LMFDB - Embedded newform 3420.2.dg.c.521.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3420,2,Mod(521,3420)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3420, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3420.521");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3420.dg (of order $6$, degree $2$, 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:	$27.3088374913$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			521.15
Character	$\chi$	$=$	3420.521
Dual form			3420.2.dg.c.2501.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 + 0.500000i) q^{5} +3.67228 q^{7} +O(q^{10})$	$q+(0.866025 + 0.500000i) q^{5} +3.67228 q^{7} -2.00000i q^{11} +(0.349493 - 0.201780i) q^{13} +(6.30917 + 3.64260i) q^{17} +(4.34971 - 0.282927i) q^{19} +(-6.22965 + 3.59669i) q^{23} +(0.500000 + 0.866025i) q^{25} +(3.94987 + 6.84138i) q^{29} -8.62396i q^{31} +(3.18029 + 1.83614i) q^{35} +1.01188i q^{37} +(4.14468 - 7.17880i) q^{41} +(-4.25035 + 7.36183i) q^{43} +(-6.28758 + 3.63013i) q^{47} +6.48563 q^{49} +(-1.47915 - 2.56197i) q^{53} +(1.00000 - 1.73205i) q^{55} +(1.04725 - 1.81389i) q^{59} +(-1.08886 - 1.88596i) q^{61} +0.403559 q^{65} +(10.2282 - 5.90527i) q^{67} +(-6.83581 + 11.8400i) q^{71} +(3.21018 - 5.56019i) q^{73} -7.34456i q^{77} +(10.0246 + 5.78769i) q^{79} -8.70472i q^{83} +(3.64260 + 6.30917i) q^{85} +(0.702112 + 1.21609i) q^{89} +(1.28343 - 0.740991i) q^{91} +(3.90842 + 1.92983i) q^{95} +(-5.04811 - 2.91453i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 16 q^{7}+O(q^{10}) $ 32 * q + 16 * q^7	$ 32 q + 16 q^{7} + 16 q^{25} - 40 q^{43} + 64 q^{49} + 32 q^{55} - 8 q^{61} - 24 q^{67} + 8 q^{73} + 120 q^{79} + 24 q^{91} + 48 q^{97}+O(q^{100}) $ 32 * q + 16 * q^7 + 16 * q^25 - 40 * q^43 + 64 * q^49 + 32 * q^55 - 8 * q^61 - 24 * q^67 + 8 * q^73 + 120 * q^79 + 24 * q^91 + 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times$.

$n$	$1711$	$1901$	$2737$	$3061$
$\chi(n)$	$1$	$-1$	$1$	$e\left(\frac{1}{6}\right)$

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
$5$	0.866025	+	0.500000i	0.387298	+	0.223607i
$5$	0.866025	+	0.500000i	0.387298	+	0.223607i
$6$		0			0
$7$	3.67228			1.38799			0.693995	−	0.719979i	$-0.255849\pi$
$7$	3.67228			1.38799			0.693995	+	0.719979i	$0.255849\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	2.00000i		−	0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$		−	2.00000i		−	0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$		0			0
$13$	0.349493	−	0.201780i	0.0969318	−	0.0559636i	−0.450750	−	0.892650i	$-0.648844\pi$
$13$	0.349493	−	0.201780i	0.0969318	−	0.0559636i	0.547682	+	0.836686i	$0.315510\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.30917	+	3.64260i	1.53020	+	0.883460i	0.999352	+	0.0359922i	$0.0114591\pi$
$17$	6.30917	+	3.64260i	1.53020	+	0.883460i	0.530846	+	0.847468i	$0.321874\pi$
$18$		0			0
$19$	4.34971	−	0.282927i	0.997891	−	0.0649079i
$19$	4.34971	−	0.282927i	0.997891	−	0.0649079i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.22965	+	3.59669i	−1.29897	+	0.749962i	−0.980227	−	0.197878i	$-0.936595\pi$
$23$	−6.22965	+	3.59669i	−1.29897	+	0.749962i	−0.318746	+	0.947840i	$0.603262\pi$
$24$		0			0
$25$	0.500000	+	0.866025i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.94987	+	6.84138i	0.733473	+	1.27041i	0.955390	+	0.295347i	$0.0954354\pi$
$29$	3.94987	+	6.84138i	0.733473	+	1.27041i	−0.221917	+	0.975066i	$0.571231\pi$
$30$		0			0
$31$		−	8.62396i		−	1.54891i	−0.632630	−	0.774454i	$-0.718024\pi$
$31$		−	8.62396i		−	1.54891i	0.632630	−	0.774454i	$-0.281976\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.18029	+	1.83614i	0.537566	+	0.310364i
$36$		0			0
$37$			1.01188i			0.166352i	0.996535	+	0.0831760i	$0.0265064\pi$
$37$			1.01188i			0.166352i	−0.996535	+	0.0831760i	$0.973494\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.14468	−	7.17880i	0.647290	−	1.12114i	−0.336477	−	0.941692i	$-0.609235\pi$
$41$	4.14468	−	7.17880i	0.647290	−	1.12114i	0.983767	−	0.179448i	$-0.0574312\pi$
$42$		0			0
$43$	−4.25035	+	7.36183i	−0.648173	+	1.12267i	0.335386	+	0.942081i	$0.391133\pi$
$43$	−4.25035	+	7.36183i	−0.648173	+	1.12267i	−0.983559	+	0.180587i	$0.942200\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.28758	+	3.63013i	−0.917138	+	0.529510i	−0.882721	−	0.469898i	$-0.844291\pi$
$47$	−6.28758	+	3.63013i	−0.917138	+	0.529510i	−0.0344168	+	0.999408i	$0.510957\pi$
$48$		0			0
$49$	6.48563			0.926518
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.47915	−	2.56197i	−0.203177	−	0.351913i	0.746373	−	0.665528i	$-0.231793\pi$
$53$	−1.47915	−	2.56197i	−0.203177	−	0.351913i	−0.949550	+	0.313614i	$0.898460\pi$
$54$		0			0
$55$	1.00000	−	1.73205i	0.134840	−	0.233550i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.04725	−	1.81389i	0.136340	−	0.236148i	−0.789769	−	0.613405i	$-0.789799\pi$
$59$	1.04725	−	1.81389i	0.136340	−	0.236148i	0.926109	+	0.377257i	$0.123133\pi$
$60$		0			0
$61$	−1.08886	−	1.88596i	−0.139414	−	0.241472i	0.787861	−	0.615853i	$-0.211189\pi$
$61$	−1.08886	−	1.88596i	−0.139414	−	0.241472i	−0.927275	+	0.374381i	$0.877855\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.403559			0.0500554
$66$		0			0
$67$	10.2282	−	5.90527i	1.24958	−	0.721444i	0.278552	−	0.960421i	$-0.410145\pi$
$67$	10.2282	−	5.90527i	1.24958	−	0.721444i	0.971025	+	0.238977i	$0.0768121\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.83581	+	11.8400i	−0.811261	+	1.40515i	0.100721	+	0.994915i	$0.467885\pi$
$71$	−6.83581	+	11.8400i	−0.811261	+	1.40515i	−0.911982	+	0.410231i	$0.865448\pi$
$72$		0			0
$73$	3.21018	−	5.56019i	0.375723	−	0.650771i	−0.614712	−	0.788752i	$-0.710728\pi$
$73$	3.21018	−	5.56019i	0.375723	−	0.650771i	0.990435	+	0.137980i	$0.0440611\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	7.34456i		−	0.836990i
