LMFDB - Embedded newform 2667.1.em.a.1187.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2667,1,Mod(11,2667)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2667, base_ring=CyclotomicField(126))

chi = DirichletCharacter(H, H._module([63, 84, 68]))

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

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

chi := DirichletCharacter("2667.11");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2667 = 3 \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2667.em (of order $126$, degree $36$, 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:	$1.33100638869$
Analytic rank:	$0$
Dimension:	$36$
Coefficient field:	$\Q(\zeta_{63})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{36} - x^{33} + x^{27} - x^{24} + x^{18} - x^{12} + x^{9} - x^{3} + 1 $ x^36 - x^33 + x^27 - x^24 + x^18 - x^12 + x^9 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{63}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{63} - \cdots)$

Embedding invariants

Embedding label			1187.1
Root			$0.456211 - 0.889872i$ of defining polynomial
Character	$\chi$	$=$	2667.1187
Dual form			2667.1.em.a.2312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.456211 + 0.889872i) q^{3} +(-0.988831 + 0.149042i) q^{4} +(-0.124344 + 0.992239i) q^{7} +(-0.583744 + 0.811938i) q^{9} +O(q^{10})$	\(q+(0.456211 + 0.889872i) q^{3} +(-0.988831 + 0.149042i) q^{4} +(-0.124344 + 0.992239i) q^{7} +(-0.583744 + 0.811938i) q^{9} +(-0.583744 - 0.811938i) q^{12} +(-1.75426 - 0.955319i) q^{13} +(0.955573 - 0.294755i) q^{16} +(-0.698237 + 1.20938i) q^{19} +(-0.939693 + 0.342020i) q^{21} +(-0.222521 - 0.974928i) q^{25} +(-0.988831 - 0.149042i) q^{27} +(-0.0249307 - 0.999689i) q^{28} +(-1.56265 - 0.315897i) q^{31} +(0.456211 - 0.889872i) q^{36} +(1.34551 + 1.12902i) q^{37} +(0.0497994 - 1.99689i) q^{39} +(-0.0185844 - 0.148300i) q^{43} +(0.698237 + 0.715867i) q^{48} +(-0.969077 - 0.246757i) q^{49} +(1.87705 + 0.683190i) q^{52} +(-1.39474 - 0.0696085i) q^{57} +(0.141740 - 0.621003i) q^{61} +(-0.733052 - 0.680173i) q^{63} +(-0.900969 + 0.433884i) q^{64} +(-1.53018 - 0.0763683i) q^{67} +(1.05187 - 0.506554i) q^{73} +(0.766044 - 0.642788i) q^{75} +(0.510189 - 1.29994i) q^{76} +(-0.509014 + 1.80914i) q^{79} +(-0.318487 - 0.947927i) q^{81} +(0.878222 - 0.478254i) q^{84} +(1.16604 - 1.62186i) q^{91} +(-0.431791 - 1.53468i) q^{93} +(-1.52344 - 1.15396i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 3 q^{4}+O(q^{10}) $ 36 * q + 3 * q^4	$ 36 q + 3 q^{4} - 6 q^{13} + 3 q^{16} - 6 q^{25} + 3 q^{27} - 6 q^{31} + 3 q^{37} + 3 q^{39} + 3 q^{52} + 3 q^{57} + 3 q^{63} - 6 q^{64} + 3 q^{67} + 3 q^{79} - 6 q^{91} + 3 q^{93}+O(q^{100}) $ 36 * q + 3 * q^4 - 6 * q^13 + 3 * q^16 - 6 * q^25 + 3 * q^27 - 6 * q^31 + 3 * q^37 + 3 * q^39 + 3 * q^52 + 3 * q^57 + 3 * q^63 - 6 * q^64 + 3 * q^67 + 3 * q^79 - 6 * q^91 + 3 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2667\mathbb{Z}\right)^\times$.

$n$	$890$	$1144$	$2416$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{43}{63}\right)$

Coefficient data

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

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

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

$2$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$2$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$3$	0.456211	+	0.889872i	0.456211	+	0.889872i
$3$	0.456211	+	0.889872i	0.456211	+	0.889872i
$4$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$5$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$5$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$6$		0			0
$7$	−0.124344	+	0.992239i	−0.124344	+	0.992239i
$7$	−0.124344	+	0.992239i	−0.124344	+	0.992239i
$8$		0			0
$9$	−0.583744	+	0.811938i	−0.583744	+	0.811938i
$10$		0			0
$11$		0			0		−0.969077	−	0.246757i	$-0.920635\pi$
$11$		0			0		0.969077	+	0.246757i	$0.0793651\pi$
$12$	−0.583744	−	0.811938i	−0.583744	−	0.811938i
$13$	−1.75426	−	0.955319i	−1.75426	−	0.955319i	−0.900969	−	0.433884i	$-0.857143\pi$
$13$	−1.75426	−	0.955319i	−1.75426	−	0.955319i	−0.853291	−	0.521435i	$-0.825397\pi$
$14$		0			0
$15$		0			0
$16$	0.955573	−	0.294755i	0.955573	−	0.294755i
$17$		0			0		0.318487	−	0.947927i	$-0.396825\pi$
$17$		0			0		−0.318487	+	0.947927i	$0.603175\pi$
$18$		0			0
$19$	−0.698237	+	1.20938i	−0.698237	+	1.20938i	0.270840	+	0.962624i	$0.412698\pi$
$19$	−0.698237	+	1.20938i	−0.698237	+	1.20938i	−0.969077	+	0.246757i	$0.920635\pi$
$20$		0			0
$21$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$22$		0			0
$23$		0			0		0.969077	−	0.246757i	$-0.0793651\pi$
$23$		0			0		−0.969077	+	0.246757i	$0.920635\pi$
$24$		0			0
$25$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$26$		0			0
$27$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$28$	−0.0249307	−	0.999689i	−0.0249307	−	0.999689i
$29$		0			0		−0.921476	−	0.388435i	$-0.873016\pi$
$29$		0			0		0.921476	+	0.388435i	$0.126984\pi$
$30$		0			0
$31$	−1.56265	−	0.315897i	−1.56265	−	0.315897i	−0.661686	−	0.749781i	$-0.730159\pi$
$31$	−1.56265	−	0.315897i	−1.56265	−	0.315897i	−0.900969	+	0.433884i	$0.857143\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.456211	−	0.889872i	0.456211	−	0.889872i
$37$	1.34551	+	1.12902i	1.34551	+	1.12902i	0.980172	+	0.198146i	$0.0634921\pi$
$37$	1.34551	+	1.12902i	1.34551	+	1.12902i	0.365341	+	0.930874i	$0.380952\pi$
$38$		0			0
$39$	0.0497994	−	1.99689i	0.0497994	−	1.99689i
$40$		0			0
$41$		0			0		0.318487	−	0.947927i	$-0.396825\pi$
