Newform orbit 12.15.c.a

• Label: 12.15.c.a
• Level: $12$
• Weight: $15$
• Character orbit: 12.c
• Self dual: yes
• Analytic conductor: $14.919$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	12.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(14.9194761782\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)

\(=\)

q - 2187 * q^3 - 1389022 * q^7 + 4782969 * q^9 + 88071962 * q^13 + 1512529226 * q^19 + 3037791114 * q^21 + 6103515625 * q^25 - 10460353203 * q^27 - 23014293166 * q^31 - 111626070166 * q^37 - 192613380894 * q^39 + 528051507962 * q^43 + 1251159043635 * q^49 - 3307901417262 * q^57 + 6214599846074 * q^61 - 6643649166318 * q^63 - 11973142765462 * q^67 + 17278443497906 * q^73 - 13348388671875 * q^75 + 38383122173618 * q^79 + 22876792454961 * q^81 - 122333892801164 * q^91 + 50332259154042 * q^93 - 11651694897022 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(7\)
\(\chi(n)\)	\(-1\)	\(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

5.1

0

0

−2187.00

0

0

0

−1.38902e6

0

4.78297e6

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	12.15.c.a	✓	1
3.b	odd	2	1	CM	12.15.c.a	✓	1
4.b	odd	2	1		48.15.e.a		1
12.b	even	2	1		48.15.e.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
12.15.c.a	✓	1	1.a	even	1	1	trivial
12.15.c.a	✓	1	3.b	odd	2	1	CM
48.15.e.a		1	4.b	odd	2	1
48.15.e.a		1	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{15}^{\mathrm{new}}(12, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T + 2187 \)
$5$	\( T \)
$7$	\( T + 1389022 \)
$11$	\( T \)