LMFDB - Embedded newform 3420.2.bj.e.1189.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3420,2,Mod(1189,3420)]

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, 0, 3, 4]))

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

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

chi := DirichletCharacter("3420.1189");

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.bj (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:	$40$
Relative dimension:	$20$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1189.9
Character	$\chi$	$=$	3420.1189
Dual form			3420.2.bj.e.2629.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.778229 + 2.09627i) q^{5} -2.19629i q^{7} +O(q^{10})$	$q+(-0.778229 + 2.09627i) q^{5} -2.19629i q^{7} +0.851781 q^{11} +(-0.197307 + 0.113916i) q^{13} +(-6.57835 - 3.79801i) q^{17} +(3.93036 + 1.88474i) q^{19} +(-1.46423 + 0.845372i) q^{23} +(-3.78872 - 3.26276i) q^{25} +(0.798103 + 1.38236i) q^{29} +5.76406 q^{31} +(4.60402 + 1.70922i) q^{35} +8.94456i q^{37} +(-3.54377 + 6.13800i) q^{41} +(6.55871 + 3.78668i) q^{43} +(-5.06032 + 2.92158i) q^{47} +2.17631 q^{49} +(4.14208 - 2.39143i) q^{53} +(-0.662880 + 1.78556i) q^{55} +(-5.33889 + 9.24723i) q^{59} +(-1.17130 - 2.02874i) q^{61} +(-0.0852477 - 0.502263i) q^{65} +(-3.70127 + 2.13693i) q^{67} +(-3.14788 + 5.45228i) q^{71} +(-7.74621 - 4.47228i) q^{73} -1.87076i q^{77} +(-0.332949 + 0.576685i) q^{79} -8.71844i q^{83} +(13.0811 - 10.8343i) q^{85} +(5.33889 + 9.24723i) q^{89} +(0.250192 + 0.433345i) q^{91} +(-7.00965 + 6.77236i) q^{95} +(-8.06614 - 4.65699i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q+O(q^{10}) $ 40 * q	$ 40 q - 8 q^{19} - 8 q^{31} - 16 q^{49} - 8 q^{61} - 8 q^{79} + 28 q^{91}+O(q^{100}) $ 40 * q - 8 * q^19 - 8 * q^31 - 16 * q^49 - 8 * q^61 - 8 * q^79 + 28 * q^91

Display 100 coefficients 10 coefficients

Character values

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

$n$	$1711$	$1901$	$2737$	$3061$
$\chi(n)$	$1$	$1$	$-1$	$e\left(\frac{2}{3}\right)$

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
$5$	−0.778229	+	2.09627i	−0.348034	+	0.937482i
$5$	−0.778229	+	2.09627i	−0.348034	+	0.937482i
$6$		0			0
$7$		−	2.19629i		−	0.830120i	−0.909794	−	0.415060i	$-0.863761\pi$
$7$		−	2.19629i		−	0.830120i	0.909794	−	0.415060i	$-0.136239\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.851781			0.256821			0.128411	−	0.991721i	$-0.459012\pi$
$11$	0.851781			0.256821			0.128411	+	0.991721i	$0.459012\pi$
$12$		0			0
$13$	−0.197307	+	0.113916i	−0.0547233	+	0.0315945i	−0.527112	−	0.849796i	$-0.676725\pi$
$13$	−0.197307	+	0.113916i	−0.0547233	+	0.0315945i	0.472389	+	0.881390i	$0.343392\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.57835	−	3.79801i	−1.59548	−	0.921153i	−0.992342	−	0.123519i	$-0.960582\pi$
$17$	−6.57835	−	3.79801i	−1.59548	−	0.921153i	−0.603142	−	0.797634i	$-0.706085\pi$
$18$		0			0
$19$	3.93036	+	1.88474i	0.901687	+	0.432389i
$19$	3.93036	+	1.88474i	0.901687	+	0.432389i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.46423	+	0.845372i	−0.305313	+	0.176272i	−0.644827	−	0.764329i	$-0.723071\pi$
$23$	−1.46423	+	0.845372i	−0.305313	+	0.176272i	0.339514	+	0.940601i	$0.389737\pi$
$24$		0			0
$25$	−3.78872	−	3.26276i	−0.757744	−	0.652552i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.798103	+	1.38236i	0.148204	+	0.256697i	0.930564	−	0.366130i	$-0.119317\pi$
$29$	0.798103	+	1.38236i	0.148204	+	0.256697i	−0.782360	+	0.622827i	$0.785984\pi$
$30$		0			0
$31$	5.76406			1.03526			0.517628	−	0.855606i	$-0.326815\pi$
$31$	5.76406			1.03526			0.517628	+	0.855606i	$0.326815\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.60402	+	1.70922i	0.778222	+	0.288910i
$36$		0			0
$37$			8.94456i			1.47048i	0.677809	+	0.735238i	$0.262930\pi$
$37$			8.94456i			1.47048i	−0.677809	+	0.735238i	$0.737070\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.54377	+	6.13800i	−0.553445	+	0.958594i	0.444578	+	0.895740i	$0.353354\pi$
$41$	−3.54377	+	6.13800i	−0.553445	+	0.958594i	−0.998023	+	0.0628541i	$0.979980\pi$
$42$		0			0
$43$	6.55871	+	3.78668i	1.00019	+	0.577463i	0.908306	−	0.418307i	$-0.137376\pi$
$43$	6.55871	+	3.78668i	1.00019	+	0.577463i	0.0918887	+	0.995769i	$0.470710\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.06032	+	2.92158i	−0.738124	+	0.426156i	−0.821387	−	0.570372i	$-0.806799\pi$
$47$	−5.06032	+	2.92158i	−0.738124	+	0.426156i	0.0832628	+	0.996528i	$0.473466\pi$
$48$		0			0
$49$	2.17631			0.310901
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.14208	−	2.39143i	0.568959	−	0.328488i	−0.187775	−	0.982212i	$-0.560127\pi$
$53$	4.14208	−	2.39143i	0.568959	−	0.328488i	0.756733	+	0.653724i	$0.226794\pi$
$54$		0			0
$55$	−0.662880	+	1.78556i	−0.0893827	+	0.240765i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.33889	+	9.24723i	−0.695065	+	1.20389i	0.275094	+	0.961417i	$0.411291\pi$
$59$	−5.33889	+	9.24723i	−0.695065	+	1.20389i	−0.970159	+	0.242470i	$0.922042\pi$
$60$		0			0
$61$	−1.17130	−	2.02874i	−0.149969	−	0.259754i	0.781247	−	0.624222i	$-0.214584\pi$
$61$	−1.17130	−	2.02874i	−0.149969	−	0.259754i	−0.931216	+	0.364468i	$0.881251\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.0852477	−	0.502263i	−0.0105737	−	0.0622980i
$66$		0			0
$67$	−3.70127	+	2.13693i	−0.452182	+	0.261067i	−0.708751	−	0.705459i	$-0.750741\pi$
$67$	−3.70127	+	2.13693i	−0.452182	+	0.261067i	0.256569	+	0.966526i	$0.417408\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.14788	+	5.45228i	−0.373584	+	0.647067i	−0.990114	−	0.140265i	$-0.955205\pi$
$71$	−3.14788	+	5.45228i	−0.373584	+	0.647067i	0.616530	+	0.787332i	$0.288538\pi$
$72$		0			0
$73$	−7.74621	−	4.47228i	−0.906626	−	0.523441i	−0.0272818	−	0.999628i	$-0.508685\pi$
$73$	−7.74621	−	4.47228i	−0.906626	−	0.523441i	−0.879344	+	0.476187i	$0.842018\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	1.87076i		−	0.213193i
