LMFDB - Embedded newform 429.2.e.a.428.6

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			428.6
Root			$-1.19709 + 0.752986i$ of defining polynomial
Character	$\chi$	$=$	429.428
Dual form			429.2.e.a.428.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.50597i q^{2} +1.73205 q^{3} -0.267949 q^{4} +1.75265 q^{5} +2.60842i q^{6} +2.60842i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.50597i q^{2} +1.73205 q^{3} -0.267949 q^{4} +1.75265 q^{5} +2.60842i q^{6} +2.60842i q^{8} +3.00000 q^{9} +2.63945i q^{10} +(-3.27050 - 0.551224i) q^{11} -0.464102 q^{12} -3.60555i q^{13} +3.03569 q^{15} -4.46410 q^{16} +4.51791i q^{18} -0.469622 q^{20} +(0.830127 - 4.92527i) q^{22} +4.51791i q^{24} -1.92820 q^{25} +5.42986 q^{26} +5.19615 q^{27} +4.57166i q^{30} -1.50597i q^{32} +(-5.66467 - 0.954747i) q^{33} -0.803848 q^{36} -6.24500i q^{39} +4.57166i q^{40} -10.1383i q^{41} +12.4900i q^{43} +(0.876327 + 0.147700i) q^{44} +5.25796 q^{45} -10.0463 q^{47} -7.73205 q^{48} -7.00000 q^{49} -2.90382i q^{50} +0.966105i q^{52} +7.82526i q^{54} +(-5.73205 - 0.966105i) q^{55} +0.469622 q^{59} -0.813410 q^{60} -7.21110i q^{61} -6.66025 q^{64} -6.31928i q^{65} +(1.43782 - 8.53083i) q^{66} +16.1177 q^{71} +7.82526i q^{72} -3.33975 q^{75} +9.40479 q^{78} +14.4222i q^{79} -7.82403 q^{80} +9.00000 q^{81} +15.2679 q^{82} -13.1502i q^{83} -18.8096 q^{86} +(1.43782 - 8.53083i) q^{88} -4.31872 q^{89} +7.91834i q^{90} -15.1294i q^{94} -2.60842i q^{96} -10.5418i q^{98} +(-9.81149 - 1.65367i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			1.50597i			1.06488i	0.846467	+	0.532441i	$0.178725\pi$
$2$			1.50597i			1.06488i	−0.846467	+	0.532441i	$0.821275\pi$
$3$	1.73205			1.00000
$3$	1.73205			1.00000
$4$	−0.267949			−0.133975
$5$	1.75265			0.783811			0.391905	−	0.920006i	$-0.371816\pi$
$5$	1.75265			0.783811			0.391905	+	0.920006i	$0.371816\pi$
$6$			2.60842i			1.06488i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$			2.60842i			0.922215i
$9$	3.00000			1.00000
$10$			2.63945i			0.834666i
$11$	−3.27050	−	0.551224i	−0.986092	−	0.166200i
$11$	−3.27050	−	0.551224i	−0.986092	−	0.166200i
$12$	−0.464102			−0.133975
$13$		−	3.60555i		−	1.00000i
$13$		−	3.60555i		−	1.00000i
$14$		0			0
$15$	3.03569			0.783811
$16$	−4.46410			−1.11603
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$			4.51791i			1.06488i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−0.469622			−0.105011
$21$		0			0
$22$	0.830127	−	4.92527i	0.176984	−	1.05007i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$			4.51791i			0.922215i
$25$	−1.92820			−0.385641
$26$	5.42986			1.06488
$27$	5.19615			1.00000
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$			4.57166i			0.834666i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		−	1.50597i		−	0.266221i
$33$	−5.66467	−	0.954747i	−0.986092	−	0.166200i
$34$		0			0
$35$		0			0
$36$	−0.803848			−0.133975
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		−	6.24500i		−	1.00000i
$40$			4.57166i			0.722842i
$41$		−	10.1383i		−	1.58333i	−0.610954	−	0.791666i	$-0.709214\pi$
$41$		−	10.1383i		−	1.58333i	0.610954	−	0.791666i	$-0.290786\pi$
$42$		0			0
$43$			12.4900i			1.90471i	0.304997	+	0.952353i	$0.401344\pi$
$43$			12.4900i			1.90471i	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0.876327	+	0.147700i	0.132111	+	0.0222666i
$45$	5.25796			0.783811
$46$		0			0
$47$	−10.0463			−1.46540			−0.732702	−	0.680550i	$-0.761741\pi$
$47$	−10.0463			−1.46540			−0.732702	+	0.680550i	$0.761741\pi$
$48$	−7.73205			−1.11603
$49$	−7.00000			−1.00000
$50$		−	2.90382i		−	0.410662i
$51$		0			0
$52$			0.966105i			0.133975i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$			7.82526i			1.06488i
$55$	−5.73205	−	0.966105i	−0.772910	−	0.130270i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.469622			0.0611396			0.0305698	−	0.999533i	$-0.490268\pi$
$59$	0.469622			0.0611396			0.0305698	+	0.999533i	$0.490268\pi$
$60$	−0.813410			−0.105011
$61$		−	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
