LMFDB - Embedded newform 429.2.e.d.428.3

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			428.3
Root			$-0.516228 + 1.31663i$ of defining polynomial
Character	$\chi$	$=$	429.428
Dual form			429.2.e.d.428.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.63326i q^{2} +(1.55701 - 0.758758i) q^{3} -4.93403 q^{4} +(-1.99800 - 4.10001i) q^{6} -4.70372 q^{7} +7.72606i q^{8} +(1.84857 - 2.36279i) q^{9} +O(q^{10})$	\(q-2.63326i q^{2} +(1.55701 - 0.758758i) q^{3} -4.93403 q^{4} +(-1.99800 - 4.10001i) q^{6} -4.70372 q^{7} +7.72606i q^{8} +(1.84857 - 2.36279i) q^{9} -3.31662i q^{11} +(-7.68235 + 3.74374i) q^{12} -3.60555 q^{13} +12.3861i q^{14} +10.4766 q^{16} +(-6.22183 - 4.86776i) q^{18} +7.09795 q^{19} +(-7.32375 + 3.56899i) q^{21} -8.73352 q^{22} -0.711862i q^{23} +(5.86221 + 12.0296i) q^{24} -5.00000 q^{25} +9.49434i q^{26} +(1.08546 - 5.08151i) q^{27} +23.2083 q^{28} -12.1355i q^{32} +(-2.51652 - 5.16402i) q^{33} +(-9.12092 + 11.6581i) q^{36} -18.6907i q^{38} +(-5.61389 + 2.73574i) q^{39} -12.1906i q^{41} +(9.39805 + 19.2853i) q^{42} +16.3643i q^{44} -1.87451 q^{46} +(16.3122 - 7.94922i) q^{48} +15.1250 q^{49} +13.1663i q^{50} +17.7899 q^{52} -14.4361i q^{53} +(-13.3809 - 2.85830i) q^{54} -36.3412i q^{56} +(11.0516 - 5.38562i) q^{57} +(-8.69516 + 11.1139i) q^{63} -11.0026 q^{64} +(-13.5982 + 6.62663i) q^{66} +(-0.540131 - 1.10838i) q^{69} +(18.2551 + 14.2822i) q^{72} +13.2927 q^{73} +(-7.78506 + 3.79379i) q^{75} -35.0215 q^{76} +15.6005i q^{77} +(7.20391 + 14.7828i) q^{78} +(-2.16556 - 8.73558i) q^{81} -32.1011 q^{82} +17.1097i q^{83} +(36.1356 - 17.6095i) q^{84} +25.6244 q^{88} +16.9595 q^{91} +3.51235i q^{92} +(-9.20790 - 18.8951i) q^{96} -39.8279i q^{98} +(-7.83649 - 6.13102i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 40 q^{4}+O(q^{10}) $ 20 * q - 40 * q^4	$ 20 q - 40 q^{4} + 80 q^{16} - 100 q^{25} + 10 q^{36} - 50 q^{42} + 70 q^{48} + 140 q^{49} - 160 q^{64} - 110 q^{66} + 130 q^{78}+O(q^{100}) $ 20 * q - 40 * q^4 + 80 * q^16 - 100 * q^25 + 10 * q^36 - 50 * q^42 + 70 * q^48 + 140 * q^49 - 160 * q^64 - 110 * q^66 + 130 * q^78

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	2.63326i		−	1.86199i	−0.365029	−	0.930996i	$-0.618941\pi$
$2$		−	2.63326i		−	1.86199i	0.365029	−	0.930996i	$-0.381059\pi$
$3$	1.55701	−	0.758758i	0.898941	−	0.438069i
$3$	1.55701	−	0.758758i	0.898941	−	0.438069i
$4$	−4.93403			−2.46702
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	−1.99800	−	4.10001i	−0.815682	−	1.67382i
$7$	−4.70372			−1.77784			−0.888919	−	0.458064i	$-0.848543\pi$
$7$	−4.70372			−1.77784			−0.888919	+	0.458064i	$0.848543\pi$
$8$			7.72606i			2.73157i
$9$	1.84857	−	2.36279i	0.616191	−	0.787597i
$10$		0			0
$11$		−	3.31662i		−	1.00000i
$11$		−	3.31662i		−	1.00000i
$12$	−7.68235	+	3.74374i	−2.21770	+	1.08072i
$13$	−3.60555			−1.00000
$13$	−3.60555			−1.00000
$14$			12.3861i			3.31032i
$15$		0			0
$16$	10.4766			2.61915
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−6.22183	−	4.86776i	−1.46650	−	1.14734i
$19$	7.09795			1.62838			0.814190	−	0.580598i	$-0.197181\pi$
$19$	7.09795			1.62838			0.814190	+	0.580598i	$0.197181\pi$
$20$		0			0
$21$	−7.32375	+	3.56899i	−1.59817	+	0.778816i
$22$	−8.73352			−1.86199
$23$		−	0.711862i		−	0.148433i	−0.997242	−	0.0742167i	$-0.976354\pi$
$23$		−	0.711862i		−	0.148433i	0.997242	−	0.0742167i	$-0.0236456\pi$
$24$	5.86221	+	12.0296i	1.19662	+	2.45552i
$25$	−5.00000			−1.00000
$26$			9.49434i			1.86199i
$27$	1.08546	−	5.08151i	0.208897	−	0.977938i
$28$	23.2083			4.38596
$29$		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			$\pi$
$32$		−	12.1355i		−	2.14527i
$33$	−2.51652	−	5.16402i	−0.438069	−	0.898941i
$34$		0			0
$35$		0			0
$36$	−9.12092	+	11.6581i	−1.52015	+	1.94301i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		−	18.6907i		−	3.03203i
$39$	−5.61389	+	2.73574i	−0.898941	+	0.438069i
$40$		0			0
$41$		−	12.1906i		−	1.90386i	−0.306320	−	0.951929i	$-0.599098\pi$
$41$		−	12.1906i		−	1.90386i	0.306320	−	0.951929i	$-0.400902\pi$
$42$	9.39805	+	19.2853i	1.45015	+	2.97579i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$			16.3643i			2.46702i
$45$		0			0
$46$	−1.87451			−0.276382
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	16.3122	−	7.94922i	2.35447	−	1.14737i
$49$	15.1250			2.16071
$50$			13.1663i			1.86199i
$51$		0			0
$52$	17.7899			2.46702
$53$		−	14.4361i		−	1.98295i	−0.130281	−	0.991477i	$-0.541588\pi$
$53$		−	14.4361i		−	1.98295i	0.130281	−	0.991477i	$-0.458412\pi$
$54$	−13.3809	−	2.85830i	−1.82091	−	0.388965i
$55$		0			0
$56$		−	36.3412i		−	4.85630i
