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Properties

Label 275.2.bo.b
Level $275$
Weight $2$
Character orbit 275.bo
Analytic conductor $2.196$
Analytic rank $0$
Dimension $208$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [275,2,Mod(87,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([9, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.87");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.bo (of order \(20\), 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: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(208\)
Relative dimension: \(26\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 208 q - 16 q^{3} - 20 q^{4} - 12 q^{5} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 208 q - 16 q^{3} - 20 q^{4} - 12 q^{5} - 20 q^{9} - 6 q^{11} - 32 q^{12} - 20 q^{14} - 24 q^{15} + 16 q^{16} - 32 q^{20} - 30 q^{22} - 16 q^{23} - 12 q^{25} - 32 q^{26} - 4 q^{27} - 12 q^{31} - 26 q^{33} - 20 q^{34} + 16 q^{36} - 32 q^{37} - 60 q^{38} - 20 q^{42} + 60 q^{44} - 176 q^{45} + 48 q^{47} - 56 q^{48} - 16 q^{53} + 38 q^{55} - 44 q^{56} + 160 q^{58} + 120 q^{59} + 116 q^{60} + 140 q^{64} - 30 q^{66} + 48 q^{67} - 20 q^{69} - 100 q^{70} - 4 q^{71} + 136 q^{75} - 90 q^{77} + 80 q^{78} - 12 q^{80} + 24 q^{81} - 60 q^{82} - 12 q^{86} - 70 q^{88} + 80 q^{89} - 12 q^{91} - 152 q^{92} + 28 q^{93} - 272 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
87.1 −2.66731 + 0.422461i −1.90947 − 0.972925i 5.03398 − 1.63564i 0.757713 − 2.10378i 5.50419 + 1.78842i 2.75509 − 2.75509i −7.92377 + 4.03736i 0.936149 + 1.28850i −1.13230 + 5.93153i
87.2 −2.49431 + 0.395060i 1.18089 + 0.601695i 4.16341 − 1.35277i −2.23183 − 0.137596i −3.18322 − 1.03429i 1.25151 − 1.25151i −5.35011 + 2.72602i −0.730883 − 1.00597i 5.62124 − 0.538501i
87.3 −2.24480 + 0.355541i −2.12156 − 1.08099i 3.01061 − 0.978206i −2.22356 − 0.236221i 5.14682 + 1.67230i −3.42923 + 3.42923i −2.36029 + 1.20263i 1.56913 + 2.15972i 5.07543 − 0.260298i
87.4 −2.24013 + 0.354802i −1.00837 − 0.513791i 2.99018 − 0.971569i 1.43354 + 1.71609i 2.44118 + 0.793186i −1.56073 + 1.56073i −2.31198 + 1.17801i −1.01052 − 1.39087i −3.82019 − 3.33563i
87.5 −1.99842 + 0.316518i 0.919759 + 0.468641i 1.99138 − 0.647039i 2.16802 − 0.547455i −1.98640 − 0.645420i 1.57190 − 1.57190i −0.169216 + 0.0862198i −1.13702 − 1.56498i −4.15933 + 1.78026i
87.6 −1.79243 + 0.283893i 2.23286 + 1.13770i 1.23010 − 0.399683i −1.17514 − 1.90238i −4.32523 − 1.40535i −0.695652 + 0.695652i 1.14255 − 0.582160i 1.92795 + 2.65360i 2.64642 + 3.07627i
87.7 −1.55950 + 0.247001i −2.68569 − 1.36843i 0.468923 − 0.152362i 0.334832 + 2.21086i 4.52634 + 1.47070i 3.02262 − 3.02262i 2.12004 − 1.08021i 3.57698 + 4.92329i −1.06825 − 3.36513i
87.8 −1.36361 + 0.215974i 2.11115 + 1.07569i −0.0893312 + 0.0290255i −0.907325 + 2.04371i −3.11111 − 1.01086i −1.63400 + 1.63400i 2.57580 − 1.31244i 1.53651 + 2.11483i 0.795846 − 2.98278i
87.9 −1.35729 + 0.214973i −0.122596 − 0.0624658i −0.106102 + 0.0344748i 0.977056 − 2.01131i 0.179826 + 0.0584291i −1.83281 + 1.83281i 2.58545 − 1.31735i −1.75223 − 2.41173i −0.893767 + 2.93996i