$64$	−6.66025			−0.832532
$65$		−	6.31928i		−	0.783811i
$66$	1.43782	−	8.53083i	0.176984	−	1.05007i
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	16.1177			1.91282			0.956408	−	0.292034i	$-0.0943319\pi$
$71$	16.1177			1.91282			0.956408	+	0.292034i	$0.0943319\pi$
$72$			7.82526i			0.922215i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$	−3.33975			−0.385641
$76$		0			0
$77$		0			0
$78$	9.40479			1.06488
$79$			14.4222i			1.62262i	0.584613	+	0.811312i	$0.301246\pi$
$79$			14.4222i			1.62262i	−0.584613	+	0.811312i	$0.698754\pi$
$80$	−7.82403			−0.874753
$81$	9.00000			1.00000
$82$	15.2679			1.68606
$83$		−	13.1502i		−	1.44342i	−0.692194	−	0.721712i	$-0.743356\pi$
$83$		−	13.1502i		−	1.44342i	0.692194	−	0.721712i	$-0.256644\pi$
$84$		0			0
$85$		0			0
$86$	−18.8096			−2.02829
$87$		0			0
$88$	1.43782	−	8.53083i	0.153272	−	0.909389i
$89$	−4.31872			−0.457783			−0.228892	−	0.973452i	$-0.573510\pi$
$89$	−4.31872			−0.457783			−0.228892	+	0.973452i	$0.573510\pi$
$90$			7.91834i			0.834666i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		−	15.1294i		−	1.56048i
$95$		0			0
$96$		−	2.60842i		−	0.266221i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		−	10.5418i		−	1.06488i
$99$	−9.81149	−	1.65367i	−0.986092	−	0.166200i
$100$	0.516660			0.0516660
$101$		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	$0.210980\pi$
$104$	9.40479			0.922215
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−1.39230			−0.133975
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	1.45493	−	8.63230i	0.138722	−	0.823058i
$111$		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
$118$			0.707238i			0.0651065i
$119$		0			0
$120$			7.91834i			0.722842i
$121$	10.3923	+	3.60555i	0.944755	+	0.327777i
$122$	10.8597			0.983192
$123$		−	17.5600i		−	1.58333i
$124$		0			0
$125$	−12.1427			−1.08608
$126$		0			0
$127$		−	14.4222i		−	1.27976i	−0.768473	−	0.639882i	$-0.778983\pi$
$127$		−	14.4222i		−	1.27976i	0.768473	−	0.639882i	$-0.221017\pi$
$128$		−	13.0421i		−	1.15277i
$129$			21.6333i			1.90471i
$130$	9.51666			0.834666
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	1.51784	+	0.255824i	0.132111	+	0.0222666i
$133$		0			0
$134$		0			0
$135$	9.10706			0.783811
$136$		0			0
$137$	18.3400			1.56689			0.783444	−	0.621463i	$-0.213461\pi$
$137$	18.3400			1.56689			0.783444	+	0.621463i	$0.213461\pi$
$138$		0			0
$139$		−	12.4900i		−	1.05939i	−0.848189	−	0.529694i	$-0.822307\pi$
$139$		−	12.4900i		−	1.05939i	0.848189	−	0.529694i	$-0.177693\pi$
$140$		0			0
$141$	−17.4007			−1.46540
$142$			24.2728i			2.03692i
$143$	−1.98747	+	11.7919i	−0.166200	+	0.986092i
$144$	−13.3923			−1.11603
$145$		0			0
$146$		0			0
$147$	−12.1244			−1.00000
$148$		0			0
$149$			0.295400i			0.0242001i	0.999927	+	0.0121001i	$0.00385166\pi$
$149$			0.295400i			0.0242001i	−0.999927	+	0.0121001i	$0.996148\pi$
$150$		−	5.02956i		−	0.410662i
$151$		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
$156$			1.67334i			0.133975i
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$	−21.7194			−1.72790
$159$		0			0
$160$		−	2.63945i		−	0.208667i
$161$		0			0
$162$			13.5537i			1.06488i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$			2.71654i			0.212126i
$165$	−9.92820	−	1.67334i	−0.772910	−	0.130270i
$166$	19.8038			1.53708
$167$			11.5361i			0.892692i	0.894860	+	0.446346i	$0.147275\pi$
$167$			11.5361i			0.892692i	−0.894860	+	0.446346i	$0.852725\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$		−	3.34668i		−	0.255182i
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$	14.5998	+	2.46072i	1.10050	+	0.185484i
$177$	0.813410			0.0611396
$178$		−	6.50386i		−	0.487485i
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	−1.40887			−0.105011
$181$	−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
$188$	2.69190			0.196327
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−11.5359			−0.832532
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		−	10.9453i		−	0.783811i
$196$	1.87564			0.133975
