LMFDB - Embedded newform 4900.2.e.d.2549.2

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:	$0$
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:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} -1.00000 q^{9} +O(q^{10})$	$q+2.00000i q^{3} -1.00000 q^{9} -1.00000 q^{11} +2.00000i q^{13} +4.00000i q^{17} -5.00000i q^{23} +4.00000i q^{27} +3.00000 q^{29} -10.0000 q^{31} -2.00000i q^{33} +5.00000i q^{37} -4.00000 q^{39} -10.0000 q^{41} +5.00000i q^{43} -4.00000i q^{47} -8.00000 q^{51} -10.0000i q^{53} +10.0000 q^{59} +10.0000 q^{61} +5.00000i q^{67} +10.0000 q^{69} +3.00000 q^{71} +10.0000i q^{73} -13.0000 q^{79} -11.0000 q^{81} +10.0000i q^{83} +6.00000i q^{87} -10.0000 q^{89} -20.0000i q^{93} +6.00000i q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} - 2 q^{11} + 6 q^{29} - 20 q^{31} - 8 q^{39} - 20 q^{41} - 16 q^{51} + 20 q^{59} + 20 q^{61} + 20 q^{69} + 6 q^{71} - 26 q^{79} - 22 q^{81} - 20 q^{89} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 2 * q^11 + 6 * q^29 - 20 * q^31 - 8 * q^39 - 20 * q^41 - 16 * q^51 + 20 * q^59 + 20 * q^61 + 20 * q^69 + 6 * q^71 - 26 * q^79 - 22 * q^81 - 20 * q^89 + 2 * 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
$3$	2.00000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	2.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000i	0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$	4.00000i	0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 5.00000i	− 1.04257i	−0.853382	−	0.521286i	$-0.825452\pi$
$23$	− 5.00000i	− 1.04257i	0.853382	−	0.521286i	$-0.174548\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.00000i	0.769800i
$28$	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
$33$	− 2.00000i	− 0.348155i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.00000i	0.821995i	0.911636	+	0.410997i	$0.134819\pi$
$37$	5.00000i	0.821995i	−0.911636	+	0.410997i	$0.865181\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	5.00000i	0.762493i	0.924473	+	0.381246i	$0.124505\pi$
$43$	5.00000i	0.762493i	−0.924473	+	0.381246i	$0.875495\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 4.00000i	− 0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$	− 4.00000i	− 0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−8.00000	−1.12022
$52$	0	0
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.00000i	0.610847i	0.952217	+	0.305424i	$0.0987981\pi$
$67$	5.00000i	0.610847i	−0.952217	+	0.305424i	$0.901202\pi$
$68$	0	0
$69$	10.0000	1.20386
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−13.0000	−1.46261	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−13.0000	−1.46261	−0.731307	+	0.682048i	$0.761089\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	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
$87$	6.00000i	0.643268i
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 20.0000i	− 2.07390i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.00000i	0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$	6.00000i	0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	−13.0000	−1.24517	−0.622587	−	0.782551i	$-0.713918\pi$
$109$	−13.0000	−1.24517	−0.622587	+	0.782551i	$0.713918\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	0	0
$113$	5.00000i	0.470360i	0.971952	+	0.235180i	$0.0755680\pi$
$113$	5.00000i	0.470360i	−0.971952	+	0.235180i	$0.924432\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	− 20.0000i	− 1.80334i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 15.0000i	− 1.33103i	−0.746382	−	0.665517i	$-0.768211\pi$
$127$	− 15.0000i	− 1.33103i	0.746382	−	0.665517i	$-0.231789\pi$
$128$	0	0
$129$	−10.0000	−0.880451
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.0000i	0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$	10.0000i	0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	− 2.00000i	− 0.167248i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−11.0000	−0.901155	−0.450578	−	0.892737i	$-0.648782\pi$
