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Properties

Label 525.3.c.a
Level $525$
Weight $3$
Character orbit 525.c
Analytic conductor $14.305$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,3,Mod(176,525)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.176");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 525.c (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: \(14.3052138789\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.65856.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 14x^{2} + 21 \) 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 21)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - \beta_1 + 1) q^{3} + (2 \beta_{2} - 3) q^{4} + (\beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{6} + \beta_{2} q^{7} + (4 \beta_{3} - 2 \beta_{2} - 5 \beta_1 + 2) q^{8} + (2 \beta_{3} + \beta_1 - 4) q^{9}+ \cdots + ( - 4 \beta_{3} + 16 \beta_1 + 26) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 12 q^{4} + 14 q^{6} - 20 q^{9} + 22 q^{12} + 36 q^{13} + 36 q^{16} - 56 q^{18} + 12 q^{19} + 14 q^{21} + 56 q^{22} - 126 q^{24} - 10 q^{27} + 56 q^{28} + 136 q^{31} - 28 q^{33} + 116 q^{36}+ \cdots + 112 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 14x^{2} + 21 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} + 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + \nu^{2} + 13\nu + 5 ) / 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{3} - 2\beta_{2} - 13\beta _1 + 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
176.1
− 3.50592i
− 1.30710i
1.30710i
3.50592i
− 3.50592i −0.822876 + 2.88494i −8.29150 0 10.1144 + 2.88494i −2.64575 15.0457i −7.64575 − 4.74789i 0
176.2 − 1.30710i 1.82288 − 2.38267i 2.29150 0 −3.11438 − 2.38267i 2.64575 − 8.22359i −2.35425 − 8.68663i 0
176.3 1.30710i 1.82288 + 2.38267i 2.29150 0 −3.11438 + 2.38267i 2.64575 8.22359i −2.35425 + 8.68663i 0
176.4 3.50592i −0.822876 − 2.88494i −8.29150 0 10.1144 − 2.88494i −2.64575 − 15.0457i −7.64575 + 4.74789i 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.3.c.a 4
3.b odd 2 1 inner 525.3.c.a 4
5.b even 2 1 21.3.b.a ✓ 4
5.c odd 4 2 525.3.f.a 8
15.d odd 2 1 21.3.b.a ✓ 4
15.e even 4 2 525.3.f.a 8
20.d odd 2 1 336.3.d.c 4
35.c odd 2 1 147.3.b.f 4
35.i odd 6 2 147.3.h.c 8
35.j even 6 2 147.3.h.e 8
40.e odd 2 1 1344.3.d.b 4
40.f even 2 1 1344.3.d.f 4
45.h odd 6 2 567.3.r.c 8
45.j even 6 2 567.3.r.c 8
60.h even 2 1 336.3.d.c 4
105.g even 2 1 147.3.b.f 4
105.o odd 6 2 147.3.h.e 8
105.p even 6 2 147.3.h.c 8
120.i odd 2 1 1344.3.d.f 4
120.m even 2 1 1344.3.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.b.a ✓ 4 5.b even 2 1
21.3.b.a ✓ 4 15.d odd 2 1
147.3.b.f 4 35.c odd 2 1
147.3.b.f 4 105.g even 2 1
147.3.h.c 8 35.i odd 6 2
147.3.h.c 8 105.p even 6 2
147.3.h.e 8 35.j even 6 2
147.3.h.e 8 105.o odd 6 2
336.3.d.c 4 20.d odd 2 1
336.3.d.c 4 60.h even 2 1
525.3.c.a 4 1.a even 1 1 trivial
525.3.c.a 4 3.b odd 2 1 inner
525.3.f.a 8 5.c odd 4 2
525.3.f.a 8 15.e even 4 2
567.3.r.c 8 45.h odd 6 2
567.3.r.c 8 45.j even 6 2
1344.3.d.b 4 40.e odd 2 1
1344.3.d.b 4 120.m even 2 1
1344.3.d.f 4 40.f even 2 1
1344.3.d.f 4 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{4} + 14T_{2}^{2} + 21 \) Copy content Toggle raw display
\( T_{13}^{2} - 18T_{13} + 74 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 14T^{2} + 21 \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 56T^{2} + 336 \) Copy content Toggle raw display
$13$ \( (T^{2} - 18 T + 74)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 168T^{2} + 3024 \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T - 166)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 672 T^{2} + 12096 \) Copy content Toggle raw display
$29$ \( T^{4} + 392 T^{2} + 27216 \) Copy content Toggle raw display
$31$ \( (T^{2} - 68 T + 1128)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 8 T - 1356)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 5432 T^{2} + 4139856 \) Copy content Toggle raw display
$43$ \( (T^{2} - 80 T + 1348)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 2688 T^{2} + 1741824 \) Copy content Toggle raw display
$53$ \( T^{4} + 11256 T^{2} + 2543184 \) Copy content Toggle raw display
$59$ \( T^{4} + 10248 T^{2} + 14606676 \) Copy content Toggle raw display
$61$ \( (T^{2} + 78 T + 1178)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 12 T - 412)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 9576 T^{2} + 22888656 \) Copy content Toggle raw display
$73$ \( (T^{2} - 16 T - 4668)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 64 T - 8048)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 13608 T^{2} + 44641044 \) Copy content Toggle raw display
$89$ \( T^{4} + 20216 T^{2} + 64754256 \) Copy content Toggle raw display
$97$ \( (T^{2} - 4 T - 444)^{2} \) Copy content Toggle raw display
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