Newform orbit 171.2.h.c

• Label: 171.2.h.c
• Level: $171$
• Weight: $2$
• Character orbit: 171.h
• Analytic conductor: $1.365$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	171.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.36544187456\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} + q^{3} + 34 q^{4} + 3 q^{5} - 7 q^{6} + q^{7} - 36 q^{8} + 17 q^{9}+O(q^{10}) \)

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	−2.61030			−0.312318	+	1.70366i	4.81368			1.00100	−	1.73379i	0.815245	−	4.44707i	0.257107	−	0.445323i	−7.34455			−2.80491	−	1.06417i	−2.61292	+	4.52572i
7.2	−2.60319			1.68991	−	0.379742i	4.77661			−1.43897	+	2.49237i	−4.39916	+	0.988542i	−1.80240	+	3.12185i	−7.22804			2.71159	−	1.28346i	3.74592	−	6.48812i
7.3	−2.39943			−1.30020	−	1.14433i	3.75726			−0.359839	+	0.623259i	3.11973	+	2.74573i	1.65862	−	2.87282i	−4.21641			0.381029	+	2.97570i	0.863408	−	1.49547i
7.4	−2.02031			0.581955	−	1.63136i	2.08166			2.09369	−	3.62638i	−1.17573	+	3.29585i	−0.976107	+	1.69067i	−0.164982			−2.32266	−	1.89875i	−4.22991	+	7.32643i
7.5	−1.69423			1.47327	+	0.910763i	0.870412			−0.0441088	+	0.0763987i	−2.49605	−	1.54304i	1.84695	−	3.19901i	1.91378			1.34102	+	2.68359i	0.0747304	−	0.129437i
7.6	−1.23359			−1.68779	−	0.389075i	−0.478252			0.275772	−	0.477650i	2.08204	+	0.479960i	−1.62156	+	2.80862i	3.05715			2.69724	+	1.31335i	−0.340189	+	0.589225i
7.7	−0.194693			−1.44110	+	0.960845i	−1.96209			0.952817	−	1.65033i	0.280573	−	0.187070i	1.69446	−	2.93489i	0.771394			1.15355	−	2.76935i	−0.185507	+	0.321308i
7.8	0.146534			1.41507	+	0.998785i	−1.97853			−1.28502	+	2.22572i	0.207356	+	0.146356i	−1.73898	+	3.01201i	−0.582990			1.00486	+	2.82671i	−0.188299	+	0.326143i
7.9	0.370889			1.73173	−	0.0335006i	−1.86244			1.77761	−	3.07890i	0.642278	−	0.0124250i	−0.124876	+	0.216291i	−1.43254			2.99776	−	0.116028i	0.659294	−	1.14193i
7.10	0.539090			−0.167752	−	1.72391i	−1.70938			0.473662	−	0.820407i	−0.0904332	−	0.929342i	1.18430	−	2.05126i	−1.99969			−2.94372	+	0.578377i	0.255347	−	0.442274i
7.11	0.791858			−1.71671	−	0.230031i	−1.37296			−1.29546	+	2.24381i	−1.35939	−	0.182152i	−0.373088	+	0.646207i	−2.67091			2.89417	+	0.789791i	−1.02582	+	1.77678i
7.12	1.60662			1.66723	−	0.469417i	0.581222			−1.87940	+	3.25521i	2.67860	−	0.754174i	2.27973	−	3.94861i	−2.27943			2.55930	−	1.56525i	−3.01948	+	5.22989i
7.13	1.77797			0.539377	+	1.64593i	1.16118			0.639786	−	1.10814i	0.958997	+	2.92641i	0.657761	−	1.13928i	−1.49140			−2.41814	+	1.77555i	1.13752	−	1.97024i
7.14	1.95703			0.933991	−	1.45865i	1.82996			0.0981173	−	0.169944i	1.82785	−	2.85462i	−2.23368	+	3.86885i	−0.332766			−1.25532	−	2.72473i	0.192018	−	0.332586i
7.15	2.09768			−1.57119	−	0.728951i	2.40028			1.44796	−	2.50795i	−3.29586	−	1.52911i	0.116480	−	0.201749i	0.839660			1.93726	+	2.29064i	3.03737	−	5.26088i
7.16	2.46808			−1.33547	+	1.10296i	4.09140			−0.957619	+	1.65865i	−3.29605	+	2.72218i	−0.324708	+	0.562412i	5.16173			0.566979	−	2.94594i	−2.36348	+	4.09366i
49.1	−2.61030			−0.312318	−	1.70366i	4.81368			1.00100	+	1.73379i	0.815245	+	4.44707i	0.257107	+	0.445323i	−7.34455			−2.80491	+	1.06417i	−2.61292	−	4.52572i
49.2	−2.60319			1.68991	+	0.379742i	4.77661			−1.43897	−	2.49237i	−4.39916	−	0.988542i	−1.80240	−	3.12185i	−7.22804			2.71159	+	1.28346i	3.74592	+	6.48812i
49.3	−2.39943			−1.30020	+	1.14433i	3.75726			−0.359839	−	0.623259i	3.11973	−	2.74573i	1.65862	+	2.87282i	−4.21641			0.381029	−	2.97570i	0.863408	+	1.49547i
49.4	−2.02031			0.581955	+	1.63136i	2.08166			2.09369	+	3.62638i	−1.17573	−	3.29585i	−0.976107	−	1.69067i	−0.164982			−2.32266	+	1.89875i	−4.22991	−	7.32643i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
171.h	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	171.2.h.c	yes	32
3.b	odd	2	1		513.2.h.c		32
9.c	even	3	1		171.2.g.c	✓	32
9.d	odd	6	1		513.2.g.c		32
19.c	even	3	1		171.2.g.c	✓	32
57.h	odd	6	1		513.2.g.c		32
171.h	even	3	1	inner	171.2.h.c	yes	32
171.j	odd	6	1		513.2.h.c		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
171.2.g.c	✓	32	9.c	even	3	1
171.2.g.c	✓	32	19.c	even	3	1
171.2.h.c	yes	32	1.a	even	1	1	trivial
171.2.h.c	yes	32	171.h	even	3	1	inner
513.2.g.c		32	9.d	odd	6	1
513.2.g.c		32	57.h	odd	6	1
513.2.h.c		32	3.b	odd	2	1
513.2.h.c		32	171.j	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(171, [\chi])\):

T2^16 + T2^15 - 24*T2^14 - 17*T2^13 + 235*T2^12 + 96*T2^11 - 1193*T2^10 - 137*T2^9 + 3270*T2^8 - 531*T2^7 - 4485*T2^6 + 1905*T2^5 + 2250*T2^4 - 1638*T2^3 + 177*T2^2 + 63*T2 - 9

T5^32 - 3*T5^31 + 49*T5^30 - 110*T5^29 + 1345*T5^28 - 2690*T5^27 + 23851*T5^26 - 41633*T5^25 + 303199*T5^24 - 493953*T5^23 + 2838685*T5^22 - 4327692*T5^21 + 20171787*T5^20 - 29715627*T5^19 + 108257049*T5^18 - 151502058*T5^17 + 441462336*T5^16 - 582107640*T5^15 + 1316027925*T5^14 - 1546493157*T5^13 + 2764861119*T5^12 - 2816662995*T5^11 + 3674796309*T5^10 - 2655256374*T5^9 + 2408204988*T5^8 - 1245372462*T5^7 + 965704140*T5^6 - 378561924*T5^5 + 170177274*T5^4 - 19394802*T5^3 + 3196179*T5^2 + 142884*T5 + 35721