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Properties

Label 688.2.bi.a
Level $688$
Weight $2$
Character orbit 688.bi
Analytic conductor $5.494$
Analytic rank $0$
Dimension $1032$
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(27,688)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(688, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([14, 7, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("688.27");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 688.bi (of order \(28\), degree \(12\), 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: \(1032\)
Relative dimension: \(86\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1032 q - 14 q^{2} - 14 q^{3} - 14 q^{4} - 14 q^{5} - 24 q^{6} - 14 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1032 q - 14 q^{2} - 14 q^{3} - 14 q^{4} - 14 q^{5} - 24 q^{6} - 14 q^{8} - 10 q^{10} - 18 q^{11} - 14 q^{12} - 10 q^{13} - 2 q^{14} + 10 q^{16} - 20 q^{17} - 14 q^{18} - 14 q^{19} + 42 q^{20} + 20 q^{21} - 14 q^{22} - 36 q^{23} - 10 q^{24} - 14 q^{26} - 14 q^{27} + 42 q^{28} - 14 q^{29} + 126 q^{30} - 14 q^{32} - 28 q^{33} - 14 q^{34} + 40 q^{35} - 10 q^{38} - 28 q^{39} + 30 q^{40} + 24 q^{44} - 14 q^{45} - 14 q^{46} - 14 q^{48} - 936 q^{49} - 14 q^{51} + 42 q^{52} + 30 q^{53} + 22 q^{54} + 196 q^{55} + 102 q^{56} - 30 q^{58} + 22 q^{59} - 42 q^{60} - 14 q^{61} - 14 q^{62} - 38 q^{64} - 28 q^{65} - 18 q^{66} - 50 q^{67} - 128 q^{68} - 14 q^{69} + 28 q^{70} - 28 q^{71} - 14 q^{72} + 116 q^{74} - 14 q^{75} - 14 q^{76} + 84 q^{77} - 148 q^{78} + 120 q^{81} + 126 q^{82} - 10 q^{83} + 136 q^{84} - 50 q^{86} - 160 q^{87} - 14 q^{88} + 310 q^{90} + 84 q^{91} + 16 q^{92} - 294 q^{94} + 66 q^{96} - 20 q^{97} + 154 q^{98} - 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 −1.41419 − 0.00818042i −0.335843 − 0.534490i 1.99987 + 0.0231373i −1.88775 − 0.212698i 0.470573 + 0.758618i 2.98976i −2.82800 − 0.0490803i 1.12876 − 2.34390i 2.66789 + 0.316238i
27.2 −1.41419 − 0.00837521i 0.512319 + 0.815350i 1.99986 + 0.0236883i 1.45609 + 0.164062i −0.717686 − 1.15735i 1.97144i −2.82798 − 0.0502489i 0.899325 − 1.86747i −2.05781 − 0.244210i
27.3 −1.40726 + 0.140039i 1.06241 + 1.69081i 1.96078 − 0.394144i −3.99512 − 0.450141i −1.73187 − 2.23064i − 2.81641i −2.70413 + 0.829249i −0.428483 + 0.889754i 5.68522 + 0.0739951i
27.4 −1.40523 + 0.159155i −0.791363 − 1.25945i 1.94934 − 0.447299i 3.50517 + 0.394937i 1.31249 + 1.64386i − 0.494466i −2.66808 + 0.938805i 0.341700 − 0.709548i −4.98842 + 0.00288755i
27.5 −1.39772 − 0.215363i −1.78613 − 2.84260i 1.90724 + 0.602033i 0.0276490 + 0.00311530i 1.88431 + 4.35783i − 3.63198i −2.53613 − 1.25222i −3.58849 + 7.45159i −0.0379746 − 0.0103089i
27.6 −1.37757 − 0.319864i 1.29665 + 2.06361i 1.79537 + 0.881268i 2.81467 + 0.317137i −1.12615 − 3.25751i 0.945696i −2.19136 − 1.78828i −1.27552 + 2.64865i −3.77595 − 1.33719i
27.7 −1.37016 − 0.350224i −1.30119 − 2.07083i 1.75469 + 0.959726i 1.31865 + 0.148577i 1.05759 + 3.29309i 3.77712i −2.06809 − 1.92951i −1.29360 + 2.68619i −1.75473 − 0.665397i