$57$	11.0516	−	5.38562i	1.46382	−	0.713343i
$58$		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
$63$	−8.69516	+	11.1139i	−1.09549	+	1.40022i
$64$	−11.0026			−1.37533
$65$		0			0
$66$	−13.5982	+	6.62663i	−1.67382	+	0.815682i
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	−0.540131	−	1.10838i	−0.0650241	−	0.133433i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	18.2551	+	14.2822i	2.15138	+	1.68317i
$73$	13.2927			1.55579			0.777896	−	0.628394i	$-0.216287\pi$
$73$	13.2927			1.55579			0.777896	+	0.628394i	$0.216287\pi$
$74$		0			0
$75$	−7.78506	+	3.79379i	−0.898941	+	0.438069i
$76$	−35.0215			−4.01724
$77$			15.6005i			1.77784i
$78$	7.20391	+	14.7828i	0.815682	+	1.67382i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−2.16556	−	8.73558i	−0.240618	−	0.970620i
$82$	−32.1011			−3.54497
$83$			17.1097i			1.87804i	0.343867	+	0.939019i	$0.388263\pi$
$83$			17.1097i			1.87804i	−0.343867	+	0.939019i	$0.611737\pi$
$84$	36.1356	−	17.6095i	3.94272	−	1.92135i
$85$		0			0
$86$		0			0
$87$		0			0
$88$	25.6244			2.73157
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	16.9595			1.77784
$92$			3.51235i			0.366188i
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−9.20790	−	18.8951i	−0.939778	−	1.92847i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		−	39.8279i		−	4.02323i
$99$	−7.83649	−	6.13102i	−0.787597	−	0.616191i
$100$	24.6702			2.46702
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	−5.22337			−0.514673			−0.257337	−	0.966322i	$-0.582845\pi$
$103$	−5.22337			−0.514673			−0.257337	+	0.966322i	$0.582845\pi$
$104$		−	27.8567i		−	2.73157i
$105$		0			0
$106$	−38.0140			−3.69225
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−5.35570	+	25.0724i	−0.515353	+	2.41259i
$109$	0.903206			0.0865115			0.0432557	−	0.999064i	$-0.486227\pi$
$109$	0.903206			0.0865115			0.0432557	+	0.999064i	$0.486227\pi$
$110$		0			0
$111$		0			0
$112$	−49.2791			−4.65643
$113$		−	5.33103i		−	0.501501i	−0.968052	−	0.250750i	$-0.919323\pi$
$113$		−	5.33103i		−	0.501501i	0.968052	−	0.250750i	$-0.0806773\pi$
$114$	−14.1817	−	29.1016i	−1.32824	−	2.72562i
$115$		0			0
$116$		0			0
$117$	−6.66512	+	8.51916i	−0.616191	+	0.787597i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	−9.24975	−	18.9810i	−0.834021	−	1.71146i
$124$		0			0
$125$		0			0
$126$	29.2657	+	22.8966i	2.60720	+	2.03979i
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$			4.70170i			0.415576i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	12.4166	+	25.4795i	1.08072	+	2.21770i
$133$	−33.3867			−2.89500
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$	−2.91864	+	1.42230i	−0.248451	+	0.121074i
$139$		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
$144$	19.3668	−	24.7541i	1.61390	−	2.06284i
$145$		0			0
$146$		−	35.0030i		−	2.89687i
$147$	23.5498	−	11.4762i	1.94235	−	0.946541i
$148$		0			0
$149$			2.35244i			0.192720i	0.995347	+	0.0963598i	$0.0307200\pi$
$149$			2.35244i			0.192720i	−0.995347	+	0.0963598i	$0.969280\pi$
$150$	9.99002	+	20.5000i	0.815682	+	1.67382i
$151$	14.4222			1.17366			0.586831	−	0.809709i	$-0.300375\pi$
$151$	14.4222			1.17366			0.586831	+	0.809709i	$0.300375\pi$
$152$			54.8391i			4.44804i
$153$		0			0
$154$	41.0800			3.31032
$155$		0			0
$156$	27.6991	−	13.4982i	2.21770	−	1.08072i
$157$	−0.644702			−0.0514528			−0.0257264	−	0.999669i	$-0.508190\pi$
$157$	−0.644702			−0.0514528			−0.0257264	+	0.999669i	$0.508190\pi$
$158$		0			0
$159$	−10.9535	−	22.4772i	−0.868671	−	1.78256i
$160$		0			0
$161$			3.34840i			0.263891i
$162$	−23.0030	+	5.70248i	−1.80729	+	0.448029i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$			60.1490i			4.69685i
$165$		0			0
$166$	45.0543			3.49689
$167$			4.65159i			0.359951i	0.983671	+	0.179976i	$0.0576019\pi$
$167$			4.65159i			0.359951i	−0.983671	+	0.179976i	$0.942398\pi$
$168$	−27.5742	−	56.5837i	−2.12739	−	4.36553i
$169$	13.0000			1.00000
$170$		0			0
$171$	13.1211	−	16.7710i	1.00339	−	1.28251i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	23.5186			1.77784
$176$		−	34.7470i		−	2.61915i
$177$		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
$181$	26.8091			1.99271			0.996353	−	0.0853299i	$-0.0271944\pi$
$181$	26.8091			1.99271			0.996353	+	0.0853299i	$0.0271944\pi$
$182$		−	44.6587i		−	3.31032i
$183$		0			0
$184$	5.49988			0.405457
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−5.10570	+	23.9020i	−0.371385	+	1.73862i
$190$		0			0
$191$			24.3197i			1.75971i	0.475241	+	0.879856i	$0.342361\pi$
$191$			24.3197i			1.75971i	−0.475241	+	0.879856i	$0.657639\pi$
$192$	−17.1312	+	8.34832i	−1.23634	+	0.602488i
