Newform orbit 19.13.f.a

• Label: 19.13.f.a
• Level: $19$
• Weight: $13$
• Character orbit: 19.f
• Analytic conductor: $17.366$
• Analytic rank: $0$
• Dimension: $114$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 19 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	19.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.3658825282\)
Analytic rank:	\(0\)
Dimension:	\(114\)
Relative dimension:	\(19\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 114 q - 6 q^{2} - 1386 q^{3} - 1392 q^{4} - 6 q^{5} + 56016 q^{6} - 131763 q^{7} - 9 q^{8} - 1362066 q^{9}+O(q^{10}) \)

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	−125.219	−	22.0794i	−639.516	+	762.146i	11343.2	+	4128.59i	152.064	−	55.3466i	96907.1	−	81314.7i	63066.5	+	109234.i	−878193.	−	507025.i	−79601.5	−	451443.i	−20263.2	+	3572.95i
2.2	−106.226	−	18.7305i	615.064	−	733.004i	7084.16	+	2578.43i	−6666.14	+	2426.28i	−79065.3	+	66343.7i	−8377.10	−	14509.6i	−321605.	−	185679.i	−66708.2	−	378321.i	753563.	−	132873.i
2.3	−92.8242	−	16.3674i	49.1377	−	58.5601i	4499.46	+	1637.67i	25909.3	−	9430.22i	−5519.65	+	4631.53i	−31927.9	−	55300.7i	−56505.5	−	32623.5i	91269.0	+	517612.i	−2.55936e6	+	451284.i
2.4	−90.2919	−	15.9209i	−39.4517	+	47.0168i	4050.18	+	1474.14i	−18256.2	+	6644.71i	4310.72	−	3617.13i	−31029.5	−	53744.6i	−17000.8	−	9815.39i	91629.6	+	519657.i	1.75418e6	−	309309.i
2.5	−68.5242	−	12.0827i	−414.801	+	494.341i	700.596	+	254.996i	−3634.58	+	1322.88i	34396.9	−	28862.4i	47041.8	+	81478.7i	201895.	+	116564.i	19970.9	+	113261.i	265041.	−	46733.9i
2.6	−61.5131	−	10.8464i	−868.138	+	1034.61i	−182.766	−	66.5213i	−2035.56	+	740.884i	64623.6	−	54225.6i	−102093.	−	176830.i	232089.	+	133997.i	−224463.	−	1.27299e6i	133250.	−	23495.5i
2.7	−48.1411	−	8.48858i	678.709	−	808.854i	−1603.47	−	583.615i	2260.55	−	822.772i	−39539.8	+	33177.9i	103172.	+	178699.i	245641.	+	141821.i	−101315.	−	574586.i	−115809.	+	20420.3i
2.8	−19.5268	−	3.44310i	−353.750	+	421.583i	−3479.54	−	1266.45i	15118.6	−	5502.73i	8359.14	−	7014.15i	55900.2	+	96821.9i	133918.	+	77317.8i	39690.8	+	225098.i	−314164.	+	55395.7i
2.9	−19.0357	−	3.35651i	628.115	−	748.559i	−3497.89	−	1273.13i	11499.3	−	4185.39i	−14469.2	+	12141.1i	−96665.3	−	167429.i	130877.	+	75562.1i	−73527.4	−	416995.i	−232945.	+	41074.5i
2.10	−15.5048	−	2.73392i	119.265	−	142.134i	−3616.06	−	1316.14i	−20464.5	+	7448.45i	−2237.76	+	1877.70i	−26822.1	−	46457.3i	108316.	+	62536.2i	86305.7	+	489464.i	337661.	−	59538.7i
2.11	26.3501	+	4.64623i	−520.087	+	619.815i	−3176.24	−	1156.06i	12842.5	−	4674.29i	−16584.1	+	13915.7i	−49981.3	−	86570.1i	−173235.	−	100017.i	−21396.9	−	121348.i	360119.	−	63498.7i
2.12	33.2458	+	5.86212i	−816.645	+	973.239i	−2778.07	−	1011.13i	−24550.7	+	8935.74i	−32855.2	+	27568.8i	93629.5	+	162171.i	−206181.	−	119039.i	−188002.	−	1.06621e6i	−868590.	+	153156.i
2.13	45.9254	+	8.09788i	180.097	−	214.631i	−1805.42	−	657.118i	−2261.02	+	822.942i	10009.1	−	8398.60i	15668.1	+	27137.9i	−243015.	−	140305.i	78652.2	+	446059.i	−110502.	+	19484.5i
2.14	62.4371	+	11.0094i	428.070	−	510.154i	−71.7903	−	26.1295i	20294.4	−	7386.57i	32343.9	−	27139.8i	52199.4	+	90412.0i	−229091.	−	132266.i	15270.6	+	86603.9i	1.34845e6	−	237768.i
2.15	62.7075	+	11.0570i	909.303	−	1083.66i	−39.0035	−	14.1961i	−24504.3	+	8918.82i	69002.2	−	57899.8i	7081.21	+	12265.0i	−228159.	−	131728.i	−255214.	−	1.44739e6i	−1.63522e6	+	288333.i
2.16	86.6858	+	15.2850i	−418.686	+	498.971i	3431.82	+	1249.08i	−7734.69	+	2815.20i	−43920.9	+	36854.1i	−67746.9	−	117341.i	−33841.4	−	19538.4i	18610.0	+	105543.i	−713518.	+	125813.i
2.17	105.726	+	18.6424i	−748.552	+	892.089i	6981.56	+	2541.08i	25880.8	−	9419.84i	−95772.4	+	80362.6i	69192.8	+	119846.i	309941.	+	178944.i	−143210.	−	812183.i	2.91189e6	−	513445.i
2.18	110.294	+	19.4478i	582.814	−	694.571i	7937.60	+	2889.05i	11803.1	−	4295.98i	77788.9	−	65272.6i	−79067.6	−	136949.i	422009.	+	243647.i	−50472.5	−	286244.i	1.38536e6	−	244277.i
2.19	111.495	+	19.6595i	45.8459	−	54.6370i	8195.57	+	2982.94i	−12813.2	+	4663.62i	6185.70	−	5190.42i	66340.0	+	114904.i	453519.	+	261839.i	91400.4	+	518357.i	−1.52029e6	+	268067.i
3.1	−76.3528	+	90.9938i	189.716	+	521.239i	−1738.85	−	9861.49i	−4101.54	+	23261.0i	−61914.9	−	22535.2i	−50249.9	−	87035.3i	608745.	+	351459.i	171409.	−	143829.i	−1.80344e6	−	2.14926e6i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.f	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	19.13.f.a	✓	114
19.f	odd	18	1	inner	19.13.f.a	✓	114

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
19.13.f.a	✓	114	1.a	even	1	1	trivial
19.13.f.a	✓	114	19.f	odd	18	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(19, [\chi])\).