LMFDB - Embedded newform 3528.2.s.n.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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1176)
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.n.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+O(q^{10})$	$q+4.00000 q^{13} +(2.00000 + 3.46410i) q^{17} +(2.00000 - 3.46410i) q^{19} +(2.00000 - 3.46410i) q^{23} +(2.50000 + 4.33013i) q^{25} -2.00000 q^{29} +(-4.00000 - 6.92820i) q^{31} +(3.00000 - 5.19615i) q^{37} -12.0000 q^{41} +4.00000 q^{43} +(-4.00000 + 6.92820i) q^{47} +(3.00000 + 5.19615i) q^{53} +(6.00000 + 10.3923i) q^{59} +(-2.00000 + 3.46410i) q^{61} +(2.00000 + 3.46410i) q^{67} +12.0000 q^{71} +(-4.00000 - 6.92820i) q^{73} +(8.00000 - 13.8564i) q^{79} +4.00000 q^{83} +(2.00000 - 3.46410i) q^{89} +16.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 8 q^{13} + 4 q^{17} + 4 q^{19} + 4 q^{23} + 5 q^{25} - 4 q^{29} - 8 q^{31} + 6 q^{37} - 24 q^{41} + 8 q^{43} - 8 q^{47} + 6 q^{53} + 12 q^{59} - 4 q^{61} + 4 q^{67} + 24 q^{71} - 8 q^{73} + 16 q^{79} + 8 q^{83} + 4 q^{89} + 32 q^{97}+O(q^{100}) $ 2 * q + 8 * q^13 + 4 * q^17 + 4 * q^19 + 4 * q^23 + 5 * q^25 - 4 * q^29 - 8 * q^31 + 6 * q^37 - 24 * q^41 + 8 * q^43 - 8 * q^47 + 6 * q^53 + 12 * q^59 - 4 * q^61 + 4 * q^67 + 24 * q^71 - 8 * q^73 + 16 * q^79 + 8 * q^83 + 4 * q^89 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		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
$13$	4.00000			1.10940			0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000			1.10940			0.554700	+	0.832050i	$0.312833\pi$
$14$		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
$19$	2.00000	−	3.46410i	0.458831	−	0.794719i	−0.540068	−	0.841621i	$-0.681602\pi$
$19$	2.00000	−	3.46410i	0.458831	−	0.794719i	0.998899	+	0.0469020i	$0.0149348\pi$
$20$		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
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		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
$31$	−4.00000	−	6.92820i	−0.718421	−	1.24434i	−0.961625	−	0.274367i	$-0.911532\pi$
$31$	−4.00000	−	6.92820i	−0.718421	−	1.24434i	0.243204	−	0.969975i	$-0.421802\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.00000	−	5.19615i	0.493197	−	0.854242i	−0.506772	−	0.862080i	$-0.669162\pi$
$37$	3.00000	−	5.19615i	0.493197	−	0.854242i	0.999969	+	0.00783774i	$0.00249486\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−12.0000			−1.87409			−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000			−1.87409			−0.937043	+	0.349215i	$0.886448\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	0.411606	+	0.911362i	$0.364968\pi$
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	−0.995066	+	0.0992202i	$0.968365\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.00000	+	5.19615i	0.412082	+	0.713746i	0.995117	−	0.0987002i	$-0.0314685\pi$
$53$	3.00000	+	5.19615i	0.412082	+	0.713746i	−0.583036	+	0.812447i	$0.698135\pi$
$54$		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	$0.118692\pi$
$59$	6.00000	+	10.3923i	0.781133	+	1.35296i	−0.150148	+	0.988663i	$0.547975\pi$
$60$		0			0
$61$	−2.00000	+	3.46410i	−0.256074	+	0.443533i	−0.965187	−	0.261562i	$-0.915762\pi$
$61$	−2.00000	+	3.46410i	−0.256074	+	0.443533i	0.709113	+	0.705095i	$0.249096\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	2.00000	+	3.46410i	0.244339	+	0.423207i	0.961946	−	0.273241i	$-0.0880957\pi$
$67$	2.00000	+	3.46410i	0.244339	+	0.423207i	−0.717607	+	0.696449i	$0.754762\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$	−4.00000	−	6.92820i	−0.468165	−	0.810885i	0.531174	−	0.847263i	$-0.321751\pi$
$73$	−4.00000	−	6.92820i	−0.468165	−	0.810885i	−0.999338	+	0.0363782i	$0.988418\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	8.00000	−	13.8564i	0.900070	−	1.55897i	0.0726692	−	0.997356i	$-0.476848\pi$
$79$	8.00000	−	13.8564i	0.900070	−	1.55897i	0.827401	−	0.561611i	$-0.189818\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.00000			0.439057			0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000			0.439057			0.219529	+	0.975606i	$0.429548\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.00000	−	3.46410i	0.212000	−	0.367194i	−0.740341	−	0.672232i	$-0.765336\pi$
$89$	2.00000	−	3.46410i	0.212000	−	0.367194i	0.952340	+	0.305038i	$0.0986691\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	16.0000			1.62455			0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000			1.62455			0.812277	+	0.583272i	$0.198228\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.00000	−	6.92820i	−0.398015	−	0.689382i	0.595466	−	0.803380i	$-0.296967\pi$