$193$	−7.04646			−0.507215			−0.253608	−	0.967307i	$-0.581617\pi$
$193$	−7.04646			−0.507215			−0.253608	+	0.967307i	$0.581617\pi$
$194$		0			0
$195$		0			0
$196$	−74.6271			−5.33051
$197$			20.4511i			1.45708i	0.685002	+	0.728541i	$0.259801\pi$
$197$			20.4511i			1.45708i	−0.685002	+	0.728541i	$0.740199\pi$
$198$	−16.1445	+	20.6355i	−1.14734	+	1.46650i
$199$	−6.90787			−0.489686			−0.244843	−	0.969563i	$-0.578736\pi$
$199$	−6.90787			−0.489686			−0.244843	+	0.969563i	$0.578736\pi$
$200$		−	38.6303i		−	2.73157i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$			13.7545i			0.958318i
$207$	−1.68198	−	1.31593i	−0.116906	−	0.0914633i
$208$	−37.7740			−2.61915
$209$		−	23.5412i		−	1.62838i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$			71.2283i			4.89198i
$213$		0			0
$214$		0			0
$215$		0			0
$216$	39.2601	+	8.38634i	2.67131	+	0.570618i
$217$		0			0
$218$		−	2.37837i		−	0.161084i
$219$	20.6969	−	10.0859i	1.39856	−	0.681544i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$			57.0819i			3.81395i
$225$	−9.24286	+	11.8140i	−0.616191	+	0.787597i
$226$	−14.0380			−0.933791
$227$			5.88143i			0.390364i	0.980767	+	0.195182i	$0.0625298\pi$
$227$			5.88143i			0.390364i	−0.980767	+	0.195182i	$0.937470\pi$
$228$	−54.5289	+	26.5729i	−3.61126	+	1.75983i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$	11.8370	+	24.2901i	0.778816	+	1.59817i
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$	22.4331	+	17.5510i	1.46650	+	1.14734i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	25.7176i		−	1.66354i	−0.555124	−	0.831768i	$-0.687329\pi$
$239$		−	25.7176i		−	1.66354i	0.555124	−	0.831768i	$-0.312671\pi$
$240$		0			0
$241$	25.8613			1.66587			0.832937	−	0.553367i	$-0.186657\pi$
$241$	25.8613			1.66587			0.832937	+	0.553367i	$0.186657\pi$
$242$			28.9658i			1.86199i
$243$	−10.0000	−	11.9583i	−0.641500	−	0.767123i
$244$		0			0
$245$		0			0
$246$	−49.9817	+	24.3569i	−3.18672	+	1.55294i
$247$	−25.5920			−1.62838
$248$		0			0
$249$	12.9821	+	26.6401i	0.822710	+	1.68824i
$250$		0			0
$251$		−	29.0654i		−	1.83459i	−0.398210	−	0.917294i	$-0.630368\pi$
$251$		−	29.0654i		−	1.83459i	0.398210	−	0.917294i	$-0.369632\pi$
$252$	42.9022	−	54.8364i	2.70259	−	3.45437i
$253$	−2.36098			−0.148433
$254$		0			0
$255$		0			0
$256$	−9.62444			−0.601527
$257$		−	17.0242i		−	1.06194i	−0.847391	−	0.530970i	$-0.821828\pi$
$257$		−	17.0242i		−	1.06194i	0.847391	−	0.530970i	$-0.178172\pi$
$258$		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	$0.5\pi$
$264$	39.8976	−	19.4428i	2.45552	−	1.19662i
$265$		0			0
$266$			87.9158i			5.39046i
$267$		0			0
$268$		0			0
$269$		−	12.7530i		−	0.777566i	−0.921329	−	0.388783i	$-0.872896\pi$
$269$		−	12.7530i		−	0.777566i	0.921329	−	0.388783i	$-0.127104\pi$
$270$		0			0
$271$	19.4874			1.18378			0.591888	−	0.806020i	$-0.298383\pi$
$271$	19.4874			1.18378			0.591888	+	0.806020i	$0.298383\pi$
$272$		0			0
$273$	26.4061	−	12.8682i	1.59817	−	0.778816i
$274$		0			0
$275$			16.5831i			1.00000i
$276$	2.66502	+	5.46877i	0.160416	+	0.329181i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	11.1479i		−	0.665030i	−0.943098	−	0.332515i	$-0.892103\pi$
$281$		−	11.1479i		−	0.665030i	0.943098	−	0.332515i	$-0.107897\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$	31.4892			1.86199
$287$			57.3413i			3.38475i
$288$	−28.6736	−	22.4333i	−1.68961	−	1.32190i
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$	−65.5866			−3.83816
$293$		−	26.5330i		−	1.55007i	−0.631916	−	0.775037i	$-0.717731\pi$
$293$		−	26.5330i		−	1.55007i	0.631916	−	0.775037i	$-0.282269\pi$
$294$	−30.2197	−	62.0125i	−1.76245	−	3.61664i
$295$		0			0
$296$		0			0
$297$	−16.8535	−	3.60007i	−0.977938	−	0.208897i
$298$	6.19458			0.358843
$299$			2.56665i			0.148433i
$300$	38.4117	−	18.7187i	2.21770	−	1.08072i
$301$		0			0
$302$		−	37.9773i		−	2.18535i
$303$		0			0
$304$	74.3625			4.26498
$305$		0			0
$306$		0			0
$307$	−28.8444			−1.64624			−0.823119	−	0.567869i	$-0.807768\pi$
$307$	−28.8444			−1.64624			−0.823119	+	0.567869i	$0.807768\pi$
$308$		−	76.9732i		−	4.38596i
$309$	−8.13284	+	3.96327i	−0.462661	+	0.225463i
$310$		0			0
$311$			15.2146i			0.862741i	0.902175	+	0.431370i	$0.141970\pi$
$311$			15.2146i			0.862741i	−0.902175	+	0.431370i	$0.858030\pi$
$312$	−21.1365	−	43.3732i	−1.19662	−	2.45552i
$313$	34.5637			1.95366			0.976828	−	0.214026i	$-0.0686579\pi$
$313$	34.5637			1.95366			0.976828	+	0.214026i	$0.0686579\pi$
$314$			1.69766i			0.0958047i
$315$		0			0
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$	−59.1883	+	28.8434i	−3.31911	+	1.61746i
$319$		0			0
$320$		0			0
$321$		0			0
