LMFDB - Embedded newform 220.2.b.a.89.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [220,2,Mod(89,220)] 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("220.89"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(220, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 220 = 2^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	220.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:	$1.75670884447$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			89.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	220.89
Dual form			220.2.b.a.89.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 2.00000i) q^{5} +2.00000i q^{7} +3.00000 q^{9} -1.00000 q^{11} +6.00000i q^{13} -2.00000i q^{17} +4.00000 q^{19} -4.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +6.00000 q^{29} -8.00000 q^{31} +(-4.00000 - 2.00000i) q^{35} +8.00000i q^{37} -2.00000 q^{41} -10.0000i q^{43} +(-3.00000 + 6.00000i) q^{45} -12.0000i q^{47} +3.00000 q^{49} -8.00000i q^{53} +(1.00000 - 2.00000i) q^{55} +2.00000 q^{61} +6.00000i q^{63} +(-12.0000 - 6.00000i) q^{65} -4.00000i q^{67} -4.00000 q^{71} +10.0000i q^{73} -2.00000i q^{77} +16.0000 q^{79} +9.00000 q^{81} +6.00000i q^{83} +(4.00000 + 2.00000i) q^{85} -2.00000 q^{89} -12.0000 q^{91} +(-4.00000 + 8.00000i) q^{95} -4.00000i q^{97} -3.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 6 q^{9} - 2 q^{11} + 8 q^{19} - 6 q^{25} + 12 q^{29} - 16 q^{31} - 8 q^{35} - 4 q^{41} - 6 q^{45} + 6 q^{49} + 2 q^{55} + 4 q^{61} - 24 q^{65} - 8 q^{71} + 32 q^{79} + 18 q^{81} + 8 q^{85}+ \cdots - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 + 6 * q^9 - 2 * q^11 + 8 * q^19 - 6 * q^25 + 12 * q^29 - 16 * q^31 - 8 * q^35 - 4 * q^41 - 6 * q^45 + 6 * q^49 + 2 * q^55 + 4 * q^61 - 24 * q^65 - 8 * q^71 + 32 * q^79 + 18 * q^81 + 8 * q^85 - 4 * q^89 - 24 * q^91 - 8 * q^95 - 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/220\mathbb{Z}\right)^\times$.

$n$	$101$	$111$	$177$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$		0			0
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$6$		0			0
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$		0			0
$9$	3.00000			1.00000
$10$		0			0
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$		0			0
$13$			6.00000i			1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$			6.00000i			1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$		0			0
$19$	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
$23$		−	4.00000i		−	0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$		−	4.00000i		−	0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	−8.00000			−1.43684			−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000			−1.43684			−0.718421	+	0.695608i	$0.755135\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.00000	−	2.00000i	−0.676123	−	0.338062i
$36$		0			0
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$		−	10.0000i		−	1.52499i	−0.646997	−	0.762493i	$-0.723975\pi$
$43$		−	10.0000i		−	1.52499i	0.646997	−	0.762493i	$-0.276025\pi$
$44$		0			0
$45$	−3.00000	+	6.00000i	−0.447214	+	0.894427i
$46$		0			0
$47$		−	12.0000i		−	1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$		−	12.0000i		−	1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	8.00000i		−	1.09888i	−0.835532	−	0.549442i	$-0.814840\pi$
$53$		−	8.00000i		−	1.09888i	0.835532	−	0.549442i	$-0.185160\pi$
$54$		0			0
$55$	1.00000	−	2.00000i	0.134840	−	0.269680i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$		0			0
$63$			6.00000i			0.755929i
$64$		0			0
$65$	−12.0000	−	6.00000i	−1.48842	−	0.744208i
$66$		0			0
$67$		−	4.00000i		−	0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$		−	4.00000i		−	0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.00000			−0.474713			−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000			−0.474713			−0.237356	+	0.971423i	$0.576281\pi$
$72$		0			0
$73$			10.0000i			1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$			10.0000i			1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	2.00000i		−	0.227921i
$78$		0			0
$79$	16.0000			1.80014			0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000			1.80014			0.900070	+	0.435745i	$0.143515\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$			6.00000i			0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$			6.00000i			0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$		0			0
$85$	4.00000	+	2.00000i	0.433861	+	0.216930i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.00000			−0.212000			−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000			−0.212000			−0.106000	+	0.994366i	$0.533804\pi$
$90$		0			0
$91$	−12.0000			−1.25794
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.00000	+	8.00000i	−0.410391	+	0.820783i
$96$		0			0
