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Properties

Label 1550.3.c.d
Level $1550$
Weight $3$
Character orbit 1550.c
Analytic conductor $42.234$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1550,3,Mod(1301,1550)]
 
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([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1550.1301");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1550 = 2 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1550.c (of order \(2\), degree \(1\), 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: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{4} - 8 q^{7} - 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{4} - 8 q^{7} - 96 q^{9} - 16 q^{14} + 96 q^{16} + 32 q^{18} - 56 q^{19} - 16 q^{28} - 32 q^{31} - 248 q^{33} - 192 q^{36} - 32 q^{38} - 104 q^{39} + 64 q^{41} + 48 q^{47} + 552 q^{49} - 400 q^{51} - 32 q^{56} - 80 q^{59} + 40 q^{62} + 16 q^{63} + 192 q^{64} - 448 q^{66} - 568 q^{67} - 8 q^{69} + 168 q^{71} + 64 q^{72} - 112 q^{76} + 624 q^{78} + 1440 q^{81} - 256 q^{82} + 296 q^{87} - 256 q^{93} - 432 q^{94} + 128 q^{97} - 640 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1301.1 −1.41421 − 5.82697i 2.00000 0 8.24058i 12.9826 −2.82843 −24.9536 0
1301.2 −1.41421 − 5.42696i 2.00000 0 7.67489i −7.66369 −2.82843 −20.4519 0
1301.3 −1.41421 − 3.58942i 2.00000 0 5.07621i −11.6269 −2.82843 −3.88397 0
1301.4 −1.41421 − 1.88454i 2.00000 0 2.66514i 1.29462 −2.82843 5.44852 0
1301.5 −1.41421 − 1.46742i 2.00000 0 2.07524i 11.9745 −2.82843 6.84669 0
1301.6 −1.41421 − 1.28940i 2.00000 0 1.82349i −6.13262 −2.82843 7.33744 0
1301.7 −1.41421 1.28940i 2.00000 0 − 1.82349i −6.13262 −2.82843 7.33744 0
1301.8 −1.41421 1.46742i 2.00000 0 − 2.07524i 11.9745 −2.82843 6.84669 0
1301.9 −1.41421 1.88454i 2.00000 0 − 2.66514i 1.29462 −2.82843 5.44852 0
1301.10 −1.41421 3.58942i 2.00000 0 − 5.07621i −11.6269 −2.82843 −3.88397 0
1301.11 −1.41421 5.42696i 2.00000 0 − 7.67489i −7.66369 −2.82843 −20.4519 0
1301.12 −1.41421 5.82697i 2.00000 0 − 8.24058i 12.9826 −2.82843 −24.9536 0
1301.13 1.41421 − 5.88545i 2.00000 0 − 8.32328i −1.81095 2.82843 −25.6385 0
1301.14 1.41421 − 4.63999i 2.00000 0 − 6.56194i 0.345649 2.82843 −12.5295 0
1301.15 1.41421 − 2.59054i 2.00000 0 − 3.66358i −11.2758 2.82843 2.28909 0
1301.16 1.41421 − 2.35136i 2.00000 0 − 3.32533i −7.27991 2.82843 3.47110 0
1301.17 1.41421 − 1.96026i 2.00000 0 − 2.77223i 10.7520 2.82843 5.15737 0
1301.18 1.41421 − 0.304421i 2.00000 0 − 0.430516i 4.44058 2.82843 8.90733 0
1301.19 1.41421 0.304421i 2.00000 0 0.430516i 4.44058 2.82843 8.90733 0
1301.20 1.41421 1.96026i 2.00000 0 2.77223i 10.7520 2.82843 5.15737 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1301.24
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1550, [\chi])\):

\( T_{3}^{24} + 156 T_{3}^{22} + 10188 T_{3}^{20} + 363508 T_{3}^{18} + 7788270 T_{3}^{16} + \cdots + 1614110976 \) Copy content Toggle raw display
\( T_{7}^{12} + 4 T_{7}^{11} - 424 T_{7}^{10} - 2104 T_{7}^{9} + 62108 T_{7}^{8} + 365728 T_{7}^{7} + \cdots + 269806144 \) Copy content Toggle raw display