# Properties

 Label 6300.2.k.a 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 420) 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.

 $$f(q)$$ $$=$$ $$q -i q^{7} +O(q^{10})$$ $$q -i q^{7} -6 q^{11} -4 i q^{13} + 6 i q^{17} -2 q^{19} + 6 q^{29} -10 q^{31} -2 i q^{37} + 6 q^{41} -4 i q^{43} - q^{49} + 12 i q^{53} + 14 q^{61} + 4 i q^{67} -6 q^{71} -4 i q^{73} + 6 i q^{77} + 16 q^{79} + 12 i q^{83} + 6 q^{89} -4 q^{91} + 16 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 12q^{11} - 4q^{19} + 12q^{29} - 20q^{31} + 12q^{41} - 2q^{49} + 28q^{61} - 12q^{71} + 32q^{79} + 12q^{89} - 8q^{91} + O(q^{100})$$

## 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.a 2
3.b odd 2 1 2100.2.k.j 2
5.b even 2 1 inner 6300.2.k.a 2
5.c odd 4 1 1260.2.a.i 1
5.c odd 4 1 6300.2.a.a 1
15.d odd 2 1 2100.2.k.j 2
15.e even 4 1 420.2.a.c 1
15.e even 4 1 2100.2.a.d 1
20.e even 4 1 5040.2.a.bc 1
35.f even 4 1 8820.2.a.b 1
60.l odd 4 1 1680.2.a.a 1
60.l odd 4 1 8400.2.a.cj 1
105.k odd 4 1 2940.2.a.f 1
105.w odd 12 2 2940.2.q.i 2
105.x even 12 2 2940.2.q.e 2
120.q odd 4 1 6720.2.a.ch 1
120.w even 4 1 6720.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.a.c 1 15.e even 4 1
1260.2.a.i 1 5.c odd 4 1
1680.2.a.a 1 60.l odd 4 1
2100.2.a.d 1 15.e even 4 1
2100.2.k.j 2 3.b odd 2 1
2100.2.k.j 2 15.d odd 2 1
2940.2.a.f 1 105.k odd 4 1
2940.2.q.e 2 105.x even 12 2
2940.2.q.i 2 105.w odd 12 2
5040.2.a.bc 1 20.e even 4 1
6300.2.a.a 1 5.c odd 4 1
6300.2.k.a 2 1.a even 1 1 trivial
6300.2.k.a 2 5.b even 2 1 inner
6720.2.a.x 1 120.w even 4 1
6720.2.a.ch 1 120.q odd 4 1
8400.2.a.cj 1 60.l odd 4 1
8820.2.a.b 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 $$S_{2}^{\mathrm{new}}(6300, [\chi])$$:

 $$T_{11} + 6$$ $$T_{13}^{2} + 16$$ $$T_{17}^{2} + 36$$ $$T_{41} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 6 + T )^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$36 + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$( 10 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$144 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -14 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$( 6 + T )^{2}$$
$73$ $$16 + T^{2}$$
$79$ $$( -16 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$256 + T^{2}$$