LMFDB - Embedded newform 429.2.e.d.428.1

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			428.1
Root			$-0.356257 + 1.36861i$ of defining polynomial
Character	$\chi$	$=$	429.428
Dual form			429.2.e.d.428.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73721i q^{2} +(-0.813662 - 1.52904i) q^{3} -5.49232 q^{4} +(-4.18530 + 2.22717i) q^{6} +0.851686 q^{7} +9.55923i q^{8} +(-1.67591 + 2.48824i) q^{9} +O(q^{10})$	\(q-2.73721i q^{2} +(-0.813662 - 1.52904i) q^{3} -5.49232 q^{4} +(-4.18530 + 2.22717i) q^{6} +0.851686 q^{7} +9.55923i q^{8} +(-1.67591 + 2.48824i) q^{9} +3.31662i q^{11} +(4.46890 + 8.39797i) q^{12} -3.60555 q^{13} -2.33124i q^{14} +15.1810 q^{16} +(6.81084 + 4.58731i) q^{18} -8.71746 q^{19} +(-0.692985 - 1.30226i) q^{21} +9.07830 q^{22} -8.87708i q^{23} +(14.6164 - 7.77798i) q^{24} -5.00000 q^{25} +9.86915i q^{26} +(5.16823 + 0.537939i) q^{27} -4.67774 q^{28} -22.4351i q^{32} +(5.07124 - 2.69861i) q^{33} +(9.20463 - 13.6662i) q^{36} +23.8615i q^{38} +(2.93370 + 5.51302i) q^{39} +7.49792i q^{41} +(-3.56456 + 1.89685i) q^{42} -18.2160i q^{44} -24.2984 q^{46} +(-12.3522 - 23.2123i) q^{48} -6.27463 q^{49} +13.6861i q^{50} +19.8029 q^{52} -10.5641i q^{53} +(1.47245 - 14.1465i) q^{54} +8.14146i q^{56} +(7.09306 + 13.3293i) q^{57} +(-1.42735 + 2.11920i) q^{63} -31.0476 q^{64} +(-7.38667 - 13.8811i) q^{66} +(-13.5734 + 7.22294i) q^{69} +(-23.7856 - 16.0204i) q^{72} -4.44237 q^{73} +(4.06831 + 7.64518i) q^{75} +47.8791 q^{76} +2.82472i q^{77} +(15.0903 - 8.03016i) q^{78} +(-3.38267 - 8.34012i) q^{81} +20.5234 q^{82} +0.671690i q^{83} +(3.80610 + 7.15243i) q^{84} -31.7044 q^{88} -3.07080 q^{91} +48.7558i q^{92} +(-34.3041 + 18.2546i) q^{96} +17.1750i q^{98} +(-8.25256 - 5.55836i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	2.73721i		−	1.93550i	−0.251912	−	0.967750i	$-0.581059\pi$
$2$		−	2.73721i		−	1.93550i	0.251912	−	0.967750i	$-0.418941\pi$
$3$	−0.813662	−	1.52904i	−0.469768	−	0.882790i
$3$	−0.813662	−	1.52904i	−0.469768	−	0.882790i
$4$	−5.49232			−2.74616
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	−4.18530	+	2.22717i	−1.70864	+	0.909236i
$7$	0.851686			0.321907			0.160954	−	0.986962i	$-0.448543\pi$
$7$	0.851686			0.321907			0.160954	+	0.986962i	$0.448543\pi$
$8$			9.55923i			3.37970i
$9$	−1.67591	+	2.48824i	−0.558636	+	0.829413i
$10$		0			0
$11$			3.31662i			1.00000i
$11$			3.31662i			1.00000i
$12$	4.46890	+	8.39797i	1.29006	+	2.42428i
$13$	−3.60555			−1.00000
$13$	−3.60555			−1.00000
$14$		−	2.33124i		−	0.623051i
$15$		0			0
$16$	15.1810			3.79524
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	6.81084	+	4.58731i	1.60533	+	1.08124i
$19$	−8.71746			−1.99992			−0.999961	−	0.00885519i	$-0.997181\pi$
$19$	−8.71746			−1.99992			−0.999961	+	0.00885519i	$0.997181\pi$
$20$		0			0
$21$	−0.692985	−	1.30226i	−0.151222	−	0.284176i
$22$	9.07830			1.93550
$23$		−	8.87708i		−	1.85100i	−0.378749	−	0.925499i	$-0.623646\pi$
$23$		−	8.87708i		−	1.85100i	0.378749	−	0.925499i	$-0.376354\pi$
$24$	14.6164	−	7.77798i	2.98356	−	1.58767i
$25$	−5.00000			−1.00000
$26$			9.86915i			1.93550i
$27$	5.16823	+	0.537939i	0.994627	+	0.103526i
$28$	−4.67774			−0.884009
$29$		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			$\pi$
$32$		−	22.4351i		−	3.96600i
$33$	5.07124	−	2.69861i	0.882790	−	0.469768i
$34$		0			0
$35$		0			0
$36$	9.20463	−	13.6662i	1.53410	−	2.27770i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$			23.8615i			3.87085i
$39$	2.93370	+	5.51302i	0.469768	+	0.882790i
$40$		0			0
$41$			7.49792i			1.17098i	0.810680	+	0.585489i	$0.199098\pi$
$41$			7.49792i			1.17098i	−0.810680	+	0.585489i	$0.800902\pi$
$42$	−3.56456	+	1.89685i	−0.550023	+	0.292690i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		−	18.2160i		−	2.74616i
$45$		0			0
$46$	−24.2984			−3.58261
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	−12.3522	−	23.2123i	−1.78288	−	3.35040i
$49$	−6.27463			−0.896376
$50$			13.6861i			1.93550i
$51$		0			0
$52$	19.8029			2.74616
$53$		−	10.5641i		−	1.45109i	−0.688175	−	0.725545i	$-0.741588\pi$
$53$		−	10.5641i		−	1.45109i	0.688175	−	0.725545i	$-0.258412\pi$
$54$	1.47245	−	14.1465i	0.200375	−	1.92510i
$55$		0			0
$56$			8.14146i			1.08795i
$57$	7.09306	+	13.3293i	0.939499	+	1.76551i
$58$		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
$63$	−1.42735	+	2.11920i	−0.179829	+	0.266994i
$64$	−31.0476			−3.88095
$65$		0			0
$66$	−7.38667	−	13.8811i	−0.909236	−	1.70864i
