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Properties

Label 806.2.bn.a
Level $806$
Weight $2$
Character orbit 806.bn
Analytic conductor $6.436$
Analytic rank $0$
Dimension $320$
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] = [806,2,Mod(151,806)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(806, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([5, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.bn (of order \(20\), degree \(8\), 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: \(6.43594240292\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(40\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q + 16 q^{7} + 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q + 16 q^{7} + 96 q^{9} - 20 q^{11} - 20 q^{13} + 12 q^{14} + 80 q^{16} + 16 q^{18} - 32 q^{19} - 60 q^{21} - 60 q^{27} - 24 q^{28} + 4 q^{31} + 52 q^{33} - 40 q^{34} - 40 q^{35} + 44 q^{39} + 28 q^{41} + 40 q^{46} - 32 q^{47} - 40 q^{53} + 4 q^{59} + 104 q^{63} + 40 q^{65} + 32 q^{66} - 32 q^{67} - 24 q^{70} + 8 q^{71} - 16 q^{72} - 48 q^{76} - 76 q^{78} + 80 q^{79} - 272 q^{81} - 40 q^{83} + 40 q^{84} + 80 q^{85} + 32 q^{87} - 80 q^{89} - 200 q^{91} - 100 q^{93} - 8 q^{94} + 52 q^{97} + 48 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1 −0.453990 + 0.891007i −3.24717 − 1.05507i −0.587785 − 0.809017i −1.92541 − 1.92541i 2.41426 − 2.41426i 0.0416892 − 0.263215i 0.987688 − 0.156434i 7.00388 + 5.08862i 2.58967 − 0.841435i
151.2 −0.453990 + 0.891007i −2.95186 − 0.959117i −0.587785 − 0.809017i 2.41996 + 2.41996i 2.19470 − 2.19470i 0.687504 − 4.34073i 0.987688 − 0.156434i 5.36652 + 3.89900i −3.25484 + 1.05756i
151.3 −0.453990 + 0.891007i −2.18571 − 0.710179i −0.587785 − 0.809017i 0.392017 + 0.392017i 1.62506 − 1.62506i −0.152066 + 0.960104i 0.987688 − 0.156434i 1.84590 + 1.34113i −0.527262 + 0.171318i
151.4 −0.453990 + 0.891007i −2.12065 − 0.689041i −0.587785 − 0.809017i 1.70128 + 1.70128i 1.57670 − 1.57670i −0.787432 + 4.97165i 0.987688 − 0.156434i 1.59533 + 1.15907i −2.28821 + 0.743486i
151.5 −0.453990 + 0.891007i −1.88539 − 0.612602i −0.587785 − 0.809017i −0.0906225 − 0.0906225i 1.40178 − 1.40178i 0.0574570 − 0.362769i 0.987688 − 0.156434i 0.752377 + 0.546634i 0.121887 − 0.0396035i
151.6 −0.453990 + 0.891007i −1.66022 − 0.539437i −0.587785 − 0.809017i −0.739883 − 0.739883i 1.23436 − 1.23436i 0.627179 − 3.95985i 0.987688 − 0.156434i 0.0382753 + 0.0278086i 0.995140 − 0.323341i
151.7 −0.453990 + 0.891007i −1.18737 − 0.385799i −0.587785 − 0.809017i −2.87060 − 2.87060i 0.882803 − 0.882803i 0.670591 − 4.23395i 0.987688 − 0.156434i −1.16605 − 0.847186i 3.86094 − 1.25450i
151.8 −0.453990 + 0.891007i −0.705862 − 0.229348i −0.587785 − 0.809017i −1.39843 − 1.39843i 0.524805 − 0.524805i −0.231683 + 1.46279i 0.987688 − 0.156434i −1.98141 − 1.43958i 1.88088 − 0.611135i