$78$		0			0
$79$	−0.332949	+	0.576685i	−0.0374597	+	0.0648822i	−0.884147	−	0.467208i	$-0.845260\pi$
$79$	−0.332949	+	0.576685i	−0.0374597	+	0.0648822i	0.846688	+	0.532090i	$0.178593\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	8.71844i		−	0.956973i	−0.878095	−	0.478487i	$-0.841186\pi$
$83$		−	8.71844i		−	0.956973i	0.878095	−	0.478487i	$-0.158814\pi$
$84$		0			0
$85$	13.0811	−	10.8343i	1.41885	−	1.17514i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.33889	+	9.24723i	0.565922	+	0.980205i	0.996963	+	0.0778729i	$0.0248128\pi$
$89$	5.33889	+	9.24723i	0.565922	+	0.980205i	−0.431042	+	0.902332i	$0.641854\pi$
$90$		0			0
$91$	0.250192	+	0.433345i	0.0262272	+	0.0454268i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−7.00965	+	6.77236i	−0.719175	+	0.694829i
$96$		0			0
$97$	−8.06614	−	4.65699i	−0.818993	−	0.472846i	0.0310763	−	0.999517i	$-0.490107\pi$
$97$	−8.06614	−	4.65699i	−0.818993	−	0.472846i	−0.850069	+	0.526671i	$0.823440\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.03264	+	5.25268i	0.301759	+	0.522661i	0.976534	−	0.215362i	$-0.0690930\pi$
$101$	3.03264	+	5.25268i	0.301759	+	0.522661i	−0.674776	+	0.738023i	$0.735760\pi$
$102$		0			0
$103$		−	10.2756i		−	1.01248i	−0.862391	−	0.506242i	$-0.831034\pi$
$103$		−	10.2756i		−	1.01248i	0.862391	−	0.506242i	$-0.168966\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			6.21742i			0.601061i	0.953772	+	0.300531i	$0.0971637\pi$
$107$			6.21742i			0.601061i	−0.953772	+	0.300531i	$0.902836\pi$
$108$		0			0
$109$	−1.75405	+	3.03811i	−0.168008	+	0.290998i	−0.937719	−	0.347394i	$-0.887067\pi$
$109$	−1.75405	+	3.03811i	−0.168008	+	0.290998i	0.769711	+	0.638392i	$0.220400\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			17.9659i			1.69009i	0.534698	+	0.845043i	$0.320425\pi$
$113$			17.9659i			1.69009i	−0.534698	+	0.845043i	$0.679575\pi$
$114$		0			0
$115$	−0.632627	−	3.72731i	−0.0589927	−	0.347574i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−8.34154	+	14.4480i	−0.764667	+	1.32444i
$120$		0			0
$121$	−10.2745			−0.934043
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.78813	−	5.40302i	0.875477	−	0.483261i
$126$		0			0
$127$	−12.6002	+	7.27472i	−1.11809	+	0.645527i	−0.940912	−	0.338652i	$-0.890029\pi$
$127$	−12.6002	+	7.27472i	−1.11809	+	0.645527i	−0.177175	+	0.984179i	$0.556696\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.18051	+	10.7050i	−0.539994	+	0.935297i	0.458910	+	0.888483i	$0.348240\pi$
$131$	−6.18051	+	10.7050i	−0.539994	+	0.935297i	−0.998904	+	0.0468142i	$0.985093\pi$
$132$		0			0
$133$	4.13943	−	8.63222i	0.358935	−	0.748508i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	12.5283	−	7.23324i	1.07037	−	0.617978i	0.142086	−	0.989854i	$-0.454619\pi$
$137$	12.5283	−	7.23324i	1.07037	−	0.617978i	0.928282	+	0.371877i	$0.121286\pi$
$138$		0			0
$139$	1.71963	+	2.97849i	0.145857	+	0.252632i	0.929692	−	0.368337i	$-0.120073\pi$
$139$	1.71963	+	2.97849i	0.145857	+	0.252632i	−0.783835	+	0.620969i	$0.786739\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.168063	+	0.0970310i	−0.0140541	+	0.00811414i
$144$		0			0
$145$	−3.51890	+	0.597254i	−0.292229	+	0.0495992i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.0236837	−	0.0410214i	0.00194025	−	0.00336060i	−0.865054	−	0.501679i	$-0.832716\pi$
$149$	0.0236837	−	0.0410214i	0.00194025	−	0.00336060i	0.866994	+	0.498319i	$0.166049\pi$
$150$		0			0
$151$	−9.94886			−0.809627			−0.404813	−	0.914399i	$-0.632664\pi$
$151$	−9.94886			−0.809627			−0.404813	+	0.914399i	$0.632664\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.48576	+	12.0830i	−0.360305	+	0.970533i
$156$		0			0
$157$	15.6549	+	9.03839i	1.24940	+	0.721342i	0.970990	−	0.239120i	$-0.0768590\pi$
$157$	15.6549	+	9.03839i	1.24940	+	0.721342i	0.278411	+	0.960462i	$0.410192\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.85668	+	3.21587i	0.146327	+	0.253446i
$162$		0			0
$163$			14.2125i			1.11321i	0.830777	+	0.556605i	$0.187896\pi$
$163$			14.2125i			1.11321i	−0.830777	+	0.556605i	$0.812104\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.18567	−	1.26190i	0.169132	−	0.0976484i	−0.413044	−	0.910711i	$-0.635535\pi$
$167$	2.18567	−	1.26190i	0.169132	−	0.0976484i	0.582177	+	0.813062i	$0.302201\pi$
$168$		0			0
$169$	−6.47405	+	11.2134i	−0.498004	+	0.862567i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−11.8605	−	6.84768i	−0.901740	−	0.520620i	−0.0239756	−	0.999713i	$-0.507632\pi$
$173$	−11.8605	−	6.84768i	−0.901740	−	0.520620i	−0.877764	+	0.479093i	$0.840966\pi$
$174$		0			0
$175$	−7.16597	+	8.32113i	−0.541696	+	0.629018i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.76670			−0.132050			−0.0660248	−	0.997818i	$-0.521032\pi$
$179$	−1.76670			−0.132050			−0.0660248	+	0.997818i	$0.521032\pi$
$180$		0			0
$181$	−5.34722	−	9.26166i	−0.397456	−	0.688414i	0.595955	−	0.803018i	$-0.296774\pi$
$181$	−5.34722	−	9.26166i	−0.397456	−	0.688414i	−0.993411	+	0.114604i	$0.963440\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−18.7502	−	6.96091i	−1.37854	−	0.511776i
$186$		0			0
$187$	−5.60331	−	3.23507i	−0.409755	−	0.236572i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−15.2844			−1.10594			−0.552971	−	0.833201i	$-0.686506\pi$
$191$	−15.2844			−1.10594			−0.552971	+	0.833201i	$0.686506\pi$
$192$		0			0
$193$	−1.60192	−	0.924867i	−0.115309	−	0.0665734i	0.441237	−	0.897391i	$-0.354540\pi$
$193$	−1.60192	−	0.924867i	−0.115309	−	0.0665734i	−0.556545	+	0.830817i	$0.687873\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			7.71391i			0.549594i	0.961502	+	0.274797i	$0.0886106\pi$
$197$			7.71391i			0.549594i	−0.961502	+	0.274797i	$0.911389\pi$
$198$		0			0
$199$	5.35147	+	9.26901i	0.379355	+	0.657063i	0.990969	−	0.134094i	$-0.0428124\pi$
$199$	5.35147	+	9.26901i	0.379355	+	0.657063i	−0.611613	+	0.791157i	$0.709479\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.03605	−	1.75287i	0.213089	−	0.123027i
