Embedded newform 90.11.g.c.73.2

• Label: 90.11.g.c.73.2
• 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			73.2
Root			\(2.29143 + 2.29143i\) of defining polynomial
Character	\(\chi\)	\(=\)	90.73
Dual form			90.11.g.c.37.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(16.0000 - 16.0000i) q^{2} -512.000i q^{4} +(-2583.39 - 1758.32i) q^{5} +(-13021.6 + 13021.6i) q^{7} +(-8192.00 - 8192.00i) q^{8} +(-69467.5 + 13201.2i) q^{10} -239729. q^{11} +(287980. + 287980. i) q^{13} +416690. i q^{14} -262144. q^{16} +(884527. - 884527. i) q^{17} +197744. i q^{19} +(-900261. + 1.32270e6i) q^{20} +(-3.83567e6 + 3.83567e6i) q^{22} +(3.55115e6 + 3.55115e6i) q^{23} +(3.58223e6 + 9.08488e6i) q^{25} +9.21535e6 q^{26} +(6.66703e6 + 6.66703e6i) q^{28} -3.57499e7i q^{29} +9.65516e6 q^{31} +(-4.19430e6 + 4.19430e6i) q^{32} -2.83049e7i q^{34} +(5.65359e7 - 1.07437e7i) q^{35} +(1.34842e7 - 1.34842e7i) q^{37} +(3.16390e6 + 3.16390e6i) q^{38} +(6.75899e6 + 3.55673e7i) q^{40} -1.08475e8 q^{41} +(8.62678e7 + 8.62678e7i) q^{43} +1.22741e8i q^{44} +1.13637e8 q^{46} +(1.76759e8 - 1.76759e8i) q^{47} -5.66463e7i q^{49} +(2.02674e8 + 8.80424e7i) q^{50} +(1.47446e8 - 1.47446e8i) q^{52} +(1.48530e8 + 1.48530e8i) q^{53} +(6.19316e8 + 4.21522e8i) q^{55} +2.13345e8 q^{56} +(-5.71999e8 - 5.71999e8i) q^{58} +7.98638e8i q^{59} -4.24449e8 q^{61} +(1.54483e8 - 1.54483e8i) q^{62} +1.34218e8i q^{64} +(-2.37604e8 - 1.25033e9i) q^{65} +(1.12768e9 - 1.12768e9i) q^{67} +(-4.52878e8 - 4.52878e8i) q^{68} +(7.32675e8 - 1.07647e9i) q^{70} +2.60778e9 q^{71} +(8.51903e8 + 8.51903e8i) q^{73} -4.31493e8i q^{74} +1.01245e8 q^{76} +(3.12165e9 - 3.12165e9i) q^{77} +4.56153e8i q^{79} +(6.77221e8 + 4.60934e8i) q^{80} +(-1.73559e9 + 1.73559e9i) q^{82} +(-7.60916e8 - 7.60916e8i) q^{83} +(-3.84037e9 + 7.29799e8i) q^{85} +2.76057e9 q^{86} +(1.96386e9 + 1.96386e9i) q^{88} -2.03858e9i q^{89} -7.49988e9 q^{91} +(1.81819e9 - 1.81819e9i) q^{92} -5.65629e9i q^{94} +(3.47698e8 - 5.10851e8i) q^{95} +(8.73239e9 - 8.73239e9i) q^{97} +(-9.06341e8 - 9.06341e8i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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
\(5\)	−2583.39	−	1758.32i	−0.826686	−	0.562663i
							\(7\)	−13021.6	+	13021.6i	−0.774769	+	0.774769i	−0.978936	−	0.204167i	\(-0.934552\pi\)
0.204167	+	0.978936i	\(0.434552\pi\)
\(11\)	−239729.			−1.48853			−0.744266	−	0.667884i	\(-0.767200\pi\)
							−0.744266	+	0.667884i	\(0.767200\pi\)
\(13\)	287980.	+	287980.i	0.775613	+	0.775613i	0.979082	−	0.203468i	\(-0.0652214\pi\)
							−0.203468	+	0.979082i	\(0.565221\pi\)
\(17\)	884527.	−	884527.i	0.622969	−	0.622969i	−0.323320	−	0.946290i	\(-0.604799\pi\)
							0.946290	+	0.323320i	\(0.104799\pi\)
\(19\)			197744.i			0.0798611i	0.999202	+	0.0399305i	\(0.0127137\pi\)
