LMFDB - Embedded newform 3528.2.s.bj.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:	no (minimal twist has level 392)
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.bj.3313.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 - 2.44949i) q^{5} +O(q^{10})$	$q+(-1.41421 - 2.44949i) q^{5} +(3.00000 - 5.19615i) q^{11} +5.65685 q^{13} +(-0.707107 + 1.22474i) q^{17} +(-2.12132 - 3.67423i) q^{19} +(2.00000 + 3.46410i) q^{23} +(-1.50000 + 2.59808i) q^{25} +6.00000 q^{29} +(1.41421 - 2.44949i) q^{31} +(-1.00000 - 1.73205i) q^{37} -1.41421 q^{41} +10.0000 q^{43} +(1.41421 + 2.44949i) q^{47} +(-1.00000 + 1.73205i) q^{53} -16.9706 q^{55} +(-0.707107 + 1.22474i) q^{59} +(-4.24264 - 7.34847i) q^{61} +(-8.00000 - 13.8564i) q^{65} +(-2.00000 + 3.46410i) q^{67} +12.0000 q^{71} +(-4.94975 + 8.57321i) q^{73} +(2.00000 + 3.46410i) q^{79} +1.41421 q^{83} +4.00000 q^{85} +(2.12132 + 3.67423i) q^{89} +(-6.00000 + 10.3923i) q^{95} -12.7279 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 12 q^{11} + 8 q^{23} - 6 q^{25} + 24 q^{29} - 4 q^{37} + 40 q^{43} - 4 q^{53} - 32 q^{65} - 8 q^{67} + 48 q^{71} + 8 q^{79} + 16 q^{85} - 24 q^{95}+O(q^{100}) $ 4 * q + 12 * q^11 + 8 * q^23 - 6 * q^25 + 24 * q^29 - 4 * q^37 + 40 * q^43 - 4 * q^53 - 32 * q^65 - 8 * q^67 + 48 * q^71 + 8 * q^79 + 16 * q^85 - 24 * q^95

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
$5$	−1.41421	−	2.44949i	−0.632456	−	1.09545i	−0.987048	−	0.160424i	$-0.948714\pi$
$5$	−1.41421	−	2.44949i	−0.632456	−	1.09545i	0.354593	−	0.935021i	$-0.384620\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.00000	−	5.19615i	0.904534	−	1.56670i	0.0829925	−	0.996550i	$-0.473552\pi$
$11$	3.00000	−	5.19615i	0.904534	−	1.56670i	0.821541	−	0.570149i	$-0.193114\pi$
$12$		0			0
$13$	5.65685			1.56893			0.784465	−	0.620174i	$-0.212938\pi$
$13$	5.65685			1.56893			0.784465	+	0.620174i	$0.212938\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.707107	+	1.22474i	−0.171499	+	0.297044i	−0.938944	−	0.344070i	$-0.888194\pi$
$17$	−0.707107	+	1.22474i	−0.171499	+	0.297044i	0.767445	+	0.641114i	$0.221528\pi$
$18$		0			0
$19$	−2.12132	−	3.67423i	−0.486664	−	0.842927i	0.513218	−	0.858258i	$-0.328453\pi$
$19$	−2.12132	−	3.67423i	−0.486664	−	0.842927i	−0.999882	+	0.0153309i	$0.995120\pi$
$20$		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
$25$	−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
$31$	1.41421	−	2.44949i	0.254000	−	0.439941i	−0.710623	−	0.703573i	$-0.751587\pi$
$31$	1.41421	−	2.44949i	0.254000	−	0.439941i	0.964623	+	0.263631i	$0.0849203\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.00000	−	1.73205i	−0.164399	−	0.284747i	0.772043	−	0.635571i	$-0.219235\pi$
$37$	−1.00000	−	1.73205i	−0.164399	−	0.284747i	−0.936442	+	0.350823i	$0.885902\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.41421			−0.220863			−0.110432	−	0.993884i	$-0.535223\pi$
$41$	−1.41421			−0.220863			−0.110432	+	0.993884i	$0.535223\pi$
$42$		0			0
$43$	10.0000			1.52499			0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000			1.52499			0.762493	+	0.646997i	$0.223975\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.41421	+	2.44949i	0.206284	+	0.357295i	0.950541	−	0.310599i	$-0.100530\pi$
$47$	1.41421	+	2.44949i	0.206284	+	0.357295i	−0.744257	+	0.667893i	$0.767196\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.00000	+	1.73205i	−0.137361	+	0.237915i	−0.926497	−	0.376303i	$-0.877195\pi$
$53$	−1.00000	+	1.73205i	−0.137361	+	0.237915i	0.789136	+	0.614218i	$0.210529\pi$
$54$		0			0
$55$	−16.9706			−2.28831
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.707107	+	1.22474i	−0.0920575	+	0.159448i	−0.908377	−	0.418153i	$-0.862678\pi$
$59$	−0.707107	+	1.22474i	−0.0920575	+	0.159448i	0.816319	+	0.577601i	$0.196011\pi$
$60$		0			0
$61$	−4.24264	−	7.34847i	−0.543214	−	0.940875i	−0.998717	−	0.0506406i	$-0.983874\pi$
$61$	−4.24264	−	7.34847i	−0.543214	−	0.940875i	0.455502	−	0.890235i	$-0.349460\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−8.00000	−	13.8564i	−0.992278	−	1.71868i
$66$		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
$73$	−4.94975	+	8.57321i	−0.579324	+	1.00342i	0.416233	+	0.909258i	$0.363350\pi$
$73$	−4.94975	+	8.57321i	−0.579324	+	1.00342i	−0.995557	+	0.0941608i	$0.969983\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	0.956325	−	0.292306i	$-0.0944227\pi$
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	−0.731307	+	0.682048i	$0.761089\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.41421			0.155230			0.0776151	−	0.996983i	$-0.475269\pi$
$83$	1.41421			0.155230			0.0776151	+	0.996983i	$0.475269\pi$
$84$		0			0
$85$	4.00000			0.433861
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.12132	+	3.67423i	0.224860	+	0.389468i	0.956277	−	0.292462i	$-0.0944744\pi$
$89$	2.12132	+	3.67423i	0.224860	+	0.389468i	−0.731418	+	0.681930i	$0.761141\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−6.00000	+	10.3923i	−0.615587	+	1.06623i
$96$		0			0
$97$	−12.7279			−1.29232			−0.646162	−	0.763200i	$-0.723627\pi$
