LMFDB - Embedded newform 660.2.c.a.529.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [660,2,Mod(529,660)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(660, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("660.529"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 660 = 2^{2} \cdot 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	660.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-2] 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:	$5.27012653340$
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			529.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	660.529
Dual form			660.2.c.a.529.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} -2.00000i q^{7} -1.00000 q^{9} +1.00000 q^{11} +2.00000i q^{13} +(-2.00000 + 1.00000i) q^{15} -6.00000i q^{17} -4.00000 q^{19} -2.00000 q^{21} +(-3.00000 + 4.00000i) q^{25} +1.00000i q^{27} -2.00000 q^{29} -8.00000 q^{31} -1.00000i q^{33} +(-4.00000 + 2.00000i) q^{35} +2.00000 q^{39} -2.00000 q^{41} +2.00000i q^{43} +(1.00000 + 2.00000i) q^{45} -8.00000i q^{47} +3.00000 q^{49} -6.00000 q^{51} -8.00000i q^{53} +(-1.00000 - 2.00000i) q^{55} +4.00000i q^{57} +8.00000 q^{59} +10.0000 q^{61} +2.00000i q^{63} +(4.00000 - 2.00000i) q^{65} -8.00000i q^{67} -12.0000 q^{71} +14.0000i q^{73} +(4.00000 + 3.00000i) q^{75} -2.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{83} +(-12.0000 + 6.00000i) q^{85} +2.00000i q^{87} +14.0000 q^{89} +4.00000 q^{91} +8.00000i q^{93} +(4.00000 + 8.00000i) q^{95} -12.0000i q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{9} + 2 q^{11} - 4 q^{15} - 8 q^{19} - 4 q^{21} - 6 q^{25} - 4 q^{29} - 16 q^{31} - 8 q^{35} + 4 q^{39} - 4 q^{41} + 2 q^{45} + 6 q^{49} - 12 q^{51} - 2 q^{55} + 16 q^{59} + 20 q^{61}+ \cdots - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^9 + 2 * q^11 - 4 * q^15 - 8 * q^19 - 4 * q^21 - 6 * q^25 - 4 * q^29 - 16 * q^31 - 8 * q^35 + 4 * q^39 - 4 * q^41 + 2 * q^45 + 6 * q^49 - 12 * q^51 - 2 * q^55 + 16 * q^59 + 20 * q^61 + 8 * q^65 - 24 * q^71 + 8 * q^75 - 16 * q^79 + 2 * q^81 - 24 * q^85 + 28 * q^89 + 8 * q^91 + 8 * q^95 - 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/660\mathbb{Z}\right)^\times$.

$n$	$221$	$331$	$397$	$541$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$		−	1.00000i		−	0.577350i
$4$		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	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$		0			0
$13$			2.00000i			0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$			2.00000i			0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$		0			0
$15$	−2.00000	+	1.00000i	−0.516398	+	0.258199i
$16$		0			0
$17$		−	6.00000i		−	1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$		−	6.00000i		−	1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$		0			0
$21$	−2.00000			−0.436436
$22$		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
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		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
$33$		−	1.00000i		−	0.174078i
$34$		0			0
$35$	−4.00000	+	2.00000i	−0.676123	+	0.338062i
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	2.00000			0.320256
$40$		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
$43$			2.00000i			0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$			2.00000i			0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$		0			0
$45$	1.00000	+	2.00000i	0.149071	+	0.298142i
$46$		0			0
$47$		−	8.00000i		−	1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$		−	8.00000i		−	1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$	−6.00000			−0.840168
$52$		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
$57$			4.00000i			0.529813i
$58$		0			0
$59$	8.00000			1.04151			0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000			1.04151			0.520756	+	0.853706i	$0.325650\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$			2.00000i			0.251976i
$64$		0			0
$65$	4.00000	−	2.00000i	0.496139	−	0.248069i
$66$		0			0
$67$		−	8.00000i		−	0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$		−	8.00000i		−	0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−12.0000			−1.42414			−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		0			0
$73$			14.0000i			1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$			14.0000i			1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$		0			0
$75$	4.00000	+	3.00000i	0.461880	+	0.346410i
$76$		0			0
$77$		−	2.00000i		−	0.227921i
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	6.00000i		−	0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$		−	6.00000i		−	0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$		0			0
$85$	−12.0000	+	6.00000i	−1.30158	+	0.650791i
$86$		0			0
$87$			2.00000i			0.214423i
$88$		0			0
$89$	14.0000			1.48400			0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000			1.48400			0.741999	+	0.670402i	$0.233878\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$		0			0
