LMFDB - Embedded newform 416.2.f.a.129.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [416,2,Mod(129,416)] 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("416.129"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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:traces = [2,0,-6,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

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	$-0.765942\pi$
$5$		−	3.00000i		−	1.34164i	0.741620	−	0.670820i	$-0.234058\pi$
$6$		0			0
$7$			1.00000i			0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$			1.00000i			0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$		0			0
$9$	6.00000			2.00000
$10$		0			0
$11$		−	4.00000i		−	1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$		−	4.00000i		−	1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$		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	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$		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	$0.0793129\pi$
$37$			3.00000i			0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$		0			0
$39$	6.00000	−	9.00000i	0.960769	−	1.44115i
$40$		0			0
$41$			12.0000i			1.87409i	0.349215	+	0.937043i	$0.386448\pi$
$41$			12.0000i			1.87409i	−0.349215	+	0.937043i	$0.613552\pi$
$42$		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	$-0.829450\pi$
$47$		−	7.00000i		−	1.02105i	0.859861	−	0.510527i	$-0.170550\pi$
$48$		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	$0.0415584\pi$
$59$			2.00000i			0.260378i	−0.991489	+	0.130189i	$0.958442\pi$
$60$		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	$0.0785709\pi$
$67$			4.00000i			0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$		0			0
$69$	18.0000			2.16695
$70$		0			0
$71$		−	11.0000i		−	1.30546i	−0.757591	−	0.652730i	$-0.773624\pi$
$71$		−	11.0000i		−	1.30546i	0.757591	−	0.652730i	$-0.226376\pi$
$72$		0			0
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$		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	$0.184923\pi$
$83$			10.0000i			1.09764i	−0.835940	+	0.548821i	$0.815077\pi$
$84$		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	$-0.896989\pi$
$89$		−	6.00000i		−	0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$		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	$-0.744889\pi$
$109$		−	15.0000i		−	1.43674i	0.695662	−	0.718370i	$-0.255111\pi$
$110$		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	$0.0825059\pi$
$137$			6.00000i			0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$		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	$-0.920959\pi$
$149$		−	6.00000i		−	0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$		0			0
$151$			3.00000i			0.244137i	0.992522	+	0.122068i	$0.0389527\pi$
$151$			3.00000i			0.244137i	−0.992522	+	0.122068i	$0.961047\pi$
$152$		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	$-0.750976\pi$
$163$		−	18.0000i		−	1.40987i	0.709273	−	0.704934i	$-0.249024\pi$
$164$		0			0
$165$	36.0000			2.80260
$166$		0			0
$167$			4.00000i			0.309529i	0.987951	+	0.154765i	$0.0494619\pi$
$167$			4.00000i			0.309529i	−0.987951	+	0.154765i	$0.950538\pi$
$168$		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	$-0.775675\pi$
