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Properties

Label 525.4.d.o
Level $525$
Weight $4$
Character orbit 525.d
Analytic conductor $30.976$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.274");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(30.9760027530\)
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: \( x^{8} + 40x^{6} + 488x^{4} + 1945x^{2} + 1936 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_1) q^{2} - 3 \beta_{4} q^{3} + ( - 3 \beta_{3} + \beta_{2} - 5) q^{4} + (3 \beta_{3} + 6) q^{6} - 7 \beta_{4} q^{7} + ( - 5 \beta_{7} + \beta_{5} + \cdots - 5 \beta_1) q^{8}+ \cdots - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_1) q^{2} - 3 \beta_{4} q^{3} + ( - 3 \beta_{3} + \beta_{2} - 5) q^{4} + (3 \beta_{3} + 6) q^{6} - 7 \beta_{4} q^{7} + ( - 5 \beta_{7} + \beta_{5} + \cdots - 5 \beta_1) q^{8}+ \cdots + (9 \beta_{6} + 27 \beta_{3} + \cdots - 99) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{4} + 36 q^{6} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{4} + 36 q^{6} - 72 q^{9} + 114 q^{11} + 84 q^{14} + 432 q^{16} + 24 q^{19} - 168 q^{21} - 558 q^{24} - 162 q^{26} - 756 q^{29} - 186 q^{31} - 1566 q^{34} + 288 q^{36} - 258 q^{39} - 930 q^{41} - 1362 q^{44} + 620 q^{46} - 392 q^{49} + 594 q^{51} - 324 q^{54} - 1302 q^{56} - 462 q^{59} - 2706 q^{61} - 6214 q^{64} - 246 q^{66} - 936 q^{69} - 3450 q^{71} + 3906 q^{74} - 6092 q^{76} - 3258 q^{79} + 648 q^{81} + 672 q^{84} - 9084 q^{86} + 1956 q^{89} - 602 q^{91} - 4960 q^{94} + 4140 q^{96} - 1026 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 40x^{6} + 488x^{4} + 1945x^{2} + 1936 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{6} - 61\nu^{4} - 323\nu^{2} + 575 ) / 147 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{6} - 61\nu^{4} - 470\nu^{2} - 895 ) / 147 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -17\nu^{7} - 592\nu^{5} - 5612\nu^{3} - 12385\nu ) / 6468 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 62\nu^{5} + 1159\nu^{3} + 5960\nu ) / 231 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 41\nu^{4} + 466\nu^{2} + 1088 ) / 21 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 41\nu^{7} + 1618\nu^{5} + 17720\nu^{3} + 42235\nu ) / 3234 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} - 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} + \beta_{5} - 8\beta_{4} - 15\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} + 29\beta_{3} - 22\beta_{2} + 159 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 61\beta_{7} - 22\beta_{5} + 258\beta_{4} + 265\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -61\beta_{6} - 723\beta_{3} + 436\beta_{2} - 2947 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -1464\beta_{7} + 436\beta_{5} - 6724\beta_{4} - 5005\beta_1 \) 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
− 4.56826i
− 2.21734i
− 3.56826i
− 1.21734i
1.21734i
3.56826i
2.21734i
4.56826i
− 5.56826i 3.00000i −23.0055 0 16.7048 7.00000i 83.5546i −9.00000 0
274.2 − 3.21734i 3.00000i −2.35129 0 9.65203 7.00000i − 18.1738i −9.00000 0
274.3 − 2.56826i − 3.00000i 1.40404 0 −7.70478 − 7.00000i − 24.1520i −9.00000 0
274.4 − 0.217342i − 3.00000i 7.95276 0 −0.652027 − 7.00000i − 3.46721i −9.00000 0
274.5 0.217342i 3.00000i 7.95276 0 −0.652027 7.00000i 3.46721i −9.00000 0
274.6 2.56826i 3.00000i 1.40404 0 −7.70478 7.00000i 24.1520i −9.00000 0
274.7 3.21734i − 3.00000i −2.35129 0 9.65203 − 7.00000i 18.1738i −9.00000 0
274.8 5.56826i − 3.00000i −23.0055 0 16.7048 − 7.00000i − 83.5546i −9.00000 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 525.4.d.o 8
5.b even 2 1 inner 525.4.d.o 8
5.c odd 4 1 525.4.a.s ✓ 4
5.c odd 4 1 525.4.a.v yes 4
15.e even 4 1 1575.4.a.bf 4
15.e even 4 1 1575.4.a.bm 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.s ✓ 4 5.c odd 4 1
525.4.a.v yes 4 5.c odd 4 1
525.4.d.o 8 1.a even 1 1 trivial
525.4.d.o 8 5.b even 2 1 inner
1575.4.a.bf 4 15.e even 4 1
1575.4.a.bm 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{8} + 48T_{2}^{6} + 596T_{2}^{4} + 2145T_{2}^{2} + 100 \) Copy content Toggle raw display
\( T_{11}^{4} - 57T_{11}^{3} - 1323T_{11}^{2} + 44829T_{11} + 99334 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 48 T^{6} + \cdots + 100 \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 57 T^{3} + \cdots + 99334)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 10502525377600 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 11989210951936 \) Copy content Toggle raw display
$19$ \( (T^{4} - 12 T^{3} + \cdots + 329386624)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 85\!\cdots\!25 \) Copy content Toggle raw display
$29$ \( (T^{4} + 378 T^{3} + \cdots + 29419291)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 93 T^{3} + \cdots + 1187132760)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 44\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{4} + 465 T^{3} + \cdots - 7122182144)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 61\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 74\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{4} + 231 T^{3} + \cdots + 323902840)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 1353 T^{3} + \cdots - 9759896064)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{4} + 1725 T^{3} + \cdots - 423829861100)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 45\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{4} + 1629 T^{3} + \cdots + 28849899226)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 51\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} - 978 T^{3} + \cdots - 108748692656)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
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