Properties

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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( -1 - \beta_{5} + \beta_{6} ) q^{5} + ( -1 - \beta_{7} ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + ( -1 - \beta_{5} + \beta_{6} ) q^{5} + ( -1 - \beta_{7} ) q^{7} + q^{8} + ( -1 - \beta_{5} + \beta_{6} ) q^{10} + ( -1 + \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{11} + ( -\beta_{1} - \beta_{3} - \beta_{6} + \beta_{7} ) q^{13} + ( -1 - \beta_{7} ) q^{14} + q^{16} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{17} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} ) q^{19} + ( -1 - \beta_{5} + \beta_{6} ) q^{20} + ( -1 + \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{22} + ( -2 + \beta_{2} - 2 \beta_{4} - \beta_{6} ) q^{23} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{25} + ( -\beta_{1} - \beta_{3} - \beta_{6} + \beta_{7} ) q^{26} + ( -1 - \beta_{7} ) q^{28} + ( -3 - \beta_{1} + 2 \beta_{2} + \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{29} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{31} + q^{32} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{34} + ( -\beta_{2} + 3 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{35} + ( -2 - \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{37} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} ) q^{38} + ( -1 - \beta_{5} + \beta_{6} ) q^{40} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} ) q^{41} + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{43} + ( -1 + \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{44} + ( -2 + \beta_{2} - 2 \beta_{4} - \beta_{6} ) q^{46} + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{47} + ( -2 \beta_{1} - \beta_{5} + \beta_{6} ) q^{49} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{50} + ( -\beta_{1} - \beta_{3} - \beta_{6} + \beta_{7} ) q^{52} + ( -2 + 2 \beta_{1} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{53} + ( -3 + 2 \beta_{1} + \beta_{3} - 2 \beta_{6} - 2 \beta_{7} ) q^{55} + ( -1 - \beta_{7} ) q^{56} + ( -3 - \beta_{1} + 2 \beta_{2} + \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{58} + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{59} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{7} ) q^{61} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{62} + q^{64} + ( -3 + \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{65} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{67} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{68} + ( -\beta_{2} + 3 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{70} + ( -5 - \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{71} + ( 3 + \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{5} + 2 \beta_{7} ) q^{73} + ( -2 - \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{74} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} ) q^{76} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{77} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{79} + ( -1 - \beta_{5} + \beta_{6} ) q^{80} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} ) q^{82} + ( -6 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{83} + ( -3 - \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{85} + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{86} + ( -1 + \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{88} + ( -8 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{89} + ( -6 + 5 \beta_{1} + \beta_{2} + \beta_{6} ) q^{91} + ( -2 + \beta_{2} - 2 \beta_{4} - \beta_{6} ) q^{92} + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{94} + ( -5 - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{6} ) q^{95} + ( 2 - \beta_{1} + \beta_{2} + \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{97} + ( -2 \beta_{1} - \beta_{5} + \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{2} + 8q^{4} - 6q^{5} - 6q^{7} + 8q^{8} + O(q^{10}) \) \( 8q + 8q^{2} + 8q^{4} - 6q^{5} - 6q^{7} + 8q^{8} - 6q^{10} - 11q^{11} - q^{13} - 6q^{14} + 8q^{16} - 16q^{17} - q^{19} - 6q^{20} - 11q^{22} - 14q^{23} + 10q^{25} - q^{26} - 6q^{28} - 21q^{29} - 6q^{31} + 8q^{32} - 16q^{34} - 8q^{35} - 14q^{37} - q^{38} - 6q^{40} - 16q^{41} - 29q^{43} - 11q^{44} - 14q^{46} - 9q^{47} - 2q^{49} + 10q^{50} - q^{52} - 11q^{53} - 22q^{55} - 6q^{56} - 21q^{58} - 21q^{59} + 3q^{61} - 6q^{62} + 8q^{64} - 24q^{65} - 20q^{67} - 16q^{68} - 8q^{70} - 32q^{71} + 13q^{73} - 14q^{74} - q^{76} - 4q^{77} + 21q^{79} - 6q^{80} - 16q^{82} - 28q^{83} - 14q^{85} - 29q^{86} - 11q^{88} - 54q^{89} - 36q^{91} - 14q^{92} - 9q^{94} - 30q^{95} + 10q^{97} - 2q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 14 x^{6} + 28 x^{5} + 43 x^{4} - 90 x^{3} - 23 x^{2} + 82 x - 28\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 13 \nu^{7} - 22 \nu^{6} - 144 \nu^{5} + 260 \nu^{4} + 57 \nu^{3} - 466 \nu^{2} + 841 \nu + 116 \)\()/194\)
\(\beta_{3}\)\(=\)\((\)\( -8 \nu^{7} + 21 \nu^{6} + 111 \nu^{5} - 257 \nu^{4} - 341 \nu^{3} + 533 \nu^{2} + 251 \nu - 243 \)\()/97\)
\(\beta_{4}\)\(=\)\((\)\( 39 \nu^{7} - 66 \nu^{6} - 626 \nu^{5} + 974 \nu^{4} + 2693 \nu^{3} - 3532 \nu^{2} - 3297 \nu + 3258 \)\()/194\)
\(\beta_{5}\)\(=\)\((\)\( 85 \nu^{7} - 114 \nu^{6} - 1240 \nu^{5} + 1506 \nu^{4} + 4387 \nu^{3} - 4196 \nu^{2} - 4425 \nu + 3370 \)\()/194\)
\(\beta_{6}\)\(=\)\((\)\( 89 \nu^{7} - 76 \nu^{6} - 1344 \nu^{5} + 1004 \nu^{4} + 5091 \nu^{3} - 2862 \nu^{2} - 5181 \nu + 2764 \)\()/194\)
\(\beta_{7}\)\(=\)\((\)\( 49 \nu^{7} - 68 \nu^{6} - 692 \nu^{5} + 883 \nu^{4} + 2222 \nu^{3} - 2234 \nu^{2} - 1889 \nu + 1355 \)\()/97\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} - \beta_{5} - \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 8 \beta_{1}\)
\(\nu^{4}\)\(=\)\(11 \beta_{7} - \beta_{6} - 10 \beta_{5} + \beta_{4} + 3 \beta_{3} - 10 \beta_{2} + 12 \beta_{1} + 31\)
\(\nu^{5}\)\(=\)\(-14 \beta_{6} + 27 \beta_{5} - 13 \beta_{4} + 16 \beta_{3} - 22 \beta_{2} + 75 \beta_{1} + 2\)
\(\nu^{6}\)\(=\)\(110 \beta_{7} - 12 \beta_{6} - 96 \beta_{5} + 11 \beta_{4} + 45 \beta_{3} - 97 \beta_{2} + 123 \beta_{1} + 288\)
\(\nu^{7}\)\(=\)\(2 \beta_{7} - 151 \beta_{6} + 292 \beta_{5} - 141 \beta_{4} + 189 \beta_{3} - 220 \beta_{2} + 735 \beta_{1} + 24\)

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
0.541762
−1.60467
0.769489
2.16537
−1.34121
3.17253
−3.14687
1.44360
1.00000 0 1.00000 −3.88234 0 −1.74188 1.00000 0 −3.88234
1.2 1.00000 0 1.00000 −3.83272 0 −2.71599 1.00000 0 −3.83272
1.3 1.00000 0 1.00000 −2.40529 0 2.01389 1.00000 0 −2.40529
1.4 1.00000 0 1.00000 −1.09191 0 1.60541 1.00000 0 −1.09191
1.5 1.00000 0 1.00000 −0.350247 0 3.21437 1.00000 0 −0.350247
1.6 1.00000 0 1.00000 0.585733 0 −1.49689 1.00000 0 0.585733
1.7 1.00000 0 1.00000 2.03557 0 −4.04095 1.00000 0 2.03557
1.8 1.00000 0 1.00000 2.94121 0 −2.83796 1.00000 0 2.94121
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):

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