LMFDB - Embedded newform 1980.4.a.l.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1980,4,Mod(1,1980)] 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("1980.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1980, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1980.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,15,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$116.823781811$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.9192.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 18x + 30 $ x^3 - x^2 - 18*x + 30
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 220)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.81626$ of defining polynomial
Character	$\chi$	$=$	1980.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000 q^{5} -22.8386 q^{7} -11.0000 q^{11} -62.7120 q^{13} +76.9530 q^{17} +150.448 q^{19} -167.037 q^{23} +25.0000 q^{25} +237.462 q^{29} +113.629 q^{31} -114.193 q^{35} +408.368 q^{37} -114.174 q^{41} -222.549 q^{43} +30.4302 q^{47} +178.600 q^{49} -377.703 q^{53} -55.0000 q^{55} +641.034 q^{59} +429.350 q^{61} -313.560 q^{65} -178.621 q^{67} -863.668 q^{71} -1027.06 q^{73} +251.224 q^{77} -785.725 q^{79} -636.536 q^{83} +384.765 q^{85} -52.0089 q^{89} +1432.25 q^{91} +752.241 q^{95} -1461.70 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 15 q^{5} - 5 q^{7} - 33 q^{11} + 2 q^{13} - 77 q^{17} + 171 q^{19} - 222 q^{23} + 75 q^{25} - 55 q^{29} + 181 q^{31} - 25 q^{35} + 317 q^{37} - 302 q^{41} - 188 q^{43} - 662 q^{47} - 268 q^{49} - 81 q^{53}+ \cdots - 1980 q^{97}+O(q^{100}) $ 3 * q + 15 * q^5 - 5 * q^7 - 33 * q^11 + 2 * q^13 - 77 * q^17 + 171 * q^19 - 222 * q^23 + 75 * q^25 - 55 * q^29 + 181 * q^31 - 25 * q^35 + 317 * q^37 - 302 * q^41 - 188 * q^43 - 662 * q^47 - 268 * q^49 - 81 * q^53 - 165 * q^55 + 42 * q^59 + 349 * q^61 + 10 * q^65 + 152 * q^67 - 927 * q^71 - 2378 * q^73 + 55 * q^77 - 1146 * q^79 + 458 * q^83 - 385 * q^85 + 875 * q^89 + 1982 * q^91 + 855 * q^95 - 1980 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−22.8386	−1.23317	−0.616583	−	0.787290i	$-0.711484\pi$
$7$	−22.8386	−1.23317	−0.616583	+	0.787290i	$0.711484\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−62.7120	−1.33794	−0.668969	−	0.743290i	$-0.733264\pi$
$13$	−62.7120	−1.33794	−0.668969	+	0.743290i	$0.733264\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	76.9530	1.09787	0.548937	−	0.835864i	$-0.315033\pi$
$17$	76.9530	1.09787	0.548937	+	0.835864i	$0.315033\pi$
$18$	0	0
$19$	150.448	1.81659	0.908294	−	0.418332i	$-0.137385\pi$
$19$	150.448	1.81659	0.908294	+	0.418332i	$0.137385\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−167.037	−1.51433	−0.757167	−	0.653222i	$-0.773417\pi$
$23$	−167.037	−1.51433	−0.757167	+	0.653222i	$0.773417\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	237.462	1.52054	0.760270	−	0.649607i	$-0.225067\pi$
$29$	237.462	1.52054	0.760270	+	0.649607i	$0.225067\pi$
$30$	0	0
$31$	113.629	0.658336	0.329168	−	0.944271i	$-0.393232\pi$
$31$	113.629	0.658336	0.329168	+	0.944271i	$0.393232\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−114.193	−0.551489
$36$	0	0
$37$	408.368	1.81447	0.907234	−	0.420626i	$-0.138190\pi$
$37$	408.368	1.81447	0.907234	+	0.420626i	$0.138190\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−114.174	−0.434901	−0.217451	−	0.976071i	$-0.569774\pi$
$41$	−114.174	−0.434901	−0.217451	+	0.976071i	$0.569774\pi$
$42$	0	0
$43$	−222.549	−0.789264	−0.394632	−	0.918839i	$-0.629128\pi$
$43$	−222.549	−0.789264	−0.394632	+	0.918839i	$0.629128\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	30.4302	0.0944405	0.0472202	−	0.998885i	$-0.484964\pi$
$47$	30.4302	0.0944405	0.0472202	+	0.998885i	$0.484964\pi$
$48$	0	0
$49$	178.600	0.520699
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−377.703	−0.978896	−0.489448	−	0.872032i	$-0.662802\pi$
$53$	−377.703	−0.978896	−0.489448	+	0.872032i	$0.662802\pi$
$54$	0	0
$55$	−55.0000	−0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	641.034	1.41450	0.707250	−	0.706964i	$-0.249936\pi$
$59$	641.034	1.41450	0.707250	+	0.706964i	$0.249936\pi$
$60$	0	0
$61$	429.350	0.901190	0.450595	−	0.892728i	$-0.351212\pi$
$61$	429.350	0.901190	0.450595	+	0.892728i	$0.351212\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−313.560	−0.598344
$66$	0	0
$67$	−178.621	−0.325702	−0.162851	−	0.986651i	$-0.552069\pi$
$67$	−178.621	−0.325702	−0.162851	+	0.986651i	$0.552069\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−863.668	−1.44364	−0.721821	−	0.692080i	$-0.756694\pi$
