LMFDB - Embedded newform 3520.2.f.a.2111.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3520,2,Mod(2111,3520)] 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("3520.2111"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3520 = 2^{6} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3520.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,2,0,-8,0,2,0,6,0,0,0,0,0,0,0,-4,0,0,0,0,0,2,0,0,0,0, 0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(33)] == traces)

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

Self dual:	no
Analytic conductor:	$28.1073415115$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 220)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2111.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	3520.2111
Dual form			3520.2.f.a.2111.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{3} +1.00000 q^{5} -4.00000 q^{7} +1.00000 q^{9} +(3.00000 + 1.41421i) q^{11} +4.24264i q^{13} +1.41421i q^{15} +1.41421i q^{17} -2.00000 q^{19} -5.65685i q^{21} -7.07107i q^{23} +1.00000 q^{25} +5.65685i q^{27} +2.82843i q^{29} +8.48528i q^{31} +(-2.00000 + 4.24264i) q^{33} -4.00000 q^{35} +10.0000 q^{37} -6.00000 q^{39} -2.82843i q^{41} -8.00000 q^{43} +1.00000 q^{45} -7.07107i q^{47} +9.00000 q^{49} -2.00000 q^{51} -6.00000 q^{53} +(3.00000 + 1.41421i) q^{55} -2.82843i q^{57} +2.82843i q^{59} +8.48528i q^{61} -4.00000 q^{63} +4.24264i q^{65} -4.24264i q^{67} +10.0000 q^{69} +5.65685i q^{71} +4.24264i q^{73} +1.41421i q^{75} +(-12.0000 - 5.65685i) q^{77} +2.00000 q^{79} -5.00000 q^{81} -12.0000 q^{83} +1.41421i q^{85} -4.00000 q^{87} -16.9706i q^{91} -12.0000 q^{93} -2.00000 q^{95} +2.00000 q^{97} +(3.00000 + 1.41421i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 8 q^{7} + 2 q^{9} + 6 q^{11} - 4 q^{19} + 2 q^{25} - 4 q^{33} - 8 q^{35} + 20 q^{37} - 12 q^{39} - 16 q^{43} + 2 q^{45} + 18 q^{49} - 4 q^{51} - 12 q^{53} + 6 q^{55} - 8 q^{63} + 20 q^{69}+ \cdots + 6 q^{99}+O(q^{100}) $ 2 * q + 2 * q^5 - 8 * q^7 + 2 * q^9 + 6 * q^11 - 4 * q^19 + 2 * q^25 - 4 * q^33 - 8 * q^35 + 20 * q^37 - 12 * q^39 - 16 * q^43 + 2 * q^45 + 18 * q^49 - 4 * q^51 - 12 * q^53 + 6 * q^55 - 8 * q^63 + 20 * q^69 - 24 * q^77 + 4 * q^79 - 10 * q^81 - 24 * q^83 - 8 * q^87 - 24 * q^93 - 4 * q^95 + 4 * q^97 + 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3520\mathbb{Z}\right)^\times$.

$n$	$321$	$1541$	$2751$	$2817$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$			1.41421i			0.816497i	0.912871	+	0.408248i	$0.133860\pi$
$3$			1.41421i			0.816497i	−0.912871	+	0.408248i	$0.866140\pi$
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	−4.00000			−1.51186			−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000			−1.51186			−0.755929	+	0.654654i	$0.772814\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$	3.00000	+	1.41421i	0.904534	+	0.426401i
$11$	3.00000	+	1.41421i	0.904534	+	0.426401i
$12$		0			0
$13$			4.24264i			1.17670i	0.808608	+	0.588348i	$0.200222\pi$
$13$			4.24264i			1.17670i	−0.808608	+	0.588348i	$0.799778\pi$
$14$		0			0
$15$			1.41421i			0.365148i
$16$		0			0
$17$			1.41421i			0.342997i	0.985184	+	0.171499i	$0.0548609\pi$
$17$			1.41421i			0.342997i	−0.985184	+	0.171499i	$0.945139\pi$
$18$		0			0
$19$	−2.00000			−0.458831			−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000			−0.458831			−0.229416	+	0.973329i	$0.573682\pi$
$20$		0			0
$21$		−	5.65685i		−	1.23443i
$22$		0			0
$23$		−	7.07107i		−	1.47442i	−0.675664	−	0.737210i	$-0.736143\pi$
$23$		−	7.07107i		−	1.47442i	0.675664	−	0.737210i	$-0.263857\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$			5.65685i			1.08866i
$28$		0			0
$29$			2.82843i			0.525226i	0.964901	+	0.262613i	$0.0845842\pi$
$29$			2.82843i			0.525226i	−0.964901	+	0.262613i	$0.915416\pi$
$30$		0			0
$31$			8.48528i			1.52400i	0.647576	+	0.762001i	$0.275783\pi$
$31$			8.48528i			1.52400i	−0.647576	+	0.762001i	$0.724217\pi$
$32$		0			0
$33$	−2.00000	+	4.24264i	−0.348155	+	0.738549i
$34$		0			0
$35$	−4.00000			−0.676123
$36$		0			0
$37$	10.0000			1.64399			0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000			1.64399			0.821995	+	0.569495i	$0.192861\pi$
$38$		0			0
$39$	−6.00000			−0.960769
$40$		0			0
$41$		−	2.82843i		−	0.441726i	−0.975305	−	0.220863i	$-0.929113\pi$
$41$		−	2.82843i		−	0.441726i	0.975305	−	0.220863i	$-0.0708874\pi$
$42$		0			0
$43$	−8.00000			−1.21999			−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000			−1.21999			−0.609994	+	0.792406i	$0.708828\pi$
$44$		0			0
$45$	1.00000			0.149071
$46$		0			0
$47$		−	7.07107i		−	1.03142i	−0.856763	−	0.515711i	$-0.827528\pi$
$47$		−	7.07107i		−	1.03142i	0.856763	−	0.515711i	$-0.172472\pi$
$48$		0			0
$49$	9.00000			1.28571
$50$		0			0
$51$	−2.00000			−0.280056
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	3.00000	+	1.41421i	0.404520	+	0.190693i
$56$		0			0
$57$		−	2.82843i		−	0.374634i
$58$		0			0
$59$			2.82843i			0.368230i	0.982905	+	0.184115i	$0.0589419\pi$
$59$			2.82843i			0.368230i	−0.982905	+	0.184115i	$0.941058\pi$
$60$		0			0
$61$			8.48528i			1.08643i	0.839594	+	0.543214i	$0.182793\pi$
$61$			8.48528i			1.08643i	−0.839594	+	0.543214i	$0.817207\pi$
$62$		0			0
$63$	−4.00000			−0.503953
$64$		0			0
