Newform orbit 99.2.e.e

• Label: 99.2.e.e
• Level: 99
• Weight: 2
• Character orbit: 99.e
• Analytic conductor: 0.791
• Analytic rank: 0
• Dimension: 8
• CM: no
• Inner twists: 2

Newspace parameters

Level:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	99.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.790518980011\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	8.0.508277025.1
Defining polynomial:	\(x^{8} - 3 x^{7} + 5 x^{6} - 15 x^{5} + 21 x^{4} + 3 x^{3} - 22 x^{2} + 3 x + 19\)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + ( -\beta_{2} - \beta_{7} ) q^{2} + ( 1 + \beta_{7} ) q^{3} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{4} + ( -1 - \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{5} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{6} + ( \beta_{1} - \beta_{4} - \beta_{6} ) q^{7} + ( 3 - 3 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{8} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8q - q^{2} + 5q^{3} - 11q^{4} - 4q^{5} + 17q^{6} - q^{7} - 5q^{9} + O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} + 5 x^{6} - 15 x^{5} + 21 x^{4} + 3 x^{3} - 22 x^{2} + 3 x + 19\):

\(\beta_{0}\)	\(=\)	\( 1 \)
\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\((\)\( -4 \nu^{7} + 79 \nu^{6} - 177 \nu^{5} + 459 \nu^{4} - 1008 \nu^{3} + 1011 \nu^{2} - 752 \nu - 478 \)\()/933\)
\(\beta_{3}\)	\(=\)	\((\)\( 35 \nu^{7} + 164 \nu^{6} - 395 \nu^{5} + 260 \nu^{4} - 2687 \nu^{3} + 2894 \nu^{2} + 1604 \nu - 2193 \)\()/1866\)
\(\beta_{4}\)	\(=\)	\((\)\( 217 \nu^{7} + 146 \nu^{6} + 39 \nu^{5} - 876 \nu^{4} - 4095 \nu^{3} + 6498 \nu^{2} + 1610 \nu - 2525 \)\()/5598\)
\(\beta_{5}\)	\(=\)	\((\)\( 241 \nu^{7} - 328 \nu^{6} + 1101 \nu^{5} - 3630 \nu^{4} + 1953 \nu^{3} - 5166 \nu^{2} + 6122 \nu + 343 \)\()/5598\)
\(\beta_{6}\)	\(=\)	\((\)\( -145 \nu^{7} + 298 \nu^{6} - 585 \nu^{5} + 1944 \nu^{4} - 2019 \nu^{3} - 438 \nu^{2} + 730 \nu + 1799 \)\()/1866\)
\(\beta_{7}\)	\(=\)	\((\)\( 701 \nu^{7} - 1016 \nu^{6} + 1863 \nu^{5} - 6966 \nu^{4} + 2181 \nu^{3} + 10122 \nu^{2} - 6296 \nu - 6265 \)\()/5598\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\)	\(46\)	\(56\)
\(\chi(n)\)	\(1\)	\(-1 + \beta_{6}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

34.1

−0.734668	+	0.348716i
−0.577806	−	2.22188i
0.947217	+	0.807294i
1.86526	+	0.199842i
−0.734668	−	0.348716i
−0.577806	+	2.22188i
0.947217	−	0.807294i
1.86526	−	0.199842i

