LMFDB - Embedded newform 3528.2.s.l.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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 504)
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.l.3313.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{5} +O(q^{10})$	$q+(-0.500000 - 0.866025i) q^{5} +(-2.50000 + 4.33013i) q^{11} -2.00000 q^{13} +(3.00000 - 5.19615i) q^{17} +(1.00000 + 1.73205i) q^{19} +(3.00000 + 5.19615i) q^{23} +(2.00000 - 3.46410i) q^{25} -3.00000 q^{29} +(2.50000 - 4.33013i) q^{31} +(1.00000 + 1.73205i) q^{37} -8.00000 q^{41} -4.00000 q^{43} +(2.00000 + 3.46410i) q^{47} +(-4.50000 + 7.79423i) q^{53} +5.00000 q^{55} +(-1.50000 + 2.59808i) q^{59} +(-6.00000 - 10.3923i) q^{61} +(1.00000 + 1.73205i) q^{65} +(-1.00000 + 1.73205i) q^{67} -8.00000 q^{71} +(-7.00000 + 12.1244i) q^{73} +(-0.500000 - 0.866025i) q^{79} -17.0000 q^{83} -6.00000 q^{85} +(9.00000 + 15.5885i) q^{89} +(1.00000 - 1.73205i) q^{95} -3.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5}+O(q^{10}) $ 2 * q - q^5	$ 2 q - q^{5} - 5 q^{11} - 4 q^{13} + 6 q^{17} + 2 q^{19} + 6 q^{23} + 4 q^{25} - 6 q^{29} + 5 q^{31} + 2 q^{37} - 16 q^{41} - 8 q^{43} + 4 q^{47} - 9 q^{53} + 10 q^{55} - 3 q^{59} - 12 q^{61} + 2 q^{65} - 2 q^{67} - 16 q^{71} - 14 q^{73} - q^{79} - 34 q^{83} - 12 q^{85} + 18 q^{89} + 2 q^{95} - 6 q^{97}+O(q^{100}) $ 2 * q - q^5 - 5 * q^11 - 4 * q^13 + 6 * q^17 + 2 * q^19 + 6 * q^23 + 4 * q^25 - 6 * q^29 + 5 * q^31 + 2 * q^37 - 16 * q^41 - 8 * q^43 + 4 * q^47 - 9 * q^53 + 10 * q^55 - 3 * q^59 - 12 * q^61 + 2 * q^65 - 2 * q^67 - 16 * q^71 - 14 * q^73 - q^79 - 34 * q^83 - 12 * q^85 + 18 * q^89 + 2 * q^95 - 6 * 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
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i	0.732294	−	0.680989i	$-0.238450\pi$
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i	−0.955901	+	0.293691i	$0.905116\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.50000	+	4.33013i	−0.753778	+	1.30558i	0.192201	+	0.981356i	$0.438437\pi$
$11$	−2.50000	+	4.33013i	−0.753778	+	1.30558i	−0.945979	+	0.324227i	$0.894896\pi$
$12$		0			0
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.00000	−	5.19615i	0.727607	−	1.26025i	−0.230285	−	0.973123i	$-0.573966\pi$
$17$	3.00000	−	5.19615i	0.727607	−	1.26025i	0.957892	−	0.287129i	$-0.0927008\pi$
$18$		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
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	0.988436	+	0.151642i	$0.0484560\pi$
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	−0.362892	+	0.931831i	$0.618211\pi$
$24$		0			0
$25$	2.00000	−	3.46410i	0.400000	−	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.00000			−0.557086			−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000			−0.557086			−0.278543	+	0.960424i	$0.589851\pi$
$30$		0			0
$31$	2.50000	−	4.33013i	0.449013	−	0.777714i	−0.549309	−	0.835619i	$-0.685109\pi$
$31$	2.50000	−	4.33013i	0.449013	−	0.777714i	0.998322	+	0.0579057i	$0.0184423\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.00000	+	1.73205i	0.164399	+	0.284747i	0.936442	−	0.350823i	$-0.114098\pi$
$37$	1.00000	+	1.73205i	0.164399	+	0.284747i	−0.772043	+	0.635571i	$0.780765\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.00000			−1.24939			−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000			−1.24939			−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		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
$53$	−4.50000	+	7.79423i	−0.618123	+	1.07062i	0.371706	+	0.928351i	$0.378773\pi$
$53$	−4.50000	+	7.79423i	−0.618123	+	1.07062i	−0.989828	+	0.142269i	$0.954560\pi$
$54$		0			0
$55$	5.00000			0.674200
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.50000	+	2.59808i	−0.195283	+	0.338241i	−0.946993	−	0.321253i	$-0.895896\pi$
$59$	−1.50000	+	2.59808i	−0.195283	+	0.338241i	0.751710	+	0.659494i	$0.229229\pi$
$60$		0			0
$61$	−6.00000	−	10.3923i	−0.768221	−	1.33060i	−0.938527	−	0.345207i	$-0.887809\pi$
$61$	−6.00000	−	10.3923i	−0.768221	−	1.33060i	0.170305	−	0.985391i	$-0.445525\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.00000	+	1.73205i	0.124035	+	0.214834i
$66$		0			0
$67$	−1.00000	+	1.73205i	−0.122169	+	0.211604i	−0.920623	−	0.390453i	$-0.872318\pi$
$67$	−1.00000	+	1.73205i	−0.122169	+	0.211604i	0.798454	+	0.602056i	$0.205652\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		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
$79$	−0.500000	−	0.866025i	−0.0562544	−	0.0974355i	0.836527	−	0.547926i	$-0.184582\pi$
$79$	−0.500000	−	0.866025i	−0.0562544	−	0.0974355i	−0.892781	+	0.450490i	$0.851249\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−17.0000			−1.86599			−0.932996	−	0.359886i	$-0.882816\pi$
$83$	−17.0000			−1.86599			−0.932996	+	0.359886i	$0.882816\pi$
$84$		0			0
$85$	−6.00000			−0.650791
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.00000	+	15.5885i	0.953998	+	1.65237i	0.736644	+	0.676280i	$0.236409\pi$
$89$	9.00000	+	15.5885i	0.953998	+	1.65237i	0.217354	+	0.976093i	$0.430258\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.00000	−	1.73205i	0.102598	−	0.177705i
$96$		0			0
$97$	−3.00000			−0.304604			−0.152302	−	0.988334i	$-0.548669\pi$
