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Properties

Label 680.2.br.a
Level $680$
Weight $2$
Character orbit 680.br
Analytic conductor $5.430$
Analytic rank $0$
Dimension $416$
Inner twists $8$

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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(189,680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(680, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 4, 4, 7])) 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("680.189"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 680.br (of order \(8\), 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: \(5.42982733745\)
Analytic rank: \(0\)
Dimension: \(416\)
Relative dimension: \(104\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 416 q - 16 q^{6} - 16 q^{9} - 4 q^{10} - 24 q^{14} - 8 q^{15} - 16 q^{16} - 20 q^{20} + 24 q^{26} - 16 q^{31} - 40 q^{34} - 8 q^{36} + 32 q^{39} + 56 q^{40} - 32 q^{41} - 24 q^{44} + 32 q^{46} - 16 q^{49}+ \cdots - 160 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
189.1 −1.40988 − 0.110686i 2.70861 − 1.12194i 1.97550 + 0.312106i 1.69735 + 1.45568i −3.94299 + 1.28200i 2.90462 + 1.20313i −2.75066 − 0.658689i 3.95650 − 3.95650i −2.23193 − 2.24020i
189.2 −1.40588 + 0.153322i 0.993360 − 0.411463i 1.95298 − 0.431104i 2.22386 − 0.233300i −1.33346 + 0.730771i −2.22932 − 0.923416i −2.67956 + 0.905515i −1.30386 + 1.30386i −3.09071 + 0.668958i
189.3 −1.40379 − 0.171400i −0.526477 + 0.218074i 1.94124 + 0.481218i −2.20426 − 0.375794i 0.776440 − 0.215892i −0.947853 − 0.392614i −2.64262 − 1.00826i −1.89170 + 1.89170i 3.02991 + 0.905346i
189.4 −1.40155 − 0.188830i 0.915206 − 0.379091i 1.92869 + 0.529309i 0.0790838 + 2.23467i −1.35429 + 0.358496i 3.01207 + 1.24764i −2.60320 − 1.10605i −1.42743 + 1.42743i 0.311132 − 3.14693i
189.5 −1.39904 + 0.206585i −2.16008 + 0.894733i 1.91465 − 0.578042i 1.10543 − 1.94371i 2.83720 − 1.69801i 4.44461 + 1.84102i −2.55926 + 1.20424i 1.74407 − 1.74407i −1.14501 + 2.94770i
189.6 −1.39620 + 0.224998i −0.349327 + 0.144696i 1.89875 − 0.628285i −0.605891 − 2.15242i 0.455174 − 0.280622i −0.459864 − 0.190482i −2.50968 + 1.30443i −2.02023 + 2.02023i 1.33023 + 2.86888i
189.7 −1.39544 − 0.229661i 3.09876 − 1.28355i 1.89451 + 0.640958i −0.179138 − 2.22888i −4.61892 + 1.07945i −3.68766 − 1.52748i −2.49648 − 1.32952i 5.83350 − 5.83350i −0.261911 + 3.15141i
189.8 −1.38778 − 0.272134i −2.09716 + 0.868672i 1.85189 + 0.755326i 1.19071 + 1.89267i 3.14680 − 0.634820i −0.787192 − 0.326065i −2.36447 − 1.55219i 1.52217 − 1.52217i −1.13738 − 2.95065i
189.9 −1.38576 + 0.282254i −2.43492 + 1.00858i 1.84066 − 0.782274i −1.03845 + 1.98031i 3.08954 − 2.08491i 2.82991 + 1.17219i −2.32992 + 1.60358i 2.79029 − 2.79029i 0.880085 − 3.03734i
189.10 −1.37975 + 0.310318i 1.71038 − 0.708465i 1.80741 − 0.856321i −1.93386 − 1.12258i −2.14005 + 1.50827i 3.10654 + 1.28677i −2.22803 + 1.74238i 0.302174 − 0.302174i 3.01659 + 0.948770i