$93$			8.00000i			0.829561i
$94$		0			0
$95$	4.00000	+	8.00000i	0.410391	+	0.820783i
$96$		0			0
$97$		−	12.0000i		−	1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$		−	12.0000i		−	1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$		0			0
$99$	−1.00000			−0.100504
$100$		0			0
$101$	14.0000			1.39305			0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000			1.39305			0.696526	+	0.717532i	$0.254728\pi$
$102$		0			0
$103$		−	4.00000i		−	0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$		−	4.00000i		−	0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$		0			0
$105$	2.00000	+	4.00000i	0.195180	+	0.390360i
$106$		0			0
$107$		−	10.0000i		−	0.966736i	−0.875417	−	0.483368i	$-0.839413\pi$
$107$		−	10.0000i		−	0.966736i	0.875417	−	0.483368i	$-0.160587\pi$
$108$		0			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			4.00000i			0.376288i	0.982141	+	0.188144i	$0.0602472\pi$
$113$			4.00000i			0.376288i	−0.982141	+	0.188144i	$0.939753\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	2.00000i		−	0.184900i
$118$		0			0
$119$	−12.0000			−1.10004
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$			2.00000i			0.180334i
$124$		0			0
$125$	11.0000	+	2.00000i	0.983870	+	0.178885i
$126$		0			0
$127$			10.0000i			0.887357i	0.896186	+	0.443678i	$0.146327\pi$
$127$			10.0000i			0.887357i	−0.896186	+	0.443678i	$0.853673\pi$
$128$		0			0
$129$	2.00000			0.176090
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$			8.00000i			0.693688i
$134$		0			0
$135$	2.00000	−	1.00000i	0.172133	−	0.0860663i
$136$		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
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$	−8.00000			−0.673722
$142$		0			0
$143$			2.00000i			0.167248i
$144$		0			0
$145$	2.00000	+	4.00000i	0.166091	+	0.332182i
$146$		0			0
$147$		−	3.00000i		−	0.247436i
$148$		0			0
$149$	10.0000			0.819232			0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000			0.819232			0.409616	+	0.912258i	$0.365663\pi$
$150$		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
$159$	−8.00000			−0.634441
$160$		0			0
$161$		0			0
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$		0			0
$165$	−2.00000	+	1.00000i	−0.155700	+	0.0778499i
$166$		0			0
$167$			2.00000i			0.154765i	0.997001	+	0.0773823i	$0.0246562\pi$
$167$			2.00000i			0.154765i	−0.997001	+	0.0773823i	$0.975344\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$		0			0
$171$	4.00000			0.305888
$172$		0			0
$173$		−	10.0000i		−	0.760286i	−0.924928	−	0.380143i	$-0.875875\pi$
$173$		−	10.0000i		−	0.760286i	0.924928	−	0.380143i	$-0.124125\pi$
$174$		0			0
$175$	8.00000	+	6.00000i	0.604743	+	0.453557i
$176$		0			0
$177$		−	8.00000i		−	0.601317i
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		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
$183$		−	10.0000i		−	0.739221i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	6.00000i		−	0.438763i
$188$		0			0
$189$	2.00000			0.145479
$190$		0			0
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$		0			0
$193$			22.0000i			1.58359i	0.610784	+	0.791797i	$0.290854\pi$
$193$			22.0000i			1.58359i	−0.610784	+	0.791797i	$0.709146\pi$
$194$		0			0
$195$	−2.00000	−	4.00000i	−0.143223	−	0.286446i
$196$		0			0
$197$		−	6.00000i		−	0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$		−	6.00000i		−	0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$		0			0
$199$	−4.00000			−0.283552			−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000			−0.283552			−0.141776	+	0.989899i	$0.545281\pi$
$200$		0			0
$201$	−8.00000			−0.564276
$202$		0			0
$203$			4.00000i			0.280745i
$204$		0			0
$205$	2.00000	+	4.00000i	0.139686	+	0.279372i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4.00000			−0.276686
$210$		0			0
$211$	28.0000			1.92760			0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000			1.92760			0.963800	+	0.266627i	$0.0859092\pi$
$212$		0			0
$213$			12.0000i			0.822226i
$214$		0			0
$215$	4.00000	−	2.00000i	0.272798	−	0.136399i
$216$		0			0
$217$			16.0000i			1.08615i
$218$		0			0
$219$	14.0000			0.946032
$220$		0			0
$221$	12.0000			0.807207
$222$		0			0
$223$		−	16.0000i		−	1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$		−	16.0000i		−	1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$		0			0
$225$	3.00000	−	4.00000i	0.200000	−	0.266667i
$226$		0			0
$227$		−	14.0000i		−	0.929213i	−0.885517	−	0.464606i	$-0.846196\pi$
$227$		−	14.0000i		−	0.929213i	0.885517	−	0.464606i	$-0.153804\pi$
$228$		0			0
$229$	−26.0000			−1.71813			−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000			−1.71813			−0.859064	+	0.511868i	$0.828954\pi$
$230$		0			0
$231$	−2.00000			−0.131590
$232$		0			0
$233$			26.0000i			1.70332i	0.524097	+	0.851658i	$0.324403\pi$
$233$			26.0000i			1.70332i	−0.524097	+	0.851658i	$0.675597\pi$
$234$		0			0
$235$	−16.0000	+	8.00000i	−1.04372	+	0.521862i
$236$		0			0
$237$			8.00000i			0.519656i
