LMFDB - Embedded newform 546.4.a.o.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 = [3,6,-9,12,-1] 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:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 218x + 456 $ x^3 - x^2 - 218*x + 456
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.11461$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -2.11461 q^{5} -6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -4.22921 q^{10} +15.6562 q^{11} -12.0000 q^{12} -13.0000 q^{13} +14.0000 q^{14} +6.34382 q^{15} +16.0000 q^{16} +137.299 q^{17} +18.0000 q^{18} -147.070 q^{19} -8.45842 q^{20} -21.0000 q^{21} +31.3124 q^{22} -82.8408 q^{23} -24.0000 q^{24} -120.528 q^{25} -26.0000 q^{26} -27.0000 q^{27} +28.0000 q^{28} +232.841 q^{29} +12.6876 q^{30} +304.203 q^{31} +32.0000 q^{32} -46.9685 q^{33} +274.598 q^{34} -14.8022 q^{35} +36.0000 q^{36} +298.445 q^{37} -294.140 q^{38} +39.0000 q^{39} -16.9168 q^{40} -90.7891 q^{41} -42.0000 q^{42} +388.674 q^{43} +62.6247 q^{44} -19.0315 q^{45} -165.682 q^{46} +43.8985 q^{47} -48.0000 q^{48} +49.0000 q^{49} -241.057 q^{50} -411.898 q^{51} -52.0000 q^{52} +458.203 q^{53} -54.0000 q^{54} -33.1067 q^{55} +56.0000 q^{56} +441.210 q^{57} +465.682 q^{58} -146.268 q^{59} +25.3753 q^{60} +210.701 q^{61} +608.406 q^{62} +63.0000 q^{63} +64.0000 q^{64} +27.4899 q^{65} -93.9371 q^{66} +357.272 q^{67} +549.197 q^{68} +248.522 q^{69} -29.6045 q^{70} +59.8985 q^{71} +72.0000 q^{72} -220.230 q^{73} +596.891 q^{74} +361.585 q^{75} -588.280 q^{76} +109.593 q^{77} +78.0000 q^{78} +1139.80 q^{79} -33.8337 q^{80} +81.0000 q^{81} -181.578 q^{82} -796.278 q^{83} -84.0000 q^{84} -290.334 q^{85} +777.349 q^{86} -698.522 q^{87} +125.249 q^{88} -577.436 q^{89} -38.0629 q^{90} -91.0000 q^{91} -331.363 q^{92} -912.609 q^{93} +87.7971 q^{94} +310.995 q^{95} -96.0000 q^{96} -105.793 q^{97} +98.0000 q^{98} +140.906 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} - q^{5} - 18 q^{6} + 21 q^{7} + 24 q^{8} + 27 q^{9} - 2 q^{10} + 63 q^{11} - 36 q^{12} - 39 q^{13} + 42 q^{14} + 3 q^{15} + 48 q^{16} - q^{17} + 54 q^{18} - 39 q^{19}+ \cdots + 567 q^{99}+O(q^{100}) $ 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 - q^5 - 18 * q^6 + 21 * q^7 + 24 * q^8 + 27 * q^9 - 2 * q^10 + 63 * q^11 - 36 * q^12 - 39 * q^13 + 42 * q^14 + 3 * q^15 + 48 * q^16 - q^17 + 54 * q^18 - 39 * q^19 - 4 * q^20 - 63 * q^21 + 126 * q^22 + 143 * q^23 - 72 * q^24 + 62 * q^25 - 78 * q^26 - 81 * q^27 + 84 * q^28 + 307 * q^29 + 6 * q^30 + 12 * q^31 + 96 * q^32 - 189 * q^33 - 2 * q^34 - 7 * q^35 + 108 * q^36 + 429 * q^37 - 78 * q^38 + 117 * q^39 - 8 * q^40 + 210 * q^41 - 126 * q^42 + 689 * q^43 + 252 * q^44 - 9 * q^45 + 286 * q^46 + 582 * q^47 - 144 * q^48 + 147 * q^49 + 124 * q^50 + 3 * q^51 - 156 * q^52 + 474 * q^53 - 162 * q^54 + 1289 * q^55 + 168 * q^56 + 117 * q^57 + 614 * q^58 - 74 * q^59 + 12 * q^60 + 1045 * q^61 + 24 * q^62 + 189 * q^63 + 192 * q^64 + 13 * q^65 - 378 * q^66 + 826 * q^67 - 4 * q^68 - 429 * q^69 - 14 * q^70 + 630 * q^71 + 216 * q^72 + 407 * q^73 + 858 * q^74 - 186 * q^75 - 156 * q^76 + 441 * q^77 + 234 * q^78 + 942 * q^79 - 16 * q^80 + 243 * q^81 + 420 * q^82 - 148 * q^83 - 252 * q^84 + 15 * q^85 + 1378 * q^86 - 921 * q^87 + 504 * q^88 - 940 * q^89 - 18 * q^90 - 273 * q^91 + 572 * q^92 - 36 * q^93 + 1164 * q^94 - 875 * q^95 - 288 * q^96 + 46 * q^97 + 294 * q^98 + 567 * 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
$5$	−2.11461	−0.189136	−0.0945680	−	0.995518i	$-0.530147\pi$
$5$	−2.11461	−0.189136	−0.0945680	+	0.995518i	$0.530147\pi$
$6$	−6.00000	−0.408248
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−4.22921	−0.133739
$11$	15.6562	0.429138	0.214569	−	0.976709i	$-0.431165\pi$
$11$	15.6562	0.429138	0.214569	+	0.976709i	$0.431165\pi$
$12$	−12.0000	−0.288675
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	14.0000	0.267261
$15$	6.34382	0.109198
$16$	16.0000	0.250000
$17$	137.299	1.95882	0.979410	−	0.201879i	$-0.0647048\pi$
$17$	137.299	1.95882	0.979410	+	0.201879i	$0.0647048\pi$
$18$	18.0000	0.235702
$19$	−147.070	−1.77580	−0.887899	−	0.460038i	$-0.847836\pi$
$19$	−147.070	−1.77580	−0.887899	+	0.460038i	$0.847836\pi$
$20$	−8.45842	−0.0945680
$21$	−21.0000	−0.218218
$22$	31.3124	0.303446
$23$	−82.8408	−0.751022	−0.375511	−	0.926818i	$-0.622533\pi$
$23$	−82.8408	−0.751022	−0.375511	+	0.926818i	$0.622533\pi$
$24$	−24.0000	−0.204124
$25$	−120.528	−0.964228
$26$	−26.0000	−0.196116
$27$	−27.0000	−0.192450
$28$	28.0000	0.188982
$29$	232.841	1.49095	0.745473	−	0.666535i	$-0.232223\pi$
$29$	232.841	1.49095	0.745473	+	0.666535i	$0.232223\pi$
$30$	12.6876	0.0772145
$31$	304.203	1.76247	0.881233	−	0.472682i	$-0.156714\pi$
$31$	304.203	1.76247	0.881233	+	0.472682i	$0.156714\pi$
$32$	32.0000	0.176777
$33$	−46.9685	−0.247763
$34$	274.598	1.38510
$35$	−14.8022	−0.0714867
$36$	36.0000	0.166667
$37$	298.445	1.32606	0.663028	−	0.748594i	$-0.269271\pi$
$37$	298.445	1.32606	0.663028	+	0.748594i	$0.269271\pi$
$38$	−294.140	−1.25568
$39$	39.0000	0.160128
$40$	−16.9168	−0.0668697
$41$	−90.7891	−0.345826	−0.172913	−	0.984937i	$-0.555318\pi$
$41$	−90.7891	−0.345826	−0.172913	+	0.984937i	$0.555318\pi$
$42$	−42.0000	−0.154303
$43$	388.674	1.37843	0.689213	−	0.724559i	$-0.257956\pi$
$43$	388.674	1.37843	0.689213	+	0.724559i	$0.257956\pi$
$44$	62.6247	0.214569
$45$	−19.0315	−0.0630454
$46$	−165.682	−0.531053
$47$	43.8985	0.136240	0.0681198	−	0.997677i	$-0.478300\pi$
$47$	43.8985	0.136240	0.0681198	+	0.997677i	$0.478300\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	−241.057	−0.681812
