LMFDB - Embedded newform 3060.2.a.r.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3060,2,Mod(1,3060)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3060, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3060.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3060.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-3,0,-2,0,0,0,2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$24.4342230185$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1016.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 2 $ x^3 - x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.85577$ of defining polynomial
Character	$\chi$	$=$	3060.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +3.15544 q^{7} -0.556108 q^{11} -5.71155 q^{13} -1.00000 q^{17} +3.15544 q^{19} +7.71155 q^{23} +1.00000 q^{25} -6.55611 q^{29} +9.71155 q^{31} -3.15544 q^{35} +0.844563 q^{37} -7.75476 q^{41} +10.3109 q^{43} -8.86698 q^{47} +2.95678 q^{49} +8.86698 q^{53} +0.556108 q^{55} +10.0224 q^{59} +8.59933 q^{61} +5.71155 q^{65} +0.288455 q^{67} -0.267653 q^{73} -1.75476 q^{77} +3.11222 q^{79} +8.31087 q^{83} +1.00000 q^{85} +10.0224 q^{89} -18.0224 q^{91} -3.15544 q^{95} +9.11222 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} - 2 q^{7} + 2 q^{11} - 2 q^{13} - 3 q^{17} - 2 q^{19} + 8 q^{23} + 3 q^{25} - 16 q^{29} + 14 q^{31} + 2 q^{35} + 14 q^{37} - 4 q^{41} + 8 q^{43} + 13 q^{49} - 2 q^{55} - 8 q^{59} + 18 q^{61}+ \cdots + 20 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 - 2 * q^7 + 2 * q^11 - 2 * q^13 - 3 * q^17 - 2 * q^19 + 8 * q^23 + 3 * q^25 - 16 * q^29 + 14 * q^31 + 2 * q^35 + 14 * q^37 - 4 * q^41 + 8 * q^43 + 13 * q^49 - 2 * q^55 - 8 * q^59 + 18 * q^61 + 2 * q^65 + 16 * q^67 + 18 * q^73 + 14 * q^77 + 2 * q^79 + 2 * q^83 + 3 * q^85 - 8 * q^89 - 16 * q^91 + 2 * q^95 + 20 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.15544	1.19264	0.596321	−	0.802746i	$-0.296628\pi$
$7$	3.15544	1.19264	0.596321	+	0.802746i	$0.296628\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.556108	−0.167673	−0.0838365	−	0.996480i	$-0.526717\pi$
$11$	−0.556108	−0.167673	−0.0838365	+	0.996480i	$0.526717\pi$
$12$	0	0
$13$	−5.71155	−1.58410	−0.792049	−	0.610458i	$-0.790985\pi$
$13$	−5.71155	−1.58410	−0.792049	+	0.610458i	$0.790985\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	3.15544	0.723907	0.361953	−	0.932196i	$-0.382110\pi$
$19$	3.15544	0.723907	0.361953	+	0.932196i	$0.382110\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.71155	1.60797	0.803984	−	0.594651i	$-0.202710\pi$
$23$	7.71155	1.60797	0.803984	+	0.594651i	$0.202710\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.55611	−1.21744	−0.608719	−	0.793386i	$-0.708316\pi$
$29$	−6.55611	−1.21744	−0.608719	+	0.793386i	$0.708316\pi$
$30$	0	0
$31$	9.71155	1.74424	0.872122	−	0.489288i	$-0.162743\pi$
$31$	9.71155	1.74424	0.872122	+	0.489288i	$0.162743\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.15544	−0.533366
$36$	0	0
$37$	0.844563	0.138845	0.0694227	−	0.997587i	$-0.477884\pi$
$37$	0.844563	0.138845	0.0694227	+	0.997587i	$0.477884\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.75476	−1.21109	−0.605545	−	0.795811i	$-0.707045\pi$
$41$	−7.75476	−1.21109	−0.605545	+	0.795811i	$0.707045\pi$
$42$	0	0
$43$	10.3109	1.57239	0.786197	−	0.617976i	$-0.212047\pi$
$43$	10.3109	1.57239	0.786197	+	0.617976i	$0.212047\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.86698	−1.29338	−0.646691	−	0.762752i	$-0.723848\pi$
$47$	−8.86698	−1.29338	−0.646691	+	0.762752i	$0.723848\pi$
$48$	0	0
$49$	2.95678	0.422397
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.86698	1.21797	0.608987	−	0.793180i	$-0.291576\pi$
$53$	8.86698	1.21797	0.608987	+	0.793180i	$0.291576\pi$
$54$	0	0
$55$	0.556108	0.0749856
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.0224	1.30481	0.652404	−	0.757871i	$-0.273761\pi$
$59$	10.0224	1.30481	0.652404	+	0.757871i	$0.273761\pi$
$60$	0	0
$61$	8.59933	1.10103	0.550516	−	0.834825i	$-0.314431\pi$
$61$	8.59933	1.10103	0.550516	+	0.834825i	$0.314431\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.71155	0.708430
$66$	0	0
$67$	0.288455	0.0352404	0.0176202	−	0.999845i	$-0.494391\pi$
$67$	0.288455	0.0352404	0.0176202	+	0.999845i	$0.494391\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−0.267653	−0.0313265	−0.0156632	−	0.999877i	$-0.504986\pi$
$73$	−0.267653	−0.0313265	−0.0156632	+	0.999877i	$0.504986\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.75476	−0.199974
