LMFDB - Embedded newform 429.2.e.a.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:	$8$
Coefficient field:	8.0.151613669376.21
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5x^{4} + 16 $ x^8 + 5*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			428.7
Root			$-0.752986 + 1.19709i$ of defining polynomial
Character	$\chi$	$=$	429.428
Dual form			429.2.e.a.428.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.39417i q^{2} -1.73205 q^{3} -3.73205 q^{4} -4.11439 q^{5} -4.14682i q^{6} -4.14682i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+2.39417i q^{2} -1.73205 q^{3} -3.73205 q^{4} -4.11439 q^{5} -4.14682i q^{6} -4.14682i q^{8} +3.00000 q^{9} -9.85055i q^{10} +(0.551224 + 3.27050i) q^{11} +6.46410 q^{12} -3.60555i q^{13} +7.12633 q^{15} +2.46410 q^{16} +7.18251i q^{18} +15.3551 q^{20} +(-7.83013 + 1.31972i) q^{22} +7.18251i q^{24} +11.9282 q^{25} +8.63230 q^{26} -5.19615 q^{27} +17.0617i q^{30} -2.39417i q^{32} +(-0.954747 - 5.66467i) q^{33} -11.1962 q^{36} +6.24500i q^{39} +17.0617i q^{40} -7.82403i q^{41} -12.4900i q^{43} +(-2.05719 - 12.2057i) q^{44} -12.3432 q^{45} +9.33123 q^{47} -4.26795 q^{48} -7.00000 q^{49} +28.5581i q^{50} +13.4561i q^{52} -12.4405i q^{54} +(-2.26795 - 13.4561i) q^{55} -15.3551 q^{59} -26.5958 q^{60} -7.21110i q^{61} +10.6603 q^{64} +14.8346i q^{65} +(13.5622 - 2.28583i) q^{66} +4.92144 q^{71} -12.4405i q^{72} -20.6603 q^{75} -14.9516 q^{78} +14.4222i q^{79} -10.1383 q^{80} +9.00000 q^{81} +18.7321 q^{82} -12.6124i q^{83} +29.9032 q^{86} +(13.5622 - 2.28583i) q^{88} -18.3671 q^{89} -29.5516i q^{90} +22.3405i q^{94} +4.14682i q^{96} -16.7592i q^{98} +(1.65367 + 9.81149i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} + 24 q^{9}+O(q^{10}) $ 8 * q - 16 * q^4 + 24 * q^9	$ 8 q - 16 q^{4} + 24 q^{9} + 24 q^{12} - 8 q^{16} - 28 q^{22} + 40 q^{25} - 48 q^{36} - 48 q^{48} - 56 q^{49} - 32 q^{55} + 16 q^{64} + 60 q^{66} - 96 q^{75} + 72 q^{81} + 136 q^{82} + 60 q^{88}+O(q^{100}) $ 8 * q - 16 * q^4 + 24 * q^9 + 24 * q^12 - 8 * q^16 - 28 * q^22 + 40 * q^25 - 48 * q^36 - 48 * q^48 - 56 * q^49 - 32 * q^55 + 16 * q^64 + 60 * q^66 - 96 * q^75 + 72 * q^81 + 136 * q^82 + 60 * q^88

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 $

$2$			2.39417i			1.69293i	0.532441	+	0.846467i	$0.321275\pi$
$2$			2.39417i			1.69293i	−0.532441	+	0.846467i	$0.678725\pi$
$3$	−1.73205			−1.00000
$3$	−1.73205			−1.00000
$4$	−3.73205			−1.86603
$5$	−4.11439			−1.84001			−0.920006	−	0.391905i	$-0.871816\pi$
$5$	−4.11439			−1.84001			−0.920006	+	0.391905i	$0.871816\pi$
$6$		−	4.14682i		−	1.69293i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$		−	4.14682i		−	1.46612i
$9$	3.00000			1.00000
$10$		−	9.85055i		−	3.11502i
$11$	0.551224	+	3.27050i	0.166200	+	0.986092i
$11$	0.551224	+	3.27050i	0.166200	+	0.986092i
$12$	6.46410			1.86603
$13$		−	3.60555i		−	1.00000i
$13$		−	3.60555i		−	1.00000i
$14$		0			0
$15$	7.12633			1.84001
$16$	2.46410			0.616025
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$			7.18251i			1.69293i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	15.3551			3.43351
$21$		0			0
$22$	−7.83013	+	1.31972i	−1.66939	+	0.281366i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$			7.18251i			1.46612i
$25$	11.9282			2.38564
$26$	8.63230			1.69293
$27$	−5.19615			−1.00000
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$			17.0617i			3.11502i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		−	2.39417i		−	0.423233i
$33$	−0.954747	−	5.66467i	−0.166200	−	0.986092i
$34$		0			0
$35$		0			0
$36$	−11.1962			−1.86603
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$			6.24500i			1.00000i
$40$			17.0617i			2.69768i
$41$		−	7.82403i		−	1.22191i	−0.791666	−	0.610954i	$-0.790786\pi$
$41$		−	7.82403i		−	1.22191i	0.791666	−	0.610954i	$-0.209214\pi$
$42$		0			0
$43$		−	12.4900i		−	1.90471i	−0.304997	−	0.952353i	$-0.598656\pi$
$43$		−	12.4900i		−	1.90471i	0.304997	−	0.952353i	$-0.401344\pi$
$44$	−2.05719	−	12.2057i	−0.310134	−	1.84007i
$45$	−12.3432			−1.84001
$46$		0			0
$47$	9.33123			1.36110			0.680550	−	0.732702i	$-0.261741\pi$
$47$	9.33123			1.36110			0.680550	+	0.732702i	$0.261741\pi$
$48$	−4.26795			−0.616025
$49$	−7.00000			−1.00000
$50$			28.5581i			4.03873i
$51$		0			0
