Newform orbit 312.2.bn.a

• Label: 312.2.bn.a
• Level: $312$
• Weight: $2$
• Character orbit: 312.bn
• Analytic conductor: $2.491$
• Analytic rank: $0$
• Dimension: $104$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	312.bn (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.49133254306\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(52\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 2 q^{3} - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
35.1	−1.40463	−	0.164360i	−1.71238	+	0.260265i	1.94597	+	0.461731i	1.12484			2.44804	−	0.0841273i	−2.84449	+	1.64227i	−2.65748	−	0.968403i	2.86452	−	0.891347i	−1.57998	−	0.184879i
35.2	−1.40122	−	0.191231i	0.596526	−	1.62609i	1.92686	+	0.535915i	−0.495602			−1.14683	+	2.16444i	−2.49990	+	1.44332i	−2.59748	−	1.11941i	−2.28831	−	1.94001i	0.694450	+	0.0947745i
35.3	−1.39321	−	0.242856i	−0.891971	−	1.48472i	1.88204	+	0.676696i	2.41329			0.882127	+	2.28514i	2.20462	−	1.27284i	−2.45773	−	1.39984i	−1.40878	+	2.64865i	−3.36220	−	0.586080i
35.4	−1.38653	+	0.278431i	1.72127	−	0.192941i	1.84495	−	0.772107i	−1.60187			−2.33288	+	0.746773i	0.132077	−	0.0762548i	−2.34311	+	1.58424i	2.92555	−	0.664207i	2.22105	−	0.446009i
35.5	−1.34582	+	0.434472i	0.814250	+	1.52872i	1.62247	−	1.16944i	3.18815			−1.76002	−	1.70362i	−4.15137	+	2.39679i	−1.67546	+	2.27878i	−1.67399	+	2.48953i	−4.29068	+	1.38516i
35.6	−1.34519	−	0.436434i	1.45626	+	0.937711i	1.61905	+	1.17417i	1.48789			−1.54969	−	1.89696i	3.71351	−	2.14399i	−1.66548	−	2.28609i	1.24140	+	2.73111i	−2.00149	−	0.649368i
35.7	−1.33082	−	0.478444i	−0.608808	+	1.62153i	1.54218	+	1.27345i	−3.23710			1.58603	−	1.86669i	−0.412669	+	0.238254i	−1.44310	−	2.43258i	−2.25871	−	1.97440i	4.30800	+	1.54877i
35.8	−1.29720	+	0.563268i	−1.61458	+	0.627008i	1.36546	−	1.46134i	−1.44919			1.74126	−	1.72280i	1.54903	−	0.894333i	−0.948145	+	2.66477i	2.21372	−	2.02471i	1.87988	−	0.816280i
35.9	−1.18109	+	0.777839i	1.40559	−	1.01209i	0.789934	−	1.83739i	3.91542			−0.872879	+	2.28869i	1.79682	−	1.03739i	0.496212	+	2.78456i	0.951350	−	2.84516i	−4.62446	+	3.04557i
35.10	−1.14209	+	0.834045i	−1.35173	−	1.08297i	0.608738	−	1.90511i	1.35437			2.44704	+	0.109444i	0.246575	−	0.142360i	0.893713	+	2.68352i	0.654353	+	2.92777i	−1.54681	+	1.12960i
35.11	−1.07976	−	0.913305i	−0.608808	+	1.62153i	0.331748	+	1.97229i	3.23710			2.13831	−	1.19483i	0.412669	−	0.238254i	1.44310	−	2.43258i	−2.25871	−	1.97440i	−3.49527	−	2.95646i
35.12	−1.05056	−	0.946748i	1.45626	+	0.937711i	0.207336	+	1.98922i	−1.48789			−0.642108	−	2.36383i	−3.71351	+	2.14399i	1.66548	−	2.28609i	1.24140	+	2.73111i	1.56312	+	1.40866i
35.13	−1.01982	+	0.979778i	1.06251	+	1.36787i	0.0800719	−	1.99840i	−3.80332			−2.42378	−	0.353955i	0.0948621	−	0.0547686i	1.87632	+	2.11646i	−0.742130	+	2.90676i	3.87871	−	3.72641i
35.14	−0.968494	+	1.03054i	0.152617	−	1.72531i	−0.124039	−	1.99615i	−1.81520			1.63020	+	1.82823i	−2.79160	+	1.61173i	2.17725	+	1.80543i	−2.95342	−	0.526623i	1.75801	−	1.87064i
35.15	−0.906922	−	1.08512i	−0.891971	−	1.48472i	−0.354985	+	1.96824i	−2.41329			−0.802154	+	2.31442i	−2.20462	+	1.27284i	2.45773	−	1.39984i	−1.40878	+	2.64865i	2.18866	+	2.61871i
35.16	−0.866223	−	1.11788i	0.596526	−	1.62609i	−0.499315	+	1.93667i	0.495602			−2.33450	+	0.741708i	2.49990	−	1.44332i	2.59748	−	1.11941i	−2.28831	−	1.94001i	−0.429302	−	0.554024i
35.17	−0.844655	−	1.13427i	−1.71238	+	0.260265i	−0.573115	+	1.91613i	−1.12484			1.74158	+	1.72246i	2.84449	−	1.64227i	2.65748	−	0.968403i	2.86452	−	0.891347i	0.950100	+	1.27586i
35.18	−0.803366	+	1.16387i	−0.521544	+	1.65166i	−0.709207	−	1.87003i	1.48897			−1.50334	−	1.93390i	3.21659	−	1.85710i	2.74624	+	0.676893i	−2.45598	−	1.72283i	−1.19619	+	1.73298i
35.19	−0.606262	+	1.27767i	−1.16961	+	1.27750i	−1.26489	−	1.54921i	1.48897			−0.923138	−	2.26888i	−3.21659	+	1.85710i	2.74624	−	0.676893i	−0.264021	−	2.98836i	−0.902708	+	1.90242i
35.20	−0.452139	−	1.33999i	1.72127	−	0.192941i	−1.59114	+	1.21172i	1.60187			−1.03679	−	2.21925i	−0.132077	+	0.0762548i	2.34311	+	1.58424i	2.92555	−	0.664207i	−0.724267	−	2.14649i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
8.d	odd	2	1	inner
13.c	even	3	1	inner
24.f	even	2	1	inner
39.i	odd	6	1	inner
104.n	odd	6	1	inner
312.bn	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	312.2.bn.a	✓	104
3.b	odd	2	1	inner	312.2.bn.a	✓	104
8.d	odd	2	1	inner	312.2.bn.a	✓	104
13.c	even	3	1	inner	312.2.bn.a	✓	104
24.f	even	2	1	inner	312.2.bn.a	✓	104
39.i	odd	6	1	inner	312.2.bn.a	✓	104
104.n	odd	6	1	inner	312.2.bn.a	✓	104
312.bn	even	6	1	inner	312.2.bn.a	✓	104

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
312.2.bn.a	✓	104	1.a	even	1	1	trivial
312.2.bn.a	✓	104	3.b	odd	2	1	inner
312.2.bn.a	✓	104	8.d	odd	2	1	inner
312.2.bn.a	✓	104	13.c	even	3	1	inner
312.2.bn.a	✓	104	24.f	even	2	1	inner
312.2.bn.a	✓	104	39.i	odd	6	1	inner
312.2.bn.a	✓	104	104.n	odd	6	1	inner
312.2.bn.a	✓	104	312.bn	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(312, [\chi])\).