LMFDB - Embedded newform 1728.3.e.u.1025.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(1025,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1728.1025");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.e (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 864)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.2
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1728.1025
Dual form			1728.3.e.u.1025.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.61037i q^{5} +9.43879 q^{7} +O(q^{10})$	$q-6.61037i q^{5} +9.43879 q^{7} +17.6969i q^{11} -20.6969 q^{13} -16.0492i q^{17} +12.2993 q^{19} -26.6969i q^{23} -18.6969 q^{25} -0.921404i q^{29} +53.7722 q^{31} -62.3939i q^{35} +64.0908 q^{37} -19.7990i q^{41} -69.7893 q^{43} -0.606123i q^{47} +40.0908 q^{49} -45.3512i q^{53} +116.983 q^{55} -71.3939i q^{59} +4.00000 q^{61} +136.814i q^{65} -7.43545 q^{67} -27.3031i q^{71} +6.30306 q^{73} +167.038i q^{77} -42.6191 q^{79} -72.3031i q^{83} -106.091 q^{85} -126.486i q^{89} -195.354 q^{91} -81.3031i q^{95} +27.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 48 q^{13} - 32 q^{25} + 160 q^{37} - 32 q^{49} + 32 q^{61} + 168 q^{73} - 496 q^{85} + 216 q^{97}+O(q^{100}) $ 8 * q - 48 * q^13 - 32 * q^25 + 160 * q^37 - 32 * q^49 + 32 * q^61 + 168 * q^73 - 496 * q^85 + 216 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 6.61037i	− 1.32207i	−0.750354	−	0.661037i	$-0.770117\pi$
$5$	− 6.61037i	− 1.32207i	0.750354	−	0.661037i	$-0.229883\pi$
$6$	0	0
$7$	9.43879	1.34840	0.674200	−	0.738549i	$-0.264489\pi$
$7$	9.43879	1.34840	0.674200	+	0.738549i	$0.264489\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	17.6969i	1.60881i	0.594080	+	0.804406i	$0.297516\pi$
$11$	17.6969i	1.60881i	−0.594080	+	0.804406i	$0.702484\pi$
$12$	0	0
$13$	−20.6969	−1.59207	−0.796036	−	0.605249i	$-0.793073\pi$
$13$	−20.6969	−1.59207	−0.796036	+	0.605249i	$0.793073\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 16.0492i	− 0.944068i	−0.881580	−	0.472034i	$-0.843520\pi$
$17$	− 16.0492i	− 0.944068i	0.881580	−	0.472034i	$-0.156480\pi$
$18$	0	0
$19$	12.2993	0.647333	0.323667	−	0.946171i	$-0.395084\pi$
$19$	12.2993	0.647333	0.323667	+	0.946171i	$0.395084\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 26.6969i	− 1.16074i	−0.814354	−	0.580368i	$-0.802909\pi$
$23$	− 26.6969i	− 1.16074i	0.814354	−	0.580368i	$-0.197091\pi$
$24$	0	0
$25$	−18.6969	−0.747878
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 0.921404i	− 0.0317725i	−0.999874	−	0.0158863i	$-0.994943\pi$
$29$	− 0.921404i	− 0.0317725i	0.999874	−	0.0158863i	$-0.00505697\pi$
$30$	0	0
$31$	53.7722	1.73459	0.867294	−	0.497796i	$-0.165857\pi$
$31$	53.7722	1.73459	0.867294	+	0.497796i	$0.165857\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 62.3939i	− 1.78268i
$36$	0	0
$37$	64.0908	1.73218	0.866092	−	0.499884i	$-0.166624\pi$
$37$	64.0908	1.73218	0.866092	+	0.499884i	$0.166624\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 19.7990i	− 0.482902i	−0.970413	−	0.241451i	$-0.922377\pi$
$41$	− 19.7990i	− 0.482902i	0.970413	−	0.241451i	$-0.0776233\pi$
$42$	0	0
$43$	−69.7893	−1.62301	−0.811503	−	0.584348i	$-0.801350\pi$
$43$	−69.7893	−1.62301	−0.811503	+	0.584348i	$0.801350\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 0.606123i	− 0.0128962i	−0.999979	−	0.00644812i	$-0.997947\pi$
$47$	− 0.606123i	− 0.0128962i	0.999979	−	0.00644812i	$-0.00205251\pi$
$48$	0	0
$49$	40.0908	0.818180
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 45.3512i	− 0.855682i	−0.903854	−	0.427841i	$-0.859274\pi$
$53$	− 45.3512i	− 0.855682i	0.903854	−	0.427841i	$-0.140726\pi$
$54$	0	0
$55$	116.983	2.12697
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 71.3939i	− 1.21007i	−0.796201	−	0.605033i	$-0.793160\pi$
$59$	− 71.3939i	− 1.21007i	0.796201	−	0.605033i	$-0.206840\pi$
$60$	0	0
$61$	4.00000	0.0655738	0.0327869	−	0.999462i	$-0.489562\pi$
$61$	4.00000	0.0655738	0.0327869	+	0.999462i	$0.489562\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	136.814i	2.10484i
$66$	0	0
$67$	−7.43545	−0.110977	−0.0554884	−	0.998459i	$-0.517672\pi$
$67$	−7.43545	−0.110977	−0.0554884	+	0.998459i	$0.517672\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 27.3031i	− 0.384550i	−0.981341	−	0.192275i	$-0.938413\pi$
$71$	− 27.3031i	− 0.384550i	0.981341	−	0.192275i	$-0.0615866\pi$
$72$	0	0
$73$	6.30306	0.0863433	0.0431717	−	0.999068i	$-0.486254\pi$
$73$	6.30306	0.0863433	0.0431717	+	0.999068i	$0.486254\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	167.038i	2.16932i
$78$	0	0
