Newform orbit 208.2.bh.a

• Label: 208.2.bh.a
• Level: $208$
• Weight: $2$
• Character orbit: 208.bh
• Analytic conductor: $1.661$
• Analytic rank: $0$
• Dimension: $104$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	208.bh (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.66088836204\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(26\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 6 q^{2} - 2 q^{3} - 2 q^{4} - 24 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
69.1	−1.41121	−	0.0921607i	−0.481707	−	1.79775i	1.98301	+	0.260116i	−1.74629	+	1.74629i	0.514106	+	2.58140i	1.98943	+	3.44580i	−2.77447	−	0.549833i	−0.401804	+	0.231982i	2.62531	−	2.30343i
69.2	−1.35555	−	0.403084i	0.386733	+	1.44331i	1.67505	+	1.09280i	2.05923	−	2.05923i	0.0575366	−	2.11237i	−1.67790	−	2.90622i	−1.83012	−	2.15654i	0.664501	−	0.383650i	−3.62144	+	1.96135i
69.3	−1.33578	+	0.464432i	−0.0531974	−	0.198535i	1.56861	−	1.24075i	−1.96959	+	1.96959i	0.163266	+	0.240493i	−2.59082	−	4.48744i	−1.51907	+	2.38588i	2.56149	−	1.47888i	1.71619	−	3.54567i
69.4	−1.30961	+	0.533774i	0.281738	+	1.05146i	1.43017	−	1.39807i	0.748325	−	0.748325i	−0.930209	−	1.22662i	0.960192	+	1.66310i	−1.12671	+	2.59432i	1.57189	−	0.907529i	−0.580580	+	1.37945i
69.5	−1.29579	−	0.566496i	0.664246	+	2.47900i	1.35816	+	1.46812i	−1.29877	+	1.29877i	0.543618	−	3.58857i	1.04828	+	1.81567i	−0.928214	−	2.67178i	−3.10614	+	1.79333i	2.41869	−	0.947190i
69.6	−1.19486	+	0.756509i	−0.703273	−	2.62465i	0.855389	−	1.80785i	1.68826	−	1.68826i	2.82588	+	2.60406i	0.114747	+	0.198748i	0.345581	+	2.80724i	−3.79611	+	2.19169i	−0.740056	+	3.29443i
69.7	−1.16081	−	0.807782i	−0.779619	−	2.90958i	0.694976	+	1.87537i	0.327738	−	0.327738i	−1.44531	+	4.00724i	−1.85803	−	3.21820i	0.708152	−	2.73834i	−5.25975	+	3.03672i	−0.645183	+	0.115702i
69.8	−0.849253	−	1.13083i	−0.181659	−	0.677959i	−0.557540	+	1.92072i	1.86702	−	1.86702i	−0.612380	+	0.781183i	1.40417	+	2.43210i	2.64549	−	1.00069i	2.17145	−	1.25369i	−3.69685	−	0.525706i
69.9	−0.821206	+	1.15136i	0.888158	+	3.31465i	−0.651241	−	1.89100i	−1.33725	+	1.33725i	−4.54570	−	1.69943i	−1.45296	−	2.51659i	2.71202	+	0.803091i	−7.60001	+	4.38787i	−0.441492	−	2.63780i
69.10	−0.688229	+	1.23545i	0.213171	+	0.795563i	−1.05268	−	1.70055i	1.28742	−	1.28742i	−1.12959	−	0.284168i	1.06242	+	1.84016i	2.82543	−	0.130173i	2.01060	−	1.16082i	0.704503	+	2.47658i
69.11	−0.682635	−	1.23855i	0.0109925	+	0.0410245i	−1.06802	+	1.69096i	−1.90442	+	1.90442i	0.0433071	−	0.0416195i	−0.0635197	−	0.110019i	2.82340	+	0.168490i	2.59651	−	1.49910i	3.65874	+	1.05870i
