LMFDB - Embedded newform 3528.2.s.b.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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 504)
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.b.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 + 3.46410i) q^{5} +O(q^{10})$	$q+(-2.00000 + 3.46410i) q^{5} +3.00000 q^{13} +(2.00000 + 3.46410i) q^{17} +(3.50000 - 6.06218i) q^{19} +(2.00000 - 3.46410i) q^{23} +(-5.50000 - 9.52628i) q^{25} +8.00000 q^{29} +(-2.50000 - 4.33013i) q^{31} +(-1.50000 + 2.59808i) q^{37} +8.00000 q^{41} +11.0000 q^{43} +(-2.00000 + 3.46410i) q^{47} +(2.00000 + 3.46410i) q^{53} +(-6.00000 - 10.3923i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(-6.00000 + 10.3923i) q^{65} +(1.50000 + 2.59808i) q^{67} -12.0000 q^{71} +(0.500000 + 0.866025i) q^{73} +(-0.500000 + 0.866025i) q^{79} +12.0000 q^{83} -16.0000 q^{85} +(-4.00000 + 6.92820i) q^{89} +(14.0000 + 24.2487i) q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5}+O(q^{10}) $ 2 * q - 4 * q^5	$ 2 q - 4 q^{5} + 6 q^{13} + 4 q^{17} + 7 q^{19} + 4 q^{23} - 11 q^{25} + 16 q^{29} - 5 q^{31} - 3 q^{37} + 16 q^{41} + 22 q^{43} - 4 q^{47} + 4 q^{53} - 12 q^{59} - 2 q^{61} - 12 q^{65} + 3 q^{67} - 24 q^{71} + q^{73} - q^{79} + 24 q^{83} - 32 q^{85} - 8 q^{89} + 28 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 + 6 * q^13 + 4 * q^17 + 7 * q^19 + 4 * q^23 - 11 * q^25 + 16 * q^29 - 5 * q^31 - 3 * q^37 + 16 * q^41 + 22 * q^43 - 4 * q^47 + 4 * q^53 - 12 * q^59 - 2 * q^61 - 12 * q^65 + 3 * q^67 - 24 * q^71 + q^73 - q^79 + 24 * q^83 - 32 * q^85 - 8 * q^89 + 28 * 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
$5$	−2.00000	+	3.46410i	−0.894427	+	1.54919i	−0.0599153	+	0.998203i	$0.519083\pi$
$5$	−2.00000	+	3.46410i	−0.894427	+	1.54919i	−0.834512	+	0.550990i	$0.814250\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		0			0
$13$	3.00000			0.832050			0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000			0.832050			0.416025	+	0.909353i	$0.363423\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.00000	+	3.46410i	0.485071	+	0.840168i	0.999853	−	0.0171533i	$-0.00546033\pi$
$17$	2.00000	+	3.46410i	0.485071	+	0.840168i	−0.514782	+	0.857321i	$0.672127\pi$
$18$		0			0
$19$	3.50000	−	6.06218i	0.802955	−	1.39076i	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	3.50000	−	6.06218i	0.802955	−	1.39076i	0.917663	−	0.397360i	$-0.130073\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	−0.578610	−	0.815604i	$-0.696405\pi$
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	0.995639	+	0.0932891i	$0.0297381\pi$
$24$		0			0
$25$	−5.50000	−	9.52628i	−1.10000	−	1.90526i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	8.00000			1.48556			0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000			1.48556			0.742781	+	0.669534i	$0.233506\pi$
$30$		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
$41$	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
$43$	11.0000			1.67748			0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000			1.67748			0.838742	+	0.544529i	$0.183292\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.00000	+	3.46410i	−0.291730	+	0.505291i	−0.974219	−	0.225605i	$-0.927564\pi$
$47$	−2.00000	+	3.46410i	−0.291730	+	0.505291i	0.682489	+	0.730896i	$0.260898\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.00000	+	3.46410i	0.274721	+	0.475831i	0.970065	−	0.242846i	$-0.0780811\pi$
$53$	2.00000	+	3.46410i	0.274721	+	0.475831i	−0.695344	+	0.718677i	$0.744748\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.00000	−	10.3923i	−0.781133	−	1.35296i	−0.931282	−	0.364299i	$-0.881308\pi$
$59$	−6.00000	−	10.3923i	−0.781133	−	1.35296i	0.150148	−	0.988663i	$-0.452025\pi$
$60$		0			0
$61$	−1.00000	+	1.73205i	−0.128037	+	0.221766i	−0.922916	−	0.385002i	$-0.874201\pi$
$61$	−1.00000	+	1.73205i	−0.128037	+	0.221766i	0.794879	+	0.606768i	$0.207534\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.00000	+	10.3923i	−0.744208	+	1.28901i
$66$		0			0
$67$	1.50000	+	2.59808i	0.183254	+	0.317406i	0.942987	−	0.332830i	$-0.108004\pi$
$67$	1.50000	+	2.59808i	0.183254	+	0.317406i	−0.759733	+	0.650236i	$0.774670\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−12.0000			−1.42414			−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		0			0
$73$	0.500000	+	0.866025i	0.0585206	+	0.101361i	0.893801	−	0.448463i	$-0.148028\pi$
$73$	0.500000	+	0.866025i	0.0585206	+	0.101361i	−0.835281	+	0.549823i	$0.814695\pi$
$74$		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
$85$	−16.0000			−1.73544
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.00000	+	6.92820i	−0.423999	+	0.734388i	−0.996326	−	0.0856373i	$-0.972707\pi$
$89$	−4.00000	+	6.92820i	−0.423999	+	0.734388i	0.572327	+	0.820025i	$0.306041\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	14.0000	+	24.2487i	1.43637	+	2.48787i
$96$		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