$97$	−12.7279			−1.29232			−0.646162	+	0.763200i	$0.723627\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.41421	−	2.44949i	0.140720	−	0.243733i	−0.787048	−	0.616891i	$-0.788392\pi$
$101$	1.41421	−	2.44949i	0.140720	−	0.243733i	0.927768	+	0.373158i	$0.121725\pi$
$102$		0			0
$103$	−7.07107	−	12.2474i	−0.696733	−	1.20678i	−0.969593	−	0.244723i	$-0.921303\pi$
$103$	−7.07107	−	12.2474i	−0.696733	−	1.20678i	0.272860	−	0.962054i	$-0.412030\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.00000	+	3.46410i	0.193347	+	0.334887i	0.946357	−	0.323122i	$-0.104732\pi$
$107$	2.00000	+	3.46410i	0.193347	+	0.334887i	−0.753010	+	0.658009i	$0.771399\pi$
$108$		0			0
$109$	5.00000	−	8.66025i	0.478913	−	0.829502i	−0.520794	−	0.853682i	$-0.674364\pi$
$109$	5.00000	−	8.66025i	0.478913	−	0.829502i	0.999708	+	0.0241802i	$0.00769755\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.00000			−0.376288			−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000			−0.376288			−0.188144	+	0.982141i	$0.560247\pi$
$114$		0			0
$115$	5.65685	−	9.79796i	0.527504	−	0.913664i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−12.5000	−	21.6506i	−1.13636	−	1.96824i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−5.65685			−0.505964
$126$		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
$131$	−10.6066	−	18.3712i	−0.926703	−	1.60510i	−0.788799	−	0.614651i	$-0.789297\pi$
$131$	−10.6066	−	18.3712i	−0.926703	−	1.60510i	−0.137904	−	0.990446i	$-0.544037\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.00000	+	3.46410i	−0.170872	+	0.295958i	−0.938725	−	0.344668i	$-0.887992\pi$
$137$	−2.00000	+	3.46410i	−0.170872	+	0.295958i	0.767853	+	0.640626i	$0.221325\pi$
$138$		0			0
$139$	21.2132			1.79928			0.899640	−	0.436632i	$-0.143829\pi$
$139$	21.2132			1.79928			0.899640	+	0.436632i	$0.143829\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	16.9706	−	29.3939i	1.41915	−	2.45804i
$144$		0			0
$145$	−8.48528	−	14.6969i	−0.704664	−	1.22051i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.00000	−	5.19615i	−0.245770	−	0.425685i	0.716578	−	0.697507i	$-0.245707\pi$
$149$	−3.00000	−	5.19615i	−0.245770	−	0.425685i	−0.962348	+	0.271821i	$0.912374\pi$
$150$		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
$157$	8.48528	−	14.6969i	0.677199	−	1.17294i	−0.298622	−	0.954372i	$-0.596527\pi$
$157$	8.48528	−	14.6969i	0.677199	−	1.17294i	0.975821	−	0.218572i	$-0.0701398\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.00000	−	1.73205i	−0.0783260	−	0.135665i	0.824202	−	0.566296i	$-0.191624\pi$
$163$	−1.00000	−	1.73205i	−0.0783260	−	0.135665i	−0.902528	+	0.430632i	$0.858291\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.82843			−0.218870			−0.109435	−	0.993994i	$-0.534904\pi$
$167$	−2.82843			−0.218870			−0.109435	+	0.993994i	$0.534904\pi$
$168$		0			0
$169$	19.0000			1.46154
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		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	$-0.564618\pi$
$179$	10.0000	−	17.3205i	0.747435	−	1.29460i	0.949048	−	0.315130i	$-0.102048\pi$
$180$		0			0
$181$	−16.9706			−1.26141			−0.630706	−	0.776022i	$-0.717235\pi$
$181$	−16.9706			−1.26141			−0.630706	+	0.776022i	$0.717235\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.82843	+	4.89898i	−0.207950	+	0.360180i
$186$		0			0
$187$	4.24264	+	7.34847i	0.310253	+	0.537373i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.00000	−	3.46410i	−0.144715	−	0.250654i	0.784552	−	0.620063i	$-0.212893\pi$
$191$	−2.00000	−	3.46410i	−0.144715	−	0.250654i	−0.929267	+	0.369410i	$0.879560\pi$
$192$		0			0
$193$	−8.00000	+	13.8564i	−0.575853	+	0.997406i	0.420096	+	0.907480i	$0.361996\pi$
$193$	−8.00000	+	13.8564i	−0.575853	+	0.997406i	−0.995948	+	0.0899262i	$0.971337\pi$
$194$		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
$199$	12.7279	−	22.0454i	0.902258	−	1.56276i	0.0777029	−	0.996977i	$-0.475241\pi$
$199$	12.7279	−	22.0454i	0.902258	−	1.56276i	0.824556	−	0.565781i	$-0.191425\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	2.00000	+	3.46410i	0.139686	+	0.241943i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−25.4558			−1.76082
$210$		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
$215$	−14.1421	−	24.4949i	−0.964486	−	1.67054i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.00000	+	6.92820i	−0.269069	+	0.466041i
$222$		0			0
$223$	−11.3137			−0.757622			−0.378811	−	0.925474i	$-0.623667\pi$
$223$	−11.3137			−0.757622			−0.378811	+	0.925474i	$0.623667\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.94975	+	8.57321i	−0.328526	+	0.569024i	−0.982220	−	0.187735i	$-0.939885\pi$
$227$	−4.94975	+	8.57321i	−0.328526	+	0.569024i	0.653693	+	0.756760i	$0.273219\pi$
$228$		0			0
$229$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$229$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4.00000	+	6.92820i	0.262049	+	0.453882i	0.966786	−	0.255586i	$-0.0822686\pi$
$233$	4.00000	+	6.92820i	0.262049	+	0.453882i	−0.704737	+	0.709468i	$0.748935\pi$
$234$		0			0
