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Properties

Label 845.2.m.j
Level $845$
Weight $2$
Character orbit 845.m
Analytic conductor $6.747$
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] = [845,2,Mod(316,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.316");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.m (of order \(6\), degree \(2\), 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: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 14 q^{3} + 34 q^{4} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 14 q^{3} + 34 q^{4} - 32 q^{9} + 6 q^{10} - 48 q^{12} - 8 q^{14} - 74 q^{16} - 2 q^{17} - 24 q^{22} + 28 q^{23} - 36 q^{25} + 88 q^{27} - 24 q^{29} - 4 q^{30} - 14 q^{35} + 6 q^{36} + 188 q^{38} + 48 q^{40} + 22 q^{42} + 78 q^{43} + 6 q^{48} + 32 q^{49} - 172 q^{51} - 32 q^{53} + 18 q^{55} - 58 q^{56} + 6 q^{61} - 20 q^{62} - 136 q^{64} - 196 q^{66} + 40 q^{68} - 26 q^{69} + 30 q^{74} + 14 q^{75} + 16 q^{77} + 156 q^{79} - 58 q^{81} - 8 q^{82} - 32 q^{87} + 84 q^{88} + 40 q^{90} - 108 q^{92} - 32 q^{94} - 8 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
316.1 −2.41570 − 1.39471i 0.907309 − 1.57151i 2.89042 + 5.00635i 1.00000i −4.38358 + 2.53086i −0.233481 + 0.134800i − 10.5463i −0.146421 − 0.253609i 1.39471 − 2.41570i
316.2 −2.28282 − 1.31799i −0.994722 + 1.72291i 2.47417 + 4.28539i 1.00000i 4.54154 − 2.62206i 2.84256 − 1.64115i − 7.77176i −0.478943 − 0.829553i 1.31799 − 2.28282i
316.3 −2.23996 − 1.29324i −0.442413 + 0.766281i 2.34494 + 4.06156i − 1.00000i 1.98197 − 1.14429i 0.743381 − 0.429191i − 6.95735i 1.10854 + 1.92005i −1.29324 + 2.23996i
316.4 −1.97522 − 1.14039i −1.60714 + 2.78365i 1.60100 + 2.77301i − 1.00000i 6.34890 − 3.66554i 1.99506 − 1.15185i − 2.74149i −3.66579 − 6.34933i −1.14039 + 1.97522i
316.5 −1.91007 − 1.10278i 0.00652833 − 0.0113074i 1.43225 + 2.48072i 1.00000i −0.0249391 + 0.0143986i −3.99148 + 2.30448i − 1.90669i 1.49991 + 2.59793i 1.10278 − 1.91007i
316.6 −1.32578 − 0.765438i −1.44363 + 2.50044i 0.171790 + 0.297550i 1.00000i 3.82786 − 2.21002i −3.34713 + 1.93247i 2.53577i −2.66813 − 4.62133i 0.765438 − 1.32578i
316.7 −0.906907 − 0.523603i 1.37934 − 2.38909i −0.451680 − 0.782333i − 1.00000i −2.50186 + 1.44445i 2.96497 − 1.71183i 3.04042i −2.30515 − 3.99264i −0.523603 + 0.906907i
316.8 −0.235017 − 0.135687i 0.159809 − 0.276797i −0.963178 − 1.66827i − 1.00000i −0.0751154 + 0.0433679i −2.92848 + 1.69076i 1.06551i 1.44892 + 2.50961i −0.135687 + 0.235017i
316.9 −0.0208077 − 0.0120133i −1.46508 + 2.53760i −0.999711 − 1.73155i 1.00000i 0.0609700 − 0.0352011i 1.44229 − 0.832707i 0.0960927i −2.79295 − 4.83752i 0.0120133 − 0.0208077i
316.10 0.0208077 + 0.0120133i −1.46508 + 2.53760i −0.999711 − 1.73155i − 1.00000i −0.0609700 + 0.0352011i −1.44229 + 0.832707i − 0.0960927i −2.79295 − 4.83752i 0.0120133 − 0.0208077i
316.11 0.235017 + 0.135687i 0.159809 − 0.276797i −0.963178 − 1.66827i 1.00000i 0.0751154 − 0.0433679i 2.92848 − 1.69076i − 1.06551i 1.44892 + 2.50961i −0.135687 + 0.235017i
