LMFDB - Embedded newform 3528.2.s.bg.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:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	3528.361
Dual form			3528.2.s.bg.3313.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 1.22474i) q^{5} +O(q^{10})$	$q+(-0.707107 - 1.22474i) q^{5} +1.41421 q^{13} +(0.707107 - 1.22474i) q^{17} +(-2.00000 - 3.46410i) q^{23} +(1.50000 - 2.59808i) q^{25} +6.00000 q^{29} +(-2.82843 + 4.89898i) q^{31} +(2.00000 + 3.46410i) q^{37} -7.07107 q^{41} +4.00000 q^{43} +(-5.65685 - 9.79796i) q^{47} +(-2.00000 + 3.46410i) q^{53} +(2.82843 - 4.89898i) q^{59} +(-2.12132 - 3.67423i) q^{61} +(-1.00000 - 1.73205i) q^{65} +(-2.00000 + 3.46410i) q^{67} +12.0000 q^{71} +(3.53553 - 6.12372i) q^{73} +(-4.00000 - 6.92820i) q^{79} -5.65685 q^{83} -2.00000 q^{85} +(-6.36396 - 11.0227i) q^{89} +4.24264 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 8 q^{23} + 6 q^{25} + 24 q^{29} + 8 q^{37} + 16 q^{43} - 8 q^{53} - 4 q^{65} - 8 q^{67} + 48 q^{71} - 16 q^{79} - 8 q^{85}+O(q^{100}) $ 4 * q - 8 * q^23 + 6 * q^25 + 24 * q^29 + 8 * q^37 + 16 * q^43 - 8 * q^53 - 4 * q^65 - 8 * q^67 + 48 * q^71 - 16 * q^79 - 8 * q^85

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
$5$	−0.707107	−	1.22474i	−0.316228	−	0.547723i	0.663470	−	0.748203i	$-0.269083\pi$
$5$	−0.707107	−	1.22474i	−0.316228	−	0.547723i	−0.979698	+	0.200480i	$0.935750\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$	1.41421			0.392232			0.196116	−	0.980581i	$-0.437167\pi$
$13$	1.41421			0.392232			0.196116	+	0.980581i	$0.437167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.707107	−	1.22474i	0.171499	−	0.297044i	−0.767445	−	0.641114i	$-0.778472\pi$
$17$	0.707107	−	1.22474i	0.171499	−	0.297044i	0.938944	+	0.344070i	$0.111806\pi$
$18$		0			0
$19$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$19$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	0.578610	−	0.815604i	$-0.303595\pi$
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	−0.995639	+	0.0932891i	$0.970262\pi$
$24$		0			0
$25$	1.50000	−	2.59808i	0.300000	−	0.519615i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	−2.82843	+	4.89898i	−0.508001	+	0.879883i	0.491957	+	0.870620i	$0.336282\pi$
$31$	−2.82843	+	4.89898i	−0.508001	+	0.879883i	−0.999957	+	0.00926296i	$0.997051\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.00000	+	3.46410i	0.328798	+	0.569495i	0.982274	−	0.187453i	$-0.0600231\pi$
$37$	2.00000	+	3.46410i	0.328798	+	0.569495i	−0.653476	+	0.756948i	$0.726690\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.07107			−1.10432			−0.552158	−	0.833740i	$-0.686195\pi$
$41$	−7.07107			−1.10432			−0.552158	+	0.833740i	$0.686195\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.65685	−	9.79796i	−0.825137	−	1.42918i	−0.901815	−	0.432123i	$-0.857765\pi$
$47$	−5.65685	−	9.79796i	−0.825137	−	1.42918i	0.0766776	−	0.997056i	$-0.475569\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.00000	+	3.46410i	−0.274721	+	0.475831i	−0.970065	−	0.242846i	$-0.921919\pi$
$53$	−2.00000	+	3.46410i	−0.274721	+	0.475831i	0.695344	+	0.718677i	$0.255252\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.82843	−	4.89898i	0.368230	−	0.637793i	−0.621059	−	0.783764i	$-0.713297\pi$
$59$	2.82843	−	4.89898i	0.368230	−	0.637793i	0.989289	+	0.145971i	$0.0466306\pi$
$60$		0			0
$61$	−2.12132	−	3.67423i	−0.271607	−	0.470438i	0.697666	−	0.716423i	$-0.254222\pi$
$61$	−2.12132	−	3.67423i	−0.271607	−	0.470438i	−0.969274	+	0.245985i	$0.920888\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.00000	−	1.73205i	−0.124035	−	0.214834i
$66$		0			0
$67$	−2.00000	+	3.46410i	−0.244339	+	0.423207i	−0.961946	−	0.273241i	$-0.911904\pi$
$67$	−2.00000	+	3.46410i	−0.244339	+	0.423207i	0.717607	+	0.696449i	$0.245238\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$	3.53553	−	6.12372i	0.413803	−	0.716728i	−0.581499	−	0.813547i	$-0.697534\pi$
$73$	3.53553	−	6.12372i	0.413803	−	0.716728i	0.995302	+	0.0968194i	$0.0308669\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−4.00000	−	6.92820i	−0.450035	−	0.779484i	0.548352	−	0.836247i	$-0.315255\pi$
$79$	−4.00000	−	6.92820i	−0.450035	−	0.779484i	−0.998388	+	0.0567635i	$0.981922\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.65685			−0.620920			−0.310460	−	0.950586i	$-0.600483\pi$
$83$	−5.65685			−0.620920			−0.310460	+	0.950586i	$0.600483\pi$
$84$		0			0
$85$	−2.00000			−0.216930
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.36396	−	11.0227i	−0.674579	−	1.16840i	−0.976592	−	0.215101i	$-0.930992\pi$
$89$	−6.36396	−	11.0227i	−0.674579	−	1.16840i	0.302013	−	0.953304i	$-0.402341\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.24264			0.430775			0.215387	−	0.976529i	$-0.430899\pi$
$97$	4.24264			0.430775			0.215387	+	0.976529i	$0.430899\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.94975	−	8.57321i	0.492518	−	0.853067i	−0.507445	−	0.861684i	$-0.669410\pi$
