Properties

Label 288.2.v.d
Level $288$
Weight $2$
Character orbit 288.v
Analytic conductor $2.300$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [288,2,Mod(37,288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(288, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 288.v (of order \(8\), 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: \(2.29969157821\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{10} + 32 q^{14} - 8 q^{16} + 32 q^{20} - 8 q^{22} + 16 q^{23} - 40 q^{26} + 40 q^{28} - 48 q^{31} - 40 q^{32} + 40 q^{34} + 48 q^{35} - 40 q^{38} + 8 q^{40} - 16 q^{43} - 8 q^{44} - 32 q^{46} - 24 q^{50} - 8 q^{52} + 32 q^{53} + 32 q^{55} - 56 q^{56} - 32 q^{58} - 64 q^{59} - 32 q^{61} - 48 q^{62} + 24 q^{64} + 16 q^{67} + 8 q^{68} - 24 q^{70} - 64 q^{71} + 32 q^{74} - 56 q^{76} + 32 q^{77} + 56 q^{80} - 40 q^{82} + 64 q^{86} - 48 q^{88} - 48 q^{91} + 80 q^{92} - 32 q^{94} + 80 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −1.40823 + 0.129991i 0 1.96620 0.366115i 2.51374 + 1.04122i 0 2.01027 + 2.01027i −2.72127 + 0.771162i 0 −3.67526 1.13951i
37.2 −1.34827 0.426820i 0 1.63565 + 1.15093i 1.46213 + 0.605634i 0 −3.54889 3.54889i −1.71405 2.24989i 0 −1.71285 1.44062i
37.3 −0.884039 1.10385i 0 −0.436951 + 1.95168i −2.14986 0.890503i 0 −1.10001 1.10001i 2.54064 1.24304i 0 0.917585 + 3.16036i
37.4 −0.605567 + 1.27800i 0 −1.26658 1.54783i −1.60930 0.666593i 0 −0.589445 0.589445i 2.74513 0.681373i 0 1.82644 1.65302i
37.5 −0.333592 1.37431i 0 −1.77743 + 0.916914i 1.20409 + 0.498752i 0 2.59422 + 2.59422i 1.85306 + 2.13686i 0 0.283762 1.82117i
37.6 0.603367 + 1.27904i 0 −1.27190 + 1.54346i −3.68816 1.52768i 0 −1.63704 1.63704i −2.74158 0.695531i 0 −0.271341 5.63906i
37.7 1.26685 0.628571i 0 1.20980 1.59260i 3.09318 + 1.28124i 0 −1.73503 1.73503i 0.531562 2.77803i 0 4.72394 0.321153i
37.8 1.29526 0.567706i 0 1.35542 1.47066i −0.825824 0.342068i 0 1.17750 + 1.17750i 0.920723 2.67437i 0 −1.26385 + 0.0257578i
109.1 −1.40823 0.129991i 0 1.96620 + 0.366115i 2.51374 1.04122i 0 2.01027 2.01027i −2.72127 0.771162i 0 −3.67526 + 1.13951i
109.2 −1.34827 + 0.426820i 0 1.63565 1.15093i 1.46213 0.605634i 0 −3.54889 + 3.54889i −1.71405 + 2.24989i 0 −1.71285 + 1.44062i
109.3 −0.884039 + 1.10385i 0 −0.436951 1.95168i −2.14986 + 0.890503i 0 −1.10001 + 1.10001i 2.54064 + 1.24304i 0 0.917585 3.16036i
109.4 −0.605567 1.27800i 0 −1.26658 + 1.54783i −1.60930 + 0.666593i 0 −0.589445 + 0.589445i 2.74513 + 0.681373i 0 1.82644 + 1.65302i
109.5 −0.333592 + 1.37431i 0 −1.77743 0.916914i 1.20409 0.498752i 0 2.59422 2.59422i 1.85306 2.13686i 0 0.283762 + 1.82117i
109.6 0.603367 1.27904i 0 −1.27190 1.54346i −3.68816 + 1.52768i 0 −1.63704 + 1.63704i −2.74158 + 0.695531i 0 −0.271341 + 5.63906i
109.7 1.26685 + 0.628571i 0 1.20980 + 1.59260i 3.09318 1.28124i 0 −1.73503 + 1.73503i 0.531562 + 2.77803i 0 4.72394 + 0.321153i
109.8 1.29526 + 0.567706i 0 1.35542 + 1.47066i −0.825824 + 0.342068i 0 1.17750 1.17750i 0.920723 + 2.67437i 0 −1.26385 0.0257578i
181.1 −1.34416 0.439595i 0 1.61351 + 1.18177i −0.184062 + 0.444366i 0 −0.134531 0.134531i −1.64931 2.29777i 0 0.442749 0.516384i
181.2 −0.890433 1.09869i 0 −0.414259 + 1.95663i 0.00259461 0.00626394i 0 −2.41880 2.41880i 2.51860 1.28710i 0 −0.00919248 + 0.00272694i
181.3 −0.525864 + 1.31281i 0 −1.44693 1.38072i 0.155637 0.375742i 0 0.709092 + 0.709092i 2.57351 1.17348i 0 0.411433 + 0.401911i
181.4 0.126318 + 1.40856i 0 −1.96809 + 0.355853i −1.36206 + 3.28830i 0 −2.73097 2.73097i −0.749846 2.72722i 0 −4.80383 1.50317i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.v.d 32
3.b odd 2 1 96.2.n.a 32
4.b odd 2 1 1152.2.v.c 32
12.b even 2 1 384.2.n.a 32
24.f even 2 1 768.2.n.b 32
24.h odd 2 1 768.2.n.a 32
32.g even 8 1 inner 288.2.v.d 32
32.h odd 8 1 1152.2.v.c 32
96.o even 8 1 384.2.n.a 32
96.o even 8 1 768.2.n.b 32
96.p odd 8 1 96.2.n.a 32
96.p odd 8 1 768.2.n.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.n.a 32 3.b odd 2 1
96.2.n.a 32 96.p odd 8 1
288.2.v.d 32 1.a even 1 1 trivial
288.2.v.d 32 32.g even 8 1 inner
384.2.n.a 32 12.b even 2 1
384.2.n.a 32 96.o even 8 1
768.2.n.a 32 24.h odd 2 1
768.2.n.a 32 96.p odd 8 1
768.2.n.b 32 24.f even 2 1
768.2.n.b 32 96.o even 8 1
1152.2.v.c 32 4.b odd 2 1
1152.2.v.c 32 32.h odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{32} + 32 T_{5}^{29} - 384 T_{5}^{27} + 160 T_{5}^{26} - 1408 T_{5}^{25} + 34560 T_{5}^{24} - 29696 T_{5}^{23} + 1152 T_{5}^{22} + 692992 T_{5}^{21} - 298496 T_{5}^{20} - 6861824 T_{5}^{19} - 4331776 T_{5}^{18} + \cdots + 4096 \) acting on \(S_{2}^{\mathrm{new}}(288, [\chi])\). Copy content Toggle raw display