Newform orbit 608.2.bs.a

• Label: 608.2.bs.a
• Level: $608$
• Weight: $2$
• Character orbit: 608.bs
• Analytic conductor: $4.855$
• Analytic rank: $0$
• Dimension: $1872$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 608 = 2^{5} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	608.bs (of order \(72\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.85490444289\)
Analytic rank:	\(0\)
Dimension:	\(1872\)
Relative dimension:	\(78\) over \(\Q(\zeta_{72})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{72}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1872 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 12 q^{7} - 12 q^{8} - 24 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−1.41386	−	0.0314833i	−0.156706	+	0.0347408i	1.99802	+	0.0890260i	−0.396204	−	0.761100i	0.222654	−	0.0441852i	0.148420	−	0.553910i	−2.82212	−	0.188775i	−2.69557	+	1.25697i	0.536216	+	1.08857i
5.2	−1.41350	+	0.0449520i	2.46504	−	0.546487i	1.99596	−	0.127079i	−0.766897	−	1.47320i	−3.45977	+	0.883267i	1.27851	−	4.77145i	−2.81557	+	0.269348i	3.05887	−	1.42637i	1.15023	+	2.04789i
5.3	−1.41014	+	0.107210i	−0.0169674	+	0.00376159i	1.97701	−	0.302363i	1.34097	+	2.57598i	0.0235233	−	0.00712347i	0.956033	−	3.56796i	−2.75546	+	0.638331i	−2.71865	+	1.26773i	−2.16713	−	3.48873i
5.4	−1.40022	+	0.198463i	−2.87045	+	0.636364i	1.92123	−	0.555782i	0.174341	+	0.334906i	3.89297	−	1.46073i	−0.216088	+	0.806452i	−2.57983	+	1.15951i	5.11562	−	2.38545i	−0.310582	−	0.434342i
5.5	−1.38078	+	0.305670i	2.85205	−	0.632284i	1.81313	−	0.844130i	1.45182	+	2.78893i	−3.74479	+	1.74483i	−0.588059	+	2.19467i	−2.24552	+	1.71978i	5.01547	−	2.33875i	−2.85715	−	3.40713i
5.6	−1.38029	−	0.307885i	−1.84888	+	0.409887i	1.81041	+	0.849943i	−1.79336	−	3.44502i	2.67819	+	0.00347873i	0.463026	−	1.72804i	−2.23722	−	1.73057i	0.531425	−	0.247808i	1.41470	+	5.30728i
5.7	−1.36444	−	0.371887i	1.61614	−	0.358289i	1.72340	+	1.01484i	0.465678	+	0.894559i	−2.33837	−	0.112156i	−0.557373	+	2.08015i	−1.97407	−	2.02559i	−0.235397	+	0.109767i	−0.302715	−	1.39375i
5.8	−1.34364	+	0.441162i	−1.55647	+	0.345060i	1.61075	−	1.18553i	−1.20654	−	2.31773i	1.93911	−	1.15029i	−0.251489	+	0.938570i	−1.64127	+	2.30353i	−0.415406	+	0.193707i	2.64365	+	2.58193i
5.9	−1.33508	−	0.466430i	1.45935	−	0.323529i	1.56489	+	1.24544i	−1.39696	−	2.68354i	−2.09925	−	0.248745i	−0.533998	+	1.99291i	−1.50834	−	2.39268i	−0.693904	+	0.323573i	0.613375	+	4.23433i
5.10	−1.32975	−	0.481423i	−1.50982	+	0.334720i	1.53646	+	1.28034i	1.92570	+	3.69923i	2.16883	+	0.281770i	−0.483496	+	1.80443i	−1.42673	−	2.44222i	−0.551390	+	0.257118i	−0.779801	−	5.84612i
5.11	−1.30489	+	0.545227i	2.34110	−	0.519010i	1.40546	−	1.42292i	−0.541983	−	1.04114i	−2.77190	+	1.95368i	0.247577	−	0.923969i	−1.05815	+	2.62304i	2.49247	−	1.16226i	1.27488	+	1.06306i
5.12	−1.23880	−	0.682192i	−1.03447	+	0.229336i	1.06923	+	1.69019i	0.411281	+	0.790064i	1.43794	+	0.421605i	0.817865	−	3.05231i	−0.171519	−	2.82322i	−1.70140	+	0.793374i	0.0294820	−	1.25930i
5.13	−1.23867	+	0.682413i	−1.65655	+	0.367249i	1.06863	−	1.69057i	1.38206	+	2.65492i	1.80131	−	1.58535i	−0.564830	+	2.10797i	−0.170009	+	2.82331i	−0.109627	+	0.0511200i	−3.52368	−	2.34544i
5.14	−1.23346	+	0.691795i	1.22417	−	0.271391i	1.04284	−	1.70660i	−1.59524	−	3.06444i	−1.32221	+	1.18162i	−1.20982	+	4.51512i	−0.105684	+	2.82645i	−1.29399	+	0.603399i	4.08763	+	2.67627i
5.15	−1.19960	−	0.748979i	−3.02012	+	0.669544i	0.878062	+	1.79694i	0.621968	+	1.19479i	4.12439	+	1.45882i	−0.164230	+	0.612915i	0.292552	−	2.81326i	5.95389	−	2.77634i	0.148761	−	1.89911i
5.16	−1.13044	+	0.849772i	0.392174	−	0.0869428i	0.555775	−	1.92123i	1.06775	+	2.05114i	−0.369446	+	0.431542i	0.0417629	−	0.155861i	1.00434	+	2.64411i	−2.57268	+	1.19966i	−2.95003	−	1.41133i
5.17	−1.07965	−	0.913432i	2.30439	−	0.510872i	0.331284	+	1.97237i	1.23464	+	2.37172i	−2.95458	−	1.55334i	0.838744	−	3.13024i	1.44396	−	2.43207i	2.33032	−	1.08664i	0.833428	−	3.68838i
5.18	−1.07640	+	0.917263i	−0.0777078	+	0.0172274i	0.317255	−	1.97468i	0.0519092	+	0.0997166i	0.0678422	−	0.0898220i	0.740746	−	2.76450i	1.46981	+	2.41654i	−2.71318	+	1.26518i	−0.147341	−	0.0597201i
5.19	−0.999963	+	1.00004i	−2.47926	+	0.549639i	−0.000147228	−	2.00000i	−0.997283	−	1.91576i	1.92951	−	3.02897i	1.26785	−	4.73167i	2.00022	+	1.99978i	3.12572	−	1.45755i	2.91308	+	0.918373i
5.20	−0.966926	−	1.03201i	−0.933017	+	0.206845i	−0.130108	+	1.99576i	−0.245292	−	0.471201i	1.11563	+	0.762883i	−0.959804	+	3.58204i	2.18546	−	1.79548i	−1.89119	+	0.881875i	−0.249107	+	0.708761i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.e	even	9	1	inner
32.g	even	8	1	inner
608.bs	even	72	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	608.2.bs.a	✓	1872
19.e	even	9	1	inner	608.2.bs.a	✓	1872
32.g	even	8	1	inner	608.2.bs.a	✓	1872
608.bs	even	72	1	inner	608.2.bs.a	✓	1872

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
608.2.bs.a	✓	1872	1.a	even	1	1	trivial
608.2.bs.a	✓	1872	19.e	even	9	1	inner
608.2.bs.a	✓	1872	32.g	even	8	1	inner
608.2.bs.a	✓	1872	608.bs	even	72	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(608, [\chi])\).