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Properties

Label 1280.4.d.ba
Level $1280$
Weight $4$
Character orbit 1280.d
Analytic conductor $75.522$
Analytic rank $0$
Dimension $6$
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] = [1280,4,Mod(641,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, 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("1280.641");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1280.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: \(75.5224448073\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.12559936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 6x^{3} + 49x^{2} - 42x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 640)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_{2}) q^{3} - 5 \beta_{2} q^{5} + (\beta_1 + 7) q^{7} + ( - 2 \beta_1 - 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_{2}) q^{3} - 5 \beta_{2} q^{5} + (\beta_1 + 7) q^{7} + ( - 2 \beta_1 - 7) q^{9} + (3 \beta_{5} + 3 \beta_{4} + 10 \beta_{2}) q^{11} + (4 \beta_{5} - 2 \beta_{4}) q^{13} + (5 \beta_{3} - 5) q^{15} + ( - 10 \beta_{3} - 4 \beta_1) q^{17} + (5 \beta_{5} - 13 \beta_{4} - 48 \beta_{2}) q^{19} + ( - 2 \beta_{5} + 14 \beta_{4} - 16 \beta_{2}) q^{21} + ( - 8 \beta_{3} - 5 \beta_1 + 53) q^{23} - 25 q^{25} + (4 \beta_{5} + 6 \beta_{4} - 2 \beta_{2}) q^{27} + ( - 18 \beta_{5} + 6 \beta_{4} + 22 \beta_{2}) q^{29} + (16 \beta_{3} + 10 \beta_1 - 50) q^{31} + ( - 34 \beta_{3} - 62) q^{33} + ( - 5 \beta_{5} - 35 \beta_{2}) q^{35} + (4 \beta_{5} + 48 \beta_{4} - 74 \beta_{2}) q^{37} + ( - 26 \beta_{3} + 12 \beta_1 + 102) q^{39} + ( - 12 \beta_{3} - 10 \beta_1 + 116) q^{41} + ( - 34 \beta_{5} + 13 \beta_{4} - 83 \beta_{2}) q^{43} + (10 \beta_{5} + 35 \beta_{2}) q^{45} + ( - 18 \beta_{3} - 19 \beta_1 - 147) q^{47} + ( - 8 \beta_{3} + 6 \beta_1 - 165) q^{49} + ( - 12 \beta_{5} + \cdots - 294 \beta_{2}) q^{51}+ \cdots + (13 \beta_{5} - 15 \beta_{4} - 790 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 40 q^{7} - 38 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 40 q^{7} - 38 q^{9} - 20 q^{15} - 12 q^{17} + 312 q^{23} - 150 q^{25} - 288 q^{31} - 440 q^{33} + 536 q^{39} + 692 q^{41} - 880 q^{47} - 1018 q^{49} + 300 q^{55} + 2536 q^{57} - 1800 q^{63} - 60 q^{65} + 40 q^{71} + 1100 q^{73} - 4144 q^{79} - 2066 q^{81} - 1736 q^{87} + 3692 q^{89} - 1620 q^{95} - 1836 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 6x^{3} + 49x^{2} - 42x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -4\nu^{5} + 46\nu^{4} - 28\nu^{3} + 12\nu^{2} + 1671 ) / 167 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 49\nu^{5} + 21\nu^{4} + 9\nu^{3} - 147\nu^{2} + 2338\nu - 1056 ) / 1002 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -10\nu^{5} - 52\nu^{4} - 70\nu^{3} + 30\nu^{2} - 1083 ) / 167 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -133\nu^{5} - 57\nu^{4} + 71\nu^{3} + 1067\nu^{2} - 5678\nu + 2580 ) / 334 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 161\nu^{5} + 69\nu^{4} + 125\nu^{3} - 1151\nu^{2} + 8350\nu - 3756 ) / 334 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + \beta_{4} + 18\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{5} + 7\beta_{4} - 7\beta_{3} - 24\beta_{2} - 7\beta _1 + 24 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{3} + 5\beta _1 - 63 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -55\beta_{5} - 43\beta_{4} - 43\beta_{3} + 276\beta_{2} - 55\beta _1 + 276 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
641.1
0.455589 + 0.455589i
1.60096 − 1.60096i
−2.05655 − 2.05655i
−2.05655 + 2.05655i
1.60096 + 1.60096i
0.455589 − 0.455589i
0 − 7.34740i 0 − 5.00000i 0 16.9921 0 −26.9842 0
641.2 0 − 6.65606i 0 5.00000i 0 12.1516 0 −17.3032 0
641.3 0 − 1.30867i 0 − 5.00000i 0 −9.14370 0 25.2874 0
641.4 0 1.30867i 0 5.00000i 0 −9.14370 0 25.2874 0
641.5 0 6.65606i 0 − 5.00000i 0 12.1516 0 −17.3032 0
641.6 0 7.34740i 0 5.00000i 0 16.9921 0 −26.9842 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 641.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.4.d.ba 6
4.b odd 2 1 1280.4.d.z 6
8.b even 2 1 inner 1280.4.d.ba 6
8.d odd 2 1 1280.4.d.z 6
16.e even 4 1 640.4.a.n yes 3
16.e even 4 1 640.4.a.o yes 3
16.f odd 4 1 640.4.a.m ✓ 3
16.f odd 4 1 640.4.a.p yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.4.a.m ✓ 3 16.f odd 4 1
640.4.a.n yes 3 16.e even 4 1
640.4.a.o yes 3 16.e even 4 1
640.4.a.p yes 3 16.f odd 4 1
1280.4.d.z 6 4.b odd 2 1
1280.4.d.z 6 8.d odd 2 1
1280.4.d.ba 6 1.a even 1 1 trivial
1280.4.d.ba 6 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1280, [\chi])\):

\( T_{3}^{6} + 100T_{3}^{4} + 2560T_{3}^{2} + 4096 \) Copy content Toggle raw display
\( T_{7}^{3} - 20T_{7}^{2} - 60T_{7} + 1888 \) Copy content Toggle raw display
\( T_{11}^{6} + 4332T_{11}^{4} + 5338416T_{11}^{2} + 1591690816 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 100 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{3} \) Copy content Toggle raw display
$7$ \( (T^{3} - 20 T^{2} + \cdots + 1888)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 1591690816 \) Copy content Toggle raw display
$13$ \( T^{6} + 6892 T^{4} + \cdots + 317980224 \) Copy content Toggle raw display
$17$ \( (T^{3} + 6 T^{2} + \cdots + 154536)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 1645637414976 \) Copy content Toggle raw display
$23$ \( (T^{3} - 156 T^{2} + \cdots + 249248)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 2278578174016 \) Copy content Toggle raw display
$31$ \( (T^{3} + 144 T^{2} + \cdots - 1385600)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 266301927937600 \) Copy content Toggle raw display
$41$ \( (T^{3} - 346 T^{2} + \cdots + 710312)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{3} + 440 T^{2} + \cdots - 14079984)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 17\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 35335796249664 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{3} - 20 T^{2} + \cdots - 67907072)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 550 T^{2} + \cdots + 367588568)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 2072 T^{2} + \cdots + 307052544)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{3} - 1846 T^{2} + \cdots + 533827256)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 918 T^{2} + \cdots - 660976152)^{2} \) Copy content Toggle raw display
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