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Properties

Label 784.2.bg.c
Level $784$
Weight $2$
Character orbit 784.bg
Analytic conductor $6.260$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [784,2,Mod(65,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [48,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(4\) over \(\Q(\zeta_{21})\)
Twist minimal: no (minimal twist has level 49)
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 14 q^{3} - 14 q^{5} + 14 q^{7} + 6 q^{9} + 3 q^{11} - 14 q^{13} + 12 q^{15} - 7 q^{17} - 21 q^{19} - 14 q^{21} - 15 q^{23} - 4 q^{25} - 7 q^{27} + 12 q^{29} - 35 q^{31} - 14 q^{33} + 15 q^{37} + 7 q^{39}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 −0.218016 − 2.90922i 0 −0.605902 + 0.413097i 0 0.310399 + 2.62748i 0 −5.44954 + 0.821385i 0
65.2 0 0.0483362 + 0.645002i 0 1.29033 − 0.879733i 0 2.25101 − 1.39030i 0 2.55280 − 0.384773i 0
65.3 0 0.0823774 + 1.09925i 0 −2.09852 + 1.43075i 0 0.301609 − 2.62850i 0 1.76493 − 0.266020i 0
65.4 0 0.173202 + 2.31121i 0 −0.392378 + 0.267519i 0 1.60453 + 2.10369i 0 −2.34522 + 0.353485i 0
81.1 0 −1.33832 − 0.412818i 0 −1.66779 + 1.54748i 0 1.77774 − 1.95950i 0 −0.858025 − 0.584992i 0
81.2 0 0.269714 + 0.0831957i 0 −1.49036 + 1.38285i 0 0.607511 + 2.57506i 0 −2.41289 − 1.64508i 0
81.3 0 0.776689 + 0.239577i 0 1.80872 − 1.67825i 0 −1.97351 − 1.76217i 0 −1.93287 − 1.31781i 0
81.4 0 3.07373 + 0.948121i 0 0.291074 − 0.270077i 0 −2.43713 + 1.02976i 0 6.07018 + 4.13858i 0
193.1 0 −0.218016 + 2.90922i 0 −0.605902 − 0.413097i 0 0.310399 − 2.62748i 0 −5.44954 − 0.821385i 0
193.2 0 0.0483362 − 0.645002i 0 1.29033 + 0.879733i 0 2.25101 + 1.39030i 0 2.55280 + 0.384773i 0
193.3 0 0.0823774 − 1.09925i 0 −2.09852 − 1.43075i 0 0.301609 + 2.62850i 0 1.76493 + 0.266020i 0
193.4 0 0.173202 − 2.31121i 0 −0.392378 − 0.267519i 0 1.60453 − 2.10369i 0 −2.34522 − 0.353485i 0
289.1 0 −1.64786 − 1.12349i 0 0.0348203 − 0.464645i 0 0.363905 + 2.62061i 0 0.357192 + 0.910110i 0
289.2 0 −0.489949 − 0.334042i 0 −0.260520 + 3.47640i 0 −1.30029 − 2.30418i 0 −0.967557 − 2.46529i 0
289.3 0 1.92585 + 1.31302i 0 −0.264371 + 3.52778i 0 −0.675443 + 2.55808i 0 0.888840 + 2.26473i 0
289.4 0 2.40355 + 1.63871i 0 0.186621 − 2.49028i 0 2.47068 − 0.946431i 0 1.99564 + 5.08481i 0
305.1 0 −2.78061 + 0.419109i 0 −0.567937 + 1.44708i 0 2.62161 + 0.356599i 0 4.68940 − 1.44649i 0
305.2 0 −0.223157 + 0.0336355i 0 0.711168 − 1.81203i 0 2.04196 + 1.68237i 0 −2.81805 + 0.869254i 0
305.3 0 1.96135 − 0.295625i 0 −1.11243 + 2.83442i 0 −2.44418 + 1.01290i 0 0.892763 − 0.275381i 0
305.4 0 2.00916 − 0.302832i 0 −0.0830372 + 0.211575i 0 −1.07894 − 2.41576i 0 1.07830 − 0.332612i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.4
Significant digits:
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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.c 48
4.b odd 2 1 49.2.g.a ✓ 48
12.b even 2 1 441.2.bb.d 48
28.d even 2 1 343.2.g.g 48
28.f even 6 1 343.2.e.c 48
28.f even 6 1 343.2.g.h 48
28.g odd 6 1 343.2.e.d 48
28.g odd 6 1 343.2.g.i 48
49.g even 21 1 inner 784.2.bg.c 48
196.j even 14 1 343.2.g.h 48
196.k odd 14 1 343.2.g.i 48
196.o odd 42 1 49.2.g.a ✓ 48
196.o odd 42 1 343.2.e.d 48
196.o odd 42 1 2401.2.a.h 24
196.p even 42 1 343.2.e.c 48
196.p even 42 1 343.2.g.g 48
196.p even 42 1 2401.2.a.i 24
588.bb even 42 1 441.2.bb.d 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.g.a ✓ 48 4.b odd 2 1
49.2.g.a ✓ 48 196.o odd 42 1
343.2.e.c 48 28.f even 6 1
343.2.e.c 48 196.p even 42 1
343.2.e.d 48 28.g odd 6 1
343.2.e.d 48 196.o odd 42 1
343.2.g.g 48 28.d even 2 1
343.2.g.g 48 196.p even 42 1
343.2.g.h 48 28.f even 6 1
343.2.g.h 48 196.j even 14 1
343.2.g.i 48 28.g odd 6 1
343.2.g.i 48 196.k odd 14 1
441.2.bb.d 48 12.b even 2 1
441.2.bb.d 48 588.bb even 42 1
784.2.bg.c 48 1.a even 1 1 trivial
784.2.bg.c 48 49.g even 21 1 inner
2401.2.a.h 24 196.o odd 42 1
2401.2.a.i 24 196.p even 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 14 T_{3}^{47} + 89 T_{3}^{46} - 329 T_{3}^{45} + 679 T_{3}^{44} + 35 T_{3}^{43} + \cdots + 4439449 \) acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\). Copy content Toggle raw display