LMFDB - Embedded newform 1029.1.t.a.113.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1029,1,Mod(8,1029)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1029, base_ring=CyclotomicField(98))

chi = DirichletCharacter(H, H._module([49, 96]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1029.8");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1029 = 3 \cdot 7^{3} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1029.t (of order $98$, degree $42$, 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:	$0.513537897999$
Analytic rank:	$0$
Dimension:	$42$
Coefficient field:	$\Q(\zeta_{98})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{42} - x^{35} + x^{28} - x^{21} + x^{14} - x^{7} + 1 $ x^42 - x^35 + x^28 - x^21 + x^14 - x^7 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{49}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{49} + \cdots)$

Embedding invariants

Embedding label			113.1
Root			$-0.926917 - 0.375267i$ of defining polynomial
Character	$\chi$	$=$	1029.113
Dual form			1029.1.t.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)

$f(q)$	$=$	$q+(-0.838088 + 0.545535i) q^{3} +(-0.672301 + 0.740278i) q^{4} +(-0.981559 + 0.191159i) q^{7} +(0.404783 - 0.914413i) q^{9} +O(q^{10})$	\(q+(-0.838088 + 0.545535i) q^{3} +(-0.672301 + 0.740278i) q^{4} +(-0.981559 + 0.191159i) q^{7} +(0.404783 - 0.914413i) q^{9} +(0.159600 - 0.987182i) q^{12} +(-0.479553 + 0.919213i) q^{13} +(-0.0960230 - 0.995379i) q^{16} -1.52289 q^{19} +(0.718349 - 0.695683i) q^{21} +(-0.462538 - 0.886599i) q^{25} +(0.159600 + 0.987182i) q^{27} +(0.518393 - 0.855143i) q^{28} +(0.299202 - 1.31089i) q^{31} +(0.404783 + 0.914413i) q^{36} +(-0.567886 - 1.91340i) q^{37} +(-0.0995552 - 1.03199i) q^{39} +(-1.52117 + 0.505064i) q^{43} +(0.623490 + 0.781831i) q^{48} +(0.926917 - 0.375267i) q^{49} +(-0.358069 - 0.972990i) q^{52} +(1.27632 - 0.830791i) q^{57} +(-0.996992 + 1.42926i) q^{61} +(-0.222521 + 0.974928i) q^{63} +(0.801414 + 0.598111i) q^{64} +(0.372984 + 1.63415i) q^{67} +(0.0182391 + 0.568763i) q^{73} +(0.871319 + 0.490718i) q^{75} +(1.02384 - 1.12736i) q^{76} +(-0.713418 + 0.894598i) q^{79} +(-0.672301 - 0.740278i) q^{81} +(0.0320516 + 0.999486i) q^{84} +(0.294994 - 0.993933i) q^{91} +(0.464378 + 1.26187i) q^{93} +(-0.430487 + 1.88609i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q+O(q^{10}) $ 42 * q	$ 42 q - 7 q^{37} - 7 q^{39} - 7 q^{48} - 7 q^{52} - 7 q^{61} - 7 q^{63}+O(q^{100}) $ 42 * q - 7 * q^37 - 7 * q^39 - 7 * q^48 - 7 * q^52 - 7 * q^61 - 7 * q^63

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1029\mathbb{Z}\right)^\times$.

$n$	$344$	$346$
$\chi(n)$	$-1$	$e\left(\frac{40}{49}\right)$

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 $

$2$		0			0		−0.404783	−	0.914413i	$-0.632653\pi$
$2$		0			0		0.404783	+	0.914413i	$0.367347\pi$
$3$	−0.838088	+	0.545535i	−0.838088	+	0.545535i
$3$	−0.838088	+	0.545535i	−0.838088	+	0.545535i
$4$	−0.672301	+	0.740278i	−0.672301	+	0.740278i
$5$		0			0		0.518393	−	0.855143i	$-0.326531\pi$
$5$		0			0		−0.518393	+	0.855143i	$0.673469\pi$
$6$		0			0
$7$	−0.981559	+	0.191159i	−0.981559	+	0.191159i
$7$	−0.981559	+	0.191159i	−0.981559	+	0.191159i
$8$		0			0
$9$	0.404783	−	0.914413i	0.404783	−	0.914413i
$10$		0			0
$11$		0			0		0.801414	−	0.598111i	$-0.204082\pi$
$11$		0			0		−0.801414	+	0.598111i	$0.795918\pi$
$12$	0.159600	−	0.987182i	0.159600	−	0.987182i
$13$	−0.479553	+	0.919213i	−0.479553	+	0.919213i	0.518393	+	0.855143i	$0.326531\pi$
$13$	−0.479553	+	0.919213i	−0.479553	+	0.919213i	−0.997945	+	0.0640702i	$0.979592\pi$
$14$		0			0
$15$		0			0
$16$	−0.0960230	−	0.995379i	−0.0960230	−	0.995379i
$17$		0			0		−0.284528	−	0.958668i	$-0.591837\pi$
$17$		0			0		0.284528	+	0.958668i	$0.408163\pi$
$18$		0			0
$19$	−1.52289			−1.52289			−0.761446	−	0.648228i	$-0.775510\pi$
$19$	−1.52289			−1.52289			−0.761446	+	0.648228i	$0.775510\pi$
$20$		0			0
$21$	0.718349	−	0.695683i	0.718349	−	0.695683i
$22$		0			0
$23$		0			0		0.926917	−	0.375267i	$-0.122449\pi$
$23$		0			0		−0.926917	+	0.375267i	$0.877551\pi$
$24$		0			0
$25$	−0.462538	−	0.886599i	−0.462538	−	0.886599i
$26$		0			0
$27$	0.159600	+	0.987182i	0.159600	+	0.987182i
$28$	0.518393	−	0.855143i	0.518393	−	0.855143i
$29$		0			0		−0.926917	−	0.375267i	$-0.877551\pi$
$29$		0			0		0.926917	+	0.375267i	$0.122449\pi$
$30$		0			0
$31$	0.299202	−	1.31089i	0.299202	−	1.31089i	−0.572117	−	0.820172i	$-0.693878\pi$
$31$	0.299202	−	1.31089i	0.299202	−	1.31089i	0.871319	−	0.490718i	$-0.163265\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.404783	+	0.914413i	0.404783	+	0.914413i
$37$	−0.567886	−	1.91340i	−0.567886	−	1.91340i	−0.345365	−	0.938468i	$-0.612245\pi$
$37$	−0.567886	−	1.91340i	−0.567886	−	1.91340i	−0.222521	−	0.974928i	$-0.571429\pi$
$38$		0			0
$39$	−0.0995552	−	1.03199i	−0.0995552	−	1.03199i
$40$		0			0
$41$		0			0		−0.345365	−	0.938468i	$-0.612245\pi$
$41$		0			0		0.345365	+	0.938468i	$0.387755\pi$
$42$		0			0
$43$	−1.52117	+	0.505064i	−1.52117	+	0.505064i	−0.949056	−	0.315108i	$-0.897959\pi$
$43$	−1.52117	+	0.505064i	−1.52117	+	0.505064i	−0.572117	+	0.820172i	$0.693878\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.0320516	−	0.999486i	$-0.489796\pi$
$47$		0			0		−0.0320516	+	0.999486i	$0.510204\pi$
$48$	0.623490	+	0.781831i	0.623490	+	0.781831i
$49$	0.926917	−	0.375267i	0.926917	−	0.375267i
$50$		0			0
$51$		0			0
$52$	−0.358069	−	0.972990i	−0.358069	−	0.972990i
