LMFDB - Embedded newform 429.2.e.d.428.7

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			428.7
Root			$-1.19153 + 0.761743i$ of defining polynomial
Character	$\chi$	$=$	429.428
Dual form			429.2.e.d.428.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.52349i q^{2} +(-1.70564 - 0.301340i) q^{3} -0.321008 q^{4} +(-0.459088 + 2.59851i) q^{6} +3.75874 q^{7} -2.55792i q^{8} +(2.81839 + 1.02795i) q^{9} +O(q^{10})$	\(q-1.52349i q^{2} +(-1.70564 - 0.301340i) q^{3} -0.321008 q^{4} +(-0.459088 + 2.59851i) q^{6} +3.75874 q^{7} -2.55792i q^{8} +(2.81839 + 1.02795i) q^{9} +3.31662i q^{11} +(0.547523 + 0.0967327i) q^{12} +3.60555 q^{13} -5.72639i q^{14} -4.53897 q^{16} +(1.56607 - 4.29377i) q^{18} +2.76726 q^{19} +(-6.41105 - 1.13266i) q^{21} +5.05283 q^{22} -9.31703i q^{23} +(-0.770804 + 4.36288i) q^{24} -5.00000 q^{25} -5.49301i q^{26} +(-4.49738 - 2.60261i) q^{27} -1.20659 q^{28} +1.79922i q^{32} +(0.999433 - 5.65695i) q^{33} +(-0.904725 - 0.329981i) q^{36} -4.21588i q^{38} +(-6.14976 - 1.08650i) q^{39} +0.0363045i q^{41} +(-1.72559 + 9.76714i) q^{42} -1.06466i q^{44} -14.1944 q^{46} +(7.74183 + 1.36777i) q^{48} +7.12816 q^{49} +7.61743i q^{50} -1.15741 q^{52} +12.7941i q^{53} +(-3.96504 + 6.85169i) q^{54} -9.61457i q^{56} +(-4.71994 - 0.833888i) q^{57} +(10.5936 + 3.86382i) q^{63} -6.33686 q^{64} +(-8.61829 - 1.52262i) q^{66} +(-2.80760 + 15.8915i) q^{69} +(2.62942 - 7.20921i) q^{72} +17.0656 q^{73} +(8.52818 + 1.50670i) q^{75} -0.888313 q^{76} +12.4663i q^{77} +(-1.65526 + 9.36907i) q^{78} +(6.88662 + 5.79435i) q^{81} +0.0553094 q^{82} -11.2461i q^{83} +(2.05800 + 0.363593i) q^{84} +8.48366 q^{88} +13.5523 q^{91} +2.99084i q^{92} +(0.542176 - 3.06881i) q^{96} -10.8596i q^{98} +(-3.40934 + 9.34754i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	1.52349i		−	1.07727i	−0.842540	−	0.538633i	$-0.818941\pi$
$2$		−	1.52349i		−	1.07727i	0.842540	−	0.538633i	$-0.181059\pi$
$3$	−1.70564	−	0.301340i	−0.984749	−	0.173979i
$3$	−1.70564	−	0.301340i	−0.984749	−	0.173979i
$4$	−0.321008			−0.160504
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	−0.459088	+	2.59851i	−0.187422	+	1.06084i
$7$	3.75874			1.42067			0.710336	−	0.703863i	$-0.248543\pi$
$7$	3.75874			1.42067			0.710336	+	0.703863i	$0.248543\pi$
$8$		−	2.55792i		−	0.904361i
$9$	2.81839	+	1.02795i	0.939463	+	0.342651i
$10$		0			0
$11$			3.31662i			1.00000i
$11$			3.31662i			1.00000i
$12$	0.547523	+	0.0967327i	0.158056	+	0.0279243i
$13$	3.60555			1.00000
$13$	3.60555			1.00000
$14$		−	5.72639i		−	1.53044i
$15$		0			0
$16$	−4.53897			−1.13474
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	1.56607	−	4.29377i	0.369127	−	1.01205i
$19$	2.76726			0.634853			0.317427	−	0.948283i	$-0.397181\pi$
$19$	2.76726			0.634853			0.317427	+	0.948283i	$0.397181\pi$
$20$		0			0
$21$	−6.41105	−	1.13266i	−1.39901	−	0.247167i
$22$	5.05283			1.07727
$23$		−	9.31703i		−	1.94274i	−0.237580	−	0.971368i	$-0.576354\pi$
$23$		−	9.31703i		−	1.94274i	0.237580	−	0.971368i	$-0.423646\pi$
$24$	−0.770804	+	4.36288i	−0.157340	+	0.890569i
$25$	−5.00000			−1.00000
$26$		−	5.49301i		−	1.07727i
$27$	−4.49738	−	2.60261i	−0.865521	−	0.500872i
$28$	−1.20659			−0.228024
$29$		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			$\pi$
$32$			1.79922i			0.318059i
$33$	0.999433	−	5.65695i	0.173979	−	0.984749i
$34$		0			0
$35$		0			0
$36$	−0.904725	−	0.329981i	−0.150788	−	0.0549969i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		−	4.21588i		−	0.683906i
$39$	−6.14976	−	1.08650i	−0.984749	−	0.173979i
$40$		0			0
$41$			0.0363045i			0.00566981i	0.999996	+	0.00283490i	$0.000902379\pi$
$41$			0.0363045i			0.00566981i	−0.999996	+	0.00283490i	$0.999098\pi$
$42$	−1.72559	+	9.76714i	−0.266265	+	1.50710i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		−	1.06466i		−	0.160504i
$45$		0			0
$46$	−14.1944			−2.09284
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	7.74183	+	1.36777i	1.11744	+	0.197421i
$49$	7.12816			1.01831
$50$			7.61743i			1.07727i
$51$		0			0
$52$	−1.15741			−0.160504
$53$			12.7941i			1.75740i	0.477376	+	0.878699i	$0.341588\pi$
$53$			12.7941i			1.75740i	−0.477376	+	0.878699i	$0.658412\pi$
$54$	−3.96504	+	6.85169i	−0.539573	+	0.932397i
$55$		0			0
$56$		−	9.61457i		−	1.28480i
$57$	−4.71994	−	0.833888i	−0.625171	−	0.110451i
$58$		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
$63$	10.5936	+	3.86382i	1.33467	+	0.486795i
$64$	−6.33686			−0.792108
$65$		0			0
$66$	−8.61829	−	1.52262i	−1.06084	−	0.187422i