$149$	−11.0000	−0.901155	−0.450578	+	0.892737i	$0.648782\pi$
$150$	0	0
$151$	−9.00000	−0.732410	−0.366205	−	0.930534i	$-0.619343\pi$
$151$	−9.00000	−0.732410	−0.366205	+	0.930534i	$0.619343\pi$
$152$	0	0
$153$	− 4.00000i	− 0.323381i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	0	0
$159$	20.0000	1.58610
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	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
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.0000i	0.912343i	0.889892	+	0.456172i	$0.150780\pi$
$173$	12.0000i	0.912343i	−0.889892	+	0.456172i	$0.849220\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	20.0000i	1.50329i
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	20.0000i	1.47844i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 4.00000i	− 0.292509i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	− 15.0000i	− 1.07972i	−0.841754	−	0.539862i	$-0.818476\pi$
$193$	− 15.0000i	− 1.07972i	0.841754	−	0.539862i	$-0.181524\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.0000i	1.06871i	0.845262	+	0.534353i	$0.179445\pi$
$197$	15.0000i	1.06871i	−0.845262	+	0.534353i	$0.820555\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	−10.0000	−0.705346
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	5.00000i	0.347524i
$208$	0	0
$209$	0	0
$210$	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
$213$	6.00000i	0.411113i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−20.0000	−1.35147
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	− 2.00000i	− 0.133930i	−0.997755	−	0.0669650i	$-0.978668\pi$
$223$	− 2.00000i	− 0.133930i	0.997755	−	0.0669650i	$-0.0213316\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 24.0000i	− 1.59294i	−0.604681	−	0.796468i	$-0.706699\pi$
$227$	− 24.0000i	− 1.59294i	0.604681	−	0.796468i	$-0.293301\pi$
$228$	0	0
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 25.0000i	− 1.63780i	−0.573933	−	0.818902i	$-0.694583\pi$
$233$	− 25.0000i	− 1.63780i	0.573933	−	0.818902i	$-0.305417\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 26.0000i	− 1.68888i
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	0	0
$243$	− 10.0000i	− 0.641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−20.0000	−1.26745
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	5.00000i	0.314347i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	30.0000i	1.87135i	0.352865	+	0.935674i	$0.385208\pi$
$257$	30.0000i	1.87135i	−0.352865	+	0.935674i	$0.614792\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3.00000	−0.185695
$262$	0	0
$263$	− 15.0000i	− 0.924940i	−0.886635	−	0.462470i	$-0.846963\pi$
$263$	− 15.0000i	− 0.924940i	0.886635	−	0.462470i	$-0.153037\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 20.0000i	− 1.22398i
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−10.0000	−0.607457	−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000	−0.607457	−0.303728	+	0.952759i	$0.598232\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	0	0
$281$	29.0000	1.72999	0.864997	−	0.501776i	$-0.167320\pi$
$281$	29.0000	1.72999	0.864997	+	0.501776i	$0.167320\pi$
$282$	0	0
$283$	− 32.0000i	− 1.90220i	−0.308879	−	0.951101i	$-0.599954\pi$
$283$	− 32.0000i	− 1.90220i	0.308879	−	0.951101i	$-0.400046\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−12.0000	−0.703452
$292$	0	0
$293$	− 18.0000i	− 1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$	− 18.0000i	− 1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 4.00000i	− 0.232104i
$298$	0	0
$299$	10.0000	0.578315
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 20.0000i	− 1.14897i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	0	0
$311$	20.0000	1.13410	0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000	1.13410	0.567048	+	0.823685i	$0.308085\pi$
