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Properties

Label 1280.3.g.f
Level $1280$
Weight $3$
Character orbit 1280.g
Analytic conductor $34.877$
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] = [1280,3,Mod(1151,1280)]
 
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([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.1151");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1280.g (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: \(34.8774738381\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 80)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{4} q^{5} + \beta_{6} q^{7} + (3 \beta_1 + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_{4} q^{5} + \beta_{6} q^{7} + (3 \beta_1 + 9) q^{9} - 2 \beta_{2} q^{11} + (6 \beta_{4} - 2 \beta_{3}) q^{13} + ( - \beta_{6} + \beta_{5}) q^{15} + 18 q^{17} + (2 \beta_{7} - 2 \beta_{2}) q^{19} + ( - 6 \beta_{4} - 3 \beta_{3}) q^{21} + ( - 3 \beta_{6} + \beta_{5}) q^{23} - 5 q^{25} + (3 \beta_{7} + 9 \beta_{2}) q^{27} + (12 \beta_{4} + 9 \beta_{3}) q^{29} + (2 \beta_{6} + 4 \beta_{5}) q^{31} + ( - 6 \beta_1 - 36) q^{33} + ( - \beta_{7} + 2 \beta_{2}) q^{35} + ( - 6 \beta_{4} + 10 \beta_{3}) q^{37} + ( - 6 \beta_{6} + 8 \beta_{5}) q^{39} + (9 \beta_1 - 12) q^{41} + ( - \beta_{7} + 6 \beta_{2}) q^{43} + (9 \beta_{4} - 15 \beta_{3}) q^{45} + ( - 9 \beta_{6} - 2 \beta_{5}) q^{47} + (9 \beta_1 + 7) q^{49} + 18 \beta_{2} q^{51} + (18 \beta_{4} + 6 \beta_{3}) q^{53} + (2 \beta_{6} - 2 \beta_{5}) q^{55} + (12 \beta_1 - 24) q^{57} - 8 \beta_{2} q^{59} + ( - 6 \beta_{4} + 32 \beta_{3}) q^{61} + ( - 3 \beta_{6} - 3 \beta_{5}) q^{63} + ( - 2 \beta_1 - 30) q^{65} + (\beta_{7} - 8 \beta_{2}) q^{67} + (30 \beta_{4} - 9 \beta_{3}) q^{69} + (6 \beta_{6} - 8 \beta_{5}) q^{71} + ( - 12 \beta_1 + 38) q^{73} - 5 \beta_{2} q^{75} + (12 \beta_{4} + 6 \beta_{3}) q^{77} + 10 \beta_{5} q^{79} + (27 \beta_1 + 99) q^{81} + (3 \beta_{7} + 12 \beta_{2}) q^{83} + 18 \beta_{4} q^{85} + ( - 12 \beta_{6} + 3 \beta_{5}) q^{87} + (12 \beta_1 + 6) q^{89} + ( - 4 \beta_{7} + 10 \beta_{2}) q^{91} + (36 \beta_{4} - 78 \beta_{3}) q^{93} + (8 \beta_{6} + 2 \beta_{5}) q^{95} + (30 \beta_1 - 26) q^{97} + ( - 6 \beta_{7} - 36 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 72 q^{9} + 144 q^{17} - 40 q^{25} - 288 q^{33} - 96 q^{41} + 56 q^{49} - 192 q^{57} - 240 q^{65} + 304 q^{73} + 792 q^{81} + 48 q^{89} - 208 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} - 9 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{7} - 8\nu^{5} + 24\nu^{3} - 17\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{7} - 8\nu^{5} + 20\nu^{3} - \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{7} - 8\nu^{5} + 22\nu^{3} - \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{6} - 16\nu^{4} + 32\nu^{2} - 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17\nu^{6} - 48\nu^{4} + 128\nu^{2} - 27 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -19\nu^{7} + 72\nu^{5} - 184\nu^{3} + 129\nu ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{4} - \beta_{3} - 2\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} - 3\beta_{5} + 2\beta _1 + 12 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{6} - 4\beta_{5} - 3\beta _1 - 14 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3\beta_{7} + 20\beta_{4} - 22\beta_{3} + 23\beta_{2} ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4\beta_{7} - 26\beta_{4} + 29\beta_{3} + 30\beta_{2} ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
1.40126 − 0.809017i
1.40126 + 0.809017i
0.535233 − 0.309017i
0.535233 + 0.309017i
−0.535233 − 0.309017i
−0.535233 + 0.309017i
−1.40126 − 0.809017i
−1.40126 + 0.809017i
0 −5.60503 0 − 2.23607i 0 − 1.32317i 0 22.4164 0
1151.2 0 −5.60503 0 2.23607i 0 1.32317i 0 22.4164 0
1151.3 0 −2.14093 0 − 2.23607i 0 − 9.06914i 0 −4.41641 0
