Embedded newform 297.2.x.a.272.16

• Label: 297.2.x.a.272.16
• Level: $297$
• Weight: $2$
• Character: 297.272
• Analytic conductor: $2.372$
• Analytic rank: $0$
• Dimension: $816$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 297 = 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	297.x (of order \(90\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.37155694003\)
Analytic rank:	\(0\)
Dimension:	\(816\)
Relative dimension:	\(34\) over \(\Q(\zeta_{90})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{90}]$

Embedding invariants

Embedding label			272.16
Character	\(\chi\)	\(=\)	297.272
Dual form			297.2.x.a.95.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.234840 - 0.481493i) q^{2} +(1.69451 + 0.358646i) q^{3} +(1.05464 - 1.34987i) q^{4} +(-0.140627 - 2.01106i) q^{5} +(-0.225253 - 0.900120i) q^{6} +(2.77243 - 0.0968153i) q^{7} +(-1.94563 - 0.413557i) q^{8} +(2.74275 + 1.21546i) q^{9} +(-0.935288 + 0.539989i) q^{10} +(-3.28509 + 0.456240i) q^{11} +(2.27122 - 1.90914i) q^{12} +(-4.91714 + 5.09185i) q^{13} +(-0.697692 - 1.31217i) q^{14} +(0.482965 - 3.45821i) q^{15} +(-0.571045 - 2.29034i) q^{16} +(0.389206 - 3.70304i) q^{17} +(-0.0588708 - 1.60605i) q^{18} +(-0.350554 + 1.64923i) q^{19} +(-2.86299 - 1.93111i) q^{20} +(4.73264 + 0.830264i) q^{21} +(0.991147 + 1.47461i) q^{22} +(-1.79865 + 4.94174i) q^{23} +(-3.14858 - 1.39857i) q^{24} +(0.926739 - 0.130245i) q^{25} +(3.60642 + 1.17180i) q^{26} +(4.21170 + 3.04329i) q^{27} +(2.79322 - 3.84453i) q^{28} +(5.13875 + 2.73232i) q^{29} +(-1.77852 + 0.579580i) q^{30} +(7.22978 - 2.07311i) q^{31} +(-4.01615 + 3.36995i) q^{32} +(-5.73026 - 0.405080i) q^{33} +(-1.87439 + 0.682222i) q^{34} +(-0.584581 - 5.56191i) q^{35} +(4.53332 - 2.42049i) q^{36} +(-4.00581 + 0.851461i) q^{37} +(0.876416 - 0.218515i) q^{38} +(-10.1583 + 6.86469i) q^{39} +(-0.558080 + 3.97095i) q^{40} +(-2.17784 + 1.15798i) q^{41} +(-0.711644 - 2.47371i) q^{42} +(0.620430 - 0.739399i) q^{43} +(-2.84872 + 4.91563i) q^{44} +(2.05866 - 5.68676i) q^{45} +(2.80180 - 0.294481i) q^{46} +(7.38871 - 5.77269i) q^{47} +(-0.146224 - 4.08581i) q^{48} +(0.694030 - 0.0485313i) q^{49} +(-0.280347 - 0.415631i) q^{50} +(1.98759 - 6.13527i) q^{51} +(1.68755 + 12.0076i) q^{52} +(-2.90818 - 4.00277i) q^{53} +(0.476246 - 2.74259i) q^{54} +(1.37950 + 6.54237i) q^{55} +(-5.43416 - 0.958189i) q^{56} +(-1.18551 + 2.66891i) q^{57} +(0.108811 - 