LMFDB - Embedded newform 2600.2.d.c.1249.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2600,2,Mod(1249,2600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2600, 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("2600.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2600 = 2^{3} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2600.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$20.7611045255$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 520)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2600.1249
Dual form			2600.2.d.c.1249.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} -1.00000 q^{9} +2.00000 q^{11} -1.00000i q^{13} +2.00000i q^{17} -2.00000 q^{19} -2.00000i q^{23} -4.00000i q^{27} +6.00000 q^{29} +2.00000 q^{31} -4.00000i q^{33} -6.00000i q^{37} -2.00000 q^{39} +2.00000 q^{41} -6.00000i q^{43} -8.00000i q^{47} +7.00000 q^{49} +4.00000 q^{51} +2.00000i q^{53} +4.00000i q^{57} -6.00000 q^{59} -14.0000 q^{61} -4.00000 q^{69} +10.0000 q^{71} +2.00000i q^{73} +4.00000 q^{79} -11.0000 q^{81} -12.0000i q^{83} -12.0000i q^{87} +6.00000 q^{89} -4.00000i q^{93} +2.00000i q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9} + 4 q^{11} - 4 q^{19} + 12 q^{29} + 4 q^{31} - 4 q^{39} + 4 q^{41} + 14 q^{49} + 8 q^{51} - 12 q^{59} - 28 q^{61} - 8 q^{69} + 20 q^{71} + 8 q^{79} - 22 q^{81} + 12 q^{89} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 + 4 * q^11 - 4 * q^19 + 12 * q^29 + 4 * q^31 - 4 * q^39 + 4 * q^41 + 14 * q^49 + 8 * q^51 - 12 * q^59 - 28 * q^61 - 8 * q^69 + 20 * q^71 + 8 * q^79 - 22 * q^81 + 12 * q^89 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times$.

$n$	$1301$	$1601$	$1951$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 2.00000i	− 0.417029i	−0.978019	−	0.208514i	$-0.933137\pi$
$23$	− 2.00000i	− 0.417029i	0.978019	−	0.208514i	$-0.0668628\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4.00000i	− 0.769800i
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	− 4.00000i	− 0.696311i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	− 6.00000i	− 0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$	− 6.00000i	− 0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.00000i	0.529813i
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	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
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 12.0000i	− 1.28654i
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 4.00000i	− 0.414781i
$94$	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
$99$	−2.00000	−0.201008
$100$	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
$103$	− 2.00000i	− 0.197066i	−0.995134	−	0.0985329i	$-0.968585\pi$
$103$	− 2.00000i	− 0.197066i	0.995134	−	0.0985329i	$-0.0314150\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.00000i	− 0.193347i	−0.995316	−	0.0966736i	$-0.969180\pi$
$107$	− 2.00000i	− 0.193347i	0.995316	−	0.0966736i	$-0.0308203\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−12.0000	−1.13899
$112$	0	0
$113$	− 2.00000i	− 0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$	− 2.00000i	− 0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000i	0.0924500i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	− 4.00000i	− 0.360668i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 10.0000i	− 0.887357i	−0.896186	−	0.443678i	$-0.853673\pi$
$127$	− 10.0000i	− 0.887357i	0.896186	−	0.443678i	$-0.146327\pi$
$128$	0	0
$129$	−12.0000	−1.05654
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	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
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	−16.0000	−1.34744
$142$	0	0
$143$	− 2.00000i	− 0.167248i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 14.0000i	− 1.15470i
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	− 2.00000i	− 0.161690i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.00000i	0.478852i	0.970915	+	0.239426i	$0.0769593\pi$
$157$	6.00000i	0.478852i	−0.970915	+	0.239426i	$0.923041\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	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
