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Properties

Label 980.2.bl.b
Level $980$
Weight $2$
Character orbit 980.bl
Analytic conductor $7.825$
Analytic rank $0$
Dimension $1920$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [980,2,Mod(59,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 21, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.59");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.bl (of order \(42\), degree \(12\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(1920\)
Relative dimension: \(160\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 1920 q - 34 q^{4} - 22 q^{5} - 260 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1920 q - 34 q^{4} - 22 q^{5} - 260 q^{9} - 2 q^{10} - 50 q^{14} - 46 q^{16} - 14 q^{20} - 4 q^{21} + 20 q^{24} - 2 q^{25} - 10 q^{26} + 9 q^{30} - 112 q^{34} - 152 q^{36} - 112 q^{40} - 56 q^{41} - 42 q^{44} - 64 q^{45} + 38 q^{46} - 56 q^{49} + 82 q^{50} - 202 q^{54} - 12 q^{56} - 40 q^{60} - 92 q^{61} + 8 q^{64} - 16 q^{65} - 102 q^{66} - 56 q^{69} + 30 q^{70} - 54 q^{74} + 238 q^{76} - 156 q^{80} - 136 q^{81} - 84 q^{84} - 64 q^{85} - 44 q^{86} + 52 q^{89} - 294 q^{90} - 58 q^{94} - 214 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 −1.41399 − 0.0249926i −0.0856684 − 0.125652i 1.99875 + 0.0706788i 1.96128 − 1.07396i 0.117994 + 0.179813i 1.57451 + 2.12624i −2.82445 − 0.149893i 1.08757 − 2.77109i −2.80007 + 1.46956i
59.2 −1.41392 + 0.0287451i −1.30270 − 1.91071i 1.99835 − 0.0812867i −1.41084 + 1.73480i 1.89684 + 2.66415i −0.196841 − 2.63842i −2.82317 + 0.172376i −0.857770 + 2.18556i 1.94494 − 2.49343i
59.3 −1.41335 − 0.0493501i 1.10356 + 1.61863i 1.99513 + 0.139498i −0.560887 − 2.16458i −1.47984 − 2.34215i 2.09550 − 1.61520i −2.81294 − 0.295620i −0.306081 + 0.779881i 0.685908 + 3.08699i
59.4 −1.40928 − 0.118000i −1.11737 − 1.63888i 1.97215 + 0.332590i −1.05778 + 1.97005i 1.38130 + 2.44150i −0.942164 + 2.47231i −2.74007 − 0.701427i −0.341393 + 0.869854i 1.72317 − 2.65154i
59.5 −1.40698 + 0.142862i −0.116846 − 0.171382i 1.95918 − 0.402007i 0.807128 + 2.08532i 0.188884 + 0.224438i 1.48590 − 2.18909i −2.69910 + 0.845508i 1.08030 − 2.75257i −1.43352 − 2.81869i
59.6 −1.39504 + 0.232089i −1.61343 − 2.36647i 1.89227 − 0.647547i 1.16444 − 1.90895i 2.80003 + 2.92686i 2.64562 − 0.0267989i −2.48950 + 1.34253i −1.90099 + 4.84365i −1.18140 + 2.93331i
59.7 −1.39235 + 0.247709i 1.82125 + 2.67129i 1.87728 − 0.689796i −1.16734 − 1.90717i −3.19752 − 3.26823i −1.76376 − 1.97209i −2.44296 + 1.42546i −2.72279 + 6.93755i 2.09778 + 2.36629i
59.8 −1.38800 + 0.271044i −0.323569 − 0.474588i 1.85307 − 0.752417i −2.08678 − 0.803334i 0.577747 + 0.571025i 2.64498 + 0.0638876i −2.36812 + 1.54662i 0.975486 − 2.48550i 3.11418 + 0.549415i
59.9 −1.38620 + 0.280102i 0.443060 + 0.649850i 1.84309 − 0.776552i 0.816771 + 2.08156i −0.796193 − 0.776718i −2.33610 + 1.24203i −2.33737 + 1.59271i 0.870020 − 2.21678i −1.71525 − 2.65667i
59.10 −1.38183 + 0.300890i 1.49324 + 2.19018i 1.81893 − 0.831561i −1.75777 + 1.38211i −2.72242 − 2.57717i 1.01084 + 2.44504i −2.26325 + 1.69638i −1.47111 + 3.74832i 2.01308 − 2.43875i
