Embedded newform 450.5.g.b.307.1

• Label: 450.5.g.b.307.1
• Level: $450$
• Weight: $5$
• Character: 450.307
• 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			307.1
Root			\(1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	450.307
Dual form			450.5.g.b.343.1

$q$-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 \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

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

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.00000	+	2.00000i	0.500000	+	0.500000i
							\(3\)		0			0
\(5\)		0			0
							\(7\)	19.0000	+	19.0000i	0.387755	+	0.387755i	0.873886	−	0.486131i	\(-0.161592\pi\)
−0.486131	+	0.873886i	\(0.661592\pi\)
\(11\)	−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	−0.350692	−	0.936491i	\(-0.614054\pi\)
							0.936491	+	0.350692i	\(0.114054\pi\)
\(17\)	−239.000	−	239.000i	−0.826990	−	0.826990i	0.160110	−	0.987099i	\(-0.448815\pi\)
							−0.987099	+	0.160110i	\(0.948815\pi\)
\(19\)			40.0000i			0.110803i	0.998464	+	0.0554017i	\(0.0176439\pi\)
							−0.998464	+	0.0554017i	\(0.982356\pi\)
\(23\)	541.000	−	541.000i	1.02268	−	1.02268i	0.0229476	−	0.999737i	\(-0.492695\pi\)
							0.999737	−	0.0229476i	\(-0.00730510\pi\)
\(29\)			200.000i			0.237812i	0.992906	+	0.118906i	\(0.0379387\pi\)
							−0.992906	+	0.118906i	\(0.962061\pi\)
\(31\)	−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	0.653732	−	0.756726i	\(-0.273203\pi\)
							−0.756726	+	0.653732i	\(0.773203\pi\)
\(41\)	−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	−0.471418	−	0.881910i	\(-0.656258\pi\)
							0.881910	+	0.471418i	\(0.156258\pi\)
\(47\)	−459.000	−	459.000i	−0.207786	−	0.207786i	0.595540	−	0.803326i	\(-0.296938\pi\)
							−0.803326	+	0.595540i	\(0.796938\pi\)
\(53\)	−1819.00	+	1819.00i	−0.647561	+	0.647561i	−0.952403	−	0.304842i	\(-0.901396\pi\)
							0.304842	+	0.952403i	\(0.401396\pi\)
\(59\)		−	4600.00i		−	1.32146i	−0.750624	−	0.660730i	\(-0.770247\pi\)
							0.750624	−	0.660730i	\(-0.229753\pi\)
\(61\)	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	−0.989865	−	0.142013i	\(-0.954642\pi\)
							−0.142013	−	0.989865i	\(-0.545358\pi\)
\(71\)	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	−0.300834	−	0.953677i	\(-0.597265\pi\)
							0.953677	+	0.300834i	\(0.0972649\pi\)
\(79\)		−	7680.00i		−	1.23057i	−0.788304	−	0.615286i	\(-0.789041\pi\)
							0.788304	−	0.615286i	\(-0.210959\pi\)
\(83\)	6081.00	−	6081.00i	0.882712	−	0.882712i	−0.111098	−	0.993809i	\(-0.535437\pi\)
							0.993809	+	0.111098i	\(0.0354367\pi\)
\(89\)			5680.00i			0.717081i	0.933514	+	0.358541i	\(0.116726\pi\)
							−0.933514	+	0.358541i	\(0.883274\pi\)
\(97\)	−561.000	−	561.000i	−0.0596238	−	0.0596238i	0.676666	−	0.736290i	\(-0.263424\pi\)
							−0.736290	+	0.676666i	\(0.763424\pi\)

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.5.c.b.3.1	✓	2	15.2	even	4
10.5.c.b.7.1	yes	2	15.14	odd	2
50.5.c.a.7.1		2	3.2	odd	2
50.5.c.a.43.1		2	15.8	even	4
80.5.p.c.17.1		2	60.59	even	2
80.5.p.c.33.1		2	60.47	odd	4
90.5.g.a.37.1		2	5.4	even	2
90.5.g.a.73.1		2	5.2	odd	4
320.5.p.d.193.1		2	120.77	even	4
320.5.p.d.257.1		2	120.29	odd	2
320.5.p.g.193.1		2	120.107	odd	4
320.5.p.g.257.1		2	120.59	even	2
400.5.p.b.193.1		2	60.23	odd	4
400.5.p.b.257.1		2	12.11	even	2
450.5.g.b.307.1		2	1.1	even	1	trivial
450.5.g.b.343.1		2	5.3	odd	4	inner