LMFDB - Embedded newform 2667.1.em.a.1031.1

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			1031.1
Root			$0.995031 - 0.0995678i$ of defining polynomial
Character	$\chi$	$=$	2667.1031
Dual form			2667.1.em.a.1922.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.995031 + 0.0995678i) q^{3} +(0.955573 - 0.294755i) q^{4} +(0.270840 + 0.962624i) q^{7} +(0.980172 + 0.198146i) q^{9} +O(q^{10})$	\(q+(0.995031 + 0.0995678i) q^{3} +(0.955573 - 0.294755i) q^{4} +(0.270840 + 0.962624i) q^{7} +(0.980172 + 0.198146i) q^{9} +(0.980172 - 0.198146i) q^{12} +(-0.375267 + 0.731986i) q^{13} +(0.826239 - 0.563320i) q^{16} +(-0.878222 - 1.52112i) q^{19} +(0.173648 + 0.984808i) q^{21} +(-0.900969 + 0.433884i) q^{25} +(0.955573 + 0.294755i) q^{27} +(0.542546 + 0.840026i) q^{28} +(-0.173643 - 1.38564i) q^{31} +(0.995031 - 0.0995678i) q^{36} +(-0.857396 - 0.312066i) q^{37} +(-0.446285 + 0.690984i) q^{39} +(-0.535631 + 1.90375i) q^{43} +(0.878222 - 0.478254i) q^{48} +(-0.853291 + 0.521435i) q^{49} +(-0.142839 + 0.810077i) q^{52} +(-0.722402 - 1.60101i) q^{57} +(-1.66044 - 0.799627i) q^{61} +(0.0747301 + 0.997204i) q^{63} +(0.623490 - 0.781831i) q^{64} +(0.772967 + 1.71307i) q^{67} +(1.22226 - 1.53266i) q^{73} +(-0.939693 + 0.342020i) q^{75} +(-1.28756 - 1.19468i) q^{76} +(-0.00865834 + 0.347188i) q^{79} +(0.921476 + 0.388435i) q^{81} +(0.456211 + 0.889872i) q^{84} +(-0.806265 - 0.162990i) q^{91} +(-0.0348151 - 1.39604i) q^{93} +(1.15382 - 1.18295i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2667\mathbb{Z}\right)^\times$.

$n$	$890$	$1144$	$2416$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{44}{63}\right)$

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		0.988831	−	0.149042i	$-0.0476190\pi$
$2$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$3$	0.995031	+	0.0995678i	0.995031	+	0.0995678i
$3$	0.995031	+	0.0995678i	0.995031	+	0.0995678i
$4$	0.955573	−	0.294755i	0.955573	−	0.294755i
$5$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$5$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$6$		0			0
$7$	0.270840	+	0.962624i	0.270840	+	0.962624i
$7$	0.270840	+	0.962624i	0.270840	+	0.962624i
$8$		0			0
$9$	0.980172	+	0.198146i	0.980172	+	0.198146i
$10$		0			0
$11$		0			0		0.853291	−	0.521435i	$-0.174603\pi$
$11$		0			0		−0.853291	+	0.521435i	$0.825397\pi$
$12$	0.980172	−	0.198146i	0.980172	−	0.198146i
$13$	−0.375267	+	0.731986i	−0.375267	+	0.731986i	−0.998757	−	0.0498459i	$-0.984127\pi$
$13$	−0.375267	+	0.731986i	−0.375267	+	0.731986i	0.623490	+	0.781831i	$0.285714\pi$
$14$		0			0
$15$		0			0
$16$	0.826239	−	0.563320i	0.826239	−	0.563320i
$17$		0			0		0.921476	−	0.388435i	$-0.126984\pi$
$17$		0			0		−0.921476	+	0.388435i	$0.873016\pi$
$18$		0			0
$19$	−0.878222	−	1.52112i	−0.878222	−	1.52112i	−0.853291	−	0.521435i	$-0.825397\pi$
$19$	−0.878222	−	1.52112i	−0.878222	−	1.52112i	−0.0249307	−	0.999689i	$-0.507937\pi$
$20$		0			0
$21$	0.173648	+	0.984808i	0.173648	+	0.984808i
$22$		0			0
$23$		0			0		−0.853291	−	0.521435i	$-0.825397\pi$
$23$		0			0		0.853291	+	0.521435i	$0.174603\pi$
$24$		0			0
$25$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$26$		0			0
$27$	0.955573	+	0.294755i	0.955573	+	0.294755i
$28$	0.542546	+	0.840026i	0.542546	+	0.840026i
$29$		0			0		0.969077	−	0.246757i	$-0.0793651\pi$
$29$		0			0		−0.969077	+	0.246757i	$0.920635\pi$
$30$		0			0
$31$	−0.173643	−	1.38564i	−0.173643	−	1.38564i	−0.797133	−	0.603804i	$-0.793651\pi$
$31$	−0.173643	−	1.38564i	−0.173643	−	1.38564i	0.623490	−	0.781831i	$-0.285714\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.995031	−	0.0995678i	0.995031	−	0.0995678i
$37$	−0.857396	−	0.312066i	−0.857396	−	0.312066i	−0.124344	−	0.992239i	$-0.539683\pi$
$37$	−0.857396	−	0.312066i	−0.857396	−	0.312066i	−0.733052	+	0.680173i	$0.761905\pi$
$38$		0			0
$39$	−0.446285	+	0.690984i	−0.446285	+	0.690984i
$40$		0			0
$41$		0			0		0.921476	−	0.388435i	$-0.126984\pi$
$41$		0			0		−0.921476	+	0.388435i	$0.873016\pi$
$42$		0			0
$43$	−0.535631	+	1.90375i	−0.535631	+	1.90375i	−0.124344	+	0.992239i	$0.539683\pi$
$43$	−0.535631	+	1.90375i	−0.535631	+	1.90375i	−0.411287	+	0.911506i	$0.634921\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$47$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$48$	0.878222	−	0.478254i	0.878222	−	0.478254i
$49$	−0.853291	+	0.521435i	−0.853291	+	0.521435i
$50$		0			0
$51$		0			0
$52$	−0.142839	+	0.810077i	−0.142839	+	0.810077i
$53$		0			0		−0.980172	−	0.198146i	$-0.936508\pi$
$53$		0			0		0.980172	+	0.198146i	$0.0634921\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.722402	−	1.60101i	−0.722402	−	1.60101i
$58$		0			0
$59$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$59$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$60$		0			0
$61$	−1.66044	−	0.799627i	−1.66044	−	0.799627i	−0.998757	−	0.0498459i	$-0.984127\pi$
$61$	−1.66044	−	0.799627i	−1.66044	−	0.799627i	−0.661686	−	0.749781i	$-0.730159\pi$
$62$		0			0
$63$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$64$	0.623490	−	0.781831i	0.623490	−	0.781831i
$65$		0			0
$66$		0			0
$67$	0.772967	+	1.71307i	0.772967	+	1.71307i	0.698237	+	0.715867i	$0.253968\pi$
