LMFDB - Embedded newform 440.2.b.c.89.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [440,2,Mod(89,440)] 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("440.89"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 440 = 2^{3} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	440.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$3.51341768894$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			89.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	440.89
Dual form			440.2.b.c.89.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(2.00000 + 1.00000i) q^{5} +1.00000i q^{7} +2.00000 q^{9} -1.00000 q^{11} +(-1.00000 + 2.00000i) q^{15} +1.00000i q^{17} -1.00000 q^{19} -1.00000 q^{21} +(3.00000 + 4.00000i) q^{25} +5.00000i q^{27} +1.00000 q^{29} -1.00000 q^{31} -1.00000i q^{33} +(-1.00000 + 2.00000i) q^{35} +1.00000i q^{37} -6.00000i q^{43} +(4.00000 + 2.00000i) q^{45} +8.00000i q^{47} +6.00000 q^{49} -1.00000 q^{51} -9.00000i q^{53} +(-2.00000 - 1.00000i) q^{55} -1.00000i q^{57} -4.00000 q^{59} -7.00000 q^{61} +2.00000i q^{63} -4.00000i q^{67} +5.00000 q^{71} -14.0000i q^{73} +(-4.00000 + 3.00000i) q^{75} -1.00000i q^{77} -4.00000 q^{79} +1.00000 q^{81} -16.0000i q^{83} +(-1.00000 + 2.00000i) q^{85} +1.00000i q^{87} +7.00000 q^{89} -1.00000i q^{93} +(-2.00000 - 1.00000i) q^{95} -16.0000i q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} + 4 q^{9} - 2 q^{11} - 2 q^{15} - 2 q^{19} - 2 q^{21} + 6 q^{25} + 2 q^{29} - 2 q^{31} - 2 q^{35} + 8 q^{45} + 12 q^{49} - 2 q^{51} - 4 q^{55} - 8 q^{59} - 14 q^{61} + 10 q^{71} - 8 q^{75}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q + 4 * q^5 + 4 * q^9 - 2 * q^11 - 2 * q^15 - 2 * q^19 - 2 * q^21 + 6 * q^25 + 2 * q^29 - 2 * q^31 - 2 * q^35 + 8 * q^45 + 12 * q^49 - 2 * q^51 - 4 * q^55 - 8 * q^59 - 14 * q^61 + 10 * q^71 - 8 * q^75 - 8 * q^79 + 2 * q^81 - 2 * q^85 + 14 * q^89 - 4 * q^95 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/440\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$221$	$321$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$			1.00000i			0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$		0			0
$5$	2.00000	+	1.00000i	0.894427	+	0.447214i
$5$	2.00000	+	1.00000i	0.894427	+	0.447214i
$6$		0			0
$7$			1.00000i			0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$			1.00000i			0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$		0			0
$9$	2.00000			0.666667
$10$		0			0
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	−1.00000	+	2.00000i	−0.258199	+	0.516398i
$16$		0			0
$17$			1.00000i			0.242536i	0.992620	+	0.121268i	$0.0386960\pi$
$17$			1.00000i			0.242536i	−0.992620	+	0.121268i	$0.961304\pi$
$18$		0			0
$19$	−1.00000			−0.229416			−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000			−0.229416			−0.114708	+	0.993399i	$0.536593\pi$
$20$		0			0
$21$	−1.00000			−0.218218
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	3.00000	+	4.00000i	0.600000	+	0.800000i
$26$		0			0
$27$			5.00000i			0.962250i
$28$		0			0
$29$	1.00000			0.185695			0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000			0.185695			0.0928477	+	0.995680i	$0.470403\pi$
$30$		0			0
$31$	−1.00000			−0.179605			−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000			−0.179605			−0.0898027	+	0.995960i	$0.528624\pi$
$32$		0			0
$33$		−	1.00000i		−	0.174078i
$34$		0			0
$35$	−1.00000	+	2.00000i	−0.169031	+	0.338062i
$36$		0			0
$37$			1.00000i			0.164399i	0.996616	+	0.0821995i	$0.0261945\pi$
$37$			1.00000i			0.164399i	−0.996616	+	0.0821995i	$0.973806\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		−	6.00000i		−	0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$		−	6.00000i		−	0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$		0			0
$45$	4.00000	+	2.00000i	0.596285	+	0.298142i
$46$		0			0
$47$			8.00000i			1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$			8.00000i			1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$		0			0
$51$	−1.00000			−0.140028
$52$		0			0
$53$		−	9.00000i		−	1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$		−	9.00000i		−	1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$		0			0
$55$	−2.00000	−	1.00000i	−0.269680	−	0.134840i
$56$		0			0
$57$		−	1.00000i		−	0.132453i
$58$		0			0
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$	−7.00000			−0.896258			−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000			−0.896258			−0.448129	+	0.893969i	$0.647910\pi$
$62$		0			0
$63$			2.00000i			0.251976i
$64$		0			0
$65$		0			0
$66$		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
$69$		0			0
$70$		0			0
$71$	5.00000			0.593391			0.296695	−	0.954972i	$-0.404115\pi$
$71$	5.00000			0.593391			0.296695	+	0.954972i	$0.404115\pi$
$72$		0			0
$73$		−	14.0000i		−	1.63858i	−0.573382	−	0.819288i	$-0.694369\pi$
$73$		−	14.0000i		−	1.63858i	0.573382	−	0.819288i	$-0.305631\pi$
$74$		0			0
$75$	−4.00000	+	3.00000i	−0.461880	+	0.346410i
$76$		0			0
$77$		−	1.00000i		−	0.113961i
$78$		0			0
$79$	−4.00000			−0.450035			−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000			−0.450035			−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	16.0000i		−	1.75623i	−0.478451	−	0.878114i	$-0.658802\pi$
