LMFDB - Embedded newform 8100.2.d.l.649.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8100,2,Mod(649,8100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8100, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8100.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8100.d (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:	$64.6788256372$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1620)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.3
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	8100.649
Dual form			8100.2.d.l.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051i q^{7} +O(q^{10})$	$q+0.732051i q^{7} +1.73205 q^{11} +1.46410i q^{13} +1.26795i q^{17} -2.46410 q^{19} -3.46410i q^{23} -4.26795 q^{29} -7.92820 q^{31} +4.19615i q^{37} +0.803848 q^{41} -6.73205i q^{43} +4.73205i q^{47} +6.46410 q^{49} -10.7321i q^{53} -4.26795 q^{59} -4.00000 q^{61} -14.3923i q^{67} +0.803848 q^{71} -10.1962i q^{73} +1.26795i q^{77} -6.39230 q^{79} +9.12436i q^{83} -5.19615 q^{89} -1.07180 q^{91} -2.73205i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 4 q^{19} - 24 q^{29} - 4 q^{31} + 24 q^{41} + 12 q^{49} - 24 q^{59} - 16 q^{61} + 24 q^{71} + 16 q^{79} - 32 q^{91}+O(q^{100}) $ 4 * q + 4 * q^19 - 24 * q^29 - 4 * q^31 + 24 * q^41 + 12 * q^49 - 24 * q^59 - 16 * q^61 + 24 * q^71 + 16 * q^79 - 32 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times$.

$n$	$4051$	$6401$	$7777$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.732051i	0.276689i	0.990384	+	0.138345i	$0.0441781\pi$
$7$	0.732051i	0.276689i	−0.990384	+	0.138345i	$0.955822\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.73205	0.522233	0.261116	−	0.965307i	$-0.415909\pi$
$11$	1.73205	0.522233	0.261116	+	0.965307i	$0.415909\pi$
$12$	0	0
$13$	1.46410i	0.406069i	0.979172	+	0.203034i	$0.0650803\pi$
$13$	1.46410i	0.406069i	−0.979172	+	0.203034i	$0.934920\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.26795i	0.307523i	0.988108	+	0.153761i	$0.0491387\pi$
$17$	1.26795i	0.307523i	−0.988108	+	0.153761i	$0.950861\pi$
$18$	0	0
$19$	−2.46410	−0.565304	−0.282652	−	0.959223i	$-0.591214\pi$
$19$	−2.46410	−0.565304	−0.282652	+	0.959223i	$0.591214\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 3.46410i	− 0.722315i	−0.932505	−	0.361158i	$-0.882382\pi$
$23$	− 3.46410i	− 0.722315i	0.932505	−	0.361158i	$-0.117618\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.26795	−0.792538	−0.396269	−	0.918134i	$-0.629695\pi$
$29$	−4.26795	−0.792538	−0.396269	+	0.918134i	$0.629695\pi$
$30$	0	0
$31$	−7.92820	−1.42395	−0.711974	−	0.702206i	$-0.752198\pi$
$31$	−7.92820	−1.42395	−0.711974	+	0.702206i	$0.752198\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.19615i	0.689843i	0.938631	+	0.344922i	$0.112095\pi$
$37$	4.19615i	0.689843i	−0.938631	+	0.344922i	$0.887905\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.803848	0.125540	0.0627700	−	0.998028i	$-0.480007\pi$
$41$	0.803848	0.125540	0.0627700	+	0.998028i	$0.480007\pi$
$42$	0	0
$43$	− 6.73205i	− 1.02663i	−0.858201	−	0.513314i	$-0.828418\pi$
$43$	− 6.73205i	− 1.02663i	0.858201	−	0.513314i	$-0.171582\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.73205i	0.690241i	0.938558	+	0.345120i	$0.112162\pi$
$47$	4.73205i	0.690241i	−0.938558	+	0.345120i	$0.887838\pi$
$48$	0	0
$49$	6.46410	0.923443
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 10.7321i	− 1.47416i	−0.675805	−	0.737080i	$-0.736204\pi$
$53$	− 10.7321i	− 1.47416i	0.675805	−	0.737080i	$-0.263796\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.26795	−0.555640	−0.277820	−	0.960633i	$-0.589612\pi$
$59$	−4.26795	−0.555640	−0.277820	+	0.960633i	$0.589612\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 14.3923i	− 1.75830i	−0.476545	−	0.879150i	$-0.658111\pi$
$67$	− 14.3923i	− 1.75830i	0.476545	−	0.879150i	$-0.341889\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0.803848	0.0953992	0.0476996	−	0.998862i	$-0.484811\pi$
$71$	0.803848	0.0953992	0.0476996	+	0.998862i	$0.484811\pi$
$72$	0	0
$73$	− 10.1962i	− 1.19337i	−0.802476	−	0.596685i	$-0.796484\pi$
$73$	− 10.1962i	− 1.19337i	0.802476	−	0.596685i	$-0.203516\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.26795i	0.144496i
$78$	0	0
$79$	−6.39230	−0.719190	−0.359595	−	0.933108i	$-0.617085\pi$
