LMFDB - Embedded newform 1379.1.ba.a.223.1

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			223.1
Root			$0.981559 + 0.191159i$ of defining polynomial
Character	$\chi$	$=$	1379.223
Dual form			1379.1.ba.a.538.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.82037 - 0.805826i) q^{2} +(1.99211 + 2.19353i) q^{4} +(0.404783 + 0.914413i) q^{7} +(-1.23147 - 3.70900i) q^{8} +(0.997945 + 0.0640702i) q^{9} +O(q^{10})$	\(q+(-1.82037 - 0.805826i) q^{2} +(1.99211 + 2.19353i) q^{4} +(0.404783 + 0.914413i) q^{7} +(-1.23147 - 3.70900i) q^{8} +(0.997945 + 0.0640702i) q^{9} +(1.72874 - 0.901883i) q^{11} -1.99076i q^{14} +(-0.462538 + 4.79469i) q^{16} +(-1.76500 - 0.920802i) q^{18} +(-3.87372 + 0.248701i) q^{22} +(-0.430487 + 0.112887i) q^{23} +(0.345365 - 0.938468i) q^{25} +(-1.19942 + 2.70951i) q^{28} +(-1.15960 + 0.987182i) q^{29} +(2.78791 - 4.95022i) q^{32} +(1.84747 + 2.31666i) q^{36} +(-0.0546424 - 0.566426i) q^{37} +(-0.319489 - 0.612401i) q^{43} +(5.42215 + 1.99540i) q^{44} +(0.874614 + 0.141401i) q^{46} +(-0.672301 + 0.740278i) q^{49} +(-1.38494 + 1.43006i) q^{50} +(-0.229297 + 1.41828i) q^{53} +(2.89308 - 2.62742i) q^{56} +(2.90640 - 0.862605i) q^{58} +(0.345365 + 0.938468i) q^{63} +(-5.20368 + 3.88361i) q^{64} +(1.18982 + 1.22858i) q^{67} +(0.0948595 - 1.47751i) q^{71} +(-0.991306 - 3.78028i) q^{72} +(-0.356971 + 1.07514i) q^{74} +(1.52446 + 1.21572i) q^{77} +(-1.61932 - 1.12957i) q^{79} +(0.991790 + 0.127877i) q^{81} +(0.0881015 + 1.37225i) q^{86} +(-5.47398 - 5.30125i) q^{88} +(-1.10520 - 0.719403i) q^{92} +(1.82037 - 0.805826i) q^{98} +(1.78297 - 0.789269i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1379\mathbb{Z}\right)^\times$.

$n$	$395$	$1184$
$\chi(n)$	$-1$	$e\left(\frac{13}{98}\right)$

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 $

$2$	−1.82037	−	0.805826i	−1.82037	−	0.805826i	−0.949056	−	0.315108i	$-0.897959\pi$
$2$	−1.82037	−	0.805826i	−1.82037	−	0.805826i	−0.871319	−	0.490718i	$-0.836735\pi$
$3$		0			0		−0.999486	−	0.0320516i	$-0.989796\pi$
$3$		0			0		0.999486	+	0.0320516i	$0.0102041\pi$
$4$	1.99211	+	2.19353i	1.99211	+	2.19353i
$5$		0			0		0.820172	−	0.572117i	$-0.193878\pi$
$5$		0			0		−0.820172	+	0.572117i	$0.806122\pi$
$6$		0			0
$7$	0.404783	+	0.914413i	0.404783	+	0.914413i
$7$	0.404783	+	0.914413i	0.404783	+	0.914413i
$8$	−1.23147	−	3.70900i	−1.23147	−	3.70900i
$9$	0.997945	+	0.0640702i	0.997945	+	0.0640702i
$10$		0			0
$11$	1.72874	−	0.901883i	1.72874	−	0.901883i	0.761446	−	0.648228i	$-0.224490\pi$
$11$	1.72874	−	0.901883i	1.72874	−	0.901883i	0.967295	−	0.253655i	$-0.0816327\pi$
$12$		0			0
$13$		0			0		−0.545535	−	0.838088i	$-0.683673\pi$
$13$		0			0		0.545535	+	0.838088i	$0.316327\pi$
$14$		−	1.99076i		−	1.99076i
$15$		0			0
$16$	−0.462538	+	4.79469i	−0.462538	+	4.79469i
$17$		0			0		−0.958668	−	0.284528i	$-0.908163\pi$
$17$		0			0		0.958668	+	0.284528i	$0.0918367\pi$
$18$	−1.76500	−	0.920802i	−1.76500	−	0.920802i
$19$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$19$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$20$		0			0
$21$		0			0
$22$	−3.87372	+	0.248701i	−3.87372	+	0.248701i
$23$	−0.430487	+	0.112887i	−0.430487	+	0.112887i	−0.462538	−	0.886599i	$-0.653061\pi$
$23$	−0.430487	+	0.112887i	−0.430487	+	0.112887i	0.0320516	+	0.999486i	$0.489796\pi$
$24$		0			0
$25$	0.345365	−	0.938468i	0.345365	−	0.938468i
$26$		0			0
$27$		0			0
$28$	−1.19942	+	2.70951i	−1.19942	+	2.70951i
$29$	−1.15960	+	0.987182i	−1.15960	+	0.987182i	−0.159600	+	0.987182i	$0.551020\pi$
$29$	−1.15960	+	0.987182i	−1.15960	+	0.987182i	−1.00000			$\pi$
$30$		0			0
$31$		0			0		0.598111	−	0.801414i	$-0.295918\pi$
$31$		0			0		−0.598111	+	0.801414i	$0.704082\pi$
$32$	2.78791	−	4.95022i	2.78791	−	4.95022i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	1.84747	+	2.31666i	1.84747	+	2.31666i
$37$	−0.0546424	−	0.566426i	−0.0546424	−	0.566426i	−0.981559	−	0.191159i	$-0.938776\pi$
$37$	−0.0546424	−	0.566426i	−0.0546424	−	0.566426i	0.926917	−	0.375267i	$-0.122449\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.284528	−	0.958668i	$-0.408163\pi$
$41$		0			0		−0.284528	+	0.958668i	$0.591837\pi$
$42$		0			0
$43$	−0.319489	−	0.612401i	−0.319489	−	0.612401i	0.672301	−	0.740278i	$-0.265306\pi$
$43$	−0.319489	−	0.612401i	−0.319489	−	0.612401i	−0.991790	+	0.127877i	$0.959184\pi$
$44$	5.42215	+	1.99540i	5.42215	+	1.99540i
$45$		0			0
$46$	0.874614	+	0.141401i	0.874614	+	0.141401i
$47$		0			0		0.718349	−	0.695683i	$-0.244898\pi$
$47$		0			0		−0.718349	+	0.695683i	$0.755102\pi$
$48$		0			0
$49$	−0.672301	+	0.740278i	−0.672301	+	0.740278i
$50$	−1.38494	+	1.43006i	−1.38494	+	1.43006i
$51$		0			0
$52$		0			0
$53$	−0.229297	+	1.41828i	−0.229297	+	1.41828i	0.572117	+	0.820172i	$0.306122\pi$
$53$	−0.229297	+	1.41828i	−0.229297	+	1.41828i	−0.801414	+	0.598111i	$0.795918\pi$
$54$		0			0
$55$		0			0
$56$	2.89308	−	2.62742i	2.89308	−	2.62742i
$57$		0			0
$58$	2.90640	−	0.862605i	2.90640	−	0.862605i
$59$		0			0		−0.981559	−	0.191159i	$-0.938776\pi$
$59$		0			0		0.981559	+	0.191159i	$0.0612245\pi$
$60$		0			0
$61$		0			0		−0.0320516	−	0.999486i	$-0.510204\pi$
$61$		0			0		0.0320516	+	0.999486i	$0.489796\pi$
$62$		0			0
$63$	0.345365	+	0.938468i	0.345365	+	0.938468i
$64$	−5.20368	+	3.88361i	−5.20368	+	3.88361i
$65$		0			0
$66$		0			0
$67$	1.18982	+	1.22858i	1.18982	+	1.22858i	0.967295	+	0.253655i	$0.0816327\pi$
