LMFDB - Embedded newform 546.4.a.g.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,-15] 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{65}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.53113$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +4.59339 q^{5} -6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -9.18677 q^{10} -45.1868 q^{11} +12.0000 q^{12} +13.0000 q^{13} -14.0000 q^{14} +13.7802 q^{15} +16.0000 q^{16} -93.5603 q^{17} -18.0000 q^{18} -35.7802 q^{19} +18.3735 q^{20} +21.0000 q^{21} +90.3735 q^{22} -79.7802 q^{23} -24.0000 q^{24} -103.901 q^{25} -26.0000 q^{26} +27.0000 q^{27} +28.0000 q^{28} -13.0331 q^{29} -27.5603 q^{30} -71.7802 q^{31} -32.0000 q^{32} -135.560 q^{33} +187.121 q^{34} +32.1537 q^{35} +36.0000 q^{36} -42.4397 q^{37} +71.5603 q^{38} +39.0000 q^{39} -36.7471 q^{40} -53.2529 q^{41} -42.0000 q^{42} -4.65952 q^{43} -180.747 q^{44} +41.3405 q^{45} +159.560 q^{46} +21.2743 q^{47} +48.0000 q^{48} +49.0000 q^{49} +207.802 q^{50} -280.681 q^{51} +52.0000 q^{52} +258.088 q^{53} -54.0000 q^{54} -207.560 q^{55} -56.0000 q^{56} -107.340 q^{57} +26.0661 q^{58} -336.000 q^{59} +55.1206 q^{60} +230.000 q^{61} +143.560 q^{62} +63.0000 q^{63} +64.0000 q^{64} +59.7140 q^{65} +271.121 q^{66} -408.043 q^{67} -374.241 q^{68} -239.340 q^{69} -64.3074 q^{70} +92.6148 q^{71} -72.0000 q^{72} +621.383 q^{73} +84.8794 q^{74} -311.702 q^{75} -143.121 q^{76} -316.307 q^{77} -78.0000 q^{78} +350.461 q^{79} +73.4942 q^{80} +81.0000 q^{81} +106.506 q^{82} -1210.94 q^{83} +84.0000 q^{84} -429.759 q^{85} +9.31904 q^{86} -39.0992 q^{87} +361.494 q^{88} -336.461 q^{89} -82.6810 q^{90} +91.0000 q^{91} -319.121 q^{92} -215.340 q^{93} -42.5487 q^{94} -164.352 q^{95} -96.0000 q^{96} -854.220 q^{97} -98.0000 q^{98} -406.681 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} - 15 q^{5} - 12 q^{6} + 14 q^{7} - 16 q^{8} + 18 q^{9} + 30 q^{10} - 42 q^{11} + 24 q^{12} + 26 q^{13} - 28 q^{14} - 45 q^{15} + 32 q^{16} - 42 q^{17} - 36 q^{18} + q^{19}+ \cdots - 378 q^{99}+O(q^{100}) $ 2 * q - 4 * q^2 + 6 * q^3 + 8 * q^4 - 15 * q^5 - 12 * q^6 + 14 * q^7 - 16 * q^8 + 18 * q^9 + 30 * q^10 - 42 * q^11 + 24 * q^12 + 26 * q^13 - 28 * q^14 - 45 * q^15 + 32 * q^16 - 42 * q^17 - 36 * q^18 + q^19 - 60 * q^20 + 42 * q^21 + 84 * q^22 - 87 * q^23 - 48 * q^24 + 155 * q^25 - 52 * q^26 + 54 * q^27 + 56 * q^28 - 147 * q^29 + 90 * q^30 - 71 * q^31 - 64 * q^32 - 126 * q^33 + 84 * q^34 - 105 * q^35 + 72 * q^36 - 230 * q^37 - 2 * q^38 + 78 * q^39 + 120 * q^40 - 300 * q^41 - 84 * q^42 - 227 * q^43 - 168 * q^44 - 135 * q^45 + 174 * q^46 - 417 * q^47 + 96 * q^48 + 98 * q^49 - 310 * q^50 - 126 * q^51 + 104 * q^52 + 105 * q^53 - 108 * q^54 - 270 * q^55 - 112 * q^56 + 3 * q^57 + 294 * q^58 - 672 * q^59 - 180 * q^60 + 460 * q^61 + 142 * q^62 + 126 * q^63 + 128 * q^64 - 195 * q^65 + 252 * q^66 + 490 * q^67 - 168 * q^68 - 261 * q^69 + 210 * q^70 - 492 * q^71 - 144 * q^72 - 281 * q^73 + 460 * q^74 + 465 * q^75 + 4 * q^76 - 294 * q^77 - 156 * q^78 + 193 * q^79 - 240 * q^80 + 162 * q^81 + 600 * q^82 - 753 * q^83 + 168 * q^84 - 1440 * q^85 + 454 * q^86 - 441 * q^87 + 336 * q^88 - 165 * q^89 + 270 * q^90 + 182 * q^91 - 348 * q^92 - 213 * q^93 + 834 * q^94 - 885 * q^95 - 192 * q^96 - 1781 * q^97 - 196 * q^98 - 378 * 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
$5$	4.59339	0.410845	0.205422	−	0.978673i	$-0.434143\pi$
$5$	4.59339	0.410845	0.205422	+	0.978673i	$0.434143\pi$
$6$	−6.00000	−0.408248
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−9.18677	−0.290511
$11$	−45.1868	−1.23857	−0.619287	−	0.785164i	$-0.712578\pi$
$11$	−45.1868	−1.23857	−0.619287	+	0.785164i	$0.712578\pi$
$12$	12.0000	0.288675
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	−14.0000	−0.267261
$15$	13.7802	0.237201
$16$	16.0000	0.250000
$17$	−93.5603	−1.33481	−0.667403	−	0.744697i	$-0.732594\pi$
$17$	−93.5603	−1.33481	−0.667403	+	0.744697i	$0.732594\pi$
$18$	−18.0000	−0.235702
$19$	−35.7802	−0.432028	−0.216014	−	0.976390i	$-0.569306\pi$
$19$	−35.7802	−0.432028	−0.216014	+	0.976390i	$0.569306\pi$
$20$	18.3735	0.205422
$21$	21.0000	0.218218
$22$	90.3735	0.875805
$23$	−79.7802	−0.723274	−0.361637	−	0.932319i	$-0.617782\pi$
$23$	−79.7802	−0.723274	−0.361637	+	0.932319i	$0.617782\pi$
$24$	−24.0000	−0.204124
$25$	−103.901	−0.831206
$26$	−26.0000	−0.196116
$27$	27.0000	0.192450
$28$	28.0000	0.188982
$29$	−13.0331	−0.0834545	−0.0417272	−	0.999129i	$-0.513286\pi$
$29$	−13.0331	−0.0834545	−0.0417272	+	0.999129i	$0.513286\pi$
$30$	−27.5603	−0.167727
$31$	−71.7802	−0.415874	−0.207937	−	0.978142i	$-0.566675\pi$
$31$	−71.7802	−0.415874	−0.207937	+	0.978142i	$0.566675\pi$
$32$	−32.0000	−0.176777
$33$	−135.560	−0.715092
$34$	187.121	0.943851
$35$	32.1537	0.155285
$36$	36.0000	0.166667
$37$	−42.4397	−0.188569	−0.0942843	−	0.995545i	$-0.530056\pi$
$37$	−42.4397	−0.188569	−0.0942843	+	0.995545i	$0.530056\pi$
$38$	71.5603	0.305490
$39$	39.0000	0.160128
$40$	−36.7471	−0.145256
$41$	−53.2529	−0.202847	−0.101423	−	0.994843i	$-0.532340\pi$
$41$	−53.2529	−0.202847	−0.101423	+	0.994843i	$0.532340\pi$
$42$	−42.0000	−0.154303
$43$	−4.65952	−0.0165249	−0.00826244	−	0.999966i	$-0.502630\pi$
$43$	−4.65952	−0.0165249	−0.00826244	+	0.999966i	$0.502630\pi$
$44$	−180.747	−0.619287
$45$	41.3405	0.136948
$46$	159.560	0.511432
$47$	21.2743	0.0660252	0.0330126	−	0.999455i	$-0.489490\pi$
$47$	21.2743	0.0660252	0.0330126	+	0.999455i	$0.489490\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	207.802	0.587752
