LMFDB - Embedded newform 416.2.f.a.129.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(129,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("416.129");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.f (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:	$3.32177672409$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	416.129
Dual form			416.2.f.a.129.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +3.00000i q^{5} -1.00000i q^{7} +6.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +3.00000i q^{5} -1.00000i q^{7} +6.00000 q^{9} +4.00000i q^{11} +(-2.00000 - 3.00000i) q^{13} -9.00000i q^{15} -5.00000 q^{17} -6.00000i q^{19} +3.00000i q^{21} -6.00000 q^{23} -4.00000 q^{25} -9.00000 q^{27} -4.00000 q^{29} -12.0000i q^{33} +3.00000 q^{35} -3.00000i q^{37} +(6.00000 + 9.00000i) q^{39} -12.0000i q^{41} -3.00000 q^{43} +18.0000i q^{45} +7.00000i q^{47} +6.00000 q^{49} +15.0000 q^{51} -2.00000 q^{53} -12.0000 q^{55} +18.0000i q^{57} -2.00000i q^{59} -12.0000 q^{61} -6.00000i q^{63} +(9.00000 - 6.00000i) q^{65} -4.00000i q^{67} +18.0000 q^{69} +11.0000i q^{71} +6.00000i q^{73} +12.0000 q^{75} +4.00000 q^{77} +6.00000 q^{79} +9.00000 q^{81} -10.0000i q^{83} -15.0000i q^{85} +12.0000 q^{87} +6.00000i q^{89} +(-3.00000 + 2.00000i) q^{91} +18.0000 q^{95} +24.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 12 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 12 * q^9	$ 2 q - 6 q^{3} + 12 q^{9} - 4 q^{13} - 10 q^{17} - 12 q^{23} - 8 q^{25} - 18 q^{27} - 8 q^{29} + 6 q^{35} + 12 q^{39} - 6 q^{43} + 12 q^{49} + 30 q^{51} - 4 q^{53} - 24 q^{55} - 24 q^{61} + 18 q^{65} + 36 q^{69} + 24 q^{75} + 8 q^{77} + 12 q^{79} + 18 q^{81} + 24 q^{87} - 6 q^{91} + 36 q^{95}+O(q^{100}) $ 2 * q - 6 * q^3 + 12 * q^9 - 4 * q^13 - 10 * q^17 - 12 * q^23 - 8 * q^25 - 18 * q^27 - 8 * q^29 + 6 * q^35 + 12 * q^39 - 6 * q^43 + 12 * q^49 + 30 * q^51 - 4 * q^53 - 24 * q^55 - 24 * q^61 + 18 * q^65 + 36 * q^69 + 24 * q^75 + 8 * q^77 + 12 * q^79 + 18 * q^81 + 24 * q^87 - 6 * q^91 + 36 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−3.00000			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$4$		0			0
$5$			3.00000i			1.34164i	0.741620	+	0.670820i	$0.234058\pi$
$5$			3.00000i			1.34164i	−0.741620	+	0.670820i	$0.765942\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$		−	1.00000i		−	0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$		0			0
$9$	6.00000			2.00000
$10$		0			0
$11$			4.00000i			1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$			4.00000i			1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$		0			0
$13$	−2.00000	−	3.00000i	−0.554700	−	0.832050i
$13$	−2.00000	−	3.00000i	−0.554700	−	0.832050i
$14$		0			0
$15$		−	9.00000i		−	2.32379i
$16$		0			0
$17$	−5.00000			−1.21268			−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000			−1.21268			−0.606339	+	0.795206i	$0.707363\pi$
$18$		0			0
$19$		−	6.00000i		−	1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$		−	6.00000i		−	1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$		0			0
$21$			3.00000i			0.654654i
$22$		0			0
$23$	−6.00000			−1.25109			−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000			−1.25109			−0.625543	+	0.780189i	$0.715123\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$		0			0
$27$	−9.00000			−1.73205
$28$		0			0
$29$	−4.00000			−0.742781			−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000			−0.742781			−0.371391	+	0.928477i	$0.621119\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		−	12.0000i		−	2.08893i
$34$		0			0
$35$	3.00000			0.507093
$36$		0			0
$37$		−	3.00000i		−	0.493197i	−0.969118	−	0.246598i	$-0.920687\pi$
$37$		−	3.00000i		−	0.493197i	0.969118	−	0.246598i	$-0.0793129\pi$
$38$		0			0
$39$	6.00000	+	9.00000i	0.960769	+	1.44115i
$40$		0			0
$41$		−	12.0000i		−	1.87409i	−0.349215	−	0.937043i	$-0.613552\pi$
$41$		−	12.0000i		−	1.87409i	0.349215	−	0.937043i	$-0.386448\pi$
$42$		0			0
$43$	−3.00000			−0.457496			−0.228748	−	0.973486i	$-0.573463\pi$
$43$	−3.00000			−0.457496			−0.228748	+	0.973486i	$0.573463\pi$
$44$		0			0
$45$			18.0000i			2.68328i
$46$		0			0
$47$			7.00000i			1.02105i	0.859861	+	0.510527i	$0.170550\pi$
