LMFDB - Embedded newform 324.2.e.d.109.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [324,2,Mod(109,324)] 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("324.109"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	324.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,3,0,-2] 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:	$2.58715302549$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	324.109
Dual form			324.2.e.d.217.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50000 - 2.59808i) q^{5} +(-1.00000 - 1.73205i) q^{7} +(-3.00000 - 5.19615i) q^{11} +(-2.50000 + 4.33013i) q^{13} +3.00000 q^{17} +2.00000 q^{19} +(3.00000 - 5.19615i) q^{23} +(-2.00000 - 3.46410i) q^{25} +(1.50000 + 2.59808i) q^{29} +(2.00000 - 3.46410i) q^{31} -6.00000 q^{35} +5.00000 q^{37} +(-3.00000 + 5.19615i) q^{41} +(5.00000 + 8.66025i) q^{43} +(1.50000 - 2.59808i) q^{49} +6.00000 q^{53} -18.0000 q^{55} +(-6.00000 + 10.3923i) q^{59} +(-2.50000 - 4.33013i) q^{61} +(7.50000 + 12.9904i) q^{65} +(-1.00000 + 1.73205i) q^{67} -6.00000 q^{71} -1.00000 q^{73} +(-6.00000 + 10.3923i) q^{77} +(5.00000 + 8.66025i) q^{79} +(4.50000 - 7.79423i) q^{85} +3.00000 q^{89} +10.0000 q^{91} +(3.00000 - 5.19615i) q^{95} +(5.00000 + 8.66025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{5} - 2 q^{7} - 6 q^{11} - 5 q^{13} + 6 q^{17} + 4 q^{19} + 6 q^{23} - 4 q^{25} + 3 q^{29} + 4 q^{31} - 12 q^{35} + 10 q^{37} - 6 q^{41} + 10 q^{43} + 3 q^{49} + 12 q^{53} - 36 q^{55} - 12 q^{59}+ \cdots + 10 q^{97}+O(q^{100}) $ 2 * q + 3 * q^5 - 2 * q^7 - 6 * q^11 - 5 * q^13 + 6 * q^17 + 4 * q^19 + 6 * q^23 - 4 * q^25 + 3 * q^29 + 4 * q^31 - 12 * q^35 + 10 * q^37 - 6 * q^41 + 10 * q^43 + 3 * q^49 + 12 * q^53 - 36 * q^55 - 12 * q^59 - 5 * q^61 + 15 * q^65 - 2 * q^67 - 12 * q^71 - 2 * q^73 - 12 * q^77 + 10 * q^79 + 9 * q^85 + 6 * q^89 + 20 * q^91 + 6 * q^95 + 10 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	1.50000	−	2.59808i	0.670820	−	1.16190i	−0.306851	−	0.951757i	$-0.599275\pi$
$5$	1.50000	−	2.59808i	0.670820	−	1.16190i	0.977672	−	0.210138i	$-0.0673912\pi$
$6$		0			0
$7$	−1.00000	−	1.73205i	−0.377964	−	0.654654i	0.612801	−	0.790237i	$-0.290043\pi$
$7$	−1.00000	−	1.73205i	−0.377964	−	0.654654i	−0.990766	+	0.135583i	$0.956709\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.00000	−	5.19615i	−0.904534	−	1.56670i	−0.821541	−	0.570149i	$-0.806886\pi$
$11$	−3.00000	−	5.19615i	−0.904534	−	1.56670i	−0.0829925	−	0.996550i	$-0.526448\pi$
$12$		0			0
$13$	−2.50000	+	4.33013i	−0.693375	+	1.20096i	0.277350	+	0.960769i	$0.410544\pi$
$13$	−2.50000	+	4.33013i	−0.693375	+	1.20096i	−0.970725	+	0.240192i	$0.922790\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.00000			0.727607			0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000			0.727607			0.363803	+	0.931476i	$0.381478\pi$
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	−0.362892	−	0.931831i	$-0.618211\pi$
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	0.988436	−	0.151642i	$-0.0484560\pi$
