Properties

Label 6012.2.a.i
Level $6012$
Weight $2$
Character orbit 6012.a
Self dual yes
Analytic conductor $48.006$
Analytic rank $0$
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: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - x^{8} - 31 x^{7} + 24 x^{6} + 293 x^{5} - 101 x^{4} - 864 x^{3} - 278 x^{2} + 24 x + 2\)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{5} + ( -\beta_{3} - \beta_{6} ) q^{7} +O(q^{10})\) \( q -\beta_{1} q^{5} + ( -\beta_{3} - \beta_{6} ) q^{7} + ( 1 + \beta_{7} ) q^{11} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{13} + ( -1 - \beta_{1} + \beta_{8} ) q^{17} + ( -1 - \beta_{3} + \beta_{8} ) q^{19} -\beta_{2} q^{23} + ( 2 + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{25} + ( -1 - \beta_{2} - \beta_{3} - \beta_{8} ) q^{29} + ( 3 + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{31} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{35} + ( 2 + \beta_{1} - \beta_{5} ) q^{37} + ( -2 - \beta_{3} - \beta_{4} - \beta_{6} ) q^{41} + ( -\beta_{3} - \beta_{4} ) q^{43} + ( 1 + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{47} + ( 3 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} ) q^{49} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{8} ) q^{53} + ( 1 - 4 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{8} ) q^{55} + ( 2 - \beta_{2} + \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{59} + ( 4 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{7} + \beta_{8} ) q^{61} + ( -3 + \beta_{6} - \beta_{8} ) q^{65} + ( 1 - 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{8} ) q^{67} + ( -1 - \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{71} + ( 5 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} ) q^{73} + ( -2 - \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{77} + ( 2 - 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{79} + ( 1 - \beta_{1} + 2 \beta_{3} - \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{83} + ( 5 + 2 \beta_{1} + 2 \beta_{4} + \beta_{7} - 2 \beta_{8} ) q^{85} + ( -5 + 2 \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{8} ) q^{89} + ( 7 - \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{91} + ( -1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} ) q^{95} + ( 6 - \beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - q^{5} + 2q^{7} + O(q^{10}) \) \( 9q - q^{5} + 2q^{7} + 9q^{11} + 10q^{13} - 7q^{17} - 2q^{19} + 3q^{23} + 18q^{25} - 5q^{29} + 12q^{31} + 6q^{35} + 15q^{37} - 14q^{41} + 6q^{43} + 3q^{47} + 27q^{49} - 9q^{53} + 19q^{55} + 9q^{59} + 30q^{61} - 28q^{65} + 16q^{67} + 3q^{71} + 32q^{73} - 18q^{77} + 24q^{79} + 3q^{83} + 37q^{85} - 46q^{89} + 33q^{91} - 11q^{95} + 43q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - x^{8} - 31 x^{7} + 24 x^{6} + 293 x^{5} - 101 x^{4} - 864 x^{3} - 278 x^{2} + 24 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 454 \nu^{8} + 294 \nu^{7} - 14998 \nu^{6} - 10080 \nu^{5} + 154841 \nu^{4} + 125150 \nu^{3} - 499492 \nu^{2} - 541382 \nu - 14306 \)\()/27853\)