151.9 −0.453990 + 0.891007i −0.613980 − 0.199494i −0.587785 − 0.809017i 2.08638 + 2.08638i 0.456492 − 0.456492i −0.256784 + 1.62127i 0.987688 − 0.156434i −2.08988 − 1.51838i −2.80617 + 0.911780i
151.10 −0.453990 + 0.891007i 0.249805 + 0.0811666i −0.587785 − 0.809017i 2.81650 + 2.81650i −0.185729 + 0.185729i 0.506242 − 3.19629i 0.987688 − 0.156434i −2.37124 − 1.72280i −3.78819 + 1.23086i
151.11 −0.453990 + 0.891007i 0.370489 + 0.120379i −0.587785 − 0.809017i −1.16018 − 1.16018i −0.275457 + 0.275457i −0.734923 + 4.64012i 0.987688 − 0.156434i −2.30428 − 1.67416i 1.56044 − 0.507016i
151.12 −0.453990 + 0.891007i 0.685264 + 0.222656i −0.587785 − 0.809017i −0.320774 − 0.320774i −0.509491 + 0.509491i 0.160164 − 1.01124i 0.987688 − 0.156434i −2.00704 − 1.45820i 0.431440 − 0.140183i
151.13 −0.453990 + 0.891007i 0.755476 + 0.245469i −0.587785 − 0.809017i 2.34475 + 2.34475i −0.561693 + 0.561693i −0.338178 + 2.13517i 0.987688 − 0.156434i −1.91656 − 1.39246i −3.15368 + 1.02469i
151.14 −0.453990 + 0.891007i 1.03017 + 0.334723i −0.587785 − 0.809017i −0.196219 − 0.196219i −0.765927 + 0.765927i 0.550369 − 3.47490i 0.987688 − 0.156434i −1.47784 − 1.07371i 0.263914 − 0.0857509i
151.15 −0.453990 + 0.891007i 1.11643 + 0.362749i −0.587785 − 0.809017i −2.02841 − 2.02841i −0.830059 + 0.830059i −0.553100 + 3.49214i 0.987688 − 0.156434i −1.31223 − 0.953391i 2.72821 − 0.886448i
151.16 −0.453990 + 0.891007i 1.99095 + 0.646898i −0.587785 − 0.809017i −1.55803 − 1.55803i −1.48026 + 1.48026i −0.111183 + 0.701982i 0.987688 − 0.156434i 1.11834 + 0.812521i 2.09555 − 0.680884i
151.17 −0.453990 + 0.891007i 2.13067 + 0.692296i −0.587785 − 0.809017i 0.720800 + 0.720800i −1.58414 + 1.58414i 0.697812 − 4.40581i 0.987688 − 0.156434i 1.63342 + 1.18675i −0.969474 + 0.315001i
151.18 −0.453990 + 0.891007i 2.50443 + 0.813740i −0.587785 − 0.809017i 1.11686 + 1.11686i −1.86204 + 1.86204i −0.185053 + 1.16838i 0.987688 − 0.156434i 3.18297 + 2.31256i −1.50218 + 0.488087i
151.19 −0.453990 + 0.891007i 2.82648 + 0.918379i −0.587785 − 0.809017i −2.95513 − 2.95513i −2.10148 + 2.10148i 0.201495 − 1.27219i 0.987688 − 0.156434i 4.71852 + 3.42821i 3.97464 − 1.29144i
151.20 −0.453990 + 0.891007i 2.89804 + 0.941632i −0.587785 − 0.809017i 1.64513 + 1.64513i −2.15468 + 2.15468i −0.323527 + 2.04267i 0.987688 − 0.156434i 5.08494 + 3.69443i −2.21269 + 0.718948i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner
31.f odd 10 1 inner
403.bn even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 806.2.bn.a ✓ 320
13.d odd 4 1 inner 806.2.bn.a ✓ 320
31.f odd 10 1 inner 806.2.bn.a ✓ 320
403.bn even 20 1 inner 806.2.bn.a ✓ 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
806.2.bn.a ✓ 320 1.a even 1 1 trivial
806.2.bn.a ✓ 320 13.d odd 4 1 inner
806.2.bn.a ✓ 320 31.f odd 10 1 inner
806.2.bn.a ✓ 320 403.bn even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(806, [\chi])\).