Newform orbit 861.2.bw.a

• Label: 861.2.bw.a
• Level: $861$
• Weight: $2$
• Character orbit: 861.bw
• Analytic conductor: $6.875$
• Analytic rank: $0$
• Dimension: $864$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 861 = 3 \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	861.bw (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.87511961403\)
Analytic rank:	\(0\)
Dimension:	\(864\)
Relative dimension:	\(108\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 864 q - 110 q^{4} - 20 q^{7} - 12 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
236.1	−1.12199	−	2.52003i	1.72469	+	0.159494i	−3.75341	+	4.16859i	−2.24815	−	0.477859i	−1.53315	−	4.52522i	−2.44829	+	1.00294i	9.46924	+	3.07674i	2.94912	+	0.550156i	1.31818	+	6.20155i
236.2	−1.11552	−	2.50551i	−0.812276	−	1.52977i	−3.69490	+	4.10361i	0.798639	+	0.169756i	−2.92674	+	3.74166i	2.21639	−	1.44486i	9.18659	+	2.98490i	−1.68041	+	2.48520i	−0.465576	−	2.19036i
236.3	−1.08688	−	2.44116i	−1.67127	+	0.454827i	−3.43972	+	3.82020i	−1.02189	−	0.217209i	2.92677	+	3.58550i	2.54630	+	0.718570i	7.98148	+	2.59334i	2.58626	−	1.52028i	0.580424	+	2.73068i
236.4	−1.07115	−	2.40584i	0.0478398	+	1.73139i	−3.30243	+	3.66773i	2.97995	+	0.633408i	4.11420	−	1.96967i	1.45575	+	2.20925i	7.35210	+	2.38884i	−2.99542	+	0.165659i	−1.66809	−	7.84775i
236.5	−1.05518	−	2.36998i	1.71788	+	0.221109i	−3.16513	+	3.51523i	2.62611	+	0.558198i	−1.28865	−	4.30465i	1.17702	−	2.36952i	6.73622	+	2.18873i	2.90222	+	0.759679i	−1.44811	−	6.81283i
236.6	−1.02469	−	2.30149i	0.742228	+	1.56496i	−2.90862	+	3.23035i	−3.16448	−	0.672631i	2.84119	−	3.31183i	1.58696	−	2.11697i	5.62309	+	1.82705i	−1.89820	+	2.32311i	1.69456	+	7.97227i
236.7	−1.02387	−	2.29966i	−0.709221	+	1.58019i	−2.90185	+	3.22283i	−2.13198	−	0.453167i	4.36006	+	0.0130485i	−1.95315	+	1.78471i	5.59438	+	1.81772i	−1.99401	−	2.24141i	1.14075	+	5.36682i
236.8	−1.02219	−	2.29588i	−1.65005	+	0.526640i	−2.88792	+	3.20736i	3.59563	+	0.764274i	2.89576	+	3.24998i	−1.89444	−	1.84692i	5.53542	+	1.79857i	2.44530	−	1.73796i	−1.92074	−	9.03635i
236.9	−1.00913	−	2.26654i	0.816124	−	1.52772i	−2.78060	+	3.08817i	−1.52541	−	0.324237i	−4.28622	−	0.308107i	−1.98234	−	1.75224i	5.08624	+	1.65262i	−1.66788	−	2.49362i	0.804444	+	3.78461i
236.10	−0.971390	−	2.18178i	−0.925710	−	1.46392i	−2.47830	+	2.75243i	−0.274571	−	0.0583618i	−2.29472	+	3.44173i	−1.84583	−	1.89550i	3.86985	+	1.25739i	−1.28612	+	2.71033i	0.139383	+	0.655744i
236.11	−0.956867	−	2.14916i	0.497308	−	1.65912i	−2.36503	+	2.62663i	−0.882689	−	0.187621i	−4.04157	+	0.518766i	0.233362	+	2.63544i	3.43325	+	1.11553i	−2.50537	−	1.65019i	0.441388	+	2.07657i
