Properties

Label 5200.2.a.cb
Level $5200$
Weight $2$
Character orbit 5200.a
Self dual yes
Analytic conductor $41.522$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(41.5222090511\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} + (\beta_{2} + \beta_1 - 1) q^{7} + (\beta_{2} + 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{3} + (\beta_{2} + \beta_1 - 1) q^{7} + (\beta_{2} + 3 \beta_1) q^{9} + (\beta_{2} + 2) q^{11} - q^{13} + ( - 2 \beta_{2} + 2) q^{17} + \beta_{2} q^{19} + ( - 2 \beta_{2} - 2 \beta_1) q^{21} + (\beta_1 - 5) q^{23} + ( - 4 \beta_{2} - 4 \beta_1 - 2) q^{27} + ( - 3 \beta_{2} - 3 \beta_1 + 3) q^{29} + ( - \beta_{2} - 2 \beta_1 + 4) q^{31} + ( - \beta_{2} - 3 \beta_1 - 1) q^{33} + ( - \beta_{2} - 3 \beta_1 + 1) q^{37} + (\beta_1 + 1) q^{39} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{41} + (2 \beta_{2} - 3 \beta_1 - 1) q^{43} + ( - \beta_{2} - \beta_1 - 3) q^{47} + ( - 2 \beta_{2} - 3) q^{49} + (2 \beta_{2} - 4) q^{51} + (4 \beta_{2} + 2 \beta_1 + 2) q^{53} + ( - \beta_{2} - \beta_1 + 1) q^{57} + ( - 3 \beta_{2} + 2 \beta_1 + 2) q^{59} + (\beta_{2} + 3 \beta_1 + 1) q^{61} + (\beta_{2} + 3 \beta_1 + 5) q^{63} + ( - 5 \beta_{2} - \beta_1 - 3) q^{67} + ( - \beta_{2} + 3 \beta_1 + 3) q^{69} + ( - \beta_{2} + 6 \beta_1 + 2) q^{71} + (\beta_{2} + 3 \beta_1 - 9) q^{73} + 2 \beta_1 q^{77} + ( - 4 \beta_{2} - 2 \beta_1 + 6) q^{79} + (5 \beta_{2} + 5 \beta_1 + 6) q^{81} + (\beta_{2} - \beta_1 - 7) q^{83} + (6 \beta_{2} + 6 \beta_1) q^{87} + (4 \beta_{2} + 4 \beta_1 + 2) q^{89} + ( - \beta_{2} - \beta_1 + 1) q^{91} + (3 \beta_{2} + \beta_1 - 1) q^{93} + (6 \beta_{2} + 4 \beta_1 - 6) q^{97} + (\beta_{2} + 8 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4 q^{3} - 2 q^{7} + 3 q^{9} + 6 q^{11} - 3 q^{13} + 6 q^{17} - 2 q^{21} - 14 q^{23} - 10 q^{27} + 6 q^{29} + 10 q^{31} - 6 q^{33} + 4 q^{39} - 4 q^{41} - 6 q^{43} - 10 q^{47} - 9 q^{49} - 12 q^{51} + 8 q^{53} + 2 q^{57} + 8 q^{59} + 6 q^{61} + 18 q^{63} - 10 q^{67} + 12 q^{69} + 12 q^{71} - 24 q^{73} + 2 q^{77} + 16 q^{79} + 23 q^{81} - 22 q^{83} + 6 q^{87} + 10 q^{89} + 2 q^{91} - 2 q^{93} - 14 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
2.17009
0.311108
−1.48119
0 −3.17009 0 0 0 1.70928 0 7.04945 0
1.2 0 −1.31111 0 0 0 −2.90321 0 −1.28100 0
1.3 0 0.481194 0 0 0 −0.806063 0 −2.76845 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(13\) \(1\)

