LMFDB - Embedded newform 3528.2.s.q.3313.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3528,2,Mod(361,3528)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3528, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3528.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3528.s (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$28.1712218331$
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 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			3313.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3528.3313
Dual form			3528.2.s.q.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{5} +O(q^{10})$	$q+(0.500000 - 0.866025i) q^{5} +(1.50000 + 2.59808i) q^{11} +6.00000 q^{13} +(2.50000 + 4.33013i) q^{17} +(0.500000 - 0.866025i) q^{19} +(-3.50000 + 6.06218i) q^{23} +(2.00000 + 3.46410i) q^{25} -2.00000 q^{29} +(-2.50000 - 4.33013i) q^{31} +(-1.50000 + 2.59808i) q^{37} -2.00000 q^{41} -4.00000 q^{43} +(-2.50000 + 4.33013i) q^{47} +(-0.500000 - 0.866025i) q^{53} +3.00000 q^{55} +(-7.50000 - 12.9904i) q^{59} +(-2.50000 + 4.33013i) q^{61} +(3.00000 - 5.19615i) q^{65} +(4.50000 + 7.79423i) q^{67} +(3.50000 + 6.06218i) q^{73} +(-0.500000 + 0.866025i) q^{79} +12.0000 q^{83} +5.00000 q^{85} +(-3.50000 + 6.06218i) q^{89} +(-0.500000 - 0.866025i) q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5}+O(q^{10}) $ 2 * q + q^5	$ 2 q + q^{5} + 3 q^{11} + 12 q^{13} + 5 q^{17} + q^{19} - 7 q^{23} + 4 q^{25} - 4 q^{29} - 5 q^{31} - 3 q^{37} - 4 q^{41} - 8 q^{43} - 5 q^{47} - q^{53} + 6 q^{55} - 15 q^{59} - 5 q^{61} + 6 q^{65} + 9 q^{67} + 7 q^{73} - q^{79} + 24 q^{83} + 10 q^{85} - 7 q^{89} - q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q + q^5 + 3 * q^11 + 12 * q^13 + 5 * q^17 + q^19 - 7 * q^23 + 4 * q^25 - 4 * q^29 - 5 * q^31 - 3 * q^37 - 4 * q^41 - 8 * q^43 - 5 * q^47 - q^53 + 6 * q^55 - 15 * q^59 - 5 * q^61 + 6 * q^65 + 9 * q^67 + 7 * q^73 - q^79 + 24 * q^83 + 10 * q^85 - 7 * q^89 - q^95 + 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$1$

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	−0.732294	−	0.680989i	$-0.761550\pi$
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i	0.955901	+	0.293691i	$0.0948835\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.50000	+	2.59808i	0.452267	+	0.783349i	0.998526	−	0.0542666i	$-0.0172821\pi$
$11$	1.50000	+	2.59808i	0.452267	+	0.783349i	−0.546259	+	0.837616i	$0.683949\pi$
$12$		0			0
$13$	6.00000			1.66410			0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000			1.66410			0.832050	+	0.554700i	$0.187167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.50000	+	4.33013i	0.606339	+	1.05021i	0.991838	+	0.127502i	$0.0406959\pi$
$17$	2.50000	+	4.33013i	0.606339	+	1.05021i	−0.385499	+	0.922708i	$0.625971\pi$
$18$		0			0
$19$	0.500000	−	0.866025i	0.114708	−	0.198680i	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	0.500000	−	0.866025i	0.114708	−	0.198680i	0.917663	+	0.397360i	$0.130073\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.50000	+	6.06218i	−0.729800	+	1.26405i	0.227167	+	0.973856i	$0.427054\pi$
$23$	−3.50000	+	6.06218i	−0.729800	+	1.26405i	−0.956967	+	0.290196i	$0.906280\pi$
$24$		0			0
$25$	2.00000	+	3.46410i	0.400000	+	0.692820i
$26$		0			0
$27$		0			0
$28$		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
$31$	−2.50000	−	4.33013i	−0.449013	−	0.777714i	0.549309	−	0.835619i	$-0.314891\pi$
$31$	−2.50000	−	4.33013i	−0.449013	−	0.777714i	−0.998322	+	0.0579057i	$0.981558\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.50000	+	2.59808i	−0.246598	+	0.427121i	−0.962580	−	0.270998i	$-0.912646\pi$
$37$	−1.50000	+	2.59808i	−0.246598	+	0.427121i	0.715981	+	0.698119i	$0.245980\pi$
$38$		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
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.50000	+	4.33013i	−0.364662	+	0.631614i	−0.988722	−	0.149763i	$-0.952149\pi$
$47$	−2.50000	+	4.33013i	−0.364662	+	0.631614i	0.624059	+	0.781377i	$0.285482\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.500000	−	0.866025i	−0.0686803	−	0.118958i	0.829640	−	0.558298i	$-0.188546\pi$
$53$	−0.500000	−	0.866025i	−0.0686803	−	0.118958i	−0.898321	+	0.439340i	$0.855212\pi$
$54$		0			0
$55$	3.00000			0.404520
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.50000	−	12.9904i	−0.976417	−	1.69120i	−0.675178	−	0.737655i	$-0.735933\pi$
$59$	−7.50000	−	12.9904i	−0.976417	−	1.69120i	−0.301239	−	0.953549i	$-0.597400\pi$
$60$		0			0
$61$	−2.50000	+	4.33013i	−0.320092	+	0.554416i	−0.980507	−	0.196485i	$-0.937047\pi$
$61$	−2.50000	+	4.33013i	−0.320092	+	0.554416i	0.660415	+	0.750901i	$0.270381\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.00000	−	5.19615i	0.372104	−	0.644503i
$66$		0			0
$67$	4.50000	+	7.79423i	0.549762	+	0.952217i	0.998290	+	0.0584478i	$0.0186151\pi$
$67$	4.50000	+	7.79423i	0.549762	+	0.952217i	−0.448528	+	0.893769i	$0.648052\pi$
$68$		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
$73$	3.50000	+	6.06218i	0.409644	+	0.709524i	0.994850	−	0.101361i	$-0.0323196\pi$
$73$	3.50000	+	6.06218i	0.409644	+	0.709524i	−0.585206	+	0.810885i	$0.698986\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.500000	+	0.866025i	−0.0562544	+	0.0974355i	−0.892781	−	0.450490i	$-0.851249\pi$
$79$	−0.500000	+	0.866025i	−0.0562544	+	0.0974355i	0.836527	+	0.547926i	$0.184582\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	12.0000			1.31717			0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000			1.31717			0.658586	+	0.752506i	$0.271155\pi$
