# Properties

 Label 2352.1.by.b Level $2352$ Weight $1$ Character orbit 2352.by Analytic conductor $1.174$ Analytic rank $0$ Dimension $6$ Projective image $D_{14}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2352.by (of order $$14$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.17380090971$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{14})$$ Defining polynomial: $$x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{14}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{14} + \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{14} q^{3} -\zeta_{14}^{5} q^{7} + \zeta_{14}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{14} q^{3} -\zeta_{14}^{5} q^{7} + \zeta_{14}^{2} q^{9} + ( -\zeta_{14}^{4} + \zeta_{14}^{6} ) q^{13} + ( \zeta_{14}^{2} - \zeta_{14}^{5} ) q^{19} -\zeta_{14}^{6} q^{21} -\zeta_{14}^{2} q^{25} + \zeta_{14}^{3} q^{27} + ( \zeta_{14}^{3} - \zeta_{14}^{4} ) q^{31} + ( -1 + \zeta_{14} ) q^{37} + ( -1 - \zeta_{14}^{5} ) q^{39} + ( \zeta_{14} + \zeta_{14}^{4} ) q^{43} -\zeta_{14}^{3} q^{49} + ( \zeta_{14}^{3} - \zeta_{14}^{6} ) q^{57} + ( -1 - \zeta_{14} ) q^{61} + q^{63} + ( \zeta_{14} + \zeta_{14}^{6} ) q^{67} + ( \zeta_{14}^{5} + \zeta_{14}^{6} ) q^{73} -\zeta_{14}^{3} q^{75} + ( -\zeta_{14}^{3} - \zeta_{14}^{4} ) q^{79} + \zeta_{14}^{4} q^{81} + ( -\zeta_{14}^{2} + \zeta_{14}^{4} ) q^{91} + ( \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{93} + ( -\zeta_{14}^{2} - \zeta_{14}^{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + q^{3} - q^{7} - q^{9} + O(q^{10})$$ $$6q + q^{3} - q^{7} - q^{9} - 2q^{19} + q^{21} + q^{25} + q^{27} + 2q^{31} - 5q^{37} - 7q^{39} - q^{49} + 2q^{57} - 7q^{61} + 6q^{63} - q^{75} - q^{81} - 2q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1471$$ $$1765$$ $$2257$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$-\zeta_{14}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
335.1
 0.900969 − 0.433884i −0.623490 + 0.781831i 0.222521 − 0.974928i 0.222521 + 0.974928i −0.623490 − 0.781831i 0.900969 + 0.433884i
0 0.900969 0.433884i 0 0 0 0.623490 + 0.781831i 0 0.623490 0.781831i 0
671.1 0 −0.623490 + 0.781831i 0 0 0 −0.222521 + 0.974928i 0 −0.222521 0.974928i 0
1007.1 0 0.222521 0.974928i 0 0 0 −0.900969 + 0.433884i 0 −0.900969 0.433884i 0
1343.1 0 0.222521 + 0.974928i 0 0 0 −0.900969 0.433884i 0 −0.900969 + 0.433884i 0
1679.1 0 −0.623490 0.781831i 0 0 0 −0.222521 0.974928i 0 −0.222521 + 0.974928i 0
2015.1 0 0.900969 + 0.433884i 0 0 0 0.623490 0.781831i 0 0.623490 + 0.781831i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2015.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
196.j even 14 1 inner
588.r odd 14 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.1.by.b yes 6
3.b odd 2 1 CM 2352.1.by.b yes 6
4.b odd 2 1 2352.1.by.a 6
12.b even 2 1 2352.1.by.a 6
49.f odd 14 1 2352.1.by.a 6
147.k even 14 1 2352.1.by.a 6
196.j even 14 1 inner 2352.1.by.b yes 6
588.r odd 14 1 inner 2352.1.by.b yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.1.by.a 6 4.b odd 2 1
2352.1.by.a 6 12.b even 2 1
2352.1.by.a 6 49.f odd 14 1
2352.1.by.a 6 147.k even 14 1
2352.1.by.b yes 6 1.a even 1 1 trivial
2352.1.by.b yes 6 3.b odd 2 1 CM
2352.1.by.b yes 6 196.j even 14 1 inner
2352.1.by.b yes 6 588.r odd 14 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{19}^{3} + T_{19}^{2} - 2 T_{19} - 1$$ acting on $$S_{1}^{\mathrm{new}}(2352, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$11$ $$T^{6}$$
$13$ $$7 + 7 T - 7 T^{3} + T^{6}$$
$17$ $$T^{6}$$
$19$ $$( -1 - 2 T + T^{2} + T^{3} )^{2}$$
$23$ $$T^{6}$$
$29$ $$T^{6}$$
$31$ $$( 1 - 2 T - T^{2} + T^{3} )^{2}$$
$37$ $$1 + 3 T + 9 T^{2} + 13 T^{3} + 11 T^{4} + 5 T^{5} + T^{6}$$
$41$ $$T^{6}$$
$43$ $$7 - 14 T + 7 T^{2} + T^{6}$$
$47$ $$T^{6}$$
$53$ $$T^{6}$$
$59$ $$T^{6}$$
$61$ $$7 + 21 T + 35 T^{2} + 35 T^{3} + 21 T^{4} + 7 T^{5} + T^{6}$$
$67$ $$7 + 14 T^{2} + 7 T^{4} + T^{6}$$
$71$ $$T^{6}$$
$73$ $$7 - 14 T + 7 T^{2} + T^{6}$$
$79$ $$7 + 14 T^{2} + 7 T^{4} + T^{6}$$
$83$ $$T^{6}$$
$89$ $$T^{6}$$
$97$ $$7 + 14 T^{2} + 7 T^{4} + T^{6}$$