# Properties

 Label 300.3.g.d Level $300$ Weight $3$ Character orbit 300.g Analytic conductor $8.174$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## 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.

 $$f(q)$$ $$=$$ $$q + ( -2 + \beta ) q^{3} -2 q^{7} + ( -1 - 4 \beta ) q^{9} +O(q^{10})$$ $$q + ( -2 + \beta ) q^{3} -2 q^{7} + ( -1 - 4 \beta ) q^{9} + 6 \beta q^{11} -8 q^{13} -6 \beta q^{17} -34 q^{19} + ( 4 - 2 \beta ) q^{21} -18 \beta q^{23} + ( 22 + 7 \beta ) q^{27} -18 \beta q^{29} + 14 q^{31} + ( -30 - 12 \beta ) q^{33} -56 q^{37} + ( 16 - 8 \beta ) q^{39} -12 \beta q^{41} -8 q^{43} + 18 \beta q^{47} -45 q^{49} + ( 30 + 12 \beta ) q^{51} + 18 \beta q^{53} + ( 68 - 34 \beta ) q^{57} + 6 \beta q^{59} -46 q^{61} + ( 2 + 8 \beta ) q^{63} -32 q^{67} + ( 90 + 36 \beta ) q^{69} + 24 \beta q^{71} + 106 q^{73} -12 \beta q^{77} -22 q^{79} + ( -79 + 8 \beta ) q^{81} + 54 \beta q^{83} + ( 90 + 36 \beta ) q^{87} + 48 \beta q^{89} + 16 q^{91} + ( -28 + 14 \beta ) q^{93} -122 q^{97} + ( 120 - 6 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{3} - 4q^{7} - 2q^{9} + O(q^{10})$$ $$2q - 4q^{3} - 4q^{7} - 2q^{9} - 16q^{13} - 68q^{19} + 8q^{21} + 44q^{27} + 28q^{31} - 60q^{33} - 112q^{37} + 32q^{39} - 16q^{43} - 90q^{49} + 60q^{51} + 136q^{57} - 92q^{61} + 4q^{63} - 64q^{67} + 180q^{69} + 212q^{73} - 44q^{79} - 158q^{81} + 180q^{87} + 32q^{91} - 56q^{93} - 244q^{97} + 240q^{99} + O(q^{100})$$

## 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 −2.00000 0 −1.00000 + 8.94427i 0
101.2 0 −2.00000 + 2.23607i 0 0 0 −2.00000 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.d 2
3.b odd 2 1 inner 300.3.g.d 2
4.b odd 2 1 1200.3.l.r 2
5.b even 2 1 60.3.g.a 2
5.c odd 4 2 300.3.b.c 4
12.b even 2 1 1200.3.l.r 2
15.d odd 2 1 60.3.g.a 2
15.e even 4 2 300.3.b.c 4
20.d odd 2 1 240.3.l.a 2
20.e even 4 2 1200.3.c.e 4
40.e odd 2 1 960.3.l.d 2
40.f even 2 1 960.3.l.a 2
45.h odd 6 2 1620.3.o.b 4
45.j even 6 2 1620.3.o.b 4
60.h even 2 1 240.3.l.a 2
60.l odd 4 2 1200.3.c.e 4
120.i odd 2 1 960.3.l.a 2
120.m even 2 1 960.3.l.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.g.a 2 5.b even 2 1
60.3.g.a 2 15.d odd 2 1
240.3.l.a 2 20.d odd 2 1
240.3.l.a 2 60.h even 2 1
300.3.b.c 4 5.c odd 4 2
300.3.b.c 4 15.e even 4 2
300.3.g.d 2 1.a even 1 1 trivial
300.3.g.d 2 3.b odd 2 1 inner
960.3.l.a 2 40.f even 2 1
960.3.l.a 2 120.i odd 2 1
960.3.l.d 2 40.e odd 2 1
960.3.l.d 2 120.m even 2 1
1200.3.c.e 4 20.e even 4 2
1200.3.c.e 4 60.l odd 4 2
1200.3.l.r 2 4.b odd 2 1
1200.3.l.r 2 12.b even 2 1
1620.3.o.b 4 45.h odd 6 2
1620.3.o.b 4 45.j even 6 2

## Hecke kernels

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

 $$T_{7} + 2$$ $$T_{11}^{2} + 180$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + 4 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 2 + T )^{2}$$
$11$ $$180 + T^{2}$$
$13$ $$( 8 + T )^{2}$$
$17$ $$180 + T^{2}$$
$19$ $$( 34 + T )^{2}$$
$23$ $$1620 + T^{2}$$
$29$ $$1620 + T^{2}$$
$31$ $$( -14 + T )^{2}$$
$37$ $$( 56 + T )^{2}$$
$41$ $$720 + T^{2}$$
$43$ $$( 8 + T )^{2}$$
$47$ $$1620 + T^{2}$$
$53$ $$1620 + T^{2}$$
$59$ $$180 + T^{2}$$
$61$ $$( 46 + T )^{2}$$
$67$ $$( 32 + T )^{2}$$
$71$ $$2880 + T^{2}$$
$73$ $$( -106 + T )^{2}$$
$79$ $$( 22 + T )^{2}$$
$83$ $$14580 + T^{2}$$
$89$ $$11520 + T^{2}$$
$97$ $$( 122 + T )^{2}$$