Properties

Label 99.2.d.a
Level 99
Weight 2
Character orbit 99.d
Analytic conductor 0.791
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.790518980011\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + q^{4} -\beta_{1} q^{5} -\beta_{3} q^{7} + \beta_{2} q^{8} +O(q^{10})\) \( q -\beta_{2} q^{2} + q^{4} -\beta_{1} q^{5} -\beta_{3} q^{7} + \beta_{2} q^{8} + \beta_{3} q^{10} + ( -2 \beta_{1} + \beta_{2} ) q^{11} -2 \beta_{3} q^{13} + 3 \beta_{1} q^{14} -5 q^{16} + 3 \beta_{3} q^{19} -\beta_{1} q^{20} + ( -3 + 2 \beta_{3} ) q^{22} + 2 \beta_{1} q^{23} + 3 q^{25} + 6 \beta_{1} q^{26} -\beta_{3} q^{28} -4 \beta_{2} q^{29} -4 q^{31} + 3 \beta_{2} q^{32} -2 \beta_{2} q^{35} + 8 q^{37} -9 \beta_{1} q^{38} -\beta_{3} q^{40} + 4 \beta_{2} q^{41} -\beta_{3} q^{43} + ( -2 \beta_{1} + \beta_{2} ) q^{44} -2 \beta_{3} q^{46} + 2 \beta_{1} q^{47} + q^{49} -3 \beta_{2} q^{50} -2 \beta_{3} q^{52} -7 \beta_{1} q^{53} + ( -4 - \beta_{3} ) q^{55} -3 \beta_{1} q^{56} + 12 q^{58} + 8 \beta_{1} q^{59} + 2 \beta_{3} q^{61} + 4 \beta_{2} q^{62} + q^{64} -4 \beta_{2} q^{65} -4 q^{67} + 6 q^{70} + 2 \beta_{1} q^{71} -8 \beta_{2} q^{74} + 3 \beta_{3} q^{76} + ( -3 \beta_{1} - 4 \beta_{2} ) q^{77} + 5 \beta_{3} q^{79} + 5 \beta_{1} q^{80} -12 q^{82} + 8 \beta_{2} q^{83} + 3 \beta_{1} q^{86} + ( 3 - 2 \beta_{3} ) q^{88} + 5 \beta_{1} q^{89} -12 q^{91} + 2 \beta_{1} q^{92} -2 \beta_{3} q^{94} + 6 \beta_{2} q^{95} -10 q^{97} -\beta_{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} + O(q^{10}) \) \( 4q + 4q^{4} - 20q^{16} - 12q^{22} + 12q^{25} - 16q^{31} + 32q^{37} + 4q^{49} - 16q^{55} + 48q^{58} + 4q^{64} - 16q^{67} + 24q^{70} - 48q^{82} + 12q^{88} - 48q^{91} - 40q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 4 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 3 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 5 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{3} + 5 \beta_{1}\)\()/2\)

Character values

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

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
98.1
0.517638i
0.517638i
1.93185i
1.93185i
−1.73205 0 1.00000 1.41421i 0 2.44949i 1.73205 0 2.44949i
98.2 −1.73205 0 1.00000 1.41421i 0 2.44949i 1.73205 0 2.44949i
98.3 1.73205 0 1.00000 1.41421i 0 2.44949i −1.73205 0 2.44949i
98.4 1.73205 0 1.00000 1.41421i 0 2.44949i −1.73205 0 2.44949i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.2.d.a 4
3.b odd 2 1 inner 99.2.d.a 4
4.b odd 2 1 1584.2.b.e 4
5.b even 2 1 2475.2.f.e 4
5.c odd 4 2 2475.2.d.a 8
8.b even 2 1 6336.2.b.s 4
8.d odd 2 1 6336.2.b.t 4
9.c even 3 2 891.2.g.c 8
9.d odd 6 2 891.2.g.c 8
11.b odd 2 1 inner 99.2.d.a 4
12.b even 2 1 1584.2.b.e 4
15.d odd 2 1 2475.2.f.e 4
15.e even 4 2 2475.2.d.a 8
24.f even 2 1 6336.2.b.t 4
24.h odd 2 1 6336.2.b.s 4
33.d even 2 1 inner 99.2.d.a 4
44.c even 2 1 1584.2.b.e 4
55.d odd 2 1 2475.2.f.e 4
55.e even 4 2 2475.2.d.a 8
88.b odd 2 1 6336.2.b.s 4
88.g even 2 1 6336.2.b.t 4
99.g even 6 2 891.2.g.c 8
99.h odd 6 2 891.2.g.c 8
132.d odd 2 1 1584.2.b.e 4
165.d even 2 1 2475.2.f.e 4
165.l odd 4 2 2475.2.d.a 8
264.m even 2 1 6336.2.b.s 4
264.p odd 2 1 6336.2.b.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.d.a 4 1.a even 1 1 trivial
99.2.d.a 4 3.b odd 2 1 inner
99.2.d.a 4 11.b odd 2 1 inner
99.2.d.a 4 33.d even 2 1 inner
891.2.g.c 8 9.c even 3 2
891.2.g.c 8 9.d odd 6 2
891.2.g.c 8 99.g even 6 2
891.2.g.c 8 99.h odd 6 2
1584.2.b.e 4 4.b odd 2 1
1584.2.b.e 4 12.b even 2 1
1584.2.b.e 4 44.c even 2 1
1584.2.b.e 4 132.d odd 2 1
2475.2.d.a 8 5.c odd 4 2
2475.2.d.a 8 15.e even 4 2
2475.2.d.a 8 55.e even 4 2
2475.2.d.a 8 165.l odd 4 2
2475.2.f.e 4 5.b even 2 1
2475.2.f.e 4 15.d odd 2 1
2475.2.f.e 4 55.d odd 2 1
2475.2.f.e 4 165.d even 2 1
6336.2.b.s 4 8.b even 2 1
6336.2.b.s 4 24.h odd 2 1
6336.2.b.s 4 88.b odd 2 1
6336.2.b.s 4 264.m even 2 1
6336.2.b.t 4 8.d odd 2 1
6336.2.b.t 4 24.f even 2 1
6336.2.b.t 4 88.g even 2 1
6336.2.b.t 4 264.p odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(99, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} + 4 T^{4} )^{2} \)
$3$ 1
$5$ \( ( 1 - 8 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 - 8 T^{2} + 49 T^{4} )^{2} \)
$11$ \( 1 + 10 T^{2} + 121 T^{4} \)
$13$ \( ( 1 - 2 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 17 T^{2} )^{4} \)
$19$ \( ( 1 + 16 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 38 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 10 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 4 T + 31 T^{2} )^{4} \)
$37$ \( ( 1 - 8 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 + 34 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 80 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 - 86 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 8 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 10 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 98 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 4 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 - 134 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 73 T^{2} )^{4} \)
$79$ \( ( 1 - 8 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 26 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 128 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 10 T + 97 T^{2} )^{4} \)
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