# Properties

 Label 1911.1.dx.a Level $1911$ Weight $1$ Character orbit 1911.dx Analytic conductor $0.954$ Analytic rank $0$ Dimension $12$ Projective image $D_{21}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1911 = 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1911.dx (of order $$42$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.953713239142$$ 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: $$D_{21}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{21} + \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{42}^{20} q^{3} + \zeta_{42}^{18} q^{4} + \zeta_{42}^{2} q^{7} -\zeta_{42}^{19} q^{9} +O(q^{10})$$ $$q + \zeta_{42}^{20} q^{3} + \zeta_{42}^{18} q^{4} + \zeta_{42}^{2} q^{7} -\zeta_{42}^{19} q^{9} -\zeta_{42}^{17} q^{12} + \zeta_{42}^{10} q^{13} -\zeta_{42}^{15} q^{16} + ( -\zeta_{42}^{5} - \zeta_{42}^{9} ) q^{19} -\zeta_{42} q^{21} -\zeta_{42}^{19} q^{25} + \zeta_{42}^{18} q^{27} + \zeta_{42}^{20} q^{28} + ( \zeta_{42}^{4} + \zeta_{42}^{10} ) q^{31} + \zeta_{42}^{16} q^{36} + ( -\zeta_{42}^{13} + \zeta_{42}^{14} ) q^{37} -\zeta_{42}^{9} q^{39} + ( \zeta_{42}^{6} + \zeta_{42}^{10} ) q^{43} + \zeta_{42}^{14} q^{48} + \zeta_{42}^{4} q^{49} -\zeta_{42}^{7} q^{52} + ( \zeta_{42}^{4} + \zeta_{42}^{8} ) q^{57} + ( \zeta_{42}^{6} + \zeta_{42}^{14} ) q^{61} + q^{63} + \zeta_{42}^{12} q^{64} + ( \zeta_{42}^{8} + \zeta_{42}^{20} ) q^{67} + ( \zeta_{42}^{2} - \zeta_{42}^{15} ) q^{73} + \zeta_{42}^{18} q^{75} + ( \zeta_{42}^{2} + \zeta_{42}^{6} ) q^{76} + ( -\zeta_{42}^{11} - \zeta_{42}^{17} ) q^{79} -\zeta_{42}^{17} q^{81} -\zeta_{42}^{19} q^{84} + \zeta_{42}^{12} q^{91} + ( -\zeta_{42}^{3} - \zeta_{42}^{9} ) q^{93} + ( \zeta_{42}^{12} + \zeta_{42}^{16} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + q^{3} - 2 q^{4} + q^{7} + q^{9} + O(q^{10})$$ $$12 q + q^{3} - 2 q^{4} + q^{7} + q^{9} + q^{12} + q^{13} - 2 q^{16} - q^{19} + q^{21} + q^{25} - 2 q^{27} + q^{28} + 2 q^{31} + q^{36} - 5 q^{37} - 2 q^{39} - q^{43} - 6 q^{48} + q^{49} - 6 q^{52} + 2 q^{57} - 8 q^{61} + 12 q^{63} - 2 q^{64} + 2 q^{67} - q^{73} - 2 q^{75} - q^{76} + 2 q^{79} + q^{81} + q^{84} - 2 q^{91} - 4 q^{93} - q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1911\mathbb{Z}\right)^\times$$.

