LMFDB - Embedded newform 2100.2.d.k.1301.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,2,Mod(1301,2100)]

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, 0, 1]))

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

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

chi := DirichletCharacter("2100.1301");

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.d (of order $2$, degree $1$, 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:	$8$
Coefficient field:	8.0.14786560000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 6x^{6} + 22x^{4} - 54x^{2} + 81 $ x^8 - 6x^6 + 22x^4 - 54*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1301.3
Root			$-1.30038 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	2100.1301
Dual form			2100.2.d.k.1301.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.30038 - 1.14412i) q^{3} +(-2.23607 - 1.41421i) q^{7} +(0.381966 + 2.97558i) q^{9} +O(q^{10})$	$q+(-1.30038 - 1.14412i) q^{3} +(-2.23607 - 1.41421i) q^{7} +(0.381966 + 2.97558i) q^{9} +1.13657i q^{11} +0.333851i q^{13} +4.20811 q^{17} +1.95440i q^{19} +(1.28970 + 4.39735i) q^{21} -2.54145i q^{23} +(2.90773 - 4.30640i) q^{27} +8.49262i q^{29} -5.11667i q^{31} +(1.30038 - 1.47797i) q^{33} -2.23607 q^{37} +(0.381966 - 0.434132i) q^{39} +5.81547 q^{41} -0.527864 q^{43} +7.42282 q^{47} +(3.00000 + 6.32456i) q^{49} +(-5.47214 - 4.81460i) q^{51} -8.22431i q^{53} +(2.23607 - 2.54145i) q^{57} +3.59416 q^{59} -11.4412i q^{61} +(3.35401 - 7.19379i) q^{63} -6.70820 q^{67} +(-2.90773 + 3.30485i) q^{69} -10.7658i q^{71} +1.41421i q^{73} +(1.60736 - 2.54145i) q^{77} +1.47214 q^{79} +(-8.70820 + 2.27314i) q^{81} +5.81547 q^{83} +(9.71660 - 11.0436i) q^{87} -15.2251 q^{89} +(0.472136 - 0.746512i) q^{91} +(-5.85410 + 6.65361i) q^{93} -9.69316i q^{97} +(-3.38197 + 0.434132i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{9}+O(q^{10}) $ 8 * q + 12 * q^9	$ 8 q + 12 q^{9} - 4 q^{21} + 12 q^{39} - 40 q^{43} + 24 q^{49} - 8 q^{51} + 20 q^{63} - 24 q^{79} - 16 q^{81} - 32 q^{91} - 20 q^{93} - 36 q^{99}+O(q^{100}) $ 8 * q + 12 * q^9 - 4 * q^21 + 12 * q^39 - 40 * q^43 + 24 * q^49 - 8 * q^51 + 20 * q^63 - 24 * q^79 - 16 * q^81 - 32 * q^91 - 20 * q^93 - 36 * 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
$3$	−1.30038	−	1.14412i	−0.750774	−	0.660560i
$3$	−1.30038	−	1.14412i	−0.750774	−	0.660560i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.23607	−	1.41421i	−0.845154	−	0.534522i
$7$	−2.23607	−	1.41421i	−0.845154	−	0.534522i
$8$		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
$13$			0.333851i			0.0925935i	0.998928	+	0.0462967i	$0.0147420\pi$
$13$			0.333851i			0.0925935i	−0.998928	+	0.0462967i	$0.985258\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.20811			1.02062			0.510308	−	0.859991i	$-0.329531\pi$
$17$	4.20811			1.02062			0.510308	+	0.859991i	$0.329531\pi$
$18$		0			0
$19$			1.95440i			0.448369i	0.974547	+	0.224184i	$0.0719718\pi$
$19$			1.95440i			0.448369i	−0.974547	+	0.224184i	$0.928028\pi$
$20$		0			0
$21$	1.28970	+	4.39735i	0.281436	+	0.959580i
$22$		0			0
$23$		−	2.54145i		−	0.529929i	−0.964258	−	0.264965i	$-0.914640\pi$
$23$		−	2.54145i		−	0.529929i	0.964258	−	0.264965i	$-0.0853603\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	2.90773	−	4.30640i	0.559594	−	0.828767i
$28$		0			0
$29$			8.49262i			1.57704i	0.615009	+	0.788520i	$0.289152\pi$
$29$			8.49262i			1.57704i	−0.615009	+	0.788520i	$0.710848\pi$
$30$		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
$33$	1.30038	−	1.47797i	0.226367	−	0.257282i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.23607			−0.367607			−0.183804	−	0.982963i	$-0.558841\pi$
$37$	−2.23607			−0.367607			−0.183804	+	0.982963i	$0.558841\pi$
$38$		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
$43$	−0.527864			−0.0804985			−0.0402493	−	0.999190i	$-0.512815\pi$
$43$	−0.527864			−0.0804985			−0.0402493	+	0.999190i	$0.512815\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.42282			1.08273			0.541365	−	0.840788i	$-0.317908\pi$
$47$	7.42282			1.08273			0.541365	+	0.840788i	$0.317908\pi$
$48$		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
$53$		−	8.22431i		−	1.12970i	−0.825195	−	0.564848i	$-0.808935\pi$
$53$		−	8.22431i		−	1.12970i	0.825195	−	0.564848i	$-0.191065\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	2.23607	−	2.54145i	0.296174	−	0.336624i
$58$		0			0
$59$	3.59416			0.467919			0.233960	−	0.972246i	$-0.424832\pi$
$59$	3.59416			0.467919			0.233960	+	0.972246i	$0.424832\pi$
$60$		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
$63$	3.35401	−	7.19379i	0.422566	−	0.906332i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−6.70820			−0.819538			−0.409769	−	0.912189i	$-0.634391\pi$
$67$	−6.70820			−0.819538			−0.409769	+	0.912189i	$0.634391\pi$