$67$	1.18982	+	1.22858i	1.18982	+	1.22858i	0.222521	+	0.974928i	$0.428571\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.0948595	−	1.47751i	0.0948595	−	1.47751i	−0.623490	−	0.781831i	$-0.714286\pi$
$71$	0.0948595	−	1.47751i	0.0948595	−	1.47751i	0.718349	−	0.695683i	$-0.244898\pi$
$72$	−0.991306	−	3.78028i	−0.991306	−	3.78028i
$73$		0			0		0.995379	−	0.0960230i	$-0.0306122\pi$
$73$		0			0		−0.995379	+	0.0960230i	$0.969388\pi$
$74$	−0.356971	+	1.07514i	−0.356971	+	1.07514i
$75$		0			0
$76$		0			0
$77$	1.52446	+	1.21572i	1.52446	+	1.21572i
$78$		0			0
$79$	−1.61932	−	1.12957i	−1.61932	−	1.12957i	−0.900969	−	0.433884i	$-0.857143\pi$
$79$	−1.61932	−	1.12957i	−1.61932	−	1.12957i	−0.718349	−	0.695683i	$-0.755102\pi$
$80$		0			0
$81$	0.991790	+	0.127877i	0.991790	+	0.127877i
$82$		0			0
$83$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$83$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$84$		0			0
$85$		0			0
$86$	0.0881015	+	1.37225i	0.0881015	+	1.37225i
$87$		0			0
$88$	−5.47398	−	5.30125i	−5.47398	−	5.30125i
$89$		0			0		−0.598111	−	0.801414i	$-0.704082\pi$
$89$		0			0		0.598111	+	0.801414i	$0.295918\pi$
$90$		0			0
$91$		0			0
$92$	−1.10520	−	0.719403i	−1.10520	−	0.719403i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.926917	−	0.375267i	$-0.122449\pi$
$97$		0			0		−0.926917	+	0.375267i	$0.877551\pi$
$98$	1.82037	−	0.805826i	1.82037	−	0.805826i
$99$	1.78297	−	0.789269i	1.78297	−	0.789269i
$100$	2.74656	−	1.11196i	2.74656	−	1.11196i
$101$		0			0		0.518393	−	0.855143i	$-0.326531\pi$
$101$		0			0		−0.518393	+	0.855143i	$0.673469\pi$
$102$		0			0
$103$		0			0		0.938468	−	0.345365i	$-0.112245\pi$
$103$		0			0		−0.938468	+	0.345365i	$0.887755\pi$
$104$		0			0
$105$		0			0
$106$	1.56030	−	2.39703i	1.56030	−	2.39703i
$107$	−0.0635769	+	0.00819733i	−0.0635769	+	0.00819733i	−0.159600	−	0.987182i	$-0.551020\pi$
$107$	−0.0635769	+	0.00819733i	−0.0635769	+	0.00819733i	0.0960230	+	0.995379i	$0.469388\pi$
$108$		0			0
$109$	0.895767	+	0.867502i	0.895767	+	0.867502i	0.991790	−	0.127877i	$-0.0408163\pi$
$109$	0.895767	+	0.867502i	0.895767	+	0.867502i	−0.0960230	+	0.995379i	$0.530612\pi$
$110$		0			0
$111$		0			0
$112$	−4.57156	+	1.51786i	−4.57156	+	1.51786i
$113$	0.220113	−	0.457070i	0.220113	−	0.457070i	−0.761446	−	0.648228i	$-0.775510\pi$
$113$	0.220113	−	0.457070i	0.220113	−	0.457070i	0.981559	+	0.191159i	$0.0612245\pi$
$114$		0			0
$115$		0			0
$116$	−4.47546	−	0.577047i	−4.47546	−	0.577047i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.60304	−	2.29807i	1.60304	−	2.29807i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	0.127548	−	1.98667i	0.127548	−	1.98667i
$127$	−0.769269	−	1.10281i	−0.769269	−	1.10281i	−0.991790	−	0.127877i	$-0.959184\pi$
$127$	−0.769269	−	1.10281i	−0.769269	−	1.10281i	0.222521	−	0.974928i	$-0.428571\pi$
$128$	7.06331	−	1.61215i	7.06331	−	1.61215i
$129$		0			0
$130$		0			0
$131$		0			0		0.999486	−	0.0320516i	$-0.0102041\pi$
$131$		0			0		−0.999486	+	0.0320516i	$0.989796\pi$
$132$		0			0
$133$		0			0
$134$	−1.17589	−	3.19526i	−1.17589	−	3.19526i
$135$		0			0
$136$		0			0
$137$	−1.00288	−	0.262985i	−1.00288	−	0.262985i	−0.284528	−	0.958668i	$-0.591837\pi$
$137$	−1.00288	−	0.262985i	−1.00288	−	0.262985i	−0.718349	+	0.695683i	$0.755102\pi$
$138$		0			0
$139$		0			0		0.958668	−	0.284528i	$-0.0918367\pi$
$139$		0			0		−0.958668	+	0.284528i	$0.908163\pi$
$140$		0			0
$141$		0			0
$142$	−1.36330	+	2.61319i	−1.36330	+	2.61319i
$143$		0			0
$144$	−0.768785	+	4.75521i	−0.768785	+	4.75521i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	1.13362	−	1.24824i	1.13362	−	1.24824i
$149$	−0.165787	+	1.28581i	−0.165787	+	1.28581i	0.672301	+	0.740278i	$0.265306\pi$
$149$	−0.165787	+	1.28581i	−0.165787	+	1.28581i	−0.838088	+	0.545535i	$0.816327\pi$
$150$		0			0
$151$	−0.740914	−	0.119785i	−0.740914	−	0.119785i	−0.222521	−	0.974928i	$-0.571429\pi$
$151$	−0.740914	−	0.119785i	−0.740914	−	0.119785i	−0.518393	+	0.855143i	$0.673469\pi$
$152$		0			0
$153$		0			0
$154$	−1.79543	−	3.44151i	−1.79543	−	3.44151i
$155$		0			0
$156$		0			0
$157$		0			0		−0.801414	−	0.598111i	$-0.795918\pi$
$157$		0			0		0.801414	+	0.598111i	$0.204082\pi$
$158$	2.03753	+	3.36112i	2.03753	+	3.36112i
$159$		0			0
$160$		0			0
$161$	−0.277479	−	0.347948i	−0.277479	−	0.347948i
$162$	−1.70238	−	1.03199i	−1.70238	−	1.03199i
$163$	0.616441	+	0.524784i	0.616441	+	0.524784i	0.900969	−	0.433884i	$-0.142857\pi$
$163$	0.616441	+	0.524784i	0.616441	+	0.524784i	−0.284528	+	0.958668i	$0.591837\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.987182	−	0.159600i	$-0.0510204\pi$
$167$		0			0		−0.987182	+	0.159600i	$0.948980\pi$
$168$		0			0
$169$	−0.404783	+	0.914413i	−0.404783	+	0.914413i
$170$		0			0
$171$		0			0
$172$	0.706864	−	1.92078i	0.706864	−	1.92078i
$173$		0			0		−0.284528	−	0.958668i	$-0.591837\pi$
$173$		0			0		0.284528	+	0.958668i	$0.408163\pi$
$174$		0			0
$175$	0.997945	−	0.0640702i	0.997945	−	0.0640702i
$176$	3.52464	+	8.70594i	3.52464	+	8.70594i
$177$		0			0
$178$		0			0
$179$	−0.338962	−	0.176836i	−0.338962	−	0.176836i	0.284528	−	0.958668i	$-0.408163\pi$
$179$	−0.338962	−	0.176836i	−0.338962	−	0.176836i	−0.623490	+	0.781831i	$0.714286\pi$
$180$		0			0
$181$		0			0		0.0960230	−	0.995379i	$-0.469388\pi$
$181$		0			0		−0.0960230	+	0.995379i	$0.530612\pi$
$182$		0			0
$183$		0			0
$184$	0.948830	+	1.45766i	0.948830	+	1.45766i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.173028	+	0.0833257i	0.173028	+	0.0833257i	0.518393	−	0.855143i	$-0.326531\pi$
$191$	0.173028	+	0.0833257i	0.173028	+	0.0833257i	−0.345365	+	0.938468i	$0.612245\pi$
