Properties

Label 9900.2.c.g
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: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
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.

\(f(q)\) \(=\) \( q + 2 i q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{7} + q^{11} + 4 i q^{13} - 6 i q^{17} - 8 q^{19} - 3 i q^{23} + 5 q^{31} - i q^{37} + 10 i q^{43} + 3 q^{49} - 6 i q^{53} + 3 q^{59} - 4 q^{61} - i q^{67} - 15 q^{71} + 4 i q^{73} + 2 i q^{77} - 2 q^{79} + 6 i q^{83} - 9 q^{89} - 8 q^{91} - 7 i q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{11} - 16 q^{19} + 10 q^{31} + 6 q^{49} + 6 q^{59} - 8 q^{61} - 30 q^{71} - 4 q^{79} - 18 q^{89} - 16 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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.g 2
3.b odd 2 1 1100.2.b.c 2
5.b even 2 1 inner 9900.2.c.g 2
5.c odd 4 1 396.2.a.c 1
5.c odd 4 1 9900.2.a.h 1
12.b even 2 1 4400.2.b.k 2
15.d odd 2 1 1100.2.b.c 2
15.e even 4 1 44.2.a.a 1
15.e even 4 1 1100.2.a.b 1
20.e even 4 1 1584.2.a.p 1
40.i odd 4 1 6336.2.a.j 1
40.k even 4 1 6336.2.a.i 1
45.k odd 12 2 3564.2.i.a 2
45.l even 12 2 3564.2.i.j 2
55.e even 4 1 4356.2.a.j 1
60.h even 2 1 4400.2.b.k 2
60.l odd 4 1 176.2.a.a 1
60.l odd 4 1 4400.2.a.v 1
105.k odd 4 1 2156.2.a.a 1
105.w odd 12 2 2156.2.i.c 2
105.x even 12 2 2156.2.i.b 2
120.q odd 4 1 704.2.a.i 1
120.w even 4 1 704.2.a.f 1
165.l odd 4 1 484.2.a.a 1
165.u odd 20 4 484.2.e.b 4
165.v even 20 4 484.2.e.a 4
195.s even 4 1 7436.2.a.d 1
240.z odd 4 1 2816.2.c.k 2
240.bb even 4 1 2816.2.c.e 2
240.bd odd 4 1 2816.2.c.k 2
240.bf even 4 1 2816.2.c.e 2
420.w even 4 1 8624.2.a.w 1
660.q even 4 1 1936.2.a.c 1
1320.bn odd 4 1 7744.2.a.m 1
1320.bt even 4 1 7744.2.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.a.a 1 15.e even 4 1
176.2.a.a 1 60.l odd 4 1
396.2.a.c 1 5.c odd 4 1
484.2.a.a 1 165.l odd 4 1
484.2.e.a 4 165.v even 20 4
484.2.e.b 4 165.u odd 20 4
704.2.a.f 1 120.w even 4 1
704.2.a.i 1 120.q odd 4 1
1100.2.a.b 1 15.e even 4 1
1100.2.b.c 2 3.b odd 2 1
1100.2.b.c 2 15.d odd 2 1
1584.2.a.p 1 20.e even 4 1
1936.2.a.c 1 660.q even 4 1
2156.2.a.a 1 105.k odd 4 1
2156.2.i.b 2 105.x even 12 2
2156.2.i.c 2 105.w odd 12 2
2816.2.c.e 2 240.bb even 4 1
2816.2.c.e 2 240.bf even 4 1
2816.2.c.k 2 240.z odd 4 1
2816.2.c.k 2 240.bd odd 4 1
3564.2.i.a 2 45.k odd 12 2
3564.2.i.j 2 45.l even 12 2
4356.2.a.j 1 55.e even 4 1
4400.2.a.v 1 60.l odd 4 1
4400.2.b.k 2 12.b even 2 1
4400.2.b.k 2 60.h even 2 1
6336.2.a.i 1 40.k even 4 1
6336.2.a.j 1 40.i odd 4 1
7436.2.a.d 1 195.s even 4 1
7744.2.a.m 1 1320.bn odd 4 1
7744.2.a.bc 1 1320.bt even 4 1
8624.2.a.w 1 420.w even 4 1
9900.2.a.h 1 5.c odd 4 1
9900.2.c.g 2 1.a even 1 1 trivial
9900.2.c.g 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 \(S_{2}^{\mathrm{new}}(9900, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 16 \) Copy content Toggle raw display
\( T_{17}^{2} + 36 \) Copy content Toggle raw display
\( T_{29} \) Copy content Toggle raw display
\( T_{41} \) Copy content Toggle raw display
\( T_{59} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( (T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 9 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 5)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 1 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 100 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 3)^{2} \) Copy content Toggle raw display
$61$ \( (T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1 \) Copy content Toggle raw display
$71$ \( (T + 15)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T + 2)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( (T + 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 49 \) Copy content Toggle raw display
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