LMFDB - Embedded newform 4320.2.q.h.1441.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4320,2,Mod(1441,4320)] 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("4320.1441"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4320, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,1,0,3,0,0,0,4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1441.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	4320.1441
Dual form			4320.2.q.h.2881.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{5} +(1.50000 + 2.59808i) q^{7} +(2.00000 + 3.46410i) q^{11} +(2.00000 - 3.46410i) q^{13} -2.00000 q^{17} -2.00000 q^{19} +(3.50000 - 6.06218i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-4.50000 - 7.79423i) q^{29} +(3.00000 - 5.19615i) q^{31} +3.00000 q^{35} +2.00000 q^{37} +(4.50000 - 7.79423i) q^{41} +(2.00000 + 3.46410i) q^{43} +(1.50000 + 2.59808i) q^{47} +(-1.00000 + 1.73205i) q^{49} -6.00000 q^{53} +4.00000 q^{55} +(3.00000 - 5.19615i) q^{59} +(6.50000 + 11.2583i) q^{61} +(-2.00000 - 3.46410i) q^{65} +(1.50000 - 2.59808i) q^{67} +8.00000 q^{73} +(-6.00000 + 10.3923i) q^{77} +(5.00000 + 8.66025i) q^{79} +(-3.50000 - 6.06218i) q^{83} +(-1.00000 + 1.73205i) q^{85} -1.00000 q^{89} +12.0000 q^{91} +(-1.00000 + 1.73205i) q^{95} +(7.00000 + 12.1244i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} + 3 q^{7} + 4 q^{11} + 4 q^{13} - 4 q^{17} - 4 q^{19} + 7 q^{23} - q^{25} - 9 q^{29} + 6 q^{31} + 6 q^{35} + 4 q^{37} + 9 q^{41} + 4 q^{43} + 3 q^{47} - 2 q^{49} - 12 q^{53} + 8 q^{55} + 6 q^{59}+ \cdots + 14 q^{97}+O(q^{100}) $ 2 * q + q^5 + 3 * q^7 + 4 * q^11 + 4 * q^13 - 4 * q^17 - 4 * q^19 + 7 * q^23 - q^25 - 9 * q^29 + 6 * q^31 + 6 * q^35 + 4 * q^37 + 9 * q^41 + 4 * q^43 + 3 * q^47 - 2 * q^49 - 12 * q^53 + 8 * q^55 + 6 * q^59 + 13 * q^61 - 4 * q^65 + 3 * q^67 + 16 * q^73 - 12 * q^77 + 10 * q^79 - 7 * q^83 - 2 * q^85 - 2 * q^89 + 24 * q^91 - 2 * q^95 + 14 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4320\mathbb{Z}\right)^\times$.

$n$	$2081$	$2431$	$3457$	$3781$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$1$

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
$3$		0			0
$4$		0			0
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$6$		0			0
$7$	1.50000	+	2.59808i	0.566947	+	0.981981i	0.996866	+	0.0791130i	$0.0252088\pi$
$7$	1.50000	+	2.59808i	0.566947	+	0.981981i	−0.429919	+	0.902867i	$0.641458\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.00000	+	3.46410i	0.603023	+	1.04447i	0.992361	+	0.123371i	$0.0393705\pi$
$11$	2.00000	+	3.46410i	0.603023	+	1.04447i	−0.389338	+	0.921095i	$0.627296\pi$
$12$		0			0
$13$	2.00000	−	3.46410i	0.554700	−	0.960769i	−0.443227	−	0.896410i	$-0.646166\pi$
$13$	2.00000	−	3.46410i	0.554700	−	0.960769i	0.997927	−	0.0643593i	$-0.0205004\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		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
$21$		0			0
$22$		0			0
$23$	3.50000	−	6.06218i	0.729800	−	1.26405i	−0.227167	−	0.973856i	$-0.572946\pi$
$23$	3.50000	−	6.06218i	0.729800	−	1.26405i	0.956967	−	0.290196i	$-0.0937204\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.50000	−	7.79423i	−0.835629	−	1.44735i	−0.893517	−	0.449029i	$-0.851770\pi$
$29$	−4.50000	−	7.79423i	−0.835629	−	1.44735i	0.0578882	−	0.998323i	$-0.481563\pi$
$30$		0			0
$31$	3.00000	−	5.19615i	0.538816	−	0.933257i	−0.460152	−	0.887840i	$-0.652205\pi$
$31$	3.00000	−	5.19615i	0.538816	−	0.933257i	0.998968	−	0.0454165i	$-0.0144615\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.00000			0.507093
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.50000	−	7.79423i	0.702782	−	1.21725i	−0.264704	−	0.964330i	$-0.585274\pi$
$41$	4.50000	−	7.79423i	0.702782	−	1.21725i	0.967486	−	0.252924i	$-0.0813924\pi$
$42$		0			0
$43$	2.00000	+	3.46410i	0.304997	+	0.528271i	0.977261	−	0.212041i	$-0.0680112\pi$
$43$	2.00000	+	3.46410i	0.304997	+	0.528271i	−0.672264	+	0.740312i	$0.734678\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.50000	+	2.59808i	0.218797	+	0.378968i	0.954441	−	0.298401i	$-0.0964533\pi$
$47$	1.50000	+	2.59808i	0.218797	+	0.378968i	−0.735643	+	0.677369i	$0.763120\pi$
$48$		0			0
$49$	−1.00000	+	1.73205i	−0.142857	+	0.247436i
$50$		0			0
$51$		0			0
$52$		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
$55$	4.00000			0.539360
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.00000	−	5.19615i	0.390567	−	0.676481i	−0.601958	−	0.798528i	$-0.705612\pi$
$59$	3.00000	−	5.19615i	0.390567	−	0.676481i	0.992524	+	0.122047i	$0.0389457\pi$
$60$		0			0
$61$	6.50000	+	11.2583i	0.832240	+	1.44148i	0.896258	+	0.443533i	$0.146275\pi$
$61$	6.50000	+	11.2583i	0.832240	+	1.44148i	−0.0640184	+	0.997949i	$0.520392\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.00000	−	3.46410i	−0.248069	−	0.429669i
$66$		0			0
$67$	1.50000	−	2.59808i	0.183254	−	0.317406i	−0.759733	−	0.650236i	$-0.774670\pi$
$67$	1.50000	−	2.59808i	0.183254	−	0.317406i	0.942987	+	0.332830i	$0.108004\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	8.00000			0.936329			0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000			0.936329			0.468165	+	0.883641i	$0.344915\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.00000	+	10.3923i	−0.683763	+	1.18431i
$78$		0			0
$79$	5.00000	+	8.66025i	0.562544	+	0.974355i	0.997274	+	0.0737937i	$0.0235106\pi$
$79$	5.00000	+	8.66025i	0.562544	+	0.974355i	−0.434730	+	0.900561i	$0.643156\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.50000	−	6.06218i	−0.384175	−	0.665410i	0.607479	−	0.794335i	$-0.292181\pi$
$83$	−3.50000	−	6.06218i	−0.384175	−	0.665410i	−0.991654	+	0.128925i	$0.958847\pi$
