Properties

Label 6042.2.a.v
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: 6.6.21848308.1
Defining polynomial: \(x^{6} - 3 x^{5} - 4 x^{4} + 9 x^{3} + 6 x^{2} - 4 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{5} - 4 \nu^{4} + 9 \nu^{2} - 3 \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - 3 \nu^{4} - 4 \nu^{3} + 9 \nu^{2} + 5 \nu - 3 \)
\(\beta_{4}\)\(=\)\( -\nu^{5} + 4 \nu^{4} + \nu^{3} - 12 \nu^{2} + \nu + 6 \)
\(\beta_{5}\)\(=\)\( 2 \nu^{5} - 7 \nu^{4} - 4 \nu^{3} + 19 \nu^{2} + \nu - 7 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{3} - \beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(3 \beta_{5} + \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 5 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(12 \beta_{5} + 4 \beta_{4} - 11 \beta_{3} - 9 \beta_{2} + 12 \beta_{1} + 9\)
\(\nu^{5}\)\(=\)\(39 \beta_{5} + 16 \beta_{4} - 35 \beta_{3} - 26 \beta_{2} + 42 \beta_{1} + 20\)

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
1.73472
−0.810057
−1.45853
3.24366
0.712460
−0.422254
−1.00000 1.00000 1.00000 −2.64265 −1.00000 −1.21612 −1.00000 1.00000 2.64265
1.2 −1.00000 1.00000 1.00000 −1.14477 −1.00000 2.60585 −1.00000 1.00000 1.14477
1.3 −1.00000 1.00000 1.00000 0.613278 −1.00000 −3.09360 −1.00000 1.00000 −0.613278
1.4 −1.00000 1.00000 1.00000 0.840706 −1.00000 −3.39143 −1.00000 1.00000 −0.840706
1.5 −1.00000 1.00000 1.00000 1.82997 −1.00000 1.67867 −1.00000 1.00000 −1.82997
1.6 −1.00000 1.00000 1.00000 3.50346 −1.00000 −3.58336 −1.00000 1.00000 −3.50346
\(n\): 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.v 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.v 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\( T_{5}^{6} - 3 T_{5}^{5} - 8 T_{5}^{4} + 23 T_{5}^{3} + 2 T_{5}^{2} - 24 T_{5} + 10 \)
\( T_{7}^{6} + 7 T_{7}^{5} + 2 T_{7}^{4} - 69 T_{7}^{3} - 90 T_{7}^{2} + 148 T_{7} + 200 \)
\( T_{11}^{6} - T_{11}^{5} - 28 T_{11}^{4} - 31 T_{11}^{3} + 102 T_{11}^{2} + 142 T_{11} - 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{6} \)
$3$ \( ( 1 - T )^{6} \)
$5$ \( 1 - 3 T + 22 T^{2} - 52 T^{3} + 217 T^{4} - 429 T^{5} + 1330 T^{6} - 2145 T^{7} + 5425 T^{8} - 6500 T^{9} + 13750 T^{10} - 9375 T^{11} + 15625 T^{12} \)
$7$ \( 1 + 7 T + 44 T^{2} + 176 T^{3} + 701 T^{4} + 2129 T^{5} + 6388 T^{6} + 14903 T^{7} + 34349 T^{8} + 60368 T^{9} + 105644 T^{10} + 117649 T^{11} + 117649 T^{12} \)
$11$ \( 1 - T + 38 T^{2} - 86 T^{3} + 685 T^{4} - 2091 T^{5} + 8518 T^{6} - 23001 T^{7} + 82885 T^{8} - 114466 T^{9} + 556358 T^{10} - 161051 T^{11} + 1771561 T^{12} \)
$13$ \( 1 + 13 T + 134 T^{2} + 920 T^{3} + 5399 T^{4} + 24779 T^{5} + 99480 T^{6} + 322127 T^{7} + 912431 T^{8} + 2021240 T^{9} + 3827174 T^{10} + 4826809 T^{11} + 4826809 T^{12} \)
$17$ \( 1 + 5 T + 60 T^{2} + 272 T^{3} + 1985 T^{4} + 7531 T^{5} + 41060 T^{6} + 128027 T^{7} + 573665 T^{8} + 1336336 T^{9} + 5011260 T^{10} + 7099285 T^{11} + 24137569 T^{12} \)
$19$ \( ( 1 - T )^{6} \)
$23$ \( 1 - 9 T + 124 T^{2} - 808 T^{3} + 6701 T^{4} - 33411 T^{5} + 200232 T^{6} - 768453 T^{7} + 3544829 T^{8} - 9830936 T^{9} + 34700284 T^{10} - 57927087 T^{11} + 148035889 T^{12} \)
