Newform orbit 171.4.t.a

• Label: 171.4.t.a
• Level: $171$
• Weight: $4$
• Character orbit: 171.t
• Analytic conductor: $10.089$
• Analytic rank: $0$
• Dimension: $116$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	171.t (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.

\(\operatorname{Tr}(f)(q) = \) \( 116 q - 6 q^{2} - 3 q^{3} + 446 q^{4} - 3 q^{5} - 53 q^{6} - 8 q^{7} - 48 q^{8} - 31 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
122.1	−5.63157			−4.07511	−	3.22390i	23.7146			2.58149	−	1.49043i	22.9492	+	18.1556i	10.4290	+	18.0636i	−88.4976			6.21297	+	26.2754i	−14.5378	+	8.39343i
122.2	−5.24125			3.71561	+	3.63239i	19.4707			6.93390	−	4.00329i	−19.4744	−	19.0383i	−13.4698	−	23.3303i	−60.1207			0.611466	+	26.9931i	−36.3423	+	20.9822i
122.3	−5.19569			0.405657	+	5.18029i	18.9952			−8.17965	+	4.72252i	−2.10767	−	26.9152i	10.3046	+	17.8481i	−57.1275			−26.6709	+	4.20284i	42.4989	−	24.5368i
122.4	−5.18898			4.58812	−	2.43908i	18.9255			−13.7582	+	7.94327i	−23.8077	+	12.6564i	2.43746	+	4.22180i	−56.6924			15.1018	−	22.3816i	71.3908	−	41.2175i
122.5	−4.94269			1.31243	−	5.02768i	16.4302			9.54211	−	5.50914i	−6.48695	+	24.8503i	−17.1560	−	29.7151i	−41.6679			−23.5550	−	13.1970i	−47.1637	+	27.2300i
122.6	−4.68308			−3.49153	+	3.84828i	13.9312			13.2223	−	7.63390i	16.3511	−	18.0218i	−2.65965	−	4.60665i	−27.7763			−2.61845	−	26.8727i	−61.9210	+	35.7501i
122.7	−4.63770			−4.63640	+	2.34603i	13.5082			−9.90447	+	5.71835i	21.5022	−	10.8802i	−2.19507	−	3.80197i	−25.5456			15.9923	−	21.7542i	45.9339	−	26.5200i
122.8	−4.44084			5.18904	−	0.271816i	11.7211			3.30635	−	1.90892i	−23.0437	+	1.20709i	2.98474	+	5.16972i	−16.5248			26.8522	−	2.82093i	−14.6830	+	8.47722i
122.9	−4.39313			−2.15128	−	4.72991i	11.2996			−16.9490	+	9.78549i	9.45083	+	20.7791i	−10.6738	−	18.4875i	−14.4954			−17.7440	+	20.3507i	74.4589	−	42.9889i
122.10	−4.17093			0.815694	−	5.13173i	9.39667			4.61421	−	2.66402i	−3.40220	+	21.4041i	9.76769	+	16.9181i	−5.82542			−25.6693	−	8.37184i	−19.2456	+	11.1114i
122.11	−4.08563			3.64705	+	3.70122i	8.69239			12.7389	−	7.35482i	−14.9005	−	15.1218i	16.7231	+	28.9653i	−2.82887			−0.398044	+	26.9971i	−52.0466	+	30.0491i
122.12	−3.88048			−4.78215	−	2.03251i	7.05811			13.4645	−	7.77373i	18.5570	+	7.88710i	−0.863566	−	1.49574i	3.65497			18.7378	+	19.4395i	−52.2487	+	30.1658i
122.13	−3.61579			0.362044	+	5.18352i	5.07392			−10.1372	+	5.85269i	−1.30908	−	18.7425i	−12.5866	−	21.8006i	10.5801			−26.7378	+	3.75333i	36.6538	−	21.1621i
122.14	−3.37854			−4.45265	−	2.67841i	3.41455			−3.32465	+	1.91949i	15.0435	+	9.04912i	−1.59457	−	2.76188i	15.4921			12.6522	+	23.8521i	11.2325	−	6.48507i
122.15	−3.20363			5.10476	+	0.970290i	2.26326			−4.58776	+	2.64875i	−16.3538	−	3.10845i	−7.29614	−	12.6373i	18.3784			25.1171	+	9.90618i	14.6975	−	8.48561i
122.16	−2.80734			3.16558	−	4.12057i	−0.118845			−2.87441	+	1.65954i	−8.88686	+	11.5678i	3.40965	+	5.90568i	22.7924			−6.95818	−	26.0880i	8.06944	−	4.65889i
122.17	−2.72125			−5.04587	+	1.24065i	−0.594773			−8.37819	+	4.83715i	13.7311	−	3.37613i	16.9653	+	29.3848i	23.3886			23.9216	−	12.5203i	22.7992	−	13.1631i
122.18	−2.57455			1.00199	+	5.09863i	−1.37171			3.89137	−	2.24668i	−2.57967	−	13.1267i	−2.51209	−	4.35107i	24.1279			−24.9920	+	10.2176i	−10.0185	+	5.78419i
122.19	−2.52657			4.01675	+	3.29631i	−1.61642			−17.9687	+	10.3742i	−10.1486	−	8.32838i	11.5762	+	20.0506i	24.2966			5.26862	+	26.4810i	45.3993	−	26.2113i
122.20	−2.15208			4.83606	−	1.90066i	−3.36853			18.1088	−	10.4551i	−10.4076	+	4.09037i	−5.96316	−	10.3285i	24.4660			19.7750	−	18.3834i	−38.9718	+	22.5004i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
171.t	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.t.a	yes	116
9.d	odd	6	1		171.4.k.a	✓	116
19.d	odd	6	1		171.4.k.a	✓	116
171.t	even	6	1	inner	171.4.t.a	yes	116

    

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

Hecke kernels

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