Properties

Label 1280.3.e.g
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

Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1280.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(34.8774738381\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1827904.1
Defining polynomial: \(x^{6} + 9 x^{4} + 14 x^{2} + 1\)
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

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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{4} + 10 \nu^{2} - 7 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{5} + 20 \nu^{3} + 38 \nu \)\()/5\)
\(\beta_{4}\)\(=\)\((\)\( 7 \nu^{5} + 2 \nu^{4} + 60 \nu^{3} + 20 \nu^{2} + 73 \nu + 23 \)\()/5\)
\(\beta_{5}\)\(=\)\((\)\( -7 \nu^{5} + 4 \nu^{4} - 60 \nu^{3} + 30 \nu^{2} - 73 \nu + 21 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + \beta_{4} - 3 \beta_{2} - 13\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{5} - \beta_{4} + 7 \beta_{3} - \beta_{2} - 12 \beta_{1} - 1\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-5 \beta_{5} - 5 \beta_{4} + 25 \beta_{2} + 79\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{5} + 5 \beta_{4} - 30 \beta_{3} + 5 \beta_{2} + 41 \beta_{1} + 5\)\()/2\)

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.

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

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 64 + 224 T^{2} + 36 T^{4} + T^{6} \)
$5$ \( 15625 + 5000 T + 675 T^{2} + 80 T^{3} + 27 T^{4} + 8 T^{5} + T^{6} \)
$7$ \( ( -8 - 40 T - 6 T^{2} + T^{3} )^{2} \)
$11$ \( ( 1600 - 272 T - 4 T^{2} + T^{3} )^{2} \)
$13$ \( ( -832 - 208 T + T^{3} )^{2} \)
$17$ \( 6553600 + 270336 T^{2} + 1088 T^{4} + T^{6} \)
$19$ \( ( 3392 - 784 T + 12 T^{2} + T^{3} )^{2} \)
$23$ \( ( -9160 - 152 T + 34 T^{2} + T^{3} )^{2} \)
$29$ \( 4494400 + 805424 T^{2} + 2380 T^{4} + T^{6} \)
$31$ \( 419430400 + 1900544 T^{2} + 2560 T^{4} + T^{6} \)
$37$ \( ( -20800 + 2704 T - 104 T^{2} + T^{3} )^{2} \)
$41$ \( ( 5000 - 100 T - 34 T^{2} + T^{3} )^{2} \)
$43$ \( 39438400 + 2459744 T^{2} + 4260 T^{4} + T^{6} \)
$47$ \( ( 22216 + 4824 T + 134 T^{2} + T^{3} )^{2} \)
$53$ \( ( -224320 - 5968 T + 32 T^{2} + T^{3} )^{2} \)
$59$ \( ( -159424 + 9968 T - 180 T^{2} + T^{3} )^{2} \)
$61$ \( 6649423936 + 11154224 T^{2} + 5964 T^{4} + T^{6} \)
$67$ \( 613057600 + 30359904 T^{2} + 14180 T^{4} + T^{6} \)
$71$ \( 14231535616 + 19738624 T^{2} + 8384 T^{4} + T^{6} \)
$73$ \( 3114532864 + 6447104 T^{2} + 4416 T^{4} + T^{6} \)
$79$ \( 37060870144 + 159514624 T^{2} + 25856 T^{4} + T^{6} \)
$83$ \( 233675560000 + 145948000 T^{2} + 24164 T^{4} + T^{6} \)
$89$ \( ( -155000 - 13940 T + 38 T^{2} + T^{3} )^{2} \)
$97$ \( 44302336 + 1384448 T^{2} + 7488 T^{4} + T^{6} \)
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