$65$			4.24264i			0.526235i
$66$		0			0
$67$		−	4.24264i		−	0.518321i	−0.965834	−	0.259161i	$-0.916554\pi$
$67$		−	4.24264i		−	0.518321i	0.965834	−	0.259161i	$-0.0834459\pi$
$68$		0			0
$69$	10.0000			1.20386
$70$		0			0
$71$			5.65685i			0.671345i	0.941979	+	0.335673i	$0.108964\pi$
$71$			5.65685i			0.671345i	−0.941979	+	0.335673i	$0.891036\pi$
$72$		0			0
$73$			4.24264i			0.496564i	0.968688	+	0.248282i	$0.0798659\pi$
$73$			4.24264i			0.496564i	−0.968688	+	0.248282i	$0.920134\pi$
$74$		0			0
$75$			1.41421i			0.163299i
$76$		0			0
$77$	−12.0000	−	5.65685i	−1.36753	−	0.644658i
$78$		0			0
$79$	2.00000			0.225018			0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000			0.225018			0.112509	+	0.993651i	$0.464111\pi$
$80$		0			0
$81$	−5.00000			−0.555556
$82$		0			0
$83$	−12.0000			−1.31717			−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000			−1.31717			−0.658586	+	0.752506i	$0.728845\pi$
$84$		0			0
$85$			1.41421i			0.153393i
$86$		0			0
$87$	−4.00000			−0.428845
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		−	16.9706i		−	1.77900i
$92$		0			0
$93$	−12.0000			−1.24434
$94$		0			0
$95$	−2.00000			−0.205196
$96$		0			0
$97$	2.00000			0.203069			0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000			0.203069			0.101535	+	0.994832i	$0.467625\pi$
$98$		0			0
$99$	3.00000	+	1.41421i	0.301511	+	0.142134i
$100$		0			0
$101$			11.3137i			1.12576i	0.826540	+	0.562878i	$0.190306\pi$
$101$			11.3137i			1.12576i	−0.826540	+	0.562878i	$0.809694\pi$
$102$		0			0
$103$			12.7279i			1.25412i	0.778971	+	0.627060i	$0.215742\pi$
$103$			12.7279i			1.25412i	−0.778971	+	0.627060i	$0.784258\pi$
$104$		0			0
$105$		−	5.65685i		−	0.552052i
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$		−	16.9706i		−	1.62549i	−0.582623	−	0.812743i	$-0.697974\pi$
$109$		−	16.9706i		−	1.62549i	0.582623	−	0.812743i	$-0.302026\pi$
$110$		0			0
$111$			14.1421i			1.34231i
$112$		0			0
$113$	−18.0000			−1.69330			−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000			−1.69330			−0.846649	+	0.532152i	$0.821383\pi$
$114$		0			0
$115$		−	7.07107i		−	0.659380i
$116$		0			0
$117$			4.24264i			0.392232i
$118$		0			0
$119$		−	5.65685i		−	0.518563i
$120$		0			0
$121$	7.00000	+	8.48528i	0.636364	+	0.771389i
$122$		0			0
$123$	4.00000			0.360668
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−16.0000			−1.41977			−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000			−1.41977			−0.709885	+	0.704317i	$0.751253\pi$
$128$		0			0
$129$		−	11.3137i		−	0.996116i
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$	8.00000			0.693688
$134$		0			0
$135$			5.65685i			0.486864i
$136$		0			0
$137$	−6.00000			−0.512615			−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000			−0.512615			−0.256307	+	0.966595i	$0.582506\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$	10.0000			0.842152
$142$		0			0
$143$	−6.00000	+	12.7279i	−0.501745	+	1.06436i
$144$		0			0
$145$			2.82843i			0.234888i
$146$		0			0
$147$			12.7279i			1.04978i
$148$		0			0
$149$			11.3137i			0.926855i	0.886135	+	0.463428i	$0.153381\pi$
$149$			11.3137i			0.926855i	−0.886135	+	0.463428i	$0.846619\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$			1.41421i			0.114332i
$154$		0			0
$155$			8.48528i			0.681554i
$156$		0			0
$157$	−14.0000			−1.11732			−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000			−1.11732			−0.558661	+	0.829396i	$0.688685\pi$
$158$		0			0
$159$		−	8.48528i		−	0.672927i
$160$		0			0
$161$			28.2843i			2.22911i
$162$		0			0
$163$			21.2132i			1.66155i	0.556611	+	0.830773i	$0.312101\pi$
$163$			21.2132i			1.66155i	−0.556611	+	0.830773i	$0.687899\pi$
$164$		0			0
$165$	−2.00000	+	4.24264i	−0.155700	+	0.330289i
$166$		0			0
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	−5.00000			−0.384615
$170$		0			0
$171$	−2.00000			−0.152944
$172$		0			0
$173$			15.5563i			1.18273i	0.806405	+	0.591364i	$0.201410\pi$
$173$			15.5563i			1.18273i	−0.806405	+	0.591364i	$0.798590\pi$
$174$		0			0
$175$	−4.00000			−0.302372
$176$		0			0
$177$	−4.00000			−0.300658
$178$		0			0
$179$			11.3137i			0.845626i	0.906217	+	0.422813i	$0.138957\pi$
$179$			11.3137i			0.845626i	−0.906217	+	0.422813i	$0.861043\pi$
$180$		0			0
$181$	−8.00000			−0.594635			−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000			−0.594635			−0.297318	+	0.954779i	$0.596092\pi$
$182$		0			0
$183$	−12.0000			−0.887066
$184$		0			0
$185$	10.0000			0.735215
$186$		0			0
$187$	−2.00000	+	4.24264i	−0.146254	+	0.310253i
$188$		0			0
$189$		−	22.6274i		−	1.64590i
$190$		0			0
$191$		−	19.7990i		−	1.43260i	−0.697790	−	0.716302i	$-0.745833\pi$
$191$		−	19.7990i		−	1.43260i	0.697790	−	0.716302i	$-0.254167\pi$
$192$		0			0
$193$		−	4.24264i		−	0.305392i	−0.988273	−	0.152696i	$-0.951204\pi$
$193$		−	4.24264i		−	0.305392i	0.988273	−	0.152696i	$-0.0487955\pi$
$194$		0			0
$195$	−6.00000			−0.429669
$196$		0			0
$197$		−	9.89949i		−	0.705310i	−0.935753	−	0.352655i	$-0.885279\pi$
$197$		−	9.89949i		−	0.705310i	0.935753	−	0.352655i	$-0.114721\pi$
$198$		0			0
$199$			8.48528i			0.601506i	0.953702	+	0.300753i	$0.0972379\pi$
$199$			8.48528i			0.601506i	−0.953702	+	0.300753i	$0.902762\pi$
$200$		0			0
$201$	6.00000			0.423207
