Embedded newform 90.5.g.a.37.1

• Label: 90.5.g.a.37.1
• Level: $90$
• Weight: $5$
• Character: 90.37
• Analytic conductor: $9.303$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	90.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.30329667755\)
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			37.1
Root			\(1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	90.37
Dual form			90.5.g.a.73.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.00000 - 2.00000i) q^{2} +8.00000i q^{4} +(15.0000 + 20.0000i) q^{5} +(-19.0000 - 19.0000i) q^{7} +(16.0000 - 16.0000i) q^{8} +(10.0000 - 70.0000i) q^{10} -202.000 q^{11} +(-99.0000 + 99.0000i) q^{13} +76.0000i q^{14} -64.0000 q^{16} +(239.000 + 239.000i) q^{17} +40.0000i q^{19} +(-160.000 + 120.000i) q^{20} +(404.000 + 404.000i) q^{22} +(-541.000 + 541.000i) q^{23} +(-175.000 + 600.000i) q^{25} +396.000 q^{26} +(152.000 - 152.000i) q^{28} +200.000i q^{29} -758.000 q^{31} +(128.000 + 128.000i) q^{32} -956.000i q^{34} +(95.0000 - 665.000i) q^{35} +(141.000 + 141.000i) q^{37} +(80.0000 - 80.0000i) q^{38} +(560.000 + 80.0000i) q^{40} -1042.00 q^{41} +(-759.000 + 759.000i) q^{43} -1616.00i q^{44} +2164.00 q^{46} +(459.000 + 459.000i) q^{47} -1679.00i q^{49} +(1550.00 - 850.000i) q^{50} +(-792.000 - 792.000i) q^{52} +(1819.00 - 1819.00i) q^{53} +(-3030.00 - 4040.00i) q^{55} -608.000 q^{56} +(400.000 - 400.000i) q^{58} -4600.00i q^{59} +2082.00 q^{61} +(1516.00 + 1516.00i) q^{62} -512.000i q^{64} +(-3465.00 - 495.000i) q^{65} +(5081.00 + 5081.00i) q^{67} +(-1912.00 + 1912.00i) q^{68} +(-1520.00 + 1140.00i) q^{70} +3478.00 q^{71} +(-3479.00 + 3479.00i) q^{73} -564.000i q^{74} -320.000 q^{76} +(3838.00 + 3838.00i) q^{77} -7680.00i q^{79} +(-960.000 - 1280.00i) q^{80} +(2084.00 + 2084.00i) q^{82} +(-6081.00 + 6081.00i) q^{83} +(-1195.00 + 8365.00i) q^{85} +3036.00 q^{86} +(-3232.00 + 3232.00i) q^{88} +5680.00i q^{89} +3762.00 q^{91} +(-4328.00 - 4328.00i) q^{92} -1836.00i q^{94} +(-800.000 + 600.000i) q^{95} +(561.000 + 561.000i) q^{97} +(-3358.00 + 3358.00i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 4 * q^2 + 30 * q^5 - 38 * q^7 + 32 * q^8 + 20 * q^10 - 404 * q^11 - 198 * q^13 - 128 * q^16 + 478 * q^17 - 320 * q^20 + 808 * q^22 - 1082 * q^23 - 350 * q^25 + 792 * q^26 + 304 * q^28 - 1516 * q^31 + 256 * q^32 + 190 * q^35 + 282 * q^37 + 160 * q^38 + 1120 * q^40 - 2084 * q^41 - 1518 * q^43 + 4328 * q^46 + 918 * q^47 + 3100 * q^50 - 1584 * q^52 + 3638 * q^53 - 6060 * q^55 - 1216 * q^56 + 800 * q^58 + 4164 * q^61 + 3032 * q^62 - 6930 * q^65 + 10162 * q^67 - 3824 * q^68 - 3040 * q^70 + 6956 * q^71 - 6958 * q^73 - 640 * q^76 + 7676 * q^77 - 1920 * q^80 + 4168 * q^82 - 12162 * q^83 - 2390 * q^85 + 6072 * q^86 - 6464 * q^88 + 7524 * q^91 - 8656 * q^92 - 1600 * q^95 + 1122 * q^97 - 6716 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\)	\(11\)	\(37\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{4}\right)\)