$51$	−280.681	−0.770651
$52$	52.0000	0.138675
$53$	258.088	0.668888	0.334444	−	0.942416i	$-0.391452\pi$
$53$	258.088	0.668888	0.334444	+	0.942416i	$0.391452\pi$
$54$	−54.0000	−0.136083
$55$	−207.560	−0.508862
$56$	−56.0000	−0.133631
$57$	−107.340	−0.249431
$58$	26.0661	0.0590112
$59$	−336.000	−0.741415	−0.370707	−	0.928750i	$-0.620885\pi$
$59$	−336.000	−0.741415	−0.370707	+	0.928750i	$0.620885\pi$
$60$	55.1206	0.118601
$61$	230.000	0.482762	0.241381	−	0.970430i	$-0.422400\pi$
$61$	230.000	0.482762	0.241381	+	0.970430i	$0.422400\pi$
$62$	143.560	0.294067
$63$	63.0000	0.125988
$64$	64.0000	0.125000
$65$	59.7140	0.113948
$66$	271.121	0.505646
$67$	−408.043	−0.744035	−0.372018	−	0.928226i	$-0.621334\pi$
$67$	−408.043	−0.744035	−0.372018	+	0.928226i	$0.621334\pi$
$68$	−374.241	−0.667403
$69$	−239.340	−0.417583
$70$	−64.3074	−0.109803
$71$	92.6148	0.154808	0.0774039	−	0.997000i	$-0.475337\pi$
$71$	92.6148	0.154808	0.0774039	+	0.997000i	$0.475337\pi$
$72$	−72.0000	−0.117851
$73$	621.383	0.996266	0.498133	−	0.867101i	$-0.334019\pi$
$73$	621.383	0.996266	0.498133	+	0.867101i	$0.334019\pi$
$74$	84.8794	0.133338
$75$	−311.702	−0.479897
$76$	−143.121	−0.216014
$77$	−316.307	−0.468137
$78$	−78.0000	−0.113228
$79$	350.461	0.499113	0.249557	−	0.968360i	$-0.419715\pi$
$79$	350.461	0.499113	0.249557	+	0.968360i	$0.419715\pi$
$80$	73.4942	0.102711
$81$	81.0000	0.111111
$82$	106.506	0.143434
$83$	−1210.94	−1.60143	−0.800713	−	0.599048i	$-0.795546\pi$
$83$	−1210.94	−1.60143	−0.800713	+	0.599048i	$0.795546\pi$
$84$	84.0000	0.109109
$85$	−429.759	−0.548399
$86$	9.31904	0.0116849
$87$	−39.0992	−0.0481825
$88$	361.494	0.437902
$89$	−336.461	−0.400728	−0.200364	−	0.979722i	$-0.564213\pi$
$89$	−336.461	−0.400728	−0.200364	+	0.979722i	$0.564213\pi$
$90$	−82.6810	−0.0968371
$91$	91.0000	0.104828
$92$	−319.121	−0.361637
$93$	−215.340	−0.240105
$94$	−42.5487	−0.0466868
$95$	−164.352	−0.177497
$96$	−96.0000	−0.102062
$97$	−854.220	−0.894153	−0.447077	−	0.894496i	$-0.647535\pi$
$97$	−854.220	−0.894153	−0.447077	+	0.894496i	$0.647535\pi$
$98$	−98.0000	−0.101015
$99$	−406.681	−0.412858

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.g.1.2	✓	2
3.2	odd	2		1638.4.a.u.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.4.a.g.1.2	✓	2	1.1	even	1	trivial
1638.4.a.u.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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +4.59339 q^{5} -6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -9.18677 q^{10} -45.1868 q^{11} +12.0000 q^{12} +13.0000 q^{13} -14.0000 q^{14} +13.7802 q^{15} +16.0000 q^{16} -93.5603 q^{17} -18.0000 q^{18} -35.7802 q^{19} +18.3735 q^{20} +21.0000 q^{21} +90.3735 q^{22} -79.7802 q^{23} -24.0000 q^{24} -103.901 q^{25} -26.0000 