# Properties

 Label 3311.1.h.a Level 3311 Weight 1 Character orbit 3311.h Self dual yes Analytic conductor 1.652 Analytic rank 0 Dimension 1 Projective image $$D_{2}$$ CM/RM discs -7, -3311, 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 $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.65240425683$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{-7}, \sqrt{473})$$ Artin image $D_4$ Artin field Galois closure of 4.0.23177.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{2} + 3q^{4} - q^{7} - 4q^{8} - q^{9} + O(q^{10})$$ $$q - 2q^{2} + 3q^{4} - q^{7} - 4q^{8} - q^{9} - q^{11} + 2q^{14} + 5q^{16} + 2q^{18} + 2q^{22} - 2q^{23} - q^{25} - 3q^{28} + 2q^{29} - 6q^{32} - 3q^{36} + q^{43} - 3q^{44} + 4q^{46} + q^{49} + 2q^{50} + 2q^{53} + 4q^{56} - 4q^{58} + q^{63} + 7q^{64} + 2q^{67} + 4q^{72} + q^{77} + q^{81} - 2q^{86} + 4q^{88} - 6q^{92} - 2q^{98} + q^{99} + O(q^{100})$$

## 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
 0
−2.00000 0 3.00000 0 0 −1.00000 −4.00000 −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 Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
473.d even 2 1 RM by $$\Q(\sqrt{473})$$
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.a 1
7.b odd 2 1 CM 3311.1.h.a 1
11.b odd 2 1 3311.1.h.f yes 1
43.b odd 2 1 3311.1.h.f yes 1
77.b even 2 1 3311.1.h.f yes 1
301.c even 2 1 3311.1.h.f yes 1
473.d even 2 1 RM 3311.1.h.a 1
3311.h odd 2 1 CM 3311.1.h.a 1

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

 $$T_{2} + 2$$ $$T_{3}$$

## Hecke characteristic polynomials

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