Properties

Label 9200.2.a.cq
Level $9200$
Weight $2$
Character orbit 9200.a
Self dual yes
Analytic conductor $73.462$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(73.4623698596\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.5744.1
Defining polynomial: \( x^{4} - 5x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + (\beta_{2} + 2) q^{7} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + (\beta_{2} + 2) q^{7} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{9} + ( - \beta_{3} + \beta_1 - 1) q^{11} + ( - \beta_1 - 3) q^{13} + ( - \beta_{3} + \beta_{2} - 3) q^{17} + (\beta_{3} + 1) q^{19} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{21} - q^{23} + (\beta_{3} - 3 \beta_{2} + 2) q^{27} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{29} + (\beta_{3} - 3 \beta_{2} + \beta_1 - 2) q^{31} + (\beta_{2} + 2 \beta_1 - 4) q^{33} + (2 \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{37} + (\beta_{3} - \beta_{2} + 3 \beta_1) q^{39} + ( - 2 \beta_{2} + 4 \beta_1 - 5) q^{41} + ( - \beta_{3} + 4 \beta_1 - 1) q^{43} + ( - 3 \beta_{3} + 3 \beta_{2} + 2) q^{47} + (4 \beta_{2} - 2 \beta_1 + 3) q^{49} + (2 \beta_{2} + 5 \beta_1 - 4) q^{51} + (2 \beta_{3} + 2 \beta_{2}) q^{53} + ( - \beta_{3} - 2 \beta_1 + 1) q^{57} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{59} + ( - 3 \beta_{3} + 2 \beta_1 - 3) q^{61} + (2 \beta_1 - 4) q^{63} + \beta_{2} q^{67} + (\beta_1 - 1) q^{69} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 3) q^{71} + ( - 3 \beta_{3} - \beta_{2} + 2 \beta_1 - 6) q^{73} + ( - 2 \beta_{2} - 2) q^{77} + (\beta_{3} - 2 \beta_{2} - 7) q^{79} + ( - \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 2) q^{81} + (\beta_{3} + 3 \beta_{2} - 6 \beta_1 + 3) q^{83} + (\beta_{3} - \beta_{2} - 2) q^{87} + (3 \beta_{3} - 2 \beta_{2} - 3) q^{89} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 7) q^{91} + (\beta_{3} - 5 \beta_{2} - 2 \beta_1 - 2) q^{93} + (3 \beta_{2} - 2 \beta_1 - 6) q^{97} + (4 \beta_{2} + 2 \beta_1 - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 6 q^{7} + 4 q^{9} - 2 q^{11} - 14 q^{13} - 14 q^{17} + 4 q^{19} - 2 q^{21} - 4 q^{23} + 14 q^{27} + 4 q^{29} - 14 q^{33} - 2 q^{37} + 8 q^{39} - 8 q^{41} + 4 q^{43} + 2 q^{47} - 10 q^{51} - 4 q^{53} - 8 q^{61} - 12 q^{63} - 2 q^{67} - 2 q^{69} + 24 q^{71} - 18 q^{73} - 4 q^{77} - 24 q^{79} + 8 q^{81} - 6 q^{83} - 6 q^{87} - 8 q^{89} - 26 q^{91} - 2 q^{93} - 34 q^{97} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} - 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} - 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + \nu^{2} + 5\nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} + \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} - 2\beta_{2} + 3\beta _1 - 1 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.37988
−0.751024
−1.92022
0.291367
0 −1.95969 0 0 0 2.28394 0 0.840379 0
1.2 0 −0.580491 0 0 0 0.315061 0 −2.66303 0
1.3 0 1.39945 0 0 0 4.60747 0 −1.04155 0
1.4 0 3.14073 0 0 0 −1.20647 0 6.86420 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(23\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9200.2.a.cq 4
4.b odd 2 1 575.2.a.i 4
5.b even 2 1 9200.2.a.ck 4
5.c odd 4 2 1840.2.e.d 8
12.b even 2 1 5175.2.a.bv 4
20.d odd 2 1 575.2.a.j 4
20.e even 4 2 115.2.b.b 8
60.h even 2 1 5175.2.a.bw 4
60.l odd 4 2 1035.2.b.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.b.b 8 20.e even 4 2
575.2.a.i 4 4.b odd 2 1
575.2.a.j 4 20.d odd 2 1
1035.2.b.e 8 60.l odd 4 2
1840.2.e.d 8 5.c odd 4 2
5175.2.a.bv 4 12.b even 2 1
5175.2.a.bw 4 60.h even 2 1
9200.2.a.ck 4 5.b even 2 1
9200.2.a.cq 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9200))\):

\( T_{3}^{4} - 2T_{3}^{3} - 6T_{3}^{2} + 6T_{3} + 5 \) Copy content Toggle raw display
\( T_{7}^{4} - 6T_{7}^{3} + 4T_{7}^{2} + 12T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{4} + 2T_{11}^{3} - 16T_{11}^{2} - 44T_{11} - 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} - 6 T^{2} + 6 T + 5 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 6 T^{3} + 4 T^{2} + 12 T - 4 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} - 16 T^{2} - 44 T - 28 \) Copy content Toggle raw display
$13$ \( T^{4} + 14 T^{3} + 66 T^{2} + 118 T + 61 \) Copy content Toggle raw display
$17$ \( T^{4} + 14 T^{3} + 58 T^{2} + 64 T + 20 \) Copy content Toggle raw display
$19$ \( T^{4} - 4 T^{3} - 6 T^{2} + 28 T - 20 \) Copy content Toggle raw display
$23$ \( (T + 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} - 22 T^{2} + 4 T + 5 \) Copy content Toggle raw display
$31$ \( T^{4} - 74 T^{2} + 256 T - 167 \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} - 72 T^{2} - 380 T - 476 \) Copy content Toggle raw display
$41$ \( T^{4} + 8 T^{3} - 94 T^{2} + \cdots + 2485 \) Copy content Toggle raw display
$43$ \( T^{4} - 4 T^{3} - 118 T^{2} + \cdots + 1964 \) Copy content Toggle raw display
$47$ \( T^{4} - 2 T^{3} - 138 T^{2} + \cdots + 4513 \) Copy content Toggle raw display
$53$ \( T^{4} + 4 T^{3} - 104 T^{2} + \cdots + 2192 \) Copy content Toggle raw display
$59$ \( T^{4} - 20 T^{2} - 16 T + 16 \) Copy content Toggle raw display
$61$ \( T^{4} + 8 T^{3} - 102 T^{2} + \cdots - 2756 \) Copy content Toggle raw display
$67$ \( T^{4} + 2 T^{3} - 8 T^{2} - 12 T + 4 \) Copy content Toggle raw display
$71$ \( T^{4} - 24 T^{3} + 66 T^{2} + \cdots - 7435 \) Copy content Toggle raw display
$73$ \( T^{4} + 18 T^{3} - 22 T^{2} + \cdots - 8339 \) Copy content Toggle raw display
$79$ \( T^{4} + 24 T^{3} + 178 T^{2} + \cdots + 28 \) Copy content Toggle raw display
$83$ \( T^{4} + 6 T^{3} - 270 T^{2} + \cdots + 14948 \) Copy content Toggle raw display
$89$ \( T^{4} + 8 T^{3} - 86 T^{2} + \cdots + 2380 \) Copy content Toggle raw display
$97$ \( T^{4} + 34 T^{3} + 348 T^{2} + \cdots - 4676 \) Copy content Toggle raw display
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