LMFDB - Embedded newform 1379.1.ba.a.174.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1379,1,Mod(41,1379)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1379, base_ring=CyclotomicField(98))

chi = DirichletCharacter(H, H._module([49, 55]))

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

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

chi := DirichletCharacter("1379.41");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1379 = 7 \cdot 197 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1379.ba (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:	$0.688210652421$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{98}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{98} - \cdots)$

Embedding invariants

Embedding label			174.1
Root			$0.949056 - 0.315108i$ of defining polynomial
Character	$\chi$	$=$	1379.174
Dual form			1379.1.ba.a.531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.85288 + 0.681876i) q^{2} +(2.20676 + 1.87864i) q^{4} +(-0.345365 - 0.938468i) q^{7} +(1.83899 + 3.26532i) q^{8} +(-0.404783 + 0.914413i) q^{9} +O(q^{10})$	\(q+(1.85288 + 0.681876i) q^{2} +(2.20676 + 1.87864i) q^{4} +(-0.345365 - 0.938468i) q^{7} +(1.83899 + 3.26532i) q^{8} +(-0.404783 + 0.914413i) q^{9} +(-1.08781 - 1.12326i) q^{11} -1.97436i q^{14} +(0.718349 + 4.44324i) q^{16} +(-1.37353 + 1.41828i) q^{18} +(-1.24966 - 2.82301i) q^{22} +(-0.119739 - 1.24122i) q^{23} +(0.997945 - 0.0640702i) q^{25} +(1.00091 - 2.71979i) q^{28} +(-1.96729 - 0.253655i) q^{29} +(-0.982350 + 5.04416i) q^{32} +(-2.61111 + 1.25744i) q^{36} +(-0.147642 + 0.913219i) q^{37} +(1.43375 - 1.38851i) q^{43} +(-0.290345 - 4.52236i) q^{44} +(0.624495 - 2.38147i) q^{46} +(-0.761446 + 0.648228i) q^{49} +(1.89276 + 0.561761i) q^{50} +(-0.550444 - 0.144343i) q^{53} +(2.42928 - 2.85357i) q^{56} +(-3.47220 - 1.81144i) q^{58} +(0.997945 + 0.0640702i) q^{63} +(-2.92642 + 4.82743i) q^{64} +(-0.719513 + 0.213548i) q^{67} +(1.18550 + 0.524784i) q^{71} +(-3.73025 + 0.359852i) q^{72} +(-0.896265 + 1.59141i) q^{74} +(-0.678448 + 1.40881i) q^{77} +(-0.507049 - 0.0162601i) q^{79} +(-0.672301 - 0.740278i) q^{81} +(3.60335 - 1.59509i) q^{86} +(1.66731 - 5.61772i) q^{88} +(2.06756 - 2.96401i) q^{92} +(-1.85288 + 0.681876i) q^{98} +(1.46745 - 0.540035i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q - 7 q^{8}+O(q^{10}) $ 42 * q - 7 * q^8	$ 42 q - 7 q^{8} - 7 q^{28} - 42 q^{29} + 7 q^{32} - 7 q^{36} + 7 q^{50} + 7 q^{56} - 7 q^{58} - 7 q^{64} + 7 q^{67} + 7 q^{71} - 7 q^{79}+O(q^{100}) $ 42 * q - 7 * q^8 - 7 * q^28 - 42 * q^29 + 7 * q^32 - 7 * q^36 + 7 * q^50 + 7 * q^56 - 7 * q^58 - 7 * q^64 + 7 * q^67 + 7 * q^71 - 7 * q^79

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1379\mathbb{Z}\right)^\times$.

$n$	$395$	$1184$
$\chi(n)$	$-1$	$e\left(\frac{11}{98}\right)$

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 $

$2$	1.85288	+	0.681876i	1.85288	+	0.681876i	0.981559	+	0.191159i	$0.0612245\pi$
$2$	1.85288	+	0.681876i	1.85288	+	0.681876i	0.871319	+	0.490718i	$0.163265\pi$
$3$		0			0		−0.545535	−	0.838088i	$-0.683673\pi$
$3$		0			0		0.545535	+	0.838088i	$0.316327\pi$
$4$	2.20676	+	1.87864i	2.20676	+	1.87864i
$5$		0			0		0.999486	−	0.0320516i	$-0.0102041\pi$
$5$		0			0		−0.999486	+	0.0320516i	$0.989796\pi$
$6$		0			0
$7$	−0.345365	−	0.938468i	−0.345365	−	0.938468i
$7$	−0.345365	−	0.938468i	−0.345365	−	0.938468i
$8$	1.83899	+	3.26532i	1.83899	+	3.26532i
$9$	−0.404783	+	0.914413i	−0.404783	+	0.914413i
$10$		0			0
$11$	−1.08781	−	1.12326i	−1.08781	−	1.12326i	−0.991790	−	0.127877i	$-0.959184\pi$
$11$	−1.08781	−	1.12326i	−1.08781	−	1.12326i	−0.0960230	−	0.995379i	$-0.530612\pi$
$12$		0			0
$13$		0			0		0.820172	−	0.572117i	$-0.193878\pi$
$13$		0			0		−0.820172	+	0.572117i	$0.806122\pi$
$14$		−	1.97436i		−	1.97436i
$15$		0			0
$16$	0.718349	+	4.44324i	0.718349	+	4.44324i
$17$		0			0		0.886599	−	0.462538i	$-0.153061\pi$
$17$		0			0		−0.886599	+	0.462538i	$0.846939\pi$
$18$	−1.37353	+	1.41828i	−1.37353	+	1.41828i
$19$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$19$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$20$		0			0
$21$		0			0
$22$	−1.24966	−	2.82301i	−1.24966	−	2.82301i
$23$	−0.119739	−	1.24122i	−0.119739	−	1.24122i	−0.838088	−	0.545535i	$-0.816327\pi$
$23$	−0.119739	−	1.24122i	−0.119739	−	1.24122i	0.718349	−	0.695683i	$-0.244898\pi$
$24$		0			0
$25$	0.997945	−	0.0640702i	0.997945	−	0.0640702i
$26$		0			0
$27$		0			0
$28$	1.00091	−	2.71979i	1.00091	−	2.71979i
$29$	−1.96729	−	0.253655i	−1.96729	−	0.253655i	−0.967295	−	0.253655i	$-0.918367\pi$
$29$	−1.96729	−	0.253655i	−1.96729	−	0.253655i	−1.00000			$\pi$
$30$		0			0
$31$		0			0		0.855143	−	0.518393i	$-0.173469\pi$
$31$		0			0		−0.855143	+	0.518393i	$0.826531\pi$
$32$	−0.982350	+	5.04416i	−0.982350	+	5.04416i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−2.61111	+	1.25744i	−2.61111	+	1.25744i
$37$	−0.147642	+	0.913219i	−0.147642	+	0.913219i	0.801414	+	0.598111i	$0.204082\pi$
$37$	−0.147642	+	0.913219i	−0.147642	+	0.913219i	−0.949056	+	0.315108i	$0.897959\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.462538	−	0.886599i	$-0.653061\pi$
$41$		0			0		0.462538	+	0.886599i	$0.346939\pi$
$42$		0			0
$43$	1.43375	−	1.38851i	1.43375	−	1.38851i	0.672301	−	0.740278i	$-0.265306\pi$
$43$	1.43375	−	1.38851i	1.43375	−	1.38851i	0.761446	−	0.648228i	$-0.224490\pi$
$44$	−0.290345	−	4.52236i	−0.290345	−	4.52236i
$45$		0			0
$46$	0.624495	−	2.38147i	0.624495	−	2.38147i
$47$		0			0		−0.284528	−	0.958668i	$-0.591837\pi$
$47$		0			0		0.284528	+	0.958668i	$0.408163\pi$
$48$		0			0
$49$	−0.761446	+	0.648228i	−0.761446	+	0.648228i
