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Properties

Label 575.2.r.a
Level $575$
Weight $2$
Character orbit 575.r
Analytic conductor $4.591$
Analytic rank $0$
Dimension $160$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [575,2,Mod(7,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([11, 38]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.r (of order \(44\), degree \(20\), 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: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(8\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q + 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q + 12 q^{6} + 152 q^{16} + 132 q^{21} - 20 q^{26} + 56 q^{31} - 184 q^{36} - 116 q^{41} - 352 q^{46} - 176 q^{51} - 352 q^{56} - 88 q^{61} - 176 q^{66} + 52 q^{71} - 176 q^{76} + 56 q^{81} - 44 q^{86} - 188 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.67924 + 0.191623i −0.633300 + 2.91124i 5.16194 − 0.742175i 0 1.13890 − 7.92124i 1.79941 − 3.29537i −8.43846 + 1.83567i −5.34533 − 2.44113i 0
7.2 −2.41425 + 0.172670i 0.356212 − 1.63748i 3.81913 − 0.549108i 0 −0.577240 + 4.01479i −1.51634 + 2.77698i −4.39530 + 0.956139i 0.174440 + 0.0796640i 0
7.3 −1.28134 + 0.0916429i −0.300302 + 1.38046i −0.346220 + 0.0497789i 0 0.258277 − 1.79636i −1.01035 + 1.85032i 2.94957 − 0.641639i 0.913396 + 0.417134i 0
7.4 −0.161364 + 0.0115410i −0.303226 + 1.39391i −1.95374 + 0.280905i 0 0.0328427 − 0.228426i 1.11522 − 2.04237i 0.628179 − 0.136652i 0.877870 + 0.400910i 0
7.5 0.161364 − 0.0115410i 0.303226 − 1.39391i −1.95374 + 0.280905i 0 0.0328427 − 0.228426i −1.11522 + 2.04237i −0.628179 + 0.136652i 0.877870 + 0.400910i 0
7.6 1.28134 − 0.0916429i 0.300302 − 1.38046i −0.346220 + 0.0497789i 0 0.258277 − 1.79636i 1.01035 − 1.85032i −2.94957 + 0.641639i 0.913396 + 0.417134i 0
7.7 2.41425 − 0.172670i −0.356212 + 1.63748i 3.81913 − 0.549108i 0 −0.577240 + 4.01479i 1.51634 − 2.77698i 4.39530 − 0.956139i 0.174440 + 0.0796640i 0
7.8 2.67924 − 0.191623i 0.633300 − 2.91124i 5.16194 − 0.742175i 0 1.13890 − 7.92124i −1.79941 + 3.29537i 8.43846 − 1.83567i −5.34533 − 2.44113i 0
43.1 −1.90141 − 1.42338i −1.77340 + 0.661444i 1.02589 + 3.49386i 0 4.31344 + 1.26654i −2.37650 − 0.516977i 1.36238 − 3.65267i 0.440187 − 0.381424i 0
43.2 −1.33671 − 1.00065i 0.542873 − 0.202481i 0.222034 + 0.756177i 0 −0.928279 − 0.272567i 0.0328047 + 0.00713623i −0.707169 + 1.89599i −2.01354 + 1.74474i 0
43.3 −0.608261 − 0.455338i −0.843617 + 0.314653i −0.400817 − 1.36506i 0 0.656413 + 0.192740i 4.08355 + 0.888322i −0.908816 + 2.43663i −1.65457 + 1.43369i 0
43.4 −0.0253234 − 0.0189569i 2.86531 − 1.06871i −0.563183 − 1.91803i 0 −0.0928188 − 0.0272540i 1.47543 + 0.320959i −0.0442071 + 0.118524i 4.80064 − 4.15978i 0
43.5 0.0253234 + 0.0189569i −2.86531 + 1.06871i −0.563183 − 1.91803i 0 −0.0928188 − 0.0272540i −1.47543 − 0.320959i 0.0442071 − 0.118524i 4.80064 − 4.15978i 0
43.6 0.608261 + 0.455338i 0.843617 − 0.314653i −0.400817 − 1.36506i 0 0.656413 + 0.192740i −4.08355 − 0.888322i 0.908816 − 2.43663i −1.65457 + 1.43369i 0
43.7 1.33671 + 1.00065i −0.542873 + 0.202481i 0.222034 + 0.756177i 0 −0.928279 − 0.272567i −0.0328047 − 0.00713623i 0.707169 − 1.89599i −2.01354 + 1.74474i 0
43.8 1.90141 + 1.42338i 1.77340 − 0.661444i 1.02589 + 3.49386i 0 4.31344 + 1.26654i 2.37650 + 0.516977i −1.36238 + 3.65267i 0.440187 − 0.381424i 0
57.1 −2.43680 − 0.530093i −0.568081 − 0.758867i 3.83772 + 1.75263i 0 0.982028 + 2.15034i 0.515610 + 0.0368771i −4.42992 − 3.31620i 0.592034 − 2.01628i 0
57.2 −1.94464 − 0.423030i 1.58231 + 2.11372i 1.78339 + 0.814448i 0 −2.18285 − 4.77978i 1.58441 + 0.113319i 0.0628268 + 0.0470315i −1.11890 + 3.81063i 0
57.3 −0.970468 − 0.211112i −0.274663 − 0.366907i −0.922025 − 0.421075i 0 0.189093 + 0.414056i −2.02452 − 0.144797i 2.39604 + 1.79365i 0.786017 − 2.67693i 0
57.4 −0.463313 − 0.100788i −1.77275 − 2.36812i −1.61476 − 0.737438i 0 0.582663 + 1.27585i 4.16008 + 0.297535i 1.43297 + 1.07271i −1.62015 + 5.51772i 0
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
23.d odd 22 1 inner
115.i odd 22 1 inner
115.l even 44 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.2.r.a ✓ 160
5.b even 2 1 inner 575.2.r.a ✓ 160
5.c odd 4 2 inner 575.2.r.a ✓ 160
23.d odd 22 1 inner 575.2.r.a ✓ 160
115.i odd 22 1 inner 575.2.r.a ✓ 160
115.l even 44 2 inner 575.2.r.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
575.2.r.a ✓ 160 1.a even 1 1 trivial
575.2.r.a ✓ 160 5.b even 2 1 inner
575.2.r.a ✓ 160 5.c odd 4 2 inner
575.2.r.a ✓ 160 23.d odd 22 1 inner
575.2.r.a ✓ 160 115.i odd 22 1 inner
575.2.r.a ✓ 160 115.l even 44 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{160} - 160 T_{2}^{156} + 11634 T_{2}^{152} - 542543 T_{2}^{148} + 19425699 T_{2}^{144} + \cdots + 14641 \) acting on \(S_{2}^{\mathrm{new}}(575, [\chi])\). Copy content Toggle raw display