Newform orbit 55.5.f.a

• Label: 55.5.f.a
• Level: $55$
• Weight: $5$
• Character orbit: 55.f
• Analytic conductor: $5.685$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 55 = 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	55.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.68534796961\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 10 q^{3} - 48 q^{5} - 56 q^{6} + 180 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
12.1	−5.38790	−	5.38790i	−5.24101	+	5.24101i			42.0589i	23.3679	−	8.88487i	56.4760			−25.1649	−	25.1649i	140.402	−	140.402i			26.0637i	−173.775	−	78.0331i
12.2	−5.12014	−	5.12014i	11.8471	−	11.8471i			36.4316i	−9.67783	−	23.0508i	−121.317			−5.09387	−	5.09387i	104.613	−	104.613i		−	199.705i	−68.4714	+	167.575i
12.3	−4.20563	−	4.20563i	4.11313	−	4.11313i			19.3746i	15.1607	+	19.8785i	−34.5965			56.8014	+	56.8014i	14.1922	−	14.1922i			47.1644i	19.8414	−	147.362i
12.4	−4.12415	−	4.12415i	−1.91974	+	1.91974i			18.0172i	−22.8841	−	10.0658i	15.8346			19.2523	+	19.2523i	8.31933	−	8.31933i			73.6292i	52.8644	+	135.890i
12.5	−3.93156	−	3.93156i	−12.0617	+	12.0617i			14.9144i	−14.4750	+	20.3832i	94.8429			−1.40891	−	1.40891i	−4.26808	+	4.26808i		−	209.970i	137.047	−	23.2283i
12.6	−3.78735	−	3.78735i	5.17377	−	5.17377i			12.6881i	−6.78218	+	24.0625i	−39.1898			−64.8489	−	64.8489i	−12.5435	+	12.5435i			27.4642i	116.819	−	65.4465i
12.7	−2.71168	−	2.71168i	−4.74142	+	4.74142i		−	1.29353i	4.46012	−	24.5989i	25.7145			−4.54884	−	4.54884i	−46.8946	+	46.8946i			36.0379i	−78.7990	+	54.6101i
12.8	−1.59824	−	1.59824i	8.15686	−	8.15686i		−	10.8912i	21.2552	−	13.1611i	−26.0733			11.9999	+	11.9999i	−42.9788	+	42.9788i		−	52.0689i	−55.0057	−	12.9364i
12.9	−1.00763	−	1.00763i	5.90446	−	5.90446i		−	13.9694i	−24.5893	−	4.51308i	−11.8990			−8.63273	−	8.63273i	−30.1980	+	30.1980i			11.2747i	20.2293	+	29.3243i
12.10	0.00572062	+	0.00572062i	−2.60834	+	2.60834i		−	15.9999i	−15.5200	+	19.5993i	−0.0298426			52.2368	+	52.2368i	0.183059	−	0.183059i			67.3931i	−0.200904	+	0.0233360i
12.11	0.742700	+	0.742700i	−11.7326	+	11.7326i		−	14.8968i	−3.42979	−	24.7636i	−17.4276			36.8201	+	36.8201i	22.9471	−	22.9471i		−	194.306i	15.8446	−	20.9393i
12.12	1.54031	+	1.54031i	9.91874	−	9.91874i		−	11.2549i	7.40565	+	23.8779i	30.5559			−14.2934	−	14.2934i	41.9810	−	41.9810i		−	115.763i	−25.3725	+	48.1865i
12.13	2.07508	+	2.07508i	1.66253	−	1.66253i		−	7.38805i	0.929262	−	24.9827i	6.89976			−54.5519	−	54.5519i	48.5322	−	48.5322i			75.4720i	53.7695	−	49.9130i
12.14	2.29964	+	2.29964i	−6.57135	+	6.57135i		−	5.42334i	−24.2372	+	6.12830i	−30.2235			−43.3847	−	43.3847i	49.2659	−	49.2659i		−	5.36536i	−69.8297	−	41.6440i
12.15	2.74859	+	2.74859i	−0.769806	+	0.769806i		−	0.890481i	24.4076	−	5.41024i	−4.23177			37.0670	+	37.0670i	46.4250	−	46.4250i			79.8148i	81.9570	+	52.2159i
12.16	3.36898	+	3.36898i	9.69022	−	9.69022i			6.70002i	−19.3894	−	15.7814i	65.2923			44.0974	+	44.0974i	31.3314	−	31.3314i		−	106.801i	−12.1552	−	118.490i
12.17	4.26269	+	4.26269i	−10.9721	+	10.9721i			20.3411i	21.2524	+	13.1658i	−93.5414			−19.1032	−	19.1032i	−18.5047	+	18.5047i		−	159.774i	34.4707	+	146.714i
12.18	4.44247	+	4.44247i	1.99881	−	1.99881i			23.4710i	−5.82448	+	24.3120i	17.7593			−0.737721	−	0.737721i	−33.1898	+	33.1898i			73.0095i	−133.881	+	82.1304i
12.19	5.03733	+	5.03733i	−5.45285	+	5.45285i			34.7494i	−17.6388	−	17.7165i	−54.9356			26.9896	+	26.9896i	−94.4468	+	94.4468i			21.5330i	0.391649	−	178.096i
12.20	5.35077	+	5.35077i	8.60532	−	8.60532i			41.2614i	22.2092	−	11.4784i	92.0902			−43.4955	−	43.4955i	−135.168	+	135.168i		−	67.1032i	180.254	+	57.4180i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	55.5.f.a	✓	40
5.c	odd	4	1	inner	55.5.f.a	✓	40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
55.5.f.a	✓	40	1.a	even	1	1	trivial
55.5.f.a	✓	40	5.c	odd	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(55, [\chi])\).