$238$		0			0
$239$	−24.0000			−1.55243			−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000			−1.55243			−0.776215	+	0.630468i	$0.782863\pi$
$240$		0			0
$241$	22.0000			1.41714			0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000			1.41714			0.708572	+	0.705638i	$0.249340\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−3.00000	−	6.00000i	−0.191663	−	0.383326i
$246$		0			0
$247$		−	8.00000i		−	0.509028i
$248$		0			0
$249$	−6.00000			−0.380235
$250$		0			0
$251$	−16.0000			−1.00991			−0.504956	−	0.863145i	$-0.668491\pi$
$251$	−16.0000			−1.00991			−0.504956	+	0.863145i	$0.668491\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	6.00000	+	12.0000i	0.375735	+	0.751469i
$256$		0			0
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	2.00000			0.123797
$262$		0			0
$263$		−	6.00000i		−	0.369976i	−0.982741	−	0.184988i	$-0.940775\pi$
$263$		−	6.00000i		−	0.369976i	0.982741	−	0.184988i	$-0.0592246\pi$
$264$		0			0
$265$	−16.0000	+	8.00000i	−0.982872	+	0.491436i
$266$		0			0
$267$		−	14.0000i		−	0.856786i
$268$		0			0
$269$	2.00000			0.121942			0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000			0.121942			0.0609711	+	0.998140i	$0.480580\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		−	4.00000i		−	0.242091i
$274$		0			0
$275$	−3.00000	+	4.00000i	−0.180907	+	0.241209i
$276$		0			0
$277$		−	26.0000i		−	1.56219i	−0.624413	−	0.781094i	$-0.714662\pi$
$277$		−	26.0000i		−	1.56219i	0.624413	−	0.781094i	$-0.285338\pi$
$278$		0			0
$279$	8.00000			0.478947
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$		−	14.0000i		−	0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$		−	14.0000i		−	0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$		0			0
$285$	8.00000	−	4.00000i	0.473879	−	0.236940i
$286$		0			0
$287$			4.00000i			0.236113i
$288$		0			0
$289$	−19.0000			−1.11765
$290$		0			0
$291$	−12.0000			−0.703452
$292$		0			0
$293$			10.0000i			0.584206i	0.956387	+	0.292103i	$0.0943550\pi$
$293$			10.0000i			0.584206i	−0.956387	+	0.292103i	$0.905645\pi$
$294$		0			0
$295$	−8.00000	−	16.0000i	−0.465778	−	0.931556i
$296$		0			0
$297$			1.00000i			0.0580259i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	4.00000			0.230556
$302$		0			0
$303$		−	14.0000i		−	0.804279i
$304$		0			0
$305$	−10.0000	−	20.0000i	−0.572598	−	1.14520i
$306$		0			0
$307$			22.0000i			1.25561i	0.778372	+	0.627803i	$0.216046\pi$
$307$			22.0000i			1.25561i	−0.778372	+	0.627803i	$0.783954\pi$
$308$		0			0
$309$	−4.00000			−0.227552
$310$		0			0
$311$	4.00000			0.226819			0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000			0.226819			0.113410	+	0.993548i	$0.463823\pi$
$312$		0			0
$313$		−	32.0000i		−	1.80875i	−0.426742	−	0.904373i	$-0.640339\pi$
$313$		−	32.0000i		−	1.80875i	0.426742	−	0.904373i	$-0.359661\pi$
$314$		0			0
$315$	4.00000	−	2.00000i	0.225374	−	0.112687i
$316$		0			0
$317$		−	4.00000i		−	0.224662i	−0.993671	−	0.112331i	$-0.964168\pi$
$317$		−	4.00000i		−	0.224662i	0.993671	−	0.112331i	$-0.0358318\pi$
$318$		0			0
$319$	−2.00000			−0.111979
$320$		0			0
$321$	−10.0000			−0.558146
$322$		0			0
$323$			24.0000i			1.33540i
$324$		0			0
$325$	−8.00000	−	6.00000i	−0.443760	−	0.332820i
$326$		0			0
$327$		−	10.0000i		−	0.553001i
$328$		0			0
$329$	−16.0000			−0.882109
$330$		0			0
$331$	28.0000			1.53902			0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000			1.53902			0.769510	+	0.638635i	$0.220501\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−16.0000	+	8.00000i	−0.874173	+	0.437087i
$336$		0			0
$337$		−	6.00000i		−	0.326841i	−0.986557	−	0.163420i	$-0.947747\pi$
$337$		−	6.00000i		−	0.326841i	0.986557	−	0.163420i	$-0.0522527\pi$
$338$		0			0
$339$	4.00000			0.217250
$340$		0			0
$341$	−8.00000			−0.433224
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			30.0000i			1.61048i	0.592946	+	0.805242i	$0.297965\pi$
$347$			30.0000i			1.61048i	−0.592946	+	0.805242i	$0.702035\pi$
$348$		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
$351$	−2.00000			−0.106752
$352$		0			0
$353$			16.0000i			0.851594i	0.904819	+	0.425797i	$0.140006\pi$
$353$			16.0000i			0.851594i	−0.904819	+	0.425797i	$0.859994\pi$
$354$		0			0
$355$	12.0000	+	24.0000i	0.636894	+	1.27379i
$356$		0			0
$357$			12.0000i			0.635107i
$358$		0			0
$359$	8.00000			0.422224			0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000			0.422224			0.211112	+	0.977462i	$0.432292\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		−	1.00000i		−	0.0524864i
$364$		0			0
$365$	28.0000	−	14.0000i	1.46559	−	0.732793i
$366$		0			0
$367$		−	24.0000i		−	1.25279i	−0.779506	−	0.626395i	$-0.784530\pi$
$367$		−	24.0000i		−	1.25279i	0.779506	−	0.626395i	$-0.215470\pi$
$368$		0			0
$369$	2.00000			0.104116
$370$		0			0
$371$	−16.0000			−0.830679
$372$		0			0
$373$			6.00000i			0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$			6.00000i			0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$		0			0
