Properties

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

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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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{2} + (\zeta_{12}^{2} - \zeta_{12} + 1) q^{3} - 2 \zeta_{12} q^{4} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} - 2 \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{2} + 2 \zeta_{12}) q^{6} + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}) q^{7} + (2 \zeta_{12}^{3} - 2) q^{8} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} - 2 \zeta_{12}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{2} + (\zeta_{12}^{2} - \zeta_{12} + 1) q^{3} - 2 \zeta_{12} q^{4} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} - 2 \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{2} + 2 \zeta_{12}) q^{6} + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}) q^{7} + (2 \zeta_{12}^{3} - 2) q^{8} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} - 2 \zeta_{12}) q^{9} + (\zeta_{12}^{2} + 3 \zeta_{12} - 1) q^{10} + ( - \zeta_{12}^{3} + \zeta_{12} - 1) q^{11} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{12} + ( - 4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{13} + (\zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12} + 2) q^{14} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 4 \zeta_{12} + 1) q^{15} + 4 \zeta_{12}^{2} q^{16} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + 3 \zeta_{12} - 3) q^{17} + (4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - \zeta_{12} - 1) q^{18} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 2 \zeta_{12} + 5) q^{19} + ( - 2 \zeta_{12}^{3} + 4) q^{20} + ( - 4 \zeta_{12}^{3} + \zeta_{12}^{2} + 5 \zeta_{12} - 3) q^{21} + (\zeta_{12}^{3} - 2 \zeta_{12} + 1) q^{22} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 3 \zeta_{12} + 1) q^{23} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2}) q^{24} + ( - 3 \zeta_{12}^{2} - 4 \zeta_{12} + 3) q^{25} + (3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 5 \zeta_{12} - 2) q^{26} - 4 \zeta_{12}^{3} q^{27} + (2 \zeta_{12}^{2} - 6) q^{28} + ( - \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 1) q^{29} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 4 \zeta_{12} - 2) q^{30} + 2 \zeta_{12} q^{31} + ( - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{32} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + 3 \zeta_{12} - 2) q^{33} + (2 \zeta_{12}^{3} - 4 \zeta_{12} + 4) q^{34} + ( - \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 3 \zeta_{12} - 4) q^{35} + ( - 2 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4) q^{36} + (3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 3 \zeta_{12}) q^{37} + ( - 5 \zeta_{12}^{2} - \zeta_{12} - 5) q^{38} + ( - 6 \zeta_{12}^{3} + \zeta_{12}^{2} + 3 \zeta_{12} - 1) q^{39} + ( - 2 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 2 \zeta_{12}) q^{40} + (6 \zeta_{12}^{3} - 12 \zeta_{12} + 1) q^{41} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{42} + ( - \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{43} + (2 \zeta_{12} - 2) q^{44} + ( - 4 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 3) q^{45} + ( - \zeta_{12}^{2} + 5 \zeta_{12} - 1) q^{46} + (6 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 7 \zeta_{12} - 1) q^{47} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4) q^{48} + ( - 8 \zeta_{12}^{2} + 3) q^{49} + (\zeta_{12}^{3} - 7) q^{50} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 8 \zeta_{12} - 6) q^{51} + (4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 2 \zeta_{12} - 8) q^{52} + (2 \zeta_{12}^{2} + 9 \zeta_{12} + 2) q^{53} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12}) q^{54} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + 3 \zeta_{12} - 