$101$	−4.00000	−	6.92820i	−0.398015	−	0.689382i	−0.993481	+	0.113998i	$0.963634\pi$
$102$		0			0
$103$	4.00000	−	6.92820i	0.394132	−	0.682656i	−0.598858	−	0.800855i	$-0.704379\pi$
$103$	4.00000	−	6.92820i	0.394132	−	0.682656i	0.992990	+	0.118199i	$0.0377120\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.00000	−	6.92820i	0.386695	−	0.669775i	−0.605308	−	0.795991i	$-0.706950\pi$
$107$	4.00000	−	6.92820i	0.386695	−	0.669775i	0.992003	+	0.126217i	$0.0402834\pi$
$108$		0			0
$109$	7.00000	+	12.1244i	0.670478	+	1.16130i	0.977769	+	0.209687i	$0.0672444\pi$
$109$	7.00000	+	12.1244i	0.670478	+	1.16130i	−0.307290	+	0.951616i	$0.599422\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.00000			−0.188144			−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000			−0.188144			−0.0940721	+	0.995565i	$0.529988\pi$
$114$		0			0
$115$		0			0
$116$		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
$125$		0			0
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.00000	+	3.46410i	−0.174741	+	0.302660i	−0.940072	−	0.340977i	$-0.889242\pi$
$131$	−2.00000	+	3.46410i	−0.174741	+	0.302660i	0.765331	+	0.643637i	$0.222575\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	11.0000	+	19.0526i	0.939793	+	1.62777i	0.765855	+	0.643013i	$0.222316\pi$
$137$	11.0000	+	19.0526i	0.939793	+	1.62777i	0.173939	+	0.984757i	$0.444351\pi$
$138$		0			0
$139$	−12.0000			−1.01783			−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000			−1.01783			−0.508913	+	0.860818i	$0.669953\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.00000	−	5.19615i	0.245770	−	0.425685i	−0.716578	−	0.697507i	$-0.754293\pi$
$149$	3.00000	−	5.19615i	0.245770	−	0.425685i	0.962348	+	0.271821i	$0.0876260\pi$
$150$		0			0
$151$	4.00000	+	6.92820i	0.325515	+	0.563809i	0.981617	−	0.190864i	$-0.0611289\pi$
$151$	4.00000	+	6.92820i	0.325515	+	0.563809i	−0.656101	+	0.754673i	$0.727796\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	10.0000	+	17.3205i	0.798087	+	1.38233i	0.920860	+	0.389892i	$0.127488\pi$
$157$	10.0000	+	17.3205i	0.798087	+	1.38233i	−0.122774	+	0.992435i	$0.539179\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	6.00000	−	10.3923i	0.469956	−	0.813988i	−0.529454	−	0.848339i	$-0.677603\pi$
$163$	6.00000	−	10.3923i	0.469956	−	0.813988i	0.999410	+	0.0343508i	$0.0109363\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	8.00000			0.619059			0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000			0.619059			0.309529	+	0.950890i	$0.399829\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$		0			0
$172$		0			0
$173$	8.00000	−	13.8564i	0.608229	−	1.05348i	−0.383304	−	0.923622i	$-0.625214\pi$
$173$	8.00000	−	13.8564i	0.608229	−	1.05348i	0.991532	−	0.129861i	$-0.0414530\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.0000	−	20.7846i	−0.896922	−	1.55351i	−0.831408	−	0.555663i	$-0.812464\pi$
$179$	−12.0000	−	20.7846i	−0.896922	−	1.55351i	−0.0655145	−	0.997852i	$-0.520869\pi$
$180$		0			0
$181$	−12.0000			−0.891953			−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000			−0.891953			−0.445976	+	0.895045i	$0.647144\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.00000	−	10.3923i	0.434145	−	0.751961i	−0.563081	−	0.826402i	$-0.690384\pi$
$191$	6.00000	−	10.3923i	0.434145	−	0.751961i	0.997225	+	0.0744412i	$0.0237173\pi$
$192$		0			0
$193$	7.00000	+	12.1244i	0.503871	+	0.872730i	0.999990	+	0.00447566i	$0.00142465\pi$
$193$	7.00000	+	12.1244i	0.503871	+	0.872730i	−0.496119	+	0.868255i	$0.665242\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.00000			−0.427482			−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000			−0.427482			−0.213741	+	0.976890i	$0.568565\pi$
$198$		0			0
$199$	4.00000	+	6.92820i	0.283552	+	0.491127i	0.972257	−	0.233915i	$-0.0751537\pi$
$199$	4.00000	+	6.92820i	0.283552	+	0.491127i	−0.688705	+	0.725042i	$0.741820\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		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
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	8.00000	+	13.8564i	0.538138	+	0.932083i
$222$		0			0
$223$	−16.0000			−1.07144			−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000			−1.07144			−0.535720	+	0.844396i	$0.679960\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.0000	−	17.3205i	−0.663723	−	1.14960i	−0.979630	−	0.200812i	$-0.935642\pi$
$227$	−10.0000	−	17.3205i	−0.663723	−	1.14960i	0.315906	−	0.948790i	$-0.397691\pi$
