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Properties

Label 405.3.l.o
Level $405$
Weight $3$
Character orbit 405.l
Analytic conductor $11.035$
Analytic rank $0$
Dimension $32$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [405,3,Mod(28,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([4, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.28");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.l (of order \(12\), degree \(4\), not 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: \(11.0354507066\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{7} - 64 q^{10} - 28 q^{13} + 20 q^{16} - 176 q^{22} - 64 q^{25} + 160 q^{28} + 208 q^{31} - 352 q^{37} + 252 q^{40} + 188 q^{43} + 376 q^{46} + 188 q^{52} - 272 q^{55} - 504 q^{58} - 296 q^{61}+ \cdots + 284 q^{97}+O(q^{100}) \) 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 −3.44665 − 0.923527i 0 7.56239 + 4.36615i 4.98625 − 0.370516i 0 5.99870 + 1.60735i −11.9402 − 11.9402i 0 −17.5280 − 3.32790i
28.2 −2.53204 − 0.678458i 0 2.48682 + 1.43577i −1.78324 + 4.67119i 0 −2.47213 − 0.662404i 2.09171 + 2.09171i 0 7.68445 − 10.6178i
28.3 −1.57943 − 0.423208i 0 −1.14859 − 0.663141i −3.80216 − 3.24709i 0 −10.5405 − 2.82431i 6.15839 + 6.15839i 0 4.63107 + 6.73767i
28.4 −0.821011 − 0.219989i 0 −2.83844 − 1.63877i 2.19088 − 4.49445i 0 9.74596 + 2.61142i 4.37397 + 4.37397i 0 −2.78747 + 3.20802i
28.5 0.821011 + 0.219989i 0 −2.83844 − 1.63877i −2.19088 + 4.49445i 0 9.74596 + 2.61142i −4.37397 − 4.37397i 0 −2.78747 + 3.20802i
28.6 1.57943 + 0.423208i 0 −1.14859 − 0.663141i 3.80216 + 3.24709i 0 −10.5405 − 2.82431i −6.15839 − 6.15839i 0 4.63107 + 6.73767i
28.7 2.53204 + 0.678458i 0 2.48682 + 1.43577i 1.78324 − 4.67119i 0 −2.47213 − 0.662404i −2.09171 − 2.09171i 0 7.68445 − 10.6178i
28.8 3.44665 + 0.923527i 0 7.56239 + 4.36615i −4.98625 + 0.370516i 0 5.99870 + 1.60735i 11.9402 + 11.9402i 0 −17.5280 − 3.32790i
217.1 −3.44665 + 0.923527i 0 7.56239 − 4.36615i 4.98625 + 0.370516i 0 5.99870 − 1.60735i −11.9402 + 11.9402i 0 −17.5280 + 3.32790i
217.2 −2.53204 + 0.678458i 0 2.48682 − 1.43577i −1.78324 − 4.67119i 0 −2.47213 + 0.662404i 2.09171 − 2.09171i 0 7.68445 + 10.6178i
217.3 −1.57943 + 0.423208i 0 −1.14859 + 0.663141i −3.80216 + 3.24709i 0 −10.5405 + 2.82431i 6.15839 − 6.15839i 0 4.63107 − 6.73767i
217.4 −0.821011 + 0.219989i 0 −2.83844 + 1.63877i 2.19088 + 4.49445i 0 9.74596 − 2.61142i 4.37397 − 4.37397i 0 −2.78747 − 3.20802i
217.5 0.821011 − 0.219989i 0 −2.83844 + 1.63877i −2.19088 − 4.49445i 0 9.74596 − 2.61142i −4.37397 + 4.37397i 0 −2.78747 − 3.20802i
217.6 1.57943 − 0.423208i 0 −1.14859 + 0.663141i 3.80216 − 3.24709i 0 −10.5405 + 2.82431i −6.15839 + 6.15839i 0 4.63107 − 6.73767i
217.7 2.53204 − 0.678458i 0 2.48682 − 1.43577i 1.78324 + 4.67119i 0 −2.47213 + 0.662404i −2.09171 + 2.09171i 0 7.68445 + 10.6178i
217.8 3.44665 − 0.923527i 0 7.56239 − 4.36615i −4.98625 − 0.370516i 0 5.99870 − 1.60735i 11.9402 − 11.9402i 0 −17.5280 + 3.32790i
298.1 −0.923527 − 3.44665i 0 −7.56239 + 4.36615i 2.17225 − 4.50348i 0 −1.60735 − 5.99870i 11.9402 + 11.9402i 0 −17.5280 − 3.32790i
298.2 −0.678458 − 2.53204i 0 −2.48682 + 1.43577i 3.15375 + 3.87993i 0 0.662404 + 2.47213i −2.09171 − 2.09171i 0 7.68445 − 10.6178i
298.3 −0.423208 − 1.57943i 0 1.14859 − 0.663141i −4.71314 + 1.66922i 0 2.82431 + 10.5405i −6.15839 − 6.15839i 0 4.63107 + 6.73767i
298.4 −0.219989 − 0.821011i 0 2.83844 − 1.63877i −2.79686 − 4.14458i 0 −2.61142 − 9.74596i −4.37397 − 4.37397i 0 −2.78747 + 3.20802i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.e even 4 1 inner
45.k odd 12 1 inner
45.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.3.l.o 32
3.b odd 2 1 inner 405.3.l.o 32
5.c odd 4 1 inner 405.3.l.o 32
9.c even 3 1 135.3.g.a ✓ 16
9.c even 3 1 inner 405.3.l.o 32
9.d odd 6 1 135.3.g.a ✓ 16
9.d odd 6 1 inner 405.3.l.o 32
15.e even 4 1 inner 405.3.l.o 32
45.h odd 6 1 675.3.g.k 16
45.j even 6 1 675.3.g.k 16
45.k odd 12 1 135.3.g.a ✓ 16
45.k odd 12 1 inner 405.3.l.o 32
45.k odd 12 1 675.3.g.k 16
45.l even 12 1 135.3.g.a ✓ 16
45.l even 12 1 inner 405.3.l.o 32
45.l even 12 1 675.3.g.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.3.g.a ✓ 16 9.c even 3 1
135.3.g.a ✓ 16 9.d odd 6 1
135.3.g.a ✓ 16 45.k odd 12 1
135.3.g.a ✓ 16 45.l even 12 1
405.3.l.o 32 1.a even 1 1 trivial
405.3.l.o 32 3.b odd 2 1 inner
405.3.l.o 32 5.c odd 4 1 inner
405.3.l.o 32 9.c even 3 1 inner
405.3.l.o 32 9.d odd 6 1 inner
405.3.l.o 32 15.e even 4 1 inner
405.3.l.o 32 45.k odd 12 1 inner
405.3.l.o 32 45.l even 12 1 inner
675.3.g.k 16 45.h odd 6 1
675.3.g.k 16 45.j even 6 1
675.3.g.k 16 45.k odd 12 1
675.3.g.k 16 45.l even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 217 T_{2}^{28} + 37825 T_{2}^{24} - 1891294 T_{2}^{20} + 72882286 T_{2}^{16} + \cdots + 815730721 \) acting on \(S_{3}^{\mathrm{new}}(405, [\chi])\). Copy content Toggle raw display