Properties

Label 4014.2.a.w
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta_{3} q^{5} + ( 1 - \beta_{2} + \beta_{6} ) q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + \beta_{3} q^{5} + ( 1 - \beta_{2} + \beta_{6} ) q^{7} - q^{8} -\beta_{3} q^{10} + ( -2 + \beta_{1} + \beta_{4} + \beta_{6} ) q^{11} + ( \beta_{3} + \beta_{4} - \beta_{6} ) q^{13} + ( -1 + \beta_{2} - \beta_{6} ) q^{14} + q^{16} + ( \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{17} -2 \beta_{1} q^{19} + \beta_{3} q^{20} + ( 2 - \beta_{1} - \beta_{4} - \beta_{6} ) q^{22} + ( -2 - \beta_{1} - \beta_{2} + \beta_{5} + \beta_{6} ) q^{23} + ( 1 - \beta_{1} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{25} + ( -\beta_{3} - \beta_{4} + \beta_{6} ) q^{26} + ( 1 - \beta_{2} + \beta_{6} ) q^{28} + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{29} + ( -\beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{31} - q^{32} + ( -\beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{34} + ( 2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{35} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{37} + 2 \beta_{1} q^{38} -\beta_{3} q^{40} + ( 6 + \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{41} + ( 2 + 2 \beta_{1} + 2 \beta_{4} ) q^{43} + ( -2 + \beta_{1} + \beta_{4} + \beta_{6} ) q^{44} + ( 2 + \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} ) q^{46} + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} - 4 \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{47} + ( 1 + 2 \beta_{1} - 2 \beta_{2} ) q^{49} + ( -1 + \beta_{1} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{50} + ( \beta_{3} + \beta_{4} - \beta_{6} ) q^{52} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{53} + ( -1 + \beta_{1} - \beta_{2} - 3 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} ) q^{55} + ( -1 + \beta_{2} - \beta_{6} ) q^{56} + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{58} + ( \beta_{1} - \beta_{2} - \beta_{3} - 4 \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{59} + ( 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{61} + ( \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{62} + q^{64} + ( 5 - 3 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} ) q^{65} + ( 4 - 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{67} + ( \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{68} + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{70} + ( -4 \beta_{2} - 2 \beta_{3} + 4 \beta_{6} ) q^{71} + ( -5 + \beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{73} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{74} -2 \beta_{1} q^{76} + ( 2 + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{77} + ( 4 - 2 \beta_{1} + 2 \beta_{5} + 2 \beta_{6} ) q^{79} + \beta_{3} q^{80} + ( -6 - \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{82} + ( 2 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{83} + ( 2 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} ) q^{85} + ( -2 - 2 \beta_{1} - 2 \beta_{4} ) q^{86} + ( 2 - \beta_{1} - \beta_{4} - \beta_{6} ) q^{88} + ( 4 - 3 \beta_{2} - \beta_{3} - 4 \beta_{4} + 2 \beta_{6} ) q^{89} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{5} + \beta_{6} ) q^{91} + ( -2 - \beta_{1} - \beta_{2} + \beta_{5} + \beta_{6} ) q^{92} + ( -2 - \beta_{1} + 2 \beta_{2} + \beta_{3} + 4 \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{94} + ( -2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{95} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{97} + ( -1 - 2 \beta_{1} + 2 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 7q^{2} + 7q^{4} - 2q^{5} + 6q^{7} - 7q^{8} + O(q^{10}) \) \( 7q - 7q^{2} + 7q^{4} - 2q^{5} + 6q^{7} - 7q^{8} + 2q^{10} - 9q^{11} - 2q^{13} - 6q^{14} + 7q^{16} + 7q^{17} - 2q^{19} - 2q^{20} + 9q^{22} - 15q^{23} + 13q^{25} + 2q^{26} + 6q^{28} - 9q^{29} - 2q^{31} - 7q^{32} - 7q^{34} + 4q^{35} + 5q^{37} + 2q^{38} + 2q^{40} + 33q^{41} + 20q^{43} - 9q^{44} + 15q^{46} + 2q^{47} + 3q^{49} - 13q^{50} - 2q^{52} + 13q^{53} - 18q^{55} - 6q^{56} + 9q^{58} - 9q^{59} + 8q^{61} + 2q^{62} + 7q^{64} + 44q^{65} + 29q^{67} + 7q^{68} - 4q^{70} - 37q^{73} - 5q^{74} - 2q^{76} + 18q^{77} + 32q^{79} - 2q^{80} - 33q^{82} + 6q^{83} - 4q^{85} - 20q^{86} + 9q^{88} + 17q^{89} - 4q^{91} - 15q^{92} - 2q^{94} + 12q^{95} + 12q^{97} - 3q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - x^{6} - 14 x^{5} + 12 x^{4} + 50 x^{3} - 36 x^{2} - 38 x + 18\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -6 \nu^{6} + 37 \nu^{5} + 92 \nu^{4} - 388 \nu^{3} - 287 \nu^{2} + 703 \nu - 98 \)\()/239\)
\(\beta_{3}\)\(=\)\((\)\( 6 \nu^{6} - 37 \nu^{5} - 92 \nu^{4} + 388 \nu^{3} + 526 \nu^{2} - 703 \nu - 858 \)\()/239\)
\(\beta_{4}\)\(=\)\((\)\( 11 \nu^{6} - 28 \nu^{5} - 89 \nu^{4} + 313 \nu^{3} - 151 \nu^{2} - 771 \nu + 817 \)\()/239\)
\(\beta_{5}\)\(=\)\((\)\( -32 \nu^{6} + 38 \nu^{5} + 411 \nu^{4} - 476 \nu^{3} - 1212 \nu^{2} + 1439 \nu + 513 \)\()/239\)
\(\beta_{6}\)\(=\)\((\)\( -47 \nu^{6} + 11 \nu^{5} + 641 \nu^{4} - 12 \nu^{3} - 2049 \nu^{2} - 269 \nu + 985 \)\()/239\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} + 7 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{5} + 4 \beta_{4} + 10 \beta_{3} + 12 \beta_{2} + \beta_{1} + 25\)
\(\nu^{5}\)\(=\)\(10 \beta_{6} - 22 \beta_{5} - 12 \beta_{4} - 13 \beta_{3} + 4 \beta_{2} + 55 \beta_{1} + 2\)
\(\nu^{6}\)\(=\)\(-3 \beta_{6} + 9 \beta_{5} + 52 \beta_{4} + 90 \beta_{3} + 121 \beta_{2} + 19 \beta_{1} + 188\)