$71$	−863.668	−1.44364	−0.721821	+	0.692080i	$0.756694\pi$
$72$	0	0
$73$	−1027.06	−1.64670	−0.823348	−	0.567537i	$-0.807896\pi$
$73$	−1027.06	−1.64670	−0.823348	+	0.567537i	$0.807896\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	251.224	0.371814
$78$	0	0
$79$	−785.725	−1.11900	−0.559500	−	0.828830i	$-0.689007\pi$
$79$	−785.725	−1.11900	−0.559500	+	0.828830i	$0.689007\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−636.536	−0.841794	−0.420897	−	0.907109i	$-0.638285\pi$
$83$	−636.536	−0.841794	−0.420897	+	0.907109i	$0.638285\pi$
$84$	0	0
$85$	384.765	0.490984
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−52.0089	−0.0619431	−0.0309716	−	0.999520i	$-0.509860\pi$
$89$	−52.0089	−0.0619431	−0.0309716	+	0.999520i	$0.509860\pi$
$90$	0	0
$91$	1432.25	1.64990
$92$	0	0
$93$	0	0
$94$	0	0
$95$	752.241	0.812403
$96$	0	0
$97$	−1461.70	−1.53003	−0.765015	−	0.644013i	$-0.777268\pi$
$97$	−1461.70	−1.53003	−0.765015	+	0.644013i	$0.777268\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1529.03	−1.50638	−0.753189	−	0.657804i	$-0.771486\pi$
$101$	−1529.03	−1.50638	−0.753189	+	0.657804i	$0.771486\pi$
$102$	0	0
$103$	−691.404	−0.661418	−0.330709	−	0.943733i	$-0.607288\pi$
$103$	−691.404	−0.661418	−0.330709	+	0.943733i	$0.607288\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−901.469	−0.814470	−0.407235	−	0.913323i	$-0.633507\pi$
$107$	−901.469	−0.814470	−0.407235	+	0.913323i	$0.633507\pi$
$108$	0	0
$109$	1157.11	1.01680	0.508400	−	0.861121i	$-0.330237\pi$
$109$	1157.11	1.01680	0.508400	+	0.861121i	$0.330237\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	250.113	0.208218	0.104109	−	0.994566i	$-0.466801\pi$
$113$	250.113	0.208218	0.104109	+	0.994566i	$0.466801\pi$
$114$	0	0
$115$	−835.186	−0.677231
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1757.50	−1.35386
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1023.01	0.714780	0.357390	−	0.933955i	$-0.383667\pi$
$127$	1023.01	0.714780	0.357390	+	0.933955i	$0.383667\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−380.179	−0.253560	−0.126780	−	0.991931i	$-0.540464\pi$
$131$	−380.179	−0.253560	−0.126780	+	0.991931i	$0.540464\pi$
$132$	0	0
$133$	−3436.02	−2.24016
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2603.29	−1.62346	−0.811731	−	0.584031i	$-0.801475\pi$
$137$	−2603.29	−1.62346	−0.811731	+	0.584031i	$0.801475\pi$
$138$	0	0
$139$	116.538	0.0711121	0.0355561	−	0.999368i	$-0.488680\pi$
$139$	116.538	0.0711121	0.0355561	+	0.999368i	$0.488680\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	689.832	0.403403
$144$	0	0
$145$	1187.31	0.680006
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−11.9283	−0.00655842	−0.00327921	−	0.999995i	$-0.501044\pi$
$149$	−11.9283	−0.00655842	−0.00327921	+	0.999995i	$0.501044\pi$
$150$	0	0
$151$	−807.857	−0.435381	−0.217690	−	0.976018i	$-0.569852\pi$
$151$	−807.857	−0.435381	−0.217690	+	0.976018i	$0.569852\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	568.146	0.294417
$156$	0	0
$157$	2393.57	1.21674	0.608368	−	0.793655i	$-0.291825\pi$
$157$	2393.57	1.21674	0.608368	+	0.793655i	$0.291825\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3814.89	1.86743
$162$	0	0
$163$	2522.57	1.21216	0.606082	−	0.795402i	$-0.292740\pi$
$163$	2522.57	1.21216	0.606082	+	0.795402i	$0.292740\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1180.17	−0.546850	−0.273425	−	0.961893i	$-0.588157\pi$
$167$	−1180.17	−0.546850	−0.273425	+	0.961893i	$0.588157\pi$
$168$	0	0
$169$	1735.80	0.790077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1573.93	0.691697	0.345848	−	0.938290i	$-0.387591\pi$
$173$	1573.93	0.691697	0.345848	+	0.938290i	$0.387591\pi$
$174$	0	0
$175$	−570.964	−0.246633
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2222.11	−0.927867	−0.463934	−	0.885870i	$-0.653562\pi$
$179$	−2222.11	−0.927867	−0.463934	+	0.885870i	$0.653562\pi$
$180$	0	0
$181$	973.312	0.399700	0.199850	−	0.979826i	$-0.435954\pi$
$181$	973.312	0.399700	0.199850	+	0.979826i	$0.435954\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2041.84	0.811455
$186$	0	0
$187$	−846.483	−0.331021
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2585.05	−0.979308	−0.489654	−	0.871917i	$-0.662877\pi$
$191$	−2585.05	−0.979308	−0.489654	+	0.871917i	$0.662877\pi$
$192$	0	0
$193$	1067.35	0.398082	0.199041	−	0.979991i	$-0.436217\pi$
$193$	1067.35	0.398082	0.199041	+	0.979991i	$0.436217\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3034.38	−1.09741	−0.548706	−	0.836015i	$-0.684880\pi$
$197$	−3034.38	−1.09741	−0.548706	+	0.836015i	$0.684880\pi$