$197$			24.3909i			1.73778i	0.495003	+	0.868891i	$0.335167\pi$
$197$			24.3909i			1.73778i	−0.495003	+	0.868891i	$0.664833\pi$
$198$	2.49038	−	14.7758i	0.176984	−	1.05007i
$199$	−24.2487			−1.71895			−0.859473	−	0.511182i	$-0.829208\pi$
$199$	−24.2487			−1.71895			−0.859473	+	0.511182i	$0.829208\pi$
$200$		−	5.02956i		−	0.355644i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		−	17.7689i		−	1.24103i
$206$			24.0955i			1.67882i
$207$		0			0
$208$			16.0955i			1.11603i
$209$		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
$213$	27.9166			1.91282
$214$		0			0
$215$			21.8906i			1.49293i
$216$			13.5537i			0.922215i
$217$		0			0
$218$		0			0
$219$		0			0
$220$	1.53590	+	0.258867i	0.103550	+	0.0174528i
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−5.78461			−0.385641
$226$		0			0
$227$			29.6078i			1.96514i	0.185901	+	0.982569i	$0.440480\pi$
$227$			29.6078i			1.96514i	−0.185901	+	0.982569i	$0.559520\pi$
$228$		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	$0.5\pi$
$234$	16.2896			1.06488
$235$	−17.6077			−1.14860
$236$	−0.125835			−0.00819115
$237$			24.9800i			1.62262i
$238$		0			0
$239$			25.7888i			1.66814i	0.551660	+	0.834069i	$0.313995\pi$
$239$			25.7888i			1.66814i	−0.551660	+	0.834069i	$0.686005\pi$
$240$	−13.5516			−0.874753
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	−5.42986	+	15.6505i	−0.349044	+	1.00605i
$243$	15.5885			1.00000
$244$			1.93221i			0.123697i
$245$	−12.2686			−0.783811
$246$	26.4449			1.68606
$247$		0			0
$248$		0			0
$249$		−	22.7768i		−	1.44342i
$250$		−	18.2866i		−	1.15655i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$	21.7194			1.36280
$255$		0			0
$256$	6.32051			0.395032
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$	−32.5791			−2.02829
$259$		0			0
$260$			1.69325i			0.105011i
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$	2.49038	−	14.7758i	0.153272	−	0.909389i
$265$		0			0
$266$		0			0
$267$	−7.48024			−0.457783
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$			13.7150i			0.834666i
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$			27.6194i			1.66855i
$275$	6.30618	+	1.06287i	0.380277	+	0.0640936i
$276$		0			0
$277$			24.9800i			1.50090i	0.660926	+	0.750451i	$0.270164\pi$
$277$			24.9800i			1.50090i	−0.660926	+	0.750451i	$0.729836\pi$
$278$	18.8096			1.12812
$279$		0			0
$280$		0			0
$281$		−	13.9573i		−	0.832621i	−0.909223	−	0.416310i	$-0.863323\pi$
$281$		−	13.9573i		−	0.832621i	0.909223	−	0.416310i	$-0.136677\pi$
$282$		−	26.2050i		−	1.56048i
$283$		−	28.8444i		−	1.71462i	−0.514799	−	0.857311i	$-0.672133\pi$
$283$		−	28.8444i		−	1.71462i	0.514799	−	0.857311i	$-0.327867\pi$
$284$	−4.31872			−0.256269
$285$		0			0
$286$	−17.7583	−	2.99307i	−1.05007	−	0.176984i
$287$		0			0
$288$		−	4.51791i		−	0.266221i
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	32.6197i		−	1.90566i	−0.303498	−	0.952832i	$-0.598154\pi$
$293$		−	32.6197i		−	1.90566i	0.303498	−	0.952832i	$-0.401846\pi$
$294$		−	18.2589i		−	1.06488i
$295$	0.823085			0.0479219
$296$		0			0
$297$	−16.9940	−	2.86424i	−0.986092	−	0.166200i
$298$	−0.444864			−0.0257703
$299$		0			0
$300$	0.894882			0.0516660
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	12.6386i		−	0.723682i
$306$		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			$\pi$
$312$	16.2896			0.922215
$313$	34.6410			1.95803			0.979013	−	0.203798i	$-0.0653285\pi$
$313$	34.6410			1.95803			0.979013	+	0.203798i	$0.0653285\pi$
$314$			3.01194i			0.169974i
$315$		0			0
$316$		−	3.86442i		−	0.217391i
$317$	31.4219			1.76483			0.882416	−	0.470470i	$-0.155916\pi$
$317$	31.4219			1.76483			0.882416	+	0.470470i	$0.155916\pi$
$318$		0			0
$319$		0			0
$320$	−11.6731			−0.652547
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−2.41154			−0.133975
$325$			6.95224i			0.385641i
$326$		0			0
$327$		0			0
$328$	26.4449			1.46017
$329$		0			0
$330$	2.52001	−	14.9516i	0.138722	−	0.823058i
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$			3.52359i			0.193382i
$333$		0			0
