Properties

Label 6042.2.a.x
Level 6042
Weight 2
Character orbit 6042.a
Self dual yes
Analytic conductor 48.246
Analytic rank 1
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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.48689336.1
Defining polynomial: \(x^{6} - 2 x^{5} - 6 x^{4} + 7 x^{3} + 9 x^{2} - 5 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 6 x^{4} + 7 x^{3} + 9 x^{2} - 5 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 3 \nu + 3 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 4 \nu^{2} + 3 \nu + 2 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 5 \nu^{3} + 5 \nu^{2} + 5 \nu - 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 2 \beta_{3} + 8 \beta_{2} + 11 \beta_{1} + 8\)
\(\nu^{5}\)\(=\)\(\beta_{5} + 2 \beta_{4} + 9 \beta_{3} + 21 \beta_{2} + 37 \beta_{1} + 12\)

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
2.95440
1.51181
0.741640
−0.290197
−1.24055
−1.67711
1.00000 1.00000 1.00000 −3.95440 1.00000 −1.81969 1.00000 1.00000 −3.95440
1.2 1.00000 1.00000 1.00000 −2.51181 1.00000 1.73805 1.00000 1.00000 −2.51181
1.3 1.00000 1.00000 1.00000 −1.74164 1.00000 1.93325 1.00000 1.00000 −1.74164
1.4 1.00000 1.00000 1.00000 −0.709803 1.00000 0.335393 1.00000 1.00000 −0.709803
1.5 1.00000 1.00000 1.00000 0.240549 1.00000 −3.02006 1.00000 1.00000 0.240549
1.6 1.00000 1.00000 1.00000 0.677113 1.00000 −5.16693 1.00000 1.00000 0.677113
\(n\): 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.x 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.x 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\( T_{5}^{6} + 8 T_{5}^{5} + 19 T_{5}^{4} + 9 T_{5}^{3} - 13 T_{5}^{2} - 6 T_{5} + 2 \)
\( T_{7}^{6} + 6 T_{7}^{5} - 5 T_{7}^{4} - 49 T_{7}^{3} + 15 T_{7}^{2} + 96 T_{7} - 32 \)
\( T_{11}^{6} - 3 T_{11}^{5} - 24 T_{11}^{4} - 3 T_{11}^{3} + 100 T_{11}^{2} + 122 T_{11} + 40 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{6} \)
$3$ \( ( 1 - T )^{6} \)
$5$ \( 1 + 8 T + 49 T^{2} + 209 T^{3} + 742 T^{4} + 2129 T^{5} + 5222 T^{6} + 10645 T^{7} + 18550 T^{8} + 26125 T^{9} + 30625 T^{10} + 25000 T^{11} + 15625 T^{12} \)
$7$ \( 1 + 6 T + 37 T^{2} + 161 T^{3} + 610 T^{4} + 2007 T^{5} + 5568 T^{6} + 14049 T^{7} + 29890 T^{8} + 55223 T^{9} + 88837 T^{10} + 100842 T^{11} + 117649 T^{12} \)
$11$ \( 1 - 3 T + 42 T^{2} - 168 T^{3} + 859 T^{4} - 3607 T^{5} + 11436 T^{6} - 39677 T^{7} + 103939 T^{8} - 223608 T^{9} + 614922 T^{10} - 483153 T^{11} + 1771561 T^{12} \)
$13$ \( 1 + 13 T + 114 T^{2} + 746 T^{3} + 4017 T^{4} + 18145 T^{5} + 70888 T^{6} + 235885 T^{7} + 678873 T^{8} + 1638962 T^{9} + 3255954 T^{10} + 4826809 T^{11} + 4826809 T^{12} \)
$17$ \( 1 + 5 T + 60 T^{2} + 274 T^{3} + 1675 T^{4} + 7673 T^{5} + 32736 T^{6} + 130441 T^{7} + 484075 T^{8} + 1346162 T^{9} + 5011260 T^{10} + 7099285 T^{11} + 24137569 T^{12} \)
$19$ \( ( 1 + T )^{6} \)
$23$ \( 1 + 12 T + 121 T^{2} + 929 T^{3} + 6032 T^{4} + 34593 T^{5} + 175436 T^{6} + 795639 T^{7} + 3190928 T^{8} + 11303143 T^{9} + 33860761 T^{10} + 77236116 T^{11} + 148035889 T^{12} \)
$29$ \( 1 + 7 T + 117 T^{2} + 519 T^{3} + 5570 T^{4} + 17518 T^{5} + 175252 T^{6} + 508022 T^{7} + 4684370 T^{8} + 12657891 T^{9} + 82751877 T^{10} + 143578043 T^{11} + 594823321 T^{12} \)
