LMFDB - Embedded newform 920.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [920,4,Mod(1,920)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("920.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	920.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$54.2817572053$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 135x^{6} + 187x^{5} + 5854x^{4} - 3309x^{3} - 85092x^{2} - 22464x + 77760 $ x^8 - 2x^7 - 135x^6 + 187x^5 + 5854x^4 - 3309x^3 - 85092x^2 - 22464*x + 77760
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.45758$ of defining polynomial
Character	$\chi$	$=$	920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.45758 q^{3} -5.00000 q^{5} +27.5388 q^{7} +14.7003 q^{9} +O(q^{10})$	$q-6.45758 q^{3} -5.00000 q^{5} +27.5388 q^{7} +14.7003 q^{9} +4.45999 q^{11} -74.7182 q^{13} +32.2879 q^{15} +22.3112 q^{17} -16.3732 q^{19} -177.834 q^{21} +23.0000 q^{23} +25.0000 q^{25} +79.4263 q^{27} +4.93873 q^{29} -16.1212 q^{31} -28.8007 q^{33} -137.694 q^{35} +253.798 q^{37} +482.499 q^{39} +391.106 q^{41} -238.748 q^{43} -73.5015 q^{45} -97.9993 q^{47} +415.386 q^{49} -144.076 q^{51} -303.841 q^{53} -22.2999 q^{55} +105.731 q^{57} +199.608 q^{59} -94.0815 q^{61} +404.829 q^{63} +373.591 q^{65} +490.878 q^{67} -148.524 q^{69} -680.386 q^{71} +754.759 q^{73} -161.439 q^{75} +122.823 q^{77} -296.299 q^{79} -909.809 q^{81} -1198.89 q^{83} -111.556 q^{85} -31.8922 q^{87} -167.603 q^{89} -2057.65 q^{91} +104.104 q^{93} +81.8659 q^{95} +652.693 q^{97} +65.5631 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{3} - 40 q^{5} - 3 q^{7} + 62 q^{9}+O(q^{10}) $ 8 * q - 6 * q^3 - 40 * q^5 - 3 * q^7 + 62 * q^9	$ 8 q - 6 q^{3} - 40 q^{5} - 3 q^{7} + 62 q^{9} - 55 q^{11} - 16 q^{13} + 30 q^{15} + 127 q^{17} + 61 q^{19} + 94 q^{21} + 184 q^{23} + 200 q^{25} - 243 q^{27} - 223 q^{29} + 102 q^{31} + 129 q^{33} + 15 q^{35} - 218 q^{37} - 403 q^{39} + 112 q^{41} - 560 q^{43} - 310 q^{45} + 379 q^{47} - 291 q^{49} - 453 q^{51} - 1006 q^{53} + 275 q^{55} + 1020 q^{57} - 860 q^{59} - 1477 q^{61} - 1206 q^{63} + 80 q^{65} - 740 q^{67} - 138 q^{69} - 1238 q^{71} - 2001 q^{73} - 150 q^{75} - 412 q^{77} - 1202 q^{79} - 2356 q^{81} - 1218 q^{83} - 635 q^{85} - 3419 q^{87} - 1318 q^{89} - 3123 q^{91} - 2445 q^{93} - 305 q^{95} + 151 q^{97} - 4075 q^{99}+O(q^{100}) $ 8 * q - 6 * q^3 - 40 * q^5 - 3 * q^7 + 62 * q^9 - 55 * q^11 - 16 * q^13 + 30 * q^15 + 127 * q^17 + 61 * q^19 + 94 * q^21 + 184 * q^23 + 200 * q^25 - 243 * q^27 - 223 * q^29 + 102 * q^31 + 129 * q^33 + 15 * q^35 - 218 * q^37 - 403 * q^39 + 112 * q^41 - 560 * q^43 - 310 * q^45 + 379 * q^47 - 291 * q^49 - 453 * q^51 - 1006 * q^53 + 275 * q^55 + 1020 * q^57 - 860 * q^59 - 1477 * q^61 - 1206 * q^63 + 80 * q^65 - 740 * q^67 - 138 * q^69 - 1238 * q^71 - 2001 * q^73 - 150 * q^75 - 412 * q^77 - 1202 * q^79 - 2356 * q^81 - 1218 * q^83 - 635 * q^85 - 3419 * q^87 - 1318 * q^89 - 3123 * q^91 - 2445 * q^93 - 305 * q^95 + 151 * q^97 - 4075 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See 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 $

$2$	0	0
$2$	0	0
$3$	−6.45758	−1.24276	−0.621381	−	0.783509i	$-0.713428\pi$
$3$	−6.45758	−1.24276	−0.621381	+	0.783509i	$0.713428\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	27.5388	1.48696	0.743478	−	0.668761i	$-0.233175\pi$
$7$	27.5388	1.48696	0.743478	+	0.668761i	$0.233175\pi$
$8$	0	0
$9$	14.7003	0.544455
$10$	0	0
$11$	4.45999	0.122249	0.0611244	−	0.998130i	$-0.480531\pi$
$11$	4.45999	0.122249	0.0611244	+	0.998130i	$0.480531\pi$
$12$	0	0
$13$	−74.7182	−1.59408	−0.797042	−	0.603923i	$-0.793603\pi$
$13$	−74.7182	−1.59408	−0.797042	+	0.603923i	$0.793603\pi$
$14$	0	0
$15$	32.2879	0.555780
$16$	0	0
$17$	22.3112	0.318309	0.159155	−	0.987254i	$-0.449123\pi$
$17$	22.3112	0.318309	0.159155	+	0.987254i	$0.449123\pi$
$18$	0	0
$19$	−16.3732	−0.197698	−0.0988491	−	0.995102i	$-0.531516\pi$
$19$	−16.3732	−0.197698	−0.0988491	+	0.995102i	$0.531516\pi$
$20$	0	0
$21$	−177.834	−1.84793
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	79.4263	0.566133
$28$	0	0
$29$	4.93873	0.0316241	0.0158121	−	0.999875i	$-0.494967\pi$
$29$	4.93873	0.0316241	0.0158121	+	0.999875i	$0.494967\pi$
$30$	0	0
$31$	−16.1212	−0.0934018	−0.0467009	−	0.998909i	$-0.514871\pi$
$31$	−16.1212	−0.0934018	−0.0467009	+	0.998909i	$0.514871\pi$
$32$	0	0
$33$	−28.8007	−0.151926
$34$	0	0
$35$	−137.694	−0.664987
$36$	0	0
$37$	253.798	1.12768	0.563839	−	0.825884i	$-0.309324\pi$
$37$	253.798	1.12768	0.563839	+	0.825884i	$0.309324\pi$
$38$	0	0
$39$	482.499	1.98107
$40$	0	0
$41$	391.106	1.48977	0.744884	−	0.667194i	$-0.232505\pi$
$41$	391.106	1.48977	0.744884	+	0.667194i	$0.232505\pi$
$42$	0	0
$43$	−238.748	−0.846715	−0.423357	−	0.905963i	$-0.639149\pi$
$43$	−238.748	−0.846715	−0.423357	+	0.905963i	$0.639149\pi$
$44$	0	0
$45$	−73.5015	−0.243488
$46$	0	0
