Properties

Label 6012.2.a.f
Level 6012
Weight 2
Character orbit 6012.a
Self dual yes
Analytic conductor 48.006
Analytic rank 0
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.161121.1
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{3} + \beta_{4} ) q^{5} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{7} +O(q^{10})\) \( q + ( 1 + \beta_{3} + \beta_{4} ) q^{5} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{7} + ( 1 + 2 \beta_{1} + \beta_{3} - \beta_{4} ) q^{11} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{13} + ( 1 + 2 \beta_{1} ) q^{17} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{19} + ( 2 + \beta_{1} - \beta_{3} + \beta_{4} ) q^{23} + ( -2 - 2 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} ) q^{25} + ( 3 - \beta_{3} - 2 \beta_{4} ) q^{29} + ( -1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{31} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{35} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{37} + ( 2 + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{41} + ( 2 - 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{43} + ( 4 - 3 \beta_{1} + 4 \beta_{3} + \beta_{4} ) q^{47} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{49} + ( 4 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{53} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{55} + ( 2 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{59} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} ) q^{61} + ( 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 6 \beta_{4} ) q^{65} + ( 1 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{67} + ( 6 - \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{71} + ( -3 + 3 \beta_{1} - \beta_{2} + \beta_{4} ) q^{73} + ( 3 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{77} + ( -4 \beta_{1} - \beta_{2} - 5 \beta_{3} + \beta_{4} ) q^{79} + ( 2 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{83} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{85} + ( 5 - 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{89} + ( 3 - 2 \beta_{1} + \beta_{2} - 5 \beta_{3} - 3 \beta_{4} ) q^{91} + ( -\beta_{1} - 4 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} ) q^{95} + ( 3 + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 7q^{5} - 2q^{7} + O(q^{10}) \) \( 5q + 7q^{5} - 2q^{7} + 5q^{11} - 8q^{13} + 7q^{17} + 2q^{19} + 13q^{23} + 2q^{25} + 11q^{29} - 12q^{31} + 12q^{35} - 7q^{37} + 12q^{41} + 19q^{47} - 9q^{49} + 21q^{53} - q^{55} + 7q^{59} - 6q^{61} - 14q^{65} + 10q^{67} + 35q^{71} - 8q^{73} + 6q^{77} + 11q^{83} + 5q^{85} + 32q^{89} + 5q^{91} + 19q^{95} + 11q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + 3 \nu + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - \nu^{3} - 6 \nu^{2} + 3 \nu + 4 \)
\(\beta_{4}\)\(=\)\( -2 \nu^{4} + 3 \nu^{3} + 11 \nu^{2} - 11 \nu - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{4} + \beta_{3} + \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} - 4 \beta_{3} + 7 \beta_{2} + 2 \beta_{1} + 15\)

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
1.31991
−2.07823
−0.261082
2.56399
−0.544588
0 0 0 −1.28459 0 −1.96468 0 0 0
1.2 0 0 0 0.614948 0 −3.46328 0 0 0
1.3 0 0 0 1.38924 0 −0.871845 0 0 0
1.4 0 0 0 1.85181 0 2.41580 0 0 0
1.5 0 0 0 4.42860 0 1.88401 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

