LMFDB - Embedded newform 980.2.e.b.589.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [980,2,Mod(589,980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(980, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("980.589");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 980 = 2^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	980.e (of order $2$, degree $1$, 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:	$7.82533939809$
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			589.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	980.589
Dual form			980.2.e.b.589.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$.

$n$	$101$	$197$	$491$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	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
$7$		0			0
$7$		0			0
$8$		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	$-0.955716\pi$
$13$		−	1.00000i		−	0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$		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	$-0.869927\pi$
$19$	−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
$27$			9.00000i			1.73205i
$28$		0			0
$29$	1.00000			0.185695			0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000			0.185695			0.0928477	+	0.995680i	$0.470403\pi$
$30$		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
$35$		0			0
$36$		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	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$		0			0
$45$	−12.0000	+	6.00000i	−1.78885	+	0.894427i
$46$		0			0
$47$			11.0000i			1.60451i	0.596978	+	0.802257i	$0.296368\pi$
$47$			11.0000i			1.60451i	−0.596978	+	0.802257i	$0.703632\pi$
$48$		0			0
$49$		0			0
$50$		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	$-0.725627\pi$
$59$	−10.0000			−1.30189			−0.650945	+	0.759125i	$0.725627\pi$
$60$		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	$0.209171\pi$
$67$			10.0000i			1.22169i	−0.791748	+	0.610847i	$0.790829\pi$
$68$		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
$77$		0			0
$78$		0			0
$79$	7.00000			0.787562			0.393781	−	0.919204i	$-0.371167\pi$
$79$	7.00000			0.787562			0.393781	+	0.919204i	$0.371167\pi$
$80$		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	$-0.360626\pi$
$89$	8.00000			0.847998			0.423999	+	0.905663i	$0.360626\pi$
$90$		0			0
$91$		0			0
$92$		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	$0.0792255\pi$
$103$			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
$111$	−30.0000			−2.84747
$112$		0			0
$113$		−	10.0000i		−	0.940721i	−0.882474	−	0.470360i	$-0.844124\pi$
$113$		−	10.0000i		−	0.940721i	0.882474	−	0.470360i	$-0.155876\pi$
$114$		0			0
$115$	2.00000	+	4.00000i	0.186501	+	0.373002i
$116$		0			0
$117$			6.00000i			0.554700i
$118$		0			0
$119$		0			0
$120$		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	$-0.527846\pi$
$131$	−2.00000			−0.174741			−0.0873704	+	0.996176i	$0.527846\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	9.00000	+	18.0000i	0.774597	+	1.54919i
$136$		0			0
$137$		−	4.00000i		−	0.341743i	−0.985293	−	0.170872i	$-0.945342\pi$
$137$		−	4.00000i		−	0.341743i	0.985293	−	0.170872i	$-0.0546583\pi$
$138$		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
$141$	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
$147$		0			0
$148$		0			0
$149$	6.00000			0.491539			0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000			0.491539			0.245770	+	0.969328i	$0.420959\pi$
$150$		0			0
$151$	9.00000			0.732410			0.366205	−	0.930534i	$-0.380657\pi$
$151$	9.00000			0.732410			0.366205	+	0.930534i	$0.380657\pi$
$152$		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	$-0.744931\pi$
$157$		−	18.0000i		−	1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$		0			0
$159$	18.0000			1.42749
$160$		0			0
$161$		0			0
$162$		0			0
$163$			6.00000i			0.469956i	0.972001	+	0.234978i	$0.0755019\pi$
$163$			6.00000i			0.469956i	−0.972001	+	0.234978i	$0.924498\pi$
$164$		0			0
$165$	−9.00000	−	18.0000i	−0.700649	−	1.40130i
$166$		0			0
$167$			3.00000i			0.232147i	0.993241	+	0.116073i	$0.0370308\pi$
$167$			3.00000i			0.232147i	−0.993241	+	0.116073i	$0.962969\pi$
$168$		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	$0.111148\pi$
$173$			9.00000i			0.684257i	−0.939653	+	0.342129i	$0.888852\pi$
$174$		0			0
$175$		0			0
$176$		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