$78$		0			0
$79$	10.0246	+	5.78769i	1.12785	+	0.651166i	0.943394	−	0.331674i	$-0.107613\pi$
$79$	10.0246	+	5.78769i	1.12785	+	0.651166i	0.184459	+	0.982840i	$0.440947\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	8.70472i		−	0.955468i	−0.878505	−	0.477734i	$-0.841458\pi$
$83$		−	8.70472i		−	0.955468i	0.878505	−	0.477734i	$-0.158542\pi$
$84$		0			0
$85$	3.64260	+	6.30917i	0.395095	+	0.684325i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.702112	+	1.21609i	0.0744237	+	0.128906i	0.900836	−	0.434161i	$-0.142955\pi$
$89$	0.702112	+	1.21609i	0.0744237	+	0.128906i	−0.826412	+	0.563066i	$0.809622\pi$
$90$		0			0
$91$	1.28343	−	0.740991i	0.134540	−	0.0776770i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.90842	+	1.92983i	0.400995	+	0.197997i
$96$		0			0
$97$	−5.04811	−	2.91453i	−0.512558	−	0.295926i	0.221326	−	0.975200i	$-0.428961\pi$
$97$	−5.04811	−	2.91453i	−0.512558	−	0.295926i	−0.733885	+	0.679274i	$0.762295\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.33244	+	4.81074i	−0.829109	+	0.478686i	−0.853547	−	0.521015i	$-0.825553\pi$
$101$	−8.33244	+	4.81074i	−0.829109	+	0.478686i	0.0244386	+	0.999701i	$0.492220\pi$
$102$		0			0
$103$		−	9.63276i		−	0.949144i	−0.880217	−	0.474572i	$-0.842603\pi$
$103$		−	9.63276i		−	0.949144i	0.880217	−	0.474572i	$-0.157397\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.24568			−0.410445			−0.205223	−	0.978715i	$-0.565792\pi$
$107$	−4.24568			−0.410445			−0.205223	+	0.978715i	$0.565792\pi$
$108$		0			0
$109$	−10.4278	−	6.02048i	−0.998800	−	0.576657i	−0.0909067	−	0.995859i	$-0.528977\pi$
$109$	−10.4278	−	6.02048i	−0.998800	−	0.576657i	−0.907893	+	0.419202i	$0.862310\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	10.4414			0.982241			0.491121	−	0.871092i	$-0.336587\pi$
$113$	10.4414			0.982241			0.491121	+	0.871092i	$0.336587\pi$
$114$		0			0
$115$	−7.19339			−0.670787
$116$		0			0
$117$		0			0
$118$		0			0
$119$	23.1690	+	13.3766i	2.12390	+	1.22623i
$120$		0			0
$121$	7.00000			0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	−5.69150	+	3.28599i	−0.505039	+	0.291584i	−0.730792	−	0.682600i	$-0.760849\pi$
$127$	−5.69150	+	3.28599i	−0.505039	+	0.291584i	0.225753	+	0.974185i	$0.427516\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.01143	+	2.89335i	0.437851	+	0.252793i	0.702685	−	0.711501i	$-0.251984\pi$
$131$	5.01143	+	2.89335i	0.437851	+	0.252793i	−0.264835	+	0.964294i	$0.585317\pi$
$132$		0			0
$133$	15.9733	−	1.03899i	1.38506	−	0.0900915i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.90997	+	1.10272i	−0.163180	+	0.0942119i	−0.579366	−	0.815068i	$-0.696700\pi$
$137$	−1.90997	+	1.10272i	−0.163180	+	0.0942119i	0.416186	+	0.909279i	$0.363366\pi$
$138$		0			0
$139$	3.85693	+	6.68040i	0.327141	+	0.566624i	0.981943	−	0.189176i	$-0.0605816\pi$
$139$	3.85693	+	6.68040i	0.327141	+	0.566624i	−0.654803	+	0.755800i	$0.727248\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.403559	−	0.698985i	−0.0337473	−	0.0584521i
$144$		0			0
$145$			7.89975i			0.656038i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.424540	−	0.245108i	−0.0347797	−	0.0200800i	0.482509	−	0.875891i	$-0.339725\pi$
$149$	−0.424540	−	0.245108i	−0.0347797	−	0.0200800i	−0.517289	+	0.855811i	$0.673059\pi$
$150$		0			0
$151$		−	1.32114i		−	0.107513i	−0.998554	−	0.0537565i	$-0.982881\pi$
$151$		−	1.32114i		−	0.107513i	0.998554	−	0.0537565i	$-0.0171195\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.31198	−	7.46857i	0.346347	−	0.599890i
$156$		0			0
$157$	−3.01010	+	5.21364i	−0.240232	+	0.416094i	−0.960780	−	0.277311i	$-0.910557\pi$
$157$	−3.01010	+	5.21364i	−0.240232	+	0.416094i	0.720548	+	0.693405i	$0.243890\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−22.8770	+	13.2081i	−1.80296	+	1.04094i
$162$		0			0
$163$	−1.39365			−0.109159			−0.0545795	−	0.998509i	$-0.517382\pi$
$163$	−1.39365			−0.109159			−0.0545795	+	0.998509i	$0.517382\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.78031	−	3.08359i	−0.137765	−	0.238615i	0.788885	−	0.614540i	$-0.210658\pi$
$167$	−1.78031	−	3.08359i	−0.137765	−	0.238615i	−0.926650	+	0.375925i	$0.877325\pi$
$168$		0			0
$169$	−6.41857	+	11.1173i	−0.493736	+	0.855176i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9.65046	+	16.7151i	−0.733711	+	1.27082i	0.221576	+	0.975143i	$0.428880\pi$
$173$	−9.65046	+	16.7151i	−0.733711	+	1.27082i	−0.955287	+	0.295681i	$0.904453\pi$
$174$		0			0
$175$	1.83614	+	3.18029i	0.138799	+	0.240407i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2.86149			0.213878			0.106939	−	0.994266i	$-0.465895\pi$
$179$	2.86149			0.213878			0.106939	+	0.994266i	$0.465895\pi$
$180$		0			0
$181$	20.7335	−	11.9705i	1.54111	−	0.889762i	0.542343	−	0.840157i	$-0.317537\pi$
$181$	20.7335	−	11.9705i	1.54111	−	0.889762i	0.998769	−	0.0496045i	$-0.0157961\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.505940	+	0.876313i	−0.0371974	+	0.0644278i
$186$		0			0
$187$	7.28520	−	12.6183i	0.532747	−	0.922744i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	7.20481i		−	0.521322i	−0.965430	−	0.260661i	$-0.916060\pi$
$191$		−	7.20481i		−	0.521322i	0.965430	−	0.260661i	$-0.0839405\pi$
$192$		0			0
$193$	7.86046	+	4.53824i	0.565808	+	0.326669i	0.755473	−	0.655179i	$-0.227407\pi$
$193$	7.86046	+	4.53824i	0.565808	+	0.326669i	−0.189665	+	0.981849i	$0.560740\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			20.5069i			1.46106i	0.682883	+	0.730528i	$0.260726\pi$
$197$			20.5069i			1.46106i	−0.682883	+	0.730528i	$0.739274\pi$
$198$		0			0
$199$	−13.0967	−	22.6842i	−0.928403	−	1.60804i	−0.785994	−	0.618234i	$-0.787848\pi$
$199$	−13.0967	−	22.6842i	−0.928403	−	1.60804i	−0.142409	−	0.989808i	$-0.545485\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	14.5050	+	25.1235i	1.01805	+	1.76332i