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	− 22.0000i	− 1.67263i	−0.548250	−	0.836315i	$-0.684706\pi$
$173$	− 22.0000i	− 1.67263i	0.548250	−	0.836315i	$-0.315294\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.0000i	0.901975i
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	28.0000i	2.06982i
$184$	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
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.0000i	1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$	22.0000i	1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.00000i	0.139010i
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	− 20.0000i	− 1.37038i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	2.00000	0.134535
$222$	0	0
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.0000i	1.59294i	0.604681	+	0.796468i	$0.293301\pi$
$227$	24.0000i	1.59294i	−0.604681	+	0.796468i	$0.706699\pi$
$228$	0	0
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 8.00000i	− 0.519656i
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	10.0000i	0.641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.00000i	0.127257i
$248$	0	0
$249$	−24.0000	−1.52094
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	− 4.00000i	− 0.251478i
$254$	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
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	− 30.0000i	− 1.84988i	−0.380114	−	0.924940i	$-0.624115\pi$
$263$	− 30.0000i	− 1.84988i	0.380114	−	0.924940i	$-0.375885\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 12.0000i	− 0.734388i
$268$	0	0
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.0000i	0.841178i	0.907251	+	0.420589i	$0.138177\pi$
$277$	14.0000i	0.841178i	−0.907251	+	0.420589i	$0.861823\pi$
$278$	0	0
$279$	−2.00000	−0.119737
$280$	0	0
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	− 22.0000i	− 1.30776i	−0.756596	−	0.653882i	$-0.773139\pi$
$283$	− 22.0000i	− 1.30776i	0.756596	−	0.653882i	$-0.226861\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	4.00000	0.234484
$292$	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
$297$	− 8.00000i	− 0.464207i
$298$	0	0
$299$	−2.00000	−0.115663
$300$	0	0
$301$	0	0
$302$	0	0
$303$	20.0000i	1.14897i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	28.0000i	1.59804i	0.601302	+	0.799022i	$0.294649\pi$
$307$	28.0000i	1.59804i	−0.601302	+	0.799022i	$0.705351\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	28.0000	1.58773	0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000	1.58773	0.793867	+	0.608091i	$0.208065\pi$
$312$	0	0
$313$	30.0000i	1.69570i	0.530236	+	0.847850i	$0.322103\pi$
$313$	30.0000i	1.69570i	−0.530236	+	0.847850i	$0.677897\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	− 4.00000i	− 0.222566i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 20.0000i	− 1.10600i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	22.0000	1.20923	0.604615	−	0.796518i	$-0.293327\pi$
$331$	22.0000	1.20923	0.604615	+	0.796518i	$0.293327\pi$
$332$	0	0
$333$	6.00000i	0.328798i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.0000i	0.980522i	0.871576	+	0.490261i	$0.163099\pi$
$337$	18.0000i	0.980522i	−0.871576	+	0.490261i	$0.836901\pi$
$338$	0	0
$339$	−4.00000	−0.217250
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 18.0000i	− 0.966291i	−0.875540	−	0.483145i	$-0.839494\pi$
$347$	− 18.0000i	− 0.966291i	0.875540	−	0.483145i	$-0.160506\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	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
$359$	−10.0000	−0.527780	−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000	−0.527780	−0.263890	+	0.964553i	$0.585006\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	14.0000i	0.734809i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.0000i	0.521996i	0.965339	+	0.260998i	$0.0840516\pi$
$367$	10.0000i	0.521996i	−0.965339	+	0.260998i	$0.915948\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 30.0000i	− 1.55334i	−0.629907	−	0.776671i	$-0.716907\pi$
$373$	− 30.0000i	− 1.55334i	0.629907	−	0.776671i	$-0.283093\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 6.00000i	− 0.309016i
$378$	0	0
$379$	30.0000	1.54100	0.770498	−	0.637442i	$-0.220007\pi$
$379$	30.0000	1.54100	0.770498	+	0.637442i	$0.220007\pi$
