LMFDB - Embedded newform 8020.2.a.c.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8020,2,Mod(1,8020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8020, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8020 = 2^{2} \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8020.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:	$64.0400224211$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.05681 q^{3} -1.00000 q^{5} +3.74444 q^{7} -1.88316 q^{9} +O(q^{10})$	$q+1.05681 q^{3} -1.00000 q^{5} +3.74444 q^{7} -1.88316 q^{9} -4.91712 q^{11} -1.25004 q^{13} -1.05681 q^{15} +3.28289 q^{17} +2.70701 q^{19} +3.95715 q^{21} +0.0904882 q^{23} +1.00000 q^{25} -5.16056 q^{27} -3.78883 q^{29} -2.75334 q^{31} -5.19645 q^{33} -3.74444 q^{35} +6.68433 q^{37} -1.32105 q^{39} -8.86049 q^{41} +9.41235 q^{43} +1.88316 q^{45} -4.56323 q^{47} +7.02080 q^{49} +3.46938 q^{51} -12.5056 q^{53} +4.91712 q^{55} +2.86079 q^{57} +6.26801 q^{59} +5.05352 q^{61} -7.05136 q^{63} +1.25004 q^{65} +2.00243 q^{67} +0.0956287 q^{69} -3.08275 q^{71} +1.79679 q^{73} +1.05681 q^{75} -18.4118 q^{77} -6.31202 q^{79} +0.195754 q^{81} -14.3011 q^{83} -3.28289 q^{85} -4.00407 q^{87} -6.23627 q^{89} -4.68069 q^{91} -2.90975 q^{93} -2.70701 q^{95} -1.66497 q^{97} +9.25971 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * 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
$3$	1.05681	0.610148	0.305074	−	0.952329i	$-0.401319\pi$
$3$	1.05681	0.610148	0.305074	+	0.952329i	$0.401319\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.74444	1.41526	0.707632	−	0.706581i	$-0.249764\pi$
$7$	3.74444	1.41526	0.707632	+	0.706581i	$0.249764\pi$
$8$	0	0
$9$	−1.88316	−0.627719
$10$	0	0
$11$	−4.91712	−1.48257	−0.741283	−	0.671192i	$-0.765783\pi$
$11$	−4.91712	−1.48257	−0.741283	+	0.671192i	$0.765783\pi$
$12$	0	0
$13$	−1.25004	−0.346698	−0.173349	−	0.984860i	$-0.555459\pi$
$13$	−1.25004	−0.346698	−0.173349	+	0.984860i	$0.555459\pi$
$14$	0	0
$15$	−1.05681	−0.272867
$16$	0	0
$17$	3.28289	0.796218	0.398109	−	0.917338i	$-0.369667\pi$
$17$	3.28289	0.796218	0.398109	+	0.917338i	$0.369667\pi$
$18$	0	0
$19$	2.70701	0.621030	0.310515	−	0.950568i	$-0.399498\pi$
$19$	2.70701	0.621030	0.310515	+	0.950568i	$0.399498\pi$
$20$	0	0
$21$	3.95715	0.863521
$22$	0	0
$23$	0.0904882	0.0188681	0.00943405	−	0.999955i	$-0.496997\pi$
$23$	0.0904882	0.0188681	0.00943405	+	0.999955i	$0.496997\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.16056	−0.993150
$28$	0	0
$29$	−3.78883	−0.703569	−0.351784	−	0.936081i	$-0.614425\pi$
$29$	−3.78883	−0.703569	−0.351784	+	0.936081i	$0.614425\pi$
$30$	0	0
$31$	−2.75334	−0.494514	−0.247257	−	0.968950i	$-0.579529\pi$
$31$	−2.75334	−0.494514	−0.247257	+	0.968950i	$0.579529\pi$
$32$	0	0
$33$	−5.19645	−0.904586
$34$	0	0
$35$	−3.74444	−0.632925
$36$	0	0
$37$	6.68433	1.09890	0.549448	−	0.835528i	$-0.314838\pi$
$37$	6.68433	1.09890	0.549448	+	0.835528i	$0.314838\pi$
$38$	0	0
$39$	−1.32105	−0.211537
$40$	0	0
$41$	−8.86049	−1.38378	−0.691888	−	0.722005i	$-0.743221\pi$
$41$	−8.86049	−1.38378	−0.691888	+	0.722005i	$0.743221\pi$
$42$	0	0
$43$	9.41235	1.43537	0.717685	−	0.696368i	$-0.245202\pi$
$43$	9.41235	1.43537	0.717685	+	0.696368i	$0.245202\pi$
$44$	0	0
$45$	1.88316	0.280725
$46$	0	0
$47$	−4.56323	−0.665616	−0.332808	−	0.942995i	$-0.607996\pi$
$47$	−4.56323	−0.665616	−0.332808	+	0.942995i	$0.607996\pi$
$48$	0	0
$49$	7.02080	1.00297
$50$	0	0
$51$	3.46938	0.485811
$52$	0	0
$53$	−12.5056	−1.71777	−0.858885	−	0.512168i	$-0.828843\pi$
$53$	−12.5056	−1.71777	−0.858885	+	0.512168i	$0.828843\pi$
$54$	0	0
$55$	4.91712	0.663024
$56$	0	0
$57$	2.86079	0.378921
$58$	0	0
$59$	6.26801	0.816026	0.408013	−	0.912976i	$-0.366222\pi$
$59$	6.26801	0.816026	0.408013	+	0.912976i	$0.366222\pi$
$60$	0	0
$61$	5.05352	0.647037	0.323519	−	0.946222i	$-0.395134\pi$
$61$	5.05352	0.647037	0.323519	+	0.946222i	$0.395134\pi$
$62$	0	0
$63$	−7.05136	−0.888388
$64$	0	0
$65$	1.25004	0.155048
$66$	0	0
$67$	2.00243	0.244635	0.122318	−	0.992491i	$-0.460967\pi$
$67$	2.00243	0.244635	0.122318	+	0.992491i	$0.460967\pi$
$68$	0	0
$69$	0.0956287	0.0115123
$70$	0	0
$71$	−3.08275	−0.365855	−0.182927	−	0.983126i	$-0.558557\pi$
$71$	−3.08275	−0.365855	−0.182927	+	0.983126i	$0.558557\pi$
$72$	0	0
$73$	1.79679	0.210299	0.105149	−	0.994456i	$-0.466468\pi$
$73$	1.79679	0.210299	0.105149	+	0.994456i	$0.466468\pi$
$74$	0	0
$75$	1.05681	0.122030
$76$	0	0
$77$	−18.4118	−2.09822
$78$	0	0
$79$	−6.31202	−0.710158	−0.355079	−	0.934836i	$-0.615546\pi$
