Newform orbit 1815.4.a.be

• Label: 1815.4.a.be
• Level: $1815$
• Weight: $4$
• Character orbit: 1815.a
• Self dual: yes
• Analytic conductor: $107.088$
• Analytic rank: $1$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1815.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(107.088466660\)
Analytic rank:	\(1\)
Dimension:	\(8\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial:	\( x^{8} - 54x^{6} + 861x^{4} - 3756x^{2} + 4356 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{3}\cdot 3 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-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 + b1 * q^2 + 3 * q^3 + (b4 + 6) * q^4 - 5 * q^5 + 3*b1 * q^6 + (-3*b2 - 4*b1) * q^7 + (b6 + b5 + 6*b2 + 6*b1) * q^8 + 9 * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{3} + 44 q^{4} - 40 q^{5} + 72 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 54x^{6} + 861x^{4} - 3756x^{2} + 4356 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{7} + 54\nu^{5} - 795\nu^{3} + 1974\nu ) / 528 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{4} - 27\nu^{2} + 66 ) / 8 \)
\(\beta_{4}\)	\(=\)	\( \nu^{2} - 14 \)
\(\beta_{5}\)	\(=\)	\( ( -4\nu^{7} + 183\nu^{5} - 2025\nu^{3} + 174\nu ) / 264 \)
\(\beta_{6}\)	\(=\)	\( ( 7\nu^{7} - 345\nu^{5} + 4674\nu^{3} - 11904\nu ) / 264 \)
\(\beta_{7}\)	\(=\)	\( ( \nu^{6} - 48\nu^{4} + 617\nu^{2} - 1314 ) / 8 \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−5.21840
−4.55910
−2.02753
−1.36824
1.36824
2.02753
4.55910
5.21840

−5.21840

3.00000

19.2316

−5.00000

−15.6552

26.0697

−58.6112

9.00000

26.0920

1.2

−4.55910

3.00000

12.7854

−5.00000

−13.6773

13.0403

−21.8172

9.00000

22.7955

1.3

−2.02753

3.00000

−3.88913

−5.00000

−6.08258

2.91396

24.1055

9.00000

10.1376

1.4

−1.36824

3.00000

−6.12793

−5.00000

−4.10471

10.6691

19.3303

9.00000

6.84118

1.5

1.36824

3.00000

−6.12793

−5.00000

4.10471

−10.6691

−19.3303

9.00000

−6.84118

1.6

2.02753

3.00000

−3.88913

−5.00000

6.08258

−2.91396

−24.1055

9.00000

−10.1376

1.7

4.55910

3.00000

12.7854

−5.00000

13.6773

−13.0403

21.8172

9.00000

−22.7955

1.8

5.21840

3.00000

19.2316

−5.00000

15.6552

−26.0697

58.6112

9.00000

−26.0920

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(5\)	\(1\)
\(11\)	\(1\)

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1815.4.a.be	✓	8
11.b	odd	2	1	inner	1815.4.a.be	✓	8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1815.4.a.be	✓	8	1.a	even	1	1	trivial
1815.4.a.be	✓	8	11.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1815))\):

\( T_{2}^{8} - 54T_{2}^{6} + 861T_{2}^{4} - 3756T_{2}^{2} + 4356 \)

\( T_{7}^{8} - 972T_{7}^{6} + 220470T_{7}^{4} - 14957868T_{7}^{2} + 111703761 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^8 - 54*T^6 + 861*T^4 - 3756*T^2 + 4356
$3$	\( (T - 3)^{8} \)
$5$	\( (T + 5)^{8} \)
$7$	T^8 - 972*T^6 + 220470*T^4 - 14957868*T^2 + 111703761
$11$	\( T^{8} \)