# Properties

 Label 300.2.e.a Level $300$ Weight $2$ Character orbit 300.e Analytic conductor $2.396$ Analytic rank $0$ Dimension $4$ CM discriminant -15 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.39551206064$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{-5})$$ Defining polynomial: $$x^{4} + x^{2} + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( \beta_{1} - \beta_{3} ) q^{3} + \beta_{2} q^{4} + ( 2 + \beta_{2} ) q^{6} + ( -\beta_{1} + 2 \beta_{3} ) q^{8} + 3 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( \beta_{1} - \beta_{3} ) q^{3} + \beta_{2} q^{4} + ( 2 + \beta_{2} ) q^{6} + ( -\beta_{1} + 2 \beta_{3} ) q^{8} + 3 q^{9} + ( \beta_{1} + 2 \beta_{3} ) q^{12} + ( -4 - \beta_{2} ) q^{16} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{17} + 3 \beta_{1} q^{18} + ( -2 - 4 \beta_{2} ) q^{19} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{23} + ( -4 + \beta_{2} ) q^{24} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{27} + ( -2 - 4 \beta_{2} ) q^{31} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{32} + ( -4 + 2 \beta_{2} ) q^{34} + 3 \beta_{2} q^{36} + ( 2 \beta_{1} - 8 \beta_{3} ) q^{38} + ( -4 - 2 \beta_{2} ) q^{46} + ( -6 \beta_{1} + 6 \beta_{3} ) q^{47} + ( -5 \beta_{1} + 2 \beta_{3} ) q^{48} + 7 q^{49} + ( 2 + 4 \beta_{2} ) q^{51} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{53} + ( 6 + 3 \beta_{2} ) q^{54} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{57} -2 q^{61} + ( 2 \beta_{1} - 8 \beta_{3} ) q^{62} + ( 4 - 3 \beta_{2} ) q^{64} + ( -6 \beta_{1} + 4 \beta_{3} ) q^{68} -6 q^{69} + ( -3 \beta_{1} + 6 \beta_{3} ) q^{72} + ( 16 + 2 \beta_{2} ) q^{76} + ( 2 + 4 \beta_{2} ) q^{79} + 9 q^{81} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{83} + ( -2 \beta_{1} - 4 \beta_{3} ) q^{92} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{93} + ( -12 - 6 \beta_{2} ) q^{94} + ( -4 - 5 \beta_{2} ) q^{96} + 7 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{4} + 6q^{6} + 12q^{9} + O(q^{10})$$ $$4q - 2q^{4} + 6q^{6} + 12q^{9} - 14q^{16} - 18q^{24} - 20q^{34} - 6q^{36} - 12q^{46} + 28q^{49} + 18q^{54} - 8q^{61} + 22q^{64} - 24q^{69} + 60q^{76} + 36q^{81} - 36q^{94} - 6q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3} - \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 −0.866025 − 1.11803i −0.866025 + 1.11803i 0.866025 − 1.11803i 0.866025 + 1.11803i
−0.866025 1.11803i −1.73205 −0.500000 + 1.93649i 0 1.50000 + 1.93649i 0 2.59808 1.11803i 3.00000 0
251.2 −0.866025 + 1.11803i −1.73205 −0.500000 1.93649i 0 1.50000 1.93649i 0 2.59808 + 1.11803i 3.00000 0
251.3 0.866025 1.11803i 1.73205 −0.500000 1.93649i 0 1.50000 1.93649i 0 −2.59808 1.11803i 3.00000 0
251.4 0.866025 + 1.11803i 1.73205 −0.500000 + 1.93649i 0 1.50000 + 1.93649i 0 −2.59808 + 1.11803i 3.00000 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.e.a 4
3.b odd 2 1 inner 300.2.e.a 4
4.b odd 2 1 inner 300.2.e.a 4
5.b even 2 1 inner 300.2.e.a 4
5.c odd 4 2 60.2.h.b 4
12.b even 2 1 inner 300.2.e.a 4
15.d odd 2 1 CM 300.2.e.a 4
15.e even 4 2 60.2.h.b 4
20.d odd 2 1 inner 300.2.e.a 4
20.e even 4 2 60.2.h.b 4
40.i odd 4 2 960.2.o.a 4
40.k even 4 2 960.2.o.a 4
60.h even 2 1 inner 300.2.e.a 4
60.l odd 4 2 60.2.h.b 4
120.q odd 4 2 960.2.o.a 4
120.w even 4 2 960.2.o.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.h.b 4 5.c odd 4 2
60.2.h.b 4 15.e even 4 2
60.2.h.b 4 20.e even 4 2
60.2.h.b 4 60.l odd 4 2
300.2.e.a 4 1.a even 1 1 trivial
300.2.e.a 4 3.b odd 2 1 inner
300.2.e.a 4 4.b odd 2 1 inner
300.2.e.a 4 5.b even 2 1 inner
300.2.e.a 4 12.b even 2 1 inner
300.2.e.a 4 15.d odd 2 1 CM
300.2.e.a 4 20.d odd 2 1 inner
300.2.e.a 4 60.h even 2 1 inner
960.2.o.a 4 40.i odd 4 2
960.2.o.a 4 40.k even 4 2
960.2.o.a 4 120.q odd 4 2
960.2.o.a 4 120.w even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(300, [\chi])$$:

 $$T_{7}$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T^{2} + T^{4}$$
$3$ $$( -3 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$( 20 + T^{2} )^{2}$$
$19$ $$( 60 + T^{2} )^{2}$$
$23$ $$( -12 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( 60 + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$( -108 + T^{2} )^{2}$$
$53$ $$( 20 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$( 2 + T )^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$( 60 + T^{2} )^{2}$$
$83$ $$( -12 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$