Properties

Label 320.6.c.g
Level 320
Weight 6
Character orbit 320.c
Analytic conductor 51.323
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(\Q(\sqrt{-11}) \)
Defining polynomial: \(x^{2} - x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -3 \beta q^{3} + ( 45 - 5 \beta ) q^{5} + 9 \beta q^{7} -153 q^{9} +O(q^{10})\) \( q -3 \beta q^{3} + ( 45 - 5 \beta ) q^{5} + 9 \beta q^{7} -153 q^{9} + 252 q^{11} + 18 \beta q^{13} + ( -660 - 135 \beta ) q^{15} -104 \beta q^{17} -220 q^{19} + 1188 q^{21} -367 \beta q^{23} + ( 925 - 450 \beta ) q^{25} -270 \beta q^{27} + 6930 q^{29} -6752 q^{31} -756 \beta q^{33} + ( 1980 + 405 \beta ) q^{35} -2106 \beta q^{37} + 2376 q^{39} -198 q^{41} -63 \beta q^{43} + ( -6885 + 765 \beta ) q^{45} + 1589 \beta q^{47} + 13243 q^{49} -13728 q^{51} + 878 \beta q^{53} + ( 11340 - 1260 \beta ) q^{55} + 660 \beta q^{57} -24660 q^{59} + 5698 q^{61} -1377 \beta q^{63} + ( 3960 + 810 \beta ) q^{65} -6579 \beta q^{67} -48444 q^{69} -53352 q^{71} + 10692 \beta q^{73} + ( -59400 - 2775 \beta ) q^{75} + 2268 \beta q^{77} -51920 q^{79} -72819 q^{81} -9323 \beta q^{83} + ( -22880 - 4680 \beta ) q^{85} -20790 \beta q^{87} -9990 q^{89} -7128 q^{91} + 20256 \beta q^{93} + ( -9900 + 1100 \beta ) q^{95} -15264 \beta q^{97} -38556 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 90q^{5} - 306q^{9} + O(q^{10}) \) \( 2q + 90q^{5} - 306q^{9} + 504q^{11} - 1320q^{15} - 440q^{19} + 2376q^{21} + 1850q^{25} + 13860q^{29} - 13504q^{31} + 3960q^{35} + 4752q^{39} - 396q^{41} - 13770q^{45} + 26486q^{49} - 27456q^{51} + 22680q^{55} - 49320q^{59} + 11396q^{61} + 7920q^{65} - 96888q^{69} - 106704q^{71} - 118800q^{75} - 103840q^{79} - 145638q^{81} - 45760q^{85} - 19980q^{89} - 14256q^{91} - 19800q^{95} - 77112q^{99} + O(q^{100}) \)

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
0.500000 + 1.65831i
0.500000 1.65831i
0 19.8997i 0 45.0000 33.1662i 0 59.6992i 0 −153.000 0
129.2 0 19.8997i 0 45.0000 + 33.1662i 0 59.6992i 0 −153.000 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
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 320.6.c.g 2
4.b odd 2 1 320.6.c.f 2
5.b even 2 1 inner 320.6.c.g 2
8.b even 2 1 80.6.c.a 2
8.d odd 2 1 5.6.b.a 2
20.d odd 2 1 320.6.c.f 2
24.f even 2 1 45.6.b.b 2
24.h odd 2 1 720.6.f.f 2
40.e odd 2 1 5.6.b.a 2
40.f even 2 1 80.6.c.a 2
40.i odd 4 2 400.6.a.t 2
40.k even 4 2 25.6.a.c 2
56.e even 2 1 245.6.b.a 2
120.i odd 2 1 720.6.f.f 2
120.m even 2 1 45.6.b.b 2
120.q odd 4 2 225.6.a.n 2
280.n even 2 1 245.6.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.b.a 2 8.d odd 2 1
5.6.b.a 2 40.e odd 2 1
25.6.a.c 2 40.k even 4 2
45.6.b.b 2 24.f even 2 1
45.6.b.b 2 120.m even 2 1
80.6.c.a 2 8.b even 2 1
80.6.c.a 2 40.f even 2 1
225.6.a.n 2 120.q odd 4 2
245.6.b.a 2 56.e even 2 1
245.6.b.a 2 280.n even 2 1
320.6.c.f 2 4.b odd 2 1
320.6.c.f 2 20.d odd 2 1
320.6.c.g 2 1.a even 1 1 trivial
320.6.c.g 2 5.b even 2 1 inner
400.6.a.t 2 40.i odd 4 2
720.6.f.f 2 24.h odd 2 1
720.6.f.f 2 120.i odd 2 1

Hecke kernels

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

\( T_{3}^{2} + 396 \)
\( T_{11} - 252 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 24 T + 243 T^{2} )( 1 + 24 T + 243 T^{2} ) \)
$5$ \( 1 - 90 T + 3125 T^{2} \)
$7$ \( 1 - 30050 T^{2} + 282475249 T^{4} \)
$11$ \( ( 1 - 252 T + 161051 T^{2} )^{2} \)
$13$ \( 1 - 728330 T^{2} + 137858491849 T^{4} \)
$17$ \( 1 - 2363810 T^{2} + 2015993900449 T^{4} \)
$19$ \( ( 1 + 220 T + 2476099 T^{2} )^{2} \)
$23$ \( 1 - 6946370 T^{2} + 41426511213649 T^{4} \)
$29$ \( ( 1 - 6930 T + 20511149 T^{2} )^{2} \)
$31$ \( ( 1 + 6752 T + 28629151 T^{2} )^{2} \)
$37$ \( 1 + 56462470 T^{2} + 4808584372417849 T^{4} \)
$41$ \( ( 1 + 198 T + 115856201 T^{2} )^{2} \)
$43$ \( 1 - 293842250 T^{2} + 21611482313284249 T^{4} \)
$47$ \( 1 - 347593490 T^{2} + 52599132235830049 T^{4} \)
$53$ \( 1 - 802472090 T^{2} + 174887470365513049 T^{4} \)
$59$ \( ( 1 + 24660 T + 714924299 T^{2} )^{2} \)
$61$ \( ( 1 - 5698 T + 844596301 T^{2} )^{2} \)
$67$ \( 1 - 795787610 T^{2} + 1822837804551761449 T^{4} \)
$71$ \( ( 1 + 53352 T + 1804229351 T^{2} )^{2} \)
$73$ \( 1 + 883886830 T^{2} + 4297625829703557649 T^{4} \)
$79$ \( ( 1 + 51920 T + 3077056399 T^{2} )^{2} \)
$83$ \( 1 - 4053674810 T^{2} + 15516041187205853449 T^{4} \)
$89$ \( ( 1 + 9990 T + 5584059449 T^{2} )^{2} \)
$97$ \( 1 - 6923133890 T^{2} + 73742412689492826049 T^{4} \)
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