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Properties

Label 465.2.j.a
Level $465$
Weight $2$
Character orbit 465.j
Analytic conductor $3.713$
Analytic rank $0$
Dimension $64$
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] = [465,2,Mod(247,465)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(465, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("465.247");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.j (of order \(4\), 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: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 8 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 8 q^{7} - 12 q^{8} - 8 q^{10} - 24 q^{16} + 40 q^{20} - 16 q^{25} - 12 q^{28} + 16 q^{31} - 64 q^{32} + 24 q^{33} - 56 q^{36} + 36 q^{38} + 16 q^{40} + 16 q^{41} + 16 q^{47} + 64 q^{50} + 16 q^{51} - 32 q^{56} - 64 q^{62} - 8 q^{63} + 16 q^{66} - 8 q^{67} - 4 q^{70} - 80 q^{71} - 12 q^{72} + 80 q^{76} + 48 q^{80} - 64 q^{81} - 44 q^{82} - 16 q^{87} - 12 q^{90} + 32 q^{93} - 40 q^{95} + 32 q^{97} + 68 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
247.1 −1.81784 − 1.81784i −0.707107 − 0.707107i 4.60908i 0.671076 − 2.13299i 2.57081i 1.57593 + 1.57593i 4.74289 − 4.74289i 1.00000i −5.09735 + 2.65753i
247.2 −1.81784 − 1.81784i 0.707107 + 0.707107i 4.60908i 0.671076 − 2.13299i − 2.57081i 1.57593 + 1.57593i 4.74289 − 4.74289i 1.00000i −5.09735 + 2.65753i
247.3 −1.54627 − 1.54627i −0.707107 − 0.707107i 2.78189i −0.906102 + 2.04426i 2.18675i −1.86120 − 1.86120i 1.20901 − 1.20901i 1.00000i 4.56204 − 1.75989i
247.4 −1.54627 − 1.54627i 0.707107 + 0.707107i 2.78189i −0.906102 + 2.04426i − 2.18675i −1.86120 − 1.86120i 1.20901 − 1.20901i 1.00000i 4.56204 − 1.75989i
247.5 −1.42305 − 1.42305i −0.707107 − 0.707107i 2.05015i 2.19075 + 0.447885i 2.01250i −0.101027 − 0.101027i 0.0713717 − 0.0713717i 1.00000i −2.48019 − 3.75492i
247.6 −1.42305 − 1.42305i 0.707107 + 0.707107i 2.05015i 2.19075 + 0.447885i − 2.01250i −0.101027 − 0.101027i 0.0713717 − 0.0713717i 1.00000i −2.48019 − 3.75492i
247.7 −1.29107 − 1.29107i −0.707107 − 0.707107i 1.33373i −0.713955 − 2.11903i 1.82585i −3.33077 − 3.33077i −0.860203 + 0.860203i 1.00000i −1.81405 + 3.65758i
247.8 −1.29107 − 1.29107i 0.707107 + 0.707107i 1.33373i −0.713955 − 2.11903i − 1.82585i −3.33077 − 3.33077i −0.860203 + 0.860203i 1.00000i −1.81405 + 3.65758i
247.9 −1.04293 − 1.04293i −0.707107 − 0.707107i 0.175410i −1.49880 − 1.65940i 1.47493i 2.55526 + 2.55526i −1.90292 + 1.90292i 1.00000i −0.167493 + 3.29378i
247.10 −1.04293 − 1.04293i 0.707107 + 0.707107i 0.175410i −1.49880 − 1.65940i − 1.47493i 2.55526 + 2.55526i −1.90292 + 1.90292i 1.00000i −0.167493 + 3.29378i
247.11 −0.691882 − 0.691882i −0.707107 − 0.707107i − 1.04260i −1.57894 + 1.58333i 0.978469i 1.79428 + 1.79428i −2.10512 + 2.10512i 1.00000i 2.18792 − 0.00303466i
247.12 −0.691882 − 0.691882i 0.707107 + 0.707107i − 1.04260i −1.57894 + 1.58333i − 0.978469i 1.79428 + 1.79428i −2.10512 + 2.10512i 1.00000i 2.18792 − 0.00303466i
247.13 −0.653006 − 0.653006i −0.707107 − 0.707107i − 1.14717i 1.02082 + 1.98946i 0.923490i −0.408469 − 0.408469i −2.05512 + 2.05512i 1.00000i 0.632528 − 1.96573i
247.14 −0.653006 − 0.653006i 0.707107 + 0.707107i − 1.14717i 1.02082 + 1.98946i − 0.923490i −0.408469 − 0.408469i −2.05512 + 2.05512i 1.00000i 0.632528 − 1.96573i
247.15 0.126336 + 0.126336i −0.707107 − 0.707107i − 1.96808i 0.312323 − 2.21415i − 0.178667i −0.830205 − 0.830205i 0.501313 − 0.501313i 1.00000i 0.319185 − 0.240270i
247.16 0.126336 + 0.126336i 0.707107 + 0.707107i − 1.96808i 0.312323 − 2.21415i 0.178667i −0.830205 − 0.830205i 0.501313 − 0.501313i 1.00000i 0.319185 − 0.240270i
247.17 0.192219 + 0.192219i −0.707107 − 0.707107i − 1.92610i 2.23575 − 0.0375717i − 0.271838i 2.53855 + 2.53855i 0.754671 − 0.754671i 1.00000i 0.436976 + 0.422532i
247.18 0.192219 + 0.192219i 0.707107 + 0.707107i − 1.92610i 2.23575 − 0.0375717i 0.271838i 2.53855 + 2.53855i 0.754671 − 0.754671i 1.00000i 0.436976 + 0.422532i
247.19 0.229071 + 0.229071i −0.707107 − 0.707107i − 1.89505i −1.69839 + 1.45447i − 0.323955i −2.82573 − 2.82573i 0.892242 − 0.892242i 1.00000i −0.722227 − 0.0558739i
247.20 0.229071 + 0.229071i 0.707107 + 0.707107i − 1.89505i −1.69839 + 1.45447i 0.323955i −2.82573 − 2.82573i 0.892242 − 0.892242i 1.00000i −0.722227 − 0.0558739i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 247.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
31.b odd 2 1 inner
155.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 465.2.j.a ✓ 64
5.c odd 4 1 inner 465.2.j.a ✓ 64
31.b odd 2 1 inner 465.2.j.a ✓ 64
155.f even 4 1 inner 465.2.j.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.j.a ✓ 64 1.a even 1 1 trivial
465.2.j.a ✓ 64 5.c odd 4 1 inner
465.2.j.a ✓ 64 31.b odd 2 1 inner
465.2.j.a ✓ 64 155.f even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(465, [\chi])\).