Newform orbit 117.3.m.a

• Label: 117.3.m.a
• Level: $117$
• Weight: $3$
• Character orbit: 117.m
• Analytic conductor: $3.188$
• Analytic rank: $0$
• Dimension: $52$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$117 = 3^{2} \cdot 13$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	117.m (of order $6$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$3.18801909302$
Analytic rank:	$0$
Dimension:	$52$
Relative dimension:	$26$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

$\operatorname{Tr}(f)(q) =$ 52 * q - 3 * q^2 - q^3 - 47 * q^4 - 3 * q^6 + 78 * q^8 + q^9 + 2 * q^10 - 3 * q^11 + 13 * q^12 - 6 * q^13 - 6 * q^14 + 24 * q^15 - 75 * q^16 - 12 * q^18 + 15 * q^19 - 6 * q^20 + 69 * q^21 + 17 * q^22 - 42 * q^24 - 92 * q^25 + 12 * q^26 + 14 * q^27 - 18 * q^28 + 24 * q^29 + 147 * q^30 - 48 * q^31 - 159 * q^32 - 120 * q^33 + 12 * q^34 - 78 * q^35 - 181 * q^36 + 27 * q^37 - 36 * q^38 - 35 * q^39 - 44 * q^40 + 66 * q^41 + 300 * q^42 + 62 * q^43 + 120 * q^44 + 117 * q^45 - 6 * q^46 - 153 * q^47 - 244 * q^48 - 146 * q^49 + 432 * q^50 - 63 * q^51 + 128 * q^52 - 132 * q^54 + 23 * q^55 - 171 * q^57 - 3 * q^58 - 3 * q^59 - 207 * q^60 + 12 * q^61 + 201 * q^62 - 192 * q^63 + 270 * q^64 - 3 * q^65 + 48 * q^66 - 48 * q^69 + 159 * q^70 - 432 * q^71 + 93 * q^72 + 346 * q^75 + 138 * q^77 - 93 * q^78 + 9 * q^79 - 84 * q^80 - 215 * q^81 - 85 * q^82 - 135 * q^83 - 246 * q^84 - 24 * q^85 - 306 * q^86 - 123 * q^87 + 145 * q^88 - 441 * q^89 + 543 * q^90 - 138 * q^91 + 864 * q^92 + 183 * q^93 + 362 * q^94 + 252 * q^95 + 747 * q^96 + 147 * q^98 + 522 * q^99

