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Properties

Label 798.2.bf.a
Level $798$
Weight $2$
Character orbit 798.bf
Analytic conductor $6.372$
Analytic rank $0$
Dimension $52$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [798,2,Mod(569,798)]
 
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([3, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.569");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.bf (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: \(52\)
Relative dimension: \(26\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 26 q^{2} - 26 q^{4} - 8 q^{7} + 52 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q - 26 q^{2} - 26 q^{4} - 8 q^{7} + 52 q^{8} + 4 q^{14} + 8 q^{15} - 26 q^{16} + 2 q^{19} + 2 q^{21} + 18 q^{25} + 6 q^{27} + 4 q^{28} - 8 q^{29} - 4 q^{30} - 26 q^{32} + 8 q^{33} + 2 q^{38} + q^{39} + 16 q^{41} + 11 q^{42} - 8 q^{43} + 14 q^{45} + 4 q^{49} - 36 q^{50} - 19 q^{51} - 24 q^{53} - 3 q^{54} + 32 q^{55} - 8 q^{56} - 4 q^{57} + 4 q^{58} + 32 q^{59} - 4 q^{60} - 20 q^{61} - 7 q^{63} + 52 q^{64} - 44 q^{65} + 8 q^{66} + 62 q^{69} - 24 q^{71} - 32 q^{73} + 16 q^{75} - 4 q^{76} - 2 q^{78} - 28 q^{81} - 8 q^{82} - 13 q^{84} + 32 q^{85} + 4 q^{86} + 34 q^{87} - 36 q^{89} - 28 q^{90} + 20 q^{93} + 30 q^{95} - 20 q^{98} - 116 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
569.1 −0.500000 + 0.866025i −1.70602 + 0.299137i −0.500000 − 0.866025i 3.21025 + 1.85344i 0.593952 − 1.62703i −0.273695 + 2.63156i 1.00000 2.82103 − 1.02067i −3.21025 + 1.85344i
569.2 −0.500000 + 0.866025i −1.69954 + 0.334025i −0.500000 − 0.866025i −3.47049 − 2.00369i 0.560495 − 1.63885i −2.24845 − 1.39444i 1.00000 2.77685 − 1.13538i 3.47049 − 2.00369i
569.3 −0.500000 + 0.866025i −1.65690 − 0.504659i −0.500000 − 0.866025i −0.0260093 − 0.0150165i 1.26550 − 1.18259i −1.30377 − 2.30221i 1.00000 2.49064 + 1.67234i 0.0260093 − 0.0150165i
569.4 −0.500000 + 0.866025i −1.61814 − 0.617744i −0.500000 − 0.866025i 1.14754 + 0.662531i 1.34405 − 1.09248i 2.24492 − 1.40012i 1.00000 2.23679 + 1.99920i −1.14754 + 0.662531i
569.5 −0.500000 + 0.866025i −1.38866 + 1.03519i −0.500000 − 0.866025i −2.20976 − 1.27580i −0.202173 − 1.72021i 2.59909 − 0.494713i 1.00000 0.856755 − 2.87506i 2.20976 − 1.27580i
569.6 −0.500000 + 0.866025i −1.21998 + 1.22949i −0.500000 − 0.866025i 0.812231 + 0.468942i −0.454783 − 1.67128i 1.87613 + 1.86551i 1.00000 −0.0233052 − 2.99991i −0.812231 + 0.468942i
569.7 −0.500000 + 0.866025i −1.14247 − 1.30183i −0.500000 − 0.866025i 1.60706 + 0.927835i 1.69865 − 0.338490i −2.30493 + 1.29897i 1.00000 −0.389534 + 2.97460i −1.60706 + 0.927835i
569.8 −0.500000 + 0.866025i −1.08023 − 1.35392i −0.500000 − 0.866025i −2.74011 − 1.58200i 1.71265 − 0.258544i 1.89471 + 1.84664i 1.00000 −0.666214 + 2.92509i 2.74011 − 1.58200i
569.9 −0.500000 + 0.866025i −1.02236 + 1.39813i −0.500000 − 0.866025i 0.295801 + 0.170781i −0.699635 − 1.58446i −0.368523 − 2.61996i 1.00000 −0.909542 − 2.85880i −0.295801 + 0.170781i
