# Properties

 Label 2352.1.dj.b Level $2352$ Weight $1$ Character orbit 2352.dj Analytic conductor $1.174$ Analytic rank $0$ Dimension $12$ Projective image $D_{42}$ 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.dj (of order $$42$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.17380090971$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\Q(\zeta_{21})$$ Defining polynomial: $$x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{42}$$ Projective field Galois closure of $$\mathbb{Q}[x]/(x^{42} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{42}^{4} q^{3} -\zeta_{42}^{20} q^{7} + \zeta_{42}^{8} q^{9} +O(q^{10})$$ $$q + \zeta_{42}^{4} q^{3} -\zeta_{42}^{20} q^{7} + \zeta_{42}^{8} q^{9} + ( -\zeta_{42}^{2} + \zeta_{42}^{10} ) q^{13} + ( \zeta_{42}^{13} + \zeta_{42}^{15} ) q^{19} + \zeta_{42}^{3} q^{21} + \zeta_{42} q^{25} + \zeta_{42}^{12} q^{27} + ( -\zeta_{42}^{5} - \zeta_{42}^{9} ) q^{31} + ( \zeta_{42}^{7} - \zeta_{42}^{18} ) q^{37} + ( -\zeta_{42}^{6} + \zeta_{42}^{14} ) q^{39} + ( -\zeta_{42}^{11} - \zeta_{42}^{16} ) q^{43} -\zeta_{42}^{19} q^{49} + ( \zeta_{42}^{17} + \zeta_{42}^{19} ) q^{57} + ( -\zeta_{42}^{11} - \zeta_{42}^{14} ) q^{61} + \zeta_{42}^{7} q^{63} + ( \zeta_{42}^{17} + \zeta_{42}^{18} ) q^{67} + ( -\zeta_{42}^{6} - \zeta_{42}^{17} ) q^{73} + \zeta_{42}^{5} q^{75} + ( -\zeta_{42}^{12} + \zeta_{42}^{16} ) q^{79} + \zeta_{42}^{16} q^{81} + ( -\zeta_{42} + \zeta_{42}^{9} ) q^{91} + ( -\zeta_{42}^{9} - \zeta_{42}^{13} ) q^{93} + ( \zeta_{42}^{6} + \zeta_{42}^{15} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + q^{3} - q^{7} + q^{9} + O(q^{10})$$ $$12q + q^{3} - q^{7} + q^{9} + q^{19} + 2q^{21} - q^{25} - 2q^{27} - q^{31} + 8q^{37} - 4q^{39} + q^{49} - 2q^{57} + 7q^{61} + 6q^{63} - 3q^{67} + 3q^{73} - q^{75} + 3q^{79} + q^{81} + 3q^{91} - q^{93} + O(q^{100})$$

