Properties

Label 6012.2.a.h
Level $6012$
Weight $2$
Character orbit 6012.a
Self dual yes
Analytic conductor $48.006$
Analytic rank $1$
Dimension $9$
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 29 x^{7} - 7 x^{6} + 266 x^{5} + 69 x^{4} - 901 x^{3} - 199 x^{2} + 875 x + 391\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2004)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{5} -\beta_{6} q^{7} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{5} -\beta_{6} q^{7} + ( -1 + \beta_{5} ) q^{11} + ( 1 - \beta_{4} + \beta_{6} + \beta_{7} ) q^{13} + ( -1 + \beta_{2} - \beta_{3} - \beta_{8} ) q^{17} + ( -\beta_{2} - \beta_{7} ) q^{19} + ( -2 - \beta_{1} - \beta_{2} - \beta_{5} + \beta_{8} ) q^{23} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{25} + ( -\beta_{1} + 2 \beta_{3} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{29} + ( 1 - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{31} + ( -2 \beta_{2} + 2 \beta_{3} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{35} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{7} + \beta_{8} ) q^{37} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{41} + ( -2 - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{43} + ( -4 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{47} + ( 2 - \beta_{4} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{49} + ( -2 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{53} + ( 1 - 2 \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} ) q^{55} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{7} + \beta_{8} ) q^{59} + ( 3 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{61} + ( -2 + 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{65} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{67} + ( -2 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{8} ) q^{71} + ( -3 + \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{73} + ( -2 + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{77} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{79} + ( -1 - \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{83} + ( 1 - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{85} + ( -4 - \beta_{2} - 2 \beta_{4} + \beta_{7} ) q^{89} + ( -4 + \beta_{1} - 2 \beta_{4} - 2 \beta_{6} + \beta_{8} ) q^{91} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{95} + ( 4 - \beta_{1} + 5 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 9q^{5} + 2q^{7} + O(q^{10}) \) \( 9q - 9q^{5} + 2q^{7} - 7q^{11} + 6q^{13} - 7q^{17} + 2q^{19} - 19q^{23} + 22q^{25} - 13q^{29} + 12q^{31} - 4q^{35} + 15q^{37} - 18q^{41} - 6q^{43} - 25q^{47} + 19q^{49} - 17q^{53} - 3q^{55} - 3q^{59} + 14q^{61} - 14q^{65} - 4q^{67} - 17q^{71} - 20q^{73} - 14q^{77} - 8q^{79} + q^{83} + 5q^{85} - 36q^{89} - 41q^{91} - 5q^{95} + 31q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 29 x^{7} - 7 x^{6} + 266 x^{5} + 69 x^{4} - 901 x^{3} - 199 x^{2} + 875 x + 391\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -1006 \nu^{8} + 2871 \nu^{7} + 22263 \nu^{6} - 53775 \nu^{5} - 157887 \nu^{4} + 312589 \nu^{3} + 454514 \nu^{2} - 550699 \nu - 575149 \)\()/51607\)