$84$		0			0
$85$	−1.00000	+	1.73205i	−0.108465	+	0.187867i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.00000			−0.106000			−0.0529999	−	0.998595i	$-0.516878\pi$
$89$	−1.00000			−0.106000			−0.0529999	+	0.998595i	$0.516878\pi$
$90$		0			0
$91$	12.0000			1.25794
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.00000	+	1.73205i	−0.102598	+	0.177705i
$96$		0			0
$97$	7.00000	+	12.1244i	0.710742	+	1.23104i	0.964579	+	0.263795i	$0.0849741\pi$
$97$	7.00000	+	12.1244i	0.710742	+	1.23104i	−0.253837	+	0.967247i	$0.581693\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.00000	+	12.1244i	0.696526	+	1.20642i	0.969664	+	0.244443i	$0.0786053\pi$
$101$	7.00000	+	12.1244i	0.696526	+	1.20642i	−0.273138	+	0.961975i	$0.588061\pi$
$102$		0			0
$103$	4.00000	−	6.92820i	0.394132	−	0.682656i	−0.598858	−	0.800855i	$-0.704379\pi$
$103$	4.00000	−	6.92820i	0.394132	−	0.682656i	0.992990	+	0.118199i	$0.0377120\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−15.0000			−1.45010			−0.725052	−	0.688694i	$-0.758184\pi$
$107$	−15.0000			−1.45010			−0.725052	+	0.688694i	$0.758184\pi$
$108$		0			0
$109$	−7.00000			−0.670478			−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000			−0.670478			−0.335239	+	0.942133i	$0.608817\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.00000	+	10.3923i	−0.564433	+	0.977626i	0.432670	+	0.901553i	$0.357572\pi$
$113$	−6.00000	+	10.3923i	−0.564433	+	0.977626i	−0.997102	+	0.0760733i	$0.975762\pi$
$114$		0			0
$115$	−3.50000	−	6.06218i	−0.326377	−	0.565301i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.00000	−	5.19615i	−0.275010	−	0.476331i
$120$		0			0
$121$	−2.50000	+	4.33013i	−0.227273	+	0.393648i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	19.0000			1.68598			0.842989	−	0.537931i	$-0.180794\pi$
$127$	19.0000			1.68598			0.842989	+	0.537931i	$0.180794\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	11.0000	−	19.0526i	0.961074	−	1.66463i	0.241264	−	0.970460i	$-0.422438\pi$
$131$	11.0000	−	19.0526i	0.961074	−	1.66463i	0.719811	−	0.694170i	$-0.244228\pi$
$132$		0			0
$133$	−3.00000	−	5.19615i	−0.260133	−	0.450564i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.00000	−	13.8564i	−0.683486	−	1.18383i	−0.973910	−	0.226935i	$-0.927130\pi$
$137$	−8.00000	−	13.8564i	−0.683486	−	1.18383i	0.290424	−	0.956898i	$-0.406204\pi$
$138$		0			0
$139$	−4.00000	+	6.92820i	−0.339276	+	0.587643i	−0.984297	−	0.176522i	$-0.943515\pi$
$139$	−4.00000	+	6.92820i	−0.339276	+	0.587643i	0.645021	+	0.764165i	$0.276849\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	16.0000			1.33799
$144$		0			0
$145$	−9.00000			−0.747409
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.50000	+	2.59808i	−0.122885	+	0.212843i	−0.920904	−	0.389789i	$-0.872548\pi$
$149$	−1.50000	+	2.59808i	−0.122885	+	0.212843i	0.798019	+	0.602632i	$0.205881\pi$
$150$		0			0
$151$	10.0000	+	17.3205i	0.813788	+	1.40952i	0.910195	+	0.414181i	$0.135932\pi$
$151$	10.0000	+	17.3205i	0.813788	+	1.40952i	−0.0964061	+	0.995342i	$0.530735\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.00000	−	5.19615i	−0.240966	−	0.417365i
$156$		0			0
$157$	7.00000	−	12.1244i	0.558661	−	0.967629i	−0.438948	−	0.898513i	$-0.644649\pi$
$157$	7.00000	−	12.1244i	0.558661	−	0.967629i	0.997609	−	0.0691164i	$-0.0220180\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	21.0000			1.65503
$162$		0			0
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.50000	−	2.59808i	0.116073	−	0.201045i	−0.802135	−	0.597143i	$-0.796303\pi$
$167$	1.50000	−	2.59808i	0.116073	−	0.201045i	0.918208	+	0.396098i	$0.129636\pi$
$168$		0			0
$169$	−1.50000	−	2.59808i	−0.115385	−	0.199852i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.00000	−	3.46410i	−0.152057	−	0.263371i	0.779926	−	0.625871i	$-0.215256\pi$
$173$	−2.00000	−	3.46410i	−0.152057	−	0.263371i	−0.931984	+	0.362500i	$0.881923\pi$
$174$		0			0
$175$	1.50000	−	2.59808i	0.113389	−	0.196396i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	24.0000			1.79384			0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000			1.79384			0.896922	+	0.442189i	$0.145798\pi$
$180$		0			0
$181$	−23.0000			−1.70958			−0.854788	−	0.518977i	$-0.826313\pi$
$181$	−23.0000			−1.70958			−0.854788	+	0.518977i	$0.826313\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.00000	−	1.73205i	0.0735215	−	0.127343i
$186$		0			0
$187$	−4.00000	−	6.92820i	−0.292509	−	0.506640i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	5.00000	+	8.66025i	0.361787	+	0.626634i	0.988255	−	0.152813i	$-0.0488333\pi$
$191$	5.00000	+	8.66025i	0.361787	+	0.626634i	−0.626468	+	0.779447i	$0.715500\pi$
$192$		0			0
$193$	−10.0000	+	17.3205i	−0.719816	+	1.24676i	0.241257	+	0.970461i	$0.422440\pi$
$193$	−10.0000	+	17.3205i	−0.719816	+	1.24676i	−0.961073	+	0.276296i	$0.910893\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	20.0000			1.42494			0.712470	−	0.701702i	$-0.247576\pi$
$197$	20.0000			1.42494			0.712470	+	0.701702i	$0.247576\pi$
$198$		0			0
$199$	14.0000			0.992434			0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000			0.992434			0.496217	+	0.868199i	$0.334722\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	13.5000	−	23.3827i	0.947514	−	1.64114i
$204$		0			0
$205$	−4.50000	−	7.79423i	−0.314294	−	0.544373i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4.00000	−	6.92820i	−0.276686	−	0.479234i
$210$		0			0
$211$	−10.0000	+	17.3205i	−0.688428	+	1.19239i	0.283918	+	0.958849i	$0.408366\pi$
$211$	−10.0000	+	17.3205i	−0.688428	+	1.19239i	−0.972346	+	0.233544i	$0.924968\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.00000			0.272798