$375$	2.00000	−	11.0000i	0.103280	−	0.568038i
$376$		0			0
$377$		−	4.00000i		−	0.206010i
$378$		0			0
$379$	−16.0000			−0.821865			−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000			−0.821865			−0.410932	+	0.911666i	$0.634797\pi$
$380$		0			0
$381$	10.0000			0.512316
$382$		0			0
$383$		−	20.0000i		−	1.02195i	−0.859595	−	0.510976i	$-0.829284\pi$
$383$		−	20.0000i		−	1.02195i	0.859595	−	0.510976i	$-0.170716\pi$
$384$		0			0
$385$	−4.00000	+	2.00000i	−0.203859	+	0.101929i
$386$		0			0
$387$		−	2.00000i		−	0.101666i
$388$		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
$393$		−	12.0000i		−	0.605320i
$394$		0			0
$395$	8.00000	+	16.0000i	0.402524	+	0.805047i
$396$		0			0
$397$		−	12.0000i		−	0.602263i	−0.953583	−	0.301131i	$-0.902636\pi$
$397$		−	12.0000i		−	0.602263i	0.953583	−	0.301131i	$-0.0973643\pi$
$398$		0			0
$399$	8.00000			0.400501
$400$		0			0
$401$	−18.0000			−0.898877			−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000			−0.898877			−0.449439	+	0.893311i	$0.648376\pi$
$402$		0			0
$403$		−	16.0000i		−	0.797017i
$404$		0			0
$405$	−1.00000	−	2.00000i	−0.0496904	−	0.0993808i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	30.0000			1.48340			0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000			1.48340			0.741702	+	0.670729i	$0.234019\pi$
$410$		0			0
$411$	−12.0000			−0.591916
$412$		0			0
$413$		−	16.0000i		−	0.787309i
$414$		0			0
$415$	−12.0000	+	6.00000i	−0.589057	+	0.294528i
$416$		0			0
$417$			4.00000i			0.195881i
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$	−34.0000			−1.65706			−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000			−1.65706			−0.828529	+	0.559946i	$0.810822\pi$
$422$		0			0
$423$			8.00000i			0.388973i
$424$		0			0
$425$	24.0000	+	18.0000i	1.16417	+	0.873128i
$426$		0			0
$427$		−	20.0000i		−	0.967868i
$428$		0			0
$429$	2.00000			0.0965609
$430$		0			0
$431$	−24.0000			−1.15604			−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000			−1.15604			−0.578020	+	0.816023i	$0.696174\pi$
$432$		0			0
$433$		−	12.0000i		−	0.576683i	−0.957528	−	0.288342i	$-0.906896\pi$
$433$		−	12.0000i		−	0.576683i	0.957528	−	0.288342i	$-0.0931039\pi$
$434$		0			0
$435$	4.00000	−	2.00000i	0.191785	−	0.0958927i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	16.0000			0.763638			0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000			0.763638			0.381819	+	0.924237i	$0.375298\pi$
$440$		0			0
$441$	−3.00000			−0.142857
$442$		0			0
$443$			28.0000i			1.33032i	0.746701	+	0.665160i	$0.231637\pi$
$443$			28.0000i			1.33032i	−0.746701	+	0.665160i	$0.768363\pi$
$444$		0			0
$445$	−14.0000	−	28.0000i	−0.663664	−	1.32733i
$446$		0			0
$447$		−	10.0000i		−	0.472984i
$448$		0			0
$449$	−22.0000			−1.03824			−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000			−1.03824			−0.519122	+	0.854700i	$0.673741\pi$
$450$		0			0
$451$	−2.00000			−0.0941763
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−4.00000	−	8.00000i	−0.187523	−	0.375046i
$456$		0			0
$457$			22.0000i			1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$			22.0000i			1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$		0			0
$459$	6.00000			0.280056
$460$		0			0
$461$	22.0000			1.02464			0.512321	−	0.858794i	$-0.328786\pi$
$461$	22.0000			1.02464			0.512321	+	0.858794i	$0.328786\pi$
$462$		0			0
$463$			16.0000i			0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$			16.0000i			0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$		0			0
$465$	16.0000	−	8.00000i	0.741982	−	0.370991i
$466$		0			0
$467$			40.0000i			1.85098i	0.378773	+	0.925490i	$0.376346\pi$
$467$			40.0000i			1.85098i	−0.378773	+	0.925490i	$0.623654\pi$
$468$		0			0
$469$	−16.0000			−0.738811
$470$		0			0
$471$	−4.00000			−0.184310
$472$		0			0
$473$			2.00000i			0.0919601i
$474$		0			0
$475$	12.0000	−	16.0000i	0.550598	−	0.734130i
$476$		0			0
$477$			8.00000i			0.366295i
$478$		0			0
$479$	32.0000			1.46212			0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000			1.46212			0.731059	+	0.682315i	$0.239027\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−24.0000	+	12.0000i	−1.08978	+	0.544892i
$486$		0			0
$487$			32.0000i			1.45006i	0.688718	+	0.725029i	$0.258174\pi$
$487$			32.0000i			1.45006i	−0.688718	+	0.725029i	$0.741826\pi$
$488$		0			0
$489$	4.00000			0.180886
$490$		0			0
$491$	−20.0000			−0.902587			−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000			−0.902587			−0.451294	+	0.892375i	$0.649037\pi$
$492$		0			0
$493$			12.0000i			0.540453i
$494$		0			0
$495$	1.00000	+	2.00000i	0.0449467	+	0.0898933i
$496$		0			0
$497$			24.0000i			1.07655i
$498$		0			0
$499$	16.0000			0.716258			0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000			0.716258			0.358129	+	0.933672i	$0.383415\pi$
$500$		0			0
$501$	2.00000			0.0893534
$502$		0			0
$503$		−	10.0000i		−	0.445878i	−0.974832	−	0.222939i	$-0.928435\pi$
