# Properties

 Label 1232.2.a.e Level $1232$ Weight $2$ Character orbit 1232.a Self dual yes Analytic conductor $9.838$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1232 = 2^{4} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1232.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.83756952902$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 154) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 4 q^{5} + q^{7} - 3 q^{9} + O(q^{10})$$ $$q - 4 q^{5} + q^{7} - 3 q^{9} + q^{11} + 2 q^{13} - 4 q^{17} + 6 q^{19} - 4 q^{23} + 11 q^{25} - 2 q^{29} + 2 q^{31} - 4 q^{35} + 10 q^{37} + 4 q^{41} + 8 q^{43} + 12 q^{45} - 2 q^{47} + q^{49} + 6 q^{53} - 4 q^{55} + 12 q^{59} - 14 q^{61} - 3 q^{63} - 8 q^{65} + 12 q^{67} + 8 q^{71} + 4 q^{73} + q^{77} + 9 q^{81} + 6 q^{83} + 16 q^{85} - 6 q^{89} + 2 q^{91} - 24 q^{95} - 14 q^{97} - 3 q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 −4.00000 0 1.00000 0 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.2.a.e 1
4.b odd 2 1 154.2.a.a 1
7.b odd 2 1 8624.2.a.r 1
8.b even 2 1 4928.2.a.w 1
8.d odd 2 1 4928.2.a.v 1
12.b even 2 1 1386.2.a.l 1
20.d odd 2 1 3850.2.a.u 1
20.e even 4 2 3850.2.c.j 2
28.d even 2 1 1078.2.a.d 1
28.f even 6 2 1078.2.e.i 2
28.g odd 6 2 1078.2.e.j 2
44.c even 2 1 1694.2.a.g 1
84.h odd 2 1 9702.2.a.ba 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.a 1 4.b odd 2 1
1078.2.a.d 1 28.d even 2 1
1078.2.e.i 2 28.f even 6 2
1078.2.e.j 2 28.g odd 6 2
1232.2.a.e 1 1.a even 1 1 trivial
1386.2.a.l 1 12.b even 2 1
1694.2.a.g 1 44.c even 2 1
3850.2.a.u 1 20.d odd 2 1
3850.2.c.j 2 20.e even 4 2
4928.2.a.v 1 8.d odd 2 1
4928.2.a.w 1 8.b even 2 1
8624.2.a.r 1 7.b odd 2 1
9702.2.a.ba 1 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1232))$$:

 $$T_{3}$$ $$T_{5} + 4$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$4 + T$$
$7$ $$-1 + T$$
$11$ $$-1 + T$$
$13$ $$-2 + T$$
$17$ $$4 + T$$
$19$ $$-6 + T$$
$23$ $$4 + T$$
$29$ $$2 + T$$
$31$ $$-2 + T$$
$37$ $$-10 + T$$
$41$ $$-4 + T$$
$43$ $$-8 + T$$
$47$ $$2 + T$$
$53$ $$-6 + T$$
$59$ $$-12 + T$$
$61$ $$14 + T$$
$67$ $$-12 + T$$
$71$ $$-8 + T$$
$73$ $$-4 + T$$
$79$ $$T$$
$83$ $$-6 + T$$
$89$ $$6 + T$$
$97$ $$14 + T$$
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