LMFDB - Embedded newform 3420.2.bb.d.2773.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3420,2,Mod(37,3420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3420, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))

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

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

chi := DirichletCharacter("3420.37");

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.bb (of order $4$, 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:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} + 28 x^{10} - 64 x^{9} + 236 x^{8} - 420 x^{7} + 946 x^{6} - 1216 x^{5} + 1896 x^{4} + \cdots + 1370 $ x^12 - 4x^11 + 28x^10 - 64x^9 + 236x^8 - 420x^7 + 946x^6 - 1216x^5 + 1896x^4 - 1564x^3 + 2284x^2 - 1088*x + 1370
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2773.4
Root			$0.344446 + 1.15131i$ of defining polynomial
Character	$\chi$	$=$	3420.2773
Dual form			3420.2.bb.d.37.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.311108 - 2.21432i) q^{5} +(0.311108 - 0.311108i) q^{7} +O(q^{10})$	$q+(-0.311108 - 2.21432i) q^{5} +(0.311108 - 0.311108i) q^{7} -2.90321 q^{11} +(2.84674 - 2.84674i) q^{13} +(2.52543 - 2.52543i) q^{17} +(4.34250 - 0.377784i) q^{19} +(4.11753 + 4.11753i) q^{23} +(-4.80642 + 1.37778i) q^{25} +2.99151 q^{29} -0.930683i q^{31} +(-0.785680 - 0.592104i) q^{35} +(-8.11992 - 8.11992i) q^{37} -2.06083i q^{41} +(2.11753 + 2.11753i) q^{43} +(2.73975 - 2.73975i) q^{47} +6.80642i q^{49} +(-0.565073 + 0.565073i) q^{53} +(0.903212 + 6.42864i) q^{55} +9.32613 q^{59} +3.52543 q^{61} +(-7.18924 - 5.41795i) q^{65} +(0.144771 + 0.144771i) q^{67} -9.61568i q^{71} +(-4.09679 - 4.09679i) q^{73} +(-0.903212 + 0.903212i) q^{77} -12.6072 q^{79} +(-9.21432 - 9.21432i) q^{83} +(-6.37778 - 4.80642i) q^{85} -7.55485 q^{89} -1.77129i q^{91} +(-2.18752 - 9.49814i) q^{95} +(-12.0421 - 12.0421i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{5} + 4 q^{7}+O(q^{10}) $ 12 * q - 4 * q^5 + 4 * q^7	$ 12 q - 4 q^{5} + 4 q^{7} - 8 q^{11} + 4 q^{17} - 4 q^{23} - 4 q^{25} - 36 q^{35} - 28 q^{43} - 20 q^{47} - 16 q^{55} + 16 q^{61} - 76 q^{73} + 16 q^{77} - 84 q^{83} - 76 q^{85} + 16 q^{95}+O(q^{100}) $ 12 * q - 4 * q^5 + 4 * q^7 - 8 * q^11 + 4 * q^17 - 4 * q^23 - 4 * q^25 - 36 * q^35 - 28 * q^43 - 20 * q^47 - 16 * q^55 + 16 * q^61 - 76 * q^73 + 16 * q^77 - 84 * q^83 - 76 * q^85 + 16 * q^95

Display 100 coefficients 10 coefficients

Character values

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

$n$	$1711$	$1901$	$2737$	$3061$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.311108	−	2.21432i	−0.139132	−	0.990274i
$5$	−0.311108	−	2.21432i	−0.139132	−	0.990274i
$6$		0			0
$7$	0.311108	−	0.311108i	0.117588	−	0.117588i	−0.645864	−	0.763452i	$-0.723503\pi$
$7$	0.311108	−	0.311108i	0.117588	−	0.117588i	0.763452	+	0.645864i	$0.223503\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.90321			−0.875351			−0.437676	−	0.899133i	$-0.644198\pi$
$11$	−2.90321			−0.875351			−0.437676	+	0.899133i	$0.644198\pi$
$12$		0			0
$13$	2.84674	−	2.84674i	0.789544	−	0.789544i	−0.191875	−	0.981419i	$-0.561457\pi$
$13$	2.84674	−	2.84674i	0.789544	−	0.789544i	0.981419	+	0.191875i	$0.0614570\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.52543	−	2.52543i	0.612506	−	0.612506i	−0.331092	−	0.943598i	$-0.607417\pi$
$17$	2.52543	−	2.52543i	0.612506	−	0.612506i	0.943598	+	0.331092i	$0.107417\pi$
$18$		0			0
$19$	4.34250	−	0.377784i	0.996237	−	0.0866697i
$19$	4.34250	−	0.377784i	0.996237	−	0.0866697i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.11753	+	4.11753i	0.858565	+	0.858565i	0.991169	−	0.132604i	$-0.0423340\pi$
$23$	4.11753	+	4.11753i	0.858565	+	0.858565i	−0.132604	+	0.991169i	$0.542334\pi$
$24$		0			0
$25$	−4.80642	+	1.37778i	−0.961285	+	0.275557i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.99151			0.555510			0.277755	−	0.960652i	$-0.410410\pi$
$29$	2.99151			0.555510			0.277755	+	0.960652i	$0.410410\pi$
$30$		0			0
$31$		−	0.930683i		−	0.167156i	−0.996501	−	0.0835778i	$-0.973365\pi$
$31$		−	0.930683i		−	0.167156i	0.996501	−	0.0835778i	$-0.0266347\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.785680	−	0.592104i	−0.132804	−	0.100084i
$36$		0			0
$37$	−8.11992	−	8.11992i	−1.33491	−	1.33491i	−0.900919	−	0.433988i	$-0.857106\pi$
$37$	−8.11992	−	8.11992i	−1.33491	−	1.33491i	−0.433988	−	0.900919i	$-0.642894\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	2.06083i		−	0.321847i	−0.986967	−	0.160924i	$-0.948553\pi$
$41$		−	2.06083i		−	0.321847i	0.986967	−	0.160924i	$-0.0514473\pi$
$42$		0			0
$43$	2.11753	+	2.11753i	0.322921	+	0.322921i	0.849886	−	0.526966i	$-0.176670\pi$
$43$	2.11753	+	2.11753i	0.322921	+	0.322921i	−0.526966	+	0.849886i	$0.676670\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.73975	−	2.73975i	0.399633	−	0.399633i	−0.478470	−	0.878104i	$-0.658809\pi$
$47$	2.73975	−	2.73975i	0.399633	−	0.399633i	0.878104	+	0.478470i	$0.158809\pi$
$48$		0			0
$49$			6.80642i			0.972346i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.565073	+	0.565073i	−0.0776188	+	0.0776188i	−0.744850	−	0.667232i	$-0.767479\pi$
$53$	−0.565073	+	0.565073i	−0.0776188	+	0.0776188i	0.667232	+	0.744850i	$0.267479\pi$
$54$		0			0
$55$	0.903212	+	6.42864i	0.121789	+	0.866838i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.32613			1.21416			0.607080	−	0.794641i	$-0.292341\pi$
$59$	9.32613			1.21416			0.607080	+	0.794641i	$0.292341\pi$
$60$		0			0
$61$	3.52543			0.451385			0.225692	−	0.974199i	$-0.427536\pi$
$61$	3.52543			0.451385			0.225692	+	0.974199i	$0.427536\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−7.18924	−	5.41795i	−0.891715	−	0.672014i
$66$		0			0
$67$	0.144771	+	0.144771i	0.0176866	+	0.0176866i	0.715895	−	0.698208i	$-0.246019\pi$
