↑

Properties

Label 560.2.r.a
Level $560$
Weight $2$
Character orbit 560.r
Analytic conductor $4.472$
Analytic rank $0$
Dimension $184$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,2,Mod(237,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.237");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.r (of order \(4\), degree \(2\), 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: \(184\)
Relative dimension: \(92\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 184 q - 4 q^{2} + 8 q^{8} - 168 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 184 q - 4 q^{2} + 8 q^{8} - 168 q^{9} - 8 q^{11} - 8 q^{15} - 8 q^{16} - 4 q^{21} - 12 q^{22} - 32 q^{28} - 44 q^{30} - 24 q^{32} + 20 q^{35} - 40 q^{36} - 8 q^{37} + 48 q^{39} - 32 q^{42} + 24 q^{44} - 8 q^{46} - 4 q^{50} - 8 q^{51} - 4 q^{56} + 24 q^{57} - 4 q^{58} + 64 q^{60} - 32 q^{63} - 8 q^{65} + 12 q^{70} - 64 q^{72} + 40 q^{74} - 28 q^{78} + 104 q^{81} - 48 q^{84} + 16 q^{85} + 8 q^{86} + 96 q^{88} - 20 q^{91} - 56 q^{92} + 16 q^{93} - 56 q^{95} - 80 q^{98} + 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
237.1 −1.41385 + 0.0320021i − 2.37744i 1.99795 − 0.0904924i −1.88118 − 1.20879i 0.0760829 + 3.36134i 2.53631 + 0.753095i −2.82191 + 0.191881i −2.65221 2.69839 + 1.64884i
237.2 −1.41385 + 0.0320021i 2.37744i 1.99795 − 0.0904924i 1.88118 + 1.20879i −0.0760829 − 3.36134i −0.753095 − 2.53631i −2.82191 + 0.191881i −2.65221 −2.69839 − 1.64884i
237.3 −1.40124 − 0.191148i − 0.303303i 1.92692 + 0.535688i −0.412090 + 2.19777i −0.0579760 + 0.425000i −2.04400 + 1.67990i −2.59768 − 1.11895i 2.90801 0.997536 − 3.00082i
237.4 −1.40124 − 0.191148i 0.303303i 1.92692 + 0.535688i 0.412090 − 2.19777i 0.0579760 − 0.425000i −1.67990 + 2.04400i −2.59768 − 1.11895i 2.90801 −0.997536 + 3.00082i
237.5 −1.39265 − 0.246023i − 2.36202i 1.87894 + 0.685249i 2.19592 + 0.421835i −0.581111 + 3.28946i 1.13552 + 2.38968i −2.44812 − 1.41658i −2.57912 −2.95436 − 1.12772i
237.6 −1.39265 − 0.246023i 2.36202i 1.87894 + 0.685249i −2.19592 − 0.421835i 0.581111 − 3.28946i −2.38968 − 1.13552i −2.44812 − 1.41658i −2.57912 2.95436 + 1.12772i
237.7 −1.34754 − 0.429109i − 2.77875i 1.63173 + 1.15648i 1.52419 − 1.63611i −1.19239 + 3.74447i −0.904951 − 2.48617i −1.70257 − 2.25860i −4.72144 −2.75598 + 1.55067i
237.8 −1.34754 − 0.429109i 2.77875i 1.63173 + 1.15648i −1.52419 + 1.63611i 1.19239 − 3.74447i 2.48617 + 0.904951i −1.70257 − 2.25860i −4.72144 2.75598 − 1.55067i
237.9 −1.33659 + 0.462100i − 0.445450i 1.57293 − 1.23527i −2.22590 − 0.213020i 0.205843 + 0.595383i −1.93702 − 1.80221i −1.53153 + 2.37790i 2.80157 3.07354 − 0.743868i