$41$		0			0		−0.318487	+	0.947927i	$0.603175\pi$
$42$		0			0
$43$	−0.0185844	−	0.148300i	−0.0185844	−	0.148300i	0.980172	−	0.198146i	$-0.0634921\pi$
$43$	−0.0185844	−	0.148300i	−0.0185844	−	0.148300i	−0.998757	+	0.0498459i	$0.984127\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$47$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$48$	0.698237	+	0.715867i	0.698237	+	0.715867i
$49$	−0.969077	−	0.246757i	−0.969077	−	0.246757i
$50$		0			0
$51$		0			0
$52$	1.87705	+	0.683190i	1.87705	+	0.683190i
$53$		0			0		0.583744	−	0.811938i	$-0.301587\pi$
$53$		0			0		−0.583744	+	0.811938i	$0.698413\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.39474	−	0.0696085i	−1.39474	−	0.0696085i
$58$		0			0
$59$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$59$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$60$		0			0
$61$	0.141740	−	0.621003i	0.141740	−	0.621003i	−0.853291	−	0.521435i	$-0.825397\pi$
$61$	0.141740	−	0.621003i	0.141740	−	0.621003i	0.995031	−	0.0995678i	$-0.0317460\pi$
$62$		0			0
$63$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$64$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$65$		0			0
$66$		0			0
$67$	−1.53018	−	0.0763683i	−1.53018	−	0.0763683i	−0.733052	−	0.680173i	$-0.761905\pi$
$67$	−1.53018	−	0.0763683i	−1.53018	−	0.0763683i	−0.797133	+	0.603804i	$0.793651\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.969077	−	0.246757i	$-0.0793651\pi$
$71$		0			0		−0.969077	+	0.246757i	$0.920635\pi$
$72$		0			0
$73$	1.05187	−	0.506554i	1.05187	−	0.506554i	0.173648	−	0.984808i	$-0.444444\pi$
$73$	1.05187	−	0.506554i	1.05187	−	0.506554i	0.878222	+	0.478254i	$0.158730\pi$
$74$		0			0
$75$	0.766044	−	0.642788i	0.766044	−	0.642788i
$76$	0.510189	−	1.29994i	0.510189	−	1.29994i
$77$		0			0
$78$		0			0
$79$	−0.509014	+	1.80914i	−0.509014	+	1.80914i	0.0747301	+	0.997204i	$0.476190\pi$
$79$	−0.509014	+	1.80914i	−0.509014	+	1.80914i	−0.583744	+	0.811938i	$0.698413\pi$
$80$		0			0
$81$	−0.318487	−	0.947927i	−0.318487	−	0.947927i
$82$		0			0
$83$		0			0		0.583744	−	0.811938i	$-0.301587\pi$
$83$		0			0		−0.583744	+	0.811938i	$0.698413\pi$
$84$	0.878222	−	0.478254i	0.878222	−	0.478254i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$89$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$90$		0			0
$91$	1.16604	−	1.62186i	1.16604	−	1.62186i
$92$		0			0
$93$	−0.431791	−	1.53468i	−0.431791	−	1.53468i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.52344	−	1.15396i	−1.52344	−	1.15396i	−0.939693	−	0.342020i	$-0.888889\pi$
$97$	−1.52344	−	1.15396i	−1.52344	−	1.15396i	−0.583744	−	0.811938i	$-0.698413\pi$
$98$		0			0
$99$		0			0
$100$	0.365341	+	0.930874i	0.365341	+	0.930874i
$101$		0			0		0.995031	−	0.0995678i	$-0.0317460\pi$
$101$		0			0		−0.995031	+	0.0995678i	$0.968254\pi$
$102$		0			0
$103$	0.286950	+	1.62737i	0.286950	+	1.62737i	0.698237	+	0.715867i	$0.253968\pi$
$103$	0.286950	+	1.62737i	0.286950	+	1.62737i	−0.411287	+	0.911506i	$0.634921\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	1.00000			1.00000
$109$	0.0425463	+	0.0259995i	0.0425463	+	0.0259995i	0.542546	−	0.840026i	$-0.317460\pi$
$109$	0.0425463	+	0.0259995i	0.0425463	+	0.0259995i	−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$	−0.390845	+	1.71241i	−0.390845	+	1.71241i
$112$	0.173648	+	0.984808i	0.173648	+	0.984808i
$113$		0			0		0.411287	−	0.911506i	$-0.365079\pi$
$113$		0			0		−0.411287	+	0.911506i	$0.634921\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.79970	−	0.866689i	1.79970	−	0.866689i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.878222	+	0.478254i	0.878222	+	0.478254i
$122$		0			0
$123$		0			0
$124$	1.59228	+	0.0794676i	1.59228	+	0.0794676i
$125$		0			0
$126$		0			0
$127$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$127$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$128$		0			0
$129$	0.123490	−	0.0841939i	0.123490	−	0.0841939i
$130$		0			0
$131$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$131$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$132$		0			0
$133$	−1.11317	−	0.843197i	−1.11317	−	0.843197i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$137$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$138$		0			0
$139$	−0.633808	+	0.0634221i	−0.633808	+	0.0634221i	−0.411287	−	0.911506i	$-0.634921\pi$
$139$	−0.633808	+	0.0634221i	−0.633808	+	0.0634221i	−0.222521	+	0.974928i	$0.571429\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.318487	+	0.947927i	−0.318487	+	0.947927i
$145$		0			0
$146$		0			0
$147$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$148$	−1.49876	−	0.915871i	−1.49876	−	0.915871i
$149$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$149$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$150$		0			0
$151$	−1.12310	+	0.942393i	−1.12310	+	0.942393i	−0.998757	−	0.0498459i	$-0.984127\pi$
$151$	−1.12310	+	0.942393i	−1.12310	+	0.942393i	−0.124344	+	0.992239i	$0.539683\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	0.248378	+	1.98201i	0.248378	+	1.98201i
$157$	−1.73181	+	0.730019i	−1.73181	+	0.730019i	−0.733052	+	0.680173i	$0.761905\pi$
$157$	−1.73181	+	0.730019i	−1.73181	+	0.730019i	−0.998757	+	0.0498459i	$0.984127\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.149274	+	0.00744998i	−0.149274	+	0.00744998i	−0.124344	−	0.992239i	$-0.539683\pi$
$163$	−0.149274	+	0.00744998i	−0.149274	+	0.00744998i	−0.0249307	+	0.999689i	$0.507937\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$167$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$168$		0			0
