Newform orbit 735.2.bj.a

• Label: 735.2.bj.a
• Level: $735$
• Weight: $2$
• Character orbit: 735.bj
• Analytic conductor: $5.869$
• Analytic rank: $0$
• Dimension: $1296$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.bj (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.86900454856\)
Analytic rank:	\(0\)
Dimension:	\(1296\)
Relative dimension:	\(108\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1296 q - 10 q^{3} - 28 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
8.1	−0.308508	+	2.73809i	−1.71725	+	0.225946i	−5.45209	−	1.24440i	−0.737112	−	2.11108i	−0.0888730	−	4.77169i	1.59193	+	2.11323i	3.26919	−	9.34282i	2.89790	−	0.776011i	6.00773	−	1.36699i
8.2	−0.305882	+	2.71478i	−1.07539	−	1.35777i	−5.32662	−	1.21577i	2.00888	+	0.982029i	4.01499	−	2.50412i	−2.63931	+	0.184449i	3.12524	−	8.93141i	−0.687082	+	2.92026i	−3.28048	+	5.15330i
8.3	−0.305503	+	2.71141i	−0.898627	+	1.48070i	−5.30857	−	1.21165i	−1.77973	+	1.35371i	−3.74025	−	2.88891i	−2.59728	−	0.504121i	3.10468	−	8.87267i	−1.38494	−	2.66119i	−3.12677	−	5.23916i
8.4	−0.298333	+	2.64778i	1.56952	+	0.732545i	−4.97188	−	1.13480i	2.00853	−	0.982754i	−2.40786	+	3.93719i	0.550334	+	2.58788i	2.72790	−	7.79588i	1.92676	+	2.29948i	2.00291	+	5.61133i
8.5	−0.293008	+	2.60052i	1.47297	−	0.911240i	−4.72700	−	1.07891i	−0.401585	+	2.19971i	1.93811	+	4.09749i	0.453378	−	2.60662i	2.46211	−	7.03631i	1.33928	−	2.68446i	−5.60273	−	1.68887i
8.6	−0.286425	+	2.54209i	0.903568	+	1.47769i	−4.43034	−	1.01120i	0.832455	+	2.07534i	−4.01523	+	1.87370i	2.31994	−	1.27195i	2.14969	−	6.14345i	−1.36713	+	2.67038i	−5.51413	+	1.52175i
8.7	−0.284737	+	2.52711i	1.00654	−	1.40956i	−4.35534	−	0.994077i	−2.23456	−	0.0821474i	3.27552	+	2.94499i	−0.882459	+	2.49425i	2.07240	−	5.92258i	−0.973747	−	2.83757i	0.843856	−	5.62358i
8.8	−0.284336	+	2.52355i	1.42735	+	0.981159i	−4.33761	−	0.990031i	−1.26745	−	1.84216i	−2.88185	+	3.32301i	−2.30172	−	1.30464i	2.05423	−	5.87066i	1.07465	+	2.80091i	5.00918	−	2.67468i
8.9	−0.275212	+	2.44257i	−0.476474	+	1.66522i	−3.94055	−	0.899405i	1.57438	−	1.58787i	−3.93630	−	1.62211i	0.156807	−	2.64110i	1.65767	−	4.73736i	−2.54595	−	1.58687i	3.44519	+	4.28253i
8.10	−0.274979	+	2.44050i	−0.990663	−	1.42077i	−3.93059	−	0.897132i	−1.38813	+	1.75302i	3.73981	−	2.02704i	2.53039	+	0.772735i	1.64799	−	4.70968i	−1.03717	+	2.81501i	−3.89656	−	3.86978i
8.11	−0.260585	+	2.31276i	−1.51161	−	0.845599i	−3.33108	−	0.760297i	−1.75234	−	1.38900i	2.34957	−	3.27563i	−1.73556	−	1.99696i	1.08903	−	3.11228i	1.56992	+	2.55643i	3.66904	−	3.69078i
