Properties

Label 6042.2.a.s
Level 6042
Weight 2
Character orbit 6042.a
Self dual yes
Analytic conductor 48.246
Analytic rank 1
Dimension 3
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: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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
1.80194
0.445042
−1.24698
1.00000 1.00000 1.00000 −3.80194 1.00000 −0.753020 1.00000 1.00000 −3.80194
1.2 1.00000 1.00000 1.00000 −2.44504 1.00000 −3.80194 1.00000 1.00000 −2.44504
1.3 1.00000 1.00000 1.00000 −0.753020 1.00000 −2.44504 1.00000 1.00000 −0.753020
\(n\): e.g. 2-40 or 990-1000
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.s 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.s 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\( T_{5}^{3} + 7 T_{5}^{2} + 14 T_{5} + 7 \)
\( T_{7}^{3} + 7 T_{7}^{2} + 14 T_{7} + 7 \)
\( T_{11}^{3} - 5 T_{11}^{2} + 6 T_{11} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{3} \)
$3$ \( ( 1 - T )^{3} \)
$5$ \( 1 + 7 T + 29 T^{2} + 77 T^{3} + 145 T^{4} + 175 T^{5} + 125 T^{6} \)
$7$ \( 1 + 7 T + 35 T^{2} + 105 T^{3} + 245 T^{4} + 343 T^{5} + 343 T^{6} \)
$11$ \( 1 - 5 T + 39 T^{2} - 111 T^{3} + 429 T^{4} - 605 T^{5} + 1331 T^{6} \)
$13$ \( 1 - T + 23 T^{2} + 3 T^{3} + 299 T^{4} - 169 T^{5} + 2197 T^{6} \)
$17$ \( 1 + 5 T + 43 T^{2} + 129 T^{3} + 731 T^{4} + 1445 T^{5} + 4913 T^{6} \)
$19$ \( ( 1 - T )^{3} \)
$23$ \( 1 - T + 11 T^{2} - 59 T^{3} + 253 T^{4} - 529 T^{5} + 12167 T^{6} \)
$29$ \( 1 + 8 T + 92 T^{2} + 421 T^{3} + 2668 T^{4} + 6728 T^{5} + 24389 T^{6} \)
$31$ \( 1 + 8 T + 49 T^{2} + 152 T^{3} + 1519 T^{4} + 7688 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 13 T + 123 T^{2} + 949 T^{3} + 4551 T^{4} + 17797 T^{5} + 50653 T^{6} \)
$41$ \( 1 + 32 T^{2} - 287 T^{3} + 1312 T^{4} + 68921 T^{6} \)
$43$ \( 1 - 9 T + 149 T^{2} - 787 T^{3} + 6407 T^{4} - 16641 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 13 T + 132 T^{2} + 873 T^{3} + 6204 T^{4} + 28717 T^{5} + 103823 T^{6} \)
$53$ \( ( 1 - T )^{3} \)
$59$ \( 1 + 12 T + 204 T^{2} + 1389 T^{3} + 12036 T^{4} + 41772 T^{5} + 205379 T^{6} \)
$61$ \( 1 + 18 T + 284 T^{2} + 2377 T^{3} + 17324 T^{4} + 66978 T^{5} + 226981 T^{6} \)
$67$ \( 1 + 11 T + 127 T^{2} + 1193 T^{3} + 8509 T^{4} + 49379 T^{5} + 300763 T^{6} \)
$71$ \( 1 + 20 T + 281 T^{2} + 2848 T^{3} + 19951 T^{4} + 100820 T^{5} + 357911 T^{6} \)
$73$ \( 1 - 7 T + 121 T^{2} - 1225 T^{3} + 8833 T^{4} - 37303 T^{5} + 389017 T^{6} \)
$79$ \( 1 - 9 T + 75 T^{2} - 693 T^{3} + 5925 T^{4} - 56169 T^{5} + 493039 T^{6} \)
$83$ \( 1 + T + 163 T^{2} + 417 T^{3} + 13529 T^{4} + 6889 T^{5} + 571787 T^{6} \)
$89$ \( 1 - 4 T + 186 T^{2} - 879 T^{3} + 16554 T^{4} - 31684 T^{5} + 704969 T^{6} \)
$97$ \( 1 + 29 T + 555 T^{2} + 6395 T^{3} + 53835 T^{4} + 272861 T^{5} + 912673 T^{6} \)
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