# Properties

 Label 320.4.c.j Level $320$ Weight $4$ Character orbit 320.c Analytic conductor $18.881$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# 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: $$8$$ Coefficient field: 8.0.359712057600.22 Defining polynomial: $$x^{8} + 17x^{6} + 82x^{4} + 96x^{2} + 9$$ x^8 + 17*x^6 + 82*x^4 + 96*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{22}$$ Twist minimal: no (minimal twist has level 160) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + (\beta_{4} - 4) q^{5} + (\beta_{3} - \beta_1) q^{7} + ( - 3 \beta_{5} - 19) q^{9}+O(q^{10})$$ q + b1 * q^3 + (b4 - 4) * q^5 + (b3 - b1) * q^7 + (-3*b5 - 19) * q^9 $$q + \beta_1 q^{3} + (\beta_{4} - 4) q^{5} + (\beta_{3} - \beta_1) q^{7} + ( - 3 \beta_{5} - 19) q^{9} - \beta_{6} q^{11} + (\beta_{5} - 2 \beta_{4} - 3 \beta_{2}) q^{13} + ( - \beta_{7} - \beta_{6} - \beta_{3} - \beta_1) q^{15} + (4 \beta_{5} - 8 \beta_{4} + \beta_{2}) q^{17} + ( - 2 \beta_{7} + \beta_{6}) q^{19} + (17 \beta_{5} + 34) q^{21} + ( - \beta_{3} + 11 \beta_1) q^{23} + ( - 5 \beta_{5} - 6 \beta_{4} + 5 \beta_{2} - 51) q^{25} + (6 \beta_{3} - 10 \beta_1) q^{27} + (16 \beta_{5} - 78) q^{29} - 2 \beta_{7} q^{31} + (14 \beta_{5} - 28 \beta_{4} + 5 \beta_{2}) q^{33} + (4 \beta_{7} - \beta_{6} - 6 \beta_{3} - 11 \beta_1) q^{35} + (7 \beta_{5} - 14 \beta_{4} + 6 \beta_{2}) q^{37} + (8 \beta_{7} + 2 \beta_{6}) q^{39} + (13 \beta_{5} - 24) q^{41} + ( - 6 \beta_{3} + 49 \beta_1) q^{43} + (15 \beta_{5} - 25 \beta_{4} - 15 \beta_{2} - 50) q^{45} + (9 \beta_{3} - 37 \beta_1) q^{47} + (17 \beta_{5} - 303) q^{49} + (6 \beta_{7} + 8 \beta_{6}) q^{51} + (15 \beta_{5} - 30 \beta_{4} + 3 \beta_{2}) q^{53} + ( - 3 \beta_{7} + 7 \beta_{6} - 8 \beta_{3} - 48 \beta_1) q^{55} + (10 \beta_{5} - 20 \beta_{4} - 45 \beta_{2}) q^{57} + ( - 6 \beta_{7} - 5 \beta_{6}) q^{59} + ( - 27 \beta_{5} - 264) q^{61} + ( - 7 \beta_{3} + 109 \beta_1) q^{63} + (65 \beta_{5} - 18 \beta_{4} + 10 \beta_{2} + 272) q^{65} + ( - 10 \beta_{3} - 51 \beta_1) q^{67} + ( - 47 \beta_{5} - 494) q^{69} + (2 \beta_{7} - 2 \beta_{6}) q^{71} + ( - 22 \beta_{5} + 44 \beta_{4} + 17 \beta_{2}) q^{73} + ( - 4 \beta_{7} + 6 \beta_{6} + 16 \beta_{3} - 99 \beta_1) q^{75} + (34 \beta_{5} - 68 \beta_{4} + 85 \beta_{2}) q^{77} + ( - 10 \beta_{7} - 12 \beta_{6}) q^{79} + (33 \beta_{5} - 125) q^{81} + (22 \beta_{3} + 117 \beta_1) q^{83} + (32 \beta_{4} - 25 \beta_{2} + 672) q^{85} + ( - 32 \beta_{3} + 18 \beta_1) q^{87} + (80 \beta_{5} + 10) q^{89} + ( - 14 \beta_{7} - 22 \beta_{6}) q^{91} + (24 \beta_{5} - 48 \beta_{4} - 40 \beta_{2}) q^{93} + (5 \beta_{7} - 15 \beta_{6} + 40 \beta_{3} - 80 \beta_1) q^{95} - 71 \beta_{2} q^{97} + (18 \beta_{7} + \beta_{6}) q^{99}+O(q^{100})$$ q + b1 * q^3 + (b4 - 4) * q^5 + (b3 - b1) * q^7 + (-3*b5 - 19) * q^9 - b6 * q^11 + (b5 - 2*b4 - 3*b2) * q^13 + (-b7 - b6 - b3 - b1) * q^15 + (4*b5 - 8*b4 + b2) * q^17 + (-2*b7 + b6) * q^19 + (17*b5 + 34) * q^21 + (-b3 + 11*b1) * q^23 + (-5*b5 - 6*b4 + 5*b2 - 51) * q^25 + (6*b3 - 10*b1) * q^27 + (16*b5 - 78) * q^29 - 2*b7 * q^31 + (14*b5 - 28*b4 + 5*b2) * q^33 + (4*b7 - b6 - 6*b3 - 11*b1) * q^35 + (7*b5 - 14*b4 + 6*b2) * q^37 + (8*b7 + 2*b6) * q^39 + (13*b5 - 24) * q^41 + (-6*b3 + 49*b1) * q^43 + (15*b5 - 25*b4 - 15*b2 - 50) * q^45 + (9*b3 - 37*b1) * q^47 + (17*b5 - 303) * q^49 + (6*b7 + 8*b6) * q^51 + (15*b5 - 30*b4 + 3*b2) * q^53 + (-3*b7 + 7*b6 - 8*b3 - 48*b1) * q^55 + (10*b5 - 20*b4 - 45*b2) * q^57 + (-6*b7 - 5*b6) * q^59 + (-27*b5 - 264) * q^61 + (-7*b3 + 109*b1) * q^63 + (65*b5 - 18*b4 + 10*b2 + 272) * q^65 + (-10*b3 - 51*b1) * q^67 + (-47*b5 - 494) * q^69 + (2*b7 - 2*b6) * q^71 + (-22*b5 + 44*b4 + 17*b2) * q^73 + (-4*b7 + 6*b6 + 16*b3 - 99*b1) * q^75 + (34*b5 - 68*b4 + 85*b2) * q^77 + (-10*b7 - 12*b6) * q^79 + (33*b5 - 125) * q^81 + (22*b3 + 117*b1) * q^83 + (32*b4 - 25*b2 + 672) * q^85 + (-32*b3 + 18*b1) * q^87 + (80*b5 + 10) * q^89 + (-14*b7 - 22*b6) * q^91 + (24*b5 - 48*b4 - 40*b2) * q^93 + (5*b7 - 15*b6 + 40*b3 - 80*b1) * q^95 - 71*b2 * q^97 + (18*b7 + b6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 32 q^{5} - 152 q^{9}+O(q^{10})$$ 8 * q - 32 * q^5 - 152 * q^9 $$8 q - 32 q^{5} - 152 q^{9} + 272 q^{21} - 408 q^{25} - 624 q^{29} - 192 q^{41} - 400 q^{45} - 2424 q^{49} - 2112 q^{61} + 2176 q^{65} - 3952 q^{69} - 1000 q^{81} + 5376 q^{85} + 80 q^{89}+O(q^{100})$$ 8 * q - 32 * q^5 - 152 * q^9 + 272 * q^21 - 408 * q^25 - 624 * q^29 - 192 * q^41 - 400 * q^45 - 2424 * q^49 - 2112 * q^61 + 2176 * q^65 - 3952 * q^69 - 1000 * q^81 + 5376 * q^85 + 80 * q^89

