LMFDB - Embedded newform 5040.2.t.s.1009.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5040,2,Mod(1009,5040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5040.1009"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5040.t (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,4,0,0,0,0,0,6,0,0,0,0,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$40.2446026187$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1009.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	5040.1009
Dual form			5040.2.t.s.1009.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 - 1.00000i) q^{5} +1.00000i q^{7} +3.00000 q^{11} +1.00000i q^{13} -5.00000i q^{17} -8.00000 q^{19} +2.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -1.00000 q^{29} +2.00000 q^{31} +(1.00000 + 2.00000i) q^{35} -10.0000i q^{37} +6.00000 q^{41} +4.00000i q^{43} -11.0000i q^{47} -1.00000 q^{49} -6.00000i q^{53} +(6.00000 - 3.00000i) q^{55} +10.0000 q^{59} +(1.00000 + 2.00000i) q^{65} -10.0000i q^{67} -10.0000i q^{73} +3.00000i q^{77} -7.00000 q^{79} +12.0000i q^{83} +(-5.00000 - 10.0000i) q^{85} +8.00000 q^{89} -1.00000 q^{91} +(-16.0000 + 8.00000i) q^{95} -3.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} + 6 q^{11} - 16 q^{19} + 6 q^{25} - 2 q^{29} + 4 q^{31} + 2 q^{35} + 12 q^{41} - 2 q^{49} + 12 q^{55} + 20 q^{59} + 2 q^{65} - 14 q^{79} - 10 q^{85} + 16 q^{89} - 2 q^{91} - 32 q^{95}+O(q^{100}) $ 2 * q + 4 * q^5 + 6 * q^11 - 16 * q^19 + 6 * q^25 - 2 * q^29 + 4 * q^31 + 2 * q^35 + 12 * q^41 - 2 * q^49 + 12 * q^55 + 20 * q^59 + 2 * q^65 - 14 * q^79 - 10 * q^85 + 16 * q^89 - 2 * q^91 - 32 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times$.

$n$	$2017$	$2801$	$3151$	$3601$	$3781$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.00000			0.904534			0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000			0.904534			0.452267	+	0.891883i	$0.350615\pi$
$12$		0			0
$13$			1.00000i			0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$			1.00000i			0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	5.00000i		−	1.21268i	−0.795206	−	0.606339i	$-0.792637\pi$
$17$		−	5.00000i		−	1.21268i	0.795206	−	0.606339i	$-0.207363\pi$
$18$		0			0
$19$	−8.00000			−1.83533			−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000			−1.83533			−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			2.00000i			0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$			2.00000i			0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.00000			−0.185695			−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000			−0.185695			−0.0928477	+	0.995680i	$0.529597\pi$
$30$		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
$33$		0			0
$34$		0			0
$35$	1.00000	+	2.00000i	0.169031	+	0.338062i
$36$		0			0
$37$		−	10.0000i		−	1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$		−	10.0000i		−	1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	11.0000i		−	1.60451i	−0.596978	−	0.802257i	$-0.703632\pi$
$47$		−	11.0000i		−	1.60451i	0.596978	−	0.802257i	$-0.296368\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$		0			0
$55$	6.00000	−	3.00000i	0.809040	−	0.404520i
$56$		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
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.00000	+	2.00000i	0.124035	+	0.248069i
$66$		0			0
$67$		−	10.0000i		−	1.22169i	−0.791748	−	0.610847i	$-0.790829\pi$
$67$		−	10.0000i		−	1.22169i	0.791748	−	0.610847i	$-0.209171\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		−	10.0000i		−	1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$		−	10.0000i		−	1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			3.00000i			0.341882i
$78$		0			0
$79$	−7.00000			−0.787562			−0.393781	−	0.919204i	$-0.628833\pi$
$79$	−7.00000			−0.787562			−0.393781	+	0.919204i	$0.628833\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			12.0000i			1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$			12.0000i			1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$		0			0
$85$	−5.00000	−	10.0000i	−0.542326	−	1.08465i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	8.00000			0.847998			0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000			0.847998			0.423999	+	0.905663i	$0.360626\pi$
$90$		0			0
$91$	−1.00000			−0.104828
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−16.0000	+	8.00000i	−1.64157	+	0.820783i
$96$		0			0
$97$		−	3.00000i		−	0.304604i	−0.988334	−	0.152302i	$-0.951331\pi$
$97$		−	3.00000i		−	0.304604i	0.988334	−	0.152302i	$-0.0486686\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	12.0000			1.19404			0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000			1.19404			0.597022	+	0.802225i	$0.296350\pi$
$102$		0			0
$103$			5.00000i			0.492665i	0.969185	+	0.246332i	$0.0792255\pi$
