Properties

Label 6012.2.a.e
Level $6012$
Weight $2$
Character orbit 6012.a
Self dual yes
Analytic conductor $48.006$
Analytic rank $1$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.149169.1
Defining polynomial: \(x^{5} - 6 x^{3} - 3 x^{2} + 4 x + 1\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{3} + \beta_{4} ) q^{5} + ( -\beta_{1} - \beta_{4} ) q^{7} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{3} + \beta_{4} ) q^{5} + ( -\beta_{1} - \beta_{4} ) q^{7} + ( -\beta_{1} + \beta_{3} - \beta_{4} ) q^{11} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{13} + ( 3 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{17} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{19} + ( -3 + \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{23} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{25} + ( -\beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{29} + ( -3 + 2 \beta_{2} + \beta_{3} ) q^{31} + ( -1 - \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{35} + ( -4 - \beta_{1} + 4 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{37} + ( 2 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{41} + ( 2 + \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{43} + ( -4 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{47} + ( -4 + \beta_{2} - \beta_{3} ) q^{49} + ( -1 + 3 \beta_{1} - \beta_{2} + 2 \beta_{4} ) q^{53} + ( -4 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{55} + ( 1 + 3 \beta_{1} - \beta_{2} - 2 \beta_{4} ) q^{59} + ( -4 - 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{61} + ( 5 + \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{65} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - 5 \beta_{4} ) q^{67} + ( -5 + 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} ) q^{71} + ( -4 + 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{73} + ( 1 + \beta_{1} - \beta_{3} + 2 \beta_{4} ) q^{77} + ( -6 - 5 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{79} + ( 1 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} ) q^{83} + ( 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{85} + ( 7 - 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{89} + ( -3 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{91} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{95} + ( -4 - 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 3q^{5} - 2q^{7} + O(q^{10}) \) \( 5q + 3q^{5} - 2q^{7} - 3q^{11} - 4q^{13} + 7q^{17} - 2q^{19} - 13q^{23} - 2q^{25} + 3q^{29} - 12q^{31} - 10q^{35} - 7q^{37} + 16q^{41} - q^{47} - 17q^{49} - 3q^{53} - 23q^{55} - q^{59} - 22q^{61} + 20q^{65} + 2q^{67} - 9q^{71} - 28q^{73} + 10q^{77} - 28q^{79} - 7q^{83} - 11q^{85} + 30q^{89} - 13q^{91} - 3q^{95} - 33q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 6 x^{3} - 3 x^{2} + 4 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - \nu^{3} - 4 \nu^{2} + \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{4} + \beta_{3} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-5 \beta_{4} + 6 \beta_{3} + \beta_{2} + 3 \beta_{1} + 14\)

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
−1.81853
2.54970
−1.25464
−0.228573
0.752046
0 0 0 −2.12560 0 0.221696 0 0 0
1.2 0 0 0 −0.951283 0 −1.28171 0 0 0
1.3 0 0 0 0.171230 0 2.92708 0 0 0
1.4 0 0 0 2.71918 0 −2.29630 0 0 0
1.5 0 0 0 3.18647 0 −1.57076 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.e 5
3.b odd 2 1 2004.2.a.a 5
12.b even 2 1 8016.2.a.t 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.2.a.a 5 3.b odd 2 1
6012.2.a.e 5 1.a even 1 1 trivial
