# Properties

 Label 7800.2.a.bw Level $7800$ Weight $2$ Character orbit 7800.a Self dual yes Analytic conductor $62.283$ Analytic rank $1$ Dimension $4$ 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.10304.1 Defining polynomial: $$x^{4} - 2x^{3} - 7x^{2} + 8x + 8$$ x^4 - 2*x^3 - 7*x^2 + 8*x + 8 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + \beta_{3} q^{7} + q^{9}+O(q^{10})$$ q + q^3 + b3 * q^7 + q^9 $$q + q^{3} + \beta_{3} q^{7} + q^{9} + ( - \beta_1 - 2) q^{11} - q^{13} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{17} + ( - \beta_{3} - \beta_{2} + \beta_1 - 3) q^{19} + \beta_{3} q^{21} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{23} + q^{27} + (\beta_{3} + 2) q^{29} + ( - 2 \beta_{3} - 4) q^{31} + ( - \beta_1 - 2) q^{33} + (2 \beta_1 - 2) q^{37} - q^{39} + (\beta_{2} - 1) q^{41} + (2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{43} + (\beta_{2} + 2 \beta_1 - 3) q^{47} + q^{49} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{51} + 2 \beta_{2} q^{53} + ( - \beta_{3} - \beta_{2} + \beta_1 - 3) q^{57} + ( - \beta_1 - 2) q^{59} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{61} + \beta_{3} q^{63} + ( - 2 \beta_{2} - 6) q^{67} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{69} + (\beta_{2} - 2 \beta_1 - 3) q^{71} + (\beta_{3} + 2 \beta_1 - 6) q^{73} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{77} + ( - 3 \beta_{2} - \beta_1 - 3) q^{79} + q^{81} + ( - 2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 3) q^{83} + (\beta_{3} + 2) q^{87} + ( - 2 \beta_{3} - 3 \beta_{2} - 1) q^{89} - \beta_{3} q^{91} + ( - 2 \beta_{3} - 4) q^{93} + (\beta_{3} - 2 \beta_1 - 2) q^{97} + ( - \beta_1 - 2) q^{99}+O(q^{100})$$ q + q^3 + b3 * q^7 + q^9 + (-b1 - 2) * q^11 - q^13 + (-b3 + b2 + b1 + 1) * q^17 + (-b3 - b2 + b1 - 3) * q^19 + b3 * q^21 + (-b3 - b2 - b1 + 1) * q^23 + q^27 + (b3 + 2) * q^29 + (-2*b3 - 4) * q^31 + (-b1 - 2) * q^33 + (2*b1 - 2) * q^37 - q^39 + (b2 - 1) * q^41 + (2*b3 + b2 - b1 + 1) * q^43 + (b2 + 2*b1 - 3) * q^47 + q^49 + (-b3 + b2 + b1 + 1) * q^51 + 2*b2 * q^53 + (-b3 - b2 + b1 - 3) * q^57 + (-b1 - 2) * q^59 + (-2*b3 + b2 + b1 + 3) * q^61 + b3 * q^63 + (-2*b2 - 6) * q^67 + (-b3 - b2 - b1 + 1) * q^69 + (b2 - 2*b1 - 3) * q^71 + (b3 + 2*b1 - 6) * q^73 + (-2*b3 - 2*b2 - 2*b1 - 2) * q^77 + (-3*b2 - b1 - 3) * q^79 + q^81 + (-2*b3 - 3*b2 + 2*b1 - 3) * q^83 + (b3 + 2) * q^87 + (-2*b3 - 3*b2 - 1) * q^89 - b3 * q^91 + (-2*b3 - 4) * q^93 + (b3 - 2*b1 - 2) * q^97 + (-b1 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 + 4 * q^9 $$4 q + 4 q^{3} + 4 q^{9} - 8 q^{11} - 4 q^{13} + 4 q^{17} - 12 q^{19} + 4 q^{23} + 4 q^{27} + 8 q^{29} - 16 q^{31} - 8 q^{33} - 8 q^{37} - 4 q^{39} - 4 q^{41} + 4 q^{43} - 12 q^{47} + 4 q^{49} + 4 q^{51} - 12 q^{57} - 8 q^{59} + 12 q^{61} - 24 q^{67} + 4 q^{69} - 12 q^{71} - 24 q^{73} - 8 q^{77} - 12 q^{79} + 4 q^{81} - 12 q^{83} + 8 q^{87} - 4 q^{89} - 16 q^{93} - 8 q^{97} - 8 q^{99}+O(q^{100})$$ 4 * q + 4 * q^3 + 4 * q^9 - 8 * q^11 - 4 * q^13 + 4 * q^17 - 12 * q^19 + 4 * q^23 + 4 * q^27 + 8 * q^29 - 16 * q^31 - 8 * q^33 - 8 * q^37 - 4 * q^39 - 4 * q^41 + 4 * q^43 - 12 * q^47 + 4 * q^49 + 4 * q^51 - 12 * q^57 - 8 * q^59 + 12 * q^61 - 24 * q^67 + 4 * q^69 - 12 * q^71 - 24 * q^73 - 8 * q^77 - 12 * q^79 + 4 * q^81 - 12 * q^83 + 8 * q^87 - 4 * q^89 - 16 * q^93 - 8 * q^97 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 7x^{2} + 8x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - \nu^{2} - 4\nu ) / 2$$ (v^3 - v^2 - 4*v) / 2 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 8\nu - 2 ) / 2$$ (-v^3 + v^2 + 8*v - 2) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{2} - \nu - 4$$ v^2 - v - 4
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta _1 + 1 ) / 2$$ (b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{3} + \beta_{2} + \beta _1 + 9 ) / 2$$ (2*b3 + b2 + b1 + 9) / 2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{3} + 5\beta_{2} + 9\beta _1 + 13 ) / 2$$ (2*b3 + 5*b2 + 9*b1 + 13) / 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
 −0.692297 1.69230 3.16053 −2.16053
0 1.00000 0 0 0 −2.82843 0 1.00000 0
1.2 0 1.00000 0 0 0 −2.82843 0 1.00000 0
1.3 0 1.00000 0 0 0 2.82843 0 1.00000 0
1.4 0 1.00000 0 0 0 2.82843 0 1.00000 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$1$$

