Properties

Label 4014.2.a.m
Level 4014
Weight 2
Character orbit 4014.a
Self dual Yes
Analytic conductor 32.052
Analytic rank 1
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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
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.

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

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

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

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
−1.53209
−0.347296
1.87939
−1.00000 0 1.00000 −0.532089 0 −2.87939 −1.00000 0 0.532089
1.2 −1.00000 0 1.00000 0.652704 0 0.532089 −1.00000 0 −0.652704
1.3 −1.00000 0 1.00000 2.87939 0 −0.652704 −1.00000 0 −2.87939
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):

\( T_{5}^{3} - 3 T_{5}^{2} + 1 \)
\( T_{7}^{3} + 3 T_{7}^{2} - 1 \)
\( T_{11}^{3} - 9 T_{11} + 9 \)