# Properties

 Label 1911.1.cv.a Level $1911$ Weight $1$ Character orbit 1911.cv Analytic conductor $0.954$ Analytic rank $0$ Dimension $12$ Projective image $D_{28}$ 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.cv (of order $$28$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.953713239142$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\Q(\zeta_{28})$$ Defining polynomial: $$x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{28}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{28} - \cdots)$$

## $q$-expansion

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

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

## Character values

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

 $$n$$ $$638$$ $$1471$$ $$1522$$ $$\chi(n)$$ $$-1$$ $$\zeta_{28}^{7}$$ $$\zeta_{28}^{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
83.1
 −0.974928 − 0.222521i 0.433884 + 0.900969i 0.974928 − 0.222521i −0.781831 − 0.623490i −0.781831 + 0.623490i 0.974928 + 0.222521i 0.433884 − 0.900969i −0.974928 + 0.222521i 0.781831 − 0.623490i −0.433884 − 0.900969i −0.433884 + 0.900969i 0.781831 + 0.623490i
0 0.781831 + 0.623490i −0.433884 + 0.900969i 0 0 −0.222521 + 0.974928i 0 0.222521 + 0.974928i 0
125.1 0 0.974928 + 0.222521i 0.781831 + 0.623490i 0 0 −0.900969 + 0.433884i 0 0.900969 + 0.433884i 0
356.1 0 −0.781831 + 0.623490i 0.433884 + 0.900969i 0 0 −0.222521 0.974928i 0 0.222521 0.974928i 0
398.1 0 −0.433884 + 0.900969i 0.974928 0.222521i 0 0 0.623490 0.781831i 0 −0.623490 0.781831i 0
629.1 0 −0.433884 0.900969i 0.974928 + 0.222521i 0 0 0.623490 + 0.781831i 0 −0.623490 + 0.781831i 0
671.1 0 −0.781831 0.623490i 0.433884 0.900969i 0 0 −0.222521 + 0.974928i 0 0.222521 + 0.974928i 0
902.1 0 0.974928 0.222521i 0.781831 0.623490i 0 0 −0.900969 0.433884i 0 0.900969 0.433884i 0
944.1 0 0.781831 0.623490i −0.433884 0.900969i 0 0 −0.222521 0.974928i 0 0.222521 0.974928i 0
1217.1 0 0.433884 + 0.900969i −0.974928 0.222521i 0 0 0.623490 + 0.781831i 0 −0.623490 + 0.781831i 0
1448.1 0 −0.974928 0.222521i −0.781831 0.623490i 0 0 −0.900969 + 0.433884i 0 0.900969 + 0.433884i 0
1490.1 0 −0.974928 + 0.222521i −0.781831 + 0.623490i 0 0 −0.900969 0.433884i 0 0.900969 0.433884i 0
1721.1 0 0.433884 0.900969i −0.974928 + 0.222521i 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 1721.1 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.bn even 28 1 inner
1911.cv odd 28 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1911.1.cv.a 12
3.b odd 2 1 CM 1911.1.cv.a 12
13.d odd 4 1 1911.1.cv.b yes 12
39.f even 4 1 1911.1.cv.b yes 12
49.f odd 14 1 1911.1.cv.b yes 12
147.k even 14 1 1911.1.cv.b yes 12
637.bn even 28 1 inner 1911.1.cv.a 12
1911.cv odd 28 1 inner 1911.1.cv.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1911.1.cv.a 12 1.a even 1 1 trivial
1911.1.cv.a 12 3.b odd 2 1 CM
1911.1.cv.a 12 637.bn even 28 1 inner
1911.1.cv.a 12 1911.cv odd 28 1 inner
1911.1.cv.b yes 12 13.d odd 4 1
1911.1.cv.b yes 12 39.f even 4 1
1911.1.cv.b yes 12 49.f odd 14 1
1911.1.cv.b yes 12 147.k even 14 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{19}^{12} - \cdots$$ acting on $$S_{1}^{\mathrm{new}}(1911, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12}$$
$5$ $$T^{12}$$
$7$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$
$11$ $$T^{12}$$
$13$ $$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12}$$
$17$ $$T^{12}$$
$19$ $$1 - 8 T + 32 T^{2} - 42 T^{3} + 26 T^{4} + 2 T^{5} + 34 T^{6} - 34 T^{7} + 17 T^{8} + 2 T^{10} - 2 T^{11} + T^{12}$$
$23$ $$T^{12}$$
$29$ $$T^{12}$$
$31$ $$1 - 8 T + 32 T^{2} - 42 T^{3} + 26 T^{4} + 2 T^{5} + 34 T^{6} - 34 T^{7} + 17 T^{8} + 2 T^{10} - 2 T^{11} + T^{12}$$
$37$ $$1 - 8 T + 11 T^{2} + 28 T^{3} - 23 T^{4} - 12 T^{5} + 41 T^{6} - 34 T^{7} + 31 T^{8} - 14 T^{9} + 9 T^{10} - 2 T^{11} + T^{12}$$
$41$ $$T^{12}$$
$43$ $$( 7 + 14 T + 7 T^{2} + T^{6} )^{2}$$
$47$ $$T^{12}$$
$53$ $$T^{12}$$
$59$ $$T^{12}$$
$61$ $$1 - 9 T^{2} + 25 T^{4} - T^{6} + 9 T^{8} + 3 T^{10} + T^{12}$$
$67$ $$1 - 8 T + 32 T^{2} - 42 T^{3} + 26 T^{4} + 2 T^{5} + 34 T^{6} - 34 T^{7} + 17 T^{8} + 2 T^{10} - 2 T^{11} + T^{12}$$
$71$ $$T^{12}$$
$73$ $$1 - 6 T + 4 T^{2} + 14 T^{3} + 61 T^{4} + 54 T^{5} + 48 T^{6} + 20 T^{7} - 4 T^{8} + 2 T^{10} + 2 T^{11} + T^{12}$$
$79$ $$( -7 + 14 T^{2} - 7 T^{4} + T^{6} )^{2}$$
$83$ $$T^{12}$$
$89$ $$T^{12}$$
$97$ $$1 + 8 T + 32 T^{2} + 42 T^{3} + 26 T^{4} - 2 T^{5} + 34 T^{6} + 34 T^{7} + 17 T^{8} + 2 T^{10} + 2 T^{11} + T^{12}$$