# Properties

 Label 1476.1.o.a Level $1476$ Weight $1$ Character orbit 1476.o Analytic conductor $0.737$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -164 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1476 = 2^{2} \cdot 3^{2} \cdot 41$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1476.o (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.736619958646$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.13284.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{2} - q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{5} - \zeta_{6}^{2} q^{6} + \zeta_{6}^{2} q^{7} + q^{8} + q^{9} +O(q^{10})$$ q + z^2 * q^2 - q^3 - z * q^4 + z * q^5 - z^2 * q^6 + z^2 * q^7 + q^8 + q^9 $$q + \zeta_{6}^{2} q^{2} - q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{5} - \zeta_{6}^{2} q^{6} + \zeta_{6}^{2} q^{7} + q^{8} + q^{9} - q^{10} - \zeta_{6}^{2} q^{11} + \zeta_{6} q^{12} - \zeta_{6} q^{14} - \zeta_{6} q^{15} + \zeta_{6}^{2} q^{16} + \zeta_{6}^{2} q^{18} + q^{19} - \zeta_{6}^{2} q^{20} - \zeta_{6}^{2} q^{21} + 2 \zeta_{6} q^{22} - q^{24} - q^{27} + q^{28} + q^{30} - \zeta_{6} q^{32} + 2 \zeta_{6}^{2} q^{33} - q^{35} - \zeta_{6} q^{36} + q^{37} + \zeta_{6}^{2} q^{38} + \zeta_{6} q^{40} - \zeta_{6} q^{41} + \zeta_{6} q^{42} - 2 q^{44} + \zeta_{6} q^{45} + \zeta_{6}^{2} q^{47} - \zeta_{6}^{2} q^{48} - \zeta_{6}^{2} q^{54} + 2 q^{55} + \zeta_{6}^{2} q^{56} - q^{57} + \zeta_{6}^{2} q^{60} + \zeta_{6}^{2} q^{61} + \zeta_{6}^{2} q^{63} + q^{64} - 2 \zeta_{6} q^{66} + \zeta_{6} q^{67} - \zeta_{6}^{2} q^{70} + q^{71} + q^{72} - q^{73} + 2 \zeta_{6}^{2} q^{74} - \zeta_{6} q^{76} + 2 \zeta_{6} q^{77} + \zeta_{6}^{2} q^{79} - q^{80} + q^{81} + q^{82} - q^{84} - 2 \zeta_{6}^{2} q^{88} - q^{90} - \zeta_{6} q^{94} + \zeta_{6} q^{95} + \zeta_{6} q^{96} - 2 \zeta_{6}^{2} q^{99} +O(q^{100})$$ q + z^2 * q^2 - q^3 - z * q^4 + z * q^5 - z^2 * q^6 + z^2 * q^7 + q^8 + q^9 - q^10 - z^2 * q^11 + z * q^12 - z * q^14 - z * q^15 + z^2 * q^16 + z^2 * q^18 + q^19 - z^2 * q^20 - z^2 * q^21 + 2*z * q^22 - q^24 - q^27 + q^28 + q^30 - z * q^32 + 2*z^2 * q^33 - q^35 - z * q^36 + q^37 + z^2 * q^38 + z * q^40 - z * q^41 + z * q^42 - 2 * q^44 + z * q^45 + z^2 * q^47 - z^2 * q^48 - z^2 * q^54 + 2 * q^55 + z^2 * q^56 - q^57 + z^2 * q^60 + z^2 * q^61 + z^2 * q^63 + q^64 - 2*z * q^66 + z * q^67 - z^2 * q^70 + q^71 + q^72 - q^73 + 2*z^2 * q^74 - z * q^76 + 2*z * q^77 + z^2 * q^79 - q^80 + q^81 + q^82 - q^84 - 2*z^2 * q^88 - q^90 - z * q^94 + z * q^95 + z * q^96 - 2*z^2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 2 q^{3} - q^{4} + q^{5} + q^{6} - q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - q^2 - 2 * q^3 - q^4 + q^5 + q^6 - q^7 + 2 * q^8 + 2 * q^9 $$2 q - q^{2} - 2 q^{3} - q^{4} + q^{5} + q^{6} - q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} + q^{12} - q^{14} - q^{15} - q^{16} - q^{18} + 2 q^{19} + q^{20} + q^{21} + 2 q^{22} - 2 q^{24} - 2 q^{27} + 2 q^{28} + 2 q^{30} - q^{32} - 2 q^{33} - 2 q^{35} - q^{36} + 4 q^{37} - q^{38} + q^{40} - q^{41} + q^{42} - 4 q^{44} + q^{45} - q^{47} + q^{48} + q^{54} + 4 q^{55} - q^{56} - 2 q^{57} - q^{60} - 2 q^{61} - q^{63} + 2 q^{64} - 2 q^{66} + 2 q^{67} + q^{70} + 2 q^{71} + 2 q^{72} - 2 q^{73} - 2 q^{74} - q^{76} + 2 q^{77} - q^{79} - 2 q^{80} + 2 q^{81} + 2 q^{82} - 2 q^{84} + 2 q^{88} - 2 q^{90} - q^{94} + q^{95} + q^{96} + 2 q^{99}+O(q^{100})$$ 2 * q - q^2 - 2 * q^3 - q^4 + q^5 + q^6 - q^7 + 2 * q^8 + 2 * q^9 - 2 * q^10 + 2 * q^11 + q^12 - q^14 - q^15 - q^16 - q^18 + 2 * q^19 + q^20 + q^21 + 2 * q^22 - 2 * q^24 - 2 * q^27 + 2 * q^28 + 2 * q^30 - q^32 - 2 * q^33 - 2 * q^35 - q^36 + 4 * q^37 - q^38 + q^40 - q^41 + q^42 - 4 * q^44 + q^45 - q^47 + q^48 + q^54 + 4 * q^55 - q^56 - 2 * q^57 - q^60 - 2 * q^61 - q^63 + 2 * q^64 - 2 * q^66 + 2 * q^67 + q^70 + 2 * q^71 + 2 * q^72 - 2 * q^73 - 2 * q^74 - q^76 + 2 * q^77 - q^79 - 2 * q^80 + 2 * q^81 + 2 * q^82 - 2 * q^84 + 2 * q^88 - 2 * q^90 - q^94 + q^95 + q^96 + 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1476\mathbb{Z}\right)^\times$$.