$503$		−	10.0000i		−	0.445878i	0.974832	−	0.222939i	$-0.0715651\pi$
$504$		0			0
$505$	−14.0000	−	28.0000i	−0.622992	−	1.24598i
$506$		0			0
$507$		−	9.00000i		−	0.399704i
$508$		0			0
$509$	−18.0000			−0.797836			−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000			−0.797836			−0.398918	+	0.916987i	$0.630614\pi$
$510$		0			0
$511$	28.0000			1.23865
$512$		0			0
$513$		−	4.00000i		−	0.176604i
$514$		0			0
$515$	−8.00000	+	4.00000i	−0.352522	+	0.176261i
$516$		0			0
$517$		−	8.00000i		−	0.351840i
$518$		0			0
$519$	−10.0000			−0.438951
$520$		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
$523$			14.0000i			0.612177i	0.952003	+	0.306089i	$0.0990204\pi$
$523$			14.0000i			0.612177i	−0.952003	+	0.306089i	$0.900980\pi$
$524$		0			0
$525$	6.00000	−	8.00000i	0.261861	−	0.349149i
$526$		0			0
$527$			48.0000i			2.09091i
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$	−8.00000			−0.347170
$532$		0			0
$533$		−	4.00000i		−	0.173259i
$534$		0			0
$535$	−20.0000	+	10.0000i	−0.864675	+	0.432338i
$536$		0			0
$537$		−	12.0000i		−	0.517838i
$538$		0			0
$539$	3.00000			0.129219
$540$		0			0
$541$	22.0000			0.945854			0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000			0.945854			0.472927	+	0.881102i	$0.343197\pi$
$542$		0			0
$543$		−	10.0000i		−	0.429141i
$544$		0			0
$545$	−10.0000	−	20.0000i	−0.428353	−	0.856706i
$546$		0			0
$547$			18.0000i			0.769624i	0.922995	+	0.384812i	$0.125734\pi$
$547$			18.0000i			0.769624i	−0.922995	+	0.384812i	$0.874266\pi$
$548$		0			0
$549$	−10.0000			−0.426790
$550$		0			0
$551$	8.00000			0.340811
$552$		0			0
$553$			16.0000i			0.680389i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	30.0000i		−	1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$		−	30.0000i		−	1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$		0			0
$559$	−4.00000			−0.169182
$560$		0			0
$561$	−6.00000			−0.253320
$562$		0			0
$563$			6.00000i			0.252870i	0.991975	+	0.126435i	$0.0403535\pi$
$563$			6.00000i			0.252870i	−0.991975	+	0.126435i	$0.959647\pi$
$564$		0			0
$565$	8.00000	−	4.00000i	0.336563	−	0.168281i
$566$		0			0
$567$		−	2.00000i		−	0.0839921i
$568$		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
$571$	20.0000			0.836974			0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000			0.836974			0.418487	+	0.908223i	$0.362561\pi$
$572$		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
$579$	22.0000			0.914289
$580$		0			0
$581$	−12.0000			−0.497844
$582$		0			0
$583$		−	8.00000i		−	0.331326i
$584$		0			0
$585$	−4.00000	+	2.00000i	−0.165380	+	0.0826898i
$586$		0			0
$587$			24.0000i			0.990586i	0.868726	+	0.495293i	$0.164939\pi$
$587$			24.0000i			0.990586i	−0.868726	+	0.495293i	$0.835061\pi$
$588$		0			0
$589$	32.0000			1.31854
$590$		0			0
$591$	−6.00000			−0.246807
$592$		0			0
$593$			6.00000i			0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$			6.00000i			0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$		0			0
$595$	12.0000	+	24.0000i	0.491952	+	0.983904i
$596$		0			0
$597$			4.00000i			0.163709i
$598$		0			0
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$		0			0
$601$	30.0000			1.22373			0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000			1.22373			0.611863	+	0.790964i	$0.290420\pi$
$602$		0			0
$603$			8.00000i			0.325785i
$604$		0			0
$605$	−1.00000	−	2.00000i	−0.0406558	−	0.0813116i
$606$		0			0
$607$			34.0000i			1.38002i	0.723801	+	0.690009i	$0.242393\pi$
$607$			34.0000i			1.38002i	−0.723801	+	0.690009i	$0.757607\pi$
$608$		0			0
$609$	4.00000			0.162088
$610$		0			0
$611$	16.0000			0.647291
$612$		0			0
$613$			38.0000i			1.53481i	0.641165	+	0.767403i	$0.278451\pi$
$613$			38.0000i			1.53481i	−0.641165	+	0.767403i	$0.721549\pi$
$614$		0			0
$615$	4.00000	−	2.00000i	0.161296	−	0.0806478i
$616$		0			0
$617$		−	20.0000i		−	0.805170i	−0.915383	−	0.402585i	$-0.868112\pi$
$617$		−	20.0000i		−	0.805170i	0.915383	−	0.402585i	$-0.131888\pi$
$618$		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
$623$		−	28.0000i		−	1.12180i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$			4.00000i			0.159745i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$		0			0
$633$		−	28.0000i		−	1.11290i
$634$		0			0
$635$	20.0000	−	10.0000i	0.793676	−	0.396838i
$636$		0			0
$637$			6.00000i			0.237729i
$638$		0			0
$639$	12.0000			0.474713
$640$		0			0
$641$	−6.00000			−0.236986			−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000			−0.236986			−0.118493	+	0.992955i	$0.537806\pi$
$642$		0			0
$643$		−	44.0000i		−	1.73519i	−0.497271	−	0.867595i	$-0.665665\pi$
$643$		−	44.0000i		−	1.73519i	0.497271	−	0.867595i	$-0.334335\pi$
$644$		0			0
$645$	−2.00000	−	4.00000i	−0.0787499	−	0.157500i
$646$		0			0
$647$			44.0000i			1.72982i	0.501928	+	0.864909i	$0.332624\pi$
$647$			44.0000i			1.72982i	−0.501928	+	0.864909i	$0.667376\pi$
$648$		0			0
$649$	8.00000			0.314027
$650$		0			0
$651$	16.0000			0.627089