$97$		−	4.00000i		−	0.406138i	−0.979164	−	0.203069i	$-0.934908\pi$
$97$		−	4.00000i		−	0.406138i	0.979164	−	0.203069i	$-0.0650917\pi$
$98$		0			0
$99$	−3.00000			−0.301511
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$			8.00000i			0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$			8.00000i			0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			18.0000i			1.74013i	0.492941	+	0.870063i	$0.335922\pi$
$107$			18.0000i			1.74013i	−0.492941	+	0.870063i	$0.664078\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	12.0000i		−	1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$		−	12.0000i		−	1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$		0			0
$115$	8.00000	+	4.00000i	0.746004	+	0.373002i
$116$		0			0
$117$			18.0000i			1.66410i
$118$		0			0
$119$	4.00000			0.366679
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.0000	−	2.00000i	0.983870	−	0.178885i
$126$		0			0
$127$		−	2.00000i		−	0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$		−	2.00000i		−	0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.00000			−0.349482			−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000			−0.349482			−0.174741	+	0.984614i	$0.555909\pi$
$132$		0			0
$133$			8.00000i			0.693688i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	12.0000i		−	1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$		−	12.0000i		−	1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$		0			0
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	6.00000i		−	0.501745i
$144$		0			0
$145$	−6.00000	+	12.0000i	−0.498273	+	0.996546i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−22.0000			−1.80231			−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000			−1.80231			−0.901155	+	0.433497i	$0.857280\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		−	6.00000i		−	0.485071i
$154$		0			0
$155$	8.00000	−	16.0000i	0.642575	−	1.28515i
$156$		0			0
$157$		−	4.00000i		−	0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$		−	4.00000i		−	0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	8.00000			0.630488
$162$		0			0
$163$			8.00000i			0.626608i	0.949653	+	0.313304i	$0.101436\pi$
$163$			8.00000i			0.626608i	−0.949653	+	0.313304i	$0.898564\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			14.0000i			1.08335i	0.840587	+	0.541676i	$0.182210\pi$
$167$			14.0000i			1.08335i	−0.840587	+	0.541676i	$0.817790\pi$
$168$		0			0
$169$	−23.0000			−1.76923
$170$		0			0
$171$	12.0000			0.917663
$172$		0			0
$173$			2.00000i			0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$			2.00000i			0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$		0			0
$175$	8.00000	−	6.00000i	0.604743	−	0.453557i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	10.0000			0.743294			0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000			0.743294			0.371647	+	0.928374i	$0.378793\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−16.0000	−	8.00000i	−1.17634	−	0.588172i
$186$		0			0
$187$			2.00000i			0.146254i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		−	14.0000i		−	1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$		−	14.0000i		−	1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	18.0000i		−	1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$		−	18.0000i		−	1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			12.0000i			0.842235i
$204$		0			0
$205$	2.00000	−	4.00000i	0.139686	−	0.279372i
$206$		0			0
$207$		−	12.0000i		−	0.834058i
$208$		0			0
$209$	−4.00000			−0.276686
$210$		0			0
$211$	−20.0000			−1.37686			−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000			−1.37686			−0.688428	+	0.725304i	$0.741699\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	20.0000	+	10.0000i	1.36399	+	0.681994i
$216$		0			0
$217$		−	16.0000i		−	1.08615i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	12.0000			0.807207
$222$		0			0
$223$		−	20.0000i		−	1.33930i	−0.742677	−	0.669650i	$-0.766444\pi$
$223$		−	20.0000i		−	1.33930i	0.742677	−	0.669650i	$-0.233556\pi$
$224$		0			0
$225$	−9.00000	−	12.0000i	−0.600000	−	0.800000i
$226$		0			0
$227$		−	18.0000i		−	1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$		−	18.0000i		−	1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$		0			0
$229$	6.00000			0.396491			0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000			0.396491			0.198246	+	0.980152i	$0.436476\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	18.0000i		−	1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$		−	18.0000i		−	1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$		0			0
$235$	24.0000	+	12.0000i	1.56559	+	0.782794i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−18.0000			−1.15948			−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000			−1.15948			−0.579741	+	0.814801i	$0.696846\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.00000	+	6.00000i	−0.191663	+	0.383326i
