Properties

Label 525.2.g.c
Level $525$
Weight $2$
Character orbit 525.g
Analytic conductor $4.192$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,2,Mod(524,525)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.524");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.g (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: \(4.19214610612\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 105)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_1 q^{3} + q^{4} + 3 q^{6} + ( - \beta_{3} - \beta_1) q^{7} + \beta_1 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_1 q^{3} + q^{4} + 3 q^{6} + ( - \beta_{3} - \beta_1) q^{7} + \beta_1 q^{8} + 3 q^{9} + \beta_{2} q^{11} - \beta_1 q^{12} + (\beta_{2} + 3) q^{14} - 5 q^{16} + 3 \beta_{3} q^{17} - 3 \beta_1 q^{18} - \beta_{2} q^{19} + (\beta_{2} + 3) q^{21} - 3 \beta_{3} q^{22} + 2 \beta_1 q^{23} - 3 q^{24} - 3 \beta_1 q^{27} + ( - \beta_{3} - \beta_1) q^{28} - 2 \beta_{2} q^{29} - \beta_{2} q^{31} + 3 \beta_1 q^{32} - 3 \beta_{3} q^{33} - 3 \beta_{2} q^{34} + 3 q^{36} + \beta_{3} q^{37} + 3 \beta_{3} q^{38} - 6 q^{41} + ( - 3 \beta_{3} - 3 \beta_1) q^{42} - 4 \beta_{3} q^{43} + \beta_{2} q^{44} - 6 q^{46} - 6 \beta_{3} q^{47} + 5 \beta_1 q^{48} + (2 \beta_{2} - 1) q^{49} - 3 \beta_{2} q^{51} + 9 q^{54} + ( - \beta_{2} - 3) q^{56} + 3 \beta_{3} q^{57} + 6 \beta_{3} q^{58} - 12 q^{59} - 2 \beta_{2} q^{61} + 3 \beta_{3} q^{62} + ( - 3 \beta_{3} - 3 \beta_1) q^{63} + q^{64} + 3 \beta_{2} q^{66} - 4 \beta_{3} q^{67} + 3 \beta_{3} q^{68} - 6 q^{69} - \beta_{2} q^{71} + 3 \beta_1 q^{72} - 4 \beta_1 q^{73} - \beta_{2} q^{74} - \beta_{2} q^{76} + ( - 3 \beta_{3} + 4 \beta_1) q^{77} - 8 q^{79} + 9 q^{81} + 6 \beta_1 q^{82} + (\beta_{2} + 3) q^{84} + 4 \beta_{2} q^{86} + 6 \beta_{3} q^{87} + 3 \beta_{3} q^{88} + 6 q^{89} + 2 \beta_1 q^{92} + 3 \beta_{3} q^{93} + 6 \beta_{2} q^{94} - 9 q^{96} + 4 \beta_1 q^{97} + ( - 6 \beta_{3} + \beta_1) q^{98} + 3 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 12 q^{6} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} + 12 q^{6} + 12 q^{9} + 12 q^{14} - 20 q^{16} + 12 q^{21} - 12 q^{24} + 12 q^{36} - 24 q^{41} - 24 q^{46} - 4 q^{49} + 36 q^{54} - 12 q^{56} - 48 q^{59} + 4 q^{64} - 24 q^{69} - 32 q^{79} + 36 q^{81} + 12 q^{84} + 24 q^{89} - 36 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\zeta_{12}^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
524.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
−1.73205 −1.73205 1.00000 0 3.00000 −1.73205 2.00000i 1.73205 3.00000 0
524.2 −1.73205 −1.73205 1.00000 0 3.00000 −1.73205 + 2.00000i 1.73205 3.00000 0
524.3 1.73205 1.73205 1.00000 0 3.00000 1.73205 2.00000i −1.73205 3.00000 0
524.4 1.73205 1.73205 1.00000 0 3.00000 1.73205 + 2.00000i −1.73205 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
5.b even 2 1 inner
21.c even 2 1 inner
105.g 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.g.c 4
3.b odd 2 1 525.2.g.b 4
5.b even 2 1 inner 525.2.g.c 4
5.c odd 4 1 105.2.b.b yes 2
5.c odd 4 1 525.2.b.b 2
7.b odd 2 1 525.2.g.b 4
15.d odd 2 1 525.2.g.b 4
15.e even 4 1 105.2.b.a 2
15.e even 4 1 525.2.b.a 2
20.e even 4 1 1680.2.f.c 2
21.c even 2 1 inner 525.2.g.c 4
35.c odd 2 1 525.2.g.b 4
35.f even 4 1 105.2.b.a 2
35.f even 4 1 525.2.b.a 2
35.k even 12 1 735.2.s.b 2
35.k even 12 1 735.2.s.d 2
35.l odd 12 1 735.2.s.a 2
35.l odd 12 1 735.2.s.f 2
60.l odd 4 1 1680.2.f.b 2
105.g even 2 1 inner 525.2.g.c 4
105.k odd 4 1 105.2.b.b yes 2
105.k odd 4 1 525.2.b.b 2
105.w odd 12 1 735.2.s.a 2
105.w odd 12 1 735.2.s.f 2
105.x even 12 1 735.2.s.b 2
105.x even 12 1 735.2.s.d 2
140.j odd 4 1 1680.2.f.b 2
420.w even 4 1 1680.2.f.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.b.a 2 15.e even 4 1
105.2.b.a 2 35.f even 4 1
105.2.b.b yes 2 5.c odd 4 1
105.2.b.b yes 2 105.k odd 4 1
525.2.b.a 2 15.e even 4 1
525.2.b.a 2 35.f even 4 1
525.2.b.b 2 5.c odd 4 1
525.2.b.b 2 105.k odd 4 1
525.2.g.b 4 3.b odd 2 1
525.2.g.b 4 7.b odd 2 1
525.2.g.b 4 15.d odd 2 1
525.2.g.b 4 35.c odd 2 1
525.2.g.c 4 1.a even 1 1 trivial
525.2.g.c 4 5.b even 2 1 inner
525.2.g.c 4 21.c even 2 1 inner
525.2.g.c 4 105.g even 2 1 inner
735.2.s.a 2 35.l odd 12 1
735.2.s.a 2 105.w odd 12 1
735.2.s.b 2 35.k even 12 1
735.2.s.b 2 105.x even 12 1
735.2.s.d 2 35.k even 12 1
735.2.s.d 2 105.x even 12 1
735.2.s.f 2 35.l odd 12 1
735.2.s.f 2 105.w odd 12 1
1680.2.f.b 2 60.l odd 4 1
1680.2.f.b 2 140.j odd 4 1
1680.2.f.c 2 20.e even 4 1
1680.2.f.c 2 420.w even 4 1

Hecke kernels

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

\( T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{41} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T + 6)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T + 12)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$79$ \( (T + 8)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T - 6)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
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