Properties

Label 320.6.c.c
Level 320
Weight 6
Character orbit 320.c
Analytic conductor 51.323
Analytic rank 0
Dimension 2
CM discriminant -4
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -41 - 38 i ) q^{5} + 243 q^{9} +O(q^{10})\) \( q + ( -41 - 38 i ) q^{5} + 243 q^{9} + 244 i q^{13} + 808 i q^{17} + ( 237 + 3116 i ) q^{25} -2950 q^{29} -11292 i q^{37} + 20950 q^{41} + ( -9963 - 9234 i ) q^{45} + 16807 q^{49} -40244 i q^{53} + 18950 q^{61} + ( 9272 - 10004 i ) q^{65} + 20144 i q^{73} + 59049 q^{81} + ( 30704 - 33128 i ) q^{85} + 51050 q^{89} -160808 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 82q^{5} + 486q^{9} + O(q^{10}) \) \( 2q - 82q^{5} + 486q^{9} + 474q^{25} - 5900q^{29} + 41900q^{41} - 19926q^{45} + 33614q^{49} + 37900q^{61} + 18544q^{65} + 118098q^{81} + 61408q^{85} + 102100q^{89} + O(q^{100}) \)

Character values

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

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

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
1.00000i
1.00000i
0 0 0 −41.0000 38.0000i 0 0 0 243.000 0
129.2 0 0 0 −41.0000 + 38.0000i 0 0 0 243.000 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 CM by \(\Q(\sqrt{-1}) \)
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.6.c.c 2
4.b odd 2 1 CM 320.6.c.c 2
5.b even 2 1 inner 320.6.c.c 2
8.b even 2 1 160.6.c.a 2
8.d odd 2 1 160.6.c.a 2
20.d odd 2 1 inner 320.6.c.c 2
40.e odd 2 1 160.6.c.a 2
40.f even 2 1 160.6.c.a 2
40.i odd 4 1 800.6.a.b 1
40.i odd 4 1 800.6.a.c 1
40.k even 4 1 800.6.a.b 1
40.k even 4 1 800.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.c.a 2 8.b even 2 1
160.6.c.a 2 8.d odd 2 1
160.6.c.a 2 40.e odd 2 1
160.6.c.a 2 40.f even 2 1
320.6.c.c 2 1.a even 1 1 trivial
320.6.c.c 2 4.b odd 2 1 CM
320.6.c.c 2 5.b even 2 1 inner
320.6.c.c 2 20.d odd 2 1 inner
800.6.a.b 1 40.i odd 4 1
800.6.a.b 1 40.k even 4 1
800.6.a.c 1 40.i odd 4 1
800.6.a.c 1 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(320, [\chi])\):

\( T_{3} \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 243 T^{2} )^{2} \)
$5$ \( 1 + 82 T + 3125 T^{2} \)
$7$ \( ( 1 - 16807 T^{2} )^{2} \)
$11$ \( ( 1 + 161051 T^{2} )^{2} \)
$13$ \( ( 1 - 1194 T + 371293 T^{2} )( 1 + 1194 T + 371293 T^{2} ) \)
$17$ \( ( 1 - 2242 T + 1419857 T^{2} )( 1 + 2242 T + 1419857 T^{2} ) \)
$19$ \( ( 1 + 2476099 T^{2} )^{2} \)
$23$ \( ( 1 - 6436343 T^{2} )^{2} \)
$29$ \( ( 1 + 2950 T + 20511149 T^{2} )^{2} \)
$31$ \( ( 1 + 28629151 T^{2} )^{2} \)
$37$ \( ( 1 - 12242 T + 69343957 T^{2} )( 1 + 12242 T + 69343957 T^{2} ) \)
$41$ \( ( 1 - 20950 T + 115856201 T^{2} )^{2} \)
$43$ \( ( 1 - 147008443 T^{2} )^{2} \)
$47$ \( ( 1 - 229345007 T^{2} )^{2} \)
$53$ \( ( 1 - 7294 T + 418195493 T^{2} )( 1 + 7294 T + 418195493 T^{2} ) \)
$59$ \( ( 1 + 714924299 T^{2} )^{2} \)
$61$ \( ( 1 - 18950 T + 844596301 T^{2} )^{2} \)
$67$ \( ( 1 - 1350125107 T^{2} )^{2} \)
$71$ \( ( 1 + 1804229351 T^{2} )^{2} \)
$73$ \( ( 1 - 88806 T + 2073071593 T^{2} )( 1 + 88806 T + 2073071593 T^{2} ) \)
$79$ \( ( 1 + 3077056399 T^{2} )^{2} \)
$83$ \( ( 1 - 3939040643 T^{2} )^{2} \)
$89$ \( ( 1 - 51050 T + 5584059449 T^{2} )^{2} \)
$97$ \( ( 1 - 92142 T + 8587340257 T^{2} )( 1 + 92142 T + 8587340257 T^{2} ) \)
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