$29$ \( 1 + 12 T + 169 T^{2} + 1399 T^{3} + 12059 T^{4} + 73997 T^{5} + 461386 T^{6} + 2145913 T^{7} + 10141619 T^{8} + 34120211 T^{9} + 119530489 T^{10} + 246133788 T^{11} + 594823321 T^{12} \)
$31$ \( 1 + 2 T + 2 T^{2} - 366 T^{3} - 193 T^{4} + 1776 T^{5} + 88604 T^{6} + 55056 T^{7} - 185473 T^{8} - 10903506 T^{9} + 1847042 T^{10} + 57258302 T^{11} + 887503681 T^{12} \)
$37$ \( 1 + 7 T + 78 T^{2} - 16 T^{3} + 599 T^{4} - 14651 T^{5} + 6304 T^{6} - 542087 T^{7} + 820031 T^{8} - 810448 T^{9} + 146184558 T^{10} + 485407699 T^{11} + 2565726409 T^{12} \)
$41$ \( 1 + 18 T + 335 T^{2} + 3655 T^{3} + 38475 T^{4} + 294995 T^{5} + 2164122 T^{6} + 12094795 T^{7} + 64676475 T^{8} + 251906255 T^{9} + 946629935 T^{10} + 2085411618 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 + 11 T + 172 T^{2} + 1702 T^{3} + 15653 T^{4} + 127691 T^{5} + 839300 T^{6} + 5490713 T^{7} + 28942397 T^{8} + 135320914 T^{9} + 588033772 T^{10} + 1617092873 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 + 5 T + 229 T^{2} + 788 T^{3} + 22341 T^{4} + 55241 T^{5} + 1295844 T^{6} + 2596327 T^{7} + 49351269 T^{8} + 81812524 T^{9} + 1117446949 T^{10} + 1146725035 T^{11} + 10779215329 T^{12} \)
$53$ \( ( 1 + T )^{6} \)
$59$ \( 1 - 2 T + 201 T^{2} - 843 T^{3} + 18383 T^{4} - 112489 T^{5} + 1175966 T^{6} - 6636851 T^{7} + 63991223 T^{8} - 173134497 T^{9} + 2435589561 T^{10} - 1429848598 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 + 12 T + 207 T^{2} + 921 T^{3} + 6025 T^{4} - 71297 T^{5} - 359682 T^{6} - 4349117 T^{7} + 22419025 T^{8} + 209049501 T^{9} + 2866089087 T^{10} + 10135155612 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 + 9 T + 296 T^{2} + 2804 T^{3} + 42239 T^{4} + 353919 T^{5} + 3609088 T^{6} + 23712573 T^{7} + 189610871 T^{8} + 843339452 T^{9} + 5964731816 T^{10} + 12151125963 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 - 18 T + 506 T^{2} - 6218 T^{3} + 96815 T^{4} - 870760 T^{5} + 9369260 T^{6} - 61823960 T^{7} + 488044415 T^{8} - 2225490598 T^{9} + 12858310586 T^{10} - 32476128318 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 + 31 T + 746 T^{2} + 12376 T^{3} + 170831 T^{4} + 1883717 T^{5} + 17768444 T^{6} + 137511341 T^{7} + 910358399 T^{8} + 4814474392 T^{9} + 21185087786 T^{10} + 64265219383 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 + 17 T + 440 T^{2} + 5688 T^{3} + 81265 T^{4} + 826839 T^{5} + 8352440 T^{6} + 65320281 T^{7} + 507174865 T^{8} + 2804405832 T^{9} + 17138035640 T^{10} + 52309958783 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 + 9 T + 252 T^{2} + 1288 T^{3} + 28135 T^{4} + 66915 T^{5} + 2204488 T^{6} + 5553945 T^{7} + 193822015 T^{8} + 736461656 T^{9} + 11959496892 T^{10} + 35451365787 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 - 18 T + 423 T^{2} - 5011 T^{3} + 74621 T^{4} - 686083 T^{5} + 7989190 T^{6} - 61061387 T^{7} + 591072941 T^{8} - 3532599659 T^{9} + 26539967943 T^{10} - 100513070082 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 + 7 T + 206 T^{2} + 1764 T^{3} + 32201 T^{4} + 259765 T^{5} + 3296352 T^{6} + 25197205 T^{7} + 302979209 T^{8} + 1609955172 T^{9} + 18237031886 T^{10} + 60111381799 T^{11} + 832972004929 T^{12} \)
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