# Properties

 Label 1232.2.q.a Level $1232$ Weight $2$ Character orbit 1232.q Analytic conductor $9.838$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1232 = 2^{4} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1232.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.83756952902$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 154) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -3 + 3 \zeta_{6} ) q^{3} + 4 \zeta_{6} q^{5} + ( 2 + \zeta_{6} ) q^{7} -6 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -3 + 3 \zeta_{6} ) q^{3} + 4 \zeta_{6} q^{5} + ( 2 + \zeta_{6} ) q^{7} -6 \zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{11} - q^{13} -12 q^{15} + ( -2 + 2 \zeta_{6} ) q^{17} + 6 \zeta_{6} q^{19} + ( -9 + 6 \zeta_{6} ) q^{21} -2 \zeta_{6} q^{23} + ( -11 + 11 \zeta_{6} ) q^{25} + 9 q^{27} + q^{29} + ( 4 - 4 \zeta_{6} ) q^{31} -3 \zeta_{6} q^{33} + ( -4 + 12 \zeta_{6} ) q^{35} + 2 \zeta_{6} q^{37} + ( 3 - 3 \zeta_{6} ) q^{39} -2 q^{41} -4 q^{43} + ( 24 - 24 \zeta_{6} ) q^{45} + 2 \zeta_{6} q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} -6 \zeta_{6} q^{51} + ( 12 - 12 \zeta_{6} ) q^{53} -4 q^{55} -18 q^{57} + ( 9 - 9 \zeta_{6} ) q^{59} + 5 \zeta_{6} q^{61} + ( 6 - 18 \zeta_{6} ) q^{63} -4 \zeta_{6} q^{65} + ( -9 + 9 \zeta_{6} ) q^{67} + 6 q^{69} -4 q^{71} + ( 2 - 2 \zeta_{6} ) q^{73} -33 \zeta_{6} q^{75} + ( -3 + 2 \zeta_{6} ) q^{77} -15 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + 6 q^{83} -8 q^{85} + ( -3 + 3 \zeta_{6} ) q^{87} -6 \zeta_{6} q^{89} + ( -2 - \zeta_{6} ) q^{91} + 12 \zeta_{6} q^{93} + ( -24 + 24 \zeta_{6} ) q^{95} -5 q^{97} + 6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{3} + 4q^{5} + 5q^{7} - 6q^{9} + O(q^{10})$$ $$2q - 3q^{3} + 4q^{5} + 5q^{7} - 6q^{9} - q^{11} - 2q^{13} - 24q^{15} - 2q^{17} + 6q^{19} - 12q^{21} - 2q^{23} - 11q^{25} + 18q^{27} + 2q^{29} + 4q^{31} - 3q^{33} + 4q^{35} + 2q^{37} + 3q^{39} - 4q^{41} - 8q^{43} + 24q^{45} + 2q^{47} + 11q^{49} - 6q^{51} + 12q^{53} - 8q^{55} - 36q^{57} + 9q^{59} + 5q^{61} - 6q^{63} - 4q^{65} - 9q^{67} + 12q^{69} - 8q^{71} + 2q^{73} - 33q^{75} - 4q^{77} - 15q^{79} - 9q^{81} + 12q^{83} - 16q^{85} - 3q^{87} - 6q^{89} - 5q^{91} + 12q^{93} - 24q^{95} - 10q^{97} + 12q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times$$.

 $$n$$ $$309$$ $$353$$ $$463$$ $$673$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
177.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −1.50000 2.59808i 0 2.00000 3.46410i 0 2.50000 0.866025i 0 −3.00000 + 5.19615i 0
529.1 0 −1.50000 + 2.59808i 0 2.00000 + 3.46410i 0 2.50000 + 0.866025i 0 −3.00000 5.19615i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.2.q.a 2
4.b odd 2 1 154.2.e.d 2
7.c even 3 1 inner 1232.2.q.a 2
7.c even 3 1 8624.2.a.bd 1
7.d odd 6 1 8624.2.a.d 1
12.b even 2 1 1386.2.k.a 2
28.d even 2 1 1078.2.e.g 2
28.f even 6 1 1078.2.a.f 1
28.f even 6 1 1078.2.e.g 2
28.g odd 6 1 154.2.e.d 2
28.g odd 6 1 1078.2.a.a 1
84.j odd 6 1 9702.2.a.bb 1
84.n even 6 1 1386.2.k.a 2
84.n even 6 1 9702.2.a.cg 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.d 2 4.b odd 2 1
154.2.e.d 2 28.g odd 6 1
1078.2.a.a 1 28.g odd 6 1
1078.2.a.f 1 28.f even 6 1
1078.2.e.g 2 28.d even 2 1
1078.2.e.g 2 28.f even 6 1
1232.2.q.a 2 1.a even 1 1 trivial
1232.2.q.a 2 7.c even 3 1 inner
1386.2.k.a 2 12.b even 2 1
1386.2.k.a 2 84.n even 6 1
8624.2.a.d 1 7.d odd 6 1
8624.2.a.bd 1 7.c even 3 1
9702.2.a.bb 1 84.j odd 6 1
9702.2.a.cg 1 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1232, [\chi])$$:

 $$T_{3}^{2} + 3 T_{3} + 9$$ $$T_{13} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + 3 T + T^{2}$$
$5$ $$16 - 4 T + T^{2}$$
$7$ $$7 - 5 T + T^{2}$$
$11$ $$1 + T + T^{2}$$
$13$ $$( 1 + T )^{2}$$
$17$ $$4 + 2 T + T^{2}$$
$19$ $$36 - 6 T + T^{2}$$
$23$ $$4 + 2 T + T^{2}$$
$29$ $$( -1 + T )^{2}$$
$31$ $$16 - 4 T + T^{2}$$
$37$ $$4 - 2 T + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$4 - 2 T + T^{2}$$
$53$ $$144 - 12 T + T^{2}$$
$59$ $$81 - 9 T + T^{2}$$
$61$ $$25 - 5 T + T^{2}$$
$67$ $$81 + 9 T + T^{2}$$
$71$ $$( 4 + T )^{2}$$
$73$ $$4 - 2 T + T^{2}$$
$79$ $$225 + 15 T + T^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$( 5 + T )^{2}$$