Newform orbit 288.2.bf.a

• Label: 288.2.bf.a
• Level: $288$
• Weight: $2$
• Character orbit: 288.bf
• 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.bf (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.

\(\operatorname{Tr}(f)(q) = \) \( 368 q - 12 q^{2} - 8 q^{3} - 4 q^{4} - 12 q^{5} - 8 q^{6} - 4 q^{7} - 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.41395	+	0.0275526i	1.72268	−	0.179925i	1.99848	−	0.0779158i	−0.211536	−	1.60678i	−2.43082	+	0.301868i	0.419720	+	1.56642i	−2.82360	+	0.165232i	2.93525	−	0.619906i	0.343371	+	2.26607i
11.2	−1.41252	+	0.0691722i	−0.126594	+	1.72742i	1.99043	−	0.195414i	0.386920	+	2.93895i	0.0593276	−	2.44877i	0.887390	+	3.31179i	−2.79801	+	0.413709i	−2.96795	−	0.437362i	−0.749827	−	4.12457i
11.3	−1.40115	+	0.191775i	−1.72586	+	0.146277i	1.92644	−	0.537410i	0.234629	+	1.78218i	2.39014	−	0.535934i	−0.927052	−	3.45980i	−2.59618	+	1.12244i	2.95721	−	0.504910i	−0.670528	−	2.45211i
11.4	−1.34246	−	0.444750i	−0.318231	−	1.70257i	1.60439	+	1.19412i	0.201801	+	1.53283i	−0.330005	+	2.42716i	0.765564	+	2.85712i	−1.62275	−	2.31661i	−2.79746	+	1.08362i	0.410817	−	2.14751i
11.5	−1.31749	−	0.514013i	−1.60170	−	0.659214i	1.47158	+	1.35442i	−0.503783	−	3.82661i	1.77138	+	1.69180i	−0.191129	−	0.713303i	−1.24261	−	2.54085i	2.13087	+	2.11172i	−1.30320	+	5.30049i
11.6	−1.31237	−	0.526966i	1.34553	−	1.09066i	1.44461	+	1.38314i	0.197197	+	1.49786i	−2.34057	+	0.722304i	−0.749169	−	2.79594i	−1.16699	−	2.57646i	0.620902	−	2.93504i	0.530525	−	2.06965i
11.7	−1.28187	+	0.597341i	−1.23148	+	1.21798i	1.28637	−	1.53142i	−0.451380	−	3.42857i	0.851043	−	2.29689i	0.762324	+	2.84503i	−0.734171	+	2.73148i	0.0330693	−	2.99982i	2.62663	+	4.12534i
11.8	−1.26591	+	0.630463i	−1.05404	−	1.37441i	1.20503	−	1.59621i	0.00886053	+	0.0673024i	2.20083	+	1.07534i	0.666754	+	2.48836i	−0.519104	+	2.78038i	−0.778009	+	2.89736i	−0.0536483	−	0.0796122i
11.9	−1.18241	−	0.775832i	−0.200716	+	1.72038i	0.796170	+	1.83470i	−0.263676	−	2.00282i	1.57205	−	1.87847i	−0.327090	−	1.22072i	0.482020	−	2.78705i	−2.91943	−	0.690617i	−1.24208	+	2.57271i
11.10	−1.14354	+	0.832059i	0.897985	−	1.48109i	0.615355	−	1.90298i	−0.289571	−	2.19951i	0.205474	+	2.44086i	−0.477791	−	1.78314i	0.879712	+	2.68814i	−1.38725	−	2.65999i	2.16126	+	2.27428i
11.11	−1.13651	+	0.841635i	1.61183	+	0.634044i	0.583300	−	1.91305i	0.335718	+	2.55003i	−2.36549	+	0.635975i	−0.254371	−	0.949325i	0.947165	+	2.66512i	2.19598	+	2.04394i	−2.52774	−	2.61558i