$322$	8.81718			0.491362
$323$		0			0
$324$	10.6850	+	43.1016i	0.593609	+	2.39454i
$325$	18.0278			1.00000
$326$		0			0
$327$	1.40630	−	0.685315i	0.0777687	−	0.0378980i
$328$	94.1856			5.20053
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		−	84.4200i		−	4.63315i
$333$		0			0
$334$	12.2488			0.670226
$335$		0			0
$336$	−76.7281	+	37.3909i	−4.18586	+	2.03984i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		−	34.2323i		−	1.86199i
$339$	−4.04496	−	8.30047i	−0.219692	−	0.450820i
$340$		0			0
$341$		0			0
$342$	−44.1622	−	34.5511i	−2.38802	−	1.86831i
$343$	−38.2176			−2.06355
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$	−35.2688			−1.88789			−0.943947	−	0.330097i	$-0.892919\pi$
$349$	−35.2688			−1.88789			−0.943947	+	0.330097i	$0.892919\pi$
$350$		−	61.9305i		−	3.31032i
$351$	−3.91369	+	18.3217i	−0.208897	+	0.977938i
$352$	−40.2489			−2.14527
$353$		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
$358$	62.9783			3.32851
$359$		−	26.9479i		−	1.42226i	−0.703062	−	0.711129i	$-0.748184\pi$
$359$		−	26.9479i		−	1.42226i	0.703062	−	0.711129i	$-0.251816\pi$
$360$		0			0
$361$	31.3808			1.65162
$362$		−	70.5952i		−	3.71040i
$363$	−17.1271	+	8.34634i	−0.898941	+	0.438069i
$364$	−83.6787			−4.38596
$365$		0			0
$366$		0			0
$367$	11.7763			0.614717			0.307358	−	0.951594i	$-0.400555\pi$
$367$	11.7763			0.614717			0.307358	+	0.951594i	$0.400555\pi$
$368$		−	7.45790i		−	0.388770i
$369$	−28.8039	−	22.5353i	−1.49947	−	1.17314i
$370$		0			0
$371$			67.9035i			3.52537i
$372$		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
$378$	62.9401	+	13.4446i	3.23729	+	0.691517i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$	64.0400			3.27657
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	3.56746	+	7.32061i	0.182051	+	0.373578i
$385$		0			0
$386$			18.5551i			0.944431i
$387$		0			0
$388$		0			0
$389$			3.77407i			0.191353i	0.995412	+	0.0956765i	$0.0305014\pi$
$389$			3.77407i			0.191353i	−0.995412	+	0.0956765i	$0.969499\pi$
$390$		0			0
$391$		0			0
$392$			116.856i			5.90214i
$393$		0			0
$394$	53.8530			2.71308
$395$		0			0
$396$	38.6655	+	30.2507i	1.94301	+	1.52015i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$			18.1902i			0.911791i
$399$	−51.9835	+	25.3325i	−2.60243	+	1.26821i
$400$	−52.3831			−2.61915
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−7.21110			−0.356566			−0.178283	−	0.983979i	$-0.557054\pi$
$409$	−7.21110			−0.356566			−0.178283	+	0.983979i	$0.557054\pi$
$410$		0			0
$411$		0			0
$412$	25.7723			1.26971
$413$		0			0
$414$	−3.46517	+	4.42908i	−0.170304	+	0.217678i
$415$		0			0
$416$			43.7551i			2.14527i
$417$		0			0
$418$	−61.9900			−3.03203
$419$			31.9128i			1.55904i	0.626376	+	0.779521i	$0.284538\pi$
$419$			31.9128i			1.55904i	−0.626376	+	0.779521i	$0.715462\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	111.534			5.41659
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	9.07343	+	18.6192i	0.438069	+	0.898941i
$430$		0			0
$431$		−	41.4910i		−	1.99855i	−0.0380545	−	0.999276i	$-0.512116\pi$
$431$		−	41.4910i		−	1.99855i	0.0380545	−	0.999276i	$-0.487884\pi$
$432$	11.3720	−	53.2371i	0.547134	−	2.56137i
$433$	−29.2433			−1.40534			−0.702672	−	0.711514i	$-0.748010\pi$
$433$	−29.2433			−1.40534			−0.702672	+	0.711514i	$0.748010\pi$
$434$		0			0
$435$		0			0
$436$	−4.45645			−0.213425
$437$		−	5.05275i		−	0.241706i
$438$	−26.5588	−	54.5001i	−1.26903	−	2.60412i
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	27.9596	−	35.7371i	1.33141	−	1.70177i
$442$		0			0
$443$			27.6416i			1.31329i	0.754198	+	0.656647i	$0.228026\pi$
$443$			27.6416i			1.31329i	−0.754198	+	0.656647i	$0.771974\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	1.78494	+	3.66278i	0.0844246	+	0.173244i
$448$	51.7532			2.44511
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	31.1092	+	24.3388i	1.46650	+	1.14734i
$451$	−40.4318			−1.90386
$452$			26.3035i			1.23721i
$453$	22.4555	−	10.9430i	1.05505	−	0.514145i
$454$	15.4873			0.726855
$455$		0			0
$456$	41.6097	+	85.3852i	1.94855	+	3.99853i
$457$	−41.6845			−1.94992			−0.974959	−	0.222386i	$-0.928615\pi$
$457$	−41.6845			−1.94992			−0.974959	+	0.222386i	$0.928615\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	41.5172i		−	1.93365i	−0.255445	−	0.966824i	$-0.582222\pi$
$461$		−	41.5172i		−	1.93365i	0.255445	−	0.966824i	$-0.417778\pi$
$462$	63.9621	−	31.1698i	2.97579	−	1.45015i
$463$		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	$0.813345\pi$
$468$	32.8859	−	42.0338i	1.52015	−	1.94301i
$469$		0			0
$470$		0			0