$79$	−42.6191	−0.539482	−0.269741	−	0.962933i	$-0.586938\pi$
$79$	−42.6191	−0.539482	−0.269741	+	0.962933i	$0.586938\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 72.3031i	− 0.871121i	−0.900159	−	0.435561i	$-0.856550\pi$
$83$	− 72.3031i	− 0.871121i	0.900159	−	0.435561i	$-0.143450\pi$
$84$	0	0
$85$	−106.091	−1.24813
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 126.486i	− 1.42119i	−0.703599	−	0.710597i	$-0.748425\pi$
$89$	− 126.486i	− 1.42119i	0.703599	−	0.710597i	$-0.251575\pi$
$90$	0	0
$91$	−195.354	−2.14675
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 81.3031i	− 0.855822i
$96$	0	0
$97$	27.0000	0.278351	0.139175	−	0.990268i	$-0.455555\pi$
$97$	27.0000	0.278351	0.139175	+	0.990268i	$0.455555\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 33.7806i	− 0.334461i	−0.985918	−	0.167231i	$-0.946518\pi$
$101$	− 33.7806i	− 0.334461i	0.985918	−	0.167231i	$-0.0534824\pi$
$102$	0	0
$103$	95.2451	0.924710	0.462355	−	0.886695i	$-0.347005\pi$
$103$	95.2451	0.924710	0.462355	+	0.886695i	$0.347005\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	96.5755i	0.902575i	0.892379	+	0.451287i	$0.149035\pi$
$107$	96.5755i	0.902575i	−0.892379	+	0.451287i	$0.850965\pi$
$108$	0	0
$109$	−90.8786	−0.833748	−0.416874	−	0.908964i	$-0.636874\pi$
$109$	−90.8786	−0.833748	−0.416874	+	0.908964i	$0.636874\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	50.1829i	0.444097i	0.975036	+	0.222048i	$0.0712743\pi$
$113$	50.1829i	0.444097i	−0.975036	+	0.222048i	$0.928726\pi$
$114$	0	0
$115$	−176.477	−1.53458
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 151.485i	− 1.27298i
$120$	0	0
$121$	−192.182	−1.58828
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 41.6655i	− 0.333324i
$126$	0	0
$127$	−134.146	−1.05627	−0.528136	−	0.849160i	$-0.677109\pi$
$127$	−134.146	−1.05627	−0.528136	+	0.849160i	$0.677109\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 46.8184i	− 0.357392i	−0.983904	−	0.178696i	$-0.942812\pi$
$131$	− 46.8184i	− 0.357392i	0.983904	−	0.178696i	$-0.0571879\pi$
$132$	0	0
$133$	116.091	0.872863
$134$	0	0
$135$	0	0
$136$	0	0
$137$	182.926i	1.33523i	0.744507	+	0.667614i	$0.232685\pi$
$137$	182.926i	1.33523i	−0.744507	+	0.667614i	$0.767315\pi$
$138$	0	0
$139$	−137.864	−0.991829	−0.495914	−	0.868371i	$-0.665167\pi$
$139$	−137.864	−0.991829	−0.495914	+	0.868371i	$0.665167\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 366.272i	− 2.56135i
$144$	0	0
$145$	−6.09082	−0.0420056
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 43.6368i	− 0.292864i	−0.989221	−	0.146432i	$-0.953221\pi$
$149$	− 43.6368i	− 0.292864i	0.989221	−	0.146432i	$-0.0467790\pi$
$150$	0	0
$151$	84.0920	0.556900	0.278450	−	0.960451i	$-0.410179\pi$
$151$	84.0920	0.556900	0.278450	+	0.960451i	$0.410179\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 355.454i	− 2.29325i
$156$	0	0
$157$	236.545	1.50666	0.753328	−	0.657645i	$-0.228447\pi$
$157$	236.545	1.50666	0.753328	+	0.657645i	$0.228447\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 251.987i	− 1.56514i
$162$	0	0
$163$	282.307	1.73194	0.865971	−	0.500094i	$-0.166701\pi$
$163$	282.307	1.73194	0.865971	+	0.500094i	$0.166701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	70.1816i	0.420249i	0.977675	+	0.210125i	$0.0673870\pi$
$167$	70.1816i	0.420249i	−0.977675	+	0.210125i	$0.932613\pi$
$168$	0	0
$169$	259.363	1.53469
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 338.586i	− 1.95715i	−0.205901	−	0.978573i	$-0.566013\pi$
$173$	− 338.586i	− 1.95715i	0.205901	−	0.978573i	$-0.433987\pi$
$174$	0	0
$175$	−176.477	−1.00844
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 348.576i	− 1.94735i	−0.227942	−	0.973675i	$-0.573200\pi$
$179$	− 348.576i	− 1.94735i	0.227942	−	0.973675i	$-0.426800\pi$
$180$	0	0
$181$	−16.1816	−0.0894013	−0.0447006	−	0.999000i	$-0.514233\pi$
$181$	−16.1816	−0.0894013	−0.0447006	+	0.999000i	$0.514233\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 423.664i	− 2.29007i
$186$	0	0
$187$	284.021	1.51883
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 171.303i	− 0.896875i	−0.893814	−	0.448437i	$-0.851981\pi$
$191$	− 171.303i	− 0.896875i	0.893814	−	0.448437i	$-0.148019\pi$
$192$	0	0
$193$	−241.091	−1.24918	−0.624588	−	0.780955i	$-0.714733\pi$
$193$	−241.091	−1.24918	−0.624588	+	0.780955i	$0.714733\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	182.165i	0.924698i	0.886698	+	0.462349i	$0.152993\pi$
$197$	182.165i	0.924698i	−0.886698	+	0.462349i	$0.847007\pi$
$198$	0	0
$199$	70.0782	0.352152	0.176076	−	0.984377i	$-0.443660\pi$
