LMFDB - Embedded newform 2240.2.g.f.449.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2240,2,Mod(449,2240)] 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("2240.449"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,4,0,0,0,-12,0,6,0,0,0,6,0,0,0,16] 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:	$17.8864900528$
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			449.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2240.449
Dual form			2240.2.g.f.449.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +(2.00000 + 1.00000i) q^{5} -1.00000i q^{7} -6.00000 q^{9} +3.00000 q^{11} +1.00000i q^{13} +(3.00000 - 6.00000i) q^{15} -5.00000i q^{17} +8.00000 q^{19} -3.00000 q^{21} +2.00000i q^{23} +(3.00000 + 4.00000i) q^{25} +9.00000i q^{27} -1.00000 q^{29} +2.00000 q^{31} -9.00000i q^{33} +(1.00000 - 2.00000i) q^{35} -10.0000i q^{37} +3.00000 q^{39} -6.00000 q^{41} +4.00000i q^{43} +(-12.0000 - 6.00000i) q^{45} -11.0000i q^{47} -1.00000 q^{49} -15.0000 q^{51} +6.00000i q^{53} +(6.00000 + 3.00000i) q^{55} -24.0000i q^{57} +10.0000 q^{59} +6.00000i q^{63} +(-1.00000 + 2.00000i) q^{65} -10.0000i q^{67} +6.00000 q^{69} +10.0000i q^{73} +(12.0000 - 9.00000i) q^{75} -3.00000i q^{77} -7.00000 q^{79} +9.00000 q^{81} -12.0000i q^{83} +(5.00000 - 10.0000i) q^{85} +3.00000i q^{87} -8.00000 q^{89} +1.00000 q^{91} -6.00000i q^{93} +(16.0000 + 8.00000i) q^{95} +3.00000i q^{97} -18.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} - 12 q^{9} + 6 q^{11} + 6 q^{15} + 16 q^{19} - 6 q^{21} + 6 q^{25} - 2 q^{29} + 4 q^{31} + 2 q^{35} + 6 q^{39} - 12 q^{41} - 24 q^{45} - 2 q^{49} - 30 q^{51} + 12 q^{55} + 20 q^{59} - 2 q^{65}+ \cdots - 36 q^{99}+O(q^{100}) $ 2 * q + 4 * q^5 - 12 * q^9 + 6 * q^11 + 6 * q^15 + 16 * q^19 - 6 * q^21 + 6 * q^25 - 2 * q^29 + 4 * q^31 + 2 * q^35 + 6 * q^39 - 12 * q^41 - 24 * q^45 - 2 * q^49 - 30 * q^51 + 12 * q^55 + 20 * q^59 - 2 * q^65 + 12 * q^69 + 24 * q^75 - 14 * q^79 + 18 * q^81 + 10 * q^85 - 16 * q^89 + 2 * q^91 + 32 * q^95 - 36 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$		−	3.00000i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$3$		−	3.00000i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$4$		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
$9$	−6.00000			−2.00000
$10$		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
$15$	3.00000	−	6.00000i	0.774597	−	1.54919i
$16$		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	$-0.130073\pi$
$19$	8.00000			1.83533			0.917663	+	0.397360i	$0.130073\pi$
$20$		0			0
$21$	−3.00000			−0.654654
$22$		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
$27$			9.00000i			1.73205i
$28$		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
$33$		−	9.00000i		−	1.56670i
$34$		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
$39$	3.00000			0.480384
$40$		0			0
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		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
$45$	−12.0000	−	6.00000i	−1.78885	−	0.894427i
$46$		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
$51$	−15.0000			−2.10042
$52$		0			0
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$	6.00000	+	3.00000i	0.809040	+	0.404520i
$56$		0			0
$57$		−	24.0000i		−	3.17888i
$58$		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
$63$			6.00000i			0.755929i
$64$		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
$69$	6.00000			0.722315
$70$		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	$0.198986\pi$
$73$			10.0000i			1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$		0			0
$75$	12.0000	−	9.00000i	1.38564	−	1.03923i
$76$		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
$81$	9.00000			1.00000
$82$		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
$85$	5.00000	−	10.0000i	0.542326	−	1.08465i
$86$		0			0
$87$			3.00000i			0.321634i
$88$		0			0
$89$	−8.00000			−0.847998			−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000			−0.847998			−0.423999	+	0.905663i	$0.639374\pi$
$90$		0			0
$91$	1.00000			0.104828
$92$		0			0
$93$		−	6.00000i		−	0.622171i
$94$		0			0