$235$	4.00000	−	6.92820i	0.260931	−	0.451946i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4.00000			0.258738			0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000			0.258738			0.129369	+	0.991596i	$0.458705\pi$
$240$		0			0
$241$	−0.707107	+	1.22474i	−0.0455488	+	0.0788928i	−0.887901	−	0.460035i	$-0.847837\pi$
$241$	−0.707107	+	1.22474i	−0.0455488	+	0.0788928i	0.842352	+	0.538927i	$0.181170\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−12.0000	−	20.7846i	−0.763542	−	1.32249i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.41421			0.0892644			0.0446322	−	0.999003i	$-0.485788\pi$
$251$	1.41421			0.0892644			0.0446322	+	0.999003i	$0.485788\pi$
$252$		0			0
$253$	24.0000			1.50887
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.36396	+	11.0227i	0.396973	+	0.687577i	0.993351	−	0.115126i	$-0.0367273\pi$
$257$	6.36396	+	11.0227i	0.396973	+	0.687577i	−0.596378	+	0.802704i	$0.703394\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.00000	+	3.46410i	−0.123325	+	0.213606i	−0.921077	−	0.389380i	$-0.872689\pi$
$263$	−2.00000	+	3.46410i	−0.123325	+	0.213606i	0.797752	+	0.602986i	$0.206023\pi$
$264$		0			0
$265$	5.65685			0.347498
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−14.1421	+	24.4949i	−0.862261	+	1.49348i	0.00747990	+	0.999972i	$0.497619\pi$
$269$	−14.1421	+	24.4949i	−0.862261	+	1.49348i	−0.869741	+	0.493508i	$0.835714\pi$
$270$		0			0
$271$	−5.65685	−	9.79796i	−0.343629	−	0.595184i	0.641474	−	0.767144i	$-0.278323\pi$
$271$	−5.65685	−	9.79796i	−0.343629	−	0.595184i	−0.985104	+	0.171961i	$0.944990\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	9.00000	+	15.5885i	0.542720	+	0.940019i
$276$		0			0
$277$	−7.00000	+	12.1244i	−0.420589	+	0.728482i	−0.995997	−	0.0893846i	$-0.971510\pi$
$277$	−7.00000	+	12.1244i	−0.420589	+	0.728482i	0.575408	+	0.817867i	$0.304843\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	32.0000			1.90896			0.954480	−	0.298275i	$-0.0964112\pi$
$281$	32.0000			1.90896			0.954480	+	0.298275i	$0.0964112\pi$
$282$		0			0
$283$	−10.6066	+	18.3712i	−0.630497	+	1.09205i	0.356953	+	0.934122i	$0.383816\pi$
$283$	−10.6066	+	18.3712i	−0.630497	+	1.09205i	−0.987450	+	0.157931i	$0.949518\pi$
$284$		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
$293$	2.82843			0.165238			0.0826192	−	0.996581i	$-0.473671\pi$
$293$	2.82843			0.165238			0.0826192	+	0.996581i	$0.473671\pi$
$294$		0			0
$295$	4.00000			0.232889
$296$		0			0
$297$		0			0
$298$		0			0
$299$	11.3137	+	19.5959i	0.654289	+	1.13326i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−12.0000	+	20.7846i	−0.687118	+	1.19012i
$306$		0			0
$307$	1.41421			0.0807134			0.0403567	−	0.999185i	$-0.487151\pi$
$307$	1.41421			0.0807134			0.0403567	+	0.999185i	$0.487151\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5.65685	−	9.79796i	0.320771	−	0.555591i	−0.659877	−	0.751374i	$-0.729391\pi$
$311$	5.65685	−	9.79796i	0.320771	−	0.555591i	0.980647	+	0.195783i	$0.0627248\pi$
$312$		0			0
$313$	10.6066	+	18.3712i	0.599521	+	1.03840i	0.992892	+	0.119020i	$0.0379754\pi$
$313$	10.6066	+	18.3712i	0.599521	+	1.03840i	−0.393371	+	0.919380i	$0.628691\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−11.0000	−	19.0526i	−0.617822	−	1.07010i	−0.989882	−	0.141890i	$-0.954682\pi$
$317$	−11.0000	−	19.0526i	−0.617822	−	1.07010i	0.372061	−	0.928208i	$-0.378651\pi$
$318$		0			0
$319$	18.0000	−	31.1769i	1.00781	−	1.74557i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6.00000			0.333849
$324$		0			0
$325$	−8.48528	+	14.6969i	−0.470679	+	0.815239i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	7.00000	+	12.1244i	0.384755	+	0.666415i	0.991735	−	0.128302i	$-0.0409527\pi$
$331$	7.00000	+	12.1244i	0.384755	+	0.666415i	−0.606980	+	0.794717i	$0.707619\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	11.3137			0.618134
$336$		0			0
$337$	−14.0000			−0.762629			−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000			−0.762629			−0.381314	+	0.924445i	$0.624528\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−8.48528	−	14.6969i	−0.459504	−	0.795884i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5.00000	−	8.66025i	0.268414	−	0.464907i	−0.700038	−	0.714105i	$-0.746834\pi$
$347$	5.00000	−	8.66025i	0.268414	−	0.464907i	0.968452	+	0.249198i	$0.0801671\pi$
$348$		0			0
$349$	−16.9706			−0.908413			−0.454207	−	0.890896i	$-0.650077\pi$
$349$	−16.9706			−0.908413			−0.454207	+	0.890896i	$0.650077\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−12.0208	+	20.8207i	−0.639803	+	1.10817i	0.345672	+	0.938355i	$0.387651\pi$
$353$	−12.0208	+	20.8207i	−0.639803	+	1.10817i	−0.985476	+	0.169817i	$0.945682\pi$
$354$		0			0
$355$	−16.9706	−	29.3939i	−0.900704	−	1.56007i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−4.00000	−	6.92820i	−0.211112	−	0.365657i	0.740951	−	0.671559i	$-0.234375\pi$
$359$	−4.00000	−	6.92820i	−0.211112	−	0.365657i	−0.952063	+	0.305903i	$0.901042\pi$
$360$		0			0
$361$	0.500000	−	0.866025i	0.0263158	−	0.0455803i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	28.0000			1.46559
