LMFDB - Embedded newform 546.4.a.j.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [546,4,Mod(1,546)] 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("546.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(546, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	546.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,4,-6,8,5] 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:	$32.2150428631$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{105}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 26 $ x^2 - x - 26
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			$-4.62348$ of defining polynomial
Character	$\chi$	$=$	546.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +7.62348 q^{5} -6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +15.2470 q^{10} -36.4939 q^{11} -12.0000 q^{12} -13.0000 q^{13} -14.0000 q^{14} -22.8704 q^{15} +16.0000 q^{16} -90.2348 q^{17} +18.0000 q^{18} -71.1052 q^{19} +30.4939 q^{20} +21.0000 q^{21} -72.9878 q^{22} +75.1052 q^{23} -24.0000 q^{24} -66.8826 q^{25} -26.0000 q^{26} -27.0000 q^{27} -28.0000 q^{28} +115.081 q^{29} -45.7409 q^{30} +159.809 q^{31} +32.0000 q^{32} +109.482 q^{33} -180.470 q^{34} -53.3643 q^{35} +36.0000 q^{36} -189.198 q^{37} -142.210 q^{38} +39.0000 q^{39} +60.9878 q^{40} -418.210 q^{41} +42.0000 q^{42} -98.3521 q^{43} -145.976 q^{44} +68.6113 q^{45} +150.210 q^{46} +155.785 q^{47} -48.0000 q^{48} +49.0000 q^{49} -133.765 q^{50} +270.704 q^{51} -52.0000 q^{52} -702.748 q^{53} -54.0000 q^{54} -278.210 q^{55} -56.0000 q^{56} +213.316 q^{57} +230.162 q^{58} -311.741 q^{59} -91.4817 q^{60} -407.037 q^{61} +319.619 q^{62} -63.0000 q^{63} +64.0000 q^{64} -99.1052 q^{65} +218.963 q^{66} +416.607 q^{67} -360.939 q^{68} -225.316 q^{69} -106.729 q^{70} -1097.42 q^{71} +72.0000 q^{72} -663.883 q^{73} -378.396 q^{74} +200.648 q^{75} -284.421 q^{76} +255.457 q^{77} +78.0000 q^{78} +973.971 q^{79} +121.976 q^{80} +81.0000 q^{81} -836.421 q^{82} +1180.58 q^{83} +84.0000 q^{84} -687.902 q^{85} -196.704 q^{86} -345.242 q^{87} -291.951 q^{88} +931.145 q^{89} +137.223 q^{90} +91.0000 q^{91} +300.421 q^{92} -479.428 q^{93} +311.570 q^{94} -542.069 q^{95} -96.0000 q^{96} +443.404 q^{97} +98.0000 q^{98} -328.445 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - 6 q^{3} + 8 q^{4} + 5 q^{5} - 12 q^{6} - 14 q^{7} + 16 q^{8} + 18 q^{9} + 10 q^{10} - 32 q^{11} - 24 q^{12} - 26 q^{13} - 28 q^{14} - 15 q^{15} + 32 q^{16} - 78 q^{17} + 36 q^{18} - 9 q^{19}+ \cdots - 288 q^{99}+O(q^{100}) $ 2 * q + 4 * q^2 - 6 * q^3 + 8 * q^4 + 5 * q^5 - 12 * q^6 - 14 * q^7 + 16 * q^8 + 18 * q^9 + 10 * q^10 - 32 * q^11 - 24 * q^12 - 26 * q^13 - 28 * q^14 - 15 * q^15 + 32 * q^16 - 78 * q^17 + 36 * q^18 - 9 * q^19 + 20 * q^20 + 42 * q^21 - 64 * q^22 + 17 * q^23 - 48 * q^24 - 185 * q^25 - 52 * q^26 - 54 * q^27 - 56 * q^28 - 67 * q^29 - 30 * q^30 - 