Properties

Label 4014.2.a.z
Level 4014
Weight 2
Character orbit 4014.a
Self dual Yes
Analytic conductor 32.052
Analytic rank 0
Dimension 8
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( -1 - \beta_{3} ) q^{5} + ( -1 + \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + ( -1 - \beta_{3} ) q^{5} + ( -1 + \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{7} + q^{8} + ( -1 - \beta_{3} ) q^{10} + ( -2 + 2 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{11} + ( 1 + \beta_{1} + \beta_{2} + \beta_{7} ) q^{13} + ( -1 + \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{14} + q^{16} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{17} + ( -2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{19} + ( -1 - \beta_{3} ) q^{20} + ( -2 + 2 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{22} + ( -2 \beta_{2} - 2 \beta_{5} + 2 \beta_{7} ) q^{23} + ( 4 + \beta_{4} + \beta_{5} ) q^{25} + ( 1 + \beta_{1} + \beta_{2} + \beta_{7} ) q^{26} + ( -1 + \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{28} -2 \beta_{1} q^{29} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{31} + q^{32} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{34} + ( 7 - 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 4 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{35} + ( 2 \beta_{1} - 2 \beta_{5} ) q^{37} + ( -2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{38} + ( -1 - \beta_{3} ) q^{40} + ( -2 + 2 \beta_{2} ) q^{41} + ( 4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{43} + ( -2 + 2 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{44} + ( -2 \beta_{2} - 2 \beta_{5} + 2 \beta_{7} ) q^{46} + ( 1 - 2 \beta_{1} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{47} + ( 5 - 2 \beta_{2} ) q^{49} + ( 4 + \beta_{4} + \beta_{5} ) q^{50} + ( 1 + \beta_{1} + \beta_{2} + \beta_{7} ) q^{52} + ( -2 \beta_{2} - 2 \beta_{5} ) q^{53} + ( 4 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{55} + ( -1 + \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{56} -2 \beta_{1} q^{58} + ( 1 + \beta_{3} + 2 \beta_{7} ) q^{59} + ( 6 - 2 \beta_{1} + 3 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{61} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{62} + q^{64} + ( 3 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{65} + ( 1 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{67} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{68} + ( 7 - 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 4 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{70} + ( 2 + 4 \beta_{2} + 2 \beta_{3} + 4 \beta_{5} ) q^{71} + ( -3 + 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{73} + ( 2 \beta_{1} - 2 \beta_{5} ) q^{74} + ( -2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{76} + ( 4 + 5 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{7} ) q^{77} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} ) q^{79} + ( -1 - \beta_{3} ) q^{80} + ( -2 + 2 \beta_{2} ) q^{82} + ( 6 + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{83} + ( -2 - 3 \beta_{2} + 2 \beta_{6} + \beta_{7} ) q^{85} + ( 4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{86} + ( -2 + 2 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{88} + ( 1 - 4 \beta_{1} - 4 \beta_{2} - \beta_{4} - \beta_{5} ) q^{89} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{91} + ( -2 \beta_{2} - 2 \beta_{5} + 2 \beta_{7} ) q^{92} + ( 