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	−2.80760	+	15.8915i	−0.337995	+	1.91311i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	2.62942	−	7.20921i	0.309881	−	0.849614i
$73$	17.0656			1.99738			0.998691	−	0.0511455i	$-0.0162872\pi$
$73$	17.0656			1.99738			0.998691	+	0.0511455i	$0.0162872\pi$
$74$		0			0
$75$	8.52818	+	1.50670i	0.984749	+	0.173979i
$76$	−0.888313			−0.101897
$77$			12.4663i			1.42067i
$78$	−1.65526	+	9.36907i	−0.187422	+	1.06084i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	6.88662	+	5.79435i	0.765180	+	0.643816i
$82$	0.0553094			0.00610790
$83$		−	11.2461i		−	1.23442i	−0.786799	−	0.617209i	$-0.788263\pi$
$83$		−	11.2461i		−	1.23442i	0.786799	−	0.617209i	$-0.211737\pi$
$84$	2.05800	+	0.363593i	0.224546	+	0.0396713i
$85$		0			0
$86$		0			0
$87$		0			0
$88$	8.48366			0.904361
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	13.5523			1.42067
$92$			2.99084i			0.311817i
$93$		0			0
$94$		0			0
$95$		0			0
$96$	0.542176	−	3.06881i	0.0553356	−	0.313209i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		−	10.8596i		−	1.09699i
$99$	−3.40934	+	9.34754i	−0.342651	+	0.939463i
$100$	1.60504			0.160504
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	−20.2683			−1.99710			−0.998548	−	0.0538681i	$-0.982845\pi$
$103$	−20.2683			−1.99710			−0.998548	+	0.0538681i	$0.982845\pi$
$104$		−	9.22271i		−	0.904361i
$105$		0			0
$106$	19.4916			1.89319
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	1.44370	+	0.835458i	0.138920	+	0.0803920i
$109$	−11.5311			−1.10448			−0.552240	−	0.833685i	$-0.686227\pi$
$109$	−11.5311			−1.10448			−0.552240	+	0.833685i	$0.686227\pi$
$110$		0			0
$111$		0			0
$112$	−17.0608			−1.61210
$113$			16.4101i			1.54374i	0.635783	+	0.771868i	$0.280677\pi$
$113$			16.4101i			1.54374i	−0.635783	+	0.771868i	$0.719323\pi$
$114$	−1.27042	+	7.19076i	−0.118985	+	0.673476i
$115$		0			0
$116$		0			0
$117$	10.1618	+	3.70634i	0.939463	+	0.342651i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	0.0109400	−	0.0619223i	0.000986427	−	0.00558334i
$124$		0			0
$125$		0			0
$126$	5.88647	−	16.1392i	0.524408	−	1.43779i
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$			13.2525i			1.17137i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	−0.320826	+	1.81593i	−0.0279243	+	0.158056i
$133$	10.4014			0.901918
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$	24.2104	+	4.27733i	2.06093	+	0.364111i
$139$		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
$144$	−12.7926	−	4.66585i	−1.06605	−	0.388821i
$145$		0			0
$146$		−	25.9993i		−	2.15171i
$147$	−12.1580	−	2.14800i	−1.00278	−	0.177164i
$148$		0			0
$149$			22.3833i			1.83371i	0.399223	+	0.916854i	$0.369280\pi$
$149$			22.3833i			1.83371i	−0.399223	+	0.916854i	$0.630720\pi$
$150$	2.29544	−	12.9926i	0.187422	−	1.06084i
$151$	−14.4222			−1.17366			−0.586831	−	0.809709i	$-0.699625\pi$
$151$	−14.4222			−1.17366			−0.586831	+	0.809709i	$0.699625\pi$
$152$		−	7.07843i		−	0.574137i
$153$		0			0
$154$	18.9923			1.53044
$155$		0			0
$156$	1.97412	+	0.348775i	0.158056	+	0.0279243i
$157$	23.6263			1.88558			0.942792	−	0.333382i	$-0.108190\pi$
$157$	23.6263			1.88558			0.942792	+	0.333382i	$0.108190\pi$
$158$		0			0
$159$	3.85536	−	21.8220i	0.305750	−	1.73060i
$160$		0			0
$161$		−	35.0203i		−	2.75999i
$162$	8.82760	−	10.4917i	0.693562	−	0.824303i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		−	0.0116540i		−	0.000910027i
$165$		0			0
$166$	−17.1332			−1.32980
$167$			18.7069i			1.44758i	0.690020	+	0.723791i	$0.257602\pi$
$167$			18.7069i			1.44758i	−0.690020	+	0.723791i	$0.742398\pi$
$168$	−2.89726	+	16.3989i	−0.223528	+	1.26521i
$169$	13.0000			1.00000
$170$		0			0
$171$	7.79922	+	2.84462i	0.596421	+	0.217533i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	−18.7937			−1.42067
$176$		−	15.0541i		−	1.13474i
$177$		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
$181$	−20.3395			−1.51182			−0.755911	−	0.654675i	$-0.772806\pi$
$181$	−20.3395			−1.51182			−0.755911	+	0.654675i	$0.772806\pi$
$182$		−	20.6468i		−	1.53044i
$183$		0			0
$184$	−23.8322			−1.75693
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−16.9045	−	9.78254i	−1.22962	−	0.711575i
$190$		0			0
$191$		−	27.3961i		−	1.98232i	−0.132689	−	0.991158i	$-0.542361\pi$
$191$		−	27.3961i		−	1.98232i	0.132689	−	0.991158i	$-0.457639\pi$
$192$	10.8084	+	1.90955i	0.780028	+	0.137810i
$193$	−23.3836			−1.68319			−0.841595	−	0.540110i	$-0.818383\pi$
$193$	−23.3836			−1.68319			−0.841595	+	0.540110i	$0.818383\pi$
