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Properties

Label 680.2.cf.a
Level $680$
Weight $2$
Character orbit 680.cf
Analytic conductor $5.430$
Analytic rank $0$
Dimension $832$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [680,2,Mod(37,680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(680, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([0, 8, 4, 1])) 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("680.37"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 680.cf (of order \(16\), degree \(8\), 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: \(832\)
Relative dimension: \(104\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 832 q - 8 q^{2} - 16 q^{6} - 16 q^{7} - 8 q^{8} - 8 q^{10} - 32 q^{12} - 32 q^{14} - 16 q^{15} - 16 q^{17} - 16 q^{18} - 8 q^{20} - 8 q^{22} - 16 q^{23} - 16 q^{25} - 48 q^{26} - 40 q^{28} - 72 q^{30} - 32 q^{31}+ \cdots - 16 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −1.41415 − 0.0131479i 0.645128 + 0.965503i 1.99965 + 0.0371862i −0.0494241 + 2.23552i −0.899616 − 1.37385i −0.178215 + 0.0354493i −2.82733 − 0.0788781i 0.632045 − 1.52589i 0.0992856 − 3.16072i
37.2 −1.41354 − 0.0437995i −0.517043 − 0.773810i 1.99616 + 0.123824i −2.20741 + 0.356844i 0.696966 + 1.11645i 2.40217 − 0.477821i −2.81622 − 0.262461i 0.816602 − 1.97145i 3.13588 − 0.407728i
37.3 −1.41350 − 0.0448075i 0.585607 + 0.876423i 1.99598 + 0.126671i 1.49958 + 1.65870i −0.788487 − 1.26507i −0.432665 + 0.0860623i −2.81566 − 0.268485i 0.722869 − 1.74516i −2.04533 − 2.41176i
37.4 −1.40593 − 0.152840i 0.159020 + 0.237990i 1.95328 + 0.429766i −0.318666 − 2.21324i −0.187196 − 0.358901i 4.61714 − 0.918406i −2.68049 − 0.902761i 1.11670 − 2.69595i 0.109749 + 3.16037i
37.5 −1.39551 − 0.229244i 1.00008 + 1.49673i 1.89489 + 0.639823i 1.25943 − 1.84765i −1.05251 − 2.31796i −0.346855 + 0.0689937i −2.49767 − 1.32727i −0.0919848 + 0.222071i −2.18111 + 2.28970i
37.6 −1.39219 + 0.248599i 1.74548 + 2.61229i 1.87640 − 0.692196i −1.88646 − 1.20052i −3.07945 − 3.20289i −1.75917 + 0.349920i −2.44023 + 1.43014i −2.62933 + 6.34776i 2.92477 + 1.20239i
37.7 −1.37681 − 0.323085i −1.73274 − 2.59324i 1.79123 + 0.889656i −1.99136 − 1.01710i 1.54783 + 4.13023i −1.15970 + 0.230678i −2.17876 − 1.80361i −2.57442 + 6.21519i 2.41312 + 2.04373i
37.8 −1.36221 + 0.379990i −1.10197 − 1.64922i 1.71121 − 1.03525i −0.610258 + 2.15118i 2.12780 + 1.82784i −4.76307 + 0.947434i −1.93764 + 2.06047i −0.357529 + 0.863152i 0.0138687 − 3.16225i
37.9 −1.36008 + 0.387549i −1.32287 − 1.97982i 1.69961 − 1.05419i 0.865885 − 2.06161i 2.56649 + 2.18003i −1.17495 + 0.233712i −1.90305 + 2.09246i −1.02164 + 2.46647i −0.378693 + 3.13952i
37.10 −1.35485 + 0.405455i 0.281224 + 0.420881i 1.67121 − 1.09866i −1.80329 − 1.32218i −0.551663 − 0.456205i −3.98768 + 0.793198i −1.81878 + 2.16611i 1.05000 − 2.53492i 2.97926 + 1.06020i