$101$	4.94975	−	8.57321i	0.492518	−	0.853067i	0.999963	+	0.00861771i	$0.00274313\pi$
$102$		0			0
$103$	5.65685	+	9.79796i	0.557386	+	0.965422i	0.997714	+	0.0675842i	$0.0215291\pi$
$103$	5.65685	+	9.79796i	0.557386	+	0.965422i	−0.440327	+	0.897837i	$0.645138\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.00000	−	13.8564i	−0.773389	−	1.33955i	−0.935695	−	0.352809i	$-0.885227\pi$
$107$	−8.00000	−	13.8564i	−0.773389	−	1.33955i	0.162306	−	0.986740i	$-0.448107\pi$
$108$		0			0
$109$	2.00000	−	3.46410i	0.191565	−	0.331801i	−0.754204	−	0.656640i	$-0.771977\pi$
$109$	2.00000	−	3.46410i	0.191565	−	0.331801i	0.945769	+	0.324840i	$0.105310\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	16.0000			1.50515			0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000			1.50515			0.752577	+	0.658505i	$0.228811\pi$
$114$		0			0
$115$	−2.82843	+	4.89898i	−0.263752	+	0.456832i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.50000	+	9.52628i	0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.3137			−1.01193
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.48528	−	14.6969i	−0.741362	−	1.28408i	−0.951875	−	0.306486i	$-0.900847\pi$
$131$	−8.48528	−	14.6969i	−0.741362	−	1.28408i	0.210513	−	0.977591i	$-0.432487\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.00000	+	1.73205i	−0.0854358	+	0.147979i	−0.905577	−	0.424182i	$-0.860562\pi$
$137$	−1.00000	+	1.73205i	−0.0854358	+	0.147979i	0.820141	+	0.572161i	$0.193895\pi$
$138$		0			0
$139$	−16.9706			−1.43942			−0.719712	−	0.694273i	$-0.755726\pi$
$139$	−16.9706			−1.43942			−0.719712	+	0.694273i	$0.755726\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−4.24264	−	7.34847i	−0.352332	−	0.610257i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.00000	−	10.3923i	−0.491539	−	0.851371i	0.508413	−	0.861113i	$-0.330232\pi$
$149$	−6.00000	−	10.3923i	−0.491539	−	0.851371i	−0.999953	+	0.00974235i	$0.996899\pi$
$150$		0			0
$151$	−4.00000	+	6.92820i	−0.325515	+	0.563809i	−0.981617	−	0.190864i	$-0.938871\pi$
$151$	−4.00000	+	6.92820i	−0.325515	+	0.563809i	0.656101	+	0.754673i	$0.272204\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.00000			0.642575
$156$		0			0
$157$	10.6066	−	18.3712i	0.846499	−	1.46618i	−0.0378141	−	0.999285i	$-0.512039\pi$
$157$	10.6066	−	18.3712i	0.846499	−	1.46618i	0.884313	−	0.466894i	$-0.154627\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−10.0000	−	17.3205i	−0.783260	−	1.35665i	−0.930033	−	0.367477i	$-0.880222\pi$
$163$	−10.0000	−	17.3205i	−0.783260	−	1.35665i	0.146772	−	0.989170i	$-0.453112\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.65685			−0.437741			−0.218870	−	0.975754i	$-0.570237\pi$
$167$	−5.65685			−0.437741			−0.218870	+	0.975754i	$0.570237\pi$
$168$		0			0
$169$	−11.0000			−0.846154
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−10.6066	−	18.3712i	−0.806405	−	1.39673i	−0.915338	−	0.402685i	$-0.868077\pi$
$173$	−10.6066	−	18.3712i	−0.806405	−	1.39673i	0.108933	−	0.994049i	$-0.465256\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.00000	+	6.92820i	−0.298974	+	0.517838i	−0.975901	−	0.218212i	$-0.929978\pi$
$179$	−4.00000	+	6.92820i	−0.298974	+	0.517838i	0.676927	+	0.736050i	$0.263311\pi$
$180$		0			0
$181$	4.24264			0.315353			0.157676	−	0.987491i	$-0.449600\pi$
$181$	4.24264			0.315353			0.157676	+	0.987491i	$0.449600\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.82843	−	4.89898i	0.207950	−	0.360180i
$186$		0			0
$187$		0			0
$188$		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
$193$	1.00000	−	1.73205i	0.0719816	−	0.124676i	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	1.00000	−	1.73205i	0.0719816	−	0.124676i	0.899770	+	0.436365i	$0.143734\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.0000			0.854965			0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000			0.854965			0.427482	+	0.904024i	$0.359401\pi$
$198$		0			0
$199$	−8.48528	+	14.6969i	−0.601506	+	1.04184i	0.391088	+	0.920353i	$0.372099\pi$
$199$	−8.48528	+	14.6969i	−0.601506	+	1.04184i	−0.992593	+	0.121485i	$0.961234\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	5.00000	+	8.66025i	0.349215	+	0.604858i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.82843	−	4.89898i	−0.192897	−	0.334108i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.00000	−	1.73205i	0.0672673	−	0.116510i
$222$		0			0
$223$	5.65685			0.378811			0.189405	−	0.981899i	$-0.439344\pi$
$223$	5.65685			0.378811			0.189405	+	0.981899i	$0.439344\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.3137	−	19.5959i	0.750917	−	1.30063i	−0.196461	−	0.980512i	$-0.562945\pi$
$227$	11.3137	−	19.5959i	0.750917	−	1.30063i	0.947379	−	0.320115i	$-0.103722\pi$
$228$		0			0
$229$	−2.12132	−	3.67423i	−0.140181	−	0.242800i	0.787384	−	0.616463i	$-0.211435\pi$
$229$	−2.12132	−	3.67423i	−0.140181	−	0.242800i	−0.927565	+	0.373663i	$0.878102\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−7.00000	−	12.1244i	−0.458585	−	0.794293i	0.540301	−	0.841472i	$-0.318310\pi$
