LMFDB - Embedded newform 3528.2.s.a.361.1

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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3528.361
Dual form			3528.2.s.a.3313.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 - 3.46410i) q^{5} +O(q^{10})$	$q+(-2.00000 - 3.46410i) q^{5} +(-1.00000 + 1.73205i) q^{17} +(1.00000 + 1.73205i) q^{19} +(4.00000 + 6.92820i) q^{23} +(-5.50000 + 9.52628i) q^{25} -2.00000 q^{29} +(-2.00000 + 3.46410i) q^{31} +(3.00000 + 5.19615i) q^{37} +2.00000 q^{41} +8.00000 q^{43} +(-2.00000 - 3.46410i) q^{47} +(-5.00000 + 8.66025i) q^{53} +(3.00000 - 5.19615i) q^{59} +(-2.00000 - 3.46410i) q^{61} +(6.00000 - 10.3923i) q^{67} +(7.00000 - 12.1244i) q^{73} +(4.00000 + 6.92820i) q^{79} -6.00000 q^{83} +8.00000 q^{85} +(5.00000 + 8.66025i) q^{89} +(4.00000 - 6.92820i) q^{95} -2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5}+O(q^{10}) $ 2 * q - 4 * q^5	$ 2 q - 4 q^{5} - 2 q^{17} + 2 q^{19} + 8 q^{23} - 11 q^{25} - 4 q^{29} - 4 q^{31} + 6 q^{37} + 4 q^{41} + 16 q^{43} - 4 q^{47} - 10 q^{53} + 6 q^{59} - 4 q^{61} + 12 q^{67} + 14 q^{73} + 8 q^{79} - 12 q^{83} + 16 q^{85} + 10 q^{89} + 8 q^{95} - 4 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 - 2 * q^17 + 2 * q^19 + 8 * q^23 - 11 * q^25 - 4 * q^29 - 4 * q^31 + 6 * q^37 + 4 * q^41 + 16 * q^43 - 4 * q^47 - 10 * q^53 + 6 * q^59 - 4 * q^61 + 12 * q^67 + 14 * q^73 + 8 * q^79 - 12 * q^83 + 16 * q^85 + 10 * q^89 + 8 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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	−0.834512	−	0.550990i	$-0.814250\pi$
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.0599153	−	0.998203i	$-0.519083\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.00000	+	1.73205i	−0.242536	+	0.420084i	−0.961436	−	0.275029i	$-0.911312\pi$
$17$	−1.00000	+	1.73205i	−0.242536	+	0.420084i	0.718900	+	0.695113i	$0.244646\pi$
$18$		0			0
$19$	1.00000	+	1.73205i	0.229416	+	0.397360i	0.957635	−	0.287984i	$-0.0929851\pi$
$19$	1.00000	+	1.73205i	0.229416	+	0.397360i	−0.728219	+	0.685344i	$0.759652\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.00000	+	6.92820i	0.834058	+	1.44463i	0.894795	+	0.446476i	$0.147321\pi$
$23$	4.00000	+	6.92820i	0.834058	+	1.44463i	−0.0607377	+	0.998154i	$0.519345\pi$
$24$		0			0
$25$	−5.50000	+	9.52628i	−1.10000	+	1.90526i
$26$		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
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	0.628619	+	0.777714i	$0.283621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.00000	+	5.19615i	0.493197	+	0.854242i	0.999969	−	0.00783774i	$-0.00249486\pi$
$37$	3.00000	+	5.19615i	0.493197	+	0.854242i	−0.506772	+	0.862080i	$0.669162\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.00000	−	3.46410i	−0.291730	−	0.505291i	0.682489	−	0.730896i	$-0.260898\pi$
$47$	−2.00000	−	3.46410i	−0.291730	−	0.505291i	−0.974219	+	0.225605i	$0.927564\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.00000	+	8.66025i	−0.686803	+	1.18958i	0.286064	+	0.958211i	$0.407653\pi$
$53$	−5.00000	+	8.66025i	−0.686803	+	1.18958i	−0.972867	+	0.231367i	$0.925680\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.00000	−	5.19615i	0.390567	−	0.676481i	−0.601958	−	0.798528i	$-0.705612\pi$
$59$	3.00000	−	5.19615i	0.390567	−	0.676481i	0.992524	+	0.122047i	$0.0389457\pi$
$60$		0			0
$61$	−2.00000	−	3.46410i	−0.256074	−	0.443533i	0.709113	−	0.705095i	$-0.249096\pi$
$61$	−2.00000	−	3.46410i	−0.256074	−	0.443533i	−0.965187	+	0.261562i	$0.915762\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.00000	−	10.3923i	0.733017	−	1.26962i	−0.222571	−	0.974916i	$-0.571445\pi$
$67$	6.00000	−	10.3923i	0.733017	−	1.26962i	0.955588	−	0.294706i	$-0.0952216\pi$
$68$		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
$73$	7.00000	−	12.1244i	0.819288	−	1.41905i	−0.0869195	−	0.996215i	$-0.527702\pi$
$73$	7.00000	−	12.1244i	0.819288	−	1.41905i	0.906208	−	0.422833i	$-0.138964\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	4.00000	+	6.92820i	0.450035	+	0.779484i	0.998388	−	0.0567635i	$-0.0180781\pi$
$79$	4.00000	+	6.92820i	0.450035	+	0.779484i	−0.548352	+	0.836247i	$0.684745\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$		0			0
$85$	8.00000			0.867722
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.00000	+	8.66025i	0.529999	+	0.917985i	0.999388	+	0.0349934i	$0.0111410\pi$
$89$	5.00000	+	8.66025i	0.529999	+	0.917985i	−0.469389	+	0.882992i	$0.655526\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.00000	−	6.92820i	0.410391	−	0.710819i
$96$		0			0
$97$	−2.00000			−0.203069			−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000			−0.203069			−0.101535	+	0.994832i	$0.532375\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.00000	−	10.3923i	0.597022	−	1.03407i	−0.396236	−	0.918149i	$-0.629684\pi$