$79$	−6.31202	−0.710158	−0.355079	+	0.934836i	$0.615546\pi$
$80$	0	0
$81$	0.195754	0.0217504
$82$	0	0
$83$	−14.3011	−1.56975	−0.784875	−	0.619655i	$-0.787273\pi$
$83$	−14.3011	−1.56975	−0.784875	+	0.619655i	$0.787273\pi$
$84$	0	0
$85$	−3.28289	−0.356080
$86$	0	0
$87$	−4.00407	−0.429281
$88$	0	0
$89$	−6.23627	−0.661043	−0.330522	−	0.943798i	$-0.607225\pi$
$89$	−6.23627	−0.661043	−0.330522	+	0.943798i	$0.607225\pi$
$90$	0	0
$91$	−4.68069	−0.490669
$92$	0	0
$93$	−2.90975	−0.301727
$94$	0	0
$95$	−2.70701	−0.277733
$96$	0	0
$97$	−1.66497	−0.169053	−0.0845263	−	0.996421i	$-0.526938\pi$
$97$	−1.66497	−0.169053	−0.0845263	+	0.996421i	$0.526938\pi$
$98$	0	0
$99$	9.25971	0.930636
$100$	0	0
$101$	−6.43910	−0.640715	−0.320357	−	0.947297i	$-0.603803\pi$
$101$	−6.43910	−0.640715	−0.320357	+	0.947297i	$0.603803\pi$
$102$	0	0
$103$	−6.37796	−0.628439	−0.314219	−	0.949350i	$-0.601743\pi$
$103$	−6.37796	−0.628439	−0.314219	+	0.949350i	$0.601743\pi$
$104$	0	0
$105$	−3.95715	−0.386178
$106$	0	0
$107$	−8.00804	−0.774167	−0.387083	−	0.922045i	$-0.626517\pi$
$107$	−8.00804	−0.774167	−0.387083	+	0.922045i	$0.626517\pi$
$108$	0	0
$109$	11.0073	1.05430	0.527152	−	0.849771i	$-0.323260\pi$
$109$	11.0073	1.05430	0.527152	+	0.849771i	$0.323260\pi$
$110$	0	0
$111$	7.06405	0.670490
$112$	0	0
$113$	−5.85815	−0.551089	−0.275544	−	0.961288i	$-0.588858\pi$
$113$	−5.85815	−0.551089	−0.275544	+	0.961288i	$0.588858\pi$
$114$	0	0
$115$	−0.0904882	−0.00843807
$116$	0	0
$117$	2.35402	0.217629
$118$	0	0
$119$	12.2926	1.12686
$120$	0	0
$121$	13.1780	1.19800
$122$	0	0
$123$	−9.36383	−0.844308
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−7.06006	−0.626479	−0.313240	−	0.949674i	$-0.601414\pi$
$127$	−7.06006	−0.626479	−0.313240	+	0.949674i	$0.601414\pi$
$128$	0	0
$129$	9.94705	0.875789
$130$	0	0
$131$	6.01709	0.525716	0.262858	−	0.964835i	$-0.415335\pi$
$131$	6.01709	0.525716	0.262858	+	0.964835i	$0.415335\pi$
$132$	0	0
$133$	10.1362	0.878922
$134$	0	0
$135$	5.16056	0.444150
$136$	0	0
$137$	−9.93574	−0.848867	−0.424434	−	0.905459i	$-0.639527\pi$
$137$	−9.93574	−0.848867	−0.424434	+	0.905459i	$0.639527\pi$
$138$	0	0
$139$	7.63141	0.647288	0.323644	−	0.946179i	$-0.395092\pi$
$139$	7.63141	0.647288	0.323644	+	0.946179i	$0.395092\pi$
$140$	0	0
$141$	−4.82246	−0.406124
$142$	0	0
$143$	6.14658	0.514003
$144$	0	0
$145$	3.78883	0.314646
$146$	0	0
$147$	7.41964	0.611961
$148$	0	0
$149$	−5.89207	−0.482697	−0.241349	−	0.970438i	$-0.577590\pi$
$149$	−5.89207	−0.482697	−0.241349	+	0.970438i	$0.577590\pi$
$150$	0	0
$151$	−3.05842	−0.248891	−0.124445	−	0.992226i	$-0.539715\pi$
$151$	−3.05842	−0.248891	−0.124445	+	0.992226i	$0.539715\pi$
$152$	0	0
$153$	−6.18220	−0.499801
$154$	0	0
$155$	2.75334	0.221153
$156$	0	0
$157$	−11.7973	−0.941526	−0.470763	−	0.882260i	$-0.656021\pi$
$157$	−11.7973	−0.941526	−0.470763	+	0.882260i	$0.656021\pi$
$158$	0	0
$159$	−13.2160	−1.04809
$160$	0	0
$161$	0.338827	0.0267033
$162$	0	0
$163$	−18.6332	−1.45946	−0.729731	−	0.683734i	$-0.760355\pi$
$163$	−18.6332	−1.45946	−0.729731	+	0.683734i	$0.760355\pi$
$164$	0	0
$165$	5.19645	0.404543
$166$	0	0
$167$	0.402683	0.0311606	0.0155803	−	0.999879i	$-0.495040\pi$
$167$	0.402683	0.0311606	0.0155803	+	0.999879i	$0.495040\pi$
$168$	0	0
$169$	−11.4374	−0.879800
$170$	0	0
$171$	−5.09772	−0.389833
$172$	0	0
$173$	13.3812	1.01735	0.508676	−	0.860958i	$-0.330135\pi$
$173$	13.3812	1.01735	0.508676	+	0.860958i	$0.330135\pi$
$174$	0	0
$175$	3.74444	0.283053
$176$	0	0
$177$	6.62409	0.497897
$178$	0	0
$179$	−6.32541	−0.472784	−0.236392	−	0.971658i	$-0.575965\pi$
$179$	−6.32541	−0.472784	−0.236392	+	0.971658i	$0.575965\pi$
$180$	0	0
$181$	−4.99761	−0.371469	−0.185735	−	0.982600i	$-0.559467\pi$
$181$	−4.99761	−0.371469	−0.185735	+	0.982600i	$0.559467\pi$
$182$	0	0
$183$	5.34060	0.394789
$184$	0	0
$185$	−6.68433	−0.491441
$186$	0	0
$187$	−16.1424	−1.18045
$188$	0	0
$189$	−19.3234	−1.40557
$190$	0	0
$191$	−22.6501	−1.63890	−0.819452	−	0.573148i	$-0.805722\pi$
$191$	−22.6501	−1.63890	−0.819452	+	0.573148i	$0.805722\pi$
$192$	0	0
$193$	−4.94284	−0.355793	−0.177897	−	0.984049i	$-0.556929\pi$
$193$	−4.94284	−0.355793	−0.177897	+	0.984049i	$0.556929\pi$
$194$	0	0
$195$	1.32105	0.0946023
$196$	0	0
$197$	0.473887	0.0337630	0.0168815	−	0.999857i	$-0.494626\pi$
$197$	0.473887	0.0337630	0.0168815	+	0.999857i	$0.494626\pi$
$198$	0	0
$199$	1.88098	0.133339	0.0666695	−	0.997775i	$-0.478763\pi$
