# Properties

 Label 300.3.f.a Level $300$ Weight $3$ Character orbit 300.f 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.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.17440793081$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 12) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{12} q^{2} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + 4 \zeta_{12}^{2} q^{4} + ( 2 + 2 \zeta_{12}^{2} ) q^{6} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{7} + 8 \zeta_{12}^{3} q^{8} + 3 q^{9} +O(q^{10})$$ $$q + 2 \zeta_{12} q^{2} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + 4 \zeta_{12}^{2} q^{4} + ( 2 + 2 \zeta_{12}^{2} ) q^{6} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{7} + 8 \zeta_{12}^{3} q^{8} + 3 q^{9} + ( 4 - 8 \zeta_{12}^{2} ) q^{11} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{12} + 2 \zeta_{12}^{3} q^{13} + ( 8 + 8 \zeta_{12}^{2} ) q^{14} + ( -16 + 16 \zeta_{12}^{2} ) q^{16} -10 \zeta_{12}^{3} q^{17} + 6 \zeta_{12} q^{18} + ( -12 + 24 \zeta_{12}^{2} ) q^{19} + 12 q^{21} + ( 8 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{22} + ( -32 \zeta_{12} + 16 \zeta_{12}^{3} ) q^{23} + ( -8 + 16 \zeta_{12}^{2} ) q^{24} + ( -4 + 4 \zeta_{12}^{2} ) q^{26} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( 16 \zeta_{12} + 16 \zeta_{12}^{3} ) q^{28} + 26 q^{29} + ( 4 - 8 \zeta_{12}^{2} ) q^{31} + ( -32 \zeta_{12} + 32 \zeta_{12}^{3} ) q^{32} -12 \zeta_{12}^{3} q^{33} + ( 20 - 20 \zeta_{12}^{2} ) q^{34} + 12 \zeta_{12}^{2} q^{36} -26 \zeta_{12}^{3} q^{37} + ( -24 \zeta_{12} + 48 \zeta_{12}^{3} ) q^{38} + ( -2 + 4 \zeta_{12}^{2} ) q^{39} + 58 q^{41} + 24 \zeta_{12} q^{42} + ( -56 \zeta_{12} + 28 \zeta_{12}^{3} ) q^{43} + ( 32 - 16 \zeta_{12}^{2} ) q^{44} + ( -32 - 32 \zeta_{12}^{2} ) q^{46} + ( -80 \zeta_{12} + 40 \zeta_{12}^{3} ) q^{47} + ( -16 \zeta_{12} + 32 \zeta_{12}^{3} ) q^{48} - q^{49} + ( 10 - 20 \zeta_{12}^{2} ) q^{51} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{52} -74 \zeta_{12}^{3} q^{53} + ( 6 + 6 \zeta_{12}^{2} ) q^{54} + ( -32 + 64 \zeta_{12}^{2} ) q^{56} + 36 \zeta_{12}^{3} q^{57} + 52 \zeta_{12} q^{58} + ( 52 - 104 \zeta_{12}^{2} ) q^{59} + 26 q^{61} + ( 8 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{62} + ( 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{63} -64 q^{64} + ( 24 - 24 \zeta_{12}^{2} ) q^{66} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{67} + ( 40 \zeta_{12} - 40 \zeta_{12}^{3} ) q^{68} -48 q^{69} + 24 \zeta_{12}^{3} q^{72} -46 \zeta_{12}^{3} q^{73} + ( 52 - 52 \zeta_{12}^{2} ) q^{74} + ( -96 + 48 \zeta_{12}^{2} ) q^{76} -48 \zeta_{12}^{3} q^{77} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{78} + ( -68 + 136 \zeta_{12}^{2} ) q^{79} + 9 q^{81} + 116 \zeta_{12} q^{82} + ( 56 \zeta_{12} - 28 \zeta_{12}^{3} ) q^{83} + 48 \zeta_{12}^{2} q^{84} + ( -56 - 56 \zeta_{12}^{2} ) q^{86} + ( 52 \zeta_{12} - 26 \zeta_{12}^{3} ) q^{87} + ( 64 \zeta_{12} - 32 \zeta_{12}^{3} ) q^{88} -82 q^{89} + ( -8 + 16 \zeta_{12}^{2} ) q^{91} + ( -64 \zeta_{12} - 64 \zeta_{12}^{3} ) q^{92} -12 \zeta_{12}^{3} q^{93} + ( -80 - 80 \zeta_{12}^{2} ) q^{94} + ( -64 + 32 \zeta_{12}^{2} ) q^{96} -2 \zeta_{12}^{3} q^{97} -2 \zeta_{12} q^{98} + ( 12 - 24 \zeta_{12}^{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{4} + 12q^{6} + 12q^{9} + O(q^{10})$$ $$4q + 8q^{4} + 12q^{6} + 12q^{9} + 48q^{14} - 32q^{16} + 48q^{21} - 8q^{26} + 104q^{29} + 40q^{34} + 24q^{36} + 232q^{41} + 96q^{44} - 192q^{46} - 4q^{49} + 36q^{54} + 104q^{61} - 256q^{64} + 48q^{66} - 192q^{69} + 104q^{74} - 288q^{76} + 36q^{81} + 96q^{84} - 336q^{86} - 328q^{89} - 480q^{94} - 192q^{96} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i