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	1.50000	−	2.59808i	0.147799	−	0.255996i	−0.782614	−	0.622507i	$-0.786114\pi$
$103$	1.50000	−	2.59808i	0.147799	−	0.255996i	0.930414	+	0.366511i	$0.119448\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.00000	+	10.3923i	−0.580042	+	1.00466i	0.415432	+	0.909624i	$0.363630\pi$
$107$	−6.00000	+	10.3923i	−0.580042	+	1.00466i	−0.995474	+	0.0950377i	$0.969703\pi$
$108$		0			0
$109$	5.50000	+	9.52628i	0.526804	+	0.912452i	0.999512	+	0.0312328i	$0.00994332\pi$
$109$	5.50000	+	9.52628i	0.526804	+	0.912452i	−0.472708	+	0.881219i	$0.656723\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	12.0000			1.12887			0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000			1.12887			0.564433	+	0.825479i	$0.309095\pi$
$114$		0			0
$115$	8.00000	+	13.8564i	0.746004	+	1.29212i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	24.0000			2.14663
$126$		0			0
$127$	−1.00000			−0.0887357			−0.0443678	−	0.999015i	$-0.514127\pi$
$127$	−1.00000			−0.0887357			−0.0443678	+	0.999015i	$0.514127\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.00000	−	6.92820i	0.349482	−	0.605320i	−0.636676	−	0.771132i	$-0.719691\pi$
$131$	4.00000	−	6.92820i	0.349482	−	0.605320i	0.986157	+	0.165812i	$0.0530244\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.00000	+	13.8564i	0.683486	+	1.18383i	0.973910	+	0.226935i	$0.0728704\pi$
$137$	8.00000	+	13.8564i	0.683486	+	1.18383i	−0.290424	+	0.956898i	$0.593796\pi$
$138$		0			0
$139$	−15.0000			−1.27228			−0.636142	−	0.771572i	$-0.719471\pi$
$139$	−15.0000			−1.27228			−0.636142	+	0.771572i	$0.719471\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−16.0000	+	27.7128i	−1.32873	+	2.30142i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.00000	+	13.8564i	−0.655386	+	1.13516i	0.326411	+	0.945228i	$0.394160\pi$
$149$	−8.00000	+	13.8564i	−0.655386	+	1.13516i	−0.981797	+	0.189933i	$0.939173\pi$
$150$		0			0
$151$	10.0000	+	17.3205i	0.813788	+	1.40952i	0.910195	+	0.414181i	$0.135932\pi$
$151$	10.0000	+	17.3205i	0.813788	+	1.40952i	−0.0964061	+	0.995342i	$0.530735\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	20.0000			1.60644
$156$		0			0
$157$	−9.00000	−	15.5885i	−0.718278	−	1.24409i	−0.961681	−	0.274169i	$-0.911597\pi$
$157$	−9.00000	−	15.5885i	−0.718278	−	1.24409i	0.243403	−	0.969925i	$-0.421736\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	4.00000	−	6.92820i	0.313304	−	0.542659i	−0.665771	−	0.746156i	$-0.731897\pi$
$163$	4.00000	−	6.92820i	0.313304	−	0.542659i	0.979076	+	0.203497i	$0.0652307\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	−4.00000			−0.307692
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.00000	+	3.46410i	−0.152057	+	0.263371i	−0.931984	−	0.362500i	$-0.881923\pi$
$173$	−2.00000	+	3.46410i	−0.152057	+	0.263371i	0.779926	+	0.625871i	$0.215256\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.0000	−	17.3205i	−0.747435	−	1.29460i	−0.949048	−	0.315130i	$-0.897952\pi$
$179$	−10.0000	−	17.3205i	−0.747435	−	1.29460i	0.201613	−	0.979465i	$-0.435382\pi$
$180$		0			0
$181$	−5.00000			−0.371647			−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000			−0.371647			−0.185824	+	0.982583i	$0.559495\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.00000	−	10.3923i	−0.441129	−	0.764057i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.00000	+	3.46410i	−0.144715	+	0.250654i	−0.929267	−	0.369410i	$-0.879560\pi$
$191$	−2.00000	+	3.46410i	−0.144715	+	0.250654i	0.784552	+	0.620063i	$0.212893\pi$
$192$		0			0
$193$	4.50000	+	7.79423i	0.323917	+	0.561041i	0.981293	−	0.192522i	$-0.0616668\pi$
$193$	4.50000	+	7.79423i	0.323917	+	0.561041i	−0.657376	+	0.753563i	$0.728333\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−12.0000			−0.854965			−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000			−0.854965			−0.427482	+	0.904024i	$0.640599\pi$
$198$		0			0
$199$	10.0000	+	17.3205i	0.708881	+	1.22782i	0.965272	+	0.261245i	$0.0841331\pi$
$199$	10.0000	+	17.3205i	0.708881	+	1.22782i	−0.256391	+	0.966573i	$0.582534\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−16.0000	+	27.7128i	−1.11749	+	1.93555i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−16.0000			−1.10149			−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000			−1.10149			−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−22.0000	+	38.1051i	−1.50039	+	2.59875i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.00000	+	10.3923i	0.403604	+	0.699062i
$222$		0			0
$223$	12.0000			0.803579			0.401790	−	0.915732i	$-0.368388\pi$
$223$	12.0000			0.803579			0.401790	+	0.915732i	$0.368388\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.00000	+	6.92820i	0.265489	+	0.459841i	0.967692	−	0.252136i	$-0.0811332\pi$
$227$	4.00000	+	6.92820i	0.265489	+	0.459841i	−0.702202	+	0.711977i	$0.747800\pi$
$228$		0			0
$229$	2.50000	−	4.33013i	0.165205	−	0.286143i	−0.771523	−	0.636201i	$-0.780505\pi$
