Newform orbit 189.2.bd.a

• Label: 189.2.bd.a
• Level: $189$
• Weight: $2$
• Character orbit: 189.bd
• Analytic conductor: $1.509$
• Analytic rank: $0$
• Dimension: $132$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	189.bd (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.50917259820\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(22\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) 132 * q - 3 * q^2 - 9 * q^3 - 3 * q^4 - 9 * q^5 + 18 * q^6 - 6 * q^7 - 18 * q^8 - 15 * q^9 - 9 * q^10 + 9 * q^11 - 9 * q^12 - 42 * q^14 - 24 * q^15 - 15 * q^16 - 9 * q^17 - 3 * q^18 - 9 * q^19 - 18 * q^20 + 15 * q^21 - 12 * q^22 + 30 * q^23 - 36 * q^24 - 3 * q^25 - 12 * q^28 + 6 * q^29 - 3 * q^30 - 9 * q^31 - 51 * q^32 - 9 * q^33 + 18 * q^34 - 9 * q^35 - 6 * q^37 - 9 * q^38 - 9 * q^39 - 9 * q^40 + 27 * q^42 - 12 * q^43 - 63 * q^45 - 6 * q^46 + 45 * q^47 + 30 * q^49 - 9 * q^50 + 33 * q^51 - 9 * q^52 + 45 * q^53 + 117 * q^54 - 51 * q^56 - 3 * q^58 - 9 * q^59 - 15 * q^60 - 63 * q^61 + 99 * q^62 - 33 * q^63 + 18 * q^64 - 102 * q^65 + 63 * q^66 - 3 * q^67 + 144 * q^68 - 108 * q^69 - 15 * q^70 + 18 * q^71 + 15 * q^72 - 33 * q^74 - 9 * q^75 - 36 * q^76 - 57 * q^77 + 66 * q^78 - 21 * q^79 - 72 * q^80 + 57 * q^81 - 18 * q^82 + 90 * q^83 + 51 * q^84 + 9 * q^85 - 33 * q^86 - 9 * q^87 + 45 * q^88 - 9 * q^89 - 81 * q^90 - 21 * q^91 + 150 * q^92 - 87 * q^93 - 9 * q^94 + 27 * q^95 - 9 * q^96 - 180 * q^98 + 96 * 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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
47.1	−2.68189	−	0.472890i	0.605371	+	1.62281i	5.08954	+	1.85244i	2.55610	+	0.930344i	−0.856127	−	4.63849i	2.58147	−	0.579686i	−8.05677	−	4.65158i	−2.26705	+	1.96481i	−6.41523	−	3.70384i
47.2	−2.47573	−	0.436538i	−1.50729	+	0.853281i	4.05929	+	1.47746i	−2.58032	−	0.939158i	4.10412	−	1.45451i	−2.62404	+	0.338290i	−5.05050	−	2.91591i	1.54382	−	2.57228i	5.97819	+	3.45151i
47.3	−2.26715	−	0.399760i	0.415691	−	1.68143i	3.10078	+	1.12859i	−1.73217	−	0.630457i	−1.61460	+	3.64587i	0.678971	−	2.55715i	−2.59137	−	1.49613i	−2.65440	−	1.39791i	3.67505	+	2.12179i
47.4	−2.06025	−	0.363278i	−1.46652	−	0.921586i	2.23328	+	0.812849i	0.338534	+	0.123216i	2.68661	+	2.43145i	1.84499	+	1.89632i	−0.682329	−	0.393943i	1.30136	+	2.70305i	−0.652704	−	0.376839i
47.5	−1.63190	−	0.287749i	1.35485	+	1.07906i	0.700923	+	0.255115i	−3.60525	−	1.31220i	−1.90049	−	2.15078i	2.03118	+	1.69538i	1.79971	+	1.03907i	0.671243	+	2.92394i	5.50583	+	3.17879i
47.6	−1.57155	−	0.277106i	1.72554	+	0.149982i	0.513586	+	0.186930i	1.78318	+	0.649025i	−2.67021	−	0.713863i	−1.28547	−	2.31248i	2.00866	+	1.15970i	2.95501	+	0.517603i	−2.62250	−	1.51410i
47.7	−1.38665	−	0.244504i	−0.458387	+	1.67029i	−0.0163608	−	0.00595485i	1.98299	+	0.721749i	1.04402	−	2.20404i	−2.18605	+	1.49037i	2.46004	+	1.42030i	−2.57976	−	1.53128i	−2.57325	−	1.48566i
47.8	−0.910598	−	0.160563i	−1.05237	−	1.37569i	−1.07598	−	0.391623i	−0.473927	−	0.172495i	0.737399	+	1.42167i	−2.46079	+	0.971863i	2.51844	+	1.45402i	−0.785044	+	2.89546i	0.403861	+	0.233169i
