Embedded newform 450.3.g.b.307.1

• Label: 450.3.g.b.307.1
• Level: $450$
• Weight: $3$
• Character: 450.307
• Analytic conductor: $12.262$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	450.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.2616118962\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			307.1
Root			\(1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	450.307
Dual form			450.3.g.b.343.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.00000i) q^{2} +2.00000i q^{4} +(-2.00000 - 2.00000i) q^{7} +(2.00000 - 2.00000i) q^{8} +8.00000 q^{11} +(-3.00000 + 3.00000i) q^{13} +4.00000i q^{14} -4.00000 q^{16} +(7.00000 + 7.00000i) q^{17} -20.0000i q^{19} +(-8.00000 - 8.00000i) q^{22} +(-2.00000 + 2.00000i) q^{23} +6.00000 q^{26} +(4.00000 - 4.00000i) q^{28} -40.0000i q^{29} +52.0000 q^{31} +(4.00000 + 4.00000i) q^{32} -14.0000i q^{34} +(3.00000 + 3.00000i) q^{37} +(-20.0000 + 20.0000i) q^{38} +8.00000 q^{41} +(42.0000 - 42.0000i) q^{43} +16.0000i q^{44} +4.00000 q^{46} +(-18.0000 - 18.0000i) q^{47} -41.0000i q^{49} +(-6.00000 - 6.00000i) q^{52} +(53.0000 - 53.0000i) q^{53} -8.00000 q^{56} +(-40.0000 + 40.0000i) q^{58} +20.0000i q^{59} -48.0000 q^{61} +(-52.0000 - 52.0000i) q^{62} -8.00000i q^{64} +(-62.0000 - 62.0000i) q^{67} +(-14.0000 + 14.0000i) q^{68} +28.0000 q^{71} +(47.0000 - 47.0000i) q^{73} -6.00000i q^{74} +40.0000 q^{76} +(-16.0000 - 16.0000i) q^{77} +(-8.00000 - 8.00000i) q^{82} +(18.0000 - 18.0000i) q^{83} -84.0000 q^{86} +(16.0000 - 16.0000i) q^{88} -80.0000i q^{89} +12.0000 q^{91} +(-4.00000 - 4.00000i) q^{92} +36.0000i q^{94} +(63.0000 + 63.0000i) q^{97} +(-41.0000 + 41.0000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 2 * q^2 - 4 * q^7 + 4 * q^8 + 16 * q^11 - 6 * q^13 - 8 * q^16 + 14 * q^17 - 16 * q^22 - 4 * q^23 + 12 * q^26 + 8 * q^28 + 104 * q^31 + 8 * q^32 + 6 * q^37 - 40 * q^38 + 16 * q^41 + 84 * q^43 + 8 * q^46 - 36 * q^47 - 12 * q^52 + 106 * q^53 - 16 * q^56 - 80 * q^58 - 96 * q^61 - 104 * q^62 - 124 * q^67 - 28 * q^68 + 56 * q^71 + 94 * q^73 + 80 * q^76 - 32 * q^77 - 16 * q^82 + 36 * q^83 - 168 * q^86 + 32 * q^88 + 24 * q^91 - 8 * q^92 + 126 * q^97 - 82 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−	1.00000i	−0.500000	−	0.500000i
							\(3\)		0			0
\(5\)		0			0
							\(7\)	−2.00000	−	2.00000i	−0.285714	−	0.285714i	0.549669	−	0.835383i	\(-0.314754\pi\)
−0.835383	+	0.549669i	\(0.814754\pi\)
\(11\)	8.00000			0.727273			0.363636	−	0.931541i	\(-0.381535\pi\)
							0.363636	+	0.931541i	\(0.381535\pi\)
\(13\)	−3.00000	+	3.00000i	−0.230769	+	0.230769i	−0.813014	−	0.582245i	\(-0.802175\pi\)
							0.582245	+	0.813014i	\(0.302175\pi\)
\(17\)	7.00000	+	7.00000i	0.411765	+	0.411765i	0.882353	−	0.470588i	\(-0.155958\pi\)
							−0.470588	+	0.882353i	\(0.655958\pi\)
\(19\)		−	20.0000i		−	1.05263i	−0.850289	−	0.526316i	\(-0.823573\pi\)
							0.850289	−	0.526316i	\(-0.176427\pi\)
\(23\)	−2.00000	+	2.00000i	−0.0869565	+	0.0869565i	−0.749247	−	0.662291i	\(-0.769584\pi\)
							0.662291	+	0.749247i	\(0.269584\pi\)
\(29\)		−	40.0000i		−	1.37931i	−0.724138	−	0.689655i	\(-0.757762\pi\)