$24$		0			0
$25$	−2.00000	−	3.46410i	−0.400000	−	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.50000	+	2.59808i	0.278543	+	0.482451i	0.971023	−	0.238987i	$-0.0768152\pi$
$29$	1.50000	+	2.59808i	0.278543	+	0.482451i	−0.692480	+	0.721437i	$0.743482\pi$
$30$		0			0
$31$	2.00000	−	3.46410i	0.359211	−	0.622171i	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	2.00000	−	3.46410i	0.359211	−	0.622171i	0.987829	+	0.155543i	$0.0497126\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.00000			−1.01419
$36$		0			0
$37$	5.00000			0.821995			0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000			0.821995			0.410997	+	0.911636i	$0.365181\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.00000	+	5.19615i	−0.468521	+	0.811503i	−0.999353	−	0.0359748i	$-0.988546\pi$
$41$	−3.00000	+	5.19615i	−0.468521	+	0.811503i	0.530831	+	0.847477i	$0.321880\pi$
$42$		0			0
$43$	5.00000	+	8.66025i	0.762493	+	1.32068i	0.941562	+	0.336840i	$0.109358\pi$
$43$	5.00000	+	8.66025i	0.762493	+	1.32068i	−0.179069	+	0.983836i	$0.557309\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	1.50000	−	2.59808i	0.214286	−	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$	−18.0000			−2.42712
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	0.150148	+	0.988663i	$0.452025\pi$
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	−0.931282	+	0.364299i	$0.881308\pi$
$60$		0			0
$61$	−2.50000	−	4.33013i	−0.320092	−	0.554416i	0.660415	−	0.750901i	$-0.270381\pi$
$61$	−2.50000	−	4.33013i	−0.320092	−	0.554416i	−0.980507	+	0.196485i	$0.937047\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	7.50000	+	12.9904i	0.930261	+	1.61126i
$66$		0			0
$67$	−1.00000	+	1.73205i	−0.122169	+	0.211604i	−0.920623	−	0.390453i	$-0.872318\pi$
$67$	−1.00000	+	1.73205i	−0.122169	+	0.211604i	0.798454	+	0.602056i	$0.205652\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$	−1.00000			−0.117041			−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000			−0.117041			−0.0585206	+	0.998286i	$0.518638\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.00000	+	10.3923i	−0.683763	+	1.18431i
$78$		0			0
$79$	5.00000	+	8.66025i	0.562544	+	0.974355i	0.997274	+	0.0737937i	$0.0235106\pi$
$79$	5.00000	+	8.66025i	0.562544	+	0.974355i	−0.434730	+	0.900561i	$0.643156\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$83$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$84$		0			0
$85$	4.50000	−	7.79423i	0.488094	−	0.845403i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.00000			0.317999			0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000			0.317999			0.159000	+	0.987279i	$0.449173\pi$
$90$		0			0
$91$	10.0000			1.04828
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.00000	−	5.19615i	0.307794	−	0.533114i
$96$		0			0
$97$	5.00000	+	8.66025i	0.507673	+	0.879316i	0.999961	+	0.00888289i	$0.00282755\pi$
$97$	5.00000	+	8.66025i	0.507673	+	0.879316i	−0.492287	+	0.870433i	$0.663839\pi$
$98$		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	324.2.e.d.109.1		2
