LMFDB - Embedded newform 3528.2.s.p.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 168)
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.p.361.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{5} +(1.50000 + 2.59808i) q^{11} -4.00000 q^{13} +(-2.00000 + 3.46410i) q^{19} +(4.00000 - 6.92820i) q^{23} +(2.00000 + 3.46410i) q^{25} +3.00000 q^{29} +(-2.50000 - 4.33013i) q^{31} +(-4.00000 + 6.92820i) q^{37} +8.00000 q^{41} +6.00000 q^{43} +(-5.00000 + 8.66025i) q^{47} +(4.50000 + 7.79423i) q^{53} +3.00000 q^{55} +(2.50000 + 4.33013i) q^{59} +(-5.00000 + 8.66025i) q^{61} +(-2.00000 + 3.46410i) q^{65} +(-3.00000 - 5.19615i) q^{67} -10.0000 q^{71} +(1.00000 + 1.73205i) q^{73} +(-5.50000 + 9.52628i) q^{79} +7.00000 q^{83} +(9.00000 - 15.5885i) q^{89} +(2.00000 + 3.46410i) q^{95} +17.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + q^{5} + 3 q^{11} - 8 q^{13} - 4 q^{19} + 8 q^{23} + 4 q^{25} + 6 q^{29} - 5 q^{31} - 8 q^{37} + 16 q^{41} + 12 q^{43} - 10 q^{47} + 9 q^{53} + 6 q^{55} + 5 q^{59} - 10 q^{61} - 4 q^{65} - 6 q^{67}+ \cdots + 34 q^{97}+O(q^{100})$ 2 * q + q^5 + 3 * q^11 - 8 * q^13 - 4 * q^19 + 8 * q^23 + 4 * q^25 + 6 * q^29 - 5 * q^31 - 8 * q^37 + 16 * q^41 + 12 * q^43 - 10 * q^47 + 9 * q^53 + 6 * q^55 + 5 * q^59 - 10 * q^61 - 4 * q^65 - 6 * q^67 - 20 * q^71 + 2 * q^73 - 11 * q^79 + 14 * q^83 + 18 * q^89 + 4 * q^95 + 34 * q^97

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
$13$	−4.00000			−1.10940			−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000			−1.10940			−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	−2.00000	+	3.46410i	−0.458831	+	0.794719i	−0.998899	−	0.0469020i	$-0.985065\pi$
$19$	−2.00000	+	3.46410i	−0.458831	+	0.794719i	0.540068	+	0.841621i	$0.318398\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.00000	−	6.92820i	0.834058	−	1.44463i	−0.0607377	−	0.998154i	$-0.519345\pi$
$23$	4.00000	−	6.92820i	0.834058	−	1.44463i	0.894795	−	0.446476i	$-0.147321\pi$
$24$		0			0
$25$	2.00000	+	3.46410i	0.400000	+	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.00000			0.557086			0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000			0.557086			0.278543	+	0.960424i	$0.410149\pi$
$30$		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
$37$	−4.00000	+	6.92820i	−0.657596	+	1.13899i	0.323640	+	0.946180i	$0.395093\pi$
$37$	−4.00000	+	6.92820i	−0.657596	+	1.13899i	−0.981236	+	0.192809i	$0.938240\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.00000			1.24939			0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000			1.24939			0.624695	+	0.780869i	$0.285223\pi$
$42$		0			0
$43$	6.00000			0.914991			0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000			0.914991			0.457496	+	0.889212i	$0.348747\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.00000	+	8.66025i	−0.729325	+	1.26323i	0.227844	+	0.973698i	$0.426832\pi$
$47$	−5.00000	+	8.66025i	−0.729325	+	1.26323i	−0.957169	+	0.289530i	$0.906501\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.50000	+	7.79423i	0.618123	+	1.07062i	0.989828	+	0.142269i	$0.0454398\pi$
$53$	4.50000	+	7.79423i	0.618123	+	1.07062i	−0.371706	+	0.928351i	$0.621227\pi$
$54$		0			0
$55$	3.00000			0.404520
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.50000	+	4.33013i	0.325472	+	0.563735i	0.981608	−	0.190909i	$-0.0611434\pi$
$59$	2.50000	+	4.33013i	0.325472	+	0.563735i	−0.656136	+	0.754643i	$0.727810\pi$
$60$		0			0
$61$	−5.00000	+	8.66025i	−0.640184	+	1.10883i	0.345207	+	0.938527i	$0.387809\pi$
$61$	−5.00000	+	8.66025i	−0.640184	+	1.10883i	−0.985391	+	0.170305i	$0.945525\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.00000	+	3.46410i	−0.248069	+	0.429669i
$66$		0			0
$67$	−3.00000	−	5.19615i	−0.366508	−	0.634811i	0.622509	−	0.782613i	$-0.286114\pi$
$67$	−3.00000	−	5.19615i	−0.366508	−	0.634811i	−0.989017	+	0.147802i	$0.952780\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−10.0000			−1.18678			−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000			−1.18678			−0.593391	+	0.804914i	$0.702211\pi$
$72$		0			0
$73$	1.00000	+	1.73205i	0.117041	+	0.202721i	0.918594	−	0.395203i	$-0.129326\pi$
$73$	1.00000	+	1.73205i	0.117041	+	0.202721i	−0.801553	+	0.597924i	$0.795992\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−5.50000	+	9.52628i	−0.618798	+	1.07179i	0.370907	+	0.928670i	$0.379047\pi$
$79$	−5.50000	+	9.52628i	−0.618798	+	1.07179i	−0.989705	+	0.143120i	$0.954286\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	7.00000			0.768350			0.384175	−	0.923260i	$-0.374486\pi$
$83$	7.00000			0.768350			0.384175	+	0.923260i	$0.374486\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.00000	−	15.5885i	0.953998	−	1.65237i	0.217354	−	0.976093i	$-0.430258\pi$
