LMFDB - Embedded newform 1296.2.i.t.865.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1296,2,Mod(433,1296)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1296.433"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1296.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$10.3486121020$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 648)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			865.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1296.865
Dual form			1296.2.i.t.433.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.86603 + 3.23205i) q^{5} +(1.73205 - 3.00000i) q^{7} +(-1.00000 + 1.73205i) q^{11} +(1.23205 + 2.13397i) q^{13} -2.26795 q^{17} +7.46410 q^{19} +(2.46410 + 4.26795i) q^{23} +(-4.46410 + 7.73205i) q^{25} +(2.13397 - 3.69615i) q^{29} +(-5.46410 - 9.46410i) q^{31} +12.9282 q^{35} -0.464102 q^{37} +(3.46410 + 6.00000i) q^{41} +(-2.26795 + 3.92820i) q^{43} +(-3.46410 + 6.00000i) q^{47} +(-2.50000 - 4.33013i) q^{49} +10.9282 q^{53} -7.46410 q^{55} +(-4.00000 - 6.92820i) q^{59} +(-5.23205 + 9.06218i) q^{61} +(-4.59808 + 7.96410i) q^{65} +(-0.267949 - 0.464102i) q^{67} -2.00000 q^{71} +1.00000 q^{73} +(3.46410 + 6.00000i) q^{77} +(0.267949 - 0.464102i) q^{79} +(1.46410 - 2.53590i) q^{83} +(-4.23205 - 7.33013i) q^{85} +5.19615 q^{89} +8.53590 q^{91} +(13.9282 + 24.1244i) q^{95} +(5.92820 - 10.2679i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 4 q^{11} - 2 q^{13} - 16 q^{17} + 16 q^{19} - 4 q^{23} - 4 q^{25} + 12 q^{29} - 8 q^{31} + 24 q^{35} + 12 q^{37} - 16 q^{43} - 10 q^{49} + 16 q^{53} - 16 q^{55} - 16 q^{59} - 14 q^{61} - 8 q^{65}+ \cdots - 4 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 - 4 * q^11 - 2 * q^13 - 16 * q^17 + 16 * q^19 - 4 * q^23 - 4 * q^25 + 12 * q^29 - 8 * q^31 + 24 * q^35 + 12 * q^37 - 16 * q^43 - 10 * q^49 + 16 * q^53 - 16 * q^55 - 16 * q^59 - 14 * q^61 - 8 * q^65 - 8 * q^67 - 8 * q^71 + 4 * q^73 + 8 * q^79 - 8 * q^83 - 10 * q^85 + 48 * q^91 + 28 * q^95 - 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.86603	+	3.23205i	0.834512	+	1.44542i	0.894427	+	0.447214i	$0.147584\pi$
$5$	1.86603	+	3.23205i	0.834512	+	1.44542i	−0.0599153	+	0.998203i	$0.519083\pi$
$6$		0			0
$7$	1.73205	−	3.00000i	0.654654	−	1.13389i	−0.327327	−	0.944911i	$-0.606148\pi$
$7$	1.73205	−	3.00000i	0.654654	−	1.13389i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	$-0.930824\pi$
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	0.674967	+	0.737848i	$0.264158\pi$
$12$		0			0
$13$	1.23205	+	2.13397i	0.341709	+	0.591858i	0.984750	−	0.173974i	$-0.0556608\pi$
$13$	1.23205	+	2.13397i	0.341709	+	0.591858i	−0.643041	+	0.765832i	$0.722327\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.26795			−0.550058			−0.275029	−	0.961436i	$-0.588688\pi$
$17$	−2.26795			−0.550058			−0.275029	+	0.961436i	$0.588688\pi$
$18$		0			0
$19$	7.46410			1.71238			0.856191	−	0.516659i	$-0.172825\pi$
$19$	7.46410			1.71238			0.856191	+	0.516659i	$0.172825\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.46410	+	4.26795i	0.513801	+	0.889929i	0.999872	+	0.0160097i	$0.00509626\pi$
