Newform orbit 117.2.bc.a

• Label: 117.2.bc.a
• Level: $117$
• Weight: $2$
• Character orbit: 117.bc
• Analytic conductor: $0.934$
• Analytic rank: $0$
• Dimension: $48$
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	$0.934249703649$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$12$ 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.

$\operatorname{Tr}(f)(q) =$ 48 * q - 6 * q^2 - 2 * q^3 - 6 * q^4 - 6 * q^5 - 2 * q^6 + 2 * q^7 - 30 * q^8 - 2 * q^9 - 12 * q^10 + 6 * q^11 - 18 * q^12 - 2 * q^13 - 12 * q^14 + 4 * q^15 + 14 * q^16 - 2 * q^18 - 4 * q^19 - 6 * q^20 + 22 * q^21 + 2 * q^22 - 12 * q^23 - 18 * q^24 + 48 * q^26 - 32 * q^27 - 6 * q^29 + 66 * q^30 + 6 * q^31 + 30 * q^32 - 56 * q^33 - 6 * q^34 - 6 * q^35 - 6 * q^36 - 6 * q^37 - 36 * q^38 - 32 * q^39 - 12 * q^40 + 18 * q^41 + 80 * q^42 - 12 * q^44 + 34 * q^45 - 12 * q^46 + 30 * q^47 + 22 * q^48 - 12 * q^50 - 16 * q^52 - 56 * q^54 - 4 * q^55 - 12 * q^56 - 2 * q^57 - 28 * q^58 + 30 * q^59 - 58 * q^60 - 4 * q^61 - 18 * q^62 - 2 * q^63 + 30 * q^65 + 32 * q^66 - 16 * q^67 - 48 * q^69 - 46 * q^70 + 48 * q^71 + 126 * q^72 - 22 * q^73 + 24 * q^75 - 18 * q^76 - 72 * q^77 + 94 * q^78 + 8 * q^79 + 54 * q^80 - 14 * q^81 - 12 * q^82 + 72 * q^83 - 110 * q^84 + 78 * q^85 + 102 * q^86 + 14 * q^87 - 6 * q^88 + 114 * q^90 - 16 * q^91 + 120 * q^92 - 44 * q^93 - 52 * q^94 - 6 * q^95 + 16 * q^96 + 48 * q^97 - 36 * q^98 - 32 * 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	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
20.1	−2.60753	−	0.698685i	−1.12895	−	1.31357i	4.57899	+	2.64368i	1.93643	+	0.518864i	2.02600	+	4.21396i	1.26724	−	1.26724i	−6.27505	−	6.27505i	−0.450943	+	2.96591i	−4.68676	−	2.70590i
20.2	−2.30434	−	0.617447i	0.895864	+	1.48237i	3.19671	+	1.84562i	−0.605981	−	0.162372i	−1.14909	−	3.96904i	−1.66339	+	1.66339i	−2.85295	−	2.85295i	−1.39486	+	2.65601i	1.29613	+	0.748322i
20.3	−1.74626	−	0.467910i	0.825450	−	1.52271i	1.09844	+	0.634187i	−3.66400	−	0.981766i	−2.15394	+	2.27281i	−0.671277	+	0.671277i	0.935276	+	0.935276i	−1.63726	−	2.51383i	5.93893	+	3.42884i
20.4	−1.55681	−	0.417145i	−1.30216	+	1.14210i	0.517589	+	0.298830i	−0.690114	−	0.184916i	2.50363	−	1.23483i	3.10547	−	3.10547i	1.59819	+	1.59819i	0.391235	−	2.97438i	0.997239	+	0.575756i
20.5	−1.03000	−	0.275986i	−1.73204	+	0.00536576i	−0.747329	−	0.431471i	2.22216	+	0.595426i	1.78548	+	0.472494i	−3.66177	+	3.66177i	2.15868	+	2.15868i	2.99994	−	0.0185875i	−2.12448	−	1.22657i
20.6	−0.724963	−	0.194253i	1.63234	+	0.579188i	−1.24421	−	0.718347i	0.862152	+	0.231013i	−1.07088	−	0.736978i	2.34485	−	2.34485i	1.82389	+	1.82389i	2.32908	+	1.89087i	−0.580153	−	0.334952i
20.7	0.0826047	+	0.0221339i	0.405260	−	1.68397i	−1.72572	−	0.996343i	1.73943	+	0.466078i	0.0707492	−	0.130134i	0.362366	−	0.362366i	−0.241441	−	0.241441i	−2.67153	−	1.36489i	0.133369	+	0.0770004i
20.8	0.425586	+	0.114035i	−1.46672	−	0.921268i	−1.56393	−	0.902936i	−2.94562	−	0.789277i	−0.519158	−	0.559337i	0.205334	−	0.205334i	−1.18572	−	1.18572i	1.30253	+	2.70248i	−1.16361	−	0.671810i
