# Properties

 Label 8400.2.a.dl 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: $$x^{3} - x^{2} - 3 x + 1$$ 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} - 2 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 5$$$$)/2$$

## 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
 −1.48119 2.17009 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.dl 3
4.b odd 2 1 4200.2.a.bn 3
5.b even 2 1 8400.2.a.di 3
5.c odd 4 2 1680.2.t.j 6
15.e even 4 2 5040.2.t.z 6
20.d odd 2 1 4200.2.a.bp 3
20.e even 4 2 840.2.t.d 6
60.l odd 4 2 2520.2.t.k 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.d 6 20.e even 4 2
1680.2.t.j 6 5.c odd 4 2
2520.2.t.k 6 60.l odd 4 2
4200.2.a.bn 3 4.b odd 2 1
4200.2.a.bp 3 20.d odd 2 1
5040.2.t.z 6 15.e even 4 2
8400.2.a.di 3 5.b even 2 1
8400.2.a.dl 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8400))$$:

 $$T_{11}^{3} - 2 T_{11}^{2} - 36 T_{11} + 104$$ $$T_{13}^{3} - 10 T_{13}^{2} + 20 T_{13} + 8$$ $$T_{17}^{3} - 8 T_{17}^{2} - 32 T_{17} + 272$$ $$T_{19}^{3} - 2 T_{19}^{2} - 60 T_{19} + 200$$ $$T_{23}^{3} + 8 T_{23}^{2} - 40 T_{23} - 304$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$T^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$104 - 36 T - 2 T^{2} + T^{3}$$
$13$ $$8 + 20 T - 10 T^{2} + T^{3}$$
$17$ $$272 - 32 T - 8 T^{2} + T^{3}$$
$19$ $$200 - 60 T - 2 T^{2} + T^{3}$$
$23$ $$-304 - 40 T + 8 T^{2} + T^{3}$$
$29$ $$( 2 + T )^{3}$$
$31$ $$-40 + 52 T - 14 T^{2} + T^{3}$$
$37$ $$320 - 16 T - 12 T^{2} + T^{3}$$
$41$ $$-8 + 20 T + 10 T^{2} + T^{3}$$
$43$ $$-64 - 48 T + 4 T^{2} + T^{3}$$
$47$ $$T^{3}$$
$53$ $$-40 + 52 T - 14 T^{2} + T^{3}$$
$59$ $$256 - 64 T - 8 T^{2} + T^{3}$$
$61$ $$104 - 212 T - 2 T^{2} + T^{3}$$
$67$ $$256 - 16 T^{2} + T^{3}$$
$71$ $$1352 - 52 T - 18 T^{2} + T^{3}$$
$73$ $$-104 + 100 T - 22 T^{2} + T^{3}$$
$79$ $$-64 - 48 T + 4 T^{2} + T^{3}$$
$83$ $$-256 - 64 T + 8 T^{2} + T^{3}$$
$89$ $$232 - 124 T + 6 T^{2} + T^{3}$$
$97$ $$200 - 60 T - 2 T^{2} + T^{3}$$