$84$		0			0
$85$	5.00000			0.542326
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.50000	+	6.06218i	−0.370999	+	0.642590i	−0.989720	−	0.143022i	$-0.954318\pi$
$89$	−3.50000	+	6.06218i	−0.370999	+	0.642590i	0.618720	+	0.785611i	$0.287651\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.500000	−	0.866025i	−0.0512989	−	0.0888523i
$96$		0			0
$97$	2.00000			0.203069			0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000			0.203069			0.101535	+	0.994832i	$0.467625\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.50000	−	2.59808i	−0.149256	−	0.258518i	0.781697	−	0.623658i	$-0.214354\pi$
$101$	−1.50000	−	2.59808i	−0.149256	−	0.258518i	−0.930953	+	0.365140i	$0.881021\pi$
$102$		0			0
$103$	−7.50000	+	12.9904i	−0.738997	+	1.27998i	0.213950	+	0.976845i	$0.431367\pi$
$103$	−7.50000	+	12.9904i	−0.738997	+	1.27998i	−0.952947	+	0.303136i	$0.901966\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.50000	−	7.79423i	0.435031	−	0.753497i	−0.562267	−	0.826956i	$-0.690071\pi$
$107$	4.50000	−	7.79423i	0.435031	−	0.753497i	0.997298	+	0.0734594i	$0.0234039\pi$
$108$		0			0
$109$	2.50000	+	4.33013i	0.239457	+	0.414751i	0.960558	−	0.278078i	$-0.0896974\pi$
$109$	2.50000	+	4.33013i	0.239457	+	0.414751i	−0.721102	+	0.692829i	$0.756364\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	18.0000			1.69330			0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000			1.69330			0.846649	+	0.532152i	$0.178617\pi$
$114$		0			0
$115$	3.50000	+	6.06218i	0.326377	+	0.565301i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.00000	−	1.73205i	0.0909091	−	0.157459i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.00000			0.804984
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.50000	+	4.33013i	−0.218426	+	0.378325i	−0.954327	−	0.298764i	$-0.903426\pi$
$131$	−2.50000	+	4.33013i	−0.218426	+	0.378325i	0.735901	+	0.677089i	$0.236759\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	5.50000	+	9.52628i	0.469897	+	0.813885i	0.999408	−	0.0344182i	$-0.0109578\pi$
$137$	5.50000	+	9.52628i	0.469897	+	0.813885i	−0.529511	+	0.848303i	$0.677624\pi$
$138$		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
$143$	9.00000	+	15.5885i	0.752618	+	1.30357i
$144$		0			0
$145$	−1.00000	+	1.73205i	−0.0830455	+	0.143839i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.50000	+	14.7224i	−0.696347	+	1.20611i	0.273377	+	0.961907i	$0.411859\pi$
$149$	−8.50000	+	14.7224i	−0.696347	+	1.20611i	−0.969724	+	0.244202i	$0.921474\pi$
$150$		0			0
$151$	2.50000	+	4.33013i	0.203447	+	0.352381i	0.949637	−	0.313353i	$-0.101452\pi$
$151$	2.50000	+	4.33013i	0.203447	+	0.352381i	−0.746190	+	0.665733i	$0.768119\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.00000			−0.401610
$156$		0			0
$157$	−4.50000	−	7.79423i	−0.359139	−	0.622047i	0.628678	−	0.777666i	$-0.283596\pi$
$157$	−4.50000	−	7.79423i	−0.359139	−	0.622047i	−0.987817	+	0.155618i	$0.950263\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−6.50000	+	11.2583i	−0.509119	+	0.881820i	0.490825	+	0.871258i	$0.336695\pi$
$163$	−6.50000	+	11.2583i	−0.509119	+	0.881820i	−0.999944	+	0.0105623i	$0.996638\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−12.0000			−0.928588			−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000			−0.928588			−0.464294	+	0.885681i	$0.653692\pi$
$168$		0			0
$169$	23.0000			1.76923
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.50000	−	11.2583i	0.494186	−	0.855955i	−0.505792	−	0.862656i	$-0.668800\pi$
$173$	6.50000	−	11.2583i	0.494186	−	0.855955i	0.999978	+	0.00670064i	$0.00213290\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−6.50000	−	11.2583i	−0.485833	−	0.841487i	0.514035	−	0.857769i	$-0.328150\pi$
$179$	−6.50000	−	11.2583i	−0.485833	−	0.841487i	−0.999867	+	0.0162823i	$0.994817\pi$
$180$		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
$185$	1.50000	+	2.59808i	0.110282	+	0.191014i
$186$		0			0
$187$	−7.50000	+	12.9904i	−0.548454	+	0.949951i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.50000	+	9.52628i	−0.397966	+	0.689297i	−0.993475	−	0.114051i	$-0.963617\pi$
$191$	−5.50000	+	9.52628i	−0.397966	+	0.689297i	0.595509	+	0.803349i	$0.296950\pi$
$192$		0			0
$193$	−1.50000	−	2.59808i	−0.107972	−	0.187014i	0.806976	−	0.590584i	$-0.201102\pi$
$193$	−1.50000	−	2.59808i	−0.107972	−	0.187014i	−0.914949	+	0.403570i	$0.867769\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	−6.50000	−	11.2583i	−0.460773	−	0.798082i	0.538227	−	0.842800i	$-0.319094\pi$
$199$	−6.50000	−	11.2583i	−0.460773	−	0.798082i	−0.999000	+	0.0447181i	$0.985761\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.00000	+	1.73205i	−0.0698430	+	0.120972i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.00000			0.207514
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.00000	+	3.46410i	−0.136399	+	0.236250i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	15.0000	+	25.9808i	1.00901	+	1.74766i
$222$		0			0
$223$	24.0000			1.60716			0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000			1.60716			0.803579	+	0.595198i	$0.202926\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.50000	−	9.52628i	−0.365048	−	0.632281i	0.623736	−	0.781635i	$-0.285614\pi$
$227$	−5.50000	−	9.52628i	−0.365048	−	0.632281i	−0.988784	+	0.149354i	$0.952281\pi$
$228$		0			0
$229$	11.5000	−	19.9186i	0.759941	−	1.31626i	−0.182939	−	0.983124i	$-0.558561\pi$
