Properties

Label 4014.2.a.r
Level 4014
Weight 2
Character orbit 4014.a
Self dual Yes
Analytic conductor 32.052
Analytic rank 1
Dimension 5
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: \(5\)
Coefficient field: 5.5.356173.1
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)  \(=\)  \( q\) \(+ q^{2}\) \(+ q^{4}\) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{5} \) \( + ( - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{7} \) \(+ q^{8}\) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \(+ q^{4}\) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{5} \) \( + ( - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{7} \) \(+ q^{8}\) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{10} \) \( + ( -3 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{11} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} \) \( + ( - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{14} \) \(+ q^{16}\) \( + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{17} \) \( + ( -1 + \beta_{2} + \beta_{3} ) q^{19} \) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{20} \) \( + ( -3 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{22} \) \( + ( -2 - \beta_{1} + 2 \beta_{4} ) q^{23} \) \( + ( \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{25} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{26} \) \( + ( - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{28} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{29} \) \( + ( -1 + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{31} \) \(+ q^{32}\) \( + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{34} \) \( + ( -3 - \beta_{1} + 3 \beta_{3} + \beta_{4} ) q^{35} \) \( + ( - \beta_{1} + 4 \beta_{3} - 2 \beta_{4} ) q^{37} \) \( + ( -1 + \beta_{2} + \beta_{3} ) q^{38} \) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{40} \) \( + ( -1 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{41} \) \( + ( 4 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{43} \) \( + ( -3 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{44} \) \( + ( -2 - \beta_{1} + 2 \beta_{4} ) q^{46} \) \( + ( -4 - \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{47} \) \( + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{49} \) \( + ( \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{50} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{52} \) \( + ( -5 + \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{53} \) \( + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{55} \) \( + ( - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{56} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{58} \) \( + ( -5 + \beta_{1} + 4 \beta_{2} + 3 \beta_{4} ) q^{59} \) \( + ( -5 + 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{61} \) \( + ( -1 + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{62} \) \(+ q^{64}\) \( + ( -1 + 3 \beta_{2} - 2 \beta_{3} ) q^{65} \) \( + ( -3 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 4 \beta_{4} ) q^{67} \) \( + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{68} \) \( + ( -3 - \beta_{1} + 3 \beta_{3} + \beta_{4} ) q^{70} \) \( + ( -2 - \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{71} \) \( + ( 3 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{73} \) \( + ( - \beta_{1} + 4 \beta_{3} - 2 \beta_{4} ) q^{74} \) \( + ( -1 + \beta_{2} + \beta_{3} ) q^{76} \) \( + ( -1 - 2 \beta_{2} + 3 \beta_{3} - 5 \beta_{4} ) q^{77} \) \( + ( -3 - \beta_{1} + \beta_{2} - 4 \beta_{3} + 5 \beta_{4} ) q^{79} \) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{80} \) \( + ( -1 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{82} \) \( + ( -3 + \beta_{2} - 6 \beta_{3} + 2 \beta_{4} ) q^{83} \) \( + ( -2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{4} ) q^{85} \) \( + ( 4 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{86} \) \( + ( -3 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{88} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{89} \) \( + ( -3 + 3 \beta_{1} - \beta_{3} + \beta_{4} ) q^{91} \) \( + ( -2 - \beta_{1} + 2 \beta_{4} ) q^{92} \) \( + ( -4 - \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{94} \) \( + ( -1 - 4 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{95} \) \( + ( -3 + 4 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{97} \) \( + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(5q \) \(\mathstrut +\mathstrut 5q^{2} \) \(\mathstrut +\mathstrut 5q^{4} \) \(\mathstrut -\mathstrut 5q^{5} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut 5q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(5q \) \(\mathstrut +\mathstrut 5q^{2} \) \(\mathstrut +\mathstrut 5q^{4} \) \(\mathstrut -\mathstrut 5q^{5} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut 5q^{8} \) \(\mathstrut -\mathstrut 5q^{10} \) \(\mathstrut -\mathstrut 9q^{11} \) \(\mathstrut -\mathstrut q^{14} \) \(\mathstrut +\mathstrut 5q^{16} \) \(\mathstrut -\mathstrut 6q^{17} \) \(\mathstrut -\mathstrut 4q^{19} \) \(\mathstrut -\mathstrut 5q^{20} \) \(\mathstrut -\mathstrut 9q^{22} \) \(\mathstrut -\mathstrut 16q^{23} \) \(\mathstrut +\mathstrut 8q^{25} \) \(\mathstrut -\mathstrut q^{28} \) \(\mathstrut -\mathstrut 8q^{29} \) \(\mathstrut -\mathstrut q^{31} \) \(\mathstrut +\mathstrut 5q^{32} \) \(\mathstrut -\mathstrut 6q^{34} \) \(\mathstrut -\mathstrut 22q^{35} \) \(\mathstrut -\mathstrut 2q^{37} \) \(\mathstrut -\mathstrut 4q^{38} \) \(\mathstrut -\mathstrut 5q^{40} \) \(\mathstrut -\mathstrut 4q^{41} \) \(\mathstrut +\mathstrut 3q^{43} \) \(\mathstrut -\mathstrut 9q^{44} \) \(\mathstrut -\mathstrut 16q^{46} \) \(\mathstrut -\mathstrut 18q^{47} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut 8q^{50} \) \(\mathstrut -\mathstrut 26q^{53} \) \(\mathstrut +\mathstrut q^{55} \) \(\mathstrut -\mathstrut q^{56} \) \(\mathstrut -\mathstrut 8q^{58} \) \(\mathstrut -\mathstrut 21q^{59} \) \(\mathstrut -\mathstrut 20q^{61} \) \(\mathstrut -\mathstrut q^{62} \) \(\mathstrut +\mathstrut 5q^{64} \) \(\mathstrut +\mathstrut 3q^{65} \) \(\mathstrut -\mathstrut 5q^{67} \) \(\mathstrut -\mathstrut 6q^{68} \) \(\mathstrut -\mathstrut 22q^{70} \) \(\mathstrut -\mathstrut 17q^{71} \) \(\mathstrut +\mathstrut 5q^{73} \) \(\mathstrut -\mathstrut 2q^{74} \) \(\mathstrut -\mathstrut 4q^{76} \) \(\mathstrut -\mathstrut 2q^{77} \) \(\mathstrut -\mathstrut 21q^{79} \) \(\mathstrut -\mathstrut 5q^{80} \) \(\mathstrut -\mathstrut 4q^{82} \) \(\mathstrut -\mathstrut 11q^{83} \) \(\mathstrut -\mathstrut 12q^{85} \) \(\mathstrut +\mathstrut 3q^{86} \) \(\mathstrut -\mathstrut 9q^{88} \) \(\mathstrut +\mathstrut 5q^{89} \) \(\mathstrut -\mathstrut 10q^{91} \) \(\mathstrut -\mathstrut 16q^{92} \) \(\mathstrut -\mathstrut 18q^{94} \) \(\mathstrut -\mathstrut 10q^{95} \) \(\mathstrut -\mathstrut 11q^{97} \) \(\mathstrut +\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5}\mathstrut -\mathstrut \) \(2\) \(x^{4}\mathstrut -\mathstrut \) \(7\) \(x^{3}\mathstrut +\mathstrut \) \(9\) \(x^{2}\mathstrut +\mathstrut \) \(14\) \(x\mathstrut -\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 1 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} - 3 \nu^{2} + 5 \nu - 2 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 7 \nu + 4 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(-\)\(5\) \(\beta_{4}\mathstrut +\mathstrut \) \(7\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(15\)

