Properties

 Label 8400.2.a.cz Level $8400$ Weight $2$ Character orbit 8400.a Self dual yes Analytic conductor $67.074$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ 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 $$\beta = 2\sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

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

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
 1.61803 −0.618034
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
 $$n$$: 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.cz 2
4.b odd 2 1 4200.2.a.bk 2
5.b even 2 1 8400.2.a.db 2
5.c odd 4 2 1680.2.t.h 4
15.e even 4 2 5040.2.t.u 4
20.d odd 2 1 4200.2.a.bj 2
20.e even 4 2 840.2.t.c 4
60.l odd 4 2 2520.2.t.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.c 4 20.e even 4 2
1680.2.t.h 4 5.c odd 4 2
2520.2.t.f 4 60.l odd 4 2
4200.2.a.bj 2 20.d odd 2 1
4200.2.a.bk 2 4.b odd 2 1
5040.2.t.u 4 15.e even 4 2
8400.2.a.cz 2 1.a even 1 1 trivial
8400.2.a.db 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8400))$$:

 $$T_{11} - 2$$ $$T_{13}^{2} - 20$$ $$T_{17}^{2} + 4 T_{17} - 16$$ $$T_{19} + 2$$ $$T_{23} - 4$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$-20 + T^{2}$$
$17$ $$-16 + 4 T + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$( -4 + T )^{2}$$
$29$ $$-4 + 8 T + T^{2}$$
$31$ $$-4 + 8 T + T^{2}$$
$37$ $$-16 - 4 T + T^{2}$$
$41$ $$44 + 16 T + T^{2}$$
$43$ $$-16 - 4 T + T^{2}$$
$47$ $$-16 - 4 T + T^{2}$$
$53$ $$( 2 + T )^{2}$$
$59$ $$T^{2}$$
$61$ $$44 + 16 T + T^{2}$$
$67$ $$16 - 12 T + T^{2}$$
$71$ $$44 + 16 T + T^{2}$$
$73$ $$124 + 24 T + T^{2}$$
$79$ $$-80 + T^{2}$$
$83$ $$-64 - 8 T + T^{2}$$
$89$ $$-164 + 8 T + T^{2}$$
$97$ $$44 + 16 T + T^{2}$$