Properties

Label 1280.2.o.t
Level $1280$
Weight $2$
Character orbit 1280.o
Analytic conductor $10.221$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1280,2,Mod(127,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.o (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: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.74906393903104.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 12x^{10} + 50x^{8} + 90x^{6} + 69x^{4} + 18x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 640)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - \beta_{6} q^{5} + (\beta_{10} - \beta_{9} - \beta_{7} + \cdots + 1) q^{7}+ \cdots + (\beta_{11} + \beta_{10} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - \beta_{6} q^{5} + (\beta_{10} - \beta_{9} - \beta_{7} + \cdots + 1) q^{7}+ \cdots + (\beta_{11} + 8 \beta_{10} + \cdots - 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{5} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{5} + 8 q^{7} + 8 q^{11} - 4 q^{13} - 16 q^{15} + 4 q^{17} - 16 q^{23} - 4 q^{25} - 24 q^{27} - 16 q^{29} + 24 q^{35} + 20 q^{37} + 16 q^{39} - 16 q^{43} + 16 q^{47} + 48 q^{51} - 4 q^{53} - 32 q^{55} - 48 q^{63} - 4 q^{65} - 32 q^{67} - 16 q^{69} - 12 q^{73} + 56 q^{75} + 32 q^{79} - 28 q^{81} - 56 q^{83} + 36 q^{85} + 32 q^{87} + 32 q^{91} + 32 q^{93} - 16 q^{95} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 12x^{10} + 50x^{8} + 90x^{6} + 69x^{4} + 18x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -\nu^{8} - 10\nu^{6} - 29\nu^{4} - 23\nu^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + 11\nu^{8} + 39\nu^{6} + 52\nu^{4} + 23\nu^{2} + 2\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} + 11\nu^{8} + 39\nu^{6} + 52\nu^{4} + 23\nu^{2} - 2\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + \nu^{10} + 11 \nu^{9} + 12 \nu^{8} + 40 \nu^{7} + 49 \nu^{6} + 60 \nu^{5} + 81 \nu^{4} + \cdots + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} - \nu^{10} + 11 \nu^{9} - 12 \nu^{8} + 40 \nu^{7} - 49 \nu^{6} + 60 \nu^{5} - 81 \nu^{4} + \cdots - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} + \nu^{10} + 12 \nu^{9} + 11 \nu^{8} + 50 \nu^{7} + 40 \nu^{6} + 89 \nu^{5} + 60 \nu^{4} + \cdots + 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{11} + \nu^{10} - 12 \nu^{9} + 11 \nu^{8} - 50 \nu^{7} + 40 \nu^{6} - 89 \nu^{5} + 60 \nu^{4} + \cdots + 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} - 2 \nu^{10} + 13 \nu^{9} - 22 \nu^{8} + 60 \nu^{7} - 79 \nu^{6} + 120 \nu^{5} - 110 \nu^{4} + \cdots - 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{11} - 2 \nu^{10} - 13 \nu^{9} - 22 \nu^{8} - 60 \nu^{7} - 79 \nu^{6} - 120 \nu^{5} - 110 \nu^{4} + \cdots - 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -2\nu^{11} - 23\nu^{9} - 89\nu^{7} - 141\nu^{5} - 86\nu^{3} - 13\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( 2\nu^{11} + 24\nu^{9} + 99\nu^{7} + 170\nu^{5} + 109\nu^{3} + 13\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} + 2\beta_{10} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 4\beta_{3} - 4\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + 5\beta_{5} - 5\beta_{4} + 6\beta_{3} + 6\beta_{2} - 5\beta _1 + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 8 \beta_{11} - 16 \beta_{10} - \beta_{9} + \beta_{8} - 6 \beta_{7} + 6 \beta_{6} + \cdots + 19 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 8 \beta_{9} - 8 \beta_{8} - 6 \beta_{7} - 6 \beta_{6} - 25 \beta_{5} + 25 \beta_{4} - 35 \beta_{3} + \cdots - 62 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 49 \beta_{11} + 102 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} + 31 \beta_{7} - 31 \beta_{6} + \cdots - 97 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 51 \beta_{9} + 51 \beta_{8} + 31 \beta_{7} + 31 \beta_{6} + 128 \beta_{5} - 128 \beta_{4} + 199 \beta_{3} + \cdots + 306 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 279 \beta_{11} - 598 \beta_{10} - 51 \beta_{9} + 51 \beta_{8} - 159 \beta_{7} + 159 \beta_{6} + \cdots + 511 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 301 \beta_{9} - 301 \beta_{8} - 159 \beta_{7} - 159 \beta_{6} - 670 \beta_{5} + 670 \beta_{4} + \cdots - 1588 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1549 \beta_{11} + 3378 \beta_{10} + 301 \beta_{9} - 301 \beta_{8} + 829 \beta_{7} + \cdots - 2734 \beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(-\beta_{10}\) \(-1\) \(-1\)

