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Properties

Label 225.4.s.a
Level $225$
Weight $4$
Character orbit 225.s
Analytic conductor $13.275$
Analytic rank $0$
Dimension $240$
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] = [225,4,Mod(8,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.8");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.s (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: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(30\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 24 q^{7} - 192 q^{10} + 108 q^{13} + 960 q^{16} - 240 q^{19} + 384 q^{22} + 144 q^{25} + 2016 q^{28} - 1320 q^{34} - 828 q^{37} - 2568 q^{40} + 96 q^{43} + 312 q^{52} + 1512 q^{55} + 3864 q^{58} + 5760 q^{64} - 3072 q^{67} - 7104 q^{70} - 9732 q^{73} - 4320 q^{79} - 11592 q^{82} - 2028 q^{85} + 1392 q^{88} + 5520 q^{94} + 23268 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −2.43300 + 4.77503i 0 −12.1792 − 16.7632i −4.50785 + 10.2313i 0 −21.9129 + 21.9129i 67.3313 − 10.6642i 0 −37.8871 − 46.4179i
8.2 −2.42229 + 4.75402i 0 −12.0309 − 16.5591i −1.41658 − 11.0902i 0 −17.4624 + 17.4624i 65.7057 − 10.4068i 0 56.1545 + 20.1293i
8.3 −2.32119 + 4.55560i 0 −10.6633 − 14.6767i 7.72381 + 8.08349i 0 16.8632 − 16.8632i 51.2134 − 8.11141i 0 −54.7536 + 16.4232i
8.4 −2.01307 + 3.95087i 0 −6.85467 − 9.43464i −8.07936 − 7.72812i 0 8.38777 − 8.38777i 16.0374 − 2.54007i 0 46.7972 − 16.3633i
8.5 −1.86071 + 3.65185i 0 −5.17150 − 7.11796i 8.42328 − 7.35176i 0 −7.50259 + 7.50259i 3.23155 − 0.511828i 0 11.1742 + 44.4401i
8.6 −1.82726 + 3.58620i 0 −4.81969 − 6.63374i 3.29327 + 10.6843i 0 20.0620 − 20.0620i 0.794081 − 0.125770i 0 −44.3338 − 7.71267i
8.7 −1.77414 + 3.48194i 0 −4.27404 − 5.88272i −10.8585 + 2.66335i 0 0.0563580 − 0.0563580i −2.81208 + 0.445390i 0 9.99079 − 42.5337i
8.8 −1.43549 + 2.81731i 0 −1.17430 − 1.61629i 9.36115 − 6.11301i 0 6.24195 − 6.24195i −18.7448 + 2.96888i 0 3.78440 + 35.1484i
8.9 −1.19202 + 2.33947i 0 0.650084 + 0.894764i 6.05072 + 9.40153i 0 −18.2415 + 18.2415i −23.6147 + 3.74021i 0 −29.2071 + 2.94865i
8.10 −0.992231 + 1.94736i 0 1.89458 + 2.60767i 10.3396 + 4.25357i 0 −5.16515 + 5.16515i −24.2273 + 3.83723i 0 −18.5425 + 15.9144i
8.11 −0.815578 + 1.60066i 0 2.80533 + 3.86121i −10.4279 + 4.03222i 0 −19.7177 + 19.7177i −22.6632 + 3.58950i 0 2.05054 − 19.9801i
8.12 −0.583474 + 1.14513i 0 3.73140 + 5.13583i −11.1310 + 1.04882i 0 17.1720 − 17.1720i −18.2135 + 2.88473i 0 5.29364 − 13.3585i
8.13 −0.497700 + 0.976791i 0 3.99587 + 5.49984i −3.35781 − 10.6642i 0 −1.14043 + 1.14043i −16.0232 + 2.53782i 0 12.0879 + 2.02770i
8.14 −0.331761 + 0.651117i 0 4.38839 + 6.04011i 0.576312 − 11.1655i 0 −7.55632 + 7.55632i −11.1629 + 1.76802i 0 7.07883 + 4.07951i
8.15 −0.306279 + 0.601107i 0 4.43476 + 6.10392i −7.34843 + 8.42618i 0 21.5348 − 21.5348i −10.3580 + 1.64055i 0 −2.81437 − 6.99796i
8.16 0.306279 − 0.601107i 0 4.43476 + 6.10392i 7.34843 − 8.42618i 0 21.5348 − 21.5348i 10.3580 − 1.64055i 0 −2.81437 − 6.99796i
8.17 0.331761 − 0.651117i 0 4.38839 + 6.04011i −0.576312 + 11.1655i 0 −7.55632 + 7.55632i 11.1629 − 1.76802i 0 7.07883 + 4.07951i
8.18 0.497700 − 0.976791i 0 3.99587 + 5.49984i 3.35781 + 10.6642i 0 −1.14043 + 1.14043i 16.0232 − 2.53782i 0 12.0879 + 2.02770i
8.19 0.583474 − 1.14513i 0 3.73140 + 5.13583i 11.1310 − 1.04882i 0 17.1720 − 17.1720i 18.2135 − 2.88473i 0 5.29364 − 13.3585i
8.20 0.815578 − 1.60066i 0 2.80533 + 3.86121i 10.4279 − 4.03222i 0 −19.7177 + 19.7177i 22.6632 − 3.58950i 0 2.05054 − 19.9801i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.f odd 20 1 inner
75.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.s.a ✓ 240
3.b odd 2 1 inner 225.4.s.a ✓ 240
25.f odd 20 1 inner 225.4.s.a ✓ 240
75.l even 20 1 inner 225.4.s.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.4.s.a ✓ 240 1.a even 1 1 trivial
225.4.s.a ✓ 240 3.b odd 2 1 inner
225.4.s.a ✓ 240 25.f odd 20 1 inner
225.4.s.a ✓ 240 75.l even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(225, [\chi])\).