$204$		0			0
$205$	7.17880	−	4.14468i	0.501389	−	0.289477i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.565854	−	8.69941i	−0.0391409	−	0.601751i
$210$		0			0
$211$	−4.78225	−	2.76104i	−0.329224	−	0.190078i	0.326273	−	0.945276i	$-0.394207\pi$
$211$	−4.78225	−	2.76104i	−0.329224	−	0.190078i	−0.655497	+	0.755198i	$0.727541\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−7.36183	+	4.25035i	−0.502072	+	0.289872i
$216$		0			0
$217$		−	31.6696i		−	2.14987i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.94001			0.197767
$222$		0			0
$223$	15.1316	+	8.73622i	1.01329	+	0.585021i	0.912152	−	0.409852i	$-0.134420\pi$
$223$	15.1316	+	8.73622i	1.01329	+	0.585021i	0.101134	+	0.994873i	$0.467753\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−17.0077			−1.12884			−0.564419	−	0.825489i	$-0.690900\pi$
$227$	−17.0077			−1.12884			−0.564419	+	0.825489i	$0.690900\pi$
$228$		0			0
$229$	−2.95271			−0.195120			−0.0975602	−	0.995230i	$-0.531104\pi$
$229$	−2.95271			−0.195120			−0.0975602	+	0.995230i	$0.531104\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5.32007	+	3.07154i	0.348529	+	0.201224i	0.664037	−	0.747699i	$-0.268842\pi$
$233$	5.32007	+	3.07154i	0.348529	+	0.201224i	−0.315508	+	0.948923i	$0.602175\pi$
$234$		0			0
$235$	−7.26027			−0.473608
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	5.46898i		−	0.353759i	−0.984233	−	0.176879i	$-0.943400\pi$
$239$		−	5.46898i		−	0.353759i	0.984233	−	0.176879i	$-0.0566003\pi$
$240$		0			0
$241$	24.8712	−	14.3594i	1.60209	−	0.924969i	0.611026	−	0.791610i	$-0.290757\pi$
$241$	24.8712	−	14.3594i	1.60209	−	0.924969i	0.991068	−	0.133359i	$-0.0425763\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	5.61672	+	3.24281i	0.358839	+	0.207176i
$246$		0			0
$247$	1.46310	−	0.976564i	0.0930950	−	0.0621373i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6.16156	+	3.55738i	−0.388914	+	0.224540i	−0.681690	−	0.731642i	$-0.738754\pi$
$251$	−6.16156	+	3.55738i	−0.388914	+	0.224540i	0.292775	+	0.956181i	$0.405421\pi$
$252$		0			0
$253$	7.19339	+	12.4593i	0.452244	+	0.783310i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.31808	+	16.1394i	0.581246	+	1.00675i	0.995332	+	0.0965094i	$0.0307678\pi$
$257$	9.31808	+	16.1394i	0.581246	+	1.00675i	−0.414086	+	0.910238i	$0.635899\pi$
$258$		0			0
$259$			3.71590i			0.230895i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	10.7794	+	6.22351i	0.664689	+	0.383758i	0.794061	−	0.607838i	$-0.207963\pi$
$263$	10.7794	+	6.22351i	0.664689	+	0.383758i	−0.129373	+	0.991596i	$0.541296\pi$
$264$		0			0
$265$		−	2.95830i		−	0.181727i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.38608	−	4.13281i	0.145482	−	0.251982i	−0.784071	−	0.620671i	$-0.786860\pi$
$269$	2.38608	−	4.13281i	0.145482	−	0.251982i	0.929553	+	0.368690i	$0.120193\pi$
$270$		0			0
$271$	13.8012	−	23.9043i	0.838361	−	1.45208i	−0.0529040	−	0.998600i	$-0.516848\pi$
$271$	13.8012	−	23.9043i	0.838361	−	1.45208i	0.891265	−	0.453484i	$-0.149819\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.73205	−	1.00000i	0.104447	−	0.0603023i
$276$		0			0
$277$	15.6688			0.941445			0.470722	−	0.882281i	$-0.343993\pi$
$277$	15.6688			0.941445			0.470722	+	0.882281i	$0.343993\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.58712	+	2.74897i	0.0946797	+	0.163990i	0.909475	−	0.415759i	$-0.136484\pi$
$281$	1.58712	+	2.74897i	0.0946797	+	0.163990i	−0.814795	+	0.579749i	$0.803151\pi$
$282$		0			0
$283$	−4.66026	+	8.07181i	−0.277024	+	0.479819i	−0.970644	−	0.240522i	$-0.922681\pi$
$283$	−4.66026	+	8.07181i	−0.277024	+	0.479819i	0.693620	+	0.720341i	$0.256015\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	15.2204	−	26.3625i	0.898433	−	1.55613i
$288$		0			0
$289$	18.0371	+	31.2411i	1.06100	+	1.83771i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	23.9046			1.39652			0.698261	−	0.715843i	$-0.253958\pi$
$293$	23.9046			1.39652			0.698261	+	0.715843i	$0.253958\pi$
$294$		0			0
$295$	1.81389	−	1.04725i	0.105609	−	0.0609731i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.45148	+	2.51404i	−0.0839412	+	0.145390i
$300$		0			0
$301$	−15.6085	+	27.0347i	−0.899658	+	1.55825i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	2.17772i		−	0.124696i
$306$		0			0
$307$	25.0076	+	14.4382i	1.42726	+	0.824029i	0.996904	−	0.0786304i	$-0.0250547\pi$
$307$	25.0076	+	14.4382i	1.42726	+	0.824029i	0.430356	+	0.902659i	$0.358388\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	25.6286i		−	1.45327i	−0.687026	−	0.726633i	$-0.741084\pi$
$311$		−	25.6286i		−	1.45327i	0.687026	−	0.726633i	$-0.258916\pi$
$312$		0			0
$313$	−12.0272	−	20.8317i	−0.679816	−	1.17748i	−0.975036	−	0.222047i	$-0.928726\pi$
$313$	−12.0272	−	20.8317i	−0.679816	−	1.17748i	0.295220	−	0.955429i	$-0.404607\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.92085	+	5.05905i	0.164051	+	0.284145i	0.936318	−	0.351154i	$-0.114210\pi$
$317$	2.92085	+	5.05905i	0.164051	+	0.284145i	−0.772267	+	0.635298i	$0.780877\pi$
$318$		0			0
$319$	13.6828	−	7.89975i	0.766088	−	0.442301i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	28.4736	+	14.0592i	1.58431	+	0.782275i
$324$		0			0
$325$	0.349493	+	0.201780i	0.0193864	+	0.0111927i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−23.0897	+	13.3309i	−1.27298	+	0.734954i
$330$		0			0
$331$			9.67560i			0.531819i	0.963998	+	0.265910i	$0.0856722\pi$
$331$			9.67560i			0.531819i	−0.963998	+	0.265910i	$0.914328\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	11.8105			0.645279
$336$		0			0
$337$	24.3715	+	14.0709i	1.32760	+	0.766492i	0.984928	−	0.172962i	$-0.0553339\pi$
$337$	24.3715	+	14.0709i	1.32760	+	0.766492i	0.342674	+	0.939454i	$0.388667\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−17.2479			−0.934027
$342$		0			0
