LMFDB - Embedded newform 546.4.a.p.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,-9,12,7] 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:	3.3.118088.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 50x - 32 $ x^3 - 50*x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$7.37163$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.57278 q^{5} -6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +13.1456 q^{10} +18.0593 q^{11} -12.0000 q^{12} +13.0000 q^{13} -14.0000 q^{14} -19.7183 q^{15} +16.0000 q^{16} -42.3389 q^{17} +18.0000 q^{18} +77.7068 q^{19} +26.2911 q^{20} +21.0000 q^{21} +36.1186 q^{22} +43.8983 q^{23} -24.0000 q^{24} -81.7985 q^{25} +26.0000 q^{26} -27.0000 q^{27} -28.0000 q^{28} +121.075 q^{29} -39.4367 q^{30} +248.214 q^{31} +32.0000 q^{32} -54.1779 q^{33} -84.6779 q^{34} -46.0095 q^{35} +36.0000 q^{36} -279.817 q^{37} +155.414 q^{38} -39.0000 q^{39} +52.5823 q^{40} +425.980 q^{41} +42.0000 q^{42} +179.185 q^{43} +72.2372 q^{44} +59.1550 q^{45} +87.7966 q^{46} -216.321 q^{47} -48.0000 q^{48} +49.0000 q^{49} -163.597 q^{50} +127.017 q^{51} +52.0000 q^{52} +713.003 q^{53} -54.0000 q^{54} +118.700 q^{55} -56.0000 q^{56} -233.121 q^{57} +242.150 q^{58} +403.004 q^{59} -78.8734 q^{60} +246.765 q^{61} +496.429 q^{62} -63.0000 q^{63} +64.0000 q^{64} +85.4462 q^{65} -108.356 q^{66} -742.301 q^{67} -169.356 q^{68} -131.695 q^{69} -92.0190 q^{70} -65.6086 q^{71} +72.0000 q^{72} +965.099 q^{73} -559.635 q^{74} +245.396 q^{75} +310.827 q^{76} -126.415 q^{77} -78.0000 q^{78} +382.593 q^{79} +105.165 q^{80} +81.0000 q^{81} +851.961 q^{82} -1347.88 q^{83} +84.0000 q^{84} -278.285 q^{85} +358.371 q^{86} -363.224 q^{87} +144.474 q^{88} +830.091 q^{89} +118.310 q^{90} -91.0000 q^{91} +175.593 q^{92} -744.643 q^{93} -432.643 q^{94} +510.750 q^{95} -96.0000 q^{96} +376.407 q^{97} +98.0000 q^{98} +162.534 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 7 q^{5} - 18 q^{6} - 21 q^{7} + 24 q^{8} + 27 q^{9} + 14 q^{10} - 47 q^{11} - 36 q^{12} + 39 q^{13} - 42 q^{14} - 21 q^{15} + 48 q^{16} + 119 q^{17} + 54 q^{18} + 101 q^{19}+ \cdots - 423 q^{99}+O(q^{100}) $ 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 + 7 * q^5 - 18 * q^6 - 21 * q^7 + 24 * q^8 + 27 * q^9 + 14 * q^10 - 47 * q^11 - 36 * q^12 + 39 * q^13 - 42 * q^14 - 21 * q^15 + 48 * q^16 + 119 * q^17 + 54 * q^18 + 101 * q^19 + 28 * q^20 + 63 * q^21 - 94 * q^22 - 27 * q^23 - 72 * q^24 + 266 * q^25 + 78 * q^26 - 81 * q^27 - 84 * q^28 + 345 * q^29 - 42 * q^30 + 152 * q^31 + 96 * q^32 + 141 * q^33 + 238 * q^34 - 49 * q^35 + 108 * q^36 + 227 * q^37 + 202 * q^38 - 117 * q^39 + 56 * q^40 + 452 * q^41 + 126 * q^42 - 415 * q^43 - 188 * q^44 + 63 * q^45 - 54 * q^46 + 240 * q^47 - 144 * q^48 + 147 * q^49 + 532 * q^50 - 357 * q^51 + 156 * q^52 + 874 * q^53 - 162 * q^54 + 1123 * q^55 - 168 * q^56 - 303 * q^57 + 690 * q^58 + 948 * q^59 - 84 * q^60 + 951 * q^61 + 304 * q^62 - 189 * q^63 + 192 * q^64 + 91 * q^65 + 282 * q^66 + 742 * q^67 + 476 * q^68 + 81 * q^69 - 98 * q^70 + 732 * q^71 + 216 * q^72 - 585 * q^73 + 454 * q^74 - 798 * q^75 + 404 * q^76 + 329 * q^77 - 234 * q^78 - 618 * q^79 + 112 * q^80 + 243 * q^81 + 904 * q^82 + 774 * q^83 + 252 * q^84 + 1273 * q^85 - 830 * q^86 - 1035 * q^87 - 376 * q^88 + 730 * q^89 + 126 * q^90 - 273 * q^91 - 108 * q^92 - 456 * q^93 + 480 * q^94 - 2617 * q^95 - 288 * q^96 + 2518 * q^97 + 294 * q^98 - 423 * 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
$5$	6.57278	0.587888	0.293944	−	0.955823i	$-0.405032\pi$
$5$	6.57278	0.587888	0.293944	+	0.955823i	$0.405032\pi$
$6$	−6.00000	−0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	13.1456	0.415699
$11$	18.0593	0.495008	0.247504	−	0.968887i	$-0.420390\pi$
$11$	18.0593	0.495008	0.247504	+	0.968887i	$0.420390\pi$
$12$	−12.0000	−0.288675
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	−14.0000	−0.267261
$15$	−19.7183	−0.339417
$16$	16.0000	0.250000
$17$	−42.3389	−0.604041	−0.302021	−	0.953301i	$-0.597661\pi$
$17$	−42.3389	−0.604041	−0.302021	+	0.953301i	$0.597661\pi$
$18$	18.0000	0.235702
$19$	77.7068	0.938272	0.469136	−	0.883126i	$-0.344565\pi$
$19$	77.7068	0.938272	0.469136	+	0.883126i	$0.344565\pi$
$20$	26.2911	0.293944
$21$	21.0000	0.218218
$22$	36.1186	0.350023
$23$	43.8983	0.397975	0.198988	−	0.980002i	$-0.436235\pi$
$23$	43.8983	0.397975	0.198988	+	0.980002i	$0.436235\pi$
$24$	−24.0000	−0.204124
$25$	−81.7985	−0.654388
$26$	26.0000	0.196116
$27$	−27.0000	−0.192450
$28$	−28.0000	−0.188982
$29$	121.075	0.775276	0.387638	−	0.921812i	$-0.373291\pi$
$29$	121.075	0.775276	0.387638	+	0.921812i	$0.373291\pi$
$30$	−39.4367	−0.240004
$31$	248.214	1.43808	0.719042	−	0.694967i	$-0.244581\pi$
$31$	248.214	1.43808	0.719042	+	0.694967i	$0.244581\pi$
$32$	32.0000	0.176777
$33$	−54.1779	−0.285793
$34$	−84.6779	−0.427122
$35$	−46.0095	−0.222201
$36$	36.0000	0.166667
$37$	−279.817	−1.24329	−0.621645	−	0.783299i	$-0.713535\pi$
$37$	−279.817	−1.24329	−0.621645	+	0.783299i	$0.713535\pi$
$38$	155.414	0.663459
$39$	−39.0000	−0.160128
$40$	52.5823	0.207850
$41$	425.980	1.62261	0.811304	−	0.584624i	$-0.198758\pi$
$41$	425.980	1.62261	0.811304	+	0.584624i	$0.198758\pi$
$42$	42.0000	0.154303
$43$	179.185	0.635477	0.317738	−	0.948178i	$-0.397077\pi$
$43$	179.185	0.635477	0.317738	+	0.948178i	$0.397077\pi$
$44$	72.2372	0.247504
$45$	59.1550	0.195963
$46$	87.7966	0.281411
$47$	−216.321	−0.671356	−0.335678	−	0.941977i	$-0.608965\pi$
