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Properties

Label 1280.4.d.bc
Level $1280$
Weight $4$
Character orbit 1280.d
Analytic conductor $75.522$
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] = [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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 52x^{5} + 231x^{4} + 324x^{3} + 242x^{2} + 66x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_1) q^{3} - 5 \beta_1 q^{5} + ( - \beta_{2} + 5) q^{7} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 27) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_1) q^{3} - 5 \beta_1 q^{5} + ( - \beta_{2} + 5) q^{7} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 27) q^{9} + (2 \beta_{7} - \beta_{6} + \cdots - 18 \beta_1) q^{11}+ \cdots + ( - 14 \beta_{7} + \beta_{6} + \cdots + 870 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 40 q^{7} - 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 40 q^{7} - 216 q^{9} + 40 q^{15} + 192 q^{17} + 88 q^{23} - 200 q^{25} - 48 q^{31} - 176 q^{33} - 368 q^{39} + 64 q^{41} - 1928 q^{47} + 2936 q^{49} - 720 q^{55} - 2032 q^{57} - 7400 q^{63} + 320 q^{65} + 1968 q^{71} - 320 q^{73} - 3552 q^{79} + 5592 q^{81} + 2416 q^{87} - 2640 q^{89} - 1280 q^{95} - 1312 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 52x^{5} + 231x^{4} + 324x^{3} + 242x^{2} + 66x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2601 \nu^{7} + 6014 \nu^{6} - 7455 \nu^{5} - 131867 \nu^{4} - 561497 \nu^{3} - 681745 \nu^{2} + \cdots - 82437 ) / 51180 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2642 \nu^{7} + 7173 \nu^{6} - 8335 \nu^{5} - 141154 \nu^{4} - 491789 \nu^{3} - 414340 \nu^{2} + \cdots - 248394 ) / 12795 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 613\nu^{7} - 1704\nu^{6} + 2609\nu^{5} + 28919\nu^{4} + 122167\nu^{3} + 102569\nu^{2} + 28866\nu - 51027 ) / 2559 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1361\nu^{7} - 3729\nu^{6} + 4870\nu^{5} + 69027\nu^{4} + 260222\nu^{3} + 218935\nu^{2} + 61647\nu - 34913 ) / 4265 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 27301 \nu^{7} + 52424 \nu^{6} - 41405 \nu^{5} - 1447987 \nu^{4} - 6392307 \nu^{3} - 8887065 \nu^{2} + \cdots - 946347 ) / 51180 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 47481 \nu^{7} - 118984 \nu^{6} + 168705 \nu^{5} + 2350687 \nu^{4} + 9808687 \nu^{3} + 11229245 \nu^{2} + \cdots + 1433127 ) / 51180 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 28941 \nu^{7} - 64664 \nu^{6} + 76605 \nu^{5} + 1478267 \nu^{4} + 6333587 \nu^{3} + 8153905 \nu^{2} + \cdots + 931227 ) / 25590 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 4\beta _1 + 4 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 3\beta_{6} - 3\beta_{5} + 64\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -8\beta_{7} + 11\beta_{6} - 13\beta_{5} - 13\beta_{4} + 8\beta_{3} - 11\beta_{2} + 160\beta _1 - 160 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -41\beta_{4} + 21\beta_{3} - 39\beta_{2} - 674 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 349\beta_{7} - 555\beta_{6} + 623\beta_{5} - 623\beta_{4} + 349\beta_{3} - 555\beta_{2} - 8916\beta _1 - 8916 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 579\beta_{7} - 988\beta_{6} + 1074\beta_{5} - 16424\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8523 \beta_{7} - 14059 \beta_{6} + 15523 \beta_{5} + 15523 \beta_{4} - 8523 \beta_{3} + 14059 \beta_{2} + \cdots + 230052 ) / 16 \) 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
−0.703738 + 0.703738i
−0.178675 − 0.178675i
3.55856 − 3.55856i
−1.67615 − 1.67615i
−1.67615 + 1.67615i
3.55856 + 3.55856i
−0.178675 + 0.178675i
−0.703738 − 0.703738i
0 − 9.86321i 0 5.00000i 0 34.2713 0 −70.2829 0
641.2 0 − 9.13763i 0 − 5.00000i 0 23.1776 0 −56.4963 0
641.3 0 − 5.50114i 0 5.00000i 0 −33.3372 0 −3.26254 0
641.4 0 − 2.22671i 0 − 5.00000i 0 −4.11169 0 22.0417 0
641.5 0 2.22671i 0 5.00000i 0 −4.11169 0 22.0417 0
641.6 0 5.50114i 0 − 5.00000i 0 −33.3372 0 −3.26254 0
641.7 0 9.13763i 0 5.00000i 0 23.1776 0 −56.4963 0
641.8 0 9.86321i 0 − 5.00000i 0 34.2713 0 −70.2829 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 641.8
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.bc 8
4.b odd 2 1 1280.4.d.bb 8
8.b even 2 1 inner 1280.4.d.bc 8
8.d odd 2 1 1280.4.d.bb 8
16.e even 4 1 640.4.a.q ✓ 4
16.e even 4 1 640.4.a.t yes 4
16.f odd 4 1 640.4.a.r yes 4
16.f odd 4 1 640.4.a.s yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.4.a.q ✓ 4 16.e even 4 1
640.4.a.r yes 4 16.f odd 4 1
640.4.a.s yes 4 16.f odd 4 1
640.4.a.t yes 4 16.e even 4 1
1280.4.d.bb 8 4.b odd 2 1
1280.4.d.bb 8 8.d odd 2 1
1280.4.d.bc 8 1.a even 1 1 trivial
1280.4.d.bc 8 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 \(S_{4}^{\mathrm{new}}(1280, [\chi])\):

\( T_{3}^{8} + 216T_{3}^{6} + 14640T_{3}^{4} + 313216T_{3}^{2} + 1218816 \) Copy content Toggle raw display
\( T_{7}^{4} - 20T_{7}^{3} - 1220T_{7}^{2} + 21872T_{7} + 108880 \) Copy content Toggle raw display
\( T_{11}^{8} + 9936T_{11}^{6} + 30267744T_{11}^{4} + 29743187200T_{11}^{2} + 2604066418944 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 216 T^{6} + \cdots + 1218816 \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 20 T^{3} + \cdots + 108880)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 2604066418944 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 66663108864 \) Copy content Toggle raw display
$17$ \( (T^{4} - 96 T^{3} + \cdots - 3269232)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 1356796313856 \) Copy content Toggle raw display
$23$ \( (T^{4} - 44 T^{3} + \cdots - 13516080)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 500338160697600 \) Copy content Toggle raw display
$31$ \( (T^{4} + 24 T^{3} + \cdots + 6237440)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} - 32 T^{3} + \cdots + 607861648)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{4} + 964 T^{3} + \cdots - 20842229424)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 39\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{4} - 984 T^{3} + \cdots + 2987836672)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 160 T^{3} + \cdots + 14073390480)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 1776 T^{3} + \cdots - 490073812992)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{4} + 1320 T^{3} + \cdots - 415967432304)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 656 T^{3} + \cdots + 433240091664)^{2} \) Copy content Toggle raw display
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