LMFDB - Embedded newform 3420.2.dg.c.2501.6

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			2501.6
Character	$\chi$	$=$	3420.2501
Dual form			3420.2.dg.c.521.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 + 0.500000i) q^{5} -2.97284 q^{7} +O(q^{10})$	$q+(-0.866025 + 0.500000i) q^{5} -2.97284 q^{7} -2.00000i q^{11} +(3.37584 + 1.94904i) q^{13} +(2.48346 - 1.43383i) q^{17} +(-0.726720 - 4.29789i) q^{19} +(-1.90892 - 1.10211i) q^{23} +(0.500000 - 0.866025i) q^{25} +(-3.07459 + 5.32534i) q^{29} +2.23956i q^{31} +(2.57455 - 1.48642i) q^{35} +9.17522i q^{37} +(-0.713661 - 1.23610i) q^{41} +(-0.927795 - 1.60699i) q^{43} +(2.64555 + 1.52741i) q^{47} +1.83777 q^{49} +(2.86386 - 4.96035i) q^{53} +(1.00000 + 1.73205i) q^{55} +(2.35896 + 4.08583i) q^{59} +(-3.26603 + 5.65693i) q^{61} -3.89808 q^{65} +(-5.99968 - 3.46392i) q^{67} +(-4.22105 - 7.31108i) q^{71} +(5.38735 + 9.33117i) q^{73} +5.94568i q^{77} +(-10.3795 + 5.99259i) q^{79} -14.1271i q^{83} +(-1.43383 + 2.48346i) q^{85} +(1.85362 - 3.21056i) q^{89} +(-10.0358 - 5.79418i) q^{91} +(2.77830 + 3.35872i) q^{95} +(-7.18400 + 4.14769i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times$.

$n$	$1711$	$1901$	$2737$	$3061$
$\chi(n)$	$1$	$-1$	$1$	$e\left(\frac{5}{6}\right)$

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
$7$	−2.97284			−1.12363			−0.561814	−	0.827264i	$-0.689896\pi$
$7$	−2.97284			−1.12363			−0.561814	+	0.827264i	$0.689896\pi$
$8$		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
$13$	3.37584	+	1.94904i	0.936289	+	0.540567i	0.888795	−	0.458305i	$-0.151543\pi$
$13$	3.37584	+	1.94904i	0.936289	+	0.540567i	0.0474940	+	0.998872i	$0.484876\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.48346	−	1.43383i	0.602328	−	0.347754i	−0.167629	−	0.985850i	$-0.553611\pi$
$17$	2.48346	−	1.43383i	0.602328	−	0.347754i	0.769957	+	0.638096i	$0.220278\pi$
$18$		0			0
$19$	−0.726720	−	4.29789i	−0.166721	−	0.986004i
$19$	−0.726720	−	4.29789i	−0.166721	−	0.986004i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.90892	−	1.10211i	−0.398037	−	0.229807i	0.287600	−	0.957751i	$-0.407143\pi$
$23$	−1.90892	−	1.10211i	−0.398037	−	0.229807i	−0.685637	+	0.727944i	$0.740476\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.07459	+	5.32534i	−0.570936	+	0.988890i	0.425534	+	0.904942i	$0.360086\pi$
$29$	−3.07459	+	5.32534i	−0.570936	+	0.988890i	−0.996470	+	0.0839480i	$0.973247\pi$
$30$		0			0
$31$			2.23956i			0.402236i	0.979567	+	0.201118i	$0.0644575\pi$
$31$			2.23956i			0.402236i	−0.979567	+	0.201118i	$0.935543\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.57455	−	1.48642i	0.435179	−	0.251251i
$36$		0			0
$37$			9.17522i			1.50840i	0.656646	+	0.754199i	$0.271974\pi$
$37$			9.17522i			1.50840i	−0.656646	+	0.754199i	$0.728026\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.713661	−	1.23610i	−0.111455	−	0.193046i	0.804902	−	0.593408i	$-0.202218\pi$
$41$	−0.713661	−	1.23610i	−0.111455	−	0.193046i	−0.916357	+	0.400362i	$0.868884\pi$
$42$		0			0
$43$	−0.927795	−	1.60699i	−0.141487	−	0.245063i	0.786570	−	0.617502i	$-0.211855\pi$
$43$	−0.927795	−	1.60699i	−0.141487	−	0.245063i	−0.928057	+	0.372438i	$0.878522\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.64555	+	1.52741i	0.385893	+	0.222796i	0.680379	−	0.732860i	$-0.261815\pi$
$47$	2.64555	+	1.52741i	0.385893	+	0.222796i	−0.294486	+	0.955656i	$0.595148\pi$
$48$		0			0
$49$	1.83777			0.262538
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.86386	−	4.96035i	0.393381	−	0.681356i	−0.599512	−	0.800366i	$-0.704639\pi$
$53$	2.86386	−	4.96035i	0.393381	−	0.681356i	0.992893	+	0.119010i	$0.0379720\pi$
$54$		0			0
$55$	1.00000	+	1.73205i	0.134840	+	0.233550i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.35896	+	4.08583i	0.307110	+	0.531930i	0.977729	−	0.209872i	$-0.0673048\pi$
$59$	2.35896	+	4.08583i	0.307110	+	0.531930i	−0.670619	+	0.741802i	$0.733971\pi$
$60$		0			0
$61$	−3.26603	+	5.65693i	−0.418173	+	0.724296i	−0.995756	−	0.0920357i	$-0.970663\pi$
$61$	−3.26603	+	5.65693i	−0.418173	+	0.724296i	0.577583	+	0.816332i	$0.303996\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.89808			−0.483498
$66$		0			0
$67$	−5.99968	−	3.46392i	−0.732977	−	0.423185i	0.0865332	−	0.996249i	$-0.472421\pi$
$67$	−5.99968	−	3.46392i	−0.732977	−	0.423185i	−0.819510	+	0.573064i	$0.805754\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.22105	−	7.31108i	−0.500947	−	0.867665i	−0.999999	−	0.00109348i	$-0.999652\pi$
$71$	−4.22105	−	7.31108i	−0.500947	−	0.867665i	0.499053	−	0.866572i	$-0.333681\pi$
$72$		0			0
$73$	5.38735	+	9.33117i	0.630542	+	1.09213i	0.987441	+	0.157988i	$0.0505006\pi$
$73$	5.38735	+	9.33117i	0.630542	+	1.09213i	−0.356899	+	0.934143i	$0.616166\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			5.94568i			0.677573i
$78$		0			0
$79$	−10.3795	+	5.99259i	−1.16778	+	0.674219i	−0.953156	−	0.302478i	$-0.902186\pi$
$79$	−10.3795	+	5.99259i	−1.16778	+	0.674219i	−0.214624	+	0.976697i	$0.568853\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	14.1271i		−	1.55066i	−0.631559	−	0.775328i	$-0.717585\pi$
$83$		−	14.1271i		−	1.55066i	0.631559	−	0.775328i	$-0.282415\pi$
$84$		0			0
$85$	−1.43383	+	2.48346i	−0.155520	+	0.269369i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.85362	−	3.21056i	0.196483	−	0.340319i	−0.750903	−	0.660413i	$-0.770381\pi$
$89$	1.85362	−	3.21056i	0.196483	−	0.340319i	0.947386	+	0.320094i	$0.103715\pi$
$90$		0			0
$91$	−10.0358	−	5.79418i	−1.05204	−	0.607396i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.77830	+	3.35872i	0.285048	+	0.344598i
$96$		0			0