$189$		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
$195$	−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
$203$		0			0
$204$		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
$217$		0			0
$218$		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	$-0.780519\pi$
$223$		−	19.0000i		−	1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$		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
$231$		0			0
$232$		0			0
$233$			16.0000i			1.04819i	0.851658	+	0.524097i	$0.175597\pi$
$233$			16.0000i			1.04819i	−0.851658	+	0.524097i	$0.824403\pi$
$234$		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	$-0.551701\pi$
$239$	−5.00000			−0.323423			−0.161712	+	0.986838i	$0.551701\pi$
$240$		0			0
$241$	18.0000			1.15948			0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000			1.15948			0.579741	+	0.814801i	$0.303154\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		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	$-0.520105\pi$
$251$	−2.00000			−0.126239			−0.0631194	+	0.998006i	$0.520105\pi$
$252$		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
$259$		0			0
$260$		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
$273$		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
$279$	−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
$287$		0			0
$288$		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	$-0.855632\pi$
$293$		−	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
$297$			27.0000i			1.56670i
$298$		0			0
$299$	2.00000			0.115663
$300$		0			0
$301$		0			0
$302$		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
$315$		0			0
$316$		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
$329$		0			0
$330$		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	$-0.795482\pi$
$337$		−	22.0000i		−	1.19842i	0.800593	−	0.599208i	$-0.204518\pi$
$338$		0			0
$339$	−30.0000			−1.62938
$340$		0			0
$341$	6.00000			0.324918
$342$		0			0
$343$		0			0
$344$		0			0
$345$	12.0000	−	6.00000i	0.646058	−	0.323029i
$346$		0			0
$347$		−	18.0000i		−	0.966291i	−0.875540	−	0.483145i	$-0.839494\pi$
$347$		−	18.0000i		−	0.966291i	0.875540	−	0.483145i	$-0.160506\pi$
$348$		0			0
$349$	36.0000			1.92704			0.963518	−	0.267644i	$-0.0862451\pi$
$349$	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
$357$		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
$363$			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	$0.165160\pi$
$367$			19.0000i			0.991792i	−0.868382	+	0.495896i	$0.834840\pi$
$368$		0			0
$369$	−36.0000			−1.87409
$370$		0			0
$371$		0			0
$372$		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
$385$		0			0
$386$		0			0
$387$			24.0000i			1.21999i
$388$		0			0
$389$	−9.00000			−0.456318			−0.228159	−	0.973624i	$-0.573271\pi$
$389$	−9.00000			−0.456318			−0.228159	+	0.973624i	$0.573271\pi$
$390$		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	$-0.859710\pi$
$397$		−	17.0000i		−	0.853206i	0.904439	−	0.426603i	$-0.140290\pi$
$398$		0			0
$399$		0			0
$400$		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	$-0.468470\pi$
$409$	4.00000			0.197787			0.0988936	+	0.995098i	$0.468470\pi$
$410$		0			0
$411$	−12.0000			−0.591916
$412$		0			0
$413$		0			0
$414$		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	$-0.484443\pi$
$419$	2.00000			0.0977064			0.0488532	+	0.998806i	$0.484443\pi$
$420$		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
$423$		−	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	$-0.850075\pi$
$431$	−37.0000			−1.78223			−0.891114	+	0.453780i	$0.850075\pi$
$432$		0			0
$433$			38.0000i			1.82616i	0.407777	+	0.913082i	$0.366304\pi$
$433$			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
$441$		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
$447$		−	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
$455$		0			0
$456$		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
$459$	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
$469$		0			0
$470$		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
$483$		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	$0.200512\pi$
$487$			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	$-0.992903\pi$
$503$		−	1.00000i		−	0.0445878i	0.999751	−	0.0222939i	$-0.00709696\pi$
$504$		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
$511$		0			0
