LMFDB - Embedded newform 2100.2.f.i.1049.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,2,Mod(1049,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2100.1049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2100.f (of order $2$, degree $1$, not 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:	$16.7685844245$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 22 x^{14} - 4 x^{13} - 80 x^{12} - 84 x^{11} + 1324 x^{10} - 3800 x^{9} + \cdots + 3204 $ x^16 - 8x^15 + 22x^14 - 4x^13 - 80x^12 - 84x^11 + 1324x^10 - 3800x^9 + 5642x^8 - 4872x^7 + 4136x^6 - 11608x^5 + 30032x^4 - 44288x^3 + 37232x^2 - 16848*x + 3204
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1049.3
Root			$0.904362 + 0.230889i$ of defining polynomial
Character	$\chi$	$=$	2100.1049
Dual form			2100.2.f.i.1049.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.14412 + 1.30038i) q^{3} +(1.41421 - 2.23607i) q^{7} +(-0.381966 - 2.97558i) q^{9} +O(q^{10})$	$q+(-1.14412 + 1.30038i) q^{3} +(1.41421 - 2.23607i) q^{7} +(-0.381966 - 2.97558i) q^{9} +1.13657i q^{11} +0.333851 q^{13} +4.20811i q^{17} -1.95440i q^{19} +(1.28970 + 4.39735i) q^{21} -2.54145 q^{23} +(4.30640 + 2.90773i) q^{27} -8.49262i q^{29} -5.11667i q^{31} +(-1.47797 - 1.30038i) q^{33} -2.23607i q^{37} +(-0.381966 + 0.434132i) q^{39} +5.81547 q^{41} +0.527864i q^{43} +7.42282i q^{47} +(-3.00000 - 6.32456i) q^{49} +(-5.47214 - 4.81460i) q^{51} -8.22431 q^{53} +(2.54145 + 2.23607i) q^{57} -3.59416 q^{59} -11.4412i q^{61} +(-7.19379 - 3.35401i) q^{63} -6.70820i q^{67} +(2.90773 - 3.30485i) q^{69} -10.7658i q^{71} +1.41421 q^{73} +(2.54145 + 1.60736i) q^{77} -1.47214 q^{79} +(-8.70820 + 2.27314i) q^{81} -5.81547i q^{83} +(11.0436 + 9.71660i) q^{87} +15.2251 q^{89} +(0.472136 - 0.746512i) q^{91} +(6.65361 + 5.85410i) q^{93} +9.69316 q^{97} +(3.38197 - 0.434132i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 24 q^{9}+O(q^{10}) $ 16 * q - 24 * q^9	$ 16 q - 24 q^{9} - 8 q^{21} - 24 q^{39} - 48 q^{49} - 16 q^{51} + 48 q^{79} - 32 q^{81} - 64 q^{91} + 72 q^{99}+O(q^{100}) $ 16 * q - 24 * q^9 - 8 * q^21 - 24 * q^39 - 48 * q^49 - 16 * q^51 + 48 * q^79 - 32 * q^81 - 64 * q^91 + 72 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$	−1.14412	+	1.30038i	−0.660560	+	0.750774i
$3$	−1.14412	+	1.30038i	−0.660560	+	0.750774i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.41421	−	2.23607i	0.534522	−	0.845154i
$7$	1.41421	−	2.23607i	0.534522	−	0.845154i
$8$		0			0
$9$	−0.381966	−	2.97558i	−0.127322	−	0.991861i
$10$		0			0
$11$			1.13657i			0.342689i	0.985211	+	0.171345i	$0.0548112\pi$
$11$			1.13657i			0.342689i	−0.985211	+	0.171345i	$0.945189\pi$
$12$		0			0
$13$	0.333851			0.0925935			0.0462967	−	0.998928i	$-0.485258\pi$
$13$	0.333851			0.0925935			0.0462967	+	0.998928i	$0.485258\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.20811i			1.02062i	0.859991	+	0.510308i	$0.170469\pi$
$17$			4.20811i			1.02062i	−0.859991	+	0.510308i	$0.829531\pi$
$18$		0			0
$19$		−	1.95440i		−	0.448369i	−0.974547	−	0.224184i	$-0.928028\pi$
$19$		−	1.95440i		−	0.448369i	0.974547	−	0.224184i	$-0.0719718\pi$
$20$		0			0
$21$	1.28970	+	4.39735i	0.281436	+	0.959580i
$22$		0			0
$23$	−2.54145			−0.529929			−0.264965	−	0.964258i	$-0.585360\pi$
$23$	−2.54145			−0.529929			−0.264965	+	0.964258i	$0.585360\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	4.30640	+	2.90773i	0.828767	+	0.559594i
$28$		0			0
$29$		−	8.49262i		−	1.57704i	−0.615009	−	0.788520i	$-0.710848\pi$
$29$		−	8.49262i		−	1.57704i	0.615009	−	0.788520i	$-0.289152\pi$
$30$		0			0
$31$		−	5.11667i		−	0.918982i	−0.888183	−	0.459491i	$-0.848032\pi$
$31$		−	5.11667i		−	0.918982i	0.888183	−	0.459491i	$-0.151968\pi$
$32$		0			0
$33$	−1.47797	−	1.30038i	−0.257282	−	0.226367i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	2.23607i		−	0.367607i	−0.982963	−	0.183804i	$-0.941159\pi$
$37$		−	2.23607i		−	0.367607i	0.982963	−	0.183804i	$-0.0588411\pi$
$38$		0			0
$39$	−0.381966	+	0.434132i	−0.0611635	+	0.0695167i
$40$		0			0
$41$	5.81547			0.908223			0.454112	−	0.890945i	$-0.349957\pi$
$41$	5.81547			0.908223			0.454112	+	0.890945i	$0.349957\pi$
$42$		0			0
$43$			0.527864i			0.0804985i	0.999190	+	0.0402493i	$0.0128152\pi$
$43$			0.527864i			0.0804985i	−0.999190	+	0.0402493i	$0.987185\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			7.42282i			1.08273i	0.840788	+	0.541365i	$0.182092\pi$
$47$			7.42282i			1.08273i	−0.840788	+	0.541365i	$0.817908\pi$
$48$		0			0
$49$	−3.00000	−	6.32456i	−0.428571	−	0.903508i
$50$		0			0
$51$	−5.47214	−	4.81460i	−0.766252	−	0.674178i
$52$		0			0
$53$	−8.22431			−1.12970			−0.564848	−	0.825195i	$-0.691065\pi$
$53$	−8.22431			−1.12970			−0.564848	+	0.825195i	$0.691065\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	2.54145	+	2.23607i	0.336624	+	0.296174i
$58$		0			0
$59$	−3.59416			−0.467919			−0.233960	−	0.972246i	$-0.575168\pi$
$59$	−3.59416			−0.467919			−0.233960	+	0.972246i	$0.575168\pi$
$60$		0			0
$61$		−	11.4412i		−	1.46490i	−0.680821	−	0.732450i	$-0.738377\pi$
$61$		−	11.4412i		−	1.46490i	0.680821	−	0.732450i	$-0.261623\pi$
$62$		0			0
$63$	−7.19379	−	3.35401i	−0.906332	−	0.422566i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	6.70820i		−	0.819538i	−0.912189	−	0.409769i	$-0.865609\pi$
$67$		−	6.70820i		−	0.819538i	0.912189	−	0.409769i	$-0.134391\pi$
$68$		0			0
$69$	2.90773	−	3.30485i	0.350050	−	0.397857i
$70$		0			0
$71$		−	10.7658i		−	1.27766i	−0.769347	−	0.638831i	$-0.779418\pi$
$71$		−	10.7658i		−	1.27766i	0.769347	−	0.638831i	$-0.220582\pi$
$72$		0			0
$73$	1.41421			0.165521			0.0827606	−	0.996569i	$-0.473626\pi$
