LMFDB - Embedded newform 546.4.a.i.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,-4,6,8,3] 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:	$32.2150428631$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{673}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 168 $ x^2 - x - 168
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			$13.4711$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +14.4711 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -28.9422 q^{10} -21.4134 q^{11} +12.0000 q^{12} +13.0000 q^{13} +14.0000 q^{14} +43.4134 q^{15} +16.0000 q^{16} -9.52888 q^{17} -18.0000 q^{18} +59.2979 q^{19} +57.8845 q^{20} -21.0000 q^{21} +42.8267 q^{22} +52.4711 q^{23} -24.0000 q^{24} +84.4134 q^{25} -26.0000 q^{26} +27.0000 q^{27} -28.0000 q^{28} +223.182 q^{29} -86.8267 q^{30} +265.884 q^{31} -32.0000 q^{32} -64.2401 q^{33} +19.0578 q^{34} -101.298 q^{35} +36.0000 q^{36} -114.240 q^{37} -118.596 q^{38} +39.0000 q^{39} -115.769 q^{40} +187.173 q^{41} +42.0000 q^{42} +47.0668 q^{43} -85.6535 q^{44} +130.240 q^{45} -104.942 q^{46} -97.1914 q^{47} +48.0000 q^{48} +49.0000 q^{49} -168.827 q^{50} -28.5866 q^{51} +52.0000 q^{52} -415.191 q^{53} -54.0000 q^{54} -309.875 q^{55} +56.0000 q^{56} +177.894 q^{57} -446.365 q^{58} +236.231 q^{59} +173.653 q^{60} -486.009 q^{61} -531.769 q^{62} -63.0000 q^{63} +64.0000 q^{64} +188.125 q^{65} +128.480 q^{66} +616.249 q^{67} -38.1155 q^{68} +157.413 q^{69} +202.596 q^{70} +804.960 q^{71} -72.0000 q^{72} -426.969 q^{73} +228.480 q^{74} +253.240 q^{75} +237.191 q^{76} +149.894 q^{77} -78.0000 q^{78} +53.1733 q^{79} +231.538 q^{80} +81.0000 q^{81} -374.347 q^{82} +231.058 q^{83} -84.0000 q^{84} -137.894 q^{85} -94.1337 q^{86} +669.547 q^{87} +171.307 q^{88} +544.809 q^{89} -260.480 q^{90} -91.0000 q^{91} +209.884 q^{92} +797.653 q^{93} +194.383 q^{94} +858.106 q^{95} -96.0000 q^{96} -757.076 q^{97} -98.0000 q^{98} -192.720 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} + 3 q^{5} - 12 q^{6} - 14 q^{7} - 16 q^{8} + 18 q^{9} - 6 q^{10} + 35 q^{11} + 24 q^{12} + 26 q^{13} + 28 q^{14} + 9 q^{15} + 32 q^{16} - 45 q^{17} - 36 q^{18} - 63 q^{19}+ \cdots + 315 q^{99}+O(q^{100}) $ 2 * q - 4 * q^2 + 6 * q^3 + 8 * q^4 + 3 * q^5 - 12 * q^6 - 14 * q^7 - 16 * q^8 + 18 * q^9 - 6 * q^10 + 35 * q^11 + 24 * q^12 + 26 * q^13 + 28 * q^14 + 9 * q^15 + 32 * q^16 - 45 * q^17 - 36 * q^18 - 63 * q^19 + 12 * q^20 - 42 * q^21 - 70 * q^22 + 79 * q^23 - 48 * q^24 + 91 * q^25 - 52 * q^26 + 54 * q^27 - 56 * q^28 + 161 * q^29 - 18 * q^30 + 428 * q^31 - 64 * q^32 + 105 * q^33 + 90 * q^34 - 21 * q^35 + 72 * q^36 + 5 * q^37 + 126 * q^38 + 78 * q^39 - 24 * q^40 + 530 * q^41 + 84 * q^42 - 295 * q^43 + 140 * q^44 + 27 * q^45 - 158 * q^46 + 532 * q^47 + 96 * q^48 + 98 * q^49 - 182 * q^50 - 135 * q^51 + 104 * q^52 - 104 * q^53 - 108 * q^54 - 957 * q^55 + 112 * q^56 - 189 * q^57 - 322 * q^58 + 680 * q^59 + 36 * q^60 - 531 * q^61 - 856 * q^62 - 126 * q^63 + 128 * q^64 + 39 * q^65 - 210 * q^66 + 558 * q^67 - 180 * q^68 + 237 * q^69 + 42 * q^70 + 676 * q^71 - 144 * q^72 + 521 * q^73 - 10 * q^74 + 273 * q^75 - 252 * q^76 - 245 * q^77 - 156 * q^78 + 262 * q^79 + 48 * q^80 + 162 * q^81 - 1060 * q^82 + 514 * q^83 - 168 * q^84 + 269 * q^85 + 590 * q^86 + 483 * q^87 - 280 * q^88 + 1816 * q^89 - 54 * q^90 - 182 * q^91 + 316 * q^92 + 1284 * q^93 - 1064 * q^94 + 2261 * q^95 - 192 * q^96 - 684 * q^97 - 196 * q^98 + 315 * 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
$5$	14.4711	1.29434	0.647168	−	0.762347i	$-0.275953\pi$
$5$	14.4711	1.29434	0.647168	+	0.762347i	$0.275953\pi$
$6$	−6.00000	−0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−28.9422	−0.915234
$11$	−21.4134	−0.586943	−0.293471	−	0.955968i	$-0.594811\pi$
$11$	−21.4134	−0.586943	−0.293471	+	0.955968i	$0.594811\pi$
$12$	12.0000	0.288675
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	14.0000	0.267261
$15$	43.4134	0.747286
$16$	16.0000	0.250000
$17$	−9.52888	−0.135947	−0.0679733	−	0.997687i	$-0.521653\pi$
$17$	−9.52888	−0.135947	−0.0679733	+	0.997687i	$0.521653\pi$
$18$	−18.0000	−0.235702
$19$	59.2979	0.715993	0.357996	−	0.933723i	$-0.383460\pi$
$19$	59.2979	0.715993	0.357996	+	0.933723i	$0.383460\pi$
$20$	57.8845	0.647168
$21$	−21.0000	−0.218218
$22$	42.8267	0.415031
$23$	52.4711	0.475695	0.237848	−	0.971303i	$-0.423558\pi$
$23$	52.4711	0.475695	0.237848	+	0.971303i	$0.423558\pi$
$24$	−24.0000	−0.204124
$25$	84.4134	0.675307
$26$	−26.0000	−0.196116
$27$	27.0000	0.192450
$28$	−28.0000	−0.188982
$29$	223.182	1.42910	0.714550	−	0.699584i	$-0.246631\pi$
$29$	223.182	1.42910	0.714550	+	0.699584i	$0.246631\pi$
$30$	−86.8267	−0.528411
$31$	265.884	1.54046	0.770230	−	0.637766i	$-0.220141\pi$
$31$	265.884	1.54046	0.770230	+	0.637766i	$0.220141\pi$
$32$	−32.0000	−0.176777
$33$	−64.2401	−0.338872
$34$	19.0578	0.0961288
$35$	−101.298	−0.489213
$36$	36.0000	0.166667
$37$	−114.240	−0.507593	−0.253797	−	0.967258i	$-0.581679\pi$
$37$	−114.240	−0.507593	−0.253797	+	0.967258i	$0.581679\pi$
$38$	−118.596	−0.506283
$39$	39.0000	0.160128
$40$	−115.769	−0.457617
$41$	187.173	0.712965	0.356482	−	0.934302i	$-0.383976\pi$
$41$	187.173	0.712965	0.356482	+	0.934302i	$0.383976\pi$
$42$	42.0000	0.154303
$43$	47.0668	0.166921	0.0834607	−	0.996511i	$-0.473403\pi$
$43$	47.0668	0.166921	0.0834607	+	0.996511i	$0.473403\pi$
$44$	−85.6535	−0.293471
$45$	130.240	0.431445
$46$	−104.942	−0.336367
$47$	−97.1914	−0.301635	−0.150817	−	0.988562i	$-0.548191\pi$
$47$	−97.1914	−0.301635	−0.150817	+	0.988562i	$0.548191\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	−168.827	−0.477514