$312$	0	0
$313$	28.0000i	1.58265i	0.611393	+	0.791327i	$0.290609\pi$
$313$	28.0000i	1.58265i	−0.611393	+	0.791327i	$0.709391\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.0000i	0.842484i	0.906948	+	0.421242i	$0.138406\pi$
$317$	15.0000i	0.842484i	−0.906948	+	0.421242i	$0.861594\pi$
$318$	0	0
$319$	−3.00000	−0.167968
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 26.0000i	− 1.43780i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	0	0
$333$	− 5.00000i	− 0.273998i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	10.0000i	0.544735i	0.962193	+	0.272367i	$0.0878066\pi$
$337$	10.0000i	0.544735i	−0.962193	+	0.272367i	$0.912193\pi$
$338$	0	0
$339$	−10.0000	−0.543125
$340$	0	0
$341$	10.0000	0.541530
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	35.0000i	1.87890i	0.342688	+	0.939449i	$0.388663\pi$
$347$	35.0000i	1.87890i	−0.342688	+	0.939449i	$0.611337\pi$
$348$	0	0
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	0	0
$351$	−8.00000	−0.427008
$352$	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
$359$	−19.0000	−1.00278	−0.501391	−	0.865221i	$-0.667178\pi$
$359$	−19.0000	−1.00278	−0.501391	+	0.865221i	$0.667178\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	− 20.0000i	− 1.04973i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 26.0000i	− 1.35719i	−0.734513	−	0.678594i	$-0.762589\pi$
$367$	− 26.0000i	− 1.35719i	0.734513	−	0.678594i	$-0.237411\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 15.0000i	− 0.776671i	−0.921518	−	0.388335i	$-0.873050\pi$
$373$	− 15.0000i	− 0.776671i	0.921518	−	0.388335i	$-0.126950\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.00000i	0.309016i
$378$	0	0
$379$	−29.0000	−1.48963	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000	−1.48963	−0.744815	+	0.667271i	$0.767462\pi$
$380$	0	0
$381$	30.0000	1.53695
$382$	0	0
$383$	− 10.0000i	− 0.510976i	−0.966812	−	0.255488i	$-0.917764\pi$
$383$	− 10.0000i	− 0.510976i	0.966812	−	0.255488i	$-0.0822362\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 5.00000i	− 0.254164i
$388$	0	0
$389$	3.00000	0.152106	0.0760530	−	0.997104i	$-0.475768\pi$
$389$	3.00000	0.152106	0.0760530	+	0.997104i	$0.475768\pi$
$390$	0	0
$391$	20.0000	1.01144
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	16.0000i	0.803017i	0.915855	+	0.401508i	$0.131514\pi$
$397$	16.0000i	0.803017i	−0.915855	+	0.401508i	$0.868486\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.00000	0.0499376	0.0249688	−	0.999688i	$-0.492051\pi$
$401$	1.00000	0.0499376	0.0249688	+	0.999688i	$0.492051\pi$
$402$	0	0
$403$	− 20.0000i	− 0.996271i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 5.00000i	− 0.247841i
$408$	0	0
$409$	−20.0000	−0.988936	−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000	−0.988936	−0.494468	+	0.869196i	$0.664637\pi$
$410$	0	0
$411$	−20.0000	−0.986527
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 20.0000i	− 0.979404i
$418$	0	0
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	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
$423$	4.00000i	0.194487i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	4.00000	0.193122
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	20.0000i	0.961139i	0.876957	+	0.480569i	$0.159570\pi$
$433$	20.0000i	0.961139i	−0.876957	+	0.480569i	$0.840430\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	20.0000i	0.950229i	0.879924	+	0.475114i	$0.157593\pi$
$443$	20.0000i	0.950229i	−0.879924	+	0.475114i	$0.842407\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 22.0000i	− 1.04056i
$448$	0	0
$449$	33.0000	1.55737	0.778683	−	0.627417i	$-0.215888\pi$
$449$	33.0000	1.55737	0.778683	+	0.627417i	$0.215888\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	0	0