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
0.409721
1.26499
−2.01250
3.08762
−0.922829
2.08198
−2.90898
−1.00000 0 1.00000 −4.32652 0 2.79833 −1.00000 0 4.32652
1.2 −1.00000 0 1.00000 −1.88711 0 −3.39932 −1.00000 0 1.88711
1.3 −1.00000 0 1.00000 −1.52490 0 0.908231 −1.00000 0 1.52490
1.4 −1.00000 0 1.00000 −0.580657 0 −1.39541 −1.00000 0 0.580657
1.5 −1.00000 0 1.00000 −0.437167 0 3.40247 −1.00000 0 0.437167
1.6 −1.00000 0 1.00000 3.23274 0 4.23794 −1.00000 0 −3.23274
1.7 −1.00000 0 1.00000 3.52361 0 −0.552246 −1.00000 0 −3.52361
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):

\( T_{5}^{7} + 2 T_{5}^{6} - 22 T_{5}^{5} - 42 T_{5}^{4} + 92 T_{5}^{3} + 256 T_{5}^{2} + 174 T_{5} + 36 \)
\( T_{7}^{7} - 6 T_{7}^{6} - 8 T_{7}^{5} + 88 T_{7}^{4} - 48 T_{7}^{3} - 224 T_{7}^{2} + 80 T_{7} + 96 \)
\( T_{11}^{7} + 9 T_{11}^{6} - 14 T_{11}^{5} - 254 T_{11}^{4} - 58 T_{11}^{3} + 2028 T_{11}^{2} + 394 T_{11} - 3494 \)