LMFDB - Embedded newform 1620.4.a.k.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(1,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1620.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.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:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 167x^{5} + 940x^{4} + 1153x^{3} - 13196x^{2} + 16629x + 714 $ x^7 - 2x^6 - 167x^5 + 940x^4 + 1153x^3 - 13196x^2 + 16629x + 714
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{7} $
Twist minimal:	no (minimal twist has level 180)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$5.54588$ of defining polynomial
Character	$\chi$	$=$	1620.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000 q^{5} +13.6186 q^{7} +O(q^{10})$	$q-5.00000 q^{5} +13.6186 q^{7} -27.8199 q^{11} +59.2596 q^{13} -87.5404 q^{17} +161.499 q^{19} +131.921 q^{23} +25.0000 q^{25} -151.333 q^{29} -228.135 q^{31} -68.0932 q^{35} +236.332 q^{37} -226.701 q^{41} -152.098 q^{43} -129.675 q^{47} -157.533 q^{49} +492.124 q^{53} +139.100 q^{55} +879.010 q^{59} -434.013 q^{61} -296.298 q^{65} +539.716 q^{67} -65.4619 q^{71} +313.792 q^{73} -378.870 q^{77} +885.759 q^{79} -744.929 q^{83} +437.702 q^{85} +401.899 q^{89} +807.035 q^{91} -807.497 q^{95} -1077.15 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 35 q^{5} + 8 q^{7}+O(q^{10}) $ 7 * q - 35 * q^5 + 8 * q^7	$ 7 q - 35 q^{5} + 8 q^{7} - 27 q^{11} + 32 q^{13} - 123 q^{17} - 67 q^{19} - 42 q^{23} + 175 q^{25} - 324 q^{29} + 98 q^{31} - 40 q^{35} + 356 q^{37} - 339 q^{41} + 119 q^{43} + 96 q^{47} + 813 q^{49} - 858 q^{53} + 135 q^{55} - 549 q^{59} + 260 q^{61} - 160 q^{65} + 881 q^{67} - 342 q^{71} + 737 q^{73} + 456 q^{77} + 1886 q^{79} + 132 q^{83} + 615 q^{85} - 396 q^{89} + 1264 q^{91} + 335 q^{95} + 1991 q^{97}+O(q^{100}) $ 7 * q - 35 * q^5 + 8 * q^7 - 27 * q^11 + 32 * q^13 - 123 * q^17 - 67 * q^19 - 42 * q^23 + 175 * q^25 - 324 * q^29 + 98 * q^31 - 40 * q^35 + 356 * q^37 - 339 * q^41 + 119 * q^43 + 96 * q^47 + 813 * q^49 - 858 * q^53 + 135 * q^55 - 549 * q^59 + 260 * q^61 - 160 * q^65 + 881 * q^67 - 342 * q^71 + 737 * q^73 + 456 * q^77 + 1886 * q^79 + 132 * q^83 + 615 * q^85 - 396 * q^89 + 1264 * q^91 + 335 * q^95 + 1991 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	13.6186	0.735337	0.367669	−	0.929957i	$-0.380156\pi$
$7$	13.6186	0.735337	0.367669	+	0.929957i	$0.380156\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−27.8199	−0.762548	−0.381274	−	0.924462i	$-0.624515\pi$
$11$	−27.8199	−0.762548	−0.381274	+	0.924462i	$0.624515\pi$
$12$	0	0
$13$	59.2596	1.26428	0.632140	−	0.774854i	$-0.282177\pi$
$13$	59.2596	1.26428	0.632140	+	0.774854i	$0.282177\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−87.5404	−1.24892	−0.624461	−	0.781056i	$-0.714681\pi$
$17$	−87.5404	−1.24892	−0.624461	+	0.781056i	$0.714681\pi$
$18$	0	0
$19$	161.499	1.95003	0.975013	−	0.222148i	$-0.0713068\pi$
$19$	161.499	1.95003	0.975013	+	0.222148i	$0.0713068\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	131.921	1.19598	0.597990	−	0.801504i	$-0.295966\pi$
$23$	131.921	1.19598	0.597990	+	0.801504i	$0.295966\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−151.333	−0.969026	−0.484513	−	0.874784i	$-0.661003\pi$
$29$	−151.333	−0.969026	−0.484513	+	0.874784i	$0.661003\pi$
$30$	0	0
$31$	−228.135	−1.32175	−0.660875	−	0.750496i	$-0.729815\pi$
$31$	−228.135	−1.32175	−0.660875	+	0.750496i	$0.729815\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−68.0932	−0.328853
