LMFDB - Embedded newform 6300.2.k.n.6049.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6300,2,Mod(6049,6300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6300, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6300.6049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6300.k (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:	$50.3057532734$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2100)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			6049.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	6300.6049
Dual form			6300.2.k.n.6049.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{7} +O(q^{10})$	$q-1.00000i q^{7} +3.00000 q^{11} -4.00000i q^{13} -6.00000i q^{17} +4.00000 q^{19} -3.00000i q^{23} +3.00000 q^{29} -10.0000 q^{31} +7.00000i q^{37} -1.00000i q^{43} -12.0000i q^{47} -1.00000 q^{49} -6.00000i q^{53} +12.0000 q^{59} -4.00000 q^{61} +7.00000i q^{67} -9.00000 q^{71} +2.00000i q^{73} -3.00000i q^{77} -17.0000 q^{79} +6.00000i q^{83} -4.00000 q^{91} -2.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 6 q^{11} + 8 q^{19} + 6 q^{29} - 20 q^{31} - 2 q^{49} + 24 q^{59} - 8 q^{61} - 18 q^{71} - 34 q^{79} - 8 q^{91}+O(q^{100}) $ 2 * q + 6 * q^11 + 8 * q^19 + 6 * q^29 - 20 * q^31 - 2 * q^49 + 24 * q^59 - 8 * q^61 - 18 * q^71 - 34 * q^79 - 8 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times$.

$n$	$2801$	$3151$	$3277$	$3601$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 3.00000i	− 0.625543i	−0.949828	−	0.312772i	$-0.898743\pi$
$23$	− 3.00000i	− 0.625543i	0.949828	−	0.312772i	$-0.101257\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.00000i	1.15079i	0.817875	+	0.575396i	$0.195152\pi$
$37$	7.00000i	1.15079i	−0.817875	+	0.575396i	$0.804848\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	− 1.00000i	− 0.152499i	−0.997089	−	0.0762493i	$-0.975706\pi$
$43$	− 1.00000i	− 0.152499i	0.997089	−	0.0762493i	$-0.0242945\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 12.0000i	− 1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	− 12.0000i	− 1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	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
$67$	7.00000i	0.855186i	0.903971	+	0.427593i	$0.140638\pi$
$67$	7.00000i	0.855186i	−0.903971	+	0.427593i	$0.859362\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	0	0
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.00000i	− 0.341882i
$78$	0	0
$79$	−17.0000	−1.91265	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−17.0000	−1.91265	−0.956325	+	0.292306i	$0.905577\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	15.0000i	1.41108i	0.708669	+	0.705541i	$0.249296\pi$
$113$	15.0000i	1.41108i	−0.708669	+	0.705541i	$0.750704\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.00000i	0.621150i	0.950549	+	0.310575i	$0.100522\pi$
$127$	7.00000i	0.621150i	−0.950549	+	0.310575i	$0.899478\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 12.0000i	− 1.00349i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.00000	0.737309	0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000	0.737309	0.368654	+	0.929567i	$0.379819\pi$
$150$	0	0
$151$	−13.0000	−1.05792	−0.528962	−	0.848645i	$-0.677419\pi$
$151$	−13.0000	−1.05792	−0.528962	+	0.848645i	$0.677419\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.00000i	0.319235i	0.987179	+	0.159617i	$0.0510260\pi$
$157$	4.00000i	0.319235i	−0.987179	+	0.159617i	$0.948974\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 24.0000i	− 1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$	− 24.0000i	− 1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 24.0000i	− 1.82469i	−0.409426	−	0.912343i	$-0.634271\pi$
$173$	− 24.0000i	− 1.82469i	0.409426	−	0.912343i	$-0.365729\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 18.0000i	− 1.31629i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	− 25.0000i	− 1.79954i	−0.436365	−	0.899770i	$-0.643734\pi$
$193$	− 25.0000i	− 1.79954i	0.436365	−	0.899770i	$-0.356266\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.00000i	− 0.213741i	−0.994273	−	0.106871i	$-0.965917\pi$
$197$	− 3.00000i	− 0.213741i	0.994273	−	0.106871i	$-0.0340831\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 3.00000i	− 0.210559i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	10.0000i	0.678844i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−24.0000	−1.61441
