Newform orbit 32.5.h.a

• Label: 32.5.h.a
• Level: $32$
• Weight: $5$
• Character orbit: 32.h
• Analytic conductor: $3.308$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	32.h (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.30783881868\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(15\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−3.89846	−	0.895564i	−5.91294	+	14.2751i	14.3959	+	6.98264i	−15.6611	−	37.8093i	35.8356	−	50.3554i	9.35938	+	9.35938i	−49.8685	−	40.1140i	−111.540	−	111.540i	27.1936	+	161.424i
3.2	−3.85182	−	1.07865i	4.50708	−	10.8811i	13.6730	+	8.30956i	−0.960017	−	2.31769i	−29.0974	+	37.0503i	−43.4088	−	43.4088i	−43.7028	−	46.7554i	−40.8081	−	40.8081i	1.19783	+	9.96283i
3.3	−3.70831	+	1.49947i	0.418560	−	1.01049i	11.5032	−	11.1210i	2.54295	+	6.13924i	−0.0369496	+	4.37484i	37.8203	+	37.8203i	−25.9819	+	58.4888i	56.4297	+	56.4297i	−18.6357	−	18.9531i
3.4	−2.40518	−	3.19611i	−1.25855	+	3.03840i	−4.43021	+	15.3744i	7.36609	+	17.7833i	12.7381	−	3.28545i	14.2479	+	14.2479i	59.7938	−	22.8188i	49.6277	+	49.6277i	39.1206	−	66.3149i
3.5	−2.36624	+	3.22504i	−4.58334	+	11.0652i	−4.80179	−	15.2625i	8.41430	+	20.3139i	−24.8403	−	40.9643i	−56.8163	−	56.8163i	60.5843	+	20.6288i	−44.1550	−	44.1550i	−85.4235	−	20.9312i
3.6	−1.38671	+	3.75194i	3.05748	−	7.38140i	−12.1541	−	10.4057i	−13.2656	−	32.0260i	23.4547	+	21.7073i	−14.4988	−	14.4988i	55.8958	−	31.1715i	12.1387	+	12.1387i	138.555	−	5.36083i
3.7	−0.0198807	−	3.99995i	6.02892	−	14.5551i	−15.9992	+	0.159043i	1.24804	+	3.01304i	−58.3395	−	23.8260i	48.9953	+	48.9953i	0.954240	+	63.9929i	−118.227	−	118.227i	12.0272	−	5.05201i
3.8	0.229468	−	3.99341i	−1.30436	+	3.14901i	−15.8947	−	1.83272i	−11.1797	−	26.9902i	12.2760	+	5.93146i	−56.0645	−	56.0645i	−10.9662	+	63.0535i	49.0607	+	49.0607i	−110.348	+	38.4517i
3.9	0.738177	+	3.93130i	2.91925	−	7.04769i	−14.9102	+	5.80398i	18.4238	+	44.4790i	29.8615	+	6.27400i	21.1801	+	21.1801i	−33.8235	−	54.3320i	16.1277	+	16.1277i	−161.260	+	105.263i
3.10	1.64108	+	3.64786i	−3.62363	+	8.74822i	−10.6137	+	11.9728i	−6.00271	−	14.4918i	−37.8589	+	1.13800i	8.54089	+	8.54089i	−61.0931	−	19.0691i	−6.12495	−	6.12495i	43.0132	−	45.6792i
3.11	2.02749	−	3.44808i	−6.35306	+	15.3377i	−7.77857	−	13.9819i	11.7276	+	28.3130i	40.0047	+	53.0028i	37.6043	+	37.6043i	−63.9818	−	1.52705i	−137.607	−	137.607i	121.403	+	16.9665i
3.12	3.34341	−	2.19581i	3.03024	−	7.31565i	6.35679	−	14.6830i	12.4171	+	29.9775i	−5.93247	−	31.1131i	−51.5981	−	51.5981i	−10.9878	−	63.0497i	12.9392	+	12.9392i	107.340	+	72.9614i
3.13	3.36240	−	2.16663i	0.302824	−	0.731083i	6.61144	−	14.5701i	−13.2203	−	31.9166i	−0.565768	−	3.11430i	47.9833	+	47.9833i	−9.33777	−	63.3151i	56.8329	+	56.8329i	−113.603	−	78.6728i
3.14	3.51631	+	1.90672i	5.09563	−	12.3019i	8.72883	+	13.4092i	−6.07195	−	14.6590i	41.3742	−	33.5415i	3.01401	+	3.01401i	5.12558	+	63.7944i	−68.0967	−	68.0967i	6.59975	−	63.1230i
3.15	3.89960	+	0.890575i	−2.61700	+	6.31800i	14.4138	+	6.94577i	3.92862	+	9.48452i	−15.8319	+	22.3070i	−7.35877	−	7.35877i	50.0221	+	39.9222i	24.2072	+	24.2072i	6.87336	+	40.4845i
11.1	−3.89846	+	0.895564i	−5.91294	−	14.2751i	14.3959	−	6.98264i	−15.6611	+	37.8093i	35.8356	+	50.3554i	9.35938	−	9.35938i	−49.8685	+	40.1140i	−111.540	+	111.540i	27.1936	−	161.424i
11.2	−3.85182	+	1.07865i	4.50708	+	10.8811i	13.6730	−	8.30956i	−0.960017	+	2.31769i	−29.0974	−	37.0503i	−43.4088	+	43.4088i	−43.7028	+	46.7554i	−40.8081	+	40.8081i	1.19783	−	9.96283i
11.3	−3.70831	−	1.49947i	0.418560	+	1.01049i	11.5032	+	11.1210i	2.54295	−	6.13924i	−0.0369496	−	4.37484i	37.8203	−	37.8203i	−25.9819	−	58.4888i	56.4297	−	56.4297i	−18.6357	+	18.9531i
11.4	−2.40518	+	3.19611i	−1.25855	−	3.03840i	−4.43021	−	15.3744i	7.36609	−	17.7833i	12.7381	+	3.28545i	14.2479	−	14.2479i	59.7938	+	22.8188i	49.6277	−	49.6277i	39.1206	+	66.3149i
11.5	−2.36624	−	3.22504i	−4.58334	−	11.0652i	−4.80179	+	15.2625i	8.41430	−	20.3139i	−24.8403	+	40.9643i	−56.8163	+	56.8163i	60.5843	−	20.6288i	−44.1550	+	44.1550i	−85.4235	+	20.9312i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
32.h	odd	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	32.5.h.a	✓	60
4.b	odd	2	1		128.5.h.a		60
32.g	even	8	1		128.5.h.a		60
32.h	odd	8	1	inner	32.5.h.a	✓	60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
32.5.h.a	✓	60	1.a	even	1	1	trivial
32.5.h.a	✓	60	32.h	odd	8	1	inner
128.5.h.a		60	4.b	odd	2	1
128.5.h.a		60	32.g	even	8	1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(32, [\chi])\).