# Properties

 Label 460.2.i.a Level $460$ Weight $2$ Character orbit 460.i Analytic conductor $3.673$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$460 = 2^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 460.i (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.67311849298$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.11574317056.3 Defining polynomial: $$x^{8} + 45x^{4} + 4$$ x^8 + 45*x^4 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - 1) q^{3} + (\beta_{6} - \beta_1) q^{5} - \beta_{2} q^{7} - \beta_{3} q^{9}+O(q^{10})$$ q + (-b3 - 1) * q^3 + (b6 - b1) * q^5 - b2 * q^7 - b3 * q^9 $$q + ( - \beta_{3} - 1) q^{3} + (\beta_{6} - \beta_1) q^{5} - \beta_{2} q^{7} - \beta_{3} q^{9} + (\beta_{2} + \beta_1) q^{11} + (\beta_{7} + \beta_{6} + 2 \beta_{5}) q^{13} + ( - \beta_{7} - \beta_{6} - \beta_{2} + \beta_1) q^{15} + ( - 2 \beta_{7} + 2 \beta_{6} - \beta_{2}) q^{17} + ( - 2 \beta_{7} + \beta_{2} - \beta_1) q^{19} + (\beta_{2} + \beta_1) q^{21} + ( - \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{3} + 2 \beta_1 - 2) q^{23} + (\beta_{7} + \beta_{6} - 2 \beta_{4} - \beta_{3} - 2) q^{25} + (4 \beta_{3} - 4) q^{27} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 5 \beta_{3}) q^{29} + (3 \beta_{5} + 3 \beta_{4} + 1) q^{31} - 2 \beta_1 q^{33} + (\beta_{5} + \beta_{4} - \beta_{3} - 2) q^{35} + ( - 2 \beta_{7} + 2 \beta_{6} - \beta_{2}) q^{37} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4}) q^{39} + (\beta_{5} + \beta_{4} + 5) q^{41} + ( - \beta_{7} - \beta_{6} - 4 \beta_1) q^{43} + ( - \beta_{7} - \beta_{2}) q^{45} + (2 \beta_{7} + 2 \beta_{6} - 4 \beta_{4} + \beta_{3} - 1) q^{47} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 4 \beta_{3}) q^{49} + ( - 4 \beta_{6} + \beta_{2} + \beta_1) q^{51} + (2 \beta_{7} + 2 \beta_{6} + \beta_1) q^{53} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} + 3 \beta_{3} + 1) q^{55} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{2}) q^{57} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3}) q^{59} + (2 \beta_{6} + 3 \beta_{2} + 3 \beta_1) q^{61} + \beta_1 q^{63} + (\beta_{7} + 3 \beta_{6} - 4 \beta_{2} + 2 \beta_1) q^{65} + (2 \beta_{7} - 2 \beta_{6} - 3 \beta_{2}) q^{67} + (2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{69} + ( - \beta_{5} - \beta_{4} + 5) q^{71} + ( - 2 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} + 3 \beta_{3} + 3) q^{73} + (2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 1) q^{75} + (\beta_{7} + \beta_{6} - 2 \beta_{4} - 3 \beta_{3} + 3) q^{77} + (6 \beta_{7} + 3 \beta_{2} - 3 \beta_1) q^{79} + 5 q^{81} - \beta_1 q^{83} + (2 \beta_{7} + 2 \beta_{6} + \beta_{5} - 3 \beta_{4} + 7 \beta_{3} - 6) q^{85} + ( - \beta_{7} - \beta_{6} + 2 \beta_{4} + 5 \beta_{3} - 5) q^{87} + (8 \beta_{7} + 5 \beta_{2} - 5 \beta_1) q^{89} + ( - 2 \beta_{6} + 4 \beta_{2} + 4 \beta_1) q^{91} + ( - 3 \beta_{7} - 3 \beta_{6} - 6 \beta_{5} - \beta_{3} - 1) q^{93} + (3 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} - 4 \beta_{4} + 5 \beta_{3} + 5) q^{95} + (5 \beta_{7} - 5 \beta_{6} + 6 \beta_{2}) q^{97} + (\beta_{2} - \beta_1) q^{99}+O(q^{100})$$ q + (-b3 - 1) * q^3 + (b6 - b1) * q^5 - b2 * q^7 - b3 * q^9 + (b2 + b1) * q^11 + (b7 + b6 + 2*b5) * q^13 + (-b7 - b6 - b2 + b1) * q^15 + (-2*b7 + 2*b6 - b2) * q^17 + (-2*b7 + b2 - b1) * q^19 + (b2 + b1) * q^21 + (-b7 - b6 - b5 - 2*b3 + 2*b1 - 2) * q^23 + (b7 + b6 - 2*b4 - b3 - 2) * q^25 + (4*b3 - 4) * q^27 + (b7 + b6 + b5 - b4 - 5*b3) * q^29 + (3*b5 + 3*b4 + 1) * q^31 - 2*b1 * q^33 + (b5 + b4 - b3 - 2) * q^35 + (-2*b7 + 2*b6 - b2) * q^37 + (-2*b7 - 2*b6 - 2*b5 + 2*b4) * q^39 + (b5 + b4 + 5) * q^41 + (-b7 - b6 - 4*b1) * q^43 + (-b7 - b2) * q^45 + (2*b7 + 2*b6 - 4*b4 + b3 - 1) * q^47 + (-b7 - b6 - b5 + b4 - 4*b3) * q^49 + (-4*b6 + b2 + b1) * q^51 + (2*b7 + 2*b6 + b1) * q^53 + (-b7 - b6 - 2*b5 + 3*b3 + 1) * q^55 + (2*b7 - 2*b6 - 2*b2) * q^57 + (-b7 - b6 - b5 + b4 - b3) * q^59 + (2*b6 + 3*b2 + 3*b1) * q^61 + b1 * q^63 + (b7 + 3*b6 - 4*b2 + 2*b1) * q^65 + (2*b7 - 2*b6 - 3*b2) * q^67 + (2*b7 + b6 + b5 - b4 + 4*b3 + 2*b2 - 2*b1) * q^69 + (-b5 - b4 + 5) * q^71 + (-2*b7 - 2*b6 - 4*b5 + 3*b3 + 3) * q^73 + (2*b5 + 2*b4 + 3*b3 + 1) * q^75 + (b7 + b6 - 2*b4 - 3*b3 + 3) * q^77 + (6*b7 + 3*b2 - 3*b1) * q^79 + 5 * q^81 - b1 * q^83 + (2*b7 + 2*b6 + b5 - 3*b4 + 7*b3 - 6) * q^85 + (-b7 - b6 + 2*b4 + 5*b3 - 5) * q^87 + (8*b7 + 5*b2 - 5*b1) * q^89 + (-2*b6 + 4*b2 + 4*b1) * q^91 + (-3*b7 - 3*b6 - 6*b5 - b3 - 1) * q^93 + (3*b7 + 3*b6 + 2*b5 - 4*b4 + 5*b3 + 5) * q^95 + (5*b7 - 5*b6 + 6*b2) * q^97 + (b2 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{3}+O(q^{10})$$ 8 * q - 8 * q^3 $$8 q - 8 q^{3} - 4 q^{13} - 14 q^{23} - 12 q^{25} - 32 q^{27} - 4 q^{31} - 20 q^{35} + 36 q^{41} + 12 q^{55} + 44 q^{71} + 32 q^{73} + 28 q^{77} + 40 q^{81} - 44 q^{85} - 44 q^{87} + 4 q^{93} + 44 q^{95}+O(q^{100})$$ 8 * q - 8 * q^3 - 4 * q^13 - 14 * q^23 - 12 * q^25 - 32 * q^27 - 4 * q^31 - 20 * q^35 + 36 * q^41 + 12 * q^55 + 44 * q^71 + 32 * q^73 + 28 * q^77 + 40 * q^81 - 44 * q^85 - 44 * q^87 + 4 * q^93 + 44 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 45x^{4} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{7} + 47\nu^{3} ) / 14$$ (v^7 + 47*v^3) / 14 $$\beta_{3}$$ $$=$$ $$( -\nu^{6} - 47\nu^{2} ) / 14$$ (-v^6 - 47*v^2) / 14 $$\beta_{4}$$ $$=$$ $$( 3\nu^{6} + 2\nu^{5} + 2\nu^{4} + 127\nu^{2} + 94\nu + 38 ) / 28$$ (3*v^6 + 2*v^5 + 2*v^4 + 127*v^2 + 94*v + 38) / 28 $$\beta_{5}$$ $$=$$ $$( -3\nu^{6} - 2\nu^{5} + 2\nu^{4} - 127\nu^{2} - 94\nu + 38 ) / 28$$ (-3*v^6 - 2*v^5 + 2*v^4 - 127*v^2 - 94*v + 38) / 28 $$\beta_{6}$$ $$=$$ $$( 7\nu^{7} + 2\nu^{5} + 315\nu^{3} + 94\nu ) / 28$$ (7*v^7 + 2*v^5 + 315*v^3 + 94*v) / 28 $$\beta_{7}$$ $$=$$ $$( -7\nu^{7} + 2\nu^{5} - 315\nu^{3} + 94\nu ) / 28$$ (-7*v^7 + 2*v^5 - 315*v^3 + 94*v) / 28
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 3\beta_{3}$$ b7 + b6 + b5 - b4 - 3*b3 $$\nu^{3}$$ $$=$$ $$\beta_{7} - \beta_{6} + 7\beta_{2}$$ b7 - b6 + 7*b2 $$\nu^{4}$$ $$=$$ $$7\beta_{5} + 7\beta_{4} - 19$$ 7*b5 + 7*b4 - 19 $$\nu^{5}$$ $$=$$ $$7\beta_{7} + 7\beta_{6} - 47\beta_1$$ 7*b7 + 7*b6 - 47*b1 $$\nu^{6}$$ $$=$$ $$-47\beta_{7} - 47\beta_{6} - 47\beta_{5} + 47\beta_{4} + 127\beta_{3}$$ -47*b7 - 47*b6 - 47*b5 + 47*b4 + 127*b3 $$\nu^{7}$$ $$=$$ $$-47\beta_{7} + 47\beta_{6} - 315\beta_{2}$$ -47*b7 + 47*b6 - 315*b2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/460\mathbb{Z}\right)^\times$$.