$36$	0	0
$37$	236.332	1.05008	0.525038	−	0.851079i	$-0.324051\pi$
$37$	236.332	1.05008	0.525038	+	0.851079i	$0.324051\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−226.701	−0.863532	−0.431766	−	0.901986i	$-0.642109\pi$
$41$	−226.701	−0.863532	−0.431766	+	0.901986i	$0.642109\pi$
$42$	0	0
$43$	−152.098	−0.539411	−0.269706	−	0.962943i	$-0.586926\pi$
$43$	−152.098	−0.539411	−0.269706	+	0.962943i	$0.586926\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−129.675	−0.402446	−0.201223	−	0.979545i	$-0.564492\pi$
$47$	−129.675	−0.402446	−0.201223	+	0.979545i	$0.564492\pi$
$48$	0	0
$49$	−157.533	−0.459279
$50$	0	0
$51$	0	0
$52$	0	0
$53$	492.124	1.27544	0.637721	−	0.770267i	$-0.279877\pi$
$53$	492.124	1.27544	0.637721	+	0.770267i	$0.279877\pi$
$54$	0	0
$55$	139.100	0.341022
$56$	0	0
$57$	0	0
$58$	0	0
$59$	879.010	1.93962	0.969809	−	0.243867i	$-0.0784162\pi$
$59$	879.010	1.93962	0.969809	+	0.243867i	$0.0784162\pi$
$60$	0	0
$61$	−434.013	−0.910979	−0.455489	−	0.890241i	$-0.650536\pi$
$61$	−434.013	−0.910979	−0.455489	+	0.890241i	$0.650536\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−296.298	−0.565403
$66$	0	0
$67$	539.716	0.984131	0.492066	−	0.870558i	$-0.336242\pi$
$67$	539.716	0.984131	0.492066	+	0.870558i	$0.336242\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−65.4619	−0.109421	−0.0547106	−	0.998502i	$-0.517424\pi$
$71$	−65.4619	−0.109421	−0.0547106	+	0.998502i	$0.517424\pi$
$72$	0	0
$73$	313.792	0.503104	0.251552	−	0.967844i	$-0.419059\pi$
$73$	313.792	0.503104	0.251552	+	0.967844i	$0.419059\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−378.870	−0.560730
$78$	0	0
$79$	885.759	1.26146	0.630732	−	0.776001i	$-0.282755\pi$
$79$	885.759	1.26146	0.630732	+	0.776001i	$0.282755\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−744.929	−0.985140	−0.492570	−	0.870273i	$-0.663942\pi$
$83$	−744.929	−0.985140	−0.492570	+	0.870273i	$0.663942\pi$
$84$	0	0
$85$	437.702	0.558535
$86$	0	0
$87$	0	0
$88$	0	0
$89$	401.899	0.478665	0.239332	−	0.970938i	$-0.423071\pi$
$89$	401.899	0.478665	0.239332	+	0.970938i	$0.423071\pi$
$90$	0	0
$91$	807.035	0.929673
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−807.497	−0.872078
$96$	0	0
$97$	−1077.15	−1.12750	−0.563750	−	0.825945i	$-0.690642\pi$
$97$	−1077.15	−1.12750	−0.563750	+	0.825945i	$0.690642\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−904.463	−0.891064	−0.445532	−	0.895266i	$-0.646985\pi$
$101$	−904.463	−0.891064	−0.445532	+	0.895266i	$0.646985\pi$
$102$	0	0
$103$	−239.332	−0.228952	−0.114476	−	0.993426i	$-0.536519\pi$
$103$	−239.332	−0.228952	−0.114476	+	0.993426i	$0.536519\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	933.267	0.843200	0.421600	−	0.906782i	$-0.361469\pi$
$107$	933.267	0.843200	0.421600	+	0.906782i	$0.361469\pi$
$108$	0	0
$109$	1182.89	1.03945	0.519725	−	0.854334i	$-0.326035\pi$
$109$	1182.89	1.03945	0.519725	+	0.854334i	$0.326035\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1599.46	1.33154	0.665772	−	0.746155i	$-0.268102\pi$
$113$	1599.46	1.33154	0.665772	+	0.746155i	$0.268102\pi$
$114$	0	0
$115$	−659.607	−0.534858
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1192.18	−0.918379
$120$	0	0
$121$	−557.051	−0.418521
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−674.489	−0.471270	−0.235635	−	0.971842i	$-0.575717\pi$