$95$	16.0000	+	8.00000i	1.64157	+	0.820783i
$96$		0			0
$97$			3.00000i			0.304604i	0.988334	+	0.152302i	$0.0486686\pi$
$97$			3.00000i			0.304604i	−0.988334	+	0.152302i	$0.951331\pi$
$98$		0			0
$99$	−18.0000			−1.80907
$100$		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	$-0.920775\pi$
$103$		−	5.00000i		−	0.492665i	0.969185	−	0.246332i	$-0.0792255\pi$
$104$		0			0
$105$	−6.00000	−	3.00000i	−0.585540	−	0.292770i
$106$		0			0
$107$			8.00000i			0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$			8.00000i			0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$		0			0
$109$	−7.00000			−0.670478			−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000			−0.670478			−0.335239	+	0.942133i	$0.608817\pi$
$110$		0			0
$111$	−30.0000			−2.84747
$112$		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
$117$		−	6.00000i		−	0.554700i
$118$		0			0
$119$	−5.00000			−0.458349
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$			18.0000i			1.62301i
$124$		0			0
$125$	2.00000	+	11.0000i	0.178885	+	0.983870i
$126$		0			0
$127$			2.00000i			0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$			2.00000i			0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$		0			0
$129$	12.0000			1.05654
$130$		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
$135$	−9.00000	+	18.0000i	−0.774597	+	1.54919i
$136$		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	$-0.639408\pi$
$139$	−10.0000			−0.848189			−0.424094	+	0.905618i	$0.639408\pi$
$140$		0			0
$141$	−33.0000			−2.77910
$142$		0			0
$143$			3.00000i			0.250873i
$144$		0			0
$145$	−2.00000	−	1.00000i	−0.166091	−	0.0830455i
$146$		0			0
$147$			3.00000i			0.247436i
$148$		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
$153$			30.0000i			2.42536i
$154$		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
$159$	18.0000			1.42749
$160$		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
$165$	9.00000	−	18.0000i	0.700649	−	1.40130i
$166$		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
$171$	−48.0000			−3.67065
$172$		0			0
$173$		−	9.00000i		−	0.684257i	−0.939653	−	0.342129i	$-0.888852\pi$
$173$		−	9.00000i		−	0.684257i	0.939653	−	0.342129i	$-0.111148\pi$
$174$		0			0
$175$	4.00000	−	3.00000i	0.302372	−	0.226779i
$176$		0			0
$177$		−	30.0000i		−	2.25494i
$178$		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
$183$		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
$189$	9.00000			0.654654
$190$		0			0
$191$	−7.00000			−0.506502			−0.253251	−	0.967401i	$-0.581500\pi$
$191$	−7.00000			−0.506502			−0.253251	+	0.967401i	$0.581500\pi$
$192$		0			0
$193$		−	8.00000i		−	0.575853i	−0.957653	−	0.287926i	$-0.907034\pi$
$193$		−	8.00000i		−	0.575853i	0.957653	−	0.287926i	$-0.0929658\pi$
$194$		0			0
$195$	6.00000	+	3.00000i	0.429669	+	0.214834i
$196$		0			0
$197$		−	10.0000i		−	0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$		−	10.0000i		−	0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$		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
$201$	−30.0000			−2.11604
$202$		0			0
$203$			1.00000i			0.0701862i
$204$		0			0
$205$	−12.0000	−	6.00000i	−0.838116	−	0.419058i
$206$		0			0
$207$		−	12.0000i		−	0.834058i
$208$		0			0
$209$	24.0000			1.66011
$210$		0			0
$211$	−3.00000			−0.206529			−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000			−0.206529			−0.103264	+	0.994654i	$0.532929\pi$
$212$		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
$219$	30.0000			2.02721
$220$		0			0
$221$	5.00000			0.336336
$222$		0			0
$223$			19.0000i			1.27233i	0.771551	+	0.636167i	$0.219481\pi$
$223$			19.0000i			1.27233i	−0.771551	+	0.636167i	$0.780519\pi$
$224$		0			0
$225$	−18.0000	−	24.0000i	−1.20000	−	1.60000i
$226$		0			0
$227$			27.0000i			1.79205i	0.444001	+	0.896026i	$0.353559\pi$
$227$			27.0000i			1.79205i	−0.444001	+	0.896026i	$0.646441\pi$
$228$		0			0
$229$	−26.0000			−1.71813			−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000			−1.71813			−0.859064	+	0.511868i	$0.828954\pi$
$230$		0			0
$231$	−9.00000			−0.592157
$232$		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
