Properties

Label 225.3.r.b
Level $225$
Weight $3$
Character orbit 225.r
Analytic conductor $6.131$
Analytic rank $0$
Dimension $80$
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] = [225,3,Mod(28,225)]
 
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([0, 7]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.28");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 225.r (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: \(6.13080594811\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(10\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 75)
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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 4 q^{2} - 4 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 4 q^{2} - 4 q^{5} - 4 q^{7} + 12 q^{8} - 4 q^{10} + 32 q^{13} + 80 q^{16} + 100 q^{17} - 100 q^{19} + 244 q^{20} - 100 q^{22} + 96 q^{23} - 16 q^{25} + 40 q^{26} + 196 q^{28} - 200 q^{29} - 636 q^{32} + 100 q^{34} - 260 q^{35} - 184 q^{37} + 564 q^{38} - 948 q^{40} - 160 q^{41} - 472 q^{43} + 700 q^{44} + 288 q^{47} - 16 q^{50} + 620 q^{52} - 304 q^{53} + 604 q^{55} + 1272 q^{58} - 800 q^{59} - 240 q^{61} - 1212 q^{62} + 100 q^{64} - 272 q^{65} - 80 q^{67} - 104 q^{68} - 260 q^{70} - 116 q^{73} + 88 q^{77} + 200 q^{79} + 164 q^{80} - 168 q^{82} + 1264 q^{83} - 212 q^{85} - 212 q^{88} + 1500 q^{89} + 1504 q^{92} - 200 q^{94} + 784 q^{95} - 260 q^{97} + 92 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1 −1.73350 3.40219i 0 −6.21871 + 8.55932i −2.10687 + 4.53443i 0 7.30480 7.30480i 24.8151 + 3.93033i 0 19.0793 0.692468i
28.2 −1.51991 2.98300i 0 −4.23700 + 5.83172i −3.83005 3.21415i 0 −3.58606 + 3.58606i 10.6092 + 1.68033i 0 −3.76645 + 16.3102i
28.3 −1.03753 2.03626i 0 −0.718759 + 0.989286i 3.06942 3.94698i 0 0.410561 0.410561i −6.26868 0.992861i 0 −11.2217 2.15504i
28.4 −0.670302 1.31554i 0 1.06979 1.47245i −0.953987 + 4.90815i 0 0.393993 0.393993i −8.48731 1.34426i 0 7.09633 2.03493i
28.5 −0.421583 0.827404i 0 1.84428 2.53843i −4.83080 + 1.28971i 0 −4.18270 + 4.18270i −6.54656 1.03687i 0 3.10370 + 3.45331i
28.6 −0.359880 0.706305i 0 1.98179 2.72770i 0.990973 4.90081i 0 8.25978 8.25978i −5.77157 0.914128i 0 −3.81810 + 1.06378i
28.7 0.775767 + 1.52253i 0 0.634862 0.873813i −4.50160 + 2.17615i 0 −3.94736 + 3.94736i 8.57385 + 1.35797i 0 −6.80544 5.16562i
28.8 1.07322 + 2.10631i 0 −0.933591 + 1.28498i −4.10427 2.85569i 0 9.10511 9.10511i 5.63093 + 0.891852i 0 1.61020 11.7096i
28.9 1.13229 + 2.22224i 0 −1.30514 + 1.79637i 4.95217 + 0.689916i 0 3.10232 3.10232i 4.38372 + 0.694313i 0 4.07413 + 11.7861i
28.10 1.47736 + 2.89947i 0 −3.87323 + 5.33104i 3.93419 3.08579i 0 −8.41509 + 8.41509i −8.32297 1.31823i 0 14.7594 + 6.84828i
37.1 −0.607139 3.83332i 0 −10.5215 + 3.41865i −2.58662 4.27895i 0 −2.30555 2.30555i 12.4449 + 24.4245i 0 −14.8321 + 12.5133i
37.2 −0.495586 3.12901i 0 −5.74087 + 1.86532i 4.93946 0.775740i 0 1.18776 + 1.18776i 2.92871 + 5.74792i 0 −4.87523 15.0712i
37.3 −0.373594 2.35878i 0 −1.62005 + 0.526386i −4.74635 1.57230i 0 −5.77573 5.77573i −2.48998 4.88686i 0 −1.93550 + 11.7830i
37.4 −0.229020 1.44598i 0 1.76583 0.573753i 2.10186 4.53676i 0 8.32198 + 8.32198i −3.89261 7.63968i 0 −7.04141 2.00024i
37.5 −0.167706 1.05885i 0 2.71118 0.880917i −4.35927 + 2.44883i 0 −2.58808 2.58808i −3.33424 6.54382i 0 3.32403 + 4.20513i
37.6 −0.0337531 0.213109i 0 3.75995 1.22168i 2.09142 + 4.54158i 0 2.53517 + 2.53517i −0.779083 1.52904i 0 0.897260 0.598992i
37.7 0.202100 + 1.27601i 0 2.21687 0.720305i −1.23939 4.84396i 0 1.27981 + 1.27981i 3.71321 + 7.28759i 0 5.93045 2.56044i
37.8 0.224472 + 1.41726i 0 1.84598 0.599796i 3.76307 + 3.29231i 0 −1.57196 1.57196i 3.87022 + 7.59573i 0 −3.82136 + 6.07229i
37.9 0.479602 + 3.02809i 0 −5.13507 + 1.66849i −3.15408 + 3.87966i 0 −5.93757 5.93757i −1.94768 3.82253i 0 −13.2607 7.69015i
37.10 0.558161 + 3.52409i 0 −8.30342 + 2.69795i 0.337675 4.98858i 0 −8.56695 8.56695i −7.66306 15.0396i 0 17.7687 1.59443i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.f odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.3.r.b 80
3.b odd 2 1 75.3.k.a 80
15.d odd 2 1 375.3.k.a 80
15.e even 4 1 375.3.k.b 80
15.e even 4 1 375.3.k.c 80
25.f odd 20 1 inner 225.3.r.b 80
75.h odd 10 1 375.3.k.b 80
75.j odd 10 1 375.3.k.c 80
75.l even 20 1 75.3.k.a 80
75.l even 20 1 375.3.k.a 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.k.a 80 3.b odd 2 1
75.3.k.a 80 75.l even 20 1
225.3.r.b 80 1.a even 1 1 trivial
225.3.r.b 80 25.f odd 20 1 inner
375.3.k.a 80 15.d odd 2 1
375.3.k.a 80 75.l even 20 1
375.3.k.b 80 15.e even 4 1
375.3.k.b 80 75.h odd 10 1
375.3.k.c 80 15.e even 4 1
375.3.k.c 80 75.j odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{80} + 4 T_{2}^{79} + 8 T_{2}^{78} - 4 T_{2}^{77} - 348 T_{2}^{76} - 580 T_{2}^{75} + 1122 T_{2}^{74} + 13404 T_{2}^{73} + 98023 T_{2}^{72} + 86076 T_{2}^{71} - 324840 T_{2}^{70} - 2770704 T_{2}^{69} + \cdots + 87\!\cdots\!41 \) acting on \(S_{3}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display