Newform orbit 32.6.g.a

• Label: 32.6.g.a
• Level: $32$
• Weight: $6$
• Character orbit: 32.g
• Analytic conductor: $5.132$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	32.g (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.13228223402\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(19\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−5.53109	+	1.18618i	7.44181	+	17.9661i	29.1860	−	13.1217i	29.7710	+	12.3316i	−62.4723	−	90.5449i	32.2015	+	32.2015i	−145.866	+	107.197i	−95.5738	+	95.5738i	−179.294	−	32.8933i
5.2	−5.42606	+	1.59933i	−9.61436	−	23.2111i	26.8843	−	17.3561i	−17.5778	−	7.28095i	89.2903	+	110.568i	−10.1181	−	10.1181i	−118.118	+	137.172i	−274.493	+	274.493i	107.023	+	11.3943i
5.3	−5.36185	−	1.80293i	0.399918	+	0.965487i	25.4989	+	19.3341i	−41.3553	−	17.1299i	−0.403590	−	5.89782i	125.552	+	125.552i	−101.863	−	149.639i	171.055	−	171.055i	190.857	+	166.409i
5.4	−4.96462	−	2.71156i	−3.34607	−	8.07812i	17.2949	+	26.9237i	90.7112	+	37.5738i	−5.29232	+	49.1779i	−136.759	−	136.759i	−12.8576	−	180.562i	117.767	−	117.767i	−348.463	−	432.508i
5.5	−3.77244	+	4.21529i	1.59693	+	3.85534i	−3.53735	−	31.8039i	−32.3386	−	13.3951i	−22.2757	−	7.81250i	−71.7275	−	71.7275i	147.407	+	105.067i	159.514	−	159.514i	178.459	−	85.7843i
5.6	−3.71766	−	4.26369i	10.1367	+	24.4721i	−4.35808	+	31.7018i	−58.4422	−	24.2075i	66.6567	−	134.198i	−163.765	−	163.765i	151.369	−	99.2751i	−324.305	+	324.305i	114.054	+	339.175i
5.7	−2.59560	−	5.02622i	−6.55463	−	15.8243i	−18.5257	+	26.0921i	−38.0662	−	15.7675i	−62.5230	+	74.0185i	29.7916	+	29.7916i	179.230	+	25.3893i	−35.6173	+	35.6173i	19.5537	+	232.255i
5.8	−1.94841	+	5.31072i	−5.15883	−	12.4545i	−24.4074	−	20.6949i	50.0856	+	20.7461i	76.1938	−	3.13064i	106.306	+	106.306i	157.460	−	89.2990i	43.3257	−	43.3257i	−207.764	+	225.568i
5.9	−0.578533	−	5.62719i	4.90532	+	11.8425i	−31.3306	+	6.51103i	56.4155	+	23.3681i	63.8021	−	34.4545i	77.5935	+	77.5935i	54.7646	+	172.536i	55.6443	−	55.6443i	98.8584	−	330.980i
5.10	0.0731871	+	5.65638i	9.04699	+	21.8414i	−31.9893	+	0.827948i	83.7809	+	34.7032i	−122.881	+	52.7718i	−34.8236	−	34.8236i	−7.02439	−	180.883i	−223.371	+	223.371i	−190.163	+	476.437i
5.11	1.01888	+	5.56434i	4.68811	+	11.3181i	−29.9238	+	11.3388i	−98.5667	−	40.8277i	−58.2012	+	37.6180i	104.908	+	104.908i	−93.5814	−	154.953i	65.7060	−	65.7060i	126.752	−	590.057i
5.12	2.01099	+	5.28734i	−7.16078	−	17.2876i	−23.9119	+	21.2655i	−8.84628	−	3.66425i	77.0054	−	72.6267i	−158.442	−	158.442i	−160.525	−	83.6652i	−75.7590	+	75.7590i	1.58435	−	54.1420i
5.13	2.44286	−	5.10220i	0.355593	+	0.858478i	−20.0649	−	24.9279i	−47.1761	−	19.5410i	5.24879	+	0.282835i	−64.1149	−	64.1149i	−176.203	+	41.4795i	171.216	−	171.216i	−214.946	+	192.966i
5.14	2.73793	−	4.95012i	−11.3810	−	27.4761i	−17.0075	−	27.1062i	62.9000	+	26.0541i	−167.170	−	18.8905i	32.1730	+	32.1730i	−180.744	+	9.97404i	−453.581	+	453.581i	301.187	−	240.029i
5.15	4.48012	+	3.45378i	−4.24511	−	10.2486i	8.14287	+	30.9466i	36.3338	+	15.0500i	16.3778	−	60.5765i	153.750	+	153.750i	−70.4017	+	166.768i	84.8142	−	84.8142i	110.801	+	192.914i
5.16	4.85839	−	2.89759i	10.7913	+	26.0525i	15.2079	−	28.1553i	−26.5895	−	11.0137i	127.918	+	95.3043i	131.646	+	131.646i	−7.69630	−	180.856i	−390.452	+	390.452i	−161.096	+	23.5365i
5.17	4.94686	+	2.74382i	6.19399	+	14.9536i	16.9429	+	27.1466i	−2.09949	−	0.869637i	−10.3893	+	90.9686i	−63.4847	−	63.4847i	9.32842	+	180.779i	−13.4180	+	13.4180i	−7.99975	−	10.0626i
5.18	5.38021	−	1.74739i	−0.512391	−	1.23702i	25.8932	−	18.8027i	50.7130	+	21.0060i	−4.91834	−	5.76009i	−40.9895	−	40.9895i	106.455	−	146.408i	170.559	−	170.559i	309.552	+	24.4011i
5.19	5.65395	−	0.181169i	−7.87640	−	19.0153i	31.9344	−	2.04864i	−91.3601	−	37.8426i	−47.9778	−	106.085i	18.5982	+	18.5982i	180.184	−	17.3684i	−127.718	+	127.718i	−523.402	−	197.409i
13.1	−5.53109	−	1.18618i	7.44181	−	17.9661i	29.1860	+	13.1217i	29.7710	−	12.3316i	−62.4723	+	90.5449i	32.2015	−	32.2015i	−145.866	−	107.197i	−95.5738	−	95.5738i	−179.294	+	32.8933i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	32.6.g.a	✓	76
4.b	odd	2	1		128.6.g.a		76
32.g	even	8	1	inner	32.6.g.a	✓	76
32.h	odd	8	1		128.6.g.a		76

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
32.6.g.a	✓	76	1.a	even	1	1	trivial
32.6.g.a	✓	76	32.g	even	8	1	inner
128.6.g.a		76	4.b	odd	2	1
128.6.g.a		76	32.h	odd	8	1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(32, [\chi])\).