0	0
$645$	−18956.4	−1.15722
$646$	0	0
$647$	−9422.28	−0.572532	−0.286266	−	0.958150i	$-0.592414\pi$
$647$	−9422.28	−0.572532	−0.286266	+	0.958150i	$0.592414\pi$
$648$	0	0
$649$	−32118.1	−1.94260
$650$	0	0
$651$	62571.4	3.76708
$652$	0	0
$653$	2650.31	0.158828	0.0794139	−	0.996842i	$-0.474695\pi$
$653$	2650.31	0.158828	0.0794139	+	0.996842i	$0.474695\pi$
$654$	0	0
$655$	21911.8	1.30712
$656$	0	0
$657$	−26357.4	−1.56515
$658$	0	0
$659$	−10595.0	−0.626286	−0.313143	−	0.949706i	$-0.601382\pi$
$659$	−10595.0	−0.626286	−0.313143	+	0.949706i	$0.601382\pi$
$660$	0	0
$661$	−769.562	−0.0452836	−0.0226418	−	0.999744i	$-0.507208\pi$
$661$	−769.562	−0.0452836	−0.0226418	+	0.999744i	$0.507208\pi$
$662$	0	0
$663$	−12711.5	−0.744604
$664$	0	0
$665$	−42777.7	−2.49451
$666$	0	0
$667$	5390.34	0.312916
$668$	0	0
$669$	25612.4	1.48017
$670$	0	0
$671$	27374.1	1.57491
$672$	0	0
$673$	−23807.2	−1.36360	−0.681798	−	0.731541i	$-0.738802\pi$
$673$	−23807.2	−1.36360	−0.681798	+	0.731541i	$0.738802\pi$
$674$	0	0
$675$	−2907.45	−0.165789
$676$	0	0
$677$	−32331.1	−1.83543	−0.917715	−	0.397238i	$-0.869969\pi$
$677$	−32331.1	−1.83543	−0.917715	+	0.397238i	$0.869969\pi$
$678$	0	0
$679$	−9526.66	−0.538438
$680$	0	0
$681$	5549.27	0.312259
$682$	0	0
$683$	185.522	0.0103936	0.00519678	−	0.999986i	$-0.498346\pi$
$683$	185.522	0.0103936	0.00519678	+	0.999986i	$0.498346\pi$
$684$	0	0
$685$	1749.61	0.0975899
$686$	0	0
$687$	14204.1	0.788821
$688$	0	0
$689$	334.441	0.0184923
$690$	0	0
$691$	13929.2	0.766849	0.383425	−	0.923572i	$-0.374745\pi$
$691$	13929.2	0.766849	0.383425	+	0.923572i	$0.374745\pi$
$692$	0	0
$693$	81671.5	4.47683
$694$	0	0
$695$	−11388.5	−0.621570
$696$	0	0
$697$	2353.66	0.127907
$698$	0	0
$699$	42572.0	2.30361
$700$	0	0
$701$	12484.8	0.672671	0.336336	−	0.941742i	$-0.390812\pi$
$701$	12484.8	0.672671	0.336336	+	0.941742i	$0.390812\pi$
$702$	0	0
$703$	−25625.9	−1.37482
$704$	0	0
$705$	52133.4	2.78504
$706$	0	0
$707$	−62369.9	−3.31777
$708$	0	0
$709$	36768.3	1.94762	0.973811	−	0.227360i	$-0.0730093\pi$
$709$	36768.3	1.94762	0.973811	+	0.227360i	$0.0730093\pi$
$710$	0	0
$711$	−39381.5	−2.07724
$712$	0	0
$713$	−5206.45	−0.273469
$714$	0	0
$715$	35220.6	1.84220
$716$	0	0
$717$	−41429.3	−2.15789
$718$	0	0
$719$	−19256.6	−0.998818	−0.499409	−	0.866366i	$-0.666449\pi$
$719$	−19256.6	−0.998818	−0.499409	+	0.866366i	$0.666449\pi$
$720$	0	0
$721$	−8439.74	−0.435940
$722$	0	0
$723$	−11835.4	−0.608801
$724$	0	0
$725$	−10811.9	−0.553856
$726$	0	0
$727$	27245.2	1.38991	0.694957	−	0.719051i	$-0.255423\pi$
$727$	27245.2	1.38991	0.694957	+	0.719051i	$0.255423\pi$
$728$	0	0
$729$	−29110.0	−1.47894
$730$	0	0
