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Properties

Label 560.2.cr.a
Level $560$
Weight $2$
Character orbit 560.cr
Analytic conductor $4.472$
Analytic rank $0$
Dimension $368$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,2,Mod(109,560)]
 
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, 6, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.cr (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: \(368\)
Relative dimension: \(92\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 368 q - 4 q^{4} - 2 q^{5} - 16 q^{6} - 8 q^{10} - 4 q^{11} + 16 q^{14} - 16 q^{15} - 4 q^{16} - 4 q^{19} - 8 q^{20} + 4 q^{21} - 4 q^{24} - 4 q^{26} - 16 q^{29} + 22 q^{30} - 56 q^{31} - 96 q^{34} + 18 q^{35}+ \cdots - 112 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 −1.41413 − 0.0158067i 0.702166 + 2.62052i 1.99950 + 0.0447052i −1.41355 + 1.73259i −0.951529 − 3.71684i 1.01362 + 2.44389i −2.82684 − 0.0948243i −3.77600 + 2.18008i 2.02633 − 2.42776i
109.2 −1.41398 + 0.0257616i −0.111122 − 0.414711i 1.99867 − 0.0728528i 0.968683 − 2.01535i 0.167807 + 0.583530i 0.251962 + 2.63373i −2.82420 + 0.154501i 2.43844 − 1.40783i −1.31778 + 2.87462i
109.3 −1.41398 − 0.0258393i −0.476445 − 1.77812i 1.99866 + 0.0730723i −1.16968 − 1.90574i 0.627737 + 2.52653i 1.31827 − 2.29394i −2.82418 − 0.154967i −0.336625 + 0.194351i 1.60466 + 2.72490i
109.4 −1.41262 + 0.0670727i 0.556398 + 2.07650i 1.99100 − 0.189497i 0.628067 − 2.14605i −0.925256 − 2.89600i −2.55518 + 0.686347i −2.79982 + 0.401229i −1.40422 + 0.810725i −0.743279 + 3.07368i
109.5 −1.41214 − 0.0765587i 0.365045 + 1.36237i 1.98828 + 0.216223i 1.04859 + 1.97496i −0.411194 − 1.95180i −1.15344 − 2.38109i −2.79117 − 0.457557i 0.875291 − 0.505349i −1.32956 − 2.86920i
109.6 −1.38543 − 0.283883i −0.832282 − 3.10612i 1.83882 + 0.786600i 1.90768 + 1.16652i 0.271291 + 4.53957i 1.46634 + 2.20224i −2.32425 − 1.61179i −6.35720 + 3.67033i −2.31179 − 2.15769i
109.7 −1.37214 − 0.342403i 0.176672 + 0.659350i 1.76552 + 0.939648i −1.89276 + 1.19058i −0.0166550 − 0.965212i 0.539354 − 2.59019i −2.10080 − 1.89385i 2.19455 − 1.26702i 3.00478 − 0.985549i
109.8 −1.35207 + 0.414632i −0.400636 − 1.49519i 1.65616 − 1.12122i −2.06107 + 0.867170i 1.16164 + 1.85548i 0.295693 + 2.62918i −1.77435 + 2.20266i 0.522982 − 0.301944i 2.42715 − 2.02706i
109.9 −1.35070 + 0.419057i 0.683112 + 2.54941i 1.64878 − 1.13204i 2.22666 − 0.204876i −1.99103 − 3.15723i 2.62002 − 0.368126i −1.75262 + 2.21998i −3.43477 + 1.98307i −2.92170 + 1.20982i
109.10 −1.32157 + 0.503445i −0.456068 − 1.70207i 1.49309 − 1.33067i 0.335144 + 2.21081i 1.45963 + 2.01980i 2.44498 − 1.01098i −1.30329 + 2.51026i −0.0909702 + 0.0525217i −1.55594 − 2.75301i
