LMFDB - Embedded newform 1027.4.a.c.1.16

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1027,4,Mod(1,1027)] 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("1027.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1027, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1027 = 13 \cdot 79 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1027.a (trivial)

Newform invariants

comment:select newform

sage:traces = [56,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$60.5949615759$
Analytic rank:	$1$
Dimension:	$56$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	1027.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.05544 q^{2} -4.05698 q^{3} +1.33571 q^{4} -8.29457 q^{5} +12.3959 q^{6} +23.0676 q^{7} +20.3623 q^{8} -10.5409 q^{9} +25.3435 q^{10} +16.3674 q^{11} -5.41895 q^{12} -13.0000 q^{13} -70.4818 q^{14} +33.6509 q^{15} -72.9016 q^{16} -103.842 q^{17} +32.2071 q^{18} +70.9148 q^{19} -11.0791 q^{20} -93.5849 q^{21} -50.0095 q^{22} -70.3996 q^{23} -82.6096 q^{24} -56.2002 q^{25} +39.7207 q^{26} +152.303 q^{27} +30.8117 q^{28} +140.117 q^{29} -102.818 q^{30} -183.552 q^{31} +59.8476 q^{32} -66.4021 q^{33} +317.284 q^{34} -191.336 q^{35} -14.0796 q^{36} +65.9484 q^{37} -216.676 q^{38} +52.7407 q^{39} -168.897 q^{40} +136.478 q^{41} +285.943 q^{42} +298.211 q^{43} +21.8620 q^{44} +87.4322 q^{45} +215.102 q^{46} -20.7823 q^{47} +295.760 q^{48} +189.116 q^{49} +171.716 q^{50} +421.286 q^{51} -17.3642 q^{52} +359.054 q^{53} -465.352 q^{54} -135.760 q^{55} +469.711 q^{56} -287.700 q^{57} -428.118 q^{58} +87.3725 q^{59} +44.9478 q^{60} +730.063 q^{61} +560.833 q^{62} -243.154 q^{63} +400.352 q^{64} +107.829 q^{65} +202.888 q^{66} +377.118 q^{67} -138.703 q^{68} +285.610 q^{69} +584.616 q^{70} -93.3450 q^{71} -214.637 q^{72} -307.914 q^{73} -201.501 q^{74} +228.003 q^{75} +94.7216 q^{76} +377.557 q^{77} -161.146 q^{78} +79.0000 q^{79} +604.687 q^{80} -333.285 q^{81} -417.002 q^{82} -1318.94 q^{83} -125.002 q^{84} +861.326 q^{85} -911.166 q^{86} -568.450 q^{87} +333.278 q^{88} -1309.58 q^{89} -267.144 q^{90} -299.879 q^{91} -94.0334 q^{92} +744.668 q^{93} +63.4992 q^{94} -588.208 q^{95} -242.801 q^{96} +1529.84 q^{97} -577.832 q^{98} -172.527 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 16 q^{2} - 17 q^{3} + 208 q^{4} - 35 q^{5} + 29 q^{6} - 37 q^{7} - 156 q^{8} + 415 q^{9} - 67 q^{10} - 114 q^{11} - 196 q^{12} - 728 q^{13} - 183 q^{14} - 17 q^{15} + 680 q^{16} - 383 q^{17} - 342 q^{18}+ \cdots - 11354 q^{99}+O(q^{100}) $ 56 * q - 16 * q^2 - 17 * q^3 + 208 * q^4 - 35 * q^5 + 29 * q^6 - 37 * q^7 - 156 * q^8 + 415 * q^9 - 67 * q^10 - 114 * q^11 - 196 * q^12 - 728 * q^13 - 183 * q^14 - 17 * q^15 + 680 * q^16 - 383 * q^17 - 342 * q^18 + 36 * q^19 - 384 * q^20 - 213 * q^21 - 148 * q^22 - 734 * q^23 + 21 * q^24 + 813 * q^25 + 208 * q^26 - 851 * q^27 - 1334 * q^28 - 1480 * q^29 - 1859 * q^30 - 244 * q^31 - 1039 * q^32 - 176 * q^33 - 615 * q^34 - 341 * q^35 + 269 * q^36 - 1231 * q^37 - 1603 * q^38 + 221 * q^39 - 671 * q^40 - 374 * q^41 - 2676 * q^42 - 1059 * q^43 - 2019 * q^44 - 994 * q^45 - 230 * q^46 - 951 * q^47 - 2664 * q^48 + 1843 * q^49 - 1135 * q^50 - 1855 * q^51 - 2704 * q^52 - 2892 * q^53 - 955 * q^54 - 2156 * q^55 - 2264 * q^56 - 2120 * q^57 - 616 * q^58 + 28 * q^59 - 171 * q^60 - 1290 * q^61 - 1158 * q^62 - 1092 * q^63 + 1498 * q^64 + 455 * q^65 + 304 * q^66 - 2820 * q^67 - 3642 * q^68 - 3252 * q^69 + 1144 * q^70 + 349 * q^71 - 5713 * q^72 - 890 * q^73 - 1519 * q^74 - 1954 * q^75 + 1678 * q^76 - 5768 * q^77 - 377 * q^78 + 4424 * q^79 - 2875 * q^80 + 1004 * q^81 - 384 * q^82 - 3482 * q^83 - 2502 * q^84 - 6769 * q^85 - 3806 * q^86 - 4820 * q^87 - 2304 * q^88 + 1890 * q^89 - 7524 * q^90 + 481 * q^91 - 9584 * q^92 - 4380 * q^93 - 1157 * q^94 - 6756 * q^95 + 647 * q^96 - 3052 * q^97 - 4070 * q^98 - 11354 * 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 $