$199$	70.0782	0.352152	0.176076	+	0.984377i	$0.443660\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 8.69694i	− 0.0428421i
$204$	0	0
$205$	−130.879	−0.638432
$206$	0	0
$207$	0	0
$208$	0	0
$209$	217.660i	1.04144i
$210$	0	0
$211$	102.681	0.486638	0.243319	−	0.969946i	$-0.421764\pi$
$211$	102.681	0.486638	0.243319	+	0.969946i	$0.421764\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	461.333i	2.14573i
$216$	0	0
$217$	507.545	2.33892
$218$	0	0
$219$	0	0
$220$	0	0
$221$	332.168i	1.50302i
$222$	0	0
$223$	205.082	0.919650	0.459825	−	0.888010i	$-0.347912\pi$
$223$	205.082	0.919650	0.459825	+	0.888010i	$0.347912\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	103.757i	0.457080i	0.973535	+	0.228540i	$0.0733952\pi$
$227$	103.757i	0.457080i	−0.973535	+	0.228540i	$0.926605\pi$
$228$	0	0
$229$	−435.576	−1.90208	−0.951038	−	0.309073i	$-0.899981\pi$
$229$	−435.576	−1.90208	−0.951038	+	0.309073i	$0.899981\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	387.344i	1.66242i	0.555959	+	0.831210i	$0.312351\pi$
$233$	387.344i	1.66242i	−0.555959	+	0.831210i	$0.687649\pi$
$234$	0	0
$235$	−4.00670	−0.0170498
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 225.303i	− 0.942691i	−0.881949	−	0.471345i	$-0.843769\pi$
$239$	− 225.303i	− 0.942691i	0.881949	−	0.471345i	$-0.156231\pi$
$240$	0	0
$241$	−127.757	−0.530113	−0.265056	−	0.964233i	$-0.585391\pi$
$241$	−127.757	−0.530113	−0.265056	+	0.964233i	$0.585391\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 265.015i	− 1.08169i
$246$	0	0
$247$	−254.558	−1.03060
$248$	0	0
$249$	0	0
$250$	0	0
$251$	54.0000i	0.215139i	0.994198	+	0.107570i	$0.0343069\pi$
$251$	54.0000i	0.215139i	−0.994198	+	0.107570i	$0.965693\pi$
$252$	0	0
$253$	472.454	1.86741
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 181.941i	− 0.707941i	−0.935257	−	0.353970i	$-0.884831\pi$
$257$	− 181.941i	− 0.707941i	0.935257	−	0.353970i	$-0.115169\pi$
$258$	0	0
$259$	604.940	2.33568
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 42.8786i	− 0.163036i	−0.996672	−	0.0815182i	$-0.974023\pi$
$263$	− 42.8786i	− 0.163036i	0.996672	−	0.0815182i	$-0.0259769\pi$
$264$	0	0
$265$	−299.788	−1.13127
$266$	0	0
$267$	0	0
$268$	0	0
$269$	140.500i	0.522305i	0.965298	+	0.261152i	$0.0841025\pi$
$269$	140.500i	0.522305i	−0.965298	+	0.261152i	$0.915898\pi$
$270$	0	0
$271$	−264.854	−0.977323	−0.488661	−	0.872474i	$-0.662515\pi$
$271$	−264.854	−0.977323	−0.488661	+	0.872474i	$0.662515\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 330.879i	− 1.20319i
$276$	0	0
$277$	91.8184	0.331474	0.165737	−	0.986170i	$-0.447000\pi$
$277$	91.8184	0.331474	0.165737	+	0.986170i	$0.447000\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	417.814i	1.48688i	0.668801	+	0.743442i	$0.266808\pi$
$281$	417.814i	1.48688i	−0.668801	+	0.743442i	$0.733192\pi$
$282$	0	0
$283$	389.851	1.37757	0.688783	−	0.724968i	$-0.258145\pi$
$283$	389.851	1.37757	0.688783	+	0.724968i	$0.258145\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 186.879i	− 0.651145i
$288$	0	0
$289$	31.4245	0.108735
$290$	0	0
$291$	0	0
$292$	0	0
$293$	108.145i	0.369095i	0.982824	+	0.184547i	$0.0590819\pi$
$293$	108.145i	0.369095i	−0.982824	+	0.184547i	$0.940918\pi$
$294$	0	0
$295$	−471.940	−1.59980
$296$	0	0
$297$	0	0
$298$	0	0
$299$	552.545i	1.84798i
$300$	0	0
$301$	−658.727	−2.18846
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 26.4415i	− 0.0866933i
$306$	0	0
$307$	−52.3468	−0.170511	−0.0852554	−	0.996359i	$-0.527171\pi$
$307$	−52.3468	−0.170511	−0.0852554	+	0.996359i	$0.527171\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 412.999i	− 1.32797i	−0.747745	−	0.663985i	$-0.768864\pi$
$311$	− 412.999i	− 1.32797i	0.747745	−	0.663985i	$-0.231136\pi$
$312$	0	0
$313$	−447.636	−1.43015	−0.715073	−	0.699050i	$-0.753607\pi$
$313$	−447.636	−1.43015	−0.715073	+	0.699050i	$0.753607\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 422.582i	− 1.33307i	−0.745476	−	0.666533i	$-0.767778\pi$
$317$	− 422.582i	− 1.33307i	0.745476	−	0.666533i	$-0.232222\pi$
$318$	0	0
$319$	16.3060	0.0511161
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 197.394i	− 0.611127i
$324$	0	0
$325$	386.969	1.19068
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 5.72107i	− 0.0173893i
$330$	0	0
$331$	211.082	0.637711	0.318855	−	0.947803i	$-0.396702\pi$
$331$	211.082	0.637711	0.318855	+	0.947803i	$0.396702\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	49.1510i	0.146719i
$336$	0	0
$337$	−113.455	−0.336662	−0.168331	−	0.985731i	$-0.553838\pi$