$67$	0.772967	+	1.71307i	0.772967	+	1.71307i	0.0747301	+	0.997204i	$0.476190\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.853291	−	0.521435i	$-0.825397\pi$
$71$		0			0		0.853291	+	0.521435i	$0.174603\pi$
$72$		0			0
$73$	1.22226	−	1.53266i	1.22226	−	1.53266i	0.456211	−	0.889872i	$-0.349206\pi$
$73$	1.22226	−	1.53266i	1.22226	−	1.53266i	0.766044	−	0.642788i	$-0.222222\pi$
$74$		0			0
$75$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$76$	−1.28756	−	1.19468i	−1.28756	−	1.19468i
$77$		0			0
$78$		0			0
$79$	−0.00865834	+	0.347188i	−0.00865834	+	0.347188i	0.980172	+	0.198146i	$0.0634921\pi$
$79$	−0.00865834	+	0.347188i	−0.00865834	+	0.347188i	−0.988831	+	0.149042i	$0.952381\pi$
$80$		0			0
$81$	0.921476	+	0.388435i	0.921476	+	0.388435i
$82$		0			0
$83$		0			0		−0.980172	−	0.198146i	$-0.936508\pi$
$83$		0			0		0.980172	+	0.198146i	$0.0634921\pi$
$84$	0.456211	+	0.889872i	0.456211	+	0.889872i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$89$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$90$		0			0
$91$	−0.806265	−	0.162990i	−0.806265	−	0.162990i
$92$		0			0
$93$	−0.0348151	−	1.39604i	−0.0348151	−	1.39604i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.15382	−	1.18295i	1.15382	−	1.18295i	0.173648	−	0.984808i	$-0.444444\pi$
$97$	1.15382	−	1.18295i	1.15382	−	1.18295i	0.980172	−	0.198146i	$-0.0634921\pi$
$98$		0			0
$99$		0			0
$100$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$101$		0			0		−0.661686	−	0.749781i	$-0.730159\pi$
$101$		0			0		0.661686	+	0.749781i	$0.269841\pi$
$102$		0			0
$103$	0.559735	+	0.469673i	0.559735	+	0.469673i	0.878222	−	0.478254i	$-0.158730\pi$
$103$	0.559735	+	0.469673i	0.559735	+	0.469673i	−0.318487	+	0.947927i	$0.603175\pi$
$104$		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
$109$	−1.08374	−	0.0540874i	−1.08374	−	0.0540874i	−0.500000	−	0.866025i	$-0.666667\pi$
$109$	−1.08374	−	0.0540874i	−1.08374	−	0.0540874i	−0.583744	+	0.811938i	$0.698413\pi$
$110$		0			0
$111$	−0.822063	−	0.395885i	−0.822063	−	0.395885i
$112$	0.766044	+	0.642788i	0.766044	+	0.642788i
$113$		0			0		0.318487	−	0.947927i	$-0.396825\pi$
$113$		0			0		−0.318487	+	0.947927i	$0.603175\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.512867	+	0.643114i	−0.512867	+	0.643114i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.456211	−	0.889872i	0.456211	−	0.889872i
$122$		0			0
$123$		0			0
$124$	−0.574352	−	1.27289i	−0.574352	−	1.27289i
$125$		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
$129$	−0.722521	+	1.84095i	−0.722521	+	1.84095i
$130$		0			0
$131$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$131$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$132$		0			0
$133$	1.22641	−	1.25738i	1.22641	−	1.25738i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$137$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$138$		0			0
$139$	−1.21946	−	1.38181i	−1.21946	−	1.38181i	−0.900969	−	0.433884i	$-0.857143\pi$
$139$	−1.21946	−	1.38181i	−1.21946	−	1.38181i	−0.318487	−	0.947927i	$-0.603175\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.921476	−	0.388435i	0.921476	−	0.388435i
$145$		0			0
$146$		0			0
$147$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$148$	−0.911287	−	0.0454804i	−0.911287	−	0.0454804i
$149$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$149$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$150$		0			0
$151$	−0.140447	+	0.0511184i	−0.140447	+	0.0511184i	−0.411287	−	0.911506i	$-0.634921\pi$
$151$	−0.140447	+	0.0511184i	−0.140447	+	0.0511184i	0.270840	+	0.962624i	$0.412698\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−0.222786	+	0.791830i	−0.222786	+	0.791830i
$157$	−0.336557	−	0.0856979i	−0.336557	−	0.0856979i	0.0747301	−	0.997204i	$-0.476190\pi$
$157$	−0.336557	−	0.0856979i	−0.336557	−	0.0856979i	−0.411287	+	0.911506i	$0.634921\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.813387	−	1.80265i	0.813387	−	1.80265i	0.270840	−	0.962624i	$-0.412698\pi$
$163$	0.813387	−	1.80265i	0.813387	−	1.80265i	0.542546	−	0.840026i	$-0.317460\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$167$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$168$		0			0
$169$	0.188766	+	0.262558i	0.188766	+	0.262558i
$170$		0			0
$171$	−0.559404	−	1.66498i	−0.559404	−	1.66498i
$172$	0.0493045	+	1.97705i	0.0493045	+	1.97705i
$173$		0			0		−0.124344	−	0.992239i	$-0.539683\pi$
$173$		0			0		0.124344	+	0.992239i	$0.460317\pi$
$174$		0			0
$175$	−0.661686	−	0.749781i	−0.661686	−	0.749781i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$179$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$180$		0			0
$181$	0.559735	+	1.42618i	0.559735	+	1.42618i	0.878222	+	0.478254i	$0.158730\pi$
$181$	0.559735	+	1.42618i	0.559735	+	1.42618i	−0.318487	+	0.947927i	$0.603175\pi$
$182$		0			0
$183$	−1.57257	−	0.960980i	−1.57257	−	0.960980i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.0249307	+	0.999689i	−0.0249307	+	0.999689i
$190$		0			0
$191$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$191$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$192$	0.698237	−	0.715867i	0.698237	−	0.715867i
$193$	−1.60138	+	1.09180i	−1.60138	+	1.09180i	−0.661686	+	0.749781i	$0.730159\pi$
$193$	−1.60138	+	1.09180i	−1.60138	+	1.09180i	−0.939693	+	0.342020i	$0.888889\pi$
