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Properties

Label 806.2.k.c
Level $806$
Weight $2$
Character orbit 806.k
Analytic conductor $6.436$
Analytic rank $0$
Dimension $28$
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(157,806)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(806, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.k (of order \(5\), degree \(4\), 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: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 7 q^{2} + q^{3} - 7 q^{4} + 2 q^{5} - 6 q^{6} - q^{7} + 7 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 7 q^{2} + q^{3} - 7 q^{4} + 2 q^{5} - 6 q^{6} - q^{7} + 7 q^{8} + 3 q^{10} + 3 q^{11} + q^{12} + 7 q^{13} + q^{14} + 6 q^{15} - 7 q^{16} - 2 q^{17} + 15 q^{18} + 7 q^{19} + 2 q^{20} + 15 q^{21} - 3 q^{22} + 18 q^{23} + 4 q^{24} + 54 q^{25} + 28 q^{26} - 26 q^{27} + 4 q^{28} + 27 q^{29} + 14 q^{30} - 33 q^{31} - 28 q^{32} + 3 q^{33} + 2 q^{34} + 8 q^{35} + 30 q^{36} - 6 q^{37} - 27 q^{38} + 4 q^{39} - 2 q^{40} - 22 q^{41} + 5 q^{42} + q^{43} - 12 q^{44} + 28 q^{45} + 27 q^{46} + 7 q^{47} - 4 q^{48} - 30 q^{49} + q^{50} - 52 q^{51} + 7 q^{52} + q^{53} + 26 q^{54} - 18 q^{55} + 6 q^{56} - 112 q^{57} + 23 q^{58} + 27 q^{59} + 6 q^{60} - 6 q^{61} - 17 q^{62} + 54 q^{63} - 7 q^{64} + 3 q^{65} + 17 q^{66} - 24 q^{67} + 8 q^{68} + 6 q^{69} - 8 q^{70} + 28 q^{71} + 15 q^{72} - 8 q^{73} - 9 q^{74} - 62 q^{75} + 27 q^{76} - 62 q^{77} + q^{78} + 40 q^{79} - 3 q^{80} - 24 q^{81} - 18 q^{82} + 3 q^{83} - 5 q^{84} - q^{85} + 4 q^{86} + 42 q^{87} - 18 q^{88} - 5 q^{89} - 23 q^{90} - 4 q^{91} + 18 q^{92} - 69 q^{93} - 12 q^{94} - 17 q^{95} - q^{96} + 29 q^{97} - 30 q^{98} + 88 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1 0.809017 + 0.587785i −2.18361 + 1.58649i 0.309017 + 0.951057i −3.79341 −2.69909 −0.909471 − 2.79906i −0.309017 + 0.951057i 1.32417 − 4.07536i −3.06893 − 2.22971i
157.2 0.809017 + 0.587785i −2.08051 + 1.51158i 0.309017 + 0.951057i −0.127652 −2.57165 0.948667 + 2.91970i −0.309017 + 0.951057i 1.11660 − 3.43653i −0.103272 − 0.0750317i
157.3 0.809017 + 0.587785i −1.32740 + 0.964410i 0.309017 + 0.951057i 4.34292 −1.64075 −1.00489 − 3.09274i −0.309017 + 0.951057i −0.0951564 + 0.292861i 3.51349 + 2.55270i
157.4 0.809017 + 0.587785i 0.217121 − 0.157748i 0.309017 + 0.951057i −0.464874 0.268376 0.660232 + 2.03199i −0.309017 + 0.951057i −0.904794 + 2.78467i −0.376091 − 0.273246i
157.5 0.809017 + 0.587785i 1.09276 − 0.793935i 0.309017 + 0.951057i −0.305662 1.35072 −1.53347 − 4.71953i −0.309017 + 0.951057i −0.363264 + 1.11801i −0.247286 − 0.179664i
157.6 0.809017 + 0.587785i 1.33132 − 0.967263i 0.309017 + 0.951057i 2.90678 1.64561 0.126739 + 0.390063i −0.309017 + 0.951057i −0.0902270 + 0.277690i 2.35164 + 1.70856i