$453$	− 18.0000i	− 0.845714i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15.0000i	0.701670i	0.936437	+	0.350835i	$0.114102\pi$
$457$	15.0000i	0.701670i	−0.936437	+	0.350835i	$0.885898\pi$
$458$	0	0
$459$	−16.0000	−0.746816
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	20.0000i	0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$	20.0000i	0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.00000i	0.185098i	0.995708	+	0.0925490i	$0.0295015\pi$
$467$	4.00000i	0.185098i	−0.995708	+	0.0925490i	$0.970499\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−20.0000	−0.921551
$472$	0	0
$473$	− 5.00000i	− 0.229900i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.0000i	0.457869i
$478$	0	0
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	0	0
$481$	−10.0000	−0.455961
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 5.00000i	− 0.226572i	−0.993562	−	0.113286i	$-0.963862\pi$
$487$	− 5.00000i	− 0.226572i	0.993562	−	0.113286i	$-0.0361376\pi$
$488$	0	0
$489$	40.0000	1.80886
$490$	0	0
$491$	−21.0000	−0.947717	−0.473858	−	0.880601i	$-0.657139\pi$
$491$	−21.0000	−0.947717	−0.473858	+	0.880601i	$0.657139\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	0	0
$503$	12.0000i	0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$	12.0000i	0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	18.0000i	0.799408i
$508$	0	0
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	4.00000i	0.175920i
$518$	0	0
$519$	−24.0000	−1.05348
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 40.0000i	− 1.74243i
$528$	0	0
$529$	−2.00000	−0.0869565
$530$	0	0
$531$	−10.0000	−0.433963
$532$	0	0
$533$	− 20.0000i	− 0.866296i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 8.00000i	− 0.345225i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	0	0
$543$	20.0000i	0.858282i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 35.0000i	− 1.49649i	−0.663421	−	0.748246i	$-0.730896\pi$
$547$	− 35.0000i	− 1.49649i	0.663421	−	0.748246i	$-0.269104\pi$
$548$	0	0
$549$	−10.0000	−0.426790
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15.0000i	0.635570i	0.948163	+	0.317785i	$0.102939\pi$
$557$	15.0000i	0.635570i	−0.948163	+	0.317785i	$0.897061\pi$
$558$	0	0
$559$	−10.0000	−0.422955
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.00000	−0.0419222	−0.0209611	−	0.999780i	$-0.506673\pi$
$569$	−1.00000	−0.0419222	−0.0209611	+	0.999780i	$0.506673\pi$
$570$	0	0
$571$	−3.00000	−0.125546	−0.0627730	−	0.998028i	$-0.519994\pi$
$571$	−3.00000	−0.125546	−0.0627730	+	0.998028i	$0.519994\pi$
$572$	0	0
$573$	− 48.0000i	− 2.00523i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 16.0000i	− 0.666089i	−0.942911	−	0.333044i	$-0.891924\pi$
$577$	− 16.0000i	− 0.666089i	0.942911	−	0.333044i	$-0.108076\pi$
$578$	0	0
$579$	30.0000	1.24676
$580$	0	0
$581$	0	0
$582$	0	0
$583$	10.0000i	0.414158i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.0000i	0.412744i	0.978474	+	0.206372i	$0.0661657\pi$
$587$	10.0000i	0.412744i	−0.978474	+	0.206372i	$0.933834\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−30.0000	−1.23404
$592$	0	0
$593$	28.0000i	1.14982i	0.818216	+	0.574911i	$0.194963\pi$
$593$	28.0000i	1.14982i	−0.818216	+	0.574911i	$0.805037\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	20.0000i	0.818546i
$598$	0	0
$599$	43.0000	1.75693	0.878466	−	0.477805i	$-0.158567\pi$
$599$	43.0000	1.75693	0.878466	+	0.477805i	$0.158567\pi$
$600$	0	0
$601$	−40.0000	−1.63163	−0.815817	−	0.578310i	$-0.803712\pi$
$601$	−40.0000	−1.63163	−0.815817	+	0.578310i	$0.803712\pi$
$602$	0	0
$603$	− 5.00000i	− 0.203616i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	34.0000i	1.38002i	0.723801	+	0.690009i	$0.242393\pi$
$607$	34.0000i	1.38002i	−0.723801	+	0.690009i	$0.757607\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	35.0000i	1.41364i	0.707395	+	0.706818i	$0.249870\pi$
