↑

Properties

Label 680.2.ba.b
Level $680$
Weight $2$
Character orbit 680.ba
Analytic conductor $5.430$
Analytic rank $0$
Dimension $140$
Inner twists $4$

Related objects

Downloads

Learn more

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(21,680)] 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([0, 2, 0, 3])) 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("680.21"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 680.ba (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [140] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.42982733745\)
Analytic rank: \(0\)
Dimension: \(140\)
Relative dimension: \(70\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 140 q - 8 q^{4} - 4 q^{6} - 12 q^{7} + 20 q^{12} + 28 q^{14} - 24 q^{16} - 16 q^{17} - 8 q^{20} + 40 q^{22} + 4 q^{23} + 48 q^{24} - 20 q^{28} + 28 q^{31} - 8 q^{34} - 56 q^{38} + 24 q^{40} + 44 q^{41}+ \cdots + 176 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −1.40821 + 0.130188i 1.50733 + 1.50733i 1.96610 − 0.366665i −0.707107 − 0.707107i −2.31888 − 1.92640i −3.09613 − 3.09613i −2.72095 + 0.772304i 1.54411i 1.08781 + 0.903697i
21.2 −1.40300 − 0.177724i −1.59244 − 1.59244i 1.93683 + 0.498695i −0.707107 − 0.707107i 1.95119 + 2.51722i 0.509104 + 0.509104i −2.62874 − 1.04389i 2.07176i 0.866402 + 1.11774i
21.3 −1.39861 − 0.209479i 0.297587 + 0.297587i 1.91224 + 0.585960i −0.707107 − 0.707107i −0.353871 − 0.478547i 2.12142 + 2.12142i −2.55173 − 1.22010i − 2.82288i 0.840845 + 1.13709i
21.4 −1.36824 − 0.357649i −0.199214 − 0.199214i 1.74417 + 0.978700i 0.707107 + 0.707107i 0.201324 + 0.343821i −1.78690 − 1.78690i −2.03642 − 1.96290i − 2.92063i −0.714598 − 1.22039i
21.5 −1.34753 − 0.429141i 1.81913 + 1.81913i 1.63168 + 1.15656i 0.707107 + 0.707107i −1.67067 − 3.23200i −0.325706 − 0.325706i −1.70240 − 2.25872i 3.61847i −0.649399 − 1.25630i
21.6 −1.34108 − 0.448889i −2.27997 − 2.27997i 1.59700 + 1.20399i 0.707107 + 0.707107i 2.03417 + 4.08107i −0.753751 − 0.753751i −1.60124 − 2.33153i 7.39649i −0.630875 − 1.26570i
21.7 −1.34037 + 0.451000i −1.45882 − 1.45882i 1.59320 − 1.20902i 0.707107 + 0.707107i 2.61330 + 1.29744i 0.765216 + 0.765216i −1.59022 + 2.33906i 1.25634i −1.26669 − 0.628882i
21.8 −1.33610 + 0.463502i 0.494360 + 0.494360i 1.57033 − 1.23857i −0.707107 − 0.707107i −0.889652 − 0.431378i −0.414696 − 0.414696i −1.52404 + 2.38271i − 2.51122i 1.27251 + 0.617020i
21.9 −1.28273 + 0.595486i −0.901457 − 0.901457i 1.29079 − 1.52770i −0.707107 − 0.707107i 1.69313 + 0.619521i 0.193573 + 0.193573i −0.746016 + 2.72827i − 1.37475i 1.32810 + 0.485955i
21.10 −1.27433 + 0.613265i 2.07444 + 2.07444i 1.24781 − 1.56300i 0.707107 + 0.707107i −3.91570 − 1.37133i 1.70925 + 1.70925i −0.631586 + 2.75701i 5.60664i −1.33473 − 0.467440i
21.11 −1.23042 − 0.697177i −0.742822 − 0.742822i 1.02789 + 1.71565i −0.707107 − 0.707107i 0.396108 + 1.43187i −3.48763 − 3.48763i −0.0686299 − 2.82759i − 1.89643i 0.377063 + 1.36302i
21.12 −1.21931 − 0.716444i 1.45367 + 1.45367i 0.973417 + 1.74713i −0.707107 − 0.707107i −0.730995 − 2.81394i 2.29587 + 2.29587i 0.0648259 − 2.82768i 1.22630i 0.355578 + 1.36878i
21.13 −1.16850 + 0.796619i 1.30395 + 1.30395i 0.730797 − 1.86170i 0.707107 + 0.707107i −2.56243 − 0.484919i −2.46360 − 2.46360i 0.629129 + 2.75757i 0.400588i −1.38955 − 0.262962i
21.14 −1.09569 − 0.894130i −1.78868 − 1.78868i 0.401063 + 1.95937i −0.707107 − 0.707107i 0.360521 + 3.55914i 1.20340 + 1.20340i 1.31250 − 2.50546i 3.39872i 0.142523 + 1.40701i
21.15 −1.07030 − 0.924369i −1.16413 − 1.16413i 0.291085 + 1.97870i 0.707107 + 0.707107i 0.169884 + 2.32206i 2.58910 + 2.58910i 1.51750 − 2.38688i − 0.289584i −0.103189 − 1.41044i
21.16 −1.06369 + 0.931965i −2.43920 − 2.43920i 0.262881 − 1.98265i −0.707107 − 0.707107i 4.86781 + 0.321308i −2.50313 − 2.50313i 1.56813 + 2.35392i 8.89942i 1.41114 + 0.0931448i
21.17 −1.04654 − 0.951189i 0.411582 + 0.411582i 0.190479 + 1.99091i 0.707107 + 0.707107i −0.0392435 − 0.822228i 0.0802668 + 0.0802668i 1.69439 − 2.26474i − 2.66120i −0.0674212 − 1.41261i
21.18 −1.03249 + 0.966421i −1.62610 − 1.62610i 0.132062 − 1.99564i 0.707107 + 0.707107i 3.25043 + 0.107432i 0.881548 + 0.881548i 1.79227 + 2.18810i 2.28843i −1.41344 − 0.0467165i
21.19 −1.02972 + 0.969373i 1.22333 + 1.22333i 0.120632 − 1.99636i −0.707107 − 0.707107i −2.44554 − 0.0738200i 0.937063 + 0.937063i 1.81100 + 2.17262i − 0.00693262i 1.41357 + 0.0426693i
21.20 −0.894223 + 1.09561i 0.255177 + 0.255177i −0.400731 − 1.95944i 0.707107 + 0.707107i −0.507760 + 0.0513899i −1.97412 − 1.97412i 2.50513 + 1.31313i − 2.86977i −1.40703 + 0.142404i
See next 80 embeddings (of 140 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.70
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
17.c even 4 1 inner
136.i 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.ba.b ✓ 140
8.b even 2 1 inner 680.2.ba.b ✓ 140
17.c even 4 1 inner 680.2.ba.b ✓ 140
136.i even 4 1 inner 680.2.ba.b ✓ 140
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
680.2.ba.b ✓ 140 1.a even 1 1 trivial
680.2.ba.b ✓ 140 8.b even 2 1 inner
680.2.ba.b ✓ 140 17.c even 4 1 inner
680.2.ba.b ✓ 140 136.i even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{140} + 1000 T_{3}^{136} + 462460 T_{3}^{132} + 131574520 T_{3}^{128} + 25844859558 T_{3}^{124} + \cdots + 31\!\cdots\!36 \) acting on \(S_{2}^{\mathrm{new}}(680, [\chi])\). Copy content Toggle raw display