$198$	0	0
$199$	−4897.23	−1.74450	−0.872250	−	0.489060i	$-0.837340\pi$
$199$	−4897.23	−1.74450	−0.872250	+	0.489060i	$0.837340\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5423.30	−1.87508
$204$	0	0
$205$	−570.869	−0.194494
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1654.93	−0.547722
$210$	0	0
$211$	2885.84	0.941560	0.470780	−	0.882251i	$-0.343973\pi$
$211$	2885.84	0.941560	0.470780	+	0.882251i	$0.343973\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1112.74	−0.352970
$216$	0	0
$217$	−2595.13	−0.811838
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4825.88	−1.46889
$222$	0	0
$223$	3844.79	1.15456	0.577279	−	0.816547i	$-0.304115\pi$
$223$	3844.79	1.15456	0.577279	+	0.816547i	$0.304115\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2285.43	−0.668235	−0.334117	−	0.942531i	$-0.608438\pi$
$227$	−2285.43	−0.668235	−0.334117	+	0.942531i	$0.608438\pi$
$228$	0	0
$229$	1891.38	0.545790	0.272895	−	0.962044i	$-0.412019\pi$
$229$	1891.38	0.545790	0.272895	+	0.962044i	$0.412019\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4140.34	1.16413	0.582066	−	0.813141i	$-0.302244\pi$
$233$	4140.34	1.16413	0.582066	+	0.813141i	$0.302244\pi$
$234$	0	0
$235$	152.151	0.0422351
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3750.56	−1.01508	−0.507539	−	0.861629i	$-0.669445\pi$
$239$	−3750.56	−1.01508	−0.507539	+	0.861629i	$0.669445\pi$
$240$	0	0
$241$	−1098.57	−0.293631	−0.146816	−	0.989164i	$-0.546902\pi$
$241$	−1098.57	−0.293631	−0.146816	+	0.989164i	$0.546902\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	892.999	0.232864
$246$	0	0
$247$	−9434.91	−2.43048
$248$	0	0
$249$	0	0
$250$	0	0
$251$	244.641	0.0615202	0.0307601	−	0.999527i	$-0.490207\pi$
$251$	244.641	0.0615202	0.0307601	+	0.999527i	$0.490207\pi$
$252$	0	0
$253$	1837.41	0.456589
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3461.10	0.840069	0.420034	−	0.907508i	$-0.362018\pi$
$257$	3461.10	0.840069	0.420034	+	0.907508i	$0.362018\pi$
$258$	0	0
$259$	−9326.54	−2.23754
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−901.642	−0.211398	−0.105699	−	0.994398i	$-0.533708\pi$
$263$	−901.642	−0.211398	−0.105699	+	0.994398i	$0.533708\pi$
$264$	0	0
$265$	−1888.52	−0.437776
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1080.02	0.244797	0.122398	−	0.992481i	$-0.460941\pi$
$269$	1080.02	0.244797	0.122398	+	0.992481i	$0.460941\pi$
$270$	0	0
$271$	−5471.36	−1.22643	−0.613213	−	0.789918i	$-0.710123\pi$
$271$	−5471.36	−1.22643	−0.613213	+	0.789918i	$0.710123\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−275.000	−0.0603023
$276$	0	0
$277$	−4917.32	−1.06662	−0.533309	−	0.845920i	$-0.679052\pi$
$277$	−4917.32	−1.06662	−0.533309	+	0.845920i	$0.679052\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4708.26	0.999542	0.499771	−	0.866157i	$-0.333417\pi$
$281$	4708.26	0.999542	0.499771	+	0.866157i	$0.333417\pi$
$282$	0	0
$283$	−1489.18	−0.312801	−0.156401	−	0.987694i	$-0.549989\pi$
$283$	−1489.18	−0.312801	−0.156401	+	0.987694i	$0.549989\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2607.56	0.536305
$288$	0	0
$289$	1008.76	0.205326
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4980.51	−0.993053	−0.496527	−	0.868022i	$-0.665391\pi$
$293$	−4980.51	−0.993053	−0.496527	+	0.868022i	$0.665391\pi$
$294$	0	0
$295$	3205.17	0.632583
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10475.2	2.02608
$300$	0	0
$301$	5082.69	0.973294
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2146.75	0.403025
$306$	0	0
$307$	−390.108	−0.0725232	−0.0362616	−	0.999342i	$-0.511545\pi$
$307$	−390.108	−0.0725232	−0.0362616	+	0.999342i	$0.511545\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2915.55	−0.531594	−0.265797	−	0.964029i	$-0.585635\pi$
$311$	−2915.55	−0.531594	−0.265797	+	0.964029i	$0.585635\pi$
$312$	0	0
$313$	−2417.23	−0.436516	−0.218258	−	0.975891i	$-0.570038\pi$
$313$	−2417.23	−0.436516	−0.218258	+	0.975891i	$0.570038\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1072.66	−0.190052	−0.0950258	−	0.995475i	$-0.530293\pi$
$317$	−1072.66	−0.190052	−0.0950258	+	0.995475i	$0.530293\pi$
$318$	0	0
$319$	−2612.09	−0.458460
$320$	0	0
$321$	0	0
$322$	0	0
$323$	11577.4	1.99438
$324$	0	0
$325$	−1567.80	−0.267588
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−694.982	−0.116461
$330$	0	0
$331$	10037.3	1.66677	0.833383	−	0.552696i	$-0.186401\pi$
$331$	10037.3	1.66677	0.833383	+	0.552696i	$0.186401\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−893.105	−0.145658
$336$	0	0
$337$	−2086.27	−0.337229	−0.168615	−	0.985682i	$-0.553929\pi$
