Embedded newform 450.5.g.b.343.1

• Label: 450.5.g.b.343.1
• Level: $450$
• Weight: $5$
• Character: 450.343
• Analytic conductor: $46.516$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(46.5164833877\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 10)
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			450.5.g.b.307.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.00000 - 2.00000i) q^{2} -8.00000i q^{4} +(19.0000 - 19.0000i) q^{7} +(-16.0000 - 16.0000i) q^{8} -202.000 q^{11} +(99.0000 + 99.0000i) q^{13} -76.0000i q^{14} -64.0000 q^{16} +(-239.000 + 239.000i) q^{17} -40.0000i q^{19} +(-404.000 + 404.000i) q^{22} +(541.000 + 541.000i) q^{23} +396.000 q^{26} +(-152.000 - 152.000i) q^{28} -200.000i q^{29} -758.000 q^{31} +(-128.000 + 128.000i) q^{32} +956.000i q^{34} +(-141.000 + 141.000i) q^{37} +(-80.0000 - 80.0000i) q^{38} -1042.00 q^{41} +(759.000 + 759.000i) q^{43} +1616.00i q^{44} +2164.00 q^{46} +(-459.000 + 459.000i) q^{47} +1679.00i q^{49} +(792.000 - 792.000i) q^{52} +(-1819.00 - 1819.00i) q^{53} -608.000 q^{56} +(-400.000 - 400.000i) q^{58} +4600.00i q^{59} +2082.00 q^{61} +(-1516.00 + 1516.00i) q^{62} +512.000i q^{64} +(-5081.00 + 5081.00i) q^{67} +(1912.00 + 1912.00i) q^{68} +3478.00 q^{71} +(3479.00 + 3479.00i) q^{73} +564.000i q^{74} -320.000 q^{76} +(-3838.00 + 3838.00i) q^{77} +7680.00i q^{79} +(-2084.00 + 2084.00i) q^{82} +(6081.00 + 6081.00i) q^{83} +3036.00 q^{86} +(3232.00 + 3232.00i) q^{88} -5680.00i q^{89} +3762.00 q^{91} +(4328.00 - 4328.00i) q^{92} +1836.00i q^{94} +(-561.000 + 561.000i) q^{97} +(3358.00 + 3358.00i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 4 * q^2 + 38 * q^7 - 32 * q^8 - 404 * q^11 + 198 * q^13 - 128 * q^16 - 478 * q^17 - 808 * q^22 + 1082 * q^23 + 792 * q^26 - 304 * q^28 - 1516 * q^31 - 256 * q^32 - 282 * q^37 - 160 * q^38 - 2084 * q^41 + 1518 * q^43 + 4328 * q^46 - 918 * q^47 + 1584 * q^52 - 3638 * q^53 - 1216 * q^56 - 800 * q^58 + 4164 * q^61 - 3032 * q^62 - 10162 * q^67 + 3824 * q^68 + 6956 * q^71 + 6958 * q^73 - 640 * q^76 - 7676 * q^77 - 4168 * q^82 + 12162 * q^83 + 6072 * q^86 + 6464 * q^88 + 7524 * q^91 + 8656 * q^92 - 1122 * q^97 + 6716 * 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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.00000	−	2.00000i	0.500000	−	0.500000i
							\(3\)		0			0
\(5\)		0			0
							\(7\)	19.0000	−	19.0000i	0.387755	−	0.387755i	−0.486131	−	0.873886i	\(-0.661592\pi\)
0.873886	+	0.486131i	\(0.161592\pi\)
\(11\)	−202.000			−1.66942			−0.834711	−	0.550689i	\(-0.814365\pi\)
							−0.834711	+	0.550689i	\(0.814365\pi\)
\(13\)	99.0000	+	99.0000i	0.585799	+	0.585799i	0.936491	−	0.350692i	\(-0.114054\pi\)
							−0.350692	+	0.936491i	\(0.614054\pi\)
\(17\)	−239.000	+	239.000i	−0.826990	+	0.826990i	−0.987099	−	0.160110i	\(-0.948815\pi\)
							0.160110	+	0.987099i	\(0.448815\pi\)
\(19\)		−	40.0000i		−	0.110803i	−0.998464	−	0.0554017i	\(-0.982356\pi\)
							0.998464	−	0.0554017i	\(-0.0176439\pi\)
\(23\)	541.000	+	541.000i	1.02268	+	1.02268i	0.999737	+	0.0229476i	\(0.00730510\pi\)
