Properties

Label 9900.2.c.d
Level $9900$
Weight $2$
Character orbit 9900.c
Analytic conductor $79.052$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{7} +O(q^{10})\) \( q + 2 i q^{7} - q^{11} -2 i q^{13} -2 q^{19} + 8 q^{31} + 2 i q^{37} -2 i q^{43} + 3 q^{49} + 6 i q^{53} -12 q^{59} + 2 q^{61} -4 i q^{67} -2 i q^{73} -2 i q^{77} + 10 q^{79} -12 i q^{83} -6 q^{89} + 4 q^{91} + 14 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 2q^{11} - 4q^{19} + 16q^{31} + 6q^{49} - 24q^{59} + 4q^{61} + 20q^{79} - 12q^{89} + 8q^{91} + O(q^{100}) \)

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
1.00000i
1.00000i
0 0 0 0 0 2.00000i 0 0 0
5149.2 0 0 0 0 0 2.00000i 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.d 2
3.b odd 2 1 3300.2.c.i 2
5.b even 2 1 inner 9900.2.c.d 2
5.c odd 4 1 1980.2.a.f 1
5.c odd 4 1 9900.2.a.e 1
15.d odd 2 1 3300.2.c.i 2
15.e even 4 1 660.2.a.d 1
15.e even 4 1 3300.2.a.b 1
20.e even 4 1 7920.2.a.y 1
60.l odd 4 1 2640.2.a.b 1
165.l odd 4 1 7260.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
660.2.a.d 1 15.e even 4 1
1980.2.a.f 1 5.c odd 4 1
2640.2.a.b 1 60.l odd 4 1
3300.2.a.b 1 15.e even 4 1
3300.2.c.i 2 3.b odd 2 1
3300.2.c.i 2 15.d odd 2 1
7260.2.a.l 1 165.l odd 4 1
7920.2.a.y 1 20.e even 4 1
9900.2.a.e 1 5.c odd 4 1
9900.2.c.d 2 1.a even 1 1 trivial
9900.2.c.d 2 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 \(S_{2}^{\mathrm{new}}(9900, [\chi])\):

\( T_{7}^{2} + 4 \)
\( T_{13}^{2} + 4 \)
\( T_{17} \)
\( T_{29} \)
\( T_{41} \)
\( T_{59} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 4 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( 12 + T )^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 4 + T^{2} \)
$79$ \( ( -10 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( 196 + T^{2} \)
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