# Properties

 Label 1250.2.a.l Level $1250$ Weight $2$ Character orbit 1250.a Self dual yes Analytic conductor $9.981$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1250 = 2 \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1250.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.98130025266$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.7625.1 Defining polynomial: $$x^{4} - x^{3} - 9 x^{2} + 4 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 50) 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + \beta_{1} q^{3} + q^{4} + \beta_{1} q^{6} + ( 1 + \beta_{3} ) q^{7} + q^{8} + ( 2 + \beta_{1} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + q^{2} + \beta_{1} q^{3} + q^{4} + \beta_{1} q^{6} + ( 1 + \beta_{3} ) q^{7} + q^{8} + ( 2 + \beta_{1} + \beta_{3} ) q^{9} + ( -1 - \beta_{3} ) q^{11} + \beta_{1} q^{12} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{13} + ( 1 + \beta_{3} ) q^{14} + q^{16} + ( 3 + \beta_{2} - \beta_{3} ) q^{17} + ( 2 + \beta_{1} + \beta_{3} ) q^{18} + ( -3 + \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{19} + ( \beta_{1} + 4 \beta_{2} ) q^{21} + ( -1 - \beta_{3} ) q^{22} + ( -1 - 2 \beta_{1} - \beta_{3} ) q^{23} + \beta_{1} q^{24} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{26} + ( 5 + 4 \beta_{2} + \beta_{3} ) q^{27} + ( 1 + \beta_{3} ) q^{28} + ( 4 + \beta_{1} + \beta_{2} ) q^{29} + ( -1 - 2 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{31} + q^{32} + ( -\beta_{1} - 4 \beta_{2} ) q^{33} + ( 3 + \beta_{2} - \beta_{3} ) q^{34} + ( 2 + \beta_{1} + \beta_{3} ) q^{36} + ( 4 - 2 \beta_{1} + \beta_{2} ) q^{37} + ( -3 + \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{38} + ( -6 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{39} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{41} + ( \beta_{1} + 4 \beta_{2} ) q^{42} + ( -1 - 4 \beta_{2} - \beta_{3} ) q^{43} + ( -1 - \beta_{3} ) q^{44} + ( -1 - 2 \beta_{1} - \beta_{3} ) q^{46} + ( 1 + \beta_{3} ) q^{47} + \beta_{1} q^{48} + ( -2 - 4 \beta_{2} + \beta_{3} ) q^{49} + ( 1 + 3 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{51} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{52} + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{53} + ( 5 + 4 \beta_{2} + \beta_{3} ) q^{54} + ( 1 + \beta_{3} ) q^{56} + ( 1 - 2 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{57} + ( 4 + \beta_{1} + \beta_{2} ) q^{58} + ( 2 - 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{59} + ( 5 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{61} + ( -1 - 2 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{62} + ( 6 + \beta_{1} + 2 \beta_{3} ) q^{63} + q^{64} + ( -\beta_{1} - 4 \beta_{2} ) q^{66} + ( 7 + 3 \beta_{3} ) q^{67} + ( 3 + \beta_{2} - \beta_{3} ) q^{68} + ( -10 - 3 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{69} + ( -3 + 4 \beta_{2} + \beta_{3} ) q^{71} + ( 2 + \beta_{1} + \beta_{3} ) q^{72} + ( 5 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{73} + ( 4 - 2 \beta_{1} + \beta_{2} ) q^{74} + ( -3 + \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{76} + ( -5 + 4 \beta_{2} - \beta_{3} ) q^{77} + ( -6 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{78} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{79} + ( -2 + 2 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{81} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{82} + ( -9 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{83} + ( \beta_{1} + 4 \beta_{2} ) q^{84} + ( -1 - 4 \beta_{2} - \beta_{3} ) q^{86} + ( 6 + 5 \beta_{1} + 2 \beta_{3} ) q^{87} + ( -1 - \beta_{3} ) q^{88} + ( -1 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{89} + ( -1 - 2 \beta_{1} + 3 \beta_{3} ) q^{91} + ( -1 - 2 \beta_{1} - \beta_{3} ) q^{92} + ( -6 - 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{93} + ( 1 + \beta_{3} ) q^{94} + \beta_{1} q^{96} + ( 8 + 5 \beta_{2} ) q^{97} + ( -2 - 4 \beta_{2} + \beta_{3} ) q^{98} + ( -6 - \beta_{1} - 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} + q^{3} + 4q^{4} + q^{6} + 2q^{7} + 4q^{8} + 7q^{9} + O(q^{10})$$ $$4q + 4q^{2} + q^{3} + 4q^{4} + q^{6} + 2q^{7} + 4q^{8} + 7q^{9} - 2q^{11} + q^{12} + 11q^{13} + 2q^{14} + 4q^{16} + 12q^{17} + 7q^{18} - 5q^{19} - 7q^{21} - 2q^{22} - 4q^{23} + q^{24} + 11q^{26} + 10q^{27} + 2q^{28} + 15q^{29} - 12q^{31} + 4q^{32} + 7q^{33} + 12q^{34} + 7q^{36} + 12q^{37} - 5q^{38} - 11q^{39} + 13q^{41} - 7q^{42} + 6q^{43} - 2q^{44} - 4q^{46} + 2q^{47} + q^{48} - 2q^{49} + 13q^{51} + 11q^{52} + 11q^{53} + 10q^{54} + 2q^{56} + 15q^{58} + 8q^{61} - 12q^{62} + 21q^{63} + 4q^{64} + 7q^{66} + 22q^{67} + 12q^{68} - 31q^{69} - 22q^{71} + 7q^{72} + 21q^{73} + 12q^{74} - 5q^{76} - 26q^{77} - 11q^{78} - 10q^{79} - 16q^{81} + 13q^{82} - 24q^{83} - 7q^{84} + 6q^{86} + 25q^{87} - 2q^{88} - 5q^{89} - 12q^{91} - 4q^{92} - 23q^{93} + 2q^{94} + q^{96} + 22q^{97} - 2q^{98} - 21q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 9 x^{2} + 4 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - \nu^{2} - 5 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - \nu - 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{1} + 5$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4 \beta_{2} + 6 \beta_{1} + 5$$