$68$		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
$73$			1.41421i			0.165521i	0.996569	+	0.0827606i	$0.0263737\pi$
$73$			1.41421i			0.165521i	−0.996569	+	0.0827606i	$0.973626\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.60736	−	2.54145i	0.183175	−	0.289625i
$78$		0			0
$79$	1.47214			0.165628			0.0828141	−	0.996565i	$-0.473609\pi$
$79$	1.47214			0.165628			0.0828141	+	0.996565i	$0.473609\pi$
$80$		0			0
$81$	−8.70820	+	2.27314i	−0.967578	+	0.252572i
$82$		0			0
$83$	5.81547			0.638330			0.319165	−	0.947699i	$-0.396598\pi$
$83$	5.81547			0.638330			0.319165	+	0.947699i	$0.396598\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	9.71660	−	11.0436i	1.04173	−	1.18400i
$88$		0			0
$89$	−15.2251			−1.61386			−0.806928	−	0.590650i	$-0.798872\pi$
$89$	−15.2251			−1.61386			−0.806928	+	0.590650i	$0.798872\pi$
$90$		0			0
$91$	0.472136	−	0.746512i	0.0494933	−	0.0782558i
$92$		0			0
$93$	−5.85410	+	6.65361i	−0.607042	+	0.689947i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	9.69316i		−	0.984192i	−0.870541	−	0.492096i	$-0.836231\pi$
$97$		−	9.69316i		−	0.984192i	0.870541	−	0.492096i	$-0.163769\pi$
$98$		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
$103$		−	15.0162i		−	1.47959i	−0.672834	−	0.739793i	$-0.734923\pi$
$103$		−	15.0162i		−	1.47959i	0.672834	−	0.739793i	$-0.265077\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	5.08290i		−	0.491383i	−0.969348	−	0.245691i	$-0.920985\pi$
$107$		−	5.08290i		−	0.491383i	0.969348	−	0.245691i	$-0.0790150\pi$
$108$		0			0
$109$	7.47214			0.715701			0.357850	−	0.933779i	$-0.383510\pi$
$109$	7.47214			0.715701			0.357850	+	0.933779i	$0.383510\pi$
$110$		0			0
$111$	2.90773	+	2.55834i	0.275990	+	0.242827i
$112$		0			0
$113$			18.9901i			1.78644i	0.449624	+	0.893218i	$0.351558\pi$
$113$			18.9901i			1.78644i	−0.449624	+	0.893218i	$0.648442\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.993400	+	0.127520i	−0.0918399	+	0.0117892i
$118$		0			0
$119$	−9.40962	−	5.95117i	−0.862579	−	0.545543i
$120$		0			0
$121$	9.70820			0.882564
$122$		0			0
$123$	−7.56231	−	6.65361i	−0.681870	−	0.599936i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−16.7082			−1.48261			−0.741307	−	0.671166i	$-0.765794\pi$
$127$	−16.7082			−1.48261			−0.741307	+	0.671166i	$0.765794\pi$
$128$		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
$133$	2.76393	−	4.37016i	0.239663	−	0.378941i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	21.5315i		−	1.83956i	−0.392431	−	0.919781i	$-0.628366\pi$
$137$		−	21.5315i		−	1.83956i	0.392431	−	0.919781i	$-0.371634\pi$
$138$		0			0
$139$		−	0.746512i		−	0.0633184i	−0.999499	−	0.0316592i	$-0.989921\pi$
$139$		−	0.746512i		−	0.0633184i	0.999499	−	0.0316592i	$-0.0100791\pi$
$140$		0			0
$141$	−9.65248	−	8.49262i	−0.812885	−	0.715208i
$142$		0			0
$143$	−0.379445			−0.0317308
$144$		0			0
$145$		0			0
$146$		0			0
$147$	3.33493	−	11.6567i	0.275061	−	0.961427i
$148$		0			0
$149$			2.54145i			0.208204i	0.994567	+	0.104102i	$0.0331968\pi$
$149$			2.54145i			0.208204i	−0.994567	+	0.104102i	$0.966803\pi$
$150$		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
$153$	1.60736	+	12.5216i	0.129947	+	1.01231i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	8.61280i		−	0.687376i	−0.939084	−	0.343688i	$-0.888324\pi$
$157$		−	8.61280i		−	0.687376i	0.939084	−	0.343688i	$-0.111676\pi$
$158$		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
$163$	8.94427			0.700569			0.350285	−	0.936643i	$-0.386085\pi$
$163$	8.94427			0.700569			0.350285	+	0.936643i	$0.386085\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	10.0236			0.775648			0.387824	−	0.921733i	$-0.373227\pi$
$167$	10.0236			0.775648			0.387824	+	0.921733i	$0.373227\pi$
$168$		0			0
$169$	12.8885			0.991426
$170$		0			0
$171$	−5.81547	+	0.746512i	−0.444720	+	0.0570872i
$172$		0			0
$173$	6.19491			0.470990			0.235495	−	0.971876i	$-0.424329\pi$
$173$	6.19491			0.470990			0.235495	+	0.971876i	$0.424329\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−4.67376	−	4.11216i	−0.351301	−	0.309089i
$178$		0			0
$179$		−	13.3072i		−	0.994628i	−0.867571	−	0.497314i	$-0.834320\pi$
$179$		−	13.3072i		−	0.994628i	0.867571	−	0.497314i	$-0.165680\pi$
$180$		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
$183$	−13.0902	+	14.8779i	−0.967653	+	1.09981i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			4.78282i			0.349755i
$188$		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
$193$	15.0000			1.07972			0.539862	−	0.841754i	$-0.318476\pi$
$193$	15.0000			1.07972			0.539862	+	0.841754i	$0.318476\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			10.7658i			0.767029i	0.923535	+	0.383515i	$0.125286\pi$
$197$			10.7658i			0.767029i	−0.923535	+	0.383515i	$0.874714\pi$
$198$		0			0