$471$	−1.00381	+	0.489173i	−0.0462530	+	0.0225399i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−35.4897			−1.62838
$476$		0			0
$477$	−34.1095	−	26.6862i	−1.56177	−	1.22188i
$478$	−67.7211			−3.09749
$479$		−	6.63325i		−	0.303081i	−0.988451	−	0.151540i	$-0.951577\pi$
$479$		−	6.63325i		−	0.303081i	0.988451	−	0.151540i	$-0.0484234\pi$
$480$		0			0
$481$		0			0
$482$		−	68.0995i		−	3.10185i
$483$	2.54062	+	5.21349i	0.115602	+	0.237222i
$484$	54.2744			2.46702
$485$		0			0
$486$	−31.4892	+	26.3326i	−1.42838	+	1.19447i
$487$		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	$0.5\pi$
$492$	45.6386	+	93.6527i	2.05754	+	4.22219i
$493$		0			0
$494$			67.3903i			3.03203i
$495$		0			0
$496$		0			0
$497$		0			0
$498$	70.1501	−	34.1853i	3.14350	−	1.53188i
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$	3.52943	+	7.24258i	0.157683	+	0.323575i
$502$	−76.5365			−3.41599
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	−85.8667	−	67.1793i	−3.82481	−	2.99241i
$505$		0			0
$506$			6.21706i			0.276382i
$507$	20.2412	−	9.86386i	0.898941	−	0.438069i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	−62.5251			−2.76595
$512$			34.7470i			1.53562i
$513$	7.70455	−	36.0683i	0.340164	−	1.59245i
$514$	−44.8290			−1.97732
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	45.3777i		−	1.98803i	−0.109232	−	0.994016i	$-0.534839\pi$
$521$		−	45.3777i		−	1.98803i	0.109232	−	0.994016i	$-0.465161\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$	36.6187	−	17.8449i	1.59817	−	0.778816i
$526$		0			0
$527$		0			0
$528$	−26.3646	−	54.1015i	−1.14737	−	2.35447i
$529$	22.4933			0.977968
$530$		0			0
$531$		0			0
$532$	164.731			7.14201
$533$			43.9540i			1.90386i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	18.1469	+	37.2383i	0.783095	+	1.60695i
$538$	−33.5820			−1.44782
$539$		−	50.1638i		−	2.16071i
$540$		0			0
$541$	21.1759			0.910421			0.455210	−	0.890384i	$-0.349564\pi$
$541$	21.1759			0.910421			0.455210	+	0.890384i	$0.349564\pi$
$542$		−	51.3154i		−	2.20418i
$543$	41.7421	−	20.3416i	1.79133	−	0.872943i
$544$		0			0
$545$		0			0
$546$	−33.8851	−	69.5341i	−1.45015	−	2.97579i
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$	43.6676			1.86199
$551$		0			0
$552$	8.56338	−	4.17308i	0.364482	−	0.177618i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$			46.4101i			1.96646i	0.182372	+	0.983230i	$0.441622\pi$
$557$			46.4101i			1.96646i	−0.182372	+	0.983230i	$0.558378\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−29.3554			−1.23828
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	10.1862	+	41.0897i	0.427780	+	1.72561i
$568$		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			$\pi$
$572$		−	59.0025i		−	2.46702i
$573$	18.4528	+	37.8661i	0.770876	+	1.58188i
$574$	150.994			6.30238
$575$			3.55931i			0.148433i
$576$	−20.3391	+	25.9969i	−0.847463	+	1.08320i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$			44.7653i			1.86199i
$579$	−10.9714	+	5.34656i	−0.455957	+	0.222195i
$580$		0			0
$581$		−	80.4794i		−	3.33885i
$582$		0			0
$583$	−47.8792			−1.98295
$584$			102.700i			4.24976i
$585$		0			0
$586$	−69.8682			−2.88623
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−116.195	+	56.6239i	−4.79181	+	2.33513i
$589$		0			0
$590$		0			0
$591$	15.5175	+	31.8426i	0.638303	+	1.30983i
$592$		0			0
$593$			31.8670i			1.30862i	0.756226	+	0.654311i	$0.227041\pi$
$593$			31.8670i			1.30862i	−0.756226	+	0.654311i	$0.772959\pi$
$594$	−9.47990	+	44.3795i	−0.388965	+	1.82091i
$595$		0			0
$596$		−	11.6070i		−	0.475443i
$597$	−10.7556	+	5.24140i	−0.440199	+	0.214516i
$598$	6.75865			0.276382
$599$			42.5299i			1.73772i	0.495054	+	0.868862i	$0.335148\pi$
$599$			42.5299i			1.73772i	−0.495054	+	0.868862i	$0.664852\pi$
$600$	−29.3111	−	60.1478i	−1.19662	−	2.45552i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	−71.1596			−2.89544
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		−	86.1371i		−	3.49332i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	25.6822			1.03729			0.518646	−	0.854989i	$-0.326436\pi$
$613$	25.6822			1.03729			0.518646	+	0.854989i	$0.326436\pi$
$614$			75.9547i			3.06528i
$615$		0			0
$616$	−120.530			−4.85630
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$	10.4363	+	21.4158i	0.419810	+	0.861472i
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$	−3.61733	−	0.772698i	−0.145159	−	0.0310073i
$622$	40.0639			1.60642
$623$		0			0
$624$	−58.8145	+	28.6613i	−2.35447	+	1.14737i
$625$	25.0000			1.00000
$626$		−	91.0150i		−	3.63769i
$627$	−17.8621	−	36.6540i	−0.713343	−	1.46382i
$628$	3.18098			0.126935
$629$		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