$194$		0			0
$195$		0			0
$196$	−0.661686	+	0.749781i	−0.661686	+	0.749781i
$197$		0			0		−0.411287	−	0.911506i	$-0.634921\pi$
$197$		0			0		0.411287	+	0.911506i	$0.365079\pi$
$198$		0			0
$199$	−0.431791	+	1.53468i	−0.431791	+	1.53468i	0.365341	+	0.930874i	$0.380952\pi$
$199$	−0.431791	+	1.53468i	−0.431791	+	1.53468i	−0.797133	+	0.603804i	$0.793651\pi$
$200$		0			0
$201$	0.598559	+	1.78152i	0.598559	+	1.78152i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.102282	+	0.816190i	0.102282	+	0.816190i
$209$		0			0
$210$		0			0
$211$	−0.172518	+	0.613166i	−0.172518	+	0.613166i	0.826239	+	0.563320i	$0.190476\pi$
$211$	−0.172518	+	0.613166i	−0.172518	+	0.613166i	−0.998757	+	0.0498459i	$0.984127\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	1.28682	−	0.542439i	1.28682	−	0.542439i
$218$		0			0
$219$	1.36879	−	1.40335i	1.36879	−	1.40335i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.84066	−	0.775904i	−1.84066	−	0.775904i	−0.939693	−	0.342020i	$-0.888889\pi$
$223$	−1.84066	−	0.775904i	−1.84066	−	0.775904i	−0.900969	−	0.433884i	$-0.857143\pi$
$224$		0			0
$225$	−0.969077	+	0.246757i	−0.969077	+	0.246757i
$226$		0			0
$227$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$227$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$228$	−1.16221	−	1.31695i	−1.16221	−	1.31695i
$229$	1.61971	+	1.10430i	1.61971	+	1.10430i	0.921476	+	0.388435i	$0.126984\pi$
$229$	1.61971	+	1.10430i	1.61971	+	1.10430i	0.698237	+	0.715867i	$0.253968\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.542546	−	0.840026i	$-0.317460\pi$
$233$		0			0		−0.542546	+	0.840026i	$0.682540\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−0.0431841	+	0.344601i	−0.0431841	+	0.344601i
$238$		0			0
$239$		0			0		−0.124344	−	0.992239i	$-0.539683\pi$
$239$		0			0		0.124344	+	0.992239i	$0.460317\pi$
$240$		0			0
$241$	1.70446	−	0.0850661i	1.70446	−	0.0850661i	0.826239	−	0.563320i	$-0.190476\pi$
$241$	1.70446	−	0.0850661i	1.70446	−	0.0850661i	0.878222	+	0.478254i	$0.158730\pi$
$242$		0			0
$243$	0.878222	+	0.478254i	0.878222	+	0.478254i
$244$	−1.82237	−	0.274678i	−1.82237	−	0.274678i
$245$		0			0
$246$		0			0
$247$	1.44301	−	0.0720176i	1.44301	−	0.0720176i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.583744	−	0.811938i	$-0.301587\pi$
$251$		0			0		−0.583744	+	0.811938i	$0.698413\pi$
$252$	0.365341	+	0.930874i	0.365341	+	0.930874i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.365341	−	0.930874i	0.365341	−	0.930874i
$257$		0			0		−0.411287	−	0.911506i	$-0.634921\pi$
$257$		0			0		0.411287	+	0.911506i	$0.365079\pi$
$258$		0			0
$259$	0.0681853	−	0.909870i	0.0681853	−	0.909870i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.661686	−	0.749781i	$-0.269841\pi$
$263$		0			0		−0.661686	+	0.749781i	$0.730159\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.24356	+	1.40913i	1.24356	+	1.40913i
$269$		0			0		0.797133	−	0.603804i	$-0.206349\pi$
$269$		0			0		−0.797133	+	0.603804i	$0.793651\pi$
$270$		0			0
$271$	−1.22128	−	0.925082i	−1.22128	−	0.925082i	−0.222521	−	0.974928i	$-0.571429\pi$
$271$	−1.22128	−	0.925082i	−1.22128	−	0.925082i	−0.998757	+	0.0498459i	$0.984127\pi$
$272$		0			0
$273$	−0.786030	−	0.242458i	−0.786030	−	0.242458i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.218403	+	0.118936i	−0.218403	+	0.118936i	−0.583744	−	0.811938i	$-0.698413\pi$
$277$	−0.218403	+	0.118936i	−0.218403	+	0.118936i	0.365341	+	0.930874i	$0.380952\pi$
$278$		0			0
$279$	0.104359	−	1.39257i	0.104359	−	1.39257i
$280$		0			0
$281$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$281$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$282$		0			0
$283$	0.870687	+	0.892671i	0.870687	+	0.892671i	0.995031	−	0.0995678i	$-0.0317460\pi$
$283$	0.870687	+	0.892671i	0.870687	+	0.892671i	−0.124344	+	0.992239i	$0.539683\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.698237	−	0.715867i	0.698237	−	0.715867i
$290$		0			0
$291$	1.26587	−	1.06219i	1.26587	−	1.06219i
$292$	0.716194	−	1.82483i	0.716194	−	1.82483i
$293$		0			0		−0.456211	−	0.889872i	$-0.650794\pi$
$293$		0			0		0.456211	+	0.889872i	$0.349206\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−0.797133	+	0.603804i	−0.797133	+	0.603804i
$301$	−1.97766			−1.97766
$302$		0			0
$303$		0			0
$304$	−1.58250	−	0.762092i	−1.58250	−	0.762092i
$305$		0			0
$306$		0			0
$307$	−1.16221	+	1.31695i	−1.16221	+	1.31695i	−0.222521	+	0.974928i	$0.571429\pi$
$307$	−1.16221	+	1.31695i	−1.16221	+	1.31695i	−0.939693	+	0.342020i	$0.888889\pi$
$308$		0			0
$309$	0.510189	+	0.523071i	0.510189	+	0.523071i
$310$		0			0
$311$		0			0		0.583744	−	0.811938i	$-0.301587\pi$
$311$		0			0		−0.583744	+	0.811938i	$0.698413\pi$
$312$		0			0
$313$	−0.630128	+	0.528741i	−0.630128	+	0.528741i	−0.900969	−	0.433884i	$-0.857143\pi$
$313$	−0.630128	+	0.528741i	−0.630128	+	0.528741i	0.270840	+	0.962624i	$0.412698\pi$
$314$		0			0
$315$		0			0
$316$	0.0940619	+	0.334316i	0.0940619	+	0.334316i
$317$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$317$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.995031	+	0.0995678i	0.995031	+	0.0995678i
$325$	0.0205073	−	0.822319i	0.0205073	−	0.822319i
$326$		0			0
