LMFDB - Embedded newform 1728.4.f.g.863.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,4,Mod(863,1728)]

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([1, 1, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1728.863");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.f (of order $2$, degree $1$, 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:	$101.955300490$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.58594980096.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 21x^{6} + 341x^{4} - 2100x^{2} + 10000 $ x^8 - 21x^6 + 341x^4 - 2100*x^2 + 10000
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			863.1
Root			$2.33962 - 1.35078i$ of defining polynomial
Character	$\chi$	$=$	1728.863
Dual form			1728.4.f.g.863.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-12.8226 q^{5} -17.2094i q^{7} +O(q^{10})$	$q-12.8226 q^{5} -17.2094i q^{7} -28.7466i q^{11} -33.2716i q^{13} -5.37188i q^{17} +50.9548 q^{19} +210.628 q^{23} +39.4187 q^{25} +83.1384 q^{29} -40.4187i q^{31} +220.669i q^{35} -264.078i q^{37} +399.769i q^{41} +208.008 q^{43} +178.397 q^{47} +46.8375 q^{49} -82.3049 q^{53} +368.606i q^{55} -554.056i q^{59} -349.230i q^{61} +426.628i q^{65} -664.263 q^{67} +399.769 q^{71} -802.212 q^{73} -494.712 q^{77} -322.653i q^{79} +148.502i q^{83} +68.8814i q^{85} +826.397i q^{89} -572.584 q^{91} -653.372 q^{95} +1115.47 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 1224 q^{23} + 8 q^{25} - 1800 q^{47} - 240 q^{49} + 432 q^{71} + 344 q^{73} - 5688 q^{95} + 1240 q^{97}+O(q^{100}) $ 8 * q + 1224 * q^23 + 8 * q^25 - 1800 * q^47 - 240 * q^49 + 432 * q^71 + 344 * q^73 - 5688 * q^95 + 1240 * 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
$5$	−12.8226	−1.14689	−0.573444	−	0.819245i	$-0.694393\pi$
$5$	−12.8226	−1.14689	−0.573444	+	0.819245i	$0.694393\pi$
$6$	0	0
$7$	− 17.2094i	− 0.929219i	−0.885516	−	0.464609i	$-0.846195\pi$
$7$	− 17.2094i	− 0.929219i	0.885516	−	0.464609i	$-0.153805\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 28.7466i	− 0.787949i	−0.919121	−	0.393974i	$-0.871100\pi$
$11$	− 28.7466i	− 0.787949i	0.919121	−	0.393974i	$-0.128900\pi$
$12$	0	0
$13$	− 33.2716i	− 0.709837i	−0.934897	−	0.354919i	$-0.884509\pi$
$13$	− 33.2716i	− 0.709837i	0.934897	−	0.354919i	$-0.115491\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 5.37188i	− 0.0766396i	−0.999266	−	0.0383198i	$-0.987799\pi$
$17$	− 5.37188i	− 0.0766396i	0.999266	−	0.0383198i	$-0.0122006\pi$
$18$	0	0
$19$	50.9548	0.615254	0.307627	−	0.951507i	$-0.400465\pi$
$19$	50.9548	0.615254	0.307627	+	0.951507i	$0.400465\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	210.628	1.90952	0.954761	−	0.297375i	$-0.0961111\pi$
$23$	210.628	1.90952	0.954761	+	0.297375i	$0.0961111\pi$
$24$	0	0
$25$	39.4187	0.315350
$26$	0	0
$27$	0	0
$28$	0	0
$29$	83.1384	0.532359	0.266180	−	0.963923i	$-0.414239\pi$
$29$	83.1384	0.532359	0.266180	+	0.963923i	$0.414239\pi$
$30$	0	0
$31$	− 40.4187i	− 0.234175i	−0.993122	−	0.117087i	$-0.962644\pi$
$31$	− 40.4187i	− 0.234175i	0.993122	−	0.117087i	$-0.0373558\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	220.669i	1.06571i
$36$	0	0
$37$	− 264.078i	− 1.17336i	−0.809820	−	0.586678i	$-0.800435\pi$
$37$	− 264.078i	− 1.17336i	0.809820	−	0.586678i	$-0.199565\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	399.769i	1.52277i	0.648303	+	0.761383i	$0.275479\pi$
$41$	399.769i	1.52277i	−0.648303	+	0.761383i	$0.724521\pi$
$42$	0	0
$43$	208.008	0.737697	0.368849	−	0.929489i	$-0.379752\pi$
$43$	208.008	0.737697	0.368849	+	0.929489i	$0.379752\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	178.397	0.553656	0.276828	−	0.960919i	$-0.410717\pi$
$47$	178.397	0.553656	0.276828	+	0.960919i	$0.410717\pi$
$48$	0	0
$49$	46.8375	0.136552
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−82.3049	−0.213310	−0.106655	−	0.994296i	$-0.534014\pi$
$53$	−82.3049	−0.213310	−0.106655	+	0.994296i	$0.534014\pi$
$54$	0	0
$55$	368.606i	0.903688i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 554.056i	− 1.22258i	−0.791409	−	0.611288i	$-0.790652\pi$
$59$	− 554.056i	− 1.22258i	0.791409	−	0.611288i	$-0.209348\pi$
$60$	0	0
$61$	− 349.230i	− 0.733022i	−0.930414	−	0.366511i	$-0.880552\pi$
$61$	− 349.230i	− 0.733022i	0.930414	−	0.366511i	$-0.119448\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	426.628i	0.814103i
$66$	0	0
$67$	−664.263	−1.21123	−0.605617	−	0.795756i	$-0.707074\pi$
$67$	−664.263	−1.21123	−0.605617	+	0.795756i	$0.707074\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	399.769	0.668223	0.334111	−	0.942534i	$-0.391564\pi$
$71$	399.769	0.668223	0.334111	+	0.942534i	$0.391564\pi$