$246$		0			0
$247$			24.0000i			1.52708i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	8.00000			0.504956			0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000			0.504956			0.252478	+	0.967603i	$0.418755\pi$
$252$		0			0
$253$			4.00000i			0.251478i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			8.00000i			0.499026i	0.968371	+	0.249513i	$0.0802706\pi$
$257$			8.00000i			0.499026i	−0.968371	+	0.249513i	$0.919729\pi$
$258$		0			0
$259$	−16.0000			−0.994192
$260$		0			0
$261$	18.0000			1.11417
$262$		0			0
$263$			22.0000i			1.35658i	0.734795	+	0.678289i	$0.237278\pi$
$263$			22.0000i			1.35658i	−0.734795	+	0.678289i	$0.762722\pi$
$264$		0			0
$265$	16.0000	+	8.00000i	0.982872	+	0.491436i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−30.0000			−1.82913			−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000			−1.82913			−0.914566	+	0.404436i	$0.867468\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.00000	+	4.00000i	0.180907	+	0.241209i
$276$		0			0
$277$			2.00000i			0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$			2.00000i			0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$		0			0
$279$	−24.0000			−1.43684
$280$		0			0
$281$	−2.00000			−0.119310			−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000			−0.119310			−0.0596550	+	0.998219i	$0.519000\pi$
$282$		0			0
$283$		−	10.0000i		−	0.594438i	−0.954809	−	0.297219i	$-0.903941\pi$
$283$		−	10.0000i		−	0.594438i	0.954809	−	0.297219i	$-0.0960592\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	4.00000i		−	0.236113i
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$			14.0000i			0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$			14.0000i			0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	24.0000			1.38796
$300$		0			0
$301$	20.0000			1.15278
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.00000	+	4.00000i	−0.114520	+	0.229039i
$306$		0			0
$307$			10.0000i			0.570730i	0.958419	+	0.285365i	$0.0921148\pi$
$307$			10.0000i			0.570730i	−0.958419	+	0.285365i	$0.907885\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	28.0000			1.58773			0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000			1.58773			0.793867	+	0.608091i	$0.208065\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	−12.0000	−	6.00000i	−0.676123	−	0.338062i
$316$		0			0
$317$			20.0000i			1.12331i	0.827371	+	0.561656i	$0.189836\pi$
$317$			20.0000i			1.12331i	−0.827371	+	0.561656i	$0.810164\pi$
$318$		0			0
$319$	−6.00000			−0.335936
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	8.00000i		−	0.445132i
$324$		0			0
$325$	24.0000	−	18.0000i	1.33128	−	0.998460i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	24.0000			1.32316
$330$		0			0
$331$	−20.0000			−1.09930			−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000			−1.09930			−0.549650	+	0.835395i	$0.685239\pi$
$332$		0			0
$333$			24.0000i			1.31519i
$334$		0			0
$335$	8.00000	+	4.00000i	0.437087	+	0.218543i
$336$		0			0
$337$		−	2.00000i		−	0.108947i	−0.998515	−	0.0544735i	$-0.982652\pi$
$337$		−	2.00000i		−	0.108947i	0.998515	−	0.0544735i	$-0.0173480\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	8.00000			0.433224
$342$		0			0
$343$			20.0000i			1.07990i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			26.0000i			1.39575i	0.716218	+	0.697877i	$0.245872\pi$
$347$			26.0000i			1.39575i	−0.716218	+	0.697877i	$0.754128\pi$
$348$		0			0
$349$	−26.0000			−1.39175			−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000			−1.39175			−0.695874	+	0.718164i	$0.744983\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			8.00000i			0.425797i	0.977074	+	0.212899i	$0.0682904\pi$
$353$			8.00000i			0.425797i	−0.977074	+	0.212899i	$0.931710\pi$
$354$		0			0
$355$	4.00000	−	8.00000i	0.212298	−	0.424596i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	16.0000			0.844448			0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000			0.844448			0.422224	+	0.906492i	$0.361250\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−20.0000	−	10.0000i	−1.04685	−	0.523424i
$366$		0			0
$367$		−	28.0000i		−	1.46159i	−0.682598	−	0.730794i	$-0.739150\pi$
$367$		−	28.0000i		−	1.46159i	0.682598	−	0.730794i	$-0.260850\pi$
$368$		0			0
$369$	−6.00000			−0.312348
$370$		0			0
$371$	16.0000			0.830679
$372$		0			0
$373$			2.00000i			0.103556i	0.998659	+	0.0517780i	$0.0164888\pi$
$373$			2.00000i			0.103556i	−0.998659	+	0.0517780i	$0.983511\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			36.0000i			1.85409i
$378$		0			0
