Newform orbit 1040.2.da.e

• Label: 1040.2.da.e
• Level: $1040$
• Weight: $2$
• Character orbit: 1040.da
• Analytic conductor: $8.304$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1040.da (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.30444181021\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	12.0.58891012706304.1
Defining polynomial:	\( x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 520)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\)	\(=\)	\( q - \beta_{11} q^{3} - \beta_{5} q^{5} + ( - \beta_{10} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3}) q^{7} + ( - \beta_{11} + \beta_{10} + \beta_{8} + \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3}) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{7} + 2 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64 \) :

\(\beta_{1}\)	\(=\)	\( ( - 3 \nu^{11} + 4 \nu^{10} + 7 \nu^{9} - 6 \nu^{8} - 13 \nu^{7} + 30 \nu^{6} - 6 \nu^{5} - 28 \nu^{4} - 4 \nu^{3} + 16 \nu^{2} - 16 \nu + 96 ) / 32 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{11} + 7 \nu^{9} - 2 \nu^{8} - 29 \nu^{7} + 10 \nu^{6} + 70 \nu^{5} - 68 \nu^{4} - 92 \nu^{3} + 160 \nu^{2} + 128 \nu - 192 ) / 32 \)
\(\beta_{3}\)	\(=\)	\( ( - 3 \nu^{11} - 2 \nu^{10} + 11 \nu^{9} + 8 \nu^{8} - 21 \nu^{7} - 4 \nu^{6} + 42 \nu^{5} - 84 \nu^{3} - 24 \nu^{2} + 64 \nu + 64 ) / 32 \)
\(\beta_{4}\)	\(=\)	\( ( 3 \nu^{11} + 4 \nu^{10} - 7 \nu^{9} - 26 \nu^{8} + 13 \nu^{7} + 50 \nu^{6} - 42 \nu^{5} - 92 \nu^{4} + 132 \nu^{3} + 160 \nu^{2} - 144 \nu - 192 ) / 32 \)
\(\beta_{5}\)	\(=\)	\( ( - 5 \nu^{11} + 2 \nu^{10} + 17 \nu^{9} - 8 \nu^{8} - 39 \nu^{7} + 44 \nu^{6} + 50 \nu^{5} - 88 \nu^{4} - 68 \nu^{3} + 120 \nu^{2} + 16 \nu - 32 ) / 32 \)
\(\beta_{6}\)	\(=\)	\( ( 3 \nu^{11} - 4 \nu^{10} - 15 \nu^{9} + 14 \nu^{8} + 37 \nu^{7} - 54 \nu^{6} - 50 \nu^{5} + 132 \nu^{4} + 52 \nu^{3} - 192 \nu^{2} - 48 \nu + 128 ) / 32 \)
\(\beta_{7}\)	\(=\)	\( ( 2 \nu^{11} - \nu^{10} - 10 \nu^{9} + \nu^{8} + 24 \nu^{7} - 11 \nu^{6} - 38 \nu^{5} + 38 \nu^{4} + 60 \nu^{3} - 60 \nu^{2} - 64 \nu + 16 ) / 16 \)
\(\beta_{8}\)	\(=\)	\( ( - \nu^{11} - 4 \nu^{10} + 17 \nu^{9} + 6 \nu^{8} - 51 \nu^{7} - 6 \nu^{6} + 122 \nu^{5} - 68 \nu^{4} - 164 \nu^{3} + 144 \nu^{2} + 224 \nu - 192 ) / 32 \)
\(\beta_{9}\)	\(=\)	\( ( \nu^{11} - 3 \nu^{10} - 5 \nu^{9} + 13 \nu^{8} + 13 \nu^{7} - 35 \nu^{6} - 12 \nu^{5} + 70 \nu^{4} - 8 \nu^{3} - 108 \nu^{2} + 16 \nu + 80 ) / 16 \)
\(\beta_{10}\)	\(=\)	\( ( - \nu^{11} - 4 \nu^{10} - 11 \nu^{9} + 30 \nu^{8} + 41 \nu^{7} - 70 \nu^{6} - 74 \nu^{5} + 204 \nu^{4} + 36 \nu^{3} - 368 \nu^{2} - 80 \nu + 416 ) / 32 \)
\(\beta_{11}\)	\(=\)	\( ( 7 \nu^{11} - 35 \nu^{9} + 2 \nu^{8} + 89 \nu^{7} - 34 \nu^{6} - 162 \nu^{5} + 140 \nu^{4} + 244 \nu^{3} - 208 \nu^{2} - 240 \nu + 128 ) / 32 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\)	\(261\)	\(417\)	\(561\)	\(911\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\beta_{9}\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

