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Properties

Label 552.2.t.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 24 q^{3} - 4 q^{4} + 11 q^{6} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 24 q^{3} - 4 q^{4} + 11 q^{6} - 24 q^{9} + 4 q^{12} - 4 q^{16} + 44 q^{17} - 55 q^{20} - 24 q^{25} + 24 q^{27} + 33 q^{34} + 7 q^{36} - 132 q^{38} + 11 q^{40} - 66 q^{42} + 77 q^{44} + 37 q^{48} - 56 q^{49} - 37 q^{50} - 55 q^{52} - 11 q^{54} + 132 q^{56} + 37 q^{58} + 11 q^{60} + 51 q^{62} - 4 q^{64} - 55 q^{66} - 132 q^{70} + 144 q^{73} + 44 q^{74} + 24 q^{75} - 55 q^{76} - 198 q^{80} - 24 q^{81} - 26 q^{82} + 44 q^{84} - 110 q^{86} + 22 q^{88} - 139 q^{92} - 117 q^{94} - 209 q^{98}+O(q^{100}) \) 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 −1.40793 − 0.133210i 0.959493 − 0.281733i 1.96451 + 0.375100i 2.14099 + 1.37593i −1.38842 + 0.268844i 0.0311079 + 0.216360i −2.71592 − 0.789806i 0.841254 − 0.540641i −2.83107 − 2.22241i
19.2 −1.40783 − 0.134195i 0.959493 − 0.281733i 1.96398 + 0.377848i −2.06189 − 1.32509i −1.38861 + 0.267873i 0.593065 + 4.12486i −2.71425 − 0.795503i 0.841254 − 0.540641i 2.72497 + 2.14220i
19.3 −1.36604 + 0.365971i 0.959493 − 0.281733i 1.73213 − 0.999862i −0.824449 − 0.529841i −1.20760 + 0.736005i −0.536086 − 3.72856i −2.00024 + 1.99976i 0.841254 − 0.540641i 1.32014 + 0.422060i
19.4 −1.23359 + 0.691558i 0.959493 − 0.281733i 1.04350 − 1.70620i −1.23207 − 0.791805i −0.988788 + 1.01109i 0.0805522 + 0.560253i −0.107310 + 2.82639i 0.841254 − 0.540641i 2.06745 + 0.124714i
19.5 −1.05818 − 0.938216i 0.959493 − 0.281733i 0.239503 + 1.98561i 0.0143700 + 0.00923506i −1.27965 − 0.602087i −0.416387 − 2.89604i 1.60949 − 2.32584i 0.841254 − 0.540641i −0.00654165 − 0.0232546i
19.6 −0.969131 − 1.02994i 0.959493 − 0.281733i −0.121568 + 1.99630i −1.74343 − 1.12044i −1.22004 − 0.715188i 0.425534 + 2.95966i 2.17389 − 1.80947i 0.841254 − 0.540641i 0.535630 + 2.88149i
19.7 −0.933371 + 1.06246i 0.959493 − 0.281733i −0.257636 − 1.98334i 2.60551 + 1.67446i −0.596234 + 1.28238i 0.418423 + 2.91019i 2.34768 + 1.57746i 0.841254 − 0.540641i −4.21095 + 1.20535i
19.8 −0.865054 − 1.11879i 0.959493 − 0.281733i −0.503365 + 1.93562i 3.59098 + 2.30779i −1.14521 − 0.829754i 0.294270 + 2.04669i 2.60098 − 1.11126i 0.841254 − 0.540641i −0.524475 − 6.01390i
19.9 −0.675020 + 1.24272i 0.959493 − 0.281733i −1.08870 − 1.67772i −0.991910 − 0.637462i −0.297563 + 1.38255i −0.126423 − 0.879292i 2.81982 − 0.220446i 0.841254 − 0.540641i 1.46175 − 0.802365i
19.10 −0.279033 − 1.38641i 0.959493 − 0.281733i −1.84428 + 0.773709i −3.59098 − 2.30779i −0.658327 − 1.25164i −0.294270 − 2.04669i 1.58729 + 2.34105i 0.841254 − 0.540641i −2.19754 + 5.62253i