$52$			13.4561i			1.86603i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		−	12.4405i		−	1.69293i
$55$	−2.26795	−	13.4561i	−0.305810	−	1.81442i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−15.3551			−1.99907			−0.999533	−	0.0305698i	$-0.990268\pi$
$59$	−15.3551			−1.99907			−0.999533	+	0.0305698i	$0.990268\pi$
$60$	−26.5958			−3.43351
$61$		−	7.21110i		−	0.923287i	−0.887066	−	0.461644i	$-0.847260\pi$
$61$		−	7.21110i		−	0.923287i	0.887066	−	0.461644i	$-0.152740\pi$
$62$		0			0
$63$		0			0
$64$	10.6603			1.33253
$65$			14.8346i			1.84001i
$66$	13.5622	−	2.28583i	1.66939	−	0.281366i
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.92144			0.584067			0.292034	−	0.956408i	$-0.405668\pi$
$71$	4.92144			0.584067			0.292034	+	0.956408i	$0.405668\pi$
$72$		−	12.4405i		−	1.46612i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$	−20.6603			−2.38564
$76$		0			0
$77$		0			0
$78$	−14.9516			−1.69293
$79$			14.4222i			1.62262i	0.584613	+	0.811312i	$0.301246\pi$
$79$			14.4222i			1.62262i	−0.584613	+	0.811312i	$0.698754\pi$
$80$	−10.1383			−1.13349
$81$	9.00000			1.00000
$82$	18.7321			2.06861
$83$		−	12.6124i		−	1.38439i	−0.721712	−	0.692194i	$-0.756644\pi$
$83$		−	12.6124i		−	1.38439i	0.721712	−	0.692194i	$-0.243356\pi$
$84$		0			0
$85$		0			0
$86$	29.9032			3.22454
$87$		0			0
$88$	13.5622	−	2.28583i	1.44573	−	0.243670i
$89$	−18.3671			−1.94690			−0.973452	−	0.228892i	$-0.926490\pi$
$89$	−18.3671			−1.94690			−0.973452	+	0.228892i	$0.926490\pi$
$90$		−	29.5516i		−	3.11502i
$91$		0			0
$92$		0			0
$93$		0			0
$94$			22.3405i			2.30425i
$95$		0			0
$96$			4.14682i			0.423233i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		−	16.7592i		−	1.69293i
$99$	1.65367	+	9.81149i	0.166200	+	0.986092i
$100$	−44.5167			−4.45167
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	16.0000			1.57653			0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000			1.57653			0.788263	+	0.615338i	$0.210980\pi$
$104$	−14.9516			−1.46612
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	19.3923			1.86603
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	32.2162	−	5.42986i	3.07169	−	0.517716i
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	10.8167i		−	1.00000i
$118$		−	36.7628i		−	3.38429i
$119$		0			0
$120$		−	29.5516i		−	2.69768i
$121$	−10.3923	+	3.60555i	−0.944755	+	0.327777i
$122$	17.2646			1.56306
$123$			13.5516i			1.22191i
$124$		0			0
$125$	−28.5053			−2.54959
$126$		0			0
$127$		−	14.4222i		−	1.27976i	−0.768473	−	0.639882i	$-0.778983\pi$
$127$		−	14.4222i		−	1.27976i	0.768473	−	0.639882i	$-0.221017\pi$
$128$			20.7341i			1.83265i
$129$			21.6333i			1.90471i
$130$	−35.5167			−3.11502
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	3.56317	+	21.1408i	0.310134	+	1.84007i
$133$		0			0
$134$		0			0
$135$	21.3790			1.84001
$136$		0			0
$137$	−14.5481			−1.24293			−0.621463	−	0.783444i	$-0.713461\pi$
$137$	−14.5481			−1.24293			−0.621463	+	0.783444i	$0.713461\pi$
$138$		0			0
$139$			12.4900i			1.05939i	0.848189	+	0.529694i	$0.177693\pi$
$139$			12.4900i			1.05939i	−0.848189	+	0.529694i	$0.822307\pi$
$140$		0			0
$141$	−16.1622			−1.36110
$142$			11.7828i			0.988787i
$143$	11.7919	−	1.98747i	0.986092	−	0.166200i
$144$	7.39230			0.616025
$145$		0			0
$146$		0			0
$147$	12.1244			1.00000
$148$		0			0
$149$		−	24.4113i		−	1.99985i	−0.0121001	−	0.999927i	$-0.503852\pi$
$149$		−	24.4113i		−	1.99985i	0.0121001	−	0.999927i	$-0.496148\pi$
$150$		−	49.4642i		−	4.03873i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		−	23.3066i		−	1.86603i
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$	−34.5292			−2.74700
$159$		0			0
$160$			9.85055i			0.778754i
$161$		0			0
$162$			21.5475i			1.69293i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$			29.1997i			2.28011i
$165$	3.92820	+	23.3066i	0.305810	+	1.81442i
$166$	30.1962			2.34368
$167$		−	23.1283i		−	1.78972i	−0.446346	−	0.894860i	$-0.647275\pi$
$167$		−	23.1283i		−	1.78972i	0.446346	−	0.894860i	$-0.352725\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$			46.6133i			3.55423i
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$	1.35827	+	8.05884i	0.102384	+	0.607458i
$177$	26.5958			1.99907
$178$		−	43.9739i		−	3.29598i
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	46.0653			3.43351