$192$		0			0
$193$	1.30063	+	1.43213i	1.30063	+	1.43213i	0.838088	+	0.545535i	$0.183673\pi$
$193$	1.30063	+	1.43213i	1.30063	+	1.43213i	0.462538	+	0.886599i	$0.346939\pi$
$194$		0			0
$195$		0			0
$196$	−2.96312			−2.96312
$197$	−0.949056	+	0.315108i	−0.949056	+	0.315108i
$197$	−0.949056	+	0.315108i	−0.949056	+	0.315108i
$198$	−3.88169			−3.88169
$199$		0			0		−0.914413	−	0.404783i	$-0.867347\pi$
$199$		0			0		0.914413	+	0.404783i	$0.132653\pi$
$200$	−3.90609	−	0.125261i	−3.90609	−	0.125261i
$201$		0			0
$202$		0			0
$203$	−1.37208	−	0.660758i	−1.37208	−	0.660758i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.436835	+	0.0850736i	−0.436835	+	0.0850736i
$208$		0			0
$209$		0			0
$210$		0			0
$211$			0.981435i			0.981435i	0.871319	+	0.490718i	$0.163265\pi$
$211$			0.981435i			0.981435i	−0.871319	+	0.490718i	$0.836735\pi$
$212$	−3.56783	+	2.32240i	−3.56783	+	2.32240i
$213$		0			0
$214$	0.122339	+	0.0363097i	0.122339	+	0.0363097i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−0.931576	−	2.30101i	−0.931576	−	2.30101i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.159600	−	0.987182i	$-0.551020\pi$
$223$		0			0		0.159600	+	0.987182i	$0.448980\pi$
$224$	5.65504	+	0.545535i	5.65504	+	0.545535i
$225$	0.404783	−	0.914413i	0.404783	−	0.914413i
$226$	−0.769007	+	0.654665i	−0.769007	+	0.654665i
$227$		0			0		0.987182	−	0.159600i	$-0.0510204\pi$
$227$		0			0		−0.987182	+	0.159600i	$0.948980\pi$
$228$		0			0
$229$		0			0		0.490718	−	0.871319i	$-0.336735\pi$
$229$		0			0		−0.490718	+	0.871319i	$0.663265\pi$
$230$		0			0
$231$		0			0
$232$	5.08947	+	3.08527i	5.08947	+	3.08527i
$233$	−0.576776	−	0.723254i	−0.576776	−	0.723254i	0.404783	−	0.914413i	$-0.367347\pi$
$233$	−0.576776	−	0.723254i	−0.576776	−	0.723254i	−0.981559	+	0.191159i	$0.938776\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.0366745	−	1.14365i	0.0366745	−	1.14365i	−0.801414	−	0.598111i	$-0.795918\pi$
$239$	0.0366745	−	1.14365i	0.0366745	−	1.14365i	0.838088	−	0.545535i	$-0.183673\pi$
$240$		0			0
$241$		0			0		−0.938468	−	0.345365i	$-0.887755\pi$
$241$		0			0		0.938468	+	0.345365i	$0.112245\pi$
$242$	−4.76997	+	2.89158i	−4.76997	+	2.89158i
$243$		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		−0.871319	−	0.490718i	$-0.836735\pi$
$251$		0			0		0.871319	+	0.490718i	$0.163265\pi$
$252$	−1.37056	+	2.62710i	−1.37056	+	2.62710i
$253$	−0.642389	+	0.583401i	−0.642389	+	0.583401i
$254$	0.511689	+	2.62742i	0.511689	+	2.62742i
$255$		0			0
$256$	−7.78359	−	1.51585i	−7.78359	−	1.51585i
$257$		0			0		−0.967295	−	0.253655i	$-0.918367\pi$
$257$		0			0		0.967295	+	0.253655i	$0.0816327\pi$
$258$		0			0
$259$	0.495828	−	0.279245i	0.495828	−	0.279245i
$260$		0			0
$261$	−1.22047	+	0.910858i	−1.22047	+	0.910858i
$262$		0			0
$263$	−0.629893	+	0.0201994i	−0.629893	+	0.0201994i	−0.345365	−	0.938468i	$-0.612245\pi$
$263$	−0.629893	+	0.0201994i	−0.629893	+	0.0201994i	−0.284528	+	0.958668i	$0.591837\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.324694	+	5.05737i	−0.324694	+	5.05737i
$269$		0			0		−0.253655	−	0.967295i	$-0.581633\pi$
$269$		0			0		0.253655	+	0.967295i	$0.418367\pi$
$270$		0			0
$271$		0			0		0.315108	−	0.949056i	$-0.397959\pi$
$271$		0			0		−0.315108	+	0.949056i	$0.602041\pi$
$272$		0			0
$273$		0			0
$274$	1.61369	+	1.28688i	1.61369	+	1.28688i
$275$	−0.249342	−	1.93385i	−0.249342	−	1.93385i
$276$		0			0
$277$	0.921046	+	1.63541i	0.921046	+	1.63541i	0.761446	+	0.648228i	$0.224490\pi$
$277$	0.921046	+	1.63541i	0.921046	+	1.63541i	0.159600	+	0.987182i	$0.448980\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.867322	+	1.80101i	−0.867322	+	1.80101i	−0.404783	+	0.914413i	$0.632653\pi$
$281$	−0.867322	+	1.80101i	−0.867322	+	1.80101i	−0.462538	+	0.886599i	$0.653061\pi$
$282$		0			0
$283$		0			0		−0.0640702	−	0.997945i	$-0.520408\pi$
$283$		0			0		0.0640702	+	0.997945i	$0.479592\pi$
$284$	3.42994	−	2.73529i	3.42994	−	2.73529i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	3.09934	−	4.76142i	3.09934	−	4.76142i
$289$	0.838088	+	0.545535i	0.838088	+	0.545535i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.518393	−	0.855143i	$-0.326531\pi$
$293$		0			0		−0.518393	+	0.855143i	$0.673469\pi$
$294$		0			0
$295$		0			0
$296$	−2.03358	+	0.900206i	−2.03358	+	0.900206i
$297$		0			0
$298$	1.33794	−	2.20707i	1.33794	−	2.20707i
$299$		0			0
$300$		0			0
$301$	0.430663	−	0.540035i	0.430663	−	0.540035i
$302$	1.25221	+	0.815101i	1.25221	+	0.815101i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$307$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$308$	0.370176	+	5.76578i	0.370176	+	5.76578i
$309$		0			0
$310$		0			0
$311$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$311$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$312$		0			0
$313$		0			0		−0.991790	−	0.127877i	$-0.959184\pi$
$313$		0			0		0.991790	+	0.127877i	$0.0408163\pi$
$314$		0			0
$315$		0			0
$316$	−0.748116	−	5.80224i	−0.748116	−	5.80224i
$317$	0.853033	+	0.680271i	0.853033	+	0.680271i	0.949056	−	0.315108i	$-0.102041\pi$
$317$	0.853033	+	0.680271i	0.853033	+	0.680271i	−0.0960230	+	0.995379i	$0.530612\pi$
$318$		0			0
$319$	−1.11433	+	2.75240i	−1.11433	+	2.75240i
$320$		0			0
$321$		0			0
$322$	0.224731	+	0.856995i	0.224731	+	0.856995i
$323$		0			0
$324$	1.69525	+	2.43027i	1.69525	+	2.43027i
$325$		0			0
$326$	−0.699269	−	1.45205i	−0.699269	−	1.45205i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.578893	−	1.57304i	−0.578893	−	1.57304i	−0.801414	−	0.598111i	$-0.795918\pi$
