Properties

Label 4026.2.a.v
Level 4026
Weight 2
Character orbit 4026.a
Self dual Yes
Analytic conductor 32.148
Analytic rank 0
Dimension 5
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: \(5\)
Coefficient field: 5.5.11492689.1
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5}\mathstrut -\mathstrut \) \(2\) \(x^{4}\mathstrut -\mathstrut \) \(9\) \(x^{3}\mathstrut +\mathstrut \) \(13\) \(x^{2}\mathstrut +\mathstrut \) \(18\) \(x\mathstrut -\mathstrut \) \(8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} - 9 \nu^{2} - \nu + 10 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{4} + 2 \nu^{3} + 7 \nu^{2} - 11 \nu - 6 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(26\)

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
−2.42642
−1.26875
0.371979
2.34840
2.97479
1.00000 −1.00000 1.00000 −1.42642 −1.00000 −4.31394 1.00000 1.00000 −1.42642
1.2 1.00000 −1.00000 1.00000 −0.268751 −1.00000 1.12152 1.00000 1.00000 −0.268751
1.3 1.00000 −1.00000 1.00000 1.37198 −1.00000 4.23361 1.00000 1.00000 1.37198
1.4 1.00000 −1.00000 1.00000 3.34840 −1.00000 0.833399 1.00000 1.00000 3.34840
1.5 1.00000 −1.00000 1.00000 3.97479 −1.00000 −1.87458 1.00000 1.00000 3.97479
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

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

Hecke kernels

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

\(T_{5}^{5} \) \(\mathstrut -\mathstrut 7 T_{5}^{4} \) \(\mathstrut +\mathstrut 9 T_{5}^{3} \) \(\mathstrut +\mathstrut 18 T_{5}^{2} \) \(\mathstrut -\mathstrut 22 T_{5} \) \(\mathstrut -\mathstrut 7 \)
\(T_{7}^{5} \) \(\mathstrut -\mathstrut 21 T_{7}^{3} \) \(\mathstrut +\mathstrut 3 T_{7}^{2} \) \(\mathstrut +\mathstrut 50 T_{7} \) \(\mathstrut -\mathstrut 32 \)
\(T_{13}^{5} \) \(\mathstrut +\mathstrut 2 T_{13}^{4} \) \(\mathstrut -\mathstrut 31 T_{13}^{3} \) \(\mathstrut -\mathstrut 23 T_{13}^{2} \) \(\mathstrut +\mathstrut 243 T_{13} \) \(\mathstrut -\mathstrut 158 \)