Newform orbit 17.13.e.a

• Label: 17.13.e.a
• Level: $17$
• Weight: $13$
• Character orbit: 17.e
• Analytic conductor: $15.538$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	17.e (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.5378948937\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(17\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 136 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

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	−111.056	+	46.0009i	−851.369	−	169.348i	7321.02	−	7321.02i	−954.538	−	1428.57i	102340.	−	20356.6i	−96128.5	+	143866.i	−287850.	+	694931.i	205164.	+	84981.6i	171722.	+	114741.i
3.2	−102.332	+	42.3875i	742.540	+	147.700i	5778.91	−	5778.91i	−12231.2	−	18305.3i	−82246.6	+	16359.9i	71225.1	−	106596.i	−172797.	+	417169.i	38563.4	+	15973.5i	2.02757e6	+	1.35478e6i
3.3	−89.7233	+	37.1646i	234.397	+	46.6245i	3772.75	−	3772.75i	10245.1	+	15332.9i	−22763.7	+	4527.97i	34036.5	−	50939.2i	−46064.5	+	111209.i	−438219.	−	181516.i	−1.48907e6	−	994963.i
3.4	−66.0119	+	27.3430i	1250.03	+	248.647i	713.617	−	713.617i	2086.62	+	3122.84i	−89315.9	+	17766.0i	−104524.	+	156432.i	84402.2	−	203765.i	1.00977e6	+	418262.i	−223129.	−	149090.i
3.5	−63.0651	+	26.1224i	−656.664	−	130.619i	398.522	−	398.522i	−11871.9	−	17767.6i	44824.7	−	8916.19i	23227.7	−	34762.7i	92275.0	−	222772.i	−76840.6	−	31828.4i	1.21284e6	+	810392.i
3.6	−57.6610	+	23.8840i	−1154.96	−	229.736i	−141.965	+	141.965i	8960.19	+	13409.9i	72083.2	−	14338.2i	67951.2	−	101696.i	102624.	−	247756.i	790168.	+	327298.i	−836934.	−	559221.i
3.7	−38.9153	+	16.1192i	−71.3658	−	14.1956i	−1641.74	+	1641.74i	−4258.30	−	6372.99i	3006.04	−	597.939i	−77511.3	+	116004.i	103450.	−	249749.i	−486096.	−	201348.i	268440.	+	179366.i
3.8	−11.6760	+	4.83634i	916.444	+	182.292i	−2783.37	+	2783.37i	−7210.29	−	10791.0i	−11582.0	+	2303.80i	94402.1	−	141283.i	38846.8	−	93784.5i	315652.	+	130747.i	136376.	+	91123.3i
3.9	−2.85762	+	1.18367i	252.743	+	50.2736i	−2889.54	+	2889.54i	9771.27	+	14623.7i	−781.750	+	155.500i	16913.1	−	25312.3i	9685.27	−	23382.3i	−429636.	−	177961.i	−45232.2	−	30223.2i
3.10	18.5280	−	7.67454i	−1031.87	−	205.253i	−2611.92	+	2611.92i	4593.87	+	6875.22i	−20693.8	+	4116.25i	−58620.3	+	87731.5i	−59783.3	+	144330.i	531649.	+	220216.i	137879.	+	92128.1i
3.11	35.8655	−	14.8560i	−825.011	−	164.105i	−1830.68	+	1830.68i	−9921.52	−	14848.6i	−32027.3	+	6370.63i	72591.5	−	108641.i	−99311.6	+	239759.i	162725.	+	67402.8i	−576430.	−	385158.i
3.12	52.4772	−	21.7368i	402.210	+	80.0045i	−614.943	+	614.943i	−11928.5	−	17852.3i	22845.9	−	4544.33i	−71722.0	+	107340.i	−107937.	+	260584.i	−335615.	−	139016.i	−1.01403e6	−	677552.i
3.13	57.0799	−	23.6432i	1210.02	+	240.689i	−197.203	+	197.203i	7596.30	+	11368.7i	74758.7	−	14870.4i	−2188.89	+	3275.91i	−103437.	+	249718.i	915242.	+	379106.i	702388.	+	469320.i
3.14	79.9124	−	33.1008i	−253.045	−	50.3338i	2394.02	−	2394.02i	13416.9	+	20079.8i	−21887.5	+	4353.70i	−41801.1	+	62559.8i	−23513.1	+	56765.6i	−429489.	−	177900.i	1.73684e6	+	1.16052e6i
3.15	83.2788	−	34.4952i	−401.148	−	79.7933i	2849.13	−	2849.13i	900.230	+	1347.29i	−36159.6	+	7192.59i	109335.	−	163631.i	−2301.59	+	5556.53i	−336435.	−	139356.i	121445.	+	81147.0i
3.16	107.125	−	44.3726i	−1217.19	−	242.114i	6610.51	−	6610.51i	−5751.92	−	8608.36i	−141134.	+	28073.3i	−94223.1	+	141015.i	233075.	−	562692.i	931938.	+	386021.i	−998150.	−	666942.i
3.17	107.325	−	44.4553i	702.054	+	139.647i	6646.00	−	6646.00i	−3536.08	−	5292.12i	81555.8	−	16222.5i	9173.07	−	13728.5i	235741.	−	569129.i	−17609.0	−	7293.88i	−614772.	−	410777.i
5.1	−44.9015	−	108.402i	−732.537	−	1096.32i	−6838.50	+	6838.50i	3290.00	−	16539.9i	−85951.0	+	128635.i	−6994.76	−	35165.0i	604351.	+	250331.i	−461931.	+	1.11520e6i	−1.94069e6	+	386027.i
5.2	−41.1443	−	99.3310i	18.2588	+	27.3263i	−5277.49	+	5277.49i	−3462.80	+	17408.7i	1963.10	−	2937.99i	130.137	+	654.244i	334497.	+	138553.i	202960.	−	489990.i	1.87170e6	−	372304.i
5.3	−39.3157	−	94.9164i	796.483	+	1192.02i	−4567.09	+	4567.09i	1313.26	−	6602.22i	81828.1	−	122464.i	−15158.4	−	76206.6i	224273.	+	92896.8i	−583155.	+	1.40786e6i	−678290.	+	134920.i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.e	odd	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	17.13.e.a	✓	136
17.e	odd	16	1	inner	17.13.e.a	✓	136

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
17.13.e.a	✓	136	1.a	even	1	1	trivial
17.13.e.a	✓	136	17.e	odd	16	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(17, [\chi])\).