$78$	0	0
$79$	3.11222	0.350152	0.175076	−	0.984555i	$-0.443983\pi$
$79$	3.11222	0.350152	0.175076	+	0.984555i	$0.443983\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.31087	0.912237	0.456118	−	0.889919i	$-0.349239\pi$
$83$	8.31087	0.912237	0.456118	+	0.889919i	$0.349239\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.0224	1.06237	0.531187	−	0.847255i	$-0.321746\pi$
$89$	10.0224	1.06237	0.531187	+	0.847255i	$0.321746\pi$
$90$	0	0
$91$	−18.0224	−1.88926
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.15544	−0.323741
$96$	0	0
$97$	9.11222	0.925205	0.462603	−	0.886566i	$-0.346916\pi$
$97$	9.11222	0.925205	0.462603	+	0.886566i	$0.346916\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−3.42309	−0.340610	−0.170305	−	0.985391i	$-0.554475\pi$
$101$	−3.42309	−0.340610	−0.170305	+	0.985391i	$0.554475\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.7340	−1.71441	−0.857203	−	0.514978i	$-0.827800\pi$
$107$	−17.7340	−1.71441	−0.857203	+	0.514978i	$0.827800\pi$
$108$	0	0
$109$	−0.310873	−0.0297763	−0.0148881	−	0.999889i	$-0.504739\pi$
$109$	−0.310873	−0.0297763	−0.0148881	+	0.999889i	$0.504739\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.599328	−0.0563801	−0.0281900	−	0.999603i	$-0.508974\pi$
$113$	−0.599328	−0.0563801	−0.0281900	+	0.999603i	$0.508974\pi$
$114$	0	0
$115$	−7.71155	−0.719105
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.15544	−0.289258
$120$	0	0
$121$	−10.6907	−0.971886
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	1.48711	0.131960	0.0659799	−	0.997821i	$-0.478983\pi$
$127$	1.48711	0.131960	0.0659799	+	0.997821i	$0.478983\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.19866	0.104727	0.0523635	−	0.998628i	$-0.483325\pi$
$131$	1.19866	0.104727	0.0523635	+	0.998628i	$0.483325\pi$
$132$	0	0
$133$	9.95678	0.863362
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.55611	0.560126	0.280063	−	0.959982i	$-0.409645\pi$
$137$	6.55611	0.560126	0.280063	+	0.959982i	$0.409645\pi$
$138$	0	0
$139$	−1.42309	−0.120705	−0.0603525	−	0.998177i	$-0.519222\pi$
$139$	−1.42309	−0.120705	−0.0603525	+	0.998177i	$0.519222\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.17624	0.265610
$144$	0	0
$145$	6.55611	0.544455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−20.8238	−1.70595	−0.852975	−	0.521953i	$-0.825204\pi$
$149$	−20.8238	−1.70595	−0.852975	+	0.521953i	$0.825204\pi$
$150$	0	0
$151$	18.5785	1.51190	0.755950	−	0.654630i	$-0.227176\pi$
$151$	18.5785	1.51190	0.755950	+	0.654630i	$0.227176\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.71155	−0.780050
$156$	0	0
$157$	0.801344	0.0639542	0.0319771	−	0.999489i	$-0.489820\pi$
$157$	0.801344	0.0639542	0.0319771	+	0.999489i	$0.489820\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	24.3333	1.91773
$162$	0	0
$163$	−11.1554	−0.873761	−0.436881	−	0.899519i	$-0.643917\pi$
$163$	−11.1554	−0.873761	−0.436881	+	0.899519i	$0.643917\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.4231	1.19347	0.596737	−	0.802437i	$-0.296463\pi$
$167$	15.4231	1.19347	0.596737	+	0.802437i	$0.296463\pi$
$168$	0	0
$169$	19.6217	1.50937
$170$	0	0
$171$	0	0
$172$	0	0
$173$	13.7980	1.04904	0.524521	−	0.851398i	$-0.324245\pi$
$173$	13.7980	1.04904	0.524521	+	0.851398i	$0.324245\pi$
$174$	0	0
$175$	3.15544	0.238529
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.42309	0.255854	0.127927	−	0.991784i	$-0.459168\pi$
$179$	3.42309	0.255854	0.127927	+	0.991784i	$0.459168\pi$
$180$	0	0
$181$	−4.59933	−0.341865	−0.170933	−	0.985283i	$-0.554678\pi$
$181$	−4.59933	−0.341865	−0.170933	+	0.985283i	$0.554678\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.844563	−0.0620935
$186$	0	0
$187$	0.556108	0.0406667
$188$	0	0
$189$	0	0
$190$	0	0
$191$	23.1346	1.67396	0.836982	−	0.547231i	$-0.184318\pi$
$191$	23.1346	1.67396	0.836982	+	0.547231i	$0.184318\pi$
$192$	0	0
$193$	25.7340	1.85237	0.926186	−	0.377068i	$-0.123068\pi$
$193$	25.7340	1.85237	0.926186	+	0.377068i	$0.123068\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−26.0448	−1.85562	−0.927809	−	0.373056i	$-0.878310\pi$
$197$	−26.0448	−1.85562	−0.927809	+	0.373056i	$0.878310\pi$
$198$	0	0
$199$	3.19866	0.226747	0.113373	−	0.993552i	$-0.463834\pi$
$199$	3.19866	0.226747	0.113373	+	0.993552i	$0.463834\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−20.6874	−1.45197
$204$	0	0
$205$	7.75476	0.541616
