Properties

Label 560.3.f.b
Level $560$
Weight $3$
Character orbit 560.f
Analytic conductor $15.259$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,3,Mod(321,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.321");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 560.f (of order \(2\), degree \(1\), 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: \(15.2588948042\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + \beta q^{5} + ( - 3 \beta + 2) q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + \beta q^{5} + ( - 3 \beta + 2) q^{7} + 4 q^{9} + q^{11} - 9 \beta q^{13} - 5 q^{15} - 3 \beta q^{17} - 6 \beta q^{19} + (2 \beta + 15) q^{21} - 8 q^{23} - 5 q^{25} + 13 \beta q^{27} + 41 q^{29} - 18 \beta q^{31} + \beta q^{33} + (2 \beta + 15) q^{35} - 28 q^{37} + 45 q^{39} + 6 \beta q^{41} + 82 q^{43} + 4 \beta q^{45} - 9 \beta q^{47} + ( - 12 \beta - 41) q^{49} + 15 q^{51} + 74 q^{53} + \beta q^{55} + 30 q^{57} + 42 \beta q^{59} + 36 \beta q^{61} + ( - 12 \beta + 8) q^{63} + 45 q^{65} - 2 q^{67} - 8 \beta q^{69} - 14 q^{71} - 30 \beta q^{73} - 5 \beta q^{75} + ( - 3 \beta + 2) q^{77} + 19 q^{79} - 29 q^{81} - 42 \beta q^{83} + 15 q^{85} + 41 \beta q^{87} + 48 \beta q^{89} + ( - 18 \beta - 135) q^{91} + 90 q^{93} + 30 q^{95} - 27 \beta q^{97} + 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{7} + 8 q^{9} + 2 q^{11} - 10 q^{15} + 30 q^{21} - 16 q^{23} - 10 q^{25} + 82 q^{29} + 30 q^{35} - 56 q^{37} + 90 q^{39} + 164 q^{43} - 82 q^{49} + 30 q^{51} + 148 q^{53} + 60 q^{57} + 16 q^{63} + 90 q^{65} - 4 q^{67} - 28 q^{71} + 4 q^{77} + 38 q^{79} - 58 q^{81} + 30 q^{85} - 270 q^{91} + 180 q^{93} + 60 q^{95} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1
2.23607i
2.23607i
0 2.23607i 0 2.23607i 0 2.00000 + 6.70820i 0 4.00000 0
321.2 0 2.23607i 0 2.23607i 0 2.00000 6.70820i 0 4.00000 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.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.3.f.b 2
4.b odd 2 1 35.3.d.b 2
7.b odd 2 1 inner 560.3.f.b 2
12.b even 2 1 315.3.h.a 2
20.d odd 2 1 175.3.d.c 2
20.e even 4 2 175.3.c.c 4
28.d even 2 1 35.3.d.b 2
28.f even 6 2 245.3.h.a 4
28.g odd 6 2 245.3.h.a 4
84.h odd 2 1 315.3.h.a 2
140.c even 2 1 175.3.d.c 2
140.j odd 4 2 175.3.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.d.b 2 4.b odd 2 1
35.3.d.b 2 28.d even 2 1
175.3.c.c 4 20.e even 4 2
175.3.c.c 4 140.j odd 4 2
175.3.d.c 2 20.d odd 2 1
175.3.d.c 2 140.c even 2 1
245.3.h.a 4 28.f even 6 2
245.3.h.a 4 28.g odd 6 2
315.3.h.a 2 12.b even 2 1
315.3.h.a 2 84.h odd 2 1
560.3.f.b 2 1.a even 1 1 trivial
560.3.f.b 2 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 5 \) acting on \(S_{3}^{\mathrm{new}}(560, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 5 \) Copy content Toggle raw display
$5$ \( T^{2} + 5 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 49 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 405 \) Copy content Toggle raw display
$17$ \( T^{2} + 45 \) Copy content Toggle raw display
$19$ \( T^{2} + 180 \) Copy content Toggle raw display
$23$ \( (T + 8)^{2} \) Copy content Toggle raw display
$29$ \( (T - 41)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 1620 \) Copy content Toggle raw display
$37$ \( (T + 28)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 180 \) Copy content Toggle raw display
$43$ \( (T - 82)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 405 \) Copy content Toggle raw display
$53$ \( (T - 74)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 8820 \) Copy content Toggle raw display
$61$ \( T^{2} + 6480 \) Copy content Toggle raw display
$67$ \( (T + 2)^{2} \) Copy content Toggle raw display
$71$ \( (T + 14)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4500 \) Copy content Toggle raw display
$79$ \( (T - 19)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 8820 \) Copy content Toggle raw display
$89$ \( T^{2} + 11520 \) Copy content Toggle raw display
$97$ \( T^{2} + 3645 \) Copy content Toggle raw display
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