Newform orbit 414.3.b.c

• Label: 414.3.b.c
• Level: $414$
• Weight: $3$
• Character orbit: 414.b
• Analytic conductor: $11.281$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	414.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.2806829445\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.1358954496.3
Defining polynomial:	\( x^{8} + 8x^{6} + 20x^{4} + 16x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index:	\( 2^{3} \)
Twist minimal:	no (minimal twist has level 138)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_{5} q^{2} + 2 q^{4} + (\beta_{6} - 2 \beta_{3} + \beta_{2}) q^{5} + (2 \beta_{7} - \beta_{6} + 3 \beta_{3} + 2 \beta_{2}) q^{7} + 2 \beta_{5} q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 8x^{6} + 20x^{4} + 16x^{2} + 1 \) :

\(\beta_{1}\)	\(=\)	\( \nu^{4} + 4\nu^{2} + 2 \)
\(\beta_{2}\)	\(=\)	\( \nu^{5} + 5\nu^{3} + 4\nu \)
\(\beta_{3}\)	\(=\)	\( \nu^{5} + 5\nu^{3} + 6\nu \)
\(\beta_{4}\)	\(=\)	\( -\nu^{6} - 6\nu^{4} - 7\nu^{2} + 2 \)
\(\beta_{5}\)	\(=\)	\( \nu^{6} + 6\nu^{4} + 9\nu^{2} + 2 \)
\(\beta_{6}\)	\(=\)	\( \nu^{7} + 7\nu^{5} + 13\nu^{3} + 5\nu \)
\(\beta_{7}\)	\(=\)	\( \nu^{7} + 7\nu^{5} + 15\nu^{3} + 11\nu \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/414\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(235\)
\(\chi(n)\)	\(1\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

91.1

		1.21752i
		1.98289i
	−	1.98289i
	−	1.21752i
		1.58671i
		0.261052i
	−	0.261052i
	−	1.58671i

−1.41421

0

2.00000

−

6.00464i

0

4.69017i

−2.82843

0

8.49185i

91.2

−1.41421

0

2.00000

−

4.92225i

0

5.13851i

−2.82843

0

6.96111i

91.3

−1.41421

0

2.00000

4.92225i

0

−

5.13851i

−2.82843

0

−

6.96111i

91.4

−1.41421

0

2.00000

6.00464i

0

−

4.69017i

−2.82843

0

−

8.49185i

91.5

1.41421

0

2.00000

−

1.69484i

0

−

4.18388i

2.82843

0

−

2.39686i

91.6

1.41421

0

2.00000

−

0.918288i

0

10.4925i

2.82843

0

−

1.29866i

91.7

1.41421

0

2.00000

0.918288i

0

−

10.4925i

2.82843

0

1.29866i

91.8

1.41421

0

2.00000

1.69484i

0

4.18388i

2.82843

0

2.39686i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	414.3.b.c		8
3.b	odd	2	1		138.3.b.a	✓	8
12.b	even	2	1		1104.3.c.c		8
23.b	odd	2	1	inner	414.3.b.c		8
69.c	even	2	1		138.3.b.a	✓	8
276.h	odd	2	1		1104.3.c.c		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
138.3.b.a	✓	8	3.b	odd	2	1
138.3.b.a	✓	8	69.c	even	2	1
414.3.b.c		8	1.a	even	1	1	trivial
414.3.b.c		8	23.b	odd	2	1	inner
1104.3.c.c		8	12.b	even	2	1
1104.3.c.c		8	276.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 64T_{5}^{6} + 1100T_{5}^{4} + 3392T_{5}^{2} + 2116 \) acting on \(S_{3}^{\mathrm{new}}(414, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} - 2)^{4} \)
$3$	\( T^{8} \)
$5$	T^8 + 64*T^6 + 1100*T^4 + 3392*T^2 + 2116
$7$	T^8 + 176*T^6 + 8684*T^4 + 167392*T^2 + 1119364
$11$	T^8 + 512*T^6 + 43712*T^4 + 1036288*T^2 + 2262016