LMFDB - Embedded newform 3420.2.dg.c.521.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("3420.521");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$27.3088374913$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			521.4
Character	$\chi$	$=$	3420.521
Dual form			3420.2.dg.c.2501.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 - 0.500000i) q^{5} +1.26808 q^{7} +O(q^{10})$	$q+(-0.866025 - 0.500000i) q^{5} +1.26808 q^{7} +2.00000i q^{11} +(-2.34801 + 1.35562i) q^{13} +(5.70200 + 3.29205i) q^{17} +(-2.58495 - 3.50971i) q^{19} +(3.28512 - 1.89667i) q^{23} +(0.500000 + 0.866025i) q^{25} +(-2.66280 - 4.61211i) q^{29} +6.39004i q^{31} +(-1.09819 - 0.634039i) q^{35} +5.95754i q^{37} +(2.22956 - 3.86171i) q^{41} +(-3.04825 + 5.27973i) q^{43} +(1.31239 - 0.757707i) q^{47} -5.39198 q^{49} +(-0.611373 - 1.05893i) q^{53} +(1.00000 - 1.73205i) q^{55} +(3.19190 - 5.52853i) q^{59} +(5.59841 + 9.69673i) q^{61} +2.71125 q^{65} +(-5.83903 + 3.37116i) q^{67} +(-1.72901 + 2.99474i) q^{71} +(-3.47709 + 6.02250i) q^{73} +2.53616i q^{77} +(6.70014 + 3.86833i) q^{79} +10.1359i q^{83} +(-3.29205 - 5.70200i) q^{85} +(4.38505 + 7.59514i) q^{89} +(-2.97746 + 1.71904i) q^{91} +(0.483776 + 4.33197i) q^{95} +(1.91940 + 1.10816i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 16 q^{7}+O(q^{10}) $ 32 * q + 16 * q^7	$ 32 q + 16 q^{7} + 16 q^{25} - 40 q^{43} + 64 q^{49} + 32 q^{55} - 8 q^{61} - 24 q^{67} + 8 q^{73} + 120 q^{79} + 24 q^{91} + 48 q^{97}+O(q^{100}) $ 32 * q + 16 * q^7 + 16 * q^25 - 40 * q^43 + 64 * q^49 + 32 * q^55 - 8 * q^61 - 24 * q^67 + 8 * q^73 + 120 * q^79 + 24 * q^91 + 48 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.866025	−	0.500000i	−0.387298	−	0.223607i
$5$	−0.866025	−	0.500000i	−0.387298	−	0.223607i
$6$		0			0
$7$	1.26808			0.479289			0.239644	−	0.970861i	$-0.422969\pi$
$7$	1.26808			0.479289			0.239644	+	0.970861i	$0.422969\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			2.00000i			0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$			2.00000i			0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$		0			0
$13$	−2.34801	+	1.35562i	−0.651220	+	0.375982i	−0.788924	−	0.614491i	$-0.789361\pi$
$13$	−2.34801	+	1.35562i	−0.651220	+	0.375982i	0.137703	+	0.990474i	$0.456028\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.70200	+	3.29205i	1.38294	+	0.798440i	0.992507	−	0.122191i	$-0.0389921\pi$
$17$	5.70200	+	3.29205i	1.38294	+	0.798440i	0.390433	+	0.920631i	$0.372325\pi$
$18$		0			0
$19$	−2.58495	−	3.50971i	−0.593028	−	0.805182i
$19$	−2.58495	−	3.50971i	−0.593028	−	0.805182i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.28512	−	1.89667i	0.684996	−	0.395482i	−0.116739	−	0.993163i	$-0.537244\pi$
$23$	3.28512	−	1.89667i	0.684996	−	0.395482i	0.801735	+	0.597680i	$0.203911\pi$
$24$		0			0
$25$	0.500000	+	0.866025i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.66280	−	4.61211i	−0.494470	−	0.856447i	0.505510	−	0.862821i	$-0.331305\pi$
$29$	−2.66280	−	4.61211i	−0.494470	−	0.856447i	−0.999980	+	0.00637360i	$0.997971\pi$
$30$		0			0
$31$			6.39004i			1.14769i	0.818965	+	0.573843i	$0.194548\pi$
$31$			6.39004i			1.14769i	−0.818965	+	0.573843i	$0.805452\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.09819	−	0.634039i	−0.185628	−	0.107172i
$36$		0			0
$37$			5.95754i			0.979414i	0.871887	+	0.489707i	$0.162896\pi$
$37$			5.95754i			0.979414i	−0.871887	+	0.489707i	$0.837104\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.22956	−	3.86171i	0.348198	−	0.603097i	−0.637731	−	0.770259i	$-0.720127\pi$
$41$	2.22956	−	3.86171i	0.348198	−	0.603097i	0.985929	+	0.167162i	$0.0534603\pi$
$42$		0			0
$43$	−3.04825	+	5.27973i	−0.464854	+	0.805151i	−0.999195	−	0.0401181i	$-0.987227\pi$
$43$	−3.04825	+	5.27973i	−0.464854	+	0.805151i	0.534341	+	0.845269i	$0.320560\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.31239	−	0.757707i	0.191431	−	0.110523i	−0.401221	−	0.915981i	$-0.631414\pi$
$47$	1.31239	−	0.757707i	0.191431	−	0.110523i	0.592652	+	0.805458i	$0.298081\pi$
$48$		0			0
$49$	−5.39198			−0.770283
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.611373	−	1.05893i	−0.0839786	−	0.145455i	0.820977	−	0.570961i	$-0.193429\pi$
$53$	−0.611373	−	1.05893i	−0.0839786	−	0.145455i	−0.904956	+	0.425506i	$0.860096\pi$
$54$		0			0
$55$	1.00000	−	1.73205i	0.134840	−	0.233550i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.19190	−	5.52853i	0.415549	−	0.719753i	−0.579937	−	0.814662i	$-0.696923\pi$
$59$	3.19190	−	5.52853i	0.415549	−	0.719753i	0.995486	+	0.0949089i	$0.0302560\pi$
$60$		0			0
$61$	5.59841	+	9.69673i	0.716803	+	1.24154i	0.962260	+	0.272132i	$0.0877287\pi$
$61$	5.59841	+	9.69673i	0.716803	+	1.24154i	−0.245457	+	0.969408i	$0.578938\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.71125			0.336289
$66$		0			0
$67$	−5.83903	+	3.37116i	−0.713351	+	0.411853i	−0.812300	−	0.583239i	$-0.801785\pi$
$67$	−5.83903	+	3.37116i	−0.713351	+	0.411853i	0.0989498	+	0.995092i	$0.468452\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.72901	+	2.99474i	−0.205196	+	0.355410i	−0.950195	−	0.311655i	$-0.899117\pi$
$71$	−1.72901	+	2.99474i	−0.205196	+	0.355410i	0.744999	+	0.667065i	$0.232450\pi$
$72$		0			0
$73$	−3.47709	+	6.02250i	−0.406963	+	0.704880i	−0.994548	−	0.104282i	$-0.966745\pi$
$73$	−3.47709	+	6.02250i	−0.406963	+	0.704880i	0.587585	+	0.809162i	$0.300079\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			2.53616i			0.289022i
$78$		0			0
$79$	6.70014	+	3.86833i	0.753824	+	0.435221i	0.827074	−	0.562093i	$-0.190004\pi$
$79$	6.70014	+	3.86833i	0.753824	+	0.435221i	−0.0732496	+	0.997314i	$0.523337\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			10.1359i			1.11256i	0.830993	+	0.556282i	$0.187773\pi$