$47$			7.00000i			1.02105i	−0.859861	+	0.510527i	$0.829450\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$		0			0
$51$	15.0000			2.10042
$52$		0			0
$53$	−2.00000			−0.274721			−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000			−0.274721			−0.137361	+	0.990521i	$0.543862\pi$
$54$		0			0
$55$	−12.0000			−1.61808
$56$		0			0
$57$			18.0000i			2.38416i
$58$		0			0
$59$		−	2.00000i		−	0.260378i	−0.991489	−	0.130189i	$-0.958442\pi$
$59$		−	2.00000i		−	0.260378i	0.991489	−	0.130189i	$-0.0415584\pi$
$60$		0			0
$61$	−12.0000			−1.53644			−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000			−1.53644			−0.768221	+	0.640184i	$0.778858\pi$
$62$		0			0
$63$		−	6.00000i		−	0.755929i
$64$		0			0
$65$	9.00000	−	6.00000i	1.11631	−	0.744208i
$66$		0			0
$67$		−	4.00000i		−	0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$		−	4.00000i		−	0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$		0			0
$69$	18.0000			2.16695
$70$		0			0
$71$			11.0000i			1.30546i	0.757591	+	0.652730i	$0.226376\pi$
$71$			11.0000i			1.30546i	−0.757591	+	0.652730i	$0.773624\pi$
$72$		0			0
$73$			6.00000i			0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$		0			0
$75$	12.0000			1.38564
$76$		0			0
$77$	4.00000			0.455842
$78$		0			0
$79$	6.00000			0.675053			0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000			0.675053			0.337526	+	0.941316i	$0.390410\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$		−	10.0000i		−	1.09764i	−0.835940	−	0.548821i	$-0.815077\pi$
$83$		−	10.0000i		−	1.09764i	0.835940	−	0.548821i	$-0.184923\pi$
$84$		0			0
$85$		−	15.0000i		−	1.62698i
$86$		0			0
$87$	12.0000			1.28654
$88$		0			0
$89$			6.00000i			0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$			6.00000i			0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$		0			0
$91$	−3.00000	+	2.00000i	−0.314485	+	0.209657i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	18.0000			1.84676
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$			24.0000i			2.41209i
$100$		0			0
$101$	−4.00000			−0.398015			−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000			−0.398015			−0.199007	+	0.979998i	$0.563772\pi$
$102$		0			0
$103$	−6.00000			−0.591198			−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000			−0.591198			−0.295599	+	0.955312i	$0.595519\pi$
$104$		0			0
$105$	−9.00000			−0.878310
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$			15.0000i			1.43674i	0.695662	+	0.718370i	$0.255111\pi$
$109$			15.0000i			1.43674i	−0.695662	+	0.718370i	$0.744889\pi$
$110$		0			0
$111$			9.00000i			0.854242i
$112$		0			0
$113$	−14.0000			−1.31701			−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000			−1.31701			−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$		−	18.0000i		−	1.67851i
$116$		0			0
$117$	−12.0000	−	18.0000i	−1.10940	−	1.66410i
$118$		0			0
$119$			5.00000i			0.458349i
$120$		0			0
$121$	−5.00000			−0.454545
$122$		0			0
$123$			36.0000i			3.24601i
$124$		0			0
$125$			3.00000i			0.268328i
$126$		0			0
$127$	−18.0000			−1.59724			−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000			−1.59724			−0.798621	+	0.601834i	$0.794437\pi$
$128$		0			0
$129$	9.00000			0.792406
$130$		0			0
$131$	−9.00000			−0.786334			−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000			−0.786334			−0.393167	+	0.919467i	$0.628621\pi$
$132$		0			0
$133$	−6.00000			−0.520266
$134$		0			0
$135$		−	27.0000i		−	2.32379i
$136$		0			0
$137$		−	6.00000i		−	0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$		−	6.00000i		−	0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$		0			0
$139$	−9.00000			−0.763370			−0.381685	−	0.924292i	$-0.624656\pi$
$139$	−9.00000			−0.763370			−0.381685	+	0.924292i	$0.624656\pi$
$140$		0			0
$141$		−	21.0000i		−	1.76852i
$142$		0			0
$143$	12.0000	−	8.00000i	1.00349	−	0.668994i
$144$		0			0
$145$		−	12.0000i		−	0.996546i
$146$		0			0
$147$	−18.0000			−1.48461
$148$		0			0
$149$			6.00000i			0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$			6.00000i			0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$		0			0
$151$		−	3.00000i		−	0.244137i	−0.992522	−	0.122068i	$-0.961047\pi$
$151$		−	3.00000i		−	0.244137i	0.992522	−	0.122068i	$-0.0389527\pi$