$199$	1.88098	0.133339	0.0666695	+	0.997775i	$0.478763\pi$
$200$	0	0
$201$	2.11618	0.149264
$202$	0	0
$203$	−14.1870	−0.995735
$204$	0	0
$205$	8.86049	0.618843
$206$	0	0
$207$	−0.170404	−0.0118439
$208$	0	0
$209$	−13.3107	−0.920719
$210$	0	0
$211$	−17.5595	−1.20885	−0.604423	−	0.796663i	$-0.706596\pi$
$211$	−17.5595	−1.20885	−0.604423	+	0.796663i	$0.706596\pi$
$212$	0	0
$213$	−3.25787	−0.223226
$214$	0	0
$215$	−9.41235	−0.641917
$216$	0	0
$217$	−10.3097	−0.699867
$218$	0	0
$219$	1.89886	0.128313
$220$	0	0
$221$	−4.10374	−0.276047
$222$	0	0
$223$	−19.6238	−1.31410	−0.657052	−	0.753845i	$-0.728197\pi$
$223$	−19.6238	−1.31410	−0.657052	+	0.753845i	$0.728197\pi$
$224$	0	0
$225$	−1.88316	−0.125544
$226$	0	0
$227$	8.77326	0.582302	0.291151	−	0.956677i	$-0.405962\pi$
$227$	8.77326	0.582302	0.291151	+	0.956677i	$0.405962\pi$
$228$	0	0
$229$	−0.782931	−0.0517375	−0.0258688	−	0.999665i	$-0.508235\pi$
$229$	−0.782931	−0.0517375	−0.0258688	+	0.999665i	$0.508235\pi$
$230$	0	0
$231$	−19.4578	−1.28023
$232$	0	0
$233$	1.17083	0.0767036	0.0383518	−	0.999264i	$-0.487789\pi$
$233$	1.17083	0.0767036	0.0383518	+	0.999264i	$0.487789\pi$
$234$	0	0
$235$	4.56323	0.297672
$236$	0	0
$237$	−6.67059	−0.433301
$238$	0	0
$239$	12.5557	0.812160	0.406080	−	0.913837i	$-0.366895\pi$
$239$	12.5557	0.812160	0.406080	+	0.913837i	$0.366895\pi$
$240$	0	0
$241$	−6.55860	−0.422476	−0.211238	−	0.977435i	$-0.567750\pi$
$241$	−6.55860	−0.422476	−0.211238	+	0.977435i	$0.567750\pi$
$242$	0	0
$243$	15.6885	1.00642
$244$	0	0
$245$	−7.02080	−0.448543
$246$	0	0
$247$	−3.38386	−0.215310
$248$	0	0
$249$	−15.1135	−0.957780
$250$	0	0
$251$	−18.2710	−1.15325	−0.576627	−	0.817008i	$-0.695631\pi$
$251$	−18.2710	−1.15325	−0.576627	+	0.817008i	$0.695631\pi$
$252$	0	0
$253$	−0.444941	−0.0279732
$254$	0	0
$255$	−3.46938	−0.217261
$256$	0	0
$257$	4.25234	0.265254	0.132627	−	0.991166i	$-0.457659\pi$
$257$	4.25234	0.265254	0.132627	+	0.991166i	$0.457659\pi$
$258$	0	0
$259$	25.0290	1.55523
$260$	0	0
$261$	7.13497	0.441644
$262$	0	0
$263$	5.25857	0.324258	0.162129	−	0.986770i	$-0.448164\pi$
$263$	5.25857	0.324258	0.162129	+	0.986770i	$0.448164\pi$
$264$	0	0
$265$	12.5056	0.768211
$266$	0	0
$267$	−6.59054	−0.403334
$268$	0	0
$269$	19.6671	1.19913	0.599563	−	0.800327i	$-0.295341\pi$
$269$	19.6671	1.19913	0.599563	+	0.800327i	$0.295341\pi$
$270$	0	0
$271$	−2.94589	−0.178950	−0.0894751	−	0.995989i	$-0.528519\pi$
$271$	−2.94589	−0.178950	−0.0894751	+	0.995989i	$0.528519\pi$
$272$	0	0
$273$	−4.94658	−0.299381
$274$	0	0
$275$	−4.91712	−0.296513
$276$	0	0
$277$	15.3319	0.921207	0.460604	−	0.887606i	$-0.347633\pi$
$277$	15.3319	0.921207	0.460604	+	0.887606i	$0.347633\pi$
$278$	0	0
$279$	5.18496	0.310416
$280$	0	0
$281$	30.8526	1.84051	0.920256	−	0.391318i	$-0.127981\pi$
$281$	30.8526	1.84051	0.920256	+	0.391318i	$0.127981\pi$
$282$	0	0
$283$	−14.3296	−0.851806	−0.425903	−	0.904769i	$-0.640044\pi$
$283$	−14.3296	−0.851806	−0.425903	+	0.904769i	$0.640044\pi$
$284$	0	0
$285$	−2.86079	−0.169458
$286$	0	0
$287$	−33.1775	−1.95841
$288$	0	0
$289$	−6.22263	−0.366037
$290$	0	0
$291$	−1.75956	−0.103147
$292$	0	0
$293$	−0.312539	−0.0182587	−0.00912935	−	0.999958i	$-0.502906\pi$
$293$	−0.312539	−0.0182587	−0.00912935	+	0.999958i	$0.502906\pi$
$294$	0	0
$295$	−6.26801	−0.364938
$296$	0	0
$297$	25.3751	1.47241
$298$	0	0
$299$	−0.113114	−0.00654153
$300$	0	0
$301$	35.2440	2.03143
$302$	0	0
$303$	−6.80490	−0.390931
$304$	0	0
$305$	−5.05352	−0.289364
$306$	0	0
$307$	−28.3896	−1.62028	−0.810141	−	0.586235i	$-0.800610\pi$
$307$	−28.3896	−1.62028	−0.810141	+	0.586235i	$0.800610\pi$
$308$	0	0
$309$	−6.74028	−0.383441
$310$	0	0
$311$	28.6739	1.62595	0.812973	−	0.582302i	$-0.197848\pi$
$311$	28.6739	1.62595	0.812973	+	0.582302i	$0.197848\pi$
$312$	0	0
$313$	−20.9787	−1.18579	−0.592893	−	0.805282i	$-0.702014\pi$
$313$	−20.9787	−1.18579	−0.592893	+	0.805282i	$0.702014\pi$
$314$	0	0
$315$	7.05136	0.397299
$316$	0	0
$317$	−6.89852	−0.387460	−0.193730	−	0.981055i	$-0.562059\pi$
$317$	−6.89852	−0.387460	−0.193730	+	0.981055i	$0.562059\pi$
$318$	0	0
$319$	18.6301	1.04309
$320$	0	0
$321$	−8.46296	−0.472356
$322$	0	0
$323$	8.88681	0.494476
$324$	0	0
$325$	−1.25004	−0.0693396
$326$	0	0
$327$	11.6326	0.643282
$328$	0	0
$329$	−17.0867	−0.942022
$330$	0	0
$331$	−1.06485	−0.0585296	−0.0292648	−	0.999572i	$-0.509317\pi$
$331$	−1.06485	−0.0585296	−0.0292648	+	0.999572i	$0.509317\pi$
$332$	0	0
$333$	−12.5876	−0.689798
$334$	0	0
$335$	−2.00243	−0.109404
$336$	0	0