$83$			10.1359i			1.11256i	−0.830993	+	0.556282i	$0.812227\pi$
$84$		0			0
$85$	−3.29205	−	5.70200i	−0.357073	−	0.618469i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.38505	+	7.59514i	0.464815	+	0.805083i	0.999193	−	0.0401626i	$-0.0127876\pi$
$89$	4.38505	+	7.59514i	0.464815	+	0.805083i	−0.534378	+	0.845245i	$0.679454\pi$
$90$		0			0
$91$	−2.97746	+	1.71904i	−0.312122	+	0.180204i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.483776	+	4.33197i	0.0496344	+	0.444451i
$96$		0			0
$97$	1.91940	+	1.10816i	0.194885	+	0.112517i	0.594267	−	0.804267i	$-0.297442\pi$
$97$	1.91940	+	1.10816i	0.194885	+	0.112517i	−0.399382	+	0.916784i	$0.630775\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.95691	+	5.17127i	−0.891246	+	0.514561i	−0.874350	−	0.485296i	$-0.838712\pi$
$101$	−8.95691	+	5.17127i	−0.891246	+	0.514561i	−0.0168958	+	0.999857i	$0.505378\pi$
$102$		0			0
$103$		−	4.40231i		−	0.433773i	−0.976197	−	0.216886i	$-0.930410\pi$
$103$		−	4.40231i		−	0.433773i	0.976197	−	0.216886i	$-0.0695901\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.18405			−0.211140			−0.105570	−	0.994412i	$-0.533667\pi$
$107$	−2.18405			−0.211140			−0.105570	+	0.994412i	$0.533667\pi$
$108$		0			0
$109$	−1.68616	−	0.973504i	−0.161505	−	0.0932447i	0.417069	−	0.908875i	$-0.363057\pi$
$109$	−1.68616	−	0.973504i	−0.161505	−	0.0932447i	−0.578574	+	0.815630i	$0.696391\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	15.1226			1.42261			0.711307	−	0.702882i	$-0.248104\pi$
$113$	15.1226			1.42261			0.711307	+	0.702882i	$0.248104\pi$
$114$		0			0
$115$	−3.79333			−0.353730
$116$		0			0
$117$		0			0
$118$		0			0
$119$	7.23059	+	4.17458i	0.662827	+	0.382683i
$120$		0			0
$121$	7.00000			0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	12.8136	−	7.39792i	1.13702	−	0.656460i	0.191330	−	0.981526i	$-0.438720\pi$
$127$	12.8136	−	7.39792i	1.13702	−	0.656460i	0.945691	+	0.325066i	$0.105387\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.81074	+	4.50953i	0.682427	+	0.394000i	0.800769	−	0.598973i	$-0.204424\pi$
$131$	7.81074	+	4.50953i	0.682427	+	0.394000i	−0.118342	+	0.992973i	$0.537758\pi$
$132$		0			0
$133$	−3.27792	−	4.45058i	−0.284231	−	0.385915i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.27920	−	3.62530i	0.536468	−	0.309730i	−0.207178	−	0.978303i	$-0.566428\pi$
$137$	6.27920	−	3.62530i	0.536468	−	0.309730i	0.743646	+	0.668573i	$0.233095\pi$
$138$		0			0
$139$	5.41280	+	9.37525i	0.459108	+	0.795198i	0.998914	−	0.0465912i	$-0.0148358\pi$
$139$	5.41280	+	9.37525i	0.459108	+	0.795198i	−0.539806	+	0.841789i	$0.681502\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.71125	−	4.69602i	−0.226726	−	0.392701i
$144$		0			0
$145$			5.32561i			0.442268i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.121558	+	0.0701813i	0.00995839	+	0.00574948i	0.504971	−	0.863136i	$-0.331503\pi$
$149$	0.121558	+	0.0701813i	0.00995839	+	0.00574948i	−0.495013	+	0.868886i	$0.664837\pi$
$150$		0			0
$151$			14.5635i			1.18516i	0.805512	+	0.592579i	$0.201890\pi$
$151$			14.5635i			1.18516i	−0.805512	+	0.592579i	$0.798110\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.19502	−	5.53394i	0.256630	−	0.444497i
$156$		0			0
$157$	−3.23094	+	5.59616i	−0.257857	+	0.446622i	−0.965668	−	0.259781i	$-0.916350\pi$
$157$	−3.23094	+	5.59616i	−0.257857	+	0.446622i	0.707810	+	0.706402i	$0.249683\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	4.16579	−	2.40512i	0.328311	−	0.189550i
$162$		0			0
$163$	8.99009			0.704158			0.352079	−	0.935970i	$-0.385475\pi$
$163$	8.99009			0.704158			0.352079	+	0.935970i	$0.385475\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.65797	−	13.2640i	−0.592592	−	1.02640i	−0.993882	−	0.110448i	$-0.964771\pi$
$167$	−7.65797	−	13.2640i	−0.592592	−	1.02640i	0.401290	−	0.915951i	$-0.368562\pi$
$168$		0			0
$169$	−2.82457	+	4.89230i	−0.217275	+	0.376331i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.40178	−	12.8203i	0.562747	−	0.974707i	−0.434508	−	0.900668i	$-0.643078\pi$
$173$	7.40178	−	12.8203i	0.562747	−	0.974707i	0.997255	−	0.0740387i	$-0.0235888\pi$
$174$		0			0
$175$	0.634039	+	1.09819i	0.0479289	+	0.0830152i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	15.7974			1.18075			0.590376	−	0.807129i	$-0.298980\pi$
$179$	15.7974			1.18075			0.590376	+	0.807129i	$0.298980\pi$
$180$		0			0
$181$	−9.12981	+	5.27110i	−0.678613	+	0.391798i	−0.799332	−	0.600889i	$-0.794813\pi$
$181$	−9.12981	+	5.27110i	−0.678613	+	0.391798i	0.120719	+	0.992687i	$0.461480\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.97877	−	5.15938i	0.219004	−	0.379325i
$186$		0			0
$187$	−6.58411	+	11.4040i	−0.481478	+	0.833944i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			19.7576i			1.42961i	0.699324	+	0.714805i	$0.253485\pi$
$191$			19.7576i			1.42961i	−0.699324	+	0.714805i	$0.746515\pi$
$192$		0			0
$193$	−14.2624	−	8.23443i	−1.02663	−	0.592727i	−0.110615	−	0.993863i	$-0.535282\pi$
$193$	−14.2624	−	8.23443i	−1.02663	−	0.592727i	−0.916018	+	0.401136i	$0.868615\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			25.4147i			1.81072i	0.424645	+	0.905360i	$0.360399\pi$
$197$			25.4147i			1.81072i	−0.424645	+	0.905360i	$0.639601\pi$
$198$		0			0
$199$	7.83627	+	13.5728i	0.555499	+	0.962152i	0.997865	+	0.0653173i	$0.0208060\pi$
$199$	7.83627	+	13.5728i	0.555499	+	0.962152i	−0.442366	+	0.896835i	$0.645861\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.37664	−	5.84852i	−0.236994	−	0.410485i
$204$		0			0
$205$	−3.86171	+	2.22956i	−0.269713	+	0.155719i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	7.01942	−	5.16989i	0.485543	−	0.357609i
$210$		0			0
