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Properties

Label 528.2.o.b
Level $528$
Weight $2$
Character orbit 528.o
Analytic conductor $4.216$
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] = [528,2,Mod(175,528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(528, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 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("528.175");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 528.o (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: \(4.21610122672\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.454201344.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 4x^{6} + 24x^{5} - 25x^{4} - 12x^{3} + 128x^{2} - 182x + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( - \beta_1 + 1) q^{5} - \beta_{7} q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + ( - \beta_1 + 1) q^{5} - \beta_{7} q^{7} - q^{9} + (\beta_{4} + \beta_{3} + \beta_{2}) q^{11} + \beta_{5} q^{13} + (\beta_{4} + \beta_{2}) q^{15} - 2 \beta_{6} q^{17} + 2 \beta_{3} q^{19} - \beta_{5} q^{21} + ( - \beta_{4} + 3 \beta_{2}) q^{23} + ( - 2 \beta_1 - 1) q^{25} - \beta_{2} q^{27} + 2 \beta_{2} q^{31} + ( - \beta_{6} + \beta_1 - 1) q^{33} + ( - 2 \beta_{7} - 2 \beta_{3}) q^{35} + (2 \beta_1 + 4) q^{37} - \beta_{7} q^{39} - 2 \beta_{5} q^{41} + 2 \beta_{3} q^{43} + (\beta_1 - 1) q^{45} + (3 \beta_{4} - \beta_{2}) q^{47} + ( - 6 \beta_1 + 9) q^{49} - 2 \beta_{3} q^{51} + (\beta_1 - 9) q^{53} + (\beta_{7} + 2 \beta_{4} + 4 \beta_{2}) q^{55} - 2 \beta_{6} q^{57} + (2 \beta_{4} + 6 \beta_{2}) q^{59} + (4 \beta_{6} + \beta_{5}) q^{61} + \beta_{7} q^{63} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{65} - 6 \beta_{4} q^{67} + ( - \beta_1 - 3) q^{69} + ( - \beta_{4} - 9 \beta_{2}) q^{71} + 2 \beta_{5} q^{73} + (2 \beta_{4} - \beta_{2}) q^{75} + (2 \beta_{6} - 2 \beta_{5} + 5 \beta_1 - 1) q^{77} + \beta_{7} q^{79} + q^{81} + 2 \beta_{7} q^{83} + 2 \beta_{5} q^{85} + 6 \beta_1 q^{89} + ( - 6 \beta_{4} - 16 \beta_{2}) q^{91} - 2 q^{93} + 2 \beta_{7} q^{95} + ( - 2 \beta_1 + 2) q^{97} + ( - \beta_{4} - \beta_{3} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{5} - 8 q^{9} - 8 q^{25} - 8 q^{33} + 32 q^{37} - 8 q^{45} + 72 q^{49} - 72 q^{53} - 24 q^{69} - 8 q^{77} + 8 q^{81} - 16 q^{93} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 4x^{6} + 24x^{5} - 25x^{4} - 12x^{3} + 128x^{2} - 182x + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -20\nu^{7} - 402\nu^{6} - 43\nu^{5} + 760\nu^{4} - 4607\nu^{3} + 5750\nu^{2} - 6903\nu - 37211 ) / 21903 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -30\nu^{7} + 34\nu^{6} + 107\nu^{5} - 330\nu^{4} - 173\nu^{3} - 342\nu^{2} - 187\nu - 1157 ) / 1911 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4786 \nu^{7} + 17697 \nu^{6} + 40087 \nu^{5} - 168580 \nu^{4} + 95639 \nu^{3} + 434146 \nu^{2} + \cdots + 1041560 ) / 284739 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -124\nu^{7} - 168\nu^{6} + 1432\nu^{5} - 1546\nu^{4} - 4336\nu^{3} + 7936\nu^{2} - 12492\nu + 3341 ) / 5811 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -720\nu^{7} + 130\nu^{6} + 5753\nu^{5} - 16446\nu^{4} + 2071\nu^{3} + 31776\nu^{2} - 153595\nu + 120604 ) / 21903 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 856\nu^{7} - 1777\nu^{6} - 1080\nu^{5} + 11278\nu^{4} - 18930\nu^{3} + 16736\nu^{2} + 29692\nu - 34032 ) / 21903 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5862 \nu^{7} + 15052 \nu^{6} + 33393 \nu^{5} - 171498 \nu^{4} + 114999 \nu^{3} + 305436 \nu^{2} + \cdots + 848796 ) / 94913 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{5} + \beta_{4} - 2\beta_{3} - \beta_{2} - \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 3\beta_{2} + 3\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 3\beta_{6} - \beta_{5} + 