$97$	−7.18400	+	4.14769i	−0.729425	+	0.421134i	−0.818212	−	0.574917i	$-0.805034\pi$
$97$	−7.18400	+	4.14769i	−0.729425	+	0.421134i	0.0887869	+	0.996051i	$0.471701\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12.9860	−	7.49749i	−1.29216	−	0.746028i	−0.313122	−	0.949713i	$-0.601375\pi$
$101$	−12.9860	−	7.49749i	−1.29216	−	0.746028i	−0.979036	+	0.203685i	$0.934708\pi$
$102$		0			0
$103$		−	2.60204i		−	0.256386i	−0.991749	−	0.128193i	$-0.959082\pi$
$103$		−	2.60204i		−	0.256386i	0.991749	−	0.128193i	$-0.0409177\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.78636			−0.656062			−0.328031	−	0.944667i	$-0.606385\pi$
$107$	−6.78636			−0.656062			−0.328031	+	0.944667i	$0.606385\pi$
$108$		0			0
$109$	−16.2180	+	9.36345i	−1.55340	+	0.896856i	−0.555539	+	0.831491i	$0.687488\pi$
$109$	−16.2180	+	9.36345i	−1.55340	+	0.896856i	−0.997861	+	0.0653654i	$0.979179\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−20.3888			−1.91802			−0.959009	−	0.283375i	$-0.908546\pi$
$113$	−20.3888			−1.91802			−0.959009	+	0.283375i	$0.908546\pi$
$114$		0			0
$115$	2.20423			0.205545
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−7.38293	+	4.26254i	−0.676792	+	0.390746i
$120$		0			0
$121$	7.00000			0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	−2.19511	−	1.26735i	−0.194785	−	0.112459i	0.399436	−	0.916761i	$-0.369206\pi$
$127$	−2.19511	−	1.26735i	−0.194785	−	0.112459i	−0.594221	+	0.804302i	$0.702539\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.08645	+	2.35931i	−0.357035	+	0.206134i	−0.667779	−	0.744359i	$-0.732755\pi$
$131$	−4.08645	+	2.35931i	−0.357035	+	0.206134i	0.310744	+	0.950494i	$0.399422\pi$
$132$		0			0
$133$	2.16042	+	12.7769i	0.187332	+	1.10790i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−12.2156	−	7.05269i	−1.04365	−	0.602552i	−0.122786	−	0.992433i	$-0.539183\pi$
$137$	−12.2156	−	7.05269i	−1.04365	−	0.602552i	−0.920865	+	0.389881i	$0.872516\pi$
$138$		0			0
$139$	−7.85765	+	13.6099i	−0.666478	+	1.15437i	0.312405	+	0.949949i	$0.398865\pi$
$139$	−7.85765	+	13.6099i	−0.666478	+	1.15437i	−0.978882	+	0.204424i	$0.934468\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.89808	−	6.75168i	0.325974	−	0.564604i
$144$		0			0
$145$		−	6.14917i		−	0.510661i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.45927	+	4.88396i	−0.693011	+	0.400110i	−0.804739	−	0.593629i	$-0.797695\pi$
$149$	−8.45927	+	4.88396i	−0.693011	+	0.400110i	0.111728	+	0.993739i	$0.464361\pi$
$150$		0			0
$151$			13.8145i			1.12421i	0.827067	+	0.562103i	$0.190008\pi$
$151$			13.8145i			1.12421i	−0.827067	+	0.562103i	$0.809992\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.11978	−	1.93951i	−0.0899427	−	0.155785i
$156$		0			0
$157$	8.78589	+	15.2176i	0.701191	+	1.21450i	0.968049	+	0.250762i	$0.0806812\pi$
$157$	8.78589	+	15.2176i	0.701191	+	1.21450i	−0.266858	+	0.963736i	$0.585986\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.67490	+	3.27641i	0.447245	+	0.258217i
$162$		0			0
$163$	6.94981			0.544351			0.272176	−	0.962248i	$-0.412257\pi$
$163$	6.94981			0.544351			0.272176	+	0.962248i	$0.412257\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.09533	+	5.36126i	−0.239523	+	0.414867i	−0.960578	−	0.278012i	$-0.910325\pi$
$167$	−3.09533	+	5.36126i	−0.239523	+	0.414867i	0.721054	+	0.692879i	$0.243658\pi$
$168$		0			0
$169$	1.09752	+	1.90097i	0.0844249	+	0.146228i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.15099	−	7.18973i	−0.315594	−	0.546625i	0.663969	−	0.747760i	$-0.268871\pi$
$173$	−4.15099	−	7.18973i	−0.315594	−	0.546625i	−0.979564	+	0.201134i	$0.935537\pi$
$174$		0			0
$175$	−1.48642	+	2.57455i	−0.112363	+	0.194618i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.41684			0.330130			0.165065	−	0.986283i	$-0.447217\pi$
$179$	4.41684			0.330130			0.165065	+	0.986283i	$0.447217\pi$
$180$		0			0
$181$	−15.6870	−	9.05691i	−1.16601	−	0.673194i	−0.213271	−	0.976993i	$-0.568412\pi$
$181$	−15.6870	−	9.05691i	−1.16601	−	0.673194i	−0.952736	+	0.303799i	$0.901745\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.58761	−	7.94598i	−0.337288	−	0.584200i
$186$		0			0
$187$	−2.86765	−	4.96692i	−0.209704	−	0.363217i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			20.3851i			1.47501i	0.675340	+	0.737506i	$0.263997\pi$
$191$			20.3851i			1.47501i	−0.675340	+	0.737506i	$0.736003\pi$
$192$		0			0
$193$	1.45584	−	0.840531i	0.104794	−	0.0605028i	−0.446687	−	0.894690i	$-0.647396\pi$
$193$	1.45584	−	0.840531i	0.104794	−	0.0605028i	0.551481	+	0.834187i	$0.314063\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	8.73471i		−	0.622322i	−0.950357	−	0.311161i	$-0.899282\pi$
$197$		−	8.73471i		−	0.622322i	0.950357	−	0.311161i	$-0.100718\pi$
$198$		0			0
$199$	0.261239	−	0.452479i	0.0185187	−	0.0320754i	−0.856618	−	0.515952i	$-0.827438\pi$
$199$	0.261239	−	0.452479i	0.0185187	−	0.0320754i	0.875136	+	0.483877i	$0.160772\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	9.14024	−	15.8314i	0.641519	−	1.11114i
$204$		0			0
$205$	1.23610	+	0.713661i	0.0863328	+	0.0498442i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.59578	+	1.45344i	−0.594583	+	0.100537i
$210$		0			0
$211$	−16.6663	+	9.62230i	−1.14736	+	0.662426i	−0.948242	−	0.317550i	$-0.897140\pi$
$211$	−16.6663	+	9.62230i	−1.14736	+	0.662426i	−0.199114	+	0.979976i	$0.563807\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.60699	+	0.927795i	0.109596	+	0.0632751i
$216$		0			0
$217$		−	6.65784i		−	0.451963i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	11.1784			0.751937
$222$		0			0
$223$	8.55573	−	4.93965i	0.572934	−	0.330784i	−0.185386	−	0.982666i	$-0.559354\pi$
