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Properties

Label 360.2.bs.a
Level $360$
Weight $2$
Character orbit 360.bs
Analytic conductor $2.875$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [360,2,Mod(113,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 0, 10, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.bs (of order \(12\), degree \(4\), 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: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 12 q^{15} + 16 q^{21} + 24 q^{23} - 12 q^{27} + 12 q^{33} + 12 q^{41} - 16 q^{45} - 36 q^{47} + 24 q^{51} - 40 q^{57} + 12 q^{61} - 44 q^{63} - 72 q^{65} - 36 q^{75} - 48 q^{77} - 20 q^{81} - 60 q^{83} + 24 q^{85} - 40 q^{87} - 84 q^{93} - 60 q^{95} + 24 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1 0 −1.72328 − 0.174114i 0 −1.74059 + 1.40368i 0 −2.54088 − 0.680826i 0 2.93937 + 0.600095i 0
113.2 0 −1.71407 + 0.248907i 0 0.528278 − 2.17277i 0 −0.980572 − 0.262743i 0 2.87609 − 0.853289i 0
113.3 0 −1.66953 − 0.461153i 0 −2.18181 − 0.489584i 0 1.65362 + 0.443086i 0 2.57468 + 1.53982i 0
113.4 0 −1.37710 + 1.05053i 0 2.15536 − 0.595338i 0 0.205487 + 0.0550600i 0 0.792791 − 2.89335i 0
113.5 0 −1.16547 − 1.28128i 0 1.29543 + 1.82260i 0 −4.79192 − 1.28399i 0 −0.283342 + 2.98659i 0
113.6 0 −1.03585 + 1.38817i 0 −0.403585 + 2.19935i 0 4.55631 + 1.22086i 0 −0.854037 − 2.87587i 0
113.7 0 −0.985769 − 1.42417i 0 2.06252 + 0.863710i 0 2.64986 + 0.710029i 0 −1.05652 + 2.80780i 0
113.8 0 −0.699583 + 1.58448i 0 −2.21911 − 0.274850i 0 −1.41974 − 0.380419i 0 −2.02117 − 2.21695i 0
113.9 0 0.254049 − 1.71332i 0 −0.831407 − 2.07576i 0 −1.77070 − 0.474457i 0 −2.87092 − 0.870532i 0
113.10 0 0.416476 − 1.68123i 0 −1.03051 + 1.98445i 0 3.03419 + 0.813008i 0 −2.65309 − 1.40039i 0
113.11 0 0.684109 + 1.59122i 0 1.43117 − 1.71807i 0 2.16574 + 0.580309i 0 −2.06399 + 2.17714i 0
113.12 0 0.771894 + 1.55054i 0 0.743858 + 2.10871i 0 0.110362 + 0.0295714i 0 −1.80836 + 2.39371i 0
113.13 0 0.836139 + 1.51686i 0 −2.22494 − 0.222822i 0 −3.21063 − 0.860287i 0 −1.60174 + 2.53662i 0
113.14 0 0.925409 − 1.46411i 0 1.76908 − 1.36761i 0 −3.63616 − 0.974307i 0 −1.28724 − 2.70980i 0
113.15 0 1.54161 − 0.789587i 0 −1.90652 − 1.16840i 0 3.69137 + 0.989099i 0 1.75310 − 2.43447i 0
113.16 0 1.60229 − 0.657781i 0 2.22211 − 0.249468i 0 2.85561 + 0.765158i 0 2.13465 − 2.10791i 0
113.17 0 1.60816 + 0.643288i 0 −0.153910 − 2.23076i 0 −1.54994 − 0.415306i 0 2.17236 + 2.06902i 0
113.18 0 1.73052 + 0.0727431i 0 0.484586 + 2.18293i 0 −1.02200 − 0.273845i 0 2.98942 + 0.251767i 0
137.1 0 −1.72328 + 0.174114i 0 −1.74059 − 1.40368i 0 −2.54088 + 0.680826i 0 2.93937 − 0.600095i 0
137.2 0 −1.71407 − 0.248907i 0 0.528278 + 2.17277i 0 −0.980572 + 0.262743i 0 2.87609 + 0.853289i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
9.d odd 6 1 inner
45.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.bs.a ✓ 72
3.b odd 2 1 1080.2.bt.a 72
4.b odd 2 1 720.2.cu.e 72
5.c odd 4 1 inner 360.2.bs.a ✓ 72
9.c even 3 1 1080.2.bt.a 72
9.d odd 6 1 inner 360.2.bs.a ✓ 72
15.e even 4 1 1080.2.bt.a 72
20.e even 4 1 720.2.cu.e 72
36.h even 6 1 720.2.cu.e 72
45.k odd 12 1 1080.2.bt.a 72
45.l even 12 1 inner 360.2.bs.a ✓ 72
180.v odd 12 1 720.2.cu.e 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bs.a ✓ 72 1.a even 1 1 trivial
360.2.bs.a ✓ 72 5.c odd 4 1 inner
360.2.bs.a ✓ 72 9.d odd 6 1 inner
360.2.bs.a ✓ 72 45.l even 12 1 inner
720.2.cu.e 72 4.b odd 2 1
720.2.cu.e 72 20.e even 4 1
720.2.cu.e 72 36.h even 6 1
720.2.cu.e 72 180.v odd 12 1
1080.2.bt.a 72 3.b odd 2 1
1080.2.bt.a 72 9.c even 3 1
1080.2.bt.a 72 15.e even 4 1
1080.2.bt.a 72 45.k odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(360, [\chi])\).