3\beta_{4} - 3\beta_{3} - 10\beta_{2} - 3\beta _1 - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6\beta_{7} - 8\beta_{6} - 2\beta_{5} + 5\beta_{4} + 12\beta_{3} - 12\beta_{2} + 4\beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{7} - 6\beta_{6} - \beta_{5} + 29\beta_{4} - 4\beta_{3} - 31\beta_{2} - 89\beta _1 - 149 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{7} - 9\beta_{4} + 45\beta_{3} + 27\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 31\beta_{7} + 212\beta_{6} + 81\beta_{5} - 111\beta_{4} - 50\beta_{3} + 211\beta_{2} - 433\beta _1 - 755 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/528\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(145\) \(353\) \(463\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
175.1
2.02641 + 1.27503i
−2.39244 + 0.0909984i
1.02715 + 1.10132i
0.338876 − 1.46735i
2.02641 − 1.27503i
−2.39244 − 0.0909984i
1.02715 − 1.10132i
0.338876 + 1.46735i
0 − 1.00000i 0 −0.732051 0 −2.36806 0 −1.00000 0
175.2 0 − 1.00000i 0 −0.732051 0 2.36806 0 −1.00000 0
175.3 0 − 1.00000i 0 2.73205 0 −5.13734 0 −1.00000 0
175.4 0 − 1.00000i 0 2.73205 0 5.13734 0 −1.00000 0
175.5 0 1.00000i 0 −0.732051 0 −2.36806 0 −1.00000 0
175.6 0 1.00000i 0 −0.732051 0 2.36806 0 −1.00000 0
175.7 0 1.00000i 0 2.73205 0 −5.13734 0 −1.00000 0
175.8 0 1.00000i 0 2.73205 0 5.13734 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 175.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.b odd 2 1 inner
44.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.2.o.b ✓ 8
3.b odd 2 1 1584.2.o.g 8
4.b odd 2 1 inner 528.2.o.b ✓ 8
8.b even 2 1 2112.2.o.e 8
8.d odd 2 1 2112.2.o.e 8
11.b odd 2 1 inner 528.2.o.b ✓ 8
12.b even 2 1 1584.2.o.g 8
33.d even 2 1 1584.2.o.g 8
44.c even 2 1 inner 528.2.o.b ✓ 8
88.b odd 2 1 2112.2.o.e 8
88.g even 2 1 2112.2.o.e 8
132.d odd 2 1 1584.2.o.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
528.2.o.b ✓ 8 1.a even 1 1 trivial
528.2.o.b ✓ 8 4.b odd 2 1 inner
528.2.o.b ✓ 8 11.b odd 2 1 inner
528.2.o.b ✓ 8 44.c even 2 1 inner
1584.2.o.g 8 3.b odd 2 1
1584.2.o.g 8 12.b even 2 1
1584.2.o.g 8 33.d even 2 1
1584.2.o.g 8 132.d odd 2 1
2112.2.o.e 8 8.b even 2 1
2112.2.o.e 8 8.d odd 2 1
2112.2.o.e 8 88.b odd 2 1
2112.2.o.e 8 88.g even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 2T_{5} - 2 \) acting on \(S_{2}^{\mathrm{new}}(528, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 2 T - 2)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 32 T^{2} + 148)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 12 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( (T^{4} + 32 T^{2} + 148)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 56 T^{2} + 592)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 56 T^{2} + 592)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 24 T^{2} + 36)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T + 4)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 128 T^{2} + 2368)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 56 T^{2} + 592)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 56 T^{2} + 676)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 18 T + 78)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 96 T^{2} + 576)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 240 T^{2} + 1332)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 108)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 168 T^{2} + 6084)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 128 T^{2} + 2368)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 32 T^{2} + 148)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 128 T^{2} + 2368)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 4 T - 8)^{4} \) Copy content Toggle raw display
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