LMFDB - Embedded newform 273.4.a.j.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [273,4,Mod(1,273)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("273.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(273, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 273 = 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	273.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,7,18,23,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$16.1075214316$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 31x^{4} + 33x^{3} + 220x^{2} - 154x - 160 $ x^6 - x^5 - 31x^4 + 33x^3 + 220x^2 - 154x - 160
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.82577$ of defining polynomial
Character	$\chi$	$=$	273.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.82577 q^{2} +3.00000 q^{3} -4.66658 q^{4} +12.8060 q^{5} -5.47730 q^{6} +7.00000 q^{7} +23.1262 q^{8} +9.00000 q^{9} -23.3807 q^{10} +13.4742 q^{11} -13.9997 q^{12} -13.0000 q^{13} -12.7804 q^{14} +38.4179 q^{15} -4.89046 q^{16} -5.82616 q^{17} -16.4319 q^{18} -44.8917 q^{19} -59.7601 q^{20} +21.0000 q^{21} -24.6007 q^{22} +106.259 q^{23} +69.3786 q^{24} +38.9932 q^{25} +23.7350 q^{26} +27.0000 q^{27} -32.6660 q^{28} +159.429 q^{29} -70.1422 q^{30} -11.2142 q^{31} -176.081 q^{32} +40.4225 q^{33} +10.6372 q^{34} +89.6419 q^{35} -41.9992 q^{36} +254.264 q^{37} +81.9617 q^{38} -39.0000 q^{39} +296.154 q^{40} +27.0457 q^{41} -38.3411 q^{42} +512.922 q^{43} -62.8782 q^{44} +115.254 q^{45} -194.003 q^{46} -249.263 q^{47} -14.6714 q^{48} +49.0000 q^{49} -71.1924 q^{50} -17.4785 q^{51} +60.6655 q^{52} -289.629 q^{53} -49.2957 q^{54} +172.550 q^{55} +161.883 q^{56} -134.675 q^{57} -291.081 q^{58} +544.510 q^{59} -179.280 q^{60} +742.989 q^{61} +20.4746 q^{62} +63.0000 q^{63} +360.606 q^{64} -166.478 q^{65} -73.8020 q^{66} +284.617 q^{67} +27.1882 q^{68} +318.776 q^{69} -163.665 q^{70} -85.0594 q^{71} +208.136 q^{72} +717.919 q^{73} -464.227 q^{74} +116.980 q^{75} +209.490 q^{76} +94.3191 q^{77} +71.2049 q^{78} -382.017 q^{79} -62.6271 q^{80} +81.0000 q^{81} -49.3790 q^{82} -1035.85 q^{83} -97.9981 q^{84} -74.6096 q^{85} -936.476 q^{86} +478.288 q^{87} +311.606 q^{88} +648.645 q^{89} -210.427 q^{90} -91.0000 q^{91} -495.864 q^{92} -33.6427 q^{93} +455.097 q^{94} -574.882 q^{95} -528.243 q^{96} -524.407 q^{97} -89.4626 q^{98} +121.267 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 7 q^{2} + 18 q^{3} + 23 q^{4} - 3 q^{5} + 21 q^{6} + 42 q^{7} + 81 q^{8} + 54 q^{9} + 41 q^{10} + 83 q^{11} + 69 q^{12} - 78 q^{13} + 49 q^{14} - 9 q^{15} + 83 q^{16} + 83 q^{17} + 63 q^{18} + 123 q^{19}+ \cdots + 747 q^{99}+O(q^{100}) $ 6 * q + 7 * q^2 + 18 * q^3 + 23 * q^4 - 3 * q^5 + 21 * q^6 + 42 * q^7 + 81 * q^8 + 54 * q^9 + 41 * q^10 + 83 * q^11 + 69 * q^12 - 78 * q^13 + 49 * q^14 - 9 * q^15 + 83 * q^16 + 83 * q^17 + 63 * q^18 + 123 * q^19 + 27 * q^20 + 126 * q^21 + 409 * q^22 + 25 * q^23 + 243 * q^24 + 341 * q^25 - 91 * q^26 + 162 * q^27 + 161 * q^28 + 131 * q^29 + 123 * q^30 + 450 * q^31 + 285 * q^32 + 249 * q^33 - 9 * q^34 - 21 * q^35 + 207 * q^36 + 609 * q^37 - 677 * q^38 - 234 * q^39 - 25 * q^40 - 312 * q^41 + 147 * q^42 - 195 * q^43 - 127 * q^44 - 27 * q^45 + 727 * q^46 + 198 * q^47 + 249 * q^48 + 294 * q^49 + 148 * q^50 + 249 * q^51 - 299 * q^52 + 790 * q^53 + 189 * q^54 - 789 * q^55 + 567 * q^56 + 369 * q^57 - 419 * q^58 + 280 * q^59 + 81 * q^60 + 1251 * q^61 - 986 * q^62 + 378 * q^63 - 585 * q^64 + 39 * q^65 + 1227 * q^66 + 974 * q^67 + 579 * q^68 + 75 * q^69 + 287 * q^70 - 778 * q^71 + 729 * q^72 + 921 * q^73 - 253 * q^74 + 1023 * q^75 - 1139 * q^76 + 581 * q^77 - 273 * q^78 + 1528 * q^79 - 1581 * q^80 + 486 * q^81 - 1380 * q^82 - 2104 * q^83 + 483 * q^84 + 925 * q^85 - 2307 * q^86 + 393 * q^87 + 717 * q^88 - 992 * q^89 + 369 * q^90 - 546 * q^91 + 149 * q^92 + 1350 * q^93 - 2646 * q^94 - 1587 * q^95 + 855 * q^96 - 986 * q^97 + 343 * q^98 + 747 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. You can download additional coefficients here.