$67$	0.144771	+	0.144771i	0.0176866	+	0.0176866i	−0.698208	+	0.715895i	$0.746019\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	9.61568i		−	1.14117i	−0.821238	−	0.570585i	$-0.806716\pi$
$71$		−	9.61568i		−	1.14117i	0.821238	−	0.570585i	$-0.193284\pi$
$72$		0			0
$73$	−4.09679	−	4.09679i	−0.479493	−	0.479493i	0.425477	−	0.904969i	$-0.360106\pi$
$73$	−4.09679	−	4.09679i	−0.479493	−	0.479493i	−0.904969	+	0.425477i	$0.860106\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.903212	+	0.903212i	−0.102931	+	0.102931i
$78$		0			0
$79$	−12.6072			−1.41842			−0.709210	−	0.704998i	$-0.750948\pi$
$79$	−12.6072			−1.41842			−0.709210	+	0.704998i	$0.750948\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−9.21432	−	9.21432i	−1.01140	−	1.01140i	−0.999934	−	0.0114688i	$-0.996349\pi$
$83$	−9.21432	−	9.21432i	−1.01140	−	1.01140i	−0.0114688	−	0.999934i	$-0.503651\pi$
$84$		0			0
$85$	−6.37778	−	4.80642i	−0.691768	−	0.521330i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.55485			−0.800812			−0.400406	−	0.916338i	$-0.631131\pi$
$89$	−7.55485			−0.800812			−0.400406	+	0.916338i	$0.631131\pi$
$90$		0			0
$91$		−	1.77129i		−	0.185681i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.18752	−	9.49814i	−0.224435	−	0.974489i
$96$		0			0
$97$	−12.0421	−	12.0421i	−1.22269	−	1.22269i	−0.966671	−	0.256020i	$-0.917589\pi$
$97$	−12.0421	−	12.0421i	−1.22269	−	1.22269i	−0.256020	−	0.966671i	$-0.582411\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8.76986			0.872634			0.436317	−	0.899793i	$-0.356283\pi$
$101$	8.76986			0.872634			0.436317	+	0.899793i	$0.356283\pi$
$102$		0			0
$103$	8.54022	−	8.54022i	0.841493	−	0.841493i	−0.147560	−	0.989053i	$-0.547142\pi$
$103$	8.54022	−	8.54022i	0.841493	−	0.841493i	0.989053	+	0.147560i	$0.0471419\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.77742	+	3.77742i	0.365177	+	0.365177i	0.865715	−	0.500538i	$-0.166864\pi$
$107$	3.77742	+	3.77742i	0.365177	+	0.365177i	−0.500538	+	0.865715i	$0.666864\pi$
$108$		0			0
$109$	−13.7373			−1.31580			−0.657899	−	0.753106i	$-0.728555\pi$
$109$	−13.7373			−1.31580			−0.657899	+	0.753106i	$0.728555\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.91606	−	1.91606i	0.180248	−	0.180248i	−0.611216	−	0.791464i	$-0.709319\pi$
$113$	1.91606	−	1.91606i	0.180248	−	0.180248i	0.791464	+	0.611216i	$0.209319\pi$
$114$		0			0
$115$	7.83654	−	10.3985i	0.730761	−	0.969668i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	1.57136i		−	0.144046i
$120$		0			0
$121$	−2.57136			−0.233760
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4.54617	+	10.2143i	0.406622	+	0.913597i
$126$		0			0
$127$	5.12841	+	5.12841i	0.455073	+	0.455073i	0.897034	−	0.441961i	$-0.145717\pi$
$127$	5.12841	+	5.12841i	0.455073	+	0.455073i	−0.441961	+	0.897034i	$0.645717\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	10.3684			0.905893			0.452946	−	0.891538i	$-0.350373\pi$
$131$	10.3684			0.905893			0.452946	+	0.891538i	$0.350373\pi$
$132$		0			0
$133$	1.23345	−	1.46852i	0.106954	−	0.127337i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−7.99063	+	7.99063i	−0.682686	+	0.682686i	−0.960605	−	0.277919i	$-0.910355\pi$
$137$	−7.99063	+	7.99063i	−0.682686	+	0.682686i	0.277919	+	0.960605i	$0.410355\pi$
$138$		0			0
$139$			10.1891i			0.864231i	0.901818	+	0.432115i	$0.142233\pi$
$139$			10.1891i			0.864231i	−0.901818	+	0.432115i	$0.857767\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−8.26469	+	8.26469i	−0.691128	+	0.691128i
$144$		0			0
$145$	−0.930683	−	6.62416i	−0.0772890	−	0.550107i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	18.0874i		−	1.48178i	−0.671627	−	0.740890i	$-0.734404\pi$
$149$		−	18.0874i		−	1.48178i	0.671627	−	0.740890i	$-0.265596\pi$
$150$		0			0
$151$		−	13.5379i		−	1.10170i	−0.834605	−	0.550848i	$-0.814304\pi$
$151$		−	13.5379i		−	1.10170i	0.834605	−	0.550848i	$-0.185696\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.06083	+	0.289543i	−0.165530	+	0.0232566i
$156$		0			0
$157$	15.1891	−	15.1891i	1.21222	−	1.21222i	0.241931	−	0.970293i	$-0.422219\pi$
$157$	15.1891	−	15.1891i	1.21222	−	1.21222i	0.970293	−	0.241931i	$-0.0777808\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.56199			0.201913
$162$		0			0
$163$	5.44938	+	5.44938i	0.426829	+	0.426829i	0.887547	−	0.460718i	$-0.152408\pi$
$163$	5.44938	+	5.44938i	0.426829	+	0.426829i	−0.460718	+	0.887547i	$0.652408\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.42644	−	2.42644i	−0.187763	−	0.187763i	0.606965	−	0.794729i	$-0.292387\pi$
$167$	−2.42644	−	2.42644i	−0.187763	−	0.187763i	−0.794729	+	0.606965i	$0.792387\pi$
$168$		0			0
$169$		−	3.20787i		−	0.246759i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.83825	−	5.83825i	0.443874	−	0.443874i	−0.449438	−	0.893312i	$-0.648376\pi$
$173$	5.83825	−	5.83825i	0.443874	−	0.443874i	0.893312	+	0.449438i	$0.148376\pi$
$174$		0			0
$175$	−1.06668	+	1.92396i	−0.0806332	+	0.145437i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−20.4516			−1.52862			−0.764311	−	0.644847i	$-0.776921\pi$
$179$	−20.4516			−1.52862			−0.764311	+	0.644847i	$0.776921\pi$
$180$		0			0
$181$			11.9660i			0.889429i	0.895672	+	0.444715i	$0.146695\pi$
$181$			11.9660i			0.889429i	−0.895672	+	0.444715i	$0.853305\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−15.4539	+	20.5063i	−1.13620	+	1.50765i
$186$		0			0
$187$	−7.33185	+	7.33185i	−0.536158	+	0.536158i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−23.0923			−1.67090			−0.835452	−	0.549564i	$-0.814794\pi$
$191$	−23.0923			−1.67090			−0.835452	+	0.549564i	$0.814794\pi$
$192$		0			0
$193$	2.42644	−	2.42644i	0.174659	−	0.174659i	−0.614364	−	0.789023i	$-0.710587\pi$
$193$	2.42644	−	2.42644i	0.174659	−	0.174659i	0.789023	+	0.614364i	$0.210587\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−13.0874	+	13.0874i	−0.932440	+	0.932440i	−0.997858	−	0.0654179i	$-0.979162\pi$
