Properties

Label 7938.2.a.ca
Level $7938$
Weight $2$
Character orbit 7938.a
Self dual yes
Analytic conductor $63.385$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7938.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.3852491245\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} -\beta_{2} q^{5} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} -\beta_{2} q^{5} + q^{8} -\beta_{2} q^{10} + \beta_{2} q^{11} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{13} + q^{16} + ( -2 + 2 \beta_{1} ) q^{17} + ( 2 - 3 \beta_{1} ) q^{19} -\beta_{2} q^{20} + \beta_{2} q^{22} + ( -2 - \beta_{1} ) q^{23} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{25} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{26} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -6 - \beta_{1} + \beta_{2} ) q^{31} + q^{32} + ( -2 + 2 \beta_{1} ) q^{34} - q^{37} + ( 2 - 3 \beta_{1} ) q^{38} -\beta_{2} q^{40} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{41} + ( 2 + 3 \beta_{1} + 3 \beta_{2} ) q^{43} + \beta_{2} q^{44} + ( -2 - \beta_{1} ) q^{46} + ( -3 - 3 \beta_{1} - 3 \beta_{2} ) q^{47} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{50} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{52} + ( 5 + \beta_{1} + \beta_{2} ) q^{53} + ( -4 + \beta_{1} + 2 \beta_{2} ) q^{55} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{58} + ( -5 - \beta_{1} - 2 \beta_{2} ) q^{59} + ( -3 - \beta_{1} - 2 \beta_{2} ) q^{61} + ( -6 - \beta_{1} + \beta_{2} ) q^{62} + q^{64} + ( -2 - \beta_{1} + 5 \beta_{2} ) q^{65} + ( -2 + \beta_{1} - 4 \beta_{2} ) q^{67} + ( -2 + 2 \beta_{1} ) q^{68} + ( -4 + 7 \beta_{1} + 2 \beta_{2} ) q^{71} + ( -8 + \beta_{1} - 4 \beta_{2} ) q^{73} - q^{74} + ( 2 - 3 \beta_{1} ) q^{76} + ( -4 \beta_{1} + \beta_{2} ) q^{79} -\beta_{2} q^{80} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{82} + ( 1 + 2 \beta_{1} + 3 \beta_{2} ) q^{83} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{85} + ( 2 + 3 \beta_{1} + 3 \beta_{2} ) q^{86} + \beta_{2} q^{88} + ( -2 - 4 \beta_{1} - \beta_{2} ) q^{89} + ( -2 - \beta_{1} ) q^{92} + ( -3 - 3 \beta_{1} - 3 \beta_{2} ) q^{94} + ( -3 + 3 \beta_{1} - 2 \beta_{2} ) q^{95} + ( -8 - 2 \beta_{1} + 2 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} + q^{5} + 3q^{8} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} + q^{5} + 3q^{8} + q^{10} - q^{11} - 8q^{13} + 3q^{16} - 4q^{17} + 3q^{19} + q^{20} - q^{22} - 7q^{23} - 2q^{25} - 8q^{26} - 5q^{29} - 20q^{31} + 3q^{32} - 4q^{34} - 3q^{37} + 3q^{38} + q^{40} + 6q^{43} - q^{44} - 7q^{46} - 9q^{47} - 2q^{50} - 8q^{52} + 15q^{53} - 13q^{55} - 5q^{58} - 14q^{59} - 8q^{61} - 20q^{62} + 3q^{64} - 12q^{65} - q^{67} - 4q^{68} - 7q^{71} - 19q^{73} - 3q^{74} + 3q^{76} - 5q^{79} + q^{80} + 2q^{83} + 2q^{85} + 6q^{86} - q^{88} - 9q^{89} - 7q^{92} - 9q^{94} - 4q^{95} - 28q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)

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
−1.69963
2.46050
0.239123
1.00000 0 1.00000 −1.58836 0 0 1.00000 0 −1.58836
1.2 1.00000 0 1.00000 −0.593579 0 0 1.00000 0 −0.593579
1.3 1.00000 0 1.00000 3.18194 0 0 1.00000 0 3.18194
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)

