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Properties

Label 525.3.s.g
Level $525$
Weight $3$
Character orbit 525.s
Analytic conductor $14.305$
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] = [525,3,Mod(124,525)]
 
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, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.124");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 525.s (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: \(14.3052138789\)
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: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_1) q^{2} + (\beta_{3} + \beta_1) q^{3} + ( - 2 \beta_{5} + 3 \beta_{2}) q^{4} + (\beta_{6} - 2 \beta_{5} + 2 \beta_{2} - 1) q^{6} + (5 \beta_{3} + 3 \beta_1) q^{7} + ( - \beta_{7} + 11 \beta_{3}) q^{8} + (3 \beta_{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_1) q^{2} + (\beta_{3} + \beta_1) q^{3} + ( - 2 \beta_{5} + 3 \beta_{2}) q^{4} + (\beta_{6} - 2 \beta_{5} + 2 \beta_{2} - 1) q^{6} + (5 \beta_{3} + 3 \beta_1) q^{7} + ( - \beta_{7} + 11 \beta_{3}) q^{8} + (3 \beta_{2} - 3) q^{9} + (2 \beta_{5} + 2 \beta_{2}) q^{11} + ( - 4 \beta_{7} + 2 \beta_{4} + \cdots - 3 \beta_1) q^{12}+ \cdots + ( - 6 \beta_{6} - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{4} - 12 q^{9} + 8 q^{11} - 8 q^{14} - 20 q^{16} - 60 q^{19} - 60 q^{21} - 132 q^{24} - 204 q^{26} + 16 q^{29} - 72 q^{36} - 60 q^{39} + 72 q^{44} + 80 q^{46} - 284 q^{49} + 72 q^{51} - 36 q^{54} - 572 q^{56} + 48 q^{59} - 468 q^{61} + 40 q^{64} + 120 q^{66} - 208 q^{71} - 340 q^{74} - 76 q^{79} - 36 q^{81} - 288 q^{84} - 296 q^{86} + 144 q^{89} - 180 q^{91} - 720 q^{94} + 180 q^{96} - 48 q^{99}+O(q^{100}) \) 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
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display

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

\(\zeta_{24}\)\(=\) \( ( \beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} - \beta_{6} + 2\beta_{5} + 2\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( 2\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} ) / 6 \) 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 \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
124.1
−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
−0.965926 + 0.258819i
−2.98735 − 1.72474i −0.866025 − 1.50000i 3.94949 + 6.84072i 0 5.97469i −2.59808 − 6.50000i − 13.4495i −1.50000 + 2.59808i 0
124.2 −1.25529 − 0.724745i 0.866025 + 1.50000i −0.949490 − 1.64456i 0 − 2.51059i 2.59808 + 6.50000i 8.55051i −1.50000 + 2.59808i 0
124.3 1.25529 + 0.724745i −0.866025 − 1.50000i −0.949490 − 1.64456i 0 − 2.51059i −2.59808 − 6.50000i − 8.55051i −1.50000 + 2.59808i 0
124.4 2.98735 + 1.72474i 0.866025 + 1.50000i 3.94949 + 6.84072i 0 5.97469i 2.59808 + 6.50000i 13.4495i −1.50000 + 2.59808i 0
199.1 −2.98735 + 1.72474i −0.866025 + 1.50000i 3.94949 − 6.84072i 0 − 5.97469i −2.59808 + 6.50000i 13.4495i −1.50000 − 2.59808i 0
199.2 −1.25529 + 0.724745i 0.866025 − 1.50000i −0.949490 + 1.64456i 0 2.51059i 2.59808 − 6.50000i − 8.55051i −1.50000 − 2.59808i 0
199.3 1.25529 − 0.724745i −0.866025 + 1.50000i −0.949490 + 1.64456i 0 2.51059i −2.59808 + 6.50000i 8.55051i −1.50000 − 2.59808i 0
199.4 2.98735 − 1.72474i 0.866025 − 1.50000i 3.94949 − 6.84072i 0 − 5.97469i 2.59808 − 6.50000i − 13.4495i −1.50000 − 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 124.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.3.s.g 8
5.b even 2 1 inner 525.3.s.g 8
5.c odd 4 1 525.3.o.j ✓ 4
5.c odd 4 1 525.3.o.k yes 4
7.d odd 6 1 inner 525.3.s.g 8
35.i odd 6 1 inner 525.3.s.g 8
35.k even 12 1 525.3.o.j ✓ 4
35.k even 12 1 525.3.o.k yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.3.o.j ✓ 4 5.c odd 4 1
525.3.o.j ✓ 4 35.k even 12 1
525.3.o.k yes 4 5.c odd 4 1
525.3.o.k yes 4 35.k even 12 1
525.3.s.g 8 1.a even 1 1 trivial
525.3.s.g 8 5.b even 2 1 inner
525.3.s.g 8 7.d odd 6 1 inner
525.3.s.g 8 35.i odd 6 1 inner

Hecke kernels

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

\( T_{2}^{8} - 14T_{2}^{6} + 171T_{2}^{4} - 350T_{2}^{2} + 625 \) Copy content Toggle raw display
\( T_{11}^{4} - 4T_{11}^{3} + 36T_{11}^{2} + 80T_{11} + 400 \) Copy content Toggle raw display
\( T_{13}^{4} - 294T_{13}^{2} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 14 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$3$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 71 T^{2} + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{3} + \cdots + 400)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 294 T^{2} + 9)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 360 T^{6} + \cdots + 1679616 \) Copy content Toggle raw display
$19$ \( (T^{4} + 30 T^{3} + \cdots + 328329)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 400 T^{2} + 160000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 860)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 1152 T^{2} + 1327104)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 2450 T^{6} + \cdots + 390625 \) Copy content Toggle raw display
$41$ \( (T^{4} + 7656 T^{2} + 10419984)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 5192 T^{2} + 5779216)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 1360488960000 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 584046595707136 \) Copy content Toggle raw display
$59$ \( (T^{4} - 24 T^{3} + \cdots + 33454656)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 234 T^{3} + \cdots + 15327225)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 56\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( (T^{2} + 52 T - 1724)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 22430753361 \) Copy content Toggle raw display
$79$ \( (T^{4} + 38 T^{3} + \cdots + 1380625)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 7056 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 72 T^{3} + \cdots + 1871424)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 22470 T^{2} + 90269001)^{2} \) Copy content Toggle raw display
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