Properties

Label 4016.2.a.c
Level 4016
Weight 2
Character orbit 4016.a
Self dual yes
Analytic conductor 32.068
Analytic rank 1
Dimension 4
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: \(4\)
Coefficient field: 4.4.725.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 251)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{5} + ( 1 + \beta_{1} - \beta_{2} ) q^{7} + ( -1 + \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{5} + ( 1 + \beta_{1} - \beta_{2} ) q^{7} + ( -1 + \beta_{2} - \beta_{3} ) q^{9} + ( 1 + \beta_{1} - \beta_{3} ) q^{11} + ( -3 - \beta_{2} + \beta_{3} ) q^{13} + q^{15} + ( 1 - 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{17} + ( 2 - 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{19} + ( -3 + \beta_{1} + 2 \beta_{3} ) q^{21} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{23} + ( -2 - 3 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{25} + ( 2 - \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{27} + ( -1 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{29} + ( 2 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{31} + ( -1 + \beta_{2} + \beta_{3} ) q^{33} + ( -2 + \beta_{1} + \beta_{2} ) q^{35} + ( -5 + 3 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -2 + \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{39} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{41} + ( 3 - 7 \beta_{1} - \beta_{2} + \beta_{3} ) q^{43} + ( 3 - 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{45} + ( -4 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{47} + ( -1 + \beta_{1} - 3 \beta_{3} ) q^{49} + ( 1 - 2 \beta_{1} - 2 \beta_{3} ) q^{51} + ( 3 + \beta_{1} - 4 \beta_{3} ) q^{53} + ( -\beta_{1} + 2 \beta_{3} ) q^{55} + ( 5 - 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{57} + ( -5 + 2 \beta_{2} + 5 \beta_{3} ) q^{59} + ( -6 + \beta_{1} + \beta_{3} ) q^{61} + ( -4 + \beta_{3} ) q^{63} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{65} + ( -8 + 5 \beta_{1} + 5 \beta_{2} ) q^{67} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{69} + ( 2 - 6 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{71} + ( 3 - 4 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{73} + ( -1 + \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{75} + ( 3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{77} + ( 6 + 7 \beta_{1} + 3 \beta_{2} - 8 \beta_{3} ) q^{79} + ( -2 - 2 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} ) q^{81} + ( -1 + 9 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{83} + ( -3 + 3 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} ) q^{85} + ( -6 - \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{87} + ( -2 + \beta_{1} + 6 \beta_{3} ) q^{89} + ( -4 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{91} + ( -8 + \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{93} + ( -2 + 4 \beta_{1} + 3 \beta_{2} - 7 \beta_{3} ) q^{95} + ( 5 + 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{97} + ( -1 - 2 \beta_{1} + 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} - 3q^{5} + 3q^{7} - 4q^{9} + O(q^{10}) \) \( 4q + 2q^{3} - 3q^{5} + 3q^{7} - 4q^{9} + 3q^{11} - 12q^{13} + 4q^{15} + q^{17} + 9q^{19} - 7q^{21} - 4q^{23} - 7q^{25} - q^{27} - 12q^{29} + 2q^{31} - 5q^{35} - 13q^{37} - 13q^{39} + q^{41} + 5q^{43} + 11q^{45} - 12q^{47} - 9q^{49} - 2q^{51} + 5q^{53} + 3q^{55} + 16q^{57} - 6q^{59} - 21q^{61} - 14q^{63} + q^{65} - 17q^{67} - 13q^{69} + 10q^{71} - 2q^{73} - 13q^{75} + 8q^{77} + 21q^{79} - 8q^{81} + q^{83} - 17q^{85} - 31q^{87} + 5q^{89} + 2q^{91} - 23q^{93} - 12q^{95} + 6q^{97} - 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 3 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 2 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 3 \beta_{1}\)

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
0.737640
−0.477260
2.09529
−1.35567
0 −1.19353 0 −0.837853 0 2.93117 0 −1.57549 0
1.2 0 −0.294963 0 −3.39026 0 0.817703 0 −2.91300 0
1.3 0 1.29496 0 0.772223 0 1.80033 0 −1.32307 0
1.4 0 2.19353 0 0.455887 0 −2.54920 0 1.81156 0
\(n\): e.g. 2-40 or 990-1000
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.c 4
4.b odd 2 1 251.2.a.a 4
12.b even 2 1 2259.2.a.f 4
20.d odd 2 1 6275.2.a.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
251.2.a.a 4 4.b odd 2 1
2259.2.a.f 4 12.b even 2 1
4016.2.a.c 4 1.a even 1 1 trivial
6275.2.a.c 4 20.d odd 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 \( T_{3}^{4} - 2 T_{3}^{3} - 2 T_{3}^{2} + 3 T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\).