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Properties

Label 688.2.x.a
Level $688$
Weight $2$
Character orbit 688.x
Analytic conductor $5.494$
Analytic rank $0$
Dimension $344$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [688,2,Mod(165,688)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(688, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("688.165");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 688.x (of order \(12\), degree \(4\), 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: \(5.49370765906\)
Analytic rank: \(0\)
Dimension: \(344\)
Relative dimension: \(86\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 344 q - 8 q^{2} - 2 q^{3} - 12 q^{4} - 2 q^{5} - 2 q^{6} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 344 q - 8 q^{2} - 2 q^{3} - 12 q^{4} - 2 q^{5} - 2 q^{6} - 8 q^{8} - 2 q^{10} + 8 q^{11} - 10 q^{12} - 2 q^{13} - 4 q^{14} - 4 q^{15} + 12 q^{16} - 4 q^{17} - 8 q^{18} - 2 q^{19} - 38 q^{20} - 20 q^{21} + 12 q^{22} - 2 q^{24} - 2 q^{26} - 20 q^{27} + 6 q^{28} + 6 q^{29} - 46 q^{30} - 4 q^{31} - 8 q^{32} - 4 q^{33} - 6 q^{34} - 28 q^{35} + 46 q^{36} - 18 q^{37} - 44 q^{38} + 18 q^{40} - 72 q^{42} + 32 q^{43} - 48 q^{44} - 16 q^{45} + 2 q^{46} - 16 q^{47} + 50 q^{48} + 144 q^{49} - 10 q^{50} + 4 q^{51} + 24 q^{52} + 14 q^{53} + 24 q^{54} + 12 q^{56} - 8 q^{58} + 24 q^{59} - 32 q^{60} - 2 q^{61} - 20 q^{63} + 48 q^{64} - 16 q^{65} - 6 q^{66} - 22 q^{67} - 46 q^{68} - 14 q^{69} + 32 q^{70} + 22 q^{72} - 72 q^{74} - 120 q^{75} - 18 q^{76} + 28 q^{77} + 112 q^{78} - 4 q^{79} - 22 q^{80} + 136 q^{81} + 48 q^{82} - 2 q^{83} + 4 q^{84} + 12 q^{85} + 42 q^{86} + 8 q^{88} - 124 q^{90} - 14 q^{91} + 74 q^{92} + 10 q^{93} + 20 q^{94} - 4 q^{95} - 60 q^{96} - 16 q^{97} - 82 q^{98} - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1 −1.41397 − 0.0264800i 2.71572 + 0.727675i 1.99860 + 0.0748835i −0.542275 + 2.02380i −3.82067 − 1.10082i 2.69688 + 1.55704i −2.82397 − 0.158806i 4.24756 + 2.45233i 0.820349 − 2.84722i
165.2 −1.41259 − 0.0676789i 0.694501 + 0.186091i 1.99084 + 0.191206i −0.872062 + 3.25458i −0.968453 − 0.309874i 2.27313 + 1.31239i −2.79931 − 0.404833i −2.15037 − 1.24152i 1.45214 − 4.53838i
165.3 −1.41166 + 0.0849033i −2.75509 − 0.738225i 1.98558 − 0.239710i −0.675921 + 2.52257i 3.95194 + 0.808208i −2.90588 − 1.67771i −2.78262 + 0.506972i 4.44748 + 2.56775i 0.739998 − 3.61841i
165.4 −1.41019 − 0.106653i −0.300791 − 0.0805966i 1.97725 + 0.300802i −0.338265 + 1.26242i 0.415575 + 0.145737i −3.20328 − 1.84941i −2.75621 − 0.635067i −2.51410 − 1.45151i 0.611658 − 1.74417i
165.5 −1.40522 − 0.159277i 2.37130 + 0.635388i 1.94926 + 0.447638i 0.583050 − 2.17597i −3.23098 − 1.27055i −0.238099 − 0.137467i −2.66783 − 0.939501i 2.62127 + 1.51339i −1.16589 + 2.96484i
165.6 −1.40403 − 0.169396i −1.53318 − 0.410814i 1.94261 + 0.475675i 0.0518819 − 0.193626i 2.08304 + 0.836510i 0.832420 + 0.480598i −2.64691 − 0.996932i −0.416209 − 0.240299i −0.105643 + 0.263068i
165.7 −1.36258 + 0.378660i −0.985825 − 0.264151i 1.71323 − 1.03191i 0.828288 − 3.09121i 1.44329 − 0.0133667i 1.10373 + 0.637240i −1.94367 + 2.05479i −1.69600 − 0.979186i 0.0419134 + 4.52566i
