Newform orbit 59.12.c.a

• Label: 59.12.c.a
• Level: $59$
• Weight: $12$
• Character orbit: 59.c
• Analytic conductor: $45.332$
• Analytic rank: $0$
• Dimension: $1512$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 59 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	59.c (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(45.3322476530\)
Analytic rank:	\(0\)
Dimension:	\(1512\)
Relative dimension:	\(54\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 1512 q - 59 q^{2} - 47 q^{3} - 54299 q^{4} + 1931 q^{5} + 31479 q^{6} + 31481 q^{7} - 249215 q^{8} - 2725807 q^{9}+O(q^{10}) \)

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	−79.6699	+	36.8592i	312.294	+	105.224i	3662.85	−	4312.24i	−10588.4	−	6370.82i	−28758.9	+	3127.72i	1530.71	−	28232.3i	−84776.7	+	305338.i	−54569.9	−	41483.0i	1.07840e6	+	117283.i
3.2	−77.4433	+	35.8291i	−584.958	−	197.095i	3387.89	−	3988.53i	2463.15	+	1482.03i	52362.8	−	5694.80i	4212.50	−	77695.0i	−72711.8	+	261884.i	162304.	+	123380.i	−243855.	−	26520.8i
3.3	−76.9065	+	35.5807i	133.930	+	45.1264i	3322.77	−	3911.87i	3777.42	+	2272.80i	−11905.7	+	1294.83i	−2692.92	+	49668.1i	−69927.5	+	251856.i	−125125.	−	95117.2i	−371375.	−	40389.5i
3.4	−74.3526	+	34.3992i	−597.870	−	201.446i	3019.16	−	3554.42i	−3119.11	−	1876.70i	51382.7	−	5588.21i	−3763.33	+	69410.5i	−57326.5	+	206471.i	175842.	+	133672.i	296471.	+	32243.1i
3.5	−71.8315	+	33.2328i	767.206	+	258.502i	2729.50	−	3213.42i	1394.52	+	839.056i	−63700.3	+	6927.83i	−3398.21	+	62676.3i	−45909.1	+	165350.i	380757.	+	289444.i	−128055.	−	13926.8i
3.6	−70.0430	+	32.4053i	481.678	+	162.296i	2530.06	−	2978.62i	6908.95	+	4156.98i	−38997.4	+	4241.23i	3800.62	−	70098.4i	−38405.4	+	138324.i	64648.5	+	49144.5i	−618631.	−	67280.2i
3.7	−69.4303	+	32.1219i	−297.456	−	100.225i	2462.90	−	2899.55i	9091.24	+	5470.02i	23871.8	−	2596.22i	333.244	−	6146.33i	−35946.2	+	129467.i	−62590.5	−	47580.1i	−806914.	−	87757.2i
3.8	−64.4556	+	29.8203i	28.5568	+	9.62189i	1939.42	−	2283.27i	−3230.49	−	1943.72i	−2127.57	+	231.387i	199.068	−	3671.59i	−18007.5	+	64857.0i	−140303.	−	106655.i	266185.	+	28949.4i
3.9	−60.6584	+	28.0636i	−437.226	−	147.319i	1566.03	−	1843.68i	−6792.02	−	4086.62i	30655.7	−	3334.01i	1283.87	−	23679.6i	−6633.81	+	23892.8i	28438.5	+	21618.4i	526679.	+	57279.8i
3.10	−53.4002	+	24.7055i	−275.546	−	92.8423i	915.367	−	1077.65i	9558.58	+	5751.20i	17007.9	−	1849.72i	−1595.15	+	29420.8i	9980.64	−	35947.0i	−73719.5	−	56040.1i	−652516.	−	70965.4i
3.11	−50.8737	+	23.5367i	699.902	+	235.825i	708.309	−	833.886i	−7652.90	−	4604.60i	−41157.1	+	4476.11i	3490.54	−	64379.2i	14304.8	−	51521.1i	293224.	+	222903.i	497708.	+	54129.0i
3.12	−50.3679	+	23.3027i	511.313	+	172.281i	668.062	−	786.503i	−5918.06	−	3560.78i	−29768.3	+	3237.50i	−1346.13	+	24827.8i	15085.6	−	54333.2i	90734.2	+	68974.4i	381056.	+	41442.3i
3.13	−49.4352	+	22.8712i	423.564	+	142.715i	594.903	−	700.374i	4861.02	+	2924.78i	−24203.1	+	2632.24i	−1186.57	+	21884.9i	16453.0	−	59258.2i	18013.5	+	13693.5i	−307199.	−	33409.9i
3.14	−44.5799	+	20.6248i	−170.583	−	57.4761i	236.134	−	277.998i	−8564.54	−	5153.11i	8790.01	−	955.971i	−4663.27	+	86009.0i	22119.5	−	79667.1i	−115230.	−	87595.9i	488088.	+	53082.8i
3.15	−43.5451	+	20.1461i	−747.042	−	251.708i	164.462	−	193.620i	2075.50	+	1248.79i	37600.9	−	4089.35i	−1192.81	+	22000.0i	23027.1	−	82936.1i	353689.	+	268868.i	−115536.	−	12565.3i
3.16	−40.7761	+	18.8650i	−542.056	−	182.640i	−19.0457	+	22.4223i	6646.83	+	3999.26i	25548.5	−	2778.56i	1066.47	−	19669.9i	24969.9	−	89933.5i	119442.	+	90797.5i	−346478.	−	37681.8i
3.17	−40.3575	+	18.6714i	−57.2392	−	19.2861i	−45.7402	+	53.8496i	93.9625	+	56.5353i	2670.13	−	290.394i	3873.53	−	71443.1i	25204.1	−	90777.0i	−138121.	−	104997.i	−4847.68	−	527.217i
3.18	−29.0805	+	13.4541i	−564.654	−	190.254i	−661.182	+	778.404i	−9616.97	−	5786.34i	18980.1	−	2064.21i	1713.93	−	31611.5i	26310.6	−	94762.1i	141612.	+	107651.i	357516.	+	38882.3i
3.19	−26.4849	+	12.2532i	499.765	+	168.391i	−774.539	+	911.858i	9441.20	+	5680.58i	−15299.6	+	1663.93i	633.827	−	11690.3i	25329.2	−	91227.4i	80384.5	+	61106.7i	−319655.	−	34764.5i
3.20	−24.5606	+	11.3630i	−44.6190	−	15.0339i	−851.740	+	1002.75i	3759.64	+	2262.10i	1266.70	−	137.762i	−3796.03	+	70013.7i	24352.2	−	87708.7i	−139261.	−	105863.i	−118043.	−	12838.0i

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	59.12.c.a	✓	1512
59.c	even	29	1	inner	59.12.c.a	✓	1512

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
59.12.c.a	✓	1512	1.a	even	1	1	trivial
59.12.c.a	✓	1512	59.c	even	29	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(59, [\chi])\).