$89$	9.00000	−	15.5885i	0.953998	−	1.65237i	0.736644	−	0.676280i	$-0.236409\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.00000	+	3.46410i	0.205196	+	0.355409i
$96$		0			0
$97$	17.0000			1.72609			0.863044	−	0.505128i	$-0.168555\pi$
$97$	17.0000			1.72609			0.863044	+	0.505128i	$0.168555\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.00000	+	1.73205i	0.0995037	+	0.172345i	0.911479	−	0.411346i	$-0.134941\pi$
$101$	1.00000	+	1.73205i	0.0995037	+	0.172345i	−0.811976	+	0.583691i	$0.801608\pi$
$102$		0			0
$103$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$103$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.50000	+	9.52628i	−0.531705	+	0.920940i	0.467610	+	0.883935i	$0.345115\pi$
$107$	−5.50000	+	9.52628i	−0.531705	+	0.920940i	−0.999315	+	0.0370053i	$0.988218\pi$
$108$		0			0
$109$	5.00000	+	8.66025i	0.478913	+	0.829502i	0.999708	−	0.0241802i	$-0.00769755\pi$
$109$	5.00000	+	8.66025i	0.478913	+	0.829502i	−0.520794	+	0.853682i	$0.674364\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.00000			0.752577			0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000			0.752577			0.376288	+	0.926503i	$0.377200\pi$
$114$		0			0
$115$	−4.00000	−	6.92820i	−0.373002	−	0.646058i
$116$		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
$127$	−7.00000			−0.621150			−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000			−0.621150			−0.310575	+	0.950549i	$0.600522\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.50000	+	12.9904i	−0.655278	+	1.13497i	0.326546	+	0.945181i	$0.394115\pi$
$131$	−7.50000	+	12.9904i	−0.655278	+	1.13497i	−0.981824	+	0.189794i	$0.939218\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−7.00000	−	12.1244i	−0.598050	−	1.03585i	−0.993109	−	0.117198i	$-0.962609\pi$
$137$	−7.00000	−	12.1244i	−0.598050	−	1.03585i	0.395058	−	0.918656i	$-0.370724\pi$
$138$		0			0
$139$	22.0000			1.86602			0.933008	−	0.359856i	$-0.117174\pi$
$139$	22.0000			1.86602			0.933008	+	0.359856i	$0.117174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.00000	−	10.3923i	−0.501745	−	0.869048i
$144$		0			0
$145$	1.50000	−	2.59808i	0.124568	−	0.215758i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.00000	−	15.5885i	0.737309	−	1.27706i	−0.216394	−	0.976306i	$-0.569430\pi$
$149$	9.00000	−	15.5885i	0.737309	−	1.27706i	0.953703	−	0.300750i	$-0.0972370\pi$
$150$		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
$157$	−2.00000	−	3.46410i	−0.159617	−	0.276465i	0.775113	−	0.631822i	$-0.217693\pi$
$157$	−2.00000	−	3.46410i	−0.159617	−	0.276465i	−0.934731	+	0.355357i	$0.884359\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−4.00000	+	6.92820i	−0.313304	+	0.542659i	−0.979076	−	0.203497i	$-0.934769\pi$
$163$	−4.00000	+	6.92820i	−0.313304	+	0.542659i	0.665771	+	0.746156i	$0.268103\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.00000			−0.154765			−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000			−0.154765			−0.0773823	+	0.997001i	$0.524656\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.00000	+	1.73205i	−0.0760286	+	0.131685i	−0.901533	−	0.432710i	$-0.857557\pi$
$173$	−1.00000	+	1.73205i	−0.0760286	+	0.131685i	0.825505	+	0.564396i	$0.190891\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.00000	+	10.3923i	0.448461	+	0.776757i	0.998286	−	0.0585225i	$-0.0186389\pi$
$179$	6.00000	+	10.3923i	0.448461	+	0.776757i	−0.549825	+	0.835280i	$0.685306\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.00000	+	6.92820i	0.294086	+	0.509372i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.00000	+	13.8564i	−0.578860	+	1.00261i	0.416751	+	0.909021i	$0.363169\pi$
$191$	−8.00000	+	13.8564i	−0.578860	+	1.00261i	−0.995610	+	0.0935936i	$0.970165\pi$
$192$		0			0
$193$	13.5000	+	23.3827i	0.971751	+	1.68312i	0.690264	+	0.723558i	$0.257494\pi$
$193$	13.5000	+	23.3827i	0.971751	+	1.68312i	0.281487	+	0.959565i	$0.409172\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	26.0000			1.85242			0.926212	−	0.377004i	$-0.123046\pi$
$197$	26.0000			1.85242			0.926212	+	0.377004i	$0.123046\pi$
$198$		0			0
$199$	6.00000	+	10.3923i	0.425329	+	0.736691i	0.996451	−	0.0841740i	$-0.0268252\pi$
$199$	6.00000	+	10.3923i	0.425329	+	0.736691i	−0.571122	+	0.820865i	$0.693492\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	4.00000	−	6.92820i	0.279372	−	0.483887i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−12.0000			−0.830057
$210$		0			0
$211$	6.00000			0.413057			0.206529	−	0.978441i	$-0.433783\pi$
$211$	6.00000			0.413057			0.206529	+	0.978441i	$0.433783\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.00000	−	5.19615i	0.204598	−	0.354375i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.00000			−0.0669650			−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000			−0.0669650			−0.0334825	+	0.999439i	$0.510660\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.5000	−	18.1865i	−0.696909	−	1.20708i	−0.969533	−	0.244962i	$-0.921225\pi$