$337$	5.75140	0.313298	0.156649	−	0.987654i	$-0.449931\pi$
$337$	5.75140	0.313298	0.156649	+	0.987654i	$0.449931\pi$
$338$	0	0
$339$	−6.19094	−0.336246
$340$	0	0
$341$	13.5385	0.733150
$342$	0	0
$343$	0.0778875	0.00420553
$344$	0	0
$345$	−0.0956287	−0.00514847
$346$	0	0
$347$	−7.99676	−0.429289	−0.214644	−	0.976692i	$-0.568859\pi$
$347$	−7.99676	−0.429289	−0.214644	+	0.976692i	$0.568859\pi$
$348$	0	0
$349$	13.0355	0.697772	0.348886	−	0.937165i	$-0.386560\pi$
$349$	13.0355	0.697772	0.348886	+	0.937165i	$0.386560\pi$
$350$	0	0
$351$	6.45089	0.344323
$352$	0	0
$353$	28.0474	1.49281	0.746406	−	0.665491i	$-0.231778\pi$
$353$	28.0474	1.49281	0.746406	+	0.665491i	$0.231778\pi$
$354$	0	0
$355$	3.08275	0.163615
$356$	0	0
$357$	12.9909	0.687551
$358$	0	0
$359$	15.7552	0.831527	0.415763	−	0.909473i	$-0.363514\pi$
$359$	15.7552	0.831527	0.415763	+	0.909473i	$0.363514\pi$
$360$	0	0
$361$	−11.6721	−0.614321
$362$	0	0
$363$	13.9267	0.730960
$364$	0	0
$365$	−1.79679	−0.0940484
$366$	0	0
$367$	2.82759	0.147599	0.0737994	−	0.997273i	$-0.476488\pi$
$367$	2.82759	0.147599	0.0737994	+	0.997273i	$0.476488\pi$
$368$	0	0
$369$	16.6857	0.868622
$370$	0	0
$371$	−46.8263	−2.43110
$372$	0	0
$373$	−3.53123	−0.182840	−0.0914200	−	0.995812i	$-0.529141\pi$
$373$	−3.53123	−0.182840	−0.0914200	+	0.995812i	$0.529141\pi$
$374$	0	0
$375$	−1.05681	−0.0545733
$376$	0	0
$377$	4.73618	0.243926
$378$	0	0
$379$	17.4053	0.894052	0.447026	−	0.894521i	$-0.352483\pi$
$379$	17.4053	0.894052	0.447026	+	0.894521i	$0.352483\pi$
$380$	0	0
$381$	−7.46113	−0.382245
$382$	0	0
$383$	15.9950	0.817309	0.408654	−	0.912689i	$-0.365998\pi$
$383$	15.9950	0.817309	0.408654	+	0.912689i	$0.365998\pi$
$384$	0	0
$385$	18.4118	0.938354
$386$	0	0
$387$	−17.7249	−0.901009
$388$	0	0
$389$	25.7786	1.30703	0.653513	−	0.756916i	$-0.273295\pi$
$389$	25.7786	1.30703	0.653513	+	0.756916i	$0.273295\pi$
$390$	0	0
$391$	0.297063	0.0150231
$392$	0	0
$393$	6.35891	0.320764
$394$	0	0
$395$	6.31202	0.317592
$396$	0	0
$397$	−13.8792	−0.696578	−0.348289	−	0.937387i	$-0.613237\pi$
$397$	−13.8792	−0.696578	−0.348289	+	0.937387i	$0.613237\pi$
$398$	0	0
$399$	10.7120	0.536273
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	3.44177	0.171447
$404$	0	0
$405$	−0.195754	−0.00972708
$406$	0	0
$407$	−32.8676	−1.62919
$408$	0	0
$409$	−28.4614	−1.40733	−0.703663	−	0.710534i	$-0.748453\pi$
$409$	−28.4614	−1.40733	−0.703663	+	0.710534i	$0.748453\pi$
$410$	0	0
$411$	−10.5002	−0.517935
$412$	0	0
$413$	23.4702	1.15489
$414$	0	0
$415$	14.3011	0.702013
$416$	0	0
$417$	8.06494	0.394942
$418$	0	0
$419$	−6.10636	−0.298315	−0.149158	−	0.988813i	$-0.547656\pi$
$419$	−6.10636	−0.298315	−0.149158	+	0.988813i	$0.547656\pi$
$420$	0	0
$421$	10.7046	0.521709	0.260855	−	0.965378i	$-0.415996\pi$
$421$	10.7046	0.521709	0.260855	+	0.965378i	$0.415996\pi$
$422$	0	0
$423$	8.59328	0.417820
$424$	0	0
$425$	3.28289	0.159244
$426$	0	0
$427$	18.9226	0.915729
$428$	0	0
$429$	6.49576	0.313618
$430$	0	0
$431$	−24.6637	−1.18801	−0.594003	−	0.804463i	$-0.702453\pi$
$431$	−24.6637	−1.18801	−0.594003	+	0.804463i	$0.702453\pi$
$432$	0	0
$433$	22.4493	1.07885	0.539423	−	0.842035i	$-0.318642\pi$
$433$	22.4493	1.07885	0.539423	+	0.842035i	$0.318642\pi$
$434$	0	0
$435$	4.00407	0.191980
$436$	0	0
$437$	0.244952	0.0117177
$438$	0	0
$439$	−8.31761	−0.396978	−0.198489	−	0.980103i	$-0.563603\pi$
$439$	−8.31761	−0.396978	−0.198489	+	0.980103i	$0.563603\pi$
$440$	0	0
$441$	−13.2213	−0.629584
$442$	0	0
$443$	−26.5092	−1.25949	−0.629745	−	0.776802i	$-0.716841\pi$
$443$	−26.5092	−1.25949	−0.629745	+	0.776802i	$0.716841\pi$
$444$	0	0
$445$	6.23627	0.295628
$446$	0	0
$447$	−6.22678	−0.294517
$448$	0	0
$449$	−23.3969	−1.10417	−0.552084	−	0.833788i	$-0.686167\pi$
$449$	−23.3969	−1.10417	−0.552084	+	0.833788i	$0.686167\pi$
$450$	0	0
$451$	43.5681	2.05154
$452$	0	0
$453$	−3.23216	−0.151860
$454$	0	0
$455$	4.68069	0.219434
$456$	0	0
$457$	−9.81149	−0.458962	−0.229481	−	0.973313i	$-0.573703\pi$
$457$	−9.81149	−0.458962	−0.229481	+	0.973313i	$0.573703\pi$
$458$	0	0
$459$	−16.9416	−0.790764
$460$	0	0
$461$	16.3078	0.759529	0.379764	−	0.925083i	$-0.376005\pi$
$461$	16.3078	0.759529	0.379764	+	0.925083i	$0.376005\pi$
$462$	0	0
$463$	18.3810	0.854238	0.427119	−	0.904195i	$-0.359529\pi$
$463$	18.3810	0.854238	0.427119	+	0.904195i	$0.359529\pi$
$464$	0	0
$465$	2.90975	0.134936
$466$	0	0
$467$	32.7463	1.51532	0.757660	−	0.652650i	$-0.226343\pi$
$467$	32.7463	1.51532	0.757660	+	0.652650i	$0.226343\pi$
