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Properties

Label 561.2.bg.a
Level $561$
Weight $2$
Character orbit 561.bg
Analytic conductor $4.480$
Analytic rank $0$
Dimension $1088$
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] = [561,2,Mod(2,561)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(561, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([20, 4, 35]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("561.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 561 = 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 561.bg (of order \(40\), degree \(16\), 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: \(4.47960755339\)
Analytic rank: \(0\)
Dimension: \(1088\)
Relative dimension: \(68\) over \(\Q(\zeta_{40})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{40}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1088 q - 12 q^{3} - 20 q^{6} - 40 q^{7} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1088 q - 12 q^{3} - 20 q^{6} - 40 q^{7} - 36 q^{9} - 80 q^{12} - 36 q^{15} + 176 q^{16} - 40 q^{18} - 40 q^{19} - 128 q^{22} - 80 q^{24} - 24 q^{25} - 12 q^{27} - 40 q^{28} - 32 q^{31} - 48 q^{33} - 96 q^{34} + 56 q^{36} - 40 q^{37} - 20 q^{39} - 200 q^{40} + 12 q^{42} - 32 q^{45} - 40 q^{46} + 60 q^{48} - 152 q^{49} - 20 q^{51} + 80 q^{52} - 80 q^{57} + 88 q^{58} - 124 q^{60} - 40 q^{61} + 40 q^{63} - 152 q^{66} - 64 q^{67} - 120 q^{69} - 56 q^{70} - 40 q^{73} + 72 q^{75} - 112 q^{78} - 40 q^{79} + 144 q^{82} - 440 q^{84} + 120 q^{85} - 8 q^{88} - 20 q^{90} - 88 q^{91} - 44 q^{93} + 200 q^{94} - 20 q^{96} - 208 q^{97} + 108 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −2.43749 + 1.24196i 1.71711 + 0.227014i 3.22332 − 4.43652i 0.243983 + 0.285667i −4.46738 + 1.57924i 1.24572 + 2.03282i −1.49092 + 9.41327i 2.89693 + 0.779616i −0.949495 − 0.393294i
2.2 −2.41004 + 1.22798i −1.44828 − 0.949994i 3.12481 − 4.30093i 1.85641 + 2.17358i 4.65699 + 0.511071i 1.36980 + 2.23531i −1.40321 + 8.85953i 1.19502 + 2.75171i −7.14314 − 2.95879i
2.3 −2.37555 + 1.21040i 0.187206 + 1.72190i 3.00259 − 4.13271i −0.468402 − 0.548428i −2.52891 − 3.86388i −0.962729 − 1.57103i −1.29641 + 8.18518i −2.92991 + 0.644700i 1.77653 + 0.735864i
2.4 −2.34006 + 1.19232i −1.73144 + 0.0459559i 2.87868 − 3.96217i −2.22548 − 2.60570i 3.99688 − 2.17197i −1.98898 − 3.24572i −1.19043 + 7.51606i 2.99578 − 0.159140i 8.31459 + 3.44402i
2.5 −2.26325 + 1.15318i 1.33056 − 1.10888i 2.61689 − 3.60185i −1.06811 − 1.25060i −1.73265 + 4.04404i −1.27911 − 2.08731i −0.974377 + 6.15197i 0.540784 − 2.95086i 3.85958 + 1.59869i
2.6 −2.15909 + 1.10011i −1.25040 + 1.19854i 2.27584 − 3.13243i 2.54319 + 2.97769i 1.38120 − 3.96333i −0.819146 − 1.33673i −0.709581 + 4.48012i 0.127006 − 2.99731i −8.76675 − 3.63131i
2.7 −2.06727 + 1.05333i −0.840155 + 1.51464i 1.98854 − 2.73699i −0.250081 − 0.292807i 0.141413 − 4.01613i 2.08400 + 3.40078i −0.502000 + 3.16950i −1.58828 − 2.54507i 0.825406 + 0.341895i
2.8 −1.95271 + 0.994954i −0.608597 − 1.62161i 1.64756 − 2.26767i 0.331698 + 0.388368i 2.80184 + 2.56100i −0.916459 − 1.49552i −0.275297 + 1.73816i −2.25922 + 1.97381i −1.03412 − 0.428345i
