Newform orbit 117.3.bb.a

• Label: 117.3.bb.a
• Level: $117$
• Weight: $3$
• Character orbit: 117.bb
• Analytic conductor: $3.188$
• Analytic rank: $0$
• Dimension: $104$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$117 = 3^{2} \cdot 13$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	117.bb (of order $12$, degree $4$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$3.18801909302$
Analytic rank:	$0$
Dimension:	$104$
Relative dimension:	$26$ over $\Q(\zeta_{12})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

$\operatorname{Tr}(f)(q) =$ 104 * q - 2 * q^2 - 2 * q^3 - 6 * q^4 - 2 * q^5 + 10 * q^6 - 6 * q^7 + 18 * q^8 - 2 * q^9 - 12 * q^10 - 14 * q^11 - 54 * q^12 - 2 * q^13 - 4 * q^14 - 44 * q^15 + 162 * q^16 - 12 * q^17 + 34 * q^18 - 36 * q^19 - 82 * q^20 + 106 * q^21 + 2 * q^22 - 66 * q^24 + 52 * q^26 + 4 * q^27 + 56 * q^28 + 74 * q^29 - 438 * q^30 - 34 * q^31 + 22 * q^32 - 68 * q^33 + 6 * q^34 - 106 * q^35 - 6 * q^36 + 14 * q^37 - 216 * q^38 - 176 * q^39 - 36 * q^40 - 26 * q^41 - 220 * q^42 + 496 * q^44 - 50 * q^45 + 118 * q^47 + 130 * q^48 + 92 * q^50 + 76 * q^52 - 208 * q^53 + 904 * q^54 - 4 * q^55 - 86 * q^57 - 76 * q^58 + 430 * q^59 - 358 * q^60 - 4 * q^61 - 282 * q^62 - 326 * q^63 + 406 * q^65 - 412 * q^66 - 72 * q^67 - 1068 * q^68 + 204 * q^69 - 106 * q^70 - 8 * q^71 - 642 * q^72 + 90 * q^73 + 356 * q^74 + 204 * q^75 - 194 * q^76 + 432 * q^77 + 646 * q^78 - 16 * q^79 - 278 * q^80 - 182 * q^81 - 12 * q^82 - 536 * q^83 + 310 * q^84 - 234 * q^85 + 414 * q^86 + 434 * q^87 - 6 * q^88 + 400 * q^89 + 798 * q^90 + 120 * q^91 + 48 * q^92 - 116 * q^93 - 244 * q^94 + 858 * q^95 + 1528 * q^96 + 344 * q^97 + 280 * q^98 - 524 * 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	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
7.1	−0.985740	+	3.67883i	1.51073	−	2.59185i	−9.09802	−	5.25275i	−1.72850	+	6.45085i	8.04580	+	8.11262i	−6.33066	+	6.33066i	17.5199	−	17.5199i	−4.43538	−	7.83118i	−22.0278	−	12.7177i
7.2	−0.954072	+	3.56064i	−1.50245	+	2.59666i	−8.30384	−	4.79422i	1.83667	−	6.85456i	−7.81234	−	7.82709i	−1.57948	+	1.57948i	14.5667	−	14.5667i	−4.48530	−	7.80270i	22.6543	+	13.0795i
7.3	−0.849385	+	3.16995i	−2.32988	−	1.88988i	−5.86301	−	3.38501i	0.483124	−	1.80304i	7.96979	−	5.78037i	6.39811	−	6.39811i	6.42800	−	6.42800i	1.85670	+	8.80640i	5.30519	+	3.06295i
7.4	−0.825818	+	3.08200i	1.49477	+	2.60109i	−5.35262	−	3.09034i	−1.67984	+	6.26926i	−9.25095	+	2.45885i	5.59737	−	5.59737i	4.91998	−	4.91998i	−4.53132	+	7.77606i	−17.9346	−	10.3545i
7.5	−0.738153	+	2.75483i	2.39100	−	1.81193i	−3.58010	−	2.06697i	1.54818	−	5.77788i	3.22664	+	7.92428i	5.82885	−	5.82885i	0.270108	−	0.270108i	2.43379	−	8.66468i	14.7743	+	8.52992i
7.6	−0.687085	+	2.56424i	2.77842	+	1.13153i	−2.63912	−	1.52369i	0.672006	−	2.50796i	−4.81053	+	6.34707i	−6.84868	+	6.84868i	−1.78820	+	1.78820i	6.43927	+	6.28775i	5.96928	+	3.44636i
7.7	−0.579256	+	2.16181i	−1.71409	−	2.46209i	−0.873789	−	0.504482i	0.365478	−	1.36398i	6.31547	−	2.27936i	−7.55072	+	7.55072i	−4.73348	+	4.73348i	−3.12377	+	8.44050i	2.73696	+	1.58019i