$237$			21.0000i			1.36410i
$238$		0			0
$239$	5.00000			0.323423			0.161712	−	0.986838i	$-0.448299\pi$
$239$	5.00000			0.323423			0.161712	+	0.986838i	$0.448299\pi$
$240$		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
$249$	−36.0000			−2.28141
$250$		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
$255$	−30.0000	−	15.0000i	−1.87867	−	0.939336i
$256$		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
$261$	6.00000			0.371391
$262$		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
$267$			24.0000i			1.46878i
$268$		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
$273$		−	3.00000i		−	0.181568i
$274$		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
$279$	−12.0000			−0.718421
$280$		0			0
$281$	15.0000			0.894825			0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000			0.894825			0.447412	+	0.894328i	$0.352346\pi$
$282$		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
$285$	24.0000	−	48.0000i	1.42164	−	2.84327i
$286$		0			0
$287$			6.00000i			0.354169i
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$	9.00000			0.527589
$292$		0			0
$293$			15.0000i			0.876309i	0.898900	+	0.438155i	$0.144368\pi$
$293$			15.0000i			0.876309i	−0.898900	+	0.438155i	$0.855632\pi$
$294$		0			0
$295$	20.0000	+	10.0000i	1.16445	+	0.582223i
$296$		0			0
$297$			27.0000i			1.56670i
$298$		0			0
$299$	−2.00000			−0.115663
$300$		0			0
$301$	4.00000			0.230556
$302$		0			0
$303$		−	36.0000i		−	2.06815i
$304$		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
$309$	−15.0000			−0.853320
$310$		0			0
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$		0			0
$313$			7.00000i			0.395663i	0.980236	+	0.197832i	$0.0633900\pi$
$313$			7.00000i			0.395663i	−0.980236	+	0.197832i	$0.936610\pi$
$314$		0			0
$315$	−6.00000	+	12.0000i	−0.338062	+	0.676123i
$316$		0			0
$317$		−	28.0000i		−	1.57264i	−0.617822	−	0.786318i	$-0.711985\pi$
$317$		−	28.0000i		−	1.57264i	0.617822	−	0.786318i	$-0.288015\pi$
$318$		0			0
$319$	−3.00000			−0.167968
$320$		0			0
$321$	24.0000			1.33955
$322$		0			0
$323$		−	40.0000i		−	2.22566i
$324$		0			0
$325$	−4.00000	+	3.00000i	−0.221880	+	0.166410i
$326$		0			0
$327$			21.0000i			1.16130i
$328$		0			0
$329$	−11.0000			−0.606450
$330$		0			0
$331$	20.0000			1.09930			0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000			1.09930			0.549650	+	0.835395i	$0.314761\pi$
$332$		0			0
$333$			60.0000i			3.28798i
$334$		0			0
$335$	10.0000	−	20.0000i	0.546358	−	1.09272i
$336$		0			0
$337$			22.0000i			1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$			22.0000i			1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$		0			0
$339$	30.0000			1.62938
$340$		0			0
$341$	6.00000			0.324918
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	12.0000	+	6.00000i	0.646058	+	0.323029i
$346$		0			0
$347$			18.0000i			0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$			18.0000i			0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$		0			0
$349$	36.0000			1.92704			0.963518	−	0.267644i	$-0.0862451\pi$
$349$	36.0000			1.92704			0.963518	+	0.267644i	$0.0862451\pi$
$350$		0			0
$351$	−9.00000			−0.480384
$352$		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
$357$			15.0000i			0.793884i
$358$		0			0
$359$	28.0000			1.47778			0.738892	−	0.673824i	$-0.235349\pi$
$359$	28.0000			1.47778			0.738892	+	0.673824i	$0.235349\pi$
$360$		0			0
$361$	45.0000			2.36842
$362$		0			0
$363$			6.00000i			0.314918i
$364$		0			0
$365$	−10.0000	+	20.0000i	−0.523424	+	1.04685i
$366$		0			0
$367$		−	19.0000i		−	0.991792i	−0.868382	−	0.495896i	$-0.834840\pi$
$367$		−	19.0000i		−	0.991792i	0.868382	−	0.495896i	$-0.165160\pi$
$368$		0			0
$369$	36.0000			1.87409
$370$		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
$375$	33.0000	−	6.00000i	1.70411	−	0.309839i
$376$		0			0
$377$		−	1.00000i		−	0.0515026i
$378$		0			0
$379$	−28.0000			−1.43826			−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000			−1.43826			−0.719132	+	0.694874i	$0.755460\pi$
$380$		0			0
$381$	6.00000			0.307389
$382$		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
