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Properties

Label 1280.2.l.e
Level $1280$
Weight $2$
Character orbit 1280.l
Analytic conductor $10.221$
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] = [1280,2,Mod(321,1280)]
 
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([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.321");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.l (of order \(4\), degree \(2\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.349241344.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 18x^{6} - 40x^{5} + 81x^{4} - 100x^{3} + 96x^{2} - 52x + 17 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_{5} + 1) q^{3} + \beta_{3} q^{5} + (\beta_{7} - 2 \beta_{6} + \cdots - \beta_{2}) q^{7}+ \cdots + (3 \beta_{6} + \beta_{5} - \beta_{4} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_{5} + 1) q^{3} + \beta_{3} q^{5} + (\beta_{7} - 2 \beta_{6} + \cdots - \beta_{2}) q^{7}+ \cdots + ( - 3 \beta_{7} - 7 \beta_{6} + \cdots - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 8 q^{11} - 8 q^{13} + 8 q^{17} - 32 q^{27} + 8 q^{29} + 48 q^{31} - 24 q^{33} + 4 q^{35} + 24 q^{37} - 20 q^{43} - 8 q^{45} - 16 q^{47} - 16 q^{49} + 40 q^{51} - 16 q^{53} - 32 q^{59} + 8 q^{61} + 24 q^{63} + 8 q^{65} - 12 q^{67} - 40 q^{69} + 4 q^{75} - 16 q^{77} + 44 q^{83} + 16 q^{85} - 40 q^{91} + 24 q^{93} + 8 q^{95} + 24 q^{97} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 18x^{6} - 40x^{5} + 81x^{4} - 100x^{3} + 96x^{2} - 52x + 17 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 17\nu^{6} + 35\nu^{5} - 209\nu^{4} + 245\nu^{3} - 536\nu^{2} + 251\nu - 212 ) / 123 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 24\nu^{6} - 88\nu^{5} + 324\nu^{4} - 616\nu^{3} + 981\nu^{2} - 856\nu + 444 ) / 123 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{7} - 18\nu^{6} + 107\nu^{5} + 3\nu^{4} + 11\nu^{3} + 648\nu^{2} - 793\nu + 651 ) / 123 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -11\nu^{7} + 59\nu^{6} - 230\nu^{5} + 653\nu^{4} - 1118\nu^{3} + 1730\nu^{2} - 1052\nu + 497 ) / 123 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8\nu^{7} - 28\nu^{6} + 130\nu^{5} - 255\nu^{4} + 500\nu^{3} - 509\nu^{2} + 370\nu - 108 ) / 41 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -22\nu^{7} + 77\nu^{6} - 337\nu^{5} + 650\nu^{4} - 1211\nu^{3} + 1205\nu^{2} - 874\nu + 256 ) / 41 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - 6\beta_{6} - 3\beta_{5} + 3\beta_{4} - 6\beta_{3} - 6\beta_{2} + 3\beta _1 - 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{7} - 13\beta_{6} - 5\beta_{5} + 9\beta_{4} - 15\beta_{3} - 11\beta_{2} - 8\beta _1 + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{7} + 29\beta_{6} + 14\beta_{5} - 4\beta_{4} + 13\beta_{3} + 23\beta_{2} - 25\beta _1 + 65 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13\beta_{7} + 60\beta_{6} + 17\beta_{5} - 28\beta_{4} + 52\beta_{3} + 35\beta_{2} + 9\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -24\beta_{7} - 63\beta_{6} - 63\beta_{5} - 49\beta_{4} + 67\beta_{3} - 87\beta_{2} + 154\beta _1 - 356 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(1\) \(-\beta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1
0.500000 − 1.01252i
0.500000 − 1.94924i
0.500000 + 0.535026i
0.500000 + 2.42673i
0.500000 + 1.01252i
0.500000 + 1.94924i
0.500000 − 0.535026i
0.500000 − 2.42673i
0 −1.21962 − 1.21962i 0 −0.707107 + 0.707107i 0 − 5.13902i 0 − 0.0250345i 0
321.2 0 −0.742133 − 0.742133i 0 0.707107 − 0.707107i 0 0.463747i 0 − 1.89848i 0
321.3 0 1.74213 + 1.74213i 0 0.707107 − 0.707107i 0 − 3.04953i 0 3.07005i 0
321.4 0 2.21962 + 2.21962i 0 −0.707107 + 0.707107i 0 − 0.275191i 0 6.85346i 0
961.1 0 −1.21962 + 1.21962i 0 −0.707107 − 0.707107i 0 5.13902i 0 0.0250345i 0
961.2 0 −0.742133 + 0.742133i 0 0.707107 + 0.707107i 0 − 0.463747i 0 1.89848i 0
961.3 0 1.74213 − 1.74213i 0 0.707107 + 0.707107i 0 3.04953i 0 − 3.07005i 0
961.4 0 2.21962 − 2.21962i 0 −0.707107 − 0.707107i 0 0.275191i 0 − 6.85346i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 321.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e 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.l.e yes 8
4.b odd 2 1 1280.2.l.b ✓ 8
8.b even 2 1 1280.2.l.c yes 8
8.d odd 2 1 1280.2.l.h yes 8
16.e even 4 1 1280.2.l.c yes 8
16.e even 4 1 inner 1280.2.l.e yes 8
16.f odd 4 1 1280.2.l.b ✓ 8
16.f odd 4 1 1280.2.l.h yes 8
32.g even 8 1 5120.2.a.i 4
32.g even 8 1 5120.2.a.p 4
32.h odd 8 1 5120.2.a.c 4
32.h odd 8 1 5120.2.a.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1280.2.l.b ✓ 8 4.b odd 2 1
1280.2.l.b ✓ 8 16.f odd 4 1
1280.2.l.c yes 8 8.b even 2 1
1280.2.l.c yes 8 16.e even 4 1
1280.2.l.e yes 8 1.a even 1 1 trivial
1280.2.l.e yes 8 16.e even 4 1 inner
1280.2.l.h yes 8 8.d odd 2 1
1280.2.l.h yes 8 16.f odd 4 1
5120.2.a.c 4 32.h odd 8 1
5120.2.a.i 4 32.g even 8 1
5120.2.a.j 4 32.h odd 8 1
5120.2.a.p 4 32.g even 8 1

