# Properties

 Label 1232.1.cd.a Level $1232$ Weight $1$ Character orbit 1232.cd Analytic conductor $0.615$ Analytic rank $0$ Dimension $4$ Projective image $D_{5}$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1232 = 2^{4} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1232.cd (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.614848095564$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.717409.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{10}^{3} q^{7} + \zeta_{10}^{4} q^{9} +O(q^{10})$$ $$q + \zeta_{10}^{3} q^{7} + \zeta_{10}^{4} q^{9} + \zeta_{10} q^{11} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{23} + \zeta_{10}^{2} q^{25} + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{29} + ( 1 - \zeta_{10} ) q^{37} + ( \zeta_{10} - \zeta_{10}^{4} ) q^{43} -\zeta_{10} q^{49} + ( 1 - \zeta_{10}^{3} ) q^{53} -\zeta_{10}^{2} q^{63} + ( \zeta_{10} - \zeta_{10}^{4} ) q^{67} + ( -1 - \zeta_{10}^{2} ) q^{71} + \zeta_{10}^{4} q^{77} + ( -1 + \zeta_{10}^{3} ) q^{79} -\zeta_{10}^{3} q^{81} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{7} - q^{9} + O(q^{10})$$ $$4 q + q^{7} - q^{9} + q^{11} + 2 q^{23} - q^{25} - 2 q^{29} + 3 q^{37} + 2 q^{43} - q^{49} + 3 q^{53} + q^{63} + 2 q^{67} - 3 q^{71} - q^{77} - 3 q^{79} - q^{81} - 4 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$309$$ $$353$$ $$463$$ $$673$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 −0.309017 − 0.951057i 0.809017 − 0.587785i 0.809017 + 0.587785i −0.309017 + 0.951057i
0 0 0 0 0 0.809017 + 0.587785i 0 0.309017 0.951057i 0
433.1 0 0 0 0 0 −0.309017 0.951057i 0 −0.809017 0.587785i 0
993.1 0 0 0 0 0 −0.309017 + 0.951057i 0 −0.809017 + 0.587785i 0
1105.1 0 0 0 0 0 0.809017 0.587785i 0 0.309017 + 0.951057i 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.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
11.c even 5 1 inner
77.j odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.1.cd.a 4
4.b odd 2 1 77.1.j.a 4
7.b odd 2 1 CM 1232.1.cd.a 4
11.c even 5 1 inner 1232.1.cd.a 4
12.b even 2 1 693.1.br.a 4
20.d odd 2 1 1925.1.bn.a 4
20.e even 4 2 1925.1.cb.a 8
28.d even 2 1 77.1.j.a 4
28.f even 6 2 539.1.u.a 8
28.g odd 6 2 539.1.u.a 8
44.c even 2 1 847.1.j.b 4
44.g even 10 1 847.1.d.b 2
44.g even 10 2 847.1.j.a 4
44.g even 10 1 847.1.j.b 4
44.h odd 10 1 77.1.j.a 4
44.h odd 10 1 847.1.d.a 2
44.h odd 10 2 847.1.j.c 4
77.j odd 10 1 inner 1232.1.cd.a 4
84.h odd 2 1 693.1.br.a 4
132.o even 10 1 693.1.br.a 4
140.c even 2 1 1925.1.bn.a 4
140.j odd 4 2 1925.1.cb.a 8
220.n odd 10 1 1925.1.bn.a 4
220.v even 20 2 1925.1.cb.a 8
308.g odd 2 1 847.1.j.b 4
308.s odd 10 1 847.1.d.b 2
308.s odd 10 2 847.1.j.a 4
308.s odd 10 1 847.1.j.b 4
308.t even 10 1 77.1.j.a 4
308.t even 10 1 847.1.d.a 2
308.t even 10 2 847.1.j.c 4
308.bb odd 30 2 539.1.u.a 8
308.be even 30 2 539.1.u.a 8
924.bk odd 10 1 693.1.br.a 4
1540.bw even 10 1 1925.1.bn.a 4
1540.ct odd 20 2 1925.1.cb.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.1.j.a 4 4.b odd 2 1
77.1.j.a 4 28.d even 2 1
77.1.j.a 4 44.h odd 10 1
77.1.j.a 4 308.t even 10 1
539.1.u.a 8 28.f even 6 2
539.1.u.a 8 28.g odd 6 2
539.1.u.a 8 308.bb odd 30 2
539.1.u.a 8 308.be even 30 2
693.1.br.a 4 12.b even 2 1
693.1.br.a 4 84.h odd 2 1
693.1.br.a 4 132.o even 10 1
693.1.br.a 4 924.bk odd 10 1
847.1.d.a 2 44.h odd 10 1
847.1.d.a 2 308.t even 10 1
847.1.d.b 2 44.g even 10 1
847.1.d.b 2 308.s odd 10 1
847.1.j.a 4 44.g even 10 2
847.1.j.a 4 308.s odd 10 2
847.1.j.b 4 44.c even 2 1
847.1.j.b 4 44.g even 10 1
847.1.j.b 4 308.g odd 2 1
847.1.j.b 4 308.s odd 10 1
847.1.j.c 4 44.h odd 10 2
847.1.j.c 4 308.t even 10 2
1232.1.cd.a 4 1.a even 1 1 trivial
1232.1.cd.a 4 7.b odd 2 1 CM
1232.1.cd.a 4 11.c even 5 1 inner
1232.1.cd.a 4 77.j odd 10 1 inner
1925.1.bn.a 4 20.d odd 2 1
1925.1.bn.a 4 140.c even 2 1
1925.1.bn.a 4 220.n odd 10 1
1925.1.bn.a 4 1540.bw even 10 1
1925.1.cb.a 8 20.e even 4 2
1925.1.cb.a 8 140.j odd 4 2
1925.1.cb.a 8 220.v even 20 2
1925.1.cb.a 8 1540.ct odd 20 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1232, [\chi])$$.

## Hecke characteristic polynomials

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