$468$	0	0
$469$	7.49795	0.346223
$470$	0	0
$471$	−12.4675	−0.574470
$472$	0	0
$473$	−46.2816	−2.12803
$474$	0	0
$475$	2.70701	0.124206
$476$	0	0
$477$	23.5499	1.07828
$478$	0	0
$479$	−17.3004	−0.790474	−0.395237	−	0.918579i	$-0.629338\pi$
$479$	−17.3004	−0.790474	−0.395237	+	0.918579i	$0.629338\pi$
$480$	0	0
$481$	−8.35566	−0.380985
$482$	0	0
$483$	0.358075	0.0162930
$484$	0	0
$485$	1.66497	0.0756026
$486$	0	0
$487$	−16.5025	−0.747799	−0.373899	−	0.927469i	$-0.621979\pi$
$487$	−16.5025	−0.747799	−0.373899	+	0.927469i	$0.621979\pi$
$488$	0	0
$489$	−19.6917	−0.890488
$490$	0	0
$491$	36.9127	1.66585	0.832924	−	0.553387i	$-0.186665\pi$
$491$	36.9127	1.66585	0.832924	+	0.553387i	$0.186665\pi$
$492$	0	0
$493$	−12.4383	−0.560194
$494$	0	0
$495$	−9.25971	−0.416193
$496$	0	0
$497$	−11.5431	−0.517781
$498$	0	0
$499$	4.32181	0.193471	0.0967355	−	0.995310i	$-0.469160\pi$
$499$	4.32181	0.193471	0.0967355	+	0.995310i	$0.469160\pi$
$500$	0	0
$501$	0.425559	0.0190126
$502$	0	0
$503$	−19.4831	−0.868708	−0.434354	−	0.900742i	$-0.643023\pi$
$503$	−19.4831	−0.868708	−0.434354	+	0.900742i	$0.643023\pi$
$504$	0	0
$505$	6.43910	0.286536
$506$	0	0
$507$	−12.0871	−0.536809
$508$	0	0
$509$	4.31667	0.191333	0.0956665	−	0.995413i	$-0.469502\pi$
$509$	4.31667	0.191333	0.0956665	+	0.995413i	$0.469502\pi$
$510$	0	0
$511$	6.72797	0.297628
$512$	0	0
$513$	−13.9697	−0.616776
$514$	0	0
$515$	6.37796	0.281046
$516$	0	0
$517$	22.4379	0.986820
$518$	0	0
$519$	14.1413	0.620735
$520$	0	0
$521$	15.0172	0.657915	0.328957	−	0.944345i	$-0.393303\pi$
$521$	15.0172	0.657915	0.328957	+	0.944345i	$0.393303\pi$
$522$	0	0
$523$	−11.8035	−0.516130	−0.258065	−	0.966128i	$-0.583085\pi$
$523$	−11.8035	−0.516130	−0.258065	+	0.966128i	$0.583085\pi$
$524$	0	0
$525$	3.95715	0.172704
$526$	0	0
$527$	−9.03890	−0.393741
$528$	0	0
$529$	−22.9918	−0.999644
$530$	0	0
$531$	−11.8037	−0.512235
$532$	0	0
$533$	11.0759	0.479752
$534$	0	0
$535$	8.00804	0.346218
$536$	0	0
$537$	−6.68475	−0.288468
$538$	0	0
$539$	−34.5221	−1.48697
$540$	0	0
$541$	−17.0641	−0.733645	−0.366822	−	0.930291i	$-0.619554\pi$
$541$	−17.0641	−0.733645	−0.366822	+	0.930291i	$0.619554\pi$
$542$	0	0
$543$	−5.28151	−0.226651
$544$	0	0
$545$	−11.0073	−0.471499
$546$	0	0
$547$	−2.19782	−0.0939719	−0.0469859	−	0.998896i	$-0.514962\pi$
$547$	−2.19782	−0.0939719	−0.0469859	+	0.998896i	$0.514962\pi$
$548$	0	0
$549$	−9.51658	−0.406158
$550$	0	0
$551$	−10.2564	−0.436938
$552$	0	0
$553$	−23.6350	−1.00506
$554$	0	0
$555$	−7.06405	−0.299852
$556$	0	0
$557$	2.02193	0.0856720	0.0428360	−	0.999082i	$-0.486361\pi$
$557$	2.02193	0.0856720	0.0428360	+	0.999082i	$0.486361\pi$
$558$	0	0
$559$	−11.7658	−0.497640
$560$	0	0
$561$	−17.0594	−0.720247
$562$	0	0
$563$	11.0051	0.463809	0.231904	−	0.972739i	$-0.425504\pi$
$563$	11.0051	0.463809	0.231904	+	0.972739i	$0.425504\pi$
$564$	0	0
$565$	5.85815	0.246454
$566$	0	0
$567$	0.732987	0.0307826
$568$	0	0
$569$	17.3530	0.727476	0.363738	−	0.931501i	$-0.381500\pi$
$569$	17.3530	0.727476	0.363738	+	0.931501i	$0.381500\pi$
$570$	0	0
$571$	36.7034	1.53599	0.767995	−	0.640456i	$-0.221255\pi$
$571$	36.7034	1.53599	0.767995	+	0.640456i	$0.221255\pi$
$572$	0	0
$573$	−23.9368	−0.999975
$574$	0	0
$575$	0.0904882	0.00377362
$576$	0	0
$577$	−10.7384	−0.447045	−0.223523	−	0.974699i	$-0.571756\pi$
$577$	−10.7384	−0.447045	−0.223523	+	0.974699i	$0.571756\pi$
$578$	0	0
$579$	−5.22363	−0.217087
$580$	0	0
$581$	−53.5495	−2.22161
$582$	0	0
$583$	61.4913	2.54671
$584$	0	0
$585$	−2.35402	−0.0973266
$586$	0	0
$587$	−42.7896	−1.76612	−0.883059	−	0.469263i	$-0.844520\pi$
$587$	−42.7896	−1.76612	−0.883059	+	0.469263i	$0.844520\pi$
$588$	0	0
$589$	−7.45330	−0.307108
$590$	0	0
$591$	0.500807	0.0206005
$592$	0	0
$593$	−6.82762	−0.280377	−0.140188	−	0.990125i	$-0.544771\pi$
$593$	−6.82762	−0.280377	−0.140188	+	0.990125i	$0.544771\pi$
$594$	0	0
$595$	−12.2926	−0.503946
$596$	0	0
$597$	1.98783	0.0813565
$598$	0	0
$599$	5.48398	0.224069	0.112035	−	0.993704i	$-0.464263\pi$
$599$	5.48398	0.224069	0.112035	+	0.993704i	$0.464263\pi$
$600$	0	0
$601$	−19.5620	−0.797949	−0.398974	−	0.916962i	$-0.630634\pi$
$601$	−19.5620	−0.797949	−0.398974	+	0.916962i	$0.630634\pi$
$602$	0	0
$603$	−3.77088	−0.153562
$604$	0	0
$605$	−13.1780	−0.535764
$606$	0	0
$607$	−43.3479	−1.75944	−0.879718	−	0.475495i	$-0.842269\pi$
$607$	−43.3479	−1.75944	−0.879718	+	0.475495i	$0.842269\pi$
$608$	0	0
$609$	−14.9930	−0.607546
$610$	0	0
$611$	5.70421	0.230768
