# Properties

 Label 462.2.g.b Level $462$ Weight $2$ Character orbit 462.g Analytic conductor $3.689$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.68908857338$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + \beta_1 q^{3} - q^{4} + ( - \beta_{3} + \beta_1) q^{5} - \beta_{3} q^{6} + (\beta_{3} + \beta_1 - 1) q^{7} + \beta_{2} q^{8} + 3 \beta_{2} q^{9}+O(q^{10})$$ q - b2 * q^2 + b1 * q^3 - q^4 + (-b3 + b1) * q^5 - b3 * q^6 + (b3 + b1 - 1) * q^7 + b2 * q^8 + 3*b2 * q^9 $$q - \beta_{2} q^{2} + \beta_1 q^{3} - q^{4} + ( - \beta_{3} + \beta_1) q^{5} - \beta_{3} q^{6} + (\beta_{3} + \beta_1 - 1) q^{7} + \beta_{2} q^{8} + 3 \beta_{2} q^{9} + ( - \beta_{3} - \beta_1) q^{10} + \beta_{2} q^{11} - \beta_1 q^{12} + (\beta_{3} + \beta_1) q^{13} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{14} + (3 \beta_{2} + 3) q^{15} + q^{16} + 3 q^{18} + ( - 3 \beta_{3} - 3 \beta_1) q^{19} + (\beta_{3} - \beta_1) q^{20} + (3 \beta_{2} - \beta_1 - 3) q^{21} + q^{22} + 6 \beta_{2} q^{23} + \beta_{3} q^{24} + q^{25} + ( - \beta_{3} + \beta_1) q^{26} + 3 \beta_{3} q^{27} + ( - \beta_{3} - \beta_1 + 1) q^{28} - 6 \beta_{2} q^{29} + ( - 3 \beta_{2} + 3) q^{30} + ( - 2 \beta_{3} - 2 \beta_1) q^{31} - \beta_{2} q^{32} + \beta_{3} q^{33} + (\beta_{3} + 6 \beta_{2} - \beta_1) q^{35} - 3 \beta_{2} q^{36} + 10 q^{37} + (3 \beta_{3} - 3 \beta_1) q^{38} + (3 \beta_{2} - 3) q^{39} + (\beta_{3} + \beta_1) q^{40} + ( - 4 \beta_{3} + 4 \beta_1) q^{41} + (\beta_{3} + 3 \beta_{2} + 3) q^{42} - 8 q^{43} - \beta_{2} q^{44} + (3 \beta_{3} + 3 \beta_1) q^{45} + 6 q^{46} + ( - 4 \beta_{3} + 4 \beta_1) q^{47} + \beta_1 q^{48} + ( - 2 \beta_{3} - 2 \beta_1 - 5) q^{49} - \beta_{2} q^{50} + ( - \beta_{3} - \beta_1) q^{52} - 6 \beta_{2} q^{53} + 3 \beta_1 q^{54} + (\beta_{3} + \beta_1) q^{55} + (\beta_{3} - \beta_{2} - \beta_1) q^{56} + ( - 9 \beta_{2} + 9) q^{57} - 6 q^{58} + (\beta_{3} - \beta_1) q^{59} + ( - 3 \beta_{2} - 3) q^{60} + ( - 3 \beta_{3} - 3 \beta_1) q^{61} + (2 \beta_{3} - 2 \beta_1) q^{62} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{63} - q^{64} + 6 \beta_{2} q^{65} + \beta_1 q^{66} - 4 q^{67} + 6 \beta_{3} q^{69} + (\beta_{3} + \beta_1 + 6) q^{70} - 3 q^{72} + ( - 2 \beta_{3} - 2 \beta_1) q^{73} - 10 \beta_{2} q^{74} + \beta_1 q^{75} + (3 \beta_{3} + 3 \beta_1) q^{76} + (\beta_{3} - \beta_{2} - \beta_1) q^{77} + (3 \beta_{2} + 3) q^{78} - 10 q^{79} + ( - \beta_{3} + \beta_1) q^{80} - 