Newform orbit 103.4.g.a

• Label: 103.4.g.a
• Level: $103$
• Weight: $4$
• Character orbit: 103.g
• Analytic conductor: $6.077$
• Analytic rank: $0$
• Dimension: $800$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 103 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	103.g (of order \(51\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.07719673059\)
Analytic rank:	\(0\)
Dimension:	\(800\)
Relative dimension:	\(25\) over \(\Q(\zeta_{51})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{51}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 800 q - 33 q^{2} - 38 q^{3} + 63 q^{4} - 31 q^{5} - 17 q^{6} - 47 q^{7} - 4 q^{8} - 568 q^{9}+O(q^{10}) \)

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	−5.50751	+	0.339692i	−4.32236	+	2.67629i	22.2779	−	2.75861i	−1.34874	−	6.15851i	22.8963	−	16.2080i	−20.1236	+	23.4883i	−78.3667	+	14.6493i	−0.514699	+	1.03365i	9.52017	+	33.4599i
2.2	−5.06447	+	0.312366i	5.81905	−	3.60300i	17.6119	−	2.18082i	3.99380	+	18.2363i	−28.3449	+	20.0650i	−5.45098	+	6.36241i	−48.6121	+	9.08717i	8.84477	−	17.7627i	−25.9229	−	91.1094i
2.3	−4.77858	+	0.294733i	3.08268	−	1.90871i	14.8086	−	1.83370i	−2.24566	−	10.2540i	−14.1683	+	10.0295i	13.3431	−	15.5741i	−32.5744	+	6.08921i	−6.17522	+	12.4015i	13.7532	+	48.3376i
2.4	−4.66618	+	0.287800i	−6.36073	+	3.93839i	13.7510	−	1.70274i	3.71359	+	16.9568i	28.5468	−	20.2079i	18.7426	−	21.8764i	−26.9111	+	5.03055i	12.9129	−	25.9327i	−22.2084	−	78.0545i
2.5	−3.68696	+	0.227404i	2.97692	−	1.84323i	5.60260	−	0.693752i	0.524885	+	2.39670i	−10.5566	+	7.47289i	−4.50066	+	5.25319i	8.54976	−	1.59823i	−6.57037	+	13.1951i	−2.48025	−	8.71717i
2.6	−3.58408	+	0.221059i	−5.29131	+	3.27624i	4.85742	−	0.601479i	−2.82312	−	12.8908i	18.2403	−	12.9120i	9.21626	−	10.7573i	10.9616	−	2.04907i	5.22927	−	10.5018i	12.9679	+	45.5776i
2.7	−3.43672	+	0.211970i	−2.96631	+	1.83666i	3.82676	−	0.473855i	1.68608	+	7.69888i	9.80506	−	6.94086i	−12.4402	+	14.5202i	14.0259	−	2.62190i	−6.60927	+	13.2732i	−7.42652	−	26.1015i
2.8	−2.20249	+	0.135845i	8.00530	−	4.95667i	−3.10687	+	0.384714i	−0.154169	−	0.703956i	−16.9582	+	12.0045i	5.47099	−	6.38577i	24.1434	−	4.51318i	27.4813	−	55.1899i	0.435183	+	1.52951i
2.9	−2.10263	+	0.129685i	3.38753	−	2.09747i	−3.53515	+	0.437746i	−4.32775	−	19.7611i	−6.85070	+	4.84951i	−19.9338	+	23.2669i	23.9423	−	4.47560i	−4.95895	+	9.95892i	11.6624	+	40.9890i
2.10	−1.66870	+	0.102922i	−1.24686	+	0.772023i	−5.16541	+	0.639616i	3.23171	+	14.7565i	2.00117	−	1.41660i	10.2023	−	11.9082i	21.7009	−	4.05660i	−11.0763	+	22.2442i	−6.91151	−	24.2914i
2.11	−0.839910	+	0.0518039i	−5.01890	+	3.10757i	−7.23660	+	0.896085i	−1.94800	−	8.89486i	4.05444	−	2.87008i	−0.231799	+	0.270557i	12.6491	−	2.36453i	3.49744	−	7.02381i	2.09694	+	7.36997i
2.12	−0.639196	+	0.0394243i	−8.46449	+	5.24099i	−7.53235	+	0.932707i	2.99479	+	13.6746i	5.20384	−	3.68373i	−16.1095	+	18.8031i	9.81393	−	1.83454i	32.1446	−	64.5551i	−2.45337	−	8.62269i
2.13	−0.636712	+	0.0392711i	1.97279	−	1.22150i	−7.53550	+	0.933098i	1.58524	+	7.23843i	−1.20813	+	0.855216i	15.9633	−	18.6325i	9.77778	−	1.82778i	−9.63511	+	19.3499i	−1.29360	−	4.54654i
2.14	0.277419	−	0.0171106i	4.87496	−	3.01845i	−7.86270	+	0.973613i	3.31155	+	15.1210i	1.30076	−	0.920787i	−21.5429	+	25.1450i	−4.35031	+	0.813213i	2.61927	−	5.26020i	1.17742	+	4.13819i
2.15	0.910883	−	0.0561814i	1.95391	−	1.20981i	−7.11281	+	0.880757i	−2.79585	−	12.7662i	1.71182	−	1.21177i	1.49014	−	1.73930i	−13.6061	+	2.54341i	−9.68081	+	19.4417i	−3.26392	−	11.4715i
2.16	1.96756	−	0.121355i	6.60500	−	4.08964i	−4.08279	+	0.505559i	−2.09586	−	9.56999i	12.4994	−	8.84817i	8.23220	−	9.60866i	−23.4737	+	4.38799i	14.8659	−	29.8548i	−5.28510	−	18.5752i
2.17	2.04121	−	0.125897i	−1.63581	+	1.01285i	−3.78868	+	0.469140i	−0.102587	−	0.468426i	−3.21151	+	2.27338i	−7.81284	+	9.11919i	−23.7566	+	4.44087i	−10.3849	+	20.8558i	−0.268375	−	0.943240i
2.18	2.25005	−	0.138778i	−7.37677	+	4.56750i	−2.89591	+	0.358592i	0.0576849	+	0.263397i	−15.9642	+	11.3008i	22.7831	−	26.5925i	−24.1937	+	4.52258i	21.5197	−	43.2175i	0.166348	+	0.584651i
2.19	3.24308	−	0.200026i	−3.20661	+	1.98545i	2.53821	−	0.314298i	2.69868	+	12.3225i	−10.0022	+	7.08038i	−4.00601	+	4.67583i	−17.3826	+	3.24937i	−5.69460	+	11.4363i	11.2169	+	39.4232i
2.20	3.53821	−	0.218230i	5.23025	−	3.23844i	4.53196	−	0.561179i	3.90841	+	17.8464i	17.7990	−	12.5997i	15.8207	−	18.4660i	−11.9640	+	2.23647i	4.83315	−	9.70627i	17.7234	+	62.2913i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
103.g	even	51	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	103.4.g.a	✓	800
103.g	even	51	1	inner	103.4.g.a	✓	800

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
103.4.g.a	✓	800	1.a	even	1	1	trivial
103.4.g.a	✓	800	103.g	even	51	1	inner

Hecke kernels

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