$194$		0			0
$195$		0			0
$196$	−2.28820			−0.163443
$197$			27.8478i			1.98407i	0.125953	+	0.992036i	$0.459801\pi$
$197$			27.8478i			1.98407i	−0.125953	+	0.992036i	$0.540199\pi$
$198$	14.2408	+	5.19408i	1.01205	+	0.369127i
$199$	−10.4901			−0.743625			−0.371813	−	0.928308i	$-0.621264\pi$
$199$	−10.4901			−0.743625			−0.371813	+	0.928308i	$0.621264\pi$
$200$			12.7896i			0.904361i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$			30.8785i			2.15141i
$207$	9.57748	−	26.2590i	0.665681	−	1.82513i
$208$	−16.3655			−1.13474
$209$			9.17797i			0.634853i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		−	4.10699i		−	0.282069i
$213$		0			0
$214$		0			0
$215$		0			0
$216$	−6.65727	+	11.5039i	−0.452970	+	0.782744i
$217$		0			0
$218$			17.5675i			1.18982i
$219$	−29.1078	−	5.14257i	−1.96692	−	0.347502i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$			6.76279i			0.451858i
$225$	−14.0919	−	5.13977i	−0.939463	−	0.342651i
$226$	25.0006			1.66302
$227$		−	12.6129i		−	0.837150i	−0.908182	−	0.418575i	$-0.862530\pi$
$227$		−	12.6129i		−	0.837150i	0.908182	−	0.418575i	$-0.137470\pi$
$228$	1.51514	+	0.267685i	0.100343	+	0.0177278i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$	3.75661	−	21.2630i	0.247167	−	1.39901i
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$	5.64656	−	15.4814i	0.369127	−	1.01205i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	30.8948i		−	1.99842i	−0.0397962	−	0.999208i	$-0.512671\pi$
$239$		−	30.8948i		−	1.99842i	0.0397962	−	0.999208i	$-0.487329\pi$
$240$		0			0
$241$	8.34864			0.537783			0.268892	−	0.963170i	$-0.413343\pi$
$241$	8.34864			0.537783			0.268892	+	0.963170i	$0.413343\pi$
$242$			16.7583i			1.07727i
$243$	−10.0000	−	11.9583i	−0.641500	−	0.767123i
$244$		0			0
$245$		0			0
$246$	−0.0943377	−	0.0166669i	−0.00601475	−	0.00106265i
$247$	9.97750			0.634853
$248$		0			0
$249$	−3.38890	+	19.1817i	−0.214763	+	1.21559i
$250$		0			0
$251$			3.01841i			0.190521i	0.995452	+	0.0952603i	$0.0303683\pi$
$251$			3.01841i			0.190521i	−0.995452	+	0.0952603i	$0.969632\pi$
$252$	−3.40063	−	1.24032i	−0.214220	−	0.0781325i
$253$	30.9011			1.94274
$254$		0			0
$255$		0			0
$256$	7.51634			0.469771
$257$		−	31.1004i		−	1.93999i	−0.243124	−	0.969995i	$-0.578172\pi$
$257$		−	31.1004i		−	1.93999i	0.243124	−	0.969995i	$-0.421828\pi$
$258$		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	$0.5\pi$
$264$	−14.4700	−	2.55647i	−0.890569	−	0.157340i
$265$		0			0
$266$		−	15.8464i		−	0.971607i
$267$		0			0
$268$		0			0
$269$			24.8018i			1.51219i	0.654461	+	0.756096i	$0.272896\pi$
$269$			24.8018i			1.51219i	−0.654461	+	0.756096i	$0.727104\pi$
$270$		0			0
$271$	31.3640			1.90523			0.952614	−	0.304181i	$-0.0983826\pi$
$271$	31.3640			1.90523			0.952614	+	0.304181i	$0.0983826\pi$
$272$		0			0
$273$	−23.1154	−	4.08387i	−1.39901	−	0.247167i
$274$		0			0
$275$		−	16.5831i		−	1.00000i
$276$	0.901261	−	5.10129i	0.0542496	−	0.307061i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			9.56596i			0.570657i	0.958430	+	0.285329i	$0.0921027\pi$
$281$			9.56596i			0.570657i	−0.958430	+	0.285329i	$0.907897\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$	18.2182			1.07727
$287$			0.136459i			0.00805494i
$288$	−1.84951	+	5.07089i	−0.108983	+	0.298805i
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$	−5.47821			−0.320588
$293$			26.5330i			1.55007i	0.631916	+	0.775037i	$0.282269\pi$
$293$			26.5330i			1.55007i	−0.631916	+	0.775037i	$0.717731\pi$
$294$	−3.27245	+	18.5226i	−0.190853	+	1.08026i
$295$		0			0
$296$		0			0
$297$	8.63188	−	14.9161i	0.500872	−	0.865521i
$298$	34.1006			1.97539
$299$		−	33.5930i		−	1.94274i
$300$	−2.73761	−	0.483663i	−0.158056	−	0.0279243i
$301$		0			0
$302$			21.9720i			1.26435i
$303$		0			0
$304$	−12.5605			−0.720395
$305$		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	$0.192232\pi$
$308$		−	4.00180i		−	0.228024i
$309$	34.5704	+	6.10766i	1.96664	+	0.347453i
$310$		0			0
$311$		−	31.0122i		−	1.75854i	−0.476322	−	0.879271i	$-0.658030\pi$
$311$		−	31.0122i		−	1.75854i	0.476322	−	0.879271i	$-0.341970\pi$
$312$	−2.77917	+	15.7306i	−0.157340	+	0.890569i
$313$	3.47839			0.196610			0.0983052	−	0.995156i	$-0.468658\pi$
$313$	3.47839			0.196610			0.0983052	+	0.995156i	$0.468658\pi$
$314$		−	35.9943i		−	2.03128i
$315$		0			0
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$	−33.2455	−	5.87359i	−1.86431	−	0.329375i
$319$		0			0
$320$		0			0
$321$		0			0
$322$	−53.3530			−2.97325
