LMFDB - Embedded newform 546.4.a.k.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1401}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 350 $ x^2 - x - 350
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			$-18.2150$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +21.2150 q^{5} +6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +42.4299 q^{10} -27.2150 q^{11} +12.0000 q^{12} -13.0000 q^{13} -14.0000 q^{14} +63.6449 q^{15} +16.0000 q^{16} +48.7850 q^{17} +18.0000 q^{18} +66.7850 q^{19} +84.8599 q^{20} -21.0000 q^{21} -54.4299 q^{22} +165.215 q^{23} +24.0000 q^{24} +325.075 q^{25} -26.0000 q^{26} +27.0000 q^{27} -28.0000 q^{28} -115.215 q^{29} +127.290 q^{30} -85.7197 q^{31} +32.0000 q^{32} -81.6449 q^{33} +97.5701 q^{34} -148.505 q^{35} +36.0000 q^{36} -366.505 q^{37} +133.570 q^{38} -39.0000 q^{39} +169.720 q^{40} -41.7197 q^{41} -42.0000 q^{42} +314.935 q^{43} -108.860 q^{44} +190.935 q^{45} +330.430 q^{46} -386.150 q^{47} +48.0000 q^{48} +49.0000 q^{49} +650.150 q^{50} +146.355 q^{51} -52.0000 q^{52} -35.7197 q^{53} +54.0000 q^{54} -577.365 q^{55} -56.0000 q^{56} +200.355 q^{57} -230.430 q^{58} -809.029 q^{59} +254.580 q^{60} +577.645 q^{61} -171.439 q^{62} -63.0000 q^{63} +64.0000 q^{64} -275.795 q^{65} -163.290 q^{66} +819.888 q^{67} +195.140 q^{68} +495.645 q^{69} -297.010 q^{70} +53.5891 q^{71} +72.0000 q^{72} +460.524 q^{73} -733.010 q^{74} +975.225 q^{75} +267.140 q^{76} +190.505 q^{77} -78.0000 q^{78} -850.729 q^{79} +339.439 q^{80} +81.0000 q^{81} -83.4395 q^{82} -1026.90 q^{83} -84.0000 q^{84} +1034.97 q^{85} +629.869 q^{86} -345.645 q^{87} -217.720 q^{88} -127.850 q^{89} +381.869 q^{90} +91.0000 q^{91} +660.860 q^{92} -257.159 q^{93} -772.299 q^{94} +1416.84 q^{95} +96.0000 q^{96} +97.4395 q^{97} +98.0000 q^{98} -244.935 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 17 q^{11} + 24 q^{12} - 26 q^{13} - 28 q^{14} + 15 q^{15} + 32 q^{16} + 135 q^{17} + 36 q^{18} + 171 q^{19}+ \cdots - 153 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 - 17 * q^11 + 24 * q^12 - 26 * q^13 - 28 * q^14 + 15 * q^15 + 32 * q^16 + 135 * q^17 + 36 * q^18 + 171 * q^19 + 20 * q^20 - 42 * q^21 - 34 * q^22 + 293 * q^23 + 48 * q^24 + 463 * q^25 - 52 * q^26 + 54 * q^27 - 56 * q^28 - 193 * q^29 + 30 * q^30 + 128 * q^31 + 64 * q^32 - 51 * q^33 + 270 * q^34 - 35 * q^35 + 72 * q^36 - 471 * q^37 + 342 * q^38 - 78 * q^39 + 40 * q^40 + 216 * q^41 - 84 * q^42 + 293 * q^43 - 68 * q^44 + 45 * q^45 + 586 * q^46 - 398 * q^47 + 96 * q^48 + 98 * q^49 + 926 * q^50 + 405 * q^51 - 104 * q^52 + 228 * q^53 + 108 * q^54 - 743 * q^55 - 112 * q^56 + 513 * q^57 - 386 * q^58 - 46 * q^59 + 60 * q^60 + 1043 * q^61 + 256 * q^62 - 126 * q^63 + 128 * q^64 - 65 * q^65 - 102 * q^66 - 82 * q^67 + 540 * q^68 + 879 * q^69 - 70 * q^70 - 866 * q^71 + 144 * q^72 - 389 * q^73 - 942 * q^74 + 1389 * q^75 + 684 * q^76 + 119 * q^77 - 156 * q^78 - 878 * q^79 + 80 * q^80 + 162 * q^81 + 432 * q^82 + 192 * q^83 - 168 * q^84 - 363 * q^85 + 586 * q^86 - 579 * q^87 - 136 * q^88 - 630 * q^89 + 90 * q^90 + 182 * q^91 + 1172 * q^92 + 384 * q^93 - 796 * q^94 - 273 * q^95 + 192 * q^96 - 404 * q^97 + 196 * q^98 - 153 * 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
$5$	21.2150	1.89752	0.948762	−	0.315991i	$-0.102337\pi$
$5$	21.2150	1.89752	0.948762	+	0.315991i	$0.102337\pi$
$6$	6.00000	0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	42.4299	1.34175
$11$	−27.2150	−0.745966	−0.372983	−	0.927838i	$-0.621665\pi$
$11$	−27.2150	−0.745966	−0.372983	+	0.927838i	$0.621665\pi$
$12$	12.0000	0.288675
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−14.0000	−0.267261
$15$	63.6449	1.09554
$16$	16.0000	0.250000
$17$	48.7850	0.696006	0.348003	−	0.937493i	$-0.386860\pi$
$17$	48.7850	0.696006	0.348003	+	0.937493i	$0.386860\pi$
$18$	18.0000	0.235702
$19$	66.7850	0.806397	0.403198	−	0.915113i	$-0.367898\pi$
$19$	66.7850	0.806397	0.403198	+	0.915113i	$0.367898\pi$
$20$	84.8599	0.948762
$21$	−21.0000	−0.218218
$22$	−54.4299	−0.527477
$23$	165.215	1.49781	0.748907	−	0.662676i	$-0.230579\pi$
$23$	165.215	1.49781	0.748907	+	0.662676i	$0.230579\pi$
$24$	24.0000	0.204124
$25$	325.075	2.60060
$26$	−26.0000	−0.196116
$27$	27.0000	0.192450
$28$	−28.0000	−0.188982
$29$	−115.215	−0.737755	−0.368877	−	0.929478i	$-0.620258\pi$
$29$	−115.215	−0.737755	−0.368877	+	0.929478i	$0.620258\pi$
$30$	127.290	0.774661
$31$	−85.7197	−0.496636	−0.248318	−	0.968679i	$-0.579878\pi$
$31$	−85.7197	−0.496636	−0.248318	+	0.968679i	$0.579878\pi$
$32$	32.0000	0.176777
$33$	−81.6449	−0.430683
$34$	97.5701	0.492151
$35$	−148.505	−0.717197
$36$	36.0000	0.166667
$37$	−366.505	−1.62846	−0.814230	−	0.580543i	$-0.802840\pi$
$37$	−366.505	−1.62846	−0.814230	+	0.580543i	$0.802840\pi$
$38$	133.570	0.570209
$39$	−39.0000	−0.160128
$40$	169.720	0.670876
$41$	−41.7197	−0.158915	−0.0794577	−	0.996838i	$-0.525319\pi$
$41$	−41.7197	−0.158915	−0.0794577	+	0.996838i	$0.525319\pi$
$42$	−42.0000	−0.154303
$43$	314.935	1.11691	0.558455	−	0.829535i	$-0.311395\pi$
$43$	314.935	1.11691	0.558455	+	0.829535i	$0.311395\pi$
$44$	−108.860	−0.372983
$45$	190.935	0.632508
$46$	330.430	1.05911
$47$	−386.150	−1.19842	−0.599210	−	0.800592i	$-0.704518\pi$
$47$	−386.150	−1.19842	−0.599210	+	0.800592i	$0.704518\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	650.150	1.83890