\(\beta_{3}\)\(=\)\((\)\( 1976 \nu^{8} - 9538 \nu^{7} - 25159 \nu^{6} + 160208 \nu^{5} - 1570 \nu^{4} - 711563 \nu^{3} + 508013 \nu^{2} + 557721 \nu - 255465 \)\()/51607\)
\(\beta_{4}\)\(=\)\((\)\( 2012 \nu^{8} - 5742 \nu^{7} - 44526 \nu^{6} + 107550 \nu^{5} + 315774 \nu^{4} - 625178 \nu^{3} - 857421 \nu^{2} + 1101398 \nu + 789049 \)\()/51607\)
\(\beta_{5}\)\(=\)\((\)\( 2980 \nu^{8} - 18354 \nu^{7} - 43479 \nu^{6} + 348793 \nu^{5} + 150155 \nu^{4} - 1846062 \nu^{3} - 203222 \nu^{2} + 2331119 \nu + 1064431 \)\()/51607\)
\(\beta_{6}\)\(=\)\((\)\( 3832 \nu^{8} - 14527 \nu^{7} - 71773 \nu^{6} + 255110 \nu^{5} + 401245 \nu^{4} - 1246613 \nu^{3} - 814593 \nu^{2} + 1505084 \nu + 875617 \)\()/51607\)
\(\beta_{7}\)\(=\)\((\)\( 3879 \nu^{8} - 3837 \nu^{7} - 87023 \nu^{6} + 28674 \nu^{5} + 546052 \nu^{4} + 57429 \nu^{3} - 948686 \nu^{2} - 418506 \nu + 172141 \)\()/51607\)
\(\beta_{8}\)\(=\)\((\)\( -3886 \nu^{8} + 8833 \nu^{7} + 75020 \nu^{6} - 124516 \nu^{5} - 361191 \nu^{4} + 420341 \nu^{3} + 179180 \nu^{2} - 75695 \nu + 292783 \)\()/51607\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + 2 \beta_{2} + 7\)
\(\nu^{3}\)\(=\)\(\beta_{8} + 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{3} + 4 \beta_{2} + 10 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(5 \beta_{8} + 4 \beta_{7} + \beta_{5} + 14 \beta_{4} + 2 \beta_{3} + 31 \beta_{2} + 5 \beta_{1} + 79\)
\(\nu^{5}\)\(=\)\(22 \beta_{8} + 38 \beta_{7} - 23 \beta_{6} + 36 \beta_{5} + 17 \beta_{4} - 12 \beta_{3} + 91 \beta_{2} + 123 \beta_{1} + 91\)
\(\nu^{6}\)\(=\)\(115 \beta_{8} + 102 \beta_{7} - 21 \beta_{6} + 38 \beta_{5} + 207 \beta_{4} + 46 \beta_{3} + 486 \beta_{2} + 163 \beta_{1} + 1059\)
\(\nu^{7}\)\(=\)\(431 \beta_{8} + 648 \beta_{7} - 397 \beta_{6} + 555 \beta_{5} + 464 \beta_{4} - 97 \beta_{3} + 1703 \beta_{2} + 1714 \beta_{1} + 2087\)
\(\nu^{8}\)\(=\)\(2125 \beta_{8} + 2069 \beta_{7} - 679 \beta_{6} + 965 \beta_{5} + 3251 \beta_{4} + 758 \beta_{3} + 7981 \beta_{2} + 3699 \beta_{1} + 15652\)

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
−3.21153
−3.19525
−2.42964
−0.907808
−0.529979
1.61974
1.79204
2.69402
4.16840
0 0 0 −4.21153 0 −5.13720 0 0 0
1.2 0 0 0 −4.19525 0 1.43344 0 0 0
1.3 0 0 0 −3.42964 0 3.44225 0 0 0
1.4 0 0 0 −1.90781 0 2.81337 0 0 0
1.5 0 0 0 −1.52998 0 −1.05249 0 0 0
1.6 0 0 0 0.619742 0 1.05844 0 0 0
1.7 0 0 0 0.792043 0 3.80237 0 0 0
1.8 0 0 0 1.69402 0 −4.12928 0 0 0
1.9 0 0 0 3.16840 0 −0.230890 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.h 9
3.b odd 2 1 2004.2.a.d 9
12.b even 2 1 8016.2.a.bb 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.2.a.d 9 3.b odd 2 1
6012.2.a.h 9 1.a even 1 1 trivial
8016.2.a.bb 9 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{9} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6012))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \)
$3$ \( T^{9} \)