$222$	0	0
$223$	2.00000i	0.133930i	0.997755	+	0.0669650i	$0.0213316\pi$
$223$	2.00000i	0.133930i	−0.997755	+	0.0669650i	$0.978668\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.00000i	0.589610i	0.955557	+	0.294805i	$0.0952546\pi$
$233$	9.00000i	0.589610i	−0.955557	+	0.294805i	$0.904745\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−16.0000	−1.03065	−0.515325	−	0.856995i	$-0.672329\pi$
$241$	−16.0000	−1.03065	−0.515325	+	0.856995i	$0.672329\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 16.0000i	− 1.01806i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	0	0
$253$	− 9.00000i	− 0.565825i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6.00000i	− 0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$	− 6.00000i	− 0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	0	0
$259$	7.00000	0.434959
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.00000i	0.184988i	0.995713	+	0.0924940i	$0.0294839\pi$
$263$	3.00000i	0.184988i	−0.995713	+	0.0924940i	$0.970516\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−15.0000	−0.894825	−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000	−0.894825	−0.447412	+	0.894328i	$0.647654\pi$
$282$	0	0
$283$	26.0000i	1.54554i	0.634686	+	0.772770i	$0.281129\pi$
$283$	26.0000i	1.54554i	−0.634686	+	0.772770i	$0.718871\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−12.0000	−0.693978
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 26.0000i	− 1.48390i	−0.670456	−	0.741949i	$-0.733902\pi$
$307$	− 26.0000i	− 1.48390i	0.670456	−	0.741949i	$-0.266098\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	0	0
$313$	20.0000i	1.13047i	0.824931	+	0.565233i	$0.191214\pi$
$313$	20.0000i	1.13047i	−0.824931	+	0.565233i	$0.808786\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3.00000i	− 0.168497i	−0.996445	−	0.0842484i	$-0.973151\pi$
$317$	− 3.00000i	− 0.168497i	0.996445	−	0.0842484i	$-0.0268489\pi$
$318$	0	0
$319$	9.00000	0.503903
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 26.0000i	− 1.41631i	−0.706057	−	0.708155i	$-0.749528\pi$
$337$	− 26.0000i	− 1.41631i	0.706057	−	0.708155i	$-0.250472\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−30.0000	−1.62459
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.00000i	0.483145i	0.970383	+	0.241573i	$0.0776632\pi$
$347$	9.00000i	0.483145i	−0.970383	+	0.241573i	$0.922337\pi$
$348$	0	0
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 18.0000i	− 0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$	− 18.0000i	− 0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	33.0000	1.74167	0.870837	−	0.491572i	$-0.163578\pi$
$359$	33.0000	1.74167	0.870837	+	0.491572i	$0.163578\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.00000i	0.208798i	0.994535	+	0.104399i	$0.0332919\pi$
$367$	4.00000i	0.208798i	−0.994535	+	0.104399i	$0.966708\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	− 37.0000i	− 1.91579i	−0.287123	−	0.957894i	$-0.592699\pi$
$373$	− 37.0000i	− 1.91579i	0.287123	−	0.957894i	$-0.407301\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12.0000i	− 0.618031i
$378$	0	0
$379$	31.0000	1.59236	0.796182	−	0.605058i	$-0.206850\pi$
$379$	31.0000	1.59236	0.796182	+	0.605058i	$0.206850\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−33.0000	−1.67317	−0.836583	−	0.547840i	$-0.815450\pi$
$389$	−33.0000	−1.67317	−0.836583	+	0.547840i	$0.815450\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.0000i	1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$	22.0000i	1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−27.0000	−1.34832	−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000	−1.34832	−0.674158	+	0.738587i	$0.735493\pi$
$402$	0	0
$403$	40.0000i	1.99254i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	21.0000i	1.04093i
$408$	0	0
$409$	−8.00000	−0.395575	−0.197787	−	0.980245i	$-0.563376\pi$
$409$	−8.00000	−0.395575	−0.197787	+	0.980245i	$0.563376\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 12.0000i	− 0.590481i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	−1.00000	−0.0487370	−0.0243685	−	0.999703i	$-0.507758\pi$
$421$	−1.00000	−0.0487370	−0.0243685	+	0.999703i	$0.507758\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.00000i	0.193574i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	− 16.0000i	− 0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$	− 16.0000i	− 0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 12.0000i	− 0.574038i
