Newform orbit 912.2.cp.a

• Label: 912.2.cp.a
• Level: $912$
• Weight: $2$
• Character orbit: 912.cp
• Analytic conductor: $7.282$
• Analytic rank: $0$
• Dimension: $1872$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.cp (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.28235666434\)
Analytic rank:	\(0\)
Dimension:	\(1872\)
Relative dimension:	\(156\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

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

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
35.1	−1.41344	−	0.0468870i	1.39453	−	1.02727i	1.99560	+	0.132544i	−0.502592	+	0.351919i	−2.01925	+	1.38659i	−0.193084	−	0.334431i	−2.81444	−	0.280910i	0.889448	−	2.86511i	0.726882	−	0.473850i
35.2	−1.41026	−	0.105681i	1.66350	+	0.482454i	1.97766	+	0.298074i	2.50717	−	1.75554i	−2.29498	−	0.856185i	−2.62025	−	4.53840i	−2.75752	−	0.629362i	2.53448	+	1.60513i	−3.72129	+	2.21081i
35.3	−1.40957	−	0.114461i	−1.29628	+	1.14876i	1.97380	+	0.322684i	−3.01939	+	2.11420i	1.95870	−	1.47089i	1.82515	+	3.16125i	−2.74528	−	0.680770i	0.360699	−	2.97824i	4.49805	−	2.63452i
35.4	−1.40933	+	0.117370i	1.13830	+	1.30548i	1.97245	−	0.330827i	−0.889415	+	0.622775i	−1.75747	−	1.70625i	1.74589	+	3.02397i	−2.74101	+	0.697752i	−0.408550	+	2.97205i	1.18039	−	0.982089i
35.5	−1.40585	−	0.153562i	−1.30967	+	1.13347i	1.95284	+	0.431772i	2.59581	−	1.81761i	2.01526	−	1.39238i	2.17690	+	3.77050i	−2.67910	−	0.906889i	0.430477	−	2.96895i	−3.92844	+	2.15667i
35.6	−1.40296	+	0.178093i	−1.65261	+	0.518551i	1.93657	−	0.499714i	−1.97280	+	1.38137i	2.22618	−	1.02182i	−1.45116	−	2.51349i	−2.62792	+	1.04597i	2.46221	−	1.71392i	2.52173	−	2.28934i
35.7	−1.40122	+	0.191258i	1.60810	+	0.643439i	1.92684	−	0.535990i	−1.90940	+	1.33698i	−2.37637	−	0.594037i	−0.518821	−	0.898624i	−2.59742	+	1.11956i	2.17197	+	2.06943i	2.41979	−	2.23859i
35.8	−1.40018	−	0.198727i	−1.72036	−	0.200890i	1.92102	+	0.556507i	1.19452	−	0.836412i	2.36890	+	0.623164i	−0.701033	−	1.21422i	−2.57918	−	1.16097i	2.91929	+	0.691207i	−1.83876	+	0.933745i
35.9	−1.40009	+	0.199356i	−0.795085	−	1.53878i	1.92051	−	0.558234i	0.330045	−	0.231100i	1.41996	+	1.99593i	−1.00889	−	1.74744i	−2.57761	+	1.16445i	−1.73568	+	2.44692i	−0.416022	+	0.389358i
35.10	−1.39854	+	0.209948i	−1.63399	−	0.574527i	1.91184	−	0.587243i	2.14105	−	1.49918i	2.40582	+	0.460448i	0.785827	+	1.36109i	−2.55050	+	1.22267i	2.33984	+	1.87754i	−2.67960	+	2.54618i
35.11	−1.39524	−	0.230882i	0.865020	−	1.50058i	1.89339	+	0.644271i	−3.40164	+	2.38186i	−1.55337	+	1.89395i	−0.488036	−	0.845302i	−2.49298	−	1.33606i	−1.50348	−	2.59606i	5.29604	−	2.53788i
