LMFDB - Embedded newform 2667.2.a.n.1.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2667,2,Mod(1,2667)] 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("2667.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2667, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2667 = 3 \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2667.a (trivial)

Newform invariants

comment:select newform

sage:traces = [16,4,16,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$21.2961022191$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 18 x^{14} + 83 x^{13} + 112 x^{12} - 668 x^{11} - 235 x^{10} + 2648 x^{9} + \cdots - 20 $ x^16 - 4x^15 - 18x^14 + 83x^13 + 112x^12 - 668x^11 - 235x^10 + 2648x^9 - 298x^8 - 5422x^7 + 2075x^6 + 5385x^5 - 3163x^4 - 1882x^3 + 1614x^2 - 200*x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$1.17466$ of defining polynomial
Character	$\chi$	$=$	2667.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.17466 q^{2} +1.00000 q^{3} -0.620175 q^{4} +3.75278 q^{5} +1.17466 q^{6} -1.00000 q^{7} -3.07781 q^{8} +1.00000 q^{9} +4.40823 q^{10} -0.236817 q^{11} -0.620175 q^{12} +2.34303 q^{13} -1.17466 q^{14} +3.75278 q^{15} -2.37503 q^{16} -5.49213 q^{17} +1.17466 q^{18} +1.29877 q^{19} -2.32738 q^{20} -1.00000 q^{21} -0.278179 q^{22} +4.33999 q^{23} -3.07781 q^{24} +9.08332 q^{25} +2.75226 q^{26} +1.00000 q^{27} +0.620175 q^{28} +3.35099 q^{29} +4.40823 q^{30} +10.2207 q^{31} +3.36577 q^{32} -0.236817 q^{33} -6.45138 q^{34} -3.75278 q^{35} -0.620175 q^{36} +5.67027 q^{37} +1.52562 q^{38} +2.34303 q^{39} -11.5503 q^{40} +5.91718 q^{41} -1.17466 q^{42} +7.58953 q^{43} +0.146868 q^{44} +3.75278 q^{45} +5.09801 q^{46} +6.69813 q^{47} -2.37503 q^{48} +1.00000 q^{49} +10.6698 q^{50} -5.49213 q^{51} -1.45309 q^{52} -11.3704 q^{53} +1.17466 q^{54} -0.888720 q^{55} +3.07781 q^{56} +1.29877 q^{57} +3.93627 q^{58} -1.67458 q^{59} -2.32738 q^{60} +6.14626 q^{61} +12.0058 q^{62} -1.00000 q^{63} +8.70370 q^{64} +8.79286 q^{65} -0.278179 q^{66} -12.3793 q^{67} +3.40608 q^{68} +4.33999 q^{69} -4.40823 q^{70} -16.3585 q^{71} -3.07781 q^{72} -2.42897 q^{73} +6.66064 q^{74} +9.08332 q^{75} -0.805466 q^{76} +0.236817 q^{77} +2.75226 q^{78} -12.9909 q^{79} -8.91297 q^{80} +1.00000 q^{81} +6.95068 q^{82} -1.57938 q^{83} +0.620175 q^{84} -20.6107 q^{85} +8.91512 q^{86} +3.35099 q^{87} +0.728878 q^{88} +3.02589 q^{89} +4.40823 q^{90} -2.34303 q^{91} -2.69155 q^{92} +10.2207 q^{93} +7.86802 q^{94} +4.87400 q^{95} +3.36577 q^{96} -1.63077 q^{97} +1.17466 q^{98} -0.236817 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{2} + 16 q^{3} + 20 q^{4} + 5 q^{5} + 4 q^{6} - 16 q^{7} + 15 q^{8} + 16 q^{9} - 4 q^{10} + q^{11} + 20 q^{12} + 20 q^{13} - 4 q^{14} + 5 q^{15} + 32 q^{16} + 3 q^{17} + 4 q^{18} + 13 q^{19}+ \cdots + q^{99}+O(q^{100}) $ 16 * q + 4 * q^2 + 16 * q^3 + 20 * q^4 + 5 * q^5 + 4 * q^6 - 16 * q^7 + 15 * q^8 + 16 * q^9 - 4 * q^10 + q^11 + 20 * q^12 + 20 * q^13 - 4 * q^14 + 5 * q^15 + 32 * q^16 + 3 * q^17 + 4 * q^18 + 13 * q^19 + 17 * q^20 - 16 * q^21 + 13 * q^22 + 5 * q^23 + 15 * q^24 + 17 * q^25 - 2 * q^26 + 16 * q^27 - 20 * q^28 + 22 * q^29 - 4 * q^30 + 26 * q^31 + 54 * q^32 + q^33 - 6 * q^34 - 5 * q^35 + 20 * q^36 + 30 * q^37 + 5 * q^38 + 20 * q^39 + 13 * q^40 + q^41 - 4 * q^42 + 31 * q^43 + 22 * q^44 + 5 * q^45 - 2 * q^46 - q^47 + 32 * q^48 + 16 * q^49 + 5 * q^50 + 3 * q^51 + 31 * q^52 + 24 * q^53 + 4 * q^54 + 8 * q^55 - 15 * q^56 + 13 * q^57 + 13 * q^58 - 17 * q^59 + 17 * q^60 + 32 * q^61 - 5 * q^62 - 16 * q^63 + 61 * q^64 - 3 * q^65 + 13 * q^66 + 16 * q^67 - 10 * q^68 + 5 * q^69 + 4 * q^70 - 10 * q^71 + 15 * q^72 + 23 * q^73 + q^74 + 17 * q^75 + 18 * q^76 - q^77 - 2 * q^78 + 48 * q^79 + 38 * q^80 + 16 * q^81 + 12 * q^82 + 9 * q^83 - 20 * q^84 + 22 * q^85 - 4 * q^86 + 22 * q^87 + 27 * q^88 + 17 * q^89 - 4 * q^90 - 20 * q^91 + 16 * q^92 + 26 * q^93 + 13 * q^94 + 22 * q^95 + 54 * q^96 + 17 * q^97 + 4 * q^98 + 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 $

