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Properties

Label 552.2.t.b
Level $552$
Weight $2$
Character orbit 552.t
Analytic conductor $4.408$
Analytic rank $0$
Dimension $240$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [552,2,Mod(19,552)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(552, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 11, 0, 15]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("552.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 552.t (of order \(22\), degree \(10\), 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: \(4.40774219157\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(24\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 4 q^{2} - 24 q^{3} - 4 q^{4} - 7 q^{6} + 4 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 4 q^{2} - 24 q^{3} - 4 q^{4} - 7 q^{6} + 4 q^{8} - 24 q^{9} - 4 q^{12} - 4 q^{16} - 44 q^{17} + 4 q^{18} + 55 q^{20} + 4 q^{24} - 24 q^{25} - 24 q^{27} + 4 q^{32} - 55 q^{34} + 7 q^{36} + 22 q^{38} + 11 q^{40} - 44 q^{42} + 77 q^{44} - 48 q^{46} + 51 q^{48} + 8 q^{49} - 33 q^{50} + 165 q^{52} - 7 q^{54} - 22 q^{56} - 59 q^{58} + 11 q^{60} + 29 q^{62} - 4 q^{64} + 55 q^{66} + 132 q^{70} + 4 q^{72} - 144 q^{73} - 66 q^{74} - 24 q^{75} - 55 q^{76} - 24 q^{81} - 26 q^{82} - 44 q^{84} - 110 q^{86} - 198 q^{88} - 47 q^{92} - 73 q^{94} + 4 q^{96} + 53 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.40456 − 0.164932i −0.959493 + 0.281733i 1.94559 + 0.463315i −1.02510 − 0.658795i 1.39414 − 0.237460i −0.612113 − 4.25734i −2.65630 − 0.971645i 0.841254 − 0.540641i 1.33117 + 1.09439i
19.2 −1.40243 + 0.182193i −0.959493 + 0.281733i 1.93361 − 0.511026i 2.78195 + 1.78785i 1.29429 − 0.569923i −0.0254555 − 0.177047i −2.61865 + 1.06897i 0.841254 − 0.540641i −4.22722 − 2.00048i
19.3 −1.34789 − 0.428007i −0.959493 + 0.281733i 1.63362 + 1.15381i −0.781202 − 0.502048i 1.41388 + 0.0309246i 0.233208 + 1.62200i −1.70810 − 2.25441i 0.841254 − 0.540641i 0.838095 + 1.01107i
19.4 −1.19352 + 0.758628i −0.959493 + 0.281733i 0.848969 − 1.81087i 0.579145 + 0.372194i 0.931442 − 1.06415i −0.117047 − 0.814077i 0.360517 + 2.80536i 0.841254 − 0.540641i −0.973576 − 0.00486464i
19.5 −1.07273 − 0.921553i −0.959493 + 0.281733i 0.301482 + 1.97715i 3.13971 + 2.01777i 1.28890 + 0.582001i −0.427596 − 2.97400i 1.49864 − 2.39877i 0.841254 − 0.540641i −1.50857 − 5.05792i
19.6 −1.05559 + 0.941132i −0.959493 + 0.281733i 0.228541 − 1.98690i −3.23381 − 2.07824i 0.747684 − 1.20040i −0.275643 − 1.91714i 1.62869 + 2.31244i 0.841254 − 0.540641i 5.36947 − 0.849668i
19.7 −0.753178 − 1.19696i −0.959493 + 0.281733i −0.865447 + 1.80305i 0.282372 + 0.181470i 1.05989 + 0.936284i 0.179788 + 1.25045i 2.81003 − 0.322111i 0.841254 − 0.540641i 0.00453619 − 0.474669i
19.8 −0.617779 + 1.27214i −0.959493 + 0.281733i −1.23670 − 1.57181i 2.92140 + 1.87747i 0.234350 − 1.39466i 0.376896 + 2.62137i 2.76357 − 0.602229i 0.841254 − 0.540641i −4.19319 + 2.55658i