$323$		0			0
$324$	−2.21066	−	1.86003i	−0.122814	−	0.103335i
$325$	−18.0278			−1.00000
$326$		0			0
$327$	19.6679	+	3.47479i	1.08764	+	0.192156i
$328$	0.0928640			0.00512756
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$			3.61008i			0.198129i
$333$		0			0
$334$	28.4996			1.55943
$335$		0			0
$336$	29.0996	+	5.14111i	1.58751	+	0.280471i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		−	19.8053i		−	1.07727i
$339$	4.94504	−	27.9897i	0.268577	−	1.52019i
$340$		0			0
$341$		0			0
$342$	4.33373	−	11.8820i	0.234341	−	0.642505i
$343$	0.481706			0.0260097
$344$		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
$349$	−0.831151			−0.0444905			−0.0222453	−	0.999753i	$-0.507081\pi$
$349$	−0.831151			−0.0444905			−0.0222453	+	0.999753i	$0.507081\pi$
$350$			28.6320i			1.53044i
$351$	−16.2155	−	9.38384i	−0.865521	−	0.500872i
$352$	−5.96732			−0.318059
$353$		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
$358$	36.4365			1.92573
$359$			33.6656i			1.77681i	0.459065	+	0.888403i	$0.348184\pi$
$359$			33.6656i			1.77681i	−0.459065	+	0.888403i	$0.651816\pi$
$360$		0			0
$361$	−11.3423			−0.596961
$362$			30.9869i			1.62864i
$363$	18.7620	+	3.31474i	0.984749	+	0.173979i
$364$	−4.35041			−0.228024
$365$		0			0
$366$		0			0
$367$	−30.9578			−1.61598			−0.807991	−	0.589195i	$-0.799445\pi$
$367$	−30.9578			−1.61598			−0.807991	+	0.589195i	$0.799445\pi$
$368$			42.2897i			2.20450i
$369$	−0.0373193	+	0.102320i	−0.00194277	+	0.00532657i
$370$		0			0
$371$			48.0896i			2.49669i
$372$		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
$378$	−14.9036	+	25.7538i	−0.766556	+	1.32463i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$	−41.7376			−2.13548
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	3.99353	−	22.6040i	0.203794	−	1.15351i
$385$		0			0
$386$			35.6246i			1.81324i
$387$		0			0
$388$		0			0
$389$			20.0262i			1.01537i	0.861543	+	0.507685i	$0.169499\pi$
$389$			20.0262i			1.01537i	−0.861543	+	0.507685i	$0.830501\pi$
$390$		0			0
$391$		0			0
$392$		−	18.2333i		−	0.920918i
$393$		0			0
$394$	42.4257			2.13738
$395$		0			0
$396$	1.09442	−	3.00063i	0.0549969	−	0.150788i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$			15.9816i			0.801083i
$399$	−17.7410	−	3.13437i	−0.888163	−	0.156915i
$400$	22.6948			1.13474
$401$		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
$412$	6.50629			0.320542
$413$		0			0
$414$	−40.0052	−	14.5912i	−1.96615	−	0.717116i
$415$		0			0
$416$			6.48716i			0.318059i
$417$		0			0
$418$	13.9825			0.683906
$419$			34.2497i			1.67321i	0.547808	+	0.836604i	$0.315462\pi$
$419$			34.2497i			1.67321i	−0.547808	+	0.836604i	$0.684538\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	32.7262			1.58932
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	3.60351	−	20.3964i	0.173979	−	0.984749i
$430$		0			0
$431$			11.3187i			0.545202i	0.962127	+	0.272601i	$0.0878840\pi$
$431$			11.3187i			0.545202i	−0.962127	+	0.272601i	$0.912116\pi$
$432$	20.4135	+	11.8132i	0.982144	+	0.568361i
$433$	41.0634			1.97338			0.986691	−	0.162607i	$-0.0519901\pi$
$433$	41.0634			1.97338			0.986691	+	0.162607i	$0.0519901\pi$
$434$		0			0
$435$		0			0
$436$	3.70158			0.177274
$437$		−	25.7827i		−	1.23335i
$438$	−7.83463	+	44.3453i	−0.374353	+	2.11890i
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	20.0899	+	7.32742i	0.956662	+	0.348925i
$442$		0			0
$443$		−	21.6525i		−	1.02874i	−0.857568	−	0.514370i	$-0.828026\pi$
$443$		−	21.6525i		−	1.02874i	0.857568	−	0.514370i	$-0.171974\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	6.74498	−	38.1777i	0.319027	−	1.80574i
$448$	−23.8186			−1.12533
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	−7.83036	+	21.4689i	−0.369127	+	1.01205i
$451$	−0.120408			−0.00566981
$452$		−	5.26779i		−	0.247776i
$453$	24.5990	+	4.34599i	1.15576	+	0.204193i
$454$	−19.2156			−0.901834
$455$		0			0
$456$	−2.13302	+	12.0732i	−0.0998877	+	0.565381i
$457$	−28.1347			−1.31609			−0.658043	−	0.752981i	$-0.728615\pi$
$457$	−28.1347			−1.31609			−0.658043	+	0.752981i	$0.728615\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	40.0357i		−	1.86465i	−0.361625	−	0.932323i	$-0.617778\pi$
$461$		−	40.0357i		−	1.86465i	0.361625	−	0.932323i	$-0.382222\pi$
$462$	−32.3939	−	5.72314i	−1.50710	−	0.266265i
$463$		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	$0.813345\pi$
$468$	−3.26203	−	1.18976i	−0.150788	−	0.0549969i
$469$		0			0
$470$		0			0