$50$	1.89276	+	0.561761i	1.89276	+	0.561761i
$51$		0			0
$52$		0			0
$53$	−0.550444	−	0.144343i	−0.550444	−	0.144343i	−0.0320516	−	0.999486i	$-0.510204\pi$
$53$	−0.550444	−	0.144343i	−0.550444	−	0.144343i	−0.518393	+	0.855143i	$0.673469\pi$
$54$		0			0
$55$		0			0
$56$	2.42928	−	2.85357i	2.42928	−	2.85357i
$57$		0			0
$58$	−3.47220	−	1.81144i	−3.47220	−	1.81144i
$59$		0			0		0.949056	−	0.315108i	$-0.102041\pi$
$59$		0			0		−0.949056	+	0.315108i	$0.897959\pi$
$60$		0			0
$61$		0			0		−0.838088	−	0.545535i	$-0.816327\pi$
$61$		0			0		0.838088	+	0.545535i	$0.183673\pi$
$62$		0			0
$63$	0.997945	+	0.0640702i	0.997945	+	0.0640702i
$64$	−2.92642	+	4.82743i	−2.92642	+	4.82743i
$65$		0			0
$66$		0			0
$67$	−0.719513	+	0.213548i	−0.719513	+	0.213548i	−0.623490	−	0.781831i	$-0.714286\pi$
$67$	−0.719513	+	0.213548i	−0.719513	+	0.213548i	−0.0960230	+	0.995379i	$0.530612\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.18550	+	0.524784i	1.18550	+	0.524784i	0.900969	−	0.433884i	$-0.142857\pi$
$71$	1.18550	+	0.524784i	1.18550	+	0.524784i	0.284528	+	0.958668i	$0.408163\pi$
$72$	−3.73025	+	0.359852i	−3.73025	+	0.359852i
$73$		0			0		−0.987182	−	0.159600i	$-0.948980\pi$
$73$		0			0		0.987182	+	0.159600i	$0.0510204\pi$
$74$	−0.896265	+	1.59141i	−0.896265	+	1.59141i
$75$		0			0
$76$		0			0
$77$	−0.678448	+	1.40881i	−0.678448	+	1.40881i
$78$		0			0
$79$	−0.507049	−	0.0162601i	−0.507049	−	0.0162601i	−0.222521	−	0.974928i	$-0.571429\pi$
$79$	−0.507049	−	0.0162601i	−0.507049	−	0.0162601i	−0.284528	+	0.958668i	$0.591837\pi$
$80$		0			0
$81$	−0.672301	−	0.740278i	−0.672301	−	0.740278i
$82$		0			0
$83$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$83$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$84$		0			0
$85$		0			0
$86$	3.60335	−	1.59509i	3.60335	−	1.59509i
$87$		0			0
$88$	1.66731	−	5.61772i	1.66731	−	5.61772i
$89$		0			0		−0.855143	−	0.518393i	$-0.826531\pi$
$89$		0			0		0.855143	+	0.518393i	$0.173469\pi$
$90$		0			0
$91$		0			0
$92$	2.06756	−	2.96401i	2.06756	−	2.96401i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.801414	−	0.598111i	$-0.795918\pi$
$97$		0			0		0.801414	+	0.598111i	$0.204082\pi$
$98$	−1.85288	+	0.681876i	−1.85288	+	0.681876i
$99$	1.46745	−	0.540035i	1.46745	−	0.540035i
$100$	2.32259	+	1.73339i	2.32259	+	1.73339i
$101$		0			0		0.926917	−	0.375267i	$-0.122449\pi$
$101$		0			0		−0.926917	+	0.375267i	$0.877551\pi$
$102$		0			0
$103$		0			0		0.0640702	−	0.997945i	$-0.479592\pi$
$103$		0			0		−0.0640702	+	0.997945i	$0.520408\pi$
$104$		0			0
$105$		0			0
$106$	−0.921481	−	0.642786i	−0.921481	−	0.642786i
$107$	−1.12689	+	1.24084i	−1.12689	+	1.24084i	−0.159600	+	0.987182i	$0.551020\pi$
$107$	−1.12689	+	1.24084i	−1.12689	+	1.24084i	−0.967295	+	0.253655i	$0.918367\pi$
$108$		0			0
$109$	−0.512701	+	1.72746i	−0.512701	+	1.72746i	0.159600	+	0.987182i	$0.448980\pi$
$109$	−0.512701	+	1.72746i	−0.512701	+	1.72746i	−0.672301	+	0.740278i	$0.734694\pi$
$110$		0			0
$111$		0			0
$112$	3.92175	−	2.20869i	3.92175	−	2.20869i
$113$	1.94085	−	0.442985i	1.94085	−	0.442985i	0.949056	−	0.315108i	$-0.102041\pi$
$113$	1.94085	−	0.442985i	1.94085	−	0.442985i	0.991790	−	0.127877i	$-0.0408163\pi$
$114$		0			0
$115$		0			0
$116$	−3.86481	−	4.25559i	−3.86481	−	4.25559i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.0463159	+	1.44430i	−0.0463159	+	1.44430i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	1.80538	+	0.799189i	1.80538	+	0.799189i
$127$	0.0488111	+	1.52211i	0.0488111	+	1.52211i	0.672301	+	0.740278i	$0.265306\pi$
$127$	0.0488111	+	1.52211i	0.0488111	+	1.52211i	−0.623490	+	0.781831i	$0.714286\pi$
$128$	−4.69622	+	3.74511i	−4.69622	+	3.74511i
$129$		0			0
$130$		0			0
$131$		0			0		0.545535	−	0.838088i	$-0.316327\pi$
$131$		0			0		−0.545535	+	0.838088i	$0.683673\pi$
$132$		0			0
$133$		0			0
$134$	−1.47878	−	0.0949410i	−1.47878	−	0.0949410i
$135$		0			0
$136$		0			0
$137$	0.178011	−	1.84527i	0.178011	−	1.84527i	−0.284528	−	0.958668i	$-0.591837\pi$
$137$	0.178011	−	1.84527i	0.178011	−	1.84527i	0.462538	−	0.886599i	$-0.346939\pi$
$138$		0			0
$139$		0			0		−0.886599	−	0.462538i	$-0.846939\pi$
$139$		0			0		0.886599	+	0.462538i	$0.153061\pi$
$140$		0			0
$141$		0			0
$142$	1.83874	+	1.78072i	1.83874	+	1.78072i
$143$		0			0
$144$	−4.35373	−	1.14168i	−4.35373	−	1.14168i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−2.04142	+	1.73788i	−2.04142	+	1.73788i
$149$	0.189329	−	0.171944i	0.189329	−	0.171944i	−0.572117	−	0.820172i	$-0.693878\pi$
$149$	0.189329	−	0.171944i	0.189329	−	0.171944i	0.761446	+	0.648228i	$0.224490\pi$
$150$		0			0
$151$	−0.303427	+	1.15710i	−0.303427	+	1.15710i	0.623490	+	0.781831i	$0.285714\pi$
$151$	−0.303427	+	1.15710i	−0.303427	+	1.15710i	−0.926917	+	0.375267i	$0.877551\pi$
$152$		0			0
$153$		0			0
$154$	−2.21772	+	2.14774i	−2.21772	+	2.14774i
$155$		0			0
$156$		0			0
$157$		0			0		−0.518393	−	0.855143i	$-0.673469\pi$
$157$		0			0		0.518393	+	0.855143i	$0.326531\pi$
$158$	−0.928412	−	0.375872i	−0.928412	−	0.375872i
$159$		0			0
$160$		0			0
$161$	−1.12349	+	0.541044i	−1.12349	+	0.541044i
$162$	−0.740914	−	1.83007i	−0.740914	−	1.83007i
$163$	0.685059	−	0.0883286i	0.685059	−	0.0883286i	0.222521	−	0.974928i	$-0.428571\pi$
$163$	0.685059	−	0.0883286i	0.685059	−	0.0883286i	0.462538	+	0.886599i	$0.346939\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.253655	−	0.967295i	$-0.581633\pi$
$167$		0			0		0.253655	+	0.967295i	$0.418367\pi$
$168$		0			0
$169$	0.345365	−	0.938468i	0.345365	−	0.938468i
$170$		0			0
$171$		0			0
$172$	5.77243	−	0.370602i	5.77243	−	0.370602i
$173$		0			0		0.462538	−	0.886599i	$-0.346939\pi$
$173$		0			0		−0.462538	+	0.886599i	$0.653061\pi$