$227$	−10.5000	−	18.1865i	−0.696909	−	1.20708i	0.272623	−	0.962121i	$-0.412109\pi$
$228$		0			0
$229$	−6.00000	+	10.3923i	−0.396491	+	0.686743i	−0.993290	−	0.115648i	$-0.963106\pi$
$229$	−6.00000	+	10.3923i	−0.396491	+	0.686743i	0.596799	+	0.802391i	$0.296439\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−12.0000	+	20.7846i	−0.786146	+	1.36165i	0.142166	+	0.989843i	$0.454593\pi$
$233$	−12.0000	+	20.7846i	−0.786146	+	1.36165i	−0.928312	+	0.371802i	$0.878740\pi$
$234$		0			0
$235$	5.00000	+	8.66025i	0.326164	+	0.564933i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	11.5000	+	19.9186i	0.740780	+	1.28307i	0.952141	+	0.305661i	$0.0988773\pi$
$241$	11.5000	+	19.9186i	0.740780	+	1.28307i	−0.211360	+	0.977408i	$0.567789\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	8.00000	−	13.8564i	0.509028	−	0.881662i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	11.0000			0.694314			0.347157	−	0.937807i	$-0.387147\pi$
$251$	11.0000			0.694314			0.347157	+	0.937807i	$0.387147\pi$
$252$		0			0
$253$	24.0000			1.50887
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.00000	+	12.1244i	−0.436648	+	0.756297i	−0.997429	−	0.0716680i	$-0.977168\pi$
$257$	−7.00000	+	12.1244i	−0.436648	+	0.756297i	0.560781	+	0.827964i	$0.310501\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−7.00000	−	12.1244i	−0.431638	−	0.747620i	0.565376	−	0.824833i	$-0.308731\pi$
$263$	−7.00000	−	12.1244i	−0.431638	−	0.747620i	−0.997015	+	0.0772134i	$0.975398\pi$
$264$		0			0
$265$	9.00000			0.552866
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.500000	−	0.866025i	−0.0304855	−	0.0528025i	0.850380	−	0.526169i	$-0.176372\pi$
$269$	−0.500000	−	0.866025i	−0.0304855	−	0.0528025i	−0.880866	+	0.473366i	$0.843039\pi$
$270$		0			0
$271$	−8.50000	+	14.7224i	−0.516338	+	0.894324i	0.483482	+	0.875354i	$0.339372\pi$
$271$	−8.50000	+	14.7224i	−0.516338	+	0.894324i	−0.999820	+	0.0189696i	$0.993961\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6.00000	+	10.3923i	−0.361814	+	0.626680i
$276$		0			0
$277$	−4.00000	−	6.92820i	−0.240337	−	0.416275i	0.720473	−	0.693482i	$-0.243925\pi$
$277$	−4.00000	−	6.92820i	−0.240337	−	0.416275i	−0.960810	+	0.277207i	$0.910591\pi$
$278$		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
$283$	−9.00000	−	15.5885i	−0.534994	−	0.926638i	−0.999164	−	0.0408910i	$-0.986980\pi$
$283$	−9.00000	−	15.5885i	−0.534994	−	0.926638i	0.464169	−	0.885747i	$-0.346353\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−19.0000			−1.10999			−0.554996	−	0.831853i	$-0.687280\pi$
$293$	−19.0000			−1.10999			−0.554996	+	0.831853i	$0.687280\pi$
$294$		0			0
$295$	5.00000			0.291111
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−16.0000	+	27.7128i	−0.925304	+	1.60267i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	5.00000	+	8.66025i	0.286299	+	0.495885i
$306$		0			0
$307$	16.0000			0.913168			0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000			0.913168			0.456584	+	0.889680i	$0.349073\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	4.50000	−	7.79423i	0.254355	−	0.440556i	−0.710365	−	0.703833i	$-0.751470\pi$
$313$	4.50000	−	7.79423i	0.254355	−	0.440556i	0.964720	+	0.263278i	$0.0848035\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.50000	−	11.2583i	0.365076	−	0.632331i	−0.623712	−	0.781654i	$-0.714376\pi$
$317$	6.50000	−	11.2583i	0.365076	−	0.632331i	0.988788	+	0.149323i	$0.0477095\pi$
$318$		0			0
$319$	4.50000	+	7.79423i	0.251952	+	0.436393i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−8.00000	−	13.8564i	−0.443760	−	0.768615i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.00000	+	3.46410i	−0.109930	+	0.190404i	−0.915742	−	0.401768i	$-0.868396\pi$
$331$	−2.00000	+	3.46410i	−0.109930	+	0.190404i	0.805812	+	0.592172i	$0.201729\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.00000			−0.327815
$336$		0			0
$337$	−15.0000			−0.817102			−0.408551	−	0.912735i	$-0.633966\pi$
$337$	−15.0000			−0.817102			−0.408551	+	0.912735i	$0.633966\pi$
$338$		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
$347$	−10.0000	−	17.3205i	−0.536828	−	0.929814i	−0.999072	−	0.0430610i	$-0.986289\pi$
$347$	−10.0000	−	17.3205i	−0.536828	−	0.929814i	0.462244	−	0.886753i	$-0.347044\pi$
$348$		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
$353$	−4.00000	−	6.92820i	−0.212899	−	0.368751i	0.739722	−	0.672913i	$-0.234957\pi$
$353$	−4.00000	−	6.92820i	−0.212899	−	0.368751i	−0.952620	+	0.304162i	$0.901624\pi$
$354$		0			0
$355$	−5.00000	+	8.66025i	−0.265372	+	0.459639i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3.00000	−	5.19615i	0.158334	−	0.274242i	−0.775934	−	0.630814i	$-0.782721\pi$
