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Properties

Label 570.2.s.a
Level $570$
Weight $2$
Character orbit 570.s
Analytic conductor $4.551$
Analytic rank $0$
Dimension $24$
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] = [570,2,Mod(221,570)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(570, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("570.221");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.s (of order \(6\), degree \(2\), 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: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{2} + 4 q^{3} - 12 q^{4} - 2 q^{6} - 12 q^{7} + 24 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{2} + 4 q^{3} - 12 q^{4} - 2 q^{6} - 12 q^{7} + 24 q^{8} - 4 q^{9} - 2 q^{12} + 18 q^{13} + 6 q^{14} - 12 q^{16} + 12 q^{17} + 2 q^{18} + 6 q^{19} - 6 q^{21} + 18 q^{22} + 4 q^{24} + 12 q^{25} + 28 q^{27} + 6 q^{28} - 12 q^{32} - 22 q^{33} - 12 q^{34} + 2 q^{36} + 6 q^{38} + 40 q^{39} + 6 q^{41} - 6 q^{42} - 22 q^{43} - 18 q^{44} + 8 q^{45} + 12 q^{47} - 2 q^{48} + 12 q^{49} - 24 q^{50} - 20 q^{51} - 18 q^{52} + 8 q^{53} + 4 q^{54} - 12 q^{56} + 26 q^{59} + 22 q^{61} - 18 q^{62} + 6 q^{63} + 24 q^{64} + 8 q^{65} + 8 q^{66} - 48 q^{67} - 64 q^{69} + 24 q^{71} - 4 q^{72} - 8 q^{73} + 30 q^{74} + 2 q^{75} - 12 q^{76} - 38 q^{78} + 18 q^{79} - 12 q^{81} + 6 q^{82} + 12 q^{84} - 22 q^{86} - 24 q^{87} + 28 q^{89} + 8 q^{90} + 18 q^{91} + 2 q^{93} - 2 q^{96} + 6 q^{97} - 6 q^{98} + 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1 −0.500000 − 0.866025i −1.62233 + 0.606673i −0.500000 + 0.866025i 0.866025 − 0.500000i 1.33656 + 1.10164i −1.76552 1.00000 2.26390 − 1.96845i −0.866025 − 0.500000i
221.2 −0.500000 − 0.866025i −1.22385 + 1.22564i −0.500000 + 0.866025i −0.866025 + 0.500000i 1.67336 + 0.447064i 3.20940 1.00000 −0.00438801 − 3.00000i 0.866025 + 0.500000i
221.3 −0.500000 − 0.866025i −1.01463 − 1.40375i −0.500000 + 0.866025i −0.866025 + 0.500000i −0.708367 + 1.58057i 2.73284 1.00000 −0.941036 + 2.84859i 0.866025 + 0.500000i
221.4 −0.500000 − 0.866025i −0.929852 − 1.46129i −0.500000 + 0.866025i 0.866025 − 0.500000i −0.800591 + 1.53592i −4.66317 1.00000 −1.27075 + 2.71757i −0.866025 − 0.500000i
221.5 −0.500000 − 0.866025i −0.641070 + 1.60905i −0.500000 + 0.866025i −0.866025 + 0.500000i 1.71401 − 0.249340i −2.43208 1.00000 −2.17806 − 2.06302i 0.866025 + 0.500000i
221.6 −0.500000 − 0.866025i 0.0903420 − 1.72969i −0.500000 + 0.866025i 0.866025 − 0.500000i −1.54313 + 0.786608i 2.34168 1.00000 −2.98368 − 0.312528i −0.866025 − 0.500000i
221.7 −0.500000 − 0.866025i 0.362568 + 1.69368i −0.500000 + 0.866025i 0.866025 − 0.500000i 1.28548 − 1.16083i −0.535070 1.00000 −2.73709 + 1.22815i −0.866025 − 0.500000i