$199$		−	24.0903i		−	1.70772i	−0.520504	−	0.853859i	$-0.674256\pi$
$199$		−	24.0903i		−	1.70772i	0.520504	−	0.853859i	$-0.325744\pi$
$200$		0			0
$201$	8.72320	+	7.67501i	0.615287	+	0.541353i
$202$		0			0
$203$	12.0104	−	18.9901i	0.842963	−	1.33284i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	7.56231	−	0.970748i	0.525617	−	0.0674717i
$208$		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
$213$	−12.3174	+	13.9996i	−0.843971	+	0.959234i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−7.23607	+	11.4412i	−0.491216	+	0.776681i
$218$		0			0
$219$	1.61803	−	1.83901i	0.109337	−	0.124269i
$220$		0			0
$221$			1.40488i			0.0945025i
$222$		0			0
$223$		−	16.8918i		−	1.13116i	−0.824695	−	0.565578i	$-0.808653\pi$
$223$		−	16.8918i		−	1.13116i	0.824695	−	0.565578i	$-0.191347\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.3964			0.756407			0.378204	−	0.925722i	$-0.376542\pi$
$227$	11.3964			0.756407			0.378204	+	0.925722i	$0.376542\pi$
$228$		0			0
$229$			10.2333i			0.676239i	0.941103	+	0.338119i	$0.109791\pi$
$229$			10.2333i			0.676239i	−0.941103	+	0.338119i	$0.890209\pi$
$230$		0			0
$231$	−4.99790	+	1.46584i	−0.328838	+	0.0964450i
$232$		0			0
$233$			15.8487i			1.03828i	0.854689	+	0.519140i	$0.173748\pi$
$233$			15.8487i			1.03828i	−0.854689	+	0.519140i	$0.826252\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.91433	−	1.68430i	−0.124349	−	0.109407i
$238$		0			0
$239$			10.4975i			0.679024i	0.940602	+	0.339512i	$0.110262\pi$
$239$			10.4975i			0.679024i	−0.940602	+	0.339512i	$0.889738\pi$
$240$		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
$243$	13.9247	+	7.00731i	0.893271	+	0.449519i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−0.652476			−0.0415160
$248$		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
$253$	2.88854			0.181601
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−24.6347			−1.53667			−0.768336	−	0.640047i	$-0.778915\pi$
$257$	−24.6347			−1.53667			−0.768336	+	0.640047i	$0.778915\pi$
$258$		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
$263$		−	15.8487i		−	0.977271i	−0.872488	−	0.488635i	$-0.837495\pi$
$263$		−	15.8487i		−	0.977271i	0.872488	−	0.488635i	$-0.162505\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	19.7984	+	17.4194i	1.21164	+	1.06605i
$268$		0			0
$269$	31.8230			1.94028			0.970142	−	0.242537i	$-0.0779796\pi$
$269$	31.8230			1.94028			0.970142	+	0.242537i	$0.0779796\pi$
$270$		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
$273$	−1.46806	+	0.430567i	−0.0888509	+	0.0260591i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	8.29180			0.498206			0.249103	−	0.968477i	$-0.419864\pi$
$277$	8.29180			0.498206			0.249103	+	0.968477i	$0.419864\pi$
$278$		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
$283$			5.91189i			0.351426i	0.984441	+	0.175713i	$0.0562230\pi$
$283$			5.91189i			0.351426i	−0.984441	+	0.175713i	$0.943777\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−13.0038	−	8.22431i	−0.767589	−	0.485466i
$288$		0			0
$289$	0.708204			0.0416591
$290$		0			0
$291$	−11.0902	+	12.6048i	−0.650117	+	0.738905i
$292$		0			0
$293$	−23.6413			−1.38114			−0.690570	−	0.723265i	$-0.742640\pi$
$293$	−23.6413			−1.38114			−0.690570	+	0.723265i	$0.742640\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	4.89453	+	3.30485i	0.284010	+	0.191767i
$298$		0			0
$299$	0.848465			0.0490680
$300$		0			0
$301$	1.18034	+	0.746512i	0.0680337	+	0.0430283i
$302$		0			0
$303$	−19.7984	−	17.4194i	−1.13739	−	1.00072i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			25.5046i			1.45562i	0.685778	+	0.727811i	$0.259462\pi$
$307$			25.5046i			1.45562i	−0.685778	+	0.727811i	$0.740538\pi$
$308$		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
$313$			4.78282i			0.270341i	0.990822	+	0.135171i	$0.0431582\pi$
$313$			4.78282i			0.270341i	−0.990822	+	0.135171i	$0.956842\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			29.1559i			1.63756i	0.574109	+	0.818779i	$0.305349\pi$
$317$			29.1559i			1.63756i	−0.574109	+	0.818779i	$0.694651\pi$
$318$		0			0
$319$	−9.65248			−0.540435
$320$		0			0
$321$	−5.81547	+	6.60970i	−0.324588	+	0.368917i
$322$		0			0
$323$			8.22431i			0.457613i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−9.71660	−	8.54904i	−0.537329	−	0.472763i
$328$		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
$333$	−0.854102	−	6.65361i	−0.0468045	−	0.364616i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−10.0000			−0.544735			−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000			−0.544735			−0.272367	+	0.962193i	$0.587807\pi$
$338$		0			0
$339$	21.7270	−	24.6943i	1.18005	−	1.34121i
$340$		0			0
$341$	5.81547			0.314925
$342$		0			0
$343$	2.23607	−	18.3848i	0.120736	−	0.992685i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			29.1559i			1.56517i	0.622544	+	0.782585i	$0.286099\pi$
