Newform orbit 340.2.bf.a

• Label: 340.2.bf.a
• Level: $340$
• Weight: $2$
• Character orbit: 340.bf
• Analytic conductor: $2.715$
• Analytic rank: $0$
• Dimension: $288$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	340.bf (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.71491366872\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(36\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) 288 * q - 16 * q^24 - 32 * q^26 - 96 * q^28 - 80 * q^32 - 64 * q^34 - 64 * q^36 - 80 * q^38 - 96 * q^42 - 32 * q^44 - 16 * q^46 + 80 * q^54 + 80 * q^56 - 160 * q^57 - 64 * q^61 + 112 * q^62 + 96 * q^64 + 208 * q^66 - 128 * q^69 + 208 * q^72 + 96 * q^74 + 112 * q^76 - 64 * q^77 - 160 * q^81 + 80 * q^82 - 128 * q^88 - 224 * q^92 - 192 * q^94 - 272 * q^98

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	−1.41293	+	0.0603381i	0.113145	−	0.0225060i	1.99272	−	0.170507i	−0.831470	−	0.555570i	−0.158508	+	0.0386263i	−1.29866	−	1.94358i	−2.80528	+	0.361150i	−2.75934	+	1.14296i	1.20833	+	0.734810i
11.2	−1.38006	−	0.308920i	1.80551	−	0.359138i	1.80914	+	0.852658i	−0.831470	−	0.555570i	−2.60266	−	0.0621262i	−0.300559	−	0.449818i	−2.23331	−	1.73560i	0.359247	−	0.148805i	0.975852	+	1.02358i
11.3	−1.34272	−	0.443977i	−2.38453	+	0.474313i	1.60577	+	1.19227i	0.831470	+	0.555570i	3.41233	+	0.421810i	−1.59437	−	2.38614i	−1.62675	−	2.31380i	2.68939	−	1.11398i	−0.869767	−	1.11513i
11.4	−1.31604	+	0.517729i	−2.96149	+	0.589077i	1.46391	−	1.36270i	−0.831470	−	0.555570i	3.59245	−	2.30850i	−1.32299	−	1.97999i	−1.22105	+	2.55128i	5.65178	−	2.34104i	1.38188	+	0.300675i
11.5	−1.31209	+	0.527661i	0.300562	−	0.0597854i	1.44315	−	1.38468i	0.831470	+	0.555570i	−0.362817	+	0.237039i	0.524425	+	0.784858i	−1.16290	+	2.57831i	−2.68488	+	1.11211i	−1.38411	−	0.290222i
11.6	−1.26338	−	0.635504i	2.38453	−	0.474313i	1.19227	+	1.60577i	0.831470	+	0.555570i	−3.31401	−	0.916143i	1.59437	+	2.38614i	−0.485817	−	2.78639i	2.68939	−	1.11398i	−0.697397	−	1.23030i
11.7	−1.19429	−	0.757411i	−1.80551	+	0.359138i	0.852658	+	1.80914i	−0.831470	−	0.555570i	2.42832	+	0.938597i	0.300559	+	0.449818i	0.351938	−	2.80645i	0.359247	−	0.148805i	0.572221	+	1.29328i
11.8	−1.15606	+	0.814564i	−0.177801	+	0.0353667i	0.672971	−	1.88338i	0.831470	+	0.555570i	0.176741	−	0.185716i	−2.15819	−	3.22996i	0.756133	+	2.72548i	−2.74128	+	1.13547i	−1.41378	+	0.0350101i
11.9	−0.956424	−	1.04175i	−0.113145	+	0.0225060i	−0.170507	+	1.99272i	−0.831470	−	0.555570i	0.131661	+	0.0963444i	1.29866	+	1.94358i	2.23900	−	1.72826i	−2.75934	+	1.14296i	0.216469	+	1.39755i
