Properties

Label 525.2.b.d
Level $525$
Weight $2$
Character orbit 525.b
Analytic conductor $4.192$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,2,Mod(251,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.b (of order \(2\), degree \(1\), 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: \(4.19214610612\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

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

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 1) q^{3} + 2 q^{4} + ( - \zeta_{6} + 3) q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 1) q^{3} + 2 q^{4} + ( - \zeta_{6} + 3) q^{7} - 3 q^{9} + ( - 4 \zeta_{6} + 2) q^{12} + ( - 6 \zeta_{6} + 3) q^{13} + 4 q^{16} + (10 \zeta_{6} - 5) q^{19} + ( - 5 \zeta_{6} + 1) q^{21} + (6 \zeta_{6} - 3) q^{27} + ( - 2 \zeta_{6} + 6) q^{28} + ( - 10 \zeta_{6} + 5) q^{31} - 6 q^{36} - 10 q^{37} - 9 q^{39} + 5 q^{43} + ( - 8 \zeta_{6} + 4) q^{48} + ( - 5 \zeta_{6} + 8) q^{49} + ( - 12 \zeta_{6} + 6) q^{52} + 15 q^{57} + (10 \zeta_{6} - 5) q^{61} + (3 \zeta_{6} - 9) q^{63} + 8 q^{64} - 5 q^{67} + (16 \zeta_{6} - 8) q^{73} + (20 \zeta_{6} - 10) q^{76} - 4 q^{79} + 9 q^{81} + ( - 10 \zeta_{6} + 2) q^{84} + ( - 15 \zeta_{6} + 3) q^{91} - 15 q^{93} + (22 \zeta_{6} - 11) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} + 5 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} + 5 q^{7} - 6 q^{9} + 8 q^{16} - 3 q^{21} + 10 q^{28} - 12 q^{36} - 20 q^{37} - 18 q^{39} + 10 q^{43} + 11 q^{49} + 30 q^{57} - 15 q^{63} + 16 q^{64} - 10 q^{67} - 8 q^{79} + 18 q^{81} - 6 q^{84} - 9 q^{91} - 30 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.73205i 2.00000 0 0 2.50000 0.866025i 0 −3.00000 0
251.2 0 1.73205i 2.00000 0 0 2.50000 + 0.866025i 0 −3.00000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.b.d yes 2
3.b odd 2 1 CM 525.2.b.d yes 2
5.b even 2 1 525.2.b.c 2
5.c odd 4 2 525.2.g.a 4
7.b odd 2 1 inner 525.2.b.d yes 2
15.d odd 2 1 525.2.b.c 2
15.e even 4 2 525.2.g.a 4
21.c even 2 1 inner 525.2.b.d yes 2
35.c odd 2 1 525.2.b.c 2
35.f even 4 2 525.2.g.a 4
105.g even 2 1 525.2.b.c 2
105.k odd 4 2 525.2.g.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.b.c 2 5.b even 2 1
525.2.b.c 2 15.d odd 2 1
525.2.b.c 2 35.c odd 2 1
525.2.b.c 2 105.g even 2 1
525.2.b.d yes 2 1.a even 1 1 trivial
525.2.b.d yes 2 3.b odd 2 1 CM
525.2.b.d yes 2 7.b odd 2 1 inner
525.2.b.d yes 2 21.c even 2 1 inner
525.2.g.a 4 5.c odd 4 2
525.2.g.a 4 15.e even 4 2
525.2.g.a 4 35.f even 4 2
525.2.g.a 4 105.k odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{37} + 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 5T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 27 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 75 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 75 \) Copy content Toggle raw display
$37$ \( (T + 10)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 5)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 75 \) Copy content Toggle raw display
$67$ \( (T + 5)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 192 \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 363 \) Copy content Toggle raw display
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