Properties

Label 9900.2.c.q
Level $9900$
Weight $2$
Character orbit 9900.c
Analytic conductor $79.052$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 9900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9900.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(79.0518980011\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 660)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{7} +O(q^{10})\) \( q + \beta_{1} q^{7} - q^{11} + \beta_{1} q^{13} + ( -\beta_{1} + \beta_{2} ) q^{17} -\beta_{3} q^{19} + 8 q^{29} + ( 2 - \beta_{3} ) q^{31} + ( -2 \beta_{1} - 3 \beta_{2} ) q^{37} -8 q^{41} + ( -\beta_{1} - 4 \beta_{2} ) q^{43} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{47} + ( -7 + \beta_{3} ) q^{49} -\beta_{2} q^{53} + 8 q^{59} -\beta_{3} q^{61} + 2 \beta_{2} q^{67} -2 \beta_{3} q^{71} + ( -\beta_{1} - 2 \beta_{2} ) q^{73} -\beta_{1} q^{77} + ( 4 - \beta_{3} ) q^{79} + ( -\beta_{1} + 3 \beta_{2} ) q^{83} + 6 q^{89} + ( -14 + \beta_{3} ) q^{91} + ( -4 \beta_{1} - 3 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{11} + 32q^{29} + 8q^{31} - 32q^{41} - 28q^{49} + 32q^{59} + 16q^{79} + 24q^{89} - 56q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{3} + 8 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( 4 \nu^{2} + 14 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 14\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{2} - 4 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9900\mathbb{Z}\right)^\times\).

\(n\) \(2377\) \(4501\) \(4951\) \(5501\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5149.1
2.30278i
1.30278i
1.30278i
2.30278i
0 0 0 0 0 4.60555i 0 0 0
5149.2 0 0 0 0 0 2.60555i 0 0 0
5149.3 0 0 0 0 0 2.60555i 0 0 0
5149.4 0 0 0 0 0 4.60555i 0 0 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9900.2.c.q 4
3.b odd 2 1 3300.2.c.l 4
5.b even 2 1 inner 9900.2.c.q 4
5.c odd 4 1 1980.2.a.h 2
5.c odd 4 1 9900.2.a.bl 2
15.d odd 2 1 3300.2.c.l 4
15.e even 4 1 660.2.a.e 2
15.e even 4 1 3300.2.a.w 2
20.e even 4 1 7920.2.a.bo 2
60.l odd 4 1 2640.2.a.bc 2
165.l odd 4 1 7260.2.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
660.2.a.e 2 15.e even 4 1
1980.2.a.h 2 5.c odd 4 1
2640.2.a.bc 2 60.l odd 4 1
3300.2.a.w 2 15.e even 4 1
3300.2.c.l 4 3.b odd 2 1
3300.2.c.l 4 15.d odd 2 1
7260.2.a.w 2 165.l odd 4 1
7920.2.a.bo 2 20.e even 4 1
9900.2.a.bl 2 5.c odd 4 1
9900.2.c.q 4 1.a even 1 1 trivial
9900.2.c.q 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(9900, [\chi])\):

\( T_{7}^{4} + 28 T_{7}^{2} + 144 \)
\( T_{13}^{4} + 28 T_{13}^{2} + 144 \)
\( T_{17}^{4} + 44 T_{17}^{2} + 16 \)
\( T_{29} - 8 \)
\( T_{41} + 8 \)
\( T_{59} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( 144 + 28 T^{2} + T^{4} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( 144 + 28 T^{2} + T^{4} \)
$17$ \( 16 + 44 T^{2} + T^{4} \)
$19$ \( ( -52 + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( ( -8 + T )^{4} \)
$31$ \( ( -48 - 4 T + T^{2} )^{2} \)
$37$ \( 1296 + 136 T^{2} + T^{4} \)
$41$ \( ( 8 + T )^{4} \)
$43$ \( 1296 + 124 T^{2} + T^{4} \)
$47$ \( 2304 + 112 T^{2} + T^{4} \)
$53$ \( ( 4 + T^{2} )^{2} \)
$59$ \( ( -8 + T )^{4} \)
$61$ \( ( -52 + T^{2} )^{2} \)
$67$ \( ( 16 + T^{2} )^{2} \)
$71$ \( ( -208 + T^{2} )^{2} \)
$73$ \( 16 + 44 T^{2} + T^{4} \)
$79$ \( ( -36 - 8 T + T^{2} )^{2} \)
$83$ \( 1296 + 124 T^{2} + T^{4} \)
$89$ \( ( -6 + T )^{4} \)
$97$ \( 41616 + 424 T^{2} + T^{4} \)
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