35.12	−1.39231	−	0.247935i	0.0360756	+	1.73168i	1.87706	+	0.690405i	1.13270	−	0.793127i	0.379114	−	2.41997i	−2.10364	−	3.64361i	−2.44227	−	1.42665i	−2.99740	+	0.124942i	−1.77372	+	0.823443i
35.13	−1.38295	−	0.295740i	0.224593	−	1.71743i	1.82508	+	0.817985i	0.0832435	−	0.0582877i	−0.818512	+	2.30869i	1.93484	+	3.35124i	−2.28207	−	1.67098i	−2.89912	−	0.771444i	−0.132359	+	0.0559903i
35.14	−1.36997	−	0.350960i	0.347180	+	1.69690i	1.75365	+	0.961612i	0.0600430	−	0.0420426i	0.119916	−	2.44655i	0.573707	+	0.993689i	−2.06497	−	1.93285i	−2.75893	+	1.17826i	−0.0970126	+	0.0365245i
35.15	−1.36684	+	0.362985i	0.458054	−	1.67039i	1.73648	−	0.992283i	3.38649	−	2.37125i	−0.0197598	+	2.44941i	−1.02470	−	1.77483i	−2.01330	+	1.98661i	−2.58037	−	1.53025i	−3.76805	+	4.47035i
35.16	−1.36548	+	0.368062i	1.47385	−	0.909820i	1.72906	−	1.00516i	2.37367	−	1.66206i	−1.67764	+	1.78481i	2.31864	+	4.01600i	−1.99103	+	2.00893i	1.34446	−	2.68187i	−2.62945	+	3.14317i
35.17	−1.33671	+	0.461738i	−0.787529	+	1.54266i	1.57360	−	1.23442i	0.180459	−	0.126359i	0.340396	−	2.42572i	0.307257	+	0.532185i	−1.53347	+	2.37665i	−1.75959	−	2.42978i	−0.182877	+	0.252230i
35.18	−1.31866	−	0.511017i	−0.803564	−	1.53437i	1.47772	+	1.34772i	−1.53868	+	1.07740i	0.275538	+	2.43394i	−2.29237	−	3.97050i	−1.25991	−	2.53232i	−1.70857	+	2.46593i	2.57956	−	0.634425i
35.19	−1.30687	+	0.540445i	0.593298	+	1.62727i	1.41584	−	1.41259i	−2.58876	+	1.81267i	−1.65481	−	1.80599i	−1.23977	−	2.14735i	−1.08689	+	2.61126i	−2.29599	+	1.93091i	2.40353	−	3.76801i
35.20	−1.28968	−	0.580284i	1.60579	−	0.649173i	1.32654	+	1.49676i	1.33440	−	0.934354i	−2.44766	−	0.0945918i	0.668958	+	1.15867i	−0.842267	−	2.70011i	2.15715	−	2.08488i	−2.26313	+	0.430688i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
16.f	odd	4	1	inner
19.e	even	9	1	inner
48.k	even	4	1	inner
57.l	odd	18	1	inner
304.bh	odd	36	1	inner
912.cp	even	36	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	912.2.cp.a	✓	1872
3.b	odd	2	1	inner	912.2.cp.a	✓	1872
16.f	odd	4	1	inner	912.2.cp.a	✓	1872
19.e	even	9	1	inner	912.2.cp.a	✓	1872
48.k	even	4	1	inner	912.2.cp.a	✓	1872
57.l	odd	18	1	inner	912.2.cp.a	✓	1872
304.bh	odd	36	1	inner	912.2.cp.a	✓	1872
912.cp	even	36	1	inner	912.2.cp.a	✓	1872

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
912.2.cp.a	✓	1872	1.a	even	1	1	trivial
912.2.cp.a	✓	1872	3.b	odd	2	1	inner
912.2.cp.a	✓	1872	16.f	odd	4	1	inner
912.2.cp.a	✓	1872	19.e	even	9	1	inner
912.2.cp.a	✓	1872	48.k	even	4	1	inner
912.2.cp.a	✓	1872	57.l	odd	18	1	inner
912.2.cp.a	✓	1872	304.bh	odd	36	1	inner
912.2.cp.a	✓	1872	912.cp	even	36	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(912, [\chi])\).