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Properties

Label 525.2.b.h
Level $525$
Weight $2$
Character orbit 525.b
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(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: \(8\)
Coefficient field: 8.0.624529833984.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} - 2x^{4} - 18x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_{7} q^{3} + (\beta_{4} - 2) q^{4} + \beta_{3} q^{6} + ( - \beta_1 - 1) q^{7} + (\beta_{6} - \beta_{2}) q^{8} + (\beta_{4} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_{7} q^{3} + (\beta_{4} - 2) q^{4} + \beta_{3} q^{6} + ( - \beta_1 - 1) q^{7} + (\beta_{6} - \beta_{2}) q^{8} + (\beta_{4} - \beta_{2}) q^{9} + (\beta_{6} - \beta_{2}) q^{11} + (\beta_{7} + \beta_{5} + \beta_1) q^{12} + \beta_1 q^{13} + ( - \beta_{5} + \beta_{3} - \beta_{2}) q^{14} + ( - \beta_{4} + 1) q^{16} + ( - \beta_{5} + \beta_{3}) q^{17} + (\beta_{6} - \beta_{4} - \beta_{2} + 4) q^{18} + ( - 2 \beta_{7} - \beta_{5} - \beta_{3}) q^{19} + (\beta_{7} + \beta_{6} + \beta_{4} - 2) q^{21} + ( - 3 \beta_{4} + 5) q^{22} + (\beta_{6} + \beta_{2}) q^{23} + (\beta_{7} + \beta_{5} + \cdots - 2 \beta_1) q^{24}+ \cdots + (\beta_{6} + 3 \beta_{4} + 4 \beta_{2} - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{4} - 8 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{4} - 8 q^{7} + 4 q^{9} + 4 q^{16} + 28 q^{18} - 12 q^{21} + 28 q^{22} + 12 q^{28} + 36 q^{36} + 32 q^{37} + 12 q^{39} - 16 q^{43} - 28 q^{46} - 40 q^{49} - 48 q^{57} + 28 q^{58} - 4 q^{63} + 8 q^{64} - 40 q^{67} - 28 q^{72} - 32 q^{79} + 16 q^{81} + 60 q^{84} - 112 q^{88} + 48 q^{91} - 36 q^{93} - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{6} - 2x^{4} - 18x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} - 7\nu^{5} - 7\nu^{3} + 63\nu ) / 54 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 2\nu^{4} + 7\nu^{2} - 18 ) / 18 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{7} + 5\nu^{5} - 22\nu^{3} + 27\nu ) / 54 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 2\nu^{4} + 11\nu^{2} + 18 ) / 18 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 2\nu^{5} + 25\nu^{3} + 36\nu ) / 54 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{6} + 5\nu^{4} - 4\nu^{2} - 45 ) / 18 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 2\nu^{5} - 2\nu^{3} - 18\nu ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{5} + \beta_{3} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + 4\beta_{5} - 2\beta_{3} - 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} + 2\beta_{4} - 2\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -13\beta_{7} + \beta_{5} + 7\beta_{3} - 11\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{6} - 3\beta_{4} + 7\beta_{2} + 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 35\beta_{7} + 28\beta_{5} + 28\beta_{3} - 8\beta_1 ) / 3 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
−0.777403 + 1.54779i
0.777403 − 1.54779i
−1.70166 + 0.323042i
1.70166 − 0.323042i
−1.70166 − 0.323042i
1.70166 + 0.323042i
−0.777403 − 1.54779i
0.777403 + 1.54779i
− 2.40651i −0.777403 − 1.54779i −3.79129 0 −3.72476 + 1.87083i −1.00000 − 2.44949i 4.31075i −1.79129 + 2.40651i 0
251.2 − 2.40651i 0.777403 + 1.54779i −3.79129 0 3.72476 − 1.87083i −1.00000 + 2.44949i 4.31075i −1.79129 + 2.40651i 0
251.3 − 1.09941i −1.70166 − 0.323042i 0.791288 0 −0.355157 + 1.87083i −1.00000 + 2.44949i − 3.06878i 2.79129 + 1.09941i 0
251.4 − 1.09941i 1.70166 + 0.323042i 0.791288 0 0.355157 − 1.87083i −1.00000 − 2.44949i − 3.06878i 2.79129 + 1.09941i 0
251.5 1.09941i −1.70166 + 0.323042i 0.791288 0 −0.355157 − 1.87083i −1.00000 − 2.44949i 3.06878i 2.79129 − 1.09941i 0
251.6 1.09941i 1.70166 − 0.323042i 0.791288 0 0.355157 + 1.87083i −1.00000 + 2.44949i 3.06878i 2.79129 − 1.09941i 0
251.7 2.40651i −0.777403 + 1.54779i −3.79129 0 −3.72476 − 1.87083i −1.00000 + 2.44949i − 4.31075i −1.79129 − 2.40651i 0
251.8 2.40651i 0.777403 − 1.54779i −3.79129 0 3.72476 + 1.87083i −1.00000 − 2.44949i − 4.31075i −1.79129 − 2.40651i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
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.h ✓ 8
3.b odd 2 1 inner 525.2.b.h ✓ 8
5.b even 2 1 525.2.b.i yes 8
5.c odd 4 2 525.2.g.f 16
7.b odd 2 1 inner 525.2.b.h ✓ 8
15.d odd 2 1 525.2.b.i yes 8
15.e even 4 2 525.2.g.f 16
21.c even 2 1 inner 525.2.b.h ✓ 8
35.c odd 2 1 525.2.b.i yes 8
35.f even 4 2 525.2.g.f 16
105.g even 2 1 525.2.b.i yes 8
105.k odd 4 2 525.2.g.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.b.h ✓ 8 1.a even 1 1 trivial
525.2.b.h ✓ 8 3.b odd 2 1 inner
525.2.b.h ✓ 8 7.b odd 2 1 inner
525.2.b.h ✓ 8 21.c even 2 1 inner
525.2.b.i yes 8 5.b even 2 1
525.2.b.i yes 8 15.d odd 2 1
525.2.b.i yes 8 35.c odd 2 1
525.2.b.i yes 8 105.g even 2 1
525.2.g.f 16 5.c odd 4 2
525.2.g.f 16 15.e even 4 2
525.2.g.f 16 35.f even 4 2
525.2.g.f 16 105.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 \(S_{2}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{4} + 7T_{2}^{2} + 7 \) Copy content Toggle raw display
\( T_{11}^{4} + 28T_{11}^{2} + 175 \) Copy content Toggle raw display
\( T_{17}^{4} - 42T_{17}^{2} + 252 \) Copy content Toggle raw display
\( T_{37}^{2} - 8T_{37} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 7 T^{2} + 7)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 7)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 28 T^{2} + 175)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 42 T^{2} + 252)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 66 T^{2} + 900)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 28 T^{2} + 7)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 28 T^{2} + 7)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 66 T^{2} + 900)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T - 5)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 126 T^{2} + 2268)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 17)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 42 T^{2} + 252)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 196 T^{2} + 2800)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 168 T^{2} + 6300)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 66 T^{2} + 900)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 10 T - 59)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 28 T^{2} + 175)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 300 T^{2} + 10404)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T - 5)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 294 T^{2} + 12348)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 168 T^{2} + 6300)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 138 T^{2} + 36)^{2} \) Copy content Toggle raw display
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