$101$	6.00000	−	10.3923i	0.597022	−	1.03407i	0.993258	−	0.115924i	$-0.0369830\pi$
$102$		0			0
$103$	6.00000	+	10.3923i	0.591198	+	1.02398i	0.994071	+	0.108729i	$0.0346780\pi$
$103$	6.00000	+	10.3923i	0.591198	+	1.02398i	−0.402874	+	0.915255i	$0.631989\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.00000	−	10.3923i	−0.580042	−	1.00466i	−0.995474	−	0.0950377i	$-0.969703\pi$
$107$	−6.00000	−	10.3923i	−0.580042	−	1.00466i	0.415432	−	0.909624i	$-0.363630\pi$
$108$		0			0
$109$	−5.00000	+	8.66025i	−0.478913	+	0.829502i	−0.999708	−	0.0241802i	$-0.992302\pi$
$109$	−5.00000	+	8.66025i	−0.478913	+	0.829502i	0.520794	+	0.853682i	$0.325636\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$	16.0000	−	27.7128i	1.49201	−	2.58423i
$116$		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
$127$	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
$131$	7.00000	+	12.1244i	0.611593	+	1.05931i	0.990972	+	0.134069i	$0.0428042\pi$
$131$	7.00000	+	12.1244i	0.611593	+	1.05931i	−0.379379	+	0.925241i	$0.623862\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.00000	−	1.73205i	0.0854358	−	0.147979i	−0.820141	−	0.572161i	$-0.806105\pi$
$137$	1.00000	−	1.73205i	0.0854358	−	0.147979i	0.905577	+	0.424182i	$0.139438\pi$
$138$		0			0
$139$	18.0000			1.52674			0.763370	−	0.645961i	$-0.223543\pi$
$139$	18.0000			1.52674			0.763370	+	0.645961i	$0.223543\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	4.00000	+	6.92820i	0.332182	+	0.575356i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.00000	−	1.73205i	−0.0819232	−	0.141895i	0.822153	−	0.569267i	$-0.192773\pi$
$149$	−1.00000	−	1.73205i	−0.0819232	−	0.141895i	−0.904076	+	0.427372i	$0.859440\pi$
$150$		0			0
$151$	−8.00000	+	13.8564i	−0.651031	+	1.12762i	0.331842	+	0.943335i	$0.392330\pi$
$151$	−8.00000	+	13.8564i	−0.651031	+	1.12762i	−0.982873	+	0.184284i	$0.941004\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	16.0000			1.28515
$156$		0			0
$157$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$157$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−8.00000	−	13.8564i	−0.626608	−	1.08532i	−0.988227	−	0.152992i	$-0.951109\pi$
$163$	−8.00000	−	13.8564i	−0.626608	−	1.08532i	0.361619	−	0.932326i	$-0.382224\pi$
$164$		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
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.00000	+	6.92820i	0.304114	+	0.526742i	0.977064	−	0.212947i	$-0.0683062\pi$
$173$	4.00000	+	6.92820i	0.304114	+	0.526742i	−0.672949	+	0.739689i	$0.734973\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2.00000	+	3.46410i	−0.149487	+	0.258919i	−0.931038	−	0.364922i	$-0.881096\pi$
$179$	−2.00000	+	3.46410i	−0.149487	+	0.258919i	0.781551	+	0.623841i	$0.214429\pi$
$180$		0			0
$181$	8.00000			0.594635			0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000			0.594635			0.297318	+	0.954779i	$0.403908\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	12.0000	−	20.7846i	0.882258	−	1.52811i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.00000	−	6.92820i	−0.289430	−	0.501307i	0.684244	−	0.729253i	$-0.260132\pi$
$191$	−4.00000	−	6.92820i	−0.289430	−	0.501307i	−0.973674	+	0.227946i	$0.926799\pi$
$192$		0			0
$193$	9.00000	−	15.5885i	0.647834	−	1.12208i	−0.335805	−	0.941932i	$-0.609008\pi$
$193$	9.00000	−	15.5885i	0.647834	−	1.12208i	0.983639	−	0.180150i	$-0.0576584\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	2.00000	−	3.46410i	0.141776	−	0.245564i	−0.786389	−	0.617731i	$-0.788052\pi$
$199$	2.00000	−	3.46410i	0.141776	−	0.245564i	0.928166	+	0.372168i	$0.121385\pi$
$200$		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
$209$		0			0
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−16.0000	−	27.7128i	−1.09119	−	1.89000i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−24.0000			−1.60716			−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000			−1.60716			−0.803579	+	0.595198i	$0.797074\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.00000	−	12.1244i	0.464606	−	0.804722i	−0.534577	−	0.845120i	$-0.679529\pi$
$227$	7.00000	−	12.1244i	0.464606	−	0.804722i	0.999184	+	0.0403978i	$0.0128625\pi$
$228$		0			0
$229$	8.00000	+	13.8564i	0.528655	+	0.915657i	0.999442	+	0.0334101i	$0.0106368\pi$
$229$	8.00000	+	13.8564i	0.528655	+	0.915657i	−0.470787	+	0.882247i	$0.656030\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	13.0000	+	22.5167i	0.851658	+	1.47512i	0.879711	+	0.475509i	$0.157736\pi$
$233$	13.0000	+	22.5167i	0.851658	+	1.47512i	−0.0280525	+	0.999606i	$0.508931\pi$
$234$		0			0
$235$	−8.00000	+	13.8564i	−0.521862	+	0.903892i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	16.0000			1.03495			0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000			1.03495			0.517477	+	0.855697i	$0.326871\pi$
$240$		0			0
$241$	1.00000	−	1.73205i	0.0644157	−	0.111571i	−0.832019	−	0.554747i	$-0.812815\pi$
