LMFDB - Embedded newform 4900.2.e.o.2549.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4900,2,Mod(2549,4900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4900.2549");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4900.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$39.1266969904$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 700)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2549.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4900.2549
Dual form			4900.2.e.o.2549.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{9} -5.00000 q^{11} -6.00000i q^{13} -4.00000i q^{17} -6.00000 q^{19} -3.00000i q^{23} +3.00000 q^{29} -2.00000 q^{31} +7.00000i q^{37} +4.00000 q^{41} +7.00000i q^{43} +2.00000i q^{47} +10.0000i q^{53} -14.0000 q^{59} -4.00000 q^{61} +3.00000i q^{67} -13.0000 q^{71} +16.0000i q^{73} -1.00000 q^{79} +9.00000 q^{81} +10.0000i q^{83} +10.0000 q^{89} +2.00000i q^{97} -15.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{9}+O(q^{10}) $ 2 * q + 6 * q^9	$ 2 q + 6 q^{9} - 10 q^{11} - 12 q^{19} + 6 q^{29} - 4 q^{31} + 8 q^{41} - 28 q^{59} - 8 q^{61} - 26 q^{71} - 2 q^{79} + 18 q^{81} + 20 q^{89} - 30 q^{99}+O(q^{100}) $ 2 * q + 6 * q^9 - 10 * q^11 - 12 * q^19 + 6 * q^29 - 4 * q^31 + 8 * q^41 - 28 * q^59 - 8 * q^61 - 26 * q^71 - 2 * q^79 + 18 * q^81 + 20 * q^89 - 30 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$	$2451$
$\chi(n)$	$1$	$-1$	$1$

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	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 6.00000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.00000i	− 0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$	− 4.00000i	− 0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	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
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	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
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	7.00000i	1.06749i	0.845645	+	0.533745i	$0.179216\pi$
$43$	7.00000i	1.06749i	−0.845645	+	0.533745i	$0.820784\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.00000i	0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$	2.00000i	0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.0000i	1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$	10.0000i	1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−14.0000	−1.82264	−0.911322	−	0.411693i	$-0.864937\pi$
$59$	−14.0000	−1.82264	−0.911322	+	0.411693i	$0.864937\pi$
$60$	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
$67$	3.00000i	0.366508i	0.983066	+	0.183254i	$0.0586631\pi$
$67$	3.00000i	0.366508i	−0.983066	+	0.183254i	$0.941337\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.0000	−1.54282	−0.771408	−	0.636341i	$-0.780447\pi$
$71$	−13.0000	−1.54282	−0.771408	+	0.636341i	$0.780447\pi$
$72$	0	0
$73$	16.0000i	1.87266i	0.351123	+	0.936329i	$0.385800\pi$
$73$	16.0000i	1.87266i	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	10.0000i	1.09764i	0.835940	+	0.548821i	$0.184923\pi$
$83$	10.0000i	1.09764i	−0.835940	+	0.548821i	$0.815077\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	0	0
$99$	−15.0000	−1.50756
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	2.00000i	0.197066i	0.995134	+	0.0985329i	$0.0314150\pi$
$103$	2.00000i	0.197066i	−0.995134	+	0.0985329i	$0.968585\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	0	0
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.0000i	1.03479i	0.855746	+	0.517396i	$0.173099\pi$