$204$		0			0
$205$	−10.1091	−	12.2055i	−0.706047	−	0.852468i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.34781	+	1.60538i	0.231573	+	0.111047i
$210$		0			0
$211$	−8.47981	+	14.6875i	−0.583774	+	1.01113i	0.411253	+	0.911521i	$0.365091\pi$
$211$	−8.47981	+	14.6875i	−0.583774	+	1.01113i	−0.995027	+	0.0996050i	$0.968242\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−13.0421	+	10.8020i	−0.889463	+	0.736687i
$216$		0			0
$217$		−	12.6595i		−	0.859386i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.73061			0.116413
$222$		0			0
$223$	23.9444	+	13.8243i	1.60344	+	0.925745i	0.990793	+	0.135386i	$0.0432274\pi$
$223$	23.9444	+	13.8243i	1.60344	+	0.925745i	0.612644	+	0.790359i	$0.290106\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	6.45083i		−	0.428157i	−0.976816	−	0.214078i	$-0.931325\pi$
$227$		−	6.45083i		−	0.428157i	0.976816	−	0.214078i	$-0.0686748\pi$
$228$		0			0
$229$	17.1452			1.13299			0.566494	−	0.824066i	$-0.308299\pi$
$229$	17.1452			1.13299			0.566494	+	0.824066i	$0.308299\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.457945	+	0.264395i	0.0300010	+	0.0173211i	0.514925	−	0.857235i	$-0.327820\pi$
$233$	0.457945	+	0.264395i	0.0300010	+	0.0173211i	−0.484924	+	0.874556i	$0.661153\pi$
$234$		0			0
$235$	−2.18634	−	12.8815i	−0.142621	−	0.840295i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6.81671			−0.440936			−0.220468	−	0.975394i	$-0.570758\pi$
$239$	−6.81671			−0.440936			−0.220468	+	0.975394i	$0.570758\pi$
$240$		0			0
$241$	−8.93999	−	15.4845i	−0.575875	−	0.997445i	−0.995946	−	0.0899536i	$-0.971328\pi$
$241$	−8.93999	−	15.4845i	−0.575875	−	0.997445i	0.420071	−	0.907491i	$-0.362005\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.69367	+	4.56214i	−0.108204	+	0.291464i
$246$		0			0
$247$	−0.990191	+	0.0758562i	−0.0630044	+	0.00482662i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	8.21999	+	14.2374i	0.518841	+	0.898659i	0.999760	+	0.0218940i	$0.00696963\pi$
$251$	8.21999	+	14.2374i	0.518841	+	0.898659i	−0.480919	+	0.876765i	$0.659697\pi$
$252$		0			0
$253$	−1.24720	+	0.720072i	−0.0784108	+	0.0452705i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.57011	−	4.37061i	0.472211	−	0.272631i	−0.244954	−	0.969535i	$-0.578773\pi$
$257$	7.57011	−	4.37061i	0.472211	−	0.272631i	0.717165	+	0.696904i	$0.245440\pi$
$258$		0			0
$259$	19.6448			1.22067
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−19.0332	−	10.9888i	−1.17364	−	0.677599i	−0.219102	−	0.975702i	$-0.570313\pi$
$263$	−19.0332	−	10.9888i	−1.17364	−	0.677599i	−0.954534	+	0.298103i	$0.903646\pi$
$264$		0			0
$265$	1.78961	+	10.5440i	0.109935	+	0.647714i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.17878	+	5.50580i	−0.193813	+	0.335695i	−0.946511	−	0.322672i	$-0.895419\pi$
$269$	−3.17878	+	5.50580i	−0.193813	+	0.335695i	0.752697	+	0.658367i	$0.228752\pi$
$270$		0			0
$271$	7.39590	−	12.8101i	0.449269	−	0.778156i	−0.549070	−	0.835776i	$-0.685018\pi$
$271$	7.39590	−	12.8101i	0.449269	−	0.778156i	0.998339	+	0.0576202i	$0.0183512\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.22716	−	2.77916i	−0.194605	−	0.167589i
$276$		0			0
$277$		−	8.26106i		−	0.496359i	−0.968714	−	0.248180i	$-0.920168\pi$
$277$		−	8.26106i		−	0.496359i	0.968714	−	0.248180i	$-0.0798323\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	11.4531	+	19.8374i	0.683235	+	1.18340i	0.973988	+	0.226600i	$0.0727610\pi$
$281$	11.4531	+	19.8374i	0.683235	+	1.18340i	−0.290753	+	0.956798i	$0.593906\pi$
$282$		0			0
$283$	−25.9233	−	14.9668i	−1.54098	−	0.889684i	−0.998777	−	0.0494347i	$-0.984258\pi$
$283$	−25.9233	−	14.9668i	−1.54098	−	0.889684i	−0.542200	−	0.840249i	$-0.682409\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	13.4808	+	7.78316i	0.795748	+	0.459425i
$288$		0			0
$289$	20.3498	+	35.2469i	1.19705	+	2.07335i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			22.4761i			1.31307i	0.754296	+	0.656534i	$0.227978\pi$
$293$			22.4761i			1.31307i	−0.754296	+	0.656534i	$0.772022\pi$
$294$		0			0
$295$	−15.2298	−	18.3882i	−0.886716	−	1.07060i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.192602	−	0.333597i	0.0111385	−	0.0192924i
$300$		0			0
$301$	8.31664	−	14.4048i	0.479363	−	0.830281i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	5.16434	−	0.876529i	0.295709	−	0.0501899i
$306$		0			0
$307$	20.6663	+	11.9317i	1.17949	+	0.680979i	0.955897	−	0.293704i	$-0.0948879\pi$
$307$	20.6663	+	11.9317i	1.17949	+	0.680979i	0.223593	+	0.974682i	$0.428221\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.5315			−0.710598			−0.355299	−	0.934753i	$-0.615621\pi$
$311$	−12.5315			−0.710598			−0.355299	+	0.934753i	$0.615621\pi$
$312$		0			0
$313$	−3.44426	+	1.98854i	−0.194681	+	0.112399i	−0.594172	−	0.804338i	$-0.702520\pi$
$313$	−3.44426	+	1.98854i	−0.194681	+	0.112399i	0.399491	+	0.916737i	$0.369187\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−18.1149	+	10.4587i	−1.01744	+	0.587417i	−0.913360	−	0.407153i	$-0.866522\pi$
$317$	−18.1149	+	10.4587i	−1.01744	+	0.587417i	−0.104076	+	0.994569i	$0.533188\pi$
$318$		0			0
$319$	0.679809	+	1.17746i	0.0380620	+	0.0659253i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−18.6970	−	27.3260i	−1.04033	−	1.52046i
$324$		0			0
$325$	1.11922	+	0.212173i	0.0620833	+	0.0117692i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.41663	+	11.1139i	0.353761	+	0.612731i
$330$		0			0
$331$	−13.5367			−0.744042			−0.372021	−	0.928224i	$-0.621335\pi$
$331$	−13.5367			−0.744042			−0.372021	+	0.928224i	$0.621335\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.59915	−	9.42189i	−0.0873709	−	0.514773i
$336$		0			0
$337$	−11.3929	−	6.57769i	−0.620610	−	0.358310i	0.156496	−	0.987679i	$-0.449980\pi$
$337$	−11.3929	−	6.57769i	−0.620610	−	0.358310i	−0.777107	+	0.629369i	$0.783313\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.90971			0.265876