$652$		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
$655$	−12.0000	−	24.0000i	−0.468879	−	0.937758i
$656$		0			0
$657$		−	14.0000i		−	0.546192i
$658$		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
$663$		−	12.0000i		−	0.466041i
$664$		0			0
$665$	16.0000	−	8.00000i	0.620453	−	0.310227i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−16.0000			−0.618596
$670$		0			0
$671$	10.0000			0.386046
$672$		0			0
$673$		−	18.0000i		−	0.693849i	−0.937893	−	0.346925i	$-0.887226\pi$
$673$		−	18.0000i		−	0.693849i	0.937893	−	0.346925i	$-0.112774\pi$
$674$		0			0
$675$	−4.00000	−	3.00000i	−0.153960	−	0.115470i
$676$		0			0
$677$		−	2.00000i		−	0.0768662i	−0.999261	−	0.0384331i	$-0.987763\pi$
$677$		−	2.00000i		−	0.0768662i	0.999261	−	0.0384331i	$-0.0122367\pi$
$678$		0			0
$679$	−24.0000			−0.921035
$680$		0			0
$681$	−14.0000			−0.536481
$682$		0			0
$683$		−	24.0000i		−	0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$		−	24.0000i		−	0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$		0			0
$685$	−24.0000	+	12.0000i	−0.916993	+	0.458496i
$686$		0			0
$687$			26.0000i			0.991962i
$688$		0			0
$689$	16.0000			0.609551
$690$		0			0
$691$	32.0000			1.21734			0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000			1.21734			0.608669	+	0.793424i	$0.291704\pi$
$692$		0			0
$693$			2.00000i			0.0759737i
$694$		0			0
$695$	4.00000	+	8.00000i	0.151729	+	0.303457i
$696$		0			0
$697$			12.0000i			0.454532i
$698$		0			0
$699$	26.0000			0.983410
$700$		0			0
$701$	−6.00000			−0.226617			−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000			−0.226617			−0.113308	+	0.993560i	$0.536145\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	8.00000	+	16.0000i	0.301297	+	0.602595i
$706$		0			0
$707$		−	28.0000i		−	1.05305i
$708$		0			0
$709$	26.0000			0.976450			0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000			0.976450			0.488225	+	0.872718i	$0.337644\pi$
$710$		0			0
$711$	8.00000			0.300023
$712$		0			0
$713$		0			0
$714$		0			0
$715$	4.00000	−	2.00000i	0.149592	−	0.0747958i
$716$		0			0
$717$			24.0000i			0.896296i
$718$		0			0
$719$	40.0000			1.49175			0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000			1.49175			0.745874	+	0.666087i	$0.232032\pi$
$720$		0			0
$721$	−8.00000			−0.297936
$722$		0			0
$723$		−	22.0000i		−	0.818189i
$724$		0			0
$725$	6.00000	−	8.00000i	0.222834	−	0.297113i
$726$		0			0
$727$		−	28.0000i		−	1.03846i	−0.854634	−	0.519231i	$-0.826218\pi$
$727$		−	28.0000i		−	1.03846i	0.854634	−	0.519231i	$-0.173782\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	12.0000			0.443836
$732$		0			0
$733$			46.0000i			1.69905i	0.527549	+	0.849524i	$0.323111\pi$
$733$			46.0000i			1.69905i	−0.527549	+	0.849524i	$0.676889\pi$
$734$		0			0
$735$	−6.00000	+	3.00000i	−0.221313	+	0.110657i
$736$		0			0
$737$		−	8.00000i		−	0.294684i
$738$		0			0
$739$	−36.0000			−1.32428			−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000			−1.32428			−0.662141	+	0.749380i	$0.730352\pi$
$740$		0			0
$741$	−8.00000			−0.293887
$742$		0			0
$743$			10.0000i			0.366864i	0.983032	+	0.183432i	$0.0587208\pi$
$743$			10.0000i			0.366864i	−0.983032	+	0.183432i	$0.941279\pi$
$744$		0			0
$745$	−10.0000	−	20.0000i	−0.366372	−	0.732743i
$746$		0			0
$747$			6.00000i			0.219529i
$748$		0			0
$749$	−20.0000			−0.730784
$750$		0			0
$751$	28.0000			1.02173			0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000			1.02173			0.510867	+	0.859660i	$0.329324\pi$
$752$		0			0
$753$			16.0000i			0.583072i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			44.0000i			1.59921i	0.600528	+	0.799604i	$0.294957\pi$
$757$			44.0000i			1.59921i	−0.600528	+	0.799604i	$0.705043\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−30.0000			−1.08750			−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000			−1.08750			−0.543750	+	0.839248i	$0.682996\pi$
$762$		0			0
$763$		−	20.0000i		−	0.724049i
$764$		0			0
$765$	12.0000	−	6.00000i	0.433861	−	0.216930i
$766$		0			0
$767$			16.0000i			0.577727i
$768$		0			0
$769$	−54.0000			−1.94729			−0.973645	−	0.228069i	$-0.926759\pi$
$769$	−54.0000			−1.94729			−0.973645	+	0.228069i	$0.926759\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	8.00000i		−	0.287740i	−0.989597	−	0.143870i	$-0.954045\pi$
$773$		−	8.00000i		−	0.287740i	0.989597	−	0.143870i	$-0.0459547\pi$
$774$		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
$781$	−12.0000			−0.429394
$782$		0			0
$783$		−	2.00000i		−	0.0714742i
$784$		0			0
$785$	−8.00000	+	4.00000i	−0.285532	+	0.142766i
$786$		0			0
$787$			54.0000i			1.92489i	0.271473	+	0.962446i	$0.412489\pi$
$787$			54.0000i			1.92489i	−0.271473	+	0.962446i	$0.587511\pi$
$788$		0			0
$789$	−6.00000			−0.213606
$790$		0			0
$791$	8.00000			0.284447
$792$		0			0
$793$			20.0000i			0.710221i
$794$		0			0
$795$	8.00000	+	16.0000i	0.283731	+	0.567462i
$796$		0			0
$797$			52.0000i			1.84193i	0.389640	+	0.920967i	$0.372599\pi$