$337$	−2086.27	−0.337229	−0.168615	+	0.985682i	$0.553929\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1249.92	−0.198496
$342$	0	0
$343$	3754.66	0.591058
$344$	0	0
$345$	0	0
$346$	0	0
$347$	208.886	0.0323159	0.0161579	−	0.999869i	$-0.494857\pi$
$347$	208.886	0.0323159	0.0161579	+	0.999869i	$0.494857\pi$
$348$	0	0
$349$	−5320.85	−0.816099	−0.408049	−	0.912960i	$-0.633791\pi$
$349$	−5320.85	−0.816099	−0.408049	+	0.912960i	$0.633791\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7759.66	−1.16999	−0.584993	−	0.811038i	$-0.698903\pi$
$353$	−7759.66	−1.16999	−0.584993	+	0.811038i	$0.698903\pi$
$354$	0	0
$355$	−4318.34	−0.645616
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5241.68	−0.770599	−0.385300	−	0.922792i	$-0.625902\pi$
$359$	−5241.68	−0.770599	−0.385300	+	0.922792i	$0.625902\pi$
$360$	0	0
$361$	15775.6	2.29999
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5135.32	−0.736425
$366$	0	0
$367$	−9066.36	−1.28954	−0.644769	−	0.764378i	$-0.723046\pi$
$367$	−9066.36	−1.28954	−0.644769	+	0.764378i	$0.723046\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8626.20	1.20714
$372$	0	0
$373$	−4012.90	−0.557051	−0.278526	−	0.960429i	$-0.589846\pi$
$373$	−4012.90	−0.557051	−0.278526	+	0.960429i	$0.589846\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−14891.8	−2.03439
$378$	0	0
$379$	−10821.4	−1.46664	−0.733319	−	0.679885i	$-0.762030\pi$
$379$	−10821.4	−1.46664	−0.733319	+	0.679885i	$0.762030\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−13383.6	−1.78556	−0.892780	−	0.450494i	$-0.851248\pi$
$383$	−13383.6	−1.78556	−0.892780	+	0.450494i	$0.851248\pi$
$384$	0	0
$385$	1256.12	0.166280
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4273.93	0.557062	0.278531	−	0.960427i	$-0.410153\pi$
$389$	4273.93	0.557062	0.278531	+	0.960427i	$0.410153\pi$
$390$	0	0
$391$	−12854.0	−1.66255
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3928.63	−0.500432
$396$	0	0
$397$	9613.91	1.21539	0.607693	−	0.794172i	$-0.292095\pi$
$397$	9613.91	1.21539	0.607693	+	0.794172i	$0.292095\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1028.18	−0.128042	−0.0640208	−	0.997949i	$-0.520392\pi$
$401$	−1028.18	−0.128042	−0.0640208	+	0.997949i	$0.520392\pi$
$402$	0	0
$403$	−7125.92	−0.880813
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4492.05	−0.547083
$408$	0	0
$409$	−1073.84	−0.129824	−0.0649118	−	0.997891i	$-0.520677\pi$
$409$	−1073.84	−0.129824	−0.0649118	+	0.997891i	$0.520677\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−14640.3	−1.74431
$414$	0	0
$415$	−3182.68	−0.376462
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8525.49	0.994028	0.497014	−	0.867743i	$-0.334430\pi$
$419$	8525.49	0.994028	0.497014	+	0.867743i	$0.334430\pi$
$420$	0	0
$421$	−8635.36	−0.999672	−0.499836	−	0.866120i	$-0.666606\pi$
$421$	−8635.36	−0.999672	−0.499836	+	0.866120i	$0.666606\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1923.83	0.219575
$426$	0	0
$427$	−9805.73	−1.11132
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11263.5	−1.25880	−0.629400	−	0.777081i	$-0.716699\pi$
$431$	−11263.5	−1.25880	−0.629400	+	0.777081i	$0.716699\pi$
$432$	0	0
$433$	6127.46	0.680063	0.340031	−	0.940414i	$-0.389562\pi$
$433$	6127.46	0.680063	0.340031	+	0.940414i	$0.389562\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−25130.4	−2.75092
$438$	0	0
$439$	−5683.18	−0.617866	−0.308933	−	0.951084i	$-0.599972\pi$
$439$	−5683.18	−0.617866	−0.308933	+	0.951084i	$0.599972\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7700.06	−0.825826	−0.412913	−	0.910771i	$-0.635489\pi$
$443$	−7700.06	−0.825826	−0.412913	+	0.910771i	$0.635489\pi$
$444$	0	0
$445$	−260.045	−0.0277018
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11433.7	1.20176	0.600879	−	0.799340i	$-0.294817\pi$
$449$	11433.7	1.20176	0.600879	+	0.799340i	$0.294817\pi$
$450$	0	0
$451$	1255.91	0.131128
$452$	0	0
$453$	0	0
$454$	0	0
$455$	7161.26	0.737858
$456$	0	0
$457$	2002.70	0.204994	0.102497	−	0.994733i	$-0.467317\pi$
$457$	2002.70	0.204994	0.102497	+	0.994733i	$0.467317\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−15252.2	−1.54093	−0.770463	−	0.637484i	$-0.779975\pi$
$461$	−15252.2	−1.54093	−0.770463	+	0.637484i	$0.779975\pi$
$462$	0	0
$463$	5351.51	0.537162	0.268581	−	0.963257i	$-0.413445\pi$
$463$	5351.51	0.537162	0.268581	+	0.963257i	$0.413445\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14310.1	1.41797	0.708985	−	0.705223i	$-0.249153\pi$
$467$	14310.1	1.41797	0.708985	+	0.705223i	$0.249153\pi$
$468$	0	0