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	−13.5734	+	7.22294i	−1.63404	+	0.869540i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−23.7856	−	16.0204i	−2.80317	−	1.88802i
$73$	−4.44237			−0.519941			−0.259970	−	0.965617i	$-0.583713\pi$
$73$	−4.44237			−0.519941			−0.259970	+	0.965617i	$0.583713\pi$
$74$		0			0
$75$	4.06831	+	7.64518i	0.469768	+	0.882790i
$76$	47.8791			5.49211
$77$			2.82472i			0.321907i
$78$	15.0903	−	8.03016i	1.70864	−	0.909236i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−3.38267	−	8.34012i	−0.375852	−	0.926680i
$82$	20.5234			2.26643
$83$			0.671690i			0.0737276i	0.999320	+	0.0368638i	$0.0117368\pi$
$83$			0.671690i			0.0737276i	−0.999320	+	0.0368638i	$0.988263\pi$
$84$	3.80610	+	7.15243i	0.415279	+	0.780394i
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−31.7044			−3.37970
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	−3.07080			−0.321907
$92$			48.7558i			5.08314i
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−34.3041	+	18.2546i	−3.50114	+	1.86310i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$			17.1750i			1.73494i
$99$	−8.25256	−	5.55836i	−0.829413	−	0.558636i
$100$	27.4616			2.74616
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	17.0401			1.67901			0.839505	−	0.543352i	$-0.182845\pi$
$103$	17.0401			1.67901			0.839505	+	0.543352i	$0.182845\pi$
$104$		−	34.4663i		−	3.37970i
$105$		0			0
$106$	−28.9161			−2.80858
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−28.3856	−	2.95453i	−2.73141	−	0.284300i
$109$	−12.9925			−1.24446			−0.622230	−	0.782835i	$-0.713773\pi$
$109$	−12.9925			−1.24446			−0.622230	+	0.782835i	$0.713773\pi$
$110$		0			0
$111$		0			0
$112$	12.9294			1.22172
$113$			7.78435i			0.732290i	0.930558	+	0.366145i	$0.119323\pi$
$113$			7.78435i			0.732290i	−0.930558	+	0.366145i	$0.880677\pi$
$114$	36.4851	−	19.4152i	3.41715	−	1.81840i
$115$		0			0
$116$		0			0
$117$	6.04257	−	8.97147i	0.558636	−	0.829413i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	11.4646	−	6.10078i	1.03373	−	0.550088i
$124$		0			0
$125$		0			0
$126$	5.80069	+	3.90695i	0.516767	+	0.348059i
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$			40.1137i			3.54558i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	−27.8529	+	14.8217i	−2.42428	+	1.29006i
$133$	−7.42454			−0.643789
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$	19.7707	+	37.1532i	1.68300	+	3.16269i
$139$		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
$144$	−25.4419	+	37.7739i	−2.12016	+	3.14783i
$145$		0			0
$146$			12.1597i			1.00635i
$147$	5.10543	+	9.59414i	0.421089	+	0.791312i
$148$		0			0
$149$		−	23.8371i		−	1.95282i	−0.215935	−	0.976408i	$-0.569280\pi$
$149$		−	23.8371i		−	1.95282i	0.215935	−	0.976408i	$-0.430720\pi$
$150$	20.9265	−	11.1358i	1.70864	−	0.909236i
$151$	14.4222			1.17366			0.586831	−	0.809709i	$-0.300375\pi$
$151$	14.4222			1.17366			0.586831	+	0.809709i	$0.300375\pi$
$152$		−	83.3322i		−	6.75913i
$153$		0			0
$154$	7.73186			0.623051
$155$		0			0
$156$	−16.1128	−	30.2793i	−1.29006	−	2.42428i
$157$	−24.0247			−1.91738			−0.958692	−	0.284447i	$-0.908190\pi$
$157$	−24.0247			−1.91738			−0.958692	+	0.284447i	$0.908190\pi$
$158$		0			0
$159$	−16.1529	+	8.59560i	−1.28101	+	0.681675i
$160$		0			0
$161$		−	7.56048i		−	0.595850i
$162$	−22.8287	+	9.25907i	−1.79359	+	0.727461i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		−	41.1810i		−	3.21570i
$165$		0			0
$166$	1.83856			0.142700
$167$		−	11.1804i		−	0.865168i	−0.901594	−	0.432584i	$-0.857602\pi$
$167$		−	11.1804i		−	0.865168i	0.901594	−	0.432584i	$-0.142398\pi$
$168$	12.4486	−	6.62440i	0.960430	−	0.511084i
$169$	13.0000			1.00000
$170$		0			0
$171$	14.6097	−	21.6911i	1.11723	−	1.65876i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	−4.25843			−0.321907
$176$			50.3496i			3.79524i
$177$		0			0
$178$		0			0
$179$		−	23.9165i		−	1.78760i	−0.448461	−	0.893802i	$-0.648028\pi$
$179$		−	23.9165i		−	1.78760i	0.448461	−	0.893802i	$-0.351972\pi$
$180$		0			0
$181$	−23.0386			−1.71244			−0.856222	−	0.516608i	$-0.827194\pi$
$181$	−23.0386			−1.71244			−0.856222	+	0.516608i	$0.827194\pi$
$182$			8.40542i			0.623051i
$183$		0			0
$184$	84.8580			6.25582
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	4.40171	+	0.458155i	0.320177	+	0.0333259i
$190$		0			0
$191$			11.9540i			0.864958i	0.901644	+	0.432479i	$0.142361\pi$
$191$			11.9540i			0.864958i	−0.901644	+	0.432479i	$0.857639\pi$
$192$	25.2623	+	47.4729i	1.82315	+	3.42606i
$193$	−27.7386			−1.99667			−0.998333	−	0.0577192i	$-0.981617\pi$
$193$	−27.7386			−1.99667			−0.998333	+	0.0577192i	$0.981617\pi$