$228$		0			0
$229$	−10.0000	+	17.3205i	−0.660819	+	1.14457i	0.319582	+	0.947559i	$0.396457\pi$
$229$	−10.0000	+	17.3205i	−0.660819	+	1.14457i	−0.980401	+	0.197013i	$0.936876\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.00000	−	5.19615i	0.196537	−	0.340411i	−0.750867	−	0.660454i	$-0.770364\pi$
$233$	3.00000	−	5.19615i	0.196537	−	0.340411i	0.947403	+	0.320043i	$0.103697\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	20.0000			1.29369			0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000			1.29369			0.646846	+	0.762620i	$0.276088\pi$
$240$		0			0
$241$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$241$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	8.00000	−	13.8564i	0.509028	−	0.881662i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		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
$263$	6.00000	+	10.3923i	0.369976	+	0.640817i	0.989561	−	0.144112i	$-0.0460326\pi$
$263$	6.00000	+	10.3923i	0.369976	+	0.640817i	−0.619586	+	0.784929i	$0.712699\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−12.0000	−	20.7846i	−0.731653	−	1.26726i	−0.956176	−	0.292791i	$-0.905416\pi$
$269$	−12.0000	−	20.7846i	−0.731653	−	1.26726i	0.224523	−	0.974469i	$-0.427917\pi$
$270$		0			0
$271$	−8.00000	+	13.8564i	−0.485965	+	0.841717i	−0.999870	−	0.0161307i	$-0.994865\pi$
$271$	−8.00000	+	13.8564i	−0.485965	+	0.841717i	0.513905	+	0.857847i	$0.328199\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	0.976231	−	0.216731i	$-0.0695395\pi$
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	−0.675810	+	0.737075i	$0.736206\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−22.0000			−1.31241			−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000			−1.31241			−0.656205	+	0.754583i	$0.727839\pi$
$282$		0			0
$283$	2.00000	+	3.46410i	0.118888	+	0.205919i	0.919327	−	0.393494i	$-0.128734\pi$
$283$	2.00000	+	3.46410i	0.118888	+	0.205919i	−0.800439	+	0.599414i	$0.795400\pi$
$284$		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	$-0.252716\pi$
$293$	24.0000			1.40209			0.701047	+	0.713115i	$0.252716\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	8.00000	−	13.8564i	0.462652	−	0.801337i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	12.0000			0.684876			0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000			0.684876			0.342438	+	0.939540i	$0.388747\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8.00000	−	13.8564i	−0.453638	−	0.785725i	0.544970	−	0.838455i	$-0.316541\pi$
$311$	−8.00000	−	13.8564i	−0.453638	−	0.785725i	−0.998609	+	0.0527306i	$0.983208\pi$
$312$		0			0
$313$	4.00000	−	6.92820i	0.226093	−	0.391605i	−0.730554	−	0.682855i	$-0.760738\pi$
$313$	4.00000	−	6.92820i	0.226093	−	0.391605i	0.956647	+	0.291250i	$0.0940712\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	0.494489	+	0.869184i	$0.335355\pi$
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	−0.999980	+	0.00635137i	$0.997978\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	16.0000			0.890264
$324$		0			0
$325$	10.0000	+	17.3205i	0.554700	+	0.960769i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.00000	+	3.46410i	−0.109930	+	0.190404i	−0.915742	−	0.401768i	$-0.868396\pi$
$331$	−2.00000	+	3.46410i	−0.109930	+	0.190404i	0.805812	+	0.592172i	$0.201729\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−18.0000			−0.980522			−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000			−0.980522			−0.490261	+	0.871576i	$0.663099\pi$
$338$		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	−0.984487	−	0.175457i	$-0.943860\pi$
$347$	−12.0000	−	20.7846i	−0.644194	−	1.11578i	0.340293	−	0.940319i	$-0.389474\pi$
$348$		0			0
$349$	−36.0000			−1.92704			−0.963518	−	0.267644i	$-0.913755\pi$
$349$	−36.0000			−1.92704			−0.963518	+	0.267644i	$0.913755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	14.0000	+	24.2487i	0.745145	+	1.29063i	0.950127	+	0.311863i	$0.100953\pi$
$353$	14.0000	+	24.2487i	0.745145	+	1.29063i	−0.204982	+	0.978766i	$0.565714\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	14.0000	−	24.2487i	0.738892	−	1.27980i	−0.214103	−	0.976811i	$-0.568683\pi$
$359$	14.0000	−	24.2487i	0.738892	−	1.27980i	0.952995	−	0.302987i	$-0.0979839\pi$
$360$		0			0
$361$	1.50000	+	2.59808i	0.0789474	+	0.136741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	8.00000	+	13.8564i	0.417597	+	0.723299i	0.995697	−	0.0926670i	$-0.0295392\pi$
