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Properties

Label 798.2.bp.d
Level $798$
Weight $2$
Character orbit 798.bp
Analytic conductor $6.372$
Analytic rank $0$
Dimension $36$
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] = [798,2,Mod(289,798)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(798, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 6, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.bp (of order \(9\), degree \(6\), 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: \(6.37206208130\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 6 q^{5} - 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 6 q^{5} - 18 q^{8} + 3 q^{10} + 3 q^{11} - 18 q^{12} + 27 q^{13} + 12 q^{14} + 3 q^{15} + 6 q^{17} - 18 q^{18} - 12 q^{19} - 6 q^{20} + 12 q^{21} - 3 q^{22} - 9 q^{23} + 6 q^{25} + 9 q^{26} - 18 q^{27} + 9 q^{29} - 6 q^{30} + 6 q^{31} - 3 q^{33} - 3 q^{34} - 6 q^{35} + 21 q^{37} - 9 q^{38} + 9 q^{39} + 3 q^{40} - 9 q^{41} + 51 q^{43} + 6 q^{44} - 6 q^{45} + 6 q^{46} - 15 q^{47} + 24 q^{49} - 9 q^{50} - 3 q^{51} - 27 q^{52} + 30 q^{53} + 27 q^{55} - 9 q^{57} + 18 q^{58} + 6 q^{59} + 3 q^{60} + 30 q^{61} - 24 q^{62} - 18 q^{64} - 84 q^{65} + 6 q^{66} + 42 q^{67} - 36 q^{68} + 6 q^{69} - 24 q^{70} + 6 q^{71} + 66 q^{73} - 12 q^{74} - 9 q^{75} - 6 q^{76} + 9 q^{77} - 27 q^{78} + 12 q^{79} - 6 q^{80} - 18 q^{82} + 30 q^{83} + 36 q^{85} - 39 q^{86} + 18 q^{87} - 6 q^{88} + 66 q^{89} + 3 q^{90} - 9 q^{91} - 18 q^{92} - 24 q^{93} + 18 q^{94} - 57 q^{95} - 18 q^{96} + 45 q^{97} + 6 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 0.766044 − 0.642788i 0.766044 − 0.642788i 0.173648 − 0.984808i −0.530514 − 3.00870i 0.173648 − 0.984808i 1.12528 − 2.39453i −0.500000 − 0.866025i 0.173648 − 0.984808i −2.34035 − 1.96379i
289.2 0.766044 − 0.642788i 0.766044 − 0.642788i 0.173648 − 0.984808i −0.371233 − 2.10537i 0.173648 − 0.984808i −2.48887 + 0.897500i −0.500000 − 0.866025i 0.173648 − 0.984808i −1.63769 − 1.37418i
289.3 0.766044 − 0.642788i 0.766044 − 0.642788i 0.173648 − 0.984808i −0.201991 − 1.14555i 0.173648 − 0.984808i 1.58268 + 2.12017i −0.500000 − 0.866025i 0.173648 − 0.984808i −0.891079 − 0.747704i
289.4 0.766044 − 0.642788i 0.766044 − 0.642788i 0.173648 − 0.984808i 0.0262105 + 0.148647i 0.173648 − 0.984808i 2.53143 − 0.769318i −0.500000 − 0.866025i 0.173648 − 0.984808i 0.115627 + 0.0970226i
289.5 0.766044 − 0.642788i 0.766044 − 0.642788i 0.173648 − 0.984808i 0.263049 + 1.49182i 0.173648 − 0.984808i −1.92115 − 1.81912i −0.500000 − 0.866025i 0.173648 − 0.984808i 1.16043 + 0.973718i
289.6 0.766044 − 0.642788i 0.766044 − 0.642788i 0.173648 − 0.984808i 0.580524 + 3.29232i 0.173648 − 0.984808i 1.81606 + 1.92404i −0.500000 − 0.866025i 0.173648 − 0.984808i 2.56097 + 2.14891i
529.1 0.173648 − 0.984808i 0.173648 − 0.984808i −0.939693 − 0.342020i −3.41192 + 1.24184i −0.939693 − 0.342020i 2.11868 + 1.58467i −0.500000 + 0.866025i −0.939693 − 0.342020i 0.630498 + 3.57573i
529.2 0.173648 − 0.984808i 0.173648 − 0.984808i −0.939693 − 0.342020i −1.94430 + 0.707666i −0.939693 − 0.342020i −2.60792 + 0.445821i −0.500000 + 0.866025i −0.939693 − 0.342020i 0.359292 + 2.03764i