							0.724138	−	0.689655i	\(-0.242238\pi\)
\(31\)	52.0000			1.67742			0.838710	−	0.544579i	\(-0.183310\pi\)
							0.838710	+	0.544579i	\(0.183310\pi\)
\(37\)	3.00000	+	3.00000i	0.0810811	+	0.0810811i	0.746484	−	0.665403i	\(-0.231740\pi\)
							−0.665403	+	0.746484i	\(0.731740\pi\)
\(41\)	8.00000			0.195122			0.0975610	−	0.995230i	\(-0.468896\pi\)
							0.0975610	+	0.995230i	\(0.468896\pi\)
\(43\)	42.0000	−	42.0000i	0.976744	−	0.976744i	−0.0229915	−	0.999736i	\(-0.507319\pi\)
							0.999736	+	0.0229915i	\(0.00731906\pi\)
\(47\)	−18.0000	−	18.0000i	−0.382979	−	0.382979i	0.489195	−	0.872174i	\(-0.337290\pi\)
							−0.872174	+	0.489195i	\(0.837290\pi\)
\(53\)	53.0000	−	53.0000i	1.00000	−	1.00000i		−	1.00000i	\(-0.5\pi\)
							1.00000			\(0\)
\(59\)			20.0000i			0.338983i	0.985532	+	0.169492i	\(0.0542125\pi\)
							−0.985532	+	0.169492i	\(0.945787\pi\)
\(61\)	−48.0000			−0.786885			−0.393443	−	0.919349i	\(-0.628716\pi\)
							−0.393443	+	0.919349i	\(0.628716\pi\)
\(67\)	−62.0000	−	62.0000i	−0.925373	−	0.925373i	0.0720294	−	0.997403i	\(-0.477052\pi\)
							−0.997403	+	0.0720294i	\(0.977052\pi\)
\(71\)	28.0000			0.394366			0.197183	−	0.980367i	\(-0.436821\pi\)
							0.197183	+	0.980367i	\(0.436821\pi\)
\(73\)	47.0000	−	47.0000i	0.643836	−	0.643836i	−0.307661	−	0.951496i	\(-0.599546\pi\)
							0.951496	+	0.307661i	\(0.0995461\pi\)
\(79\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(83\)	18.0000	−	18.0000i	0.216867	−	0.216867i	−0.590310	−	0.807177i	\(-0.700994\pi\)
							0.807177	+	0.590310i	\(0.200994\pi\)
\(89\)		−	80.0000i		−	0.898876i	−0.893311	−	0.449438i	\(-0.851624\pi\)
							0.893311	−	0.449438i	\(-0.148376\pi\)
\(97\)	63.0000	+	63.0000i	0.649485	+	0.649485i	0.952868	−	0.303384i	\(-0.0981164\pi\)
							−0.303384	+	0.952868i	\(0.598116\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.3.g.b.307.1		2
3.2	odd	2		50.3.c.c.7.1		2
5.2	odd	4		90.3.g.b.73.1		2
5.3	odd	4	inner	450.3.g.b.343.1		2
5.4	even	2		90.3.g.b.37.1		2
12.11	even	2		400.3.p.b.257.1		2
15.2	even	4		10.3.c.a.3.1	✓	2
15.8	even	4		50.3.c.c.43.1		2
15.14	odd	2		10.3.c.a.7.1	yes	2
20.7	even	4		720.3.bh.c.433.1		2
20.19	odd	2		720.3.bh.c.577.1		2
60.23	odd	4		400.3.p.b.193.1		2
60.47	odd	4		80.3.p.c.33.1		2
60.59	even	2		80.3.p.c.17.1		2
105.62	odd	4		490.3.f.b.393.1		2
105.104	even	2		490.3.f.b.197.1		2
120.29	odd	2		320.3.p.h.257.1		2
120.59	even	2		320.3.p.a.257.1		2
120.77	even	4		320.3.p.h.193.1		2
120.107	odd	4		320.3.p.a.193.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.3.c.a.3.1	✓	2	15.2	even	4
10.3.c.a.7.1	yes	2	15.14	odd	2
50.3.c.c.7.1		2	3.2	odd	2
50.3.c.c.43.1		2	15.8	even	4
80.3.p.c.17.1		2	60.59	even	2
80.3.p.c.33.1		2	60.47	odd	4
90.3.g.b.37.1		2	5.4	even	2
90.3.g.b.73.1		2	5.2	odd	4
320.3.p.a.193.1		2	120.107	odd	4
320.3.p.a.257.1		2	120.59	even	2
320.3.p.h.193.1		2	120.77	even	4
320.3.p.h.257.1		2	120.29	odd	2
400.3.p.b.193.1		2	60.23	odd	4
400.3.p.b.257.1		2	12.11	even	2
450.3.g.b.307.1		2	1.1	even	1	trivial
450.3.g.b.343.1		2	5.3	odd	4	inner
490.3.f.b.197.1		2	105.104	even	2
490.3.f.b.393.1		2	105.62	odd	4
720.3.bh.c.433.1		2	20.7	even	4
720.3.bh.c.577.1		2	20.19	odd	2