2) q^{55} + (6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12} + 2) q^{56} + ( - 5 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12} + 5) q^{57} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 4 \zeta_{12}) q^{58} + (6 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - \zeta_{12} + 1) q^{59} + ( - 4 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{60} + (6 \zeta_{12} + 6) q^{61} + ( - 2 \zeta_{12}^{3} + 2) q^{62} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 3 \zeta_{12} - 8) q^{63} - 8 \zeta_{12}^{3} q^{64} + ( - 2 \zeta_{12}^{3} + 7 \zeta_{12}^{2} + 6 \zeta_{12} + 2) q^{65} + (2 \zeta_{12}^{2} - 4 \zeta_{12} + 2) q^{66} + (4 \zeta_{12}^{2} - 6 \zeta_{12} + 4) q^{67} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 6 \zeta_{12} - 4) q^{68} + ( - \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6 \zeta_{12} + 1) q^{69} + (2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12} + 9) q^{70} + (6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{71} + (6 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 2 \zeta_{12} + 8) q^{72} + (2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{73} + ( - 6 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + \zeta_{12} + 1) q^{74} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - 7 \zeta_{12} + 6) q^{75} + (6 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 10 \zeta_{12} - 6) q^{76} + (3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{77} + (4 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 6 \zeta_{12} + 4) q^{78} + (5 \zeta_{12}^{3} + \zeta_{12}^{2} - 5 \zeta_{12} - 2) q^{79} + (4 \zeta_{12}^{2} - 8 \zeta_{12} - 4) q^{80} + (4 \zeta_{12}^{3} + \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{81} + (5 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 7 \zeta_{12} - 12) q^{82} + ( - 12 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{83} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 6 \zeta_{12} - 8) q^{84} + ( - 5 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 6 \zeta_{12} - 7) q^{85} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2 \zeta_{12} + 6) q^{86} + (6 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{87} + (2 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{88} + (6 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 6 \zeta_{12} + 8) q^{89} + (\zeta_{12}^{3} - 12 \zeta_{12}^{2} + 4 \zeta_{12} + 5) q^{90} + ( - \zeta_{12}^{3} - 8 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{91} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{92} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12}) q^{93} + (2 \zeta_{12}^{3} + 13 \zeta_{12}^{2} - \zeta_{12} - 13) q^{94} + (5 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - \zeta_{12} + 7) q^{95} + (8 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8) q^{96} + ( - 3 \zeta_{12}^{3} + 3) q^{97} + ( - 3 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 5 \zeta_{12} - 8) q^{98} + (2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 6 q^{3} + 2 q^{5} - 4 q^{6} - 8 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 6 q^{3} + 2 q^{5} - 4 q^{6} - 8 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} + 4 q^{12} + 2 q^{14} + 8 q^{15} + 8 q^{16} - 10 q^{17} - 10 q^{18} + 14 q^{19} + 16 q^{20} - 10 q^{21} + 4 q^{22} + 8 q^{23} - 8 q^{24} + 6 q^{25} - 14 q^{26} - 20 q^{28} - 4 q^{29} - 12 q^{30} + 8 q^{32} - 10 q^{33} + 16 q^{34} - 4 q^{35} - 4 q^{37} - 30 q^{38} - 2 q^{39} - 12 q^{40} + 4 q^{41} + 20 q^{42} - 8 q^{44} + 22 q^{45} - 6 q^{46} - 16 q^{47} - 4 q^{49} - 28 q^{50} - 28 q^{51} - 16 q^{52} + 12 q^{53} - 8 q^{54} - 6 q^{55} + 16 q^{56} + 28 q^{57} + 4 q^{58} + 14 q^{59} + 20 q^{60} + 24 q^{61} + 8 q^{62} - 36 q^{63} + 22 q^{65} + 12 q^{66} + 24 q^{67} - 20 q^{68} + 16 q^{69} + 30 q^{70} + 20 q^{72} - 4 q^{73} + 