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Properties

Label 680.2.bg.a
Level $680$
Weight $2$
Character orbit 680.bg
Analytic conductor $5.430$
Analytic rank $0$
Dimension $192$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [680,2,Mod(307,680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(680, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 1, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("680.307"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 680.bg (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.42982733745\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(96\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q - 12 q^{8} + 16 q^{10} - 16 q^{12} - 28 q^{18} - 20 q^{20} - 36 q^{22} - 32 q^{26} - 12 q^{28} - 4 q^{30} + 40 q^{32} - 48 q^{35} + 32 q^{36} - 16 q^{38} + 36 q^{40} + 60 q^{42} - 64 q^{43} + 48 q^{46}+ \cdots + 60 q^{98}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1 −1.41414 + 0.0144393i −1.67812 + 1.67812i 1.99958 − 0.0408384i 0.606332 + 2.15229i 2.34887 − 2.39733i −3.20134 + 3.20134i −2.82710 + 0.0866238i − 2.63219i −0.888515 − 3.03489i
307.2 −1.41408 − 0.0191244i −2.19489 + 2.19489i 1.99927 + 0.0540871i −2.07854 − 0.824419i 3.14574 − 3.06179i 0.0463784 − 0.0463784i −2.82610 − 0.114719i − 6.63511i 2.92347 + 1.20555i
307.3 −1.40922 − 0.118707i −0.0880098 + 0.0880098i 1.97182 + 0.334570i 1.15330 + 1.91570i 0.134473 − 0.113578i 0.701897 − 0.701897i −2.73901 − 0.705552i 2.98451i −1.39784 − 2.83655i
307.4 −1.40692 + 0.143433i −1.31374 + 1.31374i 1.95885 − 0.403599i 2.16430 − 0.561952i 1.65989 − 2.03676i 1.21957 − 1.21957i −2.69806 + 0.848796i − 0.451805i −2.96440 + 1.10106i
307.5 −1.40403 − 0.169411i 0.650952 − 0.650952i 1.94260 + 0.475715i −2.11540 + 0.724619i −1.02423 + 0.803678i 0.614145 − 0.614145i −2.64688 − 0.997015i 2.15252i 3.09285 − 0.659015i
307.6 −1.37584 + 0.327222i 1.78728 − 1.78728i 1.78585 − 0.900407i 0.555716 + 2.16591i −1.87417 + 3.04384i 2.95224 − 2.95224i −2.16241 + 1.82318i − 3.38873i −1.47331 − 2.79810i
307.7 −1.36957 − 0.352522i 2.29220 − 2.29220i 1.75146 + 0.965608i −0.775233 − 2.09738i −3.94738 + 2.33128i 0.167838 − 0.167838i −2.05835 − 1.93990i − 7.50837i 0.322365 + 3.14580i
307.8 −1.35624 − 0.400758i −1.60078 + 1.60078i 1.67879 + 1.08705i 0.819675 − 2.08042i 2.81258 − 1.52952i 1.76879 − 1.76879i −1.84120 − 2.14709i − 2.12502i −1.94542 + 2.49306i
307.9 −1.35479 + 0.405647i 1.77226 − 1.77226i 1.67090 − 1.09913i 2.07389 + 0.836049i −1.68213 + 3.11995i −3.46258 + 3.46258i −1.81786 + 2.16689i − 3.28182i −3.14882 − 0.291401i
307.10 −1.34539 + 0.435815i 0.0551671 − 0.0551671i 1.62013 − 1.17268i −1.08340 − 1.95608i −0.0501785 + 0.0982637i −1.51122 + 1.51122i −1.66863 + 2.28378i 2.99391i 2.31008 + 2.15952i
307.11 −1.32535 + 0.493391i 1.44782 − 1.44782i 1.51313 − 1.30784i −2.15712 + 0.588925i −1.20453 + 2.63321i −0.920819 + 0.920819i −1.36016 + 2.47991i − 1.19235i 2.56838 − 1.84484i
307.12 −1.30330 − 0.549014i 2.14739 − 2.14739i 1.39717 + 1.43106i −1.39699 + 1.74597i −3.97764 + 1.61974i −1.73358 + 1.73358i −1.03526 − 2.63216i − 6.22260i 2.77926 − 1.50855i
307.13 −1.27807 + 0.605423i −1.30295 + 1.30295i 1.26693 − 1.54755i −1.65845 − 1.49985i 0.876424 − 2.45409i 0.261675 − 0.261675i −0.682301 + 2.74490i − 0.395349i 3.02766 + 0.912850i
307.14 −1.27637 − 0.609009i −0.634491 + 0.634491i 1.25822 + 1.55464i −2.01333 + 0.972887i 1.19625 − 0.423431i 2.34167 − 2.34167i −0.659156 − 2.75055i 2.19484i 3.16224 − 0.0156248i
307.15 −1.26339 − 0.635494i 1.35688 − 1.35688i 1.19229 + 1.60575i 2.22157 − 0.254202i −2.57655 + 0.851974i 1.93753 − 1.93753i −0.485886 − 2.78638i − 0.682230i −2.96825 − 1.09064i
307.16 −1.25496 − 0.651972i −0.867353 + 0.867353i 1.14986 + 1.63640i −2.19846 − 0.408383i 1.65399 − 0.523005i −3.35494 + 3.35494i −0.376148 − 2.80330i 1.49540i 2.49273 + 1.94584i
307.17 −1.23879 + 0.682199i −1.08388 + 1.08388i 1.06921 − 1.69021i −1.43185 + 1.71750i 0.603282 − 2.08213i 3.01153 − 3.01153i −0.171470 + 2.82322i 0.650390i 0.602087 − 3.10443i
307.18 −1.21306 + 0.726977i 0.637356 − 0.637356i 0.943009 − 1.76373i 0.876428 − 2.05715i −0.309806 + 1.23649i 3.53557 − 3.53557i 0.138266 + 2.82505i 2.18755i 0.432346 + 3.13258i
307.19 −1.17894 − 0.781086i −2.14184 + 2.14184i 0.779808 + 1.84171i −0.127785 + 2.23241i 4.19807 − 0.852142i 1.63144 − 1.63144i 0.519188 − 2.78037i − 6.17495i 1.89436 − 2.53207i
307.20 −1.17397 + 0.788544i −1.95701 + 1.95701i 0.756398 − 1.85145i 1.70987 − 1.44095i 0.754277 − 3.84065i −2.67624 + 2.67624i 0.571961 + 2.76999i − 4.65975i −0.871078 + 3.03994i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.96
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.d odd 2 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 680.2.bg.a ✓ 192
5.c odd 4 1 inner 680.2.bg.a ✓ 192
8.d odd 2 1 inner 680.2.bg.a ✓ 192
40.k even 4 1 inner 680.2.bg.a ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
680.2.bg.a ✓ 192 1.a even 1 1 trivial
680.2.bg.a ✓ 192 5.c odd 4 1 inner
680.2.bg.a ✓ 192 8.d odd 2 1 inner
680.2.bg.a ✓ 192 40.k even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(680, [\chi])\).