Properties

Label 3840.2.d.r
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: \( 1 \)
Twist minimal: no (minimal twist has level 60)
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.

\(f(q)\) \(=\) \( q + q^{3} + ( -2 + i ) q^{5} + 4 i q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -2 + i ) q^{5} + 4 i q^{7} + q^{9} -4 i q^{11} + ( -2 + i ) q^{15} -4 i q^{17} + 4 i q^{21} -4 i q^{23} + ( 3 - 4 i ) q^{25} + q^{27} -6 i q^{29} + 4 q^{31} -4 i q^{33} + ( -4 - 8 i ) q^{35} -8 q^{37} + 10 q^{41} -4 q^{43} + ( -2 + i ) q^{45} + 4 i q^{47} -9 q^{49} -4 i q^{51} + 12 q^{53} + ( 4 + 8 i ) q^{55} -4 i q^{59} -2 i q^{61} + 4 i q^{63} + 4 q^{67} -4 i q^{69} + 8 i q^{73} + ( 3 - 4 i ) q^{75} + 16 q^{77} + 12 q^{79} + q^{81} + 4 q^{83} + ( 4 + 8 i ) q^{85} -6 i q^{87} -10 q^{89} + 4 q^{93} -8 i q^{97} -4 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 4q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 4q^{5} + 2q^{9} - 4q^{15} + 6q^{25} + 2q^{27} + 8q^{31} - 8q^{35} - 16q^{37} + 20q^{41} - 8q^{43} - 4q^{45} - 18q^{49} + 24q^{53} + 8q^{55} + 8q^{67} + 6q^{75} + 32q^{77} + 24q^{79} + 2q^{81} + 8q^{83} + 8q^{85} - 20q^{89} + 8q^{93} + O(q^{100}) \)

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 −2.00000 1.00000i 0 4.00000i 0 1.00000 0
2689.2 0 1.00000 0 −2.00000 + 1.00000i 0 4.00000i 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.r 2
4.b odd 2 1 3840.2.d.b 2
5.b even 2 1 3840.2.d.o 2
8.b even 2 1 3840.2.d.o 2
8.d odd 2 1 3840.2.d.be 2
16.e even 4 1 60.2.d.a 2
16.e even 4 1 960.2.f.f 2
16.f odd 4 1 240.2.f.b 2
16.f odd 4 1 960.2.f.c 2
20.d odd 2 1 3840.2.d.be 2
40.e odd 2 1 3840.2.d.b 2
40.f even 2 1 inner 3840.2.d.r 2
48.i odd 4 1 180.2.d.a 2
48.i odd 4 1 2880.2.f.l 2
48.k even 4 1 720.2.f.c 2
48.k even 4 1 2880.2.f.p 2
80.i odd 4 1 300.2.a.a 1
80.i odd 4 1 4800.2.a.bj 1
80.j even 4 1 1200.2.a.a 1
80.j even 4 1 4800.2.a.bf 1
80.k odd 4 1 240.2.f.b 2
80.k odd 4 1 960.2.f.c 2
80.q even 4 1 60.2.d.a 2
80.q even 4 1 960.2.f.f 2
80.s even 4 1 1200.2.a.s 1
80.s even 4 1 4800.2.a.bk 1
80.t odd 4 1 300.2.a.d 1
80.t odd 4 1 4800.2.a.bn 1
112.l odd 4 1 2940.2.k.c 2
112.w even 12 2 2940.2.bb.d 4
112.x odd 12 2 2940.2.bb.e 4
144.w odd 12 2 1620.2.r.d 4
144.x even 12 2 1620.2.r.c 4
240.t even 4 1 720.2.f.c 2
240.t even 4 1 2880.2.f.p 2
240.z odd 4 1 3600.2.a.bm 1
240.bb even 4 1 900.2.a.a 1
240.bd odd 4 1 3600.2.a.d 1
240.bf even 4 1 900.2.a.h 1
240.bm odd 4 1 180.2.d.a 2
240.bm odd 4 1 2880.2.f.l 2
560.bf odd 4 1 2940.2.k.c 2
560.cn odd 12 2 2940.2.bb.e 4
560.cr even 12 2 2940.2.bb.d 4
720.ce even 12 2 1620.2.r.c 4
720.ch odd 12 2 1620.2.r.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.d.a 2 16.e even 4 1
60.2.d.a 2 80.q even 4 1
180.2.d.a 2 48.i odd 4 1
180.2.d.a 2 240.bm odd 4 1
240.2.f.b 2 16.f odd 4 1
240.2.f.b 2 80.k odd 4 1
300.2.a.a 1 80.i odd 4 1
300.2.a.d 1 80.t odd 4 1
720.2.f.c 2 48.k even 4 1
720.2.f.c 2 240.t even 4 1
900.2.a.a 1 240.bb even 4 1
900.2.a.h 1 240.bf even 4 1
960.2.f.c 2 16.f odd 4 1
960.2.f.c 2 80.k odd 4 1
960.2.f.f 2 16.e even 4 1
960.2.f.f 2 80.q even 4 1
1200.2.a.a 1 80.j even 4 1
1200.2.a.s 1 80.s even 4 1
1620.2.r.c 4 144.x even 12 2
1620.2.r.c 4 720.ce even 12 2
1620.2.r.d 4 144.w odd 12 2
1620.2.r.d 4 720.ch odd 12 2
2880.2.f.l 2 48.i odd 4 1
2880.2.f.l 2 240.bm odd 4 1
2880.2.f.p 2 48.k even 4 1
2880.2.f.p 2 240.t even 4 1
2940.2.k.c 2 112.l odd 4 1
2940.2.k.c 2 560.bf odd 4 1
2940.2.bb.d 4 112.w even 12 2
2940.2.bb.d 4 560.cr even 12 2
2940.2.bb.e 4 112.x odd 12 2
2940.2.bb.e 4 560.cn odd 12 2
3600.2.a.d 1 240.bd odd 4 1
3600.2.a.bm 1 240.z odd 4 1
3840.2.d.b 2 4.b odd 2 1
3840.2.d.b 2 40.e odd 2 1
3840.2.d.o 2 5.b even 2 1
3840.2.d.o 2 8.b even 2 1
3840.2.d.r 2 1.a even 1 1 trivial
3840.2.d.r 2 40.f even 2 1 inner
3840.2.d.be 2 8.d odd 2 1
3840.2.d.be 2 20.d odd 2 1
4800.2.a.bf 1 80.j even 4 1
4800.2.a.bj 1 80.i odd 4 1
4800.2.a.bk 1 80.s even 4 1
4800.2.a.bn 1 80.t odd 4 1

Hecke kernels

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

\( T_{7}^{2} + 16 \)
\( T_{11}^{2} + 16 \)
\( T_{13} \)
\( T_{31} - 4 \)
\( T_{37} + 8 \)
\( T_{43} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( 5 + 4 T + T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( 16 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 16 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( 36 + T^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( ( 8 + T )^{2} \)
$41$ \( ( -10 + T )^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( 16 + T^{2} \)
$53$ \( ( -12 + T )^{2} \)
$59$ \( 16 + T^{2} \)
$61$ \( 4 + T^{2} \)
$67$ \( ( -4 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( 64 + T^{2} \)
$79$ \( ( -12 + T )^{2} \)
$83$ \( ( -4 + T )^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( 64 + T^{2} \)
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