$731$	−7333.99	−0.371077
$732$	0	0
$733$	−25329.1	−1.27633	−0.638165	−	0.769899i	$-0.720306\pi$
$733$	−25329.1	−1.27633	−0.638165	+	0.769899i	$0.720306\pi$
$734$	0	0
$735$	91603.5	4.59707
$736$	0	0
$737$	21598.0	1.07947
$738$	0	0
$739$	25270.3	1.25789	0.628946	−	0.777449i	$-0.283487\pi$
$739$	25270.3	1.25789	0.628946	+	0.777449i	$0.283487\pi$
$740$	0	0
$741$	29712.0	1.47300
$742$	0	0
$743$	1676.06	0.0827573	0.0413787	−	0.999144i	$-0.486825\pi$
$743$	1676.06	0.0827573	0.0413787	+	0.999144i	$0.486825\pi$
$744$	0	0
$745$	−23500.5	−1.15570
$746$	0	0
$747$	4778.16	0.234034
$748$	0	0
$749$	−10812.0	−0.527452
$750$	0	0
$751$	−16502.4	−0.801838	−0.400919	−	0.916113i	$-0.631309\pi$
$751$	−16502.4	−0.801838	−0.400919	+	0.916113i	$0.631309\pi$
$752$	0	0
$753$	57451.6	2.78041
$754$	0	0
$755$	−3429.74	−0.165326
$756$	0	0
$757$	8353.15	0.401057	0.200529	−	0.979688i	$-0.435734\pi$
$757$	8353.15	0.401057	0.200529	+	0.979688i	$0.435734\pi$
$758$	0	0
$759$	−12037.6	−0.575675
$760$	0	0
$761$	−36944.5	−1.75984	−0.879919	−	0.475123i	$-0.842404\pi$
$761$	−36944.5	−1.75984	−0.879919	+	0.475123i	$0.842404\pi$
$762$	0	0
$763$	−12395.6	−0.588139
$764$	0	0
$765$	−18249.1	−0.862481
$766$	0	0
$767$	19573.7	0.921466
$768$	0	0
$769$	8333.97	0.390807	0.195404	−	0.980723i	$-0.437398\pi$
$769$	8333.97	0.390807	0.195404	+	0.980723i	$0.437398\pi$
$770$	0	0
$771$	8358.31	0.390424
$772$	0	0
$773$	7701.50	0.358349	0.179174	−	0.983817i	$-0.442657\pi$
$773$	7701.50	0.358349	0.179174	+	0.983817i	$0.442657\pi$
$774$	0	0
$775$	10443.1	0.484035
$776$	0	0
$777$	76040.6	3.51086
$778$	0	0
$779$	−5501.47	−0.253030
$780$	0	0
$781$	−44810.0	−2.05305
$782$	0	0
$783$	14770.1	0.674128
$784$	0	0
$785$	12473.2	0.567116
$786$	0	0
$787$	−40851.1	−1.85030	−0.925148	−	0.379606i	$-0.876060\pi$
$787$	−40851.1	−1.85030	−0.925148	+	0.379606i	$0.876060\pi$
$788$	0	0
$789$	−5897.22	−0.266092
$790$	0	0
$791$	15778.5	0.709254
$792$	0	0
$793$	−16682.6	−0.747056
$794$	0	0
$795$	850.490	0.0379419
$796$	0	0
$797$	9112.62	0.405001	0.202500	−	0.979282i	$-0.435093\pi$
$797$	9112.62	0.405001	0.202500	+	0.979282i	$0.435093\pi$
$798$	0	0
$799$	20169.8	0.893060
$800$	0	0
$801$	−33070.1	−1.45877
$802$	0	0
$803$	50048.8	2.19948
$804$	0	0
$805$	−10562.1	−0.462440
$806$	0	0
$807$	2555.17	0.111458
$808$	0	0
$809$	26676.0	1.15931	0.579653	−	0.814863i	$-0.303188\pi$
$809$	26676.0	1.15931	0.579653	+	0.814863i	$0.303188\pi$
$810$	0	0
$811$	3309.05	0.143275	0.0716377	−	0.997431i	$-0.477177\pi$
$811$	3309.05	0.143275	0.0716377	+	0.997431i	$0.477177\pi$
$812$	0	0
$813$	22616.2	0.975627
$814$	0	0
$815$	30280.9	1.30147
$816$	0	0
$817$	17142.6	0.734079
$818$	0	0
$819$	−49772.9	−2.12357