$343$	−1.88892			−0.101992
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−25.3920	−	14.6601i	−1.36312	−	0.786995i	−0.373078	−	0.927800i	$-0.621698\pi$
$347$	−25.3920	−	14.6601i	−1.36312	−	0.786995i	−0.990037	+	0.140805i	$0.955031\pi$
$348$		0			0
$349$	1.69070			0.0905012			0.0452506	−	0.998976i	$-0.485591\pi$
$349$	1.69070			0.0905012			0.0452506	+	0.998976i	$0.485591\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	9.72876i		−	0.517810i	−0.965903	−	0.258905i	$-0.916638\pi$
$353$		−	9.72876i		−	0.517810i	0.965903	−	0.258905i	$-0.0833616\pi$
$354$		0			0
$355$	−11.8400	+	6.83581i	−0.628400	+	0.362807i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−26.8800	−	15.5192i	−1.41867	−	0.819072i	−0.422492	−	0.906367i	$-0.638845\pi$
$359$	−26.8800	−	15.5192i	−1.41867	−	0.819072i	−0.996183	+	0.0872946i	$0.972178\pi$
$360$		0			0
$361$	18.8399	−	2.46130i	0.991574	−	0.129542i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5.56019	−	3.21018i	0.291034	−	0.168028i
$366$		0			0
$367$	−14.8965	−	25.8014i	−0.777590	−	1.34682i	−0.933327	−	0.359026i	$-0.883109\pi$
$367$	−14.8965	−	25.8014i	−0.777590	−	1.34682i	0.155738	−	0.987798i	$-0.450224\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−5.43186	−	9.40825i	−0.282008	−	0.488452i
$372$		0			0
$373$			10.6231i			0.550043i	0.961438	+	0.275022i	$0.0886850\pi$
$373$			10.6231i			0.550043i	−0.961438	+	0.275022i	$0.911315\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.76090	+	1.59401i	0.142194	+	0.0820956i
$378$		0			0
$379$		−	11.7032i		−	0.601154i	−0.953757	−	0.300577i	$-0.902821\pi$
$379$		−	11.7032i		−	0.601154i	0.953757	−	0.300577i	$-0.0971793\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	13.4726	−	23.3352i	0.688416	−	1.19237i	−0.283934	−	0.958844i	$-0.591640\pi$
$383$	13.4726	−	23.3352i	0.688416	−	1.19237i	0.972350	−	0.233528i	$-0.0750270\pi$
$384$		0			0
$385$	3.67228	−	6.36057i	0.187157	−	0.324165i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.44611	−	1.98961i	0.174725	−	0.100877i	−0.410087	−	0.912046i	$-0.634502\pi$
$389$	3.44611	−	1.98961i	0.174725	−	0.100877i	0.584812	+	0.811169i	$0.301168\pi$
$390$		0			0
$391$	−52.4053			−2.65025
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5.78769	+	10.0246i	0.291210	+	0.504391i
$396$		0			0
$397$	−16.5650	+	28.6915i	−0.831375	+	1.43998i	0.0655724	+	0.997848i	$0.479113\pi$
$397$	−16.5650	+	28.6915i	−0.831375	+	1.43998i	−0.896948	+	0.442137i	$0.854221\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.01891	−	3.49685i	0.100819	−	0.174624i	−0.811203	−	0.584764i	$-0.801187\pi$
$401$	2.01891	−	3.49685i	0.100819	−	0.174624i	0.912022	+	0.410140i	$0.134520\pi$
$402$		0			0
$403$	−1.74014	−	3.01401i	−0.0866826	−	0.150139i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.02376			0.100314
$408$		0			0
$409$	−11.9758	+	6.91426i	−0.592167	+	0.341888i	−0.765954	−	0.642895i	$-0.777733\pi$
$409$	−11.9758	+	6.91426i	−0.592167	+	0.341888i	0.173787	+	0.984783i	$0.444400\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.84579	−	6.66110i	0.189239	−	0.327771i
$414$		0			0
$415$	4.35236	−	7.53851i	0.213649	−	0.370051i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			0.363570i			0.0177616i	0.999961	+	0.00888078i	$0.00282688\pi$
$419$			0.363570i			0.0177616i	−0.999961	+	0.00888078i	$0.997173\pi$
$420$		0			0
$421$	−19.1880	−	11.0782i	−0.935167	−	0.539919i	−0.0467247	−	0.998908i	$-0.514878\pi$
$421$	−19.1880	−	11.0782i	−0.935167	−	0.539919i	−0.888442	+	0.458989i	$0.848212\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			7.28520i			0.353384i
$426$		0			0
$427$	−3.99859	−	6.92576i	−0.193505	−	0.335161i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	14.4400	+	25.0108i	0.695550	+	1.20473i	0.969995	+	0.243126i	$0.0781728\pi$
$431$	14.4400	+	25.0108i	0.695550	+	1.20473i	−0.274444	+	0.961603i	$0.588494\pi$
$432$		0			0
$433$	−35.4318	+	20.4565i	−1.70274	+	0.983079i	−0.759779	+	0.650182i	$0.774693\pi$
$433$	−35.4318	+	20.4565i	−1.70274	+	0.983079i	−0.942963	+	0.332897i	$0.891974\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−26.0796	+	17.4071i	−1.24756	+	0.832694i
$438$		0			0
$439$	−32.1889	−	18.5843i	−1.53629	−	0.886979i	−0.999051	−	0.0435501i	$-0.986133\pi$
$439$	−32.1889	−	18.5843i	−1.53629	−	0.886979i	−0.537241	−	0.843429i	$-0.680533\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	8.71298	−	5.03044i	0.413966	−	0.239003i	−0.278526	−	0.960429i	$-0.589846\pi$
$443$	8.71298	−	5.03044i	0.413966	−	0.239003i	0.692492	+	0.721425i	$0.256513\pi$
$444$		0			0
$445$			1.40422i			0.0665666i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2.50650			−0.118289			−0.0591445	−	0.998249i	$-0.518837\pi$
$449$	−2.50650			−0.118289			−0.0591445	+	0.998249i	$0.518837\pi$
$450$		0			0
$451$	−14.3576	−	8.28936i	−0.676073	−	0.390331i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.48198			0.0694764
$456$		0			0
$457$	−25.5671			−1.19598			−0.597990	−	0.801504i	$-0.704034\pi$
$457$	−25.5671			−1.19598			−0.597990	+	0.801504i	$0.704034\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−34.8597	−	20.1263i	−1.62358	−	0.937374i	−0.985952	−	0.167030i	$-0.946582\pi$
$461$	−34.8597	−	20.1263i	−1.62358	−	0.937374i	−0.637628	−	0.770344i	$-0.720084\pi$
$462$		0			0
$463$	−16.3627			−0.760437			−0.380219	−	0.924897i	$-0.624151\pi$
$463$	−16.3627			−0.760437			−0.380219	+	0.924897i	$0.624151\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	14.3260i		−	0.662929i	−0.943468	−	0.331464i	$-0.892457\pi$
$467$		−	14.3260i		−	0.662929i	0.943468	−	0.331464i	$-0.107543\pi$
$468$		0			0
$469$	37.5609	−	21.6858i	1.73440	−	1.00136i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	14.7237	+	8.50071i	0.676994	+	0.390863i
$474$		0			0
