Embedded newform 90.5.g.b.37.1

• Label: 90.5.g.b.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.b.73.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.00000 + 2.00000i) q^{2} +8.00000i q^{4} +(15.0000 - 20.0000i) q^{5} +(29.0000 + 29.0000i) q^{7} +(-16.0000 + 16.0000i) q^{8} +(70.0000 - 10.0000i) q^{10} +118.000 q^{11} +(69.0000 - 69.0000i) q^{13} +116.000i q^{14} -64.0000 q^{16} +(271.000 + 271.000i) q^{17} +280.000i q^{19} +(160.000 + 120.000i) q^{20} +(236.000 + 236.000i) q^{22} +(-269.000 + 269.000i) q^{23} +(-175.000 - 600.000i) q^{25} +276.000 q^{26} +(-232.000 + 232.000i) q^{28} -680.000i q^{29} +202.000 q^{31} +(-128.000 - 128.000i) q^{32} +1084.00i q^{34} +(1015.00 - 145.000i) q^{35} +(-651.000 - 651.000i) q^{37} +(-560.000 + 560.000i) q^{38} +(80.0000 + 560.000i) q^{40} -1682.00 q^{41} +(1089.00 - 1089.00i) q^{43} +944.000i q^{44} -1076.00 q^{46} +(-1269.00 - 1269.00i) q^{47} -719.000i q^{49} +(850.000 - 1550.00i) q^{50} +(552.000 + 552.000i) q^{52} +(611.000 - 611.000i) q^{53} +(1770.00 - 2360.00i) q^{55} -928.000 q^{56} +(1360.00 - 1360.00i) q^{58} -1160.00i q^{59} -5598.00 q^{61} +(404.000 + 404.000i) q^{62} -512.000i q^{64} +(-345.000 - 2415.00i) q^{65} +(-751.000 - 751.000i) q^{67} +(-2168.00 + 2168.00i) q^{68} +(2320.00 + 1740.00i) q^{70} -6442.00 q^{71} +(-2951.00 + 2951.00i) q^{73} -2604.00i q^{74} -2240.00 q^{76} +(3422.00 + 3422.00i) q^{77} +10560.0i q^{79} +(-960.000 + 1280.00i) q^{80} +(-3364.00 - 3364.00i) q^{82} +(6231.00 - 6231.00i) q^{83} +(9485.00 - 1355.00i) q^{85} +4356.00 q^{86} +(-1888.00 + 1888.00i) q^{88} +14480.0i q^{89} +4002.00 q^{91} +(-2152.00 - 2152.00i) q^{92} -5076.00i q^{94} +(5600.00 + 4200.00i) q^{95} +(-7311.00 - 7311.00i) q^{97} +(1438.00 - 1438.00i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 4 * q^2 + 30 * q^5 + 58 * q^7 - 32 * q^8 + 140 * q^10 + 236 * q^11 + 138 * q^13 - 128 * q^16 + 542 * q^17 + 320 * q^20 + 472 * q^22 - 538 * q^23 - 350 * q^25 + 552 * q^26 - 464 * q^28 + 404 * q^31 - 256 * q^32 + 2030 * q^35 - 1302 * q^37 - 1120 * q^38 + 160 * q^40 - 3364 * q^41 + 2178 * q^43 - 2152 * q^46 - 2538 * q^47 + 1700 * q^50 + 1104 * q^52 + 1222 * q^53 + 3540 * q^55 - 1856 * q^56 + 2720 * q^58 - 11196 * q^61 + 808 * q^62 - 690 * q^65 - 1502 * q^67 - 4336 * q^68 + 4640 * q^70 - 12884 * q^71 - 5902 * q^73 - 4480 * q^76 + 6844 * q^77 - 1920 * q^80 - 6728 * q^82 + 12462 * q^83 + 18970 * q^85 + 8712 * q^86 - 3776 * q^88 + 8004 * q^91 - 4304 * q^92 + 11200 * q^95 - 14622 * q^97 + 2876 * 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
							\(7\)	29.0000	+	29.0000i	0.591837	+	0.591837i	0.938127	−	0.346291i	\(-0.112559\pi\)
−0.346291	+	0.938127i	\(0.612559\pi\)