$211$	−1.77252	−	1.02336i	−0.122025	−	0.0704513i	0.437745	−	0.899099i	$-0.355777\pi$
$211$	−1.77252	−	1.02336i	−0.122025	−	0.0704513i	−0.559770	+	0.828648i	$0.689111\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.27973	−	3.04825i	0.360074	−	0.207889i
$216$		0			0
$217$			8.10308i			0.550073i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−17.8511			−1.20080
$222$		0			0
$223$	6.40456	+	3.69768i	0.428881	+	0.247615i	0.698870	−	0.715249i	$-0.253687\pi$
$223$	6.40456	+	3.69768i	0.428881	+	0.247615i	−0.269989	+	0.962864i	$0.587020\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.20152			−0.278865			−0.139432	−	0.990232i	$-0.544528\pi$
$227$	−4.20152			−0.278865			−0.139432	+	0.990232i	$0.544528\pi$
$228$		0			0
$229$	−25.3882			−1.67770			−0.838848	−	0.544365i	$-0.816771\pi$
$229$	−25.3882			−1.67770			−0.838848	+	0.544365i	$0.816771\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	15.3451	+	8.85950i	1.00529	+	0.580405i	0.909809	−	0.415026i	$-0.136228\pi$
$233$	15.3451	+	8.85950i	1.00529	+	0.580405i	0.0954815	+	0.995431i	$0.469561\pi$
$234$		0			0
$235$	−1.51541			−0.0988547
$236$		0			0
$237$		0			0
$238$		0			0
$239$			3.00214i			0.194192i	0.995275	+	0.0970961i	$0.0309554\pi$
$239$			3.00214i			0.194192i	−0.995275	+	0.0970961i	$0.969045\pi$
$240$		0			0
$241$	11.4999	−	6.63946i	0.740773	−	0.427686i	−0.0815773	−	0.996667i	$-0.525996\pi$
$241$	11.4999	−	6.63946i	0.740773	−	0.427686i	0.822350	+	0.568981i	$0.192662\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.66959	+	2.69599i	0.298329	+	0.172240i
$246$		0			0
$247$	10.8273	+	4.73661i	0.688926	+	0.301383i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	5.37037	−	3.10059i	0.338975	−	0.195707i	−0.320844	−	0.947132i	$-0.603966\pi$
$251$	5.37037	−	3.10059i	0.338975	−	0.195707i	0.659819	+	0.751425i	$0.270633\pi$
$252$		0			0
$253$	3.79333	+	6.57025i	0.238485	+	0.413068i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.42138	−	4.19395i	−0.151042	−	0.261612i	0.780569	−	0.625070i	$-0.214929\pi$
$257$	−2.42138	−	4.19395i	−0.151042	−	0.261612i	−0.931611	+	0.363458i	$0.881596\pi$
$258$		0			0
$259$			7.55463i			0.469422i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.54240	−	2.04521i	−0.218434	−	0.126113i	0.386791	−	0.922167i	$-0.373583\pi$
$263$	−3.54240	−	2.04521i	−0.218434	−	0.126113i	−0.605225	+	0.796055i	$0.706917\pi$
$264$		0			0
$265$			1.22275i			0.0751127i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.79108	+	11.7625i	−0.414059	+	0.717172i	−0.995329	−	0.0965391i	$-0.969223\pi$
$269$	−6.79108	+	11.7625i	−0.414059	+	0.717172i	0.581270	+	0.813711i	$0.302556\pi$
$270$		0			0
$271$	1.06225	−	1.83987i	0.0645272	−	0.111764i	−0.831957	−	0.554840i	$-0.812779\pi$
$271$	1.06225	−	1.83987i	0.0645272	−	0.111764i	0.896484	+	0.443076i	$0.146113\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.73205	+	1.00000i	−0.104447	+	0.0603023i
$276$		0			0
$277$	−26.7307			−1.60609			−0.803046	−	0.595917i	$-0.796789\pi$
$277$	−26.7307			−1.60609			−0.803046	+	0.595917i	$0.796789\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	12.2223	+	21.1696i	0.729119	+	1.26287i	0.957256	+	0.289242i	$0.0934032\pi$
$281$	12.2223	+	21.1696i	0.729119	+	1.26287i	−0.228137	+	0.973629i	$0.573263\pi$
$282$		0			0
$283$	1.32121	−	2.28841i	0.0785380	−	0.136032i	−0.824081	−	0.566471i	$-0.808308\pi$
$283$	1.32121	−	2.28841i	0.0785380	−	0.136032i	0.902619	+	0.430440i	$0.141641\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.82725	−	4.89694i	0.166887	−	0.289057i
$288$		0			0
$289$	13.1752	+	22.8202i	0.775014	+	1.34236i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	8.44614			0.493428			0.246714	−	0.969088i	$-0.420649\pi$
$293$	8.44614			0.493428			0.246714	+	0.969088i	$0.420649\pi$
$294$		0			0
$295$	−5.52853	+	3.19190i	−0.321883	+	0.185839i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−5.14233	+	8.90678i	−0.297389	+	0.515092i
$300$		0			0
$301$	−3.86542	+	6.69511i	−0.222799	+	0.385900i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	11.1968i		−	0.641128i
$306$		0			0
$307$	11.3782	+	6.56919i	0.649386	+	0.374923i	0.788221	−	0.615392i	$-0.211002\pi$
$307$	11.3782	+	6.56919i	0.649386	+	0.374923i	−0.138835	+	0.990316i	$0.544336\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	6.00481i		−	0.340501i	−0.985401	−	0.170251i	$-0.945542\pi$
$311$		−	6.00481i		−	0.340501i	0.985401	−	0.170251i	$-0.0544577\pi$
$312$		0			0
$313$	−7.70373	−	13.3433i	−0.435441	−	0.754205i	0.561891	−	0.827211i	$-0.310074\pi$
$313$	−7.70373	−	13.3433i	−0.435441	−	0.754205i	−0.997332	+	0.0730060i	$0.976741\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	12.8176	+	22.2007i	0.719907	+	1.24692i	0.961036	+	0.276422i	$0.0891489\pi$
$317$	12.8176	+	22.2007i	0.719907	+	1.24692i	−0.241129	+	0.970493i	$0.577518\pi$
$318$		0			0
$319$	9.22422	−	5.32561i	0.516457	−	0.298177i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3.18523	−	28.5222i	−0.177231	−	1.58702i
$324$		0			0
$325$	−2.34801	−	1.35562i	−0.130244	−	0.0751964i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.66421	−	0.960832i	0.0917508	−	0.0529724i
$330$		0			0
$331$		−	28.0435i		−	1.54141i	−0.637192	−	0.770705i	$-0.719904\pi$
$331$		−	28.0435i		−	1.54141i	0.637192	−	0.770705i	$-0.280096\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.74233			0.368373
$336$		0			0
$337$	4.24823	+	2.45272i	0.231416	+	0.133608i	0.611225	−	0.791457i	$-0.290677\pi$
$337$	4.24823	+	2.45272i	0.231416	+	0.133608i	−0.379809	+	0.925065i	$0.624010\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−12.7801			−0.692081
$342$		0			0
$343$	−15.7140			−0.848476
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−21.9240	−	12.6578i	−1.17694	−	0.679508i	−0.221637	−	0.975129i	$-0.571140\pi$