\(\beta_{3}\)\(=\)\((\)\( -575 \nu^{8} + 924 \nu^{7} + 16824 \nu^{6} - 25452 \nu^{5} - 142113 \nu^{4} + 175571 \nu^{3} + 337666 \nu^{2} - 190010 \nu - 38190 \)\()/27853\)
\(\beta_{4}\)\(=\)\((\)\( -2488 \nu^{8} + 2646 \nu^{7} + 78500 \nu^{6} - 64078 \nu^{5} - 762703 \nu^{4} + 284359 \nu^{3} + 2349144 \nu^{2} + 659583 \nu - 147092 \)\()/27853\)
\(\beta_{5}\)\(=\)\((\)\( 3597 \nu^{8} - 3731 \nu^{7} - 111199 \nu^{6} + 90552 \nu^{5} + 1044226 \nu^{4} - 410322 \nu^{3} - 3026506 \nu^{2} - 792549 \nu + 6960 \)\()/27853\)
\(\beta_{6}\)\(=\)\((\)\( -577 \nu^{8} + 611 \nu^{7} + 18018 \nu^{6} - 14597 \nu^{5} - 172292 \nu^{4} + 60797 \nu^{3} + 516238 \nu^{2} + 174672 \nu - 3088 \)\()/3979\)
\(\beta_{7}\)\(=\)\((\)\( -5148 \nu^{8} + 5362 \nu^{7} + 158825 \nu^{6} - 128653 \nu^{5} - 1487567 \nu^{4} + 551542 \nu^{3} + 4318881 \nu^{2} + 1355670 \nu - 76455 \)\()/27853\)
\(\beta_{8}\)\(=\)\((\)\( -5294 \nu^{8} + 4830 \nu^{7} + 163120 \nu^{6} - 116641 \nu^{5} - 1527615 \nu^{4} + 492437 \nu^{3} + 4438630 \nu^{2} + 1359676 \nu - 110393 \)\()/27853\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 7\)
\(\nu^{3}\)\(=\)\(-2 \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} + 13 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-4 \beta_{8} + 16 \beta_{7} - 16 \beta_{6} + 10 \beta_{5} + 14 \beta_{4} + 4 \beta_{3} - 5 \beta_{2} + 2 \beta_{1} + 90\)
\(\nu^{5}\)\(=\)\(-3 \beta_{8} + 6 \beta_{7} - 42 \beta_{6} - 42 \beta_{5} + 17 \beta_{4} - 50 \beta_{3} + 22 \beta_{2} + 187 \beta_{1} + 15\)
\(\nu^{6}\)\(=\)\(-101 \beta_{8} + 246 \beta_{7} - 255 \beta_{6} + 99 \beta_{5} + 218 \beta_{4} + 96 \beta_{3} - 125 \beta_{2} + 31 \beta_{1} + 1271\)
\(\nu^{7}\)\(=\)\(-91 \beta_{8} + 150 \beta_{7} - 711 \beta_{6} - 746 \beta_{5} + 253 \beta_{4} - 943 \beta_{3} + 417 \beta_{2} + 2807 \beta_{1} - 57\)
\(\nu^{8}\)\(=\)\(-1980 \beta_{8} + 3806 \beta_{7} - 3988 \beta_{6} + 1062 \beta_{5} + 3465 \beta_{4} + 1859 \beta_{3} - 2420 \beta_{2} + 285 \beta_{1} + 18844\)

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
3.91239
3.25977
2.94172
0.111665
−0.0546093
−0.419033
−1.74256
−2.94765
−4.06168
0 0 0 −3.91239 0 1.60938 0 0 0
1.2 0 0 0 −3.25977 0 1.95900 0 0 0
1.3 0 0 0 −2.94172 0 −2.78729 0 0 0
1.4 0 0 0 −0.111665 0 −3.78430 0 0 0
1.5 0 0 0 0.0546093 0 3.75281 0 0 0
1.6 0 0 0 0.419033 0 −4.23221 0 0 0
1.7 0 0 0 1.74256 0 2.35372 0 0 0
1.8 0 0 0 2.94765 0 4.65088 0 0 0
1.9 0 0 0 4.06168 0 −1.52198 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.i 9
3.b odd 2 1 2004.2.a.c 9
12.b even 2 1 8016.2.a.bc 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.2.a.c 9 3.b odd 2 1
6012.2.a.i 9 1.a even 1 1 trivial
8016.2.a.bc 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

$p$ $F_p(T)$
$2$ \( T^{9} \)
$3$ \( T^{9} \)
$5$ \( -2 + 24 T + 278 T^{2} - 864 T^{3} + 101 T^{4} + 293 T^{5} - 24 T^{6} - 31 T^{7} + T^{8} + T^{9} \)