$2$	−3.05544	−1.08026	−0.540130	−	0.841581i	$-0.681625\pi$
$2$	−3.05544	−1.08026	−0.540130	+	0.841581i	$0.681625\pi$
$3$	−4.05698	−0.780766	−0.390383	−	0.920652i	$-0.627657\pi$
$3$	−4.05698	−0.780766	−0.390383	+	0.920652i	$0.627657\pi$
$4$	1.33571	0.166964
$5$	−8.29457	−0.741889	−0.370944	−	0.928655i	$-0.620966\pi$
$5$	−8.29457	−0.741889	−0.370944	+	0.928655i	$0.620966\pi$
$6$	12.3959	0.843431
$7$	23.0676	1.24554	0.622768	−	0.782407i	$-0.286008\pi$
$7$	23.0676	1.24554	0.622768	+	0.782407i	$0.286008\pi$
$8$	20.3623	0.899897
$9$	−10.5409	−0.390404
$10$	25.3435	0.801433
$11$	16.3674	0.448632	0.224316	−	0.974517i	$-0.427985\pi$
$11$	16.3674	0.448632	0.224316	+	0.974517i	$0.427985\pi$
$12$	−5.41895	−0.130360
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−70.4818	−1.34550
$15$	33.6509	0.579242
$16$	−72.9016	−1.13909
$17$	−103.842	−1.48150	−0.740748	−	0.671783i	$-0.765529\pi$
$17$	−103.842	−1.48150	−0.740748	+	0.671783i	$0.765529\pi$
$18$	32.2071	0.421738
$19$	70.9148	0.856262	0.428131	−	0.903717i	$-0.359172\pi$
$19$	70.9148	0.856262	0.428131	+	0.903717i	$0.359172\pi$
$20$	−11.0791	−0.123868
$21$	−93.5849	−0.972472
$22$	−50.0095	−0.484639
$23$	−70.3996	−0.638231	−0.319116	−	0.947716i	$-0.603386\pi$
$23$	−70.3996	−0.638231	−0.319116	+	0.947716i	$0.603386\pi$
$24$	−82.6096	−0.702609
$25$	−56.2002	−0.449601
$26$	39.7207	0.299610
$27$	152.303	1.08558
$28$	30.8117	0.207959
$29$	140.117	0.897207	0.448603	−	0.893731i	$-0.351922\pi$
$29$	140.117	0.897207	0.448603	+	0.893731i	$0.351922\pi$
$30$	−102.818	−0.625732
$31$	−183.552	−1.06345	−0.531725	−	0.846917i	$-0.678456\pi$
$31$	−183.552	−1.06345	−0.531725	+	0.846917i	$0.678456\pi$
$32$	59.8476	0.330614
$33$	−66.4021	−0.350276
$34$	317.284	1.60040
$35$	−191.336	−0.924048
$36$	−14.0796	−0.0651833
$37$	65.9484	0.293023	0.146511	−	0.989209i	$-0.453195\pi$
$37$	65.9484	0.293023	0.146511	+	0.989209i	$0.453195\pi$
$38$	−216.676	−0.924986
$39$	52.7407	0.216546
$40$	−168.897	−0.667623
$41$	136.478	0.519862	0.259931	−	0.965627i	$-0.416300\pi$
$41$	136.478	0.519862	0.259931	+	0.965627i	$0.416300\pi$
$42$	285.943	1.05052
$43$	298.211	1.05760	0.528799	−	0.848747i	$-0.322642\pi$
$43$	298.211	1.05760	0.528799	+	0.848747i	$0.322642\pi$
$44$	21.8620	0.0749052
