Newform orbit 49.4.e.a

• Label: 49.4.e.a
• Level: $49$
• Weight: $4$
• Character orbit: 49.e
• Analytic conductor: $2.891$
• Analytic rank: $0$
• Dimension: $78$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	49.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.89109359028\)
Analytic rank:	\(0\)
Dimension:	\(78\)
Relative dimension:	\(13\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
8.1	−3.44244	−	4.31668i	3.65176	−	1.75859i	−5.00318	+	21.9203i	−12.6967	+	6.11441i	−20.1622	−	9.70961i	−15.1671	+	10.6282i	72.0505	−	34.6977i	−6.59154	+	8.26553i	70.1015	+	33.7591i
8.2	−2.77071	−	3.47436i	−6.15987	+	2.96643i	−2.61419	+	11.4535i	1.98451	−	0.955692i	27.3737	+	13.1825i	18.5202	−	0.0393741i	15.0065	−	7.22673i	12.3100	−	15.4362i	−8.81894	−	4.24698i
8.3	−2.45388	−	3.07706i	2.79026	−	1.34372i	−1.66664	+	7.30203i	16.4856	−	7.93907i	−10.9817	−	5.28849i	−6.70370	−	17.2644i	−1.80909	+	0.871210i	−10.8543	+	13.6108i	−64.8827	−	31.2459i
8.4	−1.61233	−	2.02180i	−3.25484	+	1.56745i	0.292112	−	1.27982i	−3.53374	+	1.70176i	8.41693	+	4.05338i	−10.4442	+	15.2944i	−21.6976	+	10.4490i	−8.69714	+	10.9059i	9.13817	+	4.40071i
8.5	−1.46127	−	1.83238i	7.38411	−	3.55600i	0.557879	−	2.44423i	−0.458224	+	0.220669i	−17.3061	−	8.33419i	12.0280	+	14.0828i	−22.1867	+	10.6846i	25.0457	−	31.4064i	1.07394	+	0.517181i
8.6	−0.802750	−	1.00662i	−1.70862	+	0.822828i	1.41130	−	6.18330i	−15.9369	+	7.67483i	2.19987	+	1.05940i	0.116992	−	18.5199i	−16.6372	+	8.01205i	−14.5919	+	18.2976i	20.5190	+	9.88143i
8.7	0.153998	+	0.193108i	−8.98937	+	4.32905i	1.76659	−	7.73995i	14.2902	−	6.88179i	−2.22032	−	1.06925i	−13.6848	−	12.4791i	3.54697	−	1.70813i	45.2339	−	56.7215i	3.52959	+	1.69976i
8.8	0.521845	+	0.654373i	5.78023	−	2.78361i	1.62429	−	7.11646i	−1.28950	+	0.620991i	4.83790	+	2.32981i	−17.3557	−	6.46381i	11.5371	−	5.55599i	8.82835	−	11.0704i	−1.07928	−	0.519753i
8.9	0.777062	+	0.974405i	−0.189548	+	0.0912813i	1.43453	−	6.28508i	7.66912	−	3.69325i	−0.236235	−	0.113765i	18.3768	+	2.30053i	16.2220	−	7.81211i	−16.8066	+	21.0748i	9.55811	+	4.60294i
8.10	1.88794	+	2.36740i	−5.83533	+	2.81015i	−0.260101	+	1.13958i	−12.6471	+	6.09051i	−17.6695	−	8.50917i	−3.94107	+	18.0961i	18.6363	−	8.97477i	9.31994	−	11.6868i	−38.2955	−	18.4422i
8.11	2.59695	+	3.25648i	2.96773	−	1.42918i	−2.08030	+	9.11440i	9.52410	−	4.58657i	12.3612	+	5.95282i	−16.3066	+	8.78031i	−5.06162	+	2.43755i	−10.0694	+	12.6266i	39.6697	+	19.1039i
8.12	2.64790	+	3.32037i	6.62695	−	3.19137i	−2.23327	+	9.78459i	−17.1881	+	8.27737i	28.1441	+	13.5535i	16.3388	−	8.72030i	−7.79127	+	3.75208i	16.8974	−	21.1887i	−72.9964	−	35.1532i
8.13	3.18020	+	3.98784i	−4.68696	+	2.25712i	−4.00905	+	17.5648i	4.76744	−	2.29588i	−23.9065	−	11.5128i	7.38357	−	16.9848i	−46.0311	+	22.1674i	0.0387732	−	0.0486201i	24.3170	+	11.7104i
15.1	−1.16701	+	5.11301i	−1.24065	+	1.55573i	−17.5732	−	8.46282i	0.849213	−	1.06488i	−6.50660	−	8.15902i	−18.2375	+	3.22369i	37.6195	−	47.1733i	5.12699	+	22.4628i	4.45370	+	5.58476i
15.2	−0.892344	+	3.90961i	5.02886	−	6.30599i	−7.28105	−	3.50637i	8.76329	−	10.9888i	20.1665	+	25.2880i	3.97541	−	18.0886i	0.203423	−	0.255085i	−8.46803	−	37.1008i	35.1422	+	44.0669i
15.3	−0.846844	+	3.71026i	3.04811	−	3.82222i	−5.84116	−	2.81296i	−12.1071	+	15.1818i	11.6002	+	14.5461i	13.1709	+	13.0203i	−3.59906	+	4.51308i	0.689741	+	3.02195i	−46.0758	−	57.7772i
15.4	−0.764068	+	3.34760i	−5.99643	+	7.51929i	−3.41489	−	1.64452i	0.968529	−	1.21450i	−20.5899	−	25.8189i	18.5012	+	0.839908i	−9.01256	+	11.3014i	−14.5744	−	63.8547i	3.32563	+	4.17021i
15.5	−0.424669	+	1.86059i	0.113640	−	0.142500i	3.92628	+	1.89080i	11.1399	−	13.9690i	0.216876	+	0.271954i	1.18252	+	18.4825i	−14.7045	+	18.4389i	6.00067	+	26.2907i	21.2598	+	26.6590i
15.6	−0.379589	+	1.66309i	−2.24247	+	2.81197i	4.58597	+	2.20849i	−4.66393	+	5.84838i	−3.82533	−	4.79682i	−14.0409	−	12.0769i	−13.9224	+	17.4581i	3.12958	+	13.7116i	−7.95600	−	9.97651i
15.7	0.0613960	−	0.268993i	2.25069	−	2.82228i	7.13916	+	3.43804i	−2.36861	+	2.97014i	−0.620990	−	0.778697i	9.47466	−	15.9132i	2.73935	−	3.43503i	3.10843	+	13.6189i	0.653525	+	0.819495i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
49.e	even	7	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	49.4.e.a	✓	78
49.e	even	7	1	inner	49.4.e.a	✓	78
49.e	even	7	1		2401.4.a.d		39
49.f	odd	14	1		2401.4.a.c		39

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
49.4.e.a	✓	78	1.a	even	1	1	trivial
49.4.e.a	✓	78	49.e	even	7	1	inner
2401.4.a.c		39	49.f	odd	14	1
2401.4.a.d		39	49.e	even	7	1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(49, [\chi])\).