$152$		0			0
$153$	−30.0000			−2.42536
$154$		0			0
$155$		0			0
$156$		0			0
$157$	14.0000			1.11732			0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000			1.11732			0.558661	+	0.829396i	$0.311315\pi$
$158$		0			0
$159$	6.00000			0.475831
$160$		0			0
$161$			6.00000i			0.472866i
$162$		0			0
$163$			18.0000i			1.40987i	0.709273	+	0.704934i	$0.249024\pi$
$163$			18.0000i			1.40987i	−0.709273	+	0.704934i	$0.750976\pi$
$164$		0			0
$165$	36.0000			2.80260
$166$		0			0
$167$		−	4.00000i		−	0.309529i	−0.987951	−	0.154765i	$-0.950538\pi$
$167$		−	4.00000i		−	0.309529i	0.987951	−	0.154765i	$-0.0494619\pi$
$168$		0			0
$169$	−5.00000	+	12.0000i	−0.384615	+	0.923077i
$170$		0			0
$171$		−	36.0000i		−	2.75299i
$172$		0			0
$173$	−14.0000			−1.06440			−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000			−1.06440			−0.532200	+	0.846619i	$0.678635\pi$
$174$		0			0
$175$			4.00000i			0.302372i
$176$		0			0
$177$			6.00000i			0.450988i
$178$		0			0
$179$	21.0000			1.56961			0.784807	−	0.619740i	$-0.212762\pi$
$179$	21.0000			1.56961			0.784807	+	0.619740i	$0.212762\pi$
$180$		0			0
$181$	−6.00000			−0.445976			−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000			−0.445976			−0.222988	+	0.974821i	$0.571581\pi$
$182$		0			0
$183$	36.0000			2.66120
$184$		0			0
$185$	9.00000			0.661693
$186$		0			0
$187$		−	20.0000i		−	1.46254i
$188$		0			0
$189$			9.00000i			0.654654i
$190$		0			0
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$		0			0
$193$			18.0000i			1.29567i	0.761781	+	0.647834i	$0.224325\pi$
$193$			18.0000i			1.29567i	−0.761781	+	0.647834i	$0.775675\pi$
$194$		0			0
$195$	−27.0000	+	18.0000i	−1.93351	+	1.28901i
$196$		0			0
$197$		−	3.00000i		−	0.213741i	−0.994273	−	0.106871i	$-0.965917\pi$
$197$		−	3.00000i		−	0.213741i	0.994273	−	0.106871i	$-0.0340831\pi$
$198$		0			0
$199$	−12.0000			−0.850657			−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000			−0.850657			−0.425329	+	0.905039i	$0.639842\pi$
$200$		0			0
$201$			12.0000i			0.846415i
$202$		0			0
$203$			4.00000i			0.280745i
$204$		0			0
$205$	36.0000			2.51435
$206$		0			0
$207$	−36.0000			−2.50217
$208$		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
$213$		−	33.0000i		−	2.26112i
$214$		0			0
$215$		−	9.00000i		−	0.613795i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		−	18.0000i		−	1.21633i
$220$		0			0
$221$	10.0000	+	15.0000i	0.672673	+	1.00901i
$222$		0			0
$223$		−	21.0000i		−	1.40626i	−0.711059	−	0.703132i	$-0.751784\pi$
$223$		−	21.0000i		−	1.40626i	0.711059	−	0.703132i	$-0.248216\pi$
$224$		0			0
$225$	−24.0000			−1.60000
$226$		0			0
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$		0			0
$229$		−	9.00000i		−	0.594737i	−0.954763	−	0.297368i	$-0.903891\pi$
$229$		−	9.00000i		−	0.594737i	0.954763	−	0.297368i	$-0.0961089\pi$
$230$		0			0
$231$	−12.0000			−0.789542
$232$		0			0
$233$	19.0000			1.24473			0.622366	−	0.782727i	$-0.286172\pi$
$233$	19.0000			1.24473			0.622366	+	0.782727i	$0.286172\pi$
$234$		0			0
$235$	−21.0000			−1.36989
$236$		0			0
$237$	−18.0000			−1.16923
$238$		0			0
$239$			5.00000i			0.323423i	0.986838	+	0.161712i	$0.0517014\pi$
$239$			5.00000i			0.323423i	−0.986838	+	0.161712i	$0.948299\pi$
$240$		0			0
$241$			6.00000i			0.386494i	0.981150	+	0.193247i	$0.0619019\pi$
$241$			6.00000i			0.386494i	−0.981150	+	0.193247i	$0.938098\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			18.0000i			1.14998i
$246$		0			0
$247$	−18.0000	+	12.0000i	−1.14531	+	0.763542i
$248$		0			0
$249$			30.0000i			1.90117i
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$		−	24.0000i		−	1.50887i
$254$		0			0
$255$			45.0000i			2.81801i
$256$		0			0
$257$	−5.00000			−0.311891			−0.155946	−	0.987766i	$-0.549842\pi$
$257$	−5.00000			−0.311891			−0.155946	+	0.987766i	$0.549842\pi$
$258$		0			0
$259$	−3.00000			−0.186411
$260$		0			0
$261$	−24.0000			−1.48556
$262$		0			0
$263$	12.0000			0.739952			0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000			0.739952			0.369976	+	0.929041i	$0.379366\pi$
$264$		0			0
$265$		−	6.00000i		−	0.368577i
$266$		0			0
$267$		−	18.0000i		−	1.10158i
$268$		0			0