$47$	−216.321	−0.671356	−0.335678	+	0.941977i	$0.608965\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	−163.597	−0.462722
$51$	127.017	0.348743
$52$	52.0000	0.138675
$53$	713.003	1.84790	0.923948	−	0.382518i	$-0.124943\pi$
$53$	713.003	1.84790	0.923948	+	0.382518i	$0.124943\pi$
$54$	−54.0000	−0.136083
$55$	118.700	0.291009
$56$	−56.0000	−0.133631
$57$	−233.121	−0.541712
$58$	242.150	0.548203
$59$	403.004	0.889264	0.444632	−	0.895713i	$-0.353334\pi$
$59$	403.004	0.889264	0.444632	+	0.895713i	$0.353334\pi$
$60$	−78.8734	−0.169709
$61$	246.765	0.517950	0.258975	−	0.965884i	$-0.416615\pi$
$61$	246.765	0.517950	0.258975	+	0.965884i	$0.416615\pi$
$62$	496.429	1.01688
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	85.4462	0.163051
$66$	−108.356	−0.202086
$67$	−742.301	−1.35353	−0.676765	−	0.736199i	$-0.736619\pi$
$67$	−742.301	−1.35353	−0.676765	+	0.736199i	$0.736619\pi$
$68$	−169.356	−0.302021
$69$	−131.695	−0.229771
$70$	−92.0190	−0.157120
$71$	−65.6086	−0.109666	−0.0548331	−	0.998496i	$-0.517463\pi$
$71$	−65.6086	−0.109666	−0.0548331	+	0.998496i	$0.517463\pi$
$72$	72.0000	0.117851
$73$	965.099	1.54735	0.773673	−	0.633585i	$-0.218417\pi$
$73$	965.099	1.54735	0.773673	+	0.633585i	$0.218417\pi$
$74$	−559.635	−0.879138
$75$	245.396	0.377811
$76$	310.827	0.469136
$77$	−126.415	−0.187095
$78$	−78.0000	−0.113228
$79$	382.593	0.544874	0.272437	−	0.962174i	$-0.412170\pi$
$79$	382.593	0.544874	0.272437	+	0.962174i	$0.412170\pi$
$80$	105.165	0.146972
$81$	81.0000	0.111111
$82$	851.961	1.14736
$83$	−1347.88	−1.78252	−0.891258	−	0.453496i	$-0.850177\pi$
$83$	−1347.88	−1.78252	−0.891258	+	0.453496i	$0.850177\pi$
$84$	84.0000	0.109109
$85$	−278.285	−0.355108
$86$	358.371	0.449350
$87$	−363.224	−0.447606
$88$	144.474	0.175012
$89$	830.091	0.988646	0.494323	−	0.869278i	$-0.335416\pi$
$89$	830.091	0.988646	0.494323	+	0.869278i	$0.335416\pi$
$90$	118.310	0.138566
$91$	−91.0000	−0.104828
$92$	175.593	0.198988
$93$	−744.643	−0.830278
$94$	−432.643	−0.474720
$95$	510.750	0.551599
$96$	−96.0000	−0.102062
$97$	376.407	0.394003	0.197002	−	0.980403i	$-0.436880\pi$
$97$	376.407	0.394003	0.197002	+	0.980403i	$0.436880\pi$
$98$	98.0000	0.101015
$99$	162.534	0.165003

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.p.1.2	✓	3
3.2	odd	2		1638.4.a.v.1.2		3

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

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.57278 q^{5} -6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +13.1456 q^{10} +18.0593 q^{11} -12.0000 q^{12} +13.0000 q^{13} -14.0000 q^{14} -19.7183 q^{15} +16.0000 q^{16} -42.3389 q^{17} +18.0000 q^{18} +77.7068 q^{19} +26.2911 q^{20} +21.0000 q^{21} +36.1186 q^{22} +43.8983 q^{23} -24.0000 q^{24} -81.7985 q^{25} +26.0000 q^{26} -27.0000 q^{27} -28.0000 