$197$	−13.0874	+	13.0874i	−0.932440	+	0.932440i	0.0654179	+	0.997858i	$0.479162\pi$
$198$		0			0
$199$		−	10.2953i		−	0.729814i	−0.931044	−	0.364907i	$-0.881101\pi$
$199$		−	10.2953i		−	0.729814i	0.931044	−	0.364907i	$-0.118899\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.930683	−	0.930683i	0.0653211	−	0.0653211i
$204$		0			0
$205$	−4.56334	+	0.641140i	−0.318717	+	0.0447792i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−12.6072	+	1.09679i	−0.872057	+	0.0758664i
$210$		0			0
$211$		−	14.9576i		−	1.02972i	−0.857274	−	0.514860i	$-0.827844\pi$
$211$		−	14.9576i		−	1.02972i	0.857274	−	0.514860i	$-0.172156\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.03011	−	5.34767i	0.274851	−	0.364708i
$216$		0			0
$217$	−0.289543	−	0.289543i	−0.0196554	−	0.0196554i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	14.3785i		−	0.967201i
$222$		0			0
$223$	3.55659	−	3.55659i	0.238167	−	0.238167i	−0.577924	−	0.816091i	$-0.696137\pi$
$223$	3.55659	−	3.55659i	0.238167	−	0.238167i	0.816091	+	0.577924i	$0.196137\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	5.83825	+	5.83825i	0.387498	+	0.387498i	0.873794	−	0.486296i	$-0.161652\pi$
$227$	5.83825	+	5.83825i	0.387498	+	0.387498i	−0.486296	+	0.873794i	$0.661652\pi$
$228$		0			0
$229$			12.5763i			0.831064i	0.909579	+	0.415532i	$0.136405\pi$
$229$			12.5763i			0.831064i	−0.909579	+	0.415532i	$0.863595\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	8.74620	+	8.74620i	0.572983	+	0.572983i	0.932961	−	0.359978i	$-0.117216\pi$
$233$	8.74620	+	8.74620i	0.572983	+	0.572983i	−0.359978	+	0.932961i	$0.617216\pi$
$234$		0			0
$235$	−6.91903	−	5.21432i	−0.451348	−	0.340145i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			17.0923i			1.10561i	0.833310	+	0.552806i	$0.186443\pi$
$239$			17.0923i			1.10561i	−0.833310	+	0.552806i	$0.813557\pi$
$240$		0			0
$241$		−	25.8555i		−	1.66550i	−0.553649	−	0.832750i	$-0.686765\pi$
$241$		−	25.8555i		−	1.66550i	0.553649	−	0.832750i	$-0.313235\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	15.0716	−	2.11753i	0.962889	−	0.135284i
$246$		0			0
$247$	11.2865	−	13.4374i	0.718143	−	0.855002i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5.37778			−0.339443			−0.169721	−	0.985492i	$-0.554287\pi$
$251$	−5.37778			−0.339443			−0.169721	+	0.985492i	$0.554287\pi$
$252$		0			0
$253$	−11.9541	−	11.9541i	−0.751546	−	0.751546i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.48727	−	4.48727i	−0.279908	−	0.279908i	0.553164	−	0.833072i	$-0.313420\pi$
$257$	−4.48727	−	4.48727i	−0.279908	−	0.279908i	−0.833072	+	0.553164i	$0.813420\pi$
$258$		0			0
$259$	−5.05234			−0.313937
$260$		0			0
$261$		0			0
$262$		0			0
$263$	11.8365	+	11.8365i	0.729872	+	0.729872i	0.970594	−	0.240722i	$-0.0773842\pi$
$263$	11.8365	+	11.8365i	0.729872	+	0.729872i	−0.240722	+	0.970594i	$0.577384\pi$
$264$		0			0
$265$	1.42705	+	1.07545i	0.0876631	+	0.0660646i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.33462			−0.386229			−0.193114	−	0.981176i	$-0.561859\pi$
$269$	−6.33462			−0.386229			−0.193114	+	0.981176i	$0.561859\pi$
$270$		0			0
$271$	13.5669			0.824131			0.412066	−	0.911154i	$-0.364807\pi$
$271$	13.5669			0.824131			0.412066	+	0.911154i	$0.364807\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	13.9541	−	4.00000i	0.841462	−	0.241209i
$276$		0			0
$277$	−11.9032	+	11.9032i	−0.715195	+	0.715195i	−0.967617	−	0.252422i	$-0.918773\pi$
$277$	−11.9032	+	11.9032i	−0.715195	+	0.715195i	0.252422	+	0.967617i	$0.418773\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	26.6961i		−	1.59256i	−0.604930	−	0.796279i	$-0.706799\pi$
$281$		−	26.6961i		−	1.59256i	0.604930	−	0.796279i	$-0.293201\pi$
$282$		0			0
$283$	−13.1684	−	13.1684i	−0.782779	−	0.782779i	0.197520	−	0.980299i	$-0.436711\pi$
$283$	−13.1684	−	13.1684i	−0.782779	−	0.782779i	−0.980299	+	0.197520i	$0.936711\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.641140	−	0.641140i	−0.0378453	−	0.0378453i
$288$		0			0
$289$			4.24443i			0.249672i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	22.7879	−	22.7879i	1.33129	−	1.33129i	0.427064	−	0.904221i	$-0.359548\pi$
$293$	22.7879	−	22.7879i	1.33129	−	1.33129i	0.904221	−	0.427064i	$-0.140452\pi$
$294$		0			0
$295$	−2.90143	−	20.6510i	−0.168928	−	1.20235i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	23.4431			1.35575
$300$		0			0
$301$	1.31756			0.0759430
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.09679	−	7.80642i	−0.0628019	−	0.446995i
$306$		0			0
$307$	19.5756	+	19.5756i	1.11724	+	1.11724i	0.992145	+	0.125093i	$0.0399229\pi$
$307$	19.5756	+	19.5756i	1.11724	+	1.11724i	0.125093	+	0.992145i	$0.460077\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−30.1289			−1.70845			−0.854227	−	0.519901i	$-0.825969\pi$
$311$	−30.1289			−1.70845			−0.854227	+	0.519901i	$0.825969\pi$
$312$		0			0
$313$	11.5620	+	11.5620i	0.653522	+	0.653522i	0.953839	−	0.300317i	$-0.0970925\pi$
$313$	11.5620	+	11.5620i	0.653522	+	0.653522i	−0.300317	+	0.953839i	$0.597092\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.6011	+	10.6011i	0.595414	+	0.595414i	0.939089	−	0.343674i	$-0.111672\pi$
$317$	10.6011	+	10.6011i	0.595414	+	0.595414i	−0.343674	+	0.939089i	$0.611672\pi$
$318$		0			0
$319$	−8.68499			−0.486266
$320$		0			0
$321$		0			0
$322$		0			0
$323$	10.0126	−	11.9207i	0.557116	−	0.663287i
$324$		0			0
$325$	−9.76045	+	17.6048i	−0.541412	+	0.976541i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	1.70471i		−	0.0939839i
$330$		0			0
$331$		−	26.7862i		−	1.47230i	−0.676817	−	0.736151i	$-0.736641\pi$
$331$		−	26.7862i		−	1.47230i	0.676817	−	0.736151i	$-0.263359\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.275531	−	0.365610i	0.0150538	−	0.0199754i
$336$		0			0
$337$	10.0500	+	10.0500i	0.547458	+	0.547458i	0.925705	−	0.378247i	$-0.123473\pi$