$359$	3.00000	−	5.19615i	0.158334	−	0.274242i	0.934268	+	0.356572i	$0.116054\pi$
$360$		0			0
$361$	1.50000	+	2.59808i	0.0789474	+	0.136741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.00000			0.104685
$366$		0			0
$367$	−3.50000	−	6.06218i	−0.182699	−	0.316443i	0.760100	−	0.649806i	$-0.225150\pi$
$367$	−3.50000	−	6.06218i	−0.182699	−	0.316443i	−0.942799	+	0.333363i	$0.891817\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−8.00000	+	13.8564i	−0.414224	+	0.717458i	−0.995347	−	0.0963587i	$-0.969280\pi$
$373$	−8.00000	+	13.8564i	−0.414224	+	0.717458i	0.581122	+	0.813816i	$0.302614\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.0000			−0.618031
$378$		0			0
$379$	−16.0000			−0.821865			−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000			−0.821865			−0.410932	+	0.911666i	$0.634797\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.00000	+	1.73205i	−0.0510976	+	0.0885037i	−0.890443	−	0.455095i	$-0.849605\pi$
$383$	−1.00000	+	1.73205i	−0.0510976	+	0.0885037i	0.839345	+	0.543599i	$0.182939\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	13.0000	+	22.5167i	0.659126	+	1.14164i	0.980842	+	0.194804i	$0.0624070\pi$
$389$	13.0000	+	22.5167i	0.659126	+	1.14164i	−0.321716	+	0.946836i	$0.604260\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5.50000	+	9.52628i	0.276735	+	0.479319i
$396$		0			0
$397$	14.0000	−	24.2487i	0.702640	−	1.21701i	−0.264897	−	0.964277i	$-0.585338\pi$
$397$	14.0000	−	24.2487i	0.702640	−	1.21701i	0.967537	−	0.252731i	$-0.0813288\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−4.00000	+	6.92820i	−0.199750	+	0.345978i	−0.948447	−	0.316934i	$-0.897346\pi$
$401$	−4.00000	+	6.92820i	−0.199750	+	0.345978i	0.748697	+	0.662912i	$0.230680\pi$
$402$		0			0
$403$	10.0000	+	17.3205i	0.498135	+	0.862796i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−24.0000			−1.18964
$408$		0			0
$409$	3.50000	+	6.06218i	0.173064	+	0.299755i	0.939490	−	0.342578i	$-0.111300\pi$
$409$	3.50000	+	6.06218i	0.173064	+	0.299755i	−0.766426	+	0.642333i	$0.777967\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	3.50000	−	6.06218i	0.171808	−	0.297581i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−6.00000			−0.292422			−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000			−0.292422			−0.146211	+	0.989253i	$0.546708\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.00000	−	10.3923i	−0.289010	−	0.500580i	0.684564	−	0.728953i	$-0.259993\pi$
$431$	−6.00000	−	10.3923i	−0.289010	−	0.500580i	−0.973574	+	0.228373i	$0.926659\pi$
$432$		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
$437$	16.0000	+	27.7128i	0.765384	+	1.32568i
$438$		0			0
$439$	−4.50000	+	7.79423i	−0.214773	+	0.371998i	−0.953202	−	0.302333i	$-0.902235\pi$
$439$	−4.50000	+	7.79423i	−0.214773	+	0.371998i	0.738429	+	0.674331i	$0.235568\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.500000	+	0.866025i	−0.0237557	+	0.0411461i	−0.877659	−	0.479286i	$-0.840896\pi$
$443$	−0.500000	+	0.866025i	−0.0237557	+	0.0411461i	0.853903	+	0.520432i	$0.174229\pi$
$444$		0			0
$445$	−9.00000	−	15.5885i	−0.426641	−	0.738964i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−4.00000			−0.188772			−0.0943858	−	0.995536i	$-0.530089\pi$
$449$	−4.00000			−0.188772			−0.0943858	+	0.995536i	$0.530089\pi$
$450$		0			0
$451$	12.0000	+	20.7846i	0.565058	+	0.978709i
$452$		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	$-0.253798\pi$
$461$	30.0000			1.39724			0.698620	+	0.715493i	$0.253798\pi$
$462$		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
$467$	14.0000	−	24.2487i	0.647843	−	1.12210i	−0.335794	−	0.941935i	$-0.609005\pi$
$467$	14.0000	−	24.2487i	0.647843	−	1.12210i	0.983637	−	0.180161i	$-0.0576619\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	9.00000	+	15.5885i	0.413820	+	0.716758i
$474$		0			0
$475$	−16.0000			−0.734130
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3.00000	−	5.19615i	−0.137073	−	0.237418i	0.789314	−	0.613990i	$-0.210436\pi$
$479$	−3.00000	−	5.19615i	−0.137073	−	0.237418i	−0.926388	+	0.376571i	$0.877103\pi$
$480$		0			0
$481$	16.0000	−	27.7128i	0.729537	−	1.26360i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.50000	−	14.7224i	0.385965	−	0.668511i
$486$		0			0
$487$	−18.5000	−	32.0429i	−0.838315	−	1.45200i	−0.891303	−	0.453409i	$-0.850208\pi$
$487$	−18.5000	−	32.0429i	−0.838315	−	1.45200i	0.0529875	−	0.998595i	$-0.483126\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.00000			0.406164			0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000			0.406164			0.203082	+	0.979162i	$0.434904\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−5.00000	+	8.66025i	−0.223831	+	0.387686i	−0.955968	−	0.293471i	$-0.905190\pi$