$97$	−3.00000			−0.304604			−0.152302	+	0.988334i	$0.548669\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.00000	+	8.66025i	−0.497519	+	0.861727i	−0.999996	−	0.00286291i	$-0.999089\pi$
$101$	−5.00000	+	8.66025i	−0.497519	+	0.861727i	0.502477	+	0.864590i	$0.332422\pi$
$102$		0			0
$103$	4.00000	+	6.92820i	0.394132	+	0.682656i	0.992990	−	0.118199i	$-0.0377120\pi$
$103$	4.00000	+	6.92820i	0.394132	+	0.682656i	−0.598858	+	0.800855i	$0.704379\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.50000	−	2.59808i	−0.145010	−	0.251166i	0.784366	−	0.620298i	$-0.212988\pi$
$107$	−1.50000	−	2.59808i	−0.145010	−	0.251166i	−0.929377	+	0.369132i	$0.879655\pi$
$108$		0			0
$109$	−7.00000	+	12.1244i	−0.670478	+	1.16130i	0.307290	+	0.951616i	$0.400578\pi$
$109$	−7.00000	+	12.1244i	−0.670478	+	1.16130i	−0.977769	+	0.209687i	$0.932756\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.00000			0.752577			0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000			0.752577			0.376288	+	0.926503i	$0.377200\pi$
$114$		0			0
$115$	3.00000	−	5.19615i	0.279751	−	0.484544i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−7.00000	−	12.1244i	−0.636364	−	1.10221i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.00000			−0.804984
$126$		0			0
$127$	19.0000			1.68598			0.842989	−	0.537931i	$-0.180794\pi$
$127$	19.0000			1.68598			0.842989	+	0.537931i	$0.180794\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.50000	+	14.7224i	0.742648	+	1.28630i	0.951285	+	0.308312i	$0.0997640\pi$
$131$	8.50000	+	14.7224i	0.742648	+	1.28630i	−0.208637	+	0.977993i	$0.566903\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.00000	+	5.19615i	−0.256307	+	0.443937i	−0.965250	−	0.261329i	$-0.915839\pi$
$137$	−3.00000	+	5.19615i	−0.256307	+	0.443937i	0.708942	+	0.705266i	$0.249173\pi$
$138$		0			0
$139$	−10.0000			−0.848189			−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000			−0.848189			−0.424094	+	0.905618i	$0.639408\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	5.00000	−	8.66025i	0.418121	−	0.724207i
$144$		0			0
$145$	1.50000	+	2.59808i	0.124568	+	0.215758i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.00000	+	5.19615i	0.245770	+	0.425685i	0.962348	−	0.271821i	$-0.0876260\pi$
$149$	3.00000	+	5.19615i	0.245770	+	0.425685i	−0.716578	+	0.697507i	$0.754293\pi$
$150$		0			0
$151$	7.50000	−	12.9904i	0.610341	−	1.05714i	−0.380841	−	0.924640i	$-0.624366\pi$
$151$	7.50000	−	12.9904i	0.610341	−	1.05714i	0.991183	−	0.132502i	$-0.0423010\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.00000			−0.401610
$156$		0			0
$157$	6.00000	−	10.3923i	0.478852	−	0.829396i	−0.520854	−	0.853646i	$-0.674386\pi$
$157$	6.00000	−	10.3923i	0.478852	−	0.829396i	0.999706	+	0.0242497i	$0.00771967\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	9.00000	+	15.5885i	0.704934	+	1.22098i	0.966715	+	0.255855i	$0.0823569\pi$
$163$	9.00000	+	15.5885i	0.704934	+	1.22098i	−0.261781	+	0.965127i	$0.584310\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−22.0000			−1.70241			−0.851206	−	0.524832i	$-0.824128\pi$
$167$	−22.0000			−1.70241			−0.851206	+	0.524832i	$0.824128\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.00000	−	5.19615i	−0.228086	−	0.395056i	0.729155	−	0.684349i	$-0.239913\pi$
$173$	−3.00000	−	5.19615i	−0.228086	−	0.395056i	−0.957241	+	0.289292i	$0.906580\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.0000	+	17.3205i	−0.747435	+	1.29460i	0.201613	+	0.979465i	$0.435382\pi$
$179$	−10.0000	+	17.3205i	−0.747435	+	1.29460i	−0.949048	+	0.315130i	$0.897952\pi$
$180$		0			0
$181$	20.0000			1.48659			0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000			1.48659			0.743294	+	0.668965i	$0.233262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.00000	−	1.73205i	0.0735215	−	0.127343i
$186$		0			0
$187$	15.0000	+	25.9808i	1.09691	+	1.89990i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.00000	+	3.46410i	0.144715	+	0.250654i	0.929267	−	0.369410i	$-0.120440\pi$
$191$	2.00000	+	3.46410i	0.144715	+	0.250654i	−0.784552	+	0.620063i	$0.787107\pi$
$192$		0			0
$193$	−0.500000	+	0.866025i	−0.0359908	+	0.0623379i	−0.883460	−	0.468507i	$-0.844792\pi$
$193$	−0.500000	+	0.866025i	−0.0359908	+	0.0623379i	0.847469	+	0.530845i	$0.178125\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−18.0000			−1.28245			−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000			−1.28245			−0.641223	+	0.767354i	$0.721573\pi$
$198$		0			0
$199$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$199$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$200$		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
$209$	−10.0000			−0.691714
$210$		0			0
$211$	24.0000			1.65223			0.826114	−	0.563503i	$-0.190547\pi$
$211$	24.0000			1.65223			0.826114	+	0.563503i	$0.190547\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.00000	+	3.46410i	0.136399	+	0.236250i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.00000	+	10.3923i	−0.403604	+	0.699062i
$222$		0			0
$223$	−3.00000			−0.200895			−0.100447	−	0.994942i	$-0.532027\pi$
$223$	−3.00000			−0.200895			−0.100447	+	0.994942i	$0.532027\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.50000	−	6.06218i	0.232303	−	0.402361i	−0.726182	−	0.687502i	$-0.758707\pi$
$227$	3.50000	−	6.06218i	0.232303	−	0.402361i	0.958485	+	0.285141i	$0.0920405\pi$
$228$		0			0