$347$			29.1559i			1.56517i	−0.622544	+	0.782585i	$0.713901\pi$
$348$		0			0
$349$		−	22.4211i		−	1.20017i	−0.799935	−	0.600087i	$-0.795133\pi$
$349$		−	22.4211i		−	1.20017i	0.799935	−	0.600087i	$-0.204867\pi$
$350$		0			0
$351$	1.43769	+	0.970748i	0.0767384	+	0.0518147i
$352$		0			0
$353$	−33.0509			−1.75912			−0.879562	−	0.475784i	$-0.842164\pi$
$353$	−33.0509			−1.75912			−0.879562	+	0.475784i	$0.842164\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	5.42720	+	18.5045i	0.287238	+	0.979364i
$358$		0			0
$359$			10.7658i			0.568195i	0.958795	+	0.284098i	$0.0916940\pi$
$359$			10.7658i			0.568195i	−0.958795	+	0.284098i	$0.908306\pi$
$360$		0			0
$361$	15.1803			0.798965
$362$		0			0
$363$	−12.6243	−	11.1074i	−0.662606	−	0.582986i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	14.5247i		−	0.758183i	−0.925359	−	0.379091i	$-0.876237\pi$
$367$		−	14.5247i		−	0.758183i	0.925359	−	0.379091i	$-0.123763\pi$
$368$		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
$373$	15.6525			0.810454			0.405227	−	0.914216i	$-0.367192\pi$
$373$	15.6525			0.810454			0.405227	+	0.914216i	$0.367192\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.83527			−0.146024
$378$		0			0
$379$	−5.18034			−0.266096			−0.133048	−	0.991110i	$-0.542476\pi$
$379$	−5.18034			−0.266096			−0.133048	+	0.991110i	$0.542476\pi$
$380$		0			0
$381$	21.7270	+	19.1162i	1.11311	+	0.979354i
$382$		0			0
$383$	15.2251			0.777966			0.388983	−	0.921245i	$-0.372827\pi$
$383$	15.2251			0.777966			0.388983	+	0.921245i	$0.372827\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.201626	−	1.57070i	−0.0102492	−	0.0798434i
$388$		0			0
$389$			18.1218i			0.918812i	0.888226	+	0.459406i	$0.151938\pi$
$389$			18.1218i			0.918812i	−0.888226	+	0.459406i	$0.848062\pi$
$390$		0			0
$391$		−	10.6947i		−	0.540855i
$392$		0			0
$393$	12.2361	+	10.7658i	0.617228	+	0.543061i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			39.1853i			1.96665i	0.181844	+	0.983327i	$0.441793\pi$
$397$			39.1853i			1.96665i	−0.181844	+	0.983327i	$0.558207\pi$
$398$		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
$403$	1.70820			0.0850917
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	2.54145i		−	0.125975i
$408$		0			0
$409$		−	24.8369i		−	1.22810i	−0.789266	−	0.614052i	$-0.789539\pi$
$409$		−	24.8369i		−	1.22810i	0.789266	−	0.614052i	$-0.210461\pi$
$410$		0			0
$411$	−24.6347	+	27.9991i	−1.21514	+	1.38110i
$412$		0			0
$413$	−8.03678	−	5.08290i	−0.395464	−	0.250113i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.854102	+	0.970748i	−0.0418256	+	0.0475378i
$418$		0			0
$419$	36.2656			1.77169			0.885846	−	0.463979i	$-0.153579\pi$
$419$	36.2656			1.77169			0.885846	+	0.463979i	$0.153579\pi$
$420$		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
$423$	2.83527	+	22.0872i	0.137855	+	1.07392i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−16.1803	+	25.5834i	−0.783022	+	1.23807i
$428$		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
$433$			31.3677i			1.50744i	0.657197	+	0.753719i	$0.271742\pi$
$433$			31.3677i			1.50744i	−0.657197	+	0.753719i	$0.728258\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.96700			0.237604
$438$		0			0
$439$			18.2272i			0.869935i	0.900446	+	0.434967i	$0.143240\pi$
$439$			18.2272i			0.869935i	−0.900446	+	0.434967i	$0.856760\pi$
$440$		0			0
$441$	−17.6733	+	11.3425i	−0.841588	+	0.540120i
$442$		0			0
$443$			8.22431i			0.390749i	0.980729	+	0.195374i	$0.0625922\pi$
$443$			8.22431i			0.390749i	−0.980729	+	0.195374i	$0.937408\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	2.90773	−	3.30485i	0.137531	−	0.156314i
$448$		0			0
$449$		−	5.35121i		−	0.252539i	−0.991996	−	0.126270i	$-0.959700\pi$
$449$		−	5.35121i		−	0.252539i	0.991996	−	0.126270i	$-0.0403005\pi$
$450$		0			0
$451$			6.60970i			0.311239i
$452$		0			0
$453$	−12.5519	−	11.0436i	−0.589738	−	0.518874i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	37.3607			1.74766			0.873829	−	0.486233i	$-0.161629\pi$
$457$	37.3607			1.74766			0.873829	+	0.486233i	$0.161629\pi$
$458$		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
$463$	10.6525			0.495063			0.247531	−	0.968880i	$-0.420381\pi$
$463$	10.6525			0.495063			0.247531	+	0.968880i	$0.420381\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−25.2487			−1.16837			−0.584185	−	0.811621i	$-0.698586\pi$
$467$	−25.2487			−1.16837			−0.584185	+	0.811621i	$0.698586\pi$
$468$		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
$473$		−	0.599956i		−	0.0275860i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	24.4721	−	3.14141i	1.12050	−	0.143835i
$478$		0			0
$479$	41.2327			1.88397			0.941984	−	0.335658i	$-0.108959\pi$
$479$	41.2327			1.88397			0.941984	+	0.335658i	$0.108959\pi$
