Newform orbit 117.4.bc.a

• Label: 117.4.bc.a
• Level: $117$
• Weight: $4$
• Character orbit: 117.bc
• Analytic conductor: $6.903$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.90322347067\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(40\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 6 q^{2} - 2 q^{3} - 6 q^{4} - 6 q^{5} - 50 q^{6} - 26 q^{7} - 66 q^{8} - 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
20.1	−5.25888	−	1.40911i	5.13438	−	0.798841i	18.7420	+	10.8207i	−5.40122	−	1.44725i	−28.1268	−	3.03391i	−20.4228	+	20.4228i	−52.5164	−	52.5164i	25.7237	−	8.20310i	26.3650	+	15.2219i
20.2	−4.93223	−	1.32159i	−1.28499	+	5.03476i	15.6521	+	9.03675i	−16.2446	−	4.35273i	12.9917	−	23.1344i	14.5223	−	14.5223i	−36.3718	−	36.3718i	−23.6976	−	12.9392i	74.3696	+	42.9373i
20.3	−4.85048	−	1.29968i	−4.29507	+	2.92444i	14.9098	+	8.60815i	13.8602	+	3.71382i	24.6340	−	8.60271i	−13.4160	+	13.4160i	−32.7252	−	32.7252i	9.89529	−	25.1214i	−62.4016	−	36.0276i
20.4	−4.74226	−	1.27068i	3.54864	+	3.79567i	13.9462	+	8.05182i	17.7720	+	4.76199i	−12.0055	−	22.5093i	20.6322	−	20.6322i	−28.1324	−	28.1324i	−1.81428	+	26.9390i	−78.2284	−	45.1652i
20.5	−4.73780	−	1.26949i	−4.39181	−	2.77705i	13.9070	+	8.02918i	−10.1371	−	2.71623i	17.2821	+	18.7325i	−3.61213	+	3.61213i	−27.9489	−	27.9489i	11.5760	+	24.3925i	44.5793	+	25.7379i
20.6	−4.40958	−	1.18154i	1.85786	−	4.85266i	11.1202	+	6.42024i	−5.71245	−	1.53065i	−13.9260	+	19.2031i	18.3444	−	18.3444i	−15.6253	−	15.6253i	−20.0967	−	18.0311i	23.3810	+	13.4990i
20.7	−4.26904	−	1.14389i	0.705396	−	5.14805i	9.98803	+	5.76659i	14.1978	+	3.80429i	−8.90015	+	21.1703i	−8.90512	+	8.90512i	−11.0418	−	11.0418i	−26.0048	−	7.26283i	−56.2593	−	32.4813i
20.8	−3.38831	−	0.907896i	−5.19587	+	0.0543605i	3.72820	+	2.15248i	2.96751	+	0.795142i	17.6546	+	4.53312i	11.7767	−	11.7767i	9.16526	+	9.16526i	26.9941	−	0.564900i	−9.33296	−	5.38838i
20.9	−3.30107	−	0.884518i	2.90704	+	4.30687i	3.18647	+	1.83971i	−1.29369	−	0.346644i	−5.78683	−	16.7886i	−6.34507	+	6.34507i	10.4409	+	10.4409i	−10.0982	+	25.0405i	3.96395	+	2.28859i
20.10	−3.18581	−	0.853636i	5.15204	−	0.675630i	2.49250	+	1.43905i	5.30111	+	1.42043i	−16.9902	−	2.24554i	2.18268	−	2.18268i	11.9452	+	11.9452i	26.0870	−	6.96175i	−15.6758	−	9.05043i
20.11	−3.03682	−	0.813714i	−1.50867	+	4.97231i	1.63196	+	0.942213i	−0.653671	−	0.175151i	8.62762	−	13.8724i	−13.4182	+	13.4182i	13.5956	+	13.5956i	−22.4478	−	15.0032i	1.84256	+	1.06380i
20.12	−2.88480	−	0.772979i	5.11794	+	0.898140i	0.796351	+	0.459774i	−19.6130	−	5.25528i	−14.0700	−	6.54701i	4.79580	−	4.79580i	14.9526	+	14.9526i	25.3867	+	9.19326i	52.5172	+	30.3208i
20.13	−2.77566	−	0.743737i	−2.31409	−	4.65242i	0.222955	+	0.128723i	−8.34903	−	2.23712i	2.96295	+	14.6346i	−19.2458	+	19.2458i	15.7323	+	15.7323i	−16.2900	+	21.5322i	21.5103	+	12.4190i
20.14	−1.97465	−	0.529106i	−3.30338	−	4.01095i	−3.30891	−	1.91040i	19.5437	+	5.23671i	4.40080	+	9.66806i	4.34997	−	4.34997i	17.0875	+	17.0875i	−5.17539	+	26.4993i	−35.8211	−	20.6813i
20.15	−1.39939	−	0.374964i	−4.64612	+	2.32671i	−5.11052	−	2.95056i	−18.9784	−	5.08526i	7.37415	−	1.51383i	−6.91418	+	6.91418i	14.2406	+	14.2406i	16.1729	−	21.6203i	24.6514	+	14.2325i
20.16	−1.30283	−	0.349091i	3.93234	−	3.39657i	−5.35271	−	3.09039i	6.41146	+	1.71795i	−6.30887	+	3.05240i	8.50707	−	8.50707i	13.5247	+	13.5247i	3.92658	−	26.7130i	−7.75330	−	4.47637i
20.17	−1.07542	−	0.288157i	−1.79579	+	4.87598i	−5.85472	−	3.38022i	13.7426	+	3.68233i	3.33627	−	4.72624i	12.2275	−	12.2275i	11.6203	+	11.6203i	−20.5503	−	17.5124i	−13.7180	−	7.92008i
20.18	−0.797871	−	0.213789i	−1.76975	−	4.88549i	−6.33731	−	3.65885i	−12.3469	−	3.30833i	0.367568	+	4.27634i	22.9144	−	22.9144i	8.94679	+	8.94679i	−20.7360	+	17.2922i	9.14391	+	5.27924i
20.19	−0.763530	−	0.204587i	2.72864	−	4.42205i	−6.38708	−	3.68758i	−4.57573	−	1.22606i	−2.98809	+	2.81813i	−17.3884	+	17.3884i	8.59384	+	8.59384i	−12.1091	−	24.1323i	3.24287	+	1.87227i
20.20	−0.578769	−	0.155081i	4.51870	+	2.56542i	−6.61728	−	3.82049i	18.7972	+	5.03670i	−2.21743	−	2.18555i	−23.8934	+	23.8934i	6.62690	+	6.62690i	13.8372	+	23.1847i	−10.0982	−	5.83017i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	117.4.bc.a	yes	160
9.d	odd	6	1		117.4.x.a	✓	160
13.f	odd	12	1		117.4.x.a	✓	160
117.bc	even	12	1	inner	117.4.bc.a	yes	160

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
117.4.x.a	✓	160	9.d	odd	6	1
117.4.x.a	✓	160	13.f	odd	12	1
117.4.bc.a	yes	160	1.a	even	1	1	trivial
117.4.bc.a	yes	160	117.bc	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(117, [\chi])\).