$636$	54.0451	+	110.903i	2.14303	+	4.39760i
$637$	−54.5338			−2.16071
$638$		0			0
$639$		0			0
$640$		0			0
$641$			11.3293i			0.447480i	0.974649	+	0.223740i	$0.0718267\pi$
$641$			11.3293i			0.447480i	−0.974649	+	0.223740i	$0.928173\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		−	16.5211i		−	0.651022i
$645$		0			0
$646$		0			0
$647$			6.10950i			0.240189i	0.992762	+	0.120095i	$0.0383198\pi$
$647$			6.10950i			0.240189i	−0.992762	+	0.120095i	$0.961680\pi$
$648$	67.4916	−	16.7313i	2.65132	−	0.657266i
$649$		0			0
$650$		−	47.4717i		−	1.86199i
$651$		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	$-0.385435\pi$
$654$	−1.80461	−	3.70315i	−0.0705658	−	0.144805i
$655$		0			0
$656$		−	127.717i		−	4.98650i
$657$	24.5725	−	31.4078i	0.958664	−	1.22534i
$658$		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
$664$	−132.191			−5.13000
$665$		0			0
$666$		0			0
$667$		0			0
$668$		−	22.9511i		−	0.888005i
$669$		0			0
$670$		0			0
$671$		0			0
$672$	43.3114	+	88.8773i	1.67077	+	3.42851i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−5.42731	+	25.4076i	−0.208897	+	0.977938i
$676$	−64.1424			−2.46702
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$	−21.8573	+	10.6514i	−0.839423	+	0.409065i
$679$		0			0
$680$		0			0
$681$	4.46258	+	9.15745i	0.171007	+	0.350914i
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	−64.7398	+	82.7485i	−2.47539	+	3.16397i
$685$		0			0
$686$			100.637i			3.84232i
$687$		0			0
$688$		0			0
$689$			52.0502i			1.98295i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$	36.8607	+	28.8386i	1.40022	+	1.09549i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$			92.8717i			3.51524i
$699$		0			0
$700$	−116.042			−4.38596
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	48.2456	+	10.3057i	1.82091	+	0.388965i
$703$		0			0
$704$			36.4915i			1.37533i
$705$		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
$716$		−	118.005i		−	4.41005i
$717$	−19.5135	−	40.0427i	−0.728744	−	1.49542i
$718$	−70.9608			−2.64823
$719$			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
$721$	24.5692			0.915006
$722$		−	82.6337i		−	3.07531i
$723$	40.2664	−	19.6225i	1.49752	−	0.729769i
$724$	−132.277			−4.91604
$725$		0			0
$726$	21.9780	+	45.1001i	0.815682	+	1.67382i
$727$	39.1424			1.45171			0.725855	−	0.687848i	$-0.241444\pi$
$727$	39.1424			1.45171			0.725855	+	0.687848i	$0.241444\pi$
$728$			131.030i			4.85630i
$729$	−24.6435	−	11.0316i	−0.912724	−	0.408577i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	54.0836			1.99763			0.998813	−	0.0487194i	$-0.0155140\pi$
$733$	54.0836			1.99763			0.998813	+	0.0487194i	$0.0155140\pi$
$734$		−	31.0099i		−	1.14460i
$735$		0			0
$736$	−8.63879			−0.318430
$737$		0			0
$738$	−59.3411	+	75.8481i	−2.18438	+	2.79201i
$739$	28.2041			1.03750			0.518752	−	0.854925i	$-0.326397\pi$
$739$	28.2041			1.03750			0.518752	+	0.854925i	$0.326397\pi$
$740$		0			0
$741$	−39.8471	+	19.4181i	−1.46382	+	0.713343i
$742$	178.807			6.56422
$743$			33.1662i			1.21675i	0.793649	+	0.608376i	$0.208179\pi$
$743$			33.1662i			1.21675i	−0.793649	+	0.608376i	$0.791821\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	40.4267	+	31.6286i	1.47914	+	1.15723i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	30.4604			1.11152			0.555758	−	0.831344i	$-0.312428\pi$
$751$	30.4604			1.11152			0.555758	+	0.831344i	$0.312428\pi$
$752$		0			0
$753$	−22.0536	−	45.2551i	−0.803677	−	1.64919i
$754$		0			0
$755$		0			0
$756$	25.1917	−	117.933i	0.916214	−	4.28919i
$757$	−45.0104			−1.63593			−0.817966	−	0.575267i	$-0.804898\pi$
$757$	−45.0104			−1.63593			−0.817966	+	0.575267i	$0.804898\pi$
$758$		0			0
$759$	−3.67607	+	1.79141i	−0.133433	+	0.0650241i
$760$		0			0
$761$			16.8955i			0.612462i	0.951957	+	0.306231i	$0.0990680\pi$
$761$			16.8955i			0.612462i	−0.951957	+	0.306231i	$0.900932\pi$
$762$		0			0
$763$	−4.24843			−0.153803
$764$		−	119.994i		−	4.34124i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−14.9854	+	7.30262i	−0.540738	+	0.263511i
$769$	−30.5833			−1.10286			−0.551431	−	0.834221i	$-0.685918\pi$
$769$	−30.5833			−1.10286			−0.551431	+	0.834221i	$0.685918\pi$
$770$		0			0
$771$	−12.9172	−	26.5069i	−0.465203	−	0.954622i
$772$	34.7675			1.25131
$773$		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
$778$	9.93810			0.356298
$779$		−	86.5285i		−	3.10020i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	158.459			5.65923
$785$		0			0
$786$		0			0
$787$	−37.6115			−1.34071			−0.670353	−	0.742043i	$-0.733857\pi$
$787$	−37.6115			−1.34071			−0.670353	+	0.742043i	$0.733857\pi$