$337$	−113.455	−0.336662	−0.168331	+	0.985731i	$0.553838\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	951.604i	2.79063i
$342$	0	0
$343$	−84.0920	−0.245166
$344$	0	0
$345$	0	0
$346$	0	0
$347$	448.485i	1.29246i	0.763141	+	0.646232i	$0.223656\pi$
$347$	448.485i	1.29246i	−0.763141	+	0.646232i	$0.776344\pi$
$348$	0	0
$349$	260.454	0.746287	0.373143	−	0.927774i	$-0.378280\pi$
$349$	260.454	0.746287	0.373143	+	0.927774i	$0.378280\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 541.129i	− 1.53294i	−0.642279	−	0.766471i	$-0.722011\pi$
$353$	− 541.129i	− 1.53294i	0.642279	−	0.766471i	$-0.277989\pi$
$354$	0	0
$355$	−180.483	−0.508403
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 314.908i	− 0.877181i	−0.898687	−	0.438591i	$-0.855478\pi$
$359$	− 314.908i	− 0.877181i	0.898687	−	0.438591i	$-0.144522\pi$
$360$	0	0
$361$	−209.727	−0.580960
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 41.6655i	− 0.114152i
$366$	0	0
$367$	227.677	0.620374	0.310187	−	0.950676i	$-0.399608\pi$
$367$	227.677	0.620374	0.310187	+	0.950676i	$0.399608\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 428.060i	− 1.15380i
$372$	0	0
$373$	−165.546	−0.443823	−0.221911	−	0.975067i	$-0.571230\pi$
$373$	−165.546	−0.443823	−0.221911	+	0.975067i	$0.571230\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	19.0702i	0.0505842i
$378$	0	0
$379$	124.708	0.329044	0.164522	−	0.986373i	$-0.447392\pi$
$379$	124.708	0.329044	0.164522	+	0.986373i	$0.447392\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	382.454i	0.998575i	0.866436	+	0.499287i	$0.166405\pi$
$383$	382.454i	0.998575i	−0.866436	+	0.499287i	$0.833595\pi$
$384$	0	0
$385$	1104.18	2.86800
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 215.057i	− 0.552845i	−0.961036	−	0.276423i	$-0.910851\pi$
$389$	− 215.057i	− 0.552845i	0.961036	−	0.276423i	$-0.0891489\pi$
$390$	0	0
$391$	−428.463	−1.09581
$392$	0	0
$393$	0	0
$394$	0	0
$395$	281.728i	0.713234i
$396$	0	0
$397$	241.909	0.609343	0.304672	−	0.952457i	$-0.401453\pi$
$397$	241.909	0.609343	0.304672	+	0.952457i	$0.401453\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	325.183i	0.810929i	0.914111	+	0.405464i	$0.132890\pi$
$401$	325.183i	0.810929i	−0.914111	+	0.405464i	$0.867110\pi$
$402$	0	0
$403$	−1112.92	−2.76159
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1134.21i	2.78676i
$408$	0	0
$409$	419.302	1.02519	0.512594	−	0.858631i	$-0.328685\pi$
$409$	419.302	1.02519	0.512594	+	0.858631i	$0.328685\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 673.872i	− 1.63165i
$414$	0	0
$415$	−477.950	−1.15169
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 337.151i	− 0.804656i	−0.915496	−	0.402328i	$-0.868201\pi$
$419$	− 337.151i	− 0.804656i	0.915496	−	0.402328i	$-0.131799\pi$
$420$	0	0
$421$	422.969	1.00468	0.502339	−	0.864671i	$-0.332473\pi$
$421$	422.969	1.00468	0.502339	+	0.864671i	$0.332473\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	300.070i	0.706047i
$426$	0	0
$427$	37.7552	0.0884196
$428$	0	0
$429$	0	0
$430$	0	0
$431$	137.333i	0.318637i	0.987227	+	0.159319i	$0.0509297\pi$
$431$	137.333i	0.318637i	−0.987227	+	0.159319i	$0.949070\pi$
$432$	0	0
$433$	523.545	1.20911	0.604555	−	0.796563i	$-0.293351\pi$
$433$	523.545	1.20911	0.604555	+	0.796563i	$0.293351\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 328.354i	− 0.751383i
$438$	0	0
$439$	−566.617	−1.29070	−0.645349	−	0.763888i	$-0.723288\pi$
$439$	−566.617	−1.29070	−0.645349	+	0.763888i	$0.723288\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	592.788i	1.33812i	0.743208	+	0.669061i	$0.233303\pi$
$443$	592.788i	1.33812i	−0.743208	+	0.669061i	$0.766697\pi$
$444$	0	0
$445$	−836.120	−1.87892
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 127.151i	− 0.283187i	−0.989925	−	0.141593i	$-0.954777\pi$
$449$	− 127.151i	− 0.283187i	0.989925	−	0.141593i	$-0.0452225\pi$
$450$	0	0
$451$	350.382	0.776899
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1291.36i	2.83816i
$456$	0	0
$457$	−192.576	−0.421391	−0.210695	−	0.977552i	$-0.567573\pi$
$457$	−192.576	−0.421391	−0.210695	+	0.977552i	$0.567573\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 227.077i	− 0.492575i	−0.969197	−	0.246287i	$-0.920789\pi$
$461$	− 227.077i	− 0.492575i	0.969197	−	0.246287i	$-0.0792107\pi$
$462$	0	0
$463$	638.399	1.37883	0.689416	−	0.724365i	$-0.257867\pi$
$463$	638.399	1.37883	0.689416	+	0.724365i	$0.257867\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	604.423i	1.29427i	0.762376	+	0.647134i	$0.224033\pi$
$467$	604.423i	1.29427i	−0.762376	+	0.647134i	$0.775967\pi$