$347$	−21.9240	−	12.6578i	−1.17694	−	0.679508i	−0.955305	+	0.295621i	$0.904473\pi$
$348$		0			0
$349$	−4.99061			−0.267141			−0.133571	−	0.991039i	$-0.542644\pi$
$349$	−4.99061			−0.267141			−0.133571	+	0.991039i	$0.542644\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	8.06629i		−	0.429325i	−0.976688	−	0.214663i	$-0.931135\pi$
$353$		−	8.06629i		−	0.429325i	0.976688	−	0.214663i	$-0.0688652\pi$
$354$		0			0
$355$	2.99474	−	1.72901i	0.158944	−	0.0917664i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−18.4781	−	10.6683i	−0.975236	−	0.563053i	−0.0744075	−	0.997228i	$-0.523707\pi$
$359$	−18.4781	−	10.6683i	−0.975236	−	0.563053i	−0.900828	+	0.434175i	$0.857040\pi$
$360$		0			0
$361$	−5.63610	+	18.1448i	−0.296637	+	0.954990i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.02250	−	3.47709i	0.315232	−	0.181999i
$366$		0			0
$367$	5.23626	+	9.06947i	0.273331	+	0.473422i	0.969713	−	0.244249i	$-0.0785414\pi$
$367$	5.23626	+	9.06947i	0.273331	+	0.473422i	−0.696382	+	0.717671i	$0.745208\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.775269	−	1.34281i	−0.0402500	−	0.0697150i
$372$		0			0
$373$			8.02763i			0.415655i	0.978165	+	0.207827i	$0.0666393\pi$
$373$			8.02763i			0.415655i	−0.978165	+	0.207827i	$0.933361\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12.5046	+	7.21952i	0.644018	+	0.371824i
$378$		0			0
$379$		−	28.8830i		−	1.48362i	−0.670610	−	0.741810i	$-0.733967\pi$
$379$		−	28.8830i		−	1.48362i	0.670610	−	0.741810i	$-0.266033\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	14.1070	−	24.4340i	0.720832	−	1.24852i	−0.239835	−	0.970814i	$-0.577093\pi$
$383$	14.1070	−	24.4340i	0.720832	−	1.24852i	0.960667	−	0.277704i	$-0.0895735\pi$
$384$		0			0
$385$	1.26808	−	2.19638i	0.0646273	−	0.111938i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−8.63570	+	4.98583i	−0.437847	+	0.252791i	−0.702684	−	0.711502i	$-0.748015\pi$
$389$	−8.63570	+	4.98583i	−0.437847	+	0.252791i	0.264837	+	0.964293i	$0.414682\pi$
$390$		0			0
$391$	24.9757			1.26308
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.86833	−	6.70014i	−0.194637	−	0.337121i
$396$		0			0
$397$	4.81017	−	8.33146i	0.241416	−	0.418144i	−0.719702	−	0.694283i	$-0.755722\pi$
$397$	4.81017	−	8.33146i	0.241416	−	0.418144i	0.961118	+	0.276139i	$0.0890549\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−12.7370	+	22.0612i	−0.636057	+	1.10168i	0.350234	+	0.936662i	$0.386102\pi$
$401$	−12.7370	+	22.0612i	−0.636057	+	1.10168i	−0.986290	+	0.165020i	$0.947231\pi$
$402$		0			0
$403$	−8.66249	−	15.0039i	−0.431510	−	0.747396i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−11.9151			−0.590609
$408$		0			0
$409$	0.251615	−	0.145270i	0.0124416	−	0.00718313i	−0.493766	−	0.869595i	$-0.664380\pi$
$409$	0.251615	−	0.145270i	0.0124416	−	0.00718313i	0.506208	+	0.862411i	$0.331047\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	4.04757	−	7.01060i	0.199168	−	0.344969i
$414$		0			0
$415$	5.06797	−	8.77799i	0.248777	−	0.430895i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			8.22086i			0.401615i	0.979631	+	0.200808i	$0.0643566\pi$
$419$			8.22086i			0.401615i	−0.979631	+	0.200808i	$0.935643\pi$
$420$		0			0
$421$	20.5731	+	11.8779i	1.00267	+	0.578893i	0.909038	−	0.416713i	$-0.136818\pi$
$421$	20.5731	+	11.8779i	1.00267	+	0.578893i	0.0936346	+	0.995607i	$0.470151\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			6.58411i			0.319376i
$426$		0			0
$427$	7.09922	+	12.2962i	0.343556	+	0.595056i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−16.6514	−	28.8411i	−0.802069	−	1.38923i	−0.918252	−	0.395997i	$-0.870399\pi$
$431$	−16.6514	−	28.8411i	−0.802069	−	1.38923i	0.116182	−	0.993228i	$-0.462934\pi$
$432$		0			0
$433$	−15.3215	+	8.84588i	−0.736305	+	0.425106i	−0.820724	−	0.571325i	$-0.806430\pi$
$433$	−15.3215	+	8.84588i	−0.736305	+	0.425106i	0.0844195	+	0.996430i	$0.473096\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−15.1486	−	6.62704i	−0.724657	−	0.317014i
$438$		0			0
$439$	−9.23821	−	5.33368i	−0.440916	−	0.254563i	0.263070	−	0.964777i	$-0.415265\pi$
$439$	−9.23821	−	5.33368i	−0.440916	−	0.254563i	−0.703986	+	0.710214i	$0.748598\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	21.9833	−	12.6920i	1.04446	−	0.603017i	0.123364	−	0.992361i	$-0.460632\pi$
$443$	21.9833	−	12.6920i	1.04446	−	0.603017i	0.921092	+	0.389344i	$0.127298\pi$
$444$		0			0
$445$		−	8.77011i		−	0.415743i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−24.6307			−1.16239			−0.581196	−	0.813763i	$-0.697415\pi$
$449$	−24.6307			−1.16239			−0.581196	+	0.813763i	$0.697415\pi$
$450$		0			0
$451$	7.72341	+	4.45911i	0.363681	+	0.209971i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.43807			0.161179
$456$		0			0
$457$	−3.25585			−0.152302			−0.0761512	−	0.997096i	$-0.524263\pi$
$457$	−3.25585			−0.152302			−0.0761512	+	0.997096i	$0.524263\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	26.2340	+	15.1462i	1.22184	+	0.705429i	0.965310	−	0.261108i	$-0.0840878\pi$
$461$	26.2340	+	15.1462i	1.22184	+	0.705429i	0.256529	+	0.966537i	$0.417421\pi$
$462$		0			0
$463$	−30.8812			−1.43517			−0.717586	−	0.696470i	$-0.754753\pi$
$463$	−30.8812			−1.43517			−0.717586	+	0.696470i	$0.754753\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			26.3246i			1.21816i	0.793111	+	0.609078i	$0.208460\pi$
$467$			26.3246i			1.21816i	−0.793111	+	0.609078i	$0.791540\pi$
$468$		0			0
$469$	−7.40434	+	4.27490i	−0.341901	+	0.197397i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−10.5595	−	6.09651i	−0.485524	−	0.280318i
$474$		0			0
