Newform orbit 59.6.c.a

• Label: 59.6.c.a
• Level: $59$
• Weight: $6$
• Character orbit: 59.c
• Analytic conductor: $9.463$
• Analytic rank: $0$
• Dimension: $672$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 59 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	59.c (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.46264536897\)
Analytic rank:	\(0\)
Dimension:	\(672\)
Relative dimension:	\(24\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 672 q - 23 q^{2} - 47 q^{3} - 395 q^{4} - 85 q^{5} + 231 q^{6} - 103 q^{7} + 49 q^{8} - 1813 q^{9}+O(q^{10}) \)

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	−10.0990	+	4.67231i	−25.6776	−	8.65179i	59.4435	−	69.9823i	−19.3486	−	11.6417i	299.742	−	32.5990i	1.93986	−	35.7786i	−178.081	+	641.391i	391.035	+	297.257i	249.795	+	27.1669i
3.2	−9.89968	+	4.58008i	20.1243	+	6.78066i	56.3102	−	66.2935i	−49.3750	−	29.7080i	−230.280	+	25.0445i	6.38877	−	117.834i	−160.443	+	577.862i	165.559	+	125.854i	624.862	+	67.9578i
3.3	−8.40778	+	3.88985i	−2.65773	−	0.895493i	34.8434	−	41.0208i	12.4089	+	7.46616i	25.8289	−	2.80906i	−8.43786	+	155.627i	−54.0825	+	194.787i	−187.189	−	142.297i	−133.373	−	14.5052i
3.4	−8.17028	+	3.77997i	8.11591	+	2.73457i	31.7489	−	37.3777i	59.0864	+	35.5511i	−76.6458	+	8.33573i	4.35995	−	80.4146i	−41.0427	+	147.822i	−135.061	−	102.670i	−617.135	−	67.1174i
3.5	−6.66051	+	3.08148i	−6.64392	−	2.23860i	14.1505	−	16.6593i	−82.7043	−	49.7615i	51.1501	−	5.56291i	−4.40242	+	81.1979i	19.9123	−	71.7178i	−154.320	−	117.311i	704.192	+	76.5855i
3.6	−6.15629	+	2.84820i	25.3215	+	8.53182i	9.07125	−	10.6795i	12.2890	+	7.39405i	−180.187	+	19.5965i	−10.8383	+	199.901i	32.6427	−	117.568i	374.938	+	285.020i	−96.7143	−	10.5183i
3.7	−6.05797	+	2.80272i	−15.3761	−	5.18083i	8.12747	−	9.56839i	9.99078	+	6.01125i	107.669	−	11.7097i	9.94775	−	183.476i	34.7247	−	125.067i	16.1343	+	12.2650i	−77.3717	−	8.41468i
3.8	−5.00278	+	2.31453i	−25.2098	−	8.49417i	−1.04564	+	1.23102i	76.3732	+	45.9522i	145.779	−	15.8544i	−9.79726	+	180.700i	49.5717	−	178.541i	369.932	+	281.215i	−488.436	−	53.1206i
3.9	−3.86220	+	1.78684i	16.1955	+	5.45688i	−8.99260	+	10.5869i	−41.1262	−	24.7448i	−72.3006	+	7.86317i	7.16043	−	132.066i	52.2451	−	188.170i	39.0645	+	29.6961i	203.053	+	22.0833i
3.10	−1.95099	+	0.902625i	9.94503	+	3.35087i	−17.7247	+	20.8672i	89.4181	+	53.8011i	−22.4273	+	2.43911i	1.11564	−	20.5768i	34.1487	−	122.993i	−105.775	−	80.4083i	−223.016	−	24.2545i
3.11	−1.12460	+	0.520296i	−13.5601	−	4.56894i	−19.7223	+	23.2189i	−7.05706	−	4.24609i	17.6269	−	1.91705i	−5.62022	+	103.659i	20.7071	−	74.5802i	−30.4486	−	23.1464i	10.1456	+	1.10340i
3.12	−1.04498	+	0.483460i	4.44060	+	1.49621i	−19.8581	+	23.3788i	2.65457	+	1.59720i	−5.36370	+	0.583338i	−2.76755	+	51.0444i	19.3057	−	69.5328i	−175.970	−	133.769i	−3.54616	−	0.385668i
3.13	0.00931904	−	0.00431145i	−26.2182	−	8.83395i	−20.7163	+	24.3891i	−79.2906	−	47.7076i	−0.282416	+	0.0307146i	5.21212	−	96.1320i	−0.175808	+	0.633202i	415.906	+	316.163i	−0.944602	−	0.102732i
3.14	2.66154	−	1.23136i	−17.2843	−	5.82375i	−15.1488	+	17.8346i	71.8572	+	43.2350i	−53.1739	+	5.78301i	12.1010	−	223.190i	−43.4640	+	156.543i	71.3794	+	54.2612i	244.489	+	26.5897i
3.15	2.68028	−	1.24003i	17.1665	+	5.78405i	−15.0701	+	17.7419i	−91.0064	−	54.7567i	53.1833	−	5.78403i	−12.1210	+	223.559i	−43.6740	+	157.299i	67.7815	+	51.5262i	−311.823	−	33.9127i
3.16	2.68125	−	1.24048i	26.3857	+	8.89037i	−15.0661	+	17.7371i	23.1250	+	13.9139i	81.7748	−	8.89355i	3.04713	−	56.2010i	−43.6848	+	157.338i	423.714	+	322.099i	79.2639	+	8.62046i
3.17	4.03419	−	1.86641i	−5.06787	−	1.70756i	−7.92521	+	9.33027i	0.304780	+	0.183380i	−23.6317	+	2.57011i	−0.757280	+	13.9672i	−52.6110	+	189.488i	−170.683	−	129.750i	1.57180	+	0.170944i
3.18	4.30949	−	1.99378i	4.72737	+	1.59284i	−6.11981	+	7.20480i	−53.3270	−	32.0858i	23.5484	−	2.56104i	11.0355	−	203.538i	−52.6587	+	189.660i	−173.640	−	131.997i	−293.785	−	31.9510i
3.19	6.03200	−	2.79070i	6.66012	+	2.24406i	7.88063	−	9.27780i	66.6257	+	40.0874i	46.4363	−	5.05025i	−13.0439	+	240.581i	−35.2537	+	126.973i	−154.129	−	117.166i	513.758	+	55.8745i
3.20	6.27762	−	2.90434i	−24.6886	−	8.31855i	10.2570	−	12.0754i	10.0776	+	6.06348i	−179.145	+	19.4832i	−5.82744	+	107.481i	−29.8968	+	107.679i	346.877	+	263.689i	80.8735	+	8.79553i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
59.c	even	29	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	59.6.c.a	✓	672
59.c	even	29	1	inner	59.6.c.a	✓	672

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
59.6.c.a	✓	672	1.a	even	1	1	trivial
59.6.c.a	✓	672	59.c	even	29	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(59, [\chi])\).