# Properties

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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.d 2
4.b odd 2 1 320.4.c.c 2
5.b even 2 1 inner 320.4.c.d 2
5.c odd 4 1 1600.4.a.u 1
5.c odd 4 1 1600.4.a.bh 1
8.b even 2 1 10.4.b.a 2
8.d odd 2 1 80.4.c.a 2
20.d odd 2 1 320.4.c.c 2
20.e even 4 1 1600.4.a.t 1
20.e even 4 1 1600.4.a.bg 1
24.f even 2 1 720.4.f.f 2
24.h odd 2 1 90.4.c.b 2
40.e odd 2 1 80.4.c.a 2
40.f even 2 1 10.4.b.a 2
40.i odd 4 1 50.4.a.b 1
40.i odd 4 1 50.4.a.d 1
40.k even 4 1 400.4.a.h 1
40.k even 4 1 400.4.a.n 1
56.h odd 2 1 490.4.c.b 2
120.i odd 2 1 90.4.c.b 2
120.m even 2 1 720.4.f.f 2
120.w even 4 1 450.4.a.j 1
120.w even 4 1 450.4.a.k 1
280.c odd 2 1 490.4.c.b 2
280.s even 4 1 2450.4.a.o 1
280.s even 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.b even 2 1
10.4.b.a 2 40.f even 2 1
50.4.a.b 1 40.i odd 4 1
50.4.a.d 1 40.i odd 4 1
80.4.c.a 2 8.d odd 2 1
80.4.c.a 2 40.e odd 2 1
90.4.c.b 2 24.h odd 2 1
90.4.c.b 2 120.i odd 2 1
320.4.c.c 2 4.b odd 2 1
320.4.c.c 2 20.d odd 2 1
320.4.c.d 2 1.a even 1 1 trivial
320.4.c.d 2 5.b even 2 1 inner
400.4.a.h 1 40.k even 4 1
400.4.a.n 1 40.k even 4 1
450.4.a.j 1 120.w even 4 1
450.4.a.k 1 120.w even 4 1
490.4.c.b 2 56.h odd 2 1
490.4.c.b 2 280.c odd 2 1
720.4.f.f 2 24.f even 2 1
720.4.f.f 2 120.m even 2 1
1600.4.a.t 1 20.e even 4 1
1600.4.a.u 1 5.c odd 4 1
1600.4.a.bg 1 20.e even 4 1
1600.4.a.bh 1 5.c odd 4 1
2450.4.a.o 1 280.s even 4 1
2450.4.a.bb 1 280.s even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 4$$ $$T_{11} - 28$$

## Hecke characteristic polynomials

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