$2$	1.17466	0.830610	0.415305	−	0.909682i	$-0.363675\pi$
$2$	1.17466	0.830610	0.415305	+	0.909682i	$0.363675\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.620175	−0.310087
$5$	3.75278	1.67829	0.839146	−	0.543906i	$-0.183055\pi$
$5$	3.75278	1.67829	0.839146	+	0.543906i	$0.183055\pi$
$6$	1.17466	0.479553
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.07781	−1.08817
$9$	1.00000	0.333333
$10$	4.40823	1.39401
$11$	−0.236817	−0.0714029	−0.0357015	−	0.999362i	$-0.511367\pi$
$11$	−0.236817	−0.0714029	−0.0357015	+	0.999362i	$0.511367\pi$
$12$	−0.620175	−0.179029
$13$	2.34303	0.649839	0.324920	−	0.945742i	$-0.394663\pi$
$13$	2.34303	0.649839	0.324920	+	0.945742i	$0.394663\pi$
$14$	−1.17466	−0.313941
$15$	3.75278	0.968962
$16$	−2.37503	−0.593759
$17$	−5.49213	−1.33204	−0.666018	−	0.745935i	$-0.732003\pi$
$17$	−5.49213	−1.33204	−0.666018	+	0.745935i	$0.732003\pi$
$18$	1.17466	0.276870
$19$	1.29877	0.297959	0.148979	−	0.988840i	$-0.452401\pi$
$19$	1.29877	0.297959	0.148979	+	0.988840i	$0.452401\pi$
$20$	−2.32738	−0.520417
$21$	−1.00000	−0.218218
$22$	−0.278179	−0.0593080
$23$	4.33999	0.904950	0.452475	−	0.891777i	$-0.350541\pi$
$23$	4.33999	0.904950	0.452475	+	0.891777i	$0.350541\pi$
$24$	−3.07781	−0.628256
$25$	9.08332	1.81666
$26$	2.75226	0.539763
$27$	1.00000	0.192450
$28$	0.620175	0.117202
$29$	3.35099	0.622263	0.311131	−	0.950367i	$-0.399292\pi$
$29$	3.35099	0.622263	0.311131	+	0.950367i	$0.399292\pi$
$30$	4.40823	0.804830
$31$	10.2207	1.83569	0.917846	−	0.396937i	$-0.129927\pi$
$31$	10.2207	1.83569	0.917846	+	0.396937i	$0.129927\pi$
$32$	3.36577	0.594990
$33$	−0.236817	−0.0412245
$34$	−6.45138	−1.10640
$35$	−3.75278	−0.634335
$36$	−0.620175	−0.103362
$37$	5.67027	0.932186	0.466093	−	0.884736i	$-0.345661\pi$
$37$	5.67027	0.932186	0.466093	+	0.884736i	$0.345661\pi$
$38$	1.52562	0.247488
$39$	2.34303	0.375185
$40$	−11.5503	−1.82627
$41$	5.91718	0.924109	0.462054	−	0.886852i	$-0.347112\pi$
$41$	5.91718	0.924109	0.462054	+	0.886852i	$0.347112\pi$
$42$	−1.17466	−0.181254
$43$	7.58953	1.15739	0.578697	−	0.815543i	$-0.303562\pi$
$43$	7.58953	1.15739	0.578697	+	0.815543i	$0.303562\pi$
$44$	0.146868	0.0221411
$45$	3.75278	0.559431
$46$	5.09801	0.751660
$47$	6.69813	0.977022	0.488511	−	0.872558i	$-0.337540\pi$