$820$	0	0
$821$	−10894.5	−0.463120	−0.231560	−	0.972821i	$-0.574383\pi$
$821$	−10894.5	−0.463120	−0.231560	+	0.972821i	$0.574383\pi$
$822$	0	0
$823$	3342.49	0.141570	0.0707849	−	0.997492i	$-0.477450\pi$
$823$	3342.49	0.141570	0.0707849	+	0.997492i	$0.477450\pi$
$824$	0	0
$825$	24145.0	1.01894
$826$	0	0
$827$	38646.4	1.62499	0.812496	−	0.582967i	$-0.198108\pi$
$827$	38646.4	1.62499	0.812496	+	0.582967i	$0.198108\pi$
$828$	0	0
$829$	−1916.48	−0.0802919	−0.0401460	−	0.999194i	$-0.512782\pi$
$829$	−1916.48	−0.0802919	−0.0401460	+	0.999194i	$0.512782\pi$
$830$	0	0
$831$	−39973.6	−1.66868
$832$	0	0
$833$	35440.3	1.47411
$834$	0	0
$835$	40369.9	1.67312
$836$	0	0
$837$	−14266.3	−0.589145
$838$	0	0
$839$	−10090.7	−0.415222	−0.207611	−	0.978211i	$-0.566569\pi$
$839$	−10090.7	−0.415222	−0.207611	+	0.978211i	$0.566569\pi$
$840$	0	0
$841$	30536.8	1.25207
$842$	0	0
$843$	36178.7	1.47813
$844$	0	0
$845$	7276.27	0.296226
$846$	0	0
$847$	−108359.	−4.39581
$848$	0	0
$849$	−18982.4	−0.767345
$850$	0	0
$851$	−6327.19	−0.254869
$852$	0	0
$853$	−22716.1	−0.911824	−0.455912	−	0.890025i	$-0.650687\pi$
$853$	−22716.1	−0.911824	−0.455912	+	0.890025i	$0.650687\pi$
$854$	0	0
$855$	42655.7	1.70619
$856$	0	0
$857$	−26061.0	−1.03877	−0.519386	−	0.854540i	$-0.673839\pi$
$857$	−26061.0	−1.03877	−0.519386	+	0.854540i	$0.673839\pi$
$858$	0	0
$859$	8023.64	0.318700	0.159350	−	0.987222i	$-0.449060\pi$
$859$	8023.64	0.318700	0.159350	+	0.987222i	$0.449060\pi$
$860$	0	0
$861$	16324.7	0.646159
$862$	0	0
$863$	45634.6	1.80002	0.900011	−	0.435867i	$-0.143558\pi$
$863$	45634.6	1.80002	0.900011	+	0.435867i	$0.143558\pi$
$864$	0	0
$865$	−31036.1	−1.21995
$866$	0	0
$867$	26179.8	1.02550
$868$	0	0
$869$	74779.5	2.91912
$870$	0	0
$871$	−13162.4	−0.512046
$872$	0	0
$873$	9499.49	0.368281
$874$	0	0
$875$	−36217.2	−1.39927
$876$	0	0
$877$	−32265.7	−1.24234	−0.621172	−	0.783674i	$-0.713343\pi$
$877$	−32265.7	−1.24234	−0.621172	+	0.783674i	$0.713343\pi$
$878$	0	0
$879$	−54738.8	−2.10045
$880$	0	0
$881$	−10157.7	−0.388447	−0.194224	−	0.980957i	$-0.562219\pi$
$881$	−10157.7	−0.388447	−0.194224	+	0.980957i	$0.562219\pi$
$882$	0	0
$883$	−42435.0	−1.61727	−0.808635	−	0.588310i	$-0.799793\pi$
$883$	−42435.0	−1.61727	−0.808635	+	0.588310i	$0.799793\pi$
$884$	0	0
$885$	49776.2	1.89063
$886$	0	0
$887$	−5740.13	−0.217288	−0.108644	−	0.994081i	$-0.534651\pi$
$887$	−5740.13	−0.217288	−0.108644	+	0.994081i	$0.534651\pi$
$888$	0	0
$889$	−44202.9	−1.66762
$890$	0	0
$891$	29833.1	1.12171
$892$	0	0
$893$	−47145.1	−1.76668
$894$	0	0
$895$	−10880.8	−0.406373
$896$	0	0
$897$	7336.05	0.273070
$898$	0	0
$899$	−53052.0	−1.96817
$900$	0	0