$475$	2.41988	+	3.62549i	0.111032	+	0.166349i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−30.8484	+	17.8103i	−1.40950	+	0.813776i	−0.995340	−	0.0964282i	$-0.969258\pi$
$479$	−30.8484	+	17.8103i	−1.40950	+	0.813776i	−0.414161	+	0.910204i	$0.635925\pi$
$480$		0			0
$481$	0.204177	+	0.353644i	0.00930966	+	0.0161248i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.91453	−	5.04811i	−0.132342	−	0.229223i
$486$		0			0
$487$			13.6885i			0.620286i	0.950690	+	0.310143i	$0.100377\pi$
$487$			13.6885i			0.620286i	−0.950690	+	0.310143i	$0.899623\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−25.2921	−	14.6024i	−1.14142	−	0.658997i	−0.194636	−	0.980876i	$-0.562352\pi$
$491$	−25.2921	−	14.6024i	−1.14142	−	0.658997i	−0.946781	+	0.321878i	$0.895686\pi$
$492$		0			0
$493$			57.5513i			2.59198i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−25.1030	+	43.4796i	−1.12602	+	1.95033i
$498$		0			0
$499$	6.47728	−	11.2190i	0.289963	−	0.502231i	−0.683838	−	0.729634i	$-0.739690\pi$
$499$	6.47728	−	11.2190i	0.289963	−	0.502231i	0.973801	+	0.227404i	$0.0730237\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	28.9392	−	16.7081i	1.29034	−	0.744976i	0.311622	−	0.950206i	$-0.399128\pi$
$503$	28.9392	−	16.7081i	1.29034	−	0.744976i	0.978714	+	0.205230i	$0.0657943\pi$
$504$		0			0
$505$	−9.62147			−0.428150
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2.47534	−	4.28741i	−0.109717	−	0.190036i	0.805938	−	0.591999i	$-0.201661\pi$
$509$	−2.47534	−	4.28741i	−0.109717	−	0.190036i	−0.915656	+	0.401963i	$0.868328\pi$
$510$		0			0
$511$	11.7887	−	20.4186i	0.521500	−	0.903264i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.81638	−	8.34221i	0.212235	−	0.367602i
$516$		0			0
$517$	7.26027	+	12.5752i	0.319306	+	0.553055i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−17.1764			−0.752510			−0.376255	−	0.926516i	$-0.622788\pi$
$521$	−17.1764			−0.752510			−0.376255	+	0.926516i	$0.622788\pi$
$522$		0			0
$523$	−16.5498	+	9.55504i	−0.723673	+	0.417813i	−0.816103	−	0.577906i	$-0.803870\pi$
$523$	−16.5498	+	9.55504i	−0.723673	+	0.417813i	0.0924300	+	0.995719i	$0.470537\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	31.4136	−	54.4100i	1.36840	−	2.37014i
$528$		0			0
$529$	14.3724	−	24.8937i	0.624887	−	1.08234i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	3.34525i		−	0.144899i
$534$		0			0
$535$	−3.67687	−	2.12284i	−0.158965	−	0.0917784i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	12.9713i		−	0.558711i
$540$		0			0
$541$	12.8930	+	22.3314i	0.554314	+	0.960100i	0.997957	+	0.0638962i	$0.0203527\pi$
$541$	12.8930	+	22.3314i	0.554314	+	0.960100i	−0.443643	+	0.896204i	$0.646314\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−6.02048	−	10.4278i	−0.257889	−	0.446677i
$546$		0			0
$547$	−21.9461	+	12.6706i	−0.938346	+	0.541754i	−0.889442	−	0.457049i	$-0.848906\pi$
$547$	−21.9461	+	12.6706i	−0.938346	+	0.541754i	−0.0489047	+	0.998803i	$0.515573\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	19.1164	+	28.6405i	0.814386	+	1.22013i
$552$		0			0
$553$	36.8130	+	21.2540i	1.56545	+	0.903813i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6.01689	+	3.47386i	−0.254944	+	0.147192i	−0.622026	−	0.782997i	$-0.713690\pi$
$557$	−6.01689	+	3.47386i	−0.254944	+	0.147192i	0.367082	+	0.930189i	$0.380357\pi$
$558$		0			0
$559$			3.43054i			0.145096i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4.67227			0.196913			0.0984563	−	0.995141i	$-0.468610\pi$
$563$	4.67227			0.196913			0.0984563	+	0.995141i	$0.468610\pi$
$564$		0			0
$565$	9.04249	+	5.22068i	0.380420	+	0.219636i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−20.7950			−0.871770			−0.435885	−	0.900002i	$-0.643565\pi$
$569$	−20.7950			−0.871770			−0.435885	+	0.900002i	$0.643565\pi$
$570$		0			0
$571$	−43.6472			−1.82658			−0.913290	−	0.407310i	$-0.866467\pi$
$571$	−43.6472			−1.82658			−0.913290	+	0.407310i	$0.866467\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.22965	−	3.59669i	−0.259795	−	0.149992i
$576$		0			0
$577$	−41.1645			−1.71370			−0.856851	−	0.515564i	$-0.827582\pi$
$577$	−41.1645			−1.71370			−0.856851	+	0.515564i	$0.827582\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	31.9662i		−	1.32618i
$582$		0			0
$583$	−5.12393	+	2.95830i	−0.212212	+	0.122520i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12.3078	+	7.10590i	0.507996	+	0.293292i	0.732010	−	0.681294i	$-0.238583\pi$
$587$	12.3078	+	7.10590i	0.507996	+	0.293292i	−0.224013	+	0.974586i	$0.571916\pi$
$588$		0			0
$589$	−2.43995	−	37.5117i	−0.100536	−	1.54564i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	19.4095	−	11.2061i	0.797055	−	0.460180i	−0.0453856	−	0.998970i	$-0.514452\pi$
$593$	19.4095	−	11.2061i	0.797055	−	0.460180i	0.842440	+	0.538790i	$0.181118\pi$
$594$		0			0
$595$	13.3766	+	23.1690i	0.548389	+	0.949837i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	6.71036	+	11.6227i	0.274178	+	0.474890i	0.969927	−	0.243394i	$-0.0782610\pi$
$599$	6.71036	+	11.6227i	0.274178	+	0.474890i	−0.695749	+	0.718285i	$0.744928\pi$
$600$		0			0
$601$		−	2.44880i		−	0.0998887i	−0.998752	−	0.0499443i	$-0.984096\pi$
$601$		−	2.44880i		−	0.0998887i	0.998752	−	0.0499443i	$-0.0159044\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.06218	+	3.50000i	0.246463	+	0.142295i
$606$		0			0
$607$			0.129799i			0.00526837i	0.999997	+	0.00263419i	$0.000838488\pi$
$607$			0.129799i			0.00526837i	−0.999997	+	0.00263419i	$0.999162\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.46498	+	2.53741i	−0.0592666	+	0.102653i
$612$		0			0
$613$	−16.6213	+	28.7890i	−0.671328	+	1.16278i	0.306199	+	0.951968i	$0.400943\pi$
$613$	−16.6213	+	28.7890i	−0.671328	+	1.16278i	−0.977528	+	0.210808i	$0.932391\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	35.4816	−	20.4853i	1.42843	−	0.824707i	0.431437	−	0.902143i	$-0.358007\pi$