$53$		0			0		−0.871319	−	0.490718i	$-0.836735\pi$
$53$		0			0		0.871319	+	0.490718i	$0.163265\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.27632	−	0.830791i	1.27632	−	0.830791i
$58$		0			0
$59$		0			0		0.345365	−	0.938468i	$-0.387755\pi$
$59$		0			0		−0.345365	+	0.938468i	$0.612245\pi$
$60$		0			0
$61$	−0.996992	+	1.42926i	−0.996992	+	1.42926i	−0.0960230	+	0.995379i	$0.530612\pi$
$61$	−0.996992	+	1.42926i	−0.996992	+	1.42926i	−0.900969	+	0.433884i	$0.857143\pi$
$62$		0			0
$63$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$64$	0.801414	+	0.598111i	0.801414	+	0.598111i
$65$		0			0
$66$		0			0
$67$	0.372984	+	1.63415i	0.372984	+	1.63415i	0.718349	+	0.695683i	$0.244898\pi$
$67$	0.372984	+	1.63415i	0.372984	+	1.63415i	−0.345365	+	0.938468i	$0.612245\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.926917	−	0.375267i	$-0.122449\pi$
$71$		0			0		−0.926917	+	0.375267i	$0.877551\pi$
$72$		0			0
$73$	0.0182391	+	0.568763i	0.0182391	+	0.568763i	0.967295	+	0.253655i	$0.0816327\pi$
$73$	0.0182391	+	0.568763i	0.0182391	+	0.568763i	−0.949056	+	0.315108i	$0.897959\pi$
$74$		0			0
$75$	0.871319	+	0.490718i	0.871319	+	0.490718i
$76$	1.02384	−	1.12736i	1.02384	−	1.12736i
$77$		0			0
$78$		0			0
$79$	−0.713418	+	0.894598i	−0.713418	+	0.894598i	−0.997945	−	0.0640702i	$-0.979592\pi$
$79$	−0.713418	+	0.894598i	−0.713418	+	0.894598i	0.284528	+	0.958668i	$0.408163\pi$
$80$		0			0
$81$	−0.672301	−	0.740278i	−0.672301	−	0.740278i
$82$		0			0
$83$		0			0		−0.981559	−	0.191159i	$-0.938776\pi$
$83$		0			0		0.981559	+	0.191159i	$0.0612245\pi$
$84$	0.0320516	+	0.999486i	0.0320516	+	0.999486i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.967295	−	0.253655i	$-0.0816327\pi$
$89$		0			0		−0.967295	+	0.253655i	$0.918367\pi$
$90$		0			0
$91$	0.294994	−	0.993933i	0.294994	−	0.993933i
$92$		0			0
$93$	0.464378	+	1.26187i	0.464378	+	1.26187i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.430487	+	1.88609i	−0.430487	+	1.88609i	0.0320516	+	0.999486i	$0.489796\pi$
$97$	−0.430487	+	1.88609i	−0.430487	+	1.88609i	−0.462538	+	0.886599i	$0.653061\pi$
$98$		0			0
$99$		0			0
$100$	0.967295	+	0.253655i	0.967295	+	0.253655i
$101$		0			0		0.991790	−	0.127877i	$-0.0408163\pi$
$101$		0			0		−0.991790	+	0.127877i	$0.959184\pi$
$102$		0			0
$103$	−1.94700	+	0.251038i	−1.94700	+	0.251038i	−0.997945	−	0.0640702i	$-0.979592\pi$
$103$	−1.94700	+	0.251038i	−1.94700	+	0.251038i	−0.949056	+	0.315108i	$0.897959\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.462538	−	0.886599i	$-0.653061\pi$
$107$		0			0		0.462538	+	0.886599i	$0.346939\pi$
$108$	−0.838088	−	0.545535i	−0.838088	−	0.545535i
$109$	0.0888287	−	0.170268i	0.0888287	−	0.170268i	−0.838088	−	0.545535i	$-0.816327\pi$
$109$	0.0888287	−	0.170268i	0.0888287	−	0.170268i	0.926917	+	0.375267i	$0.122449\pi$
$110$		0			0
$111$	1.51976	+	1.29379i	1.51976	+	1.29379i
$112$	0.284528	+	0.958668i	0.284528	+	0.958668i
$113$		0			0		0.967295	−	0.253655i	$-0.0816327\pi$
$113$		0			0		−0.967295	+	0.253655i	$0.918367\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.646425	+	0.810591i	0.646425	+	0.810591i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.284528	−	0.958668i	0.284528	−	0.958668i
$122$		0			0
$123$		0			0
$124$	0.769269	+	1.10281i	0.769269	+	1.10281i
$125$		0			0
$126$		0			0
$127$	0.129207	−	0.799189i	0.129207	−	0.799189i	−0.838088	−	0.545535i	$-0.816327\pi$
$127$	0.129207	−	0.799189i	0.129207	−	0.799189i	0.967295	−	0.253655i	$-0.0816327\pi$
$128$		0			0
$129$	0.999346	−	1.25314i	0.999346	−	1.25314i
$130$		0			0
$131$		0			0		0.949056	−	0.315108i	$-0.102041\pi$
$131$		0			0		−0.949056	+	0.315108i	$0.897959\pi$
$132$		0			0
$133$	1.49481	−	0.291114i	1.49481	−	0.291114i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.518393	−	0.855143i	$-0.673469\pi$
$137$		0			0		0.518393	+	0.855143i	$0.326531\pi$
$138$		0			0
$139$	−1.88253	+	0.242725i	−1.88253	+	0.242725i	−0.981559	−	0.191159i	$-0.938776\pi$
$139$	−1.88253	+	0.242725i	−1.88253	+	0.242725i	−0.900969	+	0.433884i	$0.857143\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.949056	−	0.315108i	−0.949056	−	0.315108i
$145$		0			0
$146$		0			0
$147$	−0.572117	+	0.820172i	−0.572117	+	0.820172i
$148$	1.79824	+	0.865985i	1.79824	+	0.865985i
$149$		0			0		0.981559	−	0.191159i	$-0.0612245\pi$
$149$		0			0		−0.981559	+	0.191159i	$0.938776\pi$
$150$		0			0
$151$	−0.0639714	−	0.00410710i	−0.0639714	−	0.00410710i	0.0320516	−	0.999486i	$-0.489796\pi$
$151$	−0.0639714	−	0.00410710i	−0.0639714	−	0.00410710i	−0.0960230	+	0.995379i	$0.530612\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	0.830894	+	0.620112i	0.830894	+	0.620112i
$157$	−0.496186	+	0.480529i	−0.496186	+	0.480529i	−0.900969	−	0.433884i	$-0.857143\pi$
$157$	−0.496186	+	0.480529i	−0.496186	+	0.480529i	0.404783	+	0.914413i	$0.367347\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.42490	+	0.183721i	1.42490	+	0.183721i	0.801414	−	0.598111i	$-0.204082\pi$
$163$	1.42490	+	0.183721i	1.42490	+	0.183721i	0.623490	+	0.781831i	$0.285714\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.672301	−	0.740278i	$-0.734694\pi$
$167$		0			0		0.672301	+	0.740278i	$0.265306\pi$
$168$		0			0
$169$	−0.0428649	−	0.0614501i	−0.0428649	−	0.0614501i
$170$		0			0
$171$	−0.616441	+	1.39255i	−0.616441	+	1.39255i
$172$	0.648798	−	1.46565i	0.648798	−	1.46565i
$173$		0			0		−0.159600	−	0.987182i	$-0.551020\pi$
$173$		0			0		0.159600	+	0.987182i	$0.448980\pi$
$174$		0			0
