Properties

Label 4026.2.a.z
Level 4026
Weight 2
Character orbit 4026.a
Self dual yes
Analytic conductor 32.148
Analytic rank 0
Dimension 7
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: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 18 x^{5} - 10 x^{4} + 91 x^{3} + 90 x^{2} - 66 x - 56\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - \nu^{2} - 8 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - \nu^{5} - 13 \nu^{4} + 7 \nu^{3} + 40 \nu^{2} - 2 \nu - 8 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} + 3 \nu^{5} + 13 \nu^{4} - 25 \nu^{3} - 52 \nu^{2} + 22 \nu + 16 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} - \nu^{5} - 17 \nu^{4} + 7 \nu^{3} + 84 \nu^{2} + 6 \nu - 72 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} - 2 \nu^{5} - 13 \nu^{4} + 18 \nu^{3} + 48 \nu^{2} - 28 \nu - 28 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} - 2 \nu^{5} - 15 \nu^{4} + 18 \nu^{3} + 66 \nu^{2} - 16 \nu - 40 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} - \beta_{4} + \beta_{3} - \beta_{1} - 1\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{3} - \beta_{2} - \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(4 \beta_{6} + \beta_{5} - 4 \beta_{4} + 5 \beta_{3} - \beta_{2} - 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{6} + 11 \beta_{5} - 3 \beta_{4} + 12 \beta_{3} - 9 \beta_{2} - 12 \beta_{1} + 27\)
\(\nu^{5}\)\(=\)\(31 \beta_{6} + 15 \beta_{5} - 31 \beta_{4} + 50 \beta_{3} - 11 \beta_{2} - 28 \beta_{1} + 25\)
\(\nu^{6}\)\(=\)\(17 \beta_{6} + 111 \beta_{5} - 43 \beta_{4} + 132 \beta_{3} - 73 \beta_{2} - 124 \beta_{1} + 223\)

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
3.27906
0.868643
−0.647395
−1.70776
−2.63895
−2.20695
3.05336
−1.00000 −1.00000 1.00000 −2.92143 1.00000 −1.10194 −1.00000 1.00000 2.92143
1.2 −1.00000 −1.00000 1.00000 −2.19576 1.00000 −0.720188 −1.00000 1.00000 2.19576
1.3 −1.00000 −1.00000 1.00000 −0.757998 1.00000 −3.37265 −1.00000 1.00000 0.757998
1.4 −1.00000 −1.00000 1.00000 −0.745947 1.00000 −1.28209 −1.00000 1.00000 0.745947
1.5 −1.00000 −1.00000 1.00000 2.19404 1.00000 1.89767 −1.00000 1.00000 −2.19404
1.6 −1.00000 −1.00000 1.00000 3.06294 1.00000 4.73255 −1.00000 1.00000 −3.06294
1.7 −1.00000 −1.00000 1.00000 3.36415 1.00000 −4.15335 −1.00000 1.00000 −3.36415
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.z 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.z 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5}^{7} - 2 T_{5}^{6} - 18 T_{5}^{5} + 25 T_{5}^{4} + 104 T_{5}^{3} - 57 T_{5}^{2} - 195 T_{5} - 82 \)
\( T_{7}^{7} + 4 T_{7}^{6} - 21 T_{7}^{5} - 105 T_{7}^{4} - 36 T_{7}^{3} + 284 T_{7}^{2} + 368 T_{7} + 128 \)
\( T_{13}^{7} + 7 T_{13}^{6} - 27 T_{13}^{5} - 242 T_{13}^{4} - 192 T_{13}^{3} + 565 T_{13}^{2} - 52 T_{13} - 44 \)