$79$	−6.39230	−0.719190	−0.359595	+	0.933108i	$0.617085\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.12436i	1.00153i	0.865584	+	0.500764i	$0.166948\pi$
$83$	9.12436i	1.00153i	−0.865584	+	0.500764i	$0.833052\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.19615	−0.550791	−0.275396	−	0.961331i	$-0.588809\pi$
$89$	−5.19615	−0.550791	−0.275396	+	0.961331i	$0.588809\pi$
$90$	0	0
$91$	−1.07180	−0.112355
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 2.73205i	− 0.277398i	−0.990335	−	0.138699i	$-0.955708\pi$
$97$	− 2.73205i	− 0.277398i	0.990335	−	0.138699i	$-0.0442920\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.1244	1.80344	0.901720	−	0.432320i	$-0.142305\pi$
$101$	18.1244	1.80344	0.901720	+	0.432320i	$0.142305\pi$
$102$	0	0
$103$	2.39230i	0.235721i	0.993030	+	0.117860i	$0.0376035\pi$
$103$	2.39230i	0.235721i	−0.993030	+	0.117860i	$0.962396\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.46410i	0.334887i	0.985882	+	0.167444i	$0.0535512\pi$
$107$	3.46410i	0.334887i	−0.985882	+	0.167444i	$0.946449\pi$
$108$	0	0
$109$	−5.92820	−0.567819	−0.283909	−	0.958851i	$-0.591632\pi$
$109$	−5.92820	−0.567819	−0.283909	+	0.958851i	$0.591632\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 17.6603i	− 1.66134i	−0.556767	−	0.830668i	$-0.687959\pi$
$113$	− 17.6603i	− 1.66134i	0.556767	−	0.830668i	$-0.312041\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.928203	−0.0850883
$120$	0	0
$121$	−8.00000	−0.727273
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 6.19615i	− 0.549820i	−0.961470	−	0.274910i	$-0.911352\pi$
$127$	− 6.19615i	− 0.549820i	0.961470	−	0.274910i	$-0.0886480\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−13.7321	−1.19977	−0.599887	−	0.800084i	$-0.704788\pi$
$131$	−13.7321	−1.19977	−0.599887	+	0.800084i	$0.704788\pi$
$132$	0	0
$133$	− 1.80385i	− 0.156413i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.3923i	1.40049i	0.713903	+	0.700245i	$0.246926\pi$
$137$	16.3923i	1.40049i	−0.713903	+	0.700245i	$0.753074\pi$
$138$	0	0
$139$	5.39230	0.457369	0.228685	−	0.973501i	$-0.426558\pi$
$139$	5.39230	0.457369	0.228685	+	0.973501i	$0.426558\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.53590i	0.212062i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	3.39230	0.276062	0.138031	−	0.990428i	$-0.455923\pi$
$151$	3.39230	0.276062	0.138031	+	0.990428i	$0.455923\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.73205i	0.537276i	0.963241	+	0.268638i	$0.0865736\pi$
$157$	6.73205i	0.537276i	−0.963241	+	0.268638i	$0.913426\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.53590	0.199857
$162$	0	0
$163$	− 15.2679i	− 1.19588i	−0.801542	−	0.597939i	$-0.795986\pi$
$163$	− 15.2679i	− 1.19588i	0.801542	−	0.597939i	$-0.204014\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3.12436i	− 0.241770i	−0.992667	−	0.120885i	$-0.961427\pi$
$167$	− 3.12436i	− 0.241770i	0.992667	−	0.120885i	$-0.0385732\pi$
$168$	0	0
$169$	10.8564	0.835108
$170$	0	0
$171$	0	0
$172$	0	0
$173$	24.2487i	1.84360i	0.387671	+	0.921798i	$0.373280\pi$
$173$	24.2487i	1.84360i	−0.387671	+	0.921798i	$0.626720\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.1244	0.906217	0.453108	−	0.891455i	$-0.350315\pi$
$179$	12.1244	0.906217	0.453108	+	0.891455i	$0.350315\pi$
$180$	0	0
$181$	−9.53590	−0.708798	−0.354399	−	0.935094i	$-0.615314\pi$
$181$	−9.53590	−0.708798	−0.354399	+	0.935094i	$0.615314\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.19615i	0.160599i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.0526	1.37859	0.689297	−	0.724479i	$-0.257919\pi$
$191$	19.0526	1.37859	0.689297	+	0.724479i	$0.257919\pi$
$192$	0	0
$193$	− 20.5885i	− 1.48199i	−0.671511	−	0.740995i	$-0.734354\pi$
$193$	− 20.5885i	− 1.48199i	0.671511	−	0.740995i	$-0.265646\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 13.8564i	− 0.987228i	−0.869681	−	0.493614i	$-0.835676\pi$
$197$	− 13.8564i	− 0.987228i	0.869681	−	0.493614i	$-0.164324\pi$
$198$	0	0
$199$	11.8564	0.840478	0.420239	−	0.907413i	$-0.361946\pi$
$199$	11.8564	0.840478	0.420239	+	0.907413i	$0.361946\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 3.12436i	− 0.219287i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.26795	−0.295220
$210$	0	0
$211$	−6.07180	−0.418000	−0.209000	−	0.977916i	$-0.567021\pi$