$788$		−	100.907i		−	3.59465i
$789$		0			0
$790$		0			0
$791$			25.0756i			0.891587i
$792$	47.3686	−	60.5452i	1.68317	−	2.15138i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	34.0836			1.20806
$797$		−	47.8330i		−	1.69433i	−0.531327	−	0.847167i	$-0.678307\pi$
$797$		−	47.8330i		−	1.69433i	0.531327	−	0.847167i	$-0.321693\pi$
$798$	66.7068	+	136.886i	2.36140	+	4.84571i
$799$		0			0
$800$			60.6775i			2.14527i
$801$		0			0
$802$		0			0
$803$		−	44.0869i		−	1.55579i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−9.67646	−	19.8566i	−0.340628	−	0.698986i
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	−9.38920			−0.329699			−0.164850	−	0.986319i	$-0.552714\pi$
$811$	−9.38920			−0.329699			−0.164850	+	0.986319i	$0.552714\pi$
$812$		0			0
$813$	30.3422	−	14.7862i	1.06415	−	0.518576i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$			18.9887i			0.663923i
$819$	31.3509	−	40.0718i	1.09549	−	1.40022i
$820$		0			0
$821$			53.0660i			1.85202i	0.377504	+	0.926008i	$0.376783\pi$
$821$			53.0660i			1.85202i	−0.377504	+	0.926008i	$0.623217\pi$
$822$		0			0
$823$	52.1679			1.81846			0.909229	−	0.416296i	$-0.136672\pi$
$823$	52.1679			1.81846			0.909229	+	0.416296i	$0.136672\pi$
$824$		−	40.3560i		−	1.40587i
$825$	12.5826	+	25.8201i	0.438069	+	0.898941i
$826$		0			0
$827$		−	57.3167i		−	1.99310i	−0.0830134	−	0.996548i	$-0.526454\pi$
$827$		−	57.3167i		−	1.99310i	0.0830134	−	0.996548i	$-0.473546\pi$
$828$	8.29895	+	6.49283i	0.288408	+	0.225641i
$829$	21.5382			0.748051			0.374026	−	0.927418i	$-0.377977\pi$
$829$	21.5382			0.748051			0.374026	+	0.927418i	$0.377977\pi$
$830$		0			0
$831$		0			0
$832$	39.6705			1.37533
$833$		0			0
$834$		0			0
$835$		0			0
$836$			116.153i			4.01724i
$837$		0			0
$838$	84.0346			2.90293
$839$		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
$843$	−8.45859	−	17.3575i	−0.291329	−	0.597823i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	51.7409			1.77784
$848$		−	151.242i		−	5.19366i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−54.0739			−1.85146			−0.925728	−	0.378189i	$-0.876547\pi$
$853$	−54.0739			−1.85146			−0.925728	+	0.378189i	$0.876547\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$	49.0290	−	23.8927i	1.67382	−	0.815682i
$859$	41.8419			1.42763			0.713814	−	0.700335i	$-0.246966\pi$
$859$	41.8419			1.42763			0.713814	+	0.700335i	$0.246966\pi$
$860$		0			0
$861$	43.5082	+	89.2811i	1.48276	+	3.04269i
$862$	−109.256			−3.72129
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	−61.6667	−	13.1726i	−2.09794	−	0.448141i
$865$		0			0
$866$			77.0051i			2.61674i
$867$	−26.4692	+	12.8989i	−0.898941	+	0.438069i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$			6.97822i			0.236312i
$873$		0			0
$874$	−13.3052			−0.450055
$875$		0			0
$876$	−102.119	+	49.7643i	−3.45028	+	1.68138i
$877$	55.8804			1.88695			0.943473	−	0.331450i	$-0.107538\pi$
$877$	55.8804			1.88695			0.943473	+	0.331450i	$0.107538\pi$
$878$		0			0
$879$	−20.1321	−	41.3122i	−0.679040	−	1.39343i
$880$		0			0
$881$			12.8792i			0.433910i	0.976182	+	0.216955i	$0.0696125\pi$
$881$			12.8792i			0.433910i	−0.976182	+	0.216955i	$0.930387\pi$
$882$	−94.1050	−	73.6247i	−3.16868	−	2.47907i
$883$	4.47366			0.150551			0.0752753	−	0.997163i	$-0.476016\pi$
$883$	4.47366			0.150551			0.0752753	+	0.997163i	$0.476016\pi$
$884$		0			0
$885$		0			0
$886$	72.7875			2.44534
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−28.9726	+	7.18236i	−0.970620	+	0.240618i
$892$		0			0
$893$		0			0
$894$	9.64504	−	4.70019i	0.322578	−	0.157198i
$895$		0			0
$896$		−	22.1155i		−	0.738827i
$897$	1.94747	+	3.99631i	0.0650241	+	0.133433i
$898$		0			0
$899$		0			0
$900$	45.6046	−	58.2904i	1.52015	−	1.94301i
$901$		0			0
$902$			106.467i			3.54497i
$903$		0			0
$904$	41.1878			1.36989
$905$		0			0
$906$	−28.8156	−	59.1312i	−0.957335	−	1.96450i
$907$	29.9850			0.995637			0.497818	−	0.867281i	$-0.334135\pi$
$907$	29.9850			0.995637			0.497818	+	0.867281i	$0.334135\pi$
$908$		−	29.0192i		−	0.963035i
$909$		0			0
$910$		0			0
$911$			60.2663i			1.99671i	0.0573229	+	0.998356i	$0.481744\pi$
$911$			60.2663i			1.99671i	−0.0573229	+	0.998356i	$0.518256\pi$
$912$	115.783	−	56.4231i	3.83397	−	1.86836i
$913$	56.7466			1.87804
$914$			109.766i			3.63073i
$915$		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
$921$	−44.9111	+	21.8859i	−1.47987	+	0.721166i
$922$	−109.325			−3.60044
$923$		0			0
$924$	−58.4041	−	119.848i	−1.92135	−	3.94272i
$925$		0			0
$926$		0			0
$927$	−9.65577	+	12.3417i	−0.317137	+	0.405355i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	107.356			3.51846
