Properties

Label 1280.3.e.h
Level $1280$
Weight $3$
Character orbit 1280.e
Analytic conductor $34.877$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1280,3,Mod(639,1280)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.639");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1280.e (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: \(6\)
Coefficient field: 6.0.1827904.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 9x^{4} + 14x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{4} + 1) q^{5} + ( - \beta_{5} - \beta_{4} - 2) q^{7} + (\beta_{5} + \beta_{4} - 3 \beta_{2} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{4} + 1) q^{5} + ( - \beta_{5} - \beta_{4} - 2) q^{7} + (\beta_{5} + \beta_{4} - 3 \beta_{2} - 4) q^{9} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} + 2) q^{11} + (\beta_{5} + \beta_{4} + 3 \beta_{2} + 1) q^{13} + (\beta_{5} + \beta_{4} - 2 \beta_{3} + 4 \beta_1 - 6) q^{15} + ( - 2 \beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 2 \beta_{2} + 2) q^{17} + (2 \beta_{5} + 2 \beta_{4} + 6 \beta_{2} - 2) q^{19} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{21} + ( - 3 \beta_{5} - 3 \beta_{4} + 2 \beta_{2} + 12) q^{23} + ( - \beta_{5} + \beta_{4} + 7 \beta_{3} + 5 \beta_{2} + 6 \beta_1 + 4) q^{25} + (2 \beta_{5} - 2 \beta_{4} + 14 \beta_{3} - 2 \beta_{2} - 6 \beta_1 - 2) q^{27} + ( - \beta_{3} + 8 \beta_1) q^{29} + (4 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} - 4) q^{31} + ( - 8 \beta_{3} + 12 \beta_1) q^{33} + (4 \beta_{4} + 10 \beta_{3} + 5 \beta_1 + 16) q^{35} + (3 \beta_{5} + 3 \beta_{4} + 5 \beta_{2} - 33) q^{37} + ( - 4 \beta_{5} + 4 \beta_{4} - 16 \beta_{3} + 4 \beta_{2} + 8 \beta_1 + 4) q^{39} + (\beta_{5} + \beta_{4} + 5 \beta_{2} + 13) q^{41} + (6 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} - 6 \beta_{2} - 5 \beta_1 - 6) q^{43} + (3 \beta_{5} - 4 \beta_{4} - \beta_{3} - 15 \beta_{2} - 8 \beta_1 - 46) q^{45} + (\beta_{5} + \beta_{4} - 8 \beta_{2} + 42) q^{47} + (7 \beta_{5} + 7 \beta_{4} - 5 \beta_{2} - 12) q^{49} + ( - 4 \beta_{5} - 4 \beta_{4} + 12 \beta_{2} + 36) q^{51} + (5 \beta_{5} + 5 \beta_{4} - 17 \beta_{2} + 5) q^{53} + ( - 2 \beta_{5} + 6 \beta_{4} + 14 \beta_{3} + 10 \beta_{2} + 12 \beta_1 + 54) q^{55} + ( - 8 \beta_{5} + 8 \beta_{4} - 32 \beta_{3} + 8 \beta_{2} + 12 \beta_1 + 8) q^{57} + (2 \beta_{5} + 2 \beta_{4} + 6 \beta_{2} + 62) q^{59} + (7 \beta_{5} - 7 \beta_{4} + 8 \beta_{3} - 7 \beta_{2} - 8 \beta_1 - 7) q^{61} + ( - 7 \beta_{5} - 7 \beta_{4} + 2 \beta_{2} - 28) q^{63} + ( - 3 \beta_{5} + 3 \beta_{4} - 19 \beta_{3} + 15 \beta_{2} - 2 \beta_1 + 7) q^{65} + ( - 2 \beta_{5} + 2 \beta_{4} - 30 \beta_{3} + 2 \beta_{2} - 11 \beta_1 + 2) q^{67} + (\beta_{5} - \beta_{4} - 7 \beta_{3} - \beta_{2} + 24 \beta_1 - 1) q^{69} + (8 \beta_{5} - 8 \beta_{4} + 12 \beta_{3} - 8 \beta_{2} - 8 \beta_1 - 8) q^{71} + ( - 4 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 4) q^{73} + (8 \beta_{4} - 20 \beta_{3} + 15 \beta_1 - 48) q^{75} + (12 \beta_{5} + 12 \beta_{4} - 8 \beta_{2} + 64) q^{77} + ( - 4 \beta_{5} + 4 \beta_{4} + 44 \beta_{3} + 4 \beta_{2} + 16 \beta_1 + 4) q^{79} + (7 \beta_{5} + 7 \beta_{4} + 19 \beta_{2} + 46) q^{81} + (14 \beta_{5} - 14 \beta_{4} + 18 \beta_{3} - 14 \beta_{2} - 11 \beta_1 - 14) q^{83} + (14 \beta_{5} - 2 \beta_{4} - 8 \beta_{3} - 10 \beta_{2} - 24 \beta_1 + 42) q^{85} + (8 \beta_{5} + 8 \beta_{4} - 26 \beta_{2} - 106) q^{87} + (16 \beta_{5} + 16 \beta_{4} + 8 \beta_{2} - 10) q^{89} + (8 \beta_{2} - 32) q^{91} + (8 \beta_{5} + 8 \beta_{4} - 8 \beta_{2} - 56) q^{93} + ( - 6 \beta_{5} + 10 \beta_{4} - 38 \beta_{3} + 30 \beta_{2} - 4 \beta_1 + 10) q^{95} + ( - 6 \beta_{5} + 6 \beta_{4} + 18 \beta_{3} + 6 \beta_{2} + 8 \beta_1 + 6) q^{97} + ( - 6 \beta_{5} - 6 \beta_{4} - 34 \beta_{2} - 154) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{5} - 12 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 8 q^{5} - 12 q^{7} - 18 q^{9} + 8 q^{11} - 36 q^{15} - 24 q^{19} + 68 q^{23} + 10 q^{25} + 88 q^{35} - 208 q^{37} + 68 q^{41} - 232 q^{45} + 268 q^{47} - 62 q^{49} + 192 q^{51} + 64 q^{53} + 288 q^{55} + 360 q^{59} - 172 q^{63} - 304 q^{75} + 400 q^{77} + 238 q^{81} + 304 q^{85} - 584 q^{87} - 76 q^{89} - 208 q^{91} - 320 q^{93} - 32 q^{95} - 856 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 9x^{4} + 14x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{4} + 10\nu^{2} - 7 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{5} + 20\nu^{3} + 38\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{5} + 2\nu^{4} + 60\nu^{3} + 20\nu^{2} + 73\nu + 23 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{5} + 4\nu^{4} - 60\nu^{3} + 30\nu^{2} - 73\nu + 21 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} - 3\beta_{2} - 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} - \beta_{4} + 7\beta_{3} - \beta_{2} - 12\beta _1 - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{5} - 5\beta_{4} + 25\beta_{2} + 79 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{5} + 5\beta_{4} - 30\beta_{3} + 5\beta_{2} + 41\beta _1 + 5 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
639.1
2.65109i
1.37720i
0.273891i
0.273891i
1.37720i
2.65109i
0 5.30219i 0 4.75441 1.54778i 0 0.206625 0 −19.1132 0
639.2 0 2.75441i 0 2.54778 4.30219i 0 3.84997 0 1.41325 0
639.3 0 0.547781i 0 −3.30219 + 3.75441i 0 −10.0566 0 8.69994 0
639.4 0 0.547781i 0 −3.30219 3.75441i 0 −10.0566 0 8.69994 0
639.5 0 2.75441i 0 2.54778 + 4.30219i 0 3.84997 0 1.41325 0
639.6 0 5.30219i 0 4.75441 + 1.54778i 0 0.206625 0 −19.1132 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 639.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e 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.e.h 6
4.b odd 2 1 1280.3.e.i 6
5.b even 2 1 1280.3.e.g 6
8.b even 2 1 1280.3.e.f 6
8.d odd 2 1 1280.3.e.g 6
16.e even 4 1 160.3.h.b yes 6
16.e even 4 1 320.3.h.f 6
16.f odd 4 1 160.3.h.a 6
16.f odd 4 1 320.3.h.g 6
20.d odd 2 1 1280.3.e.f 6
40.e odd 2 1 inner 1280.3.e.h 6
40.f even 2 1 1280.3.e.i 6
48.i odd 4 1 1440.3.j.b 6
48.k even 4 1 1440.3.j.a 6
80.i odd 4 1 800.3.b.i 6
80.i odd 4 1 1600.3.b.v 6
80.j even 4 1 800.3.b.h 6
80.j even 4 1 1600.3.b.w 6
80.k odd 4 1 160.3.h.b yes 6
80.k odd 4 1 320.3.h.f 6
80.q even 4 1 160.3.h.a 6
80.q even 4 1 320.3.h.g 6
80.s even 4 1 800.3.b.i 6
80.s even 4 1 1600.3.b.v 6
80.t odd 4 1 800.3.b.h 6
80.t odd 4 1 1600.3.b.w 6
240.t even 4 1 1440.3.j.b 6
