Properties

Label 320.2.o.e
Level $320$
Weight $2$
Character orbit 320.o
Analytic conductor $2.555$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 320.o (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.55521286468\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.49787136.1
Defining polynomial: \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( -\beta_{2} - \beta_{7} ) q^{3} + \beta_{7} q^{5} -\beta_{6} q^{7} + ( -3 \beta_{1} - \beta_{5} - \beta_{6} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{2} - \beta_{7} ) q^{3} + \beta_{7} q^{5} -\beta_{6} q^{7} + ( -3 \beta_{1} - \beta_{5} - \beta_{6} ) q^{9} + ( -\beta_{2} + \beta_{3} + \beta_{7} ) q^{11} + ( \beta_{2} + \beta_{4} ) q^{13} + ( 2 + 4 \beta_{1} + \beta_{5} ) q^{15} + ( 3 + 3 \beta_{1} ) q^{17} -2 \beta_{4} q^{19} + ( -\beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{7} ) q^{21} + ( 2 + 2 \beta_{1} + \beta_{5} ) q^{23} + ( -1 - 2 \beta_{1} - \beta_{5} + \beta_{6} ) q^{25} + ( 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{27} + ( 2 \beta_{3} + 2 \beta_{7} ) q^{29} + ( -4 \beta_{1} - \beta_{5} - \beta_{6} ) q^{31} + ( -2 + 2 \beta_{1} ) q^{33} + ( -\beta_{3} - 4 \beta_{4} - 2 \beta_{7} ) q^{35} + ( 3 \beta_{2} - 3 \beta_{4} ) q^{37} + ( -4 - \beta_{5} + \beta_{6} ) q^{39} + ( -2 + \beta_{5} - \beta_{6} ) q^{41} + ( -\beta_{2} + 4 \beta_{4} + 3 \beta_{7} ) q^{43} + ( -5 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{7} ) q^{45} + ( -2 + 2 \beta_{1} + \beta_{6} ) q^{47} + ( -3 \beta_{1} + \beta_{5} + \beta_{6} ) q^{49} + ( -3 \beta_{2} - 3 \beta_{3} - 3 \beta_{7} ) q^{51} + ( \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{53} + ( 4 - 2 \beta_{1} - \beta_{5} + \beta_{6} ) q^{55} + ( 4 + 4 \beta_{1} + 2 \beta_{5} ) q^{57} + ( 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} ) q^{59} + ( 3 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} + 3 \beta_{7} ) q^{61} + ( -10 - 10 \beta_{1} - \beta_{5} ) q^{63} + ( 1 - 3 \beta_{1} + \beta_{5} - \beta_{6} ) q^{65} + ( 4 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} ) q^{67} + ( -5 \beta_{2} - 3 \beta_{3} - 3 \beta_{7} ) q^{69} + ( -8 \beta_{1} + \beta_{5} + \beta_{6} ) q^{71} + ( -5 + 5 \beta_{1} + 2 \beta_{6} ) q^{73} + ( 5 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{7} ) q^{75} + ( 2 \beta_{2} - 6 \beta_{4} - 4 \beta_{7} ) q^{77} + 12 q^{79} + ( -11 - \beta_{5} + \beta_{6} ) q^{81} + ( -\beta_{2} - 2 \beta_{4} - 3 \beta_{7} ) q^{83} + ( 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{7} ) q^{85} + ( -8 + 8 \beta_{1} + 2 \beta_{6} ) q^{87} + ( 8 \beta_{1} + 2 \beta_{5} + 2 \beta_{6} ) q^{89} + ( -\beta_{2} + 3 \beta_{3} + 3 \beta_{7} ) q^{91} + ( 2 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} ) q^{93} + ( -4 + 2 \beta_{1} - 2 \beta_{5} ) q^{95} + ( -5 - 5 \beta_{1} - 2 \beta_{5} ) q^{97} + ( -\beta_{2} + \beta_{3} - 6 \beta_{4} - \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{7} + O(q^{10}) \) \( 8 q - 4 q^{7} + 12 q^{15} + 24 q^{17} + 12 q^{23} - 16 q^{33} - 24 q^{39} - 24 q^{41} - 12 q^{47} + 40 q^{55} + 24 q^{57} - 76 q^{63} - 32 q^{73} + 96 q^{79} - 80 q^{81} - 56 q^{87} - 24 q^{95} - 32 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -3 \nu^{7} - 5 \nu^{5} + 5 \nu^{3} - 16 \nu \)\()/40\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - \nu^{5} + \nu^{3} + 8 \nu \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{7} - 2 \nu^{6} - 15 \nu^{5} + 10 \nu^{4} - 25 \nu^{3} + 30 \nu^{2} - 20 \nu + 16 \)\()/40\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{6} + 5 \nu^{4} + 15 \nu^{2} + 26 \)\()/10\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 2 \nu^{6} + 3 \nu^{5} - 6 \nu^{4} + 5 \nu^{3} - 2 \nu^{2} + 20 \nu - 16 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{6} + 3 \nu^{5} + 6 \nu^{4} + 5 \nu^{3} + 2 \nu^{2} + 20 \nu + 16 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} + \nu^{6} - 5 \nu^{5} - 5 \nu^{4} - 15 \nu^{3} - 15 \nu^{2} - 30 \nu - 8 \)\()/20\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} - 2\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} - \beta_{3} + 5 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-3 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - 4 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} - 4\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - 11 \beta_{2} - 10 \beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(5 \beta_{7} + 5 \beta_{4} - 5 \beta_{3} + 5 \beta_{2} - 9\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-7 \beta_{7} - 7 \beta_{6} - 7 \beta_{5} - 7 \beta_{3} + 13 \beta_{2} - 20 \beta_{1}\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(-1\) \(\beta_{1}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1
