# Properties

 Label 3311.1.h.n Level 3311 Weight 1 Character orbit 3311.h Self dual yes Analytic conductor 1.652 Analytic rank 0 Dimension 3 Projective image $$D_{9}$$ CM discriminant -3311 Inner twists 2

# 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: yes Analytic conductor: $$1.65240425683$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ Defining polynomial: $$x^{3} - 3 x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{9}$$ Projective field Galois closure of 9.1.120181251723841.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 + \beta - \beta^{2} ) q^{2} + \beta q^{3} + ( 1 - \beta ) q^{4} + ( -2 - \beta + \beta^{2} ) q^{5} + ( -1 - \beta + \beta^{2} ) q^{6} + q^{7} + ( 1 + \beta - \beta^{2} ) q^{8} + ( -1 + \beta^{2} ) q^{9} +O(q^{10})$$ $$q + ( 2 + \beta - \beta^{2} ) q^{2} + \beta q^{3} + ( 1 - \beta ) q^{4} + ( -2 - \beta + \beta^{2} ) q^{5} + ( -1 - \beta + \beta^{2} ) q^{6} + q^{7} + ( 1 + \beta - \beta^{2} ) q^{8} + ( -1 + \beta^{2} ) q^{9} + ( -2 + \beta ) q^{10} + q^{11} + ( \beta - \beta^{2} ) q^{12} + q^{13} + ( 2 + \beta - \beta^{2} ) q^{14} + ( 1 + \beta - \beta^{2} ) q^{15} + ( -1 - \beta + \beta^{2} ) q^{16} + ( 2 - \beta^{2} ) q^{17} + ( -1 + \beta ) q^{18} + ( -3 - 2 \beta + 2 \beta^{2} ) q^{20} + \beta q^{21} + ( 2 + \beta - \beta^{2} ) q^{22} - q^{23} + ( -1 - 2 \beta + \beta^{2} ) q^{24} + ( 1 - \beta ) q^{25} + ( 2 + \beta - \beta^{2} ) q^{26} + ( 1 + \beta ) q^{27} + ( 1 - \beta ) q^{28} -\beta q^{29} + ( -2 \beta + \beta^{2} ) q^{30} + ( -1 + \beta ) q^{32} + \beta q^{33} + ( 3 - \beta^{2} ) q^{34} + ( -2 - \beta + \beta^{2} ) q^{35} + ( -2 - 2 \beta + \beta^{2} ) q^{36} + \beta q^{39} + ( 2 \beta - \beta^{2} ) q^{40} + ( 2 - \beta^{2} ) q^{41} + ( -1 - \beta + \beta^{2} ) q^{42} + q^{43} + ( 1 - \beta ) q^{44} + ( 1 - \beta ) q^{45} + ( -2 - \beta + \beta^{2} ) q^{46} + ( 1 + 2 \beta - \beta^{2} ) q^{48} + q^{49} + ( 3 + 2 \beta - 2 \beta^{2} ) q^{50} + ( -1 - \beta ) q^{51} + ( 1 - \beta ) q^{52} + ( -2 + \beta^{2} ) q^{53} + q^{54} + ( -2 - \beta + \beta^{2} ) q^{55} + ( 1 + \beta - \beta^{2} ) q^{56} + ( 1 + \beta - \beta^{2} ) q^{58} + ( 2 + 3 \beta - 2 \beta^{2} ) q^{60} + ( -1 + \beta^{2} ) q^{63} + ( -2 - \beta + \beta^{2} ) q^{64} + ( -2 - \beta + \beta^{2} ) q^{65} + ( -1 - \beta + \beta^{2} ) q^{66} + ( 2 + \beta - \beta^{2} ) q^{67} + ( 3 + \beta - \beta^{2} ) q^{68} -\beta q^{69} + ( -2 + \beta ) q^{70} + ( \beta - \beta^{2} ) q^{72} + ( \beta - \beta^{2} ) q^{75} + q^{77} + ( -1 - \beta + \beta^{2} ) q^{78} + ( -2 \beta + \beta^{2} ) q^{80} + ( 1 + \beta ) q^{81} + ( 3 - \beta^{2} ) q^{82} + \beta q^{83} + ( \beta - \beta^{2} ) q^{84} + ( -3 + \beta^{2} ) q^{85} + ( 2 + \beta - \beta^{2} ) q^{86} -\beta^{2} q^{87} + ( 1 + \beta - \beta^{2} ) q^{88} -2 q^{89} + ( 3 + 2 \beta - 2 \beta^{2} ) q^{90} + q^{91} + ( -1 + \beta ) q^{92} + ( -\beta + \beta^{2} ) q^{96} + ( 2 + \beta - \beta^{2} ) q^{98} + ( -1 + \beta^{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{4} + 3q^{6} + 3q^{7} - 3q^{8} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{4} + 3q^{6} + 3q^{7} - 3q^{8} + 3q^{9} - 6q^{10} + 3q^{11} - 6q^{12} + 3q^{13} - 3q^{15} + 3q^{16} - 3q^{18} + 3q^{20} - 3q^{23} + 3q^{24} + 3q^{25} + 3q^{27} + 3q^{28} + 6q^{30} - 3q^{32} + 3q^{34} - 6q^{40} + 3q^{42} + 3q^{43} + 3q^{44} + 3q^{45} - 3q^{48} + 3q^{49} - 3q^{50} - 3q^{51} + 3q^{52} + 3q^{54} - 3q^{56} - 3q^{58} - 6q^{60} + 3q^{63} + 3q^{66} + 3q^{68} - 6q^{70} - 6q^{72} - 6q^{75} + 3q^{77} + 3q^{78} + 6q^{80} + 3q^{81} + 3q^{82} - 6q^{84} - 3q^{85} - 6q^{87} - 3q^{88} - 6q^{89} - 3q^{90} + 3q^{91} - 3q^{92} + 6q^{96} + 3q^{99} + O(q^{100})$$

## 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
 −1.53209 1.87939 −0.347296
−1.87939 −1.53209 2.53209 1.87939 2.87939 1.00000 −2.87939 1.34730 −3.53209
3310.2 0.347296 1.87939 −0.879385 −0.347296 0.652704 1.00000 −0.652704 2.53209 −0.120615
3310.3 1.53209 −0.347296 1.34730 −1.53209 −0.532089 1.00000 0.532089 −0.879385 −2.34730
 $$n$$: 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
3311.h odd 2 1 CM by $$\Q(\sqrt{-3311})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3311.1.h.n yes 3
7.b odd 2 1 3311.1.h.l yes 3
11.b odd 2 1 3311.1.h.k 3
43.b odd 2 1 3311.1.h.m yes 3
77.b even 2 1 3311.1.h.m yes 3
301.c even 2 1 3311.1.h.k 3
473.d even 2 1 3311.1.h.l yes 3
3311.h odd 2 1 CM 3311.1.h.n yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3311.1.h.k 3 11.b odd 2 1
3311.1.h.k 3 301.c even 2 1
3311.1.h.l yes 3 7.b odd 2 1
3311.1.h.l yes 3 473.d even 2 1
3311.1.h.m yes 3 43.b odd 2 1
3311.1.h.m yes 3 77.b even 2 1
3311.1.h.n yes 3 1.a even 1 1 trivial
3311.1.h.n yes 3 3311.h odd 2 1 CM

## Hecke kernels

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

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

## Hecke characteristic polynomials

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