Hecke kernels

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

\( T_{3}^{8} - 4T_{3}^{7} + 8T_{3}^{6} + 8T_{3}^{5} + 8T_{3}^{4} - 40T_{3}^{3} + 128T_{3}^{2} + 224T_{3} + 196 \) Copy content Toggle raw display
\( T_{13}^{8} + 8T_{13}^{7} + 32T_{13}^{6} + 16T_{13}^{5} + 1056T_{13}^{4} + 7808T_{13}^{3} + 28800T_{13}^{2} + 32640T_{13} + 18496 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{7} + \cdots + 196 \) Copy content Toggle raw display
$5$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 36 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{8} + 8 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$13$ \( T^{8} + 8 T^{7} + \cdots + 18496 \) Copy content Toggle raw display
$17$ \( (T^{4} - 4 T^{3} - 28 T^{2} + \cdots - 72)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 96 T^{5} + \cdots + 1296 \) Copy content Toggle raw display
$23$ \( T^{8} + 92 T^{6} + \cdots + 93636 \) Copy content Toggle raw display
$29$ \( T^{8} - 8 T^{7} + \cdots + 374544 \) Copy content Toggle raw display
$31$ \( (T^{2} - 12 T + 28)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} - 24 T^{7} + \cdots + 246016 \) Copy content Toggle raw display
$41$ \( T^{8} + 48 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$43$ \( T^{8} + 20 T^{7} + \cdots + 324 \) Copy content Toggle raw display
$47$ \( (T^{4} + 8 T^{3} - 10 T^{2} + \cdots - 18)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 16 T^{7} + \cdots + 5184 \) Copy content Toggle raw display
$59$ \( T^{8} + 32 T^{7} + \cdots + 20903184 \) Copy content Toggle raw display
$61$ \( T^{8} - 8 T^{7} + \cdots + 1721344 \) Copy content Toggle raw display
$67$ \( T^{8} + 12 T^{7} + \cdots + 777924 \) Copy content Toggle raw display
$71$ \( T^{8} + 304 T^{6} + \cdots + 82944 \) Copy content Toggle raw display
$73$ \( T^{8} + 184 T^{6} + \cdots + 5184 \) Copy content Toggle raw display
$79$ \( (T^{4} - 112 T^{2} + \cdots - 752)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 44 T^{7} + \cdots + 279625284 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 1184461056 \) Copy content Toggle raw display
$97$ \( (T^{4} - 12 T^{3} + \cdots - 5256)^{2} \) Copy content Toggle raw display
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