# Properties

 Label 6300.2.k.h 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: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2100) 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 $$q - i q^{7} - q^{11} + 4 i q^{13} + 2 i q^{17} + 4 q^{19} - 7 i q^{23} - 9 q^{29} - 2 q^{31} - i q^{37} - 8 q^{41} - 9 i q^{43} + 4 i q^{47} - q^{49} - 6 i q^{53} + 4 q^{59} + 4 q^{61} - 9 i q^{67} - 5 q^{71} + 10 i q^{73} + i q^{77} + 15 q^{79} + 6 i q^{83} + 8 q^{89} + 4 q^{91} - 10 i q^{97} +O(q^{100})$$ q - i * q^7 - q^11 + 4*i * q^13 + 2*i * q^17 + 4 * q^19 - 7*i * q^23 - 9 * q^29 - 2 * q^31 - i * q^37 - 8 * q^41 - 9*i * q^43 + 4*i * q^47 - q^49 - 6*i * q^53 + 4 * q^59 + 4 * q^61 - 9*i * q^67 - 5 * q^71 + 10*i * q^73 + i * q^77 + 15 * q^79 + 6*i * q^83 + 8 * q^89 + 4 * q^91 - 10*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 2 q^{11} + 8 q^{19} - 18 q^{29} - 4 q^{31} - 16 q^{41} - 2 q^{49} + 8 q^{59} + 8 q^{61} - 10 q^{71} + 30 q^{79} + 16 q^{89} + 8 q^{91}+O(q^{100})$$ 2 * q - 2 * q^11 + 8 * q^19 - 18 * q^29 - 4 * q^31 - 16 * q^41 - 2 * q^49 + 8 * q^59 + 8 * q^61 - 10 * q^71 + 30 * q^79 + 16 * q^89 + 8 * q^91

## 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.h 2
3.b odd 2 1 2100.2.k.f 2
5.b even 2 1 inner 6300.2.k.h 2
5.c odd 4 1 6300.2.a.f 1
5.c odd 4 1 6300.2.a.x 1
15.d odd 2 1 2100.2.k.f 2
15.e even 4 1 2100.2.a.g 1
15.e even 4 1 2100.2.a.m yes 1
60.l odd 4 1 8400.2.a.x 1
60.l odd 4 1 8400.2.a.bv 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.a.g 1 15.e even 4 1
2100.2.a.m yes 1 15.e even 4 1
2100.2.k.f 2 3.b odd 2 1
2100.2.k.f 2 15.d odd 2 1
6300.2.a.f 1 5.c odd 4 1
6300.2.a.x 1 5.c odd 4 1
6300.2.k.h 2 1.a even 1 1 trivial
6300.2.k.h 2 5.b even 2 1 inner
8400.2.a.x 1 60.l odd 4 1
8400.2.a.bv 1 60.l odd 4 1

## Hecke kernels

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

 $$T_{11} + 1$$ T11 + 1 $$T_{13}^{2} + 16$$ T13^2 + 16 $$T_{17}^{2} + 4$$ T17^2 + 4 $$T_{41} + 8$$ T41 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 4$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} + 49$$
$29$ $$(T + 9)^{2}$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$(T + 8)^{2}$$
$43$ $$T^{2} + 81$$
$47$ $$T^{2} + 16$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 4)^{2}$$
$61$ $$(T - 4)^{2}$$
$67$ $$T^{2} + 81$$
$71$ $$(T + 5)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 15)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T - 8)^{2}$$
$97$ $$T^{2} + 100$$