$334$	−17.3731			−0.950612
$335$		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	$-0.439572\pi$
$338$		−	19.5776i		−	1.06488i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−32.5791			−1.75655
$345$		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
$352$	−0.830127	+	4.92527i	−0.0442459	+	0.262518i
$353$	−37.4933			−1.99557			−0.997784	−	0.0665386i	$-0.978804\pi$
$353$	−37.4933			−1.99557			−0.997784	+	0.0665386i	$0.978804\pi$
$354$			1.22497i			0.0651065i
$355$	28.2487			1.49929
$356$	1.15720			0.0613313
$357$		0			0
$358$		0			0
$359$			37.8366i			1.99694i	0.0553230	+	0.998469i	$0.482381\pi$
$359$			37.8366i			1.99694i	−0.0553230	+	0.998469i	$0.517619\pi$
$360$			13.7150i			0.722842i
$361$	−19.0000			−1.00000
$362$		−	15.0597i		−	0.791521i
$363$	18.0000	+	6.24500i	0.944755	+	0.327777i
$364$		0			0
$365$		0			0
$366$	18.8096			0.983192
$367$	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
$369$		−	30.4148i		−	1.58333i
$370$		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
$375$	−21.0319			−1.08608
$376$		−	26.2050i		−	1.35142i
$377$		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
$383$	35.2710			1.80227			0.901133	−	0.433543i	$-0.142737\pi$
$383$	35.2710			1.80227			0.901133	+	0.433543i	$0.142737\pi$
$384$		−	22.5896i		−	1.15277i
$385$		0			0
$386$		0			0
$387$			37.4700i			1.90471i
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$	16.4833			0.834666
$391$		0			0
$392$		−	18.2589i		−	0.922215i
$393$		0			0
$394$	−36.7321			−1.85053
$395$			25.2771i			1.27183i
$396$	2.62898	+	0.443100i	0.132111	+	0.0222666i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		−	36.5179i		−	1.83047i
$399$		0			0
$400$	8.60770			0.430385
$401$	−25.3506			−1.26595			−0.632973	−	0.774173i	$-0.718166\pi$
$401$	−25.3506			−1.26595			−0.632973	+	0.774173i	$0.718166\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	15.7739			0.783811
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$	26.7594			1.32155
$411$	31.7657			1.56689
$412$	−4.28719			−0.211215
$413$		0			0
$414$		0			0
$415$		−	23.0478i		−	1.13137i
$416$	−5.42986			−0.266221
$417$		−	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			$\pi$
$422$	−43.4389			−2.11457
$423$	−30.1389			−1.46540
$424$		0			0
$425$		0			0
$426$			42.0416i			2.03692i
$427$		0			0
$428$		0			0
$429$	−3.44239	+	20.4242i	−0.166200	+	0.986092i
$430$	−32.9667			−1.58979
$431$			40.0415i			1.92873i	0.264578	+	0.964364i	$0.414767\pi$
$431$			40.0415i			1.92873i	−0.264578	+	0.964364i	$0.585233\pi$
$432$	−23.1962			−1.11603
$433$	20.7846			0.998845			0.499422	−	0.866359i	$-0.333546\pi$
$433$	20.7846			0.998845			0.499422	+	0.866359i	$0.333546\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$			12.4900i			0.596115i	0.954548	+	0.298057i	$0.0963387\pi$
$439$			12.4900i			0.596115i	−0.954548	+	0.298057i	$0.903661\pi$
$440$	2.52001	−	14.9516i	0.120137	−	0.712789i
$441$	−21.0000			−1.00000
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	−7.56922			−0.358815
$446$		0			0
$447$			0.511648i			0.0242001i
$448$		0			0
$449$	40.9986			1.93484			0.967422	−	0.253168i	$-0.0814726\pi$
$449$	40.9986			1.93484			0.967422	+	0.253168i	$0.0814726\pi$
$450$		−	8.71146i		−	0.410662i
$451$	−5.58846	+	33.1572i	−0.263150	+	1.56131i
$452$		0			0
$453$		0			0
$454$	−44.5885			−2.09264
$455$		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
$461$		−	24.9817i		−	1.16352i	−0.813362	−	0.581758i	$-0.802365\pi$
$461$		−	24.9817i		−	1.16352i	0.813362	−	0.581758i	$-0.197635\pi$
$462$		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			$\pi$
$468$			2.89831i			0.133975i
$469$		0			0
$470$		−	26.5167i		−	1.22312i
$471$	3.46410			0.159617
$472$			1.22497i			0.0563839i
$473$	6.88478	−	40.8485i	0.316563	−	1.87822i
$474$	−37.6191			−1.72790
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−38.8372			−1.77637
$479$			33.4268i			1.52731i	0.645626	+	0.763654i	$0.276597\pi$
$479$			33.4268i			1.52731i	−0.645626	+	0.763654i	$0.723403\pi$