$5$ \( 466 - 856 T - 506 T^{2} + 964 T^{3} + 405 T^{4} - 259 T^{5} - 126 T^{6} + 7 T^{7} + 9 T^{8} + T^{9} \)
$7$ \( -288 - 912 T + 1640 T^{2} + 480 T^{3} - 1365 T^{4} + 351 T^{5} + 109 T^{6} - 39 T^{7} - 2 T^{8} + T^{9} \)
$11$ \( -36 - 384 T - 1108 T^{2} - 520 T^{3} + 989 T^{4} + 213 T^{5} - 168 T^{6} - 23 T^{7} + 7 T^{8} + T^{9} \)
$13$ \( -5520 - 5712 T + 9600 T^{2} + 1820 T^{3} - 3243 T^{4} + 90 T^{5} + 334 T^{6} - 45 T^{7} - 6 T^{8} + T^{9} \)
$17$ \( 1562 - 8736 T - 11546 T^{2} + 7872 T^{3} + 5653 T^{4} - 263 T^{5} - 540 T^{6} - 58 T^{7} + 7 T^{8} + T^{9} \)
$19$ \( 64 + 64 T - 448 T^{2} - 700 T^{3} + 223 T^{4} + 684 T^{5} + 134 T^{6} - 69 T^{7} - 2 T^{8} + T^{9} \)
$23$ \( 288512 + 786560 T + 615744 T^{2} + 164976 T^{3} - 4015 T^{4} - 9127 T^{5} - 1212 T^{6} + 47 T^{7} + 19 T^{8} + T^{9} \)
$29$ \( 272720 + 173296 T - 488440 T^{2} - 11912 T^{3} + 52505 T^{4} + 1713 T^{5} - 1686 T^{6} - 107 T^{7} + 13 T^{8} + T^{9} \)
$31$ \( -280000 - 178400 T + 131888 T^{2} + 32480 T^{3} - 18931 T^{4} - 746 T^{5} + 895 T^{6} - 42 T^{7} - 12 T^{8} + T^{9} \)
$37$ \( -5712 - 434496 T - 544344 T^{2} + 450848 T^{3} - 81337 T^{4} - 4701 T^{5} + 2302 T^{6} - 91 T^{7} - 15 T^{8} + T^{9} \)
$41$ \( -11178 - 6318 T + 18522 T^{2} + 11862 T^{3} - 6255 T^{4} - 5714 T^{5} - 1079 T^{6} + 22 T^{7} + 18 T^{8} + T^{9} \)
$43$ \( -8589994 + 7610890 T - 142622 T^{2} - 886908 T^{3} + 70579 T^{4} + 26210 T^{5} - 1389 T^{6} - 290 T^{7} + 6 T^{8} + T^{9} \)
$47$ \( -51047340 - 22532892 T + 556332 T^{2} + 1431344 T^{3} + 131535 T^{4} - 23500 T^{5} - 3713 T^{6} + 25 T^{7} + 25 T^{8} + T^{9} \)
$53$ \( -12400550 - 7077610 T + 987166 T^{2} + 1221572 T^{3} + 198307 T^{4} - 9120 T^{5} - 4129 T^{6} - 159 T^{7} + 17 T^{8} + T^{9} \)
$59$ \( -881152 + 13443712 T - 1538720 T^{2} - 1035632 T^{3} + 68223 T^{4} + 27296 T^{5} - 829 T^{6} - 283 T^{7} + 3 T^{8} + T^{9} \)
$61$ \( -4149788 + 5249068 T + 5792052 T^{2} - 172628 T^{3} - 289205 T^{4} + 12968 T^{5} + 4170 T^{6} - 263 T^{7} - 14 T^{8} + T^{9} \)
$67$ \( 150554 + 260346 T - 729620 T^{2} - 155954 T^{3} + 50139 T^{4} + 10100 T^{5} - 972 T^{6} - 201 T^{7} + 4 T^{8} + T^{9} \)
$71$ \( -343184800 - 207451056 T - 31230720 T^{2} + 2518796 T^{3} + 858773 T^{4} + 17563 T^{5} - 6815 T^{6} - 320 T^{7} + 17 T^{8} + T^{9} \)
$73$ \( 45115056 - 5442336 T - 6872328 T^{2} + 357948 T^{3} + 312735 T^{4} + 1642 T^{5} - 4594 T^{6} - 159 T^{7} + 20 T^{8} + T^{9} \)
$79$ \( 2054818 - 3887890 T - 2543762 T^{2} - 136474 T^{3} + 128245 T^{4} + 14886 T^{5} - 1819 T^{6} - 240 T^{7} + 8 T^{8} + T^{9} \)
$83$ \( 3972576 + 1719024 T - 894312 T^{2} - 334868 T^{3} + 40425 T^{4} + 16090 T^{5} - 497 T^{6} - 259 T^{7} - T^{8} + T^{9} \)
$89$ \( 93648 - 152784 T - 1378944 T^{2} - 1004108 T^{3} - 278945 T^{4} - 30616 T^{5} + 510 T^{6} + 425 T^{7} + 36 T^{8} + T^{9} \)
$97$ \( -481925792 - 18464048 T + 55502112 T^{2} + 1278656 T^{3} - 1237713 T^{4} - 4645 T^{5} + 10451 T^{6} - 160 T^{7} - 31 T^{8} + T^{9} \)
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