Properties

Label 6171.2.a.e
Level 6171
Weight 2
Character orbit 6171.a
Self dual yes
Analytic conductor 49.276
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6171 = 3 \cdot 11^{2} \cdot 17 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6171.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.2756830873\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - 2q^{4} + 3q^{5} + 4q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} - 2q^{4} + 3q^{5} + 4q^{7} + q^{9} - 2q^{12} + q^{13} + 3q^{15} + 4q^{16} + q^{17} + q^{19} - 6q^{20} + 4q^{21} + 9q^{23} + 4q^{25} + q^{27} - 8q^{28} - 6q^{29} + 2q^{31} + 12q^{35} - 2q^{36} - 4q^{37} + q^{39} + 3q^{41} + 7q^{43} + 3q^{45} - 6q^{47} + 4q^{48} + 9q^{49} + q^{51} - 2q^{52} - 6q^{53} + q^{57} + 6q^{59} - 6q^{60} - 8q^{61} + 4q^{63} - 8q^{64} + 3q^{65} - 4q^{67} - 2q^{68} + 9q^{69} + 12q^{71} - 2q^{73} + 4q^{75} - 2q^{76} + 10q^{79} + 12q^{80} + q^{81} + 6q^{83} - 8q^{84} + 3q^{85} - 6q^{87} + 4q^{91} - 18q^{92} + 2q^{93} + 3q^{95} - 16q^{97} + O(q^{100}) \)

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
0 1.00000 −2.00000 3.00000 0 4.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 6171.2.a.e 1
11.b odd 2 1 51.2.a.a 1
33.d even 2 1 153.2.a.b 1
44.c even 2 1 816.2.a.g 1
55.d odd 2 1 1275.2.a.d 1
55.e even 4 2 1275.2.b.b 2
77.b even 2 1 2499.2.a.d 1
88.b odd 2 1 3264.2.a.a 1
88.g even 2 1 3264.2.a.r 1
132.d odd 2 1 2448.2.a.c 1
143.d odd 2 1 8619.2.a.g 1
165.d even 2 1 3825.2.a.i 1
187.b odd 2 1 867.2.a.c 1
187.f odd 4 2 867.2.d.a 2
187.i odd 8 4 867.2.e.e 4
187.m even 16 8 867.2.h.c 8
231.h odd 2 1 7497.2.a.j 1
264.m even 2 1 9792.2.a.by 1
264.p odd 2 1 9792.2.a.cd 1
561.h even 2 1 2601.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.a.a 1 11.b odd 2 1
153.2.a.b 1 33.d even 2 1
816.2.a.g 1 44.c even 2 1
867.2.a.c 1 187.b odd 2 1
867.2.d.a 2 187.f odd 4 2
867.2.e.e 4 187.i odd 8 4
867.2.h.c 8 187.m even 16 8
1275.2.a.d 1 55.d odd 2 1
1275.2.b.b 2 55.e even 4 2
2448.2.a.c 1 132.d odd 2 1
2499.2.a.d 1 77.b even 2 1
2601.2.a.f 1 561.h even 2 1
3264.2.a.a 1 88.b odd 2 1
3264.2.a.r 1 88.g even 2 1
3825.2.a.i 1 165.d even 2 1
6171.2.a.e 1 1.a even 1 1 trivial
7497.2.a.j 1 231.h odd 2 1
8619.2.a.g 1 143.d odd 2 1
9792.2.a.by 1 264.m even 2 1
9792.2.a.cd 1 264.p odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(-1\)
\(17\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6171))\):

\( T_{2} \)
\( T_{5} - 3 \)
\( T_{7} - 4 \)