$379$	16.0000			0.821865			0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000			0.821865			0.410932	+	0.911666i	$0.365203\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			8.00000i			0.408781i	0.978889	+	0.204390i	$0.0655212\pi$
$383$			8.00000i			0.408781i	−0.978889	+	0.204390i	$0.934479\pi$
$384$		0			0
$385$	4.00000	+	2.00000i	0.203859	+	0.101929i
$386$		0			0
$387$		−	30.0000i		−	1.52499i
$388$		0			0
$389$	−38.0000			−1.92668			−0.963338	−	0.268290i	$-0.913542\pi$
$389$	−38.0000			−1.92668			−0.963338	+	0.268290i	$0.913542\pi$
$390$		0			0
$391$	−8.00000			−0.404577
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−16.0000	+	32.0000i	−0.805047	+	1.61009i
$396$		0			0
$397$			20.0000i			1.00377i	0.864934	+	0.501886i	$0.167360\pi$
$397$			20.0000i			1.00377i	−0.864934	+	0.501886i	$0.832640\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	14.0000			0.699127			0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000			0.699127			0.349563	+	0.936913i	$0.386330\pi$
$402$		0			0
$403$		−	48.0000i		−	2.39105i
$404$		0			0
$405$	−9.00000	+	18.0000i	−0.447214	+	0.894427i
$406$		0			0
$407$		−	8.00000i		−	0.396545i
$408$		0			0
$409$	38.0000			1.87898			0.939490	−	0.342578i	$-0.111300\pi$
$409$	38.0000			1.87898			0.939490	+	0.342578i	$0.111300\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−12.0000	−	6.00000i	−0.589057	−	0.294528i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$	−18.0000			−0.877266			−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000			−0.877266			−0.438633	+	0.898666i	$0.644537\pi$
$422$		0			0
$423$		−	36.0000i		−	1.75038i
$424$		0			0
$425$	−8.00000	+	6.00000i	−0.388057	+	0.291043i
$426$		0			0
$427$			4.00000i			0.193574i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−32.0000			−1.54139			−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000			−1.54139			−0.770693	+	0.637207i	$0.780090\pi$
$432$		0			0
$433$		−	4.00000i		−	0.192228i	−0.995370	−	0.0961139i	$-0.969359\pi$
$433$		−	4.00000i		−	0.192228i	0.995370	−	0.0961139i	$-0.0306413\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	16.0000i		−	0.765384i
$438$		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
$443$			32.0000i			1.52037i	0.649709	+	0.760183i	$0.274891\pi$
$443$			32.0000i			1.52037i	−0.649709	+	0.760183i	$0.725109\pi$
$444$		0			0
$445$	2.00000	−	4.00000i	0.0948091	−	0.189618i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	26.0000			1.22702			0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000			1.22702			0.613508	+	0.789689i	$0.289758\pi$
$450$		0			0
$451$	2.00000			0.0941763
$452$		0			0
$453$		0			0
$454$		0			0
$455$	12.0000	−	24.0000i	0.562569	−	1.12514i
$456$		0			0
$457$		−	30.0000i		−	1.40334i	−0.712502	−	0.701670i	$-0.752438\pi$
$457$		−	30.0000i		−	1.40334i	0.712502	−	0.701670i	$-0.247562\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	14.0000			0.652045			0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000			0.652045			0.326023	+	0.945362i	$0.394291\pi$
$462$		0			0
$463$		−	36.0000i		−	1.67306i	−0.547920	−	0.836531i	$-0.684580\pi$
$463$		−	36.0000i		−	1.67306i	0.547920	−	0.836531i	$-0.315420\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	20.0000i		−	0.925490i	−0.886492	−	0.462745i	$-0.846865\pi$
$467$		−	20.0000i		−	0.925490i	0.886492	−	0.462745i	$-0.153135\pi$
$468$		0			0
$469$	8.00000			0.369406
$470$		0			0
$471$		0			0
$472$		0			0
$473$			10.0000i			0.459800i
$474$		0			0
$475$	−12.0000	−	16.0000i	−0.550598	−	0.734130i
$476$		0			0
$477$		−	24.0000i		−	1.09888i
$478$		0			0
$479$	−24.0000			−1.09659			−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000			−1.09659			−0.548294	+	0.836286i	$0.684723\pi$
$480$		0			0
$481$	−48.0000			−2.18861
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.00000	+	4.00000i	0.363261	+	0.181631i
$486$		0			0
$487$			12.0000i			0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$			12.0000i			0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−28.0000			−1.26362			−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000			−1.26362			−0.631811	+	0.775122i	$0.717688\pi$
$492$		0			0
$493$		−	12.0000i		−	0.540453i
$494$		0			0
$495$	3.00000	−	6.00000i	0.134840	−	0.269680i
$496$		0			0
$497$		−	8.00000i		−	0.358849i
$498$		0			0
$499$	−16.0000			−0.716258			−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000			−0.716258			−0.358129	+	0.933672i	$0.616585\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			18.0000i			0.802580i	0.915951	+	0.401290i	$0.131438\pi$
$503$			18.0000i			0.802580i	−0.915951	+	0.401290i	$0.868562\pi$
$504$		0			0
$505$	−6.00000	+	12.0000i	−0.266996	+	0.533993i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	30.0000			1.32973			0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000			1.32973			0.664863	+	0.746965i	$0.268490\pi$