$480$		−	4.57166i		−	0.208667i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−2.78461	−	0.966105i	−0.126573	−	0.0439138i
$485$		0			0
$486$			23.4758i			1.06488i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	18.8096			0.851469
$489$		0			0
$490$		−	18.4761i		−	0.834666i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$			4.70519i			0.212126i
$493$		0			0
$494$		0			0
$495$	−17.1962	−	2.89831i	−0.772910	−	0.130270i
$496$		0			0
$497$		0			0
$498$	34.3013			1.53708
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$	3.25364			0.145507
$501$			19.9811i			0.892692i
$502$		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
$508$			3.86442i			0.171456i
$509$	−43.5647			−1.93097			−0.965485	−	0.260457i	$-0.916127\pi$
$509$	−43.5647			−1.93097			−0.965485	+	0.260457i	$0.916127\pi$
$510$		0			0
$511$		0			0
$512$		−	16.5657i		−	0.732107i
$513$		0			0
$514$		0			0
$515$	28.0425			1.23570
$516$		−	5.79663i		−	0.255182i
$517$	32.8564	+	5.53776i	1.44502	+	0.243550i
$518$		0			0
$519$		0			0
$520$	16.4833			0.722842
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		−	12.4900i		−	0.546149i	−0.961993	−	0.273075i	$-0.911959\pi$
$523$		−	12.4900i		−	0.546149i	0.961993	−	0.273075i	$-0.0880406\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	25.2877	+	4.26209i	1.10050	+	0.185484i
$529$	23.0000			1.00000
$530$		0			0
$531$	1.40887			0.0611396
$532$		0			0
$533$	−36.5541			−1.58333
$534$		−	11.2650i		−	0.487485i
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	22.8935	+	3.85857i	0.986092	+	0.166200i
$540$	−2.44023			−0.105011
$541$		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	$-0.704275\pi$
$547$		−	37.4700i		−	1.60210i	0.598597	−	0.801050i	$-0.295725\pi$
$548$	−4.91418			−0.209923
$549$		−	21.6333i		−	0.923287i
$550$	−1.60065	+	9.49693i	−0.0682521	+	0.404950i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−37.6191			−1.59828
$555$		0			0
$556$			3.34668i			0.141931i
$557$		−	3.52359i		−	0.149299i	−0.997210	−	0.0746496i	$-0.976216\pi$
$557$		−	3.52359i		−	0.149299i	0.997210	−	0.0746496i	$-0.0237838\pi$
$558$		0			0
$559$	45.0333			1.90471
$560$		0			0
$561$		0			0
$562$	21.0192			0.886643
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	4.66251			0.196327
$565$		0			0
$566$	43.4389			1.82587
$567$		0			0
$568$			42.0416i			1.76403i
$569$		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	$-0.206248\pi$
$572$	0.532540	−	3.15964i	0.0222666	−	0.132111i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−19.9808			−0.832532
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		−	25.6015i		−	1.06488i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		−	18.9579i		−	0.783811i
$586$	49.1244			2.02931
$587$	−24.7551			−1.02175			−0.510876	−	0.859654i	$-0.670679\pi$
$587$	−24.7551			−1.02175			−0.510876	+	0.859654i	$0.670679\pi$
$588$	3.24871			0.133975
$589$		0			0
$590$			1.23954i			0.0510312i
$591$			42.2463i			1.73778i
$592$		0			0
$593$		−	39.2344i		−	1.61116i	−0.592484	−	0.805582i	$-0.701853\pi$
$593$		−	39.2344i		−	1.61116i	0.592484	−	0.805582i	$-0.298147\pi$
$594$	4.31347	−	25.5925i	0.176984	−	1.05007i
$595$		0			0
$596$		−	0.0791522i		−	0.00324220i
$597$	−42.0000			−1.71895
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		−	8.71146i		−	0.355644i
$601$			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
$605$	18.2141	+	6.31928i	0.740509	+	0.256915i
$606$		0			0
$607$			37.4700i			1.52086i	0.649420	+	0.760430i	$0.275012\pi$
$607$			37.4700i			1.52086i	−0.649420	+	0.760430i	$0.724988\pi$
$608$		0			0
$609$		0			0
$610$	19.0333			0.770637
$611$			36.2225i			1.46540i
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		−	30.7766i		−	1.24103i
$616$		0			0
$617$	16.4615			0.662714			0.331357	−	0.943506i	$-0.392494\pi$
$617$	16.4615			0.662714			0.331357	+	0.943506i	$0.392494\pi$
$618$			41.7347i			1.67882i
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$			27.8783i			1.11603i
$625$	−11.6410			−0.465641
$626$			52.1684i			2.08507i
$627$		0			0
$628$	−0.535898			−0.0213847
