LMFDB - Embedded newform 3528.2.s.bk.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:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} - 5x + 25 $ x^4 - x^3 - 4x^2 - 5x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$-1.63746 - 1.52274i$ of defining polynomial
Character	$\chi$	$=$	3528.361
Dual form			3528.2.s.bk.3313.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.63746 - 2.83616i) q^{5} +O(q^{10})$	$q+(-1.63746 - 2.83616i) q^{5} +(1.63746 - 2.83616i) q^{11} -6.27492 q^{13} +(2.00000 - 3.46410i) q^{17} +(-3.13746 - 5.43424i) q^{19} +(2.00000 + 3.46410i) q^{23} +(-2.86254 + 4.95807i) q^{25} -5.27492 q^{29} +(-0.500000 + 0.866025i) q^{31} +(1.13746 + 1.97014i) q^{37} -4.54983 q^{41} +0.274917 q^{43} +(3.00000 + 5.19615i) q^{47} +(4.63746 - 8.03231i) q^{53} -10.7251 q^{55} +(0.637459 - 1.10411i) q^{59} +(5.00000 + 8.66025i) q^{61} +(10.2749 + 17.7967i) q^{65} +(0.137459 - 0.238085i) q^{67} -2.00000 q^{71} +(-2.13746 + 3.70219i) q^{73} +(-5.77492 - 10.0025i) q^{79} +7.27492 q^{83} -13.0997 q^{85} +(5.27492 + 9.13642i) q^{89} +(-10.2749 + 17.7967i) q^{95} -8.72508 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{5}+O(q^{10}) $ 4 * q + q^5	$ 4 q + q^{5} - q^{11} - 10 q^{13} + 8 q^{17} - 5 q^{19} + 8 q^{23} - 19 q^{25} - 6 q^{29} - 2 q^{31} - 3 q^{37} + 12 q^{41} - 14 q^{43} + 12 q^{47} + 11 q^{53} - 58 q^{55} - 5 q^{59} + 20 q^{61} + 26 q^{65} - 7 q^{67} - 8 q^{71} - q^{73} - 8 q^{79} + 14 q^{83} + 8 q^{85} + 6 q^{89} - 26 q^{95} - 50 q^{97}+O(q^{100}) $ 4 * q + q^5 - q^11 - 10 * q^13 + 8 * q^17 - 5 * q^19 + 8 * q^23 - 19 * q^25 - 6 * q^29 - 2 * q^31 - 3 * q^37 + 12 * q^41 - 14 * q^43 + 12 * q^47 + 11 * q^53 - 58 * q^55 - 5 * q^59 + 20 * q^61 + 26 * q^65 - 7 * q^67 - 8 * q^71 - q^73 - 8 * q^79 + 14 * q^83 + 8 * q^85 + 6 * q^89 - 26 * q^95 - 50 * 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
$5$	−1.63746	−	2.83616i	−0.732294	−	1.26837i	−0.955901	−	0.293691i	$-0.905116\pi$
$5$	−1.63746	−	2.83616i	−0.732294	−	1.26837i	0.223607	−	0.974679i	$-0.428217\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.63746	−	2.83616i	0.493712	−	0.855135i	−0.506261	−	0.862380i	$-0.668973\pi$
$11$	1.63746	−	2.83616i	0.493712	−	0.855135i	0.999974	+	0.00724520i	$0.00230624\pi$
$12$		0			0
$13$	−6.27492			−1.74035			−0.870174	−	0.492744i	$-0.835994\pi$
$13$	−6.27492			−1.74035			−0.870174	+	0.492744i	$0.835994\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.00000	−	3.46410i	0.485071	−	0.840168i	−0.514782	−	0.857321i	$-0.672127\pi$
$17$	2.00000	−	3.46410i	0.485071	−	0.840168i	0.999853	+	0.0171533i	$0.00546033\pi$
$18$		0			0
$19$	−3.13746	−	5.43424i	−0.719782	−	1.24670i	−0.961086	−	0.276250i	$-0.910908\pi$
$19$	−3.13746	−	5.43424i	−0.719782	−	1.24670i	0.241303	−	0.970450i	$-0.422425\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	0.995639	−	0.0932891i	$-0.0297381\pi$
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	−0.578610	+	0.815604i	$0.696405\pi$
$24$		0			0
$25$	−2.86254	+	4.95807i	−0.572508	+	0.991613i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.27492			−0.979528			−0.489764	−	0.871855i	$-0.662917\pi$
$29$	−5.27492			−0.979528			−0.489764	+	0.871855i	$0.662917\pi$
$30$		0			0
$31$	−0.500000	+	0.866025i	−0.0898027	+	0.155543i	−0.907428	−	0.420208i	$-0.861957\pi$
$31$	−0.500000	+	0.866025i	−0.0898027	+	0.155543i	0.817625	+	0.575751i	$0.195290\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.13746	+	1.97014i	0.186997	+	0.323888i	0.944248	−	0.329236i	$-0.106791\pi$
$37$	1.13746	+	1.97014i	0.186997	+	0.323888i	−0.757251	+	0.653124i	$0.773458\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.54983			−0.710565			−0.355282	−	0.934759i	$-0.615615\pi$
$41$	−4.54983			−0.710565			−0.355282	+	0.934759i	$0.615615\pi$
$42$		0			0
$43$	0.274917			0.0419245			0.0209622	−	0.999780i	$-0.493327\pi$
$43$	0.274917			0.0419245			0.0209622	+	0.999780i	$0.493327\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	0.997503	−	0.0706177i	$-0.0224970\pi$
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	−0.559908	+	0.828554i	$0.689164\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.63746	−	8.03231i	0.637004	−	1.10332i	−0.349083	−	0.937092i	$-0.613507\pi$
$53$	4.63746	−	8.03231i	0.637004	−	1.10332i	0.986087	−	0.166231i	$-0.0531598\pi$
$54$		0			0
$55$	−10.7251			−1.44617
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.637459	−	1.10411i	0.0829900	−	0.143743i	−0.821543	−	0.570147i	$-0.806886\pi$
$59$	0.637459	−	1.10411i	0.0829900	−	0.143743i	0.904533	+	0.426404i	$0.140220\pi$
$60$		0			0
$61$	5.00000	+	8.66025i	0.640184	+	1.10883i	0.985391	+	0.170305i	$0.0544754\pi$
$61$	5.00000	+	8.66025i	0.640184	+	1.10883i	−0.345207	+	0.938527i	$0.612191\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	10.2749	+	17.7967i	1.27445	+	2.20741i
$66$		0			0
$67$	0.137459	−	0.238085i	0.0167932	−	0.0290867i	−0.857507	−	0.514473i	$-0.827988\pi$
$67$	0.137459	−	0.238085i	0.0167932	−	0.0290867i	0.874300	+	0.485386i	$0.161321\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$	−2.13746	+	3.70219i	−0.250171	+	0.433308i	−0.963573	−	0.267447i	$-0.913820\pi$
$73$	−2.13746	+	3.70219i	−0.250171	+	0.433308i	0.713402	+	0.700755i	$0.247153\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−5.77492	−	10.0025i	−0.649729	−	1.12536i	−0.983188	−	0.182599i	$-0.941549\pi$
$79$	−5.77492	−	10.0025i	−0.649729	−	1.12536i	0.333459	−	0.942765i	$-0.391784\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	7.27492			0.798526			0.399263	−	0.916836i	$-0.369266\pi$
