LMFDB - Embedded newform 429.2.e.d.428.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [429,2,Mod(428,429)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(429, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("429.428");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 429 = 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	429.e (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:	$3.42558224671$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 53x^{10} + 1024 $ x^20 - 53*x^10 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			428.10
Root			$1.41171 + 0.0841020i$ of defining polynomial
Character	$\chi$	$=$	429.428
Dual form			429.2.e.d.428.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.168204i q^{2} +(1.20277 + 1.24634i) q^{3} +1.97171 q^{4} +(0.209639 - 0.202310i) q^{6} -2.38069 q^{7} -0.668057i q^{8} +(-0.106712 + 2.99810i) q^{9} +O(q^{10})$	\(q-0.168204i q^{2} +(1.20277 + 1.24634i) q^{3} +1.97171 q^{4} +(0.209639 - 0.202310i) q^{6} -2.38069 q^{7} -0.668057i q^{8} +(-0.106712 + 2.99810i) q^{9} +3.31662i q^{11} +(2.37150 + 2.45741i) q^{12} +3.60555 q^{13} +0.400441i q^{14} +3.83105 q^{16} +(0.504293 + 0.0179494i) q^{18} +2.62042 q^{19} +(-2.86341 - 2.96714i) q^{21} +0.557869 q^{22} -5.04638i q^{23} +(0.832624 - 0.803516i) q^{24} -5.00000 q^{25} -0.606468i q^{26} +(-3.86500 + 3.47301i) q^{27} -4.69402 q^{28} -1.98051i q^{32} +(-4.13363 + 3.98912i) q^{33} +(-0.210406 + 5.91138i) q^{36} -0.440765i q^{38} +(4.33663 + 4.49373i) q^{39} -12.1682i q^{41} +(-0.499084 + 0.481636i) q^{42} +6.53941i q^{44} -0.848821 q^{46} +(4.60785 + 4.77477i) q^{48} -1.33233 q^{49} +0.841020i q^{50} +7.10909 q^{52} -6.26509i q^{53} +(0.584174 + 0.650108i) q^{54} +1.59043i q^{56} +(3.15175 + 3.26593i) q^{57} +(0.254049 - 7.13754i) q^{63} +7.32896 q^{64} +(0.670986 + 0.695293i) q^{66} +(6.28949 - 6.06961i) q^{69} +(2.00290 + 0.0712899i) q^{72} -14.3201 q^{73} +(-6.01383 - 6.23169i) q^{75} +5.16671 q^{76} -7.89585i q^{77} +(0.755864 - 0.729439i) q^{78} +(-8.97722 - 0.639869i) q^{81} -2.04674 q^{82} +10.1593i q^{83} +(-5.64580 - 5.85033i) q^{84} +2.21569 q^{88} -8.58369 q^{91} -9.94999i q^{92} +(2.46838 - 2.38209i) q^{96} +0.224103i q^{98} +(-9.94358 - 0.353925i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 40 q^{4}+O(q^{10}) $ 20 * q - 40 * q^4	$ 20 q - 40 q^{4} + 80 q^{16} - 100 q^{25} + 10 q^{36} - 50 q^{42} + 70 q^{48} + 140 q^{49} - 160 q^{64} - 110 q^{66} + 130 q^{78}+O(q^{100}) $ 20 * q - 40 * q^4 + 80 * q^16 - 100 * q^25 + 10 * q^36 - 50 * q^42 + 70 * q^48 + 140 * q^49 - 160 * q^64 - 110 * q^66 + 130 * q^78

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/429\mathbb{Z}\right)^\times$.

$n$	$67$	$79$	$287$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$		−	0.168204i		−	0.118938i	−0.998230	−	0.0594691i	$-0.981059\pi$
$2$		−	0.168204i		−	0.118938i	0.998230	−	0.0594691i	$-0.0189408\pi$
$3$	1.20277	+	1.24634i	0.694417	+	0.719573i
$3$	1.20277	+	1.24634i	0.694417	+	0.719573i
$4$	1.97171			0.985854
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	0.209639	−	0.202310i	0.0855847	−	0.0825927i
$7$	−2.38069			−0.899815			−0.449908	−	0.893075i	$-0.648543\pi$
$7$	−2.38069			−0.899815			−0.449908	+	0.893075i	$0.648543\pi$
$8$		−	0.668057i		−	0.236194i
$9$	−0.106712	+	2.99810i	−0.0355708	+	0.999367i
$10$		0			0
$11$			3.31662i			1.00000i
$11$			3.31662i			1.00000i
$12$	2.37150	+	2.45741i	0.684593	+	0.709394i
$13$	3.60555			1.00000
$13$	3.60555			1.00000
$14$			0.400441i			0.107022i
$15$		0			0
$16$	3.83105			0.957761
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	0.504293	+	0.0179494i	0.118863	+	0.00423072i
$19$	2.62042			0.601166			0.300583	−	0.953756i	$-0.402819\pi$
$19$	2.62042			0.601166			0.300583	+	0.953756i	$0.402819\pi$
$20$		0			0
$21$	−2.86341	−	2.96714i	−0.624847	−	0.647483i
$22$	0.557869			0.118938
$23$		−	5.04638i		−	1.05224i	−0.850409	−	0.526122i	$-0.823646\pi$
$23$		−	5.04638i		−	1.05224i	0.850409	−	0.526122i	$-0.176354\pi$
$24$	0.832624	−	0.803516i	0.169959	−	0.164017i
$25$	−5.00000			−1.00000
$26$		−	0.606468i		−	0.118938i
$27$	−3.86500	+	3.47301i	−0.743819	+	0.668381i
$28$	−4.69402			−0.887086
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		−	1.98051i		−	0.350108i
$33$	−4.13363	+	3.98912i	−0.719573	+	0.694417i
$34$		0			0
$35$		0			0
$36$	−0.210406	+	5.91138i	−0.0350676	+	0.985230i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		−	0.440765i		−	0.0715016i
$39$	4.33663	+	4.49373i	0.694417	+	0.719573i
$40$		0			0
$41$		−	12.1682i		−	1.90035i	−0.311712	−	0.950177i	$-0.600902\pi$
$41$		−	12.1682i		−	1.90035i	0.311712	−	0.950177i	$-0.399098\pi$
$42$	−0.499084	+	0.481636i	−0.0770104	+	0.0743181i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$			6.53941i			0.985854i