$269$	−4.00000			−0.243884			−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000			−0.243884			−0.121942	+	0.992537i	$0.538912\pi$
$270$		0			0
$271$			11.0000i			0.668202i	0.942537	+	0.334101i	$0.108433\pi$
$271$			11.0000i			0.668202i	−0.942537	+	0.334101i	$0.891567\pi$
$272$		0			0
$273$	9.00000	−	6.00000i	0.544705	−	0.363137i
$274$		0			0
$275$		−	16.0000i		−	0.964836i
$276$		0			0
$277$	12.0000			0.721010			0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000			0.721010			0.360505	+	0.932757i	$0.382604\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	18.0000i		−	1.07379i	−0.843649	−	0.536895i	$-0.819597\pi$
$281$		−	18.0000i		−	1.07379i	0.843649	−	0.536895i	$-0.180403\pi$
$282$		0			0
$283$	12.0000			0.713326			0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000			0.713326			0.356663	+	0.934233i	$0.383914\pi$
$284$		0			0
$285$	−54.0000			−3.19868
$286$		0			0
$287$	−12.0000			−0.708338
$288$		0			0
$289$	8.00000			0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$			9.00000i			0.525786i	0.964825	+	0.262893i	$0.0846766\pi$
$293$			9.00000i			0.525786i	−0.964825	+	0.262893i	$0.915323\pi$
$294$		0			0
$295$	6.00000			0.349334
$296$		0			0
$297$		−	36.0000i		−	2.08893i
$298$		0			0
$299$	12.0000	+	18.0000i	0.693978	+	1.04097i
$300$		0			0
$301$			3.00000i			0.172917i
$302$		0			0
$303$	12.0000			0.689382
$304$		0			0
$305$		−	36.0000i		−	2.06135i
$306$		0			0
$307$		−	2.00000i		−	0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$		−	2.00000i		−	0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$		0			0
$309$	18.0000			1.02398
$310$		0			0
$311$	−6.00000			−0.340229			−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000			−0.340229			−0.170114	+	0.985424i	$0.554414\pi$
$312$		0			0
$313$	−21.0000			−1.18699			−0.593495	−	0.804838i	$-0.702252\pi$
$313$	−21.0000			−1.18699			−0.593495	+	0.804838i	$0.702252\pi$
$314$		0			0
$315$	18.0000			1.01419
$316$		0			0
$317$			6.00000i			0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$			6.00000i			0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$		0			0
$319$		−	16.0000i		−	0.895828i
$320$		0			0
$321$	−36.0000			−2.00932
$322$		0			0
$323$			30.0000i			1.66924i
$324$		0			0
$325$	8.00000	+	12.0000i	0.443760	+	0.665640i
$326$		0			0
$327$		−	45.0000i		−	2.48851i
$328$		0			0
$329$	7.00000			0.385922
$330$		0			0
$331$		−	26.0000i		−	1.42909i	−0.699590	−	0.714545i	$-0.746634\pi$
$331$		−	26.0000i		−	1.42909i	0.699590	−	0.714545i	$-0.253366\pi$
$332$		0			0
$333$		−	18.0000i		−	0.986394i
$334$		0			0
$335$	12.0000			0.655630
$336$		0			0
$337$	13.0000			0.708155			0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000			0.708155			0.354078	+	0.935216i	$0.384795\pi$
$338$		0			0
$339$	42.0000			2.28113
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	13.0000i		−	0.701934i
$344$		0			0
$345$			54.0000i			2.90726i
$346$		0			0
$347$	−21.0000			−1.12734			−0.563670	−	0.826000i	$-0.690611\pi$
$347$	−21.0000			−1.12734			−0.563670	+	0.826000i	$0.690611\pi$
$348$		0			0
$349$			21.0000i			1.12410i	0.827102	+	0.562052i	$0.189988\pi$
$349$			21.0000i			1.12410i	−0.827102	+	0.562052i	$0.810012\pi$
$350$		0			0
$351$	18.0000	+	27.0000i	0.960769	+	1.44115i
$352$		0			0
$353$			18.0000i			0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$			18.0000i			0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$		0			0
$355$	−33.0000			−1.75146
$356$		0			0
$357$		−	15.0000i		−	0.793884i
$358$		0			0
$359$			32.0000i			1.68890i	0.535638	+	0.844448i	$0.320071\pi$
$359$			32.0000i			1.68890i	−0.535638	+	0.844448i	$0.679929\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		0			0
$363$	15.0000			0.787296
$364$		0			0
$365$	−18.0000			−0.942163
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$		−	72.0000i		−	3.74817i
$370$		0			0
$371$			2.00000i			0.103835i
$372$		0			0
$373$	12.0000			0.621336			0.310668	−	0.950518i	$-0.399447\pi$
$373$	12.0000			0.621336			0.310668	+	0.950518i	$0.399447\pi$
$374$		0			0
$375$		−	9.00000i		−	0.464758i
$376$		0			0
$377$	8.00000	+	12.0000i	0.412021	+	0.618031i
$378$		0			0
$379$			30.0000i			1.54100i	0.637442	+	0.770498i	$0.279993\pi$
