# Properties

 Label 476.2.i.b Level $476$ Weight $2$ Character orbit 476.i Analytic conductor $3.801$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$476 = 2^{2} \cdot 7 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 476.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.80087913621$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} + ( - 2 \zeta_{6} + 3) q^{7} + 2 \zeta_{6} q^{9}+O(q^{10})$$ q + (-z + 1) * q^3 + (-2*z + 3) * q^7 + 2*z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} + ( - 2 \zeta_{6} + 3) q^{7} + 2 \zeta_{6} q^{9} + ( - 5 \zeta_{6} + 5) q^{11} - 5 q^{13} + ( - \zeta_{6} + 1) q^{17} + 6 \zeta_{6} q^{19} + ( - 3 \zeta_{6} + 1) q^{21} - 4 \zeta_{6} q^{23} + ( - 5 \zeta_{6} + 5) q^{25} + 5 q^{27} + 4 q^{29} - 5 \zeta_{6} q^{33} - 8 \zeta_{6} q^{37} + (5 \zeta_{6} - 5) q^{39} + 4 q^{41} - 6 q^{43} + 6 \zeta_{6} q^{47} + ( - 8 \zeta_{6} + 5) q^{49} - \zeta_{6} q^{51} + (11 \zeta_{6} - 11) q^{53} + 6 q^{57} + (10 \zeta_{6} - 10) q^{59} + (2 \zeta_{6} + 4) q^{63} + (10 \zeta_{6} - 10) q^{67} - 4 q^{69} + 9 q^{71} + (4 \zeta_{6} - 4) q^{73} - 5 \zeta_{6} q^{75} + ( - 15 \zeta_{6} + 5) q^{77} - 9 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + 4 q^{83} + ( - 4 \zeta_{6} + 4) q^{87} + 15 \zeta_{6} q^{89} + (10 \zeta_{6} - 15) q^{91} - 18 q^{97} + 10 q^{99} +O(q^{100})$$ q + (-z + 1) * q^3 + (-2*z + 3) * q^7 + 2*z * q^9 + (-5*z + 5) * q^11 - 5 * q^13 + (-z + 1) * q^17 + 6*z * q^19 + (-3*z + 1) * q^21 - 4*z * q^23 + (-5*z + 5) * q^25 + 5 * q^27 + 4 * q^29 - 5*z * q^33 - 8*z * q^37 + (5*z - 5) * q^39 + 4 * q^41 - 6 * q^43 + 6*z * q^47 + (-8*z + 5) * q^49 - z * q^51 + (11*z - 11) * q^53 + 6 * q^57 + (10*z - 10) * q^59 + (2*z + 4) * q^63 + (10*z - 10) * q^67 - 4 * q^69 + 9 * q^71 + (4*z - 4) * q^73 - 5*z * q^75 + (-15*z + 5) * q^77 - 9*z * q^79 + (z - 1) * q^81 + 4 * q^83 + (-4*z + 4) * q^87 + 15*z * q^89 + (10*z - 15) * q^91 - 18 * q^97 + 10 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + q^3 + 4 * q^7 + 2 * q^9 $$2 q + q^{3} + 4 q^{7} + 2 q^{9} + 5 q^{11} - 10 q^{13} + q^{17} + 6 q^{19} - q^{21} - 4 q^{23} + 5 q^{25} + 10 q^{27} + 8 q^{29} - 5 q^{33} - 8 q^{37} - 5 q^{39} + 8 q^{41} - 12 q^{43} + 6 q^{47} + 2 q^{49} - q^{51} - 11 q^{53} + 12 q^{57} - 10 q^{59} + 10 q^{63} - 10 q^{67} - 8 q^{69} + 18 q^{71} - 4 q^{73} - 5 q^{75} - 5 q^{77} - 9 q^{79} - q^{81} + 8 q^{83} + 4 q^{87} + 15 q^{89} - 20 q^{91} - 36 q^{97} + 20 q^{99}+O(q^{100})$$ 2 * q + q^3 + 4 * q^7 + 2 * q^9 + 5 * q^11 - 10 * q^13 + q^17 + 6 * q^19 - q^21 - 4 * q^23 + 5 * q^25 + 10 * q^27 + 8 * q^29 - 5 * q^33 - 8 * q^37 - 5 * q^39 + 8 * q^41 - 12 * q^43 + 6 * q^47 + 2 * q^49 - q^51 - 11 * q^53 + 12 * q^57 - 10 * q^59 + 10 * q^63 - 10 * q^67 - 8 * q^69 + 18 * q^71 - 4 * q^73 - 5 * q^75 - 5 * q^77 - 9 * q^79 - q^81 + 8 * q^83 + 4 * q^87 + 15 * q^89 - 20 * q^91 - 36 * q^97 + 20 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/476\mathbb{Z}\right)^\times$$.

 $$n$$ $$239$$ $$309$$ $$409$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i 0 0 0 2.00000 1.73205i 0 1.00000 + 1.73205i 0
205.1 0 0.500000 + 0.866025i 0 0 0 2.00000 + 1.73205i 0 1.00000 1.73205i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 476.2.i.b 2
7.c even 3 1 inner 476.2.i.b 2
7.c even 3 1 3332.2.a.b 1
7.d odd 6 1 3332.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.2.i.b 2 1.a even 1 1 trivial
476.2.i.b 2 7.c even 3 1 inner
3332.2.a.b 1 7.c even 3 1
3332.2.a.e 1 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(476, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 4T + 7$$
$11$ $$T^{2} - 5T + 25$$
$13$ $$(T + 5)^{2}$$
$17$ $$T^{2} - T + 1$$
$19$ $$T^{2} - 6T + 36$$
$23$ $$T^{2} + 4T + 16$$
$29$ $$(T - 4)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 8T + 64$$
$41$ $$(T - 4)^{2}$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} - 6T + 36$$
$53$ $$T^{2} + 11T + 121$$
$59$ $$T^{2} + 10T + 100$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 10T + 100$$
$71$ $$(T - 9)^{2}$$
$73$ $$T^{2} + 4T + 16$$
$79$ $$T^{2} + 9T + 81$$
$83$ $$(T - 4)^{2}$$
$89$ $$T^{2} - 15T + 225$$
$97$ $$(T + 18)^{2}$$