19.11 −0.196694 + 1.40047i 0.959493 − 0.281733i −1.92262 − 0.550929i −3.35394 − 2.15545i 0.205831 + 1.39915i 0.412102 + 2.86624i 1.14973 − 2.58421i 0.841254 − 0.540641i 3.67834 − 4.27312i
19.12 −0.180888 + 1.40260i 0.959493 − 0.281733i −1.93456 − 0.507426i 1.54978 + 0.995986i 0.221597 + 1.39674i −0.318386 − 2.21442i 1.06165 − 2.62162i 0.841254 − 0.540641i −1.67730 + 1.99356i
19.13 −0.143733 − 1.40689i 0.959493 − 0.281733i −1.95868 + 0.404434i 1.74343 + 1.12044i −0.534278 − 1.30941i −0.425534 − 2.95966i 0.850523 + 2.69752i 0.841254 − 0.540641i 1.32574 − 2.61386i
19.14 −0.0160934 − 1.41412i 0.959493 − 0.281733i −1.99948 + 0.0455160i −0.0143700 − 0.00923506i −0.413846 − 1.35231i 0.416387 + 2.89604i 0.0965435 + 2.82678i 0.841254 − 0.540641i −0.0128282 + 0.0204696i
19.15 0.578628 + 1.29042i 0.959493 − 0.281733i −1.33038 + 1.49335i −1.60117 − 1.02901i 0.918744 + 1.07513i −0.727375 − 5.05900i −2.69685 − 0.852656i 0.841254 − 0.540641i 0.401374 − 2.66159i
19.16 0.596315 + 1.28234i 0.959493 − 0.281733i −1.28882 + 1.52936i 1.60117 + 1.02901i 0.933438 + 1.06240i 0.727375 + 5.05900i −2.72971 − 0.740725i 0.841254 − 0.540641i −0.364742 + 2.66686i
19.17 0.820516 − 1.15185i 0.959493 − 0.281733i −0.653506 − 1.89022i 2.06189 + 1.32509i 0.462767 − 1.33636i −0.593065 − 4.12486i −2.71346 − 0.798217i 0.841254 − 0.540641i 3.21812 − 1.28772i
19.18 0.821322 − 1.15127i 0.959493 − 0.281733i −0.650861 − 1.89113i −2.14099 − 1.37593i 0.463701 − 1.33603i −0.0311079 − 0.216360i −2.71178 − 0.803909i 0.841254 − 0.540641i −3.34252 + 1.33478i
19.19 1.17115 − 0.792724i 0.959493 − 0.281733i 0.743177 − 1.85679i 0.824449 + 0.529841i 0.900373 − 1.09056i 0.536086 + 3.72856i −0.601555 − 2.76372i 0.841254 − 0.540641i 1.38557 − 0.0330378i
19.20 1.17847 + 0.781800i 0.959493 − 0.281733i 0.777577 + 1.84265i −1.54978 − 0.995986i 1.35099 + 0.418119i 0.318386 + 2.21442i −0.524237 + 2.77942i 0.841254 − 0.540641i −1.04771 − 2.38536i
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.a ✓ 240
8.d odd 2 1 inner 552.2.t.a ✓ 240
23.d odd 22 1 inner 552.2.t.a ✓ 240
184.j even 22 1 inner 552.2.t.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.t.a ✓ 240 1.a even 1 1 trivial
552.2.t.a ✓ 240 8.d odd 2 1 inner
552.2.t.a ✓ 240 23.d odd 22 1 inner
552.2.t.a ✓ 240 184.j even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{240} + 72 T_{5}^{238} + 3184 T_{5}^{236} + 106556 T_{5}^{234} + 2971758 T_{5}^{232} + \cdots + 14\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(552, [\chi])\). Copy content Toggle raw display