LMFDB - Embedded newform 1331.1.d.a.1207.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1331,1,Mod(161,1331)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1331, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([7]))

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

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

chi := DirichletCharacter("1331.161");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1331 = 11^{3} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1331.d (of order $10$, degree $4$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.664255531815$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$5$ over $\Q(\zeta_{10})$
Coefficient field:	20.0.1402274470934209014892578125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + 5 x^{18} - 6 x^{17} + 20 x^{16} - 11 x^{15} + 59 x^{14} - 30 x^{13} + 179 x^{12} + \cdots + 1 $ x^20 - x^19 + 5x^18 - 6x^17 + 20x^16 - 11x^15 + 59x^14 - 30x^13 + 179x^12 - 109x^11 + 260x^10 - 128x^9 + 334x^8 + 82x^7 + 199x^6 + 22x^5 + 146x^4 - 41x^3 + 12x^2 - 3x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{11}$
Projective field:	Galois closure of 11.1.4177248169415651.1

Embedding invariants

Embedding label			1207.2
Root			$-0.404726 + 1.24562i$ of defining polynomial
Character	$\chi$	$=$	1331.1207
Dual form			1331.1.d.a.161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.404726 - 1.24562i) q^{3} +(0.309017 - 0.951057i) q^{4} +(1.55249 + 1.12795i) q^{5} +(-0.578747 + 0.420484i) q^{9} +O(q^{10})$	$q+(-0.404726 - 1.24562i) q^{3} +(0.309017 - 0.951057i) q^{4} +(1.55249 + 1.12795i) q^{5} +(-0.578747 + 0.420484i) q^{9} -1.30972 q^{12} +(0.776664 - 2.39033i) q^{15} +(-0.809017 - 0.587785i) q^{16} +(1.55249 - 1.12795i) q^{20} +0.830830 q^{23} +(0.828940 + 2.55122i) q^{25} +(-0.301590 - 0.219118i) q^{27} +(-0.672156 + 0.488350i) q^{31} +(0.221062 + 0.680358i) q^{36} +(-0.0879554 + 0.270699i) q^{37} -1.37279 q^{45} +(-0.0879554 - 0.270699i) q^{47} +(-0.404726 + 1.24562i) q^{48} +(-0.809017 - 0.587785i) q^{49} +(-1.36118 + 0.988953i) q^{53} +(0.519923 - 1.60016i) q^{59} +(-2.03333 - 1.47730i) q^{60} +(-0.809017 + 0.587785i) q^{64} -1.91899 q^{67} +(-0.336259 - 1.03490i) q^{69} +(0.230270 + 0.167301i) q^{71} +(2.84235 - 2.06509i) q^{75} +(-0.592999 - 1.82506i) q^{80} +(-0.371938 + 1.14471i) q^{81} +1.68251 q^{89} +(0.256741 - 0.790166i) q^{92} +(0.880337 + 0.639602i) q^{93} +(-0.672156 + 0.488350i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + q^{3} - 5 q^{4} + q^{5} - 4 q^{9}+O(q^{10}) $ 20 * q + q^3 - 5 * q^4 + q^5 - 4 * q^9	$ 20 q + q^{3} - 5 q^{4} + q^{5} - 4 q^{9} - 4 q^{12} + 2 q^{15} - 5 q^{16} + q^{20} - 4 q^{23} - 4 q^{25} + 2 q^{27} + q^{31} - 4 q^{36} + q^{37} - 12 q^{45} + q^{47} + q^{48} - 5 q^{49} + q^{53} + q^{59} + 2 q^{60} - 5 q^{64} - 4 q^{67} + 2 q^{69} + q^{71} + 3 q^{75} + q^{80} - 3 q^{81} - 4 q^{89} + q^{92} + 2 q^{93} + q^{97}+O(q^{100}) $ 20 * q + q^3 - 5 * q^4 + q^5 - 4 * q^9 - 4 * q^12 + 2 * q^15 - 5 * q^16 + q^20 - 4 * q^23 - 4 * q^25 + 2 * q^27 + q^31 - 4 * q^36 + q^37 - 12 * q^45 + q^47 + q^48 - 5 * q^49 + q^53 + q^59 + 2 * q^60 - 5 * q^64 - 4 * q^67 + 2 * q^69 + q^71 + 3 * q^75 + q^80 - 3 * q^81 - 4 * q^89 + q^92 + 2 * q^93 + q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1331\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{3}{10}\right)$

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		0.809017	−	0.587785i	$-0.200000\pi$
$2$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$3$	−0.404726	−	1.24562i	−0.404726	−	1.24562i	−0.921124	−	0.389270i	$-0.872727\pi$
$3$	−0.404726	−	1.24562i	−0.404726	−	1.24562i	0.516397	−	0.856349i	$-0.327273\pi$
$4$	0.309017	−	0.951057i	0.309017	−	0.951057i
$5$	1.55249	+	1.12795i	1.55249	+	1.12795i	0.941844	+	0.336049i	$0.109091\pi$
$5$	1.55249	+	1.12795i	1.55249	+	1.12795i	0.610648	+	0.791902i	$0.290909\pi$
$6$		0			0
$7$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$7$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$8$		0			0
$9$	−0.578747	+	0.420484i	−0.578747	+	0.420484i
$10$		0			0
$11$		0			0
$11$		0			0
$12$	−1.30972			−1.30972
$13$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$13$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$14$		0			0
$15$	0.776664	−	2.39033i	0.776664	−	2.39033i
$16$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$17$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$17$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$18$		0			0
$19$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$19$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$20$	1.55249	−	1.12795i	1.55249	−	1.12795i
$21$		0			0
$22$		0			0
$23$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$23$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$24$		0			0
$25$	0.828940	+	2.55122i	0.828940	+	2.55122i
$26$		0			0
$27$	−0.301590	−	0.219118i	−0.301590	−	0.219118i
