LMFDB - Embedded newform 238.2.b.a.169.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [238,2,Mod(169,238)] 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("238.169"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(238, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 238 = 2 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	238.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$1.90043956811$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			169.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	238.169
Dual form			238.2.b.a.169.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000i q^{7} -1.00000 q^{8} +3.00000 q^{9} +2.00000 q^{13} -1.00000i q^{14} +1.00000 q^{16} +(1.00000 + 4.00000i) q^{17} -3.00000 q^{18} +4.00000 q^{19} -8.00000i q^{23} +5.00000 q^{25} -2.00000 q^{26} +1.00000i q^{28} +8.00000i q^{29} +8.00000i q^{31} -1.00000 q^{32} +(-1.00000 - 4.00000i) q^{34} +3.00000 q^{36} -8.00000i q^{37} -4.00000 q^{38} -8.00000i q^{41} -4.00000 q^{43} +8.00000i q^{46} -8.00000 q^{47} -1.00000 q^{49} -5.00000 q^{50} +2.00000 q^{52} -10.0000 q^{53} -1.00000i q^{56} -8.00000i q^{58} -12.0000 q^{59} -8.00000i q^{62} +3.00000i q^{63} +1.00000 q^{64} -4.00000 q^{67} +(1.00000 + 4.00000i) q^{68} -8.00000i q^{71} -3.00000 q^{72} +8.00000i q^{73} +8.00000i q^{74} +4.00000 q^{76} -8.00000i q^{79} +9.00000 q^{81} +8.00000i q^{82} -4.00000 q^{83} +4.00000 q^{86} +6.00000 q^{89} +2.00000i q^{91} -8.00000i q^{92} +8.00000 q^{94} +8.00000i q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 6 q^{9} + 4 q^{13} + 2 q^{16} + 2 q^{17} - 6 q^{18} + 8 q^{19} + 10 q^{25} - 4 q^{26} - 2 q^{32} - 2 q^{34} + 6 q^{36} - 8 q^{38} - 8 q^{43} - 16 q^{47} - 2 q^{49} - 10 q^{50}+ \cdots + 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 6 * q^9 + 4 * q^13 + 2 * q^16 + 2 * q^17 - 6 * q^18 + 8 * q^19 + 10 * q^25 - 4 * q^26 - 2 * q^32 - 2 * q^34 + 6 * q^36 - 8 * q^38 - 8 * q^43 - 16 * q^47 - 2 * q^49 - 10 * q^50 + 4 * q^52 - 20 * q^53 - 24 * q^59 + 2 * q^64 - 8 * q^67 + 2 * q^68 - 6 * q^72 + 8 * q^76 + 18 * q^81 - 8 * q^83 + 8 * q^86 + 12 * q^89 + 16 * q^94 + 2 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/238\mathbb{Z}\right)^\times$.

$n$	$71$	$171$
$\chi(n)$	$-1$	$1$

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 $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$	1.00000			0.500000
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$	−1.00000			−0.353553
$9$	3.00000			1.00000
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		−	1.00000i		−	0.267261i
$15$		0			0
$16$	1.00000			0.250000
$17$	1.00000	+	4.00000i	0.242536	+	0.970143i
$17$	1.00000	+	4.00000i	0.242536	+	0.970143i
$18$	−3.00000			−0.707107
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	8.00000i		−	1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$		−	8.00000i		−	1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$	−2.00000			−0.392232
$27$		0			0
$28$			1.00000i			0.188982i
$29$			8.00000i			1.48556i	0.669534	+	0.742781i	$0.266494\pi$
$29$			8.00000i			1.48556i	−0.669534	+	0.742781i	$0.733506\pi$
$30$		0			0
$31$			8.00000i			1.43684i	0.695608	+	0.718421i	$0.255135\pi$
$31$			8.00000i			1.43684i	−0.695608	+	0.718421i	$0.744865\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$	−1.00000	−	4.00000i	−0.171499	−	0.685994i
$35$		0			0
$36$	3.00000			0.500000
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	−4.00000			−0.648886
$39$		0			0
$40$		0			0
$41$		−	8.00000i		−	1.24939i	−0.780869	−	0.624695i	$-0.785223\pi$
$41$		−	8.00000i		−	1.24939i	0.780869	−	0.624695i	$-0.214777\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$		0			0
$46$			8.00000i			1.17954i
$47$	−8.00000			−1.16692			−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000			−1.16692			−0.583460	+	0.812142i	$0.698301\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$	−5.00000			−0.707107
$51$		0			0
$52$	2.00000			0.277350
$53$	−10.0000			−1.37361			−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000			−1.37361			−0.686803	+	0.726844i	$0.740986\pi$
$54$		0			0
$55$		0			0
$56$		−	1.00000i		−	0.133631i
$57$		0			0
$58$		−	8.00000i		−	1.05045i
$59$	−12.0000			−1.56227			−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		−	8.00000i		−	1.01600i
$63$			3.00000i			0.377964i
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$	−4.00000			−0.488678			−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000			−0.488678			−0.244339	+	0.969690i	$0.578571\pi$
$68$	1.00000	+	4.00000i	0.121268	+	0.485071i
$69$		0			0
$70$		0			0
$71$		−	8.00000i		−	0.949425i	−0.880141	−	0.474713i	$-0.842552\pi$
$71$		−	8.00000i		−	0.949425i	0.880141	−	0.474713i	$-0.157448\pi$
$72$	−3.00000			−0.353553
$73$			8.00000i			0.936329i	0.883641	+	0.468165i	$0.155085\pi$
$73$			8.00000i			0.936329i	−0.883641	+	0.468165i	$0.844915\pi$
$74$			8.00000i			0.929981i
$75$		0			0
$76$	4.00000			0.458831
$77$		0			0
$78$		0			0
$79$		−	8.00000i		−	0.900070i	−0.893011	−	0.450035i	$-0.851411\pi$
$79$		−	8.00000i		−	0.900070i	0.893011	−	0.450035i	$-0.148589\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$			8.00000i			0.883452i
