Newform orbit 363.3.o.a

• Label: 363.3.o.a
• Level: $363$
• Weight: $3$
• Character orbit: 363.o
• Analytic conductor: $9.891$
• Analytic rank: $0$
• Dimension: $1760$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	363.o (of order \(110\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89103359628\)
Analytic rank:	\(0\)
Dimension:	\(1760\)
Relative dimension:	\(44\) over \(\Q(\zeta_{110})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{110}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1760 q - 100 q^{4} + 4 q^{5} + 30 q^{7} + 40 q^{8} - 1320 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
7.1	−3.79858	−	0.769693i	0.535233	−	1.64728i	10.1523	+	4.29040i	−5.33898	+	6.92371i	−3.30103	+	5.84536i	−0.466997	−	2.69852i	−22.4647	−	15.3610i	−2.42705	−	1.76336i	25.6097	−	22.1909i
7.2	−3.66328	−	0.742277i	0.535233	−	1.64728i	9.18414	+	3.88125i	2.50718	−	3.25136i	−3.18344	+	5.63715i	−1.43630	−	8.29960i	−18.4216	−	12.5963i	−2.42705	−	1.76336i	−11.5979	+	10.0496i
7.3	−3.57734	−	0.724864i	−0.535233	+	1.64728i	8.58746	+	3.62909i	−2.38046	+	3.08703i	3.10876	−	5.50491i	0.474761	+	2.74338i	−16.0377	−	10.9663i	−2.42705	−	1.76336i	10.7534	−	9.31787i
7.4	−3.29254	−	0.667154i	0.535233	−	1.64728i	6.71121	+	2.83618i	4.28395	−	5.55552i	−2.86126	+	5.06664i	0.955433	+	5.52092i	−9.11221	−	6.23077i	−2.42705	−	1.76336i	−17.8114	+	15.4337i
7.5	−3.12870	−	0.633958i	−0.535233	+	1.64728i	5.70240	+	2.40985i	0.0783702	−	0.101632i	2.71889	−	4.81453i	0.297321	+	1.71805i	−5.77279	−	3.94734i	−2.42705	−	1.76336i	−0.309628	+	0.268294i
7.6	−3.11831	−	0.631851i	−0.535233	+	1.64728i	5.64011	+	2.38353i	−4.08246	+	5.29422i	2.70986	−	4.79853i	−2.07375	−	11.9831i	−5.57602	−	3.81279i	−2.42705	−	1.76336i	16.0755	−	13.9295i
7.7	−3.07865	−	0.623815i	−0.535233	+	1.64728i	5.40442	+	2.28393i	5.41219	−	7.01866i	2.67539	−	4.73750i	1.91719	+	11.0784i	−4.84163	−	3.31062i	−2.42705	−	1.76336i	−21.0406	+	18.2318i
7.8	−3.06435	−	0.620919i	−0.535233	+	1.64728i	5.32023	+	2.24835i	4.71591	−	6.11570i	2.66297	−	4.71551i	−2.20107	−	12.7188i	−4.58325	−	3.13395i	−2.42705	−	1.76336i	−18.2486	+	15.8125i
7.9	−2.85534	−	0.578566i	0.535233	−	1.64728i	4.13371	+	1.74692i	0.274815	−	0.356387i	−2.48133	+	4.39386i	−0.496151	−	2.86698i	−1.17282	−	0.801957i	−2.42705	−	1.76336i	−0.990883	+	0.858605i
7.10	−2.84470	−	0.576411i	0.535233	−	1.64728i	4.07558	+	1.72236i	−2.77651	+	3.60064i	−2.47209	+	4.37750i	1.83527	+	10.6050i	−1.01726	−	0.695586i	−2.42705	−	1.76336i	9.97379	−	8.64233i
7.11	−2.53242	−	0.513135i	0.535233	−	1.64728i	2.46536	+	1.04187i	−3.18074	+	4.12486i	−2.20071	+	3.89696i	−0.794141	−	4.58890i	2.82299	+	1.93031i	−2.42705	−	1.76336i	10.1716	−	8.81374i
7.12	−2.19475	−	0.444714i	−0.535233	+	1.64728i	0.934663	+	0.394992i	−0.0905854	+	0.117473i	1.90727	−	3.37734i	0.235996	+	1.36369i	5.51840	+	3.77338i	−2.42705	−	1.76336i	0.251054	−	0.217540i
7.13	−1.79411	−	0.363533i	0.535233	−	1.64728i	−0.597834	−	0.252647i	1.80034	−	2.33473i	−1.55911	+	2.76082i	0.965524	+	5.57923i	7.02506	+	4.80361i	−2.42705	−	1.76336i	−4.07876	+	3.53426i
7.14	−1.57209	−	0.318547i	0.535233	−	1.64728i	−1.31449	−	0.555509i	0.969590	−	1.25739i	−1.36617	+	2.41918i	−2.04650	−	11.8256i	7.18591	+	4.91360i	−2.42705	−	1.76336i	−1.92482	+	1.66787i
7.15	−1.38565	−	0.280770i	−0.535233	+	1.64728i	−1.84329	−	0.778982i	3.67532	−	4.76623i	1.20415	−	2.13228i	−0.836366	−	4.83290i	7.00370	+	4.78901i	−2.42705	−	1.76336i	−6.43092	+	5.57243i
7.16	−1.35336	−	0.274226i	−0.535233	+	1.64728i	−1.92812	−	0.814829i	1.45874	−	1.89173i	1.17609	−	2.08258i	0.588777	+	3.40222i	6.94543	+	4.74917i	−2.42705	−	1.76336i	−2.49297	+	2.16017i
7.17	−1.10079	−	0.223050i	−0.535233	+	1.64728i	−2.52250	−	1.06602i	−5.91749	+	7.67393i	0.956607	−	1.69393i	1.00107	+	5.78463i	6.24754	+	4.27196i	−2.42705	−	1.76336i	8.22561	−	7.12753i
7.18	−1.01713	−	0.206096i	−0.535233	+	1.64728i	−2.69243	−	1.13783i	−1.38002	+	1.78964i	0.883898	−	1.56518i	−1.64923	−	9.53002i	5.93072	+	4.05532i	−2.42705	−	1.76336i	1.77249	−	1.53587i
7.19	−1.00987	−	0.204627i	0.535233	−	1.64728i	−2.70653	−	1.14379i	2.83787	−	3.68021i	−0.877594	+	1.55402i	1.46824	+	8.48417i	5.90144	+	4.03530i	−2.42705	−	1.76336i	−3.61895	+	3.13584i
7.20	−0.644430	−	0.130578i	0.535233	−	1.64728i	−3.28626	−	1.38878i	−4.69099	+	6.08338i	−0.560020	+	0.991666i	1.78261	+	10.3007i	4.10750	+	2.80863i	−2.42705	−	1.76336i	3.81737	−	3.30777i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
121.h	odd	110	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	363.3.o.a	✓	1760
121.h	odd	110	1	inner	363.3.o.a	✓	1760

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
363.3.o.a	✓	1760	1.a	even	1	1	trivial
363.3.o.a	✓	1760	121.h	odd	110	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(363, [\chi])\).