$103$			5.00000i			0.492665i	−0.969185	+	0.246332i	$0.920775\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	8.00000i		−	0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$		−	8.00000i		−	0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$		0			0
$109$	7.00000			0.670478			0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000			0.670478			0.335239	+	0.942133i	$0.391183\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			10.0000i			0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$			10.0000i			0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$		0			0
$115$	2.00000	+	4.00000i	0.186501	+	0.373002i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5.00000			0.458349
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$		0			0
$124$		0			0
$125$	2.00000	−	11.0000i	0.178885	−	0.983870i
$126$		0			0
$127$		−	2.00000i		−	0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$		−	2.00000i		−	0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.00000			0.174741			0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000			0.174741			0.0873704	+	0.996176i	$0.472154\pi$
$132$		0			0
$133$		−	8.00000i		−	0.693688i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			4.00000i			0.341743i	0.985293	+	0.170872i	$0.0546583\pi$
$137$			4.00000i			0.341743i	−0.985293	+	0.170872i	$0.945342\pi$
$138$		0			0
$139$	10.0000			0.848189			0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000			0.848189			0.424094	+	0.905618i	$0.360592\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			3.00000i			0.250873i
$144$		0			0
$145$	−2.00000	+	1.00000i	−0.166091	+	0.0830455i
$146$		0			0
$147$		0			0
$148$		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
$151$	−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
$153$		0			0
$154$		0			0
$155$	4.00000	−	2.00000i	0.321288	−	0.160644i
$156$		0			0
$157$			18.0000i			1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$			18.0000i			1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.00000			−0.157622
$162$		0			0
$163$		−	6.00000i		−	0.469956i	−0.972001	−	0.234978i	$-0.924498\pi$
$163$		−	6.00000i		−	0.469956i	0.972001	−	0.234978i	$-0.0755019\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	3.00000i		−	0.232147i	−0.993241	−	0.116073i	$-0.962969\pi$
$167$		−	3.00000i		−	0.232147i	0.993241	−	0.116073i	$-0.0370308\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$		0			0
$172$		0			0
$173$			9.00000i			0.684257i	0.939653	+	0.342129i	$0.111148\pi$
$173$			9.00000i			0.684257i	−0.939653	+	0.342129i	$0.888852\pi$
$174$		0			0
$175$	4.00000	+	3.00000i	0.302372	+	0.226779i
$176$		0			0
$177$		0			0
$178$		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	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−10.0000	−	20.0000i	−0.735215	−	1.47043i
$186$		0			0
$187$		−	15.0000i		−	1.09691i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.00000			0.506502			0.253251	−	0.967401i	$-0.418500\pi$
$191$	7.00000			0.506502			0.253251	+	0.967401i	$0.418500\pi$
$192$		0			0
$193$			8.00000i			0.575853i	0.957653	+	0.287926i	$0.0929658\pi$
$193$			8.00000i			0.575853i	−0.957653	+	0.287926i	$0.907034\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			10.0000i			0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$			10.0000i			0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$		0			0
$199$	18.0000			1.27599			0.637993	−	0.770042i	$-0.279765\pi$
$199$	18.0000			1.27599			0.637993	+	0.770042i	$0.279765\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	1.00000i		−	0.0701862i
$204$		0			0
$205$	12.0000	−	6.00000i	0.838116	−	0.419058i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−24.0000			−1.66011
$210$		0			0
$211$	3.00000			0.206529			0.103264	−	0.994654i	$-0.467071\pi$
$211$	3.00000			0.206529			0.103264	+	0.994654i	$0.467071\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.00000	+	8.00000i	0.272798	+	0.545595i
$216$		0			0
$217$			2.00000i			0.135769i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.00000			0.336336
$222$		0			0
$223$		−	19.0000i		−	1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$		−	19.0000i		−	1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	27.0000i		−	1.79205i	−0.444001	−	0.896026i	$-0.646441\pi$
$227$		−	27.0000i		−	1.79205i	0.444001	−	0.896026i	$-0.353559\pi$
$228$		0			0
$229$	26.0000			1.71813			0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000			1.71813			0.859064	+	0.511868i	$0.171046\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	16.0000i		−	1.04819i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$		−	16.0000i		−	1.04819i	0.851658	−	0.524097i	$-0.175597\pi$
$234$		0			0
$235$	−11.0000	−	22.0000i	−0.717561	−	1.43512i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−5.00000			−0.323423			−0.161712	−	0.986838i	$-0.551701\pi$
$239$	−5.00000			−0.323423			−0.161712	+	0.986838i	$0.551701\pi$
$240$		0			0
