LMFDB - Embedded newform 546.4.a.p.1.3

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 = [3,6,-9,12,7] 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:	$0$
Dimension:	$3$
Coefficient field:	3.3.118088.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 50x - 32 $ x^3 - 50*x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.645376$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +17.5010 q^{5} -6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +35.0020 q^{10} -3.08051 q^{11} -12.0000 q^{12} +13.0000 q^{13} -14.0000 q^{14} -52.5030 q^{15} +16.0000 q^{16} +124.540 q^{17} +18.0000 q^{18} -78.9607 q^{19} +70.0040 q^{20} +21.0000 q^{21} -6.16102 q^{22} +126.218 q^{23} -24.0000 q^{24} +181.285 q^{25} +26.0000 q^{26} -27.0000 q^{27} -28.0000 q^{28} -25.3812 q^{29} -105.006 q^{30} -215.249 q^{31} +32.0000 q^{32} +9.24154 q^{33} +249.080 q^{34} -122.507 q^{35} +36.0000 q^{36} +370.949 q^{37} -157.921 q^{38} -39.0000 q^{39} +140.008 q^{40} -215.291 q^{41} +42.0000 q^{42} -461.369 q^{43} -12.3220 q^{44} +157.509 q^{45} +252.436 q^{46} +478.874 q^{47} -48.0000 q^{48} +49.0000 q^{49} +362.569 q^{50} -373.621 q^{51} +52.0000 q^{52} +503.038 q^{53} -54.0000 q^{54} -53.9120 q^{55} -56.0000 q^{56} +236.882 q^{57} -50.7624 q^{58} +681.315 q^{59} -210.012 q^{60} -96.3991 q^{61} -430.499 q^{62} -63.0000 q^{63} +64.0000 q^{64} +227.513 q^{65} +18.4831 q^{66} +489.363 q^{67} +498.161 q^{68} -378.655 q^{69} -245.014 q^{70} +271.106 q^{71} +72.0000 q^{72} -529.647 q^{73} +741.897 q^{74} -543.854 q^{75} -315.843 q^{76} +21.5636 q^{77} -78.0000 q^{78} -910.161 q^{79} +280.016 q^{80} +81.0000 q^{81} -430.581 q^{82} +1051.22 q^{83} +84.0000 q^{84} +2179.58 q^{85} -922.738 q^{86} +76.1436 q^{87} -24.6441 q^{88} -305.279 q^{89} +315.018 q^{90} -91.0000 q^{91} +504.873 q^{92} +645.748 q^{93} +957.748 q^{94} -1381.89 q^{95} -96.0000 q^{96} +1128.48 q^{97} +98.0000 q^{98} -27.7246 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 7 q^{5} - 18 q^{6} - 21 q^{7} + 24 q^{8} + 27 q^{9} + 14 q^{10} - 47 q^{11} - 36 q^{12} + 39 q^{13} - 42 q^{14} - 21 q^{15} + 48 q^{16} + 119 q^{17} + 54 q^{18} + 101 q^{19}+ \cdots - 423 q^{99}+O(q^{100}) $ 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 + 7 * q^5 - 18 * q^6 - 21 * q^7 + 24 * q^8 + 27 * q^9 + 14 * q^10 - 47 * q^11 - 36 * q^12 + 39 * q^13 - 42 * q^14 - 21 * q^15 + 48 * q^16 + 119 * q^17 + 54 * q^18 + 101 * q^19 + 28 * q^20 + 63 * q^21 - 94 * q^22 - 27 * q^23 - 72 * q^24 + 266 * q^25 + 78 * q^26 - 81 * q^27 - 84 * q^28 + 345 * q^29 - 42 * q^30 + 152 * q^31 + 96 * q^32 + 141 * q^33 + 238 * q^34 - 49 * q^35 + 108 * q^36 + 227 * q^37 + 202 * q^38 - 117 * q^39 + 56 * q^40 + 452 * q^41 + 126 * q^42 - 415 * q^43 - 188 * q^44 + 63 * q^45 - 54 * q^46 + 240 * q^47 - 