$387$		−	24.0000i		−	1.21999i
$388$		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
$393$		−	6.00000i		−	0.302660i
$394$		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
$399$	−24.0000			−1.20150
$400$		0			0
$401$	−27.0000			−1.34832			−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000			−1.34832			−0.674158	+	0.738587i	$0.735493\pi$
$402$		0			0
$403$			2.00000i			0.0996271i
$404$		0			0
$405$	18.0000	+	9.00000i	0.894427	+	0.447214i
$406$		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
$411$	12.0000			0.591916
$412$		0			0
$413$		−	10.0000i		−	0.492068i
$414$		0			0
$415$	12.0000	−	24.0000i	0.589057	−	1.17811i
$416$		0			0
$417$			30.0000i			1.46911i
$418$		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	$-0.310618\pi$
$421$	23.0000			1.12095			0.560476	+	0.828171i	$0.310618\pi$
$422$		0			0
$423$			66.0000i			3.20903i
$424$		0			0
$425$	20.0000	−	15.0000i	0.970143	−	0.727607i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	9.00000			0.434524
$430$		0			0
$431$	37.0000			1.78223			0.891114	−	0.453780i	$-0.149925\pi$
$431$	37.0000			1.78223			0.891114	+	0.453780i	$0.149925\pi$
$432$		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
$435$	−3.00000	+	6.00000i	−0.143839	+	0.287678i
$436$		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
$441$	6.00000			0.285714
$442$		0			0
$443$		−	12.0000i		−	0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$		−	12.0000i		−	0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$		0			0
$445$	−16.0000	−	8.00000i	−0.758473	−	0.379236i
$446$		0			0
$447$			18.0000i			0.851371i
$448$		0			0
$449$	−11.0000			−0.519122			−0.259561	−	0.965727i	$-0.583578\pi$
$449$	−11.0000			−0.519122			−0.259561	+	0.965727i	$0.583578\pi$
$450$		0			0
$451$	−18.0000			−0.847587
$452$		0			0
$453$			27.0000i			1.26857i
$454$		0			0
$455$	2.00000	+	1.00000i	0.0937614	+	0.0468807i
$456$		0			0
$457$		−	22.0000i		−	1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$		−	22.0000i		−	1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$		0			0
$459$	45.0000			2.10042
$460$		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	$0.0296290\pi$
$463$			4.00000i			0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$		0			0
$465$	6.00000	−	12.0000i	0.278243	−	0.556487i
$466$		0			0
$467$		−	23.0000i		−	1.06431i	−0.846646	−	0.532157i	$-0.821382\pi$
$467$		−	23.0000i		−	1.06431i	0.846646	−	0.532157i	$-0.178618\pi$
$468$		0			0
$469$	−10.0000			−0.461757
$470$		0			0
$471$	54.0000			2.48819
$472$		0			0
$473$			12.0000i			0.551761i
$474$		0			0
$475$	24.0000	+	32.0000i	1.10120	+	1.46826i
$476$		0			0
$477$		−	36.0000i		−	1.64833i
$478$		0			0
$479$	−18.0000			−0.822441			−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000			−0.822441			−0.411220	+	0.911536i	$0.634897\pi$
$480$		0			0
$481$	10.0000			0.455961
$482$		0			0
$483$		−	6.00000i		−	0.273009i
$484$		0			0
$485$	−3.00000	+	6.00000i	−0.136223	+	0.272446i
$486$		0			0
$487$			26.0000i			1.17817i	0.808070	+	0.589086i	$0.200512\pi$
$487$			26.0000i			1.17817i	−0.808070	+	0.589086i	$0.799488\pi$
$488$		0			0
$489$	−18.0000			−0.813988
$490$		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
$495$	−36.0000	−	18.0000i	−1.61808	−	0.809040i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	29.0000			1.29822			0.649109	−	0.760695i	$-0.275142\pi$
$499$	29.0000			1.29822			0.649109	+	0.760695i	$0.275142\pi$
$500$		0			0
$501$	−9.00000			−0.402090
$502$		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
$507$		−	36.0000i		−	1.59882i
$508$		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
$513$			72.0000i			3.17888i
$514$		0			0
$515$	5.00000	−	10.0000i	0.220326	−	0.440653i
$516$		0			0
$517$		−	33.0000i		−	1.45134i
$518$		0			0
$519$	−27.0000			−1.18517
$520$		0			0
$521$	12.0000			0.525730			0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000			0.525730			0.262865	+	0.964833i	$0.415333\pi$
$522$		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
$525$	−9.00000	−	12.0000i	−0.392792	−	0.523723i
$526$		0			0
$527$		−	10.0000i		−	0.435607i
$528$		0			0