529.3 0.173648 − 0.984808i 0.173648 − 0.984808i −0.939693 − 0.342020i −0.544121 + 0.198044i −0.939693 − 0.342020i −1.16331 + 2.37628i −0.500000 + 0.866025i −0.939693 − 0.342020i 0.100550 + 0.570245i
529.4 0.173648 − 0.984808i 0.173648 − 0.984808i −0.939693 − 0.342020i 0.463649 − 0.168754i −0.939693 − 0.342020i −1.54424 − 2.14833i −0.500000 + 0.866025i −0.939693 − 0.342020i −0.0856788 − 0.485909i
529.5 0.173648 − 0.984808i 0.173648 − 0.984808i −0.939693 − 0.342020i 2.30076 − 0.837408i −0.939693 − 0.342020i 0.935291 − 2.47492i −0.500000 + 0.866025i −0.939693 − 0.342020i −0.425163 − 2.41122i
529.6 0.173648 − 0.984808i 0.173648 − 0.984808i −0.939693 − 0.342020i 2.30958 − 0.840618i −0.939693 − 0.342020i 0.903064 + 2.48686i −0.500000 + 0.866025i −0.939693 − 0.342020i −0.426793 − 2.42046i
541.1 −0.939693 − 0.342020i −0.939693 − 0.342020i 0.766044 + 0.642788i −3.28639 + 2.75761i 0.766044 + 0.642788i −2.32866 − 1.25591i −0.500000 − 0.866025i 0.766044 + 0.642788i 4.03136 − 1.46729i
541.2 −0.939693 − 0.342020i −0.939693 − 0.342020i 0.766044 + 0.642788i −1.48867 + 1.24915i 0.766044 + 0.642788i 0.537468 + 2.59058i −0.500000 − 0.866025i 0.766044 + 0.642788i 1.82613 − 0.664656i
541.3 −0.939693 − 0.342020i −0.939693 − 0.342020i 0.766044 + 0.642788i −1.23889 + 1.03956i 0.766044 + 0.642788i 2.58938 − 0.543256i −0.500000 − 0.866025i 0.766044 + 0.642788i 1.51973 − 0.553136i
541.4 −0.939693 − 0.342020i −0.939693 − 0.342020i 0.766044 + 0.642788i 0.354732 − 0.297656i 0.766044 + 0.642788i −2.43862 − 1.02621i −0.500000 − 0.866025i 0.766044 + 0.642788i −0.435143 + 0.158379i
541.5 −0.939693 − 0.342020i −0.939693 − 0.342020i 0.766044 + 0.642788i 0.865238 − 0.726021i 0.766044 + 0.642788i 2.46727 + 0.955279i −0.500000 − 0.866025i 0.766044 + 0.642788i −1.06137 + 0.386308i
541.6 −0.939693 − 0.342020i −0.939693 − 0.342020i 0.766044 + 0.642788i 2.85430 − 2.39504i 0.766044 + 0.642788i −2.11382 + 1.59115i −0.500000 − 0.866025i 0.766044 + 0.642788i −3.50131 + 1.27437i
613.1 0.766044 + 0.642788i 0.766044 + 0.642788i 0.173648 + 0.984808i −0.530514 + 3.00870i 0.173648 + 0.984808i 1.12528 + 2.39453i −0.500000 + 0.866025i 0.173648 + 0.984808i −2.34035 + 1.96379i
613.2 0.766044 + 0.642788i 0.766044 + 0.642788i 0.173648 + 0.984808i −0.371233 + 2.10537i 0.173648 + 0.984808i −2.48887 − 0.897500i −0.500000 + 0.866025i 0.173648 + 0.984808i −1.63769 + 1.37418i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.u even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.bp.d ✓ 36
7.c even 3 1 798.2.bq.d yes 36
19.e even 9 1 798.2.bq.d yes 36
133.u even 9 1 inner 798.2.bp.d ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.bp.d ✓ 36 1.a even 1 1 trivial
798.2.bp.d ✓ 36 133.u even 9 1 inner
798.2.bq.d yes 36 7.c even 3 1
798.2.bq.d yes 36 19.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{36} + 6 T_{5}^{35} + 15 T_{5}^{34} + 15 T_{5}^{33} + 132 T_{5}^{32} + 453 T_{5}^{31} + \cdots + 3884841 \) acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\). Copy content Toggle raw display