Newform orbit 16.22.e.a

• Label: 16.22.e.a
• Level: $16$
• Weight: $22$
• Character orbit: 16.e
• Analytic conductor: $44.716$
• Analytic rank: $0$
• Dimension: $82$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	16.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(44.7163750859\)
Analytic rank:	\(0\)
Dimension:	\(82\)
Relative dimension:	\(41\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 82 q - 2 q^{2} - 2 q^{3} + 1604328 q^{4} - 2 q^{5} - 142760912 q^{6} - 5437750796 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−1439.53	+	157.777i	−112866.	+	112866.i	2.04736e6	−	454251.i	−2.81638e6	−	2.81638e6i	1.44667e8	−	1.80282e8i		−	1.39005e9i	−2.87558e9	+	9.76936e8i		−	1.50171e10i	4.49863e9	+	3.60991e9i
5.2	−1428.70	+	236.579i	−28799.9	+	28799.9i	1.98521e6	−	676002.i	2.59531e7	+	2.59531e7i	3.43329e7	−	4.79599e7i			2.01795e8i	−2.67634e9	+	1.43546e9i			8.80148e9i	−4.32191e10	−	3.09392e10i
5.3	−1400.35	−	369.019i	109714.	−	109714.i	1.82480e6	+	1.03351e6i	−1.97304e7	−	1.97304e7i	−1.94125e8	+	1.13152e8i		−	1.02336e9i	−2.17397e9	−	2.12066e9i		−	1.36141e10i	2.03485e10	+	3.49103e10i
5.4	−1393.10	−	395.509i	−81293.3	+	81293.3i	1.78430e6	+	1.10197e6i	−1.56158e7	−	1.56158e7i	1.45402e8	−	8.10974e7i			1.06061e9i	−2.04986e9	−	2.24085e9i		−	2.75685e9i	1.55782e10	+	2.79306e10i
5.5	−1391.38	+	401.508i	72032.1	−	72032.1i	1.77473e6	−	1.11730e6i	−4.01913e6	−	4.01913e6i	−7.13027e7	+	1.29146e8i			8.04004e8i	−2.02073e9	+	2.26716e9i			8.31155e7i	7.20586e9	+	3.97843e9i
5.6	−1391.35	−	401.620i	12781.5	−	12781.5i	1.77455e6	+	1.11759e6i	5.32911e6	+	5.32911e6i	−2.29168e7	+	1.26502e7i		−	4.26451e8i	−2.02018e9	−	2.26765e9i			1.01336e10i	−5.27438e9	−	9.55493e9i
5.7	−1198.60	+	812.724i	−21696.3	+	21696.3i	776112.	−	1.94825e6i	−2.38583e7	−	2.38583e7i	8.37200e6	−	4.36382e7i		−	3.98700e8i	6.53148e8	+	2.96593e9i			9.51889e9i	4.79868e10	+	9.20627e9i
5.8	−1186.35	+	830.497i	117950.	−	117950.i	717703.	−	1.97052e6i	2.07522e7	+	2.07522e7i	−4.19730e7	+	2.37888e8i		−	1.29970e9i	7.85064e8	+	2.93378e9i		−	1.73642e10i	−4.18539e10	−	7.38473e9i
5.9	−1166.53	−	858.118i	132197.	−	132197.i	624420.	+	2.00204e6i	2.49441e7	+	2.49441e7i	−2.67652e8	+	4.07709e7i			1.02743e9i	9.89578e8	−	2.87126e9i		−	2.44918e10i	−7.69301e9	−	5.05030e10i
5.10	−1134.61	+	899.894i	−127691.	+	127691.i	477535.	−	2.04206e6i	6.46851e6	+	6.46851e6i	2.99713e7	−	2.59787e8i			1.09329e9i	1.29582e9	+	2.74668e9i		−	2.21494e10i	−1.31602e10	−	1.51828e9i
