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Properties

Label 784.2.bg.f
Level $784$
Weight $2$
Character orbit 784.bg
Analytic conductor $6.260$
Analytic rank $0$
Dimension $84$
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] = [784,2,Mod(65,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 0, 20]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.bg (of order \(21\), degree \(12\), not 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: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(7\) over \(\Q(\zeta_{21})\)
Twist minimal: no (minimal twist has level 392)
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q + 8 q^{3} - q^{5} - 4 q^{7} + 41 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q + 8 q^{3} - q^{5} - 4 q^{7} + 41 q^{9} - q^{11} - 2 q^{13} - q^{15} - 5 q^{17} + 13 q^{19} + 12 q^{21} + 5 q^{23} + 2 q^{25} - 4 q^{27} - 21 q^{29} - 23 q^{31} - 17 q^{33} + 22 q^{35} - 60 q^{37} + 61 q^{39} + 4 q^{41} - 22 q^{43} + 30 q^{45} - 82 q^{47} - 2 q^{49} - 3 q^{51} + 38 q^{53} - 69 q^{55} - 2 q^{57} + 6 q^{59} - 26 q^{61} + 37 q^{63} + 2 q^{65} - 43 q^{67} + 7 q^{69} + 21 q^{71} - 7 q^{73} + 80 q^{75} - 85 q^{77} + 25 q^{79} - 10 q^{81} + 17 q^{83} + 23 q^{85} - 47 q^{87} + 7 q^{89} + 3 q^{91} - 50 q^{93} + 129 q^{95} - 150 q^{97} + 94 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1 0 −0.235298 − 3.13983i 0 2.16236 − 1.47427i 0 1.78841 − 1.94977i 0 −6.83670 + 1.03047i 0
65.2 0 −0.137858 − 1.83959i 0 −2.15672 + 1.47042i 0 0.660063 − 2.56209i 0 −0.398603 + 0.0600797i 0
65.3 0 −0.100350 − 1.33908i 0 −0.499334 + 0.340440i 0 −2.50671 + 0.846419i 0 1.18344 − 0.178375i 0
65.4 0 0.0247290 + 0.329985i 0 −1.09829 + 0.748802i 0 2.39796 + 1.11795i 0 2.85821 − 0.430806i 0
65.5 0 0.0337386 + 0.450211i 0 3.66755 − 2.50050i 0 1.88168 + 1.85991i 0 2.76494 − 0.416748i 0
65.6 0 0.118310 + 1.57873i 0 1.24081 − 0.845971i 0 −2.07415 − 1.64252i 0 0.488092 − 0.0735680i 0
65.7 0 0.173239 + 2.31172i 0 −3.33484 + 2.27366i 0 −1.12122 + 2.39643i 0 −2.34753 + 0.353833i 0
81.1 0 −2.75949 − 0.851191i 0 0.0299434 − 0.0277834i 0 −2.34758 − 1.22019i 0 4.41156 + 3.00775i 0
81.2 0 −2.18142 − 0.672879i 0 1.26643 − 1.17507i 0 1.64625 + 2.07119i 0 1.82711 + 1.24570i 0
81.3 0 −0.164329 − 0.0506889i 0 −0.251564 + 0.233418i 0 −2.13516 − 1.56240i 0 −2.45428 − 1.67330i 0
81.4 0 0.146098 + 0.0450653i 0 −1.11532 + 1.03487i 0 1.93225 − 1.80733i 0 −2.45940 − 1.67679i 0
81.5 0 0.856325 + 0.264141i 0 2.06130 − 1.91261i 0 −0.234062 + 2.63538i 0 −1.81519 − 1.23758i 0
81.6 0 2.48004 + 0.764991i 0 −2.01731 + 1.87179i 0 1.91899 + 1.82140i 0 3.08666 + 2.10445i 0
81.7 0 3.02375 + 0.932703i 0 2.70398 − 2.50893i 0 0.166671 − 2.64050i 0 5.79441 + 3.95056i 0
193.1 0 −0.235298 + 3.13983i 0 2.16236 + 1.47427i 0 1.78841 + 1.94977i 0 −6.83670 − 1.03047i 0
193.2 0 −0.137858 + 1.83959i 0 −2.15672 − 1.47042i 0 0.660063 + 2.56209i 0 −0.398603 − 0.0600797i 0
193.3 0 −0.100350 + 1.33908i 0 −0.499334 − 0.340440i 0 −2.50671 − 0.846419i 0 1.18344 + 0.178375i 0
193.4 0 0.0247290 − 0.329985i 0 −1.09829 − 0.748802i 0 2.39796 − 1.11795i 0 2.85821 + 0.430806i 0
193.5 0 0.0337386 − 0.450211i 0 3.66755 + 2.50050i 0 1.88168 − 1.85991i 0 2.76494 + 0.416748i 0
193.6 0 0.118310 − 1.57873i 0 1.24081 + 0.845971i 0 −2.07415 + 1.64252i 0 0.488092 + 0.0735680i 0
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.bg.f 84
4.b odd 2 1 392.2.y.a ✓ 84
49.g even 21 1 inner 784.2.bg.f 84
196.o odd 42 1 392.2.y.a ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.y.a ✓ 84 4.b odd 2 1
392.2.y.a ✓ 84 196.o odd 42 1
784.2.bg.f 84 1.a even 1 1 trivial
784.2.bg.f 84 49.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{84} - 8 T_{3}^{83} + T_{3}^{82} + 168 T_{3}^{81} - 426 T_{3}^{80} - 1016 T_{3}^{79} + \cdots + 782376841 \) acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\). Copy content Toggle raw display