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Properties

Label 845.2.o.h
Level $845$
Weight $2$
Character orbit 845.o
Analytic conductor $6.747$
Analytic rank $0$
Dimension $24$
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] = [845,2,Mod(258,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.258");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.o (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: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{3} - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{3} - 4 q^{4} + 12 q^{10} + 16 q^{12} - 4 q^{16} - 8 q^{17} - 32 q^{22} - 8 q^{23} - 40 q^{25} - 40 q^{27} + 4 q^{30} - 4 q^{35} + 56 q^{38} - 64 q^{40} + 28 q^{42} + 8 q^{43} + 36 q^{48} + 20 q^{49} + 88 q^{53} + 12 q^{55} + 64 q^{61} - 64 q^{62} - 72 q^{64} + 144 q^{66} - 48 q^{68} - 16 q^{69} + 52 q^{75} + 120 q^{77} - 12 q^{81} + 12 q^{82} - 92 q^{87} - 12 q^{88} - 152 q^{90} - 216 q^{92} + 64 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
258.1 −1.15692 − 2.00384i 0.0840693 − 0.313751i −1.67693 + 2.90452i −1.78239 − 1.35021i −0.725969 + 0.194523i 1.45763 + 0.841565i 3.13261 2.50670 + 1.44725i −0.643527 + 5.13373i
258.2 −0.617160 − 1.06895i 0.699125 − 2.60917i 0.238226 − 0.412620i 1.12329 + 1.93345i −3.22055 + 0.862944i −2.31292 − 1.33537i −3.05674 −3.72091 − 2.14827i 1.37352 − 2.39399i
258.3 −0.175069 − 0.303228i −0.417169 + 1.55689i 0.938702 − 1.62588i −0.749198 + 2.10682i 0.545127 − 0.146066i 3.46781 + 2.00214i −1.35762 0.348184 + 0.201024i 0.770009 − 0.141661i
258.4 0.175069 + 0.303228i −0.417169 + 1.55689i 0.938702 − 1.62588i 0.749198 − 2.10682i −0.545127 + 0.146066i −3.46781 − 2.00214i 1.35762 0.348184 + 0.201024i 0.770009 − 0.141661i
258.5 0.617160 + 1.06895i 0.699125 − 2.60917i 0.238226 − 0.412620i −1.12329 − 1.93345i 3.22055 − 0.862944i 2.31292 + 1.33537i 3.05674 −3.72091 − 2.14827i 1.37352 − 2.39399i
258.6 1.15692 + 2.00384i 0.0840693 − 0.313751i −1.67693 + 2.90452i 1.78239 + 1.35021i 0.725969 − 0.194523i −1.45763 − 0.841565i −3.13261 2.50670 + 1.44725i −0.643527 + 5.13373i
357.1 −1.15692 + 2.00384i 0.0840693 + 0.313751i −1.67693 − 2.90452i −1.78239 + 1.35021i −0.725969 − 0.194523i 1.45763 − 0.841565i 3.13261 2.50670 − 1.44725i −0.643527 − 5.13373i
357.2 −0.617160 + 1.06895i 0.699125 + 2.60917i 0.238226 + 0.412620i 1.12329 − 1.93345i −3.22055 − 0.862944i −2.31292 + 1.33537i −3.05674 −3.72091 + 2.14827i 1.37352 + 2.39399i
357.3 −0.175069 + 0.303228i −0.417169 − 1.55689i 0.938702 + 1.62588i −0.749198 − 2.10682i 0.545127 + 0.146066i 3.46781 − 2.00214i −1.35762 0.348184 − 0.201024i 0.770009 + 0.141661i
357.4 0.175069 − 0.303228i −0.417169 − 1.55689i 0.938702 + 1.62588i 0.749198 + 2.10682i −0.545127 − 0.146066i −3.46781 + 2.00214i 1.35762 0.348184 − 0.201024i 0.770009 + 0.141661i
357.5 0.617160 − 1.06895i 0.699125 + 2.60917i 0.238226 + 0.412620i −1.12329 + 1.93345i 3.22055 + 0.862944i 2.31292 − 1.33537i 3.05674 −3.72091 + 2.14827i 1.37352 + 2.39399i
357.6 1.15692 − 2.00384i 0.0840693 + 0.313751i −1.67693 − 2.90452i 1.78239 − 1.35021i 0.725969 + 0.194523i −1.45763 + 0.841565i −3.13261 2.50670 − 1.44725i −0.643527 − 5.13373i
