Embedded newform 297.2.n.b.91.4

• Label: 297.2.n.b.91.4
• Level: $297$
• Weight: $2$
• Character: 297.91
• Analytic conductor: $2.372$
• Analytic rank: $0$
• Dimension: $72$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 297 = 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	297.n (of order \(15\), degree \(8\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.37155694003\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(9\) over \(\Q(\zeta_{15})\)
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			91.4
Character	\(\chi\)	\(=\)	297.91
Dual form			297.2.n.b.235.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.527858 - 0.235017i) q^{2} +(-1.11486 - 1.23818i) q^{4} +(-2.74468 + 1.22201i) q^{5} +(1.13888 - 0.242076i) q^{7} +(0.654602 + 2.01466i) q^{8} +1.73599 q^{10} +(2.54741 + 2.12385i) q^{11} +(0.412487 + 3.92455i) q^{13} +(-0.658057 - 0.139874i) q^{14} +(-0.220374 + 2.09672i) q^{16} +(-0.254185 - 0.184677i) q^{17} +(1.96794 + 6.05670i) q^{19} +(4.57300 + 2.03603i) q^{20} +(-0.845527 - 1.71977i) q^{22} +(-0.0427501 + 0.0740453i) q^{23} +(2.69431 - 2.99234i) q^{25} +(0.704604 - 2.16855i) q^{26} +(-1.56942 - 1.14025i) q^{28} +(-7.53156 + 1.60088i) q^{29} +(-0.682449 - 6.49307i) q^{31} +(2.72743 - 4.72404i) q^{32} +(0.0907715 + 0.157221i) q^{34} +(-2.83004 + 2.05614i) q^{35} +(-1.92922 + 5.93753i) q^{37} +(0.384637 - 3.65957i) q^{38} +(-4.25861 - 4.72966i) q^{40} +(5.68213 + 1.20777i) q^{41} +(3.39229 + 5.87562i) q^{43} +(-0.210303 - 5.52194i) q^{44} +(0.0399679 - 0.0290384i) q^{46} +(0.219298 - 0.243555i) q^{47} +(-5.15638 + 2.29577i) q^{49} +(-2.12546 + 0.946318i) q^{50} +(4.39943 - 4.88606i) q^{52} +(-1.96000 + 1.42402i) q^{53} +(-9.58718 - 2.71632i) q^{55} +(1.23321 + 2.13598i) q^{56} +(4.35183 + 0.925009i) q^{58} +(1.53634 + 1.70628i) q^{59} +(1.43070 - 13.6122i) q^{61} +(-1.16575 + 3.58780i) q^{62} +(0.861321 - 0.625787i) q^{64} +(-5.92799 - 10.2676i) q^{65} +(-5.83989 + 10.1150i) q^{67} +(0.0547189 + 0.520616i) q^{68} +(1.97708 - 0.420242i) q^{70} +(7.05272 + 5.12410i) q^{71} +(0.910538 - 2.80235i) q^{73} +(2.41378 - 2.68077i) q^{74} +(5.30529 - 9.18904i) q^{76} +(3.41531 + 1.80213i) q^{77} +(-8.11886 - 3.61475i) q^{79} +(-1.95735 - 6.02412i) q^{80} +(-2.71551 - 1.97293i) q^{82} +(0.518229 - 4.93062i) q^{83} +(0.923335 + 0.196261i) q^{85} +(-0.409774 - 3.89874i) q^{86} +(-2.61129 + 6.52242i) q^{88} -2.12862 q^{89} +(1.41981 + 4.36973i) q^{91} +(0.139342 - 0.0296180i) q^{92} +(-0.172998 + 0.0770236i) q^{94} +(-12.8027 - 14.2189i) q^{95} +(-0.0811587 - 0.0361342i) q^{97} +3.26138 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	72 * q + q^2 + 11 * q^4 + 8 * q^5 - 2 * q^7 - 6 * q^8 - 8 * q^10 + 2 * q^11 - 11 * q^13 + 10 * q^14 - 9 * q^16 + 20 * q^17 + 8 * q^19 + 45 * q^20 - 16 * q^22 - 20 * q^23 + 11 * q^25 + 12 * q^26 - 54 * q^28 + 23 * q^29 + 3 * q^31 - 18 * q^32 + 8 * q^34 - 18 * q^35 - 42 * q^37 + q^38 - 25 * q^40 - 10 * q^41 - 8 * q^43 - 38 * q^44 - 18 * q^46 + 34 * q^47 + q^49 - 27 * q^52 - 4 * q^53 + 18 * q^55 - 114 * q^56 + q^58 + 16 * q^59 - 3 * q^61 - 184 * q^62 + 26 * q^64 - 84 * q^65 + 10 * q^67 + 23 * q^68 - 46 * q^70 + 48 * q^71 - 40 * q^73 - 68 * q^74 + 16 * q^76 + 26 * q^77 + 19 * q^79 + 56 * q^80 + 94 * q^82 - 7 * q^83 + 25 * q^85 + 77 * q^86 + 18 * q^88 + 56 * q^89 + 20 * q^91 - 50 * q^92 - 63 * q^94 + 77 * q^95 - 33 * q^97 + 328 * q^98

