Newform orbit 676.2.f.j

• Label: 676.2.f.j
• Level: $676$
• Weight: $2$
• Character orbit: 676.f
• Analytic conductor: $5.398$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 676 = 2^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	676.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.39788717664\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(36\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 64 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
99.1	−1.41391	+	0.0293598i		−	1.05568i	1.99828	−	0.0830241i	−1.55574	−	1.55574i	0.0309945	+	1.49263i	−0.0324435	−	0.0324435i	−2.82294	+	0.176057i	1.88554			2.24535	+	2.15400i
99.2	−1.40338	−	0.174698i		−	0.433961i	1.93896	+	0.490335i	1.49858	+	1.49858i	−0.0758119	+	0.609012i	2.96675	+	2.96675i	−2.63544	−	1.02686i	2.81168			−1.84129	−	2.36489i
99.3	−1.39808	+	0.212979i			1.71187i	1.90928	−	0.595524i	2.17634	+	2.17634i	−0.364592	−	2.39334i	−1.95037	−	1.95037i	−2.54250	+	1.23923i	0.0694977			−3.50622	−	2.57919i
99.4	−1.37640	−	0.324844i		−	2.58841i	1.78895	+	0.894230i	−2.44781	−	2.44781i	−0.840828	+	3.56268i	0.893641	+	0.893641i	−2.17183	−	1.81195i	−3.69986			2.57401	+	4.16432i
99.5	−1.37390	+	0.335273i			3.21981i	1.77518	−	0.921261i	−0.993752	−	0.993752i	−1.07951	−	4.42369i	−1.03001	−	1.03001i	−2.13005	+	1.86089i	−7.36718			1.69849	+	1.03213i
99.6	−1.34564	−	0.435033i			1.38730i	1.62149	+	1.17080i	−0.271471	−	0.271471i	0.603521	−	1.86680i	−2.48002	−	2.48002i	−1.67261	−	2.28087i	1.07540			0.247204	+	0.483401i
99.7	−1.31117	+	0.529942i			0.795345i	1.43832	−	1.38969i	−2.21358	−	2.21358i	−0.421486	−	1.04283i	3.05060	+	3.05060i	−1.14943	+	2.58434i	2.36743			4.07544	+	1.72931i
99.8	−1.12461	−	0.857469i		−	2.90707i	0.529492	+	1.92864i	0.657840	+	0.657840i	−2.49272	+	3.26931i	1.77139	+	1.77139i	1.05827	−	2.62299i	−5.45103			−0.175735	−	1.30389i
99.9	−1.03639	+	0.962238i			1.64057i	0.148195	−	1.99450i	−2.49234	−	2.49234i	−1.57862	−	1.70027i	−1.98479	−	1.98479i	1.76560	+	2.20967i	0.308521			4.98125	+	0.184803i
99.10	−0.962238	+	1.03639i		−	1.64057i	−0.148195	−	1.99450i	2.49234	+	2.49234i	1.70027	+	1.57862i	−1.98479	−	1.98479i	2.20967	+	1.76560i	0.308521			−4.98125	+	0.184803i
99.11	−0.857469	−	1.12461i			2.90707i	−0.529492	+	1.92864i	0.657840	+	0.657840i	3.26931	−	2.49272i	−1.77139	−	1.77139i	2.62299	−	1.05827i	−5.45103			0.175735	−	1.30389i
99.12	−0.529942	+	1.31117i		−	0.795345i	−1.43832	−	1.38969i	2.21358	+	2.21358i	1.04283	+	0.421486i	3.05060	+	3.05060i	2.58434	−	1.14943i	2.36743			−4.07544	+	1.72931i
99.13	−0.435033	−	1.34564i		−	1.38730i	−1.62149	+	1.17080i	−0.271471	−	0.271471i	−1.86680	+	0.603521i	2.48002	+	2.48002i	2.28087	+	1.67261i	1.07540			−0.247204	+	0.483401i
99.14	−0.335273	+	1.37390i		−	3.21981i	−1.77518	−	0.921261i	0.993752	+	0.993752i	4.42369	+	1.07951i	−1.03001	−	1.03001i	1.86089	−	2.13005i	−7.36718			−1.69849	+	1.03213i
99.15	−0.324844	−	1.37640i			2.58841i	−1.78895	+	0.894230i	−2.44781	−	2.44781i	3.56268	−	0.840828i	−0.893641	−	0.893641i	1.81195	+	2.17183i	−3.69986			−2.57401	+	4.16432i
99.16	−0.212979	+	1.39808i		−	1.71187i	−1.90928	−	0.595524i	−2.17634	−	2.17634i	2.39334	+	0.364592i	−1.95037	−	1.95037i	1.23923	−	2.54250i	0.0694977			3.50622	−	2.57919i
99.17	−0.174698	−	1.40338i			0.433961i	−1.93896	+	0.490335i	1.49858	+	1.49858i	0.609012	−	0.0758119i	−2.96675	−	2.96675i	1.02686	+	2.63544i	2.81168			1.84129	−	2.36489i
99.18	−0.0293598	+	1.41391i			1.05568i	−1.99828	−	0.0830241i	1.55574	+	1.55574i	−1.49263	−	0.0309945i	−0.0324435	−	0.0324435i	0.176057	−	2.82294i	1.88554			−2.24535	+	2.15400i
99.19	0.0293598	−	1.41391i			1.05568i	−1.99828	−	0.0830241i	−1.55574	−	1.55574i	1.49263	+	0.0309945i	0.0324435	+	0.0324435i	−0.176057	+	2.82294i	1.88554			−2.24535	+	2.15400i
99.20	0.174698	+	1.40338i			0.433961i	−1.93896	+	0.490335i	−1.49858	−	1.49858i	−0.609012	+	0.0758119i	2.96675	+	2.96675i	−1.02686	−	2.63544i	2.81168			1.84129	−	2.36489i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
13.b	even	2	1	inner
13.d	odd	4	2	inner
52.b	odd	2	1	inner
52.f	even	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	676.2.f.j	✓	72
4.b	odd	2	1	inner	676.2.f.j	✓	72
13.b	even	2	1	inner	676.2.f.j	✓	72
13.c	even	3	2		676.2.l.n		144
13.d	odd	4	2	inner	676.2.f.j	✓	72
13.e	even	6	2		676.2.l.n		144
13.f	odd	12	4		676.2.l.n		144
52.b	odd	2	1	inner	676.2.f.j	✓	72
52.f	even	4	2	inner	676.2.f.j	✓	72
52.i	odd	6	2		676.2.l.n		144
52.j	odd	6	2		676.2.l.n		144
52.l	even	12	4		676.2.l.n		144

