Properties

Label 6012.2.a.b
Level $6012$
Weight $2$
Character orbit 6012.a
Self dual yes
Analytic conductor $48.006$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} + 2 \beta_{2} q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} + 2 \beta_{2} q^{7} + ( -1 - \beta_{1} - \beta_{2} ) q^{11} + 2 \beta_{1} q^{13} + ( 2 - \beta_{2} ) q^{17} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{19} -4 \beta_{1} q^{23} + ( -2 - \beta_{1} - \beta_{2} ) q^{25} -4 \beta_{2} q^{29} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{31} + ( -6 + 2 \beta_{1} + 2 \beta_{2} ) q^{35} + ( 2 + 4 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -4 - 2 \beta_{1} - 3 \beta_{2} ) q^{41} + ( -6 - 2 \beta_{1} + \beta_{2} ) q^{43} + ( 5 + \beta_{1} + \beta_{2} ) q^{47} + ( 5 - 4 \beta_{1} - 4 \beta_{2} ) q^{49} + ( -4 + 6 \beta_{1} + \beta_{2} ) q^{53} + 2 q^{55} + ( -8 + 2 \beta_{2} ) q^{59} + ( -1 + \beta_{1} + \beta_{2} ) q^{61} + ( 2 - 2 \beta_{1} ) q^{65} + ( 4 - 2 \beta_{1} - \beta_{2} ) q^{67} -4 q^{71} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{73} -4 q^{77} + ( 2 - 8 \beta_{1} - \beta_{2} ) q^{79} + ( -2 + 2 \beta_{1} - 4 \beta_{2} ) q^{83} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{85} + ( -4 - 4 \beta_{2} ) q^{89} + ( -4 + 4 \beta_{1} ) q^{91} + ( -4 + 4 \beta_{2} ) q^{95} + ( 6 + 6 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + O(q^{10}) \) \( 3q - 4q^{11} + 2q^{13} + 6q^{17} - 4q^{19} - 4q^{23} - 7q^{25} + 4q^{31} - 16q^{35} + 10q^{37} - 14q^{41} - 20q^{43} + 16q^{47} + 11q^{49} - 6q^{53} + 6q^{55} - 24q^{59} - 2q^{61} + 4q^{65} + 10q^{67} - 12q^{71} - 10q^{73} - 12q^{77} - 2q^{79} - 4q^{83} + 8q^{85} - 12q^{89} - 8q^{91} - 12q^{95} + 18q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)

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
−1.48119
2.17009
0.311108
0 0 0 −1.67513 0 3.35026 0 0 0
1.2 0 0 0 −0.539189 0 1.07838 0 0 0
1.3 0 0 0 2.21432 0 −4.42864 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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.b 3
3.b odd 2 1 6012.2.a.c yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6012.2.a.b 3 1.a even 1 1 trivial
6012.2.a.c yes 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} - 4 T_{5} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6012))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 11 T^{2} - 2 T^{3} + 55 T^{4} + 125 T^{6} \)
$7$ \( 1 + 5 T^{2} + 16 T^{3} + 35 T^{4} + 343 T^{6} \)
$11$ \( 1 + 4 T + 33 T^{2} + 84 T^{3} + 363 T^{4} + 484 T^{5} + 1331 T^{6} \)
$13$ \( 1 - 2 T + 27 T^{2} - 44 T^{3} + 351 T^{4} - 338 T^{5} + 2197 T^{6} \)
$17$ \( 1 - 6 T + 59 T^{2} - 206 T^{3} + 1003 T^{4} - 1734 T^{5} + 4913 T^{6} \)
$19$ \( 1 + 4 T + 41 T^{2} + 120 T^{3} + 779 T^{4} + 1444 T^{5} + 6859 T^{6} \)
$23$ \( 1 + 4 T + 21 T^{2} + 120 T^{3} + 483 T^{4} + 2116 T^{5} + 12167 T^{6} \)
$29$ \( 1 + 23 T^{2} - 128 T^{3} + 667 T^{4} + 24389 T^{6} \)
$31$ \( 1 - 4 T + 61 T^{2} - 280 T^{3} + 1891 T^{4} - 3844 T^{5} + 29791 T^{6} \)
$37$ \( 1 - 10 T + 91 T^{2} - 748 T^{3} + 3367 T^{4} - 13690 T^{5} + 50653 T^{6} \)
$41$ \( 1 + 14 T + 151 T^{2} + 1026 T^{3} + 6191 T^{4} + 23534 T^{5} + 68921 T^{6} \)
$43$ \( 1 + 20 T + 241 T^{2} + 1838 T^{3} + 10363 T^{4} + 36980 T^{5} + 79507 T^{6} \)
$47$ \( 1 - 16 T + 221 T^{2} - 1628 T^{3} + 10387 T^{4} - 35344 T^{5} + 103823 T^{6} \)
$53$ \( 1 + 6 T + 59 T^{2} + 170 T^{3} + 3127 T^{4} + 16854 T^{5} + 148877 T^{6} \)
$59$ \( 1 + 24 T + 353 T^{2} + 3232 T^{3} + 20827 T^{4} + 83544 T^{5} + 205379 T^{6} \)
$61$ \( 1 + 2 T + 179 T^{2} + 240 T^{3} + 10919 T^{4} + 7442 T^{5} + 226981 T^{6} \)
$67$ \( 1 - 10 T + 221 T^{2} - 1314 T^{3} + 14807 T^{4} - 44890 T^{5} + 300763 T^{6} \)
$71$ \( ( 1 + 4 T + 71 T^{2} )^{3} \)
$73$ \( 1 + 10 T + 231 T^{2} + 1420 T^{3} + 16863 T^{4} + 53290 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 2 T + 37 T^{2} + 650 T^{3} + 2923 T^{4} + 12482 T^{5} + 493039 T^{6} \)
$83$ \( 1 + 4 T + 161 T^{2} + 680 T^{3} + 13363 T^{4} + 27556 T^{5} + 571787 T^{6} \)
$89$ \( 1 + 12 T + 251 T^{2} + 1816 T^{3} + 22339 T^{4} + 95052 T^{5} + 704969 T^{6} \)
$97$ \( 1 - 18 T + 255 T^{2} - 2412 T^{3} + 24735 T^{4} - 169362 T^{5} + 912673 T^{6} \)
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