3.2	odd	2		324.2.e.a.109.1		2
4.3	odd	2		1296.2.i.p.433.1		2
9.2	odd	6		324.2.e.a.217.1		2
9.4	even	3		324.2.a.b.1.1	✓	1
9.5	odd	6		324.2.a.d.1.1	yes	1
9.7	even	3	inner	324.2.e.d.217.1		2
12.11	even	2		1296.2.i.d.433.1		2
36.7	odd	6		1296.2.i.p.865.1		2
36.11	even	6		1296.2.i.d.865.1		2
36.23	even	6		1296.2.a.j.1.1		1
36.31	odd	6		1296.2.a.a.1.1		1
45.4	even	6		8100.2.a.f.1.1		1
45.13	odd	12		8100.2.d.j.649.1		2
45.14	odd	6		8100.2.a.a.1.1		1
45.22	odd	12		8100.2.d.j.649.2		2
45.23	even	12		8100.2.d.a.649.1		2
45.32	even	12		8100.2.d.a.649.2		2
72.5	odd	6		5184.2.a.g.1.1		1
72.13	even	6		5184.2.a.bc.1.1		1
72.59	even	6		5184.2.a.d.1.1		1
72.67	odd	6		5184.2.a.z.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.2.a.b.1.1	✓	1	9.4	even	3
324.2.a.d.1.1	yes	1	9.5	odd	6
324.2.e.a.109.1		2	3.2	odd	2
324.2.e.a.217.1		2	9.2	odd	6
324.2.e.d.109.1		2	1.1	even	1	trivial
324.2.e.d.217.1		2	9.7	even	3	inner
1296.2.a.a.1.1		1	36.31	odd	6
1296.2.a.j.1.1		1	36.23	even	6
1296.2.i.d.433.1		2	12.11	even	2
1296.2.i.d.865.1		2	36.11	even	6
1296.2.i.p.433.1		2	4.3	odd	2
1296.2.i.p.865.1		2	36.7	odd	6
5184.2.a.d.1.1		1	72.59	even	6
5184.2.a.g.1.1		1	72.5	odd	6
5184.2.a.z.1.1		1	72.67	odd	6
5184.2.a.bc.1.1		1	72.13	even	6
8100.2.a.a.1.1		1	45.14	odd	6
8100.2.a.f.1.1		1	45.4	even	6
8100.2.d.a.649.1		2	45.23	even	12
8100.2.d.a.649.2		2	45.32	even	12
8100.2.d.j.649.1		2	45.13	odd	12
8100.2.d.j.649.2		2	45.22	odd	12

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

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	324.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.50000 - 2.59808i) q^{5} +(-1.00000 - 1.73205i) q^{7} +(-3.00000 - 5.19615i) q^{11} +(-2.50000 + 4.33013i) q^{13} +3.00000 q^{17} +2.00000 q^{19} +(3.00000 - 5.19615i) q^{23} +(-2.00000 - 3.46410i) q^{25} +(1.50000 + 2.59808i) q^{29} +(2.00000 - 3.46410i) q^{31} -6.00000 q^{35} +5.00000 q^{37} +(-3.00000 + 5.19615i) q^{41} +(5.00000 + 8.66025i) q^{43} +(1.50000 - 2.59808i) q^{49} +6.00000 q^{53} -18.0000 q^{55} +(-6.00000 + 10.3923i) q^{59} +(-2.50000 - 4.33013i) q^{61} +(7.50000 + 12.9904i) q^{65} +(-1.00000 + 1.73205i) q^{67} -6.00000 q^{71} -1.00000 q^{73} +(-6.00000 + 10.3923i) q^{77} +(5.00000 + 8.66025i) q^{79} +(4.50000 - 7.79423i) q^{85} +3.00000 q^{89} +10.0000 q^{91} +(3.00000 - 5.19615i) q^{95} +(5.00000 + 8.66025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{5} - 2 q^{7} - 6 q^{11} - 5 q^{13} + 6 q^{17} + 4 q^{19} + 6 q^{23} - 4 q^{25} + 3 q^{29} + 4 q^{31} - 12 q^{35} + 10 q^{37} - 6 q^{41} + 10 q^{43} + 3 q^{49} + 12 q^{53} - 36 q^{55} - 12 q^{59}+ \cdots + 10 q^{97}+O(q^{100}) \) 2 * q + 3 * q^5 - 2 * q^7 - 6 * q^11 - 5 * q^13 + 6 * q^17 + 4 * q^19 + 6 * q^23 - 4 * q^25 + 3 * q^29 + 4 * q^31 - 12 * q^35 + 10 * q^37 - 6 * q^41 + 10 * q^43 + 3 * q^49 + 12 * q^53 - 36 * q^55 - 12 * q^59 - 5 * q^61 + 15 * q^65 - 2 * q^67 - 12 * q^71 - 2 * q^73 - 12 * q^77 + 10 * q^79 + 9 * q^85 + 6 * q^89 + 20 * q^91 + 6 * q^95 + 10 * q^97

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