Properties

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

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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 30)
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.

\(f(q)\) \(=\) \( q - q^{3} + ( 1 - 2 i ) q^{5} + 2 i q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( 1 - 2 i ) q^{5} + 2 i q^{7} + q^{9} -2 i q^{11} -6 q^{13} + ( -1 + 2 i ) q^{15} + 2 i q^{17} -2 i q^{21} + 4 i q^{23} + ( -3 - 4 i ) q^{25} - q^{27} + 8 q^{31} + 2 i q^{33} + ( 4 + 2 i ) q^{35} -2 q^{37} + 6 q^{39} -2 q^{41} + 4 q^{43} + ( 1 - 2 i ) q^{45} + 8 i q^{47} + 3 q^{49} -2 i q^{51} + 6 q^{53} + ( -4 - 2 i ) q^{55} + 10 i q^{59} -2 i q^{61} + 2 i q^{63} + ( -6 + 12 i ) q^{65} + 8 q^{67} -4 i q^{69} + 12 q^{71} -4 i q^{73} + ( 3 + 4 i ) q^{75} + 4 q^{77} + q^{81} -4 q^{83} + ( 4 + 2 i ) q^{85} -10 q^{89} -12 i q^{91} -8 q^{93} -8 i q^{97} -2 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{5} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{5} + 2q^{9} - 12q^{13} - 2q^{15} - 6q^{25} - 2q^{27} + 16q^{31} + 8q^{35} - 4q^{37} + 12q^{39} - 4q^{41} + 8q^{43} + 2q^{45} + 6q^{49} + 12q^{53} - 8q^{55} - 12q^{65} + 16q^{67} + 24q^{71} + 6q^{75} + 8q^{77} + 2q^{81} - 8q^{83} + 8q^{85} - 20q^{89} - 16q^{93} + O(q^{100}) \)

