# Properties

 Label 300.1.l.a Level $300$ Weight $1$ Character orbit 300.l Analytic conductor $0.150$ Analytic rank $0$ Dimension $4$ Projective image $D_{2}$ CM/RM discs -15, -20, 12 Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 300.l (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.149719503790$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{3}, \sqrt{-5})$$ Artin image $OD_{16}:C_2$ Artin field Galois closure of 16.0.10251562500000000.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{8} q^{2} -\zeta_{8}^{3} q^{3} + \zeta_{8}^{2} q^{4} - q^{6} -\zeta_{8}^{3} q^{8} -\zeta_{8}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{8} q^{2} -\zeta_{8}^{3} q^{3} + \zeta_{8}^{2} q^{4} - q^{6} -\zeta_{8}^{3} q^{8} -\zeta_{8}^{2} q^{9} + \zeta_{8} q^{12} - q^{16} + \zeta_{8}^{3} q^{18} + 2 \zeta_{8}^{3} q^{23} -\zeta_{8}^{2} q^{24} -\zeta_{8} q^{27} + \zeta_{8} q^{32} + q^{36} + 2 q^{46} + 2 \zeta_{8} q^{47} + \zeta_{8}^{3} q^{48} + \zeta_{8}^{2} q^{49} + \zeta_{8}^{2} q^{54} -2 q^{61} -\zeta_{8}^{2} q^{64} + 2 \zeta_{8}^{2} q^{69} -\zeta_{8} q^{72} - q^{81} -2 \zeta_{8}^{3} q^{83} -2 \zeta_{8} q^{92} -2 \zeta_{8}^{2} q^{94} + q^{96} -\zeta_{8}^{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{6} + O(q^{10})$$ $$4q - 4q^{6} - 4q^{16} + 4q^{36} + 8q^{46} - 8q^{61} - 4q^{81} + 4q^{96} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{8}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i
−0.707107 0.707107i 0.707107 0.707107i 1.00000i 0 −1.00000 0 0.707107 0.707107i 1.00000i 0
107.2 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0 −1.00000 0 −0.707107 + 0.707107i 1.00000i 0
143.1 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0 −1.00000 0 0.707107 + 0.707107i 1.00000i 0
143.2 0.707107 0.707107i −0.707107 0.707107i 1.00000i 0 −1.00000 0 −0.707107 0.707107i 1.00000i 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
12.b even 2 1 RM by $$\Q(\sqrt{3})$$
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.e even 4 2 inner
20.e even 4 2 inner
60.h even 2 1 inner
60.l odd 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.1.l.a 4
3.b odd 2 1 inner 300.1.l.a 4
4.b odd 2 1 inner 300.1.l.a 4
5.b even 2 1 inner 300.1.l.a 4
5.c odd 4 2 inner 300.1.l.a 4
12.b even 2 1 RM 300.1.l.a 4
15.d odd 2 1 CM 300.1.l.a 4
15.e even 4 2 inner 300.1.l.a 4
20.d odd 2 1 CM 300.1.l.a 4
20.e even 4 2 inner 300.1.l.a 4
60.h even 2 1 inner 300.1.l.a 4
60.l odd 4 2 inner 300.1.l.a 4

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(300, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{4}$$
$3$ $$1 + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$16 + T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$16 + T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( 2 + T )^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$16 + T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$