Newform orbit 288.2.bc.a

• Label: 288.2.bc.a
• Level: $288$
• Weight: $2$
• Character orbit: 288.bc
• Analytic conductor: $2.300$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	288.bc (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29969157821\)
Analytic rank:	\(0\)
Dimension:	\(368\)
Relative dimension:	\(46\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 368 q - 4 q^{2} - 8 q^{3} - 4 q^{4} - 4 q^{5} - 8 q^{6} - 4 q^{7} - 16 q^{8} - 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
13.1	−1.41394	−	0.0276581i	−0.219683	−	1.71806i	1.99847	+	0.0782141i	−0.0865625	+	0.657507i	0.263101	+	2.43532i	−3.40174	−	0.911495i	−2.82356	−	0.165864i	−2.90348	+	0.754859i	0.140580	−	0.927284i
13.2	−1.40080	−	0.194294i	−1.43289	+	0.973053i	1.92450	+	0.544335i	0.130777	−	0.993353i	2.19625	−	1.08465i	−0.118123	−	0.0316510i	−2.59008	−	1.13642i	1.10634	−	2.78855i	−0.376196	+	1.36608i
13.3	−1.39065	+	0.257088i	1.59735	−	0.669693i	1.86781	−	0.715038i	−0.422585	+	3.20986i	−2.04918	+	1.34197i	3.16032	+	0.846806i	−2.41364	+	1.47456i	2.10302	−	2.13946i	−0.237547	−	4.57243i
13.4	−1.38932	+	0.264188i	−1.06043	−	1.36949i	1.86041	−	0.734083i	0.391027	−	2.97015i	1.83507	+	1.62250i	3.69248	+	0.989396i	−2.39076	+	1.51137i	−0.750983	+	2.90448i	0.241417	+	4.22979i
13.5	−1.34649	+	0.432388i	−0.112087	+	1.72842i	1.62608	−	1.16441i	−0.483489	+	3.67246i	−0.596424	−	2.37577i	−4.34257	−	1.16359i	−1.68603	+	2.27097i	−2.97487	−	0.387465i	−0.936914	−	5.15400i
13.6	−1.31944	−	0.508995i	0.552463	+	1.64158i	1.48185	+	1.34318i	−0.0885843	+	0.672864i	0.106613	−	2.44717i	3.68775	+	0.988129i	−1.27154	−	2.52650i	−2.38957	+	1.81383i	0.459366	−	0.842716i
13.7	−1.28406	+	0.592614i	1.54046	+	0.791819i	1.29762	−	1.52190i	0.211878	−	1.60938i	−2.44729	−	0.103844i	0.313867	+	0.0841003i	−0.764318	+	2.72320i	1.74604	+	2.43953i	0.681674	+	2.19210i
13.8	−1.25694	−	0.648153i	1.41818	−	0.994373i	1.15979	+	1.62938i	0.579315	−	4.40033i	−2.42707	+	0.330671i	1.20832	+	0.323768i	−0.401705	−	2.79976i	1.02245	−	2.82039i	−3.58025	+	5.15547i
13.9	−1.11432	+	0.870793i	−1.69615	+	0.350802i	0.483439	−	1.94069i	−0.153324	+	1.16461i	1.58459	−	1.86791i	1.36922	+	0.366881i	1.15123	+	2.58354i	2.75388	−	1.19003i	−0.843281	−	1.43127i
13.10	−1.10333	−	0.884682i	−1.59251	−	0.681109i	0.434674	+	1.95219i	0.108084	−	0.820982i	1.15450	+	2.16035i	−3.31582	−	0.888470i	1.24748	−	2.53846i	2.07218	+	2.16935i	−0.845561	+	0.810194i
13.11	−1.09136	−	0.899411i	0.499198	−	1.65855i	0.382121	+	1.96316i	−0.209198	+	1.58902i	−2.03652	+	1.36109i	1.05886	+	0.283722i	1.34865	−	2.48619i	−2.50160	−	1.65589i	1.65749	−	1.54603i