$216$		0			0
$217$	18.0000			1.22192
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.00000	+	6.92820i	−0.269069	+	0.466041i
$222$		0			0
$223$	−4.50000	−	7.79423i	−0.301342	−	0.521940i	0.675098	−	0.737728i	$-0.264101\pi$
$223$	−4.50000	−	7.79423i	−0.301342	−	0.521940i	−0.976440	+	0.215788i	$0.930768\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.00000	−	3.46410i	−0.132745	−	0.229920i	0.791989	−	0.610535i	$-0.209046\pi$
$227$	−2.00000	−	3.46410i	−0.132745	−	0.229920i	−0.924734	+	0.380615i	$0.875712\pi$
$228$		0			0
$229$	10.5000	−	18.1865i	0.693860	−	1.20180i	−0.276704	−	0.960955i	$-0.589242\pi$
$229$	10.5000	−	18.1865i	0.693860	−	1.20180i	0.970564	−	0.240845i	$-0.0774245\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$	3.00000			0.195698
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.00000	−	3.46410i	0.129369	−	0.224074i	−0.794063	−	0.607835i	$-0.792038\pi$
$239$	2.00000	−	3.46410i	0.129369	−	0.224074i	0.923432	+	0.383761i	$0.125371\pi$
$240$		0			0
$241$	−8.50000	−	14.7224i	−0.547533	−	0.948355i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	−8.50000	−	14.7224i	−0.547533	−	0.948355i	0.450910	−	0.892570i	$-0.351100\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.00000	+	1.73205i	0.0638877	+	0.110657i
$246$		0			0
$247$	−4.00000	+	6.92820i	−0.254514	+	0.440831i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−18.0000			−1.13615			−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000			−1.13615			−0.568075	+	0.822977i	$0.692312\pi$
$252$		0			0
$253$	28.0000			1.76034
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−9.00000	+	15.5885i	−0.561405	+	0.972381i	0.435970	+	0.899961i	$0.356405\pi$
$257$	−9.00000	+	15.5885i	−0.561405	+	0.972381i	−0.997374	+	0.0724199i	$0.976928\pi$
$258$		0			0
$259$	3.00000	+	5.19615i	0.186411	+	0.322873i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.00000	+	6.92820i	0.246651	+	0.427211i	0.962594	−	0.270947i	$-0.0873367\pi$
$263$	4.00000	+	6.92820i	0.246651	+	0.427211i	−0.715944	+	0.698158i	$0.754003\pi$
$264$		0			0
$265$	−3.00000	+	5.19615i	−0.184289	+	0.319197i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5.00000			−0.304855			−0.152428	−	0.988315i	$-0.548709\pi$
$269$	−5.00000			−0.304855			−0.152428	+	0.988315i	$0.548709\pi$
$270$		0			0
$271$	−2.00000			−0.121491			−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000			−0.121491			−0.0607457	+	0.998153i	$0.519348\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.00000	−	3.46410i	0.120605	−	0.208893i
$276$		0			0
$277$	−9.00000	−	15.5885i	−0.540758	−	0.936620i	−0.998861	−	0.0477206i	$-0.984804\pi$
$277$	−9.00000	−	15.5885i	−0.540758	−	0.936620i	0.458103	−	0.888899i	$-0.348529\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−11.5000	−	19.9186i	−0.686032	−	1.18824i	−0.973111	−	0.230336i	$-0.926017\pi$
$281$	−11.5000	−	19.9186i	−0.686032	−	1.18824i	0.287079	−	0.957907i	$-0.407316\pi$
$282$		0			0
$283$	11.5000	−	19.9186i	0.683604	−	1.18404i	−0.290269	−	0.956945i	$-0.593745\pi$
$283$	11.5000	−	19.9186i	0.683604	−	1.18404i	0.973873	−	0.227092i	$-0.0729218\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	27.0000			1.59376
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.00000	+	1.73205i	−0.0584206	+	0.101187i	−0.893757	−	0.448552i	$-0.851940\pi$
$293$	−1.00000	+	1.73205i	−0.0584206	+	0.101187i	0.835336	+	0.549740i	$0.185273\pi$
$294$		0			0
$295$	−3.00000	−	5.19615i	−0.174667	−	0.302532i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−14.0000	−	24.2487i	−0.809641	−	1.40234i
$300$		0			0
$301$	−6.00000	+	10.3923i	−0.345834	+	0.599002i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	13.0000			0.744378
$306$		0			0
$307$	11.0000			0.627803			0.313902	−	0.949456i	$-0.398364\pi$
$307$	11.0000			0.627803			0.313902	+	0.949456i	$0.398364\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−15.0000	+	25.9808i	−0.850572	+	1.47323i	0.0301210	+	0.999546i	$0.490411\pi$
$311$	−15.0000	+	25.9808i	−0.850572	+	1.47323i	−0.880693	+	0.473688i	$0.842923\pi$
$312$		0			0
$313$	−8.00000	−	13.8564i	−0.452187	−	0.783210i	0.546335	−	0.837567i	$-0.316023\pi$
$313$	−8.00000	−	13.8564i	−0.452187	−	0.783210i	−0.998522	+	0.0543564i	$0.982689\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$317$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$	18.0000	−	31.1769i	1.00781	−	1.74557i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.00000			0.222566
$324$		0			0
$325$	−4.00000			−0.221880
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.50000	+	7.79423i	−0.248093	+	0.429710i
$330$		0			0
$331$	3.00000	+	5.19615i	0.164895	+	0.285606i	0.936618	−	0.350352i	$-0.113938\pi$
$331$	3.00000	+	5.19615i	0.164895	+	0.285606i	−0.771723	+	0.635959i	$0.780605\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.50000	−	2.59808i	−0.0819538	−	0.141948i
$336$		0			0
$337$	8.00000	−	13.8564i	0.435788	−	0.754807i	−0.561572	−	0.827428i	$-0.689803\pi$
$337$	8.00000	−	13.8564i	0.435788	−	0.754807i	0.997360	+	0.0726214i	$0.0231365\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	24.0000			1.29967
$342$		0			0
$343$	15.0000			0.809924
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4.00000	−	6.92820i	0.214731	−	0.371925i	−0.738458	−	0.674299i	$-0.764446\pi$
$347$	4.00000	−	6.92820i	0.214731	−	0.371925i	0.953189	+	0.302374i	$0.0977791\pi$
$348$		0			0
$349$	14.5000	+	25.1147i	0.776167	+	1.34436i	0.934136	+	0.356917i	$0.116172\pi$
$349$	14.5000	+	25.1147i	0.776167	+	1.34436i	−0.157969	+	0.987444i	$0.550495\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8.00000	+	13.8564i	0.425797	+	0.737502i	0.996495	−	0.0836583i	$-0.0266604\pi$