$73$	1.41421			0.165521			0.0827606	+	0.996569i	$0.473626\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.54145	+	1.60736i	0.289625	+	0.183175i
$78$		0			0
$79$	−1.47214			−0.165628			−0.0828141	−	0.996565i	$-0.526391\pi$
$79$	−1.47214			−0.165628			−0.0828141	+	0.996565i	$0.526391\pi$
$80$		0			0
$81$	−8.70820	+	2.27314i	−0.967578	+	0.252572i
$82$		0			0
$83$		−	5.81547i		−	0.638330i	−0.947699	−	0.319165i	$-0.896598\pi$
$83$		−	5.81547i		−	0.638330i	0.947699	−	0.319165i	$-0.103402\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	11.0436	+	9.71660i	1.18400	+	1.04173i
$88$		0			0
$89$	15.2251			1.61386			0.806928	−	0.590650i	$-0.201128\pi$
$89$	15.2251			1.61386			0.806928	+	0.590650i	$0.201128\pi$
$90$		0			0
$91$	0.472136	−	0.746512i	0.0494933	−	0.0782558i
$92$		0			0
$93$	6.65361	+	5.85410i	0.689947	+	0.607042i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	9.69316			0.984192			0.492096	−	0.870541i	$-0.336231\pi$
$97$	9.69316			0.984192			0.492096	+	0.870541i	$0.336231\pi$
$98$		0			0
$99$	3.38197	−	0.434132i	0.339900	−	0.0436319i
$100$		0			0
$101$	15.2251			1.51495			0.757477	−	0.652862i	$-0.226432\pi$
$101$	15.2251			1.51495			0.757477	+	0.652862i	$0.226432\pi$
$102$		0			0
$103$	−15.0162			−1.47959			−0.739793	−	0.672834i	$-0.765077\pi$
$103$	−15.0162			−1.47959			−0.739793	+	0.672834i	$0.765077\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.08290			0.491383			0.245691	−	0.969348i	$-0.420985\pi$
$107$	5.08290			0.491383			0.245691	+	0.969348i	$0.420985\pi$
$108$		0			0
$109$	−7.47214			−0.715701			−0.357850	−	0.933779i	$-0.616490\pi$
$109$	−7.47214			−0.715701			−0.357850	+	0.933779i	$0.616490\pi$
$110$		0			0
$111$	2.90773	+	2.55834i	0.275990	+	0.242827i
$112$		0			0
$113$	18.9901			1.78644			0.893218	−	0.449624i	$-0.148442\pi$
$113$	18.9901			1.78644			0.893218	+	0.449624i	$0.148442\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.127520	−	0.993400i	−0.0117892	−	0.0918399i
$118$		0			0
$119$	9.40962	+	5.95117i	0.862579	+	0.545543i
$120$		0			0
$121$	9.70820			0.882564
$122$		0			0
$123$	−6.65361	+	7.56231i	−0.599936	+	0.681870i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	16.7082i		−	1.48261i	−0.671166	−	0.741307i	$-0.734206\pi$
$127$		−	16.7082i		−	1.48261i	0.671166	−	0.741307i	$-0.265794\pi$
$128$		0			0
$129$	−0.686423	−	0.603941i	−0.0604362	−	0.0531741i
$130$		0			0
$131$	−9.40962			−0.822123			−0.411061	−	0.911608i	$-0.634842\pi$
$131$	−9.40962			−0.822123			−0.411061	+	0.911608i	$0.634842\pi$
$132$		0			0
$133$	−4.37016	−	2.76393i	−0.378941	−	0.239663i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	21.5315			1.83956			0.919781	−	0.392431i	$-0.128366\pi$
$137$	21.5315			1.83956			0.919781	+	0.392431i	$0.128366\pi$
$138$		0			0
$139$			0.746512i			0.0633184i	0.999499	+	0.0316592i	$0.0100791\pi$
$139$			0.746512i			0.0633184i	−0.999499	+	0.0316592i	$0.989921\pi$
$140$		0			0
$141$	−9.65248	−	8.49262i	−0.812885	−	0.715208i
$142$		0			0
$143$			0.379445i			0.0317308i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	11.6567	+	3.33493i	0.961427	+	0.275061i
$148$		0			0
$149$		−	2.54145i		−	0.208204i	−0.994567	−	0.104102i	$-0.966803\pi$
$149$		−	2.54145i		−	0.208204i	0.994567	−	0.104102i	$-0.0331968\pi$
$150$		0			0
$151$	9.65248			0.785507			0.392754	−	0.919644i	$-0.371523\pi$
$151$	9.65248			0.785507			0.392754	+	0.919644i	$0.371523\pi$
$152$		0			0
$153$	12.5216	−	1.60736i	1.01231	−	0.129947i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	8.61280			0.687376			0.343688	−	0.939084i	$-0.388324\pi$
$157$	8.61280			0.687376			0.343688	+	0.939084i	$0.388324\pi$
$158$		0			0
$159$	9.40962	−	10.6947i	0.746232	−	0.848146i
$160$		0			0
$161$	−3.59416	+	5.68286i	−0.283259	+	0.447872i
$162$		0			0
$163$		−	8.94427i		−	0.700569i	−0.936643	−	0.350285i	$-0.886085\pi$
$163$		−	8.94427i		−	0.700569i	0.936643	−	0.350285i	$-0.113915\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			10.0236i			0.775648i	0.921733	+	0.387824i	$0.126773\pi$
$167$			10.0236i			0.775648i	−0.921733	+	0.387824i	$0.873227\pi$
$168$		0			0
$169$	−12.8885			−0.991426
$170$		0			0
$171$	−5.81547	+	0.746512i	−0.444720	+	0.0570872i
$172$		0			0
$173$		−	6.19491i		−	0.470990i	−0.971876	−	0.235495i	$-0.924329\pi$
$173$		−	6.19491i		−	0.470990i	0.971876	−	0.235495i	$-0.0756712\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	4.11216	−	4.67376i	0.309089	−	0.351301i
$178$		0			0
$179$			13.3072i			0.994628i	0.867571	+	0.497314i	$0.165680\pi$
$179$			13.3072i			0.994628i	−0.867571	+	0.497314i	$0.834320\pi$
$180$		0			0
$181$		−	10.9799i		−	0.816126i	−0.912954	−	0.408063i	$-0.866204\pi$
$181$		−	10.9799i		−	0.816126i	0.912954	−	0.408063i	$-0.133796\pi$
$182$		0			0
$183$	14.8779	+	13.0902i	1.09981	+	0.967653i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−4.78282			−0.349755
$188$		0			0
$189$	12.5921	−	5.51725i	0.915938	−	0.401321i
$190$		0			0
$191$		−	3.67802i		−	0.266133i	−0.991107	−	0.133066i	$-0.957518\pi$
$191$		−	3.67802i		−	0.266133i	0.991107	−	0.133066i	$-0.0424823\pi$
$192$		0			0
$193$		−	15.0000i		−	1.07972i	−0.841754	−	0.539862i	$-0.818476\pi$
$193$		−	15.0000i		−	1.07972i	0.841754	−	0.539862i	$-0.181524\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−10.7658			−0.767029			−0.383515	−	0.923535i	$-0.625286\pi$
$197$	−10.7658			−0.767029			−0.383515	+	0.923535i	$0.625286\pi$
$198$		0			0
$199$			24.0903i			1.70772i	0.520504	+	0.853859i	$0.325744\pi$
$199$			24.0903i			1.70772i	−0.520504	+	0.853859i	$0.674256\pi$
$200$		0			0
$201$	8.72320	+	7.67501i	0.615287	+	0.541353i
$202$		0			0