$83$	−4.00000			−0.439057			−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000			−0.439057			−0.219529	+	0.975606i	$0.570452\pi$
$84$		0			0
$85$		0			0
$86$	4.00000			0.431331
$87$		0			0
$88$		0			0
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		0			0
$91$			2.00000i			0.209657i
$92$		−	8.00000i		−	0.834058i
$93$		0			0
$94$	8.00000			0.825137
$95$		0			0
$96$		0			0
$97$			8.00000i			0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$			8.00000i			0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	1.00000			0.101015
$99$		0			0
$100$	5.00000			0.500000
$101$	10.0000			0.995037			0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000			0.995037			0.497519	+	0.867453i	$0.334245\pi$
$102$		0			0
$103$	−8.00000			−0.788263			−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000			−0.788263			−0.394132	+	0.919054i	$0.628955\pi$
$104$	−2.00000			−0.196116
$105$		0			0
$106$	10.0000			0.971286
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$		−	8.00000i		−	0.766261i	−0.923694	−	0.383131i	$-0.874846\pi$
$109$		−	8.00000i		−	0.766261i	0.923694	−	0.383131i	$-0.125154\pi$
$110$		0			0
$111$		0			0
$112$			1.00000i			0.0944911i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$			8.00000i			0.742781i
$117$	6.00000			0.554700
$118$	12.0000			1.10469
$119$	−4.00000	+	1.00000i	−0.366679	+	0.0916698i
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$			8.00000i			0.718421i
$125$		0			0
$126$		−	3.00000i		−	0.267261i
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$		0			0
$131$			16.0000i			1.39793i	0.715158	+	0.698963i	$0.246355\pi$
$131$			16.0000i			1.39793i	−0.715158	+	0.698963i	$0.753645\pi$
$132$		0			0
$133$			4.00000i			0.346844i
$134$	4.00000			0.345547
$135$		0			0
$136$	−1.00000	−	4.00000i	−0.0857493	−	0.342997i
$137$	6.00000			0.512615			0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000			0.512615			0.256307	+	0.966595i	$0.417494\pi$
$138$		0			0
$139$		−	16.0000i		−	1.35710i	−0.734553	−	0.678551i	$-0.762608\pi$
$139$		−	16.0000i		−	1.35710i	0.734553	−	0.678551i	$-0.237392\pi$
$140$		0			0
$141$		0			0
$142$			8.00000i			0.671345i
$143$		0			0
$144$	3.00000			0.250000
$145$		0			0
$146$		−	8.00000i		−	0.662085i
$147$		0			0
$148$		−	8.00000i		−	0.657596i
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	−4.00000			−0.324443
$153$	3.00000	+	12.0000i	0.242536	+	0.970143i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−18.0000			−1.43656			−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000			−1.43656			−0.718278	+	0.695756i	$0.755069\pi$
$158$			8.00000i			0.636446i
$159$		0			0
$160$		0			0
$161$	8.00000			0.630488
$162$	−9.00000			−0.707107
$163$		−	16.0000i		−	1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$		−	16.0000i		−	1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$		−	8.00000i		−	0.624695i
$165$		0			0
$166$	4.00000			0.310460
$167$			8.00000i			0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$			8.00000i			0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	12.0000			0.917663
$172$	−4.00000			−0.304997
$173$		−	16.0000i		−	1.21646i	−0.793762	−	0.608229i	$-0.791880\pi$
$173$		−	16.0000i		−	1.21646i	0.793762	−	0.608229i	$-0.208120\pi$
$174$		0			0
$175$			5.00000i			0.377964i
$176$		0			0
$177$		0			0
$178$	−6.00000			−0.449719
$179$	−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
$181$			16.0000i			1.18927i	0.803996	+	0.594635i	$0.202704\pi$
$181$			16.0000i			1.18927i	−0.803996	+	0.594635i	$0.797296\pi$
$182$		−	2.00000i		−	0.148250i
$183$		0			0
$184$			8.00000i			0.589768i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−8.00000			−0.583460
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		−	16.0000i		−	1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$		−	16.0000i		−	1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$		−	8.00000i		−	0.574367i
$195$		0			0
$196$	−1.00000			−0.0714286
$197$		−	8.00000i		−	0.569976i	−0.958531	−	0.284988i	$-0.908010\pi$
$197$		−	8.00000i		−	0.569976i	0.958531	−	0.284988i	$-0.0919897\pi$
$198$		0			0
$199$			24.0000i			1.70131i	0.525720	+	0.850657i	$0.323796\pi$
$199$			24.0000i			1.70131i	−0.525720	+	0.850657i	$0.676204\pi$
$200$	−5.00000			−0.353553
$201$		0			0
$202$	−10.0000			−0.703598
$203$	−8.00000			−0.561490
$204$		0			0
$205$		0			0
$206$	8.00000			0.557386
$207$		−	24.0000i		−	1.66812i
$208$	2.00000			0.138675
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	−10.0000			−0.686803
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−8.00000			−0.543075
$218$			8.00000i			0.541828i
$219$		0			0
$220$		0			0
$221$	2.00000	+	8.00000i	0.134535	+	0.538138i
$222$		0			0
$223$	−24.0000			−1.60716			−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000			−1.60716			−0.803579	+	0.595198i	$0.797074\pi$
$224$		−	1.00000i		−	0.0668153i
$225$	15.0000			1.00000
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$	−22.0000			−1.45380			−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000			−1.45380			−0.726900	+	0.686743i	$0.759040\pi$
$230$		0			0
$231$		0			0
$232$		−	8.00000i		−	0.525226i
