# Properties

 Label 3311.1.h.h Level 3311 Weight 1 Character orbit 3311.h Self dual yes Analytic conductor 1.652 Analytic rank 0 Dimension 2 Projective image $$D_{6}$$ CM discriminant -3311 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: yes Analytic conductor: $$1.65240425683$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{6}$$ Projective field Galois closure of 6.0.76739047.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} -\beta q^{3} -\beta q^{5} + \beta q^{6} + q^{7} + q^{8} + 2 q^{9} +O(q^{10})$$ $$q - q^{2} -\beta q^{3} -\beta q^{5} + \beta q^{6} + q^{7} + q^{8} + 2 q^{9} + \beta q^{10} - q^{11} - q^{14} + 3 q^{15} - q^{16} + \beta q^{17} -2 q^{18} -\beta q^{21} + q^{22} -2 q^{23} -\beta q^{24} + 2 q^{25} -\beta q^{27} + q^{29} -3 q^{30} + \beta q^{33} -\beta q^{34} -\beta q^{35} -\beta q^{40} -\beta q^{41} + \beta q^{42} - q^{43} -2 \beta q^{45} + 2 q^{46} + \beta q^{48} + q^{49} -2 q^{50} -3 q^{51} - q^{53} + \beta q^{54} + \beta q^{55} + q^{56} - q^{58} + 2 q^{63} + q^{64} -\beta q^{66} - q^{67} + 2 \beta q^{69} + \beta q^{70} + 2 q^{72} -2 \beta q^{75} - q^{77} + \beta q^{80} + q^{81} + \beta q^{82} + \beta q^{83} -3 q^{85} + q^{86} -\beta q^{87} - q^{88} + 2 \beta q^{90} - q^{98} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{7} + 2q^{8} + 4q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{7} + 2q^{8} + 4q^{9} - 2q^{11} - 2q^{14} + 6q^{15} - 2q^{16} - 4q^{18} + 2q^{22} - 4q^{23} + 4q^{25} + 2q^{29} - 6q^{30} - 2q^{43} + 4q^{46} + 2q^{49} - 4q^{50} - 6q^{51} - 2q^{53} + 2q^{56} - 2q^{58} + 4q^{63} + 2q^{64} - 2q^{67} + 4q^{72} - 2q^{77} + 2q^{81} - 6q^{85} + 2q^{86} - 2q^{88} - 2q^{98} - 4q^{99} + O(q^{100})$$

## 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
 1.73205 −1.73205
−1.00000 −1.73205 0 −1.73205 1.73205 1.00000 1.00000 2.00000 1.73205
3310.2 −1.00000 1.73205 0 1.73205 −1.73205 1.00000 1.00000 2.00000 −1.73205
 $$n$$: 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 $$\Q(\sqrt{-3311})$$
7.b odd 2 1 inner
473.d even 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.h 2
7.b odd 2 1 inner 3311.1.h.h 2
11.b odd 2 1 3311.1.h.i yes 2
43.b odd 2 1 3311.1.h.i yes 2
77.b even 2 1 3311.1.h.i yes 2
301.c even 2 1 3311.1.h.i yes 2
473.d even 2 1 inner 3311.1.h.h 2
3311.h odd 2 1 CM 3311.1.h.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3311.1.h.h 2 1.a even 1 1 trivial
3311.1.h.h 2 7.b odd 2 1 inner
3311.1.h.h 2 473.d even 2 1 inner
3311.1.h.h 2 3311.h odd 2 1 CM
3311.1.h.i yes 2 11.b odd 2 1
3311.1.h.i yes 2 43.b odd 2 1
3311.1.h.i yes 2 77.b even 2 1
3311.1.h.i 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 $$S_{1}^{\mathrm{new}}(3311, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$1 - T^{2} + T^{4}$$
$7$ $$( 1 - T )^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$( 1 + T^{2} )^{2}$$
$17$ $$1 - T^{2} + T^{4}$$
$19$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$23$ $$( 1 + T )^{4}$$
$29$ $$( 1 - T + T^{2} )^{2}$$
$31$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$37$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$41$ $$1 - T^{2} + T^{4}$$
$43$ $$( 1 + T )^{2}$$
$47$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$53$ $$( 1 + T + T^{2} )^{2}$$
$59$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$61$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$67$ $$( 1 + T + T^{2} )^{2}$$
$71$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$73$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$79$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$83$ $$1 - T^{2} + T^{4}$$
$89$ $$( 1 + T^{2} )^{2}$$
$97$ $$( 1 - T )^{2}( 1 + T )^{2}$$