$45$		0			0
$46$	−0.848821			−0.125152
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	4.60785	+	4.77477i	0.665085	+	0.689179i
$49$	−1.33233			−0.190333
$50$			0.841020i			0.118938i
$51$		0			0
$52$	7.10909			0.985854
$53$		−	6.26509i		−	0.860576i	−0.902692	−	0.430288i	$-0.858412\pi$
$53$		−	6.26509i		−	0.860576i	0.902692	−	0.430288i	$-0.141588\pi$
$54$	0.584174	+	0.650108i	0.0794961	+	0.0884684i
$55$		0			0
$56$			1.59043i			0.212531i
$57$	3.15175	+	3.26593i	0.417460	+	0.432583i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	0.254049	−	7.13754i	0.0320071	−	0.899246i
$64$	7.32896			0.916120
$65$		0			0
$66$	0.670986	+	0.695293i	0.0825927	+	0.0855847i
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	6.28949	−	6.06961i	0.757166	−	0.730695i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	2.00290	+	0.0712899i	0.236044	+	0.00840160i
$73$	−14.3201			−1.67604			−0.838021	−	0.545638i	$-0.816287\pi$
$73$	−14.3201			−1.67604			−0.838021	+	0.545638i	$0.816287\pi$
$74$		0			0
$75$	−6.01383	−	6.23169i	−0.694417	−	0.719573i
$76$	5.16671			0.592662
$77$		−	7.89585i		−	0.899815i
$78$	0.755864	−	0.729439i	0.0855847	−	0.0825927i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−8.97722	−	0.639869i	−0.997469	−	0.0710966i
$82$	−2.04674			−0.226025
$83$			10.1593i			1.11512i	0.830135	+	0.557562i	$0.188263\pi$
$83$			10.1593i			1.11512i	−0.830135	+	0.557562i	$0.811737\pi$
$84$	−5.64580	−	5.85033i	−0.616007	−	0.638323i
$85$		0			0
$86$		0			0
$87$		0			0
$88$	2.21569			0.236194
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	−8.58369			−0.899815
$92$		−	9.94999i		−	1.03736i
$93$		0			0
$94$		0			0
$95$		0			0
$96$	2.46838	−	2.38209i	0.251928	−	0.243121i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$			0.224103i			0.0226378i
$99$	−9.94358	−	0.353925i	−0.999367	−	0.0355708i
$100$	−9.85854			−0.985854
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	−7.30314			−0.719600			−0.359800	−	0.933029i	$-0.617155\pi$
$103$	−7.30314			−0.719600			−0.359800	+	0.933029i	$0.617155\pi$
$104$		−	2.40871i		−	0.236194i
$105$		0			0
$106$	−1.05381			−0.102355
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−7.62064	+	6.84776i	−0.733296	+	0.658926i
$109$	19.5610			1.87360			0.936800	−	0.349866i	$-0.113773\pi$
$109$	19.5610			1.87360			0.936800	+	0.349866i	$0.113773\pi$
$110$		0			0
$111$		0			0
$112$	−9.12052			−0.861808
$113$		−	21.2211i		−	1.99632i	−0.0606667	−	0.998158i	$-0.519323\pi$
$113$		−	21.2211i		−	1.99632i	0.0606667	−	0.998158i	$-0.480677\pi$
$114$	0.549342	−	0.530137i	0.0514506	−	0.0496519i
$115$		0			0
$116$		0			0
$117$	−0.384757	+	10.8098i	−0.0355708	+	0.999367i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	15.1657	−	14.6355i	1.36744	−	1.31964i
$124$		0			0
$125$		0			0
$126$	−1.20056	−	0.0427320i	−0.106955	−	0.00380687i
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		−	5.19378i		−	0.459070i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	−8.15031	+	7.86538i	−0.709394	+	0.684593i
$133$	−6.23841			−0.540938
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$	−1.02093	−	1.05792i	−0.0869076	−	0.0900559i
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			11.9583i			1.00000i
$144$	−0.408820	+	11.4859i	−0.0340683	+	0.957155i
$145$		0			0
$146$			2.40870i			0.199345i
$147$	−1.60248	−	1.66053i	−0.132170	−	0.136958i
$148$		0			0
$149$			16.1861i			1.32601i	0.748613	+	0.663007i	$0.230720\pi$
$149$			16.1861i			1.32601i	−0.748613	+	0.663007i	$0.769280\pi$
$150$	−1.04819	+	1.01155i	−0.0855847	+	0.0825927i
$151$	−14.4222			−1.17366			−0.586831	−	0.809709i	$-0.699625\pi$
$151$	−14.4222			−1.17366			−0.586831	+	0.809709i	$0.699625\pi$
$152$		−	1.75059i		−	0.141992i
$153$		0			0
$154$	−1.32811			−0.107022
$155$		0			0
$156$	8.55057	+	8.86033i	0.684593	+	0.709394i
$157$	15.2466			1.21681			0.608404	−	0.793628i	$-0.291810\pi$
$157$	15.2466			1.21681			0.608404	+	0.793628i	$0.291810\pi$
$158$		0			0
$159$	7.80841	−	7.53543i	0.619247	−	0.597598i
$160$		0			0
$161$			12.0139i			0.946824i
$162$	−0.107629	+	1.51000i	−0.00845609	+	0.118637i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		−	23.9921i		−	1.87347i
$165$		0			0
$166$	1.70883			0.132631
$167$		−	25.6168i		−	1.98228i	−0.132804	−	0.991142i	$-0.542398\pi$
$167$		−	25.6168i		−	1.98228i	0.132804	−	0.991142i	$-0.457602\pi$
$168$	−1.98222	+	1.91292i	−0.152931	+	0.147585i
$169$	13.0000			1.00000
$170$		0			0
$171$	−0.279632	+	7.85629i	−0.0213840	+	0.600786i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	11.9034			0.899815
$176$			12.7061i			0.957761i
$177$		0			0
$178$		0			0
$179$			23.9165i			1.78760i	0.448461	+	0.893802i	$0.351972\pi$
$179$			23.9165i			1.78760i	−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	6.10085			0.453473			0.226736	−	0.973956i	$-0.427194\pi$
