Properties

Label 4026.2.a.l
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: \(\Q(\sqrt{17}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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
2.56155
−1.56155
1.00000 −1.00000 1.00000 −3.56155 −1.00000 3.12311 1.00000 1.00000 −3.56155
1.2 1.00000 −1.00000 1.00000 0.561553 −1.00000 −5.12311 1.00000 1.00000 0.561553
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\(T_{5}^{2} \) \(\mathstrut +\mathstrut 3 T_{5} \) \(\mathstrut -\mathstrut 2 \)
\(T_{7}^{2} \) \(\mathstrut +\mathstrut 2 T_{7} \) \(\mathstrut -\mathstrut 16 \)
\(T_{13}^{2} \) \(\mathstrut -\mathstrut 5 T_{13} \) \(\mathstrut +\mathstrut 2 \)