# Properties

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

# Learn more about

## 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
 0
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.y 1
4.b odd 2 1 4200.2.a.t 1
5.b even 2 1 336.2.a.e 1
15.d odd 2 1 1008.2.a.b 1
20.d odd 2 1 168.2.a.a 1
20.e even 4 2 4200.2.t.j 2
35.c odd 2 1 2352.2.a.c 1
35.i odd 6 2 2352.2.q.w 2
35.j even 6 2 2352.2.q.d 2
40.e odd 2 1 1344.2.a.m 1
40.f even 2 1 1344.2.a.b 1
60.h even 2 1 504.2.a.e 1
80.k odd 4 2 5376.2.c.d 2
80.q even 4 2 5376.2.c.bb 2
105.g even 2 1 7056.2.a.bq 1
120.i odd 2 1 4032.2.a.bc 1
120.m even 2 1 4032.2.a.bh 1
140.c even 2 1 1176.2.a.f 1
140.p odd 6 2 1176.2.q.f 2
140.s even 6 2 1176.2.q.d 2
280.c odd 2 1 9408.2.a.da 1
280.n even 2 1 9408.2.a.be 1
420.o odd 2 1 3528.2.a.v 1
420.ba even 6 2 3528.2.s.w 2
420.be odd 6 2 3528.2.s.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.a.a 1 20.d odd 2 1
336.2.a.e 1 5.b even 2 1
504.2.a.e 1 60.h even 2 1
1008.2.a.b 1 15.d odd 2 1
1176.2.a.f 1 140.c even 2 1
1176.2.q.d 2 140.s even 6 2
1176.2.q.f 2 140.p odd 6 2
1344.2.a.b 1 40.f even 2 1
1344.2.a.m 1 40.e odd 2 1
2352.2.a.c 1 35.c odd 2 1
2352.2.q.d 2 35.j even 6 2
2352.2.q.w 2 35.i odd 6 2
3528.2.a.v 1 420.o odd 2 1
3528.2.s.g 2 420.be odd 6 2
3528.2.s.w 2 420.ba even 6 2
4032.2.a.bc 1 120.i odd 2 1
4032.2.a.bh 1 120.m even 2 1
4200.2.a.t 1 4.b odd 2 1
4200.2.t.j 2 20.e even 4 2
5376.2.c.d 2 80.k odd 4 2
5376.2.c.bb 2 80.q even 4 2
7056.2.a.bq 1 105.g even 2 1
8400.2.a.y 1 1.a even 1 1 trivial
9408.2.a.be 1 280.n even 2 1
9408.2.a.da 1 280.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8400))$$:

 $$T_{11}$$ $$T_{13} + 6$$ $$T_{17} - 2$$ $$T_{19} + 4$$ $$T_{23} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + T$$
$5$ 1
$7$ $$1 - T$$
$11$ $$1 + 11 T^{2}$$
$13$ $$1 + 6 T + 13 T^{2}$$
$17$ $$1 - 2 T + 17 T^{2}$$
$19$ $$1 + 4 T + 19 T^{2}$$
$23$ $$1 + 4 T + 23 T^{2}$$
$29$ $$1 + 10 T + 29 T^{2}$$
$31$ $$1 - 8 T + 31 T^{2}$$
$37$ $$1 + 6 T + 37 T^{2}$$
$41$ $$1 + 2 T + 41 T^{2}$$
$43$ $$1 + 4 T + 43 T^{2}$$
$47$ $$1 - 8 T + 47 T^{2}$$
$53$ $$1 - 10 T + 53 T^{2}$$
$59$ $$1 + 12 T + 59 T^{2}$$
$61$ $$1 + 2 T + 61 T^{2}$$
$67$ $$1 - 12 T + 67 T^{2}$$
$71$ $$1 - 12 T + 71 T^{2}$$
$73$ $$1 - 14 T + 73 T^{2}$$
$79$ $$1 - 8 T + 79 T^{2}$$
$83$ $$1 - 12 T + 83 T^{2}$$
$89$ $$1 + 2 T + 89 T^{2}$$
$97$ $$1 + 10 T + 97 T^{2}$$
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