$202$		0			0
$203$		−	11.3137i		−	0.794067i
$204$		0			0
$205$		−	2.82843i		−	0.197546i
$206$		0			0
$207$		−	7.07107i		−	0.491473i
$208$		0			0
$209$	−6.00000	−	2.82843i	−0.415029	−	0.195646i
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$	−8.00000			−0.548151
$214$		0			0
$215$	−8.00000			−0.545595
$216$		0			0
$217$		−	33.9411i		−	2.30407i
$218$		0			0
$219$	−6.00000			−0.405442
$220$		0			0
$221$	−6.00000			−0.403604
$222$		0			0
$223$		−	4.24264i		−	0.284108i	−0.989859	−	0.142054i	$-0.954629\pi$
$223$		−	4.24264i		−	0.284108i	0.989859	−	0.142054i	$-0.0453707\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−12.0000			−0.796468			−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000			−0.796468			−0.398234	+	0.917284i	$0.630377\pi$
$228$		0			0
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$		0			0
$231$	8.00000	−	16.9706i	0.526361	−	1.11658i
$232$		0			0
$233$		−	7.07107i		−	0.463241i	−0.972806	−	0.231621i	$-0.925597\pi$
$233$		−	7.07107i		−	0.463241i	0.972806	−	0.231621i	$-0.0744028\pi$
$234$		0			0
$235$		−	7.07107i		−	0.461266i
$236$		0			0
$237$			2.82843i			0.183726i
$238$		0			0
$239$	−18.0000			−1.16432			−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000			−1.16432			−0.582162	+	0.813073i	$0.697793\pi$
$240$		0			0
$241$		−	8.48528i		−	0.546585i	−0.961931	−	0.273293i	$-0.911887\pi$
$241$		−	8.48528i		−	0.546585i	0.961931	−	0.273293i	$-0.0881127\pi$
$242$		0			0
$243$			9.89949i			0.635053i
$244$		0			0
$245$	9.00000			0.574989
$246$		0			0
$247$		−	8.48528i		−	0.539906i
$248$		0			0
$249$		−	16.9706i		−	1.07547i
$250$		0			0
$251$		−	5.65685i		−	0.357057i	−0.983935	−	0.178529i	$-0.942866\pi$
$251$		−	5.65685i		−	0.357057i	0.983935	−	0.178529i	$-0.0571337\pi$
$252$		0			0
$253$	10.0000	−	21.2132i	0.628695	−	1.33366i
$254$		0			0
$255$	−2.00000			−0.125245
$256$		0			0
$257$	6.00000			0.374270			0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000			0.374270			0.187135	+	0.982334i	$0.440080\pi$
$258$		0			0
$259$	−40.0000			−2.48548
$260$		0			0
$261$			2.82843i			0.175075i
$262$		0			0
$263$	−24.0000			−1.47990			−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000			−1.47990			−0.739952	+	0.672660i	$0.765152\pi$
$264$		0			0
$265$	−6.00000			−0.368577
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	8.00000			0.485965			0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$		0			0
$273$	24.0000			1.45255
$274$		0			0
$275$	3.00000	+	1.41421i	0.180907	+	0.0852803i
$276$		0			0
$277$		−	21.2132i		−	1.27458i	−0.770625	−	0.637289i	$-0.780056\pi$
$277$		−	21.2132i		−	1.27458i	0.770625	−	0.637289i	$-0.219944\pi$
$278$		0			0
$279$			8.48528i			0.508001i
$280$		0			0
$281$		−	19.7990i		−	1.18111i	−0.806998	−	0.590554i	$-0.798909\pi$
$281$		−	19.7990i		−	1.18111i	0.806998	−	0.590554i	$-0.201091\pi$
$282$		0			0
$283$	4.00000			0.237775			0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000			0.237775			0.118888	+	0.992908i	$0.462067\pi$
$284$		0			0
$285$		−	2.82843i		−	0.167542i
$286$		0			0
$287$			11.3137i			0.667827i
$288$		0			0
$289$	15.0000			0.882353
$290$		0			0
$291$			2.82843i			0.165805i
$292$		0			0
$293$		−	26.8701i		−	1.56977i	−0.619644	−	0.784883i	$-0.712723\pi$
$293$		−	26.8701i		−	1.56977i	0.619644	−	0.784883i	$-0.287277\pi$
$294$		0			0
$295$			2.82843i			0.164677i
$296$		0			0
$297$	−8.00000	+	16.9706i	−0.464207	+	0.984732i
$298$		0			0
$299$	30.0000			1.73494
$300$		0			0
$301$	32.0000			1.84445
$302$		0			0
$303$	−16.0000			−0.919176
$304$		0			0
$305$			8.48528i			0.485866i
$306$		0			0
$307$	16.0000			0.913168			0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000			0.913168			0.456584	+	0.889680i	$0.349073\pi$
$308$		0			0
$309$	−18.0000			−1.02398
$310$		0			0
$311$		−	11.3137i		−	0.641542i	−0.947157	−	0.320771i	$-0.896058\pi$
$311$		−	11.3137i		−	0.641542i	0.947157	−	0.320771i	$-0.103942\pi$
$312$		0			0
$313$	−10.0000			−0.565233			−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000			−0.565233			−0.282617	+	0.959233i	$0.591202\pi$
$314$		0			0
$315$	−4.00000			−0.225374
$316$		0			0
$317$	6.00000			0.336994			0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000			0.336994			0.168497	+	0.985702i	$0.446109\pi$
$318$		0			0
$319$	−4.00000	+	8.48528i	−0.223957	+	0.475085i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	2.82843i		−	0.157378i
$324$		0			0
$325$			4.24264i			0.235339i
$326$		0			0
$327$	24.0000			1.32720
$328$		0			0
$329$			28.2843i			1.55936i
$330$		0			0
$331$			25.4558i			1.39918i	0.714545	+	0.699590i	$0.246634\pi$
$331$			25.4558i			1.39918i	−0.714545	+	0.699590i	$0.753366\pi$
$332$		0			0
$333$	10.0000			0.547997
$334$		0			0
$335$		−	4.24264i		−	0.231800i
$336$		0			0
$337$		−	4.24264i		−	0.231111i	−0.993301	−	0.115556i	$-0.963135\pi$
$337$		−	4.24264i		−	0.231111i	0.993301	−	0.115556i	$-0.0368649\pi$
$338$		0			0
$339$		−	25.4558i		−	1.38257i
$340$		0			0
$341$	−12.0000	+	25.4558i	−0.649836	+	1.37851i
$342$		0			0
$343$	−8.00000			−0.431959
$344$		0			0
$345$	10.0000			0.538382
$346$		0			0
$347$	12.0000			0.644194			0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000			0.644194			0.322097	+	0.946707i	$0.395612\pi$