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
1.73968i
1.37379i
0.594523i
0.274205i
1.09958i
2.33420i
1.73968i
1.37379i
0.594523i
0.274205i
1.09958i
2.33420i
0 −1.73968 + 1.73968i 0 −0.839394 + 2.07254i 0 1.90768 1.90768i 0 3.05295i 0
127.2 0 −1.37379 + 1.37379i 0 2.14213 0.641320i 0 −0.287197 + 0.287197i 0 0.774596i 0
127.3 0 −0.594523 + 0.594523i 0 0.267807 2.21997i 0 −0.132476 + 0.132476i 0 2.29308i 0
127.4 0 0.274205 0.274205i 0 −2.23269 0.122814i 0 −2.49551 + 2.49551i 0 2.84962i 0
127.5 0 1.09958 1.09958i 0 0.504779 + 2.17835i 0 3.36850 3.36850i 0 0.581827i 0
127.6 0 2.33420 2.33420i 0 −1.84263 1.26678i 0 1.63901 1.63901i 0 7.89698i 0
383.1 0 −1.73968 1.73968i 0 −0.839394 2.07254i 0 1.90768 + 1.90768i 0 3.05295i 0
383.2 0 −1.37379 1.37379i 0 2.14213 + 0.641320i 0 −0.287197 0.287197i 0 0.774596i 0
383.3 0 −0.594523 0.594523i 0 0.267807 + 2.21997i 0 −0.132476 0.132476i 0 2.29308i 0
383.4 0 0.274205 + 0.274205i 0 −2.23269 + 0.122814i 0 −2.49551 2.49551i 0 2.84962i 0
383.5 0 1.09958 + 1.09958i 0 0.504779 2.17835i 0 3.36850 + 3.36850i 0 0.581827i 0
383.6 0 2.33420 + 2.33420i 0 −1.84263 + 1.26678i 0 1.63901 + 1.63901i 0 7.89698i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.2.o.t 12
4.b odd 2 1 1280.2.o.s 12
5.c odd 4 1 1280.2.o.u 12
8.b even 2 1 1280.2.o.v 12
8.d odd 2 1 1280.2.o.u 12
16.e even 4 1 640.2.n.a 12
16.e even 4 1 640.2.n.b yes 12
16.f odd 4 1 640.2.n.c yes 12
16.f odd 4 1 640.2.n.d yes 12
20.e even 4 1 1280.2.o.v 12
40.i odd 4 1 1280.2.o.s 12
40.k even 4 1 inner 1280.2.o.t 12
80.i odd 4 1 640.2.n.d yes 12
80.j even 4 1 640.2.n.a 12
80.s even 4 1 640.2.n.b yes 12
80.t odd 4 1 640.2.n.c yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.n.a 12 16.e even 4 1
640.2.n.a 12 80.j even 4 1
640.2.n.b yes 12 16.e even 4 1
640.2.n.b yes 12 80.s even 4 1
640.2.n.c yes 12 16.f odd 4 1
640.2.n.c yes 12 80.t odd 4 1
640.2.n.d yes 12 16.f odd 4 1
640.2.n.d yes 12 80.i odd 4 1
1280.2.o.s 12 4.b odd 2 1
1280.2.o.s 12 40.i odd 4 1
1280.2.o.t 12 1.a even 1 1 trivial
1280.2.o.t 12 40.k even 4 1 inner
1280.2.o.u 12 5.c odd 4 1
1280.2.o.u 12 8.d odd 2 1
1280.2.o.v 12 8.b even 2 1
1280.2.o.v 12 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1280, [\chi])\):

\( T_{3}^{12} + 8 T_{3}^{9} + 88 T_{3}^{8} + 80 T_{3}^{7} + 32 T_{3}^{6} - 32 T_{3}^{5} + 592 T_{3}^{4} + \cdots + 64 \) Copy content Toggle raw display
\( T_{7}^{12} - 8 T_{7}^{11} + 32 T_{7}^{10} - 40 T_{7}^{9} + 184 T_{7}^{8} - 1232 T_{7}^{7} + 4768 T_{7}^{6} + \cdots + 64 \) Copy content Toggle raw display
\( T_{13}^{12} + 4 T_{13}^{11} + 8 T_{13}^{10} + 8 T_{13}^{9} + 1036 T_{13}^{8} + 4000 T_{13}^{7} + \cdots + 937024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 8 T^{9} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{12} + 4 T^{11} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} - 8 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T^{6} - 4 T^{5} + \cdots + 448)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 4 T^{11} + \cdots + 937024 \) Copy content Toggle raw display
$17$ \( T^{12} - 4 T^{11} + \cdots + 506944 \) Copy content Toggle raw display
$19$ \( T^{12} + 120 T^{10} + \cdots + 200704 \) Copy content Toggle raw display
$23$ \( T^{12} + 16 T^{11} + \cdots + 25482304 \) Copy content Toggle raw display
$29$ \( (T^{6} + 8 T^{5} + \cdots + 512)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 192 T^{10} + \cdots + 32444416 \) Copy content Toggle raw display
$37$ \( T^{12} - 20 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$41$ \( (T^{6} - 88 T^{4} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 16 T^{11} + \cdots + 3013696 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 476636224 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 742889536 \) Copy content Toggle raw display
$59$ \( T^{12} + 504 T^{10} + \cdots + 93392896 \) Copy content Toggle raw display
$61$ \( T^{12} + 352 T^{10} + \cdots + 58003456 \) Copy content Toggle raw display
$67$ \( T^{12} + 32 T^{11} + \cdots + 45914176 \) Copy content Toggle raw display
$71$ \( T^{12} + 256 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 1575137344 \) Copy content Toggle raw display
$79$ \( (T^{6} - 16 T^{5} + \cdots - 4096)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 18428605504 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 214228271104 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 128628387904 \) Copy content Toggle raw display
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