$367$	8.00000	+	13.8564i	0.417597	+	0.723299i	−0.578101	+	0.815966i	$0.696206\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	5.00000	−	8.66025i	0.258890	−	0.448411i	−0.707055	−	0.707159i	$-0.749977\pi$
$373$	5.00000	−	8.66025i	0.258890	−	0.448411i	0.965945	+	0.258748i	$0.0833099\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−8.00000			−0.412021
$378$		0			0
$379$	−12.0000			−0.616399			−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000			−0.616399			−0.308199	+	0.951322i	$0.599726\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	−0.377531	−	0.925997i	$-0.623227\pi$
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	0.990702	−	0.136047i	$-0.0434398\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	13.0000	+	22.5167i	0.659126	+	1.14164i	0.980842	+	0.194804i	$0.0624070\pi$
$389$	13.0000	+	22.5167i	0.659126	+	1.14164i	−0.321716	+	0.946836i	$0.604260\pi$
$390$		0			0
$391$	16.0000			0.809155
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	10.0000	−	17.3205i	0.501886	−	0.869291i	−0.498112	−	0.867113i	$-0.665973\pi$
$397$	10.0000	−	17.3205i	0.501886	−	0.869291i	0.999998	−	0.00217869i	$-0.000693499\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	−0.998350	−	0.0574304i	$-0.981709\pi$
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	0.548911	+	0.835881i	$0.315043\pi$
$402$		0			0
$403$	−16.0000	−	27.7128i	−0.797017	−	1.38047i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−16.0000	−	27.7128i	−0.791149	−	1.37031i	−0.925256	−	0.379344i	$-0.876150\pi$
$409$	−16.0000	−	27.7128i	−0.791149	−	1.37031i	0.134107	−	0.990967i	$-0.457183\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−20.0000			−0.977064			−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000			−0.977064			−0.488532	+	0.872546i	$0.662467\pi$
$420$		0			0
$421$	6.00000			0.292422			0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000			0.292422			0.146211	+	0.989253i	$0.453292\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−10.0000	+	17.3205i	−0.485071	+	0.840168i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−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
$433$	8.00000			0.384455			0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000			0.384455			0.192228	+	0.981350i	$0.438429\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.00000	−	13.8564i	−0.382692	−	0.662842i
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−16.0000	+	27.7128i	−0.760183	+	1.31668i	0.182573	+	0.983192i	$0.441557\pi$
$443$	−16.0000	+	27.7128i	−0.760183	+	1.31668i	−0.942756	+	0.333483i	$0.891776\pi$
$444$		0			0
$445$		0			0
$446$		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
$457$	−5.00000	+	8.66025i	−0.233890	+	0.405110i	−0.958950	−	0.283577i	$-0.908479\pi$
$457$	−5.00000	+	8.66025i	−0.233890	+	0.405110i	0.725059	+	0.688686i	$0.241812\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16.0000			0.745194			0.372597	−	0.927993i	$-0.378467\pi$
$461$	16.0000			0.745194			0.372597	+	0.927993i	$0.378467\pi$
$462$		0			0
$463$	16.0000			0.743583			0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000			0.743583			0.371792	+	0.928316i	$0.378744\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.0000	+	17.3205i	−0.462745	+	0.801498i	−0.999097	−	0.0424970i	$-0.986469\pi$
$467$	−10.0000	+	17.3205i	−0.462745	+	0.801498i	0.536352	+	0.843995i	$0.319802\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	20.0000			0.917663
$476$		0			0
$477$		0			0
$478$		0			0
$479$	20.0000	+	34.6410i	0.913823	+	1.58279i	0.808615	+	0.588338i	$0.200218\pi$
$479$	20.0000	+	34.6410i	0.913823	+	1.58279i	0.105208	+	0.994450i	$0.466449\pi$
$480$		0			0
$481$	12.0000	−	20.7846i	0.547153	−	0.947697i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−4.00000	−	6.92820i	−0.181257	−	0.313947i	0.761052	−	0.648691i	$-0.224683\pi$
$487$	−4.00000	−	6.92820i	−0.181257	−	0.313947i	−0.942309	+	0.334744i	$0.891350\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$	−4.00000	−	6.92820i	−0.180151	−	0.312031i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−18.0000	+	31.1769i	−0.805791	+	1.39567i	0.109965	+	0.993935i	$0.464926\pi$
$499$	−18.0000	+	31.1769i	−0.805791	+	1.39567i	−0.915756	+	0.401735i	$0.868407\pi$
$500$		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
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4.00000	−	6.92820i	0.177297	−	0.307087i	−0.763657	−	0.645622i	$-0.776598\pi$
$509$	4.00000	−	6.92820i	0.177297	−	0.307087i	0.940954	+	0.338535i	$0.109931\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	18.0000	+	31.1769i	0.788594	+	1.36589i	0.926828	+	0.375486i	$0.122524\pi$
