LMFDB - Embedded newform 546.4.a.i.1.1

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			1.1
Root			$-12.4711$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -11.4711 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +22.9422 q^{10} +56.4134 q^{11} +12.0000 q^{12} +13.0000 q^{13} +14.0000 q^{14} -34.4134 q^{15} +16.0000 q^{16} -35.4711 q^{17} -18.0000 q^{18} -122.298 q^{19} -45.8845 q^{20} -21.0000 q^{21} -112.827 q^{22} +26.5289 q^{23} -24.0000 q^{24} +6.58663 q^{25} -26.0000 q^{26} +27.0000 q^{27} -28.0000 q^{28} -62.1823 q^{29} +68.8267 q^{30} +162.116 q^{31} -32.0000 q^{32} +169.240 q^{33} +70.9422 q^{34} +80.2979 q^{35} +36.0000 q^{36} +119.240 q^{37} +244.596 q^{38} +39.0000 q^{39} +91.7690 q^{40} +342.827 q^{41} +42.0000 q^{42} -342.067 q^{43} +225.653 q^{44} -103.240 q^{45} -53.0578 q^{46} +629.191 q^{47} +48.0000 q^{48} +49.0000 q^{49} -13.1733 q^{50} -106.413 q^{51} +52.0000 q^{52} +311.191 q^{53} -54.0000 q^{54} -647.125 q^{55} +56.0000 q^{56} -366.894 q^{57} +124.365 q^{58} +443.769 q^{59} -137.653 q^{60} -44.9909 q^{61} -324.231 q^{62} -63.0000 q^{63} +64.0000 q^{64} -149.125 q^{65} -338.480 q^{66} -58.2492 q^{67} -141.884 q^{68} +79.5866 q^{69} -160.596 q^{70} -128.960 q^{71} -72.0000 q^{72} +947.969 q^{73} -238.480 q^{74} +19.7599 q^{75} -489.191 q^{76} -394.894 q^{77} -78.0000 q^{78} +208.827 q^{79} -183.538 q^{80} +81.0000 q^{81} -685.653 q^{82} +282.942 q^{83} -84.0000 q^{84} +406.894 q^{85} +684.134 q^{86} -186.547 q^{87} -451.307 q^{88} +1271.19 q^{89} +206.480 q^{90} -91.0000 q^{91} +106.116 q^{92} +486.347 q^{93} -1258.38 q^{94} +1402.89 q^{95} -96.0000 q^{96} +73.0759 q^{97} -98.0000 q^{98} +507.720 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−11.4711	−1.02601	−0.513004	−	0.858386i	$-0.671467\pi$
$5$	−11.4711	−1.02601	−0.513004	+	0.858386i	$0.671467\pi$
$6$	−6.00000	−0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	22.9422	0.725497
$11$	56.4134	1.54630	0.773149	−	0.634225i	$-0.218681\pi$
$11$	56.4134	1.54630	0.773149	+	0.634225i	$0.218681\pi$
$12$	12.0000	0.288675
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	14.0000	0.267261
$15$	−34.4134	−0.592366
$16$	16.0000	0.250000
$17$	−35.4711	−0.506059	−0.253030	−	0.967459i	$-0.581427\pi$
$17$	−35.4711	−0.506059	−0.253030	+	0.967459i	$0.581427\pi$
$18$	−18.0000	−0.235702
$19$	−122.298	−1.47669	−0.738343	−	0.674425i	$-0.764392\pi$
$19$	−122.298	−1.47669	−0.738343	+	0.674425i	$0.764392\pi$
$20$	−45.8845	−0.513004
$21$	−21.0000	−0.218218
$22$	−112.827	−1.09340
$23$	26.5289	0.240507	0.120253	−	0.992743i	$-0.461629\pi$
$23$	26.5289	0.240507	0.120253	+	0.992743i	$0.461629\pi$
$24$	−24.0000	−0.204124
