Newform orbit 810.2.w.a

• Label: 810.2.w.a
• Level: $810$
• Weight: $2$
• Character orbit: 810.w
• Analytic conductor: $6.468$
• Analytic rank: $0$
• Dimension: $1944$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	810.w (of order \(108\), degree \(36\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.46788256372\)
Analytic rank:	\(0\)
Dimension:	\(1944\)
Relative dimension:	\(54\) over \(\Q(\zeta_{108})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{108}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 1944 q + 36 q^{20} + 108 q^{30} + 108 q^{35} + 36 q^{41} + 216 q^{51} + 108 q^{67} - 144 q^{72} - 576 q^{77} - 576 q^{87} - 144 q^{92} + 36 q^{93} + 108 q^{97}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
23.1	−0.989442	+	0.144932i	−1.73193	+	0.0200796i	0.957990	−	0.286803i	2.19579	−	0.422505i	1.71074	−	0.270880i	−1.03819	−	0.0302082i	−0.906308	+	0.422618i	2.99919	−	0.0695530i	−2.11137	+	0.736284i
23.2	−0.989442	+	0.144932i	−1.72613	−	0.143043i	0.957990	−	0.286803i	−0.303554	−	2.21537i	1.72864	−	0.108639i	−2.33951	−	0.0680727i	−0.906308	+	0.422618i	2.95908	+	0.493822i	0.621426	+	2.14798i
23.3	−0.989442	+	0.144932i	−1.72037	−	0.200785i	0.957990	−	0.286803i	−1.85060	+	1.25510i	1.73131	−	0.0506719i	−2.85262	−	0.0830029i	−0.906308	+	0.422618i	2.91937	+	0.690850i	1.64916	−	1.51006i
23.4	−0.989442	+	0.144932i	−1.69878	+	0.337832i	0.957990	−	0.286803i	−0.579902	+	2.15956i	1.63189	−	0.580473i	2.86135	+	0.0832567i	−0.906308	+	0.422618i	2.77174	−	1.14781i	0.260789	−	2.22081i
23.5	−0.989442	+	0.144932i	−1.59988	−	0.663620i	0.957990	−	0.286803i	1.84421	+	1.26447i	1.67917	+	0.424740i	2.81768	+	0.0819861i	−0.906308	+	0.422618i	2.11922	+	2.12342i	−2.00800	−	0.983839i
23.6	−0.989442	+	0.144932i	−1.26007	−	1.18837i	0.957990	−	0.286803i	0.248557	−	2.22221i	1.41899	+	0.993204i	3.95411	+	0.115053i	−0.906308	+	0.422618i	0.175532	+	2.99486i	0.0761368	+	2.23477i
23.7	−0.989442	+	0.144932i	−1.21160	+	1.23775i	0.957990	−	0.286803i	−2.06476	−	0.858353i	1.01941	−	1.40029i	0.637598	+	0.0185522i	−0.906308	+	0.422618i	−0.0640749	−	2.99932i	2.16736	+	0.550041i
23.8	−0.989442	+	0.144932i	−1.00497	+	1.41068i	0.957990	−	0.286803i	−2.23559	−	0.0464100i	0.789908	−	1.54144i	3.04427	+	0.0885793i	−0.906308	+	0.422618i	−0.980063	−	2.83540i	2.21871	−	0.278088i
23.9	−0.989442	+	0.144932i	−0.865062	−	1.50056i	0.957990	−	0.286803i	2.22919	−	0.175246i	1.07341	+	1.35934i	−2.04645	−	0.0595457i	−0.906308	+	0.422618i	−1.50333	+	2.59615i	−2.18025	+	0.496477i
23.10	−0.989442	+	0.144932i	−0.835230	+	1.51736i	0.957990	−	0.286803i	0.465389	+	2.18710i	0.606497	−	1.62239i	−1.39952	−	0.0407218i	−0.906308	+	0.422618i	−1.60478	−	2.53469i	−0.777456	−	2.09656i