$337$	10.0500	+	10.0500i	0.547458	+	0.547458i	−0.378247	+	0.925705i	$0.623473\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			2.70197i			0.146320i
$342$		0			0
$343$	4.29529	+	4.29529i	0.231924	+	0.231924i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−18.0573	+	18.0573i	−0.969367	+	0.969367i	−0.999545	−	0.0301774i	$-0.990393\pi$
$347$	−18.0573	+	18.0573i	−0.969367	+	0.969367i	0.0301774	+	0.999545i	$0.490393\pi$
$348$		0			0
$349$		−	29.1941i		−	1.56272i	−0.624079	−	0.781361i	$-0.714526\pi$
$349$		−	29.1941i		−	1.56272i	0.624079	−	0.781361i	$-0.285474\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8.89384	+	8.89384i	0.473372	+	0.473372i	0.903004	−	0.429632i	$-0.141357\pi$
$353$	8.89384	+	8.89384i	0.473372	+	0.473372i	−0.429632	+	0.903004i	$0.641357\pi$
$354$		0			0
$355$	−21.2922	+	2.99151i	−1.13007	+	0.158773i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	27.9037i		−	1.47270i	−0.676601	−	0.736350i	$-0.736548\pi$
$359$		−	27.9037i		−	1.47270i	0.676601	−	0.736350i	$-0.263452\pi$
$360$		0			0
$361$	18.7146	−	3.28105i	0.984977	−	0.172687i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7.79706	+	10.3461i	−0.408117	+	0.541542i
$366$		0			0
$367$	1.89676	−	1.89676i	0.0990100	−	0.0990100i	−0.655867	−	0.754877i	$-0.727697\pi$
$367$	1.89676	−	1.89676i	0.0990100	−	0.0990100i	0.754877	+	0.655867i	$0.227697\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			0.351597i			0.0182540i
$372$		0			0
$373$	−6.05909	+	6.05909i	−0.313728	+	0.313728i	−0.846352	−	0.532624i	$-0.821206\pi$
$373$	−6.05909	+	6.05909i	−0.313728	+	0.313728i	0.532624	+	0.846352i	$0.321206\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	8.51606	−	8.51606i	0.438599	−	0.438599i
$378$		0			0
$379$	25.2144			1.29518			0.647588	−	0.761991i	$-0.275778\pi$
$379$	25.2144			1.29518			0.647588	+	0.761991i	$0.275778\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	11.3323	−	11.3323i	0.579052	−	0.579052i	−0.355590	−	0.934642i	$-0.615720\pi$
$383$	11.3323	−	11.3323i	0.579052	−	0.579052i	0.934642	+	0.355590i	$0.115720\pi$
$384$		0			0
$385$	2.28100	+	1.71900i	0.116250	+	0.0876085i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	19.1338i		−	0.970124i	−0.874480	−	0.485062i	$-0.838797\pi$
$389$		−	19.1338i		−	0.970124i	0.874480	−	0.485062i	$-0.161203\pi$
$390$		0			0
$391$	20.7971			1.05175
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.92219	+	27.9163i	0.197347	+	1.40462i
$396$		0			0
$397$	−6.71900	+	6.71900i	−0.337217	+	0.337217i	−0.855319	−	0.518102i	$-0.826639\pi$
$397$	−6.71900	+	6.71900i	−0.337217	+	0.337217i	0.518102	+	0.855319i	$0.326639\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	4.56334i		−	0.227882i	−0.993488	−	0.113941i	$-0.963653\pi$
$401$		−	4.56334i		−	0.227882i	0.993488	−	0.113941i	$-0.0363475\pi$
$402$		0			0
$403$	−2.64941	−	2.64941i	−0.131977	−	0.131977i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	23.5738	+	23.5738i	1.16851	+	1.16851i
$408$		0			0
$409$	−21.5817			−1.06715			−0.533574	−	0.845754i	$-0.679151\pi$
$409$	−21.5817			−1.06715			−0.533574	+	0.845754i	$0.679151\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.90143	−	2.90143i	0.142770	−	0.142770i
$414$		0			0
$415$	−17.5368	+	23.2701i	−0.860848	+	1.14228i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			7.58120i			0.370366i	0.982704	+	0.185183i	$0.0592878\pi$
$419$			7.58120i			0.370366i	−0.982704	+	0.185183i	$0.940712\pi$
$420$		0			0
$421$			25.0149i			1.21915i	0.792728	+	0.609576i	$0.208660\pi$
$421$			25.0149i			1.21915i	−0.792728	+	0.609576i	$0.791340\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−8.65878	+	15.6178i	−0.420013	+	0.757573i
$426$		0			0
$427$	1.09679	−	1.09679i	0.0530773	−	0.0530773i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	21.2021i		−	1.02127i	−0.859798	−	0.510635i	$-0.829410\pi$
$431$		−	21.2021i		−	1.02127i	0.859798	−	0.510635i	$-0.170590\pi$
$432$		0			0
$433$	−19.0866	+	19.0866i	−0.917243	+	0.917243i	−0.996828	−	0.0795855i	$-0.974640\pi$
$433$	−19.0866	+	19.0866i	−0.917243	+	0.917243i	0.0795855	+	0.996828i	$0.474640\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	19.4359	+	16.3248i	0.929746	+	0.780922i
$438$		0			0
$439$	38.6622			1.84524			0.922622	−	0.385705i	$-0.126042\pi$
$439$	38.6622			1.84524			0.922622	+	0.385705i	$0.126042\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.725457	−	0.725457i	−0.0344675	−	0.0344675i	0.689663	−	0.724130i	$-0.257759\pi$
$443$	−0.725457	−	0.725457i	−0.0344675	−	0.0344675i	−0.724130	+	0.689663i	$0.757759\pi$
$444$		0			0
$445$	2.35037	+	16.7288i	0.111418	+	0.793023i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−41.0746			−1.93843			−0.969215	−	0.246216i	$-0.920813\pi$
$449$	−41.0746			−1.93843			−0.969215	+	0.246216i	$0.920813\pi$
$450$		0			0
$451$			5.98302i			0.281730i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.92219	+	0.551061i	−0.183875	+	0.0258341i
$456$		0			0
$457$	15.3684	−	15.3684i	0.718904	−	0.718904i	−0.249477	−	0.968381i	$-0.580259\pi$
$457$	15.3684	−	15.3684i	0.718904	−	0.718904i	0.968381	+	0.249477i	$0.0802586\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16.9304			0.788528			0.394264	−	0.918997i	$-0.371000\pi$
$461$	16.9304			0.788528			0.394264	+	0.918997i	$0.371000\pi$
$462$		0			0
$463$	−1.49532	−	1.49532i	−0.0694932	−	0.0694932i	0.671506	−	0.740999i	$-0.265648\pi$
$463$	−1.49532	−	1.49532i	−0.0694932	−	0.0694932i	−0.740999	+	0.671506i	$0.765648\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	20.2192	−	20.2192i	0.935635	−	0.935635i	−0.0624153	−	0.998050i	$-0.519880\pi$
$467$	20.2192	−	20.2192i	0.935635	−	0.935635i	0.998050	+	0.0624153i	$0.0198803\pi$
$468$		0			0
$469$	0.0900790			0.00415946
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−6.14764	−	6.14764i	−0.282669	−	0.282669i
$474$		0			0