9 q^{81} + ( - 4 \beta_{3} - 4 \beta_1) q^{82} + (\beta_{3} - \beta_1) q^{83} + ( - 3 \beta_{2} + \beta_1 + 3) q^{84} + 8 \beta_{2} q^{86} - 6 \beta_{3} q^{87} - q^{88} + ( - 4 \beta_{3} + 4 \beta_1) q^{89} + ( - 3 \beta_{3} + 3 \beta_1) q^{90} + ( - \beta_{3} - \beta_1 - 6) q^{91} - 6 \beta_{2} q^{92} + ( - 6 \beta_{2} + 6) q^{93} + ( - 4 \beta_{3} - 4 \beta_1) q^{94} - 18 \beta_{2} q^{95} - \beta_{3} q^{96} + (2 \beta_{3} + 2 \beta_1) q^{97} + (2 \beta_{3} + 5 \beta_{2} - 2 \beta_1) q^{98} - 3 q^{99}+O(q^{100})$$ q - b2 * q^2 + b1 * q^3 - q^4 + (-b3 + b1) * q^5 - b3 * q^6 + (b3 + b1 - 1) * q^7 + b2 * q^8 + 3*b2 * q^9 + (-b3 - b1) * q^10 + b2 * q^11 - b1 * q^12 + (b3 + b1) * q^13 + (-b3 + b2 + b1) * q^14 + (3*b2 + 3) * q^15 + q^16 + 3 * q^18 + (-3*b3 - 3*b1) * q^19 + (b3 - b1) * q^20 + (3*b2 - b1 - 3) * q^21 + q^22 + 6*b2 * q^23 + b3 * q^24 + q^25 + (-b3 + b1) * q^26 + 3*b3 * q^27 + (-b3 - b1 + 1) * q^28 - 6*b2 * q^29 + (-3*b2 + 3) * q^30 + (-2*b3 - 2*b1) * q^31 - b2 * q^32 + b3 * q^33 + (b3 + 6*b2 - b1) * q^35 - 3*b2 * q^36 + 10 * q^37 + (3*b3 - 3*b1) * q^38 + (3*b2 - 3) * q^39 + (b3 + b1) * q^40 + (-4*b3 + 4*b1) * q^41 + (b3 + 3*b2 + 3) * q^42 - 8 * q^43 - b2 * q^44 + (3*b3 + 3*b1) * q^45 + 6 * q^46 + (-4*b3 + 4*b1) * q^47 + b1 * q^48 + (-2*b3 - 2*b1 - 5) * q^49 - b2 * q^50 + (-b3 - b1) * q^52 - 6*b2 * q^53 + 3*b1 * q^54 + (b3 + b1) * q^55 + (b3 - b2 - b1) * q^56 + (-9*b2 + 9) * q^57 - 6 * q^58 + (b3 - b1) * q^59 + (-3*b2 - 3) * q^60 + (-3*b3 - 3*b1) * q^61 + (2*b3 - 2*b1) * q^62 + (3*b3 - 3*b2 - 3*b1) * q^63 - q^64 + 6*b2 * q^65 + b1 * q^66 - 4 * q^67 + 6*b3 * q^69 + (b3 + b1 + 6) * q^70 - 3 * q^72 + (-2*b3 - 2*b1) * q^73 - 10*b2 * q^74 + b1 * q^75 + (3*b3 + 3*b1) * q^76 + (b3 - b2 - b1) * q^77 + (3*b2 + 3) * q^78 - 10 * q^79 + (-b3 + b1) * q^80 - 9 * q^81 + (-4*b3 - 4*b1) * q^82 + (b3 - b1) * q^83 + (-3*b2 + b1 + 3) * q^84 + 8*b2 * q^86 - 6*b3 * q^87 - q^88 + (-4*b3 + 4*b1) * q^89 + (-3*b3 + 3*b1) * q^90 + (-b3 - b1 - 6) * q^91 - 6*b2 * q^92 + (-6*b2 + 6) * q^93 + (-4*b3 - 4*b1) * q^94 - 18*b2 * q^95 - b3 * q^96 + (2*b3 + 2*b1) * q^97 + (2*b3 + 5*b2 - 2*b1) * q^98 - 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 4 q^{7}+O(q^{10})$$ 4 * q - 4 * q^4 - 4 * q^7 $$4 q - 4 q^{4} - 4 q^{7} + 12 q^{15} + 4 q^{16} + 12 q^{18} - 12 q^{21} + 4 q^{22} + 4 q^{25} + 4 q^{28} + 12 q^{30} + 40 q^{37} - 12 q^{39} + 12 q^{42} - 32 q^{43} + 24 q^{46} - 20 