Properties

Label 320.2.n.f
Level $320$
Weight $2$
Character orbit 320.n
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.n (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 160)
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.

\(f(q)\) \(=\) \( q + ( 1 + i ) q^{3} + ( -1 + 2 i ) q^{5} + ( -1 + i ) q^{7} -i q^{9} +O(q^{10})\) \( q + ( 1 + i ) q^{3} + ( -1 + 2 i ) q^{5} + ( -1 + i ) q^{7} -i q^{9} + 6 i q^{11} + ( 1 - i ) q^{13} + ( -3 + i ) q^{15} + ( 1 + i ) q^{17} -4 q^{19} -2 q^{21} + ( 5 + 5 i ) q^{23} + ( -3 - 4 i ) q^{25} + ( 4 - 4 i ) q^{27} -8 i q^{29} + 2 i q^{31} + ( -6 + 6 i ) q^{33} + ( -1 - 3 i ) q^{35} + ( 5 + 5 i ) q^{37} + 2 q^{39} + 6 q^{41} + ( -3 - 3 i ) q^{43} + ( 2 + i ) q^{45} + ( 7 - 7 i ) q^{47} + 5 i q^{49} + 2 i q^{51} + ( 1 - i ) q^{53} + ( -12 - 6 i ) q^{55} + ( -4 - 4 i ) q^{57} -4 q^{59} -2 q^{61} + ( 1 + i ) q^{63} + ( 1 + 3 i ) q^{65} + ( 7 - 7 i ) q^{67} + 10 i q^{69} -6 i q^{71} + ( 9 - 9 i ) q^{73} + ( 1 - 7 i ) q^{75} + ( -6 - 6 i ) q^{77} -8 q^{79} + 5 q^{81} + ( 5 + 5 i ) q^{83} + ( -3 + i ) q^{85} + ( 8 - 8 i ) q^{87} + 2 i q^{91} + ( -2 + 2 i ) q^{93} + ( 4 - 8 i ) q^{95} + ( -3 - 3 i ) q^{97} + 6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + O(q^{10}) \) \( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{13} - 6 q^{15} + 2 q^{17} - 8 q^{19} - 4 q^{21} + 10 q^{23} - 6 q^{25} + 8 q^{27} - 12 q^{33} - 2 q^{35} + 10 q^{37} + 4 q^{39} + 12 q^{41} - 6 q^{43} + 4 q^{45} + 14 q^{47} + 2 q^{53} - 24 q^{55} - 8 q^{57} - 8 q^{59} - 4 q^{61} + 2 q^{63} + 2 q^{65} + 14 q^{67} + 18 q^{73} + 2 q^{75} - 12 q^{77} - 16 q^{79} + 10 q^{81} + 10 q^{83} - 6 q^{85} + 16 q^{87} - 4 q^{93} + 8 q^{95} - 6 q^{97} + 12 q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
1.00000i
1.00000i
0 1.00000 1.00000i 0 −1.00000 2.00000i 0 −1.00000 1.00000i 0 1.00000i 0
127.1 0 1.00000 + 1.00000i 0 −1.00000 + 2.00000i 0 −1.00000 + 1.00000i 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
20.e 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.n.f 2
4.b odd 2 1 320.2.n.c 2
5.b even 2 1 1600.2.n.e 2
5.c odd 4 1 320.2.n.c 2
5.c odd 4 1 1600.2.n.j 2
8.b even 2 1 160.2.n.b 2
8.d odd 2 1 160.2.n.e yes 2
16.e even 4 1 1280.2.o.e 2
16.e even 4 1 1280.2.o.k 2
16.f odd 4 1 1280.2.o.d 2
16.f odd 4 1 1280.2.o.n 2
20.d odd 2 1 1600.2.n.j 2
20.e even 4 1 inner 320.2.n.f 2
20.e even 4 1 1600.2.n.e 2
24.f even 2 1 1440.2.x.e 2
24.h odd 2 1 1440.2.x.b 2
40.e odd 2 1 800.2.n.c 2
40.f even 2 1 800.2.n.h 2
40.i odd 4 1 160.2.n.e yes 2
40.i odd 4 1 800.2.n.c 2
40.k even 4 1 160.2.n.b 2
40.k even 4 1 800.2.n.h 2
80.i odd 4 1 1280.2.o.d 2
80.j even 4 1 1280.2.o.e 2
80.s even 4 1 1280.2.o.k 2
80.t odd 4 1 1280.2.o.n 2
120.q odd 4 1 1440.2.x.b 2
120.w even 4 1 1440.2.x.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.n.b 2 8.b even 2 1
160.2.n.b 2 40.k even 4 1
160.2.n.e yes 2 8.d odd 2 1
160.2.n.e yes 2 40.i odd 4 1
320.2.n.c 2 4.b odd 2 1
320.2.n.c 2 5.c odd 4 1
320.2.n.f 2 1.a even 1 1 trivial
320.2.n.f 2 20.e even 4 1 inner
800.2.n.c 2 40.e odd 2 1
800.2.n.c 2 40.i odd 4 1
800.2.n.h 2 40.f even 2 1
800.2.n.h 2 40.k even 4 1
1280.2.o.d 2 16.f odd 4 1
1280.2.o.d 2 80.i odd 4 1
1280.2.o.e 2 16.e even 4 1
1280.2.o.e 2 80.j even 4 1
1280.2.o.k 2 16.e even 4 1
1280.2.o.k 2 80.s even 4 1
1280.2.o.n 2 16.f odd 4 1
1280.2.o.n 2 80.t odd 4 1
1440.2.x.b 2 24.h odd 2 1
1440.2.x.b 2 120.q odd 4 1
1440.2.x.e 2 24.f even 2 1
1440.2.x.e 2 120.w even 4 1
1600.2.n.e 2 5.b even 2 1
1600.2.n.e 2 20.e even 4 1
1600.2.n.j 2 5.c odd 4 1
1600.2.n.j 2 20.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 \(S_{2}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{2} - 2 T_{3} + 2 \)
\( T_{7}^{2} + 2 T_{7} + 2 \)
\( T_{13}^{2} - 2 T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 2 - 2 T + T^{2} \)
$5$ \( 5 + 2 T + T^{2} \)
$7$ \( 2 + 2 T + T^{2} \)
$11$ \( 36 + T^{2} \)
$13$ \( 2 - 2 T + T^{2} \)
$17$ \( 2 - 2 T + T^{2} \)
$19$ \( ( 4 + T )^{2} \)
$23$ \( 50 - 10 T + T^{2} \)
$29$ \( 64 + T^{2} \)
$31$ \( 4 + T^{2} \)
$37$ \( 50 - 10 T + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( 18 + 6 T + T^{2} \)
$47$ \( 98 - 14 T + T^{2} \)
$53$ \( 2 - 2 T + T^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 98 - 14 T + T^{2} \)
$71$ \( 36 + T^{2} \)
$73$ \( 162 - 18 T + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 50 - 10 T + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 18 + 6 T + T^{2} \)
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