Newform orbit 378.2.w.b

• Label: 378.2.w.b
• Level: $378$
• Weight: $2$
• Character orbit: 378.w
• Analytic conductor: $3.018$
• Analytic rank: $0$
• Dimension: $72$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.w (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.01834519640\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(12\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) 72 * q + 3 * q^6 + 3 * q^7 + 36 * q^8 - 12 * q^10 - 6 * q^11 + 12 * q^13 - 3 * q^14 - 6 * q^15 + 24 * q^17 - 36 * q^19 + 18 * q^21 + 6 * q^22 - 6 * q^23 + 30 * q^25 - 18 * q^26 + 6 * q^27 + 3 * q^29 + 15 * q^30 - 9 * q^31 - 18 * q^33 - 9 * q^34 + 9 * q^35 - 3 * q^36 + 51 * q^39 - 6 * q^41 - 27 * q^42 - 24 * q^43 + 18 * q^45 + 27 * q^47 + 6 * q^48 - 69 * q^49 + 6 * q^50 + 6 * q^51 - 6 * q^52 - 15 * q^53 + 45 * q^54 - 72 * q^55 + 6 * q^56 + 57 * q^57 - 3 * q^58 + 15 * q^59 - 33 * q^60 - 18 * q^61 - 24 * q^62 - 12 * q^63 - 36 * q^64 - 90 * q^65 - 36 * q^66 - 66 * q^67 - 18 * q^68 - 39 * q^69 - 12 * q^70 + 12 * q^71 + 30 * q^73 + 9 * q^74 - 21 * q^75 - 87 * q^77 + 6 * q^78 - 45 * q^79 - 6 * q^80 - 24 * q^81 + 33 * q^82 + 18 * q^83 + 6 * q^84 + 51 * q^85 - 12 * q^86 - 18 * q^87 - 12 * q^88 + 72 * q^89 - 69 * q^90 - 30 * q^91 + 12 * q^92 - 48 * q^93 + 21 * q^95 + 48 * q^97 + 6 * q^98 + 90 * 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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
25.1	0.939693	−	0.342020i	−1.63077	+	0.583588i	0.766044	−	0.642788i	2.72763	+	0.992778i	−1.33283	+	1.10615i	−1.62553	+	2.08750i	0.500000	−	0.866025i	2.31885	−	1.90340i	2.90269
25.2	0.939693	−	0.342020i	−1.37994	−	1.04679i	0.766044	−	0.642788i	0.829059	+	0.301753i	−1.65474	−	0.511691i	1.89032	+	1.85113i	0.500000	−	0.866025i	0.808471	+	2.88901i	0.882266
25.3	0.939693	−	0.342020i	−1.28916	−	1.15675i	0.766044	−	0.642788i	−2.54535	−	0.926430i	−1.60704	−	0.646075i	−2.64478	−	0.0716849i	0.500000	−	0.866025i	0.323845	+	2.98247i	−2.70870
25.4	0.939693	−	0.342020i	−1.25782	+	1.19075i	0.766044	−	0.642788i	−1.38782	−	0.505123i	−0.774703	+	1.54914i	0.493387	−	2.59934i	0.500000	−	0.866025i	0.164223	−	2.99550i	−1.47688
25.5	0.939693	−	0.342020i	−0.902405	+	1.47840i	0.766044	−	0.642788i	−3.70320	−	1.34785i	−0.342341	+	1.69788i	−0.207321	+	2.63762i	0.500000	−	0.866025i	−1.37133	−	2.66823i	−3.94086
25.6	0.939693	−	0.342020i	−0.214796	+	1.71868i	0.766044	−	0.642788i	3.87748	+	1.41129i	0.385982	+	1.68850i	2.09775	−	1.61228i	0.500000	−	0.866025i	−2.90773	−	0.738330i	4.12633
25.7	0.939693	−	0.342020i	−0.0864462	−	1.72989i	0.766044	−	0.642788i	1.47620	+	0.537294i	−0.672891	−	1.59600i	0.815021	−	2.51709i	0.500000	−	0.866025i	−2.98505	+	0.299085i	1.57094
25.8	0.939693	−	0.342020i	1.05262	+	1.37550i	0.766044	−	0.642788i	−0.324117	−	0.117969i	1.45959	+	0.932530i	0.0472752	+	2.64533i	0.500000	−	0.866025i	−0.783993	+	2.89575i	−0.344919
25.9	0.939693	−	0.342020i	1.10671	−	1.33237i	0.766044	−	0.642788i	−2.28151	−	0.830401i	0.584268	−	1.63053i	−2.63284	−	0.261069i	0.500000	−	0.866025i	−0.550401	−	2.94908i	−2.42793
25.10	0.939693	−	0.342020i	1.47894	−	0.901527i	0.766044	−	0.642788i	1.05304	+	0.383275i	1.08140	−	1.35298i	1.66470	+	2.05639i	0.500000	−	0.866025i	1.37450	−	2.66660i	1.12062
25.11	0.939693	−	0.342020i	1.56589	+	0.740254i	0.766044	−	0.642788i	1.77013	+	0.644273i	1.72464	+	0.160044i	−2.34950	−	1.21648i	0.500000	−	0.866025i	1.90405	+	2.31832i	1.88373
25.12	0.939693	−	0.342020i	1.73083	+	0.0649612i	0.766044	−	0.642788i	−3.37095	−	1.22693i	1.64867	−	0.530936i	2.18547	−	1.49122i	0.500000	−	0.866025i	2.99156	+	0.224874i	−3.58729