$47$	6.69813	0.977022	0.488511	+	0.872558i	$0.337540\pi$
$48$	−2.37503	−0.342807
$49$	1.00000	0.142857
$50$	10.6698	1.50894
$51$	−5.49213	−0.769052
$52$	−1.45309	−0.201507
$53$	−11.3704	−1.56185	−0.780926	−	0.624624i	$-0.785252\pi$
$53$	−11.3704	−1.56185	−0.780926	+	0.624624i	$0.785252\pi$
$54$	1.17466	0.159851
$55$	−0.888720	−0.119835
$56$	3.07781	0.411290
$57$	1.29877	0.172027
$58$	3.93627	0.516858
$59$	−1.67458	−0.218011	−0.109006	−	0.994041i	$-0.534767\pi$
$59$	−1.67458	−0.218011	−0.109006	+	0.994041i	$0.534767\pi$
$60$	−2.32738	−0.300463
$61$	6.14626	0.786948	0.393474	−	0.919336i	$-0.371273\pi$
$61$	6.14626	0.786948	0.393474	+	0.919336i	$0.371273\pi$
$62$	12.0058	1.52474
$63$	−1.00000	−0.125988
$64$	8.70370	1.08796
$65$	8.79286	1.09062
$66$	−0.278179	−0.0342415
$67$	−12.3793	−1.51237	−0.756186	−	0.654357i	$-0.772940\pi$
$67$	−12.3793	−1.51237	−0.756186	+	0.654357i	$0.772940\pi$
$68$	3.40608	0.413048
$69$	4.33999	0.522473
$70$	−4.40823	−0.526885
$71$	−16.3585	−1.94140	−0.970701	−	0.240291i	$-0.922757\pi$
$71$	−16.3585	−1.94140	−0.970701	+	0.240291i	$0.922757\pi$
$72$	−3.07781	−0.362724
$73$	−2.42897	−0.284289	−0.142144	−	0.989846i	$-0.545400\pi$
$73$	−2.42897	−0.284289	−0.142144	+	0.989846i	$0.545400\pi$
$74$	6.66064	0.774283
$75$	9.08332	1.04885
$76$	−0.805466	−0.0923933
$77$	0.236817	0.0269878
$78$	2.75226	0.311632
$79$	−12.9909	−1.46159	−0.730796	−	0.682596i	$-0.760851\pi$
$79$	−12.9909	−1.46159	−0.730796	+	0.682596i	$0.760851\pi$
$80$	−8.91297	−0.996500
$81$	1.00000	0.111111
$82$	6.95068	0.767574
$83$	−1.57938	−0.173359	−0.0866795	−	0.996236i	$-0.527626\pi$
$83$	−1.57938	−0.173359	−0.0866795	+	0.996236i	$0.527626\pi$
$84$	0.620175	0.0676666
$85$	−20.6107	−2.23555
$86$	8.91512	0.961342
$87$	3.35099	0.359264
$88$	0.728878	0.0776986
$89$	3.02589	0.320743	0.160372	−	0.987057i	$-0.448731\pi$
$89$	3.02589	0.320743	0.160372	+	0.987057i	$0.448731\pi$
$90$	4.40823	0.464669
$91$	−2.34303	−0.245616
$92$	−2.69155	−0.280614
$93$	10.2207	1.05984
$94$	7.86802	0.811524
$95$	4.87400	0.500062
$96$	3.36577	0.343517
$97$	−1.63077	−0.165580	−0.0827899	−	0.996567i	$-0.526383\pi$
$97$	−1.63077	−0.165580	−0.0827899	+	0.996567i	$0.526383\pi$
$98$	1.17466	0.118659
$99$	−0.236817	−0.0238010