13.12	−1.06040	+	0.935708i	1.12868	−	1.31380i	0.248901	−	1.98445i	0.226472	−	1.72022i	0.0324819	+	2.44927i	−3.91012	−	1.04771i	1.59293	+	2.33721i	−0.452160	−	2.96573i	1.36948	+	2.03604i
13.13	−0.992813	−	1.00714i	1.67119	+	0.455112i	−0.0286462	+	1.99979i	−0.393225	+	2.98684i	−1.20082	−	2.13496i	−2.61511	−	0.700716i	2.04251	−	1.95657i	2.58575	+	1.52116i	3.39856	−	2.56934i
13.14	−0.897888	+	1.09261i	−0.186796	−	1.72195i	−0.387596	−	1.96208i	−0.364708	+	2.77023i	2.04914	+	1.34202i	1.49458	+	0.400471i	2.49181	+	1.33824i	−2.93021	+	0.643306i	−2.69932	−	2.88584i
13.15	−0.835352	−	1.14113i	0.0993087	+	1.72920i	−0.604373	+	1.90650i	0.387153	−	2.94072i	1.89029	−	1.55782i	−3.87349	−	1.03790i	2.68043	−	0.902926i	−2.98028	+	0.343450i	−3.67916	+	2.01474i
13.16	−0.750744	+	1.19849i	−0.667307	+	1.59834i	−0.872768	−	1.79952i	0.354821	−	2.69513i	−1.41463	−	1.99971i	0.937559	+	0.251218i	2.81194	+	0.304973i	−2.10940	−	2.13317i	2.96372	+	2.44861i
13.17	−0.721590	−	1.21627i	−1.43301	−	0.972870i	−0.958617	+	1.75529i	−0.00931719	+	0.0707711i	−0.149225	+	2.44494i	4.24636	+	1.13781i	2.82664	−	0.100666i	1.10705	+	2.78827i	0.0927998	−	0.0397355i
13.18	−0.515291	+	1.31699i	0.925920	+	1.46379i	−1.46895	−	1.35727i	−0.0503714	+	0.382609i	−2.40492	+	0.465155i	−2.21592	−	0.593754i	2.54446	−	1.23521i	−1.28535	+	2.71070i	−0.477938	−	0.263494i
13.19	−0.338290	−	1.37316i	1.72791	+	0.119758i	−1.77112	+	0.929050i	0.148961	−	1.13147i	−0.420086	−	2.41320i	1.82706	+	0.489558i	1.87488	+	2.11774i	2.97132	+	0.413862i	−1.60408	+	0.178219i
13.20	−0.238045	+	1.39404i	1.61798	−	0.618165i	−1.88667	−	0.663686i	0.0805221	−	0.611626i	0.476591	+	2.40268i	2.18130	+	0.584477i	1.37431	−	2.47210i	2.23575	−	2.00036i	0.833461	+	0.257845i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
32.g	even	8	1	inner
288.bc	even	24	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	288.2.bc.a	✓	368
3.b	odd	2	1		864.2.bk.a		368
9.c	even	3	1	inner	288.2.bc.a	✓	368
9.d	odd	6	1		864.2.bk.a		368
32.g	even	8	1	inner	288.2.bc.a	✓	368
96.p	odd	8	1		864.2.bk.a		368
288.bc	even	24	1	inner	288.2.bc.a	✓	368
288.be	odd	24	1		864.2.bk.a		368

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
288.2.bc.a	✓	368	1.a	even	1	1	trivial
288.2.bc.a	✓	368	9.c	even	3	1	inner
288.2.bc.a	✓	368	32.g	even	8	1	inner
288.2.bc.a	✓	368	288.bc	even	24	1	inner
864.2.bk.a		368	3.b	odd	2	1
864.2.bk.a		368	9.d	odd	6	1
864.2.bk.a		368	96.p	odd	8	1
864.2.bk.a		368	288.be	odd	24	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(288, [\chi])\).