# Properties

 Label 4600.2.a.bg Level $4600$ Weight $2$ Character orbit 4600.a Self dual yes Analytic conductor $36.731$ Analytic rank $0$ Dimension $5$ 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: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.791953.1 Defining polynomial: $$x^{5} - 2 x^{4} - 7 x^{3} + 7 x^{2} + 9 x + 2$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{3} + ( -1 + \beta_{2} - \beta_{4} ) q^{7} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{3} + ( -1 + \beta_{2} - \beta_{4} ) q^{7} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{11} + ( 1 - 2 \beta_{1} + \beta_{4} ) q^{13} + ( \beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{17} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{19} + ( -2 - 2 \beta_{4} ) q^{21} + q^{23} + ( 1 + \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{27} + ( -2 - 2 \beta_{1} + \beta_{3} - \beta_{4} ) q^{29} + ( 1 + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{31} + ( 2 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} ) q^{33} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{37} + ( 7 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{39} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{41} + ( 3 - 3 \beta_{2} + 3 \beta_{4} ) q^{43} + ( 2 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} ) q^{47} + ( -\beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{49} + ( -2 - \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{51} + ( 4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{53} + ( 4 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{57} + ( 6 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{59} + ( -6 - 2 \beta_{4} ) q^{61} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{63} + ( 2 - \beta_{1} - \beta_{3} + 2 \beta_{4} ) q^{67} + ( 1 - \beta_{1} ) q^{69} + ( \beta_{2} + 2 \beta_{4} ) q^{71} + ( -\beta_{1} - 4 \beta_{3} - 5 \beta_{4} ) q^{73} + ( 5 - 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{77} + ( 9 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{79} + ( -2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} ) q^{81} + ( 3 + 3 \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{83} + ( 3 + 3 \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{87} + ( \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} ) q^{89} + ( -5 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{91} + ( -1 - \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{93} + ( 2 + 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{97} + ( 9 - 4 \beta_{1} + \beta_{2} - 4 \beta_{3} - \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 3 q^{3} - q^{7} + 4 q^{9} + O(q^{10})$$ $$5 q + 3 q^{3} - q^{7} + 4 q^{9} - 4 q^{11} - q^{13} + 5 q^{17} + 4 q^{19} - 6 q^{21} + 5 q^{23} + 6 q^{27} - 11 q^{29} + 4 q^{31} + 13 q^{33} + 6 q^{37} + 31 q^{39} - 8 q^{41} + 3 q^{43} + 2 q^{47} - 2 q^{49} - 5 q^{51} + 18 q^{53} + 27 q^{57} + 23 q^{59} - 26 q^{61} + 5 q^{63} + 3 q^{67} + 3 q^{69} - 2 q^{71} + 4 q^{73} + 15 q^{77} + 43 q^{79} - 3 q^{81} + 30 q^{83} + 27 q^{87} + 15 q^{89} - 19 q^{91} - 15 q^{93} + 8 q^{97} + 37 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 7 x^{3} + 7 x^{2} + 9 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 6 \nu^{2} + 7 \nu + 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 7 \nu^{2} + 8 \nu + 7$$ $$\beta_{4}$$ $$=$$ $$-2 \nu^{4} + 5 \nu^{3} + 12 \nu^{2} - 20 \nu - 11$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} + 2 \beta_{2} + 6 \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{4} - 6 \beta_{3} + 11 \beta_{2} + 11 \beta_{1} + 20$$

