Embedded newform 216.5.j.a.197.35

• Label: 216.5.j.a.197.35
• Level: $216$
• Weight: $5$
• Character: 216.197
• Analytic conductor: $22.328$
• Analytic rank: $0$
• Dimension: $92$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	216.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.3279120261\)
Analytic rank:	\(0\)
Dimension:	\(92\)
Relative dimension:	\(46\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			197.35
Character	\(\chi\)	\(=\)	216.197
Dual form			216.5.j.a.125.35

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.83732 - 2.81950i) q^{2} +(0.100792 - 15.9997i) q^{4} +(21.1127 + 36.5683i) q^{5} +(-16.9047 + 29.2798i) q^{7} +(-44.8252 - 45.6804i) q^{8} +(163.008 + 44.2286i) q^{10} +(-28.4273 + 49.2375i) q^{11} +(-79.2879 + 45.7769i) q^{13} +(34.5904 + 130.739i) q^{14} +(-255.980 - 3.22527i) q^{16} +122.909i q^{17} +551.287i q^{19} +(587.209 - 334.111i) q^{20} +(58.1680 + 219.854i) q^{22} +(-595.912 + 344.050i) q^{23} +(-578.994 + 1002.85i) q^{25} +(-95.8971 + 353.436i) q^{26} +(466.763 + 273.421i) q^{28} +(665.451 - 1152.59i) q^{29} +(305.568 + 529.260i) q^{31} +(-735.390 + 712.585i) q^{32} +(346.541 + 348.731i) q^{34} -1427.62 q^{35} -1555.96i q^{37} +(1554.36 + 1564.18i) q^{38} +(724.074 - 2603.62i) q^{40} +(978.349 - 564.850i) q^{41} +(2635.89 + 1521.83i) q^{43} +(784.919 + 459.790i) q^{44} +(-720.744 + 2656.36i) q^{46} +(1412.72 + 815.637i) q^{47} +(628.964 + 1089.40i) q^{49} +(1184.74 + 4477.88i) q^{50} +(724.424 + 1273.19i) q^{52} +2783.44 q^{53} -2400.71 q^{55} +(2095.27 - 540.258i) q^{56} +(-1361.65 - 5146.52i) q^{58} +(-1220.18 - 2113.42i) q^{59} +(-4583.29 - 2646.16i) q^{61} +(2359.25 + 640.130i) q^{62} +(-77.4039 + 4095.27i) q^{64} +(-3347.97 - 1932.95i) q^{65} +(-3539.31 + 2043.42i) q^{67} +(1966.50 + 12.3882i) q^{68} +(-4050.60 + 4025.17i) q^{70} +1182.69i q^{71} +585.969 q^{73} +(-4387.05 - 4414.77i) q^{74} +(8820.42 + 55.5652i) q^{76} +(-961.108 - 1664.69i) q^{77} +(-1960.12 + 3395.02i) q^{79} +(-5286.48 - 9428.84i) q^{80} +(1183.29 - 4361.12i) q^{82} +(5665.44 - 9812.83i) q^{83} +(-4494.56 + 2594.94i) q^{85} +(11769.7 - 3113.98i) q^{86} +(3523.45 - 908.510i) q^{88} +6300.52i q^{89} -3095.37i q^{91} +(5444.63 + 9569.09i) q^{92} +(6308.04 - 1668.96i) q^{94} +(-20159.6 + 11639.2i) q^{95} +(373.253 - 646.493i) q^{97} +(4856.13 + 1317.60i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	92 * q + 3 * q^2 - q^4 - 2 * q^7 + 28 * q^10 - 852 * q^14 - q^16 - 1950 * q^20 + 31 * q^22 + 6 * q^23 - 4752 * q^25 + 508 * q^28 - 2 * q^31 - 4947 * q^32 - 387 * q^34 + 5985 * q^38 + 1024 * q^40 + 3318 * q^41 - 3648 * q^46 + 6 * q^47 - 11664 * q^49 + 8775 * q^50 + 414 * q^52 + 2492 * q^55 + 5778 * q^56 + 4126 * q^58 + 9026 * q^64 + 6 * q^65 - 5931 * q^68 - 1168 * q^70 - 8 * q^73 + 6696 * q^74 - 1071 * q^76 - 2 * q^79 - 8550 * q^82 - 2553 * q^86 - 1997 * q^88 - 11472 * q^92 + 11970 * q^94 - 3744 * q^95 - 2 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\)	\(55\)	\(109\)	\(137\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.83732	−	2.81950i	0.709330	−	0.704876i
							\(3\)		0			0
\(5\)	21.1127	+	36.5683i	0.844509	+	1.46273i								0.886047	+	0.463595i	\(0.153441\pi\)
							−0.0415382	+	0.999137i	\(0.513226\pi\)
\(7\)	−16.9047	+	29.2798i	−0.344993	+	0.597546i	−0.985353	−	0.170529i	\(-0.945452\pi\)
							0.640359	+	0.768076i	\(0.278786\pi\)
\(11\)	−28.4273	+	49.2375i	−0.234936	+	0.406922i	−0.959254	−	0.282545i	\(-0.908821\pi\)
							0.724318	+	0.689466i	\(0.242155\pi\)
\(13\)	−79.2879	+	45.7769i	−0.469159	+	0.270869i	−0.715888	−	0.698216i	\(-0.753978\pi\)
							0.246729	+	0.969085i	\(0.420644\pi\)
\(17\)			122.909i			0.425289i	0.977130	+	0.212645i	\(0.0682077\pi\)
							−0.977130	+	0.212645i	\(0.931792\pi\)
\(19\)			551.287i			1.52711i	0.645742	+	0.763556i	\(0.276548\pi\)
							−0.645742	+	0.763556i	\(0.723452\pi\)
\(23\)	−595.912	+	344.050i	−1.12649	+	0.650378i	−0.943049	−	0.332653i	\(-0.892056\pi\)
							−0.183439	+	0.983031i	\(0.558723\pi\)
\(29\)	665.451	−	1152.59i	0.791261	−	1.37050i	−0.133925	−	0.990991i	\(-0.542758\pi\)
							0.925186	−	0.379513i	\(-0.123909\pi\)
\(31\)	305.568	+	529.260i	0.317969	+	0.550739i	0.980064	−	0.198681i	\(-0.0636658\pi\)
							−0.662095	+	0.749420i	\(0.730332\pi\)
\(37\)		−	1555.96i		−	1.13657i	−0.822832	−	0.568285i	\(-0.807607\pi\)
							0.822832	−	0.568285i	\(-0.192393\pi\)
\(41\)	978.349	−	564.850i	0.582004	−	0.336020i	−0.179925	−	0.983680i	\(-0.557586\pi\)
							0.761929	+	0.647660i	\(0.224252\pi\)
\(43\)	2635.89	+	1521.83i	1.42558	+	0.823057i	0.996768	−	0.0803375i	\(-0.0255998\pi\)
							0.428810	+	0.903395i	\(0.358933\pi\)
\(47\)	1412.72	+	815.637i	0.639531	+	0.369233i	0.784434	−	0.620212i	\(-0.212954\pi\)
							−0.144903	+	0.989446i	\(0.546287\pi\)
\(53\)	2783.44			0.990901			0.495451	−	0.868636i	\(-0.335003\pi\)
							0.495451	+	0.868636i	\(0.335003\pi\)
\(59\)	−1220.18	−	2113.42i	−0.350526	−	0.607129i	0.635816	−	0.771841i	\(-0.280664\pi\)
							−0.986342	+	0.164712i	\(0.947331\pi\)
\(61\)	−4583.29	−	2646.16i	−1.23174	−	0.711143i	−0.264344	−	0.964429i	\(-0.585155\pi\)
							−0.967391	+	0.253286i	\(0.918489\pi\)
\(67\)	−3539.31	+	2043.42i	−0.788441	+	0.455207i	−0.839413	−	0.543493i	\(-0.817101\pi\)
							0.0509724	+	0.998700i	\(0.483768\pi\)
\(71\)			1182.69i			0.234615i	0.993096	+	0.117307i	\(0.0374262\pi\)
							−0.993096	+	0.117307i	\(0.962574\pi\)
\(73\)	585.969			0.109959			0.0549793	−	0.998487i	\(-0.482491\pi\)
							0.0549793	+	0.998487i	\(0.482491\pi\)
\(79\)	−1960.12	+	3395.02i	−0.314071	+	0.543986i	−0.979240	−	0.202707i	\(-0.935026\pi\)
							0.665169	+	0.746693i	\(0.268360\pi\)
\(83\)	5665.44	−	9812.83i	0.822390	−	1.42442i	−0.0815082	−	0.996673i	\(-0.525974\pi\)
							0.903898	−	0.427748i	\(-0.140693\pi\)
\(89\)			6300.52i			0.795419i	0.917511	+	0.397710i	\(0.130195\pi\)
							−0.917511	+	0.397710i	\(0.869805\pi\)
\(97\)	373.253	−	646.493i	0.0396697	−	0.0687100i	−0.845509	−	0.533961i	\(-0.820703\pi\)
							0.885179	+	0.465251i	\(0.154036\pi\)

