Newform orbit 975.2.db.a

• Label: 975.2.db.a
• Level: $975$
• Weight: $2$
• Character orbit: 975.db
• Analytic conductor: $7.785$
• Analytic rank: $0$
• Dimension: $1120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.db (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.78541419707\)
Analytic rank:	\(0\)
Dimension:	\(1120\)
Relative dimension:	\(70\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1120 q - 4 q^{2} + 140 q^{4} - 4 q^{5} + 24 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
67.1	−1.85975	+	2.06546i	−0.933580	+	0.358368i	−0.598403	−	5.69343i	−1.56487	−	1.59724i	0.996030	−	2.59475i	−2.60195	−	1.50224i	8.37535	+	6.08504i	0.743145	−	0.669131i	6.20930	−	0.261702i
67.2	−1.84991	+	2.05453i	0.933580	−	0.358368i	−0.589879	−	5.61232i	−0.166732	+	2.22984i	−0.990758	+	2.58101i	−3.35354	−	1.93617i	8.14861	+	5.92031i	0.743145	−	0.669131i	−4.27284	−	4.46755i
67.3	−1.81978	+	2.02107i	0.933580	−	0.358368i	−0.564069	−	5.36676i	2.20157	−	0.391264i	−0.974624	+	2.53898i	3.36353	+	1.94193i	7.47265	+	5.42920i	0.743145	−	0.669131i	−3.21560	+	5.16154i
67.4	−1.69723	+	1.88496i	−0.933580	+	0.358368i	−0.463446	−	4.40939i	2.17159	+	0.533111i	0.908990	−	2.36800i	−3.05769	−	1.76536i	4.99403	+	3.62838i	0.743145	−	0.669131i	−4.69058	+	3.18855i
67.5	−1.68765	+	1.87432i	0.933580	−	0.358368i	−0.455873	−	4.33734i	−0.400445	−	2.19992i	−0.903857	+	2.35463i	−0.291625	−	0.168370i	4.81800	+	3.50048i	0.743145	−	0.669131i	4.79917	+	2.96212i
67.6	−1.68120	+	1.86716i	−0.933580	+	0.358368i	−0.450803	−	4.28911i	−1.37239	+	1.76537i	0.900405	−	2.34564i	0.312300	+	0.180307i	4.70102	+	3.41549i	0.743145	−	0.669131i	−0.988966	−	5.53043i
67.7	−1.67932	+	1.86507i	−0.933580	+	0.358368i	−0.449325	−	4.27504i	2.21455	−	0.309439i	0.899396	−	2.34301i	2.56949	+	1.48349i	4.66703	+	3.39080i	0.743145	−	0.669131i	−3.14181	+	4.64994i
67.8	−1.60054	+	1.77758i	0.933580	−	0.358368i	−0.389007	−	3.70116i	−2.11276	−	0.732295i	−0.857207	+	2.23310i	0.736265	+	0.425083i	3.33144	+	2.42043i	0.743145	−	0.669131i	4.68328	−	2.58353i
67.9	−1.51756	+	1.68543i	0.933580	−	0.358368i	−0.328603	−	3.12645i	0.465630	+	2.18705i	−0.812766	+	2.11733i	2.72269	+	1.57194i	2.09843	+	1.52460i	0.743145	−	0.669131i	−4.39274	−	2.53421i
67.10	−1.49175	+	1.65676i	0.933580	−	0.358368i	−0.310469	−	2.95392i	1.19681	+	1.88882i	−0.798943	+	2.08132i	−0.0545705	−	0.0315063i	1.74985	+	1.27134i	0.743145	−	0.669131i	−4.91467	−	0.834836i
67.11	−1.42112	+	1.57831i	0.933580	−	0.358368i	−0.262434	−	2.49689i	2.08730	−	0.801974i	−0.761112	+	1.98276i	−3.10653	−	1.79356i	0.877391	+	0.637462i	0.743145	−	0.669131i	−1.70054	+	4.43411i
67.12	−1.38455	+	1.53770i	−0.933580	+	0.358368i	−0.238481	−	2.26899i	−1.87174	+	1.22335i	0.741527	−	1.93174i	−0.651694	−	0.376256i	0.471217	+	0.342359i	0.743145	−	0.669131i	0.710370	−	4.57196i
67.13	−1.27178	+	1.41246i	−0.933580	+	0.358368i	−0.168548	−	1.60363i	−0.545915	−	2.16840i	0.681131	−	1.77441i	1.32078	+	0.762555i	−0.595904	−	0.432950i	0.743145	−	0.669131i	3.75706	+	1.98665i
67.14	−1.26407	+	1.40389i	−0.933580	+	0.358368i	−0.163982	−	1.56018i	1.13343	−	1.92752i	0.677000	−	1.76365i	1.83240	+	1.05794i	−0.659054	−	0.478831i	0.743145	−	0.669131i	1.27330	+	4.02773i
67.15	−1.23925	+	1.37632i	−0.933580	+	0.358368i	−0.149476	−	1.42217i	−1.65864	−	1.49963i	0.663707	−	1.72901i	−3.49641	−	2.01865i	−0.854044	−	0.620499i	0.743145	−	0.669131i	4.11945	−	0.424416i
67.16	−1.19031	+	1.32198i	−0.933580	+	0.358368i	−0.121720	−	1.15809i	0.857331	+	2.06518i	0.637499	−	1.66074i	3.82575	+	2.20880i	−1.20246	−	0.873641i	0.743145	−	0.669131i	−3.75062	−	1.32484i
67.17	−1.11047	+	1.23330i	0.933580	−	0.358368i	−0.0788344	−	0.750059i	0.923456	−	2.03647i	−0.594739	+	1.54935i	3.42817	+	1.97925i	−1.67265	−	1.21525i	0.743145	−	0.669131i	1.48612	+	3.40035i
67.18	−1.08960	+	1.21012i	0.933580	−	0.358368i	−0.0681120	−	0.648043i	−0.767501	+	2.10022i	−0.583558	+	1.52022i	−0.898252	−	0.518606i	−1.77635	−	1.29059i	0.743145	−	0.669131i	−1.70526	−	3.21717i
67.19	−1.06953	+	1.18783i	0.933580	−	0.358368i	−0.0579971	−	0.551806i	2.23559	−	0.0462887i	−0.572811	+	1.49222i	−1.78319	−	1.02953i	−1.86876	−	1.35774i	0.743145	−	0.669131i	−2.33605	+	2.70502i
67.20	−1.00381	+	1.11484i	−0.933580	+	0.358368i	−0.0261836	−	0.249120i	−2.22057	+	0.262778i	0.537611	−	1.40052i	0.714393	+	0.412455i	−2.12330	−	1.54267i	0.743145	−	0.669131i	1.93607	−	2.73936i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
325.bn	even	60	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	975.2.db.a	yes	1120
13.f	odd	12	1		975.2.cq.a	✓	1120
25.f	odd	20	1		975.2.cq.a	✓	1120
325.bn	even	60	1	inner	975.2.db.a	yes	1120

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
975.2.cq.a	✓	1120	13.f	odd	12	1
975.2.cq.a	✓	1120	25.f	odd	20	1
975.2.db.a	yes	1120	1.a	even	1	1	trivial
975.2.db.a	yes	1120	325.bn	even	60	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(975, [\chi])\).