Embedded newform 450.3.g.e.343.1

• Label: 450.3.g.e.343.1
• Level: $450$
• Weight: $3$
• Character: 450.343
• 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 50)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			343.1
Root			\(-1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	450.343
Dual form			450.3.g.e.307.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-3.00000 + 3.00000i) q^{7} +(-2.00000 - 2.00000i) q^{8} -12.0000 q^{11} +(-12.0000 - 12.0000i) q^{13} +6.00000i q^{14} -4.00000 q^{16} +(-12.0000 + 12.0000i) q^{17} -20.0000i q^{19} +(-12.0000 + 12.0000i) q^{22} +(-3.00000 - 3.00000i) q^{23} -24.0000 q^{26} +(6.00000 + 6.00000i) q^{28} -30.0000i q^{29} -8.00000 q^{31} +(-4.00000 + 4.00000i) q^{32} +24.0000i q^{34} +(-48.0000 + 48.0000i) q^{37} +(-20.0000 - 20.0000i) q^{38} +48.0000 q^{41} +(-27.0000 - 27.0000i) q^{43} +24.0000i q^{44} -6.00000 q^{46} +(-27.0000 + 27.0000i) q^{47} +31.0000i q^{49} +(-24.0000 + 24.0000i) q^{52} +(12.0000 + 12.0000i) q^{53} +12.0000 q^{56} +(-30.0000 - 30.0000i) q^{58} -60.0000i q^{59} +32.0000 q^{61} +(-8.00000 + 8.00000i) q^{62} +8.00000i q^{64} +(-3.00000 + 3.00000i) q^{67} +(24.0000 + 24.0000i) q^{68} +48.0000 q^{71} +(-12.0000 - 12.0000i) q^{73} +96.0000i q^{74} -40.0000 q^{76} +(36.0000 - 36.0000i) q^{77} -40.0000i q^{79} +(48.0000 - 48.0000i) q^{82} +(-93.0000 - 93.0000i) q^{83} -54.0000 q^{86} +(24.0000 + 24.0000i) q^{88} +30.0000i q^{89} +72.0000 q^{91} +(-6.00000 + 6.00000i) q^{92} +54.0000i q^{94} +(12.0000 - 12.0000i) q^{97} +(31.0000 + 31.0000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 2 * q^2 - 6 * q^7 - 4 * q^8 - 24 * q^11 - 24 * q^13 - 8 * q^16 - 24 * q^17 - 24 * q^22 - 6 * q^23 - 48 * q^26 + 12 * q^28 - 16 * q^31 - 8 * q^32 - 96 * q^37 - 40 * q^38 + 96 * q^41 - 54 * q^43 - 12 * q^46 - 54 * q^47 - 48 * q^52 + 24 * q^53 + 24 * q^56 - 60 * q^58 + 64 * q^61 - 16 * q^62 - 6 * q^67 + 48 * q^68 + 96 * q^71 - 24 * q^73 - 80 * q^76 + 72 * q^77 + 96 * q^82 - 186 * q^83 - 108 * q^86 + 48 * q^88 + 144 * q^91 - 12 * q^92 + 24 * q^97 + 62 * q^98

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)

