Properties

Label 360.4.b.a
Level $360$
Weight $4$
Character orbit 360.b
Analytic conductor $21.241$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [360,4,Mod(251,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.251");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 360.b (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: \(21.2406876021\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{2} + 6 q^{4} - 120 q^{5} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{2} + 6 q^{4} - 120 q^{5} + 6 q^{8} + 30 q^{10} - 108 q^{14} - 126 q^{16} - 24 q^{19} - 30 q^{20} + 240 q^{22} + 456 q^{23} + 600 q^{25} + 168 q^{26} + 120 q^{28} + 354 q^{32} - 420 q^{34} - 420 q^{38} - 30 q^{40} + 972 q^{44} + 1272 q^{46} - 264 q^{47} - 1896 q^{49} - 150 q^{50} + 1260 q^{52} + 1056 q^{53} + 876 q^{56} - 300 q^{58} + 780 q^{62} - 2130 q^{64} - 816 q^{67} - 1092 q^{68} + 540 q^{70} + 432 q^{71} + 432 q^{73} + 3984 q^{74} + 2508 q^{76} + 630 q^{80} - 3084 q^{82} - 1800 q^{86} - 2832 q^{88} + 3600 q^{91} - 5256 q^{92} + 1236 q^{94} + 120 q^{95} - 1584 q^{97} + 1890 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1 −2.74252 0.691792i 0 7.04285 + 3.79451i −5.00000 0 25.7913i −16.6901 15.2787i 0 13.7126 + 3.45896i
251.2 −2.74252 + 0.691792i 0 7.04285 3.79451i −5.00000 0 25.7913i −16.6901 + 15.2787i 0 13.7126 3.45896i
251.3 −2.61333 1.08191i 0 5.65895 + 5.65476i −5.00000 0 24.2359i −8.67075 20.9002i 0 13.0666 + 5.40954i
251.4 −2.61333 + 1.08191i 0 5.65895 5.65476i −5.00000 0 24.2359i −8.67075 + 20.9002i 0 13.0666 5.40954i
251.5 −2.38691 1.51745i 0 3.39471 + 7.24403i −5.00000 0 27.9651i 2.88957 22.4422i 0 11.9346 + 7.58724i
251.6 −2.38691 + 1.51745i 0 3.39471 7.24403i −5.00000 0 27.9651i 2.88957 + 22.4422i 0 11.9346 7.58724i
251.7 −2.18228 1.79934i 0 1.52472 + 7.85336i −5.00000 0 12.2479i 10.8035 19.8817i 0 10.9114 + 8.99672i
251.8 −2.18228 + 1.79934i 0 1.52472 7.85336i −5.00000 0 12.2479i 10.8035 + 19.8817i 0 10.9114 8.99672i
251.9 −1.70898 2.25375i 0 −2.15877 + 7.70323i −5.00000 0 15.4755i 21.0504 8.29935i 0 8.54491 + 11.2687i
251.10 −1.70898 + 2.25375i 0 −2.15877 7.70323i −5.00000 0 15.4755i 21.0504 + 8.29935i 0 8.54491 11.2687i
251.11 −0.671247 2.74762i 0 −7.09886 + 3.68866i −5.00000 0 2.99301i 14.9001 + 17.0290i 0 3.35623 + 13.7381i
251.12 −0.671247 + 2.74762i 0 −7.09886 3.68866i −5.00000 0 2.99301i 14.9001 17.0290i 0 3.35623 13.7381i
251.13 −0.281738 2.81436i 0 −7.84125 + 1.58583i −5.00000 0 11.3970i 6.67226 + 21.6213i 0 1.40869 + 14.0718i
251.14 −0.281738 + 2.81436i 0 −7.84125 1.58583i −5.00000 0 11.3970i 6.67226 21.6213i 0 1.40869 14.0718i
251.15 0.666534 2.74877i 0 −7.11147 3.66429i −5.00000 0 22.7533i −14.8123 + 17.1054i 0 −3.33267 + 13.7438i
251.16 0.666534 + 2.74877i 0 −7.11147 + 3.66429i −5.00000 0 22.7533i −14.8123 17.1054i 0 −3.33267 13.7438i
251.17 1.51037 2.39140i 0 −3.43758 7.22378i −5.00000 0 35.1660i −22.4670 2.68996i 0 −7.55184 + 11.9570i
251.18 1.51037 + 2.39140i 0 −3.43758 + 7.22378i −5.00000 0 35.1660i −22.4670 + 2.68996i 0 −7.55184 11.9570i
251.19 2.19727 1.78101i 0 1.65600 7.82673i −5.00000 0 4.37103i −10.3008 20.1468i 0 −10.9864 + 8.90506i
251.20 2.19727 + 1.78101i 0 1.65600 + 7.82673i −5.00000 0 4.37103i −10.3008 + 20.1468i 0 −10.9864 8.90506i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.4.b.a 24
3.b odd 2 1 360.4.b.b yes 24
4.b odd 2 1 1440.4.b.a 24
8.b even 2 1 1440.4.b.b 24
8.d odd 2 1 360.4.b.b yes 24
12.b even 2 1 1440.4.b.b 24
24.f even 2 1 inner 360.4.b.a 24
24.h odd 2 1 1440.4.b.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.4.b.a 24 1.a even 1 1 trivial
360.4.b.a 24 24.f even 2 1 inner
360.4.b.b yes 24 3.b odd 2 1
360.4.b.b yes 24 8.d odd 2 1
1440.4.b.a 24 4.b odd 2 1
1440.4.b.a 24 24.h odd 2 1
1440.4.b.b 24 8.b even 2 1
1440.4.b.b 24 12.b even 2 1