## 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
 3.21042 1.72457 −0.336890 −0.514659 −2.08344
0 −2.21042 0 0 0 2.22487 0 1.88594 0
1.2 0 −0.724570 0 0 0 −2.33840 0 −2.47500 0
1.3 0 1.33689 0 0 0 3.16736 0 −1.21273 0
1.4 0 1.51466 0 0 0 −3.49880 0 −0.705809 0
1.5 0 3.08344 0 0 0 −0.555022 0 6.50759 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.bg yes 5
4.b odd 2 1 9200.2.a.cs 5
5.b even 2 1 4600.2.a.bc 5
5.c odd 4 2 4600.2.e.v 10
20.d odd 2 1 9200.2.a.cw 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4600.2.a.bc 5 5.b even 2 1
4600.2.a.bg yes 5 1.a even 1 1 trivial
4600.2.e.v 10 5.c odd 4 2
9200.2.a.cs 5 4.b odd 2 1
9200.2.a.cw 5 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4600))$$:

 $$T_{3}^{5} - 3 T_{3}^{4} - 5 T_{3}^{3} + 16 T_{3}^{2} - T_{3} - 10$$ $$T_{7}^{5} + T_{7}^{4} - 16 T_{7}^{3} - 12 T_{7}^{2} + 56 T_{7} + 32$$ $$T_{11}^{5} + 4 T_{11}^{4} - 17 T_{11}^{3} - 88 T_{11}^{2} - 92 T_{11} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$-10 - T + 16 T^{2} - 5 T^{3} - 3 T^{4} + T^{5}$$
$5$ $$T^{5}$$
$7$ $$32 + 56 T - 12 T^{2} - 16 T^{3} + T^{4} + T^{5}$$
$11$ $$-8 - 92 T - 88 T^{2} - 17 T^{3} + 4 T^{4} + T^{5}$$
$13$ $$454 + 321 T - 48 T^{2} - 37 T^{3} + T^{4} + T^{5}$$
$17$ $$-32 - 72 T + 116 T^{2} - 28 T^{3} - 5 T^{4} + T^{5}$$
$19$ $$-64 - 272 T + 264 T^{2} - 53 T^{3} - 4 T^{4} + T^{5}$$
$23$ $$( -1 + T )^{5}$$
$29$ $$-256 - 551 T - 306 T^{2} - 11 T^{3} + 11 T^{4} + T^{5}$$
$31$ $$400 + 802 T + 159 T^{2} - 75 T^{3} - 4 T^{4} + T^{5}$$
$37$ $$-5344 + 4672 T + 576 T^{2} - 140 T^{3} - 6 T^{4} + T^{5}$$
$41$ $$-2069 - 2097 T - 671 T^{2} - 52 T^{3} + 8 T^{4} + T^{5}$$
$43$ $$-7776 + 4536 T + 324 T^{2} - 144 T^{3} - 3 T^{4} + T^{5}$$
$47$ $$-5912 + 4558 T + 33 T^{2} - 167 T^{3} - 2 T^{4} + T^{5}$$
$53$ $$22144 - 15360 T + 2632 T^{2} - 52 T^{3} - 18 T^{4} + T^{5}$$
$59$ $$-5128 - 558 T + 1087 T^{2} + 53 T^{3} - 23 T^{4} + T^{5}$$
$61$ $$-256 + 880 T + 848 T^{2} + 236 T^{3} + 26 T^{4} + T^{5}$$
$67$ $$-512 + 320 T + 192 T^{2} - 60 T^{3} - 3 T^{4} + T^{5}$$
$71$ $$-8 + 666 T - 133 T^{2} - 67 T^{3} + 2 T^{4} + T^{5}$$
$73$ $$-52501 + 33791 T + 1001 T^{2} - 368 T^{3} - 4 T^{4} + T^{5}$$
$79$ $$45872 - 2080 T - 3292 T^{2} + 634 T^{3} - 43 T^{4} + T^{5}$$
$83$ $$-32960 - 8816 T + 1560 T^{2} + 169 T^{3} - 30 T^{4} + T^{5}$$
$89$ $$7120 - 12608 T + 3188 T^{2} - 154 T^{3} - 15 T^{4} + T^{5}$$
$97$ $$-1024 + 4848 T + 1512 T^{2} - 228 T^{3} - 8 T^{4} + T^{5}$$