$233$			16.0000i			1.04819i	0.851658	+	0.524097i	$0.175597\pi$
$233$			16.0000i			1.04819i	−0.851658	+	0.524097i	$0.824403\pi$
$234$	−6.00000			−0.392232
$235$		0			0
$236$	−12.0000			−0.781133
$237$		0			0
$238$	4.00000	−	1.00000i	0.259281	−	0.0648204i
$239$	−16.0000			−1.03495			−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000			−1.03495			−0.517477	+	0.855697i	$0.673129\pi$
$240$		0			0
$241$		−	24.0000i		−	1.54598i	−0.634421	−	0.772988i	$-0.718761\pi$
$241$		−	24.0000i		−	1.54598i	0.634421	−	0.772988i	$-0.281239\pi$
$242$	−11.0000			−0.707107
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	8.00000			0.509028
$248$		−	8.00000i		−	0.508001i
$249$		0			0
$250$		0			0
$251$	28.0000			1.76734			0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000			1.76734			0.883672	+	0.468106i	$0.155064\pi$
$252$			3.00000i			0.188982i
$253$		0			0
$254$	−8.00000			−0.501965
$255$		0			0
$256$	1.00000			0.0625000
$257$	14.0000			0.873296			0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000			0.873296			0.436648	+	0.899632i	$0.356166\pi$
$258$		0			0
$259$	8.00000			0.497096
$260$		0			0
$261$			24.0000i			1.48556i
$262$		−	16.0000i		−	0.988483i
$263$	16.0000			0.986602			0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000			0.986602			0.493301	+	0.869859i	$0.335790\pi$
$264$		0			0
$265$		0			0
$266$		−	4.00000i		−	0.245256i
$267$		0			0
$268$	−4.00000			−0.244339
$269$		−	16.0000i		−	0.975537i	−0.872973	−	0.487769i	$-0.837811\pi$
$269$		−	16.0000i		−	0.975537i	0.872973	−	0.487769i	$-0.162189\pi$
$270$		0			0
$271$	24.0000			1.45790			0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000			1.45790			0.728948	+	0.684569i	$0.240010\pi$
$272$	1.00000	+	4.00000i	0.0606339	+	0.242536i
$273$		0			0
$274$	−6.00000			−0.362473
$275$		0			0
$276$		0			0
$277$		−	8.00000i		−	0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$		−	8.00000i		−	0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$			16.0000i			0.959616i
$279$			24.0000i			1.43684i
$280$		0			0
$281$	10.0000			0.596550			0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000			0.596550			0.298275	+	0.954480i	$0.403589\pi$
$282$		0			0
$283$			16.0000i			0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$			16.0000i			0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$		−	8.00000i		−	0.474713i
$285$		0			0
$286$		0			0
$287$	8.00000			0.472225
$288$	−3.00000			−0.176777
$289$	−15.0000	+	8.00000i	−0.882353	+	0.470588i
$290$		0			0
$291$		0			0
$292$			8.00000i			0.468165i
$293$	−10.0000			−0.584206			−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000			−0.584206			−0.292103	+	0.956387i	$0.594355\pi$
$294$		0			0
$295$		0			0
$296$			8.00000i			0.464991i
$297$		0			0
$298$	10.0000			0.579284
$299$		−	16.0000i		−	0.925304i
$300$		0			0
$301$		−	4.00000i		−	0.230556i
$302$		0			0
$303$		0			0
$304$	4.00000			0.229416
$305$		0			0
$306$	−3.00000	−	12.0000i	−0.171499	−	0.685994i
$307$	−4.00000			−0.228292			−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000			−0.228292			−0.114146	+	0.993464i	$0.536413\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	24.0000i		−	1.36092i	−0.732787	−	0.680458i	$-0.761781\pi$
$311$		−	24.0000i		−	1.36092i	0.732787	−	0.680458i	$-0.238219\pi$
$312$		0			0
$313$		−	8.00000i		−	0.452187i	−0.974106	−	0.226093i	$-0.927405\pi$
$313$		−	8.00000i		−	0.452187i	0.974106	−	0.226093i	$-0.0725954\pi$
$314$	18.0000			1.01580
$315$		0			0
$316$		−	8.00000i		−	0.450035i
$317$			24.0000i			1.34797i	0.738743	+	0.673987i	$0.235420\pi$
$317$			24.0000i			1.34797i	−0.738743	+	0.673987i	$0.764580\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$	−8.00000			−0.445823
$323$	4.00000	+	16.0000i	0.222566	+	0.890264i
$324$	9.00000			0.500000
$325$	10.0000			0.554700
$326$			16.0000i			0.886158i
$327$		0			0
$328$			8.00000i			0.441726i
$329$		−	8.00000i		−	0.441054i
$330$		0			0
$331$	−4.00000			−0.219860			−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000			−0.219860			−0.109930	+	0.993939i	$0.535063\pi$
$332$	−4.00000			−0.219529
$333$		−	24.0000i		−	1.31519i
$334$		−	8.00000i		−	0.437741i
$335$		0			0
$336$		0			0
$337$			16.0000i			0.871576i	0.900049	+	0.435788i	$0.143530\pi$
$337$			16.0000i			0.871576i	−0.900049	+	0.435788i	$0.856470\pi$
$338$	9.00000			0.489535
$339$		0			0
$340$		0			0
$341$		0			0
$342$	−12.0000			−0.648886
$343$		−	1.00000i		−	0.0539949i
$344$	4.00000			0.215666
$345$		0			0
$346$			16.0000i			0.860165i
$347$			16.0000i			0.858925i	0.903085	+	0.429463i	$0.141297\pi$
$347$			16.0000i			0.858925i	−0.903085	+	0.429463i	$0.858703\pi$
$348$		0			0
$349$	−18.0000			−0.963518			−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000			−0.963518			−0.481759	+	0.876304i	$0.660002\pi$
$350$		−	5.00000i		−	0.267261i
$351$		0			0
$352$		0			0
$353$	18.0000			0.958043			0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000			0.958043			0.479022	+	0.877803i	$0.340992\pi$
$354$		0			0
$355$		0			0
$356$	6.00000			0.317999
$357$		0			0
$358$	4.00000			0.211407
