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Properties

Label 1550.3.d.c
Level $1550$
Weight $3$
Character orbit 1550.d
Analytic conductor $42.234$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1550,3,Mod(1549,1550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1550, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1550.1549");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1550 = 2 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1550.d (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: \(42.2344409758\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: no (minimal twist has level 310)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 96 q^{4} + 192 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 96 q^{4} + 192 q^{9} + 32 q^{14} + 192 q^{16} + 112 q^{19} - 64 q^{31} - 384 q^{36} + 208 q^{39} + 128 q^{41} - 1104 q^{49} - 800 q^{51} - 64 q^{56} + 160 q^{59} - 384 q^{64} - 896 q^{66} + 16 q^{69} + 336 q^{71} - 224 q^{76} + 2880 q^{81} + 864 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1549.1 − 1.41421i −5.82697 −2.00000 0 8.24058i 12.9826i 2.82843i 24.9536 0
1549.2 1.41421i −5.82697 −2.00000 0 − 8.24058i − 12.9826i − 2.82843i 24.9536 0
1549.3 − 1.41421i −1.46742 −2.00000 0 2.07524i 11.9745i 2.82843i −6.84669 0
1549.4 1.41421i −1.46742 −2.00000 0 − 2.07524i − 11.9745i − 2.82843i −6.84669 0
1549.5 − 1.41421i 3.58942 −2.00000 0 − 5.07621i − 11.6269i 2.82843i 3.88397 0
1549.6 1.41421i 3.58942 −2.00000 0 5.07621i 11.6269i − 2.82843i 3.88397 0
1549.7 − 1.41421i 2.59054 −2.00000 0 − 3.66358i 11.2758i 2.82843i −2.28909 0
1549.8 1.41421i 2.59054 −2.00000 0 3.66358i − 11.2758i − 2.82843i −2.28909 0
1549.9 − 1.41421i 1.96026 −2.00000 0 − 2.77223i − 10.7520i 2.82843i −5.15737 0
1549.10 1.41421i 1.96026 −2.00000 0 2.77223i 10.7520i − 2.82843i −5.15737 0
1549.11 − 1.41421i 5.42696 −2.00000 0 − 7.67489i − 7.66369i 2.82843i 20.4519 0
1549.12 1.41421i 5.42696 −2.00000 0 7.67489i 7.66369i − 2.82843i 20.4519 0
1549.13 − 1.41421i 2.35136 −2.00000 0 − 3.32533i 7.27991i 2.82843i −3.47110 0
1549.14 1.41421i 2.35136 −2.00000 0 3.32533i − 7.27991i − 2.82843i −3.47110 0
1549.15 − 1.41421i 5.88545 −2.00000 0 − 8.32328i 1.81095i 2.82843i 25.6385 0
1549.16 1.41421i 5.88545 −2.00000 0 8.32328i − 1.81095i − 2.82843i 25.6385 0
1549.17 − 1.41421i 1.28940 −2.00000 0 − 1.82349i − 6.13262i 2.82843i −7.33744 0
1549.18 1.41421i 1.28940 −2.00000 0 1.82349i 6.13262i − 2.82843i −7.33744 0
1549.19 − 1.41421i −4.63999 −2.00000 0 6.56194i − 0.345649i 2.82843i 12.5295 0
1549.20 1.41421i −4.63999 −2.00000 0 − 6.56194i 0.345649i − 2.82843i 12.5295 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1549.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.b odd 2 1 inner
155.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1550.3.d.c 48
5.b even 2 1 inner 1550.3.d.c 48
5.c odd 4 1 310.3.c.a ✓ 24
5.c odd 4 1 1550.3.c.d 24
31.b odd 2 1 inner 1550.3.d.c 48
155.c odd 2 1 inner 1550.3.d.c 48
155.f even 4 1 310.3.c.a ✓ 24
155.f even 4 1 1550.3.c.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.3.c.a ✓ 24 5.c odd 4 1
310.3.c.a ✓ 24 155.f even 4 1
1550.3.c.d 24 5.c odd 4 1
1550.3.c.d 24 155.f even 4 1
1550.3.d.c 48 1.a even 1 1 trivial
1550.3.d.c 48 5.b even 2 1 inner
1550.3.d.c 48 31.b odd 2 1 inner
1550.3.d.c 48 155.c odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 156 T_{3}^{22} + 10188 T_{3}^{20} - 363508 T_{3}^{18} + 7788270 T_{3}^{16} + \cdots + 1614110976 \) acting on \(S_{3}^{\mathrm{new}}(1550, [\chi])\). Copy content Toggle raw display