$47$	−97.9993	−0.304142	−0.152071	−	0.988370i	$-0.548594\pi$
$47$	−97.9993	−0.304142	−0.152071	+	0.988370i	$0.548594\pi$
$48$	0	0
$49$	415.386	1.21104
$50$	0	0
$51$	−144.076	−0.395583
$52$	0	0
$53$	−303.841	−0.787467	−0.393733	−	0.919225i	$-0.628817\pi$
$53$	−303.841	−0.787467	−0.393733	+	0.919225i	$0.628817\pi$
$54$	0	0
$55$	−22.2999	−0.0546713
$56$	0	0
$57$	105.731	0.245692
$58$	0	0
$59$	199.608	0.440453	0.220226	−	0.975449i	$-0.429320\pi$
$59$	199.608	0.440453	0.220226	+	0.975449i	$0.429320\pi$
$60$	0	0
$61$	−94.0815	−0.197474	−0.0987369	−	0.995114i	$-0.531480\pi$
$61$	−94.0815	−0.197474	−0.0987369	+	0.995114i	$0.531480\pi$
$62$	0	0
$63$	404.829	0.809581
$64$	0	0
$65$	373.591	0.712896
$66$	0	0
$67$	490.878	0.895080	0.447540	−	0.894264i	$-0.352300\pi$
$67$	490.878	0.895080	0.447540	+	0.894264i	$0.352300\pi$
$68$	0	0
$69$	−148.524	−0.259134
$70$	0	0
$71$	−680.386	−1.13728	−0.568641	−	0.822586i	$-0.692531\pi$
$71$	−680.386	−1.13728	−0.568641	+	0.822586i	$0.692531\pi$
$72$	0	0
$73$	754.759	1.21011	0.605054	−	0.796185i	$-0.293152\pi$
$73$	754.759	1.21011	0.605054	+	0.796185i	$0.293152\pi$
$74$	0	0
$75$	−161.439	−0.248552
$76$	0	0
$77$	122.823	0.181778
$78$	0	0
$79$	−296.299	−0.421978	−0.210989	−	0.977488i	$-0.567668\pi$
$79$	−296.299	−0.421978	−0.210989	+	0.977488i	$0.567668\pi$
$80$	0	0
$81$	−909.809	−1.24802
$82$	0	0
$83$	−1198.89	−1.58549	−0.792746	−	0.609553i	$-0.791349\pi$
$83$	−1198.89	−1.58549	−0.792746	+	0.609553i	$0.791349\pi$
$84$	0	0
$85$	−111.556	−0.142352
$86$	0	0
$87$	−31.8922	−0.0393012
$88$	0	0
$89$	−167.603	−0.199616	−0.0998081	−	0.995007i	$-0.531823\pi$
$89$	−167.603	−0.199616	−0.0998081	+	0.995007i	$0.531823\pi$
$90$	0	0
$91$	−2057.65	−2.37033
$92$	0	0
$93$	104.104	0.116076
$94$	0	0
$95$	81.8659	0.0884133
$96$	0	0
$97$	652.693	0.683205	0.341603	−	0.939844i	$-0.389030\pi$
$97$	652.693	0.683205	0.341603	+	0.939844i	$0.389030\pi$
$98$	0	0
$99$	65.5631	0.0665590
$100$	0	0
$101$	−903.783	−0.890393	−0.445197	−	0.895433i	$-0.646866\pi$
$101$	−903.783	−0.890393	−0.445197	+	0.895433i	$0.646866\pi$
$102$	0	0
$103$	−1103.00	−1.05516	−0.527580	−	0.849506i	$-0.676900\pi$
$103$	−1103.00	−1.05516	−0.527580	+	0.849506i	$0.676900\pi$
$104$	0	0
$105$	889.170	0.826420
$106$	0	0
$107$	−516.446	−0.466604	−0.233302	−	0.972404i	$-0.574953\pi$
$107$	−516.446	−0.466604	−0.233302	+	0.972404i	$0.574953\pi$
$108$	0	0
$109$	−1637.68	−1.43909	−0.719546	−	0.694444i	$-0.755650\pi$
$109$	−1637.68	−1.43909	−0.719546	+	0.694444i	$0.755650\pi$
$110$	0	0
$111$	−1638.92	−1.40144
$112$	0	0
$113$	517.541	0.430851	0.215425	−	0.976520i	$-0.430886\pi$
$113$	517.541	0.430851	0.215425	+	0.976520i	$0.430886\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	−1098.38	−0.867908
$118$	0	0
$119$	614.424	0.473312
$120$	0	0
$121$	−1311.11	−0.985055
$122$	0	0
$123$	−2525.60	−1.85143
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1669.82	−1.16672	−0.583358	−	0.812215i	$-0.698261\pi$
$127$	−1669.82	−1.16672	−0.583358	+	0.812215i	$0.698261\pi$
$128$	0	0
$129$	1541.73	1.05226
$130$	0	0
$131$	−1098.53	−0.732663	−0.366331	−	0.930484i	$-0.619386\pi$
$131$	−1098.53	−0.732663	−0.366331	+	0.930484i	$0.619386\pi$
$132$	0	0
$133$	−450.898	−0.293968
$134$	0	0
$135$	−397.131	−0.253182
$136$	0	0
$137$	−974.674	−0.607825	−0.303912	−	0.952700i	$-0.598293\pi$
$137$	−974.674	−0.607825	−0.303912	+	0.952700i	$0.598293\pi$
$138$	0	0
$139$	−1619.77	−0.988395	−0.494198	−	0.869350i	$-0.664538\pi$
$139$	−1619.77	−0.988395	−0.494198	+	0.869350i	$0.664538\pi$
$140$	0	0
$141$	632.838	0.377976
$142$	0	0
$143$	−333.242	−0.194875
$144$	0	0
$145$	−24.6936	−0.0141427
$146$	0	0
$147$	−2682.38	−1.50503
$148$	0	0
$149$	−1915.58	−1.05322	−0.526612	−	0.850106i	$-0.676538\pi$
$149$	−1915.58	−1.05322	−0.526612	+	0.850106i	$0.676538\pi$
$150$	0	0
$151$	2204.10	1.18786	0.593930	−	0.804517i	$-0.297576\pi$
$151$	2204.10	1.18786	0.593930	+	0.804517i	$0.297576\pi$
$152$	0	0
$153$	327.981	0.173305
$154$	0	0
$155$	80.6061	0.0417706
$156$	0	0
$157$	3257.16	1.65573	0.827865	−	0.560928i	$-0.189556\pi$
$157$	3257.16	1.65573	0.827865	+	0.560928i	$0.189556\pi$
$158$	0	0
$159$	1962.07	0.978633
$160$	0	0
$161$	633.392	0.310052
$162$	0	0
$163$	−2793.73	−1.34246	−0.671232	−	0.741247i	$-0.734235\pi$
$163$	−2793.73	−1.34246	−0.671232	+	0.741247i	$0.734235\pi$
$164$	0	0
$165$	144.004	0.0679434
$166$	0	0
$167$	552.871	0.256182	0.128091	−	0.991762i	$-0.459115\pi$
$167$	552.871	0.256182	0.128091	+	0.991762i	$0.459115\pi$
$168$	0	0
$169$	3385.81	1.54111
$170$	0	0
$171$	−240.691	−0.107638
$172$	0	0
$173$	3538.41	1.55503	0.777516	−	0.628863i	$-0.216480\pi$
$173$	3538.41	1.55503	0.777516	+	0.628863i	$0.216480\pi$
$174$	0	0
$175$	688.470	0.297391
$176$	0	0
$177$	−1288.98	−0.547377
$178$	0	0
$179$	−340.748	−0.142283	−0.0711415	−	0.997466i	$-0.522664\pi$
$179$	−340.748	−0.142283	−0.0711415	+	0.997466i	$0.522664\pi$