165.8 −1.35323 + 0.410816i 1.43144 + 0.383554i 1.66246 − 1.11186i 0.589198 − 2.19892i −2.09464 + 0.0690231i −4.32919 − 2.49946i −1.79292 + 2.18756i −0.696159 − 0.401928i 0.106030 + 3.21769i
165.9 −1.34166 − 0.447154i 1.24018 + 0.332306i 1.60011 + 1.19986i 1.07044 − 3.99492i −1.51531 − 1.00040i 2.17471 + 1.25557i −1.61028 − 2.32530i −1.17045 − 0.675758i −3.22250 + 4.88118i
165.10 −1.32665 + 0.489895i −0.673964 − 0.180588i 1.52001 − 1.29984i 0.171356 − 0.639509i 0.982584 − 0.0905943i 2.72999 + 1.57616i −1.37973 + 2.46908i −2.17646 − 1.25658i 0.0859628 + 0.932351i
165.11 −1.30819 + 0.537241i −3.13805 − 0.840839i 1.42274 − 1.40563i 0.818907 − 3.05620i 4.55692 − 0.585910i −2.53976 − 1.46633i −1.10606 + 2.60319i 6.54230 + 3.77720i 0.570627 + 4.43806i
165.12 −1.26781 + 0.626629i 2.55159 + 0.683697i 1.21467 − 1.58889i −0.229493 + 0.856479i −3.66335 + 0.732106i −0.267052 − 0.154183i −0.544324 + 2.77556i 3.44510 + 1.98903i −0.245742 − 1.22966i
165.13 −1.25602 − 0.649942i −1.67954 − 0.450032i 1.15515 + 1.63267i 0.948974 − 3.54162i 1.81704 + 1.65685i −2.74128 − 1.58268i −0.389746 − 2.80145i 0.0202533 + 0.0116932i −3.49377 + 3.83155i
165.14 −1.22852 − 0.700534i −2.22283 − 0.595606i 1.01850 + 1.72123i −1.06526 + 3.97559i 2.31354 + 2.28888i 2.39915 + 1.38515i −0.0454656 − 2.82806i 1.98816 + 1.14786i 4.09372 − 4.13783i
165.15 −1.22737 + 0.702539i 0.422980 + 0.113337i 1.01288 − 1.72455i −0.764108 + 2.85169i −0.598777 + 0.158053i −0.587240 − 0.339043i −0.0316143 + 2.82825i −2.43201 − 1.40412i −1.06558 − 4.03689i
165.16 −1.21134 − 0.729839i 2.99259 + 0.801861i 0.934671 + 1.76816i −0.0822268 + 0.306875i −3.03980 − 3.15543i −2.47516 − 1.42904i 0.158270 − 2.82400i 5.71451 + 3.29928i 0.323573 − 0.311716i
165.17 −1.19882 − 0.750217i −2.44927 − 0.656279i 0.874348 + 1.79875i 0.0714675 − 0.266720i 2.44388 + 2.62424i 0.105707 + 0.0610298i 0.301269 − 2.81234i 2.97012 + 1.71480i −0.285775 + 0.266134i
165.18 −1.17028 − 0.794010i 0.190191 + 0.0509614i 0.739096 + 1.85842i 0.216168 − 0.806748i −0.182112 − 0.210652i 4.34545 + 2.50885i 0.610660 − 2.76172i −2.56450 − 1.48062i −0.893542 + 0.772480i
165.19 −1.16110 − 0.807372i 1.07857 + 0.289002i 0.696301 + 1.87488i −0.643460 + 2.40143i −1.01899 − 1.20637i −2.32432 − 1.34195i 0.705250 − 2.73909i −1.51829 − 0.876583i 2.68597 − 2.26878i
165.20 −1.15702 + 0.813210i −2.84727 − 0.762922i 0.677380 − 1.88180i −0.545204 + 2.03473i 3.91475 − 1.43271i 4.19054 + 2.41941i 0.746554 + 2.72812i 4.92679 + 2.84449i −1.02385 − 2.79758i
See next 80 embeddings (of 344 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 165.86
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner
43.c even 3 1 inner
688.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.2.x.a ✓ 344
16.e even 4 1 inner 688.2.x.a ✓ 344
43.c even 3 1 inner 688.2.x.a ✓ 344
688.x even 12 1 inner 688.2.x.a ✓ 344
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
688.2.x.a ✓ 344 1.a even 1 1 trivial
688.2.x.a ✓ 344 16.e even 4 1 inner
688.2.x.a ✓ 344 43.c even 3 1 inner
688.2.x.a ✓ 344 688.x even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(688, [\chi])\).