$83$		−	16.0000i		−	1.75623i	0.478451	−	0.878114i	$-0.341198\pi$
$84$		0			0
$85$	−1.00000	+	2.00000i	−0.108465	+	0.216930i
$86$		0			0
$87$			1.00000i			0.107211i
$88$		0			0
$89$	7.00000			0.741999			0.370999	−	0.928633i	$-0.379015\pi$
$89$	7.00000			0.741999			0.370999	+	0.928633i	$0.379015\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	1.00000i		−	0.103695i
$94$		0			0
$95$	−2.00000	−	1.00000i	−0.205196	−	0.102598i
$96$		0			0
$97$		−	16.0000i		−	1.62455i	−0.583272	−	0.812277i	$-0.698228\pi$
$97$		−	16.0000i		−	1.62455i	0.583272	−	0.812277i	$-0.301772\pi$
$98$		0			0
$99$	−2.00000			−0.201008
$100$		0			0
$101$	−10.0000			−0.995037			−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000			−0.995037			−0.497519	+	0.867453i	$0.665755\pi$
$102$		0			0
$103$		−	8.00000i		−	0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$		−	8.00000i		−	0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$		0			0
$105$	−2.00000	−	1.00000i	−0.195180	−	0.0975900i
$106$		0			0
$107$			8.00000i			0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$			8.00000i			0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$		0			0
$111$	−1.00000			−0.0949158
$112$		0			0
$113$		−	2.00000i		−	0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$		−	2.00000i		−	0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.00000			−0.0916698
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$	2.00000	+	11.0000i	0.178885	+	0.983870i
$126$		0			0
$127$		−	16.0000i		−	1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$		−	16.0000i		−	1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$		0			0
$129$	6.00000			0.528271
$130$		0			0
$131$	21.0000			1.83478			0.917389	−	0.397991i	$-0.130293\pi$
$131$	21.0000			1.83478			0.917389	+	0.397991i	$0.130293\pi$
$132$		0			0
$133$		−	1.00000i		−	0.0867110i
$134$		0			0
$135$	−5.00000	+	10.0000i	−0.430331	+	0.860663i
$136$		0			0
$137$			12.0000i			1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$			12.0000i			1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$		0			0
$139$	−12.0000			−1.01783			−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000			−1.01783			−0.508913	+	0.860818i	$0.669953\pi$
$140$		0			0
$141$	−8.00000			−0.673722
$142$		0			0
$143$		0			0
$144$		0			0
$145$	2.00000	+	1.00000i	0.166091	+	0.0830455i
$146$		0			0
$147$			6.00000i			0.494872i
$148$		0			0
$149$	−17.0000			−1.39269			−0.696347	−	0.717705i	$-0.745193\pi$
$149$	−17.0000			−1.39269			−0.696347	+	0.717705i	$0.745193\pi$
$150$		0			0
$151$	10.0000			0.813788			0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000			0.813788			0.406894	+	0.913475i	$0.366612\pi$
$152$		0			0
$153$			2.00000i			0.161690i
$154$		0			0
$155$	−2.00000	−	1.00000i	−0.160644	−	0.0803219i
$156$		0			0
$157$			13.0000i			1.03751i	0.854922	+	0.518756i	$0.173605\pi$
$157$			13.0000i			1.03751i	−0.854922	+	0.518756i	$0.826395\pi$
$158$		0			0
$159$	9.00000			0.713746
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	19.0000i		−	1.48819i	−0.668071	−	0.744097i	$-0.732880\pi$
$163$		−	19.0000i		−	1.48819i	0.668071	−	0.744097i	$-0.267120\pi$
$164$		0			0
$165$	1.00000	−	2.00000i	0.0778499	−	0.155700i
$166$		0			0
$167$		−	3.00000i		−	0.232147i	−0.993241	−	0.116073i	$-0.962969\pi$
$167$		−	3.00000i		−	0.232147i	0.993241	−	0.116073i	$-0.0370308\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$	−2.00000			−0.152944
$172$		0			0
$173$			22.0000i			1.67263i	0.548250	+	0.836315i	$0.315294\pi$
$173$			22.0000i			1.67263i	−0.548250	+	0.836315i	$0.684706\pi$
$174$		0			0
$175$	−4.00000	+	3.00000i	−0.302372	+	0.226779i
$176$		0			0
$177$		−	4.00000i		−	0.300658i
$178$		0			0
$179$	10.0000			0.747435			0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000			0.747435			0.373718	+	0.927543i	$0.378083\pi$
$180$		0			0
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$		0			0
$183$		−	7.00000i		−	0.517455i
$184$		0			0
$185$	−1.00000	+	2.00000i	−0.0735215	+	0.147043i
$186$		0			0
$187$		−	1.00000i		−	0.0731272i
$188$		0			0
$189$	−5.00000			−0.363696
$190$		0			0
$191$	24.0000			1.73658			0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000			1.73658			0.868290	+	0.496058i	$0.165220\pi$
$192$		0			0
$193$		−	13.0000i		−	0.935760i	−0.883792	−	0.467880i	$-0.845018\pi$
$193$		−	13.0000i		−	0.935760i	0.883792	−	0.467880i	$-0.154982\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			18.0000i			1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$			18.0000i			1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$		0			0
$199$	−17.0000			−1.20510			−0.602549	−	0.798082i	$-0.705848\pi$
$199$	−17.0000			−1.20510			−0.602549	+	0.798082i	$0.705848\pi$
$200$		0			0
$201$	4.00000			0.282138
$202$		0			0
$203$			1.00000i			0.0701862i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.00000			0.0691714
$210$		0			0
$211$	17.0000			1.17033			0.585164	−	0.810915i	$-0.301030\pi$
$211$	17.0000			1.17033			0.585164	+	0.810915i	$0.301030\pi$
$212$		0			0
$213$			5.00000i			0.342594i
$214$		0			0
$215$	6.00000	−	12.0000i	0.409197	−	0.818393i
$216$		0			0
$217$		−	1.00000i		−	0.0678844i