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.f 5
3.b odd 2 1 2004.2.a.b 5
12.b even 2 1 8016.2.a.q 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.2.a.b 5 3.b odd 2 1
6012.2.a.f 5 1.a even 1 1 trivial
8016.2.a.q 5 12.b even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - 7 T_{5}^{4} + 11 T_{5}^{3} + 6 T_{5}^{2} - 21 T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6012))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 7 T + 36 T^{2} - 134 T^{3} + 394 T^{4} - 981 T^{5} + 1970 T^{6} - 3350 T^{7} + 4500 T^{8} - 4375 T^{9} + 3125 T^{10} \)
$7$ \( 1 + 2 T + 24 T^{2} + 41 T^{3} + 286 T^{4} + 405 T^{5} + 2002 T^{6} + 2009 T^{7} + 8232 T^{8} + 4802 T^{9} + 16807 T^{10} \)
$11$ \( 1 - 5 T + 36 T^{2} - 136 T^{3} + 628 T^{4} - 2025 T^{5} + 6908 T^{6} - 16456 T^{7} + 47916 T^{8} - 73205 T^{9} + 161051 T^{10} \)
$13$ \( 1 + 8 T + 64 T^{2} + 292 T^{3} + 1433 T^{4} + 4915 T^{5} + 18629 T^{6} + 49348 T^{7} + 140608 T^{8} + 228488 T^{9} + 371293 T^{10} \)
$17$ \( 1 - 7 T + 79 T^{2} - 402 T^{3} + 2557 T^{4} - 9625 T^{5} + 43469 T^{6} - 116178 T^{7} + 388127 T^{8} - 584647 T^{9} + 1419857 T^{10} \)
$19$ \( 1 - 2 T + 50 T^{2} - 86 T^{3} + 1389 T^{4} - 2425 T^{5} + 26391 T^{6} - 31046 T^{7} + 342950 T^{8} - 260642 T^{9} + 2476099 T^{10} \)
$23$ \( 1 - 13 T + 144 T^{2} - 1052 T^{3} + 6778 T^{4} - 34395 T^{5} + 155894 T^{6} - 556508 T^{7} + 1752048 T^{8} - 3637933 T^{9} + 6436343 T^{10} \)
$29$ \( 1 - 11 T + 166 T^{2} - 1174 T^{3} + 9958 T^{4} - 49617 T^{5} + 288782 T^{6} - 987334 T^{7} + 4048574 T^{8} - 7780091 T^{9} + 20511149 T^{10} \)
$31$ \( 1 + 12 T + 153 T^{2} + 1291 T^{3} + 9382 T^{4} + 57399 T^{5} + 290842 T^{6} + 1240651 T^{7} + 4558023 T^{8} + 11082252 T^{9} + 28629151 T^{10} \)
$37$ \( 1 + 7 T + 178 T^{2} + 920 T^{3} + 12752 T^{4} + 48883 T^{5} + 471824 T^{6} + 1259480 T^{7} + 9016234 T^{8} + 13119127 T^{9} + 69343957 T^{10} \)
$41$ \( 1 - 12 T + 193 T^{2} - 1363 T^{3} + 13060 T^{4} - 69271 T^{5} + 535460 T^{6} - 2291203 T^{7} + 13301753 T^{8} - 33909132 T^{9} + 115856201 T^{10} \)
$43$ \( 1 + 107 T^{2} - 197 T^{3} + 6754 T^{4} - 10885 T^{5} + 290422 T^{6} - 364253 T^{7} + 8507249 T^{8} + 147008443 T^{10} \)
$47$ \( 1 - 19 T + 168 T^{2} - 1313 T^{3} + 12955 T^{4} - 108573 T^{5} + 608885 T^{6} - 2900417 T^{7} + 17442264 T^{8} - 92713939 T^{9} + 229345007 T^{10} \)
$53$ \( 1 - 21 T + 350 T^{2} - 4071 T^{3} + 40465 T^{4} - 314151 T^{5} + 2144645 T^{6} - 11435439 T^{7} + 52106950 T^{8} - 165700101 T^{9} + 418195493 T^{10} \)
$59$ \( 1 - 7 T + 196 T^{2} - 789 T^{3} + 15775 T^{4} - 43795 T^{5} + 930725 T^{6} - 2746509 T^{7} + 40254284 T^{8} - 84821527 T^{9} + 714924299 T^{10} \)
$61$ \( 1 + 6 T + 158 T^{2} + 1262 T^{3} + 14037 T^{4} + 105179 T^{5} + 856257 T^{6} + 4695902 T^{7} + 35862998 T^{8} + 83075046 T^{9} + 844596301 T^{10} \)
$67$ \( 1 - 10 T + 142 T^{2} - 236 T^{3} - 205 T^{4} + 59767 T^{5} - 13735 T^{6} - 1059404 T^{7} + 42708346 T^{8} - 201511210 T^{9} + 1350125107 T^{10} \)
$71$ \( 1 - 35 T + 763 T^{2} - 11235 T^{3} + 130765 T^{4} - 1205501 T^{5} + 9284315 T^{6} - 56635635 T^{7} + 273086093 T^{8} - 889408835 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 + 8 T + 312 T^{2} + 2092 T^{3} + 42635 T^{4} + 220341 T^{5} + 3112355 T^{6} + 11148268 T^{7} + 121373304 T^{8} + 227185928 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 + 197 T^{2} + 805 T^{3} + 17746 T^{4} + 119621 T^{5} + 1401934 T^{6} + 5024005 T^{7} + 97128683 T^{8} + 3077056399 T^{10} \)
$83$ \( 1 - 11 T + 326 T^{2} - 2557 T^{3} + 45301 T^{4} - 272481 T^{5} + 3759983 T^{6} - 17615173 T^{7} + 186402562 T^{8} - 522041531 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 - 32 T + 606 T^{2} - 7762 T^{3} + 78409 T^{4} - 741105 T^{5} + 6978401 T^{6} - 61482802 T^{7} + 427211214 T^{8} - 2007751712 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 - 11 T + 505 T^{2} - 4153 T^{3} + 99605 T^{4} - 598793 T^{5} + 9661685 T^{6} - 39075577 T^{7} + 460899865 T^{8} - 973822091 T^{9} + 8587340257 T^{10} \)
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