$229$	11.5000	−	19.9186i	0.759941	−	1.31626i	0.942880	−	0.333133i	$-0.108106\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5.50000	−	9.52628i	0.360317	−	0.624087i	−0.627696	−	0.778459i	$-0.716002\pi$
$233$	5.50000	−	9.52628i	0.360317	−	0.624087i	0.988013	+	0.154371i	$0.0493352\pi$
$234$		0			0
$235$	2.50000	+	4.33013i	0.163082	+	0.282466i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−20.0000			−1.29369			−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000			−1.29369			−0.646846	+	0.762620i	$0.723912\pi$
$240$		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
$245$		0			0
$246$		0			0
$247$	3.00000	−	5.19615i	0.190885	−	0.330623i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	16.0000			1.00991			0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000			1.00991			0.504956	+	0.863145i	$0.331509\pi$
$252$		0			0
$253$	−21.0000			−1.32026
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.5000	−	18.1865i	0.654972	−	1.13444i	−0.326929	−	0.945049i	$-0.606014\pi$
$257$	10.5000	−	18.1865i	0.654972	−	1.13444i	0.981901	−	0.189396i	$-0.0606529\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4.50000	−	7.79423i	−0.277482	−	0.480613i	0.693276	−	0.720672i	$-0.256167\pi$
$263$	−4.50000	−	7.79423i	−0.277482	−	0.480613i	−0.970758	+	0.240059i	$0.922833\pi$
$264$		0			0
$265$	−1.00000			−0.0614295
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4.50000	+	7.79423i	0.274370	+	0.475223i	0.969976	−	0.243201i	$-0.0781974\pi$
$269$	4.50000	+	7.79423i	0.274370	+	0.475223i	−0.695606	+	0.718423i	$0.744864\pi$
$270$		0			0
$271$	−3.50000	+	6.06218i	−0.212610	+	0.368251i	−0.952531	−	0.304443i	$-0.901530\pi$
$271$	−3.50000	+	6.06218i	−0.212610	+	0.368251i	0.739921	+	0.672694i	$0.234863\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6.00000	+	10.3923i	−0.361814	+	0.626680i
$276$		0			0
$277$	8.50000	+	14.7224i	0.510716	+	0.884585i	0.999923	+	0.0124177i	$0.00395278\pi$
$277$	8.50000	+	14.7224i	0.510716	+	0.884585i	−0.489207	+	0.872167i	$0.662714\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$	−6.50000	−	11.2583i	−0.386385	−	0.669238i	0.605575	−	0.795788i	$-0.292943\pi$
$283$	−6.50000	−	11.2583i	−0.386385	−	0.669238i	−0.991960	+	0.126550i	$0.959610\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−4.00000	+	6.92820i	−0.235294	+	0.407541i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$		0			0
$295$	−15.0000			−0.873334
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−21.0000	+	36.3731i	−1.21446	+	2.10351i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.50000	+	4.33013i	0.143150	+	0.247942i
$306$		0			0
$307$	−4.00000			−0.228292			−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000			−0.228292			−0.114146	+	0.993464i	$0.536413\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−7.50000	−	12.9904i	−0.425286	−	0.736617i	0.571161	−	0.820838i	$-0.306493\pi$
$311$	−7.50000	−	12.9904i	−0.425286	−	0.736617i	−0.996447	+	0.0842210i	$0.973160\pi$
$312$		0			0
$313$	−0.500000	+	0.866025i	−0.0282617	+	0.0489506i	−0.879810	−	0.475325i	$-0.842331\pi$
$313$	−0.500000	+	0.866025i	−0.0282617	+	0.0489506i	0.851549	+	0.524276i	$0.175664\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.50000	−	2.59808i	0.0842484	−	0.145922i	−0.820822	−	0.571184i	$-0.806484\pi$
$317$	1.50000	−	2.59808i	0.0842484	−	0.145922i	0.905071	+	0.425261i	$0.139818\pi$
$318$		0			0
$319$	−3.00000	−	5.19615i	−0.167968	−	0.290929i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5.00000			0.278207
$324$		0			0
$325$	12.0000	+	20.7846i	0.665640	+	1.15292i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−14.5000	+	25.1147i	−0.796992	+	1.38043i	0.124574	+	0.992210i	$0.460243\pi$
$331$	−14.5000	+	25.1147i	−0.796992	+	1.38043i	−0.921567	+	0.388221i	$0.873090\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	9.00000			0.491723
$336$		0			0
$337$	−10.0000			−0.544735			−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000			−0.544735			−0.272367	+	0.962193i	$0.587807\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	7.50000	−	12.9904i	0.406148	−	0.703469i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.50000	+	12.9904i	0.402621	+	0.697360i	0.994041	−	0.109003i	$-0.0347659\pi$
$347$	7.50000	+	12.9904i	0.402621	+	0.697360i	−0.591420	+	0.806363i	$0.701433\pi$
$348$		0			0
$349$	−2.00000			−0.107058			−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000			−0.107058			−0.0535288	+	0.998566i	$0.517047\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.50000	−	2.59808i	−0.0798369	−	0.138282i	0.823343	−	0.567545i	$-0.192107\pi$
$353$	−1.50000	−	2.59808i	−0.0798369	−	0.138282i	−0.903179	+	0.429263i	$0.858773\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10.5000	−	18.1865i	0.554169	−	0.959849i	−0.443799	−	0.896126i	$-0.646370\pi$
$359$	10.5000	−	18.1865i	0.554169	−	0.959849i	0.997968	−	0.0637221i	$-0.0202971\pi$
$360$		0			0
$361$	9.00000	+	15.5885i	0.473684	+	0.820445i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7.00000			0.366397
$366$		0			0
$367$	11.5000	+	19.9186i	0.600295	+	1.03974i	0.992776	+	0.119982i	$0.0382835\pi$
$367$	11.5000	+	19.9186i	0.600295	+	1.03974i	−0.392481	+	0.919760i	$0.628383\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−5.50000	+	9.52628i	−0.284779	+	0.493252i	−0.972556	−	0.232671i	$-0.925254\pi$
$373$	−5.50000	+	9.52628i	−0.284779	+	0.493252i	0.687776	+	0.725923i	$0.258587\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.0000			−0.618031