236.12	−0.955547	−	2.14619i	1.14990	+	1.29527i	−2.35482	+	2.61529i	1.74187	+	0.370246i	1.68112	−	3.70561i	−2.64146	−	0.150671i	3.39443	+	1.10292i	−0.355452	+	2.97887i	−0.869821	−	4.09219i
236.13	−0.944147	−	2.12059i	−1.70279	−	0.317029i	−2.26722	+	2.51800i	0.854071	+	0.181538i	0.935394	+	3.91024i	−1.63195	+	2.08248i	3.06491	+	0.995850i	2.79898	+	1.07967i	−0.421400	−	1.98253i
236.14	−0.934432	−	2.09877i	−0.765177	−	1.55387i	−2.19341	+	2.43602i	−4.12960	−	0.877773i	−2.54620	+	3.05791i	−0.130098	+	2.64255i	2.79235	+	0.907289i	−1.82901	+	2.37797i	2.01659	+	9.48729i
236.15	−0.907435	−	2.03813i	1.01790	−	1.40139i	−1.99229	+	2.21266i	1.73240	+	0.368233i	−3.77989	−	0.802939i	2.39683	−	1.12036i	2.07392	+	0.673857i	−0.927774	−	2.85293i	−0.821534	−	3.86501i
236.16	−0.882457	−	1.98203i	−1.13338	+	1.30975i	−1.81146	+	2.01183i	−1.26995	−	0.269937i	3.59613	+	1.09059i	−0.0322769	−	2.64555i	1.45920	+	0.474123i	−0.430903	−	2.96889i	0.585655	+	2.75529i
236.17	−0.859146	−	1.92967i	1.58756	−	0.692569i	−1.64725	+	1.82945i	−2.32162	−	0.493476i	−2.70038	−	2.46846i	2.60017	+	0.488991i	0.927655	+	0.301413i	2.04070	−	2.19899i	1.04236	+	4.90394i
236.18	−0.825531	−	1.85417i	−1.22781	−	1.22168i	−1.41819	+	1.57506i	3.82540	+	0.813114i	−1.25161	+	3.28509i	1.94371	+	1.79499i	0.230588	+	0.0749225i	0.0150126	+	2.99996i	−1.65033	−	7.76420i
236.19	−0.802790	−	1.80310i	1.69993	−	0.332041i	−1.26842	+	1.40873i	2.93800	+	0.624491i	−1.96339	−	2.79857i	−0.459880	+	2.60548i	−0.195921	−	0.0636587i	2.77950	−	1.12889i	−1.23258	−	5.79883i
236.20	−0.782287	−	1.75705i	0.419962	+	1.68037i	−1.13698	+	1.26274i	1.20787	+	0.256740i	2.62395	−	2.05242i	−2.56649	−	0.642764i	−0.550248	−	0.178786i	−2.64726	+	1.41138i	−0.493796	−	2.32313i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.d	odd	6	1	inner
21.g	even	6	1	inner
41.f	even	10	1	inner
123.l	odd	10	1	inner
287.x	odd	30	1	inner
861.bw	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	861.2.bw.a	✓	864
3.b	odd	2	1	inner	861.2.bw.a	✓	864
7.d	odd	6	1	inner	861.2.bw.a	✓	864
21.g	even	6	1	inner	861.2.bw.a	✓	864
41.f	even	10	1	inner	861.2.bw.a	✓	864
123.l	odd	10	1	inner	861.2.bw.a	✓	864
287.x	odd	30	1	inner	861.2.bw.a	✓	864
861.bw	even	30	1	inner	861.2.bw.a	✓	864

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
861.2.bw.a	✓	864	1.a	even	1	1	trivial
861.2.bw.a	✓	864	3.b	odd	2	1	inner
861.2.bw.a	✓	864	7.d	odd	6	1	inner
861.2.bw.a	✓	864	21.g	even	6	1	inner
861.2.bw.a	✓	864	41.f	even	10	1	inner
861.2.bw.a	✓	864	123.l	odd	10	1	inner
861.2.bw.a	✓	864	287.x	odd	30	1	inner
861.2.bw.a	✓	864	861.bw	even	30	1	inner

Hecke kernels

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