Properties

Label 640.2.f.d
Level $640$
Weight $2$
Character orbit 640.f
Analytic conductor $5.110$
Analytic rank $0$
Dimension $4$
CM discriminant -40
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(5.11042572936\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{5})\)
Defining polynomial: \(x^{4} + 6 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} -\beta_{1} q^{7} -3 q^{9} +O(q^{10})\) \( q + \beta_{3} q^{5} -\beta_{1} q^{7} -3 q^{9} -\beta_{2} q^{11} -2 \beta_{3} q^{13} -\beta_{2} q^{19} + 3 \beta_{1} q^{23} + 5 q^{25} -\beta_{2} q^{35} + 2 \beta_{3} q^{37} -2 q^{41} -3 \beta_{3} q^{45} -\beta_{1} q^{47} - q^{49} + 6 \beta_{3} q^{53} -5 \beta_{1} q^{55} -\beta_{2} q^{59} + 3 \beta_{1} q^{63} -10 q^{65} -8 \beta_{3} q^{77} + 9 q^{81} + 14 q^{89} + 2 \beta_{2} q^{91} -5 \beta_{1} q^{95} + 3 \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{9} + O(q^{10}) \) \( 4q - 12q^{9} + 20q^{25} - 8q^{41} - 4q^{49} - 40q^{65} + 36q^{81} + 56q^{89} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 4 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} + 8 \nu \)
\(\beta_{3}\)\(=\)\( \nu^{2} + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 3\)
\(\nu^{3}\)\(=\)\(-\beta_{2} + 2 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
2.28825i
2.28825i
0.874032i
0.874032i
0 0 0 −2.23607 0 2.82843i 0 −3.00000 0
449.2 0 0 0 −2.23607 0 2.82843i 0 −3.00000 0
449.3 0 0 0 2.23607 0 2.82843i 0 −3.00000 0
449.4 0 0 0 2.23607 0 2.82843i 0 −3.00000 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
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 640.2.f.d 4
4.b odd 2 1 inner 640.2.f.d 4
5.b even 2 1 inner 640.2.f.d 4
5.c odd 4 2 3200.2.d.w 4
8.b even 2 1 inner 640.2.f.d 4
8.d odd 2 1 inner 640.2.f.d 4
16.e even 4 2 1280.2.c.l 4
16.f odd 4 2 1280.2.c.l 4
20.d odd 2 1 inner 640.2.f.d 4
20.e even 4 2 3200.2.d.w 4
40.e odd 2 1 CM 640.2.f.d 4
40.f even 2 1 inner 640.2.f.d 4
40.i odd 4 2 3200.2.d.w 4
40.k even 4 2 3200.2.d.w 4
80.i odd 4 2 6400.2.a.cl 4
80.j even 4 2 6400.2.a.cl 4
80.k odd 4 2 1280.2.c.l 4
80.q even 4 2 1280.2.c.l 4
80.s even 4 2 6400.2.a.cl 4
80.t odd 4 2 6400.2.a.cl 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.f.d 4 1.a even 1 1 trivial
640.2.f.d 4 4.b odd 2 1 inner
640.2.f.d 4 5.b even 2 1 inner
640.2.f.d 4 8.b even 2 1 inner
640.2.f.d 4 8.d odd 2 1 inner
640.2.f.d 4 20.d odd 2 1 inner
640.2.f.d 4 40.e odd 2 1 CM
640.2.f.d 4 40.f even 2 1 inner
1280.2.c.l 4 16.e even 4 2
1280.2.c.l 4 16.f odd 4 2
1280.2.c.l 4 80.k odd 4 2
1280.2.c.l 4 80.q even 4 2
3200.2.d.w 4 5.c odd 4 2
3200.2.d.w 4 20.e even 4 2
3200.2.d.w 4 40.i odd 4 2
3200.2.d.w 4 40.k even 4 2
6400.2.a.cl 4 80.i odd 4 2
6400.2.a.cl 4 80.j even 4 2
6400.2.a.cl 4 80.s even 4 2
6400.2.a.cl 4 80.t odd 4 2

Hecke kernels

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

\( T_{3} \)
\( T_{7}^{2} + 8 \)
\( T_{13}^{2} - 20 \)
\( T_{37}^{2} - 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -5 + T^{2} )^{2} \)
$7$ \( ( 8 + T^{2} )^{2} \)
$11$ \( ( 40 + T^{2} )^{2} \)
$13$ \( ( -20 + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( ( 40 + T^{2} )^{2} \)
$23$ \( ( 72 + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( ( -20 + T^{2} )^{2} \)
$41$ \( ( 2 + T )^{4} \)
$43$ \( T^{4} \)
$47$ \( ( 8 + T^{2} )^{2} \)
$53$ \( ( -180 + T^{2} )^{2} \)
$59$ \( ( 40 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( ( -14 + T )^{4} \)
$97$ \( T^{4} \)
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