$379$			30.0000i			1.54100i	−0.637442	+	0.770498i	$0.720007\pi$
$380$		0			0
$381$	54.0000			2.76650
$382$		0			0
$383$			11.0000i			0.562074i	0.959697	+	0.281037i	$0.0906783\pi$
$383$			11.0000i			0.562074i	−0.959697	+	0.281037i	$0.909322\pi$
$384$		0			0
$385$			12.0000i			0.611577i
$386$		0			0
$387$	−18.0000			−0.914991
$388$		0			0
$389$	−2.00000			−0.101404			−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000			−0.101404			−0.0507020	+	0.998714i	$0.516146\pi$
$390$		0			0
$391$	30.0000			1.51717
$392$		0			0
$393$	27.0000			1.36197
$394$		0			0
$395$			18.0000i			0.905678i
$396$		0			0
$397$			30.0000i			1.50566i	0.658217	+	0.752828i	$0.271311\pi$
$397$			30.0000i			1.50566i	−0.658217	+	0.752828i	$0.728689\pi$
$398$		0			0
$399$	18.0000			0.901127
$400$		0			0
$401$			6.00000i			0.299626i	0.988714	+	0.149813i	$0.0478671\pi$
$401$			6.00000i			0.299626i	−0.988714	+	0.149813i	$0.952133\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$			27.0000i			1.34164i
$406$		0			0
$407$	12.0000			0.594818
$408$		0			0
$409$		−	6.00000i		−	0.296681i	−0.988936	−	0.148340i	$-0.952607\pi$
$409$		−	6.00000i		−	0.296681i	0.988936	−	0.148340i	$-0.0473931\pi$
$410$		0			0
$411$			18.0000i			0.887875i
$412$		0			0
$413$	−2.00000			−0.0984136
$414$		0			0
$415$	30.0000			1.47264
$416$		0			0
$417$	27.0000			1.32220
$418$		0			0
$419$	−3.00000			−0.146560			−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000			−0.146560			−0.0732798	+	0.997311i	$0.523347\pi$
$420$		0			0
$421$		−	9.00000i		−	0.438633i	−0.975654	−	0.219317i	$-0.929617\pi$
$421$		−	9.00000i		−	0.438633i	0.975654	−	0.219317i	$-0.0703828\pi$
$422$		0			0
$423$			42.0000i			2.04211i
$424$		0			0
$425$	20.0000			0.970143
$426$		0			0
$427$			12.0000i			0.580721i
$428$		0			0
$429$	−36.0000	+	24.0000i	−1.73810	+	1.15873i
$430$		0			0
$431$		−	19.0000i		−	0.915198i	−0.889159	−	0.457599i	$-0.848710\pi$
$431$		−	19.0000i		−	0.915198i	0.889159	−	0.457599i	$-0.151290\pi$
$432$		0			0
$433$	−3.00000			−0.144171			−0.0720854	−	0.997398i	$-0.522965\pi$
$433$	−3.00000			−0.144171			−0.0720854	+	0.997398i	$0.522965\pi$
$434$		0			0
$435$			36.0000i			1.72607i
$436$		0			0
$437$			36.0000i			1.72211i
$438$		0			0
$439$	6.00000			0.286364			0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000			0.286364			0.143182	+	0.989696i	$0.454267\pi$
$440$		0			0
$441$	36.0000			1.71429
$442$		0			0
$443$	−39.0000			−1.85295			−0.926473	−	0.376361i	$-0.877175\pi$
$443$	−39.0000			−1.85295			−0.926473	+	0.376361i	$0.877175\pi$
$444$		0			0
$445$	−18.0000			−0.853282
$446$		0			0
$447$		−	18.0000i		−	0.851371i
$448$		0			0
$449$		−	12.0000i		−	0.566315i	−0.959073	−	0.283158i	$-0.908618\pi$
$449$		−	12.0000i		−	0.566315i	0.959073	−	0.283158i	$-0.0913819\pi$
$450$		0			0
$451$	48.0000			2.26023
$452$		0			0
$453$			9.00000i			0.422857i
$454$		0			0
$455$	−6.00000	−	9.00000i	−0.281284	−	0.421927i
$456$		0			0
$457$		−	18.0000i		−	0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$		−	18.0000i		−	0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$		0			0
$459$	45.0000			2.10042
$460$		0			0
$461$		−	33.0000i		−	1.53696i	−0.639872	−	0.768482i	$-0.721013\pi$
$461$		−	33.0000i		−	1.53696i	0.639872	−	0.768482i	$-0.278987\pi$
$462$		0			0
$463$		−	32.0000i		−	1.48717i	−0.668644	−	0.743583i	$-0.733125\pi$
$463$		−	32.0000i		−	1.48717i	0.668644	−	0.743583i	$-0.266875\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$		0			0
$469$	−4.00000			−0.184703
$470$		0			0
$471$	−42.0000			−1.93526
$472$		0			0
$473$		−	12.0000i		−	0.551761i
$474$		0			0
$475$			24.0000i			1.10120i
$476$		0			0
$477$	−12.0000			−0.549442
$478$		0			0
$479$		−	37.0000i		−	1.69057i	−0.534313	−	0.845287i	$-0.679430\pi$
$479$		−	37.0000i		−	1.69057i	0.534313	−	0.845287i	$-0.320570\pi$
$480$		0			0
$481$	−9.00000	+	6.00000i	−0.410365	+	0.273576i
$482$		0			0
$483$		−	18.0000i		−	0.819028i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			12.0000i			0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$			12.0000i			0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$		0			0
$489$		−	54.0000i		−	2.44196i
$490$		0			0
$491$	−15.0000			−0.676941			−0.338470	−	0.940977i	$-0.609909\pi$