$353$	8.00000	+	13.8564i	0.425797	+	0.737502i	−0.570697	+	0.821160i	$0.693327\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.00000			−0.105556			−0.0527780	−	0.998606i	$-0.516808\pi$
$359$	−2.00000			−0.105556			−0.0527780	+	0.998606i	$0.516808\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.00000	−	6.92820i	0.209370	−	0.362639i
$366$		0			0
$367$	12.0000	+	20.7846i	0.626395	+	1.08495i	0.988269	+	0.152721i	$0.0488036\pi$
$367$	12.0000	+	20.7846i	0.626395	+	1.08495i	−0.361874	+	0.932227i	$0.617863\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−9.00000	−	15.5885i	−0.467257	−	0.809312i
$372$		0			0
$373$	5.00000	−	8.66025i	0.258890	−	0.448411i	−0.707055	−	0.707159i	$-0.749977\pi$
$373$	5.00000	−	8.66025i	0.258890	−	0.448411i	0.965945	+	0.258748i	$0.0833099\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−36.0000			−1.85409
$378$		0			0
$379$	−26.0000			−1.33553			−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000			−1.33553			−0.667765	+	0.744372i	$0.732749\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	8.00000	−	13.8564i	0.408781	−	0.708029i	−0.585973	−	0.810331i	$-0.699287\pi$
$383$	8.00000	−	13.8564i	0.408781	−	0.708029i	0.994753	+	0.102302i	$0.0326207\pi$
$384$		0			0
$385$	6.00000	+	10.3923i	0.305788	+	0.529641i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.50000	−	2.59808i	−0.0760530	−	0.131728i	0.825491	−	0.564416i	$-0.190898\pi$
$389$	−1.50000	−	2.59808i	−0.0760530	−	0.131728i	−0.901544	+	0.432688i	$0.857565\pi$
$390$		0			0
$391$	−7.00000	+	12.1244i	−0.354005	+	0.613155i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	10.0000			0.503155
$396$		0			0
$397$	−18.0000			−0.903394			−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000			−0.903394			−0.451697	+	0.892171i	$0.649181\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5.00000	+	8.66025i	−0.249688	+	0.432472i	−0.963439	−	0.267927i	$-0.913661\pi$
$401$	−5.00000	+	8.66025i	−0.249688	+	0.432472i	0.713751	+	0.700399i	$0.246995\pi$
$402$		0			0
$403$	−12.0000	−	20.7846i	−0.597763	−	1.03536i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.00000	+	6.92820i	0.198273	+	0.343418i
$408$		0			0
$409$	5.00000	−	8.66025i	0.247234	−	0.428222i	−0.715523	−	0.698589i	$-0.753812\pi$
$409$	5.00000	−	8.66025i	0.247234	−	0.428222i	0.962757	+	0.270367i	$0.0871450\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	18.0000			0.885722
$414$		0			0
$415$	−7.00000			−0.343616
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−18.0000	+	31.1769i	−0.879358	+	1.52309i	−0.0273103	+	0.999627i	$0.508694\pi$
$419$	−18.0000	+	31.1769i	−0.879358	+	1.52309i	−0.852047	+	0.523465i	$0.824639\pi$
$420$		0			0
$421$	−5.00000	−	8.66025i	−0.243685	−	0.422075i	0.718076	−	0.695965i	$-0.245023\pi$
$421$	−5.00000	−	8.66025i	−0.243685	−	0.422075i	−0.961761	+	0.273890i	$0.911690\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.00000	+	1.73205i	0.0485071	+	0.0840168i
$426$		0			0
$427$	−19.5000	+	33.7750i	−0.943671	+	1.63449i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−10.0000			−0.481683			−0.240842	−	0.970564i	$-0.577423\pi$
$431$	−10.0000			−0.481683			−0.240842	+	0.970564i	$0.577423\pi$
$432$		0			0
$433$	−10.0000			−0.480569			−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000			−0.480569			−0.240285	+	0.970702i	$0.577241\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−7.00000	+	12.1244i	−0.334855	+	0.579987i
$438$		0			0
$439$	14.0000	+	24.2487i	0.668184	+	1.15733i	0.978412	+	0.206666i	$0.0662612\pi$
$439$	14.0000	+	24.2487i	0.668184	+	1.15733i	−0.310228	+	0.950662i	$0.600405\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6.50000	−	11.2583i	−0.308824	−	0.534899i	0.669281	−	0.743009i	$-0.266602\pi$
$443$	−6.50000	−	11.2583i	−0.308824	−	0.534899i	−0.978105	+	0.208110i	$0.933269\pi$
$444$		0			0
$445$	−0.500000	+	0.866025i	−0.0237023	+	0.0410535i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	10.0000			0.471929			0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000			0.471929			0.235965	+	0.971762i	$0.424175\pi$
$450$		0			0
$451$	36.0000			1.69517
$452$		0			0
$453$		0			0
$454$		0			0
$455$	6.00000	−	10.3923i	0.281284	−	0.487199i
$456$		0			0
$457$	−1.00000	−	1.73205i	−0.0467780	−	0.0810219i	0.841688	−	0.539964i	$-0.181562\pi$
$457$	−1.00000	−	1.73205i	−0.0467780	−	0.0810219i	−0.888466	+	0.458942i	$0.848229\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−9.50000	−	16.4545i	−0.442459	−	0.766362i	0.555412	−	0.831575i	$-0.312560\pi$
$461$	−9.50000	−	16.4545i	−0.442459	−	0.766362i	−0.997871	+	0.0652135i	$0.979227\pi$
$462$		0			0
$463$	2.00000	−	3.46410i	0.0929479	−	0.160990i	−0.815802	−	0.578331i	$-0.803704\pi$
$463$	2.00000	−	3.46410i	0.0929479	−	0.160990i	0.908750	+	0.417340i	$0.137038\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	12.0000			0.555294			0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000			0.555294			0.277647	+	0.960683i	$0.410445\pi$
$468$		0			0
$469$	9.00000			0.415581
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−8.00000	+	13.8564i	−0.367840	+	0.637118i
$474$		0			0
$475$	1.00000	+	1.73205i	0.0458831	+	0.0794719i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	13.0000	+	22.5167i	0.593985	+	1.02881i	0.993689	+	0.112168i	$0.0357796\pi$
$479$	13.0000	+	22.5167i	0.593985	+	1.02881i	−0.399704	+	0.916644i	$0.630887\pi$
$480$		0			0
$481$	4.00000	−	6.92820i	0.182384	−	0.315899i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	14.0000			0.635707
$486$		0			0
