Newform orbit 59.10.c.a

• Label: 59.10.c.a
• Level: $59$
• Weight: $10$
• Character orbit: 59.c
• Analytic conductor: $30.387$
• Analytic rank: $0$
• Dimension: $1232$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 59 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	59.c (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(30.3871143337\)
Analytic rank:	\(0\)
Dimension:	\(1232\)
Relative dimension:	\(44\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 1232 q - 11 q^{2} - 29 q^{3} - 11035 q^{4} + 1565 q^{5} - 1641 q^{6} - 9783 q^{7} + 21649 q^{8} - 337327 q^{9}+O(q^{10}) \)

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	−40.2360	+	18.6151i	−42.1892	−	14.2152i	940.948	−	1107.77i	865.109	+	520.519i	1962.14	−	213.396i	82.8680	−	1528.41i	−11166.1	+	40216.7i	−14091.6	−	10712.2i	−44498.0	−	4839.45i
3.2	−37.4449	+	17.3238i	214.561	+	72.2940i	770.540	−	907.150i	−67.0883	−	40.3657i	−9286.61	+	1009.98i	185.438	−	3420.20i	−7486.14	+	26962.7i	25140.5	+	19111.3i	3211.40	+	349.261i
3.3	−37.1520	+	17.1884i	−160.393	−	54.0425i	753.372	−	886.938i	−2225.07	−	1338.78i	6887.81	−	749.095i	82.6817	−	1524.97i	−7137.17	+	25705.8i	7135.67	+	5424.39i	105677.	+	11493.1i
3.4	−33.8225	+	15.6479i	141.218	+	47.5819i	567.640	−	668.277i	−1778.36	−	1070.00i	−5520.91	+	600.435i	−533.072	+	9831.94i	−3637.21	+	13100.1i	2009.02	+	1527.22i	76891.9	+	8362.50i
3.5	−33.6442	+	15.5655i	−201.441	−	67.8733i	558.189	−	657.151i	994.111	+	598.137i	7833.80	−	851.977i	−455.863	+	8407.90i	−3473.27	+	12509.6i	20302.1	+	15433.2i	−42756.4	−	4650.04i
3.6	−32.8427	+	15.1947i	105.748	+	35.6307i	516.306	−	607.843i	6.97146	+	4.19459i	−4014.46	+	436.599i	217.428	−	4010.23i	−2764.20	+	9955.75i	−5756.37	−	4375.88i	−292.697	−	31.8328i
3.7	−31.3400	+	14.4994i	−45.9696	−	15.4890i	440.501	−	518.598i	−407.002	−	244.885i	1665.27	−	181.109i	107.008	−	1973.65i	−1555.99	+	5604.16i	−13796.2	−	10487.6i	16306.1	+	1773.40i
3.8	−30.8889	+	14.2907i	117.598	+	39.6233i	418.439	−	492.624i	2145.95	+	1291.17i	−4198.71	+	456.637i	−605.068	+	11159.8i	−1223.29	+	4405.90i	−3410.30	−	2592.44i	−84737.9	−	9215.80i
3.9	−29.0385	+	13.4346i	−245.209	−	82.6206i	331.283	−	390.016i	481.264	+	289.567i	8230.48	−	895.119i	536.276	−	9891.04i	2.36578	−	8.52077i	37631.9	+	28607.0i	−17865.4	−	1942.98i
3.10	−25.6759	+	11.8789i	−19.7445	−	6.65270i	186.680	−	219.777i	2170.86	+	1306.16i	585.985	−	63.7297i	556.178	−	10258.1i	1692.63	−	6096.31i	−15323.9	−	11648.9i	−71254.4	−	7749.39i
3.11	−24.1670	+	11.1808i	−49.4966	−	16.6773i	127.569	−	150.186i	−976.903	−	587.783i	1382.65	−	150.372i	−436.189	+	8045.04i	2243.61	−	8080.75i	−13497.7	−	10260.7i	30180.7	+	3282.35i
3.12	−22.1891	+	10.2658i	115.769	+	39.0072i	55.5086	−	65.3497i	−2094.41	−	1260.16i	−2969.26	+	322.926i	628.336	−	11589.0i	2788.04	−	10041.6i	−3788.54	−	2879.97i	59409.7	+	6461.19i
3.13	−20.2370	+	9.36266i	243.753	+	82.1298i	−9.58296	+	11.2819i	989.062	+	595.099i	−5701.79	+	620.107i	190.778	−	3518.69i	3142.55	−	11318.4i	37000.5	+	28127.1i	−25587.4	−	2782.80i
3.14	−19.5490	+	9.04431i	−103.066	−	34.7270i	−31.0994	+	36.6131i	234.788	+	141.267i	2328.92	−	253.285i	−164.728	+	3038.23i	3227.22	−	11623.4i	−6252.86	−	4753.30i	−5867.52	−	638.132i
3.15	−18.2595	+	8.44773i	139.944	+	47.1525i	−69.4180	+	81.7251i	427.709	+	257.344i	−2953.63	+	321.226i	−102.755	+	1895.20i	3332.93	−	12004.1i	1691.38	+	1285.75i	−9983.70	−	1085.79i
3.16	−16.5991	+	7.67954i	−170.636	−	57.4939i	−114.909	+	135.281i	−1256.32	−	755.905i	3273.92	−	356.060i	352.293	−	6497.66i	3373.67	−	12150.8i	10141.6	+	7709.41i	26658.8	+	2899.32i
3.17	−10.6601	+	4.93189i	43.5929	+	14.6881i	−242.148	+	285.078i	−1115.63	−	671.255i	−537.145	+	58.4180i	−19.5763	+	361.063i	2784.20	−	10027.8i	−13984.9	−	10631.0i	15203.3	+	1653.46i
3.18	−9.89897	+	4.57975i	219.256	+	73.8760i	−254.446	+	299.557i	−1244.27	−	748.651i	−2508.74	+	272.842i	−408.185	+	7528.53i	2640.85	−	9511.47i	26946.1	+	20483.9i	15745.6	+	1712.44i
3.19	−9.76281	+	4.51676i	−247.229	−	83.3010i	−256.550	+	302.034i	−1832.96	−	1102.85i	2789.90	−	303.419i	−562.541	+	10375.5i	2613.87	−	9414.33i	38513.4	+	29277.1i	22876.2	+	2487.93i
3.20	−7.88983	+	3.65022i	−176.095	−	59.3333i	−282.537	+	332.628i	1697.90	+	1021.59i	1605.94	−	174.656i	−111.501	+	2056.51i	2205.76	−	7944.43i	11819.5	+	8984.95i	−17125.2	−	1862.48i

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	59.10.c.a	✓	1232
59.c	even	29	1	inner	59.10.c.a	✓	1232

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
59.10.c.a	✓	1232	1.a	even	1	1	trivial
59.10.c.a	✓	1232	59.c	even	29	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(59, [\chi])\).