# Properties

 Label 6300.2.k.p 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 140) 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.

 $$f(q)$$ $$=$$ $$q -i q^{7} +O(q^{10})$$ $$q -i q^{7} + 5 q^{11} + 3 i q^{13} + i q^{17} -6 q^{19} + 6 i q^{23} -9 q^{29} -4 q^{31} + 2 i q^{37} + 4 q^{41} -10 i q^{43} + i q^{47} - q^{49} + 4 i q^{53} -8 q^{59} -8 q^{61} + 12 i q^{67} -8 q^{71} -2 i q^{73} -5 i q^{77} -13 q^{79} -4 i q^{83} + 4 q^{89} + 3 q^{91} -13 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 10q^{11} - 12q^{19} - 18q^{29} - 8q^{31} + 8q^{41} - 2q^{49} - 16q^{59} - 16q^{61} - 16q^{71} - 26q^{79} + 8q^{89} + 6q^{91} + O(q^{100})$$

## 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.p 2
3.b odd 2 1 700.2.e.a 2
5.b even 2 1 inner 6300.2.k.p 2
5.c odd 4 1 1260.2.a.h 1
5.c odd 4 1 6300.2.a.bf 1
12.b even 2 1 2800.2.g.c 2
15.d odd 2 1 700.2.e.a 2
15.e even 4 1 140.2.a.b 1
15.e even 4 1 700.2.a.b 1
20.e even 4 1 5040.2.a.bd 1
21.c even 2 1 4900.2.e.a 2
35.f even 4 1 8820.2.a.n 1
60.h even 2 1 2800.2.g.c 2
60.l odd 4 1 560.2.a.a 1
60.l odd 4 1 2800.2.a.be 1
105.g even 2 1 4900.2.e.a 2
105.k odd 4 1 980.2.a.b 1
105.k odd 4 1 4900.2.a.u 1
105.w odd 12 2 980.2.i.j 2
105.x even 12 2 980.2.i.b 2
120.q odd 4 1 2240.2.a.bb 1
120.w even 4 1 2240.2.a.c 1
420.w even 4 1 3920.2.a.bl 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.a.b 1 15.e even 4 1
560.2.a.a 1 60.l odd 4 1
700.2.a.b 1 15.e even 4 1
700.2.e.a 2 3.b odd 2 1
700.2.e.a 2 15.d odd 2 1
980.2.a.b 1 105.k odd 4 1
980.2.i.b 2 105.x even 12 2
980.2.i.j 2 105.w odd 12 2
1260.2.a.h 1 5.c odd 4 1
2240.2.a.c 1 120.w even 4 1
2240.2.a.bb 1 120.q odd 4 1
2800.2.a.be 1 60.l odd 4 1
2800.2.g.c 2 12.b even 2 1
2800.2.g.c 2 60.h even 2 1
3920.2.a.bl 1 420.w even 4 1
4900.2.a.u 1 105.k odd 4 1
4900.2.e.a 2 21.c even 2 1
4900.2.e.a 2 105.g even 2 1
5040.2.a.bd 1 20.e even 4 1
6300.2.a.bf 1 5.c odd 4 1
6300.2.k.p 2 1.a even 1 1 trivial
6300.2.k.p 2 5.b even 2 1 inner
8820.2.a.n 1 35.f even 4 1

## Hecke kernels

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

 $$T_{11} - 5$$ $$T_{13}^{2} + 9$$ $$T_{17}^{2} + 1$$ $$T_{41} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( -5 + T )^{2}$$
$13$ $$9 + T^{2}$$
$17$ $$1 + T^{2}$$
$19$ $$( 6 + T )^{2}$$
$23$ $$36 + T^{2}$$
$29$ $$( 9 + T )^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( -4 + T )^{2}$$
$43$ $$100 + T^{2}$$
$47$ $$1 + T^{2}$$
$53$ $$16 + T^{2}$$
$59$ $$( 8 + T )^{2}$$
$61$ $$( 8 + T )^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$( 13 + T )^{2}$$
$83$ $$16 + T^{2}$$
$89$ $$( -4 + T )^{2}$$
$97$ $$169 + T^{2}$$