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
							\(7\)	−3.00000	+	3.00000i	−0.428571	+	0.428571i	−0.888142	−	0.459570i	\(-0.848004\pi\)
0.459570	+	0.888142i	\(0.348004\pi\)
\(11\)	−12.0000			−1.09091			−0.545455	−	0.838140i	\(-0.683643\pi\)
							−0.545455	+	0.838140i	\(0.683643\pi\)
\(13\)	−12.0000	−	12.0000i	−0.923077	−	0.923077i	0.0741688	−	0.997246i	\(-0.476370\pi\)
							−0.997246	+	0.0741688i	\(0.976370\pi\)
\(17\)	−12.0000	+	12.0000i	−0.705882	+	0.705882i	−0.965667	−	0.259784i	\(-0.916349\pi\)
							0.259784	+	0.965667i	\(0.416349\pi\)
\(19\)		−	20.0000i		−	1.05263i	−0.850289	−	0.526316i	\(-0.823573\pi\)
							0.850289	−	0.526316i	\(-0.176427\pi\)
\(23\)	−3.00000	−	3.00000i	−0.130435	−	0.130435i	0.638875	−	0.769310i	\(-0.279400\pi\)
							−0.769310	+	0.638875i	\(0.779400\pi\)
\(29\)		−	30.0000i		−	1.03448i	−0.855840	−	0.517241i	\(-0.826959\pi\)
							0.855840	−	0.517241i	\(-0.173041\pi\)
\(31\)	−8.00000			−0.258065			−0.129032	−	0.991640i	\(-0.541187\pi\)
							−0.129032	+	0.991640i	\(0.541187\pi\)
\(37\)	−48.0000	+	48.0000i	−1.29730	+	1.29730i	−0.367126	+	0.930171i	\(0.619658\pi\)
							−0.930171	+	0.367126i	\(0.880342\pi\)
\(41\)	48.0000			1.17073			0.585366	−	0.810769i	\(-0.300951\pi\)
							0.585366	+	0.810769i	\(0.300951\pi\)
\(43\)	−27.0000	−	27.0000i	−0.627907	−	0.627907i	0.319634	−	0.947541i	\(-0.396440\pi\)
							−0.947541	+	0.319634i	\(0.896440\pi\)
\(47\)	−27.0000	+	27.0000i	−0.574468	+	0.574468i	−0.933374	−	0.358906i	\(-0.883150\pi\)
							0.358906	+	0.933374i	\(0.383150\pi\)
\(53\)	12.0000	+	12.0000i	0.226415	+	0.226415i	0.811193	−	0.584778i	\(-0.198818\pi\)
							−0.584778	+	0.811193i	\(0.698818\pi\)
\(59\)		−	60.0000i		−	1.01695i	−0.861077	−	0.508475i	\(-0.830210\pi\)
							0.861077	−	0.508475i	\(-0.169790\pi\)
\(61\)	32.0000			0.524590			0.262295	−	0.964988i	\(-0.415521\pi\)
							0.262295	+	0.964988i	\(0.415521\pi\)
\(67\)	−3.00000	+	3.00000i	−0.0447761	+	0.0447761i	−0.729140	−	0.684364i	\(-0.760080\pi\)
							0.684364	+	0.729140i	\(0.260080\pi\)
\(71\)	48.0000			0.676056			0.338028	−	0.941136i	\(-0.390240\pi\)
							0.338028	+	0.941136i	\(0.390240\pi\)
\(73\)	−12.0000	−	12.0000i	−0.164384	−	0.164384i	0.620122	−	0.784505i	\(-0.287083\pi\)
							−0.784505	+	0.620122i	\(0.787083\pi\)
\(79\)		−	40.0000i		−	0.506329i	−0.967423	−	0.253165i	\(-0.918529\pi\)
							0.967423	−	0.253165i	\(-0.0814714\pi\)
\(83\)	−93.0000	−	93.0000i	−1.12048	−	1.12048i	−0.991669	−	0.128813i	\(-0.958883\pi\)
							−0.128813	−	0.991669i	\(-0.541117\pi\)
\(89\)			30.0000i			0.337079i	0.985695	+	0.168539i	\(0.0539050\pi\)
							−0.985695	+	0.168539i	\(0.946095\pi\)
\(97\)	12.0000	−	12.0000i	0.123711	−	0.123711i	−0.642540	−	0.766252i	\(-0.722120\pi\)
							0.766252	+	0.642540i	\(0.222120\pi\)

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.e.343.1		2
3.2	odd	2		50.3.c.a.43.1	yes	2
5.2	odd	4	inner	450.3.g.e.307.1		2
5.3	odd	4		450.3.g.c.307.1		2
5.4	even	2		450.3.g.c.343.1		2
12.11	even	2		400.3.p.a.193.1		2
15.2	even	4		50.3.c.a.7.1	✓	2
15.8	even	4		50.3.c.b.7.1	yes	2
15.14	odd	2		50.3.c.b.43.1	yes	2
60.23	odd	4		400.3.p.g.257.1		2
60.47	odd	4		400.3.p.a.257.1		2
60.59	even	2		400.3.p.g.193.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.3.c.a.7.1	✓	2	15.2	even	4
50.3.c.a.43.1	yes	2	3.2	odd	2
50.3.c.b.7.1	yes	2	15.8	even	4
50.3.c.b.43.1	yes	2	15.14	odd	2
400.3.p.a.193.1		2	12.11	even	2
400.3.p.a.257.1		2	60.47	odd	4
400.3.p.g.193.1		2	60.59	even	2
400.3.p.g.257.1		2	60.23	odd	4
450.3.g.c.307.1		2	5.3	odd	4
450.3.g.c.343.1		2	5.4	even	2
450.3.g.e.307.1		2	5.2	odd	4	inner
450.3.g.e.343.1		2	1.1	even	1	trivial