$211$	−6.07180	−0.418000	−0.209000	+	0.977916i	$0.567021\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 5.80385i	− 0.393991i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.85641	−0.124875
$222$	0	0
$223$	− 21.8564i	− 1.46361i	−0.681512	−	0.731807i	$-0.738677\pi$
$223$	− 21.8564i	− 1.46361i	0.681512	−	0.731807i	$-0.261323\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 15.8038i	− 1.04894i	−0.851429	−	0.524469i	$-0.824264\pi$
$227$	− 15.8038i	− 1.04894i	0.851429	−	0.524469i	$-0.175736\pi$
$228$	0	0
$229$	−9.85641	−0.651330	−0.325665	−	0.945485i	$-0.605588\pi$
$229$	−9.85641	−0.651330	−0.325665	+	0.945485i	$0.605588\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 6.58846i	− 0.431624i	−0.976435	−	0.215812i	$-0.930760\pi$
$233$	− 6.58846i	− 0.431624i	0.976435	−	0.215812i	$-0.0692399\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.4641	1.00029	0.500145	−	0.865942i	$-0.333280\pi$
$239$	15.4641	1.00029	0.500145	+	0.865942i	$0.333280\pi$
$240$	0	0
$241$	−24.3205	−1.56662	−0.783311	−	0.621630i	$-0.786471\pi$
$241$	−24.3205	−1.56662	−0.783311	+	0.621630i	$0.786471\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 3.60770i	− 0.229552i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−27.4641	−1.73352	−0.866759	−	0.498727i	$-0.833801\pi$
$251$	−27.4641	−1.73352	−0.866759	+	0.498727i	$0.833801\pi$
$252$	0	0
$253$	− 6.00000i	− 0.377217i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 8.53590i	− 0.532455i	−0.963910	−	0.266227i	$-0.914223\pi$
$257$	− 8.53590i	− 0.532455i	0.963910	−	0.266227i	$-0.0857772\pi$
$258$	0	0
$259$	−3.07180	−0.190872
$260$	0	0
$261$	0	0
$262$	0	0
$263$	24.2487i	1.49524i	0.664127	+	0.747620i	$0.268803\pi$
$263$	24.2487i	1.49524i	−0.664127	+	0.747620i	$0.731197\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−19.7321	−1.20308	−0.601542	−	0.798841i	$-0.705447\pi$
$269$	−19.7321	−1.20308	−0.601542	+	0.798841i	$0.705447\pi$
$270$	0	0
$271$	−24.7846	−1.50556	−0.752779	−	0.658273i	$-0.771287\pi$
$271$	−24.7846	−1.50556	−0.752779	+	0.658273i	$0.771287\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 19.1244i	− 1.14907i	−0.818480	−	0.574536i	$-0.805183\pi$
$277$	− 19.1244i	− 1.14907i	0.818480	−	0.574536i	$-0.194817\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.32051	0.317395	0.158697	−	0.987327i	$-0.449271\pi$
$281$	5.32051	0.317395	0.158697	+	0.987327i	$0.449271\pi$
$282$	0	0
$283$	− 23.4641i	− 1.39480i	−0.716684	−	0.697398i	$-0.754341\pi$
$283$	− 23.4641i	− 1.39480i	0.716684	−	0.697398i	$-0.245659\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.588457i	0.0347355i
$288$	0	0
$289$	15.3923	0.905430
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 6.58846i	− 0.384902i	−0.981307	−	0.192451i	$-0.938356\pi$
$293$	− 6.58846i	− 0.384902i	0.981307	−	0.192451i	$-0.0616436\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.07180	0.293310
$300$	0	0
$301$	4.92820	0.284057
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 12.1962i	− 0.696071i	−0.937481	−	0.348036i	$-0.886849\pi$
$307$	− 12.1962i	− 0.696071i	0.937481	−	0.348036i	$-0.113151\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−28.5167	−1.61703	−0.808516	−	0.588475i	$-0.799729\pi$
$311$	−28.5167	−1.61703	−0.808516	+	0.588475i	$0.799729\pi$
$312$	0	0
$313$	− 1.07180i	− 0.0605815i	−0.999541	−	0.0302908i	$-0.990357\pi$
$313$	− 1.07180i	− 0.0605815i	0.999541	−	0.0302908i	$-0.00964333\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.1244i	1.18646i	0.805032	+	0.593231i	$0.202148\pi$
$317$	21.1244i	1.18646i	−0.805032	+	0.593231i	$0.797852\pi$
$318$	0	0
$319$	−7.39230	−0.413890
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 3.12436i	− 0.173844i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.46410	−0.190982
$330$	0	0
$331$	−8.60770	−0.473122	−0.236561	−	0.971617i	$-0.576020\pi$
$331$	−8.60770	−0.473122	−0.236561	+	0.971617i	$0.576020\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	14.2487i	0.776177i	0.921622	+	0.388088i	$0.126864\pi$
$337$	14.2487i	0.776177i	−0.921622	+	0.388088i	$0.873136\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−13.7321	−0.743632
$342$	0	0
$343$	9.85641i	0.532196i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 32.1962i	− 1.72838i	−0.503166	−	0.864190i	$-0.667831\pi$
