# Properties

 Label 1104.2.m.a Level $1104$ Weight $2$ Character orbit 1104.m Analytic conductor $8.815$ Analytic rank $0$ Dimension $6$ CM discriminant -23 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1104 = 2^{4} \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1104.m (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.81548438315$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.8869743.1 Defining polynomial: $$x^{6} - 3 x^{3} + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{3}\cdot 3$$ Twist minimal: no (minimal twist has level 69) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} -\beta_{3} q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} -\beta_{3} q^{9} + ( -\beta_{2} - \beta_{3} - \beta_{4} ) q^{13} -\beta_{5} q^{23} -5 q^{25} + ( -2 + \beta_{5} ) q^{27} + ( -3 \beta_{2} - \beta_{3} + \beta_{4} ) q^{29} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{31} + ( 4 + \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{39} + ( 2 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{41} + ( \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{47} + 7 q^{49} -2 \beta_{5} q^{59} + ( -2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{69} + ( 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{71} + ( -2 \beta_{1} + 5 \beta_{2} - \beta_{3} + \beta_{4} ) q^{73} + 5 \beta_{2} q^{75} + ( 2 \beta_{1} + \beta_{2} + \beta_{4} ) q^{81} + ( -8 - \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{87} + ( -1 + 2 \beta_{1} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{93} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + O(q^{10})$$ $$6q - 30q^{25} - 12q^{27} + 24q^{39} + 42q^{49} - 48q^{87} - 6q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{3} + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + \nu^{2} + 8 \nu$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} - \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} + 7 \nu^{2}$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{5} - \nu^{2} + 4 \nu$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$2 \nu^{3} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + 2 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{4} + 3 \beta_{3} + \beta_{1}$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{5} + 3$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{4} + 12 \beta_{2} + 2 \beta_{1}$$$$)/6$$ $$\nu^{5}$$ $$=$$ $$($$$$-7 \beta_{4} - 3 \beta_{3} + 7 \beta_{1}$$$$)/6$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1104\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$277$$ $$415$$ $$737$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
689.1
 −0.261988 + 1.38973i −0.261988 − 1.38973i 1.33454 + 0.467979i 1.33454 − 0.467979i −1.07255 − 0.921756i −1.07255 + 0.921756i
0 −1.60074 0.661546i 0 0 0 0 0 2.12471 + 2.11792i 0
689.2 0 −1.60074 + 0.661546i 0 0 0 0 0 2.12471 2.11792i 0
689.3 0 0.227452 1.71705i 0 0 0 0 0 −2.89653 0.781094i 0
689.4 0 0.227452 + 1.71705i 0 0 0 0 0 −2.89653 + 0.781094i 0
689.5 0 1.37328 1.05550i 0 0 0 0 0 0.771819 2.89902i 0
689.6 0 1.37328 + 1.05550i 0 0 0 0 0 0.771819 + 2.89902i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 689.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by $$\Q(\sqrt{-23})$$
3.b odd 2 1 inner
69.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1104.2.m.a 6
3.b odd 2 1 inner 1104.2.m.a 6
4.b odd 2 1 69.2.c.a 6
12.b even 2 1 69.2.c.a 6
23.b odd 2 1 CM 1104.2.m.a 6
69.c even 2 1 inner 1104.2.m.a 6
92.b even 2 1 69.2.c.a 6
276.h odd 2 1 69.2.c.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.2.c.a 6 4.b odd 2 1
69.2.c.a 6 12.b even 2 1
69.2.c.a 6 92.b even 2 1
69.2.c.a 6 276.h odd 2 1
1104.2.m.a 6 1.a even 1 1 trivial
1104.2.m.a 6 3.b odd 2 1 inner
1104.2.m.a 6 23.b odd 2 1 CM
1104.2.m.a 6 69.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{2}^{\mathrm{new}}(1104, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$27 + 4 T^{3} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$T^{6}$$
$11$ $$T^{6}$$
$13$ $$( -74 - 39 T + T^{3} )^{2}$$
$17$ $$T^{6}$$
$19$ $$T^{6}$$
$23$ $$( 23 + T^{2} )^{3}$$
$29$ $$18032 + 7569 T^{2} + 174 T^{4} + T^{6}$$
$31$ $$( 344 - 93 T + T^{3} )^{2}$$
$37$ $$T^{6}$$
$41$ $$94208 + 15129 T^{2} + 246 T^{4} + T^{6}$$
$43$ $$T^{6}$$
$47$ $$412988 + 19881 T^{2} + 282 T^{4} + T^{6}$$
$53$ $$T^{6}$$
$59$ $$( 92 + T^{2} )^{3}$$
$61$ $$T^{6}$$
$67$ $$T^{6}$$
$71$ $$48668 + 45369 T^{2} + 426 T^{4} + T^{6}$$
$73$ $$( -1226 - 219 T + T^{3} )^{2}$$
$79$ $$T^{6}$$
$83$ $$T^{6}$$
$89$ $$T^{6}$$
$97$ $$T^{6}$$