Newform orbit 137.3.d.a

• Label: 137.3.d.a
• Level: $137$
• Weight: $3$
• Character orbit: 137.d
• Analytic conductor: $3.733$
• Analytic rank: $0$
• Dimension: $88$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 137 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	137.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.73297962174\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(22\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) 88 * q - 4 * q^2 - 4 * q^5 - 20 * q^6 - 4 * q^7 + 40 * q^9 + 48 * q^10 + 16 * q^11 + 32 * q^12 + 32 * q^13 - 352 * q^16 - 48 * q^17 - 28 * q^19 + 76 * q^20 - 64 * q^21 + 16 * q^23 - 36 * q^24 - 24 * q^25 - 16 * q^26 + 168 * q^27 + 76 * q^28 - 76 * q^29 + 140 * q^30 - 132 * q^31 + 56 * q^32 + 144 * q^33 + 264 * q^34 - 112 * q^35 - 92 * q^36 - 60 * q^37 - 8 * q^38 + 256 * q^39 + 144 * q^40 - 320 * q^41 - 196 * q^42 - 152 * q^43 + 56 * q^44 - 40 * q^45 + 144 * q^46 - 40 * q^47 - 232 * q^48 - 336 * q^50 + 172 * q^51 - 124 * q^52 + 156 * q^53 + 176 * q^54 - 340 * q^55 + 896 * q^56 - 572 * q^57 - 60 * q^58 + 296 * q^59 + 680 * q^60 - 152 * q^61 - 368 * q^62 + 152 * q^66 + 76 * q^67 - 328 * q^68 - 348 * q^69 - 536 * q^70 - 72 * q^71 + 120 * q^72 - 440 * q^73 - 1200 * q^74 + 32 * q^75 + 788 * q^76 + 804 * q^79 + 428 * q^80 - 136 * q^82 - 368 * q^83 + 1008 * q^84 - 556 * q^85 + 356 * q^86 + 1016 * q^88 - 636 * q^89 + 1168 * q^90 - 220 * q^91 + 96 * q^92 - 448 * q^93 + 32 * q^94 + 172 * q^95 - 284 * q^96 + 412 * q^97 + 336 * q^98