$229$	−10.0000	−	17.3205i	−0.660819	−	1.14457i	−0.980401	−	0.197013i	$-0.936876\pi$
$229$	−10.0000	−	17.3205i	−0.660819	−	1.14457i	0.319582	−	0.947559i	$-0.396457\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	11.0000	+	19.0526i	0.720634	+	1.24817i	0.960746	+	0.277429i	$0.0894825\pi$
$233$	11.0000	+	19.0526i	0.720634	+	1.24817i	−0.240112	+	0.970745i	$0.577184\pi$
$234$		0			0
$235$	2.00000	−	3.46410i	0.130466	−	0.225973i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.00000			0.129369			0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000			0.129369			0.0646846	+	0.997906i	$0.479396\pi$
$240$		0			0
$241$	−8.50000	+	14.7224i	−0.547533	+	0.948355i	0.450910	+	0.892570i	$0.351100\pi$
$241$	−8.50000	+	14.7224i	−0.547533	+	0.948355i	−0.998443	+	0.0557856i	$0.982234\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−2.00000	−	3.46410i	−0.127257	−	0.220416i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	15.0000			0.946792			0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000			0.946792			0.473396	+	0.880850i	$0.343028\pi$
$252$		0			0
$253$	−30.0000			−1.88608
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.00000	+	1.73205i	0.0623783	+	0.108042i	0.895528	−	0.445005i	$-0.146798\pi$
$257$	1.00000	+	1.73205i	0.0623783	+	0.108042i	−0.833150	+	0.553047i	$0.813465\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.0000	+	20.7846i	−0.739952	+	1.28163i	0.212565	+	0.977147i	$0.431818\pi$
$263$	−12.0000	+	20.7846i	−0.739952	+	1.28163i	−0.952517	+	0.304487i	$0.901515\pi$
$264$		0			0
$265$	9.00000			0.552866
$266$		0			0
$267$		0			0
$268$		0			0
$269$	12.5000	−	21.6506i	0.762138	−	1.32006i	−0.179608	−	0.983738i	$-0.557483\pi$
$269$	12.5000	−	21.6506i	0.762138	−	1.32006i	0.941746	−	0.336324i	$-0.109184\pi$
$270$		0			0
$271$	−7.50000	−	12.9904i	−0.455593	−	0.789109i	0.543130	−	0.839649i	$-0.317239\pi$
$271$	−7.50000	−	12.9904i	−0.455593	−	0.789109i	−0.998722	+	0.0505395i	$0.983906\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	10.0000	+	17.3205i	0.603023	+	1.04447i
$276$		0			0
$277$	−14.0000	+	24.2487i	−0.841178	+	1.45696i	0.0477206	+	0.998861i	$0.484804\pi$
$277$	−14.0000	+	24.2487i	−0.841178	+	1.45696i	−0.888899	+	0.458103i	$0.848529\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	−0.800439	−	0.599414i	$-0.795400\pi$
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	0.919327	+	0.393494i	$0.128734\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−9.50000	−	16.4545i	−0.558824	−	0.967911i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−21.0000			−1.22683			−0.613417	−	0.789760i	$-0.710205\pi$
$293$	−21.0000			−1.22683			−0.613417	+	0.789760i	$0.710205\pi$
$294$		0			0
$295$	3.00000			0.174667
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.00000	−	10.3923i	−0.346989	−	0.601003i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6.00000	+	10.3923i	−0.343559	+	0.595062i
$306$		0			0
$307$	−34.0000			−1.94048			−0.970241	−	0.242140i	$-0.922151\pi$
$307$	−34.0000			−1.94048			−0.970241	+	0.242140i	$0.922151\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.00000	+	3.46410i	−0.113410	+	0.196431i	−0.917143	−	0.398559i	$-0.869511\pi$
$311$	−2.00000	+	3.46410i	−0.113410	+	0.196431i	0.803733	+	0.594990i	$0.202844\pi$
$312$		0			0
$313$	−11.5000	−	19.9186i	−0.650018	−	1.12586i	−0.983118	−	0.182973i	$-0.941428\pi$
$313$	−11.5000	−	19.9186i	−0.650018	−	1.12586i	0.333099	−	0.942892i	$-0.391906\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.50000	−	7.79423i	−0.252745	−	0.437767i	0.711535	−	0.702650i	$-0.248000\pi$
$317$	−4.50000	−	7.79423i	−0.252745	−	0.437767i	−0.964281	+	0.264883i	$0.914667\pi$
$318$		0			0
$319$	7.50000	−	12.9904i	0.419919	−	0.727322i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	12.0000			0.667698
$324$		0			0
$325$	−4.00000	+	6.92820i	−0.221880	+	0.384308i
$326$		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
$335$	2.00000			0.109272
$336$		0			0
$337$	−19.0000			−1.03500			−0.517498	−	0.855684i	$-0.673136\pi$
$337$	−19.0000			−1.03500			−0.517498	+	0.855684i	$0.673136\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	12.5000	+	21.6506i	0.676913	+	1.17245i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	−0.658824	−	0.752297i	$-0.728946\pi$
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	0.980921	+	0.194409i	$0.0622790\pi$
$348$		0			0
$349$	−6.00000			−0.321173			−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000			−0.321173			−0.160586	+	0.987022i	$0.551338\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.00000	+	6.92820i	−0.212899	+	0.368751i	−0.952620	−	0.304162i	$-0.901624\pi$
$353$	−4.00000	+	6.92820i	−0.212899	+	0.368751i	0.739722	+	0.672913i	$0.234957\pi$
$354$		0			0
$355$	4.00000	+	6.92820i	0.212298	+	0.367711i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3.00000	−	5.19615i	−0.158334	−	0.274242i	0.775934	−	0.630814i	$-0.217279\pi$
$359$	−3.00000	−	5.19615i	−0.158334	−	0.274242i	−0.934268	+	0.356572i	$0.883946\pi$
$360$		0			0
$361$	7.50000	−	12.9904i	0.394737	−	0.683704i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	14.0000			0.732793
$366$		0			0
$367$	3.50000	−	6.06218i	0.182699	−	0.316443i	−0.760100	−	0.649806i	$-0.774850\pi$