$72$	0	0
$73$	−802.212	−1.28619	−0.643095	−	0.765786i	$-0.722350\pi$
$73$	−802.212	−1.28619	−0.643095	+	0.765786i	$0.722350\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−494.712	−0.732177
$78$	0	0
$79$	− 322.653i	− 0.459510i	−0.973248	−	0.229755i	$-0.926207\pi$
$79$	− 322.653i	− 0.459510i	0.973248	−	0.229755i	$-0.0737925\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	148.502i	0.196388i	0.995167	+	0.0981938i	$0.0313065\pi$
$83$	148.502i	0.196388i	−0.995167	+	0.0981938i	$0.968693\pi$
$84$	0	0
$85$	68.8814i	0.0878969i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	826.397i	0.984246i	0.870526	+	0.492123i	$0.163779\pi$
$89$	826.397i	0.984246i	−0.870526	+	0.492123i	$0.836221\pi$
$90$	0	0
$91$	−572.584	−0.659594
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−653.372	−0.705627
$96$	0	0
$97$	1115.47	1.16762	0.583808	−	0.811892i	$-0.301562\pi$
$97$	1115.47	1.16762	0.583808	+	0.811892i	$0.301562\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	998.495	0.983702	0.491851	−	0.870679i	$-0.336320\pi$
$101$	998.495	0.983702	0.491851	+	0.870679i	$0.336320\pi$
$102$	0	0
$103$	− 664.653i	− 0.635827i	−0.948120	−	0.317914i	$-0.897018\pi$
$103$	− 664.653i	− 0.635827i	0.948120	−	0.317914i	$-0.102982\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1533.94i	1.38590i	0.720984	+	0.692952i	$0.243690\pi$
$107$	1533.94i	1.38590i	−0.720984	+	0.692952i	$0.756310\pi$
$108$	0	0
$109$	− 350.838i	− 0.308295i	−0.988048	−	0.154148i	$-0.950737\pi$
$109$	− 350.838i	− 0.308295i	0.988048	−	0.154148i	$-0.0492631\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 2036.45i	− 1.69533i	−0.530528	−	0.847667i	$-0.678006\pi$
$113$	− 2036.45i	− 1.69533i	0.530528	−	0.847667i	$-0.321994\pi$
$114$	0	0
$115$	−2700.80	−2.19001
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−92.4467	−0.0712149
$120$	0	0
$121$	504.631	0.379137
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1097.37	0.785216
$126$	0	0
$127$	930.237i	0.649962i	0.945720	+	0.324981i	$0.105358\pi$
$127$	930.237i	0.649962i	−0.945720	+	0.324981i	$0.894642\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2161.23i	− 1.44143i	−0.693230	−	0.720716i	$-0.743813\pi$
$131$	− 2161.23i	− 1.44143i	0.693230	−	0.720716i	$-0.256187\pi$
$132$	0	0
$133$	− 876.899i	− 0.571705i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1333.60i	− 0.831660i	−0.909442	−	0.415830i	$-0.863491\pi$
$137$	− 1333.60i	− 0.831660i	0.909442	−	0.415830i	$-0.136509\pi$
$138$	0	0
$139$	−896.434	−0.547011	−0.273505	−	0.961870i	$-0.588183\pi$
$139$	−896.434	−0.547011	−0.273505	+	0.961870i	$0.588183\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−956.447	−0.559315
$144$	0	0
$145$	−1066.05	−0.610556
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3510.05	−1.92990	−0.964949	−	0.262436i	$-0.915474\pi$
$149$	−3510.05	−1.92990	−0.964949	+	0.262436i	$0.915474\pi$
$150$	0	0
$151$	− 57.2593i	− 0.0308589i	−0.999881	−	0.0154294i	$-0.995088\pi$
$151$	− 57.2593i	− 0.0308589i	0.999881	−	0.0154294i	$-0.00491154\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	518.273i	0.268572i
$156$	0	0
$157$	− 3194.95i	− 1.62411i	−0.583583	−	0.812054i	$-0.698350\pi$
$157$	− 3194.95i	− 1.62411i	0.583583	−	0.812054i	$-0.301650\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 3624.78i	− 1.77436i
$162$	0	0
$163$	−3329.21	−1.59978	−0.799888	−	0.600149i	$-0.795108\pi$
$163$	−3329.21	−1.59978	−0.799888	+	0.600149i	$0.795108\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−545.934	−0.252968	−0.126484	−	0.991969i	$-0.540369\pi$
$167$	−545.934	−0.252968	−0.126484	+	0.991969i	$0.540369\pi$
$168$	0	0
$169$	1090.00	0.496131
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−565.396	−0.248475	−0.124238	−	0.992252i	$-0.539649\pi$
$173$	−565.396	−0.248475	−0.124238	+	0.992252i	$0.539649\pi$
$174$	0	0
$175$	− 678.372i	− 0.293029i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	101.379i	0.0423319i	0.999776	+	0.0211660i	$0.00673784\pi$
$179$	101.379i	0.0423319i	−0.999776	+	0.0211660i	$0.993262\pi$
$180$	0	0
$181$	− 2627.48i	− 1.07900i	−0.841985	−	0.539501i	$-0.818613\pi$
$181$	− 2627.48i	− 1.07900i	0.841985	−	0.539501i	$-0.181387\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3386.17i	1.34571i
$186$	0	0
$187$	−154.424	−0.0603881
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2306.17	−0.873656	−0.436828	−	0.899545i	$-0.643898\pi$
$191$	−2306.17	−0.873656	−0.436828	+	0.899545i	$0.643898\pi$
$192$	0	0
$193$	−2712.68	−1.01173	−0.505863	−	0.862614i	$-0.668826\pi$
$193$	−2712.68	−1.01173	−0.505863	+	0.862614i	$0.668826\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4816.24	1.74184	0.870922	−	0.491422i	$-0.163523\pi$