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.75476	−0.121380
$210$	0	0
$211$	3.11222	0.214254	0.107127	−	0.994245i	$-0.465835\pi$
$211$	3.11222	0.214254	0.107127	+	0.994245i	$0.465835\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−10.3109	−0.703196
$216$	0	0
$217$	30.6442	2.08026
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.71155	0.384200
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.1346	1.53550	0.767750	−	0.640750i	$-0.221376\pi$
$227$	23.1346	1.53550	0.767750	+	0.640750i	$0.221376\pi$
$228$	0	0
$229$	5.73235	0.378804	0.189402	−	0.981900i	$-0.439345\pi$
$229$	5.73235	0.378804	0.189402	+	0.981900i	$0.439345\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.0224	−1.04966	−0.524832	−	0.851206i	$-0.675872\pi$
$233$	−16.0224	−1.04966	−0.524832	+	0.851206i	$0.675872\pi$
$234$	0	0
$235$	8.86698	0.578418
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.97758	−0.127919	−0.0639596	−	0.997952i	$-0.520373\pi$
$239$	−1.97758	−0.127919	−0.0639596	+	0.997952i	$0.520373\pi$
$240$	0	0
$241$	25.2211	1.62463	0.812316	−	0.583217i	$-0.198206\pi$
$241$	25.2211	1.62463	0.812316	+	0.583217i	$0.198206\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.95678	−0.188902
$246$	0	0
$247$	−18.0224	−1.14674
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.2469	1.53045	0.765224	−	0.643764i	$-0.222628\pi$
$251$	24.2469	1.53045	0.765224	+	0.643764i	$0.222628\pi$
$252$	0	0
$253$	−4.28845	−0.269613
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.0448	0.876093	0.438046	−	0.898952i	$-0.355671\pi$
$257$	14.0448	0.876093	0.438046	+	0.898952i	$0.355671\pi$
$258$	0	0
$259$	2.66497	0.165593
$260$	0	0
$261$	0	0
$262$	0	0
$263$	23.0914	1.42388	0.711939	−	0.702241i	$-0.247817\pi$
$263$	23.0914	1.42388	0.711939	+	0.702241i	$0.247817\pi$
$264$	0	0
$265$	−8.86698	−0.544694
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−20.8670	−1.27228	−0.636141	−	0.771573i	$-0.719470\pi$
$269$	−20.8670	−1.27228	−0.636141	+	0.771573i	$0.719470\pi$
$270$	0	0
$271$	10.2244	0.621090	0.310545	−	0.950559i	$-0.399488\pi$
$271$	10.2244	0.621090	0.310545	+	0.950559i	$0.399488\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.556108	−0.0335346
$276$	0	0
$277$	10.3109	0.619520	0.309760	−	0.950815i	$-0.399751\pi$
$277$	10.3109	0.619520	0.309760	+	0.950815i	$0.399751\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−33.2435	−1.98314	−0.991570	−	0.129570i	$-0.958640\pi$
$281$	−33.2435	−1.98314	−0.991570	+	0.129570i	$0.958640\pi$
$282$	0	0
$283$	−11.2419	−0.668260	−0.334130	−	0.942527i	$-0.608442\pi$
$283$	−11.2419	−0.668260	−0.334130	+	0.942527i	$0.608442\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−24.4697	−1.44440
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	18.6425	1.08911	0.544555	−	0.838725i	$-0.316699\pi$
$293$	18.6425	1.08911	0.544555	+	0.838725i	$0.316699\pi$
$294$	0	0
$295$	−10.0224	−0.583528
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−44.0448	−2.54718
$300$	0	0
$301$	32.5353	1.87530
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−8.59933	−0.492396
$306$	0	0
$307$	−34.8462	−1.98878	−0.994388	−	0.105792i	$-0.966262\pi$
$307$	−34.8462	−1.98878	−0.994388	+	0.105792i	$0.966262\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	11.4439	0.648924	0.324462	−	0.945899i	$-0.394817\pi$
$311$	11.4439	0.648924	0.324462	+	0.945899i	$0.394817\pi$
$312$	0	0
$313$	18.5785	1.05012	0.525060	−	0.851065i	$-0.324043\pi$
$313$	18.5785	1.05012	0.525060	+	0.851065i	$0.324043\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	25.7115	1.44410	0.722052	−	0.691839i	$-0.243199\pi$
$317$	25.7115	1.44410	0.722052	+	0.691839i	$0.243199\pi$
$318$	0	0
$319$	3.64591	0.204132
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.15544	−0.175573
$324$	0	0
$325$	−5.71155	−0.316820
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−27.9792	−1.54254
$330$	0	0
$331$	−27.6907	−1.52202	−0.761010	−	0.648740i	$-0.775296\pi$
$331$	−27.6907	−1.52202	−0.761010	+	0.648740i	$0.775296\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−0.288455	−0.0157600
$336$	0	0
$337$	−15.6907	−0.854729	−0.427365	−	0.904079i	$-0.640558\pi$
$337$	−15.6907	−0.854729	−0.427365	+	0.904079i	$0.640558\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5.40067	−0.292463
$342$	0	0
$343$	−12.7581	−0.688874
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.48711	−0.294564	−0.147282	−	0.989095i	$-0.547052\pi$