$347$	− 32.1962i	− 1.72838i	0.503166	−	0.864190i	$-0.332169\pi$
$348$	0	0
$349$	25.0000	1.33822	0.669110	−	0.743164i	$-0.266676\pi$
$349$	25.0000	1.33822	0.669110	+	0.743164i	$0.266676\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	22.9808i	1.22314i	0.791189	+	0.611571i	$0.209462\pi$
$353$	22.9808i	1.22314i	−0.791189	+	0.611571i	$0.790538\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.6603	1.40707	0.703537	−	0.710658i	$-0.251603\pi$
$359$	26.6603	1.40707	0.703537	+	0.710658i	$0.251603\pi$
$360$	0	0
$361$	−12.9282	−0.680432
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 5.60770i	− 0.292719i	−0.989231	−	0.146360i	$-0.953244\pi$
$367$	− 5.60770i	− 0.292719i	0.989231	−	0.146360i	$-0.0467557\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.85641	0.407884
$372$	0	0
$373$	− 24.0526i	− 1.24539i	−0.782463	−	0.622697i	$-0.786037\pi$
$373$	− 24.0526i	− 1.24539i	0.782463	−	0.622697i	$-0.213963\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 6.24871i	− 0.321825i
$378$	0	0
$379$	−11.4641	−0.588871	−0.294436	−	0.955671i	$-0.595132\pi$
$379$	−11.4641	−0.588871	−0.294436	+	0.955671i	$0.595132\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 30.2487i	− 1.54564i	−0.634627	−	0.772818i	$-0.718846\pi$
$383$	− 30.2487i	− 1.54564i	0.634627	−	0.772818i	$-0.281154\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	20.5359	1.04121	0.520606	−	0.853797i	$-0.325706\pi$
$389$	20.5359	1.04121	0.520606	+	0.853797i	$0.325706\pi$
$390$	0	0
$391$	4.39230	0.222128
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.67949i	0.134480i	0.997737	+	0.0672399i	$0.0214193\pi$
$397$	2.67949i	0.134480i	−0.997737	+	0.0672399i	$0.978581\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−26.7846	−1.33756	−0.668780	−	0.743461i	$-0.733183\pi$
$401$	−26.7846	−1.33756	−0.668780	+	0.743461i	$0.733183\pi$
$402$	0	0
$403$	− 11.6077i	− 0.578220i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.26795i	0.360259i
$408$	0	0
$409$	−9.85641	−0.487368	−0.243684	−	0.969855i	$-0.578356\pi$
$409$	−9.85641	−0.487368	−0.243684	+	0.969855i	$0.578356\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 3.12436i	− 0.153739i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14.5359	0.710125	0.355063	−	0.934843i	$-0.384460\pi$
$419$	14.5359	0.710125	0.355063	+	0.934843i	$0.384460\pi$
$420$	0	0
$421$	−27.7846	−1.35414	−0.677070	−	0.735919i	$-0.736750\pi$
$421$	−27.7846	−1.35414	−0.677070	+	0.735919i	$0.736750\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 2.92820i	− 0.141706i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.41154	0.116160	0.0580800	−	0.998312i	$-0.481502\pi$
$431$	2.41154	0.116160	0.0580800	+	0.998312i	$0.481502\pi$
$432$	0	0
$433$	− 4.53590i	− 0.217981i	−0.994043	−	0.108991i	$-0.965238\pi$
$433$	− 4.53590i	− 0.217981i	0.994043	−	0.108991i	$-0.0347619\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.53590i	0.408327i
$438$	0	0
$439$	−27.3923	−1.30736	−0.653682	−	0.756770i	$-0.726776\pi$
$439$	−27.3923	−1.30736	−0.653682	+	0.756770i	$0.726776\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	40.7321i	1.93524i	0.252415	+	0.967619i	$0.418775\pi$
$443$	40.7321i	1.93524i	−0.252415	+	0.967619i	$0.581225\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−0.124356	−0.00586871	−0.00293435	−	0.999996i	$-0.500934\pi$
$449$	−0.124356	−0.00586871	−0.00293435	+	0.999996i	$0.500934\pi$
$450$	0	0
$451$	1.39230	0.0655611
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11.8038i	0.552161i	0.961135	+	0.276080i	$0.0890356\pi$
$457$	11.8038i	0.552161i	−0.961135	+	0.276080i	$0.910964\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	21.5885	1.00547	0.502737	−	0.864439i	$-0.332326\pi$
$461$	21.5885	1.00547	0.502737	+	0.864439i	$0.332326\pi$
$462$	0	0
$463$	− 18.3923i	− 0.854763i	−0.904071	−	0.427381i	$-0.859436\pi$
$463$	− 18.3923i	− 0.854763i	0.904071	−	0.427381i	$-0.140564\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 15.8038i	− 0.731315i	−0.930750	−	0.365657i	$-0.880844\pi$
$467$	− 15.8038i	− 0.731315i	0.930750	−	0.365657i	$-0.119156\pi$
$468$	0	0
$469$	10.5359	0.486503
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 11.6603i	− 0.536139i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−20.6603	−0.943991	−0.471996	−	0.881601i	$-0.656466\pi$