316.12 0.906907 + 0.523603i 1.37934 − 2.38909i −0.451680 − 0.782333i 1.00000i 2.50186 − 1.44445i −2.96497 + 1.71183i − 3.04042i −2.30515 − 3.99264i −0.523603 + 0.906907i
316.13 1.32578 + 0.765438i −1.44363 + 2.50044i 0.171790 + 0.297550i − 1.00000i −3.82786 + 2.21002i 3.34713 − 1.93247i − 2.53577i −2.66813 − 4.62133i 0.765438 − 1.32578i
316.14 1.91007 + 1.10278i 0.00652833 − 0.0113074i 1.43225 + 2.48072i − 1.00000i 0.0249391 − 0.0143986i 3.99148 − 2.30448i 1.90669i 1.49991 + 2.59793i 1.10278 − 1.91007i
316.15 1.97522 + 1.14039i −1.60714 + 2.78365i 1.60100 + 2.77301i 1.00000i −6.34890 + 3.66554i −1.99506 + 1.15185i 2.74149i −3.66579 − 6.34933i −1.14039 + 1.97522i
316.16 2.23996 + 1.29324i −0.442413 + 0.766281i 2.34494 + 4.06156i 1.00000i −1.98197 + 1.14429i −0.743381 + 0.429191i 6.95735i 1.10854 + 1.92005i −1.29324 + 2.23996i
316.17 2.28282 + 1.31799i −0.994722 + 1.72291i 2.47417 + 4.28539i − 1.00000i −4.54154 + 2.62206i −2.84256 + 1.64115i 7.77176i −0.478943 − 0.829553i 1.31799 − 2.28282i
316.18 2.41570 + 1.39471i 0.907309 − 1.57151i 2.89042 + 5.00635i − 1.00000i 4.38358 − 2.53086i 0.233481 − 0.134800i 10.5463i −0.146421 − 0.253609i 1.39471 − 2.41570i
361.1 −2.41570 + 1.39471i 0.907309 + 1.57151i 2.89042 − 5.00635i − 1.00000i −4.38358 − 2.53086i −0.233481 − 0.134800i 10.5463i −0.146421 + 0.253609i 1.39471 + 2.41570i
361.2 −2.28282 + 1.31799i −0.994722 − 1.72291i 2.47417 − 4.28539i − 1.00000i 4.54154 + 2.62206i 2.84256 + 1.64115i 7.77176i −0.478943 + 0.829553i 1.31799 + 2.28282i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 316.18
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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.m.j 36
13.b even 2 1 inner 845.2.m.j 36
13.c even 3 1 845.2.c.h 18
13.c even 3 1 inner 845.2.m.j 36
13.d odd 4 1 845.2.e.o 18
13.d odd 4 1 845.2.e.p 18
13.e even 6 1 845.2.c.h 18
13.e even 6 1 inner 845.2.m.j 36
13.f odd 12 1 845.2.a.n ✓ 9
13.f odd 12 1 845.2.a.o yes 9
13.f odd 12 1 845.2.e.o 18
13.f odd 12 1 845.2.e.p 18
39.k even 12 1 7605.2.a.cp 9
39.k even 12 1 7605.2.a.cs 9
65.s odd 12 1 4225.2.a.bs 9
65.s odd 12 1 4225.2.a.bt 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.a.n ✓ 9 13.f odd 12 1
845.2.a.o yes 9 13.f odd 12 1
845.2.c.h 18 13.c even 3 1
845.2.c.h 18 13.e even 6 1
845.2.e.o 18 13.d odd 4 1
845.2.e.o 18 13.f odd 12 1
845.2.e.p 18 13.d odd 4 1
845.2.e.p 18 13.f odd 12 1
845.2.m.j 36 1.a even 1 1 trivial
845.2.m.j 36 13.b even 2 1 inner
845.2.m.j 36 13.c even 3 1 inner
845.2.m.j 36 13.e even 6 1 inner
4225.2.a.bs 9 65.s odd 12 1
4225.2.a.bt 9 65.s odd 12 1
7605.2.a.cp 9 39.k even 12 1
7605.2.a.cs 9 39.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 35 T_{2}^{34} + 718 T_{2}^{32} - 9921 T_{2}^{30} + 102934 T_{2}^{28} - 822222 T_{2}^{26} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\). Copy content Toggle raw display