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$			12.4900i			0.923287i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−34.8246			−2.53985
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−18.4641			−1.33253
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		−	25.6944i		−	1.84001i
$196$	26.1244			1.86603
$197$			13.8954i			0.990006i	0.868891	+	0.495003i	$0.164833\pi$
$197$			13.8954i			0.990006i	−0.868891	+	0.495003i	$0.835167\pi$
$198$	−23.4904	+	3.95917i	−1.66939	+	0.281366i
$199$	24.2487			1.71895			0.859473	−	0.511182i	$-0.170792\pi$
$199$	24.2487			1.71895			0.859473	+	0.511182i	$0.170792\pi$
$200$		−	49.4642i		−	3.49764i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$			32.1911i			2.24832i
$206$			38.3067i			2.66896i
$207$		0			0
$208$		−	8.88444i		−	0.616025i
$209$		0			0
$210$		0			0
$211$			28.8444i			1.98573i	0.119239	+	0.992866i	$0.461954\pi$
$211$			28.8444i			1.98573i	−0.119239	+	0.992866i	$0.538046\pi$
$212$		0			0
$213$	−8.52418			−0.584067
$214$		0			0
$215$			51.3887i			3.50468i
$216$			21.5475i			1.46612i
$217$		0			0
$218$		0			0
$219$		0			0
$220$	8.46410	+	50.2189i	0.570650	+	3.38575i
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	35.7846			2.38564
$226$		0			0
$227$			5.60175i			0.371801i	0.982569	+	0.185901i	$0.0595202\pi$
$227$			5.60175i			0.371801i	−0.982569	+	0.185901i	$0.940480\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$	25.8969			1.69293
$235$	−38.3923			−2.50444
$236$	57.3061			3.73031
$237$		−	24.9800i		−	1.62262i
$238$		0			0
$239$		−	17.0569i		−	1.10332i	−0.834069	−	0.551660i	$-0.813995\pi$
$239$		−	17.0569i		−	1.10332i	0.834069	−	0.551660i	$-0.186005\pi$
$240$	17.5600			1.13349
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	−8.63230	−	24.8809i	−0.554905	−	1.59941i
$243$	−15.5885			−1.00000
$244$			26.9122i			1.72288i
$245$	28.8007			1.84001
$246$	−32.4449			−2.06861
$247$		0			0
$248$		0			0
$249$			21.8453i			1.38439i
$250$		−	68.2466i		−	4.31629i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$	34.5292			2.16656
$255$		0			0
$256$	−28.3205			−1.77003
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$	−51.7938			−3.22454
$259$		0			0
$260$		−	55.3636i		−	3.43351i
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$	−23.4904	+	3.95917i	−1.44573	+	0.243670i
$265$		0			0
$266$		0			0
$267$	31.8127			1.94690
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$			51.1850i			3.11502i
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		−	34.8305i		−	2.10419i
$275$	6.57511	+	39.0112i	0.396494	+	2.35246i
$276$		0			0
$277$		−	24.9800i		−	1.50090i	−0.660926	−	0.750451i	$-0.729836\pi$
$277$		−	24.9800i		−	1.50090i	0.660926	−	0.750451i	$-0.270164\pi$
$278$	−29.9032			−1.79347
$279$		0			0
$280$		0			0
$281$		−	30.4827i		−	1.81845i	−0.416310	−	0.909223i	$-0.636677\pi$
$281$		−	30.4827i		−	1.81845i	0.416310	−	0.909223i	$-0.363323\pi$
$282$		−	38.6950i		−	2.30425i
$283$		−	28.8444i		−	1.71462i	−0.514799	−	0.857311i	$-0.672133\pi$
$283$		−	28.8444i		−	1.71462i	0.514799	−	0.857311i	$-0.327867\pi$
$284$	−18.3671			−1.08988
$285$		0			0
$286$	4.75833	+	28.2319i	0.281366	+	1.66939i
$287$		0			0
$288$		−	7.18251i		−	0.423233i
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	10.3901i		−	0.606996i	−0.952832	−	0.303498i	$-0.901846\pi$
$293$		−	10.3901i		−	0.606996i	0.952832	−	0.303498i	$-0.0981545\pi$
$294$			29.0278i			1.69293i
$295$	63.1769			3.67830
$296$		0			0
$297$	−2.86424	−	16.9940i	−0.166200	−	0.986092i
$298$	58.4449			3.38562
$299$		0			0
$300$	77.1051			4.45167
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$			29.6693i			1.69886i
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$	−27.7128			−1.57653
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$	25.8969			1.46612
$313$	−34.6410			−1.95803			−0.979013	−	0.203798i	$-0.934671\pi$
$313$	−34.6410			−1.95803			−0.979013	+	0.203798i	$0.934671\pi$
$314$			4.78834i			0.270222i
$315$		0			0
$316$		−	53.8244i		−	3.02786i
$317$	−16.7530			−0.940940			−0.470470	−	0.882416i	$-0.655916\pi$
$317$	−16.7530			−0.940940			−0.470470	+	0.882416i	$0.655916\pi$
$318$		0			0
$319$		0			0
$320$	−43.8604			−2.45187
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−33.5885			−1.86603
$325$		−	43.0077i		−	2.38564i
$326$		0			0
