Properties

Label 99.2.g.a
Level 99
Weight 2
Character orbit 99.g
Analytic conductor 0.791
Analytic rank 0
Dimension 4
CM discriminant -11
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 99.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.790518980011\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( 2 - 2 \beta_{2} ) q^{4} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{5} + ( 3 \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( 2 - 2 \beta_{2} ) q^{4} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{5} + ( 3 \beta_{2} + \beta_{3} ) q^{9} + ( \beta_{2} + 2 \beta_{3} ) q^{11} -2 \beta_{3} q^{12} + ( 3 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{15} -4 \beta_{2} q^{16} + ( -6 + 2 \beta_{2} - 2 \beta_{3} ) q^{20} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{23} + ( -3 + 6 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{25} + ( -3 - 2 \beta_{1} + 2 \beta_{3} ) q^{27} + ( 4 - 3 \beta_{1} - \beta_{2} + 6 \beta_{3} ) q^{31} + ( -6 + \beta_{1} - \beta_{3} ) q^{33} + ( 6 - 2 \beta_{1} + 2 \beta_{3} ) q^{36} + ( 5 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{37} + ( 2 - 4 \beta_{1} + 4 \beta_{3} ) q^{44} + ( 3 - 4 \beta_{1} - 9 \beta_{2} + \beta_{3} ) q^{45} + ( 12 - 7 \beta_{2} - 2 \beta_{3} ) q^{47} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{48} + ( -7 + 7 \beta_{2} ) q^{49} + ( 1 + 4 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} ) q^{53} + ( -4 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{55} + ( 8 - \beta_{1} + 7 \beta_{2} ) q^{59} + ( 6 + 2 \beta_{1} + 4 \beta_{3} ) q^{60} -8 q^{64} + ( -5 - 3 \beta_{1} + 8 \beta_{2} + 6 \beta_{3} ) q^{67} + ( -6 \beta_{2} - \beta_{3} ) q^{69} + ( 1 - 5 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} ) q^{71} + ( 9 - 6 \beta_{1} - 18 \beta_{2} + 3 \beta_{3} ) q^{75} + ( -8 + 4 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} ) q^{80} + ( -6 + 5 \beta_{1} + 6 \beta_{2} ) q^{81} + ( -5 + 10 \beta_{1} - 10 \beta_{3} ) q^{89} + ( 2 \beta_{2} + 4 \beta_{3} ) q^{92} + ( -18 + 3 \beta_{1} + 9 \beta_{2} - 4 \beta_{3} ) q^{93} + ( -3 + 6 \beta_{1} + 10 \beta_{2} - 3 \beta_{3} ) q^{97} + ( 3 + 5 \beta_{1} - 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{3} + 4q^{4} - 9q^{5} + 5q^{9} + O(q^{10}) \) \( 4q - q^{3} + 4q^{4} - 9q^{5} + 5q^{9} + 2q^{12} + 10q^{15} - 8q^{16} - 18q^{20} + 9q^{25} - 16q^{27} + 5q^{31} - 22q^{33} + 20q^{36} + 14q^{37} - 11q^{45} + 36q^{47} + 8q^{48} - 14q^{49} - 22q^{55} + 45q^{59} + 22q^{60} - 32q^{64} - 13q^{67} - 11q^{69} - 9q^{75} - 7q^{81} - 47q^{93} + 17q^{97} + 11q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu + 3 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-2 \beta_{3} + 2 \beta_{1} + 3\)

Character values

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

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-1\) \(1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
1.68614 + 0.396143i
−1.18614 1.26217i
1.68614 0.396143i
−1.18614 + 1.26217i
0 −1.68614 0.396143i 1.00000 1.73205i −3.68614 2.12819i 0 0 0 2.68614 + 1.33591i 0
32.2 0 1.18614 + 1.26217i 1.00000 1.73205i −0.813859 0.469882i 0 0 0 −0.186141 + 2.99422i 0
65.1 0 −1.68614 + 0.396143i 1.00000 + 1.73205i −3.68614 + 2.12819i 0 0 0 2.68614 1.33591i 0
65.2 0 1.18614 1.26217i 1.00000 + 1.73205i −0.813859 + 0.469882i 0 0 0 −0.186141 2.99422i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
9.d odd 6 1 inner
99.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.2.g.a 4
3.b odd 2 1 297.2.g.a 4
9.c even 3 1 297.2.g.a 4
9.c even 3 1 891.2.d.a 4
9.d odd 6 1 inner 99.2.g.a 4
9.d odd 6 1 891.2.d.a 4
11.b odd 2 1 CM 99.2.g.a 4
33.d even 2 1 297.2.g.a 4
99.g even 6 1 inner 99.2.g.a 4
99.g even 6 1 891.2.d.a 4
99.h odd 6 1 297.2.g.a 4
99.h odd 6 1 891.2.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.g.a 4 1.a even 1 1 trivial
99.2.g.a 4 9.d odd 6 1 inner
99.2.g.a 4 11.b odd 2 1 CM
99.2.g.a 4 99.g even 6 1 inner
297.2.g.a 4 3.b odd 2 1
297.2.g.a 4 9.c even 3 1
297.2.g.a 4 33.d even 2 1
297.2.g.a 4 99.h odd 6 1
891.2.d.a 4 9.c even 3 1
891.2.d.a 4 9.d odd 6 1
891.2.d.a 4 99.g even 6 1
891.2.d.a 4 99.h odd 6 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} + 4 T^{4} )^{2} \)
$3$ \( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \)
$5$ \( ( 1 + 3 T + 5 T^{2} )^{2}( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} ) \)
$7$ \( ( 1 + 7 T^{2} + 49 T^{4} )^{2} \)
$11$ \( 1 - 11 T^{2} + 121 T^{4} \)
$13$ \( ( 1 + 13 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 17 T^{2} )^{4} \)
$19$ \( ( 1 - 19 T^{2} )^{4} \)
$23$ \( ( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} )( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} ) \)
$29$ \( ( 1 - 29 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 5 T + 31 T^{2} )^{2}( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} ) \)
$37$ \( ( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 - 41 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 43 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 - 12 T + 47 T^{2} )^{2}( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} ) \)
$53$ \( ( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} )( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} ) \)
$59$ \( ( 1 - 15 T + 59 T^{2} )^{2}( 1 - 15 T + 166 T^{2} - 885 T^{3} + 3481 T^{4} ) \)
$61$ \( ( 1 + 61 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 13 T + 67 T^{2} )^{2}( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} ) \)
$71$ \( ( 1 - 3 T - 62 T^{2} - 213 T^{3} + 5041 T^{4} )( 1 + 3 T - 62 T^{2} + 213 T^{3} + 5041 T^{4} ) \)
$73$ \( ( 1 - 73 T^{2} )^{4} \)
$79$ \( ( 1 + 79 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 83 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 9 T + 89 T^{2} )^{2}( 1 + 9 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 17 T + 97 T^{2} )^{2}( 1 + 17 T + 192 T^{2} + 1649 T^{3} + 9409 T^{4} ) \)
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