8016.2.a.t 5 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - 3 T_{5}^{4} - 7 T_{5}^{3} + 16 T_{5}^{2} + 15 T_{5} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6012))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 3 T + 18 T^{2} - 44 T^{3} + 160 T^{4} - 293 T^{5} + 800 T^{6} - 1100 T^{7} + 2250 T^{8} - 1875 T^{9} + 3125 T^{10} \)
$7$ \( 1 + 2 T + 28 T^{2} + 37 T^{3} + 334 T^{4} + 325 T^{5} + 2338 T^{6} + 1813 T^{7} + 9604 T^{8} + 4802 T^{9} + 16807 T^{10} \)
$11$ \( 1 + 3 T + 48 T^{2} + 116 T^{3} + 994 T^{4} + 1829 T^{5} + 10934 T^{6} + 14036 T^{7} + 63888 T^{8} + 43923 T^{9} + 161051 T^{10} \)
$13$ \( 1 + 4 T + 52 T^{2} + 130 T^{3} + 1089 T^{4} + 2017 T^{5} + 14157 T^{6} + 21970 T^{7} + 114244 T^{8} + 114244 T^{9} + 371293 T^{10} \)
$17$ \( 1 - 7 T + 63 T^{2} - 258 T^{3} + 1597 T^{4} - 5353 T^{5} + 27149 T^{6} - 74562 T^{7} + 309519 T^{8} - 584647 T^{9} + 1419857 T^{10} \)
$19$ \( 1 + 2 T + 54 T^{2} + 66 T^{3} + 1489 T^{4} + 1523 T^{5} + 28291 T^{6} + 23826 T^{7} + 370386 T^{8} + 260642 T^{9} + 2476099 T^{10} \)
$23$ \( 1 + 13 T + 128 T^{2} + 964 T^{3} + 6130 T^{4} + 31369 T^{5} + 140990 T^{6} + 509956 T^{7} + 1557376 T^{8} + 3637933 T^{9} + 6436343 T^{10} \)
$29$ \( 1 - 3 T + 24 T^{2} - 38 T^{3} + 958 T^{4} - 4847 T^{5} + 27782 T^{6} - 31958 T^{7} + 585336 T^{8} - 2121843 T^{9} + 20511149 T^{10} \)
$31$ \( 1 + 12 T + 181 T^{2} + 1471 T^{3} + 11998 T^{4} + 68147 T^{5} + 371938 T^{6} + 1413631 T^{7} + 5392171 T^{8} + 11082252 T^{9} + 28629151 T^{10} \)
$37$ \( 1 + 7 T + 50 T^{2} + 162 T^{3} + 2944 T^{4} + 17933 T^{5} + 108928 T^{6} + 221778 T^{7} + 2532650 T^{8} + 13119127 T^{9} + 69343957 T^{10} \)
$41$ \( 1 - 16 T + 181 T^{2} - 967 T^{3} + 4174 T^{4} - 9063 T^{5} + 171134 T^{6} - 1625527 T^{7} + 12474701 T^{8} - 45212176 T^{9} + 115856201 T^{10} \)
$43$ \( 1 + 69 T^{2} + 337 T^{3} + 4476 T^{4} + 11541 T^{5} + 192468 T^{6} + 623113 T^{7} + 5485983 T^{8} + 147008443 T^{10} \)
$47$ \( 1 + T + 110 T^{2} - 139 T^{3} + 4573 T^{4} - 16971 T^{5} + 214931 T^{6} - 307051 T^{7} + 11420530 T^{8} + 4879681 T^{9} + 229345007 T^{10} \)
$53$ \( 1 + 3 T + 192 T^{2} + 751 T^{3} + 16843 T^{4} + 62173 T^{5} + 892679 T^{6} + 2109559 T^{7} + 28584384 T^{8} + 23671443 T^{9} + 418195493 T^{10} \)
$59$ \( 1 + T + 190 T^{2} - 59 T^{3} + 17725 T^{4} - 10735 T^{5} + 1045775 T^{6} - 205379 T^{7} + 39022010 T^{8} + 12117361 T^{9} + 714924299 T^{10} \)
$61$ \( 1 + 22 T + 354 T^{2} + 3558 T^{3} + 32581 T^{4} + 244855 T^{5} + 1987441 T^{6} + 13239318 T^{7} + 80351274 T^{8} + 304608502 T^{9} + 844596301 T^{10} \)
$67$ \( 1 - 2 T + 210 T^{2} - 50 T^{3} + 20755 T^{4} + 11445 T^{5} + 1390585 T^{6} - 224450 T^{7} + 63160230 T^{8} - 40302242 T^{9} + 1350125107 T^{10} \)
$71$ \( 1 + 9 T + 249 T^{2} + 1683 T^{3} + 25871 T^{4} + 146855 T^{5} + 1836841 T^{6} + 8484003 T^{7} + 89119839 T^{8} + 228705129 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 + 28 T + 470 T^{2} + 5474 T^{3} + 55601 T^{4} + 491083 T^{5} + 4058873 T^{6} + 29170946 T^{7} + 182837990 T^{8} + 795150748 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 + 28 T + 535 T^{2} + 7391 T^{3} + 85444 T^{4} + 807623 T^{5} + 6750076 T^{6} + 46127231 T^{7} + 263775865 T^{8} + 1090602268 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 + 7 T + 86 T^{2} + 1801 T^{3} + 18723 T^{4} + 86479 T^{5} + 1554009 T^{6} + 12407089 T^{7} + 49173682 T^{8} + 332208247 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 - 30 T + 642 T^{2} - 9704 T^{3} + 122785 T^{4} - 1246643 T^{5} + 10927865 T^{6} - 76865384 T^{7} + 452590098 T^{8} - 1882267230 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 + 33 T + 769 T^{2} + 12555 T^{3} + 168965 T^{4} + 1807507 T^{5} + 16389605 T^{6} + 118129995 T^{7} + 701845537 T^{8} + 2921466273 T^{9} + 8587340257 T^{10} \)
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