↑

Properties

Label 560.2.cq.a
Level $560$
Weight $2$
Character orbit 560.cq
Analytic conductor $4.472$
Analytic rank $0$
Dimension $256$
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] = [560,2,Mod(131,560)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(560, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 9, 0, 10])) 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("560.131"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.cq (of order \(12\), degree \(4\), 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: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(256\)
Relative dimension: \(64\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 256 q + 4 q^{4} - 8 q^{11} - 16 q^{14} - 20 q^{16} + 40 q^{18} + 40 q^{22} + 16 q^{23} - 44 q^{28} + 32 q^{29} + 20 q^{32} - 16 q^{37} - 60 q^{42} - 16 q^{43} - 48 q^{44} - 20 q^{46} - 8 q^{50} + 40 q^{51}+ \cdots - 80 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
131.1 −1.41271 − 0.0650974i −0.881367 + 3.28931i 1.99152 + 0.183928i 0.965926 − 0.258819i 1.45925 − 4.58948i 1.65139 + 2.06710i −2.80148 − 0.389481i −7.44466 − 4.29817i −1.38143 + 0.302758i
131.2 −1.41111 + 0.0936383i −0.452924 + 1.69034i 1.98246 − 0.264268i −0.965926 + 0.258819i 0.480846 − 2.42766i 2.56186 − 0.660959i −2.77273 + 0.558546i −0.0540195 − 0.0311882i 1.33879 − 0.455670i
131.3 −1.40318 − 0.176293i −0.125759 + 0.469340i 1.93784 + 0.494742i −0.965926 + 0.258819i 0.259204 − 0.636399i −2.02989 − 1.69692i −2.63193 − 1.03584i 2.39361 + 1.38195i 1.40100 − 0.192885i
131.4 −1.40292 + 0.178396i −0.435451 + 1.62512i 1.93635 − 0.500551i 0.965926 − 0.258819i 0.320985 − 2.35760i −0.451133 − 2.60701i −2.62724 + 1.04767i 0.146666 + 0.0846776i −1.30894 + 0.535419i
131.5 −1.39869 − 0.208965i 0.308027 − 1.14957i 1.91267 + 0.584555i 0.965926 − 0.258819i −0.671056 + 1.54353i −2.40355 + 1.10587i −2.55308 − 1.21729i 1.37144 + 0.791800i −1.40511 + 0.160163i
131.6 −1.37515 + 0.330105i 0.775057 − 2.89255i 1.78206 − 0.907887i 0.965926 − 0.258819i −0.110970 + 4.23354i 0.737890 + 2.54077i −2.15090 + 1.83675i −5.16806 − 2.98378i −1.24285 + 0.674772i
131.7 −1.32233 − 0.501450i 0.462065 − 1.72445i 1.49710 + 1.32616i −0.965926 + 0.258819i −1.47573 + 2.04858i 0.938017 + 2.47389i −1.31464 − 2.50434i −0.162150 − 0.0936173i 1.40705 + 0.142121i
131.8 −1.29421 + 0.570104i 0.00690551 − 0.0257717i 1.34996 − 1.47567i −0.965926 + 0.258819i 0.00575539 + 0.0372909i 0.0420874 + 2.64542i −0.905849 + 2.67945i 2.59746 + 1.49964i 1.10256 − 0.885645i
131.9 −1.27472 − 0.612435i −0.369093 + 1.37747i 1.24985 + 1.56137i 0.965926 − 0.258819i 1.31410 − 1.52985i −2.52764 + 0.781698i −0.636973 − 2.75577i 0.836876 + 0.483170i −1.38980 − 0.261644i
131.10 −1.26130 − 0.639623i 0.527774 − 1.96968i 1.18177 + 1.61351i 0.965926 − 0.258819i −1.92554 + 2.14679i 1.01863 − 2.44180i −0.458523 − 2.79101i −1.00302 − 0.579095i −1.38387 − 0.291379i
131.11 −1.23165 + 0.695008i −0.817954 + 3.05265i 1.03393 − 1.71201i −0.965926 + 0.258819i −1.11418 − 4.32828i −2.26433 − 1.36851i −0.0835767 + 2.82719i −6.05152 − 3.49385i 1.00980 − 0.990101i
131.12 −1.17180 + 0.791760i 0.282699 − 1.05505i 0.746231 − 1.85557i 0.965926 − 0.258819i 0.504077 + 1.46013i 2.60058 + 0.486822i 0.594732 + 2.76519i 1.56487 + 0.903479i −0.926949 + 1.06807i
131.13 −1.13884 − 0.838476i −0.589179 + 2.19885i 0.593914 + 1.90978i −0.965926 + 0.258819i 2.51466 − 2.01012i −1.24964 + 2.33204i 0.924933 − 2.67292i −1.88972 − 1.09103i 1.31705 + 0.515152i
131.14 −1.12144 + 0.861608i 0.282922 − 1.05588i 0.515264 − 1.93249i −0.965926 + 0.258819i 0.592473 + 1.42787i 1.08456 − 2.41324i 1.08721 + 2.61113i 1.56324 + 0.902538i 0.860229 − 1.12250i
131.15 −1.09371 − 0.896547i 0.828070 − 3.09040i 0.392407 + 1.96113i −0.965926 + 0.258819i −3.67636 + 2.63760i −2.63963 − 0.179838i 1.32906 − 2.49672i −6.26680 − 3.61814i 1.28849 + 0.582925i
131.16 −1.06468 − 0.930838i −0.713853 + 2.66413i 0.267080 + 1.98209i −0.965926 + 0.258819i 3.23990 − 2.17196i 1.78145 − 1.95613i 1.56065 − 2.35889i −3.98995 − 2.30360i 1.26932 + 0.623562i
131.17 −1.00283 − 0.997167i 0.0989001 − 0.369100i 0.0113170 + 1.99997i −0.965926 + 0.258819i −0.467234 + 0.271523i −0.933440 − 2.47562i 1.98295 − 2.01690i 2.47162 + 1.42699i 1.22674 + 0.703639i
131.18 −0.974272 + 1.02508i 0.756910 − 2.82483i −0.101589 − 1.99742i −0.965926 + 0.258819i 2.15825 + 3.52805i −2.21995 + 1.43938i 2.14649 + 1.84189i −4.80866 − 2.77628i 0.675763 − 1.24231i
131.19 −0.936556 − 1.05965i −0.198048 + 0.739127i −0.245727 + 1.98485i 0.965926 − 0.258819i 0.968701 − 0.482371i 2.08724 + 1.62586i 2.33339 − 1.59853i 2.09099 + 1.20723i −1.17890 − 0.781147i
131.20 −0.929742 + 1.06564i −0.517131 + 1.92996i −0.271161 − 1.98153i 0.965926 − 0.258819i −1.57583 − 2.34543i −0.249933 − 2.63392i 2.36370 + 1.55335i −0.859235 − 0.496080i −0.622254 + 1.26996i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 131.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
16.f odd 4 1 inner
112.v even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.cq.a ✓ 256
7.d odd 6 1 inner 560.2.cq.a ✓ 256
16.f odd 4 1 inner 560.2.cq.a ✓ 256
112.v even 12 1 inner 560.2.cq.a ✓ 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.cq.a ✓ 256 1.a even 1 1 trivial
560.2.cq.a ✓ 256 7.d odd 6 1 inner
560.2.cq.a ✓ 256 16.f odd 4 1 inner
560.2.cq.a ✓ 256 112.v even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(560, [\chi])\).