Newform orbit 16.11.f.a

• Label: 16.11.f.a
• Level: $16$
• Weight: $11$
• Character orbit: 16.f
• Analytic conductor: $10.166$
• Analytic rank: $0$
• Dimension: $38$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	16.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.1657160428\)
Analytic rank:	\(0\)
Dimension:	\(38\)
Relative dimension:	\(19\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 2 q^{2} - 2 q^{3} - 1256 q^{4} - 2 q^{5} - 17216 q^{6} - 4 q^{7} + 69124 q^{8}+O(q^{10}) \)

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	−31.9996	+	0.164143i	−221.985	+	221.985i	1023.95	−	10.5050i	2579.78	−	2579.78i	7066.97	−	7139.85i	4017.27			−32764.1	+	504.230i		−	39505.3i	−82128.4	+	82975.3i
3.2	−30.1099	−	10.8349i	127.462	−	127.462i	789.210	+	652.475i	−1086.76	+	1086.76i	−5218.91	+	2456.83i	−8881.75			−16693.5	−	28197.0i			26555.9i	44497.0	−	20947.2i
3.3	−29.1906	+	13.1115i	32.7265	−	32.7265i	680.178	−	765.463i	−1645.29	+	1645.29i	−526.211	+	1384.40i	16644.7			−9818.42	+	31262.4i			56907.0i	26454.7	−	69599.0i
3.4	−25.9865	+	18.6735i	280.546	−	280.546i	326.598	−	970.520i	3102.01	−	3102.01i	−2051.63	+	12529.2i	−27408.1			9635.89	+	31319.2i		−	98363.4i	−22685.0	+	138536.i
3.5	−19.3334	+	25.4994i	−222.022	+	222.022i	−276.439	−	985.981i	−2711.76	+	2711.76i	−1368.99	−	9953.89i	−24891.4			30486.4	+	12013.3i		−	39538.9i	−16720.7	−	121576.i
3.6	−19.3314	−	25.5009i	−196.215	+	196.215i	−276.596	+	985.936i	−1311.50	+	1311.50i	8796.79	+	1210.56i	−1120.35			30489.3	−	12006.1i		−	17952.0i	58797.6	+	8091.39i
3.7	−16.8116	−	27.2281i	183.793	−	183.793i	−458.741	+	915.496i	3496.01	−	3496.01i	−8094.18	−	1914.49i	26135.6			32639.4	−	2900.27i		−	8510.59i	−153963.	−	36416.3i
3.8	−9.61264	+	30.5221i	−107.144	+	107.144i	−839.194	−	586.796i	3434.17	−	3434.17i	−2240.31	−	4300.18i	8771.09			25977.1	−	19973.3i			36089.4i	71806.6	+	137829.i
3.9	−6.25659	+	31.3824i	196.669	−	196.669i	−945.710	−	392.694i	−1812.92	+	1812.92i	4941.46	+	7402.41i	22339.6			18240.6	−	27221.7i		−	18308.2i	−45551.0	−	68236.4i
3.10	−1.21444	−	31.9769i	287.146	−	287.146i	−1021.05	+	77.6681i	−3974.49	+	3974.49i	−9530.78	−	8833.33i	−11755.9			3723.59	+	32555.7i		−	105857.i	131919.	+	122265.i
3.11	2.46002	−	31.9053i	−46.0373	+	46.0373i	−1011.90	−	156.975i	1933.37	−	1933.37i	1355.58	+	1582.09i	−25927.9			−7497.63	+	31898.7i			54810.1i	−56928.5	−	66440.7i
3.12	12.2776	−	29.5510i	−126.065	+	126.065i	−722.523	−	725.628i	−1655.31	+	1655.31i	2177.58	+	5273.11i	29862.3			−30313.9	+	12442.3i			27264.3i	28593.0	+	69239.3i
3.13	12.5336	+	29.4433i	44.0645	−	44.0645i	−709.815	+	738.064i	−126.270	+	126.270i	1849.69	+	745.115i	−19594.2			−30627.6	−	11648.7i			55165.6i	−5300.42	−	2135.18i
3.14	17.5699	+	26.7451i	−286.288	+	286.288i	−406.599	+	939.816i	−1338.28	+	1338.28i	−12686.8	−	2626.75i	19657.4			−32279.3	+	5637.94i		−	104873.i	−59305.9	−	12279.0i
3.15	26.0928	−	18.5248i	164.886	−	164.886i	337.664	−	966.726i	897.573	−	897.573i	1247.85	−	7356.80i	594.817			−9097.83	−	31479.7i			4674.41i	6792.79	−	40047.5i
3.16	27.4041	+	16.5231i	305.982	−	305.982i	477.973	+	905.603i	155.264	−	155.264i	13440.9	−	3329.39i	8523.80			−1864.95	+	32714.9i		−	128200.i	6820.32	−	1689.43i
3.17	28.0563	−	15.3897i	−317.338	+	317.338i	550.315	−	863.556i	1564.48	−	1564.48i	−4019.61	+	13787.1i	−15496.4			2149.97	−	32697.4i		−	142358.i	19816.8	−	67970.6i
3.18	30.4523	+	9.83157i	−61.4496	+	61.4496i	830.681	+	598.787i	2595.70	−	2595.70i	−2475.42	+	1267.13i	6393.57			19409.1	+	26401.3i			51496.9i	104565.	−	53525.2i
3.19	32.0000	+	0.0155982i	−39.7294	+	39.7294i	1024.00	+	0.998283i	−4096.79	+	4096.79i	−1271.96	+	1270.72i	−7866.10			32768.0	+	47.9176i			55892.1i	−131161.	+	131033.i
11.1	−31.9996	−	0.164143i	−221.985	−	221.985i	1023.95	+	10.5050i	2579.78	+	2579.78i	7066.97	+	7139.85i	4017.27			−32764.1	−	504.230i			39505.3i	−82128.4	−	82975.3i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.f	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	16.11.f.a	✓	38
4.b	odd	2	1		64.11.f.a		38
8.b	even	2	1		128.11.f.b		38
8.d	odd	2	1		128.11.f.a		38
16.e	even	4	1		64.11.f.a		38
16.e	even	4	1		128.11.f.a		38
16.f	odd	4	1	inner	16.11.f.a	✓	38
16.f	odd	4	1		128.11.f.b		38

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
16.11.f.a	✓	38	1.a	even	1	1	trivial
16.11.f.a	✓	38	16.f	odd	4	1	inner
64.11.f.a		38	4.b	odd	2	1
64.11.f.a		38	16.e	even	4	1
128.11.f.a		38	8.d	odd	2	1
128.11.f.a		38	16.e	even	4	1
128.11.f.b		38	8.b	even	2	1
128.11.f.b		38	16.f	odd	4	1

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(16, [\chi])\).