$327$		0			0
$328$	−32.4449			−1.79147
$329$		0			0
$330$	−55.8001	+	9.40479i	−3.07169	+	0.517716i
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$			47.0700i			2.58330i
$333$		0			0
$334$	55.3731			3.02988
$335$		0			0
$336$		0			0
$337$		−	36.0555i		−	1.96407i	−0.188702	−	0.982034i	$-0.560428\pi$
$337$		−	36.0555i		−	1.96407i	0.188702	−	0.982034i	$-0.439572\pi$
$338$		−	31.1242i		−	1.69293i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−51.7938			−2.79254
$345$		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			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$			18.7350i			1.00000i
$352$	7.83013	−	1.31972i	0.417347	−	0.0703415i
$353$	2.50029			0.133077			0.0665386	−	0.997784i	$-0.478804\pi$
$353$	2.50029			0.133077			0.0665386	+	0.997784i	$0.478804\pi$
$354$			63.6750i			3.38429i
$355$	−20.2487			−1.07469
$356$	68.5468			3.63297
$357$		0			0
$358$		0			0
$359$			2.09644i			0.110646i	0.998469	+	0.0553230i	$0.0176188\pi$
$359$			2.09644i			0.110646i	−0.998469	+	0.0553230i	$0.982381\pi$
$360$			51.1850i			2.69768i
$361$	−19.0000			−1.00000
$362$		−	23.9417i		−	1.25835i
$363$	18.0000	−	6.24500i	0.944755	−	0.327777i
$364$		0			0
$365$		0			0
$366$	−29.9032			−1.56306
$367$	8.00000			0.417597			0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000			0.417597			0.208798	+	0.977959i	$0.433045\pi$
$368$		0			0
$369$		−	23.4721i		−	1.22191i
$370$		0			0
$371$		0			0
$372$		0			0
$373$			36.0555i			1.86688i	0.358729	+	0.933442i	$0.383210\pi$
$373$			36.0555i			1.86688i	−0.358729	+	0.933442i	$0.616790\pi$
$374$		0			0
$375$	49.3727			2.54959
$376$		−	38.6950i		−	1.99554i
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$			24.9800i			1.27976i
$382$		0			0
$383$	16.9692			0.867086			0.433543	−	0.901133i	$-0.357263\pi$
$383$	16.9692			0.867086			0.433543	+	0.901133i	$0.357263\pi$
$384$		−	35.9126i		−	1.83265i
$385$		0			0
$386$		0			0
$387$		−	37.4700i		−	1.90471i
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$	61.5167			3.11502
$391$		0			0
$392$			29.0278i			1.46612i
$393$		0			0
$394$	−33.2679			−1.67602
$395$		−	59.3386i		−	2.98565i
$396$	−6.17158	−	36.6170i	−0.310134	−	1.84007i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$			58.0555i			2.91006i
$399$		0			0
$400$	29.3923			1.46962
$401$	31.0056			1.54835			0.774173	−	0.632973i	$-0.218166\pi$
$401$	31.0056			1.54835			0.774173	+	0.632973i	$0.218166\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−37.0295			−1.84001
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$	−77.0710			−3.80626
$411$	25.1980			1.24293
$412$	−59.7128			−2.94184
$413$		0			0
$414$		0			0
$415$			51.8922i			2.54729i
$416$	−8.63230			−0.423233
$417$		−	21.6333i		−	1.05939i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$	−69.0584			−3.36171
$423$	27.9937			1.36110
$424$		0			0
$425$		0			0
$426$		−	20.4083i		−	0.988787i
$427$		0			0
$428$		0			0
$429$	−20.4242	+	3.44239i	−0.986092	+	0.166200i
$430$	−123.033			−5.93319
$431$		−	10.9855i		−	0.529155i	−0.964364	−	0.264578i	$-0.914767\pi$
$431$		−	10.9855i		−	0.529155i	0.964364	−	0.264578i	$-0.0852325\pi$
$432$	−12.8038			−0.616025
$433$	−20.7846			−0.998845			−0.499422	−	0.866359i	$-0.666454\pi$
$433$	−20.7846			−0.998845			−0.499422	+	0.866359i	$0.666454\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	12.4900i		−	0.596115i	−0.954548	−	0.298057i	$-0.903661\pi$
$439$		−	12.4900i		−	0.596115i	0.954548	−	0.298057i	$-0.0963387\pi$
$440$	−55.8001	+	9.40479i	−2.66016	+	0.448356i
$441$	−21.0000			−1.00000
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	75.5692			3.58232
$446$		0			0
$447$			42.2817i			1.99985i
$448$		0			0
$449$	−10.7291			−0.506336			−0.253168	−	0.967422i	$-0.581473\pi$
$449$	−10.7291			−0.506336			−0.253168	+	0.967422i	$0.581473\pi$
$450$			85.6744i			4.03873i
$451$	25.5885	−	4.31279i	1.20491	−	0.203081i
$452$		0			0
$453$		0			0
$454$	−13.4115			−0.629435
$455$		0			0
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			34.9272i			1.62672i	0.581758	+	0.813362i	$0.302365\pi$
$461$			34.9272i			1.62672i	−0.581758	+	0.813362i	$0.697635\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$			40.3683i			1.86603i
$469$		0			0
$470$		−	91.9177i		−	4.23985i
$471$	−3.46410			−0.159617
$472$			63.6750i			2.93088i
