Newform orbit 103.5.f.a

• Label: 103.5.f.a
• Level: $103$
• Weight: $5$
• Character orbit: 103.f
• Analytic conductor: $10.647$
• Analytic rank: $0$
• Dimension: $528$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 103 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	103.f (of order \(34\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.6471061976\)
Analytic rank:	\(0\)
Dimension:	\(528\)
Relative dimension:	\(33\) over \(\Q(\zeta_{34})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{34}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 528 q - 15 q^{2} - 17 q^{3} - 255 q^{4} - 17 q^{5} - 17 q^{6} - 45 q^{7} - 73 q^{8} + 662 q^{9}+O(q^{10}) \)

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	−6.43885	+	3.98677i	−12.7996	+	3.64181i	18.4327	−	37.0178i	−11.7663	−	30.3723i	67.8958	−	74.4782i	0.543717	−	5.86764i	17.7159	+	191.185i	81.6999	−	50.5864i	196.849	+	148.653i
3.2	−6.26882	+	3.88149i	−2.80686	+	0.798619i	17.1003	−	34.3421i	13.6251	+	35.1704i	14.4958	−	15.9012i	2.94554	−	31.7874i	15.2146	+	164.191i	−61.6269	+	38.1578i	−221.927	−	167.591i
3.3	−6.26112	+	3.87672i	11.7223	−	3.33528i	17.0408	−	34.2226i	−3.33407	−	8.60623i	−60.4646	+	66.3266i	−2.44210	+	26.3545i	15.1052	+	163.011i	57.4201	−	35.5530i	54.2389	+	40.9593i
3.4	−5.52223	+	3.41922i	−8.55969	+	2.43544i	11.6722	−	23.4409i	6.28575	+	16.2254i	38.9413	−	42.7165i	−6.22199	+	67.1460i	6.10445	+	65.8775i	−1.53071	+	0.947774i	−90.1896	−	68.1080i
3.5	−5.41637	+	3.35368i	5.33683	−	1.51846i	10.9581	−	22.0069i	−3.14628	−	8.12148i	−23.8139	+	26.1226i	4.95405	−	53.4627i	5.04584	+	54.4533i	−42.6915	+	26.4335i	44.2782	+	33.4374i
3.6	−4.79576	+	2.96941i	−1.61572	+	0.459712i	7.05010	−	14.1585i	−16.4981	−	42.5866i	6.38353	−	7.00240i	1.85650	−	20.0348i	−0.0954160	−	1.02970i	−66.4684	+	41.1555i	205.578	+	155.245i
3.7	−4.10206	+	2.53989i	10.1988	−	2.90181i	3.24404	−	6.51492i	13.3593	+	34.4843i	−34.4658	+	37.8072i	−3.52506	+	38.0415i	−3.88280	−	41.9021i	26.7276	−	16.5490i	−142.387	−	107.526i
3.8	−4.00627	+	2.48058i	−2.65995	+	0.756821i	2.76514	−	5.55314i	−5.04157	−	13.0138i	8.77913	−	9.63024i	−6.45216	+	69.6299i	−4.25926	−	45.9648i	−62.3650	+	38.6148i	52.4797	+	39.6308i
3.9	−3.95370	+	2.44803i	−14.5756	+	4.14712i	2.50710	−	5.03494i	3.75115	+	9.68283i	47.4754	−	52.0780i	2.28560	−	24.6656i	−4.45176	−	48.0421i	126.382	−	78.2527i	−38.5348	−	29.1001i
3.10	−3.53415	+	2.18825i	14.8424	−	4.22303i	0.569962	−	1.14464i	−9.06981	−	23.4119i	−43.2143	+	47.4039i	−1.08799	+	11.7413i	−5.64618	−	60.9320i	133.596	−	82.7192i	83.2852	+	62.8941i
3.11	−3.08589	+	1.91070i	−1.47579	+	0.419899i	−1.25989	+	2.53021i	6.08648	+	15.7110i	3.75182	−	4.11555i	7.31060	−	78.8939i	−6.30483	−	68.0399i	−66.8659	+	41.4016i	−48.8012	−	36.8529i
3.12	−2.14035	+	1.32525i	13.3666	−	3.80312i	−4.30699	+	8.64961i	0.507444	+	1.30986i	−23.5691	+	25.8540i	7.62540	−	82.2912i	−5.96086	−	64.3279i	95.3338	−	59.0282i	−2.82201	−	2.13108i
3.13	−1.57407	+	0.974621i	−3.14375	+	0.894474i	−5.60401	+	11.2544i	−0.987924	−	2.55013i	4.07670	−	4.47193i	−2.86298	+	30.8965i	−4.88083	−	52.6725i	−59.7845	+	37.0170i	4.04046	+	3.05122i
3.14	−1.30166	+	0.805955i	−10.9733	+	3.12216i	−6.08705	+	12.2244i	−13.2589	−	34.2253i	11.7672	−	12.9080i	3.33922	−	36.0359i	−4.18924	−	45.2091i	41.7970	−	25.8796i	44.8427	+	33.8636i
3.15	−0.941118	+	0.582716i	8.91406	−	2.53627i	−6.58567	+	13.2258i	−10.8661	−	28.0487i	−6.91126	+	7.58129i	−5.11752	+	55.2269i	−3.14312	−	33.9197i	4.16018	−	2.57588i	26.5707	+	20.0653i
3.16	−0.463175	+	0.286786i	−8.63215	+	2.45606i	−6.99953	+	14.0569i	17.5120	+	45.2037i	3.29383	−	3.61316i	0.0627677	−	0.677371i	−1.59357	−	17.1974i	−0.385838	+	0.238901i	−21.0749	−	15.9150i
3.17	−0.381077	+	0.235953i	4.33110	−	1.23230i	−7.04227	+	14.1428i	9.66340	+	24.9441i	−1.35971	+	1.49154i	−0.821381	+	8.86411i	−1.31507	−	14.1919i	−51.6277	+	31.9665i	−9.56812	−	7.22551i
3.18	0.388107	−	0.240306i	−13.7714	+	3.91830i	−7.03893	+	14.1361i	4.17278	+	10.7712i	−4.40318	+	4.83006i	−7.64472	+	82.4996i	1.33902	+	14.4503i	105.431	−	65.2799i	4.20786	+	3.17763i
3.19	1.09270	−	0.676568i	5.44531	−	1.54932i	−6.39557	+	12.8440i	−10.4661	−	27.0162i	4.90184	−	5.37706i	4.40555	−	47.5435i	3.59879	+	38.8371i	−41.6166	+	25.7679i	−29.7146	−	22.4394i
3.20	1.28224	−	0.793930i	14.9967	−	4.26694i	−6.11800	+	12.2866i	9.68576	+	25.0018i	15.8418	−	17.3776i	1.63203	−	17.6125i	4.13640	+	44.6388i	137.828	−	85.3394i	32.2692	+	24.3686i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
103.f	odd	34	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	103.5.f.a	✓	528
103.f	odd	34	1	inner	103.5.f.a	✓	528

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
103.5.f.a	✓	528	1.a	even	1	1	trivial
103.5.f.a	✓	528	103.f	odd	34	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(103, [\chi])\).