$197$	4816.24	1.74184	0.870922	+	0.491422i	$0.163523\pi$
$198$	0	0
$199$	82.2343i	0.0292936i	0.999893	+	0.0146468i	$0.00466240\pi$
$199$	82.2343i	0.0292936i	−0.999893	+	0.0146468i	$0.995338\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1430.76i	− 0.494678i
$204$	0	0
$205$	− 5126.07i	− 1.74644i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 1464.78i	− 0.484789i
$210$	0	0
$211$	−1687.94	−0.550722	−0.275361	−	0.961341i	$-0.588797\pi$
$211$	−1687.94	−0.550722	−0.275361	+	0.961341i	$0.588797\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2667.21	−0.846056
$216$	0	0
$217$	−695.581	−0.217600
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−178.731	−0.0544016
$222$	0	0
$223$	3379.31i	1.01478i	0.861718	+	0.507388i	$0.169389\pi$
$223$	3379.31i	1.01478i	−0.861718	+	0.507388i	$0.830611\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4987.10i	1.45818i	0.684420	+	0.729088i	$0.260055\pi$
$227$	4987.10i	1.45818i	−0.684420	+	0.729088i	$0.739945\pi$
$228$	0	0
$229$	− 5767.34i	− 1.66426i	−0.554577	−	0.832132i	$-0.687120\pi$
$229$	− 5767.34i	− 1.66426i	0.554577	−	0.832132i	$-0.312880\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3144.43i	0.884114i	0.896987	+	0.442057i	$0.145751\pi$
$233$	3144.43i	0.884114i	−0.896987	+	0.442057i	$0.854249\pi$
$234$	0	0
$235$	−2287.51	−0.634981
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3305.59	0.894647	0.447324	−	0.894372i	$-0.352377\pi$
$239$	3305.59	0.894647	0.447324	+	0.894372i	$0.352377\pi$
$240$	0	0
$241$	1831.69	0.489584	0.244792	−	0.969576i	$-0.421280\pi$
$241$	1831.69	0.489584	0.244792	+	0.969576i	$0.421280\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−600.578	−0.156610
$246$	0	0
$247$	− 1695.35i	− 0.436730i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 1414.56i	− 0.355721i	−0.984056	−	0.177860i	$-0.943082\pi$
$251$	− 1414.56i	− 0.355721i	0.984056	−	0.177860i	$-0.0569176\pi$
$252$	0	0
$253$	− 6054.85i	− 1.50461i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1328.23i	− 0.322384i	−0.986923	−	0.161192i	$-0.948466\pi$
$257$	− 1328.23i	− 0.322384i	0.986923	−	0.161192i	$-0.0515339\pi$
$258$	0	0
$259$	−4544.62	−1.09030
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2183.74	0.511996	0.255998	−	0.966677i	$-0.417596\pi$
$263$	2183.74	0.511996	0.255998	+	0.966677i	$0.417596\pi$
$264$	0	0
$265$	1055.36	0.244643
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3062.47	−0.694133	−0.347067	−	0.937840i	$-0.612822\pi$
$269$	−3062.47	−0.694133	−0.347067	+	0.937840i	$0.612822\pi$
$270$	0	0
$271$	− 6305.23i	− 1.41334i	−0.707543	−	0.706670i	$-0.750196\pi$
$271$	− 6305.23i	− 1.41334i	0.707543	−	0.706670i	$-0.249804\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1133.16i	− 0.248480i
$276$	0	0
$277$	− 445.733i	− 0.0966841i	−0.998831	−	0.0483420i	$-0.984606\pi$
$277$	− 445.733i	− 0.0966841i	0.998831	−	0.0483420i	$-0.0153937\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7864.20i	1.66953i	0.550604	+	0.834767i	$0.314397\pi$
$281$	7864.20i	1.66953i	−0.550604	+	0.834767i	$0.685603\pi$
$282$	0	0
$283$	−7392.73	−1.55283	−0.776417	−	0.630219i	$-0.782965\pi$
$283$	−7392.73	−1.55283	−0.776417	+	0.630219i	$0.782965\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6879.77	1.41498
$288$	0	0
$289$	4884.14	0.994126
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2427.07	−0.483929	−0.241965	−	0.970285i	$-0.577792\pi$
$293$	−2427.07	−0.483929	−0.241965	+	0.970285i	$0.577792\pi$
$294$	0	0
$295$	7104.43i	1.40216i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 7007.94i	− 1.35545i
$300$	0	0
$301$	− 3579.69i	− 0.685482i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4478.03i	0.840693i
$306$	0	0
$307$	−3728.66	−0.693179	−0.346590	−	0.938017i	$-0.612660\pi$
$307$	−3728.66	−0.693179	−0.346590	+	0.938017i	$0.612660\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1560.35	−0.284499	−0.142249	−	0.989831i	$-0.545434\pi$
$311$	−1560.35	−0.284499	−0.142249	+	0.989831i	$0.545434\pi$
$312$	0	0
$313$	2962.43	0.534973	0.267486	−	0.963562i	$-0.413807\pi$
$313$	2962.43	0.534973	0.267486	+	0.963562i	$0.413807\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2345.65	0.415599	0.207800	−	0.978171i	$-0.433370\pi$
$317$	2345.65	0.415599	0.207800	+	0.978171i	$0.433370\pi$
$318$	0	0
$319$	− 2389.95i	− 0.419472i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 273.723i	− 0.0471528i
$324$	0	0
$325$	− 1311.53i	− 0.223847i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 3070.10i	− 0.514468i
$330$	0	0
$331$	−6705.67	−1.11353	−0.556763	−	0.830672i	$-0.687957\pi$