$51$	146.355	0.401839
$52$	−52.0000	−0.138675
$53$	−35.7197	−0.0925752	−0.0462876	−	0.998928i	$-0.514739\pi$
$53$	−35.7197	−0.0925752	−0.0462876	+	0.998928i	$0.514739\pi$
$54$	54.0000	0.136083
$55$	−577.365	−1.41549
$56$	−56.0000	−0.133631
$57$	200.355	0.465573
$58$	−230.430	−0.521671
$59$	−809.029	−1.78520	−0.892598	−	0.450854i	$-0.851120\pi$
$59$	−809.029	−1.78520	−0.892598	+	0.450854i	$0.851120\pi$
$60$	254.580	0.547768
$61$	577.645	1.21246	0.606228	−	0.795291i	$-0.292682\pi$
$61$	577.645	1.21246	0.606228	+	0.795291i	$0.292682\pi$
$62$	−171.439	−0.351175
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	−275.795	−0.526279
$66$	−163.290	−0.304539
$67$	819.888	1.49500	0.747502	−	0.664259i	$-0.231253\pi$
$67$	819.888	1.49500	0.747502	+	0.664259i	$0.231253\pi$
$68$	195.140	0.348003
$69$	495.645	0.864763
$70$	−297.010	−0.507135
$71$	53.5891	0.0895755	0.0447878	−	0.998997i	$-0.485739\pi$
$71$	53.5891	0.0895755	0.0447878	+	0.998997i	$0.485739\pi$
$72$	72.0000	0.117851
$73$	460.524	0.738359	0.369180	−	0.929358i	$-0.379639\pi$
$73$	460.524	0.738359	0.369180	+	0.929358i	$0.379639\pi$
$74$	−733.010	−1.15149
$75$	975.225	1.50146
$76$	267.140	0.403198
$77$	190.505	0.281948
$78$	−78.0000	−0.113228
$79$	−850.729	−1.21158	−0.605788	−	0.795626i	$-0.707142\pi$
$79$	−850.729	−1.21158	−0.605788	+	0.795626i	$0.707142\pi$
$80$	339.439	0.474381
$81$	81.0000	0.111111
$82$	−83.4395	−0.112370
$83$	−1026.90	−1.35803	−0.679016	−	0.734123i	$-0.737593\pi$
$83$	−1026.90	−1.35803	−0.679016	+	0.734123i	$0.737593\pi$
$84$	−84.0000	−0.109109
$85$	1034.97	1.32069
$86$	629.869	0.789774
$87$	−345.645	−0.425943
$88$	−217.720	−0.263739
$89$	−127.850	−0.152271	−0.0761354	−	0.997097i	$-0.524258\pi$
$89$	−127.850	−0.152271	−0.0761354	+	0.997097i	$0.524258\pi$
$90$	381.869	0.447251
$91$	91.0000	0.104828
$92$	660.860	0.748907
$93$	−257.159	−0.286733
$94$	−772.299	−0.847411
$95$	1416.84	1.53016
$96$	96.0000	0.102062
$97$	97.4395	0.101995	0.0509973	−	0.998699i	$-0.483760\pi$
$97$	97.4395	0.101995	0.0509973	+	0.998699i	$0.483760\pi$
$98$	98.0000	0.101015
$99$	−244.935	−0.248655

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.k.1.2	✓	2
3.2	odd	2		1638.4.a.l.1.1		2

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

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +21.2150 q^{5} +6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +42.4299 q^{10} -27.2150 q^{11} +12.0000 q^{12} -13.0000 q^{13} -14.0000 q^{14} +63.6449 q^{15} +16.0000 q^{16} +48.7850 q^{17} +18.0000 q^{18} +66.7850 q^{19} +84.8599 q^{20} -21.0000 q^{21} -54.4299 q^{22} +165.215 q^{23} +24.0000 q^{24} +325.075 q^{25} -26.0000 q^{26} +27.0000 q^{27} -28.0000 