# Properties

 Label 153.2.a.e Level $153$ Weight $2$ Character orbit 153.a Self dual yes Analytic conductor $1.222$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$153 = 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 153.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.22171115093$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 51) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( 2 + \beta ) q^{4} + ( -1 - \beta ) q^{5} + ( 4 + \beta ) q^{8} +O(q^{10})$$ $$q + \beta q^{2} + ( 2 + \beta ) q^{4} + ( -1 - \beta ) q^{5} + ( 4 + \beta ) q^{8} + ( -4 - 2 \beta ) q^{10} + ( 1 - \beta ) q^{11} + ( 3 - \beta ) q^{13} + 3 \beta q^{16} - q^{17} + ( 3 - 3 \beta ) q^{19} + ( -6 - 4 \beta ) q^{20} -4 q^{22} + ( 5 - \beta ) q^{23} + 3 \beta q^{25} + ( -4 + 2 \beta ) q^{26} + ( -2 + 4 \beta ) q^{29} + ( -2 + 2 \beta ) q^{31} + ( 4 + \beta ) q^{32} -\beta q^{34} -2 \beta q^{37} -12 q^{38} + ( -8 - 6 \beta ) q^{40} + ( 1 + \beta ) q^{41} + ( -3 + 3 \beta ) q^{43} + ( -2 - 2 \beta ) q^{44} + ( -4 + 4 \beta ) q^{46} + ( 6 + 2 \beta ) q^{47} -7 q^{49} + ( 12 + 3 \beta ) q^{50} + 2 q^{52} + ( -2 - 4 \beta ) q^{53} + ( 3 + \beta ) q^{55} + ( 16 + 2 \beta ) q^{58} + ( -2 - 2 \beta ) q^{59} + ( 4 + 2 \beta ) q^{61} + 8 q^{62} + ( 4 - \beta ) q^{64} + ( 1 - \beta ) q^{65} + 4 q^{67} + ( -2 - \beta ) q^{68} + ( -4 + 4 \beta ) q^{71} + ( -2 - 4 \beta ) q^{73} + ( -8 - 2 \beta ) q^{74} + ( -6 - 6 \beta ) q^{76} + ( 6 - 6 \beta ) q^{79} + ( -12 - 6 \beta ) q^{80} + ( 4 + 2 \beta ) q^{82} + ( 6 - 2 \beta ) q^{83} + ( 1 + \beta ) q^{85} + 12 q^{86} -4 \beta q^{88} + ( -4 + 2 \beta ) q^{89} + ( 6 + 2 \beta ) q^{92} + ( 8 + 8 \beta ) q^{94} + ( 9 + 3 \beta ) q^{95} + ( -8 + 2 \beta ) q^{97} -7 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + 5q^{4} - 3q^{5} + 9q^{8} + O(q^{10})$$ $$2q + q^{2} + 5q^{4} - 3q^{5} + 9q^{8} - 10q^{10} + q^{11} + 5q^{13} + 3q^{16} - 2q^{17} + 3q^{19} - 16q^{20} - 8q^{22} + 9q^{23} + 3q^{25} - 6q^{26} - 2q^{31} + 9q^{32} - q^{34} - 2q^{37} - 24q^{38} - 22q^{40} + 3q^{41} - 3q^{43} - 6q^{44} - 4q^{46} + 14q^{47} - 14q^{49} + 27q^{50} + 4q^{52} - 8q^{53} + 7q^{55} + 34q^{58} - 6q^{59} + 10q^{61} + 16q^{62} + 7q^{64} + q^{65} + 8q^{67} - 5q^{68} - 4q^{71} - 8q^{73} - 18q^{74} - 18q^{76} + 6q^{79} - 30q^{80} + 10q^{82} + 10q^{83} + 3q^{85} + 24q^{86} - 4q^{88} - 6q^{89} + 14q^{92} + 24q^{94} + 21q^{95} - 14q^{97} - 7q^{98} + O(q^{100})$$

## 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
 −1.56155 2.56155
−1.56155 0 0.438447 0.561553 0 0 2.43845 0 −0.876894
1.2 2.56155 0 4.56155 −3.56155 0 0 6.56155 0 −9.12311
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$17$$ $$1$$

## 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 153.2.a.e 2
3.b odd 2 1 51.2.a.b 2
4.b odd 2 1 2448.2.a.v 2
5.b even 2 1 3825.2.a.s 2
7.b odd 2 1 7497.2.a.v 2
8.b even 2 1 9792.2.a.cy 2
8.d odd 2 1 9792.2.a.cz 2
12.b even 2 1 816.2.a.m 2
15.d odd 2 1 1275.2.a.n 2
15.e even 4 2 1275.2.b.d 4
17.b even 2 1 2601.2.a.t 2
21.c even 2 1 2499.2.a.o 2
24.f even 2 1 3264.2.a.bg 2
24.h odd 2 1 3264.2.a.bl 2
33.d even 2 1 6171.2.a.p 2
39.d odd 2 1 8619.2.a.q 2
51.c odd 2 1 867.2.a.f 2
51.f odd 4 2 867.2.d.c 4
51.g odd 8 4 867.2.e.f 8
51.i even 16 8 867.2.h.j 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.a.b 2 3.b odd 2 1
153.2.a.e 2 1.a even 1 1 trivial
816.2.a.m 2 12.b even 2 1
867.2.a.f 2 51.c odd 2 1
867.2.d.c 4 51.f odd 4 2
867.2.e.f 8 51.g odd 8 4
867.2.h.j 16 51.i even 16 8
1275.2.a.n 2 15.d odd 2 1
1275.2.b.d 4 15.e even 4 2
2448.2.a.v 2 4.b odd 2 1
2499.2.a.o 2 21.c even 2 1
2601.2.a.t 2 17.b even 2 1
3264.2.a.bg 2 24.f even 2 1
3264.2.a.bl 2 24.h odd 2 1
3825.2.a.s 2 5.b even 2 1
6171.2.a.p 2 33.d even 2 1
7497.2.a.v 2 7.b odd 2 1
8619.2.a.q 2 39.d odd 2 1
9792.2.a.cy 2 8.b even 2 1
9792.2.a.cz 2 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} - 4$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(153))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$-2 + 3 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-4 - T + T^{2}$$
$13$ $$2 - 5 T + T^{2}$$
$17$ $$( 1 + T )^{2}$$
$19$ $$-36 - 3 T + T^{2}$$
$23$ $$16 - 9 T + T^{2}$$
$29$ $$-68 + T^{2}$$
$31$ $$-16 + 2 T + T^{2}$$
$37$ $$-16 + 2 T + T^{2}$$
$41$ $$-2 - 3 T + T^{2}$$
$43$ $$-36 + 3 T + T^{2}$$
$47$ $$32 - 14 T + T^{2}$$
$53$ $$-52 + 8 T + T^{2}$$
$59$ $$-8 + 6 T + T^{2}$$
$61$ $$8 - 10 T + T^{2}$$
$67$ $$( -4 + T )^{2}$$
$71$ $$-64 + 4 T + T^{2}$$
$73$ $$-52 + 8 T + T^{2}$$
$79$ $$-144 - 6 T + T^{2}$$
$83$ $$8 - 10 T + T^{2}$$
$89$ $$-8 + 6 T + T^{2}$$
$97$ $$32 + 14 T + T^{2}$$