# Properties

 Label 6300.2.a.p Level 6300 Weight 2 Character orbit 6300.a Self dual yes Analytic conductor 50.306 Analytic rank 1 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6300.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$50.3057532734$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{7} + O(q^{10})$$ $$q - q^{7} + 6q^{11} - 2q^{13} - 4q^{19} - 6q^{23} - 6q^{29} + 8q^{31} - 2q^{37} - 12q^{41} + 4q^{43} + 12q^{47} + q^{49} - 6q^{53} - 10q^{61} - 8q^{67} - 6q^{71} + 10q^{73} - 6q^{77} - 4q^{79} - 12q^{83} - 12q^{89} + 2q^{91} + 10q^{97} + O(q^{100})$$

## 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 0 0 0 0 −1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 6300.2.a.p 1
3.b odd 2 1 2100.2.a.a 1
5.b even 2 1 252.2.a.b 1
5.c odd 4 2 6300.2.k.r 2
12.b even 2 1 8400.2.a.ct 1
15.d odd 2 1 84.2.a.b 1
15.e even 4 2 2100.2.k.a 2
20.d odd 2 1 1008.2.a.g 1
35.c odd 2 1 1764.2.a.g 1
35.i odd 6 2 1764.2.k.d 2
35.j even 6 2 1764.2.k.e 2
40.e odd 2 1 4032.2.a.t 1
40.f even 2 1 4032.2.a.u 1
45.h odd 6 2 2268.2.j.i 2
45.j even 6 2 2268.2.j.f 2
60.h even 2 1 336.2.a.b 1
105.g even 2 1 588.2.a.c 1
105.o odd 6 2 588.2.i.c 2
105.p even 6 2 588.2.i.f 2
120.i odd 2 1 1344.2.a.f 1
120.m even 2 1 1344.2.a.o 1
140.c even 2 1 7056.2.a.x 1
240.t even 4 2 5376.2.c.x 2
240.bm odd 4 2 5376.2.c.i 2
420.o odd 2 1 2352.2.a.s 1
420.ba even 6 2 2352.2.q.s 2
420.be odd 6 2 2352.2.q.g 2
840.b odd 2 1 9408.2.a.r 1
840.u even 2 1 9408.2.a.co 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.a.b 1 15.d odd 2 1
252.2.a.b 1 5.b even 2 1
336.2.a.b 1 60.h even 2 1
588.2.a.c 1 105.g even 2 1
588.2.i.c 2 105.o odd 6 2
588.2.i.f 2 105.p even 6 2
1008.2.a.g 1 20.d odd 2 1
1344.2.a.f 1 120.i odd 2 1
1344.2.a.o 1 120.m even 2 1
1764.2.a.g 1 35.c odd 2 1
1764.2.k.d 2 35.i odd 6 2
1764.2.k.e 2 35.j even 6 2
2100.2.a.a 1 3.b odd 2 1
2100.2.k.a 2 15.e even 4 2
2268.2.j.f 2 45.j even 6 2
2268.2.j.i 2 45.h odd 6 2
2352.2.a.s 1 420.o odd 2 1
2352.2.q.g 2 420.be odd 6 2
2352.2.q.s 2 420.ba even 6 2
4032.2.a.t 1 40.e odd 2 1
4032.2.a.u 1 40.f even 2 1
5376.2.c.i 2 240.bm odd 4 2
5376.2.c.x 2 240.t even 4 2
6300.2.a.p 1 1.a even 1 1 trivial
6300.2.k.r 2 5.c odd 4 2
7056.2.a.x 1 140.c even 2 1
8400.2.a.ct 1 12.b even 2 1
9408.2.a.r 1 840.b odd 2 1
9408.2.a.co 1 840.u even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6300))$$:

 $$T_{11} - 6$$ $$T_{13} + 2$$ $$T_{17}$$ $$T_{37} + 2$$

## Hecke characteristic polynomials

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