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	−37.6191			−1.49641
$633$			49.9600i			1.98573i
$634$			47.3205i			1.87934i
$635$		−	25.2771i		−	1.00309i
$636$		0			0
$637$			25.2389i			1.00000i
$638$		0			0
$639$	48.3530			1.91282
$640$		−	22.8583i		−	0.903553i
$641$		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
$645$			37.9157i			1.49293i
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$			23.4758i			0.922215i
$649$	−1.53590	−	0.258867i	−0.0602893	−	0.0101614i
$650$	−10.4699			−0.410662
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$			45.2583i			1.76704i
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$	2.66025	+	0.448371i	0.103550	+	0.0174528i
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$	34.3013			1.33115
$665$		0			0
$666$		0			0
$667$		0			0
$668$		−	3.09109i		−	0.119598i
$669$		0			0
$670$		0			0
$671$	−3.97493	+	23.5839i	−0.153450	+	0.910446i
$672$		0			0
$673$		−	49.9600i		−	1.92582i	−0.269830	−	0.962908i	$-0.586968\pi$
$673$		−	49.9600i		−	1.92582i	0.269830	−	0.962908i	$-0.413032\pi$
$674$	54.2986			2.09150
$675$	−10.0192			−0.385641
$676$	3.48334			0.133975
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$			51.2822i			1.96514i
$682$		0			0
$683$	−51.8583			−1.98430			−0.992152	−	0.125038i	$-0.960095\pi$
$683$	−51.8583			−1.98430			−0.992152	+	0.125038i	$0.960095\pi$
$684$		0			0
$685$	32.1436			1.22814
$686$		0			0
$687$		0			0
$688$		−	55.7566i		−	2.12570i
$689$		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
$695$		−	21.8906i		−	0.830359i
$696$		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	$0.5\pi$
$702$	28.2144			1.06488
$703$		0			0
$704$	21.7823	+	3.67129i	0.820953	+	0.138367i
$705$	−30.4974			−1.14860
$706$		−	56.4638i		−	2.12504i
$707$		0			0
$708$	−0.217952			−0.00819115
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$			42.5417i			1.59656i
$711$			43.2666i			1.62262i
$712$		−	11.2650i		−	0.422175i
$713$		0			0
$714$		0			0
$715$	−3.48334	+	20.6672i	−0.130270	+	0.772910i
$716$		0			0
$717$			44.6675i			1.66814i
$718$	−56.9808			−2.12650
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$	−23.4721			−0.874753
$721$		0			0
$722$		−	28.6135i		−	1.06488i
$723$		0			0
$724$	2.67949			0.0995825
$725$		0			0
$726$	−9.40479	+	27.1075i	−0.349044	+	1.00605i
$727$	51.9615			1.92715			0.963573	−	0.267445i	$-0.0861794\pi$
$727$	51.9615			1.92715			0.963573	+	0.267445i	$0.0861794\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$			3.34668i			0.123697i
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$			12.0478i			0.444691i
$735$	−21.2498			−0.783811
$736$		0			0
$737$		0			0
$738$	45.8038			1.68606
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	54.2941i		−	1.99186i	−0.0901418	−	0.995929i	$-0.528732\pi$
$743$		−	54.2941i		−	1.99186i	0.0901418	−	0.995929i	$-0.471268\pi$
$744$		0			0
$745$			0.517734i			0.0189683i
$746$	−54.2986			−1.98801
$747$		−	39.4507i		−	1.44342i
$748$		0			0
$749$		0			0
$750$		−	31.6734i		−	1.15655i
$751$	−40.0000			−1.45962			−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000			−1.45962			−0.729810	+	0.683650i	$0.760392\pi$
$752$	44.8477			1.63543
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	41.5692			1.51086			0.755429	−	0.655230i	$-0.227428\pi$
$757$	41.5692			1.51086			0.755429	+	0.655230i	$0.227428\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	27.1866i		−	0.985515i	−0.870167	−	0.492757i	$-0.835989\pi$
$761$		−	27.1866i		−	0.985515i	0.870167	−	0.492757i	$-0.164011\pi$
$762$	37.6191			1.36280
$763$		0			0
$764$		0			0
$765$		0			0
$766$			53.1172i			1.91920i
$767$		−	1.69325i		−	0.0611396i
$768$	10.9474			0.395032
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	54.0806			1.94514			0.972572	−	0.232601i	$-0.0747237\pi$
$773$	54.0806			1.94514			0.972572	+	0.232601i	$0.0747237\pi$
$774$	−56.4287			−2.02829
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$			2.93279i			0.105011i
$781$	−52.7128	−	8.88444i	−1.88621	−	0.317910i
$782$		0			0
