Newform orbit 819.2.em.a

• Label: 819.2.em.a
• Level: $819$
• Weight: $2$
• Character orbit: 819.em
• Analytic conductor: $6.540$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.em (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.53974792554\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(108\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 432 q - 6 q^{2} - 4 q^{3} - 4 q^{6} + 2 q^{7} + 12 q^{8} - 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
158.1	−1.97417	−	1.97417i	−0.0492071	−	1.73135i			5.79473i	1.13973	+	1.13973i	−3.32085	+	3.51513i	−1.87029	−	1.87137i	7.49145	−	7.49145i	−2.99516	+	0.170390i		−	4.50005i
158.2	−1.94312	−	1.94312i	−0.489082	+	1.66157i			5.55145i	2.08028	+	2.08028i	4.17897	−	2.27828i	1.54829	−	2.14541i	6.90090	−	6.90090i	−2.52160	−	1.62528i		−	8.08448i
158.3	−1.90707	−	1.90707i	1.69705	−	0.346440i			5.27383i	0.0259008	+	0.0259008i	−3.89708	−	2.57571i	−1.63768	+	2.07798i	6.24342	−	6.24342i	2.75996	−	1.17585i		−	0.0987893i
158.4	−1.90057	−	1.90057i	−1.69077	+	0.375890i			5.22436i	−2.57004	−	2.57004i	3.92784	+	2.49903i	2.22808	−	1.42676i	6.12813	−	6.12813i	2.71741	−	1.27109i			9.76909i
158.5	−1.87715	−	1.87715i	0.892617	−	1.48433i			5.04740i	−2.29084	−	2.29084i	−4.46189	+	1.11074i	2.38613	+	1.14298i	5.72042	−	5.72042i	−1.40647	−	2.64988i			8.60050i
158.6	−1.87210	−	1.87210i	−1.67301	−	0.448368i			5.00951i	2.40186	+	2.40186i	2.29265	+	3.97143i	−0.372720	+	2.61937i	5.63409	−	5.63409i	2.59793	+	1.50025i		−	8.99304i
158.7	−1.86853	−	1.86853i	0.586418	+	1.62976i			4.98281i	−0.373560	−	0.373560i	1.94951	−	4.14099i	1.28849	+	2.31080i	5.57347	−	5.57347i	−2.31223	+	1.91144i			1.39602i
158.8	−1.76685	−	1.76685i	−1.27729	+	1.16984i			4.24352i	−0.747935	−	0.747935i	4.32372	+	0.189854i	−2.63994	+	0.175253i	3.96397	−	3.96397i	0.262953	−	2.98845i			2.64298i
158.9	−1.70704	−	1.70704i	0.406889	+	1.68358i			3.82798i	−2.38041	−	2.38041i	2.17936	−	3.56852i	−2.52734	−	0.782653i	3.12044	−	3.12044i	−2.66888	+	1.37006i			8.12690i
158.10	−1.66695	−	1.66695i	1.40169	−	1.01748i			3.55747i	2.09100	+	2.09100i	−4.03264	−	0.640474i	2.41232	−	1.08661i	2.59623	−	2.59623i	0.929489	−	2.85238i		−	6.97120i
158.11	−1.64985	−	1.64985i	−1.34867	−	1.08678i			3.44399i	−1.16958	−	1.16958i	0.432077	+	4.01812i	0.191894	+	2.63878i	2.38236	−	2.38236i	0.637818	+	2.93141i			3.85926i
158.12	−1.62509	−	1.62509i	1.56172	+	0.749022i			3.28181i	0.834640	+	0.834640i	−1.32070	−	3.75515i	2.45943	+	0.975294i	2.08305	−	2.08305i	1.87793	+	2.33952i		−	2.71272i
158.13	−1.61575	−	1.61575i	−0.892747	−	1.48425i			3.22127i	0.438506	+	0.438506i	−0.955720	+	3.84062i	2.36631	−	1.18346i	1.97326	−	1.97326i	−1.40600	+	2.65012i		−	1.41703i
158.14	−1.60332	−	1.60332i	1.55079	+	0.771399i			3.14129i	2.91375	+	2.91375i	−1.24961	−	3.72322i	−2.03658	−	1.68888i	1.82986	−	1.82986i	1.80989	+	2.39255i		−	9.34337i
158.15	−1.60107	−	1.60107i	−1.63345	−	0.576067i			3.12688i	−0.180625	−	0.180625i	1.69294	+	3.53760i	−0.694020	−	2.55310i	1.80422	−	1.80422i	2.33629	+	1.88195i			0.578387i
158.16	−1.52204	−	1.52204i	0.307906	−	1.70446i			2.63322i	−1.25434	−	1.25434i	−3.06291	+	2.12562i	−0.910513	−	2.48414i	0.963780	−	0.963780i	−2.81039	−	1.04963i			3.81831i
158.17	−1.47952	−	1.47952i	−1.48746	+	0.887388i			2.37794i	1.39266	+	1.39266i	3.51363	+	0.887819i	−2.64247	+	0.131718i	0.559161	−	0.559161i	1.42509	−	2.63991i		−	4.12091i
158.18	−1.47755	−	1.47755i	1.66819	−	0.465976i			2.36630i	−0.335314	−	0.335314i	−3.15334	−	1.77633i	−2.62863	+	0.300496i	0.541219	−	0.541219i	2.56573	−	1.55468i			0.990884i
158.19	−1.46557	−	1.46557i	0.584815	−	1.63033i			2.29581i	1.88279	+	1.88279i	−3.24646	+	1.53229i	−1.34668	+	2.27738i	0.433533	−	0.433533i	−2.31598	−	1.90689i		−	5.51874i
158.20	−1.45954	−	1.45954i	−0.873658	+	1.49557i			2.26053i	0.0197941	+	0.0197941i	3.45799	−	0.907703i	1.62307	+	2.08941i	0.380256	−	0.380256i	−1.47344	−	2.61323i		−	0.0577807i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
819.em	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	819.2.em.a	✓	432
7.c	even	3	1		819.2.ge.a	yes	432
9.d	odd	6	1		819.2.fe.a	yes	432
13.f	odd	12	1		819.2.gf.a	yes	432
63.n	odd	6	1		819.2.gf.a	yes	432
91.bd	odd	12	1		819.2.fe.a	yes	432
117.bc	even	12	1		819.2.ge.a	yes	432
819.em	even	12	1	inner	819.2.em.a	✓	432

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
819.2.em.a	✓	432	1.a	even	1	1	trivial
819.2.em.a	✓	432	819.em	even	12	1	inner
819.2.fe.a	yes	432	9.d	odd	6	1
819.2.fe.a	yes	432	91.bd	odd	12	1
819.2.ge.a	yes	432	7.c	even	3	1
819.2.ge.a	yes	432	117.bc	even	12	1
819.2.gf.a	yes	432	13.f	odd	12	1
819.2.gf.a	yes	432	63.n	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(819, [\chi])\).