$487$	40.0000			1.81257			0.906287	−	0.422664i	$-0.138905\pi$
$487$	40.0000			1.81257			0.906287	+	0.422664i	$0.138905\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−18.0000	+	31.1769i	−0.812329	+	1.40699i	0.0989017	+	0.995097i	$0.468467\pi$
$491$	−18.0000	+	31.1769i	−0.812329	+	1.40699i	−0.911230	+	0.411897i	$0.864866\pi$
$492$		0			0
$493$	9.00000	+	15.5885i	0.405340	+	0.702069i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	16.0000	−	27.7128i	0.716258	−	1.24060i	−0.246214	−	0.969216i	$-0.579187\pi$
$499$	16.0000	−	27.7128i	0.716258	−	1.24060i	0.962472	−	0.271380i	$-0.0874801\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	7.00000			0.312115			0.156057	−	0.987748i	$-0.450122\pi$
$503$	7.00000			0.312115			0.156057	+	0.987748i	$0.450122\pi$
$504$		0			0
$505$	14.0000			0.622992
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6.50000	+	11.2583i	−0.288107	+	0.499017i	−0.973358	−	0.229291i	$-0.926359\pi$
$509$	−6.50000	+	11.2583i	−0.288107	+	0.499017i	0.685251	+	0.728307i	$0.259693\pi$
$510$		0			0
$511$	12.0000	+	20.7846i	0.530849	+	0.919457i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.00000	−	6.92820i	−0.176261	−	0.305293i
$516$		0			0
$517$	−6.00000	+	10.3923i	−0.263880	+	0.457053i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	21.0000			0.920027			0.460013	−	0.887912i	$-0.347845\pi$
$521$	21.0000			0.920027			0.460013	+	0.887912i	$0.347845\pi$
$522$		0			0
$523$	−41.0000			−1.79280			−0.896402	−	0.443241i	$-0.853829\pi$
$523$	−41.0000			−1.79280			−0.896402	+	0.443241i	$0.853829\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.00000	+	10.3923i	−0.261364	+	0.452696i
$528$		0			0
$529$	−13.0000	−	22.5167i	−0.565217	−	0.978985i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−18.0000	−	31.1769i	−0.779667	−	1.35042i
$534$		0			0
$535$	−7.50000	+	12.9904i	−0.324253	+	0.561623i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−8.00000			−0.344584
$540$		0			0
$541$	−3.00000			−0.128980			−0.0644900	−	0.997918i	$-0.520542\pi$
$541$	−3.00000			−0.128980			−0.0644900	+	0.997918i	$0.520542\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.50000	+	6.06218i	−0.149924	+	0.259675i
$546$		0			0
$547$	−11.5000	−	19.9186i	−0.491704	−	0.851657i	0.508250	−	0.861210i	$-0.330293\pi$
$547$	−11.5000	−	19.9186i	−0.491704	−	0.851657i	−0.999954	+	0.00955248i	$0.996959\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	9.00000	+	15.5885i	0.383413	+	0.664091i
$552$		0			0
$553$	−15.0000	+	25.9808i	−0.637865	+	1.10481i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−18.0000			−0.762684			−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000			−0.762684			−0.381342	+	0.924434i	$0.624538\pi$
$558$		0			0
$559$	16.0000			0.676728
$560$		0			0
$561$		0			0
$562$		0			0
$563$	19.5000	−	33.7750i	0.821827	−	1.42345i	−0.0824933	−	0.996592i	$-0.526288\pi$
$563$	19.5000	−	33.7750i	0.821827	−	1.42345i	0.904320	−	0.426855i	$-0.140378\pi$
$564$		0			0
$565$	6.00000	+	10.3923i	0.252422	+	0.437208i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	19.0000	+	32.9090i	0.796521	+	1.37962i	0.921869	+	0.387503i	$0.126662\pi$
$569$	19.0000	+	32.9090i	0.796521	+	1.37962i	−0.125347	+	0.992113i	$0.540004\pi$
$570$		0			0
$571$	−2.00000	+	3.46410i	−0.0836974	+	0.144968i	−0.904835	−	0.425762i	$-0.860006\pi$
$571$	−2.00000	+	3.46410i	−0.0836974	+	0.144968i	0.821138	+	0.570730i	$0.193340\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−7.00000			−0.291920
$576$		0			0
$577$	38.0000			1.58196			0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000			1.58196			0.790980	+	0.611842i	$0.209571\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	10.5000	−	18.1865i	0.435613	−	0.754505i
$582$		0			0
$583$	−12.0000	−	20.7846i	−0.496989	−	0.860811i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.50000	+	2.59808i	0.0619116	+	0.107234i	0.895320	−	0.445424i	$-0.146947\pi$
$587$	1.50000	+	2.59808i	0.0619116	+	0.107234i	−0.833408	+	0.552658i	$0.813614\pi$
$588$		0			0
$589$	−6.00000	+	10.3923i	−0.247226	+	0.428207i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	18.0000			0.739171			0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000			0.739171			0.369586	+	0.929197i	$0.379500\pi$
$594$		0			0
$595$	−6.00000			−0.245976
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−7.00000	+	12.1244i	−0.286012	+	0.495388i	−0.972854	−	0.231419i	$-0.925663\pi$
$599$	−7.00000	+	12.1244i	−0.286012	+	0.495388i	0.686842	+	0.726807i	$0.258996\pi$
$600$		0			0
$601$	−5.00000	−	8.66025i	−0.203954	−	0.353259i	0.745845	−	0.666120i	$-0.232046\pi$
$601$	−5.00000	−	8.66025i	−0.203954	−	0.353259i	−0.949799	+	0.312861i	$0.898713\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.50000	+	4.33013i	0.101639	+	0.176045i
$606$		0			0
$607$	2.50000	−	4.33013i	0.101472	−	0.175754i	−0.810819	−	0.585296i	$-0.800978\pi$
$607$	2.50000	−	4.33013i	0.101472	−	0.175754i	0.912291	+	0.409542i	$0.134311\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	12.0000			0.485468
$612$		0			0
$613$	−26.0000			−1.05013			−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000			−1.05013			−0.525065	+	0.851062i	$0.675959\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.00000	−	10.3923i	0.241551	−	0.418378i	−0.719605	−	0.694383i	$-0.755677\pi$
$617$	6.00000	−	10.3923i	0.241551	−	0.418378i	0.961156	+	0.276005i	$0.0890106\pi$
$618$		0			0
$619$	−5.00000	−	8.66025i	−0.200967	−	0.348085i	0.747873	−	0.663842i	$-0.231075\pi$
