Properties

Label 3840.2.a.br
Level 3840
Weight 2
Character orbit 3840.a
Self dual yes
Analytic conductor 30.663
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.6625543762\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 120)
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.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + ( 1 + \beta_{1} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + q^{5} + ( 1 + \beta_{1} ) q^{7} + q^{9} + ( 1 - \beta_{1} + \beta_{2} ) q^{11} -\beta_{2} q^{13} + q^{15} + ( 2 - \beta_{2} ) q^{17} + ( 1 - \beta_{1} ) q^{19} + ( 1 + \beta_{1} ) q^{21} + ( -1 + \beta_{1} ) q^{23} + q^{25} + q^{27} + 2 q^{29} + ( 2 + \beta_{2} ) q^{31} + ( 1 - \beta_{1} + \beta_{2} ) q^{33} + ( 1 + \beta_{1} ) q^{35} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{37} -\beta_{2} q^{39} + ( 4 + 2 \beta_{1} ) q^{41} + ( 2 + 2 \beta_{1} ) q^{43} + q^{45} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{47} + ( 5 + 2 \beta_{2} ) q^{49} + ( 2 - \beta_{2} ) q^{51} + 2 q^{53} + ( 1 - \beta_{1} + \beta_{2} ) q^{55} + ( 1 - \beta_{1} ) q^{57} + ( 3 + \beta_{1} - \beta_{2} ) q^{59} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{61} + ( 1 + \beta_{1} ) q^{63} -\beta_{2} q^{65} -4 q^{67} + ( -1 + \beta_{1} ) q^{69} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{71} + 6 q^{73} + q^{75} + ( -4 + 4 \beta_{1} ) q^{77} + ( -6 + \beta_{2} ) q^{79} + q^{81} + ( 6 + 2 \beta_{1} ) q^{83} + ( 2 - \beta_{2} ) q^{85} + 2 q^{87} + ( 4 - 2 \beta_{1} ) q^{89} + ( -6 - 2 \beta_{1} - 2 \beta_{2} ) q^{91} + ( 2 + \beta_{2} ) q^{93} + ( 1 - \beta_{1} ) q^{95} + ( 6 - 2 \beta_{2} ) q^{97} + ( 1 - \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} + 3q^{5} + 2q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} + 3q^{5} + 2q^{7} + 3q^{9} + 4q^{11} + 3q^{15} + 6q^{17} + 4q^{19} + 2q^{21} - 4q^{23} + 3q^{25} + 3q^{27} + 6q^{29} + 6q^{31} + 4q^{33} + 2q^{35} - 4q^{37} + 10q^{41} + 4q^{43} + 3q^{45} - 4q^{47} + 15q^{49} + 6q^{51} + 6q^{53} + 4q^{55} + 4q^{57} + 8q^{59} + 8q^{61} + 2q^{63} - 12q^{67} - 4q^{69} - 4q^{71} + 18q^{73} + 3q^{75} - 16q^{77} - 18q^{79} + 3q^{81} + 16q^{83} + 6q^{85} + 6q^{87} + 14q^{89} - 16q^{91} + 6q^{93} + 4q^{95} + 18q^{97} + 4q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 6\)\()/2\)

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
−1.81361
0.470683
2.34292
0 1.00000 0 1.00000 0 −3.62721 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 0.941367 0 1.00000 0
1.3 0 1.00000 0 1.00000 0 4.68585 0 1.00000 0
\(n\): 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 3840.2.a.br 3
4.b odd 2 1 3840.2.a.bp 3
8.b even 2 1 3840.2.a.bo 3
8.d odd 2 1 3840.2.a.bq 3
16.e even 4 2 480.2.k.b 6
16.f odd 4 2 120.2.k.b 6
48.i odd 4 2 1440.2.k.f 6
48.k even 4 2 360.2.k.f 6
80.i odd 4 2 2400.2.d.e 6
80.j even 4 2 600.2.d.e 6
80.k odd 4 2 600.2.k.c 6
80.q even 4 2 2400.2.k.c 6
80.s even 4 2 600.2.d.f 6
80.t odd 4 2 2400.2.d.f 6
240.t even 4 2 1800.2.k.p 6
240.z odd 4 2 1800.2.d.r 6
240.bb even 4 2 7200.2.d.r 6
240.bd odd 4 2 1800.2.d.q 6
240.bf even 4 2 7200.2.d.q 6
240.bm odd 4 2 7200.2.k.p 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.k.b 6 16.f odd 4 2
360.2.k.f 6 48.k even 4 2
480.2.k.b 6 16.e even 4 2
600.2.d.e 6 80.j even 4 2
600.2.d.f 6 80.s even 4 2
600.2.k.c 6 80.k odd 4 2
1440.2.k.f 6 48.i odd 4 2
1800.2.d.q 6 240.bd odd 4 2
1800.2.d.r 6 240.z odd 4 2
1800.2.k.p 6 240.t even 4 2
2400.2.d.e 6 80.i odd 4 2
2400.2.d.f 6 80.t odd 4 2
2400.2.k.c 6 80.q even 4 2
3840.2.a.bo 3 8.b even 2 1
3840.2.a.bp 3 4.b odd 2 1
3840.2.a.bq 3 8.d odd 2 1
3840.2.a.br 3 1.a even 1 1 trivial
7200.2.d.q 6 240.bf even 4 2
7200.2.d.r 6 240.bb even 4 2
7200.2.k.p 6 240.bm odd 4 2