$233$	−7.00000	−	12.1244i	−0.458585	−	0.794293i	−0.998886	+	0.0471787i	$0.984977\pi$
$234$		0			0
$235$	−8.00000	+	13.8564i	−0.521862	+	0.903892i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	20.0000			1.29369			0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000			1.29369			0.646846	+	0.762620i	$0.276088\pi$
$240$		0			0
$241$	7.77817	−	13.4722i	0.501036	−	0.867820i	−0.498963	−	0.866623i	$-0.666286\pi$
$241$	7.77817	−	13.4722i	0.501036	−	0.867820i	0.999999	−	0.00119700i	$-0.000381016\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−22.6274			−1.42823			−0.714115	−	0.700028i	$-0.753171\pi$
$251$	−22.6274			−1.42823			−0.714115	+	0.700028i	$0.753171\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.12132	−	3.67423i	−0.132324	−	0.229192i	0.792248	−	0.610199i	$-0.208911\pi$
$257$	−2.12132	−	3.67423i	−0.132324	−	0.229192i	−0.924572	+	0.381007i	$0.875577\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.00000	−	3.46410i	0.123325	−	0.213606i	−0.797752	−	0.602986i	$-0.793977\pi$
$263$	2.00000	−	3.46410i	0.123325	−	0.213606i	0.921077	+	0.389380i	$0.127311\pi$
$264$		0			0
$265$	5.65685			0.347498
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.94975	+	8.57321i	−0.301791	+	0.522718i	−0.976542	−	0.215328i	$-0.930918\pi$
$269$	−4.94975	+	8.57321i	−0.301791	+	0.522718i	0.674750	+	0.738046i	$0.264251\pi$
$270$		0			0
$271$	11.3137	+	19.5959i	0.687259	+	1.19037i	0.972721	+	0.231977i	$0.0745195\pi$
$271$	11.3137	+	19.5959i	0.687259	+	1.19037i	−0.285462	+	0.958390i	$0.592147\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	11.0000	−	19.0526i	0.660926	−	1.14476i	−0.319447	−	0.947604i	$-0.603497\pi$
$277$	11.0000	−	19.0526i	0.660926	−	1.14476i	0.980373	−	0.197153i	$-0.0631696\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.00000			−0.119310			−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000			−0.119310			−0.0596550	+	0.998219i	$0.519000\pi$
$282$		0			0
$283$	−8.48528	+	14.6969i	−0.504398	+	0.873642i	0.495589	+	0.868557i	$0.334952\pi$
$283$	−8.48528	+	14.6969i	−0.504398	+	0.873642i	−0.999987	+	0.00508540i	$0.998381\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	7.50000	+	12.9904i	0.441176	+	0.764140i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.41421			0.0826192			0.0413096	−	0.999146i	$-0.486847\pi$
$293$	1.41421			0.0826192			0.0413096	+	0.999146i	$0.486847\pi$
$294$		0			0
$295$	−8.00000			−0.465778
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.82843	−	4.89898i	−0.163572	−	0.283315i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3.00000	+	5.19615i	−0.171780	+	0.297531i
$306$		0			0
$307$	−11.3137			−0.645707			−0.322854	−	0.946449i	$-0.604642\pi$
$307$	−11.3137			−0.645707			−0.322854	+	0.946449i	$0.604642\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5.65685	+	9.79796i	−0.320771	+	0.555591i	−0.980647	−	0.195783i	$-0.937275\pi$
$311$	−5.65685	+	9.79796i	−0.320771	+	0.555591i	0.659877	+	0.751374i	$0.270609\pi$
$312$		0			0
$313$	−6.36396	−	11.0227i	−0.359712	−	0.623040i	0.628200	−	0.778052i	$-0.283792\pi$
$313$	−6.36396	−	11.0227i	−0.359712	−	0.623040i	−0.987913	+	0.155012i	$0.950459\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.00000	+	3.46410i	0.112331	+	0.194563i	0.916710	−	0.399554i	$-0.130835\pi$
$317$	2.00000	+	3.46410i	0.112331	+	0.194563i	−0.804379	+	0.594117i	$0.797502\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	2.12132	−	3.67423i	0.117670	−	0.203810i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−14.0000	−	24.2487i	−0.769510	−	1.33283i	−0.937829	−	0.347097i	$-0.887167\pi$
$331$	−14.0000	−	24.2487i	−0.769510	−	1.33283i	0.168320	−	0.985732i	$-0.446166\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.65685			0.309067
$336$		0			0
$337$	16.0000			0.871576			0.435788	−	0.900049i	$-0.356470\pi$
$337$	16.0000			0.871576			0.435788	+	0.900049i	$0.356470\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.00000	+	13.8564i	−0.429463	+	0.743851i	−0.996826	−	0.0796169i	$-0.974630\pi$
$347$	−8.00000	+	13.8564i	−0.429463	+	0.743851i	0.567363	+	0.823468i	$0.307964\pi$
$348$		0			0
$349$	12.7279			0.681310			0.340655	−	0.940188i	$-0.389351\pi$
$349$	12.7279			0.681310			0.340655	+	0.940188i	$0.389351\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	7.77817	−	13.4722i	0.413990	−	0.717053i	−0.581331	−	0.813667i	$-0.697468\pi$
$353$	7.77817	−	13.4722i	0.413990	−	0.717053i	0.995322	+	0.0966144i	$0.0308013\pi$
$354$		0			0
$355$	−8.48528	−	14.6969i	−0.450352	−	0.780033i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.00000	−	3.46410i	−0.105556	−	0.182828i	0.808409	−	0.588621i	$-0.200329\pi$
$359$	−2.00000	−	3.46410i	−0.105556	−	0.182828i	−0.913965	+	0.405793i	$0.866996\pi$
$360$		0			0
$361$	9.50000	−	16.4545i	0.500000	−	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−10.0000			−0.523424
$366$		0			0
$367$	−14.1421	+	24.4949i	−0.738213	+	1.27862i	0.215086	+	0.976595i	$0.430997\pi$