$203$	−18.9901	−	12.0104i	−1.33284	−	0.842963i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.970748	+	7.56231i	0.0674717	+	0.525617i
$208$		0			0
$209$	2.22131			0.153651
$210$		0			0
$211$	7.70820			0.530655			0.265327	−	0.964158i	$-0.414520\pi$
$211$	7.70820			0.530655			0.265327	+	0.964158i	$0.414520\pi$
$212$		0			0
$213$	13.9996	+	12.3174i	0.959234	+	0.843971i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−11.4412	−	7.23607i	−0.776681	−	0.491216i
$218$		0			0
$219$	−1.61803	+	1.83901i	−0.109337	+	0.124269i
$220$		0			0
$221$			1.40488i			0.0945025i
$222$		0			0
$223$	−16.8918			−1.13116			−0.565578	−	0.824695i	$-0.691347\pi$
$223$	−16.8918			−1.13116			−0.565578	+	0.824695i	$0.691347\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			11.3964i			0.756407i	0.925722	+	0.378204i	$0.123458\pi$
$227$			11.3964i			0.756407i	−0.925722	+	0.378204i	$0.876542\pi$
$228$		0			0
$229$		−	10.2333i		−	0.676239i	−0.941103	−	0.338119i	$-0.890209\pi$
$229$		−	10.2333i		−	0.676239i	0.941103	−	0.338119i	$-0.109791\pi$
$230$		0			0
$231$	−4.99790	+	1.46584i	−0.328838	+	0.0964450i
$232$		0			0
$233$	15.8487			1.03828			0.519140	−	0.854689i	$-0.326252\pi$
$233$	15.8487			1.03828			0.519140	+	0.854689i	$0.326252\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.68430	−	1.91433i	0.109407	−	0.124349i
$238$		0			0
$239$		−	10.4975i		−	0.679024i	−0.940602	−	0.339512i	$-0.889738\pi$
$239$		−	10.4975i		−	0.679024i	0.940602	−	0.339512i	$-0.110262\pi$
$240$		0			0
$241$		−	22.4211i		−	1.44427i	−0.691753	−	0.722135i	$-0.743161\pi$
$241$		−	22.4211i		−	1.44427i	0.691753	−	0.722135i	$-0.256839\pi$
$242$		0			0
$243$	7.00731	−	13.9247i	0.449519	−	0.893271i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	0.652476i		−	0.0415160i
$248$		0			0
$249$	7.56231	+	6.65361i	0.479242	+	0.421655i
$250$		0			0
$251$	−2.22131			−0.140208			−0.0701039	−	0.997540i	$-0.522333\pi$
$251$	−2.22131			−0.140208			−0.0701039	+	0.997540i	$0.522333\pi$
$252$		0			0
$253$		−	2.88854i		−	0.181601i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	24.6347i		−	1.53667i	−0.640047	−	0.768336i	$-0.721085\pi$
$257$		−	24.6347i		−	1.53667i	0.640047	−	0.768336i	$-0.278915\pi$
$258$		0			0
$259$	−5.00000	−	3.16228i	−0.310685	−	0.196494i
$260$		0			0
$261$	−25.2705	+	3.24389i	−1.56421	+	0.200792i
$262$		0			0
$263$	−15.8487			−0.977271			−0.488635	−	0.872488i	$-0.662505\pi$
$263$	−15.8487			−0.977271			−0.488635	+	0.872488i	$0.662505\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−17.4194	+	19.7984i	−1.06605	+	1.21164i
$268$		0			0
$269$	−31.8230			−1.94028			−0.970142	−	0.242537i	$-0.922020\pi$
$269$	−31.8230			−1.94028			−0.970142	+	0.242537i	$0.922020\pi$
$270$		0			0
$271$			22.1359i			1.34466i	0.740250	+	0.672331i	$0.234707\pi$
$271$			22.1359i			1.34466i	−0.740250	+	0.672331i	$0.765293\pi$
$272$		0			0
$273$	0.430567	+	1.46806i	0.0260591	+	0.0888509i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			8.29180i			0.498206i	0.968477	+	0.249103i	$0.0801357\pi$
$277$			8.29180i			0.498206i	−0.968477	+	0.249103i	$0.919864\pi$
$278$		0			0
$279$	−15.2251	+	1.95440i	−0.911502	+	0.117007i
$280$		0			0
$281$		−	9.36088i		−	0.558424i	−0.960230	−	0.279212i	$-0.909927\pi$
$281$		−	9.36088i		−	0.558424i	0.960230	−	0.279212i	$-0.0900732\pi$
$282$		0			0
$283$	5.91189			0.351426			0.175713	−	0.984441i	$-0.443777\pi$
$283$	5.91189			0.351426			0.175713	+	0.984441i	$0.443777\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.22431	−	13.0038i	0.485466	−	0.767589i
$288$		0			0
$289$	−0.708204			−0.0416591
$290$		0			0
$291$	−11.0902	+	12.6048i	−0.650117	+	0.738905i
$292$		0			0
$293$			23.6413i			1.38114i	0.723265	+	0.690570i	$0.242640\pi$
$293$			23.6413i			1.38114i	−0.723265	+	0.690570i	$0.757360\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−3.30485	+	4.89453i	−0.191767	+	0.284010i
$298$		0			0
$299$	−0.848465			−0.0490680
$300$		0			0
$301$	1.18034	+	0.746512i	0.0680337	+	0.0430283i
$302$		0			0
$303$	−17.4194	+	19.7984i	−1.00072	+	1.13739i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−25.5046			−1.45562			−0.727811	−	0.685778i	$-0.759462\pi$
$307$	−25.5046			−1.45562			−0.727811	+	0.685778i	$0.759462\pi$
$308$		0			0
$309$	17.1803	−	19.5267i	0.977355	−	1.11083i
$310$		0			0
$311$	−2.22131			−0.125959			−0.0629795	−	0.998015i	$-0.520060\pi$
$311$	−2.22131			−0.125959			−0.0629795	+	0.998015i	$0.520060\pi$
$312$		0			0
$313$	4.78282			0.270341			0.135171	−	0.990822i	$-0.456842\pi$
$313$	4.78282			0.270341			0.135171	+	0.990822i	$0.456842\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−29.1559			−1.63756			−0.818779	−	0.574109i	$-0.805349\pi$
$317$	−29.1559			−1.63756			−0.818779	+	0.574109i	$0.805349\pi$
$318$		0			0
$319$	9.65248			0.540435
$320$		0			0
$321$	−5.81547	+	6.60970i	−0.324588	+	0.368917i
$322$		0			0
$323$	8.22431			0.457613
$324$		0			0
$325$		0			0
$326$		0			0
$327$	8.54904	−	9.71660i	0.472763	−	0.537329i
$328$		0			0
$329$	16.5979	+	10.4975i	0.915074	+	0.578744i
$330$		0			0
$331$	29.1803			1.60390			0.801948	−	0.597393i	$-0.203797\pi$
$331$	29.1803			1.60390			0.801948	+	0.597393i	$0.203797\pi$
$332$		0			0
$333$	−6.65361	+	0.854102i	−0.364616	+	0.0468045i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	10.0000i		−	0.544735i	−0.962193	−	0.272367i	$-0.912193\pi$
$337$		−	10.0000i		−	0.544735i	0.962193	−	0.272367i	$-0.0878066\pi$
$338$		0			0
$339$	−21.7270	+	24.6943i	−1.18005	+	1.34121i
$340$		0			0
$341$	5.81547			0.314925
$342$		0			0
$343$	−18.3848	−	2.23607i	−0.992685	−	0.120736i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−29.1559			−1.56517			−0.782585	−	0.622544i	$-0.786099\pi$