$218$		0			0
$219$	14.0000			0.946032
$220$		0			0
$221$		0			0
$222$		0			0
$223$			10.0000i			0.669650i	0.942280	+	0.334825i	$0.108677\pi$
$223$			10.0000i			0.669650i	−0.942280	+	0.334825i	$0.891323\pi$
$224$		0			0
$225$	6.00000	+	8.00000i	0.400000	+	0.533333i
$226$		0			0
$227$			12.0000i			0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$			12.0000i			0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$		0			0
$229$	−12.0000			−0.792982			−0.396491	−	0.918039i	$-0.629772\pi$
$229$	−12.0000			−0.792982			−0.396491	+	0.918039i	$0.629772\pi$
$230$		0			0
$231$	1.00000			0.0657952
$232$		0			0
$233$		−	1.00000i		−	0.0655122i	−0.999463	−	0.0327561i	$-0.989572\pi$
$233$		−	1.00000i		−	0.0655122i	0.999463	−	0.0327561i	$-0.0104285\pi$
$234$		0			0
$235$	−8.00000	+	16.0000i	−0.521862	+	1.04372i
$236$		0			0
$237$		−	4.00000i		−	0.259828i
$238$		0			0
$239$	16.0000			1.03495			0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000			1.03495			0.517477	+	0.855697i	$0.326871\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$			16.0000i			1.02640i
$244$		0			0
$245$	12.0000	+	6.00000i	0.766652	+	0.383326i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	16.0000			1.01396
$250$		0			0
$251$	6.00000			0.378717			0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000			0.378717			0.189358	+	0.981908i	$0.439359\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−2.00000	−	1.00000i	−0.125245	−	0.0626224i
$256$		0			0
$257$			24.0000i			1.49708i	0.663090	+	0.748539i	$0.269245\pi$
$257$			24.0000i			1.49708i	−0.663090	+	0.748539i	$0.730755\pi$
$258$		0			0
$259$	−1.00000			−0.0621370
$260$		0			0
$261$	2.00000			0.123797
$262$		0			0
$263$			21.0000i			1.29492i	0.762101	+	0.647458i	$0.224168\pi$
$263$			21.0000i			1.29492i	−0.762101	+	0.647458i	$0.775832\pi$
$264$		0			0
$265$	9.00000	−	18.0000i	0.552866	−	1.10573i
$266$		0			0
$267$			7.00000i			0.428393i
$268$		0			0
$269$	−8.00000			−0.487769			−0.243884	−	0.969804i	$-0.578422\pi$
$269$	−8.00000			−0.487769			−0.243884	+	0.969804i	$0.578422\pi$
$270$		0			0
$271$	−6.00000			−0.364474			−0.182237	−	0.983255i	$-0.558334\pi$
$271$	−6.00000			−0.364474			−0.182237	+	0.983255i	$0.558334\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.00000	−	4.00000i	−0.180907	−	0.241209i
$276$		0			0
$277$		−	8.00000i		−	0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$		−	8.00000i		−	0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$		0			0
$279$	−2.00000			−0.119737
$280$		0			0
$281$	−22.0000			−1.31241			−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000			−1.31241			−0.656205	+	0.754583i	$0.727839\pi$
$282$		0			0
$283$			20.0000i			1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$			20.0000i			1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$		0			0
$285$	1.00000	−	2.00000i	0.0592349	−	0.118470i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	16.0000			0.941176
$290$		0			0
$291$	16.0000			0.937937
$292$		0			0
$293$		−	16.0000i		−	0.934730i	−0.884064	−	0.467365i	$-0.845203\pi$
$293$		−	16.0000i		−	0.934730i	0.884064	−	0.467365i	$-0.154797\pi$
$294$		0			0
$295$	−8.00000	−	4.00000i	−0.465778	−	0.232889i
$296$		0			0
$297$		−	5.00000i		−	0.290129i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	6.00000			0.345834
$302$		0			0
$303$		−	10.0000i		−	0.574485i
$304$		0			0
$305$	−14.0000	−	7.00000i	−0.801638	−	0.400819i
$306$		0			0
$307$		−	16.0000i		−	0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$		−	16.0000i		−	0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$		0			0
$309$	8.00000			0.455104
$310$		0			0
$311$	−13.0000			−0.737162			−0.368581	−	0.929596i	$-0.620156\pi$
$311$	−13.0000			−0.737162			−0.368581	+	0.929596i	$0.620156\pi$
$312$		0			0
$313$			18.0000i			1.01742i	0.860938	+	0.508710i	$0.169877\pi$
$313$			18.0000i			1.01742i	−0.860938	+	0.508710i	$0.830123\pi$
$314$		0			0
$315$	−2.00000	+	4.00000i	−0.112687	+	0.225374i
$316$		0			0
$317$		−	1.00000i		−	0.0561656i	−0.999606	−	0.0280828i	$-0.991060\pi$
$317$		−	1.00000i		−	0.0561656i	0.999606	−	0.0280828i	$-0.00894021\pi$
$318$		0			0
$319$	−1.00000			−0.0559893
$320$		0			0
$321$	−8.00000			−0.446516
$322$		0			0
$323$		−	1.00000i		−	0.0556415i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	10.0000i		−	0.553001i
$328$		0			0
$329$	−8.00000			−0.441054
$330$		0			0
$331$	−34.0000			−1.86881			−0.934405	−	0.356214i	$-0.884068\pi$
$331$	−34.0000			−1.86881			−0.934405	+	0.356214i	$0.884068\pi$
$332$		0			0
$333$			2.00000i			0.109599i
$334$		0			0
$335$	4.00000	−	8.00000i	0.218543	−	0.437087i
$336$		0			0
$337$		−	13.0000i		−	0.708155i	−0.935216	−	0.354078i	$-0.884795\pi$
$337$		−	13.0000i		−	0.708155i	0.935216	−	0.354078i	$-0.115205\pi$
$338$		0			0
$339$	2.00000			0.108625
$340$		0			0
$341$	1.00000			0.0541530
$342$		0			0
$343$			13.0000i			0.701934i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	18.0000i		−	0.966291i	−0.875540	−	0.483145i	$-0.839494\pi$
$347$		−	18.0000i		−	0.966291i	0.875540	−	0.483145i	$-0.160506\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			14.0000i			0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$			14.0000i			0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$		0			0