$174$		0			0
$175$	−0.404783	−	0.914413i	−0.404783	−	0.914413i
$176$	4.20947	−	5.64031i	4.20947	−	5.64031i
$177$		0			0
$178$		0			0
$179$	0.438431	−	0.452716i	0.438431	−	0.452716i	−0.462538	−	0.886599i	$-0.653061\pi$
$179$	0.438431	−	0.452716i	0.438431	−	0.452716i	0.900969	+	0.433884i	$0.142857\pi$
$180$		0			0
$181$		0			0		−0.159600	−	0.987182i	$-0.551020\pi$
$181$		0			0		0.159600	+	0.987182i	$0.448980\pi$
$182$		0			0
$183$		0			0
$184$	3.83277	−	2.67358i	3.83277	−	2.67358i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.0710286	−	0.311197i	−0.0710286	−	0.311197i	0.926917	−	0.375267i	$-0.122449\pi$
$191$	−0.0710286	−	0.311197i	−0.0710286	−	0.311197i	−0.997945	+	0.0640702i	$0.979592\pi$
$192$		0			0
$193$	−0.146233	−	0.124490i	−0.146233	−	0.124490i	0.572117	−	0.820172i	$-0.306122\pi$
$193$	−0.146233	−	0.124490i	−0.146233	−	0.124490i	−0.718349	+	0.695683i	$0.755102\pi$
$194$		0			0
$195$		0			0
$196$	−2.89811			−2.89811
$197$	0.871319	−	0.490718i	0.871319	−	0.490718i
$197$	0.871319	−	0.490718i	0.871319	−	0.490718i
$198$	3.08724			3.08724
$199$		0			0		−0.938468	−	0.345365i	$-0.887755\pi$
$199$		0			0		0.938468	+	0.345365i	$0.112245\pi$
$200$	2.04443	+	3.14079i	2.04443	+	3.14079i
$201$		0			0
$202$		0			0
$203$	0.441388	+	1.93385i	0.441388	+	1.93385i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.18345	+	0.392934i	1.18345	+	0.392934i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	0.382317i		−	0.382317i	−0.981559	−	0.191159i	$-0.938776\pi$
$211$		−	0.382317i		−	0.382317i	0.981559	−	0.191159i	$-0.0612245\pi$
$212$	−0.943526	−	1.35262i	−0.943526	−	1.35262i
$213$		0			0
$214$	−2.93410	+	1.53072i	−2.93410	+	1.53072i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−2.12789	+	2.85117i	−2.12789	+	2.85117i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.967295	−	0.253655i	$-0.0816327\pi$
$223$		0			0		−0.967295	+	0.253655i	$0.918367\pi$
$224$	5.07306	−	0.820172i	5.07306	−	0.820172i
$225$	−0.345365	+	0.938468i	−0.345365	+	0.938468i
$226$	3.89821	+	0.502619i	3.89821	+	0.502619i
$227$		0			0		−0.253655	−	0.967295i	$-0.581633\pi$
$227$		0			0		0.253655	+	0.967295i	$0.418367\pi$
$228$		0			0
$229$		0			0		0.191159	−	0.981559i	$-0.438776\pi$
$229$		0			0		−0.191159	+	0.981559i	$0.561224\pi$
$230$		0			0
$231$		0			0
$232$	−2.78958	−	6.89032i	−2.78958	−	6.89032i
$233$	−1.29442	+	0.623360i	−1.29442	+	0.623360i	−0.949056	−	0.315108i	$-0.897959\pi$
$233$	−1.29442	+	0.623360i	−1.29442	+	0.623360i	−0.345365	+	0.938468i	$0.612245\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.0537241	−	0.0349705i	0.0537241	−	0.0349705i	−0.518393	−	0.855143i	$-0.673469\pi$
$239$	0.0537241	−	0.0349705i	0.0537241	−	0.0349705i	0.572117	+	0.820172i	$0.306122\pi$
$240$		0			0
$241$		0			0		−0.0640702	−	0.997945i	$-0.520408\pi$
$241$		0			0		0.0640702	+	0.997945i	$0.479592\pi$
$242$	−1.07065	+	2.64453i	−1.07065	+	2.64453i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.981559	−	0.191159i	$-0.938776\pi$
$251$		0			0		0.981559	+	0.191159i	$0.0612245\pi$
$252$	2.08186	+	2.01617i	2.08186	+	2.01617i
$253$	−1.26395	+	1.48471i	−1.26395	+	1.48471i
$254$	−0.947449	+	2.85357i	−0.947449	+	2.85357i
$255$		0			0
$256$	−5.89765	+	1.95815i	−5.89765	+	1.95815i
$257$		0			0		0.0960230	−	0.995379i	$-0.469388\pi$
$257$		0			0		−0.0960230	+	0.995379i	$0.530612\pi$
$258$		0			0
$259$	0.908017	−	0.176836i	0.908017	−	0.176836i
$260$		0			0
$261$	1.02827	−	1.69624i	1.02827	−	1.69624i
$262$		0			0
$263$	−0.535407	+	0.822529i	−0.535407	+	0.822529i	−0.997945	−	0.0640702i	$-0.979592\pi$
$263$	−0.535407	+	0.822529i	−0.535407	+	0.822529i	0.462538	+	0.886599i	$0.346939\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.98897	−	0.880457i	−1.98897	−	0.880457i
$269$		0			0		0.995379	−	0.0960230i	$-0.0306122\pi$
$269$		0			0		−0.995379	+	0.0960230i	$0.969388\pi$
$270$		0			0
$271$		0			0		0.490718	−	0.871319i	$-0.336735\pi$
$271$		0			0		−0.490718	+	0.871319i	$0.663265\pi$
$272$		0			0
$273$		0			0
$274$	1.58808	−	3.29767i	1.58808	−	3.29767i
$275$	−1.15755	−	1.05125i	−1.15755	−	1.05125i
$276$		0			0
$277$	−0.0244952	−	0.125777i	−0.0244952	−	0.125777i	0.967295	−	0.253655i	$-0.0816327\pi$
$277$	−0.0244952	−	0.125777i	−0.0244952	−	0.125777i	−0.991790	+	0.127877i	$0.959184\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.06371	−	0.242786i	1.06371	−	0.242786i	0.345365	−	0.938468i	$-0.387755\pi$
$281$	1.06371	−	0.242786i	1.06371	−	0.242786i	0.718349	+	0.695683i	$0.244898\pi$
$282$		0			0
$283$		0			0		0.914413	−	0.404783i	$-0.132653\pi$
$283$		0			0		−0.914413	+	0.404783i	$0.867347\pi$
$284$	1.63022	+	3.38519i	1.63022	+	3.38519i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−4.21481	−	2.94007i	−4.21481	−	2.94007i
$289$	0.572117	−	0.820172i	0.572117	−	0.820172i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.926917	−	0.375267i	$-0.122449\pi$
$293$		0			0		−0.926917	+	0.375267i	$0.877551\pi$
$294$		0			0
$295$		0			0
$296$	−3.25347	+	1.19731i	−3.25347	+	1.19731i
$297$		0			0
$298$	0.468048	−	0.189492i	0.468048	−	0.189492i
$299$		0			0
$300$		0			0
$301$	−1.79824	−	0.865985i	−1.79824	−	0.865985i
$302$	−1.35121	+	1.93706i	−1.35121	+	1.93706i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$307$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$308$	−4.14382	+	1.83434i	−4.14382	+	1.83434i
$309$		0			0
$310$		0			0
$311$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$311$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$312$		0			0
$313$		0			0		−0.672301	−	0.740278i	$-0.734694\pi$
$313$		0			0		0.672301	+	0.740278i	$0.265306\pi$
$314$		0			0
$315$		0			0
