# Properties

 Label 5200.2.a.cb Level $5200$ Weight $2$ Character orbit 5200.a Self dual yes Analytic conductor $41.522$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5200 = 2^{4} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$41.5222090511$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 1) q^{3} + (\beta_{2} + \beta_1 - 1) q^{7} + (\beta_{2} + 3 \beta_1) q^{9}+O(q^{10})$$ q + (-b1 - 1) * q^3 + (b2 + b1 - 1) * q^7 + (b2 + 3*b1) * q^9 $$q + ( - \beta_1 - 1) q^{3} + (\beta_{2} + \beta_1 - 1) q^{7} + (\beta_{2} + 3 \beta_1) q^{9} + (\beta_{2} + 2) q^{11} - q^{13} + ( - 2 \beta_{2} + 2) q^{17} + \beta_{2} q^{19} + ( - 2 \beta_{2} - 2 \beta_1) q^{21} + (\beta_1 - 5) q^{23} + ( - 4 \beta_{2} - 4 \beta_1 - 2) q^{27} + ( - 3 \beta_{2} - 3 \beta_1 + 3) q^{29} + ( - \beta_{2} - 2 \beta_1 + 4) q^{31} + ( - \beta_{2} - 3 \beta_1 - 1) q^{33} + ( - \beta_{2} - 3 \beta_1 + 1) q^{37} + (\beta_1 + 1) q^{39} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{41} + (2 \beta_{2} - 3 \beta_1 - 1) q^{43} + ( - \beta_{2} - \beta_1 - 3) q^{47} + ( - 2 \beta_{2} - 3) q^{49} + (2 \beta_{2} - 4) q^{51} + (4 \beta_{2} + 2 \beta_1 + 2) q^{53} + ( - \beta_{2} - \beta_1 + 1) q^{57} + ( - 3 \beta_{2} + 2 \beta_1 + 2) q^{59} + (\beta_{2} + 3 \beta_1 + 1) q^{61} + (\beta_{2} + 3 \beta_1 + 5) q^{63} + ( - 5 \beta_{2} - \beta_1 - 3) q^{67} + ( - \beta_{2} + 3 \beta_1 + 3) q^{69} + ( - \beta_{2} + 6 \beta_1 + 2) q^{71} + (\beta_{2} + 3 \beta_1 - 9) q^{73} + 2 \beta_1 q^{77} + ( - 4 \beta_{2} - 2 \beta_1 + 6) q^{79} + (5 \beta_{2} + 5 \beta_1 + 6) q^{81} + (\beta_{2} - \beta_1 - 7) q^{83} + (6 \beta_{2} + 6 \beta_1) q^{87} + (4 \beta_{2} + 4 \beta_1 + 2) q^{89} + ( - \beta_{2} - \beta_1 + 1) q^{91} + (3 \beta_{2} + \beta_1 - 1) q^{93} + (6 \beta_{2} + 4 \beta_1 - 6) q^{97} + (\beta_{2} + 8 \beta_1) q^{99}+O(q^{100})$$ q + (-b1 - 1) * q^3 + (b2 + b1 - 1) * q^7 + (b2 + 3*b1) * q^9 + (b2 + 2) * q^11 - q^13 + (-2*b2 + 2) * q^17 + b2 * q^19 + (-2*b2 - 2*b1) * q^21 + (b1 - 5) * q^23 + (-4*b2 - 4*b1 - 2) * q^27 + (-3*b2 - 3*b1 + 3) * q^29 + (-b2 - 2*b1 + 4) * q^31 + (-b2 - 3*b1 - 1) * q^33 + (-b2 - 3*b1 + 1) * q^37 + (b1 + 1) * q^39 + (-2*b2 + 2*b1 - 2) * q^41 + (2*b2 - 3*b1 - 1) * q^43 + (-b2 - b1 - 3) * q^47 + (-2*b2 - 3) * q^49 + (2*b2 - 4) * q^51 + (4*b2 + 2*b1 + 2) * q^53 + (-b2 - b1 + 1) * q^57 + (-3*b2 + 2*b1 + 2) * q^59 + (b2 + 3*b1 + 1) * q^61 + (b2 + 3*b1 + 5) * q^63 + (-5*b2 - b1 - 3) * q^67 + (-b2 + 3*b1 + 3) * q^69 + (-b2 + 6*b1 + 2) * q^71 + (b2 + 3*b1 - 9) * q^73 + 2*b1 * q^77 + (-4*b2 - 2*b1 + 6) * q^79 + (5*b2 + 5*b1 + 6) * q^81 + (b2 - b1 - 7) * q^83 + (6*b2 + 6*b1) * q^87 + (4*b2 + 4*b1 + 2) * q^89 + (-b2 - b1 + 1) * q^91 + (3*b2 + b1 - 1) * q^93 + (6*b2 + 4*b1 - 6) * q^97 + (b2 + 8*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 4 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 4 * q^3 - 2 * q^7 + 3 * q^9 $$3 q - 4 q^{3} - 2 q^{7} + 3 q^{9} + 6 q^{11} - 3 q^{13} + 6 q^{17} - 2 q^{21} - 14 q^{23} - 10 q^{27} + 6 q^{29} + 10 q^{31} - 6 q^{33} + 4 q^{39} - 4 q^{41} - 6 q^{43} - 10 q^{47} - 9 q^{49} - 12 q^{51} + 8 q^{53} + 2 q^{57} + 8 q^{59} + 6 q^{61} + 18 q^{63} - 10 q^{67} + 12 q^{69} + 12 q^{71} - 24 q^{73} + 2 q^{77} + 16 q^{79} + 23 q^{81} - 22 q^{83} + 6 q^{87} + 10 q^{89} + 2 q^{91} - 2 q^{93} - 14 q^{97} + 8 q^{99}+O(q^{100})$$ 3 * q - 4 * q^3 - 2 * q^7 + 3 * q^9 + 6 * q^11 - 3 * q^13 + 6 * q^17 - 2 * q^21 - 14 * q^23 - 10 * q^27 + 6 * q^29 + 10 * q^31 - 6 * q^33 + 4 * q^39 - 4 * q^41 - 6 * q^43 - 10 * q^47 - 9 * q^49 - 12 * q^51 + 8 * q^53 + 2 * q^57 + 8 * q^59 + 6 * q^61 + 18 * q^63 - 10 * q^67 + 12 * q^69 + 12 * q^71 - 24 * q^73 + 2 * q^77 + 16 * q^79 + 23 * q^81 - 22 * q^83 + 6 * q^87 + 10 * q^89 + 2 * q^91 - 2 * q^93 - 14 * q^97 + 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2

