LMFDB - Embedded newform 3528.2.s.ba.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.ba.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	$0.185750\pi$
$5$	2.00000	+	3.46410i	0.894427	+	1.54919i	0.0599153	+	0.998203i	$0.480917\pi$
$6$		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	−0.718900	−	0.695113i	$-0.755354\pi$
$17$	1.00000	−	1.73205i	0.242536	−	0.420084i	0.961436	+	0.275029i	$0.0886875\pi$
$18$		0			0
$19$	−1.00000	−	1.73205i	−0.229416	−	0.397360i	0.728219	−	0.685344i	$-0.240348\pi$
$19$	−1.00000	−	1.73205i	−0.229416	−	0.397360i	−0.957635	+	0.287984i	$0.907015\pi$
$20$		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	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	2.00000	−	3.46410i	0.359211	−	0.622171i	0.987829	+	0.155543i	$0.0497126\pi$
$32$		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	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		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	0.974219	−	0.225605i	$-0.0724358\pi$
$47$	2.00000	+	3.46410i	0.291730	+	0.505291i	−0.682489	+	0.730896i	$0.739102\pi$
$48$		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	−0.992524	−	0.122047i	$-0.961054\pi$
$59$	−3.00000	+	5.19615i	−0.390567	+	0.676481i	0.601958	+	0.798528i	$0.294388\pi$
$60$		0			0
$61$	2.00000	+	3.46410i	0.256074	+	0.443533i	0.965187	−	0.261562i	$-0.0842377\pi$
$61$	2.00000	+	3.46410i	0.256074	+	0.443533i	−0.709113	+	0.705095i	$0.750904\pi$
$62$		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	$0.472298\pi$
$73$	−7.00000	+	12.1244i	−0.819288	+	1.41905i	−0.906208	+	0.422833i	$0.861036\pi$
$74$		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	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		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	$-0.988859\pi$
$89$	−5.00000	−	8.66025i	−0.529999	−	0.917985i	0.469389	−	0.882992i	$-0.344474\pi$
$90$		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	$-0.467625\pi$
$97$	2.00000			0.203069			0.101535	+	0.994832i	$0.467625\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.00000	+	10.3923i	−0.597022	+	1.03407i	0.396236	+	0.918149i	$0.370316\pi$
$101$	−6.00000	+	10.3923i	−0.597022	+	1.03407i	−0.993258	+	0.115924i	$0.963017\pi$
$102$		0			0
$103$	−6.00000	−	10.3923i	−0.591198	−	1.02398i	−0.994071	−	0.108729i	$-0.965322\pi$
$103$	−6.00000	−	10.3923i	−0.591198	−	1.02398i	0.402874	−	0.915255i	$-0.368011\pi$
$104$		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	$-0.957196\pi$
$131$	−7.00000	−	12.1244i	−0.611593	−	1.05931i	0.379379	−	0.925241i	$-0.376138\pi$
$132$		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	$-0.776457\pi$
$139$	−18.0000			−1.52674			−0.763370	+	0.645961i	$0.776457\pi$
$140$		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	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		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	0.672949	−	0.739689i	$-0.265027\pi$
$173$	−4.00000	−	6.92820i	−0.304114	−	0.526742i	−0.977064	+	0.212947i	$0.931694\pi$
$174$		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	$-0.596092\pi$
$181$	−8.00000			−0.594635			−0.297318	+	0.954779i	$0.596092\pi$
$182$		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	−0.928166	−	0.372168i	$-0.878615\pi$
$199$	−2.00000	+	3.46410i	−0.141776	+	0.245564i	0.786389	+	0.617731i	$0.211948\pi$