$347$	−29.1559			−1.56517			−0.782585	+	0.622544i	$0.786099\pi$
$348$		0			0
$349$			22.4211i			1.20017i	0.799935	+	0.600087i	$0.204867\pi$
$349$			22.4211i			1.20017i	−0.799935	+	0.600087i	$0.795133\pi$
$350$		0			0
$351$	1.43769	+	0.970748i	0.0767384	+	0.0518147i
$352$		0			0
$353$			33.0509i			1.75912i	0.475784	+	0.879562i	$0.342164\pi$
$353$			33.0509i			1.75912i	−0.475784	+	0.879562i	$0.657836\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−18.5045	+	5.42720i	−0.979364	+	0.287238i
$358$		0			0
$359$		−	10.7658i		−	0.568195i	−0.958795	−	0.284098i	$-0.908306\pi$
$359$		−	10.7658i		−	0.568195i	0.958795	−	0.284098i	$-0.0916940\pi$
$360$		0			0
$361$	15.1803			0.798965
$362$		0			0
$363$	−11.1074	+	12.6243i	−0.582986	+	0.662606i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	14.5247			0.758183			0.379091	−	0.925359i	$-0.376237\pi$
$367$	14.5247			0.758183			0.379091	+	0.925359i	$0.376237\pi$
$368$		0			0
$369$	−2.22131	−	17.3044i	−0.115637	−	0.900832i
$370$		0			0
$371$	−11.6309	+	18.3901i	−0.603848	+	0.954768i
$372$		0			0
$373$		−	15.6525i		−	0.810454i	−0.914216	−	0.405227i	$-0.867192\pi$
$373$		−	15.6525i		−	0.810454i	0.914216	−	0.405227i	$-0.132808\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	2.83527i		−	0.146024i
$378$		0			0
$379$	5.18034			0.266096			0.133048	−	0.991110i	$-0.457524\pi$
$379$	5.18034			0.266096			0.133048	+	0.991110i	$0.457524\pi$
$380$		0			0
$381$	21.7270	+	19.1162i	1.11311	+	0.979354i
$382$		0			0
$383$		−	15.2251i		−	0.777966i	−0.921245	−	0.388983i	$-0.872827\pi$
$383$		−	15.2251i		−	0.777966i	0.921245	−	0.388983i	$-0.127173\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1.57070	−	0.201626i	0.0798434	−	0.0102492i
$388$		0			0
$389$		−	18.1218i		−	0.918812i	−0.888226	−	0.459406i	$-0.848062\pi$
$389$		−	18.1218i		−	0.918812i	0.888226	−	0.459406i	$-0.151938\pi$
$390$		0			0
$391$		−	10.6947i		−	0.540855i
$392$		0			0
$393$	10.7658	−	12.2361i	0.543061	−	0.617228i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−39.1853			−1.96665			−0.983327	−	0.181844i	$-0.941793\pi$
$397$	−39.1853			−1.96665			−0.983327	+	0.181844i	$0.941793\pi$
$398$		0			0
$399$	8.59416	−	2.52058i	0.430246	−	0.126187i
$400$		0			0
$401$		−	3.94633i		−	0.197070i	−0.995134	−	0.0985352i	$-0.968584\pi$
$401$		−	3.94633i		−	0.197070i	0.995134	−	0.0985352i	$-0.0314157\pi$
$402$		0			0
$403$		−	1.70820i		−	0.0850917i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.54145			0.125975
$408$		0			0
$409$			24.8369i			1.22810i	0.789266	+	0.614052i	$0.210461\pi$
$409$			24.8369i			1.22810i	−0.789266	+	0.614052i	$0.789539\pi$
$410$		0			0
$411$	−24.6347	+	27.9991i	−1.21514	+	1.38110i
$412$		0			0
$413$	−5.08290	+	8.03678i	−0.250113	+	0.395464i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.970748	−	0.854102i	−0.0475378	−	0.0418256i
$418$		0			0
$419$	−36.2656			−1.77169			−0.885846	−	0.463979i	$-0.846421\pi$
$419$	−36.2656			−1.77169			−0.885846	+	0.463979i	$0.846421\pi$
$420$		0			0
$421$	18.8885			0.920571			0.460286	−	0.887771i	$-0.347747\pi$
$421$	18.8885			0.920571			0.460286	+	0.887771i	$0.347747\pi$
$422$		0			0
$423$	22.0872	−	2.83527i	1.07392	−	0.137855i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−25.5834	−	16.1803i	−1.23807	−	0.783022i
$428$		0			0
$429$	−0.493422	−	0.434132i	−0.0238226	−	0.0209601i
$430$		0			0
$431$			31.1607i			1.50096i	0.660894	+	0.750480i	$0.270177\pi$
$431$			31.1607i			1.50096i	−0.660894	+	0.750480i	$0.729823\pi$
$432$		0			0
$433$	31.3677			1.50744			0.753719	−	0.657197i	$-0.228258\pi$
$433$	31.3677			1.50744			0.753719	+	0.657197i	$0.228258\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			4.96700i			0.237604i
$438$		0			0
$439$		−	18.2272i		−	0.869935i	−0.900446	−	0.434967i	$-0.856760\pi$
$439$		−	18.2272i		−	0.869935i	0.900446	−	0.434967i	$-0.143240\pi$
$440$		0			0
$441$	−17.6733	+	11.3425i	−0.841588	+	0.540120i
$442$		0			0
$443$	8.22431			0.390749			0.195374	−	0.980729i	$-0.437408\pi$
$443$	8.22431			0.390749			0.195374	+	0.980729i	$0.437408\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	3.30485	+	2.90773i	0.156314	+	0.137531i
$448$		0			0
$449$			5.35121i			0.252539i	0.991996	+	0.126270i	$0.0403005\pi$
$449$			5.35121i			0.252539i	−0.991996	+	0.126270i	$0.959700\pi$
$450$		0			0
$451$			6.60970i			0.311239i
$452$		0			0
$453$	−11.0436	+	12.5519i	−0.518874	+	0.589738i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			37.3607i			1.74766i	0.486233	+	0.873829i	$0.338371\pi$
$457$			37.3607i			1.74766i	−0.486233	+	0.873829i	$0.661629\pi$
$458$		0			0
$459$	−12.2361	+	18.1218i	−0.571131	+	0.845854i
$460$		0			0
$461$	−15.2251			−0.709103			−0.354552	−	0.935036i	$-0.615366\pi$
$461$	−15.2251			−0.709103			−0.354552	+	0.935036i	$0.615366\pi$
$462$		0			0
$463$		−	10.6525i		−	0.495063i	−0.968880	−	0.247531i	$-0.920381\pi$
$463$		−	10.6525i		−	0.495063i	0.968880	−	0.247531i	$-0.0796193\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	25.2487i		−	1.16837i	−0.811621	−	0.584185i	$-0.801414\pi$
$467$		−	25.2487i		−	1.16837i	0.811621	−	0.584185i	$-0.198586\pi$
$468$		0			0
$469$	−15.0000	−	9.48683i	−0.692636	−	0.438061i
$470$		0			0
$471$	−9.85410	+	11.1999i	−0.454053	+	0.516064i
$472$		0			0
$473$	−0.599956			−0.0275860
$474$		0			0
$475$		0			0
$476$		0			0
$477$	3.14141	+	24.4721i	0.143835	+	1.12050i
$478$		0			0
$479$	−41.2327			−1.88397			−0.941984	−	0.335658i	$-0.891041\pi$
$479$	−41.2327			−1.88397			−0.941984	+	0.335658i	$0.891041\pi$
$480$		0			0
$481$		−	0.746512i		−	0.0340380i
$482$		0			0