$51$	−411.898	−1.13093
$52$	−52.0000	−0.138675
$53$	458.203	1.18753	0.593764	−	0.804639i	$-0.297641\pi$
$53$	458.203	1.18753	0.593764	+	0.804639i	$0.297641\pi$
$54$	−54.0000	−0.136083
$55$	−33.1067	−0.0811655
$56$	56.0000	0.133631
$57$	441.210	1.02526
$58$	465.682	1.05426
$59$	−146.268	−0.322753	−0.161377	−	0.986893i	$-0.551593\pi$
$59$	−146.268	−0.322753	−0.161377	+	0.986893i	$0.551593\pi$
$60$	25.3753	0.0545989
$61$	210.701	0.442254	0.221127	−	0.975245i	$-0.429027\pi$
$61$	210.701	0.442254	0.221127	+	0.975245i	$0.429027\pi$
$62$	608.406	1.24625
$63$	63.0000	0.125988
$64$	64.0000	0.125000
$65$	27.4899	0.0524569
$66$	−93.9371	−0.175195
$67$	357.272	0.651458	0.325729	−	0.945463i	$-0.394390\pi$
$67$	357.272	0.651458	0.325729	+	0.945463i	$0.394390\pi$
$68$	549.197	0.979410
$69$	248.522	0.433603
$70$	−29.6045	−0.0505487
$71$	59.8985	0.100122	0.0500609	−	0.998746i	$-0.484058\pi$
$71$	59.8985	0.100122	0.0500609	+	0.998746i	$0.484058\pi$
$72$	72.0000	0.117851
$73$	−220.230	−0.353096	−0.176548	−	0.984292i	$-0.556493\pi$
$73$	−220.230	−0.353096	−0.176548	+	0.984292i	$0.556493\pi$
$74$	596.891	0.937664
$75$	361.585	0.556697
$76$	−588.280	−0.887899
$77$	109.593	0.162199
$78$	78.0000	0.113228
$79$	1139.80	1.62325	0.811627	−	0.584177i	$-0.198582\pi$
$79$	1139.80	1.62325	0.811627	+	0.584177i	$0.198582\pi$
$80$	−33.8337	−0.0472840
$81$	81.0000	0.111111
$82$	−181.578	−0.244536
$83$	−796.278	−1.05305	−0.526523	−	0.850161i	$-0.676505\pi$
$83$	−796.278	−1.05305	−0.526523	+	0.850161i	$0.676505\pi$
$84$	−84.0000	−0.109109
$85$	−290.334	−0.370484
$86$	777.349	0.974694
$87$	−698.522	−0.860798
$88$	125.249	0.151723
$89$	−577.436	−0.687732	−0.343866	−	0.939019i	$-0.611737\pi$
$89$	−577.436	−0.687732	−0.343866	+	0.939019i	$0.611737\pi$
$90$	−38.0629	−0.0445798
$91$	−91.0000	−0.104828
$92$	−331.363	−0.375511
$93$	−912.609	−1.01756
$94$	87.7971	0.0963359
$95$	310.995	0.335868
$96$	−96.0000	−0.102062
$97$	−105.793	−0.110739	−0.0553694	−	0.998466i	$-0.517634\pi$
$97$	−105.793	−0.110739	−0.0553694	+	0.998466i	$0.517634\pi$
$98$	98.0000	0.101015
$99$	140.906	0.143046

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.o.1.2	✓	3
3.2	odd	2		1638.4.a.x.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.4.a.o.1.2	✓	3	1.1	even	1	trivial
1638.4.a.x.1.2		3	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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -2.11461 q^{5} -6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -4.22921 q^{10} +15.6562 q^{11} -12.0000 q^{12} -13.0000 q^{13} +14.0000 q^{14} +6.34382 q^{15} +16.0000 q^{16} +137.299 q^{17} +18.0000 q^{18} -147.070 q^{19} -8.45842 q^{20} -21.0000 q^{21} +31.3124 q^{22} -82.8408 q^{23} -24.0000 q^{24} -120.528 