# Properties

 Label 320.3.b.a Level $320$ Weight $3$ Character orbit 320.b Analytic conductor $8.719$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 320.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.71936845953$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 80) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} -\beta_{1} q^{5} + \beta_{3} q^{7} + ( -9 + 6 \beta_{1} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} -\beta_{1} q^{5} + \beta_{3} q^{7} + ( -9 + 6 \beta_{1} ) q^{9} + 2 \beta_{2} q^{11} + ( -4 + 6 \beta_{1} ) q^{13} + ( 2 \beta_{2} + \beta_{3} ) q^{15} + 18 q^{17} + 4 \beta_{3} q^{19} + ( 6 + 6 \beta_{1} ) q^{21} + ( -2 \beta_{2} - 3 \beta_{3} ) q^{23} + 5 q^{25} + ( -12 \beta_{2} - 6 \beta_{3} ) q^{27} + ( 18 + 12 \beta_{1} ) q^{29} + ( 8 \beta_{2} - 2 \beta_{3} ) q^{31} + ( -36 + 12 \beta_{1} ) q^{33} + ( \beta_{2} - 2 \beta_{3} ) q^{35} + ( -20 + 6 \beta_{1} ) q^{37} + ( -16 \beta_{2} - 6 \beta_{3} ) q^{39} + ( 12 + 18 \beta_{1} ) q^{41} + ( -5 \beta_{2} + 2 \beta_{3} ) q^{43} + ( -30 + 9 \beta_{1} ) q^{45} + ( -4 \beta_{2} + 9 \beta_{3} ) q^{47} + ( 7 - 18 \beta_{1} ) q^{49} + 18 \beta_{2} q^{51} + ( -12 - 18 \beta_{1} ) q^{53} + ( 4 \beta_{2} + 2 \beta_{3} ) q^{55} + ( 24 + 24 \beta_{1} ) q^{57} + 8 \beta_{2} q^{59} + ( 64 - 6 \beta_{1} ) q^{61} + ( -6 \beta_{2} + 3 \beta_{3} ) q^{63} + ( -30 + 4 \beta_{1} ) q^{65} + ( -7 \beta_{2} + 2 \beta_{3} ) q^{67} + ( 18 - 30 \beta_{1} ) q^{69} + ( 16 \beta_{2} + 6 \beta_{3} ) q^{71} + ( -38 - 24 \beta_{1} ) q^{73} + 5 \beta_{2} q^{75} + ( 12 + 12 \beta_{1} ) q^{77} + 20 \beta_{2} q^{79} + ( 99 - 54 \beta_{1} ) q^{81} + ( 15 \beta_{2} + 6 \beta_{3} ) q^{83} -18 \beta_{1} q^{85} + ( -6 \beta_{2} - 12 \beta_{3} ) q^{87} + ( -6 + 24 \beta_{1} ) q^{89} + ( -6 \beta_{2} + 8 \beta_{3} ) q^{91} + ( -156 + 36 \beta_{1} ) q^{93} + ( 4 \beta_{2} - 8 \beta_{3} ) q^{95} + ( -26 - 60 \beta_{1} ) q^{97} + ( -42 \beta_{2} - 12 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 36q^{9} + O(q^{10})$$ $$4q - 36q^{9} - 16q^{13} + 72q^{17} + 24q^{21} + 20q^{25} + 72q^{29} - 144q^{33} - 80q^{37} + 48q^{41} - 120q^{45} + 28q^{49} - 48q^{53} + 96q^{57} + 256q^{61} - 120q^{65} + 72q^{69} - 152q^{73} + 48q^{77} + 396q^{81} - 24q^{89} - 624q^{93} - 104q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 2$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} + 4 \nu + 1$$ $$\beta_{3}$$ $$=$$ $$-5 \nu^{3} + 8 \nu^{2} - 12 \nu - 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 3 \beta_{2} + 2 \beta_{1} - 6$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$\beta_{1} - 2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 0.809017 − 1.40126i −0.309017 − 0.535233i −0.309017 + 0.535233i 0.809017 + 1.40126i
