# Properties

 Label 7800.2.a.ca Level $7800$ Weight $2$ Character orbit 7800.a Self dual yes Analytic conductor $62.283$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$62.2833135766$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.504568.1 Defining polynomial: $$x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 2$$ x^5 - x^4 - 6*x^3 + 3*x^2 + 7*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1560) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + \beta_1 q^{7} + q^{9}+O(q^{10})$$ q + q^3 + b1 * q^7 + q^9 $$q + q^{3} + \beta_1 q^{7} + q^{9} + ( - \beta_{3} + 1) q^{11} + q^{13} + ( - \beta_{4} + \beta_{2} + 1) q^{17} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{19} + \beta_1 q^{21} + (\beta_{4} - \beta_{2} + 1) q^{23} + q^{27} + ( - \beta_{3} + \beta_{2}) q^{29} - 2 \beta_{4} q^{31} + ( - \beta_{3} + 1) q^{33} + (2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{37} + q^{39} + ( - \beta_{4} + 3 \beta_{3} - \beta_{2} + \beta_1) q^{41} + ( - \beta_{4} + \beta_{2} - 3 \beta_1 + 1) q^{43} + ( - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2) q^{47} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{49} + ( - \beta_{4} + \beta_{2} + 1) q^{51} + (2 \beta_{4} + \beta_{3} - 3 \beta_{2} + \beta_1 + 2) q^{53} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{57} + (2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 3) q^{59} + (\beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 5) q^{61} + \beta_1 q^{63} + ( - 2 \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 2) q^{67} + (\beta_{4} - \beta_{2} + 1) q^{69} + ( - \beta_{4} + 3 \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{71} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 4) q^{73} + (2 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 + 2) q^{77} + ( - \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{79} + q^{81} + (\beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{83} + ( - \beta_{3} + \beta_{2}) q^{87} + (\beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 2) q^{89} + \beta_1 q^{91} - 2 \beta_{4} q^{93} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{97} + ( - \beta_{3} + 1) q^{99}+O(q^{100})$$ q + q^3 + b1 * q^7 + q^9 + (-b3 + 1) * q^11 + q^13 + (-b4 + b2 + 1) * q^17 + (b4 + b3 + b1 + 1) * q^19 + b1 * q^21 + (b4 - b2 + 1) * q^23 + q^27 + (-b3 + b2) * q^29 - 2*b4 * q^31 + (-b3 + 1) * q^33 + (2*b4 - b3 - b2 + b1 + 2) * q^37 + q^39 + (-b4 + 3*b3 - b2 + b1) * q^41 + (-b4 + b2 - 3*b1 + 1) * q^43 + (-b4 - 2*b3 + 2*b2 + 2) * q^47 + (-b3 - b2 - b1 + 1) * q^49 + (-b4 + b2 + 1) * q^51 + (2*b4 + b3 - 3*b2 + b1 + 2) * q^53 + (b4 + b3 + b1 + 1) * q^57 + (2*b4 + 2*b3 + b2 + 3*b1 + 3) * q^59 + (b4 - 3*b3 + 2*b2 + 2*b1 - 5) * q^61 + b1 * q^63 + (-2*b4 + 2*b2 - 2*b1 + 2) * q^67 + (b4 - b2 + 1) * q^69 + (-b4 + 3*b3 - 3*b2 - b1 + 4) * q^71 + (-2*b4 + b3 - b2 - 2*b1 + 4) * q^73 + (2*b4 + b3 - b2 + 3*b1 + 2) * q^77 + (-b4 - 3*b3 + 2*b2 - 2*b1 + 1) * q^79 + q^81 + (b4 - 2*b3 - 2*b1) * q^83 + (-b3 + b2) * q^87 + (b4 + b3 - b2 + b1 + 2) * q^89 + b1 * q^91 - 2*b4 * q^93 + (-2*b3 + 2*b2 - b1 + 2) * q^97 + (-b3 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 5 q^{3} + q^{7} + 5 q^{9}+O(q^{10})$$ 5 * q + 5 * q^3 + q^7 + 5 * q^9 $$5 q + 5 q^{3} + q^{7} + 5 q^{9} + 5 q^{11} + 5 q^{13} + 7 q^{17} + 6 q^{19} + q^{21} + 3 q^{23} + 5 q^{27} + 2 q^{29} + 5 q^{33} + 9 q^{37} + 5 q^{39} - q^{41} + 4 q^{43} + 14 q^{47} + 2 q^{49} + 7 q^{51} + 5 q^{53} + 6 q^{57} + 20 q^{59} - 19 q^{61} + q^{63} + 12 q^{67} + 3 q^{69} + 13 q^{71} + 16 q^{73} + 11 q^{77} + 7 q^{79} + 5 q^{81} - 2 q^{83} + 2 q^{87} + 9 q^{89} + q^{91} + 13 q^{97} + 5 q^{99}+O(q^{100})$$ 5 * q + 5 * q^3 + q^7 + 5 * q^9 + 5 * q^11 + 5 * q^13 + 7 * q^17 + 6 * q^19 + q^21 + 3 * q^23 + 5 * q^27 + 2 * q^29 + 5 * q^33 + 9 * q^37 + 5 * q^39 - q^41 + 4 * q^43 + 14 * q^47 + 2 * q^49 + 7 * q^51 + 5 * q^53 + 6 * q^57 + 20 * q^59 - 19 * q^61 + q^63 + 12 * q^67 + 3 * q^69 + 13 * q^71 + 16 * q^73 + 11 * q^77 + 7 * q^79 + 5 * q^81 - 2 * q^83 + 2 * q^87 + 9 * q^89 + q^91 + 13 * q^97 + 5 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{4} - \nu^{3} - 4\nu^{2} + \nu + 1$$ v^4 - v^3 - 4*v^2 + v + 1 $$\beta_{2}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 2\nu + 4$$ v^3 - 2*v^2 - 2*v + 4 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 4\nu + 4$$ v^3 - 2*v^2 - 4*v + 4 $$\beta_{4}$$ $$=$$ $$-\nu^{4} + 2\nu^{3} + 4\nu^{2} - 5\nu - 2$$ -v^4 + 2*v^3 + 4*v^2 - 5*v - 2
 $$\nu$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{4} - \beta_{3} + \beta _1 + 5 ) / 2$$ (b4 - b3 + b1 + 5) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{4} - 2\beta_{3} + 2\beta_{2} + \beta _1 + 1$$ b4 - 2*b3 + 2*b2 + b1 + 1 $$\nu^{4}$$ $$=$$ $$( 6\beta_{4} - 7\beta_{3} + 3\beta_{2} + 8\beta _1 + 20 ) / 2$$ (6*b4 - 7*b3 + 3*b2 + 8*b1 + 20) / 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
 1.28447 −1.23118 0.271831 2.52064 −1.84576
0 1.00000 0 0 0 −3.71215 0 1.00000 0
1.2 0 1.00000 0 0 0 −2.13051 0 1.00000 0
1.3 0 1.00000 0 0 0 0.961637 0 1.00000 0
1.4 0 1.00000 0 0 0 2.45939 0 1.00000 0
1.5 0 1.00000 0 0 0 3.42163 0 1.00000 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$-1$$