$83$	7.27492			0.798526			0.399263	+	0.916836i	$0.369266\pi$
$84$		0			0
$85$	−13.0997			−1.42086
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.27492	+	9.13642i	0.559140	+	0.968459i	0.997568	+	0.0696929i	$0.0222020\pi$
$89$	5.27492	+	9.13642i	0.559140	+	0.968459i	−0.438428	+	0.898766i	$0.644465\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−10.2749	+	17.7967i	−1.05418	+	1.82590i
$96$		0			0
$97$	−8.72508			−0.885898			−0.442949	−	0.896547i	$-0.646068\pi$
$97$	−8.72508			−0.885898			−0.442949	+	0.896547i	$0.646068\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.00000	−	5.19615i	0.298511	−	0.517036i	−0.677284	−	0.735721i	$-0.736843\pi$
$101$	3.00000	−	5.19615i	0.298511	−	0.517036i	0.975796	+	0.218685i	$0.0701767\pi$
$102$		0			0
$103$	5.41238	+	9.37451i	0.533297	+	0.923698i	0.999244	+	0.0388850i	$0.0123806\pi$
$103$	5.41238	+	9.37451i	0.533297	+	0.923698i	−0.465946	+	0.884813i	$0.654286\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.91238	−	13.7046i	−0.764918	−	1.32488i	−0.940290	−	0.340375i	$-0.889446\pi$
$107$	−7.91238	−	13.7046i	−0.764918	−	1.32488i	0.175372	−	0.984502i	$-0.443887\pi$
$108$		0			0
$109$	−8.41238	+	14.5707i	−0.805759	+	1.39562i	0.110018	+	0.993930i	$0.464909\pi$
$109$	−8.41238	+	14.5707i	−0.805759	+	1.39562i	−0.915777	+	0.401687i	$0.868424\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.54983			0.428012			0.214006	−	0.976832i	$-0.431349\pi$
$113$	4.54983			0.428012			0.214006	+	0.976832i	$0.431349\pi$
$114$		0			0
$115$	6.54983	−	11.3446i	0.610775	−	1.05789i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.137459	+	0.238085i	0.0124962	+	0.0216441i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	2.37459			0.212389
$126$		0			0
$127$	−6.45017			−0.572360			−0.286180	−	0.958176i	$-0.592385\pi$
$127$	−6.45017			−0.572360			−0.286180	+	0.958176i	$0.592385\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.63746	−	6.30026i	−0.317806	−	0.550457i	0.662224	−	0.749306i	$-0.269613\pi$
$131$	−3.63746	−	6.30026i	−0.317806	−	0.550457i	−0.980030	+	0.198850i	$0.936279\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.725083	−	1.25588i	0.0619480	−	0.107297i	−0.833388	−	0.552688i	$-0.813602\pi$
$137$	0.725083	−	1.25588i	0.0619480	−	0.107297i	0.895336	+	0.445391i	$0.146935\pi$
$138$		0			0
$139$	−8.27492			−0.701869			−0.350935	−	0.936400i	$-0.614136\pi$
$139$	−8.27492			−0.701869			−0.350935	+	0.936400i	$0.614136\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−10.2749	+	17.7967i	−0.859232	+	1.48823i
$144$		0			0
$145$	8.63746	+	14.9605i	0.717302	+	1.24240i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.27492	−	12.6005i	−0.595984	−	1.03228i	−0.993407	−	0.114640i	$-0.963429\pi$
$149$	−7.27492	−	12.6005i	−0.595984	−	1.03228i	0.397423	−	0.917636i	$-0.369905\pi$
$150$		0			0
$151$	−11.1873	+	19.3770i	−0.910409	+	1.57687i	−0.0969217	+	0.995292i	$0.530900\pi$
$151$	−11.1873	+	19.3770i	−0.910409	+	1.57687i	−0.813487	+	0.581583i	$0.802434\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.27492			0.263048
$156$		0			0
$157$	−0.274917	+	0.476171i	−0.0219408	+	0.0380026i	−0.876787	−	0.480878i	$-0.840318\pi$
$157$	−0.274917	+	0.476171i	−0.0219408	+	0.0380026i	0.854847	+	0.518881i	$0.173651\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	6.00000	+	10.3923i	0.469956	+	0.813988i	0.999410	−	0.0343508i	$-0.0109363\pi$
$163$	6.00000	+	10.3923i	0.469956	+	0.813988i	−0.529454	+	0.848339i	$0.677603\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.00000			−0.464294			−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000			−0.464294			−0.232147	+	0.972681i	$0.574575\pi$
$168$		0			0
$169$	26.3746			2.02881
$170$		0			0
$171$		0			0
$172$		0			0
$173$	11.2749	+	19.5287i	0.857216	+	1.48474i	0.874574	+	0.484892i	$0.161141\pi$
$173$	11.2749	+	19.5287i	0.857216	+	1.48474i	−0.0173577	+	0.999849i	$0.505525\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.27492	+	14.3326i	−0.618496	+	1.07127i	0.371264	+	0.928527i	$0.378925\pi$
$179$	−8.27492	+	14.3326i	−0.618496	+	1.07127i	−0.989760	+	0.142740i	$0.954409\pi$
$180$		0			0
$181$	18.8248			1.39923			0.699616	−	0.714519i	$-0.253354\pi$
$181$	18.8248			1.39923			0.699616	+	0.714519i	$0.253354\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.72508	−	6.45203i	0.273874	−	0.474363i
$186$		0			0
$187$	−6.54983	−	11.3446i	−0.478971	−	0.829603i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.27492	+	3.94027i	0.164607	+	0.285108i	0.936516	−	0.350626i	$-0.114031\pi$
$191$	2.27492	+	3.94027i	0.164607	+	0.285108i	−0.771909	+	0.635734i	$0.780698\pi$
$192$		0			0
$193$	−0.225083	+	0.389855i	−0.0162018	+	0.0280624i	−0.874013	−	0.485903i	$-0.838491\pi$
$193$	−0.225083	+	0.389855i	−0.0162018	+	0.0280624i	0.857811	+	0.513966i	$0.171824\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.45017			0.103320			0.0516600	−	0.998665i	$-0.483549\pi$
$197$	1.45017			0.103320			0.0516600	+	0.998665i	$0.483549\pi$
$198$		0			0
$199$	−2.54983	+	4.41644i	−0.180753	+	0.313073i	−0.942137	−	0.335228i	$-0.891187\pi$
$199$	−2.54983	+	4.41644i	−0.180753	+	0.313073i	0.761384	+	0.648301i	$0.224520\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	7.45017	+	12.9041i	0.520342	+	0.901259i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−20.5498			−1.42146
$210$		0			0
$211$	−27.6495			−1.90347			−0.951735	−	0.306921i	$-0.900701\pi$
$211$	−27.6495			−1.90347			−0.951735	+	0.306921i	$0.900701\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.450166	−	0.779710i	−0.0307010	−	0.0531758i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.5498	+	21.7370i	−0.844193	+	1.46219i