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
−1.38363
2.75496
−1.85688
0.253142
2.23241
1.00000 0 1.00000 −4.35484 0 3.05676 1.00000 0 −4.35484
1.2 1.00000 0 1.00000 −1.54501 0 −1.28984 1.00000 0 −1.54501
1.3 1.00000 0 1.00000 −1.43386 0 −1.87103 1.00000 0 −1.43386
1.4 1.00000 0 1.00000 −0.686428 0 2.87549 1.00000 0 −0.686428
1.5 1.00000 0 1.00000 3.02013 0 −3.77138 1.00000 0 3.02013
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):

\(T_{5}^{5} \) \(\mathstrut +\mathstrut 5 T_{5}^{4} \) \(\mathstrut -\mathstrut 4 T_{5}^{3} \) \(\mathstrut -\mathstrut 41 T_{5}^{2} \) \(\mathstrut -\mathstrut 54 T_{5} \) \(\mathstrut -\mathstrut 20 \)
\(T_{7}^{5} \) \(\mathstrut +\mathstrut T_{7}^{4} \) \(\mathstrut -\mathstrut 18 T_{7}^{3} \) \(\mathstrut -\mathstrut 15 T_{7}^{2} \) \(\mathstrut +\mathstrut 72 T_{7} \) \(\mathstrut +\mathstrut 80 \)
\(T_{11}^{5} \) \(\mathstrut +\mathstrut 9 T_{11}^{4} \) \(\mathstrut +\mathstrut 6 T_{11}^{3} \) \(\mathstrut -\mathstrut 82 T_{11}^{2} \) \(\mathstrut -\mathstrut 38 T_{11} \) \(\mathstrut +\mathstrut 67 \)