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Properties

Label 525.3.f.a
Level $525$
Weight $3$
Character orbit 525.f
Analytic conductor $14.305$
Analytic rank $0$
Dimension $8$
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(449,525)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 525.f (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: \(14.3052138789\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.4337012736.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 8x^{6} + 4x^{5} + 12x^{4} - 40x^{3} + 72x^{2} + 24x + 4 \) 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 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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{3} + (2 \beta_1 + 3) q^{4} + ( - \beta_{7} + 2 \beta_1 + 3) q^{6} + ( - \beta_{6} - \beta_{4} + \cdots + \beta_{2}) q^{7} + ( - 2 \beta_{6} + 2 \beta_{3} - 3 \beta_{2}) q^{8}+ \cdots + ( - 4 \beta_{7} - 12 \beta_{5} + \cdots - 30) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{4} + 28 q^{6} + 40 q^{9} + 72 q^{16} - 24 q^{19} + 28 q^{21} + 252 q^{24} + 272 q^{31} + 232 q^{36} - 8 q^{39} - 336 q^{46} - 56 q^{49} - 168 q^{51} + 308 q^{54} - 312 q^{61} - 8 q^{64} - 56 q^{66}+ \cdots - 224 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 8x^{6} + 4x^{5} + 12x^{4} - 40x^{3} + 72x^{2} + 24x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 7\nu^{6} + 29\nu^{5} - 83\nu^{4} + 110\nu^{3} - 68\nu^{2} - 26\nu - 804 ) / 302 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 40\nu^{7} - 129\nu^{6} + 254\nu^{5} + 153\nu^{4} + 1380\nu^{3} - 1210\nu^{2} - 436\nu + 3476 ) / 2718 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -104\nu^{7} + 879\nu^{6} - 2563\nu^{5} + 2592\nu^{4} + 1848\nu^{3} + 8582\nu^{2} - 26590\nu + 15968 ) / 5436 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -379\nu^{7} + 1596\nu^{6} - 3290\nu^{5} - 1008\nu^{4} - 4242\nu^{3} + 17920\nu^{2} - 29708\nu - 4532 ) / 5436 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 51\nu^{7} - 206\nu^{6} + 422\nu^{5} + 146\nu^{4} + 778\nu^{3} - 2260\nu^{2} + 4412\nu + 672 ) / 604 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -494\nu^{7} + 1797\nu^{6} - 3001\nu^{5} - 3996\nu^{4} - 4812\nu^{3} + 16982\nu^{2} - 23698\nu - 16564 ) / 5436 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -105\nu^{7} + 433\nu^{6} - 931\nu^{5} - 194\nu^{4} - 1282\nu^{3} + 3818\nu^{2} - 8142\nu - 744 ) / 604 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{5} + 3\beta_{4} - \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} + 3\beta_{5} + 4\beta_{4} - 2\beta_{3} + 5\beta_{2} - \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} - 2\beta_{3} + 12\beta_{2} - 10\beta _1 - 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -12\beta_{7} + 26\beta_{6} - 25\beta_{5} - 37\beta_{4} + 14\beta_{3} + 11\beta_{2} - 26\beta _1 - 63 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -52\beta_{7} + 100\beta_{6} - 102\beta_{5} - 174\beta_{4} + 100\beta_{3} - 100\beta_{2} - 26\beta _1 - 26 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -126\beta_{7} + 154\beta_{6} - 238\beta_{5} - 368\beta_{4} + 280\beta_{3} - 518\beta_{2} + 154\beta _1 + 522 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−1.25296 − 1.25296i
−1.25296 + 1.25296i
−0.153548 + 0.153548i
−0.153548 − 0.153548i
1.15355 − 1.15355i
1.15355 + 1.15355i
2.25296 + 2.25296i
2.25296 − 2.25296i
−3.50592 −2.88494 − 0.822876i 8.29150 0 10.1144 + 2.88494i 2.64575i −15.0457 7.64575 + 4.74789i 0
449.2 −3.50592 −2.88494 + 0.822876i 8.29150 0 10.1144 − 2.88494i − 2.64575i −15.0457 7.64575 − 4.74789i 0
449.3 −1.30710 2.38267 − 1.82288i −2.29150 0 −3.11438 + 2.38267i 2.64575i 8.22359 2.35425 − 8.68663i 0