144 * q^48 + 147 * q^49 + 532 * q^50 - 357 * q^51 + 156 * q^52 + 874 * q^53 - 162 * q^54 + 1123 * q^55 - 168 * q^56 - 303 * q^57 + 690 * q^58 + 948 * q^59 - 84 * q^60 + 951 * q^61 + 304 * q^62 - 189 * q^63 + 192 * q^64 + 91 * q^65 + 282 * q^66 + 742 * q^67 + 476 * q^68 + 81 * q^69 - 98 * q^70 + 732 * q^71 + 216 * q^72 - 585 * q^73 + 454 * q^74 - 798 * q^75 + 404 * q^76 + 329 * q^77 - 234 * q^78 - 618 * q^79 + 112 * q^80 + 243 * q^81 + 904 * q^82 + 774 * q^83 + 252 * q^84 + 1273 * q^85 - 830 * q^86 - 1035 * q^87 - 376 * q^88 + 730 * q^89 + 126 * q^90 - 273 * q^91 - 108 * q^92 - 456 * q^93 + 480 * q^94 - 2617 * q^95 - 288 * q^96 + 2518 * q^97 + 294 * q^98 - 423 * 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
$5$	17.5010	1.56534	0.782668	−	0.622439i	$-0.213858\pi$
$5$	17.5010	1.56534	0.782668	+	0.622439i	$0.213858\pi$
$6$	−6.00000	−0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	35.0020	1.10686
$11$	−3.08051	−0.0844372	−0.0422186	−	0.999108i	$-0.513443\pi$
$11$	−3.08051	−0.0844372	−0.0422186	+	0.999108i	$0.513443\pi$
$12$	−12.0000	−0.288675
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	−14.0000	−0.267261
$15$	−52.5030	−0.903747
$16$	16.0000	0.250000
$17$	124.540	1.77679	0.888395	−	0.459079i	$-0.151821\pi$
$17$	124.540	1.77679	0.888395	+	0.459079i	$0.151821\pi$
$18$	18.0000	0.235702
$19$	−78.9607	−0.953412	−0.476706	−	0.879063i	$-0.658169\pi$
$19$	−78.9607	−0.953412	−0.476706	+	0.879063i	$0.658169\pi$
$20$	70.0040	0.782668
$21$	21.0000	0.218218
$22$	−6.16102	−0.0597061
$23$	126.218	1.14427	0.572137	−	0.820158i	$-0.306114\pi$
$23$	126.218	1.14427	0.572137	+	0.820158i	$0.306114\pi$
$24$	−24.0000	−0.204124
$25$	181.285	1.45028
$26$	26.0000	0.196116
$27$	−27.0000	−0.192450
$28$	−28.0000	−0.188982
$29$	−25.3812	−0.162523	−0.0812615	−	0.996693i	$-0.525895\pi$
$29$	−25.3812	−0.162523	−0.0812615	+	0.996693i	$0.525895\pi$
$30$	−105.006	−0.639046
$31$	−215.249	−1.24709	−0.623547	−	0.781786i	$-0.714309\pi$
$31$	−215.249	−1.24709	−0.623547	+	0.781786i	$0.714309\pi$
$32$	32.0000	0.176777
$33$	9.24154	0.0487498
$34$	249.080	1.25638
$35$	−122.507	−0.591642
$36$	36.0000	0.166667
$37$	370.949	1.64821	0.824103	−	0.566441i	$-0.191680\pi$
$37$	370.949	1.64821	0.824103	+	0.566441i	$0.191680\pi$
$38$	−157.921	−0.674164
$39$	−39.0000	−0.160128
$40$	140.008	0.553430
$41$	−215.291	−0.820067	−0.410034	−	0.912070i	$-0.634483\pi$
$41$	−215.291	−0.820067	−0.410034	+	0.912070i	$0.634483\pi$
$42$	42.0000	0.154303
$43$	−461.369	−1.63624	−0.818118	−	0.575051i	$-0.804982\pi$
$43$	−461.369	−1.63624	−0.818118	+	0.575051i	$0.804982\pi$
$44$	−12.3220	−0.0422186
$45$	157.509	0.521779
$46$	252.436	0.809124
$47$	478.874	1.48619	0.743095	−	0.669185i	$-0.233357\pi$