$28$		0			0
$29$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$29$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$30$		0			0
$31$	−0.672156	+	0.488350i	−0.672156	+	0.488350i	−0.870746	−	0.491733i	$-0.836364\pi$
$31$	−0.672156	+	0.488350i	−0.672156	+	0.488350i	0.198590	+	0.980083i	$0.436364\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.221062	+	0.680358i	0.221062	+	0.680358i
$37$	−0.0879554	+	0.270699i	−0.0879554	+	0.270699i	−0.985354	−	0.170522i	$-0.945455\pi$
$37$	−0.0879554	+	0.270699i	−0.0879554	+	0.270699i	0.897398	+	0.441221i	$0.145455\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$41$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−1.37279			−1.37279
$46$		0			0
$47$	−0.0879554	−	0.270699i	−0.0879554	−	0.270699i	0.897398	−	0.441221i	$-0.145455\pi$
$47$	−0.0879554	−	0.270699i	−0.0879554	−	0.270699i	−0.985354	+	0.170522i	$0.945455\pi$
$48$	−0.404726	+	1.24562i	−0.404726	+	1.24562i
$49$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.36118	+	0.988953i	−1.36118	+	0.988953i	−0.362808	+	0.931864i	$0.618182\pi$
$53$	−1.36118	+	0.988953i	−1.36118	+	0.988953i	−0.998369	+	0.0570888i	$0.981818\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.519923	−	1.60016i	0.519923	−	1.60016i	−0.254218	−	0.967147i	$-0.581818\pi$
$59$	0.519923	−	1.60016i	0.519923	−	1.60016i	0.774142	−	0.633012i	$-0.218182\pi$
$60$	−2.03333	−	1.47730i	−2.03333	−	1.47730i
$61$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$61$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$62$		0			0
$63$		0			0
$64$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$65$		0			0
$66$		0			0
$67$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$67$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$68$		0			0
$69$	−0.336259	−	1.03490i	−0.336259	−	1.03490i
$70$		0			0
$71$	0.230270	+	0.167301i	0.230270	+	0.167301i	0.696938	−	0.717132i	$-0.254545\pi$
$71$	0.230270	+	0.167301i	0.230270	+	0.167301i	−0.466667	+	0.884433i	$0.654545\pi$
$72$		0			0
$73$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$73$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$74$		0			0
$75$	2.84235	−	2.06509i	2.84235	−	2.06509i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$79$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$80$	−0.592999	−	1.82506i	−0.592999	−	1.82506i
$81$	−0.371938	+	1.14471i	−0.371938	+	1.14471i
$82$		0			0
$83$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$83$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$89$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$90$		0			0
$91$		0			0
$92$	0.256741	−	0.790166i	0.256741	−	0.790166i
$93$	0.880337	+	0.639602i	0.880337	+	0.639602i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.672156	+	0.488350i	−0.672156	+	0.488350i	−0.870746	−	0.491733i	$-0.836364\pi$
$97$	−0.672156	+	0.488350i	−0.672156	+	0.488350i	0.198590	+	0.980083i	$0.436364\pi$
$98$		0			0
$99$		0			0
$100$	2.68251			2.68251
$101$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$101$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$102$		0			0
$103$	−0.592999	+	1.82506i	−0.592999	+	1.82506i	−0.0285561	+	0.999592i	$0.509091\pi$
$103$	−0.592999	+	1.82506i	−0.592999	+	1.82506i	−0.564443	+	0.825472i	$0.690909\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$107$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$108$	−0.301590	+	0.219118i	−0.301590	+	0.219118i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$	0.372786			0.372786
$112$		0			0
$113$	−0.404726	−	1.24562i	−0.404726	−	1.24562i	−0.921124	−	0.389270i	$-0.872727\pi$
$113$	−0.404726	−	1.24562i	−0.404726	−	1.24562i	0.516397	−	0.856349i	$-0.327273\pi$
$114$		0			0
$115$	1.28986	+	0.937136i	1.28986	+	0.937136i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	0.256741	+	0.790166i	0.256741	+	0.790166i
$125$	−0.997725	+	3.07068i	−0.997725	+	3.07068i
$126$		0			0
$127$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$127$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−0.221062	−	0.680358i	−0.221062	−	0.680358i
$136$		0			0
$137$	−0.672156	−	0.488350i	−0.672156	−	0.488350i	0.198590	−	0.980083i	$-0.436364\pi$
$137$	−0.672156	−	0.488350i	−0.672156	−	0.488350i	−0.870746	+	0.491733i	$0.836364\pi$
$138$		0			0
$139$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$139$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$140$		0			0
$141$	−0.301590	+	0.219118i	−0.301590	+	0.219118i
$142$		0			0
$143$		0			0
$144$	0.715370			0.715370
$145$		0			0
$146$		0			0
$147$	−0.404726	+	1.24562i	−0.404726	+	1.24562i
$148$	0.230270	+	0.167301i	0.230270	+	0.167301i
$149$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$149$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$150$		0			0
$151$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$151$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.59435			−1.59435
$156$		0			0
