# Properties

 Label 6300.2.a.bc Level $6300$ Weight $2$ Character orbit 6300.a Self dual yes Analytic conductor $50.306$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6300.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$50.3057532734$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 420) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{7} + O(q^{10})$$ $$q + q^{7} + 4 q^{11} - 6 q^{13} - 2 q^{17} + 6 q^{19} - 2 q^{23} - 6 q^{29} - 2 q^{31} - 4 q^{37} - 8 q^{41} - 4 q^{43} - 4 q^{47} + q^{49} - 6 q^{53} - 4 q^{59} + 14 q^{61} + 4 q^{67} - 10 q^{73} + 4 q^{77} + 16 q^{83} - 8 q^{89} - 6 q^{91} + 10 q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 0 0 1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6300.2.a.bc 1
3.b odd 2 1 2100.2.a.e 1
5.b even 2 1 6300.2.a.n 1
5.c odd 4 2 1260.2.k.d 2
12.b even 2 1 8400.2.a.cd 1
15.d odd 2 1 2100.2.a.j 1
15.e even 4 2 420.2.k.a 2
20.e even 4 2 5040.2.t.o 2
60.h even 2 1 8400.2.a.bh 1
60.l odd 4 2 1680.2.t.a 2
105.k odd 4 2 2940.2.k.d 2
105.w odd 12 4 2940.2.bb.c 4
105.x even 12 4 2940.2.bb.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.k.a 2 15.e even 4 2
1260.2.k.d 2 5.c odd 4 2
1680.2.t.a 2 60.l odd 4 2
2100.2.a.e 1 3.b odd 2 1
2100.2.a.j 1 15.d odd 2 1
2940.2.k.d 2 105.k odd 4 2
2940.2.bb.c 4 105.w odd 12 4
2940.2.bb.h 4 105.x even 12 4
5040.2.t.o 2 20.e even 4 2
6300.2.a.n 1 5.b even 2 1
6300.2.a.bc 1 1.a even 1 1 trivial
8400.2.a.bh 1 60.h even 2 1
8400.2.a.cd 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6300))$$:

 $$T_{11} - 4$$ $$T_{13} + 6$$ $$T_{17} + 2$$ $$T_{37} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-1 + T$$
$11$ $$-4 + T$$
$13$ $$6 + T$$
$17$ $$2 + T$$
$19$ $$-6 + T$$
$23$ $$2 + T$$
$29$ $$6 + T$$
$31$ $$2 + T$$
$37$ $$4 + T$$
$41$ $$8 + T$$
$43$ $$4 + T$$
$47$ $$4 + T$$
$53$ $$6 + T$$
$59$ $$4 + T$$
$61$ $$-14 + T$$
$67$ $$-4 + T$$
$71$ $$T$$
$73$ $$10 + T$$
$79$ $$T$$
$83$ $$-16 + T$$
$89$ $$8 + T$$
$97$ $$-10 + T$$