## 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
 2.17009 0.311108 −1.48119
0 −3.17009 0 0 0 1.70928 0 7.04945 0
1.2 0 −1.31111 0 0 0 −2.90321 0 −1.28100 0
1.3 0 0.481194 0 0 0 −0.806063 0 −2.76845 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$1$$

## 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 5200.2.a.cb 3
4.b odd 2 1 325.2.a.k 3
5.b even 2 1 5200.2.a.cj 3
5.c odd 4 2 1040.2.d.c 6
12.b even 2 1 2925.2.a.bf 3
20.d odd 2 1 325.2.a.j 3
20.e even 4 2 65.2.b.a 6
52.b odd 2 1 4225.2.a.ba 3
60.h even 2 1 2925.2.a.bj 3
60.l odd 4 2 585.2.c.b 6
260.g odd 2 1 4225.2.a.bh 3
260.l odd 4 2 845.2.d.a 6
260.p even 4 2 845.2.b.c 6
260.s odd 4 2 845.2.d.b 6
260.be odd 12 4 845.2.l.d 12
260.bg even 12 4 845.2.n.g 12
260.bj even 12 4 845.2.n.f 12
260.bl odd 12 4 845.2.l.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.b.a 6 20.e even 4 2
325.2.a.j 3 20.d odd 2 1
325.2.a.k 3 4.b odd 2 1
585.2.c.b 6 60.l odd 4 2
845.2.b.c 6 260.p even 4 2
845.2.d.a 6 260.l odd 4 2
845.2.d.b 6 260.s odd 4 2
845.2.l.d 12 260.be odd 12 4
845.2.l.e 12 260.bl odd 12 4
845.2.n.f 12 260.bj even 12 4
845.2.n.g 12 260.bg even 12 4
1040.2.d.c 6 5.c odd 4 2
2925.2.a.bf 3 12.b even 2 1
2925.2.a.bj 3 60.h even 2 1
4225.2.a.ba 3 52.b odd 2 1
4225.2.a.bh 3 260.g odd 2 1
5200.2.a.cb 3 1.a even 1 1 trivial
5200.2.a.cj 3 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5200))$$:

 $$T_{3}^{3} + 4T_{3}^{2} + 2T_{3} - 2$$ T3^3 + 4*T3^2 + 2*T3 - 2 $$T_{7}^{3} + 2T_{7}^{2} - 4T_{7} - 4$$ T7^3 + 2*T7^2 - 4*T7 - 4 $$T_{11}^{3} - 6T_{11}^{2} + 8T_{11} + 2$$ T11^3 - 6*T11^2 + 8*T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + 4 T^{2} + 2 T - 2$$
$5$ $$T^{3}$$
$7$ $$T^{3} + 2 T^{2} - 4 T - 4$$
$11$ $$T^{3} - 6 T^{2} + 8 T + 2$$
$13$ $$(T + 1)^{3}$$
$17$ $$T^{3} - 6 T^{2} - 4 T + 8$$
$19$ $$T^{3} - 4T + 2$$
$23$ $$T^{3} + 14 T^{2} + 62 T + 86$$
$29$ $$T^{3} - 6 T^{2} - 36 T + 108$$
$31$ $$T^{3} - 10 T^{2} + 20 T + 26$$
$37$ $$T^{3} - 28T + 52$$
$41$ $$T^{3} + 4 T^{2} - 32 T + 32$$
$43$ $$T^{3} + 6 T^{2} - 46 T - 278$$
$47$ $$T^{3} + 10 T^{2} + 28 T + 20$$
$53$ $$T^{3} - 8 T^{2} - 40 T + 304$$
$59$ $$T^{3} - 8 T^{2} - 40 T + 262$$
$61$ $$T^{3} - 6 T^{2} - 16 T - 4$$
$67$ $$T^{3} + 10 T^{2} - 60 T - 604$$
$71$ $$T^{3} - 12 T^{2} - 88 T + 754$$
$73$ $$T^{3} + 24 T^{2} + 164 T + 236$$
$79$ $$T^{3} - 16 T^{2} + 24 T + 16$$
$83$ $$T^{3} + 22 T^{2} + 152 T + 316$$
$89$ $$T^{3} - 10 T^{2} - 52 T + 200$$
$97$ $$T^{3} + 14 T^{2} - 84 T - 200$$