$438$	0	0
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−39.0000	−1.84052	−0.920262	−	0.391303i	$-0.872024\pi$
$449$	−39.0000	−1.84052	−0.920262	+	0.391303i	$0.872024\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 35.0000i	− 1.63723i	−0.574342	−	0.818615i	$-0.694742\pi$
$457$	− 35.0000i	− 1.63723i	0.574342	−	0.818615i	$-0.305258\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.0000i	0.832941i	0.909149	+	0.416470i	$0.136733\pi$
$467$	18.0000i	0.832941i	−0.909149	+	0.416470i	$0.863267\pi$
$468$	0	0
$469$	7.00000	0.323230
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 3.00000i	− 0.137940i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	28.0000	1.27669
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 11.0000i	− 0.498458i	−0.968445	−	0.249229i	$-0.919823\pi$
$487$	− 11.0000i	− 0.498458i	0.968445	−	0.249229i	$-0.0801771\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−33.0000	−1.48927	−0.744635	−	0.667472i	$-0.767376\pi$
$491$	−33.0000	−1.48927	−0.744635	+	0.667472i	$0.767376\pi$
$492$	0	0
$493$	− 18.0000i	− 0.810679i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.00000i	0.403705i
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.00000i	0.267527i	0.991013	+	0.133763i	$0.0427062\pi$
$503$	6.00000i	0.267527i	−0.991013	+	0.133763i	$0.957294\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 36.0000i	− 1.58328i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	8.00000i	0.349816i	0.984585	+	0.174908i	$0.0559627\pi$
$523$	8.00000i	0.349816i	−0.984585	+	0.174908i	$0.944037\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	60.0000i	2.61364i
$528$	0	0
$529$	14.0000	0.608696
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	19.0000i	0.812381i	0.913788	+	0.406191i	$0.133143\pi$
$547$	19.0000i	0.812381i	−0.913788	+	0.406191i	$0.866857\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	17.0000i	0.722914i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 27.0000i	− 1.14403i	−0.820244	−	0.572013i	$-0.806163\pi$
$557$	− 27.0000i	− 1.14403i	0.820244	−	0.572013i	$-0.193837\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 42.0000i	− 1.77009i	−0.465506	−	0.885044i	$-0.654128\pi$
$563$	− 42.0000i	− 1.77009i	0.465506	−	0.885044i	$-0.345872\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	0	0
$571$	41.0000	1.71580	0.857898	−	0.513820i	$-0.171770\pi$
$571$	41.0000	1.71580	0.857898	+	0.513820i	$0.171770\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 8.00000i	− 0.333044i	−0.986038	−	0.166522i	$-0.946746\pi$
$577$	− 8.00000i	− 0.333044i	0.986038	−	0.166522i	$-0.0532537\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.00000	0.248922
$582$	0	0
$583$	− 18.0000i	− 0.745484i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.0000i	0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$	18.0000i	0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$	0	0
$589$	−40.0000	−1.64817
$590$	0	0
$591$	0	0
$592$	0	0
$593$	12.0000i	0.492781i	0.969171	+	0.246390i	$0.0792446\pi$
$593$	12.0000i	0.492781i	−0.969171	+	0.246390i	$0.920755\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−9.00000	−0.367730	−0.183865	−	0.982952i	$-0.558861\pi$
$599$	−9.00000	−0.367730	−0.183865	+	0.982952i	$0.558861\pi$
$600$	0	0
$601$	44.0000	1.79480	0.897399	−	0.441221i	$-0.145454\pi$
$601$	44.0000	1.79480	0.897399	+	0.441221i	$0.145454\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	40.0000i	1.62355i	0.583970	+	0.811775i	$0.301498\pi$
$607$	40.0000i	1.62355i	−0.583970	+	0.811775i	$0.698502\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−48.0000	−1.94187
$612$	0	0
$613$	29.0000i	1.17130i	0.810564	+	0.585649i	$0.199160\pi$
$613$	29.0000i	1.17130i	−0.810564	+	0.585649i	$0.800840\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27.0000i	1.08698i	0.839416	+	0.543490i	$0.182897\pi$
$617$	27.0000i	1.08698i	−0.839416	+	0.543490i	$0.817103\pi$
$618$	0	0
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	42.0000	1.67465
$630$	0	0
$631$	5.00000	0.199047	0.0995234	−	0.995035i	$-0.468268\pi$
$631$	5.00000	0.199047	0.0995234	+	0.995035i	$0.468268\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.00000i	0.158486i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	45.0000	1.77739	0.888697	−	0.458496i	$-0.151612\pi$