37.11 −1.34965 − 0.422420i −0.577178 − 0.863808i 1.64312 + 1.14024i 1.97239 − 1.05341i 0.414100 + 1.40965i −3.87241 + 0.770271i −1.73598 − 2.23302i 0.735021 − 1.77450i −3.10702 + 0.588563i
37.12 −1.34139 + 0.447975i −0.672289 − 1.00615i 1.59864 − 1.20182i 2.23426 + 0.0898960i 1.35253 + 1.04847i 2.33408 − 0.464277i −1.60601 + 2.32825i 0.587682 − 1.41879i −3.03728 + 0.880308i
37.13 −1.33961 + 0.453250i 1.50989 + 2.25971i 1.58913 − 1.21436i −1.01577 + 1.99204i −3.04688 − 2.34278i 1.91012 − 0.379947i −1.57841 + 2.34704i −1.67846 + 4.05216i 0.457849 − 3.12896i
37.14 −1.33544 − 0.465393i −1.38781 − 2.07701i 1.56682 + 1.24301i 2.15229 + 0.606353i 0.886721 + 3.41961i 2.34627 − 0.466702i −1.51391 − 2.38916i −1.23989 + 2.99337i −2.59206 − 1.81141i
37.15 −1.27338 − 0.615219i 1.38781 + 2.07701i 1.24301 + 1.56682i −2.15229 − 0.606353i −0.489403 − 3.49864i 2.34627 − 0.466702i −0.618894 − 2.75989i −1.23989 + 2.99337i 2.36765 + 2.09625i
37.16 −1.26524 + 0.631804i −1.84215 − 2.75697i 1.20165 − 1.59876i −0.219679 + 2.22525i 4.07262 + 2.32434i 4.13937 − 0.823371i −0.510265 + 2.78202i −3.05933 + 7.38587i −1.12798 − 2.95426i
37.17 −1.25304 − 0.655652i 0.577178 + 0.863808i 1.14024 + 1.64312i −1.97239 + 1.05341i −0.156872 − 1.46082i −3.87241 + 0.770271i −0.351454 − 2.80651i 0.735021 − 1.77450i 3.16216 − 0.0267698i
37.18 −1.20201 − 0.745099i 1.73274 + 2.59324i 0.889656 + 1.79123i 1.99136 + 1.01710i −0.150560 − 4.40816i −1.15970 + 0.230678i 0.265270 − 2.81596i −2.57442 + 6.21519i −1.63580 − 2.70632i
37.19 −1.20119 + 0.746422i 1.51859 + 2.27273i 0.885710 − 1.79319i 2.00008 − 0.999838i −3.52053 − 1.59647i 3.78525 − 0.752933i 0.274568 + 2.81507i −1.71114 + 4.13105i −1.65617 + 2.69390i
37.20 −1.18388 + 0.773579i 0.545146 + 0.815869i 0.803152 − 1.83165i 0.176872 − 2.22906i −1.27653 − 0.544179i 0.186072 − 0.0370121i 0.466089 + 2.78976i 0.779592 − 1.88210i 1.51496 + 2.77577i
See next 80 embeddings (of 832 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.104
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
85.r even 16 1 inner
680.cf even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 680.2.cf.a ✓ 832
5.c odd 4 1 680.2.ct.a yes 832
8.b even 2 1 inner 680.2.cf.a ✓ 832
17.e odd 16 1 680.2.ct.a yes 832
40.i odd 4 1 680.2.ct.a yes 832
85.r even 16 1 inner 680.2.cf.a ✓ 832
136.q odd 16 1 680.2.ct.a yes 832
680.cf even 16 1 inner 680.2.cf.a ✓ 832
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
680.2.cf.a ✓ 832 1.a even 1 1 trivial
680.2.cf.a ✓ 832 8.b even 2 1 inner
680.2.cf.a ✓ 832 85.r even 16 1 inner
680.2.cf.a ✓ 832 680.cf even 16 1 inner
680.2.ct.a yes 832 5.c odd 4 1
680.2.ct.a yes 832 17.e odd 16 1
680.2.ct.a yes 832 40.i odd 4 1
680.2.ct.a yes 832 136.q odd 16 1

Hecke kernels

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