Properties

Label 3840.2.k.r
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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3840,2,Mod(1921,3840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3840.1921");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(30.6625543762\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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.r 2
4.b odd 2 1 3840.2.k.m 2
8.b even 2 1 inner 3840.2.k.r 2
8.d odd 2 1 3840.2.k.m 2
16.e even 4 1 240.2.a.d 1
16.e even 4 1 960.2.a.a 1
16.f odd 4 1 15.2.a.a 1
16.f odd 4 1 960.2.a.l 1
48.i odd 4 1 720.2.a.c 1
48.i odd 4 1 2880.2.a.bc 1
48.k even 4 1 45.2.a.a 1
48.k even 4 1 2880.2.a.y 1
80.i odd 4 1 1200.2.f.h 2
80.i odd 4 1 4800.2.f.c 2
80.j even 4 1 75.2.b.b 2
80.j even 4 1 4800.2.f.bf 2
80.k odd 4 1 75.2.a.b 1
80.k odd 4 1 4800.2.a.t 1
80.q even 4 1 1200.2.a.e 1
80.q even 4 1 4800.2.a.bz 1
80.s even 4 1 75.2.b.b 2
80.s even 4 1 4800.2.f.bf 2
80.t odd 4 1 1200.2.f.h 2
80.t odd 4 1 4800.2.f.c 2
112.j even 4 1 735.2.a.c 1
112.u odd 12 2 735.2.i.e 2
112.v even 12 2 735.2.i.d 2
144.u even 12 2 405.2.e.c 2
144.v odd 12 2 405.2.e.f 2
176.i even 4 1 1815.2.a.d 1
208.o odd 4 1 2535.2.a.j 1
240.t even 4 1 225.2.a.b 1
240.z odd 4 1 225.2.b.b 2
240.bb even 4 1 3600.2.f.e 2
240.bd odd 4 1 225.2.b.b 2
240.bf even 4 1 3600.2.f.e 2
240.bm odd 4 1 3600.2.a.u 1
272.k odd 4 1 4335.2.a.c 1
304.m even 4 1 5415.2.a.j 1
336.v odd 4 1 2205.2.a.i 1
368.i even 4 1 7935.2.a.d 1
528.s odd 4 1 5445.2.a.c 1
560.be even 4 1 3675.2.a.j 1
624.v even 4 1 7605.2.a.g 1
880.bi even 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.f odd 4 1
45.2.a.a 1 48.k even 4 1
75.2.a.b 1 80.k odd 4 1
75.2.b.b 2 80.j even 4 1
75.2.b.b 2 80.s even 4 1
225.2.a.b 1 240.t even 4 1
225.2.b.b 2 240.z odd 4 1
225.2.b.b 2 240.bd odd 4 1
240.2.a.d 1 16.e even 4 1
405.2.e.c 2 144.u even 12 2
405.2.e.f 2 144.v odd 12 2
720.2.a.c 1 48.i odd 4 1
735.2.a.c 1 112.j even 4 1
735.2.i.d 2 112.v even 12 2
735.2.i.e 2 112.u odd 12 2
960.2.a.a 1 16.e even 4 1
960.2.a.l 1 16.f odd 4 1
1200.2.a.e 1 80.q even 4 1
1200.2.f.h 2 80.i odd 4 1
1200.2.f.h 2 80.t odd 4 1
1815.2.a.d 1 176.i even 4 1
2205.2.a.i 1 336.v odd 4 1
2535.2.a.j 1 208.o odd 4 1
2880.2.a.y 1 48.k even 4 1
2880.2.a.bc 1 48.i odd 4 1
3600.2.a.u 1 240.bm odd 4 1
3600.2.f.e 2 240.bb even 4 1
3600.2.f.e 2 240.bf even 4 1
3675.2.a.j 1 560.be even 4 1
3840.2.k.m 2 4.b odd 2 1
3840.2.k.m 2 8.d odd 2 1
3840.2.k.r 2 1.a even 1 1 trivial
3840.2.k.r 2 8.b even 2 1 inner
4335.2.a.c 1 272.k odd 4 1
4800.2.a.t 1 80.k odd 4 1
4800.2.a.bz 1 80.q even 4 1
4800.2.f.c 2 80.i odd 4 1
4800.2.f.c 2 80.t odd 4 1
4800.2.f.bf 2 80.j even 4 1
4800.2.f.bf 2 80.s even 4 1
5415.2.a.j 1 304.m even 4 1
5445.2.a.c 1 528.s odd 4 1
7605.2.a.g 1 624.v even 4 1
7935.2.a.d 1 368.i even 4 1
9075.2.a.g 1 880.bi even 4 1

Hecke kernels

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

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

Hecke characteristic polynomials

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