Properties

Label 3840.2.k.m
Level $3840$
Weight $2$
Character orbit 3840.k
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.k (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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
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.

\(f(q)\) \(=\) \( q -i q^{3} -i q^{5} - q^{9} +O(q^{10})\) \( q -i q^{3} -i q^{5} - q^{9} + 4 i q^{11} -2 i q^{13} - q^{15} + 2 q^{17} + 4 i q^{19} - q^{25} + i q^{27} -2 i q^{29} + 4 q^{33} + 10 i q^{37} -2 q^{39} -10 q^{41} -4 i q^{43} + i q^{45} + 8 q^{47} -7 q^{49} -2 i q^{51} + 10 i q^{53} + 4 q^{55} + 4 q^{57} + 4 i q^{59} -2 i q^{61} -2 q^{65} + 12 i q^{67} + 8 q^{71} -10 q^{73} + i q^{75} + q^{81} + 12 i q^{83} -2 i q^{85} -2 q^{87} + 6 q^{89} + 4 q^{95} + 2 q^{97} -4 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 2q^{15} + 4q^{17} - 2q^{25} + 8q^{33} - 4q^{39} - 20q^{41} + 16q^{47} - 14q^{49} + 8q^{55} + 8q^{57} - 4q^{65} + 16q^{71} - 20q^{73} + 2q^{81} - 4q^{87} + 12q^{89} + 8q^{95} + 4q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1921.1
1.00000i
1.00000i
0 1.00000i 0 1.00000i 0 0 0 −1.00000 0
1921.2 0 1.00000i 0 1.00000i 0 0 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
8.b 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.k.m 2
4.b odd 2 1 3840.2.k.r 2
8.b even 2 1 inner 3840.2.k.m 2
8.d odd 2 1 3840.2.k.r 2
16.e even 4 1 15.2.a.a 1
16.e even 4 1 960.2.a.l 1
16.f odd 4 1 240.2.a.d 1
16.f odd 4 1 960.2.a.a 1
48.i odd 4 1 45.2.a.a 1
48.i odd 4 1 2880.2.a.y 1
48.k even 4 1 720.2.a.c 1
48.k even 4 1 2880.2.a.bc 1
80.i odd 4 1 75.2.b.b 2
80.i odd 4 1 4800.2.f.bf 2
80.j even 4 1 1200.2.f.h 2
80.j even 4 1 4800.2.f.c 2
80.k odd 4 1 1200.2.a.e 1
80.k odd 4 1 4800.2.a.bz 1
80.q even 4 1 75.2.a.b 1
80.q even 4 1 4800.2.a.t 1
80.s even 4 1 1200.2.f.h 2
80.s even 4 1 4800.2.f.c 2
80.t odd 4 1 75.2.b.b 2
80.t odd 4 1 4800.2.f.bf 2
112.l odd 4 1 735.2.a.c 1
112.w even 12 2 735.2.i.e 2
112.x odd 12 2 735.2.i.d 2
144.w odd 12 2 405.2.e.c 2
144.x even 12 2 405.2.e.f 2
176.l odd 4 1 1815.2.a.d 1
208.p even 4 1 2535.2.a.j 1
240.t even 4 1 3600.2.a.u 1
240.z odd 4 1 3600.2.f.e 2
240.bb even 4 1 225.2.b.b 2
240.bd odd 4 1 3600.2.f.e 2
240.bf even 4 1 225.2.b.b 2
240.bm odd 4 1 225.2.a.b 1
272.r even 4 1 4335.2.a.c 1
304.j odd 4 1 5415.2.a.j 1
336.y even 4 1 2205.2.a.i 1
368.k odd 4 1 7935.2.a.d 1
528.x even 4 1 5445.2.a.c 1
560.bf odd 4 1 3675.2.a.j 1
624.bi odd 4 1 7605.2.a.g 1
880.x odd 4 1 9075.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 16.e even 4 1
45.2.a.a 1 48.i odd 4 1
75.2.a.b 1 80.q even 4 1
75.2.b.b 2 80.i odd 4 1
75.2.b.b 2 80.t odd 4 1
225.2.a.b 1 240.bm odd 4 1
225.2.b.b 2 240.bb even 4 1
225.2.b.b 2 240.bf even 4 1
240.2.a.d 1 16.f odd 4 1
405.2.e.c 2 144.w odd 12 2
405.2.e.f 2 144.x even 12 2
720.2.a.c 1 48.k even 4 1
735.2.a.c 1 112.l odd 4 1
735.2.i.d 2 112.x odd 12 2
735.2.i.e 2 112.w even 12 2
960.2.a.a 1 16.f odd 4 1
960.2.a.l 1 16.e even 4 1
1200.2.a.e 1 80.k odd 4 1
1200.2.f.h 2 80.j even 4 1
1200.2.f.h 2 80.s even 4 1
1815.2.a.d 1 176.l odd 4 1
2205.2.a.i 1 336.y even 4 1
2535.2.a.j 1 208.p even 4 1
2880.2.a.y 1 48.i odd 4 1
2880.2.a.bc 1 48.k even 4 1
3600.2.a.u 1 240.t even 4 1
3600.2.f.e 2 240.z odd 4 1
3600.2.f.e 2 240.bd odd 4 1
3675.2.a.j 1 560.bf odd 4 1
3840.2.k.m 2 1.a even 1 1 trivial
3840.2.k.m 2 8.b even 2 1 inner
3840.2.k.r 2 4.b odd 2 1
3840.2.k.r 2 8.d odd 2 1
4335.2.a.c 1 272.r even 4 1
4800.2.a.t 1 80.q even 4 1
4800.2.a.bz 1 80.k odd 4 1
4800.2.f.c 2 80.j even 4 1
4800.2.f.c 2 80.s even 4 1
4800.2.f.bf 2 80.i odd 4 1
4800.2.f.bf 2 80.t odd 4 1
5415.2.a.j 1 304.j odd 4 1
5445.2.a.c 1 528.x even 4 1
7605.2.a.g 1 624.bi odd 4 1
7935.2.a.d 1 368.k odd 4 1
9075.2.a.g 1 880.x odd 4 1

Hecke kernels

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

\( T_{7} \)
\( T_{11}^{2} + 16 \)
\( T_{13}^{2} + 4 \)
\( T_{17} - 2 \)
\( T_{23} \)
\( T_{31} \)
\( T_{47} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 16 + T^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( 16 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( 4 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 100 + T^{2} \)
$41$ \( ( 10 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( 100 + T^{2} \)
$59$ \( 16 + T^{2} \)
$61$ \( 4 + T^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( ( 10 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( ( -2 + T )^{2} \)
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