$475$	−20.3514	+	7.79882i	−0.933785	+	0.357834i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			11.2587i			0.514424i	0.966355	+	0.257212i	$0.0828039\pi$
$479$			11.2587i			0.514424i	−0.966355	+	0.257212i	$0.917196\pi$
$480$		0			0
$481$	−46.2306			−2.10793
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−22.9187	+	30.4115i	−1.04068	+	1.38091i
$486$		0			0
$487$	15.4539	+	15.4539i	0.700284	+	0.700284i	0.964471	−	0.264187i	$-0.0851036\pi$
$487$	15.4539	+	15.4539i	0.700284	+	0.700284i	−0.264187	+	0.964471i	$0.585104\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	20.9906			0.947294			0.473647	−	0.880715i	$-0.342937\pi$
$491$	20.9906			0.947294			0.473647	+	0.880715i	$0.342937\pi$
$492$		0			0
$493$	7.55485	−	7.55485i	0.340253	−	0.340253i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.99151	−	2.99151i	−0.134188	−	0.134188i
$498$		0			0
$499$			24.6780i			1.10474i	0.833599	+	0.552369i	$0.186276\pi$
$499$			24.6780i			1.10474i	−0.833599	+	0.552369i	$0.813724\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	15.0065	+	15.0065i	0.669105	+	0.669105i	0.957509	−	0.288404i	$-0.0931246\pi$
$503$	15.0065	+	15.0065i	0.669105	+	0.669105i	−0.288404	+	0.957509i	$0.593125\pi$
$504$		0			0
$505$	−2.72837	−	19.4193i	−0.121411	−	0.864146i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	34.6306			1.53497			0.767487	−	0.641065i	$-0.221507\pi$
$509$	34.6306			1.53497			0.767487	+	0.641065i	$0.221507\pi$
$510$		0			0
$511$	−2.54909			−0.112765
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−21.5677	−	16.2539i	−0.950387	−	0.716230i
$516$		0			0
$517$	−7.95407	+	7.95407i	−0.349819	+	0.349819i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			39.0418i			1.71045i	0.518255	+	0.855226i	$0.326582\pi$
$521$			39.0418i			1.71045i	−0.518255	+	0.855226i	$0.673418\pi$
$522$		0			0
$523$	25.2904	−	25.2904i	1.10587	−	1.10587i	0.112187	−	0.993687i	$-0.464215\pi$
$523$	25.2904	−	25.2904i	1.10587	−	1.10587i	0.993687	−	0.112187i	$-0.0357855\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.35037	−	2.35037i	−0.102384	−	0.102384i
$528$		0			0
$529$			10.9081i			0.474267i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−5.86665	−	5.86665i	−0.254113	−	0.254113i
$534$		0			0
$535$	7.18924	−	9.53961i	0.310818	−	0.412433i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	19.7605i		−	0.851145i
$540$		0			0
$541$	−8.85283			−0.380613			−0.190307	−	0.981725i	$-0.560948\pi$
$541$	−8.85283			−0.380613			−0.190307	+	0.981725i	$0.560948\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.27379	+	30.4189i	0.183069	+	1.30300i
$546$		0			0
$547$	11.9520	+	11.9520i	0.511032	+	0.511032i	0.914843	−	0.403810i	$-0.132314\pi$
$547$	11.9520	+	11.9520i	0.511032	+	0.511032i	−0.403810	+	0.914843i	$0.632314\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	12.9906	−	1.13015i	0.553420	−	0.0481459i
$552$		0			0
$553$	−3.92219	+	3.92219i	−0.166789	+	0.166789i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−19.8430	+	19.8430i	−0.840774	+	0.840774i	−0.988960	−	0.148185i	$-0.952657\pi$
$557$	−19.8430	+	19.8430i	−0.840774	+	0.840774i	0.148185	+	0.988960i	$0.452657\pi$
$558$		0			0
$559$	12.0561			0.509920
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10.4703	−	10.4703i	0.441270	−	0.441270i	−0.451169	−	0.892439i	$-0.648993\pi$
$563$	10.4703	−	10.4703i	0.441270	−	0.441270i	0.892439	+	0.451169i	$0.148993\pi$
$564$		0			0
$565$	−4.83887	−	3.64666i	−0.203573	−	0.153416i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−42.6744			−1.78901			−0.894503	−	0.447062i	$-0.852470\pi$
$569$	−42.6744			−1.78901			−0.894503	+	0.447062i	$0.852470\pi$
$570$		0			0
$571$	−23.2815			−0.974299			−0.487150	−	0.873319i	$-0.661963\pi$
$571$	−23.2815			−0.974299			−0.487150	+	0.873319i	$0.661963\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−25.4637	−	14.1175i	−1.06191	−	0.588742i
$576$		0			0
$577$	−14.5526	+	14.5526i	−0.605834	+	0.605834i	−0.941855	−	0.336021i	$-0.890919\pi$
$577$	−14.5526	+	14.5526i	−0.605834	+	0.605834i	0.336021	+	0.941855i	$0.390919\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−5.73329			−0.237857
$582$		0			0
$583$	1.64053	−	1.64053i	0.0679437	−	0.0679437i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.20495	−	6.20495i	0.256106	−	0.256106i	−0.567362	−	0.823468i	$-0.692036\pi$
$587$	6.20495	−	6.20495i	0.256106	−	0.256106i	0.823468	+	0.567362i	$0.192036\pi$
$588$		0			0
$589$	−0.351597	−	4.04149i	−0.0144873	−	0.166527i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−32.5210	−	32.5210i	−1.33548	−	1.33548i	−0.900387	−	0.435089i	$-0.856717\pi$
$593$	−32.5210	−	32.5210i	−1.33548	−	1.33548i	−0.435089	−	0.900387i	$-0.643283\pi$
$594$		0			0
$595$	−3.47949	+	0.488863i	−0.142645	+	0.0200414i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	20.8505			0.851929			0.425964	−	0.904740i	$-0.359935\pi$
$599$	20.8505			0.851929			0.425964	+	0.904740i	$0.359935\pi$
$600$		0			0
$601$			7.64493i			0.311843i	0.987769	+	0.155922i	$0.0498347\pi$
$601$			7.64493i			0.311843i	−0.987769	+	0.155922i	$0.950165\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.799970	+	5.69381i	0.0325234	+	0.231486i
$606$		0			0
$607$	16.0951	+	16.0951i	0.653279	+	0.653279i	0.953781	−	0.300502i	$-0.0971543\pi$
$607$	16.0951	+	16.0951i	0.653279	+	0.653279i	−0.300502	+	0.953781i	$0.597154\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	15.5987i		−	0.631056i
$612$		0			0
$613$	9.69535	+	9.69535i	0.391591	+	0.391591i	0.875254	−	0.483663i	$-0.160694\pi$
$613$	9.69535	+	9.69535i	0.391591	+	0.391591i	−0.483663	+	0.875254i	$0.660694\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	26.3319	−	26.3319i	1.06008	−	1.06008i	0.0620046	−	0.998076i	$-0.480251\pi$
$617$	26.3319	−	26.3319i	1.06008	−	1.06008i	0.998076	−	0.0620046i	$-0.0197493\pi$
$618$		0			0
