Embedded newform 450.3.g.c.343.1

• Label: 450.3.g.c.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.c.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	−0.459570	−	0.888142i	\(-0.651996\pi\)
0.888142	+	0.459570i	\(0.151996\pi\)
\(11\)	−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	0.997246	−	0.0741688i	\(-0.0236304\pi\)
							−0.0741688	+	0.997246i	\(0.523630\pi\)
\(17\)	12.0000	−	12.0000i	0.705882	−	0.705882i	−0.259784	−	0.965667i	\(-0.583651\pi\)
							0.965667	+	0.259784i	\(0.0836515\pi\)
\(19\)		−	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	0.769310	−	0.638875i	\(-0.220600\pi\)
							−0.638875	+	0.769310i	\(0.720600\pi\)
\(29\)		−	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	\(-0.380342\pi\)
							0.930171	−	0.367126i	\(-0.119658\pi\)
\(41\)	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	0.947541	−	0.319634i	\(-0.103560\pi\)
							−0.319634	+	0.947541i	\(0.603560\pi\)
\(47\)	27.0000	−	27.0000i	0.574468	−	0.574468i	−0.358906	−	0.933374i	\(-0.616850\pi\)
							0.933374	+	0.358906i	\(0.116850\pi\)
\(53\)	−12.0000	−	12.0000i	−0.226415	−	0.226415i	0.584778	−	0.811193i	\(-0.301182\pi\)
							−0.811193	+	0.584778i	\(0.801182\pi\)
\(59\)		−	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	−0.684364	−	0.729140i	\(-0.739920\pi\)
							0.729140	+	0.684364i	\(0.239920\pi\)
\(71\)	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	0.784505	−	0.620122i	\(-0.212917\pi\)
							−0.620122	+	0.784505i	\(0.712917\pi\)
\(79\)		−	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	\(0.0411167\pi\)
							0.128813	+	0.991669i	\(0.458883\pi\)
\(89\)			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	−0.766252	−	0.642540i	\(-0.777880\pi\)
							0.642540	+	0.766252i	\(0.277880\pi\)

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

    

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