189.11 −1.36915 − 0.354165i 1.54407 − 0.639573i 1.74913 + 0.969808i −1.61229 + 1.54936i −2.34057 + 0.328817i −2.60037 − 1.07711i −2.05135 − 1.94729i −0.146235 + 0.146235i 2.75619 − 1.55029i
189.12 −1.35659 + 0.399586i −1.50278 + 0.622470i 1.68066 − 1.08415i −2.13034 + 0.679452i 1.78992 − 1.44492i −3.52757 − 1.46117i −1.84676 + 2.14231i −0.250453 + 0.250453i 2.61849 − 1.77299i
189.13 −1.32984 − 0.481164i 0.337370 − 0.139743i 1.53696 + 1.27974i 1.90443 − 1.17181i −0.515889 + 0.0235062i −0.589730 − 0.244274i −1.42815 − 2.44139i −2.02703 + 2.02703i −3.09643 + 0.641987i
189.14 −1.29630 + 0.565334i −1.35560 + 0.561507i 1.36080 − 1.46569i 2.09094 + 0.792447i 1.43983 − 1.49425i −1.50301 − 0.622567i −0.935398 + 2.66928i −0.598963 + 0.598963i −3.15849 + 0.154828i
189.15 −1.26372 + 0.634824i 2.52290 − 1.04502i 1.19400 − 1.60448i −2.11060 + 0.738481i −2.52485 + 2.92222i −0.343036 − 0.142090i −0.490320 + 2.78560i 3.15165 − 3.15165i 2.19842 − 2.27310i
189.16 −1.25660 − 0.648820i −1.86646 + 0.773113i 1.15807 + 1.63061i −0.689354 − 2.12716i 2.84700 + 0.239505i 0.739929 + 0.306488i −0.397251 − 2.80039i 0.764647 − 0.764647i −0.513900 + 3.12024i
189.17 −1.22341 + 0.709418i 2.20971 − 0.915292i 0.993451 − 1.73582i 1.27033 − 1.84018i −2.05405 + 2.68738i 2.89176 + 1.19781i 0.0160243 + 2.82838i 1.92374 − 1.92374i −0.248667 + 3.15249i
189.18 −1.17301 − 0.789962i −0.850355 + 0.352229i 0.751921 + 1.85327i −2.19191 + 0.442194i 1.27573 + 0.258579i 4.44690 + 1.84197i 0.581999 − 2.76790i −1.52228 + 1.52228i 2.92046 + 1.21282i
189.19 −1.16277 + 0.804965i 0.0813899 − 0.0337128i 0.704063 − 1.87198i 1.95265 + 1.08957i −0.0675000 + 0.104716i 3.26620 + 1.35291i 0.688213 + 2.74342i −2.11583 + 2.11583i −3.14754 + 0.304895i
189.20 −1.14802 − 0.825870i 1.43999 − 0.596464i 0.635878 + 1.89622i 0.576715 − 2.16042i −2.14574 − 0.504496i 2.02012 + 0.836761i 0.836035 − 2.70204i −0.403512 + 0.403512i −2.44630 + 2.00390i
See next 80 embeddings (of 416 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 189.104
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
17.d even 8 1 inner
40.f even 2 1 inner
85.m even 8 1 inner
136.o even 8 1 inner
680.br even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 680.2.br.a ✓ 416
5.b even 2 1 inner 680.2.br.a ✓ 416
8.b even 2 1 inner 680.2.br.a ✓ 416
17.d even 8 1 inner 680.2.br.a ✓ 416
40.f even 2 1 inner 680.2.br.a ✓ 416
85.m even 8 1 inner 680.2.br.a ✓ 416
136.o even 8 1 inner 680.2.br.a ✓ 416
680.br even 8 1 inner 680.2.br.a ✓ 416
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
680.2.br.a ✓ 416 1.a even 1 1 trivial
680.2.br.a ✓ 416 5.b even 2 1 inner
680.2.br.a ✓ 416 8.b even 2 1 inner
680.2.br.a ✓ 416 17.d even 8 1 inner
680.2.br.a ✓ 416 40.f even 2 1 inner
680.2.br.a ✓ 416 85.m even 8 1 inner
680.2.br.a ✓ 416 136.o even 8 1 inner
680.2.br.a ✓ 416 680.br even 8 1 inner

Hecke kernels

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