Embedded newform 891.2.n.g.190.1

• Label: 891.2.n.g.190.1
• Level: $891$
• Weight: $2$
• Character: 891.190
• Analytic conductor: $7.115$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 891 = 3^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	891.n (of order \(15\), degree \(8\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.11467082010\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{15})\)
Twist minimal:	no (minimal twist has level 297)
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			190.1
Character	\(\chi\)	\(=\)	891.190
Dual form			891.2.n.g.136.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.13857 - 0.952153i) q^{2} +(2.32863 + 2.58621i) q^{4} +(3.27795 - 1.45944i) q^{5} +(-1.12566 + 0.239267i) q^{7} +(-1.07069 - 3.29523i) q^{8} -8.39974 q^{10} +(3.31649 - 0.0296003i) q^{11} +(0.662933 + 6.30739i) q^{13} +(2.63513 + 0.560113i) q^{14} +(-0.120292 + 1.14451i) q^{16} +(3.16415 + 2.29889i) q^{17} +(0.457371 + 1.40764i) q^{19} +(11.4075 + 5.07896i) q^{20} +(-7.12074 - 3.09451i) q^{22} +(0.623602 - 1.08011i) q^{23} +(5.26934 - 5.85220i) q^{25} +(4.58787 - 14.1200i) q^{26} +(-3.24004 - 2.35403i) q^{28} +(-5.51060 + 1.17132i) q^{29} +(0.336201 + 3.19874i) q^{31} +(-2.11781 + 3.66816i) q^{32} +(-4.57786 - 7.92909i) q^{34} +(-3.34067 + 2.42714i) q^{35} +(-3.09812 + 9.53502i) q^{37} +(0.362171 - 3.44583i) q^{38} +(-8.31883 - 9.23900i) q^{40} +(1.89935 + 0.403720i) q^{41} +(4.59169 + 7.95304i) q^{43} +(7.79944 + 8.50820i) q^{44} +(-2.36205 + 1.71613i) q^{46} +(0.166138 - 0.184515i) q^{47} +(-5.18495 + 2.30849i) q^{49} +(-16.8410 + 7.49812i) q^{50} +(-14.7685 + 16.4021i) q^{52} +(-4.71073 + 3.42255i) q^{53} +(10.8281 - 4.93724i) q^{55} +(1.99367 + 3.45313i) q^{56} +(12.9001 + 2.74200i) q^{58} +(-4.42073 - 4.90972i) q^{59} +(0.344600 - 3.27865i) q^{61} +(2.32670 - 7.16085i) q^{62} +(9.88379 - 7.18099i) q^{64} +(11.3783 + 19.7078i) q^{65} +(7.03803 - 12.1902i) q^{67} +(1.42273 + 13.5364i) q^{68} +(9.45526 - 2.00978i) q^{70} +(-4.25711 - 3.09297i) q^{71} +(0.683585 - 2.10386i) q^{73} +(15.7043 - 17.4414i) q^{74} +(-2.57540 + 4.46073i) q^{76} +(-3.72617 + 0.826846i) q^{77} +(-2.71089 - 1.20696i) q^{79} +(1.27602 + 3.92719i) q^{80} +(-3.67749 - 2.67186i) q^{82} +(1.22348 - 11.6407i) q^{83} +(13.7270 + 2.91776i) q^{85} +(-2.24714 - 21.3801i) q^{86} +(-3.64846 - 10.8969i) q^{88} +8.93456 q^{89} +(-2.25539 - 6.94137i) q^{91} +(4.24553 - 0.902415i) q^{92} +(-0.530984 + 0.236409i) q^{94} +(3.55360 + 3.94668i) q^{95} +(9.03292 + 4.02172i) q^{97} +13.2864 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	32 * q - 2 * q^4 - 8 * q^7 - 24 * q^10 - 8 * q^13 + 2 * q^16 + 20 * q^19 + 24 * q^22 + 16 * q^25 - 60 * q^28 + 6 * q^31 - 32 * q^34 + 24 * q^37 + 40 * q^40 + 80 * q^43 - 24 * q^46 - 40 * q^49 - 12 * q^52 + 112 * q^55 + 50 * q^58 + 6 * q^61 + 12 * q^64 - 64 * q^67 - 74 * q^70 - 64 * q^73 - 52 * q^76 + 4 * q^79 + 8 * q^82 - 70 * q^85 - 56 * q^88 - 188 * q^91 + 46 * q^94

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/891\mathbb{Z}\right)^\times\).