$193$		−	18.0000i		−	1.29567i	0.761781	−	0.647834i	$-0.224325\pi$
$194$		0			0
$195$	−27.0000	−	18.0000i	−1.93351	−	1.28901i
$196$		0			0
$197$			3.00000i			0.213741i	0.994273	+	0.106871i	$0.0340831\pi$
$197$			3.00000i			0.213741i	−0.994273	+	0.106871i	$0.965917\pi$
$198$		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	$0.248216\pi$
$223$			21.0000i			1.40626i	−0.711059	+	0.703132i	$0.751784\pi$
$224$		0			0
$225$	−24.0000			−1.60000
$226$		0			0
$227$			8.00000i			0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$			8.00000i			0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$		0			0
$229$			9.00000i			0.594737i	0.954763	+	0.297368i	$0.0961089\pi$
$229$			9.00000i			0.594737i	−0.954763	+	0.297368i	$0.903891\pi$
$230$		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	$-0.948299\pi$
$239$		−	5.00000i		−	0.323423i	0.986838	−	0.161712i	$-0.0517014\pi$
$240$		0			0
$241$		−	6.00000i		−	0.386494i	−0.981150	−	0.193247i	$-0.938098\pi$
$241$		−	6.00000i		−	0.386494i	0.981150	−	0.193247i	$-0.0619019\pi$
$242$		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	$-0.891567\pi$
$271$		−	11.0000i		−	0.668202i	0.942537	−	0.334101i	$-0.108433\pi$
$272$		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	$0.180403\pi$
$281$			18.0000i			1.07379i	−0.843649	+	0.536895i	$0.819597\pi$
$282$		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	$-0.915323\pi$
$293$		−	9.00000i		−	0.525786i	0.964825	−	0.262893i	$-0.0846766\pi$
$294$		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	$0.0181768\pi$
$307$			2.00000i			0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$		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	$-0.946109\pi$
$317$		−	6.00000i		−	0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$		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	$0.253366\pi$
$331$			26.0000i			1.42909i	−0.699590	+	0.714545i	$0.746634\pi$
$332$		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	$-0.810012\pi$
$349$		−	21.0000i		−	1.12410i	0.827102	−	0.562052i	$-0.189988\pi$
$350$		0			0
$351$	18.0000	−	27.0000i	0.960769	−	1.44115i
$352$		0			0
$353$		−	18.0000i		−	0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$		−	18.0000i		−	0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$		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	$-0.679929\pi$
$359$		−	32.0000i		−	1.68890i	0.535638	−	0.844448i	$-0.320071\pi$
$360$		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	$-0.720007\pi$
$379$		−	30.0000i		−	1.54100i	0.637442	−	0.770498i	$-0.279993\pi$
$380$		0			0
$381$	54.0000			2.76650
$382$		0			0
$383$		−	11.0000i		−	0.562074i	−0.959697	−	0.281037i	$-0.909322\pi$
$383$		−	11.0000i		−	0.562074i	0.959697	−	0.281037i	$-0.0906783\pi$
$384$		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	$-0.728689\pi$
$397$		−	30.0000i		−	1.50566i	0.658217	−	0.752828i	$-0.271311\pi$
$398$		0			0
$399$	18.0000			0.901127
$400$		0			0
$401$		−	6.00000i		−	0.299626i	−0.988714	−	0.149813i	$-0.952133\pi$
$401$		−	6.00000i		−	0.299626i	0.988714	−	0.149813i	$-0.0478671\pi$
$402$		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	$0.0473931\pi$
$409$			6.00000i			0.296681i	−0.988936	+	0.148340i	$0.952607\pi$