$200$		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	$-0.202926\pi$
$223$	24.0000			1.60716			0.803579	+	0.595198i	$0.202926\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.00000	+	12.1244i	−0.464606	+	0.804722i	−0.999184	−	0.0403978i	$-0.987137\pi$
$227$	−7.00000	+	12.1244i	−0.464606	+	0.804722i	0.534577	+	0.845120i	$0.320471\pi$
$228$		0			0
$229$	−8.00000	−	13.8564i	−0.528655	−	0.915657i	−0.999442	−	0.0334101i	$-0.989363\pi$
$229$	−8.00000	−	13.8564i	−0.528655	−	0.915657i	0.470787	−	0.882247i	$-0.343970\pi$
$230$		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	−0.896435	−	0.443176i	$-0.853852\pi$
$241$	−1.00000	+	1.73205i	−0.0644157	+	0.111571i	0.832019	+	0.554747i	$0.187185\pi$
$242$		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	$-0.645673\pi$
$251$	−14.0000			−0.883672			−0.441836	+	0.897096i	$0.645673\pi$
$252$		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	$-0.976928\pi$
$257$	−9.00000	−	15.5885i	−0.561405	−	0.972381i	0.435970	−	0.899961i	$-0.356405\pi$
$258$		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	$-0.572083\pi$
$269$	12.0000	−	20.7846i	0.731653	−	1.26726i	0.956176	−	0.292791i	$-0.0945841\pi$
$270$		0			0
$271$	16.0000	+	27.7128i	0.971931	+	1.68343i	0.689713	+	0.724083i	$0.257737\pi$
$271$	16.0000	+	27.7128i	0.971931	+	1.68343i	0.282218	+	0.959350i	$0.408930\pi$
$272$		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	−0.975499	−	0.220005i	$-0.929393\pi$
$283$	−5.00000	+	8.66025i	−0.297219	+	0.514799i	0.678280	+	0.734804i	$0.262726\pi$
$284$		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	$-0.613996\pi$
$293$	−12.0000			−0.701047			−0.350524	+	0.936554i	$0.613996\pi$
$294$		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	$-0.481823\pi$
$307$	2.00000			0.114146			0.0570730	+	0.998370i	$0.481823\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	−0.294384	−	0.955687i	$-0.595114\pi$
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	0.974841	−	0.222900i	$-0.0715523\pi$
$312$		0			0
$313$	7.00000	+	12.1244i	0.395663	+	0.685309i	0.993186	−	0.116543i	$-0.0371814\pi$
$313$	7.00000	+	12.1244i	0.395663	+	0.685309i	−0.597522	+	0.801852i	$0.703848\pi$
$314$		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	$-0.568687\pi$
$349$	−8.00000			−0.428230			−0.214115	+	0.976808i	$0.568687\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	15.0000	−	25.9808i	0.798369	−	1.38282i	−0.122308	−	0.992492i	$-0.539030\pi$
$353$	15.0000	−	25.9808i	0.798369	−	1.38282i	0.920677	−	0.390324i	$-0.127637\pi$
$354$		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	−0.742538	−	0.669804i	$-0.766378\pi$
$367$	4.00000	−	6.92820i	0.208798	−	0.361649i	0.951336	+	0.308155i	$0.0997115\pi$
$368$		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	0.671027	−	0.741433i	$-0.265853\pi$
$383$	−6.00000	−	10.3923i	−0.306586	−	0.531022i	−0.977613	+	0.210411i	$0.932520\pi$
$384$		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	0.748017	−	0.663679i	$-0.231006\pi$
$397$	−4.00000	−	6.92820i	−0.200754	−	0.347717i	−0.948772	+	0.315963i	$0.897673\pi$
$398$		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	−0.782274	−	0.622935i	$-0.785940\pi$
$409$	3.00000	−	5.19615i	0.148340	−	0.256933i	0.930614	+	0.366002i	$0.119274\pi$