$347$	−5.48711	−0.294564	−0.147282	+	0.989095i	$0.547052\pi$
$348$	0	0
$349$	−26.3125	−1.40848	−0.704238	−	0.709964i	$-0.748711\pi$
$349$	−26.3125	−1.40848	−0.704238	+	0.709964i	$0.748711\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.53033	0.294350	0.147175	−	0.989110i	$-0.452982\pi$
$353$	5.53033	0.294350	0.147175	+	0.989110i	$0.452982\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−18.5993	−0.981635	−0.490817	−	0.871262i	$-0.663302\pi$
$359$	−18.5993	−0.981635	−0.490817	+	0.871262i	$0.663302\pi$
$360$	0	0
$361$	−9.04322	−0.475959
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.267653	0.0140096
$366$	0	0
$367$	−5.19866	−0.271368	−0.135684	−	0.990752i	$-0.543323\pi$
$367$	−5.19866	−0.271368	−0.135684	+	0.990752i	$0.543323\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	27.9792	1.45261
$372$	0	0
$373$	−19.1571	−0.991915	−0.495957	−	0.868347i	$-0.665183\pi$
$373$	−19.1571	−0.991915	−0.495957	+	0.868347i	$0.665183\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	37.4455	1.92854
$378$	0	0
$379$	−25.4231	−1.30590	−0.652948	−	0.757403i	$-0.726468\pi$
$379$	−25.4231	−1.30590	−0.652948	+	0.757403i	$0.726468\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.02080	0.103258	0.0516291	−	0.998666i	$-0.483559\pi$
$383$	2.02080	0.103258	0.0516291	+	0.998666i	$0.483559\pi$
$384$	0	0
$385$	1.75476	0.0894311
$386$	0	0
$387$	0	0
$388$	0	0
$389$	21.1571	1.07271	0.536353	−	0.843994i	$-0.319802\pi$
$389$	21.1571	1.07271	0.536353	+	0.843994i	$0.319802\pi$
$390$	0	0
$391$	−7.71155	−0.389990
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3.11222	−0.156593
$396$	0	0
$397$	33.9152	1.70215	0.851077	−	0.525041i	$-0.175950\pi$
$397$	33.9152	1.70215	0.851077	+	0.525041i	$0.175950\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−13.4887	−0.673595	−0.336797	−	0.941577i	$-0.609344\pi$
$401$	−13.4887	−0.673595	−0.336797	+	0.941577i	$0.609344\pi$
$402$	0	0
$403$	−55.4679	−2.76305
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.469669	−0.0232806
$408$	0	0
$409$	−0.533690	−0.0263893	−0.0131946	−	0.999913i	$-0.504200\pi$
$409$	−0.533690	−0.0263893	−0.0131946	+	0.999913i	$0.504200\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	31.6251	1.55617
$414$	0	0
$415$	−8.31087	−0.407965
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−14.8670	−0.726300	−0.363150	−	0.931731i	$-0.618299\pi$
$419$	−14.8670	−0.726300	−0.363150	+	0.931731i	$0.618299\pi$
$420$	0	0
$421$	−25.2003	−1.22819	−0.614093	−	0.789234i	$-0.710478\pi$
$421$	−25.2003	−1.22819	−0.614093	+	0.789234i	$0.710478\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	27.1346	1.31314
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.09142	−0.245245	−0.122622	−	0.992453i	$-0.539130\pi$
$431$	−5.09142	−0.245245	−0.122622	+	0.992453i	$0.539130\pi$
$432$	0	0
$433$	18.5353	0.890750	0.445375	−	0.895344i	$-0.353070\pi$
$433$	18.5353	0.890750	0.445375	+	0.895344i	$0.353070\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	24.3333	1.16402
$438$	0	0
$439$	−21.2211	−1.01283	−0.506413	−	0.862291i	$-0.669029\pi$
$439$	−21.2211	−1.01283	−0.506413	+	0.862291i	$0.669029\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7.11222	−0.337912	−0.168956	−	0.985624i	$-0.554040\pi$
$443$	−7.11222	−0.337912	−0.168956	+	0.985624i	$0.554040\pi$
$444$	0	0
$445$	−10.0224	−0.475108
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−8.39731	−0.396294	−0.198147	−	0.980172i	$-0.563492\pi$
$449$	−8.39731	−0.396294	−0.198147	+	0.980172i	$0.563492\pi$
$450$	0	0
$451$	4.31249	0.203067
$452$	0	0
$453$	0	0
$454$	0	0
$455$	18.0224	0.844904
$456$	0	0
$457$	−35.4455	−1.65807	−0.829035	−	0.559196i	$-0.811110\pi$
$457$	−35.4455	−1.65807	−0.829035	+	0.559196i	$0.811110\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−31.7115	−1.47695	−0.738477	−	0.674279i	$-0.764455\pi$
$461$	−31.7115	−1.47695	−0.738477	+	0.674279i	$0.764455\pi$
$462$	0	0
$463$	−5.19866	−0.241602	−0.120801	−	0.992677i	$-0.538546\pi$
$463$	−5.19866	−0.241602	−0.120801	+	0.992677i	$0.538546\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.84120	0.362848	0.181424	−	0.983405i	$-0.441929\pi$
$467$	7.84120	0.362848	0.181424	+	0.983405i	$0.441929\pi$
$468$	0	0
$469$	0.910201	0.0420292
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−5.73396	−0.263648
$474$	0	0
$475$	3.15544	0.144781
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−15.4231	−0.704699	−0.352349	−	0.935868i	$-0.614617\pi$