$355$	10.0000	+	5.00000i	0.530745	+	0.265372i
$356$		0			0
$357$		−	1.00000i		−	0.0529256i
$358$		0			0
$359$	−20.0000			−1.05556			−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000			−1.05556			−0.527780	+	0.849381i	$0.676975\pi$
$360$		0			0
$361$	−18.0000			−0.947368
$362$		0			0
$363$			1.00000i			0.0524864i
$364$		0			0
$365$	14.0000	−	28.0000i	0.732793	−	1.46559i
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	9.00000			0.467257
$372$		0			0
$373$		−	4.00000i		−	0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$		−	4.00000i		−	0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$		0			0
$375$	−11.0000	+	2.00000i	−0.568038	+	0.103280i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−2.00000			−0.102733			−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000			−0.102733			−0.0513665	+	0.998680i	$0.516358\pi$
$380$		0			0
$381$	16.0000			0.819705
$382$		0			0
$383$		−	18.0000i		−	0.919757i	−0.887982	−	0.459879i	$-0.847893\pi$
$383$		−	18.0000i		−	0.919757i	0.887982	−	0.459879i	$-0.152107\pi$
$384$		0			0
$385$	1.00000	−	2.00000i	0.0509647	−	0.101929i
$386$		0			0
$387$		−	12.0000i		−	0.609994i
$388$		0			0
$389$	−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
$391$		0			0
$392$		0			0
$393$			21.0000i			1.05931i
$394$		0			0
$395$	−8.00000	−	4.00000i	−0.402524	−	0.201262i
$396$		0			0
$397$		−	14.0000i		−	0.702640i	−0.936255	−	0.351320i	$-0.885733\pi$
$397$		−	14.0000i		−	0.702640i	0.936255	−	0.351320i	$-0.114267\pi$
$398$		0			0
$399$	1.00000			0.0500626
$400$		0			0
$401$	−29.0000			−1.44819			−0.724095	−	0.689700i	$-0.757743\pi$
$401$	−29.0000			−1.44819			−0.724095	+	0.689700i	$0.757743\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	2.00000	+	1.00000i	0.0993808	+	0.0496904i
$406$		0			0
$407$		−	1.00000i		−	0.0495682i
$408$		0			0
$409$	32.0000			1.58230			0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000			1.58230			0.791149	+	0.611623i	$0.209483\pi$
$410$		0			0
$411$	−12.0000			−0.591916
$412$		0			0
$413$		−	4.00000i		−	0.196827i
$414$		0			0
$415$	16.0000	−	32.0000i	0.785409	−	1.57082i
$416$		0			0
$417$		−	12.0000i		−	0.587643i
$418$		0			0
$419$	−8.00000			−0.390826			−0.195413	−	0.980721i	$-0.562605\pi$
$419$	−8.00000			−0.390826			−0.195413	+	0.980721i	$0.562605\pi$
$420$		0			0
$421$	24.0000			1.16969			0.584844	−	0.811146i	$-0.301156\pi$
$421$	24.0000			1.16969			0.584844	+	0.811146i	$0.301156\pi$
$422$		0			0
$423$			16.0000i			0.777947i
$424$		0			0
$425$	−4.00000	+	3.00000i	−0.194029	+	0.145521i
$426$		0			0
$427$		−	7.00000i		−	0.338754i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$		−	26.0000i		−	1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$		−	26.0000i		−	1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$		0			0
$435$	−1.00000	+	2.00000i	−0.0479463	+	0.0958927i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−26.0000			−1.24091			−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000			−1.24091			−0.620456	+	0.784241i	$0.713053\pi$
$440$		0			0
$441$	12.0000			0.571429
$442$		0			0
$443$		−	20.0000i		−	0.950229i	−0.879924	−	0.475114i	$-0.842407\pi$
$443$		−	20.0000i		−	0.950229i	0.879924	−	0.475114i	$-0.157593\pi$
$444$		0			0
$445$	14.0000	+	7.00000i	0.663664	+	0.331832i
$446$		0			0
$447$		−	17.0000i		−	0.804072i
$448$		0			0
$449$	2.00000			0.0943858			0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000			0.0943858			0.0471929	+	0.998886i	$0.484972\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$			10.0000i			0.469841i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	1.00000i		−	0.0467780i	−0.999726	−	0.0233890i	$-0.992554\pi$
$457$		−	1.00000i		−	0.0467780i	0.999726	−	0.0233890i	$-0.00744563\pi$
$458$		0			0
$459$	−5.00000			−0.233380
$460$		0			0
$461$	9.00000			0.419172			0.209586	−	0.977790i	$-0.432788\pi$
$461$	9.00000			0.419172			0.209586	+	0.977790i	$0.432788\pi$
$462$		0			0
$463$			36.0000i			1.67306i	0.547920	+	0.836531i	$0.315420\pi$
$463$			36.0000i			1.67306i	−0.547920	+	0.836531i	$0.684580\pi$
$464$		0			0
$465$	1.00000	−	2.00000i	0.0463739	−	0.0927478i
$466$		0			0
$467$		−	5.00000i		−	0.231372i	−0.993286	−	0.115686i	$-0.963093\pi$
$467$		−	5.00000i		−	0.231372i	0.993286	−	0.115686i	$-0.0369067\pi$
$468$		0			0
$469$	4.00000			0.184703
$470$		0			0
$471$	−13.0000			−0.599008
$472$		0			0
$473$			6.00000i			0.275880i
$474$		0			0
$475$	−3.00000	−	4.00000i	−0.137649	−	0.183533i
$476$		0			0
$477$		−	18.0000i		−	0.824163i
$478$		0			0
$479$	34.0000			1.55350			0.776750	−	0.629809i	$-0.216867\pi$
$479$	34.0000			1.55350			0.776750	+	0.629809i	$0.216867\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	16.0000	−	32.0000i	0.726523	−	1.45305i
$486$		0			0
$487$			18.0000i			0.815658i	0.913058	+	0.407829i	$0.133714\pi$
$487$			18.0000i			0.815658i	−0.913058	+	0.407829i	$0.866286\pi$
$488$		0			0
$489$	19.0000			0.859210
$490$		0			0
$491$	−27.0000			−1.21849			−0.609246	−	0.792981i	$-0.708528\pi$
$491$	−27.0000			−1.21849			−0.609246	+	0.792981i	$0.708528\pi$
$492$		0			0
