LMFDB - Embedded newform 546.4.a.g.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,-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			1.1
Root			$4.53113$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -19.5934 q^{5} -6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +39.1868 q^{10} +3.18677 q^{11} +12.0000 q^{12} +13.0000 q^{13} -14.0000 q^{14} -58.7802 q^{15} +16.0000 q^{16} +51.5603 q^{17} -18.0000 q^{18} +36.7802 q^{19} -78.3735 q^{20} +21.0000 q^{21} -6.37355 q^{22} -7.21984 q^{23} -24.0000 q^{24} +258.901 q^{25} -26.0000 q^{26} +27.0000 q^{27} +28.0000 q^{28} -133.967 q^{29} +117.560 q^{30} +0.780160 q^{31} -32.0000 q^{32} +9.56032 q^{33} -103.121 q^{34} -137.154 q^{35} +36.0000 q^{36} -187.560 q^{37} -73.5603 q^{38} +39.0000 q^{39} +156.747 q^{40} -246.747 q^{41} -42.0000 q^{42} -222.340 q^{43} +12.7471 q^{44} -176.340 q^{45} +14.4397 q^{46} -438.274 q^{47} +48.0000 q^{48} +49.0000 q^{49} -517.802 q^{50} +154.681 q^{51} +52.0000 q^{52} -153.088 q^{53} -54.0000 q^{54} -62.4397 q^{55} -56.0000 q^{56} +110.340 q^{57} +267.934 q^{58} -336.000 q^{59} -235.121 q^{60} +230.000 q^{61} -1.56032 q^{62} +63.0000 q^{63} +64.0000 q^{64} -254.714 q^{65} -19.1206 q^{66} +898.043 q^{67} +206.241 q^{68} -21.6595 q^{69} +274.307 q^{70} -584.615 q^{71} -72.0000 q^{72} -902.383 q^{73} +375.121 q^{74} +776.702 q^{75} +147.121 q^{76} +22.3074 q^{77} -78.0000 q^{78} -157.461 q^{79} -313.494 q^{80} +81.0000 q^{81} +493.494 q^{82} +457.944 q^{83} +84.0000 q^{84} -1010.24 q^{85} +444.681 q^{86} -401.901 q^{87} -25.4942 q^{88} +171.461 q^{89} +352.681 q^{90} +91.0000 q^{91} -28.8794 q^{92} +2.34048 q^{93} +876.549 q^{94} -720.648 q^{95} -96.0000 q^{96} -926.780 q^{97} -98.0000 q^{98} +28.6810 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−19.5934	−1.75249	−0.876243	−	0.481870i	$-0.839958\pi$
$5$	−19.5934	−1.75249	−0.876243	+	0.481870i	$0.839958\pi$
$6$	−6.00000	−0.408248
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	39.1868	1.23919
$11$	3.18677	0.0873498	0.0436749	−	0.999046i	$-0.486093\pi$
$11$	3.18677	0.0873498	0.0436749	+	0.999046i	$0.486093\pi$
$12$	12.0000	0.288675
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	−14.0000	−0.267261
$15$	−58.7802	−1.01180
$16$	16.0000	0.250000
$17$	51.5603	0.735601	0.367800	−	0.929905i	$-0.380111\pi$
$17$	51.5603	0.735601	0.367800	+	0.929905i	$0.380111\pi$
$18$	−18.0000	−0.235702
$19$	36.7802	0.444102	0.222051	−	0.975035i	$-0.428725\pi$
$19$	36.7802	0.444102	0.222051	+	0.975035i	$0.428725\pi$
$20$	−78.3735	−0.876243
$21$	21.0000	0.218218
$22$	−6.37355	−0.0617657
$23$	−7.21984	−0.0654539	−0.0327270	−	0.999464i	$-0.510419\pi$
$23$	−7.21984	−0.0654539	−0.0327270	+	0.999464i	$0.510419\pi$