121 * q^31 + 64 * q^32 + 96 * q^33 - 156 * q^34 - 35 * q^35 + 72 * q^36 - 30 * q^37 - 18 * q^38 + 78 * q^39 + 40 * q^40 - 570 * q^41 + 84 * q^42 - 43 * q^43 - 128 * q^44 + 45 * q^45 + 34 * q^46 - 293 * q^47 - 96 * q^48 + 98 * q^49 - 370 * q^50 + 234 * q^51 - 104 * q^52 - 555 * q^53 - 108 * q^54 - 290 * q^55 - 112 * q^56 + 27 * q^57 - 134 * q^58 - 562 * q^59 - 60 * q^60 - 1060 * q^61 - 242 * q^62 - 126 * q^63 + 128 * q^64 - 65 * q^65 + 192 * q^66 - 130 * q^67 - 312 * q^68 - 51 * q^69 - 70 * q^70 - 2 * q^71 + 144 * q^72 - 1379 * q^73 - 60 * q^74 + 555 * q^75 - 36 * q^76 + 224 * q^77 + 156 * q^78 + 913 * q^79 + 80 * q^80 + 162 * q^81 - 1140 * q^82 + 363 * q^83 + 168 * q^84 - 720 * q^85 - 86 * q^86 + 201 * q^87 - 256 * q^88 + 315 * q^89 + 90 * q^90 + 182 * q^91 + 68 * q^92 + 363 * q^93 - 586 * q^94 - 705 * q^95 - 192 * q^96 - 599 * q^97 + 196 * q^98 - 288 * 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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	7.62348	0.681864	0.340932	−	0.940088i	$-0.389257\pi$
$5$	7.62348	0.681864	0.340932	+	0.940088i	$0.389257\pi$
$6$	−6.00000	−0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	15.2470	0.482151
$11$	−36.4939	−1.00030	−0.500151	−	0.865938i	$-0.666722\pi$
$11$	−36.4939	−1.00030	−0.500151	+	0.865938i	$0.666722\pi$
$12$	−12.0000	−0.288675
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−14.0000	−0.267261
$15$	−22.8704	−0.393675
$16$	16.0000	0.250000
$17$	−90.2348	−1.28736	−0.643681	−	0.765294i	$-0.722593\pi$
$17$	−90.2348	−1.28736	−0.643681	+	0.765294i	$0.722593\pi$
$18$	18.0000	0.235702
$19$	−71.1052	−0.858560	−0.429280	−	0.903171i	$-0.641233\pi$
$19$	−71.1052	−0.858560	−0.429280	+	0.903171i	$0.641233\pi$
$20$	30.4939	0.340932
$21$	21.0000	0.218218
$22$	−72.9878	−0.707321
$23$	75.1052	0.680892	0.340446	−	0.940264i	$-0.389422\pi$
$23$	75.1052	0.680892	0.340446	+	0.940264i	$0.389422\pi$
$24$	−24.0000	−0.204124
$25$	−66.8826	−0.535061
$26$	−26.0000	−0.196116
$27$	−27.0000	−0.192450
$28$	−28.0000	−0.188982
$29$	115.081	0.736895	0.368448	−	0.929648i	$-0.379889\pi$
$29$	115.081	0.736895	0.368448	+	0.929648i	$0.379889\pi$
$30$	−45.7409	−0.278370
$31$	159.809	0.925891	0.462946	−	0.886387i	$-0.346793\pi$
$31$	159.809	0.925891	0.462946	+	0.886387i	$0.346793\pi$
$32$	32.0000	0.176777
$33$	109.482	0.577525
$34$	−180.470	−0.910302
$35$	−53.3643	−0.257721
$36$	36.0000	0.166667
$37$	−189.198	−0.840648	−0.420324	−	0.907374i	$-0.638084\pi$
$37$	−189.198	−0.840648	−0.420324	+	0.907374i	$0.638084\pi$
$38$	−142.210	−0.607094
$39$	39.0000	0.160128
$40$	60.9878	0.241075
$41$	−418.210	−1.59301	−0.796506	−	0.604631i	$-0.793321\pi$
$41$	−418.210	−1.59301	−0.796506	+	0.604631i	$0.793321\pi$
$42$	42.0000	0.154303
$43$	−98.3521	−0.348804	−0.174402	−	0.984675i	$-0.555799\pi$
$43$	−98.3521	−0.348804	−0.174402	+	0.984675i	$0.555799\pi$
$44$	−145.976	−0.500151