$410$		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	$-0.280962\pi$
$419$	26.0000			1.27018			0.635092	+	0.772437i	$0.280962\pi$
$420$		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	$-0.285205\pi$
$433$	26.0000			1.24948			0.624740	+	0.780833i	$0.285205\pi$
$434$		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	$0.0274485\pi$
$439$	12.0000	+	20.7846i	0.572729	+	0.991995i	−0.423556	+	0.905870i	$0.639218\pi$
$440$		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	0.788228	−	0.615383i	$-0.210999\pi$
$467$	−3.00000	−	5.19615i	−0.138823	−	0.240449i	−0.927052	+	0.374934i	$0.877665\pi$
$468$		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	−0.908093	−	0.418769i	$-0.862462\pi$
$479$	−2.00000	+	3.46410i	−0.0913823	+	0.158279i	0.816711	+	0.577047i	$0.195795\pi$
$480$		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	$-0.752848\pi$
$503$	−32.0000			−1.42681			−0.713405	+	0.700752i	$0.752848\pi$
$504$		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	$0.0118516\pi$
$509$	12.0000	+	20.7846i	0.531891	+	0.921262i	−0.467416	+	0.884037i	$0.654815\pi$
$510$		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	−0.598714	−	0.800963i	$-0.704321\pi$
$521$	9.00000	−	15.5885i	0.394297	−	0.682943i	0.993011	+	0.118020i	$0.0376547\pi$
$522$		0			0
$523$	−17.0000	−	29.4449i	−0.743358	−	1.28753i	−0.950958	−	0.309320i	$-0.899899\pi$
$523$	−17.0000	−	29.4449i	−0.743358	−	1.28753i	0.207600	−	0.978214i	$-0.433435\pi$
$524$		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	$0.420908\pi$
$563$	−17.0000	+	29.4449i	−0.716465	+	1.24095i	−0.962392	+	0.271665i	$0.912426\pi$
$564$		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	−0.615603	−	0.788056i	$-0.711088\pi$
$577$	9.00000	−	15.5885i	0.374675	−	0.648956i	0.990278	+	0.139100i	$0.0444210\pi$
$578$		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	$-0.433834\pi$
$587$	10.0000			0.412744			0.206372	+	0.978474i	$0.433834\pi$
$588$		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	$-0.835648\pi$
$593$	−21.0000	−	36.3731i	−0.862367	−	1.49366i	0.00727173	−	0.999974i	$-0.497685\pi$
$594$		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	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$		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	$-0.941678\pi$
$607$	−16.0000	−	27.7128i	−0.649420	−	1.12483i	0.333842	−	0.942629i	$-0.391655\pi$
$608$		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	−0.799416	−	0.600777i	$-0.794858\pi$
$619$	3.00000	−	5.19615i	0.120580	−	0.208851i	0.919997	+	0.391926i	$0.128191\pi$
$620$		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	$-0.410973\pi$
$643$	14.0000			0.552106			0.276053	+	0.961142i	$0.410973\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	18.0000	−	31.1769i	0.707653	−	1.22569i	−0.258073	−	0.966126i	$-0.583087\pi$
$647$	18.0000	−	31.1769i	0.707653	−	1.22569i	0.965726	−	0.259565i	$-0.0835793\pi$
$648$		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	−0.603354	−	0.797473i	$-0.706170\pi$
$661$	10.0000	−	17.3205i	0.388955	−	0.673690i	0.992309	+	0.123784i	$0.0395028\pi$
$662$		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	0.999021	−	0.0442400i	$-0.0140866\pi$
$677$	12.0000	+	20.7846i	0.461197	+	0.798817i	−0.537823	+	0.843057i	$0.680753\pi$
