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Properties

Label 806.2.h.b
Level $806$
Weight $2$
Character orbit 806.h
Analytic conductor $6.436$
Analytic rank $0$
Dimension $38$
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] = [806,2,Mod(87,806)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(806, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.87");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.h (of order \(3\), degree \(2\), 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: \(38\)
Relative dimension: \(19\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 38 q + 19 q^{2} + 2 q^{3} - 19 q^{4} + q^{6} + 2 q^{7} - 38 q^{8} + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q + 19 q^{2} + 2 q^{3} - 19 q^{4} + q^{6} + 2 q^{7} - 38 q^{8} + 48 q^{9} - q^{12} - 2 q^{13} - 2 q^{14} + 13 q^{15} - 19 q^{16} - 11 q^{17} + 24 q^{18} - 5 q^{19} + 6 q^{21} + 10 q^{23} - 2 q^{24} - 25 q^{25} + 8 q^{26} - 28 q^{27} - 4 q^{28} + 6 q^{29} - 13 q^{30} - 5 q^{31} + 19 q^{32} - 9 q^{33} + 11 q^{34} + 8 q^{35} - 24 q^{36} - 26 q^{37} + 5 q^{38} + 21 q^{39} + 3 q^{41} - 6 q^{42} + 13 q^{43} - 9 q^{45} - 10 q^{46} - q^{48} - 9 q^{49} - 50 q^{50} - 5 q^{51} + 10 q^{52} - 9 q^{53} - 14 q^{54} - 12 q^{55} - 2 q^{56} - 22 q^{57} - 6 q^{58} + 20 q^{59} - 26 q^{60} - 6 q^{61} + 8 q^{62} + 8 q^{63} + 38 q^{64} + 9 q^{65} - 18 q^{66} - 6 q^{67} + 22 q^{68} - 15 q^{69} + 16 q^{70} + 26 q^{71} - 48 q^{72} + 4 q^{73} - 13 q^{74} - 27 q^{75} + 10 q^{76} + 16 q^{77} + 36 q^{78} - q^{79} + 118 q^{81} + 6 q^{82} - 9 q^{83} - 12 q^{84} + 22 q^{85} - 13 q^{86} + 16 q^{87} - 6 q^{89} + 9 q^{90} + 18 q^{91} - 20 q^{92} + 2 q^{93} + 33 q^{95} + q^{96} - 30 q^{97} - 18 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
87.1 0.500000 − 0.866025i −3.43623 −0.500000 − 0.866025i −1.88999 + 3.27356i −1.71812 + 2.97586i 1.80660 − 3.12912i −1.00000 8.80769 1.88999 + 3.27356i
87.2 0.500000 − 0.866025i −2.98737 −0.500000 − 0.866025i 1.06219 − 1.83976i −1.49368 + 2.58714i −0.888234 + 1.53847i −1.00000 5.92437 −1.06219 − 1.83976i
87.3 0.500000 − 0.866025i −2.85022 −0.500000 − 0.866025i 0.526984 − 0.912762i −1.42511 + 2.46836i −0.795659 + 1.37812i −1.00000 5.12376 −0.526984 − 0.912762i
87.4 0.500000 − 0.866025i −2.10210 −0.500000 − 0.866025i −0.725803 + 1.25713i −1.05105 + 1.82047i 0.613460 − 1.06254i −1.00000 1.41883 0.725803 + 1.25713i
87.5 0.500000 − 0.866025i −2.09448 −0.500000 − 0.866025i −1.19579 + 2.07117i −1.04724 + 1.81387i −1.99711 + 3.45910i −1.00000 1.38684 1.19579 + 2.07117i
87.6 0.500000 − 0.866025i −1.35261 −0.500000 − 0.866025i 1.84052 − 3.18787i −0.676304 + 1.17139i 0.0204384 − 0.0354004i −1.00000 −1.17045 −1.84052 − 3.18787i