$45$	68.6113	0.227288
$46$	150.210	0.481463
$47$	155.785	0.483481	0.241740	−	0.970341i	$-0.422282\pi$
$47$	155.785	0.483481	0.241740	+	0.970341i	$0.422282\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	−133.765	−0.378345
$51$	270.704	0.743258
$52$	−52.0000	−0.138675
$53$	−702.748	−1.82132	−0.910660	−	0.413157i	$-0.864426\pi$
$53$	−702.748	−1.82132	−0.910660	+	0.413157i	$0.864426\pi$
$54$	−54.0000	−0.136083
$55$	−278.210	−0.682070
$56$	−56.0000	−0.133631
$57$	213.316	0.495690
$58$	230.162	0.521064
$59$	−311.741	−0.687885	−0.343942	−	0.938991i	$-0.611762\pi$
$59$	−311.741	−0.687885	−0.343942	+	0.938991i	$0.611762\pi$
$60$	−91.4817	−0.196837
$61$	−407.037	−0.854356	−0.427178	−	0.904168i	$-0.640492\pi$
$61$	−407.037	−0.854356	−0.427178	+	0.904168i	$0.640492\pi$
$62$	319.619	0.654704
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	−99.1052	−0.189115
$66$	218.963	0.408372
$67$	416.607	0.759651	0.379825	−	0.925058i	$-0.375984\pi$
$67$	416.607	0.759651	0.379825	+	0.925058i	$0.375984\pi$
$68$	−360.939	−0.643681
$69$	−225.316	−0.393113
$70$	−106.729	−0.182236
$71$	−1097.42	−1.83437	−0.917185	−	0.398462i	$-0.869544\pi$
$71$	−1097.42	−1.83437	−0.917185	+	0.398462i	$0.869544\pi$
$72$	72.0000	0.117851
$73$	−663.883	−1.06441	−0.532203	−	0.846617i	$-0.678636\pi$
$73$	−663.883	−1.06441	−0.532203	+	0.846617i	$0.678636\pi$
$74$	−378.396	−0.594428
$75$	200.648	0.308918
$76$	−284.421	−0.429280
$77$	255.457	0.378079
$78$	78.0000	0.113228
$79$	973.971	1.38709	0.693546	−	0.720412i	$-0.256047\pi$
$79$	973.971	1.38709	0.693546	+	0.720412i	$0.256047\pi$
$80$	121.976	0.170466
$81$	81.0000	0.111111
$82$	−836.421	−1.12643
$83$	1180.58	1.56127	0.780634	−	0.624988i	$-0.214896\pi$
$83$	1180.58	1.56127	0.780634	+	0.624988i	$0.214896\pi$
$84$	84.0000	0.109109
$85$	−687.902	−0.877806
$86$	−196.704	−0.246641
$87$	−345.242	−0.425447
$88$	−291.951	−0.353660
$89$	931.145	1.10900	0.554501	−	0.832183i	$-0.312909\pi$
$89$	931.145	1.10900	0.554501	+	0.832183i	$0.312909\pi$
$90$	137.223	0.160717
$91$	91.0000	0.104828
$92$	300.421	0.340446
$93$	−479.428	−0.534563
$94$	311.570	0.341872
$95$	−542.069	−0.585422
$96$	−96.0000	−0.102062
$97$	443.404	0.464132	0.232066	−	0.972700i	$-0.425451\pi$
$97$	443.404	0.464132	0.232066	+	0.972700i	$0.425451\pi$
$98$	98.0000	0.101015
$99$	−328.445	−0.333434

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	546.4.a.j.1.2	✓	2
3.2	odd	2		1638.4.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.4.a.j.1.2	✓	2	1.1	even	1	trivial