$113$	11.0000i	1.03479i	−0.855746	+	0.517396i	$0.826901\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 18.0000i	− 1.66410i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 13.0000i	− 1.15356i	−0.816898	−	0.576782i	$-0.804308\pi$
$127$	− 13.0000i	− 1.15356i	0.816898	−	0.576782i	$-0.195692\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	0	0
$134$	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
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	30.0000i	2.50873i
$144$	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
$153$	− 12.0000i	− 0.970143i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 8.00000i	− 0.638470i	−0.947676	−	0.319235i	$-0.896574\pi$
$157$	− 8.00000i	− 0.638470i	0.947676	−	0.319235i	$-0.103426\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.00000i	0.464294i	0.972681	+	0.232147i	$0.0745750\pi$
$167$	6.00000i	0.464294i	−0.972681	+	0.232147i	$0.925425\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	−18.0000	−1.37649
$172$	0	0
$173$	22.0000i	1.67263i	0.548250	+	0.836315i	$0.315294\pi$
$173$	22.0000i	1.67263i	−0.548250	+	0.836315i	$0.684706\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−4.00000	−0.297318	−0.148659	−	0.988889i	$-0.547496\pi$
$181$	−4.00000	−0.297318	−0.148659	+	0.988889i	$0.547496\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	20.0000i	1.46254i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	11.0000i	0.791797i	0.918294	+	0.395899i	$0.129567\pi$
$193$	11.0000i	0.791797i	−0.918294	+	0.395899i	$0.870433\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.00000i	0.0712470i	0.999365	+	0.0356235i	$0.0113417\pi$
$197$	1.00000i	0.0712470i	−0.999365	+	0.0356235i	$0.988658\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 9.00000i	− 0.625543i
$208$	0	0
$209$	30.0000	2.07514
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−24.0000	−1.61441
$222$	0	0
$223$	− 28.0000i	− 1.87502i	−0.347960	−	0.937509i	$-0.613126\pi$
$223$	− 28.0000i	− 1.87502i	0.347960	−	0.937509i	$-0.386874\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 10.0000i	− 0.663723i	−0.943328	−	0.331862i	$-0.892323\pi$
$227$	− 10.0000i	− 0.663723i	0.943328	−	0.331862i	$-0.107677\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 27.0000i	− 1.76883i	−0.466702	−	0.884414i	$-0.654558\pi$
$233$	− 27.0000i	− 1.76883i	0.466702	−	0.884414i	$-0.345442\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	36.0000i	2.29063i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	15.0000i	0.943042i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.00000i	0.124757i	0.998053	+	0.0623783i	$0.0198685\pi$
$257$	2.00000i	0.124757i	−0.998053	+	0.0623783i	$0.980131\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	9.00000	0.557086
$262$	0	0
$263$	11.0000i	0.678289i	0.940734	+	0.339145i	$0.110138\pi$
$263$	11.0000i	0.678289i	−0.940734	+	0.339145i	$0.889862\pi$
$264$	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
$271$	−30.0000	−1.82237	−0.911185	−	0.411997i	$-0.864831\pi$
$271$	−30.0000	−1.82237	−0.911185	+	0.411997i	$0.864831\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.00000i	0.360505i	0.983620	+	0.180253i	$0.0576915\pi$
$277$	6.00000i	0.360505i	−0.983620	+	0.180253i	$0.942309\pi$
$278$	0	0
$279$	−6.00000	−0.359211
$280$	0	0
$281$	−19.0000	−1.13344	−0.566722	−	0.823909i	$-0.691789\pi$
$281$	−19.0000	−1.13344	−0.566722	+	0.823909i	$0.691789\pi$
$282$	0	0
$283$	22.0000i	1.30776i	0.756596	+	0.653882i	$0.226861\pi$
$283$	22.0000i	1.30776i	−0.756596	+	0.653882i	$0.773139\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.00000i	0.116841i	0.998292	+	0.0584206i	$0.0186065\pi$
$293$	2.00000i	0.116841i	−0.998292	+	0.0584206i	$0.981394\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−18.0000	−1.04097