$241$	1.00000	−	1.73205i	0.0644157	−	0.111571i	0.896435	+	0.443176i	$0.146148\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	14.0000			0.883672			0.441836	−	0.897096i	$-0.354327\pi$
$251$	14.0000			0.883672			0.441836	+	0.897096i	$0.354327\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.00000	+	15.5885i	0.561405	+	0.972381i	0.997374	+	0.0724199i	$0.0230722\pi$
$257$	9.00000	+	15.5885i	0.561405	+	0.972381i	−0.435970	+	0.899961i	$0.643595\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$	40.0000			2.45718
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−12.0000	+	20.7846i	−0.731653	+	1.26726i	0.224523	+	0.974469i	$0.427917\pi$
$269$	−12.0000	+	20.7846i	−0.731653	+	1.26726i	−0.956176	+	0.292791i	$0.905416\pi$
$270$		0			0
$271$	−16.0000	−	27.7128i	−0.971931	−	1.68343i	−0.689713	−	0.724083i	$-0.742263\pi$
$271$	−16.0000	−	27.7128i	−0.971931	−	1.68343i	−0.282218	−	0.959350i	$-0.591070\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−11.0000	+	19.0526i	−0.660926	+	1.14476i	0.319447	+	0.947604i	$0.396503\pi$
$277$	−11.0000	+	19.0526i	−0.660926	+	1.14476i	−0.980373	+	0.197153i	$0.936830\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$	5.00000	−	8.66025i	0.297219	−	0.514799i	−0.678280	−	0.734804i	$-0.737274\pi$
$283$	5.00000	−	8.66025i	0.297219	−	0.514799i	0.975499	+	0.220005i	$0.0706075\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	6.50000	+	11.2583i	0.382353	+	0.662255i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	12.0000			0.701047			0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000			0.701047			0.350524	+	0.936554i	$0.386004\pi$
$294$		0			0
$295$	−24.0000			−1.39733
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8.00000	+	13.8564i	−0.458079	+	0.793416i
$306$		0			0
$307$	−2.00000			−0.114146			−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000			−0.114146			−0.0570730	+	0.998370i	$0.518177\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.0000	+	20.7846i	−0.680458	+	1.17859i	0.294384	+	0.955687i	$0.404886\pi$
$311$	−12.0000	+	20.7846i	−0.680458	+	1.17859i	−0.974841	+	0.222900i	$0.928448\pi$
$312$		0			0
$313$	−7.00000	−	12.1244i	−0.395663	−	0.685309i	0.597522	−	0.801852i	$-0.296152\pi$
$313$	−7.00000	−	12.1244i	−0.395663	−	0.685309i	−0.993186	+	0.116543i	$0.962819\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.00000	+	5.19615i	0.168497	+	0.291845i	0.937892	−	0.346929i	$-0.112775\pi$
$317$	3.00000	+	5.19615i	0.168497	+	0.291845i	−0.769395	+	0.638774i	$0.779442\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.00000			−0.222566
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4.00000	−	6.92820i	−0.219860	−	0.380808i	0.734905	−	0.678170i	$-0.237227\pi$
$331$	−4.00000	−	6.92820i	−0.219860	−	0.380808i	−0.954765	+	0.297361i	$0.903893\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−48.0000			−2.62252
$336$		0			0
$337$	14.0000			0.762629			0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000			0.762629			0.381314	+	0.924445i	$0.375472\pi$
$338$		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
$347$	−12.0000	+	20.7846i	−0.644194	+	1.11578i	0.340293	+	0.940319i	$0.389474\pi$
$347$	−12.0000	+	20.7846i	−0.644194	+	1.11578i	−0.984487	+	0.175457i	$0.943860\pi$
$348$		0			0
$349$	8.00000			0.428230			0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000			0.428230			0.214115	+	0.976808i	$0.431313\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−15.0000	+	25.9808i	−0.798369	+	1.38282i	0.122308	+	0.992492i	$0.460970\pi$
$353$	−15.0000	+	25.9808i	−0.798369	+	1.38282i	−0.920677	+	0.390324i	$0.872363\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$		0			0
$361$	7.50000	−	12.9904i	0.394737	−	0.683704i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−56.0000			−2.93117
$366$		0			0
$367$	−4.00000	+	6.92820i	−0.208798	+	0.361649i	−0.951336	−	0.308155i	$-0.900289\pi$
$367$	−4.00000	+	6.92820i	−0.208798	+	0.361649i	0.742538	+	0.669804i	$0.233622\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	17.0000	+	29.4449i	0.880227	+	1.52460i	0.851089	+	0.525022i	$0.175943\pi$
$373$	17.0000	+	29.4449i	0.880227	+	1.52460i	0.0291379	+	0.999575i	$0.490724\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6.00000	+	10.3923i	0.306586	+	0.531022i	0.977613	−	0.210411i	$-0.0674801\pi$
$383$	6.00000	+	10.3923i	0.306586	+	0.531022i	−0.671027	+	0.741433i	$0.734147\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.00000	−	8.66025i	0.253510	−	0.439092i	−0.710980	−	0.703213i	$-0.751748\pi$
$389$	5.00000	−	8.66025i	0.253510	−	0.439092i	0.964490	+	0.264120i	$0.0850816\pi$
$390$		0			0
$391$	−16.0000			−0.809155
$392$		0			0
$393$		0			0
$394$		0			0
$395$	16.0000	−	27.7128i	0.805047	−	1.39438i
$396$		0			0
$397$	4.00000	+	6.92820i	0.200754	+	0.347717i	0.948772	−	0.315963i	$-0.102327\pi$
