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Properties

Label 576.2.bd.a
Level $576$
Weight $2$
Character orbit 576.bd
Analytic conductor $4.599$
Analytic rank $0$
Dimension $56$
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] = [576,2,Mod(37,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 9, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.bd (of order \(16\), degree \(8\), 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: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(7\) over \(\Q(\zeta_{16})\)
Twist minimal: no (minimal twist has level 64)
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 8 q^{2} - 8 q^{4} + 8 q^{5} - 8 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 8 q^{2} - 8 q^{4} + 8 q^{5} - 8 q^{7} + 8 q^{8} - 8 q^{10} + 8 q^{11} - 8 q^{13} + 8 q^{14} - 8 q^{16} + 8 q^{17} - 8 q^{19} + 8 q^{20} + 8 q^{23} - 8 q^{25} - 32 q^{26} + 32 q^{28} + 8 q^{29} - 32 q^{32} + 32 q^{34} + 8 q^{35} - 8 q^{37} - 32 q^{38} + 32 q^{40} + 8 q^{41} - 8 q^{43} - 8 q^{46} + 8 q^{47} - 8 q^{49} + 32 q^{50} - 56 q^{52} + 8 q^{53} + 56 q^{55} + 64 q^{56} - 80 q^{58} - 56 q^{59} - 8 q^{61} + 40 q^{62} - 104 q^{64} + 16 q^{65} + 72 q^{67} + 56 q^{68} - 104 q^{70} - 56 q^{71} - 8 q^{73} + 64 q^{74} - 72 q^{76} + 8 q^{77} + 24 q^{79} - 32 q^{80} + 72 q^{82} + 8 q^{83} - 8 q^{85} - 96 q^{86} + 72 q^{88} + 8 q^{89} - 8 q^{91} - 144 q^{92} + 88 q^{94} - 128 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −0.998819 + 1.00118i 0 −0.00472188 − 1.99999i −0.0517508 + 0.260169i 0 −0.515195 + 1.24379i 2.00707 + 1.99290i 0 −0.208786 − 0.311673i
37.2 −0.797087 − 1.16818i 0 −0.729306 + 1.86229i −0.631428 + 3.17440i 0 −0.127129 + 0.306917i 2.75681 − 0.632441i 0 4.21159 − 1.79265i
37.3 0.166454 + 1.40438i 0 −1.94459 + 0.467530i 0.573974 − 2.88556i 0 −0.410118 + 0.990113i −0.980274 − 2.65312i 0 4.14798 + 0.325767i
37.4 0.466807 − 1.33495i 0 −1.56418 − 1.24633i 0.690473 − 3.47124i 0 0.983337 − 2.37399i −2.39396 + 1.50631i 0 −4.31162 − 2.54215i
37.5 0.809383 + 1.15970i 0 −0.689800 + 1.87728i 0.159018 − 0.799435i 0 −0.742008 + 1.79137i −2.73539 + 0.719478i 0 1.05581 − 0.462637i
37.6 0.919278 − 1.07468i 0 −0.309855 − 1.97585i −0.509835 + 2.56311i 0 −1.78664 + 4.31333i −2.40824 − 1.48336i 0 2.28583 + 2.90412i
37.7 1.35786 + 0.395231i 0 1.68758 + 1.07334i −0.154331 + 0.775873i 0 1.53949 − 3.71667i 1.86729 + 2.12443i 0 −0.516209 + 0.992533i
109.1 −0.998819 − 1.00118i 0 −0.00472188 + 1.99999i −0.0517508 − 0.260169i 0 −0.515195 − 1.24379i 2.00707 − 1.99290i 0 −0.208786 + 0.311673i
109.2 −0.797087 + 1.16818i 0 −0.729306 − 1.86229i −0.631428 − 3.17440i 0 −0.127129 − 0.306917i 2.75681 + 0.632441i 0 4.21159 + 1.79265i
109.3 0.166454 − 1.40438i 0 −1.94459 − 0.467530i 0.573974 + 2.88556i 0 −0.410118 − 0.990113i −0.980274 + 2.65312i 0 4.14798 − 0.325767i
109.4 0.466807 + 1.33495i 0 −1.56418 + 1.24633i 0.690473 + 3.47124i 0 0.983337 + 2.37399i −2.39396 − 1.50631i 0 −4.31162 + 2.54215i
109.5 0.809383 − 1.15970i 0 −0.689800 − 1.87728i 0.159018 + 0.799435i 0 −0.742008 − 1.79137i −2.73539 − 0.719478i 0 1.05581 + 0.462637i
109.6 0.919278 + 1.07468i 0 −0.309855 + 1.97585i −0.509835 − 2.56311i 0 −1.78664 − 4.31333i −2.40824 + 1.48336i 0 2.28583 − 2.90412i
109.7 1.35786 − 0.395231i 0 1.68758 − 1.07334i −0.154331 − 0.775873i 0 1.53949 + 3.71667i 1.86729 − 2.12443i 0 −0.516209 − 0.992533i
181.1 −1.28112 + 0.598933i 0 1.28256 − 1.53462i −0.884671 − 1.32400i 0 −2.40727 − 0.997123i −0.723982 + 2.73420i 0 1.92636 + 1.16635i
181.2 −0.468362 − 1.33441i 0 −1.56127 + 1.24997i −0.914126 − 1.36809i 0 2.65574 + 1.10004i 2.39921 + 1.49793i 0 −1.39744 + 1.86057i
181.3 −0.325482 + 1.37625i 0 −1.78812 − 0.895889i 0.967135 + 1.44742i 0 4.53283 + 1.87756i 1.81497 − 2.16931i 0 −2.30680 + 0.859909i
181.4 −0.182356 − 1.40241i 0 −1.93349 + 0.511475i 0.787711 + 1.17889i 0 −2.16489 − 0.896725i 1.06988 + 2.61827i 0 1.50964 − 1.31967i
181.5 1.11482 + 0.870162i 0 0.485635 + 1.94014i 0.153107 + 0.229142i 0 −0.843108 − 0.349227i −1.14685 + 2.58549i 0 −0.0287035 + 0.388679i
181.6 1.12131 − 0.861781i 0 0.514666 − 1.93265i −1.82421 − 2.73012i 0 0.00395016 + 0.00163621i −1.08842 − 2.61062i 0 −4.39827 − 1.48924i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
64.i even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.bd.a 56
3.b odd 2 1 64.2.i.a ✓ 56
12.b even 2 1 256.2.i.a 56
24.f even 2 1 512.2.i.a 56
24.h odd 2 1 512.2.i.b 56
64.i even 16 1 inner 576.2.bd.a 56
192.q odd 16 1 64.2.i.a ✓ 56
192.q odd 16 1 512.2.i.b 56
192.s even 16 1 256.2.i.a 56
192.s even 16 1 512.2.i.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.i.a ✓ 56 3.b odd 2 1
64.2.i.a ✓ 56 192.q odd 16 1
256.2.i.a 56 12.b even 2 1
256.2.i.a 56 192.s even 16 1
512.2.i.a 56 24.f even 2 1
512.2.i.a 56 192.s even 16 1
512.2.i.b 56 24.h odd 2 1
512.2.i.b 56 192.q odd 16 1
576.2.bd.a 56 1.a even 1 1 trivial
576.2.bd.a 56 64.i even 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{56} - 8 T_{5}^{55} + 36 T_{5}^{54} - 88 T_{5}^{53} + 74 T_{5}^{52} + 296 T_{5}^{51} + \cdots + 191120240768 \) acting on \(S_{2}^{\mathrm{new}}(576, [\chi])\). Copy content Toggle raw display