Newform orbit 2205.2.d.m

• Label: 2205.2.d.m
• Level: $2205$
• Weight: $2$
• Character orbit: 2205.d
• Analytic conductor: $17.607$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2205.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.6070136457\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.309760000.3
Defining polynomial:	\( x^{8} - 4x^{7} + 8x^{6} - 2x^{5} - x^{4} - 2x^{3} + 18x^{2} + 6x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{6} \)
Twist minimal:	no (minimal twist has level 735)
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.

\(f(q)\)	\(=\)	\( q + \beta_{5} q^{2} + ( - \beta_1 - 2) q^{4} + \beta_{2} q^{5} + ( - \beta_{7} - \beta_{5}) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 8x^{6} - 2x^{5} - x^{4} - 2x^{3} + 18x^{2} + 6x + 1 \) :

\(\beta_{1}\)	\(=\)	\( ( 16\nu^{7} - 88\nu^{6} + 260\nu^{5} - 422\nu^{4} + 372\nu^{3} - 100\nu^{2} - 52\nu - 561 ) / 245 \)
\(\beta_{2}\)	\(=\)	\( ( -10\nu^{7} + 55\nu^{6} - 138\nu^{5} + 129\nu^{4} - 12\nu^{3} - 11\nu^{2} - 188\nu + 75 ) / 49 \)
\(\beta_{3}\)	\(=\)	\( ( -72\nu^{7} + 249\nu^{6} - 386\nu^{5} - 257\nu^{4} + 384\nu^{3} + 9\nu^{2} - 942\nu - 685 ) / 245 \)
\(\beta_{4}\)	\(=\)	\( ( -87\nu^{7} + 356\nu^{6} - 740\nu^{5} + 304\nu^{4} - 124\nu^{3} + 360\nu^{2} - 1371\nu - 548 ) / 245 \)
\(\beta_{5}\)	\(=\)	\( ( -87\nu^{7} + 356\nu^{6} - 740\nu^{5} + 304\nu^{4} - 124\nu^{3} + 360\nu^{2} - 1861\nu - 303 ) / 245 \)
\(\beta_{6}\)	\(=\)	\( ( 148\nu^{7} - 618\nu^{6} + 1278\nu^{5} - 498\nu^{4} - 38\nu^{3} - 582\nu^{2} + 2802\nu + 458 ) / 245 \)
\(\beta_{7}\)	\(=\)	\( ( 356\nu^{7} - 1468\nu^{6} + 3090\nu^{5} - 1182\nu^{4} - 298\nu^{3} - 20\nu^{2} + 6438\nu + 1054 ) / 245 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times\).

\(n\)	\(442\)	\(1081\)	\(1226\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1324.1

−0.748606	+	0.748606i
1.74861	+	1.74861i
1.16407	+	1.16407i
−0.164066	+	0.164066i
1.16407	−	1.16407i
−0.164066	−	0.164066i
−0.748606	−	0.748606i
1.74861	−	1.74861i

−

2.49721i

0

−4.23607

−1.54336

+

1.61803i

0

0

5.58394i

0

4.04057

+

3.85410i

1324.2

−

2.49721i

0

−4.23607

1.54336

−

1.61803i

0

0

5.58394i

0

−4.04057

−

3.85410i

1324.3

−

1.32813i

0

0.236068

−2.14896

+

0.618034i

0

0

−

2.96979i

0

0.820830

+

2.85410i

1324.4

−

1.32813i

0

0.236068

2.14896

−

0.618034i

0

0

−

2.96979i

0

−0.820830

−

2.85410i

1324.5

1.32813i

0

0.236068

−2.14896

−

0.618034i

0

0

2.96979i

0

0.820830

−

2.85410i

1324.6

1.32813i

0

0.236068

2.14896

+

0.618034i

0

0

2.96979i

0

−0.820830

+

2.85410i

1324.7

2.49721i

0

−4.23607

−1.54336

−

1.61803i

0

0

−

5.58394i

0

4.04057

−

3.85410i

1324.8

2.49721i

0

−4.23607

1.54336

+

1.61803i

0

0

−

5.58394i

0

−4.04057

+

3.85410i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
7.b	odd	2	1	inner
35.c	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2205.2.d.m		8
3.b	odd	2	1		735.2.d.c	✓	8
5.b	even	2	1	inner	2205.2.d.m		8
7.b	odd	2	1	inner	2205.2.d.m		8
15.d	odd	2	1		735.2.d.c	✓	8
15.e	even	4	1		3675.2.a.bt		4
15.e	even	4	1		3675.2.a.bv		4
21.c	even	2	1		735.2.d.c	✓	8
21.g	even	6	2		735.2.q.h		16
21.h	odd	6	2		735.2.q.h		16
35.c	odd	2	1	inner	2205.2.d.m		8
105.g	even	2	1		735.2.d.c	✓	8
105.k	odd	4	1		3675.2.a.bt		4
105.k	odd	4	1		3675.2.a.bv		4
105.o	odd	6	2		735.2.q.h		16
105.p	even	6	2		735.2.q.h		16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
735.2.d.c	✓	8	3.b	odd	2	1
735.2.d.c	✓	8	15.d	odd	2	1
735.2.d.c	✓	8	21.c	even	2	1
735.2.d.c	✓	8	105.g	even	2	1
735.2.q.h		16	21.g	even	6	2
735.2.q.h		16	21.h	odd	6	2
735.2.q.h		16	105.o	odd	6	2
735.2.q.h		16	105.p	even	6	2
2205.2.d.m		8	1.a	even	1	1	trivial
2205.2.d.m		8	5.b	even	2	1	inner
2205.2.d.m		8	7.b	odd	2	1	inner
2205.2.d.m		8	35.c	odd	2	1	inner
3675.2.a.bt		4	15.e	even	4	1
3675.2.a.bt		4	105.k	odd	4	1
3675.2.a.bv		4	15.e	even	4	1
3675.2.a.bv		4	105.k	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2205, [\chi])\):

\( T_{2}^{4} + 8T_{2}^{2} + 11 \)

\( T_{11}^{2} - 20 \)

\( T_{13}^{4} + 28T_{13}^{2} + 16 \)

\( T_{19}^{4} - 68T_{19}^{2} + 176 \)

\( T_{29} - 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{4} + 8 T^{2} + 11)^{2} \)
$3$	\( T^{8} \)
$5$	T^8 - 8*T^6 + 46*T^4 - 200*T^2 + 625
$7$	\( T^{8} \)
$11$	\( (T^{2} - 20)^{4} \)