$468$	0	0
$469$	−70.1816	−0.149641
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 1235.06i	− 2.61111i
$474$	0	0
$475$	−229.960	−0.484126
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 368.302i	− 0.768898i	−0.923146	−	0.384449i	$-0.874391\pi$
$479$	− 368.302i	− 0.768898i	0.923146	−	0.384449i	$-0.125609\pi$
$480$	0	0
$481$	−1326.48	−2.75776
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 178.480i	− 0.368000i
$486$	0	0
$487$	536.865	1.10239	0.551196	−	0.834376i	$-0.314172\pi$
$487$	536.865	1.10239	0.551196	+	0.834376i	$0.314172\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	405.606i	0.826082i	0.910712	+	0.413041i	$0.135533\pi$
$491$	405.606i	0.826082i	−0.910712	+	0.413041i	$0.864467\pi$
$492$	0	0
$493$	−14.7878	−0.0299954
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 257.708i	− 0.518527i
$498$	0	0
$499$	−874.090	−1.75168	−0.875842	−	0.482598i	$-0.839693\pi$
$499$	−874.090	−1.75168	−0.875842	+	0.482598i	$0.839693\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	586.515i	1.16603i	0.812460	+	0.583017i	$0.198128\pi$
$503$	586.515i	1.16603i	−0.812460	+	0.583017i	$0.801872\pi$
$504$	0	0
$505$	−223.302	−0.442182
$506$	0	0
$507$	0	0
$508$	0	0
$509$	541.096i	1.06306i	0.847040	+	0.531529i	$0.178382\pi$
$509$	541.096i	1.06306i	−0.847040	+	0.531529i	$0.821618\pi$
$510$	0	0
$511$	59.4933	0.116425
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 629.605i	− 1.22253i
$516$	0	0
$517$	10.7265	0.0207476
$518$	0	0
$519$	0	0
$520$	0	0
$521$	563.050i	1.08071i	0.841437	+	0.540355i	$0.181710\pi$
$521$	563.050i	1.08071i	−0.841437	+	0.540355i	$0.818290\pi$
$522$	0	0
$523$	−353.232	−0.675396	−0.337698	−	0.941254i	$-0.609648\pi$
$523$	−353.232	−0.675396	−0.337698	+	0.941254i	$0.609648\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 862.999i	− 1.63757i
$528$	0	0
$529$	−183.727	−0.347309
$530$	0	0
$531$	0	0
$532$	0	0
$533$	409.778i	0.768815i
$534$	0	0
$535$	638.399	1.19327
$536$	0	0
$537$	0	0
$538$	0	0
$539$	709.485i	1.31630i
$540$	0	0
$541$	−122.302	−0.226067	−0.113033	−	0.993591i	$-0.536057\pi$
$541$	−122.302	−0.226067	−0.113033	+	0.993591i	$0.536057\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	600.741i	1.10228i
$546$	0	0
$547$	−189.055	−0.345622	−0.172811	−	0.984955i	$-0.555285\pi$
$547$	−189.055	−0.345622	−0.172811	+	0.984955i	$0.555285\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 11.3326i	− 0.0205674i
$552$	0	0
$553$	−402.272	−0.727437
$554$	0	0
$555$	0	0
$556$	0	0
$557$	522.883i	0.938749i	0.882999	+	0.469375i	$0.155521\pi$
$557$	522.883i	0.938749i	−0.882999	+	0.469375i	$0.844479\pi$
$558$	0	0
$559$	1444.42	2.58394
$560$	0	0
$561$	0	0
$562$	0	0
$563$	743.574i	1.32074i	0.750942	+	0.660368i	$0.229600\pi$
$563$	743.574i	1.32074i	−0.750942	+	0.660368i	$0.770400\pi$
$564$	0	0
$565$	331.728	0.587128
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 317.984i	− 0.558848i	−0.960168	−	0.279424i	$-0.909857\pi$
$569$	− 317.984i	− 0.558848i	0.960168	−	0.279424i	$-0.0901435\pi$
$570$	0	0
$571$	1032.27	1.80782	0.903911	−	0.427720i	$-0.140683\pi$
$571$	1032.27	1.80782	0.903911	+	0.427720i	$0.140683\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	499.151i	0.868089i
$576$	0	0
$577$	374.545	0.649125	0.324562	−	0.945864i	$-0.394783\pi$
$577$	374.545	0.649125	0.324562	+	0.945864i	$0.394783\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 682.454i	− 1.17462i
$582$	0	0
$583$	802.577	1.37663
$584$	0	0
$585$	0	0
$586$	0	0
$587$	338.455i	0.576585i	0.957542	+	0.288292i	$0.0930875\pi$
$587$	338.455i	0.576585i	−0.957542	+	0.288292i	$0.906913\pi$
$588$	0	0
$589$	661.362	1.12286
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 134.371i	− 0.226596i	−0.993561	−	0.113298i	$-0.963859\pi$
$593$	− 134.371i	− 0.226596i	0.993561	−	0.113298i	$-0.0361414\pi$
$594$	0	0
$595$	−1001.37	−1.68297
$596$	0	0
$597$	0	0
$598$	0	0
$599$	615.031i	1.02676i	0.858161	+	0.513381i	$0.171607\pi$
$599$	615.031i	1.02676i	−0.858161	+	0.513381i	$0.828393\pi$
$600$	0	0
$601$	373.817	0.621992	0.310996	−	0.950411i	$-0.399337\pi$
$601$	373.817	0.621992	0.310996	+	0.950411i	$0.399337\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1270.39i	2.09982i
$606$	0	0
$607$	471.082	0.776083	0.388042	−	0.921642i	$-0.373152\pi$
$607$	471.082	0.776083	0.388042	+	0.921642i	$0.373152\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.5449i	0.0205317i
$612$	0	0
$613$	−210.727	−0.343763	−0.171881	−	0.985118i	$-0.554985\pi$
$613$	−210.727	−0.343763	−0.171881	+	0.985118i	$0.554985\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1037.73i	1.68190i	0.541114	+	0.840949i	$0.318003\pi$