$175$	0.623490	+	0.781831i	0.623490	+	0.781831i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.926917	−	0.375267i	$-0.877551\pi$
$179$		0			0		0.926917	+	0.375267i	$0.122449\pi$
$180$		0			0
$181$	0.581552	−	1.31374i	0.581552	−	1.31374i	−0.345365	−	0.938468i	$-0.612245\pi$
$181$	0.581552	−	1.31374i	0.581552	−	1.31374i	0.926917	−	0.375267i	$-0.122449\pi$
$182$		0			0
$183$	0.0558543	−	1.74174i	0.0558543	−	1.74174i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.345365	−	0.938468i	−0.345365	−	0.938468i
$190$		0			0
$191$		0			0		−0.949056	−	0.315108i	$-0.897959\pi$
$191$		0			0		0.949056	+	0.315108i	$0.102041\pi$
$192$	−0.997945	−	0.0640702i	−0.997945	−	0.0640702i
$193$	−0.190469	+	1.97441i	−0.190469	+	1.97441i	0.0320516	+	0.999486i	$0.489796\pi$
$193$	−0.190469	+	1.97441i	−0.190469	+	1.97441i	−0.222521	+	0.974928i	$0.571429\pi$
$194$		0			0
$195$		0			0
$196$	−0.345365	+	0.938468i	−0.345365	+	0.938468i
$197$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$197$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$198$		0			0
$199$	0.877949	+	0.291499i	0.877949	+	0.291499i	0.718349	−	0.695683i	$-0.244898\pi$
$199$	0.877949	+	0.291499i	0.877949	+	0.291499i	0.159600	+	0.987182i	$0.448980\pi$
$200$		0			0
$201$	−1.20408	−	1.16609i	−1.20408	−	1.16609i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.961014	+	0.389071i	0.961014	+	0.389071i
$209$		0			0
$210$		0			0
$211$	−0.857469	−	1.64361i	−0.857469	−	1.64361i	−0.761446	−	0.648228i	$-0.775510\pi$
$211$	−0.857469	−	1.64361i	−0.857469	−	1.64361i	−0.0960230	−	0.995379i	$-0.530612\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.0430966	+	1.34391i	−0.0430966	+	1.34391i
$218$		0			0
$219$	−0.325566	−	0.466723i	−0.325566	−	0.466723i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.0908211	−	0.306007i	0.0908211	−	0.306007i	−0.900969	−	0.433884i	$-0.857143\pi$
$223$	0.0908211	−	0.306007i	0.0908211	−	0.306007i	0.991790	+	0.127877i	$0.0408163\pi$
$224$		0			0
$225$	−0.997945	+	0.0640702i	−0.997945	+	0.0640702i
$226$		0			0
$227$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$227$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$228$	−0.243053	+	1.50337i	−0.243053	+	1.50337i
$229$	−0.0537241	−	0.0349705i	−0.0537241	−	0.0349705i	0.518393	−	0.855143i	$-0.326531\pi$
$229$	−0.0537241	−	0.0349705i	−0.0537241	−	0.0349705i	−0.572117	+	0.820172i	$0.693878\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.871319	−	0.490718i	$-0.163265\pi$
$233$		0			0		−0.871319	+	0.490718i	$0.836735\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.109873	−	1.13895i	0.109873	−	1.13895i
$238$		0			0
$239$		0			0		0.718349	−	0.695683i	$-0.244898\pi$
$239$		0			0		−0.718349	+	0.695683i	$0.755102\pi$
$240$		0			0
$241$	0.564383	−	1.90159i	0.564383	−	1.90159i	0.159600	−	0.987182i	$-0.448980\pi$
$241$	0.564383	−	1.90159i	0.564383	−	1.90159i	0.404783	−	0.914413i	$-0.367347\pi$
$242$		0			0
$243$	0.967295	+	0.253655i	0.967295	+	0.253655i
$244$	−0.387773	−	1.69895i	−0.387773	−	1.69895i
$245$		0			0
$246$		0			0
$247$	0.730307	−	1.39986i	0.730307	−	1.39986i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.345365	−	0.938468i	$-0.387755\pi$
$251$		0			0		−0.345365	+	0.938468i	$0.612245\pi$
$252$	−0.572117	−	0.820172i	−0.572117	−	0.820172i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.981559	+	0.191159i	−0.981559	+	0.191159i
$257$		0			0		−0.672301	−	0.740278i	$-0.734694\pi$
$257$		0			0		0.672301	+	0.740278i	$0.265306\pi$
$258$		0			0
$259$	0.923176	+	1.76956i	0.923176	+	1.76956i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$263$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.46048	−	0.822529i	−1.46048	−	0.822529i
$269$		0			0		0.761446	−	0.648228i	$-0.224490\pi$
$269$		0			0		−0.761446	+	0.648228i	$0.775510\pi$
$270$		0			0
$271$	0.129113	−	0.142167i	0.129113	−	0.142167i	−0.672301	−	0.740278i	$-0.734694\pi$
$271$	0.129113	−	0.142167i	0.129113	−	0.142167i	0.801414	+	0.598111i	$0.204082\pi$
$272$		0			0
$273$	0.294994	+	0.993933i	0.294994	+	0.993933i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.689311	−	0.0442552i	0.689311	−	0.0442552i	0.284528	−	0.958668i	$-0.408163\pi$
$277$	0.689311	−	0.0442552i	0.689311	−	0.0442552i	0.404783	+	0.914413i	$0.367347\pi$
$278$		0			0
$279$	−1.07758	−	0.804220i	−1.07758	−	0.804220i
$280$		0			0
$281$		0			0		0.981559	−	0.191159i	$-0.0612245\pi$
$281$		0			0		−0.981559	+	0.191159i	$0.938776\pi$
$282$		0			0
$283$	−1.41159	−	1.20171i	−1.41159	−	1.20171i	−0.949056	−	0.315108i	$-0.897959\pi$
$283$	−1.41159	−	1.20171i	−1.41159	−	1.20171i	−0.462538	−	0.886599i	$-0.653061\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.838088	+	0.545535i	−0.838088	+	0.545535i
$290$		0			0
$291$	−0.668140	−	1.81555i	−0.668140	−	1.81555i
$292$	−0.433305	−	0.368878i	−0.433305	−	0.368878i
$293$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$293$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−0.949056	+	0.315108i	−0.949056	+	0.315108i
$301$	1.39657	−	0.786535i	1.39657	−	0.786535i
$302$		0			0
$303$		0			0
$304$	0.146233	+	1.51585i	0.146233	+	1.51585i
$305$		0			0
$306$		0			0
$307$	0.327699	+	0.740278i	0.327699	+	0.740278i	1.00000			$0$
$307$	0.327699	+	0.740278i	0.327699	+	0.740278i	−0.672301	+	0.740278i	$0.734694\pi$
$308$		0			0
$309$	1.49481	−	1.27255i	1.49481	−	1.27255i
$310$		0			0
$311$		0			0		0.871319	−	0.490718i	$-0.163265\pi$
$311$		0			0		−0.871319	+	0.490718i	$0.836735\pi$
$312$		0			0