$342$		0			0
$343$		−	20.1538i		−	1.08820i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.21363	+	0.700688i	0.0651509	+	0.0376149i	0.532222	−	0.846605i	$-0.321357\pi$
$347$	1.21363	+	0.700688i	0.0651509	+	0.0376149i	−0.467071	+	0.884220i	$0.654691\pi$
$348$		0			0
$349$	11.4686			0.613899			0.306949	−	0.951726i	$-0.400692\pi$
$349$	11.4686			0.613899			0.306949	+	0.951726i	$0.400692\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	5.17971i		−	0.275688i	−0.990454	−	0.137844i	$-0.955983\pi$
$353$		−	5.17971i		−	0.275688i	0.990454	−	0.137844i	$-0.0440173\pi$
$354$		0			0
$355$	−8.97971	−	10.8419i	−0.476593	−	0.575430i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.29572	−	2.24425i	0.0683853	−	0.118447i	−0.829805	−	0.558053i	$-0.811549\pi$
$359$	1.29572	−	2.24425i	0.0683853	−	0.118447i	0.898191	+	0.439606i	$0.144882\pi$
$360$		0			0
$361$	11.8955	+	14.8154i	0.626080	+	0.779759i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	15.4034	−	12.7577i	0.806253	−	0.667770i
$366$		0			0
$367$	11.3728	−	6.56609i	0.593655	−	0.342747i	−0.172886	−	0.984942i	$-0.555309\pi$
$367$	11.3728	−	6.56609i	0.593655	−	0.342747i	0.766541	+	0.642195i	$0.221976\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−5.25228	−	9.09721i	−0.272685	−	0.472304i
$372$		0			0
$373$			28.8123i			1.49184i	0.666035	+	0.745921i	$0.267990\pi$
$373$			28.8123i			1.49184i	−0.666035	+	0.745921i	$0.732010\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.314944	−	0.181833i	−0.0162204	−	0.00936486i
$378$		0			0
$379$	25.2519			1.29710			0.648551	−	0.761171i	$-0.275375\pi$
$379$	25.2519			1.29710			0.648551	+	0.761171i	$0.275375\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	18.4782	+	10.6684i	0.944190	+	0.545128i	0.891271	−	0.453470i	$-0.149814\pi$
$383$	18.4782	+	10.6684i	0.944190	+	0.545128i	0.0529189	+	0.998599i	$0.483148\pi$
$384$		0			0
$385$	3.92162	+	1.45588i	0.199864	+	0.0741984i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−15.2358	−	26.3892i	−0.772486	−	1.33799i	−0.936196	−	0.351477i	$-0.885680\pi$
$389$	−15.2358	−	26.3892i	−0.772486	−	1.33799i	0.163710	−	0.986508i	$-0.447654\pi$
$390$		0			0
$391$	12.8429			0.649495
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−0.949779	−	1.14675i	−0.0477886	−	0.0576990i
$396$		0			0
$397$	4.28442	+	2.47361i	0.215029	+	0.124147i	0.603646	−	0.797252i	$-0.293714\pi$
$397$	4.28442	+	2.47361i	0.215029	+	0.124147i	−0.388617	+	0.921399i	$0.627047\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	15.4584	−	26.7747i	0.771956	−	1.33707i	−0.164533	−	0.986371i	$-0.552612\pi$
$401$	15.4584	−	26.7747i	0.771956	−	1.33707i	0.936490	−	0.350696i	$-0.114055\pi$
$402$		0			0
$403$	−1.13729	+	0.656616i	−0.0566526	+	0.0327084i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			7.61880i			0.377650i
$408$		0			0
$409$	2.31244	+	4.00526i	0.114343	+	0.198047i	0.917517	−	0.397697i	$-0.130191\pi$
$409$	2.31244	+	4.00526i	0.114343	+	0.198047i	−0.803174	+	0.595744i	$0.796857\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	20.3096	+	11.7258i	0.999371	+	0.576987i
$414$		0			0
$415$	18.2762	+	6.78494i	0.897145	+	0.333060i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−26.8070			−1.30961			−0.654803	−	0.755800i	$-0.727248\pi$
$419$	−26.8070			−1.30961			−0.654803	+	0.755800i	$0.727248\pi$
$420$		0			0
$421$	11.1453	−	19.3043i	0.543191	−	0.940834i	−0.455528	−	0.890222i	$-0.650549\pi$
$421$	11.1453	−	19.3043i	0.543191	−	0.940834i	0.998718	−	0.0506123i	$-0.0161173\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	12.5315	+	35.8532i	0.607868	+	1.73913i
$426$		0			0
$427$	−4.45571	+	2.57251i	−0.215627	+	0.124492i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−14.5079	−	25.1283i	−0.698819	−	1.21039i	−0.968876	−	0.247545i	$-0.920376\pi$
$431$	−14.5079	−	25.1283i	−0.698819	−	1.21039i	0.270058	−	0.962844i	$-0.412957\pi$
$432$		0			0
$433$	−6.47132	+	3.73622i	−0.310992	+	0.179551i	−0.647370	−	0.762176i	$-0.724131\pi$
$433$	−6.47132	+	3.73622i	−0.310992	+	0.179551i	0.336379	+	0.941727i	$0.390798\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−7.34825	+	0.562932i	−0.351515	+	0.0269287i
$438$		0			0
$439$	−10.7675	+	18.6499i	−0.513907	+	0.890112i	0.485963	+	0.873979i	$0.338469\pi$
$439$	−10.7675	+	18.6499i	−0.513907	+	0.890112i	−0.999870	+	0.0161330i	$0.994864\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	29.4833	−	17.0222i	1.40079	−	0.808748i	0.406319	−	0.913731i	$-0.366812\pi$
$443$	29.4833	−	17.0222i	1.40079	−	0.808748i	0.994474	+	0.104983i	$0.0334789\pi$
$444$		0			0
$445$	−23.5396	+	3.99531i	−1.11588	+	0.189396i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	36.9505			1.74380			0.871901	−	0.489682i	$-0.162887\pi$
$449$	36.9505			1.74380			0.871901	+	0.489682i	$0.162887\pi$
$450$		0			0
$451$	−3.01852	+	5.22823i	−0.142136	+	0.246188i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.10311	+	0.187229i	−0.0517148	+	0.00877741i
$456$		0			0
$457$			30.3030i			1.41752i	0.705452	+	0.708758i	$0.250744\pi$
$457$			30.3030i			1.41752i	−0.705452	+	0.708758i	$0.749256\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.18504	−	10.7128i	0.288066	−	0.498945i	−0.685282	−	0.728278i	$-0.740321\pi$
$461$	6.18504	−	10.7128i	0.288066	−	0.498945i	0.973348	+	0.229333i	$0.0736545\pi$
$462$		0			0
$463$			25.3893i			1.17994i	0.807425	+	0.589971i	$0.200861\pi$
$463$			25.3893i			1.17994i	−0.807425	+	0.589971i	$0.799139\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	28.5919i		−	1.32307i	−0.749912	−	0.661537i	$-0.769904\pi$
$467$		−	28.5919i		−	1.32307i	0.749912	−	0.661537i	$-0.230096\pi$
$468$		0			0
$469$	4.69331	+	8.12906i	0.216717	+	0.375365i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	5.58659	+	3.22542i	0.256871	+	0.148305i
$474$		0			0
$475$	−8.74159	−	19.9646i	−0.401092	−	0.916038i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6.22157	+	10.7761i	0.284271	+	0.492372i	0.972432	−	0.233186i	$-0.0749152\pi$