$25$	6.58663	0.0526931
$26$	−26.0000	−0.196116
$27$	27.0000	0.192450
$28$	−28.0000	−0.188982
$29$	−62.1823	−0.398171	−0.199086	−	0.979982i	$-0.563797\pi$
$29$	−62.1823	−0.398171	−0.199086	+	0.979982i	$0.563797\pi$
$30$	68.8267	0.418866
$31$	162.116	0.939252	0.469626	−	0.882866i	$-0.344389\pi$
$31$	162.116	0.939252	0.469626	+	0.882866i	$0.344389\pi$
$32$	−32.0000	−0.176777
$33$	169.240	0.892755
$34$	70.9422	0.357838
$35$	80.2979	0.387795
$36$	36.0000	0.166667
$37$	119.240	0.529809	0.264905	−	0.964275i	$-0.414659\pi$
$37$	119.240	0.529809	0.264905	+	0.964275i	$0.414659\pi$
$38$	244.596	1.04418
$39$	39.0000	0.160128
$40$	91.7690	0.362749
$41$	342.827	1.30587	0.652933	−	0.757415i	$-0.273538\pi$
$41$	342.827	1.30587	0.652933	+	0.757415i	$0.273538\pi$
$42$	42.0000	0.154303
$43$	−342.067	−1.21313	−0.606566	−	0.795033i	$-0.707454\pi$
$43$	−342.067	−1.21313	−0.606566	+	0.795033i	$0.707454\pi$
$44$	225.653	0.773149
$45$	−103.240	−0.342003
$46$	−53.0578	−0.170064
$47$	629.191	1.95270	0.976351	−	0.216191i	$-0.0693634\pi$
$47$	629.191	1.95270	0.976351	+	0.216191i	$0.0693634\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	−13.1733	−0.0372596
$51$	−106.413	−0.292174
$52$	52.0000	0.138675
$53$	311.191	0.806518	0.403259	−	0.915086i	$-0.367877\pi$
$53$	311.191	0.806518	0.403259	+	0.915086i	$0.367877\pi$
$54$	−54.0000	−0.136083
$55$	−647.125	−1.58651
$56$	56.0000	0.133631
$57$	−366.894	−0.852566
$58$	124.365	0.281550
$59$	443.769	0.979217	0.489608	−	0.871942i	$-0.337140\pi$
$59$	443.769	0.979217	0.489608	+	0.871942i	$0.337140\pi$
$60$	−137.653	−0.296183
$61$	−44.9909	−0.0944344	−0.0472172	−	0.998885i	$-0.515035\pi$
$61$	−44.9909	−0.0944344	−0.0472172	+	0.998885i	$0.515035\pi$
$62$	−324.231	−0.664151
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	−149.125	−0.284564
$66$	−338.480	−0.631273
$67$	−58.2492	−0.106213	−0.0531065	−	0.998589i	$-0.516912\pi$
$67$	−58.2492	−0.106213	−0.0531065	+	0.998589i	$0.516912\pi$
$68$	−141.884	−0.253030
$69$	79.5866	0.138857
$70$	−160.596	−0.274212
$71$	−128.960	−0.215560	−0.107780	−	0.994175i	$-0.534374\pi$
$71$	−128.960	−0.215560	−0.107780	+	0.994175i	$0.534374\pi$
$72$	−72.0000	−0.117851
$73$	947.969	1.51988	0.759941	−	0.649992i	$-0.225228\pi$
$73$	947.969	1.51988	0.759941	+	0.649992i	$0.225228\pi$
$74$	−238.480	−0.374632
$75$	19.7599	0.0304224
$76$	−489.191	−0.738343
$77$	−394.894	−0.584445
$78$	−78.0000	−0.113228
$79$	208.827	0.297403	0.148702	−	0.988882i	$-0.452491\pi$
$79$	208.827	0.297403	0.148702	+	0.988882i	$0.452491\pi$
$80$	−183.538	−0.256502
$81$	81.0000	0.111111
$82$	−685.653	−0.923387
$83$	282.942	0.374180	0.187090	−	0.982343i	$-0.440094\pi$
