Newform orbit 123.5.k.a

• Label: 123.5.k.a
• Level: $123$
• Weight: $5$
• Character orbit: 123.k
• Analytic conductor: $12.715$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 123 = 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	123.k (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.7145054593\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(54\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 216 q + 410 q^{4} - 5 q^{6} - 6 q^{7} + 128 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
59.1	−7.49407	+	2.43497i	−8.28902	−	3.50601i	37.2877	−	27.0911i	14.3395	+	19.7367i	70.6555	+	6.09076i	0.357282	−	1.09960i	−139.365	+	191.820i	56.4158	+	58.1228i	−155.520	−	112.992i
59.2	−6.78596	+	2.20489i	−6.95168	+	5.71613i	28.2435	−	20.5201i	−24.1580	−	33.2507i	34.5704	−	54.1171i	20.4888	−	63.0579i	−79.3113	+	109.163i	15.6518	−	79.4734i	237.250	+	172.372i
59.3	−6.78436	+	2.20437i	8.85257	−	1.62232i	28.2240	−	20.5059i	27.4201	+	37.7405i	−56.4828	+	30.5208i	−12.4787	+	38.4055i	−79.1915	+	108.998i	75.7361	−	28.7234i	−269.222	−	195.601i
59.4	−6.77455	+	2.20118i	0.0908123	−	8.99954i	28.1050	−	20.4195i	−14.7073	−	20.2428i	19.1944	+	61.1677i	−14.6109	+	44.9676i	−78.4615	+	107.993i	−80.9835	−	1.63454i	144.194	+	104.763i
59.5	−6.73310	+	2.18772i	7.20232	+	5.39690i	27.6043	−	20.0557i	−11.0805	−	15.2510i	−60.3009	−	20.5812i	−7.14724	+	21.9969i	−75.4055	+	103.787i	22.7469	+	77.7404i	107.971	+	78.4453i
59.6	−6.72310	+	2.18447i	−0.500016	+	8.98610i	27.4839	−	19.9682i	10.5184	+	14.4773i	−16.2682	−	61.5067i	6.57763	−	20.2439i	−74.6756	+	102.782i	−80.5000	−	8.98639i	−102.341	−	74.3552i
59.7	−6.37031	+	2.06984i	6.89447	−	5.78501i	23.3524	−	16.9665i	−4.18274	−	5.75704i	−31.9459	+	51.1228i	23.0000	−	70.7866i	−50.6510	+	69.7151i	14.0674	−	79.7691i	38.5615	+	28.0166i
59.8	−5.66298	+	1.84001i	−8.87162	+	1.51472i	15.7394	−	11.4354i	−6.35441	−	8.74610i	47.4527	−	24.9018i	−26.5561	+	81.7313i	−12.0922	+	16.6435i	76.4112	−	26.8761i	52.0779	+	37.8368i
59.9	−5.17429	+	1.68123i	−6.99913	+	5.65793i	11.0025	−	7.99379i	17.7321	+	24.4062i	26.7033	−	41.0429i	−5.40370	+	16.6309i	7.67540	−	10.5643i	16.9756	−	79.2012i	−132.784	−	96.4730i
59.10	−5.14448	+	1.67154i	−4.35218	−	7.87772i	10.7274	−	7.79390i	13.7142	+	18.8760i	35.5577	+	33.2519i	20.1381	−	61.9786i	8.71244	−	11.9916i	−43.1170	+	68.5706i	−102.104	−	74.1832i
59.11	−5.06646	+	1.64619i	−7.30651	−	5.25499i	10.0148	−	7.27615i	−10.0010	−	13.7651i	45.6688	+	14.5962i	5.47348	−	16.8456i	11.3385	−	15.6060i	25.7702	+	76.7912i	73.3295	+	53.2770i
59.12	−4.95740	+	1.61076i	2.60294	+	8.61538i	9.03700	−	6.56577i	7.00982	+	9.64819i	−26.7811	−	38.5172i	2.14304	−	6.59559i	14.7973	−	20.3667i	−67.4494	+	44.8506i	−50.2914	−	36.5388i
59.13	−4.52862	+	1.47144i	8.46275	−	3.06300i	5.39899	−	3.92260i	−9.63826	−	13.2659i	−33.8175	+	26.3235i	−20.2706	+	62.3865i	26.1033	−	35.9282i	62.2361	−	51.8427i	63.1680	+	45.8942i
59.14	−4.46942	+	1.45220i	0.858798	−	8.95893i	4.92259	−	3.57647i	18.1217	+	24.9424i	9.17187	+	41.2884i	−13.6064	+	41.8760i	27.3887	−	37.6973i	−79.5249	−	15.3878i	−117.215	−	85.1617i
59.15	−3.81052	+	1.23811i	8.27915	+	3.52926i	0.0428889	−	0.0311606i	17.9960	+	24.7693i	−35.9175	−	3.19782i	26.3672	−	81.1499i	37.5557	−	51.6910i	56.0886	+	58.4386i	−99.2413	−	72.1030i
59.16	−3.72894	+	1.21161i	8.37520	+	3.29485i	−0.507243	+	0.368534i	−19.4339	−	26.7484i	−35.2227	−	2.13888i	10.1391	−	31.2050i	38.3188	−	52.7413i	59.2879	+	55.1901i	104.876	+	76.1971i
59.17	−3.70969	+	1.20535i	−1.80379	+	8.81739i	−0.635372	+	0.461625i	−21.5341	−	29.6391i	−3.93655	−	34.8839i	−13.0303	+	40.1032i	38.4840	−	52.9687i	−74.4927	−	31.8094i	115.610	+	83.9957i
59.18	−2.79019	+	0.906586i	−8.98662	+	0.490625i	−5.98104	+	4.34548i	−7.10613	−	9.78074i	24.6295	−	9.51608i	13.2874	−	40.8943i	40.3395	−	55.5226i	80.5186	−	8.81813i	28.6945	+	20.8478i
59.19	−2.57069	+	0.835268i	6.88628	−	5.79475i	−7.03350	+	5.11014i	4.48661	+	6.17529i	−12.8623	+	20.6484i	−1.04815	+	3.22587i	39.2330	−	53.9996i	13.8416	−	79.8086i	−16.6917	−	12.1272i
59.20	−2.48159	+	0.806319i	0.975932	−	8.94693i	−7.43611	+	5.40265i	−26.6976	−	36.7461i	4.79221	+	22.9896i	10.9399	−	33.6696i	38.6365	−	53.1786i	−79.0951	−	17.4632i	95.8817	+	69.6622i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
41.d	even	5	1	inner
123.k	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	123.5.k.a	✓	216
3.b	odd	2	1	inner	123.5.k.a	✓	216
41.d	even	5	1	inner	123.5.k.a	✓	216
123.k	odd	10	1	inner	123.5.k.a	✓	216

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
123.5.k.a	✓	216	1.a	even	1	1	trivial
123.5.k.a	✓	216	3.b	odd	2	1	inner
123.5.k.a	✓	216	41.d	even	5	1	inner
123.5.k.a	✓	216	123.k	odd	10	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(123, [\chi])\).