$45$	87.4322	0.289636
$46$	215.102	0.689457
$47$	−20.7823	−0.0644982	−0.0322491	−	0.999480i	$-0.510267\pi$
$47$	−20.7823	−0.0644982	−0.0322491	+	0.999480i	$0.510267\pi$
$48$	295.760	0.889361
$49$	189.116	0.551358
$50$	171.716	0.485687
$51$	421.286	1.15670
$52$	−17.3642	−0.0463074
$53$	359.054	0.930564	0.465282	−	0.885162i	$-0.345953\pi$
$53$	359.054	0.930564	0.465282	+	0.885162i	$0.345953\pi$
$54$	−465.352	−1.17271
$55$	−135.760	−0.332835
$56$	469.711	1.12085
$57$	−287.700	−0.668540
$58$	−428.118	−0.969218
$59$	87.3725	0.192795	0.0963977	−	0.995343i	$-0.469268\pi$
$59$	87.3725	0.192795	0.0963977	+	0.995343i	$0.469268\pi$
$60$	44.9478	0.0967123
$61$	730.063	1.53238	0.766188	−	0.642617i	$-0.222151\pi$
$61$	730.063	1.53238	0.766188	+	0.642617i	$0.222151\pi$
$62$	560.833	1.14880
$63$	−243.154	−0.486262
$64$	400.352	0.781937
$65$	107.829	0.205763
$66$	202.888	0.378390
$67$	377.118	0.687646	0.343823	−	0.939034i	$-0.388278\pi$
$67$	377.118	0.687646	0.343823	+	0.939034i	$0.388278\pi$
$68$	−138.703	−0.247356
$69$	285.610	0.498310
$70$	584.616	0.998213
$71$	−93.3450	−0.156028	−0.0780142	−	0.996952i	$-0.524858\pi$
$71$	−93.3450	−0.156028	−0.0780142	+	0.996952i	$0.524858\pi$
$72$	−214.637	−0.351323
$73$	−307.914	−0.493679	−0.246840	−	0.969056i	$-0.579392\pi$
$73$	−307.914	−0.493679	−0.246840	+	0.969056i	$0.579392\pi$
$74$	−201.501	−0.316541
$75$	228.003	0.351034
$76$	94.7216	0.142965
$77$	377.557	0.558787
$78$	−161.146	−0.233926
$79$	79.0000	0.112509
$79$	79.0000	0.112509
$80$	604.687	0.845075
$81$	−333.285	−0.457181
$82$	−417.002	−0.561587
$83$	−1318.94	−1.74424	−0.872121	−	0.489291i	$-0.837256\pi$
$83$	−1318.94	−1.74424	−0.872121	+	0.489291i	$0.837256\pi$
$84$	−125.002	−0.162367
$85$	861.326	1.09910
$86$	−911.166	−1.14248
$87$	−568.450	−0.700509
$88$	333.278	0.403722
$89$	−1309.58	−1.55972	−0.779859	−	0.625955i	$-0.784709\pi$
$89$	−1309.58	−1.55972	−0.779859	+	0.625955i	$0.784709\pi$
$90$	−267.144	−0.312883
$91$	−299.879	−0.345449
$92$	−94.0334	−0.106561
$93$	744.668	0.830306
$94$	63.4992	0.0696749
$95$	−588.208	−0.635251
$96$	−242.801	−0.258133
$97$	1529.84	1.60136	0.800678	−	0.599096i	$-0.204473\pi$
$97$	1529.84	1.60136	0.800678	+	0.599096i	$0.204473\pi$
$98$	−577.832	−0.595610
$99$	−172.527	−0.175148