$180$	0	0
$181$	−1838.91	−0.755166	−0.377583	−	0.925976i	$-0.623245\pi$
$181$	−1838.91	−0.755166	−0.377583	+	0.925976i	$0.623245\pi$
$182$	0	0
$183$	607.538	0.245413
$184$	0	0
$185$	−1268.99	−0.504313
$186$	0	0
$187$	99.5076	0.0389129
$188$	0	0
$189$	2187.30	0.841815
$190$	0	0
$191$	3616.95	1.37023	0.685113	−	0.728437i	$-0.259753\pi$
$191$	3616.95	1.37023	0.685113	+	0.728437i	$0.259753\pi$
$192$	0	0
$193$	−3475.74	−1.29632	−0.648159	−	0.761505i	$-0.724461\pi$
$193$	−3475.74	−1.29632	−0.648159	+	0.761505i	$0.724461\pi$
$194$	0	0
$195$	−2412.49	−0.885960
$196$	0	0
$197$	648.236	0.234441	0.117221	−	0.993106i	$-0.462602\pi$
$197$	648.236	0.234441	0.117221	+	0.993106i	$0.462602\pi$
$198$	0	0
$199$	−4388.32	−1.56322	−0.781608	−	0.623770i	$-0.785600\pi$
$199$	−4388.32	−1.56322	−0.781608	+	0.623770i	$0.785600\pi$
$200$	0	0
$201$	−3169.89	−1.11237
$202$	0	0
$203$	136.007	0.0470236
$204$	0	0
$205$	−1955.53	−0.666245
$206$	0	0
$207$	338.107	0.113527
$208$	0	0
$209$	−73.0242	−0.0241684
$210$	0	0
$211$	−5584.88	−1.82218	−0.911088	−	0.412213i	$-0.864756\pi$
$211$	−5584.88	−1.82218	−0.911088	+	0.412213i	$0.864756\pi$
$212$	0	0
$213$	4393.65	1.41337
$214$	0	0
$215$	1193.74	0.378662
$216$	0	0
$217$	−443.959	−0.138884
$218$	0	0
$219$	−4873.91	−1.50387
$220$	0	0
$221$	−1667.05	−0.507412
$222$	0	0
$223$	−2706.19	−0.812645	−0.406322	−	0.913730i	$-0.633189\pi$
$223$	−2706.19	−0.812645	−0.406322	+	0.913730i	$0.633189\pi$
$224$	0	0
$225$	367.507	0.108891
$226$	0	0
$227$	4203.94	1.22919	0.614593	−	0.788845i	$-0.289320\pi$
$227$	4203.94	1.22919	0.614593	+	0.788845i	$0.289320\pi$
$228$	0	0
$229$	−329.562	−0.0951008	−0.0475504	−	0.998869i	$-0.515141\pi$
$229$	−329.562	−0.0951008	−0.0475504	+	0.998869i	$0.515141\pi$
$230$	0	0
$231$	−793.137	−0.225907
$232$	0	0
$233$	1986.87	0.558645	0.279322	−	0.960197i	$-0.409890\pi$
$233$	1986.87	0.558645	0.279322	+	0.960197i	$0.409890\pi$
$234$	0	0
$235$	489.996	0.136016
$236$	0	0
$237$	1913.37	0.524418
$238$	0	0
$239$	−2469.59	−0.668388	−0.334194	−	0.942504i	$-0.608464\pi$
$239$	−2469.59	−0.668388	−0.334194	+	0.942504i	$0.608464\pi$
$240$	0	0
$241$	−4028.08	−1.07664	−0.538322	−	0.842739i	$-0.680942\pi$
$241$	−4028.08	−1.07664	−0.538322	+	0.842739i	$0.680942\pi$
$242$	0	0
$243$	3730.65	0.984862
$244$	0	0
$245$	−2076.93	−0.541592
$246$	0	0
$247$	1223.37	0.315148
$248$	0	0
$249$	7741.95	1.97039
$250$	0	0
$251$	4973.95	1.25081	0.625405	−	0.780301i	$-0.284934\pi$
$251$	4973.95	1.25081	0.625405	+	0.780301i	$0.284934\pi$
$252$	0	0
$253$	102.580	0.0254906
$254$	0	0
$255$	720.381	0.176910
$256$	0	0
$257$	−2434.13	−0.590805	−0.295402	−	0.955373i	$-0.595454\pi$
$257$	−2434.13	−0.590805	−0.295402	+	0.955373i	$0.595454\pi$
$258$	0	0
$259$	6989.29	1.67681
$260$	0	0
$261$	72.6008	0.0172179
$262$	0	0
$263$	3389.97	0.794808	0.397404	−	0.917644i	$-0.369911\pi$
$263$	3389.97	0.794808	0.397404	+	0.917644i	$0.369911\pi$
$264$	0	0
$265$	1519.20	0.352166
$266$	0	0
$267$	1082.31	0.248075
$268$	0	0
$269$	−2402.71	−0.544594	−0.272297	−	0.962213i	$-0.587783\pi$
$269$	−2402.71	−0.544594	−0.272297	+	0.962213i	$0.587783\pi$
$270$	0	0
$271$	−2178.98	−0.488427	−0.244214	−	0.969721i	$-0.578530\pi$
$271$	−2178.98	−0.488427	−0.244214	+	0.969721i	$0.578530\pi$
$272$	0	0
$273$	13287.4	2.94576
$274$	0	0
$275$	111.500	0.0244498
$276$	0	0
$277$	−6640.08	−1.44030	−0.720151	−	0.693817i	$-0.755928\pi$
$277$	−6640.08	−1.44030	−0.720151	+	0.693817i	$0.755928\pi$
$278$	0	0
$279$	−236.987	−0.0508531
$280$	0	0
$281$	−6748.28	−1.43263	−0.716315	−	0.697778i	$-0.754172\pi$
$281$	−6748.28	−1.43263	−0.716315	+	0.697778i	$0.754172\pi$
$282$	0	0
$283$	−2847.47	−0.598109	−0.299054	−	0.954236i	$-0.596671\pi$
$283$	−2847.47	−0.598109	−0.299054	+	0.954236i	$0.596671\pi$
$284$	0	0
$285$	−528.655	−0.109877
$286$	0	0
$287$	10770.6	2.21522
$288$	0	0
$289$	−4415.21	−0.898679
$290$	0	0
$291$	−4214.81	−0.849061
$292$	0	0
$293$	2272.85	0.453179	0.226590	−	0.973990i	$-0.427242\pi$
$293$	2272.85	0.453179	0.226590	+	0.973990i	$0.427242\pi$
$294$	0	0
$295$	−998.038	−0.196976
$296$	0	0
$297$	354.240	0.0692091
$298$	0	0
$299$	−1718.52	−0.332390
$300$	0	0
$301$	−6574.84	−1.25903
$302$	0	0
$303$	5836.25	1.10655
$304$	0	0
$305$	470.407	0.0883129
$306$	0	0
$307$	1202.79	0.223605	0.111803	−	0.993730i	$-0.464338\pi$
$307$	1202.79	0.223605	0.111803	+	0.993730i	$0.464338\pi$
$308$	0	0
$309$	7122.68	1.31131
$310$	0	0
$311$	1727.38	0.314954	0.157477	−	0.987523i	$-0.449664\pi$
$311$	1727.38	0.314954	0.157477	+	0.987523i	$0.449664\pi$
$312$	0	0
$313$	1373.42	0.248020	0.124010	−	0.992281i	$-0.460425\pi$
$313$	1373.42	0.248020	0.124010	+	0.992281i	$0.460425\pi$
$314$	0	0
$315$	−2024.14	−0.362056
$316$	0	0
$317$	312.847	0.0554299	0.0277149	−	0.999616i	$-0.491177\pi$
$317$	312.847	0.0554299	0.0277149	+	0.999616i	$0.491177\pi$
$318$	0	0
$319$	22.0267	0.00386601
$320$	0	0
$321$	3334.99	0.579878
