Newform orbit 975.2.cy.a

• Label: 975.2.cy.a
• Level: $975$
• Weight: $2$
• Character orbit: 975.cy
• Analytic conductor: $7.785$
• Analytic rank: $0$
• Dimension: $2176$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(7.78541419707\)
Analytic rank:	\(0\)
Dimension:	\(2176\)
Relative dimension:	\(136\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 2176 q - 8 q^{3} - 20 q^{4} - 18 q^{6} - 48 q^{7} - 10 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
17.1	−1.51918	−	2.33933i	0.523404	−	1.65107i	−2.35108	+	5.28062i	0.376468	−	2.20415i	−4.65755	+	1.28386i	1.90763	+	0.511149i	10.4149	−	1.64955i	−2.45210	−	1.72836i	−5.72815	+	2.46781i
17.2	−1.50038	−	2.31038i	1.70607	+	0.298864i	−2.27324	+	5.10578i	1.78575	+	1.34577i	−1.86926	−	4.39008i	1.45541	+	0.389975i	9.76519	−	1.54665i	2.82136	+	1.01977i	0.429943	−	6.14492i
17.3	−1.45834	−	2.24564i	1.54823	+	0.776511i	−2.10268	+	4.72271i	−2.10897	+	0.743122i	−0.514082	−	4.60919i	−2.94931	−	0.790265i	8.38261	−	1.32768i	1.79406	+	2.40444i	4.74438	+	3.65228i
17.4	−1.45162	−	2.23530i	−0.374513	+	1.69108i	−2.07589	+	4.66254i	2.22741	−	0.196620i	4.32372	−	1.61765i	−4.29362	−	1.15047i	8.17063	−	1.29410i	−2.71948	−	1.26666i	−3.67286	−	4.69351i
17.5	−1.44430	−	2.22403i	−0.188913	+	1.72172i	−2.04682	+	4.59723i	−2.23205	−	0.134024i	4.10199	−	2.06653i	4.23635	+	1.13513i	7.94217	−	1.25792i	−2.92862	−	0.650509i	2.92567	+	5.15770i
17.6	−1.42442	−	2.19342i	0.712731	+	1.57861i	−1.96864	+	4.42163i	0.161777	−	2.23021i	2.44733	−	3.81193i	0.228400	+	0.0611995i	7.33636	−	1.16197i	−1.98403	+	2.25025i	−5.12222	+	2.82192i
17.7	−1.42341	−	2.19186i	−1.45397	−	0.941260i	−1.96469	+	4.41276i	−2.23557	+	0.0473798i	0.00648782	+	4.52670i	1.79263	+	0.480335i	7.30608	−	1.15717i	1.22806	+	2.73713i	3.28598	+	4.83261i
17.8	−1.39682	−	2.15092i	−1.66358	+	0.482187i	−1.86186	+	4.18181i	−0.351312	+	2.20830i	3.36087	+	2.90469i	−2.65317	−	0.710914i	6.52923	−	1.03413i	2.53499	−	1.60431i	5.24059	−	2.32896i
17.9	−1.39365	−	2.14603i	0.0479423	−	1.73139i	−1.84972	+	4.15454i	0.976954	+	2.01136i	−3.78243	+	2.31006i	−3.77852	−	1.01245i	6.43894	−	1.01983i	−2.99540	−	0.166014i	2.95491	−	4.89970i
17.10	−1.37816	−	2.12219i	−1.61079	+	0.636669i	−1.79086	+	4.02235i	−1.35883	−	1.77583i	3.57106	+	2.54097i	−1.18292	−	0.316963i	6.00574	−	0.951216i	2.18931	−	2.05108i	−1.89596	+	5.33108i
17.11	−1.34554	−	2.07195i	1.09035	−	1.34579i	−1.66902	+	3.74868i	−1.25432	+	1.85113i	−4.25551	−	0.448328i	4.35745	+	1.16757i	5.13260	−	0.812925i	−0.622295	−	2.93475i	5.52318	+	0.108123i
