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Properties

Label 525.2.r.g
Level $525$
Weight $2$
Character orbit 525.r
Analytic conductor $4.192$
Analytic rank $0$
Dimension $8$
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(424,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.424");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.r (of order \(6\), 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: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} - \beta_1 q^{3} + \beta_{7} q^{6} + ( - 2 \beta_{6} + \beta_{5} + \cdots + \beta_1) q^{7}+ \cdots + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} - \beta_1 q^{3} + \beta_{7} q^{6} + ( - 2 \beta_{6} + \beta_{5} + \cdots + \beta_1) q^{7}+ \cdots + (\beta_{7} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{9} - 8 q^{11} + 24 q^{14} + 16 q^{16} - 4 q^{19} - 8 q^{21} - 8 q^{26} - 32 q^{29} + 12 q^{31} + 48 q^{34} + 12 q^{39} - 16 q^{41} + 24 q^{46} - 20 q^{49} - 8 q^{51} - 48 q^{56} - 16 q^{61} - 64 q^{64} + 8 q^{66} - 16 q^{69} - 16 q^{71} + 8 q^{74} + 4 q^{79} - 4 q^{81} + 8 q^{86} - 32 q^{89} + 36 q^{91} + 64 q^{94} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
424.1
0.965926 + 0.258819i
0.258819 − 0.965926i
−0.965926 − 0.258819i
−0.258819 + 0.965926i
0.965926 − 0.258819i
0.258819 + 0.965926i
−0.965926 + 0.258819i
−0.258819 − 0.965926i
−1.22474 + 0.707107i −0.866025 − 0.500000i 0 0 1.41421 −0.358719 − 2.62132i − 2.82843i 0.500000 + 0.866025i 0
424.2 −1.22474 + 0.707107i 0.866025 + 0.500000i 0 0 −1.41421 −2.09077 − 1.62132i − 2.82843i 0.500000 + 0.866025i 0
424.3 1.22474 − 0.707107i −0.866025 − 0.500000i 0 0 −1.41421 2.09077 + 1.62132i 2.82843i 0.500000 + 0.866025i 0
424.4 1.22474 − 0.707107i 0.866025 + 0.500000i 0 0 1.41421 0.358719 + 2.62132i 2.82843i 0.500000 + 0.866025i 0
499.1 −1.22474 − 0.707107i −0.866025 + 0.500000i 0 0 1.41421 −0.358719 + 2.62132i 2.82843i 0.500000 − 0.866025i 0
499.2 −1.22474 − 0.707107i 0.866025 − 0.500000i 0 0 −1.41421 −2.09077 + 1.62132i 2.82843i 0.500000 − 0.866025i 0
499.3 1.22474 + 0.707107i −0.866025 + 0.500000i 0 0 −1.41421 2.09077 − 1.62132i − 2.82843i 0.500000 − 0.866025i 0
499.4 1.22474 + 0.707107i 0.866025 − 0.500000i 0 0 1.41421 0.358719 − 2.62132i − 2.82843i 0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 424.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.r.g 8
5.b even 2 1 inner 525.2.r.g 8
5.c odd 4 1 105.2.i.c ✓ 4
5.c odd 4 1 525.2.i.g 4
7.c even 3 1 inner 525.2.r.g 8
15.e even 4 1 315.2.j.d 4
20.e even 4 1 1680.2.bg.p 4
35.f even 4 1 735.2.i.j 4
35.j even 6 1 inner 525.2.r.g 8
35.k even 12 1 735.2.a.j 2
35.k even 12 1 735.2.i.j 4
35.k even 12 1 3675.2.a.x 2
35.l odd 12 1 105.2.i.c ✓ 4
35.l odd 12 1 525.2.i.g 4
35.l odd 12 1 735.2.a.i 2
35.l odd 12 1 3675.2.a.z 2
105.w odd 12 1 2205.2.a.s 2
105.x even 12 1 315.2.j.d 4
105.x even 12 1 2205.2.a.u 2
140.w even 12 1 1680.2.bg.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.c ✓ 4 5.c odd 4 1
105.2.i.c ✓ 4 35.l odd 12 1
315.2.j.d 4 15.e even 4 1
315.2.j.d 4 105.x even 12 1
525.2.i.g 4 5.c odd 4 1
525.2.i.g 4 35.l odd 12 1
525.2.r.g 8 1.a even 1 1 trivial
525.2.r.g 8 5.b even 2 1 inner
525.2.r.g 8 7.c even 3 1 inner
525.2.r.g 8 35.j even 6 1 inner
735.2.a.i 2 35.l odd 12 1
735.2.a.j 2 35.k even 12 1
735.2.i.j 4 35.f even 4 1
735.2.i.j 4 35.k even 12 1
1680.2.bg.p 4 20.e even 4 1
1680.2.bg.p 4 140.w even 12 1
2205.2.a.s 2 105.w odd 12 1
2205.2.a.u 2 105.x even 12 1
3675.2.a.x 2 35.k even 12 1
3675.2.a.z 2 35.l odd 12 1

Hecke kernels

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

\( T_{2}^{4} - 2T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} + 4T_{11}^{3} + 14T_{11}^{2} + 8T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 10 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{4} + 4 T^{3} + 14 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 22 T^{2} + 49)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 44 T^{6} + \cdots + 38416 \) Copy content Toggle raw display
$19$ \( (T^{4} + 2 T^{3} + \cdots + 961)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 44 T^{6} + \cdots + 38416 \) Copy content Toggle raw display
$29$ \( (T^{2} + 8 T - 2)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 6 T^{3} + 35 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 102 T^{6} + \cdots + 4879681 \) Copy content Toggle raw display
$41$ \( (T^{2} + 4 T - 14)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 166 T^{2} + 6241)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 264 T^{6} + \cdots + 236421376 \) Copy content Toggle raw display
$53$ \( T^{8} - 48 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$59$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 8 T^{3} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 326 T^{6} + \cdots + 671898241 \) Copy content Toggle raw display
$71$ \( (T^{2} + 4 T + 2)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} - 150 T^{6} + \cdots + 390625 \) Copy content Toggle raw display
$79$ \( (T^{4} - 2 T^{3} + \cdots + 961)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 36 T^{2} + 196)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 16 T^{3} + \cdots + 2116)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 144 T^{2} + 3136)^{2} \) Copy content Toggle raw display
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