$473$	40.8485	−	6.88478i	1.87822	−	0.316563i
$474$	59.8064			2.74700
$475$		0			0
$476$		0			0
$477$		0			0
$478$	40.8372			1.86785
$479$			28.2604i			1.29125i	0.763654	+	0.645626i	$0.223403\pi$
$479$			28.2604i			1.29125i	−0.763654	+	0.645626i	$0.776597\pi$
$480$		−	17.0617i		−	0.778754i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	38.7846	−	13.4561i	1.76294	−	0.611641i
$485$		0			0
$486$		−	37.3214i		−	1.69293i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	−29.9032			−1.35365
$489$		0			0
$490$			68.9538i			3.11502i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		−	50.5753i		−	2.28011i
$493$		0			0
$494$		0			0
$495$	−6.80385	−	40.3683i	−0.305810	−	1.81442i
$496$		0			0
$497$		0			0
$498$	−52.3013			−2.34368
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$	106.383			4.75761
$501$			40.0594i			1.78972i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	22.5167			1.00000
$508$			53.8244i			2.38807i
$509$	−11.7524			−0.520915			−0.260457	−	0.965485i	$-0.583873\pi$
$509$	−11.7524			−0.520915			−0.260457	+	0.965485i	$0.583873\pi$
$510$		0			0
$511$		0			0
$512$		−	26.3359i		−	1.16389i
$513$		0			0
$514$		0			0
$515$	−65.8302			−2.90083
$516$		−	80.7366i		−	3.55423i
$517$	5.14359	+	30.5178i	0.226215	+	1.34217i
$518$		0			0
$519$		0			0
$520$	61.5167			2.69768
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$			12.4900i			0.546149i	0.961993	+	0.273075i	$0.0880406\pi$
$523$			12.4900i			0.546149i	−0.961993	+	0.273075i	$0.911959\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−2.35259	−	13.9583i	−0.102384	−	0.607458i
$529$	23.0000			1.00000
$530$		0			0
$531$	−46.0653			−1.99907
$532$		0			0
$533$	−28.2099			−1.22191
$534$			76.1649i			3.29598i
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.85857	−	22.8935i	−0.166200	−	0.986092i
$540$	−79.7875			−3.43351
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$	17.3205			0.743294
$544$		0			0
$545$		0			0
$546$		0			0
$547$			37.4700i			1.60210i	0.598597	+	0.801050i	$0.295725\pi$
$547$			37.4700i			1.60210i	−0.598597	+	0.801050i	$0.704275\pi$
$548$	54.2941			2.31933
$549$		−	21.6333i		−	0.923287i
$550$	−93.3993	+	15.7419i	−3.98256	+	0.671238i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	59.8064			2.54093
$555$		0			0
$556$		−	46.6133i		−	1.97684i
$557$		−	47.0700i		−	1.99442i	−0.0746496	−	0.997210i	$-0.523784\pi$
$557$		−	47.0700i		−	1.99442i	0.0746496	−	0.997210i	$-0.476216\pi$
$558$		0			0
$559$	−45.0333			−1.90471
$560$		0			0
$561$		0			0
$562$	72.9808			3.07851
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	60.3180			2.53985
$565$		0			0
$566$	69.0584			2.90274
$567$		0			0
$568$		−	20.4083i		−	0.856315i
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		−	28.8444i		−	1.20710i	−0.797325	−	0.603550i	$-0.793752\pi$
$571$		−	28.8444i		−	1.20710i	0.797325	−	0.603550i	$-0.206248\pi$
$572$	−44.0081	+	7.41732i	−1.84007	+	0.310134i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	31.9808			1.33253
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		−	40.7009i		−	1.69293i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$			44.5039i			1.84001i
$586$	24.8756			1.02760
$587$	−41.6555			−1.71931			−0.859654	−	0.510876i	$-0.829321\pi$
$587$	−41.6555			−1.71931			−0.859654	+	0.510876i	$0.829321\pi$
$588$	−45.2487			−1.86603
$589$		0			0
$590$			151.256i			6.22712i
$591$		−	24.0675i		−	0.990006i
$592$		0			0
$593$			28.8559i			1.18497i	0.805582	+	0.592484i	$0.201853\pi$
$593$			28.8559i			1.18497i	−0.805582	+	0.592484i	$0.798147\pi$
$594$	40.6865	−	6.85748i	1.66939	−	0.281366i
$595$		0			0
$596$			91.1043i			3.73178i
$597$	−42.0000			−1.71895
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$			85.6744i			3.49764i
$601$			7.21110i			0.294147i	0.989126	+	0.147074i	$0.0469854\pi$
$601$			7.21110i			0.294147i	−0.989126	+	0.147074i	$0.953015\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	42.7580	−	14.8346i	1.73836	−	0.603114i
$606$		0			0
$607$		−	37.4700i		−	1.52086i	−0.649420	−	0.760430i	$-0.724988\pi$
$607$		−	37.4700i		−	1.52086i	0.649420	−	0.760430i	$-0.275012\pi$
$608$		0			0
$609$		0			0
$610$	−71.0333			−2.87606
$611$		−	33.6442i		−	1.36110i
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		−	55.7566i		−	2.24832i
$616$		0			0