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
676.2.f.j	✓	72	1.a	even	1	1	trivial
676.2.f.j	✓	72	4.b	odd	2	1	inner
676.2.f.j	✓	72	13.b	even	2	1	inner
676.2.f.j	✓	72	13.d	odd	4	2	inner
676.2.f.j	✓	72	52.b	odd	2	1	inner
676.2.f.j	✓	72	52.f	even	4	2	inner
676.2.l.n		144	13.c	even	3	2
676.2.l.n		144	13.e	even	6	2
676.2.l.n		144	13.f	odd	12	4
676.2.l.n		144	52.i	odd	6	2
676.2.l.n		144	52.j	odd	6	2
676.2.l.n		144	52.l	even	12	4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(676, [\chi])\):

T3^18 + 35*T3^16 + 490*T3^14 + 3549*T3^12 + 14469*T3^10 + 34125*T3^8 + 45913*T3^6 + 33124*T3^4 + 11102*T3^2 + 1183

T5^36 + 532*T5^32 + 110166*T5^28 + 11174646*T5^24 + 573422549*T5^20 + 13966330546*T5^16 + 145553970849*T5^12 + 456606186400*T5^8 + 273711346944*T5^4 + 5732306944

T7^36 + 974*T7^32 + 350485*T7^28 + 58464322*T7^24 + 4820317082*T7^20 + 196164328024*T7^16 + 3508288202097*T7^12 + 18098596514128*T7^8 + 26397663285760*T7^4 + 116985856