# Properties

 Label 1764.1.y.c Level 1764 Weight 1 Character orbit 1764.y Analytic conductor 0.880 Analytic rank 0 Dimension 4 Projective image $$D_{2}$$ CM/RM discs -7, -84, 12 Inner twists 16

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1764.y (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.880350682285$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{3}, \sqrt{-7})$$ Artin image $C_3\times D_4:C_2$ Artin field Galois closure of $$\mathbb{Q}[x]/(x^{24} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} -\zeta_{12}^{3} q^{8} +O(q^{10})$$ $$q -\zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} -\zeta_{12}^{3} q^{8} + 2 \zeta_{12}^{5} q^{11} + \zeta_{12}^{4} q^{16} + 2 q^{22} + 2 \zeta_{12} q^{23} + \zeta_{12}^{2} q^{25} -\zeta_{12}^{5} q^{32} + 2 \zeta_{12}^{4} q^{37} -2 \zeta_{12} q^{44} -2 \zeta_{12}^{2} q^{46} -\zeta_{12}^{3} q^{50} - q^{64} + 2 \zeta_{12}^{3} q^{71} -2 \zeta_{12}^{5} q^{74} + 2 \zeta_{12}^{2} q^{88} + 2 \zeta_{12}^{3} q^{92} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + O(q^{10})$$ $$4q + 2q^{4} - 2q^{16} + 8q^{22} + 2q^{25} - 4q^{37} - 4q^{46} - 4q^{64} + 4q^{88} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{12}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
667.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 0 1.00000i 0 0
667.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 0 1.00000i 0 0
1243.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 0 1.00000i 0 0
1243.2 0.866025 0.500000i 0 0.500000 0.866025i 0 0 0 1.00000i 0 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.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
12.b even 2 1 RM by $$\Q(\sqrt{3})$$
84.h odd 2 1 CM by $$\Q(\sqrt{-21})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
84.j odd 6 1 inner
84.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.1.y.c 4
3.b odd 2 1 inner 1764.1.y.c 4
4.b odd 2 1 inner 1764.1.y.c 4
7.b odd 2 1 CM 1764.1.y.c 4
7.c even 3 1 1764.1.g.c 2
7.c even 3 1 inner 1764.1.y.c 4
7.d odd 6 1 1764.1.g.c 2
7.d odd 6 1 inner 1764.1.y.c 4
12.b even 2 1 RM 1764.1.y.c 4
21.c even 2 1 inner 1764.1.y.c 4
21.g even 6 1 1764.1.g.c 2
21.g even 6 1 inner 1764.1.y.c 4
21.h odd 6 1 1764.1.g.c 2
21.h odd 6 1 inner 1764.1.y.c 4
28.d even 2 1 inner 1764.1.y.c 4
28.f even 6 1 1764.1.g.c 2
28.f even 6 1 inner 1764.1.y.c 4
28.g odd 6 1 1764.1.g.c 2
28.g odd 6 1 inner 1764.1.y.c 4
84.h odd 2 1 CM 1764.1.y.c 4
84.j odd 6 1 1764.1.g.c 2
84.j odd 6 1 inner 1764.1.y.c 4
84.n even 6 1 1764.1.g.c 2
84.n even 6 1 inner 1764.1.y.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.1.g.c 2 7.c even 3 1
1764.1.g.c 2 7.d odd 6 1
1764.1.g.c 2 21.g even 6 1
1764.1.g.c 2 21.h odd 6 1
1764.1.g.c 2 28.f even 6 1
1764.1.g.c 2 28.g odd 6 1
1764.1.g.c 2 84.j odd 6 1
1764.1.g.c 2 84.n even 6 1
1764.1.y.c 4 1.a even 1 1 trivial
1764.1.y.c 4 3.b odd 2 1 inner
1764.1.y.c 4 4.b odd 2 1 inner
1764.1.y.c 4 7.b odd 2 1 CM
1764.1.y.c 4 7.c even 3 1 inner
1764.1.y.c 4 7.d odd 6 1 inner
1764.1.y.c 4 12.b even 2 1 RM
1764.1.y.c 4 21.c even 2 1 inner
1764.1.y.c 4 21.g even 6 1 inner
1764.1.y.c 4 21.h odd 6 1 inner
1764.1.y.c 4 28.d even 2 1 inner
1764.1.y.c 4 28.f even 6 1 inner
1764.1.y.c 4 28.g odd 6 1 inner
1764.1.y.c 4 84.h odd 2 1 CM
1764.1.y.c 4 84.j odd 6 1 inner
1764.1.y.c 4 84.n even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1764, [\chi])$$:

 $$T_{5}$$ $$T_{11}^{4} - 4 T_{11}^{2} + 16$$ $$T_{29}$$

## Hecke characteristic polynomials

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