$378$		0			0
$379$	24.0000			1.23280			0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000			1.23280			0.616399	+	0.787434i	$0.288591\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.50000	−	2.59808i	0.0766464	−	0.132755i	−0.825155	−	0.564907i	$-0.808912\pi$
$383$	1.50000	−	2.59808i	0.0766464	−	0.132755i	0.901801	+	0.432151i	$0.142245\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−14.5000	−	25.1147i	−0.735179	−	1.27337i	−0.954645	−	0.297747i	$-0.903765\pi$
$389$	−14.5000	−	25.1147i	−0.735179	−	1.27337i	0.219465	−	0.975620i	$-0.429569\pi$
$390$		0			0
$391$	−35.0000			−1.77003
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.500000	+	0.866025i	0.0251577	+	0.0435745i
$396$		0			0
$397$	−8.50000	+	14.7224i	−0.426603	+	0.738898i	−0.996569	−	0.0827707i	$-0.973623\pi$
$397$	−8.50000	+	14.7224i	−0.426603	+	0.738898i	0.569966	+	0.821668i	$0.306956\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	13.5000	−	23.3827i	0.674158	−	1.16768i	−0.302556	−	0.953131i	$-0.597840\pi$
$401$	13.5000	−	23.3827i	0.674158	−	1.16768i	0.976714	−	0.214544i	$-0.0688266\pi$
$402$		0			0
$403$	−15.0000	−	25.9808i	−0.747203	−	1.29419i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−9.00000			−0.446113
$408$		0			0
$409$	13.5000	+	23.3827i	0.667532	+	1.15620i	0.978592	+	0.205809i	$0.0659826\pi$
$409$	13.5000	+	23.3827i	0.667532	+	1.15620i	−0.311060	+	0.950390i	$0.600684\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	6.00000	−	10.3923i	0.294528	−	0.510138i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	20.0000			0.977064			0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000			0.977064			0.488532	+	0.872546i	$0.337533\pi$
$420$		0			0
$421$	34.0000			1.65706			0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000			1.65706			0.828529	+	0.559946i	$0.189178\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−10.0000	+	17.3205i	−0.485071	+	0.840168i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.50000	+	2.59808i	0.0722525	+	0.125145i	0.899888	−	0.436121i	$-0.143648\pi$
$431$	1.50000	+	2.59808i	0.0722525	+	0.125145i	−0.827636	+	0.561266i	$0.810315\pi$
$432$		0			0
$433$	2.00000			0.0961139			0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000			0.0961139			0.0480569	+	0.998845i	$0.484697\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.50000	+	6.06218i	0.167428	+	0.289993i
$438$		0			0
$439$	10.5000	−	18.1865i	0.501138	−	0.867996i	−0.498861	−	0.866682i	$-0.666248\pi$
$439$	10.5000	−	18.1865i	0.501138	−	0.867996i	0.999999	−	0.00131415i	$-0.000418308\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−15.5000	+	26.8468i	−0.736427	+	1.27553i	0.217667	+	0.976023i	$0.430155\pi$
$443$	−15.5000	+	26.8468i	−0.736427	+	1.27553i	−0.954094	+	0.299506i	$0.903178\pi$
$444$		0			0
$445$	3.50000	+	6.06218i	0.165916	+	0.287375i
$446$		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
$451$	−3.00000	−	5.19615i	−0.141264	−	0.244677i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	8.50000	−	14.7224i	0.397613	−	0.688686i	−0.595818	−	0.803120i	$-0.703172\pi$
$457$	8.50000	−	14.7224i	0.397613	−	0.688686i	0.993431	+	0.114433i	$0.0365053\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−30.0000			−1.39724			−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000			−1.39724			−0.698620	+	0.715493i	$0.746202\pi$
$462$		0			0
$463$	−16.0000			−0.743583			−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000			−0.743583			−0.371792	+	0.928316i	$0.621256\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.50000	−	2.59808i	0.0694117	−	0.120225i	−0.829231	−	0.558906i	$-0.811221\pi$
$467$	1.50000	−	2.59808i	0.0694117	−	0.120225i	0.898642	+	0.438682i	$0.144554\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−6.00000	−	10.3923i	−0.275880	−	0.477839i
$474$		0			0
$475$	4.00000			0.183533
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−15.5000	−	26.8468i	−0.708213	−	1.22666i	−0.965519	−	0.260331i	$-0.916168\pi$
$479$	−15.5000	−	26.8468i	−0.708213	−	1.22666i	0.257306	−	0.966330i	$-0.417165\pi$
$480$		0			0
$481$	−9.00000	+	15.5885i	−0.410365	+	0.710772i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.00000	−	1.73205i	0.0454077	−	0.0786484i
$486$		0			0
$487$	−3.50000	−	6.06218i	−0.158600	−	0.274703i	0.775764	−	0.631023i	$-0.217365\pi$
$487$	−3.50000	−	6.06218i	−0.158600	−	0.274703i	−0.934364	+	0.356320i	$0.884031\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	24.0000			1.08310			0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000			1.08310			0.541552	+	0.840667i	$0.317837\pi$
$492$		0			0
$493$	−5.00000	−	8.66025i	−0.225189	−	0.390038i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	17.5000	−	30.3109i	0.783408	−	1.35690i	−0.146538	−	0.989205i	$-0.546813\pi$
$499$	17.5000	−	30.3109i	0.783408	−	1.35690i	0.929946	−	0.367697i	$-0.119854\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−32.0000			−1.42681			−0.713405	−	0.700752i	$-0.752848\pi$
$503$	−32.0000			−1.42681			−0.713405	+	0.700752i	$0.752848\pi$
$504$		0			0
$505$	−3.00000			−0.133498
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4.50000	−	7.79423i	0.199459	−	0.345473i	−0.748894	−	0.662690i	$-0.769415\pi$
$509$	4.50000	−	7.79423i	0.199459	−	0.345473i	0.948353	+	0.317217i	$0.102748\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.50000	+	12.9904i	0.330489	+	0.572425i
$516$		0			0
$517$	−15.0000			−0.659699
$518$		0			0
