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Properties

Label 798.2.bq.d
Level $798$
Weight $2$
Character orbit 798.bq
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(25,798)]
 
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, 12, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.bq (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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 3 q^{5} - 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 3 q^{5} - 18 q^{8} - 6 q^{10} - 6 q^{11} + 36 q^{12} + 27 q^{13} + 3 q^{15} - 3 q^{17} - 18 q^{18} + 6 q^{19} - 6 q^{20} - 3 q^{22} - 18 q^{23} - 21 q^{25} - 18 q^{26} - 18 q^{27} + 12 q^{28} + 9 q^{29} + 3 q^{30} - 3 q^{31} - 3 q^{33} - 3 q^{34} + 12 q^{35} + 21 q^{37} + 9 q^{38} + 9 q^{39} - 6 q^{40} - 9 q^{41} - 15 q^{42} + 51 q^{43} - 3 q^{44} + 3 q^{45} - 12 q^{46} - 6 q^{47} + 6 q^{49} - 9 q^{50} - 3 q^{51} + 27 q^{52} - 15 q^{53} + 27 q^{55} - 9 q^{57} + 18 q^{58} + 33 q^{59} + 3 q^{60} + 21 q^{61} - 24 q^{62} + 3 q^{63} - 18 q^{64} + 42 q^{65} - 3 q^{66} - 3 q^{67} + 18 q^{68} + 6 q^{69} - 33 q^{70} + 6 q^{71} - 42 q^{73} - 3 q^{74} - 9 q^{75} - 6 q^{76} + 9 q^{77} - 27 q^{78} - 24 q^{79} + 3 q^{80} + 27 q^{82} + 30 q^{83} + 36 q^{85} + 51 q^{86} - 36 q^{87} + 3 q^{88} - 42 q^{89} + 3 q^{90} + 24 q^{91} - 18 q^{92} + 21 q^{93} + 18 q^{94} - 39 q^{95} - 18 q^{96} + 45 q^{97} + 3 q^{98} + 6 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −0.939693 − 0.342020i 0.766044 − 0.642788i 0.766044 + 0.642788i −1.88279 + 1.57985i −0.939693 + 0.342020i 1.70215 − 2.02551i −0.500000 − 0.866025i 0.173648 − 0.984808i 2.30958 − 0.840618i
25.2 −0.939693 − 0.342020i 0.766044 − 0.642788i 0.766044 + 0.642788i −1.87560 + 1.57381i −0.939693 + 0.342020i −2.61099 + 0.427475i −0.500000 − 0.866025i 0.173648 − 0.984808i 2.30076 − 0.837408i
25.3 −0.939693 − 0.342020i 0.766044 − 0.642788i 0.766044 + 0.642788i −0.377970 + 0.317154i −0.939693 + 0.342020i −1.08838 + 2.41152i −0.500000 − 0.866025i 0.173648 − 0.984808i 0.463649 − 0.168754i
25.4 −0.939693 − 0.342020i 0.766044 − 0.642788i 0.766044 + 0.642788i 0.443572 − 0.372201i −0.939693 + 0.342020i 2.63957 − 0.180679i −0.500000 − 0.866025i 0.173648 − 0.984808i −0.544121 + 0.198044i
25.5 −0.939693 − 0.342020i 0.766044 − 0.642788i 0.766044 + 0.642788i 1.58501 − 1.32998i −0.939693 + 0.342020i 1.69005 + 2.03561i −0.500000 − 0.866025i 0.173648 − 0.984808i −1.94430 + 0.707666i
25.6 −0.939693 − 0.342020i 0.766044 − 0.642788i 0.766044 + 0.642788i 2.78142 − 2.33389i −0.939693 + 0.342020i 0.313023 − 2.62717i −0.500000 − 0.866025i 0.173648 − 0.984808i −3.41192 + 1.24184i
403.1 0.173648 + 0.984808i −0.939693 − 0.342020i −0.939693 + 0.342020i −3.14149 − 1.14341i 0.173648 − 0.984808i −2.57430 − 0.610734i −0.500000 − 0.866025i 0.766044 + 0.642788i 0.580524 − 3.29232i