$619$	−5.00000	−	8.66025i	−0.200967	−	0.348085i	−0.948840	+	0.315757i	$0.897742\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.50000	−	2.59808i	−0.0600962	−	0.104090i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4.00000			−0.159490
$630$		0			0
$631$	−40.0000			−1.59237			−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000			−1.59237			−0.796187	+	0.605050i	$0.793153\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	9.50000	−	16.4545i	0.376996	−	0.652976i
$636$		0			0
$637$	4.00000	+	6.92820i	0.158486	+	0.274505i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5.50000	−	9.52628i	−0.217237	−	0.376265i	0.736725	−	0.676192i	$-0.236371\pi$
$641$	−5.50000	−	9.52628i	−0.217237	−	0.376265i	−0.953962	+	0.299927i	$0.903038\pi$
$642$		0			0
$643$	16.5000	−	28.5788i	0.650696	−	1.12704i	−0.332258	−	0.943189i	$-0.607810\pi$
$643$	16.5000	−	28.5788i	0.650696	−	1.12704i	0.982954	−	0.183851i	$-0.0588563\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−35.0000			−1.37599			−0.687996	−	0.725714i	$-0.741509\pi$
$647$	−35.0000			−1.37599			−0.687996	+	0.725714i	$0.741509\pi$
$648$		0			0
$649$	24.0000			0.942082
$650$		0			0
$651$		0			0
$652$		0			0
$653$	7.00000	−	12.1244i	0.273931	−	0.474463i	−0.695934	−	0.718106i	$-0.745009\pi$
$653$	7.00000	−	12.1244i	0.273931	−	0.474463i	0.969865	+	0.243643i	$0.0783426\pi$
$654$		0			0
$655$	−11.0000	−	19.0526i	−0.429806	−	0.744445i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.00000	−	3.46410i	−0.0779089	−	0.134942i	0.824439	−	0.565951i	$-0.191491\pi$
$659$	−2.00000	−	3.46410i	−0.0779089	−	0.134942i	−0.902348	+	0.431009i	$0.858158\pi$
$660$		0			0
$661$	−11.0000	+	19.0526i	−0.427850	+	0.741059i	−0.996682	−	0.0813955i	$-0.974062\pi$
$661$	−11.0000	+	19.0526i	−0.427850	+	0.741059i	0.568831	+	0.822454i	$0.307396\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.00000			−0.232670
$666$		0			0
$667$	−63.0000			−2.43937
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−26.0000	+	45.0333i	−1.00372	+	1.73849i
$672$		0			0
$673$	−18.0000	−	31.1769i	−0.693849	−	1.20178i	−0.970567	−	0.240831i	$-0.922580\pi$
$673$	−18.0000	−	31.1769i	−0.693849	−	1.20178i	0.276718	−	0.960951i	$-0.410753\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	−21.0000	+	36.3731i	−0.805906	+	1.39587i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.0000			−0.459167			−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$		0			0
$685$	−16.0000			−0.611329
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−12.0000	+	20.7846i	−0.457164	+	0.791831i
$690$		0			0
$691$	17.0000	+	29.4449i	0.646710	+	1.12014i	0.983904	+	0.178700i	$0.0571891\pi$
$691$	17.0000	+	29.4449i	0.646710	+	1.12014i	−0.337193	+	0.941435i	$0.609478\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	4.00000	+	6.92820i	0.151729	+	0.262802i
$696$		0			0
$697$	−9.00000	+	15.5885i	−0.340899	+	0.590455i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1.00000			0.0377695			0.0188847	−	0.999822i	$-0.493988\pi$
$701$	1.00000			0.0377695			0.0188847	+	0.999822i	$0.493988\pi$
$702$		0			0
$703$	−4.00000			−0.150863
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−21.0000	+	36.3731i	−0.789786	+	1.36795i
$708$		0			0
$709$	20.5000	+	35.5070i	0.769894	+	1.33349i	0.937620	+	0.347661i	$0.113024\pi$
$709$	20.5000	+	35.5070i	0.769894	+	1.33349i	−0.167727	+	0.985834i	$0.553643\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−21.0000	−	36.3731i	−0.786456	−	1.36218i
$714$		0			0
$715$	8.00000	−	13.8564i	0.299183	−	0.518200i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2.00000			−0.0745874			−0.0372937	−	0.999304i	$-0.511874\pi$
$719$	−2.00000			−0.0745874			−0.0372937	+	0.999304i	$0.511874\pi$
$720$		0			0
$721$	24.0000			0.893807
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.50000	+	7.79423i	−0.167126	+	0.289470i
$726$		0			0
$727$	−11.5000	−	19.9186i	−0.426511	−	0.738739i	0.570049	−	0.821611i	$-0.306924\pi$
$727$	−11.5000	−	19.9186i	−0.426511	−	0.738739i	−0.996560	+	0.0828714i	$0.973591\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−4.00000	−	6.92820i	−0.147945	−	0.256249i
$732$		0			0
$733$	−17.0000	+	29.4449i	−0.627909	+	1.08757i	0.360061	+	0.932929i	$0.382756\pi$
$733$	−17.0000	+	29.4449i	−0.627909	+	1.08757i	−0.987971	+	0.154642i	$0.950578\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	12.0000			0.442026
$738$		0			0
$739$	−34.0000			−1.25071			−0.625355	−	0.780340i	$-0.715046\pi$
$739$	−34.0000			−1.25071			−0.625355	+	0.780340i	$0.715046\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	13.5000	−	23.3827i	0.495267	−	0.857828i	−0.504718	−	0.863284i	$-0.668404\pi$
$743$	13.5000	−	23.3827i	0.495267	−	0.857828i	0.999985	+	0.00545664i	$0.00173691\pi$
$744$		0			0
$745$	1.50000	+	2.59808i	0.0549557	+	0.0951861i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−22.5000	−	38.9711i	−0.822132	−	1.42397i
$750$		0			0
$751$	4.00000	−	6.92820i	0.145962	−	0.252814i	−0.783769	−	0.621052i	$-0.786706\pi$
$751$	4.00000	−	6.92820i	0.145962	−	0.252814i	0.929731	+	0.368238i	$0.120039\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	20.0000			0.727875
$756$		0			0
$757$	−8.00000			−0.290765			−0.145382	−	0.989376i	$-0.546441\pi$
$757$	−8.00000			−0.290765			−0.145382	+	0.989376i	$0.546441\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−10.5000	+	18.1865i	−0.380625	+	0.659261i	−0.991152	−	0.132734i	$-0.957624\pi$
$761$	−10.5000	+	18.1865i	−0.380625	+	0.659261i	0.610527	+	0.791995i	$0.290958\pi$
$762$		0			0
$763$	−10.5000	−	18.1865i	−0.380126	−	0.658397i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−12.0000	−	20.7846i	−0.433295	−	0.750489i
$768$		0			0