$612$	0	0
$613$	0.577104	0.0233090	0.0116545	−	0.999932i	$-0.496290\pi$
$613$	0.577104	0.0233090	0.0116545	+	0.999932i	$0.496290\pi$
$614$	0	0
$615$	9.36383	0.377586
$616$	0	0
$617$	−27.7804	−1.11840	−0.559199	−	0.829034i	$-0.688891\pi$
$617$	−27.7804	−1.11840	−0.559199	+	0.829034i	$0.688891\pi$
$618$	0	0
$619$	4.85248	0.195038	0.0975188	−	0.995234i	$-0.468909\pi$
$619$	4.85248	0.195038	0.0975188	+	0.995234i	$0.468909\pi$
$620$	0	0
$621$	−0.466970	−0.0187389
$622$	0	0
$623$	−23.3513	−0.935551
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−14.0668	−0.561775
$628$	0	0
$629$	21.9439	0.874961
$630$	0	0
$631$	−29.7232	−1.18326	−0.591631	−	0.806209i	$-0.701516\pi$
$631$	−29.7232	−1.18326	−0.591631	+	0.806209i	$0.701516\pi$
$632$	0	0
$633$	−18.5570	−0.737575
$634$	0	0
$635$	7.06006	0.280170
$636$	0	0
$637$	−8.77626	−0.347728
$638$	0	0
$639$	5.80530	0.229654
$640$	0	0
$641$	−25.1715	−0.994213	−0.497107	−	0.867690i	$-0.665604\pi$
$641$	−25.1715	−0.994213	−0.497107	+	0.867690i	$0.665604\pi$
$642$	0	0
$643$	19.2870	0.760603	0.380302	−	0.924863i	$-0.375820\pi$
$643$	19.2870	0.760603	0.380302	+	0.924863i	$0.375820\pi$
$644$	0	0
$645$	−9.94705	−0.391665
$646$	0	0
$647$	−3.41305	−0.134181	−0.0670904	−	0.997747i	$-0.521372\pi$
$647$	−3.41305	−0.134181	−0.0670904	+	0.997747i	$0.521372\pi$
$648$	0	0
$649$	−30.8206	−1.20981
$650$	0	0
$651$	−10.8954	−0.427023
$652$	0	0
$653$	22.6264	0.885441	0.442721	−	0.896660i	$-0.354013\pi$
$653$	22.6264	0.885441	0.442721	+	0.896660i	$0.354013\pi$
$654$	0	0
$655$	−6.01709	−0.235107
$656$	0	0
$657$	−3.38364	−0.132008
$658$	0	0
$659$	15.0752	0.587248	0.293624	−	0.955921i	$-0.405139\pi$
$659$	15.0752	0.587248	0.293624	+	0.955921i	$0.405139\pi$
$660$	0	0
$661$	20.8732	0.811873	0.405936	−	0.913901i	$-0.366945\pi$
$661$	20.8732	0.811873	0.405936	+	0.913901i	$0.366945\pi$
$662$	0	0
$663$	−4.33686	−0.168430
$664$	0	0
$665$	−10.1362	−0.393066
$666$	0	0
$667$	−0.342845	−0.0132750
$668$	0	0
$669$	−20.7385	−0.801798
$670$	0	0
$671$	−24.8488	−0.959276
$672$	0	0
$673$	28.3789	1.09392	0.546962	−	0.837157i	$-0.315784\pi$
$673$	28.3789	1.09392	0.546962	+	0.837157i	$0.315784\pi$
$674$	0	0
$675$	−5.16056	−0.198630
$676$	0	0
$677$	−3.70226	−0.142289	−0.0711447	−	0.997466i	$-0.522665\pi$
$677$	−3.70226	−0.142289	−0.0711447	+	0.997466i	$0.522665\pi$
$678$	0	0
$679$	−6.23439	−0.239254
$680$	0	0
$681$	9.27165	0.355291
$682$	0	0
$683$	17.0874	0.653833	0.326917	−	0.945053i	$-0.393990\pi$
$683$	17.0874	0.653833	0.326917	+	0.945053i	$0.393990\pi$
$684$	0	0
$685$	9.93574	0.379625
$686$	0	0
$687$	−0.827407	−0.0315676
$688$	0	0
$689$	15.6324	0.595548
$690$	0	0
$691$	10.6496	0.405132	0.202566	−	0.979269i	$-0.435072\pi$
$691$	10.6496	0.405132	0.202566	+	0.979269i	$0.435072\pi$
$692$	0	0
$693$	34.6724	1.31709
$694$	0	0
$695$	−7.63141	−0.289476
$696$	0	0
$697$	−29.0880	−1.10179
$698$	0	0
$699$	1.23734	0.0468006
$700$	0	0
$701$	−16.2117	−0.612309	−0.306154	−	0.951982i	$-0.599042\pi$
$701$	−16.2117	−0.612309	−0.306154	+	0.951982i	$0.599042\pi$
$702$	0	0
$703$	18.0945	0.682448
$704$	0	0
$705$	4.82246	0.181624
$706$	0	0
$707$	−24.1108	−0.906780
$708$	0	0
$709$	1.40278	0.0526827	0.0263413	−	0.999653i	$-0.491614\pi$
$709$	1.40278	0.0526827	0.0263413	+	0.999653i	$0.491614\pi$
$710$	0	0
$711$	11.8865	0.445780
$712$	0	0
$713$	−0.249145	−0.00933054
$714$	0	0
$715$	−6.14658	−0.229869
$716$	0	0
$717$	13.2690	0.495538
$718$	0	0
$719$	−33.0603	−1.23294	−0.616470	−	0.787379i	$-0.711438\pi$
$719$	−33.0603	−1.23294	−0.616470	+	0.787379i	$0.711438\pi$
$720$	0	0
$721$	−23.8819	−0.889407
$722$	0	0
$723$	−6.93118	−0.257773
$724$	0	0
$725$	−3.78883	−0.140714
$726$	0	0
$727$	−45.8719	−1.70129	−0.850647	−	0.525737i	$-0.823790\pi$
$727$	−45.8719	−1.70129	−0.850647	+	0.525737i	$0.823790\pi$
$728$	0	0
$729$	15.9925	0.592316
$730$	0	0
$731$	30.8997	1.14287
$732$	0	0
$733$	47.0247	1.73690	0.868448	−	0.495780i	$-0.165118\pi$
$733$	47.0247	1.73690	0.868448	+	0.495780i	$0.165118\pi$
$734$	0	0
$735$	−7.41964	−0.273677
$736$	0	0
$737$	−9.84616	−0.362688
$738$	0	0
$739$	26.5585	0.976969	0.488484	−	0.872573i	$-0.337550\pi$
$739$	26.5585	0.976969	0.488484	+	0.872573i	$0.337550\pi$
$740$	0	0
$741$	−3.57609	−0.131371
$742$	0	0
$743$	41.3300	1.51625	0.758125	−	0.652109i	$-0.226115\pi$
$743$	41.3300	1.51625	0.758125	+	0.652109i	$0.226115\pi$
$744$	0	0
$745$	5.89207	0.215869
$746$	0	0
$747$	26.9312	0.985361
$748$	0	0
$749$	−29.9856	−1.09565
$750$	0	0
$751$	36.3593	1.32677	0.663385	−	0.748279i	$-0.269119\pi$