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.0000i	0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$	16.0000i	0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	− 14.0000i	− 0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$313$	− 14.0000i	− 0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 19.0000i	− 1.06715i	−0.845754	−	0.533573i	$-0.820849\pi$
$317$	− 19.0000i	− 1.06715i	0.845754	−	0.533573i	$-0.179151\pi$
$318$	0	0
$319$	−15.0000	−0.839839
$320$	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
$329$	0	0
$330$	0	0
$331$	−11.0000	−0.604615	−0.302307	−	0.953211i	$-0.597757\pi$
$331$	−11.0000	−0.604615	−0.302307	+	0.953211i	$0.597757\pi$
$332$	0	0
$333$	21.0000i	1.15079i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.00000i	0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$	2.00000i	0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	10.0000	0.541530
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	29.0000i	1.55680i	0.627768	+	0.778401i	$0.283969\pi$
$347$	29.0000i	1.55680i	−0.627768	+	0.778401i	$0.716031\pi$
$348$	0	0
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.0000i	0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$	18.0000i	0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	25.0000	1.31945	0.659725	−	0.751507i	$-0.270673\pi$
$359$	25.0000	1.31945	0.659725	+	0.751507i	$0.270673\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 32.0000i	− 1.67039i	−0.549957	−	0.835193i	$-0.685356\pi$
$367$	− 32.0000i	− 1.67039i	0.549957	−	0.835193i	$-0.314644\pi$
$368$	0	0
$369$	12.0000	0.624695
$370$	0	0
$371$	0	0
$372$	0	0
$373$	23.0000i	1.19089i	0.803394	+	0.595447i	$0.203025\pi$
$373$	23.0000i	1.19089i	−0.803394	+	0.595447i	$0.796975\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 18.0000i	− 0.927047i
$378$	0	0
$379$	3.00000	0.154100	0.0770498	−	0.997027i	$-0.475450\pi$
$379$	3.00000	0.154100	0.0770498	+	0.997027i	$0.475450\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 6.00000i	− 0.306586i	−0.988181	−	0.153293i	$-0.951012\pi$
$383$	− 6.00000i	− 0.306586i	0.988181	−	0.153293i	$-0.0489878\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	21.0000i	1.06749i
$388$	0	0
$389$	19.0000	0.963338	0.481669	−	0.876353i	$-0.340031\pi$
$389$	19.0000	0.963338	0.481669	+	0.876353i	$0.340031\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 38.0000i	− 1.90717i	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	− 38.0000i	− 1.90717i	0.301131	−	0.953583i	$-0.402636\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.0000	0.649189	0.324595	−	0.945853i	$-0.394772\pi$
$401$	13.0000	0.649189	0.324595	+	0.945853i	$0.394772\pi$
$402$	0	0
$403$	12.0000i	0.597763i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 35.0000i	− 1.73489i
$408$	0	0
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	0	0
$421$	−25.0000	−1.21843	−0.609213	−	0.793007i	$-0.708514\pi$
$421$	−25.0000	−1.21843	−0.609213	+	0.793007i	$0.708514\pi$
$422$	0	0
$423$	6.00000i	0.291730i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	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
$433$	28.0000i	1.34559i	0.739827	+	0.672797i	$0.234907\pi$
$433$	28.0000i	1.34559i	−0.739827	+	0.672797i	$0.765093\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18.0000i	0.861057i
$438$	0	0
$439$	−2.00000	−0.0954548	−0.0477274	−	0.998860i	$-0.515198\pi$
$439$	−2.00000	−0.0954548	−0.0477274	+	0.998860i	$0.515198\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.0000i	0.760183i	0.924949	+	0.380091i	$0.124107\pi$