$331$	−0.578893	−	1.57304i	−0.578893	−	1.57304i	0.222521	−	0.974928i	$-0.428571\pi$
$332$		0			0
$333$	−0.0182391	−	0.568763i	−0.0182391	−	0.568763i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.265971	−	1.36571i	−0.265971	−	1.36571i	−0.838088	−	0.545535i	$-0.816327\pi$
$337$	−0.265971	−	1.36571i	−0.265971	−	1.36571i	0.572117	−	0.820172i	$-0.306122\pi$
$338$	1.47371	−	1.33839i	1.47371	−	1.33839i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.949056	−	0.315108i	−0.949056	−	0.315108i
$344$	−1.87795	+	1.93914i	−1.87795	+	1.93914i
$345$		0			0
$346$		0			0
$347$	0.229297	−	0.222062i	0.229297	−	0.222062i	−0.572117	−	0.820172i	$-0.693878\pi$
$347$	0.229297	−	0.222062i	0.229297	−	0.222062i	0.801414	+	0.598111i	$0.204082\pi$
$348$		0			0
$349$		0			0		0.855143	−	0.518393i	$-0.173469\pi$
$349$		0			0		−0.855143	+	0.518393i	$0.826531\pi$
$350$	−1.86826	−	0.687538i	−1.86826	−	0.687538i
$351$		0			0
$352$	0.355058	−	11.0720i	0.355058	−	11.0720i
$353$		0			0		0.284528	−	0.958668i	$-0.408163\pi$
$353$		0			0		−0.284528	+	0.958668i	$0.591837\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	0.474539	+	0.595053i	0.474539	+	0.595053i
$359$	−1.02294	−	0.620112i	−1.02294	−	0.620112i	−0.0960230	−	0.995379i	$-0.530612\pi$
$359$	−1.02294	−	0.620112i	−1.02294	−	0.620112i	−0.926917	+	0.375267i	$0.877551\pi$
$360$		0			0
$361$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.995379	−	0.0960230i	$-0.969388\pi$
$367$		0			0		0.995379	+	0.0960230i	$0.0306122\pi$
$368$	−0.342142	−	2.11627i	−0.342142	−	2.11627i
$369$		0			0
$370$		0			0
$371$	−1.38971	+	0.364425i	−1.38971	+	0.364425i
$372$		0			0
$373$	−0.704352	−	1.73976i	−0.704352	−	1.73976i	−0.672301	−	0.740278i	$-0.734694\pi$
$373$	−0.704352	−	1.73976i	−0.704352	−	1.73976i	−0.0320516	−	0.999486i	$-0.510204\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.958968	−	0.624219i	0.958968	−	0.624219i	0.0320516	−	0.999486i	$-0.489796\pi$
$379$	0.958968	−	0.624219i	0.958968	−	0.624219i	0.926917	+	0.375267i	$0.122449\pi$
$380$		0			0
$381$		0			0
$382$	−0.247829	−	0.291114i	−0.247829	−	0.291114i
$383$		0			0		0.886599	−	0.462538i	$-0.153061\pi$
$383$		0			0		−0.886599	+	0.462538i	$0.846939\pi$
$384$		0			0
$385$		0			0
$386$	−1.21358	−	3.65510i	−1.21358	−	3.65510i
$387$	−0.279596	−	0.631612i	−0.279596	−	0.631612i
$388$		0			0
$389$	0.105097	−	0.0733113i	0.105097	−	0.0733113i	−0.518393	−	0.855143i	$-0.673469\pi$
$389$	0.105097	−	0.0733113i	0.105097	−	0.0733113i	0.623490	+	0.781831i	$0.285714\pi$
$390$		0			0
$391$		0			0
$392$	3.57361	+	1.58193i	3.57361	+	1.58193i
$393$		0			0
$394$	1.98156	+	0.191159i	1.98156	+	0.191159i
$395$		0			0
$396$	5.28316	+	2.33870i	5.28316	+	2.33870i
$397$		0			0		−0.999486	−	0.0320516i	$-0.989796\pi$
$397$		0			0		0.999486	+	0.0320516i	$0.0102041\pi$
$398$		0			0
$399$		0			0
$400$	4.33992	+	2.09000i	4.33992	+	2.09000i
$401$	0.419673	+	0.948049i	0.419673	+	0.948049i	0.991790	+	0.127877i	$0.0408163\pi$
$401$	0.419673	+	0.948049i	0.419673	+	0.948049i	−0.572117	+	0.820172i	$0.693878\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	1.96524	+	2.30848i	1.96524	+	2.30848i
$407$	−0.605312	−	0.929922i	−0.605312	−	0.929922i
$408$		0			0
$409$		0			0		0.838088	−	0.545535i	$-0.183673\pi$
$409$		0			0		−0.838088	+	0.545535i	$0.816327\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.863758	+	0.197147i	0.863758	+	0.197147i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.345365	−	0.938468i	$-0.387755\pi$
$419$		0			0		−0.345365	+	0.938468i	$0.612245\pi$
$420$		0			0
$421$	−1.82037	−	0.175609i	−1.82037	−	0.175609i	−0.871319	−	0.490718i	$-0.836735\pi$
$421$	−1.82037	−	0.175609i	−1.82037	−	0.175609i	−0.949056	+	0.315108i	$0.897959\pi$
$422$	0.790866	−	1.78658i	0.790866	−	1.78658i
$423$		0			0
$424$	5.54278	−	0.896114i	5.54278	−	0.896114i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.144633	−	0.123128i	−0.144633	−	0.123128i
$429$		0			0
$430$		0			0
$431$	−0.191651	−	1.98667i	−0.191651	−	1.98667i	−0.159600	−	0.987182i	$-0.551020\pi$
$431$	−0.191651	−	1.98667i	−0.191651	−	1.98667i	−0.0320516	−	0.999486i	$-0.510204\pi$
$432$		0			0
$433$		0			0		−0.518393	−	0.855143i	$-0.673469\pi$
$433$		0			0		0.518393	+	0.855143i	$0.326531\pi$
$434$		0			0
$435$		0			0
$436$	−0.118429	+	3.69305i	−0.118429	+	3.69305i
$437$		0			0
$438$		0			0
$439$		0			0		0.855143	−	0.518393i	$-0.173469\pi$
$439$		0			0		−0.855143	+	0.518393i	$0.826531\pi$
$440$		0			0
$441$	−0.718349	+	0.695683i	−0.718349	+	0.695683i
$442$		0			0
$443$	−1.33356	+	1.46840i	−1.33356	+	1.46840i	−0.572117	+	0.820172i	$0.693878\pi$
$443$	−1.33356	+	1.46840i	−1.33356	+	1.46840i	−0.761446	+	0.648228i	$0.775510\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−5.65758	−	3.18629i	−5.65758	−	3.18629i
$449$	−0.147642	+	0.283002i	−0.147642	+	0.283002i	−0.949056	−	0.315108i	$-0.897959\pi$
$449$	−0.147642	+	0.283002i	−0.147642	+	0.283002i	0.801414	+	0.598111i	$0.204082\pi$
$450$	−1.47371	+	1.33839i	−1.47371	+	1.33839i
$451$		0			0
$452$	1.44109	−	0.427707i	1.44109	−	0.427707i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.668140	−	1.81555i	−0.668140	−	1.81555i	−0.572117	−	0.820172i	$-0.693878\pi$
$457$	−0.668140	−	1.81555i	−0.668140	−	1.81555i	−0.0960230	−	0.995379i	$-0.530612\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.695683	−	0.718349i	$-0.744898\pi$
$461$		0			0		0.695683	+	0.718349i	$0.255102\pi$
$462$		0			0