$359$	24.0000			1.26667			0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000			1.26667			0.633336	+	0.773877i	$0.281685\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		−	16.0000i		−	0.840941i
$363$		0			0
$364$			2.00000i			0.104828i
$365$		0			0
$366$		0			0
$367$		−	8.00000i		−	0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$		−	8.00000i		−	0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$		−	8.00000i		−	0.417029i
$369$		−	24.0000i		−	1.24939i
$370$		0			0
$371$		−	10.0000i		−	0.519174i
$372$		0			0
$373$	−22.0000			−1.13912			−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000			−1.13912			−0.569558	+	0.821951i	$0.692886\pi$
$374$		0			0
$375$		0			0
$376$	8.00000			0.412568
$377$			16.0000i			0.824042i
$378$		0			0
$379$			16.0000i			0.821865i	0.911666	+	0.410932i	$0.134797\pi$
$379$			16.0000i			0.821865i	−0.911666	+	0.410932i	$0.865203\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$			16.0000i			0.814379i
$387$	−12.0000			−0.609994
$388$			8.00000i			0.406138i
$389$	−6.00000			−0.304212			−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000			−0.304212			−0.152106	+	0.988364i	$0.548606\pi$
$390$		0			0
$391$	32.0000	−	8.00000i	1.61831	−	0.404577i
$392$	1.00000			0.0505076
$393$		0			0
$394$			8.00000i			0.403034i
$395$		0			0
$396$		0			0
$397$		−	16.0000i		−	0.803017i	−0.915855	−	0.401508i	$-0.868486\pi$
$397$		−	16.0000i		−	0.803017i	0.915855	−	0.401508i	$-0.131514\pi$
$398$		−	24.0000i		−	1.20301i
$399$		0			0
$400$	5.00000			0.250000
$401$			32.0000i			1.59800i	0.601329	+	0.799002i	$0.294638\pi$
$401$			32.0000i			1.59800i	−0.601329	+	0.799002i	$0.705362\pi$
$402$		0			0
$403$			16.0000i			0.797017i
$404$	10.0000			0.497519
$405$		0			0
$406$	8.00000			0.397033
$407$		0			0
$408$		0			0
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$		0			0
$411$		0			0
$412$	−8.00000			−0.394132
$413$		−	12.0000i		−	0.590481i
$414$			24.0000i			1.17954i
$415$		0			0
$416$	−2.00000			−0.0980581
$417$		0			0
$418$		0			0
$419$		−	32.0000i		−	1.56330i	−0.623716	−	0.781651i	$-0.714378\pi$
$419$		−	32.0000i		−	1.56330i	0.623716	−	0.781651i	$-0.285622\pi$
$420$		0			0
$421$	−38.0000			−1.85201			−0.926003	−	0.377515i	$-0.876779\pi$
$421$	−38.0000			−1.85201			−0.926003	+	0.377515i	$0.876779\pi$
$422$		0			0
$423$	−24.0000			−1.16692
$424$	10.0000			0.485643
$425$	5.00000	+	20.0000i	0.242536	+	0.970143i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$			24.0000i			1.15604i	0.816023	+	0.578020i	$0.196174\pi$
$431$			24.0000i			1.15604i	−0.816023	+	0.578020i	$0.803826\pi$
$432$		0			0
$433$	14.0000			0.672797			0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000			0.672797			0.336399	+	0.941720i	$0.390791\pi$
$434$	8.00000			0.384012
$435$		0			0
$436$		−	8.00000i		−	0.383131i
$437$		−	32.0000i		−	1.53077i
$438$		0			0
$439$		−	8.00000i		−	0.381819i	−0.981608	−	0.190910i	$-0.938856\pi$
$439$		−	8.00000i		−	0.381819i	0.981608	−	0.190910i	$-0.0611437\pi$
$440$		0			0
$441$	−3.00000			−0.142857
$442$	−2.00000	−	8.00000i	−0.0951303	−	0.380521i
$443$	20.0000			0.950229			0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000			0.950229			0.475114	+	0.879924i	$0.342407\pi$
$444$		0			0
$445$		0			0
$446$	24.0000			1.13643
$447$		0			0
$448$			1.00000i			0.0472456i
$449$			16.0000i			0.755087i	0.925992	+	0.377543i	$0.123231\pi$
$449$			16.0000i			0.755087i	−0.925992	+	0.377543i	$0.876769\pi$
$450$	−15.0000			−0.707107
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−10.0000			−0.467780			−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000			−0.467780			−0.233890	+	0.972263i	$0.575146\pi$
$458$	22.0000			1.02799
$459$		0			0
$460$		0			0
$461$	−34.0000			−1.58354			−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000			−1.58354			−0.791769	+	0.610821i	$0.790840\pi$
$462$		0			0
$463$	8.00000			0.371792			0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000			0.371792			0.185896	+	0.982569i	$0.440481\pi$
$464$			8.00000i			0.371391i
$465$		0			0
$466$		−	16.0000i		−	0.741186i
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$	6.00000			0.277350
$469$		−	4.00000i		−	0.184703i
$470$		0			0
$471$		0			0
$472$	12.0000			0.552345
$473$		0			0
$474$		0			0
$475$	20.0000			0.917663
$476$	−4.00000	+	1.00000i	−0.183340	+	0.0458349i
$477$	−30.0000			−1.37361
$478$	16.0000			0.731823
$479$			24.0000i			1.09659i	0.836286	+	0.548294i	$0.184723\pi$
$479$			24.0000i			1.09659i	−0.836286	+	0.548294i	$0.815277\pi$
$480$		0			0
$481$		−	16.0000i		−	0.729537i
$482$			24.0000i			1.09317i
$483$		0			0
$484$	11.0000			0.500000
$485$		0			0
$486$		0			0
$487$		−	8.00000i		−	0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$		−	8.00000i		−	0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	36.0000			1.62466			0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000			1.62466			0.812329	+	0.583200i	$0.198200\pi$
$492$		0			0
$493$	−32.0000	+	8.00000i	−1.44121	+	0.360302i
$494$	−8.00000			−0.359937
$495$		0			0
$496$			8.00000i			0.359211i
$497$	8.00000			0.358849
$498$		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
$501$		0			0
$502$	−28.0000			−1.24970
$503$		−	8.00000i		−	0.356702i	−0.983967	−	0.178351i	$-0.942924\pi$
