Properties

Label 320.4.c.c
Level $320$
Weight $4$
Character orbit 320.c
Analytic conductor $18.881$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 320.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.8806112018\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 i q^{3} + ( 5 - 10 i ) q^{5} + 26 i q^{7} + 23 q^{9} +O(q^{10})\) \( q + 2 i q^{3} + ( 5 - 10 i ) q^{5} + 26 i q^{7} + 23 q^{9} -28 q^{11} -12 i q^{13} + ( 20 + 10 i ) q^{15} + 64 i q^{17} + 60 q^{19} -52 q^{21} + 58 i q^{23} + ( -75 - 100 i ) q^{25} + 100 i q^{27} + 90 q^{29} + 128 q^{31} -56 i q^{33} + ( 260 + 130 i ) q^{35} + 236 i q^{37} + 24 q^{39} + 242 q^{41} + 362 i q^{43} + ( 115 - 230 i ) q^{45} + 226 i q^{47} -333 q^{49} -128 q^{51} + 108 i q^{53} + ( -140 + 280 i ) q^{55} + 120 i q^{57} + 20 q^{59} -542 q^{61} + 598 i q^{63} + ( -120 - 60 i ) q^{65} + 434 i q^{67} -116 q^{69} + 1128 q^{71} + 632 i q^{73} + ( 200 - 150 i ) q^{75} -728 i q^{77} -720 q^{79} + 421 q^{81} -478 i q^{83} + ( 640 + 320 i ) q^{85} + 180 i q^{87} + 490 q^{89} + 312 q^{91} + 256 i q^{93} + ( 300 - 600 i ) q^{95} -1456 i q^{97} -644 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{5} + 46 q^{9} + O(q^{10}) \) \( 2 q + 10 q^{5} + 46 q^{9} - 56 q^{11} + 40 q^{15} + 120 q^{19} - 104 q^{21} - 150 q^{25} + 180 q^{29} + 256 q^{31} + 520 q^{35} + 48 q^{39} + 484 q^{41} + 230 q^{45} - 666 q^{49} - 256 q^{51} - 280 q^{55} + 40 q^{59} - 1084 q^{61} - 240 q^{65} - 232 q^{69} + 2256 q^{71} + 400 q^{75} - 1440 q^{79} + 842 q^{81} + 1280 q^{85} + 980 q^{89} + 624 q^{91} + 600 q^{95} - 1288 q^{99} + O(q^{100}) \)

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
1.00000i
1.00000i
0 2.00000i 0 5.00000 + 10.0000i 0 26.0000i 0 23.0000 0
129.2 0 2.00000i 0 5.00000 10.0000i 0 26.0000i 0 23.0000 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 320.4.c.c 2
4.b odd 2 1 320.4.c.d 2
5.b even 2 1 inner 320.4.c.c 2
5.c odd 4 1 1600.4.a.t 1
5.c odd 4 1 1600.4.a.bg 1
8.b even 2 1 80.4.c.a 2
8.d odd 2 1 10.4.b.a 2
20.d odd 2 1 320.4.c.d 2
20.e even 4 1 1600.4.a.u 1
20.e even 4 1 1600.4.a.bh 1
24.f even 2 1 90.4.c.b 2
24.h odd 2 1 720.4.f.f 2
40.e odd 2 1 10.4.b.a 2
40.f even 2 1 80.4.c.a 2
40.i odd 4 1 400.4.a.h 1
40.i odd 4 1 400.4.a.n 1
40.k even 4 1 50.4.a.b 1
40.k even 4 1 50.4.a.d 1
56.e even 2 1 490.4.c.b 2
120.i odd 2 1 720.4.f.f 2
120.m even 2 1 90.4.c.b 2
120.q odd 4 1 450.4.a.j 1
120.q odd 4 1 450.4.a.k 1
280.n even 2 1 490.4.c.b 2
280.y odd 4 1 2450.4.a.o 1
280.y odd 4 1 2450.4.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.b.a 2 8.d odd 2 1
10.4.b.a 2 40.e odd 2 1
50.4.a.b 1 40.k even 4 1
50.4.a.d 1 40.k even 4 1
80.4.c.a 2 8.b even 2 1
80.4.c.a 2 40.f even 2 1
90.4.c.b 2 24.f even 2 1
90.4.c.b 2 120.m even 2 1
320.4.c.c 2 1.a even 1 1 trivial
320.4.c.c 2 5.b even 2 1 inner
320.4.c.d 2 4.b odd 2 1
320.4.c.d 2 20.d odd 2 1
400.4.a.h 1 40.i odd 4 1
400.4.a.n 1 40.i odd 4 1
450.4.a.j 1 120.q odd 4 1
450.4.a.k 1 120.q odd 4 1
490.4.c.b 2 56.e even 2 1
490.4.c.b 2 280.n even 2 1
720.4.f.f 2 24.h odd 2 1
720.4.f.f 2 120.i odd 2 1
1600.4.a.t 1 5.c odd 4 1
1600.4.a.u 1 20.e even 4 1
1600.4.a.bg 1 5.c odd 4 1
1600.4.a.bh 1 20.e even 4 1
2450.4.a.o 1 280.y odd 4 1
2450.4.a.bb 1 280.y odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{2} + 4 \)
\( T_{11} + 28 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( 125 - 10 T + T^{2} \)
$7$ \( 676 + T^{2} \)
$11$ \( ( 28 + T )^{2} \)
$13$ \( 144 + T^{2} \)
$17$ \( 4096 + T^{2} \)
$19$ \( ( -60 + T )^{2} \)
$23$ \( 3364 + T^{2} \)
$29$ \( ( -90 + T )^{2} \)
$31$ \( ( -128 + T )^{2} \)
$37$ \( 55696 + T^{2} \)
$41$ \( ( -242 + T )^{2} \)
$43$ \( 131044 + T^{2} \)
$47$ \( 51076 + T^{2} \)
$53$ \( 11664 + T^{2} \)
$59$ \( ( -20 + T )^{2} \)
$61$ \( ( 542 + T )^{2} \)
$67$ \( 188356 + T^{2} \)
$71$ \( ( -1128 + T )^{2} \)
$73$ \( 399424 + T^{2} \)
$79$ \( ( 720 + T )^{2} \)
$83$ \( 228484 + T^{2} \)
$89$ \( ( -490 + T )^{2} \)
$97$ \( 2119936 + T^{2} \)
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