$479$	−15.4231	−0.704699	−0.352349	+	0.935868i	$0.614617\pi$
$480$	0	0
$481$	−4.82376	−0.219945
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9.11222	−0.413764
$486$	0	0
$487$	19.4679	0.882176	0.441088	−	0.897464i	$-0.354593\pi$
$487$	19.4679	0.882176	0.441088	+	0.897464i	$0.354593\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15.5095	0.699935	0.349968	−	0.936762i	$-0.386193\pi$
$491$	15.5095	0.699935	0.349968	+	0.936762i	$0.386193\pi$
$492$	0	0
$493$	6.55611	0.295272
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	28.5577	1.27842	0.639210	−	0.769033i	$-0.279262\pi$
$499$	28.5577	1.27842	0.639210	+	0.769033i	$0.279262\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−16.7082	−0.744981	−0.372491	−	0.928036i	$-0.621496\pi$
$503$	−16.7082	−0.744981	−0.372491	+	0.928036i	$0.621496\pi$
$504$	0	0
$505$	3.42309	0.152326
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−12.3333	−0.546664	−0.273332	−	0.961920i	$-0.588126\pi$
$509$	−12.3333	−0.546664	−0.273332	+	0.961920i	$0.588126\pi$
$510$	0	0
$511$	−0.844563	−0.0373613
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.00000	−0.352522
$516$	0	0
$517$	4.93100	0.216865
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−11.0914	−0.485924	−0.242962	−	0.970036i	$-0.578119\pi$
$521$	−11.0914	−0.485924	−0.242962	+	0.970036i	$0.578119\pi$
$522$	0	0
$523$	1.73396	0.0758209	0.0379105	−	0.999281i	$-0.487930\pi$
$523$	1.73396	0.0758209	0.0379105	+	0.999281i	$0.487930\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.71155	−0.423042
$528$	0	0
$529$	36.4679	1.58556
$530$	0	0
$531$	0	0
$532$	0	0
$533$	44.2917	1.91849
$534$	0	0
$535$	17.7340	0.766706
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.64429	−0.0708246
$540$	0	0
$541$	−1.08980	−0.0468541	−0.0234271	−	0.999726i	$-0.507458\pi$
$541$	−1.08980	−0.0468541	−0.0234271	+	0.999726i	$0.507458\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0.310873	0.0133163
$546$	0	0
$547$	5.37987	0.230027	0.115013	−	0.993364i	$-0.463309\pi$
$547$	5.37987	0.230027	0.115013	+	0.993364i	$0.463309\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−20.6874	−0.881312
$552$	0	0
$553$	9.82040	0.417606
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.19866	−0.305017	−0.152508	−	0.988302i	$-0.548735\pi$
$557$	−7.19866	−0.305017	−0.152508	+	0.988302i	$0.548735\pi$
$558$	0	0
$559$	−58.8910	−2.49082
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−27.7132	−1.16797	−0.583985	−	0.811764i	$-0.698507\pi$
$563$	−27.7132	−1.16797	−0.583985	+	0.811764i	$0.698507\pi$
$564$	0	0
$565$	0.599328	0.0252139
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−17.7340	−0.743446	−0.371723	−	0.928344i	$-0.621233\pi$
$569$	−17.7340	−0.743446	−0.371723	+	0.928344i	$0.621233\pi$
$570$	0	0
$571$	−26.2885	−1.10014	−0.550069	−	0.835119i	$-0.685399\pi$
$571$	−26.2885	−1.10014	−0.550069	+	0.835119i	$0.685399\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.71155	0.321594
$576$	0	0
$577$	−30.0448	−1.25078	−0.625391	−	0.780311i	$-0.715061\pi$
$577$	−30.0448	−1.25078	−0.625391	+	0.780311i	$0.715061\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	26.2244	1.08797
$582$	0	0
$583$	−4.93100	−0.204221
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11.0914	0.457792	0.228896	−	0.973451i	$-0.426488\pi$
$587$	11.0914	0.457792	0.228896	+	0.973451i	$0.426488\pi$
$588$	0	0
$589$	30.6442	1.26267
$590$	0	0
$591$	0	0
$592$	0	0
$593$	24.2036	0.993924	0.496962	−	0.867772i	$-0.334449\pi$
$593$	24.2036	0.993924	0.496962	+	0.867772i	$0.334449\pi$
$594$	0	0
$595$	3.15544	0.129360
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−16.6217	−0.679146	−0.339573	−	0.940580i	$-0.610283\pi$
$599$	−16.6217	−0.679146	−0.339573	+	0.940580i	$0.610283\pi$
$600$	0	0
$601$	−38.7822	−1.58196	−0.790979	−	0.611844i	$-0.790428\pi$
$601$	−38.7822	−1.58196	−0.790979	+	0.611844i	$0.790428\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.6907	0.434641
$606$	0	0
$607$	−14.4921	−0.588216	−0.294108	−	0.955772i	$-0.595022\pi$
$607$	−14.4921	−0.588216	−0.294108	+	0.955772i	$0.595022\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	50.6442	2.04884
$612$	0	0
$613$	−43.2435	−1.74659	−0.873294	−	0.487193i	$-0.838021\pi$
$613$	−43.2435	−1.74659	−0.873294	+	0.487193i	$0.838021\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	33.8428	1.36246	0.681230	−	0.732069i	$-0.261445\pi$
$617$	33.8428	1.36246	0.681230	+	0.732069i	$0.261445\pi$