$475$	1.74702	−	3.99348i	0.0801589	−	0.183234i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.0157	−	6.93725i	0.549010	−	0.316971i	−0.199713	−	0.979855i	$-0.564001\pi$
$479$	12.0157	−	6.93725i	0.549010	−	0.316971i	0.748723	+	0.662883i	$0.230668\pi$
$480$		0			0
$481$	−8.07618	−	13.9884i	−0.368242	−	0.637814i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.10816	−	1.91940i	−0.0503191	−	0.0871553i
$486$		0			0
$487$		−	30.1906i		−	1.36807i	−0.729450	−	0.684034i	$-0.760224\pi$
$487$		−	30.1906i		−	1.36807i	0.729450	−	0.684034i	$-0.239776\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−14.7309	−	8.50491i	−0.664798	−	0.383821i	0.129305	−	0.991605i	$-0.458725\pi$
$491$	−14.7309	−	8.50491i	−0.664798	−	0.383821i	−0.794103	+	0.607784i	$0.792059\pi$
$492$		0			0
$493$		−	35.0644i		−	1.57922i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.19252	+	3.79756i	−0.0983481	+	0.170344i
$498$		0			0
$499$	15.6705	−	27.1421i	0.701507	−	1.21505i	−0.266430	−	0.963854i	$-0.585844\pi$
$499$	15.6705	−	27.1421i	0.701507	−	1.21505i	0.967937	−	0.251192i	$-0.0808226\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	31.6288	−	18.2609i	1.41026	−	0.814213i	0.414847	−	0.909891i	$-0.363835\pi$
$503$	31.6288	−	18.2609i	1.41026	−	0.814213i	0.995412	+	0.0956779i	$0.0305019\pi$
$504$		0			0
$505$	10.3425			0.460237
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−5.57140	−	9.64995i	−0.246948	−	0.427727i	0.715729	−	0.698378i	$-0.246094\pi$
$509$	−5.57140	−	9.64995i	−0.246948	−	0.427727i	−0.962678	+	0.270651i	$0.912761\pi$
$510$		0			0
$511$	−4.40922	+	7.63700i	−0.195053	+	0.337841i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.20116	+	3.81252i	−0.0969946	+	0.168000i
$516$		0			0
$517$	1.51541	+	2.62477i	0.0666478	+	0.115437i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−38.9439			−1.70616			−0.853081	−	0.521778i	$-0.825269\pi$
$521$	−38.9439			−1.70616			−0.853081	+	0.521778i	$0.825269\pi$
$522$		0			0
$523$	6.29370	−	3.63367i	0.275204	−	0.158889i	−0.356046	−	0.934468i	$-0.615875\pi$
$523$	6.29370	−	3.63367i	0.275204	−	0.158889i	0.631250	+	0.775579i	$0.282542\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−21.0364	+	36.4361i	−0.916359	+	1.58718i
$528$		0			0
$529$	−4.30531	+	7.45702i	−0.187187	+	0.324218i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			12.0898i			0.523665i
$534$		0			0
$535$	1.89144	+	1.09202i	0.0817740	+	0.0472123i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	10.7840i		−	0.464498i
$540$		0			0
$541$	−8.40607	−	14.5597i	−0.361405	−	0.625972i	0.626787	−	0.779191i	$-0.284370\pi$
$541$	−8.40607	−	14.5597i	−0.361405	−	0.625972i	−0.988192	+	0.153218i	$0.951036\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.973504	+	1.68616i	0.0417003	+	0.0722271i
$546$		0			0
$547$	38.6721	−	22.3273i	1.65350	−	0.954648i	0.677880	−	0.735173i	$-0.262899\pi$
$547$	38.6721	−	22.3273i	1.65350	−	0.954648i	0.975618	−	0.219475i	$-0.0704344\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−9.30395	+	21.2677i	−0.396362	+	0.906035i
$552$		0			0
$553$	8.49630	+	4.90534i	0.361299	+	0.208596i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.02994	+	1.74933i	−0.128383	+	0.0741217i	−0.562816	−	0.826582i	$-0.690282\pi$
$557$	−3.02994	+	1.74933i	−0.128383	+	0.0741217i	0.434433	+	0.900704i	$0.356949\pi$
$558$		0			0
$559$		−	16.5291i		−	0.699108i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	29.6976			1.25160			0.625801	−	0.779982i	$-0.284772\pi$
$563$	29.6976			1.25160			0.625801	+	0.779982i	$0.284772\pi$
$564$		0			0
$565$	−13.0965	−	7.56129i	−0.550976	−	0.318106i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−5.75315			−0.241185			−0.120592	−	0.992702i	$-0.538479\pi$
$569$	−5.75315			−0.241185			−0.120592	+	0.992702i	$0.538479\pi$
$570$		0			0
$571$	−3.91672			−0.163909			−0.0819547	−	0.996636i	$-0.526116\pi$
$571$	−3.91672			−0.163909			−0.0819547	+	0.996636i	$0.526116\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.28512	+	1.89667i	0.136999	+	0.0790965i
$576$		0			0
$577$	−32.2474			−1.34248			−0.671239	−	0.741241i	$-0.734238\pi$
$577$	−32.2474			−1.34248			−0.671239	+	0.741241i	$0.734238\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			12.8532i			0.533240i
$582$		0			0
$583$	2.11786	−	1.22275i	0.0877128	−	0.0506410i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−19.2567	−	11.1179i	−0.794810	−	0.458884i	0.0468433	−	0.998902i	$-0.485084\pi$
$587$	−19.2567	−	11.1179i	−0.794810	−	0.458884i	−0.841653	+	0.540019i	$0.818417\pi$
$588$		0			0
$589$	22.4272	−	16.5179i	0.924096	−	0.680609i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	7.29415	−	4.21128i	0.299535	−	0.172937i	−0.342699	−	0.939445i	$-0.611341\pi$
$593$	7.29415	−	4.21128i	0.299535	−	0.172937i	0.642234	+	0.766509i	$0.278008\pi$
$594$		0			0
$595$	−4.17458	−	7.23059i	−0.171141	−	0.296425i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−4.98318	−	8.63112i	−0.203607	−	0.352658i	0.746081	−	0.665855i	$-0.231933\pi$
$599$	−4.98318	−	8.63112i	−0.203607	−	0.352658i	−0.949688	+	0.313197i	$0.898600\pi$
$600$		0			0
$601$			41.3050i			1.68487i	0.538802	+	0.842433i	$0.318877\pi$
$601$			41.3050i			1.68487i	−0.538802	+	0.842433i	$0.681123\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6.06218	−	3.50000i	−0.246463	−	0.142295i
$606$		0			0
$607$		−	0.603292i		−	0.0244869i	−0.999925	−	0.0122434i	$-0.996103\pi$
$607$		−	0.603292i		−	0.0244869i	0.999925	−	0.0122434i	$-0.00389730\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.05433	+	3.55820i	−0.0831093	+	0.143950i
$612$		0			0
$613$	−8.45933	+	14.6520i	−0.341669	+	0.591788i	−0.984743	−	0.174016i	$-0.944326\pi$
$613$	−8.45933	+	14.6520i	−0.341669	+	0.591788i	0.643074	+	0.765804i	$0.277659\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	32.3657	−	18.6864i	1.30299	−	0.752284i	0.322078	−	0.946713i	$-0.395619\pi$