$529$	19.0000			0.826087
$530$		0			0
$531$	−60.0000			−2.60378
$532$		0			0
$533$		−	6.00000i		−	0.259889i
$534$		0			0
$535$	−8.00000	+	16.0000i	−0.345870	+	0.691740i
$536$		0			0
$537$			12.0000i			0.517838i
$538$		0			0
$539$	−3.00000			−0.129219
$540$		0			0
$541$	25.0000			1.07483			0.537417	−	0.843317i	$-0.319400\pi$
$541$	25.0000			1.07483			0.537417	+	0.843317i	$0.319400\pi$
$542$		0			0
$543$		−	30.0000i		−	1.28742i
$544$		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
$555$	−60.0000	−	30.0000i	−2.54686	−	1.27343i
$556$		0			0
$557$		−	20.0000i		−	0.847427i	−0.905796	−	0.423714i	$-0.860726\pi$
$557$		−	20.0000i		−	0.847427i	0.905796	−	0.423714i	$-0.139274\pi$
$558$		0			0
$559$	−4.00000			−0.169182
$560$		0			0
$561$	−45.0000			−1.89990
$562$		0			0
$563$			16.0000i			0.674320i	0.941447	+	0.337160i	$0.109466\pi$
$563$			16.0000i			0.674320i	−0.941447	+	0.337160i	$0.890534\pi$
$564$		0			0
$565$	−10.0000	+	20.0000i	−0.420703	+	0.841406i
$566$		0			0
$567$		−	9.00000i		−	0.377964i
$568$		0			0
$569$	−18.0000			−0.754599			−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000			−0.754599			−0.377300	+	0.926091i	$0.623147\pi$
$570$		0			0
$571$	20.0000			0.836974			0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000			0.836974			0.418487	+	0.908223i	$0.362561\pi$
$572$		0			0
$573$			21.0000i			0.877288i
$574$		0			0
$575$	−8.00000	+	6.00000i	−0.333623	+	0.250217i
$576$		0			0
$577$			17.0000i			0.707719i	0.935299	+	0.353860i	$0.115131\pi$
$577$			17.0000i			0.707719i	−0.935299	+	0.353860i	$0.884869\pi$
$578$		0			0
$579$	−24.0000			−0.997406
$580$		0			0
$581$	−12.0000			−0.497844
$582$		0			0
$583$			18.0000i			0.745484i
$584$		0			0
$585$	6.00000	−	12.0000i	0.248069	−	0.496139i
$586$		0			0
$587$			28.0000i			1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$			28.0000i			1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$		0			0
$589$	16.0000			0.659269
$590$		0			0
$591$	−30.0000			−1.23404
$592$		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
$597$		−	54.0000i		−	2.21007i
$598$		0			0
$599$	−21.0000			−0.858037			−0.429018	−	0.903296i	$-0.641140\pi$
$599$	−21.0000			−0.858037			−0.429018	+	0.903296i	$0.641140\pi$
$600$		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
$603$			60.0000i			2.44339i
$604$		0			0
$605$	−4.00000	−	2.00000i	−0.162623	−	0.0813116i
$606$		0			0
$607$		−	5.00000i		−	0.202944i	−0.994838	−	0.101472i	$-0.967645\pi$
$607$		−	5.00000i		−	0.202944i	0.994838	−	0.101472i	$-0.0323552\pi$
$608$		0			0
$609$	3.00000			0.121566
$610$		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
$615$	−18.0000	+	36.0000i	−0.725830	+	1.45166i
$616$		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	$-0.487203\pi$
$619$	2.00000			0.0803868			0.0401934	+	0.999192i	$0.487203\pi$
$620$		0			0
$621$	−18.0000			−0.722315
$622$		0			0
$623$			8.00000i			0.320513i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$		−	72.0000i		−	2.87540i
$628$		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
$633$			9.00000i			0.357718i
$634$		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	$-0.671640\pi$
$641$	−26.0000			−1.02694			−0.513469	+	0.858108i	$0.671640\pi$
$642$		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
$645$	24.0000	+	12.0000i	0.944999	+	0.472500i
$646$		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
$651$	−6.00000			−0.235159
$652$		0			0
$653$			36.0000i			1.40879i	0.709809	+	0.704394i	$0.248781\pi$
$653$			36.0000i			1.40879i	−0.709809	+	0.704394i	$0.751219\pi$
$654$		0			0
$655$	4.00000	+	2.00000i	0.156293	+	0.0781465i
$656$		0			0
$657$		−	60.0000i		−	2.34082i
$658$		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	$-0.683295\pi$
$661$	−28.0000			−1.08907			−0.544537	+	0.838737i	$0.683295\pi$
$662$		0			0
$663$		−	15.0000i		−	0.582552i
$664$		0			0
$665$	8.00000	−	16.0000i	0.310227	−	0.620453i
$666$		0			0
$667$		−	2.00000i		−	0.0774403i
$668$		0			0
$669$	57.0000			2.20375
$670$		0			0
$671$		0			0
$672$		0			0
$673$			16.0000i			0.616755i	0.951264	+	0.308377i	$0.0997859\pi$
