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Properties

Label 576.2.be.a
Level $576$
Weight $2$
Character orbit 576.be
Analytic conductor $4.599$
Analytic rank $0$
Dimension $128$
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(35,576)]
 
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([8, 11, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.35");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.be (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: \(128\)
Relative dimension: \(16\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 24 q^{8} - 16 q^{11} - 8 q^{14} + 16 q^{23} - 32 q^{29} - 40 q^{32} - 56 q^{38} + 24 q^{44} - 16 q^{50} - 48 q^{52} + 64 q^{55} + 56 q^{56} + 72 q^{62} - 96 q^{64} + 32 q^{67} - 40 q^{68} - 96 q^{70} + 64 q^{71} - 40 q^{74} - 16 q^{76} + 32 q^{79} + 88 q^{80} - 80 q^{82} + 80 q^{83} + 96 q^{86} - 80 q^{88} - 112 q^{92} - 96 q^{94} - 160 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 −1.40118 + 0.191528i 0 1.92663 − 0.536731i 1.17629 + 0.785972i 0 −1.13769 − 2.74664i −2.59677 + 1.12106i 0 −1.79874 − 0.876000i
35.2 −1.33450 + 0.468084i 0 1.56179 − 1.24932i 0.236987 + 0.158350i 0 1.50382 + 3.63054i −1.49943 + 2.39827i 0 −0.390381 − 0.100388i
35.3 −1.27401 − 0.613916i 0 1.24621 + 1.56427i −2.38858 − 1.59600i 0 1.26036 + 3.04279i −0.627361 − 2.75797i 0 2.06327 + 3.49970i
35.4 −1.24876 − 0.663767i 0 1.11883 + 1.65778i −0.804065 − 0.537259i 0 −1.35951 − 3.28216i −0.296774 − 2.81281i 0 0.647473 + 1.20462i
35.5 −0.879724 − 1.10729i 0 −0.452173 + 1.94821i 3.41107 + 2.27920i 0 0.0911845 + 0.220139i 2.55502 − 1.21320i 0 −0.477063 − 5.78210i
35.6 −0.831855 + 1.14369i 0 −0.616034 − 1.90276i 0.405517 + 0.270958i 0 −0.755449 − 1.82382i 2.68861 + 0.878273i 0 −0.647222 + 0.238386i
35.7 −0.422024 − 1.34978i 0 −1.64379 + 1.13928i −0.484600 − 0.323799i 0 0.472392 + 1.14045i 2.23149 + 1.73795i 0 −0.232544 + 0.790752i
35.8 −0.295514 + 1.38299i 0 −1.82534 − 0.817389i −2.89197 − 1.93235i 0 0.337427 + 0.814620i 1.66986 − 2.28289i 0 3.52705 − 3.42853i
35.9 −0.115336 + 1.40950i 0 −1.97340 − 0.325132i 2.90191 + 1.93899i 0 1.96710 + 4.74901i 0.685877 − 2.74401i 0 −3.06771 + 3.86661i
35.10 0.261379 − 1.38985i 0 −1.86336 − 0.726556i −2.30612 − 1.54090i 0 −1.51847 − 3.66590i −1.49685 + 2.39989i 0 −2.74439 + 2.80240i
35.11 0.549304 + 1.30318i 0 −1.39653 + 1.43168i −0.466253 − 0.311540i 0 −1.24280 − 3.00039i −2.63285 − 1.03350i 0 0.149877 − 0.778739i
35.12 0.961549 − 1.03703i 0 −0.150845 − 1.99430i 2.60852 + 1.74296i 0 −1.39314 − 3.36335i −2.21319 − 1.76119i 0 4.31572 − 1.02916i
35.13 1.06781 + 0.927240i 0 0.280453 + 1.98024i 2.26832 + 1.51565i 0 −0.0277850 − 0.0670789i −1.53668 + 2.37457i 0 1.01678 + 3.72171i
35.14 1.11781 − 0.866318i 0 0.498987 − 1.93675i −2.47498 − 1.65373i 0 0.718119 + 1.73369i −1.12007 − 2.59720i 0 −4.19920 + 0.295568i
35.15 1.12852 + 0.852312i 0 0.547127 + 1.92371i −2.24477 − 1.49991i 0 0.731158 + 1.76517i −1.02215 + 2.63727i 0 −1.25488 − 3.60592i
35.16 1.40998 − 0.109385i 0 1.97607 − 0.308461i 1.05270 + 0.703395i 0 0.353291 + 0.852920i 2.75247 − 0.651076i 0 1.56123 + 0.876620i
107.1 −1.41128 + 0.0910919i 0 1.98340 − 0.257112i 0.699372 + 1.04668i 0 0.134172 − 0.323921i −2.77571 + 0.543528i 0 −1.08235 − 1.41345i
107.2 −1.39520 + 0.231102i 0 1.89318 − 0.644869i −1.60913 − 2.40823i 0 0.494518 − 1.19387i −2.49234 + 1.33724i 0 2.80161 + 2.98810i
107.3 −1.19143 − 0.761906i 0 0.838999 + 1.81551i −0.832457 − 1.24586i 0 −0.891000 + 2.15106i 0.383641 − 2.80229i 0 0.0425844 + 2.11860i
107.4 −1.04735 + 0.950290i 0 0.193898 − 1.99058i 1.43801 + 2.15214i 0 0.0494276 − 0.119329i 1.68855 + 2.26910i 0 −3.55126 − 0.887520i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
192.s 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.be.a ✓ 128
3.b odd 2 1 576.2.be.b yes 128
64.j odd 16 1 576.2.be.b yes 128
192.s even 16 1 inner 576.2.be.a ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.be.a ✓ 128 1.a even 1 1 trivial
576.2.be.a ✓ 128 192.s even 16 1 inner
576.2.be.b yes 128 3.b odd 2 1
576.2.be.b yes 128 64.j odd 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{128} + 64 T_{5}^{123} - 1280 T_{5}^{122} + 4288 T_{5}^{121} - 49152 T_{5}^{119} + \cdots + 18\!\cdots\!96 \) acting on \(S_{2}^{\mathrm{new}}(576, [\chi])\). Copy content Toggle raw display