$483$	−3.27771	−	11.1757i	−0.149141	−	0.508510i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	12.2361i		−	0.554469i	−0.960802	−	0.277235i	$-0.910582\pi$
$487$		−	12.2361i		−	0.554469i	0.960802	−	0.277235i	$-0.0894179\pi$
$488$		0			0
$489$	11.6309	+	10.2333i	0.525969	+	0.462768i
$490$		0			0
$491$			14.4438i			0.651839i	0.945398	+	0.325920i	$0.105674\pi$
$491$			14.4438i			0.651839i	−0.945398	+	0.325920i	$0.894326\pi$
$492$		0			0
$493$	35.7379			1.60955
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−24.0730	−	15.2251i	−1.07982	−	0.682939i
$498$		0			0
$499$	42.6525			1.90939			0.954694	−	0.297591i	$-0.0961831\pi$
$499$	42.6525			1.90939			0.954694	+	0.297591i	$0.0961831\pi$
$500$		0			0
$501$	−13.0344	−	11.4682i	−0.582336	−	0.512362i
$502$		0			0
$503$		−	8.41622i		−	0.375261i	−0.982240	−	0.187630i	$-0.939919\pi$
$503$		−	8.41622i		−	0.375261i	0.982240	−	0.187630i	$-0.0600807\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	14.7461	−	16.7600i	0.654896	−	0.744337i
$508$		0			0
$509$	22.4134			0.993457			0.496728	−	0.867906i	$-0.334535\pi$
$509$	22.4134			0.993457			0.496728	+	0.867906i	$0.334535\pi$
$510$		0			0
$511$	2.00000	−	3.16228i	0.0884748	−	0.139891i
$512$		0			0
$513$	5.68286	−	8.41641i	0.250904	−	0.371593i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−8.43657			−0.371040
$518$		0			0
$519$	8.05573	+	7.08774i	0.353607	+	0.311117i
$520$		0			0
$521$	−4.44262			−0.194635			−0.0973174	−	0.995253i	$-0.531026\pi$
$521$	−4.44262			−0.194635			−0.0973174	+	0.995253i	$0.531026\pi$
$522$		0			0
$523$	−8.48528			−0.371035			−0.185518	−	0.982641i	$-0.559396\pi$
$523$	−8.48528			−0.371035			−0.185518	+	0.982641i	$0.559396\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	21.5315			0.937928
$528$		0			0
$529$	−16.5410			−0.719175
$530$		0			0
$531$	1.37285	+	10.6947i	0.0595764	+	0.464111i
$532$		0			0
$533$	1.94150			0.0840956
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−17.3044	−	15.2251i	−0.746741	−	0.657011i
$538$		0			0
$539$	7.18831	−	3.40972i	0.309623	−	0.146867i
$540$		0			0
$541$	19.6525			0.844926			0.422463	−	0.906380i	$-0.361166\pi$
$541$	19.6525			0.844926			0.422463	+	0.906380i	$0.361166\pi$
$542$		0			0
$543$	14.2780	+	12.5623i	0.612726	+	0.539100i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	3.29180i		−	0.140747i	−0.997521	−	0.0703735i	$-0.977581\pi$
$547$		−	3.29180i		−	0.140747i	0.997521	−	0.0703735i	$-0.0224191\pi$
$548$		0			0
$549$	−34.0443	+	4.37016i	−1.45298	+	0.186514i
$550$		0			0
$551$	−16.5979			−0.707096
$552$		0			0
$553$	−2.08191	+	3.29180i	−0.0885320	+	0.139981i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5.68286			0.240791			0.120395	−	0.992726i	$-0.461584\pi$
$557$	5.68286			0.240791			0.120395	+	0.992726i	$0.461584\pi$
$558$		0			0
$559$			0.176228i			0.00745364i
$560$		0			0
$561$	5.47214	−	6.21948i	0.231034	−	0.262587i
$562$		0			0
$563$			26.0076i			1.09609i	0.836450	+	0.548044i	$0.184627\pi$
$563$			26.0076i			1.09609i	−0.836450	+	0.548044i	$0.815373\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−7.23236	+	22.6868i	−0.303730	+	0.952758i
$568$		0			0
$569$			29.1559i			1.22228i	0.791523	+	0.611139i	$0.209288\pi$
$569$			29.1559i			1.22228i	−0.791523	+	0.611139i	$0.790712\pi$
$570$		0			0
$571$	7.00000			0.292941			0.146470	−	0.989215i	$-0.453209\pi$
$571$	7.00000			0.292941			0.146470	+	0.989215i	$0.453209\pi$
$572$		0			0
$573$	4.78282	+	4.20811i	0.199805	+	0.175796i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−7.45363			−0.310299			−0.155149	−	0.987891i	$-0.549586\pi$
$577$	−7.45363			−0.310299			−0.155149	+	0.987891i	$0.549586\pi$
$578$		0			0
$579$	19.5057	+	17.1618i	0.810628	+	0.713222i
$580$		0			0
$581$	−13.0038	−	8.22431i	−0.539488	−	0.341202i
$582$		0			0
$583$		−	9.34752i		−	0.387135i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	26.8560i		−	1.10847i	−0.832361	−	0.554233i	$-0.813012\pi$
$587$		−	26.8560i		−	1.10847i	0.832361	−	0.554233i	$-0.186988\pi$
$588$		0			0
$589$	−10.0000			−0.412043
$590$		0			0
$591$	12.3174	−	13.9996i	0.506668	−	0.575865i
$592$		0			0
$593$		−	22.7928i		−	0.935990i	−0.883731	−	0.467995i	$-0.844976\pi$
$593$		−	22.7928i		−	0.935990i	0.883731	−	0.467995i	$-0.155024\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−31.3265	−	27.5623i	−1.28211	−	1.12805i
$598$		0			0
$599$			34.5704i			1.41251i	0.707958	+	0.706255i	$0.249617\pi$
$599$			34.5704i			1.41251i	−0.707958	+	0.706255i	$0.750383\pi$
$600$		0			0
$601$		−	1.49302i		−	0.0609018i	−0.999536	−	0.0304509i	$-0.990306\pi$
$601$		−	1.49302i		−	0.0609018i	0.999536	−	0.0304509i	$-0.00969431\pi$
$602$		0			0
$603$	−19.9608	+	2.56231i	−0.812868	+	0.104345i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−2.33695			−0.0948540			−0.0474270	−	0.998875i	$-0.515102\pi$
$607$	−2.33695			−0.0948540			−0.0474270	+	0.998875i	$0.515102\pi$
$608$		0			0
$609$	37.3450	−	10.9529i	1.51330	−	0.443835i
$610$		0			0
$611$			2.47811i			0.100254i
$612$		0			0
$613$		−	2.23607i		−	0.0903139i	−0.998980	−	0.0451570i	$-0.985621\pi$
$613$		−	2.23607i		−	0.0903139i	0.998980	−	0.0451570i	$-0.0143788\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.54145			−0.102315			−0.0511575	−	0.998691i	$-0.516291\pi$
$617$	−2.54145			−0.102315			−0.0511575	+	0.998691i	$0.516291\pi$
$618$		0			0
$619$			45.5887i			1.83236i	0.400763	+	0.916182i	$0.368745\pi$
$619$			45.5887i			1.83236i	−0.400763	+	0.916182i	$0.631255\pi$
$620$		0			0
$621$	−10.9445	−	7.38987i	−0.439188	−	0.296545i
$622$		0			0
$623$	21.5315	−	34.0443i	0.862643	−	1.36396i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−2.54145	+	2.88854i	−0.101496	+	0.115357i
$628$		0			0
$629$	9.40962			0.375186
