# Properties

 Label 605.2.b.a Level $605$ Weight $2$ Character orbit 605.b Analytic conductor $4.831$ Analytic rank $0$ Dimension $2$ CM discriminant -11 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-11})$$ Defining polynomial: $$x^{2} - x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{-11})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \beta ) q^{3} + 2 q^{4} + ( 1 + \beta ) q^{5} -8 q^{9} +O(q^{10})$$ $$q + ( -1 + 2 \beta ) q^{3} + 2 q^{4} + ( 1 + \beta ) q^{5} -8 q^{9} + ( -2 + 4 \beta ) q^{12} + ( -7 + 3 \beta ) q^{15} + 4 q^{16} + ( 2 + 2 \beta ) q^{20} + ( 1 - 2 \beta ) q^{23} + ( -2 + 3 \beta ) q^{25} + ( 5 - 10 \beta ) q^{27} + 5 q^{31} -16 q^{36} + ( 3 - 6 \beta ) q^{37} + ( -8 - 8 \beta ) q^{45} + ( 2 - 4 \beta ) q^{47} + ( -4 + 8 \beta ) q^{48} + 7 q^{49} + ( -4 + 8 \beta ) q^{53} -15 q^{59} + ( -14 + 6 \beta ) q^{60} + 8 q^{64} + ( -3 + 6 \beta ) q^{67} + 11 q^{69} -3 q^{71} + ( -16 - \beta ) q^{75} + ( 4 + 4 \beta ) q^{80} + 31 q^{81} + 9 q^{89} + ( 2 - 4 \beta ) q^{92} + ( -5 + 10 \beta ) q^{93} + ( 3 - 6 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{4} + 3q^{5} - 16q^{9} + O(q^{10})$$ $$2q + 4q^{4} + 3q^{5} - 16q^{9} - 11q^{15} + 8q^{16} + 6q^{20} - q^{25} + 10q^{31} - 32q^{36} - 24q^{45} + 14q^{49} - 30q^{59} - 22q^{60} + 16q^{64} + 22q^{69} - 6q^{71} - 33q^{75} + 12q^{80} + 62q^{81} + 18q^{89} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$486$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
364.1
 0.5 − 1.65831i 0.5 + 1.65831i
0 3.31662i 2.00000 1.50000 1.65831i 0 0 0 −8.00000 0
364.2 0 3.31662i 2.00000 1.50000 + 1.65831i 0 0 0 −8.00000 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
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
5.b even 2 1 inner
55.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.b.a 2
5.b even 2 1 inner 605.2.b.a 2
5.c odd 4 2 3025.2.a.k 2
11.b odd 2 1 CM 605.2.b.a 2
11.c even 5 4 605.2.j.b 8
11.d odd 10 4 605.2.j.b 8
55.d odd 2 1 inner 605.2.b.a 2
55.e even 4 2 3025.2.a.k 2
55.h odd 10 4 605.2.j.b 8
55.j even 10 4 605.2.j.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.a 2 1.a even 1 1 trivial
605.2.b.a 2 5.b even 2 1 inner
605.2.b.a 2 11.b odd 2 1 CM
605.2.b.a 2 55.d odd 2 1 inner
605.2.j.b 8 11.c even 5 4
605.2.j.b 8 11.d odd 10 4
605.2.j.b 8 55.h odd 10 4
605.2.j.b 8 55.j even 10 4
3025.2.a.k 2 5.c odd 4 2
3025.2.a.k 2 55.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(605, [\chi])$$:

 $$T_{2}$$ $$T_{19}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$11 + T^{2}$$
$5$ $$5 - 3 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$11 + T^{2}$$
$29$ $$T^{2}$$
$31$ $$( -5 + T )^{2}$$
$37$ $$99 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$44 + T^{2}$$
$53$ $$176 + T^{2}$$
$59$ $$( 15 + T )^{2}$$
$61$ $$T^{2}$$
$67$ $$99 + T^{2}$$
$71$ $$( 3 + T )^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$( -9 + T )^{2}$$
$97$ $$99 + T^{2}$$