# Properties

 Label 605.2.a.h Level $605$ Weight $2$ Character orbit 605.a Self dual yes Analytic conductor $4.831$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# 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: $$3$$ Coefficient field: 3.3.404.1 Defining polynomial: $$x^{3} - x^{2} - 5 x - 1$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + \beta_{1} q^{3} + ( 3 + \beta_{1} - \beta_{2} ) q^{4} - q^{5} + ( 1 + 2 \beta_{1} ) q^{6} + ( -2 \beta_{1} + \beta_{2} ) q^{7} + ( -4 + \beta_{1} + 2 \beta_{2} ) q^{8} + ( \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + \beta_{1} q^{3} + ( 3 + \beta_{1} - \beta_{2} ) q^{4} - q^{5} + ( 1 + 2 \beta_{1} ) q^{6} + ( -2 \beta_{1} + \beta_{2} ) q^{7} + ( -4 + \beta_{1} + 2 \beta_{2} ) q^{8} + ( \beta_{1} + \beta_{2} ) q^{9} -\beta_{2} q^{10} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{12} + ( 2 - \beta_{1} + \beta_{2} ) q^{13} + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{14} -\beta_{1} q^{15} + ( 5 + 2 \beta_{1} - 4 \beta_{2} ) q^{16} + ( 2 - 2 \beta_{1} ) q^{17} + ( 6 + 3 \beta_{1} - \beta_{2} ) q^{18} + ( 2 - \beta_{1} - \beta_{2} ) q^{19} + ( -3 - \beta_{1} + \beta_{2} ) q^{20} + ( -5 - 2 \beta_{2} ) q^{21} + ( -2 - \beta_{1} + \beta_{2} ) q^{23} + ( 5 + \beta_{1} + \beta_{2} ) q^{24} + q^{25} + ( 4 - \beta_{1} + \beta_{2} ) q^{26} + ( 4 + \beta_{2} ) q^{27} + ( -8 - 3 \beta_{1} + 2 \beta_{2} ) q^{28} + 2 \beta_{2} q^{29} + ( -1 - 2 \beta_{1} ) q^{30} + ( 4 + \beta_{1} + \beta_{2} ) q^{31} + ( -10 - 2 \beta_{1} + 5 \beta_{2} ) q^{32} + ( -2 - 4 \beta_{1} + 2 \beta_{2} ) q^{34} + ( 2 \beta_{1} - \beta_{2} ) q^{35} + ( -2 + 3 \beta_{1} + 5 \beta_{2} ) q^{36} + ( 2 - 2 \beta_{2} ) q^{37} + ( -6 - 3 \beta_{1} + 3 \beta_{2} ) q^{38} + ( -2 + 3 \beta_{1} - \beta_{2} ) q^{39} + ( 4 - \beta_{1} - 2 \beta_{2} ) q^{40} + ( 3 + 3 \beta_{1} - 3 \beta_{2} ) q^{41} + ( -10 - 2 \beta_{1} - 3 \beta_{2} ) q^{42} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{43} + ( -\beta_{1} - \beta_{2} ) q^{45} + ( 4 - \beta_{1} - 3 \beta_{2} ) q^{46} + ( -4 + \beta_{1} - 4 \beta_{2} ) q^{47} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{48} + ( 6 - 3 \beta_{1} + 3 \beta_{2} ) q^{49} + \beta_{2} q^{50} + ( -6 - 2 \beta_{2} ) q^{51} + ( \beta_{1} + \beta_{2} ) q^{52} + ( -2 - \beta_{1} + \beta_{2} ) q^{53} + ( 5 + \beta_{1} + 3 \beta_{2} ) q^{54} + ( 1 + 2 \beta_{1} - 8 \beta_{2} ) q^{56} + ( -4 - \beta_{1} - \beta_{2} ) q^{57} + ( 10 + 2 \beta_{1} - 2 \beta_{2} ) q^{58} + ( 2 + \beta_{1} + 3 \beta_{2} ) q^{59} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{60} + ( 3 - 5 \beta_{1} - \beta_{2} ) q^{61} + ( 6 + 3 \beta_{1} + 3 \beta_{2} ) q^{62} + ( -2 - 3 \beta_{1} - 3 \beta_{2} ) q^{63} + ( 13 - 3 \beta_{1} - 7 \beta_{2} ) q^{64} + ( -2 + \beta_{1} - \beta_{2} ) q^{65} + ( 6 - \beta_{1} + 2 \beta_{2} ) q^{67} + ( 2 - 2 \beta_{1} - 4 \beta_{2} ) q^{68} + ( -2 - \beta_{1} - \beta_{2} ) q^{69} + ( -3 + 3 \beta_{1} + \beta_{2} ) q^{70} + ( 2 + \beta_{1} - \beta_{2} ) q^{71} + ( 16 + 5 \beta_{1} - 5 \beta_{2} ) q^{72} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{73} + ( -10 - 2 \beta_{1} + 4 \beta_{2} ) q^{74} + \beta_{1} q^{75} + ( 8 - \beta_{1} - 7 \beta_{2} ) q^{76} + ( -2 + 5 \beta_{1} - \beta_{2} ) q^{78} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{79} + ( -5 - 2 \beta_{1} + 4 \beta_{2} ) q^{80} + ( 1 + 3 \beta_{1} - 3 \beta_{2} ) q^{81} + ( -12 + 3 \beta_{1} + 6 \beta_{2} ) q^{82} + ( 6 + 3 \beta_{1} - 3 \beta_{2} ) q^{83} + ( -7 - 7 \beta_{1} - 3 \beta_{2} ) q^{84} + ( -2 + 2 \beta_{1} ) q^{85} + ( -3 + 3 \beta_{1} + 3 \beta_{2} ) q^{86} + ( 2 + 4 \beta_{1} ) q^{87} + ( 3 + 6 \beta_{1} - 4 \beta_{2} ) q^{89} + ( -6 - 3 \beta_{1} + \beta_{2} ) q^{90} + ( 8 - 7 \beta_{1} + 3 \beta_{2} ) q^{91} + ( -12 - 3 \beta_{1} + 5 \beta_{2} ) q^{92} + ( 4 + 7 \beta_{1} + \beta_{2} ) q^{93} + ( -19 - 2 \beta_{1} ) q^{94} + ( -2 + \beta_{1} + \beta_{2} ) q^{95} + ( -1 - 2 \beta_{1} - 2 \beta_{2} ) q^{96} + ( -2 + 7 \beta_{1} - 3 \beta_{2} ) q^{97} + ( 12 - 3 \beta_{1} + 3 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + q^{2} + q^{3} + 9q^{4} - 3q^{5} + 5q^{6} - q^{7} - 9q^{8} + 2q^{9} + O(q^{10})$$ $$3q + q^{2} + q^{3} + 9q^{4} - 3q^{5} + 5q^{6} - q^{7} - 9q^{8} + 2q^{9} - q^{10} + 9q^{12} + 6q^{13} + 5q^{14} - q^{15} + 13q^{16} + 4q^{17} + 20q^{18} + 4q^{19} - 9q^{20} - 17q^{21} - 6q^{23} + 17q^{24} + 3q^{25} + 12q^{26} + 13q^{27} - 25q^{28} + 2q^{29} - 5q^{30} + 14q^{31} - 27q^{32} - 8q^{34} + q^{35} + 2q^{36} + 4q^{37} - 18q^{38} - 4q^{39} + 9q^{40} + 9q^{41} - 35q^{42} + 7q^{43} - 2q^{45} + 8q^{46} - 15q^{47} + 7q^{48} + 18q^{49} + q^{50} - 20q^{51} + 2q^{52} - 6q^{53} + 19q^{54} - 3q^{56} - 14q^{57} + 30q^{58} + 10q^{59} - 9q^{60} + 3q^{61} + 24q^{62} - 12q^{63} + 29q^{64} - 6q^{65} + 19q^{67} - 8q^{69} - 5q^{70} + 6q^{71} + 48q^{72} - 12q^{73} - 28q^{74} + q^{75} + 16q^{76} - 2q^{78} + 2q^{79} - 13q^{80} + 3q^{81} - 27q^{82} + 18q^{83} - 31q^{84} - 4q^{85} - 3q^{86} + 10q^{87} + 11q^{89} - 20q^{90} + 20q^{91} - 34q^{92} + 20q^{93} - 59q^{94} - 4q^{95} - 7q^{96} - 2q^{97} + 36q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$