$313$	−1.67025	+	0.804348i	−1.67025	+	0.804348i	−0.672301	+	0.740278i	$0.734694\pi$
$313$	−1.67025	+	0.804348i	−1.67025	+	0.804348i	−0.997945	+	0.0640702i	$0.979592\pi$
$314$		0			0
$315$		0			0
$316$	−0.182620	−	1.12957i	−0.182620	−	1.12957i
$317$		0			0		−0.997945	−	0.0640702i	$-0.979592\pi$
$317$		0			0		0.997945	+	0.0640702i	$0.0204082\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	1.00000			1.00000
$325$	1.03679			1.03679
$326$		0			0
$327$	0.0184408	+	0.191159i	0.0184408	+	0.191159i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.313313	+	1.93795i	−0.313313	+	1.93795i	0.0320516	+	0.999486i	$0.489796\pi$
$331$	−0.313313	+	1.93795i	−0.313313	+	1.93795i	−0.345365	+	0.938468i	$0.612245\pi$
$332$		0			0
$333$	−1.97950	−	0.255229i	−1.97950	−	0.255229i
$334$		0			0
$335$		0			0
$336$	−0.761446	−	0.648228i	−0.761446	−	0.648228i
$337$	0.578893	+	0.376817i	0.578893	+	0.376817i	0.801414	−	0.598111i	$-0.204082\pi$
$337$	0.578893	+	0.376817i	0.578893	+	0.376817i	−0.222521	+	0.974928i	$0.571429\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.838088	+	0.545535i	−0.838088	+	0.545535i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.672301	−	0.740278i	$-0.265306\pi$
$347$		0			0		−0.672301	+	0.740278i	$0.734694\pi$
$348$		0			0
$349$	−0.678488	−	0.441647i	−0.678488	−	0.441647i	0.159600	−	0.987182i	$-0.448980\pi$
$349$	−0.678488	−	0.441647i	−0.678488	−	0.441647i	−0.838088	+	0.545535i	$0.816327\pi$
$350$		0			0
$351$	−0.983967	−	0.326700i	−0.983967	−	0.326700i
$352$		0			0
$353$		0			0		−0.991790	−	0.127877i	$-0.959184\pi$
$353$		0			0		0.991790	+	0.127877i	$0.0408163\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.0960230	−	0.995379i	$-0.530612\pi$
$359$		0			0		0.0960230	+	0.995379i	$0.469388\pi$
$360$		0			0
$361$	1.31920			1.31920
$362$		0			0
$363$	0.284528	+	0.958668i	0.284528	+	0.958668i
$364$	0.537462	+	0.886599i	0.537462	+	0.886599i
$365$		0			0
$366$		0			0
$367$	1.91871	−	0.503144i	1.91871	−	0.503144i	0.926917	−	0.375267i	$-0.122449\pi$
$367$	1.91871	−	0.503144i	1.91871	−	0.503144i	0.991790	−	0.127877i	$-0.0408163\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−1.24633	−	0.504585i	−1.24633	−	0.504585i
$373$	−1.74301	+	0.839387i	−1.74301	+	0.839387i	−0.761446	+	0.648228i	$0.775510\pi$
$373$	−1.74301	+	0.839387i	−1.74301	+	0.839387i	−0.981559	+	0.191159i	$0.938776\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.783090	+	1.76901i	0.783090	+	1.76901i	0.623490	+	0.781831i	$0.285714\pi$
$379$	0.783090	+	1.76901i	0.783090	+	1.76901i	0.159600	+	0.987182i	$0.448980\pi$
$380$		0			0
$381$	0.327699	+	0.740278i	0.327699	+	0.740278i
$382$		0			0
$383$		0			0		0.761446	−	0.648228i	$-0.224490\pi$
$383$		0			0		−0.761446	+	0.648228i	$0.775510\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.153908	+	1.59542i	−0.153908	+	1.59542i
$388$	−1.10681	−	1.58670i	−1.10681	−	1.58670i
$389$		0			0		−0.718349	−	0.695683i	$-0.755102\pi$
$389$		0			0		0.718349	+	0.695683i	$0.244898\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.958968	−	0.624219i	0.958968	−	0.624219i	0.0320516	−	0.999486i	$-0.489796\pi$
$397$	0.958968	−	0.624219i	0.958968	−	0.624219i	0.926917	+	0.375267i	$0.122449\pi$
$398$		0			0
$399$	−1.09397	+	1.05945i	−1.09397	+	1.05945i
$400$	−0.838088	+	0.545535i	−0.838088	+	0.545535i
$401$		0			0		−0.0320516	−	0.999486i	$-0.510204\pi$
$401$		0			0		0.0320516	+	0.999486i	$0.489796\pi$
$402$		0			0
$403$	1.06150	+	0.903671i	1.06150	+	0.903671i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.318544	+	0.0204512i	−0.318544	+	0.0204512i	−0.222521	−	0.974928i	$-0.571429\pi$
$409$	−0.318544	+	0.0204512i	−0.318544	+	0.0204512i	−0.0960230	+	0.995379i	$0.530612\pi$
$410$		0			0
$411$		0			0
$412$	1.12313	−	1.61010i	1.12313	−	1.61010i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.44531	−	1.23041i	1.44531	−	1.23041i
$418$		0			0
$419$		0			0		0.672301	−	0.740278i	$-0.265306\pi$
$419$		0			0		−0.672301	+	0.740278i	$0.734694\pi$
$420$		0			0
$421$	−1.65386	+	0.931437i	−1.65386	+	0.931437i	−0.672301	+	0.740278i	$0.734694\pi$
$421$	−1.65386	+	0.931437i	−1.65386	+	0.931437i	−0.981559	+	0.191159i	$0.938776\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.705391	−	1.59349i	0.705391	−	1.59349i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.949056	−	0.315108i	$-0.102041\pi$
$431$		0			0		−0.949056	+	0.315108i	$0.897959\pi$
$432$	0.967295	−	0.253655i	0.967295	−	0.253655i
$433$	0.0332306	−	0.0548173i	0.0332306	−	0.0548173i	−0.838088	−	0.545535i	$-0.816327\pi$
$433$	0.0332306	−	0.0548173i	0.0332306	−	0.0548173i	0.871319	+	0.490718i	$0.163265\pi$
$434$		0			0
$435$		0			0
$436$	0.0663260	+	0.180229i	0.0663260	+	0.180229i
$437$		0			0
$438$		0			0
$439$	0.775296	−	1.48610i	0.775296	−	1.48610i	−0.0960230	−	0.995379i	$-0.530612\pi$
$439$	0.775296	−	1.48610i	0.775296	−	1.48610i	0.871319	−	0.490718i	$-0.163265\pi$
$440$		0			0
$441$	0.0320516	−	0.999486i	0.0320516	−	0.999486i
$442$		0			0
$443$		0			0		−0.967295	−	0.253655i	$-0.918367\pi$
$443$		0			0		0.967295	+	0.253655i	$0.0816327\pi$
$444$	−1.97950	+	0.255229i	−1.97950	+	0.255229i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$449$		0			0		0.0960230	−	0.995379i	$-0.469388\pi$
$449$		0			0		−0.0960230	+	0.995379i	$0.530612\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	0.0558543	−	0.0314565i	0.0558543	−	0.0314565i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.55368	−	1.01133i	−1.55368	−	1.01133i	−0.981559	−	0.191159i	$-0.938776\pi$
