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Properties

Label 806.2.f.b
Level $806$
Weight $2$
Character orbit 806.f
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(191,806)]
 
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, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.f (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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q + 19 q^{2} - q^{3} - 19 q^{4} - 2 q^{6} - 4 q^{7} - 38 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q + 19 q^{2} - q^{3} - 19 q^{4} - 2 q^{6} - 4 q^{7} - 38 q^{8} - 24 q^{9} - q^{12} - 8 q^{13} - 2 q^{14} + 13 q^{15} - 19 q^{16} + 22 q^{17} + 24 q^{18} + 10 q^{19} + 6 q^{21} + 10 q^{23} + q^{24} - 25 q^{25} - 10 q^{26} - 28 q^{27} + 2 q^{28} + 6 q^{29} - 13 q^{30} - 5 q^{31} + 19 q^{32} - 9 q^{33} + 11 q^{34} + 8 q^{35} + 48 q^{36} + 13 q^{37} + 5 q^{38} + 21 q^{39} - 6 q^{41} + 12 q^{42} - 26 q^{43} + 18 q^{45} - 10 q^{46} + 2 q^{48} + 18 q^{49} + 25 q^{50} - 5 q^{51} - 2 q^{52} - 9 q^{53} - 14 q^{54} + 6 q^{55} + 4 q^{56} - 22 q^{57} - 6 q^{58} - 40 q^{59} - 26 q^{60} - 6 q^{61} - 13 q^{62} + 8 q^{63} + 38 q^{64} - 33 q^{65} - 18 q^{66} + 12 q^{67} - 11 q^{68} + 30 q^{69} + 16 q^{70} - 13 q^{71} + 24 q^{72} + 4 q^{73} + 26 q^{74} + 54 q^{75} - 5 q^{76} + 16 q^{77} + 36 q^{78} - q^{79} - 59 q^{81} - 3 q^{82} - 9 q^{83} + 6 q^{84} + 22 q^{85} - 13 q^{86} - 32 q^{87} - 6 q^{89} + 9 q^{90} + 18 q^{91} - 20 q^{92} - 13 q^{93} + 33 q^{95} + q^{96} - 30 q^{97} + 9 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1 0.500000 − 0.866025i −1.63983 − 2.84027i −0.500000 − 0.866025i 1.86842 + 3.23619i −3.27966 2.70175 −1.00000 −3.87810 + 6.71706i 3.73683
191.2 0.500000 − 0.866025i −1.41390 − 2.44895i −0.500000 − 0.866025i −0.376442 − 0.652016i −2.82781 −4.97326 −1.00000 −2.49825 + 4.32709i −0.752884
191.3 0.500000 − 0.866025i −1.37709 − 2.38519i −0.500000 − 0.866025i −0.414408 − 0.717776i −2.75417 0.270992 −1.00000 −2.29274 + 3.97114i −0.828816
191.4 0.500000 − 0.866025i −1.18279 − 2.04865i −0.500000 − 0.866025i −2.13948 − 3.70569i −2.36557 0.502130 −1.00000 −1.29797 + 2.24815i −4.27896
191.5 0.500000 − 0.866025i −0.975045 − 1.68883i −0.500000 − 0.866025i 1.14562 + 1.98428i −1.95009 −0.320435 −1.00000 −0.401426 + 0.695290i 2.29125
191.6 0.500000 − 0.866025i −0.665401 − 1.15251i −0.500000 − 0.866025i 1.79403 + 3.10735i −1.33080 −4.21519 −1.00000 0.614482 − 1.06431i 3.58805
191.7 0.500000 − 0.866025i −0.568433 − 0.984555i −0.500000 − 0.866025i 1.09873 + 1.90306i −1.13687 0.0469745 −1.00000 0.853767 − 1.47877i 2.19747
191.8 0.500000 − 0.866025i −0.477159 − 0.826463i −0.500000 − 0.866025i −1.21484 − 2.10416i −0.954317 2.39851 −1.00000 1.04464 − 1.80937i −2.42967
191.9 0.500000 − 0.866025i −0.327635 − 0.567481i −0.500000 − 0.866025i −0.134987 − 0.233804i −0.655271 4.80934 −1.00000 1.28531 − 2.22622i −0.269974
191.10 0.500000 − 0.866025i −0.154236 − 0.267145i −0.500000 − 0.866025i −1.74629 − 3.02466i −0.308472 −2.75152 −1.00000 1.45242 − 2.51567i −3.49258
191.11 0.500000 − 0.866025i 0.184513 + 0.319586i −0.500000 − 0.866025i 0.420892 + 0.729007i 0.369027 2.07058 −1.00000 1.43191 − 2.48014i 0.841784
191.12 0.500000 − 0.866025i 0.185733 + 0.321700i −0.500000 − 0.866025i 0.685994 + 1.18818i 0.371467 −3.56389 −1.00000 1.43101 − 2.47858i 1.37199
191.13 0.500000 − 0.866025i 0.499769 + 0.865625i −0.500000 − 0.866025i −0.605354 − 1.04850i 0.999537 −1.45701 −1.00000 1.00046 − 1.73285i −1.21071
191.14 0.500000 − 0.866025i 0.676304 + 1.17139i −0.500000 − 0.866025i 1.84052 + 3.18787i 1.35261 −0.0408768 −1.00000 0.585226 − 1.01364i 3.68104
191.15 0.500000 − 0.866025i 1.04724 + 1.81387i −0.500000 − 0.866025i −1.19579 − 2.07117i 2.09448 3.99422 −1.00000 −0.693421 + 1.20104i −2.39158
191.16 0.500000 − 0.866025i 1.05105 + 1.82047i −0.500000 − 0.866025i −0.725803 − 1.25713i 2.10210 −1.22692 −1.00000 −0.709417 + 1.22875i −1.45161
191.17 0.500000 − 0.866025i 1.42511 + 2.46836i −0.500000 − 0.866025i 0.526984 + 0.912762i 2.85022 1.59132 −1.00000 −2.56188 + 4.43731i 1.05397
191.18 0.500000 − 0.866025i 1.49368 + 2.58714i −0.500000 − 0.866025i 1.06219 + 1.83976i 2.98737 1.77647 −1.00000 −2.96218 + 5.13065i 2.12438
191.19 0.500000 − 0.866025i 1.71812 + 2.97586i −0.500000 − 0.866025i −1.88999 − 3.27356i 3.43623 −3.61319 −1.00000 −4.40384 + 7.62768i −3.77998
211.1 0.500000 + 0.866025i −1.63983 + 2.84027i −0.500000 + 0.866025i 1.86842 − 3.23619i −3.27966 2.70175 −1.00000 −3.87810 − 6.71706i 3.73683
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.19
Significant digits:
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Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{38} + T_{3}^{37} + 41 T_{3}^{36} + 46 T_{3}^{35} + 1000 T_{3}^{34} + 1167 T_{3}^{33} + \cdots + 674041 \) acting on \(S_{2}^{\mathrm{new}}(806, [\chi])\). Copy content Toggle raw display