$410$		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	$0.0703828\pi$
$421$			9.00000i			0.438633i	−0.975654	+	0.219317i	$0.929617\pi$
$422$		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	$0.151290\pi$
$431$			19.0000i			0.915198i	−0.889159	+	0.457599i	$0.848710\pi$
$432$		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	$0.0913819\pi$
$449$			12.0000i			0.566315i	−0.959073	+	0.283158i	$0.908618\pi$
$450$		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	$0.138322\pi$
$457$			18.0000i			0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$		0			0
$459$	45.0000			2.10042
$460$		0			0
$461$			33.0000i			1.53696i	0.639872	+	0.768482i	$0.278987\pi$
$461$			33.0000i			1.53696i	−0.639872	+	0.768482i	$0.721013\pi$
$462$		0			0
$463$			32.0000i			1.48717i	0.668644	+	0.743583i	$0.266875\pi$
$463$			32.0000i			1.48717i	−0.668644	+	0.743583i	$0.733125\pi$
$464$		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	$0.320570\pi$
$479$			37.0000i			1.69057i	−0.534313	+	0.845287i	$0.679430\pi$
$480$		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	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$		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	$0.254147\pi$
$499$			32.0000i			1.43252i	−0.697835	+	0.716258i	$0.745853\pi$
$500$		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	$0.130614\pi$
$509$			18.0000i			0.797836i	−0.916987	+	0.398918i	$0.869386\pi$
$510$		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	$0.251030\pi$
$541$			33.0000i			1.41878i	−0.704816	+	0.709390i	$0.748970\pi$
$542$		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	$-0.979756\pi$
$557$		−	3.00000i		−	0.127114i	0.997978	−	0.0635570i	$-0.0202445\pi$
$558$		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	$-0.960142\pi$
$577$		−	6.00000i		−	0.249783i	0.992170	−	0.124892i	$-0.0398583\pi$
$578$		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	$0.0131418\pi$
$587$			2.00000i			0.0825488i	−0.999148	+	0.0412744i	$0.986858\pi$
$588$		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	$0.0386643\pi$
$613$			6.00000i			0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$		0			0
$615$	−108.000			−4.35498
$616$		0			0
$617$			36.0000i			1.44931i	0.689114	+	0.724653i	$0.258000\pi$
$617$			36.0000i			1.44931i	−0.689114	+	0.724653i	$0.742000\pi$
$618$		0			0
$619$		−	44.0000i		−	1.76851i	−0.467005	−	0.884255i	$-0.654667\pi$
$619$		−	44.0000i		−	1.76851i	0.467005	−	0.884255i	$-0.345333\pi$
$620$		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	$-0.862732\pi$
$631$		−	21.0000i		−	0.835997i	0.908448	−	0.417998i	$-0.137268\pi$
$632$		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	$-0.974868\pi$
$643$		−	4.00000i		−	0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$		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	$-0.695743\pi$
$661$		−	42.0000i		−	1.63361i	0.576913	−	0.816805i	$-0.304257\pi$
$662$		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	$-0.913688\pi$
$683$		−	14.0000i		−	0.535695i	0.963461	−	0.267848i	$-0.0863124\pi$
$684$		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	$-0.951375\pi$
$691$		−	8.00000i		−	0.304334i	0.988355	−	0.152167i	$-0.0486252\pi$
$692$		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	$0.289231\pi$
