Embedded newform 90.11.g.c.73.3

• Label: 90.11.g.c.73.3
• Level: $90$
• Weight: $11$
• Character: 90.73
• Analytic conductor: $57.182$
• Analytic rank: $0$
• Dimension: $6$
• 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:	\(6\)
Relative dimension:	\(3\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial:	\( x^{6} - 1148x^{3} + 68121x^{2} - 299628x + 658952 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{8}\cdot 3^{2}\cdot 5^{6} \)
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			73.3
Root			\(10.1043 + 10.1043i\) of defining polynomial
Character	\(\chi\)	\(=\)	90.73
Dual form			90.11.g.c.37.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(16.0000 - 16.0000i) q^{2} -512.000i q^{4} +(2978.33 - 946.124i) q^{5} +(21284.7 - 21284.7i) q^{7} +(-8192.00 - 8192.00i) q^{8} +(32515.4 - 62791.3i) q^{10} -155649. q^{11} +(-358614. - 358614. i) q^{13} -681110. i q^{14} -262144. q^{16} +(609397. - 609397. i) q^{17} -335226. i q^{19} +(-484415. - 1.52491e6i) q^{20} +(-2.49038e6 + 2.49038e6i) q^{22} +(5.50132e6 + 5.50132e6i) q^{23} +(7.97532e6 - 5.63575e6i) q^{25} -1.14757e7 q^{26} +(-1.08978e7 - 1.08978e7i) q^{28} +1.33815e6i q^{29} +2.59306e7 q^{31} +(-4.19430e6 + 4.19430e6i) q^{32} -1.95007e7i q^{34} +(4.32550e7 - 8.35308e7i) q^{35} +(-5.50890e7 + 5.50890e7i) q^{37} +(-5.36362e6 - 5.36362e6i) q^{38} +(-3.21492e7 - 1.66479e7i) q^{40} -1.37064e8 q^{41} +(-9.34290e7 - 9.34290e7i) q^{43} +7.96921e7i q^{44} +1.76042e8 q^{46} +(-1.14979e8 + 1.14979e8i) q^{47} -6.23600e8i q^{49} +(3.74333e7 - 2.17777e8i) q^{50} +(-1.83611e8 + 1.83611e8i) q^{52} +(2.29978e7 + 2.29978e7i) q^{53} +(-4.63574e8 + 1.47263e8i) q^{55} -3.48728e8 q^{56} +(2.14104e7 + 2.14104e7i) q^{58} -8.73535e8i q^{59} +5.87905e8 q^{61} +(4.14890e8 - 4.14890e8i) q^{62} +1.34218e8i q^{64} +(-1.40737e9 - 7.28780e8i) q^{65} +(-6.75197e8 + 6.75197e8i) q^{67} +(-3.12011e8 - 3.12011e8i) q^{68} +(-6.44414e8 - 2.02857e9i) q^{70} +5.54597e8 q^{71} +(8.91747e8 + 8.91747e8i) q^{73} +1.76285e9i q^{74} -1.71636e8 q^{76} +(-3.31293e9 + 3.31293e9i) q^{77} +1.69149e9i q^{79} +(-7.80752e8 + 2.48021e8i) q^{80} +(-2.19302e9 + 2.19302e9i) q^{82} +(-1.96163e9 - 1.96163e9i) q^{83} +(1.23842e9 - 2.39155e9i) q^{85} -2.98973e9 q^{86} +(1.27507e9 + 1.27507e9i) q^{88} -7.73241e9i q^{89} -1.52660e10 q^{91} +(2.81667e9 - 2.81667e9i) q^{92} +3.67931e9i q^{94} +(-3.17166e8 - 9.98416e8i) q^{95} +(1.16280e10 - 1.16280e10i) q^{97} +(-9.97760e9 - 9.97760e9i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	6 * q + 96 * q^2 - 5460 * q^5 + 13512 * q^7 - 49152 * q^8 - 173280 * q^10 - 647832 * q^11 - 742902 * q^13 - 1572864 * q^16 + 755118 * q^17 - 2749440 * q^20 - 10365312 * q^22 + 15052992 * q^23 + 42644850 * q^25 - 23772864 * q^26 - 6918144 * q^28 + 153847152 * q^31 - 25165824 * q^32 + 208942440 * q^35 - 32574498 * q^37 + 17493120 * q^38 + 737280 * q^40 - 226153272 * q^41 - 63628752 * q^43 + 481695744 * q^46 + 19700448 * q^47 + 788800800 * q^50 - 380365824 * q^52 + 950001042 * q^53 - 135145080 * q^55 - 221380608 * q^56 - 310652160 * q^58 + 1603984392 * q^61 + 2461554432 * q^62 - 1398176070 * q^65 + 2502647712 * q^67 - 386620416 * q^68 + 174645120 * q^70 + 7137533808 * q^71 + 2304462438 * q^73 + 559779840 * q^76 - 597969864 * q^77 + 1431306240 * q^80 - 3618452352 * q^82 + 88180632 * q^83 + 1729841610 * q^85 - 2036120064 * q^86 + 5307039744 * q^88 - 43718253408 * q^91 + 7707131904 * q^92 + 34456200 * q^95 + 33281088582 * q^97 - 12874047264 * q^98