$519$		0			0
$520$		0			0
$521$	16.5000	+	28.5788i	0.722878	+	1.25206i	0.959841	+	0.280543i	$0.0905145\pi$
$521$	16.5000	+	28.5788i	0.722878	+	1.25206i	−0.236963	+	0.971519i	$0.576152\pi$
$522$		0			0
$523$	−3.50000	+	6.06218i	−0.153044	+	0.265081i	−0.932345	−	0.361569i	$-0.882241\pi$
$523$	−3.50000	+	6.06218i	−0.153044	+	0.265081i	0.779301	+	0.626650i	$0.215574\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	12.5000	−	21.6506i	0.544509	−	0.943116i
$528$		0			0
$529$	−13.0000	−	22.5167i	−0.565217	−	0.978985i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−12.0000			−0.519778
$534$		0			0
$535$	−4.50000	−	7.79423i	−0.194552	−	0.336974i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	20.5000	−	35.5070i	0.881364	−	1.52657i	0.0315385	−	0.999503i	$-0.489959\pi$
$541$	20.5000	−	35.5070i	0.881364	−	1.52657i	0.849825	−	0.527064i	$-0.176707\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5.00000			0.214176
$546$		0			0
$547$	−8.00000			−0.342055			−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000			−0.342055			−0.171028	+	0.985266i	$0.554709\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.00000	+	1.73205i	−0.0426014	+	0.0737878i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.50000	−	4.33013i	−0.105928	−	0.183473i	0.808189	−	0.588924i	$-0.200448\pi$
$557$	−2.50000	−	4.33013i	−0.105928	−	0.183473i	−0.914117	+	0.405450i	$0.867115\pi$
$558$		0			0
$559$	−24.0000			−1.01509
$560$		0			0
$561$		0			0
$562$		0			0
$563$	20.5000	+	35.5070i	0.863972	+	1.49644i	0.868064	+	0.496452i	$0.165364\pi$
$563$	20.5000	+	35.5070i	0.863972	+	1.49644i	−0.00409232	+	0.999992i	$0.501303\pi$
$564$		0			0
$565$	9.00000	−	15.5885i	0.378633	−	0.655811i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−16.5000	+	28.5788i	−0.691716	+	1.19809i	0.279559	+	0.960128i	$0.409812\pi$
$569$	−16.5000	+	28.5788i	−0.691716	+	1.19809i	−0.971275	+	0.237959i	$0.923522\pi$
$570$		0			0
$571$	6.50000	+	11.2583i	0.272017	+	0.471146i	0.969378	−	0.245573i	$-0.0789761\pi$
$571$	6.50000	+	11.2583i	0.272017	+	0.471146i	−0.697362	+	0.716720i	$0.745643\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−28.0000			−1.16768
$576$		0			0
$577$	−16.5000	−	28.5788i	−0.686904	−	1.18975i	−0.972834	−	0.231502i	$-0.925636\pi$
$577$	−16.5000	−	28.5788i	−0.686904	−	1.18975i	0.285930	−	0.958250i	$-0.407697\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	1.50000	−	2.59808i	0.0621237	−	0.107601i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−20.0000			−0.825488			−0.412744	−	0.910847i	$-0.635430\pi$
$587$	−20.0000			−0.825488			−0.412744	+	0.910847i	$0.635430\pi$
$588$		0			0
$589$	−5.00000			−0.206021
$590$		0			0
$591$		0			0
$592$		0			0
$593$	22.5000	−	38.9711i	0.923964	−	1.60035i	0.130746	−	0.991416i	$-0.458263\pi$
$593$	22.5000	−	38.9711i	0.923964	−	1.60035i	0.793219	−	0.608937i	$-0.208404\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8.50000	−	14.7224i	−0.347301	−	0.601542i	0.638468	−	0.769648i	$-0.279568\pi$
$599$	−8.50000	−	14.7224i	−0.347301	−	0.601542i	−0.985769	+	0.168106i	$0.946235\pi$
$600$		0			0
$601$	−22.0000			−0.897399			−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000			−0.897399			−0.448699	+	0.893683i	$0.648113\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.00000	−	1.73205i	−0.0406558	−	0.0704179i
$606$		0			0
$607$	6.50000	−	11.2583i	0.263827	−	0.456962i	−0.703429	−	0.710766i	$-0.748349\pi$
$607$	6.50000	−	11.2583i	0.263827	−	0.456962i	0.967256	+	0.253804i	$0.0816819\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−15.0000	+	25.9808i	−0.606835	+	1.05107i
$612$		0			0
$613$	−21.5000	−	37.2391i	−0.868377	−	1.50407i	−0.863655	−	0.504084i	$-0.831830\pi$
$613$	−21.5000	−	37.2391i	−0.868377	−	1.50407i	−0.00472215	−	0.999989i	$-0.501503\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−30.0000			−1.20775			−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000			−1.20775			−0.603877	+	0.797077i	$0.706378\pi$
$618$		0			0
$619$	−4.50000	−	7.79423i	−0.180870	−	0.313276i	0.761307	−	0.648392i	$-0.224558\pi$
$619$	−4.50000	−	7.79423i	−0.180870	−	0.313276i	−0.942177	+	0.335115i	$0.891225\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−5.50000	+	9.52628i	−0.220000	+	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−15.0000			−0.598089
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.00000	−	6.92820i	0.158735	−	0.274937i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−8.50000	−	14.7224i	−0.335730	−	0.581501i	0.647895	−	0.761730i	$-0.275650\pi$
$641$	−8.50000	−	14.7224i	−0.335730	−	0.581501i	−0.983625	+	0.180229i	$0.942316\pi$
$642$		0			0
$643$	44.0000			1.73519			0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000			1.73519			0.867595	+	0.497271i	$0.165665\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−7.50000	−	12.9904i	−0.294855	−	0.510705i	0.680096	−	0.733123i	$-0.261938\pi$
$647$	−7.50000	−	12.9904i	−0.294855	−	0.510705i	−0.974951	+	0.222419i	$0.928605\pi$
$648$		0			0
$649$	22.5000	−	38.9711i	0.883202	−	1.52975i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	17.5000	−	30.3109i	0.684828	−	1.18616i	−0.288663	−	0.957431i	$-0.593211\pi$
$653$	17.5000	−	30.3109i	0.684828	−	1.18616i	0.973491	−	0.228726i	$-0.0734560\pi$
$654$		0			0
$655$	2.50000	+	4.33013i	0.0976831	+	0.169192i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−44.0000			−1.71400			−0.856998	−	0.515319i	$-0.827673\pi$
