# Properties

 Label 300.2.i.b Level $300$ Weight $2$ Character orbit 300.i Analytic conductor $2.396$ Analytic rank $0$ Dimension $4$ CM discriminant -3 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.i (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.39551206064$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + 2 \beta_{3} q^{7} + 3 \beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + 2 \beta_{3} q^{7} + 3 \beta_{2} q^{9} + 4 \beta_{1} q^{13} -8 \beta_{2} q^{19} -6 q^{21} + 3 \beta_{3} q^{27} + 4 q^{31} -4 \beta_{3} q^{37} + 12 \beta_{2} q^{39} -6 \beta_{1} q^{43} -5 \beta_{2} q^{49} -8 \beta_{3} q^{57} + 14 q^{61} -6 \beta_{1} q^{63} + 2 \beta_{3} q^{67} -8 \beta_{1} q^{73} -4 \beta_{2} q^{79} -9 q^{81} -24 q^{91} + 4 \beta_{1} q^{93} -8 \beta_{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 24q^{21} + 16q^{31} + 56q^{61} - 36q^{81} - 96q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3}$$

## Character values

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

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 −1.22474 + 1.22474i 1.22474 − 1.22474i −1.22474 − 1.22474i 1.22474 + 1.22474i
0 −1.22474 + 1.22474i 0 0 0 2.44949 + 2.44949i 0 3.00000i 0
257.2 0 1.22474 1.22474i 0 0 0 −2.44949 2.44949i 0 3.00000i 0
293.1 0 −1.22474 1.22474i 0 0 0 2.44949 2.44949i 0 3.00000i 0
293.2 0 1.22474 + 1.22474i 0 0 0 −2.44949 + 2.44949i 0 3.00000i 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 $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.i.b 4
3.b odd 2 1 CM 300.2.i.b 4
4.b odd 2 1 1200.2.v.e 4
5.b even 2 1 inner 300.2.i.b 4
5.c odd 4 2 inner 300.2.i.b 4
12.b even 2 1 1200.2.v.e 4
15.d odd 2 1 inner 300.2.i.b 4
15.e even 4 2 inner 300.2.i.b 4
20.d odd 2 1 1200.2.v.e 4
20.e even 4 2 1200.2.v.e 4
60.h even 2 1 1200.2.v.e 4
60.l odd 4 2 1200.2.v.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.i.b 4 1.a even 1 1 trivial
300.2.i.b 4 3.b odd 2 1 CM
300.2.i.b 4 5.b even 2 1 inner
300.2.i.b 4 5.c odd 4 2 inner
300.2.i.b 4 15.d odd 2 1 inner
300.2.i.b 4 15.e even 4 2 inner
1200.2.v.e 4 4.b odd 2 1
1200.2.v.e 4 12.b even 2 1
1200.2.v.e 4 20.d odd 2 1
1200.2.v.e 4 20.e even 4 2
1200.2.v.e 4 60.h even 2 1
1200.2.v.e 4 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 144$$ acting on $$S_{2}^{\mathrm{new}}(300, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + T^{4}$$
$5$ $$T^{4}$$
$7$ $$144 + T^{4}$$
$11$ $$T^{4}$$
$13$ $$2304 + T^{4}$$
$17$ $$T^{4}$$
$19$ $$( 64 + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$( -4 + T )^{4}$$
$37$ $$2304 + T^{4}$$
$41$ $$T^{4}$$
$43$ $$11664 + T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( -14 + T )^{4}$$
$67$ $$144 + T^{4}$$
$71$ $$T^{4}$$
$73$ $$36864 + T^{4}$$
$79$ $$( 16 + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$36864 + T^{4}$$