$367$	−14.1421	+	24.4949i	−0.738213	+	1.27862i	−0.953299	+	0.302028i	$0.902336\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	11.0000	+	19.0526i	0.569558	+	0.986504i	0.996610	+	0.0822766i	$0.0262191\pi$
$373$	11.0000	+	19.0526i	0.569558	+	0.986504i	−0.427051	+	0.904227i	$0.640448\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	8.48528			0.437014
$378$		0			0
$379$	28.0000			1.43826			0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000			1.43826			0.719132	+	0.694874i	$0.244540\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2.82843	+	4.89898i	0.144526	+	0.250326i	0.929196	−	0.369587i	$-0.120501\pi$
$383$	2.82843	+	4.89898i	0.144526	+	0.250326i	−0.784670	+	0.619914i	$0.787168\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	9.00000	−	15.5885i	0.456318	−	0.790366i	−0.542445	−	0.840091i	$-0.682501\pi$
$389$	9.00000	−	15.5885i	0.456318	−	0.790366i	0.998763	+	0.0497253i	$0.0158346\pi$
$390$		0			0
$391$	−5.65685			−0.286079
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−5.65685	+	9.79796i	−0.284627	+	0.492989i
$396$		0			0
$397$	−4.94975	−	8.57321i	−0.248421	−	0.430277i	0.714667	−	0.699465i	$-0.246578\pi$
$397$	−4.94975	−	8.57321i	−0.248421	−	0.430277i	−0.963088	+	0.269187i	$0.913245\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.00000	−	5.19615i	−0.149813	−	0.259483i	0.781345	−	0.624099i	$-0.214534\pi$
$401$	−3.00000	−	5.19615i	−0.149813	−	0.259483i	−0.931158	+	0.364615i	$0.881200\pi$
$402$		0			0
$403$	−4.00000	+	6.92820i	−0.199254	+	0.345118i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	6.36396	−	11.0227i	0.314678	−	0.545038i	−0.664691	−	0.747118i	$-0.731437\pi$
$409$	6.36396	−	11.0227i	0.314678	−	0.545038i	0.979369	+	0.202080i	$0.0647703\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	4.00000	+	6.92820i	0.196352	+	0.340092i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5.65685			0.276355			0.138178	−	0.990407i	$-0.455875\pi$
$419$	5.65685			0.276355			0.138178	+	0.990407i	$0.455875\pi$
$420$		0			0
$421$	−6.00000			−0.292422			−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000			−0.292422			−0.146211	+	0.989253i	$0.546708\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.12132	−	3.67423i	−0.102899	−	0.178227i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6.00000	−	10.3923i	0.289010	−	0.500580i	−0.684564	−	0.728953i	$-0.740007\pi$
$431$	6.00000	−	10.3923i	0.289010	−	0.500580i	0.973574	+	0.228373i	$0.0733406\pi$
$432$		0			0
$433$	15.5563			0.747590			0.373795	−	0.927511i	$-0.378056\pi$
$433$	15.5563			0.747590			0.373795	+	0.927511i	$0.378056\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	16.9706	+	29.3939i	0.809961	+	1.40289i	0.912890	+	0.408205i	$0.133845\pi$
$439$	16.9706	+	29.3939i	0.809961	+	1.40289i	−0.102930	+	0.994689i	$0.532822\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	12.0000	+	20.7846i	0.570137	+	0.987507i	0.996551	+	0.0829786i	$0.0264433\pi$
$443$	12.0000	+	20.7846i	0.570137	+	0.987507i	−0.426414	+	0.904528i	$0.640223\pi$
$444$		0			0
$445$	−9.00000	+	15.5885i	−0.426641	+	0.738964i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−8.00000			−0.377543			−0.188772	−	0.982021i	$-0.560451\pi$
$449$	−8.00000			−0.377543			−0.188772	+	0.982021i	$0.560451\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.00000	+	5.19615i	0.140334	+	0.243066i	0.927622	−	0.373519i	$-0.121849\pi$
$457$	3.00000	+	5.19615i	0.140334	+	0.243066i	−0.787288	+	0.616585i	$0.788516\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9.89949			0.461065			0.230533	−	0.973065i	$-0.425953\pi$
$461$	9.89949			0.461065			0.230533	+	0.973065i	$0.425953\pi$
$462$		0			0
$463$	−16.0000			−0.743583			−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000			−0.743583			−0.371792	+	0.928316i	$0.621256\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	16.9706	+	29.3939i	0.785304	+	1.36019i	0.928817	+	0.370538i	$0.120827\pi$
$467$	16.9706	+	29.3939i	0.785304	+	1.36019i	−0.143513	+	0.989648i	$0.545840\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5.65685	−	9.79796i	0.258468	−	0.447680i	−0.707364	−	0.706850i	$-0.750116\pi$
$479$	5.65685	−	9.79796i	0.258468	−	0.447680i	0.965832	+	0.259170i	$0.0834489\pi$
$480$		0			0
$481$	2.82843	+	4.89898i	0.128965	+	0.223374i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.00000	−	5.19615i	−0.136223	−	0.235945i
$486$		0			0
$487$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$487$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−32.0000			−1.44414			−0.722070	−	0.691820i	$-0.756809\pi$
$491$	−32.0000			−1.44414			−0.722070	+	0.691820i	$0.756809\pi$
$492$		0			0
$493$	4.24264	−	7.34847i	0.191079	−	0.330958i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−10.0000	−	17.3205i	−0.447661	−	0.775372i	0.550572	−	0.834788i	$-0.314410\pi$