$673$			16.0000i			0.616755i	−0.951264	+	0.308377i	$0.900214\pi$
$674$		0			0
$675$	−36.0000	+	27.0000i	−1.38564	+	1.03923i
$676$		0			0
$677$			11.0000i			0.422764i	0.977403	+	0.211382i	$0.0677965\pi$
$677$			11.0000i			0.422764i	−0.977403	+	0.211382i	$0.932204\pi$
$678$		0			0
$679$	3.00000			0.115129
$680$		0			0
$681$	81.0000			3.10393
$682$		0			0
$683$		−	40.0000i		−	1.53056i	−0.643699	−	0.765279i	$-0.722601\pi$
$683$		−	40.0000i		−	1.53056i	0.643699	−	0.765279i	$-0.277399\pi$
$684$		0			0
$685$	−4.00000	+	8.00000i	−0.152832	+	0.305664i
$686$		0			0
$687$			78.0000i			2.97589i
$688$		0			0
$689$	−6.00000			−0.228582
$690$		0			0
$691$	40.0000			1.52167			0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000			1.52167			0.760836	+	0.648944i	$0.224789\pi$
$692$		0			0
$693$			18.0000i			0.683763i
$694$		0			0
$695$	−20.0000	−	10.0000i	−0.758643	−	0.379322i
$696$		0			0
$697$			30.0000i			1.13633i
$698$		0			0
$699$	−48.0000			−1.81553
$700$		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
$705$	−66.0000	−	33.0000i	−2.48570	−	1.24285i
$706$		0			0
$707$		−	12.0000i		−	0.451306i
$708$		0			0
$709$	15.0000			0.563337			0.281668	−	0.959512i	$-0.409112\pi$
$709$	15.0000			0.563337			0.281668	+	0.959512i	$0.409112\pi$
$710$		0			0
$711$	42.0000			1.57512
$712$		0			0
$713$			4.00000i			0.149801i
$714$		0			0
$715$	−3.00000	+	6.00000i	−0.112194	+	0.224387i
$716$		0			0
$717$		−	15.0000i		−	0.560185i
$718$		0			0
$719$	2.00000			0.0745874			0.0372937	−	0.999304i	$-0.488126\pi$
$719$	2.00000			0.0745874			0.0372937	+	0.999304i	$0.488126\pi$
$720$		0			0
$721$	−5.00000			−0.186210
$722$		0			0
$723$			54.0000i			2.00828i
$724$		0			0
$725$	−3.00000	−	4.00000i	−0.111417	−	0.148556i
$726$		0			0
$727$		−	28.0000i		−	1.03846i	−0.854634	−	0.519231i	$-0.826218\pi$
$727$		−	28.0000i		−	1.03846i	0.854634	−	0.519231i	$-0.173782\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		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
$735$	−3.00000	+	6.00000i	−0.110657	+	0.221313i
$736$		0			0
$737$		−	30.0000i		−	1.10506i
$738$		0			0
$739$	5.00000			0.183928			0.0919640	−	0.995762i	$-0.470686\pi$
$739$	5.00000			0.183928			0.0919640	+	0.995762i	$0.470686\pi$
$740$		0			0
$741$	24.0000			0.881662
$742$		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
$747$			72.0000i			2.63434i
$748$		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
$753$		−	6.00000i		−	0.218652i
$754$		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
$759$	18.0000			0.653359
$760$		0			0
$761$	−38.0000			−1.37750			−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000			−1.37750			−0.688749	+	0.724999i	$0.741840\pi$
$762$		0			0
$763$			7.00000i			0.253417i
$764$		0			0
$765$	−30.0000	+	60.0000i	−1.08465	+	2.16930i
$766$		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
$771$	18.0000			0.648254
$772$		0			0
$773$		−	27.0000i		−	0.971123i	−0.874203	−	0.485561i	$-0.838615\pi$
$773$		−	27.0000i		−	0.971123i	0.874203	−	0.485561i	$-0.161385\pi$
$774$		0			0
$775$	6.00000	+	8.00000i	0.215526	+	0.287368i
$776$		0			0
$777$			30.0000i			1.07624i
$778$		0			0
$779$	−48.0000			−1.71978
$780$		0			0
$781$		0			0
$782$		0			0
$783$		−	9.00000i		−	0.321634i
$784$		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
$789$	−72.0000			−2.56327
$790$		0			0
$791$	10.0000			0.355559
$792$		0			0
$793$		0			0
$794$		0			0
$795$	36.0000	+	18.0000i	1.27679	+	0.638394i
$796$		0			0
$797$		−	43.0000i		−	1.52314i	−0.648084	−	0.761569i	$-0.724429\pi$
$797$		−	43.0000i		−	1.52314i	0.648084	−	0.761569i	$-0.275571\pi$
$798$		0			0
$799$	−55.0000			−1.94576
$800$		0			0
$801$	48.0000			1.69600
$802$		0			0
$803$			30.0000i			1.05868i
$804$		0			0
$805$	4.00000	+	2.00000i	0.140981	+	0.0704907i
$806$		0			0
$807$		−	6.00000i		−	0.211210i
$808$		0			0
$809$	11.0000			0.386739			0.193370	−	0.981126i	$-0.438058\pi$
$809$	11.0000			0.386739			0.193370	+	0.981126i	$0.438058\pi$
$810$		0			0
$811$	6.00000			0.210688			0.105344	−	0.994436i	$-0.466406\pi$
$811$	6.00000			0.210688			0.105344	+	0.994436i	$0.466406\pi$
