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Properties

Label 684.2.j.a
Level $684$
Weight $2$
Character orbit 684.j
Analytic conductor $5.462$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [684,2,Mod(49,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.j (of order \(3\), 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: \(5.46176749826\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{3} - q^{7} - 2 q^{9} - q^{11} + 2 q^{13} + q^{15} + 5 q^{17} + q^{19} + 6 q^{21} + 8 q^{23} - 20 q^{25} + 7 q^{27} - 9 q^{29} + 2 q^{31} - q^{33} - 6 q^{35} + 2 q^{37} + 3 q^{39} - 25 q^{41}+ \cdots + 28 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −1.73202 + 0.0101310i 0 −1.94402 − 3.36714i 0 −0.0132784 − 0.0229989i 0 2.99979 − 0.0350943i 0
49.2 0 −1.73108 − 0.0580258i 0 0.748116 + 1.29577i 0 −1.75655 − 3.04244i 0 2.99327 + 0.200895i 0
49.3 0 −1.65259 + 0.518616i 0 1.53869 + 2.66510i 0 2.34238 + 4.05712i 0 2.46208 − 1.71411i 0
49.4 0 −1.55715 − 0.758482i 0 −0.482969 − 0.836526i 0 0.302844 + 0.524541i 0 1.84941 + 2.36214i 0
49.5 0 −1.16446 + 1.28220i 0 −0.145159 − 0.251423i 0 −0.636745 − 1.10287i 0 −0.288081 − 2.98614i 0
49.6 0 −0.961962 − 1.44036i 0 1.35066 + 2.33941i 0 −0.793446 − 1.37429i 0 −1.14926 + 2.77114i 0
49.7 0 −0.833161 + 1.51850i 0 −1.50323 − 2.60367i 0 0.886284 + 1.53509i 0 −1.61168 − 2.53031i 0
49.8 0 −0.709119 − 1.58024i 0 −0.702434 − 1.21665i 0 1.81436 + 3.14256i 0 −1.99430 + 2.24115i 0
49.9 0 −0.672276 + 1.59626i 0 1.20282 + 2.08335i 0 −1.53172 − 2.65301i 0 −2.09609 − 2.14625i 0
49.10 0 −0.404917 − 1.68406i 0 −1.80780 − 3.13120i 0 −2.29625 − 3.97722i 0 −2.67209 + 1.36380i 0
49.11 0 0.0878220 + 1.72982i 0 1.41926 + 2.45824i 0 0.648230 + 1.12277i 0 −2.98457 + 0.303833i 0
49.12 0 0.143302 − 1.72611i 0 −0.479354 − 0.830266i 0 2.11741 + 3.66746i 0 −2.95893 − 0.494710i 0
49.13 0 0.597890 + 1.62559i 0 −1.25752 − 2.17808i 0 −0.950930 − 1.64706i 0 −2.28505 + 1.94384i 0
49.14 0 0.747847 − 1.56228i 0 0.136750 + 0.236858i 0 −0.546712 − 0.946933i 0 −1.88145 − 2.33670i 0
49.15 0 1.23208 − 1.21737i 0 2.12356 + 3.67811i 0 1.38218 + 2.39401i 0 0.0360230 − 2.99978i 0
49.16 0 1.24864 + 1.20038i 0 0.745287 + 1.29087i 0 0.903772 + 1.56538i 0 0.118185 + 2.99767i 0
49.17 0 1.30529 + 1.13852i 0 −1.34080 − 2.32233i 0 −0.125175 − 0.216810i 0 0.407555 + 2.97219i 0
49.18 0 1.61687 − 0.621068i 0 −0.400523 − 0.693726i 0 −1.41378 − 2.44874i 0 2.22855 − 2.00838i 0
49.19 0 1.71780 + 0.221694i 0 1.53086 + 2.65152i 0 −2.37338 − 4.11082i 0 2.90170 + 0.761653i 0
49.20 0 1.72118 − 0.193715i 0 −0.732210 − 1.26823i 0 1.54051 + 2.66824i 0 2.92495 − 0.666840i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.j.a ✓ 40
3.b odd 2 1 2052.2.j.a 40
9.c even 3 1 684.2.l.a yes 40
9.d odd 6 1 2052.2.l.a 40
19.c even 3 1 684.2.l.a yes 40
57.h odd 6 1 2052.2.l.a 40
171.h even 3 1 inner 684.2.j.a ✓ 40
171.j odd 6 1 2052.2.j.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.2.j.a ✓ 40 1.a even 1 1 trivial
684.2.j.a ✓ 40 171.h even 3 1 inner
684.2.l.a yes 40 9.c even 3 1
684.2.l.a yes 40 19.c even 3 1
2052.2.j.a 40 3.b odd 2 1
2052.2.j.a 40 171.j odd 6 1
2052.2.l.a 40 9.d odd 6 1
2052.2.l.a 40 57.h odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(684, [\chi])\).