# Properties

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

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## 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{3} + 4 \zeta_{6} q^{5} + (2 \zeta_{6} - 3) q^{7} + 2 \zeta_{6} q^{9}+O(q^{10})$$ q + (z - 1) * q^3 + 4*z * q^5 + (2*z - 3) * q^7 + 2*z * q^9 $$q + (\zeta_{6} - 1) q^{3} + 4 \zeta_{6} q^{5} + (2 \zeta_{6} - 3) q^{7} + 2 \zeta_{6} q^{9} + ( - 3 \zeta_{6} + 3) q^{11} - 5 q^{13} - 4 q^{15} + ( - \zeta_{6} + 1) q^{17} - 6 \zeta_{6} q^{19} + ( - 3 \zeta_{6} + 1) q^{21} + 4 \zeta_{6} q^{23} + (11 \zeta_{6} - 11) q^{25} - 5 q^{27} + 8 q^{29} + 3 \zeta_{6} q^{33} + ( - 4 \zeta_{6} - 8) q^{35} + 8 \zeta_{6} q^{37} + ( - 5 \zeta_{6} + 5) q^{39} - 8 q^{41} + 10 q^{43} + (8 \zeta_{6} - 8) q^{45} + 2 \zeta_{6} q^{47} + ( - 8 \zeta_{6} + 5) q^{49} + \zeta_{6} q^{51} + (3 \zeta_{6} - 3) q^{53} + 12 q^{55} + 6 q^{57} + (2 \zeta_{6} - 2) q^{59} + 8 \zeta_{6} q^{61} + ( - 2 \zeta_{6} - 4) q^{63} - 20 \zeta_{6} q^{65} + (14 \zeta_{6} - 14) q^{67} - 4 q^{69} + 7 q^{71} + ( - 8 \zeta_{6} + 8) q^{73} - 11 \zeta_{6} q^{75} + (9 \zeta_{6} - 3) q^{77} - 15 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + 12 q^{83} + 4 q^{85} + (8 \zeta_{6} - 8) q^{87} - \zeta_{6} q^{89} + ( - 10 \zeta_{6} + 15) q^{91} + ( - 24 \zeta_{6} + 24) q^{95} + 18 q^{97} + 6 q^{99} +O(q^{100})$$ q + (z - 1) * q^3 + 4*z * q^5 + (2*z - 3) * q^7 + 2*z * q^9 + (-3*z + 3) * q^11 - 5 * q^13 - 4 * q^15 + (-z + 1) * q^17 - 6*z * q^19 + (-3*z + 1) * q^21 + 4*z * q^23 + (11*z - 11) * q^25 - 5 * q^27 + 8 * q^29 + 3*z * q^33 + (-4*z - 8) * q^35 + 8*z * q^37 + (-5*z + 5) * q^39 - 8 * q^41 + 10 * q^43 + (8*z - 8) * q^45 + 2*z * q^47 + (-8*z + 5) * q^49 + z * q^51 + (3*z - 3) * q^53 + 12 * q^55 + 6 * q^57 + (2*z - 2) * q^59 + 8*z * q^61 + (-2*z - 4) * q^63 - 20*z * q^65 + (14*z - 14) * q^67 - 4 * q^69 + 7 * q^71 + (-8*z + 8) * q^73 - 11*z * q^75 + (9*z - 3) * q^77 - 15*z * q^79 + (z - 1) * q^81 + 12 * q^83 + 4 * q^85 + (8*z - 8) * q^87 - z * q^89 + (-10*z + 15) * q^91 + (-24*z + 24) * q^95 + 18 * q^97 + 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 4 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - q^3 + 4 * q^5 - 4 * q^7 + 2 * q^9 $$2 q - q^{3} + 4 q^{5} - 4 q^{7} + 2 q^{9} + 3 q^{11} - 10 q^{13} - 8 q^{15} + q^{17} - 6 q^{19} - q^{21} + 4 q^{23} - 11 q^{25} - 10 q^{27} + 16 q^{29} + 3 q^{33} - 20 q^{35} + 8 q^{37} + 5 q^{39} - 16 q^{41} + 20 q^{43} - 8 q^{45} + 2 q^{47} + 2 q^{49} + q^{51} - 3 q^{53} + 24 q^{55} + 12 q^{57} - 2 q^{59} + 8 q^{61} - 10 q^{63} - 20 q^{65} - 14 q^{67} - 8 q^{69} + 14 q^{71} + 8 q^{73} - 11 q^{75} + 3 q^{77} - 15 q^{79} - q^{81} + 24 q^{83} + 8 q^{85} - 8 q^{87} - q^{89} + 20 q^{91} + 24 q^{95} + 36 q^{97} + 12 q^{99}+O(q^{100})$$ 2 * q - q^3 + 4 * q^5 - 4 * q^7 + 2 * q^9 + 3 * q^11 - 10 * q^13 - 8 * q^15 + q^17 - 6 * q^19 - q^21 + 4 * q^23 - 11 * q^25 - 10 * q^27 + 16 * q^29 + 3 * q^33 - 20 * q^35 + 8 * q^37 + 5 * q^39 - 16 * q^41 + 20 * q^43 - 8 * q^45 + 2 * q^47 + 2 * q^49 + q^51 - 3 * q^53 + 24 * q^55 + 12 * q^57 - 2 * q^59 + 8 * q^61 - 10 * q^63 - 20 * q^65 - 14 * q^67 - 8 * q^69 + 14 * q^71 + 8 * q^73 - 11 * q^75 + 3 * q^77 - 15 * q^79 - q^81 + 24 * q^83 + 8 * q^85 - 8 * q^87 - q^89 + 20 * q^91 + 24 * q^95 + 36 * q^97 + 12 * 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 2.00000 + 3.46410i 0 −2.00000 + 1.73205i 0 1.00000 + 1.73205i 0
205.1 0 −0.500000 0.866025i 0 2.00000 3.46410i 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.a 2
7.c even 3 1 inner 476.2.i.a 2
7.c even 3 1 3332.2.a.d 1
7.d odd 6 1 3332.2.a.c 1

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

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} - 4T + 16$$
$7$ $$T^{2} + 4T + 7$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$(T + 5)^{2}$$
$17$ $$T^{2} - T + 1$$
$19$ $$T^{2} + 6T + 36$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$(T - 8)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 8T + 64$$
$41$ $$(T + 8)^{2}$$
$43$ $$(T - 10)^{2}$$
$47$ $$T^{2} - 2T + 4$$
$53$ $$T^{2} + 3T + 9$$
$59$ $$T^{2} + 2T + 4$$
$61$ $$T^{2} - 8T + 64$$
$67$ $$T^{2} + 14T + 196$$
$71$ $$(T - 7)^{2}$$
$73$ $$T^{2} - 8T + 64$$
$79$ $$T^{2} + 15T + 225$$
$83$ $$(T - 12)^{2}$$
$89$ $$T^{2} + T + 1$$
$97$ $$(T - 18)^{2}$$
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