Newform orbit 121.2.g.a

• Label: 121.2.g.a
• Level: $121$
• Weight: $2$
• Character orbit: 121.g
• Analytic conductor: $0.966$
• Analytic rank: $0$
• Dimension: $400$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	121.g (of order \(55\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.966189864457\)
Analytic rank:	\(0\)
Dimension:	\(400\)
Relative dimension:	\(10\) over \(\Q(\zeta_{55})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{55}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 400 q - 44 q^{2} - 32 q^{3} - 34 q^{4} - 43 q^{5} - 22 q^{6} - 44 q^{7} - 44 q^{8} - 110 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−2.59904	−	0.148619i	0.517548	−	1.59285i	4.74598	+	0.544550i	0.803024	+	1.52190i	−1.58186	+	4.06297i	−0.0112634	−	0.00475995i	−7.12374	−	1.23281i	0.157740	+	0.114605i	−1.86091	−	4.07483i
4.2	−2.08449	−	0.119196i	−0.374981	+	1.15407i	2.34393	+	0.268941i	−1.51227	−	2.86606i	0.919206	−	2.36096i	2.34211	+	0.989784i	−0.739221	−	0.127927i	1.23578	+	0.897843i	2.81069	+	6.15454i
4.3	−1.94119	−	0.111001i	−0.618438	+	1.90336i	1.76894	+	0.202967i	0.816229	+	1.54693i	1.41178	−	3.62613i	−4.27807	−	1.80793i	0.420440	+	0.0727601i	−0.813251	−	0.590861i	−1.41275	−	3.09349i
4.4	−0.828838	−	0.0473947i	1.01247	−	3.11606i	−1.30224	−	0.149418i	0.0511865	+	0.0970092i	−0.986857	+	2.53472i	−0.0339504	−	0.0143476i	2.70833	+	0.468694i	−6.25768	−	4.54647i	−0.0378276	−	0.0828308i
4.5	−0.790346	−	0.0451936i	−0.0254779	+	0.0784129i	−1.36436	−	0.156546i	0.532875	+	1.00991i	0.0236801	−	0.0608219i	4.07073	+	1.72030i	2.63133	+	0.455369i	2.42155	+	1.75936i	−0.375515	−	0.822262i
4.6	−0.282035	−	0.0161273i	0.138004	−	0.424731i	−1.90768	−	0.218886i	−1.62748	−	3.08441i	−0.0457716	+	0.117563i	−3.79767	−	1.60491i	1.09122	+	0.188843i	2.26570	+	1.64613i	0.409262	+	0.896158i
4.7	0.609785	+	0.0348688i	−0.704377	+	2.16785i	−1.61634	−	0.185458i	0.903893	+	1.71307i	−0.505109	+	1.29736i	−0.168606	−	0.0712535i	−2.18283	−	0.377753i	−1.77637	−	1.29061i	0.491448	+	1.07612i
4.8	1.43269	+	0.0819240i	0.403457	−	1.24171i	0.0589155	+	0.00675993i	−0.621601	−	1.17807i	0.679753	−	1.74593i	2.51104	+	1.06117i	−2.74416	−	0.474896i	1.04798	+	0.761401i	−0.794048	−	1.73872i
4.9	1.51337	+	0.0865377i	0.615691	−	1.89490i	0.295842	+	0.0339447i	1.83167	+	3.47141i	1.09575	−	2.81441i	−3.18667	−	1.34670i	−2.54250	−	0.439998i	−0.784523	−	0.569989i	2.47160	+	5.41204i
4.10	2.13575	+	0.122126i	−0.616726	+	1.89809i	2.55954	+	0.293680i	−0.685845	−	1.29982i	−1.54898	+	3.97852i	−0.751489	−	0.317582i	1.21486	+	0.210239i	−0.795333	−	0.577843i	−1.30605	−	2.85985i
5.1	−1.31555	+	2.18160i	1.84029	+	1.33705i	−2.09536	−	3.97115i	0.0304467	−	0.0248961i	−5.33791	+	2.25582i	−2.96018	+	3.83884i	6.33320	+	0.362145i	0.671922	+	2.06796i	0.0142592	+	0.0991747i
5.2	−1.26554	+	2.09866i	−1.30803	−	0.950337i	−1.86945	−	3.54301i	−2.56800	+	2.09984i	3.64979	−	1.54242i	2.42078	−	3.13933i	4.90801	+	0.280650i	−0.119257	−	0.367035i	−1.15695	−	8.04678i
5.3	−1.05343	+	1.74691i	−1.35869	−	0.987147i	−1.00866	−	1.91163i	3.20775	−	2.62296i	3.15574	−	1.33363i	0.212285	−	0.275296i	0.328744	+	0.0187983i	−0.0554687	−	0.170715i	1.20296	+	8.36677i
5.4	−0.631909	+	1.04790i	1.62812	+	1.18290i	0.234543	+	0.444508i	0.863348	−	0.705956i	−2.26838	+	0.958627i	1.34281	−	1.74139i	−3.05739	−	0.174828i	0.324472	+	0.998622i	0.194216	+	1.35080i
5.5	−0.339119	+	0.562365i	−2.60062	−	1.88946i	0.732082	+	1.38745i	−0.891773	+	0.729199i	1.94449	−	0.821746i	−1.70268	+	2.20808i	−2.33977	−	0.133793i	2.26611	+	6.97437i	−0.107659	−	0.748787i
5.6	−0.215633	+	0.357587i	0.525350	+	0.381689i	0.851964	+	1.61465i	−2.63241	+	2.15251i	−0.249770	+	0.105554i	−0.169441	+	0.219735i	−1.59487	−	0.0911981i	−0.796745	−	2.45213i	−0.202076	−	1.40547i
5.7	0.364682	−	0.604757i	−0.929691	−	0.675460i	0.700597	+	1.32778i	0.792518	−	0.648038i	−0.747530	+	0.315909i	1.96527	−	2.54861i	2.46858	+	0.141159i	−0.518972	−	1.59723i	−0.102889	−	0.715608i
5.8	0.407235	−	0.675323i	0.0991267	+	0.0720198i	0.643113	+	1.21884i	2.11727	−	1.73128i	0.0890045	−	0.0376136i	−3.04355	+	3.94694i	2.65965	+	0.152084i	−0.922412	−	2.83889i	−0.306949	−	2.13488i
5.9	1.11220	−	1.84437i	−1.96002	−	1.42404i	−1.23139	−	2.33374i	−0.598559	+	0.489439i	−4.80638	+	2.03119i	0.115990	−	0.150419i	−1.37332	−	0.0785293i	0.886746	+	2.72912i	0.236992	+	1.64832i
5.10	1.25666	−	2.08394i	1.16081	+	0.843378i	−1.83027	−	3.46875i	−0.796654	+	0.651421i	3.21630	−	1.35922i	−1.00768	+	1.30679i	−4.66961	−	0.267018i	−0.290857	−	0.895166i	0.356397	+	2.47880i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
121.g	even	55	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	121.2.g.a	✓	400
121.g	even	55	1	inner	121.2.g.a	✓	400

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
121.2.g.a	✓	400	1.a	even	1	1	trivial
121.2.g.a	✓	400	121.g	even	55	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(121, [\chi])\).