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: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4026.2.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.k 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 - T )^{2} \)
$5$ \( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} \)
$7$ \( ( 1 - 2 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( 1 - 3 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 26 T^{2} + 289 T^{4} \)
$19$ \( 1 - 8 T + 46 T^{2} - 152 T^{3} + 361 T^{4} \)
$23$ \( 1 + 38 T^{2} + 529 T^{4} \)
$29$ \( 1 + 6 T + 35 T^{2} + 174 T^{3} + 841 T^{4} \)
$31$ \( 1 - 10 T + 85 T^{2} - 310 T^{3} + 961 T^{4} \)
$37$ \( 1 + 2 T^{2} + 1369 T^{4} \)
$41$ \( 1 + 2 T - 15 T^{2} + 82 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 + 6 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 + 6 T + 47 T^{2} )^{2} \)
$53$ \( 1 - 16 T + 162 T^{2} - 848 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 6 T - T^{2} - 354 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 + T )^{2} \)
$67$ \( 1 + 62 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 6 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 8 T + 130 T^{2} - 584 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 + 79 T^{2} )^{2} \)
$83$ \( 1 - 12 T + 74 T^{2} - 996 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 6 T + 115 T^{2} + 534 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 10 T + 187 T^{2} - 970 T^{3} + 9409 T^{4} \)
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