$512$		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	$-0.584667\pi$
$521$	−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
$531$	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
$539$		0			0
$540$		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
$543$		−	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	$0.0547087\pi$
$547$			8.00000i			0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−8.00000			−0.340811
$552$		0			0
$553$		0			0
$554$		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
$567$		0			0
$568$		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
$581$		0			0
$582$		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
$595$		0			0
$596$		0			0
$597$		−	54.0000i		−	2.21007i
$598$		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	$-0.552170\pi$
$601$	−8.00000			−0.326327			−0.163163	+	0.986599i	$0.552170\pi$
$602$		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	$0.0323552\pi$
$607$			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
$615$	−18.0000	−	36.0000i	−0.725830	−	1.45166i
$616$		0			0
$617$			34.0000i			1.36879i	0.729112	+	0.684394i	$0.239933\pi$
$617$			34.0000i			1.36879i	−0.729112	+	0.684394i	$0.760067\pi$
$618$		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
$621$	−18.0000			−0.722315
$622$		0			0
$623$		0			0
$624$		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	$-0.403490\pi$
$631$	15.0000			0.597141			0.298570	+	0.954388i	$0.403490\pi$
$632$		0			0
$633$			9.00000i			0.357718i
$634$		0			0
$635$	2.00000	+	4.00000i	0.0793676	+	0.158735i
$636$		0			0
$637$		0			0
$638$		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	$-0.843616\pi$
$647$		−	24.0000i		−	0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$		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	$0.248781\pi$
$653$			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
$665$		0			0
$666$		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	$-0.900214\pi$
$673$		−	16.0000i		−	0.616755i	0.951264	−	0.308377i	$-0.0997859\pi$
$674$		0			0
$675$	36.0000	+	27.0000i	1.38564	+	1.03923i
$676$		0			0
$677$		−	11.0000i		−	0.422764i	−0.977403	−	0.211382i	$-0.932204\pi$
$677$		−	11.0000i		−	0.422764i	0.977403	−	0.211382i	$-0.0677965\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	81.0000			3.10393
$682$		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
$687$			78.0000i			2.97589i
$688$		0			0
$689$	6.00000			0.228582
$690$		0			0
$691$	−40.0000			−1.52167			−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−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
$699$	48.0000			1.81553
$700$		0			0
$701$	−25.0000			−0.944237			−0.472118	−	0.881535i	$-0.656511\pi$
$701$	−25.0000			−0.944237			−0.472118	+	0.881535i	$0.656511\pi$
$702$		0			0
$703$			80.0000i			3.01726i
$704$		0			0
$705$	66.0000	−	33.0000i	2.48570	−	1.24285i
$706$		0			0
$707$		0			0
$708$		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
$711$	−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
$721$		0			0
$722$		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	$0.173782\pi$
$727$			28.0000i			1.03846i	−0.854634	+	0.519231i	$0.826218\pi$
$728$		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	$-0.726574\pi$
$733$		−	41.0000i		−	1.51437i	0.653201	−	0.757185i	$-0.273426\pi$
$734$		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	$-0.470686\pi$
$739$	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
$749$		0			0
$750$		0			0
$751$	13.0000			0.474377			0.237188	−	0.971464i	$-0.423774\pi$
$751$	13.0000			0.474377			0.237188	+	0.971464i	$0.423774\pi$
$752$		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	$-0.258160\pi$
$761$	38.0000			1.37750			0.688749	+	0.724999i	$0.258160\pi$
$762$		0			0
$763$		0			0
$764$		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	$-0.406848\pi$
$769$	16.0000			0.576975			0.288487	+	0.957484i	$0.406848\pi$
$770$		0			0
$771$	18.0000			0.648254
$772$		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
$783$			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
$791$		0			0
$792$		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

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:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.e (of order \(2\), degree \(1\), minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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