$51$	−28.5866	−0.0784888
$52$	52.0000	0.138675
$53$	−415.191	−1.07606	−0.538028	−	0.842927i	$-0.680830\pi$
$53$	−415.191	−1.07606	−0.538028	+	0.842927i	$0.680830\pi$
$54$	−54.0000	−0.136083
$55$	−309.875	−0.759702
$56$	56.0000	0.133631
$57$	177.894	0.413379
$58$	−446.365	−1.01053
$59$	236.231	0.521265	0.260633	−	0.965438i	$-0.416069\pi$
$59$	236.231	0.521265	0.260633	+	0.965438i	$0.416069\pi$
$60$	173.653	0.373643
$61$	−486.009	−1.02012	−0.510058	−	0.860140i	$-0.670376\pi$
$61$	−486.009	−1.02012	−0.510058	+	0.860140i	$0.670376\pi$
$62$	−531.769	−1.08927
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	188.125	0.358984
$66$	128.480	0.239618
$67$	616.249	1.12368	0.561842	−	0.827245i	$-0.310093\pi$
$67$	616.249	1.12368	0.561842	+	0.827245i	$0.310093\pi$
$68$	−38.1155	−0.0679733
$69$	157.413	0.274643
$70$	202.596	0.345926
$71$	804.960	1.34551	0.672755	−	0.739865i	$-0.265111\pi$
$71$	804.960	1.34551	0.672755	+	0.739865i	$0.265111\pi$
$72$	−72.0000	−0.117851
$73$	−426.969	−0.684562	−0.342281	−	0.939598i	$-0.611199\pi$
$73$	−426.969	−0.684562	−0.342281	+	0.939598i	$0.611199\pi$
$74$	228.480	0.358923
$75$	253.240	0.389889
$76$	237.191	0.357996
$77$	149.894	0.221844
$78$	−78.0000	−0.113228
$79$	53.1733	0.0757273	0.0378637	−	0.999283i	$-0.487945\pi$
$79$	53.1733	0.0757273	0.0378637	+	0.999283i	$0.487945\pi$
$80$	231.538	0.323584
$81$	81.0000	0.111111
$82$	−374.347	−0.504142
$83$	231.058	0.305565	0.152782	−	0.988260i	$-0.451177\pi$
$83$	231.058	0.305565	0.152782	+	0.988260i	$0.451177\pi$
$84$	−84.0000	−0.109109
$85$	−137.894	−0.175961
$86$	−94.1337	−0.118031
$87$	669.547	0.825092
$88$	171.307	0.207516
$89$	544.809	0.648872	0.324436	−	0.945908i	$-0.394826\pi$
$89$	544.809	0.648872	0.324436	+	0.945908i	$0.394826\pi$
$90$	−260.480	−0.305078
$91$	−91.0000	−0.104828
$92$	209.884	0.237848
$93$	797.653	0.889385
$94$	194.383	0.213288
$95$	858.106	0.926735
$96$	−96.0000	−0.102062
$97$	−757.076	−0.792468	−0.396234	−	0.918150i	$-0.629683\pi$
$97$	−757.076	−0.792468	−0.396234	+	0.918150i	$0.629683\pi$
$98$	−98.0000	−0.101015
$99$	−192.720	−0.195648

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.i.1.2	✓	2
3.2	odd	2		1638.4.a.q.1.1		2

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

Overview	Random
Universe	Knowledge

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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +14.4711 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -28.9422 q^{10} -21.4134 q^{11} +12.0000 q^{12} +13.0000 q^{13} +14.0000 q^{14} +43.4134 q^{15} +16.0000 q^{16} -9.52888 q^{17} -18.0000 q^{18} +59.2979 q^{19} +57.8845 q^{20} -21.0000 q^{21} +42.8267 q^{22} +52.4711 q^{23} -24.0000 q^{24} +84.4134 