Newform orbit 861.2.y.a

• Label: 861.2.y.a
• Level: $861$
• Weight: $2$
• Character orbit: 861.y
• Analytic conductor: $6.875$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 861 = 3 \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	861.y (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.87511961403\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(84\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q + 8 q^{3} + 24 q^{6} + 24 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
260.1	−1.96487	+	1.96487i	−1.04316	+	1.38269i		−	5.72147i	0.513016	+	0.513016i	−0.667139	−	4.76648i	−0.382683	−	0.923880i	7.31221	+	7.31221i	−0.823653	−	2.88472i	−2.01602
260.2	−1.93179	+	1.93179i	1.51145	−	0.845886i		−	5.46365i	2.53626	+	2.53626i	−1.28573	+	4.55388i	0.382683	+	0.923880i	6.69106	+	6.69106i	1.56895	−	2.55703i	−9.79908
260.3	−1.89745	+	1.89745i	1.70561	+	0.301477i		−	5.20066i	−2.94297	−	2.94297i	−3.80836	+	2.66428i	0.382683	+	0.923880i	6.07311	+	6.07311i	2.81822	+	1.02841i	11.1683
260.4	−1.88243	+	1.88243i	−0.780999	−	1.54598i		−	5.08709i	−2.55036	−	2.55036i	4.38037	+	1.44001i	−0.382683	−	0.923880i	5.81123	+	5.81123i	−1.78008	+	2.41481i	9.60174
260.5	−1.84841	+	1.84841i	−1.51618	−	0.837373i		−	4.83324i	0.333623	+	0.333623i	4.35033	−	1.25472i	0.382683	+	0.923880i	5.23698	+	5.23698i	1.59761	+	2.53922i	−1.23335
260.6	−1.78268	+	1.78268i	−0.176898	+	1.72299i		−	4.35591i	−0.709602	−	0.709602i	−2.75620	−	3.38690i	0.382683	+	0.923880i	4.19984	+	4.19984i	−2.93741	−	0.609588i	2.52999
260.7	−1.75196	+	1.75196i	−0.159240	−	1.72472i		−	4.13874i	1.85428	+	1.85428i	3.30062	+	2.74265i	−0.382683	−	0.923880i	3.74700	+	3.74700i	−2.94929	+	0.549288i	−6.49724
260.8	−1.74427	+	1.74427i	1.60754	+	0.644839i		−	4.08492i	0.432075	+	0.432075i	−3.92874	+	1.67921i	−0.382683	−	0.923880i	3.63666	+	3.63666i	2.16837	+	2.07321i	−1.50731
260.9	−1.63893	+	1.63893i	0.808221	−	1.53192i		−	3.37216i	−1.08014	−	1.08014i	1.18609	+	3.83532i	0.382683	+	0.923880i	2.24887	+	2.24887i	−1.69356	−	2.47626i	3.54055
260.10	−1.63245	+	1.63245i	−1.73167	−	0.0363751i		−	3.32982i	−1.72624	−	1.72624i	2.88625	−	2.76749i	−0.382683	−	0.923880i	2.17087	+	2.17087i	2.99735	+	0.125979i	5.63602
260.11	−1.54949	+	1.54949i	−1.72653	−	0.138212i		−	2.80181i	2.76790	+	2.76790i	2.88939	−	2.46107i	−0.382683	−	0.923880i	1.24239	+	1.24239i	2.96179	+	0.477253i	−8.57766
260.12	−1.54469	+	1.54469i	0.142439	+	1.72618i		−	2.77211i	−2.51811	−	2.51811i	−2.88644	−	2.44639i	−0.382683	−	0.923880i	1.19267	+	1.19267i	−2.95942	+	0.491752i	7.77939
260.13	−1.50238	+	1.50238i	−1.54379	+	0.785317i		−	2.51429i	1.95573	+	1.95573i	1.13951	−	3.49920i	0.382683	+	0.923880i	0.772663	+	0.772663i	1.76655	−	2.42472i	−5.87650
260.14	−1.43158	+	1.43158i	1.68770	−	0.389451i		−	2.09884i	−0.782949	−	0.782949i	−1.85854	+	2.97361i	−0.382683	−	0.923880i	0.141498	+	0.141498i	2.69666	−	1.31455i	2.24171
260.15	−1.38183	+	1.38183i	0.831270	+	1.51954i		−	1.81893i	−0.804227	−	0.804227i	−3.24843	−	0.951069i	0.382683	+	0.923880i	−0.250203	−	0.250203i	−1.61798	+	2.52629i	2.22262
260.16	−1.30507	+	1.30507i	−0.330688	−	1.70019i		−	1.40641i	−1.89177	−	1.89177i	2.65044	+	1.78730i	0.382683	+	0.923880i	−0.774670	−	0.774670i	−2.78129	+	1.12446i	4.93778
260.17	−1.30098	+	1.30098i	1.31012	−	1.13296i		−	1.38508i	1.28745	+	1.28745i	−0.230485	+	3.17838i	0.382683	+	0.923880i	−0.799997	−	0.799997i	0.432823	−	2.96861i	−3.34989
260.18	−1.27991	+	1.27991i	−0.935422	+	1.45773i		−	1.27634i	0.834407	+	0.834407i	−0.668512	−	3.06302i	0.382683	+	0.923880i	−0.926224	−	0.926224i	−1.24997	−	2.72719i	−2.13593
260.19	−1.17963	+	1.17963i	−1.58879	−	0.689744i		−	0.783038i	−1.86291	−	1.86291i	2.68782	−	1.06054i	0.382683	+	0.923880i	−1.43556	−	1.43556i	2.04851	+	2.19172i	4.39506
260.20	−1.17843	+	1.17843i	−1.71957	+	0.207538i		−	0.777400i	−1.25472	−	1.25472i	1.78183	−	2.27097i	−0.382683	−	0.923880i	−1.44075	−	1.44075i	2.91386	−	0.713754i	2.95721

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
41.e	odd	8	1	inner
123.i	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	861.2.y.a	✓	336
3.b	odd	2	1	inner	861.2.y.a	✓	336
41.e	odd	8	1	inner	861.2.y.a	✓	336
123.i	even	8	1	inner	861.2.y.a	✓	336

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
861.2.y.a	✓	336	1.a	even	1	1	trivial
861.2.y.a	✓	336	3.b	odd	2	1	inner
861.2.y.a	✓	336	41.e	odd	8	1	inner
861.2.y.a	✓	336	123.i	even	8	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(861, [\chi])\).