−1.23467

+

2.13851i

1.66933

−

0.461883i

−2.04881

−

3.54864i

1.21814

+

2.10988i

−1.07333

+

4.14015i

−1.16933

+

2.02534i

5.17972

2.57333

−

1.54207i

−6.01598

34.2

−1.07781

+

1.86682i

−0.635299

−

1.61133i

−1.32333

−

2.29208i

−1.81197

−

3.13842i

3.69279

+

0.550720i

1.13530

−

1.96640i

1.39396

−2.19279

+

2.04736i

7.81179

34.3

0.447217

−

0.774602i

1.22553

+

1.22396i

0.599994

+

1.03922i

−1.87447

−

3.24667i

1.49616

−

0.401921i

−0.725528

+

1.25665i

2.86218

0.00384004

+

3.00000i

−3.35317

34.4

1.36526

−

2.36469i

0.240440

+

1.71528i

−2.72785

−

4.72478i

0.468293

+

0.811107i

4.38438

+

1.77323i

0.259560

−

0.449571i

−9.43585

−2.88438

+

0.824844i

2.55736

67.1

−1.23467

−

2.13851i

1.66933

+

0.461883i

−2.04881

+

3.54864i

1.21814

−

2.10988i

−1.07333

−

4.14015i

−1.16933

−

2.02534i

5.17972

2.57333

+

1.54207i

−6.01598

67.2

−1.07781

−

1.86682i

−0.635299

+

1.61133i

−1.32333

+

2.29208i

−1.81197

+

3.13842i

3.69279

−

0.550720i

1.13530

+

1.96640i

1.39396

−2.19279

−

2.04736i

7.81179

67.3

0.447217

+

0.774602i

1.22553

−

1.22396i

0.599994

−

1.03922i

−1.87447

+

3.24667i

1.49616

+

0.401921i

−0.725528

−

1.25665i

2.86218

0.00384004

−

3.00000i

−3.35317

67.4

1.36526

+

2.36469i

0.240440

−

1.71528i

−2.72785

+

4.72478i

0.468293

−

0.811107i

4.38438

−

1.77323i

0.259560

+

0.449571i

−9.43585

−2.88438

−

0.824844i

2.55736

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	99.2.e.e	✓	8
3.b	odd	2	1		297.2.e.e		8
9.c	even	3	1	inner	99.2.e.e	✓	8
9.c	even	3	1		891.2.a.q		4
9.d	odd	6	1		297.2.e.e		8
9.d	odd	6	1		891.2.a.p		4
11.b	odd	2	1		1089.2.e.i		8
99.g	even	6	1		9801.2.a.bl		4
99.h	odd	6	1		1089.2.e.i		8
99.h	odd	6	1		9801.2.a.bi		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
99.2.e.e	✓	8	1.a	even	1	1	trivial
99.2.e.e	✓	8	9.c	even	3	1	inner
297.2.e.e		8	3.b	odd	2	1
297.2.e.e		8	9.d	odd	6	1
891.2.a.p		4	9.d	odd	6	1
891.2.a.q		4	9.c	even	3	1
1089.2.e.i		8	11.b	odd	2	1
1089.2.e.i		8	99.h	odd	6	1
9801.2.a.bi		4	99.h	odd	6	1
9801.2.a.bl		4	99.g	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(99, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( 1 + T + 2 T^{2} + 3 T^{3} + 2 T^{4} - 4 T^{5} + T^{6} - 6 T^{7} - 11 T^{8} - 12 T^{9} + 4 T^{10} - 32 T^{11} + 32 T^{12} + 96 T^{13} + 128 T^{14} + 128 T^{15} + 256 T^{16} \)
$3$	\( 1 - 5 T + 15 T^{2} - 35 T^{3} + 67 T^{4} - 105 T^{5} + 135 T^{6} - 135 T^{7} + 81 T^{8} \)
$5$	\( 1 + 4 T + 5 T^{2} - 18 T^{3} - 94 T^{4} - 232 T^{5} - 110 T^{6} + 1011 T^{7} + 3826 T^{8} + 5055 T^{9} - 2750 T^{10} - 29000 T^{11} - 58750 T^{12} - 56250 T^{13} + 78125 T^{14} + 312500 T^{15} + 390625 T^{16} \)
$7$	\( 1 + T - 21 T^{2} - 10 T^{3} + 254 T^{4} + 36 T^{5} - 2408 T^{6} - 83 T^{7} + 18573 T^{8} - 581 T^{9} - 117992 T^{10} + 12348 T^{11} + 609854 T^{12} - 168070 T^{13} - 2470629 T^{14} + 823543 T^{15} + 5764801 T^{16} \)
$11$	\( ( 1 + T + T^{2} )^{4} \)