Newform orbit 39.5.h.b

• Label: 39.5.h.b
• Level: $39$
• Weight: $5$
• Character orbit: 39.h
• Analytic conductor: $4.031$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	39.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.03142856027\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 10 q^{3} - 146 q^{4} - 102 q^{6} - 210 q^{7} + 46 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
17.1	−3.93465	+	6.81502i	2.53880	+	8.63449i	−22.9630	−	39.7731i	−14.2899			−68.8336	−	16.6717i	−10.0734	+	5.81585i	235.497			−68.1090	+	43.8426i	56.2257	−	97.3858i
17.2	−3.33303	+	5.77297i	−4.98730	−	7.49179i	−14.2181	−	24.6265i	20.0833			59.8726	−	3.82121i	26.1306	−	15.0865i	82.9006			−31.2537	+	74.7275i	−66.9380	+	115.940i
17.3	−2.73803	+	4.74242i	5.98528	−	6.72134i	−6.99367	−	12.1134i	−35.7347			15.4875	+	46.7880i	−57.4797	+	33.1859i	−11.0215			−9.35280	−	80.4582i	97.8430	−	169.469i
17.4	−2.57213	+	4.45507i	8.96733	−	0.766194i	−5.23174	−	9.06164i	33.9480			−19.6517	+	41.9208i	30.5516	−	17.6390i	−28.4813			79.8259	−	13.7414i	−87.3187	+	151.240i
17.5	−2.30428	+	3.99112i	−8.25982	+	3.57427i	−2.61937	−	4.53689i	−18.6874			4.76757	−	41.2021i	36.0084	−	20.7894i	−49.5938			55.4493	−	59.0456i	43.0610	−	74.5838i
17.6	−1.69471	+	2.93532i	−2.17375	+	8.73355i	2.25594	+	3.90740i	39.8972			−21.9519	−	21.1815i	−55.6195	+	32.1120i	−69.5232			−71.5496	−	37.9691i	−67.6141	+	117.111i
17.7	−0.853532	+	1.47836i	4.91676	+	7.53827i	6.54297	+	11.3327i	−27.3592			−15.3409	+	0.834595i	30.0740	−	17.3632i	−49.6516			−32.6509	+	74.1277i	23.3520	−	40.4468i
17.8	−0.797810	+	1.38185i	−5.68593	−	6.97640i	6.72700	+	11.6515i	−7.38171			14.1766	−	2.29125i	−52.0919	+	30.0753i	−46.9974			−16.3404	+	79.3347i	5.88920	−	10.2004i
17.9	0.797810	−	1.38185i	8.88471	+	1.43596i	6.72700	+	11.6515i	7.38171			9.07259	−	11.1317i	−52.0919	+	30.0753i	46.9974			76.8760	+	25.5162i	5.88920	−	10.2004i
17.10	0.853532	−	1.47836i	−8.98671	−	0.488907i	6.54297	+	11.3327i	27.3592			−8.39323	+	12.8683i	30.0740	−	17.3632i	49.6516			80.5219	+	8.78733i	23.3520	−	40.4468i
17.11	1.69471	−	2.93532i	−6.47660	+	6.24930i	2.25594	+	3.90740i	−39.8972			7.36774	+	29.6016i	−55.6195	+	32.1120i	69.5232			2.89261	−	80.9483i	−67.6141	+	117.111i
17.12	2.30428	−	3.99112i	1.03451	+	8.94035i	−2.61937	−	4.53689i	18.6874			38.0658	+	16.4722i	36.0084	−	20.7894i	49.5938			−78.8596	+	18.4977i	43.0610	−	74.5838i
17.13	2.57213	−	4.45507i	−3.82012	−	8.14903i	−5.23174	−	9.06164i	−33.9480			−46.1303	−	3.94151i	30.5516	−	17.6390i	28.4813			−51.8134	+	62.2605i	−87.3187	+	151.240i
17.14	2.73803	−	4.74242i	2.82821	−	8.54408i	−6.99367	−	12.1134i	35.7347			−32.7758	−	36.8065i	−57.4797	+	33.1859i	11.0215			−65.0025	−	48.3289i	97.8430	−	169.469i
17.15	3.33303	−	5.77297i	8.98173	+	0.573235i	−14.2181	−	24.6265i	−20.0833			33.2456	−	49.9406i	26.1306	−	15.0865i	−82.9006			80.3428	+	10.2973i	−66.9380	+	115.940i
17.16	3.93465	−	6.81502i	−8.74709	+	2.11858i	−22.9630	−	39.7731i	14.2899			−19.9786	+	67.9475i	−10.0734	+	5.81585i	−235.497			72.0232	−	37.0628i	56.2257	−	97.3858i
23.1	−3.93465	−	6.81502i	2.53880	−	8.63449i	−22.9630	+	39.7731i	−14.2899			−68.8336	+	16.6717i	−10.0734	−	5.81585i	235.497			−68.1090	−	43.8426i	56.2257	+	97.3858i
23.2	−3.33303	−	5.77297i	−4.98730	+	7.49179i	−14.2181	+	24.6265i	20.0833			59.8726	+	3.82121i	26.1306	+	15.0865i	82.9006			−31.2537	−	74.7275i	−66.9380	−	115.940i
23.3	−2.73803	−	4.74242i	5.98528	+	6.72134i	−6.99367	+	12.1134i	−35.7347			15.4875	−	46.7880i	−57.4797	−	33.1859i	−11.0215			−9.35280	+	80.4582i	97.8430	+	169.469i
23.4	−2.57213	−	4.45507i	8.96733	+	0.766194i	−5.23174	+	9.06164i	33.9480			−19.6517	−	41.9208i	30.5516	+	17.6390i	−28.4813			79.8259	+	13.7414i	−87.3187	−	151.240i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
13.e	even	6	1	inner
39.h	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	39.5.h.b	✓	32
3.b	odd	2	1	inner	39.5.h.b	✓	32
13.e	even	6	1	inner	39.5.h.b	✓	32
39.h	odd	6	1	inner	39.5.h.b	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
39.5.h.b	✓	32	1.a	even	1	1	trivial
39.5.h.b	✓	32	3.b	odd	2	1	inner
39.5.h.b	✓	32	13.e	even	6	1	inner
39.5.h.b	✓	32	39.h	odd	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^32 + 201*T2^30 + 24204*T2^28 + 1903567*T2^26 + 110759412*T2^24 + 4838093367*T2^22 + 164730798787*T2^20 + 4350024034626*T2^18 + 90125266197420*T2^16 + 1429541379077760*T2^14 + 17297506830627504*T2^12 + 150291069518285280*T2^10 + 935973217813488192*T2^8 + 3571801235543084544*T2^6 + 9729455765509898496*T2^4 + 15311752012313602560*T2^2 + 15626574575578137600