Newform orbit 819.2.dk.a

• Label: 819.2.dk.a
• Level: $819$
• Weight: $2$
• Character orbit: 819.dk
• Analytic conductor: $6.540$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.dk (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.53974792554\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(84\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 168 q + 84 q^{4} - 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
43.1	−2.40382	+	1.38785i	1.73042	+	0.0750435i	2.85223	−	4.94021i	−2.74435	+	1.58445i	−4.26378	+	2.22117i		−	1.00000i			10.2824i	2.98874	+	0.259714i	4.39794	−	7.61746i
43.2	−2.39366	+	1.38198i	−1.63909	−	0.559795i	2.81974	−	4.88393i	0.406151	−	0.234491i	4.69706	−	0.925237i		−	1.00000i			10.0594i	2.37326	+	1.83511i	−0.648125	+	1.12259i
43.3	−2.34108	+	1.35162i	−1.04777	+	1.37920i	2.65377	−	4.59646i	0.0842087	−	0.0486179i	0.588750	−	4.64500i			1.00000i			8.94109i	−0.804371	−	2.89015i	−0.131426	+	0.227637i
43.4	−2.32365	+	1.34156i	1.01530	+	1.40327i	2.59957	−	4.50259i	2.69046	−	1.55334i	−4.24178	−	1.89861i			1.00000i			8.58369i	−0.938317	+	2.84948i	−4.16779	+	7.21883i
43.5	−2.26853	+	1.30973i	0.820654	−	1.52530i	2.43081	−	4.21028i	2.03527	−	1.17506i	0.136058	+	4.53501i		−	1.00000i			7.49592i	−1.65305	−	2.50348i	−3.07804	+	5.33132i
43.6	−2.15989	+	1.24701i	0.380999	+	1.68963i	2.11009	−	3.65478i	−0.344331	+	0.198799i	−2.92991	−	3.17430i		−	1.00000i			5.53719i	−2.70968	+	1.28749i	0.495812	−	0.858771i
43.7	−2.11313	+	1.22002i	−1.38469	+	1.04050i	1.97688	−	3.42406i	−3.75551	+	2.16824i	1.65660	−	3.88805i		−	1.00000i			4.76725i	0.834722	−	2.88153i	5.29059	−	9.16356i
43.8	−2.09325	+	1.20854i	0.856550	−	1.50543i	1.92112	−	3.32748i	−2.34009	+	1.35105i	0.0263989	+	4.18641i			1.00000i			4.45283i	−1.53265	−	2.57895i	3.26558	−	5.65616i
43.9	−2.04803	+	1.18243i	1.08475	+	1.35030i	1.79628	−	3.11125i	−2.45760	+	1.41890i	−3.81824	−	1.48281i			1.00000i			3.76619i	−0.646620	+	2.92949i	3.35549	−	5.81188i
43.10	−1.98441	+	1.14570i	−1.71249	−	0.259546i	1.62525	−	2.81502i	1.57924	−	0.911773i	3.69565	−	1.44696i		−	1.00000i			2.86541i	2.86527	+	0.888943i	−2.08924	+	3.61866i
43.11	−1.93690	+	1.11827i	−1.56721	+	0.737459i	1.50104	−	2.59988i	0.199890	−	0.115407i	2.21085	−	3.18094i			1.00000i			2.24120i	1.91231	−	2.31151i	−0.258111	+	0.447061i
43.12	−1.93520	+	1.11729i	−0.608380	+	1.62169i	1.49667	−	2.59231i	3.57474	−	2.06388i	−0.634557	−	3.81803i		−	1.00000i			2.21970i	−2.25975	−	1.97321i	−4.61189	+	7.98803i
43.13	−1.92881	+	1.11360i	1.72725	+	0.128894i	1.48020	−	2.56379i	1.43228	−	0.826928i	−3.47507	+	1.67485i			1.00000i			2.13901i	2.96677	+	0.445262i	−1.84173	+	3.18997i
43.14	−1.82966	+	1.05635i	−1.70014	−	0.330934i	1.23177	−	2.13349i	3.43635	−	1.98398i	3.46027	−	1.19046i			1.00000i			0.979324i	2.78096	+	1.12527i	−4.19156	+	7.26000i
43.15	−1.82865	+	1.05577i	0.135757	−	1.72672i	1.22931	−	2.12923i	1.93939	−	1.11971i	1.57477	+	3.30090i		−	1.00000i			0.968412i	−2.96314	−	0.468831i	−2.36431	+	4.09511i
43.16	−1.72121	+	0.993742i	−1.11330	−	1.32686i	0.975047	−	1.68883i	−2.87698	+	1.66103i	3.23478	+	1.17749i		−	1.00000i		−	0.0991883i	−0.521140	+	2.95439i	3.30126	−	5.71796i
43.17	−1.69112	+	0.976370i	−0.403984	−	1.68428i	0.906597	−	1.57027i	1.14548	−	0.661345i	2.32767	+	2.45388i			1.00000i		−	0.364782i	−2.67359	+	1.36084i	−1.29144	+	2.23683i
43.18	−1.57147	+	0.907290i	1.36794	−	1.06242i	0.646352	−	1.11951i	−2.33854	+	1.35016i	−1.18577	+	2.91068i		−	1.00000i		−	1.28345i	0.742545	−	2.90665i	2.44997	−	4.24348i
43.19	−1.48216	+	0.855723i	1.10918	+	1.33031i	0.464525	−	0.804581i	−0.0108802	+	0.00628170i	−2.78235	−	1.02258i		−	1.00000i		−	1.83287i	−0.539450	+	2.95110i	0.0107508	−	0.0186209i
43.20	−1.46286	+	0.844585i	1.40051	+	1.01911i	0.426647	−	0.738974i	−0.617664	+	0.356609i	−2.90947	−	0.307967i		−	1.00000i		−	1.93698i	0.922843	+	2.85453i	0.602372	−	1.04334i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
117.r	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	819.2.dk.a	yes	168
9.c	even	3	1		819.2.bh.a	✓	168
13.e	even	6	1		819.2.bh.a	✓	168
117.r	even	6	1	inner	819.2.dk.a	yes	168

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
819.2.bh.a	✓	168	9.c	even	3	1
819.2.bh.a	✓	168	13.e	even	6	1
819.2.dk.a	yes	168	1.a	even	1	1	trivial
819.2.dk.a	yes	168	117.r	even	6	1	inner

Hecke kernels

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