Newform orbit 99.2.p.a

• Label: 99.2.p.a
• Level: $99$
• Weight: $2$
• Character orbit: 99.p
• Analytic conductor: $0.791$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.790518980011\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 15 q^{2} - 3 q^{3} + 5 q^{4} - 6 q^{5} - 15 q^{6} - 5 q^{7} - q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−1.66735	−	1.85178i	1.13023	+	1.31247i	−0.439972	+	4.18606i	1.05074	+	0.946093i	0.545914	−	4.28128i	1.27299	+	2.85917i	4.45339	−	3.23558i	−0.445156	+	2.96679i		−	3.52320i
2.2	−1.65440	−	1.83740i	−0.642641	−	1.60842i	−0.429934	+	4.09055i	−1.51955	−	1.36821i	−1.89212	+	3.84176i	−0.441031	−	0.990572i	4.22672	−	3.07089i	−2.17403	+	2.06727i			5.05558i
2.3	−1.02662	−	1.14018i	−0.926210	+	1.46360i	−0.0369992	+	0.352024i	−2.07594	−	1.86918i	2.61964	−	0.446523i	−1.07503	−	2.41456i	−2.04313	+	1.48442i	−1.28427	−	2.71121i			4.28588i
2.4	−0.906431	−	1.00669i	1.04844	−	1.37868i	0.0172421	−	0.164047i	2.90247	+	2.61340i	−2.33825	+	0.194222i	−0.686294	−	1.54144i	−2.37263	+	1.72381i	−0.801540	−	2.89094i		−	5.29077i
2.5	−0.536887	−	0.596273i	1.73203	+	0.00818115i	0.141763	−	1.34878i	−2.14661	−	1.93281i	−0.925027	−	1.03716i	0.542079	+	1.21753i	−2.17861	+	1.58285i	2.99987	+	0.0283400i			2.31767i
2.6	0.0350855	+	0.0389664i	−0.778272	+	1.54735i	0.208770	−	1.98631i	1.98818	+	1.79017i	−0.0876008	+	0.0239631i	1.00434	+	2.25577i	0.169565	−	0.123196i	−1.78858	−	2.40852i			0.140281i
2.7	0.148911	+	0.165382i	−1.37407	−	1.05449i	0.203880	−	1.93979i	0.397470	+	0.357884i	−0.0302197	−	0.384271i	−1.30165	−	2.92356i	0.711251	−	0.516754i	0.776111	+	2.89787i			0.119027i
2.8	0.780898	+	0.867275i	0.189274	−	1.72168i	0.0666926	−	0.634537i	−1.15400	−	1.03907i	1.64097	−	1.18030i	1.76474	+	3.96367i	2.49070	−	1.80960i	−2.92835	−	0.651738i		−	1.81224i
2.9	1.13824	+	1.26415i	0.775984	+	1.54850i	−0.0934131	+	0.888766i	−1.30752	−	1.17729i	−1.07427	+	2.74352i	−1.14161	−	2.56410i	1.52254	−	1.10619i	−1.79570	+	2.40322i		−	2.99294i
2.10	1.48406	+	1.64822i	−1.61485	+	0.626310i	−0.305125	+	2.90307i	−0.217939	−	0.196233i	−3.42883	−	1.73214i	0.539627	+	1.21202i	−1.64909	+	1.19814i	2.21547	−	2.02279i		−	0.650432i
29.1	−2.46099	+	1.09570i	−1.72861	+	0.109172i	3.51765	−	3.90675i	0.588502	−	1.32180i	4.13447	−	2.16271i	−0.300148	+	1.41209i	−2.71136	+	8.34470i	2.97616	−	0.377431i			3.89775i
29.2	−1.79897	+	0.800953i	0.666085	+	1.59885i	1.25651	−	1.39549i	−0.804185	+	1.80623i	−2.47887	−	2.34279i	0.0659410	−	0.310228i	0.0743474	−	0.228818i	−2.11266	+	2.12994i		−	3.89347i
