# Properties

 Label 6300.2.k.r Level $6300$ Weight $2$ Character orbit 6300.k Analytic conductor $50.306$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6300.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$50.3057532734$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{7}+O(q^{10})$$ q + i * q^7 $$q + i q^{7} + 6 q^{11} - 2 i q^{13} + 4 q^{19} - 6 i q^{23} + 6 q^{29} + 8 q^{31} + 2 i q^{37} - 12 q^{41} + 4 i q^{43} - 12 i q^{47} - q^{49} - 6 i q^{53} - 10 q^{61} + 8 i q^{67} - 6 q^{71} + 10 i q^{73} + 6 i q^{77} + 4 q^{79} - 12 i q^{83} + 12 q^{89} + 2 q^{91} - 10 i q^{97} +O(q^{100})$$ q + i * q^7 + 6 * q^11 - 2*i * q^13 + 4 * q^19 - 6*i * q^23 + 6 * q^29 + 8 * q^31 + 2*i * q^37 - 12 * q^41 + 4*i * q^43 - 12*i * q^47 - q^49 - 6*i * q^53 - 10 * q^61 + 8*i * q^67 - 6 * q^71 + 10*i * q^73 + 6*i * q^77 + 4 * q^79 - 12*i * q^83 + 12 * q^89 + 2 * q^91 - 10*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 12 q^{11} + 8 q^{19} + 12 q^{29} + 16 q^{31} - 24 q^{41} - 2 q^{49} - 20 q^{61} - 12 q^{71} + 8 q^{79} + 24 q^{89} + 4 q^{91}+O(q^{100})$$ 2 * q + 12 * q^11 + 8 * q^19 + 12 * q^29 + 16 * q^31 - 24 * q^41 - 2 * q^49 - 20 * q^61 - 12 * q^71 + 8 * q^79 + 24 * q^89 + 4 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times$$.

 $$n$$ $$2801$$ $$3151$$ $$3277$$ $$3601$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6049.1
 − 1.00000i 1.00000i
0 0 0 0 0 1.00000i 0 0 0
6049.2 0 0 0 0 0 1.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6300.2.k.r 2
3.b odd 2 1 2100.2.k.a 2
5.b even 2 1 inner 6300.2.k.r 2
5.c odd 4 1 252.2.a.b 1
5.c odd 4 1 6300.2.a.p 1
15.d odd 2 1 2100.2.k.a 2
15.e even 4 1 84.2.a.b 1
15.e even 4 1 2100.2.a.a 1
20.e even 4 1 1008.2.a.g 1
35.f even 4 1 1764.2.a.g 1
35.k even 12 2 1764.2.k.d 2
35.l odd 12 2 1764.2.k.e 2
40.i odd 4 1 4032.2.a.u 1
40.k even 4 1 4032.2.a.t 1
45.k odd 12 2 2268.2.j.f 2
45.l even 12 2 2268.2.j.i 2
60.l odd 4 1 336.2.a.b 1
60.l odd 4 1 8400.2.a.ct 1
105.k odd 4 1 588.2.a.c 1
105.w odd 12 2 588.2.i.f 2
105.x even 12 2 588.2.i.c 2
120.q odd 4 1 1344.2.a.o 1
120.w even 4 1 1344.2.a.f 1
140.j odd 4 1 7056.2.a.x 1
240.z odd 4 1 5376.2.c.x 2
240.bb even 4 1 5376.2.c.i 2
240.bd odd 4 1 5376.2.c.x 2
240.bf even 4 1 5376.2.c.i 2
420.w even 4 1 2352.2.a.s 1
420.bp odd 12 2 2352.2.q.s 2
420.br even 12 2 2352.2.q.g 2
840.bm even 4 1 9408.2.a.r 1
840.bp odd 4 1 9408.2.a.co 1

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

## Hecke kernels

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

 $$T_{11} - 6$$ T11 - 6 $$T_{13}^{2} + 4$$ T13^2 + 4 $$T_{17}$$ T17 $$T_{41} + 12$$ T41 + 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T - 6)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2}$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 12)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 144$$
$53$ $$T^{2} + 36$$
$59$ $$T^{2}$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T - 12)^{2}$$
$97$ $$T^{2} + 100$$