Properties

Label 320.2.q.a
Level $320$
Weight $2$
Character orbit 320.q
Analytic conductor $2.555$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 320.q (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.55521286468\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - i ) q^{3} + ( 1 + 2 i ) q^{5} -i q^{9} +O(q^{10})\) \( q + ( -1 - i ) q^{3} + ( 1 + 2 i ) q^{5} -i q^{9} + ( 3 + 3 i ) q^{11} + ( 3 + 3 i ) q^{13} + ( 1 - 3 i ) q^{15} -4 i q^{17} + ( 1 - i ) q^{19} + 8 q^{23} + ( -3 + 4 i ) q^{25} + ( -4 + 4 i ) q^{27} + ( 3 - 3 i ) q^{29} -6 i q^{33} + ( 3 - 3 i ) q^{37} -6 i q^{39} + ( -3 + 3 i ) q^{43} + ( 2 - i ) q^{45} + 2 i q^{47} -7 q^{49} + ( -4 + 4 i ) q^{51} + ( -9 + 9 i ) q^{53} + ( -3 + 9 i ) q^{55} -2 q^{57} + ( -9 - 9 i ) q^{59} + ( -5 + 5 i ) q^{61} + ( -3 + 9 i ) q^{65} + ( 3 + 3 i ) q^{67} + ( -8 - 8 i ) q^{69} -6 i q^{71} + 6 q^{73} + ( 7 - i ) q^{75} -8 q^{79} + 5 q^{81} + ( -9 - 9 i ) q^{83} + ( 8 - 4 i ) q^{85} -6 q^{87} -12 i q^{89} + ( 3 + i ) q^{95} + 12 i q^{97} + ( 3 - 3 i ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{5} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{5} + 6q^{11} + 6q^{13} + 2q^{15} + 2q^{19} + 16q^{23} - 6q^{25} - 8q^{27} + 6q^{29} + 6q^{37} - 6q^{43} + 4q^{45} - 14q^{49} - 8q^{51} - 18q^{53} - 6q^{55} - 4q^{57} - 18q^{59} - 10q^{61} - 6q^{65} + 6q^{67} - 16q^{69} + 12q^{73} + 14q^{75} - 16q^{79} + 10q^{81} - 18q^{83} + 16q^{85} - 12q^{87} + 6q^{95} + 6q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(-1\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 −1.00000 1.00000i 0 1.00000 + 2.00000i 0 0 0 1.00000i 0
209.1 0 −1.00000 + 1.00000i 0 1.00000 2.00000i 0 0 0 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.q 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.q.a 2
4.b odd 2 1 80.2.q.a 2
5.b even 2 1 320.2.q.b 2
5.c odd 4 1 1600.2.l.b 2
5.c odd 4 1 1600.2.l.c 2
8.b even 2 1 640.2.q.d 2
8.d odd 2 1 640.2.q.b 2
12.b even 2 1 720.2.bm.b 2
16.e even 4 1 320.2.q.b 2
16.e even 4 1 640.2.q.a 2
16.f odd 4 1 80.2.q.b yes 2
16.f odd 4 1 640.2.q.c 2
20.d odd 2 1 80.2.q.b yes 2
20.e even 4 1 400.2.l.a 2
20.e even 4 1 400.2.l.b 2
40.e odd 2 1 640.2.q.c 2
40.f even 2 1 640.2.q.a 2
48.k even 4 1 720.2.bm.a 2
60.h even 2 1 720.2.bm.a 2
80.i odd 4 1 1600.2.l.c 2
80.j even 4 1 400.2.l.b 2
80.k odd 4 1 80.2.q.a 2
80.k odd 4 1 640.2.q.b 2
80.q even 4 1 inner 320.2.q.a 2
80.q even 4 1 640.2.q.d 2
80.s even 4 1 400.2.l.a 2
80.t odd 4 1 1600.2.l.b 2
240.t even 4 1 720.2.bm.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.q.a 2 4.b odd 2 1
80.2.q.a 2 80.k odd 4 1
80.2.q.b yes 2 16.f odd 4 1
80.2.q.b yes 2 20.d odd 2 1
320.2.q.a 2 1.a even 1 1 trivial
320.2.q.a 2 80.q even 4 1 inner
320.2.q.b 2 5.b even 2 1
320.2.q.b 2 16.e even 4 1
400.2.l.a 2 20.e even 4 1
400.2.l.a 2 80.s even 4 1
400.2.l.b 2 20.e even 4 1
400.2.l.b 2 80.j even 4 1
640.2.q.a 2 16.e even 4 1
640.2.q.a 2 40.f even 2 1
640.2.q.b 2 8.d odd 2 1
640.2.q.b 2 80.k odd 4 1
640.2.q.c 2 16.f odd 4 1
640.2.q.c 2 40.e odd 2 1
640.2.q.d 2 8.b even 2 1
640.2.q.d 2 80.q even 4 1
720.2.bm.a 2 48.k even 4 1
720.2.bm.a 2 60.h even 2 1
720.2.bm.b 2 12.b even 2 1
720.2.bm.b 2 240.t even 4 1
1600.2.l.b 2 5.c odd 4 1
1600.2.l.b 2 80.t odd 4 1
1600.2.l.c 2 5.c odd 4 1
1600.2.l.c 2 80.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 2 T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(320, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 2 + 2 T + T^{2} \)
$5$ \( 5 - 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 18 - 6 T + T^{2} \)
$13$ \( 18 - 6 T + T^{2} \)
$17$ \( 16 + T^{2} \)
$19$ \( 2 - 2 T + T^{2} \)
$23$ \( ( -8 + T )^{2} \)
$29$ \( 18 - 6 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 18 - 6 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 18 + 6 T + T^{2} \)
$47$ \( 4 + T^{2} \)
$53$ \( 162 + 18 T + T^{2} \)
$59$ \( 162 + 18 T + T^{2} \)
$61$ \( 50 + 10 T + T^{2} \)
$67$ \( 18 - 6 T + T^{2} \)
$71$ \( 36 + T^{2} \)
$73$ \( ( -6 + T )^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 162 + 18 T + T^{2} \)
$89$ \( 144 + T^{2} \)
$97$ \( 144 + T^{2} \)
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