## 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.bw 4
5.b even 2 1 7800.2.a.bv 4
5.c odd 4 2 1560.2.l.e 8
15.e even 4 2 4680.2.l.f 8
20.e even 4 2 3120.2.l.o 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.l.e 8 5.c odd 4 2
3120.2.l.o 8 20.e even 4 2
4680.2.l.f 8 15.e even 4 2
7800.2.a.bv 4 5.b even 2 1
7800.2.a.bw 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7800))$$:

 $$T_{7}^{2} - 8$$ T7^2 - 8 $$T_{11}^{4} + 8T_{11}^{3} + 6T_{11}^{2} - 24T_{11} + 8$$ T11^4 + 8*T11^3 + 6*T11^2 - 24*T11 + 8 $$T_{17}^{4} - 4T_{17}^{3} - 44T_{17}^{2} + 224T_{17} - 224$$ T17^4 - 4*T17^3 - 44*T17^2 + 224*T17 - 224 $$T_{19}^{4} + 12T_{19}^{3} + 12T_{19}^{2} - 224T_{19} - 448$$ T19^4 + 12*T19^3 + 12*T19^2 - 224*T19 - 448

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T - 1)^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 8)^{2}$$
$11$ $$T^{4} + 8 T^{3} + 6 T^{2} - 24 T + 8$$
$13$ $$(T + 1)^{4}$$
$17$ $$T^{4} - 4 T^{3} - 44 T^{2} + 224 T - 224$$
$19$ $$T^{4} + 12 T^{3} + 12 T^{2} + \cdots - 448$$
$23$ $$T^{4} - 4 T^{3} - 44 T^{2} + 224 T - 224$$
$29$ $$(T^{2} - 4 T - 4)^{2}$$
$31$ $$(T^{2} + 8 T - 16)^{2}$$
$37$ $$T^{4} + 8 T^{3} - 48 T^{2} - 384 T - 16$$
$41$ $$T^{4} + 4 T^{3} - 14 T^{2} - 56 T - 8$$
$43$ $$T^{4} - 4 T^{3} - 68 T^{2} + 384 T - 512$$
$47$ $$T^{4} + 12 T^{3} - 30 T^{2} + \cdots - 1024$$
$53$ $$T^{4} - 80 T^{2} - 160 T + 496$$
$59$ $$T^{4} + 8 T^{3} + 6 T^{2} - 24 T + 8$$
$61$ $$T^{4} - 12 T^{3} - 44 T^{2} + \cdots - 1472$$
$67$ $$T^{4} + 24 T^{3} + 136 T^{2} + \cdots - 128$$
$71$ $$T^{4} + 12 T^{3} - 46 T^{2} + \cdots + 512$$
$73$ $$T^{4} + 24 T^{3} + 112 T^{2} + \cdots - 5008$$
$79$ $$T^{4} + 12 T^{3} - 132 T^{2} + \cdots + 6272$$
$83$ $$T^{4} + 12 T^{3} - 206 T^{2} + \cdots - 4384$$
$89$ $$T^{4} + 4 T^{3} - 190 T^{2} + \cdots + 184$$
$97$ $$T^{4} + 8 T^{3} - 48 T^{2} - 384 T - 16$$