$222$		0			0
$223$	1.27492			0.0853748			0.0426874	−	0.999088i	$-0.486408\pi$
$223$	1.27492			0.0853748			0.0426874	+	0.999088i	$0.486408\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	5.63746	−	9.76436i	0.374171	−	0.648084i	−0.616031	−	0.787722i	$-0.711261\pi$
$227$	5.63746	−	9.76436i	0.374171	−	0.648084i	0.990203	+	0.139638i	$0.0445939\pi$
$228$		0			0
$229$	9.13746	+	15.8265i	0.603820	+	1.04585i	0.992237	+	0.124363i	$0.0396889\pi$
$229$	9.13746	+	15.8265i	0.603820	+	1.04585i	−0.388416	+	0.921484i	$0.626978\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.274917	−	0.476171i	−0.0180104	−	0.0311950i	0.856880	−	0.515516i	$-0.172400\pi$
$233$	−0.274917	−	0.476171i	−0.0180104	−	0.0311950i	−0.874890	+	0.484321i	$0.839067\pi$
$234$		0			0
$235$	9.82475	−	17.0170i	0.640896	−	1.11006i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−15.4502			−0.999388			−0.499694	−	0.866202i	$-0.666554\pi$
$239$	−15.4502			−0.999388			−0.499694	+	0.866202i	$0.666554\pi$
$240$		0			0
$241$	4.91238	−	8.50848i	0.316434	−	0.548080i	−0.663307	−	0.748347i	$-0.730848\pi$
$241$	4.91238	−	8.50848i	0.316434	−	0.548080i	0.979741	+	0.200267i	$0.0641811\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	19.6873	+	34.0994i	1.25267	+	2.16969i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−18.3746			−1.15979			−0.579897	−	0.814690i	$-0.696907\pi$
$251$	−18.3746			−1.15979			−0.579897	+	0.814690i	$0.696907\pi$
$252$		0			0
$253$	13.0997			0.823569
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.54983	−	9.61260i	−0.346189	−	0.599617i	0.639380	−	0.768891i	$-0.279191\pi$
$257$	−5.54983	−	9.61260i	−0.346189	−	0.599617i	−0.985569	+	0.169274i	$0.945858\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4.72508	+	8.18408i	−0.291361	+	0.504652i	−0.974132	−	0.225980i	$-0.927441\pi$
$263$	−4.72508	+	8.18408i	−0.291361	+	0.504652i	0.682771	+	0.730633i	$0.260775\pi$
$264$		0			0
$265$	−30.3746			−1.86590
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.3625	+	17.9484i	−0.631815	+	1.09434i	0.355365	+	0.934728i	$0.384357\pi$
$269$	−10.3625	+	17.9484i	−0.631815	+	1.09434i	−0.987180	+	0.159609i	$0.948977\pi$
$270$		0			0
$271$	0.637459	+	1.10411i	0.0387229	+	0.0670699i	0.884737	−	0.466090i	$-0.154338\pi$
$271$	0.637459	+	1.10411i	0.0387229	+	0.0670699i	−0.846014	+	0.533160i	$0.821004\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	9.37459	+	16.2373i	0.565309	+	0.979144i
$276$		0			0
$277$	13.4124	−	23.2309i	0.805872	−	1.39581i	−0.109830	−	0.993950i	$-0.535030\pi$
$277$	13.4124	−	23.2309i	0.805872	−	1.39581i	0.915701	−	0.401860i	$-0.131636\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	26.5498			1.58383			0.791915	−	0.610631i	$-0.209084\pi$
$281$	26.5498			1.58383			0.791915	+	0.610631i	$0.209084\pi$
$282$		0			0
$283$	−12.9622	+	22.4512i	−0.770523	+	1.33459i	0.166753	+	0.985999i	$0.446672\pi$
$283$	−12.9622	+	22.4512i	−0.770523	+	1.33459i	−0.937276	+	0.348587i	$0.886662\pi$
$284$		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
$293$	−27.8248			−1.62554			−0.812770	−	0.582585i	$-0.802041\pi$
$293$	−27.8248			−1.62554			−0.812770	+	0.582585i	$0.802041\pi$
$294$		0			0
$295$	−4.17525			−0.243092
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−12.5498	−	21.7370i	−0.725776	−	1.25708i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	16.3746	−	28.3616i	0.937606	−	1.62398i
$306$		0			0
$307$	−11.3746			−0.649182			−0.324591	−	0.945854i	$-0.605227\pi$
$307$	−11.3746			−0.649182			−0.324591	+	0.945854i	$0.605227\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2.27492	−	3.94027i	0.128999	−	0.223432i	−0.794290	−	0.607539i	$-0.792157\pi$
$311$	2.27492	−	3.94027i	0.128999	−	0.223432i	0.923289	+	0.384106i	$0.125490\pi$
$312$		0			0
$313$	−9.77492	−	16.9307i	−0.552511	−	0.956977i	−0.998093	−	0.0617357i	$-0.980336\pi$
$313$	−9.77492	−	16.9307i	−0.552511	−	0.956977i	0.445582	−	0.895241i	$-0.352997\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−13.9124	−	24.0969i	−0.781397	−	1.35342i	−0.931128	−	0.364692i	$-0.881174\pi$
$317$	−13.9124	−	24.0969i	−0.781397	−	1.35342i	0.149731	−	0.988727i	$-0.452159\pi$
$318$		0			0
$319$	−8.63746	+	14.9605i	−0.483605	+	0.837628i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−25.0997			−1.39658
$324$		0			0
$325$	17.9622	−	31.1115i	0.996364	−	1.72575i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.587624	−	1.01779i	−0.0322987	−	0.0559431i	0.849424	−	0.527711i	$-0.176949\pi$
$331$	−0.587624	−	1.01779i	−0.0322987	−	0.0559431i	−0.881723	+	0.471768i	$0.843616\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.900331			−0.0491903
$336$		0			0
$337$	−24.0997			−1.31279			−0.656396	−	0.754416i	$-0.727920\pi$
$337$	−24.0997			−1.31279			−0.656396	+	0.754416i	$0.727920\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.63746	+	2.83616i	0.0886734	+	0.153587i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	15.0997	−	26.1534i	0.810593	−	1.40399i	−0.101857	−	0.994799i	$-0.532478\pi$
$347$	15.0997	−	26.1534i	0.810593	−	1.40399i	0.912450	−	0.409189i	$-0.134188\pi$
$348$		0			0
$349$	6.00000			0.321173			0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000			0.321173			0.160586	+	0.987022i	$0.448662\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	10.2749	−	17.7967i	0.546879	−	0.947222i	−0.451607	−	0.892217i	$-0.649149\pi$
$353$	10.2749	−	17.7967i	0.546879	−	0.947222i	0.998486	−	0.0550049i	$-0.0175174\pi$
$354$		0			0