121.1	0.939693	+	0.342020i	−1.63077	−	0.583588i	0.766044	+	0.642788i	2.72763	−	0.992778i	−1.33283	−	1.10615i	−1.62553	−	2.08750i	0.500000	+	0.866025i	2.31885	+	1.90340i	2.90269
121.2	0.939693	+	0.342020i	−1.37994	+	1.04679i	0.766044	+	0.642788i	0.829059	−	0.301753i	−1.65474	+	0.511691i	1.89032	−	1.85113i	0.500000	+	0.866025i	0.808471	−	2.88901i	0.882266
121.3	0.939693	+	0.342020i	−1.28916	+	1.15675i	0.766044	+	0.642788i	−2.54535	+	0.926430i	−1.60704	+	0.646075i	−2.64478	+	0.0716849i	0.500000	+	0.866025i	0.323845	−	2.98247i	−2.70870
121.4	0.939693	+	0.342020i	−1.25782	−	1.19075i	0.766044	+	0.642788i	−1.38782	+	0.505123i	−0.774703	−	1.54914i	0.493387	+	2.59934i	0.500000	+	0.866025i	0.164223	+	2.99550i	−1.47688
121.5	0.939693	+	0.342020i	−0.902405	−	1.47840i	0.766044	+	0.642788i	−3.70320	+	1.34785i	−0.342341	−	1.69788i	−0.207321	−	2.63762i	0.500000	+	0.866025i	−1.37133	+	2.66823i	−3.94086
121.6	0.939693	+	0.342020i	−0.214796	−	1.71868i	0.766044	+	0.642788i	3.87748	−	1.41129i	0.385982	−	1.68850i	2.09775	+	1.61228i	0.500000	+	0.866025i	−2.90773	+	0.738330i	4.12633
121.7	0.939693	+	0.342020i	−0.0864462	+	1.72989i	0.766044	+	0.642788i	1.47620	−	0.537294i	−0.672891	+	1.59600i	0.815021	+	2.51709i	0.500000	+	0.866025i	−2.98505	−	0.299085i	1.57094
121.8	0.939693	+	0.342020i	1.05262	−	1.37550i	0.766044	+	0.642788i	−0.324117	+	0.117969i	1.45959	−	0.932530i	0.0472752	−	2.64533i	0.500000	+	0.866025i	−0.783993	−	2.89575i	−0.344919

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
189.w	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	378.2.w.b	yes	72
7.c	even	3	1		378.2.v.a	✓	72
27.e	even	9	1		378.2.v.a	✓	72
189.w	even	9	1	inner	378.2.w.b	yes	72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
378.2.v.a	✓	72	7.c	even	3	1
378.2.v.a	✓	72	27.e	even	9	1
378.2.w.b	yes	72	1.a	even	1	1	trivial
378.2.w.b	yes	72	189.w	even	9	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^72 - 15*T5^70 + 41*T5^69 + 246*T5^68 - 531*T5^67 + 4291*T5^66 + 10056*T5^65 - 85170*T5^64 + 66992*T5^63 + 1442979*T5^62 - 939135*T5^61 + 16253077*T5^60 + 34301607*T5^59 - 263882943*T5^58 + 412324007*T5^57 + 4096190967*T5^56 - 3840911349*T5^55 + 13812529921*T5^54 + 68276357262*T5^53 - 320070605811*T5^52 - 550613980582*T5^51 + 3391286901339*T5^50 + 1873832114565*T5^49 - 1101638122940*T5^48 + 40587920951070*T5^47 - 19814498761593*T5^46 - 366893239643529*T5^45 + 596180626187598*T5^44 + 1405499424785109*T5^43 - 140796568547007*T5^42 + 60360733409229*T5^41 - 5957298730260531*T5^40 - 57823676578684782*T5^39 + 116297660960942049*T5^38 + 164407046434208793*T5^37 - 298865860350313851*T5^36 - 155632200141457818*T5^35 + 1648638695142496890*T5^34 - 5403881723097197340*T5^33 + 4630724665432196778*T5^32 + 7002211421692383417*T5^31 - 4135002777101786112*T5^30 - 38148730483855212234*T5^29 + 121571306152484248548*T5^28 - 256695437202966601461*T5^27 + 348633275844477830130*T5^26 - 146770351916692569336*T5^25 - 293113338893221703949*T5^24 + 454376690818097488317*T5^23 + 42748095794673514509*T5^22 - 1198812212066265517737*T5^21 + 2813108621329993470282*T5^20 - 4255697460589636800852*T5^19 + 4756100868724791430620*T5^18 - 4206471989422076360334*T5^17 + 3082686061464679497201*T5^16 - 1836670589475462702855*T5^15 + 840915635776447976028*T5^14 - 287619916478573240127*T5^13 + 73037231506301292531*T5^12 - 7797690034444260198*T5^11 - 3258493210271396880*T5^10 + 1252555404257264346*T5^9 + 412734672190806615*T5^8 - 99922780862088138*T5^7 + 296304794722237320*T5^6 - 27867572896319343*T5^5 + 23751425586749463*T5^4 - 1791831089998503*T5^3 + 62040419151354*T5^2 - 610584256602*T5 + 2678994081