$327$	−1.07297	−	0.161725i	−1.07297	−	0.161725i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.148717	+	1.98450i	0.148717	+	1.98450i	0.173648	+	0.984808i	$0.444444\pi$
$331$	0.148717	+	1.98450i	0.148717	+	1.98450i	−0.0249307	+	0.999689i	$0.507937\pi$
$332$		0			0
$333$	−0.778561	−	0.475769i	−0.778561	−	0.475769i
$334$		0			0
$335$		0			0
$336$	0.698237	+	0.715867i	0.698237	+	0.715867i
$337$	1.61971	+	0.327432i	1.61971	+	0.327432i	0.921476	−	0.388435i	$-0.126984\pi$
$337$	1.61971	+	0.327432i	1.61971	+	0.327432i	0.698237	+	0.715867i	$0.253968\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.878222	−	0.478254i	$-0.841270\pi$
$347$		0			0		0.878222	+	0.478254i	$0.158730\pi$
$348$		0			0
$349$	0.400969	+	1.75676i	0.400969	+	1.75676i	0.623490	+	0.781831i	$0.285714\pi$
$349$	0.400969	+	1.75676i	0.400969	+	1.75676i	−0.222521	+	0.974928i	$0.571429\pi$
$350$		0			0
$351$	−0.574352	+	0.588854i	−0.574352	+	0.588854i
$352$		0			0
$353$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$353$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$359$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$360$		0			0
$361$	−1.04255	+	1.80574i	−1.04255	+	1.80574i
$362$		0			0
$363$	0.542546	−	0.840026i	0.542546	−	0.840026i
$364$	−0.818487	+	0.0819019i	−0.818487	+	0.0819019i
$365$		0			0
$366$		0			0
$367$	0.894330	−	0.180793i	0.894330	−	0.180793i	0.270840	−	0.962624i	$-0.412698\pi$
$367$	0.894330	−	0.180793i	0.894330	−	0.180793i	0.623490	+	0.781831i	$0.285714\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.444758	−	1.32376i	−0.444758	−	1.32376i
$373$	−0.0872464	+	1.16422i	−0.0872464	+	1.16422i	0.766044	+	0.642788i	$0.222222\pi$
$373$	−0.0872464	+	1.16422i	−0.0872464	+	1.16422i	−0.853291	+	0.521435i	$0.825397\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.0310881	−	0.0389832i	−0.0310881	−	0.0389832i	0.766044	−	0.642788i	$-0.222222\pi$
$379$	−0.0310881	−	0.0389832i	−0.0310881	−	0.0389832i	−0.797133	+	0.603804i	$0.793651\pi$
$380$		0			0
$381$	−0.411287	−	0.911506i	−0.411287	−	0.911506i
$382$		0			0
$383$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$383$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.902230	+	1.75987i	−0.902230	+	1.75987i
$388$	0.753878	−	1.47049i	0.753878	−	1.47049i
$389$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$389$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.447558	−	0.305140i	0.447558	−	0.305140i	−0.318487	−	0.947927i	$-0.603175\pi$
$397$	0.447558	−	0.305140i	0.447558	−	0.305140i	0.766044	+	0.642788i	$0.222222\pi$
$398$		0			0
$399$	1.34551	−	1.12902i	1.34551	−	1.12902i
$400$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	1.07943	+	0.392880i	1.07943	+	0.392880i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.190506	−	0.159853i	−0.190506	−	0.159853i	0.542546	−	0.840026i	$-0.317460\pi$
$409$	−0.190506	−	0.159853i	−0.190506	−	0.159853i	−0.733052	+	0.680173i	$0.761905\pi$
$410$		0			0
$411$		0			0
$412$	0.673306	+	0.283822i	0.673306	+	0.283822i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.07581	−	1.49636i	−1.07581	−	1.49636i
$418$		0			0
$419$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$419$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$420$		0			0
$421$	0.455573	−	1.16078i	0.455573	−	1.16078i	−0.500000	−	0.866025i	$-0.666667\pi$
$421$	0.455573	−	1.16078i	0.455573	−	1.16078i	0.955573	−	0.294755i	$-0.0952381\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.320025	−	1.81495i	0.320025	−	1.81495i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$431$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$432$	0.955573	−	0.294755i	0.955573	−	0.294755i
$433$	−1.38036	−	1.15826i	−1.38036	−	1.15826i	−0.969077	−	0.246757i	$-0.920635\pi$
$433$	−1.38036	−	1.15826i	−1.38036	−	1.15826i	−0.411287	−	0.911506i	$-0.634921\pi$
$434$		0			0
$435$		0			0
$436$	−1.05154	+	0.267755i	−1.05154	+	0.267755i
$437$		0			0
$438$		0			0
$439$	−1.78596	+	0.752847i	−1.78596	+	0.752847i	−0.797133	+	0.603804i	$0.793651\pi$
$439$	−1.78596	+	0.752847i	−1.78596	+	0.752847i	−0.988831	+	0.149042i	$0.952381\pi$
$440$		0			0
$441$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$442$		0			0
$443$		0			0		−0.969077	−	0.246757i	$-0.920635\pi$
$443$		0			0		0.969077	+	0.246757i	$0.0793651\pi$
$444$	−0.902230	−	0.135989i	−0.902230	−	0.135989i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.921476	+	0.388435i	0.921476	+	0.388435i
$449$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$449$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−0.144838	+	0.0368804i	−0.144838	+	0.0368804i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.33443	−	0.411618i	1.33443	−	0.411618i	0.456211	−	0.889872i	$-0.349206\pi$
$457$	1.33443	−	0.411618i	1.33443	−	0.411618i	0.878222	+	0.478254i	$0.158730\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$461$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$462$		0			0
$463$	−1.70213	−	0.926930i	−1.70213	−	0.926930i	−0.969077	−	0.246757i	$-0.920635\pi$
$463$	−1.70213	−	0.926930i	−1.70213	−	0.926930i	−0.733052	−	0.680173i	$-0.761905\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.124344	−	0.992239i	$-0.460317\pi$
$467$		0			0		−0.124344	+	0.992239i	$0.539683\pi$
