# Properties

 Label 1440.4.f.k Level $1440$ Weight $4$ Character orbit 1440.f Analytic conductor $84.963$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1440 = 2^{5} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1440.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$84.9627504083$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} - \beta_{4} - 4) q^{5} + ( - \beta_{3} + \beta_1) q^{7}+O(q^{10})$$ q + (b5 - b4 - 4) * q^5 + (-b3 + b1) * q^7 $$q + (\beta_{5} - \beta_{4} - 4) q^{5} + ( - \beta_{3} + \beta_1) q^{7} - \beta_{6} q^{11} + (\beta_{5} - 2 \beta_{4} - 3 \beta_{2}) q^{13} + (4 \beta_{5} - 8 \beta_{4} + \beta_{2}) q^{17} + (2 \beta_{7} - \beta_{6}) q^{19} + ( - \beta_{3} + 11 \beta_1) q^{23} + ( - 11 \beta_{5} + 6 \beta_{4} - 5 \beta_{2} - 51) q^{25} + (16 \beta_{5} - 78) q^{29} - 2 \beta_{7} q^{31} + (4 \beta_{7} - \beta_{6} + 6 \beta_{3} + 11 \beta_1) q^{35} + (7 \beta_{5} - 14 \beta_{4} + 6 \beta_{2}) q^{37} + ( - 13 \beta_{5} + 24) q^{41} + ( - 6 \beta_{3} + 49 \beta_1) q^{43} + (9 \beta_{3} - 37 \beta_1) q^{47} + (17 \beta_{5} - 303) q^{49} + ( - 15 \beta_{5} + 30 \beta_{4} - 3 \beta_{2}) q^{53} + ( - 3 \beta_{7} + 7 \beta_{6} + 8 \beta_{3} + 48 \beta_1) q^{55} + ( - 6 \beta_{7} - 5 \beta_{6}) q^{59} + (27 \beta_{5} + 264) q^{61} + ( - 47 \beta_{5} - 18 \beta_{4} + 10 \beta_{2} - 272) q^{65} + ( - 10 \beta_{3} - 51 \beta_1) q^{67} + ( - 2 \beta_{7} + 2 \beta_{6}) q^{71} + (22 \beta_{5} - 44 \beta_{4} - 17 \beta_{2}) q^{73} + ( - 34 \beta_{5} + 68 \beta_{4} - 85 \beta_{2}) q^{77} + ( - 10 \beta_{7} - 12 \beta_{6}) q^{79} + ( - 22 \beta_{3} - 117 \beta_1) q^{83} + ( - 32 \beta_{5} + 32 \beta_{4} - 25 \beta_{2} - 672) q^{85} + ( - 80 \beta_{5} - 10) q^{89} + (14 \beta_{7} + 22 \beta_{6}) q^{91} + ( - 5 \beta_{7} + 15 \beta_{6} + 40 \beta_{3} - 80 \beta_1) q^{95} + 71 \beta_{2} q^{97}+O(q^{100})$$ q + (b5 - b4 - 4) * q^5 + (-b3 + b1) * q^7 - b6 * q^11 + (b5 - 2*b4 - 3*b2) * q^13 + (4*b5 - 8*b4 + b2) * q^17 + (2*b7 - b6) * q^19 + (-b3 + 11*b1) * q^23 + (-11*b5 + 6*b4 - 5*b2 - 51) * q^25 + (16*b5 - 78) * q^29 - 2*b7 * q^31 + (4*b7 - b6 + 6*b3 + 11*b1) * q^35 + (7*b5 - 14*b4 + 6*b2) * q^37 + (-13*b5 + 24) * q^41 + (-6*b3 + 49*b1) * q^43 + (9*b3 - 37*b1) * q^47 + (17*b5 - 303) * q^49 + (-15*b5 + 30*b4 - 3*b2) * q^53 + (-3*b7 + 7*b6 + 8*b3 + 48*b1) * q^55 + (-6*b7 - 5*b6) * q^59 + (27*b5 + 264) * q^61 + (-47*b5 - 18*b4 + 10*b2 - 272) * q^65 + (-10*b3 - 51*b1) * q^67 + (-2*b7 + 2*b6) * q^71 + (22*b5 - 44*b4 - 17*b2) * q^73 + (-34*b5 + 68*b4 - 85*b2) * q^77 + (-10*b7 - 12*b6) * q^79 + (-22*b3 - 117*b1) * q^83 + (-32*b5 + 32*b4 - 25*b2 - 672) * q^85 + (-80*b5 - 10) * q^89 + (14*b7 + 