$331$	−6705.67	−1.11353	−0.556763	+	0.830672i	$0.687957\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8517.57	1.38915
$336$	0	0
$337$	−12348.8	−1.99608	−0.998041	−	0.0625651i	$-0.980072\pi$
$337$	−12348.8	−1.99608	−0.998041	+	0.0625651i	$0.980072\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1161.90	−0.184518
$342$	0	0
$343$	− 6708.86i	− 1.05611i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10627.8i	1.64419i	0.569354	+	0.822093i	$0.307194\pi$
$347$	10627.8i	1.64419i	−0.569354	+	0.822093i	$0.692806\pi$
$348$	0	0
$349$	− 5775.38i	− 0.885813i	−0.896568	−	0.442907i	$-0.853947\pi$
$349$	− 5775.38i	− 0.885813i	0.896568	−	0.442907i	$-0.146053\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 4470.41i	− 0.674040i	−0.941498	−	0.337020i	$-0.890581\pi$
$353$	− 4470.41i	− 0.674040i	0.941498	−	0.337020i	$-0.109419\pi$
$354$	0	0
$355$	−5126.07	−0.766376
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6357.57	0.934651	0.467326	−	0.884085i	$-0.345218\pi$
$359$	6357.57	0.934651	0.467326	+	0.884085i	$0.345218\pi$
$360$	0	0
$361$	−4262.61	−0.621463
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10286.4	1.47511
$366$	0	0
$367$	10116.1i	1.43884i	0.694576	+	0.719419i	$0.255592\pi$
$367$	10116.1i	1.43884i	−0.694576	+	0.719419i	$0.744408\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1416.42i	0.198212i
$372$	0	0
$373$	11393.6i	1.58160i	0.612076	+	0.790799i	$0.290335\pi$
$373$	11393.6i	1.58160i	−0.612076	+	0.790799i	$0.709665\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2766.15i	− 0.377888i
$378$	0	0
$379$	−7733.63	−1.04815	−0.524076	−	0.851671i	$-0.675589\pi$
$379$	−7733.63	−1.04815	−0.524076	+	0.851671i	$0.675589\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4461.92	−0.595283	−0.297641	−	0.954678i	$-0.596200\pi$
$383$	−4461.92	−0.595283	−0.297641	+	0.954678i	$0.596200\pi$
$384$	0	0
$385$	6343.48	0.839724
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1915.57	−0.249674	−0.124837	−	0.992177i	$-0.539841\pi$
$389$	−1915.57	−0.249674	−0.124837	+	0.992177i	$0.539841\pi$
$390$	0	0
$391$	− 1131.47i	− 0.146345i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4137.25i	0.527006i
$396$	0	0
$397$	− 6336.81i	− 0.801096i	−0.916276	−	0.400548i	$-0.868820\pi$
$397$	− 6336.81i	− 0.801096i	0.916276	−	0.400548i	$-0.131180\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7255.80i	0.903584i	0.892123	+	0.451792i	$0.149215\pi$
$401$	7255.80i	0.903584i	−0.892123	+	0.451792i	$0.850785\pi$
$402$	0	0
$403$	−1344.80	−0.166226
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7591.36	−0.924545
$408$	0	0
$409$	−1825.12	−0.220651	−0.110326	−	0.993895i	$-0.535189\pi$
$409$	−1825.12	−0.220651	−0.110326	+	0.993895i	$0.535189\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−9534.96	−1.13604
$414$	0	0
$415$	− 1904.18i	− 0.225234i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 6707.50i	− 0.782059i	−0.920378	−	0.391030i	$-0.872119\pi$
$419$	− 6707.50i	− 0.782059i	0.920378	−	0.391030i	$-0.127881\pi$
$420$	0	0
$421$	− 5228.42i	− 0.605267i	−0.953107	−	0.302633i	$-0.902134\pi$
$421$	− 5228.42i	− 0.605267i	0.953107	−	0.302633i	$-0.0978658\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 211.753i	− 0.0241683i
$426$	0	0
$427$	−6010.03	−0.681138
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−14973.8	−1.67347	−0.836733	−	0.547610i	$-0.815538\pi$
$431$	−14973.8	−1.67347	−0.836733	+	0.547610i	$0.815538\pi$
$432$	0	0
$433$	5512.13	0.611769	0.305885	−	0.952069i	$-0.401048\pi$
$433$	5512.13	0.611769	0.305885	+	0.952069i	$0.401048\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10732.5	1.17484
$438$	0	0
$439$	10803.7i	1.17456i	0.809384	+	0.587280i	$0.199801\pi$
$439$	10803.7i	1.17456i	−0.809384	+	0.587280i	$0.800199\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 10661.8i	− 1.14347i	−0.820437	−	0.571736i	$-0.806270\pi$
$443$	− 10661.8i	− 1.14347i	0.820437	−	0.571736i	$-0.193730\pi$
$444$	0	0
$445$	− 10596.5i	− 1.12882i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 1971.98i	− 0.207269i	−0.994615	−	0.103634i	$-0.966953\pi$
$449$	− 1971.98i	− 0.207269i	0.994615	−	0.103634i	$-0.0330472\pi$
$450$	0	0
$451$	11492.0	1.19986
$452$	0	0
$453$	0	0
$454$	0	0
$455$	7342.00	0.756480
$456$	0	0
$457$	5583.23	0.571494	0.285747	−	0.958305i	$-0.407758\pi$
$457$	5583.23	0.571494	0.285747	+	0.958305i	$0.407758\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−8969.56	−0.906191	−0.453095	−	0.891462i	$-0.649680\pi$
$461$	−8969.56	−0.906191	−0.453095	+	0.891462i	$0.649680\pi$
$462$	0	0
$463$	− 11521.2i	− 1.15645i	−0.815879	−	0.578223i	$-0.803746\pi$
$463$	− 11521.2i	− 1.15645i	0.815879	−	0.578223i	$-0.196254\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 12869.1i	− 1.27518i	−0.770374	−	0.637592i	$-0.779930\pi$
