Newform orbit 380.2.bb.a

• Label: 380.2.bb.a
• Level: $380$
• Weight: $2$
• Character orbit: 380.bb
• Analytic conductor: $3.034$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	380.bb (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.03431527681\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(56\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q - 18 q^{4} - 12 q^{5} - 18 q^{6} - 24 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
59.1	−1.41344	−	0.0466259i	−1.17302	+	1.39795i	1.99565	+	0.131806i	−1.08439	−	1.95553i	1.72317	−	1.92123i	1.36812	+	2.36965i	−2.81460	−	0.279350i	−0.0573416	−	0.325200i	1.44155	+	2.81459i
59.2	−1.40916	−	0.119480i	1.57099	−	1.87224i	1.97145	+	0.336732i	1.82806	+	1.28771i	−2.43747	+	2.45058i	−0.614347	−	1.06408i	−2.73785	−	0.710057i	−0.516307	−	2.92812i	−2.42217	−	2.03300i
59.3	−1.39943	+	0.203928i	0.497064	−	0.592377i	1.91683	−	0.570767i	−2.20658	+	0.361956i	−0.574805	+	0.930358i	−0.647033	−	1.12069i	−2.56608	+	1.18965i	0.417106	+	2.36552i	3.01415	−	0.956516i
59.4	−1.36504	−	0.369686i	1.57099	−	1.87224i	1.72666	+	1.00927i	0.572650	−	2.16150i	−2.83661	+	1.97490i	−0.614347	−	1.06408i	−1.98385	−	2.01602i	−0.516307	−	2.92812i	−1.58076	+	2.73883i
59.5	−1.34415	−	0.439613i	−1.17302	+	1.39795i	1.61348	+	1.18181i	0.426294	+	2.19506i	2.19126	−	1.36338i	1.36812	+	2.36965i	−1.64922	−	2.29784i	−0.0573416	−	0.325200i	0.391971	−	3.13789i
59.6	−1.33052	+	0.479280i	−1.75425	+	2.09063i	1.54058	−	1.27539i	−1.37018	+	1.76709i	1.33207	−	3.62240i	−1.59127	−	2.75617i	−1.43851	+	2.43530i	−0.772405	−	4.38052i	0.976131	−	3.00785i
59.7	−1.31199	+	0.527904i	−0.108582	+	0.129403i	1.44263	−	1.38521i	1.67124	−	1.48558i	0.0741462	−	0.227097i	0.176277	+	0.305321i	−1.16146	+	2.57895i	0.515989	+	2.92632i	−1.40841	+	2.83132i
59.8	−1.31059	+	0.531372i	0.664457	−	0.791869i	1.43529	−	1.39282i	0.846472	+	2.06966i	−0.450054	+	1.39089i	1.98367	+	3.43582i	−1.14097	+	2.58809i	0.335391	+	1.90210i	−2.20913	−	2.26268i
59.9	−1.24529	−	0.670264i	0.497064	−	0.592377i	1.10149	+	1.66935i	−1.92300	+	1.14109i	−1.01604	+	0.404518i	−0.647033	−	1.12069i	−0.252775	−	2.81711i	0.417106	+	2.36552i	3.15952	−	0.132067i
59.10	−1.08636	−	0.905441i	−1.75425	+	2.09063i	0.360351	+	1.96727i	−2.18548	−	0.472930i	3.79868	−	0.682806i	−1.59127	−	2.75617i	1.38978	−	2.46344i	−0.772405	−	4.38052i	1.94601	+	2.49260i
59.11	−1.07107	+	0.923481i	2.07566	−	2.47367i	0.294366	−	1.97822i	−1.29113	+	1.82565i	0.0612219	+	4.56629i	−1.40918	−	2.44076i	1.51156	+	2.39065i	−1.28975	−	7.31455i	−0.303064	−	3.14772i