$480$		0			0
$481$		−	0.746512i		−	0.0340380i
$482$		0			0
$483$	11.1757	−	3.27771i	0.508510	−	0.149141i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−12.2361			−0.554469			−0.277235	−	0.960802i	$-0.589418\pi$
$487$	−12.2361			−0.554469			−0.277235	+	0.960802i	$0.589418\pi$
$488$		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
$493$			35.7379i			1.60955i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−15.2251	+	24.0730i	−0.682939	+	1.07982i
$498$		0			0
$499$	−42.6525			−1.90939			−0.954694	−	0.297591i	$-0.903817\pi$
$499$	−42.6525			−1.90939			−0.954694	+	0.297591i	$0.903817\pi$
$500$		0			0
$501$	−13.0344	−	11.4682i	−0.582336	−	0.512362i
$502$		0			0
$503$	8.41622			0.375261			0.187630	−	0.982240i	$-0.439919\pi$
$503$	8.41622			0.375261			0.187630	+	0.982240i	$0.439919\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−16.7600	−	14.7461i	−0.744337	−	0.654896i
$508$		0			0
$509$	−22.4134			−0.993457			−0.496728	−	0.867906i	$-0.665465\pi$
$509$	−22.4134			−0.993457			−0.496728	+	0.867906i	$0.665465\pi$
$510$		0			0
$511$	2.00000	−	3.16228i	0.0884748	−	0.139891i
$512$		0			0
$513$	8.41641	+	5.68286i	0.371593	+	0.250904i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			8.43657i			0.371040i
$518$		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
$523$		−	8.48528i		−	0.371035i	−0.982641	−	0.185518i	$-0.940604\pi$
$523$		−	8.48528i		−	0.371035i	0.982641	−	0.185518i	$-0.0593962\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	21.5315i		−	0.937928i
$528$		0			0
$529$	16.5410			0.719175
$530$		0			0
$531$	1.37285	+	10.6947i	0.0595764	+	0.464111i
$532$		0			0
$533$			1.94150i			0.0840956i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−15.2251	+	17.3044i	−0.657011	+	0.746741i
$538$		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
$543$	−12.5623	+	14.2780i	−0.539100	+	0.612726i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−3.29180			−0.140747			−0.0703735	−	0.997521i	$-0.522419\pi$
$547$	−3.29180			−0.140747			−0.0703735	+	0.997521i	$0.522419\pi$
$548$		0			0
$549$	34.0443	−	4.37016i	1.45298	−	0.186514i
$550$		0			0
$551$	−16.5979			−0.707096
$552$		0			0
$553$	−3.29180	−	2.08191i	−0.139981	−	0.0885320i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	5.68286i		−	0.240791i	−0.992726	−	0.120395i	$-0.961584\pi$
$557$		−	5.68286i		−	0.240791i	0.992726	−	0.120395i	$-0.0384162\pi$
$558$		0			0
$559$		−	0.176228i		−	0.00745364i
$560$		0			0
$561$	5.47214	−	6.21948i	0.231034	−	0.262587i
$562$		0			0
$563$	−26.0076			−1.09609			−0.548044	−	0.836450i	$-0.684627\pi$
$563$	−26.0076			−1.09609			−0.548044	+	0.836450i	$0.684627\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	22.6868	+	7.23236i	0.952758	+	0.303730i
$568$		0			0
$569$		−	29.1559i		−	1.22228i	−0.791523	−	0.611139i	$-0.790712\pi$
$569$		−	29.1559i		−	1.22228i	0.791523	−	0.611139i	$-0.209288\pi$
$570$		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
$573$	−4.20811	+	4.78282i	−0.175796	+	0.199805i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			7.45363i			0.310299i	0.987891	+	0.155149i	$0.0495859\pi$
$577$			7.45363i			0.310299i	−0.987891	+	0.155149i	$0.950414\pi$
$578$		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
$583$	9.34752			0.387135
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−26.8560			−1.10847			−0.554233	−	0.832361i	$-0.686988\pi$
$587$	−26.8560			−1.10847			−0.554233	+	0.832361i	$0.686988\pi$
$588$		0			0
$589$	10.0000			0.412043
$590$		0			0
$591$	12.3174	−	13.9996i	0.506668	−	0.575865i
$592$		0			0
$593$	22.7928			0.935990			0.467995	−	0.883731i	$-0.344976\pi$
$593$	22.7928			0.935990			0.467995	+	0.883731i	$0.344976\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−27.5623	+	31.3265i	−1.12805	+	1.28211i
$598$		0			0
$599$		−	34.5704i		−	1.41251i	−0.707958	−	0.706255i	$-0.750383\pi$
$599$		−	34.5704i		−	1.41251i	0.707958	−	0.706255i	$-0.249617\pi$
$600$		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
$603$	−2.56231	−	19.9608i	−0.104345	−	0.812868i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			2.33695i			0.0948540i	0.998875	+	0.0474270i	$0.0151022\pi$
$607$			2.33695i			0.0948540i	−0.998875	+	0.0474270i	$0.984898\pi$
$608$		0			0
$609$	−37.3450	+	10.9529i	−1.51330	+	0.443835i
$610$		0			0
$611$			2.47811i			0.100254i
$612$		0			0
$613$	2.23607			0.0903139			0.0451570	−	0.998980i	$-0.485621\pi$
$613$	2.23607			0.0903139			0.0451570	+	0.998980i	$0.485621\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			2.54145i			0.102315i	0.998691	+	0.0511575i	$0.0162911\pi$
$617$			2.54145i			0.102315i	−0.998691	+	0.0511575i	$0.983709\pi$
$618$		0			0
$619$		−	45.5887i		−	1.83236i	−0.400763	−	0.916182i	$-0.631255\pi$
$619$		−	45.5887i		−	1.83236i	0.400763	−	0.916182i	$-0.368745\pi$
$620$		0			0