$521$	18.0000	+	31.1769i	0.788594	+	1.36589i	−0.138234	+	0.990400i	$0.544143\pi$
$522$		0			0
$523$	−14.0000	+	24.2487i	−0.612177	+	1.06032i	0.378695	+	0.925521i	$0.376373\pi$
$523$	−14.0000	+	24.2487i	−0.612177	+	1.06032i	−0.990873	+	0.134801i	$0.956961\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	16.0000	−	27.7128i	0.696971	−	1.20719i
$528$		0			0
$529$	3.50000	+	6.06218i	0.152174	+	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−48.0000			−2.07911
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−15.0000	+	25.9808i	−0.644900	+	1.11700i	0.339424	+	0.940633i	$0.389768\pi$
$541$	−15.0000	+	25.9808i	−0.644900	+	1.11700i	−0.984325	+	0.176367i	$0.943566\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−28.0000			−1.19719			−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000			−1.19719			−0.598597	+	0.801050i	$0.704275\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.00000	+	6.92820i	−0.170406	+	0.295151i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−17.0000	−	29.4449i	−0.720313	−	1.24762i	−0.960874	−	0.276985i	$-0.910665\pi$
$557$	−17.0000	−	29.4449i	−0.720313	−	1.24762i	0.240561	−	0.970634i	$-0.422669\pi$
$558$		0			0
$559$	16.0000			0.676728
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.00000	−	10.3923i	−0.252870	−	0.437983i	0.711445	−	0.702742i	$-0.248041\pi$
$563$	−6.00000	−	10.3923i	−0.252870	−	0.437983i	−0.964315	+	0.264758i	$0.914708\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−5.00000	+	8.66025i	−0.209611	+	0.363057i	−0.951592	−	0.307364i	$-0.900553\pi$
$569$	−5.00000	+	8.66025i	−0.209611	+	0.363057i	0.741981	+	0.670421i	$0.233886\pi$
$570$		0			0
$571$	−6.00000	−	10.3923i	−0.251092	−	0.434904i	0.712735	−	0.701434i	$-0.247456\pi$
$571$	−6.00000	−	10.3923i	−0.251092	−	0.434904i	−0.963827	+	0.266529i	$0.914123\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	20.0000			0.834058
$576$		0			0
$577$	−16.0000	−	27.7128i	−0.666089	−	1.15370i	−0.978989	−	0.203913i	$-0.934634\pi$
$577$	−16.0000	−	27.7128i	−0.666089	−	1.15370i	0.312900	−	0.949786i	$-0.398699\pi$
$578$		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
$587$	−36.0000			−1.48588			−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000			−1.48588			−0.742940	+	0.669359i	$0.766569\pi$
$588$		0			0
$589$	−32.0000			−1.31854
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−10.0000	+	17.3205i	−0.410651	+	0.711268i	−0.994961	−	0.100262i	$-0.968032\pi$
$593$	−10.0000	+	17.3205i	−0.410651	+	0.711268i	0.584310	+	0.811530i	$0.301365\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.00000	−	10.3923i	−0.245153	−	0.424618i	0.717021	−	0.697051i	$-0.245505\pi$
$599$	−6.00000	−	10.3923i	−0.245153	−	0.424618i	−0.962175	+	0.272433i	$0.912172\pi$
$600$		0			0
$601$	16.0000			0.652654			0.326327	−	0.945257i	$-0.394189\pi$
$601$	16.0000			0.652654			0.326327	+	0.945257i	$0.394189\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$607$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−16.0000	+	27.7128i	−0.647291	+	1.12114i
$612$		0			0
$613$	−13.0000	−	22.5167i	−0.525065	−	0.909439i	−0.999574	−	0.0291886i	$-0.990708\pi$
$613$	−13.0000	−	22.5167i	−0.525065	−	0.909439i	0.474509	−	0.880251i	$-0.342626\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−22.0000			−0.885687			−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000			−0.885687			−0.442843	+	0.896599i	$0.646030\pi$
$618$		0			0
$619$	−22.0000	−	38.1051i	−0.884255	−	1.53157i	−0.846566	−	0.532284i	$-0.821334\pi$
$619$	−22.0000	−	38.1051i	−0.884255	−	1.53157i	−0.0376891	−	0.999290i	$-0.512000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	24.0000			0.956943
$630$		0			0
$631$	24.0000			0.955425			0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000			0.955425			0.477712	+	0.878516i	$0.341466\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	15.0000	+	25.9808i	0.592464	+	1.02618i	0.993899	+	0.110291i	$0.0351782\pi$
$641$	15.0000	+	25.9808i	0.592464	+	1.02618i	−0.401435	+	0.915888i	$0.631488\pi$
$642$		0			0
$643$	4.00000			0.157745			0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000			0.157745			0.0788723	+	0.996885i	$0.474868\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−4.00000	−	6.92820i	−0.157256	−	0.272376i	0.776622	−	0.629967i	$-0.216932\pi$
$647$	−4.00000	−	6.92820i	−0.157256	−	0.272376i	−0.933878	+	0.357591i	$0.883598\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.00000	−	1.73205i	0.0391330	−	0.0677804i	−0.845796	−	0.533507i	$-0.820874\pi$
