Properties

Label 8046.2.a.c
Level 8046
Weight 2
Character orbit 8046.a
Self dual yes
Analytic conductor 64.248
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8046.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.2476334663\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + 2 \beta q^{5} + ( 1 - 2 \beta ) q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + 2 \beta q^{5} + ( 1 - 2 \beta ) q^{7} - q^{8} -2 \beta q^{10} + ( -3 + \beta ) q^{11} + ( 2 + 2 \beta ) q^{13} + ( -1 + 2 \beta ) q^{14} + q^{16} + ( 5 + \beta ) q^{17} + ( -2 - 2 \beta ) q^{19} + 2 \beta q^{20} + ( 3 - \beta ) q^{22} + ( 6 - \beta ) q^{23} + 7 q^{25} + ( -2 - 2 \beta ) q^{26} + ( 1 - 2 \beta ) q^{28} -\beta q^{29} + q^{31} - q^{32} + ( -5 - \beta ) q^{34} + ( -12 + 2 \beta ) q^{35} + ( 1 + \beta ) q^{37} + ( 2 + 2 \beta ) q^{38} -2 \beta q^{40} + ( -4 + 3 \beta ) q^{41} + ( 2 + 4 \beta ) q^{43} + ( -3 + \beta ) q^{44} + ( -6 + \beta ) q^{46} + ( 4 + 2 \beta ) q^{47} + ( 6 - 4 \beta ) q^{49} -7 q^{50} + ( 2 + 2 \beta ) q^{52} + ( 8 - \beta ) q^{53} + ( 6 - 6 \beta ) q^{55} + ( -1 + 2 \beta ) q^{56} + \beta q^{58} + ( -5 + \beta ) q^{59} + ( 1 + \beta ) q^{61} - q^{62} + q^{64} + ( 12 + 4 \beta ) q^{65} + ( -1 - 3 \beta ) q^{67} + ( 5 + \beta ) q^{68} + ( 12 - 2 \beta ) q^{70} + ( -2 - 7 \beta ) q^{71} + ( -2 + 2 \beta ) q^{73} + ( -1 - \beta ) q^{74} + ( -2 - 2 \beta ) q^{76} + ( -9 + 7 \beta ) q^{77} + ( -11 - \beta ) q^{79} + 2 \beta q^{80} + ( 4 - 3 \beta ) q^{82} + ( -7 + 3 \beta ) q^{83} + ( 6 + 10 \beta ) q^{85} + ( -2 - 4 \beta ) q^{86} + ( 3 - \beta ) q^{88} + ( -6 + \beta ) q^{89} + ( -10 - 2 \beta ) q^{91} + ( 6 - \beta ) q^{92} + ( -4 - 2 \beta ) q^{94} + ( -12 - 4 \beta ) q^{95} + ( 9 + \beta ) q^{97} + ( -6 + 4 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} + 2q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} + 2q^{7} - 2q^{8} - 6q^{11} + 4q^{13} - 2q^{14} + 2q^{16} + 10q^{17} - 4q^{19} + 6q^{22} + 12q^{23} + 14q^{25} - 4q^{26} + 2q^{28} + 2q^{31} - 2q^{32} - 10q^{34} - 24q^{35} + 2q^{37} + 4q^{38} - 8q^{41} + 4q^{43} - 6q^{44} - 12q^{46} + 8q^{47} + 12q^{49} - 14q^{50} + 4q^{52} + 16q^{53} + 12q^{55} - 2q^{56} - 10q^{59} + 2q^{61} - 2q^{62} + 2q^{64} + 24q^{65} - 2q^{67} + 10q^{68} + 24q^{70} - 4q^{71} - 4q^{73} - 2q^{74} - 4q^{76} - 18q^{77} - 22q^{79} + 8q^{82} - 14q^{83} + 12q^{85} - 4q^{86} + 6q^{88} - 12q^{89} - 20q^{91} + 12q^{92} - 8q^{94} - 24q^{95} + 18q^{97} - 12q^{98} + O(q^{100}) \)

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
−1.73205
1.73205
−1.00000 0 1.00000 −3.46410 0 4.46410 −1.00000 0 3.46410
1.2 −1.00000 0 1.00000 3.46410 0 −2.46410 −1.00000 0 −3.46410
\(n\): 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 8046.2.a.c 2
3.b odd 2 1 8046.2.a.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8046.2.a.c 2 1.a even 1 1 trivial
8046.2.a.d yes 2 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(149\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

\( T_{5}^{2} - 12 \)
\( T_{11}^{2} + 6 T_{11} + 6 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( \)
$5$ \( 1 - 2 T^{2} + 25 T^{4} \)
$7$ \( 1 - 2 T + 3 T^{2} - 14 T^{3} + 49 T^{4} \)
$11$ \( 1 + 6 T + 28 T^{2} + 66 T^{3} + 121 T^{4} \)
$13$ \( 1 - 4 T + 18 T^{2} - 52 T^{3} + 169 T^{4} \)
$17$ \( 1 - 10 T + 56 T^{2} - 170 T^{3} + 289 T^{4} \)
$19$ \( 1 + 4 T + 30 T^{2} + 76 T^{3} + 361 T^{4} \)
$23$ \( 1 - 12 T + 79 T^{2} - 276 T^{3} + 529 T^{4} \)
$29$ \( 1 + 55 T^{2} + 841 T^{4} \)
$31$ \( ( 1 - T + 31 T^{2} )^{2} \)
$37$ \( 1 - 2 T + 72 T^{2} - 74 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 8 T + 71 T^{2} + 328 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 4 T + 42 T^{2} - 172 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 8 T + 98 T^{2} - 376 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 16 T + 167 T^{2} - 848 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 10 T + 140 T^{2} + 590 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 2 T + 120 T^{2} - 122 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 2 T + 108 T^{2} + 134 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 4 T - T^{2} + 284 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 4 T + 138 T^{2} + 292 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 22 T + 276 T^{2} + 1738 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 14 T + 188 T^{2} + 1162 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 12 T + 211 T^{2} + 1068 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 18 T + 272 T^{2} - 1746 T^{3} + 9409 T^{4} \)
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