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Properties

Label 1665.2.g.e
Level $1665$
Weight $2$
Character orbit 1665.g
Analytic conductor $13.295$
Analytic rank $0$
Dimension $40$
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] = [1665,2,Mod(739,1665)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1665, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1665.739");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1665 = 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1665.g (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: \(13.2950919365\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: no (minimal twist has level 555)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 44 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 44 q^{4} + 4 q^{10} - 8 q^{11} + 52 q^{16} + 8 q^{25} + 16 q^{26} - 32 q^{34} - 28 q^{40} - 8 q^{41} - 16 q^{44} - 8 q^{46} - 24 q^{49} + 92 q^{64} + 48 q^{65} - 56 q^{70} - 24 q^{71} + 68 q^{74} - 64 q^{85} - 80 q^{86} - 32 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
739.1 −2.71549 0 5.37390 0.734693 − 2.11192i 0 1.35808i −9.16180 0 −1.99505 + 5.73492i
739.2 −2.71549 0 5.37390 0.734693 + 2.11192i 0 − 1.35808i −9.16180 0 −1.99505 − 5.73492i
739.3 −2.49663 0 4.23317 2.05572 − 0.879776i 0 3.27811i −5.57542 0 −5.13238 + 2.19648i
739.4 −2.49663 0 4.23317 2.05572 + 0.879776i 0 − 3.27811i −5.57542 0 −5.13238 − 2.19648i
739.5 −2.46809 0 4.09146 −1.94703 + 1.09958i 0 1.91769i −5.16192 0 4.80544 − 2.71386i
739.6 −2.46809 0 4.09146 −1.94703 − 1.09958i 0 − 1.91769i −5.16192 0 4.80544 + 2.71386i
739.7 −1.81177 0 1.28250 −2.16949 − 0.541568i 0 − 4.65490i 1.29994 0 3.93062 + 0.981196i
739.8 −1.81177 0 1.28250 −2.16949 + 0.541568i 0 4.65490i 1.29994 0 3.93062 − 0.981196i
739.9 −1.72645 0 0.980624 −1.26006 + 1.84723i 0 − 2.83159i 1.75990 0 2.17542 − 3.18915i
739.10 −1.72645 0 0.980624 −1.26006 − 1.84723i 0 2.83159i 1.75990 0 2.17542 + 3.18915i
739.11 −1.47721 0 0.182137 −0.357763 + 2.20726i 0 − 1.62051i 2.68536 0 0.528490 − 3.26058i
739.12 −1.47721 0 0.182137 −0.357763 − 2.20726i 0 1.62051i 2.68536 0 0.528490 + 3.26058i
739.13 −1.39280 0 −0.0601015 2.20287 − 0.383884i 0 0.344118i 2.86931 0 −3.06816 + 0.534675i
739.14 −1.39280 0 −0.0601015 2.20287 + 0.383884i 0 − 0.344118i 2.86931 0 −3.06816 − 0.534675i
739.15 −0.818204 0 −1.33054 0.157839 − 2.23049i 0 − 3.66373i 2.72506 0 −0.129144 + 1.82500i
739.16 −0.818204 0 −1.33054 0.157839 + 2.23049i 0 3.66373i 2.72506 0 −0.129144 − 1.82500i
739.17 −0.414618 0 −1.82809 1.55319 − 1.60860i 0 − 0.262182i 1.58720 0 −0.643980 + 0.666956i
739.18 −0.414618 0 −1.82809 1.55319 + 1.60860i 0 0.262182i 1.58720 0 −0.643980 − 0.666956i
739.19 −0.273741 0 −1.92507 −1.93156 + 1.12653i 0 − 3.71619i 1.07445 0 0.528747 − 0.308378i
739.20 −0.273741 0 −1.92507 −1.93156 − 1.12653i 0 3.71619i 1.07445 0 0.528747 + 0.308378i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 739.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.b even 2 1 inner
185.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1665.2.g.e 40
3.b odd 2 1 555.2.g.a ✓ 40
5.b even 2 1 inner 1665.2.g.e 40
15.d odd 2 1 555.2.g.a ✓ 40
37.b even 2 1 inner 1665.2.g.e 40
111.d odd 2 1 555.2.g.a ✓ 40
185.d even 2 1 inner 1665.2.g.e 40
555.b odd 2 1 555.2.g.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.g.a ✓ 40 3.b odd 2 1
555.2.g.a ✓ 40 15.d odd 2 1
555.2.g.a ✓ 40 111.d odd 2 1
555.2.g.a ✓ 40 555.b odd 2 1
1665.2.g.e 40 1.a even 1 1 trivial
1665.2.g.e 40 5.b even 2 1 inner
1665.2.g.e 40 37.b even 2 1 inner
1665.2.g.e 40 185.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 31 T_{2}^{18} + 401 T_{2}^{16} - 2817 T_{2}^{14} + 11728 T_{2}^{12} - 29636 T_{2}^{10} + \cdots + 100 \) acting on \(S_{2}^{\mathrm{new}}(1665, [\chi])\). Copy content Toggle raw display