$24$	−24.0000	−0.204124
$25$	258.901	2.07121
$26$	−26.0000	−0.196116
$27$	27.0000	0.192450
$28$	28.0000	0.188982
$29$	−133.967	−0.857829	−0.428914	−	0.903345i	$-0.641104\pi$
$29$	−133.967	−0.857829	−0.428914	+	0.903345i	$0.641104\pi$
$30$	117.560	0.715449
$31$	0.780160	0.00452003	0.00226001	−	0.999997i	$-0.499281\pi$
$31$	0.780160	0.00452003	0.00226001	+	0.999997i	$0.499281\pi$
$32$	−32.0000	−0.176777
$33$	9.56032	0.0504315
$34$	−103.121	−0.520148
$35$	−137.154	−0.662377
$36$	36.0000	0.166667
$37$	−187.560	−0.833371	−0.416685	−	0.909051i	$-0.636808\pi$
$37$	−187.560	−0.833371	−0.416685	+	0.909051i	$0.636808\pi$
$38$	−73.5603	−0.314028
$39$	39.0000	0.160128
$40$	156.747	0.619597
$41$	−246.747	−0.939888	−0.469944	−	0.882696i	$-0.655726\pi$
$41$	−246.747	−0.939888	−0.469944	+	0.882696i	$0.655726\pi$
$42$	−42.0000	−0.154303
$43$	−222.340	−0.788526	−0.394263	−	0.918998i	$-0.629000\pi$
$43$	−222.340	−0.788526	−0.394263	+	0.918998i	$0.629000\pi$
$44$	12.7471	0.0436749
$45$	−176.340	−0.584162
$46$	14.4397	0.0462829
$47$	−438.274	−1.36019	−0.680095	−	0.733124i	$-0.738061\pi$
$47$	−438.274	−1.36019	−0.680095	+	0.733124i	$0.738061\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	−517.802	−1.46456
$51$	154.681	0.424699
$52$	52.0000	0.138675
$53$	−153.088	−0.396758	−0.198379	−	0.980125i	$-0.563568\pi$
$53$	−153.088	−0.396758	−0.198379	+	0.980125i	$0.563568\pi$
$54$	−54.0000	−0.136083
$55$	−62.4397	−0.153079
$56$	−56.0000	−0.133631
$57$	110.340	0.256403
$58$	267.934	0.606577
$59$	−336.000	−0.741415	−0.370707	−	0.928750i	$-0.620885\pi$
$59$	−336.000	−0.741415	−0.370707	+	0.928750i	$0.620885\pi$
$60$	−235.121	−0.505899
$61$	230.000	0.482762	0.241381	−	0.970430i	$-0.422400\pi$
$61$	230.000	0.482762	0.241381	+	0.970430i	$0.422400\pi$
$62$	−1.56032	−0.00319614
$63$	63.0000	0.125988
$64$	64.0000	0.125000
$65$	−254.714	−0.486052
$66$	−19.1206	−0.0356604
$67$	898.043	1.63751	0.818757	−	0.574141i	$-0.194664\pi$
$67$	898.043	1.63751	0.818757	+	0.574141i	$0.194664\pi$
$68$	206.241	0.367800
$69$	−21.6595	−0.0377899
$70$	274.307	0.468372
$71$	−584.615	−0.977197	−0.488599	−	0.872509i	$-0.662492\pi$
$71$	−584.615	−0.977197	−0.488599	+	0.872509i	$0.662492\pi$
$72$	−72.0000	−0.117851
$73$	−902.383	−1.44679	−0.723397	−	0.690432i	$-0.757420\pi$
$73$	−902.383	−1.44679	−0.723397	+	0.690432i	$0.757420\pi$
$74$	375.121	0.589282
$75$	776.702	1.19581
$76$	147.121	0.222051
$77$	22.3074	0.0330151
$78$	−78.0000	−0.113228
$79$	−157.461	−0.224250	−0.112125	−	0.993694i	$-0.535766\pi$
$79$	−157.461	−0.224250	−0.112125	+	0.993694i	$0.535766\pi$
$80$	−313.494	−0.438121
$81$	81.0000	0.111111
$82$	493.494	0.664601
