Properties

Label 6012.2.a.g
Level $6012$
Weight $2$
Character orbit 6012.a
Self dual yes
Analytic conductor $48.006$
Analytic rank $1$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 11 x^{5} - 7 x^{4} + 21 x^{3} + 17 x^{2} - 4 x - 4\)
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 a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{5} + ( -2 - \beta_{1} + \beta_{2} - \beta_{5} ) q^{7} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{5} + ( -2 - \beta_{1} + \beta_{2} - \beta_{5} ) q^{7} + ( 1 - \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{11} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{13} + ( 2 \beta_{1} + \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{17} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{6} ) q^{19} + ( 4 - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{23} + ( -1 + \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{25} + ( 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{6} ) q^{29} + ( -4 + 2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{31} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{35} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{4} ) q^{37} + ( 3 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{41} + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{43} + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{47} + ( 3 + \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{49} + ( 2 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{53} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{55} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} ) q^{59} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{61} + ( 3 \beta_{1} - 7 \beta_{2} + 5 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{65} + ( -6 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{6} ) q^{67} + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{71} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{73} + ( 5 \beta_{2} + 3 \beta_{3} + 5 \beta_{4} + \beta_{5} + 5 \beta_{6} ) q^{77} + ( \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} ) q^{79} + ( \beta_{1} + \beta_{2} + \beta_{3} + 6 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{83} + ( -4 + 2 \beta_{2} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{85} + ( -3 + 3 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{89} + ( \beta_{1} - 7 \beta_{2} + \beta_{3} - 2 \beta_{5} - 4 \beta_{6} ) q^{91} + ( 2 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{95} + ( -1 - \beta_{1} + 4 \beta_{2} + \beta_{3} - 2 \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 2q^{5} - 12q^{7} + O(q^{10}) \) \( 7q + 2q^{5} - 12q^{7} + 7q^{11} - 9q^{13} + q^{17} - 11q^{19} + 19q^{23} + 3q^{25} + 5q^{29} - 13q^{31} + 7q^{35} - 26q^{37} + 2q^{41} - 24q^{43} + 11q^{47} + 19q^{49} - 4q^{53} - 4q^{55} + 4q^{59} - 5q^{61} - 13q^{65} - 42q^{67} - 9q^{71} + 27q^{73} - 12q^{77} - 8q^{79} - 16q^{83} - 27q^{85} - 9q^{89} - 2q^{91} - 10q^{95} - 8q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 11 x^{5} - 7 x^{4} + 21 x^{3} + 17 x^{2} - 4 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 7 \nu^{6} - 6 \nu^{5} - 73 \nu^{4} + 13 \nu^{3} + 149 \nu^{2} + 5 \nu - 50 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( 9 \nu^{6} - 6 \nu^{5} - 95 \nu^{4} - \nu^{3} + 191 \nu^{2} + 35 \nu - 58 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -13 \nu^{6} + 10 \nu^{5} + 135 \nu^{4} - 11 \nu^{3} - 263 \nu^{2} - 31 \nu + 70 \)\()/4\)
\(\beta_{5}\)\(=\)\( -6 \nu^{6} + 4 \nu^{5} + 63 \nu^{4} - 123 \nu^{2} - 18 \nu + 33 \)