$493$			1.00000i			0.0450377i
$494$		0			0
$495$	−4.00000	−	2.00000i	−0.179787	−	0.0898933i
$496$		0			0
$497$			5.00000i			0.224281i
$498$		0			0
$499$	−28.0000			−1.25345			−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000			−1.25345			−0.626726	+	0.779240i	$0.715605\pi$
$500$		0			0
$501$	3.00000			0.134030
$502$		0			0
$503$			24.0000i			1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$			24.0000i			1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$		0			0
$505$	−20.0000	−	10.0000i	−0.889988	−	0.444994i
$506$		0			0
$507$			13.0000i			0.577350i
$508$		0			0
$509$	−8.00000			−0.354594			−0.177297	−	0.984157i	$-0.556735\pi$
$509$	−8.00000			−0.354594			−0.177297	+	0.984157i	$0.556735\pi$
$510$		0			0
$511$	14.0000			0.619324
$512$		0			0
$513$		−	5.00000i		−	0.220755i
$514$		0			0
$515$	8.00000	−	16.0000i	0.352522	−	0.705044i
$516$		0			0
$517$		−	8.00000i		−	0.351840i
$518$		0			0
$519$	−22.0000			−0.965693
$520$		0			0
$521$	−2.00000			−0.0876216			−0.0438108	−	0.999040i	$-0.513950\pi$
$521$	−2.00000			−0.0876216			−0.0438108	+	0.999040i	$0.513950\pi$
$522$		0			0
$523$			28.0000i			1.22435i	0.790721	+	0.612177i	$0.209706\pi$
$523$			28.0000i			1.22435i	−0.790721	+	0.612177i	$0.790294\pi$
$524$		0			0
$525$	−3.00000	−	4.00000i	−0.130931	−	0.174574i
$526$		0			0
$527$		−	1.00000i		−	0.0435607i
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$	−8.00000			−0.347170
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−8.00000	+	16.0000i	−0.345870	+	0.691740i
$536$		0			0
$537$			10.0000i			0.431532i
$538$		0			0
$539$	−6.00000			−0.258438
$540$		0			0
$541$	3.00000			0.128980			0.0644900	−	0.997918i	$-0.479458\pi$
$541$	3.00000			0.128980			0.0644900	+	0.997918i	$0.479458\pi$
$542$		0			0
$543$		−	22.0000i		−	0.944110i
$544$		0			0
$545$	−20.0000	−	10.0000i	−0.856706	−	0.428353i
$546$		0			0
$547$		−	12.0000i		−	0.513083i	−0.966533	−	0.256541i	$-0.917417\pi$
$547$		−	12.0000i		−	0.513083i	0.966533	−	0.256541i	$-0.0825830\pi$
$548$		0			0
$549$	−14.0000			−0.597505
$550$		0			0
$551$	−1.00000			−0.0426014
$552$		0			0
$553$		−	4.00000i		−	0.170097i
$554$		0			0
$555$	−2.00000	−	1.00000i	−0.0848953	−	0.0424476i
$556$		0			0
$557$			8.00000i			0.338971i	0.985533	+	0.169485i	$0.0542106\pi$
$557$			8.00000i			0.338971i	−0.985533	+	0.169485i	$0.945789\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	1.00000			0.0422200
$562$		0			0
$563$			12.0000i			0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$			12.0000i			0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$		0			0
$565$	2.00000	−	4.00000i	0.0841406	−	0.168281i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	−24.0000			−1.00613			−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000			−1.00613			−0.503066	+	0.864248i	$0.667795\pi$
$570$		0			0
$571$	7.00000			0.292941			0.146470	−	0.989215i	$-0.453209\pi$
$571$	7.00000			0.292941			0.146470	+	0.989215i	$0.453209\pi$
$572$		0			0
$573$			24.0000i			1.00261i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			4.00000i			0.166522i	0.996528	+	0.0832611i	$0.0265335\pi$
$577$			4.00000i			0.166522i	−0.996528	+	0.0832611i	$0.973466\pi$
$578$		0			0
$579$	13.0000			0.540262
$580$		0			0
$581$	16.0000			0.663792
$582$		0			0
$583$			9.00000i			0.372742i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			33.0000i			1.36206i	0.732257	+	0.681028i	$0.238467\pi$
$587$			33.0000i			1.36206i	−0.732257	+	0.681028i	$0.761533\pi$
$588$		0			0
$589$	1.00000			0.0412043
$590$		0			0
$591$	−18.0000			−0.740421
$592$		0			0
$593$			18.0000i			0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$			18.0000i			0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$		0			0
$595$	−2.00000	−	1.00000i	−0.0819920	−	0.0409960i
$596$		0			0
$597$		−	17.0000i		−	0.695764i
$598$		0			0
$599$	5.00000			0.204294			0.102147	−	0.994769i	$-0.467429\pi$
$599$	5.00000			0.204294			0.102147	+	0.994769i	$0.467429\pi$
$600$		0			0
$601$	14.0000			0.571072			0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000			0.571072			0.285536	+	0.958368i	$0.407828\pi$
$602$		0			0
$603$		−	8.00000i		−	0.325785i
$604$		0			0
$605$	2.00000	+	1.00000i	0.0813116	+	0.0406558i
$606$		0			0
$607$			3.00000i			0.121766i	0.998145	+	0.0608831i	$0.0193917\pi$
$607$			3.00000i			0.121766i	−0.998145	+	0.0608831i	$0.980608\pi$
$608$		0			0
$609$	−1.00000			−0.0405220
$610$		0			0
$611$		0			0
$612$		0			0
$613$			46.0000i			1.85792i	0.370177	+	0.928961i	$0.379297\pi$
$613$			46.0000i			1.85792i	−0.370177	+	0.928961i	$0.620703\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	8.00000i		−	0.322068i	−0.986949	−	0.161034i	$-0.948517\pi$
$617$		−	8.00000i		−	0.322068i	0.986949	−	0.161034i	$-0.0514829\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			7.00000i			0.280449i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$			1.00000i			0.0399362i
$628$		0			0
$629$	−1.00000			−0.0398726
$630$		0			0
$631$	21.0000			0.835997			0.417998	−	0.908448i	$-0.362732\pi$
$631$	21.0000			0.835997			0.417998	+	0.908448i	$0.362732\pi$
$632$		0			0
$633$			17.0000i			0.675689i
$634$		0			0