$510$		0			0
$511$	−20.0000			−0.884748
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−16.0000	−	8.00000i	−0.705044	−	0.352522i
$516$		0			0
$517$			12.0000i			0.527759i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$		0			0
$523$			34.0000i			1.48672i	0.668894	+	0.743358i	$0.266768\pi$
$523$			34.0000i			1.48672i	−0.668894	+	0.743358i	$0.733232\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			16.0000i			0.696971i
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	12.0000i		−	0.519778i
$534$		0			0
$535$	−36.0000	−	18.0000i	−1.55642	−	0.778208i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.00000			−0.129219
$540$		0			0
$541$	6.00000			0.257960			0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000			0.257960			0.128980	+	0.991647i	$0.458830\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.00000	+	4.00000i	−0.0856706	+	0.171341i
$546$		0			0
$547$		−	26.0000i		−	1.11168i	−0.831289	−	0.555840i	$-0.812397\pi$
$547$		−	26.0000i		−	1.11168i	0.831289	−	0.555840i	$-0.187603\pi$
$548$		0			0
$549$	6.00000			0.256074
$550$		0			0
$551$	24.0000			1.02243
$552$		0			0
$553$			32.0000i			1.36078i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	26.0000i		−	1.10166i	−0.834619	−	0.550828i	$-0.814312\pi$
$557$		−	26.0000i		−	1.10166i	0.834619	−	0.550828i	$-0.185688\pi$
$558$		0			0
$559$	60.0000			2.53773
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	30.0000i		−	1.26435i	−0.774826	−	0.632175i	$-0.782163\pi$
$563$		−	30.0000i		−	1.26435i	0.774826	−	0.632175i	$-0.217837\pi$
$564$		0			0
$565$	24.0000	+	12.0000i	1.00969	+	0.504844i
$566$		0			0
$567$			18.0000i			0.755929i
$568$		0			0
$569$	30.0000			1.25767			0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000			1.25767			0.628833	+	0.777541i	$0.283533\pi$
$570$		0			0
$571$	−4.00000			−0.167395			−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000			−0.167395			−0.0836974	+	0.996491i	$0.526673\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−16.0000	+	12.0000i	−0.667246	+	0.500435i
$576$		0			0
$577$			28.0000i			1.16566i	0.812596	+	0.582828i	$0.198054\pi$
$577$			28.0000i			1.16566i	−0.812596	+	0.582828i	$0.801946\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12.0000			−0.497844
$582$		0			0
$583$			8.00000i			0.331326i
$584$		0			0
$585$	−36.0000	−	18.0000i	−1.48842	−	0.744208i
$586$		0			0
$587$			12.0000i			0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$			12.0000i			0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$		0			0
$589$	−32.0000			−1.31854
$590$		0			0
$591$		0			0
$592$		0			0
$593$			2.00000i			0.0821302i	0.999156	+	0.0410651i	$0.0130751\pi$
$593$			2.00000i			0.0821302i	−0.999156	+	0.0410651i	$0.986925\pi$
$594$		0			0
$595$	−4.00000	+	8.00000i	−0.163984	+	0.327968i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	16.0000			0.653742			0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000			0.653742			0.326871	+	0.945069i	$0.394006\pi$
$600$		0			0
$601$	−26.0000			−1.06056			−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000			−1.06056			−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$		−	12.0000i		−	0.488678i
$604$		0			0
$605$	−1.00000	+	2.00000i	−0.0406558	+	0.0813116i
$606$		0			0
$607$			6.00000i			0.243532i	0.992559	+	0.121766i	$0.0388558\pi$
$607$			6.00000i			0.243532i	−0.992559	+	0.121766i	$0.961144\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	72.0000			2.91281
$612$		0			0
$613$		−	46.0000i		−	1.85792i	−0.370177	−	0.928961i	$-0.620703\pi$
$613$		−	46.0000i		−	1.85792i	0.370177	−	0.928961i	$-0.379297\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			28.0000i			1.12724i	0.826035	+	0.563619i	$0.190591\pi$
$617$			28.0000i			1.12724i	−0.826035	+	0.563619i	$0.809409\pi$
$618$		0			0
$619$	12.0000			0.482321			0.241160	−	0.970485i	$-0.422472\pi$
$619$	12.0000			0.482321			0.241160	+	0.970485i	$0.422472\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	4.00000i		−	0.160257i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	16.0000			0.637962
$630$		0			0
$631$	−12.0000			−0.477712			−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000			−0.477712			−0.238856	+	0.971055i	$0.576772\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.00000	+	2.00000i	0.158735	+	0.0793676i
$636$		0			0
$637$			18.0000i			0.713186i
$638$		0			0
$639$	−12.0000			−0.474713
$640$		0			0
$641$	10.0000			0.394976			0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000			0.394976			0.197488	+	0.980305i	$0.436722\pi$
$642$		0			0
$643$			40.0000i			1.57745i	0.614749	+	0.788723i	$0.289257\pi$
$643$			40.0000i			1.57745i	−0.614749	+	0.788723i	$0.710743\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			24.0000i			0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$			24.0000i			0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	16.0000i		−	0.626128i	−0.949732	−	0.313064i	$-0.898644\pi$