$355$	3.27492	+	5.67232i	0.173815	+	0.301056i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−9.82475	−	17.0170i	−0.518531	−	0.898121i	−0.999768	−	0.0215311i	$-0.993146\pi$
$359$	−9.82475	−	17.0170i	−0.518531	−	0.898121i	0.481238	−	0.876590i	$-0.340187\pi$
$360$		0			0
$361$	−10.1873	+	17.6449i	−0.536173	+	0.928679i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	14.0000			0.732793
$366$		0			0
$367$	11.0498	−	19.1389i	0.576797	−	0.999041i	−0.419047	−	0.907964i	$-0.637636\pi$
$367$	11.0498	−	19.1389i	0.576797	−	0.999041i	0.995844	−	0.0910767i	$-0.0290308\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	3.13746	+	5.43424i	0.162451	+	0.281374i	0.935747	−	0.352671i	$-0.114727\pi$
$373$	3.13746	+	5.43424i	0.162451	+	0.281374i	−0.773296	+	0.634045i	$0.781393\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	33.0997			1.70472
$378$		0			0
$379$	13.1752			0.676767			0.338384	−	0.941008i	$-0.390120\pi$
$379$	13.1752			0.676767			0.338384	+	0.941008i	$0.390120\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5.27492	−	9.13642i	−0.269536	−	0.466849i	0.699206	−	0.714920i	$-0.253537\pi$
$383$	−5.27492	−	9.13642i	−0.269536	−	0.466849i	−0.968742	+	0.248070i	$0.920204\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.00000	+	1.73205i	−0.0507020	+	0.0878185i	−0.890263	−	0.455448i	$-0.849479\pi$
$389$	−1.00000	+	1.73205i	−0.0507020	+	0.0878185i	0.839561	+	0.543266i	$0.182813\pi$
$390$		0			0
$391$	16.0000			0.809155
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−18.9124	+	32.7572i	−0.951585	+	1.64819i
$396$		0			0
$397$	−1.68729	−	2.92248i	−0.0846828	−	0.146675i	0.820573	−	0.571541i	$-0.193654\pi$
$397$	−1.68729	−	2.92248i	−0.0846828	−	0.146675i	−0.905256	+	0.424867i	$0.860321\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.0000	+	20.7846i	0.599251	+	1.03793i	0.992932	+	0.118686i	$0.0378683\pi$
$401$	12.0000	+	20.7846i	0.599251	+	1.03793i	−0.393680	+	0.919247i	$0.628798\pi$
$402$		0			0
$403$	3.13746	−	5.43424i	0.156288	−	0.270699i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	7.45017			0.369291
$408$		0			0
$409$	5.22508	−	9.05011i	0.258364	−	0.447499i	−0.707440	−	0.706773i	$-0.750150\pi$
$409$	5.22508	−	9.05011i	0.258364	−	0.447499i	0.965804	+	0.259274i	$0.0834834\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−11.9124	−	20.6328i	−0.584756	−	1.01283i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	28.5498			1.39475			0.697375	−	0.716706i	$-0.254351\pi$
$419$	28.5498			1.39475			0.697375	+	0.716706i	$0.254351\pi$
$420$		0			0
$421$	8.82475			0.430092			0.215046	−	0.976604i	$-0.431010\pi$
$421$	8.82475			0.430092			0.215046	+	0.976604i	$0.431010\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	11.4502	+	19.8323i	0.555415	+	0.962006i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8.82475	+	15.2849i	−0.425073	+	0.736249i	−0.996427	−	0.0844552i	$-0.973085\pi$
$431$	−8.82475	+	15.2849i	−0.425073	+	0.736249i	0.571354	+	0.820704i	$0.306418\pi$
$432$		0			0
$433$	−3.17525			−0.152593			−0.0762963	−	0.997085i	$-0.524310\pi$
$433$	−3.17525			−0.152593			−0.0762963	+	0.997085i	$0.524310\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.5498	−	21.7370i	0.600340	−	1.03982i
$438$		0			0
$439$	8.63746	+	14.9605i	0.412243	+	0.714027i	0.995135	−	0.0985236i	$-0.0314120\pi$
$439$	8.63746	+	14.9605i	0.412243	+	0.714027i	−0.582891	+	0.812550i	$0.698079\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.18729	−	5.52055i	−0.151433	−	0.262289i	0.780322	−	0.625378i	$-0.215055\pi$
$443$	−3.18729	−	5.52055i	−0.151433	−	0.262289i	−0.931754	+	0.363089i	$0.881722\pi$
$444$		0			0
$445$	17.2749	−	29.9210i	0.818910	−	1.41839i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−20.5498			−0.969807			−0.484903	−	0.874568i	$-0.661145\pi$
$449$	−20.5498			−0.969807			−0.484903	+	0.874568i	$0.661145\pi$
$450$		0			0
$451$	−7.45017	+	12.9041i	−0.350815	+	0.607629i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−18.3248	−	31.7394i	−0.857196	−	1.48471i	−0.874593	−	0.484858i	$-0.838871\pi$
$457$	−18.3248	−	31.7394i	−0.857196	−	1.48471i	0.0173972	−	0.999849i	$-0.494462\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.64950			−0.169974			−0.0849872	−	0.996382i	$-0.527085\pi$
$461$	−3.64950			−0.169974			−0.0849872	+	0.996382i	$0.527085\pi$
$462$		0			0
$463$	13.1752			0.612306			0.306153	−	0.951982i	$-0.400958\pi$
$463$	13.1752			0.612306			0.306153	+	0.951982i	$0.400958\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−20.2749	−	35.1172i	−0.938211	−	1.62503i	−0.768805	−	0.639483i	$-0.779148\pi$
$467$	−20.2749	−	35.1172i	−0.938211	−	1.62503i	−0.169406	−	0.985546i	$-0.554185\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.450166	−	0.779710i	0.0206986	−	0.0358511i
$474$		0			0
$475$	35.9244			1.64833
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2.72508	−	4.71998i	0.124512	−	0.215661i	−0.797030	−	0.603940i	$-0.793597\pi$
$479$	2.72508	−	4.71998i	0.124512	−	0.215661i	0.921542	+	0.388278i	$0.126930\pi$
$480$		0			0
$481$	−7.13746	−	12.3624i	−0.325440	−	0.563679i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	14.2870	+	24.7457i	0.648738	+	1.12365i
$486$		0			0
$487$	−0.500000	+	0.866025i	−0.0226572	+	0.0392434i	−0.877132	−	0.480250i	$-0.840546\pi$
$487$	−0.500000	+	0.866025i	−0.0226572	+	0.0392434i	0.854475	+	0.519493i	$0.173879\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−36.9244			−1.66638			−0.833188	−	0.552990i	$-0.813487\pi$
$491$	−36.9244			−1.66638			−0.833188	+	0.552990i	$0.813487\pi$
$492$		0			0