569.10 −0.500000 + 0.866025i −0.819227 + 1.52606i −0.500000 − 0.866025i 1.45289 + 0.838824i −0.911995 − 1.47250i −2.58974 + 0.541515i 1.00000 −1.65773 − 2.50038i −1.45289 + 0.838824i
569.11 −0.500000 + 0.866025i −0.632417 − 1.61247i −0.500000 − 0.866025i 2.74011 + 1.58200i 1.71265 + 0.258544i 1.89471 + 1.84664i 1.00000 −2.20010 + 2.03950i −2.74011 + 1.58200i
569.12 −0.500000 + 0.866025i −0.556186 − 1.64032i −0.500000 − 0.866025i −1.60706 − 0.927835i 1.69865 + 0.338490i −2.30493 + 1.29897i 1.00000 −2.38131 + 1.82465i 1.60706 − 0.927835i
569.13 −0.500000 + 0.866025i 0.156272 + 1.72499i −0.500000 − 0.866025i −3.05234 − 1.76227i −1.57202 − 0.727158i −2.63825 − 0.199029i 1.00000 −2.95116 + 0.539133i 3.05234 − 1.76227i
569.14 −0.500000 + 0.866025i 0.255638 + 1.71308i −0.500000 − 0.866025i −0.957576 − 0.552857i −1.61139 − 0.635152i 0.0412224 + 2.64543i 1.00000 −2.86930 + 0.875856i 0.957576 − 0.552857i
569.15 −0.500000 + 0.866025i 0.274091 − 1.71023i −0.500000 − 0.866025i −1.14754 − 0.662531i 1.34405 + 1.09248i 2.24492 − 1.40012i 1.00000 −2.84975 − 0.937515i 1.14754 − 0.662531i
569.16 −0.500000 + 0.866025i 0.391403 − 1.68725i −0.500000 − 0.866025i 0.0260093 + 0.0150165i 1.26550 + 1.18259i −1.30377 − 2.30221i 1.00000 −2.69361 − 1.32079i −0.0260093 + 0.0150165i
569.17 −0.500000 + 0.866025i 0.711087 + 1.57935i −0.500000 − 0.866025i 1.94081 + 1.12053i −1.72330 − 0.173858i 1.07130 − 2.41915i 1.00000 −1.98871 + 2.24611i −1.94081 + 1.12053i
569.18 −0.500000 + 0.866025i 1.01222 + 1.40550i −0.500000 − 0.866025i −1.94081 − 1.12053i −1.72330 + 0.173858i 1.07130 − 2.41915i 1.00000 −0.950835 + 2.84533i 1.94081 − 1.12053i
569.19 −0.500000 + 0.866025i 1.11207 − 1.32789i −0.500000 − 0.866025i −3.21025 − 1.85344i 0.593952 + 1.62703i −0.273695 + 2.63156i 1.00000 −0.526592 − 2.95342i 3.21025 − 1.85344i
569.20 −0.500000 + 0.866025i 1.13904 − 1.30483i −0.500000 − 0.866025i 3.47049 + 2.00369i 0.560495 + 1.63885i −2.24845 − 1.39444i 1.00000 −0.405164 − 2.97251i −3.47049 + 2.00369i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 569.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
57.d even 2 1 inner
399.w 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.bf.a ✓ 52
3.b odd 2 1 798.2.bf.b yes 52
7.c even 3 1 inner 798.2.bf.a ✓ 52
19.b odd 2 1 798.2.bf.b yes 52
21.h odd 6 1 798.2.bf.b yes 52
57.d even 2 1 inner 798.2.bf.a ✓ 52
133.r odd 6 1 798.2.bf.b yes 52
399.w even 6 1 inner 798.2.bf.a ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.bf.a ✓ 52 1.a even 1 1 trivial
798.2.bf.a ✓ 52 7.c even 3 1 inner
798.2.bf.a ✓ 52 57.d even 2 1 inner
798.2.bf.a ✓ 52 399.w even 6 1 inner
798.2.bf.b yes 52 3.b odd 2 1
798.2.bf.b yes 52 19.b odd 2 1
798.2.bf.b yes 52 21.h odd 6 1
798.2.bf.b yes 52 133.r odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{29}^{13} + 2 T_{29}^{12} - 190 T_{29}^{11} - 665 T_{29}^{10} + 12748 T_{29}^{9} + 64315 T_{29}^{8} + \cdots - 2926206 \) acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\). Copy content Toggle raw display