$617$	32.3657	−	18.6864i	1.30299	−	0.752284i	0.980917	+	0.194429i	$0.0622853\pi$
$618$		0			0
$619$	21.3678			0.858844			0.429422	−	0.903104i	$-0.358717\pi$
$619$	21.3678			0.858844			0.429422	+	0.903104i	$0.358717\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5.56059	+	9.63123i	0.222780	+	0.385867i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−19.6126	+	33.9699i	−0.782004	+	1.35447i
$630$		0			0
$631$	13.1511	+	22.7784i	0.523537	+	0.906793i	0.999625	+	0.0273948i	$0.00872114\pi$
$631$	13.1511	+	22.7784i	0.523537	+	0.906793i	−0.476088	+	0.879398i	$0.657946\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−14.7958			−0.587155
$636$		0			0
$637$	12.6604	−	7.30949i	0.501624	−	0.289613i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−15.6191	+	27.0531i	−0.616919	+	1.06853i	0.373126	+	0.927781i	$0.378286\pi$
$641$	−15.6191	+	27.0531i	−0.616919	+	1.06853i	−0.990045	+	0.140754i	$0.955047\pi$
$642$		0			0
$643$	13.1139	−	22.7140i	0.517163	−	0.895753i	−0.482638	−	0.875820i	$-0.660321\pi$
$643$	13.1139	−	22.7140i	0.517163	−	0.895753i	0.999801	−	0.0199332i	$-0.00634534\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	11.2918i		−	0.443925i	−0.975055	−	0.221962i	$-0.928754\pi$
$647$		−	11.2918i		−	0.443925i	0.975055	−	0.221962i	$-0.0712462\pi$
$648$		0			0
$649$	11.0571	+	6.38379i	0.434027	+	0.250586i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	24.8778i		−	0.973544i	−0.873529	−	0.486772i	$-0.838174\pi$
$653$		−	24.8778i		−	0.973544i	0.873529	−	0.486772i	$-0.161826\pi$
$654$		0			0
$655$	−4.50953	−	7.81074i	−0.176202	−	0.305191i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.57269	−	4.45603i	−0.100218	−	0.173582i	0.811557	−	0.584274i	$-0.198621\pi$
$659$	−2.57269	−	4.45603i	−0.100218	−	0.173582i	−0.911774	+	0.410692i	$0.865287\pi$
$660$		0			0
$661$	−21.2535	+	12.2707i	−0.826667	+	0.477276i	−0.852710	−	0.522384i	$-0.825043\pi$
$661$	−21.2535	+	12.2707i	−0.826667	+	0.477276i	0.0260432	+	0.999661i	$0.491709\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.613466	+	5.49328i	0.0237892	+	0.213020i
$666$		0			0
$667$	−17.4953	−	10.1009i	−0.677420	−	0.391108i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−19.3935	+	11.1968i	−0.748677	+	0.432249i
$672$		0			0
$673$		−	46.1728i		−	1.77983i	−0.456125	−	0.889916i	$-0.650763\pi$
$673$		−	46.1728i		−	1.77983i	0.456125	−	0.889916i	$-0.349237\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6.95353			0.267246			0.133623	−	0.991032i	$-0.457339\pi$
$677$	6.95353			0.267246			0.133623	+	0.991032i	$0.457339\pi$
$678$		0			0
$679$	2.43394	+	1.40524i	0.0934062	+	0.0539281i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−8.19132			−0.313432			−0.156716	−	0.987644i	$-0.550091\pi$
$683$	−8.19132			−0.313432			−0.156716	+	0.987644i	$0.550091\pi$
$684$		0			0
$685$	−7.25059			−0.277031
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.87102	+	1.65758i	0.109377	+	0.0631489i
$690$		0			0
$691$	30.2805			1.15192			0.575962	−	0.817476i	$-0.304628\pi$
$691$	30.2805			1.15192			0.575962	+	0.817476i	$0.304628\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	10.8256i		−	0.410639i
$696$		0			0
$697$	25.4259	−	14.6796i	0.963074	−	0.556031i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−7.97001	−	4.60149i	−0.301023	−	0.173796i	0.341879	−	0.939744i	$-0.388937\pi$
$701$	−7.97001	−	4.60149i	−0.301023	−	0.173796i	−0.642903	+	0.765948i	$0.722270\pi$
$702$		0			0
$703$	20.9092	−	15.3999i	0.788607	−	0.580819i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−11.3581	+	6.55758i	−0.427164	+	0.246623i
$708$		0			0
$709$	−13.2779	−	22.9980i	−0.498663	−	0.863709i	0.501336	−	0.865253i	$-0.332842\pi$
$709$	−13.2779	−	22.9980i	−0.498663	−	0.863709i	−0.999999	+	0.00154362i	$0.999509\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	12.1198	+	20.9921i	0.453890	+	0.786160i
$714$		0			0
$715$			5.42249i			0.202790i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3.58870	+	2.07194i	0.133836	+	0.0772702i	0.565423	−	0.824801i	$-0.308713\pi$
$719$	3.58870	+	2.07194i	0.133836	+	0.0772702i	−0.431587	+	0.902071i	$0.642046\pi$
$720$		0			0
$721$		−	5.58248i		−	0.207902i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.66280	−	4.61211i	0.0988940	−	0.171289i
$726$		0			0
$727$	−13.9473	+	24.1574i	−0.517276	+	0.895948i	0.482523	+	0.875883i	$0.339720\pi$
$727$	−13.9473	+	24.1574i	−0.517276	+	0.895948i	−0.999799	+	0.0200645i	$0.993613\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−34.7623	+	20.0700i	−1.28573	+	0.742317i
$732$		0			0
$733$	−43.4995			−1.60669			−0.803346	−	0.595513i	$-0.796949\pi$
$733$	−43.4995			−1.60669			−0.803346	+	0.595513i	$0.796949\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−6.74233	−	11.6781i	−0.248357	−	0.430167i
$738$		0			0
$739$	−14.9060	+	25.8180i	−0.548328	+	0.949732i	0.450061	+	0.892998i	$0.351402\pi$
$739$	−14.9060	+	25.8180i	−0.548328	+	0.949732i	−0.998389	+	0.0567342i	$0.981931\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	19.7172	−	34.1511i	0.723353	−	1.25288i	−0.236296	−	0.971681i	$-0.575933\pi$
$743$	19.7172	−	34.1511i	0.723353	−	1.25288i	0.959649	−	0.281202i	$-0.0907332\pi$
$744$		0			0
$745$	−0.0701813	−	0.121558i	−0.00257125	−	0.00445353i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.76954			−0.101197
$750$		0			0
$751$	44.9133	−	25.9307i	1.63891	−	0.946225i	0.657701	−	0.753279i	$-0.271529\pi$
$751$	44.9133	−	25.9307i	1.63891	−	0.946225i	0.981209	−	0.192946i	$-0.0618041\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	7.28173	−	12.6123i	0.265009	−	0.459010i
$756$		0			0
