Newform orbit 136.2.o.a

• Label: 136.2.o.a
• Level: $136$
• Weight: $2$
• Character orbit: 136.o
• Analytic conductor: $1.086$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 136 = 2^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	136.o (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.08596546749\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 4 q^{2} - 12 q^{6} - 8 q^{7} - 4 q^{8} - 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
53.1	−1.41119	−	0.0924010i	0.904309	+	2.18320i	1.98292	+	0.260791i	1.75957	−	0.728837i	−1.07442	−	3.16447i	0.236145	−	0.570103i	−2.77419	−	0.551251i	−1.82725	+	1.82725i	−2.55043	+	0.865943i
53.2	−1.33657	−	0.462152i	−1.18933	−	2.87130i	1.57283	+	1.23540i	3.39543	−	1.40643i	0.262644	+	4.38733i	0.358543	−	0.865599i	−1.53125	−	2.37808i	−4.70851	+	4.70851i	−5.18822	+	0.310588i
53.3	−1.31394	+	0.523025i	−0.222077	−	0.536141i	1.45289	−	1.37445i	−0.879016	+	0.364100i	0.572211	+	0.588307i	0.929939	−	2.24507i	−1.19014	+	2.56584i	1.88319	−	1.88319i	0.964543	−	0.938154i
53.4	−1.29763	−	0.562275i	−0.381356	−	0.920674i	1.36769	+	1.45925i	−3.22047	+	1.33396i	−0.0228132	+	1.40912i	−1.59047	+	3.83974i	−0.954260	−	2.66259i	1.41911	−	1.41911i	4.92903	+	0.0797992i
53.5	−0.822898	+	1.15015i	0.762752	+	1.84145i	−0.645676	−	1.89291i	−0.607426	+	0.251604i	−2.74560	−	0.638046i	−1.63175	+	3.93939i	2.70845	+	0.815049i	−0.687814	+	0.687814i	0.210468	−	0.905674i
53.6	−0.562275	−	1.29763i	0.381356	+	0.920674i	−1.36769	+	1.45925i	3.22047	−	1.33396i	0.980269	−	1.01253i	−1.59047	+	3.83974i	2.66259	+	0.954260i	1.41911	−	1.41911i	−3.54178	−	3.42892i
53.7	−0.462152	−	1.33657i	1.18933	+	2.87130i	−1.57283	+	1.23540i	−3.39543	+	1.40643i	3.28803	−	2.91660i	0.358543	−	0.865599i	2.37808	+	1.53125i	−4.70851	+	4.70851i	3.44900	+	3.88824i
53.8	−0.416332	+	1.35154i	−0.557935	−	1.34698i	−1.65334	−	1.12538i	1.95127	−	0.808242i	2.05278	−	0.193284i	0.0841835	−	0.203237i	2.20934	−	1.76602i	0.618270	−	0.618270i	0.279998	+	2.97372i
53.9	−0.0924010	−	1.41119i	−0.904309	−	2.18320i	−1.98292	+	0.260791i	−1.75957	+	0.728837i	−2.99735	+	1.47788i	0.236145	−	0.570103i	0.551251	+	2.77419i	−1.82725	+	1.82725i	1.19112	+	2.41574i
53.10	0.326141	+	1.37609i	1.10254	+	2.66176i	−1.78726	+	0.897601i	1.17621	−	0.487201i	−3.30324	+	2.38530i	1.54928	−	3.74030i	−1.81808	−	2.16670i	−3.74804	+	3.74804i	1.05404	+	1.45968i
53.11	0.523025	−	1.31394i	0.222077	+	0.536141i	−1.45289	−	1.37445i	0.879016	−	0.364100i	0.820610	−	0.0113813i	0.929939	−	2.24507i	−2.56584	+	1.19014i	1.88319	−	1.88319i	−0.0186598	−	1.34541i
53.12	0.886882	+	1.10156i	−0.254575	−	0.614599i	−0.426882	+	1.95391i	2.49546	−	1.03366i	0.451241	−	0.825507i	−0.228760	+	0.552276i	−2.53095	+	1.26265i	1.80840	−	1.80840i	3.35182	+	1.83218i
53.13	1.10156	+	0.886882i	0.254575	+	0.614599i	0.426882	+	1.95391i	−2.49546	+	1.03366i	−0.264646	+	0.902797i	−0.228760	+	0.552276i	−1.26265	+	2.53095i	1.80840	−	1.80840i	−3.66564	−	1.07455i
53.14	1.15015	−	0.822898i	−0.762752	−	1.84145i	0.645676	−	1.89291i	0.607426	−	0.251604i	−2.39260	−	1.49027i	−1.63175	+	3.93939i	−0.815049	−	2.70845i	−0.687814	+	0.687814i	0.491585	−	0.789232i
53.15	1.35154	−	0.416332i	0.557935	+	1.34698i	1.65334	−	1.12538i	−1.95127	+	0.808242i	1.31486	+	1.58821i	0.0841835	−	0.203237i	1.76602	−	2.20934i	0.618270	−	0.618270i	−2.30073	+	1.90475i
53.16	1.37609	+	0.326141i	−1.10254	−	2.66176i	1.78726	+	0.897601i	−1.17621	+	0.487201i	−0.649083	−	4.02241i	1.54928	−	3.74030i	2.16670	+	1.81808i	−3.74804	+	3.74804i	−1.77747	+	0.286825i
77.1	−1.41119	+	0.0924010i	0.904309	−	2.18320i	1.98292	−	0.260791i	1.75957	+	0.728837i	−1.07442	+	3.16447i	0.236145	+	0.570103i	−2.77419	+	0.551251i	−1.82725	−	1.82725i	−2.55043	−	0.865943i
77.2	−1.33657	+	0.462152i	−1.18933	+	2.87130i	1.57283	−	1.23540i	3.39543	+	1.40643i	0.262644	−	4.38733i	0.358543	+	0.865599i	−1.53125	+	2.37808i	−4.70851	−	4.70851i	−5.18822	−	0.310588i
77.3	−1.31394	−	0.523025i	−0.222077	+	0.536141i	1.45289	+	1.37445i	−0.879016	−	0.364100i	0.572211	−	0.588307i	0.929939	+	2.24507i	−1.19014	−	2.56584i	1.88319	+	1.88319i	0.964543	+	0.938154i
77.4	−1.29763	+	0.562275i	−0.381356	+	0.920674i	1.36769	−	1.45925i	−3.22047	−	1.33396i	−0.0228132	−	1.40912i	−1.59047	−	3.83974i	−0.954260	+	2.66259i	1.41911	+	1.41911i	4.92903	−	0.0797992i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
17.d	even	8	1	inner
136.o	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	136.2.o.a	✓	64
4.b	odd	2	1		544.2.bk.a		64
8.b	even	2	1	inner	136.2.o.a	✓	64
8.d	odd	2	1		544.2.bk.a		64
17.d	even	8	1	inner	136.2.o.a	✓	64
68.g	odd	8	1		544.2.bk.a		64
136.o	even	8	1	inner	136.2.o.a	✓	64
136.p	odd	8	1		544.2.bk.a		64

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
136.2.o.a	✓	64	1.a	even	1	1	trivial
136.2.o.a	✓	64	8.b	even	2	1	inner
136.2.o.a	✓	64	17.d	even	8	1	inner
136.2.o.a	✓	64	136.o	even	8	1	inner
544.2.bk.a		64	4.b	odd	2	1
544.2.bk.a		64	8.d	odd	2	1
544.2.bk.a		64	68.g	odd	8	1
544.2.bk.a		64	136.p	odd	8	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(136, [\chi])\).