$23$	2.46410	+	4.26795i	0.513801	+	0.889929i	−0.486071	+	0.873919i	$0.661570\pi$
$24$		0			0
$25$	−4.46410	+	7.73205i	−0.892820	+	1.54641i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.13397	−	3.69615i	0.396269	−	0.686358i	−0.596993	−	0.802246i	$-0.703638\pi$
$29$	2.13397	−	3.69615i	0.396269	−	0.686358i	0.993262	+	0.115888i	$0.0369714\pi$
$30$		0			0
$31$	−5.46410	−	9.46410i	−0.981382	−	1.69980i	−0.657027	−	0.753867i	$-0.728186\pi$
$31$	−5.46410	−	9.46410i	−0.981382	−	1.69980i	−0.324355	−	0.945935i	$-0.605147\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	12.9282			2.18527
$36$		0			0
$37$	−0.464102			−0.0762978			−0.0381489	−	0.999272i	$-0.512146\pi$
$37$	−0.464102			−0.0762978			−0.0381489	+	0.999272i	$0.512146\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.46410	+	6.00000i	0.541002	+	0.937043i	0.998847	+	0.0480106i	$0.0152881\pi$
$41$	3.46410	+	6.00000i	0.541002	+	0.937043i	−0.457845	+	0.889032i	$0.651379\pi$
$42$		0			0
$43$	−2.26795	+	3.92820i	−0.345859	+	0.599045i	−0.985509	−	0.169621i	$-0.945746\pi$
$43$	−2.26795	+	3.92820i	−0.345859	+	0.599045i	0.639650	+	0.768666i	$0.279079\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.46410	+	6.00000i	−0.505291	+	0.875190i	0.494690	+	0.869069i	$0.335282\pi$
$47$	−3.46410	+	6.00000i	−0.505291	+	0.875190i	−0.999981	+	0.00612051i	$0.998052\pi$
$48$		0			0
$49$	−2.50000	−	4.33013i	−0.357143	−	0.618590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	10.9282			1.50110			0.750552	−	0.660811i	$-0.229788\pi$
$53$	10.9282			1.50110			0.750552	+	0.660811i	$0.229788\pi$
$54$		0			0
$55$	−7.46410			−1.00646
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.00000	−	6.92820i	−0.520756	−	0.901975i	−0.999709	−	0.0241347i	$-0.992317\pi$
$59$	−4.00000	−	6.92820i	−0.520756	−	0.901975i	0.478953	−	0.877841i	$-0.341016\pi$
$60$		0			0
$61$	−5.23205	+	9.06218i	−0.669895	+	1.16029i	0.308038	+	0.951374i	$0.400328\pi$
$61$	−5.23205	+	9.06218i	−0.669895	+	1.16029i	−0.977933	+	0.208919i	$0.933006\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.59808	+	7.96410i	−0.570321	+	0.987825i
$66$		0			0
$67$	−0.267949	−	0.464102i	−0.0327352	−	0.0566990i	0.849194	−	0.528082i	$-0.177088\pi$
$67$	−0.267949	−	0.464102i	−0.0327352	−	0.0566990i	−0.881929	+	0.471383i	$0.843755\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$	1.00000			0.117041			0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000			0.117041			0.0585206	+	0.998286i	$0.481362\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.46410	+	6.00000i	0.394771	+	0.683763i
$78$		0			0
$79$	0.267949	−	0.464102i	0.0301466	−	0.0522155i	−0.850558	−	0.525880i	$-0.823736\pi$
$79$	0.267949	−	0.464102i	0.0301466	−	0.0522155i	0.880705	+	0.473665i	$0.157069\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.46410	−	2.53590i	0.160706	−	0.278351i	−0.774416	−	0.632677i	$-0.781956\pi$
$83$	1.46410	−	2.53590i	0.160706	−	0.278351i	0.935122	+	0.354326i	$0.115290\pi$
$84$		0			0
$85$	−4.23205	−	7.33013i	−0.459030	−	0.795064i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.19615			0.550791			0.275396	−	0.961331i	$-0.411191\pi$