$503$		−	8.00000i		−	0.356702i	0.983967	−	0.178351i	$-0.0570763\pi$
$504$		−	3.00000i		−	0.133631i
$505$		0			0
$506$		0			0
$507$		0			0
$508$	8.00000			0.354943
$509$	−18.0000			−0.797836			−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000			−0.797836			−0.398918	+	0.916987i	$0.630614\pi$
$510$		0			0
$511$	−8.00000			−0.353899
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	−14.0000			−0.617514
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−8.00000			−0.351500
$519$		0			0
$520$		0			0
$521$		−	24.0000i		−	1.05146i	−0.850652	−	0.525730i	$-0.823792\pi$
$521$		−	24.0000i		−	1.05146i	0.850652	−	0.525730i	$-0.176208\pi$
$522$		−	24.0000i		−	1.05045i
$523$	−4.00000			−0.174908			−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000			−0.174908			−0.0874539	+	0.996169i	$0.527873\pi$
$524$			16.0000i			0.698963i
$525$		0			0
$526$	−16.0000			−0.697633
$527$	−32.0000	+	8.00000i	−1.39394	+	0.348485i
$528$		0			0
$529$	−41.0000			−1.78261
$530$		0			0
$531$	−36.0000			−1.56227
$532$			4.00000i			0.173422i
$533$		−	16.0000i		−	0.693037i
$534$		0			0
$535$		0			0
$536$	4.00000			0.172774
$537$		0			0
$538$			16.0000i			0.689809i
$539$		0			0
$540$		0			0
$541$			8.00000i			0.343947i	0.985102	+	0.171973i	$0.0550143\pi$
$541$			8.00000i			0.343947i	−0.985102	+	0.171973i	$0.944986\pi$
$542$	−24.0000			−1.03089
$543$		0			0
$544$	−1.00000	−	4.00000i	−0.0428746	−	0.171499i
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	6.00000			0.256307
$549$		0			0
$550$		0			0
$551$			32.0000i			1.36325i
$552$		0			0
$553$	8.00000			0.340195
$554$			8.00000i			0.339887i
$555$		0			0
$556$		−	16.0000i		−	0.678551i
$557$	18.0000			0.762684			0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000			0.762684			0.381342	+	0.924434i	$0.375462\pi$
$558$		−	24.0000i		−	1.01600i
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$	−10.0000			−0.421825
$563$	12.0000			0.505740			0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000			0.505740			0.252870	+	0.967500i	$0.418626\pi$
$564$		0			0
$565$		0			0
$566$		−	16.0000i		−	0.672530i
$567$			9.00000i			0.377964i
$568$			8.00000i			0.335673i
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$			16.0000i			0.669579i	0.942293	+	0.334790i	$0.108665\pi$
$571$			16.0000i			0.669579i	−0.942293	+	0.334790i	$0.891335\pi$
$572$		0			0
$573$		0			0
$574$	−8.00000			−0.333914
$575$		−	40.0000i		−	1.66812i
$576$	3.00000			0.125000
$577$	14.0000			0.582828			0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000			0.582828			0.291414	+	0.956597i	$0.405874\pi$
$578$	15.0000	−	8.00000i	0.623918	−	0.332756i
$579$		0			0
$580$		0			0
$581$		−	4.00000i		−	0.165948i
$582$		0			0
$583$		0			0
$584$		−	8.00000i		−	0.331042i
$585$		0			0
$586$	10.0000			0.413096
$587$	−4.00000			−0.165098			−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000			−0.165098			−0.0825488	+	0.996587i	$0.526306\pi$
$588$		0			0
$589$			32.0000i			1.31854i
$590$		0			0
$591$		0			0
$592$		−	8.00000i		−	0.328798i
$593$	18.0000			0.739171			0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000			0.739171			0.369586	+	0.929197i	$0.379500\pi$
$594$		0			0
$595$		0			0
$596$	−10.0000			−0.409616
$597$		0			0
$598$			16.0000i			0.654289i
$599$	−16.0000			−0.653742			−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000			−0.653742			−0.326871	+	0.945069i	$0.605994\pi$
$600$		0			0
$601$		−	8.00000i		−	0.326327i	−0.986599	−	0.163163i	$-0.947830\pi$
$601$		−	8.00000i		−	0.326327i	0.986599	−	0.163163i	$-0.0521698\pi$
$602$			4.00000i			0.163028i
$603$	−12.0000			−0.488678
$604$		0			0
$605$		0			0
$606$		0			0
$607$			8.00000i			0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$			8.00000i			0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	−4.00000			−0.162221
$609$		0			0
$610$		0			0
$611$	−16.0000			−0.647291
$612$	3.00000	+	12.0000i	0.121268	+	0.485071i
$613$	6.00000			0.242338			0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000			0.242338			0.121169	+	0.992632i	$0.461336\pi$
$614$	4.00000			0.161427
$615$		0			0
$616$		0			0
$617$		−	48.0000i		−	1.93241i	−0.257780	−	0.966204i	$-0.582991\pi$
$617$		−	48.0000i		−	1.93241i	0.257780	−	0.966204i	$-0.417009\pi$
$618$		0			0
$619$			32.0000i			1.28619i	0.765787	+	0.643094i	$0.222350\pi$
$619$			32.0000i			1.28619i	−0.765787	+	0.643094i	$0.777650\pi$
$620$		0			0
$621$		0			0
$622$			24.0000i			0.962312i
$623$			6.00000i			0.240385i
$624$		0			0
$625$	25.0000			1.00000
$626$			8.00000i			0.319744i
$627$		0			0
$628$	−18.0000			−0.718278
$629$	32.0000	−	8.00000i	1.27592	−	0.318981i
$630$		0			0
$631$	24.0000			0.955425			0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000			0.955425			0.477712	+	0.878516i	$0.341466\pi$
$632$			8.00000i			0.318223i
$633$		0			0
$634$		−	24.0000i		−	0.953162i
$635$		0			0
$636$		0			0
$637$	−2.00000			−0.0792429
$638$		0			0
$639$		−	24.0000i		−	0.949425i
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		−	32.0000i		−	1.26196i	−0.775800	−	0.630978i	$-0.782654\pi$
$643$		−	32.0000i		−	1.26196i	0.775800	−	0.630978i	$-0.217346\pi$