$181$	6.10085			0.453473			0.226736	+	0.973956i	$0.427194\pi$
$182$			1.44381i			0.107022i
$183$		0			0
$184$	−3.37127			−0.248533
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	9.20134	−	8.26815i	0.669299	−	0.601420i
$190$		0			0
$191$			20.0082i			1.44774i	0.689935	+	0.723871i	$0.257639\pi$
$191$			20.0082i			1.44774i	−0.689935	+	0.723871i	$0.742361\pi$
$192$	8.81502	+	9.13436i	0.636169	+	0.659215i
$193$	−21.4983			−1.54748			−0.773742	−	0.633501i	$-0.781617\pi$
$193$	−21.4983			−1.54748			−0.773742	+	0.633501i	$0.781617\pi$
$194$		0			0
$195$		0			0
$196$	−2.62696			−0.187640
$197$		−	24.6075i		−	1.75321i	−0.481207	−	0.876607i	$-0.659801\pi$
$197$		−	24.6075i		−	1.75321i	0.481207	−	0.876607i	$-0.340199\pi$
$198$	−0.0595316	+	1.67255i	−0.00423072	+	0.118863i
$199$	23.8812			1.69290			0.846448	−	0.532471i	$-0.178736\pi$
$199$	23.8812			1.69290			0.846448	+	0.532471i	$0.178736\pi$
$200$			3.34028i			0.236194i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$			1.22842i			0.0855879i
$207$	15.1296	+	0.538511i	1.05158	+	0.0374291i
$208$	13.8130			0.957761
$209$			8.69096i			0.601166i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		−	12.3529i		−	0.848402i
$213$		0			0
$214$		0			0
$215$		0			0
$216$	2.32017	+	2.58204i	0.157868	+	0.175685i
$217$		0			0
$218$		−	3.29023i		−	0.222842i
$219$	−17.2237	−	17.8477i	−1.16387	−	1.20603i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$			4.71498i			0.315033i
$225$	0.533562	−	14.9905i	0.0355708	−	0.999367i
$226$	−3.56948			−0.237438
$227$			26.2896i			1.74490i	0.488703	+	0.872450i	$0.337470\pi$
$227$			26.2896i			1.74490i	−0.488703	+	0.872450i	$0.662530\pi$
$228$	6.21433	+	6.43946i	0.411554	+	0.426464i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$	9.84089	−	9.49685i	0.647483	−	0.624847i
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$	1.81825	+	0.0647176i	0.118863	+	0.00423072i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			24.2711i			1.56997i	0.619515	+	0.784984i	$0.287329\pi$
$239$			24.2711i			1.56997i	−0.619515	+	0.784984i	$0.712671\pi$
$240$		0			0
$241$	31.0211			1.99824			0.999122	−	0.0419050i	$-0.0133427\pi$
$241$	31.0211			1.99824			0.999122	+	0.0419050i	$0.0133427\pi$
$242$			1.85024i			0.118938i
$243$	−10.0000	−	11.9583i	−0.641500	−	0.767123i
$244$		0			0
$245$		0			0
$246$	−2.46175	−	2.55093i	−0.156955	−	0.162641i
$247$	9.44807			0.601166
$248$		0			0
$249$	−12.6619	+	12.2192i	−0.802414	+	0.774361i
$250$		0			0
$251$			30.9308i			1.95234i	0.217014	+	0.976169i	$0.430368\pi$
$251$			30.9308i			1.95234i	−0.217014	+	0.976169i	$0.569632\pi$
$252$	0.500910	−	14.0731i	0.0315544	−	0.886525i
$253$	16.7370			1.05224
$254$		0			0
$255$		0			0
$256$	13.7843			0.861519
$257$		−	2.19692i		−	0.137040i	−0.997650	−	0.0685200i	$-0.978172\pi$
$257$		−	2.19692i		−	0.137040i	0.997650	−	0.0685200i	$-0.0218277\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$	2.66496	+	2.76150i	0.164017	+	0.169959i
$265$		0			0
$266$			1.04932i			0.0643382i
$267$		0			0
$268$		0			0
$269$			28.0814i			1.71215i	0.516850	+	0.856076i	$0.327104\pi$
$269$			28.0814i			1.71215i	−0.516850	+	0.856076i	$0.672896\pi$
$270$		0			0
$271$	−31.2606			−1.89895			−0.949474	−	0.313845i	$-0.898383\pi$
$271$	−31.2606			−1.89895			−0.949474	+	0.313845i	$0.898383\pi$
$272$		0			0
$273$	−10.3242	−	10.6982i	−0.624847	−	0.647483i
$274$		0			0
$275$		−	16.5831i		−	1.00000i
$276$	12.4010	−	11.9675i	0.746455	−	0.720359i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	26.6260i		−	1.58837i	−0.607674	−	0.794187i	$-0.707897\pi$
$281$		−	26.6260i		−	1.58837i	0.607674	−	0.794187i	$-0.292103\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$	2.01143			0.118938
$287$			28.9687i			1.70997i
$288$	5.93777	+	0.211345i	0.349887	+	0.0124536i
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$	−28.2351			−1.65233
$293$			26.5330i			1.55007i	0.631916	+	0.775037i	$0.282269\pi$
$293$			26.5330i			1.55007i	−0.631916	+	0.775037i	$0.717731\pi$
$294$	−0.279308	+	0.269543i	−0.0162896	+	0.0157201i
$295$		0			0
$296$		0			0
$297$	−11.5187	−	12.8187i	−0.668381	−	0.743819i
$298$	2.72256			0.157714
$299$		−	18.1950i		−	1.05224i
$300$	−11.8575	−	12.2871i	−0.684593	−	0.709394i
$301$		0			0
$302$			2.42587i			0.139593i
$303$		0			0
$304$	10.0390			0.575774
$305$		0			0
$306$		0			0
$307$	28.8444			1.64624			0.823119	−	0.567869i	$-0.192232\pi$
$307$	28.8444			1.64624			0.823119	+	0.567869i	$0.192232\pi$
$308$		−	15.5683i		−	0.887086i
$309$	−8.78396	−	9.10218i	−0.499702	−	0.517805i
$310$		0			0
$311$			34.9642i			1.98264i	0.131470	+	0.991320i	$0.458030\pi$
$311$			34.9642i			1.98264i	−0.131470	+	0.991320i	$0.541970\pi$
$312$	3.00207	−	2.89712i	0.169959	−	0.164017i
