Newform orbit 33.7.g.a

• Label: 33.7.g.a
• Level: $33$
• Weight: $7$
• Character orbit: 33.g
• Analytic conductor: $7.592$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	33.g (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.59178475946\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 204 q^{4} - 224 q^{5} + 720 q^{7} - 3200 q^{8} - 2916 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
7.1	−8.29374	+	11.4154i	4.81710	−	14.8255i	−41.7471	−	128.484i	−55.3317	+	40.2008i	129.287	+	177.948i	511.454	−	166.181i	954.081	+	310.000i	−196.591	−	142.832i		−	965.045i
7.2	−6.91015	+	9.51101i	−4.81710	+	14.8255i	−22.9320	−	70.5775i	174.946	−	127.106i	−107.719	−	148.262i	472.320	−	153.466i	114.152	+	37.0902i	−196.591	−	142.832i			2542.24i
7.3	−6.32244	+	8.70209i	−4.81710	+	14.8255i	−15.9760	−	49.1691i	−92.2594	+	67.0304i	−98.5570	−	135.652i	−121.251	+	39.3968i	−125.833	−	40.8857i	−196.591	−	142.832i		−	1226.64i
7.4	−5.06354	+	6.96937i	4.81710	−	14.8255i	−3.15555	−	9.71180i	−8.79332	+	6.38872i	78.9328	+	108.642i	−242.092	+	78.6605i	−440.688	−	143.188i	−196.591	−	142.832i		−	93.6335i
7.5	−1.16939	+	1.60953i	−4.81710	+	14.8255i	18.5540	+	57.1033i	−1.49745	+	1.08796i	−18.2291	−	25.0902i	−25.7895	+	8.37953i	−234.702	−	76.2594i	−196.591	−	142.832i		−	3.68244i
7.6	1.61569	−	2.22380i	4.81710	−	14.8255i	17.4422	+	53.6817i	−125.017	+	90.8304i	−25.1861	−	34.6656i	−357.347	+	116.109i	314.870	+	102.307i	−196.591	−	142.832i			424.767i
7.7	2.37250	−	3.26547i	4.81710	−	14.8255i	14.7426	+	45.3730i	26.5586	−	19.2960i	−36.9836	−	50.9036i	457.293	−	148.583i	428.823	+	139.333i	−196.591	−	142.832i		−	132.506i
7.8	4.99839	−	6.87969i	−4.81710	+	14.8255i	−2.56917	−	7.90708i	−1.33123	+	0.967196i	77.9171	+	107.244i	525.675	−	170.802i	450.364	+	146.332i	−196.591	−	142.832i			13.9929i
7.9	6.02470	−	8.29229i	−4.81710	+	14.8255i	−12.6880	−	39.0496i	−154.787	+	112.460i	93.9158	+	129.264i	−515.310	+	167.434i	223.631	+	72.6622i	−196.591	−	142.832i			1961.08i
7.10	6.26570	−	8.62400i	4.81710	−	14.8255i	−15.3373	−	47.2032i	161.919	−	117.641i	−97.6727	−	134.435i	−310.129	+	100.767i	145.660	+	47.3278i	−196.591	−	142.832i		−	2133.50i
7.11	8.69356	−	11.9657i	4.81710	−	14.8255i	−47.8219	−	147.181i	−118.124	+	85.8219i	−135.519	−	186.526i	185.111	−	60.1462i	−1276.60	−	414.794i	−196.591	−	142.832i			2159.53i
7.12	8.96906	−	12.3449i	−4.81710	+	14.8255i	−52.1743	−	160.576i	137.717	−	100.057i	139.814	+	192.437i	−91.3553	+	29.6831i	−1521.46	−	494.352i	−196.591	−	142.832i		−	2597.52i
13.1	−14.5443	−	4.72574i	−12.6113	+	9.16267i	137.428	+	99.8469i	−61.8586	−	190.381i	226.723	−	73.6669i	168.802	−	232.336i	−951.651	−	1309.84i	75.0911	−	231.107i			3061.29i
13.2	−12.0509	−	3.91558i	12.6113	−	9.16267i	78.1159	+	56.7545i	−13.4634	−	41.4361i	−187.855	+	61.0379i	−335.737	+	462.103i	−242.477	−	333.741i	75.0911	−	231.107i			552.060i
13.3	−9.54864	−	3.10254i	−12.6113	+	9.16267i	29.7736	+	21.6318i	33.3013	+	102.491i	148.849	−	48.3638i	−90.4877	+	124.546i	160.505	+	220.916i	75.0911	−	231.107i		−	1081.97i
13.4	−9.48025	−	3.08032i	12.6113	−	9.16267i	28.6097	+	20.7861i	−12.6698	−	38.9936i	−147.782	+	48.0174i	304.303	−	418.837i	167.785	+	230.936i	75.0911	−	231.107i			408.696i
13.5	−5.17091	−	1.68013i	−12.6113	+	9.16267i	−27.8616	−	20.2427i	−25.6501	−	78.9429i	80.6065	−	26.1906i	−29.3745	+	40.4305i	314.591	+	432.997i	75.0911	−	231.107i			451.302i
13.6	−2.69988	−	0.877245i	12.6113	−	9.16267i	−45.2573	−	32.8813i	71.7465	+	220.813i	−42.0870	+	13.6749i	166.082	−	228.592i	200.136	+	275.463i	75.0911	−	231.107i		−	659.108i
13.7	−0.674572	−	0.219182i	12.6113	−	9.16267i	−51.3701	−	37.3225i	−8.69665	−	26.7655i	−10.5155	+	3.41670i	−301.986	+	415.648i	53.1545	+	73.1609i	75.0911	−	231.107i			19.9614i
13.8	4.51299	+	1.46636i	−12.6113	+	9.16267i	−33.5602	−	24.3829i	6.35291	+	19.5523i	−70.3505	+	22.8583i	355.989	−	489.977i	−294.210	−	404.946i	75.0911	−	231.107i			97.5547i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.d	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	33.7.g.a	✓	48
3.b	odd	2	1		99.7.k.b		48
11.d	odd	10	1	inner	33.7.g.a	✓	48
33.f	even	10	1		99.7.k.b		48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
33.7.g.a	✓	48	1.a	even	1	1	trivial
33.7.g.a	✓	48	11.d	odd	10	1	inner
99.7.k.b		48	3.b	odd	2	1
99.7.k.b		48	33.f	even	10	1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(33, [\chi])\).