$157$	0.519923	+	1.60016i	0.519923	+	1.60016i	0.774142	+	0.633012i	$0.218182\pi$
$157$	0.519923	+	1.60016i	0.519923	+	1.60016i	−0.254218	+	0.967147i	$0.581818\pi$
$158$		0			0
$159$	1.78276	+	1.29525i	1.78276	+	1.29525i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.05959	−	0.769835i	1.05959	−	0.769835i	0.0855750	−	0.996332i	$-0.472727\pi$
$163$	1.05959	−	0.769835i	1.05959	−	0.769835i	0.974012	+	0.226497i	$0.0727273\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$167$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$168$		0			0
$169$	0.309017	−	0.951057i	0.309017	−	0.951057i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$173$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−2.20362			−2.20362
$178$		0			0
$179$	0.256741	+	0.790166i	0.256741	+	0.790166i	0.993482	+	0.113991i	$0.0363636\pi$
$179$	0.256741	+	0.790166i	0.256741	+	0.790166i	−0.736741	+	0.676175i	$0.763636\pi$
$180$	−0.424214	+	1.30560i	−0.424214	+	1.30560i
$181$	1.05959	+	0.769835i	1.05959	+	0.769835i	0.974012	−	0.226497i	$-0.0727273\pi$
$181$	1.05959	+	0.769835i	1.05959	+	0.769835i	0.0855750	+	0.996332i	$0.472727\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.441885	+	0.321049i	−0.441885	+	0.321049i
$186$		0			0
$187$		0			0
$188$	−0.284630			−0.284630
$189$		0			0
$190$		0			0
$191$	−0.404726	+	1.24562i	−0.404726	+	1.24562i	0.516397	+	0.856349i	$0.327273\pi$
$191$	−0.404726	+	1.24562i	−0.404726	+	1.24562i	−0.921124	+	0.389270i	$0.872727\pi$
$192$	1.05959	+	0.769835i	1.05959	+	0.769835i
$193$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$193$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$194$		0			0
$195$		0			0
$196$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$199$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$200$		0			0
$201$	0.776664	+	2.39033i	0.776664	+	2.39033i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.480840	+	0.349351i	−0.480840	+	0.349351i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$211$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$212$	0.519923	+	1.60016i	0.519923	+	1.60016i
$213$	0.115197	−	0.354540i	0.115197	−	0.354540i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.0879554	−	0.270699i	−0.0879554	−	0.270699i	0.897398	−	0.441221i	$-0.145455\pi$
$223$	−0.0879554	−	0.270699i	−0.0879554	−	0.270699i	−0.985354	+	0.170522i	$0.945455\pi$
$224$		0			0
$225$	−1.55249	−	1.12795i	−1.55249	−	1.12795i
$226$		0			0
$227$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$227$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$228$		0			0
$229$	1.55249	−	1.12795i	1.55249	−	1.12795i	0.610648	−	0.791902i	$-0.290909\pi$
$229$	1.55249	−	1.12795i	1.55249	−	1.12795i	0.941844	−	0.336049i	$-0.109091\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$233$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$234$		0			0
$235$	0.168785	−	0.519467i	0.168785	−	0.519467i
$236$	−1.36118	−	0.988953i	−1.36118	−	0.988953i
$237$		0			0
$238$		0			0
$239$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$239$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$240$	−2.03333	+	1.47730i	−2.03333	+	1.47730i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	1.20362			1.20362
$244$		0			0
$245$	−0.592999	−	1.82506i	−0.592999	−	1.82506i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.05959	−	0.769835i	1.05959	−	0.769835i	0.0855750	−	0.996332i	$-0.472727\pi$
$251$	1.05959	−	0.769835i	1.05959	−	0.769835i	0.974012	+	0.226497i	$0.0727273\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.309017	+	0.951057i	0.309017	+	0.951057i
$257$	−0.0879554	+	0.270699i	−0.0879554	+	0.270699i	−0.985354	−	0.170522i	$-0.945455\pi$
$257$	−0.0879554	+	0.270699i	−0.0879554	+	0.270699i	0.897398	+	0.441221i	$0.145455\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$	−3.22871			−3.22871
$266$		0			0
$267$	−0.680955	−	2.09576i	−0.680955	−	2.09576i
$268$	−0.592999	+	1.82506i	−0.592999	+	1.82506i
$269$	−0.672156	−	0.488350i	−0.672156	−	0.488350i	0.198590	−	0.980083i	$-0.436364\pi$
$269$	−0.672156	−	0.488350i	−0.672156	−	0.488350i	−0.870746	+	0.491733i	$0.836364\pi$
$270$		0			0
$271$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$271$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	−1.08816			−1.08816
$277$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$277$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$278$		0			0
$279$	0.183665	−	0.565262i	0.183665	−	0.565262i
$280$		0			0
$281$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$281$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$282$		0			0
$283$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$283$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$284$	0.230270	−	0.167301i	0.230270	−	0.167301i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.309017	+	0.951057i	0.309017	+	0.951057i
$290$		0			0
$291$	0.880337	+	0.639602i	0.880337	+	0.639602i
$292$		0			0
