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Properties

Label 225.3.r.c
Level $225$
Weight $3$
Character orbit 225.r
Analytic conductor $6.131$
Analytic rank $0$
Dimension $80$
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] = [225,3,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: 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 20 q^{7} + 20 q^{10} - 40 q^{13} + 80 q^{16} - 100 q^{19} + 20 q^{22} + 80 q^{25} + 220 q^{28} + 100 q^{34} + 80 q^{37} + 780 q^{40} + 440 q^{43} - 580 q^{52} - 380 q^{55} - 960 q^{58} + 240 q^{61} - 1700 q^{64} - 560 q^{67} - 620 q^{70} - 260 q^{73} + 200 q^{79} - 240 q^{82} - 20 q^{85} + 100 q^{88} - 200 q^{94} - 260 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1 −1.61668 − 3.17291i 0 −5.10255 + 7.02305i 4.76008 + 1.53026i 0 −4.69233 + 4.69233i 16.4639 + 2.60762i 0 −2.84014 − 17.5772i
28.2 −1.35857 − 2.66634i 0 −2.91254 + 4.00876i −2.12483 − 4.52605i 0 5.95401 − 5.95401i 2.82296 + 0.447113i 0 −9.18126 + 11.8145i
28.3 −1.10097 − 2.16077i 0 −1.10565 + 1.52180i −2.22205 + 4.47912i 0 1.98156 − 1.98156i −5.07538 − 0.803861i 0 12.1247 − 0.130011i
28.4 −0.690887 − 1.35594i 0 0.989887 − 1.36246i −1.24802 − 4.84174i 0 −9.71480 + 9.71480i −8.54361 − 1.35318i 0 −5.70288 + 5.03734i
28.5 −0.185513 − 0.364089i 0 2.25300 − 3.10098i 4.86850 − 1.13917i 0 3.13380 − 3.13380i −3.16138 − 0.500713i 0 −1.31793 − 1.56124i
28.6 0.185513 + 0.364089i 0 2.25300 − 3.10098i −4.86850 + 1.13917i 0 3.13380 − 3.13380i 3.16138 + 0.500713i 0 −1.31793 − 1.56124i
28.7 0.690887 + 1.35594i 0 0.989887 − 1.36246i 1.24802 + 4.84174i 0 −9.71480 + 9.71480i 8.54361 + 1.35318i 0 −5.70288 + 5.03734i
28.8 1.10097 + 2.16077i 0 −1.10565 + 1.52180i 2.22205 − 4.47912i 0 1.98156 − 1.98156i 5.07538 + 0.803861i 0 12.1247 − 0.130011i
28.9 1.35857 + 2.66634i 0 −2.91254 + 4.00876i 2.12483 + 4.52605i 0 5.95401 − 5.95401i −2.82296 − 0.447113i 0 −9.18126 + 11.8145i
28.10 1.61668 + 3.17291i 0 −5.10255 + 7.02305i −4.76008 − 1.53026i 0 −4.69233 + 4.69233i −16.4639 − 2.60762i 0 −2.84014 − 17.5772i
37.1 −0.603150 − 3.80814i 0 −10.3339 + 3.35769i 4.70869 + 1.68172i 0 4.75023 + 4.75023i 12.0178 + 23.5863i 0 3.56419 − 18.9457i
37.2 −0.404496 − 2.55389i 0 −2.55450 + 0.830009i −2.53739 + 4.30832i 0 −2.01350 − 2.01350i −1.54254 − 3.02740i 0 12.0293 + 4.73751i
37.3 −0.400844 − 2.53083i 0 −2.44021 + 0.792872i −0.311348 − 4.99030i 0 0.934567 + 0.934567i −1.66841 − 3.27444i 0 −12.5048 + 2.78830i
37.4 −0.204575 − 1.29164i 0 2.17775 − 0.707594i 4.24361 + 2.64420i 0 −7.63817 − 7.63817i −3.73427 − 7.32892i 0 2.54721 − 6.02214i
37.5 −0.0649440 − 0.410040i 0 3.64031 − 1.18281i −4.80769 + 1.37334i 0 5.63713 + 5.63713i −1.47532 − 2.89547i 0 0.875357 + 1.88216i
37.6 0.0649440 + 0.410040i 0 3.64031 − 1.18281i 4.80769 − 1.37334i 0 5.63713 + 5.63713i 1.47532 + 2.89547i 0 0.875357 + 1.88216i
37.7 0.204575 + 1.29164i 0 2.17775 − 0.707594i −4.24361 − 2.64420i 0 −7.63817 − 7.63817i 3.73427 + 7.32892i 0 2.54721 − 6.02214i
37.8 0.400844 + 2.53083i 0 −2.44021 + 0.792872i 0.311348 + 4.99030i 0 0.934567 + 0.934567i 1.66841 + 3.27444i 0 −12.5048 + 2.78830i
37.9 0.404496 + 2.55389i 0 −2.55450 + 0.830009i 2.53739 − 4.30832i 0 −2.01350 − 2.01350i 1.54254 + 3.02740i 0 12.0293 + 4.73751i
37.10 0.603150 + 3.80814i 0 −10.3339 + 3.35769i −4.70869 − 1.68172i 0 4.75023 + 4.75023i −12.0178 − 23.5863i 0 3.56419 − 18.9457i
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
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 225.3.r.c ✓ 80
3.b odd 2 1 inner 225.3.r.c ✓ 80
25.f odd 20 1 inner 225.3.r.c ✓ 80
75.l even 20 1 inner 225.3.r.c ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.3.r.c ✓ 80 1.a even 1 1 trivial
225.3.r.c ✓ 80 3.b odd 2 1 inner
225.3.r.c ✓ 80 25.f odd 20 1 inner
225.3.r.c ✓ 80 75.l even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{80} - 300 T_{2}^{76} + 950 T_{2}^{74} + 66615 T_{2}^{72} - 285000 T_{2}^{70} + \cdots + 63\!\cdots\!25 \) acting on \(S_{3}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display