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	2667.2.a.n.1.11	✓	16
3.2	odd	2		8001.2.a.s.1.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.n.1.11	✓	16	1.1	even	1	trivial
8001.2.a.s.1.6		16	3.2	odd	2

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Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2667 = 3 \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2667.a (trivial)

\(f(q)\)	\(=\)	\(q+1.17466 q^{2} +1.00000 q^{3} -0.620175 q^{4} +3.75278 q^{5} +1.17466 q^{6} -1.00000 q^{7} -3.07781 q^{8} +1.00000 q^{9} +4.40823 q^{10} -0.236817 q^{11} -0.620175 q^{12} +2.34303 q^{13} -1.17466 q^{14} +3.75278 q^{15} -2.37503 q^{16} -5.49213 q^{17} +1.17466 q^{18} +1.29877 q^{19} -2.32738 q^{20} -1.00000 q^{21} -0.278179 q^{22} +4.33999 q^{23} -3.07781 q^{24} +9.08332 q^{25} +2.75226 q^{26} +1.00000 q^{27} +0.620175 q^{28} +3.35099 q^{29} +4.40823 q^{30} +10.2207 q^{31} +3.36577 q^{32} -0.236817 q^{33} -6.45138 q^{34} -3.75278 q^{35} -0.620175 q^{36} +5.67027 q^{37} +1.52562 q^{38} +2.34303 q^{39} -11.5503 q^{40} +5.91718 q^{41} -1.17466 q^{42} +7.58953 q^{43} +0.146868 q^{44} +3.75278 q^{45} +5.09801 q^{46} +6.69813 q^{47} -2.37503 q^{48} +1.00000 q^{49} +10.6698 q^{50} -5.49213 q^{51} -1.45309 q^{52} -11.3704 q^{53} +1.17466 q^{54} -0.888720 q^{55} +3.07781 q^{56} +1.29877 q^{57} +3.93627 q^{58} -1.67458 q^{59} -2.32738 q^{60} +6.14626 q^{61} +12.0058 q^{62} -1.00000 q^{63} +8.70370 q^{64} +8.79286 q^{65} -0.278179 q^{66} -12.3793 q^{67} +3.40608 q^{68} +4.33999 q^{69} -4.40823 q^{70} -16.3585 q^{71} -3.07781 q^{72} -2.42897 q^{73} +6.66064 q^{74} +9.08332 q^{75} -0.805466 q^{76} +0.236817 q^{77} +2.75226 q^{78} -12.9909 q^{79} -8.91297 q^{80} +1.00000 q^{81} +6.95068 q^{82} -1.57938 q^{83} +0.620175 q^{84} -20.6107 q^{85} +8.91512 q^{86} +3.35099 q^{87} +0.728878 q^{88} +3.02589 q^{89} +4.40823 q^{90} -2.34303 q^{91} -2.69155 q^{92} +10.2207 q^{93} +7.86802 q^{94} +4.87400 q^{95} +3.36577 q^{96} -1.63077 q^{97} +1.17466 q^{98} -0.236817 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{2} + 16 q^{3} + 20 q^{4} + 5 q^{5} + 4 q^{6} - 16 q^{7} + 15 q^{8} + 16 q^{9} - 4 q^{10} + q^{11} + 20 q^{12} + 20 q^{13} - 4 q^{14} + 5 q^{15} + 32 q^{16} + 3 q^{17} + 4 q^{18} + 13 q^{19}+ \cdots + q^{99}+O(q^{100}) \) 16 * q + 4 * q^2 + 16 * q^3 + 20 * q^4 + 5 * q^5 + 4 * q^6 - 16 * q^7 + 15 * q^8 + 16 * q^9 - 4 * q^10 + q^11 + 20 * q^12 + 20 * q^13 - 4 * q^14 + 5 * q^15 + 32 * q^16 + 3 * q^17 + 4 * q^18 + 13 * q^19 + 17 * q^20 - 16 * q^21 + 13 * q^22 + 5 * q^23 + 15 * q^24 + 17 * q^25 - 2 * q^26 + 16 * q^27 - 20 * q^28 + 22 * q^29 - 4 * q^30 + 26 * q^31 + 54 * q^32 + q^33 - 6 * q^34 - 5 * q^35 + 20 * q^36 + 30 * q^37 + 5 * q^38 + 20 * q^39 + 13 * q^40 + q^41 - 4 * q^42 + 31 * q^43 + 22 * q^44 + 5 * q^45 - 2 * q^46 - q^47 + 32 * q^48 + 16 * q^49 + 5 * q^50 + 3 * q^51 + 31 * q^52 + 24 * q^53 + 4 * q^54 + 8 * q^55 - 15 * q^56 + 13 * q^57 + 13 * q^58 - 17 * q^59 + 17 * q^60 + 32 * q^61 - 5 * q^62 - 16 * q^63 + 61 * q^64 - 3 * q^65 + 13 * q^66 + 16 * q^67 - 10 * q^68 + 5 * q^69 + 4 * q^70 - 10 * q^71 + 15 * q^72 + 23 * q^73 + q^74 + 17 * q^75 + 18 * q^76 - q^77 - 2 * q^78 + 48 * q^79 + 38 * q^80 + 16 * q^81 + 12 * q^82 + 9 * q^83 - 20 * q^84 + 22 * q^85 - 4 * q^86 + 22 * q^87 + 27 * q^88 + 17 * q^89 - 4 * q^90 - 20 * q^91 + 16 * q^92 + 26 * q^93 + 13 * q^94 + 22 * q^95 + 54 * q^96 + 17 * q^97 + 4 * q^98 + q^99

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