$293$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$293$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$294$		0			0
$295$	2.61208	−	1.89779i	2.61208	−	1.89779i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−1.08568	−	3.34138i	−1.08568	−	3.34138i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	2.51334			2.51334
$310$		0			0
$311$	−0.404726	−	1.24562i	−0.404726	−	1.24562i	−0.921124	−	0.389270i	$-0.872727\pi$
$311$	−0.404726	−	1.24562i	−0.404726	−	1.24562i	0.516397	−	0.856349i	$-0.327273\pi$
$312$		0			0
$313$	−1.36118	−	0.988953i	−1.36118	−	0.988953i	−0.998369	−	0.0570888i	$-0.981818\pi$
$313$	−1.36118	−	0.988953i	−1.36118	−	0.988953i	−0.362808	−	0.931864i	$-0.618182\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.36118	+	0.988953i	−1.36118	+	0.988953i	−0.362808	+	0.931864i	$0.618182\pi$
$317$	−1.36118	+	0.988953i	−1.36118	+	0.988953i	−0.998369	+	0.0570888i	$0.981818\pi$
$318$		0			0
$319$		0			0
$320$	−1.91899			−1.91899
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.973745	+	0.707468i	0.973745	+	0.707468i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$331$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$332$		0			0
$333$	−0.0629207	−	0.193650i	−0.0629207	−	0.193650i
$334$		0			0
$335$	−2.97921	−	2.16452i	−2.97921	−	2.16452i
$336$		0			0
$337$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$337$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$338$		0			0
$339$	−1.38776	+	1.00827i	−1.38776	+	1.00827i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	0.645276	−	1.98595i	0.645276	−	1.98595i
$346$		0			0
$347$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$347$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$348$		0			0
$349$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$349$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.00000			2.00000			1.00000			$0$
$353$	2.00000			2.00000			1.00000			$0$
$354$		0			0
$355$	0.168785	+	0.519467i	0.168785	+	0.519467i
$356$	0.519923	−	1.60016i	0.519923	−	1.60016i
$357$		0			0
$358$		0			0
$359$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$359$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$360$		0			0
$361$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.0879554	+	0.270699i	−0.0879554	+	0.270699i	−0.985354	−	0.170522i	$-0.945455\pi$
$367$	−0.0879554	+	0.270699i	−0.0879554	+	0.270699i	0.897398	+	0.441221i	$0.145455\pi$
$368$	−0.672156	−	0.488350i	−0.672156	−	0.488350i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.880337	−	0.639602i	0.880337	−	0.639602i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	4.22871			4.22871
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.36118	−	0.988953i	−1.36118	−	0.988953i	−0.998369	−	0.0570888i	$-0.981818\pi$
$379$	−1.36118	−	0.988953i	−1.36118	−	0.988953i	−0.362808	−	0.931864i	$-0.618182\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.230270	−	0.167301i	0.230270	−	0.167301i	−0.466667	−	0.884433i	$-0.654545\pi$
$383$	0.230270	−	0.167301i	0.230270	−	0.167301i	0.696938	+	0.717132i	$0.254545\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.256741	+	0.790166i	0.256741	+	0.790166i
$389$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$389$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$397$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$398$		0			0
$399$		0			0
$400$	0.828940	−	2.55122i	0.828940	−	2.55122i
$401$	1.05959	+	0.769835i	1.05959	+	0.769835i	0.974012	−	0.226497i	$-0.0727273\pi$
$401$	1.05959	+	0.769835i	1.05959	+	0.769835i	0.0855750	+	0.996332i	$0.472727\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−1.86860	+	1.35762i	−1.86860	+	1.35762i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$409$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$410$		0			0
$411$	−0.336259	+	1.03490i	−0.336259	+	1.03490i
$412$	1.55249	+	1.12795i	1.55249	+	1.12795i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$419$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$420$		0			0
$421$	−0.404726	−	1.24562i	−0.404726	−	1.24562i	−0.921124	−	0.389270i	$-0.872727\pi$
$421$	−0.404726	−	1.24562i	−0.404726	−	1.24562i	0.516397	−	0.856349i	$-0.327273\pi$
$422$		0			0
$423$	0.164728	+	0.119682i	0.164728	+	0.119682i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$431$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$432$	0.115197	+	0.354540i	0.115197	+	0.354540i
$433$	0.256741	−	0.790166i	0.256741	−	0.790166i	−0.736741	−	0.676175i	$-0.763636\pi$
$433$	0.256741	−	0.790166i	0.256741	−	0.790166i	0.993482	−	0.113991i	$-0.0363636\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	0.715370			0.715370
$442$		0			0
$443$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$443$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$444$	0.115197	−	0.354540i	0.115197	−	0.354540i