$229$	2.50000	−	4.33013i	0.165205	−	0.286143i	0.936728	+	0.350058i	$0.113838\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	14.0000	−	24.2487i	0.917170	−	1.58859i	0.113478	−	0.993540i	$-0.463801\pi$
$233$	14.0000	−	24.2487i	0.917170	−	1.58859i	0.803692	−	0.595045i	$-0.202866\pi$
$234$		0			0
$235$	−8.00000	−	13.8564i	−0.521862	−	0.903892i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	8.00000			0.517477			0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000			0.517477			0.258738	+	0.965947i	$0.416693\pi$
$240$		0			0
$241$	−1.00000	−	1.73205i	−0.0644157	−	0.111571i	0.832019	−	0.554747i	$-0.187185\pi$
$241$	−1.00000	−	1.73205i	−0.0644157	−	0.111571i	−0.896435	+	0.443176i	$0.853852\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	10.5000	−	18.1865i	0.668099	−	1.15718i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	20.0000			1.26239			0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000			1.26239			0.631194	+	0.775625i	$0.282565\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.00000	+	10.3923i	−0.374270	+	0.648254i	−0.990217	−	0.139533i	$-0.955440\pi$
$257$	−6.00000	+	10.3923i	−0.374270	+	0.648254i	0.615948	+	0.787787i	$0.288773\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.0000	+	20.7846i	0.739952	+	1.28163i	0.952517	+	0.304487i	$0.0984850\pi$
$263$	12.0000	+	20.7846i	0.739952	+	1.28163i	−0.212565	+	0.977147i	$0.568182\pi$
$264$		0			0
$265$	−16.0000			−0.982872
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	10.0000	−	17.3205i	0.607457	−	1.05215i	−0.384201	−	0.923249i	$-0.625523\pi$
$271$	10.0000	−	17.3205i	0.607457	−	1.05215i	0.991658	−	0.128897i	$-0.0411435\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−6.50000	−	11.2583i	−0.390547	−	0.676448i	0.601975	−	0.798515i	$-0.294381\pi$
$277$	−6.50000	−	11.2583i	−0.390547	−	0.676448i	−0.992522	+	0.122068i	$0.961047\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.0000			−0.715860			−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000			−0.715860			−0.357930	+	0.933748i	$0.616517\pi$
$282$		0			0
$283$	−0.500000	−	0.866025i	−0.0297219	−	0.0514799i	0.850782	−	0.525519i	$-0.176129\pi$
$283$	−0.500000	−	0.866025i	−0.0297219	−	0.0514799i	−0.880504	+	0.474039i	$0.842796\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.500000	−	0.866025i	0.0294118	−	0.0509427i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−24.0000			−1.40209			−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000			−1.40209			−0.701047	+	0.713115i	$0.747284\pi$
$294$		0			0
$295$	48.0000			2.79467
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.00000	−	10.3923i	0.346989	−	0.601003i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4.00000	−	6.92820i	−0.229039	−	0.396708i
$306$		0			0
$307$	11.0000			0.627803			0.313902	−	0.949456i	$-0.398364\pi$
$307$	11.0000			0.627803			0.313902	+	0.949456i	$0.398364\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.0000	+	20.7846i	0.680458	+	1.17859i	0.974841	+	0.222900i	$0.0715523\pi$
$311$	12.0000	+	20.7846i	0.680458	+	1.17859i	−0.294384	+	0.955687i	$0.595114\pi$
$312$		0			0
$313$	−6.50000	+	11.2583i	−0.367402	+	0.636358i	−0.989158	−	0.146852i	$-0.953086\pi$
$313$	−6.50000	+	11.2583i	−0.367402	+	0.636358i	0.621757	+	0.783210i	$0.286419\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	12.0000	−	20.7846i	0.673987	−	1.16738i	−0.302777	−	0.953062i	$-0.597914\pi$
$317$	12.0000	−	20.7846i	0.673987	−	1.16738i	0.976764	−	0.214318i	$-0.0687530\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	28.0000			1.55796
$324$		0			0
$325$	−16.5000	−	28.5788i	−0.915255	−	1.58527i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	3.50000	−	6.06218i	0.192377	−	0.333207i	−0.753660	−	0.657264i	$-0.771714\pi$
$331$	3.50000	−	6.06218i	0.192377	−	0.333207i	0.946038	+	0.324057i	$0.105047\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−12.0000			−0.655630
$336$		0			0
$337$	−19.0000			−1.03500			−0.517498	−	0.855684i	$-0.673136\pi$
$337$	−19.0000			−1.03500			−0.517498	+	0.855684i	$0.673136\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.00000	−	10.3923i	−0.322097	−	0.557888i	0.658824	−	0.752297i	$-0.271054\pi$
$347$	−6.00000	−	10.3923i	−0.322097	−	0.557888i	−0.980921	+	0.194409i	$0.937721\pi$
$348$		0			0
$349$	34.0000			1.81998			0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000			1.81998			0.909989	+	0.414632i	$0.136090\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.00000	−	10.3923i	−0.319348	−	0.553127i	0.661004	−	0.750382i	$-0.270130\pi$
$353$	−6.00000	−	10.3923i	−0.319348	−	0.553127i	−0.980352	+	0.197256i	$0.936797\pi$
$354$		0			0
$355$	24.0000	−	41.5692i	1.27379	−	2.20627i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.0000	+	20.7846i	−0.633336	+	1.09697i	0.353529	+	0.935423i	$0.384981\pi$
$359$	−12.0000	+	20.7846i	−0.633336	+	1.09697i	−0.986865	+	0.161546i	$0.948352\pi$
$360$		0			0
$361$	−15.0000	−	25.9808i	−0.789474	−	1.36741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.00000			−0.209370