 $$n$$ $$739$$ $$821$$ $$1441$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{6}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
655.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i −1.00000 −0.500000 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 1.00000 1.00000 −1.00000
1147.1 −0.500000 0.866025i −1.00000 −0.500000 + 0.866025i 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.00000 1.00000 −1.00000
 $$n$$: 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
164.d odd 2 1 CM by $$\Q(\sqrt{-41})$$
9.c even 3 1 inner
1476.o odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1476.1.o.a 2
4.b odd 2 1 1476.1.o.b yes 2
9.c even 3 1 inner 1476.1.o.a 2
36.f odd 6 1 1476.1.o.b yes 2
41.b even 2 1 1476.1.o.b yes 2
164.d odd 2 1 CM 1476.1.o.a 2
369.i even 6 1 1476.1.o.b yes 2
1476.o odd 6 1 inner 1476.1.o.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1476.1.o.a 2 1.a even 1 1 trivial
1476.1.o.a 2 9.c even 3 1 inner
1476.1.o.a 2 164.d odd 2 1 CM
1476.1.o.a 2 1476.o odd 6 1 inner
1476.1.o.b yes 2 4.b odd 2 1
1476.1.o.b yes 2 36.f odd 6 1
1476.1.o.b yes 2 41.b even 2 1
1476.1.o.b yes 2 369.i even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1476, [\chi])$$:

 $$T_{5}^{2} - T_{5} + 1$$ T5^2 - T5 + 1 $$T_{7}^{2} + T_{7} + 1$$ T7^2 + T7 + 1

## Hecke characteristic polynomials

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