11.10	−0.909173	+	1.08324i	−2.90888	+	0.578613i	−0.346809	−	1.96970i	0.831470	+	0.555570i	2.01790	−	3.67707i	1.95287	+	2.92267i	2.44896	+	1.41512i	5.35516	−	2.21818i	−1.35776	+	0.395570i
11.11	−0.721401	+	1.21638i	1.96029	−	0.389927i	−0.959162	−	1.75500i	0.831470	+	0.555570i	−0.939858	+	2.66576i	1.65022	+	2.46973i	2.82668	+	0.0993492i	0.919068	−	0.380690i	−1.27561	+	0.610594i
11.12	−0.715822	+	1.21967i	2.16217	−	0.430082i	−0.975197	−	1.74614i	−0.831470	−	0.555570i	−1.02317	+	2.94500i	−2.70443	−	4.04747i	2.82778	+	0.0605036i	1.71836	−	0.711770i	1.27280	−	0.616430i
11.13	−0.601322	+	1.28000i	−0.121832	+	0.0242338i	−1.27682	−	1.53939i	−0.831470	−	0.555570i	0.0422406	−	0.170517i	1.25782	+	1.88246i	2.73821	−	0.708671i	−2.75738	+	1.14215i	1.21111	−	0.730208i
11.14	−0.564490	−	1.29667i	2.96149	−	0.589077i	−1.36270	+	1.46391i	−0.831470	−	0.555570i	−2.43557	−	3.50755i	1.32299	+	1.97999i	2.66744	+	0.940612i	5.65178	−	2.34104i	−0.251035	+	1.39175i
11.15	−0.554673	−	1.30090i	−0.300562	+	0.0597854i	−1.38468	+	1.44315i	0.831470	+	0.555570i	0.244488	+	0.357839i	−0.524425	−	0.784858i	2.64543	+	1.00085i	−2.68488	+	1.11211i	0.261547	−	1.38982i
11.16	−0.241477	−	1.39344i	0.177801	−	0.0353667i	−1.88338	+	0.672971i	0.831470	+	0.555570i	−0.0922165	−	0.239215i	2.15819	+	3.22996i	1.39254	+	2.46188i	−2.74128	+	1.13547i	0.573375	−	1.29276i
11.17	−0.239432	+	1.39380i	−1.68900	+	0.335964i	−1.88534	−	0.667439i	0.831470	+	0.555570i	−0.0638642	−	2.43457i	−1.69902	−	2.54276i	1.38169	−	2.46798i	−0.0317728	+	0.0131607i	−0.973433	+	1.02588i
11.18	−0.0294528	+	1.41391i	−2.44045	+	0.485436i	−1.99827	−	0.0832869i	−0.831470	−	0.555570i	−0.614483	−	3.46487i	0.0736077	+	0.110162i	0.176614	−	2.82291i	2.94852	−	1.22132i	0.810014	−	1.15926i
11.19	0.123082	−	1.40885i	2.90888	−	0.578613i	−1.96970	−	0.346809i	0.831470	+	0.555570i	−0.457144	−	4.16939i	−1.95287	−	2.92267i	−0.731036	+	2.73232i	5.35516	−	2.21818i	0.885053	−	1.10303i
11.20	0.289502	+	1.38426i	3.02582	−	0.601874i	−1.83238	+	0.801494i	0.831470	+	0.555570i	1.70913	+	4.01430i	−0.629460	−	0.942053i	−1.63996	−	2.30446i	6.02172	−	2.49428i	−0.528345	+	1.31181i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
17.e	odd	16	1	inner
68.i	even	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	340.2.bf.a	✓	288
4.b	odd	2	1	inner	340.2.bf.a	✓	288
17.e	odd	16	1	inner	340.2.bf.a	✓	288
68.i	even	16	1	inner	340.2.bf.a	✓	288

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
340.2.bf.a	✓	288	1.a	even	1	1	trivial
340.2.bf.a	✓	288	4.b	odd	2	1	inner
340.2.bf.a	✓	288	17.e	odd	16	1	inner
340.2.bf.a	✓	288	68.i	even	16	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(340, [\chi])\).