$313$	−32.4139			−1.83214			−0.916072	−	0.401014i	$-0.868658\pi$
$313$	−32.4139			−1.83214			−0.916072	+	0.401014i	$0.868658\pi$
$314$		−	2.56453i		−	0.144725i
$315$		0			0
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$	−1.26749	−	1.31341i	−0.0710772	−	0.0736521i
$319$		0			0
$320$		0			0
$321$		0			0
$322$	2.02078			0.112614
$323$		0			0
$324$	−17.7005	−	1.26163i	−0.983359	−	0.0700908i
$325$	−18.0278			−1.00000
$326$		0			0
$327$	23.5272	+	24.3795i	1.30106	+	1.34819i
$328$	−8.12905			−0.448852
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$			20.0311i			1.09935i
$333$		0			0
$334$	−4.30884			−0.235769
$335$		0			0
$336$	−10.9698	−	11.3672i	−0.598454	−	0.620134i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		−	2.18665i		−	0.118938i
$339$	26.4487	−	25.5240i	1.43650	−	1.38628i
$340$		0			0
$341$		0			0
$342$	1.32146	+	0.0470351i	0.0714564	+	0.00254337i
$343$	19.8367			1.07108
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$	−35.7825			−1.91539			−0.957695	−	0.287784i	$-0.907081\pi$
$349$	−35.7825			−1.91539			−0.957695	+	0.287784i	$0.907081\pi$
$350$		−	2.00221i		−	0.107022i
$351$	−13.9354	+	12.5221i	−0.743819	+	0.668381i
$352$	6.56861			0.350108
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	4.02285			0.212614
$359$		−	6.14141i		−	0.324131i	−0.986780	−	0.162066i	$-0.948184\pi$
$359$		−	6.14141i		−	0.324131i	0.986780	−	0.162066i	$-0.0518156\pi$
$360$		0			0
$361$	−12.1334			−0.638599
$362$		−	1.02619i		−	0.0539352i
$363$	−13.2304	−	13.7097i	−0.694417	−	0.719573i
$364$	−16.9245			−0.887086
$365$		0			0
$366$		0			0
$367$	38.3144			2.00000			0.999998	−	0.00174352i	$-0.000554980\pi$
$367$	38.3144			2.00000			0.999998	+	0.00174352i	$0.000554980\pi$
$368$		−	19.3329i		−	1.00780i
$369$	36.4815	+	1.29850i	1.89915	+	0.0675971i
$370$		0			0
$371$			14.9152i			0.774359i
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$	−1.39074	−	1.54770i	−0.0715318	−	0.0796052i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$	3.36546			0.172192
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	6.47320	−	6.24690i	0.330334	−	0.318786i
$385$		0			0
$386$			3.61610i			0.184055i
$387$		0			0
$388$		0			0
$389$		−	36.1772i		−	1.83426i	−0.398593	−	0.917128i	$-0.630501\pi$
$389$		−	36.1772i		−	1.83426i	0.398593	−	0.917128i	$-0.369499\pi$
$390$		0			0
$391$		0			0
$392$			0.890072i			0.0449554i
$393$		0			0
$394$	−4.13909			−0.208524
$395$		0			0
$396$	−19.6058	−	0.697836i	−0.985230	−	0.0350676i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		−	4.01692i		−	0.201350i
$399$	−7.50334	−	7.77516i	−0.375637	−	0.389245i
$400$	−19.1552			−0.957761
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	7.21110			0.356566			0.178283	−	0.983979i	$-0.442946\pi$
$409$	7.21110			0.356566			0.178283	+	0.983979i	$0.442946\pi$
$410$		0			0
$411$		0			0
$412$	−14.3997			−0.709420
$413$		0			0
$414$	0.0905798	−	2.54485i	0.00445175	−	0.125073i
$415$		0			0
$416$		−	7.14083i		−	0.350108i
$417$		0			0
$418$	1.46185			0.0715016
$419$		−	10.7453i		−	0.524943i	−0.964940	−	0.262471i	$-0.915462\pi$
$419$		−	10.7453i		−	0.524943i	0.964940	−	0.262471i	$-0.0845375\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	−4.18543			−0.203263
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−14.9040	+	14.3830i	−0.719573	+	0.694417i
$430$		0			0
$431$		−	34.4957i		−	1.66160i	−0.556573	−	0.830799i	$-0.687884\pi$
$431$		−	34.4957i		−	1.66160i	0.556573	−	0.830799i	$-0.312116\pi$
$432$	−14.8070	+	13.3053i	−0.712401	+	0.640150i
$433$	−37.1987			−1.78766			−0.893828	−	0.448411i	$-0.851990\pi$
$433$	−37.1987			−1.78766			−0.893828	+	0.448411i	$0.851990\pi$
$434$		0			0
$435$		0			0
$436$	38.5685			1.84709
$437$		−	13.2237i		−	0.632573i
$438$	−3.00205	+	2.89710i	−0.143444	+	0.138429i
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	0.142176	−	3.99446i	0.00677029	−	0.190212i
$442$		0			0
$443$		−	41.0236i		−	1.94909i	−0.224192	−	0.974545i	$-0.571974\pi$
$443$		−	41.0236i		−	1.94909i	0.224192	−	0.974545i	$-0.428026\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−20.1733	+	19.4680i	−0.954164	+	0.920806i
$448$	−17.4480			−0.824339
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	−2.52146	−	0.0897472i	−0.118863	−	0.00423072i
$451$	40.3573			1.90035
$452$		−	41.8419i		−	1.96808i
$453$	−17.3465	−	17.9749i	−0.815011	−	0.844536i
$454$	4.42201			0.207535
$455$		0			0
$456$	2.18183	−	2.10555i	0.102173	−	0.0986014i
$457$	3.83841			0.179553			0.0897767	−	0.995962i	$-0.471385\pi$
$457$	3.83841			0.179553			0.0897767	+	0.995962i	$0.471385\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			23.2619i			1.08342i	0.840567	+	0.541708i	$0.182222\pi$
$461$			23.2619i			1.08342i	−0.840567	+	0.541708i	$0.817778\pi$
