# Properties

 Label 300.3.b.c Level $300$ Weight $3$ Character orbit 300.b Analytic conductor $8.174$ 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: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 300.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.17440793081$$ Analytic rank: $$0$$ Dimension: $$4$$ 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: $$2^{4}$$ 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 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{3} ) q^{3} + \beta_{1} q^{7} + ( 1 - 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{3} ) q^{3} + \beta_{1} q^{7} + ( 1 - 2 \beta_{2} ) q^{9} -3 \beta_{2} q^{11} -4 \beta_{1} q^{13} + 6 \beta_{3} q^{17} + 34 q^{19} + ( 4 + \beta_{2} ) q^{21} -18 \beta_{3} q^{23} + ( -11 \beta_{1} - 7 \beta_{3} ) q^{27} -9 \beta_{2} q^{29} + 14 q^{31} + ( -15 \beta_{1} - 12 \beta_{3} ) q^{33} + 28 \beta_{1} q^{37} + ( -16 - 4 \beta_{2} ) q^{39} + 6 \beta_{2} q^{41} -4 \beta_{1} q^{43} -18 \beta_{3} q^{47} + 45 q^{49} + ( 30 - 6 \beta_{2} ) q^{51} + 18 \beta_{3} q^{53} + ( -34 \beta_{1} + 34 \beta_{3} ) q^{57} + 3 \beta_{2} q^{59} -46 q^{61} + ( \beta_{1} + 8 \beta_{3} ) q^{63} + 16 \beta_{1} q^{67} + ( -90 + 18 \beta_{2} ) q^{69} -12 \beta_{2} q^{71} + 53 \beta_{1} q^{73} + 12 \beta_{3} q^{77} + 22 q^{79} + ( -79 - 4 \beta_{2} ) q^{81} + 54 \beta_{3} q^{83} + ( -45 \beta_{1} - 36 \beta_{3} ) q^{87} + 24 \beta_{2} q^{89} + 16 q^{91} + ( -14 \beta_{1} + 14 \beta_{3} ) q^{93} + 61 \beta_{1} q^{97} + ( -120 - 3 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{9} + O(q^{10})$$ $$4q + 4q^{9} + 136q^{19} + 16q^{21} + 56q^{31} - 64q^{39} + 180q^{49} + 120q^{51} - 184q^{61} - 360q^{69} + 88q^{79} - 316q^{81} + 64q^{91} - 480q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{3} + 4 \nu$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{3} + 8 \nu$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{2} + 2 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 − 1.61803i 1.61803i 0.618034i − 0.618034i
0 −2.23607 2.00000i 0 0 0 2.00000i 0 1.00000 + 8.94427i 0
149.2 0 −2.23607 + 2.00000i 0 0 0 2.00000i 0 1.00000 8.94427i 0
149.3 0 2.23607 2.00000i 0 0 0 2.00000i 0 1.00000 8.94427i 0
149.4 0 2.23607 + 2.00000i 0 0 0 2.00000i 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
5.b even 2 1 inner
15.d 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.b.c 4
3.b odd 2 1 inner 300.3.b.c 4
4.b odd 2 1 1200.3.c.e 4
5.b even 2 1 inner 300.3.b.c 4
5.c odd 4 1 60.3.g.a 2
5.c odd 4 1 300.3.g.d 2
12.b even 2 1 1200.3.c.e 4
15.d odd 2 1 inner 300.3.b.c 4
15.e even 4 1 60.3.g.a 2
15.e even 4 1 300.3.g.d 2
20.d odd 2 1 1200.3.c.e 4
20.e even 4 1 240.3.l.a 2
20.e even 4 1 1200.3.l.r 2
40.i odd 4 1 960.3.l.a 2
40.k even 4 1 960.3.l.d 2
45.k odd 12 2 1620.3.o.b 4
45.l even 12 2 1620.3.o.b 4
60.h even 2 1 1200.3.c.e 4
60.l odd 4 1 240.3.l.a 2
60.l odd 4 1 1200.3.l.r 2
120.q odd 4 1 960.3.l.d 2
120.w even 4 1 960.3.l.a 2

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

## Hecke kernels

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

 $$T_{7}^{2} + 4$$ $$T_{11}^{2} + 180$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$81 - 2 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 4 + T^{2} )^{2}$$
$11$ $$( 180 + T^{2} )^{2}$$
$13$ $$( 64 + T^{2} )^{2}$$
$17$ $$( -180 + T^{2} )^{2}$$
$19$ $$( -34 + T )^{4}$$
$23$ $$( -1620 + T^{2} )^{2}$$
$29$ $$( 1620 + T^{2} )^{2}$$
$31$ $$( -14 + T )^{4}$$
$37$ $$( 3136 + T^{2} )^{2}$$
$41$ $$( 720 + T^{2} )^{2}$$
$43$ $$( 64 + T^{2} )^{2}$$
$47$ $$( -1620 + T^{2} )^{2}$$
$53$ $$( -1620 + T^{2} )^{2}$$
$59$ $$( 180 + T^{2} )^{2}$$
$61$ $$( 46 + T )^{4}$$
$67$ $$( 1024 + T^{2} )^{2}$$
$71$ $$( 2880 + T^{2} )^{2}$$
$73$ $$( 11236 + T^{2} )^{2}$$
$79$ $$( -22 + T )^{4}$$
$83$ $$( -14580 + T^{2} )^{2}$$
$89$ $$( 11520 + T^{2} )^{2}$$
$97$ $$( 14884 + T^{2} )^{2}$$