$469$	4079.45	0.401645
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2448.04	0.237972
$474$	0	0
$475$	3761.20	0.363318
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11424.9	−1.08981	−0.544904	−	0.838499i	$-0.683434\pi$
$479$	−11424.9	−1.08981	−0.544904	+	0.838499i	$0.683434\pi$
$480$	0	0
$481$	−25609.6	−2.42765
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7308.49	−0.684250
$486$	0	0
$487$	13792.8	1.28339	0.641696	−	0.766959i	$-0.278231\pi$
$487$	13792.8	1.28339	0.641696	+	0.766959i	$0.278231\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1140.39	0.104817	0.0524087	−	0.998626i	$-0.483310\pi$
$491$	1140.39	0.104817	0.0524087	+	0.998626i	$0.483310\pi$
$492$	0	0
$493$	18273.5	1.66936
$494$	0	0
$495$	0	0
$496$	0	0
$497$	19724.9	1.78025
$498$	0	0
$499$	8974.64	0.805130	0.402565	−	0.915391i	$-0.368119\pi$
$499$	8974.64	0.805130	0.402565	+	0.915391i	$0.368119\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4452.75	−0.394709	−0.197354	−	0.980332i	$-0.563235\pi$
$503$	−4452.75	−0.394709	−0.197354	+	0.980332i	$0.563235\pi$
$504$	0	0
$505$	−7645.15	−0.673673
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5106.52	0.444681	0.222340	−	0.974969i	$-0.428630\pi$
$509$	5106.52	0.444681	0.222340	+	0.974969i	$0.428630\pi$
$510$	0	0
$511$	23456.7	2.03065
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3457.02	−0.295795
$516$	0	0
$517$	−334.732	−0.0284749
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−21931.6	−1.84422	−0.922110	−	0.386927i	$-0.873537\pi$
$521$	−21931.6	−1.84422	−0.922110	+	0.386927i	$0.873537\pi$
$522$	0	0
$523$	4592.60	0.383978	0.191989	−	0.981397i	$-0.438506\pi$
$523$	4592.60	0.383978	0.191989	+	0.981397i	$0.438506\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8744.11	0.722770
$528$	0	0
$529$	15734.4	1.29321
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7160.07	0.581871
$534$	0	0
$535$	−4507.34	−0.364242
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1964.60	−0.156997
$540$	0	0
$541$	−21130.7	−1.67926	−0.839632	−	0.543156i	$-0.817229\pi$
$541$	−21130.7	−1.67926	−0.839632	+	0.543156i	$0.817229\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5785.56	0.454727
$546$	0	0
$547$	21935.3	1.71460	0.857300	−	0.514818i	$-0.172140\pi$
$547$	21935.3	1.71460	0.857300	+	0.514818i	$0.172140\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	35725.8	2.76220
$552$	0	0
$553$	17944.8	1.37991
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7566.81	−0.575612	−0.287806	−	0.957689i	$-0.592926\pi$
$557$	−7566.81	−0.575612	−0.287806	+	0.957689i	$0.592926\pi$
$558$	0	0
$559$	13956.5	1.05599
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10750.1	−0.804730	−0.402365	−	0.915479i	$-0.631812\pi$
$563$	−10750.1	−0.804730	−0.402365	+	0.915479i	$0.631812\pi$
$564$	0	0
$565$	1250.57	0.0931181
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5780.45	0.425886	0.212943	−	0.977065i	$-0.431695\pi$
$569$	5780.45	0.425886	0.212943	+	0.977065i	$0.431695\pi$
$570$	0	0
$571$	−1803.89	−0.132207	−0.0661037	−	0.997813i	$-0.521057\pi$
$571$	−1803.89	−0.132207	−0.0661037	+	0.997813i	$0.521057\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4175.93	−0.302867
$576$	0	0
$577$	21018.1	1.51646	0.758228	−	0.651989i	$-0.226065\pi$
$577$	21018.1	1.51646	0.758228	+	0.651989i	$0.226065\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	14537.6	1.03807
$582$	0	0
$583$	4154.73	0.295148
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8126.04	0.571376	0.285688	−	0.958323i	$-0.407778\pi$
$587$	8126.04	0.571376	0.285688	+	0.958323i	$0.407778\pi$
$588$	0	0
$589$	17095.3	1.19593
$590$	0	0
$591$	0	0
$592$	0	0
$593$	22520.0	1.55951	0.779753	−	0.626087i	$-0.215345\pi$
$593$	22520.0	1.55951	0.779753	+	0.626087i	$0.215345\pi$
$594$	0	0
$595$	−8787.48	−0.605465
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−8064.41	−0.550088	−0.275044	−	0.961432i	$-0.588692\pi$
$599$	−8064.41	−0.550088	−0.275044	+	0.961432i	$0.588692\pi$
$600$	0	0
$601$	−24456.3	−1.65989	−0.829944	−	0.557846i	$-0.811628\pi$
$601$	−24456.3	−1.65989	−0.829944	+	0.557846i	$0.811628\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	605.000	0.0406558
$606$	0	0
$607$	−14488.1	−0.968785	−0.484392	−	0.874851i	$-0.660959\pi$
$607$	−14488.1	−0.968785	−0.484392	+	0.874851i	$0.660959\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1908.34	−0.126355
$612$	0	0
$613$	635.488	0.0418713	0.0209356	−	0.999781i	$-0.493335\pi$
$613$	635.488	0.0418713	0.0209356	+	0.999781i	$0.493335\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7144.52	0.466171	0.233086	−	0.972456i	$-0.425118\pi$