$617$	1037.73i	1.68190i	−0.541114	+	0.840949i	$0.681997\pi$
$618$	0	0
$619$	−75.7896	−0.122439	−0.0612194	−	0.998124i	$-0.519499\pi$
$619$	−75.7896	−0.122439	−0.0612194	+	0.998124i	$0.519499\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 1193.88i	− 1.91634i
$624$	0	0
$625$	−742.848	−1.18856
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1028.60i	− 1.63530i
$630$	0	0
$631$	−289.174	−0.458279	−0.229139	−	0.973394i	$-0.573591\pi$
$631$	−289.174	−0.458279	−0.229139	+	0.973394i	$0.573591\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	886.757i	1.39647i
$636$	0	0
$637$	−829.757	−1.30260
$638$	0	0
$639$	0	0
$640$	0	0
$641$	985.234i	1.53703i	0.639834	+	0.768513i	$0.279003\pi$
$641$	985.234i	1.53703i	−0.639834	+	0.768513i	$0.720997\pi$
$642$	0	0
$643$	743.103	1.15568	0.577840	−	0.816150i	$-0.303896\pi$
$643$	743.103	1.15568	0.577840	+	0.816150i	$0.303896\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	328.849i	0.508267i	0.967169	+	0.254134i	$0.0817903\pi$
$647$	328.849i	0.508267i	−0.967169	+	0.254134i	$0.918210\pi$
$648$	0	0
$649$	1263.45	1.94677
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 205.865i	− 0.315261i	−0.987498	−	0.157630i	$-0.949615\pi$
$653$	− 205.865i	− 0.315261i	0.987498	−	0.157630i	$-0.0503854\pi$
$654$	0	0
$655$	−309.487	−0.472499
$656$	0	0
$657$	0	0
$658$	0	0
$659$	364.151i	0.552581i	0.961074	+	0.276291i	$0.0891052\pi$
$659$	364.151i	0.552581i	−0.961074	+	0.276291i	$0.910895\pi$
$660$	0	0
$661$	−507.728	−0.768120	−0.384060	−	0.923308i	$-0.625474\pi$
$661$	−507.728	−0.768120	−0.384060	+	0.923308i	$0.625474\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 767.403i	− 1.15399i
$666$	0	0
$667$	−24.5987	−0.0368795
$668$	0	0
$669$	0	0
$670$	0	0
$671$	70.7878i	0.105496i
$672$	0	0
$673$	847.363	1.25908	0.629542	−	0.776967i	$-0.283243\pi$
$673$	847.363	1.25908	0.629542	+	0.776967i	$0.283243\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	902.076i	1.33246i	0.745746	+	0.666230i	$0.232093\pi$
$677$	902.076i	1.33246i	−0.745746	+	0.666230i	$0.767907\pi$
$678$	0	0
$679$	254.847	0.375328
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 754.604i	− 1.10484i	−0.833567	−	0.552419i	$-0.813705\pi$
$683$	− 754.604i	− 1.10484i	0.833567	−	0.552419i	$-0.186295\pi$
$684$	0	0
$685$	1209.21	1.76527
$686$	0	0
$687$	0	0
$688$	0	0
$689$	938.630i	1.36231i
$690$	0	0
$691$	420.450	0.608466	0.304233	−	0.952598i	$-0.401600\pi$
$691$	420.450	0.608466	0.304233	+	0.952598i	$0.401600\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	911.333i	1.31127i
$696$	0	0
$697$	−317.757	−0.455893
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 423.953i	− 0.604783i	−0.953184	−	0.302391i	$-0.902215\pi$
$701$	− 423.953i	− 0.604783i	0.953184	−	0.302391i	$-0.0977850\pi$
$702$	0	0
$703$	788.274	1.12130
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 318.848i	− 0.450987i
$708$	0	0
$709$	−14.3020	−0.0201721	−0.0100861	−	0.999949i	$-0.503211\pi$
$709$	−14.3020	−0.0201721	−0.0100861	+	0.999949i	$0.503211\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1435.55i	− 2.01340i
$714$	0	0
$715$	−2421.19	−3.38629
$716$	0	0
$717$	0	0
$718$	0	0
$719$	453.453i	0.630672i	0.948980	+	0.315336i	$0.102117\pi$
$719$	453.453i	0.630672i	−0.948980	+	0.315336i	$0.897883\pi$
$720$	0	0
$721$	898.999	1.24688
$722$	0	0
$723$	0	0
$724$	0	0
$725$	17.2274i	0.0237620i
$726$	0	0
$727$	818.324	1.12562	0.562809	−	0.826587i	$-0.309721\pi$
$727$	818.324	1.12562	0.562809	+	0.826587i	$0.309721\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1120.06i	1.53223i
$732$	0	0
$733$	−164.636	−0.224605	−0.112303	−	0.993674i	$-0.535823\pi$
$733$	−164.636	−0.224605	−0.112303	+	0.993674i	$0.535823\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 131.585i	− 0.178541i
$738$	0	0
$739$	392.143	0.530641	0.265320	−	0.964160i	$-0.414522\pi$
$739$	392.143	0.530641	0.265320	+	0.964160i	$0.414522\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1341.12i	1.80501i	0.430684	+	0.902503i	$0.358272\pi$
$743$	1341.12i	1.80501i	−0.430684	+	0.902503i	$0.641728\pi$
$744$	0	0
$745$	−288.455	−0.387188
$746$	0	0
$747$	0	0
$748$	0	0
$749$	911.556i	1.21703i
$750$	0	0
$751$	−330.637	−0.440262	−0.220131	−	0.975470i	$-0.570649\pi$
$751$	−330.637	−0.440262	−0.220131	+	0.975470i	$0.570649\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 555.879i	− 0.736263i
$756$	0	0
$757$	70.1816	0.0927102	0.0463551	−	0.998925i	$-0.485239\pi$