q^{28} -115.215 q^{29} +127.290 q^{30} -85.7197 q^{31} +32.0000 q^{32} -81.6449 q^{33} +97.5701 q^{34} -148.505 q^{35} +36.0000 q^{36} -366.505 q^{37} +133.570 q^{38} -39.0000 q^{39} +169.720 q^{40} -41.7197 q^{41} -42.0000 q^{42} +314.935 q^{43} -108.860 q^{44} +190.935 q^{45} +330.430 q^{46} -386.150 q^{47} +48.0000 q^{48} +49.0000 q^{49} +650.150 q^{50} +146.355 q^{51} -52.0000 q^{52} -35.7197 q^{53} +54.0000 q^{54} -577.365 q^{55} -56.0000 q^{56} +200.355 q^{57} -230.430 q^{58} -809.029 q^{59} +254.580 q^{60} +577.645 q^{61} -171.439 q^{62} -63.0000 q^{63} +64.0000 q^{64} -275.795 q^{65} -163.290 q^{66} +819.888 q^{67} +195.140 q^{68} +495.645 q^{69} -297.010 q^{70} +53.5891 q^{71} +72.0000 q^{72} +460.524 q^{73} -733.010 q^{74} +975.225 q^{75} +267.140 q^{76} +190.505 q^{77} -78.0000 q^{78} -850.729 q^{79} +339.439 q^{80} +81.0000 q^{81} -83.4395 q^{82} -1026.90 q^{83} -84.0000 q^{84} +1034.97 q^{85} +629.869 q^{86} -345.645 q^{87} -217.720 q^{88} -127.850 q^{89} +381.869 q^{90} +91.0000 q^{91} +660.860 q^{92} -257.159 q^{93} -772.299 q^{94} +1416.84 q^{95} +96.0000 q^{96} +97.4395 q^{97} +98.0000 q^{98} -244.935 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 5 q^{5} + 12 q^{6} - 14 q^{7} + 16 q^{8} + 18 q^{9} + 10 q^{10} - 17 q^{11} + 24 q^{12} - 26 q^{13} - 28 q^{14} + 15 q^{15} + 32 q^{16} + 135 q^{17} + 36 q^{18} + 171 q^{19}+ \cdots - 153 q^{99}+O(q^{100}) \) 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 - 17 * q^11 + 24 * q^12 - 26 * q^13 - 28 * q^14 + 15 * q^15 + 32 * q^16 + 135 * q^17 + 36 * q^18 + 171 * q^19 + 20 * q^20 - 42 * q^21 - 34 * q^22 + 293 * q^23 + 48 * q^24 + 463 * q^25 - 52 * q^26 + 54 * q^27 - 56 * q^28 - 193 * q^29 + 30 * q^30 + 128 * q^31 + 64 * q^32 - 51 * q^33 + 270 * q^34 - 35 * q^35 + 72 * q^36 - 471 * q^37 + 342 * q^38 - 78 * q^39 + 40 * q^40 + 216 * q^41 - 84 * q^42 + 293 * q^43 - 68 * q^44 + 45 * q^45 + 586 * q^46 - 398 * q^47 + 96 * q^48 + 98 * q^49 + 926 * q^50 + 405 * q^51 - 104 * q^52 + 228 * q^53 + 108 * q^54 - 743 * q^55 - 112 * q^56 + 513 * q^57 - 386 * q^58 - 46 * q^59 + 60 * q^60 + 1043 * q^61 + 256 * q^62 - 126 * q^63 + 128 * q^64 - 65 * q^65 - 102 * q^66 - 82 * q^67 + 540 * q^68 + 879 * q^69 - 70 * q^70 - 866 * q^71 + 144 * q^72 - 389 * q^73 - 942 * q^74 + 1389 * q^75 + 684 * q^76 + 119 * q^77 - 156 * q^78 - 878 * q^79 + 80 * q^80 + 162 * q^81 + 432 * q^82 + 192 * q^83 - 168 * q^84 - 363 * q^85 + 586 * q^86 - 579 * q^87 - 136 * q^88 - 630 * q^89 + 90 * q^90 + 182 * q^91 + 1172 * q^92 + 384 * q^93 - 796 * q^94 - 273 * q^95 + 192 * q^96 - 404 * q^97 + 196 * q^98 - 153 * q^99

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