$653$	1.00000	−	1.73205i	0.0391330	−	0.0677804i	0.884929	+	0.465727i	$0.154207\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−40.0000			−1.55818			−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000			−1.55818			−0.779089	+	0.626913i	$0.784318\pi$
$660$		0			0
$661$	−2.00000	−	3.46410i	−0.0777910	−	0.134738i	0.824506	−	0.565854i	$-0.191453\pi$
$661$	−2.00000	−	3.46410i	−0.0777910	−	0.134738i	−0.902297	+	0.431116i	$0.858120\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−4.00000	+	6.92820i	−0.154881	+	0.268261i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	14.0000			0.539660			0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000			0.539660			0.269830	+	0.962908i	$0.413032\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−16.0000	+	27.7128i	−0.614930	+	1.06509i	0.375467	+	0.926836i	$0.377482\pi$
$677$	−16.0000	+	27.7128i	−0.614930	+	1.06509i	−0.990397	+	0.138254i	$0.955851\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−24.0000	−	41.5692i	−0.918334	−	1.59060i	−0.801945	−	0.597398i	$-0.796201\pi$
$683$	−24.0000	−	41.5692i	−0.918334	−	1.59060i	−0.116390	−	0.993204i	$-0.537132\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	12.0000	+	20.7846i	0.457164	+	0.791831i
$690$		0			0
$691$	10.0000	−	17.3205i	0.380418	−	0.658903i	−0.610704	−	0.791859i	$-0.709113\pi$
$691$	10.0000	−	17.3205i	0.380418	−	0.658903i	0.991122	+	0.132956i	$0.0424468\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−24.0000	−	41.5692i	−0.909065	−	1.57455i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	46.0000			1.73740			0.868698	−	0.495342i	$-0.164957\pi$
$701$	46.0000			1.73740			0.868698	+	0.495342i	$0.164957\pi$
$702$		0			0
$703$	−12.0000	−	20.7846i	−0.452589	−	0.783906i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−13.0000	+	22.5167i	−0.488225	+	0.845631i	−0.999908	−	0.0135434i	$-0.995689\pi$
$709$	−13.0000	+	22.5167i	−0.488225	+	0.845631i	0.511683	+	0.859174i	$0.329022\pi$
$710$		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
$719$	20.0000	−	34.6410i	0.745874	−	1.29189i	−0.203911	−	0.978989i	$-0.565365\pi$
$719$	20.0000	−	34.6410i	0.745874	−	1.29189i	0.949785	−	0.312903i	$-0.101301\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5.00000	−	8.66025i	−0.185695	−	0.321634i
$726$		0			0
$727$	24.0000			0.890111			0.445055	−	0.895503i	$-0.353184\pi$
$727$	24.0000			0.890111			0.445055	+	0.895503i	$0.353184\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	8.00000	+	13.8564i	0.295891	+	0.512498i
$732$		0			0
$733$	2.00000	−	3.46410i	0.0738717	−	0.127950i	−0.826723	−	0.562609i	$-0.809798\pi$
$733$	2.00000	−	3.46410i	0.0738717	−	0.127950i	0.900595	+	0.434659i	$0.143131\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−10.0000	−	17.3205i	−0.367856	−	0.637145i	0.621374	−	0.783514i	$-0.286575\pi$
$739$	−10.0000	−	17.3205i	−0.367856	−	0.637145i	−0.989230	+	0.146369i	$0.953241\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	4.00000			0.146746			0.0733729	−	0.997305i	$-0.476624\pi$
$743$	4.00000			0.146746			0.0733729	+	0.997305i	$0.476624\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−16.0000	+	27.7128i	−0.583848	+	1.01125i	0.411170	+	0.911559i	$0.365120\pi$
$751$	−16.0000	+	27.7128i	−0.583848	+	1.01125i	−0.995018	+	0.0996961i	$0.968213\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−22.0000			−0.799604			−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000			−0.799604			−0.399802	+	0.916602i	$0.630921\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.00000	−	10.3923i	0.217500	−	0.376721i	−0.736543	−	0.676391i	$-0.763543\pi$
$761$	6.00000	−	10.3923i	0.217500	−	0.376721i	0.954043	+	0.299670i	$0.0968765\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	24.0000	+	41.5692i	0.866590	+	1.50098i
$768$		0			0
$769$	−8.00000			−0.288487			−0.144244	−	0.989542i	$-0.546075\pi$
$769$	−8.00000			−0.288487			−0.144244	+	0.989542i	$0.546075\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	12.0000	+	20.7846i	0.431610	+	0.747570i	0.997012	−	0.0772449i	$-0.0246123\pi$
$773$	12.0000	+	20.7846i	0.431610	+	0.747570i	−0.565402	+	0.824815i	$0.691279\pi$
$774$		0			0
$775$	20.0000	−	34.6410i	0.718421	−	1.24434i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−24.0000	+	41.5692i	−0.859889	+	1.48937i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	26.0000	+	45.0333i	0.926800	+	1.60526i	0.788641	+	0.614855i	$0.210785\pi$
