Properties

Label 6042.2.a.w
Level 6042
Weight 2
Character orbit 6042.a
Self dual yes
Analytic conductor 48.246
Analytic rank 0
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2456129013\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.326108912.1
Defining polynomial: \(x^{6} - 3 x^{5} - 11 x^{4} + 25 x^{3} + 12 x^{2} - 32 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + \beta_{3} q^{5} - q^{6} + ( -\beta_{1} - \beta_{3} - \beta_{5} ) q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} + \beta_{3} q^{5} - q^{6} + ( -\beta_{1} - \beta_{3} - \beta_{5} ) q^{7} + q^{8} + q^{9} + \beta_{3} q^{10} + ( -\beta_{1} - \beta_{4} - \beta_{5} ) q^{11} - q^{12} + ( 1 + \beta_{1} + \beta_{3} - \beta_{4} ) q^{13} + ( -\beta_{1} - \beta_{3} - \beta_{5} ) q^{14} -\beta_{3} q^{15} + q^{16} + 2 q^{17} + q^{18} - q^{19} + \beta_{3} q^{20} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{21} + ( -\beta_{1} - \beta_{4} - \beta_{5} ) q^{22} + ( -2 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{23} - q^{24} + ( 2 + \beta_{2} - \beta_{4} - \beta_{5} ) q^{25} + ( 1 + \beta_{1} + \beta_{3} - \beta_{4} ) q^{26} - q^{27} + ( -\beta_{1} - \beta_{3} - \beta_{5} ) q^{28} + ( 4 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{29} -\beta_{3} q^{30} + ( -3 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{31} + q^{32} + ( \beta_{1} + \beta_{4} + \beta_{5} ) q^{33} + 2 q^{34} + ( -3 - \beta_{2} - \beta_{5} ) q^{35} + q^{36} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} ) q^{37} - q^{38} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} ) q^{39} + \beta_{3} q^{40} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{41} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{42} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{5} ) q^{43} + ( -\beta_{1} - \beta_{4} - \beta_{5} ) q^{44} + \beta_{3} q^{45} + ( -2 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{46} + ( 3 - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{47} - q^{48} + ( 2 - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{49} + ( 2 + \beta_{2} - \beta_{4} - \beta_{5} ) q^{50} -2 q^{51} + ( 1 + \beta_{1} + \beta_{3} - \beta_{4} ) q^{52} - q^{53} - q^{54} + ( 5 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{55} + ( -\beta_{1} - \beta_{3} - \beta_{5} ) q^{56} + q^{57} + ( 4 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{58} + ( 1 - 2 \beta_{1} + \beta_{5} ) q^{59} -\beta_{3} q^{60} + ( 4 + 2 \beta_{1} - \beta_{2} + 2 \beta_{4} ) q^{61} + ( -3 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{62} + ( -\beta_{1} - \beta_{3} - \beta_{5} ) q^{63} + q^{64} + ( 5 - \beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{5} ) q^{65} + ( \beta_{1} + \beta_{4} + \beta_{5} ) q^{66} + ( -\beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{67} + 2 q^{68} + ( 2 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{69} + ( -3 - \beta_{2} - \beta_{5} ) q^{70} + ( 2 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{71} + q^{72} + ( 3 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{73} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} ) q^{74} + ( -2 - \beta_{2} + \beta_{4} + \beta_{5} ) q^{75} - q^{76} + ( 2 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{77} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} ) q^{78} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{79} + \beta_{3} q^{80} + q^{81} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{82} + ( -2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{83} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{84} + 2 \beta_{3} q^{85} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{5} ) q^{86} + ( -4 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{87} + ( -\beta_{1} - \beta_{4} - \beta_{5} ) q^{88} + ( 1 - 2 \beta_{1} - 4 \beta_{3} - \beta_{5} ) q^{89} + \beta_{3} q^{90} + ( -8 - 2 \beta_{4} ) q^{91} + ( -2 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{92} + ( 3 