## Character values

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

 $$n$$ $$785$$ $$1471$$ $$1765$$ $$2257$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$-\zeta_{42}^{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 0.955573 − 0.294755i 0.826239 − 0.563320i −0.733052 + 0.680173i −0.733052 − 0.680173i 0.365341 − 0.930874i 0.0747301 + 0.997204i 0.955573 + 0.294755i 0.0747301 − 0.997204i 0.826239 + 0.563320i −0.988831 + 0.149042i −0.988831 − 0.149042i 0.365341 + 0.930874i
0 0.365341 0.930874i 0 0 0 −0.955573 0.294755i 0 −0.733052 0.680173i 0
143.1 0 −0.733052 0.680173i 0 0 0 −0.826239 0.563320i 0 0.0747301 + 0.997204i 0
383.1 0 −0.988831 0.149042i 0 0 0 0.733052 + 0.680173i 0 0.955573 + 0.294755i 0
479.1 0 −0.988831 + 0.149042i 0 0 0 0.733052 0.680173i 0 0.955573 0.294755i 0
719.1 0 0.0747301 + 0.997204i 0 0 0 −0.365341 0.930874i 0 −0.988831 + 0.149042i 0
1055.1 0 0.955573 0.294755i 0 0 0 −0.0747301 + 0.997204i 0 0.826239 0.563320i 0
1151.1 0 0.365341 + 0.930874i 0 0 0 −0.955573 + 0.294755i 0 −0.733052 + 0.680173i 0
1487.1 0 0.955573 + 0.294755i 0 0 0 −0.0747301 0.997204i 0 0.826239 + 0.563320i 0
1727.1 0 −0.733052 + 0.680173i 0 0 0 −0.826239 + 0.563320i 0 0.0747301 0.997204i 0
1823.1 0 0.826239 0.563320i 0 0 0 0.988831 + 0.149042i 0 0.365341 0.930874i 0
2063.1 0 0.826239 + 0.563320i 0 0 0 0.988831 0.149042i 0 0.365341 + 0.930874i 0
2159.1 0 0.0747301 0.997204i 0 0 0 −0.365341 + 0.930874i 0 −0.988831 0.149042i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2159.1 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.p even 42 1 inner
588.bf odd 42 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.1.dj.b yes 12
3.b odd 2 1 CM 2352.1.dj.b yes 12
4.b odd 2 1 2352.1.dj.a 12
12.b even 2 1 2352.1.dj.a 12
49.h odd 42 1 2352.1.dj.a 12
147.o even 42 1 2352.1.dj.a 12
196.p even 42 1 inner 2352.1.dj.b yes 12
588.bf odd 42 1 inner 2352.1.dj.b yes 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.1.dj.a 12 4.b odd 2 1
2352.1.dj.a 12 12.b even 2 1
2352.1.dj.a 12 49.h odd 42 1
2352.1.dj.a 12 147.o even 42 1
2352.1.dj.b yes 12 1.a even 1 1 trivial
2352.1.dj.b yes 12 3.b odd 2 1 CM
2352.1.dj.b yes 12 196.p even 42 1 inner
2352.1.dj.b yes 12 588.bf odd 42 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{19}^{12} - \cdots$$ acting on $$S_{1}^{\mathrm{new}}(2352, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12}$$
$5$ $$T^{12}$$
$7$ $$1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12}$$
$11$ $$T^{12}$$
$13$ $$1 - 7 T + 16 T^{2} - 14 T^{3} + 4 T^{4} + 21 T^{5} + 15 T^{6} - 7 T^{7} + 9 T^{8} + 7 T^{9} - 3 T^{10} + T^{12}$$
$17$ $$T^{12}$$
$19$ $$1 - 8 T + 56 T^{2} - 76 T^{3} + 118 T^{4} - 49 T^{5} + 78 T^{6} - 28 T^{7} + 34 T^{8} - 6 T^{9} + 7 T^{10} - T^{11} + T^{12}$$
$23$ $$T^{12}$$
$29$ $$T^{12}$$
$31$ $$1 + 8 T + 56 T^{2} + 76 T^{3} + 118 T^{4} + 49 T^{5} + 78 T^{6} + 28 T^{7} + 34 T^{8} + 6 T^{9} + 7 T^{10} + T^{11} + T^{12}$$
$37$ $$1 + 6 T - 118 T^{3} + 349 T^{4} - 518 T^{5} + 519 T^{6} - 392 T^{7} + 230 T^{8} - 104 T^{9} + 35 T^{10} - 8 T^{11} + T^{12}$$
$41$ $$T^{12}$$
$43$ $$1 + 7 T + 30 T^{2} + 70 T^{3} + 88 T^{4} + 42 T^{5} - 13 T^{6} - 14 T^{7} + 2 T^{8} - 3 T^{10} + T^{12}$$
$47$ $$T^{12}$$
$53$ $$T^{12}$$
$59$ $$T^{12}$$
$61$ $$49 - 147 T + 196 T^{2} - 245 T^{3} + 343 T^{4} - 392 T^{5} + 357 T^{6} - 266 T^{7} + 161 T^{8} - 77 T^{9} + 28 T^{10} - 7 T^{11} + T^{12}$$
$67$ $$1 - 6 T + 10 T^{2} + 12 T^{3} - 24 T^{4} - 21 T^{5} + 36 T^{6} + 42 T^{7} + 2 T^{8} - 12 T^{9} - T^{10} + 3 T^{11} + T^{12}$$
$71$ $$T^{12}$$
$73$ $$1 - T + 3 T^{2} + 23 T^{3} + 25 T^{4} + 8 T^{6} - 7 T^{7} + 2 T^{8} - 9 T^{9} + 6 T^{10} - 3 T^{11} + T^{12}$$
$79$ $$1 + 6 T + 10 T^{2} - 12 T^{3} - 24 T^{4} + 21 T^{5} + 36 T^{6} - 42 T^{7} + 2 T^{8} + 12 T^{9} - T^{10} - 3 T^{11} + T^{12}$$
$83$ $$T^{12}$$
$89$ $$T^{12}$$
$97$ $$( 7 + 14 T^{2} + 7 T^{4} + T^{6} )^{2}$$