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Properties

Label 465.2.e.a
Level $465$
Weight $2$
Character orbit 465.e
Analytic conductor $3.713$
Analytic rank $0$
Dimension $44$
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(371,465)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(465, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("465.371");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.e (of order \(2\), degree \(1\), 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: \(44\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 52 q^{4} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 52 q^{4} + 16 q^{7} + 4 q^{10} + 68 q^{16} - 28 q^{18} - 8 q^{19} - 44 q^{25} - 56 q^{28} - 16 q^{31} + 4 q^{33} + 12 q^{36} - 20 q^{39} - 36 q^{40} - 4 q^{45} + 92 q^{49} + 8 q^{51} - 12 q^{63} - 52 q^{64} + 40 q^{66} - 32 q^{67} + 12 q^{69} - 24 q^{72} + 48 q^{76} - 88 q^{78} - 4 q^{81} + 36 q^{87} + 16 q^{90} - 20 q^{93} + 32 q^{94} + 16 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
371.1 − 2.74420i −1.15185 − 1.29354i −5.53063 1.00000i −3.54974 + 3.16090i 2.23252 9.68877i −0.346503 + 2.97992i 2.74420
371.2 − 2.74420i 1.15185 + 1.29354i −5.53063 1.00000i 3.54974 − 3.16090i 2.23252 9.68877i −0.346503 + 2.97992i 2.74420
371.3 − 2.44538i −0.0344882 − 1.73171i −3.97991 − 1.00000i −4.23469 + 0.0843369i −1.85712 4.84163i −2.99762 + 0.119447i −2.44538
371.4 − 2.44538i 0.0344882 + 1.73171i −3.97991 − 1.00000i 4.23469 − 0.0843369i −1.85712 4.84163i −2.99762 + 0.119447i −2.44538
371.5 − 2.42001i −1.32122 + 1.11999i −3.85645 1.00000i 2.71039 + 3.19737i −3.39968 4.49263i 0.491242 − 2.95951i 2.42001
371.6 − 2.42001i 1.32122 − 1.11999i −3.85645 1.00000i −2.71039 − 3.19737i −3.39968 4.49263i 0.491242 − 2.95951i 2.42001
371.7 − 2.37855i −1.59598 + 0.672939i −3.65749 − 1.00000i 1.60062 + 3.79611i 4.80431 3.94241i 2.09431 − 2.14799i −2.37855
371.8 − 2.37855i 1.59598 − 0.672939i −3.65749 − 1.00000i −1.60062 − 3.79611i 4.80431 3.94241i 2.09431 − 2.14799i −2.37855
371.9 − 1.87833i −1.66148 + 0.489360i −1.52811 1.00000i 0.919177 + 3.12081i 0.743633 − 0.886369i 2.52105 − 1.62613i 1.87833
371.10 − 1.87833i 1.66148 − 0.489360i −1.52811 1.00000i −0.919177 − 3.12081i 0.743633 − 0.886369i 2.52105 − 1.62613i 1.87833
371.11 − 1.57239i −0.222517 − 1.71770i −0.472421 1.00000i −2.70090 + 0.349884i 3.73118 − 2.40195i −2.90097 + 0.764434i 1.57239
371.12 − 1.57239i 0.222517 + 1.71770i −0.472421 1.00000i 2.70090 − 0.349884i 3.73118 − 2.40195i −2.90097 + 0.764434i 1.57239
371.13 − 1.31530i −0.652219 + 1.60456i 0.269988 − 1.00000i 2.11048 + 0.857863i 1.23320 − 2.98571i −2.14922 − 2.09305i −1.31530
371.14 − 1.31530i 0.652219 − 1.60456i 0.269988 − 1.00000i −2.11048 − 0.857863i 1.23320 − 2.98571i −2.14922 − 2.09305i −1.31530
371.15 − 1.25359i −1.67991 + 0.421766i 0.428502 − 1.00000i 0.528724 + 2.10593i −3.85670 − 3.04436i 2.64423 − 1.41706i −1.25359
371.16 − 1.25359i 1.67991 − 0.421766i 0.428502 − 1.00000i −0.528724 − 2.10593i −3.85670 − 3.04436i 2.64423 − 1.41706i −1.25359
371.17 − 0.572561i −1.21151 − 1.23784i 1.67217 − 1.00000i −0.708739 + 0.693663i 3.79443 − 2.10254i −0.0644908 + 2.99931i −0.572561
371.18 − 0.572561i 1.21151 + 1.23784i 1.67217 − 1.00000i 0.708739 − 0.693663i 3.79443 − 2.10254i −0.0644908 + 2.99931i −0.572561
371.19 − 0.552220i −1.64791 − 0.533294i 1.69505 1.00000i −0.294496 + 0.910007i −0.0110548 − 2.04048i 2.43119 + 1.75764i 0.552220
371.20 − 0.552220i 1.64791 + 0.533294i 1.69505 1.00000i 0.294496 − 0.910007i −0.0110548 − 2.04048i 2.43119 + 1.75764i 0.552220
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 371.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.b odd 2 1 inner
93.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 465.2.e.a ✓ 44
3.b odd 2 1 inner 465.2.e.a ✓ 44
31.b odd 2 1 inner 465.2.e.a ✓ 44
93.c even 2 1 inner 465.2.e.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.e.a ✓ 44 1.a even 1 1 trivial
465.2.e.a ✓ 44 3.b odd 2 1 inner
465.2.e.a ✓ 44 31.b odd 2 1 inner
465.2.e.a ✓ 44 93.c even 2 1 inner

Hecke kernels

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