Properties

Label 4014.2.a.s
Level 4014
Weight 2
Character orbit 4014.a
Self dual Yes
Analytic conductor 32.052
Analytic rank 1
Dimension 6
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4014.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0519513713\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.232773917.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( -1 + \beta_{1} ) q^{5} + ( 1 + \beta_{2} ) q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + ( -1 + \beta_{1} ) q^{5} + ( 1 + \beta_{2} ) q^{7} - q^{8} + ( 1 - \beta_{1} ) q^{10} + ( -\beta_{1} - \beta_{5} ) q^{11} + ( 1 - \beta_{2} - \beta_{5} ) q^{13} + ( -1 - \beta_{2} ) q^{14} + q^{16} + ( -2 - \beta_{3} + \beta_{4} + \beta_{5} ) q^{17} + ( -1 + \beta_{1} - \beta_{3} + \beta_{5} ) q^{19} + ( -1 + \beta_{1} ) q^{20} + ( \beta_{1} + \beta_{5} ) q^{22} + ( -\beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{23} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{25} + ( -1 + \beta_{2} + \beta_{5} ) q^{26} + ( 1 + \beta_{2} ) q^{28} + ( -1 + \beta_{3} + \beta_{5} ) q^{29} + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{31} - q^{32} + ( 2 + \beta_{3} - \beta_{4} - \beta_{5} ) q^{34} + ( -2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{35} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{37} + ( 1 - \beta_{1} + \beta_{3} - \beta_{5} ) q^{38} + ( 1 - \beta_{1} ) q^{40} + ( -2 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} ) q^{41} + ( -2 + \beta_{1} - \beta_{3} - \beta_{4} ) q^{43} + ( -\beta_{1} - \beta_{5} ) q^{44} + ( \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{46} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{47} + ( -\beta_{1} + \beta_{4} - \beta_{5} ) q^{49} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{50} + ( 1 - \beta_{2} - \beta_{5} ) q^{52} + ( -1 - \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{53} + ( -3 + \beta_{1} - \beta_{3} + 2 \beta_{5} ) q^{55} + ( -1 - \beta_{2} ) q^{56} + ( 1 - \beta_{3} - \beta_{5} ) q^{58} + ( -3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} ) q^{59} + ( -1 + 2 \beta_{2} - \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{61} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{62} + q^{64} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{65} + ( -1 - \beta_{1} + 2 \beta_{4} ) q^{67} + ( -2 - \beta_{3} + \beta_{4} + \beta_{5} ) q^{68} + ( 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{70} + ( -1 + \beta_{1} - 4 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{71} + ( -2 - 3 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{73} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{74} + ( -1 + \beta_{1} - \beta_{3} + \beta_{5} ) q^{76} + ( 2 - 3 \beta_{1} + \beta_{2} - \beta_{4} - 3 \beta_{5} ) q^{77} + ( -2 + 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{79} + ( -1 + \beta_{1} ) q^{80} + ( 2 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{82} + ( 1 - 3 \beta_{2} + 3 \beta_{3} ) q^{83} + ( -3 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{85} + ( 2 - \beta_{1} + \beta_{3} + \beta_{4} ) q^{86} + ( \beta_{1} + \beta_{5} ) q^{88} + ( 1 + 2 \beta_{1} + \beta_{2} - 4 \beta_{4} - 2 \beta_{5} ) q^{89} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{91} + ( -\beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{92} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} ) q^{94} + ( 2 - 3 \beta_{1} - \beta_{2} + \beta_{4} - 3 \beta_{5} ) q^{95} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{97} + ( \beta_{1} - \beta_{4} + \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{2} + 6q^{4} - 6q^{5} + 5q^{7} - 6q^{8} + O(q^{10}) \) \( 6q - 6q^{2} + 6q^{4} - 6q^{5} + 5q^{7} - 6q^{8} + 6q^{10} - q^{11} + 6q^{13} - 5q^{14} + 6q^{16} - 10q^{17} - 4q^{19} - 6q^{20} + q^{22} - 2q^{23} - 6q^{26} + 5q^{28} - 6q^{29} - 5q^{31} - 6q^{32} + 10q^{34} + 2q^{35} + q^{37} + 4q^{38} + 6q^{40} - 12q^{41} - 11q^{43} - q^{44} + 2q^{46} - 5q^{47} - q^{49} + 6q^{52} - 4q^{53} - 15q^{55} - 5q^{56} + 6q^{58} - q^{59} - 8q^{61} + 5q^{62} + 6q^{64} - 5q^{65} - 6q^{67} - 10q^{68} - 2q^{70} - q^{71} - 10q^{73} - q^{74} - 4q^{76} + 8q^{77} - 10q^{79} - 6q^{80} + 12q^{82} + 6q^{83} - 2q^{85} + 11q^{86} + q^{88} + 3q^{89} - 16q^{91} - 2q^{92} + 5q^{94} + 10q^{95} + 3q^{97} + q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 12 x^{4} - x^{3} + 33 x^{2} + 5 x - 22\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{5} - \nu^{4} - 10 \nu^{3} + 9 \nu^{2} + 15 \nu - 11 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - \nu^{4} - 10 \nu^{3} + 10 \nu^{2} + 15 \nu - 15 \)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{5} - 3 \nu^{4} - 31 \nu^{3} + 28 \nu^{2} + 51 \nu - 36 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -5 \nu^{5} + 7 \nu^{4} + 51 \nu^{3} - 66 \nu^{2} - 81 \nu + 86 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(-2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 6 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{5} + \beta_{4} + 10 \beta_{3} - 9 \beta_{2} + 26\)
\(\nu^{5}\)\(=\)\(\beta_{5} - 19 \beta_{4} + 11 \beta_{3} + 21 \beta_{2} + 45 \beta_{1} + 11\)

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
−2.85387
−1.36937
−1.29309
0.862793
1.72012
2.93341
−1.00000 0 1.00000 −3.85387 0 −2.71491 −1.00000 0 3.85387
1.2 −1.00000 0 1.00000 −2.36937 0 3.68274 −1.00000 0 2.36937
1.3 −1.00000 0 1.00000 −2.29309 0 0.862607 −1.00000 0 2.29309
1.4 −1.00000 0 1.00000 −0.137207 0 3.14283 −1.00000 0 0.137207
1.5 −1.00000 0 1.00000 0.720124 0 −2.15975 −1.00000 0 −0.720124
1.6 −1.00000 0 1.00000 1.93341 0 2.18648 −1.00000 0 −1.93341
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(223\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):

\( T_{5}^{6} + 6 T_{5}^{5} + 3 T_{5}^{4} - 29 T_{5}^{3} - 27 T_{5}^{2} + 26 T_{5} + 4 \)
\( T_{7}^{6} - 5 T_{7}^{5} - 8 T_{7}^{4} + 61 T_{7}^{3} - 12 T_{7}^{2} - 176 T_{7} + 128 \)
\( T_{11}^{6} + T_{11}^{5} - 26 T_{11}^{4} - 80 T_{11}^{3} - 56 T_{11}^{2} + 13 T_{11} + 4 \)