3.11593i) q^{58} +(-8.00230 + 3.23314i) q^{59} +(-4.15879 - 4.29910i) q^{60} +(-0.503892 + 1.75728i) q^{61} +(-2.69602 - 2.99424i) q^{62} +(7.72174 + 3.10423i) q^{63} +(-1.74700 - 0.777815i) q^{64} +(10.9315 + 9.17262i) q^{65} +(1.15065 + 2.85421i) q^{66} +(-0.347986 - 1.97353i) q^{67} +(-4.58817 - 4.43075i) q^{68} +(-4.82016 + 7.72876i) q^{69} +(-2.54074 + 1.58763i) q^{70} +(-7.93671 - 0.834182i) q^{71} +(-4.83371 - 3.49912i) q^{72} +(-10.1495 + 9.13863i) q^{73} +(1.35070 + 1.72881i) q^{74} +(1.61708 + 0.111670i) q^{75} +(1.85654 + 2.21254i) q^{76} +(-9.06351 + 1.58294i) q^{77} +(5.69087 + 3.27906i) q^{78} +(14.2775 - 6.96360i) q^{79} +(-4.52571 + 1.47049i) q^{80} +(6.04532 + 6.66739i) q^{81} +(1.06900 + 0.776676i) q^{82} +(-4.39047 + 4.23982i) q^{83} +(6.11197 - 5.51284i) q^{84} +(-7.50179 - 0.261968i) q^{85} +(-0.501717 - 0.125092i) q^{86} +(7.72774 + 6.47295i) q^{87} +(6.58026 + 0.470898i) q^{88} +(9.25241 + 5.34188i) q^{89} +(-3.22159 + 0.344248i) q^{90} +(-13.1394 + 14.5928i) q^{91} +(4.77381 + 7.63969i) q^{92} +(12.9945 - 0.919975i) q^{93} +(-4.51467 - 2.20195i) q^{94} +(3.36600 + 0.473061i) q^{95} +(-8.01404 + 4.27005i) q^{96} +(-2.52904 - 0.176848i) q^{97} +(-0.186353 - 0.322773i) q^{98} +(-9.56472 - 2.74155i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	816 * q - 30 * q^2 - 18 * q^3 - 18 * q^4 - 21 * q^5 - 30 * q^6 - 30 * q^7 - 45 * q^8 - 36 * q^9 - 33 * q^11 - 30 * q^12 - 30 * q^13 - 18 * q^14 - 21 * q^15 - 30 * q^16 - 45 * q^17 - 15 * q^19 - 60 * q^20 - 15 * q^22 - 84 * q^23 - 30 * q^24 - 27 * q^25 + 36 * q^27 - 60 * q^28 + 60 * q^29 - 75 * q^30 - 9 * q^31 - 63 * q^33 - 42 * q^34 - 45 * q^35 + 36 * q^36 - 9 * q^37 - 18 * q^38 - 150 * q^39 - 90 * q^40 - 30 * q^41 - 66 * q^42 - 108 * q^44 + 24 * q^45 - 15 * q^46 - 6 * q^47 + 174 * q^48 - 18 * q^49 - 105 * q^50 - 30 * q^51 - 30 * q^52 - 48 * q^55 + 54 * q^56 - 75 * q^57 - 18 * q^58 + 81 * q^59 - 72 * q^60 - 30 * q^61 - 45 * q^62 - 30 * q^63 + 51 * q^64 - 75 * q^66 + 6 * q^67 - 225 * q^68 + 9 * q^69 - 93 * q^70 - 27 * q^71 - 30 * q^72 - 15 * q^73 - 30 * q^74 + 42 * q^75 - 141 * q^77 + 30 * q^78 - 30 * q^79 + 48 * q^81 - 36 * q^82 + 15 * q^83 + 90 * q^84 - 30 * q^85 + 93 * q^86 - 108 * q^88 + 54 * q^89 - 15 * q^90 - 9 * q^91 + 276 * q^92 + 87 * q^93 - 30 * q^94 + 90 * q^95 + 90 * q^96 - 81 * q^97 + 3 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/297\mathbb{Z}\right)^\times\).