$617$	46.8724			1.88701			0.943506	−	0.331357i	$-0.107506\pi$
$617$	46.8724			1.88701			0.943506	+	0.331357i	$0.107506\pi$
$618$		−	66.3492i		−	2.66896i
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$			15.3883i			0.616025i
$625$	57.6410			2.30564
$626$		−	82.9365i		−	3.31481i
$627$		0			0
$628$	−7.46410			−0.297850
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	59.8064			2.37897
$633$		−	49.9600i		−	1.98573i
$634$		−	40.1094i		−	1.59295i
$635$			59.3386i			2.35478i
$636$		0			0
$637$			25.2389i			1.00000i
$638$		0			0
$639$	14.7643			0.584067
$640$		−	85.3083i		−	3.37211i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		−	89.0079i		−	3.50468i
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		−	37.3214i		−	1.46612i
$649$	−8.46410	−	50.2189i	−0.332245	−	1.97126i
$650$	102.968			4.03873
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		−	19.2792i		−	0.752726i
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$	−14.6603	−	86.9816i	−0.570650	−	3.38575i
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$	−52.3013			−2.02968
$665$		0			0
$666$		0			0
$667$		0			0
$668$			86.3160i			3.33966i
$669$		0			0
$670$		0			0
$671$	23.5839	−	3.97493i	0.910446	−	0.153450i
$672$		0			0
$673$			49.9600i			1.92582i	0.269830	+	0.962908i	$0.413032\pi$
$673$			49.9600i			1.92582i	−0.269830	+	0.962908i	$0.586968\pi$
$674$	86.3230			3.32504
$675$	−61.9808			−2.38564
$676$	48.5167			1.86603
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		−	9.70252i		−	0.371801i
$682$		0			0
$683$	−6.53553			−0.250075			−0.125038	−	0.992152i	$-0.539905\pi$
$683$	−6.53553			−0.250075			−0.125038	+	0.992152i	$0.539905\pi$
$684$		0			0
$685$	59.8564			2.28700
$686$		0			0
$687$		0			0
$688$		−	30.7766i		−	1.17335i
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	51.3887i		−	1.94928i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	−44.8548			−1.69293
$703$		0			0
$704$	5.87618	+	34.8643i	0.221467	+	1.31400i
$705$	66.4974			2.50444
$706$			5.98613i			0.225291i
$707$		0			0
$708$	−99.2570			−3.73031
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		−	48.4789i		−	1.81938i
$711$			43.2666i			1.62262i
$712$			76.1649i			2.85440i
$713$		0			0
$714$		0			0
$715$	−48.5167	+	8.17721i	−1.81442	+	0.305810i
$716$		0			0
$717$			29.5435i			1.10332i
$718$	−5.01924			−0.187316
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$	−30.4148			−1.13349
$721$		0			0
$722$		−	45.4892i		−	1.69293i
$723$		0			0
$724$	37.3205			1.38701
$725$		0			0
$726$	14.9516	+	43.0951i	0.554905	+	1.59941i
$727$	−51.9615			−1.92715			−0.963573	−	0.267445i	$-0.913821\pi$
$727$	−51.9615			−1.92715			−0.963573	+	0.267445i	$0.913821\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$		−	46.6133i		−	1.72288i
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$			19.1534i			0.706963i
$735$	−49.8843			−1.84001
$736$		0			0
$737$		0			0
$738$	56.1962			2.06861
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			4.91418i			0.180284i	0.995929	+	0.0901418i	$0.0287320\pi$
$743$			4.91418i			0.180284i	−0.995929	+	0.0901418i	$0.971268\pi$
$744$		0			0
$745$			100.438i			3.67975i
$746$	−86.3230			−3.16051
$747$		−	37.8371i		−	1.38439i
$748$		0			0
$749$		0			0
$750$			118.207i			4.31629i
$751$	−40.0000			−1.45962			−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000			−1.45962			−0.729810	+	0.683650i	$0.760392\pi$
$752$	22.9931			0.838472
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−41.5692			−1.51086			−0.755429	−	0.655230i	$-0.772572\pi$
$757$	−41.5692			−1.51086			−0.755429	+	0.655230i	$0.772572\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			48.0092i			1.74033i	0.492757	+	0.870167i	$0.335989\pi$
$761$			48.0092i			1.74033i	−0.492757	+	0.870167i	$0.664011\pi$
$762$	−59.8064			−2.16656
$763$		0			0
$764$		0			0
$765$		0			0
$766$			40.6272i			1.46792i
$767$			55.3636i			1.99907i
$768$	49.0526			1.77003
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12.9340			−0.465203			−0.232601	−	0.972572i	$-0.574724\pi$
$773$	−12.9340			−0.465203			−0.232601	+	0.972572i	$0.574724\pi$
$774$	89.7095			3.22454
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$			95.8926i			3.43351i
$781$	2.71281	+	16.0955i	0.0970721	+	0.575944i
$782$		0			0
$783$		0			0
$784$	−17.2487			−0.616025
