Properties

Label 560.4.q.b
Level $560$
Weight $4$
Character orbit 560.q
Analytic conductor $33.041$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,4,Mod(81,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.81");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 560.q (of order \(3\), degree \(2\), not 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: \(33.0410696032\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{3} - 5 \zeta_{6} q^{5} + (14 \zeta_{6} + 7) q^{7} + 23 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{3} - 5 \zeta_{6} q^{5} + (14 \zeta_{6} + 7) q^{7} + 23 \zeta_{6} q^{9} + (45 \zeta_{6} - 45) q^{11} + 59 q^{13} + 10 q^{15} + ( - 54 \zeta_{6} + 54) q^{17} - 121 \zeta_{6} q^{19} + (14 \zeta_{6} - 42) q^{21} + 69 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} - 100 q^{27} - 162 q^{29} + (88 \zeta_{6} - 88) q^{31} - 90 \zeta_{6} q^{33} + ( - 105 \zeta_{6} + 70) q^{35} + 259 \zeta_{6} q^{37} + (118 \zeta_{6} - 118) q^{39} + 195 q^{41} + 286 q^{43} + ( - 115 \zeta_{6} + 115) q^{45} + 45 \zeta_{6} q^{47} + (392 \zeta_{6} - 147) q^{49} + 108 \zeta_{6} q^{51} + (597 \zeta_{6} - 597) q^{53} + 225 q^{55} + 242 q^{57} + (360 \zeta_{6} - 360) q^{59} - 392 \zeta_{6} q^{61} + (483 \zeta_{6} - 322) q^{63} - 295 \zeta_{6} q^{65} + (280 \zeta_{6} - 280) q^{67} - 138 q^{69} - 48 q^{71} + (668 \zeta_{6} - 668) q^{73} - 50 \zeta_{6} q^{75} + (315 \zeta_{6} - 945) q^{77} + 782 \zeta_{6} q^{79} + (421 \zeta_{6} - 421) q^{81} - 768 q^{83} - 270 q^{85} + ( - 324 \zeta_{6} + 324) q^{87} + 1194 \zeta_{6} q^{89} + (826 \zeta_{6} + 413) q^{91} - 176 \zeta_{6} q^{93} + (605 \zeta_{6} - 605) q^{95} + 902 q^{97} - 1035 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 5 q^{5} + 28 q^{7} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 5 q^{5} + 28 q^{7} + 23 q^{9} - 45 q^{11} + 118 q^{13} + 20 q^{15} + 54 q^{17} - 121 q^{19} - 70 q^{21} + 69 q^{23} - 25 q^{25} - 200 q^{27} - 324 q^{29} - 88 q^{31} - 90 q^{33} + 35 q^{35} + 259 q^{37} - 118 q^{39} + 390 q^{41} + 572 q^{43} + 115 q^{45} + 45 q^{47} + 98 q^{49} + 108 q^{51} - 597 q^{53} + 450 q^{55} + 484 q^{57} - 360 q^{59} - 392 q^{61} - 161 q^{63} - 295 q^{65} - 280 q^{67} - 276 q^{69} - 96 q^{71} - 668 q^{73} - 50 q^{75} - 1575 q^{77} + 782 q^{79} - 421 q^{81} - 1536 q^{83} - 540 q^{85} + 324 q^{87} + 1194 q^{89} + 1652 q^{91} - 176 q^{93} - 605 q^{95} + 1804 q^{97} - 2070 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −1.00000 + 1.73205i 0 −2.50000 4.33013i 0 14.0000 + 12.1244i 0 11.5000 + 19.9186i 0
401.1 0 −1.00000 1.73205i 0 −2.50000 + 4.33013i 0 14.0000 12.1244i 0 11.5000 19.9186i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.4.q.b 2
4.b odd 2 1 35.4.e.a 2
7.c even 3 1 inner 560.4.q.b 2
12.b even 2 1 315.4.j.b 2
20.d odd 2 1 175.4.e.b 2
20.e even 4 2 175.4.k.b 4
28.d even 2 1 245.4.e.a 2
28.f even 6 1 245.4.a.f 1
28.f even 6 1 245.4.e.a 2
28.g odd 6 1 35.4.e.a 2
28.g odd 6 1 245.4.a.e 1
84.j odd 6 1 2205.4.a.g 1
84.n even 6 1 315.4.j.b 2
84.n even 6 1 2205.4.a.e 1
140.p odd 6 1 175.4.e.b 2
140.p odd 6 1 1225.4.a.b 1
140.s even 6 1 1225.4.a.a 1
140.w even 12 2 175.4.k.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.a 2 4.b odd 2 1
35.4.e.a 2 28.g odd 6 1
175.4.e.b 2 20.d odd 2 1
175.4.e.b 2 140.p odd 6 1
175.4.k.b 4 20.e even 4 2
175.4.k.b 4 140.w even 12 2
245.4.a.e 1 28.g odd 6 1
245.4.a.f 1 28.f even 6 1
245.4.e.a 2 28.d even 2 1
245.4.e.a 2 28.f even 6 1
315.4.j.b 2 12.b even 2 1
315.4.j.b 2 84.n even 6 1
560.4.q.b 2 1.a even 1 1 trivial
560.4.q.b 2 7.c even 3 1 inner
1225.4.a.a 1 140.s even 6 1
1225.4.a.b 1 140.p odd 6 1
2205.4.a.e 1 84.n even 6 1
2205.4.a.g 1 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(560, [\chi])\):

\( T_{3}^{2} + 2T_{3} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 45T_{11} + 2025 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} - 28T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} + 45T + 2025 \) Copy content Toggle raw display
$13$ \( (T - 59)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 54T + 2916 \) Copy content Toggle raw display
$19$ \( T^{2} + 121T + 14641 \) Copy content Toggle raw display
$23$ \( T^{2} - 69T + 4761 \) Copy content Toggle raw display
$29$ \( (T + 162)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 88T + 7744 \) Copy content Toggle raw display
$37$ \( T^{2} - 259T + 67081 \) Copy content Toggle raw display
$41$ \( (T - 195)^{2} \) Copy content Toggle raw display
$43$ \( (T - 286)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 45T + 2025 \) Copy content Toggle raw display
$53$ \( T^{2} + 597T + 356409 \) Copy content Toggle raw display
$59$ \( T^{2} + 360T + 129600 \) Copy content Toggle raw display
$61$ \( T^{2} + 392T + 153664 \) Copy content Toggle raw display
$67$ \( T^{2} + 280T + 78400 \) Copy content Toggle raw display
$71$ \( (T + 48)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 668T + 446224 \) Copy content Toggle raw display
$79$ \( T^{2} - 782T + 611524 \) Copy content Toggle raw display
$83$ \( (T + 768)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 1194 T + 1425636 \) Copy content Toggle raw display
$97$ \( (T - 902)^{2} \) Copy content Toggle raw display
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