$493$	−10.5498	+	18.2728i	−0.475141	+	0.822968i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−16.1375	−	27.9509i	−0.722412	−	1.25125i	−0.960030	−	0.279896i	$-0.909700\pi$
$499$	−16.1375	−	27.9509i	−0.722412	−	1.25125i	0.237619	−	0.971359i	$-0.423633\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−37.6495			−1.67871			−0.839354	−	0.543585i	$-0.817067\pi$
$503$	−37.6495			−1.67871			−0.839354	+	0.543585i	$0.817067\pi$
$504$		0			0
$505$	−19.6495			−0.874391
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.63746	+	9.76436i	0.249876	+	0.432798i	0.963491	−	0.267740i	$-0.0862768\pi$
$509$	5.63746	+	9.76436i	0.249876	+	0.432798i	−0.713615	+	0.700538i	$0.752943\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	17.7251	−	30.7007i	0.781060	−	1.35284i
$516$		0			0
$517$	19.6495			0.864184
$518$		0			0
$519$		0			0
$520$		0			0
$521$	7.27492	−	12.6005i	0.318720	−	0.552039i	−0.661501	−	0.749944i	$-0.730080\pi$
$521$	7.27492	−	12.6005i	0.318720	−	0.552039i	0.980221	+	0.197905i	$0.0634137\pi$
$522$		0			0
$523$	−8.86254	−	15.3504i	−0.387532	−	0.671225i	0.604585	−	0.796541i	$-0.293339\pi$
$523$	−8.86254	−	15.3504i	−0.387532	−	0.671225i	−0.992117	+	0.125316i	$0.960006\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.00000	+	3.46410i	0.0871214	+	0.150899i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	28.5498			1.23663
$534$		0			0
$535$	−25.9124	+	44.8816i	−1.12029	+	1.94040i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	4.13746	+	7.16629i	0.177883	+	0.308103i	0.941155	−	0.337974i	$-0.109742\pi$
$541$	4.13746	+	7.16629i	0.177883	+	0.308103i	−0.763272	+	0.646077i	$0.776408\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	55.0997			2.36021
$546$		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
$551$	16.5498	+	28.6652i	0.705047	+	1.22118i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−14.9124	+	25.8290i	−0.631858	+	1.09441i	0.355314	+	0.934747i	$0.384374\pi$
$557$	−14.9124	+	25.8290i	−0.631858	+	1.09441i	−0.987172	+	0.159663i	$0.948959\pi$
$558$		0			0
$559$	−1.72508			−0.0729632
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−7.63746	+	13.2285i	−0.321881	+	0.557513i	−0.980876	−	0.194633i	$-0.937648\pi$
$563$	−7.63746	+	13.2285i	−0.321881	+	0.557513i	0.658996	+	0.752147i	$0.270982\pi$
$564$		0			0
$565$	−7.45017	−	12.9041i	−0.313431	−	0.542878i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5.72508	+	9.91613i	0.240008	+	0.415706i	0.960716	−	0.277532i	$-0.0895166\pi$
$569$	5.72508	+	9.91613i	0.240008	+	0.415706i	−0.720708	+	0.693238i	$0.756183\pi$
$570$		0			0
$571$	−4.13746	+	7.16629i	−0.173147	+	0.299900i	−0.939519	−	0.342498i	$-0.888727\pi$
$571$	−4.13746	+	7.16629i	−0.173147	+	0.299900i	0.766371	+	0.642398i	$0.222060\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−22.9003			−0.955010
$576$		0			0
$577$	−12.5000	+	21.6506i	−0.520382	+	0.901328i	0.479337	+	0.877631i	$0.340877\pi$
$577$	−12.5000	+	21.6506i	−0.520382	+	0.901328i	−0.999719	+	0.0236970i	$0.992456\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−15.1873	−	26.3052i	−0.628993	−	1.08945i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	9.27492			0.382817			0.191408	−	0.981510i	$-0.438695\pi$
$587$	9.27492			0.382817			0.191408	+	0.981510i	$0.438695\pi$
$588$		0			0
$589$	6.27492			0.258553
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.274917	+	0.476171i	0.0112895	+	0.0195540i	0.871615	−	0.490191i	$-0.163073\pi$
$593$	0.274917	+	0.476171i	0.0112895	+	0.0195540i	−0.860325	+	0.509745i	$0.829740\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	11.2749	−	19.5287i	0.460681	−	0.797922i	−0.538314	−	0.842744i	$-0.680939\pi$
$599$	11.2749	−	19.5287i	0.460681	−	0.797922i	0.998995	+	0.0448219i	$0.0142720\pi$
$600$		0			0
$601$	−4.09967			−0.167229			−0.0836145	−	0.996498i	$-0.526646\pi$
$601$	−4.09967			−0.167229			−0.0836145	+	0.996498i	$0.526646\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.450166	−	0.779710i	0.0183018	−	0.0316997i
$606$		0			0
$607$	−3.50000	−	6.06218i	−0.142061	−	0.246056i	0.786212	−	0.617957i	$-0.212039\pi$
$607$	−3.50000	−	6.06218i	−0.142061	−	0.246056i	−0.928272	+	0.371901i	$0.878706\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−18.8248	−	32.6054i	−0.761568	−	1.31907i
$612$		0			0
$613$	4.27492	−	7.40437i	0.172662	−	0.299060i	−0.766688	−	0.642020i	$-0.778096\pi$
$613$	4.27492	−	7.40437i	0.172662	−	0.299060i	0.939350	+	0.342961i	$0.111430\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.00000			0.241551			0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000			0.241551			0.120775	+	0.992680i	$0.461462\pi$
$618$		0			0
$619$	14.4124	−	24.9630i	0.579282	−	1.00335i	−0.416280	−	0.909237i	$-0.636666\pi$
$619$	14.4124	−	24.9630i	0.579282	−	1.00335i	0.995562	−	0.0941097i	$-0.0300004\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	10.4244	+	18.0556i	0.416977	+	0.722225i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	9.09967			0.362828
$630$		0			0
$631$	19.8248			0.789211			0.394605	−	0.918851i	$-0.370881\pi$
$631$	19.8248			0.789211			0.394605	+	0.918851i	$0.370881\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	10.5619	+	18.2937i	0.419135	+	0.725964i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.82475	+	3.16056i	−0.0720734	+	0.124835i	−0.899810	−	0.436282i	$-0.856295\pi$
$641$	−1.82475	+	3.16056i	−0.0720734	+	0.124835i	0.827736	+	0.561117i	$0.189628\pi$
$642$		0			0
$643$	−5.37459			−0.211953			−0.105976	−	0.994369i	$-0.533797\pi$
$643$	−5.37459			−0.211953			−0.105976	+	0.994369i	$0.533797\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	17.0000	−	29.4449i	0.668339	−	1.15760i	−0.310029	−	0.950727i	$-0.600339\pi$