$641$	45.0000	1.77739	0.888697	+	0.458496i	$0.151612\pi$
$642$	0	0
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 42.0000i	− 1.65119i	−0.564263	−	0.825595i	$-0.690840\pi$
$647$	− 42.0000i	− 1.65119i	0.564263	−	0.825595i	$-0.309160\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 9.00000i	− 0.348481i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	− 46.0000i	− 1.77317i	−0.462566	−	0.886585i	$-0.653071\pi$
$673$	− 46.0000i	− 1.77317i	0.462566	−	0.886585i	$-0.346929\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	15.0000i	0.573959i	0.957937	+	0.286980i	$0.0926512\pi$
$683$	15.0000i	0.573959i	−0.957937	+	0.286980i	$0.907349\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	28.0000i	1.05604i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.00000i	0.225653i
$708$	0	0
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	30.0000i	1.12351i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 14.0000i	− 0.519231i	−0.965712	−	0.259616i	$-0.916404\pi$
$727$	− 14.0000i	− 0.519231i	0.965712	−	0.259616i	$-0.0835959\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−6.00000	−0.221918
$732$	0	0
$733$	− 4.00000i	− 0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$	− 4.00000i	− 0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21.0000i	0.773545i
$738$	0	0
$739$	25.0000	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 17.0000i	− 0.617876i	−0.951082	−	0.308938i	$-0.900027\pi$
$757$	− 17.0000i	− 0.617876i	0.951082	−	0.308938i	$-0.0999735\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	− 7.00000i	− 0.253417i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 48.0000i	− 1.73318i
$768$	0	0
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	42.0000i	1.51064i	0.655359	+	0.755318i	$0.272517\pi$
$773$	42.0000i	1.51064i	−0.655359	+	0.755318i	$0.727483\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−27.0000	−0.966136
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 2.00000i	− 0.0712923i	−0.999364	−	0.0356462i	$-0.988651\pi$
$787$	− 2.00000i	− 0.0712923i	0.999364	−	0.0356462i	$-0.0113489\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	15.0000	0.533339
$792$	0	0
$793$	16.0000i	0.568177i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	0	0
$799$	−72.0000	−2.54718
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.00000i	0.211735i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	45.0000	1.58212	0.791058	−	0.611741i	$-0.209531\pi$
$809$	45.0000	1.58212	0.791058	+	0.611741i	$0.209531\pi$
$810$	0	0
$811$	−22.0000	−0.772524	−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000	−0.772524	−0.386262	+	0.922389i	$0.626234\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 4.00000i	− 0.139942i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	0	0
$823$	5.00000i	0.174289i	0.996196	+	0.0871445i	$0.0277742\pi$
$823$	5.00000i	0.174289i	−0.996196	+	0.0871445i	$0.972226\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.00000i	− 0.104320i	−0.998639	−	0.0521601i	$-0.983389\pi$
$827$	− 3.00000i	− 0.104320i	0.998639	−	0.0521601i	$-0.0166106\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.00000i	0.207888i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	54.0000	1.86429	0.932144	−	0.362089i	$-0.117936\pi$
$839$	54.0000	1.86429	0.932144	+	0.362089i	$0.117936\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.00000i	0.0687208i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	21.0000	0.719871
$852$	0	0
$853$	− 4.00000i	− 0.136957i	−0.997653	−	0.0684787i	$-0.978185\pi$
$853$	− 4.00000i	− 0.136957i	0.997653	−	0.0684787i	$-0.0218145\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 12.0000i	− 0.409912i	−0.978771	−	0.204956i	$-0.934295\pi$
$857$	− 12.0000i	− 0.409912i	0.978771	−	0.204956i	$-0.0657052\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	51.0000i	1.73606i	0.496512	+	0.868030i	$0.334614\pi$
$863$	51.0000i	1.73606i	−0.496512	+	0.868030i	$0.665386\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−51.0000	−1.73006
$870$	0	0
$871$	28.0000	0.948744
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 14.0000i	− 0.472746i	−0.971662	−	0.236373i	$-0.924041\pi$
$877$	− 14.0000i	− 0.472746i	0.971662	−	0.236373i	$-0.0759588\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	54.0000	1.81931	0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000	1.81931	0.909653	+	0.415369i	$0.136347\pi$