$223$	8.55573	−	4.93965i	0.572934	−	0.330784i	0.758320	+	0.651882i	$0.226020\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.2149			−0.810734			−0.405367	−	0.914154i	$-0.632856\pi$
$227$	−12.2149			−0.810734			−0.405367	+	0.914154i	$0.632856\pi$
$228$		0			0
$229$	4.53902			0.299947			0.149973	−	0.988690i	$-0.452081\pi$
$229$	4.53902			0.299947			0.149973	+	0.988690i	$0.452081\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	18.5528	−	10.7114i	1.21543	−	0.701730i	0.251494	−	0.967859i	$-0.419078\pi$
$233$	18.5528	−	10.7114i	1.21543	−	0.701730i	0.963937	+	0.266129i	$0.0857448\pi$
$234$		0			0
$235$	−3.05482			−0.199274
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	7.94188i		−	0.513718i	−0.966449	−	0.256859i	$-0.917312\pi$
$239$		−	7.94188i		−	0.513718i	0.966449	−	0.256859i	$-0.0826876\pi$
$240$		0			0
$241$	8.27023	+	4.77482i	0.532733	+	0.307573i	0.742129	−	0.670258i	$-0.233816\pi$
$241$	8.27023	+	4.77482i	0.532733	+	0.307573i	−0.209396	+	0.977831i	$0.567150\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.59155	+	0.918883i	−0.101681	+	0.0587053i
$246$		0			0
$247$	5.92348	−	15.9254i	0.376902	−	1.01331i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	15.3541	+	8.86467i	0.969140	+	0.559533i	0.898974	−	0.438002i	$-0.144314\pi$
$251$	15.3541	+	8.86467i	0.969140	+	0.559533i	0.0701657	+	0.997535i	$0.477647\pi$
$252$		0			0
$253$	−2.20423	+	3.81783i	−0.138579	+	0.240025i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.40593	−	4.16719i	0.150078	−	0.259942i	−0.781178	−	0.624308i	$-0.785381\pi$
$257$	2.40593	−	4.16719i	0.150078	−	0.259942i	0.931256	+	0.364366i	$0.118714\pi$
$258$		0			0
$259$		−	27.2765i		−	1.69488i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−5.59921	+	3.23271i	−0.345262	+	0.199337i	−0.662597	−	0.748976i	$-0.730546\pi$
$263$	−5.59921	+	3.23271i	−0.345262	+	0.199337i	0.317334	+	0.948314i	$0.397212\pi$
$264$		0			0
$265$			5.72771i			0.351851i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.55593	+	4.42699i	0.155838	+	0.269919i	0.933364	−	0.358932i	$-0.116859\pi$
$269$	2.55593	+	4.42699i	0.155838	+	0.269919i	−0.777526	+	0.628851i	$0.783526\pi$
$270$		0			0
$271$	−7.31800	−	12.6751i	−0.444537	−	0.769960i	0.553483	−	0.832860i	$-0.313298\pi$
$271$	−7.31800	−	12.6751i	−0.444537	−	0.769960i	−0.998020	+	0.0629001i	$0.979965\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.73205	−	1.00000i	−0.104447	−	0.0603023i
$276$		0			0
$277$	−1.45596			−0.0874799			−0.0437400	−	0.999043i	$-0.513927\pi$
$277$	−1.45596			−0.0874799			−0.0437400	+	0.999043i	$0.513927\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−10.9934	+	19.0412i	−0.655813	+	1.13590i	0.325876	+	0.945412i	$0.394341\pi$
$281$	−10.9934	+	19.0412i	−0.655813	+	1.13590i	−0.981689	+	0.190489i	$0.938993\pi$
$282$		0			0
$283$	1.69391	+	2.93393i	0.100692	+	0.174404i	0.911970	−	0.410257i	$-0.134561\pi$
$283$	1.69391	+	2.93393i	0.100692	+	0.174404i	−0.811278	+	0.584661i	$0.801228\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.12160	+	3.67472i	0.125234	+	0.216912i
$288$		0			0
$289$	−4.38828	+	7.60072i	−0.258134	+	0.447101i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	16.7000			0.975625			0.487813	−	0.872948i	$-0.337795\pi$
$293$	16.7000			0.975625			0.487813	+	0.872948i	$0.337795\pi$
$294$		0			0
$295$	−4.08583	−	2.35896i	−0.237886	−	0.137344i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.29613	−	7.44112i	−0.248452	−	0.430331i
$300$		0			0
$301$	2.75818	+	4.77731i	0.158979	+	0.275360i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	6.53206i		−	0.374025i
$306$		0			0
$307$	−3.95183	+	2.28159i	−0.225543	+	0.130217i	−0.608514	−	0.793543i	$-0.708234\pi$
$307$	−3.95183	+	2.28159i	−0.225543	+	0.130217i	0.382971	+	0.923760i	$0.374901\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			18.5819i			1.05368i	0.849964	+	0.526841i	$0.176624\pi$
$311$			18.5819i			1.05368i	−0.849964	+	0.526841i	$0.823376\pi$
$312$		0			0
$313$	−2.00160	+	3.46687i	−0.113137	+	0.195959i	−0.917034	−	0.398810i	$-0.869423\pi$
$313$	−2.00160	+	3.46687i	−0.113137	+	0.195959i	0.803896	+	0.594769i	$0.202757\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.85284	+	17.0656i	−0.553391	+	0.958501i	0.444636	+	0.895711i	$0.353333\pi$
$317$	−9.85284	+	17.0656i	−0.553391	+	0.958501i	−0.998027	+	0.0627894i	$0.980000\pi$
$318$		0			0
$319$	10.6507	+	6.14917i	0.596323	+	0.344287i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−7.96722	−	9.63166i	−0.443308	−	0.535920i
$324$		0			0
$325$	3.37584	−	1.94904i	0.187258	−	0.108113i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−7.86479	−	4.54074i	−0.433600	−	0.250339i
$330$		0			0
$331$			6.20936i			0.341297i	0.985332	+	0.170649i	$0.0545863\pi$
$331$			6.20936i			0.341297i	−0.985332	+	0.170649i	$0.945414\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.92783			0.378508
$336$		0			0
$337$	−11.0175	+	6.36098i	−0.600164	+	0.346505i	−0.769106	−	0.639121i	$-0.779298\pi$
$337$	−11.0175	+	6.36098i	−0.600164	+	0.346505i	0.168942	+	0.985626i	$0.445965\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.47911			0.242558
$342$		0			0
$343$	15.3465			0.828632
$344$		0			0
$345$		0			0
$346$		0			0
$347$	29.8168	−	17.2147i	1.60065	−	0.924134i	0.609289	−	0.792948i	$-0.291455\pi$
$347$	29.8168	−	17.2147i	1.60065	−	0.924134i	0.991358	−	0.131186i	$-0.0418785\pi$
$348$		0			0
$349$	6.57003			0.351686			0.175843	−	0.984418i	$-0.443735\pi$
$349$	6.57003			0.351686			0.175843	+	0.984418i	$0.443735\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			2.97017i			0.158086i	0.996871	+	0.0790430i	$0.0251864\pi$