$348$		0			0
$349$			25.4558i			1.36262i	0.731995	+	0.681310i	$0.238589\pi$
$349$			25.4558i			1.36262i	−0.731995	+	0.681310i	$0.761411\pi$
$350$		0			0
$351$	−24.0000			−1.28103
$352$		0			0
$353$	6.00000			0.319348			0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000			0.319348			0.159674	+	0.987170i	$0.448956\pi$
$354$		0			0
$355$			5.65685i			0.300235i
$356$		0			0
$357$	8.00000			0.423405
$358$		0			0
$359$	18.0000			0.950004			0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000			0.950004			0.475002	+	0.879985i	$0.342447\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$	−12.0000	+	9.89949i	−0.629837	+	0.519589i
$364$		0			0
$365$			4.24264i			0.222070i
$366$		0			0
$367$			29.6985i			1.55025i	0.631809	+	0.775124i	$0.282313\pi$
$367$			29.6985i			1.55025i	−0.631809	+	0.775124i	$0.717687\pi$
$368$		0			0
$369$		−	2.82843i		−	0.147242i
$370$		0			0
$371$	24.0000			1.24602
$372$		0			0
$373$			21.2132i			1.09838i	0.835698	+	0.549189i	$0.185063\pi$
$373$			21.2132i			1.09838i	−0.835698	+	0.549189i	$0.814937\pi$
$374$		0			0
$375$			1.41421i			0.0730297i
$376$		0			0
$377$	−12.0000			−0.618031
$378$		0			0
$379$		−	8.48528i		−	0.435860i	−0.975964	−	0.217930i	$-0.930070\pi$
$379$		−	8.48528i		−	0.435860i	0.975964	−	0.217930i	$-0.0699304\pi$
$380$		0			0
$381$		−	22.6274i		−	1.15924i
$382$		0			0
$383$			26.8701i			1.37300i	0.727132	+	0.686498i	$0.240853\pi$
$383$			26.8701i			1.37300i	−0.727132	+	0.686498i	$0.759147\pi$
$384$		0			0
$385$	−12.0000	−	5.65685i	−0.611577	−	0.288300i
$386$		0			0
$387$	−8.00000			−0.406663
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$	10.0000			0.505722
$392$		0			0
$393$		−	16.9706i		−	0.856052i
$394$		0			0
$395$	2.00000			0.100631
$396$		0			0
$397$	−2.00000			−0.100377			−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000			−0.100377			−0.0501886	+	0.998740i	$0.515982\pi$
$398$		0			0
$399$			11.3137i			0.566394i
$400$		0			0
$401$	−12.0000			−0.599251			−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000			−0.599251			−0.299626	+	0.954057i	$0.596862\pi$
$402$		0			0
$403$	−36.0000			−1.79329
$404$		0			0
$405$	−5.00000			−0.248452
$406$		0			0
$407$	30.0000	+	14.1421i	1.48704	+	0.701000i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		−	8.48528i		−	0.418548i
$412$		0			0
$413$		−	11.3137i		−	0.556711i
$414$		0			0
$415$	−12.0000			−0.589057
$416$		0			0
$417$			5.65685i			0.277017i
$418$		0			0
$419$			11.3137i			0.552711i	0.961056	+	0.276355i	$0.0891267\pi$
$419$			11.3137i			0.552711i	−0.961056	+	0.276355i	$0.910873\pi$
$420$		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
$423$		−	7.07107i		−	0.343807i
$424$		0			0
$425$			1.41421i			0.0685994i
$426$		0			0
$427$		−	33.9411i		−	1.64253i
$428$		0			0
$429$	−18.0000	−	8.48528i	−0.869048	−	0.409673i
$430$		0			0
$431$	18.0000			0.867029			0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000			0.867029			0.433515	+	0.901146i	$0.357273\pi$
$432$		0			0
$433$	38.0000			1.82616			0.913082	−	0.407777i	$-0.133696\pi$
$433$	38.0000			1.82616			0.913082	+	0.407777i	$0.133696\pi$
$434$		0			0
$435$	−4.00000			−0.191785
$436$		0			0
$437$			14.1421i			0.676510i
$438$		0			0
$439$	32.0000			1.52728			0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000			1.52728			0.763638	+	0.645644i	$0.223411\pi$
$440$		0			0
$441$	9.00000			0.428571
$442$		0			0
$443$		−	9.89949i		−	0.470339i	−0.971954	−	0.235170i	$-0.924435\pi$
$443$		−	9.89949i		−	0.470339i	0.971954	−	0.235170i	$-0.0755646\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−16.0000			−0.756774
$448$		0			0
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$		0			0
$451$	4.00000	−	8.48528i	0.188353	−	0.399556i
$452$		0			0
$453$			11.3137i			0.531564i
$454$		0			0
$455$		−	16.9706i		−	0.795592i
$456$		0			0
$457$			29.6985i			1.38924i	0.719379	+	0.694618i	$0.244427\pi$
$457$			29.6985i			1.38924i	−0.719379	+	0.694618i	$0.755573\pi$
$458$		0			0
$459$	−8.00000			−0.373408
$460$		0			0
$461$		−	22.6274i		−	1.05386i	−0.849907	−	0.526932i	$-0.823342\pi$
$461$		−	22.6274i		−	1.05386i	0.849907	−	0.526932i	$-0.176658\pi$
$462$		0			0
$463$			12.7279i			0.591517i	0.955263	+	0.295758i	$0.0955723\pi$
$463$			12.7279i			0.591517i	−0.955263	+	0.295758i	$0.904428\pi$
$464$		0			0
$465$	−12.0000			−0.556487
$466$		0			0
$467$		−	18.3848i		−	0.850746i	−0.905018	−	0.425373i	$-0.860143\pi$
$467$		−	18.3848i		−	0.850746i	0.905018	−	0.425373i	$-0.139857\pi$
$468$		0			0
$469$			16.9706i			0.783628i
$470$		0			0
$471$		−	19.7990i		−	0.912289i
$472$		0			0
$473$	−24.0000	−	11.3137i	−1.10352	−	0.520205i
$474$		0			0
$475$	−2.00000			−0.0917663
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$	6.00000			0.274147			0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000			0.274147			0.137073	+	0.990561i	$0.456230\pi$
$480$		0			0
$481$			42.4264i			1.93448i
$482$		0			0
$483$	−40.0000			−1.82006
$484$		0			0
$485$	2.00000			0.0908153
$486$		0			0
$487$		−	4.24264i		−	0.192252i	−0.995369	−	0.0961262i	$-0.969355\pi$
$487$		−	4.24264i		−	0.192252i	0.995369	−	0.0961262i	$-0.0306452\pi$
$488$		0			0
$489$	−30.0000			−1.35665
$490$		0			0