$479$	−20.6603	−0.943991	−0.471996	+	0.881601i	$0.656466\pi$
$480$	0	0
$481$	−6.14359	−0.280124
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 37.4641i	− 1.69766i	−0.528666	−	0.848830i	$-0.677307\pi$
$487$	− 37.4641i	− 1.69766i	0.528666	−	0.848830i	$-0.322693\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.73205	0.348943	0.174471	−	0.984662i	$-0.444178\pi$
$491$	7.73205	0.348943	0.174471	+	0.984662i	$0.444178\pi$
$492$	0	0
$493$	− 5.41154i	− 0.243724i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.588457i	0.0263959i
$498$	0	0
$499$	−6.60770	−0.295801	−0.147901	−	0.989002i	$-0.547252\pi$
$499$	−6.60770	−0.295801	−0.147901	+	0.989002i	$0.547252\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 0.679492i	− 0.0302970i	−0.999885	−	0.0151485i	$-0.995178\pi$
$503$	− 0.679492i	− 0.0302970i	0.999885	−	0.0151485i	$-0.00482211\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.7846	0.655316	0.327658	−	0.944796i	$-0.393741\pi$
$509$	14.7846	0.655316	0.327658	+	0.944796i	$0.393741\pi$
$510$	0	0
$511$	7.46410	0.330192
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.19615i	0.360466i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10.3923	−0.455295	−0.227648	−	0.973744i	$-0.573103\pi$
$521$	−10.3923	−0.455295	−0.227648	+	0.973744i	$0.573103\pi$
$522$	0	0
$523$	− 24.3923i	− 1.06660i	−0.845926	−	0.533301i	$-0.820952\pi$
$523$	− 24.3923i	− 1.06660i	0.845926	−	0.533301i	$-0.179048\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 10.0526i	− 0.437896i
$528$	0	0
$529$	11.0000	0.478261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.17691i	0.0509778i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	11.1962	0.482252
$540$	0	0
$541$	−4.46410	−0.191927	−0.0959634	−	0.995385i	$-0.530593\pi$
$541$	−4.46410	−0.191927	−0.0959634	+	0.995385i	$0.530593\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 36.7846i	− 1.57280i	−0.617720	−	0.786398i	$-0.711943\pi$
$547$	− 36.7846i	− 1.57280i	0.617720	−	0.786398i	$-0.288057\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	10.5167	0.448025
$552$	0	0
$553$	− 4.67949i	− 0.198992i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 16.3923i	− 0.694564i	−0.937761	−	0.347282i	$-0.887105\pi$
$557$	− 16.3923i	− 0.694564i	0.937761	−	0.347282i	$-0.112895\pi$
$558$	0	0
$559$	9.85641	0.416882
$560$	0	0
$561$	0	0
$562$	0	0
$563$	19.2679i	0.812047i	0.913863	+	0.406024i	$0.133085\pi$
$563$	19.2679i	0.812047i	−0.913863	+	0.406024i	$0.866915\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.94744	−0.207408	−0.103704	−	0.994608i	$-0.533069\pi$
$569$	−4.94744	−0.207408	−0.103704	+	0.994608i	$0.533069\pi$
$570$	0	0
$571$	−26.8564	−1.12391	−0.561953	−	0.827169i	$-0.689950\pi$
$571$	−26.8564	−1.12391	−0.561953	+	0.827169i	$0.689950\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	22.1962i	0.924038i	0.886870	+	0.462019i	$0.152875\pi$
$577$	22.1962i	0.924038i	−0.886870	+	0.462019i	$0.847125\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6.67949	−0.277112
$582$	0	0
$583$	− 18.5885i	− 0.769855i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 14.1962i	− 0.585938i	−0.956122	−	0.292969i	$-0.905357\pi$
$587$	− 14.1962i	− 0.585938i	0.956122	−	0.292969i	$-0.0946433\pi$
$588$	0	0
$589$	19.5359	0.804963
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 36.9282i	− 1.51646i	−0.651987	−	0.758230i	$-0.726065\pi$
$593$	− 36.9282i	− 1.51646i	0.651987	−	0.758230i	$-0.273935\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−45.5885	−1.86269	−0.931347	−	0.364133i	$-0.881365\pi$
$599$	−45.5885	−1.86269	−0.931347	+	0.364133i	$0.881365\pi$
$600$	0	0
$601$	10.3205	0.420982	0.210491	−	0.977596i	$-0.432494\pi$
$601$	10.3205	0.420982	0.210491	+	0.977596i	$0.432494\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 28.5885i	− 1.16037i	−0.814485	−	0.580185i	$-0.802980\pi$
$607$	− 28.5885i	− 1.16037i	0.814485	−	0.580185i	$-0.197020\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.92820	−0.280285
$612$	0	0
$613$	− 3.60770i	− 0.145713i	−0.997342	−	0.0728567i	$-0.976788\pi$
$613$	− 3.60770i	− 0.145713i	0.997342	−	0.0728567i	$-0.0232116\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.0000i	0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$	12.0000i	0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$	0	0