$367$	3.50000	−	6.06218i	0.182699	−	0.316443i	0.942799	+	0.333363i	$0.108183\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−2.00000	−	3.46410i	−0.103556	−	0.179364i	0.809591	−	0.586994i	$-0.199689\pi$
$373$	−2.00000	−	3.46410i	−0.103556	−	0.179364i	−0.913147	+	0.407630i	$0.866355\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6.00000			0.309016
$378$		0			0
$379$	−4.00000			−0.205466			−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000			−0.205466			−0.102733	+	0.994709i	$0.532759\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.00000	−	15.5885i	−0.459879	−	0.796533i	0.539076	−	0.842257i	$-0.318774\pi$
$383$	−9.00000	−	15.5885i	−0.459879	−	0.796533i	−0.998954	+	0.0457244i	$0.985440\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7.00000	−	12.1244i	0.354914	−	0.614729i	−0.632189	−	0.774814i	$-0.717843\pi$
$389$	7.00000	−	12.1244i	0.354914	−	0.614729i	0.987103	+	0.160085i	$0.0511768\pi$
$390$		0			0
$391$	36.0000			1.82060
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−0.500000	+	0.866025i	−0.0251577	+	0.0435745i
$396$		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
$401$	−12.0000	−	20.7846i	−0.599251	−	1.03793i	−0.992932	−	0.118686i	$-0.962132\pi$
$401$	−12.0000	−	20.7846i	−0.599251	−	1.03793i	0.393680	−	0.919247i	$-0.371202\pi$
$402$		0			0
$403$	−5.00000	+	8.66025i	−0.249068	+	0.431398i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−10.0000			−0.495682
$408$		0			0
$409$	−2.50000	+	4.33013i	−0.123617	+	0.214111i	−0.921192	−	0.389109i	$-0.872783\pi$
$409$	−2.50000	+	4.33013i	−0.123617	+	0.214111i	0.797574	+	0.603220i	$0.206116\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	8.50000	+	14.7224i	0.417249	+	0.722696i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.00000			0.195413			0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000			0.195413			0.0977064	+	0.995215i	$0.468849\pi$
$420$		0			0
$421$	16.0000			0.779792			0.389896	−	0.920859i	$-0.372511\pi$
$421$	16.0000			0.779792			0.389896	+	0.920859i	$0.372511\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−12.0000	−	20.7846i	−0.582086	−	1.00820i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	16.0000	−	27.7128i	0.770693	−	1.33488i	−0.166491	−	0.986043i	$-0.553244\pi$
$431$	16.0000	−	27.7128i	0.770693	−	1.33488i	0.937184	−	0.348836i	$-0.113423\pi$
$432$		0			0
$433$	−10.0000			−0.480569			−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000			−0.480569			−0.240285	+	0.970702i	$0.577241\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.00000	+	10.3923i	−0.287019	+	0.497131i
$438$		0			0
$439$	16.5000	+	28.5788i	0.787502	+	1.36399i	0.927493	+	0.373841i	$0.121959\pi$
$439$	16.5000	+	28.5788i	0.787502	+	1.36399i	−0.139991	+	0.990153i	$0.544707\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.50000	−	7.79423i	−0.213801	−	0.370315i	0.739100	−	0.673596i	$-0.235251\pi$
$443$	−4.50000	−	7.79423i	−0.213801	−	0.370315i	−0.952901	+	0.303281i	$0.901918\pi$
$444$		0			0
$445$	9.00000	−	15.5885i	0.426641	−	0.738964i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	10.0000			0.471929			0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000			0.471929			0.235965	+	0.971762i	$0.424175\pi$
$450$		0			0
$451$	20.0000	−	34.6410i	0.941763	−	1.63118i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	14.5000	+	25.1147i	0.678281	+	1.17482i	0.975498	+	0.220008i	$0.0706083\pi$
$457$	14.5000	+	25.1147i	0.678281	+	1.17482i	−0.297217	+	0.954810i	$0.596058\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	30.0000			1.39724			0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000			1.39724			0.698620	+	0.715493i	$0.253798\pi$
$462$		0			0
$463$	20.0000			0.929479			0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000			0.929479			0.464739	+	0.885448i	$0.346148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	14.0000	+	24.2487i	0.647843	+	1.12210i	0.983637	+	0.180161i	$0.0576619\pi$
$467$	14.0000	+	24.2487i	0.647843	+	1.12210i	−0.335794	+	0.941935i	$0.609005\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10.0000	−	17.3205i	0.459800	−	0.796398i
$474$		0			0
$475$	8.00000			0.367065
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.0000	+	20.7846i	−0.548294	+	0.949673i	0.450098	+	0.892979i	$0.351389\pi$
$479$	−12.0000	+	20.7846i	−0.548294	+	0.949673i	−0.998392	+	0.0566937i	$0.981944\pi$
$480$		0			0
$481$	−2.00000	−	3.46410i	−0.0911922	−	0.157949i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.50000	+	2.59808i	0.0681115	+	0.117973i
$486$		0			0
$487$	12.5000	−	21.6506i	0.566429	−	0.981084i	−0.430486	−	0.902597i	$-0.641658\pi$
$487$	12.5000	−	21.6506i	0.566429	−	0.981084i	0.996915	−	0.0784867i	$-0.0250088\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.00000			0.406164			0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000			0.406164			0.203082	+	0.979162i	$0.434904\pi$
$492$		0			0
$493$	−9.00000	+	15.5885i	−0.405340	+	0.702069i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	17.0000	+	29.4449i	0.761025	+	1.31813i	0.942323	+	0.334705i	$0.108637\pi$
$499$	17.0000	+	29.4449i	0.761025	+	1.31813i	−0.181298	+	0.983428i	$0.558030\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−10.0000			−0.445878			−0.222939	−	0.974832i	$-0.571565\pi$
$503$	−10.0000			−0.445878			−0.222939	+	0.974832i	$0.571565\pi$
