Properties

Label 9900.2.c.x
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: \(\Q(i, \sqrt{21})\)
Defining polynomial: \( x^{4} + 11x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1100)
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.

\(f(q)\) \(=\) \( q + (3 \beta_{2} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (3 \beta_{2} + \beta_1) q^{7} + q^{11} - \beta_{2} q^{13} + ( - 2 \beta_{2} - \beta_1) q^{17} + (2 \beta_{3} - 3) q^{19} + (\beta_{2} - \beta_1) q^{23} + ( - \beta_{3} + 5) q^{29} + (2 \beta_{3} - 5) q^{31} + ( - 3 \beta_{2} - 2 \beta_1) q^{37} + (2 \beta_{3} + 5) q^{41} - 10 \beta_{2} q^{43} + ( - 7 \beta_{2} - 2 \beta_1) q^{47} + ( - 5 \beta_{3} - 2) q^{49} + ( - 3 \beta_{2} + 3 \beta_1) q^{53} + (2 \beta_{3} - 7) q^{59} + (\beta_{3} + 6) q^{61} + 4 \beta_{2} q^{67} - 6 \beta_{3} q^{71} + ( - 5 \beta_{2} + \beta_1) q^{73} + (3 \beta_{2} + \beta_1) q^{77} + ( - 7 \beta_{3} + 3) q^{79} + ( - 4 \beta_{2} - 5 \beta_1) q^{83} + (\beta_{3} + 1) q^{89} + (\beta_{3} + 2) q^{91} + (9 \beta_{2} + \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{11} - 8 q^{19} + 18 q^{29} - 16 q^{31} + 24 q^{41} - 18 q^{49} - 24 q^{59} + 26 q^{61} - 12 q^{71} - 2 q^{79} + 6 q^{89} + 10 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 11x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 6\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{2} - 6\beta_1 \) 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
1.79129i
2.79129i
2.79129i
1.79129i
0 0 0 0 0 4.79129i 0 0 0
5149.2 0 0 0 0 0 0.208712i 0 0 0
5149.3 0 0 0 0 0 0.208712i 0 0 0
5149.4 0 0 0 0 0 4.79129i 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.x 4
3.b odd 2 1 1100.2.b.d 4
5.b even 2 1 inner 9900.2.c.x 4
5.c odd 4 1 9900.2.a.bh 2
5.c odd 4 1 9900.2.a.bz 2
12.b even 2 1 4400.2.b.s 4
15.d odd 2 1 1100.2.b.d 4
15.e even 4 1 1100.2.a.g 2
15.e even 4 1 1100.2.a.h yes 2
60.h even 2 1 4400.2.b.s 4
60.l odd 4 1 4400.2.a.bi 2
60.l odd 4 1 4400.2.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1100.2.a.g 2 15.e even 4 1
1100.2.a.h yes 2 15.e even 4 1
1100.2.b.d 4 3.b odd 2 1
1100.2.b.d 4 15.d odd 2 1
4400.2.a.bi 2 60.l odd 4 1
4400.2.a.bu 2 60.l odd 4 1
4400.2.b.s 4 12.b even 2 1
4400.2.b.s 4 60.h even 2 1
9900.2.a.bh 2 5.c odd 4 1
9900.2.a.bz 2 5.c odd 4 1
9900.2.c.x 4 1.a even 1 1 trivial
9900.2.c.x 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 \(S_{2}^{\mathrm{new}}(9900, [\chi])\):

\( T_{7}^{4} + 23T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{13}^{2} + 1 \) Copy content Toggle raw display
\( T_{17}^{4} + 15T_{17}^{2} + 9 \) Copy content Toggle raw display
\( T_{29}^{2} - 9T_{29} + 15 \) Copy content Toggle raw display
\( T_{41}^{2} - 12T_{41} + 15 \) Copy content Toggle raw display
\( T_{59}^{2} + 12T_{59} + 15 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 23T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 15T^{2} + 9 \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T - 17)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 15T^{2} + 9 \) Copy content Toggle raw display
$29$ \( (T^{2} - 9 T + 15)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T - 5)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 50T^{2} + 289 \) Copy content Toggle raw display
$41$ \( (T^{2} - 12 T + 15)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 114T^{2} + 225 \) Copy content Toggle raw display
$53$ \( T^{4} + 135T^{2} + 729 \) Copy content Toggle raw display
$59$ \( (T^{2} + 12 T + 15)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 13 T + 37)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T - 180)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 71T^{2} + 625 \) Copy content Toggle raw display
$79$ \( (T^{2} + T - 257)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 267 T^{2} + 16641 \) Copy content Toggle raw display
$89$ \( (T^{2} - 3 T - 3)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 155T^{2} + 4489 \) Copy content Toggle raw display
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