Properties

 Label 1148.2.a.a Level $1148$ Weight $2$ Character orbit 1148.a Self dual yes Analytic conductor $9.167$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1148 = 2^{2} \cdot 7 \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1148.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$9.16682615204$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta ) q^{3} + ( -1 - \beta ) q^{5} + q^{7} + ( 1 + 3 \beta ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta ) q^{3} + ( -1 - \beta ) q^{5} + q^{7} + ( 1 + 3 \beta ) q^{9} + q^{11} -5 q^{13} + ( 4 + 3 \beta ) q^{15} + ( -3 + 4 \beta ) q^{17} + ( -3 + 3 \beta ) q^{19} + ( -1 - \beta ) q^{21} + ( -1 - \beta ) q^{23} + ( -1 + 3 \beta ) q^{25} + ( -7 - 4 \beta ) q^{27} + ( 8 + \beta ) q^{29} + ( 7 - 3 \beta ) q^{31} + ( -1 - \beta ) q^{33} + ( -1 - \beta ) q^{35} + ( 4 - 2 \beta ) q^{37} + ( 5 + 5 \beta ) q^{39} - q^{41} + ( -7 - 2 \beta ) q^{43} + ( -10 - 7 \beta ) q^{45} + ( -10 + 2 \beta ) q^{47} + q^{49} + ( -9 - 5 \beta ) q^{51} + ( -5 + \beta ) q^{53} + ( -1 - \beta ) q^{55} + ( -6 - 3 \beta ) q^{57} + ( -1 + 3 \beta ) q^{59} + ( -1 - 4 \beta ) q^{61} + ( 1 + 3 \beta ) q^{63} + ( 5 + 5 \beta ) q^{65} + ( -5 + 3 \beta ) q^{67} + ( 4 + 3 \beta ) q^{69} + ( -3 - 4 \beta ) q^{71} + 7 q^{73} + ( -8 - 5 \beta ) q^{75} + q^{77} + ( -12 - 2 \beta ) q^{79} + ( 16 + 6 \beta ) q^{81} + ( 3 - 6 \beta ) q^{83} + ( -9 - 5 \beta ) q^{85} + ( -11 - 10 \beta ) q^{87} + ( 3 - 5 \beta ) q^{89} -5 q^{91} + ( 2 - \beta ) q^{93} + ( -6 - 3 \beta ) q^{95} + ( -7 + 7 \beta ) q^{97} + ( 1 + 3 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{3} - 3q^{5} + 2q^{7} + 5q^{9} + O(q^{10})$$ $$2q - 3q^{3} - 3q^{5} + 2q^{7} + 5q^{9} + 2q^{11} - 10q^{13} + 11q^{15} - 2q^{17} - 3q^{19} - 3q^{21} - 3q^{23} + q^{25} - 18q^{27} + 17q^{29} + 11q^{31} - 3q^{33} - 3q^{35} + 6q^{37} + 15q^{39} - 2q^{41} - 16q^{43} - 27q^{45} - 18q^{47} + 2q^{49} - 23q^{51} - 9q^{53} - 3q^{55} - 15q^{57} + q^{59} - 6q^{61} + 5q^{63} + 15q^{65} - 7q^{67} + 11q^{69} - 10q^{71} + 14q^{73} - 21q^{75} + 2q^{77} - 26q^{79} + 38q^{81} - 23q^{85} - 32q^{87} + q^{89} - 10q^{91} + 3q^{93} - 15q^{95} - 7q^{97} + 5q^{99} + O(q^{100})$$

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
 2.30278 −1.30278
0 −3.30278 0 −3.30278 0 1.00000 0 7.90833 0
1.2 0 0.302776 0 0.302776 0 1.00000 0 −2.90833 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
$$2$$ $$-1$$
$$7$$ $$-1$$
$$41$$ $$1$$

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 1148.2.a.a 2
4.b odd 2 1 4592.2.a.o 2
7.b odd 2 1 8036.2.a.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.a.a 2 1.a even 1 1 trivial
4592.2.a.o 2 4.b odd 2 1
8036.2.a.g 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 3 T_{3} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1148))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-1 + 3 T + T^{2}$$
$5$ $$-1 + 3 T + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$( 5 + T )^{2}$$
$17$ $$-51 + 2 T + T^{2}$$
$19$ $$-27 + 3 T + T^{2}$$
$23$ $$-1 + 3 T + T^{2}$$
$29$ $$69 - 17 T + T^{2}$$
$31$ $$1 - 11 T + T^{2}$$
$37$ $$-4 - 6 T + T^{2}$$
$41$ $$( 1 + T )^{2}$$
$43$ $$51 + 16 T + T^{2}$$
$47$ $$68 + 18 T + T^{2}$$
$53$ $$17 + 9 T + T^{2}$$
$59$ $$-29 - T + T^{2}$$
$61$ $$-43 + 6 T + T^{2}$$
$67$ $$-17 + 7 T + T^{2}$$
$71$ $$-27 + 10 T + T^{2}$$
$73$ $$( -7 + T )^{2}$$
$79$ $$156 + 26 T + T^{2}$$
$83$ $$-117 + T^{2}$$
$89$ $$-81 - T + T^{2}$$
$97$ $$-147 + 7 T + T^{2}$$