$467$	− 12869.1i	− 1.27518i	0.770374	−	0.637592i	$-0.220070\pi$
$468$	0	0
$469$	11431.6i	1.12550i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 5979.54i	− 0.581268i
$474$	0	0
$475$	2008.57	0.194020
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−13860.7	−1.32216	−0.661078	−	0.750317i	$-0.729901\pi$
$479$	−13860.7	−1.32216	−0.661078	+	0.750317i	$0.729901\pi$
$480$	0	0
$481$	−8786.31	−0.832892
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−14303.2	−1.33912
$486$	0	0
$487$	− 2514.65i	− 0.233983i	−0.993133	−	0.116992i	$-0.962675\pi$
$487$	− 2514.65i	− 0.233983i	0.993133	−	0.116992i	$-0.0373251\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18593.4i	1.70897i	0.519473	+	0.854487i	$0.326128\pi$
$491$	18593.4i	1.70897i	−0.519473	+	0.854487i	$0.673872\pi$
$492$	0	0
$493$	− 446.610i	− 0.0407998i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6879.77i	− 0.620925i
$498$	0	0
$499$	16174.6	1.45105	0.725525	−	0.688196i	$-0.241597\pi$
$499$	16174.6	1.45105	0.725525	+	0.688196i	$0.241597\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12844.3	−1.13857	−0.569284	−	0.822141i	$-0.692780\pi$
$503$	−12844.3	−1.13857	−0.569284	+	0.822141i	$0.692780\pi$
$504$	0	0
$505$	−12803.3	−1.12820
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6251.37	0.544375	0.272188	−	0.962244i	$-0.412253\pi$
$509$	6251.37	0.544375	0.272188	+	0.962244i	$0.412253\pi$
$510$	0	0
$511$	13805.6i	1.19515i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8522.57i	0.729222i
$516$	0	0
$517$	− 5128.31i	− 0.436253i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 16125.0i	− 1.35595i	−0.735084	−	0.677976i	$-0.762857\pi$
$521$	− 16125.0i	− 1.35595i	0.735084	−	0.677976i	$-0.237143\pi$
$522$	0	0
$523$	−1658.32	−0.138649	−0.0693243	−	0.997594i	$-0.522084\pi$
$523$	−1658.32	−0.138649	−0.0693243	+	0.997594i	$0.522084\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−217.125	−0.0179471
$528$	0	0
$529$	32197.2	2.64627
$530$	0	0
$531$	0	0
$532$	0	0
$533$	13300.9	1.08092
$534$	0	0
$535$	− 19669.1i	− 1.58948i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1346.42i	− 0.107596i
$540$	0	0
$541$	− 13148.3i	− 1.04490i	−0.852670	−	0.522450i	$-0.825018\pi$
$541$	− 13148.3i	− 1.04490i	0.852670	−	0.522450i	$-0.174982\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4498.65i	0.353580i
$546$	0	0
$547$	11717.8	0.915939	0.457970	−	0.888968i	$-0.348577\pi$
$547$	11717.8	0.915939	0.457970	+	0.888968i	$0.348577\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4236.30	0.327536
$552$	0	0
$553$	−5552.66	−0.426986
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9502.18	0.722837	0.361418	−	0.932404i	$-0.382293\pi$
$557$	9502.18	0.722837	0.361418	+	0.932404i	$0.382293\pi$
$558$	0	0
$559$	− 6920.78i	− 0.523645i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14447.8i	1.08153i	0.841172	+	0.540767i	$0.181866\pi$
$563$	14447.8i	1.08153i	−0.841172	+	0.540767i	$0.818134\pi$
$564$	0	0
$565$	26112.5i	1.94436i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 9621.31i	− 0.708869i	−0.935081	−	0.354434i	$-0.884673\pi$
$569$	− 9621.31i	− 0.708869i	0.935081	−	0.354434i	$-0.115327\pi$
$570$	0	0
$571$	19427.4	1.42384	0.711919	−	0.702261i	$-0.247826\pi$
$571$	19427.4	1.42384	0.711919	+	0.702261i	$0.247826\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8302.70	0.602168
$576$	0	0
$577$	−18385.3	−1.32650	−0.663249	−	0.748399i	$-0.730823\pi$
$577$	−18385.3	−1.32650	−0.663249	+	0.748399i	$0.730823\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2555.62	0.182487
$582$	0	0
$583$	2365.99i	0.168078i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 17584.3i	− 1.23642i	−0.786011	−	0.618212i	$-0.787857\pi$
$587$	− 17584.3i	− 1.23642i	0.786011	−	0.618212i	$-0.212143\pi$
$588$	0	0
$589$	− 2059.53i	− 0.144077i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	10005.0i	0.692841i	0.938079	+	0.346420i	$0.112603\pi$
$593$	10005.0i	0.692841i	−0.938079	+	0.346420i	$0.887397\pi$
$594$	0	0
$595$	1185.41	0.0816755
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−11002.4	−0.750493	−0.375246	−	0.926925i	$-0.622442\pi$
$599$	−11002.4	−0.750493	−0.375246	+	0.926925i	$0.622442\pi$
$600$	0	0
$601$	17683.9	1.20024	0.600118	−	0.799912i	$-0.295120\pi$
$601$	17683.9	1.20024	0.600118	+	0.799912i	$0.295120\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6470.68	−0.434827
$606$	0	0
$607$	237.714i	0.0158954i	0.999968	+	0.00794772i	$0.00252986\pi$
$607$	237.714i	0.0158954i	−0.999968	+	0.00794772i	$0.997470\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 5935.55i	− 0.393006i
