# Properties

 Label 300.2.i.a Level $300$ Weight $2$ Character orbit 300.i Analytic conductor $2.396$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{3} ) q^{3} + ( 1 - \beta_{2} ) q^{7} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{3} ) q^{3} + ( 1 - \beta_{2} ) q^{7} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{9} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{11} + ( 3 + 3 \beta_{2} ) q^{13} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{17} + 2 \beta_{2} q^{19} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{21} + ( -1 + \beta_{2} + 2 \beta_{3} ) q^{23} + ( -3 + \beta_{1} + 4 \beta_{2} ) q^{27} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{29} + 4 q^{31} + ( -4 - 6 \beta_{2} - 2 \beta_{3} ) q^{33} + ( 3 - 3 \beta_{2} ) q^{37} + ( -3 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{39} + ( -4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{41} + ( 3 + 3 \beta_{2} ) q^{43} + ( -3 - 6 \beta_{1} - 3 \beta_{2} ) q^{47} + 5 \beta_{2} q^{49} + ( -5 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{51} + ( 1 - \beta_{2} - 2 \beta_{3} ) q^{53} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{57} + ( -4 - 4 \beta_{1} + 4 \beta_{3} ) q^{59} -6 q^{61} + ( -1 - 3 \beta_{2} - 2 \beta_{3} ) q^{63} + ( 1 - \beta_{2} ) q^{67} + ( 1 + \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{69} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{71} + ( -1 - \beta_{2} ) q^{73} + ( 2 + 4 \beta_{1} + 2 \beta_{2} ) q^{77} + 6 \beta_{2} q^{79} + ( 1 - 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{81} + ( 3 - 3 \beta_{2} - 6 \beta_{3} ) q^{83} + ( -6 - 2 \beta_{1} + 4 \beta_{2} ) q^{87} + ( -2 - 2 \beta_{1} + 2 \beta_{3} ) q^{89} + 6 q^{91} + ( -4 + 4 \beta_{3} ) q^{93} + ( -9 + 9 \beta_{2} ) q^{97} + ( 4 + 4 \beta_{1} + 10 \beta_{2} - 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} + 4q^{7} + O(q^{10})$$ $$4q - 2q^{3} + 4q^{7} + 12q^{13} - 4q^{21} - 14q^{27} + 16q^{31} - 20q^{33} + 12q^{37} + 12q^{43} - 20q^{51} + 4q^{57} - 24q^{61} - 8q^{63} + 4q^{67} - 4q^{73} + 4q^{81} - 20q^{87} + 24q^{91} - 8q^{93} - 36q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

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

## Character values

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

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

## 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
 − 0.618034i 1.61803i 0.618034i − 1.61803i
0 −1.61803 0.618034i 0 0 0 1.00000 + 1.00000i 0 2.23607 + 2.00000i 0
257.2 0 0.618034 + 1.61803i 0 0 0 1.00000 + 1.00000i 0 −2.23607 + 2.00000i 0
293.1 0 −1.61803 + 0.618034i 0 0 0 1.00000 1.00000i 0 2.23607 2.00000i 0
293.2 0 0.618034 1.61803i 0 0 0 1.00000 1.00000i 0 −2.23607 2.00000i 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 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.i.a 4
3.b odd 2 1 inner 300.2.i.a 4
4.b odd 2 1 1200.2.v.i 4
5.b even 2 1 60.2.i.a 4
5.c odd 4 1 60.2.i.a 4
5.c odd 4 1 inner 300.2.i.a 4
12.b even 2 1 1200.2.v.i 4
15.d odd 2 1 60.2.i.a 4
15.e even 4 1 60.2.i.a 4
15.e even 4 1 inner 300.2.i.a 4
20.d odd 2 1 240.2.v.b 4
20.e even 4 1 240.2.v.b 4
20.e even 4 1 1200.2.v.i 4
40.e odd 2 1 960.2.v.h 4
40.f even 2 1 960.2.v.e 4
40.i odd 4 1 960.2.v.e 4
40.k even 4 1 960.2.v.h 4
45.h odd 6 2 1620.2.x.b 8
45.j even 6 2 1620.2.x.b 8
45.k odd 12 2 1620.2.x.b 8
45.l even 12 2 1620.2.x.b 8
60.h even 2 1 240.2.v.b 4
60.l odd 4 1 240.2.v.b 4
60.l odd 4 1 1200.2.v.i 4
120.i odd 2 1 960.2.v.e 4
120.m even 2 1 960.2.v.h 4
120.q odd 4 1 960.2.v.h 4
120.w even 4 1 960.2.v.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.i.a 4 5.b even 2 1
60.2.i.a 4 5.c odd 4 1
60.2.i.a 4 15.d odd 2 1
60.2.i.a 4 15.e even 4 1
240.2.v.b 4 20.d odd 2 1
240.2.v.b 4 20.e even 4 1
240.2.v.b 4 60.h even 2 1
240.2.v.b 4 60.l odd 4 1
300.2.i.a 4 1.a even 1 1 trivial
300.2.i.a 4 3.b odd 2 1 inner
300.2.i.a 4 5.c odd 4 1 inner
300.2.i.a 4 15.e even 4 1 inner
960.2.v.e 4 40.f even 2 1
960.2.v.e 4 40.i odd 4 1
960.2.v.e 4 120.i odd 2 1
960.2.v.e 4 120.w even 4 1
960.2.v.h 4 40.e odd 2 1
960.2.v.h 4 40.k even 4 1
960.2.v.h 4 120.m even 2 1
960.2.v.h 4 120.q odd 4 1
1200.2.v.i 4 4.b odd 2 1
1200.2.v.i 4 12.b even 2 1
1200.2.v.i 4 20.e even 4 1
1200.2.v.i 4 60.l odd 4 1
1620.2.x.b 8 45.h odd 6 2
1620.2.x.b 8 45.j even 6 2
1620.2.x.b 8 45.k odd 12 2
1620.2.x.b 8 45.l even 12 2

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + 6 T + 2 T^{2} + 2 T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 2 - 2 T + T^{2} )^{2}$$
$11$ $$( 20 + T^{2} )^{2}$$
$13$ $$( 18 - 6 T + T^{2} )^{2}$$
$17$ $$100 + T^{4}$$
$19$ $$( 4 + T^{2} )^{2}$$
$23$ $$100 + T^{4}$$
$29$ $$( -20 + T^{2} )^{2}$$
$31$ $$( -4 + T )^{4}$$
$37$ $$( 18 - 6 T + T^{2} )^{2}$$
$41$ $$( 80 + T^{2} )^{2}$$
$43$ $$( 18 - 6 T + T^{2} )^{2}$$
$47$ $$8100 + T^{4}$$
$53$ $$100 + T^{4}$$
$59$ $$( -80 + T^{2} )^{2}$$
$61$ $$( 6 + T )^{4}$$
$67$ $$( 2 - 2 T + T^{2} )^{2}$$
$71$ $$( 20 + T^{2} )^{2}$$
$73$ $$( 2 + 2 T + T^{2} )^{2}$$
$79$ $$( 36 + T^{2} )^{2}$$
$83$ $$8100 + T^{4}$$
$89$ $$( -20 + T^{2} )^{2}$$
$97$ $$( 162 + 18 T + T^{2} )^{2}$$