Properties

Label 5202.2.a.c
Level $5202$
Weight $2$
Character orbit 5202.a
Self dual yes
Analytic conductor $41.538$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5202 = 2 \cdot 3^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5202.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(41.5381791315\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 102)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - 2q^{5} - q^{8} + O(q^{10}) \) \( q - q^{2} + q^{4} - 2q^{5} - q^{8} + 2q^{10} - 4q^{11} - 2q^{13} + q^{16} + 4q^{19} - 2q^{20} + 4q^{22} - q^{25} + 2q^{26} - 10q^{29} - 8q^{31} - q^{32} + 2q^{37} - 4q^{38} + 2q^{40} + 10q^{41} + 12q^{43} - 4q^{44} - 7q^{49} + q^{50} - 2q^{52} - 6q^{53} + 8q^{55} + 10q^{58} - 12q^{59} + 10q^{61} + 8q^{62} + q^{64} + 4q^{65} - 12q^{67} - 10q^{73} - 2q^{74} + 4q^{76} + 8q^{79} - 2q^{80} - 10q^{82} - 4q^{83} - 12q^{86} + 4q^{88} + 6q^{89} - 8q^{95} + 14q^{97} + 7q^{98} + O(q^{100}) \)

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 0 −1.00000 0 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(17\) \(1\)

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 5202.2.a.c 1
3.b odd 2 1 1734.2.a.j 1
17.b even 2 1 306.2.a.b 1
51.c odd 2 1 102.2.a.c 1
51.f odd 4 2 1734.2.b.b 2
51.g odd 8 4 1734.2.f.e 4
68.d odd 2 1 2448.2.a.p 1
85.c even 2 1 7650.2.a.ca 1
136.e odd 2 1 9792.2.a.l 1
136.h even 2 1 9792.2.a.k 1
204.h even 2 1 816.2.a.b 1
255.h odd 2 1 2550.2.a.c 1
255.o even 4 2 2550.2.d.m 2
357.c even 2 1 4998.2.a.be 1
408.b odd 2 1 3264.2.a.m 1
408.h even 2 1 3264.2.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.a.c 1 51.c odd 2 1
306.2.a.b 1 17.b even 2 1
816.2.a.b 1 204.h even 2 1
1734.2.a.j 1 3.b odd 2 1
1734.2.b.b 2 51.f odd 4 2
1734.2.f.e 4 51.g odd 8 4
2448.2.a.p 1 68.d odd 2 1
2550.2.a.c 1 255.h odd 2 1
2550.2.d.m 2 255.o even 4 2
3264.2.a.m 1 408.b odd 2 1
3264.2.a.bc 1 408.h even 2 1
4998.2.a.be 1 357.c even 2 1
5202.2.a.c 1 1.a even 1 1 trivial
7650.2.a.ca 1 85.c even 2 1
9792.2.a.k 1 136.h even 2 1
9792.2.a.l 1 136.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5202))\):

\( T_{5} + 2 \)
\( T_{7} \)
\( T_{23} \)
\( T_{47} \)