LMFDB - Embedded newform 3024.2.bf.c.1711.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3024,2,Mod(1711,3024)] 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("3024.1711"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3024, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 2, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.bf (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,4] 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:	$24.1467615712$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1008)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1711.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3024.1711
Dual form			3024.2.bf.c.2287.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 + 1.73205i) q^{7} -3.46410i q^{11} +(1.50000 + 0.866025i) q^{13} +(-1.50000 - 0.866025i) q^{17} +(2.50000 + 4.33013i) q^{19} -3.46410i q^{23} +5.00000 q^{25} +(1.50000 + 2.59808i) q^{29} +(0.500000 + 0.866025i) q^{31} +(-3.50000 - 6.06218i) q^{37} +(-1.50000 - 0.866025i) q^{41} +(1.50000 - 0.866025i) q^{43} +(4.50000 - 7.79423i) q^{47} +(1.00000 + 6.92820i) q^{49} +(-4.50000 + 7.79423i) q^{53} +(7.50000 + 12.9904i) q^{59} +(1.50000 + 0.866025i) q^{61} +(13.5000 - 7.79423i) q^{67} -10.3923i q^{71} +(1.50000 + 0.866025i) q^{73} +(6.00000 - 6.92820i) q^{77} +(-1.50000 - 0.866025i) q^{79} +(-4.50000 - 7.79423i) q^{83} +(-1.50000 + 0.866025i) q^{89} +(1.50000 + 4.33013i) q^{91} +(1.50000 - 0.866025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{7} + 3 q^{13} - 3 q^{17} + 5 q^{19} + 10 q^{25} + 3 q^{29} + q^{31} - 7 q^{37} - 3 q^{41} + 3 q^{43} + 9 q^{47} + 2 q^{49} - 9 q^{53} + 15 q^{59} + 3 q^{61} + 27 q^{67} + 3 q^{73} + 12 q^{77}+ \cdots + 3 q^{97}+O(q^{100}) $ 2 * q + 4 * q^7 + 3 * q^13 - 3 * q^17 + 5 * q^19 + 10 * q^25 + 3 * q^29 + q^31 - 7 * q^37 - 3 * q^41 + 3 * q^43 + 9 * q^47 + 2 * q^49 - 9 * q^53 + 15 * q^59 + 3 * q^61 + 27 * q^67 + 3 * q^73 + 12 * q^77 - 3 * q^79 - 9 * q^83 - 3 * q^89 + 3 * q^91 + 3 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{1}{6}\right)$

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
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.46410i		−	1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$		−	3.46410i		−	1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$		0			0
$13$	1.50000	+	0.866025i	0.416025	+	0.240192i	0.693375	−	0.720577i	$-0.256123\pi$
$13$	1.50000	+	0.866025i	0.416025	+	0.240192i	−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.50000	−	0.866025i	−0.363803	−	0.210042i	0.306944	−	0.951727i	$-0.400693\pi$
$17$	−1.50000	−	0.866025i	−0.363803	−	0.210042i	−0.670748	+	0.741685i	$0.734027\pi$
$18$		0			0
$19$	2.50000	+	4.33013i	0.573539	+	0.993399i	0.996199	+	0.0871106i	$0.0277634\pi$
$19$	2.50000	+	4.33013i	0.573539	+	0.993399i	−0.422659	+	0.906289i	$0.638903\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	3.46410i		−	0.722315i	−0.932505	−	0.361158i	$-0.882382\pi$