$617$	7144.52	0.466171	0.233086	+	0.972456i	$0.425118\pi$
$618$	0	0
$619$	−18138.8	−1.17780	−0.588902	−	0.808204i	$-0.700440\pi$
$619$	−18138.8	−1.17780	−0.588902	+	0.808204i	$0.700440\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1187.81	0.0763862
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	31425.2	1.99206
$630$	0	0
$631$	−4197.12	−0.264794	−0.132397	−	0.991197i	$-0.542267\pi$
$631$	−4197.12	−0.264794	−0.132397	+	0.991197i	$0.542267\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5115.03	0.319659
$636$	0	0
$637$	−11200.4	−0.696663
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6817.05	0.420058	0.210029	−	0.977695i	$-0.432644\pi$
$641$	6817.05	0.420058	0.210029	+	0.977695i	$0.432644\pi$
$642$	0	0
$643$	−20606.2	−1.26381	−0.631906	−	0.775045i	$-0.717727\pi$
$643$	−20606.2	−1.26381	−0.631906	+	0.775045i	$0.717727\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25394.7	−1.54308	−0.771538	−	0.636183i	$-0.780512\pi$
$647$	−25394.7	−1.54308	−0.771538	+	0.636183i	$0.780512\pi$
$648$	0	0
$649$	−7051.37	−0.426488
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19387.5	−1.16185	−0.580927	−	0.813956i	$-0.697310\pi$
$653$	−19387.5	−1.16185	−0.580927	+	0.813956i	$0.697310\pi$
$654$	0	0
$655$	−1900.90	−0.113396
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−18107.7	−1.07038	−0.535188	−	0.844733i	$-0.679759\pi$
$659$	−18107.7	−1.07038	−0.535188	+	0.844733i	$0.679759\pi$
$660$	0	0
$661$	6961.74	0.409653	0.204826	−	0.978798i	$-0.434337\pi$
$661$	6961.74	0.409653	0.204826	+	0.978798i	$0.434337\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−17180.1	−1.00183
$666$	0	0
$667$	−39665.1	−2.30261
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4722.85	−0.271719
$672$	0	0
$673$	6979.78	0.399778	0.199889	−	0.979819i	$-0.435942\pi$
$673$	6979.78	0.399778	0.199889	+	0.979819i	$0.435942\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2020.76	0.114718	0.0573590	−	0.998354i	$-0.481732\pi$
$677$	2020.76	0.114718	0.0573590	+	0.998354i	$0.481732\pi$
$678$	0	0
$679$	33383.1	1.88678
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8849.94	0.495803	0.247902	−	0.968785i	$-0.420259\pi$
$683$	8849.94	0.495803	0.247902	+	0.968785i	$0.420259\pi$
$684$	0	0
$685$	−13016.5	−0.726035
$686$	0	0
$687$	0	0
$688$	0	0
$689$	23686.5	1.30970
$690$	0	0
$691$	20802.0	1.14522	0.572609	−	0.819829i	$-0.305931\pi$
$691$	20802.0	1.14522	0.572609	+	0.819829i	$0.305931\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	582.688	0.0318023
$696$	0	0
$697$	−8786.01	−0.477466
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−3857.97	−0.207865	−0.103932	−	0.994584i	$-0.533143\pi$
$701$	−3857.97	−0.207865	−0.103932	+	0.994584i	$0.533143\pi$
$702$	0	0
$703$	61438.2	3.29614
$704$	0	0
$705$	0	0
$706$	0	0
$707$	34920.9	1.85762
$708$	0	0
$709$	8349.52	0.442275	0.221137	−	0.975243i	$-0.429023\pi$
$709$	8349.52	0.442275	0.221137	+	0.975243i	$0.429023\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−18980.3	−0.996941
$714$	0	0
$715$	3449.16	0.180407
$716$	0	0
$717$	0	0
$718$	0	0
$719$	23771.2	1.23298	0.616492	−	0.787361i	$-0.288553\pi$
$719$	23771.2	1.23298	0.616492	+	0.787361i	$0.288553\pi$
$720$	0	0
$721$	15790.7	0.815639
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5936.56	0.304108
$726$	0	0
$727$	3465.59	0.176797	0.0883987	−	0.996085i	$-0.471825\pi$
$727$	3465.59	0.176797	0.0883987	+	0.996085i	$0.471825\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−17125.8	−0.866512
$732$	0	0
$733$	29210.9	1.47194	0.735969	−	0.677016i	$-0.236727\pi$
$733$	29210.9	1.47194	0.735969	+	0.677016i	$0.236727\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1964.83	0.0982028
$738$	0	0
$739$	26468.5	1.31754	0.658768	−	0.752347i	$-0.271078\pi$
$739$	26468.5	1.31754	0.658768	+	0.752347i	$0.271078\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	5512.64	0.272193	0.136096	−	0.990696i	$-0.456544\pi$
$743$	5512.64	0.272193	0.136096	+	0.990696i	$0.456544\pi$
$744$	0	0
$745$	−59.6415	−0.00293302
$746$	0	0
$747$	0	0
$748$	0	0
$749$	20588.2	1.00438
$750$	0	0
$751$	−26016.3	−1.26411	−0.632056	−	0.774923i	$-0.717789\pi$
$751$	−26016.3	−1.26411	−0.632056	+	0.774923i	$0.717789\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4039.29	−0.194708
$756$	0	0
$757$	17165.7	0.824169	0.412085	−	0.911146i	$-0.364801\pi$
$757$	17165.7	0.824169	0.412085	+	0.911146i	$0.364801\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−29915.6	−1.42502	−0.712509	−	0.701663i	$-0.752441\pi$