$169$	1.62225	+	2.51173i	1.62225	+	2.51173i
$170$		0			0
$171$	−0.574352	−	1.27289i	−0.574352	−	1.27289i
$172$	0.0404799	+	0.143874i	0.0404799	+	0.143874i
$173$		0			0		−0.980172	−	0.198146i	$-0.936508\pi$
$173$		0			0		0.980172	+	0.198146i	$0.0634921\pi$
$174$		0			0
$175$	0.995031	−	0.0995678i	0.995031	−	0.0995678i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$179$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$180$		0			0
$181$	0.286950	+	0.195639i	0.286950	+	0.195639i	0.698237	−	0.715867i	$-0.253968\pi$
$181$	0.286950	+	0.195639i	0.286950	+	0.195639i	−0.411287	+	0.911506i	$0.634921\pi$
$182$		0			0
$183$	0.617276	−	0.157178i	0.617276	−	0.157178i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	0.270840	−	0.962624i	0.270840	−	0.962624i
$190$		0			0
$191$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$191$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$192$	−0.797133	−	0.603804i	−0.797133	−	0.603804i
$193$	1.76108	−	0.543220i	1.76108	−	0.543220i	0.766044	−	0.642788i	$-0.222222\pi$
$193$	1.76108	−	0.543220i	1.76108	−	0.543220i	0.995031	+	0.0995678i	$0.0317460\pi$
$194$		0			0
$195$		0			0
$196$	0.995031	+	0.0995678i	0.995031	+	0.0995678i
$197$		0			0		−0.998757	−	0.0498459i	$-0.984127\pi$
$197$		0			0		0.998757	+	0.0498459i	$0.0158730\pi$
$198$		0			0
$199$	0.164553	+	1.31310i	0.164553	+	1.31310i	0.826239	+	0.563320i	$0.190476\pi$
$199$	0.164553	+	1.31310i	0.164553	+	1.31310i	−0.661686	+	0.749781i	$0.730159\pi$
$200$		0			0
$201$	−0.630128	−	1.39651i	−0.630128	−	1.39651i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−1.95791	−	0.395800i	−1.95791	−	0.395800i
$209$		0			0
$210$		0			0
$211$	0.102282	+	0.816190i	0.102282	+	0.816190i	0.955573	+	0.294755i	$0.0952381\pi$
$211$	0.102282	+	0.816190i	0.102282	+	0.816190i	−0.853291	+	0.521435i	$0.825397\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.507752	−	1.51125i	0.507752	−	1.51125i
$218$		0			0
$219$	0.930642	+	0.704934i	0.930642	+	0.704934i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.543524	+	1.61772i	0.543524	+	1.61772i	0.766044	+	0.642788i	$0.222222\pi$
$223$	0.543524	+	1.61772i	0.543524	+	1.61772i	−0.222521	+	0.974928i	$0.571429\pi$
$224$		0			0
$225$	0.921476	+	0.388435i	0.921476	+	0.388435i
$226$		0			0
$227$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$227$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$228$	1.38953	−	0.139044i	1.38953	−	0.139044i
$229$	−1.11562	−	0.344123i	−1.11562	−	0.344123i	−0.318487	−	0.947927i	$-0.603175\pi$
$229$	−1.11562	−	0.344123i	−1.11562	−	0.344123i	−0.797133	+	0.603804i	$0.793651\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.0249307	−	0.999689i	$-0.492063\pi$
$233$		0			0		−0.0249307	+	0.999689i	$0.507937\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.84212	+	0.372393i	−1.84212	+	0.372393i
$238$		0			0
$239$		0			0		−0.980172	−	0.198146i	$-0.936508\pi$
$239$		0			0		0.980172	+	0.198146i	$0.0634921\pi$
$240$		0			0
$241$	1.65381	−	1.01062i	1.65381	−	1.01062i	0.698237	−	0.715867i	$-0.253968\pi$
$241$	1.65381	−	1.01062i	1.65381	−	1.01062i	0.955573	−	0.294755i	$-0.0952381\pi$
$242$		0			0
$243$	0.698237	−	0.715867i	0.698237	−	0.715867i
$244$	−0.0476011	+	0.635192i	−0.0476011	+	0.635192i
$245$		0			0
$246$		0			0
$247$	2.38023	−	1.45453i	2.38023	−	1.45453i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.542546	−	0.840026i	$-0.317460\pi$
$251$		0			0		−0.542546	+	0.840026i	$0.682540\pi$
$252$	0.826239	+	0.563320i	0.826239	+	0.563320i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.826239	−	0.563320i	0.826239	−	0.563320i
$257$		0			0		−0.998757	−	0.0498459i	$-0.984127\pi$
$257$		0			0		0.998757	+	0.0498459i	$0.0158730\pi$
$258$		0			0
$259$	−1.28756	+	1.19468i	−1.28756	+	1.19468i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.995031	−	0.0995678i	$-0.968254\pi$
$263$		0			0		0.995031	+	0.0995678i	$0.0317460\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.52448	−	0.152547i	1.52448	−	0.152547i
$269$		0			0		0.661686	−	0.749781i	$-0.269841\pi$
$269$		0			0		−0.661686	+	0.749781i	$0.730159\pi$
$270$		0			0
$271$	−0.229801	−	0.260396i	−0.229801	−	0.260396i	0.623490	−	0.781831i	$-0.285714\pi$
$271$	−0.229801	−	0.260396i	−0.229801	−	0.260396i	−0.853291	+	0.521435i	$0.825397\pi$
$272$		0			0
$273$	1.97520	+	0.297714i	1.97520	+	0.297714i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.36879	+	1.40335i	1.36879	+	1.40335i	0.826239	+	0.563320i	$0.190476\pi$
$277$	1.36879	+	1.40335i	1.36879	+	1.40335i	0.542546	+	0.840026i	$0.317460\pi$
$278$		0			0
$279$	1.16868	−	1.08438i	1.16868	−	1.08438i
$280$		0			0
$281$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$281$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$282$		0			0
$283$	1.43638	−	1.08802i	1.43638	−	1.08802i	0.456211	−	0.889872i	$-0.349206\pi$
$283$	1.43638	−	1.08802i	1.43638	−	1.08802i	0.980172	−	0.198146i	$-0.0634921\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.797133	−	0.603804i	−0.797133	−	0.603804i
$290$		0			0
$291$	0.331867	−	1.88211i	0.331867	−	1.88211i
$292$	−0.964623	+	0.657669i	−0.964623	+	0.657669i
$293$		0			0		0.878222	−	0.478254i	$-0.158730\pi$
$293$		0			0		−0.878222	+	0.478254i	$0.841270\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−0.661686	+	0.749781i	−0.661686	+	0.749781i
$301$	0.149460			0.149460
$302$		0			0
$303$		0			0
$304$	−0.310745	+	1.36146i	−0.310745	+	1.36146i
$305$		0			0
$306$		0			0