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3840))\):

\( T_{7}^{3} - 2 T_{7}^{2} - 16 T_{7} + 16 \)
\( T_{11}^{3} - 4 T_{11}^{2} - 24 T_{11} + 64 \)
\( T_{13}^{3} - 28 T_{13} - 16 \)
\( T_{17}^{3} - 6 T_{17}^{2} - 16 T_{17} + 32 \)
\( T_{19}^{3} - 4 T_{19}^{2} - 12 T_{19} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T )^{3} \)
$5$ \( ( 1 - T )^{3} \)
$7$ \( 1 - 2 T + 5 T^{2} - 12 T^{3} + 35 T^{4} - 98 T^{5} + 343 T^{6} \)
$11$ \( 1 - 4 T + 9 T^{2} - 24 T^{3} + 99 T^{4} - 484 T^{5} + 1331 T^{6} \)
$13$ \( 1 + 11 T^{2} - 16 T^{3} + 143 T^{4} + 2197 T^{6} \)
$17$ \( 1 - 6 T + 35 T^{2} - 172 T^{3} + 595 T^{4} - 1734 T^{5} + 4913 T^{6} \)
$19$ \( 1 - 4 T + 45 T^{2} - 136 T^{3} + 855 T^{4} - 1444 T^{5} + 6859 T^{6} \)
$23$ \( 1 + 4 T + 57 T^{2} + 168 T^{3} + 1311 T^{4} + 2116 T^{5} + 12167 T^{6} \)
$29$ \( ( 1 - 2 T + 29 T^{2} )^{3} \)
$31$ \( 1 - 6 T + 77 T^{2} - 308 T^{3} + 2387 T^{4} - 5766 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 4 T + 51 T^{2} + 40 T^{3} + 1887 T^{4} + 5476 T^{5} + 50653 T^{6} \)
$41$ \( 1 - 10 T + 87 T^{2} - 588 T^{3} + 3567 T^{4} - 16810 T^{5} + 68921 T^{6} \)
$43$ \( 1 - 4 T + 65 T^{2} - 216 T^{3} + 2795 T^{4} - 7396 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 4 T + 49 T^{2} - 120 T^{3} + 2303 T^{4} + 8836 T^{5} + 103823 T^{6} \)
$53$ \( ( 1 - 2 T + 53 T^{2} )^{3} \)
$59$ \( 1 - 8 T + 169 T^{2} - 912 T^{3} + 9971 T^{4} - 27848 T^{5} + 205379 T^{6} \)
$61$ \( 1 - 8 T + 87 T^{2} - 464 T^{3} + 5307 T^{4} - 29768 T^{5} + 226981 T^{6} \)
$67$ \( ( 1 + 4 T + 67 T^{2} )^{3} \)
$71$ \( 1 + 4 T + 101 T^{2} + 632 T^{3} + 7171 T^{4} + 20164 T^{5} + 357911 T^{6} \)
$73$ \( ( 1 - 6 T + 73 T^{2} )^{3} \)
$79$ \( 1 + 18 T + 317 T^{2} + 2908 T^{3} + 25043 T^{4} + 112338 T^{5} + 493039 T^{6} \)
$83$ \( 1 - 16 T + 265 T^{2} - 2400 T^{3} + 21995 T^{4} - 110224 T^{5} + 571787 T^{6} \)
$89$ \( 1 - 14 T + 263 T^{2} - 2308 T^{3} + 23407 T^{4} - 110894 T^{5} + 704969 T^{6} \)
$97$ \( 1 - 18 T + 287 T^{2} - 3164 T^{3} + 27839 T^{4} - 169362 T^{5} + 912673 T^{6} \)
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