$457$	−1.55368	−	1.01133i	−1.55368	−	1.01133i	−0.572117	−	0.820172i	$-0.693878\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.997945	−	0.0640702i	$-0.0204082\pi$
$461$		0			0		−0.997945	+	0.0640702i	$0.979592\pi$
$462$		0			0
$463$	0.564383	−	1.90159i	0.564383	−	1.90159i	0.159600	−	0.987182i	$-0.448980\pi$
$463$	0.564383	−	1.90159i	0.564383	−	1.90159i	0.404783	−	0.914413i	$-0.367347\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.572117	−	0.820172i	$-0.693878\pi$
$467$		0			0		0.572117	+	0.820172i	$0.306122\pi$
$468$	−1.03465	−	0.0664271i	−1.03465	−	0.0664271i
$469$	−0.678488	−	1.53272i	−0.678488	−	1.53272i
$470$		0			0
$471$	0.153702	−	0.673412i	0.153702	−	0.673412i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.704396	+	1.35019i	0.704396	+	1.35019i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.0320516	−	0.999486i	$-0.489796\pi$
$479$		0			0		−0.0320516	+	0.999486i	$0.510204\pi$
$480$		0			0
$481$	2.03115	+	0.395566i	2.03115	+	0.395566i
$482$		0			0
$483$		0			0
$484$	0.518393	+	0.855143i	0.518393	+	0.855143i
$485$		0			0
$486$		0			0
$487$	−1.75939	−	0.584158i	−1.75939	−	0.584158i	−0.761446	−	0.648228i	$-0.775510\pi$
$487$	−1.75939	−	0.584158i	−1.75939	−	0.584158i	−0.997945	+	0.0640702i	$0.979592\pi$
$488$		0			0
$489$	−1.29442	+	0.623360i	−1.29442	+	0.623360i
$490$		0			0
$491$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$491$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.33356	−	0.171944i	−1.33356	−	0.171944i
$497$		0			0
$498$		0			0
$499$	1.39657	+	1.04229i	1.39657	+	1.04229i	0.991790	+	0.127877i	$0.0408163\pi$
$499$	1.39657	+	1.04229i	1.39657	+	1.04229i	0.404783	+	0.914413i	$0.367347\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.0320516	−	0.999486i	$-0.489796\pi$
$503$		0			0		−0.0320516	+	0.999486i	$0.510204\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.0694478	+	0.0281163i	0.0694478	+	0.0281163i
$508$	0.504757	+	0.632945i	0.504757	+	0.632945i
$509$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$509$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$510$		0			0
$511$	−0.126627	−	0.554788i	−0.126627	−	0.554788i
$512$		0			0
$513$	−0.243053	−	1.50337i	−0.243053	−	1.50337i
$514$		0			0
$515$		0			0
$516$	0.255811	+	1.58228i	0.255811	+	1.58228i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$521$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$522$		0			0
$523$	0.316579	+	0.0408184i	0.316579	+	0.0408184i	0.284528	−	0.958668i	$-0.408163\pi$
$523$	0.316579	+	0.0408184i	0.316579	+	0.0408184i	0.0320516	+	0.999486i	$0.489796\pi$
$524$		0			0
$525$	−0.949056	−	0.315108i	−0.949056	−	0.315108i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.718349	−	0.695683i	0.718349	−	0.695683i
$530$		0			0
$531$		0			0
$532$	−0.789456	+	1.30229i	−0.789456	+	1.30229i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.741369	+	0.553298i	−0.741369	+	0.553298i	−0.900969	−	0.433884i	$-0.857143\pi$
$541$	−0.741369	+	0.553298i	−0.741369	+	0.553298i	0.159600	+	0.987182i	$0.448980\pi$
$542$		0			0
$543$	0.229297	+	1.41828i	0.229297	+	1.41828i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.316579	−	0.0408184i	0.316579	−	0.0408184i	0.0320516	−	0.999486i	$-0.489796\pi$
$547$	0.316579	−	0.0408184i	0.316579	−	0.0408184i	0.284528	+	0.958668i	$0.408163\pi$
$548$		0			0
$549$	0.903370	+	1.49020i	0.903370	+	1.49020i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.529252	−	1.01448i	0.529252	−	1.01448i
$554$		0			0
$555$		0			0
$556$	1.08594	−	1.55678i	1.08594	−	1.55678i
$557$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$557$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$558$		0			0
$559$	0.265221	−	1.64049i	0.265221	−	1.64049i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.801414	−	0.598111i	$-0.795918\pi$
$563$		0			0		0.801414	+	0.598111i	$0.204082\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.801414	+	0.598111i	0.801414	+	0.598111i
$568$		0			0
$569$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$569$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$570$		0			0
$571$	0.278125	−	1.72030i	0.278125	−	1.72030i	−0.345365	−	0.938468i	$-0.612245\pi$
$571$	0.278125	−	1.72030i	0.278125	−	1.72030i	0.623490	−	0.781831i	$-0.285714\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.871319	−	0.490718i	0.871319	−	0.490718i
$577$	0.319489	−	0.612401i	0.319489	−	0.612401i	−0.672301	−	0.740278i	$-0.734694\pi$
$577$	0.319489	−	0.612401i	0.319489	−	0.612401i	0.991790	+	0.127877i	$0.0408163\pi$
$578$		0			0
$579$	−0.917482	−	1.75864i	−0.917482	−	1.75864i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$587$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$588$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$589$	−0.455652	+	1.99634i	−0.455652	+	1.99634i
$590$		0			0
$591$		0			0
$592$	−1.85002	+	0.748992i	−1.85002	+	0.748992i
$593$		0			0		−0.345365	−	0.938468i	$-0.612245\pi$
$593$		0			0		0.345365	+	0.938468i	$0.387755\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.894822	+	0.234650i	−0.894822	+	0.234650i
$598$		0			0
$599$		0			0		0.981559	−	0.191159i	$-0.0612245\pi$
$599$		0			0		−0.981559	+	0.191159i	$0.938776\pi$
$600$		0			0
$601$	1.20620	+	0.316302i	1.20620	+	0.316302i	0.801414	−	0.598111i	$-0.204082\pi$
$601$	1.20620	+	0.316302i	1.20620	+	0.316302i	0.404783	+	0.914413i	$0.367347\pi$
$602$		0			0
$603$	1.64527	+	0.320416i	1.64527	+	0.320416i
$604$	0.0460485	−	0.0445954i	0.0460485	−	0.0445954i
