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Properties

Label 840.2.p.b
Level $840$
Weight $2$
Character orbit 840.p
Analytic conductor $6.707$
Analytic rank $0$
Dimension $72$
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] = [840,2,Mod(659,840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("840.659");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.p (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: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(72\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 2 q^{4} + 72 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 2 q^{4} + 72 q^{7} + 2 q^{10} + 10 q^{16} + 10 q^{18} + 8 q^{19} + 6 q^{24} - 2 q^{28} + 4 q^{30} + 12 q^{34} + 18 q^{36} - 26 q^{40} - 20 q^{45} - 16 q^{46} - 8 q^{48} + 72 q^{49} - 12 q^{52} - 34 q^{54} + 8 q^{55} - 32 q^{58} - 52 q^{60} - 26 q^{64} - 38 q^{66} + 2 q^{70} - 32 q^{72} - 28 q^{75} - 16 q^{76} + 14 q^{78} - 60 q^{82} - 32 q^{85} + 24 q^{87} - 12 q^{88} - 12 q^{90} + 40 q^{93} + 32 q^{94} - 46 q^{96} - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
659.1 −1.41360 − 0.0417205i −0.409296 + 1.68300i 1.99652 + 0.117952i 1.92125 + 1.14404i 0.648796 − 2.36200i 1.00000 −2.81735 − 0.250033i −2.66495 − 1.37769i −2.66814 − 1.69736i
659.2 −1.41360 + 0.0417205i −0.409296 − 1.68300i 1.99652 − 0.117952i 1.92125 − 1.14404i 0.648796 + 2.36200i 1.00000 −2.81735 + 0.250033i −2.66495 + 1.37769i −2.66814 + 1.69736i
659.3 −1.40767 − 0.135926i −1.47813 + 0.902852i 1.96305 + 0.382677i −1.00079 + 1.99960i 2.20343 − 1.07000i 1.00000 −2.71130 − 0.805511i 1.36972 − 2.66906i 1.68058 − 2.67874i
659.4 −1.40767 + 0.135926i −1.47813 − 0.902852i 1.96305 − 0.382677i −1.00079 − 1.99960i 2.20343 + 1.07000i 1.00000 −2.71130 + 0.805511i 1.36972 + 2.66906i 1.68058 + 2.67874i
659.5 −1.40487 − 0.162289i 1.47315 + 0.910943i 1.94732 + 0.455989i 1.79299 + 1.33611i −1.92176 − 1.51883i 1.00000 −2.66174 − 0.956635i 1.34036 + 2.68392i −2.30208 − 2.16805i
659.6 −1.40487 + 0.162289i 1.47315 − 0.910943i 1.94732 − 0.455989i 1.79299 − 1.33611i −1.92176 + 1.51883i 1.00000 −2.66174 + 0.956635i 1.34036 − 2.68392i −2.30208 + 2.16805i
659.7 −1.33938 − 0.453938i 1.62751 − 0.592625i 1.58788 + 1.21599i −1.59464 − 1.56752i −2.44887 + 0.0549611i 1.00000 −1.57479 − 2.34948i 2.29759 − 1.92901i 1.42427 + 2.82338i
659.8 −1.33938 + 0.453938i 1.62751 + 0.592625i 1.58788 − 1.21599i −1.59464 + 1.56752i −2.44887 − 0.0549611i 1.00000 −1.57479 + 2.34948i 2.29759 + 1.92901i 1.42427 − 2.82338i
659.9 −1.25293 − 0.655872i −0.794734 + 1.53896i 1.13966 + 1.64352i −0.151252 − 2.23095i 2.00511 − 1.40696i 1.00000 −0.349978 − 2.80669i −1.73679 − 2.44613i −1.27371 + 2.89442i
659.10 −1.25293 + 0.655872i −0.794734 − 1.53896i 1.13966 − 1.64352i −0.151252 + 2.23095i 2.00511 + 1.40696i 1.00000 −0.349978 + 2.80669i −1.73679 + 2.44613i −1.27371 − 2.89442i