$307$	1.38953	+	0.139044i	1.38953	+	0.139044i	0.766044	−	0.642788i	$-0.222222\pi$
$307$	1.38953	+	0.139044i	1.38953	+	0.139044i	0.623490	+	0.781831i	$0.285714\pi$
$308$		0			0
$309$	−1.31724	+	0.997773i	−1.31724	+	0.997773i
$310$		0			0
$311$		0			0		0.542546	−	0.840026i	$-0.317460\pi$
$311$		0			0		−0.542546	+	0.840026i	$0.682540\pi$
$312$		0			0
$313$	−0.346865	+	1.96717i	−0.346865	+	1.96717i	−0.124344	+	0.992239i	$0.539683\pi$
$313$	−0.346865	+	1.96717i	−0.346865	+	1.96717i	−0.222521	+	0.974928i	$0.571429\pi$
$314$		0			0
$315$		0			0
$316$	0.233690	−	1.86480i	0.233690	−	1.86480i
$317$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$317$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.456211	+	0.889872i	0.456211	+	0.889872i
$325$	−0.541008	+	1.92286i	−0.541008	+	1.92286i
$326$		0			0
$327$	−0.00372615	+	0.0497220i	−0.00372615	+	0.0497220i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.668852	−	0.620604i	−0.668852	−	0.620604i	0.270840	−	0.962624i	$-0.412698\pi$
$331$	−0.668852	−	0.620604i	−0.668852	−	0.620604i	−0.939693	+	0.342020i	$0.888889\pi$
$332$		0			0
$333$	−1.70213	+	0.433415i	−1.70213	+	0.433415i
$334$		0			0
$335$		0			0
$336$	−0.797133	+	0.603804i	−0.797133	+	0.603804i
$337$	−1.11562	+	1.55173i	−1.11562	+	1.55173i	−0.318487	+	0.947927i	$0.603175\pi$
$337$	−1.11562	+	1.55173i	−1.11562	+	1.55173i	−0.797133	+	0.603804i	$0.793651\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.365341	−	0.930874i	0.365341	−	0.930874i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.698237	−	0.715867i	$-0.253968\pi$
$347$		0			0		−0.698237	+	0.715867i	$0.746032\pi$
$348$		0			0
$349$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	−0.900969	−	0.433884i	$-0.857143\pi$
$349$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	0.623490	+	0.781831i	$0.285714\pi$
$350$		0			0
$351$	1.59228	+	1.20611i	1.59228	+	1.20611i
$352$		0			0
$353$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$353$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$359$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$360$		0			0
$361$	−0.475069	−	0.822844i	−0.475069	−	0.822844i
$362$		0			0
$363$	−0.0249307	+	0.999689i	−0.0249307	+	0.999689i
$364$	−0.911287	+	1.77753i	−0.911287	+	1.77753i
$365$		0			0
$366$		0			0
$367$	−1.02531	−	1.42612i	−1.02531	−	1.42612i	−0.900969	−	0.433884i	$-0.857143\pi$
$367$	−1.02531	−	1.42612i	−1.02531	−	1.42612i	−0.124344	−	0.992239i	$-0.539683\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.655701	+	1.45318i	0.655701	+	1.45318i
$373$	−0.795429	+	0.738050i	−0.795429	+	0.738050i	−0.969077	−	0.246757i	$-0.920635\pi$
$373$	−0.795429	+	0.738050i	−0.795429	+	0.738050i	0.173648	+	0.984808i	$0.444444\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.488038	−	0.235027i	−0.488038	−	0.235027i	0.173648	−	0.984808i	$-0.444444\pi$
$379$	−0.488038	−	0.235027i	−0.488038	−	0.235027i	−0.661686	+	0.749781i	$0.730159\pi$
$380$		0			0
$381$	−0.998757	−	0.0498459i	−0.998757	−	0.0498459i
$382$		0			0
$383$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$383$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.131259	+	0.0714799i	0.131259	+	0.0714799i
$388$	1.67841	+	0.914013i	1.67841	+	0.914013i
$389$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$389$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.237639	+	0.0733019i	−0.237639	+	0.0733019i	−0.411287	−	0.911506i	$-0.634921\pi$
$397$	−0.237639	+	0.0733019i	−0.237639	+	0.0733019i	0.173648	+	0.984808i	$0.444444\pi$
$398$		0			0
$399$	0.242495	−	1.37526i	0.242495	−	1.37526i
$400$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	2.43952	+	2.04700i	2.43952	+	2.04700i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.340410	+	1.93056i	0.340410	+	1.93056i	0.365341	+	0.930874i	$0.380952\pi$
$409$	0.340410	+	1.93056i	0.340410	+	1.93056i	−0.0249307	+	0.999689i	$0.507937\pi$
$410$		0			0
$411$		0			0
$412$	−0.526292	−	1.56643i	−0.526292	−	1.56643i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.345587	−	0.535074i	−0.345587	−	0.535074i
$418$		0			0
$419$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$419$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$420$		0			0
$421$	−1.48883	+	1.01507i	−1.48883	+	1.01507i	−0.500000	+	0.866025i	$0.666667\pi$
$421$	−1.48883	+	1.01507i	−1.48883	+	1.01507i	−0.988831	+	0.149042i	$0.952381\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.598559	+	0.217858i	0.598559	+	0.217858i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$431$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$432$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$433$	−0.0772807	−	0.438281i	−0.0772807	−	0.438281i	−0.998757	−	0.0498459i	$-0.984127\pi$
$433$	−0.0772807	−	0.438281i	−0.0772807	−	0.438281i	0.921476	−	0.388435i	$-0.126984\pi$
$434$		0			0
$435$		0			0
$436$	−0.0459461	−	0.0193679i	−0.0459461	−	0.0193679i
$437$		0			0
$438$		0			0
$439$	−0.586956	+	1.74699i	−0.586956	+	1.74699i	0.0747301	+	0.997204i	$0.476190\pi$
$439$	−0.586956	+	1.74699i	−0.586956	+	1.74699i	−0.661686	+	0.749781i	$0.730159\pi$
$440$		0			0
$441$	0.766044	−	0.642788i	0.766044	−	0.642788i
$442$		0			0
$443$		0			0		0.921476	−	0.388435i	$-0.126984\pi$
$443$		0			0		−0.921476	+	0.388435i	$0.873016\pi$
$444$	0.131259	−	1.75153i	0.131259	−	1.75153i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.318487	−	0.947927i	−0.318487	−	0.947927i
$449$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$449$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−1.35098	−	0.569486i	−1.35098	−	0.569486i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.57646	−	0.237613i	1.57646	−	0.237613i	0.698237	−	0.715867i	$-0.253968\pi$