$499$	−10.0000	−	17.3205i	−0.447661	−	0.775372i	−0.998233	+	0.0594153i	$0.981076\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	5.65685			0.252227			0.126113	−	0.992016i	$-0.459750\pi$
$503$	5.65685			0.252227			0.126113	+	0.992016i	$0.459750\pi$
$504$		0			0
$505$	−14.0000			−0.622992
$506$		0			0
$507$		0			0
$508$		0			0
$509$	12.0208	+	20.8207i	0.532813	+	0.922860i	0.999266	+	0.0383134i	$0.0121985\pi$
$509$	12.0208	+	20.8207i	0.532813	+	0.922860i	−0.466453	+	0.884546i	$0.654468\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8.00000	−	13.8564i	0.352522	−	0.610586i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−20.5061	+	35.5176i	−0.898388	+	1.55605i	−0.0688342	+	0.997628i	$0.521928\pi$
$521$	−20.5061	+	35.5176i	−0.898388	+	1.55605i	−0.829554	+	0.558426i	$0.811405\pi$
$522$		0			0
$523$	−11.3137	−	19.5959i	−0.494714	−	0.856870i	0.505268	−	0.862963i	$-0.331394\pi$
$523$	−11.3137	−	19.5959i	−0.494714	−	0.856870i	−0.999981	+	0.00609311i	$0.998060\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.00000	+	6.92820i	0.174243	+	0.301797i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−10.0000			−0.433148
$534$		0			0
$535$	−11.3137	+	19.5959i	−0.489134	+	0.847205i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.00000	+	1.73205i	0.0429934	+	0.0744667i	0.886721	−	0.462304i	$-0.152977\pi$
$541$	1.00000	+	1.73205i	0.0429934	+	0.0744667i	−0.843728	+	0.536771i	$0.819644\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−5.65685			−0.242313
$546$		0			0
$547$	20.0000			0.855138			0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000			0.855138			0.427569	+	0.903983i	$0.359370\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−18.0000	+	31.1769i	−0.762684	+	1.32101i	0.178778	+	0.983890i	$0.442786\pi$
$557$	−18.0000	+	31.1769i	−0.762684	+	1.32101i	−0.941462	+	0.337119i	$0.890548\pi$
$558$		0			0
$559$	5.65685			0.239259
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−8.48528	+	14.6969i	−0.357612	+	0.619402i	−0.987561	−	0.157234i	$-0.949742\pi$
$563$	−8.48528	+	14.6969i	−0.357612	+	0.619402i	0.629949	+	0.776636i	$0.283076\pi$
$564$		0			0
$565$	−11.3137	−	19.5959i	−0.475971	−	0.824406i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	23.0000	+	39.8372i	0.964210	+	1.67006i	0.711722	+	0.702461i	$0.247915\pi$
$569$	23.0000	+	39.8372i	0.964210	+	1.67006i	0.252488	+	0.967600i	$0.418751\pi$
$570$		0			0
$571$	2.00000	−	3.46410i	0.0836974	−	0.144968i	−0.821138	−	0.570730i	$-0.806660\pi$
$571$	2.00000	−	3.46410i	0.0836974	−	0.144968i	0.904835	+	0.425762i	$0.139994\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−12.0000			−0.500435
$576$		0			0
$577$	6.36396	−	11.0227i	0.264935	−	0.458881i	−0.702611	−	0.711574i	$-0.747983\pi$
$577$	6.36396	−	11.0227i	0.264935	−	0.458881i	0.967547	+	0.252693i	$0.0813161\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	33.9411			1.40090			0.700450	−	0.713701i	$-0.252983\pi$
$587$	33.9411			1.40090			0.700450	+	0.713701i	$0.252983\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	20.5061	+	35.5176i	0.842084	+	1.45853i	0.888129	+	0.459594i	$0.152005\pi$
$593$	20.5061	+	35.5176i	0.842084	+	1.45853i	−0.0460448	+	0.998939i	$0.514662\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.00000	+	10.3923i	−0.245153	+	0.424618i	−0.962175	−	0.272433i	$-0.912172\pi$
$599$	−6.00000	+	10.3923i	−0.245153	+	0.424618i	0.717021	+	0.697051i	$0.245505\pi$
$600$		0			0
$601$	−26.8701			−1.09605			−0.548026	−	0.836461i	$-0.684621\pi$
$601$	−26.8701			−1.09605			−0.548026	+	0.836461i	$0.684621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.77817	−	13.4722i	0.316228	−	0.547723i
$606$		0			0
$607$	19.7990	+	34.2929i	0.803616	+	1.39190i	0.917221	+	0.398378i	$0.130427\pi$
$607$	19.7990	+	34.2929i	0.803616	+	1.39190i	−0.113605	+	0.993526i	$0.536240\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.00000	−	13.8564i	−0.323645	−	0.560570i
$612$		0			0
$613$	6.00000	−	10.3923i	0.242338	−	0.419741i	−0.719042	−	0.694967i	$-0.755419\pi$
$613$	6.00000	−	10.3923i	0.242338	−	0.419741i	0.961380	+	0.275225i	$0.0887525\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−14.0000			−0.563619			−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000			−0.563619			−0.281809	+	0.959470i	$0.590935\pi$
$618$		0			0
$619$	16.9706	−	29.3939i	0.682105	−	1.18144i	−0.292233	−	0.956347i	$-0.594398\pi$
$619$	16.9706	−	29.3939i	0.682105	−	1.18144i	0.974337	−	0.225092i	$-0.0722684\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.500000	+	0.866025i	0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5.65685			0.225554
$630$		0			0
$631$	−32.0000			−1.27390			−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000			−1.27390			−0.636950	+	0.770905i	$0.719804\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5.65685	+	9.79796i	0.224485	+	0.388820i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	21.0000	−	36.3731i	0.829450	−	1.43665i	−0.0690201	−	0.997615i	$-0.521987\pi$
$641$	21.0000	−	36.3731i	0.829450	−	1.43665i	0.898470	−	0.439034i	$-0.144679\pi$
