Properties

Label 1568.2.t.e
Level $1568$
Weight $2$
Character orbit 1568.t
Analytic conductor $12.521$
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] = [1568,2,Mod(177,1568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1568, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1568.177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1568.t (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: \(12.5205430369\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.432972864.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{6} + 4x^{5} - 6x^{4} + 8x^{3} + 4x^{2} - 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 56)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} - \beta_{3} q^{5} + ( - \beta_{7} + \beta_{4} + 2 \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} - \beta_{3} q^{5} + ( - \beta_{7} + \beta_{4} + 2 \beta_1 + 2) q^{9} + ( - \beta_{6} - \beta_{2}) q^{11} + (\beta_{6} - \beta_{3}) q^{13} + ( - \beta_{4} - 1) q^{15} - 2 \beta_1 q^{17} + ( - \beta_{5} + \beta_{2}) q^{19} + ( - \beta_{7} + \beta_{4} + \beta_1 + 1) q^{23} + ( - \beta_{7} - 2 \beta_1) q^{25} + (2 \beta_{6} - 2 \beta_{5} - 2 \beta_{3}) q^{27} - 2 \beta_{5} q^{29} + (2 \beta_{7} - 2 \beta_1) q^{31} + (4 \beta_1 + 4) q^{33} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{37} + ( - \beta_{7} + \beta_1) q^{39} + ( - 2 \beta_{4} + 4) q^{41} + (\beta_{6} + \beta_{5} - \beta_{3}) q^{43} + (\beta_{6} + 4 \beta_{2}) q^{45} + (2 \beta_{5} - 2 \beta_{2}) q^{51} + (2 \beta_{6} + 2 \beta_{2}) q^{53} + ( - 2 \beta_{4} + 6) q^{55} + ( - \beta_{4} - 5) q^{57} + ( - 2 \beta_{6} - \beta_{2}) q^{59} + \beta_{3} q^{61} + ( - \beta_{7} + \beta_{4} - 7 \beta_1 - 7) q^{65} + (5 \beta_{6} - \beta_{2}) q^{67} + (2 \beta_{6} - 4 \beta_{5} - 2 \beta_{3}) q^{69} + 8 q^{71} + 6 \beta_1 q^{73} + ( - \beta_{5} - 2 \beta_{3} + \beta_{2}) q^{75} + ( - \beta_{7} + 6 \beta_1) q^{81} + ( - 2 \beta_{6} - 3 \beta_{5} + 2 \beta_{3}) q^{83} + (2 \beta_{6} - 2 \beta_{3}) q^{85} + ( - 2 \beta_{7} + 10 \beta_1) q^{87} + (2 \beta_{7} - 2 \beta_{4} - 8 \beta_1 - 8) q^{89} + (8 \beta_{5} + 4 \beta_{3} - 8 \beta_{2}) q^{93} + (\beta_{7} - \beta_1) q^{95} + ( - 2 \beta_{4} - 4) q^{97} + (3 \beta_{6} - \beta_{5} - 3 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} - 8 q^{15} + 8 q^{17} + 4 q^{23} + 8 q^{25} + 8 q^{31} + 16 q^{33} - 4 q^{39} + 32 q^{41} + 48 q^{55} - 40 q^{57} - 28 q^{65} + 64 q^{71} - 24 q^{73} - 24 q^{81} - 40 q^{87} - 32 q^{89} + 4 q^{95} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + x^{6} + 4x^{5} - 6x^{4} + 8x^{3} + 4x^{2} - 8x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + \nu^{6} - \nu^{5} + 2\nu^{3} - 4\nu^{2} + 4\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - \nu^{4} + \nu^{3} - 2\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - \nu^{6} + \nu^{5} + 4\nu^{4} - 2\nu^{3} + 4\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + \nu^{5} - \nu^{4} + 2\nu^{3} + 2\nu^{2} - 4\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 2\nu^{5} - 2\nu^{4} - \nu^{3} + 4\nu^{2} - 8\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - \nu^{6} - \nu^{5} + 6\nu^{4} - 8\nu^{3} + 8\nu - 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} - 3\nu^{6} + 11\nu^{5} - 6\nu^{3} + 36\nu^{2} - 12\nu + 24 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} - 3\beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{6} - \beta_{5} - \beta_{4} + 3\beta_{3} - 7 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{7} + \beta_{5} + \beta_{4} + 5\beta_{3} - \beta_{2} + 7\beta _1 + 7 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} + 3\beta_{6} + 9\beta_{2} + 9\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5\beta_{6} - \beta_{5} + 7\beta_{4} - 5\beta_{3} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -\beta_{7} - 7\beta_{5} + \beta_{4} - 3\beta_{3} + 7\beta_{2} - 41\beta _1 - 41 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(-1\) \(1\) \(-1 - \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
0.630783 + 1.26575i
0.121053 1.40902i
1.15972 0.809347i
−1.41156 + 0.0865986i
0.630783 1.26575i
0.121053 + 1.40902i
1.15972 + 0.809347i
−1.41156 0.0865986i
0 −2.61578 + 1.51022i 0 1.46890 + 0.848071i 0 0 0 3.06155 5.30277i 0