47.9	−0.877614	−	0.154747i	−0.824649	+	1.52314i	−1.13313	−	0.412424i	−1.30288	−	0.474210i	0.959425	−	1.20912i	1.81190	−	1.92796i	2.47415	+	1.42845i	−1.63991	−	2.51211i	1.07004	+	0.617790i
47.10	−0.313923	−	0.0553531i	1.24200	−	1.20725i	−1.78390	−	0.649287i	−2.68563	−	0.977491i	−0.456717	+	0.310234i	−2.48320	+	0.913082i	1.07619	+	0.621337i	0.0851152	−	2.99879i	0.788975	+	0.455515i
47.11	−0.0147002	−	0.00259205i	−1.71867	+	0.214901i	−1.87918	−	0.683964i	1.54651	+	0.562885i	0.0258218	+	0.00129578i	2.21011	+	1.45445i	0.0517058	+	0.0298524i	2.90764	−	0.738686i	−0.0212751	−	0.0122832i
47.12	0.0159182	+	0.00280680i	−0.402650	−	1.68460i	−1.87914	−	0.683951i	3.75147	+	1.36542i	−0.00168110	−	0.0279459i	0.157938	−	2.64103i	−0.0559891	−	0.0323253i	−2.67575	+	1.35661i	0.0558840	+	0.0322646i
47.13	0.245063	+	0.0432112i	1.36915	+	1.06086i	−1.82120	−	0.662861i	1.99870	+	0.727467i	0.289688	+	0.319140i	0.302346	+	2.62842i	−0.848674	−	0.489982i	0.749157	+	2.90496i	0.458373	+	0.264642i
47.14	0.882178	+	0.155552i	1.65967	−	0.495491i	−1.12534	−	0.409592i	0.123522	+	0.0449584i	1.54119	−	0.178947i	1.69913	−	2.02805i	−2.48059	−	1.43217i	2.50898	−	1.64470i	0.101975	+	0.0588754i
47.15	0.892534	+	0.157378i	−1.71459	+	0.245341i	−1.10754	−	0.403110i	−1.14226	−	0.415747i	−1.56894	−	0.0508626i	−1.98483	−	1.74942i	−2.49483	−	1.44039i	2.87962	−	0.841317i	−0.954073	−	0.550834i
47.16	1.10401	+	0.194666i	−0.867806	−	1.49897i	−0.698446	−	0.254214i	−3.85196	−	1.40200i	−0.666265	−	1.82381i	2.61259	+	0.417595i	−2.66330	−	1.53766i	−1.49383	+	2.60163i	−3.97967	−	2.29767i
47.17	1.65422	+	0.291684i	−0.549790	+	1.64248i	0.771989	+	0.280981i	3.52188	+	1.28186i	−1.38856	+	2.55666i	−2.17331	−	1.50888i	−1.71431	−	0.989760i	−2.39546	−	1.80604i	5.45209	+	3.14776i
47.18	1.66985	+	0.294440i	0.685602	−	1.59058i	0.822331	+	0.299304i	1.39543	+	0.507895i	1.61319	−	2.45417i	−0.356869	+	2.62157i	−1.65184	−	0.953692i	−2.05990	−	2.18101i	2.18062	+	1.25898i
47.19	1.87320	+	0.330296i	0.583821	+	1.63069i	1.52040	+	0.553380i	−0.848304	−	0.308757i	0.555004	+	3.24744i	2.48993	−	0.894564i	−0.629295	−	0.363324i	−2.31831	+	1.90406i	−1.48706	−	0.858555i
47.20	2.22112	+	0.391643i	1.67689	+	0.433617i	2.90060	+	1.05573i	−3.20807	−	1.16764i	3.55476	+	1.61986i	−2.62457	+	0.334082i	2.12267	+	1.22552i	2.62395	+	1.45426i	−6.66821	−	3.84989i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
189.bd	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	189.2.bd.a	yes	132
3.b	odd	2	1		567.2.bd.a		132
7.d	odd	6	1		189.2.ba.a	✓	132
21.g	even	6	1		567.2.ba.a		132
27.e	even	9	1		567.2.ba.a		132
27.f	odd	18	1		189.2.ba.a	✓	132
189.z	odd	18	1		567.2.bd.a		132
189.bd	even	18	1	inner	189.2.bd.a	yes	132

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
189.2.ba.a	✓	132	7.d	odd	6	1
189.2.ba.a	✓	132	27.f	odd	18	1
189.2.bd.a	yes	132	1.a	even	1	1	trivial
189.2.bd.a	yes	132	189.bd	even	18	1	inner
567.2.ba.a		132	21.g	even	6	1
567.2.ba.a		132	27.e	even	9	1
567.2.bd.a		132	3.b	odd	2	1
567.2.bd.a		132	189.z	odd	18	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(189, [\chi])\).