Properties

Label 4026.2.a.r
Level 4026
Weight 2
Character orbit 4026.a
Self dual Yes
Analytic conductor 32.148
Analytic rank 1
Dimension 4
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: \(4\)
Coefficient field: 4.4.7537.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 4 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\)

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
2.15976
−2.04717
1.37933
−0.491918
−1.00000 1.00000 1.00000 −1.66454 −1.00000 −2.43525 −1.00000 1.00000 1.66454
1.2 −1.00000 1.00000 1.00000 −1.19091 −1.00000 −0.609175 −1.00000 1.00000 1.19091
1.3 −1.00000 1.00000 1.00000 1.09744 −1.00000 1.89307 −1.00000 1.00000 −1.09744
1.4 −1.00000 1.00000 1.00000 2.75802 −1.00000 −2.84864 −1.00000 1.00000 −2.75802
\(n\): e.g. 2-40 or 990-1000
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\(T_{5}^{4} \) \(\mathstrut -\mathstrut T_{5}^{3} \) \(\mathstrut -\mathstrut 6 T_{5}^{2} \) \(\mathstrut +\mathstrut T_{5} \) \(\mathstrut +\mathstrut 6 \)
\(T_{7}^{4} \) \(\mathstrut +\mathstrut 4 T_{7}^{3} \) \(\mathstrut -\mathstrut T_{7}^{2} \) \(\mathstrut -\mathstrut 15 T_{7} \) \(\mathstrut -\mathstrut 8 \)
\(T_{13}^{4} \) \(\mathstrut +\mathstrut 3 T_{13}^{3} \) \(\mathstrut -\mathstrut 32 T_{13}^{2} \) \(\mathstrut -\mathstrut 155 T_{13} \) \(\mathstrut -\mathstrut 178 \)