## 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 7800.2.a.ca 5
5.b even 2 1 7800.2.a.bz 5
5.c odd 4 2 1560.2.l.f 10
15.e even 4 2 4680.2.l.h 10
20.e even 4 2 3120.2.l.q 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.l.f 10 5.c odd 4 2
3120.2.l.q 10 20.e even 4 2
4680.2.l.h 10 15.e even 4 2
7800.2.a.bz 5 5.b even 2 1
7800.2.a.ca 5 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7800))$$:

 $$T_{7}^{5} - T_{7}^{4} - 18T_{7}^{3} + 20T_{7}^{2} + 64T_{7} - 64$$ T7^5 - T7^4 - 18*T7^3 + 20*T7^2 + 64*T7 - 64 $$T_{11}^{5} - 5T_{11}^{4} - 10T_{11}^{3} + 66T_{11}^{2} + 8T_{11} - 184$$ T11^5 - 5*T11^4 - 10*T11^3 + 66*T11^2 + 8*T11 - 184 $$T_{17}^{5} - 7T_{17}^{4} - 6T_{17}^{3} + 24T_{17}^{2} + 8T_{17} - 16$$ T17^5 - 7*T17^4 - 6*T17^3 + 24*T17^2 + 8*T17 - 16 $$T_{19}^{5} - 6T_{19}^{4} - 28T_{19}^{3} + 160T_{19}^{2} - 96T_{19} - 128$$ T19^5 - 6*T19^4 - 28*T19^3 + 160*T19^2 - 96*T19 - 128

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$(T - 1)^{5}$$
$5$ $$T^{5}$$
$7$ $$T^{5} - T^{4} - 18 T^{3} + 20 T^{2} + \cdots - 64$$
$11$ $$T^{5} - 5 T^{4} - 10 T^{3} + 66 T^{2} + \cdots - 184$$
$13$ $$(T - 1)^{5}$$
$17$ $$T^{5} - 7 T^{4} - 6 T^{3} + 24 T^{2} + \cdots - 16$$
$19$ $$T^{5} - 6 T^{4} - 28 T^{3} + 160 T^{2} + \cdots - 128$$
$23$ $$T^{5} - 3 T^{4} - 22 T^{3} + 100 T^{2} + \cdots + 32$$
$29$ $$T^{5} - 2 T^{4} - 24 T^{3} + 24 T^{2} + \cdots - 64$$
$31$ $$T^{5} - 88 T^{3} + 64 T^{2} + \cdots - 1024$$
$37$ $$T^{5} - 9 T^{4} - 72 T^{3} + 752 T^{2} + \cdots - 368$$
$41$ $$T^{5} + T^{4} - 186 T^{3} + \cdots - 29816$$
$43$ $$T^{5} - 4 T^{4} - 180 T^{3} + \cdots - 20224$$
$47$ $$T^{5} - 14 T^{4} - 22 T^{3} + \cdots - 6400$$
$53$ $$T^{5} - 5 T^{4} - 172 T^{3} + \cdots + 944$$
$59$ $$T^{5} - 20 T^{4} - 106 T^{3} + \cdots - 45088$$
$61$ $$T^{5} + 19 T^{4} - 104 T^{3} + \cdots - 62912$$
$67$ $$T^{5} - 12 T^{4} - 112 T^{3} + \cdots - 4096$$
$71$ $$T^{5} - 13 T^{4} - 210 T^{3} + \cdots + 3104$$
$73$ $$T^{5} - 16 T^{4} - 36 T^{3} + \cdots + 13792$$
$79$ $$T^{5} - 7 T^{4} - 192 T^{3} + \cdots - 2752$$
$83$ $$T^{5} + 2 T^{4} - 150 T^{3} + \cdots - 1024$$
$89$ $$T^{5} - 9 T^{4} - 10 T^{3} + 206 T^{2} + \cdots - 200$$
$97$ $$T^{5} - 13 T^{4} - 62 T^{3} + \cdots - 1552$$