$479$	6.22157	+	10.7761i	0.284271	+	0.492372i	−0.688161	+	0.725558i	$0.741582\pi$
$480$		0			0
$481$	−1.01892	−	1.76483i	−0.0464589	−	0.0804692i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	16.0396	−	13.2846i	0.728322	−	0.603224i
$486$		0			0
$487$		−	18.2489i		−	0.826937i	−0.910518	−	0.413469i	$-0.864317\pi$
$487$		−	18.2489i		−	0.826937i	0.910518	−	0.413469i	$-0.135683\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−9.23347	+	15.9928i	−0.416701	+	0.721747i	−0.995605	−	0.0936485i	$-0.970147\pi$
$491$	−9.23347	+	15.9928i	−0.416701	+	0.721747i	0.578905	+	0.815395i	$0.303480\pi$
$492$		0			0
$493$		−	12.1248i		−	0.546075i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	11.9748	+	6.91365i	0.537143	+	0.310120i
$498$		0			0
$499$	0.746715	−	1.29335i	0.0334276	−	0.0578982i	−0.848828	−	0.528670i	$-0.822691\pi$
$499$	0.746715	−	1.29335i	0.0334276	−	0.0578982i	0.882255	+	0.470771i	$0.156024\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−9.34173	+	5.39345i	−0.416527	+	0.240482i	−0.693590	−	0.720370i	$-0.743972\pi$
$503$	−9.34173	+	5.39345i	−0.416527	+	0.240482i	0.277063	+	0.960852i	$0.410639\pi$
$504$		0			0
$505$	−13.3711	+	2.26945i	−0.595008	+	0.100989i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2.28888	−	3.96446i	−0.101453	−	0.175722i	0.810831	−	0.585281i	$-0.199016\pi$
$509$	−2.28888	−	3.96446i	−0.101453	−	0.175722i	−0.912283	+	0.409559i	$0.865682\pi$
$510$		0			0
$511$	−9.82242	+	17.0129i	−0.434518	+	0.752608i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	21.5404	+	7.99676i	0.949186	+	0.352379i
$516$		0			0
$517$	−4.31028	+	2.48854i	−0.189566	+	0.109446i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	29.1613			1.27758			0.638791	−	0.769380i	$-0.279435\pi$
$521$	29.1613			1.27758			0.638791	+	0.769380i	$0.279435\pi$
$522$		0			0
$523$	27.7312	−	16.0106i	1.21260	−	0.700097i	0.249277	−	0.968432i	$-0.419807\pi$
$523$	27.7312	−	16.0106i	1.21260	−	0.700097i	0.963325	+	0.268336i	$0.0864736\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−37.9180	−	21.8920i	−1.65173	−	0.953629i
$528$		0			0
$529$	−10.0707	+	17.4430i	−0.437856	+	0.758389i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	1.61476i		−	0.0699432i
$534$		0			0
$535$	−13.0334	−	4.83858i	−0.563484	−	0.209190i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.85374			0.0798462
$540$		0			0
$541$	−5.36073	−	9.28505i	−0.230476	−	0.399196i	0.727473	−	0.686137i	$-0.240695\pi$
$541$	−5.36073	−	9.28505i	−0.230476	−	0.399196i	−0.957948	+	0.286941i	$0.907362\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−5.00365	−	6.04132i	−0.214333	−	0.258782i
$546$		0			0
$547$	14.7394	−	8.50982i	0.630213	−	0.363854i	−0.150622	−	0.988591i	$-0.548127\pi$
$547$	14.7394	−	8.50982i	0.630213	−	0.363854i	0.780835	+	0.624738i	$0.214794\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	0.531456	+	6.93738i	0.0226408	+	0.295542i
$552$		0			0
$553$	1.26657	+	0.731253i	0.0538600	+	0.0310961i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−19.3042	+	11.1453i	−0.817946	+	0.472241i	−0.849708	−	0.527254i	$-0.823222\pi$
$557$	−19.3042	+	11.1453i	−0.817946	+	0.472241i	0.0317616	+	0.999495i	$0.489888\pi$
$558$		0			0
$559$	−1.72544			−0.0729785
$560$		0			0
$561$		0			0
$562$		0			0
$563$			45.7030i			1.92615i	0.269234	+	0.963075i	$0.413230\pi$
$563$			45.7030i			1.92615i	−0.269234	+	0.963075i	$0.586770\pi$
$564$		0			0
$565$	−37.6613	−	13.9816i	−1.58442	−	0.588208i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12.2807			0.514835			0.257417	−	0.966300i	$-0.417128\pi$
$569$	12.2807			0.514835			0.257417	+	0.966300i	$0.417128\pi$
$570$		0			0
$571$	−16.5574			−0.692907			−0.346453	−	0.938067i	$-0.612614\pi$
$571$	−16.5574			−0.692907			−0.346453	+	0.938067i	$0.612614\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	8.30579	+	1.57454i	0.346376	+	0.0656630i
$576$		0			0
$577$		−	29.0257i		−	1.20836i	−0.796849	−	0.604179i	$-0.793501\pi$
$577$		−	29.0257i		−	1.20836i	0.796849	−	0.604179i	$-0.206499\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−19.1482			−0.794402
$582$		0			0
$583$	3.52814	−	2.03698i	0.146121	−	0.0843629i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−8.69083	−	5.01765i	−0.358709	−	0.207101i	0.309805	−	0.950800i	$-0.399736\pi$
$587$	−8.69083	−	5.01765i	−0.358709	−	0.207101i	−0.668514	+	0.743699i	$0.733069\pi$
$588$		0			0
$589$	22.6548	+	10.8637i	0.933477	+	0.447633i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−13.2446	+	7.64676i	−0.543890	+	0.314015i	−0.746654	−	0.665213i	$-0.768341\pi$
$593$	−13.2446	+	7.64676i	−0.543890	+	0.314015i	0.202764	+	0.979228i	$0.435007\pi$
$594$		0			0
$595$	−23.7953	−	28.7300i	−0.975510	−	1.17781i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−13.0303	−	22.5691i	−0.532402	−	0.922147i	−0.999284	−	0.0378276i	$-0.987956\pi$
$599$	−13.0303	−	22.5691i	−0.532402	−	0.922147i	0.466883	−	0.884319i	$-0.345377\pi$
$600$		0			0
$601$	15.7129			0.640942			0.320471	−	0.947258i	$-0.396159\pi$
$601$	15.7129			0.640942			0.320471	+	0.947258i	$0.396159\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.99589	−	21.5381i	0.325079	−	0.875648i
$606$		0			0
$607$		−	26.1236i		−	1.06032i	−0.847896	−	0.530162i	$-0.822131\pi$
$607$		−	26.1236i		−	1.06032i	0.847896	−	0.530162i	$-0.177869\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	0.665626	−	1.15290i	0.0269284	−	0.0466413i
$612$		0			0
$613$	−33.1549	−	19.1420i	−1.33912	−	0.773139i	−0.352439	−	0.935835i	$-0.614648\pi$
$613$	−33.1549	−	19.1420i	−1.33912	−	0.773139i	−0.986676	+	0.162696i	$0.947981\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	15.0154	−	8.66916i	0.604498	−	0.349007i	−0.166311	−	0.986073i	$-0.553186\pi$
$617$	15.0154	−	8.66916i	0.604498	−	0.349007i	0.770809	+	0.637066i	$0.219852\pi$
$618$		0			0
