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Properties

Label 845.2.t.h
Level $845$
Weight $2$
Character orbit 845.t
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(188,845)]
 
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, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.188");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.t (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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
188.1 −2.00384 − 1.15692i −0.313751 + 0.0840693i 1.67693 + 2.90452i 1.35021 − 1.78239i 0.725969 + 0.194523i 0.841565 + 1.45763i − 3.13261i −2.50670 + 1.44725i −4.76770 + 2.00955i
188.2 −1.06895 − 0.617160i −2.60917 + 0.699125i −0.238226 − 0.412620i −1.93345 + 1.12329i 3.22055 + 0.862944i −1.33537 − 2.31292i 3.05674i 3.72091 − 2.14827i 2.76002 − 0.00749302i
188.3 −0.303228 − 0.175069i 1.55689 − 0.417169i −0.938702 − 1.62588i −2.10682 − 0.749198i −0.545127 − 0.146066i 2.00214 + 3.46781i 1.35762i −0.348184 + 0.201024i 0.507686 + 0.596017i
188.4 0.303228 + 0.175069i 1.55689 − 0.417169i −0.938702 − 1.62588i 2.10682 + 0.749198i 0.545127 + 0.146066i −2.00214 − 3.46781i − 1.35762i −0.348184 + 0.201024i 0.507686 + 0.596017i
188.5 1.06895 + 0.617160i −2.60917 + 0.699125i −0.238226 − 0.412620i 1.93345 − 1.12329i −3.22055 − 0.862944i 1.33537 + 2.31292i − 3.05674i 3.72091 − 2.14827i 2.76002 − 0.00749302i
188.6 2.00384 + 1.15692i −0.313751 + 0.0840693i 1.67693 + 2.90452i −1.35021 + 1.78239i −0.725969 − 0.194523i −0.841565 − 1.45763i 3.13261i −2.50670 + 1.44725i −4.76770 + 2.00955i
418.1 −2.00384 + 1.15692i 0.0840693 − 0.313751i 1.67693 − 2.90452i −1.35021 + 1.78239i 0.194523 + 0.725969i −0.841565 + 1.45763i 3.13261i 2.50670 + 1.44725i 0.643527 − 5.13373i
418.2 −1.06895 + 0.617160i 0.699125 − 2.60917i −0.238226 + 0.412620i 1.93345 − 1.12329i 0.862944 + 3.22055i 1.33537 − 2.31292i − 3.05674i −3.72091 − 2.14827i −1.37352 + 2.39399i
418.3 −0.303228 + 0.175069i −0.417169 + 1.55689i −0.938702 + 1.62588i 2.10682 + 0.749198i −0.146066 − 0.545127i −2.00214 + 3.46781i − 1.35762i 0.348184 + 0.201024i −0.770009 + 0.141661i
418.4 0.303228 − 0.175069i −0.417169 + 1.55689i −0.938702 + 1.62588i −2.10682 − 0.749198i 0.146066 + 0.545127i 2.00214 − 3.46781i 1.35762i 0.348184 + 0.201024i −0.770009 + 0.141661i
418.5 1.06895 − 0.617160i 0.699125 − 2.60917i −0.238226 + 0.412620i −1.93345 + 1.12329i −0.862944 − 3.22055i −1.33537 + 2.31292i 3.05674i −3.72091 − 2.14827i −1.37352 + 2.39399i
418.6 2.00384 − 1.15692i 0.0840693 − 0.313751i 1.67693 − 2.90452i 1.35021 − 1.78239i −0.194523 − 0.725969i 0.841565 − 1.45763i − 3.13261i 2.50670 + 1.44725i 0.643527 − 5.13373i
427.1 −2.00384 + 1.15692i −0.313751 − 0.0840693i 1.67693 − 2.90452i 1.35021 + 1.78239i 0.725969 − 0.194523i 0.841565 − 1.45763i 3.13261i −2.50670 − 1.44725i −4.76770 − 2.00955i
427.2 −1.06895 + 0.617160i −2.60917 − 0.699125i −0.238226 + 0.412620i −1.93345 − 1.12329i 3.22055 − 0.862944i −1.33537 + 2.31292i − 3.05674i 3.72091 + 2.14827i 2.76002 + 0.00749302i