22*b6) * q^91 + (-5*b7 + 15*b6 + 40*b3 - 80*b1) * q^95 + 71*b2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 32 q^{5}+O(q^{10})$$ 8 * q - 32 * q^5 $$8 q - 32 q^{5} - 408 q^{25} - 624 q^{29} + 192 q^{41} - 2424 q^{49} + 2112 q^{61} - 2176 q^{65} - 5376 q^{85} - 80 q^{89}+O(q^{100})$$ 8 * q - 32 * q^5 - 408 * q^25 - 624 * q^29 + 192 * q^41 - 2424 * q^49 + 2112 * q^61 - 2176 * q^65 - 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 $$\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$641$$ $$901$$ $$991$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 − 2.47107i − 0.320221i 0.320221i 2.47107i 1.24759i − 3.03888i 3.03888i − 1.24759i
0 0 0 −8.58258 7.16515i 0 28.3162i 0 0 0
289.2 0 0 0 −8.58258 7.16515i 0 28.3162i 0 0 0
289.3 0 0 0 −8.58258 + 7.16515i 0 28.3162i 0 0 0
289.4 0 0 0 −8.58258 + 7.16515i 0 28.3162i 0 0 0
289.5 0 0 0 0.582576 11.1652i 0 22.1403i 0 0 0
289.6 0 0 0 0.582576 11.1652i 0 22.1403i 0 0 0
289.7 0 0 0 0.582576 + 11.1652i 0 22.1403i 0 0 0
289.8 0 0 0 0.582576 + 11.1652i 0 22.1403i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.8 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 1440.4.f.k 8
3.b odd 2 1 160.4.c.d 8
4.b odd 2 1 inner 1440.4.f.k 8
5.b even 2 1 inner 1440.4.f.k 8
12.b even 2 1 160.4.c.d 8
15.d odd 2 1 160.4.c.d 8
15.e even 4 1 800.4.a.y 4
15.e even 4 1 800.4.a.z 4
20.d odd 2 1 inner 1440.4.f.k 8
24.f even 2 1 320.4.c.j 8
24.h odd 2 1 320.4.c.j 8
60.h even 2 1 160.4.c.d 8
60.l odd 4 1 800.4.a.y 4
60.l odd 4 1 800.4.a.z 4
120.i odd 2 1 320.4.c.j 8
120.m even 2 1 320.4.c.j 8
120.q odd 4 1 1600.4.a.cu 4
120.q odd 4 1 1600.4.a.cv 4
120.w even 4 1 1600.4.a.cu 4
120.w even 4 1 1600.4.a.cv 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.c.d 8 3.b odd 2 1
160.4.c.d 8 12.b even 2 1
160.4.c.d 8 15.d odd 2 1
160.4.c.d 8 60.h even 2 1
320.4.c.j 8 24.f even 2 1
320.4.c.j 8 24.h odd 2 1
320.4.c.j 8 120.i odd 2 1
320.4.c.j 8 120.m even 2 1
800.4.a.y 4 15.e even 4 1
800.4.a.y 4 60.l odd 4 1
800.4.a.z 4 15.e even 4 1
800.4.a.z 4 60.l odd 4 1
1440.4.f.k 8 1.a even 1 1 trivial
1440.4.f.k 8 4.b odd 2 1 inner
1440.4.f.k 8 5.b even 2 1 inner
1440.4.f.k 8 20.d odd 2 1 inner
1600.4.a.cu 4 120.q odd 4 1
1600.4.a.cu 4 120.w even 4 1
1600.4.a.cv 4 120.q odd 4 1
1600.4.a.cv 4 120.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1440, [\chi])$$:

 $$T_{7}^{4} + 1292T_{7}^{2} + 393040$$ T7^4 + 1292*T7^2 + 393040 $$T_{11}^{4} - 4992T_{11}^{2} + 3133440$$ T11^4 - 4992*T11^2 + 3133440 $$T_{17}^{2} + 5376$$ T17^2 + 5376 $$T_{29}^{2} + 156T_{29} - 15420$$ T29^2 + 156*T29 - 15420

## Hecke characteristic polynomials

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