q^{28} +121.075 q^{29} -39.4367 q^{30} +248.214 q^{31} +32.0000 q^{32} -54.1779 q^{33} -84.6779 q^{34} -46.0095 q^{35} +36.0000 q^{36} -279.817 q^{37} +155.414 q^{38} -39.0000 q^{39} +52.5823 q^{40} +425.980 q^{41} +42.0000 q^{42} +179.185 q^{43} +72.2372 q^{44} +59.1550 q^{45} +87.7966 q^{46} -216.321 q^{47} -48.0000 q^{48} +49.0000 q^{49} -163.597 q^{50} +127.017 q^{51} +52.0000 q^{52} +713.003 q^{53} -54.0000 q^{54} +118.700 q^{55} -56.0000 q^{56} -233.121 q^{57} +242.150 q^{58} +403.004 q^{59} -78.8734 q^{60} +246.765 q^{61} +496.429 q^{62} -63.0000 q^{63} +64.0000 q^{64} +85.4462 q^{65} -108.356 q^{66} -742.301 q^{67} -169.356 q^{68} -131.695 q^{69} -92.0190 q^{70} -65.6086 q^{71} +72.0000 q^{72} +965.099 q^{73} -559.635 q^{74} +245.396 q^{75} +310.827 q^{76} -126.415 q^{77} -78.0000 q^{78} +382.593 q^{79} +105.165 q^{80} +81.0000 q^{81} +851.961 q^{82} -1347.88 q^{83} +84.0000 q^{84} -278.285 q^{85} +358.371 q^{86} -363.224 q^{87} +144.474 q^{88} +830.091 q^{89} +118.310 q^{90} -91.0000 q^{91} +175.593 q^{92} -744.643 q^{93} -432.643 q^{94} +510.750 q^{95} -96.0000 q^{96} +376.407 q^{97} +98.0000 q^{98} +162.534 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 7 q^{5} - 18 q^{6} - 21 q^{7} + 24 q^{8} + 27 q^{9} + 14 q^{10} - 47 q^{11} - 36 q^{12} + 39 q^{13} - 42 q^{14} - 21 q^{15} + 48 q^{16} + 119 q^{17} + 54 q^{18} + 101 q^{19}+ \cdots - 423 q^{99}+O(q^{100}) \) 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 + 7 * q^5 - 18 * q^6 - 21 * q^7 + 24 * q^8 + 27 * q^9 + 14 * q^10 - 47 * q^11 - 36 * q^12 + 39 * q^13 - 42 * q^14 - 21 * q^15 + 48 * q^16 + 119 * q^17 + 54 * q^18 + 101 * q^19 + 28 * q^20 + 63 * q^21 - 94 * q^22 - 27 * q^23 - 72 * q^24 + 266 * q^25 + 78 * q^26 - 81 * q^27 - 84 * q^28 + 345 * q^29 - 42 * q^30 + 152 * q^31 + 96 * q^32 + 141 * q^33 + 238 * q^34 - 49 * q^35 + 108 * q^36 + 227 * q^37 + 202 * q^38 - 117 * q^39 + 56 * q^40 + 452 * q^41 + 126 * q^42 - 415 * q^43 - 188 * q^44 + 63 * q^45 - 54 * q^46 + 240 * q^47 - 144 * q^48 + 147 * q^49 + 532 * q^50 - 357 * q^51 + 156 * q^52 + 874 * q^53 - 162 * q^54 + 1123 * q^55 - 168 * q^56 - 303 * q^57 + 690 * q^58 + 948 * q^59 - 84 * q^60 + 951 * q^61 + 304 * q^62 - 189 * q^63 + 192 * q^64 + 91 * q^65 + 282 * q^66 + 742 * q^67 + 476 * q^68 + 81 * q^69 - 98 * q^70 + 732 * q^71 + 216 * q^72 - 585 * q^73 + 454 * q^74 - 798 * q^75 + 404 * q^76 + 329 * q^77 - 234 * q^78 - 618 * q^79 + 112 * q^80 + 243 * q^81 + 904 * q^82 + 774 * q^83 + 252 * q^84 + 1273 * q^85 - 830 * q^86 - 1035 * q^87 - 376 * q^88 + 730 * q^89 + 126 * q^90 - 273 * q^91 - 108 * q^92 - 456 * q^93 + 480 * q^94 - 2617 * q^95 - 288 * q^96 + 2518 * q^97 + 294 * q^98 - 423 * q^99

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