$23$		−	3.46410i		−	0.722315i	0.932505	−	0.361158i	$-0.117618\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		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
$31$	0.500000	+	0.866025i	0.0898027	+	0.155543i	0.907428	−	0.420208i	$-0.138043\pi$
$31$	0.500000	+	0.866025i	0.0898027	+	0.155543i	−0.817625	+	0.575751i	$0.804710\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.50000	−	6.06218i	−0.575396	−	0.996616i	−0.995998	−	0.0893706i	$-0.971514\pi$
$37$	−3.50000	−	6.06218i	−0.575396	−	0.996616i	0.420602	−	0.907245i	$-0.361819\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.50000	−	0.866025i	−0.234261	−	0.135250i	0.378275	−	0.925693i	$-0.376517\pi$
$41$	−1.50000	−	0.866025i	−0.234261	−	0.135250i	−0.612536	+	0.790443i	$0.709851\pi$
$42$		0			0
$43$	1.50000	−	0.866025i	0.228748	−	0.132068i	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	1.50000	−	0.866025i	0.228748	−	0.132068i	0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.50000	−	7.79423i	0.656392	−	1.13691i	−0.325150	−	0.945662i	$-0.605415\pi$
$47$	4.50000	−	7.79423i	0.656392	−	1.13691i	0.981543	−	0.191243i	$-0.0612518\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.50000	+	7.79423i	−0.618123	+	1.07062i	0.371706	+	0.928351i	$0.378773\pi$
$53$	−4.50000	+	7.79423i	−0.618123	+	1.07062i	−0.989828	+	0.142269i	$0.954560\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.50000	+	12.9904i	0.976417	+	1.69120i	0.675178	+	0.737655i	$0.264067\pi$
$59$	7.50000	+	12.9904i	0.976417	+	1.69120i	0.301239	+	0.953549i	$0.402600\pi$
$60$		0			0
$61$	1.50000	+	0.866025i	0.192055	+	0.110883i	0.592944	−	0.805243i	$-0.297965\pi$
$61$	1.50000	+	0.866025i	0.192055	+	0.110883i	−0.400889	+	0.916127i	$0.631299\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	13.5000	−	7.79423i	1.64929	−	0.952217i	0.671932	−	0.740613i	$-0.265465\pi$
$67$	13.5000	−	7.79423i	1.64929	−	0.952217i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.3923i		−	1.23334i	−0.787222	−	0.616670i	$-0.788481\pi$
$71$		−	10.3923i		−	1.23334i	0.787222	−	0.616670i	$-0.211519\pi$
$72$		0			0
$73$	1.50000	+	0.866025i	0.175562	+	0.101361i	0.585206	−	0.810885i	$-0.301014\pi$
$73$	1.50000	+	0.866025i	0.175562	+	0.101361i	−0.409644	+	0.912245i	$0.634347\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.00000	−	6.92820i	0.683763	−	0.789542i
$78$		0			0
$79$	−1.50000	−	0.866025i	−0.168763	−	0.0974355i	0.413239	−	0.910622i	$-0.364397\pi$
$79$	−1.50000	−	0.866025i	−0.168763	−	0.0974355i	−0.582003	+	0.813187i	$0.697731\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.50000	−	7.79423i	−0.493939	−	0.855528i	0.506036	−	0.862512i	$-0.331110\pi$
$83$	−4.50000	−	7.79423i	−0.493939	−	0.855528i	−0.999976	+	0.00698436i	$0.997777\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.50000	+	0.866025i	−0.159000	+	0.0917985i	−0.577389	−	0.816469i	$-0.695928\pi$
$89$	−1.50000	+	0.866025i	−0.159000	+	0.0917985i	0.418389	+	0.908268i	$0.362595\pi$
$90$		0			0
