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Properties

Label 1280.3.g.c
Level $1280$
Weight $3$
Character orbit 1280.g
Analytic conductor $34.877$
Analytic rank $0$
Dimension $4$
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] = [1280,3,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: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 160)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 1) q^{3} + \beta_{2} q^{5} + (\beta_{2} - \beta_1) q^{7} + ( - 2 \beta_{3} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 1) q^{3} + \beta_{2} q^{5} + (\beta_{2} - \beta_1) q^{7} + ( - 2 \beta_{3} - 3) q^{9} + (6 \beta_{3} - 2) q^{11} + ( - 6 \beta_{2} + 8 \beta_1) q^{13} + (\beta_{2} - 5 \beta_1) q^{15} + (4 \beta_{3} - 2) q^{17} + ( - 8 \beta_{3} - 12) q^{19} + (2 \beta_{2} - 6 \beta_1) q^{21} + ( - 9 \beta_{2} + \beta_1) q^{23} - 5 q^{25} + (10 \beta_{3} - 2) q^{27} + (4 \beta_{2} - 30 \beta_1) q^{29} + (2 \beta_{2} + 30 \beta_1) q^{31} + (8 \beta_{3} - 32) q^{33} + (\beta_{3} - 5) q^{35} + ( - 14 \beta_{2} + 12 \beta_1) q^{37} + ( - 14 \beta_{2} + 38 \beta_1) q^{39} + 26 \beta_{3} q^{41} + (\beta_{3} - 65) q^{43} + ( - 3 \beta_{2} - 10 \beta_1) q^{45} + (19 \beta_{2} + 21 \beta_1) q^{47} + (2 \beta_{3} + 43) q^{49} + (6 \beta_{3} - 22) q^{51} + ( - 26 \beta_{2} - 40 \beta_1) q^{53} + ( - 2 \beta_{2} + 30 \beta_1) q^{55} + (4 \beta_{3} + 28) q^{57} + ( - 20 \beta_{3} + 64) q^{59} + ( - 22 \beta_{2} + 48 \beta_1) q^{61} + ( - \beta_{2} - 7 \beta_1) q^{63} + ( - 8 \beta_{3} + 30) q^{65} + (17 \beta_{3} - 33) q^{67} + ( - 10 \beta_{2} + 46 \beta_1) q^{69} + ( - 10 \beta_{2} - 62 \beta_1) q^{71} + (20 \beta_{3} + 26) q^{73} + (5 \beta_{3} - 5) q^{75} + ( - 8 \beta_{2} + 32 \beta_1) q^{77} + ( - 36 \beta_{2} - 44 \beta_1) q^{79} + (30 \beta_{3} - 25) q^{81} + (7 \beta_{3} + 145) q^{83} + ( - 2 \beta_{2} + 20 \beta_1) q^{85} + (34 \beta_{2} - 50 \beta_1) q^{87} + (16 \beta_{3} - 82) q^{89} + ( - 14 \beta_{3} + 38) q^{91} + ( - 28 \beta_{2} + 20 \beta_1) q^{93} + ( - 12 \beta_{2} - 40 \beta_1) q^{95} + (48 \beta_{3} + 26) q^{97} + ( - 14 \beta_{3} - 54) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 12 q^{9} - 8 q^{11} - 8 q^{17} - 48 q^{19} - 20 q^{25} - 8 q^{27} - 128 q^{33} - 20 q^{35} - 260 q^{43} + 172 q^{49} - 88 q^{51} + 112 q^{57} + 256 q^{59} + 120 q^{65} - 132 q^{67} + 104 q^{73} - 20 q^{75} - 100 q^{81} + 580 q^{83} - 328 q^{89} + 152 q^{91} + 104 q^{97} - 216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{2} + 2\beta_1 \) 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
− 0.618034i
0.618034i
− 1.61803i
1.61803i
0 −1.23607 0 − 2.23607i 0 − 1.23607i 0 −7.47214 0
1151.2 0 −1.23607 0 2.23607i 0 1.23607i 0 −7.47214 0
1151.3 0 3.23607 0 − 2.23607i 0 − 3.23607i 0 1.47214 0
1151.4 0 3.23607 0 2.23607i 0 3.23607i 0 1.47214 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.c 4