237.10 −1.33659 + 0.462100i 0.445450i 1.57293 − 1.23527i 2.22590 + 0.213020i −0.205843 − 0.595383i 1.80221 + 1.93702i −1.53153 + 2.37790i 2.80157 −3.07354 + 0.743868i
237.11 −1.33522 + 0.466046i − 0.958223i 1.56560 − 1.24454i 0.666131 − 2.13454i 0.446576 + 1.27943i 0.729292 − 2.54325i −1.51040 + 2.39138i 2.08181 0.105366 + 3.16052i
237.12 −1.33522 + 0.466046i 0.958223i 1.56560 − 1.24454i −0.666131 + 2.13454i −0.446576 − 1.27943i 2.54325 − 0.729292i −1.51040 + 2.39138i 2.08181 −0.105366 − 3.16052i
237.13 −1.31458 + 0.521428i − 2.31341i 1.45623 − 1.37091i 1.44475 + 1.70666i 1.20628 + 3.04115i −2.55188 − 0.698510i −1.19949 + 2.56149i −2.35185 −2.78914 − 1.49020i
237.14 −1.31458 + 0.521428i 2.31341i 1.45623 − 1.37091i −1.44475 − 1.70666i −1.20628 − 3.04115i 0.698510 + 2.55188i −1.19949 + 2.56149i −2.35185 2.78914 + 1.49020i
237.15 −1.24845 − 0.664358i − 1.05190i 1.11726 + 1.65884i −2.04569 + 0.902849i −0.698839 + 1.31325i 0.442290 − 2.60852i −0.292782 − 2.81323i 1.89350 3.15376 + 0.231910i
237.16 −1.24845 − 0.664358i 1.05190i 1.11726 + 1.65884i 2.04569 − 0.902849i 0.698839 − 1.31325i 2.60852 − 0.442290i −0.292782 − 2.81323i 1.89350 −3.15376 − 0.231910i
237.17 −1.18156 + 0.777113i − 2.96802i 0.792190 − 1.83642i −0.835108 + 2.07427i 2.30649 + 3.50691i 2.64524 − 0.0522070i 0.491081 + 2.78547i −5.80916 −0.625209 − 3.09986i
237.18 −1.18156 + 0.777113i 2.96802i 0.792190 − 1.83642i 0.835108 − 2.07427i −2.30649 − 3.50691i 0.0522070 − 2.64524i 0.491081 + 2.78547i −5.80916 0.625209 + 3.09986i
237.19 −1.13458 − 0.844232i − 3.14518i 0.574546 + 1.91570i −1.64124 + 1.51866i −2.65526 + 3.56847i −1.68783 + 2.03745i 0.965423 − 2.65856i −6.89218 3.14422 − 0.337448i
237.20 −1.13458 − 0.844232i 3.14518i 0.574546 + 1.91570i 1.64124 − 1.51866i 2.65526 − 3.56847i −2.03745 + 1.68783i 0.965423 − 2.65856i −6.89218 −3.14422 + 0.337448i
See next 80 embeddings (of 184 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 237.92
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
80.t odd 4 1 inner
560.r even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.r.a ✓ 184
5.c odd 4 1 560.2.bn.a yes 184
7.b odd 2 1 inner 560.2.r.a ✓ 184
16.e even 4 1 560.2.bn.a yes 184
35.f even 4 1 560.2.bn.a yes 184
80.t odd 4 1 inner 560.2.r.a ✓ 184
112.l odd 4 1 560.2.bn.a yes 184
560.r even 4 1 inner 560.2.r.a ✓ 184
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.r.a ✓ 184 1.a even 1 1 trivial
560.2.r.a ✓ 184 7.b odd 2 1 inner
560.2.r.a ✓ 184 80.t odd 4 1 inner
560.2.r.a ✓ 184 560.r even 4 1 inner
560.2.bn.a yes 184 5.c odd 4 1
560.2.bn.a yes 184 16.e even 4 1
560.2.bn.a yes 184 35.f even 4 1
560.2.bn.a yes 184 112.l odd 4 1

Hecke kernels

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