$366$		0			0
$367$	8.50000	+	14.7224i	0.443696	+	0.768505i	0.997960	−	0.0638362i	$-0.0203335\pi$
$367$	8.50000	+	14.7224i	0.443696	+	0.768505i	−0.554264	+	0.832341i	$0.687000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	0.500000	−	0.866025i	0.0258890	−	0.0448411i	−0.852791	−	0.522253i	$-0.825092\pi$
$373$	0.500000	−	0.866025i	0.0258890	−	0.0448411i	0.878680	+	0.477412i	$0.158425\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	24.0000			1.23606
$378$		0			0
$379$	21.0000			1.07870			0.539349	−	0.842082i	$-0.318670\pi$
$379$	21.0000			1.07870			0.539349	+	0.842082i	$0.318670\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.00000	+	10.3923i	−0.306586	+	0.531022i	−0.977613	−	0.210411i	$-0.932520\pi$
$383$	−6.00000	+	10.3923i	−0.306586	+	0.531022i	0.671027	+	0.741433i	$0.265853\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.00000	−	3.46410i	−0.101404	−	0.175637i	0.810859	−	0.585241i	$-0.199000\pi$
$389$	−2.00000	−	3.46410i	−0.101404	−	0.175637i	−0.912263	+	0.409604i	$0.865667\pi$
$390$		0			0
$391$	16.0000			0.809155
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.00000	−	3.46410i	−0.100631	−	0.174298i
$396$		0			0
$397$	3.50000	−	6.06218i	0.175660	−	0.304252i	−0.764730	−	0.644351i	$-0.777127\pi$
$397$	3.50000	−	6.06218i	0.175660	−	0.304252i	0.940389	+	0.340099i	$0.110461\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−18.0000	+	31.1769i	−0.898877	+	1.55690i	−0.0699455	+	0.997551i	$0.522283\pi$
$401$	−18.0000	+	31.1769i	−0.898877	+	1.55690i	−0.828932	+	0.559350i	$0.811051\pi$
$402$		0			0
$403$	−7.50000	−	12.9904i	−0.373602	−	0.647097i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−7.50000	−	12.9904i	−0.370851	−	0.642333i	0.618846	−	0.785513i	$-0.287601\pi$
$409$	−7.50000	−	12.9904i	−0.370851	−	0.642333i	−0.989697	+	0.143180i	$0.954267\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−24.0000	+	41.5692i	−1.17811	+	2.04055i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	16.0000			0.781651			0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000			0.781651			0.390826	+	0.920465i	$0.372190\pi$
$420$		0			0
$421$	1.00000			0.0487370			0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000			0.0487370			0.0243685	+	0.999703i	$0.492242\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	22.0000	−	38.1051i	1.06716	−	1.84837i
$426$		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
$433$	5.00000			0.240285			0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000			0.240285			0.120142	+	0.992757i	$0.461665\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−14.0000	−	24.2487i	−0.669711	−	1.15997i
$438$		0			0
$439$	−6.00000	+	10.3923i	−0.286364	+	0.495998i	−0.972939	−	0.231062i	$-0.925780\pi$
$439$	−6.00000	+	10.3923i	−0.286364	+	0.495998i	0.686575	+	0.727059i	$0.259113\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.00000	−	3.46410i	0.0950229	−	0.164584i	−0.814595	−	0.580030i	$-0.803041\pi$
$443$	2.00000	−	3.46410i	0.0950229	−	0.164584i	0.909618	+	0.415445i	$0.136374\pi$
$444$		0			0
$445$	−16.0000	−	27.7128i	−0.758473	−	1.31371i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−20.0000			−0.943858			−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000			−0.943858			−0.471929	+	0.881636i	$0.656442\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.500000	+	0.866025i	−0.0233890	+	0.0405110i	−0.877483	−	0.479608i	$-0.840779\pi$
$457$	−0.500000	+	0.866025i	−0.0233890	+	0.0405110i	0.854094	+	0.520119i	$0.174112\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	5.00000			0.232370			0.116185	−	0.993228i	$-0.462933\pi$
$463$	5.00000			0.232370			0.116185	+	0.993228i	$0.462933\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6.00000	−	10.3923i	0.277647	−	0.480899i	−0.693153	−	0.720791i	$-0.743779\pi$
$467$	6.00000	−	10.3923i	0.277647	−	0.480899i	0.970799	+	0.239892i	$0.0771121\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−77.0000			−3.53300
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2.00000	+	3.46410i	0.0913823	+	0.158279i	0.908093	−	0.418769i	$-0.137538\pi$
$479$	2.00000	+	3.46410i	0.0913823	+	0.158279i	−0.816711	+	0.577047i	$0.804205\pi$
$480$		0			0
$481$	−4.50000	+	7.79423i	−0.205182	+	0.355386i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.00000	+	6.92820i	−0.181631	+	0.314594i
$486$		0			0
$487$	17.5000	+	30.3109i	0.793001	+	1.37352i	0.924101	+	0.382148i	$0.124816\pi$
$487$	17.5000	+	30.3109i	0.793001	+	1.37352i	−0.131100	+	0.991369i	$0.541851\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−24.0000			−1.08310			−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000			−1.08310			−0.541552	+	0.840667i	$0.682163\pi$
$492$		0			0
$493$	16.0000	+	27.7128i	0.720604	+	1.24812i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	14.5000	−	25.1147i	0.649109	−	1.12429i	−0.334227	−	0.942493i	$-0.608475\pi$