$491$	−15.0000			−0.676941			−0.338470	+	0.940977i	$0.609909\pi$
$492$		0			0
$493$	20.0000			0.900755
$494$		0			0
$495$	−72.0000			−3.23616
$496$		0			0
$497$	11.0000			0.493417
$498$		0			0
$499$		−	32.0000i		−	1.43252i	−0.697835	−	0.716258i	$-0.745853\pi$
$499$		−	32.0000i		−	1.43252i	0.697835	−	0.716258i	$-0.254147\pi$
$500$		0			0
$501$			12.0000i			0.536120i
$502$		0			0
$503$	6.00000			0.267527			0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000			0.267527			0.133763	+	0.991013i	$0.457294\pi$
$504$		0			0
$505$		−	12.0000i		−	0.533993i
$506$		0			0
$507$	15.0000	−	36.0000i	0.666173	−	1.59882i
$508$		0			0
$509$		−	18.0000i		−	0.797836i	−0.916987	−	0.398918i	$-0.869386\pi$
$509$		−	18.0000i		−	0.797836i	0.916987	−	0.398918i	$-0.130614\pi$
$510$		0			0
$511$	6.00000			0.265424
$512$		0			0
$513$			54.0000i			2.38416i
$514$		0			0
$515$		−	18.0000i		−	0.793175i
$516$		0			0
$517$	−28.0000			−1.23144
$518$		0			0
$519$	42.0000			1.84360
$520$		0			0
$521$	11.0000			0.481919			0.240959	−	0.970535i	$-0.422538\pi$
$521$	11.0000			0.481919			0.240959	+	0.970535i	$0.422538\pi$
$522$		0			0
$523$	−24.0000			−1.04945			−0.524723	−	0.851273i	$-0.675831\pi$
$523$	−24.0000			−1.04945			−0.524723	+	0.851273i	$0.675831\pi$
$524$		0			0
$525$		−	12.0000i		−	0.523723i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$		−	12.0000i		−	0.520756i
$532$		0			0
$533$	−36.0000	+	24.0000i	−1.55933	+	1.03956i
$534$		0			0
$535$			36.0000i			1.55642i
$536$		0			0
$537$	−63.0000			−2.71865
$538$		0			0
$539$			24.0000i			1.03375i
$540$		0			0
$541$		−	33.0000i		−	1.41878i	−0.704816	−	0.709390i	$-0.748970\pi$
$541$		−	33.0000i		−	1.41878i	0.704816	−	0.709390i	$-0.251030\pi$
$542$		0			0
$543$	18.0000			0.772454
$544$		0			0
$545$	−45.0000			−1.92759
$546$		0			0
$547$	33.0000			1.41098			0.705489	−	0.708721i	$-0.250727\pi$
$547$	33.0000			1.41098			0.705489	+	0.708721i	$0.250727\pi$
$548$		0			0
$549$	−72.0000			−3.07289
$550$		0			0
$551$			24.0000i			1.02243i
$552$		0			0
$553$		−	6.00000i		−	0.255146i
$554$		0			0
$555$	−27.0000			−1.14609
$556$		0			0
$557$			3.00000i			0.127114i	0.997978	+	0.0635570i	$0.0202445\pi$
$557$			3.00000i			0.127114i	−0.997978	+	0.0635570i	$0.979756\pi$
$558$		0			0
$559$	6.00000	+	9.00000i	0.253773	+	0.380659i
$560$		0			0
$561$			60.0000i			2.53320i
$562$		0			0
$563$	−3.00000			−0.126435			−0.0632175	−	0.998000i	$-0.520136\pi$
$563$	−3.00000			−0.126435			−0.0632175	+	0.998000i	$0.520136\pi$
$564$		0			0
$565$		−	42.0000i		−	1.76695i
$566$		0			0
$567$		−	9.00000i		−	0.377964i
$568$		0			0
$569$	−11.0000			−0.461144			−0.230572	−	0.973055i	$-0.574060\pi$
$569$	−11.0000			−0.461144			−0.230572	+	0.973055i	$0.574060\pi$
$570$		0			0
$571$	−21.0000			−0.878823			−0.439411	−	0.898286i	$-0.644813\pi$
$571$	−21.0000			−0.878823			−0.439411	+	0.898286i	$0.644813\pi$
$572$		0			0
$573$	72.0000			3.00784
$574$		0			0
$575$	24.0000			1.00087
$576$		0			0
$577$			6.00000i			0.249783i	0.992170	+	0.124892i	$0.0398583\pi$
$577$			6.00000i			0.249783i	−0.992170	+	0.124892i	$0.960142\pi$
$578$		0			0
$579$		−	54.0000i		−	2.24416i
$580$		0			0
$581$	−10.0000			−0.414870
$582$		0			0
$583$		−	8.00000i		−	0.331326i
$584$		0			0
$585$	54.0000	−	36.0000i	2.23263	−	1.48842i
$586$		0			0
$587$		−	2.00000i		−	0.0825488i	−0.999148	−	0.0412744i	$-0.986858\pi$
$587$		−	2.00000i		−	0.0825488i	0.999148	−	0.0412744i	$-0.0131418\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$			9.00000i			0.370211i
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$	−15.0000			−0.614940
$596$		0			0
$597$	36.0000			1.47338
$598$		0			0
$599$	−30.0000			−1.22577			−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000			−1.22577			−0.612883	+	0.790173i	$0.709990\pi$
$600$		0			0
$601$	−3.00000			−0.122373			−0.0611863	−	0.998126i	$-0.519488\pi$
$601$	−3.00000			−0.122373			−0.0611863	+	0.998126i	$0.519488\pi$
$602$		0			0
$603$		−	24.0000i		−	0.977356i
$604$		0			0
$605$		−	15.0000i		−	0.609837i
$606$		0			0
$607$	−30.0000			−1.21766			−0.608831	−	0.793300i	$-0.708361\pi$
$607$	−30.0000			−1.21766			−0.608831	+	0.793300i	$0.708361\pi$
$608$		0			0