8.12	−0.250984	+	2.22754i	1.73154	+	0.0420748i	−2.94909	−	0.673111i	−2.01597	−	0.967408i	−0.528312	+	3.84652i	2.53334	−	0.763026i	0.758826	−	2.16860i	2.99646	+	0.145709i	2.66092	−	4.24785i
8.13	−0.246619	+	2.18881i	0.122710	−	1.72770i	−2.78020	−	0.634562i	0.419076	−	2.19645i	3.75134	+	0.694672i	−2.04799	+	1.67504i	0.619602	−	1.77072i	−2.96988	−	0.424011i	4.70424	+	1.45896i
8.14	−0.243132	+	2.15785i	1.39212	−	1.03054i	−2.64736	−	0.604244i	1.69820	+	1.45469i	1.88529	+	3.25454i	0.350420	+	2.62244i	0.513121	−	1.46642i	0.875978	−	2.86926i	−3.55190	+	3.31078i
8.15	−0.240517	+	2.13464i	0.0724443	−	1.73054i	−2.54900	−	0.581793i	2.23601	−	0.0160129i	3.67665	+	0.570865i	2.64575	+	0.00110943i	0.436019	−	1.24607i	−2.98950	−	0.250735i	−0.503616	+	4.77694i
8.16	−0.238352	+	2.11543i	0.579043	+	1.63239i	−2.46839	−	0.563395i	−1.99114	+	1.01753i	−3.59124	+	0.835843i	−0.00150049	+	2.64575i	0.373962	−	1.06872i	−2.32942	+	1.89045i	−1.67793	−	4.45465i
8.17	−0.234629	+	2.08239i	−1.52431	+	0.822485i	−2.33145	−	0.532138i	2.07171	−	0.841447i	−1.35509	−	3.36719i	−1.99920	+	1.73297i	0.270900	−	0.774189i	1.64704	−	2.50744i	1.26614	+	4.51153i
8.18	−0.230694	+	2.04746i	−0.0870773	+	1.72986i	−2.18904	−	0.499633i	−0.477801	−	2.18442i	−3.52174	−	0.577356i	2.08432	+	1.62961i	0.166951	−	0.477118i	−2.98484	−	0.301263i	4.58276	−	0.474348i
8.19	−0.230010	+	2.04139i	−1.53999	+	0.792746i	−2.16453	−	0.494040i	−2.18873	+	0.457647i	−1.26409	−	3.32606i	2.26769	−	1.36294i	0.149401	−	0.426964i	1.74311	−	2.44163i	−0.430806	−	4.57333i
8.20	−0.229138	+	2.03365i	−1.39184	−	1.03091i	−2.13339	−	0.486933i	1.08600	−	1.95464i	2.41544	−	2.59430i	0.908957	−	2.48471i	0.127246	−	0.363648i	0.874447	+	2.86973i	3.72621	+	2.65643i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
15.e	even	4	1	inner
49.e	even	7	1	inner
147.l	odd	14	1	inner
245.r	odd	28	1	inner
735.bj	even	28	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	735.2.bj.a	✓	1296
3.b	odd	2	1	inner	735.2.bj.a	✓	1296
5.c	odd	4	1	inner	735.2.bj.a	✓	1296
15.e	even	4	1	inner	735.2.bj.a	✓	1296
49.e	even	7	1	inner	735.2.bj.a	✓	1296
147.l	odd	14	1	inner	735.2.bj.a	✓	1296
245.r	odd	28	1	inner	735.2.bj.a	✓	1296
735.bj	even	28	1	inner	735.2.bj.a	✓	1296

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
735.2.bj.a	✓	1296	1.a	even	1	1	trivial
735.2.bj.a	✓	1296	3.b	odd	2	1	inner
735.2.bj.a	✓	1296	5.c	odd	4	1	inner
735.2.bj.a	✓	1296	15.e	even	4	1	inner
735.2.bj.a	✓	1296	49.e	even	7	1	inner
735.2.bj.a	✓	1296	147.l	odd	14	1	inner
735.2.bj.a	✓	1296	245.r	odd	28	1	inner
735.2.bj.a	✓	1296	735.bj	even	28	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(735, [\chi])\).