$322$	0	0
$323$	−365.305	−0.0629292
$324$	0	0
$325$	−1867.96	−0.318817
$326$	0	0
$327$	10575.4	1.78845
$328$	0	0
$329$	−2698.78	−0.452245
$330$	0	0
$331$	−3274.83	−0.543809	−0.271904	−	0.962324i	$-0.587653\pi$
$331$	−3274.83	−0.543809	−0.271904	+	0.962324i	$0.587653\pi$
$332$	0	0
$333$	3730.90	0.613971
$334$	0	0
$335$	−2454.39	−0.400292
$336$	0	0
$337$	5346.19	0.864171	0.432086	−	0.901833i	$-0.357778\pi$
$337$	5346.19	0.864171	0.432086	+	0.901833i	$0.357778\pi$
$338$	0	0
$339$	−3342.06	−0.535445
$340$	0	0
$341$	−71.9004	−0.0114183
$342$	0	0
$343$	1993.41	0.313802
$344$	0	0
$345$	742.621	0.115888
$346$	0	0
$347$	−1912.73	−0.295910	−0.147955	−	0.988994i	$-0.547269\pi$
$347$	−1912.73	−0.295910	−0.147955	+	0.988994i	$0.547269\pi$
$348$	0	0
$349$	−606.341	−0.0929992	−0.0464996	−	0.998918i	$-0.514807\pi$
$349$	−606.341	−0.0929992	−0.0464996	+	0.998918i	$0.514807\pi$
$350$	0	0
$351$	−5934.59	−0.902464
$352$	0	0
$353$	8234.32	1.24155	0.620777	−	0.783987i	$-0.286817\pi$
$353$	8234.32	1.24155	0.620777	+	0.783987i	$0.286817\pi$
$354$	0	0
$355$	3401.93	0.508608
$356$	0	0
$357$	−3967.69	−0.588214
$358$	0	0
$359$	2095.73	0.308101	0.154051	−	0.988063i	$-0.450768\pi$
$359$	2095.73	0.308101	0.154051	+	0.988063i	$0.450768\pi$
$360$	0	0
$361$	−6590.92	−0.960915
$362$	0	0
$363$	8466.58	1.22419
$364$	0	0
$365$	−3773.79	−0.541176
$366$	0	0
$367$	−448.222	−0.0637520	−0.0318760	−	0.999492i	$-0.510148\pi$
$367$	−448.222	−0.0637520	−0.0318760	+	0.999492i	$0.510148\pi$
$368$	0	0
$369$	5749.38	0.811113
$370$	0	0
$371$	−8367.41	−1.17093
$372$	0	0
$373$	6893.22	0.956883	0.478442	−	0.878119i	$-0.341202\pi$
$373$	6893.22	0.956883	0.478442	+	0.878119i	$0.341202\pi$
$374$	0	0
$375$	807.197	0.111156
$376$	0	0
$377$	−369.013	−0.0504115
$378$	0	0
$379$	−10925.2	−1.48072	−0.740358	−	0.672212i	$-0.765344\pi$
$379$	−10925.2	−1.48072	−0.740358	+	0.672212i	$0.765344\pi$
$380$	0	0
$381$	10783.0	1.44995
$382$	0	0
$383$	−2466.28	−0.329036	−0.164518	−	0.986374i	$-0.552607\pi$
$383$	−2466.28	−0.329036	−0.164518	+	0.986374i	$0.552607\pi$
$384$	0	0
$385$	−614.113	−0.0812938
$386$	0	0
$387$	−3509.67	−0.460998
$388$	0	0
$389$	512.054	0.0667408	0.0333704	−	0.999443i	$-0.489376\pi$
$389$	512.054	0.0667408	0.0333704	+	0.999443i	$0.489376\pi$
$390$	0	0
$391$	513.158	0.0663721
$392$	0	0
$393$	7093.83	0.910525
$394$	0	0
$395$	1481.49	0.188714
$396$	0	0
$397$	−663.014	−0.0838179	−0.0419090	−	0.999121i	$-0.513344\pi$
$397$	−663.014	−0.0838179	−0.0419090	+	0.999121i	$0.513344\pi$
$398$	0	0
$399$	2911.71	0.365332
$400$	0	0
$401$	10986.7	1.36821	0.684103	−	0.729385i	$-0.260194\pi$
$401$	10986.7	1.36821	0.684103	+	0.729385i	$0.260194\pi$
$402$	0	0
$403$	1204.55	0.148890
$404$	0	0
$405$	4549.05	0.558133
$406$	0	0
$407$	1131.94	0.137857
$408$	0	0
$409$	12642.6	1.52845	0.764224	−	0.644951i	$-0.223122\pi$
$409$	12642.6	1.52845	0.764224	+	0.644951i	$0.223122\pi$
$410$	0	0
$411$	6294.03	0.755381
$412$	0	0
$413$	5496.96	0.654933
$414$	0	0
$415$	5994.47	0.709053
$416$	0	0
$417$	10459.8	1.22834
$418$	0	0
$419$	−3303.90	−0.385218	−0.192609	−	0.981276i	$-0.561695\pi$
$419$	−3303.90	−0.385218	−0.192609	+	0.981276i	$0.561695\pi$
$420$	0	0
$421$	439.444	0.0508722	0.0254361	−	0.999676i	$-0.491903\pi$
$421$	439.444	0.0508722	0.0254361	+	0.999676i	$0.491903\pi$
$422$	0	0
$423$	−1440.62	−0.165592
$424$	0	0
$425$	557.780	0.0636619
$426$	0	0
$427$	−2590.89	−0.293635
$428$	0	0
$429$	2151.94	0.242183
$430$	0	0
$431$	−11905.9	−1.33059	−0.665295	−	0.746580i	$-0.731694\pi$
$431$	−11905.9	−1.33059	−0.665295	+	0.746580i	$0.731694\pi$
$432$	0	0
$433$	6273.22	0.696239	0.348120	−	0.937450i	$-0.386820\pi$
$433$	6273.22	0.696239	0.348120	+	0.937450i	$0.386820\pi$
$434$	0	0
$435$	159.461	0.0175760
$436$	0	0
$437$	−376.583	−0.0412229
$438$	0	0
$439$	−14126.3	−1.53579	−0.767893	−	0.640578i	$-0.778695\pi$
$439$	−14126.3	−1.53579	−0.767893	+	0.640578i	$0.778695\pi$
$440$	0	0
$441$	6106.29	0.659355
$442$	0	0
$443$	−10377.6	−1.11299	−0.556495	−	0.830851i	$-0.687854\pi$
$443$	−10377.6	−1.11299	−0.556495	+	0.830851i	$0.687854\pi$
$444$	0	0
$445$	838.013	0.0892711
$446$	0	0
$447$	12370.0	1.30891
$448$	0	0
$449$	−12308.3	−1.29368	−0.646841	−	0.762625i	$-0.723910\pi$
$449$	−12308.3	−1.29368	−0.646841	+	0.762625i	$0.723910\pi$
$450$	0	0
$451$	1744.33	0.182122
$452$	0	0
$453$	−14233.1	−1.47623
$454$	0	0
$455$	10288.3	1.06005
$456$	0	0
$457$	−6142.92	−0.628783	−0.314391	−	0.949293i	$-0.601800\pi$
$457$	−6142.92	−0.628783	−0.314391	+	0.949293i	$0.601800\pi$
$458$	0	0
$459$	1772.10	0.180206
$460$	0	0
$461$	7547.94	0.762565	0.381283	−	0.924459i	$-0.375482\pi$
$461$	7547.94	0.762565	0.381283	+	0.924459i	$0.375482\pi$
$462$	0	0
$463$	502.260	0.0504147	0.0252074	−	0.999682i	$-0.491975\pi$
$463$	502.260	0.0504147	0.0252074	+	0.999682i	$0.491975\pi$
$464$	0	0
$465$	−520.520	−0.0519108
$466$	0	0