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 5200.2.a.cb 3
4.b odd 2 1 325.2.a.k 3
5.b even 2 1 5200.2.a.cj 3
5.c odd 4 2 1040.2.d.c 6
12.b even 2 1 2925.2.a.bf 3
20.d odd 2 1 325.2.a.j 3
20.e even 4 2 65.2.b.a 6
52.b odd 2 1 4225.2.a.ba 3
60.h even 2 1 2925.2.a.bj 3
60.l odd 4 2 585.2.c.b 6
260.g odd 2 1 4225.2.a.bh 3
260.l odd 4 2 845.2.d.a 6
260.p even 4 2 845.2.b.c 6
260.s odd 4 2 845.2.d.b 6
260.be odd 12 4 845.2.l.d 12
260.bg even 12 4 845.2.n.g 12
260.bj even 12 4 845.2.n.f 12
260.bl odd 12 4 845.2.l.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.b.a 6 20.e even 4 2
325.2.a.j 3 20.d odd 2 1
325.2.a.k 3 4.b odd 2 1
585.2.c.b 6 60.l odd 4 2
845.2.b.c 6 260.p even 4 2
845.2.d.a 6 260.l odd 4 2
845.2.d.b 6 260.s odd 4 2
845.2.l.d 12 260.be odd 12 4
845.2.l.e 12 260.bl odd 12 4
845.2.n.f 12 260.bj even 12 4
845.2.n.g 12 260.bg even 12 4
1040.2.d.c 6 5.c odd 4 2
2925.2.a.bf 3 12.b even 2 1
2925.2.a.bj 3 60.h even 2 1
4225.2.a.ba 3 52.b odd 2 1
4225.2.a.bh 3 260.g odd 2 1
5200.2.a.cb 3 1.a even 1 1 trivial
5200.2.a.cj 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5200))\):

\( T_{3}^{3} + 4T_{3}^{2} + 2T_{3} - 2 \) Copy content Toggle raw display
\( T_{7}^{3} + 2T_{7}^{2} - 4T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{3} - 6T_{11}^{2} + 8T_{11} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 4 T^{2} + 2 T - 2 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 2 T^{2} - 4 T - 4 \) Copy content Toggle raw display
$11$ \( T^{3} - 6 T^{2} + 8 T + 2 \) Copy content Toggle raw display
$13$ \( (T + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 6 T^{2} - 4 T + 8 \) Copy content Toggle raw display
$19$ \( T^{3} - 4T + 2 \) Copy content Toggle raw display
$23$ \( T^{3} + 14 T^{2} + 62 T + 86 \) Copy content Toggle raw display
$29$ \( T^{3} - 6 T^{2} - 36 T + 108 \) Copy content Toggle raw display
$31$ \( T^{3} - 10 T^{2} + 20 T + 26 \) Copy content Toggle raw display
$37$ \( T^{3} - 28T + 52 \) Copy content Toggle raw display
$41$ \( T^{3} + 4 T^{2} - 32 T + 32 \) Copy content Toggle raw display
$43$ \( T^{3} + 6 T^{2} - 46 T - 278 \) Copy content Toggle raw display
$47$ \( T^{3} + 10 T^{2} + 28 T + 20 \) Copy content Toggle raw display
$53$ \( T^{3} - 8 T^{2} - 40 T + 304 \) Copy content Toggle raw display
$59$ \( T^{3} - 8 T^{2} - 40 T + 262 \) Copy content Toggle raw display
$61$ \( T^{3} - 6 T^{2} - 16 T - 4 \) Copy content Toggle raw display
$67$ \( T^{3} + 10 T^{2} - 60 T - 604 \) Copy content Toggle raw display
$71$ \( T^{3} - 12 T^{2} - 88 T + 754 \) Copy content Toggle raw display
$73$ \( T^{3} + 24 T^{2} + 164 T + 236 \) Copy content Toggle raw display
$79$ \( T^{3} - 16 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$83$ \( T^{3} + 22 T^{2} + 152 T + 316 \) Copy content Toggle raw display
$89$ \( T^{3} - 10 T^{2} - 52 T + 200 \) Copy content Toggle raw display
$97$ \( T^{3} + 14 T^{2} - 84 T - 200 \) Copy content Toggle raw display
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