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
\(5\)	15.0000	+	20.0000i	0.600000	+	0.800000i
							\(7\)	−19.0000	−	19.0000i	−0.387755	−	0.387755i	0.486131	−	0.873886i	\(-0.338408\pi\)
−0.873886	+	0.486131i	\(0.838408\pi\)
\(11\)	−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	\(-0.885946\pi\)
							0.350692	+	0.936491i	\(0.385946\pi\)
\(17\)	239.000	+	239.000i	0.826990	+	0.826990i	0.987099	−	0.160110i	\(-0.0511848\pi\)
							−0.160110	+	0.987099i	\(0.551185\pi\)
\(19\)			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	\(0.507305\pi\)
							−0.999737	+	0.0229476i	\(0.992695\pi\)
\(29\)			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	0.756726	−	0.653732i	\(-0.226797\pi\)
							−0.653732	+	0.756726i	\(0.726797\pi\)
\(41\)	−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	\(-0.843742\pi\)
							0.471418	+	0.881910i	\(0.343742\pi\)
\(47\)	459.000	+	459.000i	0.207786	+	0.207786i	0.803326	−	0.595540i	\(-0.203062\pi\)
							−0.595540	+	0.803326i	\(0.703062\pi\)
\(53\)	1819.00	−	1819.00i	0.647561	−	0.647561i	−0.304842	−	0.952403i	\(-0.598604\pi\)
							0.952403	+	0.304842i	\(0.0986035\pi\)
\(59\)		−	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	\(0.0453575\pi\)
							0.142013	+	0.989865i	\(0.454642\pi\)
\(71\)	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	\(-0.902735\pi\)
							0.300834	+	0.953677i	\(0.402735\pi\)
\(79\)		−	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	−0.993809	−	0.111098i	\(-0.964563\pi\)
							0.111098	+	0.993809i	\(0.464563\pi\)
\(89\)			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	0.736290	−	0.676666i	\(-0.236576\pi\)
							−0.676666	+	0.736290i	\(0.736576\pi\)

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	90.5.g.a.37.1		2
3.2	odd	2		10.5.c.b.7.1	yes	2
5.2	odd	4		450.5.g.b.343.1		2
5.3	odd	4	inner	90.5.g.a.73.1		2
5.4	even	2		450.5.g.b.307.1		2
12.11	even	2		80.5.p.c.17.1		2
15.2	even	4		50.5.c.a.43.1		2
15.8	even	4		10.5.c.b.3.1	✓	2
15.14	odd	2		50.5.c.a.7.1		2
24.5	odd	2		320.5.p.d.257.1		2
24.11	even	2		320.5.p.g.257.1		2
60.23	odd	4		80.5.p.c.33.1		2
60.47	odd	4		400.5.p.b.193.1		2
60.59	even	2		400.5.p.b.257.1		2
120.53	even	4		320.5.p.d.193.1		2
120.83	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	15.8	even	4
10.5.c.b.7.1	yes	2	3.2	odd	2
50.5.c.a.7.1		2	15.14	odd	2
50.5.c.a.43.1		2	15.2	even	4
80.5.p.c.17.1		2	12.11	even	2
80.5.p.c.33.1		2	60.23	odd	4
90.5.g.a.37.1		2	1.1	even	1	trivial
90.5.g.a.73.1		2	5.3	odd	4	inner
320.5.p.d.193.1		2	120.53	even	4
320.5.p.d.257.1		2	24.5	odd	2
320.5.p.g.193.1		2	120.83	odd	4
320.5.p.g.257.1		2	24.11	even	2
400.5.p.b.193.1		2	60.47	odd	4
400.5.p.b.257.1		2	60.59	even	2
450.5.g.b.307.1		2	5.4	even	2
450.5.g.b.343.1		2	5.2	odd	4