403.2 0.173648 + 0.984808i −0.939693 − 0.342020i −0.939693 + 0.342020i −1.42348 − 0.518104i 0.173648 − 0.984808i 2.53598 + 0.754206i −0.500000 − 0.866025i 0.766044 + 0.642788i 0.263049 − 1.49182i
403.3 0.173648 + 0.984808i −0.939693 − 0.342020i −0.939693 + 0.342020i −0.141838 − 0.0516247i 0.173648 − 0.984808i −0.599468 − 2.57694i −0.500000 − 0.866025i 0.766044 + 0.642788i 0.0262105 − 0.148647i
403.4 0.173648 + 0.984808i −0.939693 − 0.342020i −0.939693 + 0.342020i 1.09307 + 0.397845i 0.173648 − 0.984808i −2.62746 − 0.310561i −0.500000 − 0.866025i 0.766044 + 0.642788i −0.201991 + 1.14555i
403.5 0.173648 + 0.984808i −0.939693 − 0.342020i −0.939693 + 0.342020i 2.00892 + 0.731187i 0.173648 − 0.984808i 0.467179 + 2.60418i −0.500000 − 0.866025i 0.766044 + 0.642788i −0.371233 + 2.10537i
403.6 0.173648 + 0.984808i −0.939693 − 0.342020i −0.939693 + 0.342020i 2.87086 + 1.04491i 0.173648 − 0.984808i 1.51108 − 2.17178i −0.500000 − 0.866025i 0.766044 + 0.642788i −0.530514 + 3.00870i
415.1 −0.939693 + 0.342020i 0.766044 + 0.642788i 0.766044 − 0.642788i −1.88279 − 1.57985i −0.939693 − 0.342020i 1.70215 + 2.02551i −0.500000 + 0.866025i 0.173648 + 0.984808i 2.30958 + 0.840618i
415.2 −0.939693 + 0.342020i 0.766044 + 0.642788i 0.766044 − 0.642788i −1.87560 − 1.57381i −0.939693 − 0.342020i −2.61099 − 0.427475i −0.500000 + 0.866025i 0.173648 + 0.984808i 2.30076 + 0.837408i
415.3 −0.939693 + 0.342020i 0.766044 + 0.642788i 0.766044 − 0.642788i −0.377970 − 0.317154i −0.939693 − 0.342020i −1.08838 − 2.41152i −0.500000 + 0.866025i 0.173648 + 0.984808i 0.463649 + 0.168754i
415.4 −0.939693 + 0.342020i 0.766044 + 0.642788i 0.766044 − 0.642788i 0.443572 + 0.372201i −0.939693 − 0.342020i 2.63957 + 0.180679i −0.500000 + 0.866025i 0.173648 + 0.984808i −0.544121 − 0.198044i
415.5 −0.939693 + 0.342020i 0.766044 + 0.642788i 0.766044 − 0.642788i 1.58501 + 1.32998i −0.939693 − 0.342020i 1.69005 − 2.03561i −0.500000 + 0.866025i 0.173648 + 0.984808i −1.94430 − 0.707666i
415.6 −0.939693 + 0.342020i 0.766044 + 0.642788i 0.766044 − 0.642788i 2.78142 + 2.33389i −0.939693 − 0.342020i 0.313023 + 2.62717i −0.500000 + 0.866025i 0.173648 + 0.984808i −3.41192 − 1.24184i
499.1 0.173648 − 0.984808i −0.939693 + 0.342020i −0.939693 − 0.342020i −3.14149 + 1.14341i 0.173648 + 0.984808i −2.57430 + 0.610734i −0.500000 + 0.866025i 0.766044 − 0.642788i 0.580524 + 3.29232i
499.2 0.173648 − 0.984808i −0.939693 + 0.342020i −0.939693 − 0.342020i −1.42348 + 0.518104i 0.173648 + 0.984808i 2.53598 − 0.754206i −0.500000 + 0.866025i 0.766044 − 0.642788i 0.263049 + 1.49182i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.6
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{36} - 3 T_{5}^{35} + 15 T_{5}^{34} - 39 T_{5}^{33} + 6 T_{5}^{32} + 480 T_{5}^{31} + \cdots + 3884841 \) acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\). Copy content Toggle raw display