$659$	−44.0000			−1.71400			−0.856998	+	0.515319i	$0.827673\pi$
$660$		0			0
$661$	11.5000	+	19.9186i	0.447298	+	0.774743i	0.998209	−	0.0598209i	$-0.0190530\pi$
$661$	11.5000	+	19.9186i	0.447298	+	0.774743i	−0.550911	+	0.834564i	$0.685720\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	7.00000	−	12.1244i	0.271041	−	0.469457i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−15.0000			−0.579069
$672$		0			0
$673$	−26.0000			−1.00223			−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000			−1.00223			−0.501113	+	0.865382i	$0.667076\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−19.5000	+	33.7750i	−0.749446	+	1.29808i	0.198643	+	0.980072i	$0.436347\pi$
$677$	−19.5000	+	33.7750i	−0.749446	+	1.29808i	−0.948089	+	0.318006i	$0.896987\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	13.5000	+	23.3827i	0.516563	+	0.894714i	0.999815	+	0.0192323i	$0.00612219\pi$
$683$	13.5000	+	23.3827i	0.516563	+	0.894714i	−0.483252	+	0.875481i	$0.660544\pi$
$684$		0			0
$685$	11.0000			0.420288
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.00000	−	5.19615i	−0.114291	−	0.197958i
$690$		0			0
$691$	2.50000	−	4.33013i	0.0951045	−	0.164726i	−0.814548	−	0.580097i	$-0.803015\pi$
$691$	2.50000	−	4.33013i	0.0951045	−	0.164726i	0.909652	+	0.415371i	$0.136348\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.00000	−	10.3923i	0.227593	−	0.394203i
$696$		0			0
$697$	−5.00000	−	8.66025i	−0.189389	−	0.328031i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	50.0000			1.88847			0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000			1.88847			0.944237	+	0.329267i	$0.106802\pi$
$702$		0			0
$703$	1.50000	+	2.59808i	0.0565736	+	0.0979883i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	14.5000	−	25.1147i	0.544559	−	0.943204i	−0.454076	−	0.890963i	$-0.650030\pi$
$709$	14.5000	−	25.1147i	0.544559	−	0.943204i	0.998635	−	0.0522406i	$-0.0166363\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	35.0000			1.31076
$714$		0			0
$715$	18.0000			0.673162
$716$		0			0
$717$		0			0
$718$		0			0
$719$	19.5000	−	33.7750i	0.727227	−	1.25959i	−0.230823	−	0.972996i	$-0.574142\pi$
$719$	19.5000	−	33.7750i	0.727227	−	1.25959i	0.958051	−	0.286599i	$-0.0925247\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.00000	−	6.92820i	−0.148556	−	0.257307i
$726$		0			0
$727$	−32.0000			−1.18681			−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000			−1.18681			−0.593407	+	0.804902i	$0.702218\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−10.0000	−	17.3205i	−0.369863	−	0.640622i
$732$		0			0
$733$	9.50000	−	16.4545i	0.350891	−	0.607760i	−0.635515	−	0.772088i	$-0.719212\pi$
$733$	9.50000	−	16.4545i	0.350891	−	0.607760i	0.986406	+	0.164328i	$0.0525456\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−13.5000	+	23.3827i	−0.497279	+	0.861312i
$738$		0			0
$739$	2.50000	+	4.33013i	0.0919640	+	0.159286i	0.908337	−	0.418238i	$-0.137352\pi$
$739$	2.50000	+	4.33013i	0.0919640	+	0.159286i	−0.816373	+	0.577524i	$0.804019\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	16.0000			0.586983			0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000			0.586983			0.293492	+	0.955962i	$0.405183\pi$
$744$		0			0
$745$	8.50000	+	14.7224i	0.311416	+	0.539388i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−26.5000	+	45.8993i	−0.966999	+	1.67489i	−0.262852	+	0.964836i	$0.584663\pi$
$751$	−26.5000	+	45.8993i	−0.966999	+	1.67489i	−0.704146	+	0.710055i	$0.748670\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	5.00000			0.181969
$756$		0			0
$757$	10.0000			0.363456			0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000			0.363456			0.181728	+	0.983349i	$0.441831\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.50000	+	2.59808i	−0.0543750	+	0.0941802i	−0.891932	−	0.452170i	$-0.850650\pi$
$761$	−1.50000	+	2.59808i	−0.0543750	+	0.0941802i	0.837557	+	0.546350i	$0.183983\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−45.0000	−	77.9423i	−1.62486	−	2.81433i
$768$		0			0
$769$	−22.0000			−0.793340			−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000			−0.793340			−0.396670	+	0.917961i	$0.629834\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−9.50000	−	16.4545i	−0.341691	−	0.591827i	0.643056	−	0.765819i	$-0.277666\pi$
$773$	−9.50000	−	16.4545i	−0.341691	−	0.591827i	−0.984747	+	0.173993i	$0.944333\pi$
$774$		0			0
$775$	10.0000	−	17.3205i	0.359211	−	0.622171i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.00000	+	1.73205i	−0.0358287	+	0.0620572i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−9.00000			−0.321224
$786$		0			0
$787$	−8.50000	−	14.7224i	−0.302992	−	0.524798i	0.673820	−	0.738896i	$-0.264652\pi$
$787$	−8.50000	−	14.7224i	−0.302992	−	0.524798i	−0.976812	+	0.214097i	$0.931319\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−15.0000	+	25.9808i	−0.532666	+	0.922604i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	18.0000			0.637593			0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000			0.637593			0.318796	+	0.947823i	$0.396721\pi$
$798$		0			0
$799$	−25.0000			−0.884436
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−10.5000	+	18.1865i	−0.370537	+	0.641789i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	19.5000	+	33.7750i	0.685583	+	1.18747i	0.973253	+	0.229736i	$0.0737862\pi$
$809$	19.5000	+	33.7750i	0.685583	+	1.18747i	−0.287670	+	0.957730i	$0.592880\pi$
$810$		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