$769$	23.5000	−	40.7032i	0.847432	−	1.46779i	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	23.5000	−	40.7032i	0.847432	−	1.46779i	0.883493	−	0.468445i	$-0.155186\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−44.0000			−1.58257			−0.791285	−	0.611448i	$-0.790588\pi$
$773$	−44.0000			−1.58257			−0.791285	+	0.611448i	$0.790588\pi$
$774$		0			0
$775$	−6.00000			−0.215526
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−9.00000	+	15.5885i	−0.322458	+	0.558514i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−7.00000	−	12.1244i	−0.249841	−	0.432737i
$786$		0			0
$787$	−20.0000	+	34.6410i	−0.712923	+	1.23482i	0.250832	+	0.968031i	$0.419296\pi$
$787$	−20.0000	+	34.6410i	−0.712923	+	1.23482i	−0.963755	+	0.266788i	$0.914038\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−36.0000			−1.28001
$792$		0			0
$793$	52.0000			1.84657
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−14.0000	+	24.2487i	−0.495905	+	0.858933i	−0.999989	−	0.00472155i	$-0.998497\pi$
$797$	−14.0000	+	24.2487i	−0.495905	+	0.858933i	0.504083	+	0.863655i	$0.331830\pi$
$798$		0			0
$799$	−3.00000	−	5.19615i	−0.106132	−	0.183827i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	16.0000	+	27.7128i	0.564628	+	0.977964i
$804$		0			0
$805$	10.5000	−	18.1865i	0.370076	−	0.640991i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	38.0000			1.33601			0.668004	−	0.744157i	$-0.267149\pi$
$809$	38.0000			1.33601			0.668004	+	0.744157i	$0.267149\pi$
$810$		0			0
$811$	−30.0000			−1.05344			−0.526721	−	0.850038i	$-0.676579\pi$
$811$	−30.0000			−1.05344			−0.526721	+	0.850038i	$0.676579\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.00000	+	3.46410i	−0.0700569	+	0.121342i
$816$		0			0
$817$	−4.00000	−	6.92820i	−0.139942	−	0.242387i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−16.5000	−	28.5788i	−0.575854	−	0.997408i	−0.995948	−	0.0899279i	$-0.971336\pi$
$821$	−16.5000	−	28.5788i	−0.575854	−	0.997408i	0.420094	−	0.907480i	$-0.361997\pi$
$822$		0			0
$823$	9.50000	−	16.4545i	0.331149	−	0.573567i	−0.651588	−	0.758573i	$-0.725897\pi$
$823$	9.50000	−	16.4545i	0.331149	−	0.573567i	0.982737	+	0.185006i	$0.0592303\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−45.0000			−1.56480			−0.782402	−	0.622774i	$-0.786006\pi$
$827$	−45.0000			−1.56480			−0.782402	+	0.622774i	$0.786006\pi$
$828$		0			0
$829$	21.0000			0.729360			0.364680	−	0.931133i	$-0.381178\pi$
$829$	21.0000			0.729360			0.364680	+	0.931133i	$0.381178\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.00000	−	3.46410i	0.0692959	−	0.120024i
$834$		0			0
$835$	−1.50000	−	2.59808i	−0.0519096	−	0.0899101i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	21.0000	+	36.3731i	0.725001	+	1.25574i	0.958974	+	0.283495i	$0.0914938\pi$
$839$	21.0000	+	36.3731i	0.725001	+	1.25574i	−0.233973	+	0.972243i	$0.575173\pi$
$840$		0			0
$841$	−26.0000	+	45.0333i	−0.896552	+	1.55287i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.00000			−0.103203
$846$		0			0
$847$	−15.0000			−0.515406
$848$		0			0
$849$		0			0
$850$		0			0
$851$	7.00000	−	12.1244i	0.239957	−	0.415618i
$852$		0			0
$853$	6.00000	+	10.3923i	0.205436	+	0.355826i	0.950272	−	0.311422i	$-0.100805\pi$
$853$	6.00000	+	10.3923i	0.205436	+	0.355826i	−0.744836	+	0.667248i	$0.767472\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−16.0000	−	27.7128i	−0.546550	−	0.946652i	−0.998508	−	0.0546125i	$-0.982608\pi$
$857$	−16.0000	−	27.7128i	−0.546550	−	0.946652i	0.451958	−	0.892039i	$-0.350726\pi$
$858$		0			0
$859$	−10.0000	+	17.3205i	−0.341196	+	0.590968i	−0.984655	−	0.174512i	$-0.944165\pi$
$859$	−10.0000	+	17.3205i	−0.341196	+	0.590968i	0.643459	+	0.765480i	$0.277499\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	35.0000			1.19141			0.595707	−	0.803202i	$-0.296872\pi$
$863$	35.0000			1.19141			0.595707	+	0.803202i	$0.296872\pi$
$864$		0			0
$865$	−4.00000			−0.136004
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−20.0000	+	34.6410i	−0.678454	+	1.17512i
$870$		0			0
$871$	−6.00000	−	10.3923i	−0.203302	−	0.352130i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.50000	−	2.59808i	−0.0507093	−	0.0878310i
$876$		0			0
$877$	16.0000	−	27.7128i	0.540282	−	0.935795i	−0.458606	−	0.888640i	$-0.651651\pi$
$877$	16.0000	−	27.7128i	0.540282	−	0.935795i	0.998888	−	0.0471555i	$-0.0150156\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−33.0000			−1.11180			−0.555899	−	0.831250i	$-0.687626\pi$
$881$	−33.0000			−1.11180			−0.555899	+	0.831250i	$0.687626\pi$
$882$		0			0
$883$	−29.0000			−0.975928			−0.487964	−	0.872864i	$-0.662260\pi$
$883$	−29.0000			−0.975928			−0.487964	+	0.872864i	$0.662260\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−4.00000	+	6.92820i	−0.134307	+	0.232626i	−0.925332	−	0.379157i	$-0.876214\pi$
$887$	−4.00000	+	6.92820i	−0.134307	+	0.232626i	0.791026	+	0.611783i	$0.209547\pi$
$888$		0			0
$889$	28.5000	+	49.3634i	0.955859	+	1.65560i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−3.00000	−	5.19615i	−0.100391	−	0.173883i
$894$		0			0
$895$	12.0000	−	20.7846i	0.401116	−	0.694753i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−54.0000			−1.80100
$900$		0			0
$901$	12.0000			0.399778
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−11.5000	+	19.9186i	−0.382273	+	0.662116i
$906$		0			0
$907$	19.5000	+	33.7750i	0.647487	+	1.12148i	0.983721	+	0.179702i	$0.0575133\pi$
$907$	19.5000	+	33.7750i	0.647487	+	1.12148i	−0.336234	+	0.941778i	$0.609153\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	17.0000	+	29.4449i	0.563235	+	0.975552i	0.997211	+	0.0746276i	$0.0237768\pi$