$499$	14.5000	−	25.1147i	0.649109	−	1.12429i	0.983336	−	0.181797i	$-0.0581915\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−40.0000			−1.78351			−0.891756	−	0.452517i	$-0.850526\pi$
$503$	−40.0000			−1.78351			−0.891756	+	0.452517i	$0.850526\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6.00000	−	10.3923i	0.265945	−	0.460631i	−0.701866	−	0.712309i	$-0.747649\pi$
$509$	6.00000	−	10.3923i	0.265945	−	0.460631i	0.967811	+	0.251679i	$0.0809826\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.00000	+	10.3923i	0.264392	+	0.457940i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	12.0000	+	20.7846i	0.525730	+	0.910590i	0.999551	+	0.0299693i	$0.00954094\pi$
$521$	12.0000	+	20.7846i	0.525730	+	0.910590i	−0.473821	+	0.880621i	$0.657126\pi$
$522$		0			0
$523$	14.5000	−	25.1147i	0.634041	−	1.09819i	−0.352677	−	0.935745i	$-0.614728\pi$
$523$	14.5000	−	25.1147i	0.634041	−	1.09819i	0.986718	−	0.162446i	$-0.0519382\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	10.0000	−	17.3205i	0.435607	−	0.754493i
$528$		0			0
$529$	3.50000	+	6.06218i	0.152174	+	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	24.0000			1.03956
$534$		0			0
$535$	−24.0000	−	41.5692i	−1.03761	−	1.79719i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	11.5000	−	19.9186i	0.494424	−	0.856367i	−0.505556	−	0.862794i	$-0.668712\pi$
$541$	11.5000	−	19.9186i	0.494424	−	0.856367i	0.999979	+	0.00642713i	$0.00204583\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−44.0000			−1.88475
$546$		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
$551$	28.0000	−	48.4974i	1.19284	−	2.06606i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.00000	+	6.92820i	0.169485	+	0.293557i	0.938239	−	0.345988i	$-0.112456\pi$
$557$	4.00000	+	6.92820i	0.169485	+	0.293557i	−0.768754	+	0.639545i	$0.779123\pi$
$558$		0			0
$559$	33.0000			1.39575
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10.0000	−	17.3205i	−0.421450	−	0.729972i	0.574632	−	0.818412i	$-0.305145\pi$
$563$	−10.0000	−	17.3205i	−0.421450	−	0.729972i	−0.996082	+	0.0884397i	$0.971812\pi$
$564$		0			0
$565$	−24.0000	+	41.5692i	−1.00969	+	1.74883i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12.0000	−	20.7846i	0.503066	−	0.871336i	−0.496928	−	0.867792i	$-0.665539\pi$
$569$	12.0000	−	20.7846i	0.503066	−	0.871336i	0.999994	−	0.00354413i	$-0.00112814\pi$
$570$		0			0
$571$	12.5000	+	21.6506i	0.523109	+	0.906051i	0.999638	+	0.0268925i	$0.00856117\pi$
$571$	12.5000	+	21.6506i	0.523109	+	0.906051i	−0.476530	+	0.879158i	$0.658105\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−44.0000			−1.83493
$576$		0			0
$577$	13.5000	+	23.3827i	0.562012	+	0.973434i	0.997321	+	0.0731526i	$0.0233060\pi$
$577$	13.5000	+	23.3827i	0.562012	+	0.973434i	−0.435308	+	0.900281i	$0.643361\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−16.0000			−0.660391			−0.330195	−	0.943913i	$-0.607115\pi$
$587$	−16.0000			−0.660391			−0.330195	+	0.943913i	$0.607115\pi$
$588$		0			0
$589$	−35.0000			−1.44215
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−12.0000	+	20.7846i	−0.492781	+	0.853522i	−0.999965	−	0.00831589i	$-0.997353\pi$
$593$	−12.0000	+	20.7846i	−0.492781	+	0.853522i	0.507184	+	0.861838i	$0.330686\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8.00000	−	13.8564i	−0.326871	−	0.566157i	0.655018	−	0.755613i	$-0.272661\pi$
$599$	−8.00000	−	13.8564i	−0.326871	−	0.566157i	−0.981889	+	0.189456i	$0.939328\pi$
$600$		0			0
$601$	41.0000			1.67242			0.836212	−	0.548406i	$-0.184765\pi$
$601$	41.0000			1.67242			0.836212	+	0.548406i	$0.184765\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	22.0000	+	38.1051i	0.894427	+	1.54919i
$606$		0			0
$607$	−17.5000	+	30.3109i	−0.710303	+	1.23028i	0.254440	+	0.967088i	$0.418109\pi$
$607$	−17.5000	+	30.3109i	−0.710303	+	1.23028i	−0.964743	+	0.263193i	$0.915225\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−6.00000	+	10.3923i	−0.242734	+	0.420428i
$612$		0			0
$613$	13.0000	+	22.5167i	0.525065	+	0.909439i	0.999574	+	0.0291886i	$0.00929235\pi$
$613$	13.0000	+	22.5167i	0.525065	+	0.909439i	−0.474509	+	0.880251i	$0.657374\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$	4.50000	+	7.79423i	0.180870	+	0.313276i	0.942177	−	0.335115i	$-0.108775\pi$
$619$	4.50000	+	7.79423i	0.180870	+	0.313276i	−0.761307	+	0.648392i	$0.775442\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−20.5000	+	35.5070i	−0.820000	+	1.42028i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.0000			−0.478471
$630$		0			0
$631$	−4.00000			−0.159237			−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000			−0.159237			−0.0796187	+	0.996825i	$0.525370\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.00000	−	3.46410i	0.0793676	−	0.137469i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−14.0000	−	24.2487i	−0.552967	−	0.957767i	−0.998059	−	0.0622816i	$-0.980162\pi$
$641$	−14.0000	−	24.2487i	−0.552967	−	0.957767i	0.445092	−	0.895485i	$-0.353171\pi$