Character values

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

\(n\)	\(56\)	\(244\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{4}{5}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.527858	−	0.235017i	−0.373252	−	0.166182i	0.211533	−	0.977371i	\(-0.432154\pi\)
							−0.584785	+	0.811188i	\(0.698821\pi\)
\(3\)		0			0
							\(5\)	−2.74468	+	1.22201i	−1.22746	+	0.546500i	−0.915009	−	0.403434i	\(-0.867816\pi\)
−0.312450	+	0.949934i	\(0.601150\pi\)
\(7\)	1.13888	−	0.242076i	0.430455	−	0.0914960i	0.0124114	−	0.999923i	\(-0.496049\pi\)
							0.418044	+	0.908427i	\(0.362716\pi\)
\(11\)	2.54741	+	2.12385i	0.768072	+	0.640364i
							\(13\)	0.412487	+	3.92455i	0.114403	+	1.08848i	0.889595	+	0.456750i	\(0.150986\pi\)
−0.775192	+	0.631726i	\(0.782347\pi\)
\(17\)	−0.254185	−	0.184677i	−0.0616490	−	0.0447906i	0.556534	−	0.830825i	\(-0.312131\pi\)
							−0.618183	+	0.786034i	\(0.712131\pi\)
\(19\)	1.96794	+	6.05670i	0.451477	+	1.38950i	0.875222	+	0.483721i	\(0.160715\pi\)
							−0.423746	+	0.905781i	\(0.639285\pi\)
\(23\)	−0.0427501	+	0.0740453i	−0.00891401	+	0.0154395i	−0.870448	−	0.492260i	\(-0.836171\pi\)
							0.861534	+	0.507700i	\(0.169504\pi\)
\(29\)	−7.53156	+	1.60088i	−1.39858	+	0.297276i	−0.844663	−	0.535298i	\(-0.820199\pi\)
							−0.553913	+	0.832575i	\(0.686866\pi\)
\(31\)	−0.682449	−	6.49307i	−0.122571	−	1.16619i	−0.866937	−	0.498418i	\(-0.833915\pi\)
							0.744366	−	0.667772i	\(-0.232752\pi\)
\(37\)	−1.92922	+	5.93753i	−0.317162	+	0.976124i	0.657694	+	0.753286i	\(0.271532\pi\)
							−0.974855	+	0.222838i	\(0.928468\pi\)
\(41\)	5.68213	+	1.20777i	0.887399	+	0.188623i	0.628990	−	0.777413i	\(-0.283468\pi\)
							0.258409	+	0.966036i	\(0.416802\pi\)
\(43\)	3.39229	+	5.87562i	0.517320	+	0.896024i	0.999798	+	0.0201157i	\(0.00640347\pi\)
							−0.482478	+	0.875908i	\(0.660263\pi\)
\(47\)	0.219298	−	0.243555i	0.0319879	−	0.0355262i	−0.726939	−	0.686703i	\(-0.759058\pi\)
							0.758926	+	0.651176i	\(0.225724\pi\)
\(53\)	−1.96000	+	1.42402i	−0.269227	+	0.195605i	−0.714205	−	0.699937i	\(-0.753211\pi\)
							0.444978	+	0.895541i	\(0.353211\pi\)
\(59\)	1.53634	+	1.70628i	0.200015	+	0.222139i	0.834805	−	0.550545i	\(-0.185580\pi\)
							−0.634790	+	0.772685i	\(0.718913\pi\)
\(61\)	1.43070	−	13.6122i	0.183183	−	1.74287i	−0.387656	−	0.921804i	\(-0.626715\pi\)
							0.570839	−	0.821062i	\(-0.306618\pi\)
\(67\)	−5.83989	+	10.1150i	−0.713456	+	1.23574i	0.250096	+	0.968221i	\(0.419538\pi\)
							−0.963552	+	0.267521i	\(0.913795\pi\)
\(71\)	7.05272	+	5.12410i	0.837004	+	0.608119i	0.921532	−	0.388302i	\(-0.126938\pi\)
							−0.0845279	+	0.996421i	\(0.526938\pi\)
\(73\)	0.910538	−	2.80235i	0.106570	−	0.327990i	−0.883525	−	0.468383i	\(-0.844837\pi\)
							0.990096	+	0.140393i	\(0.0448367\pi\)
\(79\)	−8.11886	−	3.61475i	−0.913443	−	0.406691i	−0.104464	−	0.994529i	\(-0.533313\pi\)
							−0.808979	+	0.587838i	\(0.799979\pi\)
\(83\)	0.518229	−	4.93062i	0.0568831	−	0.541206i	−0.928558	−	0.371188i	\(-0.878951\pi\)
							0.985441	−	0.170018i	\(-0.0543827\pi\)
\(89\)	−2.12862			−0.225634			−0.112817	−	0.993616i	\(-0.535987\pi\)
							−0.112817	+	0.993616i	\(0.535987\pi\)
\(97\)	−0.0811587	−	0.0361342i	−0.00824041	−	0.00366887i	0.402612	−	0.915371i	\(-0.368102\pi\)
							−0.410853	+	0.911702i	\(0.634769\pi\)

