Newform orbit 213.3.j.a

• Label: 213.3.j.a
• Level: $213$
• Weight: $3$
• Character orbit: 213.j
• Analytic conductor: $5.804$
• Analytic rank: $0$
• Dimension: $144$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 213 = 3 \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	213.j (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.80382963087\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(24\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 4 q^{2} - 36 q^{4} - 8 q^{8} - 72 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
34.1	−2.42860	+	3.04537i	−0.385418	−	1.68862i	−2.48610	−	10.8923i	−9.75692			6.07852	+	2.92726i	5.42995	−	4.33024i	25.1711	+	12.1218i	−2.70291	+	1.30165i	23.6957	−	29.7135i
34.2	−2.39559	+	3.00397i	0.385418	+	1.68862i	−2.39493	−	10.4929i	5.20744			−5.99589	−	2.88747i	−0.102654	+	0.0818642i	23.4107	+	11.2740i	−2.70291	+	1.30165i	−12.4749	+	15.6430i
34.3	−2.18472	+	2.73956i	−0.385418	−	1.68862i	−1.84207	−	8.07064i	4.25154			5.46812	+	2.63331i	−8.53054	+	6.80288i	13.5063	+	6.50431i	−2.70291	+	1.30165i	−9.28845	+	11.6473i
34.4	−1.98172	+	2.48500i	0.385418	+	1.68862i	−1.35792	−	5.94943i	−6.08751			−4.96002	−	2.38862i	−4.20266	+	3.35151i	6.02067	+	2.89940i	−2.70291	+	1.30165i	12.0638	−	15.1275i
34.5	−1.79867	+	2.25546i	0.385418	+	1.68862i	−0.961803	−	4.21393i	−1.09754			−4.50186	−	2.16798i	9.97914	−	7.95810i	0.837712	+	0.403421i	−2.70291	+	1.30165i	1.97411	−	2.47545i
34.6	−1.69583	+	2.12651i	−0.385418	−	1.68862i	−0.756099	−	3.31268i	0.186531			4.24447	+	2.04403i	1.34043	−	1.06896i	−1.47553	−	0.710580i	−2.70291	+	1.30165i	−0.316326	+	0.396660i
34.7	−1.34543	+	1.68712i	0.385418	+	1.68862i	−0.146095	−	0.640084i	4.08062			−3.36746	−	1.62168i	−0.902285	+	0.719548i	−6.50036	−	3.13041i	−2.70291	+	1.30165i	−5.49018	+	6.88447i
34.8	−1.21070	+	1.51818i	−0.385418	−	1.68862i	0.0510327	+	0.223589i	−1.25078			3.03025	+	1.45929i	4.25364	−	3.39216i	−7.39931	−	3.56332i	−2.70291	+	1.30165i	1.51433	−	1.89891i
34.9	−0.803415	+	1.00745i	−0.385418	−	1.68862i	0.520603	+	2.28091i	8.58393			2.01086	+	0.968377i	4.82536	−	3.84810i	−7.36004	−	3.54441i	−2.70291	+	1.30165i	−6.89646	+	8.64788i
34.10	−0.772816	+	0.969080i	0.385418	+	1.68862i	0.548211	+	2.40187i	−0.309598			−1.93427	−	0.931495i	−7.33519	+	5.84962i	−7.21828	−	3.47614i	−2.70291	+	1.30165i	0.239262	−	0.300025i
34.11	−0.458604	+	0.575072i	−0.385418	−	1.68862i	0.769694	+	3.37225i	3.33310			1.14783	+	0.552768i	−8.89631	+	7.09457i	−4.94308	−	2.38046i	−2.70291	+	1.30165i	−1.52858	+	1.91677i
34.12	−0.0556856	+	0.0698275i	0.385418	+	1.68862i	0.888309	+	3.89193i	−6.61485			−0.139375	−	0.0671193i	−0.0493627	+	0.0393655i	−0.643102	−	0.309702i	−2.70291	+	1.30165i	0.368352	−	0.461898i
34.13	0.0968684	−	0.121469i	0.385418	+	1.68862i	0.884712	+	3.87618i	4.20004			0.242451	+	0.116758i	5.97089	−	4.76163i	1.11645	+	0.537655i	−2.70291	+	1.30165i	0.406851	−	0.510175i
34.14	0.467420	−	0.586127i	−0.385418	−	1.68862i	0.765021	+	3.35178i	−5.08529			−1.16990	−	0.563394i	−1.97059	+	1.57149i	5.02392	+	2.41939i	−2.70291	+	1.30165i	−2.37697	+	2.98062i
34.15	0.813447	−	1.02003i	0.385418	+	1.68862i	0.511318	+	2.24023i	9.08311			2.03596	+	0.980469i	−7.96643	+	6.35301i	7.40289	+	3.56504i	−2.70291	+	1.30165i	7.38863	−	9.26505i
34.16	0.870853	−	1.09202i	−0.385418	−	1.68862i	0.455972	+	1.99774i	−3.04398			−2.17965	−	1.04966i	8.48465	−	6.76628i	7.61233	+	3.66590i	−2.70291	+	1.30165i	−2.65086	+	3.32407i
34.17	1.00379	−	1.25871i	−0.385418	−	1.68862i	0.313321	+	1.37275i	6.56215			−2.51237	−	1.20989i	−0.0330237	+	0.0263355i	7.84447	+	3.77770i	−2.70291	+	1.30165i	6.58702	−	8.25986i
34.18	1.21977	−	1.52954i	0.385418	+	1.68862i	0.0384235	+	0.168344i	−4.83659			3.05294	+	1.47022i	−6.28200	+	5.00973i	7.35482	+	3.54189i	−2.70291	+	1.30165i	−5.89952	+	7.39776i
34.19	1.58468	−	1.98713i	0.385418	+	1.68862i	−0.547383	−	2.39824i	0.731478			3.96628	+	1.91006i	6.86390	−	5.47378i	3.52669	+	1.69837i	−2.70291	+	1.30165i	1.15916	−	1.45354i
34.20	1.78116	−	2.23351i	−0.385418	−	1.68862i	−0.925930	−	4.05676i	−6.92444			−4.45805	−	2.14688i	−8.70690	+	6.94352i	−0.414622	−	0.199672i	−2.70291	+	1.30165i	−12.3335	+	15.4658i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
71.f	odd	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	213.3.j.a	✓	144
71.f	odd	14	1	inner	213.3.j.a	✓	144

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
213.3.j.a	✓	144	1.a	even	1	1	trivial
213.3.j.a	✓	144	71.f	odd	14	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(213, [\chi])\).