$491$	30.0000			1.35388			0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000			1.35388			0.676941	+	0.736038i	$0.263305\pi$
$492$		0			0
$493$	−4.00000			−0.180151
$494$		0			0
$495$	3.00000	+	1.41421i	0.134840	+	0.0635642i
$496$		0			0
$497$		−	22.6274i		−	1.01498i
$498$		0			0
$499$			8.48528i			0.379853i	0.981798	+	0.189927i	$0.0608250\pi$
$499$			8.48528i			0.379853i	−0.981798	+	0.189927i	$0.939175\pi$
$500$		0			0
$501$			16.9706i			0.758189i
$502$		0			0
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		0			0
$505$			11.3137i			0.503453i
$506$		0			0
$507$		−	7.07107i		−	0.314037i
$508$		0			0
$509$	−30.0000			−1.32973			−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000			−1.32973			−0.664863	+	0.746965i	$0.731510\pi$
$510$		0			0
$511$		−	16.9706i		−	0.750733i
$512$		0			0
$513$		−	11.3137i		−	0.499512i
$514$		0			0
$515$			12.7279i			0.560859i
$516$		0			0
$517$	10.0000	−	21.2132i	0.439799	−	0.932956i
$518$		0			0
$519$	−22.0000			−0.965693
$520$		0			0
$521$	36.0000			1.57719			0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000			1.57719			0.788594	+	0.614914i	$0.210809\pi$
$522$		0			0
$523$	4.00000			0.174908			0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000			0.174908			0.0874539	+	0.996169i	$0.472127\pi$
$524$		0			0
$525$		−	5.65685i		−	0.246885i
$526$		0			0
$527$	−12.0000			−0.522728
$528$		0			0
$529$	−27.0000			−1.17391
$530$		0			0
$531$			2.82843i			0.122743i
$532$		0			0
$533$	12.0000			0.519778
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−16.0000			−0.690451
$538$		0			0
$539$	27.0000	+	12.7279i	1.16297	+	0.548230i
$540$		0			0
$541$			33.9411i			1.45924i	0.683851	+	0.729621i	$0.260304\pi$
$541$			33.9411i			1.45924i	−0.683851	+	0.729621i	$0.739696\pi$
$542$		0			0
$543$		−	11.3137i		−	0.485518i
$544$		0			0
$545$		−	16.9706i		−	0.726939i
$546$		0			0
$547$	16.0000			0.684111			0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000			0.684111			0.342055	+	0.939680i	$0.388877\pi$
$548$		0			0
$549$			8.48528i			0.362143i
$550$		0			0
$551$		−	5.65685i		−	0.240990i
$552$		0			0
$553$	−8.00000			−0.340195
$554$		0			0
$555$			14.1421i			0.600300i
$556$		0			0
$557$			32.5269i			1.37821i	0.724662	+	0.689105i	$0.241996\pi$
$557$			32.5269i			1.37821i	−0.724662	+	0.689105i	$0.758004\pi$
$558$		0			0
$559$		−	33.9411i		−	1.43556i
$560$		0			0
$561$	−6.00000	−	2.82843i	−0.253320	−	0.119416i
$562$		0			0
$563$	−12.0000			−0.505740			−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000			−0.505740			−0.252870	+	0.967500i	$0.581374\pi$
$564$		0			0
$565$	−18.0000			−0.757266
$566$		0			0
$567$	20.0000			0.839921
$568$		0			0
$569$			22.6274i			0.948591i	0.880366	+	0.474295i	$0.157297\pi$
$569$			22.6274i			0.948591i	−0.880366	+	0.474295i	$0.842703\pi$
$570$		0			0
$571$	22.0000			0.920671			0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000			0.920671			0.460336	+	0.887745i	$0.347729\pi$
$572$		0			0
$573$	28.0000			1.16972
$574$		0			0
$575$		−	7.07107i		−	0.294884i
$576$		0			0
$577$	26.0000			1.08239			0.541197	−	0.840896i	$-0.317971\pi$
$577$	26.0000			1.08239			0.541197	+	0.840896i	$0.317971\pi$
$578$		0			0
$579$	6.00000			0.249351
$580$		0			0
$581$	48.0000			1.99138
$582$		0			0
$583$	−18.0000	−	8.48528i	−0.745484	−	0.351424i
$584$		0			0
$585$			4.24264i			0.175412i
$586$		0			0
$587$			7.07107i			0.291854i	0.989295	+	0.145927i	$0.0466165\pi$
$587$			7.07107i			0.291854i	−0.989295	+	0.145927i	$0.953384\pi$
$588$		0			0
$589$		−	16.9706i		−	0.699260i
$590$		0			0
$591$	14.0000			0.575883
$592$		0			0
$593$			18.3848i			0.754972i	0.926015	+	0.377486i	$0.123211\pi$
$593$			18.3848i			0.754972i	−0.926015	+	0.377486i	$0.876789\pi$
$594$		0			0
$595$		−	5.65685i		−	0.231908i
$596$		0			0
$597$	−12.0000			−0.491127
$598$		0			0
$599$			5.65685i			0.231133i	0.993300	+	0.115566i	$0.0368683\pi$
$599$			5.65685i			0.231133i	−0.993300	+	0.115566i	$0.963132\pi$
$600$		0			0
$601$			25.4558i			1.03837i	0.854663	+	0.519183i	$0.173764\pi$
$601$			25.4558i			1.03837i	−0.854663	+	0.519183i	$0.826236\pi$
$602$		0			0
$603$		−	4.24264i		−	0.172774i
$604$		0			0
$605$	7.00000	+	8.48528i	0.284590	+	0.344976i
$606$		0			0
$607$	32.0000			1.29884			0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000			1.29884			0.649420	+	0.760430i	$0.275012\pi$
$608$		0			0
$609$	16.0000			0.648353
$610$		0			0
$611$	30.0000			1.21367
$612$		0			0
$613$		−	21.2132i		−	0.856793i	−0.903591	−	0.428397i	$-0.859079\pi$
$613$		−	21.2132i		−	0.856793i	0.903591	−	0.428397i	$-0.140921\pi$
$614$		0			0
$615$	4.00000			0.161296
$616$		0			0
$617$	18.0000			0.724653			0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000			0.724653			0.362326	+	0.932051i	$0.381983\pi$
$618$		0			0
$619$		−	33.9411i		−	1.36421i	−0.731255	−	0.682105i	$-0.761065\pi$
$619$		−	33.9411i		−	1.36421i	0.731255	−	0.682105i	$-0.238935\pi$
$620$		0			0
$621$	40.0000			1.60514
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	4.00000	−	8.48528i	0.159745	−	0.338869i
$628$		0			0
$629$			14.1421i			0.563884i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$			5.65685i			0.224840i
$634$		0			0
$635$	−16.0000			−0.634941
$636$		0			0
$637$			38.1838i			1.51290i
$638$		0			0