−1.73205 1.00000i −1.73205 2.00000 + 3.46410i 0 3.00000 + 1.73205i −6.92820 8.00000i 3.00000 0
199.2 −1.73205 + 1.00000i −1.73205 2.00000 3.46410i 0 3.00000 1.73205i −6.92820 8.00000i 3.00000 0
199.3 1.73205 1.00000i 1.73205 2.00000 3.46410i 0 3.00000 1.73205i 6.92820 8.00000i 3.00000 0
199.4 1.73205 + 1.00000i 1.73205 2.00000 + 3.46410i 0 3.00000 + 1.73205i 6.92820 8.00000i 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
4.b odd 2 1 inner
5.b even 2 1 inner
20.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.f.a 4
3.b odd 2 1 900.3.f.c 4
4.b odd 2 1 inner 300.3.f.a 4
5.b even 2 1 inner 300.3.f.a 4
5.c odd 4 1 12.3.d.a 2
5.c odd 4 1 300.3.c.b 2
12.b even 2 1 900.3.f.c 4
15.d odd 2 1 900.3.f.c 4
15.e even 4 1 36.3.d.c 2
15.e even 4 1 900.3.c.e 2
20.d odd 2 1 inner 300.3.f.a 4
20.e even 4 1 12.3.d.a 2
20.e even 4 1 300.3.c.b 2
35.f even 4 1 588.3.g.b 2
40.i odd 4 1 192.3.g.b 2
40.k even 4 1 192.3.g.b 2
45.k odd 12 1 324.3.f.d 2
45.k odd 12 1 324.3.f.j 2
45.l even 12 1 324.3.f.a 2
45.l even 12 1 324.3.f.g 2
60.h even 2 1 900.3.f.c 4
60.l odd 4 1 36.3.d.c 2
60.l odd 4 1 900.3.c.e 2
80.i odd 4 1 768.3.b.c 4
80.j even 4 1 768.3.b.c 4
80.s even 4 1 768.3.b.c 4
80.t odd 4 1 768.3.b.c 4
120.q odd 4 1 576.3.g.e 2
120.w even 4 1 576.3.g.e 2
140.j odd 4 1 588.3.g.b 2
180.v odd 12 1 324.3.f.a 2
180.v odd 12 1 324.3.f.g 2
180.x even 12 1 324.3.f.d 2
180.x even 12 1 324.3.f.j 2
240.z odd 4 1 2304.3.b.l 4
240.bb even 4 1 2304.3.b.l 4
240.bd odd 4 1 2304.3.b.l 4
240.bf even 4 1 2304.3.b.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.d.a 2 5.c odd 4 1
12.3.d.a 2 20.e even 4 1
36.3.d.c 2 15.e even 4 1
36.3.d.c 2 60.l odd 4 1
192.3.g.b 2 40.i odd 4 1
192.3.g.b 2 40.k even 4 1
300.3.c.b 2 5.c odd 4 1
300.3.c.b 2 20.e even 4 1
300.3.f.a 4 1.a even 1 1 trivial
300.3.f.a 4 4.b odd 2 1 inner
300.3.f.a 4 5.b even 2 1 inner
300.3.f.a 4 20.d odd 2 1 inner
324.3.f.a 2 45.l even 12 1
324.3.f.a 2 180.v odd 12 1
324.3.f.d 2 45.k odd 12 1
324.3.f.d 2 180.x even 12 1
324.3.f.g 2 45.l even 12 1
324.3.f.g 2 180.v odd 12 1
324.3.f.j 2 45.k odd 12 1
324.3.f.j 2 180.x even 12 1
576.3.g.e 2 120.q odd 4 1
576.3.g.e 2 120.w even 4 1
588.3.g.b 2 35.f even 4 1
588.3.g.b 2 140.j odd 4 1
768.3.b.c 4 80.i odd 4 1
768.3.b.c 4 80.j even 4 1
768.3.b.c 4 80.s even 4 1
768.3.b.c 4 80.t odd 4 1
900.3.c.e 2 15.e even 4 1
900.3.c.e 2 60.l odd 4 1
900.3.f.c 4 3.b odd 2 1
900.3.f.c 4 12.b even 2 1
900.3.f.c 4 15.d odd 2 1
900.3.f.c 4 60.h even 2 1
2304.3.b.l 4 240.z odd 4 1
2304.3.b.l 4 240.bb even 4 1
2304.3.b.l 4 240.bd odd 4 1
2304.3.b.l 4 240.bf even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - 48$$ acting on $$S_{3}^{\mathrm{new}}(300, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 - 4 T^{2} + T^{4}$$
$3$ $$( -3 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( -48 + T^{2} )^{2}$$
$11$ $$( 48 + T^{2} )^{2}$$
$13$ $$( 4 + T^{2} )^{2}$$
$17$ $$( 100 + T^{2} )^{2}$$
$19$ $$( 432 + T^{2} )^{2}$$
$23$ $$( -768 + T^{2} )^{2}$$
$29$ $$( -26 + T )^{4}$$
$31$ $$( 48 + T^{2} )^{2}$$
$37$ $$( 676 + T^{2} )^{2}$$
$41$ $$( -58 + T )^{4}$$
$43$ $$( -2352 + T^{2} )^{2}$$
$47$ $$( -4800 + T^{2} )^{2}$$
$53$ $$( 5476 + T^{2} )^{2}$$
$59$ $$( 8112 + T^{2} )^{2}$$
$61$ $$( -26 + T )^{4}$$
$67$ $$( -48 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$( 2116 + T^{2} )^{2}$$
$79$ $$( 13872 + T^{2} )^{2}$$
$83$ $$( -2352 + T^{2} )^{2}$$
$89$ $$( 82 + T )^{4}$$
$97$ $$( 4 + T^{2} )^{2}$$