Properties

Label 6012.2.a.k
Level $6012$
Weight $2$
Character orbit 6012.a
Self dual yes
Analytic conductor $48.006$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 4 x^{9} - 26 x^{8} + 82 x^{7} + 211 x^{6} - 340 x^{5} - 593 x^{4} + 192 x^{3} + 423 x^{2} + 126 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{3} ) q^{5} + \beta_{1} q^{7} +O(q^{10})\) \( q + ( 1 + \beta_{3} ) q^{5} + \beta_{1} q^{7} + ( \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{11} -\beta_{9} q^{13} + ( -\beta_{3} + \beta_{5} + \beta_{7} - \beta_{9} ) q^{17} + ( -\beta_{1} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{19} + ( 3 - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{23} + ( 2 + 2 \beta_{2} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{25} + ( 1 - \beta_{2} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{29} + ( -\beta_{1} + \beta_{2} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{31} + ( -1 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{35} + ( -\beta_{1} - \beta_{4} - \beta_{7} + \beta_{9} ) q^{37} + ( -3 + \beta_{1} - \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + \beta_{8} + \beta_{9} ) q^{41} + ( 2 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + 2 \beta_{8} ) q^{43} + ( 5 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{7} + \beta_{9} ) q^{47} + ( 1 + \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{8} ) q^{49} + ( 3 - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{8} - \beta_{9} ) q^{53} + ( -1 + 2 \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{55} + ( 1 - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{9} ) q^{59} + ( -2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} - 3 \beta_{7} - \beta_{8} + \beta_{9} ) q^{61} + ( 1 + 3 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{65} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{7} - \beta_{9} ) q^{67} + ( 4 - 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} - 2 \beta_{9} ) q^{71} + ( 2 + \beta_{1} + 2 \beta_{3} + \beta_{4} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{73} + ( 2 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} - 2 \beta_{8} + \beta_{9} ) q^{77} + ( -\beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} ) q^{79} + ( 4 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{83} + ( -5 + 2 \beta_{1} + \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{85} + ( 1 - 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{89} + ( 5 + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{91} + ( 2 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} ) q^{95} + ( -1 - 3 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 6q^{5} + 4q^{7} + O(q^{10}) \) \( 10q + 6q^{5} + 4q^{7} + 8q^{11} - 2q^{13} + 6q^{17} + 20q^{23} + 24q^{25} + 8q^{29} - 4q^{31} - 4q^{37} - 14q^{41} + 20q^{43} + 48q^{47} - 2q^{49} + 22q^{53} - 6q^{55} + 2q^{59} - 8q^{61} + 28q^{65} - 6q^{67} + 20q^{71} + 20q^{73} + 24q^{77} - 4q^{79} + 46q^{83} - 18q^{85} - 8q^{89} + 28q^{91} + 36q^{95} - 34q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 4 x^{9} - 26 x^{8} + 82 x^{7} + 211 x^{6} - 340 x^{5} - 593 x^{4} + 192 x^{3} + 423 x^{2} + 126 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -1438 \nu^{9} + 6681 \nu^{8} + 32441 \nu^{7} - 136919 \nu^{6} - 197274 \nu^{5} + 584470 \nu^{4} + 319314 \nu^{3} - 469017 \nu^{2} + 48114 \nu + 66348 \)\()/44982\)
