Properties

Label 4026.2.a.p
Level 4026
Weight 2
Character orbit 4026.a
Self dual Yes
Analytic conductor 32.148
Analytic rank 1
Dimension 3
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: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
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 root \(\beta\) of the polynomial \(x^{3} - x^{2} - 2 x + 1\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.80194
−1.24698
0.445042
1.00000 1.00000 1.00000 −3.24698 1.00000 −0.307979 1.00000 1.00000 −3.24698
1.2 1.00000 1.00000 1.00000 −1.55496 1.00000 −0.643104 1.00000 1.00000 −1.55496
1.3 1.00000 1.00000 1.00000 −0.198062 1.00000 −5.04892 1.00000 1.00000 −0.198062
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5}^{3} + 5 T_{5}^{2} + 6 T_{5} + 1 \)
\( T_{7}^{3} + 6 T_{7}^{2} + 5 T_{7} + 1 \)
\( T_{13}^{3} + 9 T_{13}^{2} + 20 T_{13} + 13 \)