$613$	35.0000i	1.41364i	−0.707395	+	0.706818i	$0.750130\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	45.0000i	1.81163i	0.423672	+	0.905816i	$0.360741\pi$
$617$	45.0000i	1.81163i	−0.423672	+	0.905816i	$0.639259\pi$
$618$	0	0
$619$	30.0000	1.20580	0.602901	−	0.797816i	$-0.294011\pi$
$619$	30.0000	1.20580	0.602901	+	0.797816i	$0.294011\pi$
$620$	0	0
$621$	20.0000	0.802572
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−20.0000	−0.797452
$630$	0	0
$631$	1.00000	0.0398094	0.0199047	−	0.999802i	$-0.493664\pi$
$631$	1.00000	0.0398094	0.0199047	+	0.999802i	$0.493664\pi$
$632$	0	0
$633$	− 8.00000i	− 0.317971i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−3.00000	−0.118678
$640$	0	0
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	12.0000i	0.473234i	0.971603	+	0.236617i	$0.0760386\pi$
$643$	12.0000i	0.473234i	−0.971603	+	0.236617i	$0.923961\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20.0000i	0.786281i	0.919478	+	0.393141i	$0.128611\pi$
$647$	20.0000i	0.786281i	−0.919478	+	0.393141i	$0.871389\pi$
$648$	0	0
$649$	−10.0000	−0.392534
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.0000i	0.391330i	0.980671	+	0.195665i	$0.0626866\pi$
$653$	10.0000i	0.391330i	−0.980671	+	0.195665i	$0.937313\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 10.0000i	− 0.390137i
$658$	0	0
$659$	−8.00000	−0.311636	−0.155818	−	0.987786i	$-0.549801\pi$
$659$	−8.00000	−0.311636	−0.155818	+	0.987786i	$0.549801\pi$
$660$	0	0
$661$	−40.0000	−1.55582	−0.777910	−	0.628376i	$-0.783720\pi$
$661$	−40.0000	−1.55582	−0.777910	+	0.628376i	$0.783720\pi$
$662$	0	0
$663$	− 16.0000i	− 0.621389i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 15.0000i	− 0.580802i
$668$	0	0
$669$	4.00000	0.154649
$670$	0	0
$671$	−10.0000	−0.386046
$672$	0	0
$673$	− 10.0000i	− 0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$	− 10.0000i	− 0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 44.0000i	− 1.69106i	−0.533930	−	0.845529i	$-0.679285\pi$
$677$	− 44.0000i	− 1.69106i	0.533930	−	0.845529i	$-0.320715\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	48.0000	1.83936
$682$	0	0
$683$	− 15.0000i	− 0.573959i	−0.957937	−	0.286980i	$-0.907349\pi$
$683$	− 15.0000i	− 0.573959i	0.957937	−	0.286980i	$-0.0926512\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 40.0000i	− 1.52610i
$688$	0	0
$689$	20.0000	0.761939
$690$	0	0
$691$	10.0000	0.380418	0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000	0.380418	0.190209	+	0.981744i	$0.439083\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 40.0000i	− 1.51511i
$698$	0	0
$699$	50.0000	1.89117
$700$	0	0
$701$	34.0000	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	42.0000	1.57734	0.788672	−	0.614815i	$-0.210769\pi$
$709$	42.0000	1.57734	0.788672	+	0.614815i	$0.210769\pi$
$710$	0	0
$711$	13.0000	0.487538
$712$	0	0
$713$	50.0000i	1.87251i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	16.0000i	0.597531i
$718$	0	0
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	40.0000i	1.48762i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	40.0000i	1.48352i	0.670667	+	0.741759i	$0.266008\pi$
$727$	40.0000i	1.48352i	−0.670667	+	0.741759i	$0.733992\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−20.0000	−0.739727
$732$	0	0
$733$	− 48.0000i	− 1.77292i	−0.462805	−	0.886460i	$-0.653157\pi$
$733$	− 48.0000i	− 1.77292i	0.462805	−	0.886460i	$-0.346843\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5.00000i	− 0.184177i
$738$	0	0
$739$	−3.00000	−0.110357	−0.0551784	−	0.998477i	$-0.517573\pi$
$739$	−3.00000	−0.110357	−0.0551784	+	0.998477i	$0.517573\pi$
$740$	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
$747$	− 10.0000i	− 0.365881i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	0	0
$753$	40.0000i	1.45768i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	25.0000i	0.908640i	0.890838	+	0.454320i	$0.150118\pi$
