Properties

Label 4014.2.a.v
Level 4014
Weight 2
Character orbit 4014.a
Self dual Yes
Analytic conductor 32.052
Analytic rank 0
Dimension 7
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( -1 - \beta_{4} ) q^{5} -\beta_{3} q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + ( -1 - \beta_{4} ) q^{5} -\beta_{3} q^{7} - q^{8} + ( 1 + \beta_{4} ) q^{10} + ( \beta_{2} - \beta_{6} ) q^{11} + ( 1 + \beta_{2} - \beta_{5} ) q^{13} + \beta_{3} q^{14} + q^{16} + ( -3 - \beta_{3} - \beta_{5} ) q^{17} + ( \beta_{1} - \beta_{6} ) q^{19} + ( -1 - \beta_{4} ) q^{20} + ( -\beta_{2} + \beta_{6} ) q^{22} + ( -1 + \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} ) q^{23} + ( 3 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{25} + ( -1 - \beta_{2} + \beta_{5} ) q^{26} -\beta_{3} q^{28} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} ) q^{29} + ( 1 - 2 \beta_{4} - \beta_{5} ) q^{31} - q^{32} + ( 3 + \beta_{3} + \beta_{5} ) q^{34} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{35} + ( 2 - 2 \beta_{1} - \beta_{4} - \beta_{5} ) q^{37} + ( -\beta_{1} + \beta_{6} ) q^{38} + ( 1 + \beta_{4} ) q^{40} + ( -4 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{41} + ( 1 + 4 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{43} + ( \beta_{2} - \beta_{6} ) q^{44} + ( 1 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} ) q^{46} + ( -2 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{47} + ( 3 + \beta_{1} - 3 \beta_{2} - 2 \beta_{4} + \beta_{6} ) q^{49} + ( -3 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{50} + ( 1 + \beta_{2} - \beta_{5} ) q^{52} + ( -\beta_{1} + 2 \beta_{5} + \beta_{6} ) q^{53} + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{55} + \beta_{3} q^{56} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{58} + ( 2 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{59} + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} ) q^{61} + ( -1 + 2 \beta_{4} + \beta_{5} ) q^{62} + q^{64} + ( -1 - 2 \beta_{2} + 3 \beta_{5} ) q^{65} + ( -1 + 2 \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{67} + ( -3 - \beta_{3} - \beta_{5} ) q^{68} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{70} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{71} + ( 3 - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{73} + ( -2 + 2 \beta_{1} + \beta_{4} + \beta_{5} ) q^{74} + ( \beta_{1} - \beta_{6} ) q^{76} + ( -\beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{77} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{79} + ( -1 - \beta_{4} ) q^{80} + ( 4 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{82} + ( 3 \beta_{1} - 3 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{83} + ( 3 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{85} + ( -1 - 4 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{86} + ( -\beta_{2} + \beta_{6} ) q^{88} + ( 2 \beta_{1} + \beta_{3} - 2 \beta_{4} + 2 \beta_{6} ) q^{89} + ( -1 - \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{91} + ( -1 + \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} ) q^{92} + ( 2 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{94} + ( 2 - 3 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 3 \beta_{6} ) q^{95} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{97} + ( -3 - \beta_{1} + 3 \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 7q^{2} + 7q^{4} - 6q^{5} + 3q^{7} - 7q^{8} + O(q^{10}) \) \( 7q - 7q^{2} + 7q^{4} - 6q^{5} + 3q^{7} - 7q^{8} + 6q^{10} + q^{11} + 8q^{13} - 3q^{14} + 7q^{16} - 16q^{17} + 2q^{19} - 6q^{20} - q^{22} - 8q^{23} + 19q^{25} - 8q^{26} + 3q^{28} - 4q^{29} + 11q^{31} - 7q^{32} + 16q^{34} + 17q^{37} - 2q^{38} + 6q^{40} - 18q^{41} - q^{43} + q^{44} + 8q^{46} - 5q^{47} + 24q^{49} - 19q^{50} + 8q^{52} - 6q^{53} + 21q^{55} - 3q^{56} + 4q^{58} + 7q^{59} + 24q^{61} - 11q^{62} + 7q^{64} - 11q^{65} - 4q^{67} - 16q^{68} + 7q^{71} + 28q^{73} - 17q^{74} + 2q^{76} + 2q^{77} - 6q^{80} + 18q^{82} + 2q^{83} + 4q^{85} + q^{86} - q^{88} - 5q^{89} + 2q^{91} - 8q^{92} + 5q^{94} + 14q^{95} + 23q^{97} - 24q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 18 x^{5} - 8 x^{4} + 51 x^{3} + 47 x^{2} - 2 x - 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{6} - \nu^{5} - 17 \nu^{4} + 9 \nu^{3} + 42 \nu^{2} + 5 \nu - 8 \)
\(\beta_{3}\)\(=\)\((\)\( -9 \nu^{6} + 7 \nu^{5} + 156 \nu^{4} - 49 \nu^{3} - 413 \nu^{2} - 104 \nu + 88 \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( -21 \nu^{6} + 16 \nu^{5} + 366 \nu^{4} - 112 \nu^{3} - 989 \nu^{2} - 221 \nu + 226 \)\()/6\)
\(\beta_{5}\)\(=\)\((\)\( 11 \nu^{6} - 9 \nu^{5} - 191 \nu^{4} + 68 \nu^{3} + 512 \nu^{2} + 104 \nu - 123 \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( -14 \nu^{6} + 10 \nu^{5} + 245 \nu^{4} - 63 \nu^{3} - 670 \nu^{2} - 181 \nu + 148 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{5} - \beta_{3} - 2 \beta_{2} + \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(2 \beta_{6} - \beta_{5} - 4 \beta_{4} - \beta_{2} + 13 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(17 \beta_{6} + 11 \beta_{5} - 4 \beta_{4} - 18 \beta_{3} - 29 \beta_{2} + 18 \beta_{1} + 59\)
\(\nu^{5}\)\(=\)\(40 \beta_{6} - 17 \beta_{5} - 70 \beta_{4} - 11 \beta_{3} - 29 \beta_{2} + 188 \beta_{1} + 57\)
\(\nu^{6}\)\(=\)\(269 \beta_{6} + 137 \beta_{5} - 102 \beta_{4} - 275 \beta_{3} - 428 \beta_{2} + 330 \beta_{1} + 831\)