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 7938.2.a.ca 3
3.b odd 2 1 7938.2.a.bv 3
7.b odd 2 1 7938.2.a.bz 3
7.c even 3 2 1134.2.g.l 6
9.c even 3 2 2646.2.f.l 6
9.d odd 6 2 882.2.f.n 6
21.c even 2 1 7938.2.a.bw 3
21.h odd 6 2 1134.2.g.m 6
63.g even 3 2 378.2.h.c 6
63.h even 3 2 378.2.e.d 6
63.i even 6 2 882.2.e.o 6
63.j odd 6 2 126.2.e.c 6
63.k odd 6 2 2646.2.h.o 6
63.l odd 6 2 2646.2.f.m 6
63.n odd 6 2 126.2.h.d yes 6
63.o even 6 2 882.2.f.o 6
63.s even 6 2 882.2.h.p 6
63.t odd 6 2 2646.2.e.p 6
252.o even 6 2 1008.2.t.h 6
252.u odd 6 2 3024.2.q.g 6
252.bb even 6 2 1008.2.q.g 6
252.bl odd 6 2 3024.2.t.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.c 6 63.j odd 6 2
126.2.h.d yes 6 63.n odd 6 2
378.2.e.d 6 63.h even 3 2
378.2.h.c 6 63.g even 3 2
882.2.e.o 6 63.i even 6 2
882.2.f.n 6 9.d odd 6 2
882.2.f.o 6 63.o even 6 2
882.2.h.p 6 63.s even 6 2
1008.2.q.g 6 252.bb even 6 2
1008.2.t.h 6 252.o even 6 2
1134.2.g.l 6 7.c even 3 2
1134.2.g.m 6 21.h odd 6 2
2646.2.e.p 6 63.t odd 6 2
2646.2.f.l 6 9.c even 3 2
2646.2.f.m 6 63.l odd 6 2
2646.2.h.o 6 63.k odd 6 2
3024.2.q.g 6 252.u odd 6 2
3024.2.t.h 6 252.bl odd 6 2
7938.2.a.bv 3 3.b odd 2 1
7938.2.a.bw 3 21.c even 2 1
7938.2.a.bz 3 7.b odd 2 1
7938.2.a.ca 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7938))\):

\( T_{5}^{3} - T_{5}^{2} - 6 T_{5} - 3 \)
\( T_{11}^{3} + T_{11}^{2} - 6 T_{11} + 3 \)
\( T_{13}^{3} + 8 T_{13}^{2} + T_{13} - 69 \)
\( T_{17}^{3} + 4 T_{17}^{2} - 12 T_{17} - 24 \)
\( T_{23}^{3} + 7 T_{23}^{2} + 12 T_{23} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{3} \)
$3$ 1
$5$ \( 1 - T + 9 T^{2} - 13 T^{3} + 45 T^{4} - 25 T^{5} + 125 T^{6} \)
$7$ 1
$11$ \( 1 + T + 27 T^{2} + 25 T^{3} + 297 T^{4} + 121 T^{5} + 1331 T^{6} \)
$13$ \( 1 + 8 T + 40 T^{2} + 139 T^{3} + 520 T^{4} + 1352 T^{5} + 2197 T^{6} \)
$17$ \( 1 + 4 T + 39 T^{2} + 112 T^{3} + 663 T^{4} + 1156 T^{5} + 4913 T^{6} \)
$19$ \( 1 - 3 T + 21 T^{2} - 65 T^{3} + 399 T^{4} - 1083 T^{5} + 6859 T^{6} \)
$23$ \( 1 + 7 T + 81 T^{2} + 325 T^{3} + 1863 T^{4} + 3703 T^{5} + 12167 T^{6} \)
$29$ \( 1 + 5 T + 21 T^{2} - 73 T^{3} + 609 T^{4} + 4205 T^{5} + 24389 T^{6} \)
$31$ \( 1 + 20 T + 214 T^{2} + 1441 T^{3} + 6634 T^{4} + 19220 T^{5} + 29791 T^{6} \)
$37$ \( ( 1 + T + 37 T^{2} )^{3} \)
$41$ \( 1 + 90 T^{2} - 9 T^{3} + 3690 T^{4} + 68921 T^{6} \)
$43$ \( 1 - 6 T + 60 T^{2} - 389 T^{3} + 2580 T^{4} - 11094 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 9 T + 87 T^{2} + 657 T^{3} + 4089 T^{4} + 19881 T^{5} + 103823 T^{6} \)
$53$ \( 1 - 15 T + 225 T^{2} - 1671 T^{3} + 11925 T^{4} - 42135 T^{5} + 148877 T^{6} \)
$59$ \( 1 + 14 T + 216 T^{2} + 1589 T^{3} + 12744 T^{4} + 48734 T^{5} + 205379 T^{6} \)
$61$ \( 1 + 8 T + 178 T^{2} + 883 T^{3} + 10858 T^{4} + 29768 T^{5} + 226981 T^{6} \)
$67$ \( 1 + T + 89 T^{2} - 77 T^{3} + 5963 T^{4} + 4489 T^{5} + 300763 T^{6} \)
$71$ \( 1 + 7 T + 15 T^{2} - 599 T^{3} + 1065 T^{4} + 35287 T^{5} + 357911 T^{6} \)
$73$ \( 1 + 19 T + 227 T^{2} + 2143 T^{3} + 16571 T^{4} + 101251 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 5 T + 163 T^{2} + 469 T^{3} + 12877 T^{4} + 31205 T^{5} + 493039 T^{6} \)
$83$ \( 1 - 2 T + 186 T^{2} - 185 T^{3} + 15438 T^{4} - 13778 T^{5} + 571787 T^{6} \)
$89$ \( 1 + 9 T + 225 T^{2} + 1611 T^{3} + 20025 T^{4} + 71289 T^{5} + 704969 T^{6} \)
$97$ \( 1 + 28 T + 503 T^{2} + 5680 T^{3} + 48791 T^{4} + 263452 T^{5} + 912673 T^{6} \)
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