# Properties

 Label 605.2.a.m Level $605$ Weight $2$ Character orbit 605.a Self dual yes Analytic conductor $4.831$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.6.27433728.1 Defining polynomial: $$x^{6} - 9 x^{4} + 15 x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 9 x^{4} + 15 x^{2} - 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{4} - 8 \nu^{2} + 7$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} - 8 \nu^{3} + 9 \nu$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{5} + 10 \nu^{3} - 19 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{3} + 8 \beta_{2} + 17$$ $$\nu^{5}$$ $$=$$ $$8 \beta_{5} + 10 \beta_{4} + 31 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.62383 −1.37268 −0.480901 0.480901 1.37268 2.62383
−2.62383 1.33988 4.88448 1.00000 −3.51561 2.62383 −7.56839 −1.20473 −2.62383
1.2 −1.37268 3.26180 −0.115749 1.00000 −4.47741 1.37268 2.90425 7.63935 −1.37268
1.3 −0.480901 −1.60168 −1.76873 1.00000 0.770249 0.480901 1.81239 −0.434624 −0.480901
1.4 0.480901 −1.60168 −1.76873 1.00000 −0.770249 −0.480901 −1.81239 −0.434624 0.480901
1.5 1.37268 3.26180 −0.115749 1.00000 4.47741 −1.37268 −2.90425 7.63935 1.37268
1.6 2.62383 1.33988 4.88448 1.00000 3.51561 −2.62383 7.56839 −1.20473 2.62383
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.m 6
3.b odd 2 1 5445.2.a.bx 6
4.b odd 2 1 9680.2.a.cw 6
5.b even 2 1 3025.2.a.bg 6
11.b odd 2 1 inner 605.2.a.m 6
11.c even 5 4 605.2.g.q 24
11.d odd 10 4 605.2.g.q 24
33.d even 2 1 5445.2.a.bx 6
44.c even 2 1 9680.2.a.cw 6
55.d odd 2 1 3025.2.a.bg 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.m 6 1.a even 1 1 trivial
605.2.a.m 6 11.b odd 2 1 inner
605.2.g.q 24 11.c even 5 4
605.2.g.q 24 11.d odd 10 4
3025.2.a.bg 6 5.b even 2 1
3025.2.a.bg 6 55.d odd 2 1
5445.2.a.bx 6 3.b odd 2 1
5445.2.a.bx 6 33.d even 2 1
9680.2.a.cw 6 4.b odd 2 1
9680.2.a.cw 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(605))$$:

 $$T_{2}^{6} - 9 T_{2}^{4} + 15 T_{2}^{2} - 3$$ $$T_{3}^{3} - 3 T_{3}^{2} - 3 T_{3} + 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + 15 T^{2} - 9 T^{4} + T^{6}$$
$3$ $$( 7 - 3 T - 3 T^{2} + T^{3} )^{2}$$
$5$ $$( -1 + T )^{6}$$
$7$ $$-3 + 15 T^{2} - 9 T^{4} + T^{6}$$
$11$ $$T^{6}$$
$13$ $$-3888 + 1296 T^{2} - 72 T^{4} + T^{6}$$
$17$ $$-768 + 384 T^{2} - 48 T^{4} + T^{6}$$
$19$ $$-48 + 96 T^{2} - 36 T^{4} + T^{6}$$
$23$ $$( 84 - 12 T - 6 T^{2} + T^{3} )^{2}$$
$29$ $$-6912 + 5184 T^{2} - 144 T^{4} + T^{6}$$
$31$ $$( -124 - 48 T + T^{3} )^{2}$$
$37$ $$( -16 - 24 T + T^{3} )^{2}$$
$41$ $$-7203 + 2499 T^{2} - 105 T^{4} + T^{6}$$
$43$ $$-43923 + 4695 T^{2} - 129 T^{4} + T^{6}$$
$47$ $$( -303 + 141 T - 21 T^{2} + T^{3} )^{2}$$
$53$ $$( 732 - 84 T - 12 T^{2} + T^{3} )^{2}$$
$59$ $$( 12 - 12 T + T^{3} )^{2}$$
$61$ $$-1083 + 339 T^{2} - 33 T^{4} + T^{6}$$
$67$ $$( -97 + 69 T - 15 T^{2} + T^{3} )^{2}$$
$71$ $$( -12 - 12 T + T^{3} )^{2}$$
$73$ $$-49152 + 6144 T^{2} - 192 T^{4} + T^{6}$$
$79$ $$-101568 + 11184 T^{2} - 276 T^{4} + T^{6}$$
$83$ $$-40368 + 14640 T^{2} - 312 T^{4} + T^{6}$$
$89$ $$( -147 - 21 T + 15 T^{2} + T^{3} )^{2}$$
$97$ $$( 76 - 36 T - 12 T^{2} + T^{3} )^{2}$$