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.

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

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

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):

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