$463$	1.59922	−	0.365011i	1.59922	−	0.365011i	0.672301	−	0.740278i	$-0.265306\pi$
$463$	1.59922	−	0.365011i	1.59922	−	0.365011i	0.926917	+	0.375267i	$0.122449\pi$
$464$	−4.19687	−	6.01654i	−4.19687	−	6.01654i
$465$		0			0
$466$	0.467131	+	1.78137i	0.467131	+	1.78137i
$467$		0			0		0.995379	−	0.0960230i	$-0.0306122\pi$
$467$		0			0		−0.995379	+	0.0960230i	$0.969388\pi$
$468$		0			0
$469$	−0.641814	+	1.58529i	−0.641814	+	1.58529i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.10463	−	0.770541i	−1.10463	−	0.770541i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.319696	+	1.40068i	−0.319696	+	1.40068i
$478$	−0.988340	+	2.05231i	−0.988340	+	2.05231i
$479$		0			0		0.949056	−	0.315108i	$-0.102041\pi$
$479$		0			0		−0.949056	+	0.315108i	$0.897959\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	8.23431	−	1.06170i	8.23431	−	1.06170i
$485$		0			0
$486$		0			0
$487$	−0.119739	+	0.150148i	−0.119739	+	0.150148i	−0.838088	−	0.545535i	$-0.816327\pi$
$487$	−0.119739	+	0.150148i	−0.119739	+	0.150148i	0.718349	+	0.695683i	$0.244898\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.24633	+	0.504585i	−1.24633	+	0.504585i	−0.900969	−	0.433884i	$-0.857143\pi$
$491$	−1.24633	+	0.504585i	−1.24633	+	0.504585i	−0.345365	+	0.938468i	$0.612245\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.38946	−	0.511332i	1.38946	−	0.511332i
$498$		0			0
$499$	−1.59078	−	1.03549i	−1.59078	−	1.03549i	−0.967295	−	0.253655i	$-0.918367\pi$
$499$	−1.59078	−	1.03549i	−1.59078	−	1.03549i	−0.623490	−	0.781831i	$-0.714286\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.718349	−	0.695683i	$-0.755102\pi$
$503$		0			0		0.718349	+	0.695683i	$0.244898\pi$
$504$	3.05547	−	2.43666i	3.05547	−	2.43666i
$505$		0			0
$506$	1.63951	−	0.544354i	1.63951	−	0.544354i
$507$		0			0
$508$	0.886571	−	3.88432i	0.886571	−	3.88432i
$509$		0			0		0.648228	−	0.761446i	$-0.275510\pi$
$509$		0			0		−0.648228	+	0.761446i	$0.724490\pi$
$510$		0			0
$511$		0			0
$512$	7.00543	+	4.88668i	7.00543	+	4.88668i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−1.12762	+	0.108780i	−1.12762	+	0.108780i
$519$		0			0
$520$		0			0
$521$		0			0		−0.572117	−	0.820172i	$-0.693878\pi$
$521$		0			0		0.572117	+	0.820172i	$0.306122\pi$
$522$	2.95570	−	0.674619i	2.95570	−	0.674619i
$523$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$523$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$524$		0			0
$525$		0			0
$526$	1.16292	+	0.470813i	1.16292	+	0.470813i
$527$		0			0
$528$		0			0
$529$	−0.698743	+	0.393525i	−0.698743	+	0.393525i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	3.09159	−	5.92599i	3.09159	−	5.92599i
$537$		0			0
$538$		0			0
$539$	−0.494590	+	1.88609i	−0.494590	+	1.88609i
$540$		0			0
$541$	1.08781	−	1.12326i	1.08781	−	1.12326i	0.0960230	−	0.995379i	$-0.469388\pi$
$541$	1.08781	−	1.12326i	1.08781	−	1.12326i	0.991790	−	0.127877i	$-0.0408163\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.38946	−	0.511332i	−1.38946	−	0.511332i	−0.462538	−	0.886599i	$-0.653061\pi$
$547$	−1.38946	−	0.511332i	−1.38946	−	0.511332i	−0.926917	+	0.375267i	$0.877551\pi$
$548$	−1.42097	−	2.72374i	−1.42097	−	2.72374i
$549$		0			0
$550$	−1.10445	+	3.72125i	−1.10445	+	3.72125i
$551$		0			0
$552$		0			0
$553$	0.377417	−	1.93795i	0.377417	−	1.93795i
$554$	−0.358793	−	3.71926i	−0.358793	−	3.71926i
$555$		0			0
$556$		0			0
$557$	0.704396	+	0.599661i	0.704396	+	0.599661i	0.926917	−	0.375267i	$-0.122449\pi$
$557$	0.704396	+	0.599661i	0.704396	+	0.599661i	−0.222521	+	0.974928i	$0.571429\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	3.03015	−	2.57961i	3.03015	−	2.57961i
$563$		0			0		0.404783	−	0.914413i	$-0.367347\pi$
$563$		0			0		−0.404783	+	0.914413i	$0.632653\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.284528	+	0.958668i	0.284528	+	0.958668i
$568$	−5.59692	+	1.46768i	−5.59692	+	1.46768i
$569$	−1.59953	+	0.102693i	−1.59953	+	0.102693i	−0.838088	−	0.545535i	$-0.816327\pi$
$569$	−1.59953	+	0.102693i	−1.59953	+	0.102693i	−0.761446	+	0.648228i	$0.775510\pi$
$570$		0			0
$571$	−1.92486	−	0.439337i	−1.92486	−	0.439337i	−0.997945	−	0.0640702i	$-0.979592\pi$
$571$	−1.92486	−	0.439337i	−1.92486	−	0.439337i	−0.926917	−	0.375267i	$-0.877551\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.0427343	+	0.442985i	−0.0427343	+	0.442985i
$576$	−5.44181	+	3.54223i	−5.44181	+	3.54223i
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	−1.08603	−	1.66843i	−1.08603	−	1.66843i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	0.882730	+	2.65864i	0.882730	+	2.65864i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.672301	−	0.740278i	$-0.734694\pi$
$587$		0			0		0.672301	+	0.740278i	$0.265306\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	2.74111			2.74111
$593$		0			0		−0.914413	−	0.404783i	$-0.867347\pi$
$593$		0			0		0.914413	+	0.404783i	$0.132653\pi$
$594$		0			0
$595$		0			0
$596$	−3.15074	+	2.19782i	−3.15074	+	2.19782i
$597$		0			0
$598$		0			0
$599$	−0.0805903	−	0.242725i	−0.0805903	−	0.242725i	0.900969	−	0.433884i	$-0.142857\pi$
$599$	−0.0805903	−	0.242725i	−0.0805903	−	0.242725i	−0.981559	+	0.191159i	$0.938776\pi$
$600$		0			0
$601$		0			0		0.981559	−	0.191159i	$-0.0612245\pi$
$601$		0			0		−0.981559	+	0.191159i	$0.938776\pi$
$602$	−1.21914	+	0.636026i	−1.21914	+	0.636026i
$603$	1.10866	+	1.30229i	1.10866	+	1.30229i
$604$	−1.21323	−	1.86384i	−1.21323	−	1.86384i
$605$		0			0
$606$		0			0
$607$		0			0		0.0960230	−	0.995379i	$-0.469388\pi$