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	1027.4.a.c.1.16	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1027.4.a.c.1.16	✓	56	1.1	even	1	trivial

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Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1027 = 13 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1027.a (trivial)

\(f(q)\)	\(=\)	\(q-3.05544 q^{2} -4.05698 q^{3} +1.33571 q^{4} -8.29457 q^{5} +12.3959 q^{6} +23.0676 q^{7} +20.3623 q^{8} -10.5409 q^{9} +25.3435 q^{10} +16.3674 q^{11} -5.41895 q^{12} -13.0000 q^{13} -70.4818 q^{14} +33.6509 q^{15} -72.9016 q^{16} -103.842 q^{17} +32.2071 q^{18} +70.9148 q^{19} -11.0791 q^{20} -93.5849 q^{21} -50.0095 q^{22} -70.3996 q^{23} -82.6096 q^{24} -56.2002 q^{25} +39.7207 q^{26} +152.303 q^{27} +30.8117 q^{28} +140.117 q^{29} -102.818 q^{30} -183.552 q^{31} +59.8476 q^{32} -66.4021 q^{33} +317.284 q^{34} -191.336 q^{35} -14.0796 q^{36} +65.9484 q^{37} -216.676 q^{38} +52.7407 q^{39} -168.897 q^{40} +136.478 q^{41} +285.943 q^{42} +298.211 q^{43} +21.8620 q^{44} +87.4322 q^{45} +215.102 q^{46} -20.7823 q^{47} +295.760 q^{48} +189.116 q^{49} +171.716 q^{50} +421.286 q^{51} -17.3642 q^{52} +359.054 q^{53} -465.352 q^{54} -135.760 q^{55} +469.711 q^{56} -287.700 q^{57} -428.118 q^{58} +87.3725 q^{59} +44.9478 q^{60} +730.063 q^{61} +560.833 q^{62} -243.154 q^{63} +400.352 q^{64} +107.829 q^{65} +202.888 q^{66} +377.118 q^{67} -138.703 q^{68} +285.610 q^{69} +584.616 q^{70} -93.3450 q^{71} -214.637 q^{72} -307.914 q^{73} -201.501 q^{74} +228.003 q^{75} +94.7216 q^{76} +377.557 q^{77} -161.146 q^{78} +79.0000 q^{79} +604.687 q^{80} -333.285 q^{81} -417.002 q^{82} -1318.94 q^{83} -125.002 q^{84} +861.326 q^{85} -911.166 q^{86} -568.450 q^{87} +333.278 q^{88} -1309.58 q^{89} -267.144 q^{90} -299.879 q^{91} -94.0334 q^{92} +744.668 q^{93} +63.4992 q^{94} -588.208 q^{95} -242.801 q^{96} +1529.84 q^{97} -577.832 q^{98} -172.527 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 16 q^{2} - 17 q^{3} + 208 q^{4} - 35 q^{5} + 29 q^{6} - 37 q^{7} - 156 q^{8} + 415 q^{9} - 67 q^{10} - 114 q^{11} - 196 q^{12} - 728 q^{13} - 183 q^{14} - 17 q^{15} + 680 q^{16} - 383 q^{17} - 342 q^{18}+ \cdots - 11354 q^{99}+O(q^{100}) \) 56 * q - 16 * q^2 - 17 * q^3 + 208 * q^4 - 35 * q^5 + 29 * q^6 - 37 * q^7 - 156 * q^8 + 415 * q^9 - 67 * q^10 - 114 * q^11 - 196 * q^12 - 728 * q^13 - 183 * q^14 - 17 * q^15 + 680 * q^16 - 383 * q^17 - 342 * q^18 + 36 * q^19 - 384 * q^20 - 213 * q^21 - 148 * q^22 - 734 * q^23 + 21 * q^24 + 813 * q^25 + 208 * q^26 - 851 * q^27 - 1334 * q^28 - 1480 * q^29 - 1859 * q^30 - 244 * q^31 - 1039 * q^32 - 176 * q^33 - 615 * q^34 - 341 * q^35 + 269 * q^36 - 1231 * q^37 - 1603 * q^38 + 221 * q^39 - 671 * q^40 - 374 * q^41 - 2676 * q^42 - 1059 * q^43 - 2019 * q^44 - 994 * q^45 - 230 * q^46 - 951 * q^47 - 2664 * q^48 + 1843 * q^49 - 1135 * q^50 - 1855 * q^51 - 2704 * q^52 - 2892 * q^53 - 955 * q^54 - 2156 * q^55 - 2264 * q^56 - 2120 * q^57 - 616 * q^58 + 28 * q^59 - 171 * q^60 - 1290 * q^61 - 1158 * q^62 - 1092 * q^63 + 1498 * q^64 + 455 * q^65 + 304 * q^66 - 2820 * q^67 - 3642 * q^68 - 3252 * q^69 + 1144 * q^70 + 349 * q^71 - 5713 * q^72 - 890 * q^73 - 1519 * q^74 - 1954 * q^75 + 1678 * q^76 - 5768 * q^77 - 377 * q^78 + 4424 * q^79 - 2875 * q^80 + 1004 * q^81 - 384 * q^82 - 3482 * q^83 - 2502 * q^84 - 6769 * q^85 - 3806 * q^86 - 4820 * q^87 - 2304 * q^88 + 1890 * q^89 - 7524 * q^90 + 481 * q^91 - 9584 * q^92 - 4380 * q^93 - 1157 * q^94 - 6756 * q^95 + 647 * q^96 - 3052 * q^97 - 4070 * q^98 - 11354 * q^99

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