$467$	−6993.42	−0.692970	−0.346485	−	0.938055i	$-0.612625\pi$
$467$	−6993.42	−0.692970	−0.346485	+	0.938055i	$0.612625\pi$
$468$	0	0
$469$	13518.2	1.33094
$470$	0	0
$471$	−21033.3	−2.05768
$472$	0	0
$473$	−1064.81	−0.103510
$474$	0	0
$475$	−409.329	−0.0395396
$476$	0	0
$477$	−4466.55	−0.428740
$478$	0	0
$479$	−13908.7	−1.32673	−0.663366	−	0.748295i	$-0.730873\pi$
$479$	−13908.7	−1.32673	−0.663366	+	0.748295i	$0.730873\pi$
$480$	0	0
$481$	−18963.3	−1.79762
$482$	0	0
$483$	−4090.18	−0.385320
$484$	0	0
$485$	−3263.46	−0.305539
$486$	0	0
$487$	−1322.34	−0.123041	−0.0615203	−	0.998106i	$-0.519595\pi$
$487$	−1322.34	−0.123041	−0.0615203	+	0.998106i	$0.519595\pi$
$488$	0	0
$489$	18040.7	1.66836
$490$	0	0
$491$	15302.9	1.40654	0.703271	−	0.710922i	$-0.251722\pi$
$491$	15302.9	1.40654	0.703271	+	0.710922i	$0.251722\pi$
$492$	0	0
$493$	110.189	0.0100663
$494$	0	0
$495$	−327.816	−0.0297661
$496$	0	0
$497$	−18737.0	−1.69109
$498$	0	0
$499$	−4356.18	−0.390801	−0.195400	−	0.980724i	$-0.562601\pi$
$499$	−4356.18	−0.390801	−0.195400	+	0.980724i	$0.562601\pi$
$500$	0	0
$501$	−3570.21	−0.318374
$502$	0	0
$503$	16979.2	1.50510	0.752548	−	0.658537i	$-0.228824\pi$
$503$	16979.2	1.50510	0.752548	+	0.658537i	$0.228824\pi$
$504$	0	0
$505$	4518.91	0.398196
$506$	0	0
$507$	−21864.1	−1.91523
$508$	0	0
$509$	11226.9	0.977651	0.488825	−	0.872382i	$-0.337426\pi$
$509$	11226.9	0.977651	0.488825	+	0.872382i	$0.337426\pi$
$510$	0	0
$511$	20785.1	1.79938
$512$	0	0
$513$	−1300.46	−0.111923
$514$	0	0
$515$	5514.98	0.471882
$516$	0	0
$517$	−437.075	−0.0371809
$518$	0	0
$519$	−22849.6	−1.93253
$520$	0	0
$521$	1363.54	0.114660	0.0573298	−	0.998355i	$-0.481741\pi$
$521$	1363.54	0.114660	0.0573298	+	0.998355i	$0.481741\pi$
$522$	0	0
$523$	6005.47	0.502105	0.251053	−	0.967973i	$-0.419223\pi$
$523$	6005.47	0.502105	0.251053	+	0.967973i	$0.419223\pi$
$524$	0	0
$525$	−4445.85	−0.369586
$526$	0	0
$527$	−359.684	−0.0297307
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	2934.29	0.239807
$532$	0	0
$533$	−29222.8	−2.37482
$534$	0	0
$535$	2582.23	0.208672
$536$	0	0
$537$	2200.40	0.176824
$538$	0	0
$539$	1852.61	0.148048
$540$	0	0
$541$	8091.00	0.642993	0.321497	−	0.946911i	$-0.395814\pi$
$541$	8091.00	0.642993	0.321497	+	0.946911i	$0.395814\pi$
$542$	0	0
$543$	11874.9	0.938491
$544$	0	0
$545$	8188.39	0.643582
$546$	0	0
$547$	12121.2	0.947472	0.473736	−	0.880667i	$-0.342905\pi$
$547$	12121.2	0.947472	0.473736	+	0.880667i	$0.342905\pi$
$548$	0	0
$549$	−1383.03	−0.107516
$550$	0	0
$551$	−80.8627	−0.00625203
$552$	0	0
$553$	−8159.72	−0.627462
$554$	0	0
$555$	8194.60	0.626741
$556$	0	0
$557$	8951.71	0.680963	0.340481	−	0.940251i	$-0.389410\pi$
$557$	8951.71	0.680963	0.340481	+	0.940251i	$0.389410\pi$
$558$	0	0
$559$	17838.8	1.34974
$560$	0	0
$561$	−642.578	−0.0483595
$562$	0	0
$563$	15120.3	1.13187	0.565935	−	0.824450i	$-0.308515\pi$
$563$	15120.3	1.13187	0.565935	+	0.824450i	$0.308515\pi$
$564$	0	0
$565$	−2587.70	−0.192682
$566$	0	0
$567$	−25055.1	−1.85576
$568$	0	0
$569$	−11297.7	−0.832376	−0.416188	−	0.909278i	$-0.636634\pi$
$569$	−11297.7	−0.832376	−0.416188	+	0.909278i	$0.636634\pi$
$570$	0	0
$571$	−2604.16	−0.190860	−0.0954298	−	0.995436i	$-0.530423\pi$
$571$	−2604.16	−0.190860	−0.0954298	+	0.995436i	$0.530423\pi$
$572$	0	0
$573$	−23356.7	−1.70286
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	−15787.0	−1.13903	−0.569517	−	0.821979i	$-0.692870\pi$
$577$	−15787.0	−1.13903	−0.569517	+	0.821979i	$0.692870\pi$
$578$	0	0
$579$	22444.9	1.61101
$580$	0	0
$581$	−33016.1	−2.35755
$582$	0	0
$583$	−1355.13	−0.0962668
$584$	0	0
$585$	5491.90	0.388140
$586$	0	0
$587$	−23689.0	−1.66567	−0.832837	−	0.553519i	$-0.813285\pi$
$587$	−23689.0	−1.66567	−0.832837	+	0.553519i	$0.813285\pi$
$588$	0	0
$589$	263.956	0.0184654
$590$	0	0
$591$	−4186.03	−0.291354
$592$	0	0
$593$	23044.0	1.59579	0.797896	−	0.602795i	$-0.205946\pi$
$593$	23044.0	1.59579	0.797896	+	0.602795i	$0.205946\pi$
$594$	0	0
$595$	−3072.12	−0.211672
$596$	0	0
$597$	28337.9	1.94270
$598$	0	0
$599$	−8536.57	−0.582295	−0.291148	−	0.956678i	$-0.594037\pi$
$599$	−8536.57	−0.582295	−0.291148	+	0.956678i	$0.594037\pi$
$600$	0	0
$601$	18586.8	1.26151	0.630757	−	0.775981i	$-0.282745\pi$
$601$	18586.8	1.26151	0.630757	+	0.775981i	$0.282745\pi$
$602$	0	0
$603$	7216.06	0.487331
$604$	0	0
$605$	6555.54	0.440530
$606$	0	0
$607$	−17511.5	−1.17095	−0.585477	−	0.810689i	$-0.699093\pi$
$607$	−17511.5	−1.17095	−0.585477	+	0.810689i	$0.699093\pi$
$608$	0	0
$609$	−878.274	−0.0584392
$610$	0	0
$611$	7322.33	0.484828
$612$	0	0
$613$	11573.8	0.762580	0.381290	−	0.924456i	$-0.375480\pi$
$613$	11573.8	0.762580	0.381290	+	0.924456i	$0.375480\pi$
$614$	0	0
$615$	12628.0	0.827983
$616$	0	0
$617$	14114.6	0.920960	0.460480	−	0.887670i	$-0.347677\pi$
$617$	14114.6	0.920960	0.460480	+	0.887670i	$0.347677\pi$
$618$	0	0
$619$	22877.5	1.48550	0.742749	−	0.669570i	$-0.233521\pi$