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 3.80385i	− 0.152398i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.32051	−0.212143
$630$	0	0
$631$	−13.9282	−0.554473	−0.277237	−	0.960802i	$-0.589419\pi$
$631$	−13.9282	−0.554473	−0.277237	+	0.960802i	$0.589419\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	9.46410i	0.374981i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	23.1962	0.916193	0.458096	−	0.888902i	$-0.348531\pi$
$641$	23.1962	0.916193	0.458096	+	0.888902i	$0.348531\pi$
$642$	0	0
$643$	16.5885i	0.654185i	0.944992	+	0.327092i	$0.106069\pi$
$643$	16.5885i	0.654185i	−0.944992	+	0.327092i	$0.893931\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	48.2487i	1.89685i	0.316998	+	0.948426i	$0.397325\pi$
$647$	48.2487i	1.89685i	−0.316998	+	0.948426i	$0.602675\pi$
$648$	0	0
$649$	−7.39230	−0.290173
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 9.46410i	− 0.370359i	−0.982705	−	0.185179i	$-0.940713\pi$
$653$	− 9.46410i	− 0.370359i	0.982705	−	0.185179i	$-0.0592866\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−9.46410	−0.368669	−0.184335	−	0.982864i	$-0.559013\pi$
$659$	−9.46410	−0.368669	−0.184335	+	0.982864i	$0.559013\pi$
$660$	0	0
$661$	−5.39230	−0.209736	−0.104868	−	0.994486i	$-0.533442\pi$
$661$	−5.39230	−0.209736	−0.104868	+	0.994486i	$0.533442\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	14.7846i	0.572462i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.92820	−0.267460
$672$	0	0
$673$	17.6077i	0.678727i	0.940655	+	0.339363i	$0.110212\pi$
$673$	17.6077i	0.678727i	−0.940655	+	0.339363i	$0.889788\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.6410i	1.10076i	0.834913	+	0.550382i	$0.185518\pi$
$677$	28.6410i	1.10076i	−0.834913	+	0.550382i	$0.814482\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	14.5359i	0.556201i	0.960552	+	0.278100i	$0.0897048\pi$
$683$	14.5359i	0.556201i	−0.960552	+	0.278100i	$0.910295\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	15.7128	0.598610
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.01924i	0.0386064i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	42.1244	1.59101	0.795507	−	0.605944i	$-0.207204\pi$
$701$	42.1244	1.59101	0.795507	+	0.605944i	$0.207204\pi$
$702$	0	0
$703$	− 10.3397i	− 0.389971i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.2679i	0.498993i
$708$	0	0
$709$	−17.4641	−0.655878	−0.327939	−	0.944699i	$-0.606354\pi$
$709$	−17.4641	−0.655878	−0.327939	+	0.944699i	$0.606354\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	27.4641i	1.02854i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6.80385	−0.253741	−0.126870	−	0.991919i	$-0.540493\pi$
$719$	−6.80385	−0.253741	−0.126870	+	0.991919i	$0.540493\pi$
$720$	0	0
$721$	−1.75129	−0.0652214
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	33.1769i	1.23046i	0.788346	+	0.615232i	$0.210938\pi$
$727$	33.1769i	1.23046i	−0.788346	+	0.615232i	$0.789062\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.53590	0.315712
$732$	0	0
$733$	39.5692i	1.46152i	0.682633	+	0.730761i	$0.260835\pi$
$733$	39.5692i	1.46152i	−0.682633	+	0.730761i	$0.739165\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 24.9282i	− 0.918242i
$738$	0	0
$739$	−36.1769	−1.33079	−0.665395	−	0.746492i	$-0.731737\pi$
$739$	−36.1769	−1.33079	−0.665395	+	0.746492i	$0.731737\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 36.5885i	− 1.34230i	−0.741321	−	0.671150i	$-0.765801\pi$
$743$	− 36.5885i	− 1.34230i	0.741321	−	0.671150i	$-0.234199\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.53590	−0.0926597
$750$	0	0
$751$	−0.784610	−0.0286308	−0.0143154	−	0.999898i	$-0.504557\pi$
$751$	−0.784610	−0.0286308	−0.0143154	+	0.999898i	$0.504557\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0.392305i	0.0142586i	0.999975	+	0.00712928i	$0.00226934\pi$
$757$	0.392305i	0.0142586i	−0.999975	+	0.00712928i	$0.997731\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5.87564	0.212992	0.106496	−	0.994313i	$-0.466037\pi$
$761$	5.87564	0.212992	0.106496	+	0.994313i	$0.466037\pi$
$762$	0	0
$763$	− 4.33975i	− 0.157109i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 6.24871i	− 0.225628i
$768$	0	0
$769$	13.2487	0.477761	0.238880	−	0.971049i	$-0.423220\pi$