$194$		0			0
$195$		0			0
$196$	34.4623			2.46159
$197$			5.24283i			0.373536i	0.982404	+	0.186768i	$0.0598013\pi$
$197$			5.24283i			0.373536i	−0.982404	+	0.186768i	$0.940199\pi$
$198$	−15.2144	+	22.5890i	−1.08124	+	1.60533i
$199$	21.6673			1.53595			0.767977	−	0.640478i	$-0.221264\pi$
$199$	21.6673			1.53595			0.767977	+	0.640478i	$0.221264\pi$
$200$		−	47.7961i		−	3.37970i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		−	46.6423i		−	3.24973i
$207$	22.0883	+	14.8772i	1.53524	+	1.03403i
$208$	−54.7358			−3.79524
$209$		−	28.9125i		−	1.99992i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$			58.0214i			3.98493i
$213$		0			0
$214$		0			0
$215$		0			0
$216$	−5.14228	+	49.4043i	−0.349888	+	3.36154i
$217$		0			0
$218$			35.5633i			2.40865i
$219$	3.61459	+	6.79255i	0.244251	+	0.458998i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		−	19.1076i		−	1.27668i
$225$	8.37954	−	12.4412i	0.558636	−	0.829413i
$226$	21.3074			1.41735
$227$			22.1293i			1.46877i	0.678732	+	0.734386i	$0.262530\pi$
$227$			22.1293i			1.46877i	−0.678732	+	0.734386i	$0.737470\pi$
$228$	−38.9574	−	73.2089i	−2.58002	−	4.84838i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$	4.31911	−	2.29837i	0.284176	−	0.151222i
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$	−24.5568	−	16.5398i	−1.60533	−	1.08124i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	10.7173i		−	0.693241i	−0.938005	−	0.346621i	$-0.887329\pi$
$239$		−	10.7173i		−	0.693241i	0.938005	−	0.346621i	$-0.112671\pi$
$240$		0			0
$241$	24.3318			1.56735			0.783675	−	0.621171i	$-0.213343\pi$
$241$	24.3318			1.56735			0.783675	+	0.621171i	$0.213343\pi$
$242$			30.1093i			1.93550i
$243$	−10.0000	+	11.9583i	−0.641500	+	0.767123i
$244$		0			0
$245$		0			0
$246$	−16.6991	−	31.3810i	−1.06470	−	2.00078i
$247$	31.4312			1.99992
$248$		0			0
$249$	1.02704	−	0.546529i	0.0650860	−	0.0346349i
$250$		0			0
$251$			20.9818i			1.32436i	0.749345	+	0.662179i	$0.230368\pi$
$251$			20.9818i			1.32436i	−0.749345	+	0.662179i	$0.769632\pi$
$252$	7.83945	−	11.6393i	0.493839	−	0.733209i
$253$	29.4419			1.85100
$254$		0			0
$255$		0			0
$256$	47.7044			2.98152
$257$		−	20.5789i		−	1.28368i	−0.766841	−	0.641838i	$-0.778172\pi$
$257$		−	20.5789i		−	1.28368i	0.766841	−	0.641838i	$-0.221828\pi$
$258$		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	$0.5\pi$
$264$	25.7967	+	48.4772i	1.58767	+	2.98356i
$265$		0			0
$266$			20.3225i			1.24605i
$267$		0			0
$268$		0			0
$269$			32.6836i			1.99275i	0.0850479	+	0.996377i	$0.472896\pi$
$269$			32.6836i			1.99275i	−0.0850479	+	0.996377i	$0.527104\pi$
$270$		0			0
$271$	−0.167292			−0.0101623			−0.00508114	−	0.999987i	$-0.501617\pi$
$271$	−0.167292			−0.0101623			−0.00508114	+	0.999987i	$0.501617\pi$
$272$		0			0
$273$	2.49859	+	4.69536i	0.151222	+	0.284176i
$274$		0			0
$275$		−	16.5831i		−	1.00000i
$276$	74.5494	−	39.6707i	4.48735	−	2.38790i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	27.6037i		−	1.64670i	−0.567535	−	0.823349i	$-0.692103\pi$
$281$		−	27.6037i		−	1.64670i	0.567535	−	0.823349i	$-0.307897\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$	−32.7323			−1.93550
$287$			6.38588i			0.376946i
$288$	55.8238	+	37.5991i	3.28945	+	2.21555i
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$	24.3990			1.42784
$293$			26.5330i			1.55007i	0.631916	+	0.775037i	$0.282269\pi$
$293$			26.5330i			1.55007i	−0.631916	+	0.775037i	$0.717731\pi$
$294$	26.2612	−	13.9746i	1.53158	−	0.815017i
$295$		0			0
$296$		0			0
$297$	−1.78414	+	17.1411i	−0.103526	+	0.994627i
$298$	−65.2473			−3.77968
$299$			32.0068i			1.85100i
$300$	−22.3445	−	41.9898i	−1.29006	−	2.42428i
$301$		0			0
$302$		−	39.4766i		−	2.27162i
$303$		0			0
$304$	−132.340			−7.59019
$305$		0			0
$306$		0			0
$307$	−28.8444			−1.64624			−0.823119	−	0.567869i	$-0.807768\pi$
$307$	−28.8444			−1.64624			−0.823119	+	0.567869i	$0.807768\pi$
$308$		−	15.5143i		−	0.884009i
$309$	−13.8649	−	26.0549i	−0.788746	−	1.48221i
$310$		0			0
$311$		−	6.39449i		−	0.362598i	−0.983428	−	0.181299i	$-0.941970\pi$
$311$		−	6.39449i		−	0.362598i	0.983428	−	0.181299i	$-0.0580302\pi$
$312$	−52.7002	+	28.0439i	−2.98356	+	1.58767i
$313$	17.8831			1.01082			0.505408	−	0.862881i	$-0.331342\pi$
$313$	17.8831			1.01082			0.505408	+	0.862881i	$0.331342\pi$
$314$			65.7608i			3.71110i
$315$		0			0
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$	23.5280	+	44.2138i	1.31938	+	2.47939i
$319$		0			0
$320$		0			0
$321$		0			0
$322$	−20.6946			−1.15327
$323$		0			0