							−0.999202	+	0.0399305i	\(0.987286\pi\)
\(23\)	3.55115e6	+	3.55115e6i	0.551734	+	0.551734i	0.926941	−	0.375207i	\(-0.122428\pi\)
							−0.375207	+	0.926941i	\(0.622428\pi\)
\(29\)		−	3.57499e7i		−	1.74295i	−0.490439	−	0.871476i	\(-0.663163\pi\)
							0.490439	−	0.871476i	\(-0.336837\pi\)
\(31\)	9.65516e6			0.337249			0.168625	−	0.985680i	\(-0.446067\pi\)
							0.168625	+	0.985680i	\(0.446067\pi\)
\(37\)	1.34842e7	−	1.34842e7i	0.194453	−	0.194453i	−0.603164	−	0.797617i	\(-0.706094\pi\)
							0.797617	+	0.603164i	\(0.206094\pi\)
\(41\)	−1.08475e8			−0.936286			−0.468143	−	0.883653i	\(-0.655077\pi\)
							−0.468143	+	0.883653i	\(0.655077\pi\)
\(43\)	8.62678e7	+	8.62678e7i	0.586822	+	0.586822i	0.936769	−	0.349947i	\(-0.113801\pi\)
							−0.349947	+	0.936769i	\(0.613801\pi\)
\(47\)	1.76759e8	−	1.76759e8i	0.770713	−	0.770713i	−0.207518	−	0.978231i	\(-0.566539\pi\)
							0.978231	+	0.207518i	\(0.0665387\pi\)
\(53\)	1.48530e8	+	1.48530e8i	0.355169	+	0.355169i	0.862029	−	0.506860i	\(-0.169194\pi\)
							−0.506860	+	0.862029i	\(0.669194\pi\)
\(59\)			7.98638e8i			1.11709i	0.829473	+	0.558547i	\(0.188641\pi\)
							−0.829473	+	0.558547i	\(0.811359\pi\)
\(61\)	−4.24449e8			−0.502546			−0.251273	−	0.967916i	\(-0.580849\pi\)
							−0.251273	+	0.967916i	\(0.580849\pi\)
\(67\)	1.12768e9	−	1.12768e9i	0.835239	−	0.835239i	−0.152989	−	0.988228i	\(-0.548890\pi\)
							0.988228	+	0.152989i	\(0.0488899\pi\)
\(71\)	2.60778e9			1.44537			0.722685	−	0.691178i	\(-0.242908\pi\)
							0.722685	+	0.691178i	\(0.242908\pi\)
\(73\)	8.51903e8	+	8.51903e8i	0.410938	+	0.410938i	0.882065	−	0.471128i	\(-0.156153\pi\)
							−0.471128	+	0.882065i	\(0.656153\pi\)
\(79\)			4.56153e8i			0.148243i	0.997249	+	0.0741217i	\(0.0236153\pi\)
							−0.997249	+	0.0741217i	\(0.976385\pi\)
\(83\)	−7.60916e8	−	7.60916e8i	−0.193173	−	0.193173i	0.603893	−	0.797066i	\(-0.293616\pi\)
							−0.797066	+	0.603893i	\(0.793616\pi\)
\(89\)		−	2.03858e9i		−	0.365072i	−0.983199	−	0.182536i	\(-0.941569\pi\)
							0.983199	−	0.182536i	\(-0.0584306\pi\)
\(97\)	8.73239e9	−	8.73239e9i	1.01689	−	1.01689i	0.0170366	−	0.999855i	\(-0.494577\pi\)
							0.999855	−	0.0170366i	\(-0.00542316\pi\)

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.73.2		6
3.2	odd	2		10.11.c.c.3.1	✓	6
5.2	odd	4	inner	90.11.g.c.37.2		6
12.11	even	2		80.11.p.c.33.3		6
15.2	even	4		10.11.c.c.7.1	yes	6
15.8	even	4		50.11.c.e.7.3		6
15.14	odd	2		50.11.c.e.43.3		6
60.47	odd	4		80.11.p.c.17.3		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.11.c.c.3.1	✓	6	3.2	odd	2
10.11.c.c.7.1	yes	6	15.2	even	4
50.11.c.e.7.3		6	15.8	even	4
50.11.c.e.43.3		6	15.14	odd	2
80.11.p.c.17.3		6	60.47	odd	4
80.11.p.c.33.3		6	12.11	even	2
90.11.g.c.37.2		6	5.2	odd	4	inner
90.11.g.c.73.2		6	1.1	even	1	trivial