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Properties

Label 3528.2.bl.b
Level $3528$
Weight $2$
Character orbit 3528.bl
Analytic conductor $28.171$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3528,2,Mod(521,3528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3528, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3528.521");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3528.bl (of order \(6\), degree \(2\), 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: \(28.1712218331\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{22} \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{5} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{11} - \beta_{9} q^{13} + (\beta_{14} - \beta_{6} + \beta_{4}) q^{17} + \beta_{10} q^{19} + (3 \beta_{7} - \beta_1) q^{23} + (2 \beta_{8} - 3 \beta_{2}) q^{25} + ( - \beta_{7} + \beta_{5} - 2 \beta_{3}) q^{29} + ( - 2 \beta_{11} + 2 \beta_{9}) q^{31} + (4 \beta_{2} - 4) q^{37} + ( - \beta_{15} - \beta_{6}) q^{41} + (3 \beta_{13} + 2) q^{43} + ( - \beta_{15} + \beta_{14}) q^{47} + (\beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{53} + ( - \beta_{12} + 2 \beta_{9}) q^{55} + ( - 8 \beta_{7} + 4 \beta_1) q^{65} + (2 \beta_{8} + 4 \beta_{2}) q^{67} + (\beta_{7} - \beta_{5} - 3 \beta_{3}) q^{71} + ( - 2 \beta_{12} + \beta_{11} + \cdots - \beta_{9}) q^{73}+ \cdots + (2 \beta_{12} - \beta_{9}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{25} - 32 q^{37} + 32 q^{43} + 32 q^{67} + 64 q^{79} - 128 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{48}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{48}^{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{48}^{12} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\zeta_{48}^{7} + 2\zeta_{48} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{48}^{14} + \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{48}^{15} + 2\zeta_{48}^{9} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{48}^{14} + \zeta_{48}^{10} + \zeta_{48}^{6} \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 2\zeta_{48}^{14} + 2\zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 2\zeta_{48}^{15} - 2\zeta_{48}^{13} + 2\zeta_{48}^{9} + 2\zeta_{48}^{5} - 2\zeta_{48}^{3} \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( 2\zeta_{48}^{13} - 2\zeta_{48}^{11} + 2\zeta_{48}^{7} + 2\zeta_{48}^{3} + 2\zeta_{48} \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( -2\zeta_{48}^{13} + 2\zeta_{48}^{11} + 2\zeta_{48}^{7} - 2\zeta_{48}^{3} + 2\zeta_{48} \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( 2\zeta_{48}^{15} + 2\zeta_{48}^{13} + 2\zeta_{48}^{9} - 2\zeta_{48}^{5} + 2\zeta_{48}^{3} \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( -2\zeta_{48}^{10} + 2\zeta_{48}^{6} + 2\zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( 4\zeta_{48}^{11} + 4\zeta_{48}^{5} \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( -4\zeta_{48}^{13} + 4\zeta_{48}^{5} + 4\zeta_{48}^{3} \) Copy content Toggle raw display

Display \(\zeta_{48}^j\) in terms of \(\beta_i\)