59.12	−1.05231	−	0.944795i	−0.108582	+	0.129403i	0.214726	+	1.98844i	2.23516	+	0.0637665i	0.236522	−	0.0335848i	0.176277	+	0.305321i	1.65271	−	2.29533i	0.515989	+	2.92632i	−2.29184	−	2.17887i
59.13	−1.04981	−	0.947574i	0.664457	−	0.791869i	0.204208	+	1.98955i	−0.681915	−	2.12955i	−1.44791	+	0.201691i	1.98367	+	3.43582i	1.67086	−	2.28215i	0.335391	+	1.90210i	−1.30203	+	2.88179i
59.14	−0.997360	+	1.00263i	0.300395	−	0.357997i	−0.0105474	−	1.99997i	−1.26326	−	1.84504i	0.0593378	+	0.658238i	−1.68031	−	2.91038i	2.01576	+	1.98412i	0.483020	+	2.73934i	3.10982	+	0.573580i
59.15	−0.934972	+	1.06105i	−0.912584	+	1.08758i	−0.251654	−	1.98410i	1.86792	+	1.22918i	−0.300731	−	1.98515i	−0.869878	−	1.50667i	2.34052	+	1.58806i	0.170934	+	0.969416i	−3.05067	+	0.832706i
59.16	−0.906917	+	1.08513i	1.82738	−	2.17779i	−0.355003	−	1.96824i	1.74742	−	1.39518i	0.705894	+	3.95801i	1.71831	+	2.97619i	2.45775	+	1.39981i	−0.882491	−	5.00485i	−0.0708181	+	3.16148i
59.17	−0.777543	+	1.18128i	−1.85226	+	2.20744i	−0.790854	−	1.83699i	−0.543418	−	2.16903i	−1.16739	−	3.90441i	0.0958109	+	0.165949i	2.78493	+	0.494121i	−0.920966	−	5.22306i	2.98477	+	1.04458i
59.18	−0.690624	−	1.23411i	2.07566	−	2.47367i	−1.04608	+	1.70462i	−2.16257	−	0.568604i	−4.48629	−	0.853220i	−1.40918	−	2.44076i	2.82614	+	0.113728i	−1.28975	−	7.31455i	0.791797	+	3.06154i
59.19	−0.660906	+	1.25028i	0.742048	−	0.884338i	−1.12641	−	1.65264i	−2.17942	−	0.500136i	0.615247	+	1.51223i	1.69793	+	2.94091i	2.81071	−	0.316087i	0.289526	+	1.64198i	2.06570	−	2.39434i
59.20	−0.594291	−	1.28328i	0.300395	−	0.357997i	−1.29364	+	1.52529i	0.218254	+	2.22539i	−0.637934	−	0.172738i	−1.68031	−	2.91038i	2.72617	+	0.753640i	0.483020	+	2.73934i	2.72610	−	1.60261i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
19.f	odd	18	1	inner
20.d	odd	2	1	inner
76.k	even	18	1	inner
95.o	odd	18	1	inner
380.bb	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	380.2.bb.a	✓	336
4.b	odd	2	1	inner	380.2.bb.a	✓	336
5.b	even	2	1	inner	380.2.bb.a	✓	336
19.f	odd	18	1	inner	380.2.bb.a	✓	336
20.d	odd	2	1	inner	380.2.bb.a	✓	336
76.k	even	18	1	inner	380.2.bb.a	✓	336
95.o	odd	18	1	inner	380.2.bb.a	✓	336
380.bb	even	18	1	inner	380.2.bb.a	✓	336

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
380.2.bb.a	✓	336	1.a	even	1	1	trivial
380.2.bb.a	✓	336	4.b	odd	2	1	inner
380.2.bb.a	✓	336	5.b	even	2	1	inner
380.2.bb.a	✓	336	19.f	odd	18	1	inner
380.2.bb.a	✓	336	20.d	odd	2	1	inner
380.2.bb.a	✓	336	76.k	even	18	1	inner
380.2.bb.a	✓	336	95.o	odd	18	1	inner
380.2.bb.a	✓	336	380.bb	even	18	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(380, [\chi])\).