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
10.1	−2.74569	−	2.74569i	1.11310	−	2.68726i			11.0776i	−2.48087	+	1.02761i	−10.4346	+	4.32215i	9.24005	+	9.24005i	19.4329	−	19.4329i	0.381588	+	0.381588i	9.63318	+	3.99019i
10.2	−2.35901	−	2.35901i	−1.59197	+	3.84335i			7.12982i	−4.70862	+	1.95037i	12.8220	−	5.31103i	1.68449	+	1.68449i	7.38327	−	7.38327i	−5.87304	−	5.87304i	15.7086	+	6.50672i
10.3	−2.31991	−	2.31991i	1.21568	−	2.93490i			6.76395i	4.40571	−	1.82490i	−9.62897	+	3.98845i	−7.00584	−	7.00584i	6.41211	−	6.41211i	−0.771827	−	0.771827i	−14.4545	−	5.98723i
10.4	−2.14088	−	2.14088i	−0.290258	+	0.700744i			5.16677i	−1.77400	+	0.734816i	2.12162	−	0.878804i	−4.15373	−	4.15373i	2.49792	−	2.49792i	5.95717	+	5.95717i	5.37109	+	2.22478i
10.5	−1.70641	−	1.70641i	0.0284420	−	0.0686651i			1.82370i	6.65316	−	2.75583i	−0.165705	+	0.0686373i	7.64239	+	7.64239i	−3.71367	+	3.71367i	6.36006	+	6.36006i	−16.0556	−	6.65046i
10.6	−1.57680	−	1.57680i	1.95541	−	4.72077i			0.972597i	−4.90750	+	2.03276i	−10.5270	+	4.36043i	−0.543799	−	0.543799i	−4.77361	+	4.77361i	−12.0981	−	12.0981i	10.9434	+	4.53291i
10.7	−1.45935	−	1.45935i	−1.94475	+	4.69503i			0.259432i	4.36448	−	1.80783i	9.68980	−	4.01364i	0.766925	+	0.766925i	−5.45882	+	5.45882i	−11.8973	−	11.8973i	−9.00758	−	3.73106i
10.8	−1.02507	−	1.02507i	−0.185120	+	0.446919i		−	1.89845i	−3.09027	+	1.28003i	0.647887	−	0.268364i	−3.05744	−	3.05744i	−6.04634	+	6.04634i	6.19849	+	6.19849i	4.47988	+	1.85563i
10.9	−0.709687	−	0.709687i	−0.207519	+	0.500994i		−	2.99269i	−7.62865	+	3.15989i	0.502822	−	0.208276i	4.97757	+	4.97757i	−4.96262	+	4.96262i	6.15603	+	6.15603i	7.65648	+	3.17142i
10.10	−0.594480	−	0.594480i	1.50190	−	3.62590i		−	3.29319i	6.49916	−	2.69204i	−3.04837	+	1.26268i	−2.07726	−	2.07726i	−4.33565	+	4.33565i	−4.52750	−	4.52750i	−5.46398	−	2.26325i
10.11	−0.382949	−	0.382949i	−1.10591	+	2.66990i		−	3.70670i	3.91519	−	1.62172i	1.44594	−	0.598928i	−6.17621	−	6.17621i	−2.95127	+	2.95127i	0.458635	+	0.458635i	−2.12035	−	0.878279i
10.12	0.0578059	+	0.0578059i	1.41134	−	3.40727i		−	3.99332i	0.923205	−	0.382404i	0.278544	−	0.115377i	4.56680	+	4.56680i	0.462061	−	0.462061i	−3.25366	−	3.25366i	0.0754718	+	0.0312615i
10.13	0.445682	+	0.445682i	−2.01641	+	4.86805i		−	3.60273i	−5.38408	+	2.23016i	−3.06829	+	1.27093i	−1.81171	−	1.81171i	3.38841	−	3.38841i	−13.2681	−	13.2681i	−3.39354	−	1.40565i
10.14	0.674958	+	0.674958i	−0.849895	+	2.05183i		−	3.08886i	1.54527	−	0.640074i	−1.95854	+	0.811255i	5.54784	+	5.54784i	4.78469	−	4.78469i	2.87628	+	2.87628i	1.47502	+	0.610973i
10.15	0.718616	+	0.718616i	1.06804	−	2.57847i		−	2.96718i	−6.40764	+	2.65413i	2.62044	−	1.08542i	−9.32927	−	9.32927i	5.00673	−	5.00673i	0.856156	+	0.856156i	−6.51194	−	2.69733i
10.16	1.36656	+	1.36656i	−0.335573	+	0.810145i		−	0.265023i	8.96412	−	3.71306i	−1.56569	+	0.648532i	−6.34788	−	6.34788i	5.82841	−	5.82841i	5.82024	+	5.82024i	17.3241	+	7.17589i
10.17	1.47630	+	1.47630i	0.741483	−	1.79010i			0.358900i	0.876108	−	0.362896i	3.73736	−	1.54807i	0.880212	+	0.880212i	5.37534	−	5.37534i	3.70930	+	3.70930i	1.82914	+	0.757653i
10.18	2.08041	+	2.08041i	2.11068	−	5.09564i			4.65620i	−0.118135	+	0.0489332i	14.9921	−	6.20993i	0.543298	+	0.543298i	−1.36516	+	1.36516i	−15.1466	−	15.1466i	−0.347570	−	0.143968i
10.19	2.13301	+	2.13301i	−2.07735	+	5.01517i			5.09946i	5.40918	−	2.24056i	−15.1284	+	6.26639i	7.25414	+	7.25414i	−2.34515	+	2.34515i	−14.4726	−	14.4726i	16.3170	+	6.75870i
10.20	2.13507	+	2.13507i	0.164961	−	0.398251i			5.11704i	−8.78772	+	3.63999i	1.20250	−	0.498090i	7.20046	+	7.20046i	−2.38496	+	2.38496i	6.23257	+	6.23257i	−26.5340	−	10.9908i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
137.d	odd	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	137.3.d.a	✓	88
137.d	odd	8	1	inner	137.3.d.a	✓	88

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
137.3.d.a	✓	88	1.a	even	1	1	trivial
137.3.d.a	✓	88	137.d	odd	8	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(137, [\chi])\).