q^{49} + 36 q^{57} - 24 q^{58} - 12 q^{60} - 4 q^{64} - 16 q^{67} + 24 q^{70} - 12 q^{72} + 12 q^{78} - 40 q^{79} - 36 q^{81} + 12 q^{84} - 4 q^{88} - 24 q^{91} + 24 q^{93} - 12 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 - 4 * q^7 + 12 * q^15 + 4 * q^16 + 12 * q^18 - 12 * q^21 + 4 * q^22 + 4 * q^25 + 4 * q^28 + 12 * q^30 + 40 * q^37 - 12 * q^39 + 12 * q^42 - 32 * q^43 + 24 * q^46 - 20 * q^49 + 36 * q^57 - 24 * q^58 - 12 * q^60 - 4 * q^64 - 16 * q^67 + 24 * q^70 - 12 * q^72 + 12 * q^78 - 40 * q^79 - 36 * q^81 + 12 * q^84 - 4 * q^88 - 24 * q^91 + 24 * q^93 - 12 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 3$$ (v^3) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2}$$ 3*b2 $$\nu^{3}$$ $$=$$ $$3\beta_{3}$$ 3*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/462\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
419.1
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
1.00000i −1.22474 1.22474i −1.00000 −2.44949 −1.22474 + 1.22474i −1.00000 2.44949i 1.00000i 3.00000i 2.44949i
419.2 1.00000i 1.22474 + 1.22474i −1.00000 2.44949 1.22474 1.22474i −1.00000 + 2.44949i 1.00000i 3.00000i 2.44949i
419.3 1.00000i −1.22474 + 1.22474i −1.00000 −2.44949 −1.22474 1.22474i −1.00000 + 2.44949i 1.00000i 3.00000i 2.44949i
419.4 1.00000i 1.22474 1.22474i −1.00000 2.44949 1.22474 + 1.22474i −1.00000 2.44949i 1.00000i 3.00000i 2.44949i
 $$n$$: 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
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.g.b 4
3.b odd 2 1 inner 462.2.g.b 4
7.b odd 2 1 inner 462.2.g.b 4
21.c even 2 1 inner 462.2.g.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.g.b 4 1.a even 1 1 trivial
462.2.g.b 4 3.b odd 2 1 inner
462.2.g.b 4 7.b odd 2 1 inner
462.2.g.b 4 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 6$$ acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4} + 9$$
$5$ $$(T^{2} - 6)^{2}$$
$7$ $$(T^{2} + 2 T + 7)^{2}$$
$11$ $$(T^{2} + 1)^{2}$$
$13$ $$(T^{2} + 6)^{2}$$
$17$ $$T^{4}$$
$19$ $$(T^{2} + 54)^{2}$$
$23$ $$(T^{2} + 36)^{2}$$
$29$ $$(T^{2} + 36)^{2}$$
$31$ $$(T^{2} + 24)^{2}$$
$37$ $$(T - 10)^{4}$$
$41$ $$(T^{2} - 96)^{2}$$
$43$ $$(T + 8)^{4}$$
$47$ $$(T^{2} - 96)^{2}$$
$53$ $$(T^{2} + 36)^{2}$$
$59$ $$(T^{2} - 6)^{2}$$
$61$ $$(T^{2} + 54)^{2}$$
$67$ $$(T + 4)^{4}$$
$71$ $$T^{4}$$
$73$ $$(T^{2} + 24)^{2}$$
$79$ $$(T + 10)^{4}$$
$83$ $$(T^{2} - 6)^{2}$$
$89$ $$(T^{2} - 96)^{2}$$
$97$ $$(T^{2} + 24)^{2}$$