$353$			2.97017i			0.158086i	−0.996871	+	0.0790430i	$0.974814\pi$
$354$		0			0
$355$	7.31108	+	4.22105i	0.388032	+	0.224030i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12.3106	−	7.10751i	0.649727	−	0.375120i	−0.138625	−	0.990345i	$-0.544268\pi$
$359$	12.3106	−	7.10751i	0.649727	−	0.375120i	0.788352	+	0.615225i	$0.210935\pi$
$360$		0			0
$361$	−17.9438	+	6.24673i	−0.944408	+	0.328775i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−9.33117	−	5.38735i	−0.488416	−	0.281987i
$366$		0			0
$367$	−16.6502	+	28.8389i	−0.869131	+	1.50538i	−0.00624540	+	0.999980i	$0.501988\pi$
$367$	−16.6502	+	28.8389i	−0.869131	+	1.50538i	−0.862886	+	0.505399i	$0.831345\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−8.51378	+	14.7463i	−0.442014	+	0.765590i
$372$		0			0
$373$		−	24.7334i		−	1.28065i	−0.768106	−	0.640323i	$-0.778801\pi$
$373$		−	24.7334i		−	1.28065i	0.768106	−	0.640323i	$-0.221199\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−20.7586	+	11.9850i	−1.06912	+	0.617258i
$378$		0			0
$379$			24.4245i			1.25460i	0.778777	+	0.627301i	$0.215841\pi$
$379$			24.4245i			1.25460i	−0.778777	+	0.627301i	$0.784159\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−16.7012	−	28.9274i	−0.853394	−	1.47812i	−0.878127	−	0.478428i	$-0.841207\pi$
$383$	−16.7012	−	28.9274i	−0.853394	−	1.47812i	0.0247331	−	0.999694i	$-0.492126\pi$
$384$		0			0
$385$	−2.97284	−	5.14911i	−0.151510	−	0.262423i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	25.5154	+	14.7313i	1.29368	+	0.746909i	0.979305	−	0.202389i	$-0.0648704\pi$
$389$	25.5154	+	14.7313i	1.29368	+	0.746909i	0.314379	+	0.949298i	$0.398204\pi$
$390$		0			0
$391$	−6.32096			−0.319665
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5.99259	−	10.3795i	0.301520	−	0.522247i
$396$		0			0
$397$	3.85171	+	6.67135i	0.193312	+	0.334825i	0.946346	−	0.323156i	$-0.104744\pi$
$397$	3.85171	+	6.67135i	0.193312	+	0.334825i	−0.753034	+	0.657981i	$0.771411\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.437503	+	0.757777i	0.0218478	+	0.0378416i	0.876743	−	0.480960i	$-0.159712\pi$
$401$	0.437503	+	0.757777i	0.0218478	+	0.0378416i	−0.854895	+	0.518801i	$0.826378\pi$
$402$		0			0
$403$	−4.36499	+	7.56038i	−0.217435	+	0.376609i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	18.3504			0.909598
$408$		0			0
$409$	2.93016	+	1.69173i	0.144887	+	0.0836506i	0.570691	−	0.821165i	$-0.306675\pi$
$409$	2.93016	+	1.69173i	0.144887	+	0.0836506i	−0.425804	+	0.904815i	$0.640009\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−7.01279	−	12.1465i	−0.345077	−	0.597691i
$414$		0			0
$415$	7.06357	+	12.2345i	0.346737	+	0.600566i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	13.1569i		−	0.642758i	−0.946951	−	0.321379i	$-0.895854\pi$
$419$		−	13.1569i		−	0.642758i	0.946951	−	0.321379i	$-0.104146\pi$
$420$		0			0
$421$	10.0146	−	5.78194i	0.488083	−	0.281795i	−0.235696	−	0.971827i	$-0.575737\pi$
$421$	10.0146	−	5.78194i	0.488083	−	0.281795i	0.723779	+	0.690032i	$0.242404\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	2.86765i		−	0.139102i
$426$		0			0
$427$	9.70938	−	16.8171i	0.469870	−	0.813839i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7.10235	+	12.3016i	−0.342108	+	0.592549i	−0.984824	−	0.173556i	$-0.944474\pi$
$431$	−7.10235	+	12.3016i	−0.342108	+	0.592549i	0.642716	+	0.766105i	$0.277808\pi$
$432$		0			0
$433$	9.67008	+	5.58302i	0.464714	+	0.268303i	0.714024	−	0.700121i	$-0.246871\pi$
$433$	9.67008	+	5.58302i	0.464714	+	0.268303i	−0.249310	+	0.968424i	$0.580204\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.34952	+	9.00525i	−0.160229	+	0.430780i
$438$		0			0
$439$	23.9147	−	13.8071i	1.14138	−	0.658979i	0.194612	−	0.980880i	$-0.437655\pi$
$439$	23.9147	−	13.8071i	1.14138	−	0.658979i	0.946773	+	0.321902i	$0.104322\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−14.3882	−	8.30706i	−0.683606	−	0.394680i	0.117606	−	0.993060i	$-0.462478\pi$
$443$	−14.3882	−	8.30706i	−0.683606	−	0.394680i	−0.801212	+	0.598380i	$0.795811\pi$
$444$		0			0
$445$			3.70724i			0.175740i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−11.5842			−0.546694			−0.273347	−	0.961915i	$-0.588131\pi$
$449$	−11.5842			−0.546694			−0.273347	+	0.961915i	$0.588131\pi$
$450$		0			0
$451$	−2.47219	+	1.42732i	−0.116411	+	0.0672100i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	11.5884			0.543271
$456$		0			0
$457$	−1.37218			−0.0641879			−0.0320940	−	0.999485i	$-0.510218\pi$
$457$	−1.37218			−0.0641879			−0.0320940	+	0.999485i	$0.510218\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−33.3391	+	19.2483i	−1.55276	+	0.896484i	−0.554839	+	0.831957i	$0.687220\pi$
$461$	−33.3391	+	19.2483i	−1.55276	+	0.896484i	−0.997916	+	0.0645263i	$0.979446\pi$
$462$		0			0
$463$	1.94543			0.0904118			0.0452059	−	0.998978i	$-0.485606\pi$
$463$	1.94543			0.0904118			0.0452059	+	0.998978i	$0.485606\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	8.24251i		−	0.381418i	−0.981647	−	0.190709i	$-0.938921\pi$
$467$		−	8.24251i		−	0.381418i	0.981647	−	0.190709i	$-0.0610787\pi$
$468$		0			0
$469$	17.8361	+	10.2977i	0.823593	+	0.475502i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.21397	+	1.85559i	−0.147779	+	0.0853201i
$474$		0			0
$475$	−4.08544	−	1.51959i	−0.187453	−	0.0697235i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−15.6984	−	9.06347i	−0.717278	−	0.414120i	0.0964723	−	0.995336i	$-0.469244\pi$
$479$	−15.6984	−	9.06347i	−0.717278	−	0.414120i	−0.813750	+	0.581215i	$0.802577\pi$
$480$		0			0
$481$	−17.8829	+	30.9741i	−0.815390	+	1.41230i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.14769	−	7.18400i	0.188337	−	0.326209i