$316$	−1.08839	−	0.988443i	−1.08839	−	0.988443i
$317$	−0.711719	+	1.47790i	−0.711719	+	1.47790i	0.159600	+	0.987182i	$0.448980\pi$
$317$	−0.711719	+	1.47790i	−0.711719	+	1.47790i	−0.871319	+	0.490718i	$0.836735\pi$
$318$		0			0
$319$	1.85513	+	2.48571i	1.85513	+	2.48571i
$320$		0			0
$321$		0			0
$322$	−2.45061	+	0.236408i	−2.45061	+	0.236408i
$323$		0			0
$324$	−0.0928890	−	2.89662i	−0.0928890	−	2.89662i
$325$		0			0
$326$	1.32956	+	0.303463i	1.32956	+	0.303463i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.14188	−	0.0733113i	−1.14188	−	0.0733113i	−0.518393	−	0.855143i	$-0.673469\pi$
$331$	−1.14188	−	0.0733113i	−1.14188	−	0.0733113i	−0.623490	+	0.781831i	$0.714286\pi$
$332$		0			0
$333$	−0.775296	−	0.504662i	−0.775296	−	0.504662i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.604168	+	1.81966i	−0.604168	+	1.81966i	−0.0320516	+	0.999486i	$0.510204\pi$
$337$	−0.604168	+	1.81966i	−0.604168	+	1.81966i	−0.572117	+	0.820172i	$0.693878\pi$
$338$	1.27984	−	1.50337i	1.27984	−	1.50337i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.871319	+	0.490718i	0.871319	+	0.490718i
$344$	7.17057	+	2.12819i	7.17057	+	2.12819i
$345$		0			0
$346$		0			0
$347$	0.550444	+	1.85463i	0.550444	+	1.85463i	0.518393	+	0.855143i	$0.326531\pi$
$347$	0.550444	+	1.85463i	0.550444	+	1.85463i	0.0320516	+	0.999486i	$0.489796\pi$
$348$		0			0
$349$		0			0		0.375267	−	0.926917i	$-0.377551\pi$
$349$		0			0		−0.375267	+	0.926917i	$0.622449\pi$
$350$	−0.126498	−	1.97031i	−0.126498	−	1.97031i
$351$		0			0
$352$	6.73450	−	4.38367i	6.73450	−	4.38367i
$353$		0			0		−0.462538	−	0.886599i	$-0.653061\pi$
$353$		0			0		0.462538	+	0.886599i	$0.346939\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	1.12105	−	0.539871i	1.12105	−	0.539871i
$359$	−0.641814	−	1.58529i	−0.641814	−	1.58529i	−0.801414	−	0.598111i	$-0.795918\pi$
$359$	−0.641814	−	1.58529i	−0.641814	−	1.58529i	0.159600	−	0.987182i	$-0.448980\pi$
$360$		0			0
$361$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.987182	−	0.159600i	$-0.0510204\pi$
$367$		0			0		−0.987182	+	0.159600i	$0.948980\pi$
$368$	5.42902	−	1.42366i	5.42902	−	1.42366i
$369$		0			0
$370$		0			0
$371$	0.0546424	+	0.566426i	0.0546424	+	0.566426i
$372$		0			0
$373$	0.0766421	−	0.102693i	0.0766421	−	0.102693i	−0.761446	−	0.648228i	$-0.775510\pi$
$373$	0.0766421	−	0.102693i	0.0766421	−	0.102693i	0.838088	+	0.545535i	$0.183673\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.0366745	−	0.0525756i	−0.0366745	−	0.0525756i	0.801414	−	0.598111i	$-0.204082\pi$
$379$	−0.0366745	−	0.0525756i	−0.0366745	−	0.0525756i	−0.838088	+	0.545535i	$0.816327\pi$
$380$		0			0
$381$		0			0
$382$	0.0805903	−	0.625042i	0.0805903	−	0.625042i
$383$		0			0		−0.695683	−	0.718349i	$-0.744898\pi$
$383$		0			0		0.695683	+	0.718349i	$0.255102\pi$
$384$		0			0
$385$		0			0
$386$	−0.186065	−	0.330377i	−0.186065	−	0.330377i
$387$	0.689311	+	1.87308i	0.689311	+	1.87308i
$388$		0			0
$389$	−1.82789	+	0.0586167i	−1.82789	+	0.0586167i	−0.926917	−	0.375267i	$-0.877551\pi$
$389$	−1.82789	+	0.0586167i	−1.82789	+	0.0586167i	−0.900969	+	0.433884i	$0.857143\pi$
$390$		0			0
$391$		0			0
$392$	−3.51697	−	1.29428i	−3.51697	−	1.29428i
$393$		0			0
$394$	1.94906	−	0.315108i	1.94906	−	0.315108i
$395$		0			0
$396$	4.25283	+	1.56508i	4.25283	+	1.56508i
$397$		0			0		−0.545535	−	0.838088i	$-0.683673\pi$
$397$		0			0		0.545535	+	0.838088i	$0.316327\pi$
$398$		0			0
$399$		0			0
$400$	1.00155	+	4.38809i	1.00155	+	4.38809i
$401$	−0.640249	−	1.73976i	−0.640249	−	1.73976i	−0.672301	−	0.740278i	$-0.734694\pi$
$401$	−0.640249	−	1.73976i	−0.640249	−	1.73976i	0.0320516	−	0.999486i	$-0.489796\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−0.500806	+	3.88416i	−0.500806	+	3.88416i
$407$	1.18639	−	0.827571i	1.18639	−	0.827571i
$408$		0			0
$409$		0			0		−0.572117	−	0.820172i	$-0.693878\pi$
$409$		0			0		0.572117	+	0.820172i	$0.306122\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	1.92486	+	1.53503i	1.92486	+	1.53503i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.997945	−	0.0640702i	$-0.0204082\pi$
$419$		0			0		−0.997945	+	0.0640702i	$0.979592\pi$
$420$		0			0
$421$	1.85288	−	0.299559i	1.85288	−	0.299559i	0.871319	−	0.490718i	$-0.163265\pi$
$421$	1.85288	−	0.299559i	1.85288	−	0.299559i	0.981559	+	0.191159i	$0.0612245\pi$
$422$	0.260693	−	0.708387i	0.260693	−	0.708387i
$423$		0			0
$424$	−0.540936	−	2.06282i	−0.540936	−	2.06282i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−4.81786	+	0.621195i	−4.81786	+	0.621195i
$429$		0			0
$430$		0			0
$431$	−0.129207	+	0.799189i	−0.129207	+	0.799189i	0.838088	+	0.545535i	$0.183673\pi$
$431$	−0.129207	+	0.799189i	−0.129207	+	0.799189i	−0.967295	+	0.253655i	$0.918367\pi$
$432$		0			0
$433$		0			0		−0.926917	−	0.375267i	$-0.877551\pi$
$433$		0			0		0.926917	+	0.375267i	$0.122449\pi$
$434$		0			0
$435$		0			0
$436$	−4.37668	+	2.84890i	−4.37668	+	2.84890i
$437$		0			0
$438$		0			0
$439$		0			0		0.375267	−	0.926917i	$-0.377551\pi$
$439$		0			0		−0.375267	+	0.926917i	$0.622449\pi$
$440$		0			0
$441$	−0.284528	−	0.958668i	−0.284528	−	0.958668i
$442$		0			0
$443$	1.02384	−	0.871609i	1.02384	−	0.871609i	0.0320516	−	0.999486i	$-0.489796\pi$
$443$	1.02384	−	0.871609i	1.02384	−	0.871609i	0.991790	+	0.127877i	$0.0408163\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	5.54107	+	1.07912i	5.54107	+	1.07912i
$449$	1.38971	+	1.34586i	1.38971	+	1.34586i	0.871319	+	0.490718i	$0.163265\pi$
$449$	1.38971	+	1.34586i	1.38971	+	1.34586i	0.518393	+	0.855143i	$0.326531\pi$
$450$	−1.27984	+	1.50337i	−1.27984	+	1.50337i
$451$		0			0