$468$	−0.300520	+	0.765713i	−0.300520	+	0.765713i
$469$	−1.43969	+	1.20805i	−1.43969	+	1.20805i
$470$		0			0
$471$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	1.45124	+	0.989440i	1.45124	+	0.989440i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.698237	−	0.715867i	$-0.746032\pi$
$479$		0			0		0.698237	+	0.715867i	$0.253968\pi$
$480$		0			0
$481$	0.550181	−	0.510493i	0.550181	−	0.510493i
$482$		0			0
$483$		0			0
$484$	0.173648	−	0.984808i	0.173648	−	0.984808i
$485$		0			0
$486$		0			0
$487$	−0.603736	−	1.17763i	−0.603736	−	1.17763i	−0.969077	−	0.246757i	$-0.920635\pi$
$487$	−0.603736	−	1.17763i	−0.603736	−	1.17763i	0.365341	−	0.930874i	$-0.380952\pi$
$488$		0			0
$489$	0.988831	−	1.71271i	0.988831	−	1.71271i
$490$		0			0
$491$		0			0		0.124344	−	0.992239i	$-0.460317\pi$
$491$		0			0		−0.124344	+	0.992239i	$0.539683\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.924027	−	1.04705i	−0.924027	−	1.04705i
$497$		0			0
$498$		0			0
$499$	−1.76621	−	0.357047i	−1.76621	−	0.357047i	−0.797133	−	0.603804i	$-0.793651\pi$
$499$	−1.76621	−	0.357047i	−1.76621	−	0.357047i	−0.969077	+	0.246757i	$0.920635\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$503$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.161686	+	0.280048i	0.161686	+	0.280048i
$508$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	1.80641	+	0.761466i	1.80641	+	0.761466i
$512$		0			0
$513$	−0.390845	−	1.71241i	−0.390845	−	1.71241i
$514$		0			0
$515$		0			0
$516$	−0.147791	+	1.97213i	−0.147791	+	1.97213i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.998757	−	0.0498459i	$-0.0158730\pi$
$521$		0			0		−0.998757	+	0.0498459i	$0.984127\pi$
$522$		0			0
$523$	−0.586956	−	0.247423i	−0.586956	−	0.247423i	0.0747301	−	0.997204i	$-0.476190\pi$
$523$	−0.586956	−	0.247423i	−0.586956	−	0.247423i	−0.661686	+	0.749781i	$0.730159\pi$
$524$		0			0
$525$	−0.583744	−	0.811938i	−0.583744	−	0.811938i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.456211	+	0.889872i	0.456211	+	0.889872i
$530$		0			0
$531$		0			0
$532$	0.801308	−	1.56301i	0.801308	−	1.56301i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.79970	−	0.866689i	1.79970	−	0.866689i	0.878222	−	0.478254i	$-0.158730\pi$
$541$	1.79970	−	0.866689i	1.79970	−	0.866689i	0.921476	−	0.388435i	$-0.126984\pi$
$542$		0			0
$543$	0.414952	+	1.47483i	0.414952	+	1.47483i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.416256	+	0.811938i	0.416256	+	0.811938i	1.00000			$0$
$547$	0.416256	+	0.811938i	0.416256	+	0.811938i	−0.583744	+	0.811938i	$0.698413\pi$
$548$		0			0
$549$	−1.46908	−	1.11278i	−1.46908	−	1.11278i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.336557	+	0.0856979i	−0.336557	+	0.0856979i
$554$		0			0
$555$		0			0
$556$	−1.57257	−	0.960980i	−1.57257	−	0.960980i
$557$		0			0		0.995031	−	0.0995678i	$-0.0317460\pi$
$557$		0			0		−0.995031	+	0.0995678i	$0.968254\pi$
$558$		0			0
$559$	−1.19251	−	1.10649i	−1.19251	−	1.10649i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.270840	−	0.962624i	$-0.412698\pi$
$563$		0			0		−0.270840	+	0.962624i	$0.587302\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.124344	+	0.992239i	−0.124344	+	0.992239i
$568$		0			0
$569$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$569$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$570$		0			0
$571$	−0.535631	−	0.0807334i	−0.535631	−	0.0807334i	−0.124344	−	0.992239i	$-0.539683\pi$
$571$	−0.535631	−	0.0807334i	−0.535631	−	0.0807334i	−0.411287	+	0.911506i	$0.634921\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.766044	−	0.642788i	0.766044	−	0.642788i
$577$	0.0553382	−	0.441588i	0.0553382	−	0.441588i	−0.939693	−	0.342020i	$-0.888889\pi$
$577$	0.0553382	−	0.441588i	0.0553382	−	0.441588i	0.995031	−	0.0995678i	$-0.0317460\pi$
$578$		0			0
$579$	−1.70213	+	0.926930i	−1.70213	+	0.926930i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.878222	−	0.478254i	$-0.158730\pi$
$587$		0			0		−0.878222	+	0.478254i	$0.841270\pi$
$588$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$589$	−1.95523	+	1.48103i	−1.95523	+	1.48103i
$590$		0			0
$591$		0			0
$592$	−0.884207	+	0.225147i	−0.884207	+	0.225147i
$593$		0			0		0.995031	−	0.0995678i	$-0.0317460\pi$
$593$		0			0		−0.995031	+	0.0995678i	$0.968254\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.582450	+	1.48406i	−0.582450	+	1.48406i
$598$		0			0
$599$		0			0		−0.583744	−	0.811938i	$-0.698413\pi$
$599$		0			0		0.583744	+	0.811938i	$0.301587\pi$
$600$		0			0
$601$	0.996206	−	0.608769i	0.996206	−	0.608769i	0.0747301	−	0.997204i	$-0.476190\pi$
$601$	0.996206	−	0.608769i	0.996206	−	0.608769i	0.921476	+	0.388435i	$0.126984\pi$
$602$		0			0
$603$	0.418203	+	1.83227i	0.418203	+	1.83227i
$604$	−0.119140	+	0.0902447i	−0.119140	+	0.0902447i
$605$		0			0
$606$		0			0
$607$	−0.254586	+	1.44383i	−0.254586	+	1.44383i	0.542546	+	0.840026i	$0.317460\pi$
$607$	−0.254586	+	1.44383i	−0.254586	+	1.44383i	−0.797133	+	0.603804i	$0.793651\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.598559	+	0.217858i	0.598559	+	0.217858i	0.623490	−	0.781831i	$-0.285714\pi$
$613$	0.598559	+	0.217858i	0.598559	+	0.217858i	−0.0249307	+	0.999689i	$0.507937\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.542546	−	0.840026i	$-0.317460\pi$