$769$	13.2487	0.477761	0.238880	+	0.971049i	$0.423220\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0.339746i	0.0122198i	0.999981	+	0.00610991i	$0.00194486\pi$
$773$	0.339746i	0.0122198i	−0.999981	+	0.00610991i	$0.998055\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.98076	−0.0709682
$780$	0	0
$781$	1.39230	0.0498206
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 25.8038i	− 0.919808i	−0.887969	−	0.459904i	$-0.847884\pi$
$787$	− 25.8038i	− 0.919808i	0.887969	−	0.459904i	$-0.152116\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12.9282	0.459674
$792$	0	0
$793$	− 5.85641i	− 0.207967i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	11.3205i	0.400993i	0.979694	+	0.200496i	$0.0642555\pi$
$797$	11.3205i	0.400993i	−0.979694	+	0.200496i	$0.935744\pi$
$798$	0	0
$799$	−6.00000	−0.212265
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 17.6603i	− 0.623217i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−39.5885	−1.39186	−0.695928	−	0.718112i	$-0.745007\pi$
$809$	−39.5885	−1.39186	−0.695928	+	0.718112i	$0.745007\pi$
$810$	0	0
$811$	23.2487	0.816373	0.408186	−	0.912899i	$-0.366161\pi$
$811$	23.2487	0.816373	0.408186	+	0.912899i	$0.366161\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	16.5885i	0.580357i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−20.6603	−0.721048	−0.360524	−	0.932750i	$-0.617402\pi$
$821$	−20.6603	−0.721048	−0.360524	+	0.932750i	$0.617402\pi$
$822$	0	0
$823$	3.07180i	0.107076i	0.998566	+	0.0535381i	$0.0170498\pi$
$823$	3.07180i	0.107076i	−0.998566	+	0.0535381i	$0.982950\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38.5359i	1.34002i	0.742350	+	0.670012i	$0.233711\pi$
$827$	38.5359i	1.34002i	−0.742350	+	0.670012i	$0.766289\pi$
$828$	0	0
$829$	10.2154	0.354795	0.177398	−	0.984139i	$-0.443232\pi$
$829$	10.2154	0.354795	0.177398	+	0.984139i	$0.443232\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	8.19615i	0.283980i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−39.5885	−1.36675	−0.683373	−	0.730070i	$-0.739488\pi$
$839$	−39.5885	−1.36675	−0.683373	+	0.730070i	$0.739488\pi$
$840$	0	0
$841$	−10.7846	−0.371883
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 5.85641i	− 0.201229i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	14.5359	0.498284
$852$	0	0
$853$	− 10.1962i	− 0.349110i	−0.984647	−	0.174555i	$-0.944151\pi$
$853$	− 10.1962i	− 0.349110i	0.984647	−	0.174555i	$-0.0558486\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	42.9282i	1.46640i	0.680013	+	0.733200i	$0.261974\pi$
$857$	42.9282i	1.46640i	−0.680013	+	0.733200i	$0.738026\pi$
$858$	0	0
$859$	33.7846	1.15272	0.576358	−	0.817197i	$-0.304473\pi$
$859$	33.7846	1.15272	0.576358	+	0.817197i	$0.304473\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 4.48334i	− 0.152615i	−0.997084	−	0.0763073i	$-0.975687\pi$
$863$	− 4.48334i	− 0.152615i	0.997084	−	0.0763073i	$-0.0243130\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−11.0718	−0.375585
$870$	0	0
$871$	21.0718	0.713991
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 34.2487i	− 1.15650i	−0.815861	−	0.578248i	$-0.803736\pi$
$877$	− 34.2487i	− 1.15650i	0.815861	−	0.578248i	$-0.196264\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−17.1962	−0.579353	−0.289677	−	0.957125i	$-0.593548\pi$
$881$	−17.1962	−0.579353	−0.289677	+	0.957125i	$0.593548\pi$
$882$	0	0
$883$	− 32.8372i	− 1.10506i	−0.833493	−	0.552529i	$-0.813663\pi$
$883$	− 32.8372i	− 1.10506i	0.833493	−	0.552529i	$-0.186337\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 16.7321i	− 0.561807i	−0.959736	−	0.280904i	$-0.909366\pi$
$887$	− 16.7321i	− 0.561807i	0.959736	−	0.280904i	$-0.0906341\pi$
$888$	0	0
$889$	4.53590	0.152129
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 11.6603i	− 0.390196i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	33.8372	1.12853
$900$	0	0
$901$	13.6077	0.453338
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 42.7846i	− 1.42064i	−0.703879	−	0.710320i	$-0.748550\pi$
$907$	− 42.7846i	− 1.42064i	0.703879	−	0.710320i	$-0.251450\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	17.1962	0.569734	0.284867	−	0.958567i	$-0.408051\pi$
$911$	17.1962	0.569734	0.284867	+	0.958567i	$0.408051\pi$
$912$	0	0
$913$	15.8038i	0.523031i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 10.0526i	− 0.331965i
$918$	0	0