$647$	17.0000	−	29.4449i	0.668339	−	1.15760i	0.978368	−	0.206870i	$-0.0663277\pi$
$648$		0			0
$649$	−2.08762	−	3.61587i	−0.0819464	−	0.141935i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.46221	−	5.99672i	−0.135487	−	0.234670i	0.790297	−	0.612725i	$-0.209926\pi$
$653$	−3.46221	−	5.99672i	−0.135487	−	0.234670i	−0.925783	+	0.378055i	$0.876593\pi$
$654$		0			0
$655$	−11.9124	+	20.6328i	−0.465455	+	0.806192i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	42.1993			1.64385			0.821926	−	0.569594i	$-0.192899\pi$
$659$	42.1993			1.64385			0.821926	+	0.569594i	$0.192899\pi$
$660$		0			0
$661$	3.58762	−	6.21395i	0.139542	−	0.241695i	−0.787781	−	0.615955i	$-0.788770\pi$
$661$	3.58762	−	6.21395i	0.139542	−	0.241695i	0.927323	+	0.374261i	$0.122104\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−10.5498	−	18.2728i	−0.408491	−	0.707528i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	32.7492			1.26427
$672$		0			0
$673$	41.5498			1.60163			0.800814	−	0.598913i	$-0.204400\pi$
$673$	41.5498			1.60163			0.800814	+	0.598913i	$0.204400\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6.63746	+	11.4964i	0.255098	+	0.441843i	0.964922	−	0.262536i	$-0.0845588\pi$
$677$	6.63746	+	11.4964i	0.255098	+	0.441843i	−0.709824	+	0.704379i	$0.751226\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	6.08762	−	10.5441i	0.232936	−	0.403458i	−0.725735	−	0.687975i	$-0.758500\pi$
$683$	6.08762	−	10.5441i	0.232936	−	0.403458i	0.958671	+	0.284517i	$0.0918332\pi$
$684$		0			0
$685$	−4.74917			−0.181457
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−29.0997	+	50.4021i	−1.10861	+	1.92017i
$690$		0			0
$691$	−5.41238	−	9.37451i	−0.205896	−	0.356623i	0.744522	−	0.667598i	$-0.232678\pi$
$691$	−5.41238	−	9.37451i	−0.205896	−	0.356623i	−0.950418	+	0.310975i	$0.899344\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	13.5498	+	23.4690i	0.513975	+	0.890230i
$696$		0			0
$697$	−9.09967	+	15.7611i	−0.344675	+	0.596994i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	12.9244			0.488149			0.244074	−	0.969757i	$-0.421516\pi$
$701$	12.9244			0.488149			0.244074	+	0.969757i	$0.421516\pi$
$702$		0			0
$703$	7.13746	−	12.3624i	0.269194	−	0.466258i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	14.0997	+	24.4213i	0.529524	+	0.917163i	0.999407	+	0.0344340i	$0.0109628\pi$
$709$	14.0997	+	24.4213i	0.529524	+	0.917163i	−0.469883	+	0.882729i	$0.655704\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.00000			−0.149801
$714$		0			0
$715$	67.2990			2.51684
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−14.0997	−	24.4213i	−0.525829	−	0.910762i	−0.999547	−	0.0300860i	$-0.990422\pi$
$719$	−14.0997	−	24.4213i	−0.525829	−	0.910762i	0.473718	−	0.880676i	$-0.342911\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	15.0997	−	26.1534i	0.560788	−	0.971313i
$726$		0			0
$727$	−31.5498			−1.17012			−0.585059	−	0.810991i	$-0.698929\pi$
$727$	−31.5498			−1.17012			−0.585059	+	0.810991i	$0.698929\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.549834	−	0.952341i	0.0203364	−	0.0352236i
$732$		0			0
$733$	2.96221	+	5.13070i	0.109412	+	0.189507i	0.915532	−	0.402245i	$-0.131770\pi$
$733$	2.96221	+	5.13070i	0.109412	+	0.189507i	−0.806120	+	0.591752i	$0.798437\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.450166	−	0.779710i	−0.0165821	−	0.0287210i
$738$		0			0
$739$	−0.687293	+	1.19043i	−0.0252825	+	0.0437905i	−0.878390	−	0.477945i	$-0.841382\pi$
$739$	−0.687293	+	1.19043i	−0.0252825	+	0.0437905i	0.853107	+	0.521735i	$0.174715\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	44.1993			1.62152			0.810758	−	0.585381i	$-0.199055\pi$
$743$	44.1993			1.62152			0.810758	+	0.585381i	$0.199055\pi$
$744$		0			0
$745$	−23.8248	+	41.2657i	−0.872871	+	1.51186i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	5.22508	+	9.05011i	0.190666	+	0.330243i	0.945471	−	0.325706i	$-0.105602\pi$
$751$	5.22508	+	9.05011i	0.190666	+	0.330243i	−0.754805	+	0.655949i	$0.772269\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	73.2749			2.66675
$756$		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
$761$	12.5498	+	21.7370i	0.454931	+	0.787964i	0.998684	−	0.0512814i	$-0.0163305\pi$
$761$	12.5498	+	21.7370i	0.454931	+	0.787964i	−0.543753	+	0.839245i	$0.682997\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.00000	+	6.92820i	−0.144432	+	0.250163i
$768$		0			0
$769$	32.6495			1.17737			0.588686	−	0.808362i	$-0.299646\pi$
$769$	32.6495			1.17737			0.588686	+	0.808362i	$0.299646\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.54983	+	2.68439i	−0.0557437	+	0.0965509i	−0.892551	−	0.450947i	$-0.851086\pi$
$773$	−1.54983	+	2.68439i	−0.0557437	+	0.0965509i	0.836807	+	0.547498i	$0.184420\pi$
$774$		0			0
$775$	−2.86254	−	4.95807i	−0.102826	−	0.178099i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	14.2749	+	24.7249i	0.511452	+	0.885861i
$780$		0			0
$781$	−3.27492	+	5.67232i	−0.117186	+	0.202972i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.80066			0.0642684
$786$		0			0
$787$	1.27492	−	2.20822i	0.0454459	−	0.0787146i	−0.842408	−	0.538841i	$-0.818862\pi$
$787$	1.27492	−	2.20822i	0.0454459	−	0.0787146i	0.887854	+	0.460126i	$0.152196\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−31.3746	−	54.3424i	−1.11414	−	1.92975i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−31.4743			−1.11488			−0.557438	−	0.830219i	$-0.688215\pi$
$797$	−31.4743			−1.11488			−0.557438	+	0.830219i	$0.688215\pi$
$798$		0			0
$799$	24.0000			0.849059