14 q^{74} + 26 q^{75} - 4 q^{76} + 2 q^{77} + 4 q^{78} - 6 q^{79} - 8 q^{80} - 2 q^{81} - 38 q^{82} - 36 q^{84} - 24 q^{85} + 16 q^{86} - 4 q^{87} + 12 q^{88} + 24 q^{89} - 4 q^{90} - 32 q^{91} + 20 q^{92} - 4 q^{93} - 26 q^{94} + 44 q^{95} + 16 q^{96} + 12 q^{97} - 22 q^{98} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(\zeta_{12}^{2}\) \(\zeta_{12}^{3}\) \(1\) \(-\zeta_{12}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.366025 1.36603i 0.633975 + 0.366025i −1.73205 1.00000i −1.23205 + 1.86603i 0.732051 0.732051i 1.73205 2.00000i −2.00000 + 2.00000i −1.23205 2.13397i 2.09808 + 2.36603i
213.1 −1.36603 0.366025i 2.36603 + 1.36603i 1.73205 + 1.00000i 2.23205 0.133975i −2.73205 2.73205i −1.73205 + 2.00000i −2.00000 2.00000i 2.23205 + 3.86603i −3.09808 0.633975i
397.1 −1.36603 + 0.366025i 2.36603 1.36603i 1.73205 1.00000i 2.23205 + 0.133975i −2.73205 + 2.73205i −1.73205 2.00000i −2.00000 + 2.00000i 2.23205 3.86603i −3.09808 + 0.633975i
453.1 0.366025 + 1.36603i 0.633975 0.366025i −1.73205 + 1.00000i −1.23205 1.86603i 0.732051 + 0.732051i 1.73205 + 2.00000i −2.00000 2.00000i −1.23205 + 2.13397i 2.09808 2.36603i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.b yes 4
5.c odd 4 1 560.2.ch.b yes 4
7.d odd 6 1 560.2.cz.a yes 4
16.e even 4 1 560.2.ch.a 4
35.k even 12 1 560.2.ch.a 4
80.t odd 4 1 560.2.cz.a yes 4
112.x odd 12 1 560.2.ch.b yes 4
560.cz even 12 1 inner 560.2.cz.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.ch.a 4 16.e even 4 1
560.2.ch.a 4 35.k even 12 1
560.2.ch.b yes 4 5.c odd 4 1
560.2.ch.b yes 4 112.x odd 12 1
560.2.cz.a yes 4 7.d odd 6 1
560.2.cz.a yes 4 80.t odd 4 1
560.2.cz.b yes 4 1.a even 1 1 trivial
560.2.cz.b yes 4 560.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 \(S_{2}^{\mathrm{new}}(560, [\chi])\):

\( T_{3}^{4} - 6T_{3}^{3} + 14T_{3}^{2} - 12T_{3} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} + 4T_{11}^{3} + 5T_{11}^{2} + 2T_{11} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} - 6 T^{3} + 14 T^{2} - 12 T + 4 \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} - T^{2} - 10 T + 25 \) Copy content Toggle raw display
$7$ \( T^{4} + 2T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} + 5 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$13$ \( T^{4} + 38T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} + 10 T^{3} + 26 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$19$ \( T^{4} - 14 T^{3} + 113 T^{2} + \cdots + 1369 \) Copy content Toggle raw display
$23$ \( T^{4} - 8 T^{3} + 17 T^{2} - 22 T + 121 \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} + 8 T^{2} - 88 T + 484 \) Copy content Toggle raw display
$31$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} + 39 T^{2} - 92 T + 529 \) Copy content Toggle raw display
$41$ \( (T^{2} - 2 T - 107)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 56T^{2} + 676 \) Copy content Toggle raw display
$47$ \( T^{4} + 16 T^{3} + 89 T^{2} + \cdots + 6889 \) Copy content Toggle raw display
$53$ \( T^{4} - 12 T^{3} - 21 T^{2} + \cdots + 4761 \) Copy content Toggle raw display
$59$ \( T^{4} - 14 T^{3} + 170 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$61$ \( T^{4} - 24 T^{3} + 180 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$67$ \( T^{4} - 24 T^{3} + 204 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$71$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$73$ \( T^{4} + 4 T^{3} + 8 T^{2} + 32 T + 64 \) Copy content Toggle raw display
$79$ \( T^{4} + 6 T^{3} - 10 T^{2} - 132 T + 484 \) Copy content Toggle raw display
$83$ \( T^{4} + 312 T^{2} + 17424 \) Copy content Toggle raw display
$89$ \( T^{4} - 24 T^{3} + 204 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$97$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
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