Newform orbit 370.2.w.b

• Label: 370.2.w.b
• Level: $370$
• Weight: $2$
• Character orbit: 370.w
• Analytic conductor: $2.954$
• Analytic rank: $0$
• Dimension: $48$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$370 = 2 \cdot 5 \cdot 37$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	370.w (of order $18$, degree $6$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$2.95446487479$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$8$ 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.

$\operatorname{Tr}(f)(q) =$ 48 * q + 12 * q^7 + 24 * q^9 - 24 * q^10 + 18 * q^11 + 24 * q^17 - 18 * q^21 + 12 * q^22 - 12 * q^26 + 24 * q^27 + 6 * q^28 - 18 * q^29 - 12 * q^33 - 12 * q^34 + 6 * q^35 - 84 * q^36 - 6 * q^37 + 12 * q^38 - 42 * q^39 - 6 * q^41 - 54 * q^42 + 6 * q^44 + 36 * q^45 + 36 * q^46 - 30 * q^47 + 6 * q^48 + 18 * q^49 + 18 * q^52 + 54 * q^54 - 6 * q^55 - 6 * q^56 - 36 * q^57 - 12 * q^58 + 42 * q^59 - 24 * q^61 + 54 * q^62 - 24 * q^63 + 24 * q^64 - 36 * q^66 + 60 * q^67 - 102 * q^69 - 6 * q^70 + 60 * q^71 - 48 * q^73 + 30 * q^74 + 12 * q^75 - 126 * q^77 + 30 * q^78 + 36 * q^81 + 90 * q^83 + 12 * q^84 + 6 * q^85 + 36 * q^86 - 30 * q^87 - 36 * q^88 - 6 * q^89 - 12 * q^90 + 84 * q^91 - 6 * q^92 - 54 * q^93 - 18 * q^94 - 24 * q^95 + 36 * q^97 - 24 * q^98 - 72 * 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	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
21.1	−0.342020	+	0.939693i	−2.79539	+	1.01744i	−0.766044	−	0.642788i	0.984808	+	0.173648i		−	2.97480i	0.125624	−	0.712447i	0.866025	−	0.500000i	4.48091	−	3.75993i	−0.500000	+	0.866025i
21.2	−0.342020	+	0.939693i	−1.18324	+	0.430663i	−0.766044	−	0.642788i	0.984808	+	0.173648i		−	1.25917i	0.165700	−	0.939732i	0.866025	−	0.500000i	−1.08356	+	0.909211i	−0.500000	+	0.866025i
21.3	−0.342020	+	0.939693i	1.67405	−	0.609304i	−0.766044	−	0.642788i	0.984808	+	0.173648i			1.78149i	0.521989	−	2.96035i	0.866025	−	0.500000i	0.133056	−	0.111647i	−0.500000	+	0.866025i
21.4	−0.342020	+	0.939693i	3.24427	−	1.18082i	−0.766044	−	0.642788i	0.984808	+	0.173648i			3.45248i	−0.802595	+	4.55174i	0.866025	−	0.500000i	6.83285	−	5.73344i	−0.500000	+	0.866025i
21.5	0.342020	−	0.939693i	−2.52763	+	0.919981i	−0.766044	−	0.642788i	−0.984808	−	0.173648i			2.68984i	0.420839	−	2.38669i	−0.866025	+	0.500000i	3.24439	−	2.72237i	−0.500000	+	0.866025i
21.6	0.342020	−	0.939693i	−0.384909	+	0.140095i	−0.766044	−	0.642788i	−0.984808	−	0.173648i			0.409612i	−0.723927	+	4.10559i	−0.866025	+	0.500000i	−2.16961	+	1.82051i	−0.500000	+	0.866025i
21.7	0.342020	−	0.939693i	0.815392	−	0.296779i	−0.766044	−	0.642788i	−0.984808	−	0.173648i		−	0.867722i	0.629880	−	3.57223i	−0.866025	+	0.500000i	−1.72135	+	1.44438i	−0.500000	+	0.866025i
21.8	0.342020	−	0.939693i	3.03684	−	1.10532i	−0.766044	−	0.642788i	−0.984808	−	0.173648i		−	3.23173i	0.130401	−	0.739542i	−0.866025	+	0.500000i	5.70251	−	4.78497i	−0.500000	+	0.866025i
41.1	−0.984808	−	0.173648i	−0.396937	−	2.25114i	0.939693	+	0.342020i	0.642788	+	0.766044i			2.28587i	−1.58779	+	1.33231i	−0.866025	−	0.500000i	−2.09099	+	0.761058i	−0.500000	−	0.866025i