$499$	−5.00000	+	8.66025i	−0.223831	+	0.387686i	0.732137	+	0.681157i	$0.238523\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−12.0000			−0.535054			−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000			−0.535054			−0.267527	+	0.963550i	$0.586206\pi$
$504$		0			0
$505$	2.00000			0.0889988
$506$		0			0
$507$		0			0
$508$		0			0
$509$	19.5000	−	33.7750i	0.864322	−	1.49705i	−0.00339621	−	0.999994i	$-0.501081\pi$
$509$	19.5000	−	33.7750i	0.864322	−	1.49705i	0.867719	−	0.497056i	$-0.165586\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−30.0000			−1.31940
$518$		0			0
$519$		0			0
$520$		0			0
$521$	9.00000	+	15.5885i	0.394297	+	0.682943i	0.993011	−	0.118020i	$-0.0376547\pi$
$521$	9.00000	+	15.5885i	0.394297	+	0.682943i	−0.598714	+	0.800963i	$0.704321\pi$
$522$		0			0
$523$	−6.00000	+	10.3923i	−0.262362	+	0.454424i	−0.966869	−	0.255273i	$-0.917835\pi$
$523$	−6.00000	+	10.3923i	−0.262362	+	0.454424i	0.704507	+	0.709697i	$0.251168\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−20.5000	−	35.5070i	−0.891304	−	1.54378i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−32.0000			−1.38607
$534$		0			0
$535$	5.50000	+	9.52628i	0.237786	+	0.411857i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−17.0000	+	29.4449i	−0.730887	+	1.26593i	0.225617	+	0.974216i	$0.427560\pi$
$541$	−17.0000	+	29.4449i	−0.730887	+	1.26593i	−0.956504	+	0.291718i	$0.905773\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10.0000			0.428353
$546$		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
$551$	−6.00000	+	10.3923i	−0.255609	+	0.442727i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	7.50000	+	12.9904i	0.317785	+	0.550420i	0.980026	−	0.198871i	$-0.0637276\pi$
$557$	7.50000	+	12.9904i	0.317785	+	0.550420i	−0.662240	+	0.749291i	$0.730394\pi$
$558$		0			0
$559$	−24.0000			−1.01509
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−19.5000	−	33.7750i	−0.821827	−	1.42345i	−0.904320	−	0.426855i	$-0.859622\pi$
$563$	−19.5000	−	33.7750i	−0.821827	−	1.42345i	0.0824933	−	0.996592i	$-0.473712\pi$
$564$		0			0
$565$	4.00000	−	6.92820i	0.168281	−	0.291472i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.00000	−	10.3923i	0.251533	−	0.435668i	−0.712415	−	0.701758i	$-0.752399\pi$
$569$	6.00000	−	10.3923i	0.251533	−	0.435668i	0.963948	+	0.266090i	$0.0857319\pi$
$570$		0			0
$571$	−11.0000	−	19.0526i	−0.460336	−	0.797325i	0.538642	−	0.842535i	$-0.318938\pi$
$571$	−11.0000	−	19.0526i	−0.460336	−	0.797325i	−0.998978	+	0.0452101i	$0.985604\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	32.0000			1.33449
$576$		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
$583$	−13.5000	+	23.3827i	−0.559113	+	0.968412i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	45.0000			1.85735			0.928674	−	0.370896i	$-0.120949\pi$
$587$	45.0000			1.85735			0.928674	+	0.370896i	$0.120949\pi$
$588$		0			0
$589$	20.0000			0.824086
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−21.0000	−	36.3731i	−0.858037	−	1.48616i	−0.873799	−	0.486287i	$-0.838351\pi$
$599$	−21.0000	−	36.3731i	−0.858037	−	1.48616i	0.0157622	−	0.999876i	$-0.494983\pi$
$600$		0			0
$601$	13.0000			0.530281			0.265141	−	0.964210i	$-0.414582\pi$
$601$	13.0000			0.530281			0.265141	+	0.964210i	$0.414582\pi$
$602$		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
$611$	20.0000	−	34.6410i	0.809113	−	1.40143i
$612$		0			0
$613$	−4.00000	−	6.92820i	−0.161558	−	0.279827i	0.773869	−	0.633345i	$-0.218319\pi$
$613$	−4.00000	−	6.92820i	−0.161558	−	0.279827i	−0.935428	+	0.353518i	$0.884985\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	30.0000			1.20775			0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000			1.20775			0.603877	+	0.797077i	$0.293622\pi$
$618$		0			0
$619$	3.00000	+	5.19615i	0.120580	+	0.208851i	0.919997	−	0.391926i	$-0.128191\pi$
$619$	3.00000	+	5.19615i	0.120580	+	0.208851i	−0.799416	+	0.600777i	$0.794858\pi$
$620$		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
$629$		0			0
$630$		0			0
$631$	−11.0000			−0.437903			−0.218952	−	0.975736i	$-0.570264\pi$
$631$	−11.0000			−0.437903			−0.218952	+	0.975736i	$0.570264\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.50000	+	6.06218i	−0.138893	+	0.240570i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−11.0000	−	19.0526i	−0.434474	−	0.752531i	0.562779	−	0.826608i	$-0.309732\pi$
$641$	−11.0000	−	19.0526i	−0.434474	−	0.752531i	−0.997253	+	0.0740768i	$0.976399\pi$
$642$		0			0
$643$	−46.0000			−1.81406			−0.907031	−	0.421063i	$-0.861657\pi$
$643$	−46.0000			−1.81406			−0.907031	+	0.421063i	$0.861657\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	25.0000	+	43.3013i	0.982851	+	1.70235i	0.651120	+	0.758975i	$0.274299\pi$
$647$	25.0000	+	43.3013i	0.982851	+	1.70235i	0.331731	+	0.943374i	$0.392367\pi$