$504$		0			0
$505$	10.0000			0.444994
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6.50000	+	11.2583i	0.288107	+	0.499017i	0.973358	−	0.229291i	$-0.0736406\pi$
$509$	6.50000	+	11.2583i	0.288107	+	0.499017i	−0.685251	+	0.728307i	$0.740307\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.00000	−	6.92820i	0.176261	−	0.305293i
$516$		0			0
$517$	−20.0000			−0.879599
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−12.0000	+	20.7846i	−0.525730	+	0.910590i	0.473821	+	0.880621i	$0.342874\pi$
$521$	−12.0000	+	20.7846i	−0.525730	+	0.910590i	−0.999551	+	0.0299693i	$0.990459\pi$
$522$		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
$527$	−15.0000	−	25.9808i	−0.653410	−	1.13174i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	16.0000			0.693037
$534$		0			0
$535$	−1.50000	+	2.59808i	−0.0648507	+	0.112325i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−16.0000	−	27.7128i	−0.687894	−	1.19147i	−0.972518	−	0.232828i	$-0.925202\pi$
$541$	−16.0000	−	27.7128i	−0.687894	−	1.19147i	0.284624	−	0.958639i	$-0.408131\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	14.0000			0.599694
$546$		0			0
$547$	−18.0000			−0.769624			−0.384812	−	0.922995i	$-0.625734\pi$
$547$	−18.0000			−0.769624			−0.384812	+	0.922995i	$0.625734\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3.00000	−	5.19615i	−0.127804	−	0.221364i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	18.5000	−	32.0429i	0.783870	−	1.35770i	−0.145802	−	0.989314i	$-0.546576\pi$
$557$	18.5000	−	32.0429i	0.783870	−	1.35770i	0.929672	−	0.368389i	$-0.120091\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.50000	−	4.33013i	0.105362	−	0.182493i	−0.808524	−	0.588463i	$-0.799733\pi$
$563$	2.50000	−	4.33013i	0.105362	−	0.182493i	0.913886	+	0.405970i	$0.133066\pi$
$564$		0			0
$565$	−4.00000	−	6.92820i	−0.168281	−	0.291472i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−7.00000	−	12.1244i	−0.293455	−	0.508279i	0.681169	−	0.732126i	$-0.261472\pi$
$569$	−7.00000	−	12.1244i	−0.293455	−	0.508279i	−0.974624	+	0.223847i	$0.928139\pi$
$570$		0			0
$571$	−20.0000	+	34.6410i	−0.836974	+	1.44968i	0.0554391	+	0.998462i	$0.482344\pi$
$571$	−20.0000	+	34.6410i	−0.836974	+	1.44968i	−0.892413	+	0.451219i	$0.850989\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	24.0000			1.00087
$576$		0			0
$577$	3.50000	−	6.06218i	0.145707	−	0.252372i	−0.783930	−	0.620850i	$-0.786788\pi$
$577$	3.50000	−	6.06218i	0.145707	−	0.252372i	0.929636	+	0.368478i	$0.120121\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−22.5000	−	38.9711i	−0.931855	−	1.61402i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	21.0000			0.866763			0.433381	−	0.901211i	$-0.357320\pi$
$587$	21.0000			0.866763			0.433381	+	0.901211i	$0.357320\pi$
$588$		0			0
$589$	10.0000			0.412043
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−13.0000	−	22.5167i	−0.533846	−	0.924648i	−0.999218	−	0.0395334i	$-0.987413\pi$
$593$	−13.0000	−	22.5167i	−0.533846	−	0.924648i	0.465372	−	0.885115i	$-0.345920\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	3.00000	−	5.19615i	0.122577	−	0.212309i	−0.798206	−	0.602384i	$-0.794218\pi$
$599$	3.00000	−	5.19615i	0.122577	−	0.212309i	0.920783	+	0.390075i	$0.127551\pi$
$600$		0			0
$601$	1.00000			0.0407909			0.0203954	−	0.999792i	$-0.493507\pi$
$601$	1.00000			0.0407909			0.0203954	+	0.999792i	$0.493507\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.00000	+	12.1244i	−0.284590	+	0.492925i
$606$		0			0
$607$	−12.5000	−	21.6506i	−0.507359	−	0.878772i	−0.999964	−	0.00851879i	$-0.997288\pi$
$607$	−12.5000	−	21.6506i	−0.507359	−	0.878772i	0.492604	−	0.870253i	$-0.336045\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−4.00000	−	6.92820i	−0.161823	−	0.280285i
$612$		0			0
$613$	3.00000	−	5.19615i	0.121169	−	0.209871i	−0.799060	−	0.601251i	$-0.794669\pi$
$613$	3.00000	−	5.19615i	0.121169	−	0.209871i	0.920229	+	0.391381i	$0.128002\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−20.0000			−0.805170			−0.402585	−	0.915383i	$-0.631888\pi$
$617$	−20.0000			−0.805170			−0.402585	+	0.915383i	$0.631888\pi$
$618$		0			0
$619$	2.00000	−	3.46410i	0.0803868	−	0.139234i	−0.823029	−	0.567999i	$-0.807718\pi$
$619$	2.00000	−	3.46410i	0.0803868	−	0.139234i	0.903416	+	0.428765i	$0.141051\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−5.50000	−	9.52628i	−0.220000	−	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	12.0000			0.478471
$630$		0			0
$631$	−9.00000			−0.358284			−0.179142	−	0.983823i	$-0.557332\pi$
$631$	−9.00000			−0.358284			−0.179142	+	0.983823i	$0.557332\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−9.50000	−	16.4545i	−0.376996	−	0.652976i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.00000	+	10.3923i	−0.236986	+	0.410471i	−0.959848	−	0.280521i	$-0.909493\pi$
$641$	−6.00000	+	10.3923i	−0.236986	+	0.410471i	0.722862	+	0.690992i	$0.242826\pi$
$642$		0			0
$643$	−22.0000			−0.867595			−0.433798	−	0.901010i	$-0.642827\pi$
$643$	−22.0000			−0.867595			−0.433798	+	0.901010i	$0.642827\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	14.0000	−	24.2487i	0.550397	−	0.953315i	−0.447849	−	0.894109i	$-0.647810\pi$
$647$	14.0000	−	24.2487i	0.550397	−	0.953315i	0.998246	−	0.0592060i	$-0.0188569\pi$