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 17x^{6} + 82x^{4} + 96x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( 8\nu^{7} + 130\nu^{5} + 572\nu^{3} + 420\nu ) / 27$$ (8*v^7 + 130*v^5 + 572*v^3 + 420*v) / 27 $$\beta_{2}$$ $$=$$ $$( 64\nu^{7} + 1040\nu^{5} + 5008\nu^{3} + 7248\nu ) / 135$$ (64*v^7 + 1040*v^5 + 5008*v^3 + 7248*v) / 135 $$\beta_{3}$$ $$=$$ $$( 8\nu^{7} + 148\nu^{5} + 788\nu^{3} + 996\nu ) / 9$$ (8*v^7 + 148*v^5 + 788*v^3 + 996*v) / 9 $$\beta_{4}$$ $$=$$ $$( -64\nu^{7} - 30\nu^{6} - 1040\nu^{5} - 420\nu^{4} - 4468\nu^{3} - 1470\nu^{2} - 3468\nu - 765 ) / 135$$ (-64*v^7 - 30*v^6 - 1040*v^5 - 420*v^4 - 4468*v^3 - 1470*v^2 - 3468*v - 765) / 135 $$\beta_{5}$$ $$=$$ $$( -4\nu^{6} - 56\nu^{4} - 196\nu^{2} - 102 ) / 9$$ (-4*v^6 - 56*v^4 - 196*v^2 - 102) / 9 $$\beta_{6}$$ $$=$$ $$( 16\nu^{6} + 80\nu^{4} - 1088\nu^{2} - 3048 ) / 45$$ (16*v^6 + 80*v^4 - 1088*v^2 - 3048) / 45 $$\beta_{7}$$ $$=$$ $$( 16\nu^{6} + 320\nu^{4} + 1552\nu^{2} + 672 ) / 15$$ (16*v^6 + 320*v^4 + 1552*v^2 + 672) / 15
 $$\nu$$ $$=$$ $$( 2\beta_{5} - 4\beta_{4} + \beta_{2} - 8\beta_1 ) / 32$$ (2*b5 - 4*b4 + b2 - 8*b1) / 32 $$\nu^{2}$$ $$=$$ $$( -\beta_{7} - 2\beta_{6} - 4\beta_{5} - 136 ) / 32$$ (-b7 - 2*b6 - 4*b5 - 136) / 32 $$\nu^{3}$$ $$=$$ $$( -18\beta_{5} + 36\beta_{4} + \beta_{2} + 56\beta_1 ) / 32$$ (-18*b5 + 36*b4 + b2 + 56*b1) / 32 $$\nu^{4}$$ $$=$$ $$( 13\beta_{7} + 16\beta_{6} + 44\beta_{5} + 1000 ) / 32$$ (13*b7 + 16*b6 + 44*b5 + 1000) / 32 $$\nu^{5}$$ $$=$$ $$( 38\beta_{5} - 76\beta_{4} + 4\beta_{3} - 11\beta_{2} - 116\beta_1 ) / 8$$ (38*b5 - 76*b4 + 4*b3 - 11*b2 - 116*b1) / 8 $$\nu^{6}$$ $$=$$ $$( -133\beta_{7} - 126\beta_{6} - 492\beta_{5} - 8152 ) / 32$$ (-133*b7 - 126*b6 - 492*b5 - 8152) / 32 $$\nu^{7}$$ $$=$$ $$( -1288\beta_{5} + 2576\beta_{4} - 260\beta_{3} + 591\beta_{2} + 4064\beta_1 ) / 32$$ (-1288*b5 + 2576*b4 - 260*b3 + 591*b2 + 4064*b1) / 32