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	297.2.n.b.91.4		72
3.2	odd	2		99.2.m.b.58.6	yes	72
9.2	odd	6		99.2.m.b.25.4	yes	72
9.4	even	3		891.2.f.e.487.6		36
9.5	odd	6		891.2.f.f.487.4		36
9.7	even	3	inner	297.2.n.b.289.6		72
11.4	even	5	inner	297.2.n.b.37.6		72
33.2	even	10		1089.2.e.o.364.11		36
33.20	odd	10		1089.2.e.p.364.8		36
33.26	odd	10		99.2.m.b.4.4	✓	72
99.2	even	30		1089.2.e.o.727.11		36
99.4	even	15		891.2.f.e.730.6		36
99.13	odd	30		9801.2.a.cn.1.11		18
99.20	odd	30		1089.2.e.p.727.8		36
99.31	even	15		9801.2.a.cp.1.8		18
99.59	odd	30		891.2.f.f.730.4		36
99.68	even	30		9801.2.a.co.1.8		18
99.70	even	15	inner	297.2.n.b.235.4		72
99.86	odd	30		9801.2.a.cm.1.11		18
99.92	odd	30		99.2.m.b.70.6	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.m.b.4.4	✓	72	33.26	odd	10
99.2.m.b.25.4	yes	72	9.2	odd	6
99.2.m.b.58.6	yes	72	3.2	odd	2
99.2.m.b.70.6	yes	72	99.92	odd	30
297.2.n.b.37.6		72	11.4	even	5	inner
297.2.n.b.91.4		72	1.1	even	1	trivial
297.2.n.b.235.4		72	99.70	even	15	inner
297.2.n.b.289.6		72	9.7	even	3	inner
891.2.f.e.487.6		36	9.4	even	3
891.2.f.e.730.6		36	99.4	even	15
891.2.f.f.487.4		36	9.5	odd	6
891.2.f.f.730.4		36	99.59	odd	30
1089.2.e.o.364.11		36	33.2	even	10
1089.2.e.o.727.11		36	99.2	even	30
1089.2.e.p.364.8		36	33.20	odd	10
1089.2.e.p.727.8		36	99.20	odd	30
9801.2.a.cm.1.11		18	99.86	odd	30
9801.2.a.cn.1.11		18	99.13	odd	30
9801.2.a.co.1.8		18	99.68	even	30
9801.2.a.cp.1.8		18	99.31	even	15