$618$	0	0
$619$	−42.1313	−1.69340	−0.846699	−	0.532071i	$-0.821414\pi$
$619$	−42.1313	−1.69340	−0.846699	+	0.532071i	$0.821414\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	31.6251	1.26703
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−0.844563	−0.0336749
$630$	0	0
$631$	−25.4663	−1.01380	−0.506899	−	0.862006i	$-0.669208\pi$
$631$	−25.4663	−1.01380	−0.506899	+	0.862006i	$0.669208\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.48711	−0.0590142
$636$	0	0
$637$	−16.8878	−0.669118
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−36.2901	−1.43337	−0.716686	−	0.697396i	$-0.754342\pi$
$641$	−36.2901	−1.43337	−0.716686	+	0.697396i	$0.754342\pi$
$642$	0	0
$643$	32.9758	1.30044	0.650220	−	0.759746i	$-0.274677\pi$
$643$	32.9758	1.30044	0.650220	+	0.759746i	$0.274677\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.1571	1.06765	0.533827	−	0.845594i	$-0.320753\pi$
$647$	27.1571	1.06765	0.533827	+	0.845594i	$0.320753\pi$
$648$	0	0
$649$	−5.57355	−0.218781
$650$	0	0
$651$	0	0
$652$	0	0
$653$	33.5095	1.31133	0.655665	−	0.755052i	$-0.272389\pi$
$653$	33.5095	1.31133	0.655665	+	0.755052i	$0.272389\pi$
$654$	0	0
$655$	−1.19866	−0.0468354
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−39.5095	−1.53907	−0.769536	−	0.638603i	$-0.779513\pi$
$659$	−39.5095	−1.53907	−0.769536	+	0.638603i	$0.779513\pi$
$660$	0	0
$661$	43.4247	1.68903	0.844513	−	0.535536i	$-0.179890\pi$
$661$	43.4247	1.68903	0.844513	+	0.535536i	$0.179890\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−9.95678	−0.386107
$666$	0	0
$667$	−50.5577	−1.95760
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.78216	−0.184613
$672$	0	0
$673$	36.7082	1.41500	0.707498	−	0.706715i	$-0.249824\pi$
$673$	36.7082	1.41500	0.707498	+	0.706715i	$0.249824\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−34.0897	−1.31017	−0.655086	−	0.755554i	$-0.727368\pi$
$677$	−34.0897	−1.31017	−0.655086	+	0.755554i	$0.727368\pi$
$678$	0	0
$679$	28.7530	1.10344
$680$	0	0
$681$	0	0
$682$	0	0
$683$	50.3109	1.92509	0.962546	−	0.271118i	$-0.0873935\pi$
$683$	50.3109	1.92509	0.962546	+	0.271118i	$0.0873935\pi$
$684$	0	0
$685$	−6.55611	−0.250496
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−50.6442	−1.92939
$690$	0	0
$691$	−47.6924	−1.81430	−0.907151	−	0.420804i	$-0.861748\pi$
$691$	−47.6924	−1.81430	−0.907151	+	0.420804i	$0.861748\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.42309	0.0539809
$696$	0	0
$697$	7.75476	0.293733
$698$	0	0
$699$	0	0
$700$	0	0
$701$	24.1604	0.912526	0.456263	−	0.889845i	$-0.349188\pi$
$701$	24.1604	0.912526	0.456263	+	0.889845i	$0.349188\pi$
$702$	0	0
$703$	2.66497	0.100511
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10.8013	−0.406226
$708$	0	0
$709$	−12.3109	−0.462345	−0.231172	−	0.972913i	$-0.574256\pi$
$709$	−12.3109	−0.462345	−0.231172	+	0.972913i	$0.574256\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	74.8910	2.80469
$714$	0	0
$715$	−3.17624	−0.118785
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.86698	−0.106920	−0.0534602	−	0.998570i	$-0.517025\pi$
$719$	−2.86698	−0.106920	−0.0534602	+	0.998570i	$0.517025\pi$
$720$	0	0
$721$	25.2435	0.940117
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6.55611	−0.243488
$726$	0	0
$727$	−0.576910	−0.0213964	−0.0106982	−	0.999943i	$-0.503405\pi$
$727$	−0.576910	−0.0213964	−0.0106982	+	0.999943i	$0.503405\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−10.3109	−0.381361
$732$	0	0
$733$	−3.48711	−0.128799	−0.0643997	−	0.997924i	$-0.520513\pi$
$733$	−3.48711	−0.128799	−0.0643997	+	0.997924i	$0.520513\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.160412	−0.00590886
$738$	0	0
$739$	−24.5769	−0.904076	−0.452038	−	0.891999i	$-0.649303\pi$
$739$	−24.5769	−0.904076	−0.452038	+	0.891999i	$0.649303\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	26.8910	0.986536	0.493268	−	0.869877i	$-0.335802\pi$
$743$	26.8910	0.986536	0.493268	+	0.869877i	$0.335802\pi$
$744$	0	0
$745$	20.8238	0.762924
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−55.9584	−2.04468
$750$	0	0
$751$	−13.7564	−0.501977	−0.250989	−	0.967990i	$-0.580756\pi$
$751$	−13.7564	−0.501977	−0.250989	+	0.967990i	$0.580756\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−18.5785	−0.676142
$756$	0	0
$757$	−46.4937	−1.68984	−0.844921	−	0.534891i	$-0.820353\pi$