$324$	18.5787	+	45.8066i	1.03215	+	2.54481i
$325$	18.0278			1.00000
$326$		0			0
$327$	10.5715	+	19.8661i	0.584607	+	1.09860i
$328$	−71.6744			−3.95755
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		−	3.68914i		−	0.202468i
$333$		0			0
$334$	−30.6032			−1.67453
$335$		0			0
$336$	−10.5202	−	19.7696i	−0.573923	−	1.07852i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		−	35.5837i		−	1.93550i
$339$	11.9026	−	6.33384i	0.646459	−	0.344007i
$340$		0			0
$341$		0			0
$342$	−59.3732	−	39.9897i	−3.21053	−	2.16240i
$343$	−11.3058			−0.610457
$344$		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
$349$	−22.6285			−1.21127			−0.605637	−	0.795741i	$-0.707081\pi$
$349$	−22.6285			−1.21127			−0.605637	+	0.795741i	$0.707081\pi$
$350$			11.6562i			0.623051i
$351$	−18.6343	−	1.93957i	−0.994627	−	0.103526i
$352$	74.4087			3.96600
$353$		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
$358$	−65.4646			−3.45991
$359$		−	17.0109i		−	0.897802i	−0.893582	−	0.448901i	$-0.851816\pi$
$359$		−	17.0109i		−	0.897802i	0.893582	−	0.448901i	$-0.148184\pi$
$360$		0			0
$361$	56.9940			2.99969
$362$			63.0614i			3.31444i
$363$	8.95028	+	16.8194i	0.469768	+	0.882790i
$364$	16.8658			0.884009
$365$		0			0
$366$		0			0
$367$	11.9033			0.621349			0.310675	−	0.950516i	$-0.399445\pi$
$367$	11.9033			0.621349			0.310675	+	0.950516i	$0.399445\pi$
$368$		−	134.763i		−	7.02499i
$369$	−18.6566	−	12.5658i	−0.971225	−	0.654151i
$370$		0			0
$371$		−	8.99729i		−	0.467116i
$372$		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
$378$	1.25407	−	12.0484i	0.0645022	−	0.619703i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$	32.7205			1.67413
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	61.3353	−	32.6390i	3.13000	−	1.66560i
$385$		0			0
$386$			75.9263i			3.86455i
$387$		0			0
$388$		0			0
$389$			26.1328i			1.32499i	0.749068	+	0.662493i	$0.230501\pi$
$389$			26.1328i			1.32499i	−0.749068	+	0.662493i	$0.769499\pi$
$390$		0			0
$391$		0			0
$392$		−	59.9806i		−	3.02948i
$393$		0			0
$394$	14.3507			0.722979
$395$		0			0
$396$	45.3257	+	30.5283i	2.27770	+	1.53410i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		−	59.3079i		−	2.97284i
$399$	6.04106	+	11.3524i	0.302431	+	0.568330i
$400$	−75.9049			−3.79524
$401$		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	$-0.557054\pi$
$409$	−7.21110			−0.356566			−0.178283	+	0.983979i	$0.557054\pi$
$410$		0			0
$411$		0			0
$412$	−93.5897			−4.61083
$413$		0			0
$414$	40.7219	−	60.4603i	2.00137	−	2.97146i
$415$		0			0
$416$			80.8908i			3.96600i
$417$		0			0
$418$	−79.1397			−3.87085
$419$			14.5265i			0.709667i	0.934929	+	0.354834i	$0.115462\pi$
$419$			14.5265i			0.709667i	−0.934929	+	0.354834i	$0.884538\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	100.985			4.90424
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−18.2846	+	9.72998i	−0.882790	+	0.469768i
$430$		0			0
$431$			14.3242i			0.689970i	0.938608	+	0.344985i	$0.112116\pi$
$431$			14.3242i			0.689970i	−0.938608	+	0.344985i	$0.887884\pi$
$432$	78.4588	+	8.16644i	3.77485	+	0.392908i
$433$	6.25326			0.300513			0.150256	−	0.988647i	$-0.451990\pi$
$433$	6.25326			0.300513			0.150256	+	0.988647i	$0.451990\pi$
$434$		0			0
$435$		0			0
$436$	71.3592			3.41749
$437$			77.3855i			3.70185i
$438$	18.5927	−	9.89390i	0.888391	−	0.472749i
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	10.5157	−	15.6128i	0.500748	−	0.743466i
$442$		0			0
$443$		−	38.7359i		−	1.84040i	−0.391448	−	0.920200i	$-0.628026\pi$
$443$		−	38.7359i		−	1.84040i	0.391448	−	0.920200i	$-0.371974\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−36.4479	+	19.3954i	−1.72393	+	0.917370i
$448$	−26.4428			−1.24931
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	−34.0542	−	22.9366i	−1.60533	−	1.08124i
$451$	−24.8678			−1.17098
$452$		−	42.7542i		−	2.01099i
$453$	−11.7348	−	22.0521i	−0.551349	−	1.03610i
$454$	60.5725			2.84281
$455$		0			0
$456$	−127.418	+	67.8042i	−5.96689	+	3.17522i
$457$	39.3122			1.83895			0.919474	−	0.393152i	$-0.128615\pi$
$457$	39.3122			1.83895			0.919474	+	0.393152i	$0.128615\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	27.1405i		−	1.26406i	−0.774944	−	0.632030i	$-0.782222\pi$
$461$		−	27.1405i		−	1.26406i	0.774944	−	0.632030i	$-0.217778\pi$
$462$	−6.29112	−	11.8223i	−0.292690	−	0.550023i
$463$		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	$-0.813345\pi$
$467$		−	23.9165i		−	1.10672i	0.832941	−	0.553362i	$-0.186655\pi$
$468$	−33.1878	+	49.2742i	−1.53410	+	2.27770i
$469$		0			0