$397$	4.00000	+	6.92820i	0.200754	+	0.347717i	−0.748017	+	0.663679i	$0.768994\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	15.0000	+	25.9808i	0.749064	+	1.29742i	0.948272	+	0.317460i	$0.102830\pi$
$401$	15.0000	+	25.9808i	0.749064	+	1.29742i	−0.199207	+	0.979957i	$0.563837\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−3.00000	+	5.19615i	−0.148340	+	0.256933i	−0.930614	−	0.366002i	$-0.880726\pi$
$409$	−3.00000	+	5.19615i	−0.148340	+	0.256933i	0.782274	+	0.622935i	$0.214060\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	12.0000	+	20.7846i	0.589057	+	1.02028i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−26.0000			−1.27018			−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000			−1.27018			−0.635092	+	0.772437i	$0.719038\pi$
$420$		0			0
$421$	22.0000			1.07221			0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000			1.07221			0.536107	+	0.844150i	$0.319894\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−11.0000	−	19.0526i	−0.533578	−	0.924185i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$431$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$432$		0			0
$433$	−26.0000			−1.24948			−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000			−1.24948			−0.624740	+	0.780833i	$0.714795\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.00000	+	13.8564i	−0.382692	+	0.662842i
$438$		0			0
$439$	−12.0000	−	20.7846i	−0.572729	−	0.991995i	−0.996284	−	0.0861252i	$-0.972552\pi$
$439$	−12.0000	−	20.7846i	−0.572729	−	0.991995i	0.423556	−	0.905870i	$-0.360782\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.00000	−	3.46410i	−0.0950229	−	0.164584i	0.814595	−	0.580030i	$-0.196959\pi$
$443$	−2.00000	−	3.46410i	−0.0950229	−	0.164584i	−0.909618	+	0.415445i	$0.863626\pi$
$444$		0			0
$445$	20.0000	−	34.6410i	0.948091	−	1.64214i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	14.0000			0.660701			0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000			0.660701			0.330350	+	0.943858i	$0.392833\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−11.0000	−	19.0526i	−0.514558	−	0.891241i	−0.999857	−	0.0168929i	$-0.994623\pi$
$457$	−11.0000	−	19.0526i	−0.514558	−	0.891241i	0.485299	−	0.874348i	$-0.338711\pi$
$458$		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
$463$	−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
$467$	3.00000	+	5.19615i	0.138823	+	0.240449i	0.927052	−	0.374934i	$-0.122335\pi$
$467$	3.00000	+	5.19615i	0.138823	+	0.240449i	−0.788228	+	0.615383i	$0.789001\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−22.0000			−1.00943
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2.00000	−	3.46410i	0.0913823	−	0.158279i	−0.816711	−	0.577047i	$-0.804205\pi$
$479$	2.00000	−	3.46410i	0.0913823	−	0.158279i	0.908093	+	0.418769i	$0.137538\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.00000	+	6.92820i	0.181631	+	0.314594i
$486$		0			0
$487$	4.00000	−	6.92820i	0.181257	−	0.313947i	−0.761052	−	0.648691i	$-0.775317\pi$
$487$	4.00000	−	6.92820i	0.181257	−	0.313947i	0.942309	+	0.334744i	$0.108650\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	36.0000			1.62466			0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000			1.62466			0.812329	+	0.583200i	$0.198200\pi$
$492$		0			0
$493$	2.00000	−	3.46410i	0.0900755	−	0.156015i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−2.00000	−	3.46410i	−0.0895323	−	0.155074i	0.817781	−	0.575529i	$-0.195204\pi$
$499$	−2.00000	−	3.46410i	−0.0895323	−	0.155074i	−0.907314	+	0.420455i	$0.861871\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	32.0000			1.42681			0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000			1.42681			0.713405	+	0.700752i	$0.247152\pi$
$504$		0			0
$505$	−48.0000			−2.13597
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−12.0000	−	20.7846i	−0.531891	−	0.921262i	−0.999307	−	0.0372243i	$-0.988148\pi$
$509$	−12.0000	−	20.7846i	−0.531891	−	0.921262i	0.467416	−	0.884037i	$-0.345185\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	24.0000	−	41.5692i	1.05757	−	1.83176i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9.00000	+	15.5885i	−0.394297	+	0.682943i	−0.993011	−	0.118020i	$-0.962345\pi$
$521$	−9.00000	+	15.5885i	−0.394297	+	0.682943i	0.598714	+	0.800963i	$0.295679\pi$
$522$		0			0
$523$	17.0000	+	29.4449i	0.743358	+	1.28753i	0.950958	+	0.309320i	$0.100101\pi$
$523$	17.0000	+	29.4449i	0.743358	+	1.28753i	−0.207600	+	0.978214i	$0.566565\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.00000	−	6.92820i	−0.174243	−	0.301797i
$528$		0			0
$529$	−20.5000	+	35.5070i	−0.891304	+	1.54378i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		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
$541$	−11.0000	−	19.0526i	−0.472927	−	0.819133i	0.526593	−	0.850118i	$-0.323469\pi$