 $$n$$ $$231$$ $$277$$ $$281$$ $$\chi(n)$$ $$1$$ $$-\beta_{3}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
137.1
 1.83051 + 1.83051i 0.386289 + 0.386289i −0.386289 − 0.386289i −1.83051 − 1.83051i 1.83051 − 1.83051i 0.386289 − 0.386289i −0.386289 + 0.386289i −1.83051 + 1.83051i
0 −1.00000 + 1.00000i 0 −1.83051 1.28422i 0 1.83051 1.83051i 0 1.00000i 0
137.2 0 −1.00000 + 1.00000i 0 −0.386289 + 2.20245i 0 0.386289 0.386289i 0 1.00000i 0
137.3 0 −1.00000 + 1.00000i 0 0.386289 2.20245i 0 −0.386289 + 0.386289i 0 1.00000i 0
137.4 0 −1.00000 + 1.00000i 0 1.83051 + 1.28422i 0 −1.83051 + 1.83051i 0 1.00000i 0
413.1 0 −1.00000 1.00000i 0 −1.83051 + 1.28422i 0 1.83051 + 1.83051i 0 1.00000i 0
413.2 0 −1.00000 1.00000i 0 −0.386289 2.20245i 0 0.386289 + 0.386289i 0 1.00000i 0
413.3 0 −1.00000 1.00000i 0 0.386289 + 2.20245i 0 −0.386289 0.386289i 0 1.00000i 0
413.4 0 −1.00000 1.00000i 0 1.83051 1.28422i 0 −1.83051 1.83051i 0 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 413.4 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.c odd 4 1 inner
23.b odd 2 1 inner
115.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.i.a 8
5.b even 2 1 2300.2.i.c 8
5.c odd 4 1 inner 460.2.i.a 8
5.c odd 4 1 2300.2.i.c 8
23.b odd 2 1 inner 460.2.i.a 8
115.c odd 2 1 2300.2.i.c 8
115.e even 4 1 inner 460.2.i.a 8
115.e even 4 1 2300.2.i.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.i.a 8 1.a even 1 1 trivial
460.2.i.a 8 5.c odd 4 1 inner
460.2.i.a 8 23.b odd 2 1 inner
460.2.i.a 8 115.e even 4 1 inner
2300.2.i.c 8 5.b even 2 1
2300.2.i.c 8 5.c odd 4 1
2300.2.i.c 8 115.c odd 2 1
2300.2.i.c 8 115.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 2T_{3} + 2$$ acting on $$S_{2}^{\mathrm{new}}(460, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{2} + 2 T + 2)^{4}$$
$5$ $$T^{8} + 6 T^{6} + 18 T^{4} + 150 T^{2} + \cdots + 625$$
$7$ $$T^{8} + 45T^{4} + 4$$
$11$ $$(T^{4} + 14 T^{2} + 8)^{2}$$
$13$ $$(T^{4} + 2 T^{3} + 2 T^{2} - 40 T + 400)^{2}$$
$17$ $$T^{8} + 2109 T^{4} + 2500$$
$19$ $$(T^{4} - 58 T^{2} + 800)^{2}$$
$23$ $$T^{8} + 14 T^{7} + 98 T^{6} + \cdots + 279841$$
$29$ $$(T^{4} + 81 T^{2} + 400)^{2}$$
$31$ $$(T^{2} + T - 92)^{4}$$
$37$ $$T^{8} + 2109 T^{4} + 2500$$
$41$ $$(T^{2} - 9 T + 10)^{4}$$
$43$ $$T^{8} + 16500 T^{4} + \cdots + 17909824$$
$47$ $$(T^{4} + 6724)^{2}$$
$53$ $$T^{8} + 4125 T^{4} + \cdots + 1119364$$
$59$ $$(T^{4} + 21 T^{2} + 100)^{2}$$
$61$ $$(T^{4} + 202 T^{2} + 8192)^{2}$$
$67$ $$T^{8} + 13965 T^{4} + \cdots + 48469444$$
$71$ $$(T^{2} - 11 T + 20)^{4}$$
$73$ $$(T^{4} - 16 T^{3} + 128 T^{2} + 800 T + 2500)^{2}$$
$79$ $$(T^{4} - 234 T^{2} + 10368)^{2}$$
$83$ $$T^{8} + 45T^{4} + 4$$
$89$ $$(T^{4} - 478 T^{2} + 55112)^{2}$$
$97$ $$T^{8} + 69540 T^{4} + \cdots + 945685504$$
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