$452$	5.11518	+	2.66859i	5.11518	+	2.66859i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.191651	+	0.0123044i	0.191651	+	0.0123044i	0.159600	−	0.987182i	$-0.448980\pi$
$457$	0.191651	+	0.0123044i	0.191651	+	0.0123044i	0.0320516	+	0.999486i	$0.489796\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.958668	−	0.284528i	$-0.0918367\pi$
$461$		0			0		−0.958668	+	0.284528i	$0.908163\pi$
$462$		0			0
$463$	1.56286	−	1.24634i	1.56286	−	1.24634i	0.761446	−	0.648228i	$-0.224490\pi$
$463$	1.56286	−	1.24634i	1.56286	−	1.24634i	0.801414	−	0.598111i	$-0.204082\pi$
$464$	−0.286156	−	8.92339i	−0.286156	−	8.92339i
$465$		0			0
$466$	−2.82346	+	0.272376i	−2.82346	+	0.272376i
$467$		0			0		−0.987182	−	0.159600i	$-0.948980\pi$
$467$		0			0		0.987182	+	0.159600i	$0.0510204\pi$
$468$		0			0
$469$	0.448902	+	0.601488i	0.448902	+	0.601488i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.11930	−	0.100030i	−3.11930	−	0.100030i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.354800	−	0.444905i	0.354800	−	0.444905i
$478$	0.123390	−	0.0281629i	0.123390	−	0.0281629i
$479$		0			0		0.871319	−	0.490718i	$-0.163265\pi$
$479$		0			0		−0.871319	+	0.490718i	$0.836735\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−2.81552	+	3.10020i	−2.81552	+	3.10020i
$485$		0			0
$486$		0			0
$487$	−0.287589	−	0.138496i	−0.287589	−	0.138496i	0.284528	−	0.958668i	$-0.408163\pi$
$487$	−0.287589	−	0.138496i	−0.287589	−	0.138496i	−0.572117	+	0.820172i	$0.693878\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.22047	−	0.910858i	−1.22047	−	0.910858i	−0.222521	−	0.974928i	$-0.571429\pi$
$491$	−1.22047	−	0.910858i	−1.22047	−	0.910858i	−0.997945	+	0.0640702i	$0.979592\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.0830643	−	1.29379i	0.0830643	−	1.29379i
$498$		0			0
$499$	0.996992	−	1.42926i	0.996992	−	1.42926i	0.0960230	−	0.995379i	$-0.469388\pi$
$499$	0.996992	−	1.42926i	0.996992	−	1.42926i	0.900969	−	0.433884i	$-0.142857\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.284528	−	0.958668i	$-0.408163\pi$
$503$		0			0		−0.284528	+	0.958668i	$0.591837\pi$
$504$	1.62601	+	3.37644i	1.62601	+	3.37644i
$505$		0			0
$506$	−3.35434	+	1.88913i	−3.35434	+	1.88913i
$507$		0			0
$508$	−2.75178	+	3.45062i	−2.75178	+	3.45062i
$509$		0			0		−0.127877	−	0.991790i	$-0.540816\pi$
$509$		0			0		0.127877	+	0.991790i	$0.459184\pi$
$510$		0			0
$511$		0			0
$512$	−6.25923	−	0.200721i	−6.25923	−	0.200721i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	1.80303	+	0.291499i	1.80303	+	0.291499i
$519$		0			0
$520$		0			0
$521$		0			0		−0.0320516	−	0.999486i	$-0.510204\pi$
$521$		0			0		0.0320516	+	0.999486i	$0.489796\pi$
$522$	3.06189	−	2.44178i	3.06189	−	2.44178i
$523$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$523$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$524$		0			0
$525$		0			0
$526$	−1.55291	+	1.15896i	−1.55291	+	1.15896i
$527$		0			0
$528$		0			0
$529$	−0.544724	+	0.106085i	−0.544724	+	0.106085i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−2.02048	−	1.95673i	−2.02048	−	1.95673i
$537$		0			0
$538$		0			0
$539$	1.55644	+	0.150148i	1.55644	+	0.150148i
$540$		0			0
$541$	−0.831901	−	0.246904i	−0.831901	−	0.246904i	−0.159600	−	0.987182i	$-0.551020\pi$
$541$	−0.831901	−	0.246904i	−0.831901	−	0.246904i	−0.672301	+	0.740278i	$0.734694\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.0830643	−	1.29379i	−0.0830643	−	1.29379i	−0.801414	−	0.598111i	$-0.795918\pi$
$547$	−0.0830643	−	1.29379i	−0.0830643	−	1.29379i	0.718349	−	0.695683i	$-0.244898\pi$
$548$	3.85942	−	3.73764i	3.85942	−	3.73764i
$549$		0			0
$550$	−1.42797	−	2.73714i	−1.42797	−	2.73714i
$551$		0			0
$552$		0			0
$553$	0.159857	+	0.481465i	0.159857	+	0.481465i
$554$	0.0403781	−	0.249753i	0.0403781	−	0.249753i
$555$		0			0
$556$		0			0
$557$	1.42490	−	0.183721i	1.42490	−	0.183721i	0.623490	−	0.781831i	$-0.285714\pi$
$557$	1.42490	−	0.183721i	1.42490	−	0.183721i	0.801414	+	0.598111i	$0.204082\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	2.13648	+	0.275469i	2.13648	+	0.275469i
$563$		0			0		0.345365	−	0.938468i	$-0.387755\pi$
$563$		0			0		−0.345365	+	0.938468i	$0.612245\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.462538	+	0.886599i	−0.462538	+	0.886599i
$568$	0.466533	+	4.83610i	0.466533	+	4.83610i
$569$	0.419673	+	0.948049i	0.419673	+	0.948049i	0.991790	+	0.127877i	$0.0408163\pi$
$569$	0.419673	+	0.948049i	0.419673	+	0.948049i	−0.572117	+	0.820172i	$0.693878\pi$
$570$		0			0
$571$	−0.396630	−	0.316302i	−0.396630	−	0.316302i	0.404783	−	0.914413i	$-0.367347\pi$
$571$	−0.396630	−	0.316302i	−0.396630	−	0.316302i	−0.801414	+	0.598111i	$0.795918\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.199018	−	1.23100i	−0.199018	−	1.23100i
$576$	−3.22970	−	4.63001i	−3.22970	−	4.63001i
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	1.61932	−	1.12957i	1.61932	−	1.12957i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	0.436646	+	0.775309i	0.436646	+	0.775309i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.761446	−	0.648228i	$-0.775510\pi$
$587$		0			0		0.761446	+	0.648228i	$0.224490\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−4.16371			−4.16371
$593$		0			0		−0.938468	−	0.345365i	$-0.887755\pi$
$593$		0			0		0.938468	+	0.345365i	$0.112245\pi$
$594$		0			0
$595$		0			0
$596$	0.740824	−	0.0237568i	0.740824	−	0.0237568i
$597$		0			0
$598$		0			0
$599$	−0.726535	−	1.29004i	−0.726535	−	1.29004i	−0.949056	−	0.315108i	$-0.897959\pi$
$599$	−0.726535	−	1.29004i	−0.726535	−	1.29004i	0.222521	−	0.974928i	$-0.428571\pi$
$600$		0			0
$601$		0			0		−0.949056	−	0.315108i	$-0.897959\pi$
$601$		0			0		0.949056	+	0.315108i	$0.102041\pi$
$602$	−2.74142	−	2.83074i	−2.74142	−	2.83074i