$619$	22877.5	1.48550	0.742749	+	0.669570i	$0.233521\pi$
$620$	0	0
$621$	1826.80	0.118047
$622$	0	0
$623$	−4615.58	−0.296821
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	471.559	0.0300355
$628$	0	0
$629$	5662.54	0.358951
$630$	0	0
$631$	464.046	0.0292764	0.0146382	−	0.999893i	$-0.495340\pi$
$631$	464.046	0.0292764	0.0146382	+	0.999893i	$0.495340\pi$
$632$	0	0
$633$	36064.8	2.26453
$634$	0	0
$635$	8349.12	0.521771
$636$	0	0
$637$	−31036.9	−1.93050
$638$	0	0
$639$	−10001.9	−0.619199
$640$	0	0
$641$	22469.7	1.38456	0.692278	−	0.721631i	$-0.256607\pi$
$641$	22469.7	1.38456	0.692278	+	0.721631i	$0.256607\pi$
$642$	0	0
$643$	20710.5	1.27020	0.635102	−	0.772428i	$-0.280958\pi$
$643$	20710.5	1.27020	0.635102	+	0.772428i	$0.280958\pi$
$644$	0	0
$645$	−7708.67	−0.470587
$646$	0	0
$647$	2664.24	0.161889	0.0809445	−	0.996719i	$-0.474206\pi$
$647$	2664.24	0.161889	0.0809445	+	0.996719i	$0.474206\pi$
$648$	0	0
$649$	890.247	0.0538448
$650$	0	0
$651$	2866.90	0.172600
$652$	0	0
$653$	22096.7	1.32421	0.662107	−	0.749410i	$-0.269663\pi$
$653$	22096.7	1.32421	0.662107	+	0.749410i	$0.269663\pi$
$654$	0	0
$655$	5492.64	0.327657
$656$	0	0
$657$	11095.2	0.658849
$658$	0	0
$659$	−10465.6	−0.618635	−0.309318	−	0.950959i	$-0.600101\pi$
$659$	−10465.6	−0.618635	−0.309318	+	0.950959i	$0.600101\pi$
$660$	0	0
$661$	−26205.0	−1.54199	−0.770994	−	0.636842i	$-0.780240\pi$
$661$	−26205.0	−1.54199	−0.770994	+	0.636842i	$0.780240\pi$
$662$	0	0
$663$	10765.1	0.630592
$664$	0	0
$665$	2254.49	0.131467
$666$	0	0
$667$	113.591	0.00659408
$668$	0	0
$669$	17475.4	1.00992
$670$	0	0
$671$	−419.602	−0.0241409
$672$	0	0
$673$	−8840.15	−0.506334	−0.253167	−	0.967423i	$-0.581472\pi$
$673$	−8840.15	−0.506334	−0.253167	+	0.967423i	$0.581472\pi$
$674$	0	0
$675$	1985.66	0.113227
$676$	0	0
$677$	−24164.9	−1.37183	−0.685917	−	0.727680i	$-0.740599\pi$
$677$	−24164.9	−1.37183	−0.685917	+	0.727680i	$0.740599\pi$
$678$	0	0
$679$	17974.4	1.01590
$680$	0	0
$681$	−27147.2	−1.52758
$682$	0	0
$683$	−17967.5	−1.00660	−0.503300	−	0.864112i	$-0.667881\pi$
$683$	−17967.5	−1.00660	−0.503300	+	0.864112i	$0.667881\pi$
$684$	0	0
$685$	4873.37	0.271828
$686$	0	0
$687$	2128.17	0.118188
$688$	0	0
$689$	22702.4	1.25529
$690$	0	0
$691$	−24393.2	−1.34292	−0.671461	−	0.741040i	$-0.734333\pi$
$691$	−24393.2	−1.34292	−0.671461	+	0.741040i	$0.734333\pi$
$692$	0	0
$693$	1805.53	0.0989703
$694$	0	0
$695$	8098.84	0.442024
$696$	0	0
$697$	8726.05	0.474207
$698$	0	0
$699$	−12830.4	−0.694262
$700$	0	0
$701$	3868.54	0.208435	0.104217	−	0.994555i	$-0.466766\pi$
$701$	3868.54	0.208435	0.104217	+	0.994555i	$0.466766\pi$
$702$	0	0
$703$	−4155.48	−0.222940
$704$	0	0
$705$	−3164.19	−0.169036
$706$	0	0
$707$	−24889.1	−1.32398
$708$	0	0
$709$	−36236.1	−1.91943	−0.959715	−	0.280975i	$-0.909342\pi$
$709$	−36236.1	−1.91943	−0.959715	+	0.280975i	$0.909342\pi$
$710$	0	0
$711$	−4355.68	−0.229748
$712$	0	0
$713$	−370.788	−0.0194756
$714$	0	0
$715$	1666.21	0.0871507
$716$	0	0
$717$	15947.6	0.830646
$718$	0	0
$719$	−11442.3	−0.593498	−0.296749	−	0.954955i	$-0.595902\pi$
$719$	−11442.3	−0.593498	−0.296749	+	0.954955i	$0.595902\pi$
$720$	0	0
$721$	−30375.2	−1.56897
$722$	0	0
$723$	26011.6	1.33801
$724$	0	0
$725$	123.468	0.00632482
$726$	0	0
$727$	29014.1	1.48015	0.740077	−	0.672522i	$-0.234789\pi$
$727$	29014.1	1.48015	0.740077	+	0.672522i	$0.234789\pi$
$728$	0	0
$729$	473.871	0.0240752
$730$	0	0
$731$	−5326.76	−0.269517
$732$	0	0
$733$	23849.1	1.20176	0.600878	−	0.799341i	$-0.294818\pi$
$733$	23849.1	1.20176	0.600878	+	0.799341i	$0.294818\pi$
$734$	0	0
$735$	13411.9	0.673070
$736$	0	0
$737$	2189.31	0.109422
$738$	0	0
$739$	25041.9	1.24653	0.623263	−	0.782012i	$-0.285807\pi$
$739$	25041.9	1.24653	0.623263	+	0.782012i	$0.285807\pi$
$740$	0	0
$741$	−7900.04	−0.391653
$742$	0	0
$743$	23499.4	1.16031	0.580153	−	0.814507i	$-0.302993\pi$
$743$	23499.4	1.16031	0.580153	+	0.814507i	$0.302993\pi$
$744$	0	0
$745$	9577.90	0.471016
$746$	0	0
$747$	−17624.1	−0.863229
$748$	0	0
$749$	−14222.3	−0.693820
$750$	0	0
$751$	−12264.7	−0.595931	−0.297966	−	0.954577i	$-0.596308\pi$
$751$	−12264.7	−0.595931	−0.297966	+	0.954577i	$0.596308\pi$
$752$	0	0
$753$	−32119.7	−1.55446
$754$	0	0
$755$	−11020.5	−0.531227
$756$	0	0
$757$	−10449.3	−0.501701	−0.250850	−	0.968026i	$-0.580710\pi$
$757$	−10449.3	−0.501701	−0.250850	+	0.968026i	$0.580710\pi$
$758$	0	0
$759$	−662.416	−0.0316788
$760$	0	0
$761$	15896.0	0.757201	0.378601	−	0.925560i	$-0.376405\pi$
$761$	15896.0	0.757201	0.378601	+	0.925560i	$0.376405\pi$
$762$	0	0
$763$	−45099.7	−2.13987
$764$	0	0
$765$	−1639.91	−0.0775045
$766$	0	0
$767$	−14914.3	−0.702119
$768$	0	0
$769$	−39682.8	−1.86085	−0.930427	−	0.366476i	$-0.880564\pi$
$769$	−39682.8	−1.86085	−0.930427	+	0.366476i	$0.880564\pi$
$770$	0	0
$771$	15718.6	0.734229
$772$	0	0
$773$	−30184.0	−1.40445	−0.702226	−	0.711954i	$-0.747810\pi$