$462$	−1.59741	−	1.65528i	−0.0743181	−	0.0770104i
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			23.9165i			1.10672i	0.832941	+	0.553362i	$0.186655\pi$
$467$			23.9165i			1.10672i	−0.832941	+	0.553362i	$0.813345\pi$
$468$	−0.758628	+	21.3138i	−0.0350676	+	0.985230i
$469$		0			0
$470$		0			0
$471$	18.3380	+	19.0023i	0.844972	+	0.875582i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−13.1021			−0.601166
$476$		0			0
$477$	18.7834	+	0.668562i	0.860031	+	0.0306114i
$478$	4.08250			0.186729
$479$			6.63325i			0.303081i	0.988451	+	0.151540i	$0.0484234\pi$
$479$			6.63325i			0.303081i	−0.988451	+	0.151540i	$0.951577\pi$
$480$		0			0
$481$		0			0
$482$		−	5.21787i		−	0.237667i
$483$	−14.9733	+	14.4498i	−0.681309	+	0.657491i
$484$	−21.6888			−0.985854
$485$		0			0
$486$	−2.01143	+	1.68204i	−0.0912402	+	0.0762989i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$	29.9023	−	28.8569i	1.34810	−	1.30097i
$493$		0			0
$494$		−	1.58920i		−	0.0715016i
$495$		0			0
$496$		0			0
$497$		0			0
$498$	2.05532	+	2.12978i	0.0921011	+	0.0954376i
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$	31.9271	−	30.8109i	1.42640	−	1.37653i
$502$	5.20269			0.232207
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	−4.76828	−	0.169719i	−0.212396	−	0.00755989i
$505$		0			0
$506$		−	2.81522i		−	0.125152i
$507$	15.6359	+	16.2024i	0.694417	+	0.719573i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	34.0917			1.50813
$512$		−	12.7061i		−	0.561537i
$513$	−10.1279	+	9.10076i	−0.447159	+	0.401808i
$514$	−0.369530			−0.0162993
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			33.7803i			1.47994i	0.672639	+	0.739971i	$0.265161\pi$
$521$			33.7803i			1.47994i	−0.672639	+	0.739971i	$0.734839\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$	14.3170	+	14.8357i	0.624847	+	0.647483i
$526$		0			0
$527$		0			0
$528$	−15.8361	+	15.2825i	−0.689179	+	0.665085i
$529$	−2.46597			−0.107216
$530$		0			0
$531$		0			0
$532$	−12.3003			−0.533286
$533$		−	43.8731i		−	1.90035i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−29.8080	+	28.7660i	−1.28631	+	1.24134i
$538$	4.72340			0.203640
$539$		−	4.41884i		−	0.190333i
$540$		0			0
$541$	−7.21421			−0.310163			−0.155081	−	0.987902i	$-0.549564\pi$
$541$	−7.21421			−0.310163			−0.155081	+	0.987902i	$0.549564\pi$
$542$			5.25816i			0.225857i
$543$	7.33789	+	7.60372i	0.314899	+	0.326307i
$544$		0			0
$545$		0			0
$546$	−1.79947	+	1.73656i	−0.0770104	+	0.0743181i
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$	−2.78935			−0.118938
$551$		0			0
$552$	−4.05485	−	4.20174i	−0.172586	−	0.178838i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$			32.4867i			1.37651i	0.725470	+	0.688254i	$0.241622\pi$
$557$			32.4867i			1.37651i	−0.725470	+	0.688254i	$0.758378\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−4.47860			−0.188918
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	21.3720	+	1.52333i	0.897538	+	0.0639738i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$			23.5782i			0.985854i
$573$	−24.9370	+	24.0652i	−1.04176	+	1.00534i
$574$	4.87265			0.203380
$575$			25.2319i			1.05224i
$576$	−0.782091	+	21.9730i	−0.0325871	+	0.915540i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$			2.85947i			0.118938i
$579$	−25.8574	−	26.7942i	−1.07460	−	1.11353i
$580$		0			0
$581$		−	24.1860i		−	1.00341i
$582$		0			0
$583$	20.7789			0.860576
$584$			9.56665i			0.395871i
$585$		0			0
$586$	4.46296			0.184363
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−3.15962	−	3.27408i	−0.130301	−	0.135021i
$589$		0			0
$590$		0			0
$591$	30.6693	−	29.5971i	1.26157	−	1.21746i
$592$		0			0
$593$			4.13248i			0.169701i	0.996394	+	0.0848503i	$0.0270412\pi$
$593$			4.13248i			0.169701i	−0.996394	+	0.0848503i	$0.972959\pi$
$594$	−2.15616	+	1.93749i	−0.0884684	+	0.0794961i
$595$		0			0
$596$			31.9142i			1.30726i
$597$	28.7235	+	29.7641i	1.17558	+	1.21816i
$598$	−3.06047			−0.125152
$599$		−	9.90389i		−	0.404662i	−0.979317	−	0.202331i	$-0.935148\pi$
$599$		−	9.90389i		−	0.404662i	0.979317	−	0.202331i	$-0.0648517\pi$
$600$	−4.16312	+	4.01758i	−0.169959	+	0.164017i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	−28.4364			−1.15706
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		−	5.18978i		−	0.210473i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−48.2012			−1.94683			−0.973413	−	0.229056i	$-0.926436\pi$
$613$	−48.2012			−1.94683			−0.973413	+	0.229056i	$0.926436\pi$
$614$		−	4.85174i		−	0.195800i
$615$		0			0
$616$	−5.27487			−0.212531
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$	−1.53102	+	1.47750i	−0.0615867	+	0.0594337i
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$	17.5261	+	19.5042i	0.703300	+	0.782678i
$622$	5.88112			0.235812
$623$		0			0