$619$	32.1118			1.29068			0.645342	−	0.763894i	$-0.276715\pi$
$619$	32.1118			1.29068			0.645342	+	0.763894i	$0.276715\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	20.3096	−	11.7258i	0.813687	−	0.469783i
$624$		0			0
$625$	3.70880	+	24.7234i	0.148352	+	0.988935i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	33.9715	−	58.8404i	1.35453	−	2.34612i
$630$		0			0
$631$	0.172064	+	0.298024i	0.00684978	+	0.0118642i	0.869430	−	0.494056i	$-0.164486\pi$
$631$	0.172064	+	0.298024i	0.00684978	+	0.0118642i	−0.862580	+	0.505920i	$0.831153\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.44398	−	32.0748i	−0.216038	−	1.27285i
$636$		0			0
$637$	−0.429402	+	0.247915i	−0.0170135	+	0.00982277i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	15.6410	−	27.0909i	0.617781	−	1.07003i	−0.372109	−	0.928189i	$-0.621365\pi$
$641$	15.6410	−	27.0909i	0.617781	−	1.07003i	0.989890	−	0.141839i	$-0.0453015\pi$
$642$		0			0
$643$	5.11932	+	2.95564i	0.201886	+	0.116559i	0.597535	−	0.801843i	$-0.296147\pi$
$643$	5.11932	+	2.95564i	0.201886	+	0.116559i	−0.395649	+	0.918402i	$0.629480\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			19.1643i			0.753425i	0.926330	+	0.376712i	$0.122945\pi$
$647$			19.1643i			0.753425i	−0.926330	+	0.376712i	$0.877055\pi$
$648$		0			0
$649$	−4.54757	+	7.87661i	−0.178508	+	0.309184i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	42.5209i		−	1.66397i	−0.554799	−	0.831985i	$-0.687205\pi$
$653$		−	42.5209i		−	1.66397i	0.554799	−	0.831985i	$-0.312795\pi$
$654$		0			0
$655$	−17.6307	−	21.2870i	−0.688887	−	0.831750i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−18.4989	−	32.0411i	−0.720616	−	1.24814i	−0.960753	−	0.277405i	$-0.910526\pi$
$659$	−18.4989	−	32.0411i	−0.720616	−	1.24814i	0.240137	−	0.970739i	$-0.422808\pi$
$660$		0			0
$661$	−11.1793	−	19.3631i	−0.434824	−	0.753137i	0.562457	−	0.826826i	$-0.309856\pi$
$661$	−11.1793	−	19.3631i	−0.434824	−	0.753137i	−0.997281	+	0.0736893i	$0.976523\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	14.8741	+	15.3952i	0.576791	+	0.597001i
$666$		0			0
$667$	−2.33721	−	1.34939i	−0.0904971	−	0.0522485i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−0.997687	−	1.72804i	−0.0385153	−	0.0667104i
$672$		0			0
$673$			4.94723i			0.190702i	0.995444	+	0.0953508i	$0.0303973\pi$
$673$			4.94723i			0.190702i	−0.995444	+	0.0953508i	$0.969603\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	4.58590i		−	0.176250i	−0.996109	−	0.0881252i	$-0.971912\pi$
$677$		−	4.58590i		−	0.176250i	0.996109	−	0.0881252i	$-0.0280875\pi$
$678$		0			0
$679$	−10.2281	+	17.7156i	−0.392518	+	0.679862i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			26.4117i			1.01062i	0.862939	+	0.505308i	$0.168621\pi$
$683$			26.4117i			1.01062i	−0.862939	+	0.505308i	$0.831379\pi$
$684$		0			0
$685$	5.41293	+	31.8919i	0.206817	+	1.21853i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.544843	+	0.943695i	−0.0207568	+	0.0359519i
$690$		0			0
$691$	−43.1599			−1.64188			−0.820939	−	0.571016i	$-0.806550\pi$
$691$	−43.1599			−1.64188			−0.820939	+	0.571016i	$0.806550\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−7.58199	+	1.28687i	−0.287601	+	0.0488137i
$696$		0			0
$697$	46.6244	−	26.9186i	1.76602	−	1.01961i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	13.5715	−	23.5065i	0.512588	−	0.887829i	−0.487305	−	0.873232i	$-0.662020\pi$
$701$	13.5715	−	23.5065i	0.512588	−	0.887829i	0.999893	−	0.0145974i	$-0.00464665\pi$
$702$		0			0
$703$	−16.8582	+	35.1554i	−0.635818	+	1.32591i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	11.5364	−	6.66055i	0.433871	−	0.250496i
$708$		0			0
$709$	10.3352	+	17.9011i	0.388146	+	0.672288i	0.992200	−	0.124655i	$-0.0397824\pi$
$709$	10.3352	+	17.9011i	0.388146	+	0.672288i	−0.604054	+	0.796943i	$0.706449\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−8.43989	+	4.87277i	−0.316076	+	0.182487i
$714$		0			0
$715$	−0.0726123	−	0.427818i	−0.00271555	−	0.0159995i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	15.5398	−	26.9157i	0.579537	−	1.00379i	−0.415995	−	0.909367i	$-0.636567\pi$
$719$	15.5398	−	26.9157i	0.579537	−	1.00379i	0.995532	−	0.0944208i	$-0.0300999\pi$
$720$		0			0
$721$	−22.5682			−0.840483
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.48650	−	7.84138i	0.0552074	−	0.291221i
$726$		0			0
$727$	−20.0463	−	11.5737i	−0.743475	−	0.429246i	0.0798562	−	0.996806i	$-0.474554\pi$
$727$	−20.0463	−	11.5737i	−0.743475	−	0.429246i	−0.823332	+	0.567561i	$0.807887\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−28.7637	−	49.8202i	−1.06386	−	1.84267i
$732$		0			0
$733$		−	17.8489i		−	0.659266i	−0.944109	−	0.329633i	$-0.893075\pi$
$733$		−	17.8489i		−	0.659266i	0.944109	−	0.329633i	$-0.106925\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.15267	+	1.82019i	−0.116130	+	0.0670477i
$738$		0			0
$739$	−17.7079	+	30.6710i	−0.651396	+	1.12825i	0.331388	+	0.943494i	$0.392483\pi$
$739$	−17.7079	+	30.6710i	−0.651396	+	1.12825i	−0.982784	+	0.184756i	$0.940850\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−31.3859	−	18.1207i	−1.15144	−	0.664782i	−0.202200	−	0.979344i	$-0.564809\pi$
$743$	−31.3859	−	18.1207i	−1.15144	−	0.664782i	−0.949237	+	0.314562i	$0.898142\pi$
$744$		0			0
$745$	0.0675607	+	0.0815716i	0.00247523	+	0.00298855i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	13.6553			0.498953
$750$		0			0
$751$	−20.1202	−	34.8491i	−0.734195	−	1.27166i	−0.955076	−	0.296362i	$-0.904226\pi$
$751$	−20.1202	−	34.8491i	−0.734195	−	1.27166i	0.220881	−	0.975301i	$-0.429107\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	7.74249	−	20.8555i	0.281778	−	0.759010i
$756$		0			0
$757$	26.1071	+	15.0730i	0.948880	+	0.547836i	0.892733	−	0.450587i	$-0.148785\pi$
$757$	26.1071	+	15.0730i	0.948880	+	0.547836i	0.0561469	+	0.998423i	$0.482118\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−15.6964			−0.568994			−0.284497	−	0.958677i	$-0.591827\pi$