$380$	0	0
$381$	−20.0000	−1.02463
$382$	0	0
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.00000i	0.304997i
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	0	0
$393$	8.00000i	0.403547i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 30.0000i	− 1.50566i	−0.658217	−	0.752828i	$-0.728689\pi$
$397$	− 30.0000i	− 1.50566i	0.658217	−	0.752828i	$-0.271311\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	− 2.00000i	− 0.0996271i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 12.0000i	− 0.594818i
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	20.0000	0.986527
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 32.0000i	− 1.56705i
$418$	0	0
$419$	8.00000	0.390826	0.195413	−	0.980721i	$-0.437395\pi$
$419$	8.00000	0.390826	0.195413	+	0.980721i	$0.437395\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	8.00000i	0.388973i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	0	0
$433$	38.0000i	1.82616i	0.407777	+	0.913082i	$0.366304\pi$
$433$	38.0000i	1.82616i	−0.407777	+	0.913082i	$0.633696\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.00000i	0.191346i
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	0	0
$443$	38.0000i	1.80543i	0.430234	+	0.902717i	$0.358431\pi$
$443$	38.0000i	1.80543i	−0.430234	+	0.902717i	$0.641569\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	12.0000i	0.567581i
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	0	0
$453$	20.0000i	0.939682i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.0000i	1.21623i	0.793849	+	0.608114i	$0.208074\pi$
$457$	26.0000i	1.21623i	−0.793849	+	0.608114i	$0.791926\pi$
$458$	0	0
$459$	8.00000	0.373408
$460$	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
$463$	4.00000i	0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$	4.00000i	0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6.00000i	− 0.277647i	−0.990317	−	0.138823i	$-0.955668\pi$
$467$	− 6.00000i	− 0.277647i	0.990317	−	0.138823i	$-0.0443321\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	12.0000	0.552931
$472$	0	0
$473$	− 12.0000i	− 0.551761i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 2.00000i	− 0.0915737i
$478$	0	0
$479$	14.0000	0.639676	0.319838	−	0.947472i	$-0.396371\pi$
$479$	14.0000	0.639676	0.319838	+	0.947472i	$0.396371\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.00000i	0.181257i	0.995885	+	0.0906287i	$0.0288876\pi$
$487$	4.00000i	0.181257i	−0.995885	+	0.0906287i	$0.971112\pi$
$488$	0	0
$489$	−32.0000	−1.44709
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	38.0000	1.70111	0.850557	−	0.525883i	$-0.176265\pi$
$499$	38.0000	1.70111	0.850557	+	0.525883i	$0.176265\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	0	0
$503$	10.0000i	0.445878i	0.974832	+	0.222939i	$0.0715651\pi$
$503$	10.0000i	0.445878i	−0.974832	+	0.222939i	$0.928435\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.00000i	0.0888231i
$508$	0	0
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	8.00000i	0.353209i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 16.0000i	− 0.703679i
$518$	0	0
$519$	−44.0000	−1.93139
$520$	0	0
$521$	38.0000	1.66481	0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000	1.66481	0.832405	+	0.554168i	$0.186963\pi$
$522$	0	0
$523$	− 34.0000i	− 1.48672i	−0.668894	−	0.743358i	$-0.733232\pi$
$523$	− 34.0000i	− 1.48672i	0.668894	−	0.743358i	$-0.266768\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.00000i	0.174243i
$528$	0	0
$529$	19.0000	0.826087
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	− 2.00000i	− 0.0866296i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 8.00000i	− 0.345225i
$538$	0	0
$539$	14.0000	0.603023
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	0	0
$543$	28.0000i	1.20160i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	22.0000i	0.940652i	0.882493	+	0.470326i	$0.155864\pi$
$547$	22.0000i	0.940652i	−0.882493	+	0.470326i	$0.844136\pi$
$548$	0	0
$549$	14.0000	0.597505
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 14.0000i	− 0.593199i	−0.955002	−	0.296600i	$-0.904147\pi$
$557$	− 14.0000i	− 0.593199i	0.955002	−	0.296600i	$-0.0958526\pi$
$558$	0	0
$559$	−6.00000	−0.253773