\(11\)	118.000			0.975207			0.487603	−	0.873065i	\(-0.337871\pi\)
							0.487603	+	0.873065i	\(0.337871\pi\)
\(13\)	69.0000	−	69.0000i	0.408284	−	0.408284i	−0.472856	−	0.881140i	\(-0.656777\pi\)
							0.881140	+	0.472856i	\(0.156777\pi\)
\(17\)	271.000	+	271.000i	0.937716	+	0.937716i	0.998171	−	0.0604547i	\(-0.0192551\pi\)
							−0.0604547	+	0.998171i	\(0.519255\pi\)
\(19\)			280.000i			0.775623i	0.921739	+	0.387812i	\(0.126769\pi\)
							−0.921739	+	0.387812i	\(0.873231\pi\)
\(23\)	−269.000	+	269.000i	−0.508507	+	0.508507i	−0.914068	−	0.405561i	\(-0.867076\pi\)
							0.405561	+	0.914068i	\(0.367076\pi\)
\(29\)		−	680.000i		−	0.808561i	−0.914635	−	0.404281i	\(-0.867522\pi\)
							0.914635	−	0.404281i	\(-0.132478\pi\)
\(31\)	202.000			0.210198			0.105099	−	0.994462i	\(-0.466484\pi\)
							0.105099	+	0.994462i	\(0.466484\pi\)
\(37\)	−651.000	−	651.000i	−0.475530	−	0.475530i	0.428169	−	0.903699i	\(-0.359159\pi\)
							−0.903699	+	0.428169i	\(0.859159\pi\)
\(41\)	−1682.00			−1.00059			−0.500297	−	0.865854i	\(-0.666776\pi\)
							−0.500297	+	0.865854i	\(0.666776\pi\)
\(43\)	1089.00	−	1089.00i	0.588967	−	0.588967i	−0.348385	−	0.937352i	\(-0.613270\pi\)
							0.937352	+	0.348385i	\(0.113270\pi\)
\(47\)	−1269.00	−	1269.00i	−0.574468	−	0.574468i	0.358906	−	0.933374i	\(-0.383150\pi\)
							−0.933374	+	0.358906i	\(0.883150\pi\)
\(53\)	611.000	−	611.000i	0.217515	−	0.217515i	−0.589935	−	0.807450i	\(-0.700847\pi\)
							0.807450	+	0.589935i	\(0.200847\pi\)
\(59\)		−	1160.00i		−	0.333238i	−0.986021	−	0.166619i	\(-0.946715\pi\)
							0.986021	−	0.166619i	\(-0.0532849\pi\)
\(61\)	−5598.00			−1.50443			−0.752217	−	0.658915i	\(-0.771016\pi\)
							−0.752217	+	0.658915i	\(0.771016\pi\)
\(67\)	−751.000	−	751.000i	−0.167298	−	0.167298i	0.618493	−	0.785791i	\(-0.287744\pi\)
							−0.785791	+	0.618493i	\(0.787744\pi\)
\(71\)	−6442.00			−1.27792			−0.638961	−	0.769240i	\(-0.720635\pi\)
							−0.638961	+	0.769240i	\(0.720635\pi\)
\(73\)	−2951.00	+	2951.00i	−0.553762	+	0.553762i	−0.927525	−	0.373762i	\(-0.878068\pi\)
							0.373762	+	0.927525i	\(0.378068\pi\)
\(79\)			10560.0i			1.69204i	0.533154	+	0.846018i	\(0.321007\pi\)
							−0.533154	+	0.846018i	\(0.678993\pi\)
\(83\)	6231.00	−	6231.00i	0.904485	−	0.904485i	−0.0913348	−	0.995820i	\(-0.529113\pi\)
							0.995820	+	0.0913348i	\(0.0291134\pi\)
\(89\)			14480.0i			1.82805i	0.405656	+	0.914026i	\(0.367043\pi\)
							−0.405656	+	0.914026i	\(0.632957\pi\)
\(97\)	−7311.00	−	7311.00i	−0.777022	−	0.777022i	0.202301	−	0.979323i	\(-0.435158\pi\)
							−0.979323	+	0.202301i	\(0.935158\pi\)

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

    

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