488.1 −1.15692 + 2.00384i −0.313751 + 0.0840693i −1.67693 − 2.90452i −1.78239 − 1.35021i 0.194523 − 0.725969i −1.45763 + 0.841565i 3.13261 −2.50670 + 1.44725i 4.76770 − 2.00955i
488.2 −0.617160 + 1.06895i −2.60917 + 0.699125i 0.238226 + 0.412620i 1.12329 + 1.93345i 0.862944 − 3.22055i 2.31292 − 1.33537i −3.05674 3.72091 − 2.14827i −2.76002 + 0.00749302i
488.3 −0.175069 + 0.303228i 1.55689 − 0.417169i 0.938702 + 1.62588i −0.749198 + 2.10682i −0.146066 + 0.545127i −3.46781 + 2.00214i −1.35762 −0.348184 + 0.201024i −0.507686 − 0.596017i
488.4 0.175069 − 0.303228i 1.55689 − 0.417169i 0.938702 + 1.62588i 0.749198 − 2.10682i 0.146066 − 0.545127i 3.46781 − 2.00214i 1.35762 −0.348184 + 0.201024i −0.507686 − 0.596017i
488.5 0.617160 − 1.06895i −2.60917 + 0.699125i 0.238226 + 0.412620i −1.12329 − 1.93345i −0.862944 + 3.22055i −2.31292 + 1.33537i 3.05674 3.72091 − 2.14827i −2.76002 + 0.00749302i
488.6 1.15692 − 2.00384i −0.313751 + 0.0840693i −1.67693 − 2.90452i 1.78239 + 1.35021i −0.194523 + 0.725969i 1.45763 − 0.841565i −3.13261 −2.50670 + 1.44725i 4.76770 − 2.00955i
587.1 −1.15692 − 2.00384i −0.313751 − 0.0840693i −1.67693 + 2.90452i −1.78239 + 1.35021i 0.194523 + 0.725969i −1.45763 − 0.841565i 3.13261 −2.50670 − 1.44725i 4.76770 + 2.00955i
587.2 −0.617160 − 1.06895i −2.60917 − 0.699125i 0.238226 − 0.412620i 1.12329 − 1.93345i 0.862944 + 3.22055i 2.31292 + 1.33537i −3.05674 3.72091 + 2.14827i −2.76002 − 0.00749302i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 258.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner
65.f even 4 1 inner
65.k even 4 1 inner
65.o even 12 1 inner
65.t even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.o.h 24
5.c odd 4 1 845.2.t.h 24
13.b even 2 1 inner 845.2.o.h 24
13.c even 3 1 845.2.k.c yes 12
13.c even 3 1 inner 845.2.o.h 24
13.d odd 4 2 845.2.t.h 24
13.e even 6 1 845.2.k.c yes 12
13.e even 6 1 inner 845.2.o.h 24
13.f odd 12 2 845.2.f.c ✓ 12
13.f odd 12 2 845.2.t.h 24
65.f even 4 1 inner 845.2.o.h 24
65.h odd 4 1 845.2.t.h 24
65.k even 4 1 inner 845.2.o.h 24
65.o even 12 1 845.2.k.c yes 12
65.o even 12 1 inner 845.2.o.h 24
65.q odd 12 1 845.2.f.c ✓ 12
65.q odd 12 1 845.2.t.h 24
65.r odd 12 1 845.2.f.c ✓ 12
65.r odd 12 1 845.2.t.h 24
65.t even 12 1 845.2.k.c yes 12
65.t even 12 1 inner 845.2.o.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.f.c ✓ 12 13.f odd 12 2
845.2.f.c ✓ 12 65.q odd 12 1
845.2.f.c ✓ 12 65.r odd 12 1
845.2.k.c yes 12 13.c even 3 1
845.2.k.c yes 12 13.e even 6 1
845.2.k.c yes 12 65.o even 12 1
845.2.k.c yes 12 65.t even 12 1
845.2.o.h 24 1.a even 1 1 trivial
845.2.o.h 24 13.b even 2 1 inner
845.2.o.h 24 13.c even 3 1 inner
845.2.o.h 24 13.e even 6 1 inner
845.2.o.h 24 65.f even 4 1 inner
845.2.o.h 24 65.k even 4 1 inner
845.2.o.h 24 65.o even 12 1 inner
845.2.o.h 24 65.t even 12 1 inner
845.2.t.h 24 5.c odd 4 1
845.2.t.h 24 13.d odd 4 2
845.2.t.h 24 13.f odd 12 2
845.2.t.h 24 65.h odd 4 1
845.2.t.h 24 65.q odd 12 1
845.2.t.h 24 65.r odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\):

\( T_{2}^{12} + 7T_{2}^{10} + 40T_{2}^{8} + 61T_{2}^{6} + 74T_{2}^{4} + 9T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{12} + 2 T_{3}^{11} + 2 T_{3}^{10} + 16 T_{3}^{9} - 68 T_{3}^{7} - 8 T_{3}^{6} - 124 T_{3}^{5} + \cdots + 4 \) Copy content Toggle raw display