# Properties

 Label 1224.1.m.b Level $1224$ Weight $1$ Character orbit 1224.m Analytic conductor $0.611$ Analytic rank $0$ Dimension $2$ Projective image $S_{4}$ CM/RM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.610855575463$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$S_{4}$$ Projective field Galois closure of 4.2.7344.1 Artin image $\GL(2,3)$ Artin field Galois closure of 8.2.11002604544.4

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{5} -\beta q^{7} +O(q^{10})$$ $$q + q^{5} -\beta q^{7} + q^{11} - q^{13} + q^{17} - q^{19} - q^{23} -\beta q^{31} -\beta q^{35} + \beta q^{37} + q^{41} + q^{43} + \beta q^{47} - q^{49} + \beta q^{53} + q^{55} -\beta q^{61} - q^{65} + \beta q^{73} -\beta q^{77} + \beta q^{83} + q^{85} -\beta q^{89} + \beta q^{91} - q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} + O(q^{10})$$ $$2q + 2q^{5} + 2q^{11} - 2q^{13} + 2q^{17} - 2q^{19} - 2q^{23} + 2q^{41} + 2q^{43} - 2q^{49} + 2q^{55} - 2q^{65} + 2q^{85} - 2q^{95} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
305.1
 1.41421i − 1.41421i
0 0 0 1.00000 0 1.41421i 0 0 0
305.2 0 0 0 1.00000 0 1.41421i 0 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
51.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.1.m.b yes 2
3.b odd 2 1 1224.1.m.a 2
4.b odd 2 1 2448.1.m.b 2
12.b even 2 1 2448.1.m.a 2
17.b even 2 1 1224.1.m.a 2
51.c odd 2 1 inner 1224.1.m.b yes 2
68.d odd 2 1 2448.1.m.a 2
204.h even 2 1 2448.1.m.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.1.m.a 2 3.b odd 2 1
1224.1.m.a 2 17.b even 2 1
1224.1.m.b yes 2 1.a even 1 1 trivial
1224.1.m.b yes 2 51.c odd 2 1 inner
2448.1.m.a 2 12.b even 2 1
2448.1.m.a 2 68.d odd 2 1
2448.1.m.b 2 4.b odd 2 1
2448.1.m.b 2 204.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 1$$ acting on $$S_{1}^{\mathrm{new}}(1224, [\chi])$$.

## Hecke characteristic polynomials

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