## 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
 −2.33275 −1.34841 1.71472 2.96645
1.00000 −2.33275 1.00000 0 −2.33275 3.77447 1.00000 2.44172 0
1.2 1.00000 −1.34841 1.00000 0 −1.34841 −0.833366 1.00000 −1.18178 0
1.3 1.00000 1.71472 1.00000 0 1.71472 −2.77447 1.00000 −0.0597522 0
1.4 1.00000 2.96645 1.00000 0 2.96645 1.83337 1.00000 5.79981 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$

## 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 1250.2.a.l 4
4.b odd 2 1 10000.2.a.t 4
5.b even 2 1 1250.2.a.f 4
5.c odd 4 2 1250.2.b.e 8
20.d odd 2 1 10000.2.a.x 4
25.d even 5 2 50.2.d.b 8
25.e even 10 2 250.2.d.d 8
25.f odd 20 4 250.2.e.c 16
75.j odd 10 2 450.2.h.e 8
100.j odd 10 2 400.2.u.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.b 8 25.d even 5 2
250.2.d.d 8 25.e even 10 2
250.2.e.c 16 25.f odd 20 4
400.2.u.d 8 100.j odd 10 2
450.2.h.e 8 75.j odd 10 2
1250.2.a.f 4 5.b even 2 1
1250.2.a.l 4 1.a even 1 1 trivial
1250.2.b.e 8 5.c odd 4 2
10000.2.a.t 4 4.b odd 2 1
10000.2.a.x 4 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - T_{3}^{3} - 9 T_{3}^{2} + 4 T_{3} + 16$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1250))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{4}$$
$3$ $$16 + 4 T - 9 T^{2} - T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$16 + 12 T - 11 T^{2} - 2 T^{3} + T^{4}$$
$11$ $$16 - 12 T - 11 T^{2} + 2 T^{3} + T^{4}$$
$13$ $$-199 + 59 T + 26 T^{2} - 11 T^{3} + T^{4}$$
$17$ $$-109 + 2 T + 39 T^{2} - 12 T^{3} + T^{4}$$
$19$ $$80 - 20 T - 35 T^{2} + 5 T^{3} + T^{4}$$
$23$ $$16 + 4 T - 29 T^{2} + 4 T^{3} + T^{4}$$
$29$ $$5 - 105 T + 70 T^{2} - 15 T^{3} + T^{4}$$
$31$ $$-1264 - 432 T - T^{2} + 12 T^{3} + T^{4}$$
$37$ $$71 + 102 T + 19 T^{2} - 12 T^{3} + T^{4}$$
$41$ $$-89 + 23 T + 34 T^{2} - 13 T^{3} + T^{4}$$
$43$ $$176 + 64 T - 39 T^{2} - 6 T^{3} + T^{4}$$
$47$ $$16 + 12 T - 11 T^{2} - 2 T^{3} + T^{4}$$
$53$ $$256 + 464 T - 49 T^{2} - 11 T^{3} + T^{4}$$
$59$ $$-320 - 560 T - 140 T^{2} + T^{4}$$
$61$ $$-1709 + 958 T - 101 T^{2} - 8 T^{3} + T^{4}$$
$67$ $$-944 + 572 T + 69 T^{2} - 22 T^{3} + T^{4}$$
$71$ $$-64 + 168 T + 129 T^{2} + 22 T^{3} + T^{4}$$
$73$ $$-1084 + 214 T + 81 T^{2} - 21 T^{3} + T^{4}$$
$79$ $$-320 - 240 T - 20 T^{2} + 10 T^{3} + T^{4}$$
$83$ $$-3664 - 456 T + 131 T^{2} + 24 T^{3} + T^{4}$$
$89$ $$-3100 - 1500 T - 165 T^{2} + 5 T^{3} + T^{4}$$
$97$ $$( -1 - 11 T + T^{2} )^{2}$$