157.7 0.809017 + 0.587785i 2.64130 − 1.91901i 0.309017 + 0.951057i −0.940065 3.26482 0.903176 + 2.77969i −0.309017 + 0.951057i 2.36678 − 7.28420i −0.760528 − 0.552556i
287.1 −0.309017 + 0.951057i −0.656869 − 2.02163i −0.809017 − 0.587785i −1.73248 2.12567 0.597248 + 0.433926i 0.809017 − 0.587785i −1.22848 + 0.892540i 0.535366 − 1.64769i
287.2 −0.309017 + 0.951057i −0.403893 − 1.24306i −0.809017 − 0.587785i 2.72499 1.30703 −1.30342 − 0.946989i 0.809017 − 0.587785i 1.04499 − 0.759233i −0.842068 + 2.59162i
287.3 −0.309017 + 0.951057i −0.353174 − 1.08696i −0.809017 − 0.587785i −2.87504 1.14289 −1.14822 − 0.834227i 0.809017 − 0.587785i 1.37031 − 0.995586i 0.888437 − 2.73433i
287.4 −0.309017 + 0.951057i 0.00132149 + 0.00406714i −0.809017 − 0.587785i 2.29597 −0.00427644 2.93258 + 2.13064i 0.809017 − 0.587785i 2.42704 − 1.76335i −0.709495 + 2.18360i
287.5 −0.309017 + 0.951057i 0.590112 + 1.81618i −0.809017 − 0.587785i 3.65779 −1.90964 −1.83636 − 1.33419i 0.809017 − 0.587785i −0.523215 + 0.380138i −1.13032 + 3.47877i
287.6 −0.309017 + 0.951057i 0.778878 + 2.39714i −0.809017 − 0.587785i −0.648799 −2.52050 3.22415 + 2.34248i 0.809017 − 0.587785i −2.71258 + 1.97080i 0.200490 − 0.617044i
287.7 −0.309017 + 0.951057i 0.852641 + 2.62416i −0.809017 − 0.587785i −4.04047 −2.75921 −2.15697 − 1.56713i 0.809017 − 0.587785i −3.73217 + 2.71158i 1.24857 − 3.84271i
469.1 −0.309017 − 0.951057i −0.656869 + 2.02163i −0.809017 + 0.587785i −1.73248 2.12567 0.597248 − 0.433926i 0.809017 + 0.587785i −1.22848 − 0.892540i 0.535366 + 1.64769i
469.2 −0.309017 − 0.951057i −0.403893 + 1.24306i −0.809017 + 0.587785i 2.72499 1.30703 −1.30342 + 0.946989i 0.809017 + 0.587785i 1.04499 + 0.759233i −0.842068 − 2.59162i
469.3 −0.309017 − 0.951057i −0.353174 + 1.08696i −0.809017 + 0.587785i −2.87504 1.14289 −1.14822 + 0.834227i 0.809017 + 0.587785i 1.37031 + 0.995586i 0.888437 + 2.73433i
469.4 −0.309017 − 0.951057i 0.00132149 − 0.00406714i −0.809017 + 0.587785i 2.29597 −0.00427644 2.93258 − 2.13064i 0.809017 + 0.587785i 2.42704 + 1.76335i −0.709495 − 2.18360i
469.5 −0.309017 − 0.951057i 0.590112 − 1.81618i −0.809017 + 0.587785i 3.65779 −1.90964 −1.83636 + 1.33419i 0.809017 + 0.587785i −0.523215 − 0.380138i −1.13032 − 3.47877i
469.6 −0.309017 − 0.951057i 0.778878 − 2.39714i −0.809017 + 0.587785i −0.648799 −2.52050 3.22415 − 2.34248i 0.809017 + 0.587785i −2.71258 − 1.97080i 0.200490 + 0.617044i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.7
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 806.2.k.c ✓ 28
31.d even 5 1 inner 806.2.k.c ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
806.2.k.c ✓ 28 1.a even 1 1 trivial
806.2.k.c ✓ 28 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} - T_{3}^{27} + 11 T_{3}^{26} + 6 T_{3}^{25} + 115 T_{3}^{24} + 8 T_{3}^{23} + 1447 T_{3}^{22} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(806, [\chi])\). Copy content Toggle raw display