$617$	35.4816	−	20.4853i	1.42843	−	0.824707i	0.996997	+	0.0774365i	$0.0246735\pi$
$618$		0			0
$619$	2.00714			0.0806739			0.0403370	−	0.999186i	$-0.487157\pi$
$619$	2.00714			0.0806739			0.0403370	+	0.999186i	$0.487157\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.57835	+	4.46583i	0.103299	+	0.178920i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3.68587	+	6.38412i	−0.146965	+	0.254551i
$630$		0			0
$631$	4.79819	+	8.31072i	0.191013	+	0.330844i	0.945586	−	0.325372i	$-0.105489\pi$
$631$	4.79819	+	8.31072i	0.191013	+	0.330844i	−0.754573	+	0.656216i	$0.772156\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.57197			−0.260801
$636$		0			0
$637$	2.26668	−	1.30867i	0.0898091	−	0.0518513i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	12.4457	−	21.5566i	0.491576	−	0.851435i	−0.508377	−	0.861135i	$-0.669754\pi$
$641$	12.4457	−	21.5566i	0.491576	−	0.851435i	0.999953	+	0.00969979i	$0.00308759\pi$
$642$		0			0
$643$	−12.3543	+	21.3982i	−0.487205	+	0.843863i	−0.999892	−	0.0147121i	$-0.995317\pi$
$643$	−12.3543	+	21.3982i	−0.487205	+	0.843863i	0.512687	+	0.858576i	$0.328650\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	18.8219i		−	0.739964i	−0.929039	−	0.369982i	$-0.879364\pi$
$647$		−	18.8219i		−	0.739964i	0.929039	−	0.369982i	$-0.120636\pi$
$648$		0			0
$649$	−3.62777	−	2.09450i	−0.142403	−	0.0822161i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	11.8686i		−	0.464456i	−0.972661	−	0.232228i	$-0.925398\pi$
$653$		−	11.8686i		−	0.464456i	0.972661	−	0.232228i	$-0.0746015\pi$
$654$		0			0
$655$	2.89335	+	5.01143i	0.113053	+	0.195813i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−16.6473	−	28.8339i	−0.648486	−	1.12321i	−0.983485	−	0.180991i	$-0.942069\pi$
$659$	−16.6473	−	28.8339i	−0.648486	−	1.12321i	0.334999	−	0.942218i	$-0.391264\pi$
$660$		0			0
$661$	39.6297	−	22.8802i	1.54142	−	0.889937i	0.542666	−	0.839948i	$-0.317415\pi$
$661$	39.6297	−	22.8802i	1.54142	−	0.889937i	0.998750	−	0.0499888i	$-0.0159186\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	14.3528	+	7.08688i	0.556578	+	0.274817i
$666$		0			0
$667$	−49.2127	−	28.4130i	−1.90552	−	1.10015i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.77192	+	2.17772i	−0.145613	+	0.0840698i
$672$		0			0
$673$			5.78500i			0.222995i	0.993765	+	0.111498i	$0.0355647\pi$
$673$			5.78500i			0.222995i	−0.993765	+	0.111498i	$0.964435\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−46.5081			−1.78745			−0.893725	−	0.448615i	$-0.851918\pi$
$677$	−46.5081			−1.78745			−0.893725	+	0.448615i	$0.851918\pi$
$678$		0			0
$679$	−18.5381	−	10.7030i	−0.711426	−	0.410742i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	21.0026			0.803642			0.401821	−	0.915718i	$-0.368377\pi$
$683$	21.0026			0.803642			0.401821	+	0.915718i	$0.368377\pi$
$684$		0			0
$685$	−2.20544			−0.0842657
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.03391	−	0.596926i	−0.0393887	−	0.0227411i
$690$		0			0
$691$	−22.4934			−0.855690			−0.427845	−	0.903852i	$-0.640727\pi$
$691$	−22.4934			−0.855690			−0.427845	+	0.903852i	$0.640727\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			7.71386i			0.292603i
$696$		0			0
$697$	52.2990	−	30.1948i	1.98097	−	1.14371i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−39.3523	−	22.7200i	−1.48631	−	0.858124i	−0.486436	−	0.873716i	$-0.661703\pi$
$701$	−39.3523	−	22.7200i	−1.48631	−	0.858124i	−0.999878	+	0.0155920i	$0.995037\pi$
$702$		0			0
$703$	0.286288	+	4.40138i	0.0107976	+	0.166001i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−30.5990	+	17.6664i	−1.15080	+	0.664412i
$708$		0			0
$709$	3.43728	+	5.95355i	0.129090	+	0.223590i	0.923324	−	0.384021i	$-0.125461\pi$
$709$	3.43728	+	5.95355i	0.129090	+	0.223590i	−0.794234	+	0.607611i	$0.792128\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	31.0177	+	53.7243i	1.16162	+	2.01199i
$714$		0			0
$715$		−	0.807119i		−	0.0301845i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9.91615	+	5.72509i	0.369810	+	0.213510i	0.673376	−	0.739301i	$-0.264844\pi$
$719$	9.91615	+	5.72509i	0.369810	+	0.213510i	−0.303565	+	0.952811i	$0.598177\pi$
$720$		0			0
$721$		−	35.3742i		−	1.31740i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.94987	+	6.84138i	−0.146695	+	0.254083i
$726$		0			0
$727$	9.66778	−	16.7451i	0.358558	−	0.621041i	−0.629162	−	0.777274i	$-0.716602\pi$
$727$	9.66778	−	16.7451i	0.358558	−	0.621041i	0.987720	+	0.156233i	$0.0499351\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−53.6324	+	30.9647i	−1.98367	+	1.14527i
$732$		0			0
$733$	−25.0625			−0.925706			−0.462853	−	0.886435i	$-0.653174\pi$
$733$	−25.0625			−0.925706			−0.462853	+	0.886435i	$0.653174\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−11.8105	−	20.4565i	−0.435047	−	0.753524i
$738$		0			0
$739$	−22.3502	+	38.7116i	−0.822164	+	1.42403i	0.0819030	+	0.996640i	$0.473900\pi$
$739$	−22.3502	+	38.7116i	−0.822164	+	1.42403i	−0.904067	+	0.427390i	$0.859433\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.76587	+	3.05857i	−0.0647833	+	0.112208i	−0.896598	−	0.442846i	$-0.853969\pi$
$743$	−1.76587	+	3.05857i	−0.0647833	+	0.112208i	0.831815	+	0.555054i	$0.187302\pi$
$744$		0			0
$745$	−0.245108	−	0.424540i	−0.00898007	−	0.0155539i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−15.5913			−0.569694
$750$		0			0
$751$	12.1483	−	7.01382i	0.443298	−	0.255938i	−0.261698	−	0.965150i	$-0.584282\pi$
$751$	12.1483	−	7.01382i	0.443298	−	0.255938i	0.704995	+	0.709212i	$0.250949\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.660571	−	1.14414i	0.0240406	−	0.0416396i
$756$		0			0
$757$	1.27629	−	2.21059i	0.0463874	−	0.0803454i	−0.841899	−	0.539634i	$-0.818562\pi$
$757$	1.27629	−	2.21059i	0.0463874	−	0.0803454i	0.888287	+	0.459289i	$0.151896\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			27.4469i			0.994948i	0.867479	+	0.497474i	$0.165739\pi$
