# Properties

 Label 8400.2.a.dh Level $8400$ Weight $2$ Character orbit 8400.a Self dual yes Analytic conductor $67.074$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$67.0743376979$$ Analytic rank: $$0$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 840) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{7} + q^{9}+O(q^{10})$$ q - q^3 - q^7 + q^9 $$q - q^{3} - q^{7} + q^{9} + ( - \beta_{2} - \beta_1 - 1) q^{11} + (\beta_1 + 1) q^{13} + \beta_{2} q^{17} + ( - \beta_1 - 1) q^{19} + q^{21} + ( - \beta_1 - 3) q^{23} - q^{27} + ( - 2 \beta_{2} - 2 \beta_1) q^{29} + (\beta_1 + 1) q^{31} + (\beta_{2} + \beta_1 + 1) q^{33} + ( - 2 \beta_1 - 2) q^{37} + ( - \beta_1 - 1) q^{39} + (\beta_1 + 5) q^{41} + ( - 2 \beta_{2} - 4) q^{43} + (2 \beta_{2} + 2 \beta_1 - 2) q^{47} + q^{49} - \beta_{2} q^{51} + ( - 2 \beta_{2} + \beta_1 + 5) q^{53} + (\beta_1 + 1) q^{57} + (2 \beta_{2} + 2 \beta_1 - 2) q^{59} + (2 \beta_{2} - 2 \beta_1) q^{61} - q^{63} + (\beta_1 + 3) q^{69} + (\beta_{2} + 3 \beta_1 + 3) q^{71} + ( - 2 \beta_{2} - 3 \beta_1 + 1) q^{73} + (\beta_{2} + \beta_1 + 1) q^{77} + ( - 2 \beta_1 + 6) q^{79} + q^{81} + (2 \beta_{2} + 2 \beta_1 - 2) q^{83} + (2 \beta_{2} + 2 \beta_1) q^{87} + (2 \beta_{2} + \beta_1 + 1) q^{89} + ( - \beta_1 - 1) q^{91} + ( - \beta_1 - 1) q^{93} + ( - 2 \beta_{2} - \beta_1 - 5) q^{97} + ( - \beta_{2} - \beta_1 - 1) q^{99}+O(q^{100})$$ q - q^3 - q^7 + q^9 + (-b2 - b1 - 1) * q^11 + (b1 + 1) * q^13 + b2 * q^17 + (-b1 - 1) * q^19 + q^21 + (-b1 - 3) * q^23 - q^27 + (-2*b2 - 2*b1) * q^29 + (b1 + 1) * q^31 + (b2 + b1 + 1) * q^33 + (-2*b1 - 2) * q^37 + (-b1 - 1) * q^39 + (b1 + 5) * q^41 + (-2*b2 - 4) * q^43 + (2*b2 + 2*b1 - 2) * q^47 + q^49 - b2 * q^51 + (-2*b2 + b1 + 5) * q^53 + (b1 + 1) * q^57 + (2*b2 + 2*b1 - 2) * q^59 + (2*b2 - 2*b1) * q^61 - q^63 + (b1 + 3) * q^69 + (b2 + 3*b1 + 3) * q^71 + (-2*b2 - 3*b1 + 1) * q^73 + (b2 + b1 + 1) * q^77 + (-2*b1 + 6) * q^79 + q^81 + (2*b2 + 2*b1 - 2) * q^83 + (2*b2 + 2*b1) * q^87 + (2*b2 + b1 + 1) * q^89 + (-b1 - 1) * q^91 + (-b1 - 1) * q^93 + (-2*b2 - b1 - 5) * q^97 + (-b2 - b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 $$3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{19} + 3 q^{21} - 8 q^{23} - 3 q^{27} + 2 q^{29} + 2 q^{31} + 2 q^{33} - 4 q^{37} - 2 q^{39} + 14 q^{41} - 12 q^{43} - 8 q^{47} + 3 q^{49} + 14 q^{53} + 2 q^{57} - 8 q^{59} + 2 q^{61} - 3 q^{63} + 8 q^{69} + 6 q^{71} + 6 q^{73} + 2 q^{77} + 20 q^{79} + 3 q^{81} - 8 q^{83} - 2 q^{87} + 2 q^{89} - 2 q^{91} - 2 q^{93} - 14 q^{97} - 2 q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 - 2 * q^19 + 3 * q^21 - 8 * q^23 - 3 * q^27 + 2 * q^29 + 2 * q^31 + 2 * q^33 - 4 * q^37 - 2 * q^39 + 14 * q^41 - 12 * q^43 - 8 * q^47 + 3 * q^49 + 14 * q^53 + 2 * q^57 - 8 * q^59 + 2 * q^61 - 3 * q^63 + 8 * q^69 + 6 * q^71 + 6 * q^73 + 2 * q^77 + 20 * q^79 + 3 * q^81 - 8 * q^83 - 2 * q^87 + 2 * q^89 - 2 * q^91 - 2 * q^93 - 14 * q^97 - 2 * q^99