$617$		0			0		−0.542546	+	0.840026i	$0.682540\pi$
$618$		0			0
$619$	0.0546039	+	0.728639i	0.0546039	+	0.728639i	0.955573	+	0.294755i	$0.0952381\pi$
$619$	0.0546039	+	0.728639i	0.0546039	+	0.728639i	−0.900969	+	0.433884i	$0.857143\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	0.0205073	+	0.822319i	0.0205073	+	0.822319i
$625$	0.623490	−	0.781831i	0.623490	−	0.781831i
$626$		0			0
$627$		0			0
$628$	−0.346865	+	0.0173113i	−0.346865	+	0.0173113i
$629$		0			0
$630$		0			0
$631$	0.379750	−	1.66379i	0.379750	−	1.66379i	−0.318487	−	0.947927i	$-0.603175\pi$
$631$	0.379750	−	1.66379i	0.379750	−	1.66379i	0.698237	−	0.715867i	$-0.253968\pi$
$632$		0			0
$633$	−0.232712	+	0.592942i	−0.232712	+	0.592942i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.0614710	−	0.820274i	−0.0614710	−	0.820274i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.542546	−	0.840026i	$-0.682540\pi$
$641$		0			0		0.542546	+	0.840026i	$0.317460\pi$
$642$		0			0
$643$	−1.72188	+	0.829215i	−1.72188	+	0.829215i	−0.733052	+	0.680173i	$0.761905\pi$
$643$	−1.72188	+	0.829215i	−1.72188	+	0.829215i	−0.988831	+	0.149042i	$0.952381\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.318487	−	0.947927i	$-0.396825\pi$
$647$		0			0		−0.318487	+	0.947927i	$0.603175\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	1.33443	−	0.411618i	1.33443	−	0.411618i
$652$	0.245910	−	1.96231i	0.245910	−	1.96231i
$653$		0			0		−0.456211	−	0.889872i	$-0.650794\pi$
$653$		0			0		0.456211	+	0.889872i	$0.349206\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.50171	−	1.26009i	1.50171	−	1.26009i
$658$		0			0
$659$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$659$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$660$		0			0
$661$	1.12922	+	1.27956i	1.12922	+	1.27956i	0.955573	+	0.294755i	$0.0952381\pi$
$661$	1.12922	+	1.27956i	1.12922	+	1.27956i	0.173648	+	0.984808i	$0.444444\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−1.75426	−	0.955319i	−1.75426	−	0.955319i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.78181	−	0.268565i	1.78181	−	0.268565i	0.826239	−	0.563320i	$-0.190476\pi$
$673$	1.78181	−	0.268565i	1.78181	−	0.268565i	0.955573	+	0.294755i	$0.0952381\pi$
$674$		0			0
$675$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$676$	0.257770	+	0.195253i	0.257770	+	0.195253i
$677$		0			0		0.995031	−	0.0995678i	$-0.0317460\pi$
$677$		0			0		−0.995031	+	0.0995678i	$0.968254\pi$
$678$		0			0
$679$	1.45124	+	0.790304i	1.45124	+	0.790304i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.853291	−	0.521435i	$-0.825397\pi$
$683$		0			0		0.853291	+	0.521435i	$0.174603\pi$
$684$	−1.02531	−	1.42612i	−1.02531	−	1.42612i
$685$		0			0
$686$		0			0
$687$	1.50171	+	1.26009i	1.50171	+	1.26009i
$688$	0.629859	+	1.87468i	0.629859	+	1.87468i
$689$		0			0
$690$		0			0
$691$	1.22641	−	0.667869i	1.22641	−	0.667869i	0.270840	−	0.962624i	$-0.412698\pi$
$691$	1.22641	−	0.667869i	1.22641	−	0.667869i	0.955573	+	0.294755i	$0.0952381\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.853291	−	0.521435i	−0.853291	−	0.521435i
$701$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$701$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$702$		0			0
$703$	0.278291	+	1.57827i	0.278291	+	1.57827i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.0792036	+	0.235738i	0.0792036	+	0.235738i	0.980172	−	0.198146i	$-0.0634921\pi$
$709$	0.0792036	+	0.235738i	0.0792036	+	0.235738i	−0.900969	+	0.433884i	$0.857143\pi$
$710$		0			0
$711$	−0.0772807	+	0.338589i	−0.0772807	+	0.338589i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.698237	−	0.715867i	$-0.746032\pi$
$719$		0			0		0.698237	+	0.715867i	$0.253968\pi$
$720$		0			0
$721$	−0.300520	+	0.666021i	−0.300520	+	0.666021i
$722$		0			0
$723$	1.70446	+	0.0850661i	1.70446	+	0.0850661i
$724$	0.955242	+	1.19784i	0.955242	+	1.19784i
$725$		0			0
$726$		0			0
$727$	−1.92852	−	0.491062i	−1.92852	−	0.491062i	−0.988831	−	0.149042i	$-0.952381\pi$
$727$	−1.92852	−	0.491062i	−1.92852	−	0.491062i	−0.939693	−	0.342020i	$-0.888889\pi$
$728$		0			0
$729$	0.826239	+	0.563320i	0.826239	+	0.563320i
$730$		0			0
$731$		0			0
$732$	−1.78596	−	0.454762i	−1.78596	−	0.454762i
$733$	−0.358423	+	1.27391i	−0.358423	+	1.27391i	0.542546	+	0.840026i	$0.317460\pi$
$733$	−0.358423	+	1.27391i	−0.358423	+	1.27391i	−0.900969	+	0.433884i	$0.857143\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−0.390845	+	0.212843i	−0.390845	+	0.212843i	−0.661686	−	0.749781i	$-0.730159\pi$
$739$	−0.390845	+	0.212843i	−0.390845	+	0.212843i	0.270840	+	0.962624i	$0.412698\pi$
$740$		0			0
$741$	1.44301	+	0.0720176i	1.44301	+	0.0720176i
$742$		0			0
$743$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$743$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.34551	+	0.732728i	1.34551	+	0.732728i	0.980172	−	0.198146i	$-0.0634921\pi$
$751$	1.34551	+	0.732728i	1.34551	+	0.732728i	0.365341	+	0.930874i	$0.380952\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.270840	+	0.962624i	0.270840	+	0.962624i
$757$	−0.914101	+	0.848162i	−0.914101	+	0.848162i	−0.988831	−	0.149042i	$-0.952381\pi$