$241$	−18.0000			−1.15948			−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000			−1.15948			−0.579741	+	0.814801i	$0.696846\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.00000	+	1.00000i	−0.127775	+	0.0638877i
$246$		0			0
$247$		−	8.00000i		−	0.509028i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.00000			0.126239			0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000			0.126239			0.0631194	+	0.998006i	$0.479895\pi$
$252$		0			0
$253$			6.00000i			0.377217i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			6.00000i			0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$			6.00000i			0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$		0			0
$259$	10.0000			0.621370
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	24.0000i		−	1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$		−	24.0000i		−	1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$		0			0
$265$	−6.00000	−	12.0000i	−0.368577	−	0.737154i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.00000			0.121942			0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000			0.121942			0.0609711	+	0.998140i	$0.480580\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	9.00000	−	12.0000i	0.542720	−	0.723627i
$276$		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
$279$		0			0
$280$		0			0
$281$	−15.0000			−0.894825			−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000			−0.894825			−0.447412	+	0.894328i	$0.647654\pi$
$282$		0			0
$283$			7.00000i			0.416107i	0.978117	+	0.208053i	$0.0667128\pi$
$283$			7.00000i			0.416107i	−0.978117	+	0.208053i	$0.933287\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			6.00000i			0.354169i
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	15.0000i		−	0.876309i	−0.898900	−	0.438155i	$-0.855632\pi$
$293$		−	15.0000i		−	0.876309i	0.898900	−	0.438155i	$-0.144368\pi$
$294$		0			0
$295$	20.0000	−	10.0000i	1.16445	−	0.582223i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.00000			−0.115663
$300$		0			0
$301$	−4.00000			−0.230556
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	19.0000i		−	1.08439i	−0.840254	−	0.542194i	$-0.817594\pi$
$307$		−	19.0000i		−	1.08439i	0.840254	−	0.542194i	$-0.182406\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.0000			0.680458			0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000			0.680458			0.340229	+	0.940343i	$0.389495\pi$
$312$		0			0
$313$		−	7.00000i		−	0.395663i	−0.980236	−	0.197832i	$-0.936610\pi$
$313$		−	7.00000i		−	0.395663i	0.980236	−	0.197832i	$-0.0633900\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			28.0000i			1.57264i	0.617822	+	0.786318i	$0.288015\pi$
$317$			28.0000i			1.57264i	−0.617822	+	0.786318i	$0.711985\pi$
$318$		0			0
$319$	−3.00000			−0.167968
$320$		0			0
$321$		0			0
$322$		0			0
$323$			40.0000i			2.22566i
$324$		0			0
$325$	4.00000	+	3.00000i	0.221880	+	0.166410i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	11.0000			0.606450
$330$		0			0
$331$	−20.0000			−1.09930			−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000			−1.09930			−0.549650	+	0.835395i	$0.685239\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−10.0000	−	20.0000i	−0.546358	−	1.09272i
$336$		0			0
$337$		−	22.0000i		−	1.19842i	−0.800593	−	0.599208i	$-0.795482\pi$
$337$		−	22.0000i		−	1.19842i	0.800593	−	0.599208i	$-0.204518\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.00000			0.324918
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		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
$349$	−36.0000			−1.92704			−0.963518	−	0.267644i	$-0.913755\pi$
$349$	−36.0000			−1.92704			−0.963518	+	0.267644i	$0.913755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			9.00000i			0.479022i	0.970894	+	0.239511i	$0.0769871\pi$
$353$			9.00000i			0.479022i	−0.970894	+	0.239511i	$0.923013\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−28.0000			−1.47778			−0.738892	−	0.673824i	$-0.764651\pi$
$359$	−28.0000			−1.47778			−0.738892	+	0.673824i	$0.764651\pi$
$360$		0			0
$361$	45.0000			2.36842
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−10.0000	−	20.0000i	−0.523424	−	1.04685i
$366$		0			0
$367$			19.0000i			0.991792i	0.868382	+	0.495896i	$0.165160\pi$
$367$			19.0000i			0.991792i	−0.868382	+	0.495896i	$0.834840\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	6.00000			0.311504
$372$		0			0
$373$			32.0000i			1.65690i	0.560065	+	0.828449i	$0.310776\pi$
$373$			32.0000i			1.65690i	−0.560065	+	0.828449i	$0.689224\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	1.00000i		−	0.0515026i
$378$		0			0
$379$	28.0000			1.43826			0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000			1.43826			0.719132	+	0.694874i	$0.244540\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$	3.00000	+	6.00000i	0.152894	+	0.305788i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	9.00000			0.456318			0.228159	−	0.973624i	$-0.426729\pi$