$815$	6.50000	+	11.2583i	0.227685	+	0.394362i
$816$		0			0
$817$	−2.00000	+	3.46410i	−0.0699711	+	0.121194i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−4.50000	+	7.79423i	−0.157051	+	0.272020i	−0.933804	−	0.357785i	$-0.883532\pi$
$821$	−4.50000	+	7.79423i	−0.157051	+	0.272020i	0.776753	+	0.629805i	$0.216865\pi$
$822$		0			0
$823$	−23.5000	−	40.7032i	−0.819159	−	1.41882i	−0.906303	−	0.422628i	$-0.861108\pi$
$823$	−23.5000	−	40.7032i	−0.819159	−	1.41882i	0.0871445	−	0.996196i	$-0.472226\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	36.0000			1.25184			0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000			1.25184			0.625921	+	0.779886i	$0.284723\pi$
$828$		0			0
$829$	5.50000	+	9.52628i	0.191023	+	0.330861i	0.945589	−	0.325362i	$-0.105486\pi$
$829$	5.50000	+	9.52628i	0.191023	+	0.330861i	−0.754567	+	0.656223i	$0.772153\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−6.00000	+	10.3923i	−0.207639	+	0.359641i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$		0			0
$844$		0			0
$845$	11.5000	−	19.9186i	0.395612	−	0.685220i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−10.5000	−	18.1865i	−0.359935	−	0.623426i
$852$		0			0
$853$	−34.0000			−1.16414			−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000			−1.16414			−0.582069	+	0.813139i	$0.697757\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−19.5000	−	33.7750i	−0.666107	−	1.15373i	−0.978984	−	0.203938i	$-0.934626\pi$
$857$	−19.5000	−	33.7750i	−0.666107	−	1.15373i	0.312877	−	0.949794i	$-0.398707\pi$
$858$		0			0
$859$	0.500000	−	0.866025i	0.0170598	−	0.0295484i	−0.857369	−	0.514701i	$-0.827903\pi$
$859$	0.500000	−	0.866025i	0.0170598	−	0.0295484i	0.874429	+	0.485153i	$0.161236\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	8.50000	−	14.7224i	0.289343	−	0.501157i	−0.684310	−	0.729191i	$-0.739896\pi$
$863$	8.50000	−	14.7224i	0.289343	−	0.501157i	0.973653	+	0.228034i	$0.0732297\pi$
$864$		0			0
$865$	−6.50000	−	11.2583i	−0.221007	−	0.382795i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.00000			−0.101768
$870$		0			0
$871$	27.0000	+	46.7654i	0.914860	+	1.58458i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−11.5000	+	19.9186i	−0.388327	+	0.672603i	−0.992225	−	0.124459i	$-0.960280\pi$
$877$	−11.5000	+	19.9186i	−0.388327	+	0.672603i	0.603897	+	0.797062i	$0.293614\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−14.0000			−0.471672			−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000			−0.471672			−0.235836	+	0.971793i	$0.575783\pi$
$882$		0			0
$883$	−20.0000			−0.673054			−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000			−0.673054			−0.336527	+	0.941674i	$0.609252\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.50000	−	2.59808i	0.0503651	−	0.0872349i	−0.839744	−	0.542983i	$-0.817295\pi$
$887$	1.50000	−	2.59808i	0.0503651	−	0.0872349i	0.890109	+	0.455748i	$0.150628\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.50000	+	4.33013i	0.0836593	+	0.144902i
$894$		0			0
$895$	−13.0000			−0.434542
$896$		0			0
$897$		0			0
$898$		0			0
$899$	5.00000	+	8.66025i	0.166759	+	0.288836i
$900$		0			0
$901$	2.50000	−	4.33013i	0.0832871	−	0.144257i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	5.00000	−	8.66025i	0.166206	−	0.287877i
$906$		0			0
$907$	0.500000	+	0.866025i	0.0166022	+	0.0287559i	0.874207	−	0.485553i	$-0.161382\pi$
$907$	0.500000	+	0.866025i	0.0166022	+	0.0287559i	−0.857605	+	0.514309i	$0.828048\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−16.0000			−0.530104			−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000			−0.530104			−0.265052	+	0.964234i	$0.585389\pi$
$912$		0			0
$913$	18.0000	+	31.1769i	0.595713	+	1.03181i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	5.50000	−	9.52628i	0.181428	−	0.314243i	−0.760939	−	0.648824i	$-0.775261\pi$
$919$	5.50000	−	9.52628i	0.181428	−	0.314243i	0.942367	+	0.334581i	$0.108595\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−12.0000			−0.394558
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.50000	+	6.06218i	−0.114831	+	0.198894i	−0.917712	−	0.397246i	$-0.869966\pi$
$929$	−3.50000	+	6.06218i	−0.114831	+	0.198894i	0.802881	+	0.596139i	$0.203299\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	7.50000	+	12.9904i	0.245276	+	0.424831i
$936$		0			0
$937$	6.00000			0.196011			0.0980057	−	0.995186i	$-0.468754\pi$
$937$	6.00000			0.196011			0.0980057	+	0.995186i	$0.468754\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	16.5000	+	28.5788i	0.537885	+	0.931644i	0.999018	+	0.0443125i	$0.0141097\pi$
$941$	16.5000	+	28.5788i	0.537885	+	0.931644i	−0.461133	+	0.887331i	$0.652557\pi$
$942$		0			0
$943$	7.00000	−	12.1244i	0.227951	−	0.394823i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	18.5000	−	32.0429i	0.601169	−	1.04126i	−0.391475	−	0.920189i	$-0.628035\pi$
$947$	18.5000	−	32.0429i	0.601169	−	1.04126i	0.992644	−	0.121067i	$-0.0386316\pi$
$948$		0			0
$949$	21.0000	+	36.3731i	0.681689	+	1.18072i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−22.0000			−0.712650			−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000			−0.712650			−0.356325	+	0.934362i	$0.615970\pi$
$954$		0			0
$955$	5.50000	+	9.52628i	0.177976	+	0.308263i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	3.00000	−	5.19615i	0.0967742	−	0.167618i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3.00000			−0.0965734
$966$		0			0