q^{26} +27.0000 q^{27} +28.0000 q^{28} -13.0331 q^{29} -27.5603 q^{30} -71.7802 q^{31} -32.0000 q^{32} -135.560 q^{33} +187.121 q^{34} +32.1537 q^{35} +36.0000 q^{36} -42.4397 q^{37} +71.5603 q^{38} +39.0000 q^{39} -36.7471 q^{40} -53.2529 q^{41} -42.0000 q^{42} -4.65952 q^{43} -180.747 q^{44} +41.3405 q^{45} +159.560 q^{46} +21.2743 q^{47} +48.0000 q^{48} +49.0000 q^{49} +207.802 q^{50} -280.681 q^{51} +52.0000 q^{52} +258.088 q^{53} -54.0000 q^{54} -207.560 q^{55} -56.0000 q^{56} -107.340 q^{57} +26.0661 q^{58} -336.000 q^{59} +55.1206 q^{60} +230.000 q^{61} +143.560 q^{62} +63.0000 q^{63} +64.0000 q^{64} +59.7140 q^{65} +271.121 q^{66} -408.043 q^{67} -374.241 q^{68} -239.340 q^{69} -64.3074 q^{70} +92.6148 q^{71} -72.0000 q^{72} +621.383 q^{73} +84.8794 q^{74} -311.702 q^{75} -143.121 q^{76} -316.307 q^{77} -78.0000 q^{78} +350.461 q^{79} +73.4942 q^{80} +81.0000 q^{81} +106.506 q^{82} -1210.94 q^{83} +84.0000 q^{84} -429.759 q^{85} +9.31904 q^{86} -39.0992 q^{87} +361.494 q^{88} -336.461 q^{89} -82.6810 q^{90} +91.0000 q^{91} -319.121 q^{92} -215.340 q^{93} -42.5487 q^{94} -164.352 q^{95} -96.0000 q^{96} -854.220 q^{97} -98.0000 q^{98} -406.681 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} - 15 q^{5} - 12 q^{6} + 14 q^{7} - 16 q^{8} + 18 q^{9} + 30 q^{10} - 42 q^{11} + 24 q^{12} + 26 q^{13} - 28 q^{14} - 45 q^{15} + 32 q^{16} - 42 q^{17} - 36 q^{18} + q^{19}+ \cdots - 378 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 + 6 * q^3 + 8 * q^4 - 15 * q^5 - 12 * q^6 + 14 * q^7 - 16 * q^8 + 18 * q^9 + 30 * q^10 - 42 * q^11 + 24 * q^12 + 26 * q^13 - 28 * q^14 - 45 * q^15 + 32 * q^16 - 42 * q^17 - 36 * q^18 + q^19 - 60 * q^20 + 42 * q^21 + 84 * q^22 - 87 * q^23 - 48 * q^24 + 155 * q^25 - 52 * q^26 + 54 * q^27 + 56 * q^28 - 147 * q^29 + 90 * q^30 - 71 * q^31 - 64 * q^32 - 126 * q^33 + 84 * q^34 - 105 * q^35 + 72 * q^36 - 230 * q^37 - 2 * q^38 + 78 * q^39 + 120 * q^40 - 300 * q^41 - 84 * q^42 - 227 * q^43 - 168 * q^44 - 135 * q^45 + 174 * q^46 - 417 * q^47 + 96 * q^48 + 98 * q^49 - 310 * q^50 - 126 * q^51 + 104 * q^52 + 105 * q^53 - 108 * q^54 - 270 * q^55 - 112 * q^56 + 3 * q^57 + 294 * q^58 - 672 * q^59 - 180 * q^60 + 460 * q^61 + 142 * q^62 + 126 * q^63 + 128 * q^64 - 195 * q^65 + 252 * q^66 + 490 * q^67 - 168 * q^68 - 261 * q^69 + 210 * q^70 - 492 * q^71 - 144 * q^72 - 281 * q^73 + 460 * q^74 + 465 * q^75 + 4 * q^76 - 294 * q^77 - 156 * q^78 + 193 * q^79 - 240 * q^80 + 162 * q^81 + 600 * q^82 - 753 * q^83 + 168 * q^84 - 1440 * q^85 + 454 * q^86 - 441 * q^87 + 336 * q^88 - 165 * q^89 + 270 * q^90 + 182 * q^91 - 348 * q^92 - 213 * q^93 + 834 * q^94 - 885 * q^95 - 192 * q^96 - 1781 * q^97 - 196 * q^98 - 378 * q^99

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