$642$		0			0
$643$	−39.5980			−1.56159			−0.780796	−	0.624786i	$-0.785186\pi$
$643$	−39.5980			−1.56159			−0.780796	+	0.624786i	$0.785186\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.48528	+	14.6969i	−0.333591	+	0.577796i	−0.983213	−	0.182461i	$-0.941594\pi$
$647$	−8.48528	+	14.6969i	−0.333591	+	0.577796i	0.649622	+	0.760257i	$0.274927\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	13.0000	+	22.5167i	0.508729	+	0.881145i	0.999949	+	0.0101092i	$0.00321793\pi$
$653$	13.0000	+	22.5167i	0.508729	+	0.881145i	−0.491220	+	0.871036i	$0.663449\pi$
$654$		0			0
$655$	−12.0000	+	20.7846i	−0.468879	+	0.812122i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−24.0000			−0.934907			−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000			−0.934907			−0.467454	+	0.884018i	$0.654829\pi$
$660$		0			0
$661$	4.94975	−	8.57321i	0.192523	−	0.333459i	−0.753563	−	0.657376i	$-0.771666\pi$
$661$	4.94975	−	8.57321i	0.192523	−	0.333459i	0.946086	+	0.323917i	$0.105000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−12.0000	−	20.7846i	−0.464642	−	0.804783i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−8.00000			−0.308377			−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000			−0.308377			−0.154189	+	0.988041i	$0.549276\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14.8492	+	25.7196i	0.570703	+	0.988486i	0.996494	+	0.0836647i	$0.0266625\pi$
$677$	14.8492	+	25.7196i	0.570703	+	0.988486i	−0.425791	+	0.904821i	$0.640004\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	4.00000	−	6.92820i	0.153056	−	0.265100i	−0.779294	−	0.626659i	$-0.784422\pi$
$683$	4.00000	−	6.92820i	0.153056	−	0.265100i	0.932349	+	0.361559i	$0.117755\pi$
$684$		0			0
$685$	2.82843			0.108069
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2.82843	+	4.89898i	−0.107754	+	0.186636i
$690$		0			0
$691$	−14.1421	−	24.4949i	−0.537992	−	0.931830i	−0.999012	−	0.0444400i	$-0.985850\pi$
$691$	−14.1421	−	24.4949i	−0.537992	−	0.931830i	0.461020	−	0.887390i	$-0.347484\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	12.0000	+	20.7846i	0.455186	+	0.788405i
$696$		0			0
$697$	−5.00000	+	8.66025i	−0.189389	+	0.328031i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−22.0000			−0.830929			−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000			−0.830929			−0.415464	+	0.909610i	$0.636381\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−2.00000	−	3.46410i	−0.0751116	−	0.130097i	0.826023	−	0.563636i	$-0.190598\pi$
$709$	−2.00000	−	3.46410i	−0.0751116	−	0.130097i	−0.901135	+	0.433539i	$0.857265\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	22.6274			0.847403
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	19.7990	+	34.2929i	0.738378	+	1.27891i	0.953225	+	0.302260i	$0.0977411\pi$
$719$	19.7990	+	34.2929i	0.738378	+	1.27891i	−0.214848	+	0.976648i	$0.568926\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	9.00000	−	15.5885i	0.334252	−	0.578941i
$726$		0			0
$727$	33.9411			1.25881			0.629403	−	0.777079i	$-0.283299\pi$
$727$	33.9411			1.25881			0.629403	+	0.777079i	$0.283299\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.82843	−	4.89898i	0.104613	−	0.181195i
$732$		0			0
$733$	14.8492	+	25.7196i	0.548469	+	0.949977i	0.998380	+	0.0569030i	$0.0181226\pi$
$733$	14.8492	+	25.7196i	0.548469	+	0.949977i	−0.449910	+	0.893074i	$0.648544\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−6.00000	+	10.3923i	−0.220714	+	0.382287i	−0.955025	−	0.296526i	$-0.904172\pi$
$739$	−6.00000	+	10.3923i	−0.220714	+	0.382287i	0.734311	+	0.678813i	$0.237505\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−36.0000			−1.32071			−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000			−1.32071			−0.660356	+	0.750953i	$0.729595\pi$
$744$		0			0
$745$	−8.48528	+	14.6969i	−0.310877	+	0.538454i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	12.0000	+	20.7846i	0.437886	+	0.758441i	0.997526	−	0.0702946i	$-0.0223939\pi$
$751$	12.0000	+	20.7846i	0.437886	+	0.758441i	−0.559640	+	0.828736i	$0.689061\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	11.3137			0.411748
$756$		0			0
$757$	−4.00000			−0.145382			−0.0726912	−	0.997354i	$-0.523159\pi$
$757$	−4.00000			−0.145382			−0.0726912	+	0.997354i	$0.523159\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	7.77817	+	13.4722i	0.281959	+	0.488367i	0.971867	−	0.235530i	$-0.0756826\pi$
$761$	7.77817	+	13.4722i	0.281959	+	0.488367i	−0.689908	+	0.723897i	$0.742349\pi$
$762$		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
$769$	38.1838			1.37694			0.688471	−	0.725264i	$-0.258282\pi$
$769$	38.1838			1.37694			0.688471	+	0.725264i	$0.258282\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−16.2635	+	28.1691i	−0.584956	+	1.01317i	0.409925	+	0.912119i	$0.365555\pi$
$773$	−16.2635	+	28.1691i	−0.584956	+	1.01317i	−0.994881	+	0.101054i	$0.967778\pi$
$774$		0			0
$775$	8.48528	+	14.6969i	0.304800	+	0.527930i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−30.0000			−1.07075