$882$	0	0
$883$	41.0000i	1.37976i	0.723924	+	0.689880i	$0.242337\pi$
$883$	41.0000i	1.37976i	−0.723924	+	0.689880i	$0.757663\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12.0000i	0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$	12.0000i	0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$	0	0
$889$	7.00000	0.234772
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 48.0000i	− 1.60626i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−30.0000	−1.00056
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 8.00000i	− 0.265636i	−0.991140	−	0.132818i	$-0.957597\pi$
$907$	− 8.00000i	− 0.265636i	0.991140	−	0.132818i	$-0.0424025\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−27.0000	−0.894550	−0.447275	−	0.894397i	$-0.647605\pi$
$911$	−27.0000	−0.894550	−0.447275	+	0.894397i	$0.647605\pi$
$912$	0	0
$913$	18.0000i	0.595713i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 6.00000i	− 0.198137i
$918$	0	0
$919$	19.0000	0.626752	0.313376	−	0.949629i	$-0.398540\pi$
$919$	19.0000	0.626752	0.313376	+	0.949629i	$0.398540\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	36.0000i	1.18495i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	52.0000i	1.69877i	0.527777	+	0.849383i	$0.323026\pi$
$937$	52.0000i	1.69877i	−0.527777	+	0.849383i	$0.676974\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	60.0000i	1.94974i	0.222779	+	0.974869i	$0.428487\pi$
$947$	60.0000i	1.94974i	−0.222779	+	0.974869i	$0.571513\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 57.0000i	− 1.84641i	−0.384307	−	0.923206i	$-0.625559\pi$
$953$	− 57.0000i	− 1.84641i	0.384307	−	0.923206i	$-0.374441\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−18.0000	−0.581250
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	40.0000i	1.28631i	0.765735	+	0.643157i	$0.222376\pi$
$967$	40.0000i	1.28631i	−0.765735	+	0.643157i	$0.777624\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	0	0
$973$	14.0000i	0.448819i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 45.0000i	− 1.43968i	−0.694141	−	0.719839i	$-0.744216\pi$
$977$	− 45.0000i	− 1.43968i	0.694141	−	0.719839i	$-0.255784\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	42.0000i	1.33959i	0.742545	+	0.669796i	$0.233618\pi$
$983$	42.0000i	1.33959i	−0.742545	+	0.669796i	$0.766382\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.00000	−0.0953945
$990$	0	0
$991$	−19.0000	−0.603555	−0.301777	−	0.953378i	$-0.597580\pi$
$991$	−19.0000	−0.603555	−0.301777	+	0.953378i	$0.597580\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.0000i	0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$	28.0000i	0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6300.2.k.n.6049.1		2
3.2	odd	2		2100.2.k.b.1849.2		2
5.2	odd	4		6300.2.a.bb.1.1		1
5.3	odd	4		6300.2.a.m.1.1		1
5.4	even	2	inner	6300.2.k.n.6049.2		2
15.2	even	4		2100.2.a.o.1.1	yes	1
15.8	even	4		2100.2.a.b.1.1	✓	1
15.14	odd	2		2100.2.k.b.1849.1		2
60.23	odd	4		8400.2.a.cs.1.1		1
60.47	odd	4		8400.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2100.2.a.b.1.1	✓	1	15.8	even	4
2100.2.a.o.1.1	yes	1	15.2	even	4
2100.2.k.b.1849.1		2	15.14	odd	2
2100.2.k.b.1849.2		2	3.2	odd	2
6300.2.a.m.1.1		1	5.3	odd	4
6300.2.a.bb.1.1		1	5.2	odd	4
6300.2.k.n.6049.1		2	1.1	even	1	trivial
6300.2.k.n.6049.2		2	5.4	even	2	inner
8400.2.a.i.1.1		1	60.47	odd	4
8400.2.a.cs.1.1		1	60.23	odd	4

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:	\( N \)	\(=\)	\( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6300.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{7} +O(q^{10})\)	\(q-1.00000i q^{7} +3.00000 q^{11} -4.00000i q^{13} -6.00000i q^{17} +4.00000 q^{19} -3.00000i q^{23} +3.00000 q^{29} -10.0000 q^{31} +7.00000i q^{37} -1.00000i q^{43} -12.0000i q^{47} -1.00000 q^{49} -6.00000i q^{53} +12.0000 q^{59} -4.00000 q^{61} +7.00000i q^{67} -9.00000 q^{71} +2.00000i q^{73} -3.00000i q^{77} -17.0000 q^{79} +6.00000i q^{83} -4.00000 q^{91} -2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 6 q^{11} + 8 q^{19} + 6 q^{29} - 20 q^{31} - 2 q^{49} + 24 q^{59} - 8 q^{61} - 18 q^{71} - 34 q^{79} - 8 q^{91}+O(q^{100}) \) 2 * q + 6 * q^11 + 8 * q^19 + 6 * q^29 - 20 * q^31 - 2 * q^49 + 24 * q^59 - 8 * q^61 - 18 * q^71 - 34 * q^79 - 8 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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