19.9 −0.562315 + 1.29761i −0.959493 + 0.281733i −1.36760 − 1.45934i −1.62830 − 1.04644i 0.173957 − 1.40347i 0.707273 + 4.91919i 2.66268 − 0.954016i 0.841254 − 0.540641i 2.27350 − 1.52447i
19.10 −0.411379 − 1.35306i −0.959493 + 0.281733i −1.66154 + 1.11324i −0.282372 − 0.181470i 0.775916 + 1.18235i −0.179788 − 1.25045i 2.18980 + 1.79019i 0.841254 − 0.540641i −0.129377 + 0.456719i
19.11 −0.218084 + 1.39730i −0.959493 + 0.281733i −1.90488 − 0.609457i −0.659581 − 0.423887i −0.184414 − 1.40214i −0.167612 − 1.16577i 1.26702 − 2.52877i 0.841254 − 0.540641i 0.736140 − 0.829187i
19.12 0.00602308 − 1.41420i −0.959493 + 0.281733i −1.99993 − 0.0170357i −3.13971 − 2.01777i 0.392647 + 1.35861i 0.427596 + 2.97400i −0.0361376 + 2.82820i 0.841254 − 0.540641i −2.87244 + 4.42803i
19.13 0.175979 + 1.40322i −0.959493 + 0.281733i −1.93806 + 0.493874i 1.87195 + 1.20303i −0.564184 − 1.29680i −0.493720 − 3.43390i −1.03407 − 2.63262i 0.841254 − 0.540641i −1.35870 + 2.83848i
19.14 0.492284 + 1.32577i −0.959493 + 0.281733i −1.51531 + 1.30531i 2.38725 + 1.53419i −0.845855 − 1.13337i 0.401276 + 2.79093i −2.47650 − 1.36637i 0.841254 − 0.540641i −0.858775 + 3.92019i
19.15 0.559215 − 1.29895i −0.959493 + 0.281733i −1.37456 − 1.45279i 0.781202 + 0.502048i −0.170606 + 1.40389i −0.233208 − 1.62200i −2.65578 + 0.973064i 0.841254 − 0.540641i 1.08900 − 0.733992i
19.16 0.679570 + 1.24024i −0.959493 + 0.281733i −1.07637 + 1.68565i −2.38725 − 1.53419i −1.00146 − 0.998540i −0.401276 − 2.79093i −2.82208 − 0.189432i 0.841254 − 0.540641i 0.280457 − 4.00334i
19.17 0.795146 − 1.16951i −0.959493 + 0.281733i −0.735486 − 1.85986i 1.02510 + 0.658795i −0.433449 + 1.34615i 0.612113 + 4.25734i −2.75993 − 0.618702i 0.841254 − 0.540641i 1.58557 − 0.675027i
19.18 0.945243 + 1.05191i −0.959493 + 0.281733i −0.213032 + 1.98862i −1.87195 − 1.20303i −1.20331 − 0.742995i 0.493720 + 3.43390i −2.29322 + 1.65564i 0.841254 − 0.540641i −0.503969 − 3.10629i
19.19 1.05609 − 0.940573i −0.959493 + 0.281733i 0.230643 − 1.98666i −2.78195 − 1.78785i −0.748319 + 1.20001i 0.0254555 + 0.177047i −1.62502 − 2.31502i 0.841254 − 0.540641i −4.61959 + 0.728501i
19.20 1.19882 + 0.750218i −0.959493 + 0.281733i 0.874346 + 1.79875i 0.659581 + 0.423887i −1.36162 − 0.382082i 0.167612 + 1.16577i −0.301274 + 2.81234i 0.841254 − 0.540641i 0.472712 + 1.00299i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
23.d odd 22 1 inner
184.j even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 552.2.t.b ✓ 240
8.d odd 2 1 inner 552.2.t.b ✓ 240
23.d odd 22 1 inner 552.2.t.b ✓ 240
184.j even 22 1 inner 552.2.t.b ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.t.b ✓ 240 1.a even 1 1 trivial
552.2.t.b ✓ 240 8.d odd 2 1 inner
552.2.t.b ✓ 240 23.d odd 22 1 inner
552.2.t.b ✓ 240 184.j even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{240} + 72 T_{5}^{238} + 2872 T_{5}^{236} + 87380 T_{5}^{234} + 2287006 T_{5}^{232} + \cdots + 14\!\cdots\!41 \) acting on \(S_{2}^{\mathrm{new}}(552, [\chi])\). Copy content Toggle raw display