$91$	1.50000	+	4.33013i	0.157243	+	0.453921i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.50000	−	0.866025i	0.152302	−	0.0879316i	−0.421912	−	0.906637i	$-0.638641\pi$
$97$	1.50000	−	0.866025i	0.152302	−	0.0879316i	0.574214	+	0.818705i	$0.305308\pi$
$98$		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	3024.2.bf.c.1711.1		2
3.2	odd	2		1008.2.bf.c.31.1	yes	2
4.3	odd	2		3024.2.bf.b.1711.1		2
7.5	odd	6		3024.2.cz.a.1279.1		2
9.2	odd	6		1008.2.cz.d.367.1	yes	2
9.7	even	3		3024.2.cz.b.2719.1		2
12.11	even	2		1008.2.bf.b.31.1	✓	2
21.5	even	6		1008.2.cz.a.607.1	yes	2
28.19	even	6		3024.2.cz.b.1279.1		2
36.7	odd	6		3024.2.cz.a.2719.1		2
36.11	even	6		1008.2.cz.a.367.1	yes	2
63.47	even	6		1008.2.bf.b.943.1	yes	2
63.61	odd	6		3024.2.bf.b.2287.1		2
84.47	odd	6		1008.2.cz.d.607.1	yes	2
252.47	odd	6		1008.2.bf.c.943.1	yes	2
252.187	even	6	inner	3024.2.bf.c.2287.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.bf.b.31.1	✓	2	12.11	even	2
1008.2.bf.b.943.1	yes	2	63.47	even	6
1008.2.bf.c.31.1	yes	2	3.2	odd	2
1008.2.bf.c.943.1	yes	2	252.47	odd	6
1008.2.cz.a.367.1	yes	2	36.11	even	6
1008.2.cz.a.607.1	yes	2	21.5	even	6
1008.2.cz.d.367.1	yes	2	9.2	odd	6
1008.2.cz.d.607.1	yes	2	84.47	odd	6
3024.2.bf.b.1711.1		2	4.3	odd	2
3024.2.bf.b.2287.1		2	63.61	odd	6
3024.2.bf.c.1711.1		2	1.1	even	1	trivial
3024.2.bf.c.2287.1		2	252.187	even	6	inner
3024.2.cz.a.1279.1		2	7.5	odd	6
3024.2.cz.a.2719.1		2	36.7	odd	6
3024.2.cz.b.1279.1		2	28.19	even	6
3024.2.cz.b.2719.1		2	9.7	even	3

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Level:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.bf (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 + 1.73205i) q^{7} -3.46410i q^{11} +(1.50000 + 0.866025i) q^{13} +(-1.50000 - 0.866025i) q^{17} +(2.50000 + 4.33013i) q^{19} -3.46410i q^{23} +5.00000 q^{25} +(1.50000 + 2.59808i) q^{29} +(0.500000 + 0.866025i) q^{31} +(-3.50000 - 6.06218i) q^{37} +(-1.50000 - 0.866025i) q^{41} +(1.50000 - 0.866025i) q^{43} +(4.50000 - 7.79423i) q^{47} +(1.00000 + 6.92820i) q^{49} +(-4.50000 + 7.79423i) q^{53} +(7.50000 + 12.9904i) q^{59} +(1.50000 + 0.866025i) q^{61} +(13.5000 - 7.79423i) q^{67} -10.3923i q^{71} +(1.50000 + 0.866025i) q^{73} +(6.00000 - 6.92820i) q^{77} +(-1.50000 - 0.866025i) q^{79} +(-4.50000 - 7.79423i) q^{83} +(-1.50000 + 0.866025i) q^{89} +(1.50000 + 4.33013i) q^{91} +(1.50000 - 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{7} + 3 q^{13} - 3 q^{17} + 5 q^{19} + 10 q^{25} + 3 q^{29} + q^{31} - 7 q^{37} - 3 q^{41} + 3 q^{43} + 9 q^{47} + 2 q^{49} - 9 q^{53} + 15 q^{59} + 3 q^{61} + 27 q^{67} + 3 q^{73} + 12 q^{77}+ \cdots + 3 q^{97}+O(q^{100}) \) 2 * q + 4 * q^7 + 3 * q^13 - 3 * q^17 + 5 * q^19 + 10 * q^25 + 3 * q^29 + q^31 - 7 * q^37 - 3 * q^41 + 3 * q^43 + 9 * q^47 + 2 * q^49 - 9 * q^53 + 15 * q^59 + 3 * q^61 + 27 * q^67 + 3 * q^73 + 12 * q^77 - 3 * q^79 - 9 * q^83 - 3 * q^89 + 3 * q^91 + 3 * q^97

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