# Properties

 Label 1224.1.bv.a Level $1224$ Weight $1$ Character orbit 1224.bv Analytic conductor $0.611$ Analytic rank $0$ Dimension $4$ Projective image $D_{8}$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1224 = 2^{3} \cdot 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1224.bv (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.610855575463$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 136) Projective image $$D_{8}$$ Projective field Galois closure of 8.0.1680747204608.3

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} -\zeta_{8}^{3} q^{8} +O(q^{10})$$ $$q -\zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} -\zeta_{8}^{3} q^{8} + ( 1 - \zeta_{8} ) q^{11} - q^{16} + \zeta_{8} q^{17} + ( -\zeta_{8} + \zeta_{8}^{2} ) q^{22} -\zeta_{8}^{3} q^{25} + \zeta_{8} q^{32} -\zeta_{8}^{2} q^{34} + ( \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{41} + ( 1 - \zeta_{8}^{2} ) q^{43} + ( \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{44} + \zeta_{8} q^{49} - q^{50} + ( 1 - \zeta_{8}^{2} ) q^{59} -\zeta_{8}^{2} q^{64} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{67} + \zeta_{8}^{3} q^{68} + ( \zeta_{8} - \zeta_{8}^{2} ) q^{73} + ( -1 - \zeta_{8}^{3} ) q^{82} + ( -1 - \zeta_{8}^{2} ) q^{83} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{86} + ( -1 - \zeta_{8}^{3} ) q^{88} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{89} + ( -1 + \zeta_{8}^{3} ) q^{97} -\zeta_{8}^{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 4q^{11} - 4q^{16} + 4q^{43} - 4q^{50} + 4q^{59} - 4q^{82} - 4q^{83} - 4q^{88} - 4q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$137$$ $$613$$ $$649$$ $$919$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{8}^{3}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
0.707107 + 0.707107i 0 1.00000i 0 0 0 −0.707107 + 0.707107i 0 0
451.1 0.707107 0.707107i 0 1.00000i 0 0 0 −0.707107 0.707107i 0 0
739.1 −0.707107 + 0.707107i 0 1.00000i 0 0 0 0.707107 + 0.707107i 0 0
1171.1 −0.707107 0.707107i 0 1.00000i 0 0 0 0.707107 0.707107i 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
17.d even 8 1 inner
136.p odd 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.1.bv.a 4
3.b odd 2 1 136.1.p.a 4
8.d odd 2 1 CM 1224.1.bv.a 4
12.b even 2 1 544.1.bl.a 4
15.d odd 2 1 3400.1.ce.a 4
15.e even 4 1 3400.1.br.a 4
15.e even 4 1 3400.1.br.b 4
17.d even 8 1 inner 1224.1.bv.a 4
24.f even 2 1 136.1.p.a 4
24.h odd 2 1 544.1.bl.a 4
51.c odd 2 1 2312.1.p.b 4
51.f odd 4 1 2312.1.p.a 4
51.f odd 4 1 2312.1.p.d 4
51.g odd 8 1 136.1.p.a 4
51.g odd 8 1 2312.1.p.a 4
51.g odd 8 1 2312.1.p.b 4
51.g odd 8 1 2312.1.p.d 4
51.i even 16 2 2312.1.e.b 4
