Properties

Label 670.3.d.a
Level $670$
Weight $3$
Character orbit 670.d
Analytic conductor $18.256$
Analytic rank $0$
Dimension $68$
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] = [670,3,Mod(669,670)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(670, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("670.669");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 670 = 2 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 670.d (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: \(18.2561777121\)
Analytic rank: \(0\)
Dimension: \(68\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 136 q^{4} + 8 q^{6} + 148 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q + 136 q^{4} + 8 q^{6} + 148 q^{9} - 24 q^{14} - 12 q^{15} + 272 q^{16} - 96 q^{19} + 24 q^{21} + 16 q^{24} - 68 q^{25} + 32 q^{29} + 4 q^{35} + 296 q^{36} + 192 q^{39} + 260 q^{49} - 80 q^{54} - 136 q^{55} - 48 q^{56} - 200 q^{59} - 24 q^{60} + 544 q^{64} + 68 q^{65} - 376 q^{71} - 192 q^{76} + 60 q^{81} + 48 q^{84} + 520 q^{86} - 192 q^{89} + 384 q^{90} + 24 q^{91} + 32 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
669.1 −1.41421 −1.77494 2.00000 −3.68387 + 3.38070i 2.51014 −10.8562 −2.82843 −5.84960 5.20977 4.78103i
669.2 −1.41421 −0.0124828 2.00000 4.79667 1.41135i 0.0176534 −9.71384 −2.82843 −8.99984 −6.78352 + 1.99596i
669.3 −1.41421 −2.51077 2.00000 −1.59553 + 4.73860i 3.55077 12.6787 −2.82843 −2.69602 2.25641 6.70139i
669.4 −1.41421 −3.99608 2.00000 4.47252 2.23529i 5.65132 2.78599 −2.82843 6.96868 −6.32511 + 3.16118i
669.5 −1.41421 5.26432 2.00000 −3.36480 + 3.69839i −7.44488 5.58736 −2.82843 18.7131 4.75855 5.23032i
669.6 −1.41421 −4.71040 2.00000 2.12438 + 4.52626i 6.66152 −9.43300 −2.82843 13.1879 −3.00433 6.40109i
669.7 −1.41421 4.35599 2.00000 −3.67481 + 3.39055i −6.16030 −9.33106 −2.82843 9.97462 5.19696 4.79496i
669.8 −1.41421 −0.138900 2.00000 0.630186 4.96013i 0.196434 8.09234 −2.82843 −8.98071 −0.891217 + 7.01468i
669.9 −1.41421 1.81538 2.00000 −4.91531 0.916390i −2.56734 6.65615 −2.82843 −5.70439 6.95129 + 1.29597i
669.10 −1.41421 2.60604 2.00000 4.19101 2.72680i −3.68550 7.60396 −2.82843 −2.20856 −5.92698 + 3.85628i
669.11 −1.41421 4.88427 2.00000 3.13203 + 3.89749i −6.90740 5.12422 −2.82843 14.8561 −4.42935 5.51188i
669.12 −1.41421 −3.95933 2.00000 −4.95609 0.661209i 5.59934 −0.640014 −2.82843 6.67633 7.00897 + 0.935091i
669.13 −1.41421 −0.496387 2.00000 −2.14002 4.51888i 0.701998 −2.51848 −2.82843 −8.75360 3.02645 + 6.39067i
669.14 −1.41421 1.48160 2.00000 −0.518984 + 4.97299i −2.09530 −3.34558 −2.82843 −6.80485 0.733954 7.03287i
669.15 −1.41421 −2.06083 2.00000 4.62161 1.90807i 2.91446 4.51265 −2.82843 −4.75297 −6.53594 + 2.69843i
669.16 −1.41421 −5.38418 2.00000 −2.05371 + 4.55876i 7.61437 4.57967 −2.82843 19.9893 2.90439 6.44706i
669.17 −1.41421 3.22250 2.00000 2.93470 4.04815i −4.55730 −7.54023 −2.82843 1.38449 −4.15030 + 5.72495i
669.18 −1.41421 3.22250 2.00000 2.93470 + 4.04815i −4.55730 −7.54023 −2.82843 1.38449 −4.15030 5.72495i
669.19 −1.41421 −5.38418 2.00000 −2.05371 4.55876i 7.61437 4.57967 −2.82843 19.9893 2.90439 + 6.44706i
669.20 −1.41421 −2.06083 2.00000 4.62161 + 1.90807i 2.91446 4.51265 −2.82843 −4.75297 −6.53594 2.69843i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 669.68
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
67.b odd 2 1 inner
335.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 670.3.d.a 68
5.b even 2 1 inner 670.3.d.a 68
67.b odd 2 1 inner 670.3.d.a 68
335.d odd 2 1 inner 670.3.d.a 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
670.3.d.a 68 1.a even 1 1 trivial
670.3.d.a 68 5.b even 2 1 inner
670.3.d.a 68 67.b odd 2 1 inner
670.3.d.a 68 335.d odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(670, [\chi])\).