$773$	−30184.0	−1.40445	−0.702226	+	0.711954i	$0.747810\pi$
$774$	0	0
$775$	−403.030	−0.0186804
$776$	0	0
$777$	−45133.9	−2.08387
$778$	0	0
$779$	−6403.65	−0.294525
$780$	0	0
$781$	−3034.51	−0.139031
$782$	0	0
$783$	392.265	0.0179035
$784$	0	0
$785$	−16285.8	−0.740465
$786$	0	0
$787$	−11449.7	−0.518598	−0.259299	−	0.965797i	$-0.583492\pi$
$787$	−11449.7	−0.518598	−0.259299	+	0.965797i	$0.583492\pi$
$788$	0	0
$789$	−21891.0	−0.987756
$790$	0	0
$791$	14252.5	0.640656
$792$	0	0
$793$	7029.60	0.314790
$794$	0	0
$795$	−9810.37	−0.437658
$796$	0	0
$797$	−19158.0	−0.851455	−0.425727	−	0.904851i	$-0.639982\pi$
$797$	−19158.0	−0.851455	−0.425727	+	0.904851i	$0.639982\pi$
$798$	0	0
$799$	−2186.48	−0.0968112
$800$	0	0
$801$	−2463.81	−0.108682
$802$	0	0
$803$	3366.21	0.147934
$804$	0	0
$805$	−3166.96	−0.138659
$806$	0	0
$807$	15515.7	0.676800
$808$	0	0
$809$	6143.12	0.266972	0.133486	−	0.991051i	$-0.457383\pi$
$809$	6143.12	0.266972	0.133486	+	0.991051i	$0.457383\pi$
$810$	0	0
$811$	35089.4	1.51930	0.759651	−	0.650331i	$-0.225370\pi$
$811$	35089.4	1.51930	0.759651	+	0.650331i	$0.225370\pi$
$812$	0	0
$813$	14070.9	0.606998
$814$	0	0
$815$	13968.6	0.600368
$816$	0	0
$817$	3909.06	0.167394
$818$	0	0
$819$	−30248.1	−1.29054
$820$	0	0
$821$	−23493.3	−0.998685	−0.499342	−	0.866405i	$-0.666425\pi$
$821$	−23493.3	−0.998685	−0.499342	+	0.866405i	$0.666425\pi$
$822$	0	0
$823$	−44309.1	−1.87669	−0.938346	−	0.345698i	$-0.887642\pi$
$823$	−44309.1	−1.87669	−0.938346	+	0.345698i	$0.887642\pi$
$824$	0	0
$825$	−720.018	−0.0303852
$826$	0	0
$827$	−6489.29	−0.272860	−0.136430	−	0.990650i	$-0.543563\pi$
$827$	−6489.29	−0.272860	−0.136430	+	0.990650i	$0.543563\pi$
$828$	0	0
$829$	3792.65	0.158895	0.0794477	−	0.996839i	$-0.474684\pi$
$829$	3792.65	0.158895	0.0794477	+	0.996839i	$0.474684\pi$
$830$	0	0
$831$	42878.8	1.78995
$832$	0	0
$833$	9267.75	0.385484
$834$	0	0
$835$	−2764.36	−0.114568
$836$	0	0
$837$	−1280.45	−0.0528779
$838$	0	0
$839$	−35100.0	−1.44432	−0.722160	−	0.691726i	$-0.756851\pi$
$839$	−35100.0	−1.44432	−0.722160	+	0.691726i	$0.756851\pi$
$840$	0	0
$841$	−24364.6	−0.999000
$842$	0	0
$843$	43577.6	1.78042
$844$	0	0
$845$	−16929.1	−0.689204
$846$	0	0
$847$	−36106.4	−1.46473
$848$	0	0
$849$	18387.8	0.743306
$850$	0	0
$851$	5837.35	0.235137
$852$	0	0
$853$	−11185.3	−0.448976	−0.224488	−	0.974477i	$-0.572071\pi$
$853$	−11185.3	−0.448976	−0.224488	+	0.974477i	$0.572071\pi$
$854$	0	0
$855$	1203.45	0.0481371
$856$	0	0
$857$	−32830.4	−1.30859	−0.654297	−	0.756238i	$-0.727035\pi$
$857$	−32830.4	−1.30859	−0.654297	+	0.756238i	$0.727035\pi$
$858$	0	0
$859$	26751.1	1.06256	0.531278	−	0.847197i	$-0.321712\pi$
$859$	26751.1	1.06256	0.531278	+	0.847197i	$0.321712\pi$
$860$	0	0
$861$	−69551.9	−2.75299
$862$	0	0
$863$	7026.27	0.277146	0.138573	−	0.990352i	$-0.455748\pi$
$863$	7026.27	0.277146	0.138573	+	0.990352i	$0.455748\pi$
$864$	0	0
$865$	−17692.1	−0.695431
$866$	0	0
$867$	28511.6	1.11684
$868$	0	0
$869$	−1321.49	−0.0515863
$870$	0	0
$871$	−36677.6	−1.42683
$872$	0	0
$873$	9594.78	0.371975
$874$	0	0
$875$	−3442.35	−0.132997
$876$	0	0
$877$	21465.4	0.826492	0.413246	−	0.910619i	$-0.364395\pi$
$877$	21465.4	0.826492	0.413246	+	0.910619i	$0.364395\pi$
$878$	0	0
$879$	−14677.1	−0.563194
$880$	0	0
$881$	42173.6	1.61279	0.806393	−	0.591381i	$-0.201417\pi$
$881$	42173.6	1.61279	0.806393	+	0.591381i	$0.201417\pi$
$882$	0	0
$883$	−17092.2	−0.651415	−0.325708	−	0.945471i	$-0.605603\pi$
$883$	−17092.2	−0.651415	−0.325708	+	0.945471i	$0.605603\pi$
$884$	0	0
$885$	6444.91	0.244795
$886$	0	0
$887$	17105.6	0.647521	0.323760	−	0.946139i	$-0.395053\pi$
$887$	17105.6	0.647521	0.323760	+	0.946139i	$0.395053\pi$
$888$	0	0
$889$	−45985.0	−1.73485
$890$	0	0
$891$	−4057.74	−0.152569
$892$	0	0
$893$	1604.56	0.0601283
$894$	0	0
$895$	1703.74	0.0636309
$896$	0	0
$897$	11097.5	0.413081
$898$	0	0
$899$	−79.6183	−0.00295375
$900$	0	0
$901$	−6779.05	−0.250658
$902$	0	0
$903$	42457.5	1.56467
$904$	0	0
$905$	9194.55	0.337721
$906$	0	0
$907$	−24336.5	−0.890938	−0.445469	−	0.895297i	$-0.646963\pi$
$907$	−24336.5	−0.890938	−0.445469	+	0.895297i	$0.646963\pi$
$908$	0	0
$909$	−13285.9	−0.484779
$910$	0	0
$911$	14007.4	0.509424	0.254712	−	0.967017i	$-0.418019\pi$
$911$	14007.4	0.509424	0.254712	+	0.967017i	$0.418019\pi$
$912$	0	0
$913$	−5347.05	−0.193824
$914$	0	0
$915$	−3037.69	−0.109752
$916$	0	0
$917$	−30252.1	−1.08944
$918$	0	0
$919$	−27299.3	−0.979894	−0.489947	−	0.871752i	$-0.662984\pi$
$919$	−27299.3	−0.979894	−0.489947	+	0.871752i	$0.662984\pi$
$920$	0	0
$921$	−7767.10	−0.277888
$922$	0	0
$923$	50837.2	1.81292
$924$	0	0
$925$	6344.95	0.225536
$926$	0	0
$927$	−16214.4	−0.574487
$928$	0	0
$929$	13103.0	0.462752	0.231376	−	0.972864i	$-0.425677\pi$
$929$	13103.0	0.462752	0.231376	+	0.972864i	$0.425677\pi$
$930$	0	0
$931$	−6801.18	−0.239420
$932$	0	0