$471$	−40.2979	−	7.11956i	−1.85683	−	0.328052i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−13.8363			−0.634853
$476$		0			0
$477$	−13.1517	+	36.0586i	−0.602175	+	1.65101i
$478$	−47.0677			−2.15283
$479$			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
$482$		−	12.7190i		−	0.579336i
$483$	−10.5530	+	59.7320i	−0.480180	+	2.71790i
$484$	3.53109			0.160504
$485$		0			0
$486$	−18.2182	+	15.2349i	−0.826396	+	0.691067i
$487$		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	$0.5\pi$
$492$	−0.00351183	+	0.0198775i	−0.000158326	+	0.000896149i
$493$		0			0
$494$		−	15.2006i		−	0.683906i
$495$		0			0
$496$		0			0
$497$		0			0
$498$	29.2231	+	5.16294i	1.30952	+	0.231357i
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$	5.63714	−	31.9071i	0.251849	−	1.42550i
$502$	4.59851			0.205241
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	9.88333	−	27.0976i	0.440238	−	1.20702i
$505$		0			0
$506$		−	47.0774i		−	2.09284i
$507$	−22.1733	−	3.91742i	−0.984749	−	0.173979i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	64.1454			2.83762
$512$			15.0541i			0.665302i
$513$	−12.4454	−	7.20210i	−0.549479	−	0.317980i
$514$	−47.3810			−2.08989
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	18.7650i		−	0.822108i	−0.911611	−	0.411054i	$-0.865161\pi$
$521$		−	18.7650i		−	0.822108i	0.911611	−	0.411054i	$-0.134839\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$	32.0552	+	5.66331i	1.39901	+	0.247167i
$526$		0			0
$527$		0			0
$528$	−4.53640	+	25.6767i	−0.197421	+	1.11744i
$529$	−63.8071			−2.77422
$530$		0			0
$531$		0			0
$532$	−3.33894			−0.144761
$533$			0.130898i			0.00566981i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	7.20701	−	40.7929i	0.311006	−	1.76034i
$538$	37.7852			1.62903
$539$			23.6414i			1.01831i
$540$		0			0
$541$	−45.9361			−1.97495			−0.987473	−	0.157787i	$-0.949564\pi$
$541$	−45.9361			−1.97495			−0.987473	+	0.157787i	$0.949564\pi$
$542$		−	47.7826i		−	2.05244i
$543$	34.6917	+	6.12910i	1.48877	+	0.263025i
$544$		0			0
$545$		0			0
$546$	−6.22171	+	35.2159i	−0.266265	+	1.50710i
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$	−25.2641			−1.07727
$551$		0			0
$552$	40.6491	+	7.18161i	1.73014	+	0.305670i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	22.5285i		−	0.954562i	−0.878751	−	0.477281i	$-0.841622\pi$
$557$		−	22.5285i		−	0.954562i	0.878751	−	0.477281i	$-0.158378\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	14.5736			0.614750
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	25.8850	+	21.7795i	1.08707	+	0.914651i
$568$		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			$\pi$
$572$		−	3.83870i		−	0.160504i
$573$	−8.25556	+	46.7279i	−0.344881	+	1.95208i
$574$	0.207894			0.00867732
$575$			46.5852i			1.94274i
$576$	−17.8597	−	6.51400i	−0.744156	−	0.271417i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$			25.8993i			1.07727i
$579$	39.8839	+	7.04643i	1.65752	+	0.292839i
$580$		0			0
$581$		−	42.2712i		−	1.75370i
$582$		0			0
$583$	−42.4331			−1.75740
$584$		−	43.6525i		−	1.80636i
$585$		0			0
$586$	40.4226			1.66984
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	3.90283	+	0.689526i	0.160950	+	0.0284356i
$589$		0			0
$590$		0			0
$591$	8.39166	−	47.4982i	0.345187	−	1.95381i
$592$		0			0
$593$		−	44.8754i		−	1.84281i	−0.388600	−	0.921407i	$-0.627041\pi$
$593$		−	44.8754i		−	1.84281i	0.388600	−	0.921407i	$-0.372959\pi$
$594$	−22.7245	−	13.1505i	−0.932397	−	0.539573i
$595$		0			0
$596$		−	7.18521i		−	0.294318i
$597$	17.8923	+	3.16110i	0.732285	+	0.129375i
$598$	−51.1785			−2.09284
$599$		−	20.1640i		−	0.823878i	−0.911211	−	0.411939i	$-0.864852\pi$
$599$		−	20.1640i		−	0.823878i	0.911211	−	0.411939i	$-0.135148\pi$
$600$	3.85402	−	21.8144i	0.157340	−	0.890569i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	4.62964			0.188378
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$			4.97890i			0.201921i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	45.6624			1.84429			0.922144	−	0.386848i	$-0.126436\pi$
$613$	45.6624			1.84429			0.922144	+	0.386848i	$0.126436\pi$
$614$		−	43.9440i		−	1.77344i
$615$		0			0
$616$	31.8879			1.28480
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$	9.30493	−	52.6674i	0.374299	−	2.11860i
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$	−24.2486	+	41.9022i	−0.973063	+	1.68148i
$622$	−47.2467			−1.89442
$623$		0			0
$624$	27.9136	+	4.93158i	1.11744	+	0.197421i
$625$	25.0000			1.00000
$626$		−	5.29928i		−	0.211802i
$627$	2.76569	−	15.6543i	0.110451	−	0.625171i
$628$	−7.58423			−0.302644