$486$		0			0
$487$		−	20.4087i		−	0.924808i	−0.886669	−	0.462404i	$-0.846987\pi$
$487$		−	20.4087i		−	0.924808i	0.886669	−	0.462404i	$-0.153013\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	10.7564	−	6.21020i	0.485428	−	0.280262i	−0.237248	−	0.971449i	$-0.576245\pi$
$491$	10.7564	−	6.21020i	0.485428	−	0.280262i	0.722676	+	0.691187i	$0.242912\pi$
$492$		0			0
$493$			17.6337i			0.794182i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	12.5485	+	21.7346i	0.562877	+	0.974932i
$498$		0			0
$499$	−19.4867	−	33.7519i	−0.872344	−	1.51094i	−0.859565	−	0.511026i	$-0.829265\pi$
$499$	−19.4867	−	33.7519i	−0.872344	−	1.51094i	−0.0127793	−	0.999918i	$-0.504068\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.40178	−	1.96402i	−0.151678	−	0.0875712i	0.422240	−	0.906484i	$-0.361244\pi$
$503$	−3.40178	−	1.96402i	−0.151678	−	0.0875712i	−0.573918	+	0.818913i	$0.694577\pi$
$504$		0			0
$505$	14.9950			0.667268
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4.20064	−	7.27572i	0.186190	−	0.322491i	−0.757787	−	0.652502i	$-0.773719\pi$
$509$	4.20064	−	7.27572i	0.186190	−	0.322491i	0.943977	+	0.330011i	$0.107053\pi$
$510$		0			0
$511$	−16.0157	−	27.7401i	−0.708494	−	1.22715i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.30102	+	2.25343i	0.0573297	+	0.0992980i
$516$		0			0
$517$	3.05482	−	5.29110i	0.134351	−	0.232702i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6.90278			0.302416			0.151208	−	0.988502i	$-0.451684\pi$
$521$	6.90278			0.302416			0.151208	+	0.988502i	$0.451684\pi$
$522$		0			0
$523$	−28.2569	−	16.3141i	−1.23559	−	0.713367i	−0.267398	−	0.963586i	$-0.586164\pi$
$523$	−28.2569	−	16.3141i	−1.23559	−	0.713367i	−0.968189	+	0.250219i	$0.919497\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.21114	+	5.56185i	0.139879	+	0.242278i
$528$		0			0
$529$	−9.07069	−	15.7109i	−0.394378	−	0.683082i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	5.56382i		−	0.240996i
$534$		0			0
$535$	5.87716	−	3.39318i	0.254092	−	0.146700i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	3.67553i		−	0.158316i
$540$		0			0
$541$	12.1206	−	20.9935i	0.521105	−	0.902581i	−0.478594	−	0.878037i	$-0.658853\pi$
$541$	12.1206	−	20.9935i	0.521105	−	0.902581i	0.999699	−	0.0245440i	$-0.00781339\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	9.36345	−	16.2180i	0.401086	−	0.694702i
$546$		0			0
$547$	14.8212	+	8.55704i	0.633710	+	0.365873i	0.782187	−	0.623043i	$-0.214104\pi$
$547$	14.8212	+	8.55704i	0.633710	+	0.365873i	−0.148477	+	0.988916i	$0.547437\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	25.1221	+	9.34420i	1.07024	+	0.398077i
$552$		0			0
$553$	30.8565	−	17.8150i	1.31215	−	0.757570i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	6.68768	+	3.86114i	0.283366	+	0.163602i	0.634946	−	0.772556i	$-0.281022\pi$
$557$	6.68768	+	3.86114i	0.283366	+	0.163602i	−0.351580	+	0.936158i	$0.614356\pi$
$558$		0			0
$559$		−	7.23324i		−	0.305933i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	28.9219			1.21891			0.609457	−	0.792819i	$-0.291387\pi$
$563$	28.9219			1.21891			0.609457	+	0.792819i	$0.291387\pi$
$564$		0			0
$565$	17.6572	−	10.1944i	0.742845	−	0.428882i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4.86955			−0.204142			−0.102071	−	0.994777i	$-0.532547\pi$
$569$	−4.86955			−0.204142			−0.102071	+	0.994777i	$0.532547\pi$
$570$		0			0
$571$	−13.2347			−0.553855			−0.276928	−	0.960891i	$-0.589316\pi$
$571$	−13.2347			−0.553855			−0.276928	+	0.960891i	$0.589316\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.90892	+	1.10211i	−0.0796074	+	0.0459613i
$576$		0			0
$577$	−35.1003			−1.46125			−0.730623	−	0.682781i	$-0.760770\pi$
$577$	−35.1003			−1.46125			−0.730623	+	0.682781i	$0.760770\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			41.9977i			1.74236i
$582$		0			0
$583$	−9.92069	−	5.72771i	−0.410873	−	0.237218i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−23.9454	+	13.8249i	−0.988331	+	0.570613i	−0.904775	−	0.425890i	$-0.859961\pi$
$587$	−23.9454	+	13.8249i	−0.988331	+	0.570613i	−0.0835557	+	0.996503i	$0.526628\pi$
$588$		0			0
$589$	9.62537	−	1.62753i	0.396606	−	0.0670613i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	24.0708	+	13.8973i	0.988470	+	0.570693i	0.904817	−	0.425801i	$-0.140008\pi$
$593$	24.0708	+	13.8973i	0.988470	+	0.570693i	0.0836535	+	0.996495i	$0.473341\pi$
$594$		0			0
$595$	4.26254	−	7.38293i	0.174747	−	0.302671i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	20.2808	−	35.1273i	0.828649	−	1.43526i	−0.0704489	−	0.997515i	$-0.522443\pi$
$599$	20.2808	−	35.1273i	0.828649	−	1.43526i	0.899098	−	0.437747i	$-0.144224\pi$
$600$		0			0
$601$		−	20.9030i		−	0.852649i	−0.904570	−	0.426325i	$-0.859808\pi$
$601$		−	20.9030i		−	0.852649i	0.904570	−	0.426325i	$-0.140192\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6.06218	+	3.50000i	−0.246463	+	0.142295i
$606$		0			0
$607$		−	15.4979i		−	0.629041i	−0.949251	−	0.314520i	$-0.898156\pi$
$607$		−	15.4979i		−	0.629041i	0.949251	−	0.314520i	$-0.101844\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.95397	+	10.3126i	0.240872	+	0.417202i
$612$		0			0
$613$	16.2663	+	28.1740i	0.656989	+	1.13794i	0.981391	+	0.192019i	$0.0615036\pi$
$613$	16.2663	+	28.1740i	0.656989	+	1.13794i	−0.324402	+	0.945919i	$0.605163\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	34.7302	+	20.0515i	1.39819	+	0.807243i	0.994203	−	0.107523i	$-0.0342921\pi$
$617$	34.7302	+	20.0515i	1.39819	+	0.807243i	0.403983	+	0.914766i	$0.367625\pi$
$618$		0			0
$619$	−9.52721			−0.382931			−0.191466	−	0.981499i	$-0.561324\pi$