$607$		0			0		−0.0960230	+	0.995379i	$0.530612\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.73906	−	0.111651i	1.73906	−	0.111651i	0.838088	−	0.545535i	$-0.183673\pi$
$613$	1.73906	−	0.111651i	1.73906	−	0.111651i	0.900969	+	0.433884i	$0.142857\pi$
$614$		0			0
$615$		0			0
$616$	2.63176	−	7.15134i	2.63176	−	7.15134i
$617$	0.255811	+	1.58228i	0.255811	+	1.58228i	0.718349	+	0.695683i	$0.244898\pi$
$617$	0.255811	+	1.58228i	0.255811	+	1.58228i	−0.462538	+	0.886599i	$0.653061\pi$
$618$		0			0
$619$		0			0		0.404783	−	0.914413i	$-0.367347\pi$
$619$		0			0		−0.404783	+	0.914413i	$0.632653\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.761446	−	0.648228i	−0.761446	−	0.648228i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.48569	−	1.10880i	−1.48569	−	1.10880i	−0.967295	−	0.253655i	$-0.918367\pi$
$631$	−1.48569	−	1.10880i	−1.48569	−	1.10880i	−0.518393	−	0.855143i	$-0.673469\pi$
$632$	−2.19541	+	7.39708i	−2.19541	+	7.39708i
$633$		0			0
$634$	−1.00466	−	1.92574i	−1.00466	−	1.92574i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	4.24645	−	4.11246i	4.24645	−	4.11246i
$639$	0.189329	−	1.46840i	0.189329	−	1.46840i
$640$		0			0
$641$	0.901922	−	0.931309i	0.901922	−	0.931309i	−0.0960230	−	0.995379i	$-0.530612\pi$
$641$	0.901922	−	0.931309i	0.901922	−	0.931309i	0.997945	+	0.0640702i	$0.0204082\pi$
$642$		0			0
$643$		0			0		0.253655	−	0.967295i	$-0.418367\pi$
$643$		0			0		−0.253655	+	0.967295i	$0.581633\pi$
$644$	0.210466	−	1.30181i	0.210466	−	1.30181i
$645$		0			0
$646$		0			0
$647$		0			0		0.740278	−	0.672301i	$-0.234694\pi$
$647$		0			0		−0.740278	+	0.672301i	$0.765306\pi$
$648$	−0.747066	−	3.83603i	−0.747066	−	3.83603i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.0768867	+	2.39761i	0.0768867	+	2.39761i
$653$	−1.08652	+	0.611915i	−1.08652	+	0.611915i	−0.926917	−	0.375267i	$-0.877551\pi$
$653$	−1.08652	+	0.611915i	−1.08652	+	0.611915i	−0.159600	+	0.987182i	$0.551020\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.793498	+	1.64771i	0.793498	+	1.64771i	0.761446	+	0.648228i	$0.224490\pi$
$659$	0.793498	+	1.64771i	0.793498	+	1.64771i	0.0320516	+	0.999486i	$0.489796\pi$
$660$		0			0
$661$		0			0		−0.572117	−	0.820172i	$-0.693878\pi$
$661$		0			0		0.572117	+	0.820172i	$0.306122\pi$
$662$	−0.213793	+	3.33001i	−0.213793	+	3.33001i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.425122	+	1.05006i	−0.425122	+	1.05006i
$667$	0.387752	−	0.555872i	0.387752	−	0.555872i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.01361	+	1.19064i	−1.01361	+	1.19064i	−0.0320516	+	0.999486i	$0.510204\pi$
$673$	−1.01361	+	1.19064i	−1.01361	+	1.19064i	−0.981559	+	0.191159i	$0.938776\pi$
$674$	−0.616354	+	2.70043i	−0.616354	+	2.70043i
$675$		0			0
$676$	−2.81216	+	0.933703i	−2.81216	+	0.933703i
$677$		0			0		−0.0640702	−	0.997945i	$-0.520408\pi$
$677$		0			0		0.0640702	+	0.997945i	$0.479592\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.34331	+	0.874398i	1.34331	+	0.874398i	0.997945	−	0.0640702i	$-0.0204082\pi$
$683$	1.34331	+	0.874398i	1.34331	+	0.874398i	0.345365	+	0.938468i	$0.387755\pi$
$684$		0			0
$685$		0			0
$686$	1.47371	+	1.33839i	1.47371	+	1.33839i
$687$		0			0
$688$	3.08405	−	1.24859i	3.08405	−	1.24859i
$689$		0			0
$690$		0			0
$691$		0			0		0.926917	−	0.375267i	$-0.122449\pi$
$691$		0			0		−0.926917	+	0.375267i	$0.877551\pi$
$692$		0			0
$693$	1.44344	+	1.31089i	1.44344	+	1.31089i
$694$	−0.596349	+	0.219462i	−0.596349	+	0.219462i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	2.12855	+	2.06139i	2.12855	+	2.06139i
$701$	1.28247	−	1.02274i	1.28247	−	1.02274i	0.284528	−	0.958668i	$-0.408163\pi$
$701$	1.28247	−	1.02274i	1.28247	−	1.02274i	0.997945	−	0.0640702i	$-0.0204082\pi$
$702$		0			0
$703$		0			0
$704$	−5.49326	+	11.4069i	−5.49326	+	11.4069i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.105097	−	0.0733113i	−0.105097	−	0.0733113i	0.518393	−	0.855143i	$-0.326531\pi$
$709$	−0.105097	−	0.0733113i	−0.105097	−	0.0733113i	−0.623490	+	0.781831i	$0.714286\pi$
$710$		0			0
$711$	−1.54362	−	1.23100i	−1.54362	−	1.23100i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.287353	−	1.09580i	−0.287353	−	1.09580i
$717$		0			0
$718$	1.36243	+	1.95315i	1.36243	+	1.95315i
$719$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$719$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$720$		0			0
$721$		0			0
$722$	1.98974	−	0.0638069i	1.98974	−	0.0638069i
$723$		0			0
$724$		0			0
$725$	0.525954	+	1.42919i	0.525954	+	1.42919i
$726$		0			0
$727$		0			0		−0.0320516	−	0.999486i	$-0.510204\pi$
$727$		0			0		0.0320516	+	0.999486i	$0.489796\pi$
$728$		0			0
$729$	0.981559	+	0.191159i	0.981559	+	0.191159i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.462538	−	0.886599i	$-0.346939\pi$
$733$		0			0		−0.462538	+	0.886599i	$0.653061\pi$
$734$		0			0
$735$		0			0
$736$	−0.641344	+	2.44572i	−0.641344	+	2.44572i
$737$	3.16492	+	1.05083i	3.16492	+	1.05083i
$738$		0			0
$739$	−0.544272	+	0.599304i	−0.544272	+	0.599304i	−0.949056	−	0.315108i	$-0.897959\pi$
$739$	−0.544272	+	0.599304i	−0.544272	+	0.599304i	0.404783	+	0.914413i	$0.367347\pi$
$740$		0			0
$741$		0			0
$742$	2.82346	+	0.456475i	2.82346	+	0.456475i
$743$	1.60505	−	0.972990i	1.60505	−	0.972990i	0.623490	−	0.781831i	$-0.285714\pi$
$743$	1.60505	−	0.972990i	1.60505	−	0.972990i	0.981559	−	0.191159i	$-0.0612245\pi$
$744$		0			0
$745$		0			0
$746$	−0.119762	+	3.73461i	−0.119762	+	3.73461i
$747$		0			0
$748$		0			0
$749$	−0.0332306	−	0.0548173i	−0.0332306	−	0.0548173i