$648$		0			0
$649$	−7.50000	−	12.9904i	−0.294401	−	0.509917i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	14.5000	+	25.1147i	0.567429	+	0.982816i	0.996819	+	0.0796966i	$0.0253951\pi$
$653$	14.5000	+	25.1147i	0.567429	+	0.982816i	−0.429390	+	0.903119i	$0.641272\pi$
$654$		0			0
$655$	8.50000	−	14.7224i	0.332122	−	0.575253i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−36.0000			−1.40236			−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000			−1.40236			−0.701180	+	0.712984i	$0.747343\pi$
$660$		0			0
$661$	−20.0000	+	34.6410i	−0.777910	+	1.34738i	0.155235	+	0.987878i	$0.450387\pi$
$661$	−20.0000	+	34.6410i	−0.777910	+	1.34738i	−0.933144	+	0.359502i	$0.882947\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−9.00000	−	15.5885i	−0.348481	−	0.603587i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	60.0000			2.31627
$672$		0			0
$673$	1.00000			0.0385472			0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000			0.0385472			0.0192736	+	0.999814i	$0.493865\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−22.5000	−	38.9711i	−0.864745	−	1.49778i	−0.867300	−	0.497786i	$-0.834147\pi$
$677$	−22.5000	−	38.9711i	−0.864745	−	1.49778i	0.00255466	−	0.999997i	$-0.499187\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	10.5000	−	18.1865i	0.401771	−	0.695888i	−0.592168	−	0.805814i	$-0.701728\pi$
$683$	10.5000	−	18.1865i	0.401771	−	0.695888i	0.993940	+	0.109926i	$0.0350613\pi$
$684$		0			0
$685$	6.00000			0.229248
$686$		0			0
$687$		0			0
$688$		0			0
$689$	9.00000	−	15.5885i	0.342873	−	0.593873i
$690$		0			0
$691$	−4.00000	−	6.92820i	−0.152167	−	0.263561i	0.779857	−	0.625958i	$-0.215292\pi$
$691$	−4.00000	−	6.92820i	−0.152167	−	0.263561i	−0.932024	+	0.362397i	$0.881959\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5.00000	+	8.66025i	0.189661	+	0.328502i
$696$		0			0
$697$	−24.0000	+	41.5692i	−0.909065	+	1.57455i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	45.0000			1.69963			0.849813	−	0.527084i	$-0.176715\pi$
$701$	45.0000			1.69963			0.849813	+	0.527084i	$0.176715\pi$
$702$		0			0
$703$	−2.00000	+	3.46410i	−0.0754314	+	0.130651i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	9.00000	+	15.5885i	0.338002	+	0.585437i	0.984057	−	0.177854i	$-0.0569156\pi$
$709$	9.00000	+	15.5885i	0.338002	+	0.585437i	−0.646055	+	0.763291i	$0.723582\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	30.0000			1.12351
$714$		0			0
$715$	−10.0000			−0.373979
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−5.00000	−	8.66025i	−0.186469	−	0.322973i	0.757602	−	0.652717i	$-0.226371\pi$
$719$	−5.00000	−	8.66025i	−0.186469	−	0.322973i	−0.944070	+	0.329744i	$0.893038\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6.00000	+	10.3923i	−0.222834	+	0.385961i
$726$		0			0
$727$	35.0000			1.29808			0.649039	−	0.760755i	$-0.275171\pi$
$727$	35.0000			1.29808			0.649039	+	0.760755i	$0.275171\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12.0000	+	20.7846i	−0.443836	+	0.768747i
$732$		0			0
$733$	10.0000	+	17.3205i	0.369358	+	0.639748i	0.989465	−	0.144770i	$-0.0462441\pi$
$733$	10.0000	+	17.3205i	0.369358	+	0.639748i	−0.620107	+	0.784517i	$0.712911\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.00000	−	8.66025i	−0.184177	−	0.319005i
$738$		0			0
$739$	−17.0000	+	29.4449i	−0.625355	+	1.08315i	0.363117	+	0.931744i	$0.381713\pi$
$739$	−17.0000	+	29.4449i	−0.625355	+	1.08315i	−0.988472	+	0.151403i	$0.951621\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	2.00000			0.0733729			0.0366864	−	0.999327i	$-0.488320\pi$
$743$	2.00000			0.0733729			0.0366864	+	0.999327i	$0.488320\pi$
$744$		0			0
$745$	3.00000	−	5.19615i	0.109911	−	0.190372i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	14.5000	+	25.1147i	0.529113	+	0.916450i	0.999424	+	0.0339490i	$0.0108084\pi$
$751$	14.5000	+	25.1147i	0.529113	+	0.916450i	−0.470311	+	0.882501i	$0.655858\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−15.0000			−0.545906
$756$		0			0
$757$	−20.0000			−0.726912			−0.363456	−	0.931611i	$-0.618403\pi$
$757$	−20.0000			−0.726912			−0.363456	+	0.931611i	$0.618403\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.00000	−	15.5885i	−0.326250	−	0.565081i	0.655515	−	0.755182i	$-0.272452\pi$
$761$	−9.00000	−	15.5885i	−0.326250	−	0.565081i	−0.981764	+	0.190101i	$0.939118\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.00000	−	5.19615i	0.108324	−	0.187622i
$768$		0			0
$769$	19.0000			0.685158			0.342579	−	0.939489i	$-0.388700\pi$
$769$	19.0000			0.685158			0.342579	+	0.939489i	$0.388700\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	19.0000	−	32.9090i	0.683383	−	1.18365i	−0.290560	−	0.956857i	$-0.593841\pi$
$773$	19.0000	−	32.9090i	0.683383	−	1.18365i	0.973942	−	0.226796i	$-0.0728252\pi$
$774$		0			0
$775$	−10.0000	−	17.3205i	−0.359211	−	0.622171i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−8.00000	−	13.8564i	−0.286630	−	0.496457i
$780$		0			0
$781$	20.0000	−	34.6410i	0.715656	−	1.23955i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−12.0000			−0.428298
$786$		0			0
$787$	−20.0000	+	34.6410i	−0.712923	+	1.23482i	0.250832	+	0.968031i	$0.419296\pi$