\(n\)	\(244\)	\(650\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{3}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.13857	−	0.952153i	−1.51220	−	0.673274i	−0.527822	−	0.849355i	\(-0.676991\pi\)
							−0.984376	+	0.176081i	\(0.943658\pi\)
\(3\)		0			0
							\(5\)	3.27795	−	1.45944i	1.46594	−	0.652680i	0.490202	−	0.871609i	\(-0.336923\pi\)
0.975741	+	0.218929i	\(0.0702562\pi\)
\(7\)	−1.12566	+	0.239267i	−0.425460	+	0.0904343i	−0.415665	−	0.909518i	\(-0.636451\pi\)
							−0.00979483	+	0.999952i	\(0.503118\pi\)
\(11\)	3.31649	−	0.0296003i	0.999960	−	0.00892484i
							\(13\)	0.662933	+	6.30739i	0.183865	+	1.74935i	0.565233	+	0.824931i	\(0.308786\pi\)
−0.381368	+	0.924423i	\(0.624547\pi\)
\(17\)	3.16415	+	2.29889i	0.767418	+	0.557562i	0.901177	−	0.433452i	\(-0.142705\pi\)
							−0.133758	+	0.991014i	\(0.542705\pi\)
\(19\)	0.457371	+	1.40764i	0.104928	+	0.322935i	0.989714	−	0.143063i	\(-0.0456952\pi\)
							−0.884786	+	0.465998i	\(0.845695\pi\)
\(23\)	0.623602	−	1.08011i	0.130030	−	0.225219i	−0.793658	−	0.608364i	\(-0.791826\pi\)
							0.923688	+	0.383146i	\(0.125159\pi\)
\(29\)	−5.51060	+	1.17132i	−1.02329	+	0.217508i	−0.688847	−	0.724907i	\(-0.741883\pi\)
							−0.334447	+	0.942415i	\(0.608549\pi\)
\(31\)	0.336201	+	3.19874i	0.0603835	+	0.574511i	0.982325	+	0.187182i	\(0.0599354\pi\)
							−0.921942	+	0.387329i	\(0.873398\pi\)
\(37\)	−3.09812	+	9.53502i	−0.509327	+	1.56755i	0.284045	+	0.958811i	\(0.408324\pi\)
							−0.793372	+	0.608737i	\(0.791676\pi\)
\(41\)	1.89935	+	0.403720i	0.296629	+	0.0630504i	0.353822	−	0.935313i	\(-0.384882\pi\)
							−0.0571934	+	0.998363i	\(0.518215\pi\)
\(43\)	4.59169	+	7.95304i	0.700226	+	1.21283i	0.968387	+	0.249453i	\(0.0802507\pi\)
							−0.268161	+	0.963374i	\(0.586416\pi\)
\(47\)	0.166138	−	0.184515i	0.0242337	−	0.0269142i	−0.730906	−	0.682478i	\(-0.760902\pi\)
							0.755140	+	0.655564i	\(0.227569\pi\)
\(53\)	−4.71073	+	3.42255i	−0.647069	+	0.470123i	−0.862271	−	0.506446i	\(-0.830959\pi\)
							0.215203	+	0.976569i	\(0.430959\pi\)
\(59\)	−4.42073	−	4.90972i	−0.575530	−	0.639191i	0.383147	−	0.923687i	\(-0.374840\pi\)
							−0.958677	+	0.284496i	\(0.908174\pi\)
\(61\)	0.344600	−	3.27865i	0.0441215	−	0.419788i	−0.950060	−	0.312068i	\(-0.898978\pi\)
							0.994181	−	0.107720i	\(-0.0343550\pi\)
\(67\)	7.03803	−	12.1902i	0.859832	−	1.48927i	−0.0122574	−	0.999925i	\(-0.503902\pi\)
							0.872089	−	0.489347i	\(-0.162765\pi\)
\(71\)	−4.25711	−	3.09297i	−0.505226	−	0.367068i	0.305784	−	0.952101i	\(-0.401082\pi\)
							−0.811009	+	0.585033i	\(0.801082\pi\)
\(73\)	0.683585	−	2.10386i	0.0800075	−	0.246238i	−0.903050	−	0.429536i	\(-0.858677\pi\)
							0.983057	+	0.183298i	\(0.0586773\pi\)
\(79\)	−2.71089	−	1.20696i	−0.304998	−	0.135794i	0.248530	−	0.968624i	\(-0.420052\pi\)
							−0.553529	+	0.832830i	\(0.686719\pi\)
\(83\)	1.22348	−	11.6407i	0.134295	−	1.27773i	−0.695037	−	0.718974i	\(-0.744612\pi\)
							0.829332	−	0.558756i	\(-0.188721\pi\)
\(89\)	8.93456			0.947061			0.473531	−	0.880777i	\(-0.342979\pi\)
							0.473531	+	0.880777i	\(0.342979\pi\)
\(97\)	9.03292	+	4.02172i	0.917154	+	0.408343i	0.810356	−	0.585937i	\(-0.199274\pi\)
							0.106798	+	0.994281i	\(0.465940\pi\)

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	891.2.n.g.190.1		32
3.2	odd	2	inner	891.2.n.g.190.4		32
9.2	odd	6	inner	891.2.n.g.784.1		32
9.4	even	3		297.2.f.c.190.4	yes	16
9.5	odd	6		297.2.f.c.190.1	yes	16
9.7	even	3	inner	891.2.n.g.784.4		32
11.4	even	5	inner	891.2.n.g.433.4		32
33.26	odd	10	inner	891.2.n.g.433.1		32
99.4	even	15		297.2.f.c.136.4	yes	16
99.13	odd	30		3267.2.a.bh.1.8		8
99.31	even	15		3267.2.a.bg.1.1		8
99.59	odd	30		297.2.f.c.136.1	✓	16
99.68	even	30		3267.2.a.bh.1.1		8
99.70	even	15	inner	891.2.n.g.136.1		32
99.86	odd	30		3267.2.a.bg.1.8		8
99.92	odd	30	inner	891.2.n.g.136.4		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.f.c.136.1	✓	16	99.59	odd	30
297.2.f.c.136.4	yes	16	99.4	even	15
297.2.f.c.190.1	yes	16	9.5	odd	6
297.2.f.c.190.4	yes	16	9.4	even	3
891.2.n.g.136.1		32	99.70	even	15	inner
891.2.n.g.136.4		32	99.92	odd	30	inner
891.2.n.g.190.1		32	1.1	even	1	trivial
891.2.n.g.190.4		32	3.2	odd	2	inner
891.2.n.g.433.1		32	33.26	odd	10	inner
891.2.n.g.433.4		32	11.4	even	5	inner
891.2.n.g.784.1		32	9.2	odd	6	inner
891.2.n.g.784.4		32	9.7	even	3	inner
3267.2.a.bg.1.1		8	99.31	even	15
3267.2.a.bg.1.8		8	99.86	odd	30
3267.2.a.bh.1.1		8	99.68	even	30
3267.2.a.bh.1.8		8	99.13	odd	30