$644$	8.00000			0.315244
$645$		0			0
$646$	−4.00000	−	16.0000i	−0.157378	−	0.629512i
$647$	48.0000			1.88707			0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000			1.88707			0.943537	+	0.331266i	$0.107476\pi$
$648$	−9.00000			−0.353553
$649$		0			0
$650$	−10.0000			−0.392232
$651$		0			0
$652$		−	16.0000i		−	0.626608i
$653$			24.0000i			0.939193i	0.882881	+	0.469596i	$0.155601\pi$
$653$			24.0000i			0.939193i	−0.882881	+	0.469596i	$0.844399\pi$
$654$		0			0
$655$		0			0
$656$		−	8.00000i		−	0.312348i
$657$			24.0000i			0.936329i
$658$			8.00000i			0.311872i
$659$	−28.0000			−1.09073			−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000			−1.09073			−0.545363	+	0.838200i	$0.683608\pi$
$660$		0			0
$661$	−26.0000			−1.01128			−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000			−1.01128			−0.505641	+	0.862744i	$0.668744\pi$
$662$	4.00000			0.155464
$663$		0			0
$664$	4.00000			0.155230
$665$		0			0
$666$			24.0000i			0.929981i
$667$	64.0000			2.47809
$668$			8.00000i			0.309529i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$			32.0000i			1.23351i	0.787155	+	0.616755i	$0.211553\pi$
$673$			32.0000i			1.23351i	−0.787155	+	0.616755i	$0.788447\pi$
$674$		−	16.0000i		−	0.616297i
$675$		0			0
$676$	−9.00000			−0.346154
$677$			16.0000i			0.614930i	0.951559	+	0.307465i	$0.0994807\pi$
$677$			16.0000i			0.614930i	−0.951559	+	0.307465i	$0.900519\pi$
$678$		0			0
$679$	−8.00000			−0.307012
$680$		0			0
$681$		0			0
$682$		0			0
$683$			48.0000i			1.83667i	0.395805	+	0.918334i	$0.370466\pi$
$683$			48.0000i			1.83667i	−0.395805	+	0.918334i	$0.629534\pi$
$684$	12.0000			0.458831
$685$		0			0
$686$			1.00000i			0.0381802i
$687$		0			0
$688$	−4.00000			−0.152499
$689$	−20.0000			−0.761939
$690$		0			0
$691$			16.0000i			0.608669i	0.952565	+	0.304334i	$0.0984340\pi$
$691$			16.0000i			0.608669i	−0.952565	+	0.304334i	$0.901566\pi$
$692$		−	16.0000i		−	0.608229i
$693$		0			0
$694$		−	16.0000i		−	0.607352i
$695$		0			0
$696$		0			0
$697$	32.0000	−	8.00000i	1.21209	−	0.303022i
$698$	18.0000			0.681310
$699$		0			0
$700$			5.00000i			0.188982i
$701$	34.0000			1.28416			0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000			1.28416			0.642081	+	0.766637i	$0.278071\pi$
$702$		0			0
$703$		−	32.0000i		−	1.20690i
$704$		0			0
$705$		0			0
$706$	−18.0000			−0.677439
$707$			10.0000i			0.376089i
$708$		0			0
$709$		−	24.0000i		−	0.901339i	−0.892691	−	0.450669i	$-0.851185\pi$
$709$		−	24.0000i		−	0.901339i	0.892691	−	0.450669i	$-0.148815\pi$
$710$		0			0
$711$		−	24.0000i		−	0.900070i
$712$	−6.00000			−0.224860
$713$	64.0000			2.39682
$714$		0			0
$715$		0			0
$716$	−4.00000			−0.149487
$717$		0			0
$718$	−24.0000			−0.895672
$719$			24.0000i			0.895049i	0.894272	+	0.447524i	$0.147694\pi$
$719$			24.0000i			0.895049i	−0.894272	+	0.447524i	$0.852306\pi$
$720$		0			0
$721$		−	8.00000i		−	0.297936i
$722$	3.00000			0.111648
$723$		0			0
$724$			16.0000i			0.594635i
$725$			40.0000i			1.48556i
$726$		0			0
$727$	32.0000			1.18681			0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000			1.18681			0.593407	+	0.804902i	$0.297782\pi$
$728$		−	2.00000i		−	0.0741249i
$729$	27.0000			1.00000
$730$		0			0
$731$	−4.00000	−	16.0000i	−0.147945	−	0.591781i
$732$		0			0
$733$	46.0000			1.69905			0.849524	−	0.527549i	$-0.176889\pi$
$733$	46.0000			1.69905			0.849524	+	0.527549i	$0.176889\pi$
$734$			8.00000i			0.295285i
$735$		0			0
$736$			8.00000i			0.294884i
$737$		0			0
$738$			24.0000i			0.883452i
$739$	44.0000			1.61857			0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000			1.61857			0.809283	+	0.587419i	$0.199856\pi$
$740$		0			0
$741$		0			0
$742$			10.0000i			0.367112i
$743$		−	24.0000i		−	0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$		−	24.0000i		−	0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$		0			0
$745$		0			0
$746$	22.0000			0.805477
$747$	−12.0000			−0.439057
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	8.00000i		−	0.291924i	−0.989290	−	0.145962i	$-0.953372\pi$
$751$		−	8.00000i		−	0.291924i	0.989290	−	0.145962i	$-0.0466277\pi$
$752$	−8.00000			−0.291730
$753$		0			0
$754$		−	16.0000i		−	0.582686i
$755$		0			0
$756$		0			0
$757$	42.0000			1.52652			0.763258	−	0.646094i	$-0.223599\pi$
$757$	42.0000			1.52652			0.763258	+	0.646094i	$0.223599\pi$
$758$		−	16.0000i		−	0.581146i
$759$		0			0
$760$		0			0
$761$	−42.0000			−1.52250			−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000			−1.52250			−0.761249	+	0.648459i	$0.775414\pi$
$762$		0			0
$763$	8.00000			0.289619
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−24.0000			−0.866590
$768$		0			0
$769$	−50.0000			−1.80305			−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000			−1.80305			−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$		0			0
$772$		−	16.0000i		−	0.575853i
$773$	10.0000			0.359675			0.179838	−	0.983696i	$-0.442443\pi$
$773$	10.0000			0.359675			0.179838	+	0.983696i	$0.442443\pi$
$774$	12.0000			0.431331
$775$			40.0000i			1.43684i
$776$		−	8.00000i		−	0.287183i
$777$		0			0
$778$	6.00000			0.215110
$779$		−	32.0000i		−	1.14652i