$443$	16.0000i	0.760183i	−0.924949	+	0.380091i	$0.875893\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−27.0000	−1.27421	−0.637104	−	0.770778i	$-0.719868\pi$
$449$	−27.0000	−1.27421	−0.637104	+	0.770778i	$0.719868\pi$
$450$	0	0
$451$	−20.0000	−0.941763
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 31.0000i	− 1.45012i	−0.688686	−	0.725059i	$-0.741812\pi$
$457$	− 31.0000i	− 1.45012i	0.688686	−	0.725059i	$-0.258188\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	26.0000	1.21094	0.605470	−	0.795868i	$-0.292985\pi$
$461$	26.0000	1.21094	0.605470	+	0.795868i	$0.292985\pi$
$462$	0	0
$463$	8.00000i	0.371792i	0.982569	+	0.185896i	$0.0595187\pi$
$463$	8.00000i	0.371792i	−0.982569	+	0.185896i	$0.940481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.0000i	0.647843i	0.946084	+	0.323921i	$0.105001\pi$
$467$	14.0000i	0.647843i	−0.946084	+	0.323921i	$0.894999\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 35.0000i	− 1.60930i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	30.0000i	1.37361i
$478$	0	0
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	42.0000	1.91504
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 23.0000i	− 1.04223i	−0.853487	−	0.521115i	$-0.825516\pi$
$487$	− 23.0000i	− 1.04223i	0.853487	−	0.521115i	$-0.174484\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 18.0000i	− 0.802580i	−0.915951	−	0.401290i	$-0.868562\pi$
$503$	− 18.0000i	− 0.802580i	0.915951	−	0.401290i	$-0.131438\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 10.0000i	− 0.439799i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	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
$527$	8.00000i	0.348485i
$528$	0	0
$529$	14.0000	0.608696
$530$	0	0
$531$	−42.0000	−1.82264
$532$	0	0
$533$	− 24.0000i	− 1.03956i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−21.0000	−0.902861	−0.451430	−	0.892306i	$-0.649086\pi$
$541$	−21.0000	−0.902861	−0.451430	+	0.892306i	$0.649086\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	27.0000i	1.15444i	0.816590	+	0.577218i	$0.195862\pi$
$547$	27.0000i	1.15444i	−0.816590	+	0.577218i	$0.804138\pi$
$548$	0	0
$549$	−12.0000	−0.512148
$550$	0	0
$551$	−18.0000	−0.766826
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 31.0000i	− 1.31351i	−0.754103	−	0.656756i	$-0.771928\pi$
$557$	− 31.0000i	− 1.31351i	0.754103	−	0.656756i	$-0.228072\pi$
$558$	0	0
$559$	42.0000	1.77641
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 26.0000i	− 1.09577i	−0.836554	−	0.547885i	$-0.815433\pi$
$563$	− 26.0000i	− 1.09577i	0.836554	−	0.547885i	$-0.184567\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.00000	0.125767	0.0628833	−	0.998021i	$-0.479970\pi$
$569$	3.00000	0.125767	0.0628833	+	0.998021i	$0.479970\pi$
$570$	0	0
$571$	−15.0000	−0.627730	−0.313865	−	0.949468i	$-0.601624\pi$
$571$	−15.0000	−0.627730	−0.313865	+	0.949468i	$0.601624\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	34.0000i	1.41544i	0.706494	+	0.707719i	$0.250276\pi$
$577$	34.0000i	1.41544i	−0.706494	+	0.707719i	$0.749724\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 50.0000i	− 2.07079i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	0	0
$591$	0	0
$592$	0	0
$593$	24.0000i	0.985562i	0.870153	+	0.492781i	$0.164020\pi$
$593$	24.0000i	0.985562i	−0.870153	+	0.492781i	$0.835980\pi$
$594$	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
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	9.00000i	0.366508i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	10.0000i	0.405887i	0.979190	+	0.202944i	$0.0650509\pi$
$607$	10.0000i	0.405887i	−0.979190	+	0.202944i	$0.934949\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	9.00000i	0.363507i	0.983344	+	0.181753i	$0.0581772\pi$