$619$			38.0054i			1.52757i	0.645473	+	0.763783i	$0.276660\pi$
$619$			38.0054i			1.52757i	−0.645473	+	0.763783i	$0.723340\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.35037	+	2.35037i	−0.0941657	+	0.0941657i
$624$		0			0
$625$	21.2034	−	13.2444i	0.848137	−	0.529777i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−41.0125			−1.63528
$630$		0			0
$631$	25.0968			0.999087			0.499544	−	0.866289i	$-0.333501\pi$
$631$	25.0968			0.999087			0.499544	+	0.866289i	$0.333501\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	9.76045	−	12.9514i	0.387332	−	0.513962i
$636$		0			0
$637$	19.3761	+	19.3761i	0.767710	+	0.767710i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			10.1667i			0.401562i	0.979636	+	0.200781i	$0.0643480\pi$
$641$			10.1667i			0.401562i	−0.979636	+	0.200781i	$0.935652\pi$
$642$		0			0
$643$	31.1082	+	31.1082i	1.22679	+	1.22679i	0.965173	+	0.261614i	$0.0842546\pi$
$643$	31.1082	+	31.1082i	1.22679	+	1.22679i	0.261614	+	0.965173i	$0.415745\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−24.6336	+	24.6336i	−0.968446	+	0.968446i	−0.999517	−	0.0310708i	$-0.990108\pi$
$647$	−24.6336	+	24.6336i	−0.968446	+	0.968446i	0.0310708	+	0.999517i	$0.490108\pi$
$648$		0			0
$649$	−27.0757			−1.06282
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−4.09234	−	4.09234i	−0.160146	−	0.160146i	0.622486	−	0.782631i	$-0.286123\pi$
$653$	−4.09234	−	4.09234i	−0.160146	−	0.160146i	−0.782631	+	0.622486i	$0.786123\pi$
$654$		0			0
$655$	−3.22570	−	22.9590i	−0.126038	−	0.897082i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9.90522			−0.385853			−0.192926	−	0.981213i	$-0.561798\pi$
$659$	−9.90522			−0.385853			−0.192926	+	0.981213i	$0.561798\pi$
$660$		0			0
$661$			22.1328i			0.860866i	0.902623	+	0.430433i	$0.141639\pi$
$661$			22.1328i			0.860866i	−0.902623	+	0.430433i	$0.858361\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.63550	−	2.27439i	−0.140979	−	0.0881972i
$666$		0			0
$667$	12.3176	+	12.3176i	0.476941	+	0.476941i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−10.2351			−0.395120
$672$		0			0
$673$	−27.1304	+	27.1304i	−1.04580	+	1.04580i	−0.0469019	+	0.998899i	$0.514935\pi$
$673$	−27.1304	+	27.1304i	−1.04580	+	1.04580i	−0.998899	+	0.0469019i	$0.985065\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	21.5677	+	21.5677i	0.828915	+	0.828915i	0.987367	−	0.158452i	$-0.0506503\pi$
$677$	21.5677	+	21.5677i	0.828915	+	0.828915i	−0.158452	+	0.987367i	$0.550650\pi$
$678$		0			0
$679$	−7.49279			−0.287547
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−18.0872	+	18.0872i	−0.692087	+	0.692087i	−0.962691	−	0.270604i	$-0.912777\pi$
$683$	−18.0872	+	18.0872i	−0.692087	+	0.692087i	0.270604	+	0.962691i	$0.412777\pi$
$684$		0			0
$685$	20.1798	+	15.2079i	0.771029	+	0.581063i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			3.21723i			0.122567i
$690$		0			0
$691$	9.09679			0.346058			0.173029	−	0.984917i	$-0.444645\pi$
$691$	9.09679			0.346058			0.173029	+	0.984917i	$0.444645\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	22.5620	−	3.16992i	0.855825	−	0.120242i
$696$		0			0
$697$	−5.20448	−	5.20448i	−0.197134	−	0.197134i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	14.0687			0.531367			0.265683	−	0.964060i	$-0.414402\pi$
$701$	14.0687			0.531367			0.265683	+	0.964060i	$0.414402\pi$
$702$		0			0
$703$	−38.3283	−	32.1932i	−1.44558	−	1.21419i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.72837	−	2.72837i	0.102611	−	0.102611i
$708$		0			0
$709$			40.6035i			1.52490i	0.647050	+	0.762448i	$0.276003\pi$
$709$			40.6035i			1.52490i	−0.647050	+	0.762448i	$0.723997\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3.83212	−	3.83212i	0.143514	−	0.143514i
$714$		0			0
$715$	20.8719	+	15.7295i	0.780564	+	0.588248i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			33.5067i			1.24959i	0.780789	+	0.624794i	$0.214817\pi$
$719$			33.5067i			1.24959i	−0.780789	+	0.624794i	$0.785183\pi$
$720$		0			0
$721$		−	5.31386i		−	0.197898i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−14.3785	+	4.12166i	−0.534003	+	0.153075i
$726$		0			0
$727$	2.59210	−	2.59210i	0.0961358	−	0.0961358i	−0.657403	−	0.753539i	$-0.728345\pi$
$727$	2.59210	−	2.59210i	0.0961358	−	0.0961358i	0.753539	+	0.657403i	$0.228345\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	10.6953			0.395582
$732$		0			0
$733$	15.1334	+	15.1334i	0.558963	+	0.558963i	0.929012	−	0.370049i	$-0.120659\pi$
$733$	15.1334	+	15.1334i	0.558963	+	0.558963i	−0.370049	+	0.929012i	$0.620659\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.420302	−	0.420302i	−0.0154820	−	0.0154820i
$738$		0			0
$739$		−	13.4291i		−	0.493998i	−0.969016	−	0.246999i	$-0.920556\pi$
$739$		−	13.4291i		−	0.493998i	0.969016	−	0.246999i	$-0.0794444\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	23.7186	−	23.7186i	0.870152	−	0.870152i	−0.122337	−	0.992489i	$-0.539039\pi$
$743$	23.7186	−	23.7186i	0.870152	−	0.870152i	0.992489	+	0.122337i	$0.0390388\pi$
$744$		0			0
$745$	−40.0513	+	5.62714i	−1.46737	+	0.206162i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.35037			0.0858807
$750$		0			0
$751$			20.3615i			0.743002i	0.928433	+	0.371501i	$0.121157\pi$
$751$			20.3615i			0.743002i	−0.928433	+	0.371501i	$0.878843\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−29.9772	+	4.21174i	−1.09098	+	0.153281i
$756$		0			0
$757$	−3.67799	+	3.67799i	−0.133679	+	0.133679i	−0.770780	−	0.637101i	$-0.780133\pi$
$757$	−3.67799	+	3.67799i	−0.133679	+	0.133679i	0.637101	+	0.770780i	$0.280133\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	15.9126			0.576831			0.288415	−	0.957505i	$-0.406872\pi$
$761$	15.9126			0.576831			0.288415	+	0.957505i	$0.406872\pi$
$762$		0			0
$763$	−4.27379	+	4.27379i	−0.154722	+	0.154722i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	26.5491	−	26.5491i	0.958632	−	0.958632i