$639$			5.65685i			0.223782i
$640$		0			0
$641$	18.0000			0.710957			0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000			0.710957			0.355479	+	0.934684i	$0.384318\pi$
$642$		0			0
$643$			21.2132i			0.836567i	0.908317	+	0.418284i	$0.137368\pi$
$643$			21.2132i			0.836567i	−0.908317	+	0.418284i	$0.862632\pi$
$644$		0			0
$645$		−	11.3137i		−	0.445477i
$646$		0			0
$647$		−	32.5269i		−	1.27876i	−0.768889	−	0.639382i	$-0.779190\pi$
$647$		−	32.5269i		−	1.27876i	0.768889	−	0.639382i	$-0.220810\pi$
$648$		0			0
$649$	−4.00000	+	8.48528i	−0.157014	+	0.333076i
$650$		0			0
$651$	48.0000			1.88127
$652$		0			0
$653$	−6.00000			−0.234798			−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000			−0.234798			−0.117399	+	0.993085i	$0.537456\pi$
$654$		0			0
$655$	−12.0000			−0.468879
$656$		0			0
$657$			4.24264i			0.165521i
$658$		0			0
$659$	36.0000			1.40236			0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000			1.40236			0.701180	+	0.712984i	$0.252657\pi$
$660$		0			0
$661$	−20.0000			−0.777910			−0.388955	−	0.921257i	$-0.627164\pi$
$661$	−20.0000			−0.777910			−0.388955	+	0.921257i	$0.627164\pi$
$662$		0			0
$663$		−	8.48528i		−	0.329541i
$664$		0			0
$665$	8.00000			0.310227
$666$		0			0
$667$	20.0000			0.774403
$668$		0			0
$669$	6.00000			0.231973
$670$		0			0
$671$	−12.0000	+	25.4558i	−0.463255	+	0.982712i
$672$		0			0
$673$			29.6985i			1.14479i	0.819977	+	0.572396i	$0.193986\pi$
$673$			29.6985i			1.14479i	−0.819977	+	0.572396i	$0.806014\pi$
$674$		0			0
$675$			5.65685i			0.217732i
$676$		0			0
$677$		−	35.3553i		−	1.35882i	−0.733761	−	0.679408i	$-0.762237\pi$
$677$		−	35.3553i		−	1.35882i	0.733761	−	0.679408i	$-0.237763\pi$
$678$		0			0
$679$	−8.00000			−0.307012
$680$		0			0
$681$		−	16.9706i		−	0.650313i
$682$		0			0
$683$		−	26.8701i		−	1.02815i	−0.857744	−	0.514077i	$-0.828135\pi$
$683$		−	26.8701i		−	1.02815i	0.857744	−	0.514077i	$-0.171865\pi$
$684$		0			0
$685$	−6.00000			−0.229248
$686$		0			0
$687$			14.1421i			0.539556i
$688$		0			0
$689$		−	25.4558i		−	0.969790i
$690$		0			0
$691$			33.9411i			1.29118i	0.763684	+	0.645591i	$0.223389\pi$
$691$			33.9411i			1.29118i	−0.763684	+	0.645591i	$0.776611\pi$
$692$		0			0
$693$	−12.0000	−	5.65685i	−0.455842	−	0.214886i
$694$		0			0
$695$	4.00000			0.151729
$696$		0			0
$697$	4.00000			0.151511
$698$		0			0
$699$	10.0000			0.378235
$700$		0			0
$701$		−	14.1421i		−	0.534141i	−0.963677	−	0.267071i	$-0.913944\pi$
$701$		−	14.1421i		−	0.534141i	0.963677	−	0.267071i	$-0.0860557\pi$
$702$		0			0
$703$	−20.0000			−0.754314
$704$		0			0
$705$	10.0000			0.376622
$706$		0			0
$707$		−	45.2548i		−	1.70198i
$708$		0			0
$709$	4.00000			0.150223			0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000			0.150223			0.0751116	+	0.997175i	$0.476069\pi$
$710$		0			0
$711$	2.00000			0.0750059
$712$		0			0
$713$	60.0000			2.24702
$714$		0			0
$715$	−6.00000	+	12.7279i	−0.224387	+	0.475997i
$716$		0			0
$717$		−	25.4558i		−	0.950666i
$718$		0			0
$719$			5.65685i			0.210965i	0.994421	+	0.105483i	$0.0336387\pi$
$719$			5.65685i			0.210965i	−0.994421	+	0.105483i	$0.966361\pi$
$720$		0			0
$721$		−	50.9117i		−	1.89605i
$722$		0			0
$723$	12.0000			0.446285
$724$		0			0
$725$			2.82843i			0.105045i
$726$		0			0
$727$		−	38.1838i		−	1.41616i	−0.706133	−	0.708079i	$-0.749562\pi$
$727$		−	38.1838i		−	1.41616i	0.706133	−	0.708079i	$-0.250438\pi$
$728$		0			0
$729$	−29.0000			−1.07407
$730$		0			0
$731$		−	11.3137i		−	0.418453i
$732$		0			0
$733$			29.6985i			1.09694i	0.836171	+	0.548469i	$0.184789\pi$
$733$			29.6985i			1.09694i	−0.836171	+	0.548469i	$0.815211\pi$
$734$		0			0
$735$			12.7279i			0.469476i
$736$		0			0
$737$	6.00000	−	12.7279i	0.221013	−	0.468839i
$738$		0			0
$739$	−44.0000			−1.61857			−0.809283	−	0.587419i	$-0.800144\pi$
$739$	−44.0000			−1.61857			−0.809283	+	0.587419i	$0.800144\pi$
$740$		0			0
$741$	12.0000			0.440831
$742$		0			0
$743$	36.0000			1.32071			0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000			1.32071			0.660356	+	0.750953i	$0.270405\pi$
$744$		0			0
$745$			11.3137i			0.414502i
$746$		0			0
$747$	−12.0000			−0.439057
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	50.9117i		−	1.85779i	−0.370338	−	0.928897i	$-0.620758\pi$
$751$		−	50.9117i		−	1.85779i	0.370338	−	0.928897i	$-0.379242\pi$
$752$		0			0
$753$	8.00000			0.291536
$754$		0			0
$755$	8.00000			0.291150
$756$		0			0
$757$	34.0000			1.23575			0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000			1.23575			0.617876	+	0.786276i	$0.287994\pi$
$758$		0			0
$759$	30.0000	+	14.1421i	1.08893	+	0.513327i
$760$		0			0
$761$		−	11.3137i		−	0.410122i	−0.978749	−	0.205061i	$-0.934261\pi$
$761$		−	11.3137i		−	0.410122i	0.978749	−	0.205061i	$-0.0657392\pi$
$762$		0			0
$763$			67.8823i			2.45750i
$764$		0			0
$765$			1.41421i			0.0511310i
$766$		0			0
$767$	−12.0000			−0.433295
$768$		0			0
$769$		−	25.4558i		−	0.917961i	−0.888446	−	0.458981i	$-0.848215\pi$
$769$		−	25.4558i		−	0.917961i	0.888446	−	0.458981i	$-0.151785\pi$
$770$		0			0
$771$			8.48528i			0.305590i
$772$		0			0
$773$	18.0000			0.647415			0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000			0.647415			0.323708	+	0.946157i	$0.395071\pi$
$774$		0			0
