# Properties

 Label 3311.1.h.j Level 3311 Weight 1 Character orbit 3311.h Analytic conductor 1.652 Analytic rank 0 Dimension 2 Projective image $$D_{6}$$ RM disc. 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 $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$1.65240425683$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Projective image $$D_{6}$$ Projective field Galois closure of 6.0.76739047.3

## $q$-expansion

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

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

## 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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
473.d Even 1 RM by $$\Q(\sqrt{473})$$ yes
7.b Odd 1 yes
3311.h Odd 1 yes

## Hecke kernels

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

 $$T_{2}$$ $$\mathstrut -\mathstrut 1$$ $$T_{3}$$