$653$		−	16.0000i		−	0.626128i	0.949732	−	0.313064i	$-0.101356\pi$
$654$		0			0
$655$	4.00000	−	8.00000i	0.156293	−	0.312586i
$656$		0			0
$657$			30.0000i			1.17041i
$658$		0			0
$659$	20.0000			0.779089			0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000			0.779089			0.389545	+	0.921008i	$0.372632\pi$
$660$		0			0
$661$	10.0000			0.388955			0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000			0.388955			0.194477	+	0.980907i	$0.437699\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−16.0000	−	8.00000i	−0.620453	−	0.310227i
$666$		0			0
$667$		−	24.0000i		−	0.929284i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.00000			−0.0772091
$672$		0			0
$673$			26.0000i			1.00223i	0.865382	+	0.501113i	$0.167076\pi$
$673$			26.0000i			1.00223i	−0.865382	+	0.501113i	$0.832924\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	6.00000i		−	0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$		−	6.00000i		−	0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$		0			0
$679$	8.00000			0.307012
$680$		0			0
$681$		0			0
$682$		0			0
$683$			28.0000i			1.07139i	0.844411	+	0.535695i	$0.179950\pi$
$683$			28.0000i			1.07139i	−0.844411	+	0.535695i	$0.820050\pi$
$684$		0			0
$685$	24.0000	+	12.0000i	0.916993	+	0.458496i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	48.0000			1.82865
$690$		0			0
$691$	−32.0000			−1.21734			−0.608669	−	0.793424i	$-0.708296\pi$
$691$	−32.0000			−1.21734			−0.608669	+	0.793424i	$0.708296\pi$
$692$		0			0
$693$		−	6.00000i		−	0.227921i
$694$		0			0
$695$	−12.0000	+	24.0000i	−0.455186	+	0.910372i
$696$		0			0
$697$			4.00000i			0.151511i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−38.0000			−1.43524			−0.717620	−	0.696435i	$-0.754769\pi$
$701$	−38.0000			−1.43524			−0.717620	+	0.696435i	$0.754769\pi$
$702$		0			0
$703$			32.0000i			1.20690i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			12.0000i			0.451306i
$708$		0			0
$709$	−6.00000			−0.225335			−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000			−0.225335			−0.112667	+	0.993633i	$0.535939\pi$
$710$		0			0
$711$	48.0000			1.80014
$712$		0			0
$713$			32.0000i			1.19841i
$714$		0			0
$715$	12.0000	+	6.00000i	0.448775	+	0.224387i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−48.0000			−1.79010			−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000			−1.79010			−0.895049	+	0.445968i	$0.852860\pi$
$720$		0			0
$721$	−16.0000			−0.595871
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−18.0000	−	24.0000i	−0.668503	−	0.891338i
$726$		0			0
$727$		−	48.0000i		−	1.78022i	−0.455744	−	0.890111i	$-0.650627\pi$
$727$		−	48.0000i		−	1.78022i	0.455744	−	0.890111i	$-0.349373\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$	−20.0000			−0.739727
$732$		0			0
$733$		−	6.00000i		−	0.221615i	−0.993842	−	0.110808i	$-0.964656\pi$
$733$		−	6.00000i		−	0.221615i	0.993842	−	0.110808i	$-0.0353437\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			4.00000i			0.147342i
$738$		0			0
$739$	28.0000			1.03000			0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000			1.03000			0.514998	+	0.857191i	$0.327793\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	42.0000i		−	1.54083i	−0.637542	−	0.770415i	$-0.720049\pi$
$743$		−	42.0000i		−	1.54083i	0.637542	−	0.770415i	$-0.279951\pi$
$744$		0			0
$745$	22.0000	−	44.0000i	0.806018	−	1.61204i
$746$		0			0
$747$			18.0000i			0.658586i
$748$		0			0
$749$	−36.0000			−1.31541
$750$		0			0
$751$	−4.00000			−0.145962			−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000			−0.145962			−0.0729810	+	0.997333i	$0.523251\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	20.0000i		−	0.726912i	−0.931611	−	0.363456i	$-0.881597\pi$
$757$		−	20.0000i		−	0.726912i	0.931611	−	0.363456i	$-0.118403\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−22.0000			−0.797499			−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000			−0.797499			−0.398750	+	0.917060i	$0.630556\pi$
$762$		0			0
$763$			4.00000i			0.144810i
$764$		0			0
$765$	12.0000	+	6.00000i	0.433861	+	0.216930i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	10.0000			0.360609			0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000			0.360609			0.180305	+	0.983611i	$0.442292\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	24.0000i		−	0.863220i	−0.902060	−	0.431610i	$-0.857946\pi$
$773$		−	24.0000i		−	0.863220i	0.902060	−	0.431610i	$-0.142054\pi$
$774$		0			0
$775$	24.0000	+	32.0000i	0.862105	+	1.14947i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−8.00000			−0.286630
$780$		0			0
$781$	4.00000			0.143131
$782$		0			0