\(n\)	\(56\)	\(244\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{3}{10}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.234840	−	0.481493i	−0.166057	−	0.340467i	0.799497	−	0.600670i	\(-0.205099\pi\)
							−0.965554	+	0.260203i	\(0.916210\pi\)
\(3\)	1.69451	+	0.358646i	0.978327	+	0.207064i
							\(5\)	−0.140627	−	2.01106i	−0.0628904	−	0.899375i	−0.921903	−	0.387421i	\(-0.873366\pi\)
0.859013	−	0.511954i	\(-0.171078\pi\)
\(7\)	2.77243	−	0.0968153i	1.04788	−	0.0365927i	0.494220	−	0.869337i	\(-0.335454\pi\)
							0.553658	+	0.832744i	\(0.313231\pi\)
\(11\)	−3.28509	+	0.456240i	−0.990493	+	0.137562i
							\(13\)	−4.91714	+	5.09185i	−1.36377	+	1.41222i	−0.544568	+	0.838717i	\(0.683306\pi\)
−0.819201	+	0.573507i	\(0.805583\pi\)
\(17\)	0.389206	−	3.70304i	0.0943962	−	0.898120i	−0.840168	−	0.542326i	\(-0.817544\pi\)
							0.934564	−	0.355794i	\(-0.115790\pi\)
\(19\)	−0.350554	+	1.64923i	−0.0804227	+	0.378359i	−0.999890	−	0.0148065i	\(-0.995287\pi\)
							0.919468	+	0.393166i	\(0.128620\pi\)
\(23\)	−1.79865	+	4.94174i	−0.375044	+	1.03042i	0.598340	+	0.801242i	\(0.295827\pi\)
							−0.973384	+	0.229181i	\(0.926395\pi\)
\(29\)	5.13875	+	2.73232i	0.954242	+	0.507380i	0.872146	−	0.489245i	\(-0.162728\pi\)
							0.0820957	+	0.996624i	\(0.473839\pi\)
\(31\)	7.22978	−	2.07311i	1.29851	−	0.372341i	0.445989	−	0.895039i	\(-0.352852\pi\)
							0.852518	+	0.522698i	\(0.175074\pi\)
\(37\)	−4.00581	+	0.851461i	−0.658551	+	0.139979i	−0.525053	−	0.851070i	\(-0.675954\pi\)
							−0.133498	+	0.991049i	\(0.542621\pi\)
\(41\)	−2.17784	+	1.15798i	−0.340122	+	0.180846i	−0.630742	−	0.775992i	\(-0.717250\pi\)
							0.290620	+	0.956839i	\(0.406138\pi\)
\(43\)	0.620430	−	0.739399i	0.0946146	−	0.112757i	−0.716660	−	0.697423i	\(-0.754330\pi\)
							0.811274	+	0.584666i	\(0.198774\pi\)
\(47\)	7.38871	−	5.77269i	1.07775	−	0.842034i	0.0896734	−	0.995971i	\(-0.471418\pi\)
							0.988081	+	0.153937i	\(0.0491954\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	297.2.x.a.272.16	yes	816
3.2	odd	2		891.2.bb.a.8.19		816
11.7	odd	10	inner	297.2.x.a.29.16	✓	816
27.13	even	9		891.2.bb.a.800.19		816
27.14	odd	18	inner	297.2.x.a.41.16	yes	816
33.29	even	10		891.2.bb.a.656.19		816
297.40	odd	90		891.2.bb.a.557.19		816
297.95	even	90	inner	297.2.x.a.95.16	yes	816

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.x.a.29.16	✓	816	11.7	odd	10	inner
297.2.x.a.41.16	yes	816	27.14	odd	18	inner
297.2.x.a.95.16	yes	816	297.95	even	90	inner
297.2.x.a.272.16	yes	816	1.1	even	1	trivial
891.2.bb.a.8.19		816	3.2	odd	2
891.2.bb.a.557.19		816	297.40	odd	90
891.2.bb.a.656.19		816	33.29	even	10
891.2.bb.a.800.19		816	27.13	even	9