$605$		0			0
$606$		0			0
$607$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	−0.900969	−	0.433884i	$-0.857143\pi$
$607$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	0.623490	+	0.781831i	$0.285714\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.0182391	+	0.568763i	0.0182391	+	0.568763i	0.967295	+	0.253655i	$0.0816327\pi$
$613$	0.0182391	+	0.568763i	0.0182391	+	0.568763i	−0.949056	+	0.315108i	$0.897959\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.572117	−	0.820172i	$-0.306122\pi$
$617$		0			0		−0.572117	+	0.820172i	$0.693878\pi$
$618$		0			0
$619$	0.400969	+	1.75676i	0.400969	+	1.75676i	0.623490	+	0.781831i	$0.285714\pi$
$619$	0.400969	+	1.75676i	0.400969	+	1.75676i	−0.222521	+	0.974928i	$0.571429\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−1.01767	+	0.198190i	−1.01767	+	0.198190i
$625$	−0.572117	+	0.820172i	−0.572117	+	0.820172i
$626$		0			0
$627$		0			0
$628$	−0.0221390	−	0.690375i	−0.0221390	−	0.690375i
$629$		0			0
$630$		0			0
$631$	−0.0546424	−	0.566426i	−0.0546424	−	0.566426i	−0.981559	−	0.191159i	$-0.938776\pi$
$631$	−0.0546424	−	0.566426i	−0.0546424	−	0.566426i	0.926917	−	0.375267i	$-0.122449\pi$
$632$		0			0
$633$	1.61528	+	0.909709i	1.61528	+	0.909709i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.0995552	+	1.03199i	−0.0995552	+	1.03199i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.572117	−	0.820172i	$-0.693878\pi$
$641$		0			0		0.572117	+	0.820172i	$0.306122\pi$
$642$		0			0
$643$	1.59078	−	0.528177i	1.59078	−	0.528177i	0.623490	−	0.781831i	$-0.285714\pi$
$643$	1.59078	−	0.528177i	1.59078	−	0.528177i	0.967295	+	0.253655i	$0.0816327\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.0960230	−	0.995379i	$-0.530612\pi$
$647$		0			0		0.0960230	+	0.995379i	$0.469388\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−0.697032	−	1.14983i	−0.697032	−	1.14983i
$652$	−1.09397	+	0.931309i	−1.09397	+	0.931309i
$653$		0			0		0.345365	−	0.938468i	$-0.387755\pi$
$653$		0			0		−0.345365	+	0.938468i	$0.612245\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	0.527467	+	0.213548i	0.527467	+	0.213548i
$658$		0			0
$659$		0			0		−0.159600	−	0.987182i	$-0.551020\pi$
$659$		0			0		0.159600	+	0.987182i	$0.448980\pi$
$660$		0			0
$661$	0.833465	+	1.59760i	0.833465	+	1.59760i	0.801414	+	0.598111i	$0.204082\pi$
$661$	0.833465	+	1.59760i	0.833465	+	1.59760i	0.0320516	+	0.999486i	$0.489796\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	0.0908211	+	0.306007i	0.0908211	+	0.306007i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.664528	+	1.27378i	−0.664528	+	1.27378i	0.284528	+	0.958668i	$0.408163\pi$
$673$	−0.664528	+	1.27378i	−0.664528	+	1.27378i	−0.949056	+	0.315108i	$0.897959\pi$
$674$		0			0
$675$	0.801414	−	0.598111i	0.801414	−	0.598111i
$676$	0.0743083	+	0.00958099i	0.0743083	+	0.00958099i
$677$		0			0		0.404783	−	0.914413i	$-0.367347\pi$
$677$		0			0		−0.404783	+	0.914413i	$0.632653\pi$
$678$		0			0
$679$	0.0620067	−	1.93360i	0.0620067	−	1.93360i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.838088	−	0.545535i	$-0.183673\pi$
$683$		0			0		−0.838088	+	0.545535i	$0.816327\pi$
$684$	−0.616441	−	1.39255i	−0.616441	−	1.39255i
$685$		0			0
$686$		0			0
$687$	0.0641032			0.0641032
$688$	0.648798	+	1.46565i	0.648798	+	1.46565i
$689$		0			0
$690$		0			0
$691$	−0.230706	+	0.380574i	−0.230706	+	0.380574i	−0.949056	−	0.315108i	$-0.897959\pi$
$691$	−0.230706	+	0.380574i	−0.230706	+	0.380574i	0.718349	+	0.695683i	$0.244898\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.997945	−	0.0640702i	−0.997945	−	0.0640702i
$701$		0			0		−0.0320516	−	0.999486i	$-0.510204\pi$
$701$		0			0		0.0320516	+	0.999486i	$0.489796\pi$
$702$		0			0
$703$	0.864829	+	2.91390i	0.864829	+	2.91390i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.750401	−	0.303804i	0.750401	−	0.303804i	0.0320516	−	0.999486i	$-0.489796\pi$
$709$	0.750401	−	0.303804i	0.750401	−	0.303804i	0.718349	+	0.695683i	$0.244898\pi$
$710$		0			0
$711$	0.529252	+	1.01448i	0.529252	+	1.01448i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.345365	−	0.938468i	$-0.387755\pi$
$719$		0			0		−0.345365	+	0.938468i	$0.612245\pi$
$720$		0			0
$721$	1.86311	−	0.618595i	1.86311	−	0.618595i
$722$		0			0
$723$	0.564383	+	1.90159i	0.564383	+	1.90159i
$724$	0.581552	+	1.31374i	0.581552	+	1.31374i
$725$		0			0
$726$		0			0
$727$	−0.430663	−	1.17025i	−0.430663	−	1.17025i	−0.949056	−	0.315108i	$-0.897959\pi$
$727$	−0.430663	−	1.17025i	−0.430663	−	1.17025i	0.518393	−	0.855143i	$-0.326531\pi$
$728$		0			0
$729$	−0.949056	+	0.315108i	−0.949056	+	0.315108i
$730$		0			0
$731$		0			0
$732$	1.25182	+	1.21232i	1.25182	+	1.21232i
$733$	0.0513731	−	1.60200i	0.0513731	−	1.60200i	−0.572117	−	0.820172i	$-0.693878\pi$
$733$	0.0513731	−	1.60200i	0.0513731	−	1.60200i	0.623490	−	0.781831i	$-0.285714\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.25182	+	0.705013i	1.25182	+	0.705013i	0.967295	−	0.253655i	$-0.0816327\pi$
$739$	1.25182	+	0.705013i	1.25182	+	0.705013i	0.284528	+	0.958668i	$0.408163\pi$
$740$		0			0
$741$	0.151612	+	1.57162i	0.151612	+	1.57162i
$742$		0			0
$743$		0			0		0.838088	−	0.545535i	$-0.183673\pi$
$743$		0			0		−0.838088	+	0.545535i	$0.816327\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.419673	+	0.692295i	0.419673	+	0.692295i	0.991790	−	0.127877i	$-0.0408163\pi$
$751$	0.419673	+	0.692295i	0.419673	+	0.692295i	−0.572117	+	0.820172i	$0.693878\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.926917	+	0.375267i	0.926917	+	0.375267i