$775$			8.48528i			0.304800i
$776$		0			0
$777$		−	56.5685i		−	2.02939i
$778$		0			0
$779$			5.65685i			0.202678i
$780$		0			0
$781$	−8.00000	+	16.9706i	−0.286263	+	0.607254i
$782$		0			0
$783$	−16.0000			−0.571793
$784$		0			0
$785$	−14.0000			−0.499681
$786$		0			0
$787$	40.0000			1.42585			0.712923	−	0.701242i	$-0.247371\pi$
$787$	40.0000			1.42585			0.712923	+	0.701242i	$0.247371\pi$
$788$		0			0
$789$		−	33.9411i		−	1.20834i
$790$		0			0
$791$	72.0000			2.56003
$792$		0			0
$793$	−36.0000			−1.27840
$794$		0			0
$795$		−	8.48528i		−	0.300942i
$796$		0			0
$797$	−6.00000			−0.212531			−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000			−0.212531			−0.106265	+	0.994338i	$0.533889\pi$
$798$		0			0
$799$	10.0000			0.353775
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.00000	+	12.7279i	−0.211735	+	0.449159i
$804$		0			0
$805$			28.2843i			0.996890i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			48.0833i			1.69052i	0.534357	+	0.845259i	$0.320554\pi$
$809$			48.0833i			1.69052i	−0.534357	+	0.845259i	$0.679446\pi$
$810$		0			0
$811$	22.0000			0.772524			0.386262	−	0.922389i	$-0.373766\pi$
$811$	22.0000			0.772524			0.386262	+	0.922389i	$0.373766\pi$
$812$		0			0
$813$			11.3137i			0.396789i
$814$		0			0
$815$			21.2132i			0.743066i
$816$		0			0
$817$	16.0000			0.559769
$818$		0			0
$819$		−	16.9706i		−	0.592999i
$820$		0			0
$821$		−	48.0833i		−	1.67812i	−0.544041	−	0.839059i	$-0.683106\pi$
$821$		−	48.0833i		−	1.67812i	0.544041	−	0.839059i	$-0.316894\pi$
$822$		0			0
$823$			4.24264i			0.147889i	0.997262	+	0.0739446i	$0.0235588\pi$
$823$			4.24264i			0.147889i	−0.997262	+	0.0739446i	$0.976441\pi$
$824$		0			0
$825$	−2.00000	+	4.24264i	−0.0696311	+	0.147710i
$826$		0			0
$827$	12.0000			0.417281			0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000			0.417281			0.208640	+	0.977992i	$0.433096\pi$
$828$		0			0
$829$	34.0000			1.18087			0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000			1.18087			0.590434	+	0.807086i	$0.298956\pi$
$830$		0			0
$831$	30.0000			1.04069
$832$		0			0
$833$			12.7279i			0.440996i
$834$		0			0
$835$	12.0000			0.415277
$836$		0			0
$837$	−48.0000			−1.65912
$838$		0			0
$839$		−	53.7401i		−	1.85531i	−0.373432	−	0.927657i	$-0.621819\pi$
$839$		−	53.7401i		−	1.85531i	0.373432	−	0.927657i	$-0.378181\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$		0			0
$843$	28.0000			0.964371
$844$		0			0
$845$	−5.00000			−0.172005
$846$		0			0
$847$	−28.0000	−	33.9411i	−0.962091	−	1.16623i
$848$		0			0
$849$			5.65685i			0.194143i
$850$		0			0
$851$		−	70.7107i		−	2.42393i
$852$		0			0
$853$		−	4.24264i		−	0.145265i	−0.997359	−	0.0726326i	$-0.976860\pi$
$853$		−	4.24264i		−	0.145265i	0.997359	−	0.0726326i	$-0.0231401\pi$
$854$		0			0
$855$	−2.00000			−0.0683986
$856$		0			0
$857$			1.41421i			0.0483086i	0.999708	+	0.0241543i	$0.00768930\pi$
$857$			1.41421i			0.0483086i	−0.999708	+	0.0241543i	$0.992311\pi$
$858$		0			0
$859$		−	16.9706i		−	0.579028i	−0.957174	−	0.289514i	$-0.906506\pi$
$859$		−	16.9706i		−	0.579028i	0.957174	−	0.289514i	$-0.0934937\pi$
$860$		0			0
$861$	−16.0000			−0.545279
$862$		0			0
$863$		−	7.07107i		−	0.240702i	−0.992731	−	0.120351i	$-0.961598\pi$
$863$		−	7.07107i		−	0.240702i	0.992731	−	0.120351i	$-0.0384020\pi$
$864$		0			0
$865$			15.5563i			0.528932i
$866$		0			0
$867$			21.2132i			0.720438i
$868$		0			0
$869$	6.00000	+	2.82843i	0.203536	+	0.0959478i
$870$		0			0
$871$	18.0000			0.609907
$872$		0			0
$873$	2.00000			0.0676897
$874$		0			0
$875$	−4.00000			−0.135225
$876$		0			0
$877$		−	12.7279i		−	0.429791i	−0.976637	−	0.214896i	$-0.931059\pi$
$877$		−	12.7279i		−	0.429791i	0.976637	−	0.214896i	$-0.0689412\pi$
$878$		0			0
$879$	38.0000			1.28171
$880$		0			0
$881$	−24.0000			−0.808581			−0.404290	−	0.914631i	$-0.632481\pi$
$881$	−24.0000			−0.808581			−0.404290	+	0.914631i	$0.632481\pi$
$882$		0			0
$883$		−	12.7279i		−	0.428329i	−0.976798	−	0.214164i	$-0.931297\pi$
$883$		−	12.7279i		−	0.428329i	0.976798	−	0.214164i	$-0.0687028\pi$
$884$		0			0
$885$	−4.00000			−0.134459
$886$		0			0
$887$	48.0000			1.61168			0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000			1.61168			0.805841	+	0.592132i	$0.201714\pi$
$888$		0			0
$889$	64.0000			2.14649
$890$		0			0
$891$	−15.0000	−	7.07107i	−0.502519	−	0.236890i
$892$		0			0
$893$			14.1421i			0.473249i
$894$		0			0
$895$			11.3137i			0.378176i
$896$		0			0
$897$			42.4264i			1.41658i
$898$		0			0
$899$	−24.0000			−0.800445
$900$		0			0
$901$		−	8.48528i		−	0.282686i
$902$		0			0
$903$			45.2548i			1.50599i
$904$		0			0
$905$	−8.00000			−0.265929
$906$		0			0
$907$		−	29.6985i		−	0.986122i	−0.869995	−	0.493061i	$-0.835878\pi$
$907$		−	29.6985i		−	0.986122i	0.869995	−	0.493061i	$-0.164122\pi$
$908$		0			0
$909$			11.3137i			0.375252i
$910$		0			0
$911$			14.1421i			0.468550i	0.972170	+	0.234275i	$0.0752716\pi$
$911$			14.1421i			0.468550i	−0.972170	+	0.234275i	$0.924728\pi$
$912$		0			0
$913$	−36.0000	−	16.9706i	−1.19143	−	0.561644i
$914$		0			0
$915$	−12.0000			−0.396708
$916$		0			0
$917$	48.0000			1.58510
$918$		0			0
$919$	−40.0000			−1.31948			−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000			−1.31948			−0.659739	+	0.751495i	$0.729333\pi$
