Newform orbit 32.7.h.a

• Label: 32.7.h.a
• Level: $32$
• Weight: $7$
• Character orbit: 32.h
• Analytic conductor: $7.362$
• Analytic rank: $0$
• Dimension: $92$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	32.h (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.36173067584\)
Analytic rank:	\(0\)
Dimension:	\(92\)
Relative dimension:	\(23\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 92 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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	−7.90592	−	1.22331i	−2.30421	+	5.56286i	61.0070	+	19.3427i	−24.6192	−	59.4359i	25.0220	−	41.1608i	−184.356	−	184.356i	−458.654	−	227.552i	489.845	+	489.845i	121.929	+	500.012i
3.2	−7.86726	+	1.45129i	−13.7231	+	33.1306i	59.7875	−	22.8354i	61.4842	+	148.436i	59.8812	−	280.563i	332.829	+	332.829i	−437.223	+	266.421i	−393.829	−	393.829i	−699.136	−	1078.55i
3.3	−7.67362	+	2.26176i	17.2727	−	41.7000i	53.7689	−	34.7117i	−86.9420	−	209.897i	−38.2289	+	359.056i	327.807	+	327.807i	−334.093	+	387.977i	−925.059	−	925.059i	1141.89	+	1414.02i
3.4	−7.07639	+	3.73158i	8.05802	−	19.4538i	36.1506	−	52.8122i	47.3798	+	114.385i	15.5716	+	167.732i	−370.791	−	370.791i	−58.7432	+	508.619i	201.963	+	201.963i	−762.114	−	632.631i
3.5	−7.07479	−	3.73461i	15.4738	−	37.3570i	36.1053	+	52.8432i	83.8100	+	202.335i	−248.988	+	206.504i	245.584	+	245.584i	−58.0884	−	508.694i	−640.625	−	640.625i	162.706	−	1744.48i
3.6	−6.15032	−	5.11601i	−1.46788	+	3.54377i	11.6529	+	62.9302i	−21.2394	−	51.2765i	27.1578	−	14.2856i	192.560	+	192.560i	250.282	−	446.657i	505.077	+	505.077i	−131.702	+	424.028i
3.7	−5.73127	+	5.58145i	−12.5303	+	30.2508i	1.69493	−	63.9776i	−55.2565	−	133.401i	−97.0288	−	243.313i	−54.1486	−	54.1486i	347.373	+	376.133i	−242.623	−	242.623i	1061.26	+	456.146i
3.8	−4.87881	−	6.34013i	−18.1109	+	43.7235i	−16.3944	+	61.8646i	26.7176	+	64.5021i	365.572	−	98.4935i	−302.204	−	302.204i	472.214	−	197.883i	−1068.26	−	1068.26i	278.601	−	484.087i
3.9	−3.45440	+	7.21575i	6.10682	−	14.7432i	−40.1342	−	49.8523i	16.7341	+	40.3996i	85.2876	+	94.9941i	159.216	+	159.216i	498.361	−	117.389i	335.413	+	335.413i	−349.320	−	18.8077i
3.10	−2.94181	−	7.43947i	12.3232	−	29.7509i	−46.6915	+	43.7710i	−32.1164	−	77.5358i	−257.584	−	4.15689i	−226.721	−	226.721i	462.991	+	218.595i	−217.773	−	217.773i	−482.345	+	467.024i
3.11	−0.0786463	−	7.99961i	−10.3203	+	24.9155i	−63.9876	+	1.25828i	−65.9372	−	159.187i	200.126	+	80.5992i	442.549	+	442.549i	15.0981	+	511.777i	1.20838	+	1.20838i	−1268.24	+	539.992i
3.12	0.312634	+	7.99389i	−13.5002	+	32.5923i	−63.8045	+	4.99832i	58.5180	+	141.275i	−264.760	−	97.7294i	24.7024	+	24.7024i	−59.9035	−	508.484i	−364.523	−	364.523i	−1111.04	+	511.954i
3.13	0.340220	−	7.99276i	−1.13717	+	2.74537i	−63.7685	−	5.43860i	77.9902	+	188.285i	21.5562	+	10.0232i	−9.79715	−	9.79715i	−65.1647	+	507.836i	509.237	+	509.237i	1531.45	−	559.299i
3.14	1.59216	+	7.83996i	−3.90316	+	9.42306i	−58.9300	+	24.9650i	−65.3898	−	157.865i	−80.0909	−	15.5976i	−135.273	−	135.273i	−289.551	−	422.261i	441.921	+	441.921i	1133.54	−	764.001i
3.15	1.73705	+	7.80914i	19.2736	−	46.5306i	−57.9653	+	27.1297i	−6.47841	−	15.6403i	396.843	+	69.6843i	−298.115	−	298.115i	−312.548	−	405.534i	−1278.14	−	1278.14i	110.884	−	77.7587i
3.16	4.76168	−	6.42856i	11.3331	−	27.3605i	−18.6528	−	61.2215i	−12.4480	−	30.0521i	−121.924	−	203.138i	53.5733	+	53.5733i	−482.385	−	171.607i	−104.679	−	104.679i	−252.465	−	63.0759i
3.17	4.95962	−	6.27711i	−12.6791	+	30.6101i	−14.8042	−	62.2642i	−24.1545	−	58.3141i	129.259	+	231.403i	−334.211	−	334.211i	−464.263	−	215.879i	−260.739	−	260.739i	−485.842	−	137.596i
3.18	5.50965	+	5.80033i	6.14474	−	14.8347i	−3.28761	+	63.9155i	−3.38655	−	8.17585i	119.902	−	46.0925i	414.432	+	414.432i	−388.844	+	333.083i	333.170	+	333.170i	28.7639	−	64.6891i
3.19	6.37330	+	4.83539i	−1.88835	+	4.55888i	17.2380	+	61.6348i	58.7942	+	141.942i	−34.0790	+	19.9242i	−404.754	−	404.754i	−188.165	+	476.170i	498.263	+	498.263i	−311.631	+	1188.93i
3.20	7.09295	+	3.70000i	−19.0234	+	45.9265i	36.6200	+	52.4879i	−33.7350	−	81.4434i	−304.860	+	255.368i	135.315	+	135.315i	65.5382	+	507.788i	−1231.87	−	1231.87i	62.0604	−	702.494i

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	32.7.h.a	✓	92
4.b	odd	2	1		128.7.h.a		92
32.g	even	8	1		128.7.h.a		92
32.h	odd	8	1	inner	32.7.h.a	✓	92

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
32.7.h.a	✓	92	1.a	even	1	1	trivial
32.7.h.a	✓	92	32.h	odd	8	1	inner
128.7.h.a		92	4.b	odd	2	1
128.7.h.a		92	32.g	even	8	1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(32, [\chi])\).