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Properties

Label 560.2.cz.c
Level $560$
Weight $2$
Character orbit 560.cz
Analytic conductor $4.472$
Analytic rank $0$
Dimension $360$
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] = [560,2,Mod(157,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 9, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.cz (of order \(12\), degree \(4\), 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: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(360\)
Relative dimension: \(90\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 360 q + 2 q^{2} - 24 q^{3} - 6 q^{5} - 8 q^{7} - 4 q^{8} + 164 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 360 q + 2 q^{2} - 24 q^{3} - 6 q^{5} - 8 q^{7} - 4 q^{8} + 164 q^{9} - 6 q^{10} - 2 q^{11} - 6 q^{12} - 4 q^{14} - 32 q^{15} - 20 q^{16} + 8 q^{18} - 30 q^{19} - 4 q^{21} - 8 q^{22} - 10 q^{23} - 12 q^{24} - 12 q^{25} - 2 q^{28} + 8 q^{29} + 2 q^{30} - 24 q^{31} + 2 q^{32} - 24 q^{35} + 16 q^{36} + 4 q^{37} + 72 q^{38} + 28 q^{39} - 36 q^{40} - 50 q^{42} - 20 q^{44} - 72 q^{45} - 4 q^{46} - 6 q^{47} + 48 q^{50} + 16 q^{51} - 36 q^{52} + 24 q^{54} - 8 q^{56} - 80 q^{57} - 46 q^{58} - 84 q^{59} - 20 q^{60} - 48 q^{61} + 58 q^{63} - 36 q^{65} - 108 q^{66} - 48 q^{68} - 16 q^{70} - 72 q^{72} + 52 q^{74} - 42 q^{75} + 152 q^{78} + 108 q^{80} - 124 q^{81} - 6 q^{82} - 28 q^{84} + 20 q^{85} + 8 q^{86} + 12 q^{88} - 48 q^{89} + 28 q^{91} + 4 q^{92} - 20 q^{93} + 72 q^{94} - 26 q^{95} - 60 q^{96} - 10 q^{98} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1 −1.41398 − 0.0258858i 1.02040 + 0.589127i 1.99866 + 0.0732039i 0.861688 − 2.06337i −1.42757 − 0.859426i −2.19408 + 1.47851i −2.82416 − 0.155246i −0.805859 − 1.39579i −1.27182 + 2.89525i
157.2 −1.40809 + 0.131435i −2.81127 − 1.62309i 1.96545 − 0.370144i 0.127144 + 2.23245i 4.17186 + 1.91596i −0.817989 + 2.51613i −2.71889 + 0.779526i 3.76884 + 6.52782i −0.472452 − 3.12679i
157.3 −1.40768 − 0.135774i −2.26829 − 1.30960i 1.96313 + 0.382254i 1.38826 − 1.75292i 3.01521 + 2.15147i 2.57765 + 0.596439i −2.71156 − 0.804635i 1.93008 + 3.34300i −2.19223 + 2.27906i
157.4 −1.40697 + 0.142943i −1.19030 − 0.687220i 1.95913 − 0.402233i −1.99224 − 1.01537i 1.77295 + 0.796754i −2.38239 + 1.15075i −2.69895 + 0.845974i −0.555456 − 0.962078i 2.94817 + 1.14382i
157.5 −1.40695 + 0.143118i 1.86035 + 1.07407i 1.95903 − 0.402721i −2.22299 + 0.241479i −2.77114 − 1.24492i 0.273794 − 2.63155i −2.69863 + 0.846983i 0.807260 + 1.39821i 3.09308 − 0.657899i
157.6 −1.40162 − 0.188291i 0.138072 + 0.0797158i 1.92909 + 0.527825i 1.69801 + 1.45491i −0.178515 − 0.137729i 2.53589 − 0.754486i −2.60448 − 1.10304i −1.48729 − 2.57606i −2.10602 − 2.35895i
157.7 −1.37173 + 0.344014i 1.29779 + 0.749277i 1.76331 − 0.943791i 0.490009 + 2.18172i −2.03798 − 0.581352i −1.56843 − 2.13073i −2.09411 + 1.90123i −0.377169 − 0.653275i −1.42270 − 2.82417i
157.8 −1.36015 + 0.387300i 2.03628 + 1.17565i 1.70000 − 1.05357i 1.90288 − 1.17433i −3.22497 − 0.810400i 1.81150 + 1.92834i −1.90420 + 2.09142i 1.26429 + 2.18981i −2.13337 + 2.33425i
157.9 −1.35881 + 0.391962i −1.58492 − 0.915057i 1.69273 − 1.06520i 2.19842 + 0.408601i 2.51228 + 0.622158i −2.07850 − 1.63702i −1.88258 + 2.11090i 0.174657 + 0.302515i −3.14739 + 0.306485i