$389$	9.00000			0.456318			0.228159	+	0.973624i	$0.426729\pi$
$390$		0			0
$391$	10.0000			0.505722
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−14.0000	+	7.00000i	−0.704416	+	0.352208i
$396$		0			0
$397$			17.0000i			0.853206i	0.904439	+	0.426603i	$0.140290\pi$
$397$			17.0000i			0.853206i	−0.904439	+	0.426603i	$0.859710\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	27.0000			1.34832			0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000			1.34832			0.674158	+	0.738587i	$0.264507\pi$
$402$		0			0
$403$			2.00000i			0.0996271i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	30.0000i		−	1.48704i
$408$		0			0
$409$	−4.00000			−0.197787			−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000			−0.197787			−0.0988936	+	0.995098i	$0.531530\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			10.0000i			0.492068i
$414$		0			0
$415$	12.0000	+	24.0000i	0.589057	+	1.17811i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.00000			−0.0977064			−0.0488532	−	0.998806i	$-0.515557\pi$
$419$	−2.00000			−0.0977064			−0.0488532	+	0.998806i	$0.515557\pi$
$420$		0			0
$421$	−23.0000			−1.12095			−0.560476	−	0.828171i	$-0.689382\pi$
$421$	−23.0000			−1.12095			−0.560476	+	0.828171i	$0.689382\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−20.0000	−	15.0000i	−0.970143	−	0.727607i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−37.0000			−1.78223			−0.891114	−	0.453780i	$-0.850075\pi$
$431$	−37.0000			−1.78223			−0.891114	+	0.453780i	$0.850075\pi$
$432$		0			0
$433$		−	38.0000i		−	1.82616i	−0.407777	−	0.913082i	$-0.633696\pi$
$433$		−	38.0000i		−	1.82616i	0.407777	−	0.913082i	$-0.366304\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	16.0000i		−	0.765384i
$438$		0			0
$439$	26.0000			1.24091			0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000			1.24091			0.620456	+	0.784241i	$0.286947\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			12.0000i			0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$			12.0000i			0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$		0			0
$445$	16.0000	−	8.00000i	0.758473	−	0.379236i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	11.0000			0.519122			0.259561	−	0.965727i	$-0.416422\pi$
$449$	11.0000			0.519122			0.259561	+	0.965727i	$0.416422\pi$
$450$		0			0
$451$	18.0000			0.847587
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.00000	+	1.00000i	−0.0937614	+	0.0468807i
$456$		0			0
$457$			22.0000i			1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$			22.0000i			1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−28.0000			−1.30409			−0.652045	−	0.758180i	$-0.726089\pi$
$461$	−28.0000			−1.30409			−0.652045	+	0.758180i	$0.726089\pi$
$462$		0			0
$463$		−	4.00000i		−	0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$		−	4.00000i		−	0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			23.0000i			1.06431i	0.846646	+	0.532157i	$0.178618\pi$
$467$			23.0000i			1.06431i	−0.846646	+	0.532157i	$0.821382\pi$
$468$		0			0
$469$	10.0000			0.461757
$470$		0			0
$471$		0			0
$472$		0			0
$473$			12.0000i			0.551761i
$474$		0			0
$475$	−24.0000	+	32.0000i	−1.10120	+	1.46826i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	18.0000			0.822441			0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000			0.822441			0.411220	+	0.911536i	$0.365103\pi$
$480$		0			0
$481$	10.0000			0.455961
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.00000	−	6.00000i	−0.136223	−	0.272446i
$486$		0			0
$487$		−	26.0000i		−	1.17817i	−0.808070	−	0.589086i	$-0.799488\pi$
$487$		−	26.0000i		−	1.17817i	0.808070	−	0.589086i	$-0.200512\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	33.0000			1.48927			0.744635	−	0.667472i	$-0.232624\pi$
$491$	33.0000			1.48927			0.744635	+	0.667472i	$0.232624\pi$
$492$		0			0
$493$			5.00000i			0.225189i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−29.0000			−1.29822			−0.649109	−	0.760695i	$-0.724858\pi$
$499$	−29.0000			−1.29822			−0.649109	+	0.760695i	$0.724858\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			1.00000i			0.0445878i	0.999751	+	0.0222939i	$0.00709696\pi$
$503$			1.00000i			0.0445878i	−0.999751	+	0.0222939i	$0.992903\pi$
$504$		0			0
$505$	24.0000	−	12.0000i	1.06799	−	0.533993i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	26.0000			1.15243			0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000			1.15243			0.576215	+	0.817298i	$0.304529\pi$
$510$		0			0
$511$	10.0000			0.442374
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.00000	+	10.0000i	0.220326	+	0.440653i
$516$		0			0
$517$		−	33.0000i		−	1.45134i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−12.0000			−0.525730			−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000			−0.525730			−0.262865	+	0.964833i	$0.584667\pi$
$522$		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
$527$		−	10.0000i		−	0.435607i
$528$		0			0