$709$			42.0000i			1.57734i	−0.614815	+	0.788672i	$0.710769\pi$
$710$		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	$-0.791394\pi$
$733$		−	33.0000i		−	1.21888i	0.792831	−	0.609441i	$-0.208606\pi$
$734$		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	$-0.904919\pi$
$739$		−	16.0000i		−	0.588570i	0.955718	−	0.294285i	$-0.0950814\pi$
$740$		0			0
$741$	54.0000	+	36.0000i	1.98374	+	1.32249i
$742$		0			0
$743$			11.0000i			0.403551i	0.979432	+	0.201775i	$0.0646711\pi$
$743$			11.0000i			0.403551i	−0.979432	+	0.201775i	$0.935329\pi$
$744$		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	$0.182996\pi$
$761$			30.0000i			1.08750i	−0.839248	+	0.543750i	$0.817004\pi$
$762$		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	$0.332975\pi$
$769$			48.0000i			1.73092i	−0.500974	+	0.865462i	$0.667025\pi$
$770$		0			0
$771$	15.0000			0.540212
$772$		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
$775$		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	$-0.806812\pi$
$787$		−	32.0000i		−	1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$		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		−	0.638394i
$796$		0			0
$797$	50.0000			1.77109			0.885545	−	0.464553i	$-0.153785\pi$
$797$	50.0000			1.77109			0.885545	+	0.464553i	$0.153785\pi$
$798$		0			0
$799$			35.0000i			1.23821i
$800$		0			0
$801$		−	36.0000i		−	1.27200i
$802$		0			0
$803$	−24.0000			−0.846942
$804$		0			0
$805$	−18.0000			−0.634417
$806$		0			0
$807$	12.0000			0.422420
$808$		0			0
$809$	−25.0000			−0.878953			−0.439477	−	0.898254i	$-0.644836\pi$
$809$	−25.0000			−0.878953			−0.439477	+	0.898254i	$0.644836\pi$
$810$		0			0
$811$		−	24.0000i		−	0.842754i	−0.906886	−	0.421377i	$-0.861547\pi$
$811$		−	24.0000i		−	0.842754i	0.906886	−	0.421377i	$-0.138453\pi$
$812$		0			0
$813$			33.0000i			1.15736i
$814$		0			0
$815$	−54.0000			−1.89154
$816$		0			0
$817$		−	18.0000i		−	0.629740i
$818$		0			0
$819$	−18.0000	−	12.0000i	−0.628971	−	0.419314i
$820$		0			0
$821$			45.0000i			1.57051i	0.619172	+	0.785255i	$0.287468\pi$
$821$			45.0000i			1.57051i	−0.619172	+	0.785255i	$0.712532\pi$
$822$		0			0
$823$	30.0000			1.04573			0.522867	−	0.852414i	$-0.324862\pi$
$823$	30.0000			1.04573			0.522867	+	0.852414i	$0.324862\pi$
$824$		0			0
$825$		−	48.0000i		−	1.67115i
$826$		0			0
$827$			14.0000i			0.486828i	0.969923	+	0.243414i	$0.0782673\pi$
$827$			14.0000i			0.486828i	−0.969923	+	0.243414i	$0.921733\pi$
$828$		0			0
$829$	−52.0000			−1.80603			−0.903017	−	0.429604i	$-0.858653\pi$
$829$	−52.0000			−1.80603			−0.903017	+	0.429604i	$0.858653\pi$
$830$		0			0
$831$	−36.0000			−1.24883
$832$		0			0
$833$	−30.0000			−1.03944
$834$		0			0
$835$	12.0000			0.415277
$836$		0			0
$837$		0			0
$838$		0			0
$839$			56.0000i			1.93333i	0.256036	+	0.966667i	$0.417584\pi$
$839$			56.0000i			1.93333i	−0.256036	+	0.966667i	$0.582416\pi$
$840$		0			0
$841$	−13.0000			−0.448276
$842$		0			0
$843$		−	54.0000i		−	1.85986i
$844$		0			0
$845$	−36.0000	+	15.0000i	−1.23844	+	0.516016i
$846$		0			0
$847$		−	5.00000i		−	0.171802i
$848$		0			0
$849$	−36.0000			−1.23552
$850$		0			0
$851$		−	18.0000i		−	0.617032i
$852$		0			0
$853$			33.0000i			1.12990i	0.825126	+	0.564949i	$0.191104\pi$