41.2	−0.984808	−	0.173648i	−0.240758	−	1.36541i	0.939693	+	0.342020i	0.642788	+	0.766044i			1.38647i	2.42202	−	2.03232i	−0.866025	−	0.500000i	1.01271	−	0.368596i	−0.500000	−	0.866025i
41.3	−0.984808	−	0.173648i	0.105668	+	0.599271i	0.939693	+	0.342020i	0.642788	+	0.766044i		−	0.608515i	−2.12489	+	1.78299i	−0.866025	−	0.500000i	2.47112	−	0.899413i	−0.500000	−	0.866025i
41.4	−0.984808	−	0.173648i	0.358379	+	2.03247i	0.939693	+	0.342020i	0.642788	+	0.766044i		−	2.06382i	3.75435	−	3.15027i	−0.866025	−	0.500000i	−1.18341	+	0.430725i	−0.500000	−	0.866025i
41.5	0.984808	+	0.173648i	−0.568495	−	3.22410i	0.939693	+	0.342020i	−0.642788	−	0.766044i		−	3.27383i	2.50060	−	2.09825i	0.866025	+	0.500000i	−7.25253	+	2.63970i	−0.500000	−	0.866025i
41.6	0.984808	+	0.173648i	−0.192543	−	1.09196i	0.939693	+	0.342020i	−0.642788	−	0.766044i		−	1.10881i	−0.750891	+	0.630073i	0.866025	+	0.500000i	1.66377	−	0.605561i	−0.500000	−	0.866025i
41.7	0.984808	+	0.173648i	0.168670	+	0.956577i	0.939693	+	0.342020i	−0.642788	−	0.766044i			0.971334i	2.87596	−	2.41322i	0.866025	+	0.500000i	1.93249	−	0.703368i	−0.500000	−	0.866025i
41.8	0.984808	+	0.173648i	0.418719	+	2.37467i	0.939693	+	0.342020i	−0.642788	−	0.766044i			2.41131i	−3.20998	+	2.69349i	0.866025	+	0.500000i	−2.64467	+	0.962583i	−0.500000	−	0.866025i
141.1	−0.342020	−	0.939693i	−2.79539	−	1.01744i	−0.766044	+	0.642788i	0.984808	−	0.173648i			2.97480i	0.125624	+	0.712447i	0.866025	+	0.500000i	4.48091	+	3.75993i	−0.500000	−	0.866025i
141.2	−0.342020	−	0.939693i	−1.18324	−	0.430663i	−0.766044	+	0.642788i	0.984808	−	0.173648i			1.25917i	0.165700	+	0.939732i	0.866025	+	0.500000i	−1.08356	−	0.909211i	−0.500000	−	0.866025i
141.3	−0.342020	−	0.939693i	1.67405	+	0.609304i	−0.766044	+	0.642788i	0.984808	−	0.173648i		−	1.78149i	0.521989	+	2.96035i	0.866025	+	0.500000i	0.133056	+	0.111647i	−0.500000	−	0.866025i
141.4	−0.342020	−	0.939693i	3.24427	+	1.18082i	−0.766044	+	0.642788i	0.984808	−	0.173648i		−	3.45248i	−0.802595	−	4.55174i	0.866025	+	0.500000i	6.83285	+	5.73344i	−0.500000	−	0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
37.h	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	370.2.w.b	✓	48
37.h	even	18	1	inner	370.2.w.b	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
370.2.w.b	✓	48	1.a	even	1	1	trivial
370.2.w.b	✓	48	37.h	even	18	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^48 - 12*T3^46 - 20*T3^45 + 90*T3^44 + 450*T3^43 + 1006*T3^42 - 4074*T3^41 - 15048*T3^40 + 15680*T3^39 + 179511*T3^38 + 258426*T3^37 + 150894*T3^36 - 1597986*T3^35 - 5467104*T3^34 - 836606*T3^33 + 61344453*T3^32 + 163517022*T3^31 + 331226120*T3^30 + 322873062*T3^29 + 448861725*T3^28 + 1482806430*T3^27 + 10676520657*T3^26 + 18310023252*T3^25 + 30339513046*T3^24 + 5498967000*T3^23 - 19341656091*T3^22 - 177576922930*T3^21 + 222019872396*T3^20 + 564200624322*T3^19 + 1416433273809*T3^18 + 744120967554*T3^17 + 1814022237330*T3^16 - 991186473710*T3^15 + 2929362859047*T3^14 + 2184533364330*T3^13 + 1232113343877*T3^12 + 2682410886636*T3^11 + 812743000116*T3^10 - 5612452103648*T3^9 + 7632361559808*T3^8 - 8325994472064*T3^7 + 7082419647680*T3^6 - 1083676035072*T3^5 + 158792788224*T3^4 + 769978007552*T3^3 - 159643779072*T3^2 - 35004997632*T3 + 71464259584