$757$	70.1816	0.0927102	0.0463551	+	0.998925i	$0.485239\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	761.704i	1.00093i	0.865758	+	0.500463i	$0.166837\pi$
$761$	761.704i	1.00093i	−0.865758	+	0.500463i	$0.833163\pi$
$762$	0	0
$763$	−857.784	−1.12423
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1477.63i	1.92651i
$768$	0	0
$769$	−853.272	−1.10959	−0.554794	−	0.831988i	$-0.687203\pi$
$769$	−853.272	−1.10959	−0.554794	+	0.831988i	$0.687203\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 487.925i	− 0.631209i	−0.948891	−	0.315605i	$-0.897793\pi$
$773$	− 487.925i	− 0.631209i	0.948891	−	0.315605i	$-0.102207\pi$
$774$	0	0
$775$	−1005.38	−1.29726
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 243.514i	− 0.312599i
$780$	0	0
$781$	483.181	0.618669
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1563.65i	− 1.99191i
$786$	0	0
$787$	492.532	0.625834	0.312917	−	0.949780i	$-0.398694\pi$
$787$	492.532	0.625834	0.312917	+	0.949780i	$0.398694\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	473.666i	0.598820i
$792$	0	0
$793$	−82.7878	−0.104398
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 361.085i	− 0.453055i	−0.974005	−	0.226528i	$-0.927263\pi$
$797$	− 361.085i	− 0.453055i	0.974005	−	0.226528i	$-0.0727374\pi$
$798$	0	0
$799$	−9.72777	−0.0121749
$800$	0	0
$801$	0	0
$802$	0	0
$803$	111.545i	0.138910i
$804$	0	0
$805$	−1665.73	−2.06922
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 579.548i	− 0.716376i	−0.933649	−	0.358188i	$-0.883395\pi$
$809$	− 579.548i	− 0.716376i	0.933649	−	0.358188i	$-0.116605\pi$
$810$	0	0
$811$	1406.37	1.73412	0.867059	−	0.498205i	$-0.166007\pi$
$811$	1406.37	1.73412	0.867059	+	0.498205i	$0.166007\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1866.15i	− 2.28975i
$816$	0	0
$817$	−858.361	−1.05063
$818$	0	0
$819$	0	0
$820$	0	0
$821$	111.702i	0.136056i	0.997683	+	0.0680280i	$0.0216707\pi$
$821$	111.702i	0.136056i	−0.997683	+	0.0680280i	$0.978329\pi$
$822$	0	0
$823$	−161.895	−0.196713	−0.0983564	−	0.995151i	$-0.531359\pi$
$823$	−161.895	−0.196713	−0.0983564	+	0.995151i	$0.531359\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	592.788i	0.716793i	0.933569	+	0.358396i	$0.116676\pi$
$827$	592.788i	0.716793i	−0.933569	+	0.358396i	$0.883324\pi$
$828$	0	0
$829$	950.363	1.14640	0.573199	−	0.819416i	$-0.305702\pi$
$829$	950.363	1.14640	0.573199	+	0.819416i	$0.305702\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 643.424i	− 0.772418i
$834$	0	0
$835$	463.926	0.555600
$836$	0	0
$837$	0	0
$838$	0	0
$839$	326.636i	0.389316i	0.980871	+	0.194658i	$0.0623596\pi$
$839$	326.636i	0.389316i	−0.980871	+	0.194658i	$0.937640\pi$
$840$	0	0
$841$	840.151	0.998991
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1714.49i	− 2.02898i
$846$	0	0
$847$	−1813.96	−2.14163
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1711.03i	− 2.01061i
$852$	0	0
$853$	906.727	1.06299	0.531493	−	0.847063i	$-0.321631\pi$
$853$	906.727	1.06299	0.531493	+	0.847063i	$0.321631\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	715.483i	0.834869i	0.908707	+	0.417435i	$0.137071\pi$
$857$	715.483i	0.834869i	−0.908707	+	0.417435i	$0.862929\pi$
$858$	0	0
$859$	49.7558	0.0579229	0.0289615	−	0.999581i	$-0.490780\pi$
$859$	49.7558	0.0579229	0.0289615	+	0.999581i	$0.490780\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1595.76i	− 1.84908i	−0.381086	−	0.924539i	$-0.624450\pi$
$863$	− 1595.76i	− 1.84908i	0.381086	−	0.924539i	$-0.375550\pi$
$864$	0	0
$865$	−2238.18	−2.58749
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 754.227i	− 0.867925i
$870$	0	0
$871$	153.891	0.176683
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 393.272i	− 0.449454i
$876$	0	0
$877$	494.182	0.563491	0.281746	−	0.959489i	$-0.409087\pi$
$877$	494.182	0.563491	0.281746	+	0.959489i	$0.409087\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	728.383i	0.826768i	0.910557	+	0.413384i	$0.135653\pi$
$881$	728.383i	0.826768i	−0.910557	+	0.413384i	$0.864347\pi$
$882$	0	0
$883$	−1378.34	−1.56098	−0.780489	−	0.625170i	$-0.785030\pi$
$883$	−1378.34	−1.56098	−0.780489	+	0.625170i	$0.785030\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1329.00i	− 1.49831i	−0.662397	−	0.749153i	$-0.730461\pi$
$887$	− 1329.00i	− 1.49831i	0.662397	−	0.749153i	$-0.269539\pi$
$888$	0	0
$889$	−1266.18	−1.42428
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 7.45491i	− 0.00834816i
$894$	0	0
$895$	−2304.21	−2.57454
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 49.5459i	− 0.0551123i
$900$	0	0
$901$	−727.848	−0.807822
$902$	0	0
$903$	0	0
$904$	0	0