$797$			52.0000i			1.84193i	−0.389640	+	0.920967i	$0.627401\pi$
$798$		0			0
$799$	−48.0000			−1.69812
$800$		0			0
$801$	−14.0000			−0.494666
$802$		0			0
$803$			14.0000i			0.494049i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	2.00000i		−	0.0704033i
$808$		0			0
$809$	10.0000			0.351581			0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000			0.351581			0.175791	+	0.984428i	$0.443752\pi$
$810$		0			0
$811$	−12.0000			−0.421377			−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000			−0.421377			−0.210688	+	0.977553i	$0.567571\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8.00000	−	4.00000i	0.280228	−	0.140114i
$816$		0			0
$817$		−	8.00000i		−	0.279885i
$818$		0			0
$819$	−4.00000			−0.139771
$820$		0			0
$821$	−14.0000			−0.488603			−0.244302	−	0.969699i	$-0.578559\pi$
$821$	−14.0000			−0.488603			−0.244302	+	0.969699i	$0.578559\pi$
$822$		0			0
$823$		−	28.0000i		−	0.976019i	−0.872838	−	0.488009i	$-0.837723\pi$
$823$		−	28.0000i		−	0.976019i	0.872838	−	0.488009i	$-0.162277\pi$
$824$		0			0
$825$	4.00000	+	3.00000i	0.139262	+	0.104447i
$826$		0			0
$827$		−	18.0000i		−	0.625921i	−0.949766	−	0.312961i	$-0.898679\pi$
$827$		−	18.0000i		−	0.625921i	0.949766	−	0.312961i	$-0.101321\pi$
$828$		0			0
$829$	22.0000			0.764092			0.382046	−	0.924143i	$-0.375220\pi$
$829$	22.0000			0.764092			0.382046	+	0.924143i	$0.375220\pi$
$830$		0			0
$831$	−26.0000			−0.901930
$832$		0			0
$833$		−	18.0000i		−	0.623663i
$834$		0			0
$835$	4.00000	−	2.00000i	0.138426	−	0.0692129i
$836$		0			0
$837$		−	8.00000i		−	0.276520i
$838$		0			0
$839$	−20.0000			−0.690477			−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000			−0.690477			−0.345238	+	0.938515i	$0.612202\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$			18.0000i			0.619953i
$844$		0			0
$845$	−9.00000	−	18.0000i	−0.309609	−	0.619219i
$846$		0			0
$847$		−	2.00000i		−	0.0687208i
$848$		0			0
$849$	−14.0000			−0.480479
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	34.0000i		−	1.16414i	−0.813139	−	0.582069i	$-0.802243\pi$
$853$		−	34.0000i		−	1.16414i	0.813139	−	0.582069i	$-0.197757\pi$
$854$		0			0
$855$	−4.00000	−	8.00000i	−0.136797	−	0.273594i
$856$		0			0
$857$			42.0000i			1.43469i	0.696717	+	0.717346i	$0.254643\pi$
$857$			42.0000i			1.43469i	−0.696717	+	0.717346i	$0.745357\pi$
$858$		0			0
$859$	−12.0000			−0.409435			−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000			−0.409435			−0.204717	+	0.978821i	$0.565628\pi$
$860$		0			0
$861$	4.00000			0.136320
$862$		0			0
$863$		−	12.0000i		−	0.408485i	−0.978920	−	0.204242i	$-0.934527\pi$
$863$		−	12.0000i		−	0.408485i	0.978920	−	0.204242i	$-0.0654731\pi$
$864$		0			0
$865$	−20.0000	+	10.0000i	−0.680020	+	0.340010i
$866$		0			0
$867$			19.0000i			0.645274i
$868$		0			0
$869$	−8.00000			−0.271381
$870$		0			0
$871$	16.0000			0.542139
$872$		0			0
$873$			12.0000i			0.406138i
$874$		0			0
$875$	4.00000	−	22.0000i	0.135225	−	0.743736i
$876$		0			0
$877$			42.0000i			1.41824i	0.705088	+	0.709120i	$0.250907\pi$
$877$			42.0000i			1.41824i	−0.705088	+	0.709120i	$0.749093\pi$
$878$		0			0
$879$	10.0000			0.337292
$880$		0			0
$881$	−34.0000			−1.14549			−0.572745	−	0.819734i	$-0.694121\pi$
$881$	−34.0000			−1.14549			−0.572745	+	0.819734i	$0.694121\pi$
$882$		0			0
$883$			36.0000i			1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$			36.0000i			1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$		0			0
$885$	−16.0000	+	8.00000i	−0.537834	+	0.268917i
$886$		0			0
$887$		−	38.0000i		−	1.27592i	−0.770072	−	0.637958i	$-0.779780\pi$
$887$		−	38.0000i		−	1.27592i	0.770072	−	0.637958i	$-0.220220\pi$
$888$		0			0
$889$	20.0000			0.670778
$890$		0			0
$891$	1.00000			0.0335013
$892$		0			0
$893$			32.0000i			1.07084i
$894$		0			0
$895$	−12.0000	−	24.0000i	−0.401116	−	0.802232i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	16.0000			0.533630
$900$		0			0
$901$	−48.0000			−1.59911
$902$		0			0
$903$		−	4.00000i		−	0.133112i
$904$		0			0
$905$	−10.0000	−	20.0000i	−0.332411	−	0.664822i
$906$		0			0
$907$		−	40.0000i		−	1.32818i	−0.747653	−	0.664089i	$-0.768820\pi$
$907$		−	40.0000i		−	1.32818i	0.747653	−	0.664089i	$-0.231180\pi$
$908$		0			0
$909$	−14.0000			−0.464351
$910$		0			0
$911$	8.00000			0.265052			0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000			0.265052			0.132526	+	0.991180i	$0.457691\pi$
$912$		0			0
$913$		−	6.00000i		−	0.198571i
$914$		0			0
$915$	−20.0000	+	10.0000i	−0.661180	+	0.330590i
$916$		0			0
$917$		−	24.0000i		−	0.792550i
$918$		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
$921$	22.0000			0.724925
$922$		0			0
$923$		−	24.0000i		−	0.789970i
$924$		0			0
$925$		0			0
$926$		0			0
$927$			4.00000i			0.131377i
$928$		0			0
$929$	14.0000			0.459325			0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000			0.459325			0.229663	+	0.973270i	$0.426238\pi$
$930$		0			0
$931$	−12.0000			−0.393284