$751$	36.3593	1.32677	0.663385	+	0.748279i	$0.269119\pi$
$752$	0	0
$753$	−19.3089	−0.703656
$754$	0	0
$755$	3.05842	0.111307
$756$	0	0
$757$	−12.6905	−0.461246	−0.230623	−	0.973043i	$-0.574076\pi$
$757$	−12.6905	−0.461246	−0.230623	+	0.973043i	$0.574076\pi$
$758$	0	0
$759$	−0.470217	−0.0170678
$760$	0	0
$761$	42.6066	1.54449	0.772244	−	0.635326i	$-0.219134\pi$
$761$	42.6066	1.54449	0.772244	+	0.635326i	$0.219134\pi$
$762$	0	0
$763$	41.2160	1.49212
$764$	0	0
$765$	6.18220	0.223518
$766$	0	0
$767$	−7.83525	−0.282915
$768$	0	0
$769$	2.00878	0.0724385	0.0362193	−	0.999344i	$-0.488469\pi$
$769$	2.00878	0.0724385	0.0362193	+	0.999344i	$0.488469\pi$
$770$	0	0
$771$	4.49390	0.161844
$772$	0	0
$773$	−47.7404	−1.71710	−0.858551	−	0.512729i	$-0.828635\pi$
$773$	−47.7404	−1.71710	−0.858551	+	0.512729i	$0.828635\pi$
$774$	0	0
$775$	−2.75334	−0.0989027
$776$	0	0
$777$	26.4509	0.948920
$778$	0	0
$779$	−23.9854	−0.859367
$780$	0	0
$781$	15.1582	0.542404
$782$	0	0
$783$	19.5525	0.698749
$784$	0	0
$785$	11.7973	0.421063
$786$	0	0
$787$	29.1783	1.04009	0.520047	−	0.854138i	$-0.325914\pi$
$787$	29.1783	1.04009	0.520047	+	0.854138i	$0.325914\pi$
$788$	0	0
$789$	5.55730	0.197845
$790$	0	0
$791$	−21.9355	−0.779936
$792$	0	0
$793$	−6.31709	−0.224327
$794$	0	0
$795$	13.2160	0.468722
$796$	0	0
$797$	−45.1039	−1.59766	−0.798832	−	0.601554i	$-0.794548\pi$
$797$	−45.1039	−1.59766	−0.798832	+	0.601554i	$0.794548\pi$
$798$	0	0
$799$	−14.9806	−0.529975
$800$	0	0
$801$	11.7439	0.414949
$802$	0	0
$803$	−8.83504	−0.311782
$804$	0	0
$805$	−0.338827	−0.0119421
$806$	0	0
$807$	20.7844	0.731645
$808$	0	0
$809$	−25.4823	−0.895909	−0.447955	−	0.894056i	$-0.647847\pi$
$809$	−25.4823	−0.895909	−0.447955	+	0.894056i	$0.647847\pi$
$810$	0	0
$811$	7.05225	0.247638	0.123819	−	0.992305i	$-0.460486\pi$
$811$	7.05225	0.247638	0.123819	+	0.992305i	$0.460486\pi$
$812$	0	0
$813$	−3.11324	−0.109186
$814$	0	0
$815$	18.6332	0.652691
$816$	0	0
$817$	25.4793	0.891409
$818$	0	0
$819$	8.81447	0.308002
$820$	0	0
$821$	−35.1278	−1.22597	−0.612983	−	0.790096i	$-0.710031\pi$
$821$	−35.1278	−1.22597	−0.612983	+	0.790096i	$0.710031\pi$
$822$	0	0
$823$	6.95512	0.242440	0.121220	−	0.992626i	$-0.461319\pi$
$823$	6.95512	0.242440	0.121220	+	0.992626i	$0.461319\pi$
$824$	0	0
$825$	−5.19645	−0.180917
$826$	0	0
$827$	−34.2821	−1.19211	−0.596054	−	0.802945i	$-0.703265\pi$
$827$	−34.2821	−1.19211	−0.596054	+	0.802945i	$0.703265\pi$
$828$	0	0
$829$	−39.4277	−1.36938	−0.684691	−	0.728834i	$-0.740063\pi$
$829$	−39.4277	−1.36938	−0.684691	+	0.728834i	$0.740063\pi$
$830$	0	0
$831$	16.2029	0.562073
$832$	0	0
$833$	23.0485	0.798584
$834$	0	0
$835$	−0.402683	−0.0139354
$836$	0	0
$837$	14.2088	0.491126
$838$	0	0
$839$	11.6032	0.400587	0.200294	−	0.979736i	$-0.435810\pi$
$839$	11.6032	0.400587	0.200294	+	0.979736i	$0.435810\pi$
$840$	0	0
$841$	−14.6447	−0.504991
$842$	0	0
$843$	32.6053	1.12298
$844$	0	0
$845$	11.4374	0.393459
$846$	0	0
$847$	49.3444	1.69549
$848$	0	0
$849$	−15.1436	−0.519728
$850$	0	0
$851$	0.604853	0.0207341
$852$	0	0
$853$	−30.6383	−1.04904	−0.524518	−	0.851400i	$-0.675754\pi$
$853$	−30.6383	−1.04904	−0.524518	+	0.851400i	$0.675754\pi$
$854$	0	0
$855$	5.09772	0.174338
$856$	0	0
$857$	−19.8084	−0.676640	−0.338320	−	0.941031i	$-0.609859\pi$
$857$	−19.8084	−0.676640	−0.338320	+	0.941031i	$0.609859\pi$
$858$	0	0
$859$	38.4445	1.31171	0.655854	−	0.754887i	$-0.272308\pi$
$859$	38.4445	1.31171	0.655854	+	0.754887i	$0.272308\pi$
$860$	0	0
$861$	−35.0623	−1.19492
$862$	0	0
$863$	37.7715	1.28576	0.642878	−	0.765969i	$-0.277740\pi$
$863$	37.7715	1.28576	0.642878	+	0.765969i	$0.277740\pi$
$864$	0	0
$865$	−13.3812	−0.454973
$866$	0	0
$867$	−6.57612	−0.223337
$868$	0	0
$869$	31.0369	1.05286
$870$	0	0
$871$	−2.50311	−0.0848145
$872$	0	0
$873$	3.13541	0.106118
$874$	0	0
$875$	−3.74444	−0.126585
$876$	0	0
$877$	20.8955	0.705591	0.352796	−	0.935700i	$-0.385231\pi$
$877$	20.8955	0.705591	0.352796	+	0.935700i	$0.385231\pi$
$878$	0	0
$879$	−0.330293	−0.0111405
$880$	0	0
$881$	18.2753	0.615709	0.307855	−	0.951433i	$-0.400389\pi$
$881$	18.2753	0.615709	0.307855	+	0.951433i	$0.400389\pi$
$882$	0	0
$883$	51.5596	1.73512	0.867559	−	0.497334i	$-0.165688\pi$
$883$	51.5596	1.73512	0.867559	+	0.497334i	$0.165688\pi$
$884$	0	0
$885$	−6.62409	−0.222666
$886$	0	0
$887$	−35.8071	−1.20229	−0.601143	−	0.799142i	$-0.705288\pi$
$887$	−35.8071	−1.20229	−0.601143	+	0.799142i	$0.705288\pi$
$888$	0	0
