Newform orbit 684.2.bt.a

• Label: 684.2.bt.a
• Level: $684$
• Weight: $2$
• Character orbit: 684.bt
• 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.bt (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.

\(\operatorname{Tr}(f)(q) = \) \( 696 q - 3 q^{2} - 3 q^{4} - 6 q^{5} - 6 q^{6} - 18 q^{8} - 18 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
67.1	−1.41416	+	0.0126159i	−0.699467	−	1.58453i	1.99968	−	0.0356818i	0.193873	−	0.162679i	1.00915	+	2.23195i		−	0.203894i	−2.82741	+	0.0756876i	−2.02149	+	2.21666i	−0.272115	+	0.232500i
67.2	−1.41226	−	0.0742616i	1.33474	−	1.10384i	1.98897	+	0.209754i	−2.70198	+	2.26723i	−1.96698	+	1.45979i		−	4.22244i	−2.79337	−	0.443931i	0.563075	−	2.94668i	3.98427	−	3.00127i
67.3	−1.41052	−	0.102089i	0.425233	−	1.67904i	1.97916	+	0.287997i	3.29015	−	2.76076i	−0.771212	+	2.32492i		−	0.518154i	−2.76225	−	0.608276i	−2.63835	−	1.42797i	−4.92267	+	3.55823i
67.4	−1.40940	−	0.116607i	1.71544	+	0.239317i	1.97281	+	0.328691i	2.17933	−	1.82867i	−2.38983	−	0.537324i		−	2.40654i	−2.74214	−	0.693300i	2.88546	+	0.821065i	−3.28477	+	2.32320i
67.5	−1.40807	+	0.131631i	−1.73203	+	0.00774749i	1.96535	−	0.370694i	−2.15420	+	1.80759i	2.43781	−	0.238899i		−	2.09358i	−2.71856	+	0.780666i	2.99988	−	0.0268378i	2.79533	−	2.82878i
67.6	−1.40710	−	0.141671i	−0.601110	+	1.62440i	1.95986	+	0.398689i	−0.895313	+	0.751257i	1.07595	−	2.20053i		−	2.21333i	−2.70123	−	0.838650i	−2.27733	−	1.95288i	1.36623	−	0.930254i
67.7	−1.40111	+	0.192059i	−0.497535	+	1.65905i	1.92623	−	0.538192i	0.976110	−	0.819053i	0.378467	−	2.42008i			4.59654i	−2.59549	+	1.12402i	−2.50492	−	1.65088i	−1.21033	+	1.33506i
67.8	−1.38734	−	0.274395i	1.03332	−	1.39006i	1.84941	+	0.761359i	−0.902090	+	0.756944i	−1.81498	+	1.64494i			3.70336i	−2.35685	−	1.56373i	−0.864519	−	2.87274i	1.45921	−	0.802607i
67.9	−1.36260	+	0.378576i	1.67593	−	0.437331i	1.71336	−	1.03169i	1.10869	−	0.930303i	−2.11806	+	1.23037i			0.509749i	−1.94405	+	2.05442i	2.61748	−	1.46587i	−1.15851	+	1.68735i
67.10	−1.34053	−	0.450547i	0.820815	+	1.52521i	1.59402	+	1.20794i	−0.729987	+	0.612532i	−0.413145	−	2.41440i		−	1.73018i	−1.59258	−	2.33745i	−1.65253	+	2.50383i	1.25454	−	0.492221i
67.11	−1.33305	+	0.472205i	−0.431483	−	1.67745i	1.55405	−	1.25894i	−2.15251	+	1.80617i	1.36729	+	2.03237i			1.97511i	−1.47714	+	2.41206i	−2.62764	+	1.44758i	2.01652	−	3.42414i
67.12	−1.33162	+	0.476211i	1.27590	+	1.17136i	1.54645	−	1.26827i	−0.976984	+	0.819787i	−2.25683	−	0.952219i		−	2.76703i	−1.45532	+	2.42529i	0.255819	+	2.98907i	0.910584	−	1.55690i
67.13	−1.32967	−	0.481643i	−1.69819	−	0.340807i	1.53604	+	1.28085i	−1.91519	+	1.60703i	2.09388	+	1.27108i			3.98610i	−1.42551	−	2.44293i	2.76770	+	1.15751i	3.32059	−	1.21439i
67.14	−1.32416	+	0.496580i	0.695449	+	1.58630i	1.50682	−	1.31511i	−1.88940	+	1.58540i	−1.70861	−	1.75518i			3.45988i	−1.34222	+	2.48967i	−2.03270	+	2.20638i	1.71460	−	3.03757i
67.15	−1.32029	+	0.506776i	−1.72825	+	0.114615i	1.48636	−	1.33819i	2.22735	−	1.86897i	2.22372	−	1.02716i			0.197123i	−1.28427	+	2.52005i	2.97373	−	0.396169i	−1.99361	+	3.59636i
67.16	−1.30702	−	0.540088i	−1.49120	+	0.881091i	1.41661	+	1.41181i	2.12846	−	1.78599i	2.42490	−	0.346226i		−	4.47847i	−1.08904	−	2.61036i	1.44736	−	2.62777i	−3.74653	+	1.18477i
67.17	−1.29352	−	0.571672i	−1.43825	−	0.965116i	1.34638	+	1.47894i	2.30313	−	1.93255i	1.30867	+	2.07060i			2.90676i	−0.896104	−	2.68272i	1.13710	+	2.77615i	−4.08393	+	1.18316i
67.18	−1.28853	−	0.582838i	1.02404	+	1.39691i	1.32060	+	1.50200i	1.84626	−	1.54920i	−0.505330	−	2.39680i			2.41789i	−0.826205	−	2.70507i	−0.902693	+	2.86097i	−3.28189	+	0.920111i
67.19	−1.23533	+	0.688443i	−0.0305498	+	1.73178i	1.05209	−	1.70091i	2.38927	−	2.00484i	−1.15449	−	2.16036i		−	3.16722i	−0.128703	+	2.82550i	−2.99813	−	0.105811i	−1.57133	+	4.12152i
67.20	−1.21030	−	0.731558i	−0.721548	+	1.57460i	0.929646	+	1.77081i	−2.54082	+	2.13200i	2.02520	−	1.37788i			1.23465i	0.170299	−	2.82330i	−1.95874	−	2.27230i	4.63483	−	0.721600i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
171.be	odd	18	1	inner
684.bt	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.bt.a	✓	696
4.b	odd	2	1	inner	684.2.bt.a	✓	696
9.c	even	3	1		684.2.cc.a	yes	696
19.f	odd	18	1		684.2.cc.a	yes	696
36.f	odd	6	1		684.2.cc.a	yes	696
76.k	even	18	1		684.2.cc.a	yes	696
171.be	odd	18	1	inner	684.2.bt.a	✓	696
684.bt	even	18	1	inner	684.2.bt.a	✓	696

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
684.2.bt.a	✓	696	1.a	even	1	1	trivial
684.2.bt.a	✓	696	4.b	odd	2	1	inner
684.2.bt.a	✓	696	171.be	odd	18	1	inner
684.2.bt.a	✓	696	684.bt	even	18	1	inner
684.2.cc.a	yes	696	9.c	even	3	1
684.2.cc.a	yes	696	19.f	odd	18	1
684.2.cc.a	yes	696	36.f	odd	6	1
684.2.cc.a	yes	696	76.k	even	18	1

Hecke kernels

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