$932$		0			0
$933$	11.5442	+	23.6893i	0.377940	+	0.775553i
$934$	62.9783			2.06071
$935$		0			0
$936$	−65.8196	−	51.4951i	−2.15138	−	1.68317i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$	53.8161	−	26.2255i	1.75622	−	0.855837i
$940$		0			0
$941$		−	42.7470i		−	1.39351i	−0.717308	−	0.696756i	$-0.754626\pi$
$941$		−	42.7470i		−	1.39351i	0.717308	−	0.696756i	$-0.245374\pi$
$942$	1.28812	+	2.64328i	0.0419691	+	0.0861228i
$943$	−8.67804			−0.282596
$944$		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
$949$	−47.9275			−1.55579
$950$			93.4535i			3.03203i
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$	−70.2716	+	89.8191i	−2.27513	+	2.90800i
$955$		0			0
$956$			126.892i			4.10397i
$957$		0			0
$958$	−17.4670			−0.564334
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$	−127.601			−4.10974
$965$		0			0
$966$	13.7285	−	6.69011i	0.441706	−	0.215251i
$967$	62.0751			1.99620			0.998100	−	0.0616096i	$-0.0196234\pi$
$967$	62.0751			1.99620			0.998100	+	0.0616096i	$0.0196234\pi$
$968$		−	84.9866i		−	2.73157i
$969$		0			0
$970$		0			0
$971$		−	62.2970i		−	1.99921i	−0.0281505	−	0.999604i	$-0.508962\pi$
$971$		−	62.2970i		−	1.99921i	0.0281505	−	0.999604i	$-0.491038\pi$
$972$	49.3403	+	59.0025i	1.58259	+	1.89250i
$973$		0			0
$974$		0			0
$975$	28.0694	−	13.6787i	0.898941	−	0.438069i
$976$		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
$981$	1.66964	−	2.13409i	0.0533075	−	0.0681362i
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	146.648	−	71.4641i	4.67497	−	2.27819i
$985$		0			0
$986$		0			0
$987$		0			0
$988$	126.272			4.01724
$989$		0			0
$990$		0			0
$991$	−62.9603			−2.00000			−1.00000		0.000626381i	$-0.999801\pi$
$991$	−62.9603			−2.00000			−1.00000		0.000626381i	$0.999801\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	−64.0544	−	131.443i	−2.02964	−	4.16493i
$997$		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.428.3	✓	20
3.2	odd	2	inner	429.2.e.d.428.18	yes	20
11.10	odd	2	inner	429.2.e.d.428.17	yes	20
13.12	even	2	inner	429.2.e.d.428.17	yes	20
33.32	even	2	inner	429.2.e.d.428.4	yes	20
39.38	odd	2	inner	429.2.e.d.428.4	yes	20
143.142	odd	2	CM	429.2.e.d.428.3	✓	20
429.428	even	2	inner	429.2.e.d.428.18	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.e.d.428.3	✓	20	1.1	even	1	trivial
429.2.e.d.428.3	✓	20	143.142	odd	2	CM
429.2.e.d.428.4	yes	20	33.32	even	2	inner
429.2.e.d.428.4	yes	20	39.38	odd	2	inner
429.2.e.d.428.17	yes	20	11.10	odd	2	inner
429.2.e.d.428.17	yes	20	13.12	even	2	inner
429.2.e.d.428.18	yes	20	3.2	odd	2	inner
429.2.e.d.428.18	yes	20	429.428	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.63326i q^{2} +(1.55701 - 0.758758i) q^{3} -4.93403 q^{4} +(-1.99800 - 4.10001i) q^{6} -4.70372 q^{7} +7.72606i q^{8} +(1.84857 - 2.36279i) q^{9} +O(q^{10})\)	\(q-2.63326i q^{2} +(1.55701 - 0.758758i) q^{3} -4.93403 q^{4} +(-1.99800 - 4.10001i) q^{6} -4.70372 q^{7} +7.72606i q^{8} +(1.84857 - 2.36279i) q^{9} -3.31662i q^{11} +(-7.68235 + 3.74374i) q^{12} -3.60555 q^{13} +12.3861i q^{14} +10.4766 q^{16} +(-6.22183 - 4.86776i) q^{18} +7.09795 q^{19} +(-7.32375 + 3.56899i) q^{21} -8.73352 q^{22} -0.711862i q^{23} +(5.86221 + 12.0296i) q^{24} -5.00000 q^{25} +9.49434i q^{26} +(1.08546 - 5.08151i) q^{27} +23.2083 q^{28} -12.1355i q^{32} +(-2.51652 - 5.16402i) q^{33} +(-9.12092 + 11.6581i) q^{36} -18.6907i q^{38} +(-5.61389 + 2.73574i) q^{39} -12.1906i q^{41} +(9.39805 + 19.2853i) q^{42} +16.3643i q^{44} -1.87451 q^{46} +(16.3122 - 7.94922i) q^{48} +15.1250 q^{49} +13.1663i q^{50} +17.7899 q^{52} -14.4361i q^{53} +(-13.3809 - 2.85830i) q^{54} -36.3412i q^{56} +(11.0516 - 5.38562i) q^{57} +(-8.69516 + 11.1139i) q^{63} -11.0026 q^{64} +(-13.5982 + 6.62663i) q^{66} +(-0.540131 - 1.10838i) q^{69} +(18.2551 + 14.2822i) q^{72} +13.2927 q^{73} +(-7.78506 + 3.79379i) q^{75} -35.0215 q^{76} +15.6005i q^{77} +(7.20391 + 14.7828i) q^{78} +(-2.16556 - 8.73558i) q^{81} -32.1011 q^{82} +17.1097i q^{83} +(36.1356 - 17.6095i) q^{84} +25.6244 q^{88} +16.9595 q^{91} +3.51235i q^{92} +(-9.20790 - 18.8951i) q^{96} -39.8279i q^{98} +(-7.83649 - 6.13102i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 40 q^{4}+O(q^{10}) \) 20 * q - 40 * q^4	\( 20 q - 40 q^{4} + 80 q^{16} - 100 q^{25} + 10 q^{36} - 50 q^{42} + 70 q^{48} + 140 q^{49} - 160 q^{64} - 110 q^{66} + 130 q^{78}+O(q^{100}) \) 20 * q - 40 * q^4 + 80 * q^16 - 100 * q^25 + 10 * q^36 - 50 * q^42 + 70 * q^48 + 140 * q^49 - 160 * q^64 - 110 * q^66 + 130 * q^78

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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