$757$	−46.4937	−1.68984	−0.844921	+	0.534891i	$0.820353\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	44.2917	1.60557	0.802786	−	0.596267i	$-0.203350\pi$
$761$	44.2917	1.60557	0.802786	+	0.596267i	$0.203350\pi$
$762$	0	0
$763$	−0.980941	−0.0355124
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−57.2435	−2.06694
$768$	0	0
$769$	−0.620130	−0.0223625	−0.0111812	−	0.999937i	$-0.503559\pi$
$769$	−0.620130	−0.0223625	−0.0111812	+	0.999937i	$0.503559\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−25.4887	−0.916766	−0.458383	−	0.888755i	$-0.651571\pi$
$773$	−25.4887	−0.916766	−0.458383	+	0.888755i	$0.651571\pi$
$774$	0	0
$775$	9.71155	0.348849
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−24.4697	−0.876717
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−0.801344	−0.0286012
$786$	0	0
$787$	30.0465	1.07104	0.535520	−	0.844522i	$-0.320116\pi$
$787$	30.0465	1.07104	0.535520	+	0.844522i	$0.320116\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.89114	−0.0672413
$792$	0	0
$793$	−49.1155	−1.74414
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−24.2901	−0.860398	−0.430199	−	0.902734i	$-0.641557\pi$
$797$	−24.2901	−0.860398	−0.430199	+	0.902734i	$0.641557\pi$
$798$	0	0
$799$	8.86698	0.313691
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0.148844	0.00525260
$804$	0	0
$805$	−24.3333	−0.857636
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.37825	−0.0484568	−0.0242284	−	0.999706i	$-0.507713\pi$
$809$	−1.37825	−0.0484568	−0.0242284	+	0.999706i	$0.507713\pi$
$810$	0	0
$811$	14.2469	0.500275	0.250137	−	0.968210i	$-0.419524\pi$
$811$	14.2469	0.500275	0.250137	+	0.968210i	$0.419524\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	11.1554	0.390758
$816$	0	0
$817$	32.5353	1.13827
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−12.8462	−0.448335	−0.224167	−	0.974551i	$-0.571966\pi$
$821$	−12.8462	−0.448335	−0.224167	+	0.974551i	$0.571966\pi$
$822$	0	0
$823$	1.01744	0.0354658	0.0177329	−	0.999843i	$-0.494355\pi$
$823$	1.01744	0.0354658	0.0177329	+	0.999843i	$0.494355\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.51289	0.226475	0.113238	−	0.993568i	$-0.463878\pi$
$827$	6.51289	0.226475	0.113238	+	0.993568i	$0.463878\pi$
$828$	0	0
$829$	−18.3541	−0.637464	−0.318732	−	0.947845i	$-0.603257\pi$
$829$	−18.3541	−0.637464	−0.318732	+	0.947845i	$0.603257\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.95678	−0.102446
$834$	0	0
$835$	−15.4231	−0.533738
$836$	0	0
$837$	0	0
$838$	0	0
$839$	32.6009	1.12551	0.562755	−	0.826624i	$-0.309742\pi$
$839$	32.6009	1.12551	0.562755	+	0.826624i	$0.309742\pi$
$840$	0	0
$841$	13.9826	0.482157
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−19.6217	−0.675009
$846$	0	0
$847$	−33.7340	−1.15911
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.51289	0.223259
$852$	0	0
$853$	14.7597	0.505364	0.252682	−	0.967549i	$-0.418687\pi$
$853$	14.7597	0.505364	0.252682	+	0.967549i	$0.418687\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14.5769	0.497938	0.248969	−	0.968511i	$-0.419908\pi$
$857$	14.5769	0.497938	0.248969	+	0.968511i	$0.419908\pi$
$858$	0	0
$859$	−28.9758	−0.988643	−0.494321	−	0.869279i	$-0.664583\pi$
$859$	−28.9758	−0.988643	−0.494321	+	0.869279i	$0.664583\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−9.97920	−0.339696	−0.169848	−	0.985470i	$-0.554328\pi$
$863$	−9.97920	−0.339696	−0.169848	+	0.985470i	$0.554328\pi$
$864$	0	0
$865$	−13.7980	−0.469146
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.73073	−0.0587110
$870$	0	0
$871$	−1.64752	−0.0558242
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.15544	−0.106673
$876$	0	0
$877$	−7.73235	−0.261103	−0.130551	−	0.991442i	$-0.541675\pi$
$877$	−7.73235	−0.261103	−0.130551	+	0.991442i	$0.541675\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−40.8254	−1.37544	−0.687721	−	0.725975i	$-0.741389\pi$
$881$	−40.8254	−1.37544	−0.687721	+	0.725975i	$0.741389\pi$
$882$	0	0
$883$	−29.6184	−0.996738	−0.498369	−	0.866965i	$-0.666068\pi$
$883$	−29.6184	−0.996738	−0.498369	+	0.866965i	$0.666068\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.60269	0.322427	0.161213	−	0.986920i	$-0.448459\pi$
$887$	9.60269	0.322427	0.161213	+	0.986920i	$0.448459\pi$
$888$	0	0
$889$	4.69249	0.157381
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−27.9792	−0.936288
$894$	0	0
$895$	−3.42309	−0.114421
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−63.6699	−2.12351
$900$	0	0
$901$	−8.86698	−0.295402