$757$	25.4728	−	44.1202i	0.925824	−	1.60357i	0.135593	−	0.990765i	$-0.456706\pi$
$757$	25.4728	−	44.1202i	0.925824	−	1.60357i	0.790231	−	0.612810i	$-0.209961\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	18.8261i		−	0.682445i	−0.939982	−	0.341223i	$-0.889159\pi$
$761$		−	18.8261i		−	0.682445i	0.939982	−	0.341223i	$-0.110841\pi$
$762$		0			0
$763$	−2.13818	−	1.23448i	−0.0774073	−	0.0446911i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			17.3080i			0.624957i
$768$		0			0
$769$	12.5028	+	21.6555i	0.450863	+	0.780918i	0.998440	−	0.0558377i	$-0.0177829\pi$
$769$	12.5028	+	21.6555i	0.450863	+	0.780918i	−0.547577	+	0.836755i	$0.684450\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−19.1634	−	33.1920i	−0.689259	−	1.19383i	−0.972078	−	0.234658i	$-0.924603\pi$
$773$	−19.1634	−	33.1920i	−0.689259	−	1.19383i	0.282819	−	0.959173i	$-0.408730\pi$
$774$		0			0
$775$	−5.53394	+	3.19502i	−0.198785	+	0.114769i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−19.3167	+	2.15721i	−0.692094	+	0.0772901i
$780$		0			0
$781$	−5.98947	−	3.45802i	−0.214320	−	0.123738i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	5.59616	−	3.23094i	0.199735	−	0.115317i
$786$		0			0
$787$			53.6699i			1.91313i	0.291526	+	0.956563i	$0.405837\pi$
$787$			53.6699i			1.91313i	−0.291526	+	0.956563i	$0.594163\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	19.1766			0.681842
$792$		0			0
$793$	−26.2902	−	15.1787i	−0.933594	−	0.539011i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	5.93487			0.210224			0.105112	−	0.994460i	$-0.466480\pi$
$797$	5.93487			0.210224			0.105112	+	0.994460i	$0.466480\pi$
$798$		0			0
$799$	9.97765			0.352984
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−12.0450	−	6.95418i	−0.425059	−	0.245408i
$804$		0			0
$805$	−4.81024			−0.169539
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	28.1195i		−	0.988628i	−0.869283	−	0.494314i	$-0.835419\pi$
$809$		−	28.1195i		−	0.988628i	0.869283	−	0.494314i	$-0.164581\pi$
$810$		0			0
$811$	30.9308	−	17.8579i	1.08613	−	0.627076i	0.153585	−	0.988135i	$-0.450918\pi$
$811$	30.9308	−	17.8579i	1.08613	−	0.627076i	0.932543	+	0.361059i	$0.117585\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7.78565	−	4.49504i	−0.272719	−	0.157455i
$816$		0			0
$817$	26.4099	−	2.94934i	0.923965	−	0.103184i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−8.84237	+	5.10515i	−0.308601	+	0.178171i	−0.646300	−	0.763083i	$-0.723685\pi$
$821$	−8.84237	+	5.10515i	−0.308601	+	0.178171i	0.337699	+	0.941254i	$0.390351\pi$
$822$		0			0
$823$	−21.6230	−	37.4521i	−0.753729	−	1.30550i	−0.946004	−	0.324156i	$-0.894920\pi$
$823$	−21.6230	−	37.4521i	−0.753729	−	1.30550i	0.192275	−	0.981341i	$-0.438413\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−13.5877	−	23.5346i	−0.472490	−	0.818377i	0.527014	−	0.849856i	$-0.323311\pi$
$827$	−13.5877	−	23.5346i	−0.472490	−	0.818377i	−0.999504	+	0.0314797i	$0.989978\pi$
$828$		0			0
$829$		−	13.7887i		−	0.478902i	−0.970908	−	0.239451i	$-0.923033\pi$
$829$		−	13.7887i		−	0.478902i	0.970908	−	0.239451i	$-0.0769674\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−30.7451	−	17.7507i	−1.06525	−	0.615025i
$834$		0			0
$835$			15.3159i			0.530030i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−27.1648	+	47.0509i	−0.937834	+	1.62438i	−0.168334	+	0.985730i	$0.553839\pi$
$839$	−27.1648	+	47.0509i	−0.937834	+	1.62438i	−0.769500	+	0.638646i	$0.779495\pi$
$840$		0			0
$841$	0.318958	−	0.552451i	0.0109985	−	0.0190500i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.89230	−	2.82457i	0.168300	−	0.0971682i
$846$		0			0
$847$	8.87655			0.305002
$848$		0			0
$849$		0			0
$850$		0			0
$851$	11.2995	+	19.5713i	0.387341	+	0.670894i
$852$		0			0
$853$	0.854807	−	1.48057i	0.0292680	−	0.0506937i	−0.851020	−	0.525133i	$-0.824016\pi$
$853$	0.854807	−	1.48057i	0.0292680	−	0.0506937i	0.880288	+	0.474439i	$0.157349\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.70878	+	6.42380i	−0.126690	+	0.219433i	−0.922392	−	0.386255i	$-0.873769\pi$
$857$	−3.70878	+	6.42380i	−0.126690	+	0.219433i	0.795703	+	0.605688i	$0.207102\pi$
$858$		0			0
$859$	−9.66137	−	16.7340i	−0.329642	−	0.570956i	0.652799	−	0.757531i	$-0.273595\pi$
$859$	−9.66137	−	16.7340i	−0.329642	−	0.570956i	−0.982441	+	0.186575i	$0.940261\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−7.54288			−0.256763			−0.128381	−	0.991725i	$-0.540978\pi$
$863$	−7.54288			−0.256763			−0.128381	+	0.991725i	$0.540978\pi$
$864$		0			0
$865$	−12.8203	+	7.40178i	−0.435902	+	0.251668i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7.73665	+	13.4003i	−0.262448	+	0.454573i
$870$		0			0
$871$	9.14006	−	15.8310i	0.309699	−	0.536414i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	1.26808i		−	0.0428689i
$876$		0			0
$877$	−35.1238	−	20.2788i	−1.18605	−	0.684765i	−0.228642	−	0.973511i	$-0.573429\pi$
$877$	−35.1238	−	20.2788i	−1.18605	−	0.684765i	−0.957406	+	0.288746i	$0.906762\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			5.85598i			0.197293i	0.995123	+	0.0986465i	$0.0314513\pi$
$881$			5.85598i			0.197293i	−0.995123	+	0.0986465i	$0.968549\pi$
$882$		0			0
$883$	−21.6326	−	37.4688i	−0.727995	−	1.26092i	−0.957729	−	0.287671i	$-0.907119\pi$
$883$	−21.6326	−	37.4688i	−0.727995	−	1.26092i	0.229734	−	0.973253i	$-0.426214\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−26.4389	−	45.7935i	−0.887731	−	1.53759i	−0.842552	−	0.538616i	$-0.818948\pi$
$887$	−26.4389	−	45.7935i	−0.887731	−	1.53759i	−0.0451789	−	0.998979i	$-0.514386\pi$
$888$		0			0
$889$	16.2486	−	9.38115i	0.544961	−	0.314634i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.05178	−	2.64746i	−0.202515	−	0.0885939i
$894$		0			0
