Newform orbit 684.2.bs.a

• Label: 684.2.bs.a
• Level: $684$
• Weight: $2$
• Character orbit: 684.bs
• Analytic conductor: $5.462$
• Analytic rank: $0$
• Dimension: $696$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	684.bs (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.46176749826\)
Analytic rank:	\(0\)
Dimension:	\(696\)
Relative dimension:	\(116\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 696 q - 9 q^{2} - 3 q^{4} - 18 q^{5} - 6 q^{6} - 18 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
23.1	−1.41202	+	0.0786649i	−1.38928	+	1.03436i	1.98762	−	0.222153i	−4.15853	+	0.733261i	1.88033	−	1.56983i		−	3.64991i	−2.78910	+	0.470042i	0.860198	−	2.87403i	5.81426	−	1.36251i
23.2	−1.40911	+	0.120010i	−0.775266	+	1.54886i	1.97120	−	0.338215i	3.87264	−	0.682851i	0.906558	−	2.27556i			0.103868i	−2.73705	+	0.713147i	−1.79793	−	2.40155i	−5.37504	+	1.42697i
23.3	−1.40868	−	0.125026i	−0.226717	+	1.71715i	1.96874	+	0.352243i	1.04087	−	0.183533i	0.534060	−	2.39056i			1.52016i	−2.72927	−	0.742339i	−2.89720	−	0.778615i	−1.48919	+	0.128403i
23.4	−1.40831	+	0.129087i	−0.356216	−	1.69503i	1.96667	−	0.363589i	−1.53473	+	0.270614i	0.720469	+	2.34114i		−	0.923751i	−2.72275	+	0.765918i	−2.74622	+	1.20759i	2.12644	−	0.579223i
23.5	−1.40681	+	0.144565i	1.40038	+	1.01929i	1.95820	−	0.406750i	−1.83392	+	0.323370i	−2.11741	−	1.23149i		−	0.495207i	−2.69601	+	0.855305i	0.922103	+	2.85477i	2.53322	−	0.720039i
23.6	−1.40179	−	0.187006i	1.60726	−	0.645527i	1.93006	+	0.524287i	−0.300003	+	0.0528987i	−2.37377	+	0.604329i		−	4.64420i	−2.60750	−	1.09588i	2.16659	−	2.07506i	0.430435	−	0.0180508i
23.7	−1.39710	−	0.219372i	1.71213	−	0.261969i	1.90375	+	0.612968i	1.82315	−	0.321470i	−2.44947	−	0.00959719i			3.37875i	−2.52525	−	1.27401i	2.86274	−	0.897047i	−2.61763	+	0.0491764i
23.8	−1.37799	−	0.318047i	−1.72498	−	0.156369i	1.79769	+	0.876530i	0.975388	−	0.171987i	2.32726	+	0.764099i		−	0.0214153i	−2.19842	−	1.77960i	2.95110	+	0.539468i	−1.39877	−	0.0732235i
23.9	−1.37141	+	0.345309i	−1.40038	−	1.01929i	1.76152	−	0.947120i	−1.83392	+	0.323370i	2.27246	+	0.914298i			0.495207i	−2.08872	+	1.90716i	0.922103	+	2.85477i	2.40339	−	1.07674i
23.10	−1.36862	−	0.356187i	−1.19183	−	1.25680i	1.74626	+	0.974972i	2.67723	−	0.472068i	1.18351	+	2.14460i		−	4.72128i	−2.04270	−	1.95637i	−0.159081	+	2.99578i	−3.83226	−	0.307512i
23.11	−1.36753	+	0.360368i	0.356216	+	1.69503i	1.74027	−	0.985628i	−1.53473	+	0.270614i	−1.09797	−	2.18963i			0.923751i	−2.02468	+	1.97501i	−2.74622	+	1.20759i	2.00127	−	0.923141i
23.12	−1.36518	+	0.369172i	0.775266	−	1.54886i	1.72742	−	1.00797i	3.87264	−	0.682851i	−0.486581	+	2.40067i		−	0.103868i	−1.98613	+	2.01378i	−1.79793	−	2.40155i	−5.03475	+	2.36188i
23.13	−1.35377	+	0.409020i	1.38928	−	1.03436i	1.66541	−	1.10744i	−4.15853	+	0.733261i	−1.45770	+	1.96853i			3.64991i	−1.80162	+	2.18041i	0.860198	−	2.87403i	5.32979	−	2.69359i
23.14	−1.35302	−	0.411502i	0.219865	−	1.71804i	1.66133	+	1.11354i	−0.859989	+	0.151639i	−1.00446	+	2.23407i			3.05338i	−1.78959	−	2.19029i	−2.90332	−	0.755474i	1.22598	+	0.148716i
23.15	−1.28409	−	0.592547i	−0.707659	+	1.58089i	1.29778	+	1.52177i	−3.14727	+	0.554948i	1.84545	−	1.61069i			5.13171i	−0.764742	−	2.72308i	−1.99844	−	2.23747i	4.37021	+	1.15230i
23.16	−1.28096	+	0.599282i	0.226717	−	1.71715i	1.28172	−	1.53531i	1.04087	−	0.183533i	0.738640	+	2.33547i		−	1.52016i	−0.721752	+	2.73479i	−2.89720	−	0.778615i	−1.22332	+	0.858872i
23.17	−1.25666	−	0.648697i	−1.62202	−	0.607495i	1.15839	+	1.63038i	−2.64599	+	0.466559i	1.64425	+	1.81561i			1.11605i	−0.398073	−	2.80027i	2.26190	+	1.97074i	3.62776	+	1.13014i
23.18	−1.25330	+	0.655170i	−1.60726	+	0.645527i	1.14150	−	1.64224i	−0.300003	+	0.0528987i	1.59145	−	1.86207i			4.64420i	−0.354695	+	2.80610i	2.16659	−	2.07506i	0.341336	−	0.262851i
23.19	−1.24014	−	0.679748i	1.64569	+	0.540095i	1.07589	+	1.68596i	−3.25234	+	0.573476i	−1.67376	−	1.78845i		−	0.838984i	−0.188220	−	2.82216i	2.41659	+	1.77766i	4.42317	+	1.49958i
23.20	−1.23781	+	0.683977i	−1.71213	+	0.261969i	1.06435	−	1.69327i	1.82315	−	0.321470i	1.94011	−	1.49532i		−	3.37875i	−0.159307	+	2.82394i	2.86274	−	0.897047i	−2.03683	+	1.64491i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
171.bf	odd	18	1	inner
684.bs	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	684.2.bs.a	✓	696
4.b	odd	2	1	inner	684.2.bs.a	✓	696
9.d	odd	6	1		684.2.ch.a	yes	696
19.e	even	9	1		684.2.ch.a	yes	696
36.h	even	6	1		684.2.ch.a	yes	696
76.l	odd	18	1		684.2.ch.a	yes	696
171.bf	odd	18	1	inner	684.2.bs.a	✓	696
684.bs	even	18	1	inner	684.2.bs.a	✓	696

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
684.2.bs.a	✓	696	1.a	even	1	1	trivial
684.2.bs.a	✓	696	4.b	odd	2	1	inner
684.2.bs.a	✓	696	171.bf	odd	18	1	inner
684.2.bs.a	✓	696	684.bs	even	18	1	inner
684.2.ch.a	yes	696	9.d	odd	6	1
684.2.ch.a	yes	696	19.e	even	9	1
684.2.ch.a	yes	696	36.h	even	6	1
684.2.ch.a	yes	696	76.l	odd	18	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(684, [\chi])\).