$642$		0			0
$643$	−7.00000			−0.276053			−0.138027	−	0.990429i	$-0.544076\pi$
$643$	−7.00000			−0.276053			−0.138027	+	0.990429i	$0.544076\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−24.0000	−	41.5692i	−0.943537	−	1.63425i	−0.758654	−	0.651494i	$-0.774142\pi$
$647$	−24.0000	−	41.5692i	−0.943537	−	1.63425i	−0.184884	−	0.982760i	$-0.559191\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	8.00000	−	13.8564i	0.313064	−	0.542243i	−0.665960	−	0.745988i	$-0.731978\pi$
$653$	8.00000	−	13.8564i	0.313064	−	0.542243i	0.979024	+	0.203744i	$0.0653112\pi$
$654$		0			0
$655$	16.0000	+	27.7128i	0.625172	+	1.08283i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−4.00000			−0.155818			−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000			−0.155818			−0.0779089	+	0.996960i	$0.524824\pi$
$660$		0			0
$661$	17.5000	+	30.3109i	0.680671	+	1.17896i	0.974776	+	0.223184i	$0.0716450\pi$
$661$	17.5000	+	30.3109i	0.680671	+	1.17896i	−0.294105	+	0.955773i	$0.595022\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	16.0000	−	27.7128i	0.619522	−	1.07304i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−29.0000			−1.11787			−0.558934	−	0.829212i	$-0.688789\pi$
$673$	−29.0000			−1.11787			−0.558934	+	0.829212i	$0.688789\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−18.0000	−	31.1769i	−0.688751	−	1.19295i	−0.972242	−	0.233977i	$-0.924826\pi$
$683$	−18.0000	−	31.1769i	−0.688751	−	1.19295i	0.283491	−	0.958975i	$-0.408507\pi$
$684$		0			0
$685$	−64.0000			−2.44531
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6.00000	+	10.3923i	0.228582	+	0.395915i
$690$		0			0
$691$	8.50000	−	14.7224i	0.323355	−	0.560068i	−0.657823	−	0.753173i	$-0.728522\pi$
$691$	8.50000	−	14.7224i	0.323355	−	0.560068i	0.981178	+	0.193105i	$0.0618558\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	30.0000	−	51.9615i	1.13796	−	1.97101i
$696$		0			0
$697$	16.0000	+	27.7128i	0.606043	+	1.04970i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−20.0000			−0.755390			−0.377695	−	0.925930i	$-0.623283\pi$
$701$	−20.0000			−0.755390			−0.377695	+	0.925930i	$0.623283\pi$
$702$		0			0
$703$	10.5000	+	18.1865i	0.396015	+	0.685918i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	19.0000	−	32.9090i	0.713560	−	1.23592i	−0.249952	−	0.968258i	$-0.580415\pi$
$709$	19.0000	−	32.9090i	0.713560	−	1.23592i	0.963512	−	0.267664i	$-0.0862517\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−20.0000			−0.749006
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−44.0000	−	76.2102i	−1.63412	−	2.83038i
$726$		0			0
$727$	−5.00000			−0.185440			−0.0927199	−	0.995692i	$-0.529556\pi$
$727$	−5.00000			−0.185440			−0.0927199	+	0.995692i	$0.529556\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	22.0000	+	38.1051i	0.813699	+	1.40937i
$732$		0			0
$733$	−17.5000	+	30.3109i	−0.646377	+	1.11956i	0.337604	+	0.941288i	$0.390383\pi$
$733$	−17.5000	+	30.3109i	−0.646377	+	1.11956i	−0.983982	+	0.178270i	$0.942950\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−9.50000	−	16.4545i	−0.349463	−	0.605288i	0.636691	−	0.771119i	$-0.280303\pi$
$739$	−9.50000	−	16.4545i	−0.349463	−	0.605288i	−0.986154	+	0.165831i	$0.946969\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	8.00000			0.293492			0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000			0.293492			0.146746	+	0.989174i	$0.453120\pi$
$744$		0			0
$745$	−32.0000	−	55.4256i	−1.17239	−	2.03064i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−5.50000	+	9.52628i	−0.200698	+	0.347619i	−0.948753	−	0.316017i	$-0.897654\pi$
$751$	−5.50000	+	9.52628i	−0.200698	+	0.347619i	0.748056	+	0.663636i	$0.230988\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−80.0000			−2.91150
$756$		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
$761$	−6.00000	+	10.3923i	−0.217500	+	0.376721i	−0.954043	−	0.299670i	$-0.903123\pi$
$761$	−6.00000	+	10.3923i	−0.217500	+	0.376721i	0.736543	+	0.676391i	$0.236457\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−18.0000	−	31.1769i	−0.649942	−	1.12573i
$768$		0			0
$769$	−31.0000			−1.11789			−0.558944	−	0.829205i	$-0.688793\pi$
$769$	−31.0000			−1.11789			−0.558944	+	0.829205i	$0.688793\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4.00000	−	6.92820i	−0.143870	−	0.249190i	0.785081	−	0.619393i	$-0.212621\pi$
$773$	−4.00000	−	6.92820i	−0.143870	−	0.249190i	−0.928951	+	0.370203i	$0.879288\pi$
$774$		0			0
$775$	−27.5000	+	47.6314i	−0.987829	+	1.71097i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	28.0000	−	48.4974i	1.00320	−	1.73760i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	72.0000			2.56979
$786$		0			0
$787$	20.0000	+	34.6410i	0.712923	+	1.23482i	0.963755	+	0.266788i	$0.0859624\pi$
$787$	20.0000	+	34.6410i	0.712923	+	1.23482i	−0.250832	+	0.968031i	$0.580704\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−3.00000	+	5.19615i	−0.106533	+	0.184521i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12.0000			0.425062			0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000			0.425062			0.212531	+	0.977154i	$0.431829\pi$