$619$	−9.52721			−0.382931			−0.191466	+	0.981499i	$0.561324\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5.51051	+	9.54448i	−0.220774	+	0.382392i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	13.1557	+	22.7863i	0.524552	+	0.908550i
$630$		0			0
$631$	17.1854	−	29.7661i	0.684142	−	1.18497i	−0.289564	−	0.957159i	$-0.593510\pi$
$631$	17.1854	−	29.7661i	0.684142	−	1.18497i	0.973706	−	0.227810i	$-0.0731564\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.53470			0.100586
$636$		0			0
$637$	6.20400	+	3.58188i	0.245811	+	0.141919i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−14.2294	−	24.6461i	−0.562029	−	0.973463i	−0.997319	−	0.0731731i	$-0.976687\pi$
$641$	−14.2294	−	24.6461i	−0.562029	−	0.973463i	0.435290	−	0.900290i	$-0.356646\pi$
$642$		0			0
$643$	−0.248144	−	0.429798i	−0.00978585	−	0.0169496i	0.861091	−	0.508451i	$-0.169782\pi$
$643$	−0.248144	−	0.429798i	−0.00978585	−	0.0169496i	−0.870877	+	0.491501i	$0.836448\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			32.8611i			1.29190i	0.763379	+	0.645951i	$0.223539\pi$
$647$			32.8611i			1.29190i	−0.763379	+	0.645951i	$0.776461\pi$
$648$		0			0
$649$	8.17166	−	4.71791i	0.320766	−	0.185194i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	32.7995i		−	1.28354i	−0.766896	−	0.641772i	$-0.778200\pi$
$653$		−	32.7995i		−	1.28354i	0.766896	−	0.641772i	$-0.221800\pi$
$654$		0			0
$655$	2.35931	−	4.08645i	0.0921860	−	0.159671i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−20.8042	+	36.0339i	−0.810415	+	1.40368i	0.102159	+	0.994768i	$0.467425\pi$
$659$	−20.8042	+	36.0339i	−0.810415	+	1.40368i	−0.912574	+	0.408912i	$0.865908\pi$
$660$		0			0
$661$	−2.81964	−	1.62792i	−0.109671	−	0.0633188i	0.444161	−	0.895947i	$-0.353502\pi$
$661$	−2.81964	−	1.62792i	−0.109671	−	0.0633188i	−0.553832	+	0.832628i	$0.686835\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−8.25945	−	9.98494i	−0.320288	−	0.387199i
$666$		0			0
$667$	11.7383	−	6.77709i	0.454507	−	0.262410i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	11.3139	+	6.53206i	0.436767	+	0.252168i
$672$		0			0
$673$		−	10.0849i		−	0.388743i	−0.980928	−	0.194372i	$-0.937733\pi$
$673$		−	10.0849i		−	0.388743i	0.980928	−	0.194372i	$-0.0622667\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−22.7912			−0.875938			−0.437969	−	0.898990i	$-0.644302\pi$
$677$	−22.7912			−0.875938			−0.437969	+	0.898990i	$0.644302\pi$
$678$		0			0
$679$	21.3569	−	12.3304i	0.819602	−	0.473197i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	13.2050			0.505275			0.252638	−	0.967561i	$-0.418702\pi$
$683$	13.2050			0.505275			0.252638	+	0.967561i	$0.418702\pi$
$684$		0			0
$685$	14.1054			0.538939
$686$		0			0
$687$		0			0
$688$		0			0
$689$	19.3358	−	11.1635i	0.736637	−	0.425297i
$690$		0			0
$691$	−13.3076			−0.506244			−0.253122	−	0.967434i	$-0.581457\pi$
$691$	−13.3076			−0.506244			−0.253122	+	0.967434i	$0.581457\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	15.7153i		−	0.596116i
$696$		0			0
$697$	−3.54470	−	2.04653i	−0.134265	−	0.0775180i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−45.1733	+	26.0808i	−1.70617	+	0.985059i	−0.766977	+	0.641675i	$0.778240\pi$
$701$	−45.1733	+	26.0808i	−1.70617	+	0.985059i	−0.939195	+	0.343384i	$0.888427\pi$
$702$		0			0
$703$	39.4341	−	6.66782i	1.48729	−	0.251482i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	38.6054	+	22.2888i	1.45190	+	0.838257i
$708$		0			0
$709$	10.2313	−	17.7211i	0.384244	−	0.665531i	−0.607420	−	0.794381i	$-0.707795\pi$
$709$	10.2313	−	17.7211i	0.384244	−	0.665531i	0.991664	+	0.128850i	$0.0411287\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.46825	−	4.27513i	0.0924365	−	0.160105i
$714$		0			0
$715$			7.79616i			0.291560i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2.39080	+	1.38033i	−0.0891618	+	0.0514776i	−0.543918	−	0.839138i	$-0.683060\pi$
$719$	−2.39080	+	1.38033i	−0.0891618	+	0.0514776i	0.454756	+	0.890616i	$0.349726\pi$
$720$		0			0
$721$			7.73543i			0.288083i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.07459	+	5.32534i	0.114187	+	0.197778i
$726$		0			0
$727$	−11.3432	−	19.6470i	−0.420697	−	0.728668i	0.575311	−	0.817935i	$-0.304881\pi$
$727$	−11.3432	−	19.6470i	−0.420697	−	0.728668i	−0.996008	+	0.0892664i	$0.971548\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−4.60828	−	2.66059i	−0.170444	−	0.0984056i
$732$		0			0
$733$	45.5596			1.68278			0.841390	−	0.540428i	$-0.181738\pi$
$733$	45.5596			1.68278			0.841390	+	0.540428i	$0.181738\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−6.92783	+	11.9994i	−0.255190	+	0.442002i
$738$		0			0
$739$	−1.28159	−	2.21977i	−0.0471439	−	0.0816556i	0.841491	−	0.540272i	$-0.181679\pi$
$739$	−1.28159	−	2.21977i	−0.0471439	−	0.0816556i	−0.888634	+	0.458616i	$0.848345\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	8.08151	+	13.9976i	0.296482	+	0.513521i	0.975328	−	0.220759i	$-0.0708533\pi$
$743$	8.08151	+	13.9976i	0.296482	+	0.513521i	−0.678847	+	0.734280i	$0.737520\pi$
$744$		0			0
$745$	4.88396	−	8.45927i	0.178935	−	0.309924i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	20.1748			0.737170
$750$		0			0
$751$	−29.5716	−	17.0731i	−1.07908	−	0.623008i	−0.148433	−	0.988922i	$-0.547423\pi$
$751$	−29.5716	−	17.0731i	−1.07908	−	0.623008i	−0.930648	+	0.365915i	$0.880756\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−6.90724	−	11.9637i	−0.251380	−	0.435403i
$756$		0			0
$757$	−0.157002	−	0.271936i	−0.00570634	−	0.00988367i	0.863158	−	0.504934i	$-0.168483\pi$