$7$ \( -8800 + 4592 T + 5976 T^{2} - 3192 T^{3} - 1125 T^{4} + 611 T^{5} + 81 T^{6} - 43 T^{7} - 2 T^{8} + T^{9} \)
$11$ \( -170748 - 1512 T + 74412 T^{2} - 2376 T^{3} - 9565 T^{4} + 583 T^{5} + 492 T^{6} - 43 T^{7} - 9 T^{8} + T^{9} \)
$13$ \( 272 + 1392 T + 1632 T^{2} - 1364 T^{3} - 2669 T^{4} - 250 T^{5} + 416 T^{6} - 25 T^{7} - 10 T^{8} + T^{9} \)
$17$ \( 190 - 3488 T + 13666 T^{2} - 12204 T^{3} + 1965 T^{4} + 1197 T^{5} - 260 T^{6} - 50 T^{7} + 7 T^{8} + T^{9} \)
$19$ \( 147008 + 88576 T - 51520 T^{2} - 29372 T^{3} + 4539 T^{4} + 2652 T^{5} - 162 T^{6} - 89 T^{7} + 2 T^{8} + T^{9} \)
$23$ \( 256 + 1024 T + 672 T^{2} - 1320 T^{3} - 895 T^{4} + 729 T^{5} + 72 T^{6} - 61 T^{7} - 3 T^{8} + T^{9} \)
$29$ \( -16 - 48 T + 2392 T^{2} - 2520 T^{3} - 9813 T^{4} + 4547 T^{5} - 142 T^{6} - 121 T^{7} + 5 T^{8} + T^{9} \)
$31$ \( -2138048 + 194720 T + 505136 T^{2} - 38656 T^{3} - 39759 T^{4} + 2666 T^{5} + 1187 T^{6} - 86 T^{7} - 12 T^{8} + T^{9} \)
$37$ \( 1936 - 7456 T + 3000 T^{2} + 9792 T^{3} - 1343 T^{4} - 1965 T^{5} + 468 T^{6} + 29 T^{7} - 15 T^{8} + T^{9} \)
$41$ \( -1455166 - 988974 T + 57394 T^{2} + 169094 T^{3} + 30361 T^{4} - 3876 T^{5} - 1387 T^{6} - 42 T^{7} + 14 T^{8} + T^{9} \)
$43$ \( 4286 - 5322 T - 21494 T^{2} - 13796 T^{3} - 143 T^{4} + 1810 T^{5} + 225 T^{6} - 68 T^{7} - 6 T^{8} + T^{9} \)
$47$ \( 97276 - 295228 T - 318812 T^{2} - 61704 T^{3} + 20285 T^{4} + 6888 T^{5} - 43 T^{6} - 145 T^{7} - 3 T^{8} + T^{9} \)
$53$ \( -11282146 + 16800314 T - 3005002 T^{2} - 1005912 T^{3} + 172807 T^{4} + 26248 T^{5} - 2607 T^{6} - 305 T^{7} + 9 T^{8} + T^{9} \)
$59$ \( -43725824 + 12935168 T + 7296512 T^{2} - 1285368 T^{3} - 232785 T^{4} + 33556 T^{5} + 2553 T^{6} - 321 T^{7} - 9 T^{8} + T^{9} \)
$61$ \( -1198252 + 1471884 T - 414372 T^{2} - 171636 T^{3} + 124055 T^{4} - 24772 T^{5} + 1066 T^{6} + 237 T^{7} - 30 T^{8} + T^{9} \)
$67$ \( 789266 + 17470 T - 984536 T^{2} - 529502 T^{3} - 55927 T^{4} + 13236 T^{5} + 1958 T^{6} - 149 T^{7} - 16 T^{8} + T^{9} \)
$71$ \( -82528 - 60720 T + 507968 T^{2} + 85388 T^{3} - 116251 T^{4} + 15059 T^{5} + 1319 T^{6} - 250 T^{7} - 3 T^{8} + T^{9} \)
$73$ \( 127893904 - 64157920 T - 12060248 T^{2} + 7049020 T^{3} - 519771 T^{4} - 67630 T^{5} + 8760 T^{6} + 7 T^{7} - 32 T^{8} + T^{9} \)
$79$ \( -405486 + 874026 T + 106742 T^{2} - 376150 T^{3} - 115957 T^{4} + 1978 T^{5} + 2795 T^{6} - 18 T^{7} - 24 T^{8} + T^{9} \)
$83$ \( 6165280 + 11343632 T - 670920 T^{2} - 1995772 T^{3} - 71983 T^{4} + 52208 T^{5} + 1149 T^{6} - 435 T^{7} - 3 T^{8} + T^{9} \)
$89$ \( -1336363088 - 203947472 T + 45941936 T^{2} + 9261156 T^{3} - 260331 T^{4} - 133052 T^{5} - 5008 T^{6} + 533 T^{7} + 46 T^{8} + T^{9} \)
$97$ \( 68517472 - 24594864 T - 6846880 T^{2} + 2081264 T^{3} + 139659 T^{4} - 59301 T^{5} + 1495 T^{6} + 512 T^{7} - 43 T^{8} + T^{9} \)
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