449.4 −1.30710 2.38267 + 1.82288i −2.29150 0 −3.11438 − 2.38267i − 2.64575i 8.22359 2.35425 + 8.68663i 0
449.5 1.30710 −2.38267 − 1.82288i −2.29150 0 −3.11438 − 2.38267i 2.64575i −8.22359 2.35425 + 8.68663i 0
449.6 1.30710 −2.38267 + 1.82288i −2.29150 0 −3.11438 + 2.38267i − 2.64575i −8.22359 2.35425 − 8.68663i 0
449.7 3.50592 2.88494 − 0.822876i 8.29150 0 10.1144 − 2.88494i 2.64575i 15.0457 7.64575 − 4.74789i 0
449.8 3.50592 2.88494 + 0.822876i 8.29150 0 10.1144 + 2.88494i − 2.64575i 15.0457 7.64575 + 4.74789i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d 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.f.a 8
3.b odd 2 1 inner 525.3.f.a 8
5.b even 2 1 inner 525.3.f.a 8
5.c odd 4 1 21.3.b.a ✓ 4
5.c odd 4 1 525.3.c.a 4
15.d odd 2 1 inner 525.3.f.a 8
15.e even 4 1 21.3.b.a ✓ 4
15.e even 4 1 525.3.c.a 4
20.e even 4 1 336.3.d.c 4
35.f even 4 1 147.3.b.f 4
35.k even 12 2 147.3.h.c 8
35.l odd 12 2 147.3.h.e 8
40.i odd 4 1 1344.3.d.f 4
40.k even 4 1 1344.3.d.b 4
45.k odd 12 2 567.3.r.c 8
45.l even 12 2 567.3.r.c 8
60.l odd 4 1 336.3.d.c 4
105.k odd 4 1 147.3.b.f 4
105.w odd 12 2 147.3.h.c 8
105.x even 12 2 147.3.h.e 8
120.q odd 4 1 1344.3.d.b 4
120.w even 4 1 1344.3.d.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.b.a ✓ 4 5.c odd 4 1
21.3.b.a ✓ 4 15.e even 4 1
147.3.b.f 4 35.f even 4 1
147.3.b.f 4 105.k odd 4 1
147.3.h.c 8 35.k even 12 2
147.3.h.c 8 105.w odd 12 2
147.3.h.e 8 35.l odd 12 2
147.3.h.e 8 105.x even 12 2
336.3.d.c 4 20.e even 4 1
336.3.d.c 4 60.l odd 4 1
525.3.c.a 4 5.c odd 4 1
525.3.c.a 4 15.e even 4 1
525.3.f.a 8 1.a even 1 1 trivial
525.3.f.a 8 3.b odd 2 1 inner
525.3.f.a 8 5.b even 2 1 inner
525.3.f.a 8 15.d odd 2 1 inner
567.3.r.c 8 45.k odd 12 2
567.3.r.c 8 45.l even 12 2
1344.3.d.b 4 40.k even 4 1
1344.3.d.b 4 120.q odd 4 1
1344.3.d.f 4 40.i odd 4 1
1344.3.d.f 4 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 14T_{2}^{2} + 21 \) acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 14 T^{2} + 21)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 20 T^{6} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 56 T^{2} + 336)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 176 T^{2} + 5476)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 168 T^{2} + 3024)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T - 166)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 672 T^{2} + 12096)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 392 T^{2} + 27216)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 68 T + 1128)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2776 T^{2} + 1838736)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 5432 T^{2} + 4139856)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 3704 T^{2} + 1817104)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 2688 T^{2} + 1741824)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 11256 T^{2} + 2543184)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 10248 T^{2} + 14606676)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 78 T + 1178)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 968 T^{2} + 169744)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 9576 T^{2} + 22888656)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 9592 T^{2} + 21790224)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 64 T - 8048)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 13608 T^{2} + 44641044)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 20216 T^{2} + 64754256)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 904 T^{2} + 197136)^{2} \) Copy content Toggle raw display
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