$757$	−0.157002	−	0.271936i	−0.00570634	−	0.00988367i	−0.868864	+	0.495050i	$0.835150\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	12.0298i		−	0.436079i	−0.975940	−	0.218040i	$-0.930034\pi$
$761$		−	12.0298i		−	0.436079i	0.975940	−	0.218040i	$-0.0699662\pi$
$762$		0			0
$763$	48.2134	−	27.8360i	1.74544	−	1.00773i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			18.3908i			0.664053i
$768$		0			0
$769$	−11.9487	+	20.6957i	−0.430880	+	0.746305i	−0.996949	−	0.0780520i	$-0.975130\pi$
$769$	−11.9487	+	20.6957i	−0.430880	+	0.746305i	0.566070	+	0.824357i	$0.308463\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−2.06645	+	3.57919i	−0.0743249	+	0.128734i	−0.900792	−	0.434250i	$-0.857014\pi$
$773$	−2.06645	+	3.57919i	−0.0743249	+	0.128734i	0.826468	+	0.562984i	$0.190347\pi$
$774$		0			0
$775$	1.93951	+	1.11978i	0.0696693	+	0.0402236i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.79398	+	3.96553i	−0.171762	+	0.142080i
$780$		0			0
$781$	−14.6222	+	8.44210i	−0.523222	+	0.302082i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−15.2176	−	8.78589i	−0.543140	−	0.313582i
$786$		0			0
$787$		−	9.87801i		−	0.352113i	−0.984380	−	0.176056i	$-0.943666\pi$
$787$		−	9.87801i		−	0.352113i	0.984380	−	0.176056i	$-0.0563341\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	60.6127			2.15514
$792$		0			0
$793$	−22.0512	+	12.7313i	−0.783061	+	0.452100i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.506639			−0.0179461			−0.00897303	−	0.999960i	$-0.502856\pi$
$797$	−0.506639			−0.0179461			−0.00897303	+	0.999960i	$0.502856\pi$
$798$		0			0
$799$	8.76017			0.309912
$800$		0			0
$801$		0			0
$802$		0			0
$803$	18.6623	−	10.7747i	0.658580	−	0.380231i
$804$		0			0
$805$	−6.55281			−0.230956
$806$		0			0
$807$		0			0
$808$		0			0
$809$			34.1081i			1.19918i	0.800308	+	0.599589i	$0.204669\pi$
$809$			34.1081i			1.19918i	−0.800308	+	0.599589i	$0.795331\pi$
$810$		0			0
$811$	24.3475	+	14.0570i	0.854956	+	0.493609i	0.862320	−	0.506364i	$-0.169011\pi$
$811$	24.3475	+	14.0570i	0.854956	+	0.493609i	−0.00736416	+	0.999973i	$0.502344\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.01871	+	3.47490i	−0.210826	+	0.121721i
$816$		0			0
$817$	−6.23241	+	5.15539i	−0.218044	+	0.180364i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	46.6459	+	26.9310i	1.62795	+	0.939898i	0.984703	+	0.174241i	$0.0557473\pi$
$821$	46.6459	+	26.9310i	1.62795	+	0.939898i	0.643249	+	0.765657i	$0.277586\pi$
$822$		0			0
$823$	13.0862	−	22.6660i	0.456157	−	0.790088i	−0.542597	−	0.839993i	$-0.682559\pi$
$823$	13.0862	−	22.6660i	0.456157	−	0.790088i	0.998754	+	0.0499058i	$0.0158921\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	4.07571	−	7.05934i	0.141726	−	0.245477i	−0.786420	−	0.617692i	$-0.788068\pi$
$827$	4.07571	−	7.05934i	0.141726	−	0.245477i	0.928147	+	0.372214i	$0.121401\pi$
$828$		0			0
$829$			32.0175i			1.11201i	0.831177	+	0.556007i	$0.187667\pi$
$829$			32.0175i			1.11201i	−0.831177	+	0.556007i	$0.812333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	4.56402	−	2.63504i	0.158134	−	0.0912987i
$834$		0			0
$835$		−	6.19065i		−	0.214236i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.23126	+	2.13260i	0.0425076	+	0.0736254i	0.886496	−	0.462735i	$-0.153132\pi$
$839$	1.23126	+	2.13260i	0.0425076	+	0.0736254i	−0.843989	+	0.536361i	$0.819799\pi$
$840$		0			0
$841$	−4.40615	−	7.63168i	−0.151936	−	0.263161i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.90097	−	1.09752i	−0.0653952	−	0.0377560i
$846$		0			0
$847$	−20.8099			−0.715035
$848$		0			0
$849$		0			0
$850$		0			0
$851$	10.1121	−	17.5147i	0.346640	−	0.600398i
$852$		0			0
$853$	12.8987	+	22.3412i	0.441642	+	0.764946i	0.997812	−	0.0661224i	$-0.0210628\pi$
$853$	12.8987	+	22.3412i	0.441642	+	0.764946i	−0.556169	+	0.831069i	$0.687729\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−22.2512	−	38.5403i	−0.760088	−	1.31651i	−0.942805	−	0.333346i	$-0.891822\pi$
$857$	−22.2512	−	38.5403i	−0.760088	−	1.31651i	0.182717	−	0.983166i	$-0.441511\pi$
$858$		0			0
$859$	26.6752	−	46.2027i	0.910145	−	1.57642i	0.0962863	−	0.995354i	$-0.469304\pi$
$859$	26.6752	−	46.2027i	0.910145	−	1.57642i	0.813858	−	0.581063i	$-0.197363\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−10.8121			−0.368049			−0.184025	−	0.982922i	$-0.558913\pi$
$863$	−10.8121			−0.368049			−0.184025	+	0.982922i	$0.558913\pi$
$864$		0			0
$865$	7.18973	+	4.15099i	0.244458	+	0.141138i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	11.9852	+	20.7589i	0.406569	+	0.704198i
$870$		0			0
$871$	−13.5026	−	23.3872i	−0.457519	−	0.792446i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	2.97284i		−	0.100500i
$876$		0			0
$877$	25.6612	−	14.8155i	0.866516	−	0.500283i	0.000327217	−	1.00000i	$-0.499896\pi$
$877$	25.6612	−	14.8155i	0.866516	−	0.500283i	0.866189	+	0.499717i	$0.166563\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	16.8924i		−	0.569119i	−0.958658	−	0.284560i	$-0.908153\pi$
$881$		−	16.8924i		−	0.569119i	0.958658	−	0.284560i	$-0.0918473\pi$
$882$		0			0
$883$	17.3264	−	30.0102i	0.583079	−	1.00992i	−0.412033	−	0.911169i	$-0.635181\pi$
$883$	17.3264	−	30.0102i	0.583079	−	1.00992i	0.995112	−	0.0987533i	$-0.0314855\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−0.818456	+	1.41761i	−0.0274811	+	0.0475986i	−0.879439	−	0.476012i	$-0.842082\pi$
$887$	−0.818456	+	1.41761i	−0.0274811	+	0.0475986i	0.851958	+	0.523610i	$0.175415\pi$
$888$		0			0
$889$	6.52572	+	3.76763i	0.218866	+	0.126362i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.64207	−	12.4803i	0.155341	−	0.417637i
$894$		0			0