$366$		0			0
$367$	2.82843	−	4.89898i	0.147643	−	0.255725i	−0.782713	−	0.622383i	$-0.786165\pi$
$367$	2.82843	−	4.89898i	0.147643	−	0.255725i	0.930356	+	0.366658i	$0.119498\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−13.0000	−	22.5167i	−0.673114	−	1.16587i	−0.977016	−	0.213165i	$-0.931623\pi$
$373$	−13.0000	−	22.5167i	−0.673114	−	1.16587i	0.303902	−	0.952703i	$-0.401711\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	33.9411			1.74806
$378$		0			0
$379$	−2.00000			−0.102733			−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000			−0.102733			−0.0513665	+	0.998680i	$0.516358\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.07107	−	12.2474i	−0.361315	−	0.625815i	0.626863	−	0.779130i	$-0.284339\pi$
$383$	−7.07107	−	12.2474i	−0.361315	−	0.625815i	−0.988177	+	0.153314i	$0.951005\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−15.0000	+	25.9808i	−0.760530	+	1.31728i	0.182047	+	0.983290i	$0.441728\pi$
$389$	−15.0000	+	25.9808i	−0.760530	+	1.31728i	−0.942578	+	0.333987i	$0.891606\pi$
$390$		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
$397$	14.1421	+	24.4949i	0.709773	+	1.22936i	0.964941	+	0.262467i	$0.0845360\pi$
$397$	14.1421	+	24.4949i	0.709773	+	1.22936i	−0.255168	+	0.966897i	$0.582131\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	0.548911	−	0.835881i	$-0.315043\pi$
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	−0.998350	+	0.0574304i	$0.981709\pi$
$402$		0			0
$403$	8.00000	−	13.8564i	0.398508	−	0.690237i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−12.0000			−0.594818
$408$		0			0
$409$	−2.12132	+	3.67423i	−0.104893	+	0.181679i	−0.913694	−	0.406402i	$-0.866783\pi$
$409$	−2.12132	+	3.67423i	−0.104893	+	0.181679i	0.808802	+	0.588081i	$0.200117\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−2.00000	−	3.46410i	−0.0981761	−	0.170046i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	32.5269			1.58904			0.794522	−	0.607236i	$-0.207722\pi$
$419$	32.5269			1.58904			0.794522	+	0.607236i	$0.207722\pi$
$420$		0			0
$421$	−18.0000			−0.877266			−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000			−0.877266			−0.438633	+	0.898666i	$0.644537\pi$
$422$		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
$431$	18.0000	−	31.1769i	0.867029	−	1.50174i	0.00201168	−	0.999998i	$-0.499360\pi$
$431$	18.0000	−	31.1769i	0.867029	−	1.50174i	0.865018	−	0.501741i	$-0.167307\pi$
$432$		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
$437$	8.48528	−	14.6969i	0.405906	−	0.703050i
$438$		0			0
$439$	−8.48528	−	14.6969i	−0.404980	−	0.701447i	0.589339	−	0.807886i	$-0.299388\pi$
$439$	−8.48528	−	14.6969i	−0.404980	−	0.701447i	−0.994319	+	0.106439i	$0.966055\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	18.0000	+	31.1769i	0.855206	+	1.48126i	0.876454	+	0.481486i	$0.159903\pi$
$443$	18.0000	+	31.1769i	0.855206	+	1.48126i	−0.0212481	+	0.999774i	$0.506764\pi$
$444$		0			0
$445$	6.00000	−	10.3923i	0.284427	−	0.492642i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.00000			0.0943858			0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000			0.0943858			0.0471929	+	0.998886i	$0.484972\pi$
$450$		0			0
$451$	−4.24264	+	7.34847i	−0.199778	+	0.346026i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	12.0000	+	20.7846i	0.561336	+	0.972263i	0.997380	+	0.0723376i	$0.0230459\pi$
$457$	12.0000	+	20.7846i	0.561336	+	0.972263i	−0.436044	+	0.899925i	$0.643621\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−22.6274			−1.05386			−0.526932	−	0.849907i	$-0.676658\pi$
$461$	−22.6274			−1.05386			−0.526932	+	0.849907i	$0.676658\pi$
$462$		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
$467$	10.6066	+	18.3712i	0.490815	+	0.850117i	0.999944	−	0.0105737i	$-0.00336576\pi$
$467$	10.6066	+	18.3712i	0.490815	+	0.850117i	−0.509129	+	0.860690i	$0.670032\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	30.0000	−	51.9615i	1.37940	−	2.38919i
$474$		0			0
$475$	12.7279			0.583997
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9.89949	+	17.1464i	−0.452319	+	0.783440i	−0.998530	−	0.0542078i	$-0.982737\pi$
$479$	−9.89949	+	17.1464i	−0.452319	+	0.783440i	0.546210	+	0.837648i	$0.316070\pi$
$480$		0			0
$481$	−5.65685	−	9.79796i	−0.257930	−	0.446748i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	18.0000	+	31.1769i	0.817338	+	1.41567i
$486$		0			0
$487$	−18.0000	+	31.1769i	−0.815658	+	1.41276i	0.0931967	+	0.995648i	$0.470291\pi$
$487$	−18.0000	+	31.1769i	−0.815658	+	1.41276i	−0.908855	+	0.417113i	$0.863042\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−28.0000			−1.26362			−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000			−1.26362			−0.631811	+	0.775122i	$0.717688\pi$
$492$		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
$499$	14.0000	+	24.2487i	0.626726	+	1.08552i	0.988204	+	0.153141i	$0.0489388\pi$
$499$	14.0000	+	24.2487i	0.626726	+	1.08552i	−0.361478	+	0.932381i	$0.617728\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−5.65685			−0.252227			−0.126113	−	0.992016i	$-0.540250\pi$
