# Properties

 Label 6300.2.k.g 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: $$x^{2} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q -i q^{7} +O(q^{10})$$ $$q -i q^{7} -2 q^{11} + 6 i q^{13} + 4 i q^{17} + 4 q^{19} + 2 i q^{23} -2 q^{29} + 2 i q^{37} + 4 i q^{43} -12 i q^{47} - q^{49} -6 i q^{53} -8 q^{59} + 6 q^{61} -8 i q^{67} -14 q^{71} + 2 i q^{73} + 2 i q^{77} -12 q^{79} -4 i q^{83} + 6 q^{91} -2 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 4q^{11} + 8q^{19} - 4q^{29} - 2q^{49} - 16q^{59} + 12q^{61} - 28q^{71} - 24q^{79} + 12q^{91} + O(q^{100})$$

## 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.g 2
3.b odd 2 1 2100.2.k.i 2
5.b even 2 1 inner 6300.2.k.g 2
5.c odd 4 1 252.2.a.a 1
5.c odd 4 1 6300.2.a.w 1
15.d odd 2 1 2100.2.k.i 2
15.e even 4 1 84.2.a.a 1
15.e even 4 1 2100.2.a.r 1
20.e even 4 1 1008.2.a.a 1
35.f even 4 1 1764.2.a.k 1
35.k even 12 2 1764.2.k.a 2
35.l odd 12 2 1764.2.k.k 2
40.i odd 4 1 4032.2.a.bm 1
40.k even 4 1 4032.2.a.bn 1
45.k odd 12 2 2268.2.j.n 2
45.l even 12 2 2268.2.j.a 2
60.l odd 4 1 336.2.a.f 1
60.l odd 4 1 8400.2.a.e 1
105.k odd 4 1 588.2.a.d 1
105.w odd 12 2 588.2.i.d 2
105.x even 12 2 588.2.i.e 2
120.q odd 4 1 1344.2.a.a 1
120.w even 4 1 1344.2.a.k 1
140.j odd 4 1 7056.2.a.cd 1
240.z odd 4 1 5376.2.c.p 2
240.bb even 4 1 5376.2.c.q 2
240.bd odd 4 1 5376.2.c.p 2
240.bf even 4 1 5376.2.c.q 2
420.w even 4 1 2352.2.a.a 1
420.bp odd 12 2 2352.2.q.b 2
420.br even 12 2 2352.2.q.z 2
840.bm even 4 1 9408.2.a.df 1
840.bp odd 4 1 9408.2.a.bn 1

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

## Hecke kernels

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

 $$T_{11} + 2$$ $$T_{13}^{2} + 36$$ $$T_{17}^{2} + 16$$ $$T_{41}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ 1
$7$ $$1 + T^{2}$$
$11$ $$( 1 + 2 T + 11 T^{2} )^{2}$$
$13$ $$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$
$17$ $$1 - 18 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 4 T + 19 T^{2} )^{2}$$
$23$ $$1 - 42 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 2 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 31 T^{2} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$1 - 70 T^{2} + 1849 T^{4}$$
$47$ $$1 + 50 T^{2} + 2209 T^{4}$$
$53$ $$1 - 70 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 8 T + 59 T^{2} )^{2}$$
$61$ $$( 1 - 6 T + 61 T^{2} )^{2}$$
$67$ $$1 - 70 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 14 T + 71 T^{2} )^{2}$$
$73$ $$1 - 142 T^{2} + 5329 T^{4}$$
$79$ $$( 1 + 12 T + 79 T^{2} )^{2}$$
$83$ $$1 - 150 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 89 T^{2} )^{2}$$
$97$ $$1 - 190 T^{2} + 9409 T^{4}$$