23.11	−0.989442	+	0.144932i	−0.714289	+	1.57791i	0.957990	−	0.286803i	1.79217	−	1.33721i	0.478058	−	1.66477i	−3.01476	−	0.0877205i	−0.906308	+	0.422618i	−1.97958	−	2.25416i	−1.57945	+	1.58283i
23.12	−0.989442	+	0.144932i	−0.686142	−	1.59035i	0.957990	−	0.286803i	−2.10154	−	0.763888i	0.909390	+	1.47411i	−2.89740	−	0.0843058i	−0.906308	+	0.422618i	−2.05842	+	2.18241i	2.19006	+	0.451242i
23.13	−0.989442	+	0.144932i	−0.107384	−	1.72872i	0.957990	−	0.286803i	0.876175	+	2.05726i	0.356797	+	1.69490i	−3.77684	−	0.109895i	−0.906308	+	0.422618i	−2.97694	+	0.371275i	−1.16509	−	1.90855i
23.14	−0.989442	+	0.144932i	0.000411202		1.73205i	0.957990	−	0.286803i	2.15694	+	0.589579i	−0.251436	−	1.71370i	4.85974	+	0.141404i	−0.906308	+	0.422618i	−3.00000	+	0.00142444i	−2.21962	−	0.270744i
23.15	−0.989442	+	0.144932i	0.128776	−	1.72726i	0.957990	−	0.286803i	−1.90840	+	1.16533i	0.122918	+	1.72768i	3.82371	+	0.111259i	−0.906308	+	0.422618i	−2.96683	−	0.444858i	1.71936	−	1.42962i
23.16	−0.989442	+	0.144932i	0.148656	−	1.72566i	0.957990	−	0.286803i	0.753503	−	2.10529i	0.103016	+	1.72898i	−0.00438391		0.000127559i	−0.906308	+	0.422618i	−2.95580	−	0.513061i	−0.440424	+	2.19227i
23.17	−0.989442	+	0.144932i	0.212730	+	1.71894i	0.957990	−	0.286803i	−0.709703	−	2.12045i	−0.459612	−	1.66996i	0.0329090		0.000957555i	−0.906308	+	0.422618i	−2.90949	+	0.731338i	1.00953	+	1.99521i
23.18	−0.989442	+	0.144932i	0.691018	+	1.58824i	0.957990	−	0.286803i	0.152632	+	2.23085i	−0.913908	−	1.47132i	0.335576	+	0.00976425i	−0.906308	+	0.422618i	−2.04499	+	2.19500i	−0.474342	−	2.18518i
23.19	−0.989442	+	0.144932i	0.934576	−	1.45828i	0.957990	−	0.286803i	1.98946	+	1.02081i	−0.713358	+	1.57833i	3.55193	+	0.103350i	−0.906308	+	0.422618i	−1.25313	−	2.72574i	−2.11640	−	0.721693i
23.20	−0.989442	+	0.144932i	1.19124	−	1.25736i	0.957990	−	0.286803i	−0.817927	+	2.08110i	−0.996433	+	1.41673i	−2.78353	−	0.0809924i	−0.906308	+	0.422618i	−0.161888	−	2.99563i	0.507673	−	2.17767i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
81.h	odd	54	1	inner
405.x	even	108	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	810.2.w.a	✓	1944
5.c	odd	4	1	inner	810.2.w.a	✓	1944
81.h	odd	54	1	inner	810.2.w.a	✓	1944
405.x	even	108	1	inner	810.2.w.a	✓	1944

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
810.2.w.a	✓	1944	1.a	even	1	1	trivial
810.2.w.a	✓	1944	5.c	odd	4	1	inner
810.2.w.a	✓	1944	81.h	odd	54	1	inner
810.2.w.a	✓	1944	405.x	even	108	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(810, [\chi])\).