$853$			33.0000i			1.12990i	−0.825126	+	0.564949i	$0.808896\pi$
$854$		0			0
$855$	108.000			3.69352
$856$		0			0
$857$	10.0000			0.341593			0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000			0.341593			0.170797	+	0.985306i	$0.445366\pi$
$858$		0			0
$859$	48.0000			1.63774			0.818869	−	0.573980i	$-0.194601\pi$
$859$	48.0000			1.63774			0.818869	+	0.573980i	$0.194601\pi$
$860$		0			0
$861$	36.0000			1.22688
$862$		0			0
$863$		−	29.0000i		−	0.987171i	−0.869697	−	0.493586i	$-0.835686\pi$
$863$		−	29.0000i		−	0.987171i	0.869697	−	0.493586i	$-0.164314\pi$
$864$		0			0
$865$			42.0000i			1.42804i
$866$		0			0
$867$	−24.0000			−0.815083
$868$		0			0
$869$		−	24.0000i		−	0.814144i
$870$		0			0
$871$	−12.0000	−	8.00000i	−0.406604	−	0.271070i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.00000			0.101419
$876$		0			0
$877$			27.0000i			0.911725i	0.890050	+	0.455863i	$0.150669\pi$
$877$			27.0000i			0.911725i	−0.890050	+	0.455863i	$0.849331\pi$
$878$		0			0
$879$			27.0000i			0.910687i
$880$		0			0
$881$	−41.0000			−1.38133			−0.690663	−	0.723177i	$-0.742681\pi$
$881$	−41.0000			−1.38133			−0.690663	+	0.723177i	$0.742681\pi$
$882$		0			0
$883$	−9.00000			−0.302874			−0.151437	−	0.988467i	$-0.548390\pi$
$883$	−9.00000			−0.302874			−0.151437	+	0.988467i	$0.548390\pi$
$884$		0			0
$885$	−18.0000			−0.605063
$886$		0			0
$887$	−24.0000			−0.805841			−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000			−0.805841			−0.402921	+	0.915235i	$0.632005\pi$
$888$		0			0
$889$		−	18.0000i		−	0.603701i
$890$		0			0
$891$		−	36.0000i		−	1.20605i
$892$		0			0
$893$	42.0000			1.40548
$894$		0			0
$895$		−	63.0000i		−	2.10586i
$896$		0			0
$897$	−36.0000	+	54.0000i	−1.20201	+	1.80301i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	10.0000			0.333148
$902$		0			0
$903$			9.00000i			0.299501i
$904$		0			0
$905$			18.0000i			0.598340i
$906$		0			0
$907$	21.0000			0.697294			0.348647	−	0.937254i	$-0.386641\pi$
$907$	21.0000			0.697294			0.348647	+	0.937254i	$0.386641\pi$
$908$		0			0
$909$	−24.0000			−0.796030
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	40.0000			1.32381
$914$		0			0
$915$		−	108.000i		−	3.57037i
$916$		0			0
$917$		−	9.00000i		−	0.297206i
$918$		0			0
$919$	36.0000			1.18753			0.593765	−	0.804638i	$-0.297641\pi$
$919$	36.0000			1.18753			0.593765	+	0.804638i	$0.297641\pi$
$920$		0			0
$921$		−	6.00000i		−	0.197707i
$922$		0			0
$923$	33.0000	+	22.0000i	1.08621	+	0.724139i
$924$		0			0
$925$		−	12.0000i		−	0.394558i
$926$		0			0
$927$	−36.0000			−1.18240
$928$		0			0
$929$		−	36.0000i		−	1.18112i	−0.806993	−	0.590561i	$-0.798907\pi$
$929$		−	36.0000i		−	1.18112i	0.806993	−	0.590561i	$-0.201093\pi$
$930$		0			0
$931$			36.0000i			1.17985i
$932$		0			0
$933$	18.0000			0.589294
$934$		0			0
$935$	60.0000			1.96221
$936$		0			0
$937$	54.0000			1.76410			0.882052	−	0.471153i	$-0.156162\pi$
$937$	54.0000			1.76410			0.882052	+	0.471153i	$0.156162\pi$
$938$		0			0
$939$	63.0000			2.05593
$940$		0			0
$941$			3.00000i			0.0977972i	0.998804	+	0.0488986i	$0.0155711\pi$
$941$			3.00000i			0.0977972i	−0.998804	+	0.0488986i	$0.984429\pi$
$942$		0			0
$943$		−	72.0000i		−	2.34464i
$944$		0			0
$945$	−27.0000			−0.878310