$787$	26.0000	+	45.0333i	0.926800	+	1.60526i	0.138159	+	0.990410i	$0.455881\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−8.00000	+	13.8564i	−0.284088	+	0.492055i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−16.0000			−0.566749			−0.283375	−	0.959009i	$-0.591454\pi$
$797$	−16.0000			−0.566749			−0.283375	+	0.959009i	$0.591454\pi$
$798$		0			0
$799$	−32.0000			−1.13208
$800$		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
$809$	−19.0000	−	32.9090i	−0.668004	−	1.15702i	−0.978461	−	0.206430i	$-0.933815\pi$
$809$	−19.0000	−	32.9090i	−0.668004	−	1.15702i	0.310457	−	0.950587i	$-0.399518\pi$
$810$		0			0
$811$	−20.0000			−0.702295			−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000			−0.702295			−0.351147	+	0.936320i	$0.614208\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	8.00000	−	13.8564i	0.279885	−	0.484774i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−13.0000	+	22.5167i	−0.453703	+	0.785837i	−0.998613	−	0.0526580i	$-0.983231\pi$
$821$	−13.0000	+	22.5167i	−0.453703	+	0.785837i	0.544909	+	0.838495i	$0.316564\pi$
$822$		0			0
$823$	4.00000	+	6.92820i	0.139431	+	0.241502i	0.927281	−	0.374365i	$-0.122139\pi$
$823$	4.00000	+	6.92820i	0.139431	+	0.241502i	−0.787850	+	0.615867i	$0.788806\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	8.00000			0.278187			0.139094	−	0.990279i	$-0.455581\pi$
$827$	8.00000			0.278187			0.139094	+	0.990279i	$0.455581\pi$
$828$		0			0
$829$	10.0000	+	17.3205i	0.347314	+	0.601566i	0.985771	−	0.168091i	$-0.0537604\pi$
$829$	10.0000	+	17.3205i	0.347314	+	0.601566i	−0.638457	+	0.769657i	$0.720427\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	24.0000			0.828572			0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000			0.828572			0.414286	+	0.910147i	$0.364031\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−12.0000	−	20.7846i	−0.411355	−	0.712487i
$852$		0			0
$853$	−4.00000			−0.136957			−0.0684787	−	0.997653i	$-0.521815\pi$
$853$	−4.00000			−0.136957			−0.0684787	+	0.997653i	$0.521815\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−18.0000	−	31.1769i	−0.614868	−	1.06498i	−0.990408	−	0.138177i	$-0.955876\pi$
$857$	−18.0000	−	31.1769i	−0.614868	−	1.06498i	0.375539	−	0.926806i	$-0.377458\pi$
$858$		0			0
$859$	−18.0000	+	31.1769i	−0.614152	+	1.06374i	0.376381	+	0.926465i	$0.377169\pi$
$859$	−18.0000	+	31.1769i	−0.614152	+	1.06374i	−0.990533	+	0.137277i	$0.956165\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	10.0000	−	17.3205i	0.340404	−	0.589597i	−0.644104	−	0.764938i	$-0.722770\pi$
$863$	10.0000	−	17.3205i	0.340404	−	0.589597i	0.984508	+	0.175341i	$0.0561028\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	8.00000	+	13.8564i	0.271070	+	0.469506i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	7.00000	−	12.1244i	0.236373	−	0.409410i	−0.723298	−	0.690536i	$-0.757375\pi$
$877$	7.00000	−	12.1244i	0.236373	−	0.409410i	0.959671	+	0.281126i	$0.0907079\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−12.0000			−0.404290			−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000			−0.404290			−0.202145	+	0.979356i	$0.564791\pi$
$882$		0			0
$883$	−44.0000			−1.48072			−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000			−1.48072			−0.740359	+	0.672212i	$0.765344\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−28.0000	+	48.4974i	−0.940148	+	1.62838i	−0.174962	+	0.984575i	$0.555980\pi$
$887$	−28.0000	+	48.4974i	−0.940148	+	1.62838i	−0.765186	+	0.643809i	$0.777353\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	16.0000	+	27.7128i	0.535420	+	0.927374i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8.00000	+	13.8564i	0.266815	+	0.462137i
$900$		0			0
$901$	−12.0000	+	20.7846i	−0.399778	+	0.692436i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−22.0000	−	38.1051i	−0.730498	−	1.26526i	−0.956671	−	0.291172i	$-0.905955\pi$
$907$	−22.0000	−	38.1051i	−0.730498	−	1.26526i	0.226173	−	0.974087i	$-0.427379\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−36.0000			−1.19273			−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000			−1.19273			−0.596367	+	0.802712i	$0.703390\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$919$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	48.0000			1.57994
$924$		0			0
$925$	30.0000			0.986394
$926$		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
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$	−24.0000	+	41.5692i	−0.781548	+	1.35368i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−20.0000	+	34.6410i	−0.649913	+	1.12568i	0.333231	+	0.942845i	$0.391861\pi$
