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Properties

Label 784.2.u.f
Level $784$
Weight $2$
Character orbit 784.u
Analytic conductor $6.260$
Analytic rank $0$
Dimension $42$
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(113,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.u (of order \(7\), degree \(6\), 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: \(42\)
Relative dimension: \(7\) over \(\Q(\zeta_{7})\)
Twist minimal: no (minimal twist has level 392)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 5 q^{3} + 4 q^{5} + q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q - 5 q^{3} + 4 q^{5} + q^{7} - 2 q^{9} - 2 q^{11} + 14 q^{13} + q^{15} + 2 q^{17} + 26 q^{19} - 9 q^{21} + 10 q^{23} - 17 q^{25} - 11 q^{27} + 21 q^{29} + 32 q^{31} + 14 q^{33} - 25 q^{35} + 6 q^{37} + 17 q^{39} + 2 q^{41} + 22 q^{43} - 24 q^{45} - 11 q^{47} + 41 q^{49} - 6 q^{51} - 29 q^{53} + 63 q^{55} + 2 q^{57} + 27 q^{59} + 17 q^{61} - 34 q^{63} + 4 q^{65} - 86 q^{67} + 89 q^{69} - 21 q^{71} + 28 q^{73} - 41 q^{75} - 35 q^{77} + 50 q^{79} - 20 q^{81} + 55 q^{83} - 23 q^{85} - 10 q^{87} - 10 q^{89} + 63 q^{91} - 79 q^{93} + 69 q^{95} - 96 q^{97} - 94 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1 0 −1.84570 + 2.31444i 0 0.101846 − 0.127711i 0 1.88075 + 1.86086i 0 −1.28244 − 5.61873i 0
113.2 0 −1.22867 + 1.54070i 0 1.69839 − 2.12971i 0 −2.43170 − 1.04250i 0 −0.196573 − 0.861244i 0
113.3 0 −0.921166 + 1.15511i 0 −2.23802 + 2.80639i 0 1.64258 − 2.07411i 0 0.181840 + 0.796694i 0
113.4 0 −0.0714866 + 0.0896413i 0 1.38008 − 1.73057i 0 2.33937 + 1.23585i 0 0.664638 + 2.91197i 0
113.5 0 0.162528 − 0.203804i 0 −0.622878 + 0.781064i 0 −2.35862 + 1.19872i 0 0.652442 + 2.85854i 0
113.6 0 1.44889 − 1.81685i 0 0.799185 − 1.00215i 0 0.480566 − 2.60174i 0 −0.534104 − 2.34006i 0
113.7 0 1.67812 − 2.10430i 0 −2.08811 + 2.61840i 0 1.71849 + 2.01166i 0 −0.944420 − 4.13777i 0
225.1 0 −0.659432 − 2.88916i 0 0.423495 + 1.85545i 0 2.15523 + 1.53460i 0 −5.20950 + 2.50876i 0
225.2 0 −0.301716 − 1.32190i 0 −0.856276 − 3.75159i 0 0.648507 − 2.56504i 0 1.04651 − 0.503972i 0
225.3 0 −0.284478 − 1.24638i 0 0.437465 + 1.91666i 0 −1.37017 − 2.26332i 0 1.23038 − 0.592517i 0
225.4 0 −0.220159 − 0.964580i 0 −0.150203 − 0.658081i 0 −0.828814 + 2.51258i 0 1.82096 − 0.876929i 0
225.5 0 0.384316 + 1.68380i 0 −0.419945 − 1.83990i 0 −2.57784 + 0.595623i 0 0.0154284 − 0.00742991i 0
225.6 0 0.414931 + 1.81793i 0 −0.300295 − 1.31568i 0 2.62249 + 0.350041i 0 −0.429806 + 0.206984i 0
225.7 0 0.567507 + 2.48641i 0 0.909831 + 3.98623i 0 −1.44045 + 2.21925i 0 −3.15726 + 1.52046i 0
337.1 0 −2.89639 − 1.39483i 0 −2.06790 − 0.995846i 0 −2.61451 + 0.405358i 0 4.57304 + 5.73441i 0
337.2 0 −2.27666 − 1.09638i 0 2.01217 + 0.969012i 0 2.56866 − 0.634011i 0 2.11065 + 2.64667i 0
337.3 0 −0.805030 − 0.387682i 0 3.56435 + 1.71650i 0 −2.63621 + 0.224522i 0 −1.37269 − 1.72130i 0
337.4 0 −0.636649 − 0.306594i 0 −1.28687 − 0.619724i 0 1.64956 + 2.06857i 0 −1.55915 − 1.95511i 0
337.5 0 0.928435 + 0.447111i 0 −0.587115 − 0.282740i 0 −2.17796 − 1.50216i 0 −1.20839 − 1.51527i 0
337.6 0 2.01889 + 0.972248i 0 2.40901 + 1.16012i 0 2.28083 − 1.34082i 0 1.26019 + 1.58023i 0
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.u.f 42
4.b odd 2 1 392.2.q.b ✓ 42
49.e even 7 1 inner 784.2.u.f 42
196.k odd 14 1 392.2.q.b ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.q.b ✓ 42 4.b odd 2 1
392.2.q.b ✓ 42 196.k odd 14 1
784.2.u.f 42 1.a even 1 1 trivial
784.2.u.f 42 49.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{42} + 5 T_{3}^{41} + 24 T_{3}^{40} + 78 T_{3}^{39} + 291 T_{3}^{38} + 818 T_{3}^{37} + \cdots + 3265249 \) acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\). Copy content Toggle raw display