$629$		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
$636$	−1.23760	+	7.00504i	−0.0490741	+	0.277768i
$637$	25.7009			1.01831
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	43.4359i		−	1.71561i	−0.513973	−	0.857807i	$-0.671827\pi$
$641$		−	43.4359i		−	1.71561i	0.513973	−	0.857807i	$-0.328173\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$			11.2418i			0.442989i
$645$		0			0
$646$		0			0
$647$		−	34.6283i		−	1.36138i	−0.732572	−	0.680690i	$-0.761680\pi$
$647$		−	34.6283i		−	1.36138i	0.732572	−	0.680690i	$-0.238320\pi$
$648$	14.8215	−	17.6154i	0.582242	−	0.691999i
$649$		0			0
$650$			27.4650i			1.07727i
$651$		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	$-0.385435\pi$
$654$	5.29379	−	29.9637i	0.207004	−	1.17168i
$655$		0			0
$656$		−	0.164785i		−	0.00643377i
$657$	48.0976	+	17.5427i	1.87647	+	0.684406i
$658$		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
$664$	−28.7666			−1.11636
$665$		0			0
$666$		0			0
$667$		0			0
$668$		−	6.00506i		−	0.232343i
$669$		0			0
$670$		0			0
$671$		0			0
$672$	2.03790	−	11.5349i	0.0786137	−	0.444967i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	22.4869	+	13.0130i	0.865521	+	0.500872i
$676$	−4.17310			−0.160504
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$	−42.6419	−	7.53369i	−1.63765	−	0.289330i
$679$		0			0
$680$		0			0
$681$	−3.80078	+	21.5131i	−0.145646	+	0.824383i
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	−2.50361	−	0.913145i	−0.0957280	−	0.0349150i
$685$		0			0
$686$		−	0.733873i		−	0.0280194i
$687$		0			0
$688$		0			0
$689$			46.1296i			1.75740i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$	−12.8148	+	35.1350i	−0.486795	+	1.33467i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$			1.26625i			0.0479282i
$699$		0			0
$700$	6.03294			0.228024
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	−14.2961	+	24.7041i	−0.539573	+	0.932397i
$703$		0			0
$704$		−	21.0170i		−	0.792108i
$705$		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
$716$		−	7.67740i		−	0.286918i
$717$	−9.30984	+	52.6952i	−0.347682	+	1.96794i
$718$	51.2891			1.91409
$719$			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
$721$	−76.1834			−2.83722
$722$			17.2798i			0.643087i
$723$	−14.2397	−	2.51578i	−0.529582	−	0.0935629i
$724$	6.52913			0.242653
$725$		0			0
$726$	5.04996	−	28.5836i	0.187422	−	1.06084i
$727$	47.3730			1.75697			0.878484	−	0.477772i	$-0.158556\pi$
$727$	47.3730			1.75697			0.878484	+	0.477772i	$0.158556\pi$
$728$		−	34.6658i		−	1.28480i
$729$	13.4529	+	23.4098i	0.498254	+	0.867031i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−14.2038			−0.524630			−0.262315	−	0.964982i	$-0.584486\pi$
$733$	−14.2038			−0.524630			−0.262315	+	0.964982i	$0.584486\pi$
$734$			47.1637i			1.74084i
$735$		0			0
$736$	16.7633			0.617905
$737$		0			0
$738$	0.155883	+	0.0568555i	0.00573814	+	0.00209288i
$739$	35.4910			1.30556			0.652779	−	0.757549i	$-0.273603\pi$
$739$	35.4910			1.30556			0.652779	+	0.757549i	$0.273603\pi$
$740$		0			0
$741$	−17.0180	−	3.00662i	−0.625171	−	0.110451i
$742$	73.2638			2.68960
$743$		−	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
$747$	11.5605	−	31.6958i	0.422975	−	1.15969i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−51.4254			−1.87654			−0.938270	−	0.345905i	$-0.887572\pi$
$751$	−51.4254			−1.87654			−0.938270	+	0.345905i	$0.887572\pi$
$752$		0			0
$753$	0.909569	−	5.14831i	0.0331466	−	0.187615i
$754$		0			0
$755$		0			0
$756$	5.42648	+	3.14027i	0.197359	+	0.114211i
$757$	−44.0150			−1.59975			−0.799876	−	0.600165i	$-0.795102\pi$
$757$	−44.0150			−1.59975			−0.799876	+	0.600165i	$0.795102\pi$
$758$		0			0
$759$	−52.7060	−	9.31175i	−1.91311	−	0.337995i
$760$		0			0
$761$			44.7302i			1.62147i	0.585414	+	0.810734i	$0.300932\pi$
$761$			44.7302i			1.62147i	−0.585414	+	0.810734i	$0.699068\pi$
$762$		0			0
$763$	−43.3425			−1.56910
$764$			8.79438i			0.318170i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−12.8201	−	2.26498i	−0.462607	−	0.0817303i
$769$	53.4536			1.92758			0.963792	−	0.266654i	$-0.0859179\pi$
$769$	53.4536			1.92758			0.963792	+	0.266654i	$0.0859179\pi$
$770$		0			0
$771$	−9.37181	+	53.0460i	−0.337517	+	1.91040i
$772$	7.50633			0.270159
$773$		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
$778$	30.5097			1.09382
$779$			0.100464i			0.00359950i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−32.3545			−1.15552
$785$		0			0
$786$		0			0
$787$	−27.9735			−0.997148			−0.498574	−	0.866847i	$-0.666143\pi$