$612$	0	0
$613$	22429.5i	1.47784i	0.673790	+	0.738922i	$0.264665\pi$
$613$	22429.5i	1.47784i	−0.673790	+	0.738922i	$0.735335\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 16485.8i	− 1.07568i	−0.843047	−	0.537840i	$-0.819240\pi$
$617$	− 16485.8i	− 1.07568i	0.843047	−	0.537840i	$-0.180760\pi$
$618$	0	0
$619$	5726.03	0.371807	0.185904	−	0.982568i	$-0.440479\pi$
$619$	5726.03	0.371807	0.185904	+	0.982568i	$0.440479\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14221.8	0.914580
$624$	0	0
$625$	−18998.5	−1.21590
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1418.60	−0.0899255
$630$	0	0
$631$	18543.9i	1.16992i	0.811060	+	0.584962i	$0.198891\pi$
$631$	18543.9i	1.16992i	−0.811060	+	0.584962i	$0.801109\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 11928.0i	− 0.745433i
$636$	0	0
$637$	− 1558.36i	− 0.0969300i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24637.0i	1.51810i	0.651032	+	0.759050i	$0.274336\pi$
$641$	24637.0i	1.51810i	−0.651032	+	0.759050i	$0.725664\pi$
$642$	0	0
$643$	−8950.99	−0.548977	−0.274489	−	0.961590i	$-0.588509\pi$
$643$	−8950.99	−0.548977	−0.274489	+	0.961590i	$0.588509\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	30126.3	1.83058	0.915292	−	0.402791i	$-0.131960\pi$
$647$	30126.3	1.83058	0.915292	+	0.402791i	$0.131960\pi$
$648$	0	0
$649$	−15927.2	−0.963327
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10405.4	−0.623578	−0.311789	−	0.950151i	$-0.600928\pi$
$653$	−10405.4	−0.623578	−0.311789	+	0.950151i	$0.600928\pi$
$654$	0	0
$655$	27712.6i	1.65316i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	23235.2i	1.37347i	0.726910	+	0.686733i	$0.240956\pi$
$659$	23235.2i	1.37347i	−0.726910	+	0.686733i	$0.759044\pi$
$660$	0	0
$661$	29301.5i	1.72420i	0.506739	+	0.862099i	$0.330851\pi$
$661$	29301.5i	1.72420i	−0.506739	+	0.862099i	$0.669149\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	11244.1i	0.655682i
$666$	0	0
$667$	17511.3	1.01655
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10039.2	−0.577584
$672$	0	0
$673$	19148.5	1.09676	0.548380	−	0.836229i	$-0.315245\pi$
$673$	19148.5	1.09676	0.548380	+	0.836229i	$0.315245\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−29994.5	−1.70278	−0.851391	−	0.524532i	$-0.824240\pi$
$677$	−29994.5	−1.70278	−0.851391	+	0.524532i	$0.824240\pi$
$678$	0	0
$679$	− 19196.5i	− 1.08497i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 5269.84i	− 0.295234i	−0.989045	−	0.147617i	$-0.952840\pi$
$683$	− 5269.84i	− 0.295234i	0.989045	−	0.147617i	$-0.0471603\pi$
$684$	0	0
$685$	17100.2i	0.953820i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2738.42i	0.151416i
$690$	0	0
$691$	20435.8	1.12506	0.562528	−	0.826778i	$-0.309829\pi$
$691$	20435.8	1.12506	0.562528	+	0.826778i	$0.309829\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11494.6	0.627360
$696$	0	0
$697$	2147.51	0.116704
$698$	0	0
$699$	0	0
$700$	0	0
$701$	12475.4	0.672165	0.336082	−	0.941833i	$-0.390898\pi$
$701$	12475.4	0.672165	0.336082	+	0.941833i	$0.390898\pi$
$702$	0	0
$703$	− 13456.0i	− 0.721912i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 17183.5i	− 0.914075i
$708$	0	0
$709$	25405.1i	1.34571i	0.739773	+	0.672856i	$0.234933\pi$
$709$	25405.1i	1.34571i	−0.739773	+	0.672856i	$0.765067\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 8513.32i	− 0.447162i
$714$	0	0
$715$	12264.1	0.641472
$716$	0	0
$717$	0	0
$718$	0	0
$719$	4498.65	0.233340	0.116670	−	0.993171i	$-0.462778\pi$
$719$	4498.65	0.233340	0.116670	+	0.993171i	$0.462778\pi$
$720$	0	0
$721$	−11438.3	−0.590823
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3277.21	0.167879
$726$	0	0
$727$	25698.4i	1.31101i	0.755193	+	0.655503i	$0.227543\pi$
$727$	25698.4i	1.31101i	−0.755193	+	0.655503i	$0.772457\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1117.40i	− 0.0565368i
$732$	0	0
$733$	16107.3i	0.811644i	0.913952	+	0.405822i	$0.133015\pi$
$733$	16107.3i	0.811644i	−0.913952	+	0.405822i	$0.866985\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	19095.3i	0.954390i
$738$	0	0
$739$	−799.199	−0.0397822	−0.0198911	−	0.999802i	$-0.506332\pi$
$739$	−799.199	−0.0397822	−0.0198911	+	0.999802i	$0.506332\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17026.9	0.840722	0.420361	−	0.907357i	$-0.361903\pi$
$743$	17026.9	0.840722	0.420361	+	0.907357i	$0.361903\pi$
$744$	0	0
$745$	45008.0	2.21338
$746$	0	0
$747$	0	0
$748$	0	0
$749$	26398.2	1.28781
$750$	0	0
$751$	− 32026.8i	− 1.55616i	−0.628167	−	0.778078i	$-0.716195\pi$
$751$	− 32026.8i	− 1.55616i	0.628167	−	0.778078i	$-0.283805\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	734.212i	0.0353917i
$756$	0	0