q^{25} -26.0000 q^{26} -27.0000 q^{27} +28.0000 q^{28} +232.841 q^{29} +12.6876 q^{30} +304.203 q^{31} +32.0000 q^{32} -46.9685 q^{33} +274.598 q^{34} -14.8022 q^{35} +36.0000 q^{36} +298.445 q^{37} -294.140 q^{38} +39.0000 q^{39} -16.9168 q^{40} -90.7891 q^{41} -42.0000 q^{42} +388.674 q^{43} +62.6247 q^{44} -19.0315 q^{45} -165.682 q^{46} +43.8985 q^{47} -48.0000 q^{48} +49.0000 q^{49} -241.057 q^{50} -411.898 q^{51} -52.0000 q^{52} +458.203 q^{53} -54.0000 q^{54} -33.1067 q^{55} +56.0000 q^{56} +441.210 q^{57} +465.682 q^{58} -146.268 q^{59} +25.3753 q^{60} +210.701 q^{61} +608.406 q^{62} +63.0000 q^{63} +64.0000 q^{64} +27.4899 q^{65} -93.9371 q^{66} +357.272 q^{67} +549.197 q^{68} +248.522 q^{69} -29.6045 q^{70} +59.8985 q^{71} +72.0000 q^{72} -220.230 q^{73} +596.891 q^{74} +361.585 q^{75} -588.280 q^{76} +109.593 q^{77} +78.0000 q^{78} +1139.80 q^{79} -33.8337 q^{80} +81.0000 q^{81} -181.578 q^{82} -796.278 q^{83} -84.0000 q^{84} -290.334 q^{85} +777.349 q^{86} -698.522 q^{87} +125.249 q^{88} -577.436 q^{89} -38.0629 q^{90} -91.0000 q^{91} -331.363 q^{92} -912.609 q^{93} +87.7971 q^{94} +310.995 q^{95} -96.0000 q^{96} -105.793 q^{97} +98.0000 q^{98} +140.906 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} - q^{5} - 18 q^{6} + 21 q^{7} + 24 q^{8} + 27 q^{9} - 2 q^{10} + 63 q^{11} - 36 q^{12} - 39 q^{13} + 42 q^{14} + 3 q^{15} + 48 q^{16} - q^{17} + 54 q^{18} - 39 q^{19}+ \cdots + 567 q^{99}+O(q^{100}) \) 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 - q^5 - 18 * q^6 + 21 * q^7 + 24 * q^8 + 27 * q^9 - 2 * q^10 + 63 * q^11 - 36 * q^12 - 39 * q^13 + 42 * q^14 + 3 * q^15 + 48 * q^16 - q^17 + 54 * q^18 - 39 * q^19 - 4 * q^20 - 63 * q^21 + 126 * q^22 + 143 * q^23 - 72 * q^24 + 62 * q^25 - 78 * q^26 - 81 * q^27 + 84 * q^28 + 307 * q^29 + 6 * q^30 + 12 * q^31 + 96 * q^32 - 189 * q^33 - 2 * q^34 - 7 * q^35 + 108 * q^36 + 429 * q^37 - 78 * q^38 + 117 * q^39 - 8 * q^40 + 210 * q^41 - 126 * q^42 + 689 * q^43 + 252 * q^44 - 9 * q^45 + 286 * q^46 + 582 * q^47 - 144 * q^48 + 147 * q^49 + 124 * q^50 + 3 * q^51 - 156 * q^52 + 474 * q^53 - 162 * q^54 + 1289 * q^55 + 168 * q^56 + 117 * q^57 + 614 * q^58 - 74 * q^59 + 12 * q^60 + 1045 * q^61 + 24 * q^62 + 189 * q^63 + 192 * q^64 + 13 * q^65 - 378 * q^66 + 826 * q^67 - 4 * q^68 - 429 * q^69 - 14 * q^70 + 630 * q^71 + 216 * q^72 + 407 * q^73 + 858 * q^74 - 186 * q^75 - 156 * q^76 + 441 * q^77 + 234 * q^78 + 942 * q^79 - 16 * q^80 + 243 * q^81 + 420 * q^82 - 148 * q^83 - 252 * q^84 + 15 * q^85 + 1378 * q^86 - 921 * q^87 + 504 * q^88 - 940 * q^89 - 18 * q^90 - 273 * q^91 + 572 * q^92 - 36 * q^93 + 1164 * q^94 - 875 * q^95 - 288 * q^96 + 46 * q^97 + 294 * q^98 + 567 * q^99

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