Properties

Label 320.2.a.g
Level $320$
Weight $2$
Character orbit 320.a
Self dual yes
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.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.55521286468\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - q^{5} + \beta q^{7} + 5 q^{9} +O(q^{10})\) \( q + \beta q^{3} - q^{5} + \beta q^{7} + 5 q^{9} -2 \beta q^{11} + 2 q^{13} -\beta q^{15} + 2 q^{17} + 8 q^{21} -\beta q^{23} + q^{25} + 2 \beta q^{27} -6 q^{29} -2 \beta q^{31} -16 q^{33} -\beta q^{35} + 10 q^{37} + 2 \beta q^{39} + 2 q^{41} -3 \beta q^{43} -5 q^{45} + \beta q^{47} + q^{49} + 2 \beta q^{51} -6 q^{53} + 2 \beta q^{55} + 4 \beta q^{59} + 2 q^{61} + 5 \beta q^{63} -2 q^{65} -\beta q^{67} -8 q^{69} -2 \beta q^{71} -6 q^{73} + \beta q^{75} -16 q^{77} -4 \beta q^{79} + q^{81} + \beta q^{83} -2 q^{85} -6 \beta q^{87} + 10 q^{89} + 2 \beta q^{91} -16 q^{93} + 2 q^{97} -10 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + 10q^{9} + O(q^{10}) \) \( 2q - 2q^{5} + 10q^{9} + 4q^{13} + 4q^{17} + 16q^{21} + 2q^{25} - 12q^{29} - 32q^{33} + 20q^{37} + 4q^{41} - 10q^{45} + 2q^{49} - 12q^{53} + 4q^{61} - 4q^{65} - 16q^{69} - 12q^{73} - 32q^{77} + 2q^{81} - 4q^{85} + 20q^{89} - 32q^{93} + 4q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
0 −2.82843 0 −1.00000 0 −2.82843 0 5.00000 0
1.2 0 2.82843 0 −1.00000 0 2.82843 0 5.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.2.a.g 2
3.b odd 2 1 2880.2.a.bk 2
4.b odd 2 1 inner 320.2.a.g 2
5.b even 2 1 1600.2.a.bc 2
5.c odd 4 2 1600.2.c.n 4
8.b even 2 1 160.2.a.c 2
8.d odd 2 1 160.2.a.c 2
12.b even 2 1 2880.2.a.bk 2
16.e even 4 2 1280.2.d.l 4
16.f odd 4 2 1280.2.d.l 4
20.d odd 2 1 1600.2.a.bc 2
20.e even 4 2 1600.2.c.n 4
24.f even 2 1 1440.2.a.o 2
24.h odd 2 1 1440.2.a.o 2
40.e odd 2 1 800.2.a.m 2
40.f even 2 1 800.2.a.m 2
40.i odd 4 2 800.2.c.f 4
40.k even 4 2 800.2.c.f 4
56.e even 2 1 7840.2.a.bf 2
56.h odd 2 1 7840.2.a.bf 2
120.i odd 2 1 7200.2.a.cm 2
120.m even 2 1 7200.2.a.cm 2
120.q odd 4 2 7200.2.f.bh 4
120.w even 4 2 7200.2.f.bh 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.a.c 2 8.b even 2 1
160.2.a.c 2 8.d odd 2 1
320.2.a.g 2 1.a even 1 1 trivial
320.2.a.g 2 4.b odd 2 1 inner
800.2.a.m 2 40.e odd 2 1
800.2.a.m 2 40.f even 2 1
800.2.c.f 4 40.i odd 4 2
800.2.c.f 4 40.k even 4 2
1280.2.d.l 4 16.e even 4 2
1280.2.d.l 4 16.f odd 4 2
1440.2.a.o 2 24.f even 2 1
1440.2.a.o 2 24.h odd 2 1
1600.2.a.bc 2 5.b even 2 1
1600.2.a.bc 2 20.d odd 2 1
1600.2.c.n 4 5.c odd 4 2
1600.2.c.n 4 20.e even 4 2
2880.2.a.bk 2 3.b odd 2 1
2880.2.a.bk 2 12.b even 2 1
7200.2.a.cm 2 120.i odd 2 1
7200.2.a.cm 2 120.m even 2 1
7200.2.f.bh 4 120.q odd 4 2
7200.2.f.bh 4 120.w even 4 2
7840.2.a.bf 2 56.e even 2 1
7840.2.a.bf 2 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(320))\):

\( T_{3}^{2} - 8 \)
\( T_{7}^{2} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -8 + T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -8 + T^{2} \)
$11$ \( -32 + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( T^{2} \)
$23$ \( -8 + T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( -32 + T^{2} \)
$37$ \( ( -10 + T )^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( -72 + T^{2} \)
$47$ \( -8 + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( -128 + T^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( -8 + T^{2} \)
$71$ \( -32 + T^{2} \)
$73$ \( ( 6 + T )^{2} \)
$79$ \( -128 + T^{2} \)
$83$ \( -8 + T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( ( -2 + T )^{2} \)
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