# Properties

 Label 605.2.a.f Level $605$ Weight $2$ Character orbit 605.a Self dual yes Analytic conductor $4.831$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 2 q^{3} + q^{4} + q^{5} + 2 \beta q^{6} + 2 \beta q^{7} - \beta q^{8} + q^{9} +O(q^{10})$$ q + b * q^2 + 2 * q^3 + q^4 + q^5 + 2*b * q^6 + 2*b * q^7 - b * q^8 + q^9 $$q + \beta q^{2} + 2 q^{3} + q^{4} + q^{5} + 2 \beta q^{6} + 2 \beta q^{7} - \beta q^{8} + q^{9} + \beta q^{10} + 2 q^{12} + 6 q^{14} + 2 q^{15} - 5 q^{16} - 4 \beta q^{17} + \beta q^{18} - 4 \beta q^{19} + q^{20} + 4 \beta q^{21} + 6 q^{23} - 2 \beta q^{24} + q^{25} - 4 q^{27} + 2 \beta q^{28} + 2 \beta q^{30} + 4 q^{31} - 3 \beta q^{32} - 12 q^{34} + 2 \beta q^{35} + q^{36} + 10 q^{37} - 12 q^{38} - \beta q^{40} + 4 \beta q^{41} + 12 q^{42} - 2 \beta q^{43} + q^{45} + 6 \beta q^{46} - 6 q^{47} - 10 q^{48} + 5 q^{49} + \beta q^{50} - 8 \beta q^{51} - 6 q^{53} - 4 \beta q^{54} - 6 q^{56} - 8 \beta q^{57} + 2 q^{60} + 4 \beta q^{61} + 4 \beta q^{62} + 2 \beta q^{63} + q^{64} + 10 q^{67} - 4 \beta q^{68} + 12 q^{69} + 6 q^{70} - \beta q^{72} - 4 \beta q^{73} + 10 \beta q^{74} + 2 q^{75} - 4 \beta q^{76} - 4 \beta q^{79} - 5 q^{80} - 11 q^{81} + 12 q^{82} - 10 \beta q^{83} + 4 \beta q^{84} - 4 \beta q^{85} - 6 q^{86} - 6 q^{89} + \beta q^{90} + 6 q^{92} + 8 q^{93} - 6 \beta q^{94} - 4 \beta q^{95} - 6 \beta q^{96} - 10 q^{97} + 5 \beta q^{98} +O(q^{100})$$ q + b * q^2 + 2 * q^3 + q^4 + q^5 + 2*b * q^6 + 2*b * q^7 - b * q^8 + q^9 + b * q^10 + 2 * q^12 + 6 * q^14 + 2 * q^15 - 5 * q^16 - 4*b * q^17 + b * q^18 - 4*b * q^19 + q^20 + 4*b * q^21 + 6 * q^23 - 2*b * q^24 + q^25 - 4 * q^27 + 2*b * q^28 + 2*b * q^30 + 4 * q^31 - 3*b * q^32 - 12 * q^34 + 2*b * q^35 + q^36 + 10 * q^37 - 12 * q^38 - b * q^40 + 4*b * q^41 + 12 * q^42 - 2*b * q^43 + q^45 + 6*b * q^46 - 6 * q^47 - 10 * q^48 + 5 * q^49 + b * q^50 - 8*b * q^51 - 6 * q^53 - 4*b * q^54 - 6 * q^56 - 8*b * q^57 + 2 * q^60 + 4*b * q^61 + 4*b * q^62 + 2*b * q^63 + q^64 + 10 * q^67 - 4*b * q^68 + 12 * q^69 + 6 * q^70 - b * q^72 - 4*b * q^73 + 10*b * q^74 + 2 * q^75 - 4*b * q^76 - 4*b * q^79 - 5 * q^80 - 11 * q^81 + 12 * q^82 - 10*b * q^83 + 4*b * q^84 - 4*b * q^85 - 6 * q^86 - 6 * q^89 + b * q^90 + 6 * q^92 + 8 * q^93 - 6*b * q^94 - 4*b * q^95 - 6*b * q^96 - 10 * q^97 + 5*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q + 4 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^9 $$2 q + 4 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{9} + 4 q^{12} + 12 q^{14} + 4 q^{15} - 10 q^{16} + 2 q^{20} + 12 q^{23} + 2 q^{25} - 8 q^{27} + 8 q^{31} - 24 q^{34} + 2 q^{36} + 20 q^{37} - 24 q^{38} + 24 q^{42} + 2 q^{45} - 12 q^{47} - 20 q^{48} + 10 q^{49} - 12 q^{53} - 12 q^{56} + 4 q^{60} + 2 q^{64} + 20 q^{67} + 24 q^{69} + 12 q^{70} + 4 q^{75} - 10 q^{80} - 22 q^{81} + 24 q^{82} - 12 q^{86} - 12 q^{89} + 12 q^{92} + 16 q^{93} - 20 q^{97}+O(q^{100})$$ 2 * q + 4 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^9 + 4 * q^12 + 12 * q^14 + 4 * q^15 - 10 * q^16 + 2 * q^20 + 12 * q^23 + 2 * q^25 - 8 * q^27 + 8 * q^31 - 24 * q^34 + 2 * q^36 + 20 * q^37 - 24 * q^38 + 24 * q^42 + 2 * q^45 - 12 * q^47 - 20 * q^48 + 10 * q^49 - 12 * q^53 - 12 * q^56 + 4 * q^60 + 2 * q^64 + 20 * q^67 + 24 * q^69 + 12 * q^70 + 4 * q^75 - 10 * q^80 - 22 * q^81 + 24 * q^82 - 12 * q^86 - 12 * q^89 + 12 * q^92 + 16 * q^93 - 20 * q^97

## 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 2.00000 1.00000 1.00000 −3.46410 −3.46410 1.73205 1.00000 −1.73205
1.2 1.73205 2.00000 1.00000 1.00000 3.46410 3.46410 −1.73205 1.00000 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.f 2
3.b odd 2 1 5445.2.a.r 2
4.b odd 2 1 9680.2.a.bg 2
5.b even 2 1 3025.2.a.j 2
11.b odd 2 1 inner 605.2.a.f 2
11.c even 5 4 605.2.g.h 8
11.d odd 10 4 605.2.g.h 8
33.d even 2 1 5445.2.a.r 2
44.c even 2 1 9680.2.a.bg 2
55.d odd 2 1 3025.2.a.j 2

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

 $$T_{2}^{2} - 3$$ T2^2 - 3 $$T_{3} - 2$$ T3 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3$$
$3$ $$(T - 2)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 12$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 48$$
$19$ $$T^{2} - 48$$
$23$ $$(T - 6)^{2}$$
$29$ $$T^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$(T - 10)^{2}$$
$41$ $$T^{2} - 48$$
$43$ $$T^{2} - 12$$
$47$ $$(T + 6)^{2}$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 48$$
$67$ $$(T - 10)^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 48$$
$79$ $$T^{2} - 48$$
$83$ $$T^{2} - 300$$
$89$ $$(T + 6)^{2}$$
$97$ $$(T + 10)^{2}$$