$635$	16.0000	−	32.0000i	0.634941	−	1.26988i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	10.0000			0.395594
$640$		0			0
$641$	−1.00000			−0.0394976			−0.0197488	−	0.999805i	$-0.506287\pi$
$641$	−1.00000			−0.0394976			−0.0197488	+	0.999805i	$0.506287\pi$
$642$		0			0
$643$			1.00000i			0.0394362i	0.999806	+	0.0197181i	$0.00627687\pi$
$643$			1.00000i			0.0394362i	−0.999806	+	0.0197181i	$0.993723\pi$
$644$		0			0
$645$	12.0000	+	6.00000i	0.472500	+	0.236250i
$646$		0			0
$647$			48.0000i			1.88707i	0.331266	+	0.943537i	$0.392524\pi$
$647$			48.0000i			1.88707i	−0.331266	+	0.943537i	$0.607476\pi$
$648$		0			0
$649$	4.00000			0.157014
$650$		0			0
$651$	1.00000			0.0391931
$652$		0			0
$653$		−	27.0000i		−	1.05659i	−0.849060	−	0.528296i	$-0.822831\pi$
$653$		−	27.0000i		−	1.05659i	0.849060	−	0.528296i	$-0.177169\pi$
$654$		0			0
$655$	42.0000	+	21.0000i	1.64108	+	0.820538i
$656$		0			0
$657$		−	28.0000i		−	1.09238i
$658$		0			0
$659$	33.0000			1.28550			0.642749	−	0.766077i	$-0.277794\pi$
$659$	33.0000			1.28550			0.642749	+	0.766077i	$0.277794\pi$
$660$		0			0
$661$	28.0000			1.08907			0.544537	−	0.838737i	$-0.316705\pi$
$661$	28.0000			1.08907			0.544537	+	0.838737i	$0.316705\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.00000	−	2.00000i	0.0387783	−	0.0775567i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−10.0000			−0.386622
$670$		0			0
$671$	7.00000			0.270232
$672$		0			0
$673$			35.0000i			1.34915i	0.738206	+	0.674575i	$0.235673\pi$
$673$			35.0000i			1.34915i	−0.738206	+	0.674575i	$0.764327\pi$
$674$		0			0
$675$	−20.0000	+	15.0000i	−0.769800	+	0.577350i
$676$		0			0
$677$			36.0000i			1.38359i	0.722093	+	0.691796i	$0.243180\pi$
$677$			36.0000i			1.38359i	−0.722093	+	0.691796i	$0.756820\pi$
$678$		0			0
$679$	16.0000			0.614024
$680$		0			0
$681$	−12.0000			−0.459841
$682$		0			0
$683$			15.0000i			0.573959i	0.957937	+	0.286980i	$0.0926512\pi$
$683$			15.0000i			0.573959i	−0.957937	+	0.286980i	$0.907349\pi$
$684$		0			0
$685$	−12.0000	+	24.0000i	−0.458496	+	0.916993i
$686$		0			0
$687$		−	12.0000i		−	0.457829i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	40.0000			1.52167			0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000			1.52167			0.760836	+	0.648944i	$0.224789\pi$
$692$		0			0
$693$		−	2.00000i		−	0.0759737i
$694$		0			0
$695$	−24.0000	−	12.0000i	−0.910372	−	0.455186i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	1.00000			0.0378235
$700$		0			0
$701$	33.0000			1.24639			0.623196	−	0.782065i	$-0.285834\pi$
$701$	33.0000			1.24639			0.623196	+	0.782065i	$0.285834\pi$
$702$		0			0
$703$		−	1.00000i		−	0.0377157i
$704$		0			0
$705$	−16.0000	−	8.00000i	−0.602595	−	0.301297i
$706$		0			0
$707$		−	10.0000i		−	0.376089i
$708$		0			0
$709$	−18.0000			−0.676004			−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000			−0.676004			−0.338002	+	0.941145i	$0.609751\pi$
$710$		0			0
$711$	−8.00000			−0.300023
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$			16.0000i			0.597531i
$718$		0			0
$719$	41.0000			1.52904			0.764521	−	0.644599i	$-0.222976\pi$
$719$	41.0000			1.52904			0.764521	+	0.644599i	$0.222976\pi$
$720$		0			0
$721$	8.00000			0.297936
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.00000	+	4.00000i	0.111417	+	0.148556i
$726$		0			0
$727$		−	18.0000i		−	0.667583i	−0.942647	−	0.333792i	$-0.891672\pi$
$727$		−	18.0000i		−	0.667583i	0.942647	−	0.333792i	$-0.108328\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$	6.00000			0.221918
$732$		0			0
$733$			6.00000i			0.221615i	0.993842	+	0.110808i	$0.0353437\pi$
$733$			6.00000i			0.221615i	−0.993842	+	0.110808i	$0.964656\pi$
$734$		0			0
$735$	−6.00000	+	12.0000i	−0.221313	+	0.442627i
$736$		0			0
$737$			4.00000i			0.147342i
$738$		0			0
$739$	16.0000			0.588570			0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000			0.588570			0.294285	+	0.955718i	$0.404919\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	21.0000i		−	0.770415i	−0.922830	−	0.385208i	$-0.874130\pi$
$743$		−	21.0000i		−	0.770415i	0.922830	−	0.385208i	$-0.125870\pi$
$744$		0			0
$745$	−34.0000	−	17.0000i	−1.24566	−	0.622832i
$746$		0			0
$747$		−	32.0000i		−	1.17082i
$748$		0			0
$749$	−8.00000			−0.292314
$750$		0			0
$751$	−29.0000			−1.05823			−0.529113	−	0.848552i	$-0.677475\pi$
$751$	−29.0000			−1.05823			−0.529113	+	0.848552i	$0.677475\pi$
$752$		0			0
$753$			6.00000i			0.218652i
$754$		0			0
$755$	20.0000	+	10.0000i	0.727875	+	0.363937i
$756$		0			0
$757$			10.0000i			0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$			10.0000i			0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−20.0000			−0.724999			−0.362500	−	0.931984i	$-0.618077\pi$
$761$	−20.0000			−0.724999			−0.362500	+	0.931984i	$0.618077\pi$
$762$		0			0
$763$		−	10.0000i		−	0.362024i
$764$		0			0
$765$	−2.00000	+	4.00000i	−0.0723102	+	0.144620i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	22.0000			0.793340			0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000			0.793340			0.396670	+	0.917961i	$0.370166\pi$
$770$		0			0
$771$	−24.0000			−0.864339
$772$		0			0
$773$		−	45.0000i		−	1.61854i	−0.587439	−	0.809269i	$-0.699864\pi$
$773$		−	45.0000i		−	1.61854i	0.587439	−	0.809269i	$-0.300136\pi$