$630$		0			0
$631$	−25.4721			−1.01403			−0.507015	−	0.861937i	$-0.669251\pi$
$631$	−25.4721			−1.01403			−0.507015	+	0.861937i	$0.669251\pi$
$632$		0			0
$633$	−8.81913	+	10.0236i	−0.350529	+	0.398401i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1.00155	−	2.11146i	−0.0396829	−	0.0836589i
$638$		0			0
$639$	−32.0344	+	4.11216i	−1.26726	+	0.162674i
$640$		0			0
$641$		−	37.3802i		−	1.47643i	−0.674566	−	0.738215i	$-0.735669\pi$
$641$		−	37.3802i		−	1.47643i	0.674566	−	0.738215i	$-0.264331\pi$
$642$		0			0
$643$	−7.40492			−0.292021			−0.146011	−	0.989283i	$-0.546643\pi$
$643$	−7.40492			−0.292021			−0.146011	+	0.989283i	$0.546643\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			37.6385i			1.47972i	0.672760	+	0.739861i	$0.265109\pi$
$647$			37.6385i			1.47972i	−0.672760	+	0.739861i	$0.734891\pi$
$648$		0			0
$649$		−	4.08502i		−	0.160351i
$650$		0			0
$651$	22.4998	−	6.59897i	0.881836	−	0.258634i
$652$		0			0
$653$	−13.3072			−0.520752			−0.260376	−	0.965507i	$-0.583846\pi$
$653$	−13.3072			−0.520752			−0.260376	+	0.965507i	$0.583846\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−0.540182	−	4.20811i	−0.0210745	−	0.164174i
$658$		0			0
$659$			26.9461i			1.04967i	0.851204	+	0.524835i	$0.175873\pi$
$659$			26.9461i			1.04967i	−0.851204	+	0.524835i	$0.824127\pi$
$660$		0			0
$661$		−	12.1877i		−	0.474048i	−0.971504	−	0.237024i	$-0.923828\pi$
$661$		−	12.1877i		−	0.474048i	0.971504	−	0.237024i	$-0.0761720\pi$
$662$		0			0
$663$	−1.82688	−	1.60736i	−0.0709500	−	0.0624245i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			21.5836i			0.835720i
$668$		0			0
$669$	19.3262	−	21.9657i	0.747196	−	0.849242i
$670$		0			0
$671$	13.0038			0.502005
$672$		0			0
$673$			31.3050i			1.20672i	0.797470	+	0.603359i	$0.206171\pi$
$673$			31.3050i			1.20672i	−0.797470	+	0.603359i	$0.793829\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	31.2091i		−	1.19946i	−0.800202	−	0.599731i	$-0.795274\pi$
$677$		−	31.2091i		−	1.19946i	0.800202	−	0.599731i	$-0.204726\pi$
$678$		0			0
$679$	13.7082	−	21.6746i	0.526073	−	0.831794i
$680$		0			0
$681$	−14.8197	−	13.0389i	−0.567891	−	0.499652i
$682$		0			0
$683$	32.2973			1.23582			0.617911	−	0.786248i	$-0.287979\pi$
$683$	32.2973			1.23582			0.617911	+	0.786248i	$0.287979\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	13.3072	+	11.7082i	0.507702	+	0.446696i
$688$		0			0
$689$	−2.74569			−0.104603
$690$		0			0
$691$			12.9343i			0.492042i	0.969264	+	0.246021i	$0.0791232\pi$
$691$			12.9343i			0.492042i	−0.969264	+	0.246021i	$0.920877\pi$
$692$		0			0
$693$	3.81207	−	8.17626i	0.144809	−	0.310590i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			24.4721i			0.926948i
$698$		0			0
$699$	−18.1328	+	20.6093i	−0.685846	+	0.779514i
$700$		0			0
$701$		−	6.48779i		−	0.245040i	−0.992466	−	0.122520i	$-0.960902\pi$
$701$		−	6.48779i		−	0.245040i	0.992466	−	0.122520i	$-0.0390976\pi$
$702$		0			0
$703$	−4.37016			−0.164824
$704$		0			0
$705$		0			0
$706$		0			0
$707$	21.5315	−	34.0443i	0.809777	−	1.28037i
$708$		0			0
$709$	−23.3050			−0.875236			−0.437618	−	0.899161i	$-0.644178\pi$
$709$	−23.3050			−0.875236			−0.437618	+	0.899161i	$0.644178\pi$
$710$		0			0
$711$	0.562306	+	4.38046i	0.0210881	+	0.164280i
$712$		0			0
$713$			13.0038i			0.486995i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	13.6507	+	12.0104i	0.509793	+	0.448536i
$718$		0			0
$719$	31.8230			1.18680			0.593399	−	0.804908i	$-0.297786\pi$
$719$	31.8230			1.18680			0.593399	+	0.804908i	$0.297786\pi$
$720$		0			0
$721$	−21.2361	+	33.5772i	−0.790872	+	1.25048i
$722$		0			0
$723$	29.1559	+	25.6525i	1.08432	+	0.954026i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	30.0810			1.11564			0.557822	−	0.829961i	$-0.311637\pi$
$727$	30.0810			1.11564			0.557822	+	0.829961i	$0.311637\pi$
$728$		0			0
$729$	10.0902	+	25.0437i	0.373710	+	0.927546i
$730$		0			0
$731$	−2.22131			−0.0821581
$732$		0			0
$733$	−35.1189			−1.29715			−0.648573	−	0.761152i	$-0.724634\pi$
$733$	−35.1189			−1.29715			−0.648573	+	0.761152i	$0.724634\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	7.62436			0.280847
$738$		0			0
$739$	36.5967			1.34623			0.673117	−	0.739536i	$-0.264955\pi$
$739$	36.5967			1.34623			0.673117	+	0.739536i	$0.264955\pi$
$740$		0			0
$741$	0.848465	+	0.746512i	0.0311691	+	0.0274238i
$742$		0			0
$743$	−13.3072			−0.488194			−0.244097	−	0.969751i	$-0.578492\pi$
$743$	−13.3072			−0.488194			−0.244097	+	0.969751i	$0.578492\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−17.3044	+	2.22131i	−0.633135	+	0.0812735i
$748$		0			0
$749$	7.18831	−	11.3657i	0.262655	−	0.415294i
$750$		0			0
$751$	−1.12461			−0.0410377			−0.0205188	−	0.999789i	$-0.506532\pi$
$751$	−1.12461			−0.0410377			−0.0205188	+	0.999789i	$0.506532\pi$
$752$		0			0
$753$	2.54145	−	2.88854i	0.0926157	−	0.105264i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	22.2361i		−	0.808184i	−0.914718	−	0.404092i	$-0.867588\pi$
$757$		−	22.2361i		−	0.808184i	0.914718	−	0.404092i	$-0.132412\pi$
$758$		0			0
$759$	3.75620	+	3.30485i	0.136341	+	0.119958i
$760$		0			0
$761$	7.18831			0.260576			0.130288	−	0.991476i	$-0.458410\pi$
$761$	7.18831			0.260576			0.130288	+	0.991476i	$0.458410\pi$
$762$		0			0
$763$	−10.5672	+	16.7082i	−0.382558	+	0.604878i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.19991			−0.0433263
$768$		0			0
$769$			3.62365i			0.130672i	0.997863	+	0.0653360i	$0.0208119\pi$
$769$			3.62365i			0.130672i	−0.997863	+	0.0653360i	$0.979188\pi$
$770$		0			0
$771$	32.0344	+	28.1851i	1.15369	+	1.01506i
$772$		0			0
$773$			23.4068i			0.841884i	0.907087	+	0.420942i	$0.138300\pi$