$457$	1.57646	−	0.237613i	1.57646	−	0.237613i	0.878222	+	0.478254i	$0.158730\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$461$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$462$		0			0
$463$	1.28682	−	1.31931i	1.28682	−	1.31931i	0.365341	−	0.930874i	$-0.380952\pi$
$463$	1.28682	−	1.31931i	1.28682	−	1.31931i	0.921476	−	0.388435i	$-0.126984\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.980172	−	0.198146i	$-0.0634921\pi$
$467$		0			0		−0.980172	+	0.198146i	$0.936508\pi$
$468$	−1.65042	+	1.12524i	−1.65042	+	1.12524i
$469$	0.266044	−	1.50881i	0.266044	−	1.50881i
$470$		0			0
$471$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	1.33443	+	0.411618i	1.33443	+	0.411618i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.797133	−	0.603804i	$-0.206349\pi$
$479$		0			0		−0.797133	+	0.603804i	$0.793651\pi$
$480$		0			0
$481$	−1.28181	−	3.26599i	−1.28181	−	3.26599i
$482$		0			0
$483$		0			0
$484$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$485$		0			0
$486$		0			0
$487$	1.74771	−	0.951755i	1.74771	−	0.951755i	0.826239	−	0.563320i	$-0.190476\pi$
$487$	1.74771	−	0.951755i	1.74771	−	0.951755i	0.921476	−	0.388435i	$-0.126984\pi$
$488$		0			0
$489$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i
$490$		0			0
$491$		0			0		0.980172	−	0.198146i	$-0.0634921\pi$
$491$		0			0		−0.980172	+	0.198146i	$0.936508\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.58634	+	0.158738i	−1.58634	+	0.158738i
$497$		0			0
$498$		0			0
$499$	0.259790	−	0.361346i	0.259790	−	0.361346i	−0.661686	−	0.749781i	$-0.730159\pi$
$499$	0.259790	−	0.361346i	0.259790	−	0.361346i	0.921476	+	0.388435i	$0.126984\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$503$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.49503	+	2.58947i	−1.49503	+	2.58947i
$508$	0.365341	−	0.930874i	0.365341	−	0.930874i
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$	0.371829	+	1.10669i	0.371829	+	1.10669i
$512$		0			0
$513$	0.870687	−	1.09181i	0.870687	−	1.09181i
$514$		0			0
$515$		0			0
$516$	−0.109562	+	0.101659i	−0.109562	+	0.101659i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.853291	−	0.521435i	$-0.174603\pi$
$521$		0			0		−0.853291	+	0.521435i	$0.825397\pi$
$522$		0			0
$523$	0.261979	+	0.779741i	0.261979	+	0.779741i	0.995031	+	0.0995678i	$0.0317460\pi$
$523$	0.261979	+	0.779741i	0.261979	+	0.779741i	−0.733052	+	0.680173i	$0.761905\pi$
$524$		0			0
$525$	0.542546	+	0.840026i	0.542546	+	0.840026i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.878222	−	0.478254i	0.878222	−	0.478254i
$530$		0			0
$531$		0			0
$532$	1.22641	+	0.667869i	1.22641	+	0.667869i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.379750	+	1.66379i	0.379750	+	1.66379i	0.698237	+	0.715867i	$0.253968\pi$
$541$	0.379750	+	1.66379i	0.379750	+	1.66379i	−0.318487	+	0.947927i	$0.603175\pi$
$542$		0			0
$543$	−0.0431841	+	0.344601i	−0.0431841	+	0.344601i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.54255	−	0.840026i	1.54255	−	0.840026i	0.542546	−	0.840026i	$-0.317460\pi$
$547$	1.54255	−	0.840026i	1.54255	−	0.840026i	1.00000			$0$
$548$		0			0
$549$	0.421476	+	0.477591i	0.421476	+	0.477591i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.73181	−	0.730019i	−1.73181	−	0.730019i
$554$		0			0
$555$		0			0
$556$	0.617276	−	0.157178i	0.617276	−	0.157178i
$557$		0			0		0.456211	−	0.889872i	$-0.349206\pi$
$557$		0			0		−0.456211	+	0.889872i	$0.650794\pi$
$558$		0			0
$559$	−0.109072	+	0.277911i	−0.109072	+	0.277911i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.124344	−	0.992239i	$-0.539683\pi$
$563$		0			0		0.124344	+	0.992239i	$0.460317\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.980172	−	0.198146i	0.980172	−	0.198146i
$568$		0			0
$569$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$569$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$570$		0			0
$571$	−0.0185844	+	0.247992i	−0.0185844	+	0.247992i	0.980172	+	0.198146i	$0.0634921\pi$
$571$	−0.0185844	+	0.247992i	−0.0185844	+	0.247992i	−0.998757	+	0.0498459i	$0.984127\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.173648	−	0.984808i	0.173648	−	0.984808i
$577$	1.22226	−	0.247084i	1.22226	−	0.247084i	0.456211	−	0.889872i	$-0.349206\pi$
$577$	1.22226	−	0.247084i	1.22226	−	0.247084i	0.766044	+	0.642788i	$0.222222\pi$
$578$		0			0
$579$	1.28682	+	1.31931i	1.28682	+	1.31931i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.698237	−	0.715867i	$-0.746032\pi$
$587$		0			0		0.698237	+	0.715867i	$0.253968\pi$
$588$	0.365341	+	0.930874i	0.365341	+	0.930874i
$589$	1.47314	−	1.66927i	1.47314	−	1.66927i
$590$		0			0
$591$		0			0
$592$	1.61852	+	0.682264i	1.61852	+	0.682264i
$593$		0			0		0.456211	−	0.889872i	$-0.349206\pi$
$593$		0			0		−0.456211	+	0.889872i	$0.650794\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1.09342	+	0.745482i	−1.09342	+	0.745482i
$598$		0			0
$599$		0			0		−0.542546	−	0.840026i	$-0.682540\pi$
$599$		0			0		0.542546	+	0.840026i	$0.317460\pi$
$600$		0			0
$601$	−1.05154	−	0.267755i	−1.05154	−	0.267755i	−0.318487	−	0.947927i	$-0.603175\pi$
$601$	−1.05154	−	0.267755i	−1.05154	−	0.267755i	−0.733052	+	0.680173i	$0.761905\pi$
$602$		0			0
$603$	0.955242	−	1.19784i	0.955242	−	1.19784i
$604$	0.970100	−	1.09926i	0.970100	−	1.09926i
$605$		0			0
$606$		0			0
$607$	−0.686617	−	0.249908i	−0.686617	−	0.249908i	−0.0249307	−	0.999689i	$-0.507937\pi$
