Newform orbit 61.4.g.a

• Label: 61.4.g.a
• Level: $61$
• Weight: $4$
• Character orbit: 61.g
• Analytic conductor: $3.599$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	61.g (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.59911651035\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 5 q^{2} - 3 q^{3} + 33 q^{4} - 26 q^{5} - 35 q^{6} + 45 q^{7} - 5 q^{8} - 51 q^{9}+O(q^{10}) \)

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	−4.85236	−	1.57663i	0.307214	−	0.945507i	14.5875	+	10.5984i	−13.0281	−	9.46549i	−2.98142	+	4.10358i	−15.3133	+	4.97559i	−30.0826	−	41.4051i	21.0439	+	15.2893i	48.2936	+	66.4704i
3.2	−4.34900	−	1.41308i	1.67142	−	5.14411i	10.4449	+	7.58865i	13.1686	+	9.56756i	−14.5380	+	20.0099i	9.26579	−	3.01064i	−13.1988	−	18.1666i	−1.82477	−	1.32577i	−43.7506	−	60.2176i
3.3	−4.17464	−	1.35642i	−2.48370	+	7.64403i	9.11560	+	6.62287i	5.24583	+	3.81132i	20.7371	−	28.5421i	−13.1023	+	4.25719i	−8.43039	−	11.6034i	−30.4190	−	22.1007i	−16.7297	−	23.0264i
3.4	−2.64844	−	0.860532i	2.11274	−	6.50235i	−0.198391	−	0.144139i	−7.27194	−	5.28337i	−11.1910	+	15.4030i	0.669168	−	0.217426i	13.4960	+	18.5757i	−15.9734	−	11.6054i	14.7128	+	20.2505i
3.5	−2.53128	−	0.822462i	−1.35525	+	4.17102i	−0.741211	−	0.538521i	−6.44412	−	4.68193i	6.86102	−	9.44338i	24.1379	−	7.84287i	13.9486	+	19.1986i	6.28273	+	4.56467i	12.4612	+	17.1513i
3.6	−1.55797	−	0.506216i	−0.451246	+	1.38879i	−4.30111	−	3.12494i	9.62864	+	6.99562i	1.40606	−	1.93527i	−11.5797	+	3.76246i	12.8222	+	17.6482i	20.1183	+	14.6168i	−11.4599	−	15.7731i
3.7	0.552802	+	0.179616i	−0.190651	+	0.586762i	−6.19881	−	4.50370i	−8.64055	−	6.27772i	−0.210784	+	0.290119i	−26.9049	+	8.74194i	−5.35098	−	7.36499i	21.5355	+	15.6465i	−3.64893	−	5.02232i
3.8	0.566527	+	0.184076i	2.88056	−	8.86545i	−6.18507	−	4.49371i	8.02286	+	5.82895i	3.26383	−	4.49228i	−12.1266	+	3.94017i	−5.47789	−	7.53967i	−48.4551	−	35.2047i	3.47220	+	4.77907i
3.9	0.913598	+	0.296846i	−2.89680	+	8.91544i	−5.72559	−	4.15989i	−6.65918	−	4.83817i	−5.29303	+	7.28522i	4.36223	−	1.41737i	−8.51312	−	11.7173i	−49.2501	−	35.7823i	−4.64762	−	6.39690i
3.10	1.33493	+	0.433746i	0.367985	−	1.13254i	−4.87823	−	3.54424i	10.4853	+	7.61798i	0.982470	−	1.35225i	33.6990	−	10.9495i	−11.5751	−	15.9317i	20.6962	+	15.0367i	10.6928	+	14.7174i
3.11	2.04842	+	0.665573i	1.48023	−	4.55567i	−2.71909	−	1.97553i	−14.4070	−	10.4673i	6.06426	−	8.34673i	20.3161	−	6.60110i	−14.3829	−	19.7964i	3.28042	+	2.38336i	−22.5448	−	31.0303i
3.12	3.49287	+	1.13490i	−1.60499	+	4.93965i	4.44000	+	3.22585i	7.56571	+	5.49681i	−11.2120	+	15.4320i	−5.43549	+	1.76610i	−5.42237	−	7.46325i	0.0193460	+	0.0140557i	20.1877	+	27.7860i
3.13	4.25579	+	1.38279i	1.64291	−	5.05635i	9.72750	+	7.06744i	1.31028	+	0.951974i	13.9837	−	19.2470i	−9.52309	+	3.09424i	10.5836	+	14.5671i	−1.02407	−	0.744033i	4.25989	+	5.86324i
3.14	5.13973	+	1.67000i	−1.67141	+	5.14407i	17.1558	+	12.4644i	−15.5386	−	11.2895i	−17.1812	+	23.6479i	17.8163	−	5.78888i	41.9485	+	57.7371i	−1.82437	−	1.32548i	−61.0109	−	83.9743i
27.1	−3.02843	−	4.16827i	6.51237	−	4.73151i	−5.73100	+	17.6382i	5.04273	−	15.5199i	−39.4445	−	12.8163i	6.75688	−	9.30004i	51.6759	−	16.7905i	11.6803	−	35.9482i	−79.9628	+	25.9815i
27.2	−2.73035	−	3.75801i	−6.37323	+	4.63043i	−4.19566	+	12.9129i	−0.339836	+	1.04591i	34.8024	+	11.3080i	12.2949	−	16.9225i	24.6401	−	8.00605i	10.8338	−	33.3430i	4.85841	−	1.57859i
27.3	−2.63724	−	3.62986i	1.01836	−	0.739880i	−3.74866	+	11.5372i	−3.38601	+	10.4211i	−5.37132	−	1.74525i	−9.13545	+	12.5739i	17.6272	−	5.72744i	−7.85383	+	24.1716i	46.7567	−	15.1922i
27.4	−1.90219	−	2.61814i	−3.43440	+	2.49524i	−0.764204	+	2.35198i	6.57191	−	20.2263i	13.0658	+	4.24533i	−17.2829	+	23.7878i	−17.0110	+	5.52722i	−2.77456	+	8.53922i	−65.4563	+	21.2680i
27.5	−1.54664	−	2.12876i	2.62276	−	1.90555i	0.332594	−	1.02362i	−0.143809	+	0.442599i	−8.11292	−	2.63605i	16.6148	−	22.8683i	−22.7135	+	7.38007i	−5.09569	+	15.6829i	1.16461	−	0.378404i
27.6	−0.582091	−	0.801180i	−3.31811	+	2.41075i	2.16908	−	6.67573i	−3.99493	+	12.2951i	3.86288	+	1.25513i	−4.30231	+	5.92162i	−14.1458	+	4.59625i	−3.14532	+	9.68030i	12.1760	−	3.95623i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
61.g	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	61.4.g.a	✓	56
61.g	even	10	1	inner	61.4.g.a	✓	56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
61.4.g.a	✓	56	1.a	even	1	1	trivial
61.4.g.a	✓	56	61.g	even	10	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(61, [\chi])\).