## 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
 −0.210756 −1.65544 2.86620
−2.74483 −0.210756 5.53407 −1.00000 0.578488 −2.32331 −9.70041 −2.95558 2.74483
1.2 1.39593 −1.65544 −0.0513742 −1.00000 −2.31088 4.70682 −2.86358 −0.259511 −1.39593
1.3 2.34889 2.86620 3.51730 −1.00000 6.73240 −3.38350 3.56399 5.21509 −2.34889
 $$n$$: 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.a.h yes 3
3.b odd 2 1 5445.2.a.bb 3
4.b odd 2 1 9680.2.a.cb 3
5.b even 2 1 3025.2.a.p 3
11.b odd 2 1 605.2.a.g 3
11.c even 5 4 605.2.g.o 12
11.d odd 10 4 605.2.g.p 12
33.d even 2 1 5445.2.a.bd 3
44.c even 2 1 9680.2.a.bz 3
55.d odd 2 1 3025.2.a.u 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.g 3 11.b odd 2 1
605.2.a.h yes 3 1.a even 1 1 trivial
605.2.g.o 12 11.c even 5 4
605.2.g.p 12 11.d odd 10 4
3025.2.a.p 3 5.b even 2 1
3025.2.a.u 3 55.d odd 2 1
5445.2.a.bb 3 3.b odd 2 1
5445.2.a.bd 3 33.d even 2 1
9680.2.a.bz 3 44.c even 2 1
9680.2.a.cb 3 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(605))$$:

 $$T_{2}^{3} - T_{2}^{2} - 7 T_{2} + 9$$ $$T_{3}^{3} - T_{3}^{2} - 5 T_{3} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 - 7 T - T^{2} + T^{3}$$
$3$ $$-1 - 5 T - T^{2} + T^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$-37 - 19 T + T^{2} + T^{3}$$
$11$ $$T^{3}$$
$13$ $$4 + 4 T - 6 T^{2} + T^{3}$$
$17$ $$48 - 16 T - 4 T^{2} + T^{3}$$
$19$ $$36 - 12 T - 4 T^{2} + T^{3}$$
$23$ $$-12 + 4 T + 6 T^{2} + T^{3}$$
$29$ $$72 - 28 T - 2 T^{2} + T^{3}$$
$31$ $$-36 + 48 T - 14 T^{2} + T^{3}$$
$37$ $$-16 - 24 T - 4 T^{2} + T^{3}$$
$41$ $$297 - 45 T - 9 T^{2} + T^{3}$$
$43$ $$63 - 3 T - 7 T^{2} + T^{3}$$
$47$ $$-801 - 29 T + 15 T^{2} + T^{3}$$
$53$ $$-12 + 4 T + 6 T^{2} + T^{3}$$
$59$ $$348 - 52 T - 10 T^{2} + T^{3}$$
$61$ $$919 - 161 T - 3 T^{2} + T^{3}$$
$67$ $$-59 + 95 T - 19 T^{2} + T^{3}$$
$71$ $$12 + 4 T - 6 T^{2} + T^{3}$$
$73$ $$-32 + 16 T + 12 T^{2} + T^{3}$$
$79$ $$296 - 76 T - 2 T^{2} + T^{3}$$
$83$ $$324 + 36 T - 18 T^{2} + T^{3}$$
$89$ $$1719 - 157 T - 11 T^{2} + T^{3}$$
$97$ $$932 - 228 T + 2 T^{2} + T^{3}$$