$613$	9.00000i	0.363507i	−0.983344	+	0.181753i	$0.941823\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 17.0000i	− 0.684394i	−0.939628	−	0.342197i	$-0.888829\pi$
$617$	− 17.0000i	− 0.684394i	0.939628	−	0.342197i	$-0.111171\pi$
$618$	0	0
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	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
$629$	28.0000	1.11643
$630$	0	0
$631$	29.0000	1.15447	0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000	1.15447	0.577236	+	0.816577i	$0.304131\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−39.0000	−1.54282
$640$	0	0
$641$	−7.00000	−0.276483	−0.138242	−	0.990399i	$-0.544145\pi$
$641$	−7.00000	−0.276483	−0.138242	+	0.990399i	$0.544145\pi$
$642$	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
$647$	− 14.0000i	− 0.550397i	−0.961387	−	0.275198i	$-0.911256\pi$
$647$	− 14.0000i	− 0.550397i	0.961387	−	0.275198i	$-0.0887435\pi$
$648$	0	0
$649$	70.0000	2.74774
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14.0000i	0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$	14.0000i	0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	48.0000i	1.87266i
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	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
$671$	20.0000	0.772091
$672$	0	0
$673$	− 38.0000i	− 1.46479i	−0.680879	−	0.732396i	$-0.738402\pi$
$673$	− 38.0000i	− 1.46479i	0.680879	−	0.732396i	$-0.261598\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 37.0000i	− 1.41577i	−0.706330	−	0.707883i	$-0.749650\pi$
$683$	− 37.0000i	− 1.41577i	0.706330	−	0.707883i	$-0.250350\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	60.0000	2.28582
$690$	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
$697$	− 16.0000i	− 0.606043i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	0	0
$703$	− 42.0000i	− 1.58406i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	−3.00000	−0.112509
$712$	0	0
$713$	6.00000i	0.224702i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	48.0000i	1.78022i	0.455744	+	0.890111i	$0.349373\pi$
$727$	48.0000i	1.78022i	−0.455744	+	0.890111i	$0.650627\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	28.0000	1.03562
$732$	0	0
$733$	20.0000i	0.738717i	0.929287	+	0.369358i	$0.120423\pi$
$733$	20.0000i	0.738717i	−0.929287	+	0.369358i	$0.879577\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 15.0000i	− 0.552532i
$738$	0	0
$739$	−7.00000	−0.257499	−0.128750	−	0.991677i	$-0.541096\pi$
$739$	−7.00000	−0.257499	−0.128750	+	0.991677i	$0.541096\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 36.0000i	− 1.32071i	−0.750953	−	0.660356i	$-0.770405\pi$
$743$	− 36.0000i	− 1.32071i	0.750953	−	0.660356i	$-0.229595\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	30.0000i	1.09764i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	11.0000i	0.399802i	0.979816	+	0.199901i	$0.0640620\pi$
$757$	11.0000i	0.399802i	−0.979816	+	0.199901i	$0.935938\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	84.0000i	3.03306i
$768$	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
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	65.0000	2.32588
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	28.0000i	0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$	28.0000i	0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	24.0000i	0.852265i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 48.0000i	− 1.70025i	−0.526583	−	0.850124i	$-0.676527\pi$
$797$	− 48.0000i	− 1.70025i	0.526583	−	0.850124i	$-0.323473\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	30.0000	1.06000