$895$	−3.82509	+	2.20842i	−0.127859	+	0.0738193i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−11.9264	−	6.88571i	−0.397767	−	0.229651i
$900$		0			0
$901$		−	16.4251i		−	0.547200i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	18.1138			0.602123
$906$		0			0
$907$	−38.2028	+	22.0564i	−1.26850	+	0.732371i	−0.974705	−	0.223496i	$-0.928253\pi$
$907$	−38.2028	+	22.0564i	−1.26850	+	0.732371i	−0.293799	+	0.955867i	$0.594920\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−38.5215			−1.27628			−0.638138	−	0.769922i	$-0.720295\pi$
$911$	−38.5215			−1.27628			−0.638138	+	0.769922i	$0.720295\pi$
$912$		0			0
$913$	−28.2543			−0.935081
$914$		0			0
$915$		0			0
$916$		0			0
$917$	12.1484	−	7.01386i	0.401174	−	0.231618i
$918$		0			0
$919$	−27.4693			−0.906128			−0.453064	−	0.891478i	$-0.649669\pi$
$919$	−27.4693			−0.906128			−0.453064	+	0.891478i	$0.649669\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	32.9080i		−	1.08318i
$924$		0			0
$925$	7.94598	+	4.58761i	0.261262	+	0.150840i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	10.0307	−	5.79121i	0.329096	−	0.190004i	−0.326344	−	0.945251i	$-0.605817\pi$
$929$	10.0307	−	5.79121i	0.329096	−	0.190004i	0.655440	+	0.755248i	$0.272483\pi$
$930$		0			0
$931$	−1.33554	−	7.89852i	−0.0437706	−	0.258864i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4.96692	+	2.86765i	0.162436	+	0.0937823i
$936$		0			0
$937$	17.3072	−	29.9770i	0.565403	−	0.979307i	−0.431609	−	0.902061i	$-0.642054\pi$
$937$	17.3072	−	29.9770i	0.565403	−	0.979307i	0.997012	−	0.0772460i	$-0.0246127\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−23.8959	+	41.3888i	−0.778983	+	1.34924i	0.153546	+	0.988141i	$0.450931\pi$
$941$	−23.8959	+	41.3888i	−0.778983	+	1.34924i	−0.932529	+	0.361096i	$0.882403\pi$
$942$		0			0
$943$			3.14614i			0.102452i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	30.0733	−	17.3628i	0.977251	−	0.564216i	0.0758117	−	0.997122i	$-0.475845\pi$
$947$	30.0733	−	17.3628i	0.977251	−	0.564216i	0.901439	+	0.432906i	$0.142512\pi$
$948$		0			0
$949$			42.0007i			1.36340i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−11.3669	−	19.6881i	−0.368211	−	0.637760i	0.621075	−	0.783751i	$-0.286696\pi$
$953$	−11.3669	−	19.6881i	−0.368211	−	0.637760i	−0.989286	+	0.145991i	$0.953363\pi$
$954$		0			0
$955$	−10.1925	−	17.6540i	−0.329823	−	0.571270i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	36.3151	+	20.9665i	1.17267	+	0.677044i
$960$		0			0
$961$	25.9844			0.838206
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.840531	+	1.45584i	−0.0270577	+	0.0468652i
$966$		0			0
$967$	14.4750	+	25.0714i	0.465483	+	0.806240i	0.999223	−	0.0394081i	$-0.0125472\pi$
$967$	14.4750	+	25.0714i	0.465483	+	0.806240i	−0.533740	+	0.845649i	$0.679214\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	27.4674	+	47.5750i	0.881472	+	1.52675i	0.849705	+	0.527258i	$0.176780\pi$
$971$	27.4674	+	47.5750i	0.881472	+	1.52675i	0.0317667	+	0.999495i	$0.489887\pi$
$972$		0			0
$973$	23.3595	−	40.4599i	0.748872	−	1.29708i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−48.2205			−1.54271			−0.771355	−	0.636405i	$-0.780421\pi$
$977$	−48.2205			−1.54271			−0.771355	+	0.636405i	$0.780421\pi$
$978$		0			0
$979$	−6.42112	−	3.70724i	−0.205220	−	0.118484i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−2.99110	−	5.18074i	−0.0954013	−	0.165240i	0.814375	−	0.580339i	$-0.197080\pi$
$983$	−2.99110	−	5.18074i	−0.0954013	−	0.165240i	−0.909776	+	0.415100i	$0.863747\pi$
$984$		0			0
$985$	4.36735	+	7.56448i	0.139155	+	0.241024i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			4.09014i			0.130059i
$990$		0			0
$991$	23.7739	−	13.7259i	0.755204	−	0.436017i	−0.0723675	−	0.997378i	$-0.523055\pi$
$991$	23.7739	−	13.7259i	0.755204	−	0.436017i	0.827571	+	0.561361i	$0.189722\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			0.522477i			0.0165636i
$996$		0			0
$997$	11.6770	−	20.2252i	0.369816	−	0.640540i	−0.619721	−	0.784822i	$-0.712754\pi$
$997$	11.6770	−	20.2252i	0.369816	−	0.640540i	0.989537	+	0.144283i	$0.0460874\pi$
$998$		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.2501.6	yes	32
3.2	odd	2	inner	3420.2.dg.c.2501.13	yes	32
19.8	odd	6	inner	3420.2.dg.c.521.13	yes	32
57.8	even	6	inner	3420.2.dg.c.521.6	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3420.2.dg.c.521.6	✓	32	57.8	even	6	inner
3420.2.dg.c.521.13	yes	32	19.8	odd	6	inner
3420.2.dg.c.2501.6	yes	32	1.1	even	1	trivial
3420.2.dg.c.2501.13	yes	32	3.2	odd	2	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)

\(f(q)\)	\(=\)	\(q+(-0.866025 + 0.500000i) q^{5} -2.97284 q^{7} +O(q^{10})\)	\(q+(-0.866025 + 0.500000i) q^{5} -2.97284 q^{7} -2.00000i q^{11} +(3.37584 + 1.94904i) q^{13} +(2.48346 - 1.43383i) q^{17} +(-0.726720 - 4.29789i) q^{19} +(-1.90892 - 1.10211i) q^{23} +(0.500000 - 0.866025i) q^{25} +(-3.07459 + 5.32534i) q^{29} +2.23956i q^{31} +(2.57455 - 1.48642i) q^{35} +9.17522i q^{37} +(-0.713661 - 1.23610i) q^{41} +(-0.927795 - 1.60699i) q^{43} +(2.64555 + 1.52741i) q^{47} +1.83777 q^{49} +(2.86386 - 4.96035i) q^{53} +(1.00000 + 1.73205i) q^{55} +(2.35896 + 4.08583i) q^{59} +(-3.26603 + 5.65693i) q^{61} -3.89808 q^{65} +(-5.99968 - 3.46392i) q^{67} +(-4.22105 - 7.31108i) q^{71} +(5.38735 + 9.33117i) q^{73} +5.94568i q^{77} +(-10.3795 + 5.99259i) q^{79} -14.1271i q^{83} +(-1.43383 + 2.48346i) q^{85} +(1.85362 - 3.21056i) q^{89} +(-10.0358 - 5.79418i) q^{91} +(2.77830 + 3.35872i) q^{95} +(-7.18400 + 4.14769i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 16 q^{7}+O(q^{10}) \) 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

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