$901$	329.044	0.0121665
$902$	0	0
$903$	−50867.6	−1.87461
$904$	0	0
$905$	20025.3	0.735540
$906$	0	0
$907$	1897.29	0.0694583	0.0347291	−	0.999397i	$-0.488943\pi$
$907$	1897.29	0.0694583	0.0347291	+	0.999397i	$0.488943\pi$
$908$	0	0
$909$	62192.0	2.26928
$910$	0	0
$911$	−25938.8	−0.943349	−0.471674	−	0.881773i	$-0.656350\pi$
$911$	−25938.8	−0.943349	−0.471674	+	0.881773i	$0.656350\pi$
$912$	0	0
$913$	−9073.00	−0.328885
$914$	0	0
$915$	−42424.1	−1.53278
$916$	0	0
$917$	58798.2	2.11743
$918$	0	0
$919$	13202.4	0.473892	0.236946	−	0.971523i	$-0.423854\pi$
$919$	13202.4	0.473892	0.236946	+	0.971523i	$0.423854\pi$
$920$	0	0
$921$	7899.66	0.282631
$922$	0	0
$923$	27308.5	0.973857
$924$	0	0
$925$	12691.1	0.451113
$926$	0	0
$927$	8415.67	0.298174
$928$	0	0
$929$	−7627.97	−0.269393	−0.134696	−	0.990887i	$-0.543006\pi$
$929$	−7627.97	−0.269393	−0.134696	+	0.990887i	$0.543006\pi$
$930$	0	0
$931$	−82838.5	−2.91614
$932$	0	0
$933$	−54357.1	−1.90737
$934$	0	0
$935$	34652.3	1.21203
$936$	0	0
$937$	−12830.6	−0.447341	−0.223670	−	0.974665i	$-0.571804\pi$
$937$	−12830.6	−0.447341	−0.223670	+	0.974665i	$0.571804\pi$
$938$	0	0
$939$	−52443.4	−1.82260
$940$	0	0
$941$	34812.2	1.20600	0.602999	−	0.797742i	$-0.293972\pi$
$941$	34812.2	1.20600	0.602999	+	0.797742i	$0.293972\pi$
$942$	0	0
$943$	−1358.34	−0.0469075
$944$	0	0
$945$	−28941.3	−0.996254
$946$	0	0
$947$	36412.9	1.24948	0.624742	−	0.780832i	$-0.285204\pi$
$947$	36412.9	1.24948	0.624742	+	0.780832i	$0.285204\pi$
$948$	0	0
$949$	−30501.1	−1.04332
$950$	0	0
$951$	14370.2	0.489996
$952$	0	0
$953$	20178.9	0.685896	0.342948	−	0.939354i	$-0.388575\pi$
$953$	20178.9	0.685896	0.342948	+	0.939354i	$0.388575\pi$
$954$	0	0
$955$	−16715.7	−0.566397
$956$	0	0
$957$	−122659.	−4.14317
$958$	0	0
$959$	4694.91	0.158088
$960$	0	0
$961$	21451.2	0.720056
$962$	0	0
$963$	10781.2	0.360766
$964$	0	0
$965$	21328.0	0.711474
$966$	0	0
$967$	46524.3	1.54718	0.773589	−	0.633688i	$-0.218460\pi$
$967$	46524.3	1.54718	0.773589	+	0.633688i	$0.218460\pi$
$968$	0	0
$969$	29232.5	0.969127
$970$	0	0
$971$	10042.0	0.331888	0.165944	−	0.986135i	$-0.446933\pi$
$971$	10042.0	0.331888	0.165944	+	0.986135i	$0.446933\pi$
$972$	0	0
$973$	−30560.0	−1.00690
$974$	0	0
$975$	−14714.7	−0.483329
$976$	0	0
$977$	27740.9	0.908402	0.454201	−	0.890899i	$-0.349925\pi$
$977$	27740.9	0.908402	0.454201	+	0.890899i	$0.349925\pi$
$978$	0	0
$979$	62795.2	2.04999
$980$	0	0
$981$	12360.2	0.402275
$982$	0	0
$983$	−13613.6	−0.441716	−0.220858	−	0.975306i	$-0.570886\pi$
$983$	−13613.6	−0.441716	−0.220858	+	0.975306i	$0.570886\pi$
$984$	0	0
$985$	29525.2	0.955076
$986$	0	0
$987$	139895.	4.51156
$988$	0	0