$757$	−0.640249	+	0.259208i	−0.640249	+	0.259208i	−0.672301	−	0.740278i	$-0.734694\pi$
$757$	−0.640249	+	0.259208i	−0.640249	+	0.259208i	0.0320516	+	0.999486i	$0.489796\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.871319	−	0.490718i	$-0.836735\pi$
$761$		0			0		0.871319	+	0.490718i	$0.163265\pi$
$762$		0			0
$763$	−0.0546424	+	0.184108i	−0.0546424	+	0.184108i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.718349	−	0.695683i	0.718349	−	0.695683i
$769$	−1.22398	−	0.238371i	−1.22398	−	0.238371i	−0.462538	−	0.886599i	$-0.653061\pi$
$769$	−1.22398	−	0.238371i	−1.22398	−	0.238371i	−0.761446	+	0.648228i	$0.775510\pi$
$770$		0			0
$771$		0			0
$772$	−1.33356	−	1.46840i	−1.33356	−	1.46840i
$773$		0			0		0.981559	−	0.191159i	$-0.0612245\pi$
$773$		0			0		−0.981559	+	0.191159i	$0.938776\pi$
$774$		0			0
$775$	−1.30063	+	0.341064i	−1.30063	+	0.341064i
$776$		0			0
$777$	−1.73906	−	0.979419i	−1.73906	−	0.979419i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.462538	−	0.886599i	−0.462538	−	0.886599i
$785$		0			0
$786$		0			0
$787$	−1.13484	+	0.146321i	−1.13484	+	0.146321i	−0.672301	−	0.740278i	$-0.734694\pi$
$787$	−1.13484	+	0.146321i	−1.13484	+	0.146321i	−0.462538	+	0.886599i	$0.653061\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.835687	−	1.60185i	−0.835687	−	1.60185i
$794$		0			0
$795$		0			0
$796$	−0.806037	+	0.453951i	−0.806037	+	0.453951i
$797$		0			0		−0.761446	−	0.648228i	$-0.775510\pi$
$797$		0			0		0.761446	+	0.648228i	$0.224490\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	1.67273	−	0.107393i	1.67273	−	0.107393i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.801414	−	0.598111i	$-0.795918\pi$
$809$		0			0		0.801414	+	0.598111i	$0.204082\pi$
$810$		0			0
$811$	1.89421	+	0.121612i	1.89421	+	0.121612i	0.967295	−	0.253655i	$-0.0816327\pi$
$811$	1.89421	+	0.121612i	1.89421	+	0.121612i	0.926917	+	0.375267i	$0.122449\pi$
$812$		0			0
$813$	−0.0306505	+	0.189584i	−0.0306505	+	0.189584i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	2.31658	−	0.769158i	2.31658	−	0.769158i
$818$		0			0
$819$	−0.789456	−	0.672074i	−0.789456	−	0.672074i
$820$		0			0
$821$		0			0		−0.926917	−	0.375267i	$-0.877551\pi$
$821$		0			0		0.926917	+	0.375267i	$0.122449\pi$
$822$		0			0
$823$	−0.479553	−	0.791073i	−0.479553	−	0.791073i	0.518393	−	0.855143i	$-0.326531\pi$
$823$	−0.479553	−	0.791073i	−0.479553	−	0.791073i	−0.997945	+	0.0640702i	$0.979592\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.518393	−	0.855143i	$-0.673469\pi$
$827$		0			0		0.518393	+	0.855143i	$0.326531\pi$
$828$		0			0
$829$	−0.318544	−	1.97031i	−0.318544	−	1.97031i	−0.222521	−	0.974928i	$-0.571429\pi$
$829$	−0.318544	−	1.97031i	−0.318544	−	1.97031i	−0.0960230	−	0.995379i	$-0.530612\pi$
$830$		0			0
$831$	−0.553561	+	0.413133i	−0.553561	+	0.413133i
$832$	−0.934111	+	0.449844i	−0.934111	+	0.449844i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.34184	+	0.0861489i	1.34184	+	0.0861489i
$838$		0			0
$839$		0			0		−0.991790	−	0.127877i	$-0.959184\pi$
$839$		0			0		0.991790	+	0.127877i	$0.0408163\pi$
$840$		0			0
$841$	0.718349	+	0.695683i	0.718349	+	0.695683i
$842$		0			0
$843$		0			0
$844$	1.79320	+	0.470233i	1.79320	+	0.470233i
$845$		0			0
$846$		0			0
$847$	−0.0960230	+	0.995379i	−0.0960230	+	0.995379i
$848$		0			0
$849$	1.83861	+	0.237063i	1.83861	+	0.237063i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.621930	+	0.684814i	0.621930	+	0.684814i	0.967295	−	0.253655i	$-0.0816327\pi$
$853$	0.621930	+	0.684814i	0.621930	+	0.684814i	−0.345365	+	0.938468i	$0.612245\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.404783	−	0.914413i	$-0.367347\pi$
$857$		0			0		−0.404783	+	0.914413i	$0.632653\pi$
$858$		0			0
$859$	−0.313313	−	1.93795i	−0.313313	−	1.93795i	−0.345365	−	0.938468i	$-0.612245\pi$
$859$	−0.313313	−	1.93795i	−0.313313	−	1.93795i	0.0320516	−	0.999486i	$-0.489796\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$863$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.404783	−	0.914413i	0.404783	−	0.914413i
$868$	−0.965894	−	0.935416i	−0.965894	−	0.935416i
$869$		0			0
$870$		0			0
$871$	−1.68100	−	0.440810i	−1.68100	−	0.440810i
$872$		0			0
$873$	1.55041	+	1.15710i	1.55041	+	1.15710i
$874$		0			0
$875$		0			0
$876$	0.564383	+	0.0727692i	0.564383	+	0.0727692i
$877$	1.86311	+	0.618595i	1.86311	+	0.618595i	0.991790	+	0.127877i	$0.0408163\pi$
$877$	1.86311	+	0.618595i	1.86311	+	0.618595i	0.871319	+	0.490718i	$0.163265\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$881$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$882$		0			0
$883$	0.173028	−	0.0833257i	0.173028	−	0.0833257i	−0.345365	−	0.938468i	$-0.612245\pi$
$883$	0.173028	−	0.0833257i	0.173028	−	0.0833257i	0.518393	+	0.855143i	$0.326531\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.718349	−	0.695683i	$-0.755102\pi$
$887$		0			0		0.718349	+	0.695683i	$0.244898\pi$
$888$		0			0
$889$	0.0259479	+	0.809151i	0.0259479	+	0.809151i
$890$		0			0
$891$		0			0
$892$	0.165471	+	0.272961i	0.165471	+	0.272961i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.623490	−	0.781831i	0.623490	−	0.781831i
$901$		0			0
$902$		0			0
$903$	−0.741369	+	1.42107i	−0.741369	+	1.42107i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.86311	+	0.362840i	1.86311	+	0.362840i	0.991790	−	0.127877i	$-0.0408163\pi$