$603$	0.0959762	−	0.744372i	0.0959762	−	0.744372i
$604$	−2.84336	+	1.98340i	−2.84336	+	1.98340i
$605$		0			0
$606$		0			0
$607$		0			0		−0.159600	−	0.987182i	$-0.551020\pi$
$607$		0			0		0.159600	+	0.987182i	$0.448980\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.794638	+	1.79510i	0.794638	+	1.79510i	0.572117	+	0.820172i	$0.306122\pi$
$613$	0.794638	+	1.79510i	0.794638	+	1.79510i	0.222521	+	0.974928i	$0.428571\pi$
$614$		0			0
$615$		0			0
$616$	−5.84788	+	0.375447i	−5.84788	+	0.375447i
$617$	1.00288	−	0.262985i	1.00288	−	0.262985i	0.284528	−	0.958668i	$-0.408163\pi$
$617$	1.00288	−	0.262985i	1.00288	−	0.262985i	0.718349	+	0.695683i	$0.244898\pi$
$618$		0			0
$619$		0			0		0.345365	−	0.938468i	$-0.387755\pi$
$619$		0			0		−0.345365	+	0.938468i	$0.612245\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.991790	−	0.127877i	0.991790	−	0.127877i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.830894	−	1.37065i	−0.830894	−	1.37065i	−0.926917	−	0.375267i	$-0.877551\pi$
$631$	−0.830894	−	1.37065i	−0.830894	−	1.37065i	0.0960230	−	0.995379i	$-0.469388\pi$
$632$	−0.879365	−	1.68558i	−0.879365	−	1.68558i
$633$		0			0
$634$	−2.32647	+	2.25306i	−2.32647	+	2.25306i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	1.74239	+	5.87068i	1.74239	+	5.87068i
$639$	−0.959738	+	0.871609i	−0.959738	+	0.871609i
$640$		0			0
$641$	−0.245183	−	0.0727692i	−0.245183	−	0.0727692i	0.159600	−	0.987182i	$-0.448980\pi$
$641$	−0.245183	−	0.0727692i	−0.245183	−	0.0727692i	−0.404783	+	0.914413i	$0.632653\pi$
$642$		0			0
$643$		0			0		−0.995379	−	0.0960230i	$-0.969388\pi$
$643$		0			0		0.995379	+	0.0960230i	$0.0306122\pi$
$644$	−3.49569	−	0.916679i	−3.49569	−	0.916679i
$645$		0			0
$646$		0			0
$647$		0			0		0.648228	−	0.761446i	$-0.275510\pi$
$647$		0			0		−0.648228	+	0.761446i	$0.724490\pi$
$648$	1.18089	−	3.55665i	1.18089	−	3.55665i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	1.67770	+	1.09206i	1.67770	+	1.09206i
$653$	−1.76871	+	0.344456i	−1.76871	+	0.344456i	−0.967295	−	0.253655i	$-0.918367\pi$
$653$	−1.76871	+	0.344456i	−1.76871	+	0.344456i	−0.801414	+	0.598111i	$0.795918\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.82988	−	0.417658i	−1.82988	−	0.417658i	−0.838088	−	0.545535i	$-0.816327\pi$
$659$	−1.82988	−	0.417658i	−1.82988	−	0.417658i	−0.991790	+	0.127877i	$0.959184\pi$
$660$		0			0
$661$		0			0		−0.0320516	−	0.999486i	$-0.510204\pi$
$661$		0			0		0.0320516	+	0.999486i	$0.489796\pi$
$662$	−2.06578	−	0.914459i	−2.06578	−	0.914459i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−1.09241	−	1.46373i	−1.09241	−	1.46373i
$667$	−0.0792791	+	2.47221i	−0.0792791	+	2.47221i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.110968	−	0.860643i	−0.110968	−	0.860643i	−0.949056	−	0.315108i	$-0.897959\pi$
$673$	−0.110968	−	0.860643i	−0.110968	−	0.860643i	0.838088	−	0.545535i	$-0.183673\pi$
$674$	−2.36023	+	2.95964i	−2.36023	+	2.95964i
$675$		0			0
$676$	2.52518	−	1.42215i	2.52518	−	1.42215i
$677$		0			0		0.914413	−	0.404783i	$-0.132653\pi$
$677$		0			0		−0.914413	+	0.404783i	$0.867347\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.593162	−	0.850342i	0.593162	−	0.850342i	−0.404783	−	0.914413i	$-0.632653\pi$
$683$	0.593162	−	0.850342i	0.593162	−	0.850342i	0.997945	+	0.0640702i	$0.0204082\pi$
$684$		0			0
$685$		0			0
$686$	1.27984	+	1.50337i	1.27984	+	1.50337i
$687$		0			0
$688$	7.19940	+	5.37306i	7.19940	+	5.37306i
$689$		0			0
$690$		0			0
$691$		0			0		−0.801414	−	0.598111i	$-0.795918\pi$
$691$		0			0		0.801414	+	0.598111i	$0.204082\pi$
$692$		0			0
$693$	−1.01361	−	1.19064i	−1.01361	−	1.19064i
$694$	−0.244722	+	3.81174i	−0.244722	+	3.81174i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.824593	−	2.77833i	0.824593	−	2.77833i
$701$	−0.867322	−	1.80101i	−0.867322	−	1.80101i	−0.462538	−	0.886599i	$-0.653061\pi$
$701$	−0.867322	−	1.80101i	−0.867322	−	1.80101i	−0.404783	−	0.914413i	$-0.632653\pi$
$702$		0			0
$703$		0			0
$704$	8.60583	−	1.96422i	8.60583	−	1.96422i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.82789	+	0.0586167i	1.82789	+	0.0586167i	0.926917	−	0.375267i	$-0.122449\pi$
$709$	1.82789	+	0.0586167i	1.82789	+	0.0586167i	0.900969	+	0.433884i	$0.142857\pi$
$710$		0			0
$711$	0.220113	−	0.457070i	0.220113	−	0.457070i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	1.81800	−	0.175380i	1.81800	−	0.175380i
$717$		0			0
$718$	−0.108229	−	3.37499i	−0.108229	−	3.37499i
$719$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$719$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$720$		0			0
$721$		0			0
$722$	−1.07708	+	1.65469i	−1.07708	+	1.65469i
$723$		0			0
$724$		0			0
$725$	−1.97950	−	0.127088i	−1.97950	−	0.127088i
$726$		0			0
$727$		0			0		−0.838088	−	0.545535i	$-0.816327\pi$
$727$		0			0		0.838088	+	0.545535i	$0.183673\pi$
$728$		0			0
$729$	0.949056	−	0.315108i	0.949056	−	0.315108i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.718349	−	0.695683i	$-0.755102\pi$
$733$		0			0		0.718349	+	0.695683i	$0.244898\pi$
$734$		0			0
$735$		0			0
$736$	6.37853	+	0.615329i	6.37853	+	0.615329i
$737$	1.02256	+	0.575897i	1.02256	+	0.575897i
$738$		0			0
$739$	0.525954	−	0.447751i	0.525954	−	0.447751i	−0.345365	−	0.938468i	$-0.612245\pi$
$739$	0.525954	−	0.447751i	0.525954	−	0.447751i	0.871319	+	0.490718i	$0.163265\pi$
$740$		0			0
$741$		0			0
$742$	−0.284986	+	1.08678i	−0.284986	+	1.08678i
$743$	0.0480869	−	0.118776i	0.0480869	−	0.118776i	−0.900969	−	0.433884i	$-0.857143\pi$
$743$	0.0480869	−	0.118776i	0.0480869	−	0.118776i	0.949056	+	0.315108i	$0.102041\pi$
$744$		0			0
$745$		0			0
$746$	0.212033	−	0.138018i	0.212033	−	0.138018i
