# Properties

 Label 3311.1.h.o 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 disc. -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 $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: Yes Analytic conductor: $$1.65240425683$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{36})^+$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

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

Basis of coefficient ring in terms of $$\nu = \zeta_{36} + \zeta_{36}^{-1}$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 5 \nu^{2} + 4$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - 5 \nu^{3} + 4 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 5 \beta_{2} + 6$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + 5 \beta_{3} + 11 \beta_{1}$$

## 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
 1.96962 −1.96962 0.684040 −0.684040 1.28558 −1.28558
−1.53209 −1.96962 1.34730 1.28558 3.01763 −1.00000 −0.532089 2.87939 −1.96962
3310.2 −1.53209 1.96962 1.34730 −1.28558 −3.01763 −1.00000 −0.532089 2.87939 1.96962
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 1.87939 −1.28558 2.53209 −0.684040 −2.41609 −1.00000 2.87939 0.652704 −1.28558
3310.6 1.87939 1.28558 2.53209 0.684040 2.41609 −1.00000 2.87939 0.652704 1.28558
 $$n$$: 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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3311.h Odd 1 CM by $$\Q(\sqrt{-3311})$$ yes
7.b Odd 1 yes
473.d Even 1 yes

## Hecke kernels

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

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