$785$	−8.22878			−0.293698
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		−	51.8583i		−	1.84738i
$789$		0			0
$790$	142.067			5.05450
$791$		0			0
$792$	40.6865	−	6.85748i	1.44573	−	0.243670i
$793$	−26.0000			−0.923287
$794$		0			0
$795$		0			0
$796$	−90.4974			−3.20760
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		−	28.5581i		−	1.00968i
$801$	−55.1012			−1.94690
$802$			74.2327i			2.62125i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		−	88.6549i		−	3.11502i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		−	120.139i		−	4.19543i
$821$			0.125835i			0.00439167i	0.999998	+	0.00219583i	$0.000698956\pi$
$821$			0.125835i			0.00439167i	−0.999998	+	0.00219583i	$0.999301\pi$
$822$			60.3283i			2.10419i
$823$	−56.0000			−1.95204			−0.976019	−	0.217687i	$-0.930149\pi$
$823$	−56.0000			−1.95204			−0.976019	+	0.217687i	$0.930149\pi$
$824$		−	66.3492i		−	2.31138i
$825$	−11.3884	−	67.5693i	−0.396494	−	2.35246i
$826$		0			0
$827$			27.5728i			0.958802i	0.877596	+	0.479401i	$0.159146\pi$
$827$			27.5728i			0.958802i	−0.877596	+	0.479401i	$0.840854\pi$
$828$		0			0
$829$	27.7128			0.962506			0.481253	−	0.876582i	$-0.340182\pi$
$829$	27.7128			0.962506			0.481253	+	0.876582i	$0.340182\pi$
$830$	−124.239			−4.31239
$831$			43.2666i			1.50090i
$832$		−	38.4361i		−	1.33253i
$833$		0			0
$834$	51.7938			1.79347
$835$			95.1588i			3.29311i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	44.4512			1.53463			0.767314	−	0.641272i	$-0.221593\pi$
$839$	44.4512			1.53463			0.767314	+	0.641272i	$0.221593\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$			52.7976i			1.81845i
$844$		−	107.649i		−	3.70542i
$845$	53.4871			1.84001
$846$			67.0216i			2.30425i
$847$		0			0
$848$		0			0
$849$			49.9600i			1.71462i
$850$		0			0
$851$		0			0
$852$	31.8127			1.08988
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$	−8.24167	−	48.8991i	−0.281366	−	1.66939i
$859$	10.3923			0.354581			0.177290	−	0.984159i	$-0.443267\pi$
$859$	10.3923			0.354581			0.177290	+	0.984159i	$0.443267\pi$
$860$		−	191.785i		−	6.53982i
$861$		0			0
$862$	26.3013			0.895825
$863$	−47.6794			−1.62303			−0.811513	−	0.584334i	$-0.801356\pi$
$863$	−47.6794			−1.62303			−0.811513	+	0.584334i	$0.801356\pi$
$864$			12.4405i			0.423233i
$865$		0			0
$866$		−	49.7619i		−	1.69098i
$867$	29.4449			1.00000
$868$		0			0
$869$	−47.1678	+	7.94986i	−1.60006	+	0.269681i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$	29.9032			1.00918
$879$			17.9962i			0.606996i
$880$	−5.58846	−	33.1572i	−0.188387	−	1.11773i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		−	50.2776i		−	1.69293i
$883$	51.9615			1.74864			0.874322	−	0.485346i	$-0.161306\pi$
$883$	51.9615			1.74864			0.874322	+	0.485346i	$0.161306\pi$
$884$		0			0
$885$	−109.426			−3.67830
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$			180.926i			6.06464i
$891$	4.96101	+	29.4345i	0.166200	+	0.986092i
$892$		0			0
$893$		0			0
$894$	−101.229			−3.38562
$895$		0			0
$896$		0			0
$897$		0			0
$898$		−	25.6872i		−	0.857194i
$899$		0			0
$900$	−133.550			−4.45167
$901$		0			0
$902$	10.3256	+	61.2631i	0.343803	+	2.03984i
$903$		0			0
$904$		0			0
$905$	41.1439			1.36767
$906$		0			0
$907$	−17.3205			−0.575118			−0.287559	−	0.957763i	$-0.592844\pi$
$907$	−17.3205			−0.575118			−0.287559	+	0.957763i	$0.592844\pi$
$908$		−	20.9060i		−	0.693790i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	41.2487	−	6.95224i	1.36513	−	0.230085i
$914$		0			0
$915$		−	51.3887i		−	1.69886i
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	14.4222i		−	0.475745i	−0.971296	−	0.237872i	$-0.923550\pi$
$919$		−	14.4222i		−	0.475745i	0.971296	−	0.237872i	$-0.0764500\pi$
$920$		0			0
$921$		0			0
$922$	−83.6218			−2.75394
$923$		−	17.7445i		−	0.584067i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	48.0000			1.57653
$928$		0			0
$929$	−59.5110			−1.95249			−0.976245	−	0.216667i	$-0.930481\pi$
$929$	−59.5110			−1.95249			−0.976245	+	0.216667i	$0.930481\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−44.8548			−1.46612
$937$		−	50.4777i		−	1.64904i	−0.565836	−	0.824518i	$-0.691447\pi$
$937$		−	50.4777i		−	1.64904i	0.565836	−	0.824518i	$-0.308553\pi$
$938$		0			0
$939$	60.0000			1.95803
$940$	143.282			4.67334