$747$		0			0
$748$		0			0
$749$	1.55368	+	0.629014i	1.55368	+	0.629014i
$750$		0			0
$751$	0.308760	−	1.90979i	0.308760	−	1.90979i	−0.0960230	−	0.995379i	$-0.530612\pi$
$751$	0.308760	−	1.90979i	0.308760	−	1.90979i	0.404783	−	0.914413i	$-0.367347\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.46254	+	0.886599i	−1.46254	+	0.886599i	−0.462538	+	0.886599i	$0.653061\pi$
$757$	−1.46254	+	0.886599i	−1.46254	+	0.886599i	−1.00000			$\pi$
$758$	−0.0321033	−	0.122424i	−0.0321033	−	0.122424i
$759$		0			0
$760$		0			0
$761$		0			0		0.987182	−	0.159600i	$-0.0510204\pi$
$761$		0			0		−0.987182	+	0.159600i	$0.948980\pi$
$762$		0			0
$763$	1.79824	−	0.115451i	1.79824	−	0.115451i
$764$	0.427883	−	0.820172i	0.427883	−	0.820172i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$769$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$770$		0			0
$771$		0			0
$772$	−0.0888287	−	0.549437i	−0.0888287	−	0.549437i
$773$		0			0		−0.572117	−	0.820172i	$-0.693878\pi$
$773$		0			0		0.572117	+	0.820172i	$0.306122\pi$
$774$			3.94061i			3.94061i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−3.42682	−	1.13778i	−3.42682	−	1.13778i
$779$		0			0
$780$		0			0
$781$	−0.700131	−	1.90248i	−0.700131	−	1.90248i
$782$		0			0
$783$		0			0
$784$	−3.42722	−	2.91764i	−3.42722	−	2.91764i
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	2.84467	+	0.553999i	2.84467	+	0.553999i
$789$		0			0
$790$		0			0
$791$	−1.08603	−	1.66843i	−1.08603	−	1.66843i
$792$	4.46202	+	3.79857i	4.46202	+	3.79857i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.404783	−	0.914413i	$-0.367347\pi$
$797$		0			0		−0.404783	+	0.914413i	$0.632653\pi$
$798$		0			0
$799$		0			0
$800$	−0.657151	+	5.09674i	−0.657151	+	5.09674i
$801$		0			0
$802$		−	3.66014i		−	3.66014i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.228668	+	0.306394i	−0.228668	+	0.306394i	−0.900969	−	0.433884i	$-0.857143\pi$
$809$	−0.228668	+	0.306394i	−0.228668	+	0.306394i	0.672301	+	0.740278i	$0.265306\pi$
$810$		0			0
$811$		0			0		−0.0960230	−	0.995379i	$-0.530612\pi$
$811$		0			0		0.0960230	+	0.995379i	$0.469388\pi$
$812$	−2.65896	+	5.09674i	−2.65896	+	5.09674i
$813$		0			0
$814$	2.76253	−	0.724420i	2.76253	−	0.724420i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.0710286	+	0.311197i	−0.0710286	+	0.311197i	−0.997945	−	0.0640702i	$-0.979592\pi$
$821$	−0.0710286	+	0.311197i	−0.0710286	+	0.311197i	0.926917	+	0.375267i	$0.122449\pi$
$822$		0			0
$823$	−0.731717	−	1.80735i	−0.731717	−	1.80735i	−0.572117	−	0.820172i	$-0.693878\pi$
$823$	−0.731717	−	1.80735i	−0.731717	−	1.80735i	−0.159600	−	0.987182i	$-0.551020\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.178011	+	0.0720685i	0.178011	+	0.0720685i	0.462538	−	0.886599i	$-0.346939\pi$
$827$	0.178011	+	0.0720685i	0.178011	+	0.0720685i	−0.284528	+	0.958668i	$0.591837\pi$
$828$	1.87341	+	3.09039i	1.87341	+	3.09039i
$829$		0			0		−0.462538	−	0.886599i	$-0.653061\pi$
$829$		0			0		0.462538	+	0.886599i	$0.346939\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.871319	−	0.490718i	$-0.836735\pi$
$839$		0			0		0.871319	+	0.490718i	$0.163265\pi$
$840$		0			0
$841$	2.83861	+	0.744372i	2.83861	+	0.744372i
$842$	3.63742	+	0.708387i	3.63742	+	0.708387i
$843$		0			0
$844$	0.718236	−	0.843681i	0.718236	−	0.843681i
$845$		0			0
$846$		0			0
$847$	1.37143	−	0.455345i	1.37143	−	0.455345i
$848$	0.245942	−	2.54945i	0.245942	−	2.54945i
$849$		0			0
$850$		0			0
$851$	1.15118	+	0.0739083i	1.15118	+	0.0739083i
$852$		0			0
$853$		0			0		0.801414	−	0.598111i	$-0.204082\pi$
$853$		0			0		−0.801414	+	0.598111i	$0.795918\pi$
$854$		0			0
$855$		0			0
$856$	−6.12408	−	1.39778i	−6.12408	−	1.39778i
$857$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$857$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$858$		0			0
$859$		0			0		−0.914413	−	0.404783i	$-0.867347\pi$
$859$		0			0		0.914413	+	0.404783i	$0.132653\pi$
$860$		0			0
$861$		0			0
$862$	−0.784353	+	1.39270i	−0.784353	+	1.39270i
$863$	−0.519021	−	0.695441i	−0.519021	−	0.695441i	0.462538	−	0.886599i	$-0.346939\pi$
$863$	−0.519021	−	0.695441i	−0.519021	−	0.695441i	−0.981559	+	0.191159i	$0.938776\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	0.533310	+	0.587233i	0.533310	+	0.587233i
$870$		0			0
$871$		0			0
$872$	−6.58357	+	1.50266i	−6.58357	+	1.50266i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.46254	−	0.886599i	−1.46254	−	0.886599i	−0.462538	−	0.886599i	$-0.653061\pi$
$877$	−1.46254	−	0.886599i	−1.46254	−	0.886599i	−1.00000			$\pi$
$878$		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	$0.142857\pi$
$882$	0.126498	−	1.97031i	0.126498	−	1.97031i
$883$	−1.18550	−	1.39255i	−1.18550	−	1.39255i	−0.900969	−	0.433884i	$-0.857143\pi$
$883$	−1.18550	−	1.39255i	−1.18550	−	1.39255i	−0.284528	−	0.958668i	$-0.591837\pi$
$884$		0			0
$885$		0			0
$886$	2.49138	−	0.916852i	2.49138	−	0.916852i
$887$		0			0		0.938468	−	0.345365i	$-0.112245\pi$
$887$		0			0		−0.938468	+	0.345365i	$0.887755\pi$
$888$		0			0
$889$	1.41159	−	0.571491i	1.41159	−	0.571491i
$890$		0			0
$891$	−0.100184	+	1.56045i	−0.100184	+	1.56045i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	5.13658	+	3.11383i	5.13658	+	3.11383i
$897$		0			0
$898$	1.65726	+	3.44133i	1.65726	+	3.44133i
$899$		0			0
$900$	−2.52518	+	1.42215i	−2.52518	+	1.42215i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	5.01569	+	5.52284i	5.01569	+	5.52284i
$905$		0			0
$906$		0			0
$907$	0.642389	+	0.583401i	0.642389	+	0.583401i	0.926917	−	0.375267i	$-0.122449\pi$
$907$	0.642389	+	0.583401i	0.642389	+	0.583401i	−0.284528	+	0.958668i	$0.591837\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.940870	+	1.67061i	−0.940870	+	1.67061i	−0.222521	+	0.974928i	$0.571429\pi$