							0.0229476	+	0.999737i	\(0.492695\pi\)
\(29\)		−	200.000i		−	0.237812i	−0.992906	−	0.118906i	\(-0.962061\pi\)
							0.992906	−	0.118906i	\(-0.0379387\pi\)
\(31\)	−758.000			−0.788762			−0.394381	−	0.918947i	\(-0.629041\pi\)
							−0.394381	+	0.918947i	\(0.629041\pi\)
\(37\)	−141.000	+	141.000i	−0.102995	+	0.102995i	−0.756726	−	0.653732i	\(-0.773203\pi\)
							0.653732	+	0.756726i	\(0.273203\pi\)
\(41\)	−1042.00			−0.619869			−0.309935	−	0.950758i	\(-0.600307\pi\)
							−0.309935	+	0.950758i	\(0.600307\pi\)
\(43\)	759.000	+	759.000i	0.410492	+	0.410492i	0.881910	−	0.471418i	\(-0.156258\pi\)
							−0.471418	+	0.881910i	\(0.656258\pi\)
\(47\)	−459.000	+	459.000i	−0.207786	+	0.207786i	−0.803326	−	0.595540i	\(-0.796938\pi\)
							0.595540	+	0.803326i	\(0.296938\pi\)
\(53\)	−1819.00	−	1819.00i	−0.647561	−	0.647561i	0.304842	−	0.952403i	\(-0.401396\pi\)
							−0.952403	+	0.304842i	\(0.901396\pi\)
\(59\)			4600.00i			1.32146i	0.750624	+	0.660730i	\(0.229753\pi\)
							−0.750624	+	0.660730i	\(0.770247\pi\)
\(61\)	2082.00			0.559527			0.279764	−	0.960069i	\(-0.409744\pi\)
							0.279764	+	0.960069i	\(0.409744\pi\)
\(67\)	−5081.00	+	5081.00i	−1.13188	+	1.13188i	−0.142013	+	0.989865i	\(0.545358\pi\)
							−0.989865	+	0.142013i	\(0.954642\pi\)
\(71\)	3478.00			0.689942			0.344971	−	0.938613i	\(-0.387889\pi\)
							0.344971	+	0.938613i	\(0.387889\pi\)
\(73\)	3479.00	+	3479.00i	0.652843	+	0.652843i	0.953677	−	0.300834i	\(-0.0972649\pi\)
							−0.300834	+	0.953677i	\(0.597265\pi\)
\(79\)			7680.00i			1.23057i	0.788304	+	0.615286i	\(0.210959\pi\)
							−0.788304	+	0.615286i	\(0.789041\pi\)
\(83\)	6081.00	+	6081.00i	0.882712	+	0.882712i	0.993809	−	0.111098i	\(-0.0354367\pi\)
							−0.111098	+	0.993809i	\(0.535437\pi\)
\(89\)		−	5680.00i		−	0.717081i	−0.933514	−	0.358541i	\(-0.883274\pi\)
							0.933514	−	0.358541i	\(-0.116726\pi\)
\(97\)	−561.000	+	561.000i	−0.0596238	+	0.0596238i	−0.736290	−	0.676666i	\(-0.763424\pi\)
							0.676666	+	0.736290i	\(0.263424\pi\)

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	450.5.g.b.343.1		2
3.2	odd	2		50.5.c.a.43.1		2
5.2	odd	4	inner	450.5.g.b.307.1		2
5.3	odd	4		90.5.g.a.37.1		2
5.4	even	2		90.5.g.a.73.1		2
12.11	even	2		400.5.p.b.193.1		2
15.2	even	4		50.5.c.a.7.1		2
15.8	even	4		10.5.c.b.7.1	yes	2
15.14	odd	2		10.5.c.b.3.1	✓	2
60.23	odd	4		80.5.p.c.17.1		2
60.47	odd	4		400.5.p.b.257.1		2
60.59	even	2		80.5.p.c.33.1		2
120.29	odd	2		320.5.p.d.193.1		2
120.53	even	4		320.5.p.d.257.1		2
120.59	even	2		320.5.p.g.193.1		2
120.83	odd	4		320.5.p.g.257.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.5.c.b.3.1	✓	2	15.14	odd	2
10.5.c.b.7.1	yes	2	15.8	even	4
50.5.c.a.7.1		2	15.2	even	4
50.5.c.a.43.1		2	3.2	odd	2
80.5.p.c.17.1		2	60.23	odd	4
80.5.p.c.33.1		2	60.59	even	2
90.5.g.a.37.1		2	5.3	odd	4
90.5.g.a.73.1		2	5.4	even	2
320.5.p.d.193.1		2	120.29	odd	2
320.5.p.d.257.1		2	120.53	even	4
320.5.p.g.193.1		2	120.59	even	2
320.5.p.g.257.1		2	120.83	odd	4
400.5.p.b.193.1		2	12.11	even	2
400.5.p.b.257.1		2	60.47	odd	4
450.5.g.b.307.1		2	5.2	odd	4	inner
450.5.g.b.343.1		2	1.1	even	1	trivial