# Properties

 Label 1232.3.m.a Level $1232$ Weight $3$ Character orbit 1232.m Self dual yes Analytic conductor $33.570$ Analytic rank $0$ Dimension $4$ CM discriminant -308 Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1232 = 2^{4} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1232.m (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$33.5695685692$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{7}, \sqrt{11})$$ Defining polynomial: $$x^{4} - 9 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -7 q^{7} + ( 9 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -7 q^{7} + ( 9 + \beta_{2} ) q^{9} -11 q^{11} + ( 2 \beta_{1} - \beta_{3} ) q^{13} + ( 4 \beta_{1} + \beta_{3} ) q^{17} -7 \beta_{1} q^{21} + 25 q^{25} + ( 17 \beta_{1} + \beta_{3} ) q^{27} + ( 7 \beta_{1} - 2 \beta_{3} ) q^{31} -11 \beta_{1} q^{33} -3 \beta_{2} q^{37} + ( 34 + \beta_{2} ) q^{39} + ( 11 \beta_{1} + 2 \beta_{3} ) q^{41} -3 \beta_{2} q^{43} + ( \beta_{1} + 4 \beta_{3} ) q^{47} + 49 q^{49} + ( 74 + 5 \beta_{2} ) q^{51} -3 \beta_{2} q^{53} + ( -2 \beta_{1} - 5 \beta_{3} ) q^{59} + ( 19 \beta_{1} - 2 \beta_{3} ) q^{61} + ( -63 - 7 \beta_{2} ) q^{63} + ( -14 \beta_{1} - 5 \beta_{3} ) q^{73} + 25 \beta_{1} q^{75} + 77 q^{77} + 9 \beta_{2} q^{79} + ( 227 + 9 \beta_{2} ) q^{81} + ( -14 \beta_{1} + 7 \beta_{3} ) q^{91} + ( 122 + 5 \beta_{2} ) q^{93} + ( -99 - 11 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 28 q^{7} + 36 q^{9} + O(q^{10})$$ $$4 q - 28 q^{7} + 36 q^{9} - 44 q^{11} + 100 q^{25} + 136 q^{39} + 196 q^{49} + 296 q^{51} - 252 q^{63} + 308 q^{77} + 908 q^{81} + 488 q^{93} - 396 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 9 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$4 \nu^{2} - 18$$ $$\beta_{3}$$ $$=$$ $$8 \nu^{3} - 70 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + 18$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{3} + 35 \beta_{1}$$$$)/8$$

## Character values

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

 $$n$$ $$309$$ $$353$$ $$463$$ $$673$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1231.1
 −2.98119 −0.335437 0.335437 2.98119
0 −5.96238 0 0 0 −7.00000 0 26.5499 0
1231.2 0 −0.670873 0 0 0 −7.00000 0 −8.54993 0
1231.3 0 0.670873 0 0 0 −7.00000 0 −8.54993 0
1231.4 0 5.96238 0 0 0 −7.00000 0 26.5499 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
308.g odd 2 1 CM by $$\Q(\sqrt{-77})$$
7.b odd 2 1 inner
44.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.3.m.a 4
4.b odd 2 1 1232.3.m.b yes 4
7.b odd 2 1 inner 1232.3.m.a 4
11.b odd 2 1 1232.3.m.b yes 4
28.d even 2 1 1232.3.m.b yes 4
44.c even 2 1 inner 1232.3.m.a 4
77.b even 2 1 1232.3.m.b yes 4
308.g odd 2 1 CM 1232.3.m.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1232.3.m.a 4 1.a even 1 1 trivial
1232.3.m.a 4 7.b odd 2 1 inner
1232.3.m.a 4 44.c even 2 1 inner
1232.3.m.a 4 308.g odd 2 1 CM
1232.3.m.b yes 4 4.b odd 2 1
1232.3.m.b yes 4 11.b odd 2 1
1232.3.m.b yes 4 28.d even 2 1
1232.3.m.b yes 4 77.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1232, [\chi])$$:

 $$T_{3}^{4} - 36 T_{3}^{2} + 16$$ $$T_{107} + 38$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$16 - 36 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 7 + T )^{4}$$
$11$ $$( 11 + T )^{4}$$
$13$ $$44944 - 676 T^{2} + T^{4}$$
$17$ $$309136 - 1156 T^{2} + T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$3225616 - 3844 T^{2} + T^{4}$$
$37$ $$( -2772 + T^{2} )^{2}$$
$41$ $$7907344 - 6724 T^{2} + T^{4}$$
$43$ $$( -2772 + T^{2} )^{2}$$
$47$ $$3083536 - 8836 T^{2} + T^{4}$$
$53$ $$( -2772 + T^{2} )^{2}$$
$59$ $$10523536 - 13924 T^{2} + T^{4}$$
$61$ $$39790864 - 14884 T^{2} + T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$113124496 - 21316 T^{2} + T^{4}$$
$79$ $$( -24948 + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$
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