# Properties

 Label 300.3.b.d 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: $$\Q(i, \sqrt{35})$$ Defining polynomial: $$x^{4} - 17 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -8 \beta_{2} q^{7} + ( 9 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -8 \beta_{2} q^{7} + ( 9 + \beta_{3} ) q^{9} + ( -3 - 6 \beta_{3} ) q^{11} + 2 \beta_{2} q^{13} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{17} -11 q^{19} -8 \beta_{3} q^{21} + ( 12 \beta_{1} - 6 \beta_{2} ) q^{23} + ( 8 \beta_{1} + 9 \beta_{2} ) q^{27} + ( 6 + 12 \beta_{3} ) q^{29} -46 q^{31} + ( 3 \beta_{1} - 54 \beta_{2} ) q^{33} + 16 \beta_{2} q^{37} + 2 \beta_{3} q^{39} + ( -9 - 18 \beta_{3} ) q^{41} + 62 \beta_{2} q^{43} + ( 12 \beta_{1} - 6 \beta_{2} ) q^{47} -15 q^{49} + ( 54 + 3 \beta_{3} ) q^{51} + ( -12 \beta_{1} + 6 \beta_{2} ) q^{53} -11 \beta_{1} q^{57} + ( -12 - 24 \beta_{3} ) q^{59} -16 q^{61} + ( 8 \beta_{1} - 72 \beta_{2} ) q^{63} -113 \beta_{2} q^{67} + ( 108 + 6 \beta_{3} ) q^{69} + ( 18 + 36 \beta_{3} ) q^{71} + 101 \beta_{2} q^{73} + ( -48 \beta_{1} + 24 \beta_{2} ) q^{77} -68 q^{79} + ( 72 + 17 \beta_{3} ) q^{81} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{83} + ( -6 \beta_{1} + 108 \beta_{2} ) q^{87} + ( 9 + 18 \beta_{3} ) q^{89} + 16 q^{91} -46 \beta_{1} q^{93} + 22 \beta_{2} q^{97} + ( 27 - 51 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 34q^{9} + O(q^{10})$$ $$4q + 34q^{9} - 44q^{19} + 16q^{21} - 184q^{31} - 4q^{39} - 60q^{49} + 210q^{51} - 64q^{61} + 420q^{69} - 272q^{79} + 254q^{81} + 64q^{91} + 210q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 17 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 8 \nu$$$$)/9$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - 9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 9$$ $$\nu^{3}$$ $$=$$ $$9 \beta_{2} + 8 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 −2.95804 − 0.500000i −2.95804 + 0.500000i 2.95804 − 0.500000i 2.95804 + 0.500000i
0 −2.95804 0.500000i 0 0 0 8.00000i 0 8.50000 + 2.95804i 0
149.2 0 −2.95804 + 0.500000i 0 0 0 8.00000i 0 8.50000 2.95804i 0
149.3 0 2.95804 0.500000i 0 0 0 8.00000i 0 8.50000 2.95804i 0
149.4 0 2.95804 + 0.500000i 0 0 0 8.00000i 0 8.50000 + 2.95804i 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.d 4
3.b odd 2 1 inner 300.3.b.d 4
4.b odd 2 1 1200.3.c.j 4
5.b even 2 1 inner 300.3.b.d 4
5.c odd 4 1 300.3.g.f 2
5.c odd 4 1 300.3.g.g yes 2
12.b even 2 1 1200.3.c.j 4
15.d odd 2 1 inner 300.3.b.d 4
15.e even 4 1 300.3.g.f 2
15.e even 4 1 300.3.g.g yes 2
20.d odd 2 1 1200.3.c.j 4
20.e even 4 1 1200.3.l.k 2
20.e even 4 1 1200.3.l.m 2
60.h even 2 1 1200.3.c.j 4
60.l odd 4 1 1200.3.l.k 2
60.l odd 4 1 1200.3.l.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.b.d 4 1.a even 1 1 trivial
300.3.b.d 4 3.b odd 2 1 inner
300.3.b.d 4 5.b even 2 1 inner
300.3.b.d 4 15.d odd 2 1 inner
300.3.g.f 2 5.c odd 4 1
300.3.g.f 2 15.e even 4 1
300.3.g.g yes 2 5.c odd 4 1
300.3.g.g yes 2 15.e even 4 1
1200.3.c.j 4 4.b odd 2 1
1200.3.c.j 4 12.b even 2 1
1200.3.c.j 4 20.d odd 2 1
1200.3.c.j 4 60.h even 2 1
1200.3.l.k 2 20.e even 4 1
1200.3.l.k 2 60.l odd 4 1
1200.3.l.m 2 20.e even 4 1
1200.3.l.m 2 60.l odd 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 64$$ $$T_{11}^{2} + 315$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$81 - 17 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 64 + T^{2} )^{2}$$
$11$ $$( 315 + T^{2} )^{2}$$
$13$ $$( 4 + T^{2} )^{2}$$
$17$ $$( -315 + T^{2} )^{2}$$
$19$ $$( 11 + T )^{4}$$
$23$ $$( -1260 + T^{2} )^{2}$$
$29$ $$( 1260 + T^{2} )^{2}$$
$31$ $$( 46 + T )^{4}$$
$37$ $$( 256 + T^{2} )^{2}$$
$41$ $$( 2835 + T^{2} )^{2}$$
$43$ $$( 3844 + T^{2} )^{2}$$
$47$ $$( -1260 + T^{2} )^{2}$$
$53$ $$( -1260 + T^{2} )^{2}$$
$59$ $$( 5040 + T^{2} )^{2}$$
$61$ $$( 16 + T )^{4}$$
$67$ $$( 12769 + T^{2} )^{2}$$
$71$ $$( 11340 + T^{2} )^{2}$$
$73$ $$( 10201 + T^{2} )^{2}$$
$79$ $$( 68 + T )^{4}$$
$83$ $$( -315 + T^{2} )^{2}$$
$89$ $$( 2835 + T^{2} )^{2}$$
$97$ $$( 484 + T^{2} )^{2}$$