87.7 0.500000 − 0.866025i −0.999537 −0.500000 − 0.866025i −0.605354 + 1.04850i −0.499769 + 0.865625i 0.728503 − 1.26181i −1.00000 −2.00093 0.605354 + 1.04850i
87.8 0.500000 − 0.866025i −0.371467 −0.500000 − 0.866025i 0.685994 − 1.18818i −0.185733 + 0.321700i 1.78194 − 3.08642i −1.00000 −2.86201 −0.685994 − 1.18818i
87.9 0.500000 − 0.866025i −0.369027 −0.500000 − 0.866025i 0.420892 − 0.729007i −0.184513 + 0.319586i −1.03529 + 1.79318i −1.00000 −2.86382 −0.420892 − 0.729007i
87.10 0.500000 − 0.866025i 0.308472 −0.500000 − 0.866025i −1.74629 + 3.02466i 0.154236 − 0.267145i 1.37576 − 2.38288i −1.00000 −2.90484 1.74629 + 3.02466i
87.11 0.500000 − 0.866025i 0.655271 −0.500000 − 0.866025i −0.134987 + 0.233804i 0.327635 − 0.567481i −2.40467 + 4.16501i −1.00000 −2.57062 0.134987 + 0.233804i
87.12 0.500000 − 0.866025i 0.954317 −0.500000 − 0.866025i −1.21484 + 2.10416i 0.477159 − 0.826463i −1.19925 + 2.07717i −1.00000 −2.08928 1.21484 + 2.10416i
87.13 0.500000 − 0.866025i 1.13687 −0.500000 − 0.866025i 1.09873 − 1.90306i 0.568433 − 0.984555i −0.0234872 + 0.0406811i −1.00000 −1.70753 −1.09873 − 1.90306i
87.14 0.500000 − 0.866025i 1.33080 −0.500000 − 0.866025i 1.79403 − 3.10735i 0.665401 − 1.15251i 2.10759 − 3.65046i −1.00000 −1.22896 −1.79403 − 3.10735i
87.15 0.500000 − 0.866025i 1.95009 −0.500000 − 0.866025i 1.14562 − 1.98428i 0.975045 − 1.68883i 0.160217 − 0.277505i −1.00000 0.802851 −1.14562 − 1.98428i
87.16 0.500000 − 0.866025i 2.36557 −0.500000 − 0.866025i −2.13948 + 3.70569i 1.18279 − 2.04865i −0.251065 + 0.434857i −1.00000 2.59594 2.13948 + 3.70569i
87.17 0.500000 − 0.866025i 2.75417 −0.500000 − 0.866025i −0.414408 + 0.717776i 1.37709 − 2.38519i −0.135496 + 0.234686i −1.00000 4.58548 0.414408 + 0.717776i
87.18 0.500000 − 0.866025i 2.82781 −0.500000 − 0.866025i −0.376442 + 0.652016i 1.41390 − 2.44895i 2.48663 − 4.30697i −1.00000 4.99649 0.376442 + 0.652016i
87.19 0.500000 − 0.866025i 3.27966 −0.500000 − 0.866025i 1.86842 − 3.23619i 1.63983 − 2.84027i −1.35088 + 2.33978i −1.00000 7.75620 −1.86842 − 3.23619i
315.1 0.500000 + 0.866025i −3.43623 −0.500000 + 0.866025i −1.88999 − 3.27356i −1.71812 − 2.97586i 1.80660 + 3.12912i −1.00000 8.80769 1.88999 − 3.27356i
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 87.19
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 806.2.h.b yes 38
13.c even 3 1 806.2.f.b ✓ 38
31.c even 3 1 806.2.f.b ✓ 38
403.g even 3 1 inner 806.2.h.b yes 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
806.2.f.b ✓ 38 13.c even 3 1
806.2.f.b ✓ 38 31.c even 3 1
806.2.h.b yes 38 1.a even 1 1 trivial
806.2.h.b yes 38 403.g even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{19} - T_{3}^{18} - 40 T_{3}^{17} + 43 T_{3}^{16} + 643 T_{3}^{15} - 733 T_{3}^{14} - 5332 T_{3}^{13} + \cdots + 821 \) acting on \(S_{2}^{\mathrm{new}}(806, [\chi])\). Copy content Toggle raw display