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Properties

Label 240.5.c.f
Level $240$
Weight $5$
Character orbit 240.c
Analytic conductor $24.809$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,5,Mod(209,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.209");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 240.c (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: \(24.8087911401\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 120)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{9} + 128 q^{15} + 808 q^{21} + 456 q^{25} + 1056 q^{31} - 1216 q^{39} + 2584 q^{45} - 4536 q^{49} - 1056 q^{51} + 864 q^{55} + 288 q^{61} + 184 q^{69} - 4576 q^{75} + 27744 q^{79} + 10504 q^{81} - 2784 q^{85} + 3840 q^{91} - 8800 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1 0 −8.62424 − 2.57342i 0 16.5314 − 18.7540i 0 − 73.9225i 0 67.7550 + 44.3876i 0
209.2 0 −8.62424 + 2.57342i 0 16.5314 + 18.7540i 0 73.9225i 0 67.7550 − 44.3876i 0
209.3 0 −8.41878 − 3.18186i 0 −16.9150 − 18.4088i 0 16.8440i 0 60.7516 + 53.5747i 0
209.4 0 −8.41878 + 3.18186i 0 −16.9150 + 18.4088i 0 − 16.8440i 0 60.7516 − 53.5747i 0
209.5 0 −7.04122 − 5.60546i 0 −24.5755 − 4.58768i 0 30.8149i 0 18.1576 + 78.9386i 0
209.6 0 −7.04122 + 5.60546i 0 −24.5755 + 4.58768i 0 − 30.8149i 0 18.1576 − 78.9386i 0
209.7 0 −6.61251 − 6.10530i 0 17.9095 + 17.4428i 0 − 21.0743i 0 6.45058 + 80.7427i 0
209.8 0 −6.61251 + 6.10530i 0 17.9095 − 17.4428i 0 21.0743i 0 6.45058 − 80.7427i 0
209.9 0 −1.22991 − 8.91557i 0 20.8936 − 13.7280i 0 79.2407i 0 −77.9747 + 21.9306i 0
209.10 0 −1.22991 + 8.91557i 0 20.8936 + 13.7280i 0 − 79.2407i 0 −77.9747 − 21.9306i 0
209.11 0 −0.964350 − 8.94819i 0 −3.36971 + 24.7719i 0 − 46.0321i 0 −79.1401 + 17.2584i 0
209.12 0 −0.964350 + 8.94819i 0 −3.36971 − 24.7719i 0 46.0321i 0 −79.1401 − 17.2584i 0
209.13 0 0.964350 − 8.94819i 0 3.36971 − 24.7719i 0 − 46.0321i 0 −79.1401 − 17.2584i 0
209.14 0 0.964350 + 8.94819i 0 3.36971 + 24.7719i 0 46.0321i 0 −79.1401 + 17.2584i 0
209.15 0 1.22991 − 8.91557i 0 −20.8936 + 13.7280i 0 79.2407i 0 −77.9747 − 21.9306i 0
209.16 0 1.22991 + 8.91557i 0 −20.8936 − 13.7280i 0 − 79.2407i 0 −77.9747 + 21.9306i 0
209.17 0 6.61251 − 6.10530i 0 −17.9095 − 17.4428i 0 − 21.0743i 0 6.45058 − 80.7427i 0
209.18 0 6.61251 + 6.10530i 0 −17.9095 + 17.4428i 0 21.0743i 0 6.45058 + 80.7427i 0
209.19 0 7.04122 − 5.60546i 0 24.5755 + 4.58768i 0 30.8149i 0 18.1576 − 78.9386i 0
209.20 0 7.04122 + 5.60546i 0 24.5755 − 4.58768i 0 − 30.8149i 0 18.1576 + 78.9386i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.5.c.f 24
3.b odd 2 1 inner 240.5.c.f 24
4.b odd 2 1 120.5.c.a ✓ 24
5.b even 2 1 inner 240.5.c.f 24
12.b even 2 1 120.5.c.a ✓ 24
15.d odd 2 1 inner 240.5.c.f 24
20.d odd 2 1 120.5.c.a ✓ 24
20.e even 4 2 600.5.l.e 24
60.h even 2 1 120.5.c.a ✓ 24
60.l odd 4 2 600.5.l.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.5.c.a ✓ 24 4.b odd 2 1
120.5.c.a ✓ 24 12.b even 2 1
120.5.c.a ✓ 24 20.d odd 2 1
120.5.c.a ✓ 24 60.h even 2 1
240.5.c.f 24 1.a even 1 1 trivial
240.5.c.f 24 3.b odd 2 1 inner
240.5.c.f 24 5.b even 2 1 inner
240.5.c.f 24 15.d odd 2 1 inner
600.5.l.e 24 20.e even 4 2
600.5.l.e 24 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(240, [\chi])\):

\( T_{7}^{12} + 15540 T_{7}^{10} + 83266980 T_{7}^{8} + 183450603008 T_{7}^{6} + 171988922572800 T_{7}^{4} + \cdots + 86\!\cdots\!36 \) Copy content Toggle raw display
\( T_{17}^{12} - 690672 T_{17}^{10} + 181997089632 T_{17}^{8} + \cdots + 44\!\cdots\!84 \) Copy content Toggle raw display