\(\beta_{3}\)\(=\)\((\)\( 1499 \nu^{9} - 6987 \nu^{8} - 34945 \nu^{7} + 148276 \nu^{6} + 234612 \nu^{5} - 710756 \nu^{4} - 561669 \nu^{3} + 837054 \nu^{2} + 435294 \nu - 151056 \)\()/44982\)
\(\beta_{4}\)\(=\)\((\)\( -2840 \nu^{9} + 9945 \nu^{8} + 81184 \nu^{7} - 204046 \nu^{6} - 752934 \nu^{5} + 832712 \nu^{4} + 2390865 \nu^{3} - 463230 \nu^{2} - 1880118 \nu - 239175 \)\()/44982\)
\(\beta_{5}\)\(=\)\((\)\( -1167 \nu^{9} + 5185 \nu^{8} + 27571 \nu^{7} - 105787 \nu^{6} - 188846 \nu^{5} + 438577 \nu^{4} + 436038 \nu^{3} - 257619 \nu^{2} - 276453 \nu - 92196 \)\()/14994\)
\(\beta_{6}\)\(=\)\((\)\( 426 \nu^{9} - 1700 \nu^{8} - 11178 \nu^{7} + 35355 \nu^{6} + 91532 \nu^{5} - 154395 \nu^{4} - 254713 \nu^{3} + 119881 \nu^{2} + 168813 \nu + 29004 \)\()/4998\)
\(\beta_{7}\)\(=\)\((\)\( 1849 \nu^{9} - 8415 \nu^{8} - 43331 \nu^{7} + 174848 \nu^{6} + 291942 \nu^{5} - 775366 \nu^{4} - 667089 \nu^{3} + 651792 \nu^{2} + 460620 \nu + 60750 \)\()/14994\)
\(\beta_{8}\)\(=\)\((\)\( 2419 \nu^{9} - 9622 \nu^{8} - 63438 \nu^{7} + 198654 \nu^{6} + 521801 \nu^{5} - 846911 \nu^{4} - 1490655 \nu^{3} + 595053 \nu^{2} + 1098198 \nu + 204129 \)\()/14994\)
\(\beta_{9}\)\(=\)\((\)\( 14219 \nu^{9} - 61455 \nu^{8} - 349141 \nu^{7} + 1274200 \nu^{6} + 2574117 \nu^{5} - 5573576 \nu^{4} - 6555282 \nu^{3} + 4419333 \nu^{2} + 4422843 \nu + 679113 \)\()/44982\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-2 \beta_{8} + \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{1} + 8\)
\(\nu^{3}\)\(=\)\(3 \beta_{9} - 3 \beta_{8} - 6 \beta_{7} + \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + \beta_{2} + 18 \beta_{1} + 7\)
\(\nu^{4}\)\(=\)\(5 \beta_{9} - 43 \beta_{8} - 11 \beta_{7} + 25 \beta_{6} - 26 \beta_{5} - 34 \beta_{4} + 21 \beta_{3} + 9 \beta_{2} + 38 \beta_{1} + 126\)
\(\nu^{5}\)\(=\)\(65 \beta_{9} - 93 \beta_{8} - 146 \beta_{7} + 46 \beta_{6} - 79 \beta_{5} - 18 \beta_{4} + 77 \beta_{3} + 41 \beta_{2} + 354 \beta_{1} + 226\)
\(\nu^{6}\)\(=\)\(138 \beta_{9} - 866 \beta_{8} - 381 \beta_{7} + 571 \beta_{6} - 625 \beta_{5} - 584 \beta_{4} + 520 \beta_{3} + 264 \beta_{2} + 1030 \beta_{1} + 2345\)
\(\nu^{7}\)\(=\)\(1227 \beta_{9} - 2433 \beta_{8} - 3207 \beta_{7} + 1585 \beta_{6} - 2366 \beta_{5} - 684 \beta_{4} + 2249 \beta_{3} + 1165 \beta_{2} + 7317 \beta_{1} + 6043\)
\(\nu^{8}\)\(=\)\(3143 \beta_{9} - 17755 \beta_{8} - 10508 \beta_{7} + 13075 \beta_{6} - 15044 \beta_{5} - 10450 \beta_{4} + 13002 \beta_{3} + 6621 \beta_{2} + 25514 \beta_{1} + 46794\)
\(\nu^{9}\)\(=\)\(22925 \beta_{9} - 59655 \beta_{8} - 70667 \beta_{7} + 45883 \beta_{6} - 63610 \beta_{5} - 19074 \beta_{4} + 59723 \beta_{3} + 30131 \beta_{2} + 156123 \beta_{1} + 149656\)

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
−0.291281
−0.108196
4.25727
−0.617293
2.09707
−1.19255
−3.81111
−2.20804
4.79792
1.07621
0 0 0 −3.41200 0 −0.291281 0 0 0
1.2 0 0 0 −3.17375 0 −0.108196 0 0 0
1.3 0 0 0 −1.39178 0 4.25727 0 0 0
1.4 0 0 0 −0.860385 0 −0.617293 0 0 0
1.5 0 0 0 0.399927 0 2.09707 0 0 0
1.6 0 0 0 0.553287 0 −1.19255 0 0 0
1.7 0 0 0 2.67502 0 −3.81111 0 0 0
1.8 0 0 0 3.55061 0 −2.20804 0 0 0
1.9 0 0 0 3.67303 0 4.79792 0 0 0
1.10 0 0 0 3.98604 0 1.07621 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.k yes 10
3.b odd 2 1 6012.2.a.j 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6012.2.a.j 10 3.b odd 2 1
6012.2.a.k yes 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{10} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6012))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \)
