Properties

Label 6042.2.a.u
Level 6042
Weight 2
Character orbit 6042.a
Self dual yes
Analytic conductor 48.246
Analytic rank 0
Dimension 5
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: \(5\)
Coefficient field: 5.5.14377697.1
Defining polynomial: \(x^{5} - 2 x^{4} - 9 x^{3} + 5 x^{2} + 19 x + 6\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 9 x^{3} + 5 x^{2} + 19 x + 6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 3 \nu^{2} - 3 \nu + 5 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 3 \nu^{3} - 6 \nu^{2} + 10 \nu + 9 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 3 \beta_{2} + 9 \beta_{1} + 7\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 3 \beta_{3} + 15 \beta_{2} + 29 \beta_{1} + 36\)

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
−1.84112
−1.28745
−0.375482
1.83975
3.66430
1.00000 −1.00000 1.00000 −1.84112 −1.00000 −3.07196 1.00000 1.00000 −1.84112
1.2 1.00000 −1.00000 1.00000 −1.28745 −1.00000 −0.232422 1.00000 1.00000 −1.28745
1.3 1.00000 −1.00000 1.00000 −0.375482 −1.00000 3.10805 1.00000 1.00000 −0.375482
1.4 1.00000 −1.00000 1.00000 1.83975 −1.00000 4.29482 1.00000 1.00000 1.83975
1.5 1.00000 −1.00000 1.00000 3.66430 −1.00000 −2.09848 1.00000 1.00000 3.66430
\(n\): 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 6042.2.a.u 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.u 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\( T_{5}^{5} - 2 T_{5}^{4} - 9 T_{5}^{3} + 5 T_{5}^{2} + 19 T_{5} + 6 \)
\( T_{7}^{5} - 2 T_{7}^{4} - 19 T_{7}^{3} + 17 T_{7}^{2} + 91 T_{7} + 20 \)
\( T_{11}^{5} + 3 T_{11}^{4} - 24 T_{11}^{3} - 77 T_{11}^{2} + 44 T_{11} + 192 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{5} \)
$3$ \( ( 1 + T )^{5} \)
$5$ \( 1 - 2 T + 16 T^{2} - 35 T^{3} + 134 T^{4} - 244 T^{5} + 670 T^{6} - 875 T^{7} + 2000 T^{8} - 1250 T^{9} + 3125 T^{10} \)
$7$ \( 1 - 2 T + 16 T^{2} - 39 T^{3} + 182 T^{4} - 330 T^{5} + 1274 T^{6} - 1911 T^{7} + 5488 T^{8} - 4802 T^{9} + 16807 T^{10} \)
$11$ \( 1 + 3 T + 31 T^{2} + 55 T^{3} + 462 T^{4} + 676 T^{5} + 5082 T^{6} + 6655 T^{7} + 41261 T^{8} + 43923 T^{9} + 161051 T^{10} \)
$13$ \( 1 - 7 T + 61 T^{2} - 279 T^{3} + 1416 T^{4} - 4848 T^{5} + 18408 T^{6} - 47151 T^{7} + 134017 T^{8} - 199927 T^{9} + 371293 T^{10} \)
$17$ \( 1 - 11 T + 89 T^{2} - 509 T^{3} + 2652 T^{4} - 11484 T^{5} + 45084 T^{6} - 147101 T^{7} + 437257 T^{8} - 918731 T^{9} + 1419857 T^{10} \)
$19$ \( ( 1 - T )^{5} \)
$23$ \( 1 - 4 T + 38 T^{2} - 105 T^{3} - 292 T^{4} - 514 T^{5} - 6716 T^{6} - 55545 T^{7} + 462346 T^{8} - 1119364 T^{9} + 6436343 T^{10} \)
$29$ \( 1 - 5 T + 106 T^{2} - 470 T^{3} + 5456 T^{4} - 18596 T^{5} + 158224 T^{6} - 395270 T^{7} + 2585234 T^{8} - 3536405 T^{9} + 20511149 T^{10} \)
$31$ \( 1 - 5 T + 135 T^{2} - 496 T^{3} + 7630 T^{4} - 21206 T^{5} + 236530 T^{6} - 476656 T^{7} + 4021785 T^{8} - 4617605 T^{9} + 28629151 T^{10} \)
$37$ \( 1 + 3 T + 161 T^{2} + 367 T^{3} + 11070 T^{4} + 19136 T^{5} + 409590 T^{6} + 502423 T^{7} + 8155133 T^{8} + 5622483 T^{9} + 69343957 T^{10} \)
$41$ \( 1 - 24 T + 400 T^{2} - 4519 T^{3} + 41023 T^{4} - 289010 T^{5} + 1681943 T^{6} - 7596439 T^{7} + 27568400 T^{8} - 67818264 T^{9} + 115856201 T^{10} \)
$43$ \( 1 - 6 T + 126 T^{2} - 111 T^{3} + 4236 T^{4} + 15222 T^{5} + 182148 T^{6} - 205239 T^{7} + 10017882 T^{8} - 20512806 T^{9} + 147008443 T^{10} \)
$47$ \( 1 + 23 T + 378 T^{2} + 4421 T^{3} + 41517 T^{4} + 314152 T^{5} + 1951299 T^{6} + 9765989 T^{7} + 39245094 T^{8} + 112232663 T^{9} + 229345007 T^{10} \)
$53$ \( ( 1 - T )^{5} \)
$59$ \( 1 - 11 T + 270 T^{2} - 2240 T^{3} + 30618 T^{4} - 188132 T^{5} + 1806462 T^{6} - 7797440 T^{7} + 55452330 T^{8} - 133290971 T^{9} + 714924299 T^{10} \)
$61$ \( 1 - 16 T + 288 T^{2} - 3261 T^{3} + 33585 T^{4} - 281770 T^{5} + 2048685 T^{6} - 12134181 T^{7} + 65370528 T^{8} - 221533456 T^{9} + 844596301 T^{10} \)
$67$ \( 1 - 4 T + 246 T^{2} - 657 T^{3} + 27996 T^{4} - 56490 T^{5} + 1875732 T^{6} - 2949273 T^{7} + 73987698 T^{8} - 80604484 T^{9} + 1350125107 T^{10} \)
$71$ \( 1 + 10 T + 191 T^{2} + 1624 T^{3} + 13254 T^{4} + 128892 T^{5} + 941034 T^{6} + 8186584 T^{7} + 68361001 T^{8} + 254116810 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 - 9 T + 251 T^{2} - 2005 T^{3} + 32202 T^{4} - 192248 T^{5} + 2350746 T^{6} - 10684645 T^{7} + 97643267 T^{8} - 255584169 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 + 17 T + 415 T^{2} + 4563 T^{3} + 64494 T^{4} + 512600 T^{5} + 5095026 T^{6} + 28477683 T^{7} + 204611185 T^{8} + 662151377 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 - 9 T + 329 T^{2} - 2539 T^{3} + 47834 T^{4} - 299752 T^{5} + 3970222 T^{6} - 17491171 T^{7} + 188117923 T^{8} - 427124889 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 + 5 T + 380 T^{2} + 1486 T^{3} + 62174 T^{4} + 185418 T^{5} + 5533486 T^{6} + 11770606 T^{7} + 267888220 T^{8} + 313711205 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 - 13 T + 435 T^{2} - 3685 T^{3} + 73316 T^{4} - 461868 T^{5} + 7111652 T^{6} - 34672165 T^{7} + 397012755 T^{8} - 1150880653 T^{9} + 8587340257 T^{10} \)
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