$609$		−	12.0000i		−	0.486265i
$610$		0			0
$611$	21.0000	−	14.0000i	0.849569	−	0.566379i
$612$		0			0
$613$		−	6.00000i		−	0.242338i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$		−	6.00000i		−	0.242338i	0.992632	−	0.121169i	$-0.0386643\pi$
$614$		0			0
$615$	−108.000			−4.35498
$616$		0			0
$617$		−	36.0000i		−	1.44931i	−0.689114	−	0.724653i	$-0.742000\pi$
$617$		−	36.0000i		−	1.44931i	0.689114	−	0.724653i	$-0.258000\pi$
$618$		0			0
$619$			44.0000i			1.76851i	0.467005	+	0.884255i	$0.345333\pi$
$619$			44.0000i			1.76851i	−0.467005	+	0.884255i	$0.654667\pi$
$620$		0			0
$621$	54.0000			2.16695
$622$		0			0
$623$	6.00000			0.240385
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$	−72.0000			−2.87540
$628$		0			0
$629$			15.0000i			0.598089i
$630$		0			0
$631$			21.0000i			0.835997i	0.908448	+	0.417998i	$0.137268\pi$
$631$			21.0000i			0.835997i	−0.908448	+	0.417998i	$0.862732\pi$
$632$		0			0
$633$	−9.00000			−0.357718
$634$		0			0
$635$		−	54.0000i		−	2.14292i
$636$		0			0
$637$	−12.0000	−	18.0000i	−0.475457	−	0.713186i
$638$		0			0
$639$			66.0000i			2.61092i
$640$		0			0
$641$	−10.0000			−0.394976			−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000			−0.394976			−0.197488	+	0.980305i	$0.563278\pi$
$642$		0			0
$643$			4.00000i			0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$			4.00000i			0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$		0			0
$645$			27.0000i			1.06312i
$646$		0			0
$647$	−18.0000			−0.707653			−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000			−0.707653			−0.353827	+	0.935311i	$0.615120\pi$
$648$		0			0
$649$	8.00000			0.314027
$650$		0			0
$651$		0			0
$652$		0			0
$653$	28.0000			1.09572			0.547862	−	0.836569i	$-0.315442\pi$
$653$	28.0000			1.09572			0.547862	+	0.836569i	$0.315442\pi$
$654$		0			0
$655$		−	27.0000i		−	1.05498i
$656$		0			0
$657$			36.0000i			1.40449i
$658$		0			0
$659$	24.0000			0.934907			0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000			0.934907			0.467454	+	0.884018i	$0.345171\pi$
$660$		0			0
$661$			42.0000i			1.63361i	0.576913	+	0.816805i	$0.304257\pi$
$661$			42.0000i			1.63361i	−0.576913	+	0.816805i	$0.695743\pi$
$662$		0			0
$663$	−30.0000	−	45.0000i	−1.16510	−	1.74766i
$664$		0			0
$665$		−	18.0000i		−	0.698010i
$666$		0			0
$667$	24.0000			0.929284
$668$		0			0
$669$			63.0000i			2.43572i
$670$		0			0
$671$		−	48.0000i		−	1.85302i
$672$		0			0
$673$	−1.00000			−0.0385472			−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000			−0.0385472			−0.0192736	+	0.999814i	$0.506135\pi$
$674$		0			0
$675$	36.0000			1.38564
$676$		0			0
$677$	−14.0000			−0.538064			−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000			−0.538064			−0.269032	+	0.963131i	$0.586704\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$			24.0000i			0.919682i
$682$		0			0
$683$			14.0000i			0.535695i	0.963461	+	0.267848i	$0.0863124\pi$
$683$			14.0000i			0.535695i	−0.963461	+	0.267848i	$0.913688\pi$
$684$		0			0
$685$	18.0000			0.687745
$686$		0			0
$687$			27.0000i			1.03011i
$688$		0			0
$689$	4.00000	+	6.00000i	0.152388	+	0.228582i
$690$		0			0
$691$			8.00000i			0.304334i	0.988355	+	0.152167i	$0.0486252\pi$
$691$			8.00000i			0.304334i	−0.988355	+	0.152167i	$0.951375\pi$
$692$		0			0
$693$	24.0000			0.911685
$694$		0			0
$695$		−	27.0000i		−	1.02417i
$696$		0			0
$697$			60.0000i			2.27266i
$698$		0			0
$699$	−57.0000			−2.15594
$700$		0			0
$701$	−26.0000			−0.982006			−0.491003	−	0.871158i	$-0.663370\pi$
$701$	−26.0000			−0.982006			−0.491003	+	0.871158i	$0.663370\pi$
$702$		0			0
$703$	−18.0000			−0.678883
$704$		0			0
$705$	63.0000			2.37272
$706$		0			0
$707$			4.00000i			0.150435i
$708$		0			0
$709$		−	42.0000i		−	1.57734i	−0.614815	−	0.788672i	$-0.710769\pi$
$709$		−	42.0000i		−	1.57734i	0.614815	−	0.788672i	$-0.289231\pi$
$710$		0			0
$711$	36.0000			1.35011
$712$		0			0
$713$		0			0
$714$		0			0
$715$	24.0000	+	36.0000i	0.897549	+	1.34632i
$716$		0			0
$717$		−	15.0000i		−	0.560185i
$718$		0			0
$719$	48.0000			1.79010			0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000			1.79010			0.895049	+	0.445968i	$0.147140\pi$
$720$		0			0
$721$			6.00000i			0.223452i
$722$		0			0