51.i even 16 2 2312.1.f.c 4
51.i even 16 4 2312.1.j.c 8
120.m even 2 1 3400.1.ce.a 4
120.q odd 4 1 3400.1.br.a 4
120.q odd 4 1 3400.1.br.b 4
136.p odd 8 1 inner 1224.1.bv.a 4
204.p even 8 1 544.1.bl.a 4
255.v even 8 1 3400.1.br.b 4
255.y odd 8 1 3400.1.ce.a 4
255.ba even 8 1 3400.1.br.a 4
408.h even 2 1 2312.1.p.b 4
408.q even 4 1 2312.1.p.a 4
408.q even 4 1 2312.1.p.d 4
408.bd even 8 1 136.1.p.a 4
408.bd even 8 1 2312.1.p.a 4
408.bd even 8 1 2312.1.p.b 4
408.bd even 8 1 2312.1.p.d 4
408.be odd 8 1 544.1.bl.a 4
408.bg odd 16 2 2312.1.e.b 4
408.bg odd 16 2 2312.1.f.c 4
408.bg odd 16 4 2312.1.j.c 8
2040.dh even 8 1 3400.1.ce.a 4
2040.dp odd 8 1 3400.1.br.a 4
2040.dv odd 8 1 3400.1.br.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.1.p.a 4 3.b odd 2 1
136.1.p.a 4 24.f even 2 1
136.1.p.a 4 51.g odd 8 1
136.1.p.a 4 408.bd even 8 1
544.1.bl.a 4 12.b even 2 1
544.1.bl.a 4 24.h odd 2 1
544.1.bl.a 4 204.p even 8 1
544.1.bl.a 4 408.be odd 8 1
1224.1.bv.a 4 1.a even 1 1 trivial
1224.1.bv.a 4 8.d odd 2 1 CM
1224.1.bv.a 4 17.d even 8 1 inner
1224.1.bv.a 4 136.p odd 8 1 inner
2312.1.e.b 4 51.i even 16 2
2312.1.e.b 4 408.bg odd 16 2
2312.1.f.c 4 51.i even 16 2
2312.1.f.c 4 408.bg odd 16 2
2312.1.j.c 8 51.i even 16 4
2312.1.j.c 8 408.bg odd 16 4
2312.1.p.a 4 51.f odd 4 1
2312.1.p.a 4 51.g odd 8 1
2312.1.p.a 4 408.q even 4 1
2312.1.p.a 4 408.bd even 8 1
2312.1.p.b 4 51.c odd 2 1
2312.1.p.b 4 51.g odd 8 1
2312.1.p.b 4 408.h even 2 1
2312.1.p.b 4 408.bd even 8 1
2312.1.p.d 4 51.f odd 4 1
2312.1.p.d 4 51.g odd 8 1
2312.1.p.d 4 408.q even 4 1
2312.1.p.d 4 408.bd even 8 1
3400.1.br.a 4 15.e even 4 1
3400.1.br.a 4 120.q odd 4 1
3400.1.br.a 4 255.ba even 8 1
3400.1.br.a 4 2040.dp odd 8 1
3400.1.br.b 4 15.e even 4 1
3400.1.br.b 4 120.q odd 4 1
3400.1.br.b 4 255.v even 8 1
3400.1.br.b 4 2040.dv odd 8 1
3400.1.ce.a 4 15.d odd 2 1
3400.1.ce.a 4 120.m even 2 1
3400.1.ce.a 4 255.y odd 8 1
3400.1.ce.a 4 2040.dh even 8 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1224, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{4}$$
$3$ 1
$5$ $$1 + T^{8}$$
$7$ $$1 + T^{8}$$
$11$ $$( 1 - T )^{4}( 1 + T^{4} )$$
$13$ $$( 1 + T^{2} )^{4}$$
$17$ $$1 + T^{4}$$
$19$ $$( 1 + T^{4} )^{2}$$
$23$ $$1 + T^{8}$$
$29$ $$1 + T^{8}$$
$31$ $$1 + T^{8}$$
$37$ $$1 + T^{8}$$
$41$ $$( 1 + T^{2} )^{2}( 1 + T^{4} )$$
$43$ $$( 1 - T )^{4}( 1 + T^{2} )^{2}$$
$47$ $$( 1 + T^{2} )^{4}$$
$53$ $$( 1 + T^{4} )^{2}$$
$59$ $$( 1 - T )^{4}( 1 + T^{2} )^{2}$$
$61$ $$1 + T^{8}$$
$67$ $$( 1 + T^{4} )^{2}$$
$71$ $$1 + T^{8}$$
$73$ $$( 1 + T^{2} )^{2}( 1 + T^{4} )$$
$79$ $$1 + T^{8}$$
$83$ $$( 1 + T )^{4}( 1 + T^{2} )^{2}$$
$89$ $$( 1 + T^{4} )^{2}$$
$97$ $$( 1 + T )^{4}( 1 + T^{4} )$$