Properties

Label 4026.2.a.k
Level 4026
Weight 2
Character orbit 4026.a
Self dual Yes
Analytic conductor 32.148
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.41421
1.41421
−1.00000 1.00000 1.00000 −4.41421 −1.00000 2.00000 −1.00000 1.00000 4.41421
1.2 −1.00000 1.00000 1.00000 −1.58579 −1.00000 2.00000 −1.00000 1.00000 1.58579
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\(T_{5}^{2} \) \(\mathstrut +\mathstrut 6 T_{5} \) \(\mathstrut +\mathstrut 7 \)
\(T_{7} \) \(\mathstrut -\mathstrut 2 \)
\(T_{13} \) \(\mathstrut -\mathstrut 3 \)