$773$			23.4068i			0.841884i	−0.907087	+	0.420942i	$0.861700\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	9.83277	−	2.88386i	0.352749	−	0.103458i
$778$		0			0
$779$		−	11.3657i		−	0.407219i
$780$		0			0
$781$	12.2361			0.437841
$782$		0			0
$783$	24.6943	−	36.5726i	0.882502	−	1.30700i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	7.32611			0.261148			0.130574	−	0.991439i	$-0.458318\pi$
$787$	7.32611			0.261148			0.130574	+	0.991439i	$0.458318\pi$
$788$		0			0
$789$	18.1328	−	20.6093i	0.645546	−	0.733709i
$790$		0			0
$791$	26.8560	−	42.4631i	0.954890	−	1.50981i
$792$		0			0
$793$		−	3.81966i		−	0.135640i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			34.6583i			1.22766i	0.789438	+	0.613830i	$0.210372\pi$
$797$			34.6583i			1.22766i	−0.789438	+	0.613830i	$0.789628\pi$
$798$		0			0
$799$	−31.2361			−1.10505
$800$		0			0
$801$	−5.81547	−	45.3035i	−0.205479	−	1.60072i
$802$		0			0
$803$			1.60736i			0.0567223i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	36.4095	−	41.3820i	1.28167	−	1.45671i
$808$		0			0
$809$		−	21.7998i		−	0.766442i	−0.923657	−	0.383221i	$-0.874815\pi$
$809$		−	21.7998i		−	0.766442i	0.923657	−	0.383221i	$-0.125185\pi$
$810$		0			0
$811$			5.40182i			0.189683i	0.995492	+	0.0948417i	$0.0302345\pi$
$811$			5.40182i			0.189683i	−0.995492	+	0.0948417i	$0.969766\pi$
$812$		0			0
$813$	−28.7851	−	25.3262i	−1.00954	−	0.888230i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1.03165			0.0360930
$818$		0			0
$819$	−2.40165	−	1.11974i	−0.0839205	−	0.0391268i
$820$		0			0
$821$			36.5753i			1.27649i	0.769835	+	0.638243i	$0.220338\pi$
$821$			36.5753i			1.27649i	−0.769835	+	0.638243i	$0.779662\pi$
$822$		0			0
$823$		−	17.3607i		−	0.605155i	−0.953125	−	0.302578i	$-0.902153\pi$
$823$		−	17.3607i		−	0.605155i	0.953125	−	0.302578i	$-0.0978472\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−7.62436			−0.265125			−0.132562	−	0.991175i	$-0.542321\pi$
$827$	−7.62436			−0.265125			−0.132562	+	0.991175i	$0.542321\pi$
$828$		0			0
$829$			33.5772i			1.16618i	0.812406	+	0.583092i	$0.198157\pi$
$829$			33.5772i			1.16618i	−0.812406	+	0.583092i	$0.801843\pi$
$830$		0			0
$831$	−10.7825	−	9.48683i	−0.374040	−	0.329095i
$832$		0			0
$833$	26.6144	−	12.6243i	0.922136	−	0.437407i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	14.8779	−	22.0344i	0.514256	−	0.761622i
$838$		0			0
$839$	19.6677			0.679005			0.339502	−	0.940605i	$-0.389741\pi$
$839$	19.6677			0.679005			0.339502	+	0.940605i	$0.389741\pi$
$840$		0			0
$841$	−43.1246			−1.48706
$842$		0			0
$843$	12.1727	+	10.7100i	0.419250	+	0.368872i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	13.7295	−	21.7082i	0.471750	−	0.745903i
$848$		0			0
$849$	−6.76393	+	7.68770i	−0.232138	+	0.263841i
$850$		0			0
$851$			5.68286i			0.194806i
$852$		0			0
$853$	23.1375			0.792213			0.396106	−	0.918205i	$-0.370361\pi$
$853$	23.1375			0.792213			0.396106	+	0.918205i	$0.370361\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			24.6347i			0.841506i	0.907175	+	0.420753i	$0.138234\pi$
$857$			24.6347i			0.841506i	−0.907175	+	0.420753i	$0.861766\pi$
$858$		0			0
$859$		−	52.6598i		−	1.79673i	−0.439252	−	0.898364i	$-0.644757\pi$
$859$		−	52.6598i		−	1.79673i	0.439252	−	0.898364i	$-0.355243\pi$
$860$		0			0
$861$	7.50021	+	25.5726i	0.255606	+	0.871513i
$862$		0			0
$863$	−40.5216			−1.37937			−0.689686	−	0.724109i	$-0.742251\pi$
$863$	−40.5216			−1.37937			−0.689686	+	0.724109i	$0.742251\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.810272	−	0.920933i	0.0275183	−	0.0312765i
$868$		0			0
$869$		−	1.67319i		−	0.0567590i
$870$		0			0
$871$		−	2.23954i		−	0.0758838i
$872$		0			0
$873$	−3.70246	−	28.8428i	−0.125309	−	0.976182i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			15.5279i			0.524339i	0.965022	+	0.262169i	$0.0844379\pi$
$877$			15.5279i			0.524339i	−0.965022	+	0.262169i	$0.915562\pi$
$878$		0			0
$879$	−30.7426	−	27.0486i	−1.03692	−	0.912326i
$880$		0			0
$881$	−1.37285			−0.0462523			−0.0231262	−	0.999733i	$-0.507362\pi$
$881$	−1.37285			−0.0462523			−0.0231262	+	0.999733i	$0.507362\pi$
$882$		0			0
$883$		−	7.11146i		−	0.239320i	−0.992815	−	0.119660i	$-0.961820\pi$
$883$		−	7.11146i		−	0.239320i	0.992815	−	0.119660i	$-0.0381804\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			47.8966i			1.60821i	0.594487	+	0.804105i	$0.297355\pi$
$887$			47.8966i			1.60821i	−0.594487	+	0.804105i	$0.702645\pi$
$888$		0			0
$889$	−37.3607	−	23.6290i	−1.25304	−	0.792490i
$890$		0			0
$891$	−2.58359	−	9.89750i	−0.0865536	−	0.331579i
$892$		0			0
$893$	14.5071			0.485463
$894$		0			0
$895$		0			0
$896$		0			0
$897$	0.970748	−	1.10333i	0.0324123	−	0.0368390i
$898$		0			0
$899$	−43.4540			−1.44927
$900$		0			0
$901$		−	34.6088i		−	1.15299i
$902$		0			0
$903$	−2.32120	+	0.680786i	−0.0772448	+	0.0226551i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			21.3050i			0.707419i	0.935355	+	0.353710i	$0.115080\pi$
$907$			21.3050i			0.707419i	−0.935355	+	0.353710i	$0.884920\pi$
$908$		0			0
$909$	−5.81547	−	45.3035i	−0.192887	−	1.50262i
$910$		0			0
$911$			35.4387i			1.17414i	0.809537	+	0.587068i	$0.199718\pi$
$911$			35.4387i			1.17414i	−0.809537	+	0.587068i	$0.800282\pi$
$912$		0			0
$913$	6.60970			0.218749
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−13.3072	+	21.0406i	−0.439443	+	0.694820i
$918$		0			0
$919$	34.3050			1.13162			0.565808	−	0.824537i	$-0.308564\pi$
$919$	34.3050			1.13162			0.565808	+	0.824537i	$0.308564\pi$
$920$		0			0
$921$	29.1803	−	33.1656i	0.961525	−	1.09284i
$922$		0			0
$923$		−	3.59416i		−	0.118303i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	5.73567	+	44.6819i	0.188384	+	1.46755i