$761$	−15.6964			−0.568994			−0.284497	+	0.958677i	$0.591827\pi$
$762$		0			0
$763$	6.67257	+	3.85241i	0.241563	+	0.139467i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	2.43273i		−	0.0878408i
$768$		0			0
$769$	9.71335	+	16.8240i	0.350272	+	0.606689i	0.986297	−	0.164979i	$-0.0527557\pi$
$769$	9.71335	+	16.8240i	0.350272	+	0.606689i	−0.636025	+	0.771669i	$0.719422\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	42.5491	−	24.5657i	1.53039	−	0.883568i	0.531042	−	0.847346i	$-0.321801\pi$
$773$	42.5491	−	24.5657i	1.53039	−	0.883568i	0.999344	−	0.0362225i	$-0.0115325\pi$
$774$		0			0
$775$	−21.8384	−	18.8067i	−0.784459	−	0.675558i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−25.4968	+	17.4455i	−0.913519	+	0.625049i
$780$		0			0
$781$	−2.68130	+	4.64415i	−0.0959445	+	0.166181i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−31.1301	+	25.7831i	−1.11108	+	0.920239i
$786$		0			0
$787$			24.8250i			0.884917i	0.896789	+	0.442459i	$0.145894\pi$
$787$			24.8250i			0.884917i	−0.896789	+	0.442459i	$0.854106\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	39.4582			1.40297
$792$		0			0
$793$	0.462211	+	0.266858i	0.0164136	+	0.00947639i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			29.8866i			1.05864i	0.848423	+	0.529319i	$0.177553\pi$
$797$			29.8866i			1.05864i	−0.848423	+	0.529319i	$0.822447\pi$
$798$		0			0
$799$	44.3848			1.57022
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.59807	−	3.80940i	−0.232841	−	0.134431i
$804$		0			0
$805$	−8.18626	+	1.38943i	−0.288528	+	0.0489710i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	46.8929			1.64867			0.824333	−	0.566105i	$-0.191550\pi$
$809$	46.8929			1.64867			0.824333	+	0.566105i	$0.191550\pi$
$810$		0			0
$811$	0.623347	+	1.07967i	0.0218887	+	0.0379123i	0.876762	−	0.480924i	$-0.159699\pi$
$811$	0.623347	+	1.07967i	0.0218887	+	0.0379123i	−0.854874	+	0.518836i	$0.826365\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−29.7933	−	11.0606i	−1.04361	−	0.387435i
$816$		0			0
$817$	18.6412	+	27.2445i	0.652174	+	0.953164i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.101045	+	0.175014i	0.00352648	+	0.00610805i	0.867783	−	0.496943i	$-0.165544\pi$
$821$	0.101045	+	0.175014i	0.00352648	+	0.00610805i	−0.864257	+	0.503051i	$0.832211\pi$
$822$		0			0
$823$	12.8967	−	7.44589i	0.449549	−	0.259548i	−0.258090	−	0.966121i	$-0.583093\pi$
$823$	12.8967	−	7.44589i	0.449549	−	0.259548i	0.707640	+	0.706573i	$0.249760\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.72867	−	1.57540i	0.0948851	−	0.0547819i	−0.451807	−	0.892116i	$-0.649220\pi$
$827$	2.72867	−	1.57540i	0.0948851	−	0.0547819i	0.546692	+	0.837334i	$0.315887\pi$
$828$		0			0
$829$	40.7012			1.41361			0.706805	−	0.707409i	$-0.250136\pi$
$829$	40.7012			1.41361			0.706805	+	0.707409i	$0.250136\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−14.3165	−	8.26565i	−0.496038	−	0.286388i
$834$		0			0
$835$	0.944328	+	5.56380i	0.0326798	+	0.192543i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−13.1206	+	22.7255i	−0.452973	+	0.784572i	−0.998569	−	0.0534763i	$-0.982970\pi$
$839$	−13.1206	+	22.7255i	−0.452973	+	0.784572i	0.545596	+	0.838048i	$0.316303\pi$
$840$		0			0
$841$	13.2261	−	22.9082i	0.456071	−	0.789938i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−18.4680	−	22.2979i	−0.635319	−	0.767072i
$846$		0			0
$847$			22.5657i			0.775367i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7.56148	−	13.0969i	−0.259204	−	0.448955i
$852$		0			0
$853$	2.85321	+	1.64730i	0.0976919	+	0.0564025i	0.548050	−	0.836446i	$-0.315370\pi$
$853$	2.85321	+	1.64730i	0.0976919	+	0.0564025i	−0.450358	+	0.892848i	$0.648704\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	18.0460	+	10.4188i	0.616439	+	0.355901i	0.775481	−	0.631371i	$-0.217507\pi$
$857$	18.0460	+	10.4188i	0.616439	+	0.355901i	−0.159042	+	0.987272i	$0.550841\pi$
$858$		0			0
$859$	14.7172	+	25.4910i	0.502145	+	0.869740i	0.999997	+	0.00247823i	$0.000788846\pi$
$859$	14.7172	+	25.4910i	0.502145	+	0.869740i	−0.497852	+	0.867262i	$0.665878\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			49.0615i			1.67007i	0.550195	+	0.835036i	$0.314553\pi$
$863$			49.0615i			1.67007i	−0.550195	+	0.835036i	$0.685447\pi$
$864$		0			0
$865$	23.5848	−	19.5339i	0.801908	−	0.664171i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.283600	+	0.491209i	−0.00962047	+	0.0166631i
$870$		0			0
$871$	0.486859	−	0.843264i	0.0164966	−	0.0285729i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−11.8666	−	21.4976i	−0.401164	−	0.726750i
$876$		0			0
$877$	−42.4480	−	24.5074i	−1.43337	−	0.827556i	−0.435993	−	0.899950i	$-0.643603\pi$
$877$	−42.4480	−	24.5074i	−1.43337	−	0.827556i	−0.997376	+	0.0723944i	$0.976936\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	43.8746			1.47817			0.739086	−	0.673611i	$-0.235258\pi$
$881$	43.8746			1.47817			0.739086	+	0.673611i	$0.235258\pi$
$882$		0			0
$883$	−48.3690	+	27.9259i	−1.62775	+	0.939780i	−0.642984	+	0.765880i	$0.722304\pi$
$883$	−48.3690	+	27.9259i	−1.62775	+	0.939780i	−0.984763	+	0.173900i	$0.944363\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	20.5390	−	11.8582i	0.689633	−	0.398160i	−0.113841	−	0.993499i	$-0.536316\pi$
$887$	20.5390	−	11.8582i	0.689633	−	0.398160i	0.803475	+	0.595339i	$0.202982\pi$
$888$		0			0
$889$	15.9774	+	27.6737i	0.535865	+	0.928145i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−25.3953	+	1.94548i	−0.849822	+	0.0651029i
$894$		0			0
$895$	1.37490	−	3.70349i	0.0459578	−	0.123794i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4.60031	+	7.96798i	0.153429	+	0.265747i
$900$		0			0
$901$	−36.3308			−1.21035
$902$		0			0
$903$		0			0
$904$		0			0
$905$	23.5763	−	4.00155i	0.783704	−	0.133016i
$906$		0			0
$907$	−21.7205	−	12.5403i	−0.721216	−	0.416394i	0.0939839	−	0.995574i	$-0.470040\pi$