0 5.60503i 0 2.23607 0 1.32317i 0 −22.4164 0
191.2 0 2.14093i 0 −2.23607 0 9.06914i 0 4.41641 0
191.3 0 2.14093i 0 −2.23607 0 9.06914i 0 4.41641 0
191.4 0 5.60503i 0 2.23607 0 1.32317i 0 −22.4164 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.3.b.a 4
3.b odd 2 1 2880.3.e.b 4
4.b odd 2 1 inner 320.3.b.a 4
5.b even 2 1 1600.3.b.k 4
5.c odd 4 2 1600.3.h.p 8
8.b even 2 1 80.3.b.a 4
8.d odd 2 1 80.3.b.a 4
12.b even 2 1 2880.3.e.b 4
16.e even 4 2 1280.3.g.f 8
16.f odd 4 2 1280.3.g.f 8
20.d odd 2 1 1600.3.b.k 4
20.e even 4 2 1600.3.h.p 8
24.f even 2 1 720.3.e.c 4
24.h odd 2 1 720.3.e.c 4
40.e odd 2 1 400.3.b.g 4
40.f even 2 1 400.3.b.g 4
40.i odd 4 2 400.3.h.d 8
40.k even 4 2 400.3.h.d 8
120.i odd 2 1 3600.3.e.bb 4
120.m even 2 1 3600.3.e.bb 4
120.q odd 4 2 3600.3.j.k 8
120.w even 4 2 3600.3.j.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.3.b.a 4 8.b even 2 1
80.3.b.a 4 8.d odd 2 1
320.3.b.a 4 1.a even 1 1 trivial
320.3.b.a 4 4.b odd 2 1 inner
400.3.b.g 4 40.e odd 2 1
400.3.b.g 4 40.f even 2 1
400.3.h.d 8 40.i odd 4 2
400.3.h.d 8 40.k even 4 2
720.3.e.c 4 24.f even 2 1
720.3.e.c 4 24.h odd 2 1
1280.3.g.f 8 16.e even 4 2
1280.3.g.f 8 16.f odd 4 2
1600.3.b.k 4 5.b even 2 1
1600.3.b.k 4 20.d odd 2 1
1600.3.h.p 8 5.c odd 4 2
1600.3.h.p 8 20.e even 4 2
2880.3.e.b 4 3.b odd 2 1
2880.3.e.b 4 12.b even 2 1
3600.3.e.bb 4 120.i odd 2 1
3600.3.e.bb 4 120.m even 2 1
3600.3.j.k 8 120.q odd 4 2
3600.3.j.k 8 120.w even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 36 T_{3}^{2} + 144$$ acting on $$S_{3}^{\mathrm{new}}(320, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$144 + 36 T^{2} + T^{4}$$
$5$ $$( -5 + T^{2} )^{2}$$
$7$ $$144 + 84 T^{2} + T^{4}$$
$11$ $$2304 + 144 T^{2} + T^{4}$$
$13$ $$( -164 + 8 T + T^{2} )^{2}$$
$17$ $$( -18 + T )^{4}$$
$19$ $$36864 + 1344 T^{2} + T^{4}$$
$23$ $$121104 + 756 T^{2} + T^{4}$$
$29$ $$( -396 - 36 T + T^{2} )^{2}$$
$31$ $$2214144 + 3024 T^{2} + T^{4}$$
$37$ $$( 220 + 40 T + T^{2} )^{2}$$
$41$ $$( -1476 - 24 T + T^{2} )^{2}$$
$43$ $$535824 + 1476 T^{2} + T^{4}$$
$47$ $$898704 + 8244 T^{2} + T^{4}$$
$53$ $$( -1476 + 24 T + T^{2} )^{2}$$
$59$ $$589824 + 2304 T^{2} + T^{4}$$
$61$ $$( 3916 - 128 T + T^{2} )^{2}$$
$67$ $$1468944 + 2436 T^{2} + T^{4}$$
$71$ $$3873024 + 9936 T^{2} + T^{4}$$
$73$ $$( -1436 + 76 T + T^{2} )^{2}$$
$79$ $$23040000 + 14400 T^{2} + T^{4}$$
$83$ $$4210704 + 8964 T^{2} + T^{4}$$
$89$ $$( -2844 + 12 T + T^{2} )^{2}$$
$97$ $$( -17324 + 52 T + T^{2} )^{2}$$
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