$503$	−5.65685			−0.252227			−0.126113	+	0.992016i	$0.540250\pi$
$504$		0			0
$505$	−8.00000			−0.355995
$506$		0			0
$507$		0			0
$508$		0			0
$509$	19.7990	+	34.2929i	0.877575	+	1.52000i	0.853994	+	0.520282i	$0.174173\pi$
$509$	19.7990	+	34.2929i	0.877575	+	1.52000i	0.0235804	+	0.999722i	$0.492493\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−20.0000	+	34.6410i	−0.881305	+	1.52647i
$516$		0			0
$517$	16.9706			0.746364
$518$		0			0
$519$		0			0
$520$		0			0
$521$	16.2635	−	28.1691i	0.712515	−	1.23411i	−0.251395	−	0.967885i	$-0.580889\pi$
$521$	16.2635	−	28.1691i	0.712515	−	1.23411i	0.963910	−	0.266228i	$-0.0857773\pi$
$522$		0			0
$523$	16.2635	+	28.1691i	0.711151	+	1.23175i	0.964425	+	0.264356i	$0.0851592\pi$
$523$	16.2635	+	28.1691i	0.711151	+	1.23175i	−0.253274	+	0.967395i	$0.581507\pi$
$524$		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
$533$	−8.00000			−0.346518
$534$		0			0
$535$	5.65685	−	9.79796i	0.244567	−	0.423603i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−5.00000	−	8.66025i	−0.214967	−	0.372333i	0.738296	−	0.674477i	$-0.235631\pi$
$541$	−5.00000	−	8.66025i	−0.214967	−	0.372333i	−0.953262	+	0.302144i	$0.902298\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−28.2843			−1.21157
$546$		0			0
$547$	14.0000			0.598597			0.299298	−	0.954160i	$-0.403247\pi$
$547$	14.0000			0.598597			0.299298	+	0.954160i	$0.403247\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−12.7279	−	22.0454i	−0.542228	−	0.939166i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	9.00000	−	15.5885i	0.381342	−	0.660504i	−0.609912	−	0.792469i	$-0.708795\pi$
$557$	9.00000	−	15.5885i	0.381342	−	0.660504i	0.991254	+	0.131965i	$0.0421286\pi$
$558$		0			0
$559$	56.5685			2.39259
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10.6066	−	18.3712i	0.447015	−	0.774253i	−0.551175	−	0.834390i	$-0.685820\pi$
$563$	10.6066	−	18.3712i	0.447015	−	0.774253i	0.998190	+	0.0601369i	$0.0191537\pi$
$564$		0			0
$565$	5.65685	+	9.79796i	0.237986	+	0.412203i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11.0000	−	19.0526i	−0.461144	−	0.798725i	0.537874	−	0.843025i	$-0.319228\pi$
$569$	−11.0000	−	19.0526i	−0.461144	−	0.798725i	−0.999018	+	0.0443003i	$0.985894\pi$
$570$		0			0
$571$	5.00000	−	8.66025i	0.209243	−	0.362420i	−0.742233	−	0.670142i	$-0.766233\pi$
$571$	5.00000	−	8.66025i	0.209243	−	0.362420i	0.951476	+	0.307722i	$0.0995665\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−12.0000			−0.500435
$576$		0			0
$577$	10.6066	−	18.3712i	0.441559	−	0.764802i	−0.556247	−	0.831017i	$-0.687759\pi$
$577$	10.6066	−	18.3712i	0.441559	−	0.764802i	0.997805	+	0.0662152i	$0.0210924\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	6.00000	+	10.3923i	0.248495	+	0.430405i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.24264			−0.175113			−0.0875563	−	0.996160i	$-0.527906\pi$
$587$	−4.24264			−0.175113			−0.0875563	+	0.996160i	$0.527906\pi$
$588$		0			0
$589$	−12.0000			−0.494451
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.53553	−	6.12372i	−0.145187	−	0.251471i	0.784256	−	0.620438i	$-0.213045\pi$
$593$	−3.53553	−	6.12372i	−0.145187	−	0.251471i	−0.929443	+	0.368967i	$0.879712\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		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
$605$	−35.3553	+	61.2372i	−1.43740	+	2.48965i
$606$		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
$613$	−21.0000	+	36.3731i	−0.848182	+	1.46909i	0.0346469	+	0.999400i	$0.488969\pi$
$613$	−21.0000	+	36.3731i	−0.848182	+	1.46909i	−0.882829	+	0.469695i	$0.844364\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−22.0000			−0.885687			−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000			−0.885687			−0.442843	+	0.896599i	$0.646030\pi$
$618$		0			0
$619$	2.12132	−	3.67423i	0.0852631	−	0.147680i	−0.820240	−	0.572019i	$-0.806160\pi$
$619$	2.12132	−	3.67423i	0.0852631	−	0.147680i	0.905503	+	0.424339i	$0.139494\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	15.5000	+	26.8468i	0.620000	+	1.07387i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.82843			0.112777
$630$		0			0
$631$	28.0000			1.11466			0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000			1.11466			0.557331	+	0.830290i	$0.311825\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	11.3137	+	19.5959i	0.448971	+	0.777640i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.00000	+	5.19615i	−0.118493	+	0.205236i	−0.919171	−	0.393860i	$-0.871140\pi$
$641$	−3.00000	+	5.19615i	−0.118493	+	0.205236i	0.800678	+	0.599095i	$0.204473\pi$
$642$		0			0
$643$	−35.3553			−1.39428			−0.697139	−	0.716936i	$-0.745544\pi$
$643$	−35.3553			−1.39428			−0.697139	+	0.716936i	$0.745544\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	21.2132	−	36.7423i	0.833977	−	1.44449i	−0.0608835	−	0.998145i	$-0.519392\pi$
$647$	21.2132	−	36.7423i	0.833977	−	1.44449i	0.894861	−	0.446346i	$-0.147275\pi$
$648$		0			0
$649$	4.24264	+	7.34847i	0.166538	+	0.288453i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−13.0000	−	22.5167i	−0.508729	−	0.881145i	−0.999949	−	0.0101092i	$-0.996782\pi$
