Properties

Label 640.2.d.d
Level $640$
Weight $2$
Character orbit 640.d
Analytic conductor $5.110$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 640.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.11042572936\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{3} + \zeta_{8}^{2} q^{5} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{7} + q^{9} +O(q^{10})\) \( q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{3} + \zeta_{8}^{2} q^{5} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{7} + q^{9} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{11} + 2 \zeta_{8}^{2} q^{13} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{15} + 6 q^{17} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{19} + 6 \zeta_{8}^{2} q^{21} + ( 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{23} - q^{25} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{27} + 4 \zeta_{8}^{2} q^{29} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{31} + 8 q^{33} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{35} + 2 \zeta_{8}^{2} q^{37} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{39} -8 q^{41} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{43} + \zeta_{8}^{2} q^{45} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{47} + 11 q^{49} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{51} -2 \zeta_{8}^{2} q^{53} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{55} + 4 q^{57} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{59} + 14 \zeta_{8}^{2} q^{61} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{63} -2 q^{65} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{67} -10 \zeta_{8}^{2} q^{69} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{71} -6 q^{73} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{75} -24 \zeta_{8}^{2} q^{77} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{79} -5 q^{81} + ( 9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{83} + 6 \zeta_{8}^{2} q^{85} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{87} -6 q^{89} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{91} + 4 \zeta_{8}^{2} q^{93} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{95} + 10 q^{97} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{9} + 24q^{17} - 4q^{25} + 32q^{33} - 32q^{41} + 44q^{49} + 16q^{57} - 8q^{65} - 24q^{73} - 20q^{81} - 24q^{89} + 40q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/640\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0 1.41421i 0 1.00000i 0 4.24264 0 1.00000 0
321.2 0 1.41421i 0 1.00000i 0 −4.24264 0 1.00000 0
321.3 0 1.41421i 0 1.00000i 0 −4.24264 0 1.00000 0
321.4 0 1.41421i 0 1.00000i 0 4.24264 0 1.00000 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 640.2.d.d 4
3.b odd 2 1 5760.2.k.o 4
4.b odd 2 1 inner 640.2.d.d 4
5.b even 2 1 3200.2.d.s 4
5.c odd 4 1 3200.2.f.i 4
5.c odd 4 1 3200.2.f.j 4
8.b even 2 1 inner 640.2.d.d 4
8.d odd 2 1 inner 640.2.d.d 4
12.b even 2 1 5760.2.k.o 4
16.e even 4 1 1280.2.a.f 2
16.e even 4 1 1280.2.a.j 2
16.f odd 4 1 1280.2.a.f 2
16.f odd 4 1 1280.2.a.j 2
20.d odd 2 1 3200.2.d.s 4
20.e even 4 1 3200.2.f.i 4
20.e even 4 1 3200.2.f.j 4
24.f even 2 1 5760.2.k.o 4
24.h odd 2 1 5760.2.k.o 4
40.e odd 2 1 3200.2.d.s 4
40.f even 2 1 3200.2.d.s 4
40.i odd 4 1 3200.2.f.i 4
40.i odd 4 1 3200.2.f.j 4
40.k even 4 1 3200.2.f.i 4
40.k even 4 1 3200.2.f.j 4
80.k odd 4 1 6400.2.a.bl 2
80.k odd 4 1 6400.2.a.bn 2
80.q even 4 1 6400.2.a.bl 2
80.q even 4 1 6400.2.a.bn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.d.d 4 1.a even 1 1 trivial
640.2.d.d 4 4.b odd 2 1 inner
640.2.d.d 4 8.b even 2 1 inner
640.2.d.d 4 8.d odd 2 1 inner
1280.2.a.f 2 16.e even 4 1
1280.2.a.f 2 16.f odd 4 1
1280.2.a.j 2 16.e even 4 1
1280.2.a.j 2 16.f odd 4 1
3200.2.d.s 4 5.b even 2 1
3200.2.d.s 4 20.d odd 2 1
3200.2.d.s 4 40.e odd 2 1
3200.2.d.s 4 40.f even 2 1
3200.2.f.i 4 5.c odd 4 1
3200.2.f.i 4 20.e even 4 1
3200.2.f.i 4 40.i odd 4 1
3200.2.f.i 4 40.k even 4 1
3200.2.f.j 4 5.c odd 4 1
3200.2.f.j 4 20.e even 4 1
3200.2.f.j 4 40.i odd 4 1
3200.2.f.j 4 40.k even 4 1
5760.2.k.o 4 3.b odd 2 1
5760.2.k.o 4 12.b even 2 1
5760.2.k.o 4 24.f even 2 1
5760.2.k.o 4 24.h odd 2 1
6400.2.a.bl 2 80.k odd 4 1
6400.2.a.bl 2 80.q even 4 1
6400.2.a.bn 2 80.k odd 4 1
6400.2.a.bn 2 80.q even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(640, [\chi])\):

\( T_{3}^{2} + 2 \)
\( T_{7}^{2} - 18 \)
\( T_{11}^{2} + 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 2 + T^{2} )^{2} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( ( -18 + T^{2} )^{2} \)
$11$ \( ( 32 + T^{2} )^{2} \)
$13$ \( ( 4 + T^{2} )^{2} \)
$17$ \( ( -6 + T )^{4} \)
$19$ \( ( 8 + T^{2} )^{2} \)
$23$ \( ( -50 + T^{2} )^{2} \)
$29$ \( ( 16 + T^{2} )^{2} \)
$31$ \( ( -8 + T^{2} )^{2} \)
$37$ \( ( 4 + T^{2} )^{2} \)
$41$ \( ( 8 + T )^{4} \)
$43$ \( ( 2 + T^{2} )^{2} \)
$47$ \( ( -2 + T^{2} )^{2} \)
$53$ \( ( 4 + T^{2} )^{2} \)
$59$ \( ( 8 + T^{2} )^{2} \)
$61$ \( ( 196 + T^{2} )^{2} \)
$67$ \( ( 18 + T^{2} )^{2} \)
$71$ \( ( -8 + T^{2} )^{2} \)
$73$ \( ( 6 + T )^{4} \)
$79$ \( ( -288 + T^{2} )^{2} \)
$83$ \( ( 162 + T^{2} )^{2} \)
$89$ \( ( 6 + T )^{4} \)
$97$ \( ( -10 + T )^{4} \)
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