$941$			57.5859i			1.87725i	0.344943	+	0.938624i	$0.387898\pi$
$941$			57.5859i			1.87725i	−0.344943	+	0.938624i	$0.612102\pi$
$942$		−	8.29365i		−	0.270222i
$943$		0			0
$944$	−37.8366			−1.23147
$945$		0			0
$946$	16.4833	+	97.7983i	0.535920	+	3.17970i
$947$	20.7882			0.675526			0.337763	−	0.941231i	$-0.390330\pi$
$947$	20.7882			0.675526			0.337763	+	0.941231i	$0.390330\pi$
$948$			93.2266i			3.02786i
$949$		0			0
$950$		0			0
$951$	29.0170			0.940940
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$			63.6573i			2.05882i
$957$		0			0
$958$	−67.6603			−2.18600
$959$		0			0
$960$	75.9685			2.45187
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	14.9516	+	43.0951i	0.480562	+	1.38513i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	58.1769			1.86603
$973$		0			0
$974$		0			0
$975$			74.4916i			2.38564i
$976$		−	17.7689i		−	0.568768i
$977$	35.4154			1.13304			0.566520	−	0.824048i	$-0.308289\pi$
$977$	35.4154			1.13304			0.566520	+	0.824048i	$0.308289\pi$
$978$		0			0
$979$	−10.1244	−	60.0694i	−0.323576	−	1.91983i
$980$	−107.486			−3.43351
$981$		0			0
$982$		0			0
$983$	12.5594			0.400583			0.200292	−	0.979736i	$-0.435811\pi$
$983$	12.5594			0.400583			0.200292	+	0.979736i	$0.435811\pi$
$984$	56.1962			1.79147
$985$		−	57.1711i		−	1.82162i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	96.6486	−	16.2896i	3.07169	−	0.517716i
$991$	−8.00000			−0.254128			−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000			−0.254128			−0.127064	+	0.991894i	$0.540555\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−99.7686			−3.16288
$996$		−	81.5276i		−	2.58330i
$997$			24.9800i			0.791124i	0.918439	+	0.395562i	$0.129450\pi$
$997$			24.9800i			0.791124i	−0.918439	+	0.395562i	$0.870550\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.e.a.428.7	yes	8
3.2	odd	2	inner	429.2.e.a.428.2	yes	8
11.10	odd	2	inner	429.2.e.a.428.1	✓	8
13.12	even	2	inner	429.2.e.a.428.2	yes	8
33.32	even	2	inner	429.2.e.a.428.8	yes	8
39.38	odd	2	CM	429.2.e.a.428.7	yes	8
143.142	odd	2	inner	429.2.e.a.428.8	yes	8
429.428	even	2	inner	429.2.e.a.428.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.e.a.428.1	✓	8	11.10	odd	2	inner
429.2.e.a.428.1	✓	8	429.428	even	2	inner
429.2.e.a.428.2	yes	8	3.2	odd	2	inner
429.2.e.a.428.2	yes	8	13.12	even	2	inner
429.2.e.a.428.7	yes	8	1.1	even	1	trivial
429.2.e.a.428.7	yes	8	39.38	odd	2	CM
429.2.e.a.428.8	yes	8	33.32	even	2	inner
429.2.e.a.428.8	yes	8	143.142	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.39417i q^{2} -1.73205 q^{3} -3.73205 q^{4} -4.11439 q^{5} -4.14682i q^{6} -4.14682i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+2.39417i q^{2} -1.73205 q^{3} -3.73205 q^{4} -4.11439 q^{5} -4.14682i q^{6} -4.14682i q^{8} +3.00000 q^{9} -9.85055i q^{10} +(0.551224 + 3.27050i) q^{11} +6.46410 q^{12} -3.60555i q^{13} +7.12633 q^{15} +2.46410 q^{16} +7.18251i q^{18} +15.3551 q^{20} +(-7.83013 + 1.31972i) q^{22} +7.18251i q^{24} +11.9282 q^{25} +8.63230 q^{26} -5.19615 q^{27} +17.0617i q^{30} -2.39417i q^{32} +(-0.954747 - 5.66467i) q^{33} -11.1962 q^{36} +6.24500i q^{39} +17.0617i q^{40} -7.82403i q^{41} -12.4900i q^{43} +(-2.05719 - 12.2057i) q^{44} -12.3432 q^{45} +9.33123 q^{47} -4.26795 q^{48} -7.00000 q^{49} +28.5581i q^{50} +13.4561i q^{52} -12.4405i q^{54} +(-2.26795 - 13.4561i) q^{55} -15.3551 q^{59} -26.5958 q^{60} -7.21110i q^{61} +10.6603 q^{64} +14.8346i q^{65} +(13.5622 - 2.28583i) q^{66} +4.92144 q^{71} -12.4405i q^{72} -20.6603 q^{75} -14.9516 q^{78} +14.4222i q^{79} -10.1383 q^{80} +9.00000 q^{81} +18.7321 q^{82} -12.6124i q^{83} +29.9032 q^{86} +(13.5622 - 2.28583i) q^{88} -18.3671 q^{89} -29.5516i q^{90} +22.3405i q^{94} +4.14682i q^{96} -16.7592i q^{98} +(1.65367 + 9.81149i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} + 24 q^{9}+O(q^{10}) \) 8 * q - 16 * q^4 + 24 * q^9	\( 8 q - 16 q^{4} + 24 q^{9} + 24 q^{12} - 8 q^{16} - 28 q^{22} + 40 q^{25} - 48 q^{36} - 48 q^{48} - 56 q^{49} - 32 q^{55} + 16 q^{64} + 60 q^{66} - 96 q^{75} + 72 q^{81} + 136 q^{82} + 60 q^{88}+O(q^{100}) \) 8 * q - 16 * q^4 + 24 * q^9 + 24 * q^12 - 8 * q^16 - 28 * q^22 + 40 * q^25 - 48 * q^36 - 48 * q^48 - 56 * q^49 - 32 * q^55 + 16 * q^64 + 60 * q^66 - 96 * q^75 + 72 * q^81 + 136 * q^82 + 60 * q^88

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