$529$	19.0000			0.826087
$530$		0			0
$531$		0			0
$532$		0			0
$533$			6.00000i			0.259889i
$534$		0			0
$535$	−8.00000	−	16.0000i	−0.345870	−	0.691740i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.00000			−0.129219
$540$		0			0
$541$	−25.0000			−1.07483			−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000			−1.07483			−0.537417	+	0.843317i	$0.680600\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	14.0000	−	7.00000i	0.599694	−	0.299847i
$546$		0			0
$547$		−	8.00000i		−	0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$		−	8.00000i		−	0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	8.00000			0.340811
$552$		0			0
$553$		−	7.00000i		−	0.297670i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			20.0000i			0.847427i	0.905796	+	0.423714i	$0.139274\pi$
$557$			20.0000i			0.847427i	−0.905796	+	0.423714i	$0.860726\pi$
$558$		0			0
$559$	−4.00000			−0.169182
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	16.0000i		−	0.674320i	−0.941447	−	0.337160i	$-0.890534\pi$
$563$		−	16.0000i		−	0.674320i	0.941447	−	0.337160i	$-0.109466\pi$
$564$		0			0
$565$	10.0000	+	20.0000i	0.420703	+	0.841406i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18.0000			0.754599			0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000			0.754599			0.377300	+	0.926091i	$0.376853\pi$
$570$		0			0
$571$	−20.0000			−0.836974			−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000			−0.836974			−0.418487	+	0.908223i	$0.637439\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	8.00000	+	6.00000i	0.333623	+	0.250217i
$576$		0			0
$577$		−	17.0000i		−	0.707719i	−0.935299	−	0.353860i	$-0.884869\pi$
$577$		−	17.0000i		−	0.707719i	0.935299	−	0.353860i	$-0.115131\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12.0000			−0.497844
$582$		0			0
$583$		−	18.0000i		−	0.745484i
$584$		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
$589$	−16.0000			−0.659269
$590$		0			0
$591$		0			0
$592$		0			0
$593$			3.00000i			0.123195i	0.998101	+	0.0615976i	$0.0196196\pi$
$593$			3.00000i			0.123195i	−0.998101	+	0.0615976i	$0.980380\pi$
$594$		0			0
$595$	10.0000	−	5.00000i	0.409960	−	0.204980i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	21.0000			0.858037			0.429018	−	0.903296i	$-0.358860\pi$
$599$	21.0000			0.858037			0.429018	+	0.903296i	$0.358860\pi$
$600$		0			0
$601$	8.00000			0.326327			0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000			0.326327			0.163163	+	0.986599i	$0.447830\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.00000	+	2.00000i	−0.162623	+	0.0813116i
$606$		0			0
$607$			5.00000i			0.202944i	0.994838	+	0.101472i	$0.0323552\pi$
$607$			5.00000i			0.202944i	−0.994838	+	0.101472i	$0.967645\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	11.0000			0.445012
$612$		0			0
$613$		−	12.0000i		−	0.484675i	−0.970192	−	0.242338i	$-0.922086\pi$
$613$		−	12.0000i		−	0.484675i	0.970192	−	0.242338i	$-0.0779142\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	34.0000i		−	1.36879i	−0.729112	−	0.684394i	$-0.760067\pi$
$617$		−	34.0000i		−	1.36879i	0.729112	−	0.684394i	$-0.239933\pi$
$618$		0			0
$619$	−2.00000			−0.0803868			−0.0401934	−	0.999192i	$-0.512797\pi$
$619$	−2.00000			−0.0803868			−0.0401934	+	0.999192i	$0.512797\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			8.00000i			0.320513i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−50.0000			−1.99363
$630$		0			0
$631$	−15.0000			−0.597141			−0.298570	−	0.954388i	$-0.596510\pi$
$631$	−15.0000			−0.597141			−0.298570	+	0.954388i	$0.596510\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.00000	−	4.00000i	−0.0793676	−	0.158735i
$636$		0			0
$637$		−	1.00000i		−	0.0396214i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	26.0000			1.02694			0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000			1.02694			0.513469	+	0.858108i	$0.328360\pi$
$642$		0			0
$643$			5.00000i			0.197181i	0.995128	+	0.0985904i	$0.0314334\pi$
$643$			5.00000i			0.197181i	−0.995128	+	0.0985904i	$0.968567\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			24.0000i			0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$			24.0000i			0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$		0			0
$649$	30.0000			1.17760
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	36.0000i		−	1.40879i	−0.709809	−	0.704394i	$-0.751219\pi$
$653$		−	36.0000i		−	1.40879i	0.709809	−	0.704394i	$-0.248781\pi$
$654$		0			0
$655$	4.00000	−	2.00000i	0.156293	−	0.0781465i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−39.0000			−1.51922			−0.759612	−	0.650376i	$-0.774611\pi$
$659$	−39.0000			−1.51922			−0.759612	+	0.650376i	$0.774611\pi$
$660$		0			0
$661$	28.0000			1.08907			0.544537	−	0.838737i	$-0.316705\pi$
$661$	28.0000			1.08907			0.544537	+	0.838737i	$0.316705\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−8.00000	−	16.0000i	−0.310227	−	0.620453i
