Properties

Label 4026.2.a.x
Level 4026
Weight 2
Character orbit 4026.a
Self dual yes
Analytic conductor 32.148
Analytic rank 1
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4026.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1477718538\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.46101901.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + ( -1 + \beta_{3} ) q^{5} - q^{6} + ( \beta_{2} - \beta_{5} ) q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} + ( -1 + \beta_{3} ) q^{5} - q^{6} + ( \beta_{2} - \beta_{5} ) q^{7} + q^{8} + q^{9} + ( -1 + \beta_{3} ) q^{10} - q^{11} - q^{12} + ( \beta_{1} - \beta_{2} + \beta_{5} ) q^{13} + ( \beta_{2} - \beta_{5} ) q^{14} + ( 1 - \beta_{3} ) q^{15} + q^{16} + ( -2 - \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{17} + q^{18} + ( -2 \beta_{4} - \beta_{5} ) q^{19} + ( -1 + \beta_{3} ) q^{20} + ( -\beta_{2} + \beta_{5} ) q^{21} - q^{22} + ( -3 \beta_{1} + \beta_{4} - \beta_{5} ) q^{23} - q^{24} + ( 1 - 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{25} + ( \beta_{1} - \beta_{2} + \beta_{5} ) q^{26} - q^{27} + ( \beta_{2} - \beta_{5} ) q^{28} + ( -1 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{29} + ( 1 - \beta_{3} ) q^{30} + ( -1 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{31} + q^{32} + q^{33} + ( -2 - \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{34} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{35} + q^{36} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{37} + ( -2 \beta_{4} - \beta_{5} ) q^{38} + ( -\beta_{1} + \beta_{2} - \beta_{5} ) q^{39} + ( -1 + \beta_{3} ) q^{40} + ( -4 + \beta_{1} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{41} + ( -\beta_{2} + \beta_{5} ) q^{42} + ( 1 + 3 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} ) q^{43} - q^{44} + ( -1 + \beta_{3} ) q^{45} + ( -3 \beta_{1} + \beta_{4} - \beta_{5} ) q^{46} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{47} - q^{48} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{49} + ( 1 - 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{50} + ( 2 + \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{51} + ( \beta_{1} - \beta_{2} + \beta_{5} ) q^{52} + ( -3 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{53} - q^{54} + ( 1 - \beta_{3} ) q^{55} + ( \beta_{2} - \beta_{5} ) q^{56} + ( 2 \beta_{4} + \beta_{5} ) q^{57} + ( -1 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{58} + ( -4 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} ) q^{59} + ( 1 - \beta_{3} ) q^{60} + q^{61} + ( -1 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{62} + ( \beta_{2} - \beta_{5} ) q^{63} + q^{64} + ( 1 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{65} + q^{66} + ( 4 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{67} + ( -2 - \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{68} + ( 3 \beta_{1} - \beta_{4} + \beta_{5} ) q^{69} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{70} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{71} + q^{72} + ( -4 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{73} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{74} + ( -1 + 3 \beta_{3} + \beta_{4} - \beta_{5} ) q^{75} + ( -2 \beta_{4} - \beta_{5} ) q^{76} + ( -\beta_{2} + \beta_{5} ) q^{77} + ( -\beta_{1} + \beta_{2} - \beta_{5} ) q^{78} + ( -6 + \beta_{1} + 5 \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{79} + ( -1 + \beta_{3} ) q^{80} + q^{81} + ( -4 + \beta_{1} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{82} + ( -4 + 7 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{83} + ( -\beta_{2} + \beta_{5} ) q^{84} + ( -3 - 4 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} ) q^{85} + ( 1 + 3 