$895$	−13.6809	−	7.89869i	−0.457303	−	0.264024i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	29.4716	−	17.0154i	0.982933	−	0.567496i
$900$		0			0
$901$		−	8.05070i		−	0.268208i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	10.5422			0.350434
$906$		0			0
$907$	3.85509	+	2.22574i	0.128006	+	0.0739045i	0.562636	−	0.826705i	$-0.309787\pi$
$907$	3.85509	+	2.22574i	0.128006	+	0.0739045i	−0.434630	+	0.900609i	$0.643121\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−11.0418			−0.365830			−0.182915	−	0.983129i	$-0.558553\pi$
$911$	−11.0418			−0.365830			−0.182915	+	0.983129i	$0.558553\pi$
$912$		0			0
$913$	−20.2719			−0.670902
$914$		0			0
$915$		0			0
$916$		0			0
$917$	9.90463	+	5.71844i	0.327080	+	0.188840i
$918$		0			0
$919$	−23.8749			−0.787560			−0.393780	−	0.919205i	$-0.628833\pi$
$919$	−23.8749			−0.787560			−0.393780	+	0.919205i	$0.628833\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	9.37556i		−	0.308600i
$924$		0			0
$925$	−5.15938	+	2.97877i	−0.169639	+	0.0979414i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	13.5192	+	7.80532i	0.443551	+	0.256084i	0.705103	−	0.709105i	$-0.250901\pi$
$929$	13.5192	+	7.80532i	0.443551	+	0.256084i	−0.261552	+	0.965189i	$0.584234\pi$
$930$		0			0
$931$	13.9380	+	18.9243i	0.456799	+	0.620218i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	11.4040	−	6.58411i	0.372951	−	0.215323i
$936$		0			0
$937$	−5.33279	−	9.23666i	−0.174215	−	0.301748i	0.765675	−	0.643228i	$-0.222405\pi$
$937$	−5.33279	−	9.23666i	−0.174215	−	0.301748i	−0.939889	+	0.341480i	$0.889072\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	16.4822	+	28.5479i	0.537303	+	0.930636i	0.999048	+	0.0436236i	$0.0138902\pi$
$941$	16.4822	+	28.5479i	0.537303	+	0.930636i	−0.461745	+	0.887013i	$0.652776\pi$
$942$		0			0
$943$		−	16.9149i		−	0.550825i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−3.37463	−	1.94834i	−0.109661	−	0.0633127i	0.444166	−	0.895944i	$-0.353500\pi$
$947$	−3.37463	−	1.94834i	−0.109661	−	0.0633127i	−0.553827	+	0.832632i	$0.686833\pi$
$948$		0			0
$949$		−	18.8545i		−	0.612043i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−4.17994	+	7.23986i	−0.135401	+	0.234522i	−0.925751	−	0.378134i	$-0.876566\pi$
$953$	−4.17994	+	7.23986i	−0.135401	+	0.234522i	0.790349	+	0.612656i	$0.209899\pi$
$954$		0			0
$955$	9.87880	−	17.1106i	0.319671	−	0.553686i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7.96251	−	4.59716i	0.257123	−	0.148450i
$960$		0			0
$961$	−9.83267			−0.317183
$962$		0			0
$963$		0			0
$964$		0			0
$965$	8.23443	+	14.2624i	0.265076	+	0.459124i
$966$		0			0
$967$	7.03685	−	12.1882i	0.226290	−	0.391946i	−0.730416	−	0.683003i	$-0.760674\pi$
$967$	7.03685	−	12.1882i	0.226290	−	0.391946i	0.956706	+	0.291057i	$0.0940070\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−9.58167	+	16.5959i	−0.307490	+	0.532589i	−0.977813	−	0.209481i	$-0.932823\pi$
$971$	−9.58167	+	16.5959i	−0.307490	+	0.532589i	0.670322	+	0.742070i	$0.266156\pi$
$972$		0			0
$973$	6.86386	+	11.8885i	0.220045	+	0.381129i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−21.5162			−0.688364			−0.344182	−	0.938903i	$-0.611844\pi$
$977$	−21.5162			−0.688364			−0.344182	+	0.938903i	$0.611844\pi$
$978$		0			0
$979$	−15.1903	+	8.77011i	−0.485483	+	0.280294i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−5.14739	+	8.91553i	−0.164176	+	0.284361i	−0.936362	−	0.351035i	$-0.885830\pi$
$983$	−5.14739	+	8.91553i	−0.164176	+	0.284361i	0.772186	+	0.635396i	$0.219163\pi$
$984$		0			0
$985$	12.7073	−	22.0097i	0.404889	−	0.701289i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			23.1261i			0.735366i
$990$		0			0
$991$	36.9202	+	21.3159i	1.17281	+	0.677121i	0.954340	−	0.298723i	$-0.0965606\pi$
$991$	36.9202	+	21.3159i	1.17281	+	0.677121i	0.218468	+	0.975844i	$0.429894\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	15.6725i		−	0.496853i
$996$		0			0
$997$	23.6134	+	40.8996i	0.747844	+	1.29530i	0.948854	+	0.315715i	$0.102244\pi$
$997$	23.6134	+	40.8996i	0.747844	+	1.29530i	−0.201010	+	0.979589i	$0.564422\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.2.dg.c.521.4	✓	32
3.2	odd	2	inner	3420.2.dg.c.521.10	yes	32
19.12	odd	6	inner	3420.2.dg.c.2501.10	yes	32
57.50	even	6	inner	3420.2.dg.c.2501.3	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3420.2.dg.c.521.4	✓	32	1.1	even	1	trivial
3420.2.dg.c.521.10	yes	32	3.2	odd	2	inner
3420.2.dg.c.2501.3	yes	32	57.50	even	6	inner
3420.2.dg.c.2501.10	yes	32	19.12	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.dg (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 - 0.500000i) q^{5} +1.26808 q^{7} +O(q^{10})\)	\(q+(-0.866025 - 0.500000i) q^{5} +1.26808 q^{7} +2.00000i q^{11} +(-2.34801 + 1.35562i) q^{13} +(5.70200 + 3.29205i) q^{17} +(-2.58495 - 3.50971i) q^{19} +(3.28512 - 1.89667i) q^{23} +(0.500000 + 0.866025i) q^{25} +(-2.66280 - 4.61211i) q^{29} +6.39004i q^{31} +(-1.09819 - 0.634039i) q^{35} +5.95754i q^{37} +(2.22956 - 3.86171i) q^{41} +(-3.04825 + 5.27973i) q^{43} +(1.31239 - 0.757707i) q^{47} -5.39198 q^{49} +(-0.611373 - 1.05893i) q^{53} +(1.00000 - 1.73205i) q^{55} +(3.19190 - 5.52853i) q^{59} +(5.59841 + 9.69673i) q^{61} +2.71125 q^{65} +(-5.83903 + 3.37116i) q^{67} +(-1.72901 + 2.99474i) q^{71} +(-3.47709 + 6.02250i) q^{73} +2.53616i q^{77} +(6.70014 + 3.86833i) q^{79} +10.1359i q^{83} +(-3.29205 - 5.70200i) q^{85} +(4.38505 + 7.59514i) q^{89} +(-2.97746 + 1.71904i) q^{91} +(0.483776 + 4.33197i) q^{95} +(1.91940 + 1.10816i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 16 q^{7}+O(q^{10}) \) 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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