$648$		0			0
$649$	−7.50000	+	12.9904i	−0.294401	+	0.509917i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	7.50000	−	12.9904i	0.293498	−	0.508353i	−0.681137	−	0.732156i	$-0.738514\pi$
$653$	7.50000	−	12.9904i	0.293498	−	0.508353i	0.974634	+	0.223803i	$0.0718474\pi$
$654$		0			0
$655$	7.50000	+	12.9904i	0.293049	+	0.507576i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−24.0000			−0.934907			−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000			−0.934907			−0.467454	+	0.884018i	$0.654829\pi$
$660$		0			0
$661$	19.0000	+	32.9090i	0.739014	+	1.28001i	0.952940	+	0.303160i	$0.0980418\pi$
$661$	19.0000	+	32.9090i	0.739014	+	1.28001i	−0.213925	+	0.976850i	$0.568625\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	12.0000	−	20.7846i	0.464642	−	0.804783i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−30.0000			−1.15814
$672$		0			0
$673$	29.0000			1.11787			0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000			1.11787			0.558934	+	0.829212i	$0.311211\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−9.50000	+	16.4545i	−0.365115	+	0.632397i	−0.988795	−	0.149283i	$-0.952304\pi$
$677$	−9.50000	+	16.4545i	−0.365115	+	0.632397i	0.623680	+	0.781680i	$0.285637\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	8.50000	+	14.7224i	0.325243	+	0.563338i	0.981562	−	0.191146i	$-0.0612204\pi$
$683$	8.50000	+	14.7224i	0.325243	+	0.563338i	−0.656318	+	0.754484i	$0.727887\pi$
$684$		0			0
$685$	−14.0000			−0.534913
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−18.0000	−	31.1769i	−0.685745	−	1.18775i
$690$		0			0
$691$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$691$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	11.0000	−	19.0526i	0.417254	−	0.722705i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−5.00000			−0.188847			−0.0944237	−	0.995532i	$-0.530101\pi$
$701$	−5.00000			−0.188847			−0.0944237	+	0.995532i	$0.530101\pi$
$702$		0			0
$703$	−16.0000	−	27.7128i	−0.603451	−	1.04521i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−3.00000	+	5.19615i	−0.112667	+	0.195146i	−0.916845	−	0.399244i	$-0.869273\pi$
$709$	−3.00000	+	5.19615i	−0.112667	+	0.195146i	0.804178	+	0.594389i	$0.202606\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−40.0000			−1.49801
$714$		0			0
$715$	−12.0000			−0.448775
$716$		0			0
$717$		0			0
$718$		0			0
$719$	17.0000	−	29.4449i	0.633993	−	1.09811i	−0.352735	−	0.935723i	$-0.614748\pi$
$719$	17.0000	−	29.4449i	0.633993	−	1.09811i	0.986728	−	0.162385i	$-0.0519185\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6.00000	+	10.3923i	0.222834	+	0.385961i
$726$		0			0
$727$	−47.0000			−1.74313			−0.871567	−	0.490277i	$-0.836896\pi$
$727$	−47.0000			−1.74313			−0.871567	+	0.490277i	$0.836896\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	7.00000	−	12.1244i	0.258551	−	0.447823i	−0.707303	−	0.706910i	$-0.750088\pi$
$733$	7.00000	−	12.1244i	0.258551	−	0.447823i	0.965854	+	0.259087i	$0.0834217\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	9.00000	−	15.5885i	0.331519	−	0.574208i
$738$		0			0
$739$	−5.00000	−	8.66025i	−0.183928	−	0.318573i	0.759287	−	0.650756i	$-0.225548\pi$
$739$	−5.00000	−	8.66025i	−0.183928	−	0.318573i	−0.943215	+	0.332184i	$0.892215\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−34.0000			−1.24734			−0.623670	−	0.781688i	$-0.714359\pi$
$743$	−34.0000			−1.24734			−0.623670	+	0.781688i	$0.714359\pi$
$744$		0			0
$745$	−9.00000	−	15.5885i	−0.329734	−	0.571117i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	13.5000	−	23.3827i	0.492622	−	0.853246i	−0.507342	−	0.861745i	$-0.669372\pi$
$751$	13.5000	−	23.3827i	0.492622	−	0.853246i	0.999964	+	0.00849853i	$0.00270520\pi$
$752$		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	$-0.558169\pi$
$757$	−10.0000			−0.363456			−0.181728	+	0.983349i	$0.558169\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.00000	−	10.3923i	0.217500	−	0.376721i	−0.736543	−	0.676391i	$-0.763543\pi$
$761$	6.00000	−	10.3923i	0.217500	−	0.376721i	0.954043	+	0.299670i	$0.0968765\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−10.0000	−	17.3205i	−0.361079	−	0.625407i
$768$		0			0
$769$	−37.0000			−1.33425			−0.667127	−	0.744944i	$-0.732476\pi$
$769$	−37.0000			−1.33425			−0.667127	+	0.744944i	$0.732476\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	13.0000	+	22.5167i	0.467578	+	0.809868i	0.999314	−	0.0370420i	$-0.0117935\pi$
$773$	13.0000	+	22.5167i	0.467578	+	0.809868i	−0.531736	+	0.846910i	$0.678460\pi$
$774$		0			0
$775$	10.0000	−	17.3205i	0.359211	−	0.622171i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−16.0000	+	27.7128i	−0.573259	+	0.992915i
$780$		0			0
$781$	−15.0000	−	25.9808i	−0.536742	−	0.929665i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.00000			−0.142766