$989$	4232.60	0.136086
$990$	0	0
$991$	35332.8	1.13258	0.566289	−	0.824207i	$-0.308379\pi$
$991$	35332.8	1.13258	0.566289	+	0.824207i	$0.308379\pi$
$992$	0	0
$993$	−14913.9	−0.476616
$994$	0	0
$995$	−24064.6	−0.766734
$996$	0	0
$997$	−57488.9	−1.82617	−0.913085	−	0.407769i	$-0.866307\pi$
$997$	−57488.9	−1.82617	−0.913085	+	0.407769i	$0.866307\pi$
$998$	0	0
$999$	−17337.2	−0.549074

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.4.a.bg.1.3		8
4.3	odd	2		1472.4.a.bh.1.6		8
8.3	odd	2		736.4.a.e.1.3	✓	8
8.5	even	2		736.4.a.f.1.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
736.4.a.e.1.3	✓	8	8.3	odd	2
736.4.a.f.1.6	yes	8	8.5	even	2
1472.4.a.bg.1.3		8	1.1	even	1	trivial
1472.4.a.bh.1.6		8	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:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

\(f(q)\)	\(=\)	\(q-7.87424 q^{3} -13.0818 q^{5} -35.1038 q^{7} +35.0036 q^{9} -66.4666 q^{11} +40.5066 q^{13} +103.009 q^{15} +39.8530 q^{17} -93.1529 q^{19} +276.415 q^{21} -23.0000 q^{23} +46.1334 q^{25} -63.0226 q^{27} -234.363 q^{29} +226.367 q^{31} +523.374 q^{33} +459.220 q^{35} +275.095 q^{37} -318.959 q^{39} +59.0584 q^{41} -184.026 q^{43} -457.911 q^{45} +506.104 q^{47} +889.275 q^{49} -313.812 q^{51} +8.25645 q^{53} +869.503 q^{55} +733.509 q^{57} +483.221 q^{59} -411.848 q^{61} -1228.76 q^{63} -529.899 q^{65} -324.945 q^{67} +181.108 q^{69} +674.173 q^{71} -752.991 q^{73} -363.265 q^{75} +2333.23 q^{77} -1125.07 q^{79} -448.843 q^{81} +136.505 q^{83} -521.349 q^{85} +1845.43 q^{87} -944.763 q^{89} -1421.94 q^{91} -1782.47 q^{93} +1218.61 q^{95} +271.386 q^{97} -2326.57 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{3} + 12 q^{5} - 14 q^{7} + 90 q^{9} - 88 q^{11} + 30 q^{13} + 30 q^{15} + 58 q^{17} - 190 q^{19} + 66 q^{21} - 184 q^{23} + 28 q^{25} - 432 q^{27} - 190 q^{29} - 60 q^{31} + 346 q^{33} - 192 q^{35}+ \cdots - 5986 q^{99}+O(q^{100}) \) 8 * q - 12 * q^3 + 12 * q^5 - 14 * q^7 + 90 * q^9 - 88 * q^11 + 30 * q^13 + 30 * q^15 + 58 * q^17 - 190 * q^19 + 66 * q^21 - 184 * q^23 + 28 * q^25 - 432 * q^27 - 190 * q^29 - 60 * q^31 + 346 * q^33 - 192 * q^35 + 156 * q^37 + 156 * q^39 + 282 * q^41 - 810 * q^43 - 358 * q^45 + 564 * q^47 + 752 * q^49 - 1614 * q^51 + 230 * q^53 + 1292 * q^55 + 796 * q^57 - 1916 * q^59 - 22 * q^61 + 1036 * q^63 + 222 * q^65 - 2292 * q^67 + 276 * q^69 + 2376 * q^71 - 630 * q^73 - 3952 * q^75 + 948 * q^77 + 892 * q^79 - 1044 * q^81 - 2800 * q^83 - 288 * q^85 + 2756 * q^87 - 308 * q^89 - 4610 * q^91 + 982 * q^93 + 1300 * q^95 + 854 * q^97 - 5986 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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