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Properties

Label 798.2.be.b
Level $798$
Weight $2$
Character orbit 798.be
Analytic conductor $6.372$
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] = [798,2,Mod(493,798)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(798, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.493");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.be (of order \(6\), 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: \(6.37206208130\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 14 q^{3} + 14 q^{4} - 6 q^{5} - 2 q^{7} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 14 q^{3} + 14 q^{4} - 6 q^{5} - 2 q^{7} - 14 q^{9} + 4 q^{10} + 6 q^{11} - 14 q^{12} - 8 q^{13} - 4 q^{14} - 14 q^{16} - 12 q^{17} - 6 q^{19} - 4 q^{21} + 6 q^{23} + 20 q^{25} + 12 q^{26} - 28 q^{27} - 4 q^{28} - 4 q^{30} - 4 q^{31} - 6 q^{33} - 8 q^{34} - 10 q^{35} - 28 q^{36} - 12 q^{37} - 12 q^{38} - 4 q^{39} - 4 q^{40} - 40 q^{41} - 8 q^{42} + 52 q^{43} - 6 q^{44} + 6 q^{45} - 36 q^{46} - 6 q^{47} - 28 q^{48} + 18 q^{49} - 12 q^{51} - 4 q^{52} - 36 q^{53} - 8 q^{56} - 12 q^{57} + 4 q^{58} - 32 q^{59} + 6 q^{60} + 6 q^{61} - 2 q^{63} - 28 q^{64} + 60 q^{65} + 12 q^{67} - 12 q^{68} + 12 q^{69} + 12 q^{70} + 30 q^{73} + 12 q^{74} - 20 q^{75} - 12 q^{76} + 28 q^{77} + 36 q^{79} + 6 q^{80} - 14 q^{81} - 2 q^{84} + 64 q^{85} + 24 q^{86} - 16 q^{89} - 8 q^{90} + 20 q^{91} + 12 q^{92} + 4 q^{93} - 22 q^{95} + 16 q^{97} + 8 q^{98} - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
493.1 −0.866025 + 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i −3.40054 + 1.96330i 1.00000i 0.223574 − 2.63629i 1.00000i −0.500000 − 0.866025i 1.96330 − 3.40054i
493.2 −0.866025 + 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i −2.72411 + 1.57277i 1.00000i −0.413227 + 2.61328i 1.00000i −0.500000 − 0.866025i 1.57277 − 2.72411i
493.3 −0.866025 + 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i −1.23273 + 0.711717i 1.00000i 2.49274 + 0.886713i 1.00000i −0.500000 − 0.866025i 0.711717 − 1.23273i
493.4 −0.866025 + 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i −0.114418 + 0.0660594i 1.00000i −2.62389 − 0.339393i 1.00000i −0.500000 − 0.866025i 0.0660594 − 0.114418i
493.5 −0.866025 + 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i 0.247308 − 0.142783i 1.00000i −0.779658 + 2.52827i 1.00000i −0.500000 − 0.866025i −0.142783 + 0.247308i
493.6 −0.866025 + 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i 1.94282 − 1.12169i 1.00000i −2.03542 − 1.69029i 1.00000i −0.500000 − 0.866025i −1.12169 + 1.94282i
493.7 −0.866025 + 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i 2.04963 − 1.18335i 1.00000i 2.63588 − 0.228315i 1.00000i −0.500000 − 0.866025i −1.18335 + 2.04963i
493.8 0.866025 − 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i −3.04056 + 1.75547i − 1.00000i −0.775540 − 2.52953i − 1.00000i −0.500000 − 0.866025i −1.75547 + 3.04056i
493.9 0.866025 − 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i −2.81074 + 1.62278i − 1.00000i 2.61522 − 0.400774i − 1.00000i −0.500000 − 0.866025i −1.62278 + 2.81074i
493.10 0.866025 − 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i −1.36428 + 0.787665i − 1.00000i −2.58854 + 0.547225i − 1.00000i −0.500000 − 0.866025i −0.787665 + 1.36428i
493.11 0.866025 − 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i 0.612231 − 0.353472i − 1.00000i 0.810021 + 2.51870i − 1.00000i −0.500000 − 0.866025i 0.353472 − 0.612231i
493.12 0.866025 − 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i 1.02150 − 0.589763i − 1.00000i −0.486091 − 2.60071i − 1.00000i −0.500000 − 0.866025i 0.589763 − 1.02150i
493.13 0.866025 − 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i 2.00675 − 1.15860i − 1.00000i 2.55528 − 0.685945i − 1.00000i −0.500000 − 0.866025i 1.15860 − 2.00675i
493.14 0.866025 − 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i 3.80714 − 2.19805i − 1.00000i −2.63035 + 0.285014i − 1.00000i −0.500000 − 0.866025i 2.19805 − 3.80714i
607.1 −0.866025 − 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −3.40054 − 1.96330i − 1.00000i 0.223574 + 2.63629i − 1.00000i −0.500000 + 0.866025i 1.96330 + 3.40054i
607.2 −0.866025 − 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −2.72411 − 1.57277i − 1.00000i −0.413227 − 2.61328i − 1.00000i −0.500000 + 0.866025i 1.57277 + 2.72411i
607.3 −0.866025 − 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −1.23273 − 0.711717i − 1.00000i 2.49274 − 0.886713i − 1.00000i −0.500000 + 0.866025i 0.711717 + 1.23273i
607.4 −0.866025 − 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −0.114418 − 0.0660594i − 1.00000i −2.62389 + 0.339393i − 1.00000i −0.500000 + 0.866025i 0.0660594 + 0.114418i
607.5 −0.866025 − 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i 0.247308 + 0.142783i − 1.00000i −0.779658 − 2.52827i − 1.00000i −0.500000 + 0.866025i −0.142783 − 0.247308i
607.6 −0.866025 − 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i 1.94282 + 1.12169i − 1.00000i −2.03542 + 1.69029i − 1.00000i −0.500000 + 0.866025i −1.12169 − 1.94282i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 493.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.be.b yes 28
7.d odd 6 1 798.2.be.a ✓ 28
19.b odd 2 1 798.2.be.a ✓ 28
133.o even 6 1 inner 798.2.be.b yes 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.be.a ✓ 28 7.d odd 6 1
798.2.be.a ✓ 28 19.b odd 2 1
798.2.be.b yes 28 1.a even 1 1 trivial
798.2.be.b yes 28 133.o even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{14} + 4 T_{13}^{13} - 96 T_{13}^{12} - 352 T_{13}^{11} + 3228 T_{13}^{10} + 10304 T_{13}^{9} + \cdots + 1075456 \) acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\). Copy content Toggle raw display