$786$		0			0
$787$	−11.0000	−	19.0526i	−0.392108	−	0.679150i	0.600620	−	0.799535i	$-0.294921\pi$
$787$	−11.0000	−	19.0526i	−0.392108	−	0.679150i	−0.992727	+	0.120384i	$0.961587\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	20.0000	−	34.6410i	0.710221	−	1.23014i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	3.00000			0.106265			0.0531327	−	0.998587i	$-0.483079\pi$
$797$	3.00000			0.106265			0.0531327	+	0.998587i	$0.483079\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.00000	+	5.19615i	−0.105868	+	0.183368i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−8.00000	−	13.8564i	−0.281265	−	0.487165i	0.690432	−	0.723398i	$-0.257421\pi$
$809$	−8.00000	−	13.8564i	−0.281265	−	0.487165i	−0.971697	+	0.236232i	$0.924087\pi$
$810$		0			0
$811$	−2.00000			−0.0702295			−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000			−0.0702295			−0.0351147	+	0.999383i	$0.511180\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.00000	+	6.92820i	0.140114	+	0.242684i
$816$		0			0
$817$	−12.0000	+	20.7846i	−0.419827	+	0.727161i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−19.5000	+	33.7750i	−0.680555	+	1.17876i	0.294257	+	0.955726i	$0.404928\pi$
$821$	−19.5000	+	33.7750i	−0.680555	+	1.17876i	−0.974812	+	0.223029i	$0.928406\pi$
$822$		0			0
$823$	−16.0000	−	27.7128i	−0.557725	−	0.966008i	−0.997686	−	0.0679910i	$-0.978341\pi$
$823$	−16.0000	−	27.7128i	−0.557725	−	0.966008i	0.439961	−	0.898017i	$-0.354992\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	41.0000			1.42571			0.712855	−	0.701312i	$-0.247402\pi$
$827$	41.0000			1.42571			0.712855	+	0.701312i	$0.247402\pi$
$828$		0			0
$829$	−2.00000	−	3.46410i	−0.0694629	−	0.120313i	0.829202	−	0.558949i	$-0.188795\pi$
$829$	−2.00000	−	3.46410i	−0.0694629	−	0.120313i	−0.898665	+	0.438636i	$0.855462\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−1.00000	+	1.73205i	−0.0346064	+	0.0599401i
$836$		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
$841$	−20.0000			−0.689655
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.50000	−	2.59808i	0.0516016	−	0.0893765i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	32.0000	+	55.4256i	1.09695	+	1.89997i
$852$		0			0
$853$	−14.0000			−0.479351			−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000			−0.479351			−0.239675	+	0.970853i	$0.577041\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−7.00000	−	12.1244i	−0.239115	−	0.414160i	0.721345	−	0.692576i	$-0.243524\pi$
$857$	−7.00000	−	12.1244i	−0.239115	−	0.414160i	−0.960461	+	0.278416i	$0.910191\pi$
$858$		0			0
$859$	−7.00000	+	12.1244i	−0.238837	+	0.413678i	−0.960381	−	0.278691i	$-0.910099\pi$
$859$	−7.00000	+	12.1244i	−0.238837	+	0.413678i	0.721544	+	0.692369i	$0.243433\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	11.0000	−	19.0526i	0.374444	−	0.648557i	−0.615799	−	0.787903i	$-0.711167\pi$
$863$	11.0000	−	19.0526i	0.374444	−	0.648557i	0.990244	+	0.139346i	$0.0445001\pi$
$864$		0			0
$865$	1.00000	+	1.73205i	0.0340010	+	0.0588915i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−33.0000			−1.11945
$870$		0			0
$871$	12.0000	+	20.7846i	0.406604	+	0.704260i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		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
$881$	−54.0000			−1.81931			−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000			−1.81931			−0.909653	+	0.415369i	$0.863653\pi$
$882$		0			0
$883$	40.0000			1.34611			0.673054	−	0.739594i	$-0.264982\pi$
$883$	40.0000			1.34611			0.673054	+	0.739594i	$0.264982\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	24.0000	−	41.5692i	0.805841	−	1.39576i	−0.109881	−	0.993945i	$-0.535047\pi$
$887$	24.0000	−	41.5692i	0.805841	−	1.39576i	0.915722	−	0.401813i	$-0.131620\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−20.0000	−	34.6410i	−0.669274	−	1.15922i
$894$		0			0
$895$	12.0000			0.401116
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−7.50000	−	12.9904i	−0.250139	−	0.433253i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	8.00000	+	13.8564i	0.265636	+	0.460094i	0.967730	−	0.251990i	$-0.0810849\pi$
$907$	8.00000	+	13.8564i	0.265636	+	0.460094i	−0.702094	+	0.712084i	$0.747752\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−6.00000			−0.198789			−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000			−0.198789			−0.0993944	+	0.995048i	$0.531691\pi$
$912$		0			0
$913$	10.5000	+	18.1865i	0.347499	+	0.601886i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	8.00000	−	13.8564i	0.263896	−	0.457081i	−0.703378	−	0.710816i	$-0.748326\pi$
$919$	8.00000	−	13.8564i	0.263896	−	0.457081i	0.967274	+	0.253735i	$0.0816592\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	40.0000			1.31662
$924$		0			0
$925$	−32.0000			−1.05215