$911$	17.0000	+	29.4449i	0.563235	+	0.975552i	−0.433976	+	0.900924i	$0.642890\pi$
$912$		0			0
$913$	14.0000	−	24.2487i	0.463332	−	0.802515i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	66.0000			2.17951
$918$		0			0
$919$	−34.0000			−1.12156			−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000			−1.12156			−0.560778	+	0.827966i	$0.689498\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.00000	−	1.73205i	−0.0328798	−	0.0569495i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−9.00000	−	15.5885i	−0.295280	−	0.511441i	0.679770	−	0.733426i	$-0.262080\pi$
$929$	−9.00000	−	15.5885i	−0.295280	−	0.511441i	−0.975050	+	0.221985i	$0.928746\pi$
$930$		0			0
$931$	2.00000	−	3.46410i	0.0655474	−	0.113531i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−8.00000			−0.261628
$936$		0			0
$937$	12.0000			0.392023			0.196011	−	0.980602i	$-0.437201\pi$
$937$	12.0000			0.392023			0.196011	+	0.980602i	$0.437201\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.50000	+	6.06218i	−0.114097	+	0.197621i	−0.917418	−	0.397924i	$-0.869731\pi$
$941$	−3.50000	+	6.06218i	−0.114097	+	0.197621i	0.803322	+	0.595545i	$0.203064\pi$
$942$		0			0
$943$	−31.5000	−	54.5596i	−1.02578	−	1.77671i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	8.50000	+	14.7224i	0.276213	+	0.478415i	0.970440	−	0.241341i	$-0.0775872\pi$
$947$	8.50000	+	14.7224i	0.276213	+	0.478415i	−0.694228	+	0.719756i	$0.744254\pi$
$948$		0			0
$949$	16.0000	−	27.7128i	0.519382	−	0.899596i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−24.0000			−0.777436			−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000			−0.777436			−0.388718	+	0.921357i	$0.627082\pi$
$954$		0			0
$955$	10.0000			0.323592
$956$		0			0
$957$		0			0
$958$		0			0
$959$	24.0000	−	41.5692i	0.775000	−	1.34234i
$960$		0			0
$961$	−2.50000	−	4.33013i	−0.0806452	−	0.139682i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	10.0000	+	17.3205i	0.321911	+	0.557567i
$966$		0			0
$967$	−14.5000	+	25.1147i	−0.466289	+	0.807635i	−0.999259	−	0.0384986i	$-0.987742\pi$
$967$	−14.5000	+	25.1147i	−0.466289	+	0.807635i	0.532970	+	0.846134i	$0.321076\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−10.0000			−0.320915			−0.160458	−	0.987043i	$-0.551297\pi$
$971$	−10.0000			−0.320915			−0.160458	+	0.987043i	$0.551297\pi$
$972$		0			0
$973$	−24.0000			−0.769405
$974$		0			0
$975$		0			0
$976$		0			0
$977$	24.0000	−	41.5692i	0.767828	−	1.32992i	−0.170910	−	0.985287i	$-0.554671\pi$
$977$	24.0000	−	41.5692i	0.767828	−	1.32992i	0.938738	−	0.344631i	$-0.111996\pi$
$978$		0			0
$979$	−2.00000	−	3.46410i	−0.0639203	−	0.110713i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−30.5000	−	52.8275i	−0.972799	−	1.68494i	−0.687015	−	0.726643i	$-0.741079\pi$
$983$	−30.5000	−	52.8275i	−0.972799	−	1.68494i	−0.285784	−	0.958294i	$-0.592254\pi$
$984$		0			0
$985$	10.0000	−	17.3205i	0.318626	−	0.551877i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	28.0000			0.890348
$990$		0			0
$991$	38.0000			1.20711			0.603555	−	0.797321i	$-0.293750\pi$
$991$	38.0000			1.20711			0.603555	+	0.797321i	$0.293750\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	7.00000	−	12.1244i	0.221915	−	0.384368i
$996$		0			0
$997$	21.0000	+	36.3731i	0.665077	+	1.15195i	0.979265	+	0.202586i	$0.0649345\pi$
$997$	21.0000	+	36.3731i	0.665077	+	1.15195i	−0.314188	+	0.949361i	$0.601732\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.q.h.1441.1		2
3.2	odd	2		1440.2.q.a.481.1	✓	2
4.3	odd	2		4320.2.q.g.1441.1		2
9.2	odd	6		1440.2.q.a.961.1	yes	2
9.7	even	3	inner	4320.2.q.h.2881.1		2
12.11	even	2		1440.2.q.e.481.1	yes	2
36.7	odd	6		4320.2.q.g.2881.1		2
36.11	even	6		1440.2.q.e.961.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.q.a.481.1	✓	2	3.2	odd	2
1440.2.q.a.961.1	yes	2	9.2	odd	6
1440.2.q.e.481.1	yes	2	12.11	even	2
1440.2.q.e.961.1	yes	2	36.11	even	6
4320.2.q.g.1441.1		2	4.3	odd	2
4320.2.q.g.2881.1		2	36.7	odd	6
4320.2.q.h.1441.1		2	1.1	even	1	trivial
4320.2.q.h.2881.1		2	9.7	even	3	inner

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:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{5} +(1.50000 + 2.59808i) q^{7} +(2.00000 + 3.46410i) q^{11} +(2.00000 - 3.46410i) q^{13} -2.00000 q^{17} -2.00000 q^{19} +(3.50000 - 6.06218i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-4.50000 - 7.79423i) q^{29} +(3.00000 - 5.19615i) q^{31} +3.00000 q^{35} +2.00000 q^{37} +(4.50000 - 7.79423i) q^{41} +(2.00000 + 3.46410i) q^{43} +(1.50000 + 2.59808i) q^{47} +(-1.00000 + 1.73205i) q^{49} -6.00000 q^{53} +4.00000 q^{55} +(3.00000 - 5.19615i) q^{59} +(6.50000 + 11.2583i) q^{61} +(-2.00000 - 3.46410i) q^{65} +(1.50000 - 2.59808i) q^{67} +8.00000 q^{73} +(-6.00000 + 10.3923i) q^{77} +(5.00000 + 8.66025i) q^{79} +(-3.50000 - 6.06218i) q^{83} +(-1.00000 + 1.73205i) q^{85} -1.00000 q^{89} +12.0000 q^{91} +(-1.00000 + 1.73205i) q^{95} +(7.00000 + 12.1244i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} + 3 q^{7} + 4 q^{11} + 4 q^{13} - 4 q^{17} - 4 q^{19} + 7 q^{23} - q^{25} - 9 q^{29} + 6 q^{31} + 6 q^{35} + 4 q^{37} + 9 q^{41} + 4 q^{43} + 3 q^{47} - 2 q^{49} - 12 q^{53} + 8 q^{55} + 6 q^{59}+ \cdots + 14 q^{97}+O(q^{100}) \) 2 * q + q^5 + 3 * q^7 + 4 * q^11 + 4 * q^13 - 4 * q^17 - 4 * q^19 + 7 * q^23 - q^25 - 9 * q^29 + 6 * q^31 + 6 * q^35 + 4 * q^37 + 9 * q^41 + 4 * q^43 + 3 * q^47 - 2 * q^49 - 12 * q^53 + 8 * q^55 + 6 * q^59 + 13 * q^61 - 4 * q^65 + 3 * q^67 + 16 * q^73 - 12 * q^77 + 10 * q^79 - 7 * q^83 - 2 * q^85 - 2 * q^89 + 24 * q^91 - 2 * q^95 + 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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