59.11 −1.38145 − 0.302636i −1.11737 − 1.63888i 1.81682 + 0.836154i −1.33958 − 1.79040i 1.04761 + 2.60219i −0.942164 + 2.47231i −2.25681 − 1.70494i −0.341393 + 0.869854i 1.30873 + 2.87875i
59.12 −1.36511 − 0.369435i 1.10356 + 1.61863i 1.72704 + 1.00864i −0.232008 + 2.22400i −0.908501 − 2.61729i 2.09550 − 1.61520i −1.98496 − 2.01493i −0.306081 + 0.779881i 1.13834 − 2.95029i
59.13 −1.36086 + 0.384786i 1.67254 + 2.45317i 1.70388 − 1.04728i 2.04256 + 0.909917i −3.22005 − 2.69485i 2.63898 + 0.189203i −1.91576 + 2.08083i −2.12463 + 5.41346i −3.12976 − 0.452322i
59.14 −1.35854 − 0.392899i −0.0856684 − 0.125652i 1.69126 + 1.06754i 2.09944 + 0.769656i 0.0670151 + 0.204363i 1.57451 + 2.12624i −1.87821 − 2.11479i 1.08757 − 2.77109i −2.54977 − 1.87048i
59.15 −1.34668 + 0.431812i −0.516599 − 0.757711i 1.62708 − 1.16302i 0.496082 − 2.18034i 1.02288 + 0.797319i −2.29749 − 1.31207i −1.68894 + 2.26881i 0.788771 − 2.00976i 0.273437 + 3.15043i
59.16 −1.34263 − 0.444229i −1.30270 − 1.91071i 1.60532 + 1.19287i −1.65364 − 1.50515i 0.900256 + 3.14408i −0.196841 − 2.63842i −1.62545 − 2.31472i −0.857770 + 2.18556i 1.55159 + 2.75546i
59.17 −1.31005 + 0.532703i 1.01817 + 1.49338i 1.43245 − 1.39573i 0.779858 − 2.09567i −2.12938 − 1.41402i −0.382858 + 2.61790i −1.13307 + 2.59155i −0.0974946 + 0.248412i 0.0947163 + 3.16086i
59.18 −1.30236 − 0.551229i −0.116846 − 0.171382i 1.39229 + 1.43580i 0.487312 − 2.18232i 0.0577053 + 0.287610i 1.48590 − 2.18909i −1.02181 − 2.63740i 1.08030 − 2.75257i −1.83762 + 2.57355i
59.19 −1.29210 + 0.574875i −1.52198 − 2.23233i 1.33904 − 1.48559i 1.76070 + 1.37838i 3.24985 + 2.00944i −0.817448 + 2.51630i −0.876137 + 2.68931i −1.57086 + 4.00248i −3.06739 − 0.768823i
59.20 −1.26465 − 0.632973i −1.61343 − 2.36647i 1.19869 + 1.60098i 1.43595 + 1.71407i 0.542518 + 4.01402i 2.64562 − 0.0267989i −0.502547 − 2.78342i −1.90099 + 4.84365i −0.731016 − 3.07662i
See next 80 embeddings (of 1920 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.160
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner
49.h odd 42 1 inner
196.p even 42 1 inner
245.u odd 42 1 inner
980.bl even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.bl.b ✓ 1920
4.b odd 2 1 inner 980.2.bl.b ✓ 1920
5.b even 2 1 inner 980.2.bl.b ✓ 1920
20.d odd 2 1 inner 980.2.bl.b ✓ 1920
49.h odd 42 1 inner 980.2.bl.b ✓ 1920
196.p even 42 1 inner 980.2.bl.b ✓ 1920
245.u odd 42 1 inner 980.2.bl.b ✓ 1920
980.bl even 42 1 inner 980.2.bl.b ✓ 1920
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.bl.b ✓ 1920 1.a even 1 1 trivial
980.2.bl.b ✓ 1920 4.b odd 2 1 inner
980.2.bl.b ✓ 1920 5.b even 2 1 inner
980.2.bl.b ✓ 1920 20.d odd 2 1 inner
980.2.bl.b ✓ 1920 49.h odd 42 1 inner
980.2.bl.b ✓ 1920 196.p even 42 1 inner
980.2.bl.b ✓ 1920 245.u odd 42 1 inner
980.2.bl.b ✓ 1920 980.bl even 42 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{960} + 185 T_{3}^{958} + 16603 T_{3}^{956} + 955290 T_{3}^{954} + 39140587 T_{3}^{952} + \cdots + 53\!\cdots\!36 \) acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\). Copy content Toggle raw display