Properties

Label 1280.4.a.q
Level $1280$
Weight $4$
Character orbit 1280.a
Self dual yes
Analytic conductor $75.522$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1280,4,Mod(1,1280)]
 
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, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1280.a (trivial)

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: yes
Analytic conductor: \(75.5224448073\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.190224.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 17x^{2} + 18x + 78 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 320)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - 5 q^{5} + ( - 3 \beta_{2} + \beta_1 - 8) q^{7} + (\beta_{3} - \beta_{2} + \beta_1 + 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - 5 q^{5} + ( - 3 \beta_{2} + \beta_1 - 8) q^{7} + (\beta_{3} - \beta_{2} + \beta_1 + 13) q^{9} + ( - 2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 4) q^{11} + (\beta_{3} - 3 \beta_{2} - 7 \beta_1 - 20) q^{13} - 5 \beta_1 q^{15} + ( - 3 \beta_{3} + 13 \beta_{2} - 3 \beta_1 - 12) q^{17} + (2 \beta_{3} + 23 \beta_{2} + 4 \beta_1 + 52) q^{19} + (4 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 58) q^{21} + ( - 2 \beta_{3} + 17 \beta_{2} + 3 \beta_1 - 52) q^{23} + 25 q^{25} + (2 \beta_{3} - 32 \beta_{2} - 6 \beta_1 + 36) q^{27} + (2 \beta_{3} + 20 \beta_{2} - 14 \beta_1 - 40) q^{29} + ( - 8 \beta_{3} + 2 \beta_1 - 8) q^{31} + (5 \beta_{3} + 65 \beta_{2} - 3 \beta_1 + 118) q^{33} + (15 \beta_{2} - 5 \beta_1 + 40) q^{35} + ( - 8 \beta_{3} + 12 \beta_{2} - 32 \beta_1 + 54) q^{37} + ( - 4 \beta_{3} - 22 \beta_{2} - 18 \beta_1 - 272) q^{39} + ( - \beta_{3} - 79 \beta_{2} - 33 \beta_1 - 26) q^{41} + (6 \beta_{3} - 46 \beta_{2} + 5 \beta_1 + 76) q^{43} + ( - 5 \beta_{3} + 5 \beta_{2} - 5 \beta_1 - 65) q^{45} + (16 \beta_{3} + 23 \beta_{2} + 9 \beta_1 + 152) q^{47} + (7 \beta_{3} + 53 \beta_{2} - 9 \beta_1 - 95) q^{49} + ( - 16 \beta_{3} + 86 \beta_{2} - 46 \beta_1 - 168) q^{51} + (\beta_{3} - 51 \beta_{2} - 47 \beta_1 + 180) q^{53} + (10 \beta_{3} + 15 \beta_{2} - 10 \beta_1 - 20) q^{55} + ( - 19 \beta_{3} - 91 \beta_{2} + 45 \beta_1 + 2) q^{57} + (2 \beta_{3} - 87 \beta_{2} - 16 \beta_1 - 12) q^{59} + ( - 22 \beta_{3} - 134 \beta_{2} - 62 \beta_1 - 382) q^{61} + ( - 6 \beta_{3} - 45 \beta_{2} + 49 \beta_1 + 4) q^{63} + ( - 5 \beta_{3} + 15 \beta_{2} + 35 \beta_1 + 100) q^{65} + (6 \beta_{3} + 34 \beta_{2} + 55 \beta_1 - 4) q^{67} + ( - 14 \beta_{3} + 44 \beta_{2} - 78 \beta_1 + 38) q^{69} + (28 \beta_{3} - 36 \beta_{2} + 70 \beta_1 - 168) q^{71} + (19 \beta_{3} + 11 \beta_{2} - 21 \beta_1 - 72) q^{73} + 25 \beta_1 q^{75} + (33 \beta_{3} + 53 \beta_{2} - 79 \beta_1 + 194) q^{77} + (20 \beta_{3} - 2 \beta_{2} - 72 \beta_1 - 464) q^{79} + ( - \beta_{3} + \beta_{2} + 47 \beta_1 - 419) q^{81} + (30 \beta_{3} + 60 \beta_{2} + 29 \beta_1 + 668) q^{83} + (15 \beta_{3} - 65 \beta_{2} + 15 \beta_1 + 60) q^{85} + ( - 34 \beta_{3} - 70 \beta_{2} - 62 \beta_1 - 700) q^{87} + (30 \beta_{3} - 50 \beta_{2} + 78 \beta_1 - 162) q^{89} + ( - 36 \beta_{3} + 56 \beta_{2} + 50 \beta_1 - 112) q^{91} + (2 \beta_{3} + 254 \beta_{2} - 54 \beta_1 + 160) q^{93} + ( - 10 \beta_{3} - 115 \beta_{2} - 20 \beta_1 - 260) q^{95} + ( - 39 \beta_{3} - 131 \beta_{2} - 47 \beta_1 - 112) q^{97} + ( - 14 \beta_{3} - 141 \beta_{2} + 26 \beta_1 - 668) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{5} - 32 q^{7} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 20 q^{5} - 32 q^{7} + 52 q^{9} + 16 q^{11} - 80 q^{13} - 48 q^{17} + 208 q^{19} + 232 q^{21} - 208 q^{23} + 100 q^{25} + 144 q^{27} - 160 q^{29} - 32 q^{31} + 472 q^{33} + 160 q^{35} + 216 q^{37} - 1088 q^{39} - 104 q^{41} + 304 q^{43} - 260 q^{45} + 608 q^{47} - 380 q^{49} - 672 q^{51} + 720 q^{53} - 80 q^{55} + 8 q^{57} - 48 q^{59} - 1528 q^{61} + 16 q^{63} + 400 q^{65} - 16 q^{67} + 152 q^{69} - 672 q^{71} - 288 q^{73} + 776 q^{77} - 1856 q^{79} - 1676 q^{81} + 2672 q^{83} + 240 q^{85} - 2800 q^{87} - 648 q^{89} - 448 q^{91} + 640 q^{93} - 1040 q^{95} - 448 q^{97} - 2672 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 17x^{2} + 18x + 78 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{2} + 2\nu + 18 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{3} - 5\nu^{2} - 37\nu + 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{2} + 2\beta _1 + 38 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 8\beta_{2} + 21\beta _1 + 56 ) / 4 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.24189
−2.81392
3.24189
3.81392
0 −7.21583 0 −5.00000 0 −25.6081 0 25.0682 0
1.2 0 −4.89578 0 −5.00000 0 −2.50348 0 −3.03129 0
1.3 0 3.75172 0 −5.00000 0 −14.6406 0 −12.9246 0
1.4 0 8.35989 0 −5.00000 0 10.7522 0 42.8877 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.4.a.q 4
4.b odd 2 1 1280.4.a.u 4
8.b even 2 1 1280.4.a.v 4
8.d odd 2 1 1280.4.a.z 4
16.e even 4 2 320.4.d.d yes 8
16.f odd 4 2 320.4.d.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.4.d.c 8 16.f odd 4 2
320.4.d.d yes 8 16.e even 4 2
1280.4.a.q 4 1.a even 1 1 trivial
1280.4.a.u 4 4.b odd 2 1
1280.4.a.v 4 8.b even 2 1
1280.4.a.z 4 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1280))\):