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−21.0000	+	36.3731i	−0.688988	+	1.19336i	0.283178	+	0.959067i	$0.408611\pi$
$929$	−21.0000	+	36.3731i	−0.688988	+	1.19336i	−0.972166	+	0.234294i	$0.924722\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−19.0000			−0.620703			−0.310351	−	0.950622i	$-0.600447\pi$
$937$	−19.0000			−0.620703			−0.310351	+	0.950622i	$0.600447\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	6.50000	+	11.2583i	0.211894	+	0.367011i	0.952307	−	0.305141i	$-0.0987035\pi$
$941$	6.50000	+	11.2583i	0.211894	+	0.367011i	−0.740413	+	0.672152i	$0.765370\pi$
$942$		0			0
$943$	32.0000	−	55.4256i	1.04206	−	1.80491i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−24.0000	+	41.5692i	−0.779895	+	1.35082i	0.152106	+	0.988364i	$0.451394\pi$
$947$	−24.0000	+	41.5692i	−0.779895	+	1.35082i	−0.932002	+	0.362454i	$0.881939\pi$
$948$		0			0
$949$	−4.00000	−	6.92820i	−0.129845	−	0.224899i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	38.0000			1.23094			0.615470	−	0.788160i	$-0.288966\pi$
$953$	38.0000			1.23094			0.615470	+	0.788160i	$0.288966\pi$
$954$		0			0
$955$	8.00000	+	13.8564i	0.258874	+	0.448383i
$956$		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
$965$	27.0000			0.869161
$966$		0			0
$967$	3.00000			0.0964735			0.0482367	−	0.998836i	$-0.484640\pi$
$967$	3.00000			0.0964735			0.0482367	+	0.998836i	$0.484640\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−12.5000	+	21.6506i	−0.401144	+	0.694802i	−0.993864	−	0.110607i	$-0.964721\pi$
$971$	−12.5000	+	21.6506i	−0.401144	+	0.694802i	0.592720	+	0.805408i	$0.298054\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−11.0000	−	19.0526i	−0.351921	−	0.609545i	0.634665	−	0.772787i	$-0.281138\pi$
$977$	−11.0000	−	19.0526i	−0.351921	−	0.609545i	−0.986586	+	0.163242i	$0.947805\pi$
$978$		0			0
$979$	54.0000			1.72585
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−22.0000	−	38.1051i	−0.701691	−	1.21536i	−0.967872	−	0.251442i	$-0.919095\pi$
$983$	−22.0000	−	38.1051i	−0.701691	−	1.21536i	0.266181	−	0.963923i	$-0.414238\pi$
$984$		0			0
$985$	13.0000	−	22.5167i	0.414214	−	0.717440i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	24.0000	−	41.5692i	0.763156	−	1.32182i
$990$		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
$995$	12.0000			0.380426
$996$		0			0
$997$	−13.0000	−	22.5167i	−0.411714	−	0.713110i	0.583363	−	0.812211i	$-0.301736\pi$
$997$	−13.0000	−	22.5167i	−0.411714	−	0.713110i	−0.995077	+	0.0991016i	$0.968403\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.s.p.3313.1		2
3.2	odd	2		1176.2.q.g.961.1		2
7.2	even	3		3528.2.a.i.1.1		1
7.3	odd	6		504.2.s.d.361.1		2
7.4	even	3	inner	3528.2.s.p.361.1		2
7.5	odd	6		3528.2.a.q.1.1		1
7.6	odd	2		504.2.s.d.289.1		2
12.11	even	2		2352.2.q.f.961.1		2
21.2	odd	6		1176.2.a.c.1.1		1
21.5	even	6		1176.2.a.g.1.1		1
21.11	odd	6		1176.2.q.g.361.1		2
21.17	even	6		168.2.q.a.25.1	✓	2
21.20	even	2		168.2.q.a.121.1	yes	2
28.3	even	6		1008.2.s.f.865.1		2
28.19	even	6		7056.2.a.bk.1.1		1
28.23	odd	6		7056.2.a.t.1.1		1
28.27	even	2		1008.2.s.f.289.1		2
84.11	even	6		2352.2.q.f.1537.1		2
84.23	even	6		2352.2.a.u.1.1		1
84.47	odd	6		2352.2.a.g.1.1		1
84.59	odd	6		336.2.q.e.193.1		2
84.83	odd	2		336.2.q.e.289.1		2
168.5	even	6		9408.2.a.ba.1.1		1
168.59	odd	6		1344.2.q.d.193.1		2
168.83	odd	2		1344.2.q.d.961.1		2
168.101	even	6		1344.2.q.o.193.1		2
168.107	even	6		9408.2.a.p.1.1		1
168.125	even	2		1344.2.q.o.961.1		2
168.131	odd	6		9408.2.a.cq.1.1		1
168.149	odd	6		9408.2.a.cf.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.q.a.25.1	✓	2	21.17	even	6
168.2.q.a.121.1	yes	2	21.20	even	2
336.2.q.e.193.1		2	84.59	odd	6
336.2.q.e.289.1		2	84.83	odd	2
504.2.s.d.289.1		2	7.6	odd	2
504.2.s.d.361.1		2	7.3	odd	6
1008.2.s.f.289.1		2	28.27	even	2
1008.2.s.f.865.1		2	28.3	even	6
1176.2.a.c.1.1		1	21.2	odd	6
1176.2.a.g.1.1		1	21.5	even	6
1176.2.q.g.361.1		2	21.11	odd	6
1176.2.q.g.961.1		2	3.2	odd	2
1344.2.q.d.193.1		2	168.59	odd	6
1344.2.q.d.961.1		2	168.83	odd	2
1344.2.q.o.193.1		2	168.101	even	6
1344.2.q.o.961.1		2	168.125	even	2
2352.2.a.g.1.1		1	84.47	odd	6
2352.2.a.u.1.1		1	84.23	even	6
2352.2.q.f.961.1		2	12.11	even	2
2352.2.q.f.1537.1		2	84.11	even	6
3528.2.a.i.1.1		1	7.2	even	3
3528.2.a.q.1.1		1	7.5	odd	6
3528.2.s.p.361.1		2	7.4	even	3	inner
3528.2.s.p.3313.1		2	1.1	even	1	trivial
7056.2.a.t.1.1		1	28.23	odd	6
7056.2.a.bk.1.1		1	28.19	even	6
9408.2.a.p.1.1		1	168.107	even	6
9408.2.a.ba.1.1		1	168.5	even	6
9408.2.a.cf.1.1		1	168.149	odd	6
9408.2.a.cq.1.1		1	168.131	odd	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3528.2.s.p.3313.1

Level	$3528$
Weight	$2$
Character	3528.3313
Analytic conductor	$28.171$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$