641.1

−1.12906	−	0.851598i
0.759479	−	1.19298i
−1.30089	+	0.554694i
−1.08105	+	0.911778i
1.40744	+	0.138282i
1.34408	+	0.439820i
−1.12906	+	0.851598i
0.759479	+	1.19298i
−1.30089	−	0.554694i
−1.08105	−	0.911778i
1.40744	−	0.138282i
1.34408	−	0.439820i

0

−1.30204

+

2.25519i

0

−

1.00000i

0

−4.29663

+

2.48066i

0

−1.89060

−

3.27461i

0

641.2

0

−0.653409

+

1.13174i

0

1.00000i

0

−2.51573

+

1.45245i

0

0.646115

+

1.11910i

0

641.3

0

−0.170066

+

0.294562i

0

1.00000i

0

4.07999

−

2.35558i

0

1.44216

+

2.49789i

0

641.4

0

0.249100

−

0.431454i

0

−

1.00000i

0

−1.15921

+

0.669268i

0

1.37590

+

2.38313i

0

641.5

0

0.823474

−

1.42630i

0

1.00000i

0

−1.33221

+

0.769155i

0

0.143781

+

0.249036i

0

641.6

0

1.05294

−

1.82374i

0

−

1.00000i

0

2.22378

−

1.28390i

0

−0.717351

−

1.24249i

0

881.1

0

−1.30204

−

2.25519i

0

1.00000i

0

−4.29663

−

2.48066i

0

−1.89060

+

3.27461i

0

881.2

0

−0.653409

−

1.13174i

0

−

1.00000i

0

−2.51573

−

1.45245i

0

0.646115

−

1.11910i

0

881.3

0

−0.170066

−

0.294562i

0

−

1.00000i

0

4.07999

+

2.35558i

0

1.44216

−

2.49789i

0

881.4

0

0.249100

+

0.431454i

0

1.00000i

0

−1.15921

−

0.669268i

0

1.37590

−

2.38313i

0

881.5

0

0.823474

+

1.42630i

0

−

1.00000i

0

−1.33221

−

0.769155i

0

0.143781

−

0.249036i

0

881.6

0

1.05294

+

1.82374i

0

1.00000i

0

2.22378

+

1.28390i

0

−0.717351

+

1.24249i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.e	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1040.2.da.e		12
4.b	odd	2	1		520.2.bu.a	✓	12
13.e	even	6	1	inner	1040.2.da.e		12
52.i	odd	6	1		520.2.bu.a	✓	12
52.l	even	12	1		6760.2.a.bg		6
52.l	even	12	1		6760.2.a.bj		6

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
520.2.bu.a	✓	12	4.b	odd	2	1
520.2.bu.a	✓	12	52.i	odd	6	1
1040.2.da.e		12	1.a	even	1	1	trivial
1040.2.da.e		12	13.e	even	6	1	inner
6760.2.a.bg		6	52.l	even	12	1
6760.2.a.bj		6	52.l	even	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 8 T_{3}^{10} - 4 T_{3}^{9} + 51 T_{3}^{8} - 18 T_{3}^{7} + 104 T_{3}^{6} - 6 T_{3}^{5} + 157 T_{3}^{4} - 18 T_{3}^{3} + 30 T_{3}^{2} + 4 T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{12} \)
$3$	T^12 + 8*T^10 - 4*T^9 + 51*T^8 - 18*T^7 + 104*T^6 - 6*T^5 + 157*T^4 - 18*T^3 + 30*T^2 + 4*T + 4
$5$	\( (T^{2} + 1)^{6} \)
$7$	T^12 + 6*T^11 - 15*T^10 - 162*T^9 + 310*T^8 + 3702*T^7 + 4781*T^6 - 16098*T^5 - 30122*T^4 + 69222*T^3 + 261649*T^2 + 299406*T + 128881
$11$	T^12 - 19*T^10 + 290*T^8 + 336*T^7 - 1179*T^6 - 1704*T^5 + 4290*T^4 + 8520*T^3 + 5581*T^2 + 1320*T + 121