Properties

Label 980.2.bk.a
Level $980$
Weight $2$
Character orbit 980.bk
Analytic conductor $7.825$
Analytic rank $0$
Dimension $1968$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [980,2,Mod(43,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([14, 21, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.bk (of order \(28\), 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: \(1968\)
Relative dimension: \(164\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1968 q - 10 q^{2} - 20 q^{5} - 20 q^{6} - 22 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1968 q - 10 q^{2} - 20 q^{5} - 20 q^{6} - 22 q^{8} - 10 q^{10} + 2 q^{12} - 20 q^{13} - 4 q^{16} - 20 q^{17} - 44 q^{18} - 26 q^{20} - 56 q^{21} - 18 q^{22} - 20 q^{25} - 4 q^{26} - 14 q^{28} - 4 q^{30} - 10 q^{32} - 28 q^{33} - 52 q^{36} - 20 q^{37} - 42 q^{38} + 30 q^{40} - 24 q^{41} + 6 q^{42} - 60 q^{45} - 20 q^{46} - 4 q^{48} + 192 q^{50} - 22 q^{52} - 20 q^{53} - 4 q^{56} - 80 q^{57} - 50 q^{58} - 34 q^{60} - 40 q^{61} - 26 q^{62} - 20 q^{65} - 256 q^{66} - 84 q^{68} - 154 q^{70} + 120 q^{72} - 36 q^{73} - 148 q^{76} - 28 q^{77} + 86 q^{78} - 40 q^{80} + 24 q^{81} + 38 q^{82} + 28 q^{85} - 108 q^{86} - 152 q^{88} - 10 q^{90} + 42 q^{92} + 40 q^{93} + 104 q^{96} - 128 q^{97} - 280 q^{98}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.41226 0.0742447i 1.98344 0.694037i 1.98898 + 0.209706i 1.71471 1.43519i −2.85267 + 0.832902i 1.76581 1.97026i −2.79339 0.443831i 1.10686 0.882694i −2.52818 + 1.89955i
43.2 −1.41030 + 0.105184i −1.21166 + 0.423978i 1.97787 0.296680i 2.23335 + 0.110254i 1.66420 0.725381i 0.246144 2.63428i −2.75818 + 0.626446i −1.05713 + 0.843035i −3.16128 + 0.0794211i
43.3 −1.40933 0.117431i 0.998651 0.349443i 1.97242 + 0.330997i −1.20433 1.88403i −1.44846 + 0.375208i −1.03437 2.43518i −2.74092 0.698107i −1.47030 + 1.17253i 1.47606 + 2.79665i
43.4 −1.40577 0.154339i 0.258629 0.0904982i 1.95236 + 0.433929i −2.21946 0.272005i −0.377539 + 0.0873028i 2.07765 1.63810i −2.67759 0.911328i −2.28680 + 1.82366i 3.07806 + 0.724925i
43.5 −1.40555 + 0.156312i 2.81238 0.984094i 1.95113 0.439409i −2.13943 + 0.650274i −3.79911 + 1.82280i 2.61319 + 0.413788i −2.67373 + 0.922596i 4.59555 3.66483i 2.90542 1.24841i
43.6 −1.40487 + 0.162324i −1.05229 + 0.368211i 1.94730 0.456088i 0.731251 2.11312i 1.41855 0.688098i −2.31427 + 1.28225i −2.66167 + 0.956836i −1.37377 + 1.09554i −0.684300 + 3.08735i
43.7 −1.40060 0.195769i −1.56249 + 0.546739i 1.92335 + 0.548388i 1.59083 + 1.57139i 2.29546 0.459874i −0.889736 + 2.49166i −2.58648 1.14460i −0.203042 + 0.161920i −1.92048 2.51232i
43.8 −1.39416 + 0.237318i 1.38411 0.484323i 1.88736 0.661720i 1.88572 + 1.20168i −1.81474 + 1.00370i 1.49236 + 2.18469i −2.47424 + 1.37045i −0.664290 + 0.529753i −2.91418 1.22782i
43.9 −1.39351 + 0.241111i −2.14193 + 0.749494i 1.88373 0.671980i −1.35940 + 1.77540i 2.80409 1.56087i 2.35594 + 1.20397i −2.46297 + 1.39060i 1.68063 1.34026i 1.46626 2.80180i