$678$		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	0.967901	−	0.251330i	$-0.0808679\pi$
$691$	7.00000	+	12.1244i	0.266293	+	0.461232i	−0.701609	+	0.712562i	$0.747535\pi$
$692$		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	$-0.932409\pi$
$719$	−18.0000	−	31.1769i	−0.671287	−	1.16270i	0.306253	−	0.951950i	$-0.400925\pi$
$720$		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	$-0.620944\pi$
$727$	−20.0000			−0.741759			−0.370879	+	0.928681i	$0.620944\pi$
$728$		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	$0.243377\pi$
$733$	26.0000	+	45.0333i	0.960332	+	1.66334i	0.238667	+	0.971102i	$0.423290\pi$
$734$		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	$0.108747\pi$
$761$	21.0000	+	36.3731i	0.761249	+	1.31852i	−0.180957	+	0.983491i	$0.557920\pi$
$762$		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	$-0.344689\pi$
$769$	26.0000			0.937584			0.468792	+	0.883309i	$0.344689\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−2.00000	+	3.46410i	−0.0719350	+	0.124595i	−0.899749	−	0.436407i	$-0.856251\pi$
$773$	−2.00000	+	3.46410i	−0.0719350	+	0.124595i	0.827814	+	0.561002i	$0.189584\pi$
$774$		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	−0.600620	−	0.799535i	$-0.705079\pi$
$787$	11.0000	−	19.0526i	0.392108	−	0.679150i	0.992727	+	0.120384i	$0.0384127\pi$
$788$		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	$-0.639748\pi$
$797$	−24.0000			−0.850124			−0.425062	+	0.905164i	$0.639748\pi$
$798$		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	$-0.676579\pi$
$811$	−30.0000			−1.05344			−0.526721	+	0.850038i	$0.676579\pi$
$812$		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	−0.638457	−	0.769657i	$-0.720427\pi$
$829$	10.0000	−	17.3205i	0.347314	−	0.601566i	0.985771	+	0.168091i	$0.0537604\pi$
$830$		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	$-0.286552\pi$
$839$	36.0000			1.24286			0.621429	+	0.783470i	$0.286552\pi$
$840$		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	$-0.315454\pi$
$853$	32.0000			1.09566			0.547830	+	0.836590i	$0.315454\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.00000	+	5.19615i	−0.102478	+	0.177497i	−0.912705	−	0.408619i	$-0.866010\pi$
$857$	−3.00000	+	5.19615i	−0.102478	+	0.177497i	0.810227	+	0.586116i	$0.199344\pi$
$858$		0			0
$859$	−7.00000	−	12.1244i	−0.238837	−	0.413678i	0.721544	−	0.692369i	$-0.243433\pi$
$859$	−7.00000	−	12.1244i	−0.238837	−	0.413678i	−0.960381	+	0.278691i	$0.910099\pi$
$860$		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	$-0.446122\pi$
$881$	10.0000			0.336909			0.168454	+	0.985709i	$0.446122\pi$
$882$		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	$-0.960087\pi$
$887$	−18.0000	−	31.1769i	−0.604381	−	1.04682i	0.387768	−	0.921757i	$-0.373246\pi$
$888$		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	0.881964	−	0.471317i	$-0.156221\pi$
$929$	1.00000	+	1.73205i	0.0328089	+	0.0568267i	−0.849155	+	0.528144i	$0.822888\pi$
$930$		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	$-0.594992\pi$
$937$	−18.0000			−0.588034			−0.294017	+	0.955800i	$0.594992\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	6.00000	−	10.3923i	0.195594	−	0.338779i	−0.751501	−	0.659732i	$-0.770670\pi$