$757$	25.0000i	0.908640i	−0.890838	+	0.454320i	$0.849882\pi$
$758$	0	0
$759$	−10.0000	−0.362977
$760$	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
$767$	20.0000i	0.722158i
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	−60.0000	−2.16085
$772$	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
$781$	−3.00000	−0.107348
$782$	0	0
$783$	12.0000i	0.428845i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	6.00000i	0.213877i	0.994266	+	0.106938i	$0.0341048\pi$
$787$	6.00000i	0.213877i	−0.994266	+	0.106938i	$0.965895\pi$
$788$	0	0
$789$	30.0000	1.06803
$790$	0	0
$791$	0	0
$792$	0	0
$793$	20.0000i	0.710221i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.00000i	0.141687i	0.997487	+	0.0708436i	$0.0225691\pi$
$797$	4.00000i	0.141687i	−0.997487	+	0.0708436i	$0.977431\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	10.0000	0.353333
$802$	0	0
$803$	− 10.0000i	− 0.352892i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 20.0000i	− 0.704033i
$808$	0	0
$809$	13.0000	0.457056	0.228528	−	0.973537i	$-0.426609\pi$
$809$	13.0000	0.457056	0.228528	+	0.973537i	$0.426609\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	− 20.0000i	− 0.701431i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	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
$823$	55.0000i	1.91718i	0.284791	+	0.958590i	$0.408076\pi$
$823$	55.0000i	1.91718i	−0.284791	+	0.958590i	$0.591924\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	35.0000i	1.21707i	0.793527	+	0.608535i	$0.208242\pi$
$827$	35.0000i	1.21707i	−0.793527	+	0.608535i	$0.791758\pi$
$828$	0	0
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	0	0
$831$	20.0000	0.693792
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 40.0000i	− 1.38260i
$838$	0	0
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	58.0000i	1.99763i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	64.0000	2.19647
$850$	0	0
$851$	25.0000	0.856989
$852$	0	0
$853$	20.0000i	0.684787i	0.939557	+	0.342393i	$0.111238\pi$
$853$	20.0000i	0.684787i	−0.939557	+	0.342393i	$0.888762\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	30.0000i	1.02478i	0.858753	+	0.512390i	$0.171240\pi$
$857$	30.0000i	1.02478i	−0.858753	+	0.512390i	$0.828760\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	45.0000i	1.53182i	0.642949	+	0.765909i	$0.277711\pi$
$863$	45.0000i	1.53182i	−0.642949	+	0.765909i	$0.722289\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.00000i	0.0679236i
$868$	0	0
$869$	13.0000	0.440995
$870$	0	0
$871$	−10.0000	−0.338837
$872$	0	0
$873$	− 6.00000i	− 0.203069i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	30.0000i	1.01303i	0.862232	+	0.506514i	$0.169066\pi$
$877$	30.0000i	1.01303i	−0.862232	+	0.506514i	$0.830934\pi$
$878$	0	0
$879$	36.0000	1.21425
$880$	0	0
$881$	−50.0000	−1.68454	−0.842271	−	0.539054i	$-0.818782\pi$
$881$	−50.0000	−1.68454	−0.842271	+	0.539054i	$0.818782\pi$
$882$	0	0
$883$	− 25.0000i	− 0.841317i	−0.907219	−	0.420658i	$-0.861799\pi$
$883$	− 25.0000i	− 0.841317i	0.907219	−	0.420658i	$-0.138201\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 10.0000i	− 0.335767i	−0.985807	−	0.167884i	$-0.946307\pi$
$887$	− 10.0000i	− 0.335767i	0.985807	−	0.167884i	$-0.0536933\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	11.0000	0.368514
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	20.0000i	0.667781i
$898$	0	0
$899$	−30.0000	−1.00056
$900$	0	0
$901$	40.0000	1.33259
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 20.0000i	− 0.664089i	−0.943264	−	0.332045i	$-0.892262\pi$
$907$	− 20.0000i	− 0.664089i	0.943264	−	0.332045i	$-0.107738\pi$
$908$	0	0
$909$	10.0000	0.331679
$910$	0	0
$911$	13.0000	0.430709	0.215355	−	0.976536i	$-0.430909\pi$
$911$	13.0000	0.430709	0.215355	+	0.976536i	$0.430909\pi$
$912$	0	0