659.11 −1.24511 − 0.670607i −0.257633 − 1.71278i 1.10057 + 1.66995i −2.23353 + 0.106484i −0.827824 + 2.30536i 1.00000 −0.250448 − 2.81732i −2.86725 + 0.882537i 2.85239 + 1.36524i
659.12 −1.24511 + 0.670607i −0.257633 + 1.71278i 1.10057 − 1.66995i −2.23353 − 0.106484i −0.827824 − 2.30536i 1.00000 −0.250448 + 2.81732i −2.86725 − 0.882537i 2.85239 − 1.36524i
659.13 −1.18366 − 0.773912i −1.72910 + 0.101130i 0.802120 + 1.83210i 2.07129 − 0.842472i 2.12493 + 1.21847i 1.00000 0.468448 − 2.78936i 2.97955 − 0.349726i −3.10371 − 0.605792i
659.14 −1.18366 + 0.773912i −1.72910 − 0.101130i 0.802120 − 1.83210i 2.07129 + 0.842472i 2.12493 − 1.21847i 1.00000 0.468448 + 2.78936i 2.97955 + 0.349726i −3.10371 + 0.605792i
659.15 −1.17414 − 0.788285i 1.15131 + 1.29402i 0.757213 + 1.85112i −2.21259 + 0.323174i −0.331739 − 2.42692i 1.00000 0.570132 − 2.77037i −0.348984 + 2.97963i 2.85265 + 1.36470i
659.16 −1.17414 + 0.788285i 1.15131 − 1.29402i 0.757213 − 1.85112i −2.21259 − 0.323174i −0.331739 + 2.42692i 1.00000 0.570132 + 2.77037i −0.348984 − 2.97963i 2.85265 − 1.36470i
659.17 −1.02029 − 0.979292i −1.48418 − 0.892866i 0.0819726 + 1.99832i 0.199154 + 2.22718i 0.639913 + 2.36443i 1.00000 1.87330 − 2.11914i 1.40558 + 2.65035i 1.97787 − 2.46739i
659.18 −1.02029 + 0.979292i −1.48418 + 0.892866i 0.0819726 − 1.99832i 0.199154 − 2.22718i 0.639913 − 2.36443i 1.00000 1.87330 + 2.11914i 1.40558 − 2.65035i 1.97787 + 2.46739i
659.19 −0.970630 − 1.02853i 1.72413 + 0.165426i −0.115754 + 1.99665i 1.20030 + 1.88661i −1.50335 − 1.93389i 1.00000 2.16597 − 1.81895i 2.94527 + 0.570434i 0.775391 − 3.06574i
659.20 −0.970630 + 1.02853i 1.72413 − 0.165426i −0.115754 − 1.99665i 1.20030 − 1.88661i −1.50335 + 1.93389i 1.00000 2.16597 + 1.81895i 2.94527 − 0.570434i 0.775391 + 3.06574i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 659.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
40.e odd 2 1 inner
120.m even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.2.p.b yes 72
3.b odd 2 1 inner 840.2.p.b yes 72
5.b even 2 1 840.2.p.a ✓ 72
8.d odd 2 1 840.2.p.a ✓ 72
15.d odd 2 1 840.2.p.a ✓ 72
24.f even 2 1 840.2.p.a ✓ 72
40.e odd 2 1 inner 840.2.p.b yes 72
120.m even 2 1 inner 840.2.p.b yes 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.p.a ✓ 72 5.b even 2 1
840.2.p.a ✓ 72 8.d odd 2 1
840.2.p.a ✓ 72 15.d odd 2 1
840.2.p.a ✓ 72 24.f even 2 1
840.2.p.b yes 72 1.a even 1 1 trivial
840.2.p.b yes 72 3.b odd 2 1 inner
840.2.p.b yes 72 40.e odd 2 1 inner
840.2.p.b yes 72 120.m even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{18} - 122 T_{13}^{16} + 12 T_{13}^{15} + 5721 T_{13}^{14} - 500 T_{13}^{13} - 136464 T_{13}^{12} + \cdots - 7405568 \) acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\). Copy content Toggle raw display