# Properties

 Label 605.2.a.l Level $605$ Weight $2$ Character orbit 605.a Self dual yes Analytic conductor $4.831$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.2525.1 Defining polynomial: $$x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 5$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{2} - \beta_{3} ) q^{2} + ( -1 + \beta_{1} ) q^{3} + ( 2 - \beta_{1} - \beta_{3} ) q^{4} + q^{5} + ( -2 + \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{6} + ( 3 + \beta_{3} ) q^{7} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{8} + ( 1 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{2} - \beta_{3} ) q^{2} + ( -1 + \beta_{1} ) q^{3} + ( 2 - \beta_{1} - \beta_{3} ) q^{4} + q^{5} + ( -2 + \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{6} + ( 3 + \beta_{3} ) q^{7} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{8} + ( 1 - \beta_{1} + \beta_{2} ) q^{9} + ( 1 + \beta_{2} - \beta_{3} ) q^{10} + ( -6 + 2 \beta_{1} - 3 \beta_{2} ) q^{12} + ( 1 - \beta_{2} - \beta_{3} ) q^{13} + ( 1 + 4 \beta_{2} - 2 \beta_{3} ) q^{14} + ( -1 + \beta_{1} ) q^{15} + ( 3 + 4 \beta_{2} - \beta_{3} ) q^{16} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{17} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{18} + ( 3 + \beta_{1} + \beta_{2} ) q^{19} + ( 2 - \beta_{1} - \beta_{3} ) q^{20} + ( -2 + 3 \beta_{1} + 2 \beta_{2} ) q^{21} + ( 1 - 3 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{23} + ( -7 + 3 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{24} + q^{25} + ( 2 + \beta_{1} - 3 \beta_{3} ) q^{26} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{27} + ( 3 - 4 \beta_{1} - \beta_{2} - \beta_{3} ) q^{28} + ( -1 + \beta_{1} + 3 \beta_{2} ) q^{29} + ( -2 + \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{30} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{31} + ( 3 - 2 \beta_{1} + 2 \beta_{3} ) q^{32} + ( -4 + 2 \beta_{1} - \beta_{2} ) q^{34} + ( 3 + \beta_{3} ) q^{35} + ( 6 - 3 \beta_{1} + 5 \beta_{2} ) q^{36} + ( -\beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{37} + ( 3 + \beta_{2} - 2 \beta_{3} ) q^{38} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{39} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{40} + ( -2 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{41} + ( -3 + \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{42} + ( 5 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{43} + ( 1 - \beta_{1} + \beta_{2} ) q^{45} + ( 10 - 7 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{46} + ( -2 + 3 \beta_{1} + \beta_{3} ) q^{47} + ( -4 + 3 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{48} + ( 4 + \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{49} + ( 1 + \beta_{2} - \beta_{3} ) q^{50} + ( 4 + 4 \beta_{2} - \beta_{3} ) q^{51} + ( 5 + \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{52} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{53} + ( -3 + 4 \beta_{2} ) q^{54} + ( 6 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{56} + ( 3 \beta_{1} + \beta_{3} ) q^{57} + ( 1 - 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{58} + ( -3 - 2 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{59} + ( -6 + 2 \beta_{1} - 3 \beta_{2} ) q^{60} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{61} + ( -2 + 3 \beta_{1} - 6 \beta_{2} - \beta_{3} ) q^{62} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{63} + ( -5 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{64} + ( 1 - \beta_{2} - \beta_{3} ) q^{65} + ( -2 - 5 \beta_{2} + 3 \beta_{3} ) q^{67} + ( -7 + \beta_{1} - 6 \beta_{2} + \beta_{3} ) q^{68} + ( -11 + \beta_{1} - 9 \beta_{2} + 4 \beta_{3} ) q^{69} + ( 1 + 4 \beta_{2} - 2 \beta_{3} ) q^{70} + ( -8 + 2 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} ) q^{71} + ( 8 - 4 \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{72} + ( -6 + 2 \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{73} + ( -6 + 5 \beta_{2} + 2 \beta_{3} ) q^{74} + ( -1 + \beta_{1} ) q^{75} + ( 2 - 3 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{76} + ( -2 + 2 \beta_{1} - 5 \beta_{2} ) q^{78} + ( 1 - 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{79} + ( 3 + 4 \beta_{2} - \beta_{3} ) q^{80} + ( -7 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{81} + ( -10 + 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{82} + ( 6 - \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{83} + ( -16 + 3 \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{84} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{85} + ( 2 + \beta_{1} + 10 \beta_{2} - 7 \beta_{3} ) q^{86} + ( 4 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{87} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{89} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{90} + ( 1 - 2 \beta_{1} - 2 \beta_{2} ) q^{91} + ( 17 - 7 \beta_{1} + 18 \beta_{2} ) q^{92} + ( 6 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{93} + ( -7 + 3 \beta_{1} - 7 \beta_{2} + 3 \beta_{3} ) q^{94} + ( 3 + \beta_{1} + \beta_{2} ) q^{95} + ( -7 + 3 \beta_{1} + 2 \beta_{2} ) q^{96} + ( 1 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{97} + ( -8 + 2 \beta_{1} + 7 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 3q^{2} - 2q^{3} + 7q^{4} + 4q^{5} - q^{6} + 11q^{7} + 9q^{8} + O(q^{10})$$ $$4q + 3q^{2} - 2q^{3} + 7q^{4} + 4q^{5} - q^{6} + 11q^{7} + 9q^{8} + 3q^{10} - 14q^{12} + 7q^{13} - 2q^{14} - 2q^{15} + 5q^{16} + 3q^{17} + 2q^{18} + 12q^{19} + 7q^{20} - 6q^{21} - 9q^{23} - 15q^{24} + 4q^{25} + 13q^{26} - 5q^{27} + 7q^{28} - 8q^{29} - q^{30} + 3q^{31} + 6q^{32} - 10q^{34} + 11q^{35} + 8q^{36} - 3q^{37} + 12q^{38} - 3q^{39} + 9q^{40} - 7q^{41} + 2q^{42} + 21q^{43} + 12q^{46} - 3q^{47} - 2q^{48} + 15q^{49} + 3q^{50} + 9q^{51} + 27q^{52} - 11q^{53} - 20q^{54} + 15q^{56} + 5q^{57} + 2q^{58} - 7q^{59} - 14q^{60} + 4q^{61} + 11q^{62} + 3q^{63} - 27q^{64} + 7q^{65} - q^{67} - 15q^{68} - 28q^{69} - 2q^{70} - 15q^{71} + 13q^{72} - 9q^{73} - 36q^{74} - 2q^{75} + 8q^{76} + 6q^{78} + 6q^{79} + 5q^{80} - 20q^{81} - 44q^{82} + 15q^{83} - 47q^{84} + 3q^{85} - 3q^{86} + 15q^{87} + 2q^{90} + 4q^{91} + 18q^{92} + 21q^{93} - 11q^{94} + 12q^{95} - 26q^{96} + 6q^{97} - 42q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 5$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4 \beta_{1} + 3$$