$783$		0			0
$784$		0			0
$785$	8.00000	+	4.00000i	0.285532	+	0.142766i
$786$		0			0
$787$		−	6.00000i		−	0.213877i	−0.994266	−	0.106938i	$-0.965895\pi$
$787$		−	6.00000i		−	0.213877i	0.994266	−	0.106938i	$-0.0341048\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	24.0000			0.853342
$792$		0			0
$793$			12.0000i			0.426132i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	52.0000i		−	1.84193i	−0.389640	−	0.920967i	$-0.627401\pi$
$797$		−	52.0000i		−	1.84193i	0.389640	−	0.920967i	$-0.372599\pi$
$798$		0			0
$799$	−24.0000			−0.849059
$800$		0			0
$801$	−6.00000			−0.212000
$802$		0			0
$803$		−	10.0000i		−	0.352892i
$804$		0			0
$805$	−8.00000	+	16.0000i	−0.281963	+	0.563926i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	2.00000			0.0703163			0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000			0.0703163			0.0351581	+	0.999382i	$0.488807\pi$
$810$		0			0
$811$	44.0000			1.54505			0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000			1.54505			0.772524	+	0.634985i	$0.218994\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−16.0000	−	8.00000i	−0.560456	−	0.280228i
$816$		0			0
$817$		−	40.0000i		−	1.39942i
$818$		0			0
$819$	−36.0000			−1.25794
$820$		0			0
$821$	−46.0000			−1.60541			−0.802706	−	0.596376i	$-0.796607\pi$
$821$	−46.0000			−1.60541			−0.802706	+	0.596376i	$0.796607\pi$
$822$		0			0
$823$			32.0000i			1.11545i	0.830026	+	0.557725i	$0.188326\pi$
$823$			32.0000i			1.11545i	−0.830026	+	0.557725i	$0.811674\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	6.00000i		−	0.208640i	−0.994544	−	0.104320i	$-0.966733\pi$
$827$		−	6.00000i		−	0.208640i	0.994544	−	0.104320i	$-0.0332667\pi$
$828$		0			0
$829$	−26.0000			−0.903017			−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000			−0.903017			−0.451509	+	0.892267i	$0.649114\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	6.00000i		−	0.207888i
$834$		0			0
$835$	−28.0000	−	14.0000i	−0.968980	−	0.484490i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−12.0000			−0.414286			−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000			−0.414286			−0.207143	+	0.978311i	$0.566417\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$	23.0000	−	46.0000i	0.791224	−	1.58245i
$846$		0			0
$847$			2.00000i			0.0687208i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	32.0000			1.09695
$852$		0			0
$853$		−	38.0000i		−	1.30110i	−0.759465	−	0.650548i	$-0.774539\pi$
$853$		−	38.0000i		−	1.30110i	0.759465	−	0.650548i	$-0.225461\pi$
$854$		0			0
$855$	−12.0000	+	24.0000i	−0.410391	+	0.820783i
$856$		0			0
$857$		−	2.00000i		−	0.0683187i	−0.999416	−	0.0341593i	$-0.989125\pi$
$857$		−	2.00000i		−	0.0683187i	0.999416	−	0.0341593i	$-0.0108754\pi$
$858$		0			0
$859$	20.0000			0.682391			0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000			0.682391			0.341196	+	0.939992i	$0.389168\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	−4.00000	−	2.00000i	−0.136004	−	0.0680020i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−16.0000			−0.542763
$870$		0			0
$871$	24.0000			0.813209
$872$		0			0
$873$		−	12.0000i		−	0.406138i
$874$		0			0
$875$	4.00000	+	22.0000i	0.135225	+	0.743736i
$876$		0			0
$877$		−	2.00000i		−	0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$		−	2.00000i		−	0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$		−	40.0000i		−	1.34611i	−0.739594	−	0.673054i	$-0.764982\pi$
$883$		−	40.0000i		−	1.34611i	0.739594	−	0.673054i	$-0.235018\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			38.0000i			1.27592i	0.770072	+	0.637958i	$0.220220\pi$
$887$			38.0000i			1.27592i	−0.770072	+	0.637958i	$0.779780\pi$
$888$		0			0
$889$	4.00000			0.134156
$890$		0			0
$891$	−9.00000			−0.301511
$892$		0			0
$893$		−	48.0000i		−	1.60626i
$894$		0			0
$895$	12.0000	−	24.0000i	0.401116	−	0.802232i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−48.0000			−1.60089
$900$		0			0
$901$	−16.0000			−0.533037
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10.0000	+	20.0000i	−0.332411	+	0.664822i
$906$		0			0
$907$			28.0000i			0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$			28.0000i			0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$		0			0
$909$	18.0000			0.597022
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		−	6.00000i		−	0.198571i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	8.00000i		−	0.264183i
$918$		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
$921$		0			0
$922$		0			0
$923$		−	24.0000i		−	0.789970i
$924$		0			0
$925$	32.0000	−	24.0000i	1.05215	−	0.789115i
$926$		0			0
$927$			24.0000i			0.788263i
$928$		0			0
$929$	−2.00000			−0.0656179			−0.0328089	−	0.999462i	$-0.510445\pi$
$929$	−2.00000			−0.0656179			−0.0328089	+	0.999462i	$0.510445\pi$
$930$		0			0