\(\zeta_{48}\)\(=\) \( ( \beta_{11} + \beta_{10} + 2\beta_{4} ) / 8 \) Copy content Toggle raw display
\(\zeta_{48}^{2}\)\(=\) \( ( \beta_{8} + 2\beta_{5} ) / 4 \) Copy content Toggle raw display
\(\zeta_{48}^{3}\)\(=\) \( ( \beta_{15} + \beta_{12} - \beta_{9} ) / 8 \) Copy content Toggle raw display
\(\zeta_{48}^{4}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{5}\)\(=\) \( ( \beta_{14} - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} ) / 8 \) Copy content Toggle raw display
\(\zeta_{48}^{6}\)\(=\) \( ( \beta_{13} + 2\beta_{7} - 2\beta_{5} ) / 4 \) Copy content Toggle raw display
\(\zeta_{48}^{7}\)\(=\) \( ( \beta_{11} + \beta_{10} - 2\beta_{4} ) / 8 \) Copy content Toggle raw display
\(\zeta_{48}^{8}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{48}^{9}\)\(=\) \( ( \beta_{12} + \beta_{9} + 2\beta_{6} ) / 8 \) Copy content Toggle raw display
\(\zeta_{48}^{10}\)\(=\) \( ( -\beta_{13} + \beta_{8} + 2\beta_{7} ) / 4 \) Copy content Toggle raw display
\(\zeta_{48}^{11}\)\(=\) \( ( \beta_{14} + \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} ) / 8 \) Copy content Toggle raw display
\(\zeta_{48}^{12}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{13}\)\(=\) \( ( -\beta_{15} + \beta_{14} - \beta_{11} + \beta_{10} ) / 8 \) Copy content Toggle raw display
\(\zeta_{48}^{14}\)\(=\) \( ( \beta_{8} - 2\beta_{5} ) / 4 \) Copy content Toggle raw display
\(\zeta_{48}^{15}\)\(=\) \( ( \beta_{12} + \beta_{9} - 2\beta_{6} ) / 8 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{48}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(-1\) \(1 - \beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
521.1
−0.793353 − 0.608761i
−0.130526 − 0.991445i
0.608761 − 0.793353i
−0.991445 + 0.130526i
−0.608761 + 0.793353i
0.991445 − 0.130526i
0.793353 + 0.608761i
0.130526 + 0.991445i
−0.793353 + 0.608761i
−0.130526 + 0.991445i
0.608761 + 0.793353i
−0.991445 − 0.130526i
−0.608761 − 0.793353i
0.991445 + 0.130526i
0.793353 − 0.608761i
0.130526 − 0.991445i
0 0 0 −1.84776 − 3.20041i 0 0 0 0 0
521.2 0 0 0 −1.84776 − 3.20041i 0 0 0 0 0
521.3 0 0 0 −0.765367 − 1.32565i 0 0 0 0 0
521.4 0 0 0 −0.765367 − 1.32565i 0 0 0 0 0
521.5 0 0 0 0.765367 + 1.32565i 0 0 0 0 0
521.6 0 0 0 0.765367 + 1.32565i 0 0 0 0 0
521.7 0 0 0 1.84776 + 3.20041i 0 0 0 0 0
521.8 0 0 0 1.84776 + 3.20041i 0 0 0 0 0
1097.1 0 0 0 −1.84776 + 3.20041i 0 0 0 0 0
1097.2 0 0 0 −1.84776 + 3.20041i 0 0 0 0 0
1097.3 0 0 0 −0.765367 + 1.32565i 0 0 0 0 0
1097.4 0 0 0 −0.765367 + 1.32565i 0 0 0 0 0
1097.5 0 0 0 0.765367 − 1.32565i 0 0 0 0 0
1097.6 0 0 0 0.765367 − 1.32565i 0 0 0 0 0
1097.7 0 0 0 1.84776 − 3.20041i 0 0 0 0 0
1097.8 0 0 0 1.84776 − 3.20041i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 521.8
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
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.2.bl.b 16
3.b odd 2 1 inner 3528.2.bl.b 16
7.b odd 2 1 inner 3528.2.bl.b 16
7.c even 3 1 504.2.k.a ✓ 8
7.c even 3 1 inner 3528.2.bl.b 16
7.d odd 6 1 504.2.k.a ✓ 8
7.d odd 6 1 inner 3528.2.bl.b 16
21.c even 2 1 inner 3528.2.bl.b 16
21.g even 6 1 504.2.k.a ✓ 8
21.g even 6 1 inner 3528.2.bl.b 16
21.h odd 6 1 504.2.k.a ✓ 8
21.h odd 6 1 inner 3528.2.bl.b 16
28.f even 6 1 1008.2.k.c 8
28.g odd 6 1 1008.2.k.c 8
56.j odd 6 1 4032.2.k.f 8
56.k odd 6 1 4032.2.k.e 8
56.m even 6 1 4032.2.k.e 8
56.p even 6 1 4032.2.k.f 8
84.j odd 6 1 1008.2.k.c 8
84.n even 6 1 1008.2.k.c 8
168.s odd 6 1 4032.2.k.f 8
168.v even 6 1 4032.2.k.e 8
168.ba even 6 1 4032.2.k.f 8
168.be odd 6 1 4032.2.k.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.k.a ✓ 8 7.c even 3 1
504.2.k.a ✓ 8 7.d odd 6 1
504.2.k.a ✓ 8 21.g even 6 1
504.2.k.a ✓ 8 21.h odd 6 1
1008.2.k.c 8 28.f even 6 1
1008.2.k.c 8 28.g odd 6 1
1008.2.k.c 8 84.j odd 6 1
1008.2.k.c 8 84.n even 6 1
3528.2.bl.b 16 1.a even 1 1 trivial
3528.2.bl.b 16 3.b odd 2 1 inner
3528.2.bl.b 16 7.b odd 2 1 inner
3528.2.bl.b 16 7.c even 3 1 inner
3528.2.bl.b 16 7.d odd 6 1 inner
3528.2.bl.b 16 21.c even 2 1 inner
3528.2.bl.b 16 21.g even 6 1 inner
3528.2.bl.b 16 21.h odd 6 1 inner
4032.2.k.e 8 56.k odd 6 1
4032.2.k.e 8 56.m even 6 1
4032.2.k.e 8 168.v even 6 1
4032.2.k.e 8 168.be odd 6 1
4032.2.k.f 8 56.j odd 6 1
4032.2.k.f 8 56.p even 6 1
4032.2.k.f 8 168.s odd 6 1
4032.2.k.f 8 168.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 16T_{5}^{6} + 224T_{5}^{4} + 512T_{5}^{2} + 1024 \) acting on \(S_{2}^{\mathrm{new}}(3528, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 16 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 12 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 32 T^{2} + 128)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 80 T^{6} + \cdots + 2458624)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 32 T^{6} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 44 T^{6} + \cdots + 38416)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 36 T^{2} + 196)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 128 T^{6} + \cdots + 4194304)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4 T + 16)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} - 80 T^{2} + 32)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 4 T - 68)^{8} \) Copy content Toggle raw display
$47$ \( (T^{8} + 64 T^{6} + \cdots + 262144)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 36 T^{6} + \cdots + 38416)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( (T^{4} - 8 T^{3} + \cdots + 256)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 76 T^{2} + 1156)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 160 T^{6} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 16 T^{3} + \cdots + 1024)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 320 T^{2} + 25088)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + 80 T^{6} + \cdots + 2458624)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 160 T^{2} + 6272)^{4} \) Copy content Toggle raw display
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