Newform orbit 77.3.o.a

• Label: 77.3.o.a
• Level: $77$
• Weight: $3$
• Character orbit: 77.o
• Analytic conductor: $2.098$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	77.o (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.09809803557\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 5 q^{2} - 3 q^{3} - 21 q^{4} - q^{5} - 20 q^{6} - 25 q^{7} + 20 q^{8} + 21 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−2.69667	+	2.42809i	−4.20124	−	1.87051i	0.958281	−	9.11744i	4.39298	+	0.933757i	15.8711	−	5.15685i	−6.58845	+	2.36481i	11.0222	+	15.1707i	8.12944	+	9.02866i	−14.1137	+	8.14852i
2.2	−2.56152	+	2.30640i	2.50619	+	1.11583i	0.823777	−	7.83771i	−8.51266	−	1.80942i	−8.99320	+	2.92207i	−4.37697	+	5.46279i	7.86274	+	10.8221i	−0.986263	−	1.09536i	25.9786	−	14.9988i
2.3	−2.38512	+	2.14758i	2.33105	+	1.03785i	0.658623	−	6.26638i	7.26118	+	1.54341i	−7.78872	+	2.53071i	6.98406	+	0.472086i	4.34064	+	5.97438i	−1.66550	−	1.84972i	−20.6334	+	11.9127i
2.4	−1.81020	+	1.62991i	−1.74873	−	0.778586i	0.202101	−	1.92286i	−3.37943	−	0.718320i	4.43459	−	1.44088i	5.90556	−	3.75824i	−2.95881	−	4.07246i	−3.57031	−	3.96523i	7.28826	−	4.20788i
2.5	−1.12351	+	1.01162i	−1.92416	−	0.856692i	−0.179199	+	1.70496i	−0.593107	−	0.126069i	3.02846	−	0.984007i	−5.56558	−	4.24550i	−5.07797	−	6.98923i	−3.05370	−	3.39148i	0.793896	−	0.458356i
2.6	−0.989117	+	0.890605i	4.72138	+	2.10209i	−0.232938	+	2.21626i	2.18305	+	0.464022i	−6.54214	+	2.12567i	−4.29301	−	5.52902i	−4.87275	−	6.70677i	11.8505	+	13.1613i	−2.57255	+	1.48527i
2.7	−0.415666	+	0.374267i	2.01314	+	0.896306i	−0.385412	+	3.66695i	−0.925869	−	0.196799i	−1.17225	+	0.380887i	1.08192	+	6.91588i	−2.52729	−	3.47851i	−2.77282	−	3.07953i	0.458508	−	0.264719i
2.8	−0.0281720	+	0.0253662i	−4.67194	−	2.08008i	−0.417964	+	3.97666i	6.95673	+	1.47870i	0.184382	−	0.0599092i	6.93803	+	0.929347i	−0.178227	−	0.245309i	11.4781	+	12.7478i	−0.233494	+	0.134808i
2.9	0.609425	−	0.548729i	−3.43154	−	1.52782i	−0.347818	+	3.30927i	−6.89374	−	1.46531i	−2.92962	+	0.951892i	−2.94579	+	6.34999i	3.53201	+	4.86139i	3.41903	+	3.79722i	−5.00527	+	2.88980i
2.10	1.05190	−	0.947132i	0.0320824	+	0.0142840i	−0.208686	+	1.98552i	8.44301	+	1.79462i	0.0472762	−	0.0153610i	−6.25973	−	3.13301i	4.98899	+	6.86676i	−6.02135	−	6.68739i	10.5809	−	6.10890i
2.11	1.12394	−	1.01200i	2.24502	+	0.999546i	−0.179016	+	1.70322i	−0.919807	−	0.195511i	3.53481	−	1.14853i	6.57321	−	2.40684i	5.07836	+	6.98976i	−1.98117	−	2.20031i	−1.23167	+	0.711104i
2.12	2.15421	−	1.93966i	4.27278	+	1.90236i	0.460229	−	4.37879i	−6.93791	−	1.47470i	12.8944	−	4.18965i	−6.92888	−	0.995298i	−0.686495	−	0.944879i	8.61549	+	9.56847i	−17.8061	+	10.2804i
2.13	2.30353	−	2.07410i	−3.51540	−	1.56516i	0.586210	−	5.57742i	−2.33898	−	0.497165i	−11.3441	+	3.68592i	3.63950	−	5.97947i	−2.92996	−	4.03275i	3.88613	+	4.31598i	−6.41907	+	3.70605i
2.14	2.47982	−	2.23284i	−0.106777	−	0.0475403i	0.745818	−	7.09599i	2.24271	+	0.476702i	−0.370938	+	0.120525i	−0.987183	+	6.93004i	−6.14912	−	8.46354i	−6.01303	−	6.67815i	6.62590	−	3.82547i
18.1	−3.71680	−	0.390651i	−2.03180	−	2.25655i	9.74941	+	2.07230i	5.40367	+	2.40587i	6.67029	+	9.18086i	−2.62160	−	6.49055i	−21.2096	−	6.89142i	−0.0230200	+	0.219021i	−19.1445	−	11.0531i
18.2	−3.19130	−	0.335419i	0.502267	+	0.557824i	6.15928	+	1.30919i	−4.39136	−	1.95516i	−1.41578	−	1.94865i	5.17640	+	4.71221i	−7.00966	−	2.27758i	0.881861	−	8.39034i	13.3583	+	7.71244i
18.3	−3.05232	−	0.320811i	3.41137	+	3.78871i	5.30112	+	1.12679i	2.71599	+	1.20924i	−9.19712	−	12.6587i	−6.93407	−	0.958481i	−4.14354	−	1.34632i	−1.77612	+	16.8987i	−7.90212	−	4.56229i
18.4	−2.14499	−	0.225448i	−2.30053	−	2.55499i	0.637569	+	0.135520i	−0.464065	−	0.206615i	4.35859	+	5.99909i	−1.93603	+	6.72695i	6.86795	+	2.23153i	−0.294814	+	2.80497i	0.948835	+	0.547810i
18.5	−1.39933	−	0.147076i	1.30261	+	1.44669i	−1.97609	−	0.420030i	−5.97949	−	2.66224i	−1.61001	−	2.21598i	−3.29542	−	6.17578i	8.05613	+	2.61760i	0.544626	−	5.18177i	7.97576	+	4.60481i
18.6	−1.09182	−	0.114755i	−0.273374	−	0.303612i	−2.73369	−	0.581064i	6.23664	+	2.77673i	0.263633	+	0.362860i	5.03938	−	4.85847i	7.09441	+	2.30511i	0.923309	−	8.78470i	−6.49063	−	3.74737i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
11.d	odd	10	1	inner
77.o	odd	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	77.3.o.a	✓	112
7.c	even	3	1	inner	77.3.o.a	✓	112
11.d	odd	10	1	inner	77.3.o.a	✓	112
77.o	odd	30	1	inner	77.3.o.a	✓	112

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
77.3.o.a	✓	112	1.a	even	1	1	trivial
77.3.o.a	✓	112	7.c	even	3	1	inner
77.3.o.a	✓	112	11.d	odd	10	1	inner
77.3.o.a	✓	112	77.o	odd	30	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(77, [\chi])\).