Newform orbit 845.2.d.a

• Label: 845.2.d.a
• Level: $845$
• Weight: $2$
• Character orbit: 845.d
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	6.0.350464.1
Defining polynomial:	\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - \beta_1 - 1) q^{2} - \beta_{2} q^{3} + ( - \beta_{3} + \beta_1 + 2) q^{4} + ( - \beta_{4} - \beta_{3}) q^{5} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{6} + ( - \beta_{3} - \beta_1 + 1) q^{7} + (3 \beta_{3} - 4) q^{8} + ( - 3 \beta_{3} - \beta_1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 10 q^{4} - 2 q^{5} + 4 q^{7} - 18 q^{8} - 6 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) :

\(\beta_{1}\)	\(=\)	\( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \)
\(\beta_{2}\)	\(=\)	\( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} + 19\nu^{2} - 38\nu + 14 ) / 23 \)
\(\beta_{3}\)	\(=\)	\( ( 5\nu^{5} - 17\nu^{4} + 20\nu^{3} + 5\nu^{2} - 10\nu - 29 ) / 23 \)
\(\beta_{4}\)	\(=\)	\( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \)
\(\beta_{5}\)	\(=\)	\( ( 18\nu^{5} - 29\nu^{4} + 26\nu^{3} + 41\nu^{2} + 102\nu - 40 ) / 23 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

844.1

0.403032	−	0.403032i
0.403032	+	0.403032i
−0.854638	−	0.854638i
−0.854638	+	0.854638i
1.45161	+	1.45161i
1.45161	−	1.45161i

−2.67513

−

0.481194i

5.15633

1.48119

−

1.67513i

1.28726i

0.806063

−8.44358

2.76845

−3.96239

+

4.48119i

844.2

−2.67513

0.481194i

5.15633

1.48119

+

1.67513i

−

1.28726i

0.806063

−8.44358

2.76845

−3.96239

−

4.48119i

844.3

−1.53919

−

3.17009i

0.369102

−2.17009

+

0.539189i

4.87936i

−1.70928

2.51026

−7.04945

3.34017

−

0.829914i

844.4

−1.53919

3.17009i

0.369102

−2.17009

−

0.539189i

−

4.87936i

−1.70928

2.51026

−7.04945

3.34017

+

0.829914i

844.5

1.21432

−

1.31111i

−0.525428

−0.311108

−

2.21432i

−

1.59210i

2.90321

−3.06668

1.28100

−0.377784

−

2.68889i

844.6

1.21432

1.31111i

−0.525428

−0.311108

+

2.21432i

1.59210i

2.90321

−3.06668

1.28100

−0.377784

+

2.68889i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
65.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	845.2.d.a		6
5.b	even	2	1		845.2.d.b		6
13.b	even	2	1		845.2.d.b		6
13.c	even	3	2		845.2.l.e		12
13.d	odd	4	1		65.2.b.a	✓	6
13.d	odd	4	1		845.2.b.c		6
13.e	even	6	2		845.2.l.d		12
13.f	odd	12	2		845.2.n.f		12
13.f	odd	12	2		845.2.n.g		12
39.f	even	4	1		585.2.c.b		6
52.f	even	4	1		1040.2.d.c		6
65.d	even	2	1	inner	845.2.d.a		6
65.f	even	4	1		325.2.a.k		3
65.f	even	4	1		4225.2.a.bh		3
65.g	odd	4	1		65.2.b.a	✓	6
65.g	odd	4	1		845.2.b.c		6
65.k	even	4	1		325.2.a.j		3
65.k	even	4	1		4225.2.a.ba		3
65.l	even	6	2		845.2.l.e		12
65.n	even	6	2		845.2.l.d		12
65.s	odd	12	2		845.2.n.f		12
65.s	odd	12	2		845.2.n.g		12
195.j	odd	4	1		2925.2.a.bj		3
195.n	even	4	1		585.2.c.b		6
195.u	odd	4	1		2925.2.a.bf		3
260.l	odd	4	1		5200.2.a.cb		3
260.s	odd	4	1		5200.2.a.cj		3
260.u	even	4	1		1040.2.d.c		6

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.2.b.a	✓	6	13.d	odd	4	1
65.2.b.a	✓	6	65.g	odd	4	1
325.2.a.j		3	65.k	even	4	1
325.2.a.k		3	65.f	even	4	1
585.2.c.b		6	39.f	even	4	1
585.2.c.b		6	195.n	even	4	1
845.2.b.c		6	13.d	odd	4	1
845.2.b.c		6	65.g	odd	4	1
845.2.d.a		6	1.a	even	1	1	trivial
845.2.d.a		6	65.d	even	2	1	inner
845.2.d.b		6	5.b	even	2	1
845.2.d.b		6	13.b	even	2	1
845.2.l.d		12	13.e	even	6	2
845.2.l.d		12	65.n	even	6	2
845.2.l.e		12	13.c	even	3	2
845.2.l.e		12	65.l	even	6	2
845.2.n.f		12	13.f	odd	12	2
845.2.n.f		12	65.s	odd	12	2
845.2.n.g		12	13.f	odd	12	2
845.2.n.g		12	65.s	odd	12	2
1040.2.d.c		6	52.f	even	4	1
1040.2.d.c		6	260.u	even	4	1
2925.2.a.bf		3	195.u	odd	4	1
2925.2.a.bj		3	195.j	odd	4	1
4225.2.a.ba		3	65.k	even	4	1
4225.2.a.bh		3	65.f	even	4	1
5200.2.a.cb		3	260.l	odd	4	1
5200.2.a.cj		3	260.s	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 3T_{2}^{2} - T_{2} - 5 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{3} + 3 T^{2} - T - 5)^{2} \)
$3$	T^6 + 12*T^4 + 20*T^2 + 4
$5$	T^6 + 2*T^5 + 3*T^4 + 12*T^3 + 15*T^2 + 50*T + 125
$7$	\( (T^{3} - 2 T^{2} - 4 T + 4)^{2} \)
$11$	T^6 + 20*T^4 + 88*T^2 + 4