$761$			27.4469i			0.994948i	−0.867479	+	0.497474i	$0.834261\pi$
$762$		0			0
$763$	−38.2937	−	22.1089i	−1.38632	−	0.800395i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	0.845253i		−	0.0305203i
$768$		0			0
$769$	10.7552	+	18.6286i	0.387844	+	0.671766i	0.992159	−	0.124980i	$-0.0398865\pi$
$769$	10.7552	+	18.6286i	0.387844	+	0.671766i	−0.604315	+	0.796745i	$0.706553\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−20.1222	−	34.8527i	−0.723747	−	1.25357i	−0.959488	−	0.281750i	$-0.909085\pi$
$773$	−20.1222	−	34.8527i	−0.723747	−	1.25357i	0.235741	−	0.971816i	$-0.424248\pi$
$774$		0			0
$775$	7.46857	−	4.31198i	0.268279	−	0.154891i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	15.9971	−	32.3983i	0.573155	−	1.16079i
$780$		0			0
$781$	23.6799	+	13.6716i	0.847335	+	0.489209i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−5.21364	+	3.01010i	−0.186083	+	0.107435i
$786$		0			0
$787$		−	45.9138i		−	1.63665i	−0.574755	−	0.818325i	$-0.694903\pi$
$787$		−	45.9138i		−	1.63665i	0.574755	−	0.818325i	$-0.305097\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	38.3436			1.36334
$792$		0			0
$793$	−0.761096	−	0.439419i	−0.0270273	−	0.0156042i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−7.97105			−0.282349			−0.141175	−	0.989985i	$-0.545088\pi$
$797$	−7.97105			−0.282349			−0.141175	+	0.989985i	$0.545088\pi$
$798$		0			0
$799$	−52.8925			−1.87120
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−11.1204	−	6.42036i	−0.392430	−	0.226569i
$804$		0			0
$805$	−26.4161			−0.931046
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1.96070i		−	0.0689344i	−0.999406	−	0.0344672i	$-0.989027\pi$
$809$		−	1.96070i		−	0.0689344i	0.999406	−	0.0344672i	$-0.0109734\pi$
$810$		0			0
$811$	6.97419	−	4.02655i	0.244897	−	0.141391i	−0.372529	−	0.928021i	$-0.621509\pi$
$811$	6.97419	−	4.02655i	0.244897	−	0.141391i	0.617425	+	0.786630i	$0.288176\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.20694	−	0.696825i	−0.0422771	−	0.0244087i
$816$		0			0
$817$	−16.4049	+	33.2243i	−0.573936	+	1.16237i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−24.4029	+	14.0890i	−0.851666	+	0.491710i	−0.861213	−	0.508245i	$-0.830294\pi$
$821$	−24.4029	+	14.0890i	−0.851666	+	0.491710i	0.00954669	+	0.999954i	$0.496961\pi$
$822$		0			0
$823$	14.6656	+	25.4016i	0.511212	+	0.885444i	0.999916	+	0.0129947i	$0.00413647\pi$
$823$	14.6656	+	25.4016i	0.511212	+	0.885444i	−0.488704	+	0.872450i	$0.662530\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−17.5537	−	30.4040i	−0.610404	−	1.05725i	−0.991172	−	0.132580i	$-0.957674\pi$
$827$	−17.5537	−	30.4040i	−0.610404	−	1.05725i	0.380769	−	0.924670i	$-0.375659\pi$
$828$		0			0
$829$			13.8108i			0.479670i	0.970814	+	0.239835i	$0.0770933\pi$
$829$			13.8108i			0.479670i	−0.970814	+	0.239835i	$0.922907\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	40.9189	+	23.6245i	1.41776	+	0.818542i
$834$		0			0
$835$		−	3.56063i		−	0.123221i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−23.0929	+	39.9981i	−0.797257	+	1.38089i	0.124140	+	0.992265i	$0.460383\pi$
$839$	−23.0929	+	39.9981i	−0.797257	+	1.38089i	−0.921396	+	0.388624i	$0.872950\pi$
$840$		0			0
$841$	−16.7030	+	28.9305i	−0.575966	+	0.997603i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−11.1173	+	6.41857i	−0.382446	+	0.220806i
$846$		0			0
$847$	25.7059			0.883267
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.63942	−	6.30366i	−0.124758	−	0.216087i
$852$		0			0
$853$	−0.0464977	+	0.0805364i	−0.00159205	+	0.00275751i	−0.866820	−	0.498621i	$-0.833840\pi$
$853$	−0.0464977	+	0.0805364i	−0.00159205	+	0.00275751i	0.865228	+	0.501378i	$0.167173\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−14.2043	+	24.6026i	−0.485211	+	0.840410i	−0.999856	−	0.0169937i	$-0.994590\pi$
$857$	−14.2043	+	24.6026i	−0.485211	+	0.840410i	0.514645	+	0.857404i	$0.327924\pi$
$858$		0			0
$859$	12.7778	+	22.1318i	0.435973	+	0.755127i	0.997374	−	0.0724166i	$-0.0230711\pi$
$859$	12.7778	+	22.1318i	0.435973	+	0.755127i	−0.561402	+	0.827543i	$0.689738\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	51.3618			1.74838			0.874188	−	0.485588i	$-0.161395\pi$
$863$	51.3618			1.74838			0.874188	+	0.485588i	$0.161395\pi$
$864$		0			0
$865$	−16.7151	+	9.65046i	−0.568330	+	0.328125i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	11.5754	−	20.0492i	0.392668	−	0.680121i
$870$		0			0
$871$	2.38313	−	4.12770i	0.0807492	−	0.139862i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			3.67228i			0.124146i
$876$		0			0
$877$	−15.9432	−	9.20479i	−0.538363	−	0.310824i	0.206052	−	0.978541i	$-0.433938\pi$
$877$	−15.9432	−	9.20479i	−0.538363	−	0.310824i	−0.744415	+	0.667717i	$0.767272\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	46.1776i		−	1.55576i	−0.628411	−	0.777881i	$-0.716294\pi$
$881$		−	46.1776i		−	1.55576i	0.628411	−	0.777881i	$-0.283706\pi$
$882$		0			0
$883$	6.47655	+	11.2177i	0.217953	+	0.377506i	0.954182	−	0.299227i	$-0.0967287\pi$
$883$	6.47655	+	11.2177i	0.217953	+	0.377506i	−0.736229	+	0.676733i	$0.763395\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−16.9560	−	29.3687i	−0.569328	−	0.986105i	−0.996633	−	0.0819973i	$-0.973870\pi$
$887$	−16.9560	−	29.3687i	−0.569328	−	0.986105i	0.427305	−	0.904108i	$-0.359463\pi$
$888$		0			0
$889$	−20.9008	+	12.0671i	−0.700989	+	0.404716i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−26.3221	+	17.5689i	−0.880834	+	0.587922i
$894$		0			0
$895$	2.47813	+	1.43075i	0.0828346	+	0.0478246i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	58.9998	−	34.0636i	1.96775	−	1.13608i
$900$		0			0
$901$		−	21.5518i		−	0.717996i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	23.9410			0.795827