$624$	16.6138	+	17.2157i	0.665085	+	0.689179i
$625$	25.0000			1.00000
$626$			5.45215i			0.217912i
$627$	−10.8319	+	10.4532i	−0.432583	+	0.417460i
$628$	30.0617			1.19959
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	15.3959	−	14.8577i	0.610487	−	0.589144i
$637$	−4.80378			−0.190333
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	38.1741i		−	1.50779i	−0.656996	−	0.753894i	$-0.728173\pi$
$641$		−	38.1741i		−	1.50779i	0.656996	−	0.753894i	$-0.271827\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$			23.6878i			0.933430i
$645$		0			0
$646$		0			0
$647$			49.9203i			1.96257i	0.192564	+	0.981284i	$0.438320\pi$
$647$			49.9203i			1.96257i	−0.192564	+	0.981284i	$0.561680\pi$
$648$	−0.427469	+	5.99730i	−0.0167926	+	0.235596i
$649$		0			0
$650$			3.03234i			0.118938i
$651$		0			0
$652$		0			0
$653$		−	47.8330i		−	1.87185i	−0.352197	−	0.935926i	$-0.614565\pi$
$653$		−	47.8330i		−	1.87185i	0.352197	−	0.935926i	$-0.385435\pi$
$654$	4.10074	−	3.95737i	0.160351	−	0.154746i
$655$		0			0
$656$		−	46.6169i		−	1.82008i
$657$	1.52813	−	42.9331i	0.0596181	−	1.67498i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$	6.78697			0.263386
$665$		0			0
$666$		0			0
$667$		0			0
$668$		−	50.5088i		−	1.95424i
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−5.87645	+	5.67101i	−0.226689	+	0.218764i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	19.3250	−	17.3651i	0.743819	−	0.668381i
$676$	25.6322			0.985854
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$	−4.29324	−	4.44877i	−0.164881	−	0.170854i
$679$		0			0
$680$		0			0
$681$	−32.7657	+	31.6202i	−1.25558	+	1.21169i
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	−0.551352	+	15.4903i	−0.0210815	+	0.592287i
$685$		0			0
$686$		−	3.33661i		−	0.127392i
$687$		0			0
$688$		0			0
$689$		−	22.5891i		−	0.860576i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$	23.6725	+	0.842584i	0.899246	+	0.0320071i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$			6.01875i			0.227813i
$699$		0			0
$700$	23.4701			0.887086
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	2.10627	+	2.34400i	0.0794961	+	0.0884684i
$703$		0			0
$704$			24.3074i			0.916120i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$			47.1564i			1.76232i
$717$	−30.2500	+	29.1925i	−1.12971	+	1.09021i
$718$	−1.03301			−0.0385516
$719$			23.9165i			0.891936i	0.895049	+	0.445968i	$0.147140\pi$
$719$			23.9165i			0.891936i	−0.895049	+	0.445968i	$0.852860\pi$
$720$		0			0
$721$	17.3865			0.647507
$722$			2.04088i			0.0759538i
$723$	37.3111	+	38.6627i	1.38761	+	1.43788i
$724$	12.0291			0.447058
$725$		0			0
$726$	−2.30603	+	2.22541i	−0.0855847	+	0.0825927i
$727$	−9.86424			−0.365844			−0.182922	−	0.983127i	$-0.558556\pi$
$727$	−9.86424			−0.365844			−0.182922	+	0.983127i	$0.558556\pi$
$728$			5.73439i			0.212531i
$729$	2.87637	−	26.8463i	0.106532	−	0.994309i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	45.3052			1.67339			0.836693	−	0.547672i	$-0.184486\pi$
$733$	45.3052			1.67339			0.836693	+	0.547672i	$0.184486\pi$
$734$		−	6.44464i		−	0.237876i
$735$		0			0
$736$	−9.99441			−0.368399
$737$		0			0
$738$	0.218412	−	6.13633i	0.00803987	−	0.225882i
$739$	50.1387			1.84438			0.922191	−	0.386734i	$-0.126397\pi$
$739$	50.1387			1.84438			0.922191	+	0.386734i	$0.126397\pi$
$740$		0			0
$741$	11.3638	+	11.7755i	0.417460	+	0.432583i
$742$	2.50880			0.0921009
$743$		−	33.1662i		−	1.21675i	−0.793649	−	0.608376i	$-0.791821\pi$
$743$		−	33.1662i		−	1.21675i	0.793649	−	0.608376i	$-0.208179\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−30.4585	−	1.08412i	−1.11442	−	0.0396659i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	52.7476			1.92479			0.962394	−	0.271658i	$-0.0875720\pi$
$751$	52.7476			1.92479			0.962394	+	0.271658i	$0.0875720\pi$
$752$		0			0
$753$	−38.5502	+	37.2025i	−1.40485	+	1.35574i
$754$		0			0
$755$		0			0
$756$	18.1424	−	16.3024i	0.659831	−	0.592912i
$757$	17.8077			0.647230			0.323615	−	0.946189i	$-0.395102\pi$
$757$	17.8077			0.647230			0.323615	+	0.946189i	$0.395102\pi$
$758$		0			0
$759$	20.1306	+	20.8599i	0.730695	+	0.757166i
$760$		0			0
$761$			44.5403i			1.61459i	0.590151	+	0.807293i	$0.299068\pi$
$761$			44.5403i			1.61459i	−0.590151	+	0.807293i	$0.700932\pi$
$762$		0			0
$763$	−46.5685			−1.68589
$764$			39.4503i			1.42726i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	16.5793	+	17.1799i	0.598253	+	0.619926i
$769$	2.45283			0.0884514			0.0442257	−	0.999022i	$-0.485918\pi$
$769$	2.45283			0.0884514			0.0442257	+	0.999022i	$0.485918\pi$
$770$		0			0
$771$	2.73810	−	2.64238i	0.0986103	−	0.0951628i
$772$	−42.3884			−1.52559
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−6.08514			−0.218163
$779$		−	31.8858i		−	1.14243i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−5.10421			−0.182293