$933$	−11154.7	−0.391412
$934$	0	0
$935$	−497.538	−0.0174024
$936$	0	0
$937$	−29281.2	−1.02089	−0.510445	−	0.859910i	$-0.670519\pi$
$937$	−29281.2	−1.02089	−0.510445	+	0.859910i	$0.670519\pi$
$938$	0	0
$939$	−8868.96	−0.308230
$940$	0	0
$941$	32885.4	1.13925	0.569624	−	0.821906i	$-0.307089\pi$
$941$	32885.4	1.13925	0.569624	+	0.821906i	$0.307089\pi$
$942$	0	0
$943$	8995.44	0.310638
$944$	0	0
$945$	−10936.5	−0.376471
$946$	0	0
$947$	−18348.1	−0.629601	−0.314801	−	0.949158i	$-0.601938\pi$
$947$	−18348.1	−0.629601	−0.314801	+	0.949158i	$0.601938\pi$
$948$	0	0
$949$	−56394.2	−1.92901
$950$	0	0
$951$	−2020.24	−0.0688861
$952$	0	0
$953$	14297.7	0.485990	0.242995	−	0.970028i	$-0.421870\pi$
$953$	14297.7	0.485990	0.242995	+	0.970028i	$0.421870\pi$
$954$	0	0
$955$	−18084.7	−0.612784
$956$	0	0
$957$	−142.239	−0.00480453
$958$	0	0
$959$	−26841.4	−0.903809
$960$	0	0
$961$	−29531.1	−0.991276
$962$	0	0
$963$	−7591.90	−0.254045
$964$	0	0
$965$	17378.7	0.579731
$966$	0	0
$967$	8790.92	0.292344	0.146172	−	0.989259i	$-0.453305\pi$
$967$	8790.92	0.292344	0.146172	+	0.989259i	$0.453305\pi$
$968$	0	0
$969$	2358.99	0.0782060
$970$	0	0
$971$	36908.3	1.21982	0.609910	−	0.792471i	$-0.291206\pi$
$971$	36908.3	1.21982	0.609910	+	0.792471i	$0.291206\pi$
$972$	0	0
$973$	−44606.5	−1.46970
$974$	0	0
$975$	12062.5	0.396213
$976$	0	0
$977$	−30231.0	−0.989943	−0.494972	−	0.868909i	$-0.664822\pi$
$977$	−30231.0	−0.989943	−0.494972	+	0.868909i	$0.664822\pi$
$978$	0	0
$979$	−747.506	−0.0244028
$980$	0	0
$981$	−24074.3	−0.783522
$982$	0	0
$983$	42303.4	1.37260	0.686302	−	0.727317i	$-0.259233\pi$
$983$	42303.4	1.37260	0.686302	+	0.727317i	$0.259233\pi$
$984$	0	0
$985$	−3241.18	−0.104845
$986$	0	0
$987$	17427.6	0.562033
$988$	0	0
$989$	−5491.21	−0.176552
$990$	0	0
$991$	−11944.1	−0.382862	−0.191431	−	0.981506i	$-0.561313\pi$
$991$	−11944.1	−0.382862	−0.191431	+	0.981506i	$0.561313\pi$
$992$	0	0
$993$	21147.4	0.675825
$994$	0	0
$995$	21941.6	0.699092
$996$	0	0
$997$	−24292.3	−0.771659	−0.385830	−	0.922570i	$-0.626085\pi$
$997$	−24292.3	−0.771659	−0.385830	+	0.922570i	$0.626085\pi$
$998$	0	0
$999$	20158.2	0.638416

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.4.a.d.1.2	✓	8
4.3	odd	2		1840.4.a.w.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.d.1.2	✓	8	1.1	even	1	trivial
1840.4.a.w.1.7		8	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	920.a (trivial)

\(f(q)\)	\(=\)	\(q-6.45758 q^{3} -5.00000 q^{5} +27.5388 q^{7} +14.7003 q^{9} +O(q^{10})\)	\(q-6.45758 q^{3} -5.00000 q^{5} +27.5388 q^{7} +14.7003 q^{9} +4.45999 q^{11} -74.7182 q^{13} +32.2879 q^{15} +22.3112 q^{17} -16.3732 q^{19} -177.834 q^{21} +23.0000 q^{23} +25.0000 q^{25} +79.4263 q^{27} +4.93873 q^{29} -16.1212 q^{31} -28.8007 q^{33} -137.694 q^{35} +253.798 q^{37} +482.499 q^{39} +391.106 q^{41} -238.748 q^{43} -73.5015 q^{45} -97.9993 q^{47} +415.386 q^{49} -144.076 q^{51} -303.841 q^{53} -22.2999 q^{55} +105.731 q^{57} +199.608 q^{59} -94.0815 q^{61} +404.829 q^{63} +373.591 q^{65} +490.878 q^{67} -148.524 q^{69} -680.386 q^{71} +754.759 q^{73} -161.439 q^{75} +122.823 q^{77} -296.299 q^{79} -909.809 q^{81} -1198.89 q^{83} -111.556 q^{85} -31.8922 q^{87} -167.603 q^{89} -2057.65 q^{91} +104.104 q^{93} +81.8659 q^{95} +652.693 q^{97} +65.5631 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{3} - 40 q^{5} - 3 q^{7} + 62 q^{9}+O(q^{10}) \) 8 * q - 6 * q^3 - 40 * q^5 - 3 * q^7 + 62 * q^9	\( 8 q - 6 q^{3} - 40 q^{5} - 3 q^{7} + 62 q^{9} - 55 q^{11} - 16 q^{13} + 30 q^{15} + 127 q^{17} + 61 q^{19} + 94 q^{21} + 184 q^{23} + 200 q^{25} - 243 q^{27} - 223 q^{29} + 102 q^{31} + 129 q^{33} + 15 q^{35} - 218 q^{37} - 403 q^{39} + 112 q^{41} - 560 q^{43} - 310 q^{45} + 379 q^{47} - 291 q^{49} - 453 q^{51} - 1006 q^{53} + 275 q^{55} + 1020 q^{57} - 860 q^{59} - 1477 q^{61} - 1206 q^{63} + 80 q^{65} - 740 q^{67} - 138 q^{69} - 1238 q^{71} - 2001 q^{73} - 150 q^{75} - 412 q^{77} - 1202 q^{79} - 2356 q^{81} - 1218 q^{83} - 635 q^{85} - 3419 q^{87} - 1318 q^{89} - 3123 q^{91} - 2445 q^{93} - 305 q^{95} + 151 q^{97} - 4075 q^{99}+O(q^{100}) \) 8 * q - 6 * q^3 - 40 * q^5 - 3 * q^7 + 62 * q^9 - 55 * q^11 - 16 * q^13 + 30 * q^15 + 127 * q^17 + 61 * q^19 + 94 * q^21 + 184 * q^23 + 200 * q^25 - 243 * q^27 - 223 * q^29 + 102 * q^31 + 129 * q^33 + 15 * q^35 - 218 * q^37 - 403 * q^39 + 112 * q^41 - 560 * q^43 - 310 * q^45 + 379 * q^47 - 291 * q^49 - 453 * q^51 - 1006 * q^53 + 275 * q^55 + 1020 * q^57 - 860 * q^59 - 1477 * q^61 - 1206 * q^63 + 80 * q^65 - 740 * q^67 - 138 * q^69 - 1238 * q^71 - 2001 * q^73 - 150 * q^75 - 412 * q^77 - 1202 * q^79 - 2356 * q^81 - 1218 * q^83 - 635 * q^85 - 3419 * q^87 - 1318 * q^89 - 3123 * q^91 - 2445 * q^93 - 305 * q^95 + 151 * q^97 - 4075 * q^99

Introduction

L-functions

Modular forms

Varieties

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