$947$	−20.0000	+	34.6410i	−0.649913	+	1.12568i	−0.983143	+	0.182836i	$0.941472\pi$
$948$		0			0
$949$	−16.0000	−	27.7128i	−0.519382	−	0.899596i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−16.5000	+	28.5788i	−0.532258	+	0.921898i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	16.0000			0.514525			0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000			0.514525			0.257263	+	0.966342i	$0.417179\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	14.0000	−	24.2487i	0.449281	−	0.778178i	−0.549058	−	0.835784i	$-0.685013\pi$
$971$	14.0000	−	24.2487i	0.449281	−	0.778178i	0.998339	+	0.0576061i	$0.0183467\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−17.0000	−	29.4449i	−0.543878	−	0.942025i	−0.998677	−	0.0514302i	$-0.983622\pi$
$977$	−17.0000	−	29.4449i	−0.543878	−	0.942025i	0.454798	−	0.890594i	$-0.349711\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−28.0000	−	48.4974i	−0.893061	−	1.54683i	−0.836186	−	0.548446i	$-0.815220\pi$
$983$	−28.0000	−	48.4974i	−0.893061	−	1.54683i	−0.0568755	−	0.998381i	$-0.518114\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	8.00000	−	13.8564i	0.254385	−	0.440608i
$990$		0			0
$991$	4.00000	+	6.92820i	0.127064	+	0.220082i	0.922538	−	0.385906i	$-0.126111\pi$
$991$	4.00000	+	6.92820i	0.127064	+	0.220082i	−0.795474	+	0.605988i	$0.792778\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	14.0000	+	24.2487i	0.443384	+	0.767964i	0.997938	−	0.0641836i	$-0.0204443\pi$
$997$	14.0000	+	24.2487i	0.443384	+	0.767964i	−0.554554	+	0.832148i	$0.687111\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.s.n.3313.1		2
3.2	odd	2		1176.2.q.c.961.1		2
7.2	even	3		3528.2.a.n.1.1		1
7.3	odd	6		3528.2.s.m.361.1		2
7.4	even	3	inner	3528.2.s.n.361.1		2
7.5	odd	6		3528.2.a.m.1.1		1
7.6	odd	2		3528.2.s.m.3313.1		2
12.11	even	2		2352.2.q.t.961.1		2
21.2	odd	6		1176.2.a.h.1.1	yes	1
21.5	even	6		1176.2.a.b.1.1	✓	1
21.11	odd	6		1176.2.q.c.361.1		2
21.17	even	6		1176.2.q.h.361.1		2
21.20	even	2		1176.2.q.h.961.1		2
28.19	even	6		7056.2.a.ba.1.1		1
28.23	odd	6		7056.2.a.bc.1.1		1
84.11	even	6		2352.2.q.t.1537.1		2
84.23	even	6		2352.2.a.h.1.1		1
84.47	odd	6		2352.2.a.r.1.1		1
84.59	odd	6		2352.2.q.h.1537.1		2
84.83	odd	2		2352.2.q.h.961.1		2
168.5	even	6		9408.2.a.cl.1.1		1
168.107	even	6		9408.2.a.ck.1.1		1
168.131	odd	6		9408.2.a.v.1.1		1
168.149	odd	6		9408.2.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.2.a.b.1.1	✓	1	21.5	even	6
1176.2.a.h.1.1	yes	1	21.2	odd	6
1176.2.q.c.361.1		2	21.11	odd	6
1176.2.q.c.961.1		2	3.2	odd	2
1176.2.q.h.361.1		2	21.17	even	6
1176.2.q.h.961.1		2	21.20	even	2
2352.2.a.h.1.1		1	84.23	even	6
2352.2.a.r.1.1		1	84.47	odd	6
2352.2.q.h.961.1		2	84.83	odd	2
2352.2.q.h.1537.1		2	84.59	odd	6
2352.2.q.t.961.1		2	12.11	even	2
2352.2.q.t.1537.1		2	84.11	even	6
3528.2.a.m.1.1		1	7.5	odd	6
3528.2.a.n.1.1		1	7.2	even	3
3528.2.s.m.361.1		2	7.3	odd	6
3528.2.s.m.3313.1		2	7.6	odd	2
3528.2.s.n.361.1		2	7.4	even	3	inner
3528.2.s.n.3313.1		2	1.1	even	1	trivial
7056.2.a.ba.1.1		1	28.19	even	6
7056.2.a.bc.1.1		1	28.23	odd	6
9408.2.a.u.1.1		1	168.149	odd	6
9408.2.a.v.1.1		1	168.131	odd	6
9408.2.a.ck.1.1		1	168.107	even	6
9408.2.a.cl.1.1		1	168.5	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)

\(f(q)\)	\(=\)	\(q+O(q^{10})\)	\(q+4.00000 q^{13} +(2.00000 + 3.46410i) q^{17} +(2.00000 - 3.46410i) q^{19} +(2.00000 - 3.46410i) q^{23} +(2.50000 + 4.33013i) q^{25} -2.00000 q^{29} +(-4.00000 - 6.92820i) q^{31} +(3.00000 - 5.19615i) q^{37} -12.0000 q^{41} +4.00000 q^{43} +(-4.00000 + 6.92820i) q^{47} +(3.00000 + 5.19615i) q^{53} +(6.00000 + 10.3923i) q^{59} +(-2.00000 + 3.46410i) q^{61} +(2.00000 + 3.46410i) q^{67} +12.0000 q^{71} +(-4.00000 - 6.92820i) q^{73} +(8.00000 - 13.8564i) q^{79} +4.00000 q^{83} +(2.00000 - 3.46410i) q^{89} +16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 8 q^{13} + 4 q^{17} + 4 q^{19} + 4 q^{23} + 5 q^{25} - 4 q^{29} - 8 q^{31} + 6 q^{37} - 24 q^{41} + 8 q^{43} - 8 q^{47} + 6 q^{53} + 12 q^{59} - 4 q^{61} + 4 q^{67} + 24 q^{71} - 8 q^{73} + 16 q^{79} + 8 q^{83} + 4 q^{89} + 32 q^{97}+O(q^{100}) \) 2 * q + 8 * q^13 + 4 * q^17 + 4 * q^19 + 4 * q^23 + 5 * q^25 - 4 * q^29 - 8 * q^31 + 6 * q^37 - 24 * q^41 + 8 * q^43 - 8 * q^47 + 6 * q^53 + 12 * q^59 - 4 * q^61 + 4 * q^67 + 24 * q^71 - 8 * q^73 + 16 * q^79 + 8 * q^83 + 4 * q^89 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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