240.bm odd 4 1 1440.3.j.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.h.a 6 16.f odd 4 1
160.3.h.a 6 80.q even 4 1
160.3.h.b yes 6 16.e even 4 1
160.3.h.b yes 6 80.k odd 4 1
320.3.h.f 6 16.e even 4 1
320.3.h.f 6 80.k odd 4 1
320.3.h.g 6 16.f odd 4 1
320.3.h.g 6 80.q even 4 1
800.3.b.h 6 80.j even 4 1
800.3.b.h 6 80.t odd 4 1
800.3.b.i 6 80.i odd 4 1
800.3.b.i 6 80.s even 4 1
1280.3.e.f 6 8.b even 2 1
1280.3.e.f 6 20.d odd 2 1
1280.3.e.g 6 5.b even 2 1
1280.3.e.g 6 8.d odd 2 1
1280.3.e.h 6 1.a even 1 1 trivial
1280.3.e.h 6 40.e odd 2 1 inner
1280.3.e.i 6 4.b odd 2 1
1280.3.e.i 6 40.f even 2 1
1440.3.j.a 6 48.k even 4 1
1440.3.j.a 6 240.bm odd 4 1
1440.3.j.b 6 48.i odd 4 1
1440.3.j.b 6 240.t even 4 1
1600.3.b.v 6 80.i odd 4 1
1600.3.b.v 6 80.s even 4 1
1600.3.b.w 6 80.j even 4 1
1600.3.b.w 6 80.t odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1280, [\chi])\):

\( T_{3}^{6} + 36T_{3}^{4} + 224T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{3} + 6T_{7}^{2} - 40T_{7} + 8 \) Copy content Toggle raw display
\( T_{11}^{3} - 4T_{11}^{2} - 272T_{11} + 1600 \) Copy content Toggle raw display
\( T_{13}^{3} - 208T_{13} + 832 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 36 T^{4} + 224 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{6} - 8 T^{5} + 27 T^{4} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{3} + 6 T^{2} - 40 T + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 4 T^{2} - 272 T + 1600)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} - 208 T + 832)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 1088 T^{4} + \cdots + 6553600 \) Copy content Toggle raw display
$19$ \( (T^{3} + 12 T^{2} - 784 T + 3392)^{2} \) Copy content Toggle raw display
$23$ \( (T^{3} - 34 T^{2} - 152 T + 9160)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + 2380 T^{4} + \cdots + 4494400 \) Copy content Toggle raw display
$31$ \( T^{6} + 2560 T^{4} + \cdots + 419430400 \) Copy content Toggle raw display
$37$ \( (T^{3} + 104 T^{2} + 2704 T + 20800)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 34 T^{2} - 100 T + 5000)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 4260 T^{4} + \cdots + 39438400 \) Copy content Toggle raw display
$47$ \( (T^{3} - 134 T^{2} + 4824 T - 22216)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 32 T^{2} - 5968 T + 224320)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} - 180 T^{2} + 9968 T - 159424)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + 5964 T^{4} + \cdots + 6649423936 \) Copy content Toggle raw display
$67$ \( T^{6} + 14180 T^{4} + \cdots + 613057600 \) Copy content Toggle raw display
$71$ \( T^{6} + 8384 T^{4} + \cdots + 14231535616 \) Copy content Toggle raw display
$73$ \( T^{6} + 4416 T^{4} + \cdots + 3114532864 \) Copy content Toggle raw display
$79$ \( T^{6} + 25856 T^{4} + \cdots + 37060870144 \) Copy content Toggle raw display
$83$ \( T^{6} + 24164 T^{4} + \cdots + 233675560000 \) Copy content Toggle raw display
$89$ \( (T^{3} + 38 T^{2} - 13940 T - 155000)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 7488 T^{4} + \cdots + 44302336 \) Copy content Toggle raw display
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