$666$		0			0
$667$		−	2.00000i		−	0.0774403i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	16.0000i		−	0.616755i	−0.951264	−	0.308377i	$-0.900214\pi$
$673$		−	16.0000i		−	0.616755i	0.951264	−	0.308377i	$-0.0997859\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	11.0000i		−	0.422764i	−0.977403	−	0.211382i	$-0.932204\pi$
$677$		−	11.0000i		−	0.422764i	0.977403	−	0.211382i	$-0.0677965\pi$
$678$		0			0
$679$	3.00000			0.115129
$680$		0			0
$681$		0			0
$682$		0			0
$683$			40.0000i			1.53056i	0.643699	+	0.765279i	$0.277399\pi$
$683$			40.0000i			1.53056i	−0.643699	+	0.765279i	$0.722601\pi$
$684$		0			0
$685$	4.00000	+	8.00000i	0.152832	+	0.305664i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6.00000			0.228582
$690$		0			0
$691$	−40.0000			−1.52167			−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000			−1.52167			−0.760836	+	0.648944i	$0.775211\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	20.0000	−	10.0000i	0.758643	−	0.379322i
$696$		0			0
$697$		−	30.0000i		−	1.13633i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	25.0000			0.944237			0.472118	−	0.881535i	$-0.343489\pi$
$701$	25.0000			0.944237			0.472118	+	0.881535i	$0.343489\pi$
$702$		0			0
$703$			80.0000i			3.01726i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			12.0000i			0.451306i
$708$		0			0
$709$	−15.0000			−0.563337			−0.281668	−	0.959512i	$-0.590888\pi$
$709$	−15.0000			−0.563337			−0.281668	+	0.959512i	$0.590888\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			4.00000i			0.149801i
$714$		0			0
$715$	3.00000	+	6.00000i	0.112194	+	0.224387i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2.00000			−0.0745874			−0.0372937	−	0.999304i	$-0.511874\pi$
$719$	−2.00000			−0.0745874			−0.0372937	+	0.999304i	$0.511874\pi$
$720$		0			0
$721$	−5.00000			−0.186210
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.00000	+	4.00000i	−0.111417	+	0.148556i
$726$		0			0
$727$			28.0000i			1.03846i	0.854634	+	0.519231i	$0.173782\pi$
$727$			28.0000i			1.03846i	−0.854634	+	0.519231i	$0.826218\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	20.0000			0.739727
$732$		0			0
$733$			41.0000i			1.51437i	0.653201	+	0.757185i	$0.273426\pi$
$733$			41.0000i			1.51437i	−0.653201	+	0.757185i	$0.726574\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	30.0000i		−	1.10506i
$738$		0			0
$739$	−5.00000			−0.183928			−0.0919640	−	0.995762i	$-0.529314\pi$
$739$	−5.00000			−0.183928			−0.0919640	+	0.995762i	$0.529314\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			30.0000i			1.10059i	0.834969	+	0.550297i	$0.185485\pi$
$743$			30.0000i			1.10059i	−0.834969	+	0.550297i	$0.814515\pi$
$744$		0			0
$745$	−12.0000	+	6.00000i	−0.439646	+	0.219823i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	8.00000			0.292314
$750$		0			0
$751$	−13.0000			−0.474377			−0.237188	−	0.971464i	$-0.576226\pi$
$751$	−13.0000			−0.474377			−0.237188	+	0.971464i	$0.576226\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−18.0000	+	9.00000i	−0.655087	+	0.327544i
$756$		0			0
$757$			48.0000i			1.74459i	0.488980	+	0.872295i	$0.337369\pi$
$757$			48.0000i			1.74459i	−0.488980	+	0.872295i	$0.662631\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	38.0000			1.37750			0.688749	−	0.724999i	$-0.258160\pi$
$761$	38.0000			1.37750			0.688749	+	0.724999i	$0.258160\pi$
$762$		0			0
$763$			7.00000i			0.253417i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			10.0000i			0.361079i
$768$		0			0
$769$	−16.0000			−0.576975			−0.288487	−	0.957484i	$-0.593152\pi$
$769$	−16.0000			−0.576975			−0.288487	+	0.957484i	$0.593152\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			27.0000i			0.971123i	0.874203	+	0.485561i	$0.161385\pi$
$773$			27.0000i			0.971123i	−0.874203	+	0.485561i	$0.838615\pi$
$774$		0			0
$775$	6.00000	−	8.00000i	0.215526	−	0.287368i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−48.0000			−1.71978
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	18.0000	+	36.0000i	0.642448	+	1.28490i
$786$		0			0
$787$			3.00000i			0.106938i	0.998569	+	0.0534692i	$0.0170279\pi$
$787$			3.00000i			0.106938i	−0.998569	+	0.0534692i	$0.982972\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−10.0000			−0.355559
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$			43.0000i			1.52314i	0.648084	+	0.761569i	$0.275571\pi$
$797$			43.0000i			1.52314i	−0.648084	+	0.761569i	$0.724429\pi$
$798$		0			0
$799$	−55.0000			−1.94576
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	30.0000i		−	1.05868i
$804$		0			0
$805$	−4.00000	+	2.00000i	−0.140981	+	0.0704907i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−11.0000			−0.386739			−0.193370	−	0.981126i	$-0.561942\pi$
$809$	−11.0000			−0.386739			−0.193370	+	0.981126i	$0.561942\pi$
$810$		0			0
$811$	−6.00000			−0.210688			−0.105344	−	0.994436i	$-0.533594\pi$