$920$		0			0
$921$			22.6274i			0.745599i
$922$		0			0
$923$	−24.0000			−0.789970
$924$		0			0
$925$	10.0000			0.328798
$926$		0			0
$927$			12.7279i			0.418040i
$928$		0			0
$929$	−12.0000			−0.393707			−0.196854	−	0.980433i	$-0.563072\pi$
$929$	−12.0000			−0.393707			−0.196854	+	0.980433i	$0.563072\pi$
$930$		0			0
$931$	−18.0000			−0.589926
$932$		0			0
$933$	16.0000			0.523816
$934$		0			0
$935$	−2.00000	+	4.24264i	−0.0654070	+	0.138749i
$936$		0			0
$937$			55.1543i			1.80181i	0.434013	+	0.900907i	$0.357097\pi$
$937$			55.1543i			1.80181i	−0.434013	+	0.900907i	$0.642903\pi$
$938$		0			0
$939$		−	14.1421i		−	0.461511i
$940$		0			0
$941$		−	5.65685i		−	0.184408i	−0.995740	−	0.0922041i	$-0.970609\pi$
$941$		−	5.65685i		−	0.184408i	0.995740	−	0.0922041i	$-0.0293912\pi$
$942$		0			0
$943$	−20.0000			−0.651290
$944$		0			0
$945$		−	22.6274i		−	0.736070i
$946$		0			0
$947$			41.0122i			1.33272i	0.745631	+	0.666359i	$0.232148\pi$
$947$			41.0122i			1.33272i	−0.745631	+	0.666359i	$0.767852\pi$
$948$		0			0
$949$	−18.0000			−0.584305
$950$		0			0
$951$			8.48528i			0.275154i
$952$		0			0
$953$		−	15.5563i		−	0.503920i	−0.967738	−	0.251960i	$-0.918925\pi$
$953$		−	15.5563i		−	0.503920i	0.967738	−	0.251960i	$-0.0810751\pi$
$954$		0			0
$955$		−	19.7990i		−	0.640680i
$956$		0			0
$957$	−12.0000	−	5.65685i	−0.387905	−	0.182860i
$958$		0			0
$959$	24.0000			0.775000
$960$		0			0
$961$	−41.0000			−1.32258
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	4.24264i		−	0.136575i
$966$		0			0
$967$	32.0000			1.02905			0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000			1.02905			0.514525	+	0.857475i	$0.327968\pi$
$968$		0			0
$969$	4.00000			0.128499
$970$		0			0
$971$		−	31.1127i		−	0.998454i	−0.866471	−	0.499227i	$-0.833617\pi$
$971$		−	31.1127i		−	0.998454i	0.866471	−	0.499227i	$-0.166383\pi$
$972$		0			0
$973$	−16.0000			−0.512936
$974$		0			0
$975$	−6.00000			−0.192154
$976$		0			0
$977$	−18.0000			−0.575871			−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000			−0.575871			−0.287936	+	0.957650i	$0.592969\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		−	16.9706i		−	0.541828i
$982$		0			0
$983$			35.3553i			1.12766i	0.825891	+	0.563830i	$0.190673\pi$
$983$			35.3553i			1.12766i	−0.825891	+	0.563830i	$0.809327\pi$
$984$		0			0
$985$		−	9.89949i		−	0.315424i
$986$		0			0
$987$	−40.0000			−1.27321
$988$		0			0
$989$			56.5685i			1.79878i
$990$		0			0
$991$			8.48528i			0.269544i	0.990877	+	0.134772i	$0.0430302\pi$
$991$			8.48528i			0.269544i	−0.990877	+	0.134772i	$0.956970\pi$
$992$		0			0
$993$	−36.0000			−1.14243
$994$		0			0
$995$			8.48528i			0.269002i
$996$		0			0
$997$			46.6690i			1.47802i	0.673693	+	0.739012i	$0.264707\pi$
$997$			46.6690i			1.47802i	−0.673693	+	0.739012i	$0.735293\pi$
$998$		0			0
$999$			56.5685i			1.78975i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3520.2.f.a.2111.2		2
4.3	odd	2		3520.2.f.d.2111.1		2
8.3	odd	2		220.2.d.b.131.1	yes	2
8.5	even	2		220.2.d.a.131.1	✓	2
11.10	odd	2		3520.2.f.d.2111.2		2
44.43	even	2	inner	3520.2.f.a.2111.1		2
88.21	odd	2		220.2.d.b.131.2	yes	2
88.43	even	2		220.2.d.a.131.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.2.d.a.131.1	✓	2	8.5	even	2
220.2.d.a.131.2	yes	2	88.43	even	2
220.2.d.b.131.1	yes	2	8.3	odd	2
220.2.d.b.131.2	yes	2	88.21	odd	2
3520.2.f.a.2111.1		2	44.43	even	2	inner
3520.2.f.a.2111.2		2	1.1	even	1	trivial
3520.2.f.d.2111.1		2	4.3	odd	2
3520.2.f.d.2111.2		2	11.10	odd	2

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:	\( N \)	\(=\)	\( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3520.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{3} +1.00000 q^{5} -4.00000 q^{7} +1.00000 q^{9} +(3.00000 + 1.41421i) q^{11} +4.24264i q^{13} +1.41421i q^{15} +1.41421i q^{17} -2.00000 q^{19} -5.65685i q^{21} -7.07107i q^{23} +1.00000 q^{25} +5.65685i q^{27} +2.82843i q^{29} +8.48528i q^{31} +(-2.00000 + 4.24264i) q^{33} -4.00000 q^{35} +10.0000 q^{37} -6.00000 q^{39} -2.82843i q^{41} -8.00000 q^{43} +1.00000 q^{45} -7.07107i q^{47} +9.00000 q^{49} -2.00000 q^{51} -6.00000 q^{53} +(3.00000 + 1.41421i) q^{55} -2.82843i q^{57} +2.82843i q^{59} +8.48528i q^{61} -4.00000 q^{63} +4.24264i q^{65} -4.24264i q^{67} +10.0000 q^{69} +5.65685i q^{71} +4.24264i q^{73} +1.41421i q^{75} +(-12.0000 - 5.65685i) q^{77} +2.00000 q^{79} -5.00000 q^{81} -12.0000 q^{83} +1.41421i q^{85} -4.00000 q^{87} -16.9706i q^{91} -12.0000 q^{93} -2.00000 q^{95} +2.00000 q^{97} +(3.00000 + 1.41421i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 8 q^{7} + 2 q^{9} + 6 q^{11} - 4 q^{19} + 2 q^{25} - 4 q^{33} - 8 q^{35} + 20 q^{37} - 12 q^{39} - 16 q^{43} + 2 q^{45} + 18 q^{49} - 4 q^{51} - 12 q^{53} + 6 q^{55} - 8 q^{63} + 20 q^{69}+ \cdots + 6 q^{99}+O(q^{100}) \) 2 * q + 2 * q^5 - 8 * q^7 + 2 * q^9 + 6 * q^11 - 4 * q^19 + 2 * q^25 - 4 * q^33 - 8 * q^35 + 20 * q^37 - 12 * q^39 - 16 * q^43 + 2 * q^45 + 18 * q^49 - 4 * q^51 - 12 * q^53 + 6 * q^55 - 8 * q^63 + 20 * q^69 - 24 * q^77 + 4 * q^79 - 10 * q^81 - 24 * q^83 - 8 * q^87 - 24 * q^93 - 4 * q^95 + 4 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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