$783$		0			0
$784$	31.2487			1.11603
$785$	3.50531			0.125110
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		−	6.53553i		−	0.232819i
$789$		0			0
$790$	−38.0666			−1.35435
$791$		0			0
$792$	4.31347	−	25.5925i	0.153272	−	0.909389i
$793$	−26.0000			−0.923287
$794$		0			0
$795$		0			0
$796$	6.49742			0.230295
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$			2.90382i			0.102665i
$801$	−12.9562			−0.457783
$802$		−	38.1772i		−	1.34808i
$803$		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	$0.5\pi$
$810$			23.7550i			0.834666i
$811$		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
$820$			4.76116i			0.166267i
$821$		−	57.3061i		−	2.00000i	−0.00219583	−	0.999998i	$-0.500699\pi$
$821$		−	57.3061i		−	2.00000i	0.00219583	−	0.999998i	$-0.499301\pi$
$822$			47.8383i			1.66855i
$823$	−56.0000			−1.95204			−0.976019	−	0.217687i	$-0.930149\pi$
$823$	−56.0000			−1.95204			−0.976019	+	0.217687i	$0.930149\pi$
$824$			41.7347i			1.45390i
$825$	10.9226	+	1.84095i	0.380277	+	0.0640936i
$826$		0			0
$827$		−	50.4751i		−	1.75519i	−0.479401	−	0.877596i	$-0.659146\pi$
$827$		−	50.4751i		−	1.75519i	0.479401	−	0.877596i	$-0.340854\pi$
$828$		0			0
$829$	−27.7128			−0.962506			−0.481253	−	0.876582i	$-0.659818\pi$
$829$	−27.7128			−0.962506			−0.481253	+	0.876582i	$0.659818\pi$
$830$	34.7093			1.20478
$831$			43.2666i			1.50090i
$832$			24.0139i			0.832532i
$833$		0			0
$834$	32.5791			1.12812
$835$			20.2188i			0.699702i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−37.1495			−1.28254			−0.641272	−	0.767314i	$-0.721593\pi$
$839$	−37.1495			−1.28254			−0.641272	+	0.767314i	$0.721593\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		−	24.1747i		−	0.832621i
$844$		−	7.72884i		−	0.266038i
$845$	−22.7845			−0.783811
$846$		−	45.3883i		−	1.56048i
$847$		0			0
$848$		0			0
$849$		−	49.9600i		−	1.71462i
$850$		0			0
$851$		0			0
$852$	−7.48024			−0.256269
$853$		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	$0.5\pi$
$858$	−30.7583	−	5.18414i	−1.05007	−	0.176984i
$859$	−10.3923			−0.354581			−0.177290	−	0.984159i	$-0.556733\pi$
$859$	−10.3923			−0.354581			−0.177290	+	0.984159i	$0.556733\pi$
$860$		−	5.86558i		−	0.200015i
$861$		0			0
$862$	−60.3013			−2.05387
$863$	−34.3318			−1.16867			−0.584334	−	0.811513i	$-0.698644\pi$
$863$	−34.3318			−1.16867			−0.584334	+	0.811513i	$0.698644\pi$
$864$		−	7.82526i		−	0.266221i
$865$		0			0
$866$			31.3010i			1.06365i
$867$	−29.4449			−1.00000
$868$		0			0
$869$	7.94986	−	47.1678i	0.269681	−	1.60006i
$870$		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	$0.5\pi$
$878$	−18.8096			−0.634792
$879$		−	56.4990i		−	1.90566i
$880$	25.5885	+	4.31279i	0.862587	+	0.145384i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		−	31.6254i		−	1.06488i
$883$	−51.9615			−1.74864			−0.874322	−	0.485346i	$-0.838694\pi$
$883$	−51.9615			−1.74864			−0.874322	+	0.485346i	$0.838694\pi$
$884$		0			0
$885$	1.42563			0.0479219
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		−	11.3990i		−	0.382096i
$891$	−29.4345	−	4.96101i	−0.986092	−	0.166200i
$892$		0			0
$893$		0			0
$894$	−0.770527			−0.0257703
$895$		0			0
$896$		0			0
$897$		0			0
$898$			61.7427i			2.06038i
$899$		0			0
$900$	1.54998			0.0516660
$901$		0			0
$902$	−49.9338	−	8.41606i	−1.66261	−	0.280224i
$903$		0			0
$904$		0			0
$905$	−17.5265			−0.582602
$906$		0			0
$907$	17.3205			0.575118			0.287559	−	0.957763i	$-0.407156\pi$
$907$	17.3205			0.575118			0.287559	+	0.957763i	$0.407156\pi$
$908$		−	7.93338i		−	0.263278i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	−7.24871	+	43.0077i	−0.239897	+	1.42335i
$914$		0			0
$915$		−	21.8906i		−	0.723682i
$916$		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
$922$	37.6218			1.23901
$923$		−	58.1131i		−	1.91282i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	48.0000			1.57653
$928$		0			0
$929$	13.2078			0.433335			0.216667	−	0.976245i	$-0.430481\pi$
$929$	13.2078			0.433335			0.216667	+	0.976245i	$0.430481\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	28.2144			0.922215