$607$	−0.686617	−	0.249908i	−0.686617	−	0.249908i	−0.661686	+	0.749781i	$0.730159\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.630128	−	0.528741i	−0.630128	−	0.528741i	0.270840	−	0.962624i	$-0.412698\pi$
$613$	−0.630128	−	0.528741i	−0.630128	−	0.528741i	−0.900969	+	0.433884i	$0.857143\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.0249307	−	0.999689i	$-0.492063\pi$
$617$		0			0		−0.0249307	+	0.999689i	$0.507937\pi$
$618$		0			0
$619$	−1.21135	−	1.12397i	−1.21135	−	1.12397i	−0.988831	−	0.149042i	$-0.952381\pi$
$619$	−1.21135	−	1.12397i	−1.21135	−	1.12397i	−0.222521	−	0.974928i	$-0.571429\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−0.541008	−	1.92286i	−0.541008	−	1.92286i
$625$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$626$		0			0
$627$		0			0
$628$	1.60366	−	0.979978i	1.60366	−	0.979978i
$629$		0			0
$630$		0			0
$631$	−1.20842	−	1.51531i	−1.20842	−	1.51531i	−0.797133	−	0.603804i	$-0.793651\pi$
$631$	−1.20842	−	1.51531i	−1.20842	−	1.51531i	−0.411287	−	0.911506i	$-0.634921\pi$
$632$		0			0
$633$	−0.679643	+	0.463373i	−0.679643	+	0.463373i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.46428	+	1.35865i	1.46428	+	1.35865i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.0249307	−	0.999689i	$-0.507937\pi$
$641$		0			0		0.0249307	+	0.999689i	$0.492063\pi$
$642$		0			0
$643$	0.440071	+	1.92808i	0.440071	+	1.92808i	0.365341	+	0.930874i	$0.380952\pi$
$643$	0.440071	+	1.92808i	0.440071	+	1.92808i	0.0747301	+	0.997204i	$0.476190\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.411287	−	0.911506i	$-0.365079\pi$
$647$		0			0		−0.411287	+	0.911506i	$0.634921\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	1.57646	−	0.237613i	1.57646	−	0.237613i
$652$	0.146497	−	0.0296150i	0.146497	−	0.0296150i
$653$		0			0		0.878222	−	0.478254i	$-0.158730\pi$
$653$		0			0		−0.878222	+	0.478254i	$0.841270\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−0.202732	+	1.14975i	−0.202732	+	1.14975i
$658$		0			0
$659$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$659$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$660$		0			0
$661$	−1.92852	+	0.192978i	−1.92852	+	0.192978i	−0.988831	−	0.149042i	$-0.952381\pi$
$661$	−1.92852	+	0.192978i	−1.92852	+	0.192978i	−0.939693	+	0.342020i	$0.888889\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−1.19160	+	1.22169i	−1.19160	+	1.22169i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.0332580	−	0.443797i	−0.0332580	−	0.443797i	−0.988831	−	0.149042i	$-0.952381\pi$
$673$	−0.0332580	−	0.443797i	−0.0332580	−	0.443797i	0.955573	−	0.294755i	$-0.0952381\pi$
$674$		0			0
$675$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$676$	−1.97848	−	2.24189i	−1.97848	−	2.24189i
$677$		0			0		0.456211	−	0.889872i	$-0.349206\pi$
$677$		0			0		−0.456211	+	0.889872i	$0.650794\pi$
$678$		0			0
$679$	1.33443	−	1.36813i	1.33443	−	1.36813i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.969077	−	0.246757i	$-0.0793651\pi$
$683$		0			0		−0.969077	+	0.246757i	$0.920635\pi$
$684$	0.757652	+	1.17307i	0.757652	+	1.17307i
$685$		0			0
$686$		0			0
$687$	−0.202732	−	1.14975i	−0.202732	−	1.14975i
$688$	−0.0614710	−	0.136234i	−0.0614710	−	0.136234i
$689$		0			0
$690$		0			0
$691$	−1.11317	−	1.14128i	−1.11317	−	1.14128i	−0.988831	−	0.149042i	$-0.952381\pi$
$691$	−1.11317	−	1.14128i	−1.11317	−	1.14128i	−0.124344	−	0.992239i	$-0.539683\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.969077	+	0.246757i	−0.969077	+	0.246757i
$701$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$701$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$702$		0			0
$703$	−2.30490	+	0.838916i	−2.30490	+	0.838916i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.806265	−	1.78687i	−0.806265	−	1.78687i	−0.583744	−	0.811938i	$-0.698413\pi$
$709$	−0.806265	−	1.78687i	−0.806265	−	1.78687i	−0.222521	−	0.974928i	$-0.571429\pi$
$710$		0			0
$711$	−1.17178	−	1.46936i	−1.17178	−	1.46936i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.797133	−	0.603804i	$-0.206349\pi$
$719$		0			0		−0.797133	+	0.603804i	$0.793651\pi$
$720$		0			0
$721$	−1.65042	+	0.0823692i	−1.65042	+	0.0823692i
$722$		0			0
$723$	1.65381	+	1.01062i	1.65381	+	1.01062i
$724$	−0.312903	−	0.150686i	−0.312903	−	0.150686i
$725$		0			0
$726$		0			0
$727$	0.840775	−	0.354416i	0.840775	−	0.354416i	0.0747301	−	0.997204i	$-0.476190\pi$
$727$	0.840775	−	0.354416i	0.840775	−	0.354416i	0.766044	+	0.642788i	$0.222222\pi$
$728$		0			0
$729$	0.955573	+	0.294755i	0.955573	+	0.294755i
$730$		0			0
$731$		0			0
$732$	−0.586956	+	0.247423i	−0.586956	+	0.247423i
$733$	−0.247452	−	1.97462i	−0.247452	−	1.97462i	−0.222521	−	0.974928i	$-0.571429\pi$
$733$	−0.247452	−	1.97462i	−0.247452	−	1.97462i	−0.0249307	−	0.999689i	$-0.507937\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.870687	+	0.892671i	0.870687	+	0.892671i	0.995031	−	0.0995678i	$-0.0317460\pi$
$739$	0.870687	+	0.892671i	0.870687	+	0.892671i	−0.124344	+	0.992239i	$0.539683\pi$
$740$		0			0
$741$	2.38023	+	1.45453i	2.38023	+	1.45453i
$742$		0			0
$743$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$743$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.242495	−	0.248618i	0.242495	−	0.248618i	−0.583744	−	0.811938i	$-0.698413\pi$
$751$	0.242495	−	0.248618i	0.242495	−	0.248618i	0.826239	+	0.563320i	$0.190476\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.124344	+	0.992239i	−0.124344	+	0.992239i
$757$	−0.658322	−	1.67738i	−0.658322	−	1.67738i	−0.733052	−	0.680173i	$-0.761905\pi$