43.10 −1.39184 0.250568i −0.428666 + 0.149997i 1.87443 + 0.697500i −0.746109 2.10792i 0.634219 0.101361i 1.68855 + 2.03686i −2.43414 1.44048i −2.18424 + 1.74187i 0.510288 + 3.12083i
43.11 −1.38528 0.284609i −0.414553 + 0.145058i 1.83800 + 0.788526i −1.55889 + 1.60308i 0.615556 0.0829608i −1.96396 1.77281i −2.32171 1.61544i −2.19468 + 1.75020i 2.61574 1.77704i
43.12 −1.36731 + 0.361193i −2.75717 + 0.964776i 1.73908 0.987726i −0.559737 2.16488i 3.42144 2.31502i 2.51037 0.835501i −2.02110 + 1.97867i 4.32571 3.44964i 1.54727 + 2.75789i
43.13 −1.36674 0.363362i 2.20437 0.771341i 1.73594 + 0.993239i −1.41697 + 1.72979i −3.29306 + 0.253238i −0.756704 + 2.53523i −2.01166 1.98827i 1.91877 1.53016i 2.56517 1.84930i
43.14 −1.36143 + 0.382773i 3.08329 1.07889i 1.70697 1.04224i 1.91380 1.15645i −3.78471 + 2.64903i −1.66528 + 2.05593i −1.92498 + 2.07231i 5.99720 4.78261i −2.16284 + 2.30698i
43.15 −1.35823 0.393966i −3.16048 + 1.10590i 1.68958 + 1.07019i 0.255185 + 2.22146i 4.72835 0.256947i 0.216393 2.63689i −1.87322 2.11921i 6.42013 5.11988i 0.528578 3.11779i
43.16 −1.35733 + 0.397058i 1.33271 0.466334i 1.68469 1.07788i 1.41300 + 1.73304i −1.62376 + 1.16213i −2.51513 0.821039i −1.85870 + 2.13195i −0.786858 + 0.627498i −2.60603 1.79126i
43.17 −1.35462 + 0.406221i 0.871746 0.305037i 1.66997 1.10055i −2.06822 0.849992i −1.05697 + 0.767330i −2.43982 + 1.02337i −1.81510 + 2.16920i −1.67860 + 1.33864i 3.14692 + 0.311259i
43.18 −1.34319 0.442547i 2.84445 0.995317i 1.60830 + 1.18885i 0.866001 + 2.06156i −4.26111 + 0.0780940i −1.64391 2.07306i −1.63413 2.30859i 4.75476 3.79179i −0.250864 3.15231i
43.19 −1.33432 + 0.468610i −2.46734 + 0.863358i 1.56081 1.25055i −2.23559 0.0463024i 2.88763 2.30821i −2.62139 + 0.358211i −1.49659 + 2.40004i 2.99686 2.38992i 3.00468 0.985838i
43.20 −1.33030 0.479898i 1.61150 0.563887i 1.53940 + 1.27682i 1.35364 1.77979i −2.41438 0.0232149i −2.64196 + 0.141657i −1.43512 2.43730i −0.0665419 + 0.0530654i −2.65486 + 1.71805i
See next 80 embeddings (of 1968 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.164
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner
49.e even 7 1 inner
196.k odd 14 1 inner
245.r odd 28 1 inner
980.bk even 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.bk.a 1968
4.b odd 2 1 inner 980.2.bk.a 1968
5.c odd 4 1 inner 980.2.bk.a 1968
20.e even 4 1 inner 980.2.bk.a 1968
49.e even 7 1 inner 980.2.bk.a 1968
196.k odd 14 1 inner 980.2.bk.a 1968
245.r odd 28 1 inner 980.2.bk.a 1968
980.bk even 28 1 inner 980.2.bk.a 1968
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.bk.a 1968 1.a even 1 1 trivial
980.2.bk.a 1968 4.b odd 2 1 inner
980.2.bk.a 1968 5.c odd 4 1 inner
980.2.bk.a 1968 20.e even 4 1 inner
980.2.bk.a 1968 49.e even 7 1 inner
980.2.bk.a 1968 196.k odd 14 1 inner
980.2.bk.a 1968 245.r odd 28 1 inner
980.2.bk.a 1968 980.bk even 28 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(980, [\chi])\).