$798$		0			0
$799$	−16.0000			−0.566039
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	6.00000	+	10.3923i	0.210949	+	0.365374i	0.952012	−	0.306062i	$-0.0990113\pi$
$809$	6.00000	+	10.3923i	0.210949	+	0.365374i	−0.741063	+	0.671436i	$0.765678\pi$
$810$		0			0
$811$	−24.0000			−0.842754			−0.421377	−	0.906886i	$-0.638453\pi$
$811$	−24.0000			−0.842754			−0.421377	+	0.906886i	$0.638453\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	16.0000	+	27.7128i	0.560456	+	0.970737i
$816$		0			0
$817$	38.5000	−	66.6840i	1.34694	−	2.33298i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−18.0000	+	31.1769i	−0.628204	+	1.08808i	0.359708	+	0.933065i	$0.382876\pi$
$821$	−18.0000	+	31.1769i	−0.628204	+	1.08808i	−0.987912	+	0.155017i	$0.950457\pi$
$822$		0			0
$823$	2.00000	+	3.46410i	0.0697156	+	0.120751i	0.898776	−	0.438408i	$-0.144457\pi$
$823$	2.00000	+	3.46410i	0.0697156	+	0.120751i	−0.829060	+	0.559159i	$0.811124\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−36.0000			−1.25184			−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000			−1.25184			−0.625921	+	0.779886i	$0.715277\pi$
$828$		0			0
$829$	−9.50000	−	16.4545i	−0.329949	−	0.571488i	0.652553	−	0.757743i	$-0.273698\pi$
$829$	−9.50000	−	16.4545i	−0.329949	−	0.571488i	−0.982501	+	0.186256i	$0.940365\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−24.0000	+	41.5692i	−0.830554	+	1.43856i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	35.0000			1.20690
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8.00000	−	13.8564i	0.275208	−	0.476675i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.00000	+	10.3923i	0.205677	+	0.356244i
$852$		0			0
$853$	17.0000			0.582069			0.291034	−	0.956713i	$-0.406001\pi$
$853$	17.0000			0.582069			0.291034	+	0.956713i	$0.406001\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	24.0000	+	41.5692i	0.819824	+	1.41998i	0.905811	+	0.423681i	$0.139262\pi$
$857$	24.0000	+	41.5692i	0.819824	+	1.41998i	−0.0859870	+	0.996296i	$0.527404\pi$
$858$		0			0
$859$	−4.00000	+	6.92820i	−0.136478	+	0.236387i	−0.926161	−	0.377128i	$-0.876912\pi$
$859$	−4.00000	+	6.92820i	−0.136478	+	0.236387i	0.789683	+	0.613515i	$0.210245\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−4.00000	+	6.92820i	−0.136162	+	0.235839i	−0.926041	−	0.377424i	$-0.876810\pi$
$863$	−4.00000	+	6.92820i	−0.136162	+	0.235839i	0.789879	+	0.613263i	$0.210143\pi$
$864$		0			0
$865$	−8.00000	−	13.8564i	−0.272008	−	0.471132i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	4.50000	+	7.79423i	0.152477	+	0.264097i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−7.00000	+	12.1244i	−0.236373	+	0.409410i	−0.959671	−	0.281126i	$-0.909292\pi$
$877$	−7.00000	+	12.1244i	−0.236373	+	0.409410i	0.723298	+	0.690536i	$0.242625\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−28.0000			−0.943344			−0.471672	−	0.881774i	$-0.656349\pi$
$881$	−28.0000			−0.943344			−0.471672	+	0.881774i	$0.656349\pi$
$882$		0			0
$883$	−47.0000			−1.58168			−0.790838	−	0.612026i	$-0.790355\pi$
$883$	−47.0000			−1.58168			−0.790838	+	0.612026i	$0.790355\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	12.0000	−	20.7846i	0.402921	−	0.697879i	−0.591156	−	0.806557i	$-0.701328\pi$
$887$	12.0000	−	20.7846i	0.402921	−	0.697879i	0.994077	+	0.108678i	$0.0346618\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	14.0000	+	24.2487i	0.468492	+	0.811452i
$894$		0			0
$895$	80.0000			2.67411
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−20.0000	−	34.6410i	−0.667037	−	1.15534i
$900$		0			0
$901$	−8.00000	+	13.8564i	−0.266519	+	0.461624i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	10.0000	−	17.3205i	0.332411	−	0.575753i
$906$		0			0
$907$	−26.5000	−	45.8993i	−0.879918	−	1.52406i	−0.851430	−	0.524469i	$-0.824264\pi$
$907$	−26.5000	−	45.8993i	−0.879918	−	1.52406i	−0.0284883	−	0.999594i	$-0.509069\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	4.00000			0.132526			0.0662630	−	0.997802i	$-0.478892\pi$
$911$	4.00000			0.132526			0.0662630	+	0.997802i	$0.478892\pi$
$912$		0			0
$913$		0			0
$914$		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
$923$	−36.0000			−1.18495
$924$		0			0
$925$	33.0000			1.08503
$926$		0			0
$927$		0			0
$928$		0			0
$929$	14.0000	−	24.2487i	0.459325	−	0.795574i	−0.539600	−	0.841921i	$-0.681425\pi$
$929$	14.0000	−	24.2487i	0.459325	−	0.795574i	0.998925	+	0.0463469i	$0.0147580\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	45.0000			1.47009			0.735043	−	0.678021i	$-0.237162\pi$
$937$	45.0000			1.47009			0.735043	+	0.678021i	$0.237162\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	18.0000	+	31.1769i	0.586783	+	1.01634i	0.994651	+	0.103297i	$0.0329393\pi$
$941$	18.0000	+	31.1769i	0.586783	+	1.01634i	−0.407867	+	0.913041i	$0.633727\pi$
