Newform orbit 99.3.n.a

• Label: 99.3.n.a
• Level: $99$
• Weight: $3$
• Character orbit: 99.n
• Analytic conductor: $2.698$
• Analytic rank: $0$
• Dimension: $176$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	99.n (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.69755461717\)
Analytic rank:	\(0\)
Dimension:	\(176\)
Relative dimension:	\(22\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 176 q - 9 q^{2} - 3 q^{3} - 43 q^{4} - 18 q^{5} + 14 q^{6} - 3 q^{7} - q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−2.74959	−	2.47574i	−0.0649714	+	2.99930i	1.01283	+	9.63647i	2.63494	−	2.37251i	7.60413	−	8.08598i	−10.0678	−	4.48246i	12.3735	−	17.0306i	−8.99156	−	0.389737i	−13.1187
5.2	−2.66752	−	2.40185i	−2.48757	−	1.67691i	0.928690	+	8.83589i	−2.42406	+	2.18263i	2.60797	+	10.4479i	4.96959	+	2.21260i	10.3057	−	14.1846i	3.37597	+	8.34283i	11.7086
5.3	−2.64610	−	2.38256i	2.79583	−	1.08782i	0.907141	+	8.63087i	3.60917	−	3.24971i	−9.98982	−	3.78274i	9.48228	+	4.22178i	9.79150	−	13.4768i	6.63330	−	6.08271i	−17.2928
5.4	−2.00944	−	1.80930i	2.47600	−	1.69393i	0.346137	+	3.29328i	−6.05625	+	5.45307i	−8.04021	−	1.07600i	−8.71198	−	3.87882i	−1.09440	+	1.50631i	3.26119	−	8.38836i	22.0359
5.5	−1.78528	−	1.60747i	1.60633	+	2.53372i	0.185141	+	1.76149i	−2.94966	+	2.65589i	1.20514	−	7.10552i	4.53509	+	2.01915i	−3.14719	+	4.33174i	−3.83943	+	8.13995i	9.53524
5.6	−1.75104	−	1.57664i	−2.48719	+	1.67746i	0.162223	+	1.54345i	−1.13559	+	1.02249i	6.99994	+	0.984104i	4.46838	+	1.98945i	−3.39048	+	4.66660i	3.37222	−	8.34435i	3.60058
5.7	−1.56878	−	1.41254i	−2.96282	−	0.470866i	0.0477020	+	0.453854i	5.70469	−	5.13653i	3.98291	+	4.92378i	−8.32159	−	3.70501i	−4.39702	+	6.05198i	8.55657	+	2.79018i	−16.2050
5.8	−1.09871	−	0.989279i	0.331959	−	2.98158i	−0.189633	−	1.80424i	3.60942	−	3.24994i	−3.31434	+	2.94747i	5.20171	+	2.31595i	−5.05259	+	6.95430i	−8.77961	−	1.97952i	−7.18079
5.9	−0.922273	−	0.830419i	2.92552	+	0.664328i	−0.257121	−	2.44634i	3.29951	−	2.97089i	−2.14646	−	3.04210i	−4.94691	−	2.20251i	−4.71221	+	6.48581i	8.11734	+	3.88701i	−5.51014
5.10	−0.665101	−	0.598859i	−1.66057	−	2.49850i	−0.334387	−	3.18148i	−5.53540	+	4.98410i	−0.391809	+	2.65620i	0.905787	+	0.403282i	−3.78709	+	5.21248i	−3.48504	+	8.29786i	6.66637
5.11	−0.0924912	−	0.0832794i	−1.74419	+	2.44086i	−0.416495	−	3.96268i	−3.28729	+	2.95989i	0.364596	−	0.0805036i	−5.97920	−	2.66211i	−0.584109	+	0.803957i	−2.91564	−	8.51464i	0.550542
5.12	0.343624	+	0.309400i	0.447017	+	2.96651i	−0.395765	−	3.76545i	6.53439	−	5.88359i	−0.764233	+	1.15767i	1.65967	+	0.738931i	2.11619	−	2.91268i	−8.60035	+	2.65216i	4.06576
5.13	0.615077	+	0.553818i	1.93721	−	2.29068i	−0.346508	−	3.29681i	−0.483423	+	0.435276i	2.46015	−	0.336088i	−2.11950	−	0.943661i	3.55866	−	4.89808i	−1.49447	−	8.87505i	−0.538406
5.14	0.651581	+	0.586686i	2.92562	+	0.663883i	−0.337757	−	3.21354i	−3.85686	+	3.47273i	1.51679	+	2.14899i	11.4192	+	5.08417i	3.72672	−	5.12938i	8.11852	+	3.88454i	−4.55045
5.15	0.764016	+	0.687923i	−2.93951	+	0.599386i	−0.307631	−	2.92692i	2.53336	−	2.28105i	−2.65817	−	1.56422i	9.22752	+	4.10836i	4.19563	−	5.77479i	8.28147	−	3.52381i	3.50471
5.16	1.07957	+	0.972045i	−2.18293	−	2.05787i	−0.197524	−	1.87932i	1.32239	−	1.19069i	−0.356278	−	4.34351i	−10.9658	−	4.88228i	5.02904	−	6.92188i	0.530374	+	8.98436i	2.58501
5.17	1.91836	+	1.72730i	0.309614	+	2.98398i	0.278432	+	2.64910i	−2.05611	+	1.85133i	−4.56029	+	6.25916i	0.820347	+	0.365242i	2.02760	−	2.79075i	−8.80828	+	1.84777i	−7.14217
5.18	2.03984	+	1.83668i	2.81777	+	1.02965i	0.369441	+	3.51500i	2.07703	−	1.87016i	3.85666	+	7.27567i	−8.11140	−	3.61143i	0.751263	−	1.03402i	6.87964	+	5.80263i	7.67171
5.19	2.10463	+	1.89502i	−1.05732	−	2.80750i	0.420264	+	3.99854i	3.64737	−	3.28411i	3.09501	−	7.91240i	7.00784	+	3.12009i	−0.0342347	+	0.0471200i	−6.76416	+	5.93685i	13.8998
5.20	2.18392	+	1.96641i	−2.99989	+	0.0258802i	0.484626	+	4.61091i	−6.81263	+	6.13412i	−6.60242	−	5.84250i	1.75949	+	0.783375i	−1.09914	+	1.51283i	8.99866	−	0.155275i	−26.9405

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.d	odd	6	1	inner
11.c	even	5	1	inner
99.n	odd	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	99.3.n.a	✓	176
3.b	odd	2	1		297.3.r.a		176
9.c	even	3	1		297.3.r.a		176
9.d	odd	6	1	inner	99.3.n.a	✓	176
11.c	even	5	1	inner	99.3.n.a	✓	176
33.h	odd	10	1		297.3.r.a		176
99.m	even	15	1		297.3.r.a		176
99.n	odd	30	1	inner	99.3.n.a	✓	176

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
99.3.n.a	✓	176	1.a	even	1	1	trivial
99.3.n.a	✓	176	9.d	odd	6	1	inner
99.3.n.a	✓	176	11.c	even	5	1	inner
99.3.n.a	✓	176	99.n	odd	30	1	inner
297.3.r.a		176	3.b	odd	2	1
297.3.r.a		176	9.c	even	3	1
297.3.r.a		176	33.h	odd	10	1
297.3.r.a		176	99.m	even	15	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(99, [\chi])\).