# Properties

 Label 3311.1.h.g Level 3311 Weight 1 Character orbit 3311.h Analytic conductor 1.652 Analytic rank 0 Dimension 2 Projective image $$D_{6}$$ RM discriminant 473 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3311 = 7 \cdot 11 \cdot 43$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3311.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.65240425683$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{6}$$ Projective field Galois closure of 6.0.76739047.3

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - q^{2} + \zeta_{6}^{2} q^{7} + q^{8} - q^{9} +O(q^{10})$$ $$q - q^{2} + \zeta_{6}^{2} q^{7} + q^{8} - q^{9} - q^{11} -\zeta_{6}^{2} q^{14} - q^{16} + q^{18} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{19} + q^{22} + q^{23} - q^{25} + q^{29} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{31} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{38} - q^{43} - q^{46} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{47} -\zeta_{6} q^{49} + q^{50} - q^{53} + \zeta_{6}^{2} q^{56} - q^{58} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{59} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{61} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{62} -\zeta_{6}^{2} q^{63} + q^{64} - q^{67} - q^{72} -\zeta_{6}^{2} q^{77} + q^{81} + q^{86} - q^{88} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{94} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{97} + \zeta_{6} q^{98} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - q^{7} + 2q^{8} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} - q^{7} + 2q^{8} - 2q^{9} - 2q^{11} + q^{14} - 2q^{16} + 2q^{18} + 2q^{22} + 2q^{23} - 2q^{25} + 2q^{29} - 2q^{43} - 2q^{46} - q^{49} + 2q^{50} - 2q^{53} - q^{56} - 2q^{58} + q^{63} + 2q^{64} - 2q^{67} - 2q^{72} + q^{77} + 2q^{81} + 2q^{86} - 2q^{88} + q^{98} + 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3311\mathbb{Z}\right)^\times$$.

 $$n$$ $$904$$ $$1893$$ $$2927$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3310.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 0 0 0 0 −0.500000 0.866025i 1.00000 −1.00000 0
3310.2 −1.00000 0 0 0 0 −0.500000 + 0.866025i 1.00000 −1.00000 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
473.d even 2 1 RM by $$\Q(\sqrt{473})$$
7.b odd 2 1 inner
3311.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3311.1.h.g 2
7.b odd 2 1 inner 3311.1.h.g 2
11.b odd 2 1 3311.1.h.j yes 2
43.b odd 2 1 3311.1.h.j yes 2
77.b even 2 1 3311.1.h.j yes 2
301.c even 2 1 3311.1.h.j yes 2
473.d even 2 1 RM 3311.1.h.g 2
3311.h odd 2 1 inner 3311.1.h.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3311.1.h.g 2 1.a even 1 1 trivial
3311.1.h.g 2 7.b odd 2 1 inner
3311.1.h.g 2 473.d even 2 1 RM
3311.1.h.g 2 3311.h odd 2 1 inner
3311.1.h.j yes 2 11.b odd 2 1
3311.1.h.j yes 2 43.b odd 2 1
3311.1.h.j yes 2 77.b even 2 1
3311.1.h.j yes 2 301.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3311, [\chi])$$:

 $$T_{2} + 1$$ $$T_{3}$$

## Hecke characteristic polynomials

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