$83$	282.942	0.374180	0.187090	+	0.982343i	$0.440094\pi$
$84$	−84.0000	−0.109109
$85$	406.894	0.519221
$86$	684.134	0.857814
$87$	−186.547	−0.229884
$88$	−451.307	−0.546699
$89$	1271.19	1.51400	0.757000	−	0.653415i	$-0.226664\pi$
$89$	1271.19	1.51400	0.757000	+	0.653415i	$0.226664\pi$
$90$	206.480	0.241832
$91$	−91.0000	−0.104828
$92$	106.116	0.120253
$93$	486.347	0.542277
$94$	−1258.38	−1.38077
$95$	1402.89	1.51509
$96$	−96.0000	−0.102062
$97$	73.0759	0.0764921	0.0382460	−	0.999268i	$-0.487823\pi$
$97$	73.0759	0.0764921	0.0382460	+	0.999268i	$0.487823\pi$
$98$	−98.0000	−0.101015
$99$	507.720	0.515432

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.1.1	✓	2
3.2	odd	2		1638.4.a.q.1.2		2

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

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -11.4711 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +22.9422 q^{10} +56.4134 q^{11} +12.0000 q^{12} +13.0000 q^{13} +14.0000 q^{14} -34.4134 q^{15} +16.0000 q^{16} -35.4711 q^{17} -18.0000 q^{18} -122.298 q^{19} -45.8845 q^{20} -21.0000 q^{21} -112.827 q^{22} +26.5289 q^{23} -24.0000 q^{24} +6.58663 q^{25} -26.0000 q^{26} +27.0000 q^{27} -28.0000 q^{28} -62.1823 q^{29} +68.8267 q^{30} +162.116 q^{31} -32.0000 q^{32} +169.240 q^{33} +70.9422 q^{34} +80.2979 q^{35} +36.0000 q^{36} +119.240 q^{37} +244.596 q^{38} +39.0000 q^{39} +91.7690 q^{40} +342.827 q^{41} +42.0000 q^{42} -342.067 q^{43} +225.653 q^{44} -103.240 q^{45} -53.0578 q^{46} +629.191 q^{47} +48.0000 q^{48} +49.0000 q^{49} -13.1733 q^{50} -106.413 q^{51} +52.0000 q^{52} +311.191 q^{53} -54.0000 q^{54} -647.125 q^{55} +56.0000 q^{56} -366.894 q^{57} +124.365 q^{58} +443.769 q^{59} -137.653 q^{60} -44.9909 q^{61} -324.231 q^{62} -63.0000 q^{63} +64.0000 q^{64} -149.125 q^{65} -338.480 q^{66} -58.2492 q^{67} -141.884 q^{68} +79.5866 q^{69} -160.596 q^{70} -128.960 q^{71} -72.0000 q^{72} +947.969 q^{73} -238.480 q^{74} +19.7599 q^{75} -489.191 q^{76} -394.894 q^{77} -78.0000 q^{78} +208.827 q^{79} -183.538 q^{80} +81.0000 q^{81} -685.653 q^{82} +282.942 q^{83} -84.0000 q^{84} +406.894 q^{85} +684.134 q^{86} -186.547 q^{87} -451.307 q^{88} +1271.19 q^{89} +206.480 q^{90} -91.0000 q^{91} +106.116 q^{92} +486.347 q^{93} -1258.38 q^{94} +1402.89 q^{95} -96.0000 q^{96} +73.0759 q^{97} -98.0000 q^{98} +507.720 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} + 3 q^{5} - 12 q^{6} - 14 q^{7} - 16 q^{8} + 18 q^{9} - 6 q^{10} + 35 q^{11} + 24 q^{12} + 26 q^{13} + 28 q^{14} + 9 q^{15} + 32 q^{16} - 45 q^{17} - 36 q^{18} - 63 q^{19}+ \cdots + 315 q^{99}+O(q^{100}) \) 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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