Properties

Label 6012.2.a.a
Level 6012
Weight 2
Character orbit 6012.a
Self dual yes
Analytic conductor 48.006
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6012.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.0060616952\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 668)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{5} + ( 2 - \beta ) q^{7} +O(q^{10})\) \( q + 3 q^{5} + ( 2 - \beta ) q^{7} + ( 5 - \beta ) q^{13} -\beta q^{17} + 2 q^{19} + \beta q^{23} + 4 q^{25} + ( -3 - 2 \beta ) q^{29} + ( 2 + 2 \beta ) q^{31} + ( 6 - 3 \beta ) q^{35} + ( 5 - 2 \beta ) q^{37} + ( -3 + 4 \beta ) q^{41} + ( 5 + 2 \beta ) q^{43} + ( 3 - 2 \beta ) q^{47} -3 \beta q^{49} + 2 \beta q^{53} + ( 6 - 6 \beta ) q^{59} + ( -7 + 6 \beta ) q^{61} + ( 15 - 3 \beta ) q^{65} + ( -1 + 4 \beta ) q^{67} + ( 3 - 3 \beta ) q^{71} + ( -10 + 3 \beta ) q^{73} + ( 11 - 2 \beta ) q^{79} + ( -3 + 2 \beta ) q^{83} -3 \beta q^{85} + 6 \beta q^{89} + ( 13 - 6 \beta ) q^{91} + 6 q^{95} + ( -7 + 5 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{5} + 3q^{7} + O(q^{10}) \) \( 2q + 6q^{5} + 3q^{7} + 9q^{13} - q^{17} + 4q^{19} + q^{23} + 8q^{25} - 8q^{29} + 6q^{31} + 9q^{35} + 8q^{37} - 2q^{41} + 12q^{43} + 4q^{47} - 3q^{49} + 2q^{53} + 6q^{59} - 8q^{61} + 27q^{65} + 2q^{67} + 3q^{71} - 17q^{73} + 20q^{79} - 4q^{83} - 3q^{85} + 6q^{89} + 20q^{91} + 12q^{95} - 9q^{97} + O(q^{100}) \)

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
2.30278
−1.30278
0 0 0 3.00000 0 −0.302776 0 0 0
1.2 0 0 0 3.00000 0 3.30278 0 0 0
\(n\): 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 6012.2.a.a 2
3.b odd 2 1 668.2.a.a 2
12.b even 2 1 2672.2.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
668.2.a.a 2 3.b odd 2 1
2672.2.a.c 2 12.b even 2 1
6012.2.a.a 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(167\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6012))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 3 T + 5 T^{2} )^{2} \)
$7$ \( 1 - 3 T + 13 T^{2} - 21 T^{3} + 49 T^{4} \)
$11$ \( ( 1 + 11 T^{2} )^{2} \)
$13$ \( 1 - 9 T + 43 T^{2} - 117 T^{3} + 169 T^{4} \)
$17$ \( 1 + T + 31 T^{2} + 17 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 2 T + 19 T^{2} )^{2} \)
$23$ \( 1 - T + 43 T^{2} - 23 T^{3} + 529 T^{4} \)
$29$ \( 1 + 8 T + 61 T^{2} + 232 T^{3} + 841 T^{4} \)
$31$ \( 1 - 6 T + 58 T^{2} - 186 T^{3} + 961 T^{4} \)
$37$ \( 1 - 8 T + 77 T^{2} - 296 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 2 T + 31 T^{2} + 82 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 12 T + 109 T^{2} - 516 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 4 T + 85 T^{2} - 188 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 2 T + 94 T^{2} - 106 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 6 T + 10 T^{2} - 354 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 8 T + 21 T^{2} + 488 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 2 T + 83 T^{2} - 134 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 3 T + 115 T^{2} - 213 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 17 T + 189 T^{2} + 1241 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 20 T + 245 T^{2} - 1580 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 4 T + 157 T^{2} + 332 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 6 T + 70 T^{2} - 534 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 9 T + 133 T^{2} + 873 T^{3} + 9409 T^{4} \)
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