27.8 −1.36457 − 0.371428i 0.142745 + 0.227178i 1.72408 + 1.01368i −0.671324 − 0.0756401i −0.110405 − 0.363019i − 2.91667i −1.97612 − 2.02360i 1.27042 − 2.63805i 0.887972 + 0.352565i
27.9 −1.35485 + 0.405438i 0.624444 + 0.993797i 1.67124 − 1.09861i 0.629114 + 0.0708842i −1.24895 − 1.09327i − 3.36367i −1.81886 + 2.16604i 0.703949 − 1.46176i −0.881095 + 0.159029i
27.10 −1.30271 + 0.550405i −1.21176 − 1.92851i 1.39411 − 1.43404i −1.58240 − 0.178294i 2.64004 + 1.84533i − 3.53511i −1.02682 + 2.63546i −0.949134 + 1.97090i 2.15955 − 0.638697i
27.11 −1.28838 + 0.583155i −1.03086 − 1.64061i 1.31986 − 1.50265i −2.30742 − 0.259984i 2.28487 + 1.51258i 1.66905i −0.824204 + 2.70568i −0.327260 + 0.679563i 3.12445 − 1.01063i
27.12 −1.28679 − 0.586662i −0.651680 − 1.03714i 1.31166 + 1.50982i −3.49872 − 0.394211i 0.230123 + 1.71690i − 2.48873i −0.802071 − 2.71232i 0.650673 − 1.35114i 4.27085 + 2.55983i
27.13 −1.27632 + 0.609095i 1.51293 + 2.40781i 1.25801 − 1.55480i 0.371479 + 0.0418557i −3.39757 − 2.15163i 4.69104i −0.658602 + 2.75068i −2.20696 + 4.58280i −0.499622 + 0.172845i
27.14 −1.24809 − 0.665031i 1.67965 + 2.67314i 1.11547 + 1.66004i 0.895047 + 0.100848i −0.318630 − 4.45335i − 4.71181i −0.288226 − 2.81370i −3.02283 + 6.27697i −1.05003 − 0.721101i
27.15 −1.21164 − 0.729333i 1.32193 + 2.10385i 0.936146 + 1.76738i −3.18754 − 0.359149i −0.0673043 − 3.51324i 3.24462i 0.154736 − 2.82419i −1.37701 + 2.85939i 3.60021 + 2.75993i
27.16 −1.19807 + 0.751420i 1.28307 + 2.04200i 0.870735 − 1.80051i 3.71676 + 0.418778i −3.07161 − 1.48233i − 2.20804i 0.309736 + 2.81142i −1.22183 + 2.53716i −4.76761 + 2.29112i
27.17 −1.19632 + 0.754195i −1.58296 − 2.51926i 0.862380 − 1.80452i 2.09825 + 0.236416i 3.79374 + 1.81999i 2.53164i 0.329276 + 2.80920i −2.53927 + 5.27285i −2.68849 + 1.29966i
27.18 −1.16491 + 0.801869i 0.666143 + 1.06016i 0.714011 − 1.86820i −3.59251 − 0.404778i −1.62610 − 0.700828i 1.62247i 0.666300 + 2.74883i 0.621457 − 1.29047i 4.50951 − 2.40919i
27.19 −1.16427 + 0.802788i 0.219611 + 0.349509i 0.711062 − 1.86933i 0.0530869 + 0.00598146i −0.536268 − 0.230622i 2.39123i 0.672804 + 2.74724i 1.22772 − 2.54939i −0.0666095 + 0.0356535i
27.20 −1.16410 − 0.803038i −0.368028 − 0.585713i 0.710258 + 1.86963i 3.61653 + 0.407485i −0.0419288 + 0.977369i − 2.89325i 0.674577 − 2.74681i 1.09404 − 2.27179i −3.88278 − 3.37857i
See next 80 embeddings (of 1032 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.86
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner
43.f odd 14 1 inner
688.bi even 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.2.bi.a ✓ 1032
16.f odd 4 1 inner 688.2.bi.a ✓ 1032
43.f odd 14 1 inner 688.2.bi.a ✓ 1032
688.bi even 28 1 inner 688.2.bi.a ✓ 1032
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
688.2.bi.a ✓ 1032 1.a even 1 1 trivial
688.2.bi.a ✓ 1032 16.f odd 4 1 inner
688.2.bi.a ✓ 1032 43.f odd 14 1 inner
688.2.bi.a ✓ 1032 688.bi even 28 1 inner

Hecke kernels

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