$3$ \( T^{10} \)
$5$ \( 399 - 1196 T - 587 T^{2} + 2728 T^{3} + 404 T^{4} - 1182 T^{5} + 55 T^{6} + 152 T^{7} - 19 T^{8} - 6 T^{9} + T^{10} \)
$7$ \( 9 + 126 T + 423 T^{2} + 192 T^{3} - 593 T^{4} - 340 T^{5} + 211 T^{6} + 82 T^{7} - 26 T^{8} - 4 T^{9} + T^{10} \)
$11$ \( -50841 + 135692 T - 117001 T^{2} + 25488 T^{3} + 14188 T^{4} - 6884 T^{5} - 75 T^{6} + 424 T^{7} - 37 T^{8} - 8 T^{9} + T^{10} \)
$13$ \( 9289 - 40164 T + 27274 T^{2} + 10028 T^{3} - 10490 T^{4} - 246 T^{5} + 1223 T^{6} - 48 T^{7} - 58 T^{8} + 2 T^{9} + T^{10} \)
$17$ \( -655837 - 807334 T + 160245 T^{2} + 211136 T^{3} - 26850 T^{4} - 19124 T^{5} + 2602 T^{6} + 640 T^{7} - 97 T^{8} - 6 T^{9} + T^{10} \)
$19$ \( -62927 + 43712 T + 143460 T^{2} - 21682 T^{3} - 51688 T^{4} + 642 T^{5} + 4201 T^{6} - 118 T^{8} + T^{10} \)
$23$ \( -2401 + 26180 T - 5891 T^{2} - 45624 T^{3} + 29236 T^{4} + 2284 T^{5} - 3673 T^{6} + 370 T^{7} + 97 T^{8} - 20 T^{9} + T^{10} \)
$29$ \( 199927 - 333944 T - 265733 T^{2} + 229740 T^{3} + 5686 T^{4} - 24948 T^{5} + 1995 T^{6} + 860 T^{7} - 99 T^{8} - 8 T^{9} + T^{10} \)
$31$ \( -496399 + 4595216 T + 1518122 T^{2} - 687628 T^{3} - 205391 T^{4} + 34786 T^{5} + 9424 T^{6} - 658 T^{7} - 168 T^{8} + 4 T^{9} + T^{10} \)
$37$ \( -7450623 - 716688 T + 3703563 T^{2} - 214038 T^{3} - 310140 T^{4} + 18522 T^{5} + 10635 T^{6} - 478 T^{7} - 167 T^{8} + 4 T^{9} + T^{10} \)
$41$ \( 1233827 + 190930 T - 1522102 T^{2} - 256052 T^{3} + 319559 T^{4} + 95488 T^{5} - 872 T^{6} - 2310 T^{7} - 120 T^{8} + 14 T^{9} + T^{10} \)
$43$ \( 1501993 - 5296254 T - 789550 T^{2} + 1089960 T^{3} + 112427 T^{4} - 68326 T^{5} - 3522 T^{6} + 1884 T^{7} - 6 T^{8} - 20 T^{9} + T^{10} \)
$47$ \( -720657 - 3637008 T + 1262754 T^{2} + 953100 T^{3} - 469629 T^{4} + 2472 T^{5} + 34667 T^{6} - 8456 T^{7} + 913 T^{8} - 48 T^{9} + T^{10} \)
$53$ \( -3701117 + 6393632 T - 3374086 T^{2} + 162682 T^{3} + 325717 T^{4} - 57362 T^{5} - 8589 T^{6} + 2110 T^{7} + 21 T^{8} - 22 T^{9} + T^{10} \)
$59$ \( 304311 - 932152 T + 493274 T^{2} + 482238 T^{3} - 229391 T^{4} - 51366 T^{5} + 17571 T^{6} + 702 T^{7} - 275 T^{8} - 2 T^{9} + T^{10} \)
$61$ \( -64024443 + 48023712 T + 16933248 T^{2} - 8452494 T^{3} - 1769864 T^{4} + 234042 T^{5} + 41937 T^{6} - 2376 T^{7} - 354 T^{8} + 8 T^{9} + T^{10} \)
$67$ \( 92973717 + 160633388 T + 24725470 T^{2} - 9253972 T^{3} - 1620746 T^{4} + 196782 T^{5} + 34935 T^{6} - 1808 T^{7} - 314 T^{8} + 6 T^{9} + T^{10} \)
$71$ \( 609118083 - 674212374 T - 56505321 T^{2} + 42448404 T^{3} + 443937 T^{4} - 893378 T^{5} + 20292 T^{6} + 7364 T^{7} - 297 T^{8} - 20 T^{9} + T^{10} \)
$73$ \( 9538237 + 5055418 T - 6826738 T^{2} + 567014 T^{3} + 627484 T^{4} - 107602 T^{5} - 14133 T^{6} + 3650 T^{7} - 78 T^{8} - 20 T^{9} + T^{10} \)
$79$ \( 189 - 9810 T - 51318 T^{2} - 52740 T^{3} - 6585 T^{4} + 10382 T^{5} + 2686 T^{6} - 420 T^{7} - 106 T^{8} + 4 T^{9} + T^{10} \)
$83$ \( 3298771 - 4595294 T - 2006974 T^{2} + 4257930 T^{3} - 1845175 T^{4} + 307022 T^{5} - 4125 T^{6} - 5584 T^{7} + 797 T^{8} - 46 T^{9} + T^{10} \)
$89$ \( 11998287 + 349555696 T + 87857450 T^{2} - 24275920 T^{3} - 4684834 T^{4} + 469740 T^{5} + 74129 T^{6} - 3356 T^{7} - 462 T^{8} + 8 T^{9} + T^{10} \)
$97$ \( 95931549 + 57882378 T - 379507249 T^{2} - 37122510 T^{3} + 10866707 T^{4} + 1270124 T^{5} - 70448 T^{6} - 11934 T^{7} - 51 T^{8} + 34 T^{9} + T^{10} \)
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