$919$	−30.6077	−1.00965	−0.504827	−	0.863220i	$-0.668444\pi$
$919$	−30.6077	−1.00965	−0.504827	+	0.863220i	$0.668444\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.17691i	0.0387386i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	46.5167	1.52616	0.763081	−	0.646303i	$-0.223686\pi$
$929$	46.5167	1.52616	0.763081	+	0.646303i	$0.223686\pi$
$930$	0	0
$931$	−15.9282	−0.522026
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	32.9282i	1.07572i	0.843035	+	0.537859i	$0.180767\pi$
$937$	32.9282i	1.07572i	−0.843035	+	0.537859i	$0.819233\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	15.4641	0.504115	0.252058	−	0.967712i	$-0.418893\pi$
$941$	15.4641	0.504115	0.252058	+	0.967712i	$0.418893\pi$
$942$	0	0
$943$	− 2.78461i	− 0.0906794i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 16.6410i	− 0.540760i	−0.962754	−	0.270380i	$-0.912851\pi$
$947$	− 16.6410i	− 0.540760i	0.962754	−	0.270380i	$-0.0871494\pi$
$948$	0	0
$949$	14.9282	0.484590
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 9.46410i	− 0.306572i	−0.988182	−	0.153286i	$-0.951014\pi$
$953$	− 9.46410i	− 0.306572i	0.988182	−	0.153286i	$-0.0489856\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12.0000	−0.387500
$960$	0	0
$961$	31.8564	1.02763
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	38.5885i	1.24092i	0.784238	+	0.620461i	$0.213054\pi$
$967$	38.5885i	1.24092i	−0.784238	+	0.620461i	$0.786946\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−25.7321	−0.825781	−0.412890	−	0.910781i	$-0.635481\pi$
$971$	−25.7321	−0.825781	−0.412890	+	0.910781i	$0.635481\pi$
$972$	0	0
$973$	3.94744i	0.126549i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	16.3923i	0.524436i	0.965009	+	0.262218i	$0.0844540\pi$
$977$	16.3923i	0.524436i	−0.965009	+	0.262218i	$0.915546\pi$
$978$	0	0
$979$	−9.00000	−0.287641
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 49.2679i	− 1.57140i	−0.618605	−	0.785702i	$-0.712302\pi$
$983$	− 49.2679i	− 1.57140i	0.618605	−	0.785702i	$-0.287698\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−23.3205	−0.741549
$990$	0	0
$991$	14.2154	0.451567	0.225783	−	0.974178i	$-0.427506\pi$
$991$	14.2154	0.451567	0.225783	+	0.974178i	$0.427506\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	41.8038i	1.32394i	0.749530	+	0.661971i	$0.230280\pi$
$997$	41.8038i	1.32394i	−0.749530	+	0.661971i	$0.769720\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8100.2.d.l.649.3		4
3.2	odd	2		8100.2.d.m.649.3		4
5.2	odd	4		8100.2.a.s.1.1		2
5.3	odd	4		1620.2.a.h.1.2	yes	2
5.4	even	2	inner	8100.2.d.l.649.2		4
15.2	even	4		8100.2.a.t.1.1		2
15.8	even	4		1620.2.a.g.1.2	✓	2
15.14	odd	2		8100.2.d.m.649.2		4
20.3	even	4		6480.2.a.bp.1.1		2
45.13	odd	12		1620.2.i.m.541.1		4
45.23	even	12		1620.2.i.n.541.1		4
45.38	even	12		1620.2.i.n.1081.1		4
45.43	odd	12		1620.2.i.m.1081.1		4
60.23	odd	4		6480.2.a.bh.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.2.a.g.1.2	✓	2	15.8	even	4
1620.2.a.h.1.2	yes	2	5.3	odd	4
1620.2.i.m.541.1		4	45.13	odd	12
1620.2.i.m.1081.1		4	45.43	odd	12
1620.2.i.n.541.1		4	45.23	even	12
1620.2.i.n.1081.1		4	45.38	even	12
6480.2.a.bh.1.1		2	60.23	odd	4
6480.2.a.bp.1.1		2	20.3	even	4
8100.2.a.s.1.1		2	5.2	odd	4
8100.2.a.t.1.1		2	15.2	even	4
8100.2.d.l.649.2		4	5.4	even	2	inner
8100.2.d.l.649.3		4	1.1	even	1	trivial
8100.2.d.m.649.2		4	15.14	odd	2
8100.2.d.m.649.3		4	3.2	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:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.732051i q^{7} +O(q^{10})\)	\(q+0.732051i q^{7} +1.73205 q^{11} +1.46410i q^{13} +1.26795i q^{17} -2.46410 q^{19} -3.46410i q^{23} -4.26795 q^{29} -7.92820 q^{31} +4.19615i q^{37} +0.803848 q^{41} -6.73205i q^{43} +4.73205i q^{47} +6.46410 q^{49} -10.7321i q^{53} -4.26795 q^{59} -4.00000 q^{61} -14.3923i q^{67} +0.803848 q^{71} -10.1962i q^{73} +1.26795i q^{77} -6.39230 q^{79} +9.12436i q^{83} -5.19615 q^{89} -1.07180 q^{91} -2.73205i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 4 q^{19} - 24 q^{29} - 4 q^{31} + 24 q^{41} + 12 q^{49} - 24 q^{59} - 16 q^{61} + 24 q^{71} + 16 q^{79} - 32 q^{91}+O(q^{100}) \) 4 * q + 4 * q^19 - 24 * q^29 - 4 * q^31 + 24 * q^41 + 12 * q^49 - 24 * q^59 - 16 * q^61 + 24 * q^71 + 16 * q^79 - 32 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more