$946$		0			0
$947$		−	20.0000i		−	0.649913i	−0.945729	−	0.324956i	$-0.894650\pi$
$947$		−	20.0000i		−	0.649913i	0.945729	−	0.324956i	$-0.105350\pi$
$948$		0			0
$949$	18.0000	+	12.0000i	0.584305	+	0.389536i
$950$		0			0
$951$			18.0000i			0.583690i
$952$		0			0
$953$	41.0000			1.32812			0.664060	−	0.747679i	$-0.268832\pi$
$953$	41.0000			1.32812			0.664060	+	0.747679i	$0.268832\pi$
$954$		0			0
$955$			72.0000i			2.32987i
$956$		0			0
$957$		−	48.0000i		−	1.55162i
$958$		0			0
$959$	−6.00000			−0.193750
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	72.0000			2.32017
$964$		0			0
$965$	−54.0000			−1.73832
$966$		0			0
$967$			33.0000i			1.06121i	0.847620	+	0.530604i	$0.178035\pi$
$967$			33.0000i			1.06121i	−0.847620	+	0.530604i	$0.821965\pi$
$968$		0			0
$969$			90.0000i			2.89122i
$970$		0			0
$971$	−15.0000			−0.481373			−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000			−0.481373			−0.240686	+	0.970603i	$0.577373\pi$
$972$		0			0
$973$		−	9.00000i		−	0.288527i
$974$		0			0
$975$	−24.0000	+	36.0000i	−0.768615	+	1.15292i
$976$		0			0
$977$			12.0000i			0.383914i	0.981403	+	0.191957i	$0.0614834\pi$
$977$			12.0000i			0.383914i	−0.981403	+	0.191957i	$0.938517\pi$
$978$		0			0
$979$	−24.0000			−0.767043
$980$		0			0
$981$		−	90.0000i		−	2.87348i
$982$		0			0
$983$		−	35.0000i		−	1.11633i	−0.829731	−	0.558163i	$-0.811506\pi$
$983$		−	35.0000i		−	1.11633i	0.829731	−	0.558163i	$-0.188494\pi$
$984$		0			0
$985$	9.00000			0.286764
$986$		0			0
$987$	−21.0000			−0.668437
$988$		0			0
$989$	18.0000			0.572367
$990$		0			0
$991$	54.0000			1.71537			0.857683	−	0.514178i	$-0.171903\pi$
$991$	54.0000			1.71537			0.857683	+	0.514178i	$0.171903\pi$
$992$		0			0
$993$		−	78.0000i		−	2.47526i
$994$		0			0
$995$			36.0000i			1.14128i
$996$		0			0
$997$	28.0000			0.886769			0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000			0.886769			0.443384	+	0.896332i	$0.353778\pi$
$998$		0			0
$999$		−	27.0000i		−	0.854242i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.f.a.129.1	✓	2
3.2	odd	2		3744.2.c.c.3457.2		2
4.3	odd	2		416.2.f.c.129.1	yes	2
8.3	odd	2		832.2.f.a.129.2		2
8.5	even	2		832.2.f.e.129.2		2
12.11	even	2		3744.2.c.b.3457.2		2
13.5	odd	4		5408.2.a.a.1.1		1
13.8	odd	4		5408.2.a.b.1.1		1
13.12	even	2	inner	416.2.f.a.129.2	yes	2
39.38	odd	2		3744.2.c.c.3457.1		2
52.31	even	4		5408.2.a.l.1.1		1
52.47	even	4		5408.2.a.m.1.1		1
52.51	odd	2		416.2.f.c.129.2	yes	2
104.51	odd	2		832.2.f.a.129.1		2
104.77	even	2		832.2.f.e.129.1		2
156.155	even	2		3744.2.c.b.3457.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.f.a.129.1	✓	2	1.1	even	1	trivial
416.2.f.a.129.2	yes	2	13.12	even	2	inner
416.2.f.c.129.1	yes	2	4.3	odd	2
416.2.f.c.129.2	yes	2	52.51	odd	2
832.2.f.a.129.1		2	104.51	odd	2
832.2.f.a.129.2		2	8.3	odd	2
832.2.f.e.129.1		2	104.77	even	2
832.2.f.e.129.2		2	8.5	even	2
3744.2.c.b.3457.1		2	156.155	even	2
3744.2.c.b.3457.2		2	12.11	even	2
3744.2.c.c.3457.1		2	39.38	odd	2
3744.2.c.c.3457.2		2	3.2	odd	2
5408.2.a.a.1.1		1	13.5	odd	4
5408.2.a.b.1.1		1	13.8	odd	4
5408.2.a.l.1.1		1	52.31	even	4
5408.2.a.m.1.1		1	52.47	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

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