# Properties

 Label 1224.1.n.c Level $1224$ Weight $1$ Character orbit 1224.n Self dual yes Analytic conductor $0.611$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -136 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1224 = 2^{3} \cdot 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1224.n (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.610855575463$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{4}$$ Projective field Galois closure of 4.0.9792.1 Artin image $D_8$ Artin field Galois closure of 8.0.14670139392.3

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} -\beta q^{5} + \beta q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} -\beta q^{5} + \beta q^{7} + q^{8} -\beta q^{10} + \beta q^{14} + q^{16} - q^{17} -\beta q^{20} + \beta q^{23} + q^{25} + \beta q^{28} + \beta q^{29} -\beta q^{31} + q^{32} - q^{34} -2 q^{35} -\beta q^{37} -\beta q^{40} -2 q^{43} + \beta q^{46} + q^{49} + q^{50} + \beta q^{56} + \beta q^{58} + \beta q^{61} -\beta q^{62} + q^{64} - q^{68} -2 q^{70} -\beta q^{71} -\beta q^{74} -\beta q^{79} -\beta q^{80} + \beta q^{85} -2 q^{86} -2 q^{89} + \beta q^{92} + q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{4} + 2q^{8} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{4} + 2q^{8} + 2q^{16} - 2q^{17} + 2q^{25} + 2q^{32} - 2q^{34} - 4q^{35} - 4q^{43} + 2q^{49} + 2q^{50} + 2q^{64} - 2q^{68} - 4q^{70} - 4q^{86} - 4q^{89} + 2q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$$.

 $$n$$ $$137$$ $$613$$ $$649$$ $$919$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
883.1
 1.41421 −1.41421
1.00000 0 1.00000 −1.41421 0 1.41421 1.00000 0 −1.41421
883.2 1.00000 0 1.00000 1.41421 0 −1.41421 1.00000 0 1.41421
 $$n$$: 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
136.e odd 2 1 CM by $$\Q(\sqrt{-34})$$
8.d odd 2 1 inner
17.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.1.n.c yes 2
3.b odd 2 1 1224.1.n.b 2
8.d odd 2 1 inner 1224.1.n.c yes 2
17.b even 2 1 inner 1224.1.n.c yes 2
24.f even 2 1 1224.1.n.b 2
51.c odd 2 1 1224.1.n.b 2
136.e odd 2 1 CM 1224.1.n.c yes 2
408.h even 2 1 1224.1.n.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.1.n.b 2 3.b odd 2 1
1224.1.n.b 2 24.f even 2 1
1224.1.n.b 2 51.c odd 2 1
1224.1.n.b 2 408.h even 2 1
1224.1.n.c yes 2 1.a even 1 1 trivial
1224.1.n.c yes 2 8.d odd 2 1 inner
1224.1.n.c yes 2 17.b even 2 1 inner
1224.1.n.c yes 2 136.e odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1224, [\chi])$$:

 $$T_{5}^{2} - 2$$ $$T_{89} + 2$$

## Hecke characteristic polynomials

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