29.3	−1.65764	+	0.738027i	−0.00460448	−	1.73204i	0.864813	−	0.960472i	−0.488756	+	1.09776i	1.28593	+	2.86770i	1.00654	−	4.73540i	0.396737	−	1.22103i	−2.99996	+	0.0159503i		−	2.18041i
29.4	−1.31398	+	0.585020i	1.60559	−	0.649686i	0.0460231	−	0.0511138i	1.45099	−	3.25897i	−1.72962	+	1.79297i	−0.449955	+	2.11687i	0.858363	−	2.64177i	2.15582	−	2.08625i			5.13106i
29.5	−0.571041	+	0.254244i	−1.42001	−	0.991749i	−1.07681	+	1.19592i	−0.518875	+	1.16541i	1.06303	+	0.205300i	−1.00684	+	4.73680i	0.697171	−	2.14567i	1.03287	+	2.81659i		−	0.797419i
29.6	0.337994	−	0.150485i	−1.17234	+	1.27500i	−1.24667	+	1.38456i	−0.966595	+	2.17101i	−0.204377	+	0.607361i	0.327889	−	1.54259i	−0.441671	+	1.35932i	−0.251238	−	2.98946i			0.879247i
29.7	0.471493	−	0.209922i	1.16254	+	1.28394i	−1.16002	+	1.28834i	0.661751	−	1.48632i	0.817657	+	0.361323i	0.171735	−	0.807949i	−0.595467	+	1.83266i	−0.296983	+	2.98526i		−	0.839703i
29.8	1.05233	−	0.468527i	1.16424	−	1.28240i	−0.450383	+	0.500201i	−0.381354	+	0.856535i	0.624322	−	1.89498i	0.180225	−	0.847891i	−0.951517	+	2.92847i	−0.289100	−	2.98604i			1.08003i
29.9	1.93906	−	0.863327i	−1.27072	+	1.17698i	1.67637	−	1.86180i	1.21148	−	2.72104i	−1.44788	+	3.37928i	−0.664357	+	3.12555i	0.331430	−	1.02004i	0.229440	−	2.99121i		−	6.32217i
29.10	2.14162	−	0.953512i	−1.25398	−	1.19479i	2.33911	−	2.59784i	−1.18835	+	2.66909i	−3.82481	−	1.36311i	0.273495	−	1.28669i	1.08356	−	3.33484i	0.144932	+	2.99650i			6.84930i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.d	odd	6	1	inner
11.d	odd	10	1	inner
99.p	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	99.2.p.a	✓	80
3.b	odd	2	1		297.2.t.a		80
9.c	even	3	1		297.2.t.a		80
9.c	even	3	1		891.2.k.a		80
9.d	odd	6	1	inner	99.2.p.a	✓	80
9.d	odd	6	1		891.2.k.a		80
11.d	odd	10	1	inner	99.2.p.a	✓	80
33.f	even	10	1		297.2.t.a		80
99.o	odd	30	1		297.2.t.a		80
99.o	odd	30	1		891.2.k.a		80
99.p	even	30	1	inner	99.2.p.a	✓	80
99.p	even	30	1		891.2.k.a		80

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
99.2.p.a	✓	80	1.a	even	1	1	trivial
99.2.p.a	✓	80	9.d	odd	6	1	inner
99.2.p.a	✓	80	11.d	odd	10	1	inner
99.2.p.a	✓	80	99.p	even	30	1	inner
297.2.t.a		80	3.b	odd	2	1
297.2.t.a		80	9.c	even	3	1
297.2.t.a		80	33.f	even	10	1
297.2.t.a		80	99.o	odd	30	1
891.2.k.a		80	9.c	even	3	1
891.2.k.a		80	9.d	odd	6	1
891.2.k.a		80	99.o	odd	30	1
891.2.k.a		80	99.p	even	30	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(99, [\chi])\).