Newform orbit 103.4.e.a

• Label: 103.4.e.a
• Level: $103$
• Weight: $4$
• Character orbit: 103.e
• Analytic conductor: $6.077$
• Analytic rank: $0$
• Dimension: $400$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 103 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	103.e (of order \(17\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.07719673059\)
Analytic rank:	\(0\)
Dimension:	\(400\)
Relative dimension:	\(25\) over \(\Q(\zeta_{17})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{17}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 400 q - 15 q^{2} - 19 q^{3} - 111 q^{4} - 17 q^{5} - 19 q^{6} - 7 q^{7} - 41 q^{8} - 194 q^{9}+O(q^{10}) \)

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	−5.26078	+	0.983411i	0.485613	+	5.24060i	19.2490	−	7.45709i	−0.632154	−	0.837108i	−7.70837	−	27.0921i	−8.60631	−	5.32880i	−57.5290	+	35.6205i	−0.687777	+	0.128568i	4.14885	+	3.78217i
8.2	−5.22411	+	0.976555i	−0.676420	−	7.29974i	18.8779	−	7.31333i	3.07225	+	4.06832i	10.6623	+	37.4741i	13.4823	+	8.34787i	−55.3298	+	34.2587i	−26.2883	+	4.91414i	−20.0227	−	18.2531i
8.3	−4.48542	+	0.838470i	−0.411989	−	4.44607i	11.9562	−	4.63185i	−12.1481	−	16.0866i	5.57585	+	19.5971i	−20.1023	−	12.4468i	−18.7077	+	11.5833i	6.94244	−	1.29777i	67.9773	+	61.9695i
8.4	−4.12025	+	0.770208i	0.543155	+	5.86158i	8.92345	−	3.45696i	2.95177	+	3.90878i	−6.75257	−	23.7328i	26.6631	+	16.5091i	−5.59396	+	3.46363i	−7.52281	+	1.40626i	−15.1726	−	13.8317i
8.5	−3.93886	+	0.736300i	0.0108574	+	0.117170i	7.51268	−	2.91043i	9.80317	+	12.9815i	−0.129038	−	0.453520i	−17.6246	−	10.9127i	−0.193299	+	0.119686i	26.5267	−	4.95869i	−48.1716	−	43.9142i
8.6	−3.40502	+	0.636509i	−0.450711	−	4.86395i	3.72927	−	1.44473i	1.68296	+	2.22859i	4.63063	+	16.2750i	9.86271	+	6.10673i	11.7826	−	7.29547i	3.08543	−	0.576766i	−7.14903	−	6.51720i
8.7	−3.26993	+	0.611256i	0.255035	+	2.75227i	2.85902	−	1.10759i	−11.6921	−	15.4828i	−2.51629	−	8.84383i	18.4540	+	11.4262i	13.9547	−	8.64037i	19.0303	−	3.55739i	47.6962	+	43.4808i
8.8	−3.05581	+	0.571229i	0.891800	+	9.62406i	1.55188	−	0.601201i	−5.37803	−	7.12166i	−8.22272	−	28.8999i	−19.3012	−	11.9508i	16.7460	−	10.3687i	−65.2869	+	12.2042i	20.5023	+	18.6903i
8.9	−1.58841	+	0.296925i	−0.778945	−	8.40616i	−5.02490	+	1.94666i	−4.97706	−	6.59070i	3.73328	+	13.1211i	−3.85349	−	2.38598i	18.3947	−	11.3895i	−43.5164	+	8.13463i	9.86256	+	8.99091i
8.10	−1.58839	+	0.296921i	0.0860448	+	0.928571i	−5.02496	+	1.94668i	−0.528150	−	0.699384i	−0.412385	−	1.44938i	−15.2586	−	9.44770i	18.3945	−	11.3894i	25.6854	−	4.80144i	1.04657	+	0.954074i
8.11	−1.27733	+	0.238775i	0.631381	+	6.81368i	−5.88522	+	2.27994i	10.8721	+	14.3970i	−2.43342	−	8.55257i	8.48745	+	5.25521i	15.8115	−	9.79009i	−19.4873	+	3.64282i	−17.3250	−	15.7938i
8.12	−0.518246	+	0.0968769i	−0.734583	−	7.92741i	−7.20058	+	2.78952i	13.2790	+	17.5842i	1.14868	+	4.03718i	9.13699	+	5.65739i	7.04746	−	4.36361i	−35.7639	+	6.68544i	−8.58529	−	7.82652i
8.13	−0.105256	+	0.0196758i	−0.0266856	−	0.287984i	−7.44909	+	2.88579i	−4.97982	−	6.59434i	0.00847513	+	0.0297870i	16.2287	+	10.0484i	1.45561	−	0.901274i	26.4581	−	4.94587i	0.653905	+	0.596113i
8.14	−0.0194008	+	0.00362664i	0.597946	+	6.45287i	−7.45941	+	2.88979i	−4.13970	−	5.48185i	−0.0350029	−	0.123022i	2.04900	+	1.26869i	0.268484	−	0.166238i	−14.7417	+	2.75570i	0.100194	+	0.0913392i
8.15	1.03541	−	0.193552i	−0.344709	−	3.72000i	−6.42516	+	2.48912i	4.16853	+	5.52002i	−1.07693	−	3.78502i	−25.0781	−	15.5277i	−13.3355	+	8.25701i	12.8207	−	2.39660i	5.38457	+	4.90868i
8.16	1.87073	−	0.349700i	−0.0722607	−	0.779817i	−4.08244	+	1.58155i	4.97182	+	6.58375i	−0.407882	−	1.43356i	18.9114	+	11.7094i	−20.0287	+	12.4012i	25.9374	−	4.84854i	11.6033	+	10.5778i
8.17	2.00633	−	0.375049i	−0.648204	−	6.99524i	−3.57507	+	1.38499i	−9.03723	−	11.9672i	−3.92407	−	13.7917i	21.0819	+	13.0533i	−20.5363	+	12.7155i	−21.9729	+	4.10745i	−22.6200	−	20.6209i
8.18	2.33529	−	0.436541i	0.305713	+	3.29917i	−2.19679	+	0.851040i	−13.2104	−	17.4934i	2.15415	+	7.57104i	−19.8827	−	12.3109i	−20.9177	+	12.9517i	15.7492	−	2.94404i	−38.4867	−	35.0853i
8.19	2.54979	−	0.476639i	0.724436	+	7.81790i	−1.18551	+	0.459269i	2.91587	+	3.86124i	5.57348	+	19.5888i	−5.55780	−	3.44125i	−20.4473	+	12.6605i	−34.0546	+	6.36590i	9.27529	+	8.45554i
8.20	2.90753	−	0.543512i	−0.731831	−	7.89772i	0.698568	−	0.270627i	−1.21903	−	1.61426i	−6.42033	−	22.5651i	−20.8823	−	12.9298i	−18.2348	+	11.2905i	−35.2981	+	6.59836i	−4.42175	−	4.03096i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
103.e	even	17	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	103.4.e.a	✓	400
103.e	even	17	1	inner	103.4.e.a	✓	400

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
103.4.e.a	✓	400	1.a	even	1	1	trivial
103.4.e.a	✓	400	103.e	even	17	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(103, [\chi])\).