$750$		0			0
$751$	−0.0306505	−	0.317725i	−0.0306505	−	0.317725i	−0.997945	−	0.0640702i	$-0.979592\pi$
$751$	−0.0306505	−	0.317725i	−0.0306505	−	0.317725i	0.967295	−	0.253655i	$-0.0816327\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.715472	+	0.958668i	−0.715472	+	0.958668i	0.284528	+	0.958668i	$0.408163\pi$
$757$	−0.715472	+	0.958668i	−0.715472	+	0.958668i	−1.00000			$\pi$
$758$	−2.24869	+	0.363551i	−2.24869	+	0.363551i
$759$		0			0
$760$		0			0
$761$		0			0		−0.995379	−	0.0960230i	$-0.969388\pi$
$761$		0			0		0.995379	+	0.0960230i	$0.0306122\pi$
$762$		0			0
$763$	−0.430663	+	1.17025i	−0.430663	+	1.17025i
$764$	0.161912	+	0.545535i	0.161912	+	0.545535i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$769$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$770$		0			0
$771$		0			0
$772$	−0.550444	+	5.70593i	−0.550444	+	5.70593i
$773$		0			0		0.838088	−	0.545535i	$-0.183673\pi$
$773$		0			0		−0.838088	+	0.545535i	$0.816327\pi$
$774$			1.37508i			1.37508i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−0.250392	+	0.0487639i	−0.250392	+	0.0487639i
$779$		0			0
$780$		0			0
$781$	−1.16856	−	2.63979i	−1.16856	−	2.63979i
$782$		0			0
$783$		0			0
$784$	−3.23844	−	3.56588i	−3.23844	−	3.56588i
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	−2.58182	−	1.45405i	−2.58182	−	1.45405i
$789$		0			0
$790$		0			0
$791$	0.507049	+	0.0162601i	0.507049	+	0.0162601i
$792$	−5.12308	−	5.64108i	−5.12308	−	5.64108i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.997945	−	0.0640702i	$-0.979592\pi$
$797$		0			0		0.997945	+	0.0640702i	$0.0204082\pi$
$798$		0			0
$799$		0			0
$800$	−3.68278	−	4.32600i	−3.68278	−	4.32600i
$801$		0			0
$802$		−	2.06399i		−	2.06399i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.368300	−	0.909709i	−0.368300	−	0.909709i	−0.991790	−	0.127877i	$-0.959184\pi$
$809$	−0.368300	−	0.909709i	−0.368300	−	0.909709i	0.623490	−	0.781831i	$-0.285714\pi$
$810$		0			0
$811$		0			0		0.967295	−	0.253655i	$-0.0816327\pi$
$811$		0			0		−0.967295	+	0.253655i	$0.918367\pi$
$812$	−1.28393	−	4.32600i	−1.28393	−	4.32600i
$813$		0			0
$814$	0.352540	+	2.18058i	0.352540	+	2.18058i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.173028	−	0.0833257i	0.173028	−	0.0833257i	−0.345365	−	0.938468i	$-0.612245\pi$
$821$	0.173028	−	0.0833257i	0.173028	−	0.0833257i	0.518393	+	0.855143i	$0.326531\pi$
$822$		0			0
$823$	−0.742065	−	0.449844i	−0.742065	−	0.449844i	0.0960230	−	0.995379i	$-0.469388\pi$
$823$	−0.742065	−	0.449844i	−0.742065	−	0.449844i	−0.838088	+	0.545535i	$0.816327\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.00288	−	1.65435i	−1.00288	−	1.65435i	−0.718349	−	0.695683i	$-0.755102\pi$
$827$	−1.00288	−	1.65435i	−1.00288	−	1.65435i	−0.284528	−	0.958668i	$-0.591837\pi$
$828$	−1.05683	−	0.788735i	−1.05683	−	0.788735i
$829$		0			0		0.284528	−	0.958668i	$-0.408163\pi$
$829$		0			0		−0.284528	+	0.958668i	$0.591837\pi$
$830$		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		−0.949056	−	0.315108i	$-0.897959\pi$
$839$		0			0		0.949056	+	0.315108i	$0.102041\pi$
$840$		0			0
$841$	0.210544	−	1.30229i	0.210544	−	1.30229i
$842$	3.17225	+	1.78658i	3.17225	+	1.78658i
$843$		0			0
$844$	−2.15281	+	1.95512i	−2.15281	+	1.95512i
$845$		0			0
$846$		0			0
$847$	2.75027	+	0.535615i	2.75027	+	0.535615i
$848$	−6.69417	−	1.75542i	−6.69417	−	1.75542i
$849$		0			0
$850$		0			0
$851$	0.0874649	+	0.237670i	0.0874649	+	0.237670i
$852$		0			0
$853$		0			0		−0.926917	−	0.375267i	$-0.877551\pi$
$853$		0			0		0.926917	+	0.375267i	$0.122449\pi$
$854$		0			0
$855$		0			0
$856$	0.108697	+	0.225712i	0.108697	+	0.225712i
$857$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$857$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$858$		0			0
$859$		0			0		0.0640702	−	0.997945i	$-0.479592\pi$
$859$		0			0		−0.0640702	+	0.997945i	$0.520408\pi$
$860$		0			0
$861$		0			0
$862$	−1.25203	+	3.77092i	−1.25203	+	3.77092i
$863$	0.586791	−	1.44939i	0.586791	−	1.44939i	−0.284528	−	0.958668i	$-0.591837\pi$
$863$	0.586791	−	1.44939i	0.586791	−	1.44939i	0.871319	−	0.490718i	$-0.163265\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.81812	−	0.492292i	−3.81812	−	0.492292i
$870$		0			0
$871$		0			0
$872$	2.11445	−	4.39070i	2.11445	−	4.39070i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.715472	−	0.958668i	−0.715472	−	0.958668i	0.284528	−	0.958668i	$-0.408163\pi$
$877$	−0.715472	−	0.958668i	−0.715472	−	0.958668i	−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$881$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$882$	1.86826	−	0.687538i	1.86826	−	0.687538i
$883$	−0.0948595	−	0.0861489i	−0.0948595	−	0.0861489i	0.623490	−	0.781831i	$-0.285714\pi$
$883$	−0.0948595	−	0.0861489i	−0.0948595	−	0.0861489i	−0.718349	+	0.695683i	$0.755102\pi$
$884$		0			0
$885$		0			0
$886$	3.61086	−	1.59842i	3.61086	−	1.59842i
$887$		0			0		0.914413	−	0.404783i	$-0.132653\pi$
$887$		0			0		−0.914413	+	0.404783i	$0.867347\pi$
$888$		0			0
$889$	0.697032	−	1.14983i	0.697032	−	1.14983i
$890$		0			0
$891$	1.82988	−	0.673412i	1.82988	−	0.673412i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	4.33328	+	5.80621i	4.33328	+	5.80621i
$897$		0			0
$898$	0.496815	−	0.396196i	0.496815	−	0.396196i
$899$		0			0
$900$	2.81216	−	0.933703i	2.81216	−	0.933703i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−1.96633	−	0.253531i	−1.96633	−	0.253531i
$905$		0			0
$906$		0			0
