Newform orbit 32.3.h.a

• Label: 32.3.h.a
• Level: $32$
• Weight: $3$
• Character orbit: 32.h
• Analytic conductor: $0.872$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	32.h (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.871936845953\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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	−1.96275	−	0.384217i	−1.10785	+	2.67458i	3.70475	+	1.50824i	2.95565	+	7.13556i	3.20204	−	4.82387i	−4.18452	−	4.18452i	−6.69200	−	4.38373i	0.437918	+	0.437918i	−3.05958	−	15.1409i
3.2	−1.46783	+	1.35848i	2.10187	−	5.07436i	0.309042	−	3.98804i	1.74699	+	4.21761i	3.80826	+	10.3037i	−0.392379	−	0.392379i	4.96407	+	6.27359i	−14.9674	−	14.9674i	−8.29385	−	3.81747i
3.3	−1.44490	−	1.38284i	0.936461	−	2.26082i	0.175499	+	3.99615i	−3.18221	−	7.68254i	−4.47945	+	1.97169i	3.67370	+	3.67370i	5.27246	−	6.01674i	2.12963	+	2.12963i	−6.02574	+	15.5010i
3.4	−0.658450	+	1.88850i	−1.31872	+	3.18367i	−3.13289	−	2.48697i	−0.659338	−	1.59178i	−5.14406	−	4.58669i	9.54718	+	9.54718i	6.75950	−	4.27892i	−2.03276	−	2.03276i	3.44023	−	0.197052i
3.5	0.682385	−	1.87999i	0.299792	−	0.723762i	−3.06870	−	2.56575i	1.34740	+	3.25291i	−1.15609	−	1.05749i	0.583225	+	0.583225i	−6.91761	+	4.01829i	5.93000	+	5.93000i	7.03487	−	0.313357i
3.6	1.20513	+	1.59614i	0.527719	−	1.27403i	−1.09531	+	3.84712i	−0.642823	−	1.55191i	2.66949	−	0.693061i	−4.95044	−	4.95044i	−7.46052	+	2.88803i	5.01930	+	5.01930i	1.70238	−	2.89629i
3.7	1.93931	−	0.488972i	−1.73217	+	4.18183i	3.52181	−	1.89653i	−1.85856	−	4.48696i	−1.31441	+	8.95683i	−5.27676	−	5.27676i	5.90252	−	5.40002i	−8.12333	−	8.12333i	−5.79831	−	7.79280i
11.1	−1.96275	+	0.384217i	−1.10785	−	2.67458i	3.70475	−	1.50824i	2.95565	−	7.13556i	3.20204	+	4.82387i	−4.18452	+	4.18452i	−6.69200	+	4.38373i	0.437918	−	0.437918i	−3.05958	+	15.1409i
11.2	−1.46783	−	1.35848i	2.10187	+	5.07436i	0.309042	+	3.98804i	1.74699	−	4.21761i	3.80826	−	10.3037i	−0.392379	+	0.392379i	4.96407	−	6.27359i	−14.9674	+	14.9674i	−8.29385	+	3.81747i
11.3	−1.44490	+	1.38284i	0.936461	+	2.26082i	0.175499	−	3.99615i	−3.18221	+	7.68254i	−4.47945	−	1.97169i	3.67370	−	3.67370i	5.27246	+	6.01674i	2.12963	−	2.12963i	−6.02574	−	15.5010i
11.4	−0.658450	−	1.88850i	−1.31872	−	3.18367i	−3.13289	+	2.48697i	−0.659338	+	1.59178i	−5.14406	+	4.58669i	9.54718	−	9.54718i	6.75950	+	4.27892i	−2.03276	+	2.03276i	3.44023	+	0.197052i
11.5	0.682385	+	1.87999i	0.299792	+	0.723762i	−3.06870	+	2.56575i	1.34740	−	3.25291i	−1.15609	+	1.05749i	0.583225	−	0.583225i	−6.91761	−	4.01829i	5.93000	−	5.93000i	7.03487	+	0.313357i
11.6	1.20513	−	1.59614i	0.527719	+	1.27403i	−1.09531	−	3.84712i	−0.642823	+	1.55191i	2.66949	+	0.693061i	−4.95044	+	4.95044i	−7.46052	−	2.88803i	5.01930	−	5.01930i	1.70238	+	2.89629i
11.7	1.93931	+	0.488972i	−1.73217	−	4.18183i	3.52181	+	1.89653i	−1.85856	+	4.48696i	−1.31441	−	8.95683i	−5.27676	+	5.27676i	5.90252	+	5.40002i	−8.12333	+	8.12333i	−5.79831	+	7.79280i
19.1	−1.98676	−	0.229757i	1.58190	+	0.655246i	3.89442	+	0.912943i	4.18866	−	1.73500i	−2.99232	−	1.66527i	3.93197	+	3.93197i	−7.52753	−	2.70857i	−4.29089	−	4.29089i	−8.72048	+	2.48465i
19.2	−1.62478	+	1.16623i	−4.68670	−	1.94129i	1.27980	−	3.78974i	−4.51028	+	1.86822i	9.87885	−	2.31161i	−3.85317	−	3.85317i	2.34032	+	7.65003i	11.8326	+	11.8326i	5.14942	−	8.29547i
19.3	−0.360897	+	1.96717i	2.49683	+	1.03422i	−3.73951	−	1.41989i	−0.452310	+	0.187353i	−2.93558	+	4.53843i	0.429965	+	0.429965i	4.14274	−	6.84381i	−1.19943	−	1.19943i	−0.205317	−	0.957385i
19.4	−0.108191	−	1.99707i	4.35131	+	1.80237i	−3.97659	+	0.432130i	−2.81639	+	1.16659i	3.12870	−	8.88489i	−6.23443	−	6.23443i	1.29322	+	7.89478i	9.32143	+	9.32143i	2.63447	+	5.49832i
19.5	0.345994	−	1.96984i	−3.70255	−	1.53365i	−3.76058	−	1.36311i	7.20074	−	2.98264i	−4.30210	+	6.76281i	4.26150	+	4.26150i	−3.98625	+	6.93612i	4.99283	+	4.99283i	−3.38393	−	15.2163i
19.6	1.61758	+	1.17620i	−1.37292	−	0.568682i	1.23313	+	3.80518i	2.28872	−	0.948019i	−1.55193	−	2.53471i	−6.37744	−	6.37744i	−2.48095	+	7.60558i	−4.80245	−	4.80245i	4.81725	+	1.15849i

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	32.3.h.a	✓	28
3.b	odd	2	1		288.3.u.a		28
4.b	odd	2	1		128.3.h.a		28
8.b	even	2	1		256.3.h.b		28
8.d	odd	2	1		256.3.h.a		28
32.g	even	8	1		128.3.h.a		28
32.g	even	8	1		256.3.h.a		28
32.h	odd	8	1	inner	32.3.h.a	✓	28
32.h	odd	8	1		256.3.h.b		28
96.o	even	8	1		288.3.u.a		28

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
32.3.h.a	✓	28	1.a	even	1	1	trivial
32.3.h.a	✓	28	32.h	odd	8	1	inner
128.3.h.a		28	4.b	odd	2	1
128.3.h.a		28	32.g	even	8	1
256.3.h.a		28	8.d	odd	2	1
256.3.h.a		28	32.g	even	8	1
256.3.h.b		28	8.b	even	2	1
256.3.h.b		28	32.h	odd	8	1
288.3.u.a		28	3.b	odd	2	1
288.3.u.a		28	96.o	even	8	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(32, [\chi])\).