$761$	−29915.6	−1.42502	−0.712509	+	0.701663i	$0.752441\pi$
$762$	0	0
$763$	−26426.8	−1.25388
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−40200.5	−1.89251
$768$	0	0
$769$	−31347.7	−1.47000	−0.734998	−	0.678069i	$-0.762817\pi$
$769$	−31347.7	−1.47000	−0.734998	+	0.678069i	$0.762817\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	17975.5	0.836396	0.418198	−	0.908356i	$-0.362662\pi$
$773$	17975.5	0.836396	0.418198	+	0.908356i	$0.362662\pi$
$774$	0	0
$775$	2840.73	0.131667
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−17177.2	−0.790036
$780$	0	0
$781$	9500.35	0.435274
$782$	0	0
$783$	0	0
$784$	0	0
$785$	11967.8	0.544141
$786$	0	0
$787$	−34294.6	−1.55333	−0.776666	−	0.629913i	$-0.783091\pi$
$787$	−34294.6	−1.55333	−0.776666	+	0.629913i	$0.783091\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5712.23	−0.256768
$792$	0	0
$793$	−26925.4	−1.20574
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23365.3	1.03845	0.519223	−	0.854639i	$-0.326221\pi$
$797$	23365.3	1.03845	0.519223	+	0.854639i	$0.326221\pi$
$798$	0	0
$799$	2341.70	0.103684
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11297.7	0.496497
$804$	0	0
$805$	19074.5	0.835138
$806$	0	0
$807$	0	0
$808$	0	0
$809$	17229.3	0.748766	0.374383	−	0.927274i	$-0.377855\pi$
$809$	17229.3	0.748766	0.374383	+	0.927274i	$0.377855\pi$
$810$	0	0
$811$	17092.0	0.740050	0.370025	−	0.929022i	$-0.379349\pi$
$811$	17092.0	0.740050	0.370025	+	0.929022i	$0.379349\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	12612.8	0.542096
$816$	0	0
$817$	−33482.1	−1.43377
$818$	0	0
$819$	0	0
$820$	0	0
$821$	37033.7	1.57428	0.787141	−	0.616773i	$-0.211561\pi$
$821$	37033.7	1.57428	0.787141	+	0.616773i	$0.211561\pi$
$822$	0	0
$823$	−28429.5	−1.20412	−0.602060	−	0.798451i	$-0.705653\pi$
$823$	−28429.5	−1.20412	−0.602060	+	0.798451i	$0.705653\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13912.7	0.584994	0.292497	−	0.956266i	$-0.405514\pi$
$827$	13912.7	0.584994	0.292497	+	0.956266i	$0.405514\pi$
$828$	0	0
$829$	21308.0	0.892710	0.446355	−	0.894856i	$-0.352722\pi$
$829$	21308.0	0.892710	0.446355	+	0.894856i	$0.352722\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	13743.8	0.571662
$834$	0	0
$835$	−5900.83	−0.244559
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−29990.4	−1.23407	−0.617034	−	0.786936i	$-0.711666\pi$
$839$	−29990.4	−1.23407	−0.617034	+	0.786936i	$0.711666\pi$
$840$	0	0
$841$	31999.4	1.31204
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8679.00	0.353333
$846$	0	0
$847$	−2763.47	−0.112106
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−68212.7	−2.74771
$852$	0	0
$853$	−46095.0	−1.85025	−0.925125	−	0.379663i	$-0.876040\pi$
$853$	−46095.0	−1.85025	−0.925125	+	0.379663i	$0.876040\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27650.8	−1.10214	−0.551070	−	0.834459i	$-0.685780\pi$
$857$	−27650.8	−1.10214	−0.551070	+	0.834459i	$0.685780\pi$
$858$	0	0
$859$	−12233.6	−0.485918	−0.242959	−	0.970037i	$-0.578118\pi$
$859$	−12233.6	−0.485918	−0.242959	+	0.970037i	$0.578118\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−25358.2	−1.00023	−0.500117	−	0.865958i	$-0.666710\pi$
$863$	−25358.2	−1.00023	−0.500117	+	0.865958i	$0.666710\pi$
$864$	0	0
$865$	7869.64	0.309336
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8642.98	0.337391
$870$	0	0
$871$	11201.7	0.435769
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2854.82	−0.110298
$876$	0	0
$877$	43605.7	1.67897	0.839487	−	0.543379i	$-0.182855\pi$
$877$	43605.7	1.67897	0.839487	+	0.543379i	$0.182855\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	26843.9	1.02655	0.513277	−	0.858223i	$-0.328431\pi$
$881$	26843.9	1.02655	0.513277	+	0.858223i	$0.328431\pi$
$882$	0	0
$883$	26559.8	1.01224	0.506120	−	0.862463i	$-0.331079\pi$
$883$	26559.8	1.01224	0.506120	+	0.862463i	$0.331079\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	6892.43	0.260908	0.130454	−	0.991454i	$-0.458357\pi$
$887$	6892.43	0.260908	0.130454	+	0.991454i	$0.458357\pi$
$888$	0	0
$889$	−23364.0	−0.881443
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4578.17	0.171559
$894$	0	0
$895$	−11110.5	−0.414955
$896$	0	0
$897$	0	0
$898$	0	0
$899$	26982.7	1.00103
$900$	0	0
$901$	−29065.4	−1.07470
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4866.56	0.178751
$906$	0	0
$907$	−28702.2	−1.05076	−0.525380	−	0.850868i	$-0.676077\pi$
$907$	−28702.2	−1.05076	−0.525380	+	0.850868i	$0.676077\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	39940.6	1.45257	0.726285	−	0.687393i	$-0.241245\pi$