$786$		0			0
$787$	2.82843	−	4.89898i	0.100823	−	0.174630i	−0.811201	−	0.584767i	$-0.801186\pi$
$787$	2.82843	−	4.89898i	0.100823	−	0.174630i	0.912024	+	0.410137i	$0.134519\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−3.00000	−	5.19615i	−0.106533	−	0.184521i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	24.0416			0.851598			0.425799	−	0.904818i	$-0.359993\pi$
$797$	24.0416			0.851598			0.425799	+	0.904818i	$0.359993\pi$
$798$		0			0
$799$	−16.0000			−0.566039
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−20.0000	+	34.6410i	−0.703163	+	1.21791i	0.264188	+	0.964471i	$0.414896\pi$
$809$	−20.0000	+	34.6410i	−0.703163	+	1.21791i	−0.967351	+	0.253442i	$0.918437\pi$
$810$		0			0
$811$	−11.3137			−0.397278			−0.198639	−	0.980073i	$-0.563652\pi$
$811$	−11.3137			−0.397278			−0.198639	+	0.980073i	$0.563652\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−14.1421	+	24.4949i	−0.495377	+	0.858019i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	18.0000	+	31.1769i	0.628204	+	1.08808i	0.987912	+	0.155017i	$0.0495431\pi$
$821$	18.0000	+	31.1769i	0.628204	+	1.08808i	−0.359708	+	0.933065i	$0.617124\pi$
$822$		0			0
$823$	16.0000	−	27.7128i	0.557725	−	0.966008i	−0.439961	−	0.898017i	$-0.645008\pi$
$823$	16.0000	−	27.7128i	0.557725	−	0.966008i	0.997686	−	0.0679910i	$-0.0216589\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	8.00000			0.278187			0.139094	−	0.990279i	$-0.455581\pi$
$827$	8.00000			0.278187			0.139094	+	0.990279i	$0.455581\pi$
$828$		0			0
$829$	2.12132	−	3.67423i	0.0736765	−	0.127611i	−0.826833	−	0.562447i	$-0.809860\pi$
$829$	2.12132	−	3.67423i	0.0736765	−	0.127611i	0.900510	+	0.434835i	$0.143193\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	4.00000	+	6.92820i	0.138426	+	0.239760i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	39.5980			1.36707			0.683537	−	0.729916i	$-0.260441\pi$
$839$	39.5980			1.36707			0.683537	+	0.729916i	$0.260441\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7.77817	+	13.4722i	0.267577	+	0.463458i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	8.00000	−	13.8564i	0.274236	−	0.474991i
$852$		0			0
$853$	−38.1838			−1.30739			−0.653694	−	0.756759i	$-0.726781\pi$
$853$	−38.1838			−1.30739			−0.653694	+	0.756759i	$0.726781\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	13.4350	−	23.2702i	0.458932	−	0.794893i	−0.539973	−	0.841682i	$-0.681566\pi$
$857$	13.4350	−	23.2702i	0.458932	−	0.794893i	0.998905	+	0.0467891i	$0.0148989\pi$
$858$		0			0
$859$	−28.2843	−	48.9898i	−0.965047	−	1.67151i	−0.709489	−	0.704716i	$-0.751074\pi$
$859$	−28.2843	−	48.9898i	−0.965047	−	1.67151i	−0.255558	−	0.966794i	$-0.582259\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−10.0000	−	17.3205i	−0.340404	−	0.589597i	0.644104	−	0.764938i	$-0.277230\pi$
$863$	−10.0000	−	17.3205i	−0.340404	−	0.589597i	−0.984508	+	0.175341i	$0.943897\pi$
$864$		0			0
$865$	−15.0000	+	25.9808i	−0.510015	+	0.883372i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−2.82843	+	4.89898i	−0.0958376	+	0.165996i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−14.0000	−	24.2487i	−0.472746	−	0.818821i	0.526767	−	0.850010i	$-0.323404\pi$
$877$	−14.0000	−	24.2487i	−0.472746	−	0.818821i	−0.999514	+	0.0311889i	$0.990071\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−29.6985			−1.00057			−0.500284	−	0.865862i	$-0.666771\pi$
$881$	−29.6985			−1.00057			−0.500284	+	0.865862i	$0.666771\pi$
$882$		0			0
$883$	44.0000			1.48072			0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000			1.48072			0.740359	+	0.672212i	$0.234656\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	14.1421	+	24.4949i	0.474846	+	0.822458i	0.999585	−	0.0288053i	$-0.00917026\pi$
$887$	14.1421	+	24.4949i	0.474846	+	0.822458i	−0.524739	+	0.851263i	$0.675837\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	11.3137			0.378176
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−16.9706	+	29.3939i	−0.566000	+	0.980341i
$900$		0			0
$901$	2.82843	+	4.89898i	0.0942286	+	0.163209i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3.00000	−	5.19615i	−0.0997234	−	0.172726i
$906$		0			0
$907$	−2.00000	+	3.46410i	−0.0664089	+	0.115024i	−0.897318	−	0.441384i	$-0.854488\pi$
$907$	−2.00000	+	3.46410i	−0.0664089	+	0.115024i	0.830909	+	0.556408i	$0.187821\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−60.0000			−1.98789			−0.993944	−	0.109885i	$-0.964952\pi$
$911$	−60.0000			−1.98789			−0.993944	+	0.109885i	$0.964952\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	16.0000	+	27.7128i	0.527791	+	0.914161i	0.999475	+	0.0323936i	$0.0103130\pi$
$919$	16.0000	+	27.7128i	0.527791	+	0.914161i	−0.471684	+	0.881768i	$0.656354\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	16.9706			0.558593
$924$		0			0
$925$	12.0000			0.394558
$926$		0			0
$927$		0			0
$928$		0			0
$929$	4.94975	+	8.57321i	0.162396	+	0.281278i	0.935727	−	0.352724i	$-0.114744\pi$