221.8 −0.500000 − 0.866025i 0.691758 − 1.58791i −0.500000 + 0.866025i −0.866025 + 0.500000i −1.72105 + 0.194877i −1.96058 1.00000 −2.04294 − 2.19691i 0.866025 + 0.500000i
221.9 −0.500000 − 0.866025i 1.37489 + 1.05342i −0.500000 + 0.866025i 0.866025 − 0.500000i 0.224845 − 1.71739i −1.74360 1.00000 0.780619 + 2.89666i −0.866025 − 0.500000i
221.10 −0.500000 − 0.866025i 1.56078 + 0.750973i −0.500000 + 0.866025i −0.866025 + 0.500000i −0.130029 − 1.72716i −4.16200 1.00000 1.87208 + 2.34421i 0.866025 + 0.500000i
221.11 −0.500000 − 0.866025i 1.62701 − 0.593994i −0.500000 + 0.866025i −0.866025 + 0.500000i −1.32792 − 1.11204i −0.387589 1.00000 2.29434 − 1.93287i 0.866025 + 0.500000i
221.12 −0.500000 − 0.866025i 1.72438 − 0.162784i −0.500000 + 0.866025i 0.866025 − 0.500000i −1.00317 − 1.41197i 3.36569 1.00000 2.94700 − 0.561404i −0.866025 − 0.500000i
521.1 −0.500000 + 0.866025i −1.62233 − 0.606673i −0.500000 − 0.866025i 0.866025 + 0.500000i 1.33656 − 1.10164i −1.76552 1.00000 2.26390 + 1.96845i −0.866025 + 0.500000i
521.2 −0.500000 + 0.866025i −1.22385 − 1.22564i −0.500000 − 0.866025i −0.866025 − 0.500000i 1.67336 − 0.447064i 3.20940 1.00000 −0.00438801 + 3.00000i 0.866025 − 0.500000i
521.3 −0.500000 + 0.866025i −1.01463 + 1.40375i −0.500000 − 0.866025i −0.866025 − 0.500000i −0.708367 − 1.58057i 2.73284 1.00000 −0.941036 − 2.84859i 0.866025 − 0.500000i
521.4 −0.500000 + 0.866025i −0.929852 + 1.46129i −0.500000 − 0.866025i 0.866025 + 0.500000i −0.800591 − 1.53592i −4.66317 1.00000 −1.27075 − 2.71757i −0.866025 + 0.500000i
521.5 −0.500000 + 0.866025i −0.641070 − 1.60905i −0.500000 − 0.866025i −0.866025 − 0.500000i 1.71401 + 0.249340i −2.43208 1.00000 −2.17806 + 2.06302i 0.866025 − 0.500000i
521.6 −0.500000 + 0.866025i 0.0903420 + 1.72969i −0.500000 − 0.866025i 0.866025 + 0.500000i −1.54313 − 0.786608i 2.34168 1.00000 −2.98368 + 0.312528i −0.866025 + 0.500000i
521.7 −0.500000 + 0.866025i 0.362568 − 1.69368i −0.500000 − 0.866025i 0.866025 + 0.500000i 1.28548 + 1.16083i −0.535070 1.00000 −2.73709 − 1.22815i −0.866025 + 0.500000i
521.8 −0.500000 + 0.866025i 0.691758 + 1.58791i −0.500000 − 0.866025i −0.866025 − 0.500000i −1.72105 − 0.194877i −1.96058 1.00000 −2.04294 + 2.19691i 0.866025 − 0.500000i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 221.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.s.a ✓ 24
3.b odd 2 1 570.2.s.b yes 24
19.d odd 6 1 570.2.s.b yes 24
57.f even 6 1 inner 570.2.s.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.s.a ✓ 24 1.a even 1 1 trivial
570.2.s.a ✓ 24 57.f even 6 1 inner
570.2.s.b yes 24 3.b odd 2 1
570.2.s.b yes 24 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{24} - 12 T_{17}^{23} - 40 T_{17}^{22} + 1056 T_{17}^{21} + 812 T_{17}^{20} + \cdots + 18547561529344 \) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\). Copy content Toggle raw display