$757$	39862.3i	1.91390i	0.290261	+	0.956948i	$0.406258\pi$
$757$	39862.3i	1.91390i	−0.290261	+	0.956948i	$0.593742\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22217.8i	1.05834i	0.848517	+	0.529168i	$0.177496\pi$
$761$	22217.8i	1.05834i	−0.848517	+	0.529168i	$0.822504\pi$
$762$	0	0
$763$	−6037.70	−0.286474
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−18434.3	−0.867829
$768$	0	0
$769$	−26842.6	−1.25874	−0.629369	−	0.777106i	$-0.716687\pi$
$769$	−26842.6	−1.25874	−0.629369	+	0.777106i	$0.716687\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	36697.0	1.70750	0.853752	−	0.520680i	$-0.174321\pi$
$773$	36697.0	1.70750	0.853752	+	0.520680i	$0.174321\pi$
$774$	0	0
$775$	− 1593.26i	− 0.0738470i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	20370.1i	0.936887i
$780$	0	0
$781$	− 11492.0i	− 0.526525i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	40967.5i	1.86267i
$786$	0	0
$787$	8335.44	0.377543	0.188772	−	0.982021i	$-0.439549\pi$
$787$	8335.44	0.377543	0.188772	+	0.982021i	$0.439549\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−35046.0	−1.57534
$792$	0	0
$793$	−11619.4	−0.520326
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−35046.2	−1.55759	−0.778796	−	0.627277i	$-0.784169\pi$
$797$	−35046.2	−1.55759	−0.778796	+	0.627277i	$0.784169\pi$
$798$	0	0
$799$	− 958.327i	− 0.0424320i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	23060.9i	1.01345i
$804$	0	0
$805$	46479.0i	2.03499i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6462.51i	0.280853i	0.990091	+	0.140426i	$0.0448473\pi$
$809$	6462.51i	0.280853i	−0.990091	+	0.140426i	$0.955153\pi$
$810$	0	0
$811$	42136.5	1.82443	0.912214	−	0.409713i	$-0.134371\pi$
$811$	42136.5	1.82443	0.912214	+	0.409713i	$0.134371\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	42689.0	1.83476
$816$	0	0
$817$	10599.0	0.453871
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5323.70	0.226307	0.113154	−	0.993577i	$-0.463905\pi$
$821$	5323.70	0.226307	0.113154	+	0.993577i	$0.463905\pi$
$822$	0	0
$823$	− 12239.0i	− 0.518379i	−0.965826	−	0.259190i	$-0.916545\pi$
$823$	− 12239.0i	− 0.518379i	0.965826	−	0.259190i	$-0.0834555\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33925.6i	1.42649i	0.700915	+	0.713245i	$0.252775\pi$
$827$	33925.6i	1.42649i	−0.700915	+	0.713245i	$0.747225\pi$
$828$	0	0
$829$	− 28954.3i	− 1.21306i	−0.795061	−	0.606529i	$-0.792561\pi$
$829$	− 28954.3i	− 1.21306i	0.795061	−	0.606529i	$-0.207439\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 251.605i	− 0.0104653i
$834$	0	0
$835$	7000.29	0.290126
$836$	0	0
$837$	0	0
$838$	0	0
$839$	7786.74	0.320415	0.160208	−	0.987083i	$-0.448784\pi$
$839$	7786.74	0.320415	0.160208	+	0.987083i	$0.448784\pi$
$840$	0	0
$841$	−17477.0	−0.716594
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−13976.6	−0.569006
$846$	0	0
$847$	− 8684.39i	− 0.352301i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 55622.3i	− 2.24055i
$852$	0	0
$853$	39947.8i	1.60350i	0.597658	+	0.801751i	$0.296098\pi$
$853$	39947.8i	1.60350i	−0.597658	+	0.801751i	$0.703902\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 38134.2i	− 1.52000i	−0.649924	−	0.759999i	$-0.725199\pi$
$857$	− 38134.2i	− 1.52000i	0.649924	−	0.759999i	$-0.274801\pi$
$858$	0	0
$859$	29831.8	1.18492	0.592461	−	0.805599i	$-0.298156\pi$
$859$	29831.8	1.18492	0.592461	+	0.805599i	$0.298156\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4059.65	0.160130	0.0800649	−	0.996790i	$-0.474487\pi$
$863$	4059.65	0.160130	0.0800649	+	0.996790i	$0.474487\pi$
$864$	0	0
$865$	7249.84	0.284973
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−9275.19	−0.362070
$870$	0	0
$871$	22101.1i	0.859779i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 18885.1i	− 0.729638i
$876$	0	0
$877$	− 21003.1i	− 0.808692i	−0.914606	−	0.404346i	$-0.867499\pi$
$877$	− 21003.1i	− 0.808692i	0.914606	−	0.404346i	$-0.132501\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 49825.5i	− 1.90541i	−0.303898	−	0.952704i	$-0.598288\pi$
$881$	− 49825.5i	− 1.90541i	0.303898	−	0.952704i	$-0.401712\pi$
$882$	0	0
$883$	9171.08	0.349526	0.174763	−	0.984611i	$-0.444084\pi$
$883$	9171.08	0.349526	0.174763	+	0.984611i	$0.444084\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−26060.2	−0.986488	−0.493244	−	0.869891i	$-0.664189\pi$
$887$	−26060.2	−0.986488	−0.493244	+	0.869891i	$0.664189\pi$
$888$	0	0
$889$	16008.8	0.603957
$890$	0	0
$891$	0	0
$892$	0	0
$893$	9090.17	0.340639
$894$	0	0
$895$	− 1299.94i	− 0.0485499i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 3360.35i	− 0.124665i
$900$	0	0
$901$	442.132i	0.0163480i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	33691.1i	1.23749i