$911$	39940.6	1.45257	0.726285	+	0.687393i	$0.241245\pi$
$912$	0	0
$913$	7001.89	0.253810
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8682.75	0.312682
$918$	0	0
$919$	4701.26	0.168749	0.0843745	−	0.996434i	$-0.473111\pi$
$919$	4701.26	0.168749	0.0843745	+	0.996434i	$0.473111\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	54162.4	1.93150
$924$	0	0
$925$	10209.2	0.362894
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−10844.7	−0.382997	−0.191498	−	0.981493i	$-0.561335\pi$
$929$	−10844.7	−0.382997	−0.191498	+	0.981493i	$0.561335\pi$
$930$	0	0
$931$	26870.0	0.945896
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4232.42	−0.148037
$936$	0	0
$937$	−1486.92	−0.0518417	−0.0259209	−	0.999664i	$-0.508252\pi$
$937$	−1486.92	−0.0518417	−0.0259209	+	0.999664i	$0.508252\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	28867.6	1.00006	0.500030	−	0.866008i	$-0.333322\pi$
$941$	28867.6	1.00006	0.500030	+	0.866008i	$0.333322\pi$
$942$	0	0
$943$	19071.3	0.658585
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−27363.1	−0.938945	−0.469473	−	0.882947i	$-0.655556\pi$
$947$	−27363.1	−0.938945	−0.469473	+	0.882947i	$0.655556\pi$
$948$	0	0
$949$	64409.3	2.20318
$950$	0	0
$951$	0	0
$952$	0	0
$953$	16239.2	0.551981	0.275991	−	0.961160i	$-0.410994\pi$
$953$	16239.2	0.551981	0.275991	+	0.961160i	$0.410994\pi$
$954$	0	0
$955$	−12925.3	−0.437960
$956$	0	0
$957$	0	0
$958$	0	0
$959$	59455.5	2.00200
$960$	0	0
$961$	−16879.4	−0.566593
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5336.77	0.178028
$966$	0	0
$967$	−36670.3	−1.21948	−0.609740	−	0.792601i	$-0.708726\pi$
$967$	−36670.3	−1.21948	−0.609740	+	0.792601i	$0.708726\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	14074.8	0.465171	0.232585	−	0.972576i	$-0.425281\pi$
$971$	14074.8	0.465171	0.232585	+	0.972576i	$0.425281\pi$
$972$	0	0
$973$	−2661.55	−0.0876931
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9580.28	−0.313716	−0.156858	−	0.987621i	$-0.550136\pi$
$977$	−9580.28	−0.313716	−0.156858	+	0.987621i	$0.550136\pi$
$978$	0	0
$979$	572.098	0.0186766
$980$	0	0
$981$	0	0
$982$	0	0
$983$	44780.4	1.45297	0.726487	−	0.687180i	$-0.241152\pi$
$983$	44780.4	1.45297	0.726487	+	0.687180i	$0.241152\pi$
$984$	0	0
$985$	−15171.9	−0.490778
$986$	0	0
$987$	0	0
$988$	0	0
$989$	37173.9	1.19521
$990$	0	0
$991$	−21668.9	−0.694586	−0.347293	−	0.937757i	$-0.612899\pi$
$991$	−21668.9	−0.694586	−0.347293	+	0.937757i	$0.612899\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−24486.2	−0.780164
$996$	0	0
$997$	11881.8	0.377433	0.188716	−	0.982032i	$-0.439567\pi$
$997$	11881.8	0.377433	0.188716	+	0.982032i	$0.439567\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1980.4.a.l.1.1		3
3.2	odd	2		220.4.a.f.1.1	✓	3
12.11	even	2		880.4.a.w.1.3		3
15.2	even	4		1100.4.b.h.749.6		6
15.8	even	4		1100.4.b.h.749.1		6
15.14	odd	2		1100.4.a.i.1.3		3
33.32	even	2		2420.4.a.i.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.4.a.f.1.1	✓	3	3.2	odd	2
880.4.a.w.1.3		3	12.11	even	2
1100.4.a.i.1.3		3	15.14	odd	2
1100.4.b.h.749.1		6	15.8	even	4
1100.4.b.h.749.6		6	15.2	even	4
1980.4.a.l.1.1		3	1.1	even	1	trivial
2420.4.a.i.1.1		3	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1980.a (trivial)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} -22.8386 q^{7} -11.0000 q^{11} -62.7120 q^{13} +76.9530 q^{17} +150.448 q^{19} -167.037 q^{23} +25.0000 q^{25} +237.462 q^{29} +113.629 q^{31} -114.193 q^{35} +408.368 q^{37} -114.174 q^{41} -222.549 q^{43} +30.4302 q^{47} +178.600 q^{49} -377.703 q^{53} -55.0000 q^{55} +641.034 q^{59} +429.350 q^{61} -313.560 q^{65} -178.621 q^{67} -863.668 q^{71} -1027.06 q^{73} +251.224 q^{77} -785.725 q^{79} -636.536 q^{83} +384.765 q^{85} -52.0089 q^{89} +1432.25 q^{91} +752.241 q^{95} -1461.70 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 15 q^{5} - 5 q^{7} - 33 q^{11} + 2 q^{13} - 77 q^{17} + 171 q^{19} - 222 q^{23} + 75 q^{25} - 55 q^{29} + 181 q^{31} - 25 q^{35} + 317 q^{37} - 302 q^{41} - 188 q^{43} - 662 q^{47} - 268 q^{49} - 81 q^{53}+ \cdots - 1980 q^{97}+O(q^{100}) \) 3 * q + 15 * q^5 - 5 * q^7 - 33 * q^11 + 2 * q^13 - 77 * q^17 + 171 * q^19 - 222 * q^23 + 75 * q^25 - 55 * q^29 + 181 * q^31 - 25 * q^35 + 317 * q^37 - 302 * q^41 - 188 * q^43 - 662 * q^47 - 268 * q^49 - 81 * q^53 - 165 * q^55 + 42 * q^59 + 349 * q^61 + 10 * q^65 + 152 * q^67 - 927 * q^71 - 2378 * q^73 + 55 * q^77 - 1146 * q^79 + 458 * q^83 - 385 * q^85 + 875 * q^89 + 1982 * q^91 + 855 * q^95 - 1980 * q^97

Introduction

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