$723$		−	18.0000i		−	0.669427i
$724$		0			0
$725$	16.0000			0.594225
$726$		0			0
$727$	24.0000			0.890111			0.445055	−	0.895503i	$-0.353184\pi$
$727$	24.0000			0.890111			0.445055	+	0.895503i	$0.353184\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	15.0000			0.554795
$732$		0			0
$733$			33.0000i			1.21888i	0.792831	+	0.609441i	$0.208606\pi$
$733$			33.0000i			1.21888i	−0.792831	+	0.609441i	$0.791394\pi$
$734$		0			0
$735$		−	54.0000i		−	1.99182i
$736$		0			0
$737$	16.0000			0.589368
$738$		0			0
$739$			16.0000i			0.588570i	0.955718	+	0.294285i	$0.0950814\pi$
$739$			16.0000i			0.588570i	−0.955718	+	0.294285i	$0.904919\pi$
$740$		0			0
$741$	54.0000	−	36.0000i	1.98374	−	1.32249i
$742$		0			0
$743$		−	11.0000i		−	0.403551i	−0.979432	−	0.201775i	$-0.935329\pi$
$743$		−	11.0000i		−	0.403551i	0.979432	−	0.201775i	$-0.0646711\pi$
$744$		0			0
$745$	−18.0000			−0.659469
$746$		0			0
$747$		−	60.0000i		−	2.19529i
$748$		0			0
$749$		−	12.0000i		−	0.438470i
$750$		0			0
$751$	36.0000			1.31366			0.656829	−	0.754039i	$-0.271897\pi$
$751$	36.0000			1.31366			0.656829	+	0.754039i	$0.271897\pi$
$752$		0			0
$753$	−36.0000			−1.31191
$754$		0			0
$755$	9.00000			0.327544
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$		0			0
$759$			72.0000i			2.61343i
$760$		0			0
$761$		−	30.0000i		−	1.08750i	−0.839248	−	0.543750i	$-0.817004\pi$
$761$		−	30.0000i		−	1.08750i	0.839248	−	0.543750i	$-0.182996\pi$
$762$		0			0
$763$	15.0000			0.543036
$764$		0			0
$765$		−	90.0000i		−	3.25396i
$766$		0			0
$767$	−6.00000	+	4.00000i	−0.216647	+	0.144432i
$768$		0			0
$769$		−	48.0000i		−	1.73092i	−0.500974	−	0.865462i	$-0.667025\pi$
$769$		−	48.0000i		−	1.73092i	0.500974	−	0.865462i	$-0.332975\pi$
$770$		0			0
$771$	15.0000			0.540212
$772$		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
$775$		0			0
$776$		0			0
$777$	9.00000			0.322873
$778$		0			0
$779$	−72.0000			−2.57967
$780$		0			0
$781$	−44.0000			−1.57444
$782$		0			0
$783$	36.0000			1.28654
$784$		0			0
$785$			42.0000i			1.49904i
$786$		0			0
$787$			32.0000i			1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$			32.0000i			1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$		0			0
$789$	−36.0000			−1.28163
$790$		0			0
$791$			14.0000i			0.497783i
$792$		0			0
$793$	24.0000	+	36.0000i	0.852265	+	1.27840i
$794$		0			0
$795$			18.0000i

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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +3.00000i q^{5} -1.00000i q^{7} +6.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +3.00000i q^{5} -1.00000i q^{7} +6.00000 q^{9} +4.00000i q^{11} +(-2.00000 - 3.00000i) q^{13} -9.00000i q^{15} -5.00000 q^{17} -6.00000i q^{19} +3.00000i q^{21} -6.00000 q^{23} -4.00000 q^{25} -9.00000 q^{27} -4.00000 q^{29} -12.0000i q^{33} +3.00000 q^{35} -3.00000i q^{37} +(6.00000 + 9.00000i) q^{39} -12.0000i q^{41} -3.00000 q^{43} +18.0000i q^{45} +7.00000i q^{47} +6.00000 q^{49} +15.0000 q^{51} -2.00000 q^{53} -12.0000 q^{55} +18.0000i q^{57} -2.00000i q^{59} -12.0000 q^{61} -6.00000i q^{63} +(9.00000 - 6.00000i) q^{65} -4.00000i q^{67} +18.0000 q^{69} +11.0000i q^{71} +6.00000i q^{73} +12.0000 q^{75} +4.00000 q^{77} +6.00000 q^{79} +9.00000 q^{81} -10.0000i q^{83} -15.0000i q^{85} +12.0000 q^{87} +6.00000i q^{89} +(-3.00000 + 2.00000i) q^{91} +18.0000 q^{95} +24.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 12 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 12 * q^9	\( 2 q - 6 q^{3} + 12 q^{9} - 4 q^{13} - 10 q^{17} - 12 q^{23} - 8 q^{25} - 18 q^{27} - 8 q^{29} + 6 q^{35} + 12 q^{39} - 6 q^{43} + 12 q^{49} + 30 q^{51} - 4 q^{53} - 24 q^{55} - 24 q^{61} + 18 q^{65} + 36 q^{69} + 24 q^{75} + 8 q^{77} + 12 q^{79} + 18 q^{81} + 24 q^{87} - 6 q^{91} + 36 q^{95}+O(q^{100}) \) 2 * q - 6 * q^3 + 12 * q^9 - 4 * q^13 - 10 * q^17 - 12 * q^23 - 8 * q^25 - 18 * q^27 - 8 * q^29 + 6 * q^35 + 12 * q^39 - 6 * q^43 + 12 * q^49 + 30 * q^51 - 4 * q^53 - 24 * q^55 - 24 * q^61 + 18 * q^65 + 36 * q^69 + 24 * q^75 + 8 * q^77 + 12 * q^79 + 18 * q^81 + 24 * q^87 - 6 * q^91 + 36 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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