$541$	−11.0000	−	19.0526i	−0.472927	−	0.819133i	−0.999520	+	0.0309841i	$0.990136\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	40.0000			1.71341
$546$		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
$551$	−2.00000	−	3.46410i	−0.0852029	−	0.147576i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−13.0000	+	22.5167i	−0.550828	+	0.954062i	0.447387	+	0.894340i	$0.352355\pi$
$557$	−13.0000	+	22.5167i	−0.550828	+	0.954062i	−0.998215	+	0.0597213i	$0.980979\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	17.0000	−	29.4449i	0.716465	−	1.24095i	−0.245927	−	0.969288i	$-0.579092\pi$
$563$	17.0000	−	29.4449i	0.716465	−	1.24095i	0.962392	−	0.271665i	$-0.0875742\pi$
$564$		0			0
$565$	12.0000	+	20.7846i	0.504844	+	0.874415i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−21.0000	−	36.3731i	−0.880366	−	1.52484i	−0.850935	−	0.525271i	$-0.823964\pi$
$569$	−21.0000	−	36.3731i	−0.880366	−	1.52484i	−0.0294311	−	0.999567i	$-0.509370\pi$
$570$		0			0
$571$	8.00000	−	13.8564i	0.334790	−	0.579873i	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	8.00000	−	13.8564i	0.334790	−	0.579873i	0.983444	+	0.181210i	$0.0580014\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−88.0000			−3.66985
$576$		0			0
$577$	−9.00000	+	15.5885i	−0.374675	+	0.648956i	−0.990278	−	0.139100i	$-0.955579\pi$
$577$	−9.00000	+	15.5885i	−0.374675	+	0.648956i	0.615603	+	0.788056i	$0.288912\pi$
$578$		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
$587$	−10.0000			−0.412744			−0.206372	−	0.978474i	$-0.566166\pi$
$587$	−10.0000			−0.412744			−0.206372	+	0.978474i	$0.566166\pi$
$588$		0			0
$589$	−8.00000			−0.329634
$590$		0			0
$591$		0			0
$592$		0			0
$593$	21.0000	+	36.3731i	0.862367	+	1.49366i	0.869638	+	0.493689i	$0.164352\pi$
$593$	21.0000	+	36.3731i	0.862367	+	1.49366i	−0.00727173	+	0.999974i	$0.502315\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	20.0000	−	34.6410i	0.817178	−	1.41539i	−0.0905757	−	0.995890i	$-0.528871\pi$
$599$	20.0000	−	34.6410i	0.817178	−	1.41539i	0.907754	−	0.419504i	$-0.137796\pi$
$600$		0			0
$601$	10.0000			0.407909			0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000			0.407909			0.203954	+	0.978980i	$0.434621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	22.0000	−	38.1051i	0.894427	−	1.54919i
$606$		0			0
$607$	16.0000	+	27.7128i	0.649420	+	1.12483i	0.983262	+	0.182199i	$0.0583216\pi$
$607$	16.0000	+	27.7128i	0.649420	+	1.12483i	−0.333842	+	0.942629i	$0.608345\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−5.00000	+	8.66025i	−0.201948	+	0.349784i	−0.949156	−	0.314806i	$-0.898061\pi$
$613$	−5.00000	+	8.66025i	−0.201948	+	0.349784i	0.747208	+	0.664590i	$0.231394\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6.00000			−0.241551			−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000			−0.241551			−0.120775	+	0.992680i	$0.538538\pi$
$618$		0			0
$619$	−3.00000	+	5.19615i	−0.120580	+	0.208851i	−0.919997	−	0.391926i	$-0.871809\pi$
$619$	−3.00000	+	5.19615i	−0.120580	+	0.208851i	0.799416	+	0.600777i	$0.205142\pi$
$620$		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
$631$	32.0000			1.27390			0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000			1.27390			0.636950	+	0.770905i	$0.280196\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−16.0000	−	27.7128i	−0.634941	−	1.09975i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.00000	+	1.73205i	−0.0394976	+	0.0684119i	−0.885098	−	0.465404i	$-0.845909\pi$
$641$	−1.00000	+	1.73205i	−0.0394976	+	0.0684119i	0.845601	+	0.533816i	$0.179242\pi$
$642$		0			0
$643$	−14.0000			−0.552106			−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000			−0.552106			−0.276053	+	0.961142i	$0.589027\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−18.0000	+	31.1769i	−0.707653	+	1.22569i	0.258073	+	0.966126i	$0.416913\pi$
$647$	−18.0000	+	31.1769i	−0.707653	+	1.22569i	−0.965726	+	0.259565i	$0.916421\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−11.0000	−	19.0526i	−0.430463	−	0.745584i	0.566450	−	0.824096i	$-0.308316\pi$
$653$	−11.0000	−	19.0526i	−0.430463	−	0.745584i	−0.996913	+	0.0785119i	$0.974983\pi$
$654$		0			0
$655$	28.0000	−	48.4974i	1.09405	−	1.89495i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	40.0000			1.55818			0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000			1.55818			0.779089	+	0.626913i	$0.215682\pi$
$660$		0			0
$661$	−10.0000	+	17.3205i	−0.388955	+	0.673690i	−0.992309	−	0.123784i	$-0.960497\pi$
$661$	−10.0000	+	17.3205i	−0.388955	+	0.673690i	0.603354	+	0.797473i	$0.293830\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−8.00000	−	13.8564i	−0.309761	−	0.536522i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	10.0000			0.385472			0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000			0.385472			0.192736	+	0.981251i	$0.438264\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−12.0000	−	20.7846i	−0.461197	−	0.798817i	0.537823	−	0.843057i	$-0.319247\pi$