$621$	−10.9445	−	7.38987i	−0.439188	−	0.296545i
$622$		0			0
$623$	34.0443	+	21.5315i	1.36396	+	0.862643i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	2.88854	+	2.54145i	0.115357	+	0.101496i
$628$		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
$633$	−10.0236	−	8.81913i	−0.398401	−	0.350529i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−2.11146	+	1.00155i	−0.0836589	+	0.0396829i
$638$		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
$643$		−	7.40492i		−	0.292021i	−0.989283	−	0.146011i	$-0.953357\pi$
$643$		−	7.40492i		−	0.292021i	0.989283	−	0.146011i	$-0.0466434\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	37.6385			1.47972			0.739861	−	0.672760i	$-0.234891\pi$
$647$	37.6385			1.47972			0.739861	+	0.672760i	$0.234891\pi$
$648$		0			0
$649$			4.08502i			0.160351i
$650$		0			0
$651$	22.4998	−	6.59897i	0.881836	−	0.258634i
$652$		0			0
$653$		−	13.3072i		−	0.520752i	−0.965507	−	0.260376i	$-0.916154\pi$
$653$		−	13.3072i		−	0.520752i	0.965507	−	0.260376i	$-0.0838465\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−4.20811	+	0.540182i	−0.164174	+	0.0210745i
$658$		0			0
$659$		−	26.9461i		−	1.04967i	−0.851204	−	0.524835i	$-0.824127\pi$
$659$		−	26.9461i		−	1.04967i	0.851204	−	0.524835i	$-0.175873\pi$
$660$		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
$663$	1.60736	−	1.82688i	0.0624245	−	0.0709500i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	21.5836			0.835720
$668$		0			0
$669$	−19.3262	+	21.9657i	−0.747196	+	0.849242i
$670$		0			0
$671$	13.0038			0.502005
$672$		0			0
$673$	−31.3050			−1.20672			−0.603359	−	0.797470i	$-0.706171\pi$
$673$	−31.3050			−1.20672			−0.603359	+	0.797470i	$0.706171\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−31.2091			−1.19946			−0.599731	−	0.800202i	$-0.704726\pi$
$677$	−31.2091			−1.19946			−0.599731	+	0.800202i	$0.704726\pi$
$678$		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
$683$			32.2973i			1.23582i	0.786248	+	0.617911i	$0.212021\pi$
$683$			32.2973i			1.23582i	−0.786248	+	0.617911i	$0.787979\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	11.7082	−	13.3072i	0.446696	−	0.507702i
$688$		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
$693$	8.17626	+	3.81207i	0.310590	+	0.144809i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	24.4721			0.926948
$698$		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
$703$		−	4.37016i		−	0.164824i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−34.0443	−	21.5315i	−1.28037	−	0.809777i
$708$		0			0
$709$	23.3050			0.875236			0.437618	−	0.899161i	$-0.355822\pi$
$709$	23.3050			0.875236			0.437618	+	0.899161i	$0.355822\pi$
$710$		0			0
$711$	0.562306	+	4.38046i	0.0210881	+	0.164280i
$712$		0			0
$713$	−13.0038			−0.486995
$714$		0			0
$715$		0			0
$716$		0			0
$717$	12.0104	−	13.6507i	0.448536	−	0.509793i
$718$		0			0
$719$	−31.8230			−1.18680			−0.593399	−	0.804908i	$-0.702214\pi$
$719$	−31.8230			−1.18680			−0.593399	+	0.804908i	$0.702214\pi$
$720$		0			0
$721$	−21.2361	+	33.5772i	−0.790872	+	1.25048i
$722$		0			0
$723$	−25.6525	+	29.1559i	−0.954026	+	1.08432i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	30.0810i		−	1.11564i	−0.829961	−	0.557822i	$-0.811637\pi$
$727$		−	30.0810i		−	1.11564i	0.829961	−	0.557822i	$-0.188363\pi$
$728$		0			0
$729$	−10.0902	−	25.0437i	−0.373710	−	0.927546i
$730$		0			0
$731$	−2.22131			−0.0821581
$732$		0			0
$733$		−	35.1189i		−	1.29715i	−0.761152	−	0.648573i	$-0.775366\pi$
$733$		−	35.1189i		−	1.29715i	0.761152	−	0.648573i	$-0.224634\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	7.62436i		−	0.280847i
$738$		0			0
$739$	−36.5967			−1.34623			−0.673117	−	0.739536i	$-0.735045\pi$
$739$	−36.5967			−1.34623			−0.673117	+	0.739536i	$0.735045\pi$
$740$		0			0
$741$	0.848465	+	0.746512i	0.0311691	+	0.0274238i
$742$		0			0
$743$		−	13.3072i		−	0.488194i	−0.969751	−	0.244097i	$-0.921508\pi$
$743$		−	13.3072i		−	0.488194i	0.969751	−	0.244097i	$-0.0784916\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	2.22131	+	17.3044i	0.0812735	+	0.633135i
$748$		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
$753$	2.88854	+	2.54145i	0.105264	+	0.0926157i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−22.2361			−0.808184			−0.404092	−	0.914718i	$-0.632412\pi$
$757$	−22.2361			−0.808184			−0.404092	+	0.914718i	$0.632412\pi$
$758$		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
$763$	−16.7082	−	10.5672i	−0.604878	−	0.382558i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			1.19991i			0.0433263i
$768$		0			0
$769$		−	3.62365i		−	0.130672i	−0.997863	−	0.0653360i	$-0.979188\pi$
$769$		−	3.62365i		−	0.130672i	0.997863	−	0.0653360i	$-0.0208119\pi$
$770$		0			0
$771$	32.0344	+	28.1851i	1.15369	+	1.01506i
$772$		0			0