5.11	−1089.16	−	954.398i	17071.7	−	17071.7i	275400.	+	2.07899e6i	3.05605e6	+	3.05605e6i	−3.48871e7	+	2.30067e6i			3.52028e7i	1.68423e9	−	2.52720e9i			9.87747e9i	−4.11849e8	−	6.24523e9i
5.12	−971.098	−	1074.30i	−113951.	+	113951.i	−211090.	+	2.08650e6i	2.44217e7	+	2.44217e7i	2.33076e8	+	1.17601e7i			1.34162e7i	2.44652e9	−	1.79942e9i		−	1.55095e10i	2.52039e9	−	4.99522e10i
5.13	−840.196	−	1179.50i	−72314.6	+	72314.6i	−685292.	+	1.98202e6i	−1.66664e7	−	1.66664e7i	1.46054e8	+	2.45367e7i		−	4.57359e8i	2.91358e9	−	8.56988e8i			1.53935e6i	−5.65498e9	+	3.36611e10i
5.14	−810.478	+	1200.12i	−26628.3	+	26628.3i	−783402.	−	1.94533e6i	1.05871e7	+	1.05871e7i	−1.03754e7	−	5.35386e7i		−	2.00853e8i	2.96956e9	+	6.36479e8i			9.04222e9i	−2.12864e10	+	4.12514e9i
5.15	−687.568	−	1274.52i	80161.3	−	80161.3i	−1.15165e6	+	1.75264e6i	−2.92075e7	−	2.92075e7i	−1.57284e8	−	4.70509e7i			9.19049e8i	3.02561e9	+	2.62745e8i		−	2.39133e9i	−1.71434e10	+	5.73077e10i
5.16	−591.628	+	1321.79i	96038.8	−	96038.8i	−1.39711e6	−	1.56402e6i	−7.03831e6	−	7.03831e6i	7.01239e7	+	1.83762e8i			4.28554e8i	2.89387e9	−	9.21365e8i		−	7.98656e9i	1.34672e10	−	5.13911e9i
5.17	−410.878	−	1388.64i	57368.5	−	57368.5i	−1.75951e6	+	1.14113e6i	9.04515e6	+	9.04515e6i	−1.03236e8	−	5.60929e7i		−	8.24638e8i	2.30756e9	+	1.97447e9i			3.87806e9i	8.84403e9	−	1.62769e10i
5.18	−172.809	+	1437.81i	−93060.3	+	93060.3i	−2.03743e6	−	496931.i	−1.84176e7	−	1.84176e7i	−1.17721e8	−	1.49884e8i			6.32962e8i	1.06657e9	−	2.84355e9i		−	6.86010e9i	2.96637e10	−	2.32983e10i
5.19	−54.2492	−	1447.14i	−21619.3	+	21619.3i	−2.09127e6	+	157012.i	1.11303e7	+	1.11303e7i	3.24590e7	+	3.01133e7i			1.47909e9i	3.40668e8	+	3.01783e9i			9.52556e9i	1.55033e10	−	1.67109e10i
5.20	−26.6184	+	1447.91i	−103140.	+	103140.i	−2.09573e6	−	77082.0i	8.68960e6	+	8.68960e6i	−1.46592e8	−	1.52083e8i		−	1.15126e9i	1.67393e8	−	3.03238e9i		−	1.08153e10i	−1.28131e10	+	1.23505e10i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	16.22.e.a	✓	82
4.b	odd	2	1		64.22.e.a		82
16.e	even	4	1	inner	16.22.e.a	✓	82
16.f	odd	4	1		64.22.e.a		82

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
16.22.e.a	✓	82	1.a	even	1	1	trivial
16.22.e.a	✓	82	16.e	even	4	1	inner
64.22.e.a		82	4.b	odd	2	1
64.22.e.a		82	16.f	odd	4	1

Hecke kernels

This newform subspace is the entire newspace \(S_{22}^{\mathrm{new}}(16, [\chi])\).