$931$	12.0000			0.393284
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−4.00000	−	2.00000i	−0.130814	−	0.0654070i
$936$		0			0
$937$			34.0000i			1.11073i	0.831606	+	0.555366i	$0.187422\pi$
$937$			34.0000i			1.11073i	−0.831606	+	0.555366i	$0.812578\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−26.0000			−0.847576			−0.423788	−	0.905761i	$-0.639300\pi$
$941$	−26.0000			−0.847576			−0.423788	+	0.905761i	$0.639300\pi$
$942$		0			0
$943$			8.00000i			0.260516i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	12.0000i		−	0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$		−	12.0000i		−	0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$		0			0
$949$	−60.0000			−1.94768
$950$		0			0
$951$		0			0
$952$		0			0
$953$			22.0000i			0.712650i	0.934362	+	0.356325i	$0.115970\pi$
$953$			22.0000i			0.712650i	−0.934362	+	0.356325i	$0.884030\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	24.0000			0.775000
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$			54.0000i			1.74013i
$964$		0			0
$965$	28.0000	+	14.0000i	0.901352	+	0.450676i
$966$		0			0
$967$		−	26.0000i		−	0.836104i	−0.908423	−	0.418052i	$-0.862713\pi$
$967$		−	26.0000i		−	0.836104i	0.908423	−	0.418052i	$-0.137287\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	20.0000			0.641831			0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000			0.641831			0.320915	+	0.947108i	$0.396010\pi$
$972$		0			0
$973$			24.0000i			0.769405i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	44.0000i		−	1.40768i	−0.710356	−	0.703842i	$-0.751466\pi$
$977$		−	44.0000i		−	1.40768i	0.710356	−	0.703842i	$-0.248534\pi$
$978$		0			0
$979$	2.00000			0.0639203
$980$		0			0
$981$	6.00000			0.191565
$982$		0			0
$983$		−	16.0000i		−	0.510321i	−0.966899	−	0.255160i	$-0.917872\pi$
$983$		−	16.0000i		−	0.510321i	0.966899	−	0.255160i	$-0.0821283\pi$
$984$		0			0
$985$	36.0000	+	18.0000i	1.14706	+	0.573528i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−40.0000			−1.27193
$990$		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
$993$		0			0
$994$		0			0
$995$	20.0000	−	40.0000i	0.634043	−	1.26809i
$996$		0			0
$997$			22.0000i			0.696747i	0.937356	+	0.348373i	$0.113266\pi$
$997$			22.0000i			0.696747i	−0.937356	+	0.348373i	$0.886734\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	220.2.b.a.89.2	yes	2
3.2	odd	2		1980.2.c.d.1189.1		2
4.3	odd	2		880.2.b.c.529.2		2
5.2	odd	4		1100.2.a.c.1.1		1
5.3	odd	4		1100.2.a.d.1.1		1
5.4	even	2	inner	220.2.b.a.89.1	✓	2
11.10	odd	2		2420.2.b.a.969.2		2
15.2	even	4		9900.2.a.i.1.1		1
15.8	even	4		9900.2.a.x.1.1		1
15.14	odd	2		1980.2.c.d.1189.2		2
20.3	even	4		4400.2.a.n.1.1		1
20.7	even	4		4400.2.a.q.1.1		1
20.19	odd	2		880.2.b.c.529.1		2
55.54	odd	2		2420.2.b.a.969.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.2.b.a.89.1	✓	2	5.4	even	2	inner
220.2.b.a.89.2	yes	2	1.1	even	1	trivial
880.2.b.c.529.1		2	20.19	odd	2
880.2.b.c.529.2		2	4.3	odd	2
1100.2.a.c.1.1		1	5.2	odd	4
1100.2.a.d.1.1		1	5.3	odd	4
1980.2.c.d.1189.1		2	3.2	odd	2
1980.2.c.d.1189.2		2	15.14	odd	2
2420.2.b.a.969.1		2	55.54	odd	2
2420.2.b.a.969.2		2	11.10	odd	2
4400.2.a.n.1.1		1	20.3	even	4
4400.2.a.q.1.1		1	20.7	even	4
9900.2.a.i.1.1		1	15.2	even	4
9900.2.a.x.1.1		1	15.8	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:	\( N \)	\(=\)	\( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	220.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 2.00000i) q^{5} +2.00000i q^{7} +3.00000 q^{9} -1.00000 q^{11} +6.00000i q^{13} -2.00000i q^{17} +4.00000 q^{19} -4.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +6.00000 q^{29} -8.00000 q^{31} +(-4.00000 - 2.00000i) q^{35} +8.00000i q^{37} -2.00000 q^{41} -10.0000i q^{43} +(-3.00000 + 6.00000i) q^{45} -12.0000i q^{47} +3.00000 q^{49} -8.00000i q^{53} +(1.00000 - 2.00000i) q^{55} +2.00000 q^{61} +6.00000i q^{63} +(-12.0000 - 6.00000i) q^{65} -4.00000i q^{67} -4.00000 q^{71} +10.0000i q^{73} -2.00000i q^{77} +16.0000 q^{79} +9.00000 q^{81} +6.00000i q^{83} +(4.00000 + 2.00000i) q^{85} -2.00000 q^{89} -12.0000 q^{91} +(-4.00000 + 8.00000i) q^{95} -4.00000i q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 6 q^{9} - 2 q^{11} + 8 q^{19} - 6 q^{25} + 12 q^{29} - 16 q^{31} - 8 q^{35} - 4 q^{41} - 6 q^{45} + 6 q^{49} + 2 q^{55} + 4 q^{61} - 24 q^{65} - 8 q^{71} + 32 q^{79} + 18 q^{81} + 8 q^{85}+ \cdots - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 + 6 * q^9 - 2 * q^11 + 8 * q^19 - 6 * q^25 + 12 * q^29 - 16 * q^31 - 8 * q^35 - 4 * q^41 - 6 * q^45 + 6 * q^49 + 2 * q^55 + 4 * q^61 - 24 * q^65 - 8 * q^71 + 32 * q^79 + 18 * q^81 + 8 * q^85 - 4 * q^89 - 24 * q^91 - 8 * q^95 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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