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Properties

Label 804.2.o.d
Level $804$
Weight $2$
Character orbit 804.o
Analytic conductor $6.420$
Analytic rank $0$
Dimension $36$
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] = [804,2,Mod(365,804)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(804, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("804.365");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 804.o (of order \(6\), degree \(2\), 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: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{9} - 36 q^{13} + 18 q^{15} + 16 q^{21} + 76 q^{25} + 6 q^{31} + 4 q^{33} + 42 q^{37} - 21 q^{39} + 2 q^{49} + 18 q^{51} + 20 q^{55} + 18 q^{57} - 24 q^{61} - 12 q^{63} - 8 q^{67} + 3 q^{69} + 14 q^{73} + 72 q^{79} - 12 q^{81} - 18 q^{85} - 21 q^{87} - 68 q^{91} + 9 q^{93} - 48 q^{97} + 15 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
365.1 0 −1.72171 − 0.188968i 0 3.09374 0 −1.38610 − 0.800262i 0 2.92858 + 0.650697i 0
365.2 0 −1.71192 + 0.263286i 0 −0.454473 0 −0.768091 − 0.443457i 0 2.86136 − 0.901449i 0
365.3 0 −1.53706 + 0.798403i 0 −3.20386 0 2.52764 + 1.45934i 0 1.72511 − 2.45439i 0
365.4 0 −1.41646 − 0.996822i 0 0.835859 0 2.58614 + 1.49311i 0 1.01269 + 2.82391i 0
365.5 0 −1.34511 − 1.09118i 0 −3.88149 0 −3.35972 − 1.93974i 0 0.618641 + 2.93552i 0
365.6 0 −0.870385 + 1.49747i 0 −2.31311 0 −0.381576 − 0.220303i 0 −1.48486 − 2.60676i 0
365.7 0 −0.869231 + 1.49814i 0 2.66956 0 0.767031 + 0.442845i 0 −1.48888 − 2.60447i 0
365.8 0 −0.536215 + 1.64696i 0 1.65208 0 −3.18562 − 1.83922i 0 −2.42495 − 1.76625i 0
365.9 0 −0.355178 − 1.69524i 0 −3.60368 0 3.20029 + 1.84769i 0 −2.74770 + 1.20423i 0
365.10 0 0.355178 − 1.69524i 0 3.60368 0 3.20029 + 1.84769i 0 −2.74770 − 1.20423i 0
365.11 0 0.536215 + 1.64696i 0 −1.65208 0 −3.18562 − 1.83922i 0 −2.42495 + 1.76625i 0
365.12 0 0.869231 + 1.49814i 0 −2.66956 0 0.767031 + 0.442845i 0 −1.48888 + 2.60447i 0
365.13 0 0.870385 + 1.49747i 0 2.31311 0 −0.381576 − 0.220303i 0 −1.48486 + 2.60676i 0
365.14 0 1.34511 − 1.09118i 0 3.88149 0 −3.35972 − 1.93974i 0 0.618641 − 2.93552i 0
365.15 0 1.41646 − 0.996822i 0 −0.835859 0 2.58614 + 1.49311i 0 1.01269 − 2.82391i 0
365.16 0 1.53706 + 0.798403i 0 3.20386 0 2.52764 + 1.45934i 0 1.72511 + 2.45439i 0
365.17 0 1.71192 + 0.263286i 0 0.454473 0 −0.768091 − 0.443457i 0 2.86136 + 0.901449i 0
365.18 0 1.72171 − 0.188968i 0 −3.09374 0 −1.38610 − 0.800262i 0 2.92858 − 0.650697i 0
641.1 0 −1.72171 + 0.188968i 0 3.09374 0 −1.38610 + 0.800262i 0 2.92858 − 0.650697i 0
641.2 0 −1.71192 − 0.263286i 0 −0.454473 0 −0.768091 + 0.443457i 0 2.86136 + 0.901449i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 365.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
67.d odd 6 1 inner
201.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 804.2.o.d ✓ 36
3.b odd 2 1 inner 804.2.o.d ✓ 36
67.d odd 6 1 inner 804.2.o.d ✓ 36
201.f even 6 1 inner 804.2.o.d ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
804.2.o.d ✓ 36 1.a even 1 1 trivial
804.2.o.d ✓ 36 3.b odd 2 1 inner
804.2.o.d ✓ 36 67.d odd 6 1 inner
804.2.o.d ✓ 36 201.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} - 64 T_{5}^{16} + 1708 T_{5}^{14} - 24632 T_{5}^{12} + 207688 T_{5}^{10} - 1032063 T_{5}^{8} + \cdots - 288684 \) acting on \(S_{2}^{\mathrm{new}}(804, [\chi])\). Copy content Toggle raw display