$773$	−23.4068			−0.841884			−0.420942	−	0.907087i	$-0.638300\pi$
$773$	−23.4068			−0.841884			−0.420942	+	0.907087i	$0.638300\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−2.88386	−	9.83277i	−0.103458	−	0.352749i
$778$		0			0
$779$			11.3657i			0.407219i
$780$		0			0
$781$	12.2361			0.437841
$782$		0			0
$783$	36.5726	+	24.6943i	1.30700	+	0.882502i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	7.32611i		−	0.261148i	−0.991439	−	0.130574i	$-0.958318\pi$
$787$		−	7.32611i		−	0.261148i	0.991439	−	0.130574i	$-0.0416819\pi$
$788$		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
$793$	3.81966			0.135640
$794$		0			0
$795$		0			0
$796$		0			0
$797$	34.6583			1.22766			0.613830	−	0.789438i	$-0.289628\pi$
$797$	34.6583			1.22766			0.613830	+	0.789438i	$0.289628\pi$
$798$		0			0
$799$	31.2361			1.10505
$800$		0			0
$801$	−5.81547	−	45.3035i	−0.205479	−	1.60072i
$802$		0			0
$803$	−1.60736			−0.0567223
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−41.3820	−	36.4095i	−1.45671	−	1.28167i
$808$		0			0
$809$			21.7998i			0.766442i	0.923657	+	0.383221i	$0.125185\pi$
$809$			21.7998i			0.766442i	−0.923657	+	0.383221i	$0.874815\pi$
$810$		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
$813$	25.3262	−	28.7851i	0.888230	−	1.00954i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	1.03165i		−	0.0360930i
$818$		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
$823$	17.3607			0.605155			0.302578	−	0.953125i	$-0.402153\pi$
$823$	17.3607			0.605155			0.302578	+	0.953125i	$0.402153\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			7.62436i			0.265125i	0.991175	+	0.132562i	$0.0423205\pi$
$827$			7.62436i			0.265125i	−0.991175	+	0.132562i	$0.957679\pi$
$828$		0			0
$829$		−	33.5772i		−	1.16618i	−0.812406	−	0.583092i	$-0.801843\pi$
$829$		−	33.5772i		−	1.16618i	0.812406	−	0.583092i	$-0.198157\pi$
$830$		0			0
$831$	−10.7825	−	9.48683i	−0.374040	−	0.329095i
$832$		0			0
$833$	12.6243	+	26.6144i	0.437407	+	0.922136i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−22.0344	−	14.8779i	−0.761622	−	0.514256i
$838$		0			0
$839$	−19.6677			−0.679005			−0.339502	−	0.940605i	$-0.610259\pi$
$839$	−19.6677			−0.679005			−0.339502	+	0.940605i	$0.610259\pi$
$840$		0			0
$841$	−43.1246			−1.48706
$842$		0			0
$843$	−10.7100	+	12.1727i	−0.368872	+	0.419250i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−21.7082	−	13.7295i	−0.745903	−	0.471750i
$848$		0			0
$849$	6.76393	−	7.68770i	0.232138	−	0.263841i
$850$		0			0
$851$			5.68286i			0.194806i
$852$		0			0
$853$			23.1375i			0.792213i	0.918205	+	0.396106i	$0.129639\pi$
$853$			23.1375i			0.792213i	−0.918205	+	0.396106i	$0.870361\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	24.6347			0.841506			0.420753	−	0.907175i	$-0.361766\pi$
$857$	24.6347			0.841506			0.420753	+	0.907175i	$0.361766\pi$
$858$		0			0
$859$			52.6598i			1.79673i	0.439252	+	0.898364i	$0.355243\pi$
$859$			52.6598i			1.79673i	−0.439252	+	0.898364i	$0.644757\pi$
$860$		0			0
$861$	7.50021	+	25.5726i	0.255606	+	0.871513i
$862$		0			0
$863$		−	40.5216i		−	1.37937i	−0.724109	−	0.689686i	$-0.757749\pi$
$863$		−	40.5216i		−	1.37937i	0.724109	−	0.689686i	$-0.242251\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.920933	−	0.810272i	−0.0312765	−	0.0275183i
$868$		0			0
$869$			1.67319i			0.0567590i
$870$		0			0
$871$		−	2.23954i		−	0.0758838i
$872$		0			0
$873$	28.8428	−	3.70246i	0.976182	−	0.125309i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	15.5279			0.524339			0.262169	−	0.965022i	$-0.415562\pi$
$877$	15.5279			0.524339			0.262169	+	0.965022i	$0.415562\pi$
$878$		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
$883$	7.11146			0.239320			0.119660	−	0.992815i	$-0.461820\pi$
$883$	7.11146			0.239320			0.119660	+	0.992815i	$0.461820\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	47.8966			1.60821			0.804105	−	0.594487i	$-0.202645\pi$
$887$	47.8966			1.60821			0.804105	+	0.594487i	$0.202645\pi$
$888$		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
$893$			14.5071i			0.485463i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−1.10333	−	0.970748i	−0.0368390	−	0.0324123i
$898$		0			0
$899$	43.4540			1.44927
$900$		0			0
$901$		−	34.6088i		−	1.15299i
$902$		0			0
$903$	−0.680786	−	2.32120i	−0.0226551	−	0.0772448i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	21.3050			0.707419			0.353710	−	0.935355i	$-0.384920\pi$
$907$	21.3050			0.707419			0.353710	+	0.935355i	$0.384920\pi$
$908$		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
$913$			6.60970i			0.218749i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	21.0406	+	13.3072i	0.694820	+	0.439443i
$918$		0			0
$919$	−34.3050			−1.13162			−0.565808	−	0.824537i	$-0.691436\pi$
$919$	−34.3050			−1.13162			−0.565808	+	0.824537i	$0.691436\pi$