$653$	−13.0000	−	22.5167i	−0.508729	−	0.881145i	0.491220	−	0.871036i	$-0.336551\pi$
$654$		0			0
$655$	−30.0000	+	51.9615i	−1.17220	+	2.03030i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6.00000			−0.233727			−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000			−0.233727			−0.116863	+	0.993148i	$0.537284\pi$
$660$		0			0
$661$	−18.3848	+	31.8434i	−0.715085	+	1.23856i	0.247842	+	0.968801i	$0.420279\pi$
$661$	−18.3848	+	31.8434i	−0.715085	+	1.23856i	−0.962927	+	0.269763i	$0.913055\pi$
$662$		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
$671$	−50.9117			−1.96542
$672$		0			0
$673$	−44.0000			−1.69608			−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000			−1.69608			−0.848038	+	0.529936i	$0.822216\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	16.9706	+	29.3939i	0.652232	+	1.12970i	0.982580	+	0.185839i	$0.0595004\pi$
$677$	16.9706	+	29.3939i	0.652232	+	1.12970i	−0.330348	+	0.943859i	$0.607166\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	2.00000	−	3.46410i	0.0765279	−	0.132550i	−0.825222	−	0.564809i	$-0.808950\pi$
$683$	2.00000	−	3.46410i	0.0765279	−	0.132550i	0.901750	+	0.432259i	$0.142283\pi$
$684$		0			0
$685$	11.3137			0.432275
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.65685	+	9.79796i	−0.215509	+	0.373273i
$690$		0			0
$691$	21.9203	+	37.9671i	0.833888	+	1.44434i	0.894933	+	0.446201i	$0.147223\pi$
$691$	21.9203	+	37.9671i	0.833888	+	1.44434i	−0.0610448	+	0.998135i	$0.519443\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−30.0000	−	51.9615i	−1.13796	−	1.97101i
$696$		0			0
$697$	1.00000	−	1.73205i	0.0378777	−	0.0656061i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−38.0000			−1.43524			−0.717620	−	0.696435i	$-0.754769\pi$
$701$	−38.0000			−1.43524			−0.717620	+	0.696435i	$0.754769\pi$
$702$		0			0
$703$	−4.24264	+	7.34847i	−0.160014	+	0.277153i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−17.0000	−	29.4449i	−0.638448	−	1.10583i	−0.985773	−	0.168080i	$-0.946243\pi$
$709$	−17.0000	−	29.4449i	−0.638448	−	1.10583i	0.347325	−	0.937745i	$-0.387090\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	11.3137			0.423702
$714$		0			0
$715$	−96.0000			−3.59020
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.41421	+	2.44949i	0.0527413	+	0.0913506i	0.891191	−	0.453629i	$-0.149871\pi$
$719$	1.41421	+	2.44949i	0.0527413	+	0.0913506i	−0.838449	+	0.544979i	$0.816537\pi$
$720$		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
$727$	25.4558			0.944105			0.472052	−	0.881570i	$-0.343513\pi$
$727$	25.4558			0.944105			0.472052	+	0.881570i	$0.343513\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−7.07107	+	12.2474i	−0.261533	+	0.452988i
$732$		0			0
$733$	4.24264	+	7.34847i	0.156706	+	0.271422i	0.933679	−	0.358112i	$-0.116579\pi$
$733$	4.24264	+	7.34847i	0.156706	+	0.271422i	−0.776973	+	0.629534i	$0.783246\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	12.0000	+	20.7846i	0.442026	+	0.765611i
$738$		0			0
$739$	3.00000	−	5.19615i	0.110357	−	0.191144i	−0.805557	−	0.592518i	$-0.798134\pi$
$739$	3.00000	−	5.19615i	0.110357	−	0.191144i	0.915914	+	0.401374i	$0.131467\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	24.0000			0.880475			0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000			0.880475			0.440237	+	0.897881i	$0.354894\pi$
$744$		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
$751$	6.00000	+	10.3923i	0.218943	+	0.379221i	0.954485	−	0.298259i	$-0.0964058\pi$
$751$	6.00000	+	10.3923i	0.218943	+	0.379221i	−0.735542	+	0.677479i	$0.763072\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	22.6274			0.823496
$756$		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
$761$	13.4350	+	23.2702i	0.487019	+	0.843542i	0.999889	−	0.0149244i	$-0.00475076\pi$
$761$	13.4350	+	23.2702i	0.487019	+	0.843542i	−0.512869	+	0.858467i	$0.671417\pi$
$762$		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
$769$	29.6985			1.07095			0.535477	−	0.844550i	$-0.320132\pi$
$769$	29.6985			1.07095			0.535477	+	0.844550i	$0.320132\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	9.89949	−	17.1464i	0.356060	−	0.616714i	−0.631239	−	0.775589i	$-0.717453\pi$
$773$	9.89949	−	17.1464i	0.356060	−	0.616714i	0.987299	+	0.158874i	$0.0507865\pi$
$774$		0			0
$775$	4.24264	+	7.34847i	0.152400	+	0.263965i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.00000	+	5.19615i	0.107486	+	0.186171i
$780$		0			0
$781$	36.0000	−	62.3538i	1.28818	−	2.23120i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−48.0000			−1.71319
$786$		0			0
$787$	−16.2635	+	28.1691i	−0.579730	+	1.00412i	0.415780	+	0.909465i	$0.363508\pi$
$787$	−16.2635	+	28.1691i	−0.579730	+	1.00412i	−0.995510	+	0.0946561i	$0.969825\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−24.0000	−	41.5692i	−0.852265	−	1.47617i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	39.5980			1.40263			0.701316	−	0.712850i	$-0.252596\pi$
$797$	39.5980			1.40263			0.701316	+	0.712850i	$0.252596\pi$