$932$		0			0
$933$		−	4.00000i		−	0.130954i
$934$		0			0
$935$	−12.0000	+	6.00000i	−0.392442	+	0.196221i
$936$		0			0
$937$		−	58.0000i		−	1.89478i	−0.320085	−	0.947389i	$-0.603712\pi$
$937$		−	58.0000i		−	1.89478i	0.320085	−	0.947389i	$-0.396288\pi$
$938$		0			0
$939$	−32.0000			−1.04428
$940$		0			0
$941$	46.0000			1.49956			0.749779	−	0.661689i	$-0.230160\pi$
$941$	46.0000			1.49956			0.749779	+	0.661689i	$0.230160\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−2.00000	−	4.00000i	−0.0650600	−	0.130120i
$946$		0			0
$947$		−	48.0000i		−	1.55979i	−0.625910	−	0.779895i	$-0.715272\pi$
$947$		−	48.0000i		−	1.55979i	0.625910	−	0.779895i	$-0.284728\pi$
$948$		0			0
$949$	−28.0000			−0.908918
$950$		0			0
$951$	−4.00000			−0.129709
$952$		0			0
$953$			34.0000i			1.10137i	0.834714	+	0.550684i	$0.185633\pi$
$953$			34.0000i			1.10137i	−0.834714	+	0.550684i	$0.814367\pi$
$954$		0			0
$955$	24.0000	+	48.0000i	0.776622	+	1.55324i
$956$		0			0
$957$			2.00000i			0.0646508i
$958$		0			0
$959$	−24.0000			−0.775000
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$			10.0000i			0.322245i
$964$		0			0
$965$	44.0000	−	22.0000i	1.41641	−	0.708205i
$966$		0			0
$967$			34.0000i			1.09337i	0.837340	+	0.546683i	$0.184110\pi$
$967$			34.0000i			1.09337i	−0.837340	+	0.546683i	$0.815890\pi$
$968$		0			0
$969$	24.0000			0.770991
$970$		0			0
$971$	60.0000			1.92549			0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000			1.92549			0.962746	+	0.270408i	$0.0871586\pi$
$972$		0			0
$973$			8.00000i			0.256468i
$974$		0			0
$975$	−6.00000	+	8.00000i	−0.192154	+	0.256205i
$976$		0			0
$977$		−	12.0000i		−	0.383914i	−0.981403	−	0.191957i	$-0.938517\pi$
$977$		−	12.0000i		−	0.383914i	0.981403	−	0.191957i	$-0.0614834\pi$
$978$		0			0
$979$	14.0000			0.447442
$980$		0			0
$981$	−10.0000			−0.319275
$982$		0			0
$983$		−	28.0000i		−	0.893061i	−0.894768	−	0.446531i	$-0.852659\pi$
$983$		−	28.0000i		−	0.893061i	0.894768	−	0.446531i	$-0.147341\pi$
$984$		0			0
$985$	−12.0000	+	6.00000i	−0.382352	+	0.191176i
$986$		0			0
$987$			16.0000i			0.509286i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−56.0000			−1.77890			−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000			−1.77890			−0.889449	+	0.457034i	$0.848912\pi$
$992$		0			0
$993$		−	28.0000i		−	0.888553i
$994$		0			0
$995$	4.00000	+	8.00000i	0.126809	+	0.253617i
$996$		0			0
$997$			2.00000i			0.0633406i	0.999498	+	0.0316703i	$0.0100827\pi$
$997$			2.00000i			0.0633406i	−0.999498	+	0.0316703i	$0.989917\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	660.2.c.a.529.1	✓	2
3.2	odd	2		1980.2.c.c.1189.2		2
4.3	odd	2		2640.2.d.a.529.2		2
5.2	odd	4		3300.2.a.g.1.1		1
5.3	odd	4		3300.2.a.l.1.1		1
5.4	even	2	inner	660.2.c.a.529.2	yes	2
15.2	even	4		9900.2.a.v.1.1		1
15.8	even	4		9900.2.a.f.1.1		1
15.14	odd	2		1980.2.c.c.1189.1		2
20.19	odd	2		2640.2.d.a.529.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
660.2.c.a.529.1	✓	2	1.1	even	1	trivial
660.2.c.a.529.2	yes	2	5.4	even	2	inner
1980.2.c.c.1189.1		2	15.14	odd	2
1980.2.c.c.1189.2		2	3.2	odd	2
2640.2.d.a.529.1		2	20.19	odd	2
2640.2.d.a.529.2		2	4.3	odd	2
3300.2.a.g.1.1		1	5.2	odd	4
3300.2.a.l.1.1		1	5.3	odd	4
9900.2.a.f.1.1		1	15.8	even	4
9900.2.a.v.1.1		1	15.2	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:	\( N \)	\(=\)	\( 660 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	660.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} -2.00000i q^{7} -1.00000 q^{9} +1.00000 q^{11} +2.00000i q^{13} +(-2.00000 + 1.00000i) q^{15} -6.00000i q^{17} -4.00000 q^{19} -2.00000 q^{21} +(-3.00000 + 4.00000i) q^{25} +1.00000i q^{27} -2.00000 q^{29} -8.00000 q^{31} -1.00000i q^{33} +(-4.00000 + 2.00000i) q^{35} +2.00000 q^{39} -2.00000 q^{41} +2.00000i q^{43} +(1.00000 + 2.00000i) q^{45} -8.00000i q^{47} +3.00000 q^{49} -6.00000 q^{51} -8.00000i q^{53} +(-1.00000 - 2.00000i) q^{55} +4.00000i q^{57} +8.00000 q^{59} +10.0000 q^{61} +2.00000i q^{63} +(4.00000 - 2.00000i) q^{65} -8.00000i q^{67} -12.0000 q^{71} +14.0000i q^{73} +(4.00000 + 3.00000i) q^{75} -2.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{83} +(-12.0000 + 6.00000i) q^{85} +2.00000i q^{87} +14.0000 q^{89} +4.00000 q^{91} +8.00000i q^{93} +(4.00000 + 8.00000i) q^{95} -12.0000i q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{9} + 2 q^{11} - 4 q^{15} - 8 q^{19} - 4 q^{21} - 6 q^{25} - 4 q^{29} - 16 q^{31} - 8 q^{35} + 4 q^{39} - 4 q^{41} + 2 q^{45} + 6 q^{49} - 12 q^{51} - 2 q^{55} + 16 q^{59} + 20 q^{61}+ \cdots - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^9 + 2 * q^11 - 4 * q^15 - 8 * q^19 - 4 * q^21 - 6 * q^25 - 4 * q^29 - 16 * q^31 - 8 * q^35 + 4 * q^39 - 4 * q^41 + 2 * q^45 + 6 * q^49 - 12 * q^51 - 2 * q^55 + 16 * q^59 + 20 * q^61 + 8 * q^65 - 24 * q^71 + 8 * q^75 - 16 * q^79 + 2 * q^81 - 24 * q^85 + 28 * q^89 + 8 * q^91 + 8 * q^95 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more