Embedded newform 90.11.g.a.37.1

• Label: 90.11.g.a.37.1
• Level: $90$
• Weight: $11$
• Character: 90.37
• Analytic conductor: $57.182$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	90.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(57.1821527406\)
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			90.11.g.a.73.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-16.0000 - 16.0000i) q^{2} +512.000i q^{4} +(-2925.00 - 1100.00i) q^{5} +(6953.00 + 6953.00i) q^{7} +(8192.00 - 8192.00i) q^{8} +(29200.0 + 64400.0i) q^{10} -75242.0 q^{11} +(109857. - 109857. i) q^{13} -222496. i q^{14} -262144. q^{16} +(1.52893e6 + 1.52893e6i) q^{17} -4.03868e6i q^{19} +(563200. - 1.49760e6i) q^{20} +(1.20387e6 + 1.20387e6i) q^{22} +(712423. - 712423. i) q^{23} +(7.34562e6 + 6.43500e6i) q^{25} -3.51542e6 q^{26} +(-3.55994e6 + 3.55994e6i) q^{28} -446120. i q^{29} -2.90807e7 q^{31} +(4.19430e6 + 4.19430e6i) q^{32} -4.89257e7i q^{34} +(-1.26892e7 - 2.79858e7i) q^{35} +(-911847. - 911847. i) q^{37} +(-6.46189e7 + 6.46189e7i) q^{38} +(-3.29728e7 + 1.49504e7i) q^{40} +1.63946e8 q^{41} +(1.18423e8 - 1.18423e8i) q^{43} -3.85239e7i q^{44} -2.27975e7 q^{46} +(-2.76320e8 - 2.76320e8i) q^{47} -1.85787e8i q^{49} +(-1.45700e7 - 2.20490e8i) q^{50} +(5.62468e7 + 5.62468e7i) q^{52} +(-3.08460e8 + 3.08460e8i) q^{53} +(2.20083e8 + 8.27662e7i) q^{55} +1.13918e8 q^{56} +(-7.13792e6 + 7.13792e6i) q^{58} +9.40888e8i q^{59} -1.35361e9 q^{61} +(4.65291e8 + 4.65291e8i) q^{62} -1.34218e8i q^{64} +(-4.42174e8 + 2.00489e8i) q^{65} +(8.53571e8 + 8.53571e8i) q^{67} +(-7.82811e8 + 7.82811e8i) q^{68} +(-2.44746e8 + 6.50801e8i) q^{70} -2.82701e9 q^{71} +(-2.75330e9 + 2.75330e9i) q^{73} +2.91791e7i q^{74} +2.06780e9 q^{76} +(-5.23158e8 - 5.23158e8i) q^{77} +3.32450e9i q^{79} +(7.66771e8 + 2.88358e8i) q^{80} +(-2.62313e9 - 2.62313e9i) q^{82} +(-1.34634e9 + 1.34634e9i) q^{83} +(-2.79029e9 - 6.15393e9i) q^{85} -3.78953e9 q^{86} +(-6.16382e8 + 6.16382e8i) q^{88} +2.66745e9i q^{89} +1.52767e9 q^{91} +(3.64761e8 + 3.64761e8i) q^{92} +8.84225e9i q^{94} +(-4.44255e9 + 1.18131e10i) q^{95} +(-5.26563e8 - 5.26563e8i) q^{97} +(-2.97259e9 + 2.97259e9i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 32 * q^2 - 5850 * q^5 + 13906 * q^7 + 16384 * q^8 + 58400 * q^10 - 150484 * q^11 + 219714 * q^13 - 524288 * q^16 + 3057854 * q^17 + 1126400 * q^20 + 2407744 * q^22 + 1424846 * q^23 + 14691250 * q^25 - 7030848 * q^26 - 7119872 * q^28 - 58161436 * q^31 + 8388608 * q^32 - 25378450 * q^35 - 1823694 * q^37 - 129237760 * q^38 - 65945600 * q^40 + 327891356 * q^41 + 236845554 * q^43 - 45595072 * q^46 - 552640626 * q^47 - 29140000 * q^50 + 112493568 * q^52 - 616920194 * q^53 + 440165700 * q^55 + 227835904 * q^56 - 14275840 * q^58 - 2707220076 * q^61 + 930582976 * q^62 - 884348850 * q^65 + 1707141826 * q^67 - 1565621248 * q^68 - 489491200 * q^70 - 5654029124 * q^71 - 5506594366 * q^73 + 4135608320 * q^76 - 1046315252 * q^77 + 1533542400 * q^80 - 5246261696 * q^82 - 2692678194 * q^83 - 5580583550 * q^85 - 7579057728 * q^86 - 1232764928 * q^88 + 3055342884 * q^91 + 729521152 * q^92 - 8885096000 * q^95 - 1053125694 * q^97 - 5945178592 * 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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−16.0000	−	16.0000i	−0.500000	−	0.500000i
							\(3\)		0			0
\(5\)	−2925.00	−	1100.00i	−0.936000	−	0.352000i