$907$	−21.7205	−	12.5403i	−0.721216	−	0.416394i	−0.815200	+	0.579179i	$0.803373\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	3.00341			0.0995074			0.0497537	−	0.998762i	$-0.484156\pi$
$911$	3.00341			0.0995074			0.0497537	+	0.998762i	$0.484156\pi$
$912$		0			0
$913$		−	7.42620i		−	0.245771i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	23.5112	+	13.5742i	0.776409	+	0.448260i
$918$		0			0
$919$	18.5467			0.611799			0.305899	−	0.952064i	$-0.401043\pi$
$919$	18.5467			0.611799			0.305899	+	0.952064i	$0.401043\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	1.43437i		−	0.0472128i
$924$		0			0
$925$	29.1839	−	33.8884i	0.959562	−	1.11424i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−4.47449	+	7.75005i	−0.146803	+	0.254271i	−0.930044	−	0.367447i	$-0.880232\pi$
$929$	−4.47449	+	7.75005i	−0.146803	+	0.254271i	0.783241	+	0.621718i	$0.213565\pi$
$930$		0			0
$931$	8.55369	+	4.10178i	0.280336	+	0.134430i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	11.1423	−	9.22844i	0.364391	−	0.301802i
$936$		0			0
$937$	−27.4136	+	15.8272i	−0.895563	+	0.517054i	−0.875758	−	0.482750i	$-0.839638\pi$
$937$	−27.4136	+	15.8272i	−0.895563	+	0.517054i	−0.0198050	+	0.999804i	$0.506305\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−14.9011	−	25.8095i	−0.485763	−	0.841366i	0.514103	−	0.857728i	$-0.328125\pi$
$941$	−14.9011	−	25.8095i	−0.485763	−	0.841366i	−0.999866	+	0.0163621i	$0.994792\pi$
$942$		0			0
$943$		−	11.9832i		−	0.390228i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	50.1907	+	28.9776i	1.63098	+	0.941646i	0.983792	+	0.179314i	$0.0573879\pi$
$947$	50.1907	+	28.9776i	1.63098	+	0.941646i	0.647187	+	0.762332i	$0.275945\pi$
$948$		0			0
$949$	2.03785			0.0661513
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−26.6329	−	15.3765i	−0.862725	−	0.498094i	0.00219907	−	0.999998i	$-0.499300\pi$
$953$	−26.6329	−	15.3765i	−0.862725	−	0.498094i	−0.864924	+	0.501903i	$0.832633\pi$
$954$		0			0
$955$	11.8948	−	32.0403i	0.384906	−	1.03680i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−15.8863	−	27.5159i	−0.512995	−	0.888534i
$960$		0			0
$961$	2.22438			0.0717541
$962$		0			0
$963$		0			0
$964$		0			0
$965$	3.18543	−	2.63830i	0.102543	−	0.0849298i
$966$		0			0
$967$	3.12068	+	1.80172i	0.100354	+	0.0579395i	0.549337	−	0.835601i	$-0.314880\pi$
$967$	3.12068	+	1.80172i	0.100354	+	0.0579395i	−0.448983	+	0.893540i	$0.648214\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	15.9543	−	27.6337i	0.511998	−	0.886807i	−0.487905	−	0.872897i	$-0.662239\pi$
$971$	15.9543	−	27.6337i	0.511998	−	0.886807i	0.999903	−	0.0139100i	$-0.00442783\pi$
$972$		0			0
$973$	6.54162	−	3.77681i	0.209715	−	0.121079i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	40.9910i		−	1.31142i	−0.755014	−	0.655709i	$-0.772370\pi$
$977$		−	40.9910i		−	1.31142i	0.755014	−	0.655709i	$-0.227630\pi$
$978$		0			0
$979$	4.54757	+	7.87661i	0.145341	+	0.251738i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	39.9335	+	23.0556i	1.27368	+	0.735361i	0.975679	−	0.219204i	$-0.0703460\pi$
$983$	39.9335	+	23.0556i	1.27368	+	0.735361i	0.298003	+	0.954565i	$0.403679\pi$
$984$		0			0
$985$	−16.1705	−	6.00319i	−0.515234	−	0.191278i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−12.8046			−0.407163
$990$		0			0
$991$	1.26185	−	2.18559i	0.0400840	−	0.0694276i	−0.845287	−	0.534312i	$-0.820571\pi$
$991$	1.26185	−	2.18559i	0.0400840	−	0.0694276i	0.885371	+	0.464884i	$0.153904\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−23.5950	+	4.00472i	−0.748013	+	0.126958i
$996$		0			0
$997$	0.194307	−	0.112183i	0.00615376	−	0.00355288i	−0.496920	−	0.867796i	$-0.665536\pi$
$997$	0.194307	−	0.112183i	0.00615376	−	0.00355288i	0.503074	+	0.864244i	$0.332202\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.2.bj.e.1189.9	yes	40
3.2	odd	2	inner	3420.2.bj.e.1189.12	yes	40
5.4	even	2	inner	3420.2.bj.e.1189.5	✓	40
15.14	odd	2	inner	3420.2.bj.e.1189.16	yes	40
19.7	even	3	inner	3420.2.bj.e.2629.5	yes	40
57.26	odd	6	inner	3420.2.bj.e.2629.16	yes	40
95.64	even	6	inner	3420.2.bj.e.2629.9	yes	40
285.254	odd	6	inner	3420.2.bj.e.2629.12	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3420.2.bj.e.1189.5	✓	40	5.4	even	2	inner
3420.2.bj.e.1189.9	yes	40	1.1	even	1	trivial
3420.2.bj.e.1189.12	yes	40	3.2	odd	2	inner
3420.2.bj.e.1189.16	yes	40	15.14	odd	2	inner
3420.2.bj.e.2629.5	yes	40	19.7	even	3	inner
3420.2.bj.e.2629.9	yes	40	95.64	even	6	inner
3420.2.bj.e.2629.12	yes	40	285.254	odd	6	inner
3420.2.bj.e.2629.16	yes	40	57.26	odd	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.bj (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.778229 + 2.09627i) q^{5} -2.19629i q^{7} +O(q^{10})\)	\(q+(-0.778229 + 2.09627i) q^{5} -2.19629i q^{7} +0.851781 q^{11} +(-0.197307 + 0.113916i) q^{13} +(-6.57835 - 3.79801i) q^{17} +(3.93036 + 1.88474i) q^{19} +(-1.46423 + 0.845372i) q^{23} +(-3.78872 - 3.26276i) q^{25} +(0.798103 + 1.38236i) q^{29} +5.76406 q^{31} +(4.60402 + 1.70922i) q^{35} +8.94456i q^{37} +(-3.54377 + 6.13800i) q^{41} +(6.55871 + 3.78668i) q^{43} +(-5.06032 + 2.92158i) q^{47} +2.17631 q^{49} +(4.14208 - 2.39143i) q^{53} +(-0.662880 + 1.78556i) q^{55} +(-5.33889 + 9.24723i) q^{59} +(-1.17130 - 2.02874i) q^{61} +(-0.0852477 - 0.502263i) q^{65} +(-3.70127 + 2.13693i) q^{67} +(-3.14788 + 5.45228i) q^{71} +(-7.74621 - 4.47228i) q^{73} -1.87076i q^{77} +(-0.332949 + 0.576685i) q^{79} -8.71844i q^{83} +(13.0811 - 10.8343i) q^{85} +(5.33889 + 9.24723i) q^{89} +(0.250192 + 0.433345i) q^{91} +(-7.00965 + 6.77236i) q^{95} +(-8.06614 - 4.65699i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q+O(q^{10}) \) 40 * q	\( 40 q - 8 q^{19} - 8 q^{31} - 16 q^{49} - 8 q^{61} - 8 q^{79} + 28 q^{91}+O(q^{100}) \) 40 * q - 8 * q^19 - 8 * q^31 - 16 * q^49 - 8 * q^61 - 8 * q^79 + 28 * q^91

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