$928$		0			0
$929$	15.2251			0.499519			0.249760	−	0.968308i	$-0.419648\pi$
$929$	15.2251			0.499519			0.249760	+	0.968308i	$0.419648\pi$
$930$		0			0
$931$	−12.3607	+	5.86319i	−0.405105	+	0.192158i
$932$		0			0
$933$	2.54145	−	2.88854i	0.0832034	−	0.0945667i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	44.3020			1.44728			0.723641	−	0.690176i	$-0.242467\pi$
$937$	44.3020			1.44728			0.723641	+	0.690176i	$0.242467\pi$
$938$		0			0
$939$	−5.47214	+	6.21948i	−0.178576	+	0.202965i
$940$		0			0
$941$	36.2656			1.18223			0.591113	−	0.806589i	$-0.298689\pi$
$941$	36.2656			1.18223			0.591113	+	0.806589i	$0.298689\pi$
$942$		0			0
$943$	−14.7797			−0.481294
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.28282			0.204164			0.102082	−	0.994776i	$-0.467450\pi$
$947$	6.28282			0.204164			0.102082	+	0.994776i	$0.467450\pi$
$948$		0			0
$949$	0.472136			0.0153262
$950$		0			0
$951$	33.3579	−	37.9137i	1.08170	−	1.22944i
$952$		0			0
$953$	−47.5460			−1.54017			−0.770083	−	0.637944i	$-0.779785\pi$
$953$	−47.5460			−1.54017			−0.770083	+	0.637944i	$0.779785\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−11.0436	+	12.5519i	−0.356989	+	0.405744i
$958$		0			0
$959$	30.4502	−	48.1460i	0.983288	−	1.55471i
$960$		0			0
$961$	4.81966			0.155473
$962$		0			0
$963$	−1.94150	−	15.1246i	−0.0625639	−	0.487384i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	44.0689i		−	1.41716i	−0.705631	−	0.708580i	$-0.749336\pi$
$967$		−	44.0689i		−	1.41716i	0.705631	−	0.708580i	$-0.250664\pi$
$968$		0			0
$969$	−9.40962	+	10.6947i	−0.302281	+	0.343564i
$970$		0			0
$971$	−47.0481			−1.50985			−0.754923	−	0.655813i	$-0.772326\pi$
$971$	−47.0481			−1.50985			−0.754923	+	0.655813i	$0.772326\pi$
$972$		0			0
$973$	1.66925	+	1.05573i	0.0535138	+	0.0338451i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	7.62436			0.243925			0.121962	−	0.992535i	$-0.461081\pi$
$977$	7.62436			0.243925			0.121962	+	0.992535i	$0.461081\pi$
$978$		0			0
$979$			17.3044i			0.553051i
$980$		0			0
$981$	2.85410	+	22.2340i	0.0911245	+	0.709876i
$982$		0			0
$983$			4.58756i			0.146320i	0.997320	+	0.0731602i	$0.0233084\pi$
$983$			4.58756i			0.146320i	−0.997320	+	0.0731602i	$0.976692\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−32.6407	+	9.57321i	−1.03897	+	0.304719i
$988$		0			0
$989$		−	1.34154i		−	0.0426585i
$990$		0			0
$991$	−31.7214			−1.00766			−0.503831	−	0.863802i	$-0.668077\pi$
$991$	−31.7214			−1.00766			−0.503831	+	0.863802i	$0.668077\pi$
$992$		0			0
$993$	−33.3859	+	37.9455i	−1.05947	+	1.20416i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	60.1134			1.90381			0.951905	−	0.306394i	$-0.0991224\pi$
$997$	60.1134			1.90381			0.951905	+	0.306394i	$0.0991224\pi$
$998$		0			0
$999$	6.50189	−	9.62940i	0.205711	−	0.304661i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.2.f.i.1049.3		16
3.2	odd	2	inner	2100.2.f.i.1049.1		16
5.2	odd	4		2100.2.d.l.1301.6	yes	8
5.3	odd	4		2100.2.d.k.1301.3	✓	8
5.4	even	2	inner	2100.2.f.i.1049.14		16
7.6	odd	2	inner	2100.2.f.i.1049.13		16
15.2	even	4		2100.2.d.l.1301.4	yes	8
15.8	even	4		2100.2.d.k.1301.5	yes	8
15.14	odd	2	inner	2100.2.f.i.1049.16		16
21.20	even	2	inner	2100.2.f.i.1049.15		16
35.13	even	4		2100.2.d.k.1301.6	yes	8
35.27	even	4		2100.2.d.l.1301.3	yes	8
35.34	odd	2	inner	2100.2.f.i.1049.4		16
105.62	odd	4		2100.2.d.l.1301.5	yes	8
105.83	odd	4		2100.2.d.k.1301.4	yes	8
105.104	even	2	inner	2100.2.f.i.1049.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2100.2.d.k.1301.3	✓	8	5.3	odd	4
2100.2.d.k.1301.4	yes	8	105.83	odd	4
2100.2.d.k.1301.5	yes	8	15.8	even	4
2100.2.d.k.1301.6	yes	8	35.13	even	4
2100.2.d.l.1301.3	yes	8	35.27	even	4
2100.2.d.l.1301.4	yes	8	15.2	even	4
2100.2.d.l.1301.5	yes	8	105.62	odd	4
2100.2.d.l.1301.6	yes	8	5.2	odd	4
2100.2.f.i.1049.1		16	3.2	odd	2	inner
2100.2.f.i.1049.2		16	105.104	even	2	inner
2100.2.f.i.1049.3		16	1.1	even	1	trivial
2100.2.f.i.1049.4		16	35.34	odd	2	inner
2100.2.f.i.1049.13		16	7.6	odd	2	inner
2100.2.f.i.1049.14		16	5.4	even	2	inner
2100.2.f.i.1049.15		16	21.20	even	2	inner
2100.2.f.i.1049.16		16	15.14	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.14412 + 1.30038i) q^{3} +(1.41421 - 2.23607i) q^{7} +(-0.381966 - 2.97558i) q^{9} +O(q^{10})\)	\(q+(-1.14412 + 1.30038i) q^{3} +(1.41421 - 2.23607i) q^{7} +(-0.381966 - 2.97558i) q^{9} +1.13657i q^{11} +0.333851 q^{13} +4.20811i q^{17} -1.95440i q^{19} +(1.28970 + 4.39735i) q^{21} -2.54145 q^{23} +(4.30640 + 2.90773i) q^{27} -8.49262i q^{29} -5.11667i q^{31} +(-1.47797 - 1.30038i) q^{33} -2.23607i q^{37} +(-0.381966 + 0.434132i) q^{39} +5.81547 q^{41} +0.527864i q^{43} +7.42282i q^{47} +(-3.00000 - 6.32456i) q^{49} +(-5.47214 - 4.81460i) q^{51} -8.22431 q^{53} +(2.54145 + 2.23607i) q^{57} -3.59416 q^{59} -11.4412i q^{61} +(-7.19379 - 3.35401i) q^{63} -6.70820i q^{67} +(2.90773 - 3.30485i) q^{69} -10.7658i q^{71} +1.41421 q^{73} +(2.54145 + 1.60736i) q^{77} -1.47214 q^{79} +(-8.70820 + 2.27314i) q^{81} -5.81547i q^{83} +(11.0436 + 9.71660i) q^{87} +15.2251 q^{89} +(0.472136 - 0.746512i) q^{91} +(6.65361 + 5.85410i) q^{93} +9.69316 q^{97} +(3.38197 - 0.434132i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 24 q^{9}+O(q^{10}) \) 16 * q - 24 * q^9	\( 16 q - 24 q^{9} - 8 q^{21} - 24 q^{39} - 48 q^{49} - 16 q^{51} + 48 q^{79} - 32 q^{81} - 64 q^{91} + 72 q^{99}+O(q^{100}) \) 16 * q - 24 * q^9 - 8 * q^21 - 24 * q^39 - 48 * q^49 - 16 * q^51 + 48 * q^79 - 32 * q^81 - 64 * q^91 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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