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.82577	−0.645506	−0.322753	−	0.946483i	$-0.604608\pi$
$2$	−1.82577	−0.645506	−0.322753	+	0.946483i	$0.604608\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−4.66658	−0.583322
$5$	12.8060	1.14540	0.572701	−	0.819764i	$-0.305896\pi$
$5$	12.8060	1.14540	0.572701	+	0.819764i	$0.305896\pi$
$6$	−5.47730	−0.372683
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	23.1262	1.02204
$9$	9.00000	0.333333
$10$	−23.3807	−0.739364
$11$	13.4742	0.369328	0.184664	−	0.982802i	$-0.440880\pi$
$11$	13.4742	0.369328	0.184664	+	0.982802i	$0.440880\pi$
$12$	−13.9997	−0.336781
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−12.7804	−0.243978
$15$	38.4179	0.661298
$16$	−4.89046	−0.0764134
$17$	−5.82616	−0.0831206	−0.0415603	−	0.999136i	$-0.513233\pi$
$17$	−5.82616	−0.0831206	−0.0415603	+	0.999136i	$0.513233\pi$
$18$	−16.4319	−0.215169
$19$	−44.8917	−0.542045	−0.271022	−	0.962573i	$-0.587362\pi$
$19$	−44.8917	−0.542045	−0.271022	+	0.962573i	$0.587362\pi$
$20$	−59.7601	−0.668138
$21$	21.0000	0.218218
$22$	−24.6007	−0.238404
$23$	106.259	0.963324	0.481662	−	0.876357i	$-0.340033\pi$
$23$	106.259	0.963324	0.481662	+	0.876357i	$0.340033\pi$
$24$	69.3786	0.590077
$25$	38.9932	0.311945
$26$	23.7350	0.179031
$27$	27.0000	0.192450
$28$	−32.6660	−0.220475
$29$	159.429	1.02087	0.510436	−	0.859916i	$-0.329484\pi$
$29$	159.429	1.02087	0.510436	+	0.859916i	$0.329484\pi$
$30$	−70.1422	−0.426872
$31$	−11.2142	−0.0649721	−0.0324860	−	0.999472i	$-0.510342\pi$
$31$	−11.2142	−0.0649721	−0.0324860	+	0.999472i	$0.510342\pi$
$32$	−176.081	−0.972719
$33$	40.4225	0.213232
$34$	10.6372	0.0536548
$35$	89.6419	0.432921
$36$	−41.9992	−0.194441
$37$	254.264	1.12975	0.564875	−	0.825176i	$-0.308924\pi$
$37$	254.264	1.12975	0.564875	+	0.825176i	$0.308924\pi$
$38$	81.9617	0.349893
$39$	−39.0000	−0.160128
$40$	296.154	1.17065
$41$	27.0457	0.103020	0.0515100	−	0.998672i	$-0.483597\pi$
$41$	27.0457	0.103020	0.0515100	+	0.998672i	$0.483597\pi$
$42$	−38.3411	−0.140861
$43$	512.922	1.81907	0.909534	−	0.415630i	$-0.136439\pi$
$43$	512.922	1.81907	0.909534	+	0.415630i	$0.136439\pi$
$44$	−62.8782	−0.215437
$45$	115.254	0.381801
$46$	−194.003	−0.621831
$47$	−249.263	−0.773591	−0.386796	−	0.922165i	$-0.626418\pi$
$47$	−249.263	−0.773591	−0.386796	+	0.922165i	$0.626418\pi$
$48$	−14.6714	−0.0441173
$49$	49.0000	0.142857
$50$	−71.1924	−0.201363
$51$	−17.4785	−0.0479897
$52$	60.6655	0.161784
$53$	−289.629	−0.750634	−0.375317	−	0.926897i	$-0.622466\pi$
$53$	−289.629	−0.750634	−0.375317	+	0.926897i	$0.622466\pi$
$54$	−49.2957	−0.124228
$55$	172.550	0.423029
$56$	161.883	0.386296
$57$	−134.675	−0.312950
$58$	−291.081	−0.658979
