Properties

Label 4014.2.a.o
Level 4014
Weight 2
Character orbit 4014.a
Self dual Yes
Analytic conductor 32.052
Analytic rank 0
Dimension 3
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta_{1} q^{5} + ( \beta_{1} + \beta_{2} ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + \beta_{1} q^{5} + ( \beta_{1} + \beta_{2} ) q^{7} + q^{8} + \beta_{1} q^{10} + ( 3 - \beta_{1} + \beta_{2} ) q^{11} + ( -2 - \beta_{1} ) q^{13} + ( \beta_{1} + \beta_{2} ) q^{14} + q^{16} + ( 3 - \beta_{1} ) q^{17} + ( -3 - 3 \beta_{2} ) q^{19} + \beta_{1} q^{20} + ( 3 - \beta_{1} + \beta_{2} ) q^{22} + ( 3 - \beta_{1} + \beta_{2} ) q^{23} + ( -2 + \beta_{2} ) q^{25} + ( -2 - \beta_{1} ) q^{26} + ( \beta_{1} + \beta_{2} ) q^{28} + ( 4 + 3 \beta_{1} - \beta_{2} ) q^{29} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{31} + q^{32} + ( 3 - \beta_{1} ) q^{34} + ( 3 + \beta_{1} + 2 \beta_{2} ) q^{35} + ( 3 - \beta_{1} + \beta_{2} ) q^{37} + ( -3 - 3 \beta_{2} ) q^{38} + \beta_{1} q^{40} + ( 2 - 3 \beta_{2} ) q^{41} + ( 4 + 3 \beta_{2} ) q^{43} + ( 3 - \beta_{1} + \beta_{2} ) q^{44} + ( 3 - \beta_{1} + \beta_{2} ) q^{46} + ( 4 - 3 \beta_{1} ) q^{47} + ( -1 + 3 \beta_{1} + 2 \beta_{2} ) q^{49} + ( -2 + \beta_{2} ) q^{50} + ( -2 - \beta_{1} ) q^{52} + ( 3 + 2 \beta_{1} + 3 \beta_{2} ) q^{53} + ( -3 + 4 \beta_{1} ) q^{55} + ( \beta_{1} + \beta_{2} ) q^{56} + ( 4 + 3 \beta_{1} - \beta_{2} ) q^{58} + ( 1 + 5 \beta_{1} - 4 \beta_{2} ) q^{59} + ( 2 - 3 \beta_{1} - 3 \beta_{2} ) q^{61} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{62} + q^{64} + ( -3 - 2 \beta_{1} - \beta_{2} ) q^{65} + ( -2 - 5 \beta_{1} + 2 \beta_{2} ) q^{67} + ( 3 - \beta_{1} ) q^{68} + ( 3 + \beta_{1} + 2 \beta_{2} ) q^{70} + ( 5 - 4 \beta_{1} - 2 \beta_{2} ) q^{71} + ( 9 - 3 \beta_{1} + 2 \beta_{2} ) q^{73} + ( 3 - \beta_{1} + \beta_{2} ) q^{74} + ( -3 - 3 \beta_{2} ) q^{76} + ( 4 \beta_{1} + \beta_{2} ) q^{77} + ( 7 - 5 \beta_{1} - 5 \beta_{2} ) q^{79} + \beta_{1} q^{80} + ( 2 - 3 \beta_{2} ) q^{82} + ( -1 + 4 \beta_{1} - 5 \beta_{2} ) q^{83} + ( -3 + 3 \beta_{1} - \beta_{2} ) q^{85} + ( 4 + 3 \beta_{2} ) q^{86} + ( 3 - \beta_{1} + \beta_{2} ) q^{88} + ( 2 + 3 \beta_{1} + 3 \beta_{2} ) q^{89} + ( -3 - 3 \beta_{1} - 4 \beta_{2} ) q^{91} + ( 3 - \beta_{1} + \beta_{2} ) q^{92} + ( 4 - 3 \beta_{1} ) q^{94} + ( -6 \beta_{1} - 3 \beta_{2} ) q^{95} + ( -9 - 5 \beta_{2} ) q^{97} + ( -1 + 3 \beta_{1} + 2 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} + q^{5} + q^{7} + 3q^{8} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} + q^{5} + q^{7} + 3q^{8} + q^{10} + 8q^{11} - 7q^{13} + q^{14} + 3q^{16} + 8q^{17} - 9q^{19} + q^{20} + 8q^{22} + 8q^{23} - 6q^{25} - 7q^{26} + q^{28} + 15q^{29} + q^{31} + 3q^{32} + 8q^{34} + 10q^{35} + 8q^{37} - 9q^{38} + q^{40} + 6q^{41} + 12q^{43} + 8q^{44} + 8q^{46} + 9q^{47} - 6q^{50} - 7q^{52} + 11q^{53} - 5q^{55} + q^{56} + 15q^{58} + 8q^{59} + 3q^{61} + q^{62} + 3q^{64} - 11q^{65} - 11q^{67} + 8q^{68} + 10q^{70} + 11q^{71} + 24q^{73} + 8q^{74} - 9q^{76} + 4q^{77} + 16q^{79} + q^{80} + 6q^{82} + q^{83} - 6q^{85} + 12q^{86} + 8q^{88} + 9q^{89} - 12q^{91} + 8q^{92} + 9q^{94} - 6q^{95} - 27q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)

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
−1.91223
0.713538
2.19869
1.00000 0 1.00000 −1.91223 0 −1.25561 1.00000 0 −1.91223
1.2 1.00000 0 1.00000 0.713538 0 −1.77733 1.00000 0 0.713538
1.3 1.00000 0 1.00000 2.19869 0 4.03293 1.00000 0 2.19869
\(n\): 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
\(2\) \(-1\)
\(3\) \(-1\)
\(223\) \(-1\)

Hecke kernels

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

\( T_{5}^{3} - T_{5}^{2} - 4 T_{5} + 3 \)
\( T_{7}^{3} - T_{7}^{2} - 10 T_{7} - 9 \)
\( T_{11}^{3} - 8 T_{11}^{2} + 13 T_{11} + 3 \)