$445$	2.61208	+	1.89779i	2.61208	+	1.89779i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.05959	−	0.769835i	1.05959	−	0.769835i	0.0855750	−	0.996332i	$-0.472727\pi$
$449$	1.05959	−	0.769835i	1.05959	−	0.769835i	0.974012	+	0.226497i	$0.0727273\pi$
$450$		0			0
$451$		0			0
$452$	−1.30972			−1.30972
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$457$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$458$		0			0
$459$		0			0
$460$	1.28986	−	0.937136i	1.28986	−	0.937136i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$463$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$464$		0			0
$465$	0.645276	+	1.98595i	0.645276	+	1.98595i
$466$		0			0
$467$	−1.36118	−	0.988953i	−1.36118	−	0.988953i	−0.998369	−	0.0570888i	$-0.981818\pi$
$467$	−1.36118	−	0.988953i	−1.36118	−	0.988953i	−0.362808	−	0.931864i	$-0.618182\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	1.78276	−	1.29525i	1.78276	−	1.29525i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.371938	−	1.14471i	0.371938	−	1.14471i
$478$		0			0
$479$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$479$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.59435			−1.59435
$486$		0			0
$487$	0.519923	+	1.60016i	0.519923	+	1.60016i	0.774142	+	0.633012i	$0.218182\pi$
$487$	0.519923	+	1.60016i	0.519923	+	1.60016i	−0.254218	+	0.967147i	$0.581818\pi$
$488$		0			0
$489$	−1.38776	−	1.00827i	−1.38776	−	1.00827i
$490$		0			0
$491$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$491$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.830830			0.830830
$497$		0			0
$498$		0			0
$499$	−0.592999	+	1.82506i	−0.592999	+	1.82506i	−0.0285561	+	0.999592i	$0.509091\pi$
$499$	−0.592999	+	1.82506i	−0.592999	+	1.82506i	−0.564443	+	0.825472i	$0.690909\pi$
$500$	2.61208	+	1.89779i	2.61208	+	1.89779i
$501$		0			0
$502$		0			0
$503$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$503$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.30972			−1.30972
$508$		0			0
$509$	−0.404726	−	1.24562i	−0.404726	−	1.24562i	−0.921124	−	0.389270i	$-0.872727\pi$
$509$	−0.404726	−	1.24562i	−0.404726	−	1.24562i	0.516397	−	0.856349i	$-0.327273\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.97921	+	2.16452i	−2.97921	+	2.16452i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.256741	−	0.790166i	0.256741	−	0.790166i	−0.736741	−	0.676175i	$-0.763636\pi$
$521$	0.256741	−	0.790166i	0.256741	−	0.790166i	0.993482	−	0.113991i	$-0.0363636\pi$
$522$		0			0
$523$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$523$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.309721			−0.309721
$530$		0			0
$531$	0.371938	+	1.14471i	0.371938	+	1.14471i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.880337	−	0.639602i	0.880337	−	0.639602i
$538$		0			0
$539$		0			0
$540$	−0.715370			−0.715370
$541$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$541$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$542$		0			0
$543$	0.530079	−	1.63141i	0.530079	−	1.63141i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$547$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$548$	−0.672156	+	0.488350i	−0.672156	+	0.488350i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	0.578747	+	0.420484i	0.578747	+	0.420484i
$556$		0			0
$557$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$557$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$563$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$564$	0.115197	+	0.354540i	0.115197	+	0.354540i
$565$	0.776664	−	2.39033i	0.776664	−	2.39033i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$569$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	1.71537			1.71537
$574$		0			0
$575$	0.688708	+	2.11963i	0.688708	+	2.11963i
$576$	0.221062	−	0.680358i	0.221062	−	0.680358i
$577$	1.05959	+	0.769835i	1.05959	+	0.769835i	0.974012	−	0.226497i	$-0.0727273\pi$
$577$	1.05959	+	0.769835i	1.05959	+	0.769835i	0.0855750	+	0.996332i	$0.472727\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	0.256741	−	0.790166i	0.256741	−	0.790166i	−0.736741	−	0.676175i	$-0.763636\pi$
$587$	0.256741	−	0.790166i	0.256741	−	0.790166i	0.993482	−	0.113991i	$-0.0363636\pi$
$588$	1.05959	+	0.769835i	1.05959	+	0.769835i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.230270	−	0.167301i	0.230270	−	0.167301i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.680955	−	2.09576i	−0.680955	−	2.09576i
$598$		0			0
$599$	1.55249	+	1.12795i	1.55249	+	1.12795i	0.941844	+	0.336049i	$0.109091\pi$
$599$	1.55249	+	1.12795i	1.55249	+	1.12795i	0.610648	+	0.791902i	$0.290909\pi$
$600$		0			0
$601$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$601$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$602$		0			0
$603$	1.11061	−	0.806903i	1.11061	−	0.806903i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$607$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$613$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$617$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$618$		0			0