 $$n$$ $$638$$ $$1471$$ $$1522$$ $$\chi(n)$$ $$-1$$ $$\zeta_{42}^{14}$$ $$-\zeta_{42}^{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
74.1
 −0.988831 + 0.149042i 0.365341 − 0.930874i 0.955573 + 0.294755i 0.955573 − 0.294755i −0.733052 − 0.680173i 0.826239 + 0.563320i 0.365341 + 0.930874i 0.0747301 + 0.997204i 0.0747301 − 0.997204i −0.733052 + 0.680173i −0.988831 − 0.149042i 0.826239 − 0.563320i
0 −0.988831 0.149042i −0.900969 0.433884i 0 0 0.955573 0.294755i 0 0.955573 + 0.294755i 0
107.1 0 0.365341 + 0.930874i −0.900969 0.433884i 0 0 −0.733052 0.680173i 0 −0.733052 + 0.680173i 0
347.1 0 0.955573 0.294755i 0.623490 0.781831i 0 0 0.826239 + 0.563320i 0 0.826239 0.563320i 0
380.1 0 0.955573 + 0.294755i 0.623490 + 0.781831i 0 0 0.826239 0.563320i 0 0.826239 + 0.563320i 0
620.1 0 −0.733052 + 0.680173i 0.623490 + 0.781831i 0 0 0.0747301 + 0.997204i 0 0.0747301 0.997204i 0
653.1 0 0.826239 0.563320i −0.222521 0.974928i 0 0 0.365341 + 0.930874i 0 0.365341 0.930874i 0
893.1 0 0.365341 0.930874i −0.900969 + 0.433884i 0 0 −0.733052 + 0.680173i 0 −0.733052 0.680173i 0
926.1 0 0.0747301 0.997204i −0.222521 + 0.974928i 0 0 −0.988831 + 0.149042i 0 −0.988831 0.149042i 0
1166.1 0 0.0747301 + 0.997204i −0.222521 0.974928i 0 0 −0.988831 0.149042i 0 −0.988831 + 0.149042i 0
1199.1 0 −0.733052 0.680173i 0.623490 0.781831i 0 0 0.0747301 0.997204i 0 0.0747301 + 0.997204i 0
1472.1 0 −0.988831 + 0.149042i −0.900969 + 0.433884i 0 0 0.955573 + 0.294755i 0 0.955573 0.294755i 0
1712.1 0 0.826239 + 0.563320i −0.222521 + 0.974928i 0 0 0.365341 0.930874i 0 0.365341 + 0.930874i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1712.1 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 $$\Q(\sqrt{-3})$$
637.bi even 21 1 inner
1911.dx odd 42 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1911.1.dx.a yes 12
3.b odd 2 1 CM 1911.1.dx.a yes 12
13.c even 3 1 1911.1.dd.a 12
39.i odd 6 1 1911.1.dd.a 12
49.g even 21 1 1911.1.dd.a 12
147.n odd 42 1 1911.1.dd.a 12
637.bi even 21 1 inner 1911.1.dx.a yes 12
1911.dx odd 42 1 inner 1911.1.dx.a yes 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1911.1.dd.a 12 13.c even 3 1
1911.1.dd.a 12 39.i odd 6 1
1911.1.dd.a 12 49.g even 21 1
1911.1.dd.a 12 147.n odd 42 1
1911.1.dx.a yes 12 1.a even 1 1 trivial
1911.1.dx.a yes 12 3.b odd 2 1 CM
1911.1.dx.a yes 12 637.bi even 21 1 inner
1911.1.dx.a yes 12 1911.dx odd 42 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1911, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12}$$
$5$ $$T^{12}$$
$7$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12}$$
$11$ $$T^{12}$$
$13$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12}$$
$17$ $$T^{12}$$
$19$ $$1 + 8 T + 56 T^{2} + 76 T^{3} + 118 T^{4} + 49 T^{5} + 78 T^{6} + 28 T^{7} + 34 T^{8} + 6 T^{9} + 7 T^{10} + T^{11} + T^{12}$$
$23$ $$T^{12}$$
$29$ $$T^{12}$$
$31$ $$( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} )^{2}$$
$37$ $$1 + 12 T + 45 T^{2} + 10 T^{3} + 61 T^{4} + 92 T^{5} + 105 T^{6} + 92 T^{7} + 68 T^{8} + 38 T^{9} + 17 T^{10} + 5 T^{11} + T^{12}$$
$41$ $$T^{12}$$
$43$ $$1 - 13 T + 49 T^{2} - 29 T^{3} + 69 T^{4} - 20 T^{6} - 21 T^{7} + 6 T^{8} - T^{9} + T^{11} + T^{12}$$
$47$ $$T^{12}$$
$53$ $$T^{12}$$
$59$ $$T^{12}$$
$61$ $$1 - 6 T + 118 T^{3} + 349 T^{4} + 518 T^{5} + 519 T^{6} + 392 T^{7} + 230 T^{8} + 104 T^{9} + 35 T^{10} + 8 T^{11} + T^{12}$$
$67$ $$( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} )^{2}$$
$71$ $$T^{12}$$
$73$ $$1 + 8 T + 28 T^{2} - 71 T^{3} + 48 T^{4} + 21 T^{5} + 22 T^{6} - 15 T^{8} - T^{9} + T^{11} + T^{12}$$
$79$ $$( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} )^{2}$$
$83$ $$T^{12}$$
$89$ $$T^{12}$$
$97$ $$1 + 8 T + 56 T^{2} + 76 T^{3} + 118 T^{4} + 49 T^{5} + 78 T^{6} + 28 T^{7} + 34 T^{8} + 6 T^{9} + 7 T^{10} + T^{11} + T^{12}$$