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	− 6.00000i	− 0.252870i	−0.991975	−	0.126435i	$-0.959647\pi$
$563$	− 6.00000i	− 0.252870i	0.991975	−	0.126435i	$-0.0403535\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	48.0000i	2.00523i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	22.0000i	0.915872i	0.888985	+	0.457936i	$0.151411\pi$
$577$	22.0000i	0.915872i	−0.888985	+	0.457936i	$0.848589\pi$
$578$	0	0
$579$	28.0000	1.16364
$580$	0	0
$581$	0	0
$582$	0	0
$583$	4.00000i	0.165663i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 40.0000i	− 1.65098i	−0.564419	−	0.825488i	$-0.690900\pi$
$587$	− 40.0000i	− 1.65098i	0.564419	−	0.825488i	$-0.309100\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	44.0000	1.80992
$592$	0	0
$593$	30.0000i	1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$	30.0000i	1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	18.0000i	0.730597i	0.930890	+	0.365299i	$0.119033\pi$
$607$	18.0000i	0.730597i	−0.930890	+	0.365299i	$0.880967\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	0	0
$619$	−30.0000	−1.20580	−0.602901	−	0.797816i	$-0.705989\pi$
$619$	−30.0000	−1.20580	−0.602901	+	0.797816i	$0.705989\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8.00000i	0.319489i
$628$	0	0
$629$	12.0000	0.478471
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	0	0
$633$	− 40.0000i	− 1.58986i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 7.00000i	− 0.277350i
$638$	0	0
$639$	−10.0000	−0.395594
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	20.0000i	0.788723i	0.918955	+	0.394362i	$0.129034\pi$
$643$	20.0000i	0.788723i	−0.918955	+	0.394362i	$0.870966\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.00000i	0.235884i	0.993020	+	0.117942i	$0.0376297\pi$
$647$	6.00000i	0.235884i	−0.993020	+	0.117942i	$0.962370\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 6.00000i	− 0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$	− 6.00000i	− 0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.00000i	− 0.0780274i
$658$	0	0
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	− 4.00000i	− 0.155347i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 12.0000i	− 0.464642i
$668$	0	0
$669$	32.0000	1.23719
$670$	0	0
$671$	−28.0000	−1.08093
$672$	0	0
$673$	− 2.00000i	− 0.0770943i	−0.999257	−	0.0385472i	$-0.987727\pi$
$673$	− 2.00000i	− 0.0770943i	0.999257	−	0.0385472i	$-0.0122730\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 2.00000i	− 0.0768662i	−0.999261	−	0.0384331i	$-0.987763\pi$
$677$	− 2.00000i	− 0.0768662i	0.999261	−	0.0384331i	$-0.0122367\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	48.0000	1.83936
$682$	0	0
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	44.0000i	1.67870i
$688$	0	0
$689$	2.00000	0.0761939
$690$	0	0
$691$	14.0000	0.532585	0.266293	−	0.963892i	$-0.414201\pi$
$691$	14.0000	0.532585	0.266293	+	0.963892i	$0.414201\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	4.00000i	0.151511i
$698$	0	0
$699$	−36.0000	−1.36165
$700$	0	0
$701$	−50.0000	−1.88847	−0.944237	−	0.329267i	$-0.893198\pi$
$701$	−50.0000	−1.88847	−0.944237	+	0.329267i	$0.893198\pi$
$702$	0	0
$703$	12.0000i	0.452589i
$704$	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
$711$	−4.00000	−0.150012
$712$	0	0
$713$	− 4.00000i	− 0.149801i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	12.0000i	0.448148i
$718$	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
$723$	− 4.00000i	− 0.148762i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	6.00000i	0.222528i	0.993791	+	0.111264i	$0.0354899\pi$
$727$	6.00000i	0.222528i	−0.993791	+	0.111264i	$0.964510\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	12.0000	0.443836
$732$	0	0
$733$	2.00000i	0.0738717i	0.999318	+	0.0369358i	$0.0117597\pi$
$733$	2.00000i	0.0738717i	−0.999318	+	0.0369358i	$0.988240\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	18.0000	0.662141	0.331070	−	0.943606i	$-0.392590\pi$
$739$	18.0000	0.662141	0.331070	+	0.943606i	$0.392590\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	0	0
$743$	32.0000i	1.17397i	0.809599	+	0.586983i	$0.199684\pi$