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	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
23.1	−1.93775	+	3.35628i	2.99297	−	0.205259i	−5.50976	−	9.54319i	2.28307	−	3.95440i	−5.11072	+	10.4430i		−	7.95332i	27.2042			8.91574	−	1.22867i	8.84806	+	15.3253i
23.2	−1.93591	+	3.35310i	−1.58366	+	2.54795i	−5.49551	−	9.51849i	−4.14167	+	7.17357i	−5.47770	−	10.2428i			2.09604i	27.0680			−3.98407	−	8.07014i	−16.0358	−	27.7748i
23.3	−1.70867	+	2.95950i	−2.46748	−	1.70633i	−3.83911	−	6.64954i	2.84214	−	4.92273i	9.26599	−	4.38696i			10.4396i	12.5698			3.17690	+	8.42065i	9.71257	+	16.8227i
23.4	−1.53167	+	2.65294i	0.626364	−	2.93388i	−2.69205	−	4.66277i	−1.85161	+	3.20708i	6.82402	+	6.15546i			3.36992i	4.24001			−8.21534	−	3.67536i	−5.67212	−	9.82439i
23.5	−1.37577	+	2.38291i	−2.45641	+	1.72222i	−1.78551	−	3.09260i	3.47982	−	6.02723i	−0.724439	−	8.22280i		−	9.53043i	−1.18034			3.06790	−	8.46097i	9.57490	+	16.5842i
23.6	−1.37367	+	2.37927i	1.51461	+	2.58959i	−1.77395	−	3.07257i	1.55246	−	2.68894i	−8.24190	+	0.0464192i			10.5101i	−1.24207			−4.41192	+	7.84442i	4.26515	+	7.38745i
23.7	−1.21581	+	2.10585i	−2.69201	−	1.32405i	−0.956401	−	1.65653i	−3.20934	+	5.55874i	6.06122	−	4.05917i		−	7.16022i	−5.07529			5.49381	+	7.12868i	−7.80391	−	13.5168i
23.8	−1.10435	+	1.91278i	2.90392	+	0.753140i	−0.439159	−	0.760646i	−3.11152	+	5.38931i	−4.64753	+	4.72285i		−	3.20199i	−6.89483			7.86556	+	4.37413i	−6.87238	−	11.9033i
23.9	−0.946118	+	1.63872i	1.13673	−	2.77630i	0.209721	+	0.363247i	2.45061	−	4.24459i	3.47411	+	4.48950i		−	6.31973i	−8.36263			−6.41568	−	6.31182i	4.63714	+	8.03176i
23.10	−0.620888	+	1.07541i	−2.03795	+	2.20153i	1.22900	+	2.12868i	−0.320482	+	0.555091i	−1.10221	−	3.55854i			5.50883i	−8.01938			−0.693489	−	8.97324i	−0.397967	−	0.689299i
23.11	−0.323897	+	0.561007i	2.87676	−	0.851031i	1.79018	+	3.10068i	1.12097	−	1.94158i	−0.454341	+	1.88953i			7.69849i	−4.91052			7.55149	−	4.89642i	0.726158	+	1.25774i
23.12	−0.239751	+	0.415260i	−2.67694	−	1.35425i	1.88504	+	3.26498i	0.131616	−	0.227965i	1.20417	−	0.786943i			0.552346i	−3.72576			5.33200	+	7.25050i	0.0631100	+	0.109310i
23.13	−0.138316	+	0.239570i	0.191257	+	2.99390i	1.96174	+	3.39783i	−2.47988	+	4.29528i	−0.743703	−	0.368285i		−	6.85336i	−2.19189			−8.92684	+	1.14521i	−0.686015	−	1.18821i
23.14	0.000244066	0	0.000422734i	1.90499	+	2.31755i	2.00000	+	3.46410i	4.59055	−	7.95106i	0.00144465		0.000239669i		−	5.76773i	0.00390505			−1.74205	+	8.82979i	−0.00224079	−	0.00388116i
23.15	0.120411	−	0.208558i	−0.240500	−	2.99034i	1.97100	+	3.41388i	−4.12967	+	7.15280i	−0.652619	−	0.309912i			10.8968i	1.91261			−8.88432	+	1.43836i	0.994515	+	1.72255i
23.16	0.587917	−	1.01830i	−0.850106	−	2.87703i	1.30871	+	2.26675i	0.331589	−	0.574329i	−3.42948	−	0.825792i		−	12.4699i	7.78098			−7.55464	+	4.89157i	−0.389894	−	0.675315i
23.17	0.594188	−	1.02916i	−2.98008	+	0.345181i	1.29388	+	2.24107i	3.27957	−	5.68038i	−1.41548	+	3.27209i			2.41808i	7.82874			8.76170	−	2.05733i	−3.89736	−	6.75042i
23.18	0.670962	−	1.16214i	2.55159	−	1.57778i	1.09962	+	1.90460i	−0.383399	+	0.664067i	−0.121583	−	4.02393i		−	5.48499i	8.31891			4.02121	−	8.05170i	0.514492	+	0.891127i
23.19	0.927144	−	1.60586i	2.67307	+	1.36187i	0.280807	+	0.486372i	−2.88924	+	5.00431i	4.66530	−	3.02993i		−	2.33066i	8.45855			5.29062	+	7.28075i	5.35749	+	9.27944i
23.20	1.01844	−	1.76399i	−0.178787	+	2.99467i	−0.0744351	−	0.128925i	−0.0620677	+	0.107504i	5.10047	+	3.36526i			8.93133i	7.84428			−8.93607	−	1.07081i	0.126424	+	0.218973i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
117.m	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	117.3.m.a	✓	52
3.b	odd	2	1		351.3.m.a		52
9.c	even	3	1		351.3.v.a		52
9.d	odd	6	1		117.3.v.a	yes	52
13.e	even	6	1		117.3.v.a	yes	52
39.h	odd	6	1		351.3.v.a		52
117.m	odd	6	1	inner	117.3.m.a	✓	52
117.r	even	6	1		351.3.m.a		52

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
117.3.m.a	✓	52	1.a	even	1	1	trivial
117.3.m.a	✓	52	117.m	odd	6	1	inner
117.3.v.a	yes	52	9.d	odd	6	1
117.3.v.a	yes	52	13.e	even	6	1
351.3.m.a		52	3.b	odd	2	1
351.3.m.a		52	117.r	even	6	1
351.3.v.a		52	9.c	even	3	1
351.3.v.a		52	39.h	odd	6	1

Hecke kernels

This newform subspace is the entire newspace $S_{3}^{\mathrm{new}}(117, [\chi])$.