$787$	−27.9735			−0.997148			−0.498574	+	0.866847i	$0.666143\pi$
$788$		−	8.93936i		−	0.318452i
$789$		0			0
$790$		0			0
$791$			61.6815i			2.19314i
$792$	23.9102	+	8.72081i	0.849614	+	0.309881i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	3.36742			0.119355
$797$		−	47.8330i		−	1.69433i	−0.531327	−	0.847167i	$-0.678307\pi$
$797$		−	47.8330i		−	1.69433i	0.531327	−	0.847167i	$-0.321693\pi$
$798$	−4.77517	+	27.0282i	−0.169039	+	0.956789i
$799$		0			0
$800$		−	8.99608i		−	0.318059i
$801$		0			0
$802$		0			0
$803$			56.6003i			1.99738i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	7.47378	−	42.3028i	0.263089	−	1.48913i
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	−50.5260			−1.77421			−0.887103	−	0.461571i	$-0.847286\pi$
$811$	−50.5260			−1.77421			−0.887103	+	0.461571i	$0.847286\pi$
$812$		0			0
$813$	−53.4956	−	9.45125i	−1.87617	−	0.331470i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		−	10.9860i		−	0.384117i
$819$	38.1958	+	13.9312i	1.33467	+	0.486795i
$820$		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
$823$	−6.59556			−0.229907			−0.114953	−	0.993371i	$-0.536672\pi$
$823$	−6.59556			−0.229907			−0.114953	+	0.993371i	$0.536672\pi$
$824$			51.8447i			1.80610i
$825$	−4.99716	+	28.2848i	−0.173979	+	0.984749i
$826$		0			0
$827$		−	49.1766i		−	1.71004i	−0.518597	−	0.855019i	$-0.673546\pi$
$827$		−	49.1766i		−	1.71004i	0.518597	−	0.855019i	$-0.326454\pi$
$828$	−3.07445	+	8.42935i	−0.106844	+	0.292940i
$829$	57.4470			1.99522			0.997608	−	0.0691315i	$-0.0220228\pi$
$829$	57.4470			1.99522			0.997608	+	0.0691315i	$0.0220228\pi$
$830$		0			0
$831$		0			0
$832$	−22.8479			−0.792108
$833$		0			0
$834$		0			0
$835$		0			0
$836$		−	2.94620i		−	0.101897i
$837$		0			0
$838$	52.1790			1.80249
$839$		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
$843$	2.88261	−	16.3160i	0.0992823	−	0.561954i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−41.3462			−1.42067
$848$		−	58.0718i		−	1.99419i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−56.7315			−1.94245			−0.971224	−	0.238168i	$-0.923453\pi$
$853$	−56.7315			−1.94245			−0.971224	+	0.238168i	$0.923453\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$	−31.0737	−	5.48989i	−1.06084	−	0.187422i
$859$	−9.72119			−0.331683			−0.165841	−	0.986152i	$-0.553034\pi$
$859$	−9.72119			−0.331683			−0.165841	+	0.986152i	$0.553034\pi$
$860$		0			0
$861$	0.0411207	−	0.232750i	0.00140139	−	0.00793210i
$862$	17.2439			0.587328
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	4.68265	−	8.09175i	0.159307	−	0.275287i
$865$		0			0
$866$		−	62.5595i		−	2.12586i
$867$	28.9958	+	5.12279i	0.984749	+	0.173979i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$			29.4957i			0.998850i
$873$		0			0
$874$	−39.2795			−1.32865
$875$		0			0
$876$	9.34383	+	1.65081i	0.315699	+	0.0557755i
$877$	33.6692			1.13693			0.568464	−	0.822708i	$-0.307538\pi$
$877$	33.6692			1.13693			0.568464	+	0.822708i	$0.307538\pi$
$878$		0			0
$879$	7.99546	−	45.2556i	0.269680	−	1.52643i
$880$		0			0
$881$			23.6423i			0.796529i	0.917271	+	0.398265i	$0.130387\pi$
$881$			23.6423i			0.796529i	−0.917271	+	0.398265i	$0.869613\pi$
$882$	11.1632	−	30.6067i	0.375885	−	1.03058i
$883$	31.2141			1.05044			0.525219	−	0.850967i	$-0.323984\pi$
$883$	31.2141			1.05044			0.525219	+	0.850967i	$0.323984\pi$
$884$		0			0
$885$		0			0
$886$	−32.9872			−1.10823
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−19.2177	+	22.8403i	−0.643816	+	0.765180i
$892$		0			0
$893$		0			0
$894$	−58.1632	−	10.2759i	−1.94527	−	0.343677i
$895$		0			0
$896$			49.8129i			1.66413i
$897$	−10.1229	+	57.2975i	−0.337995	+	1.91311i
$898$		0			0
$899$		0			0
$900$	4.52363	+	1.64991i	0.150788	+	0.0549969i
$901$		0			0
$902$			0.183440i			0.00610790i
$903$		0			0
$904$	41.9758			1.39609
$905$		0			0
$906$	6.62106	−	37.4763i	0.219970	−	1.24507i
$907$	−40.4162			−1.34200			−0.670999	−	0.741458i	$-0.734135\pi$
$907$	−40.4162			−1.34200			−0.670999	+	0.741458i	$0.734135\pi$
$908$			4.04885i			0.134366i
$909$		0			0
$910$		0			0
$911$			21.9143i			0.726052i	0.931779	+	0.363026i	$0.118256\pi$
$911$			21.9143i			0.726052i	−0.931779	+	0.363026i	$0.881744\pi$
$912$	21.4237	+	3.78499i	0.709409	+	0.125334i
$913$	37.2990			1.23442
$914$			42.8628i			1.41778i
$915$		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
$921$	−49.1981	−	8.69198i	−1.62113	−	0.286411i
$922$	−60.9938			−2.00872
$923$		0			0
$924$	−1.20590	+	6.82561i	−0.0396713	+	0.224546i