157.10 −1.33402 − 0.469460i −1.58749 − 0.916536i 1.55921 + 1.25254i −1.16312 + 1.90975i 1.68746 + 1.96794i 0.640293 − 2.56710i −1.49200 − 2.40290i 0.180076 + 0.311902i 2.44817 − 2.00161i
157.11 −1.32958 − 0.481883i 0.689402 + 0.398026i 1.53558 + 1.28141i 1.03726 + 1.98093i −0.724815 − 0.861420i −1.14153 + 2.38682i −1.42419 − 2.44370i −1.18315 − 2.04928i −0.424538 − 3.13365i
157.12 −1.31953 − 0.508770i 2.04100 + 1.17837i 1.48231 + 1.34267i −1.50799 − 1.65105i −2.09363 − 2.59329i 2.45399 + 0.988914i −1.27283 − 2.52585i 1.27711 + 2.21202i 1.14983 + 2.94583i
157.13 −1.30919 + 0.534817i −0.886317 − 0.511715i 1.42794 − 1.40035i −0.672423 − 2.13257i 1.43403 + 0.195914i 2.39326 − 1.12797i −1.12051 + 2.59701i −0.976295 − 1.69099i 2.02086 + 2.43231i
157.14 −1.29449 − 0.569466i 2.95419 + 1.70560i 1.35142 + 1.47434i 2.23531 − 0.0580249i −2.85289 − 3.89019i −1.25390 − 2.32975i −0.909815 − 2.67810i 4.31815 + 7.47925i −2.92664 − 1.19782i
157.15 −1.26510 − 0.632072i −1.02960 − 0.594438i 1.20097 + 1.59927i −2.17918 + 0.501194i 0.926818 + 1.40280i 1.72211 + 2.00856i −0.508498 − 2.78234i −0.793287 − 1.37401i 3.07367 + 0.743332i
157.16 −1.24382 − 0.672989i −2.55888 − 1.47737i 1.09417 + 1.67415i −0.419370 − 2.19639i 2.18853 + 3.55967i −1.32606 − 2.28945i −0.234267 − 2.81871i 2.86524 + 4.96273i −0.956525 + 3.01414i
157.17 −1.14220 + 0.833898i 2.56466 + 1.48071i 0.609228 − 1.90495i −1.96999 − 1.05790i −4.16411 + 0.447406i −2.38885 + 1.13728i 0.892678 + 2.68386i 2.88499 + 4.99696i 3.13229 − 0.434436i
157.18 −1.10415 + 0.883656i 0.838053 + 0.483850i 0.438305 − 1.95138i 0.854769 − 2.06625i −1.35290 + 0.206306i −0.160837 − 2.64086i 1.24039 + 2.54193i −1.03178 − 1.78709i 0.882054 + 3.03677i
157.19 −1.10250 − 0.885711i 1.24762 + 0.720311i 0.431031 + 1.95300i −1.92908 + 1.13078i −0.737514 − 1.89917i −2.63815 − 0.200473i 1.25458 − 2.53496i −0.462303 − 0.800732i 3.12836 + 0.461922i
157.20 −1.09307 + 0.897326i −0.623698 − 0.360092i 0.389613 − 1.96168i 2.21506 + 0.305784i 1.00487 − 0.166053i 0.529295 + 2.59227i 1.33439 + 2.49387i −1.24067 − 2.14890i −2.69561 + 1.65339i
See next 80 embeddings (of 360 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
80.t odd 4 1 inner
560.cz even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.cz.c yes 360
5.c odd 4 1 560.2.ch.c ✓ 360
7.d odd 6 1 inner 560.2.cz.c yes 360
16.e even 4 1 560.2.ch.c ✓ 360
35.k even 12 1 560.2.ch.c ✓ 360
80.t odd 4 1 inner 560.2.cz.c yes 360
112.x odd 12 1 560.2.ch.c ✓ 360
560.cz even 12 1 inner 560.2.cz.c yes 360
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.ch.c ✓ 360 5.c odd 4 1
560.2.ch.c ✓ 360 16.e even 4 1
560.2.ch.c ✓ 360 35.k even 12 1
560.2.ch.c ✓ 360 112.x odd 12 1
560.2.cz.c yes 360 1.a even 1 1 trivial
560.2.cz.c yes 360 7.d odd 6 1 inner
560.2.cz.c yes 360 80.t odd 4 1 inner
560.2.cz.c yes 360 560.cz even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(560, [\chi])\):

\( T_{3}^{180} + 12 T_{3}^{179} - 104 T_{3}^{178} - 1824 T_{3}^{177} + 4285 T_{3}^{176} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
\( T_{11}^{360} + 2 T_{11}^{359} + 2 T_{11}^{358} - 40 T_{11}^{357} - 11821 T_{11}^{356} + \cdots + 40\!\cdots\!36 \) Copy content Toggle raw display