Properties

Label 4016.2.a.f
Level 4016
Weight 2
Character orbit 4016.a
Self dual yes
Analytic conductor 32.068
Analytic rank 1
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0679214517\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.60853001.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 502)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + ( -1 + \beta_{1} - \beta_{3} ) q^{5} + ( -1 + \beta_{5} ) q^{7} + ( 3 - \beta_{1} + \beta_{5} ) q^{9} +O(q^{10})\) \( q + \beta_{4} q^{3} + ( -1 + \beta_{1} - \beta_{3} ) q^{5} + ( -1 + \beta_{5} ) q^{7} + ( 3 - \beta_{1} + \beta_{5} ) q^{9} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{11} + ( -1 + \beta_{1} + \beta_{3} ) q^{13} + ( -1 + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{15} + ( 2 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{17} + ( -\beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{19} + ( -1 - \beta_{1} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{21} + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{23} + ( 1 - 2 \beta_{1} + \beta_{4} ) q^{25} + ( -1 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{27} + ( 3 \beta_{2} - \beta_{4} - \beta_{5} ) q^{29} + ( -2 \beta_{1} + \beta_{2} - \beta_{5} ) q^{31} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{33} + ( -\beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{35} + ( -1 - \beta_{1} + 2 \beta_{3} ) q^{37} + ( 1 + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{39} + ( 1 - 3 \beta_{2} + \beta_{3} - \beta_{5} ) q^{41} + ( -1 + 3 \beta_{1} - 2 \beta_{4} ) q^{43} + ( -7 + 4 \beta_{1} - 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{45} + ( -2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{47} + ( -1 + 2 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{49} + ( 5 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{51} + ( 1 - \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{5} ) q^{53} + ( 6 - 2 \beta_{1} ) q^{55} + ( -4 + 2 \beta_{5} ) q^{57} + ( -1 + \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{5} ) q^{59} + ( -11 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{61} + ( 2 + 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} ) q^{63} + ( 2 - 2 \beta_{1} + 4 \beta_{3} - \beta_{4} ) q^{65} + ( 3 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 4 \beta_{5} ) q^{67} + ( -6 + 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{69} + ( -2 - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} ) q^{71} + ( 1 - \beta_{1} + 4 \beta_{2} - 3 \beta_{4} + 2 \beta_{5} ) q^{73} + ( 6 - \beta_{1} - 4 \beta_{2} + 3 \beta_{4} + \beta_{5} ) q^{75} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} + 3 \beta_{5} ) q^{77} + ( -5 - \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{79} + ( -2 \beta_{2} - \beta_{3} + 3 \beta_{5} ) q^{81} + ( -4 + 4 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{83} + ( -6 + 3 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{85} + ( -2 + 8 \beta_{1} - 6 \beta_{3} + 4 \beta_{5} ) q^{87} + ( 5 - \beta_{1} + 4 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{89} + ( 2 - \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + \beta_{4} - \beta_{5} ) q^{91} + ( 2 + 3 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{93} + ( -2 - 2 \beta_{2} + 4 \beta_{3} ) q^{95} + ( 2 - 2 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} ) q^{97} + ( -5 + 5 \beta_{1} - \beta_{2} - 5 \beta_{3} + 4 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{3} - q^{5} - 6q^{7} + 15q^{9} + O(q^{10}) \) \( 6q - q^{3} - q^{5} - 6q^{7} + 15q^{9} - q^{11} - 5q^{13} - 2q^{15} + 8q^{17} - 3q^{19} - 14q^{21} - 18q^{23} - q^{25} - 16q^{27} + q^{29} - 6q^{31} - 16q^{33} - 6q^{35} - 13q^{37} + 6q^{39} + 4q^{41} + 5q^{43} - 23q^{45} - 8q^{47} + 8q^{49} + 16q^{51} - 3q^{53} + 30q^{55} - 24q^{57} + 5q^{59} - 61q^{61} + 27q^{63} - q^{65} + 13q^{67} - 21q^{69} - 22q^{71} + 6q^{73} + 30q^{75} + 4q^{77} - 28q^{79} + 2q^{81} - 14q^{83} - 16q^{85} + 24q^{87} + 18q^{89} + 16q^{91} + 27q^{93} - 20q^{95} + 16q^{97} - 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 8 x^{4} + 15 x^{3} + 20 x^{2} - 12 x - 5\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + 4 \nu^{4} + 4 \nu^{3} - 15 \nu^{2} - 9 \nu + 1 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - 4 \nu^{4} - 4 \nu^{3} + 19 \nu^{2} + 5 \nu - 17 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - 6 \nu^{4} + 6 \nu^{3} + 17 \nu^{2} - 27 \nu - 3 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} - 3 \nu^{4} - 7 \nu^{3} + 12 \nu^{2} + 15 \nu - 4 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} + \beta_{3} + 4 \beta_{2} + 7 \beta_{1} + 6\)
\(\nu^{4}\)\(=\)\(5 \beta_{5} + 3 \beta_{4} + 6 \beta_{3} + 19 \beta_{2} + 18 \beta_{1} + 33\)
\(\nu^{5}\)\(=\)\(24 \beta_{5} + 16 \beta_{4} + 13 \beta_{3} + 73 \beta_{2} + 76 \beta_{1} + 97\)

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
0.730357
−1.76567
2.15902
−0.303212
3.75626
−1.57676
0 −3.20334 0 1.15583 0 3.99175 0 7.26139 0
1.2 0 −2.70836 0 −2.61208 0 −1.43044 0 4.33524 0
1.3 0 −1.28113 0 0.633107 0 −3.19968 0 −1.35870 0
1.4 0 1.63228 0 2.87032 0 −4.63886 0 −0.335646 0
1.5 0 1.68934 0 0.420498 0 −0.389856 0 −0.146119 0
1.6 0 2.87121 0 −3.46767 0 −0.332919 0 5.24384 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4016.2.a.f 6
4.b odd 2 1 502.2.a.e 6
12.b even 2 1 4518.2.a.x 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
502.2.a.e 6 4.b odd 2 1
4016.2.a.f 6 1.a even 1 1 trivial
4518.2.a.x 6 12.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(251\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + T_{3}^{5} - 16 T_{3}^{4} - 9 T_{3}^{3} + 74 T_{3}^{2} + 8 T_{3} - 88 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\).