$470$		0			0
$471$	19.5480	+	36.7347i	0.900725	+	1.69265i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	43.5873			1.99992
$476$		0			0
$477$	26.2860	+	17.7044i	1.20355	+	0.810630i
$478$	−29.3354			−1.34177
$479$			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
$482$		−	66.6013i		−	3.03361i
$483$	−11.5603	+	6.15168i	−0.526010	+	0.279911i
$484$	60.4156			2.74616
$485$		0			0
$486$	32.7323	+	27.3721i	1.48477	+	1.24162i
$487$		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	$0.5\pi$
$492$	−62.9673	+	33.5074i	−2.83879	+	1.51063i
$493$		0			0
$494$		−	86.0339i		−	3.87085i
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−1.49597	−	2.81122i	−0.0670358	−	0.125974i
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$	−17.0953	+	9.09710i	−0.763762	+	0.406428i
$502$	57.4316			2.56330
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	−20.2579	−	13.6443i	−0.902359	−	0.607767i
$505$		0			0
$506$		−	80.5888i		−	3.58261i
$507$	−10.5776	−	19.8775i	−0.469768	−	0.882790i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	−3.78351			−0.167373
$512$		−	50.3496i		−	2.22516i
$513$	−45.0538	−	4.68946i	−1.98918	−	0.207045i
$514$	−56.3287			−2.48455
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			9.27999i			0.406564i	0.979120	+	0.203282i	$0.0651608\pi$
$521$			9.27999i			0.406564i	−0.979120	+	0.203282i	$0.934839\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$	3.46492	+	6.51130i	0.151222	+	0.284176i
$526$		0			0
$527$		0			0
$528$	76.9864	−	40.9676i	3.35040	−	1.78288i
$529$	−55.8025			−2.42620
$530$		0			0
$531$		0			0
$532$	40.7780			1.76795
$533$		−	27.0341i		−	1.17098i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−36.5692	+	19.4600i	−1.57808	+	0.839760i
$538$	89.4619			3.85698
$539$		−	20.8106i		−	0.896376i
$540$		0			0
$541$	−32.8487			−1.41228			−0.706138	−	0.708075i	$-0.749564\pi$
$541$	−32.8487			−1.41228			−0.706138	+	0.708075i	$0.749564\pi$
$542$			0.457914i			0.0196691i
$543$	18.7456	+	35.2268i	0.804452	+	1.51173i
$544$		0			0
$545$		0			0
$546$	12.8522	−	6.83917i	0.550023	−	0.292690i
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$	−45.3915			−1.93550
$551$		0			0
$552$	−69.0458	−	129.751i	−2.93878	−	5.52257i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	6.15454i		−	0.260776i	−0.991463	−	0.130388i	$-0.958378\pi$
$557$		−	6.15454i		−	0.260776i	0.991463	−	0.130388i	$-0.0416224\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−75.5572			−3.18719
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−2.88097	−	7.10316i	−0.120989	−	0.298305i
$568$		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			$\pi$
$572$			65.6786i			2.74616i
$573$	18.2780	−	9.72648i	0.763576	−	0.406330i
$574$	17.4795			0.729580
$575$			44.3854i			1.85100i
$576$	52.0329	−	77.2538i	2.16804	−	3.21891i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$			46.5326i			1.93550i
$579$	22.5698	+	42.4133i	0.937970	+	1.76264i
$580$		0			0
$581$			0.572069i			0.0237334i
$582$		0			0
$583$	35.0371			1.45109
$584$		−	42.4657i		−	1.75724i
$585$		0			0
$586$	72.6264			3.00017
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−28.0407	−	52.6941i	−1.15638	−	2.17307i
$589$		0			0
$590$		0			0
$591$	8.01648	−	4.26589i	0.329754	−	0.175475i
$592$		0			0
$593$			25.1805i			1.03404i	0.855973	+	0.517020i	$0.172959\pi$
$593$			25.1805i			1.03404i	−0.855973	+	0.517020i	$0.827041\pi$
$594$	46.9188	+	4.88357i	1.92510	+	0.200375i
$595$		0			0
$596$			130.921i			5.36275i
$597$	−17.6299	−	33.1301i	−0.721542	−	1.35592i
$598$	87.6093			3.58261
$599$			48.6508i			1.98782i	0.110197	+	0.993910i	$0.464852\pi$
$599$			48.6508i			1.98782i	−0.110197	+	0.993910i	$0.535148\pi$
$600$	−73.0821	+	38.8899i	−2.98356	+	1.58767i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	−79.2114			−3.22307
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$			195.577i			7.93169i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	4.10779			0.165912			0.0829560	−	0.996553i	$-0.473564\pi$
$613$	4.10779			0.165912			0.0829560	+	0.996553i	$0.473564\pi$
$614$			78.9532i			3.18629i
$615$		0			0
$616$	−27.0022			−1.08795
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$	−71.3179	+	37.9511i	−2.86882	+	1.52662i
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$	4.77532	−	45.8788i	0.191627	−	1.84105i
$622$	−17.5031			−0.701809
$623$		0			0
$624$	44.5364	+	83.6930i	1.78288	+	3.35040i
$625$	25.0000			1.00000
$626$		−	48.9500i		−	1.95643i
$627$	−44.2083	+	23.5250i	−1.76551	+	0.939499i
$628$	131.952			5.26545