1638.4.a.m.1.1		2	3.2	odd	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	546.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +7.62348 q^{5} -6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +15.2470 q^{10} -36.4939 q^{11} -12.0000 q^{12} -13.0000 q^{13} -14.0000 q^{14} -22.8704 q^{15} +16.0000 q^{16} -90.2348 q^{17} +18.0000 q^{18} -71.1052 q^{19} +30.4939 q^{20} +21.0000 q^{21} -72.9878 q^{22} +75.1052 q^{23} -24.0000 q^{24} -66.8826 q^{25} -26.0000 q^{26} -27.0000 q^{27} -28.0000 q^{28} +115.081 q^{29} -45.7409 q^{30} +159.809 q^{31} +32.0000 q^{32} +109.482 q^{33} -180.470 q^{34} -53.3643 q^{35} +36.0000 q^{36} -189.198 q^{37} -142.210 q^{38} +39.0000 q^{39} +60.9878 q^{40} -418.210 q^{41} +42.0000 q^{42} -98.3521 q^{43} -145.976 q^{44} +68.6113 q^{45} +150.210 q^{46} +155.785 q^{47} -48.0000 q^{48} +49.0000 q^{49} -133.765 q^{50} +270.704 q^{51} -52.0000 q^{52} -702.748 q^{53} -54.0000 q^{54} -278.210 q^{55} -56.0000 q^{56} +213.316 q^{57} +230.162 q^{58} -311.741 q^{59} -91.4817 q^{60} -407.037 q^{61} +319.619 q^{62} -63.0000 q^{63} +64.0000 q^{64} -99.1052 q^{65} +218.963 q^{66} +416.607 q^{67} -360.939 q^{68} -225.316 q^{69} -106.729 q^{70} -1097.42 q^{71} +72.0000 q^{72} -663.883 q^{73} -378.396 q^{74} +200.648 q^{75} -284.421 q^{76} +255.457 q^{77} +78.0000 q^{78} +973.971 q^{79} +121.976 q^{80} +81.0000 q^{81} -836.421 q^{82} +1180.58 q^{83} +84.0000 q^{84} -687.902 q^{85} -196.704 q^{86} -345.242 q^{87} -291.951 q^{88} +931.145 q^{89} +137.223 q^{90} +91.0000 q^{91} +300.421 q^{92} -479.428 q^{93} +311.570 q^{94} -542.069 q^{95} -96.0000 q^{96} +443.404 q^{97} +98.0000 q^{98} -328.445 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - 6 q^{3} + 8 q^{4} + 5 q^{5} - 12 q^{6} - 14 q^{7} + 16 q^{8} + 18 q^{9} + 10 q^{10} - 32 q^{11} - 24 q^{12} - 26 q^{13} - 28 q^{14} - 15 q^{15} + 32 q^{16} - 78 q^{17} + 36 q^{18} - 9 q^{19}+ \cdots - 288 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 - 6 * q^3 + 8 * q^4 + 5 * q^5 - 12 * q^6 - 14 * q^7 + 16 * q^8 + 18 * q^9 + 10 * q^10 - 32 * q^11 - 24 * q^12 - 26 * q^13 - 28 * q^14 - 15 * q^15 + 32 * q^16 - 78 * q^17 + 36 * q^18 - 9 * q^19 + 20 * q^20 + 42 * q^21 - 64 * q^22 + 17 * q^23 - 48 * q^24 - 185 * q^25 - 52 * q^26 - 54 * q^27 - 56 * q^28 - 67 * q^29 - 30 * q^30 - 121 * q^31 + 64 * q^32 + 96 * q^33 - 156 * q^34 - 35 * q^35 + 72 * q^36 - 30 * q^37 - 18 * q^38 + 78 * q^39 + 40 * q^40 - 570 * q^41 + 84 * q^42 - 43 * q^43 - 128 * q^44 + 45 * q^45 + 34 * q^46 - 293 * q^47 - 96 * q^48 + 98 * q^49 - 370 * q^50 + 234 * q^51 - 104 * q^52 - 555 * q^53 - 108 * q^54 - 290 * q^55 - 112 * q^56 + 27 * q^57 - 134 * q^58 - 562 * q^59 - 60 * q^60 - 1060 * q^61 - 242 * q^62 - 126 * q^63 + 128 * q^64 - 65 * q^65 + 192 * q^66 - 130 * q^67 - 312 * q^68 - 51 * q^69 - 70 * q^70 - 2 * q^71 + 144 * q^72 - 1379 * q^73 - 60 * q^74 + 555 * q^75 - 36 * q^76 + 224 * q^77 + 156 * q^78 + 913 * q^79 + 80 * q^80 + 162 * q^81 - 1140 * q^82 + 363 * q^83 + 168 * q^84 - 720 * q^85 - 86 * q^86 + 201 * q^87 - 256 * q^88 + 315 * q^89 + 90 * q^90 + 182 * q^91 + 68 * q^92 + 363 * q^93 - 586 * q^94 - 705 * q^95 - 192 * q^96 - 599 * q^97 + 196 * q^98 - 288 * q^99

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