# Properties

 Label 4600.2.a.x Level $4600$ Weight $2$ Character orbit 4600.a Self dual yes Analytic conductor $36.731$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4600 = 2^{3} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4600.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.7311849298$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.2597.1 Defining polynomial: $$x^{3} - x^{2} - 9 x + 8$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( -1 + \beta_{1} ) q^{7} + ( 3 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( -1 + \beta_{1} ) q^{7} + ( 3 + \beta_{2} ) q^{9} + ( 2 + \beta_{1} ) q^{11} + \beta_{2} q^{13} + ( -3 - \beta_{2} ) q^{17} + ( -4 - \beta_{2} ) q^{19} + ( -6 + \beta_{1} - \beta_{2} ) q^{21} - q^{23} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{27} + ( 5 - \beta_{1} - \beta_{2} ) q^{29} + ( -3 + \beta_{1} ) q^{31} + ( -6 - 2 \beta_{1} - \beta_{2} ) q^{33} + ( -3 + 3 \beta_{1} + \beta_{2} ) q^{37} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{39} + ( 3 - \beta_{2} ) q^{41} -8 q^{43} -2 \beta_{2} q^{47} + ( -2 \beta_{1} + \beta_{2} ) q^{49} + ( -2 + 6 \beta_{1} + \beta_{2} ) q^{51} + ( 1 - 3 \beta_{1} - \beta_{2} ) q^{53} + ( -2 + 7 \beta_{1} + \beta_{2} ) q^{57} + ( -5 - \beta_{1} - \beta_{2} ) q^{59} + ( 4 - \beta_{1} + 2 \beta_{2} ) q^{61} + ( -5 + 6 \beta_{1} ) q^{63} + ( -3 + 3 \beta_{1} + \beta_{2} ) q^{67} + \beta_{1} q^{69} + ( -7 - 2 \beta_{1} + \beta_{2} ) q^{71} + ( -6 + 4 \beta_{1} + 2 \beta_{2} ) q^{73} + ( 4 + \beta_{1} + \beta_{2} ) q^{77} -4 \beta_{1} q^{79} + ( 7 + \beta_{1} + \beta_{2} ) q^{81} + ( 5 + \beta_{1} - \beta_{2} ) q^{83} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{87} + ( -4 + 4 \beta_{1} ) q^{89} + ( -2 + 3 \beta_{1} ) q^{91} + ( -6 + 3 \beta_{1} - \beta_{2} ) q^{93} + ( 2 - 3 \beta_{1} ) q^{97} + ( 4 + 6 \beta_{1} + 3 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - q^{3} - 2q^{7} + 10q^{9} + O(q^{10})$$ $$3q - q^{3} - 2q^{7} + 10q^{9} + 7q^{11} + q^{13} - 10q^{17} - 13q^{19} - 18q^{21} - 3q^{23} + 2q^{27} + 13q^{29} - 8q^{31} - 21q^{33} - 5q^{37} + 2q^{39} + 8q^{41} - 24q^{43} - 2q^{47} - q^{49} + q^{51} - q^{53} + 2q^{57} - 17q^{59} + 13q^{61} - 9q^{63} - 5q^{67} + q^{69} - 22q^{71} - 12q^{73} + 14q^{77} - 4q^{79} + 23q^{81} + 15q^{83} + 12q^{87} - 8q^{89} - 3q^{91} - 16q^{93} + 3q^{97} + 21q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 9 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 6$$

## 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
 3.07912 0.878468 −2.95759
0 −3.07912 0 0 0 2.07912 0 6.48097 0
1.2 0 −0.878468 0 0 0 −0.121532 0 −2.22829 0
1.3 0 2.95759 0 0 0 −3.95759 0 5.74732 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
$$2$$ $$1$$
$$5$$ $$1$$
$$23$$ $$1$$

## 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 4600.2.a.x 3
4.b odd 2 1 9200.2.a.ce 3
5.b even 2 1 920.2.a.h 3
5.c odd 4 2 4600.2.e.p 6
15.d odd 2 1 8280.2.a.bj 3
20.d odd 2 1 1840.2.a.s 3
40.e odd 2 1 7360.2.a.cc 3
40.f even 2 1 7360.2.a.by 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.h 3 5.b even 2 1
1840.2.a.s 3 20.d odd 2 1
4600.2.a.x 3 1.a even 1 1 trivial
4600.2.e.p 6 5.c odd 4 2
7360.2.a.by 3 40.f even 2 1
7360.2.a.cc 3 40.e odd 2 1
8280.2.a.bj 3 15.d odd 2 1
9200.2.a.ce 3 4.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4600))$$:

 $$T_{3}^{3} + T_{3}^{2} - 9 T_{3} - 8$$ $$T_{7}^{3} + 2 T_{7}^{2} - 8 T_{7} - 1$$ $$T_{11}^{3} - 7 T_{11}^{2} + 7 T_{11} + 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-8 - 9 T + T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$-1 - 8 T + 2 T^{2} + T^{3}$$
$11$ $$14 + 7 T - 7 T^{2} + T^{3}$$
$13$ $$50 - 23 T - T^{2} + T^{3}$$
$17$ $$-83 + 10 T + 10 T^{2} + T^{3}$$
$19$ $$-62 + 33 T + 13 T^{2} + T^{3}$$
$23$ $$( 1 + T )^{3}$$
$29$ $$76 + 26 T - 13 T^{2} + T^{3}$$
$31$ $$-1 + 12 T + 8 T^{2} + T^{3}$$
$37$ $$-496 - 92 T + 5 T^{2} + T^{3}$$
$41$ $$1 - 2 T - 8 T^{2} + T^{3}$$
$43$ $$( 8 + T )^{3}$$
$47$ $$-400 - 92 T + 2 T^{2} + T^{3}$$
$53$ $$300 - 100 T + T^{2} + T^{3}$$
$59$ $$36 + 66 T + 17 T^{2} + T^{3}$$
$61$ $$720 - 51 T - 13 T^{2} + T^{3}$$
$67$ $$-496 - 92 T + 5 T^{2} + T^{3}$$
$71$ $$-225 + 96 T + 22 T^{2} + T^{3}$$
$73$ $$-2120 - 176 T + 12 T^{2} + T^{3}$$
$79$ $$-512 - 144 T + 4 T^{2} + T^{3}$$
$83$ $$36 + 40 T - 15 T^{2} + T^{3}$$
$89$ $$-64 - 128 T + 8 T^{2} + T^{3}$$
$97$ $$-50 - 81 T - 3 T^{2} + T^{3}$$