$768$		0			0
$769$		−	27.0277i		−	0.974643i	−0.873223	−	0.487321i	$-0.837974\pi$
$769$		−	27.0277i		−	0.974643i	0.873223	−	0.487321i	$-0.162026\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−8.89182	+	8.89182i	−0.319817	+	0.319817i	−0.848697	−	0.528880i	$-0.822612\pi$
$773$	−8.89182	+	8.89182i	−0.319817	+	0.319817i	0.528880	+	0.848697i	$0.322612\pi$
$774$		0			0
$775$	1.28228	+	4.47326i	0.0460609	+	0.160684i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.778549	−	8.94914i	−0.0278944	−	0.320636i
$780$		0			0
$781$			27.9163i			0.998925i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−38.3590	−	28.9081i	−1.36909	−	1.03178i
$786$		0			0
$787$	−20.9266	−	20.9266i	−0.745952	−	0.745952i	0.227764	−	0.973716i	$-0.426858\pi$
$787$	−20.9266	−	20.9266i	−0.745952	−	0.745952i	−0.973716	+	0.227764i	$0.926858\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1.19220i		−	0.0423898i
$792$		0			0
$793$	10.0360	−	10.0360i	0.356388	−	0.356388i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	35.3951	+	35.3951i	1.25376	+	1.25376i	0.954021	+	0.299738i	$0.0968993\pi$
$797$	35.3951	+	35.3951i	1.25376	+	1.25376i	0.299738	+	0.954021i	$0.403101\pi$
$798$		0			0
$799$		−	13.8381i		−	0.489556i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	11.8938	+	11.8938i	0.419725	+	0.419725i
$804$		0			0
$805$	−0.797056	−	5.67307i	−0.0280925	−	0.199949i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			18.3555i			0.645345i	0.946511	+	0.322673i	$0.104581\pi$
$809$			18.3555i			0.645345i	−0.946511	+	0.322673i	$0.895419\pi$
$810$		0			0
$811$		−	1.77129i		−	0.0621983i	−0.999516	−	0.0310991i	$-0.990099\pi$
$811$		−	1.77129i		−	0.0621983i	0.999516	−	0.0310991i	$-0.00990076\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	10.3713	−	13.7620i	0.363292	−	0.482063i
$816$		0			0
$817$	9.99534	+	8.39540i	0.349693	+	0.293718i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−18.7654			−0.654917			−0.327459	−	0.944865i	$-0.606192\pi$
$821$	−18.7654			−0.654917			−0.327459	+	0.944865i	$0.606192\pi$
$822$		0			0
$823$	−25.6844	−	25.6844i	−0.895304	−	0.895304i	0.0997127	−	0.995016i	$-0.468208\pi$
$823$	−25.6844	−	25.6844i	−0.895304	−	0.895304i	−0.995016	+	0.0997127i	$0.968208\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	19.7063	+	19.7063i	0.685257	+	0.685257i	0.961180	−	0.275923i	$-0.0889835\pi$
$827$	19.7063	+	19.7063i	0.685257	+	0.685257i	−0.275923	+	0.961180i	$0.588983\pi$
$828$		0			0
$829$	−14.1790			−0.492457			−0.246229	−	0.969212i	$-0.579191\pi$
$829$	−14.1790			−0.492457			−0.246229	+	0.969212i	$0.579191\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	17.1891	+	17.1891i	0.595568	+	0.595568i
$834$		0			0
$835$	−4.61803	+	6.12780i	−0.159813	+	0.212061i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	19.0319			0.657054			0.328527	−	0.944495i	$-0.393448\pi$
$839$	19.0319			0.657054			0.328527	+	0.944495i	$0.393448\pi$
$840$		0			0
$841$	−20.0509			−0.691409
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−7.10324	+	0.997992i	−0.244359	+	0.0343320i
$846$		0			0
$847$	−0.799970	+	0.799970i	−0.0274873	+	0.0274873i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	66.8681i		−	2.29221i
$852$		0			0
$853$	−11.9906	−	11.9906i	−0.410551	−	0.410551i	0.471379	−	0.881931i	$-0.343756\pi$
$853$	−11.9906	−	11.9906i	−0.410551	−	0.410551i	−0.881931	+	0.471379i	$0.843756\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	26.5107	+	26.5107i	0.905587	+	0.905587i	0.995912	−	0.0903251i	$-0.0287906\pi$
$857$	26.5107	+	26.5107i	0.905587	+	0.905587i	−0.0903251	+	0.995912i	$0.528791\pi$
$858$		0			0
$859$		−	44.9719i		−	1.53442i	−0.641395	−	0.767211i	$-0.721644\pi$
$859$		−	44.9719i		−	1.53442i	0.641395	−	0.767211i	$-0.278356\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−29.4121	+	29.4121i	−1.00120	+	1.00120i	−0.00120035	+	0.999999i	$0.500382\pi$
$863$	−29.4121	+	29.4121i	−1.00120	+	1.00120i	−0.999999	+	0.00120035i	$0.999618\pi$
$864$		0			0
$865$	−14.7441	−	11.1114i	−0.501314	−	0.377800i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	36.6013			1.24162
$870$		0			0
$871$	0.824253			0.0279287
$872$		0			0
$873$		0			0
$874$		0			0
$875$	4.59210	+	1.76341i	0.155241	+	0.0596140i
$876$		0			0
$877$	6.98977	+	6.98977i	0.236028	+	0.236028i	0.815203	−	0.579175i	$-0.196625\pi$
$877$	6.98977	+	6.98977i	0.236028	+	0.236028i	−0.579175	+	0.815203i	$0.696625\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−15.5254			−0.523065			−0.261532	−	0.965195i	$-0.584228\pi$
$881$	−15.5254			−0.523065			−0.261532	+	0.965195i	$0.584228\pi$
$882$		0			0
$883$	3.40790	+	3.40790i	0.114685	+	0.114685i	0.762120	−	0.647435i	$-0.224158\pi$
$883$	3.40790	+	3.40790i	0.114685	+	0.114685i	−0.647435	+	0.762120i	$0.724158\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	23.4077	+	23.4077i	0.785954	+	0.785954i	0.980828	−	0.194874i	$-0.0624299\pi$
$887$	23.4077	+	23.4077i	0.785954	+	0.785954i	−0.194874	+	0.980828i	$0.562430\pi$
$888$		0			0
$889$	3.19098			0.107022
$890$		0			0
$891$		0			0
$892$		0			0
$893$	10.8623	−	12.9324i	0.363493	−	0.432766i
$894$		0			0
$895$	6.36265	+	45.2863i	0.212680	+	1.51376i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	2.78415i		−	0.0928566i
$900$		0			0
$901$			2.85410i			0.0950840i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	26.4967	−	3.72273i	0.880779	−	0.123748i
$906$		0			0
$907$	22.0781	+	22.0781i	0.733091	+	0.733091i	0.971231	−	0.238140i	$-0.0765376\pi$
$907$	22.0781	+	22.0781i	0.733091	+	0.733091i	−0.238140	+	0.971231i	$0.576538\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	4.67272i		−	0.154814i	−0.997000	−	0.0774071i	$-0.975336\pi$
$911$		−	4.67272i		−	0.154814i	0.997000	−	0.0774071i	$-0.0246641\pi$
$912$		0			0
$913$	26.7511	+	26.7511i	0.885333	+	0.885333i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	3.22570	−	3.22570i	0.106522	−	0.106522i
