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Properties

Label 805.2.d.f
Level $805$
Weight $2$
Character orbit 805.d
Analytic conductor $6.428$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [805,2,Mod(804,805)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(805, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("805.804");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 805 = 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 805.d (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: \(6.42795736271\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 56 q^{4} + 176 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 56 q^{4} + 176 q^{9} + 24 q^{16} - 40 q^{25} + 48 q^{29} - 24 q^{35} - 216 q^{36} + 8 q^{39} - 32 q^{49} - 8 q^{50} - 32 q^{64} + 56 q^{70} + 32 q^{71} - 64 q^{81} - 16 q^{85} - 80 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
804.1 − 2.65018i −2.36000 −5.02343 −0.662648 + 2.13563i 6.25442i 0.885182 + 2.49328i 8.01263i 2.56960 5.65979 + 1.75613i
804.2 − 2.65018i −2.36000 −5.02343 0.662648 − 2.13563i 6.25442i −0.885182 − 2.49328i 8.01263i 2.56960 −5.65979 − 1.75613i
804.3 − 2.65018i 2.36000 −5.02343 −0.662648 + 2.13563i − 6.25442i −0.885182 + 2.49328i 8.01263i 2.56960 5.65979 + 1.75613i
804.4 − 2.65018i 2.36000 −5.02343 0.662648 − 2.13563i − 6.25442i 0.885182 − 2.49328i 8.01263i 2.56960 −5.65979 − 1.75613i
804.5 − 2.21088i −3.27173 −2.88800 −2.04091 + 0.913618i 7.23341i 0.849085 − 2.50580i 1.96327i 7.70421 2.01990 + 4.51221i
804.6 − 2.21088i −3.27173 −2.88800 2.04091 − 0.913618i 7.23341i −0.849085 + 2.50580i 1.96327i 7.70421 −2.01990 − 4.51221i
804.7 − 2.21088i 3.27173 −2.88800 −2.04091 + 0.913618i − 7.23341i −0.849085 − 2.50580i 1.96327i 7.70421 2.01990 + 4.51221i
804.8 − 2.21088i 3.27173 −2.88800 2.04091 − 0.913618i − 7.23341i 0.849085 + 2.50580i 1.96327i 7.70421 −2.01990 − 4.51221i
804.9 − 1.90358i −1.81660 −1.62361 −1.10425 − 1.94439i 3.45804i 2.63803 + 0.201935i − 0.716491i 0.300029 −3.70129 + 2.10202i
804.10 − 1.90358i −1.81660 −1.62361 1.10425 + 1.94439i 3.45804i −2.63803 − 0.201935i − 0.716491i 0.300029 3.70129 − 2.10202i
804.11 − 1.90358i 1.81660 −1.62361 −1.10425 − 1.94439i − 3.45804i −2.63803 + 0.201935i − 0.716491i 0.300029 −3.70129 + 2.10202i
804.12 − 1.90358i 1.81660 −1.62361 1.10425 + 1.94439i − 3.45804i 2.63803 − 0.201935i − 0.716491i 0.300029 3.70129 − 2.10202i
804.13 − 1.56554i −2.65468 −0.450922 −1.99685 − 1.00627i 4.15602i −2.42121 − 1.06665i − 2.42515i 4.04734 −1.57535 + 3.12616i
804.14 − 1.56554i −2.65468 −0.450922 1.99685 + 1.00627i 4.15602i 2.42121 + 1.06665i − 2.42515i 4.04734 1.57535 − 3.12616i
804.15 − 1.56554i 2.65468 −0.450922 −1.99685 − 1.00627i − 4.15602i 2.42121 − 1.06665i − 2.42515i 4.04734 −1.57535 + 3.12616i
804.16 − 1.56554i 2.65468 −0.450922 1.99685 + 1.00627i − 4.15602i −2.42121 + 1.06665i − 2.42515i 4.04734 1.57535 − 3.12616i
804.17 − 0.871276i −2.78799 1.24088 −0.596168 + 2.15513i 2.42911i −2.08691 + 1.62628i − 2.82370i 4.77289 1.87771 + 0.519426i
804.18 − 0.871276i −2.78799 1.24088 0.596168 − 2.15513i 2.42911i 2.08691 − 1.62628i − 2.82370i 4.77289 −1.87771 − 0.519426i
804.19 − 0.871276i 2.78799 1.24088 −0.596168 + 2.15513i − 2.42911i 2.08691 + 1.62628i − 2.82370i 4.77289 1.87771 + 0.519426i
804.20 − 0.871276i 2.78799 1.24088 0.596168 − 2.15513i − 2.42911i −2.08691 − 1.62628i − 2.82370i 4.77289 −1.87771 − 0.519426i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 804.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
23.b odd 2 1 inner
35.c odd 2 1 inner
115.c odd 2 1 inner
161.c even 2 1 inner
805.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 805.2.d.f ✓ 48
5.b even 2 1 inner 805.2.d.f ✓ 48
7.b odd 2 1 inner 805.2.d.f ✓ 48
23.b odd 2 1 inner 805.2.d.f ✓ 48
35.c odd 2 1 inner 805.2.d.f ✓ 48
115.c odd 2 1 inner 805.2.d.f ✓ 48
161.c even 2 1 inner 805.2.d.f ✓ 48
805.d even 2 1 inner 805.2.d.f ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.d.f ✓ 48 1.a even 1 1 trivial
805.2.d.f ✓ 48 5.b even 2 1 inner
805.2.d.f ✓ 48 7.b odd 2 1 inner
805.2.d.f ✓ 48 23.b odd 2 1 inner
805.2.d.f ✓ 48 35.c odd 2 1 inner
805.2.d.f ✓ 48 115.c odd 2 1 inner
805.2.d.f ✓ 48 161.c even 2 1 inner
805.2.d.f ✓ 48 805.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(805, [\chi])\):

\( T_{2}^{12} + 19T_{2}^{10} + 134T_{2}^{8} + 435T_{2}^{6} + 646T_{2}^{4} + 370T_{2}^{2} + 59 \) Copy content Toggle raw display
\( T_{3}^{12} - 40T_{3}^{10} + 651T_{3}^{8} - 5517T_{3}^{6} + 25652T_{3}^{4} - 61922T_{3}^{2} + 60416 \) Copy content Toggle raw display