4.b odd 2 1 1280.3.g.b 4
8.b even 2 1 1280.3.g.b 4
8.d odd 2 1 inner 1280.3.g.c 4
16.e even 4 1 160.3.b.b ✓ 4
16.e even 4 1 320.3.b.d 4
16.f odd 4 1 160.3.b.b ✓ 4
16.f odd 4 1 320.3.b.d 4
48.i odd 4 1 1440.3.e.a 4
48.i odd 4 1 2880.3.e.h 4
48.k even 4 1 1440.3.e.a 4
48.k even 4 1 2880.3.e.h 4
80.i odd 4 1 800.3.h.e 4
80.i odd 4 1 1600.3.h.f 4
80.j even 4 1 800.3.h.e 4
80.j even 4 1 1600.3.h.f 4
80.k odd 4 1 800.3.b.g 4
80.k odd 4 1 1600.3.b.u 4
80.q even 4 1 800.3.b.g 4
80.q even 4 1 1600.3.b.u 4
80.s even 4 1 800.3.h.h 4
80.s even 4 1 1600.3.h.k 4
80.t odd 4 1 800.3.h.h 4
80.t odd 4 1 1600.3.h.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.b.b ✓ 4 16.e even 4 1
160.3.b.b ✓ 4 16.f odd 4 1
320.3.b.d 4 16.e even 4 1
320.3.b.d 4 16.f odd 4 1
800.3.b.g 4 80.k odd 4 1
800.3.b.g 4 80.q even 4 1
800.3.h.e 4 80.i odd 4 1
800.3.h.e 4 80.j even 4 1
800.3.h.h 4 80.s even 4 1
800.3.h.h 4 80.t odd 4 1
1280.3.g.b 4 4.b odd 2 1
1280.3.g.b 4 8.b even 2 1
1280.3.g.c 4 1.a even 1 1 trivial
1280.3.g.c 4 8.d odd 2 1 inner
1440.3.e.a 4 48.i odd 4 1
1440.3.e.a 4 48.k even 4 1
1600.3.b.u 4 80.k odd 4 1
1600.3.b.u 4 80.q even 4 1
1600.3.h.f 4 80.i odd 4 1
1600.3.h.f 4 80.j even 4 1
1600.3.h.k 4 80.s even 4 1
1600.3.h.k 4 80.t odd 4 1
2880.3.e.h 4 48.i odd 4 1
2880.3.e.h 4 48.k even 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2 T - 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} + 4 T - 176)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 488 T^{2} + 13456 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T - 76)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 24 T - 176)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 812 T^{2} + 163216 \) Copy content Toggle raw display
$29$ \( T^{4} + 1960 T^{2} + 672400 \) Copy content Toggle raw display
$31$ \( T^{4} + 1840 T^{2} + 774400 \) Copy content Toggle raw display
$37$ \( T^{4} + 2248 T^{2} + 698896 \) Copy content Toggle raw display
$41$ \( (T^{2} - 3380)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 130 T + 4220)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 4492 T^{2} + 1860496 \) Copy content Toggle raw display
$53$ \( T^{4} + 9960 T^{2} + 3168400 \) Copy content Toggle raw display
$59$ \( (T^{2} - 128 T + 2096)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 9448 T^{2} + 13456 \) Copy content Toggle raw display
$67$ \( (T^{2} + 66 T - 356)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 8688 T^{2} + 11182336 \) Copy content Toggle raw display
$73$ \( (T^{2} - 52 T - 1324)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 16832 T^{2} + 20647936 \) Copy content Toggle raw display
$83$ \( (T^{2} - 290 T + 20780)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 164 T + 5444)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 52 T - 10844)^{2} \) Copy content Toggle raw display
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