## 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
 −0.777484 2.46673 1.77748 −1.46673
−1.87603 −1.77748 1.51949 1.00000 3.33461 4.25800 0.901454 0.159450 −1.87603
1.2 0.0935099 1.46673 −1.99126 1.00000 0.137154 4.52452 −0.373222 −0.848698 0.0935099
1.3 2.25800 0.777484 3.09855 1.00000 1.75556 0.123970 2.48051 −2.39552 2.25800
1.4 2.52452 −2.46673 4.37322 1.00000 −6.22732 2.09351 5.99126 3.08477 2.52452
 $$n$$: 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.l 4
3.b odd 2 1 5445.2.a.bg 4
4.b odd 2 1 9680.2.a.cs 4
5.b even 2 1 3025.2.a.v 4
11.b odd 2 1 605.2.a.i 4
11.c even 5 2 55.2.g.a 8
11.c even 5 2 605.2.g.j 8
11.d odd 10 2 605.2.g.g 8
11.d odd 10 2 605.2.g.n 8
33.d even 2 1 5445.2.a.bu 4
33.h odd 10 2 495.2.n.f 8
44.c even 2 1 9680.2.a.cv 4
44.h odd 10 2 880.2.bo.e 8
55.d odd 2 1 3025.2.a.be 4
55.j even 10 2 275.2.h.b 8
55.k odd 20 4 275.2.z.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.g.a 8 11.c even 5 2
275.2.h.b 8 55.j even 10 2
275.2.z.b 16 55.k odd 20 4
495.2.n.f 8 33.h odd 10 2
605.2.a.i 4 11.b odd 2 1
605.2.a.l 4 1.a even 1 1 trivial
605.2.g.g 8 11.d odd 10 2
605.2.g.j 8 11.c even 5 2
605.2.g.n 8 11.d odd 10 2
880.2.bo.e 8 44.h odd 10 2
3025.2.a.v 4 5.b even 2 1
3025.2.a.be 4 55.d odd 2 1
5445.2.a.bg 4 3.b odd 2 1
5445.2.a.bu 4 33.d even 2 1
9680.2.a.cs 4 4.b odd 2 1
9680.2.a.cv 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(605))$$:

 $$T_{2}^{4} - 3 T_{2}^{3} - 3 T_{2}^{2} + 11 T_{2} - 1$$ $$T_{3}^{4} + 2 T_{3}^{3} - 4 T_{3}^{2} - 5 T_{3} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + 11 T - 3 T^{2} - 3 T^{3} + T^{4}$$
$3$ $$5 - 5 T - 4 T^{2} + 2 T^{3} + T^{4}$$
$5$ $$( -1 + T )^{4}$$
$7$ $$5 - 45 T + 39 T^{2} - 11 T^{3} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$-11 + 9 T + 7 T^{2} - 7 T^{3} + T^{4}$$
$17$ $$-11 + 26 T - 8 T^{2} - 3 T^{3} + T^{4}$$
$19$ $$25 - 65 T + 46 T^{2} - 12 T^{3} + T^{4}$$
$23$ $$-1669 - 706 T - 54 T^{2} + 9 T^{3} + T^{4}$$
$29$ $$-55 - 95 T - 4 T^{2} + 8 T^{3} + T^{4}$$
$31$ $$101 + 3 T - 31 T^{2} - 3 T^{3} + T^{4}$$
$37$ $$151 - 28 T - 56 T^{2} + 3 T^{3} + T^{4}$$
$41$ $$-499 - 382 T - 46 T^{2} + 7 T^{3} + T^{4}$$
$43$ $$-59 - 191 T + 121 T^{2} - 21 T^{3} + T^{4}$$
$47$ $$71 - 133 T - 51 T^{2} + 3 T^{3} + T^{4}$$
$53$ $$-1 + 27 T + 33 T^{2} + 11 T^{3} + T^{4}$$
$59$ $$-3025 - 1195 T - 109 T^{2} + 7 T^{3} + T^{4}$$
$61$ $$55 + 90 T - 41 T^{2} - 4 T^{3} + T^{4}$$
$67$ $$-101 - 238 T - 82 T^{2} + T^{3} + T^{4}$$
$71$ $$-7799 - 2270 T - 98 T^{2} + 15 T^{3} + T^{4}$$
$73$ $$-389 - 596 T - 74 T^{2} + 9 T^{3} + T^{4}$$
$79$ $$-2155 + 935 T - 96 T^{2} - 6 T^{3} + T^{4}$$
$83$ $$-29 + 25 T + 43 T^{2} - 15 T^{3} + T^{4}$$
$89$ $$725 - 400 T - 150 T^{2} + T^{4}$$
$97$ $$-25 - 90 T - 56 T^{2} - 6 T^{3} + T^{4}$$