\( T_{3}^{4} - 80T_{3}^{2} - 48T_{3} + 1108 \) Copy content Toggle raw display
\( T_{7}^{4} + 32T_{7}^{3} + 16T_{7}^{2} - 4176T_{7} - 10092 \) Copy content Toggle raw display
\( T_{13}^{4} + 80T_{13}^{3} - 2456T_{13}^{2} - 162240T_{13} + 1789584 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 80 T^{2} - 48 T + 1108 \) Copy content Toggle raw display
$5$ \( (T + 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 32 T^{3} + 16 T^{2} + \cdots - 10092 \) Copy content Toggle raw display
$11$ \( T^{4} - 16 T^{3} - 4088 T^{2} + \cdots + 601872 \) Copy content Toggle raw display
$13$ \( T^{4} + 80 T^{3} - 2456 T^{2} + \cdots + 1789584 \) Copy content Toggle raw display
$17$ \( T^{4} + 48 T^{3} - 13272 T^{2} + \cdots + 269712 \) Copy content Toggle raw display
$19$ \( T^{4} - 208 T^{3} + \cdots - 44774256 \) Copy content Toggle raw display
$23$ \( T^{4} + 208 T^{3} + \cdots - 49970700 \) Copy content Toggle raw display
$29$ \( T^{4} + 160 T^{3} + \cdots - 176947200 \) Copy content Toggle raw display
$31$ \( T^{4} + 32 T^{3} + \cdots + 574916928 \) Copy content Toggle raw display
$37$ \( T^{4} - 216 T^{3} + \cdots - 3057259248 \) Copy content Toggle raw display
$41$ \( T^{4} + 104 T^{3} + \cdots - 656734656 \) Copy content Toggle raw display
$43$ \( T^{4} - 304 T^{3} + \cdots - 179889132 \) Copy content Toggle raw display
$47$ \( T^{4} - 608 T^{3} + \cdots + 7324274484 \) Copy content Toggle raw display
$53$ \( T^{4} - 720 T^{3} + \cdots + 756896400 \) Copy content Toggle raw display
$59$ \( T^{4} + 48 T^{3} + \cdots + 4213048464 \) Copy content Toggle raw display
$61$ \( T^{4} + 1528 T^{3} + \cdots - 266067505776 \) Copy content Toggle raw display
$67$ \( T^{4} + 16 T^{3} + \cdots + 5101138068 \) Copy content Toggle raw display
$71$ \( T^{4} + 672 T^{3} + \cdots - 102344105664 \) Copy content Toggle raw display
$73$ \( T^{4} + 288 T^{3} + \cdots + 5501446672 \) Copy content Toggle raw display
$79$ \( T^{4} + 1856 T^{3} + \cdots - 176297524992 \) Copy content Toggle raw display
$83$ \( T^{4} - 2672 T^{3} + \cdots - 39883946700 \) Copy content Toggle raw display
$89$ \( T^{4} + 648 T^{3} + \cdots - 174173282928 \) Copy content Toggle raw display
$97$ \( T^{4} + 448 T^{3} + \cdots + 269085830928 \) Copy content Toggle raw display
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