Newform orbit 77.3.p.a

• Label: 77.3.p.a
• Level: $77$
• Weight: $3$
• Character orbit: 77.p
• 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.p (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.

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 5 q^{2} - 9 q^{3} + 27 q^{4} - 15 q^{5} - 23 q^{7} - 72 q^{8} - 27 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−3.47800	−	1.54850i	0.405840	−	1.90933i	7.02207	+	7.79880i	4.28573	+	0.450449i	−4.36811	+	6.01219i	2.91525	+	6.36407i	−7.64037	−	23.5147i	4.74108	+	2.11087i	−14.2082	−	8.20313i
3.2	−3.14125	−	1.39857i	−0.904545	+	4.25555i	5.23491	+	5.81396i	−2.97205	−	0.312375i	8.79310	−	12.1027i	−0.982273	−	6.93074i	−4.06266	−	12.5036i	−9.06958	−	4.03804i	8.89907	+	5.13788i
3.3	−2.13423	−	0.950219i	0.866650	−	4.07727i	0.975483	+	1.08338i	−7.57520	−	0.796185i	−5.72392	+	7.87830i	6.99898	−	0.119278i	1.83525	+	5.64833i	−7.65111	−	3.40649i	15.4106	+	8.89733i
3.4	−2.11301	−	0.940772i	−0.286000	+	1.34552i	0.903232	+	1.00314i	0.485505	+	0.0510287i	1.87015	−	2.57404i	−4.83539	+	5.06153i	1.89419	+	5.82970i	6.49328	+	2.89099i	−0.977871	−	0.564574i
3.5	−1.63381	−	0.727421i	0.994951	−	4.68088i	−0.536317	−	0.595640i	7.45896	+	0.783969i	−5.03053	+	6.92393i	−5.98810	−	3.62528i	2.65358	+	8.16689i	−12.6988	−	5.65386i	−11.6163	−	6.70666i
3.6	−1.39802	−	0.622439i	−0.407108	+	1.91529i	−1.10949	−	1.23221i	3.14966	+	0.331043i	1.76130	−	2.42422i	6.37630	−	2.88839i	2.67570	+	8.23495i	4.71931	+	2.10117i	−4.19724	−	2.42328i
3.7	−0.169983	−	0.0756813i	0.0447743	−	0.210647i	−2.65336	−	2.94685i	−7.93821	−	0.834340i	−0.0235529	+	0.0324177i	−6.97723	+	0.564195i	0.457998	+	1.40957i	8.17954	+	3.64177i	1.28622	+	0.742598i
3.8	0.333205	+	0.148352i	−1.16799	+	5.49496i	−2.58751	−	2.87372i	−5.30093	−	0.557150i	−1.20437	+	1.65767i	4.75117	+	5.14066i	−0.886687	−	2.72894i	−20.6085	−	9.17550i	−1.68364	−	0.972049i
3.9	0.710777	+	0.316458i	0.455331	−	2.14216i	−2.27146	−	2.52272i	1.67507	+	0.176057i	1.00154	−	1.37851i	1.90441	−	6.73596i	−1.77788	−	5.47176i	3.84038	+	1.70985i	1.13489	+	0.655228i
3.10	1.25182	+	0.557345i	−0.248510	+	1.16915i	−1.42011	−	1.57719i	8.69060	+	0.913418i	−0.962708	+	1.32505i	−0.656163	+	6.96918i	−2.59244	−	7.97872i	6.91676	+	3.07954i	10.3699	+	5.98709i
3.11	1.35964	+	0.605349i	0.923126	−	4.34296i	−1.19436	−	1.32647i	−1.52806	−	0.160606i	3.88412	−	5.34604i	3.55163	+	6.03207i	−2.66057	−	8.18838i	−9.78727	−	4.35757i	−1.98039	−	1.14338i
3.12	2.43731	+	1.08516i	−0.812940	+	3.82458i	2.08639	+	2.31717i	2.08180	+	0.218806i	−6.13168	+	8.43953i	−4.71907	−	5.17015i	−0.727112	−	2.23782i	−5.74466	−	2.55769i	4.83655	+	2.79239i
3.13	2.86109	+	1.27384i	−0.315131	+	1.48258i	3.88664	+	4.31656i	−4.70921	−	0.494958i	−2.79018	+	3.84036i	6.97032	+	0.643947i	1.75026	+	5.38674i	6.12319	+	2.72622i	−12.8430	−	7.41490i
3.14	3.07171	+	1.36761i	0.674115	−	3.17146i	4.88852	+	5.42926i	−1.65544	−	0.173994i	6.40802	−	8.81989i	−6.98372	+	0.477162i	3.43485	+	10.5714i	−1.38183	−	0.615230i	−4.84708	−	2.79846i
5.1	−3.67332	−	0.780787i	4.52797	+	0.475909i	9.22944	+	4.10921i	−2.23369	+	2.01122i	−16.2611	−	5.28355i	−0.790956	+	6.95517i	−18.5416	−	13.4712i	11.4727	+	2.43860i	9.77538	−	5.64382i
5.2	−3.42306	−	0.727594i	−4.13871	−	0.434996i	7.53377	+	3.35425i	−6.41701	+	5.77790i	13.8506	+	4.50032i	−0.243758	−	6.99575i	−12.0233	−	8.73544i	8.13640	+	1.72945i	26.1698	−	15.1091i
5.3	−2.88156	−	0.612495i	−0.771458	−	0.0810835i	4.27406	+	1.90294i	3.48954	−	3.14199i	2.17334	+	0.706161i	−6.97684	+	0.568896i	−1.61719	−	1.17496i	−8.21476	−	1.74610i	−11.9798	+	6.91652i
5.4	−2.08046	−	0.442214i	2.46033	+	0.258591i	0.478559	+	0.213068i	1.04600	−	0.941822i	−5.00425	−	1.62598i	5.96372	−	3.66525i	5.98150	+	4.34582i	−2.81699	−	0.598769i	−2.59264	+	1.49686i
5.5	−1.85255	−	0.393773i	−4.21020	−	0.442510i	−0.377280	−	0.167976i	1.58739	−	1.42930i	7.62537	+	2.47763i	4.15912	+	5.63043i	6.76171	+	4.91267i	8.72662	+	1.85490i	−3.50355	+	2.02278i
5.6	−0.979296	−	0.208156i	0.761101	+	0.0799950i	−2.73849	−	1.21925i	−5.47727	+	4.93175i	−0.728692	−	0.236766i	−0.164579	+	6.99806i	5.66787	+	4.11795i	−8.23045	−	1.74944i	6.39044	−	3.68952i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner
11.c	even	5	1	inner
77.p	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.p.a	✓	112
7.d	odd	6	1	inner	77.3.p.a	✓	112
11.c	even	5	1	inner	77.3.p.a	✓	112
77.p	odd	30	1	inner	77.3.p.a	✓	112

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
77.3.p.a	✓	112	1.a	even	1	1	trivial
77.3.p.a	✓	112	7.d	odd	6	1	inner
77.3.p.a	✓	112	11.c	even	5	1	inner
77.3.p.a	✓	112	77.p	odd	30	1	inner

Hecke kernels

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