$802$	0	0
$803$	− 80.0000i	− 2.82314i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−39.0000	−1.37117	−0.685583	−	0.727994i	$-0.740453\pi$
$809$	−39.0000	−1.37117	−0.685583	+	0.727994i	$0.740453\pi$
$810$	0	0
$811$	−14.0000	−0.491606	−0.245803	−	0.969320i	$-0.579052\pi$
$811$	−14.0000	−0.491606	−0.245803	+	0.969320i	$0.579052\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 42.0000i	− 1.46939i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	1.00000i	0.0348578i	0.999848	+	0.0174289i	$0.00554807\pi$
$823$	1.00000i	0.0348578i	−0.999848	+	0.0174289i	$0.994452\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 47.0000i	− 1.63435i	−0.576390	−	0.817175i	$-0.695539\pi$
$827$	− 47.0000i	− 1.63435i	0.576390	−	0.817175i	$-0.304461\pi$
$828$	0	0
$829$	32.0000	1.11141	0.555703	−	0.831381i	$-0.312449\pi$
$829$	32.0000	1.11141	0.555703	+	0.831381i	$0.312449\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	21.0000	0.719871
$852$	0	0
$853$	16.0000i	0.547830i	0.961754	+	0.273915i	$0.0883186\pi$
$853$	16.0000i	0.547830i	−0.961754	+	0.273915i	$0.911681\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12.0000i	0.409912i	0.978771	+	0.204956i	$0.0657052\pi$
$857$	12.0000i	0.409912i	−0.978771	+	0.204956i	$0.934295\pi$
$858$	0	0
$859$	6.00000	0.204717	0.102359	−	0.994748i	$-0.467361\pi$
$859$	6.00000	0.204717	0.102359	+	0.994748i	$0.467361\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	43.0000i	1.46374i	0.681446	+	0.731869i	$0.261351\pi$
$863$	43.0000i	1.46374i	−0.681446	+	0.731869i	$0.738649\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.00000	0.169613
$870$	0	0
$871$	18.0000	0.609907
$872$	0	0
$873$	6.00000i	0.203069i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10.0000i	0.337676i	0.985644	+	0.168838i	$0.0540015\pi$
$877$	10.0000i	0.337676i	−0.985644	+	0.168838i	$0.945999\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	0	0
$883$	− 27.0000i	− 0.908622i	−0.890843	−	0.454311i	$-0.849885\pi$
$883$	− 27.0000i	− 0.908622i	0.890843	−	0.454311i	$-0.150115\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	44.0000i	1.47738i	0.674048	+	0.738688i	$0.264554\pi$
$887$	44.0000i	1.47738i	−0.674048	+	0.738688i	$0.735446\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−45.0000	−1.50756
$892$	0	0
$893$	− 12.0000i	− 0.401565i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−6.00000	−0.200111
$900$	0	0
$901$	40.0000	1.33259
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	28.0000i	0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$	28.0000i	0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$	0	0
$909$	−36.0000	−1.19404
$910$	0	0
$911$	57.0000	1.88849	0.944247	−	0.329238i	$-0.106792\pi$
$911$	57.0000	1.88849	0.944247	+	0.329238i	$0.106792\pi$
$912$	0	0
$913$	− 50.0000i	− 1.65476i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−41.0000	−1.35247	−0.676233	−	0.736688i	$-0.736389\pi$
$919$	−41.0000	−1.35247	−0.676233	+	0.736688i	$0.736389\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	78.0000i	2.56740i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.00000i	0.197066i
$928$	0	0
$929$	−20.0000	−0.656179	−0.328089	−	0.944647i	$-0.606405\pi$
$929$	−20.0000	−0.656179	−0.328089	+	0.944647i	$0.606405\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 4.00000i	− 0.130674i	−0.997863	−	0.0653372i	$-0.979188\pi$
$937$	− 4.00000i	− 0.130674i	0.997863	−	0.0653372i	$-0.0208123\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	22.0000	0.717180	0.358590	−	0.933495i	$-0.383258\pi$
$941$	22.0000	0.717180	0.358590	+	0.933495i	$0.383258\pi$
$942$	0	0