$906$	0	0
$907$	35645.0	1.30493	0.652467	−	0.757817i	$-0.273734\pi$
$907$	35645.0	1.30493	0.652467	+	0.757817i	$0.273734\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	10782.5	0.392141	0.196070	−	0.980590i	$-0.437182\pi$
$911$	10782.5	0.392141	0.196070	+	0.980590i	$0.437182\pi$
$912$	0	0
$913$	4268.92	0.154743
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−37193.4	−1.33941
$918$	0	0
$919$	− 20191.1i	− 0.724748i	−0.932033	−	0.362374i	$-0.881966\pi$
$919$	− 20191.1i	− 0.724748i	0.932033	−	0.362374i	$-0.118034\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 13300.9i	− 0.474329i
$924$	0	0
$925$	− 10409.6i	− 0.370018i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 37161.0i	− 1.31239i	−0.754590	−	0.656196i	$-0.772164\pi$
$929$	− 37161.0i	− 1.31239i	0.754590	−	0.656196i	$-0.227836\pi$
$930$	0	0
$931$	2386.59	0.0840144
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1980.11	0.0692583
$936$	0	0
$937$	6784.39	0.236538	0.118269	−	0.992982i	$-0.462265\pi$
$937$	6784.39	0.236538	0.118269	+	0.992982i	$0.462265\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−10527.9	−0.364718	−0.182359	−	0.983232i	$-0.558373\pi$
$941$	−10527.9	−0.364718	−0.182359	+	0.983232i	$0.558373\pi$
$942$	0	0
$943$	84202.5i	2.90775i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 20262.1i	− 0.695281i	−0.937628	−	0.347640i	$-0.886983\pi$
$947$	− 20262.1i	− 0.695281i	0.937628	−	0.347640i	$-0.113017\pi$
$948$	0	0
$949$	26690.9i	0.912985i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 21691.3i	− 0.737304i	−0.929567	−	0.368652i	$-0.879819\pi$
$953$	− 21691.3i	− 0.737304i	0.929567	−	0.368652i	$-0.120181\pi$
$954$	0	0
$955$	29571.0	1.00198
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−22950.5	−0.772794
$960$	0	0
$961$	28157.3	0.945162
$962$	0	0
$963$	0	0
$964$	0	0
$965$	34783.6	1.16034
$966$	0	0
$967$	− 22236.5i	− 0.739481i	−0.929135	−	0.369741i	$-0.879447\pi$
$967$	− 22236.5i	− 0.739481i	0.929135	−	0.369741i	$-0.120553\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	27549.5i	0.910511i	0.890361	+	0.455255i	$0.150452\pi$
$971$	27549.5i	0.910511i	−0.890361	+	0.455255i	$0.849548\pi$
$972$	0	0
$973$	15427.1i	0.508293i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 34859.6i	− 1.14151i	−0.821120	−	0.570756i	$-0.806650\pi$
$977$	− 34859.6i	− 1.14151i	0.821120	−	0.570756i	$-0.193350\pi$
$978$	0	0
$979$	23756.1	0.775535
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−52096.9	−1.69037	−0.845184	−	0.534475i	$-0.820509\pi$
$983$	−52096.9	−1.69037	−0.845184	+	0.534475i	$0.820509\pi$
$984$	0	0
$985$	−61756.7	−1.99770
$986$	0	0
$987$	0	0
$988$	0	0
$989$	43812.4	1.40865
$990$	0	0
$991$	− 53076.9i	− 1.70135i	−0.525688	−	0.850677i	$-0.676192\pi$
$991$	− 53076.9i	− 1.70135i	0.525688	−	0.850677i	$-0.323808\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1054.46i	− 0.0335965i
$996$	0	0
$997$	2814.79i	0.0894136i	0.999000	+	0.0447068i	$0.0142354\pi$
$997$	2814.79i	0.0894136i	−0.999000	+	0.0447068i	$0.985765\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.f.g.863.1	yes	8
3.2	odd	2		1728.4.f.c.863.7	yes	8
4.3	odd	2		1728.4.f.c.863.2	yes	8
8.3	odd	2		1728.4.f.c.863.8	yes	8
8.5	even	2	inner	1728.4.f.g.863.7	yes	8
12.11	even	2	inner	1728.4.f.g.863.8	yes	8
24.5	odd	2		1728.4.f.c.863.1	✓	8
24.11	even	2	inner	1728.4.f.g.863.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.4.f.c.863.1	✓	8	24.5	odd	2
1728.4.f.c.863.2	yes	8	4.3	odd	2
1728.4.f.c.863.7	yes	8	3.2	odd	2
1728.4.f.c.863.8	yes	8	8.3	odd	2
1728.4.f.g.863.1	yes	8	1.1	even	1	trivial
1728.4.f.g.863.2	yes	8	24.11	even	2	inner
1728.4.f.g.863.7	yes	8	8.5	even	2	inner
1728.4.f.g.863.8	yes	8	12.11	even	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-12.8226 q^{5} -17.2094i q^{7} +O(q^{10})\)	\(q-12.8226 q^{5} -17.2094i q^{7} -28.7466i q^{11} -33.2716i q^{13} -5.37188i q^{17} +50.9548 q^{19} +210.628 q^{23} +39.4187 q^{25} +83.1384 q^{29} -40.4187i q^{31} +220.669i q^{35} -264.078i q^{37} +399.769i q^{41} +208.008 q^{43} +178.397 q^{47} +46.8375 q^{49} -82.3049 q^{53} +368.606i q^{55} -554.056i q^{59} -349.230i q^{61} +426.628i q^{65} -664.263 q^{67} +399.769 q^{71} -802.212 q^{73} -494.712 q^{77} -322.653i q^{79} +148.502i q^{83} +68.8814i q^{85} +826.397i q^{89} -572.584 q^{91} -653.372 q^{95} +1115.47 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 1224 q^{23} + 8 q^{25} - 1800 q^{47} - 240 q^{49} + 432 q^{71} + 344 q^{73} - 5688 q^{95} + 1240 q^{97}+O(q^{100}) \) 8 * q + 1224 * q^23 + 8 * q^25 - 1800 * q^47 - 240 * q^49 + 432 * q^71 + 344 * q^73 - 5688 * q^95 + 1240 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more