$942$		0			0
$943$	16.0000	−	27.7128i	0.521032	−	0.902453i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	4.00000	−	6.92820i	0.129983	−	0.225136i	−0.793687	−	0.608326i	$-0.791841\pi$
$947$	4.00000	−	6.92820i	0.129983	−	0.225136i	0.923670	+	0.383190i	$0.125175\pi$
$948$		0			0
$949$	1.50000	+	2.59808i	0.0486921	+	0.0843371i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−20.0000			−0.647864			−0.323932	−	0.946080i	$-0.605005\pi$
$953$	−20.0000			−0.647864			−0.323932	+	0.946080i	$0.605005\pi$
$954$		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
$965$	−36.0000			−1.15888
$966$		0			0
$967$	57.0000			1.83300			0.916498	−	0.400039i	$-0.131003\pi$
$967$	57.0000			1.83300			0.916498	+	0.400039i	$0.131003\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−10.0000	+	17.3205i	−0.320915	+	0.555842i	−0.980677	−	0.195633i	$-0.937324\pi$
$971$	−10.0000	+	17.3205i	−0.320915	+	0.555842i	0.659762	+	0.751475i	$0.270657\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−24.0000	−	41.5692i	−0.767828	−	1.32992i	−0.938738	−	0.344631i	$-0.888004\pi$
$977$	−24.0000	−	41.5692i	−0.767828	−	1.32992i	0.170910	−	0.985287i	$-0.445329\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−6.00000	−	10.3923i	−0.191370	−	0.331463i	0.754334	−	0.656490i	$-0.227960\pi$
$983$	−6.00000	−	10.3923i	−0.191370	−	0.331463i	−0.945705	+	0.325027i	$0.894626\pi$
$984$		0			0
$985$	24.0000	−	41.5692i	0.764704	−	1.32451i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	22.0000	−	38.1051i	0.699559	−	1.21167i
$990$		0			0
$991$	−1.50000	−	2.59808i	−0.0476491	−	0.0825306i	0.841217	−	0.540697i	$-0.181840\pi$
$991$	−1.50000	−	2.59808i	−0.0476491	−	0.0825306i	−0.888866	+	0.458167i	$0.848506\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−80.0000			−2.53617
$996$		0			0
$997$	−14.5000	−	25.1147i	−0.459220	−	0.795392i	0.539700	−	0.841857i	$-0.318538\pi$
$997$	−14.5000	−	25.1147i	−0.459220	−	0.795392i	−0.998920	+	0.0464655i	$0.985204\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.s.b.3313.1		2
3.2	odd	2		3528.2.s.bb.3313.1		2
7.2	even	3		3528.2.a.ba.1.1		1
7.3	odd	6		504.2.s.h.361.1	yes	2
7.4	even	3	inner	3528.2.s.b.361.1		2
7.5	odd	6		3528.2.a.a.1.1		1
7.6	odd	2		504.2.s.h.289.1	yes	2
21.2	odd	6		3528.2.a.c.1.1		1
21.5	even	6		3528.2.a.z.1.1		1
21.11	odd	6		3528.2.s.bb.361.1		2
21.17	even	6		504.2.s.a.361.1	yes	2
21.20	even	2		504.2.s.a.289.1	✓	2
28.3	even	6		1008.2.s.q.865.1		2
28.19	even	6		7056.2.a.b.1.1		1
28.23	odd	6		7056.2.a.cc.1.1		1
28.27	even	2		1008.2.s.q.289.1		2
84.23	even	6		7056.2.a.d.1.1		1
84.47	odd	6		7056.2.a.cb.1.1		1
84.59	odd	6		1008.2.s.a.865.1		2
84.83	odd	2		1008.2.s.a.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.s.a.289.1	✓	2	21.20	even	2
504.2.s.a.361.1	yes	2	21.17	even	6
504.2.s.h.289.1	yes	2	7.6	odd	2
504.2.s.h.361.1	yes	2	7.3	odd	6
1008.2.s.a.289.1		2	84.83	odd	2
1008.2.s.a.865.1		2	84.59	odd	6
1008.2.s.q.289.1		2	28.27	even	2
1008.2.s.q.865.1		2	28.3	even	6
3528.2.a.a.1.1		1	7.5	odd	6
3528.2.a.c.1.1		1	21.2	odd	6
3528.2.a.z.1.1		1	21.5	even	6
3528.2.a.ba.1.1		1	7.2	even	3
3528.2.s.b.361.1		2	7.4	even	3	inner
3528.2.s.b.3313.1		2	1.1	even	1	trivial
3528.2.s.bb.361.1		2	21.11	odd	6
3528.2.s.bb.3313.1		2	3.2	odd	2
7056.2.a.b.1.1		1	28.19	even	6
7056.2.a.d.1.1		1	84.23	even	6
7056.2.a.cb.1.1		1	84.47	odd	6
7056.2.a.cc.1.1		1	28.23	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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+(-2.00000 + 3.46410i) q^{5} +O(q^{10})\)	\(q+(-2.00000 + 3.46410i) q^{5} +3.00000 q^{13} +(2.00000 + 3.46410i) q^{17} +(3.50000 - 6.06218i) q^{19} +(2.00000 - 3.46410i) q^{23} +(-5.50000 - 9.52628i) q^{25} +8.00000 q^{29} +(-2.50000 - 4.33013i) q^{31} +(-1.50000 + 2.59808i) q^{37} +8.00000 q^{41} +11.0000 q^{43} +(-2.00000 + 3.46410i) q^{47} +(2.00000 + 3.46410i) q^{53} +(-6.00000 - 10.3923i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(-6.00000 + 10.3923i) q^{65} +(1.50000 + 2.59808i) q^{67} -12.0000 q^{71} +(0.500000 + 0.866025i) q^{73} +(-0.500000 + 0.866025i) q^{79} +12.0000 q^{83} -16.0000 q^{85} +(-4.00000 + 6.92820i) q^{89} +(14.0000 + 24.2487i) q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5}+O(q^{10}) \) 2 * q - 4 * q^5	\( 2 q - 4 q^{5} + 6 q^{13} + 4 q^{17} + 7 q^{19} + 4 q^{23} - 11 q^{25} + 16 q^{29} - 5 q^{31} - 3 q^{37} + 16 q^{41} + 22 q^{43} - 4 q^{47} + 4 q^{53} - 12 q^{59} - 2 q^{61} - 12 q^{65} + 3 q^{67} - 24 q^{71} + q^{73} - q^{79} + 24 q^{83} - 32 q^{85} - 8 q^{89} + 28 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 + 6 * q^13 + 4 * q^17 + 7 * q^19 + 4 * q^23 - 11 * q^25 + 16 * q^29 - 5 * q^31 - 3 * q^37 + 16 * q^41 + 22 * q^43 - 4 * q^47 + 4 * q^53 - 12 * q^59 - 2 * q^61 - 12 * q^65 + 3 * q^67 - 24 * q^71 + q^73 - q^79 + 24 * q^83 - 32 * q^85 - 8 * q^89 + 28 * q^95 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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