Newform orbit 59.9.d.a

• Label: 59.9.d.a
• Level: $59$
• Weight: $9$
• Character orbit: 59.d
• Analytic conductor: $24.035$
• Analytic rank: $0$
• Dimension: $1092$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 59 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	59.d (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(24.0353379936\)
Analytic rank:	\(0\)
Dimension:	\(1092\)
Relative dimension:	\(39\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1092 q - 29 q^{2} - 69 q^{3} + 4837 q^{4} + 267 q^{5} - 29 q^{6} + 133 q^{7} - 29 q^{8} - 67324 q^{9}+O(q^{10}) \)

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	−31.4628	−	1.70587i	16.2275	−	7.50766i	732.501	+	79.6642i	104.481	+	35.2036i	−523.371	+	208.530i	1125.81	+	1660.44i	−14950.6	−	2451.03i	−4040.53	+	4756.88i	−3227.20	−	1285.83i
2.2	−28.4746	−	1.54385i	−119.279	+	55.1845i	553.919	+	60.2424i	−79.9912	−	26.9522i	3481.62	−	1387.21i	−717.790	−	1058.66i	−8475.59	−	1389.50i	6934.71	−	8164.17i	2236.11	+	890.946i
2.3	−27.3512	−	1.48294i	21.5513	−	9.97071i	491.388	+	53.4417i	772.045	+	260.132i	−604.240	+	240.751i	−2199.62	−	3244.20i	−6440.97	−	1055.94i	−3882.46	+	4570.78i	−20730.6	−	8259.81i
2.4	−27.0223	−	1.46511i	145.312	−	67.2287i	473.561	+	51.5029i	494.411	+	166.587i	−4025.18	+	1603.78i	337.680	+	498.041i	−5884.66	−	964.741i	12348.5	−	14537.8i	−13116.1	−	5225.93i
2.5	−26.7619	−	1.45099i	−42.7275	+	19.7678i	459.594	+	49.9838i	−956.724	−	322.358i	1172.15	−	467.027i	−405.976	−	598.770i	−5456.35	−	894.523i	−2812.63	+	3311.29i	25136.0	+	10015.1i
2.6	−26.0354	−	1.41160i	93.0591	−	43.0537i	421.351	+	45.8247i	−815.822	−	274.883i	−2483.61	+	989.559i	−247.819	−	365.505i	−4318.45	−	707.974i	2558.86	−	3012.53i	20852.3	+	8308.30i
2.7	−24.2002	−	1.31209i	−81.2413	+	37.5862i	329.427	+	35.8274i	295.006	+	99.3991i	2015.37	−	802.997i	2546.28	+	3755.48i	−1802.57	−	295.517i	939.923	−	1106.56i	−7008.77	−	2792.55i
2.8	−20.7654	−	1.12587i	54.9805	−	25.4367i	175.437	+	19.0799i	571.044	+	192.407i	−1170.33	+	466.303i	1328.38	+	1959.22i	1632.08	+	267.567i	−1871.67	+	2203.50i	−11641.3	−	4638.34i
2.9	−20.1375	−	1.09182i	−10.6628	+	4.93312i	149.829	+	16.2948i	−163.807	−	55.1929i	220.108	−	87.6990i	−404.556	−	596.675i	2095.38	+	343.520i	−4158.14	+	4895.34i	3238.40	+	1290.30i
2.10	−19.1814	−	1.03998i	−99.0477	+	45.8244i	112.345	+	12.2182i	1083.10	+	364.941i	1947.53	−	775.966i	−409.494	−	603.958i	2710.65	+	444.388i	3463.08	−	4077.05i	−20395.9	−	8126.48i
2.11	−16.8638	−	0.914326i	−83.2870	+	38.5327i	29.0510	+	3.15948i	−404.110	−	136.161i	1439.76	−	573.654i	−1608.46	−	2372.31i	3779.49	+	619.616i	1204.45	−	1417.99i	6690.32	+	2665.67i
2.12	−13.3307	−	0.722770i	79.1048	−	36.5978i	−77.3137	−	8.40837i	−247.112	−	83.2618i	−1080.97	+	430.700i	−2434.78	−	3591.03i	4397.22	+	720.888i	670.666	−	789.570i	3234.00	+	1288.55i
2.13	−13.3015	−	0.721189i	−44.8833	+	20.7652i	−78.0885	−	8.49263i	−1052.56	−	354.650i	611.993	−	243.840i	2248.08	+	3315.67i	4397.84	+	720.990i	−2664.18	+	3136.52i	13744.9	+	5476.49i
2.14	−12.9639	−	0.702883i	86.4021	−	39.9739i	−86.9304	−	9.45425i	−329.792	−	111.120i	−1148.21	+	457.487i	1157.20	+	1706.74i	4400.16	+	721.370i	1619.91	−	1907.11i	4197.29	+	1672.36i
2.15	−8.81682	−	0.478034i	122.780	−	56.8039i	−176.991	−	19.2490i	897.778	+	302.497i	−1109.68	+	442.137i	−1038.67	−	1531.92i	3781.94	+	620.018i	7600.65	−	8948.18i	−7770.95	−	3096.23i
2.16	−7.92647	−	0.429761i	−130.236	+	60.2534i	−191.855	−	20.8655i	−306.377	−	103.230i	1058.20	−	421.627i	164.992	+	243.346i	3517.15	+	576.608i	9083.34	−	10693.7i	2384.12	+	949.921i
2.17	−5.98923	−	0.324727i	−28.1213	+	13.0103i	−218.734	−	23.7887i	382.303	+	128.813i	172.650	−	68.7900i	−485.370	−	715.868i	2817.59	+	461.921i	−3625.96	+	4268.81i	−2247.87	−	895.634i
2.18	−5.62841	−	0.305164i	−17.6461	+	8.16398i	−222.913	−	24.2433i	755.419	+	254.530i	101.811	−	40.5653i	209.830	+	309.476i	2671.23	+	437.926i	−4002.77	+	4712.42i	−4174.14	−	1663.13i
2.19	0.953623	+	0.0517039i	−14.4103	+	6.66692i	−253.593	−	27.5799i	−887.316	−	298.971i	−14.0867	+	5.61266i	−2183.24	−	3220.04i	−481.671	−	78.9660i	−4084.29	+	4808.40i	−830.707	−	330.984i
2.20	1.59040	+	0.0862291i	127.422	−	58.9518i	−251.977	−	27.4042i	−725.333	−	244.393i	207.736	−	82.7695i	14.9458	+	22.0434i	−800.752	−	131.277i	8513.59	−	10023.0i	−1132.50	−	451.228i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
59.d	odd	58	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	59.9.d.a	✓	1092
59.d	odd	58	1	inner	59.9.d.a	✓	1092

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
59.9.d.a	✓	1092	1.a	even	1	1	trivial
59.9.d.a	✓	1092	59.d	odd	58	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(59, [\chi])\).