11.12	−0.979102	−	1.02047i	1.47070	+	0.914907i	−0.0827170	+	1.99829i	−0.257196	−	1.95359i	−0.506327	−	2.39659i	1.34022	+	5.00178i	2.12018	−	1.87212i	1.32589	+	2.69110i	−1.74176	+	2.17523i
11.13	−0.939466	+	1.05707i	−0.494102	+	1.66008i	−0.234807	−	1.98617i	−0.0359748	−	0.273256i	−1.29063	−	2.08189i	−1.13217	−	4.22533i	2.32012	+	1.61773i	−2.51173	−	1.64050i	0.322648	+	0.218686i
11.14	−0.931130	−	1.06442i	−1.32517	+	1.11531i	−0.265992	+	1.98223i	0.240456	+	1.82644i	2.42107	+	0.372039i	−0.327296	−	1.22149i	2.35761	−	1.56259i	0.512151	−	2.95596i	1.72021	−	1.95660i
11.15	−0.749302	−	1.19939i	0.475486	−	1.66551i	−0.877093	+	1.79742i	−0.384557	−	2.92100i	−2.35388	+	0.677673i	−0.163982	−	0.611989i	2.81302	−	0.294828i	−2.54783	−	1.58385i	−3.21528	+	2.64995i
11.16	−0.747170	−	1.20072i	−1.18400	−	1.26417i	−0.883475	+	1.79429i	0.396975	+	3.01532i	−0.633273	+	2.36621i	−0.375906	−	1.40290i	2.81455	−	0.279828i	−0.196274	+	2.99357i	3.32396	−	2.72962i
11.17	−0.564220	+	1.29679i	0.888753	−	1.48665i	−1.36331	−	1.46335i	0.132037	+	1.00292i	1.42641	+	1.99132i	1.09866	+	4.10026i	2.66685	−	0.942277i	−1.42024	−	2.64252i	−1.37507	−	0.394642i
11.18	−0.514835	+	1.31717i	−0.894226	−	1.48336i	−1.46989	−	1.35625i	0.446317	+	3.39012i	2.41422	−	0.414163i	−0.836733	−	3.12273i	2.54317	−	1.23785i	−1.40072	+	2.65292i	−4.69515	−	1.15747i
11.19	−0.484847	+	1.32850i	0.622180	+	1.61644i	−1.52985	−	1.28824i	−0.0345958	−	0.262781i	−2.44912	+	0.0428399i	0.779563	+	2.90937i	2.45318	−	1.40781i	−2.22579	+	2.01144i	0.365879	+	0.0814479i
11.20	−0.370153	−	1.36491i	1.60090	+	0.661153i	−1.72597	+	1.01045i	−0.155947	−	1.18454i	0.309839	−	2.42981i	−1.23982	−	4.62708i	2.01805	+	1.98178i	2.12575	+	2.11688i	−1.55907	+	0.651315i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.d	odd	6	1	inner
32.h	odd	8	1	inner
288.bf	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.bf.a	✓	368
3.b	odd	2	1		864.2.bn.a		368
9.c	even	3	1		864.2.bn.a		368
9.d	odd	6	1	inner	288.2.bf.a	✓	368
32.h	odd	8	1	inner	288.2.bf.a	✓	368
96.o	even	8	1		864.2.bn.a		368
288.bd	odd	24	1		864.2.bn.a		368
288.bf	even	24	1	inner	288.2.bf.a	✓	368

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
288.2.bf.a	✓	368	1.a	even	1	1	trivial
288.2.bf.a	✓	368	9.d	odd	6	1	inner
288.2.bf.a	✓	368	32.h	odd	8	1	inner
288.2.bf.a	✓	368	288.bf	even	24	1	inner
864.2.bn.a		368	3.b	odd	2	1
864.2.bn.a		368	9.c	even	3	1
864.2.bn.a		368	96.o	even	8	1
864.2.bn.a		368	288.bd	odd	24	1

Hecke kernels

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