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.

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

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

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

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

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