$780$		0			0
$781$		0			0
$782$	−32.0000	+	8.00000i	−1.14432	+	0.286079i
$783$		0			0
$784$	−1.00000			−0.0357143
$785$		0			0
$786$		0			0
$787$			48.0000i			1.71102i	0.517790	+	0.855508i	$0.326755\pi$
$787$			48.0000i			1.71102i	−0.517790	+	0.855508i	$0.673245\pi$
$788$		−	8.00000i		−	0.284988i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$			16.0000i			0.567819i
$795$		0			0
$796$			24.0000i			0.850657i
$797$	−18.0000			−0.637593			−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000			−0.637593			−0.318796	+	0.947823i	$0.603279\pi$
$798$		0			0
$799$	−8.00000	−	32.0000i	−0.283020	−	1.13208i
$800$	−5.00000			−0.176777
$801$	18.0000			0.635999
$802$		−	32.0000i		−	1.12996i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		−	16.0000i		−	0.563576i
$807$		0			0
$808$	−10.0000			−0.351799
$809$			48.0000i			1.68759i	0.536666	+	0.843795i	$0.319684\pi$
$809$			48.0000i			1.68759i	−0.536666	+	0.843795i	$0.680316\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	−8.00000			−0.280745
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−16.0000			−0.559769
$818$	−10.0000			−0.349642
$819$			6.00000i			0.209657i
$820$		0			0
$821$		−	24.0000i		−	0.837606i	−0.908077	−	0.418803i	$-0.862450\pi$
$821$		−	24.0000i		−	0.837606i	0.908077	−	0.418803i	$-0.137550\pi$
$822$		0			0
$823$			24.0000i			0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$			24.0000i			0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$	8.00000			0.278693
$825$		0			0
$826$			12.0000i			0.417533i
$827$		−	32.0000i		−	1.11275i	−0.830932	−	0.556375i	$-0.812192\pi$
$827$		−	32.0000i		−	1.11275i	0.830932	−	0.556375i	$-0.187808\pi$
$828$		−	24.0000i		−	0.834058i
$829$	46.0000			1.59765			0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000			1.59765			0.798823	+	0.601566i	$0.205456\pi$
$830$		0			0
$831$		0			0
$832$	2.00000			0.0693375
$833$	−1.00000	−	4.00000i	−0.0346479	−	0.138592i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$			32.0000i			1.10542i
$839$		−	24.0000i		−	0.828572i	−0.910147	−	0.414286i	$-0.864031\pi$
$839$		−	24.0000i		−	0.828572i	0.910147	−	0.414286i	$-0.135969\pi$
$840$		0			0
$841$	−35.0000			−1.20690
$842$	38.0000			1.30957
$843$		0			0
$844$		0			0
$845$		0			0
$846$	24.0000			0.825137
$847$			11.0000i			0.377964i
$848$	−10.0000			−0.343401
$849$		0			0
$850$	−5.00000	−	20.0000i	−0.171499	−	0.685994i
$851$	−64.0000			−2.19389
$852$		0			0
$853$		−	16.0000i		−	0.547830i	−0.961754	−	0.273915i	$-0.911681\pi$
$853$		−	16.0000i		−	0.547830i	0.961754	−	0.273915i	$-0.0883186\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	24.0000i		−	0.819824i	−0.912125	−	0.409912i	$-0.865559\pi$
$857$		−	24.0000i		−	0.819824i	0.912125	−	0.409912i	$-0.134441\pi$
$858$		0			0
$859$	36.0000			1.22830			0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000			1.22830			0.614152	+	0.789188i	$0.289498\pi$
$860$		0			0
$861$		0			0
$862$		−	24.0000i		−	0.817443i
$863$	−24.0000			−0.816970			−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000			−0.816970			−0.408485	+	0.912765i	$0.633943\pi$
$864$		0			0
$865$		0			0
$866$	−14.0000			−0.475739
$867$		0			0
$868$	−8.00000			−0.271538
$869$		0			0
$870$		0			0
$871$	−8.00000			−0.271070
$872$			8.00000i			0.270914i
$873$			24.0000i			0.812277i
$874$			32.0000i			1.08242i
$875$		0			0
$876$		0			0
$877$			24.0000i			0.810422i	0.914223	+	0.405211i	$0.132802\pi$
$877$			24.0000i			0.810422i	−0.914223	+	0.405211i	$0.867198\pi$
$878$			8.00000i			0.269987i
$879$		0			0
$880$		0			0
$881$		−	24.0000i		−	0.808581i	−0.914631	−	0.404290i	$-0.867519\pi$
$881$		−	24.0000i		−	0.808581i	0.914631	−	0.404290i	$-0.132481\pi$
$882$	3.00000			0.101015
$883$	−52.0000			−1.74994			−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000			−1.74994			−0.874970	+	0.484178i	$0.839119\pi$
$884$	2.00000	+	8.00000i	0.0672673	+	0.269069i
$885$		0			0
$886$	−20.0000			−0.671913
$887$			56.0000i			1.88030i	0.340766	+	0.940148i	$0.389313\pi$
$887$			56.0000i			1.88030i	−0.340766	+	0.940148i	$0.610687\pi$
$888$		0			0
$889$			8.00000i			0.268311i
$890$		0			0
$891$		0			0
$892$	−24.0000			−0.803579
$893$	−32.0000			−1.07084
$894$		0			0
$895$		0			0
$896$		−	1.00000i		−	0.0334077i
$897$		0			0
$898$		−	16.0000i		−	0.533927i
$899$	−64.0000			−2.13452
$900$	15.0000			0.500000
$901$	−10.0000	−	40.0000i	−0.333148	−	1.33259i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	32.0000i		−	1.06254i	−0.847202	−	0.531271i	$-0.821714\pi$
$907$		−	32.0000i		−	1.06254i	0.847202	−	0.531271i	$-0.178286\pi$
$908$		0			0
$909$	30.0000			0.995037
$910$		0			0
$911$		−	24.0000i		−	0.795155i	−0.917568	−	0.397578i	$-0.869851\pi$
$911$		−	24.0000i		−	0.795155i	0.917568	−	0.397578i	$-0.130149\pi$
$912$		0			0
$913$		0			0
$914$	10.0000			0.330771
$915$		0			0
$916$	−22.0000			−0.726900
$917$	−16.0000			−0.528367
$918$		0			0
$919$	8.00000			0.263896			0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000			0.263896			0.131948	+	0.991257i	$0.457877\pi$
$920$		0			0
$921$		0			0
$922$	34.0000			1.11973
$923$		−	16.0000i		−	0.526646i
$924$		0			0
$925$		−	40.0000i		−	1.31519i
