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

Related objects

Downloads

Learn more about

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: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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.

Atkin-Lehner signs

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

Hecke kernels

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

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