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: \( 1 \)
Twist minimal: yes
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.

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

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
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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4026.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.l 2 1.a even 1 1 trivial

Atkin-Lehner signs

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

Hecke kernels

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} \)
$7$ \( 1 + 2 T - 2 T^{2} + 14 T^{3} + 49 T^{4} \)
$11$ \( ( 1 - T )^{2} \)
$13$ \( 1 - 5 T + 28 T^{2} - 65 T^{3} + 169 T^{4} \)
$17$ \( 1 - 2 T + 18 T^{2} - 34 T^{3} + 289 T^{4} \)
$19$ \( ( 1 + 4 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 + 23 T^{2} )^{2} \)
$29$ \( 1 - 5 T + 60 T^{2} - 145 T^{3} + 841 T^{4} \)
$31$ \( 1 - 3 T + 26 T^{2} - 93 T^{3} + 961 T^{4} \)
$37$ \( ( 1 + 6 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 19 T + 168 T^{2} - 779 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 4 T + 22 T^{2} - 172 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 4 T + 30 T^{2} - 188 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 2 T + 90 T^{2} + 106 T^{3} + 2809 T^{4} \)
$59$ \( 1 + T + 114 T^{2} + 59 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 - T )^{2} \)
$67$ \( ( 1 - 4 T + 67 T^{2} )^{2} \)
$71$ \( 1 + 2 T + 126 T^{2} + 142 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 2 T + 130 T^{2} - 146 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 + 79 T^{2} )^{2} \)
$83$ \( 1 - 4 T + 102 T^{2} - 332 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 7 T + 84 T^{2} - 623 T^{3} + 7921 T^{4} \)
$97$ \( 1 + T + 156 T^{2} + 97 T^{3} + 9409 T^{4} \)
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