$911$	−0.940870	+	1.67061i	−0.940870	+	1.67061i	−0.718349	+	0.695683i	$0.755102\pi$
$912$		0			0
$913$		0			0
$914$	0.346717	+	0.153481i	0.346717	+	0.153481i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	0.343805	−	0.528177i	0.343805	−	0.528177i	−0.623490	−	0.781831i	$-0.714286\pi$
$919$	0.343805	−	0.528177i	0.343805	−	0.528177i	0.967295	+	0.253655i	$0.0816327\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.0888287	+	0.920802i	−0.0888287	+	0.920802i
$926$	3.74564	−	1.24364i	3.74564	−	1.24364i
$927$		0			0
$928$	3.21205	−	9.67417i	3.21205	−	9.67417i
$929$		0			0		0.648228	−	0.761446i	$-0.275510\pi$
$929$		0			0		−0.648228	+	0.761446i	$0.724490\pi$
$930$		0			0
$931$		0			0
$932$	−4.02754	−	1.05614i	−4.02754	−	1.05614i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.740278	−	0.672301i	$-0.234694\pi$
$937$		0			0		−0.740278	+	0.672301i	$0.765306\pi$
$938$	0.421621	+	1.42058i	0.421621	+	1.42058i
$939$		0			0
$940$		0			0
$941$		0			0		−0.0640702	−	0.997945i	$-0.520408\pi$
$941$		0			0		0.0640702	+	0.997945i	$0.479592\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−5.71147	−	2.31232i	−5.71147	−	2.31232i
$947$	−0.376939	−	1.13528i	−0.376939	−	1.13528i	−0.949056	−	0.315108i	$-0.897959\pi$
$947$	−0.376939	−	1.13528i	−0.376939	−	1.13528i	0.572117	−	0.820172i	$-0.306122\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.372732	+	1.91390i	−0.372732	+	1.91390i	0.0320516	+	0.999486i	$0.489796\pi$
$953$	−0.372732	+	1.91390i	−0.372732	+	1.91390i	−0.404783	+	0.914413i	$0.632653\pi$
$954$	0.960772	−	0.582425i	0.960772	−	0.582425i
$955$		0			0
$956$	0.184253	+	0.0237568i	0.184253	+	0.0237568i
$957$		0			0
$958$		0			0
$959$	−1.79320	+	0.470233i	−1.79320	+	0.470233i
$960$		0			0
$961$	0.462538	−	0.886599i	0.462538	−	0.886599i
$962$		0			0
$963$	−0.678488	−	1.53272i	−0.678488	−	1.53272i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.759038	+	0.783769i	−0.759038	+	0.783769i	−0.981559	−	0.191159i	$-0.938776\pi$
$967$	−0.759038	+	0.783769i	−0.759038	+	0.783769i	0.222521	+	0.974928i	$0.428571\pi$
$968$	−4.80128	+	2.50482i	−4.80128	+	2.50482i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$	−0.438431	−	0.452716i	−0.438431	−	0.452716i
$975$		0			0
$976$		0			0
$977$	0.682767	+	1.21232i	0.682767	+	1.21232i	0.967295	+	0.253655i	$0.0816327\pi$
$977$	0.682767	+	1.21232i	0.682767	+	1.21232i	−0.284528	+	0.958668i	$0.591837\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1.37208	−	1.16807i	−1.37208	−	1.16807i
$982$	−1.64028	−	2.51991i	−1.64028	−	2.51991i
$983$		0			0		−0.938468	−	0.345365i	$-0.887755\pi$
$983$		0			0		0.938468	+	0.345365i	$0.112245\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.89511	−	1.61333i	−1.89511	−	1.61333i
$990$		0			0
$991$	0.372984	+	1.63415i	0.372984	+	1.63415i	0.718349	+	0.695683i	$0.244898\pi$
$991$	0.372984	+	1.63415i	0.372984	+	1.63415i	−0.345365	+	0.938468i	$0.612245\pi$
$992$		0			0
$993$		0			0
$994$	1.03611	−	2.34060i	1.03611	−	2.34060i
$995$		0			0
$996$		0			0
$997$		0			0		0.127877	−	0.991790i	$-0.459184\pi$
$997$		0			0		−0.127877	+	0.991790i	$0.540816\pi$
$998$	2.82188	−	1.96842i	2.82188	−	1.96842i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1379.1.ba.a.174.1	✓	42
7.6	odd	2	CM	1379.1.ba.a.174.1	✓	42
197.137	even	98	inner	1379.1.ba.a.531.1	yes	42
1379.531	odd	98	inner	1379.1.ba.a.531.1	yes	42

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1379.1.ba.a.174.1	✓	42	1.1	even	1	trivial
1379.1.ba.a.174.1	✓	42	7.6	odd	2	CM
1379.1.ba.a.531.1	yes	42	197.137	even	98	inner
1379.1.ba.a.531.1	yes	42	1379.531	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:	\( N \)	\(=\)	\( 1379 = 7 \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1379.ba (of order \(98\), degree \(42\), minimal)

\(f(q)\)	\(=\)	\(q+(1.85288 + 0.681876i) q^{2} +(2.20676 + 1.87864i) q^{4} +(-0.345365 - 0.938468i) q^{7} +(1.83899 + 3.26532i) q^{8} +(-0.404783 + 0.914413i) q^{9} +O(q^{10})\)	\(q+(1.85288 + 0.681876i) q^{2} +(2.20676 + 1.87864i) q^{4} +(-0.345365 - 0.938468i) q^{7} +(1.83899 + 3.26532i) q^{8} +(-0.404783 + 0.914413i) q^{9} +(-1.08781 - 1.12326i) q^{11} -1.97436i q^{14} +(0.718349 + 4.44324i) q^{16} +(-1.37353 + 1.41828i) q^{18} +(-1.24966 - 2.82301i) q^{22} +(-0.119739 - 1.24122i) q^{23} +(0.997945 - 0.0640702i) q^{25} +(1.00091 - 2.71979i) q^{28} +(-1.96729 - 0.253655i) q^{29} +(-0.982350 + 5.04416i) q^{32} +(-2.61111 + 1.25744i) q^{36} +(-0.147642 + 0.913219i) q^{37} +(1.43375 - 1.38851i) q^{43} +(-0.290345 - 4.52236i) q^{44} +(0.624495 - 2.38147i) q^{46} +(-0.761446 + 0.648228i) q^{49} +(1.89276 + 0.561761i) q^{50} +(-0.550444 - 0.144343i) q^{53} +(2.42928 - 2.85357i) q^{56} +(-3.47220 - 1.81144i) q^{58} +(0.997945 + 0.0640702i) q^{63} +(-2.92642 + 4.82743i) q^{64} +(-0.719513 + 0.213548i) q^{67} +(1.18550 + 0.524784i) q^{71} +(-3.73025 + 0.359852i) q^{72} +(-0.896265 + 1.59141i) q^{74} +(-0.678448 + 1.40881i) q^{77} +(-0.507049 - 0.0162601i) q^{79} +(-0.672301 - 0.740278i) q^{81} +(3.60335 - 1.59509i) q^{86} +(1.66731 - 5.61772i) q^{88} +(2.06756 - 2.96401i) q^{92} +(-1.85288 + 0.681876i) q^{98} +(1.46745 - 0.540035i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 7 q^{8}+O(q^{10}) \) 42 * q - 7 * q^8	\( 42 q - 7 q^{8} - 7 q^{28} - 42 q^{29} + 7 q^{32} - 7 q^{36} + 7 q^{50} + 7 q^{56} - 7 q^{58} - 7 q^{64} + 7 q^{67} + 7 q^{71} - 7 q^{79}+O(q^{100}) \) 42 * q - 7 * q^8 - 7 * q^28 - 42 * q^29 + 7 * q^32 - 7 * q^36 + 7 * q^50 + 7 * q^56 - 7 * q^58 - 7 * q^64 + 7 * q^67 + 7 * q^71 - 7 * q^79

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