Newform orbit 171.4.l.a

• Label: 171.4.l.a
• Level: $171$
• Weight: $4$
• Character orbit: 171.l
• Analytic conductor: $10.089$
• Analytic rank: $0$
• Dimension: $116$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	171.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.0893266110\)
Analytic rank:	\(0\)
Dimension:	\(116\)
Relative dimension:	\(58\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 116 q - 226 q^{4} - 6 q^{5} - 8 q^{6} + 10 q^{7} - 34 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
56.1	−2.75861	+	4.77805i	4.54987	−	2.50972i	−11.2198	−	19.4333i	−3.73372	+	2.15566i	−0.559733	+	28.6628i	−9.31085	+	16.1269i	79.6666			14.4026	−	22.8378i		−	23.7865i
56.2	−2.67571	+	4.63446i	−4.51992	+	2.56327i	−10.3188	−	17.8727i	−17.9248	+	10.3489i	0.214591	−	27.8060i	4.89510	−	8.47857i	67.6294			13.8593	−	23.1715i		−	110.763i
56.3	−2.65899	+	4.60550i	3.25709	+	4.04863i	−10.1404	−	17.5637i	5.44572	−	3.14409i	−27.3065	+	4.23527i	15.4046	−	26.6815i	65.3093			−5.78277	+	26.3735i			33.4404i
56.4	−2.55733	+	4.42942i	−5.14946	−	0.695034i	−9.07984	−	15.7268i	13.8107	−	7.97359i	16.2474	−	21.0317i	−7.49309	+	12.9784i	51.9633			26.0339	+	7.15810i			81.5643i
56.5	−2.42511	+	4.20042i	−0.642741	−	5.15625i	−7.76235	−	13.4448i	6.93757	−	4.00541i	23.2171	+	9.80470i	8.28054	−	14.3423i	36.4965			−26.1738	+	6.62826i			38.8543i
56.6	−2.39983	+	4.15664i	−0.0363177	+	5.19603i	−7.51841	−	13.0223i	0.384609	−	0.222054i	−21.5108	−	12.6206i	−11.5914	+	20.0769i	33.7744			−26.9974	−	0.377416i			2.13157i
56.7	−2.33859	+	4.05055i	−0.999995	−	5.09902i	−6.93796	−	12.0169i	−12.0396	+	6.95107i	22.9924	+	7.87397i	−3.03908	+	5.26384i	27.4827			−25.0000	+	10.1980i		−	65.0227i
56.8	−2.18132	+	3.77816i	4.92328	−	1.66171i	−5.51631	−	9.55453i	9.75307	−	5.63094i	−4.46104	+	22.2257i	4.85247	−	8.40472i	13.2302			21.4774	−	16.3622i			49.1315i
56.9	−2.18102	+	3.77764i	−3.46472	+	3.87243i	−5.51371	−	9.55003i	5.63334	−	3.25241i	−7.07199	−	21.5343i	4.38374	−	7.59287i	13.2058			−2.99136	−	26.8338i			28.3743i
56.10	−2.02925	+	3.51476i	4.12835	+	3.15542i	−4.23569	−	7.33643i	−11.6513	+	6.72688i	−19.4680	+	8.10704i	−3.96007	+	6.85904i	1.91307			7.08663	+	26.0534i		−	54.6020i
56.11	−1.85368	+	3.21067i	−5.00176	−	1.40796i	−2.87225	−	4.97488i	−3.62597	+	2.09346i	13.7922	−	13.4491i	11.3243	−	19.6143i	−8.36195			23.0353	+	14.0846i		−	15.5224i
56.12	−1.85230	+	3.20827i	−4.33182	−	2.86973i	−2.86201	−	4.95715i	−7.57540	+	4.37366i	17.2307	−	8.58206i	−17.4964	+	30.3047i	−8.43157			10.5293	+	24.8623i		−	32.4053i
56.13	−1.81476	+	3.14326i	3.70981	−	3.63831i	−2.58674	−	4.48037i	−16.0089	+	9.24272i	4.70374	+	18.2636i	13.7561	−	23.8263i	−10.2589			0.525393	−	26.9949i		−	67.0935i
56.14	−1.72831	+	2.99351i	1.58636	−	4.94808i	−1.97408	−	3.41921i	15.1202	−	8.72964i	12.0704	+	13.3006i	−12.9519	+	22.4334i	−14.0056			−21.9669	−	15.6989i			60.3499i
56.15	−1.57999	+	2.73662i	5.18445	−	0.348511i	−0.992737	−	1.71947i	0.387263	−	0.223586i	−7.23764	+	14.7385i	−7.01217	+	12.1454i	−19.0058			26.7571	−	3.61368i			1.41306i
56.16	−1.51027	+	2.61586i	−0.358631	+	5.18376i	−0.561832	−	0.973122i	15.6579	−	9.04008i	−13.0184	−	8.76701i	−1.99843	+	3.46139i	−20.7702			−26.7428	−	3.71811i			54.6119i
56.17	−1.33488	+	2.31208i	4.49417	+	2.60814i	0.436184	+	0.755493i	12.4799	−	7.20528i	−12.0294	+	6.90934i	6.11766	−	10.5961i	−23.6871			13.3952	+	23.4429i			38.4728i
56.18	−1.27055	+	2.20066i	−4.35384	+	2.83621i	0.771399	+	1.33610i	−0.272591	+	0.157381i	−0.709768	−	13.1849i	−0.141798	+	0.245602i	−24.2492			10.9118	−	24.6968i		−	0.799841i
56.19	−1.21956	+	2.11235i	0.359110	+	5.18373i	1.02533	+	1.77592i	−9.50440	+	5.48737i	−11.3878	−	5.56332i	18.3181	−	31.7279i	−24.5148			−26.7421	+	3.72306i		−	26.7688i
56.20	−1.00639	+	1.74312i	−2.73921	+	4.41551i	1.97436	+	3.41970i	−16.9817	+	9.80440i	−4.94005	−	9.21847i	−11.4051	+	19.7541i	−24.0501			−11.9935	−	24.1900i		−	39.4682i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.d	odd	6	1	inner
19.b	odd	2	1	inner
171.l	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	171.4.l.a	✓	116
9.d	odd	6	1	inner	171.4.l.a	✓	116
19.b	odd	2	1	inner	171.4.l.a	✓	116
171.l	even	6	1	inner	171.4.l.a	✓	116

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
171.4.l.a	✓	116	1.a	even	1	1	trivial
171.4.l.a	✓	116	9.d	odd	6	1	inner
171.4.l.a	✓	116	19.b	odd	2	1	inner
171.4.l.a	✓	116	171.l	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(171, [\chi])\).