Newform orbit 99.3.o.a

• Label: 99.3.o.a
• Level: $99$
• Weight: $3$
• Character orbit: 99.o
• 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.o (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.

\(\operatorname{Tr}(f)(q) = \) \( 176 q - 5 q^{2} - 9 q^{3} - 43 q^{4} - 6 q^{5} - 30 q^{6} - 5 q^{7} - 20 q^{8} + 23 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
7.1	−1.54394	−	3.46775i	1.72598	+	2.45377i	−6.96503	+	7.73545i	−5.06331	−	2.25433i	5.84427	−	9.77375i	−1.76302	+	8.29433i	23.1377	+	7.51788i	−3.04201	+	8.47031i			21.0389i
7.2	−1.31348	−	2.95012i	2.92797	−	0.653437i	−4.30144	+	4.77723i	6.39511	+	2.84729i	−5.77354	−	7.77958i	2.15374	−	10.1326i	7.45822	+	2.42332i	8.14604	−	3.82649i		−	22.6061i
7.3	−1.30322	−	2.92708i	−2.75305	+	1.19194i	−4.19291	+	4.65670i	4.50724	+	2.00675i	7.07675	+	6.50504i	−1.71994	+	8.09170i	6.90574	+	2.24381i	6.15855	−	6.56295i		−	15.8083i
7.4	−1.24198	−	2.78954i	0.953766	−	2.84435i	−3.56248	+	3.95653i	−3.54883	−	1.58004i	−9.11898	+	0.872068i	−0.636125	+	2.99273i	3.84512	+	1.24935i	−7.18066	−	5.42569i			11.8620i
7.5	−1.05861	−	2.37768i	−0.860464	+	2.87395i	−1.85620	+	2.06152i	0.208677	+	0.0929091i	7.74425	−	0.996492i	2.61432	−	12.2994i	−3.03463	−	0.986011i	−7.51920	−	4.94587i		−	0.594523i
7.6	−0.876409	−	1.96845i	−2.58625	−	1.52030i	−0.430165	+	0.477747i	−2.85772	−	1.27234i	−0.726020	+	6.42330i	0.214678	−	1.00998i	−6.87967	−	2.23534i	4.37737	+	7.86375i			6.74036i
7.7	−0.628401	−	1.41141i	2.53549	+	1.60352i	1.07933	−	1.19872i	4.71531	+	2.09939i	0.669920	−	4.58628i	−1.70312	+	8.01256i	−8.24759	−	2.67981i	3.85745	+	8.13143i		−	7.97450i
7.8	−0.474480	−	1.06570i	2.88776	+	0.812934i	1.76594	−	1.96127i	−8.38535	−	3.73340i	−0.503839	−	3.46321i	1.59485	−	7.50317i	−7.36587	−	2.39332i	7.67828	+	4.69511i			10.7077i
7.9	−0.396168	−	0.889807i	−1.17325	−	2.76107i	2.04171	−	2.26755i	6.96534	+	3.10117i	−1.99201	+	2.13781i	0.616656	−	2.90114i	−6.53192	−	2.12235i	−6.24698	+	6.47883i		−	7.42639i
7.10	−0.267778	−	0.601438i	−2.00139	+	2.23483i	2.38650	−	2.65048i	−7.40803	−	3.29827i	1.88004	+	0.605276i	−1.88617	+	8.87372i	−4.73769	−	1.53937i	−0.988888	−	8.94551i			5.33867i
7.11	−0.157628	−	0.354039i	1.97741	−	2.25607i	2.57603	−	2.86097i	0.305660	+	0.136089i	−1.11043	−	0.344461i	−0.576329	+	2.71141i	−2.89325	−	0.940074i	−1.17968	−	8.92235i		−	0.129667i
7.12	0.0317491	+	0.0713096i	−0.657068	+	2.92716i	2.67245	−	2.96805i	3.92580	+	1.74788i	−0.229596	+	0.0460793i	−0.503515	+	2.36885i	0.593449	+	0.192823i	−8.13652	−	3.84669i			0.335441i
7.13	0.299266	+	0.672161i	−2.98544	+	0.295224i	2.31428	−	2.57027i	−0.505407	−	0.225022i	−1.09188	−	1.91835i	2.02231	−	9.51422i	5.21926	+	1.69584i	8.82569	−	1.76274i		−	0.407056i
7.14	0.562956	+	1.26442i	2.10046	+	2.14198i	1.39468	−	1.54895i	1.32019	+	0.587789i	−1.52590	+	3.86170i	1.22466	−	5.76159i	8.00903	+	2.60229i	−0.176150	+	8.99828i			2.00018i
7.15	0.688807	+	1.54708i	−0.653667	−	2.92792i	0.757506	−	0.841295i	−6.94007	−	3.08992i	4.07949	−	3.02805i	1.38170	−	6.50039i	8.26577	+	2.68571i	−8.14544	+	3.82777i		−	12.8652i
7.16	0.745362	+	1.67411i	−2.89657	−	0.780953i	0.429438	−	0.476940i	4.34516	+	1.93459i	−0.851591	−	5.43127i	−2.68620	+	12.6376i	8.08995	+	2.62858i	7.78022	+	4.52417i			8.71626i
7.17	0.808969	+	1.81697i	2.81328	−	1.04185i	0.0295576	−	0.0328271i	−1.94462	−	0.865799i	4.16887	+	4.26884i	−1.69839	+	7.99028i	7.64988	+	2.48560i	6.82911	−	5.86202i		−	4.23372i
7.18	1.10256	+	2.47639i	0.614835	+	2.93632i	−2.24033	+	2.48814i	−5.91792	−	2.63483i	−6.59357	+	4.76003i	−1.14408	+	5.38250i	1.68060	+	0.546060i	−8.24396	+	3.61071i		−	17.5601i
7.19	1.14899	+	2.58067i	0.373306	−	2.97668i	−2.66318	+	2.95776i	5.24558	+	2.33548i	8.11077	−	2.45680i	0.617689	−	2.90600i	0.0535807	+	0.0174094i	−8.72129	−	2.22243i			16.2206i
7.20	1.33293	+	2.99382i	−1.55606	+	2.56489i	−4.50972	+	5.00855i	6.03198	+	2.68561i	−9.75295	−	1.23974i	0.905192	−	4.25859i	−8.53887	−	2.77445i	−4.15734	−	7.98226i			21.6384i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
11.d	odd	10	1	inner
99.o	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.o.a	✓	176
3.b	odd	2	1		297.3.s.a		176
9.c	even	3	1	inner	99.3.o.a	✓	176
9.d	odd	6	1		297.3.s.a		176
11.d	odd	10	1	inner	99.3.o.a	✓	176
33.f	even	10	1		297.3.s.a		176
99.o	odd	30	1	inner	99.3.o.a	✓	176
99.p	even	30	1		297.3.s.a		176

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
99.3.o.a	✓	176	1.a	even	1	1	trivial
99.3.o.a	✓	176	9.c	even	3	1	inner
99.3.o.a	✓	176	11.d	odd	10	1	inner
99.3.o.a	✓	176	99.o	odd	30	1	inner
297.3.s.a		176	3.b	odd	2	1
297.3.s.a		176	9.d	odd	6	1
297.3.s.a		176	33.f	even	10	1
297.3.s.a		176	99.p	even	30	1

Hecke kernels

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