$905$	106.967i	0.118195i
$906$	0	0
$907$	940.152	1.03655	0.518276	−	0.855214i	$-0.326574\pi$
$907$	940.152	1.03655	0.518276	+	0.855214i	$0.326574\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 962.486i	− 1.05652i	−0.849084	−	0.528258i	$-0.822845\pi$
$911$	− 962.486i	− 1.05652i	0.849084	−	0.528258i	$-0.177155\pi$
$912$	0	0
$913$	1279.54	1.40147
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 441.909i	− 0.481907i
$918$	0	0
$919$	−441.610	−0.480533	−0.240267	−	0.970707i	$-0.577235\pi$
$919$	−441.610	−0.480533	−0.240267	+	0.970707i	$0.577235\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	565.090i	0.612232i
$924$	0	0
$925$	−1198.30	−1.29546
$926$	0	0
$927$	0	0
$928$	0	0
$929$	401.078i	0.431731i	0.976423	+	0.215866i	$0.0692573\pi$
$929$	401.078i	0.431731i	−0.976423	+	0.215866i	$0.930743\pi$
$930$	0	0
$931$	493.090	0.529635
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1877.48i	− 2.00800i
$936$	0	0
$937$	493.000	0.526147	0.263074	−	0.964776i	$-0.415264\pi$
$937$	493.000	0.526147	0.263074	+	0.964776i	$0.415264\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	556.439i	0.591328i	0.955292	+	0.295664i	$0.0955408\pi$
$941$	556.439i	0.591328i	−0.955292	+	0.295664i	$0.904459\pi$
$942$	0	0
$943$	−528.572	−0.560522
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1295.91i	− 1.36844i	−0.729278	−	0.684218i	$-0.760144\pi$
$947$	− 1295.91i	− 1.36844i	0.729278	−	0.684218i	$-0.239856\pi$
$948$	0	0
$949$	−130.454	−0.137465
$950$	0	0
$951$	0	0
$952$	0	0
$953$	838.672i	0.880034i	0.897990	+	0.440017i	$0.145028\pi$
$953$	838.672i	0.880034i	−0.897990	+	0.440017i	$0.854972\pi$
$954$	0	0
$955$	−1132.38	−1.18573
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1726.60i	1.80042i
$960$	0	0
$961$	1930.45	2.00880
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1593.70i	1.65150i
$966$	0	0
$967$	−96.6705	−0.0999695	−0.0499848	−	0.998750i	$-0.515917\pi$
$967$	−96.6705	−0.0999695	−0.0499848	+	0.998750i	$0.515917\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1218.66i	1.25506i	0.778592	+	0.627531i	$0.215934\pi$
$971$	1218.66i	1.25506i	−0.778592	+	0.627531i	$0.784066\pi$
$972$	0	0
$973$	−1301.27	−1.33738
$974$	0	0
$975$	0	0
$976$	0	0
$977$	315.692i	0.323124i	0.986863	+	0.161562i	$0.0516532\pi$
$977$	315.692i	0.323124i	−0.986863	+	0.161562i	$0.948347\pi$
$978$	0	0
$979$	2238.42	2.28643
$980$	0	0
$981$	0	0
$982$	0	0
$983$	40.2429i	0.0409388i	0.999790	+	0.0204694i	$0.00651607\pi$
$983$	40.2429i	0.0409388i	−0.999790	+	0.0204694i	$0.993484\pi$
$984$	0	0
$985$	1204.18	1.22252
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1863.16i	1.88388i
$990$	0	0
$991$	173.635	0.175212	0.0876062	−	0.996155i	$-0.472078\pi$
$991$	173.635	0.175212	0.0876062	+	0.996155i	$0.472078\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 463.243i	− 0.465571i
$996$	0	0
$997$	818.454	0.820917	0.410458	−	0.911879i	$-0.365369\pi$
$997$	818.454	0.820917	0.410458	+	0.911879i	$0.365369\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.e.u.1025.2		8
3.2	odd	2	inner	1728.3.e.u.1025.8		8
4.3	odd	2	inner	1728.3.e.u.1025.1		8
8.3	odd	2		864.3.e.f.161.7	yes	8
8.5	even	2		864.3.e.f.161.8	yes	8
12.11	even	2	inner	1728.3.e.u.1025.7		8
24.5	odd	2		864.3.e.f.161.2	yes	8
24.11	even	2		864.3.e.f.161.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.3.e.f.161.1	✓	8	24.11	even	2
864.3.e.f.161.2	yes	8	24.5	odd	2
864.3.e.f.161.7	yes	8	8.3	odd	2
864.3.e.f.161.8	yes	8	8.5	even	2
1728.3.e.u.1025.1		8	4.3	odd	2	inner
1728.3.e.u.1025.2		8	1.1	even	1	trivial
1728.3.e.u.1025.7		8	12.11	even	2	inner
1728.3.e.u.1025.8		8	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-6.61037i q^{5} +9.43879 q^{7} +O(q^{10})\)	\(q-6.61037i q^{5} +9.43879 q^{7} +17.6969i q^{11} -20.6969 q^{13} -16.0492i q^{17} +12.2993 q^{19} -26.6969i q^{23} -18.6969 q^{25} -0.921404i q^{29} +53.7722 q^{31} -62.3939i q^{35} +64.0908 q^{37} -19.7990i q^{41} -69.7893 q^{43} -0.606123i q^{47} +40.0908 q^{49} -45.3512i q^{53} +116.983 q^{55} -71.3939i q^{59} +4.00000 q^{61} +136.814i q^{65} -7.43545 q^{67} -27.3031i q^{71} +6.30306 q^{73} +167.038i q^{77} -42.6191 q^{79} -72.3031i q^{83} -106.091 q^{85} -126.486i q^{89} -195.354 q^{91} -81.3031i q^{95} +27.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 48 q^{13} - 32 q^{25} + 160 q^{37} - 32 q^{49} + 32 q^{61} + 168 q^{73} - 496 q^{85} + 216 q^{97}+O(q^{100}) \) 8 * q - 48 * q^13 - 32 * q^25 + 160 * q^37 - 32 * q^49 + 32 * q^61 + 168 * q^73 - 496 * q^85 + 216 * q^97

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