1.09445 0.895644i
0.228425 + 1.39564i
−0.228425 + 1.39564i
−1.09445 0.895644i
1.09445 + 0.895644i
0.228425 1.39564i
−0.228425 1.39564i
−1.09445 + 0.895644i
0 −2.18890 2.18890i 0 0.456850 + 2.18890i 0 1.79129 + 1.79129i 0 6.58258i 0
223.2 0 −0.456850 0.456850i 0 2.18890 + 0.456850i 0 −2.79129 2.79129i 0 2.58258i 0
223.3 0 0.456850 + 0.456850i 0 −2.18890 0.456850i 0 −2.79129 2.79129i 0 2.58258i 0
223.4 0 2.18890 + 2.18890i 0 −0.456850 2.18890i 0 1.79129 + 1.79129i 0 6.58258i 0
287.1 0 −2.18890 + 2.18890i 0 0.456850 2.18890i 0 1.79129 1.79129i 0 6.58258i 0
287.2 0 −0.456850 + 0.456850i 0 2.18890 0.456850i 0 −2.79129 + 2.79129i 0 2.58258i 0
287.3 0 0.456850 0.456850i 0 −2.18890 + 0.456850i 0 −2.79129 + 2.79129i 0 2.58258i 0
287.4 0 2.18890 2.18890i 0 −0.456850 + 2.18890i 0 1.79129 1.79129i 0 6.58258i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 287.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
20.e even 4 1 inner
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 320.2.o.e 8
4.b odd 2 1 320.2.o.f yes 8
5.b even 2 1 1600.2.o.l 8
5.c odd 4 1 320.2.o.f yes 8
5.c odd 4 1 1600.2.o.e 8
8.b even 2 1 inner 320.2.o.e 8
8.d odd 2 1 320.2.o.f yes 8
16.e even 4 2 1280.2.n.p 8
16.f odd 4 2 1280.2.n.n 8
20.d odd 2 1 1600.2.o.e 8
20.e even 4 1 inner 320.2.o.e 8
20.e even 4 1 1600.2.o.l 8
40.e odd 2 1 1600.2.o.e 8
40.f even 2 1 1600.2.o.l 8
40.i odd 4 1 320.2.o.f yes 8
40.i odd 4 1 1600.2.o.e 8
40.k even 4 1 inner 320.2.o.e 8
40.k even 4 1 1600.2.o.l 8
80.i odd 4 1 1280.2.n.n 8
80.j even 4 1 1280.2.n.p 8
80.s even 4 1 1280.2.n.p 8
80.t odd 4 1 1280.2.n.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.o.e 8 1.a even 1 1 trivial
320.2.o.e 8 8.b even 2 1 inner
320.2.o.e 8 20.e even 4 1 inner
320.2.o.e 8 40.k even 4 1 inner
320.2.o.f yes 8 4.b odd 2 1
320.2.o.f yes 8 5.c odd 4 1
320.2.o.f yes 8 8.d odd 2 1
320.2.o.f yes 8 40.i odd 4 1
1280.2.n.n 8 16.f odd 4 2
1280.2.n.n 8 80.i odd 4 1
1280.2.n.n 8 80.t odd 4 1
1280.2.n.p 8 16.e even 4 2
1280.2.n.p 8 80.j even 4 1
1280.2.n.p 8 80.s even 4 1
1600.2.o.e 8 5.c odd 4 1
1600.2.o.e 8 20.d odd 2 1
1600.2.o.e 8 40.e odd 2 1
1600.2.o.e 8 40.i odd 4 1
1600.2.o.l 8 5.b even 2 1
1600.2.o.l 8 20.e even 4 1
1600.2.o.l 8 40.f even 2 1
1600.2.o.l 8 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{8} + 92 T_{3}^{4} + 16 \)
\( T_{7}^{4} + 2 T_{7}^{3} + 2 T_{7}^{2} - 20 T_{7} + 100 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 16 + 92 T^{4} + T^{8} \)
$5$ \( 625 - 34 T^{4} + T^{8} \)
$7$ \( ( 100 - 20 T + 2 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$11$ \( ( 16 - 20 T^{2} + T^{4} )^{2} \)
$13$ \( ( 36 + T^{4} )^{2} \)
$17$ \( ( 18 - 6 T + T^{2} )^{4} \)
$19$ \( ( 12 + T^{2} )^{4} \)
$23$ \( ( 36 + 36 T + 18 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$29$ \( ( -28 + T^{2} )^{4} \)
$31$ \( ( 144 + 60 T^{2} + T^{4} )^{2} \)
$37$ \( ( 2916 + T^{4} )^{2} \)
$41$ \( ( -12 + 6 T + T^{2} )^{4} \)
$43$ \( 1296 + 18972 T^{4} + T^{8} \)
$47$ \( ( 36 - 36 T + 18 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$53$ \( ( 196 + T^{4} )^{2} \)
$59$ \( ( 28 + T^{2} )^{4} \)
$61$ \( ( 3600 + 132 T^{2} + T^{4} )^{2} \)
$67$ \( 1296 + 18972 T^{4} + T^{8} \)
$71$ \( ( 3600 + 204 T^{2} + T^{4} )^{2} \)
$73$ \( ( 100 - 160 T + 128 T^{2} + 16 T^{3} + T^{4} )^{2} \)
$79$ \( ( -12 + T )^{8} \)
$83$ \( 810000 + 2556 T^{4} + T^{8} \)
$89$ \( ( 2304 + 240 T^{2} + T^{4} )^{2} \)
$97$ \( ( 100 - 160 T + 128 T^{2} + 16 T^{3} + T^{4} )^{2} \)
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