$787$	−20.0000	+	34.6410i	−0.712923	+	1.23482i	−0.963755	+	0.266788i	$0.914038\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	12.0000	+	20.7846i	0.426132	+	0.738083i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	33.0000			1.16892			0.584460	−	0.811423i	$-0.301306\pi$
$797$	33.0000			1.16892			0.584460	+	0.811423i	$0.301306\pi$
$798$		0			0
$799$	24.0000			0.849059
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−35.0000	−	60.6218i	−1.23512	−	2.13930i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	14.0000	−	24.2487i	0.492214	−	0.852539i	−0.507746	−	0.861507i	$-0.669521\pi$
$809$	14.0000	−	24.2487i	0.492214	−	0.852539i	0.999960	+	0.00896753i	$0.00285449\pi$
$810$		0			0
$811$	−34.0000			−1.19390			−0.596951	−	0.802278i	$-0.703621\pi$
$811$	−34.0000			−1.19390			−0.596951	+	0.802278i	$0.703621\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	9.00000	−	15.5885i	0.315256	−	0.546040i
$816$		0			0
$817$	−4.00000	−	6.92820i	−0.139942	−	0.242387i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−14.5000	−	25.1147i	−0.506053	−	0.876510i	−0.999975	−	0.00700413i	$-0.997770\pi$
$821$	−14.5000	−	25.1147i	−0.506053	−	0.876510i	0.493922	−	0.869506i	$-0.335563\pi$
$822$		0			0
$823$	12.0000	−	20.7846i	0.418294	−	0.724506i	−0.577474	−	0.816409i	$-0.695962\pi$
$823$	12.0000	−	20.7846i	0.418294	−	0.724506i	0.995768	+	0.0919029i	$0.0292950\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	41.0000			1.42571			0.712855	−	0.701312i	$-0.247402\pi$
$827$	41.0000			1.42571			0.712855	+	0.701312i	$0.247402\pi$
$828$		0			0
$829$	13.0000	−	22.5167i	0.451509	−	0.782036i	−0.546971	−	0.837151i	$-0.684219\pi$
$829$	13.0000	−	22.5167i	0.451509	−	0.782036i	0.998480	+	0.0551154i	$0.0175527\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	11.0000	+	19.0526i	0.380671	+	0.659341i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	30.0000			1.03572			0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000			1.03572			0.517858	+	0.855467i	$0.326730\pi$
$840$		0			0
$841$	−20.0000			−0.689655
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.50000	+	7.79423i	0.154805	+	0.268130i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.00000	+	10.3923i	−0.205677	+	0.356244i
$852$		0			0
$853$	22.0000			0.753266			0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000			0.753266			0.376633	+	0.926363i	$0.377082\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6.00000	−	10.3923i	0.204956	−	0.354994i	−0.745163	−	0.666883i	$-0.767628\pi$
$857$	6.00000	−	10.3923i	0.204956	−	0.354994i	0.950119	+	0.311888i	$0.100962\pi$
$858$		0			0
$859$	16.0000	+	27.7128i	0.545913	+	0.945549i	0.998549	+	0.0538535i	$0.0171504\pi$
$859$	16.0000	+	27.7128i	0.545913	+	0.945549i	−0.452636	+	0.891695i	$0.649516\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	19.0000	+	32.9090i	0.646768	+	1.12023i	0.983890	+	0.178774i	$0.0572129\pi$
$863$	19.0000	+	32.9090i	0.646768	+	1.12023i	−0.337123	+	0.941461i	$0.609454\pi$
$864$		0			0
$865$	−3.00000	+	5.19615i	−0.102003	+	0.176674i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	5.00000			0.169613
$870$		0			0
$871$	2.00000	−	3.46410i	0.0677674	−	0.117377i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−12.0000	−	20.7846i	−0.405211	−	0.701846i	0.589135	−	0.808035i	$-0.299469\pi$
$877$	−12.0000	−	20.7846i	−0.405211	−	0.701846i	−0.994346	+	0.106188i	$0.966135\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$	28.0000			0.942275			0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000			0.942275			0.471138	+	0.882060i	$0.343844\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	13.0000	+	22.5167i	0.436497	+	0.756035i	0.997417	−	0.0718351i	$-0.0228855\pi$
$887$	13.0000	+	22.5167i	0.436497	+	0.756035i	−0.560919	+	0.827871i	$0.689552\pi$
$888$		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
$895$	20.0000			0.668526
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−7.50000	+	12.9904i	−0.250139	+	0.433253i
$900$		0			0
$901$	27.0000	+	46.7654i	0.899500	+	1.55798i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10.0000	−	17.3205i	−0.332411	−	0.575753i
$906$		0			0
$907$	11.0000	−	19.0526i	0.365249	−	0.632630i	−0.623567	−	0.781770i	$-0.714317\pi$
$907$	11.0000	−	19.0526i	0.365249	−	0.632630i	0.988816	+	0.149140i	$0.0476505\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−24.0000			−0.795155			−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000			−0.795155			−0.397578	+	0.917568i	$0.630149\pi$
$912$		0			0
$913$	42.5000	−	73.6122i	1.40654	−	2.43621i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−22.0000	−	38.1051i	−0.725713	−	1.25697i	−0.958680	−	0.284487i	$-0.908177\pi$
$919$	−22.0000	−	38.1051i	−0.725713	−	1.25697i	0.232967	−	0.972485i	$-0.425157\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	16.0000			0.526646
$924$		0			0
$925$	8.00000			0.263038
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−4.00000	−	6.92820i	−0.131236	−	0.227307i	0.792917	−	0.609329i	$-0.208561\pi$
$929$	−4.00000	−	6.92820i	−0.131236	−	0.227307i	−0.924153	+	0.382022i	$0.875228\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	15.0000	−	25.9808i	0.490552	−	0.849662i
$936$		0			0
$937$	5.00000			0.163343			0.0816714	−	0.996659i	$-0.473974\pi$
$937$	5.00000			0.163343			0.0816714	+	0.996659i	$0.473974\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−10.5000	+	18.1865i	−0.342290	+	0.592864i	−0.984858	−	0.173365i	$-0.944536\pi$