1151.4 0 −2.14093 0 2.23607i 0 9.06914i 0 −4.41641 0
1151.5 0 2.14093 0 − 2.23607i 0 9.06914i 0 −4.41641 0
1151.6 0 2.14093 0 2.23607i 0 − 9.06914i 0 −4.41641 0
1151.7 0 5.60503 0 − 2.23607i 0 1.32317i 0 22.4164 0
1151.8 0 5.60503 0 2.23607i 0 − 1.32317i 0 22.4164 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1151.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.3.g.f 8
4.b odd 2 1 inner 1280.3.g.f 8
8.b even 2 1 inner 1280.3.g.f 8
8.d odd 2 1 inner 1280.3.g.f 8
16.e even 4 1 80.3.b.a ✓ 4
16.e even 4 1 320.3.b.a 4
16.f odd 4 1 80.3.b.a ✓ 4
16.f odd 4 1 320.3.b.a 4
48.i odd 4 1 720.3.e.c 4
48.i odd 4 1 2880.3.e.b 4
48.k even 4 1 720.3.e.c 4
48.k even 4 1 2880.3.e.b 4
80.i odd 4 1 400.3.h.d 8
80.i odd 4 1 1600.3.h.p 8
80.j even 4 1 400.3.h.d 8
80.j even 4 1 1600.3.h.p 8
80.k odd 4 1 400.3.b.g 4
80.k odd 4 1 1600.3.b.k 4
80.q even 4 1 400.3.b.g 4
80.q even 4 1 1600.3.b.k 4
80.s even 4 1 400.3.h.d 8
80.s even 4 1 1600.3.h.p 8
80.t odd 4 1 400.3.h.d 8
80.t odd 4 1 1600.3.h.p 8
240.t even 4 1 3600.3.e.bb 4
240.z odd 4 1 3600.3.j.k 8
240.bb even 4 1 3600.3.j.k 8
240.bd odd 4 1 3600.3.j.k 8
240.bf even 4 1 3600.3.j.k 8
240.bm odd 4 1 3600.3.e.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.3.b.a ✓ 4 16.e even 4 1
80.3.b.a ✓ 4 16.f odd 4 1
320.3.b.a 4 16.e even 4 1
320.3.b.a 4 16.f odd 4 1
400.3.b.g 4 80.k odd 4 1
400.3.b.g 4 80.q even 4 1
400.3.h.d 8 80.i odd 4 1
400.3.h.d 8 80.j even 4 1
400.3.h.d 8 80.s even 4 1
400.3.h.d 8 80.t odd 4 1
720.3.e.c 4 48.i odd 4 1
720.3.e.c 4 48.k even 4 1
1280.3.g.f 8 1.a even 1 1 trivial
1280.3.g.f 8 4.b odd 2 1 inner
1280.3.g.f 8 8.b even 2 1 inner
1280.3.g.f 8 8.d odd 2 1 inner
1600.3.b.k 4 80.k odd 4 1
1600.3.b.k 4 80.q even 4 1
1600.3.h.p 8 80.i odd 4 1
1600.3.h.p 8 80.j even 4 1
1600.3.h.p 8 80.s even 4 1
1600.3.h.p 8 80.t odd 4 1
2880.3.e.b 4 48.i odd 4 1
2880.3.e.b 4 48.k even 4 1
3600.3.e.bb 4 240.t even 4 1
3600.3.e.bb 4 240.bm odd 4 1
3600.3.j.k 8 240.z odd 4 1
3600.3.j.k 8 240.bb even 4 1
3600.3.j.k 8 240.bd odd 4 1
3600.3.j.k 8 240.bf even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 36T_{3}^{2} + 144 \) acting on \(S_{3}^{\mathrm{new}}(1280, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 36 T^{2} + 144)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 84 T^{2} + 144)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 144 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 392 T^{2} + 26896)^{2} \) Copy content Toggle raw display
$17$ \( (T - 18)^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} - 1344 T^{2} + 36864)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 756 T^{2} + 121104)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 2088 T^{2} + 156816)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 3024 T^{2} + 2214144)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 1160 T^{2} + 48400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 24 T - 1476)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 1476 T^{2} + 535824)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 8244 T^{2} + 898704)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 3528 T^{2} + 2178576)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 2304 T^{2} + 589824)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 8552 T^{2} + 15335056)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 2436 T^{2} + 1468944)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 9936 T^{2} + 3873024)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 76 T - 1436)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 14400 T^{2} + 23040000)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 8964 T^{2} + 4210704)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 12 T - 2844)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 52 T - 17324)^{4} \) Copy content Toggle raw display
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