## 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
 3.03888i 1.24759i 2.47107i − 0.320221i 0.320221i − 2.47107i − 1.24759i − 3.03888i
0 8.57295i 0 0.582576 11.1652i 0 22.1403i 0 −46.4955 0
129.2 0 8.57295i 0 0.582576 + 11.1652i 0 22.1403i 0 −46.4955 0
129.3 0 4.30169i 0 −8.58258 7.16515i 0 28.3162i 0 8.49545 0
129.4 0 4.30169i 0 −8.58258 + 7.16515i 0 28.3162i 0 8.49545 0
129.5 0 4.30169i 0 −8.58258 7.16515i 0 28.3162i 0 8.49545 0
129.6 0 4.30169i 0 −8.58258 + 7.16515i 0 28.3162i 0 8.49545 0
129.7 0 8.57295i 0 0.582576 11.1652i 0 22.1403i 0 −46.4955 0
129.8 0 8.57295i 0 0.582576 + 11.1652i 0 22.1403i 0 −46.4955 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 129.8 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
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 320.4.c.j 8
4.b odd 2 1 inner 320.4.c.j 8
5.b even 2 1 inner 320.4.c.j 8
5.c odd 4 1 1600.4.a.cu 4
5.c odd 4 1 1600.4.a.cv 4
8.b even 2 1 160.4.c.d 8
8.d odd 2 1 160.4.c.d 8
20.d odd 2 1 inner 320.4.c.j 8
20.e even 4 1 1600.4.a.cu 4
20.e even 4 1 1600.4.a.cv 4
24.f even 2 1 1440.4.f.k 8
24.h odd 2 1 1440.4.f.k 8
40.e odd 2 1 160.4.c.d 8
40.f even 2 1 160.4.c.d 8
40.i odd 4 1 800.4.a.y 4
40.i odd 4 1 800.4.a.z 4
40.k even 4 1 800.4.a.y 4
40.k even 4 1 800.4.a.z 4
120.i odd 2 1 1440.4.f.k 8
120.m even 2 1 1440.4.f.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.c.d 8 8.b even 2 1
160.4.c.d 8 8.d odd 2 1
160.4.c.d 8 40.e odd 2 1
160.4.c.d 8 40.f even 2 1
320.4.c.j 8 1.a even 1 1 trivial
320.4.c.j 8 4.b odd 2 1 inner
320.4.c.j 8 5.b even 2 1 inner
320.4.c.j 8 20.d odd 2 1 inner
800.4.a.y 4 40.i odd 4 1
800.4.a.y 4 40.k even 4 1
800.4.a.z 4 40.i odd 4 1
800.4.a.z 4 40.k even 4 1
1440.4.f.k 8 24.f even 2 1
1440.4.f.k 8 24.h odd 2 1
1440.4.f.k 8 120.i odd 2 1
1440.4.f.k 8 120.m even 2 1
1600.4.a.cu 4 5.c odd 4 1
1600.4.a.cu 4 20.e even 4 1
1600.4.a.cv 4 5.c odd 4 1
1600.4.a.cv 4 20.e even 4 1