20.9	0.863432	+	0.231356i	0.168175	+	1.72387i	−1.04006	−	0.600479i	2.81220	+	0.753527i	−0.253619	+	1.52735i	−0.653306	+	0.653306i	−2.02325	−	2.02325i	−2.94343	+	0.579822i	2.25381	+	1.30124i
20.10	1.30411	+	0.349436i	1.57241	−	0.726303i	−0.153448	−	0.0885935i	−0.236109	−	0.0632653i	2.30440	−	0.397723i	−2.19861	+	2.19861i	−2.07851	−	2.07851i	1.94497	−	2.28410i	−0.285806	−	0.165010i
20.11	1.86810	+	0.500557i	1.15921	+	1.28694i	1.50720	+	0.870184i	−4.03603	−	1.08145i	1.52135	+	2.98439i	1.89104	−	1.89104i	−0.355058	−	0.355058i	−0.312441	+	2.98369i	−6.99840	−	4.04053i
20.12	2.19401	+	0.587883i	−1.52885	+	0.814015i	2.73602	+	1.57964i	0.239470	+	0.0641657i	−3.83286	+	0.887173i	−0.693990	+	0.693990i	1.86196	+	1.86196i	1.67476	−	2.48901i	0.487677	+	0.281560i
41.1	−2.60753	+	0.698685i	−1.12895	+	1.31357i	4.57899	−	2.64368i	1.93643	−	0.518864i	2.02600	−	4.21396i	1.26724	+	1.26724i	−6.27505	+	6.27505i	−0.450943	−	2.96591i	−4.68676	+	2.70590i
41.2	−2.30434	+	0.617447i	0.895864	−	1.48237i	3.19671	−	1.84562i	−0.605981	+	0.162372i	−1.14909	+	3.96904i	−1.66339	−	1.66339i	−2.85295	+	2.85295i	−1.39486	−	2.65601i	1.29613	−	0.748322i
41.3	−1.74626	+	0.467910i	0.825450	+	1.52271i	1.09844	−	0.634187i	−3.66400	+	0.981766i	−2.15394	−	2.27281i	−0.671277	−	0.671277i	0.935276	−	0.935276i	−1.63726	+	2.51383i	5.93893	−	3.42884i
41.4	−1.55681	+	0.417145i	−1.30216	−	1.14210i	0.517589	−	0.298830i	−0.690114	+	0.184916i	2.50363	+	1.23483i	3.10547	+	3.10547i	1.59819	−	1.59819i	0.391235	+	2.97438i	0.997239	−	0.575756i
41.5	−1.03000	+	0.275986i	−1.73204	−	0.00536576i	−0.747329	+	0.431471i	2.22216	−	0.595426i	1.78548	−	0.472494i	−3.66177	−	3.66177i	2.15868	−	2.15868i	2.99994	+	0.0185875i	−2.12448	+	1.22657i
41.6	−0.724963	+	0.194253i	1.63234	−	0.579188i	−1.24421	+	0.718347i	0.862152	−	0.231013i	−1.07088	+	0.736978i	2.34485	+	2.34485i	1.82389	−	1.82389i	2.32908	−	1.89087i	−0.580153	+	0.334952i
41.7	0.0826047	−	0.0221339i	0.405260	+	1.68397i	−1.72572	+	0.996343i	1.73943	−	0.466078i	0.0707492	+	0.130134i	0.362366	+	0.362366i	−0.241441	+	0.241441i	−2.67153	+	1.36489i	0.133369	−	0.0770004i
41.8	0.425586	−	0.114035i	−1.46672	+	0.921268i	−1.56393	+	0.902936i	−2.94562	+	0.789277i	−0.519158	+	0.559337i	0.205334	+	0.205334i	−1.18572	+	1.18572i	1.30253	−	2.70248i	−1.16361	+	0.671810i

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	117.2.bc.a	yes	48
3.b	odd	2	1		351.2.bf.a		48
9.c	even	3	1		351.2.ba.a		48
9.d	odd	6	1		117.2.x.a	✓	48
13.f	odd	12	1		117.2.x.a	✓	48
39.k	even	12	1		351.2.ba.a		48
117.bb	odd	12	1		351.2.bf.a		48
117.bc	even	12	1	inner	117.2.bc.a	yes	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
117.2.x.a	✓	48	9.d	odd	6	1
117.2.x.a	✓	48	13.f	odd	12	1
117.2.bc.a	yes	48	1.a	even	1	1	trivial
117.2.bc.a	yes	48	117.bc	even	12	1	inner
351.2.ba.a		48	9.c	even	3	1
351.2.ba.a		48	39.k	even	12	1
351.2.bf.a		48	3.b	odd	2	1
351.2.bf.a		48	117.bb	odd	12	1

Hecke kernels

This newform subspace is the entire newspace $S_{2}^{\mathrm{new}}(117, [\chi])$.