Properties

Label 640.2.c.a
Level $640$
Weight $2$
Character orbit 640.c
Analytic conductor $5.110$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(5.11042572936\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + (\beta_{5} + \beta_{2}) q^{5} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{7} + ( - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + (\beta_{5} + \beta_{2}) q^{5} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{7} + ( - \beta_1 - 1) q^{9} + (\beta_{5} + \beta_{3} + \beta_1) q^{11} + (\beta_{5} - 2 \beta_{4} - \beta_{3}) q^{13} + ( - \beta_{4} - \beta_{2} - \beta_1 - 2) q^{15} - 2 \beta_{2} q^{17} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{19} + ( - \beta_{5} - \beta_{3} - 2 \beta_1 - 2) q^{21} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2}) q^{23} + ( - 2 \beta_{3} - \beta_1 - 1) q^{25} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2}) q^{27} + 2 q^{29} + (2 \beta_{5} + 2 \beta_{3} + 4) q^{31} + (2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} + 2 \beta_{2}) q^{33} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 4) q^{35} + (\beta_{5} - \beta_{3} - 2 \beta_{2}) q^{37} + ( - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{39} + (2 \beta_{5} + 2 \beta_{3} + \beta_1 + 2) q^{41} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 3 \beta_{2}) q^{43} + (2 \beta_{4} + \beta_{3} - 3 \beta_{2} + 2) q^{45} + ( - \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2}) q^{47} + ( - \beta_1 - 1) q^{49} + (2 \beta_1 + 8) q^{51} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3}) q^{53} + ( - \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{55} + (2 \beta_{5} - 2 \beta_{3} + 6 \beta_{2}) q^{57} + (\beta_{5} + \beta_{3} - \beta_1) q^{59} + ( - \beta_{5} - \beta_{3} - 2 \beta_1 - 4) q^{61} + ( - \beta_{5} + 3 \beta_{4} + \beta_{3} - 5 \beta_{2}) q^{63} + ( - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{2} - \beta_1 - 2) q^{65} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 5 \beta_{2}) q^{67} + (\beta_{5} + \beta_{3} + 2) q^{69} + (2 \beta_1 - 4) q^{71} + ( - 2 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} - 6 \beta_{2}) q^{73} + ( - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 4) q^{75} + (2 \beta_{4} + 6 \beta_{2}) q^{77} + (2 \beta_{5} + 2 \beta_{3} - 4) q^{79} + (2 \beta_{5} + 2 \beta_{3} + \beta_1 + 1) q^{81} + (2 \beta_{5} - 6 \beta_{4} - 2 \beta_{3} + \beta_{2}) q^{83} + (2 \beta_{4} + 2 \beta_{2} + 2 \beta_1 + 4) q^{85} + 2 \beta_{2} q^{87} + ( - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_1 + 2) q^{89} + (2 \beta_{5} + 2 \beta_{3} - 8) q^{91} - 4 \beta_{4} q^{93} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 - 6) q^{95} + (4 \beta_{4} - 2 \beta_{2}) q^{97} + ( - \beta_{5} - \beta_{3} - 3 \beta_1 - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{5} - 6 q^{9} - 4 q^{11} - 12 q^{15} + 4 q^{19} - 8 q^{21} - 2 q^{25} + 12 q^{29} + 16 q^{31} - 20 q^{35} + 8 q^{39} + 4 q^{41} + 10 q^{45} - 6 q^{49} + 48 q^{51} + 20 q^{55} - 4 q^{59} - 20 q^{61} - 8 q^{65} + 8 q^{69} - 24 q^{71} + 24 q^{75} - 32 q^{79} - 2 q^{81} + 24 q^{85} + 20 q^{89} - 56 q^{91} - 36 q^{95} - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{5} - 16\nu^{4} + 8\nu^{3} + 2\nu^{2} - 4\nu - 76 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{5} - 9\nu^{4} + 16\nu^{3} + 4\nu^{2} + 38\nu - 14 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\nu^{5} - 12\nu^{4} + 6\nu^{3} + 36\nu^{2} + 112\nu - 11 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 60\nu^{2} - 64\nu + 26 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{5} + 2\nu^{4} - 2\nu^{3} - 2\nu^{2} - 4\nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{3} + 2\beta_{2} + \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 2\beta_{4} - \beta_{3} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{4} - 2\beta_{3} + 4\beta_{2} - 2\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} - \beta_{3} - 5\beta _1 - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{5} + 3\beta_{4} + \beta_{3} - 8\beta_{2} - 4\beta _1 - 9 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
1.45161 1.45161i
−0.854638 0.854638i
0.403032 0.403032i
0.403032 + 0.403032i
−0.854638 + 0.854638i
1.45161 + 1.45161i
0 2.90321i 0 −0.311108 2.21432i 0 3.52543i 0 −5.42864 0
129.2 0 1.70928i 0 −2.17009 0.539189i 0 2.63090i 0 0.0783777 0
129.3 0 0.806063i 0 1.48119 + 1.67513i 0 2.15633i 0 2.35026 0
129.4 0 0.806063i 0 1.48119 1.67513i 0 2.15633i 0 2.35026 0
129.5 0 1.70928i 0 −2.17009 + 0.539189i 0 2.63090i 0 0.0783777 0
129.6 0 2.90321i 0 −0.311108 + 2.21432i 0 3.52543i 0 −5.42864 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b 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.c.a 6
4.b odd 2 1 640.2.c.b yes 6
5.b even 2 1 inner 640.2.c.a 6
5.c odd 4 1 3200.2.a.bq 3
5.c odd 4 1 3200.2.a.bs 3
8.b even 2 1 640.2.c.d yes 6
8.d odd 2 1 640.2.c.c yes 6
16.e even 4 1 1280.2.f.i 6
16.e even 4 1 1280.2.f.l 6
16.f odd 4 1 1280.2.f.j 6
16.f odd 4 1 1280.2.f.k 6
20.d odd 2 1 640.2.c.b yes 6
20.e even 4 1 3200.2.a.br 3
20.e even 4 1 3200.2.a.bt 3
40.e odd 2 1 640.2.c.c yes 6
40.f even 2 1 640.2.c.d yes 6
40.i odd 4 1 3200.2.a.bp 3
40.i odd 4 1 3200.2.a.bv 3
40.k even 4 1 3200.2.a.bo 3
40.k even 4 1 3200.2.a.bu 3
80.k odd 4 1 1280.2.f.j 6
80.k odd 4 1 1280.2.f.k 6
80.q even 4 1 1280.2.f.i 6
80.q even 4 1 1280.2.f.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.c.a 6 1.a even 1 1 trivial
640.2.c.a 6 5.b even 2 1 inner
640.2.c.b yes 6 4.b odd 2 1
640.2.c.b yes 6 20.d odd 2 1
640.2.c.c yes 6 8.d odd 2 1
640.2.c.c yes 6 40.e odd 2 1
640.2.c.d yes 6 8.b even 2 1
640.2.c.d yes 6 40.f even 2 1
1280.2.f.i 6 16.e even 4 1
1280.2.f.i 6 80.q even 4 1
1280.2.f.j 6 16.f odd 4 1
1280.2.f.j 6 80.k odd 4 1
1280.2.f.k 6 16.f odd 4 1
1280.2.f.k 6 80.k odd 4 1
1280.2.f.l 6 16.e even 4 1
1280.2.f.l 6 80.q even 4 1
3200.2.a.bo 3 40.k even 4 1
3200.2.a.bp 3 40.i odd 4 1
3200.2.a.bq 3 5.c odd 4 1
3200.2.a.br 3 20.e even 4 1
3200.2.a.bs 3 5.c odd 4 1
3200.2.a.bt 3 20.e even 4 1
3200.2.a.bu 3 40.k even 4 1
3200.2.a.bv 3 40.i odd 4 1