$757$	−0.658322	−	1.67738i	−0.658322	−	1.67738i	0.0747301	−	0.997204i	$-0.476190\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$	−0.0310881	+	0.0389832i	−0.0310881	+	0.0389832i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.878222	+	0.478254i	0.878222	+	0.478254i
$769$	−0.0381960	−	1.53161i	−0.0381960	−	1.53161i	−0.661686	−	0.749781i	$-0.730159\pi$
$769$	−0.0381960	−	1.53161i	−0.0381960	−	1.53161i	0.623490	−	0.781831i	$-0.285714\pi$
$770$		0			0
$771$		0			0
$772$	−1.66044	+	0.799627i	−1.66044	+	0.799627i
$773$		0			0		−0.969077	−	0.246757i	$-0.920635\pi$
$773$		0			0		0.969077	+	0.246757i	$0.0793651\pi$
$774$		0			0
$775$	0.0397461	+	1.59377i	0.0397461	+	1.59377i
$776$		0			0
$777$	−1.65052	−	0.600739i	−1.65052	−	0.600739i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.998757	+	0.0498459i	−0.998757	+	0.0498459i
$785$		0			0
$786$		0			0
$787$	−0.436218	−	1.91120i	−0.436218	−	1.91120i	−0.411287	−	0.911506i	$-0.634921\pi$
$787$	−0.436218	−	1.91120i	−0.436218	−	1.91120i	−0.0249307	−	0.999689i	$-0.507937\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.841905	+	0.953994i	−0.841905	+	0.953994i
$794$		0			0
$795$		0			0
$796$	−0.358423	−	1.27391i	−0.358423	−	1.27391i
$797$		0			0		0.921476	−	0.388435i	$-0.126984\pi$
$797$		0			0		−0.921476	+	0.388435i	$0.873016\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	0.831229	+	1.28699i	0.831229	+	1.28699i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$809$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$810$		0			0
$811$	−1.98759	+	0.0991964i	−1.98759	+	0.0991964i	−0.998757	−	0.0498459i	$-0.984127\pi$
$811$	−1.98759	+	0.0991964i	−1.98759	+	0.0991964i	−0.988831	+	0.149042i	$0.952381\pi$
$812$		0			0
$813$	0.126882	−	0.323289i	0.126882	−	0.323289i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.192328	+	0.0810730i	0.192328	+	0.0810730i
$818$		0			0
$819$	0.636181	+	1.89350i	0.636181	+	1.89350i
$820$		0			0
$821$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$821$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$822$		0			0
$823$	1.33443	−	0.411618i	1.33443	−	0.411618i	0.456211	−	0.889872i	$-0.349206\pi$
$823$	1.33443	−	0.411618i	1.33443	−	0.411618i	0.878222	+	0.478254i	$0.158730\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.124344	−	0.992239i	$-0.460317\pi$
$827$		0			0		−0.124344	+	0.992239i	$0.539683\pi$
$828$		0			0
$829$	−0.541008	−	0.0270006i	−0.541008	−	0.0270006i	−0.222521	−	0.974928i	$-0.571429\pi$
$829$	−0.541008	−	0.0270006i	−0.541008	−	0.0270006i	−0.318487	+	0.947927i	$0.603175\pi$
$830$		0			0
$831$	−0.624344	+	1.85826i	−0.624344	+	1.85826i
$832$	1.99503	+	0.0995678i	1.99503	+	0.0995678i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.49812	+	0.545271i	1.49812	+	0.545271i
$838$		0			0
$839$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$839$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$840$		0			0
$841$	0.698237	+	0.715867i	0.698237	+	0.715867i
$842$		0			0
$843$		0			0
$844$	−0.222786	−	0.791830i	−0.222786	−	0.791830i
$845$		0			0
$846$		0			0
$847$	−0.583744	+	0.811938i	−0.583744	+	0.811938i
$848$		0			0
$849$	1.62349	+	0.781831i	1.62349	+	0.781831i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.416256	−	0.811938i	0.416256	−	0.811938i	−0.583744	−	0.811938i	$-0.698413\pi$
$853$	0.416256	−	0.811938i	0.416256	−	0.811938i	1.00000			$0$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$857$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$858$		0			0
$859$	−0.0673546	+	0.537477i	−0.0673546	+	0.537477i	0.921476	+	0.388435i	$0.126984\pi$
$859$	−0.0673546	+	0.537477i	−0.0673546	+	0.537477i	−0.988831	+	0.149042i	$0.952381\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.411287	−	0.911506i	$-0.365079\pi$
$863$		0			0		−0.411287	+	0.911506i	$0.634921\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.173648	−	0.984808i	0.173648	−	0.984808i
$868$	−0.276841	+	1.57004i	−0.276841	+	1.57004i
$869$		0			0
$870$		0			0
$871$	2.61138	+	1.59578i	2.61138	+	1.59578i
$872$		0			0
$873$	1.82624	−	0.563320i	1.82624	−	0.563320i
$874$		0			0
$875$		0			0
$876$	−1.02531	−	0.558355i	−1.02531	−	0.558355i
$877$	−0.426531	−	0.593269i	−0.426531	−	0.593269i	0.542546	−	0.840026i	$-0.317460\pi$
$877$	−0.426531	−	0.593269i	−0.426531	−	0.593269i	−0.969077	+	0.246757i	$0.920635\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$881$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$882$		0			0
$883$	−1.67274	+	1.02219i	−1.67274	+	1.02219i	−0.733052	+	0.680173i	$0.761905\pi$
$883$	−1.67274	+	1.02219i	−1.67274	+	1.02219i	−0.939693	+	0.342020i	$0.888889\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$887$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$888$		0			0
$889$	−0.797133	−	0.603804i	−0.797133	−	0.603804i
$890$		0			0
$891$		0			0
$892$	−0.778561	−	1.51864i	−0.778561	−	1.51864i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.969077	−	0.246757i	−0.969077	−	0.246757i
$901$		0			0
$902$		0			0
$903$	0.0681853	+	0.133000i	0.0681853	+	0.133000i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.45621	+	0.889872i	1.45621	+	0.889872i	1.00000			$0$
$907$	1.45621	+	0.889872i	1.45621	+	0.889872i	0.456211	+	0.889872i	$0.349206\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$911$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$912$	−1.35329	+	0.344590i	−1.35329	+	0.344590i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.15445	+	0.174005i	1.15445	+	0.174005i