427.3 −0.303228 + 0.175069i 1.55689 + 0.417169i −0.938702 + 1.62588i −2.10682 + 0.749198i −0.545127 + 0.146066i 2.00214 − 3.46781i − 1.35762i −0.348184 − 0.201024i 0.507686 − 0.596017i
427.4 0.303228 − 0.175069i 1.55689 + 0.417169i −0.938702 + 1.62588i 2.10682 − 0.749198i 0.545127 − 0.146066i −2.00214 + 3.46781i 1.35762i −0.348184 − 0.201024i 0.507686 − 0.596017i
427.5 1.06895 − 0.617160i −2.60917 − 0.699125i −0.238226 + 0.412620i 1.93345 + 1.12329i −3.22055 + 0.862944i 1.33537 − 2.31292i 3.05674i 3.72091 + 2.14827i 2.76002 + 0.00749302i
427.6 2.00384 − 1.15692i −0.313751 − 0.0840693i 1.67693 − 2.90452i −1.35021 − 1.78239i −0.725969 + 0.194523i −0.841565 + 1.45763i − 3.13261i −2.50670 − 1.44725i −4.76770 − 2.00955i
657.1 −2.00384 − 1.15692i 0.0840693 + 0.313751i 1.67693 + 2.90452i −1.35021 − 1.78239i 0.194523 − 0.725969i −0.841565 − 1.45763i − 3.13261i 2.50670 − 1.44725i 0.643527 + 5.13373i
657.2 −1.06895 − 0.617160i 0.699125 + 2.60917i −0.238226 − 0.412620i 1.93345 + 1.12329i 0.862944 − 3.22055i 1.33537 + 2.31292i 3.05674i −3.72091 + 2.14827i −1.37352 − 2.39399i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 188.6
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.t.h 24
5.c odd 4 1 845.2.o.h 24
13.b even 2 1 inner 845.2.t.h 24
13.c even 3 1 845.2.f.c ✓ 12
13.c even 3 1 inner 845.2.t.h 24
13.d odd 4 2 845.2.o.h 24
13.e even 6 1 845.2.f.c ✓ 12
13.e even 6 1 inner 845.2.t.h 24
13.f odd 12 2 845.2.k.c yes 12
13.f odd 12 2 845.2.o.h 24
65.f even 4 1 inner 845.2.t.h 24
65.h odd 4 1 845.2.o.h 24
65.k even 4 1 inner 845.2.t.h 24
65.o even 12 1 845.2.f.c ✓ 12
65.o even 12 1 inner 845.2.t.h 24
65.q odd 12 1 845.2.k.c yes 12
65.q odd 12 1 845.2.o.h 24
65.r odd 12 1 845.2.k.c yes 12
65.r odd 12 1 845.2.o.h 24
65.t even 12 1 845.2.f.c ✓ 12
65.t even 12 1 inner 845.2.t.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.f.c ✓ 12 13.c even 3 1
845.2.f.c ✓ 12 13.e even 6 1
845.2.f.c ✓ 12 65.o even 12 1
845.2.f.c ✓ 12 65.t even 12 1
845.2.k.c yes 12 13.f odd 12 2
845.2.k.c yes 12 65.q odd 12 1
845.2.k.c yes 12 65.r odd 12 1
845.2.o.h 24 5.c odd 4 1
845.2.o.h 24 13.d odd 4 2
845.2.o.h 24 13.f odd 12 2
845.2.o.h 24 65.h odd 4 1
845.2.o.h 24 65.q odd 12 1
845.2.o.h 24 65.r odd 12 1
845.2.t.h 24 1.a even 1 1 trivial
845.2.t.h 24 13.b even 2 1 inner
845.2.t.h 24 13.c even 3 1 inner
845.2.t.h 24 13.e even 6 1 inner
845.2.t.h 24 65.f even 4 1 inner
845.2.t.h 24 65.k even 4 1 inner
845.2.t.h 24 65.o even 12 1 inner
845.2.t.h 24 65.t even 12 1 inner

Hecke kernels

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

\( T_{2}^{12} - 7T_{2}^{10} + 40T_{2}^{8} - 61T_{2}^{6} + 74T_{2}^{4} - 9T_{2}^{2} + 1 \) 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
\( T_{7}^{12} + 26T_{7}^{10} + 496T_{7}^{8} + 4032T_{7}^{6} + 23976T_{7}^{4} + 58320T_{7}^{2} + 104976 \) Copy content Toggle raw display