Properties

Label 315.5.f.a
Level $315$
Weight $5$
Character orbit 315.f
Analytic conductor $32.562$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,5,Mod(134,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.134");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 315.f (of order \(2\), degree \(1\), 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: \(32.5615383714\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 440 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 440 q^{4} - 432 q^{10} + 3800 q^{16} + 2448 q^{19} - 584 q^{25} + 1600 q^{31} + 11088 q^{34} - 15152 q^{40} + 7904 q^{46} - 16464 q^{49} - 31688 q^{55} - 2688 q^{61} + 60328 q^{64} + 9016 q^{70} + 38944 q^{76} + 4704 q^{79} + 34568 q^{85} - 1168 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
134.1 −7.75434 0 44.1298 21.9687 + 11.9322i 0 18.5203i −218.128 0 −170.353 92.5261i
134.2 −7.75434 0 44.1298 21.9687 11.9322i 0 18.5203i −218.128 0 −170.353 + 92.5261i
134.3 −7.26389 0 36.7640 −21.6988 + 12.4163i 0 18.5203i −150.828 0 157.617 90.1902i
134.4 −7.26389 0 36.7640 −21.6988 12.4163i 0 18.5203i −150.828 0 157.617 + 90.1902i
134.5 −7.05397 0 33.7584 −2.15368 + 24.9071i 0 18.5203i −125.267 0 15.1920 175.694i
134.6 −7.05397 0 33.7584 −2.15368 24.9071i 0 18.5203i −125.267 0 15.1920 + 175.694i
134.7 −6.05730 0 20.6909 21.8682 + 12.1154i 0 18.5203i −28.4144 0 −132.462 73.3869i
134.8 −6.05730 0 20.6909 21.8682 12.1154i 0 18.5203i −28.4144 0 −132.462 + 73.3869i
134.9 −5.50164 0 14.2681 15.7292 + 19.4317i 0 18.5203i 9.52837 0 −86.5364 106.907i
134.10 −5.50164 0 14.2681 15.7292 19.4317i 0 18.5203i 9.52837 0 −86.5364 + 106.907i
134.11 −5.17669 0 10.7981 −5.01029 24.4928i 0 18.5203i 26.9285 0 25.9367 + 126.792i
134.12 −5.17669 0 10.7981 −5.01029 + 24.4928i 0 18.5203i 26.9285 0 25.9367 126.792i
134.13 −3.82553 0 −1.36532 21.7643 12.3010i 0 18.5203i 66.4316 0 −83.2601 + 47.0578i
134.14 −3.82553 0 −1.36532 21.7643 + 12.3010i 0 18.5203i 66.4316 0 −83.2601 47.0578i
134.15 −3.57740 0 −3.20217 −24.6015 + 4.44567i 0 18.5203i 68.6940 0 88.0097 15.9040i
134.16 −3.57740 0 −3.20217 −24.6015 4.44567i 0 18.5203i 68.6940 0 88.0097 + 15.9040i
134.17 −3.05766 0 −6.65074 −23.4191 8.74905i 0 18.5203i 69.2582 0 71.6075 + 26.7516i
134.18 −3.05766 0 −6.65074 −23.4191 + 8.74905i 0 18.5203i 69.2582 0 71.6075 26.7516i
134.19 −2.67247 0 −8.85789 −7.43665 + 23.8683i 0 18.5203i 66.4320 0 19.8743 63.7874i
134.20 −2.67247 0 −8.85789 −7.43665 23.8683i 0 18.5203i 66.4320 0 19.8743 + 63.7874i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 134.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.5.f.a 48
3.b odd 2 1 inner 315.5.f.a 48
5.b even 2 1 inner 315.5.f.a 48
15.d odd 2 1 inner 315.5.f.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.5.f.a 48 1.a even 1 1 trivial
315.5.f.a 48 3.b odd 2 1 inner
315.5.f.a 48 5.b even 2 1 inner
315.5.f.a 48 15.d odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(315, [\chi])\).