$917$		0			0
$918$		0			0
$919$	0.145170	−	1.15843i	0.145170	−	1.15843i	−0.733052	−	0.680173i	$-0.761905\pi$
$919$	0.145170	−	1.15843i	0.145170	−	1.15843i	0.878222	−	0.478254i	$-0.158730\pi$
$920$		0			0
$921$	0.510189	+	1.29994i	0.510189	+	1.29994i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.801308	−	1.56301i	0.801308	−	1.56301i
$926$		0			0
$927$	−1.48883	−	0.716983i	−1.48883	−	0.716983i
$928$		0			0
$929$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$929$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$930$		0			0
$931$	0.975069	−	0.999689i	0.975069	−	0.999689i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.574352	−	0.588854i	−0.574352	−	0.588854i	0.365341	−	0.930874i	$-0.380952\pi$
$937$	−0.574352	−	0.588854i	−0.574352	−	0.588854i	−0.939693	+	0.342020i	$0.888889\pi$
$938$		0			0
$939$	−1.90877	+	0.588778i	−1.90877	+	0.588778i
$940$		0			0
$941$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$941$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.318487	−	0.947927i	$-0.396825\pi$
$947$		0			0		−0.318487	+	0.947927i	$0.603175\pi$
$948$	1.76604	−	0.642788i	1.76604	−	0.642788i
$949$	−2.32917	−	0.116244i	−2.32917	−	0.116244i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$953$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1.42062	+	0.598843i	1.42062	+	0.598843i
$962$		0			0
$963$		0			0
$964$	−1.48471	+	1.24582i	−1.48471	+	1.24582i
$965$		0			0
$966$		0			0
$967$	0.821552	−	0.0410019i	0.821552	−	0.0410019i	0.365341	−	0.930874i	$-0.380952\pi$
$967$	0.821552	−	0.0410019i	0.821552	−	0.0410019i	0.456211	+	0.889872i	$0.349206\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.969077	−	0.246757i	$-0.920635\pi$
$971$		0			0		0.969077	+	0.246757i	$0.0793651\pi$
$972$	−0.583744	+	0.811938i	−0.583744	+	0.811938i
$973$	0.0158802	−	0.636775i	0.0158802	−	0.636775i
$974$		0			0
$975$	−1.95791	+	0.395800i	−1.95791	+	0.395800i
$976$	−0.0476011	−	0.635192i	−0.0476011	−	0.635192i
$977$		0			0		0.0249307	−	0.999689i	$-0.492063\pi$
$977$		0			0		−0.0249307	+	0.999689i	$0.507937\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.0459461	+	0.0193679i	−0.0459461	+	0.0193679i
$982$		0			0
$983$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$983$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−2.13686	+	1.79304i	−2.13686	+	1.79304i
$989$		0			0
$990$		0			0
$991$	0.431280	+	1.88956i	0.431280	+	1.88956i	0.456211	+	0.889872i	$0.349206\pi$
$991$	0.431280	+	1.88956i	0.431280	+	1.88956i	−0.0249307	+	0.999689i	$0.507937\pi$
$992$		0			0
$993$	0.247121	−	0.878319i	0.247121	−	0.878319i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.541681			0.541681			0.270840	−	0.962624i	$-0.412698\pi$
$997$	0.541681			0.541681			0.270840	+	0.962624i	$0.412698\pi$
$998$		0			0
$999$	−1.16221	−	1.31695i	−1.16221	−	1.31695i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2667.1.em.a.1187.1	yes	36
3.2	odd	2	CM	2667.1.em.a.1187.1	yes	36
7.2	even	3		2667.1.ej.a.44.1	✓	36
21.2	odd	6		2667.1.ej.a.44.1	✓	36
127.26	even	63		2667.1.ej.a.788.1	yes	36
381.26	odd	126		2667.1.ej.a.788.1	yes	36
889.534	even	63	inner	2667.1.em.a.2312.1	yes	36
2667.2312	odd	126	inner	2667.1.em.a.2312.1	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.1.ej.a.44.1	✓	36	7.2	even	3
2667.1.ej.a.44.1	✓	36	21.2	odd	6
2667.1.ej.a.788.1	yes	36	127.26	even	63
2667.1.ej.a.788.1	yes	36	381.26	odd	126
2667.1.em.a.1187.1	yes	36	1.1	even	1	trivial
2667.1.em.a.1187.1	yes	36	3.2	odd	2	CM
2667.1.em.a.2312.1	yes	36	889.534	even	63	inner
2667.1.em.a.2312.1	yes	36	2667.2312	odd	126	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:	\( N \)	\(=\)	\( 2667 = 3 \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2667.em (of order \(126\), degree \(36\), minimal)

\(f(q)\)	\(=\)	\(q+(0.456211 + 0.889872i) q^{3} +(-0.988831 + 0.149042i) q^{4} +(-0.124344 + 0.992239i) q^{7} +(-0.583744 + 0.811938i) q^{9} +O(q^{10})\)	\(q+(0.456211 + 0.889872i) q^{3} +(-0.988831 + 0.149042i) q^{4} +(-0.124344 + 0.992239i) q^{7} +(-0.583744 + 0.811938i) q^{9} +(-0.583744 - 0.811938i) q^{12} +(-1.75426 - 0.955319i) q^{13} +(0.955573 - 0.294755i) q^{16} +(-0.698237 + 1.20938i) q^{19} +(-0.939693 + 0.342020i) q^{21} +(-0.222521 - 0.974928i) q^{25} +(-0.988831 - 0.149042i) q^{27} +(-0.0249307 - 0.999689i) q^{28} +(-1.56265 - 0.315897i) q^{31} +(0.456211 - 0.889872i) q^{36} +(1.34551 + 1.12902i) q^{37} +(0.0497994 - 1.99689i) q^{39} +(-0.0185844 - 0.148300i) q^{43} +(0.698237 + 0.715867i) q^{48} +(-0.969077 - 0.246757i) q^{49} +(1.87705 + 0.683190i) q^{52} +(-1.39474 - 0.0696085i) q^{57} +(0.141740 - 0.621003i) q^{61} +(-0.733052 - 0.680173i) q^{63} +(-0.900969 + 0.433884i) q^{64} +(-1.53018 - 0.0763683i) q^{67} +(1.05187 - 0.506554i) q^{73} +(0.766044 - 0.642788i) q^{75} +(0.510189 - 1.29994i) q^{76} +(-0.509014 + 1.80914i) q^{79} +(-0.318487 - 0.947927i) q^{81} +(0.878222 - 0.478254i) q^{84} +(1.16604 - 1.62186i) q^{91} +(-0.431791 - 1.53468i) q^{93} +(-1.52344 - 1.15396i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 3 q^{4}+O(q^{10}) \) 36 * q + 3 * q^4	\( 36 q + 3 q^{4} - 6 q^{13} + 3 q^{16} - 6 q^{25} + 3 q^{27} - 6 q^{31} + 3 q^{37} + 3 q^{39} + 3 q^{52} + 3 q^{57} + 3 q^{63} - 6 q^{64} + 3 q^{67} + 3 q^{79} - 6 q^{91} + 3 q^{93}+O(q^{100}) \) 36 * q + 3 * q^4 - 6 * q^13 + 3 * q^16 - 6 * q^25 + 3 * q^27 - 6 * q^31 + 3 * q^37 + 3 * q^39 + 3 * q^52 + 3 * q^57 + 3 * q^63 - 6 * q^64 + 3 * q^67 + 3 * q^79 - 6 * q^91 + 3 * q^93

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