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.j 2
4.b odd 2 1 3840.2.d.y 2
5.b even 2 1 3840.2.d.x 2
8.b even 2 1 3840.2.d.x 2
8.d odd 2 1 3840.2.d.g 2
16.e even 4 1 240.2.f.a 2
16.e even 4 1 960.2.f.i 2
16.f odd 4 1 30.2.c.a 2
16.f odd 4 1 960.2.f.h 2
20.d odd 2 1 3840.2.d.g 2
40.e odd 2 1 3840.2.d.y 2
40.f even 2 1 inner 3840.2.d.j 2
48.i odd 4 1 720.2.f.f 2
48.i odd 4 1 2880.2.f.c 2
48.k even 4 1 90.2.c.a 2
48.k even 4 1 2880.2.f.e 2
80.i odd 4 1 1200.2.a.m 1
80.i odd 4 1 4800.2.a.cj 1
80.j even 4 1 150.2.a.c 1
80.j even 4 1 4800.2.a.cg 1
80.k odd 4 1 30.2.c.a 2
80.k odd 4 1 960.2.f.h 2
80.q even 4 1 240.2.f.a 2
80.q even 4 1 960.2.f.i 2
80.s even 4 1 150.2.a.a 1
80.s even 4 1 4800.2.a.l 1
80.t odd 4 1 1200.2.a.g 1
80.t odd 4 1 4800.2.a.m 1
112.j even 4 1 1470.2.g.g 2
112.u odd 12 2 1470.2.n.h 4
112.v even 12 2 1470.2.n.a 4
144.u even 12 2 810.2.i.b 4
144.v odd 12 2 810.2.i.e 4
240.t even 4 1 90.2.c.a 2
240.t even 4 1 2880.2.f.e 2
240.z odd 4 1 450.2.a.f 1
240.bb even 4 1 3600.2.a.o 1
240.bd odd 4 1 450.2.a.b 1
240.bf even 4 1 3600.2.a.bg 1
240.bm odd 4 1 720.2.f.f 2
240.bm odd 4 1 2880.2.f.c 2
560.u odd 4 1 7350.2.a.bg 1
560.be even 4 1 1470.2.g.g 2
560.bm odd 4 1 7350.2.a.cc 1
560.co even 12 2 1470.2.n.a 4
560.cs odd 12 2 1470.2.n.h 4
720.cz odd 12 2 810.2.i.e 4
720.da even 12 2 810.2.i.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.c.a 2 16.f odd 4 1
30.2.c.a 2 80.k odd 4 1
90.2.c.a 2 48.k even 4 1
90.2.c.a 2 240.t even 4 1
150.2.a.a 1 80.s even 4 1
150.2.a.c 1 80.j even 4 1
240.2.f.a 2 16.e even 4 1
240.2.f.a 2 80.q even 4 1
450.2.a.b 1 240.bd odd 4 1
450.2.a.f 1 240.z odd 4 1
720.2.f.f 2 48.i odd 4 1
720.2.f.f 2 240.bm odd 4 1
810.2.i.b 4 144.u even 12 2
810.2.i.b 4 720.da even 12 2
810.2.i.e 4 144.v odd 12 2
810.2.i.e 4 720.cz odd 12 2
960.2.f.h 2 16.f odd 4 1
960.2.f.h 2 80.k odd 4 1
960.2.f.i 2 16.e even 4 1
960.2.f.i 2 80.q even 4 1
1200.2.a.g 1 80.t odd 4 1
1200.2.a.m 1 80.i odd 4 1
1470.2.g.g 2 112.j even 4 1
1470.2.g.g 2 560.be even 4 1
1470.2.n.a 4 112.v even 12 2
1470.2.n.a 4 560.co even 12 2
1470.2.n.h 4 112.u odd 12 2
1470.2.n.h 4 560.cs odd 12 2
2880.2.f.c 2 48.i odd 4 1
2880.2.f.c 2 240.bm odd 4 1
2880.2.f.e 2 48.k even 4 1
2880.2.f.e 2 240.t even 4 1
3600.2.a.o 1 240.bb even 4 1
3600.2.a.bg 1 240.bf even 4 1
3840.2.d.g 2 8.d odd 2 1
3840.2.d.g 2 20.d odd 2 1
3840.2.d.j 2 1.a even 1 1 trivial
3840.2.d.j 2 40.f even 2 1 inner
3840.2.d.x 2 5.b even 2 1
3840.2.d.x 2 8.b even 2 1
3840.2.d.y 2 4.b odd 2 1
3840.2.d.y 2 40.e odd 2 1
4800.2.a.l 1 80.s even 4 1
4800.2.a.m 1 80.t odd 4 1
4800.2.a.cg 1 80.j even 4 1
4800.2.a.cj 1 80.i odd 4 1
7350.2.a.bg 1 560.u odd 4 1
7350.2.a.cc 1 560.bm odd 4 1

Hecke kernels

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

\( T_{7}^{2} + 4 \)
\( T_{11}^{2} + 4 \)
\( T_{13} + 6 \)
\( T_{31} - 8 \)
\( T_{37} + 2 \)
\( T_{43} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T )^{2} \)
$5$ \( 1 - 2 T + 5 T^{2} \)
$7$ \( 1 - 10 T^{2} + 49 T^{4} \)
$11$ \( 1 - 18 T^{2} + 121 T^{4} \)
$13$ \( ( 1 + 6 T + 13 T^{2} )^{2} \)
$17$ \( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \)
$19$ \( ( 1 - 19 T^{2} )^{2} \)
$23$ \( 1 - 30 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 29 T^{2} )^{2} \)
$31$ \( ( 1 - 8 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 + 2 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 + 2 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 4 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 30 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 6 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 18 T^{2} + 3481 T^{4} \)
$61$ \( 1 - 118 T^{2} + 3721 T^{4} \)
$67$ \( ( 1 - 8 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 130 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 79 T^{2} )^{2} \)
$83$ \( ( 1 + 4 T + 83 T^{2} )^{2} \)
$89$ \( ( 1 + 10 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \)
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