$677$	−12.0000	−	20.7846i	−0.461197	−	0.798817i	−0.999021	+	0.0442400i	$0.985913\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	6.00000	−	10.3923i	0.229584	−	0.397650i	−0.728101	−	0.685470i	$-0.759597\pi$
$683$	6.00000	−	10.3923i	0.229584	−	0.397650i	0.957685	+	0.287819i	$0.0929302\pi$
$684$		0			0
$685$	−8.00000			−0.305664
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−7.00000	−	12.1244i	−0.266293	−	0.461232i	0.701609	−	0.712562i	$-0.252465\pi$
$691$	−7.00000	−	12.1244i	−0.266293	−	0.461232i	−0.967901	+	0.251330i	$0.919132\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−36.0000	−	62.3538i	−1.36556	−	2.36522i
$696$		0			0
$697$	−2.00000	+	3.46410i	−0.0757554	+	0.131212i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−10.0000			−0.377695			−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000			−0.377695			−0.188847	+	0.982006i	$0.560475\pi$
$702$		0			0
$703$	−6.00000	+	10.3923i	−0.226294	+	0.391953i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−5.00000	−	8.66025i	−0.187779	−	0.325243i	0.756730	−	0.653727i	$-0.226796\pi$
$709$	−5.00000	−	8.66025i	−0.187779	−	0.325243i	−0.944509	+	0.328484i	$0.893462\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−32.0000			−1.19841
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	18.0000	+	31.1769i	0.671287	+	1.16270i	0.977539	+	0.210752i	$0.0675914\pi$
$719$	18.0000	+	31.1769i	0.671287	+	1.16270i	−0.306253	+	0.951950i	$0.599075\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	11.0000	−	19.0526i	0.408530	−	0.707594i
$726$		0			0
$727$	20.0000			0.741759			0.370879	−	0.928681i	$-0.379056\pi$
$727$	20.0000			0.741759			0.370879	+	0.928681i	$0.379056\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.00000	+	13.8564i	−0.295891	+	0.512498i
$732$		0			0
$733$	−26.0000	−	45.0333i	−0.960332	−	1.66334i	−0.721665	−	0.692242i	$-0.756623\pi$
$733$	−26.0000	−	45.0333i	−0.960332	−	1.66334i	−0.238667	−	0.971102i	$-0.576710\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−8.00000	+	13.8564i	−0.294285	+	0.509716i	−0.974818	−	0.223001i	$-0.928415\pi$
$739$	−8.00000	+	13.8564i	−0.294285	+	0.509716i	0.680534	+	0.732717i	$0.261748\pi$
$740$		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
$745$	−4.00000	+	6.92820i	−0.146549	+	0.253830i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−16.0000	−	27.7128i	−0.583848	−	1.01125i	−0.995018	−	0.0996961i	$-0.968213\pi$
$751$	−16.0000	−	27.7128i	−0.583848	−	1.01125i	0.411170	−	0.911559i	$-0.365120\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	64.0000			2.32920
$756$		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
$761$	−21.0000	−	36.3731i	−0.761249	−	1.31852i	−0.942207	−	0.335032i	$-0.891253\pi$
$761$	−21.0000	−	36.3731i	−0.761249	−	1.31852i	0.180957	−	0.983491i	$-0.442080\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−26.0000			−0.937584			−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000			−0.937584			−0.468792	+	0.883309i	$0.655311\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	2.00000	−	3.46410i	0.0719350	−	0.124595i	−0.827814	−	0.561002i	$-0.810416\pi$
$773$	2.00000	−	3.46410i	0.0719350	−	0.124595i	0.899749	+	0.436407i	$0.143749\pi$
$774$		0			0
$775$	−22.0000	−	38.1051i	−0.790263	−	1.36878i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.00000	+	3.46410i	0.0716574	+	0.124114i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−11.0000	+	19.0526i	−0.392108	+	0.679150i	−0.992727	−	0.120384i	$-0.961587\pi$
$787$	−11.0000	+	19.0526i	−0.392108	+	0.679150i	0.600620	+	0.799535i	$0.294921\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	24.0000			0.850124			0.425062	−	0.905164i	$-0.360252\pi$
$797$	24.0000			0.850124			0.425062	+	0.905164i	$0.360252\pi$
$798$		0			0
$799$	8.00000			0.283020
$800$		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
$809$	3.00000	−	5.19615i	0.105474	−	0.182687i	−0.808458	−	0.588555i	$-0.799697\pi$
$809$	3.00000	−	5.19615i	0.105474	−	0.182687i	0.913932	+	0.405868i	$0.133031\pi$
$810$		0			0
$811$	30.0000			1.05344			0.526721	−	0.850038i	$-0.323421\pi$
$811$	30.0000			1.05344			0.526721	+	0.850038i	$0.323421\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−32.0000	+	55.4256i	−1.12091	+	1.94147i
$816$		0			0
$817$	8.00000	+	13.8564i	0.279885	+	0.484774i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3.00000	+	5.19615i	0.104701	+	0.181347i	0.913616	−	0.406578i	$-0.133278\pi$
$821$	3.00000	+	5.19615i	0.104701	+	0.181347i	−0.808915	+	0.587925i	$0.799945\pi$
$822$		0			0
$823$	−4.00000	+	6.92820i	−0.139431	+	0.241502i	−0.927281	−	0.374365i	$-0.877861\pi$
$823$	−4.00000	+	6.92820i	−0.139431	+	0.241502i	0.787850	+	0.615867i	$0.211194\pi$
$824$		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