$812$		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
$819$	−6.00000			−0.209657
$820$		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	$-0.898425\pi$
$823$		−	18.0000i		−	0.627441i	0.949515	−	0.313720i	$-0.101575\pi$
$824$		0			0
$825$	36.0000	−	27.0000i	1.25336	−	0.940019i
$826$		0			0
$827$		−	22.0000i		−	0.765015i	−0.923952	−	0.382507i	$-0.875061\pi$
$827$		−	22.0000i		−	0.765015i	0.923952	−	0.382507i	$-0.124939\pi$
$828$		0			0
$829$	34.0000			1.18087			0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000			1.18087			0.590434	+	0.807086i	$0.298956\pi$
$830$		0			0
$831$	42.0000			1.45696
$832$		0			0
$833$			5.00000i			0.173240i
$834$		0			0
$835$	3.00000	−	6.00000i	0.103819	−	0.207639i
$836$		0			0
$837$			18.0000i			0.622171i
$838$		0			0
$839$	−46.0000			−1.58810			−0.794048	−	0.607855i	$-0.792030\pi$
$839$	−46.0000			−1.58810			−0.794048	+	0.607855i	$0.792030\pi$
$840$		0			0
$841$	−28.0000			−0.965517
$842$		0			0
$843$		−	45.0000i		−	1.54988i
$844$		0			0
$845$	24.0000	+	12.0000i	0.825625	+	0.412813i
$846$		0			0
$847$			2.00000i			0.0687208i
$848$		0			0
$849$	21.0000			0.720718
$850$		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
$855$	−96.0000	−	48.0000i	−3.28313	−	1.64157i
$856$		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	$-0.634274\pi$
$859$	−24.0000			−0.818869			−0.409435	+	0.912339i	$0.634274\pi$
$860$		0			0
$861$	18.0000			0.613438
$862$		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
$867$			24.0000i			0.815083i
$868$		0			0
$869$	−21.0000			−0.712376
$870$		0			0
$871$	10.0000			0.338837
$872$		0			0
$873$		−	18.0000i		−	0.609208i
$874$		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
$879$	45.0000			1.51781
$880$		0			0
$881$	−48.0000			−1.61716			−0.808581	−	0.588386i	$-0.799764\pi$
$881$	−48.0000			−1.61716			−0.808581	+	0.588386i	$0.799764\pi$
$882$		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
$885$	30.0000	−	60.0000i	1.00844	−	2.01688i
$886$		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
$891$	27.0000			0.904534
$892$		0			0
$893$		−	88.0000i		−	2.94481i
$894$		0			0
$895$	−8.00000	−	4.00000i	−0.267411	−	0.133705i
$896$		0			0
$897$			6.00000i			0.200334i
$898$		0			0
$899$	−2.00000			−0.0667037
$900$		0			0
$901$	30.0000			0.999445
$902$		0			0
$903$		−	12.0000i		−	0.399335i
$904$		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
$909$	−72.0000			−2.38809
$910$		0			0
$911$	52.0000			1.72284			0.861418	−	0.507896i	$-0.169577\pi$
$911$	52.0000			1.72284			0.861418	+	0.507896i	$0.169577\pi$
$912$		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
$921$	−57.0000			−1.87821
$922$		0			0
$923$		0			0
$924$		0			0
$925$	40.0000	−	30.0000i	1.31519	−	0.986394i
$926$		0			0
$927$			30.0000i			0.985329i
$928$		0			0
$929$	18.0000			0.590561			0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000			0.590561			0.295280	+	0.955411i	$0.404587\pi$
$930$		0			0
$931$	−8.00000			−0.262189
$932$		0			0
$933$			36.0000i			1.17859i
$934$		0			0
$935$	15.0000	−	30.0000i	0.490552	−	0.981105i
$936$		0			0
$937$			7.00000i			0.228680i	0.993442	+	0.114340i	$0.0364753\pi$
$937$			7.00000i			0.228680i	−0.993442	+	0.114340i	$0.963525\pi$
$938$		0			0
$939$	21.0000			0.685309
$940$		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
$945$	18.0000	+	9.00000i	0.585540	+	0.292770i
$946$		0			0
$947$			6.00000i			0.194974i	0.995237	+	0.0974869i	$0.0310804\pi$
$947$			6.00000i			0.194974i	−0.995237	+	0.0974869i	$0.968920\pi$
$948$		0			0
$949$	−10.0000			−0.324614
$950$		0			0
$951$	−84.0000			−2.72389
$952$		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
$957$			9.00000i			0.290929i
$958$		0			0
$959$	4.00000			0.129167
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$		−	48.0000i		−	1.54678i
$964$		0			0
$965$	8.00000	−	16.0000i	0.257529	−	0.515058i
$966$		0			0
$967$		−	8.00000i		−	0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$		−	8.00000i		−	0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$		0			0
$969$	−120.000			−3.85496
$970$		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
