Properties

Label 4002.2.a.x
Level 4002
Weight 2
Character orbit 4002.a
Self dual yes
Analytic conductor 31.956
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9561308889\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

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

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
2.44949
−2.44949
1.00000 1.00000 1.00000 4.00000 1.00000 −0.449490 1.00000 1.00000 4.00000
1.2 1.00000 1.00000 1.00000 4.00000 1.00000 4.44949 1.00000 1.00000 4.00000
\(n\): 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 4002.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.x 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(1\)
\(29\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\( T_{5} - 4 \)
\( T_{7}^{2} - 4 T_{7} - 2 \)
\( T_{11}^{2} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( ( 1 - T )^{2} \)
$5$ \( ( 1 - 4 T + 5 T^{2} )^{2} \)
$7$ \( 1 - 4 T + 12 T^{2} - 28 T^{3} + 49 T^{4} \)
$11$ \( 1 - 2 T^{2} + 121 T^{4} \)
$13$ \( 1 + 4 T + 6 T^{2} + 52 T^{3} + 169 T^{4} \)
$17$ \( 1 + 10 T^{2} + 289 T^{4} \)
$19$ \( 1 + 4 T + 36 T^{2} + 76 T^{3} + 361 T^{4} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( ( 1 - T )^{2} \)
$31$ \( ( 1 + 6 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 + 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 8 T + 74 T^{2} + 328 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 20 T + 180 T^{2} - 860 T^{3} + 1849 T^{4} \)
$47$ \( ( 1 - 4 T + 47 T^{2} )^{2} \)
$53$ \( 1 + 8 T + 98 T^{2} + 424 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 94 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 4 T + 102 T^{2} + 244 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 2 T + 67 T^{2} )^{2} \)
$71$ \( 1 + 12 T + 154 T^{2} + 852 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 4 T + 126 T^{2} + 292 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 - 10 T + 79 T^{2} )^{2} \)
$83$ \( 1 + 8 T + 176 T^{2} + 664 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 16 T + 218 T^{2} - 1424 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 4 T - 96 T^{2} + 388 T^{3} + 9409 T^{4} \)
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