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Properties

Label 684.3.g.c
Level $684$
Weight $3$
Character orbit 684.g
Analytic conductor $18.638$
Analytic rank $0$
Dimension $36$
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] = [684,3,Mod(343,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.343");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.g (of order \(2\), degree \(1\), 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: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: no (minimal twist has level 228)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 4 q^{2} + 12 q^{4} - 8 q^{5} + 20 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 4 q^{2} + 12 q^{4} - 8 q^{5} + 20 q^{8} + 8 q^{10} - 24 q^{13} + 12 q^{14} + 4 q^{16} + 40 q^{17} + 80 q^{20} + 12 q^{22} + 284 q^{25} + 112 q^{26} - 48 q^{28} - 104 q^{29} - 44 q^{32} + 140 q^{34} - 184 q^{37} + 180 q^{40} + 200 q^{41} - 96 q^{44} - 28 q^{46} - 332 q^{49} - 176 q^{50} + 276 q^{52} - 264 q^{53} + 192 q^{56} - 184 q^{58} + 40 q^{61} + 240 q^{62} - 372 q^{64} - 176 q^{65} + 104 q^{68} - 60 q^{70} + 424 q^{73} + 104 q^{74} + 400 q^{77} - 704 q^{80} + 528 q^{82} - 128 q^{85} - 668 q^{86} - 496 q^{88} + 520 q^{89} + 456 q^{92} - 32 q^{94} - 440 q^{97} + 472 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
343.1 −1.99932 − 0.0520666i 0 3.99458 + 0.208196i −6.84194 0 12.2496i −7.97561 − 0.624234i 0 13.6792 + 0.356236i
343.2 −1.99932 + 0.0520666i 0 3.99458 − 0.208196i −6.84194 0 − 12.2496i −7.97561 + 0.624234i 0 13.6792 − 0.356236i
343.3 −1.95612 − 0.416662i 0 3.65279 + 1.63008i −1.84551 0 − 7.45193i −6.46608 − 4.71060i 0 3.61003 + 0.768954i
343.4 −1.95612 + 0.416662i 0 3.65279 − 1.63008i −1.84551 0 7.45193i −6.46608 + 4.71060i 0 3.61003 − 0.768954i
343.5 −1.79512 − 0.881776i 0 2.44494 + 3.16579i 9.55892 0 12.8591i −1.59745 − 7.83889i 0 −17.1594 − 8.42882i
343.6 −1.79512 + 0.881776i 0 2.44494 − 3.16579i 9.55892 0 − 12.8591i −1.59745 + 7.83889i 0 −17.1594 + 8.42882i
343.7 −1.78242 − 0.907175i 0 2.35407 + 3.23394i 3.42194 0 1.09734i −1.26219 − 7.89980i 0 −6.09935 − 3.10430i
343.8 −1.78242 + 0.907175i 0 2.35407 − 3.23394i 3.42194 0 − 1.09734i −1.26219 + 7.89980i 0 −6.09935 + 3.10430i
343.9 −1.54492 − 1.27013i 0 0.773559 + 3.92449i −7.21023 0 4.48213i 3.78951 − 7.04554i 0 11.1392 + 9.15790i
343.10 −1.54492 + 1.27013i 0 0.773559 − 3.92449i −7.21023 0 − 4.48213i 3.78951 + 7.04554i 0 11.1392 − 9.15790i
343.11 −1.28823 − 1.52986i 0 −0.680940 + 3.94161i −3.40651 0 4.01056i 6.90732 − 4.03595i 0 4.38835 + 5.21148i
343.12 −1.28823 + 1.52986i 0 −0.680940 − 3.94161i −3.40651 0 − 4.01056i 6.90732 + 4.03595i 0 4.38835 − 5.21148i
343.13 −1.26893 − 1.54590i 0 −0.779622 + 3.92329i 6.46378 0 − 11.6492i 7.05430 − 3.77317i 0 −8.20210 − 9.99237i
343.14 −1.26893 + 1.54590i 0 −0.779622 − 3.92329i 6.46378 0 11.6492i 7.05430 + 3.77317i 0 −8.20210 + 9.99237i
343.15 −0.895709 − 1.78821i 0 −2.39541 + 3.20344i 5.60753 0 − 0.452718i 7.87402 + 1.41416i 0 −5.02271 − 10.0275i
343.16 −0.895709 + 1.78821i 0 −2.39541 − 3.20344i 5.60753 0 0.452718i 7.87402 − 1.41416i 0 −5.02271 + 10.0275i
343.17 −0.234708 − 1.98618i 0 −3.88982 + 0.932347i 0.816108 0 3.29246i 2.76478 + 7.50706i 0 −0.191547 − 1.62094i
343.18 −0.234708 + 1.98618i 0 −3.88982 − 0.932347i 0.816108 0 − 3.29246i 2.76478 − 7.50706i 0 −0.191547 + 1.62094i
343.19 −0.157997 − 1.99375i 0 −3.95007 + 0.630012i −9.57494 0 − 3.50228i 1.88018 + 7.77592i 0 1.51281 + 19.0900i
343.20 −0.157997 + 1.99375i 0 −3.95007 − 0.630012i −9.57494 0 3.50228i 1.88018 − 7.77592i 0 1.51281 − 19.0900i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 343.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.g.c 36
3.b odd 2 1 228.3.g.a ✓ 36
4.b odd 2 1 inner 684.3.g.c 36
12.b even 2 1 228.3.g.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.g.a ✓ 36 3.b odd 2 1
228.3.g.a ✓ 36 12.b even 2 1
684.3.g.c 36 1.a even 1 1 trivial
684.3.g.c 36 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} + 4 T_{5}^{17} - 288 T_{5}^{16} - 1144 T_{5}^{15} + 32454 T_{5}^{14} + 126816 T_{5}^{13} + \cdots - 18460196864 \) acting on \(S_{3}^{\mathrm{new}}(684, [\chi])\). Copy content Toggle raw display