$798$		0			0
$799$	−4.00000			−0.141510
$800$		0			0
$801$		0			0
$802$		0			0
$803$	29.6985	+	51.4393i	1.04804	+	1.81525i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−16.0000	+	27.7128i	−0.562530	+	0.974331i	0.434745	+	0.900554i	$0.356839\pi$
$809$	−16.0000	+	27.7128i	−0.562530	+	0.974331i	−0.997275	+	0.0737769i	$0.976495\pi$
$810$		0			0
$811$	−7.07107			−0.248299			−0.124149	−	0.992264i	$-0.539620\pi$
$811$	−7.07107			−0.248299			−0.124149	+	0.992264i	$0.539620\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.82843	+	4.89898i	−0.0990755	+	0.171604i
$816$		0			0
$817$	−21.2132	−	36.7423i	−0.742156	−	1.28545i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9.00000	−	15.5885i	−0.314102	−	0.544041i	0.665144	−	0.746715i	$-0.268370\pi$
$821$	−9.00000	−	15.5885i	−0.314102	−	0.544041i	−0.979246	+	0.202674i	$0.935037\pi$
$822$		0			0
$823$	28.0000	−	48.4974i	0.976019	−	1.69051i	0.299487	−	0.954100i	$-0.403185\pi$
$823$	28.0000	−	48.4974i	0.976019	−	1.69051i	0.676532	−	0.736413i	$-0.263482\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	4.00000			0.139094			0.0695468	−	0.997579i	$-0.477845\pi$
$827$	4.00000			0.139094			0.0695468	+	0.997579i	$0.477845\pi$
$828$		0			0
$829$	12.7279	−	22.0454i	0.442059	−	0.765669i	−0.555783	−	0.831327i	$-0.687582\pi$
$829$	12.7279	−	22.0454i	0.442059	−	0.765669i	0.997842	+	0.0656587i	$0.0209148\pi$
$830$		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
$839$	−31.1127			−1.07413			−0.537065	−	0.843541i	$-0.680467\pi$
$839$	−31.1127			−1.07413			−0.537065	+	0.843541i	$0.680467\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−26.8701	−	46.5403i	−0.924358	−	1.60104i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4.00000	−	6.92820i	0.137118	−	0.237496i
$852$		0			0
$853$	16.9706			0.581061			0.290531	−	0.956866i	$-0.406168\pi$
$853$	16.9706			0.581061			0.290531	+	0.956866i	$0.406168\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3.53553	−	6.12372i	0.120772	−	0.209182i	−0.799301	−	0.600931i	$-0.794796\pi$
$857$	3.53553	−	6.12372i	0.120772	−	0.209182i	0.920072	+	0.391749i	$0.128130\pi$
$858$		0			0
$859$	7.77817	+	13.4722i	0.265388	+	0.459665i	0.967665	−	0.252238i	$-0.0811667\pi$
$859$	7.77817	+	13.4722i	0.265388	+	0.459665i	−0.702277	+	0.711904i	$0.747833\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	22.0000	+	38.1051i	0.748889	+	1.29711i	0.948356	+	0.317209i	$0.102746\pi$
$863$	22.0000	+	38.1051i	0.748889	+	1.29711i	−0.199467	+	0.979905i	$0.563921\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	24.0000			0.814144
$870$		0			0
$871$	−11.3137	+	19.5959i	−0.383350	+	0.663982i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	19.0000	+	32.9090i	0.641584	+	1.11126i	0.985079	+	0.172102i	$0.0550559\pi$
$877$	19.0000	+	32.9090i	0.641584	+	1.11126i	−0.343495	+	0.939155i	$0.611611\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	38.1838			1.28644			0.643222	−	0.765680i	$-0.277597\pi$
$881$	38.1838			1.28644			0.643222	+	0.765680i	$0.277597\pi$
$882$		0			0
$883$	−28.0000			−0.942275			−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000			−0.942275			−0.471138	+	0.882060i	$0.656156\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−18.3848	−	31.8434i	−0.617300	−	1.06920i	−0.989976	−	0.141234i	$-0.954893\pi$
$887$	−18.3848	−	31.8434i	−0.617300	−	1.06920i	0.372676	−	0.927962i	$-0.378440\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	6.00000	−	10.3923i	0.200782	−	0.347765i
$894$		0			0
$895$	−56.5685			−1.89088
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8.48528	−	14.6969i	0.283000	−	0.490170i
$900$		0			0
$901$	−1.41421	−	2.44949i	−0.0471143	−	0.0816043i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	24.0000	+	41.5692i	0.797787	+	1.38181i
$906$		0			0
$907$	10.0000	−	17.3205i	0.332045	−	0.575118i	−0.650868	−	0.759191i	$-0.725595\pi$
$907$	10.0000	−	17.3205i	0.332045	−	0.575118i	0.982913	+	0.184073i	$0.0589282\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	48.0000			1.59031			0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000			1.59031			0.795155	+	0.606406i	$0.207389\pi$
$912$		0			0
$913$	4.24264	−	7.34847i	0.140411	−	0.243199i
$914$		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
$923$	67.8823			2.23437
$924$		0			0
$925$	6.00000			0.197279
$926$		0			0
$927$		0			0
$928$		0			0
$929$	16.2635	+	28.1691i	0.533587	+	0.924199i	0.999230	+	0.0392269i	$0.0124895\pi$
$929$	16.2635	+	28.1691i	0.533587	+	0.924199i	−0.465644	+	0.884972i	$0.654177\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	12.0000	−	20.7846i	0.392442	−	0.679729i
$936$		0			0
$937$	21.2132			0.693005			0.346503	−	0.938049i	$-0.387369\pi$
$937$	21.2132			0.693005			0.346503	+	0.938049i	$0.387369\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	15.5563	−	26.9444i	0.507122	−	0.878362i	−0.492844	−	0.870118i	$-0.664043\pi$
$941$	15.5563	−	26.9444i	0.507122	−	0.878362i	0.999966	−	0.00824396i	$-0.00262416\pi$
$942$		0			0
