# Properties

 Label 8400.2.a.dg Level $8400$ Weight $2$ Character orbit 8400.a Self dual yes Analytic conductor $67.074$ Analytic rank $1$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 105) 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.

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

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.dg 3
4.b odd 2 1 525.2.a.j 3
5.b even 2 1 8400.2.a.dj 3
5.c odd 4 2 1680.2.t.k 6
12.b even 2 1 1575.2.a.x 3
15.e even 4 2 5040.2.t.v 6
20.d odd 2 1 525.2.a.k 3
20.e even 4 2 105.2.d.b 6
28.d even 2 1 3675.2.a.bi 3
60.h even 2 1 1575.2.a.w 3
60.l odd 4 2 315.2.d.e 6
140.c even 2 1 3675.2.a.bj 3
140.j odd 4 2 735.2.d.b 6
140.w even 12 4 735.2.q.e 12
140.x odd 12 4 735.2.q.f 12
420.w even 4 2 2205.2.d.l 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.b 6 20.e even 4 2
315.2.d.e 6 60.l odd 4 2
525.2.a.j 3 4.b odd 2 1
525.2.a.k 3 20.d odd 2 1
735.2.d.b 6 140.j odd 4 2
735.2.q.e 12 140.w even 12 4
735.2.q.f 12 140.x odd 12 4
1575.2.a.w 3 60.h even 2 1
1575.2.a.x 3 12.b even 2 1
1680.2.t.k 6 5.c odd 4 2
2205.2.d.l 6 420.w even 4 2
3675.2.a.bi 3 28.d even 2 1
3675.2.a.bj 3 140.c even 2 1
5040.2.t.v 6 15.e even 4 2
8400.2.a.dg 3 1.a even 1 1 trivial
8400.2.a.dj 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8400))$$:

 $$T_{11} + 2$$ $$T_{13}^{3} - 6 T_{13}^{2} - 4 T_{13} + 8$$ $$T_{17}^{3} - 16 T_{17} + 16$$ $$T_{19}^{3} + 6 T_{19}^{2} - 4 T_{19} - 40$$ $$T_{23}^{3} - 4 T_{23}^{2} - 8 T_{23} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$( 2 + T )^{3}$$
$13$ $$8 - 4 T - 6 T^{2} + T^{3}$$
$17$ $$16 - 16 T + T^{3}$$
$19$ $$-40 - 4 T + 6 T^{2} + T^{3}$$
$23$ $$16 - 8 T - 4 T^{2} + T^{3}$$
$29$ $$40 - 52 T - 2 T^{2} + T^{3}$$
$31$ $$-184 - 52 T + 2 T^{2} + T^{3}$$
$37$ $$64 - 80 T - 4 T^{2} + T^{3}$$
$41$ $$200 - 60 T - 2 T^{2} + T^{3}$$
$43$ $$832 - 144 T - 4 T^{2} + T^{3}$$
$47$ $$128 - 32 T - 8 T^{2} + T^{3}$$
$53$ $$-296 + 12 T + 14 T^{2} + T^{3}$$
$59$ $$-1280 - 64 T + 16 T^{2} + T^{3}$$
$61$ $$-248 - 52 T + 6 T^{2} + T^{3}$$
$67$ $$128 - 32 T - 8 T^{2} + T^{3}$$
$71$ $$( 2 + T )^{3}$$
$73$ $$-104 + 92 T - 18 T^{2} + T^{3}$$
$79$ $$-320 - 16 T + 12 T^{2} + T^{3}$$
$83$ $$256 - 64 T - 8 T^{2} + T^{3}$$
$89$ $$-40 + 52 T - 14 T^{2} + T^{3}$$
$97$ $$1864 - 36 T - 22 T^{2} + T^{3}$$