$889$	−26.4359	−0.886633
$890$	0	0
$891$	−0.962544	−0.0322464
$892$	0	0
$893$	−12.3527	−0.413368
$894$	0	0
$895$	6.32541	0.211435
$896$	0	0
$897$	−0.119539	−0.00399131
$898$	0	0
$899$	10.4319	0.347924
$900$	0	0
$901$	−41.0544	−1.36772
$902$	0	0
$903$	37.2461	1.23947
$904$	0	0
$905$	4.99761	0.166126
$906$	0	0
$907$	27.2510	0.904855	0.452427	−	0.891801i	$-0.350558\pi$
$907$	27.2510	0.904855	0.452427	+	0.891801i	$0.350558\pi$
$908$	0	0
$909$	12.1258	0.402189
$910$	0	0
$911$	−41.3107	−1.36869	−0.684343	−	0.729160i	$-0.739911\pi$
$911$	−41.3107	−1.36869	−0.684343	+	0.729160i	$0.739911\pi$
$912$	0	0
$913$	70.3202	2.32726
$914$	0	0
$915$	−5.34060	−0.176555
$916$	0	0
$917$	22.5306	0.744026
$918$	0	0
$919$	2.71698	0.0896248	0.0448124	−	0.998995i	$-0.485731\pi$
$919$	2.71698	0.0896248	0.0448124	+	0.998995i	$0.485731\pi$
$920$	0	0
$921$	−30.0024	−0.988613
$922$	0	0
$923$	3.85355	0.126841
$924$	0	0
$925$	6.68433	0.219779
$926$	0	0
$927$	12.0107	0.394483
$928$	0	0
$929$	−10.9603	−0.359596	−0.179798	−	0.983704i	$-0.557544\pi$
$929$	−10.9603	−0.359596	−0.179798	+	0.983704i	$0.557544\pi$
$930$	0	0
$931$	19.0054	0.622876
$932$	0	0
$933$	30.3028	0.992068
$934$	0	0
$935$	16.1424	0.527912
$936$	0	0
$937$	3.00923	0.0983073	0.0491536	−	0.998791i	$-0.484348\pi$
$937$	3.00923	0.0983073	0.0491536	+	0.998791i	$0.484348\pi$
$938$	0	0
$939$	−22.1704	−0.723505
$940$	0	0
$941$	−13.4967	−0.439978	−0.219989	−	0.975502i	$-0.570602\pi$
$941$	−13.4967	−0.439978	−0.219989	+	0.975502i	$0.570602\pi$
$942$	0	0
$943$	−0.801770	−0.0261092
$944$	0	0
$945$	19.3234	0.628590
$946$	0	0
$947$	−44.6148	−1.44979	−0.724893	−	0.688862i	$-0.758111\pi$
$947$	−44.6148	−1.44979	−0.724893	+	0.688862i	$0.758111\pi$
$948$	0	0
$949$	−2.24606	−0.0729101
$950$	0	0
$951$	−7.29041	−0.236408
$952$	0	0
$953$	19.3972	0.628337	0.314168	−	0.949367i	$-0.398274\pi$
$953$	19.3972	0.628337	0.314168	+	0.949367i	$0.398274\pi$
$954$	0	0
$955$	22.6501	0.732940
$956$	0	0
$957$	19.6885	0.636438
$958$	0	0
$959$	−37.2037	−1.20137
$960$	0	0
$961$	−23.4191	−0.755456
$962$	0	0
$963$	15.0804	0.485959
$964$	0	0
$965$	4.94284	0.159116
$966$	0	0
$967$	−5.54759	−0.178399	−0.0891993	−	0.996014i	$-0.528431\pi$
$967$	−5.54759	−0.178399	−0.0891993	+	0.996014i	$0.528431\pi$
$968$	0	0
$969$	9.39165	0.301703
$970$	0	0
$971$	53.0446	1.70228	0.851140	−	0.524938i	$-0.175912\pi$
$971$	53.0446	1.70228	0.851140	+	0.524938i	$0.175912\pi$
$972$	0	0
$973$	28.5753	0.916083
$974$	0	0
$975$	−1.32105	−0.0423074
$976$	0	0
$977$	37.7604	1.20806	0.604032	−	0.796960i	$-0.293560\pi$
$977$	37.7604	1.20806	0.604032	+	0.796960i	$0.293560\pi$
$978$	0	0
$979$	30.6645	0.980041
$980$	0	0
$981$	−20.7284	−0.661807
$982$	0	0
$983$	1.18374	0.0377556	0.0188778	−	0.999822i	$-0.493991\pi$
$983$	1.18374	0.0377556	0.0188778	+	0.999822i	$0.493991\pi$
$984$	0	0
$985$	−0.473887	−0.0150993
$986$	0	0
$987$	−18.0574	−0.574773
$988$	0	0
$989$	0.851707	0.0270827
$990$	0	0
$991$	−44.3559	−1.40901	−0.704505	−	0.709699i	$-0.748831\pi$
$991$	−44.3559	−1.40901	−0.704505	+	0.709699i	$0.748831\pi$
$992$	0	0
$993$	−1.12535	−0.0357118
$994$	0	0
$995$	−1.88098	−0.0596310
$996$	0	0
$997$	40.8454	1.29359	0.646794	−	0.762665i	$-0.276110\pi$
$997$	40.8454	1.29359	0.646794	+	0.762665i	$0.276110\pi$
$998$	0	0
$999$	−34.4949	−1.09137

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.c.1.18	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.c.1.18	✓	28	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8020.a (trivial)

\(f(q)\)	\(=\)	\(q+1.05681 q^{3} -1.00000 q^{5} +3.74444 q^{7} -1.88316 q^{9} +O(q^{10})\)	\(q+1.05681 q^{3} -1.00000 q^{5} +3.74444 q^{7} -1.88316 q^{9} -4.91712 q^{11} -1.25004 q^{13} -1.05681 q^{15} +3.28289 q^{17} +2.70701 q^{19} +3.95715 q^{21} +0.0904882 q^{23} +1.00000 q^{25} -5.16056 q^{27} -3.78883 q^{29} -2.75334 q^{31} -5.19645 q^{33} -3.74444 q^{35} +6.68433 q^{37} -1.32105 q^{39} -8.86049 q^{41} +9.41235 q^{43} +1.88316 q^{45} -4.56323 q^{47} +7.02080 q^{49} +3.46938 q^{51} -12.5056 q^{53} +4.91712 q^{55} +2.86079 q^{57} +6.26801 q^{59} +5.05352 q^{61} -7.05136 q^{63} +1.25004 q^{65} +2.00243 q^{67} +0.0956287 q^{69} -3.08275 q^{71} +1.79679 q^{73} +1.05681 q^{75} -18.4118 q^{77} -6.31202 q^{79} +0.195754 q^{81} -14.3011 q^{83} -3.28289 q^{85} -4.00407 q^{87} -6.23627 q^{89} -4.68069 q^{91} -2.90975 q^{93} -2.70701 q^{95} -1.66497 q^{97} +9.25971 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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