$937$		−	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
$940$	4.71797			0.153883
$941$		−	21.1627i		−	0.689886i	−0.938624	−	0.344943i	$-0.887898\pi$
$941$		−	21.1627i		−	0.689886i	0.938624	−	0.344943i	$-0.112102\pi$
$942$			5.21684i			0.169974i
$943$		0			0
$944$	−2.09644			−0.0682334
$945$		0			0
$946$	61.5167	+	10.3683i	2.00008	+	0.337102i
$947$	57.9297			1.88246			0.941231	−	0.337763i	$-0.109670\pi$
$947$	57.9297			1.88246			0.941231	+	0.337763i	$0.109670\pi$
$948$		−	6.69337i		−	0.217391i
$949$		0			0
$950$		0			0
$951$	54.4244			1.76483
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		−	6.91008i		−	0.223488i
$957$		0			0
$958$	−50.3397			−1.62640
$959$		0			0
$960$	−20.2184			−0.652547
$961$	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	$0.5\pi$
$968$	−9.40479	+	27.1075i	−0.302281	+	0.871267i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	−4.17691			−0.133975
$973$		0			0
$974$		0			0
$975$			12.0416i			0.385641i
$976$			32.1911i			1.03041i
$977$	−51.5145			−1.64810			−0.824048	−	0.566520i	$-0.808289\pi$
$977$	−51.5145			−1.64810			−0.824048	+	0.566520i	$0.808289\pi$
$978$		0			0
$979$	14.1244	+	2.38058i	0.451416	+	0.0760837i
$980$	3.28736			0.105011
$981$		0			0
$982$		0			0
$983$	61.4350			1.95947			0.979736	−	0.200292i	$-0.0641890\pi$
$983$	61.4350			1.95947			0.979736	+	0.200292i	$0.0641890\pi$
$984$	45.8038			1.46017
$985$			42.7489i			1.36209i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	4.36478	−	25.8969i	0.138722	−	0.823058i
$991$	−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
$995$	−42.4996			−1.34733
$996$			6.10304i			0.193382i
$997$		−	24.9800i		−	0.791124i	−0.918439	−	0.395562i	$-0.870550\pi$
$997$		−	24.9800i		−	0.791124i	0.918439	−	0.395562i	$-0.129450\pi$
$998$		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.428.6	yes	8
3.2	odd	2	inner	429.2.e.a.428.3	✓	8
11.10	odd	2	inner	429.2.e.a.428.4	yes	8
13.12	even	2	inner	429.2.e.a.428.3	✓	8
33.32	even	2	inner	429.2.e.a.428.5	yes	8
39.38	odd	2	CM	429.2.e.a.428.6	yes	8
143.142	odd	2	inner	429.2.e.a.428.5	yes	8
429.428	even	2	inner	429.2.e.a.428.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.e.a.428.3	✓	8	3.2	odd	2	inner
429.2.e.a.428.3	✓	8	13.12	even	2	inner
429.2.e.a.428.4	yes	8	11.10	odd	2	inner
429.2.e.a.428.4	yes	8	429.428	even	2	inner
429.2.e.a.428.5	yes	8	33.32	even	2	inner
429.2.e.a.428.5	yes	8	143.142	odd	2	inner
429.2.e.a.428.6	yes	8	1.1	even	1	trivial
429.2.e.a.428.6	yes	8	39.38	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q+1.50597i q^{2} +1.73205 q^{3} -0.267949 q^{4} +1.75265 q^{5} +2.60842i q^{6} +2.60842i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.50597i q^{2} +1.73205 q^{3} -0.267949 q^{4} +1.75265 q^{5} +2.60842i q^{6} +2.60842i q^{8} +3.00000 q^{9} +2.63945i q^{10} +(-3.27050 - 0.551224i) q^{11} -0.464102 q^{12} -3.60555i q^{13} +3.03569 q^{15} -4.46410 q^{16} +4.51791i q^{18} -0.469622 q^{20} +(0.830127 - 4.92527i) q^{22} +4.51791i q^{24} -1.92820 q^{25} +5.42986 q^{26} +5.19615 q^{27} +4.57166i q^{30} -1.50597i q^{32} +(-5.66467 - 0.954747i) q^{33} -0.803848 q^{36} -6.24500i q^{39} +4.57166i q^{40} -10.1383i q^{41} +12.4900i q^{43} +(0.876327 + 0.147700i) q^{44} +5.25796 q^{45} -10.0463 q^{47} -7.73205 q^{48} -7.00000 q^{49} -2.90382i q^{50} +0.966105i q^{52} +7.82526i q^{54} +(-5.73205 - 0.966105i) q^{55} +0.469622 q^{59} -0.813410 q^{60} -7.21110i q^{61} -6.66025 q^{64} -6.31928i q^{65} +(1.43782 - 8.53083i) q^{66} +16.1177 q^{71} +7.82526i q^{72} -3.33975 q^{75} +9.40479 q^{78} +14.4222i q^{79} -7.82403 q^{80} +9.00000 q^{81} +15.2679 q^{82} -13.1502i q^{83} -18.8096 q^{86} +(1.43782 - 8.53083i) q^{88} -4.31872 q^{89} +7.91834i q^{90} -15.1294i q^{94} -2.60842i q^{96} -10.5418i q^{98} +(-9.81149 - 1.65367i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} + 24 q^{9}+O(q^{10}) \) 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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