$907$	−0.199957	−	1.55083i	−0.199957	−	1.55083i	−0.718349	−	0.695683i	$-0.755102\pi$
$907$	−0.199957	−	1.55083i	−0.199957	−	1.55083i	0.518393	−	0.855143i	$-0.326531\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.438431	+	1.32048i	−0.438431	+	1.32048i	0.462538	+	0.886599i	$0.346939\pi$
$911$	−0.438431	+	1.32048i	−0.438431	+	1.32048i	−0.900969	+	0.433884i	$0.857143\pi$
$912$		0			0
$913$		0			0
$914$	−0.246754	+	3.84339i	−0.246754	+	3.84339i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	0.382121	−	0.0122539i	0.382121	−	0.0122539i	0.159600	−	0.987182i	$-0.448980\pi$
$919$	0.382121	−	0.0122539i	0.382121	−	0.0122539i	0.222521	+	0.974928i	$0.428571\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.550444	−	0.144343i	−0.550444	−	0.144343i
$926$	−3.20531	−	0.624234i	−3.20531	−	0.624234i
$927$		0			0
$928$	1.65390	+	8.49244i	1.65390	+	8.49244i
$929$		0			0		0.740278	−	0.672301i	$-0.234694\pi$
$929$		0			0		−0.740278	+	0.672301i	$0.765306\pi$
$930$		0			0
$931$		0			0
$932$	0.437481	−	2.70598i	0.437481	−	2.70598i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.127877	−	0.991790i	$-0.459184\pi$
$937$		0			0		−0.127877	+	0.991790i	$0.540816\pi$
$938$	2.44581	−	2.36864i	2.44581	−	2.36864i
$939$		0			0
$940$		0			0
$941$		0			0		−0.938468	−	0.345365i	$-0.887755\pi$
$941$		0			0		0.938468	+	0.345365i	$0.112245\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	1.38991	+	2.29281i	1.38991	+	2.29281i
$947$	−0.143471	+	0.736694i	−0.143471	+	0.736694i	0.838088	+	0.545535i	$0.183673\pi$
$947$	−0.143471	+	0.736694i	−0.143471	+	0.736694i	−0.981559	+	0.191159i	$0.938776\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.425829	−	0.756102i	0.425829	−	0.756102i	−0.572117	−	0.820172i	$-0.693878\pi$
$953$	0.425829	−	0.756102i	0.425829	−	0.756102i	0.997945	+	0.0640702i	$0.0204082\pi$
$954$	1.71067	−	2.29214i	1.71067	−	2.29214i
$955$		0			0
$956$	2.58168	−	2.19782i	2.58168	−	2.19782i
$957$		0			0
$958$		0			0
$959$	−0.165471	−	1.02350i	−0.165471	−	1.02350i
$960$		0			0
$961$	−0.284528	−	0.958668i	−0.284528	−	0.958668i
$962$		0			0
$963$	−0.0639714	+	0.00410710i	−0.0639714	+	0.00410710i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1.77229	+	0.924601i	1.77229	+	0.924601i	0.900969	+	0.433884i	$0.142857\pi$
$967$	1.77229	+	0.924601i	1.77229	+	0.924601i	0.871319	+	0.490718i	$0.163265\pi$
$968$	−10.4976	−	3.11564i	−10.4976	−	3.11564i
$969$		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
$974$	0.338962	−	0.176836i	0.338962	−	0.176836i
$975$		0			0
$976$		0			0
$977$	−0.558749	−	1.68286i	−0.558749	−	1.68286i	−0.718349	−	0.695683i	$-0.755102\pi$
$977$	−0.558749	−	1.68286i	−0.558749	−	1.68286i	0.159600	−	0.987182i	$-0.448980\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.838345	+	0.923112i	0.838345	+	0.923112i
$982$	2.67540	+	0.0857949i	2.67540	+	0.0857949i
$983$		0			0		−0.914413	−	0.404783i	$-0.867347\pi$
$983$		0			0		0.914413	+	0.404783i	$0.132653\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.206668	+	0.227564i	0.206668	+	0.227564i
$990$		0			0
$991$	−0.0577549	−	0.0278133i	−0.0577549	−	0.0278133i	0.404783	−	0.914413i	$-0.367347\pi$
$991$	−0.0577549	−	0.0278133i	−0.0577549	−	0.0278133i	−0.462538	+	0.886599i	$0.653061\pi$
$992$		0			0
$993$		0			0
$994$	−2.94137	−	0.188842i	−2.94137	−	0.188842i
$995$		0			0
$996$		0			0
$997$		0			0		−0.648228	−	0.761446i	$-0.724490\pi$
$997$		0			0		0.648228	+	0.761446i	$0.275510\pi$
$998$	2.06140	+	3.16687i	2.06140	+	3.16687i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1379.1.ba.a.223.1	✓	42
7.6	odd	2	CM	1379.1.ba.a.223.1	✓	42
197.144	even	98	inner	1379.1.ba.a.538.1	yes	42
1379.538	odd	98	inner	1379.1.ba.a.538.1	yes	42

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1379.1.ba.a.223.1	✓	42	1.1	even	1	trivial
1379.1.ba.a.223.1	✓	42	7.6	odd	2	CM
1379.1.ba.a.538.1	yes	42	197.144	even	98	inner
1379.1.ba.a.538.1	yes	42	1379.538	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)

\(f(q)\)	\(=\)	\(q+(-1.82037 - 0.805826i) q^{2} +(1.99211 + 2.19353i) q^{4} +(0.404783 + 0.914413i) q^{7} +(-1.23147 - 3.70900i) q^{8} +(0.997945 + 0.0640702i) q^{9} +O(q^{10})\)	\(q+(-1.82037 - 0.805826i) q^{2} +(1.99211 + 2.19353i) q^{4} +(0.404783 + 0.914413i) q^{7} +(-1.23147 - 3.70900i) q^{8} +(0.997945 + 0.0640702i) q^{9} +(1.72874 - 0.901883i) q^{11} -1.99076i q^{14} +(-0.462538 + 4.79469i) q^{16} +(-1.76500 - 0.920802i) q^{18} +(-3.87372 + 0.248701i) q^{22} +(-0.430487 + 0.112887i) q^{23} +(0.345365 - 0.938468i) q^{25} +(-1.19942 + 2.70951i) q^{28} +(-1.15960 + 0.987182i) q^{29} +(2.78791 - 4.95022i) q^{32} +(1.84747 + 2.31666i) q^{36} +(-0.0546424 - 0.566426i) q^{37} +(-0.319489 - 0.612401i) q^{43} +(5.42215 + 1.99540i) q^{44} +(0.874614 + 0.141401i) q^{46} +(-0.672301 + 0.740278i) q^{49} +(-1.38494 + 1.43006i) q^{50} +(-0.229297 + 1.41828i) q^{53} +(2.89308 - 2.62742i) q^{56} +(2.90640 - 0.862605i) q^{58} +(0.345365 + 0.938468i) q^{63} +(-5.20368 + 3.88361i) q^{64} +(1.18982 + 1.22858i) q^{67} +(0.0948595 - 1.47751i) q^{71} +(-0.991306 - 3.78028i) q^{72} +(-0.356971 + 1.07514i) q^{74} +(1.52446 + 1.21572i) q^{77} +(-1.61932 - 1.12957i) q^{79} +(0.991790 + 0.127877i) q^{81} +(0.0881015 + 1.37225i) q^{86} +(-5.47398 - 5.30125i) q^{88} +(-1.10520 - 0.719403i) q^{92} +(1.82037 - 0.805826i) q^{98} +(1.78297 - 0.789269i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 7 q^{8}+O(q^{10}) \) 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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