109.11 −1.30660 + 0.541096i 0.176005 + 0.656861i 1.41443 − 1.41400i −2.08402 − 0.810465i −0.585395 − 0.763022i −2.59981 − 0.490918i −1.08299 + 2.61288i 2.19759 − 1.26878i 3.16153 − 0.0686997i
109.12 −1.30452 − 0.546096i 0.00957310 + 0.0357273i 1.40356 + 1.42479i 2.15151 − 0.609097i 0.00702222 − 0.0518349i 2.61560 − 0.398306i −1.05290 − 2.62515i 2.59689 − 1.49932i −3.13932 − 0.380351i
109.13 −1.27536 + 0.611118i −0.522208 − 1.94891i 1.25307 − 1.55879i 2.23422 − 0.0908072i 1.85701 + 2.16642i −1.13277 − 2.39099i −0.645505 + 2.75378i −0.927462 + 0.535470i −2.79394 + 1.48119i
109.14 −1.25773 − 0.646612i 0.734175 + 2.73998i 1.16379 + 1.62653i −0.310069 − 2.21447i 0.848306 − 3.92089i −0.269749 − 2.63196i −0.411999 − 2.79826i −4.37039 + 2.52325i −1.04191 + 2.98570i
109.15 −1.24211 − 0.676133i −0.124199 − 0.463517i 1.08569 + 1.67967i 0.0754865 + 2.23479i −0.159130 + 0.659716i −1.51467 + 2.16928i −0.212869 − 2.82041i 2.39865 − 1.38486i 1.41725 − 2.82690i
109.16 −1.19960 + 0.748973i −0.849916 − 3.17193i 0.878078 − 1.79694i 0.0978968 − 2.23392i 3.39525 + 3.16848i −2.01880 + 1.71010i 0.292515 + 2.81326i −6.74071 + 3.89175i 1.55571 + 2.75314i
109.17 −1.19364 − 0.758440i −0.0643864 − 0.240293i 0.849537 + 1.81060i −2.16970 − 0.540741i −0.105394 + 0.335656i −2.37316 + 1.16967i 0.359196 − 2.80553i 2.54448 − 1.46906i 2.17971 + 2.29104i
109.18 −1.17827 − 0.782092i −0.520713 − 1.94333i 0.776663 + 1.84304i 0.750852 − 2.10623i −0.906318 + 2.69702i −2.58556 − 0.561118i 0.526305 − 2.77903i −0.907297 + 0.523828i −2.53198 + 1.89449i
109.19 −1.17009 − 0.794288i 0.542884 + 2.02607i 0.738214 + 1.85877i −1.69801 − 1.45491i 0.974060 − 2.80189i 2.27024 + 1.35868i 0.612626 − 2.76128i −1.21216 + 0.699839i 0.831209 + 3.05108i
109.20 −1.10426 − 0.883519i −0.704473 − 2.62913i 0.438788 + 1.95127i −2.16543 − 0.557594i −1.54496 + 3.52566i 2.48520 + 0.907612i 1.23945 − 2.54239i −3.81797 + 2.20431i 1.89856 + 2.52893i
See next 80 embeddings (of 368 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.92
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
16.e even 4 1 inner
35.j even 6 1 inner
80.q even 4 1 inner
112.w even 12 1 inner
560.cr 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.cr.a ✓ 368
5.b even 2 1 inner 560.2.cr.a ✓ 368
7.c even 3 1 inner 560.2.cr.a ✓ 368
16.e even 4 1 inner 560.2.cr.a ✓ 368
35.j even 6 1 inner 560.2.cr.a ✓ 368
80.q even 4 1 inner 560.2.cr.a ✓ 368
112.w even 12 1 inner 560.2.cr.a ✓ 368
560.cr even 12 1 inner 560.2.cr.a ✓ 368
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.cr.a ✓ 368 1.a even 1 1 trivial
560.2.cr.a ✓ 368 5.b even 2 1 inner
560.2.cr.a ✓ 368 7.c even 3 1 inner
560.2.cr.a ✓ 368 16.e even 4 1 inner
560.2.cr.a ✓ 368 35.j even 6 1 inner
560.2.cr.a ✓ 368 80.q even 4 1 inner
560.2.cr.a ✓ 368 112.w even 12 1 inner
560.2.cr.a ✓ 368 560.cr even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(560, [\chi])\).