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} ) q^{86} + ( 1 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{87} - q^{88} + ( -3 + 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} ) q^{89} + ( -1 + \beta_{3} ) q^{90} + ( -4 - \beta_{1} + 2 \beta_{4} ) q^{91} + ( -3 \beta_{1} + \beta_{4} - \beta_{5} ) q^{92} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{93} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{94} + ( -5 + 5 \beta_{1} - 6 \beta_{2} + \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{95} - q^{96} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{5} ) q^{97} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{98} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} - 6q^{3} + 6q^{4} - 6q^{5} - 6q^{6} + q^{7} + 6q^{8} + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{2} - 6q^{3} + 6q^{4} - 6q^{5} - 6q^{6} + q^{7} + 6q^{8} + 6q^{9} - 6q^{10} - 6q^{11} - 6q^{12} + 2q^{13} + q^{14} + 6q^{15} + 6q^{16} - 13q^{17} + 6q^{18} + q^{19} - 6q^{20} - q^{21} - 6q^{22} - 11q^{23} - 6q^{24} + 8q^{25} + 2q^{26} - 6q^{27} + q^{28} - 14q^{29} + 6q^{30} - 5q^{31} + 6q^{32} + 6q^{33} - 13q^{34} - 13q^{35} + 6q^{36} - 6q^{37} + q^{38} - 2q^{39} - 6q^{40} - 25q^{41} - q^{42} + 19q^{43} - 6q^{44} - 6q^{45} - 11q^{46} - 10q^{47} - 6q^{48} - 5q^{49} + 8q^{50} + 13q^{51} + 2q^{52} - 17q^{53} - 6q^{54} + 6q^{55} + q^{56} - q^{57} - 14q^{58} - 14q^{59} + 6q^{60} + 6q^{61} - 5q^{62} + q^{63} + 6q^{64} + 6q^{65} + 6q^{66} + 12q^{67} - 13q^{68} + 11q^{69} - 13q^{70} + 6q^{71} + 6q^{72} - 29q^{73} - 6q^{74} - 8q^{75} + q^{76} - q^{77} - 2q^{78} - 24q^{79} - 6q^{80} + 6q^{81} - 25q^{82} - 9q^{83} - q^{84} - 22q^{85} + 19q^{86} + 14q^{87} - 6q^{88} - 4q^{89} - 6q^{90} - 29q^{91} - 11q^{92} + 5q^{93} - 10q^{94} - 27q^{95} - 6q^{96} - 5q^{97} - 5q^{98} - 6q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( -\nu^{5} + 3 \nu^{4} + 5 \nu^{3} - 12 \nu^{2} - 5 \nu + 4 \)
\(\beta_{3}\)\(=\)\( -\nu^{5} + 4 \nu^{4} + 2 \nu^{3} - 16 \nu^{2} + 4 \nu + 6 \)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{5} + 10 \nu^{4} + 11 \nu^{3} - 37 \nu^{2} - 3 \nu + 8 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{5} + 10 \nu^{4} + 11 \nu^{3} - 39 \nu^{2} - \nu + 14 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{4} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(-3 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 6 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(-13 \beta_{5} + 7 \beta_{4} + 4 \beta_{3} + 5 \beta_{2} + 13 \beta_{1} + 19\)
\(\nu^{5}\)\(=\)\(-42 \beta_{5} + 14 \beta_{4} + 17 \beta_{3} + 24 \beta_{2} + 52 \beta_{1} + 40\)

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
−0.654438
1.90193
3.29072
−1.86911
−0.371781
0.702675
1.00000 −1.00000 1.00000 −4.17718 −1.00000 2.87039 1.00000 1.00000 −4.17718
1.2 1.00000 −1.00000 1.00000 −4.05628 −1.00000 −1.59539 1.00000 1.00000 −4.05628
1.3 1.00000 −1.00000 1.00000 −0.656263 −1.00000 4.00423 1.00000 1.00000 −0.656263
1.4 1.00000 −1.00000 1.00000 0.199215 −1.00000 −0.938542 1.00000 1.00000 0.199215
1.5 1.00000 −1.00000 1.00000 1.28209 −1.00000 −0.306395 1.00000 1.00000 1.28209
1.6 1.00000 −1.00000 1.00000 1.40842 −1.00000 −3.03430 1.00000 1.00000 1.40842
\(n\): 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 4026.2.a.x 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.x 6 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(11\) \(1\)
\(61\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5}^{6} + 6 T_{5}^{5} - T_{5}^{4} - 33 T_{5}^{3} + 17 T_{5}^{2} + 18 T_{5} - 4 \)
\( T_{7}^{6} - T_{7}^{5} - 18 T_{7}^{4} + 76 T_{7}^{2} + 75 T_{7} + 16 \)
\( T_{13}^{6} - 2 T_{13}^{5} - 18 T_{13}^{4} + 22 T_{13}^{3} + 69 T_{13}^{2} + 35 T_{13} + 2 \)