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Properties

Label 684.2.bb.b
Level $684$
Weight $2$
Character orbit 684.bb
Analytic conductor $5.462$
Analytic rank $0$
Dimension $38$
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] = [684,2,Mod(293,684)]
 
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, 5, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.293");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.bb (of order \(6\), 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: \(38\)
Relative dimension: \(19\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 38 q + 3 q^{3} + 2 q^{7} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q + 3 q^{3} + 2 q^{7} - 5 q^{9} + 6 q^{11} - 3 q^{15} + 6 q^{17} + 7 q^{19} - 3 q^{21} + 25 q^{25} - 9 q^{27} - 3 q^{29} + 21 q^{31} + 24 q^{33} + 3 q^{41} + 10 q^{43} - 4 q^{45} + 18 q^{47} - 27 q^{49} + 3 q^{51} + 21 q^{53} - 33 q^{57} + 48 q^{59} + 5 q^{61} + 20 q^{63} + 6 q^{65} - 57 q^{69} + 6 q^{71} + 7 q^{73} - 6 q^{75} - 51 q^{77} - 41 q^{81} - 12 q^{83} + 54 q^{87} - 21 q^{89} - 15 q^{91} - 6 q^{93} - 24 q^{95} - 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1 0 −1.72938 − 0.0960612i 0 1.05931 − 0.611591i 0 2.38880 + 4.13752i 0 2.98154 + 0.332254i 0
293.2 0 −1.60398 − 0.653645i 0 2.00804 − 1.15934i 0 −2.15763 − 3.73712i 0 2.14550 + 2.09686i 0
293.3 0 −1.49209 − 0.879585i 0 −3.53228 + 2.03936i 0 −0.972187 − 1.68388i 0 1.45266 + 2.62484i 0
293.4 0 −1.48059 + 0.898810i 0 −1.50724 + 0.870203i 0 −0.708633 − 1.22739i 0 1.38428 − 2.66153i 0
293.5 0 −1.06805 − 1.36355i 0 2.06506 − 1.19226i 0 0.509777 + 0.882960i 0 −0.718546 + 2.91268i 0
293.6 0 −0.952802 + 1.44643i 0 3.17233 − 1.83155i 0 1.47573 + 2.55605i 0 −1.18434 − 2.75633i 0
293.7 0 −0.703965 − 1.58254i 0 −1.16412 + 0.672102i 0 0.397917 + 0.689212i 0 −2.00887 + 2.22811i 0
293.8 0 −0.260338 + 1.71237i 0 −2.84857 + 1.64462i 0 2.02877 + 3.51393i 0 −2.86445 − 0.891591i 0
293.9 0 −0.237467 + 1.71570i 0 1.87333 − 1.08157i 0 −0.579619 − 1.00393i 0 −2.88722 − 0.814842i 0
293.10 0 0.172027 + 1.72349i 0 0.641245 − 0.370223i 0 −2.42092 − 4.19315i 0 −2.94081 + 0.592973i 0
293.11 0 0.280199 − 1.70924i 0 0.886675 − 0.511922i 0 −0.304367 − 0.527179i 0 −2.84298 − 0.957852i 0
293.12 0 0.929362 − 1.46160i 0 −1.17392 + 0.677766i 0 2.35216 + 4.07406i 0 −1.27257 − 2.71672i 0
293.13 0 1.14906 + 1.29602i 0 −0.760180 + 0.438890i 0 1.16089 + 2.01071i 0 −0.359317 + 2.97840i 0
293.14 0 1.19778 − 1.25113i 0 −3.79749 + 2.19248i 0 −0.000153367 0 0.000265639i 0 −0.130652 − 2.99715i 0
293.15 0 1.23067 + 1.21879i 0 −2.15981 + 1.24697i 0 −1.05912 − 1.83445i 0 0.0291010 + 2.99986i 0
293.16 0 1.30531 − 1.13849i 0 2.85106 − 1.64606i 0 −1.68772 − 2.92322i 0 0.407672 − 2.97217i 0
293.17 0 1.34541 + 1.09081i 0 2.89435 − 1.67106i 0 1.19821 + 2.07537i 0 0.620272 + 2.93518i 0
293.18 0 1.70135 − 0.324667i 0 1.14537 − 0.661282i 0 0.853104 + 1.47762i 0 2.78918 − 1.10474i 0
293.19 0 1.71749 + 0.224123i 0 −1.65318 + 0.954464i 0 −1.47502 − 2.55481i 0 2.89954 + 0.769858i 0
677.1 0 −1.72938 + 0.0960612i 0 1.05931 + 0.611591i 0 2.38880 − 4.13752i 0 2.98154 − 0.332254i 0
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 293.19
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.t even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.bb.b yes 38
3.b odd 2 1 2052.2.bb.b 38
9.c even 3 1 2052.2.n.b 38
9.d odd 6 1 684.2.n.b ✓ 38
19.d odd 6 1 684.2.n.b ✓ 38
57.f even 6 1 2052.2.n.b 38
171.i odd 6 1 2052.2.bb.b 38
171.t even 6 1 inner 684.2.bb.b yes 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.2.n.b ✓ 38 9.d odd 6 1
684.2.n.b ✓ 38 19.d odd 6 1
684.2.bb.b yes 38 1.a even 1 1 trivial
684.2.bb.b yes 38 171.t even 6 1 inner
2052.2.n.b 38 9.c even 3 1
2052.2.n.b 38 57.f even 6 1
2052.2.bb.b 38 3.b odd 2 1
2052.2.bb.b 38 171.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{38} - 60 T_{5}^{36} + 2136 T_{5}^{34} - 561 T_{5}^{33} - 49879 T_{5}^{32} + 24717 T_{5}^{31} + \cdots + 212403681675 \) acting on \(S_{2}^{\mathrm{new}}(684, [\chi])\). Copy content Toggle raw display