$629$		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
$636$	88.7169	−	47.2098i	3.51785	−	1.87199i
$637$	22.6235			0.896376
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	50.4377i		−	1.99217i	−0.0883934	−	0.996086i	$-0.528173\pi$
$641$		−	50.4377i		−	1.99217i	0.0883934	−	0.996086i	$-0.471827\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$			41.5246i			1.63630i
$645$		0			0
$646$		0			0
$647$		−	24.7429i		−	0.972745i	−0.873752	−	0.486372i	$-0.838320\pi$
$647$		−	24.7429i		−	0.972745i	0.873752	−	0.486372i	$-0.161680\pi$
$648$	79.7251	−	32.3357i	3.13190	−	1.27027i
$649$		0			0
$650$		−	49.3458i		−	1.93550i
$651$		0			0
$652$		0			0
$653$			47.8330i			1.87185i	0.352197	+	0.935926i	$0.385435\pi$
$653$			47.8330i			1.87185i	−0.352197	+	0.935926i	$0.614565\pi$
$654$	54.3776	−	28.9365i	2.12633	−	1.13151i
$655$		0			0
$656$			113.826i			4.44415i
$657$	7.44501	−	11.0537i	0.290457	−	0.431245i
$658$		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
$664$	−6.42084			−0.249177
$665$		0			0
$666$		0			0
$667$		0			0
$668$			61.4066i			2.37589i
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−29.2163	+	15.5472i	−1.12704	+	0.599745i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−25.8412	−	2.68969i	−0.994627	−	0.103526i
$676$	−71.4002			−2.74616
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$	−17.3370	−	32.5798i	−0.665825	−	1.25122i
$679$		0			0
$680$		0			0
$681$	33.8365	−	18.0058i	1.29662	−	0.689982i
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	−80.2409	+	119.135i	−3.06809	+	4.55523i
$685$		0			0
$686$			30.9464i			1.18154i
$687$		0			0
$688$		0			0
$689$			38.0894i			1.45109i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$	−7.02859	−	4.73398i	−0.266994	−	0.179829i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$			61.9388i			2.34442i
$699$		0			0
$700$	23.3887			0.884009
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	−5.30900	+	51.0061i	−0.200375	+	1.92510i
$703$		0			0
$704$		−	102.973i		−	3.88095i
$705$		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
$716$			131.357i			4.90905i
$717$	−16.3871	+	8.72022i	−0.611986	+	0.325663i
$718$	−46.5625			−1.73770
$719$		−	23.9165i		−	0.891936i	−0.895049	−	0.445968i	$-0.852860\pi$
$719$		−	23.9165i		−	0.891936i	0.895049	−	0.445968i	$-0.147140\pi$
$720$		0			0
$721$	14.5128			0.540485
$722$		−	156.005i		−	5.80589i
$723$	−19.7979	−	37.2043i	−0.736291	−	1.38364i
$724$	126.535			4.70265
$725$		0			0
$726$	46.0383	−	24.4988i	1.70864	−	0.909236i
$727$	−23.1817			−0.859761			−0.429881	−	0.902886i	$-0.641444\pi$
$727$	−23.1817			−0.859761			−0.429881	+	0.902886i	$0.641444\pi$
$728$		−	29.3545i		−	1.08795i
$729$	26.4212	+	5.56038i	0.978565	+	0.205940i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	19.2217			0.709970			0.354985	−	0.934872i	$-0.384486\pi$
$733$	19.2217			0.709970			0.354985	+	0.934872i	$0.384486\pi$
$734$		−	32.5820i		−	1.20262i
$735$		0			0
$736$	−199.158			−7.34106
$737$		0			0
$738$	−34.3953	+	51.0671i	−1.26611	+	1.87981i
$739$	52.9221			1.94677			0.973385	−	0.229176i	$-0.0736031\pi$
$739$	52.9221			1.94677			0.973385	+	0.229176i	$0.0736031\pi$
$740$		0			0
$741$	−25.5744	−	48.0595i	−0.939499	−	1.76551i
$742$	−24.6275			−0.904103
$743$		−	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
$747$	−1.67133	−	1.12569i	−0.0611506	−	0.0411869i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	2.13940			0.0780677			0.0390338	−	0.999238i	$-0.487572\pi$
$751$	2.13940			0.0780677			0.0390338	+	0.999238i	$0.487572\pi$
$752$		0			0
$753$	32.0819	−	17.0721i	1.16913	−	0.622141i
$754$		0			0
$755$		0			0
$756$	−24.1756	−	2.51634i	−0.879259	−	0.0915182i
$757$	16.1970			0.588691			0.294346	−	0.955699i	$-0.404898\pi$
$757$	16.1970			0.588691			0.294346	+	0.955699i	$0.404898\pi$
$758$		0			0
$759$	−23.9558	−	45.0178i	−0.869540	−	1.63404i
$760$		0			0
$761$		−	55.1722i		−	1.99999i	−0.00292787	−	0.999996i	$-0.500932\pi$
$761$		−	55.1722i		−	1.99999i	0.00292787	−	0.999996i	$-0.499068\pi$
$762$		0			0
$763$	−11.0656			−0.400600
$764$		−	65.6550i		−	2.37531i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−38.8152	−	72.9417i	−1.40062	−	2.63206i
$769$	34.5521			1.24598			0.622989	−	0.782230i	$-0.285918\pi$
$769$	34.5521			1.24598			0.622989	+	0.782230i	$0.285918\pi$
$770$		0			0
$771$	−31.4659	+	16.7443i	−1.13322	+	0.603030i
$772$	152.349			5.48317
$773$		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
$778$	71.5310			2.56451
$779$		−	65.3628i		−	2.34187i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−95.2550			−3.40197
