# Properties

 Label 300.3.g.e Level $300$ Weight $3$ Character orbit 300.g Analytic conductor $8.174$ Analytic rank $0$ Dimension $2$ CM no 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: no Analytic conductor: $$8.17440793081$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ Defining polynomial: $$x^{2} + 5$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + \beta ) q^{3} + 8 q^{7} + ( -1 - 4 \beta ) q^{9} +O(q^{10})$$ $$q + ( -2 + \beta ) q^{3} + 8 q^{7} + ( -1 - 4 \beta ) q^{9} -4 \beta q^{11} + 12 q^{13} + 14 \beta q^{17} + 6 q^{19} + ( -16 + 8 \beta ) q^{21} + 2 \beta q^{23} + ( 22 + 7 \beta ) q^{27} + 12 \beta q^{29} + 34 q^{31} + ( 20 + 8 \beta ) q^{33} + 44 q^{37} + ( -24 + 12 \beta ) q^{39} + 8 \beta q^{41} -28 q^{43} -2 \beta q^{47} + 15 q^{49} + ( -70 - 28 \beta ) q^{51} + 18 \beta q^{53} + ( -12 + 6 \beta ) q^{57} -44 \beta q^{59} + 74 q^{61} + ( -8 - 32 \beta ) q^{63} -92 q^{67} + ( -10 - 4 \beta ) q^{69} + 24 \beta q^{71} + 56 q^{73} -32 \beta q^{77} + 78 q^{79} + ( -79 + 8 \beta ) q^{81} -46 \beta q^{83} + ( -60 - 24 \beta ) q^{87} + 8 \beta q^{89} + 96 q^{91} + ( -68 + 34 \beta ) q^{93} -32 q^{97} + ( -80 + 4 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{3} + 16q^{7} - 2q^{9} + O(q^{10})$$ $$2q - 4q^{3} + 16q^{7} - 2q^{9} + 24q^{13} + 12q^{19} - 32q^{21} + 44q^{27} + 68q^{31} + 40q^{33} + 88q^{37} - 48q^{39} - 56q^{43} + 30q^{49} - 140q^{51} - 24q^{57} + 148q^{61} - 16q^{63} - 184q^{67} - 20q^{69} + 112q^{73} + 156q^{79} - 158q^{81} - 120q^{87} + 192q^{91} - 136q^{93} - 64q^{97} - 160q^{99} + O(q^{100})$$

## 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
 − 2.23607i 2.23607i
0 −2.00000 2.23607i 0 0 0 8.00000 0 −1.00000 + 8.94427i 0
101.2 0 −2.00000 + 2.23607i 0 0 0 8.00000 0 −1.00000 8.94427i 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.3.g.e 2
3.b odd 2 1 inner 300.3.g.e 2
4.b odd 2 1 1200.3.l.q 2
5.b even 2 1 300.3.g.h 2
5.c odd 4 2 60.3.b.a 4
12.b even 2 1 1200.3.l.q 2
15.d odd 2 1 300.3.g.h 2
15.e even 4 2 60.3.b.a 4
20.d odd 2 1 1200.3.l.h 2
20.e even 4 2 240.3.c.d 4
40.i odd 4 2 960.3.c.h 4
40.k even 4 2 960.3.c.g 4
45.k odd 12 4 1620.3.t.b 8
45.l even 12 4 1620.3.t.b 8
60.h even 2 1 1200.3.l.h 2
60.l odd 4 2 240.3.c.d 4
120.q odd 4 2 960.3.c.g 4
120.w even 4 2 960.3.c.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.b.a 4 5.c odd 4 2
60.3.b.a 4 15.e even 4 2
240.3.c.d 4 20.e even 4 2
240.3.c.d 4 60.l odd 4 2
300.3.g.e 2 1.a even 1 1 trivial
300.3.g.e 2 3.b odd 2 1 inner
300.3.g.h 2 5.b even 2 1
300.3.g.h 2 15.d odd 2 1
960.3.c.g 4 40.k even 4 2
960.3.c.g 4 120.q odd 4 2
960.3.c.h 4 40.i odd 4 2
960.3.c.h 4 120.w even 4 2
1200.3.l.h 2 20.d odd 2 1
1200.3.l.h 2 60.h even 2 1
1200.3.l.q 2 4.b odd 2 1
1200.3.l.q 2 12.b even 2 1
1620.3.t.b 8 45.k odd 12 4
1620.3.t.b 8 45.l even 12 4

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(300, [\chi])$$:

 $$T_{7} - 8$$ $$T_{11}^{2} + 80$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + 4 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -8 + T )^{2}$$
$11$ $$80 + T^{2}$$
$13$ $$( -12 + T )^{2}$$
$17$ $$980 + T^{2}$$
$19$ $$( -6 + T )^{2}$$
$23$ $$20 + T^{2}$$
$29$ $$720 + T^{2}$$
$31$ $$( -34 + T )^{2}$$
$37$ $$( -44 + T )^{2}$$
$41$ $$320 + T^{2}$$
$43$ $$( 28 + T )^{2}$$
$47$ $$20 + T^{2}$$
$53$ $$1620 + T^{2}$$
$59$ $$9680 + T^{2}$$
$61$ $$( -74 + T )^{2}$$
$67$ $$( 92 + T )^{2}$$
$71$ $$2880 + T^{2}$$
$73$ $$( -56 + T )^{2}$$
$79$ $$( -78 + T )^{2}$$
$83$ $$10580 + T^{2}$$
$89$ $$320 + T^{2}$$
$97$ $$( 32 + T )^{2}$$
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