Newform orbit 315.2.cc.a

• Label: 315.2.cc.a
• Level: $315$
• Weight: $2$
• Character orbit: 315.cc
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.cc (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(36\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 4 q^{3}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
92.1	−0.713154	−	2.66153i	1.69514	+	0.355654i	−4.84309	+	2.79616i	1.19095	+	1.89252i	−0.262317	−	4.76530i	0.965926	−	0.258819i	6.99916	+	6.99916i	2.74702	+	1.20577i	4.18768	−	4.51940i
92.2	−0.679005	−	2.53408i	0.297629	+	1.70629i	−4.22847	+	2.44131i	0.0454078	−	2.23561i	4.12178	−	1.91279i	−0.965926	+	0.258819i	5.34747	+	5.34747i	−2.82283	+	1.01568i	−5.69604	+	1.40292i
92.3	−0.646610	−	2.41318i	−1.26505	+	1.18307i	−3.67328	+	2.12077i	−2.08010	+	0.820487i	3.67295	+	2.28781i	0.965926	−	0.258819i	3.95984	+	3.95984i	0.200708	−	2.99328i	3.32499	+	4.48911i
92.4	−0.626308	−	2.33741i	−1.18606	−	1.26224i	−3.33919	+	1.92788i	−0.366266	+	2.20587i	−2.20755	+	3.56287i	−0.965926	+	0.258819i	3.17541	+	3.17541i	−0.186522	+	2.99420i	5.38542	−	0.525436i
92.5	−0.616561	−	2.30104i	1.18411	−	1.26408i	−3.18258	+	1.83746i	−2.21436	+	0.310826i	−3.63877	−	1.94529i	−0.965926	+	0.258819i	2.82137	+	2.82137i	−0.195791	−	2.99360i	2.08051	+	4.90368i
92.6	−0.540409	−	2.01684i	1.29595	−	1.14913i	−2.04353	+	1.17983i	−0.0970501	−	2.23396i	−3.01795	−	1.99273i	0.965926	−	0.258819i	0.531022	+	0.531022i	0.358997	−	2.97844i	−4.45308	+	1.40299i
92.7	−0.522977	−	1.95178i	0.00508642	−	1.73204i	−1.80388	+	1.04147i	2.08954	+	0.796124i	−3.38322	+	0.895892i	0.965926	−	0.258819i	0.118505	+	0.118505i	−2.99995	−	0.0176198i	0.461075	−	4.49468i
92.8	−0.484811	−	1.80934i	0.413768	+	1.68190i	−1.30661	+	0.754373i	1.61436	+	1.54720i	2.84253	−	1.56405i	−0.965926	+	0.258819i	−0.650680	−	0.650680i	−2.65759	+	1.39183i	2.01675	−	3.67103i
92.9	−0.372518	−	1.39026i	−1.46969	+	0.916517i	−0.0619937	+	0.0357921i	−0.217588	+	2.22546i	1.82168	+	1.70183i	−0.965926	+	0.258819i	−1.96262	−	1.96262i	1.31999	−	2.69400i	3.17501	−	0.526519i
92.10	−0.352695	−	1.31628i	−0.672895	−	1.59600i	0.123861	−	0.0715111i	−0.441766	−	2.19200i	−1.86345	+	1.44862i	−0.965926	+	0.258819i	−2.06498	−	2.06498i	−2.09442	+	2.14788i	−2.72946	+	1.35459i
92.11	−0.303557	−	1.13289i	−0.471331	+	1.66669i	0.540756	−	0.312205i	−0.436585	−	2.19303i	2.03125	+	0.0280322i	0.965926	−	0.258819i	−2.17651	−	2.17651i	−2.55569	−	1.57112i	−2.35194	+	1.16031i
92.12	−0.292475	−	1.09153i	−1.63819	−	0.562423i	0.626149	−	0.361507i	−2.16876	−	0.544513i	−0.134772	+	1.95264i	0.965926	−	0.258819i	−2.17584	−	2.17584i	2.36736	+	1.84272i	0.0399539	+	2.52653i
92.13	−0.270870	−	1.01090i	1.36595	+	1.06499i	0.783502	−	0.452355i	2.05901	−	0.872052i	0.706600	−	1.66931i	0.965926	−	0.258819i	−2.14957	−	2.14957i	0.731614	+	2.90942i	−1.43928	−	1.84524i
92.14	−0.231820	−	0.865164i	1.63235	−	0.579155i	1.03728	−	0.598875i	−0.767760	+	2.10013i	−0.879477	−	1.27799i	0.965926	−	0.258819i	−2.02528	−	2.02528i	2.32916	−	1.89077i	1.99494	+	0.177387i
92.15	−0.220022	−	0.821133i	0.953832	−	1.44575i	1.10620	−	0.638666i	1.96491	+	1.06730i	−1.39702	−	0.465126i	−0.965926	+	0.258819i	−1.97004	−	1.97004i	−1.18041	−	2.75801i	0.444073	−	1.84828i
92.16	−0.165403	−	0.617291i	−1.49090	+	0.881605i	1.37836	−	0.795797i	1.71710	−	1.43233i	0.790805	+	0.774498i	−0.965926	+	0.258819i	−1.62300	−	1.62300i	1.44555	−	2.62876i	−1.16818	−	0.823041i
92.17	−0.0549668	−	0.205139i	1.73204	−	0.00719422i	1.69299	−	0.977448i	−1.03445	−	1.98240i	−0.0966803	−	0.354913i	−0.965926	+	0.258819i	−0.593915	−	0.593915i	2.99990	−	0.0249213i	−0.349807	+	0.321173i
92.18	−0.0242815	−	0.0906199i	−0.235889	+	1.71591i	1.72443	−	0.995599i	−0.436480	+	2.19305i	0.161224	−	0.0202887i	0.965926	−	0.258819i	−0.264770	−	0.264770i	−2.88871	−	0.809530i	0.209333	−	0.0136969i
92.19	0.0573303	+	0.213960i	−1.05605	−	1.37287i	1.68956	−	0.975467i	1.86575	−	1.23247i	0.233194	−	0.304659i	0.965926	−	0.258819i	0.618832	+	0.618832i	−0.769518	+	2.89963i	0.370664	+	0.328536i
92.20	0.0867243	+	0.323660i	−1.73132	+	0.0504811i	1.63482	−	0.943862i	−2.20059	+	0.396767i	−0.166486	−	0.555979i	−0.965926	+	0.258819i	0.921139	+	0.921139i	2.99490	−	0.174797i	−0.319262	−	0.677831i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
9.d	odd	6	1	inner
45.l	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	315.2.cc.a	✓	144
3.b	odd	2	1		945.2.cf.a		144
5.c	odd	4	1	inner	315.2.cc.a	✓	144
9.c	even	3	1		945.2.cf.a		144
9.d	odd	6	1	inner	315.2.cc.a	✓	144
15.e	even	4	1		945.2.cf.a		144
45.k	odd	12	1		945.2.cf.a		144
45.l	even	12	1	inner	315.2.cc.a	✓	144

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
315.2.cc.a	✓	144	1.a	even	1	1	trivial
315.2.cc.a	✓	144	5.c	odd	4	1	inner
315.2.cc.a	✓	144	9.d	odd	6	1	inner
315.2.cc.a	✓	144	45.l	even	12	1	inner
945.2.cf.a		144	3.b	odd	2	1
945.2.cf.a		144	9.c	even	3	1
945.2.cf.a		144	15.e	even	4	1
945.2.cf.a		144	45.k	odd	12	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(315, [\chi])\).