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{93} + ( 3 - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{94} -\beta_{3} q^{95} - q^{96} + ( 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{97} + ( 2 - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{98} + ( -\beta_{1} - \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} - 6q^{3} + 6q^{4} - q^{5} - 6q^{6} + q^{7} + 6q^{8} + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{2} - 6q^{3} + 6q^{4} - q^{5} - 6q^{6} + q^{7} + 6q^{8} + 6q^{9} - q^{10} + 2q^{11} - 6q^{12} + 6q^{13} + q^{14} + q^{15} + 6q^{16} + 12q^{17} + 6q^{18} - 6q^{19} - q^{20} - q^{21} + 2q^{22} - 9q^{23} - 6q^{24} + 11q^{25} + 6q^{26} - 6q^{27} + q^{28} + 21q^{29} + q^{30} + 5q^{31} + 6q^{32} - 2q^{33} + 12q^{34} - 17q^{35} + 6q^{36} - 8q^{37} - 6q^{38} - 6q^{39} - q^{40} + 16q^{41} - q^{42} - 5q^{43} + 2q^{44} - q^{45} - 9q^{46} + 16q^{47} - 6q^{48} + 11q^{49} + 11q^{50} - 12q^{51} + 6q^{52} - 6q^{53} - 6q^{54} + 22q^{55} + q^{56} + 6q^{57} + 21q^{58} + 9q^{59} + q^{60} + 20q^{61} + 5q^{62} + q^{63} + 6q^{64} + 22q^{65} - 2q^{66} - 3q^{67} + 12q^{68} + 9q^{69} - 17q^{70} + 10q^{71} + 6q^{72} + 18q^{73} - 8q^{74} - 11q^{75} - 6q^{76} + 16q^{77} - 6q^{78} - 14q^{79} - q^{80} + 6q^{81} + 16q^{82} + 2q^{83} - q^{84} - 2q^{85} - 5q^{86} - 21q^{87} + 2q^{88} + 11q^{89} - q^{90} - 44q^{91} - 9q^{92} - 5q^{93} + 16q^{94} + q^{95} - 6q^{96} + 11q^{98} + 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 11 x^{4} + 25 x^{3} + 12 x^{2} - 32 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{5} + 5 \nu^{4} + 22 \nu^{3} - 29 \nu^{2} - 26 \nu + 6 \)\()/10\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 16 \nu^{3} - 13 \nu^{2} + 38 \nu + 12 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( 4 \nu^{5} - 5 \nu^{4} - 54 \nu^{3} + 3 \nu^{2} + 82 \nu + 28 \)\()/10\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{5} - 5 \nu^{4} - 38 \nu^{3} + 21 \nu^{2} + 49 \nu - 14 \)\()/5\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 13 \nu^{3} + 12 \nu^{2} + 24 \nu - 10 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{2} - \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{4} - 3 \beta_{3} + \beta_{2} + \beta_{1} + 11\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-4 \beta_{5} + 6 \beta_{4} - 15 \beta_{3} + 11 \beta_{2} - 11 \beta_{1} + 19\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-8 \beta_{5} + 34 \beta_{4} - 53 \beta_{3} + 27 \beta_{2} + 3 \beta_{1} + 137\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-64 \beta_{5} + 122 \beta_{4} - 241 \beta_{3} + 161 \beta_{2} - 125 \beta_{1} + 385\)\()/2\)

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
1.53707
4.13445
−2.86940
−1.25042
1.31489
0.133418
1.00000 −1.00000 1.00000 −2.85609 −1.00000 1.31587 1.00000 1.00000 −2.85609
1.2 1.00000 −1.00000 1.00000 −2.67572 −1.00000 2.79372 1.00000 1.00000 −2.67572
1.3 1.00000 −1.00000 1.00000 −2.38494 −1.00000 −1.10454 1.00000 1.00000 −2.38494
1.4 1.00000 −1.00000 1.00000 1.12805 −1.00000 3.91138 1.00000 1.00000 1.12805
1.5 1.00000 −1.00000 1.00000 1.90230 −1.00000 −5.13108 1.00000 1.00000 1.90230
1.6 1.00000 −1.00000 1.00000 3.88640 −1.00000 −0.785350 1.00000 1.00000 3.88640
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 6042.2.a.w 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.w 6 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(19\) \(1\)
\(53\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\( T_{5}^{6} + T_{5}^{5} - 20 T_{5}^{4} - 24 T_{5}^{3} + 98 T_{5}^{2} + 80 T_{5} - 152 \)
\( T_{7}^{6} - T_{7}^{5} - 26 T_{7}^{4} + 44 T_{7}^{3} + 72 T_{7}^{2} - 64 T_{7} - 64 \)
\( T_{11}^{6} - 2 T_{11}^{5} - 40 T_{11}^{4} + 66 T_{11}^{3} + 456 T_{11}^{2} - 400 T_{11} - 1664 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{6} \)
$3$ \( ( 1 + T )^{6} \)
$5$ \( 1 + T + 10 T^{2} + T^{3} + 73 T^{4} - 30 T^{5} + 328 T^{6} - 150 T^{7} + 1825 T^{8} + 125 T^{9} + 6250 T^{10} + 3125 T^{11} + 15625 T^{12} \)