$929$	4.94975	+	8.57321i	0.162396	+	0.281278i	−0.773332	+	0.634002i	$0.781411\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−38.1838			−1.24741			−0.623705	−	0.781660i	$-0.714373\pi$
$937$	−38.1838			−1.24741			−0.623705	+	0.781660i	$0.714373\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−13.4350	+	23.2702i	−0.437969	+	0.758585i	−0.997533	−	0.0702023i	$-0.977636\pi$
$941$	−13.4350	+	23.2702i	−0.437969	+	0.758585i	0.559563	+	0.828788i	$0.310969\pi$
$942$		0			0
$943$	14.1421	+	24.4949i	0.460531	+	0.797664i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	24.0000	+	41.5692i	0.779895	+	1.35082i	0.932002	+	0.362454i	$0.118061\pi$
$947$	24.0000	+	41.5692i	0.779895	+	1.35082i	−0.152106	+	0.988364i	$0.548606\pi$
$948$		0			0
$949$	5.00000	−	8.66025i	0.162307	−	0.281124i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−8.00000			−0.259145			−0.129573	−	0.991570i	$-0.541361\pi$
$953$	−8.00000			−0.259145			−0.129573	+	0.991570i	$0.541361\pi$
$954$		0			0
$955$	2.82843	−	4.89898i	0.0915258	−	0.158527i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.500000	−	0.866025i	−0.0161290	−	0.0279363i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.82843			−0.0910503
$966$		0			0
$967$	−48.0000			−1.54358			−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000			−1.54358			−0.771788	+	0.635880i	$0.780637\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	11.3137	+	19.5959i	0.363074	+	0.628863i	0.988465	−	0.151449i	$-0.0483938\pi$
$971$	11.3137	+	19.5959i	0.363074	+	0.628863i	−0.625391	+	0.780312i	$0.715060\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3.00000	−	5.19615i	0.0959785	−	0.166240i	−0.814038	−	0.580812i	$-0.802735\pi$
$977$	3.00000	−	5.19615i	0.0959785	−	0.166240i	0.910017	+	0.414572i	$0.136069\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	22.6274	−	39.1918i	0.721703	−	1.25003i	−0.238614	−	0.971114i	$-0.576693\pi$
$983$	22.6274	−	39.1918i	0.721703	−	1.25003i	0.960317	−	0.278911i	$-0.0899735\pi$
$984$		0			0
$985$	−8.48528	−	14.6969i	−0.270364	−	0.468283i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−8.00000	−	13.8564i	−0.254385	−	0.440608i
$990$		0			0
$991$	16.0000	−	27.7128i	0.508257	−	0.880327i	−0.491698	−	0.870766i	$-0.663623\pi$
$991$	16.0000	−	27.7128i	0.508257	−	0.880327i	0.999954	−	0.00956046i	$-0.00304324\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	24.0000			0.760851
$996$		0			0
$997$	−23.3345	+	40.4166i	−0.739012	+	1.28001i	0.213929	+	0.976849i	$0.431374\pi$
$997$	−23.3345	+	40.4166i	−0.739012	+	1.28001i	−0.952941	+	0.303157i	$0.901959\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.s.bg.361.1		4
3.2	odd	2		3528.2.s.bh.361.2		4
7.2	even	3	inner	3528.2.s.bg.3313.1		4
7.3	odd	6		3528.2.a.bh.1.1	yes	2
7.4	even	3		3528.2.a.bh.1.2	yes	2
7.5	odd	6	inner	3528.2.s.bg.3313.2		4
7.6	odd	2	inner	3528.2.s.bg.361.2		4
21.2	odd	6		3528.2.s.bh.3313.2		4
21.5	even	6		3528.2.s.bh.3313.1		4
21.11	odd	6		3528.2.a.bg.1.1	✓	2
21.17	even	6		3528.2.a.bg.1.2	yes	2
21.20	even	2		3528.2.s.bh.361.1		4
28.3	even	6		7056.2.a.cn.1.1		2
28.11	odd	6		7056.2.a.cn.1.2		2
84.11	even	6		7056.2.a.cp.1.1		2
84.59	odd	6		7056.2.a.cp.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3528.2.a.bg.1.1	✓	2	21.11	odd	6
3528.2.a.bg.1.2	yes	2	21.17	even	6
3528.2.a.bh.1.1	yes	2	7.3	odd	6
3528.2.a.bh.1.2	yes	2	7.4	even	3
3528.2.s.bg.361.1		4	1.1	even	1	trivial
3528.2.s.bg.361.2		4	7.6	odd	2	inner
3528.2.s.bg.3313.1		4	7.2	even	3	inner
3528.2.s.bg.3313.2		4	7.5	odd	6	inner
3528.2.s.bh.361.1		4	21.20	even	2
3528.2.s.bh.361.2		4	3.2	odd	2
3528.2.s.bh.3313.1		4	21.5	even	6
3528.2.s.bh.3313.2		4	21.2	odd	6
7056.2.a.cn.1.1		2	28.3	even	6
7056.2.a.cn.1.2		2	28.11	odd	6
7056.2.a.cp.1.1		2	84.11	even	6
7056.2.a.cp.1.2		2	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)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 1.22474i) q^{5} +O(q^{10})\)	\(q+(-0.707107 - 1.22474i) q^{5} +1.41421 q^{13} +(0.707107 - 1.22474i) q^{17} +(-2.00000 - 3.46410i) q^{23} +(1.50000 - 2.59808i) q^{25} +6.00000 q^{29} +(-2.82843 + 4.89898i) q^{31} +(2.00000 + 3.46410i) q^{37} -7.07107 q^{41} +4.00000 q^{43} +(-5.65685 - 9.79796i) q^{47} +(-2.00000 + 3.46410i) q^{53} +(2.82843 - 4.89898i) q^{59} +(-2.12132 - 3.67423i) q^{61} +(-1.00000 - 1.73205i) q^{65} +(-2.00000 + 3.46410i) q^{67} +12.0000 q^{71} +(3.53553 - 6.12372i) q^{73} +(-4.00000 - 6.92820i) q^{79} -5.65685 q^{83} -2.00000 q^{85} +(-6.36396 - 11.0227i) q^{89} +4.24264 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 8 q^{23} + 6 q^{25} + 24 q^{29} + 8 q^{37} + 16 q^{43} - 8 q^{53} - 4 q^{65} - 8 q^{67} + 48 q^{71} - 16 q^{79} - 8 q^{85}+O(q^{100}) \) 4 * q - 8 * q^23 + 6 * q^25 + 24 * q^29 + 8 * q^37 + 16 * q^43 - 8 * q^53 - 4 * q^65 - 8 * q^67 + 48 * q^71 - 16 * q^79 - 8 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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