Hecke kernels

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

\( T_{11}^{3} + 2T_{11}^{2} - 20T_{11} - 8 \) Copy content Toggle raw display
\( T_{29} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 12 T^{4} + 32 T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{6} + 2 T^{5} + 3 T^{4} + 12 T^{3} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 24 T^{4} + 176 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$11$ \( (T^{3} + 2 T^{2} - 20 T - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 48 T^{4} + 320 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{6} + 48 T^{4} + 512 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( (T^{3} - 2 T^{2} - 36 T + 104)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 40 T^{4} + 80 T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T - 2)^{6} \) Copy content Toggle raw display
$31$ \( (T^{3} - 8 T^{2} - 32 T + 128)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 128 T^{4} + 5376 T^{2} + \cdots + 73984 \) Copy content Toggle raw display
$41$ \( (T^{3} - 2 T^{2} - 52 T + 184)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 156 T^{4} + 3200 T^{2} + \cdots + 10000 \) Copy content Toggle raw display
$47$ \( T^{6} + 72 T^{4} + 1712 T^{2} + \cdots + 13456 \) Copy content Toggle raw display
$53$ \( T^{6} + 112 T^{4} + 2624 T^{2} + \cdots + 6400 \) Copy content Toggle raw display
$59$ \( (T^{3} + 2 T^{2} - 36 T - 104)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 10 T^{2} - 28 T - 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 220 T^{4} + 15680 T^{2} + \cdots + 364816 \) Copy content Toggle raw display
$71$ \( (T^{3} + 12 T^{2} - 16 T - 320)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 400 T^{4} + 47360 T^{2} + \cdots + 1401856 \) Copy content Toggle raw display
$79$ \( (T^{3} + 16 T^{2} + 32 T - 128)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 380 T^{4} + 36160 T^{2} + \cdots + 274576 \) Copy content Toggle raw display
$89$ \( (T^{3} - 10 T^{2} - 116 T + 1096)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 304 T^{4} + 23552 T^{2} + \cdots + 369664 \) Copy content Toggle raw display
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