$774$		0			0
$775$	−3.00000	−	4.00000i	−0.107763	−	0.143684i
$776$		0			0
$777$		−	1.00000i		−	0.0358748i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−5.00000			−0.178914
$782$		0			0
$783$			5.00000i			0.178685i
$784$		0			0
$785$	−13.0000	+	26.0000i	−0.463990	+	0.927980i
$786$		0			0
$787$			30.0000i			1.06938i	0.845047	+	0.534692i	$0.179572\pi$
$787$			30.0000i			1.06938i	−0.845047	+	0.534692i	$0.820428\pi$
$788$		0			0
$789$	−21.0000			−0.747620
$790$		0			0
$791$	2.00000			0.0711118
$792$		0			0
$793$		0			0
$794$		0			0
$795$	18.0000	+	9.00000i	0.638394	+	0.319197i
$796$		0			0
$797$		−	54.0000i		−	1.91278i	−0.292096	−	0.956389i	$-0.594353\pi$
$797$		−	54.0000i		−	1.91278i	0.292096	−	0.956389i	$-0.405647\pi$
$798$		0			0
$799$	−8.00000			−0.283020
$800$		0			0
$801$	14.0000			0.494666
$802$		0			0
$803$			14.0000i			0.494049i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	8.00000i		−	0.281613i
$808$		0			0
$809$	−34.0000			−1.19538			−0.597688	−	0.801729i	$-0.703914\pi$
$809$	−34.0000			−1.19538			−0.597688	+	0.801729i	$0.703914\pi$
$810$		0			0
$811$	−17.0000			−0.596951			−0.298475	−	0.954417i	$-0.596478\pi$
$811$	−17.0000			−0.596951			−0.298475	+	0.954417i	$0.596478\pi$
$812$		0			0
$813$		−	6.00000i		−	0.210429i
$814$		0			0
$815$	19.0000	−	38.0000i	0.665541	−	1.33108i
$816$		0			0
$817$			6.00000i			0.209913i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	42.0000			1.46581			0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000			1.46581			0.732905	+	0.680331i	$0.238164\pi$
$822$		0			0
$823$		−	2.00000i		−	0.0697156i	−0.999392	−	0.0348578i	$-0.988902\pi$
$823$		−	2.00000i		−	0.0697156i	0.999392	−	0.0348578i	$-0.0110978\pi$
$824$		0			0
$825$	4.00000	−	3.00000i	0.139262	−	0.104447i
$826$		0			0
$827$		−	14.0000i		−	0.486828i	−0.969923	−	0.243414i	$-0.921733\pi$
$827$		−	14.0000i		−	0.486828i	0.969923	−	0.243414i	$-0.0782673\pi$
$828$		0			0
$829$	−4.00000			−0.138926			−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000			−0.138926			−0.0694629	+	0.997585i	$0.522129\pi$
$830$		0			0
$831$	8.00000			0.277517
$832$		0			0
$833$			6.00000i			0.207888i
$834$		0			0
$835$	3.00000	−	6.00000i	0.103819	−	0.207639i
$836$		0			0
$837$		−	5.00000i		−	0.172825i
$838$		0			0
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$		0			0
$841$	−28.0000			−0.965517
$842$		0			0
$843$		−	22.0000i		−	0.757720i
$844$		0			0
$845$	26.0000	+	13.0000i	0.894427	+	0.447214i
$846$		0			0
$847$			1.00000i			0.0343604i
$848$		0			0
$849$	−20.0000			−0.686398
$850$		0			0
$851$		0			0
$852$		0			0
$853$			16.0000i			0.547830i	0.961754	+	0.273915i	$0.0883186\pi$
$853$			16.0000i			0.547830i	−0.961754	+	0.273915i	$0.911681\pi$
$854$		0			0
$855$	−4.00000	−	2.00000i	−0.136797	−	0.0683986i
$856$		0			0
$857$			53.0000i			1.81045i	0.424937	+	0.905223i	$0.360296\pi$
$857$			53.0000i			1.81045i	−0.424937	+	0.905223i	$0.639704\pi$
$858$		0			0
$859$	−2.00000			−0.0682391			−0.0341196	−	0.999418i	$-0.510863\pi$
$859$	−2.00000			−0.0682391			−0.0341196	+	0.999418i	$0.510863\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	52.0000i		−	1.77010i	−0.465495	−	0.885050i	$-0.654124\pi$
$863$		−	52.0000i		−	1.77010i	0.465495	−	0.885050i	$-0.345876\pi$
$864$		0			0
$865$	−22.0000	+	44.0000i	−0.748022	+	1.49604i
$866$		0			0
$867$			16.0000i			0.543388i
$868$		0			0
$869$	4.00000			0.135691
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	32.0000i		−	1.08304i
$874$		0			0
$875$	−11.0000	+	2.00000i	−0.371868	+	0.0676123i
$876$		0			0
$877$		−	38.0000i		−	1.28317i	−0.767052	−	0.641584i	$-0.778277\pi$
$877$		−	38.0000i		−	1.28317i	0.767052	−	0.641584i	$-0.221723\pi$
$878$		0			0
$879$	16.0000			0.539667
$880$		0			0
$881$	−10.0000			−0.336909			−0.168454	−	0.985709i	$-0.553878\pi$
$881$	−10.0000			−0.336909			−0.168454	+	0.985709i	$0.553878\pi$
$882$		0			0
$883$		−	41.0000i		−	1.37976i	−0.723924	−	0.689880i	$-0.757663\pi$
$883$		−	41.0000i		−	1.37976i	0.723924	−	0.689880i	$-0.242337\pi$
$884$		0			0
$885$	4.00000	−	8.00000i	0.134459	−	0.268917i
$886$		0			0
$887$			48.0000i			1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$			48.0000i			1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$		0			0
$889$	16.0000			0.536623
$890$		0			0
$891$	−1.00000			−0.0335013
$892$		0			0
$893$		−	8.00000i		−	0.267710i
$894$		0			0
$895$	20.0000	+	10.0000i	0.668526	+	0.334263i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.00000			−0.0333519
$900$		0			0
$901$	9.00000			0.299833
$902$		0			0
$903$			6.00000i			0.199667i
$904$		0			0
$905$	−44.0000	−	22.0000i	−1.46261	−	0.731305i
$906$		0			0
$907$			7.00000i			0.232431i	0.993224	+	0.116216i	$0.0370764\pi$
$907$			7.00000i			0.232431i	−0.993224	+	0.116216i	$0.962924\pi$
$908$		0			0
$909$	−20.0000			−0.663358
$910$		0			0
$911$	−27.0000			−0.894550			−0.447275	−	0.894397i	$-0.647605\pi$
$911$	−27.0000			−0.894550			−0.447275	+	0.894397i	$0.647605\pi$
$912$		0			0
$913$			16.0000i			0.529523i
$914$		0			0
$915$	7.00000	−	14.0000i	0.231413	−	0.462826i
$916$		0			0
$917$			21.0000i			0.693481i
$918$		0			0