$800$		0			0
$801$		0			0
$802$		0			0
$803$	7.00000	+	12.1244i	0.247025	+	0.427859i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−14.8248	+	25.6772i	−0.521211	+	0.902763i	0.478485	+	0.878096i	$0.341186\pi$
$809$	−14.8248	+	25.6772i	−0.521211	+	0.902763i	−0.999696	+	0.0246675i	$0.992147\pi$
$810$		0			0
$811$	42.5498			1.49413			0.747063	−	0.664753i	$-0.231463\pi$
$811$	42.5498			1.49413			0.747063	+	0.664753i	$0.231463\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	19.6495	−	34.0339i	0.688292	−	1.19216i
$816$		0			0
$817$	−0.862541	−	1.49397i	−0.0301765	−	0.0522672i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6.36254	+	11.0202i	0.222054	+	0.384609i	0.955432	−	0.295213i	$-0.0953905\pi$
$821$	6.36254	+	11.0202i	0.222054	+	0.384609i	−0.733377	+	0.679822i	$0.762057\pi$
$822$		0			0
$823$	17.0997	−	29.6175i	0.596057	−	1.03240i	−0.397340	−	0.917672i	$-0.630067\pi$
$823$	17.0997	−	29.6175i	0.596057	−	1.03240i	0.993397	−	0.114729i	$-0.0366001\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	44.0241			1.53087			0.765434	−	0.643515i	$-0.222524\pi$
$827$	44.0241			1.53087			0.765434	+	0.643515i	$0.222524\pi$
$828$		0			0
$829$	15.1375	−	26.2188i	0.525746	−	0.910618i	−0.473805	−	0.880630i	$-0.657120\pi$
$829$	15.1375	−	26.2188i	0.525746	−	0.910618i	0.999550	−	0.0299882i	$-0.00954697\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	9.82475	+	17.0170i	0.340000	+	0.588896i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−30.1993			−1.04260			−0.521298	−	0.853374i	$-0.674552\pi$
$839$	−30.1993			−1.04260			−0.521298	+	0.853374i	$0.674552\pi$
$840$		0			0
$841$	−1.17525			−0.0405258
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−43.1873	−	74.8026i	−1.48569	−	2.57329i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−4.54983	+	7.88054i	−0.155966	+	0.270142i
$852$		0			0
$853$	−13.3746			−0.457937			−0.228969	−	0.973434i	$-0.573535\pi$
$853$	−13.3746			−0.457937			−0.228969	+	0.973434i	$0.573535\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6.72508	−	11.6482i	0.229724	−	0.397894i	−0.728002	−	0.685575i	$-0.759551\pi$
$857$	6.72508	−	11.6482i	0.229724	−	0.397894i	0.957726	+	0.287681i	$0.0928842\pi$
$858$		0			0
$859$	−27.8248	−	48.1939i	−0.949368	−	1.64435i	−0.746759	−	0.665094i	$-0.768391\pi$
$859$	−27.8248	−	48.1939i	−0.949368	−	1.64435i	−0.202609	−	0.979260i	$-0.564942\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	9.54983	+	16.5408i	0.325080	+	0.563055i	0.981529	−	0.191316i	$-0.0612754\pi$
$863$	9.54983	+	16.5408i	0.325080	+	0.563055i	−0.656448	+	0.754371i	$0.727942\pi$
$864$		0			0
$865$	36.9244	−	63.9550i	1.25547	−	2.17453i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−37.8248			−1.28312
$870$		0			0
$871$	−0.862541	+	1.49397i	−0.0292261	+	0.0506211i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−23.3746	−	40.4860i	−0.789304	−	1.36711i	−0.926394	−	0.376556i	$-0.877108\pi$
$877$	−23.3746	−	40.4860i	−0.789304	−	1.36711i	0.137090	−	0.990559i	$-0.456225\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	9.45017			0.318384			0.159192	−	0.987248i	$-0.449111\pi$
$881$	9.45017			0.318384			0.159192	+	0.987248i	$0.449111\pi$
$882$		0			0
$883$	7.37459			0.248175			0.124087	−	0.992271i	$-0.460400\pi$
$883$	7.37459			0.248175			0.124087	+	0.992271i	$0.460400\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−16.8248	−	29.1413i	−0.564920	−	0.978470i	−0.997057	−	0.0766621i	$-0.975574\pi$
$887$	−16.8248	−	29.1413i	−0.564920	−	0.978470i	0.432137	−	0.901808i	$-0.357760\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	18.8248	−	32.6054i	0.629946	−	1.09110i
$894$		0			0
$895$	54.1993			1.81168
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2.63746	−	4.56821i	0.0879642	−	0.152358i
$900$		0			0
$901$	−18.5498	−	32.1293i	−0.617985	−	1.07038i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−30.8248	−	53.3900i	−1.02465	−	1.77474i
$906$		0			0
$907$	14.8625	−	25.7427i	0.493503	−	0.854772i	−0.506469	−	0.862258i	$-0.669050\pi$
$907$	14.8625	−	25.7427i	0.493503	−	0.854772i	0.999972	+	0.00748639i	$0.00238301\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−53.8488			−1.78409			−0.892046	−	0.451945i	$-0.850730\pi$
$911$	−53.8488			−1.78409			−0.892046	+	0.451945i	$0.850730\pi$
$912$		0			0
$913$	11.9124	−	20.6328i	0.394242	−	0.682847i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	1.41238	+	2.44631i	0.0465900	+	0.0806962i	0.888380	−	0.459109i	$-0.151831\pi$
$919$	1.41238	+	2.44631i	0.0465900	+	0.0806962i	−0.841790	+	0.539805i	$0.818498\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	12.5498			0.413083
$924$		0			0
$925$	−13.0241			−0.428229
$926$		0			0
$927$		0			0
$928$		0			0
$929$	20.0997	+	34.8136i	0.659449	+	1.14220i	0.980759	+	0.195225i	$0.0625436\pi$
$929$	20.0997	+	34.8136i	0.659449	+	1.14220i	−0.321310	+	0.946974i	$0.604123\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−21.4502	+	37.1528i	−0.701495	+	1.21503i
$936$		0			0
$937$	6.09967			0.199267			0.0996337	−	0.995024i	$-0.468233\pi$
$937$	6.09967			0.199267			0.0996337	+	0.995024i	$0.468233\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−25.9124	+	44.8816i	−0.844719	+	1.46310i	0.0411454	+	0.999153i	$0.486899\pi$
$941$	−25.9124	+	44.8816i	−0.844719	+	1.46310i	−0.885865	+	0.463944i	$0.846434\pi$
$942$		0			0
$943$	−9.09967	−	15.7611i	−0.296326	−	0.513252i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9.72508	−	16.8443i	−0.316023	−	0.547367i	0.663632	−	0.748060i	$-0.269014\pi$
$947$	−9.72508	−	16.8443i	−0.316023	−	0.547367i	−0.979654	+	0.200692i	$0.935681\pi$