$785$		0			0
$786$		0			0
$787$	−51.2187			−1.82575			−0.912875	−	0.408240i	$-0.866143\pi$
$787$	−51.2187			−1.82575			−0.912875	+	0.408240i	$0.866143\pi$
$788$		−	28.7953i		−	1.02579i
$789$		0			0
$790$		0			0
$791$			6.62983i			0.235729i
$792$	53.1336	−	78.8881i	1.88802	−	2.80317i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−119.004			−4.21798
$797$			47.8330i			1.69433i	0.531327	+	0.847167i	$0.321693\pi$
$797$			47.8330i			1.69433i	−0.531327	+	0.847167i	$0.678307\pi$
$798$	31.0739	−	16.5357i	1.10000	−	0.585356i
$799$		0			0
$800$			112.175i			3.96600i
$801$		0			0
$802$		0			0
$803$		−	14.7337i		−	0.519941i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	49.9744	−	26.5934i	1.75918	−	0.936132i
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	−56.3288			−1.97797			−0.988986	−	0.148008i	$-0.952714\pi$
$811$	−56.3288			−1.97797			−0.988986	+	0.148008i	$0.952714\pi$
$812$		0			0
$813$	0.136119	+	0.255796i	0.00477391	+	0.00897115i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$			19.7383i			0.690134i
$819$	5.14637	−	7.64088i	0.179829	−	0.266994i
$820$		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
$823$	38.8371			1.35378			0.676888	−	0.736086i	$-0.263328\pi$
$823$	38.8371			1.35378			0.676888	+	0.736086i	$0.263328\pi$
$824$			162.890i			5.67455i
$825$	−25.3562	+	13.4931i	−0.882790	+	0.469768i
$826$		0			0
$827$		−	43.5638i		−	1.51486i	−0.652916	−	0.757431i	$-0.726454\pi$
$827$		−	43.5638i		−	1.51486i	0.652916	−	0.757431i	$-0.273546\pi$
$828$	−121.316	−	81.7102i	−4.21602	−	2.83963i
$829$	−44.1356			−1.53289			−0.766447	−	0.642308i	$-0.777977\pi$
$829$	−44.1356			−1.53289			−0.766447	+	0.642308i	$0.777977\pi$
$830$		0			0
$831$		0			0
$832$	111.944			3.88095
$833$		0			0
$834$		0			0
$835$		0			0
$836$			158.797i			5.49211i
$837$		0			0
$838$	39.7622			1.37356
$839$		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
$843$	−42.2071	+	22.4601i	−1.45369	+	0.773567i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−9.36855			−0.321907
$848$		−	160.373i		−	5.50724i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	30.7620			1.05327			0.526636	−	0.850091i	$-0.323453\pi$
$853$	30.7620			1.05327			0.526636	+	0.850091i	$0.323453\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$	26.6330	+	50.0489i	0.909236	+	1.70864i
$859$	−57.9805			−1.97827			−0.989135	−	0.147013i	$-0.953034\pi$
$859$	−57.9805			−1.97827			−0.989135	+	0.147013i	$0.953034\pi$
$860$		0			0
$861$	9.76424	−	5.19595i	0.332764	−	0.177077i
$862$	39.2082			1.33544
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	12.0687	−	115.950i	0.410586	−	3.94469i
$865$		0			0
$866$		−	17.1165i		−	0.581642i
$867$	13.8323	+	25.9936i	0.469768	+	0.882790i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		−	124.199i		−	4.20590i
$873$		0			0
$874$	211.821			7.16494
$875$		0			0
$876$	−19.8525	−	37.3069i	−0.670754	−	1.26048i
$877$	−56.7471			−1.91621			−0.958107	−	0.286411i	$-0.907538\pi$
$877$	−56.7471			−1.91621			−0.958107	+	0.286411i	$0.907538\pi$
$878$		0			0
$879$	40.5699	−	21.5889i	1.36839	−	0.728175i
$880$		0			0
$881$			44.4812i			1.49861i	0.662224	+	0.749306i	$0.269613\pi$
$881$			44.4812i			1.49861i	−0.662224	+	0.749306i	$0.730387\pi$
$882$	−42.7355	−	28.7837i	−1.43898	−	0.969197i
$883$	−38.4526			−1.29403			−0.647017	−	0.762476i	$-0.723984\pi$
$883$	−38.4526			−1.29403			−0.647017	+	0.762476i	$0.723984\pi$
$884$		0			0
$885$		0			0
$886$	−106.028			−3.56210
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	27.6610	−	11.2190i	0.926680	−	0.375852i
$892$		0			0
$893$		0			0
$894$	53.0893	+	99.7655i	1.77557	+	3.33666i
$895$		0			0
$896$			34.1643i			1.14135i
$897$	48.9395	−	26.0427i	1.63404	−	0.869540i
$898$		0			0
$899$		0			0
$900$	−46.0231	+	68.3311i	−1.53410	+	2.27770i
$901$		0			0
$902$			68.0684i			2.26643i
$903$		0			0
$904$	−74.4124			−2.47492
$905$		0			0
$906$	−60.3612	+	32.1206i	−2.00537	+	1.06714i
$907$	58.9480			1.95734			0.978668	−	0.205449i	$-0.0658654\pi$
$907$	58.9480			1.95734			0.978668	+	0.205449i	$0.0658654\pi$
$908$		−	121.541i		−	4.03349i
$909$		0			0
$910$		0			0
$911$		−	15.3323i		−	0.507983i	−0.967206	−	0.253992i	$-0.918256\pi$
$911$		−	15.3323i		−	0.507983i	0.967206	−	0.253992i	$-0.0817436\pi$
$912$	107.680	+	202.352i	3.56563	+	6.70054i
$913$	−2.22774			−0.0737276
$914$		−	107.606i		−	3.55928i
$915$		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
$921$	23.4696	+	44.1042i	0.773350	+	1.45328i
$922$	−74.2893			−2.44659