$785$		0			0
$786$		0			0
$787$	−54.9001			−1.95698			−0.978488	−	0.206302i	$-0.933857\pi$
$787$	−54.9001			−1.95698			−0.978488	+	0.206302i	$0.933857\pi$
$788$		−	48.5189i		−	1.72841i
$789$		0			0
$790$		0			0
$791$			50.5209i			1.79632i
$792$	−0.236442	+	6.64288i	−0.00840160	+	0.236044i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	47.0868			1.66895
$797$		−	47.8330i		−	1.69433i	−0.531327	−	0.847167i	$-0.678307\pi$
$797$		−	47.8330i		−	1.69433i	0.531327	−	0.847167i	$-0.321693\pi$
$798$	−1.30781	+	1.26209i	−0.0462961	+	0.0446775i
$799$		0			0
$800$			9.90255i			0.350108i
$801$		0			0
$802$		0			0
$803$		−	47.4944i		−	1.67604i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−34.9989	+	33.7753i	−1.23202	+	1.18895i
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	−40.6160			−1.42622			−0.713110	−	0.701052i	$-0.752714\pi$
$811$	−40.6160			−1.42622			−0.713110	+	0.701052i	$0.752714\pi$
$812$		0			0
$813$	−37.5992	−	38.9613i	−1.31866	−	1.36643i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		−	1.21294i		−	0.0424093i
$819$	0.915986	−	25.7348i	0.0320071	−	0.899246i
$820$		0			0
$821$		−	53.0660i		−	1.85202i	−0.377504	−	0.926008i	$-0.623217\pi$
$821$		−	53.0660i		−	1.85202i	0.377504	−	0.926008i	$-0.376783\pi$
$822$		0			0
$823$	−56.2442			−1.96055			−0.980274	−	0.197641i	$-0.936672\pi$
$823$	−56.2442			−1.96055			−0.980274	+	0.197641i	$0.936672\pi$
$824$			4.87891i			0.169965i
$825$	20.6682	−	19.9456i	0.719573	−	0.694417i
$826$		0			0
$827$			22.2527i			0.773802i	0.922121	+	0.386901i	$0.126454\pi$
$827$			22.2527i			0.773802i	−0.922121	+	0.386901i	$0.873546\pi$
$828$	29.8311	+	1.06179i	1.03670	+	0.0368996i
$829$	13.9660			0.485059			0.242530	−	0.970144i	$-0.422023\pi$
$829$	13.9660			0.485059			0.242530	+	0.970144i	$0.422023\pi$
$830$		0			0
$831$		0			0
$832$	26.4249			0.916120
$833$		0			0
$834$		0			0
$835$		0			0
$836$			17.1360i			0.592662i
$837$		0			0
$838$	−1.80740			−0.0624357
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$	33.1850	−	32.0248i	1.14295	−	1.10299i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	26.1876			0.899815
$848$		−	24.0018i		−	0.824226i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	37.7195			1.29149			0.645745	−	0.763553i	$-0.276547\pi$
$853$	37.7195			1.29149			0.645745	+	0.763553i	$0.276547\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$	2.41927	+	2.50692i	0.0825927	+	0.0855847i
$859$	−26.1127			−0.890955			−0.445477	−	0.895293i	$-0.646966\pi$
$859$	−26.1127			−0.890955			−0.445477	+	0.895293i	$0.646966\pi$
$860$		0			0
$861$	−36.1047	+	34.8425i	−1.23045	+	1.18743i
$862$	−5.80231			−0.197627
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	6.87834	+	7.65466i	0.234006	+	0.260417i
$865$		0			0
$866$			6.25697i			0.212620i
$867$	−20.4470	−	21.1877i	−0.694417	−	0.719573i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		−	13.0678i		−	0.442533i
$873$		0			0
$874$	−2.22427			−0.0752371
$875$		0			0
$876$	−33.9601	−	35.1904i	−1.14741	−	1.18897i
$877$	1.40243			0.0473567			0.0236784	−	0.999720i	$-0.492462\pi$
$877$	1.40243			0.0473567			0.0236784	+	0.999720i	$0.492462\pi$
$878$		0			0
$879$	−33.0691	+	31.9130i	−1.11539	+	1.07640i
$880$		0			0
$881$		−	51.1332i		−	1.72272i	−0.507993	−	0.861361i	$-0.669613\pi$
$881$		−	51.1332i		−	1.72272i	0.507993	−	0.861361i	$-0.330387\pi$
$882$	−0.671884	−	0.0239146i	−0.0226235	−	0.000805246i
$883$	−54.9791			−1.85019			−0.925097	−	0.379731i	$-0.876016\pi$
$883$	−54.9791			−1.85019			−0.925097	+	0.379731i	$0.876016\pi$
$884$		0			0
$885$		0			0
$886$	−6.90033			−0.231821
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	2.12221	−	29.7741i	0.0710966	−	0.997469i
$892$		0			0
$893$		0			0
$894$	3.27460	+	3.39323i	0.109519	+	0.113486i
$895$		0			0
$896$			12.3648i			0.413078i
$897$	22.6771	−	21.8843i	0.757166	−	0.730695i
$898$		0			0
$899$		0			0
$900$	1.05203	−	29.5569i	0.0350676	−	0.985230i
$901$		0			0
$902$		−	6.78827i		−	0.226025i
$903$		0			0
$904$	−14.1769			−0.471518
$905$		0			0
$906$	−3.02345	+	2.91775i	−0.100448	+	0.0969359i
$907$	−54.9636			−1.82504			−0.912519	−	0.409035i	$-0.865865\pi$
$907$	−54.9636			−1.82504			−0.912519	+	0.409035i	$0.865865\pi$
$908$			51.8354i			1.72022i
$909$		0			0
$910$		0			0
$911$		−	46.7225i		−	1.54799i	−0.633194	−	0.773993i	$-0.718256\pi$
$911$		−	46.7225i		−	1.54799i	0.633194	−	0.773993i	$-0.281744\pi$
$912$	12.0745	+	12.5119i	0.399827	+	0.414311i
$913$	−33.6945			−1.11512
$914$		−	0.645636i		−	0.0213558i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	34.6930	+	35.9499i	1.14317	+	1.18459i
$922$	3.91275			0.128859
$923$		0			0
$924$	19.4033	−	18.7250i	0.638323	−	0.616007i
$925$		0			0
$926$		0			0
