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Properties

Label 784.2.bp.a
Level $784$
Weight $2$
Character orbit 784.bp
Analytic conductor $6.260$
Analytic rank $0$
Dimension $108$
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(47,784)]
 
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([21, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.bp (of order \(42\), degree \(12\), 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: \(108\)
Relative dimension: \(9\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 108 q - q^{3} - 3 q^{5} + 4 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 108 q - q^{3} - 3 q^{5} + 4 q^{7} + 8 q^{9} - 18 q^{11} + 16 q^{17} - 7 q^{19} + 5 q^{21} - 54 q^{23} - 4 q^{25} + 53 q^{27} - 16 q^{29} - 5 q^{31} - 3 q^{33} + 9 q^{35} + 2 q^{37} + 43 q^{39} - 28 q^{41} - 111 q^{45} + 60 q^{47} - 58 q^{49} + 3 q^{51} - 70 q^{53} - 69 q^{55} + 31 q^{57} - 9 q^{59} - 8 q^{61} - 93 q^{63} - 8 q^{65} + 21 q^{67} - 56 q^{69} + 63 q^{71} - 24 q^{73} - 2 q^{75} - 18 q^{77} - 21 q^{79} + 45 q^{81} - 60 q^{83} + 6 q^{85} + 6 q^{87} + 7 q^{89} + 66 q^{93} - 15 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 0 −1.07795 + 2.74656i 0 0.464656 + 3.08279i 0 1.08303 − 2.41393i 0 −4.18248 − 3.88077i 0
47.2 0 −0.907277 + 2.31170i 0 −0.334776 − 2.22109i 0 2.38229 + 1.15095i 0 −2.32167 − 2.15419i 0
47.3 0 −0.736560 + 1.87672i 0 0.278231 + 1.84594i 0 −2.36774 + 1.18060i 0 −0.780415 − 0.724120i 0
47.4 0 −0.187670 + 0.478176i 0 −0.521072 − 3.45709i 0 −2.39683 + 1.12035i 0 2.00572 + 1.86104i 0
47.5 0 0.0224589 − 0.0572243i 0 −0.283077 − 1.87809i 0 0.576303 − 2.58222i 0 2.19639 + 2.03795i 0
47.6 0 0.182256 − 0.464381i 0 0.225508 + 1.49615i 0 2.40562 + 1.10136i 0 2.01672 + 1.87125i 0
47.7 0 0.445155 − 1.13424i 0 0.420701 + 2.79117i 0 −1.41956 − 2.23267i 0 1.11083 + 1.03070i 0
47.8 0 0.743685 − 1.89488i 0 −0.00920263 − 0.0610555i 0 0.155534 + 2.64118i 0 −0.838341 − 0.777867i 0
47.9 0 1.15056 − 2.93157i 0 −0.408885 − 2.71278i 0 1.96741 − 1.76898i 0 −5.07117 − 4.70536i 0
143.1 0 −2.01411 − 1.86882i 0 0.966431 + 3.13309i 0 −0.244130 + 2.63446i 0 0.339959 + 4.53643i 0
143.2 0 −1.71479 − 1.59109i 0 0.413375 + 1.34013i 0 −0.510015 − 2.59613i 0 0.184737 + 2.46515i 0
143.3 0 −1.21435 − 1.12675i 0 −0.508734 − 1.64928i 0 1.14700 − 2.38420i 0 −0.0191159 − 0.255084i 0
143.4 0 −0.187256 − 0.173748i 0 0.112931 + 0.366114i 0 −2.28327 + 1.33667i 0 −0.219314 − 2.92654i 0
143.5 0 −0.0326407 − 0.0302861i 0 0.563927 + 1.82821i 0 2.59856 + 0.497481i 0 −0.224042 − 2.98963i 0
143.6 0 0.442247 + 0.410345i 0 −1.11720 − 3.62188i 0 1.47725 + 2.19493i 0 −0.196991 − 2.62866i 0
143.7 0 1.48865 + 1.38126i 0 −0.596652 − 1.93430i 0 −1.63551 − 2.07969i 0 0.0839937 + 1.12082i 0
143.8 0 1.86775 + 1.73302i 0 0.806741 + 2.61539i 0 −1.97055 + 1.76548i 0 0.260945 + 3.48207i 0
143.9 0 2.09755 + 1.94624i 0 −0.0920567 − 0.298440i 0 2.48787 − 0.900276i 0 0.387669 + 5.17308i 0
159.1 0 −2.01411 + 1.86882i 0 0.966431 − 3.13309i 0 −0.244130 − 2.63446i 0 0.339959 − 4.53643i 0
159.2 0 −1.71479 + 1.59109i 0 0.413375 − 1.34013i 0 −0.510015 + 2.59613i 0 0.184737 − 2.46515i 0
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
196.p even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.bp.a ✓ 108
4.b odd 2 1 784.2.bp.b yes 108
49.h odd 42 1 784.2.bp.b yes 108
196.p even 42 1 inner 784.2.bp.a ✓ 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
784.2.bp.a ✓ 108 1.a even 1 1 trivial
784.2.bp.a ✓ 108 196.p even 42 1 inner
784.2.bp.b yes 108 4.b odd 2 1
784.2.bp.b yes 108 49.h odd 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{108} + T_{3}^{107} - 17 T_{3}^{106} - 39 T_{3}^{105} + 53 T_{3}^{104} + 407 T_{3}^{103} + \cdots + 3098821560409 \) acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\). Copy content Toggle raw display