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Properties

Label 960.2.w.f
Level $960$
Weight $2$
Character orbit 960.w
Analytic conductor $7.666$
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] = [960,2,Mod(127,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("960.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.w (of order \(4\), 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: \(7.66563859404\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.1698758656.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 480)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + \beta_{4} q^{5} + (\beta_{6} + \beta_{5} + \beta_{3} + \cdots + 1) q^{7}+ \cdots - \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + \beta_{4} q^{5} + (\beta_{6} + \beta_{5} + \beta_{3} + \cdots + 1) q^{7}+ \cdots + (\beta_{4} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{7} + 12 q^{13} + 4 q^{15} - 4 q^{17} - 8 q^{19} - 4 q^{25} + 4 q^{33} - 32 q^{35} - 12 q^{37} - 24 q^{39} + 8 q^{41} + 24 q^{43} - 4 q^{45} + 24 q^{47} - 4 q^{53} - 4 q^{55} - 8 q^{57} + 16 q^{59} - 24 q^{61} + 4 q^{63} + 4 q^{65} + 24 q^{67} - 16 q^{73} + 32 q^{77} + 16 q^{79} - 8 q^{81} - 16 q^{83} + 44 q^{85} + 4 q^{87} - 40 q^{95} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 2\nu^{6} + 18\nu^{5} + 28\nu^{4} + 89\nu^{3} + 74\nu^{2} + 104\nu - 16 ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 18\nu^{5} + 8\nu^{4} + 105\nu^{3} + 72\nu^{2} + 248\nu + 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 18\nu^{5} - 28\nu^{4} + 89\nu^{3} - 74\nu^{2} + 104\nu + 16 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 18\nu^{5} - 8\nu^{4} + 105\nu^{3} - 72\nu^{2} + 248\nu - 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} - 46\nu^{5} - 179\nu^{3} - 168\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 6\nu^{6} + 10\nu^{5} - 92\nu^{4} - 15\nu^{3} - 358\nu^{2} - 120\nu - 336 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 6\nu^{6} + 10\nu^{5} + 92\nu^{4} - 15\nu^{3} + 358\nu^{2} - 120\nu + 336 ) / 64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + 2\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{4} + 3\beta_{3} - \beta_{2} - 3\beta _1 - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{7} - 9\beta_{6} - 18\beta_{5} - 5\beta_{4} - 13\beta_{3} - 5\beta_{2} - 13\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -9\beta_{7} + 9\beta_{6} - 17\beta_{4} - 27\beta_{3} + 17\beta_{2} + 27\beta _1 + 74 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 81\beta_{7} + 81\beta_{6} + 178\beta_{5} + 37\beta_{4} + 149\beta_{3} + 37\beta_{2} + 149\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 89\beta_{7} - 89\beta_{6} + 201\beta_{4} + 235\beta_{3} - 201\beta_{2} - 235\beta _1 - 650 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -761\beta_{7} - 761\beta_{6} - 1810\beta_{5} - 325\beta_{4} - 1565\beta_{3} - 325\beta_{2} - 1565\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(-\beta_{5}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
− 3.16053i
2.16053i
0.692297i
− 1.69230i
3.16053i
− 2.16053i
− 0.692297i
1.69230i
0 −0.707107 − 0.707107i 0 −2.23483 + 0.0743018i 0 3.16053 − 3.16053i 0 1.00000i 0
127.2 0 −0.707107 − 0.707107i 0 1.52773 + 1.63280i 0 −2.16053 + 2.16053i 0 1.00000i 0
127.3 0 0.707107 + 0.707107i 0 −0.489528 + 2.18183i 0 −0.692297 + 0.692297i 0 1.00000i 0
127.4 0 0.707107 + 0.707107i 0 1.19663 − 1.88893i 0 1.69230 − 1.69230i 0 1.00000i 0
703.1 0 −0.707107 + 0.707107i 0 −2.23483 − 0.0743018i 0 3.16053 + 3.16053i 0 − 1.00000i 0
703.2 0 −0.707107 + 0.707107i 0 1.52773 − 1.63280i 0 −2.16053 − 2.16053i 0 − 1.00000i 0
703.3 0 0.707107 − 0.707107i 0 −0.489528 − 2.18183i 0 −0.692297 − 0.692297i 0 − 1.00000i 0
703.4 0 0.707107 − 0.707107i 0 1.19663 + 1.88893i 0 1.69230 + 1.69230i 0 − 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.2.w.f 8
4.b odd 2 1 960.2.w.e 8
5.c odd 4 1 960.2.w.e 8
8.b even 2 1 480.2.w.d yes 8
8.d odd 2 1 480.2.w.c ✓ 8
20.e even 4 1 inner 960.2.w.f 8
24.f even 2 1 1440.2.x.q 8
24.h odd 2 1 1440.2.x.r 8
40.e odd 2 1 2400.2.w.j 8
40.f even 2 1 2400.2.w.i 8
40.i odd 4 1 480.2.w.c ✓ 8
40.i odd 4 1 2400.2.w.j 8
40.k even 4 1 480.2.w.d yes 8
40.k even 4 1 2400.2.w.i 8
120.q odd 4 1 1440.2.x.r 8
120.w even 4 1 1440.2.x.q 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.w.c ✓ 8 8.d odd 2 1
480.2.w.c ✓ 8 40.i odd 4 1
480.2.w.d yes 8 8.b even 2 1
480.2.w.d yes 8 40.k even 4 1
960.2.w.e 8 4.b odd 2 1
960.2.w.e 8 5.c odd 4 1
960.2.w.f 8 1.a even 1 1 trivial
960.2.w.f 8 20.e even 4 1 inner
1440.2.x.q 8 24.f even 2 1
1440.2.x.q 8 120.w even 4 1
1440.2.x.r 8 24.h odd 2 1
1440.2.x.r 8 120.q odd 4 1
2400.2.w.i 8 40.f even 2 1
2400.2.w.i 8 40.k even 4 1
2400.2.w.j 8 40.e odd 2 1
2400.2.w.j 8 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 4T_{7}^{7} + 8T_{7}^{6} + 24T_{7}^{5} + 132T_{7}^{4} - 320T_{7}^{3} + 512T_{7}^{2} + 1024T_{7} + 1024 \) acting on \(S_{2}^{\mathrm{new}}(960, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( T^{8} + 36 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( T^{8} - 12 T^{7} + \cdots + 141376 \) Copy content Toggle raw display
$17$ \( T^{8} + 4 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( (T^{4} + 4 T^{3} + \cdots + 128)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 192 T^{5} + \cdots + 4096 \) Copy content Toggle raw display
$29$ \( T^{8} + 196 T^{6} + \cdots + 2458624 \) Copy content Toggle raw display
$31$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} + 12 T^{7} + \cdots + 4426816 \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} + \cdots + 128)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 24 T^{7} + \cdots + 10240000 \) Copy content Toggle raw display
$47$ \( T^{8} - 24 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{8} + 4 T^{7} + \cdots + 8111104 \) Copy content Toggle raw display
$59$ \( (T^{4} - 8 T^{3} + \cdots - 2848)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 12 T^{3} + \cdots - 1600)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 24 T^{7} + \cdots + 36192256 \) Copy content Toggle raw display
$71$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + 16 T^{7} + \cdots + 795664 \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 16 T^{7} + \cdots + 6885376 \) Copy content Toggle raw display
$89$ \( T^{8} + 584 T^{6} + \cdots + 154157056 \) Copy content Toggle raw display
$97$ \( T^{8} - 32 T^{7} + \cdots + 258064 \) Copy content Toggle raw display
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