# Properties

 Label 300.3.g.a Level $300$ Weight $3$ Character orbit 300.g Self dual yes Analytic conductor $8.174$ Analytic rank $0$ Dimension $1$ CM discriminant -3 Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 300.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.17440793081$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3q^{3} - 13q^{7} + 9q^{9} + O(q^{10})$$ $$q - 3q^{3} - 13q^{7} + 9q^{9} + 23q^{13} + 11q^{19} + 39q^{21} - 27q^{27} + 59q^{31} + 26q^{37} - 69q^{39} + 83q^{43} + 120q^{49} - 33q^{57} - 121q^{61} - 117q^{63} - 13q^{67} - 46q^{73} - 142q^{79} + 81q^{81} - 299q^{91} - 177q^{93} + 167q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 0
0 −3.00000 0 0 0 −13.0000 0 9.00000 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.3.g.a 1
3.b odd 2 1 CM 300.3.g.a 1
4.b odd 2 1 1200.3.l.e 1
5.b even 2 1 300.3.g.c yes 1
5.c odd 4 2 300.3.b.b 2
12.b even 2 1 1200.3.l.e 1
15.d odd 2 1 300.3.g.c yes 1
15.e even 4 2 300.3.b.b 2
20.d odd 2 1 1200.3.l.a 1
20.e even 4 2 1200.3.c.b 2
60.h even 2 1 1200.3.l.a 1
60.l odd 4 2 1200.3.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.b.b 2 5.c odd 4 2
300.3.b.b 2 15.e even 4 2
300.3.g.a 1 1.a even 1 1 trivial
300.3.g.a 1 3.b odd 2 1 CM
300.3.g.c yes 1 5.b even 2 1
300.3.g.c yes 1 15.d odd 2 1
1200.3.c.b 2 20.e even 4 2
1200.3.c.b 2 60.l odd 4 2
1200.3.l.a 1 20.d odd 2 1
1200.3.l.a 1 60.h even 2 1
1200.3.l.e 1 4.b odd 2 1
1200.3.l.e 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 $$S_{3}^{\mathrm{new}}(300, [\chi])$$:

 $$T_{7} + 13$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$3 + T$$
$5$ $$T$$
$7$ $$13 + T$$
$11$ $$T$$
$13$ $$-23 + T$$
$17$ $$T$$
$19$ $$-11 + T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$-59 + T$$
$37$ $$-26 + T$$
$41$ $$T$$
$43$ $$-83 + T$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$121 + T$$
$67$ $$13 + T$$
$71$ $$T$$
$73$ $$46 + T$$
$79$ $$142 + T$$
$83$ $$T$$
$89$ $$T$$
$97$ $$-167 + T$$
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