$829$	−10.0000	+	17.3205i	−0.347314	+	0.601566i	−0.985771	−	0.168091i	$-0.946240\pi$
$829$	−10.0000	+	17.3205i	−0.347314	+	0.601566i	0.638457	+	0.769657i	$0.279573\pi$
$830$		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
$839$	−36.0000			−1.24286			−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000			−1.24286			−0.621429	+	0.783470i	$0.713448\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$		0			0
$844$		0			0
$845$	26.0000	+	45.0333i	0.894427	+	1.54919i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−24.0000	+	41.5692i	−0.822709	+	1.42497i
$852$		0			0
$853$	−32.0000			−1.09566			−0.547830	−	0.836590i	$-0.684546\pi$
$853$	−32.0000			−1.09566			−0.547830	+	0.836590i	$0.684546\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3.00000	−	5.19615i	0.102478	−	0.177497i	−0.810227	−	0.586116i	$-0.800656\pi$
$857$	3.00000	−	5.19615i	0.102478	−	0.177497i	0.912705	+	0.408619i	$0.133990\pi$
$858$		0			0
$859$	7.00000	+	12.1244i	0.238837	+	0.413678i	0.960381	−	0.278691i	$-0.0899005\pi$
$859$	7.00000	+	12.1244i	0.238837	+	0.413678i	−0.721544	+	0.692369i	$0.756567\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4.00000	+	6.92820i	0.136162	+	0.235839i	0.926041	−	0.377424i	$-0.123190\pi$
$863$	4.00000	+	6.92820i	0.136162	+	0.235839i	−0.789879	+	0.613263i	$0.789857\pi$
$864$		0			0
$865$	16.0000	−	27.7128i	0.544016	−	0.942264i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.00000	−	1.73205i	−0.0337676	−	0.0584872i	0.848648	−	0.528958i	$-0.177417\pi$
$877$	−1.00000	−	1.73205i	−0.0337676	−	0.0584872i	−0.882415	+	0.470471i	$0.844084\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−10.0000			−0.336909			−0.168454	−	0.985709i	$-0.553878\pi$
$881$	−10.0000			−0.336909			−0.168454	+	0.985709i	$0.553878\pi$
$882$		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
$887$	18.0000	+	31.1769i	0.604381	+	1.04682i	0.992149	+	0.125061i	$0.0399128\pi$
$887$	18.0000	+	31.1769i	0.604381	+	1.04682i	−0.387768	+	0.921757i	$0.626754\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.00000	−	6.92820i	0.133855	−	0.231843i
$894$		0			0
$895$	16.0000			0.534821
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4.00000	−	6.92820i	0.133407	−	0.231069i
$900$		0			0
$901$	−10.0000	−	17.3205i	−0.333148	−	0.577030i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−16.0000	−	27.7128i	−0.531858	−	0.921205i
$906$		0			0
$907$	14.0000	−	24.2487i	0.464862	−	0.805165i	−0.534333	−	0.845274i	$-0.679437\pi$
$907$	14.0000	−	24.2487i	0.464862	−	0.805165i	0.999195	+	0.0401089i	$0.0127705\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	8.00000			0.265052			0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000			0.265052			0.132526	+	0.991180i	$0.457691\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−20.0000	−	34.6410i	−0.659739	−	1.14270i	−0.980683	−	0.195603i	$-0.937333\pi$
$919$	−20.0000	−	34.6410i	−0.659739	−	1.14270i	0.320944	−	0.947098i	$-0.396000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−66.0000			−2.17007
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.00000	−	1.73205i	−0.0328089	−	0.0568267i	0.849155	−	0.528144i	$-0.177112\pi$
$929$	−1.00000	−	1.73205i	−0.0328089	−	0.0568267i	−0.881964	+	0.471317i	$0.843779\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	18.0000			0.588034			0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000			0.588034			0.294017	+	0.955800i	$0.405008\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−6.00000	+	10.3923i	−0.195594	+	0.338779i	−0.947095	−	0.320953i	$-0.895997\pi$
$941$	−6.00000	+	10.3923i	−0.195594	+	0.338779i	0.751501	+	0.659732i	$0.229330\pi$
$942$		0			0
$943$	8.00000	+	13.8564i	0.260516	+	0.451227i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	20.0000	+	34.6410i	0.649913	+	1.12568i	0.983143	+	0.182836i	$0.0585279\pi$
$947$	20.0000	+	34.6410i	0.649913	+	1.12568i	−0.333231	+	0.942845i	$0.608139\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−10.0000			−0.323932			−0.161966	−	0.986796i	$-0.551783\pi$
$953$	−10.0000			−0.323932			−0.161966	+	0.986796i	$0.551783\pi$
$954$		0			0
$955$	−16.0000	+	27.7128i	−0.517748	+	0.896766i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	7.50000	+	12.9904i	0.241935	+	0.419045i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−72.0000			−2.31776
$966$		0			0
$967$	−24.0000			−0.771788			−0.385894	−	0.922543i	$-0.626107\pi$
$967$	−24.0000			−0.771788			−0.385894	+	0.922543i	$0.626107\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−13.0000	−	22.5167i	−0.417190	−	0.722594i	0.578466	−	0.815707i	$-0.303652\pi$
$971$	−13.0000	−	22.5167i	−0.417190	−	0.722594i	−0.995656	+	0.0931127i	$0.970318\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−27.0000	+	46.7654i	−0.863807	+	1.49616i	0.00442082	+	0.999990i	$0.498593\pi$