$906$		0			0
$907$	11.3084	+	6.52893i	0.375491	+	0.216790i	0.675854	−	0.737035i	$-0.263775\pi$
$907$	11.3084	+	6.52893i	0.375491	+	0.216790i	−0.300364	+	0.953825i	$0.597108\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−4.55511			−0.150917			−0.0754587	−	0.997149i	$-0.524042\pi$
$911$	−4.55511			−0.150917			−0.0754587	+	0.997149i	$0.524042\pi$
$912$		0			0
$913$	−17.4094			−0.576169
$914$		0			0
$915$		0			0
$916$		0			0
$917$	18.4034	+	10.6252i	0.607733	+	0.350875i
$918$		0			0
$919$	7.58903			0.250339			0.125169	−	0.992135i	$-0.460053\pi$
$919$	7.58903			0.250339			0.125169	+	0.992135i	$0.460053\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			5.51731i			0.181604i
$924$		0			0
$925$	−0.876313	+	0.505940i	−0.0288130	+	0.0166352i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6.79558	−	3.92343i	−0.222956	−	0.128724i	0.384362	−	0.923182i	$-0.374421\pi$
$929$	−6.79558	−	3.92343i	−0.222956	−	0.128724i	−0.607318	+	0.794459i	$0.707755\pi$
$930$		0			0
$931$	28.2106	−	1.83496i	0.924564	−	0.0601383i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	12.6183	−	7.28520i	0.412664	−	0.238252i
$936$		0			0
$937$	−9.98707	−	17.2981i	−0.326263	−	0.565105i	0.655504	−	0.755192i	$-0.272456\pi$
$937$	−9.98707	−	17.2981i	−0.326263	−	0.565105i	−0.981767	+	0.190087i	$0.939123\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	11.4032	+	19.7509i	0.371732	+	0.643860i	0.989832	−	0.142240i	$-0.0454305\pi$
$941$	11.4032	+	19.7509i	0.371732	+	0.643860i	−0.618100	+	0.786100i	$0.712097\pi$
$942$		0			0
$943$			59.6286i			1.94177i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	18.5575	+	10.7142i	0.603039	+	0.348165i	0.770236	−	0.637759i	$-0.220138\pi$
$947$	18.5575	+	10.7142i	0.603039	+	0.348165i	−0.167197	+	0.985924i	$0.553472\pi$
$948$		0			0
$949$		−	2.59100i		−	0.0841073i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−12.9690	+	22.4630i	−0.420107	+	0.727647i	−0.995950	−	0.0899139i	$-0.971341\pi$
$953$	−12.9690	+	22.4630i	−0.420107	+	0.727647i	0.575843	+	0.817561i	$0.304674\pi$
$954$		0			0
$955$	3.60241	−	6.23955i	0.116571	−	0.201907i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−7.01394	+	4.04950i	−0.226492	+	0.130765i
$960$		0			0
$961$	−43.3727			−1.39912
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4.53824	+	7.86046i	0.146091	+	0.253037i
$966$		0			0
$967$	17.9490	−	31.0885i	0.577200	−	0.999740i	−0.418599	−	0.908171i	$-0.637479\pi$
$967$	17.9490	−	31.0885i	0.577200	−	0.999740i	0.995799	−	0.0915685i	$-0.0291881\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	23.7854	−	41.1975i	0.763310	−	1.32209i	−0.177826	−	0.984062i	$-0.556906\pi$
$971$	23.7854	−	41.1975i	0.763310	−	1.32209i	0.941136	−	0.338029i	$-0.109760\pi$
$972$		0			0
$973$	14.1637	+	24.5323i	0.454068	+	0.786469i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	27.5432			0.881185			0.440592	−	0.897707i	$-0.354768\pi$
$977$	27.5432			0.881185			0.440592	+	0.897707i	$0.354768\pi$
$978$		0			0
$979$	2.43219	−	1.40422i	0.0777331	−	0.0448792i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.57553	−	2.72890i	0.0502517	−	0.0870385i	−0.839805	−	0.542888i	$-0.817331\pi$
$983$	1.57553	−	2.72890i	0.0502517	−	0.0870385i	0.890057	+	0.455849i	$0.150664\pi$
$984$		0			0
$985$	−10.2534	+	17.7595i	−0.326702	+	0.565864i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	61.1489i		−	1.94442i
$990$		0			0
$991$	−4.62637	−	2.67104i	−0.146961	−	0.0848482i	0.424716	−	0.905327i	$-0.360374\pi$
$991$	−4.62637	−	2.67104i	−0.146961	−	0.0848482i	−0.571678	+	0.820478i	$0.693707\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	26.1935i		−	0.830389i
$996$		0			0
$997$	18.9111	+	32.7551i	0.598922	+	1.03736i	0.992981	+	0.118278i	$0.0377374\pi$
$997$	18.9111	+	32.7551i	0.598922	+	1.03736i	−0.394058	+	0.919085i	$0.628929\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.2.dg.c.521.15	yes	32
3.2	odd	2	inner	3420.2.dg.c.521.7	✓	32
19.12	odd	6	inner	3420.2.dg.c.2501.7	yes	32
57.50	even	6	inner	3420.2.dg.c.2501.15	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3420.2.dg.c.521.7	✓	32	3.2	odd	2	inner
3420.2.dg.c.521.15	yes	32	1.1	even	1	trivial
3420.2.dg.c.2501.7	yes	32	19.12	odd	6	inner
3420.2.dg.c.2501.15	yes	32	57.50	even	6	inner

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:	\( N \)	\(=\)	\( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3420.dg (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 + 0.500000i) q^{5} +3.67228 q^{7} +O(q^{10})\)	\(q+(0.866025 + 0.500000i) q^{5} +3.67228 q^{7} -2.00000i q^{11} +(0.349493 - 0.201780i) q^{13} +(6.30917 + 3.64260i) q^{17} +(4.34971 - 0.282927i) q^{19} +(-6.22965 + 3.59669i) q^{23} +(0.500000 + 0.866025i) q^{25} +(3.94987 + 6.84138i) q^{29} -8.62396i q^{31} +(3.18029 + 1.83614i) q^{35} +1.01188i q^{37} +(4.14468 - 7.17880i) q^{41} +(-4.25035 + 7.36183i) q^{43} +(-6.28758 + 3.63013i) q^{47} +6.48563 q^{49} +(-1.47915 - 2.56197i) q^{53} +(1.00000 - 1.73205i) q^{55} +(1.04725 - 1.81389i) q^{59} +(-1.08886 - 1.88596i) q^{61} +0.403559 q^{65} +(10.2282 - 5.90527i) q^{67} +(-6.83581 + 11.8400i) q^{71} +(3.21018 - 5.56019i) q^{73} -7.34456i q^{77} +(10.0246 + 5.78769i) q^{79} -8.70472i q^{83} +(3.64260 + 6.30917i) q^{85} +(0.702112 + 1.21609i) q^{89} +(1.28343 - 0.740991i) q^{91} +(3.90842 + 1.92983i) q^{95} +(-5.04811 - 2.91453i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 16 q^{7}+O(q^{10}) \) 32 * q + 16 * q^7	\( 32 q + 16 q^{7} + 16 q^{25} - 40 q^{43} + 64 q^{49} + 32 q^{55} - 8 q^{61} - 24 q^{67} + 8 q^{73} + 120 q^{79} + 24 q^{91} + 48 q^{97}+O(q^{100}) \) 32 * q + 16 * q^7 + 16 * q^25 - 40 * q^43 + 64 * q^49 + 32 * q^55 - 8 * q^61 - 24 * q^67 + 8 * q^73 + 120 * q^79 + 24 * q^91 + 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more