177.2 0 −0.810969 + 0.468213i 0 −2.88831 1.66757i 0 0 0 −1.06155 + 1.83866i 0
177.3 0 0.810969 0.468213i 0 2.88831 + 1.66757i 0 0 0 −1.06155 + 1.83866i 0
177.4 0 2.61578 1.51022i 0 −1.46890 0.848071i 0 0 0 3.06155 5.30277i 0
753.1 0 −2.61578 1.51022i 0 1.46890 0.848071i 0 0 0 3.06155 + 5.30277i 0
753.2 0 −0.810969 0.468213i 0 −2.88831 + 1.66757i 0 0 0 −1.06155 1.83866i 0
753.3 0 0.810969 + 0.468213i 0 2.88831 1.66757i 0 0 0 −1.06155 1.83866i 0
753.4 0 2.61578 + 1.51022i 0 −1.46890 + 0.848071i 0 0 0 3.06155 + 5.30277i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 177.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
8.b even 2 1 inner
56.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.2.t.e 8
4.b odd 2 1 392.2.p.e 8
7.b odd 2 1 1568.2.t.d 8
7.c even 3 1 1568.2.b.d 4
7.c even 3 1 inner 1568.2.t.e 8
7.d odd 6 1 224.2.b.b 4
7.d odd 6 1 1568.2.t.d 8
8.b even 2 1 inner 1568.2.t.e 8
8.d odd 2 1 392.2.p.e 8
21.g even 6 1 2016.2.c.c 4
28.d even 2 1 392.2.p.f 8
28.f even 6 1 56.2.b.b 4
28.f even 6 1 392.2.p.f 8
28.g odd 6 1 392.2.b.c 4
28.g odd 6 1 392.2.p.e 8
56.e even 2 1 392.2.p.f 8
56.h odd 2 1 1568.2.t.d 8
56.j odd 6 1 224.2.b.b 4
56.j odd 6 1 1568.2.t.d 8
56.k odd 6 1 392.2.b.c 4
56.k odd 6 1 392.2.p.e 8
56.m even 6 1 56.2.b.b 4
56.m even 6 1 392.2.p.f 8
56.p even 6 1 1568.2.b.d 4
56.p even 6 1 inner 1568.2.t.e 8
84.j odd 6 1 504.2.c.d 4
112.v even 12 2 1792.2.a.x 4
112.x odd 12 2 1792.2.a.v 4
168.ba even 6 1 2016.2.c.c 4
168.be odd 6 1 504.2.c.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.b.b 4 28.f even 6 1
56.2.b.b 4 56.m even 6 1
224.2.b.b 4 7.d odd 6 1
224.2.b.b 4 56.j odd 6 1
392.2.b.c 4 28.g odd 6 1
392.2.b.c 4 56.k odd 6 1
392.2.p.e 8 4.b odd 2 1
392.2.p.e 8 8.d odd 2 1
392.2.p.e 8 28.g odd 6 1
392.2.p.e 8 56.k odd 6 1
392.2.p.f 8 28.d even 2 1
392.2.p.f 8 28.f even 6 1
392.2.p.f 8 56.e even 2 1
392.2.p.f 8 56.m even 6 1
504.2.c.d 4 84.j odd 6 1
504.2.c.d 4 168.be odd 6 1
1568.2.b.d 4 7.c even 3 1
1568.2.b.d 4 56.p even 6 1
1568.2.t.d 8 7.b odd 2 1
1568.2.t.d 8 7.d odd 6 1
1568.2.t.d 8 56.h odd 2 1
1568.2.t.d 8 56.j odd 6 1
1568.2.t.e 8 1.a even 1 1 trivial
1568.2.t.e 8 7.c even 3 1 inner
1568.2.t.e 8 8.b even 2 1 inner
1568.2.t.e 8 56.p even 6 1 inner
1792.2.a.v 4 112.x odd 12 2
1792.2.a.x 4 112.v even 12 2
2016.2.c.c 4 21.g even 6 1
2016.2.c.c 4 168.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1568, [\chi])\):

\( T_{3}^{8} - 10T_{3}^{6} + 92T_{3}^{4} - 80T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{17}^{2} - 2T_{17} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 10 T^{6} + 92 T^{4} - 80 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{8} - 14 T^{6} + 164 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 20 T^{6} + 368 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( (T^{4} + 14 T^{2} + 32)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T + 4)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} - 10 T^{6} + 92 T^{4} - 80 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( (T^{4} - 2 T^{3} + 20 T^{2} + 32 T + 256)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 40 T^{2} + 128)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} + 80 T^{2} + 256 T + 4096)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 40 T^{6} + 1472 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$41$ \( (T^{2} - 8 T - 52)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 20 T^{2} + 32)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} - 80 T^{6} + 5888 T^{4} + \cdots + 262144 \) Copy content Toggle raw display
$59$ \( T^{8} - 58 T^{6} + 3356 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$61$ \( T^{8} - 14 T^{6} + 164 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$67$ \( T^{8} - 380 T^{6} + \cdots + 1073741824 \) Copy content Toggle raw display
$71$ \( (T - 8)^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} + 6 T + 36)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} + 122 T^{2} + 2888)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 16 T^{3} + 260 T^{2} - 64 T + 16)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T - 52)^{4} \) Copy content Toggle raw display
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