$7$ \( 1 - T + 16 T^{2} + 9 T^{3} + 79 T^{4} + 370 T^{5} + 160 T^{6} + 2590 T^{7} + 3871 T^{8} + 3087 T^{9} + 38416 T^{10} - 16807 T^{11} + 117649 T^{12} \)
$11$ \( 1 - 2 T + 26 T^{2} - 44 T^{3} + 511 T^{4} - 642 T^{5} + 5948 T^{6} - 7062 T^{7} + 61831 T^{8} - 58564 T^{9} + 380666 T^{10} - 322102 T^{11} + 1771561 T^{12} \)
$13$ \( 1 - 6 T + 32 T^{2} - 82 T^{3} + 255 T^{4} - 520 T^{5} + 2640 T^{6} - 6760 T^{7} + 43095 T^{8} - 180154 T^{9} + 913952 T^{10} - 2227758 T^{11} + 4826809 T^{12} \)
$17$ \( ( 1 - 2 T + 17 T^{2} )^{6} \)
$19$ \( ( 1 + T )^{6} \)
$23$ \( 1 + 9 T + 132 T^{2} + 815 T^{3} + 6819 T^{4} + 31982 T^{5} + 198288 T^{6} + 735586 T^{7} + 3607251 T^{8} + 9916105 T^{9} + 36939012 T^{10} + 57927087 T^{11} + 148035889 T^{12} \)
$29$ \( 1 - 21 T + 318 T^{2} - 3297 T^{3} + 28251 T^{4} - 193910 T^{5} + 1147740 T^{6} - 5623390 T^{7} + 23759091 T^{8} - 80410533 T^{9} + 224915358 T^{10} - 430734129 T^{11} + 594823321 T^{12} \)
$31$ \( 1 - 5 T + 68 T^{2} - 287 T^{3} + 1915 T^{4} - 2522 T^{5} + 47456 T^{6} - 78182 T^{7} + 1840315 T^{8} - 8550017 T^{9} + 62799428 T^{10} - 143145755 T^{11} + 887503681 T^{12} \)
$37$ \( 1 + 8 T + 152 T^{2} + 1200 T^{3} + 12159 T^{4} + 75296 T^{5} + 586416 T^{6} + 2785952 T^{7} + 16645671 T^{8} + 60783600 T^{9} + 284872472 T^{10} + 554751656 T^{11} + 2565726409 T^{12} \)
$41$ \( 1 - 16 T + 266 T^{2} - 2640 T^{3} + 25423 T^{4} - 185824 T^{5} + 1326956 T^{6} - 7618784 T^{7} + 42736063 T^{8} - 181951440 T^{9} + 751652426 T^{10} - 1853699216 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 + 5 T + 152 T^{2} + 187 T^{3} + 7935 T^{4} - 20550 T^{5} + 282368 T^{6} - 883650 T^{7} + 14671815 T^{8} + 14867809 T^{9} + 519657752 T^{10} + 735042215 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 - 16 T + 268 T^{2} - 2826 T^{3} + 28891 T^{4} - 229834 T^{5} + 1744960 T^{6} - 10802198 T^{7} + 63820219 T^{8} - 293403798 T^{9} + 1307754508 T^{10} - 3669520112 T^{11} + 10779215329 T^{12} \)
$53$ \( ( 1 + T )^{6} \)
$59$ \( 1 - 9 T + 246 T^{2} - 1569 T^{3} + 26375 T^{4} - 131052 T^{5} + 1811860 T^{6} - 7732068 T^{7} + 91811375 T^{8} - 322239651 T^{9} + 2980870806 T^{10} - 6434318691 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 - 20 T + 332 T^{2} - 4264 T^{3} + 46263 T^{4} - 441732 T^{5} + 3581480 T^{6} - 26945652 T^{7} + 172144623 T^{8} - 967846984 T^{9} + 4596819212 T^{10} - 16891926020 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 + 3 T + 202 T^{2} + 921 T^{3} + 25575 T^{4} + 99674 T^{5} + 2102252 T^{6} + 6678158 T^{7} + 114806175 T^{8} + 277002723 T^{9} + 4070526442 T^{10} + 4050375321 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 - 10 T + 218 T^{2} - 2858 T^{3} + 32159 T^{4} - 309040 T^{5} + 3104620 T^{6} - 21941840 T^{7} + 162113519 T^{8} - 1022909638 T^{9} + 5539746458 T^{10} - 18042293510 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 - 18 T + 514 T^{2} - 6386 T^{3} + 100495 T^{4} - 916236 T^{5} + 9970812 T^{6} - 66885228 T^{7} + 535537855 T^{8} - 2484262562 T^{9} + 14596695874 T^{10} - 37315288674 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 + 14 T + 358 T^{2} + 4374 T^{3} + 61487 T^{4} + 622456 T^{5} + 6179732 T^{6} + 49174024 T^{7} + 383740367 T^{8} + 2156552586 T^{9} + 13944128998 T^{10} + 43078789586 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 - 2 T + 96 T^{2} - 678 T^{3} + 12559 T^{4} - 29724 T^{5} + 1156368 T^{6} - 2467092 T^{7} + 86518951 T^{8} - 387671586 T^{9} + 4555998816 T^{10} - 7878081286 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 - 11 T + 344 T^{2} - 2821 T^{3} + 57651 T^{4} - 396424 T^{5} + 6342248 T^{6} - 35281736 T^{7} + 456653571 T^{8} - 1988717549 T^{9} + 21583330904 T^{10} - 61424653939 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 + 210 T^{2} + 496 T^{3} + 30031 T^{4} + 117264 T^{5} + 2844476 T^{6} + 11374608 T^{7} + 282561679 T^{8} + 452685808 T^{9} + 18591149010 T^{10} + 832972004929 T^{12} \)
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