$943$	−2.82843	−	4.89898i	−0.0921063	−	0.159533i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−21.0000	−	36.3731i	−0.682408	−	1.18197i	−0.974244	−	0.225497i	$-0.927599\pi$
$947$	−21.0000	−	36.3731i	−0.682408	−	1.18197i	0.291835	−	0.956469i	$-0.405734\pi$
$948$		0			0
$949$	−28.0000	+	48.4974i	−0.908918	+	1.57429i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	26.0000			0.842223			0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000			0.842223			0.421111	+	0.907009i	$0.361640\pi$
$954$		0			0
$955$	−5.65685	+	9.79796i	−0.183052	+	0.317055i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	11.5000	+	19.9186i	0.370968	+	0.642535i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	45.2548			1.45680
$966$		0			0
$967$	60.0000			1.92947			0.964735	−	0.263223i	$-0.0847856\pi$
$967$	60.0000			1.92947			0.964735	+	0.263223i	$0.0847856\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	16.2635	+	28.1691i	0.521919	+	0.903990i	0.999675	+	0.0254978i	$0.00811707\pi$
$971$	16.2635	+	28.1691i	0.521919	+	0.903990i	−0.477756	+	0.878493i	$0.658550\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	6.00000	−	10.3923i	0.191957	−	0.332479i	−0.753942	−	0.656941i	$-0.771850\pi$
$977$	6.00000	−	10.3923i	0.191957	−	0.332479i	0.945899	+	0.324462i	$0.105183\pi$
$978$		0			0
$979$	25.4558			0.813572
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−26.8701	+	46.5403i	−0.857022	+	1.48441i	0.0177349	+	0.999843i	$0.494355\pi$
$983$	−26.8701	+	46.5403i	−0.857022	+	1.48441i	−0.874757	+	0.484562i	$0.838979\pi$
$984$		0			0
$985$	25.4558	+	44.0908i	0.811091	+	1.40485i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	20.0000	+	34.6410i	0.635963	+	1.10152i
$990$		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
$995$	−72.0000			−2.28255
$996$		0			0
$997$	−4.24264	+	7.34847i	−0.134366	+	0.232728i	−0.925355	−	0.379102i	$-0.876233\pi$
$997$	−4.24264	+	7.34847i	−0.134366	+	0.232728i	0.790989	+	0.611830i	$0.209566\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.s.bj.361.1		4
3.2	odd	2		392.2.i.h.361.2		4
7.2	even	3	inner	3528.2.s.bj.3313.1		4
7.3	odd	6		3528.2.a.be.1.1		2
7.4	even	3		3528.2.a.be.1.2		2
7.5	odd	6	inner	3528.2.s.bj.3313.2		4
7.6	odd	2	inner	3528.2.s.bj.361.2		4
12.11	even	2		784.2.i.n.753.1		4
21.2	odd	6		392.2.i.h.177.2		4
21.5	even	6		392.2.i.h.177.1		4
21.11	odd	6		392.2.a.g.1.1	✓	2
21.17	even	6		392.2.a.g.1.2	yes	2
21.20	even	2		392.2.i.h.361.1		4
28.3	even	6		7056.2.a.ct.1.1		2
28.11	odd	6		7056.2.a.ct.1.2		2
84.11	even	6		784.2.a.k.1.2		2
84.23	even	6		784.2.i.n.177.1		4
84.47	odd	6		784.2.i.n.177.2		4
84.59	odd	6		784.2.a.k.1.1		2
84.83	odd	2		784.2.i.n.753.2		4
105.59	even	6		9800.2.a.bv.1.1		2
105.74	odd	6		9800.2.a.bv.1.2		2
168.11	even	6		3136.2.a.bp.1.1		2
168.53	odd	6		3136.2.a.bk.1.2		2
168.59	odd	6		3136.2.a.bp.1.2		2
168.101	even	6		3136.2.a.bk.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.a.g.1.1	✓	2	21.11	odd	6
392.2.a.g.1.2	yes	2	21.17	even	6
392.2.i.h.177.1		4	21.5	even	6
392.2.i.h.177.2		4	21.2	odd	6
392.2.i.h.361.1		4	21.20	even	2
392.2.i.h.361.2		4	3.2	odd	2
784.2.a.k.1.1		2	84.59	odd	6
784.2.a.k.1.2		2	84.11	even	6
784.2.i.n.177.1		4	84.23	even	6
784.2.i.n.177.2		4	84.47	odd	6
784.2.i.n.753.1		4	12.11	even	2
784.2.i.n.753.2		4	84.83	odd	2
3136.2.a.bk.1.1		2	168.101	even	6
3136.2.a.bk.1.2		2	168.53	odd	6
3136.2.a.bp.1.1		2	168.11	even	6
3136.2.a.bp.1.2		2	168.59	odd	6
3528.2.a.be.1.1		2	7.3	odd	6
3528.2.a.be.1.2		2	7.4	even	3
3528.2.s.bj.361.1		4	1.1	even	1	trivial
3528.2.s.bj.361.2		4	7.6	odd	2	inner
3528.2.s.bj.3313.1		4	7.2	even	3	inner
3528.2.s.bj.3313.2		4	7.5	odd	6	inner
7056.2.a.ct.1.1		2	28.3	even	6
7056.2.a.ct.1.2		2	28.11	odd	6
9800.2.a.bv.1.1		2	105.59	even	6
9800.2.a.bv.1.2		2	105.74	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3528.s (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.41421 - 2.44949i) q^{5} +O(q^{10})\)	\(q+(-1.41421 - 2.44949i) q^{5} +(3.00000 - 5.19615i) q^{11} +5.65685 q^{13} +(-0.707107 + 1.22474i) q^{17} +(-2.12132 - 3.67423i) q^{19} +(2.00000 + 3.46410i) q^{23} +(-1.50000 + 2.59808i) q^{25} +6.00000 q^{29} +(1.41421 - 2.44949i) q^{31} +(-1.00000 - 1.73205i) q^{37} -1.41421 q^{41} +10.0000 q^{43} +(1.41421 + 2.44949i) q^{47} +(-1.00000 + 1.73205i) q^{53} -16.9706 q^{55} +(-0.707107 + 1.22474i) q^{59} +(-4.24264 - 7.34847i) q^{61} +(-8.00000 - 13.8564i) q^{65} +(-2.00000 + 3.46410i) q^{67} +12.0000 q^{71} +(-4.94975 + 8.57321i) q^{73} +(2.00000 + 3.46410i) q^{79} +1.41421 q^{83} +4.00000 q^{85} +(2.12132 + 3.67423i) q^{89} +(-6.00000 + 10.3923i) q^{95} -12.7279 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 12 q^{11} + 8 q^{23} - 6 q^{25} + 24 q^{29} - 4 q^{37} + 40 q^{43} - 4 q^{53} - 32 q^{65} - 8 q^{67} + 48 q^{71} + 8 q^{79} + 16 q^{85} - 24 q^{95}+O(q^{100}) \) 4 * q + 12 * q^11 + 8 * q^23 - 6 * q^25 + 24 * q^29 - 4 * q^37 + 40 * q^43 - 4 * q^53 - 32 * q^65 - 8 * q^67 + 48 * q^71 + 8 * q^79 + 16 * q^85 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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