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Properties

Label 240.5.bj.b
Level $240$
Weight $5$
Character orbit 240.bj
Analytic conductor $24.809$
Analytic rank $0$
Dimension $64$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,5,Mod(47,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 2, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.47");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 240.bj (of order \(4\), degree \(2\), 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: \(64\)
Relative dimension: \(32\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 200 q^{13} - 824 q^{21} - 1176 q^{25} + 1120 q^{33} - 8680 q^{37} - 2312 q^{45} - 6520 q^{57} - 1600 q^{61} - 80 q^{73} - 7856 q^{81} - 24328 q^{85} - 14880 q^{93} + 15280 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 0 −8.99259 + 0.365243i 0 −17.6456 − 17.7097i 0 50.6834 − 50.6834i 0 80.7332 − 6.56896i 0
47.2 0 −8.97181 + 0.711810i 0 16.8580 − 18.4610i 0 −58.6941 + 58.6941i 0 79.9867 − 12.7724i 0
47.3 0 −8.71589 − 2.24348i 0 −24.2580 − 6.04549i 0 −12.1404 + 12.1404i 0 70.9336 + 39.1078i 0
47.4 0 −8.60028 + 2.65238i 0 21.3619 + 12.9873i 0 −6.41375 + 6.41375i 0 66.9297 − 45.6225i 0
47.5 0 −8.01800 + 4.08800i 0 −3.50867 + 24.7526i 0 6.77507 − 6.77507i 0 47.5766 − 65.5551i 0
47.6 0 −7.52604 − 4.93545i 0 4.14186 + 24.6545i 0 36.3058 − 36.3058i 0 32.2826 + 74.2889i 0
47.7 0 −6.53776 + 6.18528i 0 22.9467 − 9.92222i 0 58.0567 − 58.0567i 0 4.48459 − 80.8758i 0
47.8 0 −6.38581 − 6.34204i 0 15.1705 − 19.8710i 0 −13.9666 + 13.9666i 0 0.557094 + 80.9981i 0
47.9 0 −6.34204 − 6.38581i 0 −15.1705 + 19.8710i 0 −13.9666 + 13.9666i 0 −0.557094 + 80.9981i 0
47.10 0 −6.18528 + 6.53776i 0 −22.9467 + 9.92222i 0 −58.0567 + 58.0567i 0 −4.48459 − 80.8758i 0
47.11 0 −4.93545 − 7.52604i 0 −4.14186 − 24.6545i 0 36.3058 − 36.3058i 0 −32.2826 + 74.2889i 0
47.12 0 −4.08800 + 8.01800i 0 3.50867 − 24.7526i 0 −6.77507 + 6.77507i 0 −47.5766 − 65.5551i 0
47.13 0 −2.65238 + 8.60028i 0 −21.3619 − 12.9873i 0 6.41375 − 6.41375i 0 −66.9297 − 45.6225i 0
47.14 0 −2.24348 − 8.71589i 0 24.2580 + 6.04549i 0 −12.1404 + 12.1404i 0 −70.9336 + 39.1078i 0
47.15 0 −0.711810 + 8.97181i 0 −16.8580 + 18.4610i 0 58.6941 − 58.6941i 0 −79.9867 − 12.7724i 0
47.16 0 −0.365243 + 8.99259i 0 17.6456 + 17.7097i 0 −50.6834 + 50.6834i 0 −80.7332 − 6.56896i 0
47.17 0 0.365243 − 8.99259i 0 17.6456 + 17.7097i 0 50.6834 − 50.6834i 0 −80.7332 − 6.56896i 0
47.18 0 0.711810 − 8.97181i 0 −16.8580 + 18.4610i 0 −58.6941 + 58.6941i 0 −79.9867 − 12.7724i 0
47.19 0 2.24348 + 8.71589i 0 24.2580 + 6.04549i 0 12.1404 − 12.1404i 0 −70.9336 + 39.1078i 0
47.20 0 2.65238 − 8.60028i 0 −21.3619 − 12.9873i 0 −6.41375 + 6.41375i 0 −66.9297 − 45.6225i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.c odd 4 1 inner
12.b even 2 1 inner
15.e even 4 1 inner
20.e even 4 1 inner
60.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.5.bj.b ✓ 64
3.b odd 2 1 inner 240.5.bj.b ✓ 64
4.b odd 2 1 inner 240.5.bj.b ✓ 64
5.c odd 4 1 inner 240.5.bj.b ✓ 64
12.b even 2 1 inner 240.5.bj.b ✓ 64
15.e even 4 1 inner 240.5.bj.b ✓ 64
20.e even 4 1 inner 240.5.bj.b ✓ 64
60.l odd 4 1 inner 240.5.bj.b ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.5.bj.b ✓ 64 1.a even 1 1 trivial
240.5.bj.b ✓ 64 3.b odd 2 1 inner
240.5.bj.b ✓ 64 4.b odd 2 1 inner
240.5.bj.b ✓ 64 5.c odd 4 1 inner
240.5.bj.b ✓ 64 12.b even 2 1 inner
240.5.bj.b ✓ 64 15.e even 4 1 inner
240.5.bj.b ✓ 64 20.e even 4 1 inner
240.5.bj.b ✓ 64 60.l odd 4 1 inner