							\(7\)	6953.00	+	6953.00i	0.413697	+	0.413697i	0.883024	−	0.469328i	\(-0.155504\pi\)
−0.469328	+	0.883024i	\(0.655504\pi\)
\(11\)	−75242.0			−0.467194			−0.233597	−	0.972334i	\(-0.575050\pi\)
							−0.233597	+	0.972334i	\(0.575050\pi\)
\(13\)	109857.	−	109857.i	0.295877	−	0.295877i	−0.543520	−	0.839396i	\(-0.682909\pi\)
							0.839396	+	0.543520i	\(0.182909\pi\)
\(17\)	1.52893e6	+	1.52893e6i	1.07682	+	1.07682i	0.996793	+	0.0800247i	\(0.0254999\pi\)
							0.0800247	+	0.996793i	\(0.474500\pi\)
\(19\)		−	4.03868e6i		−	1.63107i	−0.578711	−	0.815533i	\(-0.696444\pi\)
							0.578711	−	0.815533i	\(-0.303556\pi\)
\(23\)	712423.	−	712423.i	0.110688	−	0.110688i	−0.649594	−	0.760281i	\(-0.725061\pi\)
							0.760281	+	0.649594i	\(0.225061\pi\)
\(29\)		−	446120.i		−	0.0217501i	−0.999941	−	0.0108751i	\(-0.996538\pi\)
							0.999941	−	0.0108751i	\(-0.00346171\pi\)
\(31\)	−2.90807e7			−1.01577			−0.507886	−	0.861424i	\(-0.669573\pi\)
							−0.507886	+	0.861424i	\(0.669573\pi\)
\(37\)	−911847.	−	911847.i	−0.0131496	−	0.0131496i	0.700501	−	0.713651i	\(-0.252960\pi\)
							−0.713651	+	0.700501i	\(0.752960\pi\)
\(41\)	1.63946e8			1.41508			0.707540	−	0.706674i	\(-0.249805\pi\)
							0.707540	+	0.706674i	\(0.249805\pi\)
\(43\)	1.18423e8	−	1.18423e8i	0.805551	−	0.805551i	−0.178406	−	0.983957i	\(-0.557094\pi\)
							0.983957	+	0.178406i	\(0.0570941\pi\)
\(47\)	−2.76320e8	−	2.76320e8i	−1.20482	−	1.20482i	−0.972682	−	0.232142i	\(-0.925427\pi\)
							−0.232142	−	0.972682i	\(-0.574573\pi\)
\(53\)	−3.08460e8	+	3.08460e8i	−0.737598	+	0.737598i	−0.972113	−	0.234515i	\(-0.924650\pi\)
							0.234515	+	0.972113i	\(0.424650\pi\)
\(59\)			9.40888e8i			1.31607i	0.752989	+	0.658034i	\(0.228612\pi\)
							−0.752989	+	0.658034i	\(0.771388\pi\)
\(61\)	−1.35361e9			−1.60267			−0.801336	−	0.598215i	\(-0.795877\pi\)
							−0.801336	+	0.598215i	\(0.795877\pi\)
\(67\)	8.53571e8	+	8.53571e8i	0.632216	+	0.632216i	0.948623	−	0.316407i	\(-0.102477\pi\)
							−0.316407	+	0.948623i	\(0.602477\pi\)
\(71\)	−2.82701e9			−1.56688			−0.783441	−	0.621466i	\(-0.786537\pi\)
							−0.783441	+	0.621466i	\(0.786537\pi\)
\(73\)	−2.75330e9	+	2.75330e9i	−1.32812	+	1.32812i	−0.421119	+	0.907005i	\(0.638363\pi\)
							−0.907005	+	0.421119i	\(0.861637\pi\)
\(79\)			3.32450e9i			1.08042i	0.841532	+	0.540208i	\(0.181654\pi\)
							−0.841532	+	0.540208i	\(0.818346\pi\)
\(83\)	−1.34634e9	+	1.34634e9i	−0.341794	+	0.341794i	−0.857041	−	0.515248i	\(-0.827700\pi\)
							0.515248	+	0.857041i	\(0.327700\pi\)
\(89\)			2.66745e9i			0.477690i	0.971058	+	0.238845i	\(0.0767688\pi\)
							−0.971058	+	0.238845i	\(0.923231\pi\)
\(97\)	−5.26563e8	−	5.26563e8i	−0.0613185	−	0.0613185i	0.675783	−	0.737101i	\(-0.263806\pi\)
							−0.737101	+	0.675783i	\(0.763806\pi\)

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	90.11.g.a.37.1		2
3.2	odd	2		10.11.c.b.7.1	yes	2
5.3	odd	4	inner	90.11.g.a.73.1		2
12.11	even	2		80.11.p.a.17.1		2
15.2	even	4		50.11.c.b.43.1		2
15.8	even	4		10.11.c.b.3.1	✓	2
15.14	odd	2		50.11.c.b.7.1		2
60.23	odd	4		80.11.p.a.33.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.11.c.b.3.1	✓	2	15.8	even	4
10.11.c.b.7.1	yes	2	3.2	odd	2
50.11.c.b.7.1		2	15.14	odd	2
50.11.c.b.43.1		2	15.2	even	4
80.11.p.a.17.1		2	12.11	even	2
80.11.p.a.33.1		2	60.23	odd	4
90.11.g.a.37.1		2	1.1	even	1	trivial
90.11.g.a.73.1		2	5.3	odd	4	inner