69.12	−0.242149	+	1.39333i	−0.377650	−	1.40941i	−1.88273	−	0.674785i	0.102773	−	0.102773i	2.05522	−	0.184904i	−0.783859	−	1.35768i	1.39610	−	2.45986i	0.754265	−	0.435475i	0.118310	+	0.168083i
69.13	0.0473076	−	1.41342i	−0.678610	−	2.53261i	−1.99552	−	0.133731i	−2.66902	+	2.66902i	−3.61175	+	0.839351i	0.621243	+	1.07602i	−0.283422	+	2.81419i	−3.35552	+	1.93731i	3.64619	+	3.89872i
69.14	0.0993186	−	1.41072i	0.179015	+	0.668093i	−1.98027	−	0.280222i	1.40192	−	1.40192i	0.960273	−	0.186186i	−1.03954	−	1.80053i	−0.591993	+	2.76578i	2.18377	−	1.26080i	−1.83848	−	2.11695i
69.15	0.232775	+	1.39493i	0.270689	+	1.01022i	−1.89163	+	0.649406i	−2.22668	+	2.22668i	−1.34618	+	0.612745i	0.0371602	+	0.0643633i	−1.34620	−	2.48752i	1.65079	−	0.953087i	−3.62437	−	2.58774i
69.16	0.253710	−	1.39127i	0.632763	+	2.36150i	−1.87126	−	0.705959i	−0.610101	+	0.610101i	3.44603	−	0.281206i	1.82980	+	3.16931i	−1.45694	+	2.42432i	−2.57824	+	1.48855i	0.694026	+	1.00360i
69.17	0.431628	+	1.34674i	0.732034	+	2.73199i	−1.62740	+	1.16258i	2.46847	−	2.46847i	−3.36330	+	2.16506i	1.47353	+	2.55223i	−2.26811	−	1.68987i	−4.32980	+	2.49981i	4.38983	+	2.25891i
69.18	0.656813	−	1.25244i	−0.688281	−	2.56870i	−1.13719	−	1.64523i	2.27566	−	2.27566i	−3.66920	−	0.825127i	1.76194	+	3.05178i	−2.80747	+	0.343652i	−3.52640	+	2.03597i	−1.35543	−	4.34480i
69.19	0.773419	+	1.18399i	−0.345357	−	1.28889i	−0.803646	+	1.83143i	2.90860	−	2.90860i	1.25892	−	1.40575i	−1.63618	−	2.83394i	−2.78995	+	0.464962i	1.05611	−	0.609747i	5.69331	+	1.19418i
69.20	0.901381	−	1.08973i	−0.141304	−	0.527354i	−0.375024	−	1.96452i	−0.665012	+	0.665012i	−0.702042	−	0.321364i	−1.49168	−	2.58367i	−2.47884	−	1.36211i	2.33994	−	1.35097i	0.125255	+	1.32411i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.e	even	6	1	inner
16.e	even	4	1	inner
208.bh	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	208.2.bh.a	✓	104
4.b	odd	2	1		832.2.bp.a		104
13.e	even	6	1	inner	208.2.bh.a	✓	104
16.e	even	4	1	inner	208.2.bh.a	✓	104
16.f	odd	4	1		832.2.bp.a		104
52.i	odd	6	1		832.2.bp.a		104
208.bh	even	12	1	inner	208.2.bh.a	✓	104
208.bi	odd	12	1		832.2.bp.a		104

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
208.2.bh.a	✓	104	1.a	even	1	1	trivial
208.2.bh.a	✓	104	13.e	even	6	1	inner
208.2.bh.a	✓	104	16.e	even	4	1	inner
208.2.bh.a	✓	104	208.bh	even	12	1	inner
832.2.bp.a		104	4.b	odd	2	1
832.2.bp.a		104	16.f	odd	4	1
832.2.bp.a		104	52.i	odd	6	1
832.2.bp.a		104	208.bi	odd	12	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(208, [\chi])\).