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	216.5.j.a.197.35		92
3.2	odd	2		72.5.j.a.29.12	yes	92
4.3	odd	2		864.5.n.a.305.45		92
8.3	odd	2		864.5.n.a.305.2		92
8.5	even	2	inner	216.5.j.a.197.20		92
9.4	even	3		72.5.j.a.5.27	yes	92
9.5	odd	6	inner	216.5.j.a.125.20		92
12.11	even	2		288.5.n.a.209.36		92
24.5	odd	2		72.5.j.a.29.27	yes	92
24.11	even	2		288.5.n.a.209.11		92
36.23	even	6		864.5.n.a.17.2		92
36.31	odd	6		288.5.n.a.113.11		92
72.5	odd	6	inner	216.5.j.a.125.35		92
72.13	even	6		72.5.j.a.5.12	✓	92
72.59	even	6		864.5.n.a.17.45		92
72.67	odd	6		288.5.n.a.113.36		92

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.5.j.a.5.12	✓	92	72.13	even	6
72.5.j.a.5.27	yes	92	9.4	even	3
72.5.j.a.29.12	yes	92	3.2	odd	2
72.5.j.a.29.27	yes	92	24.5	odd	2
216.5.j.a.125.20		92	9.5	odd	6	inner
216.5.j.a.125.35		92	72.5	odd	6	inner
216.5.j.a.197.20		92	8.5	even	2	inner
216.5.j.a.197.35		92	1.1	even	1	trivial
288.5.n.a.113.11		92	36.31	odd	6
288.5.n.a.113.36		92	72.67	odd	6
288.5.n.a.209.11		92	24.11	even	2
288.5.n.a.209.36		92	12.11	even	2
864.5.n.a.17.2		92	36.23	even	6
864.5.n.a.17.45		92	72.59	even	6
864.5.n.a.305.2		92	8.3	odd	2
864.5.n.a.305.45		92	4.3	odd	2