2.9 −1.91214 + 0.974284i 1.63463 + 0.572703i 1.53148 − 2.10790i 1.35996 + 1.59231i −3.68361 + 0.497505i −1.74686 − 2.85061i −0.203279 + 1.28345i 2.34402 + 1.87231i −4.15180 − 1.71973i
2.10 −1.87568 + 0.955704i 0.943755 − 1.45235i 1.42922 − 1.96715i −1.81116 − 2.12059i −0.382158 + 3.62609i 1.54065 + 2.51410i −0.142108 + 0.897233i −1.21865 − 2.74133i 5.42381 + 2.24661i
2.11 −1.85245 + 0.943869i 0.324197 − 1.70144i 1.36510 − 1.87890i 1.44642 + 1.69354i 1.00538 + 3.45783i −0.301367 − 0.491787i −0.104871 + 0.662128i −2.78979 − 1.10320i −4.27790 − 1.77197i
2.12 −1.84900 + 0.942112i −1.72785 − 0.120498i 1.35565 − 1.86589i 0.0239927 + 0.0280918i 3.30832 − 1.40503i 0.280926 + 0.458430i −0.0994566 + 0.627944i 2.97096 + 0.416406i −0.0708280 − 0.0293379i
2.13 −1.82621 + 0.930503i 1.14675 + 1.29806i 1.29365 − 1.78056i −2.66718 − 3.12287i −3.30206 − 1.30348i −0.00134955 − 0.00220227i −0.0644120 + 0.406681i −0.369920 + 2.97711i 7.77669 + 3.22121i
2.14 −1.78521 + 0.909609i 1.65385 − 0.514557i 1.18401 − 1.62965i 2.40270 + 2.81321i −2.48443 + 2.42295i 0.0552864 + 0.0902191i −0.00450014 + 0.0284128i 2.47046 − 1.70200i −6.84825 − 2.83664i
2.15 −1.67962 + 0.855810i 1.32729 + 1.11279i 0.913146 − 1.25684i −0.168342 − 0.197103i −3.18168 − 0.733168i 2.27429 + 3.71130i 0.131659 − 0.831260i 0.523378 + 2.95399i 0.451433 + 0.186990i
2.16 −1.41191 + 0.719405i 0.0781856 + 1.73029i 0.300380 − 0.413438i 1.68040 + 1.96749i −1.35517 − 2.38676i 1.24094 + 2.02503i 0.369099 − 2.33040i −2.98777 + 0.270567i −3.78800 − 1.56904i
2.17 −1.38159 + 0.703955i 0.283208 + 1.70874i 0.237665 − 0.327118i −0.0536469 − 0.0628124i −1.59415 − 2.16141i −1.46189 − 2.38558i 0.387053 − 2.44376i −2.83959 + 0.967857i 0.118335 + 0.0490160i
2.18 −1.32163 + 0.673407i −1.62344 − 0.603692i 0.117672 − 0.161961i −0.756509 − 0.885758i 2.55212 − 0.295374i 0.0657892 + 0.107358i 0.417627 − 2.63679i 2.27111 + 1.96012i 1.59630 + 0.661210i
2.19 −1.28628 + 0.655390i −0.982136 + 1.42668i 0.0493988 − 0.0679917i −2.40795 − 2.81934i 0.328267 − 2.47878i −0.560142 − 0.914069i 0.432685 − 2.73186i −1.07082 − 2.80238i 4.94506 + 2.04831i
2.20 −1.17383 + 0.598098i −0.762473 − 1.55520i −0.155406 + 0.213898i −1.90960 − 2.23586i 1.82518 + 1.36951i −2.17642 − 3.55160i 0.466670 − 2.94644i −1.83727 + 2.37159i 3.57882 + 1.48239i
See next 80 embeddings (of 1088 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.68
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
17.d even 8 1 inner
33.f even 10 1 inner
51.g odd 8 1 inner
187.q odd 40 1 inner
561.bg even 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 561.2.bg.a ✓ 1088
3.b odd 2 1 inner 561.2.bg.a ✓ 1088
11.d odd 10 1 inner 561.2.bg.a ✓ 1088
17.d even 8 1 inner 561.2.bg.a ✓ 1088
33.f even 10 1 inner 561.2.bg.a ✓ 1088
51.g odd 8 1 inner 561.2.bg.a ✓ 1088
187.q odd 40 1 inner 561.2.bg.a ✓ 1088
561.bg even 40 1 inner 561.2.bg.a ✓ 1088
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
561.2.bg.a ✓ 1088 1.a even 1 1 trivial
561.2.bg.a ✓ 1088 3.b odd 2 1 inner
561.2.bg.a ✓ 1088 11.d odd 10 1 inner
561.2.bg.a ✓ 1088 17.d even 8 1 inner
561.2.bg.a ✓ 1088 33.f even 10 1 inner
561.2.bg.a ✓ 1088 51.g odd 8 1 inner
561.2.bg.a ✓ 1088 187.q odd 40 1 inner
561.2.bg.a ✓ 1088 561.bg even 40 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(561, [\chi])\).