$47$	478.874	1.48619	0.743095	+	0.669185i	$0.233357\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	362.569	1.02550
$51$	−373.621	−1.02583
$52$	52.0000	0.138675
$53$	503.038	1.30373	0.651864	−	0.758336i	$-0.273987\pi$
$53$	503.038	1.30373	0.651864	+	0.758336i	$0.273987\pi$
$54$	−54.0000	−0.136083
$55$	−53.9120	−0.132173
$56$	−56.0000	−0.133631
$57$	236.882	0.550453
$58$	−50.7624	−0.114921
$59$	681.315	1.50338	0.751692	−	0.659515i	$-0.229238\pi$
$59$	681.315	1.50338	0.751692	+	0.659515i	$0.229238\pi$
$60$	−210.012	−0.451874
$61$	−96.3991	−0.202338	−0.101169	−	0.994869i	$-0.532258\pi$
$61$	−96.3991	−0.202338	−0.101169	+	0.994869i	$0.532258\pi$
$62$	−430.499	−0.881829
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	227.513	0.434146
$66$	18.4831	0.0344713
$67$	489.363	0.892316	0.446158	−	0.894954i	$-0.352792\pi$
$67$	489.363	0.892316	0.446158	+	0.894954i	$0.352792\pi$
$68$	498.161	0.888395
$69$	−378.655	−0.660647
$70$	−245.014	−0.418354
$71$	271.106	0.453160	0.226580	−	0.973993i	$-0.427246\pi$
$71$	271.106	0.453160	0.226580	+	0.973993i	$0.427246\pi$
$72$	72.0000	0.117851
$73$	−529.647	−0.849185	−0.424593	−	0.905384i	$-0.639583\pi$
$73$	−529.647	−0.849185	−0.424593	+	0.905384i	$0.639583\pi$
$74$	741.897	1.16546
$75$	−543.854	−0.837318
$76$	−315.843	−0.476706
$77$	21.5636	0.0319143
$78$	−78.0000	−0.113228
$79$	−910.161	−1.29622	−0.648108	−	0.761548i	$-0.724440\pi$
$79$	−910.161	−1.29622	−0.648108	+	0.761548i	$0.724440\pi$
$80$	280.016	0.391334
$81$	81.0000	0.111111
$82$	−430.581	−0.579875
$83$	1051.22	1.39020	0.695100	−	0.718913i	$-0.255360\pi$
$83$	1051.22	1.39020	0.695100	+	0.718913i	$0.255360\pi$
$84$	84.0000	0.109109
$85$	2179.58	2.78128
$86$	−922.738	−1.15699
$87$	76.1436	0.0938327
$88$	−24.6441	−0.0298531
$89$	−305.279	−0.363590	−0.181795	−	0.983336i	$-0.558191\pi$
$89$	−305.279	−0.363590	−0.181795	+	0.983336i	$0.558191\pi$
$90$	315.018	0.368953
$91$	−91.0000	−0.104828
$92$	504.873	0.572137
$93$	645.748	0.720010
$94$	957.748	1.05090
$95$	−1381.89	−1.49241
$96$	−96.0000	−0.102062
$97$	1128.48	1.18124	0.590619	−	0.806951i	$-0.298884\pi$
$97$	1128.48	1.18124	0.590619	+	0.806951i	$0.298884\pi$
$98$	98.0000	0.101015
$99$	−27.7246	−0.0281457

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.p.1.3	✓	3
3.2	odd	2		1638.4.a.v.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.4.a.p.1.3	✓	3	1.1	even	1	trivial
1638.4.a.v.1.1		3	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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +17.5010 q^{5} -6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +35.0020 q^{10} -3.08051 q^{11} -12.0000 q^{12} +13.0000 q^{13} -14.0000 q^{14} -52.5030 q^{15} +16.0000 q^{16} +124.540 q^{17} +18.0000 q^{18} -78.9607 q^{19} +70.0040 q^{20} +21.0000 q^{21} -6.16102 q^{22} +126.218 q^{23} -24.0000 q^{24} +181.285 