# Properties

 Label 6300.2.a.d 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 140) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{7} + O(q^{10})$$ $$q - q^{7} - 3q^{11} + q^{13} - 3q^{17} + 2q^{19} - 6q^{23} + 9q^{29} + 8q^{31} + 10q^{37} - 2q^{43} - 3q^{47} + q^{49} - 12q^{59} + 8q^{61} - 8q^{67} - 14q^{73} + 3q^{77} + 5q^{79} - 12q^{83} - 12q^{89} - q^{91} - 17q^{97} + O(q^{100})$$

## 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.d 1
3.b odd 2 1 700.2.a.d 1
5.b even 2 1 1260.2.a.c 1
5.c odd 4 2 6300.2.k.c 2
12.b even 2 1 2800.2.a.y 1
15.d odd 2 1 140.2.a.a 1
15.e even 4 2 700.2.e.c 2
20.d odd 2 1 5040.2.a.h 1
21.c even 2 1 4900.2.a.p 1
35.c odd 2 1 8820.2.a.r 1
60.h even 2 1 560.2.a.c 1
60.l odd 4 2 2800.2.g.j 2
105.g even 2 1 980.2.a.c 1
105.k odd 4 2 4900.2.e.l 2
105.o odd 6 2 980.2.i.d 2
105.p even 6 2 980.2.i.h 2
120.i odd 2 1 2240.2.a.g 1
120.m even 2 1 2240.2.a.r 1
420.o odd 2 1 3920.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.a.a 1 15.d odd 2 1
560.2.a.c 1 60.h even 2 1
700.2.a.d 1 3.b odd 2 1
700.2.e.c 2 15.e even 4 2
980.2.a.c 1 105.g even 2 1
980.2.i.d 2 105.o odd 6 2
980.2.i.h 2 105.p even 6 2
1260.2.a.c 1 5.b even 2 1
2240.2.a.g 1 120.i odd 2 1
2240.2.a.r 1 120.m even 2 1
2800.2.a.y 1 12.b even 2 1
2800.2.g.j 2 60.l odd 4 2
3920.2.a.u 1 420.o odd 2 1
4900.2.a.p 1 21.c even 2 1
4900.2.e.l 2 105.k odd 4 2
5040.2.a.h 1 20.d odd 2 1
6300.2.a.d 1 1.a even 1 1 trivial
6300.2.k.c 2 5.c odd 4 2
8820.2.a.r 1 35.c odd 2 1

## Hecke kernels

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

 $$T_{11} + 3$$ $$T_{13} - 1$$ $$T_{17} + 3$$ $$T_{37} - 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$1 + T$$
$11$ $$3 + T$$
$13$ $$-1 + T$$
$17$ $$3 + T$$
$19$ $$-2 + T$$
$23$ $$6 + T$$
$29$ $$-9 + T$$
$31$ $$-8 + T$$
$37$ $$-10 + T$$
$41$ $$T$$
$43$ $$2 + T$$
$47$ $$3 + T$$
$53$ $$T$$
$59$ $$12 + T$$
$61$ $$-8 + T$$
$67$ $$8 + T$$
$71$ $$T$$
$73$ $$14 + T$$
$79$ $$-5 + T$$
$83$ $$12 + T$$
$89$ $$12 + T$$
$97$ $$17 + T$$