$948$		0			0
$949$	13.4124	−	23.2309i	0.435384	−	0.754107i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−55.6495			−1.80266			−0.901332	−	0.433129i	$-0.857410\pi$
$953$	−55.6495			−1.80266			−0.901332	+	0.433129i	$0.857410\pi$
$954$		0			0
$955$	7.45017	−	12.9041i	0.241082	−	0.417566i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.0000	+	25.9808i	0.483871	+	0.838089i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.47425			0.0474579
$966$		0			0
$967$	−53.5498			−1.72205			−0.861023	−	0.508566i	$-0.830176\pi$
$967$	−53.5498			−1.72205			−0.861023	+	0.508566i	$0.830176\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−28.2870	−	48.9945i	−0.907772	−	1.57231i	−0.817153	−	0.576421i	$-0.804449\pi$
$971$	−28.2870	−	48.9945i	−0.907772	−	1.57231i	−0.0906191	−	0.995886i	$-0.528885\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	4.45017	−	7.70791i	0.142373	−	0.246598i	−0.786017	−	0.618205i	$-0.787860\pi$
$977$	4.45017	−	7.70791i	0.142373	−	0.246598i	0.928390	+	0.371608i	$0.121193\pi$
$978$		0			0
$979$	34.5498			1.10422
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−19.6495	+	34.0339i	−0.626722	+	1.08551i	0.361483	+	0.932379i	$0.382270\pi$
$983$	−19.6495	+	34.0339i	−0.626722	+	1.08551i	−0.988205	+	0.153135i	$0.951063\pi$
$984$		0			0
$985$	−2.37459	−	4.11290i	−0.0756606	−	0.131048i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.549834	+	0.952341i	0.0174837	+	0.0302827i
$990$		0			0
$991$	7.04983	−	12.2107i	0.223945	−	0.387885i	−0.732057	−	0.681243i	$-0.761440\pi$
$991$	7.04983	−	12.2107i	0.223945	−	0.387885i	0.956003	+	0.293358i	$0.0947729\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	16.7010			0.529457
$996$		0			0
$997$	−22.4124	+	38.8194i	−0.709807	+	1.22942i	0.255121	+	0.966909i	$0.417885\pi$
$997$	−22.4124	+	38.8194i	−0.709807	+	1.22942i	−0.964928	+	0.262513i	$0.915449\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.s.bk.361.1		4
3.2	odd	2		1176.2.q.l.361.2		4
7.2	even	3	inner	3528.2.s.bk.3313.1		4
7.3	odd	6		3528.2.a.bk.1.1		2
7.4	even	3		3528.2.a.bd.1.2		2
7.5	odd	6		504.2.s.i.289.2		4
7.6	odd	2		504.2.s.i.361.2		4
12.11	even	2		2352.2.q.bf.1537.2		4
21.2	odd	6		1176.2.q.l.961.2		4
21.5	even	6		168.2.q.c.121.1	yes	4
21.11	odd	6		1176.2.a.n.1.1		2
21.17	even	6		1176.2.a.k.1.2		2
21.20	even	2		168.2.q.c.25.1	✓	4
28.3	even	6		7056.2.a.cu.1.1		2
28.11	odd	6		7056.2.a.ch.1.2		2
28.19	even	6		1008.2.s.r.289.2		4
28.27	even	2		1008.2.s.r.865.2		4
84.11	even	6		2352.2.a.ba.1.1		2
84.23	even	6		2352.2.q.bf.961.2		4
84.47	odd	6		336.2.q.g.289.1		4
84.59	odd	6		2352.2.a.bf.1.2		2
84.83	odd	2		336.2.q.g.193.1		4
168.5	even	6		1344.2.q.w.961.2		4
168.11	even	6		9408.2.a.dw.1.2		2
168.53	odd	6		9408.2.a.dj.1.2		2
168.59	odd	6		9408.2.a.dp.1.1		2
168.83	odd	2		1344.2.q.x.193.2		4
168.101	even	6		9408.2.a.ec.1.1		2
168.125	even	2		1344.2.q.w.193.2		4
168.131	odd	6		1344.2.q.x.961.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.q.c.25.1	✓	4	21.20	even	2
168.2.q.c.121.1	yes	4	21.5	even	6
336.2.q.g.193.1		4	84.83	odd	2
336.2.q.g.289.1		4	84.47	odd	6
504.2.s.i.289.2		4	7.5	odd	6
504.2.s.i.361.2		4	7.6	odd	2
1008.2.s.r.289.2		4	28.19	even	6
1008.2.s.r.865.2		4	28.27	even	2
1176.2.a.k.1.2		2	21.17	even	6
1176.2.a.n.1.1		2	21.11	odd	6
1176.2.q.l.361.2		4	3.2	odd	2
1176.2.q.l.961.2		4	21.2	odd	6
1344.2.q.w.193.2		4	168.125	even	2
1344.2.q.w.961.2		4	168.5	even	6
1344.2.q.x.193.2		4	168.83	odd	2
1344.2.q.x.961.2		4	168.131	odd	6
2352.2.a.ba.1.1		2	84.11	even	6
2352.2.a.bf.1.2		2	84.59	odd	6
2352.2.q.bf.961.2		4	84.23	even	6
2352.2.q.bf.1537.2		4	12.11	even	2
3528.2.a.bd.1.2		2	7.4	even	3
3528.2.a.bk.1.1		2	7.3	odd	6
3528.2.s.bk.361.1		4	1.1	even	1	trivial
3528.2.s.bk.3313.1		4	7.2	even	3	inner
7056.2.a.ch.1.2		2	28.11	odd	6
7056.2.a.cu.1.1		2	28.3	even	6
9408.2.a.dj.1.2		2	168.53	odd	6
9408.2.a.dp.1.1		2	168.59	odd	6
9408.2.a.dw.1.2		2	168.11	even	6
9408.2.a.ec.1.1		2	168.101	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)

\(f(q)\)	\(=\)	\(q+(-1.63746 - 2.83616i) q^{5} +O(q^{10})\)	\(q+(-1.63746 - 2.83616i) q^{5} +(1.63746 - 2.83616i) q^{11} -6.27492 q^{13} +(2.00000 - 3.46410i) q^{17} +(-3.13746 - 5.43424i) q^{19} +(2.00000 + 3.46410i) q^{23} +(-2.86254 + 4.95807i) q^{25} -5.27492 q^{29} +(-0.500000 + 0.866025i) q^{31} +(1.13746 + 1.97014i) q^{37} -4.54983 q^{41} +0.274917 q^{43} +(3.00000 + 5.19615i) q^{47} +(4.63746 - 8.03231i) q^{53} -10.7251 q^{55} +(0.637459 - 1.10411i) q^{59} +(5.00000 + 8.66025i) q^{61} +(10.2749 + 17.7967i) q^{65} +(0.137459 - 0.238085i) q^{67} -2.00000 q^{71} +(-2.13746 + 3.70219i) q^{73} +(-5.77492 - 10.0025i) q^{79} +7.27492 q^{83} -13.0997 q^{85} +(5.27492 + 9.13642i) q^{89} +(-10.2749 + 17.7967i) q^{95} -8.72508 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{5}+O(q^{10}) \) 4 * q + q^5	\( 4 q + q^{5} - q^{11} - 10 q^{13} + 8 q^{17} - 5 q^{19} + 8 q^{23} - 19 q^{25} - 6 q^{29} - 2 q^{31} - 3 q^{37} + 12 q^{41} - 14 q^{43} + 12 q^{47} + 11 q^{53} - 58 q^{55} - 5 q^{59} + 20 q^{61} + 26 q^{65} - 7 q^{67} - 8 q^{71} - q^{73} - 8 q^{79} + 14 q^{83} + 8 q^{85} + 6 q^{89} - 26 q^{95} - 50 q^{97}+O(q^{100}) \) 4 * q + q^5 - q^11 - 10 * q^13 + 8 * q^17 - 5 * q^19 + 8 * q^23 - 19 * q^25 - 6 * q^29 - 2 * q^31 - 3 * q^37 + 12 * q^41 - 14 * q^43 + 12 * q^47 + 11 * q^53 - 58 * q^55 - 5 * q^59 + 20 * q^61 + 26 * q^65 - 7 * q^67 - 8 * q^71 - q^73 - 8 * q^79 + 14 * q^83 + 8 * q^85 + 6 * q^89 - 26 * q^95 - 50 * q^97

Introduction

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