87.10 −0.844088 + 0.133690i −0.722586 − 0.368176i −1.20750 + 0.392341i −1.57488 + 1.58737i 0.659148 + 0.214170i −0.0311351 + 0.0311351i 2.48971 − 1.26857i −1.37678 − 1.89497i 1.11712 − 1.55043i
87.11 −0.720267 + 0.114079i 1.47974 + 0.753965i −1.39634 + 0.453699i 1.68545 + 1.46944i −1.15182 − 0.374249i 2.54899 − 2.54899i 2.25351 − 1.14822i −0.142191 − 0.195710i −1.38160 − 0.866118i
87.12 −0.380582 + 0.0602782i −1.67990 − 0.855951i −1.76090 + 0.572152i 2.22863 − 0.182268i 0.690934 + 0.224498i −1.31308 + 1.31308i 1.32234 − 0.673763i 0.326049 + 0.448768i −0.837188 + 0.203706i
87.13 −0.177015 + 0.0280364i 0.737985 + 0.376022i −1.87156 + 0.608108i −1.66349 − 1.49426i −0.141177 − 0.0458711i 3.40396 − 3.40396i 0.633621 − 0.322846i −1.36013 − 1.87205i 0.336356 + 0.217867i
87.14 0.177015 − 0.0280364i 0.737985 + 0.376022i −1.87156 + 0.608108i −1.66349 − 1.49426i 0.141177 + 0.0458711i −3.40396 + 3.40396i −0.633621 + 0.322846i −1.36013 − 1.87205i −0.336356 − 0.217867i
87.15 0.380582 − 0.0602782i −1.67990 − 0.855951i −1.76090 + 0.572152i 2.22863 − 0.182268i −0.690934 − 0.224498i 1.31308 − 1.31308i −1.32234 + 0.673763i 0.326049 + 0.448768i 0.837188 − 0.203706i
87.16 0.720267 − 0.114079i 1.47974 + 0.753965i −1.39634 + 0.453699i 1.68545 + 1.46944i 1.15182 + 0.374249i −2.54899 + 2.54899i −2.25351 + 1.14822i −0.142191 − 0.195710i 1.38160 + 0.866118i
87.17 0.844088 − 0.133690i −0.722586 − 0.368176i −1.20750 + 0.392341i −1.57488 + 1.58737i −0.659148 − 0.214170i 0.0311351 − 0.0311351i −2.48971 + 1.26857i −1.37678 − 1.89497i −1.11712 + 1.55043i
87.18 1.35729 − 0.214973i −0.122596 − 0.0624658i −0.106102 + 0.0344748i 0.977056 − 2.01131i −0.179826 − 0.0584291i 1.83281 − 1.83281i −2.58545 + 1.31735i −1.75223 − 2.41173i 0.893767 − 2.93996i
87.19 1.36361 − 0.215974i 2.11115 + 1.07569i −0.0893312 + 0.0290255i −0.907325 + 2.04371i 3.11111 + 1.01086i 1.63400 − 1.63400i −2.57580 + 1.31244i 1.53651 + 2.11483i −0.795846 + 2.98278i
87.20 1.55950 − 0.247001i −2.68569 − 1.36843i 0.468923 − 0.152362i 0.334832 + 2.21086i −4.52634 − 1.47070i −3.02262 + 3.02262i −2.12004 + 1.08021i 3.57698 + 4.92329i 1.06825 + 3.36513i
See next 80 embeddings (of 208 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 87.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
25.f odd 20 1 inner
275.bo even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.2.bo.b ✓ 208
11.b odd 2 1 inner 275.2.bo.b ✓ 208
25.f odd 20 1 inner 275.2.bo.b ✓ 208
275.bo even 20 1 inner 275.2.bo.b ✓ 208
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.bo.b ✓ 208 1.a even 1 1 trivial
275.2.bo.b ✓ 208 11.b odd 2 1 inner
275.2.bo.b ✓ 208 25.f odd 20 1 inner
275.2.bo.b ✓ 208 275.bo even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{208} + 10 T_{2}^{206} - 148 T_{2}^{204} - 1970 T_{2}^{202} + 12148 T_{2}^{200} + \cdots + 41\!\cdots\!25 \) acting on \(S_{2}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display