$757$	−0.914101	+	0.848162i	−0.914101	+	0.848162i	0.0747301	+	0.997204i	$0.476190\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$	−0.241456	−	1.05789i	−0.241456	−	1.05789i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.456211	−	0.889872i	0.456211	−	0.889872i
$769$	−1.01965	−	1.57873i	−1.01965	−	1.57873i	−0.797133	−	0.603804i	$-0.793651\pi$
$769$	−1.01965	−	1.57873i	−1.01965	−	1.57873i	−0.222521	−	0.974928i	$-0.571429\pi$
$770$		0			0
$771$		0			0
$772$	−1.20842	+	1.51531i	−1.20842	+	1.51531i
$773$		0			0		0.853291	−	0.521435i	$-0.174603\pi$
$773$		0			0		−0.853291	+	0.521435i	$0.825397\pi$
$774$		0			0
$775$	0.757652	+	1.17307i	0.757652	+	1.17307i
$776$		0			0
$777$	0.158440	−	0.898560i	0.158440	−	0.898560i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.411287	+	0.911506i	−0.411287	+	0.911506i
$785$		0			0
$786$		0			0
$787$	0.224060	−	0.107901i	0.224060	−	0.107901i	−0.318487	−	0.947927i	$-0.603175\pi$
$787$	0.224060	−	0.107901i	0.224060	−	0.107901i	0.542546	+	0.840026i	$0.317460\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.20843	−	0.915346i	1.20843	−	0.915346i
$794$		0			0
$795$		0			0
$796$	0.0397461	+	1.59377i	0.0397461	+	1.59377i
$797$		0			0		−0.969077	−	0.246757i	$-0.920635\pi$
$797$		0			0		0.969077	+	0.246757i	$0.0793651\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	1.09708	+	1.52594i	1.09708	+	1.52594i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$809$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$810$		0			0
$811$	0.544286	−	1.20626i	0.544286	−	1.20626i	−0.411287	−	0.911506i	$-0.634921\pi$
$811$	0.544286	−	1.20626i	0.544286	−	1.20626i	0.955573	−	0.294755i	$-0.0952381\pi$
$812$		0			0
$813$	−1.12310	−	1.04209i	−1.12310	−	1.04209i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	3.36624	−	0.857149i	3.36624	−	0.857149i
$818$		0			0
$819$	−0.757983	−	0.319516i	−0.757983	−	0.319516i
$820$		0			0
$821$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$821$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$822$		0			0
$823$	1.45124	−	0.989440i	1.45124	−	0.989440i	0.456211	−	0.889872i	$-0.349206\pi$
$823$	1.45124	−	0.989440i	1.45124	−	0.989440i	0.995031	−	0.0995678i	$-0.0317460\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.270840	−	0.962624i	$-0.587302\pi$
$827$		0			0		0.270840	+	0.962624i	$0.412698\pi$
$828$		0			0
$829$	0.0205073	+	0.0454489i	0.0205073	+	0.0454489i	0.921476	−	0.388435i	$-0.126984\pi$
$829$	0.0205073	+	0.0454489i	0.0205073	+	0.0454489i	−0.900969	+	0.433884i	$0.857143\pi$
$830$		0			0
$831$	−0.229160	+	0.0965988i	−0.229160	+	0.0965988i
$832$	0.338314	+	0.749781i	0.338314	+	0.749781i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.242495	−	1.37526i	0.242495	−	1.37526i
$838$		0			0
$839$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$839$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$840$		0			0
$841$	0.878222	−	0.478254i	0.878222	−	0.478254i
$842$		0			0
$843$		0			0
$844$	0.0158802	+	0.636775i	0.0158802	+	0.636775i
$845$		0			0
$846$		0			0
$847$	0.980172	+	0.198146i	0.980172	+	0.198146i
$848$		0			0
$849$	0.777479	+	0.974928i	0.777479	+	0.974928i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.98017	−	0.198146i	1.98017	−	0.198146i	0.980172	−	0.198146i	$-0.0634921\pi$
$853$	1.98017	−	0.198146i	1.98017	−	0.198146i	1.00000			$0$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$857$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$858$		0			0
$859$	−0.0135045	−	0.0479978i	−0.0135045	−	0.0479978i	0.955573	−	0.294755i	$-0.0952381\pi$
$859$	−0.0135045	−	0.0479978i	−0.0135045	−	0.0479978i	−0.969077	+	0.246757i	$0.920635\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.318487	−	0.947927i	$-0.396825\pi$
$863$		0			0		−0.318487	+	0.947927i	$0.603175\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.766044	−	0.642788i	0.766044	−	0.642788i
$868$	1.06976	−	0.897636i	1.06976	−	0.897636i
$869$		0			0
$870$		0			0
$871$	−1.54401	−	0.0770584i	−1.54401	−	0.0770584i
$872$		0			0
$873$	1.36534	−	0.930874i	1.36534	−	0.930874i
$874$		0			0
$875$		0			0
$876$	0.894330	−	1.74446i	0.894330	−	1.74446i
$877$	−1.43703	+	0.290503i	−1.43703	+	0.290503i	−0.853291	−	0.521435i	$-0.825397\pi$
$877$	−1.43703	+	0.290503i	−1.43703	+	0.290503i	−0.583744	+	0.811938i	$0.698413\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$881$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$882$		0			0
$883$	0.248378	−	0.0123960i	0.248378	−	0.0123960i	0.0747301	−	0.997204i	$-0.476190\pi$
$883$	0.248378	−	0.0123960i	0.248378	−	0.0123960i	0.173648	+	0.984808i	$0.444444\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$887$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$888$		0			0
$889$	0.698237	−	0.715867i	0.698237	−	0.715867i
$890$		0			0
$891$		0			0
$892$	−1.98759	−	0.198888i	−1.98759	−	0.198888i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.853291	+	0.521435i	−0.853291	+	0.521435i
$901$		0			0
$902$		0			0
$903$	−1.96783	−	0.196912i	−1.96783	−	0.196912i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.99503	+	0.0995678i	1.99503	+	0.0995678i	1.00000			$0$