$59$	544.510	1.20151	0.600756	−	0.799433i	$-0.294866\pi$
$59$	544.510	1.20151	0.600756	+	0.799433i	$0.294866\pi$
$60$	−179.280	−0.385750
$61$	742.989	1.55951	0.779754	−	0.626086i	$-0.215344\pi$
$61$	742.989	1.55951	0.779754	+	0.626086i	$0.215344\pi$
$62$	20.4746	0.0419399
$63$	63.0000	0.125988
$64$	360.606	0.704309
$65$	−166.478	−0.317677
$66$	−73.8020	−0.137642
$67$	284.617	0.518978	0.259489	−	0.965746i	$-0.416446\pi$
$67$	284.617	0.518978	0.259489	+	0.965746i	$0.416446\pi$
$68$	27.1882	0.0484861
$69$	318.776	0.556175
$70$	−163.665	−0.279453
$71$	−85.0594	−0.142179	−0.0710894	−	0.997470i	$-0.522648\pi$
$71$	−85.0594	−0.142179	−0.0710894	+	0.997470i	$0.522648\pi$
$72$	208.136	0.340681
$73$	717.919	1.15104	0.575521	−	0.817787i	$-0.304799\pi$
$73$	717.919	1.15104	0.575521	+	0.817787i	$0.304799\pi$
$74$	−464.227	−0.729261
$75$	116.980	0.180102
$76$	209.490	0.316187
$77$	94.3191	0.139593
$78$	71.2049	0.103364
$79$	−382.017	−0.544053	−0.272027	−	0.962290i	$-0.587694\pi$
$79$	−382.017	−0.544053	−0.272027	+	0.962290i	$0.587694\pi$
$80$	−62.6271	−0.0875241
$81$	81.0000	0.111111
$82$	−49.3790	−0.0665000
$83$	−1035.85	−1.36987	−0.684933	−	0.728606i	$-0.740169\pi$
$83$	−1035.85	−1.36987	−0.684933	+	0.728606i	$0.740169\pi$
$84$	−97.9981	−0.127291
$85$	−74.6096	−0.0952065
$86$	−936.476	−1.17422
$87$	478.288	0.589400
$88$	311.606	0.377470
$89$	648.645	0.772542	0.386271	−	0.922385i	$-0.373763\pi$
$89$	648.645	0.772542	0.386271	+	0.922385i	$0.373763\pi$
$90$	−210.427	−0.246455
$91$	−91.0000	−0.104828
$92$	−495.864	−0.561928
$93$	−33.6427	−0.0375117
$94$	455.097	0.499358
$95$	−574.882	−0.620859
$96$	−528.243	−0.561599
$97$	−524.407	−0.548922	−0.274461	−	0.961598i	$-0.588499\pi$
$97$	−524.407	−0.548922	−0.274461	+	0.961598i	$0.588499\pi$
$98$	−89.4626	−0.0922151
$99$	121.267	0.123109

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.4.a.j.1.2	✓	6
3.2	odd	2		819.4.a.k.1.5		6
7.6	odd	2		1911.4.a.q.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.4.a.j.1.2	✓	6	1.1	even	1	trivial
819.4.a.k.1.5		6	3.2	odd	2
1911.4.a.q.1.2		6	7.6	odd	2

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	273.a (trivial)

\(f(q)\)	\(=\)	\(q-1.82577 q^{2} +3.00000 q^{3} -4.66658 q^{4} +12.8060 q^{5} -5.47730 q^{6} +7.00000 q^{7} +23.1262 q^{8} +9.00000 q^{9} -23.3807 q^{10} +13.4742 q^{11} -13.9997 q^{12} -13.0000 q^{13} -12.7804 q^{14} +38.4179 q^{15} -4.89046 q^{16} -5.82616 q^{17} -16.4319 q^{18} -44.8917 q^{19} -59.7601 q^{20} +21.0000 q^{21} -24.6007 q^{22} +106.259 q^{23} +69.3786 q^{24} +38.9932 q^{25} +23.7350 q^{26} +27.0000 q^{27} -32.6660 q^{28} +159.429 q^{29} -70.1422 q^{30} -11.2142 q^{31} -176.081 q^{32} +40.4225 