$919$	26.0000			0.857661			0.428830	−	0.903385i	$-0.358926\pi$
$919$	26.0000			0.857661			0.428830	+	0.903385i	$0.358926\pi$
$920$		0			0
$921$	16.0000			0.527218
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−4.00000	+	3.00000i	−0.131519	+	0.0986394i
$926$		0			0
$927$		−	16.0000i		−	0.525509i
$928$		0			0
$929$	−15.0000			−0.492134			−0.246067	−	0.969253i	$-0.579138\pi$
$929$	−15.0000			−0.492134			−0.246067	+	0.969253i	$0.579138\pi$
$930$		0			0
$931$	−6.00000			−0.196642
$932$		0			0
$933$		−	13.0000i		−	0.425601i
$934$		0			0
$935$	1.00000	−	2.00000i	0.0327035	−	0.0654070i
$936$		0			0
$937$			10.0000i			0.326686i	0.986569	+	0.163343i	$0.0522277\pi$
$937$			10.0000i			0.326686i	−0.986569	+	0.163343i	$0.947772\pi$
$938$		0			0
$939$	−18.0000			−0.587408
$940$		0			0
$941$	27.0000			0.880175			0.440087	−	0.897955i	$-0.354947\pi$
$941$	27.0000			0.880175			0.440087	+	0.897955i	$0.354947\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−10.0000	−	5.00000i	−0.325300	−	0.162650i
$946$		0			0
$947$		−	35.0000i		−	1.13735i	−0.822563	−	0.568674i	$-0.807457\pi$
$947$		−	35.0000i		−	1.13735i	0.822563	−	0.568674i	$-0.192543\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	1.00000			0.0324272
$952$		0			0
$953$		−	39.0000i		−	1.26333i	−0.775240	−	0.631667i	$-0.782371\pi$
$953$		−	39.0000i		−	1.26333i	0.775240	−	0.631667i	$-0.217629\pi$
$954$		0			0
$955$	48.0000	+	24.0000i	1.55324	+	0.776622i
$956$		0			0
$957$		−	1.00000i		−	0.0323254i
$958$		0			0
$959$	−12.0000			−0.387500
$960$		0			0
$961$	−30.0000			−0.967742
$962$		0			0
$963$			16.0000i			0.515593i
$964$		0			0
$965$	13.0000	−	26.0000i	0.418485	−	0.836970i
$966$		0			0
$967$		−	25.0000i		−	0.803946i	−0.915652	−	0.401973i	$-0.868325\pi$
$967$		−	25.0000i		−	0.803946i	0.915652	−	0.401973i	$-0.131675\pi$
$968$		0			0
$969$	1.00000			0.0321246
$970$		0			0
$971$	−48.0000			−1.54039			−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000			−1.54039			−0.770197	+	0.637806i	$0.779842\pi$
$972$		0			0
$973$		−	12.0000i		−	0.384702i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$	−7.00000			−0.223721
$980$		0			0
$981$	−20.0000			−0.638551
$982$		0			0
$983$		−	42.0000i		−	1.33959i	−0.742545	−	0.669796i	$-0.766382\pi$
$983$		−	42.0000i		−	1.33959i	0.742545	−	0.669796i	$-0.233618\pi$
$984$		0			0
$985$	−18.0000	+	36.0000i	−0.573528	+	1.14706i
$986$		0			0
$987$		−	8.00000i		−	0.254643i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	24.0000			0.762385			0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000			0.762385			0.381193	+	0.924496i	$0.375513\pi$
$992$		0			0
$993$		−	34.0000i		−	1.07896i
$994$		0			0
$995$	−34.0000	−	17.0000i	−1.07787	−	0.538936i
$996$		0			0
$997$			10.0000i			0.316703i	0.987383	+	0.158352i	$0.0506179\pi$
$997$			10.0000i			0.316703i	−0.987383	+	0.158352i	$0.949382\pi$
$998$		0			0
$999$	−5.00000			−0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	440.2.b.c.89.2	yes	2
3.2	odd	2		3960.2.d.a.3169.1		2
4.3	odd	2		880.2.b.g.529.1		2
5.2	odd	4		2200.2.a.h.1.1		1
5.3	odd	4		2200.2.a.d.1.1		1
5.4	even	2	inner	440.2.b.c.89.1	✓	2
15.14	odd	2		3960.2.d.a.3169.2		2
20.3	even	4		4400.2.a.u.1.1		1
20.7	even	4		4400.2.a.j.1.1		1
20.19	odd	2		880.2.b.g.529.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.b.c.89.1	✓	2	5.4	even	2	inner
440.2.b.c.89.2	yes	2	1.1	even	1	trivial
880.2.b.g.529.1		2	4.3	odd	2
880.2.b.g.529.2		2	20.19	odd	2
2200.2.a.d.1.1		1	5.3	odd	4
2200.2.a.h.1.1		1	5.2	odd	4
3960.2.d.a.3169.1		2	3.2	odd	2
3960.2.d.a.3169.2		2	15.14	odd	2
4400.2.a.j.1.1		1	20.7	even	4
4400.2.a.u.1.1		1	20.3	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:	\( N \)	\(=\)	\( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	440.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(2.00000 + 1.00000i) q^{5} +1.00000i q^{7} +2.00000 q^{9} -1.00000 q^{11} +(-1.00000 + 2.00000i) q^{15} +1.00000i q^{17} -1.00000 q^{19} -1.00000 q^{21} +(3.00000 + 4.00000i) q^{25} +5.00000i q^{27} +1.00000 q^{29} -1.00000 q^{31} -1.00000i q^{33} +(-1.00000 + 2.00000i) q^{35} +1.00000i q^{37} -6.00000i q^{43} +(4.00000 + 2.00000i) q^{45} +8.00000i q^{47} +6.00000 q^{49} -1.00000 q^{51} -9.00000i q^{53} +(-2.00000 - 1.00000i) q^{55} -1.00000i q^{57} -4.00000 q^{59} -7.00000 q^{61} +2.00000i q^{63} -4.00000i q^{67} +5.00000 q^{71} -14.0000i q^{73} +(-4.00000 + 3.00000i) q^{75} -1.00000i q^{77} -4.00000 q^{79} +1.00000 q^{81} -16.0000i q^{83} +(-1.00000 + 2.00000i) q^{85} +1.00000i q^{87} +7.00000 q^{89} -1.00000i q^{93} +(-2.00000 - 1.00000i) q^{95} -16.0000i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} + 4 q^{9} - 2 q^{11} - 2 q^{15} - 2 q^{19} - 2 q^{21} + 6 q^{25} + 2 q^{29} - 2 q^{31} - 2 q^{35} + 8 q^{45} + 12 q^{49} - 2 q^{51} - 4 q^{55} - 8 q^{59} - 14 q^{61} + 10 q^{71} - 8 q^{75}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q + 4 * q^5 + 4 * q^9 - 2 * q^11 - 2 * q^15 - 2 * q^19 - 2 * q^21 + 6 * q^25 + 2 * q^29 - 2 * q^31 - 2 * q^35 + 8 * q^45 + 12 * q^49 - 2 * q^51 - 4 * q^55 - 8 * q^59 - 14 * q^61 + 10 * q^71 - 8 * q^75 - 8 * q^79 + 2 * q^81 - 2 * q^85 + 14 * q^89 - 4 * q^95 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more