$927$	0.779335	−	21.8956i	0.0255967	−	0.719144i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−3.49127			−0.114422
$932$		0			0
$933$	−43.5772	+	42.0538i	−1.42665	+	1.37678i
$934$	4.02285			0.131632
$935$		0			0
$936$	7.22157	+	0.257040i	0.236044	+	0.00840160i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$	−38.9863	−	40.3987i	−1.27227	−	1.31836i
$940$		0			0
$941$		−	28.6444i		−	0.933782i	−0.884315	−	0.466891i	$-0.845374\pi$
$941$		−	28.6444i		−	0.933782i	0.884315	−	0.466891i	$-0.154626\pi$
$942$	3.19627	−	3.08453i	0.104140	−	0.100499i
$943$	−61.4054			−1.99963
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$	−51.6319			−1.67604
$950$			2.20383i			0.0715016i
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$	0.112455	−	3.15944i	0.00364086	−	0.102291i
$955$		0			0
$956$			47.8556i			1.54776i
$957$		0			0
$958$	1.11574			0.0360479
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$	61.1645			1.96998
$965$		0			0
$966$	2.43052	+	2.51857i	0.0782007	+	0.0810337i
$967$	−15.5381			−0.499671			−0.249836	−	0.968288i	$-0.580377\pi$
$967$	−15.5381			−0.499671			−0.249836	+	0.968288i	$0.580377\pi$
$968$			7.34863i			0.236194i
$969$		0			0
$970$		0			0
$971$		−	17.5823i		−	0.564244i	−0.959379	−	0.282122i	$-0.908962\pi$
$971$		−	17.5823i		−	0.564244i	0.959379	−	0.282122i	$-0.0910382\pi$
$972$	−19.7171	−	23.5782i	−0.632425	−	0.756271i
$973$		0			0
$974$		0			0
$975$	−21.6832	−	22.4687i	−0.694417	−	0.719573i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−2.08740	+	58.6457i	−0.0666454	+	1.87241i
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	−9.77734	−	10.1315i	−0.311690	−	0.322982i
$985$		0			0
$986$		0			0
$987$		0			0
$988$	18.6288			0.592662
$989$		0			0
$990$		0			0
$991$	−19.4183			−0.616842			−0.308421	−	0.951250i	$-0.599801\pi$
$991$	−19.4183			−0.616842			−0.308421	+	0.951250i	$0.599801\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	−24.9655	+	24.0927i	−0.791062	+	0.763407i
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.e.d.428.10	yes	20
3.2	odd	2	inner	429.2.e.d.428.11	yes	20
11.10	odd	2	inner	429.2.e.d.428.12	yes	20
13.12	even	2	inner	429.2.e.d.428.12	yes	20
33.32	even	2	inner	429.2.e.d.428.9	✓	20
39.38	odd	2	inner	429.2.e.d.428.9	✓	20
143.142	odd	2	CM	429.2.e.d.428.10	yes	20
429.428	even	2	inner	429.2.e.d.428.11	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.e.d.428.9	✓	20	33.32	even	2	inner
429.2.e.d.428.9	✓	20	39.38	odd	2	inner
429.2.e.d.428.10	yes	20	1.1	even	1	trivial
429.2.e.d.428.10	yes	20	143.142	odd	2	CM
429.2.e.d.428.11	yes	20	3.2	odd	2	inner
429.2.e.d.428.11	yes	20	429.428	even	2	inner
429.2.e.d.428.12	yes	20	11.10	odd	2	inner
429.2.e.d.428.12	yes	20	13.12	even	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:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.168204i q^{2} +(1.20277 + 1.24634i) q^{3} +1.97171 q^{4} +(0.209639 - 0.202310i) q^{6} -2.38069 q^{7} -0.668057i q^{8} +(-0.106712 + 2.99810i) q^{9} +O(q^{10})\)	\(q-0.168204i q^{2} +(1.20277 + 1.24634i) q^{3} +1.97171 q^{4} +(0.209639 - 0.202310i) q^{6} -2.38069 q^{7} -0.668057i q^{8} +(-0.106712 + 2.99810i) q^{9} +3.31662i q^{11} +(2.37150 + 2.45741i) q^{12} +3.60555 q^{13} +0.400441i q^{14} +3.83105 q^{16} +(0.504293 + 0.0179494i) q^{18} +2.62042 q^{19} +(-2.86341 - 2.96714i) q^{21} +0.557869 q^{22} -5.04638i q^{23} +(0.832624 - 0.803516i) q^{24} -5.00000 q^{25} -0.606468i q^{26} +(-3.86500 + 3.47301i) q^{27} -4.69402 q^{28} -1.98051i q^{32} +(-4.13363 + 3.98912i) q^{33} +(-0.210406 + 5.91138i) q^{36} -0.440765i q^{38} +(4.33663 + 4.49373i) q^{39} -12.1682i q^{41} +(-0.499084 + 0.481636i) q^{42} +6.53941i q^{44} -0.848821 q^{46} +(4.60785 + 4.77477i) q^{48} -1.33233 q^{49} +0.841020i q^{50} +7.10909 q^{52} -6.26509i q^{53} +(0.584174 + 0.650108i) q^{54} +1.59043i q^{56} +(3.15175 + 3.26593i) q^{57} +(0.254049 - 7.13754i) q^{63} +7.32896 q^{64} +(0.670986 + 0.695293i) q^{66} +(6.28949 - 6.06961i) q^{69} +(2.00290 + 0.0712899i) q^{72} -14.3201 q^{73} +(-6.01383 - 6.23169i) q^{75} +5.16671 q^{76} -7.89585i q^{77} +(0.755864 - 0.729439i) q^{78} +(-8.97722 - 0.639869i) q^{81} -2.04674 q^{82} +10.1593i q^{83} +(-5.64580 - 5.85033i) q^{84} +2.21569 q^{88} -8.58369 q^{91} -9.94999i q^{92} +(2.46838 - 2.38209i) q^{96} +0.224103i q^{98} +(-9.94358 - 0.353925i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 40 q^{4}+O(q^{10}) \) 20 * q - 40 * q^4	\( 20 q - 40 q^{4} + 80 q^{16} - 100 q^{25} + 10 q^{36} - 50 q^{42} + 70 q^{48} + 140 q^{49} - 160 q^{64} - 110 q^{66} + 130 q^{78}+O(q^{100}) \) 20 * q - 40 * q^4 + 80 * q^16 - 100 * q^25 + 10 * q^36 - 50 * q^42 + 70 * q^48 + 140 * q^49 - 160 * q^64 - 110 * q^66 + 130 * q^78

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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