$941$	6.00000	−	10.3923i	0.195594	−	0.338779i	0.947095	+	0.320953i	$0.104003\pi$
$942$		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	0.995656	−	0.0931127i	$-0.0296817\pi$
$971$	13.0000	+	22.5167i	0.417190	+	0.722594i	−0.578466	+	0.815707i	$0.696348\pi$
$972$		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	−0.754334	−	0.656490i	$-0.772040\pi$
$983$	6.00000	−	10.3923i	0.191370	−	0.331463i	0.945705	+	0.325027i	$0.105374\pi$
$984$		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	$-0.525392\pi$
$997$	26.0000	−	45.0333i	0.823428	−	1.42622i	0.903115	−	0.429400i	$-0.141275\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.s.ba.361.1		2
3.2	odd	2		392.2.i.e.361.1		2
7.2	even	3	inner	3528.2.s.ba.3313.1		2
7.3	odd	6		504.2.a.h.1.1		1
7.4	even	3		3528.2.a.b.1.1		1
7.5	odd	6		3528.2.s.a.3313.1		2
7.6	odd	2		3528.2.s.a.361.1		2
12.11	even	2		784.2.i.b.753.1		2
21.2	odd	6		392.2.i.e.177.1		2
21.5	even	6		392.2.i.a.177.1		2
21.11	odd	6		392.2.a.b.1.1		1
21.17	even	6		56.2.a.b.1.1	✓	1
21.20	even	2		392.2.i.a.361.1		2
28.3	even	6		1008.2.a.m.1.1		1
28.11	odd	6		7056.2.a.c.1.1		1
56.3	even	6		4032.2.a.a.1.1		1
56.45	odd	6		4032.2.a.d.1.1		1
84.11	even	6		784.2.a.i.1.1		1
84.23	even	6		784.2.i.b.177.1		2
84.47	odd	6		784.2.i.j.177.1		2
84.59	odd	6		112.2.a.a.1.1		1
84.83	odd	2		784.2.i.j.753.1		2
105.17	odd	12		1400.2.g.b.449.1		2
105.38	odd	12		1400.2.g.b.449.2		2
105.59	even	6		1400.2.a.a.1.1		1
105.74	odd	6		9800.2.a.bj.1.1		1
168.11	even	6		3136.2.a.c.1.1		1
168.53	odd	6		3136.2.a.w.1.1		1
168.59	odd	6		448.2.a.h.1.1		1
168.101	even	6		448.2.a.c.1.1		1
231.164	odd	6		6776.2.a.h.1.1		1
273.38	even	6		9464.2.a.h.1.1		1
336.59	odd	12		1792.2.b.h.897.2		2
336.101	even	12		1792.2.b.a.897.1		2
336.227	odd	12		1792.2.b.h.897.1		2
336.269	even	12		1792.2.b.a.897.2		2
420.59	odd	6		2800.2.a.bd.1.1		1
420.143	even	12		2800.2.g.g.449.1		2
420.227	even	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	21.17	even	6
112.2.a.a.1.1		1	84.59	odd	6
392.2.a.b.1.1		1	21.11	odd	6
392.2.i.a.177.1		2	21.5	even	6
392.2.i.a.361.1		2	21.20	even	2
392.2.i.e.177.1		2	21.2	odd	6
392.2.i.e.361.1		2	3.2	odd	2
448.2.a.c.1.1		1	168.101	even	6
448.2.a.h.1.1		1	168.59	odd	6
504.2.a.h.1.1		1	7.3	odd	6
784.2.a.i.1.1		1	84.11	even	6
784.2.i.b.177.1		2	84.23	even	6
784.2.i.b.753.1		2	12.11	even	2
784.2.i.j.177.1		2	84.47	odd	6
784.2.i.j.753.1		2	84.83	odd	2
1008.2.a.m.1.1		1	28.3	even	6
1400.2.a.a.1.1		1	105.59	even	6
1400.2.g.b.449.1		2	105.17	odd	12
1400.2.g.b.449.2		2	105.38	odd	12
1792.2.b.a.897.1		2	336.101	even	12
1792.2.b.a.897.2		2	336.269	even	12
1792.2.b.h.897.1		2	336.227	odd	12
1792.2.b.h.897.2		2	336.59	odd	12
2800.2.a.bd.1.1		1	420.59	odd	6
2800.2.g.g.449.1		2	420.143	even	12
2800.2.g.g.449.2		2	420.227	even	12
3136.2.a.c.1.1		1	168.11	even	6
3136.2.a.w.1.1		1	168.53	odd	6
3528.2.a.b.1.1		1	7.4	even	3
3528.2.s.a.361.1		2	7.6	odd	2
3528.2.s.a.3313.1		2	7.5	odd	6
3528.2.s.ba.361.1		2	1.1	even	1	trivial
3528.2.s.ba.3313.1		2	7.2	even	3	inner
4032.2.a.a.1.1		1	56.3	even	6
4032.2.a.d.1.1		1	56.45	odd	6
6776.2.a.h.1.1		1	231.164	odd	6
7056.2.a.c.1.1		1	28.11	odd	6
9464.2.a.h.1.1		1	273.38	even	6
9800.2.a.bj.1.1		1	105.74	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)

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