Character values

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

\(n\)	\(11\)	\(37\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	16.0000	−	16.0000i	0.500000	−	0.500000i
							\(3\)		0			0
\(5\)	2978.33	−	946.124i	0.953067	−	0.302760i
							\(7\)	21284.7	−	21284.7i	1.26642	−	1.26642i	0.318492	−	0.947926i	\(-0.396824\pi\)
0.947926	−	0.318492i	\(-0.103176\pi\)
\(11\)	−155649.			−0.966456			−0.483228	−	0.875495i	\(-0.660536\pi\)
							−0.483228	+	0.875495i	\(0.660536\pi\)
\(13\)	−358614.	−	358614.i	−0.965853	−	0.965853i	0.0335834	−	0.999436i	\(-0.489308\pi\)
							−0.999436	+	0.0335834i	\(0.989308\pi\)
\(17\)	609397.	−	609397.i	0.429196	−	0.429196i	−0.459158	−	0.888354i	\(-0.651849\pi\)
							0.888354	+	0.459158i	\(0.151849\pi\)
\(19\)		−	335226.i		−	0.135385i	−0.997706	−	0.0676924i	\(-0.978436\pi\)
							0.997706	−	0.0676924i	\(-0.0215637\pi\)
\(23\)	5.50132e6	+	5.50132e6i	0.854727	+	0.854727i	0.990711	−	0.135984i	\(-0.0434196\pi\)
							−0.135984	+	0.990711i	\(0.543420\pi\)
\(29\)			1.33815e6i			0.0652402i	0.999468	+	0.0326201i	\(0.0103851\pi\)
							−0.999468	+	0.0326201i	\(0.989615\pi\)
\(31\)	2.59306e7			0.905743			0.452871	−	0.891576i	\(-0.350400\pi\)
							0.452871	+	0.891576i	\(0.350400\pi\)
\(37\)	−5.50890e7	+	5.50890e7i	−0.794431	+	0.794431i	−0.982211	−	0.187780i	\(-0.939871\pi\)
							0.187780	+	0.982211i	\(0.439871\pi\)
\(41\)	−1.37064e8			−1.18305			−0.591525	−	0.806287i	\(-0.701474\pi\)
							−0.591525	+	0.806287i	\(0.701474\pi\)
\(43\)	−9.34290e7	−	9.34290e7i	−0.635535	−	0.635535i	0.313916	−	0.949451i	\(-0.398359\pi\)
							−0.949451	+	0.313916i	\(0.898359\pi\)
\(47\)	−1.14979e8	+	1.14979e8i	−0.501334	+	0.501334i	−0.911852	−	0.410518i	\(-0.865348\pi\)
							0.410518	+	0.911852i	\(0.365348\pi\)
\(53\)	2.29978e7	+	2.29978e7i	0.0549930	+	0.0549930i	0.734068	−	0.679075i	\(-0.237619\pi\)
							−0.679075	+	0.734068i	\(0.737619\pi\)
\(59\)		−	8.73535e8i		−	1.22186i	−0.791686	−	0.610929i	\(-0.790796\pi\)
							0.791686	−	0.610929i	\(-0.209204\pi\)
\(61\)	5.87905e8			0.696079			0.348039	−	0.937480i	\(-0.386848\pi\)
							0.348039	+	0.937480i	\(0.386848\pi\)
\(67\)	−6.75197e8	+	6.75197e8i	−0.500100	+	0.500100i	−0.911469	−	0.411369i	\(-0.865051\pi\)
							0.411369	+	0.911469i	\(0.365051\pi\)
\(71\)	5.54597e8			0.307387			0.153694	−	0.988119i	\(-0.450883\pi\)
							0.153694	+	0.988119i	\(0.450883\pi\)
\(73\)	8.91747e8	+	8.91747e8i	0.430158	+	0.430158i	0.888682	−	0.458524i	\(-0.151622\pi\)
							−0.458524	+	0.888682i	\(0.651622\pi\)
\(79\)			1.69149e9i			0.549711i	0.961486	+	0.274856i	\(0.0886300\pi\)
							−0.961486	+	0.274856i	\(0.911370\pi\)
\(83\)	−1.96163e9	−	1.96163e9i	−0.497996	−	0.497996i	0.412818	−	0.910814i	\(-0.364545\pi\)
							−0.910814	+	0.412818i	\(0.864545\pi\)
\(89\)		−	7.73241e9i		−	1.38473i	−0.721548	−	0.692364i	\(-0.756569\pi\)
							0.721548	−	0.692364i	\(-0.243431\pi\)
\(97\)	1.16280e10	−	1.16280e10i	1.35408	−	1.35408i	0.473039	−	0.881041i	\(-0.343157\pi\)
							0.881041	−	0.473039i	\(-0.156843\pi\)

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.c.73.3		6
3.2	odd	2		10.11.c.c.3.2	✓	6
5.2	odd	4	inner	90.11.g.c.37.3		6
12.11	even	2		80.11.p.c.33.2		6
15.2	even	4		10.11.c.c.7.2	yes	6
15.8	even	4		50.11.c.e.7.2		6
15.14	odd	2		50.11.c.e.43.2		6
60.47	odd	4		80.11.p.c.17.2		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.11.c.c.3.2	✓	6	3.2	odd	2
10.11.c.c.7.2	yes	6	15.2	even	4
50.11.c.e.7.2		6	15.8	even	4
50.11.c.e.43.2		6	15.14	odd	2
80.11.p.c.17.2		6	60.47	odd	4
80.11.p.c.33.2		6	12.11	even	2
90.11.g.c.37.3		6	5.2	odd	4	inner
90.11.g.c.73.3		6	1.1	even	1	trivial