$619$	−0.592999	−	1.82506i	−0.592999	−	1.82506i	−0.564443	−	0.825472i	$-0.690909\pi$
$619$	−0.592999	−	1.82506i	−0.592999	−	1.82506i	−0.0285561	−	0.999592i	$-0.509091\pi$
$620$	−0.492682	+	1.51632i	−0.492682	+	1.51632i
$621$	−0.250570	−	0.182050i	−0.250570	−	0.182050i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−2.84235	+	2.06509i	−2.84235	+	2.06509i
$626$		0			0
$627$		0			0
$628$	1.68251			1.68251
$629$		0			0
$630$		0			0
$631$	−0.404726	+	1.24562i	−0.404726	+	1.24562i	0.516397	+	0.856349i	$0.327273\pi$
$631$	−0.404726	+	1.24562i	−0.404726	+	1.24562i	−0.921124	+	0.389270i	$0.872727\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	1.78276	−	1.29525i	1.78276	−	1.29525i
$637$		0			0
$638$		0			0
$639$	−0.203616			−0.203616
$640$		0			0
$641$	−0.592999	−	1.82506i	−0.592999	−	1.82506i	−0.564443	−	0.825472i	$-0.690909\pi$
$641$	−0.592999	−	1.82506i	−0.592999	−	1.82506i	−0.0285561	−	0.999592i	$-0.509091\pi$
$642$		0			0
$643$	−1.36118	−	0.988953i	−1.36118	−	0.988953i	−0.998369	−	0.0570888i	$-0.981818\pi$
$643$	−1.36118	−	0.988953i	−1.36118	−	0.988953i	−0.362808	−	0.931864i	$-0.618182\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.55249	−	1.12795i	1.55249	−	1.12795i	0.610648	−	0.791902i	$-0.290909\pi$
$647$	1.55249	−	1.12795i	1.55249	−	1.12795i	0.941844	−	0.336049i	$-0.109091\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−0.404726	−	1.24562i	−0.404726	−	1.24562i
$653$	0.519923	−	1.60016i	0.519923	−	1.60016i	−0.254218	−	0.967147i	$-0.581818\pi$
$653$	0.519923	−	1.60016i	0.519923	−	1.60016i	0.774142	−	0.633012i	$-0.218182\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$661$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.301590	+	0.219118i	−0.301590	+	0.219118i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$673$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$674$		0			0
$675$	0.309017	−	0.951057i	0.309017	−	0.951057i
$676$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$677$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$677$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$683$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$684$		0			0
$685$	−0.492682	−	1.51632i	−0.492682	−	1.51632i
$686$		0			0
$687$	−2.03333	−	1.47730i	−2.03333	−	1.47730i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−1.36118	+	0.988953i	−1.36118	+	0.988953i	−0.362808	+	0.931864i	$0.618182\pi$
$691$	−1.36118	+	0.988953i	−1.36118	+	0.988953i	−0.998369	+	0.0570888i	$0.981818\pi$
$692$		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			0
$701$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$701$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−0.715370			−0.715370
$706$		0			0
$707$		0			0
$708$	−0.680955	+	2.09576i	−0.680955	+	2.09576i
$709$	−0.672156	−	0.488350i	−0.672156	−	0.488350i	0.198590	−	0.980083i	$-0.436364\pi$
$709$	−0.672156	−	0.488350i	−0.672156	−	0.488350i	−0.870746	+	0.491733i	$0.836364\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.558447	+	0.405736i	−0.558447	+	0.405736i
$714$		0			0
$715$		0			0
$716$	0.830830			0.830830
$717$		0			0
$718$		0			0
$719$	−0.592999	+	1.82506i	−0.592999	+	1.82506i	−0.0285561	+	0.999592i	$0.509091\pi$
$719$	−0.592999	+	1.82506i	−0.592999	+	1.82506i	−0.564443	+	0.825472i	$0.690909\pi$
$720$	1.11061	+	0.806903i	1.11061	+	0.806903i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	1.05959	−	0.769835i	1.05959	−	0.769835i
$725$		0			0
$726$		0			0
$727$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$727$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$728$		0			0
$729$	−0.115197	−	0.354540i	−0.115197	−	0.354540i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$733$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$734$		0			0
$735$	−2.03333	+	1.47730i	−2.03333	+	1.47730i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$739$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$740$	0.168785	+	0.519467i	0.168785	+	0.519467i
$741$		0			0
$742$		0			0
$743$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$743$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.592999	−	1.82506i	−0.592999	−	1.82506i	−0.564443	−	0.825472i	$-0.690909\pi$
$751$	−0.592999	−	1.82506i	−0.592999	−	1.82506i	−0.0285561	−	0.999592i	$-0.509091\pi$
$752$	−0.0879554	+	0.270699i	−0.0879554	+	0.270699i
$753$	−1.38776	−	1.00827i	−1.38776	−	1.00827i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.230270	−	0.167301i	0.230270	−	0.167301i	−0.466667	−	0.884433i	$-0.654545\pi$
$757$	0.230270	−	0.167301i	0.230270	−	0.167301i	0.696938	+	0.717132i	$0.254545\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$761$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$762$		0			0
$763$		0			0
$764$	1.05959	+	0.769835i	1.05959	+	0.769835i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	1.05959	−	0.769835i	1.05959	−	0.769835i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	0.372786			0.372786
