# Properties

 Label 6300.2.k.f Level $6300$ Weight $2$ Character orbit 6300.k Analytic conductor $50.306$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

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## 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: $$1$$ 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 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 $$q - i q^{7} - 2 q^{11} - 4 i q^{13} - 2 i q^{17} + 2 q^{19} + 4 i q^{23} - 2 q^{29} - 6 q^{31} - 6 i q^{37} - 6 q^{41} + 4 i q^{43} - q^{49} + 8 i q^{53} - 10 q^{61} - 12 i q^{67} + 14 q^{71} - 4 i q^{73} + 2 i q^{77} + 8 q^{79} + 12 i q^{83} - 14 q^{89} - 4 q^{91} + 8 i q^{97} +O(q^{100})$$ q - i * q^7 - 2 * q^11 - 4*i * q^13 - 2*i * q^17 + 2 * q^19 + 4*i * q^23 - 2 * q^29 - 6 * q^31 - 6*i * q^37 - 6 * q^41 + 4*i * q^43 - q^49 + 8*i * q^53 - 10 * q^61 - 12*i * q^67 + 14 * q^71 - 4*i * q^73 + 2*i * q^77 + 8 * q^79 + 12*i * q^83 - 14 * q^89 - 4 * q^91 + 8*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 4 q^{11} + 4 q^{19} - 4 q^{29} - 12 q^{31} - 12 q^{41} - 2 q^{49} - 20 q^{61} + 28 q^{71} + 16 q^{79} - 28 q^{89} - 8 q^{91}+O(q^{100})$$ 2 * q - 4 * q^11 + 4 * q^19 - 4 * q^29 - 12 * q^31 - 12 * q^41 - 2 * q^49 - 20 * q^61 + 28 * q^71 + 16 * q^79 - 28 * 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.f 2
3.b odd 2 1 2100.2.k.h 2
5.b even 2 1 inner 6300.2.k.f 2
5.c odd 4 1 1260.2.a.a 1
5.c odd 4 1 6300.2.a.u 1
15.d odd 2 1 2100.2.k.h 2
15.e even 4 1 420.2.a.d 1
15.e even 4 1 2100.2.a.h 1
20.e even 4 1 5040.2.a.r 1
35.f even 4 1 8820.2.a.s 1
60.l odd 4 1 1680.2.a.i 1
60.l odd 4 1 8400.2.a.bq 1
105.k odd 4 1 2940.2.a.c 1
105.w odd 12 2 2940.2.q.l 2
105.x even 12 2 2940.2.q.b 2
120.q odd 4 1 6720.2.a.bs 1
120.w even 4 1 6720.2.a.e 1

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

 $$T_{11} + 2$$ T11 + 2 $$T_{13}^{2} + 16$$ T13^2 + 16 $$T_{17}^{2} + 4$$ T17^2 + 4 $$T_{41} + 6$$ T41 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 4$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T + 2)^{2}$$
$31$ $$(T + 6)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 64$$
$59$ $$T^{2}$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$(T - 14)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T + 14)^{2}$$
$97$ $$T^{2} + 64$$
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