$743$	32.0000i	1.17397i	−0.809599	+	0.586983i	$0.800316\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.0000i	0.439057i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	46.0000i	1.67190i	0.548807	+	0.835949i	$0.315082\pi$
$757$	46.0000i	1.67190i	−0.548807	+	0.835949i	$0.684918\pi$
$758$	0	0
$759$	−8.00000	−0.290382
$760$	0	0
$761$	26.0000	0.942499	0.471250	−	0.882000i	$-0.343803\pi$
$761$	26.0000	0.942499	0.471250	+	0.882000i	$0.343803\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.00000i	0.216647i
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	0	0
$773$	− 42.0000i	− 1.51064i	−0.655359	−	0.755318i	$-0.727483\pi$
$773$	− 42.0000i	− 1.51064i	0.655359	−	0.755318i	$-0.272517\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.00000	−0.143315
$780$	0	0
$781$	20.0000	0.715656
$782$	0	0
$783$	− 24.0000i	− 0.857690i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	44.0000i	1.56843i	0.620489	+	0.784215i	$0.286934\pi$
$787$	44.0000i	1.56843i	−0.620489	+	0.784215i	$0.713066\pi$
$788$	0	0
$789$	−60.0000	−2.13606
$790$	0	0
$791$	0	0
$792$	0	0
$793$	14.0000i	0.497155i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 2.00000i	− 0.0708436i	−0.999372	−	0.0354218i	$-0.988723\pi$
$797$	− 2.00000i	− 0.0708436i	0.999372	−	0.0354218i	$-0.0112775\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	4.00000i	0.141157i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	28.0000i	0.985647i
$808$	0	0
$809$	50.0000	1.75791	0.878953	−	0.476908i	$-0.158243\pi$
$809$	50.0000	1.75791	0.878953	+	0.476908i	$0.158243\pi$
$810$	0	0
$811$	−42.0000	−1.47482	−0.737410	−	0.675446i	$-0.763951\pi$
$811$	−42.0000	−1.47482	−0.737410	+	0.675446i	$0.763951\pi$
$812$	0	0
$813$	4.00000i	0.140286i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	12.0000i	0.419827i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	− 26.0000i	− 0.906303i	−0.891434	−	0.453152i	$-0.850300\pi$
$823$	− 26.0000i	− 0.906303i	0.891434	−	0.453152i	$-0.149700\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	28.0000	0.971309
$832$	0	0
$833$	14.0000i	0.485071i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 8.00000i	− 0.276520i
$838$	0	0
$839$	−34.0000	−1.17381	−0.586905	−	0.809656i	$-0.699654\pi$
$839$	−34.0000	−1.17381	−0.586905	+	0.809656i	$0.699654\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	− 52.0000i	− 1.79098i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−44.0000	−1.51008
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	46.0000i	1.57501i	0.616308	+	0.787505i	$0.288628\pi$
$853$	46.0000i	1.57501i	−0.616308	+	0.787505i	$0.711372\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.0000i	0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$	18.0000i	0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$	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
$863$	− 8.00000i	− 0.272323i	−0.990687	−	0.136162i	$-0.956523\pi$
$863$	− 8.00000i	− 0.272323i	0.990687	−	0.136162i	$-0.0434766\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 26.0000i	− 0.883006i
$868$	0	0
$869$	8.00000	0.271381
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 2.00000i	− 0.0676897i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46.0000i	1.55331i	0.629926	+	0.776655i	$0.283085\pi$
$877$	46.0000i	1.55331i	−0.629926	+	0.776655i	$0.716915\pi$
$878$	0	0
$879$	−36.0000	−1.21425
$880$	0	0
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	0	0
$883$	26.0000i	0.874970i	0.899226	+	0.437485i	$0.144131\pi$
$883$	26.0000i	0.874970i	−0.899226	+	0.437485i	$0.855869\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 22.0000i	− 0.738688i	−0.929293	−	0.369344i	$-0.879582\pi$
$887$	− 22.0000i	− 0.738688i	0.929293	−	0.369344i	$-0.120418\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−22.0000	−0.737028
$892$	0	0
$893$	16.0000i	0.535420i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.00000i	0.133556i
$898$	0	0
$899$	12.0000	0.400222
$900$	0	0
$901$	−4.00000	−0.133259
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	18.0000i	0.597680i	0.954303	+	0.298840i	$0.0965997\pi$
$907$	18.0000i	0.597680i	−0.954303	+	0.298840i	$0.903400\pi$
$908$	0	0
$909$	10.0000	0.331679