$83$	457.944	0.605613	0.302806	−	0.953052i	$-0.402076\pi$
$83$	457.944	0.605613	0.302806	+	0.953052i	$0.402076\pi$
$84$	84.0000	0.109109
$85$	−1010.24	−1.28913
$86$	444.681	0.557572
$87$	−401.901	−0.495268
$88$	−25.4942	−0.0308828
$89$	171.461	0.204212	0.102106	−	0.994774i	$-0.467442\pi$
$89$	171.461	0.204212	0.102106	+	0.994774i	$0.467442\pi$
$90$	352.681	0.413065
$91$	91.0000	0.104828
$92$	−28.8794	−0.0327270
$93$	2.34048	0.00260964
$94$	876.549	0.961799
$95$	−720.648	−0.778283
$96$	−96.0000	−0.102062
$97$	−926.780	−0.970106	−0.485053	−	0.874485i	$-0.661200\pi$
$97$	−926.780	−0.970106	−0.485053	+	0.874485i	$0.661200\pi$
$98$	−98.0000	−0.101015
$99$	28.6810	0.0291166

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.1.1	✓	2
3.2	odd	2		1638.4.a.u.1.2		2

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

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -19.5934 q^{5} -6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +39.1868 q^{10} +3.18677 q^{11} +12.0000 q^{12} +13.0000 q^{13} -14.0000 q^{14} -58.7802 q^{15} +16.0000 q^{16} +51.5603 q^{17} -18.0000 q^{18} +36.7802 q^{19} -78.3735 q^{20} +21.0000 q^{21} -6.37355 q^{22} -7.21984 q^{23} -24.0000 q^{24} +258.901 q^{25} -26.0000 q^{26} +27.0000 q^{27} +28.0000 q^{28} -133.967 q^{29} +117.560 q^{30} +0.780160 q^{31} -32.0000 q^{32} +9.56032 q^{33} -103.121 q^{34} -137.154 q^{35} +36.0000 q^{36} -187.560 q^{37} -73.5603 q^{38} +39.0000 q^{39} +156.747 q^{40} -246.747 q^{41} -42.0000 q^{42} -222.340 q^{43} +12.7471 q^{44} -176.340 q^{45} +14.4397 q^{46} -438.274 q^{47} +48.0000 q^{48} +49.0000 q^{49} -517.802 q^{50} +154.681 q^{51} +52.0000 q^{52} -153.088 q^{53} -54.0000 q^{54} -62.4397 q^{55} -56.0000 q^{56} +110.340 q^{57} +267.934 q^{58} -336.000 q^{59} -235.121 q^{60} +230.000 q^{61} -1.56032 q^{62} +63.0000 q^{63} +64.0000 q^{64} -254.714 q^{65} -19.1206 q^{66} +898.043 q^{67} +206.241 q^{68} -21.6595 q^{69} +274.307 q^{70} -584.615 q^{71} -72.0000 q^{72} -902.383 q^{73} +375.121 q^{74} +776.702 q^{75} +147.121 q^{76} +22.3074 q^{77} -78.0000 q^{78} -157.461 q^{79} -313.494 q^{80} +81.0000 q^{81} +493.494 q^{82} +457.944 q^{83} +84.0000 q^{84} -1010.24 q^{85} +444.681 q^{86} -401.901 q^{87} -25.4942 q^{88} +171.461 q^{89} +352.681 q^{90} +91.0000 q^{91} -28.8794 q^{92} +2.34048 q^{93} +876.549 q^{94} -720.648 q^{95} -96.0000 q^{96} -926.780 q^{97} -98.0000 q^{98} +28.6810 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} - 15 q^{5} - 12 q^{6} + 14 q^{7} - 16 q^{8} + 18 q^{9} + 30 q^{10} - 42 q^{11} + 24 q^{12} + 26 q^{13} - 28 q^{14} - 45 q^{15} + 32 q^{16} - 42 q^{17} - 36 q^{18} + q^{19}+ \cdots - 378 q^{99}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more