q^{25} -26.0000 q^{26} +27.0000 q^{27} -28.0000 q^{28} +223.182 q^{29} -86.8267 q^{30} +265.884 q^{31} -32.0000 q^{32} -64.2401 q^{33} +19.0578 q^{34} -101.298 q^{35} +36.0000 q^{36} -114.240 q^{37} -118.596 q^{38} +39.0000 q^{39} -115.769 q^{40} +187.173 q^{41} +42.0000 q^{42} +47.0668 q^{43} -85.6535 q^{44} +130.240 q^{45} -104.942 q^{46} -97.1914 q^{47} +48.0000 q^{48} +49.0000 q^{49} -168.827 q^{50} -28.5866 q^{51} +52.0000 q^{52} -415.191 q^{53} -54.0000 q^{54} -309.875 q^{55} +56.0000 q^{56} +177.894 q^{57} -446.365 q^{58} +236.231 q^{59} +173.653 q^{60} -486.009 q^{61} -531.769 q^{62} -63.0000 q^{63} +64.0000 q^{64} +188.125 q^{65} +128.480 q^{66} +616.249 q^{67} -38.1155 q^{68} +157.413 q^{69} +202.596 q^{70} +804.960 q^{71} -72.0000 q^{72} -426.969 q^{73} +228.480 q^{74} +253.240 q^{75} +237.191 q^{76} +149.894 q^{77} -78.0000 q^{78} +53.1733 q^{79} +231.538 q^{80} +81.0000 q^{81} -374.347 q^{82} +231.058 q^{83} -84.0000 q^{84} -137.894 q^{85} -94.1337 q^{86} +669.547 q^{87} +171.307 q^{88} +544.809 q^{89} -260.480 q^{90} -91.0000 q^{91} +209.884 q^{92} +797.653 q^{93} +194.383 q^{94} +858.106 q^{95} -96.0000 q^{96} -757.076 q^{97} -98.0000 q^{98} -192.720 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} + 3 q^{5} - 12 q^{6} - 14 q^{7} - 16 q^{8} + 18 q^{9} - 6 q^{10} + 35 q^{11} + 24 q^{12} + 26 q^{13} + 28 q^{14} + 9 q^{15} + 32 q^{16} - 45 q^{17} - 36 q^{18} - 63 q^{19}+ \cdots + 315 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 + 6 * q^3 + 8 * q^4 + 3 * q^5 - 12 * q^6 - 14 * q^7 - 16 * q^8 + 18 * q^9 - 6 * q^10 + 35 * q^11 + 24 * q^12 + 26 * q^13 + 28 * q^14 + 9 * q^15 + 32 * q^16 - 45 * q^17 - 36 * q^18 - 63 * q^19 + 12 * q^20 - 42 * q^21 - 70 * q^22 + 79 * q^23 - 48 * q^24 + 91 * q^25 - 52 * q^26 + 54 * q^27 - 56 * q^28 + 161 * q^29 - 18 * q^30 + 428 * q^31 - 64 * q^32 + 105 * q^33 + 90 * q^34 - 21 * q^35 + 72 * q^36 + 5 * q^37 + 126 * q^38 + 78 * q^39 - 24 * q^40 + 530 * q^41 + 84 * q^42 - 295 * q^43 + 140 * q^44 + 27 * q^45 - 158 * q^46 + 532 * q^47 + 96 * q^48 + 98 * q^49 - 182 * q^50 - 135 * q^51 + 104 * q^52 - 104 * q^53 - 108 * q^54 - 957 * q^55 + 112 * q^56 - 189 * q^57 - 322 * q^58 + 680 * q^59 + 36 * q^60 - 531 * q^61 - 856 * q^62 - 126 * q^63 + 128 * q^64 + 39 * q^65 - 210 * q^66 + 558 * q^67 - 180 * q^68 + 237 * q^69 + 42 * q^70 + 676 * q^71 - 144 * q^72 + 521 * q^73 - 10 * q^74 + 273 * q^75 - 252 * q^76 - 245 * q^77 - 156 * q^78 + 262 * q^79 + 48 * q^80 + 162 * q^81 - 1060 * q^82 + 514 * q^83 - 168 * q^84 + 269 * q^85 + 590 * q^86 + 483 * q^87 - 280 * q^88 + 1816 * q^89 - 54 * q^90 - 182 * q^91 + 316 * q^92 + 1284 * q^93 - 1064 * q^94 + 2261 * q^95 - 192 * q^96 - 684 * q^97 - 196 * q^98 + 315 * q^99

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