$975$	9.00000	+	12.0000i	0.288231	+	0.384308i
$976$		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
$981$	42.0000			1.34096
$982$		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
$987$			33.0000i			1.05040i
$988$		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
$993$		−	60.0000i		−	1.90404i
$994$		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
$999$	90.0000			2.84747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.g.f.449.1		2
4.3	odd	2		2240.2.g.e.449.2		2
5.4	even	2	inner	2240.2.g.f.449.2		2
8.3	odd	2		140.2.e.a.29.1	✓	2
8.5	even	2		560.2.g.a.449.2		2
20.19	odd	2		2240.2.g.e.449.1		2
24.5	odd	2		5040.2.t.s.1009.2		2
24.11	even	2		1260.2.k.c.1009.2		2
40.3	even	4		700.2.a.j.1.1		1
40.13	odd	4		2800.2.a.a.1.1		1
40.19	odd	2		140.2.e.a.29.2	yes	2
40.27	even	4		700.2.a.a.1.1		1
40.29	even	2		560.2.g.a.449.1		2
40.37	odd	4		2800.2.a.bf.1.1		1
56.3	even	6		980.2.q.c.569.2		4
56.11	odd	6		980.2.q.f.569.1		4
56.19	even	6		980.2.q.c.949.1		4
56.27	even	2		980.2.e.b.589.2		2
56.51	odd	6		980.2.q.f.949.2		4
120.29	odd	2		5040.2.t.s.1009.1		2
120.59	even	2		1260.2.k.c.1009.1		2
120.83	odd	4		6300.2.a.t.1.1		1
120.107	odd	4		6300.2.a.c.1.1		1
280.19	even	6		980.2.q.c.949.2		4
280.27	odd	4		4900.2.a.w.1.1		1
280.59	even	6		980.2.q.c.569.1		4
280.83	odd	4		4900.2.a.b.1.1		1
280.139	even	2		980.2.e.b.589.1		2
280.179	odd	6		980.2.q.f.569.2		4
280.219	odd	6		980.2.q.f.949.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.e.a.29.1	✓	2	8.3	odd	2
140.2.e.a.29.2	yes	2	40.19	odd	2
560.2.g.a.449.1		2	40.29	even	2
560.2.g.a.449.2		2	8.5	even	2
700.2.a.a.1.1		1	40.27	even	4
700.2.a.j.1.1		1	40.3	even	4
980.2.e.b.589.1		2	280.139	even	2
980.2.e.b.589.2		2	56.27	even	2
980.2.q.c.569.1		4	280.59	even	6
980.2.q.c.569.2		4	56.3	even	6
980.2.q.c.949.1		4	56.19	even	6
980.2.q.c.949.2		4	280.19	even	6
980.2.q.f.569.1		4	56.11	odd	6
980.2.q.f.569.2		4	280.179	odd	6
980.2.q.f.949.1		4	280.219	odd	6
980.2.q.f.949.2		4	56.51	odd	6
1260.2.k.c.1009.1		2	120.59	even	2
1260.2.k.c.1009.2		2	24.11	even	2
2240.2.g.e.449.1		2	20.19	odd	2
2240.2.g.e.449.2		2	4.3	odd	2
2240.2.g.f.449.1		2	1.1	even	1	trivial
2240.2.g.f.449.2		2	5.4	even	2	inner
2800.2.a.a.1.1		1	40.13	odd	4
2800.2.a.bf.1.1		1	40.37	odd	4
4900.2.a.b.1.1		1	280.83	odd	4
4900.2.a.w.1.1		1	280.27	odd	4
5040.2.t.s.1009.1		2	120.29	odd	2
5040.2.t.s.1009.2		2	24.5	odd	2
6300.2.a.c.1.1		1	120.107	odd	4
6300.2.a.t.1.1		1	120.83	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +(2.00000 + 1.00000i) q^{5} -1.00000i q^{7} -6.00000 q^{9} +3.00000 q^{11} +1.00000i q^{13} +(3.00000 - 6.00000i) q^{15} -5.00000i q^{17} +8.00000 q^{19} -3.00000 q^{21} +2.00000i q^{23} +(3.00000 + 4.00000i) q^{25} +9.00000i q^{27} -1.00000 q^{29} +2.00000 q^{31} -9.00000i q^{33} +(1.00000 - 2.00000i) q^{35} -10.0000i q^{37} +3.00000 q^{39} -6.00000 q^{41} +4.00000i q^{43} +(-12.0000 - 6.00000i) q^{45} -11.0000i q^{47} -1.00000 q^{49} -15.0000 q^{51} +6.00000i q^{53} +(6.00000 + 3.00000i) q^{55} -24.0000i q^{57} +10.0000 q^{59} +6.00000i q^{63} +(-1.00000 + 2.00000i) q^{65} -10.0000i q^{67} +6.00000 q^{69} +10.0000i q^{73} +(12.0000 - 9.00000i) q^{75} -3.00000i q^{77} -7.00000 q^{79} +9.00000 q^{81} -12.0000i q^{83} +(5.00000 - 10.0000i) q^{85} +3.00000i q^{87} -8.00000 q^{89} +1.00000 q^{91} -6.00000i q^{93} +(16.0000 + 8.00000i) q^{95} +3.00000i q^{97} -18.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} - 12 q^{9} + 6 q^{11} + 6 q^{15} + 16 q^{19} - 6 q^{21} + 6 q^{25} - 2 q^{29} + 4 q^{31} + 2 q^{35} + 6 q^{39} - 12 q^{41} - 24 q^{45} - 2 q^{49} - 30 q^{51} + 12 q^{55} + 20 q^{59} - 2 q^{65}+ \cdots - 36 q^{99}+O(q^{100}) \) 2 * q + 4 * q^5 - 12 * q^9 + 6 * q^11 + 6 * q^15 + 16 * q^19 - 6 * q^21 + 6 * q^25 - 2 * q^29 + 4 * q^31 + 2 * q^35 + 6 * q^39 - 12 * q^41 - 24 * q^45 - 2 * q^49 - 30 * q^51 + 12 * q^55 + 20 * q^59 - 2 * q^65 + 12 * q^69 + 24 * q^75 - 14 * q^79 + 18 * q^81 + 10 * q^85 - 16 * q^89 + 2 * q^91 + 32 * q^95 - 36 * q^99

Introduction

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Database

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