$918$		0			0
$919$		−	16.5303i		−	0.545286i	−0.962115	−	0.272643i	$-0.912102\pi$
$919$		−	16.5303i		−	0.545286i	0.962115	−	0.272643i	$-0.0878978\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−27.3733	−	27.3733i	−0.901004	−	0.901004i
$924$		0			0
$925$	50.2153	+	27.8403i	1.65107	+	0.915383i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			33.1655i			1.08812i	0.839045	+	0.544062i	$0.183114\pi$
$929$			33.1655i			1.08812i	−0.839045	+	0.544062i	$0.816886\pi$
$930$		0			0
$931$	2.57136	+	29.5569i	0.0842729	+	0.968687i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	18.5161	+	13.9541i	0.605540	+	0.456347i
$936$		0			0
$937$	32.6686	−	32.6686i	1.06724	−	1.06724i	0.0696672	−	0.997570i	$-0.477806\pi$
$937$	32.6686	−	32.6686i	1.06724	−	1.06724i	0.997570	−	0.0696672i	$-0.0221937\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	3.86014i		−	0.125837i	−0.998019	−	0.0629185i	$-0.979959\pi$
$941$		−	3.86014i		−	0.125837i	0.998019	−	0.0629185i	$-0.0200408\pi$
$942$		0			0
$943$	8.48553	−	8.48553i	0.276327	−	0.276327i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−36.1733	+	36.1733i	−1.17547	+	1.17547i	−0.194590	+	0.980885i	$0.562338\pi$
$947$	−36.1733	+	36.1733i	−1.17547	+	1.17547i	−0.980885	+	0.194590i	$0.937662\pi$
$948$		0			0
$949$	−23.3250			−0.757161
$950$		0			0
$951$		0			0
$952$		0			0
$953$	13.5239	−	13.5239i	0.438081	−	0.438081i	−0.453285	−	0.891366i	$-0.649748\pi$
$953$	13.5239	−	13.5239i	0.438081	−	0.438081i	0.891366	+	0.453285i	$0.149748\pi$
$954$		0			0
$955$	7.18421	+	51.1338i	0.232476	+	1.65465i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			4.97190i			0.160551i
$960$		0			0
$961$	30.1338			0.972059
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−6.12780	−	4.61803i	−0.197261	−	0.148660i
$966$		0			0
$967$	−21.0163	+	21.0163i	−0.675839	+	0.675839i	−0.959056	−	0.283217i	$-0.908598\pi$
$967$	−21.0163	+	21.0163i	−0.675839	+	0.675839i	0.283217	+	0.959056i	$0.408598\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			29.4261i			0.944329i	0.881510	+	0.472164i	$0.156527\pi$
$971$			29.4261i			0.944329i	−0.881510	+	0.472164i	$0.843473\pi$
$972$		0			0
$973$	3.16992	+	3.16992i	0.101623	+	0.101623i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−10.4917	−	10.4917i	−0.335658	−	0.335658i	0.519072	−	0.854730i	$-0.326278\pi$
$977$	−10.4917	−	10.4917i	−0.335658	−	0.335658i	−0.854730	+	0.519072i	$0.826278\pi$
$978$		0			0
$979$	21.9333			0.700992
$980$		0			0
$981$		0			0
$982$		0			0
$983$	4.06697	−	4.06697i	0.129716	−	0.129716i	−0.639268	−	0.768984i	$-0.720763\pi$
$983$	4.06697	−	4.06697i	0.129716	−	0.129716i	0.768984	+	0.639268i	$0.220763\pi$
$984$		0			0
$985$	33.0513	+	24.9081i	1.05310	+	0.793639i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			17.4380i			0.554496i
$990$		0			0
$991$		−	16.8189i		−	0.534271i	−0.963659	−	0.267135i	$-0.913923\pi$
$991$		−	16.8189i		−	0.534271i	0.963659	−	0.267135i	$-0.0860771\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−22.7971	+	3.20294i	−0.722715	+	0.101540i
$996$		0			0
$997$	17.1847	−	17.1847i	0.544244	−	0.544244i	−0.380526	−	0.924770i	$-0.624257\pi$
$997$	17.1847	−	17.1847i	0.544244	−	0.544244i	0.924770	+	0.380526i	$0.124257\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.2.bb.d.2773.4		12
3.2	odd	2		380.2.l.b.113.4	yes	12
5.2	odd	4	inner	3420.2.bb.d.37.3		12
15.2	even	4		380.2.l.b.37.3	✓	12
15.8	even	4		1900.2.l.b.1557.4		12
15.14	odd	2		1900.2.l.b.493.3		12
19.18	odd	2	inner	3420.2.bb.d.2773.3		12
57.56	even	2		380.2.l.b.113.3	yes	12
95.37	even	4	inner	3420.2.bb.d.37.4		12
285.113	odd	4		1900.2.l.b.1557.3		12
285.227	odd	4		380.2.l.b.37.4	yes	12
285.284	even	2		1900.2.l.b.493.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.l.b.37.3	✓	12	15.2	even	4
380.2.l.b.37.4	yes	12	285.227	odd	4
380.2.l.b.113.3	yes	12	57.56	even	2
380.2.l.b.113.4	yes	12	3.2	odd	2
1900.2.l.b.493.3		12	15.14	odd	2
1900.2.l.b.493.4		12	285.284	even	2
1900.2.l.b.1557.3		12	285.113	odd	4
1900.2.l.b.1557.4		12	15.8	even	4
3420.2.bb.d.37.3		12	5.2	odd	4	inner
3420.2.bb.d.37.4		12	95.37	even	4	inner
3420.2.bb.d.2773.3		12	19.18	odd	2	inner
3420.2.bb.d.2773.4		12	1.1	even	1	trivial

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.bb (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.311108 - 2.21432i) q^{5} +(0.311108 - 0.311108i) q^{7} +O(q^{10})\)	\(q+(-0.311108 - 2.21432i) q^{5} +(0.311108 - 0.311108i) q^{7} -2.90321 q^{11} +(2.84674 - 2.84674i) q^{13} +(2.52543 - 2.52543i) q^{17} +(4.34250 - 0.377784i) q^{19} +(4.11753 + 4.11753i) q^{23} +(-4.80642 + 1.37778i) q^{25} +2.99151 q^{29} -0.930683i q^{31} +(-0.785680 - 0.592104i) q^{35} +(-8.11992 - 8.11992i) q^{37} -2.06083i q^{41} +(2.11753 + 2.11753i) q^{43} +(2.73975 - 2.73975i) q^{47} +6.80642i q^{49} +(-0.565073 + 0.565073i) q^{53} +(0.903212 + 6.42864i) q^{55} +9.32613 q^{59} +3.52543 q^{61} +(-7.18924 - 5.41795i) q^{65} +(0.144771 + 0.144771i) q^{67} -9.61568i q^{71} +(-4.09679 - 4.09679i) q^{73} +(-0.903212 + 0.903212i) q^{77} -12.6072 q^{79} +(-9.21432 - 9.21432i) q^{83} +(-6.37778 - 4.80642i) q^{85} -7.55485 q^{89} -1.77129i q^{91} +(-2.18752 - 9.49814i) q^{95} +(-12.0421 - 12.0421i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{5} + 4 q^{7}+O(q^{10}) \) 12 * q - 4 * q^5 + 4 * q^7	\( 12 q - 4 q^{5} + 4 q^{7} - 8 q^{11} + 4 q^{17} - 4 q^{23} - 4 q^{25} - 36 q^{35} - 28 q^{43} - 20 q^{47} - 16 q^{55} + 16 q^{61} - 76 q^{73} + 16 q^{77} - 84 q^{83} - 76 q^{85} + 16 q^{95}+O(q^{100}) \) 12 * q - 4 * q^5 + 4 * q^7 - 8 * q^11 + 4 * q^17 - 4 * q^23 - 4 * q^25 - 36 * q^35 - 28 * q^43 - 20 * q^47 - 16 * q^55 + 16 * q^61 - 76 * q^73 + 16 * q^77 - 84 * q^83 - 76 * q^85 + 16 * q^95

Introduction

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