$920$		0			0
$921$	29.1803	−	33.1656i	0.961525	−	1.09284i
$922$		0			0
$923$	3.59416			0.118303
$924$		0			0
$925$		0			0
$926$		0			0
$927$	44.6819	−	5.73567i	1.46755	−	0.188384i
$928$		0			0
$929$	−15.2251			−0.499519			−0.249760	−	0.968308i	$-0.580352\pi$
$929$	−15.2251			−0.499519			−0.249760	+	0.968308i	$0.580352\pi$
$930$		0			0
$931$	−12.3607	+	5.86319i	−0.405105	+	0.192158i
$932$		0			0
$933$	2.88854	+	2.54145i	0.0945667	+	0.0832034i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	44.3020i		−	1.44728i	−0.690176	−	0.723641i	$-0.742467\pi$
$937$		−	44.3020i		−	1.44728i	0.690176	−	0.723641i	$-0.257533\pi$
$938$		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
$943$		−	14.7797i		−	0.481294i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	6.28282i		−	0.204164i	−0.994776	−	0.102082i	$-0.967450\pi$
$947$		−	6.28282i		−	0.204164i	0.994776	−	0.102082i	$-0.0325504\pi$
$948$		0			0
$949$	−0.472136			−0.0153262
$950$		0			0
$951$	33.3579	−	37.9137i	1.08170	−	1.22944i
$952$		0			0
$953$		−	47.5460i		−	1.54017i	−0.637944	−	0.770083i	$-0.720215\pi$
$953$		−	47.5460i		−	1.54017i	0.637944	−	0.770083i	$-0.279785\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	12.5519	+	11.0436i	0.405744	+	0.356989i
$958$		0			0
$959$	−30.4502	+	48.1460i	−0.983288	+	1.55471i
$960$		0			0
$961$	4.81966			0.155473
$962$		0			0
$963$	15.1246	−	1.94150i	0.487384	−	0.0625639i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−44.0689			−1.41716			−0.708580	−	0.705631i	$-0.750664\pi$
$967$	−44.0689			−1.41716			−0.708580	+	0.705631i	$0.750664\pi$
$968$		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
$973$	−1.05573	+	1.66925i	−0.0338451	+	0.0535138i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	7.62436i		−	0.243925i	−0.992535	−	0.121962i	$-0.961081\pi$
$977$		−	7.62436i		−	0.243925i	0.992535	−	0.121962i	$-0.0389187\pi$
$978$		0			0
$979$		−	17.3044i		−	0.553051i
$980$		0			0
$981$	2.85410	+	22.2340i	0.0911245	+	0.709876i
$982$		0			0
$983$	−4.58756			−0.146320			−0.0731602	−	0.997320i	$-0.523308\pi$
$983$	−4.58756			−0.146320			−0.0731602	+	0.997320i	$0.523308\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	9.57321	+	32.6407i	0.304719	+	1.03897i
$988$		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
$993$	−37.9455	−	33.3859i	−1.20416	−	1.05947i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	60.1134i		−	1.90381i	−0.306394	−	0.951905i	$-0.599122\pi$
$997$		−	60.1134i		−	1.90381i	0.306394	−	0.951905i	$-0.400878\pi$
$998$		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.d.k.1301.3	✓	8
3.2	odd	2	inner	2100.2.d.k.1301.5	yes	8
5.2	odd	4		2100.2.f.i.1049.3		16
5.3	odd	4		2100.2.f.i.1049.14		16
5.4	even	2		2100.2.d.l.1301.6	yes	8
7.6	odd	2	inner	2100.2.d.k.1301.6	yes	8
15.2	even	4		2100.2.f.i.1049.1		16
15.8	even	4		2100.2.f.i.1049.16		16
15.14	odd	2		2100.2.d.l.1301.4	yes	8
21.20	even	2	inner	2100.2.d.k.1301.4	yes	8
35.13	even	4		2100.2.f.i.1049.4		16
35.27	even	4		2100.2.f.i.1049.13		16
35.34	odd	2		2100.2.d.l.1301.3	yes	8
105.62	odd	4		2100.2.f.i.1049.15		16
105.83	odd	4		2100.2.f.i.1049.2		16
105.104	even	2		2100.2.d.l.1301.5	yes	8

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

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.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.30038 - 1.14412i) q^{3} +(-2.23607 - 1.41421i) q^{7} +(0.381966 + 2.97558i) q^{9} +O(q^{10})\)	\(q+(-1.30038 - 1.14412i) q^{3} +(-2.23607 - 1.41421i) q^{7} +(0.381966 + 2.97558i) q^{9} +1.13657i q^{11} +0.333851i q^{13} +4.20811 q^{17} +1.95440i q^{19} +(1.28970 + 4.39735i) q^{21} -2.54145i q^{23} +(2.90773 - 4.30640i) q^{27} +8.49262i q^{29} -5.11667i q^{31} +(1.30038 - 1.47797i) q^{33} -2.23607 q^{37} +(0.381966 - 0.434132i) q^{39} +5.81547 q^{41} -0.527864 q^{43} +7.42282 q^{47} +(3.00000 + 6.32456i) q^{49} +(-5.47214 - 4.81460i) q^{51} -8.22431i q^{53} +(2.23607 - 2.54145i) q^{57} +3.59416 q^{59} -11.4412i q^{61} +(3.35401 - 7.19379i) q^{63} -6.70820 q^{67} +(-2.90773 + 3.30485i) q^{69} -10.7658i q^{71} +1.41421i q^{73} +(1.60736 - 2.54145i) q^{77} +1.47214 q^{79} +(-8.70820 + 2.27314i) q^{81} +5.81547 q^{83} +(9.71660 - 11.0436i) q^{87} -15.2251 q^{89} +(0.472136 - 0.746512i) q^{91} +(-5.85410 + 6.65361i) q^{93} -9.69316i q^{97} +(-3.38197 + 0.434132i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{9}+O(q^{10}) \) 8 * q + 12 * q^9	\( 8 q + 12 q^{9} - 4 q^{21} + 12 q^{39} - 40 q^{43} + 24 q^{49} - 8 q^{51} + 20 q^{63} - 24 q^{79} - 16 q^{81} - 32 q^{91} - 20 q^{93} - 36 q^{99}+O(q^{100}) \) 8 * q + 12 * q^9 - 4 * q^21 + 12 * q^39 - 40 * q^43 + 24 * q^49 - 8 * q^51 + 20 * q^63 - 24 * q^79 - 16 * q^81 - 32 * q^91 - 20 * q^93 - 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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