$967$	48.0000			1.54358			0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000			1.54358			0.771788	+	0.635880i	$0.219363\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	17.5000	−	30.3109i	0.561602	−	0.972723i	−0.435755	−	0.900065i	$-0.643519\pi$
$971$	17.5000	−	30.3109i	0.561602	−	0.972723i	0.997357	−	0.0726575i	$-0.0231480\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.50000	+	2.59808i	0.0479893	+	0.0831198i	0.889022	−	0.457864i	$-0.151385\pi$
$977$	1.50000	+	2.59808i	0.0479893	+	0.0831198i	−0.841033	+	0.540984i	$0.818052\pi$
$978$		0			0
$979$	−21.0000			−0.671163
$980$		0			0
$981$		0			0
$982$		0			0
$983$	10.5000	+	18.1865i	0.334898	+	0.580060i	0.983465	−	0.181097i	$-0.0579648\pi$
$983$	10.5000	+	18.1865i	0.334898	+	0.580060i	−0.648567	+	0.761157i	$0.724631\pi$
$984$		0			0
$985$	3.00000	−	5.19615i	0.0955879	−	0.165563i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	14.0000	−	24.2487i	0.445174	−	0.771064i
$990$		0			0
$991$	−7.50000	−	12.9904i	−0.238245	−	0.412653i	0.721966	−	0.691929i	$-0.243239\pi$
$991$	−7.50000	−	12.9904i	−0.238245	−	0.412653i	−0.960211	+	0.279276i	$0.909906\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−13.0000			−0.412128
$996$		0			0
$997$	−20.5000	−	35.5070i	−0.649242	−	1.12452i	−0.983304	−	0.181968i	$-0.941753\pi$
$997$	−20.5000	−	35.5070i	−0.649242	−	1.12452i	0.334063	−	0.942551i	$-0.391580\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.s.q.3313.1		2
3.2	odd	2		392.2.i.b.177.1		2
7.2	even	3		3528.2.a.j.1.1		1
7.3	odd	6		504.2.s.c.361.1		2
7.4	even	3	inner	3528.2.s.q.361.1		2
7.5	odd	6		3528.2.a.p.1.1		1
7.6	odd	2		504.2.s.c.289.1		2
12.11	even	2		784.2.i.h.177.1		2
21.2	odd	6		392.2.a.e.1.1		1
21.5	even	6		392.2.a.c.1.1		1
21.11	odd	6		392.2.i.b.361.1		2
21.17	even	6		56.2.i.b.25.1	yes	2
21.20	even	2		56.2.i.b.9.1	✓	2
28.3	even	6		1008.2.s.g.865.1		2
28.19	even	6		7056.2.a.bj.1.1		1
28.23	odd	6		7056.2.a.u.1.1		1
28.27	even	2		1008.2.s.g.289.1		2
84.11	even	6		784.2.i.h.753.1		2
84.23	even	6		784.2.a.c.1.1		1
84.47	odd	6		784.2.a.h.1.1		1
84.59	odd	6		112.2.i.a.81.1		2
84.83	odd	2		112.2.i.a.65.1		2
105.17	odd	12		1400.2.bh.a.249.1		4
105.38	odd	12		1400.2.bh.a.249.2		4
105.44	odd	6		9800.2.a.s.1.1		1
105.59	even	6		1400.2.q.d.1201.1		2
105.62	odd	4		1400.2.bh.a.849.2		4
105.83	odd	4		1400.2.bh.a.849.1		4
105.89	even	6		9800.2.a.be.1.1		1
105.104	even	2		1400.2.q.d.401.1		2
168.5	even	6		3136.2.a.u.1.1		1
168.59	odd	6		448.2.i.d.193.1		2
168.83	odd	2		448.2.i.d.65.1		2
168.101	even	6		448.2.i.b.193.1		2
168.107	even	6		3136.2.a.t.1.1		1
168.125	even	2		448.2.i.b.65.1		2
168.131	odd	6		3136.2.a.j.1.1		1
168.149	odd	6		3136.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.i.b.9.1	✓	2	21.20	even	2
56.2.i.b.25.1	yes	2	21.17	even	6
112.2.i.a.65.1		2	84.83	odd	2
112.2.i.a.81.1		2	84.59	odd	6
392.2.a.c.1.1		1	21.5	even	6
392.2.a.e.1.1		1	21.2	odd	6
392.2.i.b.177.1		2	3.2	odd	2
392.2.i.b.361.1		2	21.11	odd	6
448.2.i.b.65.1		2	168.125	even	2
448.2.i.b.193.1		2	168.101	even	6
448.2.i.d.65.1		2	168.83	odd	2
448.2.i.d.193.1		2	168.59	odd	6
504.2.s.c.289.1		2	7.6	odd	2
504.2.s.c.361.1		2	7.3	odd	6
784.2.a.c.1.1		1	84.23	even	6
784.2.a.h.1.1		1	84.47	odd	6
784.2.i.h.177.1		2	12.11	even	2
784.2.i.h.753.1		2	84.11	even	6
1008.2.s.g.289.1		2	28.27	even	2
1008.2.s.g.865.1		2	28.3	even	6
1400.2.q.d.401.1		2	105.104	even	2
1400.2.q.d.1201.1		2	105.59	even	6
1400.2.bh.a.249.1		4	105.17	odd	12
1400.2.bh.a.249.2		4	105.38	odd	12
1400.2.bh.a.849.1		4	105.83	odd	4
1400.2.bh.a.849.2		4	105.62	odd	4
3136.2.a.i.1.1		1	168.149	odd	6
3136.2.a.j.1.1		1	168.131	odd	6
3136.2.a.t.1.1		1	168.107	even	6
3136.2.a.u.1.1		1	168.5	even	6
3528.2.a.j.1.1		1	7.2	even	3
3528.2.a.p.1.1		1	7.5	odd	6
3528.2.s.q.361.1		2	7.4	even	3	inner
3528.2.s.q.3313.1		2	1.1	even	1	trivial
7056.2.a.u.1.1		1	28.23	odd	6
7056.2.a.bj.1.1		1	28.19	even	6
9800.2.a.s.1.1		1	105.44	odd	6
9800.2.a.be.1.1		1	105.89	even	6

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3528.s (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{5} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{5} +(1.50000 + 2.59808i) q^{11} +6.00000 q^{13} +(2.50000 + 4.33013i) q^{17} +(0.500000 - 0.866025i) q^{19} +(-3.50000 + 6.06218i) q^{23} +(2.00000 + 3.46410i) q^{25} -2.00000 q^{29} +(-2.50000 - 4.33013i) q^{31} +(-1.50000 + 2.59808i) q^{37} -2.00000 q^{41} -4.00000 q^{43} +(-2.50000 + 4.33013i) q^{47} +(-0.500000 - 0.866025i) q^{53} +3.00000 q^{55} +(-7.50000 - 12.9904i) q^{59} +(-2.50000 + 4.33013i) q^{61} +(3.00000 - 5.19615i) q^{65} +(4.50000 + 7.79423i) q^{67} +(3.50000 + 6.06218i) q^{73} +(-0.500000 + 0.866025i) q^{79} +12.0000 q^{83} +5.00000 q^{85} +(-3.50000 + 6.06218i) q^{89} +(-0.500000 - 0.866025i) q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5}+O(q^{10}) \) 2 * q + q^5	\( 2 q + q^{5} + 3 q^{11} + 12 q^{13} + 5 q^{17} + q^{19} - 7 q^{23} + 4 q^{25} - 4 q^{29} - 5 q^{31} - 3 q^{37} - 4 q^{41} - 8 q^{43} - 5 q^{47} - q^{53} + 6 q^{55} - 15 q^{59} - 5 q^{61} + 6 q^{65} + 9 q^{67} + 7 q^{73} - q^{79} + 24 q^{83} + 10 q^{85} - 7 q^{89} - q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q + q^5 + 3 * q^11 + 12 * q^13 + 5 * q^17 + q^19 - 7 * q^23 + 4 * q^25 - 4 * q^29 - 5 * q^31 - 3 * q^37 - 4 * q^41 - 8 * q^43 - 5 * q^47 - q^53 + 6 * q^55 - 15 * q^59 - 5 * q^61 + 6 * q^65 + 9 * q^67 + 7 * q^73 - q^79 + 24 * q^83 + 10 * q^85 - 7 * q^89 - q^95 + 4 * q^97

Introduction

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