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Properties

Label 805.2.f.b
Level $805$
Weight $2$
Character orbit 805.f
Analytic conductor $6.428$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [805,2,Mod(321,805)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(805, 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("805.321");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 805 = 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 805.f (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: \(6.42795736271\)
Analytic rank: \(0\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} + 34 q^{4} + 32 q^{5} + 3 q^{7} - 6 q^{8} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} + 34 q^{4} + 32 q^{5} + 3 q^{7} - 6 q^{8} - 28 q^{9} - 2 q^{10} - 6 q^{14} + 46 q^{16} + 2 q^{17} - 6 q^{18} - 8 q^{19} + 34 q^{20} + 16 q^{21} + 8 q^{23} + 32 q^{25} + 10 q^{28} - 6 q^{29} - 34 q^{32} + 20 q^{33} + 16 q^{34} + 3 q^{35} - 70 q^{36} - 52 q^{38} + 24 q^{39} - 6 q^{40} + 50 q^{42} - 28 q^{45} - 16 q^{46} + q^{49} - 2 q^{50} - 62 q^{56} - 20 q^{58} - 8 q^{61} - 45 q^{63} + 62 q^{64} - 112 q^{66} + 40 q^{68} - 32 q^{69} - 6 q^{70} + 10 q^{71} - 46 q^{72} - 36 q^{76} - 14 q^{77} - 20 q^{78} + 46 q^{80} + 72 q^{81} + 22 q^{83} + 32 q^{84} + 2 q^{85} + 12 q^{89} - 6 q^{90} - 18 q^{91} + 60 q^{92} + 44 q^{93} - 8 q^{95} - 12 q^{97} - 62 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1 −2.76345 − 2.75568i 5.63664 1.00000 7.61517i 2.58354 − 0.570366i −10.0497 −4.59376 −2.76345
321.2 −2.76345 2.75568i 5.63664 1.00000 − 7.61517i 2.58354 + 0.570366i −10.0497 −4.59376 −2.76345
321.3 −2.54257 − 0.210550i 4.46464 1.00000 0.535336i −1.16018 − 2.37781i −6.26651 2.95567 −2.54257
321.4 −2.54257 0.210550i 4.46464 1.00000 − 0.535336i −1.16018 + 2.37781i −6.26651 2.95567 −2.54257
321.5 −2.11248 − 2.08839i 2.46258 1.00000 4.41169i −1.44237 + 2.21801i −0.977187 −1.36137 −2.11248
321.6 −2.11248 2.08839i 2.46258 1.00000 − 4.41169i −1.44237 − 2.21801i −0.977187 −1.36137 −2.11248
321.7 −1.79961 − 0.324087i 1.23860 1.00000 0.583230i 2.55575 − 0.684205i 1.37023 2.89497 −1.79961
321.8 −1.79961 0.324087i 1.23860 1.00000 − 0.583230i 2.55575 + 0.684205i 1.37023 2.89497 −1.79961
321.9 −1.36599 − 1.74192i −0.134075 1.00000 2.37945i −2.62831 − 0.303333i 2.91512 −0.0342922 −1.36599
321.10 −1.36599 1.74192i −0.134075 1.00000 − 2.37945i −2.62831 + 0.303333i 2.91512 −0.0342922 −1.36599
321.11 −1.26309 − 3.32302i −0.404601 1.00000 4.19727i 1.35871 − 2.27022i 3.03723 −8.04243 −1.26309
321.12 −1.26309 3.32302i −0.404601 1.00000 − 4.19727i 1.35871 + 2.27022i 3.03723 −8.04243 −1.26309
321.13 −0.583678 − 1.83942i −1.65932 1.00000 1.07363i −2.41185 + 1.08765i 2.13587 −0.383471 −0.583678
321.14 −0.583678 1.83942i −1.65932 1.00000 − 1.07363i −2.41185 − 1.08765i 2.13587 −0.383471 −0.583678
321.15 −0.406198 − 0.956972i −1.83500 1.00000 0.388721i 0.522086 − 2.59373i 1.55777 2.08420 −0.406198
321.16 −0.406198 0.956972i −1.83500 1.00000 − 0.388721i 0.522086 + 2.59373i 1.55777 2.08420 −0.406198
321.17 0.142628 − 1.38627i −1.97966 1.00000 − 0.197720i 2.57651 + 0.601339i −0.567609 1.07826 0.142628
321.18 0.142628 1.38627i −1.97966 1.00000 0.197720i 2.57651 − 0.601339i −0.567609 1.07826 0.142628
321.19 0.515830 − 2.80728i −1.73392 1.00000 − 1.44808i 0.154402 + 2.64124i −1.92607 −4.88081 0.515830
321.20 0.515830 2.80728i −1.73392 1.00000 1.44808i 0.154402 − 2.64124i −1.92607 −4.88081 0.515830
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 321.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
161.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 805.2.f.b yes 32
7.b odd 2 1 805.2.f.a ✓ 32
23.b odd 2 1 805.2.f.a ✓ 32
161.c even 2 1 inner 805.2.f.b yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.f.a ✓ 32 7.b odd 2 1
805.2.f.a ✓ 32 23.b odd 2 1
805.2.f.b yes 32 1.a even 1 1 trivial
805.2.f.b yes 32 161.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{16} - T_{17}^{15} - 134 T_{17}^{14} + 40 T_{17}^{13} + 6699 T_{17}^{12} + 1583 T_{17}^{11} + \cdots + 1673568 \) acting on \(S_{2}^{\mathrm{new}}(805, [\chi])\). Copy content Toggle raw display