$89$	5.19615			0.550791			0.275396	+	0.961331i	$0.411191\pi$
$90$		0			0
$91$	8.53590			0.894805
$92$		0			0
$93$		0			0
$94$		0			0
$95$	13.9282	+	24.1244i	1.42900	+	2.47511i
$96$		0			0
$97$	5.92820	−	10.2679i	0.601918	−	1.04255i	−0.390613	−	0.920555i	$-0.627737\pi$
$97$	5.92820	−	10.2679i	0.601918	−	1.04255i	0.992530	−	0.121997i	$-0.0389299\pi$
$98$		0			0
$99$		0			0

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.2.i.t.865.2		4
3.2	odd	2		1296.2.i.r.865.1		4
4.3	odd	2		648.2.i.j.217.2		4
9.2	odd	6		1296.2.a.q.1.2		2
9.4	even	3	inner	1296.2.i.t.433.2		4
9.5	odd	6		1296.2.i.r.433.1		4
9.7	even	3		1296.2.a.m.1.1		2
12.11	even	2		648.2.i.i.217.1		4
36.7	odd	6		648.2.a.e.1.1	✓	2
36.11	even	6		648.2.a.h.1.2	yes	2
36.23	even	6		648.2.i.i.433.1		4
36.31	odd	6		648.2.i.j.433.2		4
72.11	even	6		5184.2.a.bg.1.1		2
72.29	odd	6		5184.2.a.bi.1.1		2
72.43	odd	6		5184.2.a.cb.1.2		2
72.61	even	6		5184.2.a.bz.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.a.e.1.1	✓	2	36.7	odd	6
648.2.a.h.1.2	yes	2	36.11	even	6
648.2.i.i.217.1		4	12.11	even	2
648.2.i.i.433.1		4	36.23	even	6
648.2.i.j.217.2		4	4.3	odd	2
648.2.i.j.433.2		4	36.31	odd	6
1296.2.a.m.1.1		2	9.7	even	3
1296.2.a.q.1.2		2	9.2	odd	6
1296.2.i.r.433.1		4	9.5	odd	6
1296.2.i.r.865.1		4	3.2	odd	2
1296.2.i.t.433.2		4	9.4	even	3	inner
1296.2.i.t.865.2		4	1.1	even	1	trivial
5184.2.a.bg.1.1		2	72.11	even	6
5184.2.a.bi.1.1		2	72.29	odd	6
5184.2.a.bz.1.2		2	72.61	even	6
5184.2.a.cb.1.2		2	72.43	odd	6

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Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1296.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.86603 + 3.23205i) q^{5} +(1.73205 - 3.00000i) q^{7} +(-1.00000 + 1.73205i) q^{11} +(1.23205 + 2.13397i) q^{13} -2.26795 q^{17} +7.46410 q^{19} +(2.46410 + 4.26795i) q^{23} +(-4.46410 + 7.73205i) q^{25} +(2.13397 - 3.69615i) q^{29} +(-5.46410 - 9.46410i) q^{31} +12.9282 q^{35} -0.464102 q^{37} +(3.46410 + 6.00000i) q^{41} +(-2.26795 + 3.92820i) q^{43} +(-3.46410 + 6.00000i) q^{47} +(-2.50000 - 4.33013i) q^{49} +10.9282 q^{53} -7.46410 q^{55} +(-4.00000 - 6.92820i) q^{59} +(-5.23205 + 9.06218i) q^{61} +(-4.59808 + 7.96410i) q^{65} +(-0.267949 - 0.464102i) q^{67} -2.00000 q^{71} +1.00000 q^{73} +(3.46410 + 6.00000i) q^{77} +(0.267949 - 0.464102i) q^{79} +(1.46410 - 2.53590i) q^{83} +(-4.23205 - 7.33013i) q^{85} +5.19615 q^{89} +8.53590 q^{91} +(13.9282 + 24.1244i) q^{95} +(5.92820 - 10.2679i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 4 q^{11} - 2 q^{13} - 16 q^{17} + 16 q^{19} - 4 q^{23} - 4 q^{25} + 12 q^{29} - 8 q^{31} + 24 q^{35} + 12 q^{37} - 16 q^{43} - 10 q^{49} + 16 q^{53} - 16 q^{55} - 16 q^{59} - 14 q^{61} - 8 q^{65}+ \cdots - 4 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 - 4 * q^11 - 2 * q^13 - 16 * q^17 + 16 * q^19 - 4 * q^23 - 4 * q^25 + 12 * q^29 - 8 * q^31 + 24 * q^35 + 12 * q^37 - 16 * q^43 - 10 * q^49 + 16 * q^53 - 16 * q^55 - 16 * q^59 - 14 * q^61 - 8 * q^65 - 8 * q^67 - 8 * q^71 + 4 * q^73 + 8 * q^79 - 8 * q^83 - 10 * q^85 + 48 * q^91 + 28 * q^95 - 4 * q^97

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