$926$	−8.00000			−0.262896
$927$	−24.0000			−0.788263
$928$		−	8.00000i		−	0.262613i
$929$			8.00000i			0.262471i	0.991351	+	0.131236i	$0.0418945\pi$
$929$			8.00000i			0.262471i	−0.991351	+	0.131236i	$0.958106\pi$
$930$		0			0
$931$	−4.00000			−0.131095
$932$			16.0000i			0.524097i
$933$		0			0
$934$	12.0000			0.392652
$935$		0			0
$936$	−6.00000			−0.196116
$937$	−38.0000			−1.24141			−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000			−1.24141			−0.620703	+	0.784046i	$0.713153\pi$
$938$			4.00000i			0.130605i
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$	−64.0000			−2.08413
$944$	−12.0000			−0.390567
$945$		0			0
$946$		0			0
$947$		−	32.0000i		−	1.03986i	−0.854209	−	0.519930i	$-0.825958\pi$
$947$		−	32.0000i		−	1.03986i	0.854209	−	0.519930i	$-0.174042\pi$
$948$		0			0
$949$			16.0000i			0.519382i
$950$	−20.0000			−0.648886
$951$		0			0
$952$	4.00000	−	1.00000i	0.129641	−	0.0324102i
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$	30.0000			0.971286
$955$		0			0
$956$	−16.0000			−0.517477
$957$		0			0
$958$		−	24.0000i		−	0.775405i
$959$			6.00000i			0.193750i
$960$		0			0
$961$	−33.0000			−1.06452
$962$			16.0000i			0.515861i
$963$		0			0
$964$		−	24.0000i		−	0.772988i
$965$		0			0
$966$		0			0
$967$	56.0000			1.80084			0.900419	−	0.435023i	$-0.143260\pi$
$967$	56.0000			1.80084			0.900419	+	0.435023i	$0.143260\pi$
$968$	−11.0000			−0.353553
$969$		0			0
$970$		0			0
$971$	−28.0000			−0.898563			−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000			−0.898563			−0.449281	+	0.893390i	$0.648320\pi$
$972$		0			0
$973$	16.0000			0.512936
$974$			8.00000i			0.256337i
$975$		0			0
$976$		0			0
$977$	18.0000			0.575871			0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000			0.575871			0.287936	+	0.957650i	$0.407031\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		−	24.0000i		−	0.766261i
$982$	−36.0000			−1.14881
$983$		−	8.00000i		−	0.255160i	−0.991828	−	0.127580i	$-0.959279\pi$
$983$		−	8.00000i		−	0.255160i	0.991828	−	0.127580i	$-0.0407210\pi$
$984$		0			0
$985$		0			0
$986$	32.0000	−	8.00000i	1.01909	−	0.254772i
$987$		0			0
$988$	8.00000			0.254514
$989$			32.0000i			1.01754i
$990$		0			0
$991$			56.0000i			1.77890i	0.457034	+	0.889449i	$0.348912\pi$
$991$			56.0000i			1.77890i	−0.457034	+	0.889449i	$0.651088\pi$
$992$		−	8.00000i		−	0.254000i
$993$		0			0
$994$	−8.00000			−0.253745
$995$		0			0
$996$		0			0
$997$			32.0000i			1.01345i	0.862108	+	0.506725i	$0.169144\pi$
$997$			32.0000i			1.01345i	−0.862108	+	0.506725i	$0.830856\pi$
$998$			32.0000i			1.01294i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	238.2.b.a.169.2	yes	2
3.2	odd	2		2142.2.b.e.883.2		2
4.3	odd	2		1904.2.c.b.1121.1		2
7.6	odd	2		1666.2.b.d.883.1		2
17.4	even	4		4046.2.a.n.1.1		1
17.13	even	4		4046.2.a.o.1.1		1
17.16	even	2	inner	238.2.b.a.169.1	✓	2
51.50	odd	2		2142.2.b.e.883.1		2
68.67	odd	2		1904.2.c.b.1121.2		2
119.118	odd	2		1666.2.b.d.883.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
238.2.b.a.169.1	✓	2	17.16	even	2	inner
238.2.b.a.169.2	yes	2	1.1	even	1	trivial
1666.2.b.d.883.1		2	7.6	odd	2
1666.2.b.d.883.2		2	119.118	odd	2
1904.2.c.b.1121.1		2	4.3	odd	2
1904.2.c.b.1121.2		2	68.67	odd	2
2142.2.b.e.883.1		2	51.50	odd	2
2142.2.b.e.883.2		2	3.2	odd	2
4046.2.a.n.1.1		1	17.4	even	4
4046.2.a.o.1.1		1	17.13	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:	\( N \)	\(=\)	\( 238 = 2 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	238.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000i q^{7} -1.00000 q^{8} +3.00000 q^{9} +2.00000 q^{13} -1.00000i q^{14} +1.00000 q^{16} +(1.00000 + 4.00000i) q^{17} -3.00000 q^{18} +4.00000 q^{19} -8.00000i q^{23} +5.00000 q^{25} -2.00000 q^{26} +1.00000i q^{28} +8.00000i q^{29} +8.00000i q^{31} -1.00000 q^{32} +(-1.00000 - 4.00000i) q^{34} +3.00000 q^{36} -8.00000i q^{37} -4.00000 q^{38} -8.00000i q^{41} -4.00000 q^{43} +8.00000i q^{46} -8.00000 q^{47} -1.00000 q^{49} -5.00000 q^{50} +2.00000 q^{52} -10.0000 q^{53} -1.00000i q^{56} -8.00000i q^{58} -12.0000 q^{59} -8.00000i q^{62} +3.00000i q^{63} +1.00000 q^{64} -4.00000 q^{67} +(1.00000 + 4.00000i) q^{68} -8.00000i q^{71} -3.00000 q^{72} +8.00000i q^{73} +8.00000i q^{74} +4.00000 q^{76} -8.00000i q^{79} +9.00000 q^{81} +8.00000i q^{82} -4.00000 q^{83} +4.00000 q^{86} +6.00000 q^{89} +2.00000i q^{91} -8.00000i q^{92} +8.00000 q^{94} +8.00000i q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 6 q^{9} + 4 q^{13} + 2 q^{16} + 2 q^{17} - 6 q^{18} + 8 q^{19} + 10 q^{25} - 4 q^{26} - 2 q^{32} - 2 q^{34} + 6 q^{36} - 8 q^{38} - 8 q^{43} - 16 q^{47} - 2 q^{49} - 10 q^{50}+ \cdots + 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 6 * q^9 + 4 * q^13 + 2 * q^16 + 2 * q^17 - 6 * q^18 + 8 * q^19 + 10 * q^25 - 4 * q^26 - 2 * q^32 - 2 * q^34 + 6 * q^36 - 8 * q^38 - 8 * q^43 - 16 * q^47 - 2 * q^49 - 10 * q^50 + 4 * q^52 - 20 * q^53 - 24 * q^59 + 2 * q^64 - 8 * q^67 + 2 * q^68 - 6 * q^72 + 8 * q^76 + 18 * q^81 - 8 * q^83 + 8 * q^86 + 12 * q^89 + 16 * q^94 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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