Properties

Label 225.3.r.a
Level $225$
Weight $3$
Character orbit 225.r
Analytic conductor $6.131$
Analytic rank $0$
Dimension $32$
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: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 25)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 10 q^{2} - 10 q^{4} + 10 q^{5} - 10 q^{7} + 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 10 q^{2} - 10 q^{4} + 10 q^{5} - 10 q^{7} + 10 q^{8} - 10 q^{10} + 6 q^{11} - 10 q^{13} + 10 q^{14} + 2 q^{16} - 60 q^{17} + 90 q^{19} - 130 q^{20} + 70 q^{22} - 10 q^{23} - 40 q^{25} - 4 q^{26} - 250 q^{28} + 110 q^{29} - 6 q^{31} + 290 q^{32} - 260 q^{34} + 120 q^{35} + 50 q^{37} - 320 q^{38} + 440 q^{40} + 86 q^{41} + 230 q^{43} - 340 q^{44} - 6 q^{46} - 70 q^{47} + 100 q^{50} - 320 q^{52} + 190 q^{53} - 250 q^{55} + 70 q^{56} - 640 q^{58} + 260 q^{59} + 114 q^{61} - 60 q^{62} + 340 q^{64} - 360 q^{65} + 270 q^{67} - 710 q^{68} + 310 q^{70} + 66 q^{71} + 30 q^{73} - 80 q^{76} + 250 q^{77} - 210 q^{79} + 850 q^{80} + 30 q^{82} + 600 q^{85} + 6 q^{86} + 190 q^{88} + 10 q^{89} - 6 q^{91} + 30 q^{92} + 790 q^{94} - 310 q^{95} + 270 q^{97} - 170 q^{98}+O(q^{100}) \) 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 −1.29583 2.54321i 0 −2.43759 + 3.35505i 4.45624 + 2.26758i 0 −3.40272 + 3.40272i 0.414625 + 0.0656701i 0 −0.00760491 14.2715i
28.2 0.259330 + 0.508965i 0 2.15935 2.97209i 3.73307 + 3.32629i 0 1.66138 1.66138i 4.32944 + 0.685716i 0 −0.724866 + 2.76261i
28.3 0.395527 + 0.776265i 0 1.90500 2.62200i −1.22928 4.84653i 0 −5.60844 + 5.60844i 6.23083 + 0.986866i 0 3.27598 2.87118i
28.4 1.69523 + 3.32707i 0 −5.84445 + 8.04419i −2.26962 + 4.45520i 0 1.68463 1.68463i −21.9189 3.47161i 0 −18.6703 + 0.00137996i
37.1 −0.463000 2.92327i 0 −4.52691 + 1.47088i −0.953911 + 4.90816i 0 4.44588 + 4.44588i 1.02103 + 2.00389i 0 14.7895 + 0.516059i
37.2 −0.0933465 0.589367i 0 3.46559 1.12604i 3.31432 3.74370i 0 −7.64532 7.64532i −2.07076 4.06409i 0 −2.51579 1.60389i
37.3 0.312579 + 1.97355i 0 0.00704800 0.00229003i −4.93389 0.810386i 0 3.91191 + 3.91191i 3.63528 + 7.13464i 0 0.0571038 9.99057i
37.4 0.513943 + 3.24491i 0 −6.46108 + 2.09933i 4.99960 + 0.0628765i 0 7.51823 + 7.51823i −4.16668 8.17758i 0 2.36548 + 16.2556i
73.1 −0.463000 + 2.92327i 0 −4.52691 1.47088i −0.953911 4.90816i 0 4.44588 4.44588i 1.02103 2.00389i 0 14.7895 0.516059i
73.2 −0.0933465 + 0.589367i 0 3.46559 + 1.12604i 3.31432 + 3.74370i 0 −7.64532 + 7.64532i −2.07076 + 4.06409i 0 −2.51579 + 1.60389i
73.3 0.312579 1.97355i 0 0.00704800 + 0.00229003i −4.93389 + 0.810386i 0 3.91191 3.91191i 3.63528 7.13464i 0 0.0571038 + 9.99057i
73.4 0.513943 3.24491i 0 −6.46108 2.09933i 4.99960 0.0628765i 0 7.51823 7.51823i −4.16668 + 8.17758i 0 2.36548 16.2556i
127.1 −1.80600 0.286042i 0 −0.624420 0.202886i 3.20727 3.83580i 0 3.62927 + 3.62927i 7.58652 + 3.86553i 0 −6.88953 + 6.01004i
127.2 −0.287585 0.0455490i 0 −3.72360 1.20987i −2.36408 + 4.40581i 0 −2.38950 2.38950i 2.05348 + 1.04630i 0 0.880552 1.15936i
127.3 1.86717 + 0.295731i 0 −0.405347 0.131705i −4.99561 0.209511i 0 −3.57009 3.57009i −7.45551 3.79877i 0 −9.26571 1.86855i
127.4 3.57427 + 0.566108i 0 8.65068 + 2.81078i 0.872190 4.92334i 0 −0.574149 0.574149i 16.4311 + 8.37205i 0 5.90458 17.1036i
163.1 −1.80600 + 0.286042i 0 −0.624420 + 0.202886i 3.20727 + 3.83580i 0 3.62927 3.62927i 7.58652 3.86553i 0 −6.88953 6.01004i
163.2 −0.287585 + 0.0455490i 0 −3.72360 + 1.20987i −2.36408 4.40581i 0 −2.38950 + 2.38950i 2.05348 1.04630i 0 0.880552 + 1.15936i
163.3 1.86717 0.295731i 0 −0.405347 + 0.131705i −4.99561 + 0.209511i 0 −3.57009 + 3.57009i −7.45551 + 3.79877i 0 −9.26571 + 1.86855i
163.4 3.57427 0.566108i 0 8.65068 2.81078i 0.872190 + 4.92334i 0 −0.574149 + 0.574149i 16.4311 8.37205i 0 5.90458 + 17.1036i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.4
Significant digits:
Format:

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.a 32
3.b odd 2 1 25.3.f.a 32
12.b even 2 1 400.3.bg.c 32
15.d odd 2 1 125.3.f.c 32
15.e even 4 1 125.3.f.a 32
15.e even 4 1 125.3.f.b 32
25.f odd 20 1 inner 225.3.r.a 32
75.h odd 10 1 125.3.f.b 32
75.j odd 10 1 125.3.f.a 32
75.l even 20 1 25.3.f.a 32
75.l even 20 1 125.3.f.c 32
300.u odd 20 1 400.3.bg.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.3.f.a 32 3.b odd 2 1
25.3.f.a 32 75.l even 20 1
125.3.f.a 32 15.e even 4 1
125.3.f.a 32 75.j odd 10 1
125.3.f.b 32 15.e even 4 1
125.3.f.b 32 75.h odd 10 1
125.3.f.c 32 15.d odd 2 1
125.3.f.c 32 75.l even 20 1
225.3.r.a 32 1.a even 1 1 trivial
225.3.r.a 32 25.f odd 20 1 inner
400.3.bg.c 32 12.b even 2 1
400.3.bg.c 32 300.u odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 10 T_{2}^{31} + 55 T_{2}^{30} - 220 T_{2}^{29} + 612 T_{2}^{28} - 1380 T_{2}^{27} + 2695 T_{2}^{26} - 5200 T_{2}^{25} + 24863 T_{2}^{24} - 106160 T_{2}^{23} + 401985 T_{2}^{22} - 1356130 T_{2}^{21} + \cdots + 12952801 \) acting on \(S_{3}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display