## Hecke kernels

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

 $$T_{3}^{4} + 92T_{3}^{2} + 1360$$ T3^4 + 92*T3^2 + 1360 $$T_{11}^{4} - 4992T_{11}^{2} + 3133440$$ T11^4 - 4992*T11^2 + 3133440

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} + 92 T^{2} + 1360)^{2}$$
$5$ $$(T^{4} + 16 T^{3} + 230 T^{2} + \cdots + 15625)^{2}$$
$7$ $$(T^{4} + 1292 T^{2} + 393040)^{2}$$
$11$ $$(T^{4} - 4992 T^{2} + 3133440)^{2}$$
$13$ $$(T^{4} + 6080 T^{2} + 5607424)^{2}$$
$17$ $$(T^{2} + 5376)^{4}$$
$19$ $$(T^{4} - 30080 T^{2} + \cdots + 217600000)^{2}$$
$23$ $$(T^{4} + 11852 T^{2} + 2514640)^{2}$$
$29$ $$(T^{2} + 156 T - 15420)^{4}$$
$31$ $$(T^{4} - 23552 T^{2} + 89128960)^{2}$$
$37$ $$(T^{4} + 42176 T^{2} + \cdots + 140185600)^{2}$$
$41$ $$(T^{2} + 48 T - 13620)^{4}$$
$43$ $$(T^{4} + 251708 T^{2} + \cdots + 57154000)^{2}$$
$47$ $$(T^{4} + 211052 T^{2} + \cdots + 3485953360)^{2}$$
$53$ $$(T^{4} + 151488 T^{2} + \cdots + 5693607936)^{2}$$
$59$ $$(T^{4} - 313728 T^{2} + \cdots + 4289679360)^{2}$$
$61$ $$(T^{2} + 528 T + 8460)^{4}$$
$67$ $$(T^{4} + 388572 T^{2} + \cdots + 27064708560)^{2}$$
$71$ $$(T^{4} - 46592 T^{2} + \cdots + 272957440)^{2}$$
$73$ $$(T^{4} + 584448 T^{2} + \cdots + 1090584576)^{2}$$
$79$ $$(T^{4} - 1215488 T^{2} + \cdots + 196885872640)^{2}$$
$83$ $$(T^{4} + 1986972 T^{2} + \cdots + 739915650000)^{2}$$
$89$ $$(T^{2} - 20 T - 537500)^{4}$$
$97$ $$(T^{2} + 1290496)^{4}$$