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
2.01737
−3.57857
0.369925
−0.718922
4.01465
−1.31546
−0.788998
−1.00000 0 1.00000 −3.79108 0 −2.04653 −1.00000 0 3.79108
1.2 −1.00000 0 1.00000 −3.27125 0 3.11004 −1.00000 0 3.27125
1.3 −1.00000 0 1.00000 −2.69148 0 2.17419 −1.00000 0 2.69148
1.4 −1.00000 0 1.00000 −2.18886 0 −2.20076 −1.00000 0 2.18886
1.5 −1.00000 0 1.00000 0.603045 0 4.55050 −1.00000 0 −0.603045
1.6 −1.00000 0 1.00000 1.60399 0 −4.86544 −1.00000 0 −1.60399
1.7 −1.00000 0 1.00000 3.73564 0 2.27800 −1.00000 0 −3.73564
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):

\( T_{5}^{7} + 6 T_{5}^{6} - 9 T_{5}^{5} - 105 T_{5}^{4} - 91 T_{5}^{3} + 316 T_{5}^{2} + 304 T_{5} - 264 \)
\( T_{7}^{7} - 3 T_{7}^{6} - 32 T_{7}^{5} + 101 T_{7}^{4} + 224 T_{7}^{3} - 736 T_{7}^{2} - 448 T_{7} + 1536 \)
\( T_{11}^{7} - T_{11}^{6} - 48 T_{11}^{5} + 8 T_{11}^{4} + 624 T_{11}^{3} + 305 T_{11}^{2} - 2384 T_{11} - 2512 \)