$811$	−6.00000			−0.210688			−0.105344	+	0.994436i	$0.533594\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.00000	−	12.0000i	−0.210171	−	0.420342i
$816$		0			0
$817$		−	32.0000i		−	1.11954i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−45.0000			−1.57051			−0.785255	−	0.619172i	$-0.787468\pi$
$821$	−45.0000			−1.57051			−0.785255	+	0.619172i	$0.787468\pi$
$822$		0			0
$823$			18.0000i			0.627441i	0.949515	+	0.313720i	$0.101575\pi$
$823$			18.0000i			0.627441i	−0.949515	+	0.313720i	$0.898425\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			22.0000i			0.765015i	0.923952	+	0.382507i	$0.124939\pi$
$827$			22.0000i			0.765015i	−0.923952	+	0.382507i	$0.875061\pi$
$828$		0			0
$829$	−34.0000			−1.18087			−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000			−1.18087			−0.590434	+	0.807086i	$0.701044\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			5.00000i			0.173240i
$834$		0			0
$835$	−3.00000	−	6.00000i	−0.103819	−	0.207639i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	46.0000			1.58810			0.794048	−	0.607855i	$-0.207970\pi$
$839$	46.0000			1.58810			0.794048	+	0.607855i	$0.207970\pi$
$840$		0			0
$841$	−28.0000			−0.965517
$842$		0			0
$843$		0			0
$844$		0			0
$845$	24.0000	−	12.0000i	0.825625	−	0.412813i
$846$		0			0
$847$		−	2.00000i		−	0.0687208i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	20.0000			0.685591
$852$		0			0
$853$			26.0000i			0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$			26.0000i			0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	22.0000i		−	0.751506i	−0.926720	−	0.375753i	$-0.877384\pi$
$857$		−	22.0000i		−	0.751506i	0.926720	−	0.375753i	$-0.122616\pi$
$858$		0			0
$859$	24.0000			0.818869			0.409435	−	0.912339i	$-0.365726\pi$
$859$	24.0000			0.818869			0.409435	+	0.912339i	$0.365726\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			46.0000i			1.56586i	0.622111	+	0.782929i	$0.286275\pi$
$863$			46.0000i			1.56586i	−0.622111	+	0.782929i	$0.713725\pi$
$864$		0			0
$865$	9.00000	+	18.0000i	0.306009	+	0.612018i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−21.0000			−0.712376
$870$		0			0
$871$	10.0000			0.338837
$872$		0			0
$873$		0			0
$874$		0			0
$875$	11.0000	+	2.00000i	0.371868	+	0.0676123i
$876$		0			0
$877$		−	40.0000i		−	1.35070i	−0.737496	−	0.675352i	$-0.763992\pi$
$877$		−	40.0000i		−	1.35070i	0.737496	−	0.675352i	$-0.236008\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	48.0000			1.61716			0.808581	−	0.588386i	$-0.200236\pi$
$881$	48.0000			1.61716			0.808581	+	0.588386i	$0.200236\pi$
$882$		0			0
$883$		−	4.00000i		−	0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$		−	4.00000i		−	0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	12.0000i		−	0.402921i	−0.979497	−	0.201460i	$-0.935431\pi$
$887$		−	12.0000i		−	0.402921i	0.979497	−	0.201460i	$-0.0645687\pi$
$888$		0			0
$889$	2.00000			0.0670778
$890$		0			0
$891$		0			0
$892$		0			0
$893$			88.0000i			2.94481i
$894$		0			0
$895$	−8.00000	+	4.00000i	−0.267411	+	0.133705i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.00000			−0.0667037
$900$		0			0
$901$	−30.0000			−0.999445
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−20.0000	+	10.0000i	−0.664822	+	0.332411i
$906$		0			0
$907$			38.0000i			1.26177i	0.775877	+	0.630885i	$0.217308\pi$
$907$			38.0000i			1.26177i	−0.775877	+	0.630885i	$0.782692\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−52.0000			−1.72284			−0.861418	−	0.507896i	$-0.830423\pi$
$911$	−52.0000			−1.72284			−0.861418	+	0.507896i	$0.830423\pi$
$912$		0			0
$913$			36.0000i			1.19143i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			2.00000i			0.0660458i
$918$		0			0
$919$	−55.0000			−1.81428			−0.907141	−	0.420826i	$-0.861740\pi$
$919$	−55.0000			−1.81428			−0.907141	+	0.420826i	$0.861740\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−40.0000	−	30.0000i	−1.31519	−	0.986394i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−18.0000			−0.590561			−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000			−0.590561			−0.295280	+	0.955411i	$0.595413\pi$
$930$		0			0
$931$	8.00000			0.262189
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−15.0000	−	30.0000i	−0.490552	−	0.981105i
$936$		0			0
$937$		−	7.00000i		−	0.228680i	−0.993442	−	0.114340i	$-0.963525\pi$
$937$		−	7.00000i		−	0.228680i	0.993442	−	0.114340i	$-0.0364753\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	40.0000			1.30396			0.651981	−	0.758235i	$-0.273938\pi$
$941$	40.0000			1.30396			0.651981	+	0.758235i	$0.273938\pi$
$942$		0			0
$943$			12.0000i			0.390774i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	6.00000i		−	0.194974i	−0.995237	−	0.0974869i	$-0.968920\pi$
$947$		−	6.00000i		−	0.194974i	0.995237	−	0.0974869i	$-0.0310804\pi$
$948$		0			0
$949$	10.0000			0.324614
$950$		0			0
$951$		0			0
$952$		0			0
