# Properties

 Label 605.2.a.e Level $605$ Weight $2$ Character orbit 605.a Self dual yes Analytic conductor $4.831$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.83094932229$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - q^{3} + q^{4} - q^{5} -\beta q^{6} -\beta q^{7} -\beta q^{8} -2 q^{9} +O(q^{10})$$ $$q + \beta q^{2} - q^{3} + q^{4} - q^{5} -\beta q^{6} -\beta q^{7} -\beta q^{8} -2 q^{9} -\beta q^{10} - q^{12} + 2 \beta q^{13} -3 q^{14} + q^{15} -5 q^{16} -4 \beta q^{17} -2 \beta q^{18} -2 \beta q^{19} - q^{20} + \beta q^{21} + \beta q^{24} + q^{25} + 6 q^{26} + 5 q^{27} -\beta q^{28} + \beta q^{30} -8 q^{31} -3 \beta q^{32} -12 q^{34} + \beta q^{35} -2 q^{36} -8 q^{37} -6 q^{38} -2 \beta q^{39} + \beta q^{40} + 7 \beta q^{41} + 3 q^{42} + 5 \beta q^{43} + 2 q^{45} + 9 q^{47} + 5 q^{48} -4 q^{49} + \beta q^{50} + 4 \beta q^{51} + 2 \beta q^{52} + 6 q^{53} + 5 \beta q^{54} + 3 q^{56} + 2 \beta q^{57} -12 q^{59} + q^{60} + 5 \beta q^{61} -8 \beta q^{62} + 2 \beta q^{63} + q^{64} -2 \beta q^{65} -5 q^{67} -4 \beta q^{68} + 3 q^{70} -12 q^{71} + 2 \beta q^{72} -8 \beta q^{74} - q^{75} -2 \beta q^{76} -6 q^{78} -6 \beta q^{79} + 5 q^{80} + q^{81} + 21 q^{82} + 2 \beta q^{83} + \beta q^{84} + 4 \beta q^{85} + 15 q^{86} + 3 q^{89} + 2 \beta q^{90} -6 q^{91} + 8 q^{93} + 9 \beta q^{94} + 2 \beta q^{95} + 3 \beta q^{96} -10 q^{97} -4 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{4} - 2q^{5} - 4q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{4} - 2q^{5} - 4q^{9} - 2q^{12} - 6q^{14} + 2q^{15} - 10q^{16} - 2q^{20} + 2q^{25} + 12q^{26} + 10q^{27} - 16q^{31} - 24q^{34} - 4q^{36} - 16q^{37} - 12q^{38} + 6q^{42} + 4q^{45} + 18q^{47} + 10q^{48} - 8q^{49} + 12q^{53} + 6q^{56} - 24q^{59} + 2q^{60} + 2q^{64} - 10q^{67} + 6q^{70} - 24q^{71} - 2q^{75} - 12q^{78} + 10q^{80} + 2q^{81} + 42q^{82} + 30q^{86} + 6q^{89} - 12q^{91} + 16q^{93} - 20q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
−1.73205 −1.00000 1.00000 −1.00000 1.73205 1.73205 1.73205 −2.00000 1.73205
1.2 1.73205 −1.00000 1.00000 −1.00000 −1.73205 −1.73205 −1.73205 −2.00000 −1.73205
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$11$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b 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.a.e 2
3.b odd 2 1 5445.2.a.u 2
4.b odd 2 1 9680.2.a.bu 2
5.b even 2 1 3025.2.a.l 2
11.b odd 2 1 inner 605.2.a.e 2
11.c even 5 4 605.2.g.i 8
11.d odd 10 4 605.2.g.i 8
33.d even 2 1 5445.2.a.u 2
44.c even 2 1 9680.2.a.bu 2
55.d odd 2 1 3025.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.e 2 1.a even 1 1 trivial
605.2.a.e 2 11.b odd 2 1 inner
605.2.g.i 8 11.c even 5 4
605.2.g.i 8 11.d odd 10 4
3025.2.a.l 2 5.b even 2 1
3025.2.a.l 2 55.d odd 2 1
5445.2.a.u 2 3.b odd 2 1
5445.2.a.u 2 33.d even 2 1
9680.2.a.bu 2 4.b odd 2 1
9680.2.a.bu 2 44.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(605))$$:

 $$T_{2}^{2} - 3$$ $$T_{3} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$-3 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$-12 + T^{2}$$
$17$ $$-48 + T^{2}$$
$19$ $$-12 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$( 8 + T )^{2}$$
$41$ $$-147 + T^{2}$$
$43$ $$-75 + T^{2}$$
$47$ $$( -9 + T )^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$( 12 + T )^{2}$$
$61$ $$-75 + T^{2}$$
$67$ $$( 5 + T )^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$T^{2}$$
$79$ $$-108 + T^{2}$$
$83$ $$-12 + T^{2}$$
$89$ $$( -3 + T )^{2}$$
$97$ $$( 10 + T )^{2}$$