$923$		0			0
$924$	−23.7219	+	12.6234i	−0.780394	+	0.415279i
$925$		0			0
$926$		0			0
$927$	−28.5576	+	42.3998i	−0.937956	+	1.39259i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	54.6988			1.79268
$932$		0			0
$933$	−9.77741	+	5.20295i	−0.320098	+	0.170337i
$934$	−65.4646			−2.14207
$935$		0			0
$936$	85.7604	+	57.7623i	2.80317	+	1.88802i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$	−14.5508	−	27.3440i	−0.474849	−	0.892337i
$940$		0			0
$941$		−	60.4502i		−	1.97062i	−0.170771	−	0.985311i	$-0.554626\pi$
$941$		−	60.4502i		−	1.97062i	0.170771	−	0.985311i	$-0.445374\pi$
$942$	100.551	−	53.5071i	3.27612	−	1.74335i
$943$	66.5596			2.16748
$944$		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
$949$	16.0172			0.519941
$950$		−	119.308i		−	3.87085i
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$	48.4608	−	71.9503i	1.56898	−	2.32948i
$955$		0			0
$956$			58.8626i			1.90375i
$957$		0			0
$958$	18.1566			0.586613
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$	−133.638			−4.30420
$965$		0			0
$966$	16.8384	+	31.6429i	0.541768	+	1.01809i
$967$	−52.4720			−1.68739			−0.843693	−	0.536825i	$-0.819623\pi$
$967$	−52.4720			−1.68739			−0.843693	+	0.536825i	$0.819623\pi$
$968$		−	105.152i		−	3.37970i
$969$		0			0
$970$		0			0
$971$		−	51.4306i		−	1.65049i	−0.564778	−	0.825243i	$-0.691038\pi$
$971$		−	51.4306i		−	1.65049i	0.564778	−	0.825243i	$-0.308962\pi$
$972$	54.9232	−	65.6786i	1.76166	−	2.10664i
$973$		0			0
$974$		0			0
$975$	−14.6685	−	27.5651i	−0.469768	−	0.882790i
$976$		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
$981$	21.7743	−	32.3285i	0.695200	−	1.03217i
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	58.3187	+	109.593i	1.85913	+	3.49369i
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−172.631			−5.49211
$989$		0			0
$990$		0			0
$991$	50.9591			1.61877			0.809385	−	0.587278i	$-0.199801\pi$
$991$	50.9591			1.61877			0.809385	+	0.587278i	$0.199801\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	−5.64083	+	3.00171i	−0.178737	+	0.0951130i
$997$		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.428.1	✓	20
3.2	odd	2	inner	429.2.e.d.428.20	yes	20
11.10	odd	2	inner	429.2.e.d.428.19	yes	20
13.12	even	2	inner	429.2.e.d.428.19	yes	20
33.32	even	2	inner	429.2.e.d.428.2	yes	20
39.38	odd	2	inner	429.2.e.d.428.2	yes	20
143.142	odd	2	CM	429.2.e.d.428.1	✓	20
429.428	even	2	inner	429.2.e.d.428.20	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.e.d.428.1	✓	20	1.1	even	1	trivial
429.2.e.d.428.1	✓	20	143.142	odd	2	CM
429.2.e.d.428.2	yes	20	33.32	even	2	inner
429.2.e.d.428.2	yes	20	39.38	odd	2	inner
429.2.e.d.428.19	yes	20	11.10	odd	2	inner
429.2.e.d.428.19	yes	20	13.12	even	2	inner
429.2.e.d.428.20	yes	20	3.2	odd	2	inner
429.2.e.d.428.20	yes	20	429.428	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)

\(f(q)\)	\(=\)	\(q-2.73721i q^{2} +(-0.813662 - 1.52904i) q^{3} -5.49232 q^{4} +(-4.18530 + 2.22717i) q^{6} +0.851686 q^{7} +9.55923i q^{8} +(-1.67591 + 2.48824i) q^{9} +O(q^{10})\)	\(q-2.73721i q^{2} +(-0.813662 - 1.52904i) q^{3} -5.49232 q^{4} +(-4.18530 + 2.22717i) q^{6} +0.851686 q^{7} +9.55923i q^{8} +(-1.67591 + 2.48824i) q^{9} +3.31662i q^{11} +(4.46890 + 8.39797i) q^{12} -3.60555 q^{13} -2.33124i q^{14} +15.1810 q^{16} +(6.81084 + 4.58731i) q^{18} -8.71746 q^{19} +(-0.692985 - 1.30226i) q^{21} +9.07830 q^{22} -8.87708i q^{23} +(14.6164 - 7.77798i) q^{24} -5.00000 q^{25} +9.86915i q^{26} +(5.16823 + 0.537939i) q^{27} -4.67774 q^{28} -22.4351i q^{32} +(5.07124 - 2.69861i) q^{33} +(9.20463 - 13.6662i) q^{36} +23.8615i q^{38} +(2.93370 + 5.51302i) q^{39} +7.49792i q^{41} +(-3.56456 + 1.89685i) q^{42} -18.2160i q^{44} -24.2984 q^{46} +(-12.3522 - 23.2123i) q^{48} -6.27463 q^{49} +13.6861i q^{50} +19.8029 q^{52} -10.5641i q^{53} +(1.47245 - 14.1465i) q^{54} +8.14146i q^{56} +(7.09306 + 13.3293i) q^{57} +(-1.42735 + 2.11920i) q^{63} -31.0476 q^{64} +(-7.38667 - 13.8811i) q^{66} +(-13.5734 + 7.22294i) q^{69} +(-23.7856 - 16.0204i) q^{72} -4.44237 q^{73} +(4.06831 + 7.64518i) q^{75} +47.8791 q^{76} +2.82472i q^{77} +(15.0903 - 8.03016i) q^{78} +(-3.38267 - 8.34012i) q^{81} +20.5234 q^{82} +0.671690i q^{83} +(3.80610 + 7.15243i) q^{84} -31.7044 q^{88} -3.07080 q^{91} +48.7558i q^{92} +(-34.3041 + 18.2546i) q^{96} +17.1750i q^{98} +(-8.25256 - 5.55836i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 40 q^{4}+O(q^{10}) \) 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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