$907$	1.99503	+	0.0995678i	1.99503	+	0.0995678i	0.995031	+	0.0995678i	$0.0317460\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$911$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$912$	−1.49876	−	0.915871i	−1.49876	−	0.915871i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.87325	+	0.577822i	1.87325	+	0.577822i
$917$		0			0
$918$		0			0
$919$	0.530941	+	1.88708i	0.530941	+	1.88708i	0.456211	+	0.889872i	$0.349206\pi$
$919$	0.530941	+	1.88708i	0.530941	+	1.88708i	0.0747301	+	0.997204i	$0.476190\pi$
$920$		0			0
$921$	−1.28756	+	1.19468i	−1.28756	+	1.19468i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.907887	−	0.0908478i	0.907887	−	0.0908478i
$926$		0			0
$927$	0.455573	+	0.571270i	0.455573	+	0.571270i
$928$		0			0
$929$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$929$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$930$		0			0
$931$	1.54255	+	0.840026i	1.54255	+	0.840026i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.559404	+	0.304635i	−0.559404	+	0.304635i	−0.733052	−	0.680173i	$-0.761905\pi$
$937$	−0.559404	+	0.304635i	−0.559404	+	0.304635i	0.173648	+	0.984808i	$0.444444\pi$
$938$		0			0
$939$	−0.679643	+	0.463373i	−0.679643	+	0.463373i
$940$		0			0
$941$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$941$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.921476	−	0.388435i	$-0.126984\pi$
$947$		0			0		−0.921476	+	0.388435i	$0.873016\pi$
$948$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$949$	0.663212	+	1.46983i	0.663212	+	1.46983i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$953$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.920758	+	0.234454i	−0.920758	+	0.234454i
$962$		0			0
$963$		0			0
$964$	1.60366	−	0.583685i	1.60366	−	0.583685i
$965$		0			0
$966$		0			0
$967$	0.261979	−	0.580605i	0.261979	−	0.580605i	−0.733052	−	0.680173i	$-0.761905\pi$
$967$	0.261979	−	0.580605i	0.261979	−	0.580605i	0.995031	+	0.0995678i	$0.0317460\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.853291	−	0.521435i	$-0.174603\pi$
$971$		0			0		−0.853291	+	0.521435i	$0.825397\pi$
$972$	0.980172	+	0.198146i	0.980172	+	0.198146i
$973$	0.999887	−	1.54813i	0.999887	−	1.54813i
$974$		0			0
$975$	0.102282	−	0.816190i	0.102282	−	0.816190i
$976$	−1.82237	+	0.274678i	−1.82237	+	0.274678i
$977$		0			0		0.542546	−	0.840026i	$-0.317460\pi$
$977$		0			0		−0.542546	+	0.840026i	$0.682540\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1.05154	−	0.267755i	−1.05154	−	0.267755i
$982$		0			0
$983$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$983$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	1.35767	−	0.494152i	1.35767	−	0.494152i
$989$		0			0
$990$		0			0
$991$	1.53758	−	0.740458i	1.53758	−	0.740458i	0.542546	−	0.840026i	$-0.317460\pi$
$991$	1.53758	−	0.740458i	1.53758	−	0.740458i	0.995031	+	0.0995678i	$0.0317460\pi$
$992$		0			0
$993$	−0.0496136	+	1.98944i	−0.0496136	+	1.98944i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.0498614			−0.0498614			−0.0249307	−	0.999689i	$-0.507937\pi$
$997$	−0.0498614			−0.0498614			−0.0249307	+	0.999689i	$0.507937\pi$
$998$		0			0
$999$	−0.727321	−	0.550924i	−0.727321	−	0.550924i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2667.1.em.a.1031.1	yes	36
3.2	odd	2	CM	2667.1.em.a.1031.1	yes	36
7.4	even	3		2667.1.ej.a.2174.1	yes	36
21.11	odd	6		2667.1.ej.a.2174.1	yes	36
127.17	even	63		2667.1.ej.a.779.1	✓	36
381.17	odd	126		2667.1.ej.a.779.1	✓	36
889.144	even	63	inner	2667.1.em.a.1922.1	yes	36
2667.1922	odd	126	inner	2667.1.em.a.1922.1	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.1.ej.a.779.1	✓	36	127.17	even	63
2667.1.ej.a.779.1	✓	36	381.17	odd	126
2667.1.ej.a.2174.1	yes	36	7.4	even	3
2667.1.ej.a.2174.1	yes	36	21.11	odd	6
2667.1.em.a.1031.1	yes	36	1.1	even	1	trivial
2667.1.em.a.1031.1	yes	36	3.2	odd	2	CM
2667.1.em.a.1922.1	yes	36	889.144	even	63	inner
2667.1.em.a.1922.1	yes	36	2667.1922	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)

\(f(q)\)	\(=\)	\(q+(0.995031 + 0.0995678i) q^{3} +(0.955573 - 0.294755i) q^{4} +(0.270840 + 0.962624i) q^{7} +(0.980172 + 0.198146i) q^{9} +O(q^{10})\)	\(q+(0.995031 + 0.0995678i) q^{3} +(0.955573 - 0.294755i) q^{4} +(0.270840 + 0.962624i) q^{7} +(0.980172 + 0.198146i) q^{9} +(0.980172 - 0.198146i) q^{12} +(-0.375267 + 0.731986i) q^{13} +(0.826239 - 0.563320i) q^{16} +(-0.878222 - 1.52112i) q^{19} +(0.173648 + 0.984808i) q^{21} +(-0.900969 + 0.433884i) q^{25} +(0.955573 + 0.294755i) q^{27} +(0.542546 + 0.840026i) q^{28} +(-0.173643 - 1.38564i) q^{31} +(0.995031 - 0.0995678i) q^{36} +(-0.857396 - 0.312066i) q^{37} +(-0.446285 + 0.690984i) q^{39} +(-0.535631 + 1.90375i) q^{43} +(0.878222 - 0.478254i) q^{48} +(-0.853291 + 0.521435i) q^{49} +(-0.142839 + 0.810077i) q^{52} +(-0.722402 - 1.60101i) q^{57} +(-1.66044 - 0.799627i) q^{61} +(0.0747301 + 0.997204i) q^{63} +(0.623490 - 0.781831i) q^{64} +(0.772967 + 1.71307i) q^{67} +(1.22226 - 1.53266i) q^{73} +(-0.939693 + 0.342020i) q^{75} +(-1.28756 - 1.19468i) q^{76} +(-0.00865834 + 0.347188i) q^{79} +(0.921476 + 0.388435i) q^{81} +(0.456211 + 0.889872i) q^{84} +(-0.806265 - 0.162990i) q^{91} +(-0.0348151 - 1.39604i) q^{93} +(1.15382 - 1.18295i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 3 q^{4}+O(q^{10}) \) 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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