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

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$2\nu^{2} - 2\nu - 4$$ 2*v^2 - 2*v - 4
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + \beta _1 + 5 ) / 2$$ (b2 + b1 + 5) / 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 −1.48119 0.311108
0 −1.00000 0 0 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 0 0 −1.00000 0 1.00000 0
1.3 0 −1.00000 0 0 0 −1.00000 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
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## 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 8400.2.a.dh 3
4.b odd 2 1 4200.2.a.bq 3
5.b even 2 1 8400.2.a.dk 3
5.c odd 4 2 1680.2.t.i 6
15.e even 4 2 5040.2.t.ba 6
20.d odd 2 1 4200.2.a.bo 3
20.e even 4 2 840.2.t.e 6
60.l odd 4 2 2520.2.t.j 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.e 6 20.e even 4 2
1680.2.t.i 6 5.c odd 4 2
2520.2.t.j 6 60.l odd 4 2
4200.2.a.bo 3 20.d odd 2 1
4200.2.a.bq 3 4.b odd 2 1
5040.2.t.ba 6 15.e even 4 2
8400.2.a.dh 3 1.a even 1 1 trivial
8400.2.a.dk 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8400))$$:

 $$T_{11}^{3} + 2T_{11}^{2} - 20T_{11} - 8$$ T11^3 + 2*T11^2 - 20*T11 - 8 $$T_{13}^{3} - 2T_{13}^{2} - 12T_{13} + 8$$ T13^3 - 2*T13^2 - 12*T13 + 8 $$T_{17}^{3} - 16T_{17} + 16$$ T17^3 - 16*T17 + 16 $$T_{19}^{3} + 2T_{19}^{2} - 12T_{19} - 8$$ T19^3 + 2*T19^2 - 12*T19 - 8 $$T_{23}^{3} + 8T_{23}^{2} + 8T_{23} - 16$$ T23^3 + 8*T23^2 + 8*T23 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T + 1)^{3}$$
$5$ $$T^{3}$$
$7$ $$(T + 1)^{3}$$
$11$ $$T^{3} + 2 T^{2} - 20 T - 8$$
$13$ $$T^{3} - 2 T^{2} - 12 T + 8$$
$17$ $$T^{3} - 16T + 16$$
$19$ $$T^{3} + 2 T^{2} - 12 T - 8$$
$23$ $$T^{3} + 8 T^{2} + 8 T - 16$$
$29$ $$T^{3} - 2 T^{2} - 84 T + 104$$
$31$ $$T^{3} - 2 T^{2} - 12 T + 8$$
$37$ $$T^{3} + 4 T^{2} - 48 T - 64$$
$41$ $$T^{3} - 14 T^{2} + 52 T - 40$$
$43$ $$T^{3} + 12 T^{2} - 16 T - 320$$
$47$ $$T^{3} + 8 T^{2} - 64 T - 256$$
$53$ $$T^{3} - 14 T^{2} - 28 T + 472$$
$59$ $$T^{3} + 8 T^{2} - 64 T - 256$$
$61$ $$T^{3} - 2 T^{2} - 148 T - 536$$
$67$ $$T^{3}$$
$71$ $$T^{3} - 6 T^{2} - 100 T - 200$$
$73$ $$T^{3} - 6 T^{2} - 124 T + 760$$
$79$ $$T^{3} - 20 T^{2} + 80 T + 64$$
$83$ $$T^{3} + 8 T^{2} - 64 T - 256$$
$89$ $$T^{3} - 2 T^{2} - 60 T + 200$$
$97$ $$T^{3} + 14 T^{2} + 4 T - 344$$