$907$	1.86311	+	0.362840i	1.86311	+	0.362840i	0.871319	+	0.490718i	$0.163265\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.997945	−	0.0640702i	$-0.0204082\pi$
$911$		0			0		−0.997945	+	0.0640702i	$0.979592\pi$
$912$	−0.949508	−	1.19064i	−0.949508	−	1.19064i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.0620067	−	0.0162601i	0.0620067	−	0.0162601i
$917$		0			0
$918$		0			0
$919$	−1.65386	+	0.931437i	−1.65386	+	0.931437i	−0.672301	+	0.740278i	$0.734694\pi$
$919$	−1.65386	+	0.931437i	−1.65386	+	0.931437i	−0.981559	+	0.191159i	$0.938776\pi$
$920$		0			0
$921$	−0.678488	−	0.441647i	−0.678488	−	0.441647i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.43375	+	1.38851i	−1.43375	+	1.38851i
$926$		0			0
$927$	−0.558561	+	1.88198i	−0.558561	+	1.88198i
$928$		0			0
$929$		0			0		−0.967295	−	0.253655i	$-0.918367\pi$
$929$		0			0		0.967295	+	0.253655i	$0.0816327\pi$
$930$		0			0
$931$	−1.41159	+	0.571491i	−1.41159	+	0.571491i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.668140	+	1.81555i	−0.668140	+	1.81555i	−0.0960230	+	0.995379i	$0.530612\pi$
$937$	−0.668140	+	1.81555i	−0.668140	+	1.81555i	−0.572117	+	0.820172i	$0.693878\pi$
$938$		0			0
$939$	0.961014	−	1.58529i	0.961014	−	1.58529i
$940$		0			0
$941$		0			0		0.949056	−	0.315108i	$-0.102041\pi$
$941$		0			0		−0.949056	+	0.315108i	$0.897959\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.718349	−	0.695683i	$-0.244898\pi$
$947$		0			0		−0.718349	+	0.695683i	$0.755102\pi$
$948$	0.769269	+	0.847051i	0.769269	+	0.847051i
$949$	−0.531561	−	0.255986i	−0.531561	−	0.255986i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.672301	−	0.740278i	$-0.265306\pi$
$953$		0			0		−0.672301	+	0.740278i	$0.734694\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.727941	−	0.350558i	−0.727941	−	0.350558i
$962$		0			0
$963$		0			0
$964$	1.02827	+	1.69624i	1.02827	+	1.69624i
$965$		0			0
$966$		0			0
$967$	−1.89891	+	0.369814i	−1.89891	+	0.369814i	−0.997945	−	0.0640702i	$-0.979592\pi$
$967$	−1.89891	+	0.369814i	−1.89891	+	0.369814i	−0.900969	+	0.433884i	$0.857143\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.0320516	−	0.999486i	$-0.510204\pi$
$971$		0			0		0.0320516	+	0.999486i	$0.489796\pi$
$972$	−0.838088	+	0.545535i	−0.838088	+	0.545535i
$973$	1.80141	−	0.598111i	1.80141	−	0.598111i
$974$		0			0
$975$	−0.868917	+	0.565602i	−0.868917	+	0.565602i
$976$	1.51839	+	0.855143i	1.51839	+	0.855143i
$977$		0			0		−0.345365	−	0.938468i	$-0.612245\pi$
$977$		0			0		0.345365	+	0.938468i	$0.387755\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.119739	−	0.150148i	−0.119739	−	0.150148i
$982$		0			0
$983$		0			0		−0.718349	−	0.695683i	$-0.755102\pi$
$983$		0			0		0.718349	+	0.695683i	$0.244898\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.545301	+	1.48176i	0.545301	+	1.48176i
$989$		0			0
$990$		0			0
$991$	−0.374456	−	0.845902i	−0.374456	−	0.845902i	−0.997945	−	0.0640702i	$-0.979592\pi$
$991$	−0.374456	−	0.845902i	−0.374456	−	0.845902i	0.623490	−	0.781831i	$-0.285714\pi$
$992$		0			0
$993$	−0.794638	−	1.79510i	−0.794638	−	1.79510i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.57006	+	0.884242i	−1.57006	+	0.884242i	−0.572117	+	0.820172i	$0.693878\pi$
$997$	−1.57006	+	0.884242i	−1.57006	+	0.884242i	−0.997945	+	0.0640702i	$0.979592\pi$
$998$		0			0
$999$	1.79824	−	0.865985i	1.79824	−	0.865985i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1029.1.t.a.113.1	✓	42
3.2	odd	2	CM	1029.1.t.a.113.1	✓	42
343.85	even	49	inner	1029.1.t.a.428.1	yes	42
1029.428	odd	98	inner	1029.1.t.a.428.1	yes	42

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1029.1.t.a.113.1	✓	42	1.1	even	1	trivial
1029.1.t.a.113.1	✓	42	3.2	odd	2	CM
1029.1.t.a.428.1	yes	42	343.85	even	49	inner
1029.1.t.a.428.1	yes	42	1029.428	odd	98	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:	\( N \)	\(=\)	\( 1029 = 3 \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1029.t (of order \(98\), degree \(42\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.838088 + 0.545535i) q^{3} +(-0.672301 + 0.740278i) q^{4} +(-0.981559 + 0.191159i) q^{7} +(0.404783 - 0.914413i) q^{9} +O(q^{10})\)	\(q+(-0.838088 + 0.545535i) q^{3} +(-0.672301 + 0.740278i) q^{4} +(-0.981559 + 0.191159i) q^{7} +(0.404783 - 0.914413i) q^{9} +(0.159600 - 0.987182i) q^{12} +(-0.479553 + 0.919213i) q^{13} +(-0.0960230 - 0.995379i) q^{16} -1.52289 q^{19} +(0.718349 - 0.695683i) q^{21} +(-0.462538 - 0.886599i) q^{25} +(0.159600 + 0.987182i) q^{27} +(0.518393 - 0.855143i) q^{28} +(0.299202 - 1.31089i) q^{31} +(0.404783 + 0.914413i) q^{36} +(-0.567886 - 1.91340i) q^{37} +(-0.0995552 - 1.03199i) q^{39} +(-1.52117 + 0.505064i) q^{43} +(0.623490 + 0.781831i) q^{48} +(0.926917 - 0.375267i) q^{49} +(-0.358069 - 0.972990i) q^{52} +(1.27632 - 0.830791i) q^{57} +(-0.996992 + 1.42926i) q^{61} +(-0.222521 + 0.974928i) q^{63} +(0.801414 + 0.598111i) q^{64} +(0.372984 + 1.63415i) q^{67} +(0.0182391 + 0.568763i) q^{73} +(0.871319 + 0.490718i) q^{75} +(1.02384 - 1.12736i) q^{76} +(-0.713418 + 0.894598i) q^{79} +(-0.672301 - 0.740278i) q^{81} +(0.0320516 + 0.999486i) q^{84} +(0.294994 - 0.993933i) q^{91} +(0.464378 + 1.26187i) q^{93} +(-0.430487 + 1.88609i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q+O(q^{10}) \) 42 * q	\( 42 q - 7 q^{37} - 7 q^{39} - 7 q^{48} - 7 q^{52} - 7 q^{61} - 7 q^{63}+O(q^{100}) \) 42 * q - 7 * q^37 - 7 * q^39 - 7 * q^48 - 7 * q^52 - 7 * q^61 - 7 * q^63

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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