Properties

 Label 608.2.a.h Level $608$ Weight $2$ Character orbit 608.a Self dual yes Analytic conductor $4.855$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$608 = 2^{5} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 608.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$4.85490444289$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + ( -1 + \beta ) q^{5} + 3 q^{7} + ( 1 + \beta ) q^{9} +O(q^{10})$$ $$q + \beta q^{3} + ( -1 + \beta ) q^{5} + 3 q^{7} + ( 1 + \beta ) q^{9} + ( -1 - \beta ) q^{11} + \beta q^{13} + 4 q^{15} + ( -3 - 2 \beta ) q^{17} + q^{19} + 3 \beta q^{21} + ( 4 - \beta ) q^{23} -\beta q^{25} + ( 4 - \beta ) q^{27} -3 \beta q^{29} + ( 6 - 2 \beta ) q^{31} + ( -4 - 2 \beta ) q^{33} + ( -3 + 3 \beta ) q^{35} + ( 4 - 2 \beta ) q^{37} + ( 4 + \beta ) q^{39} + 4 q^{41} + ( 7 + \beta ) q^{43} + ( 3 + \beta ) q^{45} + ( 1 + 3 \beta ) q^{47} + 2 q^{49} + ( -8 - 5 \beta ) q^{51} + ( -6 - \beta ) q^{53} + ( -3 - \beta ) q^{55} + \beta q^{57} + ( -6 - \beta ) q^{59} + ( 7 - 5 \beta ) q^{61} + ( 3 + 3 \beta ) q^{63} + 4 q^{65} + ( -2 - \beta ) q^{67} + ( -4 + 3 \beta ) q^{69} + ( 2 + 4 \beta ) q^{71} + ( -3 + 4 \beta ) q^{73} + ( -4 - \beta ) q^{75} + ( -3 - 3 \beta ) q^{77} + 10 q^{79} -7 q^{81} + ( -8 + 6 \beta ) q^{83} + ( -5 - 3 \beta ) q^{85} + ( -12 - 3 \beta ) q^{87} + ( -6 + 6 \beta ) q^{89} + 3 \beta q^{91} + ( -8 + 4 \beta ) q^{93} + ( -1 + \beta ) q^{95} + ( 4 - 2 \beta ) q^{97} + ( -5 - 3 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - q^{5} + 6 q^{7} + 3 q^{9} + O(q^{10})$$ $$2 q + q^{3} - q^{5} + 6 q^{7} + 3 q^{9} - 3 q^{11} + q^{13} + 8 q^{15} - 8 q^{17} + 2 q^{19} + 3 q^{21} + 7 q^{23} - q^{25} + 7 q^{27} - 3 q^{29} + 10 q^{31} - 10 q^{33} - 3 q^{35} + 6 q^{37} + 9 q^{39} + 8 q^{41} + 15 q^{43} + 7 q^{45} + 5 q^{47} + 4 q^{49} - 21 q^{51} - 13 q^{53} - 7 q^{55} + q^{57} - 13 q^{59} + 9 q^{61} + 9 q^{63} + 8 q^{65} - 5 q^{67} - 5 q^{69} + 8 q^{71} - 2 q^{73} - 9 q^{75} - 9 q^{77} + 20 q^{79} - 14 q^{81} - 10 q^{83} - 13 q^{85} - 27 q^{87} - 6 q^{89} + 3 q^{91} - 12 q^{93} - q^{95} + 6 q^{97} - 13 q^{99} + O(q^{100})$$

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
 −1.56155 2.56155
0 −1.56155 0 −2.56155 0 3.00000 0 −0.561553 0
1.2 0 2.56155 0 1.56155 0 3.00000 0 3.56155 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
$$2$$ $$1$$
$$19$$ $$-1$$

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 608.2.a.h yes 2
3.b odd 2 1 5472.2.a.bf 2
4.b odd 2 1 608.2.a.g 2
8.b even 2 1 1216.2.a.s 2
8.d odd 2 1 1216.2.a.t 2
12.b even 2 1 5472.2.a.bc 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.a.g 2 4.b odd 2 1
608.2.a.h yes 2 1.a even 1 1 trivial
1216.2.a.s 2 8.b even 2 1
1216.2.a.t 2 8.d odd 2 1
5472.2.a.bc 2 12.b even 2 1
5472.2.a.bf 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(608))$$:

 $$T_{3}^{2} - T_{3} - 4$$ $$T_{5}^{2} + T_{5} - 4$$ $$T_{7} - 3$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-4 - T + T^{2}$$
$5$ $$-4 + T + T^{2}$$
$7$ $$( -3 + T )^{2}$$
$11$ $$-2 + 3 T + T^{2}$$
$13$ $$-4 - T + T^{2}$$
$17$ $$-1 + 8 T + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$8 - 7 T + T^{2}$$
$29$ $$-36 + 3 T + T^{2}$$
$31$ $$8 - 10 T + T^{2}$$
$37$ $$-8 - 6 T + T^{2}$$
$41$ $$( -4 + T )^{2}$$
$43$ $$52 - 15 T + T^{2}$$
$47$ $$-32 - 5 T + T^{2}$$
$53$ $$38 + 13 T + T^{2}$$
$59$ $$38 + 13 T + T^{2}$$
$61$ $$-86 - 9 T + T^{2}$$
$67$ $$2 + 5 T + T^{2}$$
$71$ $$-52 - 8 T + T^{2}$$
$73$ $$-67 + 2 T + T^{2}$$
$79$ $$( -10 + T )^{2}$$
$83$ $$-128 + 10 T + T^{2}$$
$89$ $$-144 + 6 T + T^{2}$$
$97$ $$-8 - 6 T + T^{2}$$