Properties

Label 6137.2.a.b
Level $6137$
Weight $2$
Character orbit 6137.a
Self dual yes
Analytic conductor $49.004$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6137 = 17 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6137.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.0041917205\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{4} - 2 q^{5} + 4 q^{7} - 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{4} - 2 q^{5} + 4 q^{7} - 3 q^{8} - 3 q^{9} - 2 q^{10} + 2 q^{13} + 4 q^{14} - q^{16} + q^{17} - 3 q^{18} + 2 q^{20} + 4 q^{23} - q^{25} + 2 q^{26} - 4 q^{28} - 6 q^{29} - 4 q^{31} + 5 q^{32} + q^{34} - 8 q^{35} + 3 q^{36} + 2 q^{37} + 6 q^{40} + 6 q^{41} + 4 q^{43} + 6 q^{45} + 4 q^{46} + 9 q^{49} - q^{50} - 2 q^{52} - 6 q^{53} - 12 q^{56} - 6 q^{58} + 12 q^{59} - 10 q^{61} - 4 q^{62} - 12 q^{63} + 7 q^{64} - 4 q^{65} - 4 q^{67} - q^{68} - 8 q^{70} + 4 q^{71} + 9 q^{72} - 6 q^{73} + 2 q^{74} - 12 q^{79} + 2 q^{80} + 9 q^{81} + 6 q^{82} - 4 q^{83} - 2 q^{85} + 4 q^{86} - 10 q^{89} + 6 q^{90} + 8 q^{91} - 4 q^{92} - 2 q^{97} + 9 q^{98}+O(q^{100}) \) 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
0
1.00000 0 −1.00000 −2.00000 0 4.00000 −3.00000 −3.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \(-1\)
\(19\) \(-1\)

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 6137.2.a.b 1
19.b odd 2 1 17.2.a.a 1
57.d even 2 1 153.2.a.c 1
76.d even 2 1 272.2.a.b 1
95.d odd 2 1 425.2.a.d 1
95.g even 4 2 425.2.b.b 2
133.c even 2 1 833.2.a.a 1
133.o even 6 2 833.2.e.a 2
133.r odd 6 2 833.2.e.b 2
152.b even 2 1 1088.2.a.h 1
152.g odd 2 1 1088.2.a.i 1
209.d even 2 1 2057.2.a.e 1
228.b odd 2 1 2448.2.a.o 1
247.d odd 2 1 2873.2.a.c 1
285.b even 2 1 3825.2.a.d 1
323.c odd 2 1 289.2.a.a 1
323.g odd 4 2 289.2.b.a 2
323.l odd 8 4 289.2.c.a 4
323.p even 16 8 289.2.d.d 8
380.d even 2 1 6800.2.a.n 1
399.h odd 2 1 7497.2.a.l 1
437.b even 2 1 8993.2.a.a 1
456.l odd 2 1 9792.2.a.i 1
456.p even 2 1 9792.2.a.n 1
969.h even 2 1 2601.2.a.g 1
1292.c even 2 1 4624.2.a.d 1
1615.g odd 2 1 7225.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.a.a 1 19.b odd 2 1
153.2.a.c 1 57.d even 2 1
272.2.a.b 1 76.d even 2 1
289.2.a.a 1 323.c odd 2 1
289.2.b.a 2 323.g odd 4 2
289.2.c.a 4 323.l odd 8 4
289.2.d.d 8 323.p even 16 8
425.2.a.d 1 95.d odd 2 1
425.2.b.b 2 95.g even 4 2
833.2.a.a 1 133.c even 2 1
833.2.e.a 2 133.o even 6 2
833.2.e.b 2 133.r odd 6 2
1088.2.a.h 1 152.b even 2 1
1088.2.a.i 1 152.g odd 2 1
2057.2.a.e 1 209.d even 2 1
2448.2.a.o 1 228.b odd 2 1
2601.2.a.g 1 969.h even 2 1
2873.2.a.c 1 247.d odd 2 1
3825.2.a.d 1 285.b even 2 1
4624.2.a.d 1 1292.c even 2 1
6137.2.a.b 1 1.a even 1 1 trivial
6800.2.a.n 1 380.d even 2 1
7225.2.a.g 1 1615.g odd 2 1
7497.2.a.l 1 399.h odd 2 1
8993.2.a.a 1 437.b even 2 1
9792.2.a.i 1 456.l odd 2 1
9792.2.a.n 1 456.p even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6137))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T - 1 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T - 4 \) Copy content Toggle raw display
$73$ \( T + 6 \) Copy content Toggle raw display
$79$ \( T + 12 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T + 10 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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