Newform orbit 348.2.b.b

• Label: 348.2.b.b
• Level: $348$
• Weight: $2$
• Character orbit: 348.b
• Analytic conductor: $2.779$
• Analytic rank: $0$
• Dimension: $12$
• CM discriminant: -116
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	348.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.77879399034\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	\( x^{12} - 4x^{6} + 729 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_{3} q^{2} - \beta_{11} q^{3} - 2 q^{4} + \beta_{9} q^{5} + \beta_{2} q^{6} - 2 \beta_{3} q^{8} + \beta_{10} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 24 q^{4}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{6} + 729 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{7} + 23\nu ) / 15 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{8} + 23\nu^{2} ) / 45 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{9} + 23\nu^{3} ) / 135 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{6} - 2 ) / 5 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{7} + 22\nu ) / 15 \)
\(\beta_{6}\)	\(=\)	\( ( -\nu^{11} + 247\nu^{5} - 1215\nu ) / 1215 \)
\(\beta_{7}\)	\(=\)	\( ( \nu^{10} - 9\nu^{8} + 158\nu^{4} + 198\nu^{2} ) / 405 \)
\(\beta_{8}\)	\(=\)	\( ( -\nu^{9} + 22\nu^{3} ) / 45 \)
\(\beta_{9}\)	\(=\)	\( ( -2\nu^{10} + 9\nu^{8} + 89\nu^{4} - 198\nu^{2} ) / 405 \)
\(\beta_{10}\)	\(=\)	\( ( -\nu^{10} + 4\nu^{4} ) / 81 \)
\(\beta_{11}\)	\(=\)	\( ( \nu^{11} - 4\nu^{5} ) / 243 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/348\mathbb{Z}\right)^\times\).

\(n\)	\(175\)	\(205\)	\(233\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

347.1

−1.67844	−	0.427580i
−1.20952	−	1.23979i
−0.468927	+	1.66737i
0.468927	+	1.66737i
1.20952	−	1.23979i
1.67844	−	0.427580i
−1.67844	+	0.427580i
−1.20952	+	1.23979i
−0.468927	−	1.66737i
0.468927	−	1.66737i
1.20952	+	1.23979i
1.67844	+	0.427580i

−

1.41421i

−1.67844

+

0.427580i

−2.00000

1.87668i

0.604690

+

2.37368i

0

2.82843i

2.63435

−

1.43534i

2.65402

347.2

−

1.41421i

−1.20952

+

1.23979i

−2.00000

−

2.57714i

1.75332

+

1.71052i

0

2.82843i

−0.0741344

−

2.99908i

−3.64462

347.3

−

1.41421i

−0.468927

−

1.66737i

−2.00000

4.45381i

−2.35801

+

0.663162i

0

2.82843i

−2.56022

+

1.56374i

6.29864

347.4

−

1.41421i

0.468927

−

1.66737i

−2.00000

−

4.45381i

−2.35801

−

0.663162i

0

2.82843i

−2.56022

−

1.56374i

−6.29864

347.5

−

1.41421i

1.20952

+

1.23979i

−2.00000

2.57714i

1.75332

−

1.71052i

0

2.82843i

−0.0741344

+

2.99908i

3.64462

347.6

−

1.41421i

1.67844

+

0.427580i

−2.00000

−

1.87668i

0.604690

−

2.37368i

0

2.82843i

2.63435

+

1.43534i

−2.65402

347.7

1.41421i

−1.67844

−

0.427580i

−2.00000

−

1.87668i

0.604690

−

2.37368i

0

−

2.82843i

2.63435

+

1.43534i

2.65402

347.8

1.41421i

−1.20952

−

1.23979i

−2.00000

2.57714i

1.75332

−

1.71052i

0

−

2.82843i

−0.0741344

+

2.99908i

−3.64462

347.9

1.41421i

−0.468927

+

1.66737i

−2.00000

−

4.45381i

−2.35801

−

0.663162i

0

−

2.82843i

−2.56022

−

1.56374i

6.29864

347.10

1.41421i

0.468927

+

1.66737i

−2.00000

4.45381i

−2.35801

+

0.663162i

0

−

2.82843i

−2.56022

+

1.56374i

−6.29864

347.11

1.41421i

1.20952

−

1.23979i

−2.00000

−

2.57714i

1.75332

+

1.71052i

0

−

2.82843i

−0.0741344

−

2.99908i

3.64462

347.12

1.41421i

1.67844

−

0.427580i

−2.00000

1.87668i

0.604690

+

2.37368i

0

−

2.82843i

2.63435

−

1.43534i

−2.65402

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
116.d	odd	2	1	CM by \(\Q(\sqrt{-29}) \)
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner
29.b	even	2	1	inner
87.d	odd	2	1	inner
348.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	348.2.b.b	✓	12
3.b	odd	2	1	inner	348.2.b.b	✓	12
4.b	odd	2	1	inner	348.2.b.b	✓	12
12.b	even	2	1	inner	348.2.b.b	✓	12
29.b	even	2	1	inner	348.2.b.b	✓	12
87.d	odd	2	1	inner	348.2.b.b	✓	12
116.d	odd	2	1	CM	348.2.b.b	✓	12
348.b	even	2	1	inner	348.2.b.b	✓	12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
348.2.b.b	✓	12	1.a	even	1	1	trivial
348.2.b.b	✓	12	3.b	odd	2	1	inner
348.2.b.b	✓	12	4.b	odd	2	1	inner
348.2.b.b	✓	12	12.b	even	2	1	inner
348.2.b.b	✓	12	29.b	even	2	1	inner
348.2.b.b	✓	12	87.d	odd	2	1	inner
348.2.b.b	✓	12	116.d	odd	2	1	CM
348.2.b.b	✓	12	348.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 30T_{5}^{4} + 225T_{5}^{2} + 464 \) acting on \(S_{2}^{\mathrm{new}}(348, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} + 2)^{6} \)
$3$	\( T^{12} - 4T^{6} + 729 \)
$5$	(T^6 + 30*T^4 + 225*T^2 + 464)^2
$7$	\( T^{12} \)
$11$	(T^6 + 66*T^4 + 1089*T^2 + 4802)^2