Properties

Label 3840.2.d.e
Level $3840$
Weight $2$
Character orbit 3840.d
Analytic conductor $30.663$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.6625543762\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 480)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( -1 + 2 i ) q^{5} + 2 i q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( -1 + 2 i ) q^{5} + 2 i q^{7} + q^{9} -6 i q^{11} -2 q^{13} + ( 1 - 2 i ) q^{15} -6 i q^{17} + 4 i q^{19} -2 i q^{21} + 8 i q^{23} + ( -3 - 4 i ) q^{25} - q^{27} + 8 q^{31} + 6 i q^{33} + ( -4 - 2 i ) q^{35} + 2 q^{37} + 2 q^{39} + 6 q^{41} + 4 q^{43} + ( -1 + 2 i ) q^{45} + 4 i q^{47} + 3 q^{49} + 6 i q^{51} -6 q^{53} + ( 12 + 6 i ) q^{55} -4 i q^{57} + 6 i q^{59} -6 i q^{61} + 2 i q^{63} + ( 2 - 4 i ) q^{65} -8 i q^{69} -4 q^{71} + 12 i q^{73} + ( 3 + 4 i ) q^{75} + 12 q^{77} -8 q^{79} + q^{81} -12 q^{83} + ( 12 + 6 i ) q^{85} + 14 q^{89} -4 i q^{91} -8 q^{93} + ( -8 - 4 i ) q^{95} + 8 i q^{97} -6 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{5} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{5} + 2q^{9} - 4q^{13} + 2q^{15} - 6q^{25} - 2q^{27} + 16q^{31} - 8q^{35} + 4q^{37} + 4q^{39} + 12q^{41} + 8q^{43} - 2q^{45} + 6q^{49} - 12q^{53} + 24q^{55} + 4q^{65} - 8q^{71} + 6q^{75} + 24q^{77} - 16q^{79} + 2q^{81} - 24q^{83} + 24q^{85} + 28q^{89} - 16q^{93} - 16q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(1537\) \(2561\) \(2821\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2689.1
1.00000i
1.00000i
0 −1.00000 0 −1.00000 2.00000i 0 2.00000i 0 1.00000 0
2689.2 0 −1.00000 0 −1.00000 + 2.00000i 0 2.00000i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3840.2.d.e 2
4.b odd 2 1 3840.2.d.u 2
5.b even 2 1 3840.2.d.bc 2
8.b even 2 1 3840.2.d.bc 2
8.d odd 2 1 3840.2.d.k 2
16.e even 4 1 480.2.f.b yes 2
16.e even 4 1 960.2.f.g 2
16.f odd 4 1 480.2.f.a 2
16.f odd 4 1 960.2.f.j 2
20.d odd 2 1 3840.2.d.k 2
40.e odd 2 1 3840.2.d.u 2
40.f even 2 1 inner 3840.2.d.e 2
48.i odd 4 1 1440.2.f.e 2
48.i odd 4 1 2880.2.f.g 2
48.k even 4 1 1440.2.f.g 2
48.k even 4 1 2880.2.f.a 2
80.i odd 4 1 2400.2.a.bf 1
80.i odd 4 1 4800.2.a.bt 1
80.j even 4 1 2400.2.a.be 1
80.j even 4 1 4800.2.a.bu 1
80.k odd 4 1 480.2.f.a 2
80.k odd 4 1 960.2.f.j 2
80.q even 4 1 480.2.f.b yes 2
80.q even 4 1 960.2.f.g 2
80.s even 4 1 2400.2.a.c 1
80.s even 4 1 4800.2.a.ba 1
80.t odd 4 1 2400.2.a.d 1
80.t odd 4 1 4800.2.a.z 1
240.t even 4 1 1440.2.f.g 2
240.t even 4 1 2880.2.f.a 2
240.z odd 4 1 7200.2.a.p 1
240.bb even 4 1 7200.2.a.bl 1
240.bd odd 4 1 7200.2.a.br 1
240.bf even 4 1 7200.2.a.j 1
240.bm odd 4 1 1440.2.f.e 2
240.bm odd 4 1 2880.2.f.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.f.a 2 16.f odd 4 1
480.2.f.a 2 80.k odd 4 1
480.2.f.b yes 2 16.e even 4 1
480.2.f.b yes 2 80.q even 4 1
960.2.f.g 2 16.e even 4 1
960.2.f.g 2 80.q even 4 1
960.2.f.j 2 16.f odd 4 1
960.2.f.j 2 80.k odd 4 1
1440.2.f.e 2 48.i odd 4 1
1440.2.f.e 2 240.bm odd 4 1
1440.2.f.g 2 48.k even 4 1
1440.2.f.g 2 240.t even 4 1
2400.2.a.c 1 80.s even 4 1
2400.2.a.d 1 80.t odd 4 1
2400.2.a.be 1 80.j even 4 1
2400.2.a.bf 1 80.i odd 4 1
2880.2.f.a 2 48.k even 4 1
2880.2.f.a 2 240.t even 4 1
2880.2.f.g 2 48.i odd 4 1
2880.2.f.g 2 240.bm odd 4 1
3840.2.d.e 2 1.a even 1 1 trivial
3840.2.d.e 2 40.f even 2 1 inner
3840.2.d.k 2 8.d odd 2 1
3840.2.d.k 2 20.d odd 2 1
3840.2.d.u 2 4.b odd 2 1
3840.2.d.u 2 40.e odd 2 1
3840.2.d.bc 2 5.b even 2 1
3840.2.d.bc 2 8.b even 2 1
4800.2.a.z 1 80.t odd 4 1
4800.2.a.ba 1 80.s even 4 1
4800.2.a.bt 1 80.i odd 4 1
4800.2.a.bu 1 80.j even 4 1
7200.2.a.j 1 240.bf even 4 1
7200.2.a.p 1 240.z odd 4 1
7200.2.a.bl 1 240.bb even 4 1
7200.2.a.br 1 240.bd odd 4 1

Hecke kernels

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

\( T_{7}^{2} + 4 \)
\( T_{11}^{2} + 36 \)
\( T_{13} + 2 \)
\( T_{31} - 8 \)
\( T_{37} - 2 \)
\( T_{43} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 5 + 2 T + T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( 36 + T^{2} \)
$13$ \( ( 2 + T )^{2} \)
$17$ \( 36 + T^{2} \)
$19$ \( 16 + T^{2} \)
$23$ \( 64 + T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( 16 + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( 36 + T^{2} \)
$61$ \( 36 + T^{2} \)
$67$ \( T^{2} \)
$71$ \( ( 4 + T )^{2} \)
$73$ \( 144 + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( ( 12 + T )^{2} \)
$89$ \( ( -14 + T )^{2} \)
$97$ \( 64 + T^{2} \)
show more
show less