7.8	−0.533716	+	1.99185i	−1.28337	+	2.71163i	−0.218531	−	0.126169i	−0.459354	+	1.71433i	−4.71622	−	4.00353i	−2.37210	+	2.37210i	−5.46461	+	5.46461i	−5.70591	−	6.96007i	−3.16954	−	1.82993i
7.9	−0.412698	+	1.54021i	−2.94208	+	0.586681i	1.26217	+	0.728715i	−1.42093	+	5.30297i	0.310577	−	4.77354i	2.67174	−	2.67174i	−6.15332	+	6.15332i	8.31161	−	3.45212i	−7.58129	−	4.37706i
7.10	−0.326843	+	1.21980i	0.212026	−	2.99250i	2.08303	+	1.20264i	−2.20981	+	8.24713i	3.58094	+	1.23671i	4.91549	−	4.91549i	−5.71960	+	5.71960i	−8.91009	−	1.26898i	−9.33755	−	5.39104i
7.11	−0.227045	+	0.847343i	−2.93694	+	0.611861i	2.79766	+	1.61523i	2.29731	−	8.57367i	0.148361	−	2.62752i	0.760861	−	0.760861i	−4.48504	+	4.48504i	8.25125	−	3.59400i	6.74325	+	3.89322i
7.12	−0.203464	+	0.759340i	2.94965	−	0.547339i	2.92890	+	1.69100i	−0.980067	+	3.65766i	−0.184532	+	2.35115i	−1.50909	+	1.50909i	−4.10347	+	4.10347i	8.40084	−	3.22892i	−2.57800	−	1.48841i
7.13	−0.150066	+	0.560053i	1.88011	+	2.33777i	3.17296	+	1.83191i	1.19056	−	4.44321i	−1.59142	+	0.702140i	8.23071	−	8.23071i	−3.14207	+	3.14207i	−1.93038	+	8.79054i	2.30977	+	1.33355i
7.14	−0.00774623	+	0.0289093i	1.19499	−	2.75172i	3.46333	+	1.99955i	1.76908	−	6.60228i	0.0702937	+	0.0558619i	−3.33116	+	3.33116i	−0.169286	+	0.169286i	−6.14397	−	6.57659i	0.177164	+	0.102286i
7.15	0.107592	−	0.401541i	−0.191564	+	2.99388i	3.31444	+	1.91359i	−0.322695	+	1.20431i	1.18155	+	0.399039i	−3.52152	+	3.52152i	2.30079	−	2.30079i	−8.92661	−	1.14704i	0.448861	+	0.259150i
7.16	0.224300	−	0.837100i	−2.71604	−	1.27402i	2.81368	+	1.62448i	−1.33823	+	4.99435i	−1.67569	+	1.98783i	−8.46705	+	8.46705i	4.44216	−	4.44216i	5.75373	+	6.92059i	3.88061	+	2.24047i
7.17	0.254432	−	0.949552i	−1.28144	−	2.71254i	2.62719	+	1.51681i	0.323014	−	1.20550i	−2.90174	+	0.526641i	5.37476	−	5.37476i	4.88921	−	4.88921i	−5.71580	+	6.95195i	−1.06250	−	0.613437i
7.18	0.449246	−	1.67661i	−2.83824	+	0.971803i	0.854911	+	0.493583i	−0.762199	+	2.84456i	0.354267	+	5.19519i	6.90561	−	6.90561i	6.12106	−	6.12106i	7.11120	−	5.51642i	4.42680	+	2.55582i
7.19	0.524404	−	1.95710i	2.63868	+	1.42737i	−0.0911449	−	0.0526225i	2.10179	−	7.84399i	4.17724	−	4.41564i	−7.76479	+	7.76479i	5.58001	−	5.58001i	4.92522	+	7.53274i	−14.2493	−	8.22684i
7.20	0.535255	−	1.99760i	2.51826	−	1.63045i	−0.239808	−	0.138453i	−0.486029	+	1.81388i	−1.90908	−	5.90318i	0.855863	−	0.855863i	5.44445	−	5.44445i	3.68326	−	8.21180i	3.36327	+	1.94178i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
117.bb	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	117.3.bb.a	yes	104
3.b	odd	2	1		351.3.be.a		104
9.c	even	3	1		117.3.w.a	✓	104
9.d	odd	6	1		351.3.z.a		104
13.f	odd	12	1		117.3.w.a	✓	104
39.k	even	12	1		351.3.z.a		104
117.bb	odd	12	1	inner	117.3.bb.a	yes	104
117.bc	even	12	1		351.3.be.a		104

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
117.3.w.a	✓	104	9.c	even	3	1
117.3.w.a	✓	104	13.f	odd	12	1
117.3.bb.a	yes	104	1.a	even	1	1	trivial
117.3.bb.a	yes	104	117.bb	odd	12	1	inner
351.3.z.a		104	9.d	odd	6	1
351.3.z.a		104	39.k	even	12	1
351.3.be.a		104	3.b	odd	2	1
351.3.be.a		104	117.bc	even	12	1

Hecke kernels

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