1 - 2 \beta_{1} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{94} + ( 10 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 4 \beta_{5} - 2 \beta_{6} ) q^{95} + ( 2 + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} ) q^{97} + ( 5 - 2 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{2} + 8q^{4} - 4q^{5} + 4q^{7} + 8q^{8} + O(q^{10}) \) \( 8q + 8q^{2} + 8q^{4} - 4q^{5} + 4q^{7} + 8q^{8} - 4q^{10} - 2q^{11} + 10q^{13} + 4q^{14} + 8q^{16} - 8q^{17} + 16q^{19} - 4q^{20} - 2q^{22} + 4q^{23} + 24q^{25} + 10q^{26} + 4q^{28} - 8q^{29} + 12q^{31} + 8q^{32} - 8q^{34} + 24q^{35} + 16q^{37} + 16q^{38} - 4q^{40} - 16q^{41} + 12q^{43} - 2q^{44} + 4q^{46} + 4q^{47} + 40q^{49} + 24q^{50} + 10q^{52} + 8q^{53} + 8q^{55} + 4q^{56} - 8q^{58} + 26q^{61} + 12q^{62} + 8q^{64} + 24q^{65} - 10q^{67} - 8q^{68} + 24q^{70} - 8q^{71} + 8q^{73} + 16q^{74} + 16q^{76} + 56q^{77} + 16q^{79} - 4q^{80} - 16q^{82} + 44q^{83} - 18q^{85} + 12q^{86} - 2q^{88} + 4q^{91} + 4q^{92} + 4q^{94} + 48q^{95} + 12q^{97} + 40q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} - 12 x^{6} + 46 x^{5} + 54 x^{4} - 148 x^{3} - 98 x^{2} + 126 x + 69\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{6} - 11 \nu^{5} + 4 \nu^{4} + 64 \nu^{3} - 97 \nu^{2} - 63 \nu + 108 \)\()/33\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - 7 \nu^{6} + 2 \nu^{5} + 62 \nu^{4} - 47 \nu^{3} - 165 \nu^{2} + 27 \nu + 84 \)\()/33\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} - 4 \nu^{6} - 9 \nu^{5} + 35 \nu^{4} + 27 \nu^{3} - 52 \nu^{2} - 29 \nu - 40 \)\()/11\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 4 \nu^{6} - 9 \nu^{5} + 35 \nu^{4} + 27 \nu^{3} - 63 \nu^{2} - 7 \nu + 4 \)\()/11\)
\(\beta_{6}\)\(=\)\((\)\( 4 \nu^{7} - 15 \nu^{6} - 47 \nu^{5} + 142 \nu^{4} + 228 \nu^{3} - 284 \nu^{2} - 351 \nu + 48 \)\()/33\)
\(\beta_{7}\)\(=\)\((\)\( 2 \nu^{7} - 12 \nu^{6} - 7 \nu^{5} + 128 \nu^{4} - 63 \nu^{3} - 394 \nu^{2} + 222 \nu + 309 \)\()/33\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{4} + 2 \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{7} - \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 9 \beta_{1} + 5\)
\(\nu^{4}\)\(=\)\(-2 \beta_{7} - \beta_{6} - 9 \beta_{5} + 10 \beta_{4} + 5 \beta_{3} + 4 \beta_{2} + 27 \beta_{1} + 34\)
\(\nu^{5}\)\(=\)\(-16 \beta_{7} - 6 \beta_{6} - 11 \beta_{5} + 19 \beta_{4} + 32 \beta_{3} + 19 \beta_{2} + 97 \beta_{1} + 88\)
\(\nu^{6}\)\(=\)\(-52 \beta_{7} - 31 \beta_{6} - 59 \beta_{5} + 101 \beta_{4} + 102 \beta_{3} + 81 \beta_{2} + 320 \beta_{1} + 396\)
\(\nu^{7}\)\(=\)\(-255 \beta_{7} - 143 \beta_{6} - 45 \beta_{5} + 261 \beta_{4} + 467 \beta_{3} + 328 \beta_{2} + 1098 \beta_{1} + 1299\)

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
−2.00550
3.63363
3.34519
−0.517925
−1.07351
1.23047
−2.36314
1.75078
1.00000 0 1.00000 −3.95370 0 −4.39233 1.00000 0 −3.95370
1.2 1.00000 0 1.00000 −3.28836 0 −3.44951 1.00000 0 −3.28836
1.3 1.00000 0 1.00000 −2.47654 0 4.22714 1.00000 0 −2.47654
1.4 1.00000 0 1.00000 −2.10687 0 2.35547 1.00000 0 −2.10687
1.5 1.00000 0 1.00000 −0.701327 0 3.38691 1.00000 0 −0.701327
1.6 1.00000 0 1.00000 1.80043 0 −3.56810 1.00000 0 1.80043
1.7 1.00000 0 1.00000 2.80312 0 1.62712 1.00000 0 2.80312
1.8 1.00000 0 1.00000 3.92325 0 3.81331 1.00000 0 3.92325
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):

\(T_{5}^{8} + \cdots\)
\(T_{7}^{8} - \cdots\)
\(T_{11}^{8} + \cdots\)