$772$		0			0
$773$	−0.592999	−	1.82506i	−0.592999	−	1.82506i	−0.564443	−	0.825472i	$-0.690909\pi$
$773$	−0.592999	−	1.82506i	−0.592999	−	1.82506i	−0.0285561	−	0.999592i	$-0.509091\pi$
$774$		0			0
$775$	−1.80306	−	1.31000i	−1.80306	−	1.31000i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.309017	+	0.951057i	0.309017	+	0.951057i
$785$	−0.997725	+	3.07068i	−0.997725	+	3.07068i
$786$		0			0
$787$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$787$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	1.30674	+	4.02174i	1.30674	+	4.02174i
$796$	0.519923	−	1.60016i	0.519923	−	1.60016i
$797$	0.230270	+	0.167301i	0.230270	+	0.167301i	0.696938	−	0.717132i	$-0.254545\pi$
$797$	0.230270	+	0.167301i	0.230270	+	0.167301i	−0.466667	+	0.884433i	$0.654545\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−0.973745	+	0.707468i	−0.973745	+	0.707468i
$802$		0			0
$803$		0			0
$804$	2.51334			2.51334
$805$		0			0
$806$		0			0
$807$	−0.336259	+	1.03490i	−0.336259	+	1.03490i
$808$		0			0
$809$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$809$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$810$		0			0
$811$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$811$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.51334			2.51334
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$821$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$822$		0			0
$823$	1.55249	−	1.12795i	1.55249	−	1.12795i	0.610648	−	0.791902i	$-0.290909\pi$
$823$	1.55249	−	1.12795i	1.55249	−	1.12795i	0.941844	−	0.336049i	$-0.109091\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$827$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$828$	0.183665	+	0.565262i	0.183665	+	0.565262i
$829$	−0.404726	+	1.24562i	−0.404726	+	1.24562i	0.516397	+	0.856349i	$0.327273\pi$
$829$	−0.404726	+	1.24562i	−0.404726	+	1.24562i	−0.921124	+	0.389270i	$0.872727\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.309721			0.309721
$838$		0			0
$839$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$839$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$840$		0			0
$841$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.55249	−	1.12795i	1.55249	−	1.12795i
$846$		0			0
$847$		0			0
$848$	1.68251			1.68251
$849$		0			0
$850$		0			0
$851$	−0.0730760	+	0.224905i	−0.0730760	+	0.224905i
$852$	−0.301590	−	0.219118i	−0.301590	−	0.219118i
$853$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$853$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$859$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.05959	+	0.769835i	1.05959	+	0.769835i	0.974012	−	0.226497i	$-0.0727273\pi$
$863$	1.05959	+	0.769835i	1.05959	+	0.769835i	0.0855750	+	0.996332i	$0.472727\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	1.05959	−	0.769835i	1.05959	−	0.769835i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.183665	−	0.565262i	0.183665	−	0.565262i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$877$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$881$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$882$		0			0
$883$	0.256741	+	0.790166i	0.256741	+	0.790166i	0.993482	+	0.113991i	$0.0363636\pi$
$883$	0.256741	+	0.790166i	0.256741	+	0.790166i	−0.736741	+	0.676175i	$0.763636\pi$
$884$		0			0
$885$	−3.42110	−	2.48557i	−3.42110	−	2.48557i
$886$		0			0
$887$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$887$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−0.284630			−0.284630
$893$		0			0
$894$		0			0
$895$	−0.492682	+	1.51632i	−0.492682	+	1.51632i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−1.55249	+	1.12795i	−1.55249	+	1.12795i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.776664	+	2.39033i	0.776664	+	2.39033i
$906$		0			0
$907$	−0.672156	−	0.488350i	−0.672156	−	0.488350i	0.198590	−	0.980083i	$-0.436364\pi$
$907$	−0.672156	−	0.488350i	−0.672156	−	0.488350i	−0.870746	+	0.491733i	$0.836364\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.55249	−	1.12795i	1.55249	−	1.12795i	0.610648	−	0.791902i	$-0.290909\pi$
$911$	1.55249	−	1.12795i	1.55249	−	1.12795i	0.941844	−	0.336049i	$-0.109091\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.592999	−	1.82506i	−0.592999	−	1.82506i
$917$		0			0
$918$		0			0
$919$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$919$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.763521			−0.763521
$926$		0			0
$927$	−0.424214	−	1.30560i	−0.424214	−	1.30560i
$928$		0			0
$929$	0.230270	+	0.167301i	0.230270	+	0.167301i	0.696938	−	0.717132i	$-0.254545\pi$
$929$	0.230270	+	0.167301i	0.230270	+	0.167301i	−0.466667	+	0.884433i	$0.654545\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−1.38776	+	1.00827i	−1.38776	+	1.00827i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$937$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$938$		0			0
$939$	−0.680955	+	2.09576i	−0.680955	+	2.09576i