$925$		0			0
$926$		0			0
$927$	−57.1240	−	20.8349i	−1.87620	−	0.684308i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	19.7255			0.646476
$932$		0			0
$933$	−9.34524	+	52.8956i	−0.305949	+	1.73172i
$934$	36.4365			1.19224
$935$		0			0
$936$	9.48052	−	25.9932i	0.309881	−	0.849614i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$	−5.93287	−	1.04818i	−0.193612	−	0.0342061i
$940$		0			0
$941$		−	8.71587i		−	0.284129i	−0.989857	−	0.142065i	$-0.954626\pi$
$941$		−	8.71587i		−	0.284129i	0.989857	−	0.142065i	$-0.0453741\pi$
$942$	−10.8465	+	61.3932i	−0.353399	+	2.00030i
$943$	0.338250			0.0110149
$944$		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
$949$	61.5311			1.99738
$950$			21.0794i			0.683906i
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$	54.9348	+	20.0364i	1.77858	+	0.648703i
$955$		0			0
$956$			9.91746i			0.320754i
$957$		0			0
$958$	10.1057			0.326499
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$	−2.67998			−0.0863164
$965$		0			0
$966$	91.0008	+	16.0774i	2.92790	+	0.517282i
$967$	47.9676			1.54253			0.771267	−	0.636512i	$-0.219623\pi$
$967$	47.9676			1.54253			0.771267	+	0.636512i	$0.219623\pi$
$968$			28.1371i			0.904361i
$969$		0			0
$970$		0			0
$971$			49.3682i			1.58430i	0.610327	+	0.792150i	$0.291038\pi$
$971$			49.3682i			1.58430i	−0.610327	+	0.792150i	$0.708962\pi$
$972$	3.21008	+	3.83870i	0.102963	+	0.123126i
$973$		0			0
$974$		0			0
$975$	30.7488	+	5.43249i	0.984749	+	0.173979i
$976$		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
$981$	−32.4992	−	11.8535i	−1.03762	−	0.378452i
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	−0.158392	−	0.0279837i	−0.00504936	−	0.000892087i
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−3.20286			−0.101897
$989$		0			0
$990$		0			0
$991$	50.9128			1.61730			0.808649	−	0.588292i	$-0.200199\pi$
$991$	50.9128			1.61730			0.808649	+	0.588292i	$0.200199\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	1.08786	−	6.15749i	0.0344703	−	0.195108i
$997$		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.428.7	✓	20
3.2	odd	2	inner	429.2.e.d.428.14	yes	20
11.10	odd	2	inner	429.2.e.d.428.13	yes	20
13.12	even	2	inner	429.2.e.d.428.13	yes	20
33.32	even	2	inner	429.2.e.d.428.8	yes	20
39.38	odd	2	inner	429.2.e.d.428.8	yes	20
143.142	odd	2	CM	429.2.e.d.428.7	✓	20
429.428	even	2	inner	429.2.e.d.428.14	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.e.d.428.7	✓	20	1.1	even	1	trivial
429.2.e.d.428.7	✓	20	143.142	odd	2	CM
429.2.e.d.428.8	yes	20	33.32	even	2	inner
429.2.e.d.428.8	yes	20	39.38	odd	2	inner
429.2.e.d.428.13	yes	20	11.10	odd	2	inner
429.2.e.d.428.13	yes	20	13.12	even	2	inner
429.2.e.d.428.14	yes	20	3.2	odd	2	inner
429.2.e.d.428.14	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)

\(f(q)\)	\(=\)	\(q-1.52349i q^{2} +(-1.70564 - 0.301340i) q^{3} -0.321008 q^{4} +(-0.459088 + 2.59851i) q^{6} +3.75874 q^{7} -2.55792i q^{8} +(2.81839 + 1.02795i) q^{9} +O(q^{10})\)	\(q-1.52349i q^{2} +(-1.70564 - 0.301340i) q^{3} -0.321008 q^{4} +(-0.459088 + 2.59851i) q^{6} +3.75874 q^{7} -2.55792i q^{8} +(2.81839 + 1.02795i) q^{9} +3.31662i q^{11} +(0.547523 + 0.0967327i) q^{12} +3.60555 q^{13} -5.72639i q^{14} -4.53897 q^{16} +(1.56607 - 4.29377i) q^{18} +2.76726 q^{19} +(-6.41105 - 1.13266i) q^{21} +5.05283 q^{22} -9.31703i q^{23} +(-0.770804 + 4.36288i) q^{24} -5.00000 q^{25} -5.49301i q^{26} +(-4.49738 - 2.60261i) q^{27} -1.20659 q^{28} +1.79922i q^{32} +(0.999433 - 5.65695i) q^{33} +(-0.904725 - 0.329981i) q^{36} -4.21588i q^{38} +(-6.14976 - 1.08650i) q^{39} +0.0363045i q^{41} +(-1.72559 + 9.76714i) q^{42} -1.06466i q^{44} -14.1944 q^{46} +(7.74183 + 1.36777i) q^{48} +7.12816 q^{49} +7.61743i q^{50} -1.15741 q^{52} +12.7941i q^{53} +(-3.96504 + 6.85169i) q^{54} -9.61457i q^{56} +(-4.71994 - 0.833888i) q^{57} +(10.5936 + 3.86382i) q^{63} -6.33686 q^{64} +(-8.61829 - 1.52262i) q^{66} +(-2.80760 + 15.8915i) q^{69} +(2.62942 - 7.20921i) q^{72} +17.0656 q^{73} +(8.52818 + 1.50670i) q^{75} -0.888313 q^{76} +12.4663i q^{77} +(-1.65526 + 9.36907i) q^{78} +(6.88662 + 5.79435i) q^{81} +0.0553094 q^{82} -11.2461i q^{83} +(2.05800 + 0.363593i) q^{84} +8.48366 q^{88} +13.5523 q^{91} +2.99084i q^{92} +(0.542176 - 3.06881i) q^{96} -10.8596i q^{98} +(-3.40934 + 9.34754i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 40 q^{4}+O(q^{10}) \) 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

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