q^{33} +10.6372 q^{34} +89.6419 q^{35} -41.9992 q^{36} +254.264 q^{37} +81.9617 q^{38} -39.0000 q^{39} +296.154 q^{40} +27.0457 q^{41} -38.3411 q^{42} +512.922 q^{43} -62.8782 q^{44} +115.254 q^{45} -194.003 q^{46} -249.263 q^{47} -14.6714 q^{48} +49.0000 q^{49} -71.1924 q^{50} -17.4785 q^{51} +60.6655 q^{52} -289.629 q^{53} -49.2957 q^{54} +172.550 q^{55} +161.883 q^{56} -134.675 q^{57} -291.081 q^{58} +544.510 q^{59} -179.280 q^{60} +742.989 q^{61} +20.4746 q^{62} +63.0000 q^{63} +360.606 q^{64} -166.478 q^{65} -73.8020 q^{66} +284.617 q^{67} +27.1882 q^{68} +318.776 q^{69} -163.665 q^{70} -85.0594 q^{71} +208.136 q^{72} +717.919 q^{73} -464.227 q^{74} +116.980 q^{75} +209.490 q^{76} +94.3191 q^{77} +71.2049 q^{78} -382.017 q^{79} -62.6271 q^{80} +81.0000 q^{81} -49.3790 q^{82} -1035.85 q^{83} -97.9981 q^{84} -74.6096 q^{85} -936.476 q^{86} +478.288 q^{87} +311.606 q^{88} +648.645 q^{89} -210.427 q^{90} -91.0000 q^{91} -495.864 q^{92} -33.6427 q^{93} +455.097 q^{94} -574.882 q^{95} -528.243 q^{96} -524.407 q^{97} -89.4626 q^{98} +121.267 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 7 q^{2} + 18 q^{3} + 23 q^{4} - 3 q^{5} + 21 q^{6} + 42 q^{7} + 81 q^{8} + 54 q^{9} + 41 q^{10} + 83 q^{11} + 69 q^{12} - 78 q^{13} + 49 q^{14} - 9 q^{15} + 83 q^{16} + 83 q^{17} + 63 q^{18} + 123 q^{19}+ \cdots + 747 q^{99}+O(q^{100}) \) 6 * q + 7 * q^2 + 18 * q^3 + 23 * q^4 - 3 * q^5 + 21 * q^6 + 42 * q^7 + 81 * q^8 + 54 * q^9 + 41 * q^10 + 83 * q^11 + 69 * q^12 - 78 * q^13 + 49 * q^14 - 9 * q^15 + 83 * q^16 + 83 * q^17 + 63 * q^18 + 123 * q^19 + 27 * q^20 + 126 * q^21 + 409 * q^22 + 25 * q^23 + 243 * q^24 + 341 * q^25 - 91 * q^26 + 162 * q^27 + 161 * q^28 + 131 * q^29 + 123 * q^30 + 450 * q^31 + 285 * q^32 + 249 * q^33 - 9 * q^34 - 21 * q^35 + 207 * q^36 + 609 * q^37 - 677 * q^38 - 234 * q^39 - 25 * q^40 - 312 * q^41 + 147 * q^42 - 195 * q^43 - 127 * q^44 - 27 * q^45 + 727 * q^46 + 198 * q^47 + 249 * q^48 + 294 * q^49 + 148 * q^50 + 249 * q^51 - 299 * q^52 + 790 * q^53 + 189 * q^54 - 789 * q^55 + 567 * q^56 + 369 * q^57 - 419 * q^58 + 280 * q^59 + 81 * q^60 + 1251 * q^61 - 986 * q^62 + 378 * q^63 - 585 * q^64 + 39 * q^65 + 1227 * q^66 + 974 * q^67 + 579 * q^68 + 75 * q^69 + 287 * q^70 - 778 * q^71 + 729 * q^72 + 921 * q^73 - 253 * q^74 + 1023 * q^75 - 1139 * q^76 + 581 * q^77 - 273 * q^78 + 1528 * q^79 - 1581 * q^80 + 486 * q^81 - 1380 * q^82 - 2104 * q^83 + 483 * q^84 + 925 * q^85 - 2307 * q^86 + 393 * q^87 + 717 * q^88 - 992 * q^89 + 369 * q^90 - 546 * q^91 + 149 * q^92 + 1350 * q^93 - 2646 * q^94 - 1587 * q^95 + 855 * q^96 - 986 * q^97 + 343 * q^98 + 747 * q^99

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