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	− 24.0000i	− 0.794284i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	56.0000	1.84526
$922$	0	0
$923$	− 10.0000i	− 0.329154i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.00000i	0.0656886i
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	−14.0000	−0.458831
$932$	0	0
$933$	− 56.0000i	− 1.83336i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.00000i	0.0653372i	0.999466	+	0.0326686i	$0.0104006\pi$
$937$	2.00000i	0.0653372i	−0.999466	+	0.0326686i	$0.989599\pi$
$938$	0	0
$939$	60.0000	1.95803
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	− 4.00000i	− 0.130258i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 36.0000i	− 1.16984i	−0.811090	−	0.584921i	$-0.801125\pi$
$947$	− 36.0000i	− 1.16984i	0.811090	−	0.584921i	$-0.198875\pi$
$948$	0	0
$949$	2.00000	0.0649227
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	14.0000i	0.453504i	0.973952	+	0.226752i	$0.0728108\pi$
$953$	14.0000i	0.453504i	−0.973952	+	0.226752i	$0.927189\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 24.0000i	− 0.775810i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	2.00000i	0.0644491i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 12.0000i	− 0.385894i	−0.981209	−	0.192947i	$-0.938195\pi$
$967$	− 12.0000i	− 0.385894i	0.981209	−	0.192947i	$-0.0618045\pi$
$968$	0	0
$969$	−8.00000	−0.256997
$970$	0	0
$971$	−52.0000	−1.66876	−0.834380	−	0.551190i	$-0.814174\pi$
$971$	−52.0000	−1.66876	−0.834380	+	0.551190i	$0.814174\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	38.0000i	1.21573i	0.794041	+	0.607864i	$0.207973\pi$
$977$	38.0000i	1.21573i	−0.794041	+	0.607864i	$0.792027\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	−10.0000	−0.319275
$982$	0	0
$983$	52.0000i	1.65854i	0.558846	+	0.829271i	$0.311244\pi$
$983$	52.0000i	1.65854i	−0.558846	+	0.829271i	$0.688756\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−12.0000	−0.381578
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	0	0
$993$	− 44.0000i	− 1.39630i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	14.0000i	0.443384i	0.975117	+	0.221692i	$0.0711580\pi$
$997$	14.0000i	0.443384i	−0.975117	+	0.221692i	$0.928842\pi$
$998$	0	0
$999$	−24.0000	−0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2600.2.d.c.1249.1		2
5.2	odd	4		2600.2.a.c.1.1		1
5.3	odd	4		520.2.a.b.1.1	✓	1
5.4	even	2	inner	2600.2.d.c.1249.2		2
15.8	even	4		4680.2.a.e.1.1		1
20.3	even	4		1040.2.a.c.1.1		1
20.7	even	4		5200.2.a.bf.1.1		1
40.3	even	4		4160.2.a.p.1.1		1
40.13	odd	4		4160.2.a.a.1.1		1
60.23	odd	4		9360.2.a.n.1.1		1
65.38	odd	4		6760.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.a.b.1.1	✓	1	5.3	odd	4
1040.2.a.c.1.1		1	20.3	even	4
2600.2.a.c.1.1		1	5.2	odd	4
2600.2.d.c.1249.1		2	1.1	even	1	trivial
2600.2.d.c.1249.2		2	5.4	even	2	inner
4160.2.a.a.1.1		1	40.13	odd	4
4160.2.a.p.1.1		1	40.3	even	4
4680.2.a.e.1.1		1	15.8	even	4
5200.2.a.bf.1.1		1	20.7	even	4
6760.2.a.k.1.1		1	65.38	odd	4
9360.2.a.n.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:	\( N \)	\(=\)	\( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2600.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} -1.00000 q^{9} +2.00000 q^{11} -1.00000i q^{13} +2.00000i q^{17} -2.00000 q^{19} -2.00000i q^{23} -4.00000i q^{27} +6.00000 q^{29} +2.00000 q^{31} -4.00000i q^{33} -6.00000i q^{37} -2.00000 q^{39} +2.00000 q^{41} -6.00000i q^{43} -8.00000i q^{47} +7.00000 q^{49} +4.00000 q^{51} +2.00000i q^{53} +4.00000i q^{57} -6.00000 q^{59} -14.0000 q^{61} -4.00000 q^{69} +10.0000 q^{71} +2.00000i q^{73} +4.00000 q^{79} -11.0000 q^{81} -12.0000i q^{83} -12.0000i q^{87} +6.00000 q^{89} -4.00000i q^{93} +2.00000i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} + 4 q^{11} - 4 q^{19} + 12 q^{29} + 4 q^{31} - 4 q^{39} + 4 q^{41} + 14 q^{49} + 8 q^{51} - 12 q^{59} - 28 q^{61} - 8 q^{69} + 20 q^{71} + 8 q^{79} - 22 q^{81} + 12 q^{89} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 + 4 * q^11 - 4 * q^19 + 12 * q^29 + 4 * q^31 - 4 * q^39 + 4 * q^41 + 14 * q^49 + 8 * q^51 - 12 * q^59 - 28 * q^61 - 8 * q^69 + 20 * q^71 + 8 * q^79 - 22 * q^81 + 12 * q^89 - 4 * q^99

Introduction

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