$953$			42.0000i			1.36051i	0.732974	+	0.680257i	$0.238132\pi$
$953$			42.0000i			1.36051i	−0.732974	+	0.680257i	$0.761868\pi$
$954$		0			0
$955$	14.0000	−	7.00000i	0.453029	−	0.226515i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.00000			−0.129167
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$		0			0
$964$		0			0
$965$	8.00000	+	16.0000i	0.257529	+	0.515058i
$966$		0			0
$967$			8.00000i			0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$			8.00000i			0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	18.0000			0.577647			0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000			0.577647			0.288824	+	0.957382i	$0.406736\pi$
$972$		0			0
$973$			10.0000i			0.320585i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	60.0000i		−	1.91957i	−0.280736	−	0.959785i	$-0.590579\pi$
$977$		−	60.0000i		−	1.91957i	0.280736	−	0.959785i	$-0.409421\pi$
$978$		0			0
$979$	24.0000			0.767043
$980$		0			0
$981$		0			0
$982$		0			0
$983$			13.0000i			0.414636i	0.978274	+	0.207318i	$0.0664734\pi$
$983$			13.0000i			0.414636i	−0.978274	+	0.207318i	$0.933527\pi$
$984$		0			0
$985$	10.0000	+	20.0000i	0.318626	+	0.637253i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−8.00000			−0.254385
$990$		0			0
$991$	32.0000			1.01651			0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000			1.01651			0.508257	+	0.861206i	$0.330290\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	36.0000	−	18.0000i	1.14128	−	0.570638i
$996$		0			0
$997$			53.0000i			1.67853i	0.543725	+	0.839263i	$0.317013\pi$
$997$			53.0000i			1.67853i	−0.543725	+	0.839263i	$0.682987\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.t.s.1009.1		2
3.2	odd	2		560.2.g.a.449.1		2
4.3	odd	2		1260.2.k.c.1009.1		2
5.4	even	2	inner	5040.2.t.s.1009.2		2
12.11	even	2		140.2.e.a.29.2	yes	2
15.2	even	4		2800.2.a.a.1.1		1
15.8	even	4		2800.2.a.bf.1.1		1
15.14	odd	2		560.2.g.a.449.2		2
20.3	even	4		6300.2.a.c.1.1		1
20.7	even	4		6300.2.a.t.1.1		1
20.19	odd	2		1260.2.k.c.1009.2		2
24.5	odd	2		2240.2.g.f.449.2		2
24.11	even	2		2240.2.g.e.449.1		2
60.23	odd	4		700.2.a.a.1.1		1
60.47	odd	4		700.2.a.j.1.1		1
60.59	even	2		140.2.e.a.29.1	✓	2
84.11	even	6		980.2.q.f.569.2		4
84.23	even	6		980.2.q.f.949.1		4
84.47	odd	6		980.2.q.c.949.2		4
84.59	odd	6		980.2.q.c.569.1		4
84.83	odd	2		980.2.e.b.589.1		2
120.29	odd	2		2240.2.g.f.449.1		2
120.59	even	2		2240.2.g.e.449.2		2
420.59	odd	6		980.2.q.c.569.2		4
420.83	even	4		4900.2.a.w.1.1		1
420.167	even	4		4900.2.a.b.1.1		1
420.179	even	6		980.2.q.f.569.1		4
420.299	odd	6		980.2.q.c.949.1		4
420.359	even	6		980.2.q.f.949.2		4
420.419	odd	2		980.2.e.b.589.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.e.a.29.1	✓	2	60.59	even	2
140.2.e.a.29.2	yes	2	12.11	even	2
560.2.g.a.449.1		2	3.2	odd	2
560.2.g.a.449.2		2	15.14	odd	2
700.2.a.a.1.1		1	60.23	odd	4
700.2.a.j.1.1		1	60.47	odd	4
980.2.e.b.589.1		2	84.83	odd	2
980.2.e.b.589.2		2	420.419	odd	2
980.2.q.c.569.1		4	84.59	odd	6
980.2.q.c.569.2		4	420.59	odd	6
980.2.q.c.949.1		4	420.299	odd	6
980.2.q.c.949.2		4	84.47	odd	6
980.2.q.f.569.1		4	420.179	even	6
980.2.q.f.569.2		4	84.11	even	6
980.2.q.f.949.1		4	84.23	even	6
980.2.q.f.949.2		4	420.359	even	6
1260.2.k.c.1009.1		2	4.3	odd	2
1260.2.k.c.1009.2		2	20.19	odd	2
2240.2.g.e.449.1		2	24.11	even	2
2240.2.g.e.449.2		2	120.59	even	2
2240.2.g.f.449.1		2	120.29	odd	2
2240.2.g.f.449.2		2	24.5	odd	2
2800.2.a.a.1.1		1	15.2	even	4
2800.2.a.bf.1.1		1	15.8	even	4
4900.2.a.b.1.1		1	420.167	even	4
4900.2.a.w.1.1		1	420.83	even	4
5040.2.t.s.1009.1		2	1.1	even	1	trivial
5040.2.t.s.1009.2		2	5.4	even	2	inner
6300.2.a.c.1.1		1	20.3	even	4
6300.2.a.t.1.1		1	20.7	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5040.t (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 - 1.00000i) q^{5} +1.00000i q^{7} +3.00000 q^{11} +1.00000i q^{13} -5.00000i q^{17} -8.00000 q^{19} +2.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -1.00000 q^{29} +2.00000 q^{31} +(1.00000 + 2.00000i) q^{35} -10.0000i q^{37} +6.00000 q^{41} +4.00000i q^{43} -11.0000i q^{47} -1.00000 q^{49} -6.00000i q^{53} +(6.00000 - 3.00000i) q^{55} +10.0000 q^{59} +(1.00000 + 2.00000i) q^{65} -10.0000i q^{67} -10.0000i q^{73} +3.00000i q^{77} -7.00000 q^{79} +12.0000i q^{83} +(-5.00000 - 10.0000i) q^{85} +8.00000 q^{89} -1.00000 q^{91} +(-16.0000 + 8.00000i) q^{95} -3.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} + 6 q^{11} - 16 q^{19} + 6 q^{25} - 2 q^{29} + 4 q^{31} + 2 q^{35} + 12 q^{41} - 2 q^{49} + 12 q^{55} + 20 q^{59} + 2 q^{65} - 14 q^{79} - 10 q^{85} + 16 q^{89} - 2 q^{91} - 32 q^{95}+O(q^{100}) \) 2 * q + 4 * q^5 + 6 * q^11 - 16 * q^19 + 6 * q^25 - 2 * q^29 + 4 * q^31 + 2 * q^35 + 12 * q^41 - 2 * q^49 + 12 * q^55 + 20 * q^59 + 2 * q^65 - 14 * q^79 - 10 * q^85 + 16 * q^89 - 2 * q^91 - 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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