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Properties

Label 820.2.s.c
Level $820$
Weight $2$
Character orbit 820.s
Analytic conductor $6.548$
Analytic rank $0$
Dimension $240$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [820,2,Mod(583,820)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(820, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 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("820.583"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 820.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [240,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.54773296574\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(120\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 4 q^{6} - 12 q^{8} + 240 q^{9} - 20 q^{10} + 8 q^{14} + 8 q^{16} - 12 q^{18} - 16 q^{20} - 12 q^{24} - 16 q^{25} - 30 q^{30} - 24 q^{33} + 20 q^{34} - 8 q^{37} - 4 q^{40} - 16 q^{41} + 84 q^{42}+ \cdots - 12 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
583.1 −1.41406 − 0.0211309i −1.52685 1.99911 + 0.0597605i 1.60139 − 1.56062i 2.15906 + 0.0322637i − 3.05913i −2.82559 − 0.126748i −0.668720 −2.29743 + 2.17297i
583.2 −1.41166 + 0.0849858i −0.897148 1.98555 − 0.239942i −0.904744 − 2.04486i 1.26647 − 0.0762448i 3.82669i −2.78253 + 0.507459i −2.19513 1.45097 + 2.80975i
583.3 −1.40983 + 0.111253i 1.20792 1.97525 − 0.313697i −2.18102 − 0.493096i −1.70296 + 0.134385i − 1.66183i −2.74986 + 0.662012i −1.54093 3.12973 + 0.452536i
583.4 −1.40557 + 0.156153i 1.16025 1.95123 − 0.438966i 1.36425 + 1.77168i −1.63081 + 0.181176i 1.14648i −2.67404 + 0.921686i −1.65382 −2.19419 − 2.27718i
583.5 −1.39720 + 0.218732i 0.467001 1.90431 − 0.611223i 1.33090 − 1.79686i −0.652492 + 0.102148i − 1.64743i −2.52700 + 1.27053i −2.78191 −1.46649 + 2.80168i
583.6 −1.39711 − 0.219268i −2.16299 1.90384 + 0.612684i −0.498284 + 2.17984i 3.02194 + 0.474275i 0.295068i −2.52554 − 1.27344i 1.67853 1.17413 − 2.93623i
583.7 −1.38579 + 0.282096i −3.17733 1.84084 − 0.781854i −2.19587 + 0.422089i 4.40312 − 0.896313i 0.985785i −2.33047 + 1.60278i 7.09542 2.92395 − 1.20437i
583.8 −1.37024 − 0.349906i 2.06782 1.75513 + 0.958912i −0.600491 + 2.15393i −2.83341 − 0.723542i − 3.38406i −2.06943 − 1.92807i 1.27587 1.57649 − 2.74129i
583.9 −1.36750 + 0.360496i 3.39311 1.74009 − 0.985952i −1.17956 + 1.89964i −4.64006 + 1.22320i 0.149808i −2.02413 + 1.97558i 8.51321 0.928236 − 3.02297i
583.10 −1.36670 + 0.363517i 2.89040 1.73571 − 0.993633i 2.03850 − 0.918980i −3.95030 + 1.05071i 3.97005i −2.01099 + 1.98895i 5.35442 −2.45194 + 1.99699i
583.11 −1.35926 − 0.390402i 2.27020 1.69517 + 1.06132i −0.515280 − 2.17589i −3.08579 − 0.886291i − 0.797143i −1.88984 − 2.10440i 2.15382 −0.149072 + 3.15876i
583.12 −1.35700 − 0.398171i −3.14018 1.68292 + 1.08064i 0.999859 − 2.00007i 4.26124 + 1.25033i 1.77374i −1.85345 − 2.13652i 6.86075 −2.15318 + 2.31599i
583.13 −1.33944 − 0.453773i −0.433002 1.58818 + 1.21560i 2.22449 + 0.227258i 0.579978 + 0.196485i 2.39586i −1.57566 − 2.34889i −2.81251 −2.87644 − 1.31381i
583.14 −1.33694 − 0.461086i −2.00537 1.57480 + 1.23289i −1.66826 − 1.48893i 2.68106 + 0.924650i − 4.57903i −1.53694 − 2.37441i 1.02153 1.54384 + 2.75981i
583.15 −1.31886 + 0.510493i −2.58515 1.47879 − 1.34654i 1.64672 + 1.51272i 3.40946 − 1.31970i − 5.22815i −1.26293 + 2.53081i 3.68301 −2.94403 − 1.15443i
583.16 −1.30141 − 0.553469i 1.18609 1.38734 + 1.44058i −1.20902 + 1.88103i −1.54359 − 0.656465i 3.79382i −1.00819 − 2.64264i −1.59319 2.61453 − 1.77883i
583.17 −1.29904 + 0.559020i −2.19452 1.37499 − 1.45238i 2.19252 + 0.439175i 2.85077 − 1.22678i 3.90646i −0.974259 + 2.65534i 1.81594 −3.09367 + 0.655157i
583.18 −1.29553 − 0.567098i 3.12002 1.35680 + 1.46939i −1.84259 − 1.26683i −4.04209 − 1.76936i 3.18739i −0.924488 − 2.67307i 6.73454 1.66871 + 2.68615i
583.19 −1.27523 + 0.611379i 1.22240 1.25243 − 1.55930i 1.86406 + 1.23502i −1.55885 + 0.747352i − 2.99230i −0.643820 + 2.75418i −1.50573 −3.13217 − 0.435297i
583.20 −1.27255 − 0.616948i 2.04335 1.23875 + 1.57019i 2.20077 − 0.395737i −2.60025 − 1.26064i 1.91480i −0.607645 − 2.76238i 1.17526 −3.04473 − 0.854166i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 583.120
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
205.f odd 4 1 inner
820.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 820.2.s.c yes 240
4.b odd 2 1 inner 820.2.s.c yes 240
5.c odd 4 1 820.2.j.c ✓ 240
20.e even 4 1 820.2.j.c ✓ 240
41.c even 4 1 820.2.j.c ✓ 240
164.e odd 4 1 820.2.j.c ✓ 240
205.f odd 4 1 inner 820.2.s.c yes 240
820.s even 4 1 inner 820.2.s.c yes 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
820.2.j.c ✓ 240 5.c odd 4 1
820.2.j.c ✓ 240 20.e even 4 1
820.2.j.c ✓ 240 41.c even 4 1
820.2.j.c ✓ 240 164.e odd 4 1
820.2.s.c yes 240 1.a even 1 1 trivial
820.2.s.c yes 240 4.b odd 2 1 inner
820.2.s.c yes 240 205.f odd 4 1 inner
820.2.s.c yes 240 820.s even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(820, [\chi])\):

\( T_{3}^{120} - 240 T_{3}^{118} + 27984 T_{3}^{116} - 2112528 T_{3}^{114} + 116091792 T_{3}^{112} + \cdots + 40\!\cdots\!72 \) Copy content Toggle raw display
\( T_{13}^{120} + 804 T_{13}^{118} + 312792 T_{13}^{116} + 78459264 T_{13}^{114} + 14264927322 T_{13}^{112} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display