$940$	−0.441885	−	0.321049i	−0.441885	−	0.321049i
$941$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$941$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$942$		0			0
$943$		0			0
$944$	−1.36118	+	0.988953i	−1.36118	+	0.988953i
$945$		0			0
$946$		0			0
$947$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$947$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	1.78276	+	1.29525i	1.78276	+	1.29525i
$952$		0			0
$953$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$953$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$954$		0			0
$955$	−2.03333	+	1.47730i	−2.03333	+	1.47730i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	0.776664	+	2.39033i	0.776664	+	2.39033i
$961$	−0.0957092	+	0.294563i	−0.0957092	+	0.294563i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−0.0879554	−	0.270699i	−0.0879554	−	0.270699i	0.897398	−	0.441221i	$-0.145455\pi$
$971$	−0.0879554	−	0.270699i	−0.0879554	−	0.270699i	−0.985354	+	0.170522i	$0.945455\pi$
$972$	0.371938	−	1.14471i	0.371938	−	1.14471i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.05959	−	0.769835i	1.05959	−	0.769835i	0.0855750	−	0.996332i	$-0.472727\pi$
$977$	1.05959	−	0.769835i	1.05959	−	0.769835i	0.974012	+	0.226497i	$0.0727273\pi$
$978$		0			0
$979$		0			0
$980$	−1.91899			−1.91899
$981$		0			0
$982$		0			0
$983$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$983$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$991$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$992$		0			0
$993$	0.776664	+	2.39033i	0.776664	+	2.39033i
$994$		0			0
$995$	2.61208	+	1.89779i	2.61208	+	1.89779i
$996$		0			0
$997$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$997$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$998$		0			0
$999$	0.0858414	−	0.0623675i	0.0858414	−	0.0623675i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1331.1.d.a.1207.2		20
11.2	odd	10		1331.1.b.a.1330.2	✓	5
11.3	even	5	inner	1331.1.d.a.699.4		20
11.4	even	5	inner	1331.1.d.a.161.2		20
11.5	even	5	inner	1331.1.d.a.596.4		20
11.6	odd	10	inner	1331.1.d.a.596.4		20
11.7	odd	10	inner	1331.1.d.a.161.2		20
11.8	odd	10	inner	1331.1.d.a.699.4		20
11.9	even	5		1331.1.b.a.1330.2	✓	5
11.10	odd	2	CM	1331.1.d.a.1207.2		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1331.1.b.a.1330.2	✓	5	11.2	odd	10
1331.1.b.a.1330.2	✓	5	11.9	even	5
1331.1.d.a.161.2		20	11.4	even	5	inner
1331.1.d.a.161.2		20	11.7	odd	10	inner
1331.1.d.a.596.4		20	11.5	even	5	inner
1331.1.d.a.596.4		20	11.6	odd	10	inner
1331.1.d.a.699.4		20	11.3	even	5	inner
1331.1.d.a.699.4		20	11.8	odd	10	inner
1331.1.d.a.1207.2		20	1.1	even	1	trivial
1331.1.d.a.1207.2		20	11.10	odd	2	CM

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:	\( N \)	\(=\)	\( 1331 = 11^{3} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1331.d (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.404726 - 1.24562i) q^{3} +(0.309017 - 0.951057i) q^{4} +(1.55249 + 1.12795i) q^{5} +(-0.578747 + 0.420484i) q^{9} +O(q^{10})\)	\(q+(-0.404726 - 1.24562i) q^{3} +(0.309017 - 0.951057i) q^{4} +(1.55249 + 1.12795i) q^{5} +(-0.578747 + 0.420484i) q^{9} -1.30972 q^{12} +(0.776664 - 2.39033i) q^{15} +(-0.809017 - 0.587785i) q^{16} +(1.55249 - 1.12795i) q^{20} +0.830830 q^{23} +(0.828940 + 2.55122i) q^{25} +(-0.301590 - 0.219118i) q^{27} +(-0.672156 + 0.488350i) q^{31} +(0.221062 + 0.680358i) q^{36} +(-0.0879554 + 0.270699i) q^{37} -1.37279 q^{45} +(-0.0879554 - 0.270699i) q^{47} +(-0.404726 + 1.24562i) q^{48} +(-0.809017 - 0.587785i) q^{49} +(-1.36118 + 0.988953i) q^{53} +(0.519923 - 1.60016i) q^{59} +(-2.03333 - 1.47730i) q^{60} +(-0.809017 + 0.587785i) q^{64} -1.91899 q^{67} +(-0.336259 - 1.03490i) q^{69} +(0.230270 + 0.167301i) q^{71} +(2.84235 - 2.06509i) q^{75} +(-0.592999 - 1.82506i) q^{80} +(-0.371938 + 1.14471i) q^{81} +1.68251 q^{89} +(0.256741 - 0.790166i) q^{92} +(0.880337 + 0.639602i) q^{93} +(-0.672156 + 0.488350i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + q^{3} - 5 q^{4} + q^{5} - 4 q^{9}+O(q^{10}) \) 20 * q + q^3 - 5 * q^4 + q^5 - 4 * q^9	\( 20 q + q^{3} - 5 q^{4} + q^{5} - 4 q^{9} - 4 q^{12} + 2 q^{15} - 5 q^{16} + q^{20} - 4 q^{23} - 4 q^{25} + 2 q^{27} + q^{31} - 4 q^{36} + q^{37} - 12 q^{45} + q^{47} + q^{48} - 5 q^{49} + q^{53} + q^{59} + 2 q^{60} - 5 q^{64} - 4 q^{67} + 2 q^{69} + q^{71} + 3 q^{75} + q^{80} - 3 q^{81} - 4 q^{89} + q^{92} + 2 q^{93} + q^{97}+O(q^{100}) \) 20 * q + q^3 - 5 * q^4 + q^5 - 4 * q^9 - 4 * q^12 + 2 * q^15 - 5 * q^16 + q^20 - 4 * q^23 - 4 * q^25 + 2 * q^27 + q^31 - 4 * q^36 + q^37 - 12 * q^45 + q^47 + q^48 - 5 * q^49 + q^53 + q^59 + 2 * q^60 - 5 * q^64 - 4 * q^67 + 2 * q^69 + q^71 + 3 * q^75 + q^80 - 3 * q^81 - 4 * q^89 + q^92 + 2 * q^93 + q^97

Introduction

L-functions

Modular forms

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Database

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