Properties

Label 4014.2.a.j
Level 4014
Weight 2
Character orbit 4014.a
Self dual Yes
Analytic conductor 32.052
Analytic rank 1
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

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

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
2.30278
−1.30278
−1.00000 0 1.00000 0 0 −4.60555 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 2.60555 −1.00000 0 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
\(2\) \(1\)
\(3\) \(1\)
\(223\) \(1\)

Hecke kernels

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

\(T_{5} \)
\(T_{7}^{2} \) \(\mathstrut +\mathstrut 2 T_{7} \) \(\mathstrut -\mathstrut 12 \)
\(T_{11}^{2} \) \(\mathstrut +\mathstrut 9 T_{11} \) \(\mathstrut +\mathstrut 17 \)