17.12	−1.30347	−	2.00717i	−1.00262	−	1.41236i	−1.51622	+	3.40548i	0.0609743	+	2.23524i	−1.52796	+	3.85339i	1.04627	+	0.280348i	4.08409	−	0.646856i	−0.989519	+	2.83211i	4.40702	−	3.03595i
17.13	−1.30063	−	2.00279i	1.45291	−	0.942898i	−1.50606	+	3.38267i	−1.35983	−	1.77507i	−3.77812	−	1.68351i	−2.48159	−	0.664939i	4.01629	−	0.636119i	1.22189	−	2.73989i	−1.78646	+	5.03215i
17.14	−1.25795	−	1.93708i	−1.02817	−	1.39387i	−1.35635	+	3.04640i	1.99011	−	1.01954i	−1.40664	+	3.74506i	1.06210	+	0.284588i	3.04481	−	0.482250i	−0.885732	+	2.86627i	−4.47838	−	2.57247i
17.15	−1.25355	−	1.93029i	−1.15092	+	1.29437i	−1.34118	+	3.01234i	1.34376	−	1.78726i	3.94124	+	0.599066i	2.81600	+	0.754546i	2.94938	−	0.467136i	−0.350765	−	2.97942i	−5.13441	−	0.353435i
17.16	−1.25055	−	1.92569i	1.09865	−	1.33902i	−1.33090	+	2.98926i	2.20786	+	0.354075i	−3.95245	−	0.441148i	−1.14282	−	0.306217i	2.88505	−	0.456947i	−0.585924	−	2.94223i	−2.07921	−	4.69443i
17.17	−1.23049	−	1.89479i	−0.806555	+	1.53280i	−1.26265	+	2.83597i	1.70793	+	1.44326i	3.89679	−	0.357842i	1.88129	+	0.504089i	2.46433	−	0.390311i	−1.69894	−	2.47257i	0.633080	−	5.01208i
17.18	−1.21744	−	1.87469i	0.783086	+	1.54492i	−1.21884	+	2.73755i	0.443284	+	2.19169i	1.94289	−	3.34889i	1.21164	+	0.324657i	2.20034	−	0.348500i	−1.77355	+	2.41961i	3.56907	−	3.49927i
17.19	−1.21232	−	1.86681i	−1.39432	−	1.02755i	−1.20179	+	2.69927i	−0.743591	−	2.10881i	−0.227872	+	3.84867i	−4.09486	−	1.09721i	2.09897	−	0.332444i	0.888283	+	2.86548i	−3.03528	+	3.94470i
17.20	−1.19938	−	1.84688i	1.71520	+	0.241024i	−1.15900	+	2.60315i	1.78338	−	1.34890i	−1.61203	−	3.45685i	2.86872	+	0.768672i	1.84770	−	0.292646i	2.88381	+	0.826808i	−4.63023	−	1.67585i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
13.e	even	6	1	inner
25.f	odd	20	1	inner
39.h	odd	6	1	inner
75.l	even	20	1	inner
325.bk	odd	60	1	inner
975.cy	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.cy.a	✓	2176
3.b	odd	2	1	inner	975.2.cy.a	✓	2176
13.e	even	6	1	inner	975.2.cy.a	✓	2176
25.f	odd	20	1	inner	975.2.cy.a	✓	2176
39.h	odd	6	1	inner	975.2.cy.a	✓	2176
75.l	even	20	1	inner	975.2.cy.a	✓	2176
325.bk	odd	60	1	inner	975.2.cy.a	✓	2176
975.cy	even	60	1	inner	975.2.cy.a	✓	2176

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
975.2.cy.a	✓	2176	1.a	even	1	1	trivial
975.2.cy.a	✓	2176	3.b	odd	2	1	inner
975.2.cy.a	✓	2176	13.e	even	6	1	inner
975.2.cy.a	✓	2176	25.f	odd	20	1	inner
975.2.cy.a	✓	2176	39.h	odd	6	1	inner
975.2.cy.a	✓	2176	75.l	even	20	1	inner
975.2.cy.a	✓	2176	325.bk	odd	60	1	inner
975.2.cy.a	✓	2176	975.cy	even	60	1	inner

Hecke kernels

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