q^{25} +26.0000 q^{26} -27.0000 q^{27} -28.0000 q^{28} -25.3812 q^{29} -105.006 q^{30} -215.249 q^{31} +32.0000 q^{32} +9.24154 q^{33} +249.080 q^{34} -122.507 q^{35} +36.0000 q^{36} +370.949 q^{37} -157.921 q^{38} -39.0000 q^{39} +140.008 q^{40} -215.291 q^{41} +42.0000 q^{42} -461.369 q^{43} -12.3220 q^{44} +157.509 q^{45} +252.436 q^{46} +478.874 q^{47} -48.0000 q^{48} +49.0000 q^{49} +362.569 q^{50} -373.621 q^{51} +52.0000 q^{52} +503.038 q^{53} -54.0000 q^{54} -53.9120 q^{55} -56.0000 q^{56} +236.882 q^{57} -50.7624 q^{58} +681.315 q^{59} -210.012 q^{60} -96.3991 q^{61} -430.499 q^{62} -63.0000 q^{63} +64.0000 q^{64} +227.513 q^{65} +18.4831 q^{66} +489.363 q^{67} +498.161 q^{68} -378.655 q^{69} -245.014 q^{70} +271.106 q^{71} +72.0000 q^{72} -529.647 q^{73} +741.897 q^{74} -543.854 q^{75} -315.843 q^{76} +21.5636 q^{77} -78.0000 q^{78} -910.161 q^{79} +280.016 q^{80} +81.0000 q^{81} -430.581 q^{82} +1051.22 q^{83} +84.0000 q^{84} +2179.58 q^{85} -922.738 q^{86} +76.1436 q^{87} -24.6441 q^{88} -305.279 q^{89} +315.018 q^{90} -91.0000 q^{91} +504.873 q^{92} +645.748 q^{93} +957.748 q^{94} -1381.89 q^{95} -96.0000 q^{96} +1128.48 q^{97} +98.0000 q^{98} -27.7246 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 7 q^{5} - 18 q^{6} - 21 q^{7} + 24 q^{8} + 27 q^{9} + 14 q^{10} - 47 q^{11} - 36 q^{12} + 39 q^{13} - 42 q^{14} - 21 q^{15} + 48 q^{16} + 119 q^{17} + 54 q^{18} + 101 q^{19}+ \cdots - 423 q^{99}+O(q^{100}) \) 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 + 7 * q^5 - 18 * q^6 - 21 * q^7 + 24 * q^8 + 27 * q^9 + 14 * q^10 - 47 * q^11 - 36 * q^12 + 39 * q^13 - 42 * q^14 - 21 * q^15 + 48 * q^16 + 119 * q^17 + 54 * q^18 + 101 * q^19 + 28 * q^20 + 63 * q^21 - 94 * q^22 - 27 * q^23 - 72 * q^24 + 266 * q^25 + 78 * q^26 - 81 * q^27 - 84 * q^28 + 345 * q^29 - 42 * q^30 + 152 * q^31 + 96 * q^32 + 141 * q^33 + 238 * q^34 - 49 * q^35 + 108 * q^36 + 227 * q^37 + 202 * q^38 - 117 * q^39 + 56 * q^40 + 452 * q^41 + 126 * q^42 - 415 * q^43 - 188 * q^44 + 63 * q^45 - 54 * q^46 + 240 * q^47 - 144 * q^48 + 147 * q^49 + 532 * q^50 - 357 * q^51 + 156 * q^52 + 874 * q^53 - 162 * q^54 + 1123 * q^55 - 168 * q^56 - 303 * q^57 + 690 * q^58 + 948 * q^59 - 84 * q^60 + 951 * q^61 + 304 * q^62 - 189 * q^63 + 192 * q^64 + 91 * q^65 + 282 * q^66 + 742 * q^67 + 476 * q^68 + 81 * q^69 - 98 * q^70 + 732 * q^71 + 216 * q^72 - 585 * q^73 + 454 * q^74 - 798 * q^75 + 404 * q^76 + 329 * q^77 - 234 * q^78 - 618 * q^79 + 112 * q^80 + 243 * q^81 + 904 * q^82 + 774 * q^83 + 252 * q^84 + 1273 * q^85 - 830 * q^86 - 1035 * q^87 - 376 * q^88 + 730 * q^89 + 126 * q^90 - 273 * q^91 - 108 * q^92 - 456 * q^93 + 480 * q^94 - 2617 * q^95 - 288 * q^96 + 2518 * q^97 + 294 * q^98 - 423 * q^99

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