# Properties

 Label 3311.1.h.p Level 3311 Weight 1 Character orbit 3311.h Self dual yes Analytic conductor 1.652 Analytic rank 0 Dimension 6 Projective image $$D_{18}$$ 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: $$6$$ Coefficient field: $$\Q(\zeta_{36})^+$$ Defining polynomial: $$x^{6} - 6 x^{4} + 9 x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{18}$$ Projective field Galois closure of $$\mathbb{Q}[x]/(x^{18} - \cdots)$$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.28558 −1.28558 0.684040 −0.684040 1.96962 −1.96962
−1.87939 −1.28558 2.53209 −0.684040 2.41609 1.00000 −2.87939 0.652704 1.28558
3310.2 −1.87939 1.28558 2.53209 0.684040 −2.41609 1.00000 −2.87939 0.652704 −1.28558
3310.3 0.347296 −0.684040 −0.879385 1.96962 −0.237565 1.00000 −0.652704 −0.532089 0.684040
3310.4 0.347296 0.684040 −0.879385 −1.96962 0.237565 1.00000 −0.652704 −0.532089 −0.684040
3310.5 1.53209 −1.96962 1.34730 1.28558 −3.01763 1.00000 0.532089 2.87939 1.96962
3310.6 1.53209 1.96962 1.34730 −1.28558 3.01763 1.00000 0.532089 2.87939 −1.96962
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3310.6 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.p yes 6
7.b odd 2 1 inner 3311.1.h.p yes 6
11.b odd 2 1 3311.1.h.o 6
43.b odd 2 1 3311.1.h.o 6
77.b even 2 1 3311.1.h.o 6
301.c even 2 1 3311.1.h.o 6
473.d even 2 1 inner 3311.1.h.p yes 6
3311.h odd 2 1 CM 3311.1.h.p yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3311.1.h.o 6 11.b odd 2 1
3311.1.h.o 6 43.b odd 2 1
3311.1.h.o 6 77.b even 2 1
3311.1.h.o 6 301.c even 2 1
3311.1.h.p yes 6 1.a even 1 1 trivial
3311.1.h.p yes 6 7.b odd 2 1 inner
3311.1.h.p yes 6 473.d even 2 1 inner
3311.1.h.p yes 6 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 $$S_{1}^{\mathrm{new}}(3311, [\chi])$$:

 $$T_{2}^{3} - 3 T_{2} + 1$$ $$T_{3}^{6} - 6 T_{3}^{4} + 9 T_{3}^{2} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{3} + T^{6} )^{2}$$
$3$ $$1 - T^{6} + T^{12}$$
$5$ $$1 - T^{6} + T^{12}$$
$7$ $$( 1 - T )^{6}$$
$11$ $$( 1 + T )^{6}$$
$13$ $$( 1 - T^{2} + T^{4} )^{3}$$
$17$ $$1 - T^{6} + T^{12}$$
$19$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$23$ $$( 1 - T + T^{2} )^{6}$$
$29$ $$( 1 - T^{3} + T^{6} )^{2}$$
$31$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$37$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$41$ $$1 - T^{6} + T^{12}$$
$43$ $$( 1 + T )^{6}$$
$47$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$53$ $$( 1 + T^{3} + T^{6} )^{2}$$
$59$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$61$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$67$ $$( 1 + T^{3} + T^{6} )^{2}$$
$71$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$73$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$79$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$83$ $$1 - T^{6} + T^{12}$$
$89$ $$( 1 + T^{2} )^{6}$$
$97$ $$( 1 - T )^{6}( 1 + T )^{6}$$