$913$	− 10.0000i	− 0.330952i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−17.0000	−0.560778	−0.280389	−	0.959886i	$-0.590464\pi$
$919$	−17.0000	−0.560778	−0.280389	+	0.959886i	$0.590464\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	0	0
$923$	6.00000i	0.197492i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8.00000i	0.262754i
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	40.0000i	1.30954i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	54.0000i	1.76410i	0.471153	+	0.882052i	$0.343838\pi$
$937$	54.0000i	1.76410i	−0.471153	+	0.882052i	$0.656162\pi$
$938$	0	0
$939$	−56.0000	−1.82749
$940$	0	0
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	0	0
$943$	50.0000i	1.62822i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 60.0000i	− 1.94974i	−0.222779	−	0.974869i	$-0.571513\pi$
$947$	− 60.0000i	− 1.94974i	0.222779	−	0.974869i	$-0.428487\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	−30.0000	−0.972817
$952$	0	0
$953$	− 35.0000i	− 1.13376i	−0.823800	−	0.566881i	$-0.808150\pi$
$953$	− 35.0000i	− 1.13376i	0.823800	−	0.566881i	$-0.191850\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 6.00000i	− 0.193952i
$958$	0	0
$959$	0	0
$960$	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	$-0.777624\pi$
$967$	− 40.0000i	− 1.28631i	0.765735	−	0.643157i	$-0.222376\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	45.0000i	1.43968i	0.694141	+	0.719839i	$0.255784\pi$
$977$	45.0000i	1.43968i	−0.694141	+	0.719839i	$0.744216\pi$
$978$	0	0
$979$	10.0000	0.319601
$980$	0	0
$981$	13.0000	0.415058
$982$	0	0
$983$	− 22.0000i	− 0.701691i	−0.936433	−	0.350846i	$-0.885894\pi$
$983$	− 22.0000i	− 0.701691i	0.936433	−	0.350846i	$-0.114106\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	25.0000	0.794954
$990$	0	0
$991$	33.0000	1.04828	0.524140	−	0.851632i	$-0.324387\pi$
$991$	33.0000	1.04828	0.524140	+	0.851632i	$0.324387\pi$
$992$	0	0
$993$	− 14.0000i	− 0.444277i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	36.0000i	1.14013i	0.821599	+	0.570066i	$0.193082\pi$
$997$	36.0000i	1.14013i	−0.821599	+	0.570066i	$0.806918\pi$
$998$	0	0
$999$	−20.0000	−0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.e.d.2549.2		2
5.2	odd	4		4900.2.a.r.1.1	yes	1
5.3	odd	4		4900.2.a.d.1.1	yes	1
5.4	even	2	inner	4900.2.e.d.2549.1		2
7.6	odd	2		4900.2.e.e.2549.1		2
35.13	even	4		4900.2.a.s.1.1	yes	1
35.27	even	4		4900.2.a.c.1.1	✓	1
35.34	odd	2		4900.2.e.e.2549.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4900.2.a.c.1.1	✓	1	35.27	even	4
4900.2.a.d.1.1	yes	1	5.3	odd	4
4900.2.a.r.1.1	yes	1	5.2	odd	4
4900.2.a.s.1.1	yes	1	35.13	even	4
4900.2.e.d.2549.1		2	5.4	even	2	inner
4900.2.e.d.2549.2		2	1.1	even	1	trivial
4900.2.e.e.2549.1		2	7.6	odd	2
4900.2.e.e.2549.2		2	35.34	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:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} -1.00000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{3} -1.00000 q^{9} -1.00000 q^{11} +2.00000i q^{13} +4.00000i q^{17} -5.00000i q^{23} +4.00000i q^{27} +3.00000 q^{29} -10.0000 q^{31} -2.00000i q^{33} +5.00000i q^{37} -4.00000 q^{39} -10.0000 q^{41} +5.00000i q^{43} -4.00000i q^{47} -8.00000 q^{51} -10.0000i q^{53} +10.0000 q^{59} +10.0000 q^{61} +5.00000i q^{67} +10.0000 q^{69} +3.00000 q^{71} +10.0000i q^{73} -13.0000 q^{79} -11.0000 q^{81} +10.0000i q^{83} +6.00000i q^{87} -10.0000 q^{89} -20.0000i q^{93} +6.00000i q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} - 2 q^{11} + 6 q^{29} - 20 q^{31} - 8 q^{39} - 20 q^{41} - 16 q^{51} + 20 q^{59} + 20 q^{61} + 20 q^{69} + 6 q^{71} - 26 q^{79} - 22 q^{81} - 20 q^{89} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 2 * q^11 + 6 * q^29 - 20 * q^31 - 8 * q^39 - 20 * q^41 - 16 * q^51 + 20 * q^59 + 20 * q^61 + 20 * q^69 + 6 * q^71 - 26 * q^79 - 22 * q^81 - 20 * q^89 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more