↑

Properties

Label 525.6.d.m
Level $525$
Weight $6$
Character orbit 525.d
Analytic conductor $84.202$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,6,Mod(274,525)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.274");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 525.d (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: \(84.2015054018\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 121x^{6} + 4400x^{4} + 47104x^{2} + 153664 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + ( - 4 \beta_{2} + \beta_1) q^{2} - 9 \beta_{2} q^{3} + (7 \beta_{4} + \beta_{3} - 14) q^{4} + (9 \beta_{4} - 36) q^{6} + 49 \beta_{2} q^{7} + ( - 9 \beta_{6} + 4 \beta_{5} + \cdots - 21 \beta_1) q^{8}+ \cdots - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 4 \beta_{2} + \beta_1) q^{2} - 9 \beta_{2} q^{3} + (7 \beta_{4} + \beta_{3} - 14) q^{4} + (9 \beta_{4} - 36) q^{6} + 49 \beta_{2} q^{7} + ( - 9 \beta_{6} + 4 \beta_{5} + \cdots - 21 \beta_1) q^{8}+ \cdots + ( - 1053 \beta_{7} + 891 \beta_{4} + \cdots - 3726) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 98 q^{4} - 270 q^{6} - 648 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 98 q^{4} - 270 q^{6} - 648 q^{9} + 372 q^{11} + 1470 q^{14} + 5634 q^{16} + 5484 q^{19} + 3528 q^{21} + 9774 q^{24} + 9618 q^{26} - 1140 q^{29} - 18828 q^{31} - 4290 q^{34} + 7938 q^{36} + 10980 q^{39} - 34008 q^{41} - 69180 q^{44} - 60970 q^{46} - 19208 q^{49} - 2916 q^{51} + 21870 q^{54} - 53214 q^{56} - 19560 q^{59} - 120816 q^{61} - 200290 q^{64} - 42624 q^{66} - 50868 q^{69} + 2028 q^{71} - 283368 q^{74} + 5200 q^{76} - 298320 q^{79} + 52488 q^{81} - 43218 q^{84} + 192570 q^{86} + 15924 q^{89} - 59780 q^{91} - 465184 q^{94} - 211518 q^{96} - 30132 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 121x^{6} + 4400x^{4} + 47104x^{2} + 153664 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -13\nu^{7} - 1475\nu^{5} - 46126\nu^{3} - 267392\nu ) / 2352 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 113\nu^{4} - 3496\nu^{2} - 19664 ) / 24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 113\nu^{4} - 3520\nu^{2} - 20384 ) / 24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 269\nu^{7} + 30589\nu^{5} + 959768\nu^{3} + 5614192\nu ) / 4704 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -73\nu^{7} - 8294\nu^{5} - 259705\nu^{3} - 1505444\nu ) / 588 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\nu^{6} + 2159\nu^{4} + 67636\nu^{2} + 392768 ) / 48 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} - 30 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{6} + 4\beta_{5} - 26\beta_{2} - 49\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{7} + 101\beta_{4} - 63\beta_{3} + 1434 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -235\beta_{6} - 244\beta_{5} + 2754\beta_{2} + 2641\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -452\beta_{7} - 7917\beta_{4} + 3599\beta_{3} - 76826 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 16019\beta_{6} + 13492\beta_{5} - 220402\beta_{2} - 146361\beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1
− 6.77213i
− 2.72516i
− 7.76014i
2.73715i
− 2.73715i
7.76014i
2.72516i
6.77213i
− 10.7721i − 9.00000i −84.0388 0 −96.9492 49.0000i 560.569i −81.0000 0
274.2 − 6.72516i − 9.00000i −13.2278 0 −60.5264 49.0000i − 126.246i −81.0000 0
274.3 − 3.76014i 9.00000i 17.8614 0 33.8412 − 49.0000i − 187.486i −81.0000 0
274.4 − 1.26285i − 9.00000i 30.4052 0 −11.3656 49.0000i − 78.8082i −81.0000 0
274.5 1.26285i 9.00000i 30.4052 0 −11.3656 − 49.0000i 78.8082i −81.0000 0
274.6 3.76014i − 9.00000i 17.8614 0 33.8412 49.0000i 187.486i −81.0000 0
274.7 6.72516i 9.00000i −13.2278 0 −60.5264 − 49.0000i 126.246i −81.0000 0
274.8 10.7721i 9.00000i −84.0388 0 −96.9492 − 49.0000i − 560.569i −81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 274.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.6.d.m 8
5.b even 2 1 inner 525.6.d.m 8
5.c odd 4 1 525.6.a.k ✓ 4
5.c odd 4 1 525.6.a.p yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.6.a.k ✓ 4 5.c odd 4 1
525.6.a.p yes 4 5.c odd 4 1
525.6.d.m 8 1.a even 1 1 trivial
525.6.d.m 8 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 177T_{2}^{6} + 7808T_{2}^{4} + 86208T_{2}^{2} + 118336 \) acting on \(S_{6}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 177 T^{6} + \cdots + 118336 \) Copy content Toggle raw display
$3$ \( (T^{2} + 81)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2401)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 186 T^{3} + \cdots - 9038084300)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 91\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( (T^{4} - 2742 T^{3} + \cdots - 304545071360)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 31\!\cdots\!29 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 11538421263305)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 11\!\cdots\!56)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 38\!\cdots\!09 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots - 31\!\cdots\!60)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 33\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 24\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 36\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 77\!\cdots\!40)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 37\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 16\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 43\!\cdots\!64 \) Copy content Toggle raw display
show more
show less