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.59933	0.152887
$906$	0	0
$907$	7.77718	0.258237	0.129119	−	0.991629i	$-0.458785\pi$
$907$	7.77718	0.258237	0.129119	+	0.991629i	$0.458785\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	51.5336	1.70738	0.853692	−	0.520779i	$-0.174358\pi$
$911$	51.5336	1.70738	0.853692	+	0.520779i	$0.174358\pi$
$912$	0	0
$913$	−4.62175	−0.152957
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.78228	0.124902
$918$	0	0
$919$	39.8220	1.31361	0.656804	−	0.754062i	$-0.271908\pi$
$919$	39.8220	1.31361	0.656804	+	0.754062i	$0.271908\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0.844563	0.0277691
$926$	0	0
$927$	0	0
$928$	0	0
$929$	39.7132	1.30295	0.651473	−	0.758672i	$-0.274151\pi$
$929$	39.7132	1.30295	0.651473	+	0.758672i	$0.274151\pi$
$930$	0	0
$931$	9.32993	0.305776
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−0.556108	−0.0181867
$936$	0	0
$937$	−17.7980	−0.581435	−0.290717	−	0.956809i	$-0.593894\pi$
$937$	−17.7980	−0.581435	−0.290717	+	0.956809i	$0.593894\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	6.17288	0.201230	0.100615	−	0.994925i	$-0.467919\pi$
$941$	6.17288	0.201230	0.100615	+	0.994925i	$0.467919\pi$
$942$	0	0
$943$	−59.8012	−1.94740
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−6.35248	−0.206428	−0.103214	−	0.994659i	$-0.532913\pi$
$947$	−6.35248	−0.206428	−0.103214	+	0.994659i	$0.532913\pi$
$948$	0	0
$949$	1.52871	0.0496242
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−0.290072	−0.00939634	−0.00469817	−	0.999989i	$-0.501495\pi$
$953$	−0.290072	−0.00939634	−0.00469817	+	0.999989i	$0.501495\pi$
$954$	0	0
$955$	−23.1346	−0.748619
$956$	0	0
$957$	0	0
$958$	0	0
$959$	20.6874	0.668030
$960$	0	0
$961$	63.3141	2.04239
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−25.7340	−0.828406
$966$	0	0
$967$	35.1762	1.13119	0.565596	−	0.824683i	$-0.308646\pi$
$967$	35.1762	1.13119	0.565596	+	0.824683i	$0.308646\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−18.6858	−0.599655	−0.299827	−	0.953993i	$-0.596929\pi$
$971$	−18.6858	−0.599655	−0.299827	+	0.953993i	$0.596929\pi$
$972$	0	0
$973$	−4.49047	−0.143958
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−27.2435	−0.871597	−0.435798	−	0.900044i	$-0.643534\pi$
$977$	−27.2435	−0.871597	−0.435798	+	0.900044i	$0.643534\pi$
$978$	0	0
$979$	−5.57355	−0.178131
$980$	0	0
$981$	0	0
$982$	0	0
$983$	40.8686	1.30351	0.651753	−	0.758431i	$-0.274034\pi$
$983$	40.8686	1.30351	0.651753	+	0.758431i	$0.274034\pi$
$984$	0	0
$985$	26.0448	0.829857
$986$	0	0
$987$	0	0
$988$	0	0
$989$	79.5128	2.52836
$990$	0	0
$991$	−29.7980	−0.946564	−0.473282	−	0.880911i	$-0.656931\pi$
$991$	−29.7980	−0.946564	−0.473282	+	0.880911i	$0.656931\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3.19866	−0.101404
$996$	0	0
$997$	9.33503	0.295643	0.147822	−	0.989014i	$-0.452774\pi$
$997$	9.33503	0.295643	0.147822	+	0.989014i	$0.452774\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3060.2.a.r.1.3	✓	3
3.2	odd	2		3060.2.a.t.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3060.2.a.r.1.3	✓	3	1.1	even	1	trivial
3060.2.a.t.1.3	yes	3	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3060.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +3.15544 q^{7} -0.556108 q^{11} -5.71155 q^{13} -1.00000 q^{17} +3.15544 q^{19} +7.71155 q^{23} +1.00000 q^{25} -6.55611 q^{29} +9.71155 q^{31} -3.15544 q^{35} +0.844563 q^{37} -7.75476 q^{41} +10.3109 q^{43} -8.86698 q^{47} +2.95678 q^{49} +8.86698 q^{53} +0.556108 q^{55} +10.0224 q^{59} +8.59933 q^{61} +5.71155 q^{65} +0.288455 q^{67} -0.267653 q^{73} -1.75476 q^{77} +3.11222 q^{79} +8.31087 q^{83} +1.00000 q^{85} +10.0224 q^{89} -18.0224 q^{91} -3.15544 q^{95} +9.11222 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} - 2 q^{7} + 2 q^{11} - 2 q^{13} - 3 q^{17} - 2 q^{19} + 8 q^{23} + 3 q^{25} - 16 q^{29} + 14 q^{31} + 2 q^{35} + 14 q^{37} - 4 q^{41} + 8 q^{43} + 13 q^{49} - 2 q^{55} - 8 q^{59} + 18 q^{61}+ \cdots + 20 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 - 2 * q^7 + 2 * q^11 - 2 * q^13 - 3 * q^17 - 2 * q^19 + 8 * q^23 + 3 * q^25 - 16 * q^29 + 14 * q^31 + 2 * q^35 + 14 * q^37 - 4 * q^41 + 8 * q^43 + 13 * q^49 - 2 * q^55 - 8 * q^59 + 18 * q^61 + 2 * q^65 + 16 * q^67 + 18 * q^73 + 14 * q^77 + 2 * q^79 + 2 * q^83 + 3 * q^85 - 8 * q^89 - 16 * q^91 + 2 * q^95 + 20 * q^97

Introduction

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