$31$ \( 1 + 17 T + 278 T^{2} + 2823 T^{3} + 25911 T^{4} + 180758 T^{5} + 1131684 T^{6} + 5603498 T^{7} + 24900471 T^{8} + 84099993 T^{9} + 256738838 T^{10} + 486695567 T^{11} + 887503681 T^{12} \)
$37$ \( 1 + 15 T + 206 T^{2} + 1494 T^{3} + 10791 T^{4} + 48459 T^{5} + 328340 T^{6} + 1792983 T^{7} + 14772879 T^{8} + 75675582 T^{9} + 386077166 T^{10} + 1040159355 T^{11} + 2565726409 T^{12} \)
$41$ \( 1 + 12 T + 253 T^{2} + 2095 T^{3} + 25373 T^{4} + 159093 T^{5} + 1375802 T^{6} + 6522813 T^{7} + 42652013 T^{8} + 144389495 T^{9} + 714917533 T^{10} + 1390274412 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 + 4 T + 213 T^{2} + 633 T^{3} + 20088 T^{4} + 46213 T^{5} + 1100572 T^{6} + 1987159 T^{7} + 37142712 T^{8} + 50327931 T^{9} + 728204613 T^{10} + 588033772 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 + 29 T + 547 T^{2} + 7432 T^{3} + 81225 T^{4} + 723457 T^{5} + 5427798 T^{6} + 34002479 T^{7} + 179426025 T^{8} + 771612536 T^{9} + 2669185507 T^{10} + 6651005203 T^{11} + 10779215329 T^{12} \)
$53$ \( ( 1 + T )^{6} \)
$59$ \( 1 - 9 T + 325 T^{2} - 2181 T^{3} + 44044 T^{4} - 228320 T^{5} + 3345116 T^{6} - 13470880 T^{7} + 153317164 T^{8} - 447931599 T^{9} + 3938142325 T^{10} - 6434318691 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 + 16 T + 251 T^{2} + 1349 T^{3} + 5849 T^{4} - 90945 T^{5} - 692634 T^{6} - 5547645 T^{7} + 21764129 T^{8} + 306197369 T^{9} + 3475306091 T^{10} + 13513540816 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 + 8 T + 55 T^{2} + 57 T^{3} + 6436 T^{4} + 40449 T^{5} + 546624 T^{6} + 2710083 T^{7} + 28891204 T^{8} + 17143491 T^{9} + 1108311655 T^{10} + 10801000856 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 + 2 T + 218 T^{2} - 210 T^{3} + 24127 T^{4} - 52900 T^{5} + 1968492 T^{6} - 3755900 T^{7} + 121624207 T^{8} - 75161310 T^{9} + 5539746458 T^{10} + 3608458702 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 + 17 T + 246 T^{2} + 2732 T^{3} + 27179 T^{4} + 243467 T^{5} + 2203628 T^{6} + 17773091 T^{7} + 144836891 T^{8} + 1062794444 T^{9} + 6985967286 T^{10} + 35242217081 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 + 31 T + 740 T^{2} + 12130 T^{3} + 166991 T^{4} + 1863487 T^{5} + 18081240 T^{6} + 147215473 T^{7} + 1042190831 T^{8} + 5980563070 T^{9} + 28823059940 T^{10} + 95388748369 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 + 5 T + 280 T^{2} + 1936 T^{3} + 41907 T^{4} + 285151 T^{5} + 4219320 T^{6} + 23667533 T^{7} + 288697323 T^{8} + 1106979632 T^{9} + 13288329880 T^{10} + 19695203215 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 + 5 T + 385 T^{2} + 1459 T^{3} + 71568 T^{4} + 219484 T^{5} + 8022420 T^{6} + 19534076 T^{7} + 566890128 T^{8} + 1028549771 T^{9} + 24155762785 T^{10} + 27920297245 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 - 7 T + 268 T^{2} - 2358 T^{3} + 39909 T^{4} - 386763 T^{5} + 4139276 T^{6} - 37516011 T^{7} + 375503781 T^{8} - 2152082934 T^{9} + 23725847308 T^{10} - 60111381799 T^{11} + 832972004929 T^{12} \)
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