# Properties

 Label 6300.2.k.d 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 700) 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.

 $$f(q)$$ $$=$$ $$q -i q^{7} +O(q^{10})$$ $$q -i q^{7} -3 q^{11} -4 i q^{13} -2 q^{19} + 3 i q^{23} + 9 q^{29} + 8 q^{31} -5 i q^{37} + 6 q^{41} + 11 i q^{43} + 6 i q^{47} - q^{49} -6 i q^{53} -10 q^{61} -5 i q^{67} -15 q^{71} -10 i q^{73} + 3 i q^{77} + 7 q^{79} -12 i q^{83} -12 q^{89} -4 q^{91} -8 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + O(q^{10})$$ $$2 q - 6 q^{11} - 4 q^{19} + 18 q^{29} + 16 q^{31} + 12 q^{41} - 2 q^{49} - 20 q^{61} - 30 q^{71} + 14 q^{79} - 24 q^{89} - 8 q^{91} + O(q^{100})$$

## 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.d 2
3.b odd 2 1 700.2.e.b 2
5.b even 2 1 inner 6300.2.k.d 2
5.c odd 4 1 6300.2.a.e 1
5.c odd 4 1 6300.2.a.s 1
12.b even 2 1 2800.2.g.e 2
15.d odd 2 1 700.2.e.b 2
15.e even 4 1 700.2.a.c 1
15.e even 4 1 700.2.a.i yes 1
21.c even 2 1 4900.2.e.g 2
60.h even 2 1 2800.2.g.e 2
60.l odd 4 1 2800.2.a.e 1
60.l odd 4 1 2800.2.a.ba 1
105.g even 2 1 4900.2.e.g 2
105.k odd 4 1 4900.2.a.f 1
105.k odd 4 1 4900.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.a.c 1 15.e even 4 1
700.2.a.i yes 1 15.e even 4 1
700.2.e.b 2 3.b odd 2 1
700.2.e.b 2 15.d odd 2 1
2800.2.a.e 1 60.l odd 4 1
2800.2.a.ba 1 60.l odd 4 1
2800.2.g.e 2 12.b even 2 1
2800.2.g.e 2 60.h even 2 1
4900.2.a.f 1 105.k odd 4 1
4900.2.a.t 1 105.k odd 4 1
4900.2.e.g 2 21.c even 2 1
4900.2.e.g 2 105.g even 2 1
6300.2.a.e 1 5.c odd 4 1
6300.2.a.s 1 5.c odd 4 1
6300.2.k.d 2 1.a even 1 1 trivial
6300.2.k.d 2 5.b even 2 1 inner

## Hecke kernels

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

 $$T_{11} + 3$$ $$T_{13}^{2} + 16$$ $$T_{17}$$ $$T_{41} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 3 + T )^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$9 + T^{2}$$
$29$ $$( -9 + T )^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$25 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$121 + T^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( 10 + T )^{2}$$
$67$ $$25 + T^{2}$$
$71$ $$( 15 + T )^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( -7 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( 12 + T )^{2}$$
$97$ $$64 + T^{2}$$