$941$	−10.5000	+	18.1865i	−0.342290	+	0.592864i	0.642567	+	0.766229i	$0.277869\pi$
$942$		0			0
$943$	−24.0000	−	41.5692i	−0.781548	−	1.35368i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−14.0000	−	24.2487i	−0.454939	−	0.787977i	0.543746	−	0.839250i	$-0.317006\pi$
$947$	−14.0000	−	24.2487i	−0.454939	−	0.787977i	−0.998685	+	0.0512727i	$0.983672\pi$
$948$		0			0
$949$	14.0000	−	24.2487i	0.454459	−	0.787146i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	60.0000			1.94359			0.971795	−	0.235826i	$-0.0757795\pi$
$953$	60.0000			1.94359			0.971795	+	0.235826i	$0.0757795\pi$
$954$		0			0
$955$	2.00000	−	3.46410i	0.0647185	−	0.112096i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	3.00000	+	5.19615i	0.0967742	+	0.167618i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.00000			0.0321911
$966$		0			0
$967$	17.0000			0.546683			0.273342	−	0.961917i	$-0.411871\pi$
$967$	17.0000			0.546683			0.273342	+	0.961917i	$0.411871\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	7.50000	+	12.9904i	0.240686	+	0.416881i	0.960910	−	0.276861i	$-0.0892941\pi$
$971$	7.50000	+	12.9904i	0.240686	+	0.416881i	−0.720224	+	0.693742i	$0.755961\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	14.0000	−	24.2487i	0.447900	−	0.775785i	−0.550349	−	0.834934i	$-0.685506\pi$
$977$	14.0000	−	24.2487i	0.447900	−	0.775785i	0.998249	+	0.0591494i	$0.0188388\pi$
$978$		0			0
$979$	−90.0000			−2.87641
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−24.0000	+	41.5692i	−0.765481	+	1.32585i	0.174511	+	0.984655i	$0.444166\pi$
$983$	−24.0000	+	41.5692i	−0.765481	+	1.32585i	−0.939992	+	0.341197i	$0.889168\pi$
$984$		0			0
$985$	9.00000	+	15.5885i	0.286764	+	0.496690i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−12.0000	−	20.7846i	−0.381578	−	0.660912i
$990$		0			0
$991$	3.50000	−	6.06218i	0.111181	−	0.192571i	−0.805066	−	0.593186i	$-0.797870\pi$
$991$	3.50000	−	6.06218i	0.111181	−	0.192571i	0.916247	+	0.400614i	$0.131203\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−7.00000	+	12.1244i	−0.221692	+	0.383982i	−0.955322	−	0.295567i	$-0.904491\pi$
$997$	−7.00000	+	12.1244i	−0.221692	+	0.383982i	0.733630	+	0.679549i	$0.237825\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.s.l.361.1		2
3.2	odd	2		3528.2.s.r.361.1		2
7.2	even	3	inner	3528.2.s.l.3313.1		2
7.3	odd	6		3528.2.a.l.1.1		1
7.4	even	3		3528.2.a.s.1.1		1
7.5	odd	6		504.2.s.f.289.1	yes	2
7.6	odd	2		504.2.s.f.361.1	yes	2
21.2	odd	6		3528.2.s.r.3313.1		2
21.5	even	6		504.2.s.b.289.1	✓	2
21.11	odd	6		3528.2.a.h.1.1		1
21.17	even	6		3528.2.a.o.1.1		1
21.20	even	2		504.2.s.b.361.1	yes	2
28.3	even	6		7056.2.a.r.1.1		1
28.11	odd	6		7056.2.a.bh.1.1		1
28.19	even	6		1008.2.s.l.289.1		2
28.27	even	2		1008.2.s.l.865.1		2
84.11	even	6		7056.2.a.v.1.1		1
84.47	odd	6		1008.2.s.h.289.1		2
84.59	odd	6		7056.2.a.bm.1.1		1
84.83	odd	2		1008.2.s.h.865.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.s.b.289.1	✓	2	21.5	even	6
504.2.s.b.361.1	yes	2	21.20	even	2
504.2.s.f.289.1	yes	2	7.5	odd	6
504.2.s.f.361.1	yes	2	7.6	odd	2
1008.2.s.h.289.1		2	84.47	odd	6
1008.2.s.h.865.1		2	84.83	odd	2
1008.2.s.l.289.1		2	28.19	even	6
1008.2.s.l.865.1		2	28.27	even	2
3528.2.a.h.1.1		1	21.11	odd	6
3528.2.a.l.1.1		1	7.3	odd	6
3528.2.a.o.1.1		1	21.17	even	6
3528.2.a.s.1.1		1	7.4	even	3
3528.2.s.l.361.1		2	1.1	even	1	trivial
3528.2.s.l.3313.1		2	7.2	even	3	inner
3528.2.s.r.361.1		2	3.2	odd	2
3528.2.s.r.3313.1		2	21.2	odd	6
7056.2.a.r.1.1		1	28.3	even	6
7056.2.a.v.1.1		1	84.11	even	6
7056.2.a.bh.1.1		1	28.11	odd	6
7056.2.a.bm.1.1		1	84.59	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)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{5} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{5} +(-2.50000 + 4.33013i) q^{11} -2.00000 q^{13} +(3.00000 - 5.19615i) q^{17} +(1.00000 + 1.73205i) q^{19} +(3.00000 + 5.19615i) q^{23} +(2.00000 - 3.46410i) q^{25} -3.00000 q^{29} +(2.50000 - 4.33013i) q^{31} +(1.00000 + 1.73205i) q^{37} -8.00000 q^{41} -4.00000 q^{43} +(2.00000 + 3.46410i) q^{47} +(-4.50000 + 7.79423i) q^{53} +5.00000 q^{55} +(-1.50000 + 2.59808i) q^{59} +(-6.00000 - 10.3923i) q^{61} +(1.00000 + 1.73205i) q^{65} +(-1.00000 + 1.73205i) q^{67} -8.00000 q^{71} +(-7.00000 + 12.1244i) q^{73} +(-0.500000 - 0.866025i) q^{79} -17.0000 q^{83} -6.00000 q^{85} +(9.00000 + 15.5885i) q^{89} +(1.00000 - 1.73205i) q^{95} -3.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5}+O(q^{10}) \) 2 * q - q^5	\( 2 q - q^{5} - 5 q^{11} - 4 q^{13} + 6 q^{17} + 2 q^{19} + 6 q^{23} + 4 q^{25} - 6 q^{29} + 5 q^{31} + 2 q^{37} - 16 q^{41} - 8 q^{43} + 4 q^{47} - 9 q^{53} + 10 q^{55} - 3 q^{59} - 12 q^{61} + 2 q^{65} - 2 q^{67} - 16 q^{71} - 14 q^{73} - q^{79} - 34 q^{83} - 12 q^{85} + 18 q^{89} + 2 q^{95} - 6 q^{97}+O(q^{100}) \) 2 * q - q^5 - 5 * q^11 - 4 * q^13 + 6 * q^17 + 2 * q^19 + 6 * q^23 + 4 * q^25 - 6 * q^29 + 5 * q^31 + 2 * q^37 - 16 * q^41 - 8 * q^43 + 4 * q^47 - 9 * q^53 + 10 * q^55 - 3 * q^59 - 12 * q^61 + 2 * q^65 - 2 * q^67 - 16 * q^71 - 14 * q^73 - q^79 - 34 * q^83 - 12 * q^85 + 18 * q^89 + 2 * q^95 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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