$943$	− 12.0000i	− 0.390774i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.00000i	0.129983i	0.997886	+	0.0649913i	$0.0207020\pi$
$947$	4.00000i	0.129983i	−0.997886	+	0.0649913i	$0.979298\pi$
$948$	0	0
$949$	96.0000	3.11629
$950$	0	0
$951$	0	0
$952$	0	0
$953$	15.0000i	0.485898i	0.970039	+	0.242949i	$0.0781147\pi$
$953$	15.0000i	0.485898i	−0.970039	+	0.242949i	$0.921885\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	− 36.0000i	− 1.16008i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	56.0000i	1.80084i	0.435023	+	0.900419i	$0.356740\pi$
$967$	56.0000i	1.80084i	−0.435023	+	0.900419i	$0.643260\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−30.0000	−0.962746	−0.481373	−	0.876516i	$-0.659862\pi$
$971$	−30.0000	−0.962746	−0.481373	+	0.876516i	$0.659862\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 9.00000i	− 0.287936i	−0.989582	−	0.143968i	$-0.954014\pi$
$977$	− 9.00000i	− 0.287936i	0.989582	−	0.143968i	$-0.0459862\pi$
$978$	0	0
$979$	−50.0000	−1.59801
$980$	0	0
$981$	−27.0000	−0.862044
$982$	0	0
$983$	− 50.0000i	− 1.59475i	−0.603483	−	0.797376i	$-0.706221\pi$
$983$	− 50.0000i	− 1.59475i	0.603483	−	0.797376i	$-0.293779\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	21.0000	0.667761
$990$	0	0
$991$	−11.0000	−0.349427	−0.174713	−	0.984619i	$-0.555900\pi$
$991$	−11.0000	−0.349427	−0.174713	+	0.984619i	$0.555900\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.00000i	0.0633406i	0.999498	+	0.0316703i	$0.0100827\pi$
$997$	2.00000i	0.0633406i	−0.999498	+	0.0316703i	$0.989917\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.e.o.2549.1		2
5.2	odd	4		4900.2.a.j.1.1		1
5.3	odd	4		4900.2.a.k.1.1		1
5.4	even	2	inner	4900.2.e.o.2549.2		2
7.6	odd	2		700.2.e.d.449.2		2
21.20	even	2		6300.2.k.q.6049.2		2
28.27	even	2		2800.2.g.q.449.1		2
35.13	even	4		700.2.a.g.1.1	yes	1
35.27	even	4		700.2.a.e.1.1	✓	1
35.34	odd	2		700.2.e.d.449.1		2
105.62	odd	4		6300.2.a.o.1.1		1
105.83	odd	4		6300.2.a.be.1.1		1
105.104	even	2		6300.2.k.q.6049.1		2
140.27	odd	4		2800.2.a.t.1.1		1
140.83	odd	4		2800.2.a.q.1.1		1
140.139	even	2		2800.2.g.q.449.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.a.e.1.1	✓	1	35.27	even	4
700.2.a.g.1.1	yes	1	35.13	even	4
700.2.e.d.449.1		2	35.34	odd	2
700.2.e.d.449.2		2	7.6	odd	2
2800.2.a.q.1.1		1	140.83	odd	4
2800.2.a.t.1.1		1	140.27	odd	4
2800.2.g.q.449.1		2	28.27	even	2
2800.2.g.q.449.2		2	140.139	even	2
4900.2.a.j.1.1		1	5.2	odd	4
4900.2.a.k.1.1		1	5.3	odd	4
4900.2.e.o.2549.1		2	1.1	even	1	trivial
4900.2.e.o.2549.2		2	5.4	even	2	inner
6300.2.a.o.1.1		1	105.62	odd	4
6300.2.a.be.1.1		1	105.83	odd	4
6300.2.k.q.6049.1		2	105.104	even	2
6300.2.k.q.6049.2		2	21.20	even	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:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{9} -5.00000 q^{11} -6.00000i q^{13} -4.00000i q^{17} -6.00000 q^{19} -3.00000i q^{23} +3.00000 q^{29} -2.00000 q^{31} +7.00000i q^{37} +4.00000 q^{41} +7.00000i q^{43} +2.00000i q^{47} +10.0000i q^{53} -14.0000 q^{59} -4.00000 q^{61} +3.00000i q^{67} -13.0000 q^{71} +16.0000i q^{73} -1.00000 q^{79} +9.00000 q^{81} +10.0000i q^{83} +10.0000 q^{89} +2.00000i q^{97} -15.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{9}+O(q^{10}) \) 2 * q + 6 * q^9	\( 2 q + 6 q^{9} - 10 q^{11} - 12 q^{19} + 6 q^{29} - 4 q^{31} + 8 q^{41} - 28 q^{59} - 8 q^{61} - 26 q^{71} - 2 q^{79} + 18 q^{81} + 20 q^{89} - 30 q^{99}+O(q^{100}) \) 2 * q + 6 * q^9 - 10 * q^11 - 12 * q^19 + 6 * q^29 - 4 * q^31 + 8 * q^41 - 28 * q^59 - 8 * q^61 - 26 * q^71 - 2 * q^79 + 18 * q^81 + 20 * q^89 - 30 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more