\(\beta_{6}\)\(=\)\((\)\( 27 \nu^{6} - 18 \nu^{5} - 285 \nu^{4} + \nu^{3} + 565 \nu^{2} + 81 \nu - 154 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{6} - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 6 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(-11 \beta_{6} - 12 \beta_{5} + 9 \beta_{4} + 7 \beta_{3} + 9 \beta_{2} + 4 \beta_{1} + 29\)
\(\nu^{5}\)\(=\)\(-18 \beta_{6} - 29 \beta_{5} + 27 \beta_{4} - 3 \beta_{3} + 24 \beta_{2} + 48 \beta_{1} + 48\)
\(\nu^{6}\)\(=\)\(-107 \beta_{6} - 125 \beta_{5} + 92 \beta_{4} + 51 \beta_{3} + 90 \beta_{2} + 71 \beta_{1} + 260\)

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
0.495342
1.47217
3.27771
−2.38961
−1.47685
−0.721798
−0.656969
0 0 0 −4.03761 0 −3.68648 0 0 0
1.2 0 0 0 −1.35854 0 −3.73540 0 0 0
1.3 0 0 0 −0.610181 0 −0.184306 0 0 0
1.4 0 0 0 0.836956 0 1.84506 0 0 0
1.5 0 0 0 1.35423 0 −2.86706 0 0 0
1.6 0 0 0 2.77086 0 1.71364 0 0 0
1.7 0 0 0 3.04428 0 −5.08545 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.g 7
3.b odd 2 1 668.2.a.c 7
12.b even 2 1 2672.2.a.k 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
668.2.a.c 7 3.b odd 2 1
2672.2.a.k 7 12.b even 2 1
6012.2.a.g 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{7} - 2 T_{5}^{6} - 17 T_{5}^{5} + 42 T_{5}^{4} + 28 T_{5}^{3} - 88 T_{5}^{2} + 32 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6012))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \)
$3$ \( T^{7} \)
$5$ \( 32 - 88 T^{2} + 28 T^{3} + 42 T^{4} - 17 T^{5} - 2 T^{6} + T^{7} \)
$7$ \( 117 + 630 T - 81 T^{2} - 306 T^{3} - 38 T^{4} + 38 T^{5} + 12 T^{6} + T^{7} \)
$11$ \( 1744 + 1752 T - 2159 T^{2} + 43 T^{3} + 251 T^{4} - 28 T^{5} - 7 T^{6} + T^{7} \)
$13$ \( 1184 + 3632 T + 1808 T^{2} - 396 T^{3} - 346 T^{4} - 23 T^{5} + 9 T^{6} + T^{7} \)
$17$ \( 32 - 128 T - 1032 T^{2} + 648 T^{3} + 76 T^{4} - 53 T^{5} - T^{6} + T^{7} \)
$19$ \( 1996 - 4628 T + 2629 T^{2} + 149 T^{3} - 311 T^{4} - 10 T^{5} + 11 T^{6} + T^{7} \)
$23$ \( -56288 + 37104 T - 392 T^{2} - 3716 T^{3} + 548 T^{4} + 71 T^{5} - 19 T^{6} + T^{7} \)
$29$ \( 2237 - 903 T - 1420 T^{2} + 487 T^{3} + 168 T^{4} - 41 T^{5} - 5 T^{6} + T^{7} \)
$31$ \( -48068 + 207806 T + 74057 T^{2} - 623 T^{3} - 2113 T^{4} - 122 T^{5} + 13 T^{6} + T^{7} \)
$37$ \( 547488 + 179424 T - 27144 T^{2} - 15912 T^{3} - 1340 T^{4} + 145 T^{5} + 26 T^{6} + T^{7} \)
$41$ \( 36896 - 21296 T - 8200 T^{2} + 4872 T^{3} + 226 T^{4} - 141 T^{5} - 2 T^{6} + T^{7} \)
$43$ \( -32 - 160 T + 1080 T^{2} + 1636 T^{3} + 870 T^{4} + 213 T^{5} + 24 T^{6} + T^{7} \)
$47$ \( 482877 - 64341 T - 65700 T^{2} + 5463 T^{3} + 1978 T^{4} - 175 T^{5} - 11 T^{6} + T^{7} \)
$53$ \( 11392 - 12416 T - 7872 T^{2} + 6144 T^{3} - 352 T^{4} - 164 T^{5} + 4 T^{6} + T^{7} \)
$59$ \( 4736 + 3904 T - 6144 T^{2} + 960 T^{3} + 472 T^{4} - 92 T^{5} - 4 T^{6} + T^{7} \)
$61$ \( 36603 + 170757 T + 96984 T^{2} + 10407 T^{3} - 1432 T^{4} - 217 T^{5} + 5 T^{6} + T^{7} \)
$67$ \( -162784 - 332736 T - 140072 T^{2} - 14212 T^{3} + 2374 T^{4} + 591 T^{5} + 42 T^{6} + T^{7} \)
$71$ \( -64512 + 87552 T + 64320 T^{2} + 6384 T^{3} - 1588 T^{4} - 177 T^{5} + 9 T^{6} + T^{7} \)
$73$ \( 71072 + 34064 T - 19648 T^{2} - 5844 T^{3} + 1610 T^{4} + 103 T^{5} - 27 T^{6} + T^{7} \)
$79$ \( -380448 - 212544 T + 25128 T^{2} + 15588 T^{3} - 938 T^{4} - 239 T^{5} + 8 T^{6} + T^{7} \)
$83$ \( 147296 + 2051296 T + 571880 T^{2} + 17180 T^{3} - 6142 T^{4} - 343 T^{5} + 16 T^{6} + T^{7} \)
$89$ \( 369764 + 576542 T + 256249 T^{2} + 28649 T^{3} - 2989 T^{4} - 362 T^{5} + 9 T^{6} + T^{7} \)
$97$ \( 1674729 - 778356 T + 25527 T^{2} + 23938 T^{3} - 1244 T^{4} - 262 T^{5} + 8 T^{6} + T^{7} \)
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