$977$	−27.0000	+	46.7654i	−0.863807	+	1.49616i	−0.868227	+	0.496167i	$0.834741\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−6.00000	+	10.3923i	−0.191370	+	0.331463i	−0.945705	−	0.325027i	$-0.894626\pi$
$983$	−6.00000	+	10.3923i	−0.191370	+	0.331463i	0.754334	+	0.656490i	$0.227960\pi$
$984$		0			0
$985$	−36.0000	−	62.3538i	−1.14706	−	1.98676i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	32.0000	+	55.4256i	1.01754	+	1.76243i
$990$		0			0
$991$	24.0000	−	41.5692i	0.762385	−	1.32049i	−0.179233	−	0.983807i	$-0.557362\pi$
$991$	24.0000	−	41.5692i	0.762385	−	1.32049i	0.941618	−	0.336683i	$-0.109305\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−16.0000			−0.507234
$996$		0			0
$997$	−26.0000	+	45.0333i	−0.823428	+	1.42622i	0.0796863	+	0.996820i	$0.474608\pi$
$997$	−26.0000	+	45.0333i	−0.823428	+	1.42622i	−0.903115	+	0.429400i	$0.858725\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.s.a.361.1		2
3.2	odd	2		392.2.i.a.361.1		2
7.2	even	3	inner	3528.2.s.a.3313.1		2
7.3	odd	6		3528.2.a.b.1.1		1
7.4	even	3		504.2.a.h.1.1		1
7.5	odd	6		3528.2.s.ba.3313.1		2
7.6	odd	2		3528.2.s.ba.361.1		2
12.11	even	2		784.2.i.j.753.1		2
21.2	odd	6		392.2.i.a.177.1		2
21.5	even	6		392.2.i.e.177.1		2
21.11	odd	6		56.2.a.b.1.1	✓	1
21.17	even	6		392.2.a.b.1.1		1
21.20	even	2		392.2.i.e.361.1		2
28.3	even	6		7056.2.a.c.1.1		1
28.11	odd	6		1008.2.a.m.1.1		1
56.11	odd	6		4032.2.a.a.1.1		1
56.53	even	6		4032.2.a.d.1.1		1
84.11	even	6		112.2.a.a.1.1		1
84.23	even	6		784.2.i.j.177.1		2
84.47	odd	6		784.2.i.b.177.1		2
84.59	odd	6		784.2.a.i.1.1		1
84.83	odd	2		784.2.i.b.753.1		2
105.32	even	12		1400.2.g.b.449.1		2
105.53	even	12		1400.2.g.b.449.2		2
105.59	even	6		9800.2.a.bj.1.1		1
105.74	odd	6		1400.2.a.a.1.1		1
168.11	even	6		448.2.a.h.1.1		1
168.53	odd	6		448.2.a.c.1.1		1
168.59	odd	6		3136.2.a.c.1.1		1
168.101	even	6		3136.2.a.w.1.1		1
231.32	even	6		6776.2.a.h.1.1		1
273.116	odd	6		9464.2.a.h.1.1		1
336.11	even	12		1792.2.b.h.897.2		2
336.53	odd	12		1792.2.b.a.897.1		2
336.179	even	12		1792.2.b.h.897.1		2
336.221	odd	12		1792.2.b.a.897.2		2
420.179	even	6		2800.2.a.bd.1.1		1
420.263	odd	12		2800.2.g.g.449.1		2
420.347	odd	12		2800.2.g.g.449.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.a.b.1.1	✓	1	21.11	odd	6
112.2.a.a.1.1		1	84.11	even	6
392.2.a.b.1.1		1	21.17	even	6
392.2.i.a.177.1		2	21.2	odd	6
392.2.i.a.361.1		2	3.2	odd	2
392.2.i.e.177.1		2	21.5	even	6
392.2.i.e.361.1		2	21.20	even	2
448.2.a.c.1.1		1	168.53	odd	6
448.2.a.h.1.1		1	168.11	even	6
504.2.a.h.1.1		1	7.4	even	3
784.2.a.i.1.1		1	84.59	odd	6
784.2.i.b.177.1		2	84.47	odd	6
784.2.i.b.753.1		2	84.83	odd	2
784.2.i.j.177.1		2	84.23	even	6
784.2.i.j.753.1		2	12.11	even	2
1008.2.a.m.1.1		1	28.11	odd	6
1400.2.a.a.1.1		1	105.74	odd	6
1400.2.g.b.449.1		2	105.32	even	12
1400.2.g.b.449.2		2	105.53	even	12
1792.2.b.a.897.1		2	336.53	odd	12
1792.2.b.a.897.2		2	336.221	odd	12
1792.2.b.h.897.1		2	336.179	even	12
1792.2.b.h.897.2		2	336.11	even	12
2800.2.a.bd.1.1		1	420.179	even	6
2800.2.g.g.449.1		2	420.263	odd	12
2800.2.g.g.449.2		2	420.347	odd	12
3136.2.a.c.1.1		1	168.59	odd	6
3136.2.a.w.1.1		1	168.101	even	6
3528.2.a.b.1.1		1	7.3	odd	6
3528.2.s.a.361.1		2	1.1	even	1	trivial
3528.2.s.a.3313.1		2	7.2	even	3	inner
3528.2.s.ba.361.1		2	7.6	odd	2
3528.2.s.ba.3313.1		2	7.5	odd	6
4032.2.a.a.1.1		1	56.11	odd	6
4032.2.a.d.1.1		1	56.53	even	6
6776.2.a.h.1.1		1	231.32	even	6
7056.2.a.c.1.1		1	28.3	even	6
9464.2.a.h.1.1		1	273.116	odd	6
9800.2.a.bj.1.1		1	105.59	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)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 3.46410i) q^{5} +O(q^{10})\)	\(q+(-2.00000 - 3.46410i) q^{5} +(-1.00000 + 1.73205i) q^{17} +(1.00000 + 1.73205i) q^{19} +(4.00000 + 6.92820i) q^{23} +(-5.50000 + 9.52628i) q^{25} -2.00000 q^{29} +(-2.00000 + 3.46410i) q^{31} +(3.00000 + 5.19615i) q^{37} +2.00000 q^{41} +8.00000 q^{43} +(-2.00000 - 3.46410i) q^{47} +(-5.00000 + 8.66025i) q^{53} +(3.00000 - 5.19615i) q^{59} +(-2.00000 - 3.46410i) q^{61} +(6.00000 - 10.3923i) q^{67} +(7.00000 - 12.1244i) q^{73} +(4.00000 + 6.92820i) q^{79} -6.00000 q^{83} +8.00000 q^{85} +(5.00000 + 8.66025i) q^{89} +(4.00000 - 6.92820i) q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5}+O(q^{10}) \) 2 * q - 4 * q^5	\( 2 q - 4 q^{5} - 2 q^{17} + 2 q^{19} + 8 q^{23} - 11 q^{25} - 4 q^{29} - 4 q^{31} + 6 q^{37} + 4 q^{41} + 16 q^{43} - 4 q^{47} - 10 q^{53} + 6 q^{59} - 4 q^{61} + 12 q^{67} + 14 q^{73} + 8 q^{79} - 12 q^{83} + 16 q^{85} + 10 q^{89} + 8 q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 - 2 * q^17 + 2 * q^19 + 8 * q^23 - 11 * q^25 - 4 * q^29 - 4 * q^31 + 6 * q^37 + 4 * q^41 + 16 * q^43 - 4 * q^47 - 10 * q^53 + 6 * q^59 - 4 * q^61 + 12 * q^67 + 14 * q^73 + 8 * q^79 - 12 * q^83 + 16 * q^85 + 10 * q^89 + 8 * q^95 - 4 * q^97

Introduction

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