$127$	−674.489	−0.471270	−0.235635	+	0.971842i	$0.575717\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	446.886	0.298051	0.149025	−	0.988833i	$-0.452386\pi$
$131$	446.886	0.298051	0.149025	+	0.988833i	$0.452386\pi$
$132$	0	0
$133$	2199.40	1.43393
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1835.59	1.14471	0.572353	−	0.820007i	$-0.306031\pi$
$137$	1835.59	1.14471	0.572353	+	0.820007i	$0.306031\pi$
$138$	0	0
$139$	131.710	0.0803703	0.0401851	−	0.999192i	$-0.487205\pi$
$139$	131.710	0.0803703	0.0401851	+	0.999192i	$0.487205\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1648.60	−0.964075
$144$	0	0
$145$	756.663	0.433362
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−426.756	−0.234639	−0.117320	−	0.993094i	$-0.537430\pi$
$149$	−426.756	−0.234639	−0.117320	+	0.993094i	$0.537430\pi$
$150$	0	0
$151$	16.3620	0.00881804	0.00440902	−	0.999990i	$-0.498597\pi$
$151$	16.3620	0.00881804	0.00440902	+	0.999990i	$0.498597\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1140.68	0.591105
$156$	0	0
$157$	1450.21	0.737192	0.368596	−	0.929590i	$-0.379839\pi$
$157$	1450.21	0.737192	0.368596	+	0.929590i	$0.379839\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1796.59	0.879448
$162$	0	0
$163$	2990.94	1.43723	0.718615	−	0.695408i	$-0.244776\pi$
$163$	2990.94	1.43723	0.718615	+	0.695408i	$0.244776\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2809.35	1.30176	0.650880	−	0.759180i	$-0.274400\pi$
$167$	2809.35	1.30176	0.650880	+	0.759180i	$0.274400\pi$
$168$	0	0
$169$	1314.70	0.598405
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1156.42	0.508214	0.254107	−	0.967176i	$-0.418218\pi$
$173$	1156.42	0.508214	0.254107	+	0.967176i	$0.418218\pi$
$174$	0	0
$175$	340.466	0.147067
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3271.49	−1.36605	−0.683025	−	0.730395i	$-0.739336\pi$
$179$	−3271.49	−1.36605	−0.683025	+	0.730395i	$0.739336\pi$
$180$	0	0
$181$	3919.89	1.60974	0.804871	−	0.593450i	$-0.202234\pi$
$181$	3919.89	1.60974	0.804871	+	0.593450i	$0.202234\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1181.66	−0.469608
$186$	0	0
$187$	2435.37	0.952362
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3342.25	1.26616	0.633080	−	0.774086i	$-0.281790\pi$
$191$	3342.25	1.26616	0.633080	+	0.774086i	$0.281790\pi$
$192$	0	0
$193$	1198.07	0.446834	0.223417	−	0.974723i	$-0.428279\pi$
$193$	1198.07	0.446834	0.223417	+	0.974723i	$0.428279\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2152.33	−0.778412	−0.389206	−	0.921151i	$-0.627251\pi$
$197$	−2152.33	−0.778412	−0.389206	+	0.921151i	$0.627251\pi$
$198$	0	0
$199$	−4276.77	−1.52348	−0.761738	−	0.647885i	$-0.775654\pi$
$199$	−4276.77	−1.52348	−0.761738	+	0.647885i	$0.775654\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2060.94	−0.712561
$204$	0	0
$205$	1133.51	0.386183
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4492.90	−1.48699
$210$	0	0
$211$	3812.36	1.24386	0.621928	−	0.783074i	$-0.286349\pi$
$211$	3812.36	1.24386	0.621928	+	0.783074i	$0.286349\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	760.489	0.241232
$216$	0	0
$217$	−3106.89	−0.971933
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5187.61	−1.57899
$222$	0	0
$223$	3038.85	0.912540	0.456270	−	0.889841i	$-0.349185\pi$
$223$	3038.85	0.912540	0.456270	+	0.889841i	$0.349185\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2079.42	0.608000	0.304000	−	0.952672i	$-0.401678\pi$
$227$	2079.42	0.608000	0.304000	+	0.952672i	$0.401678\pi$
$228$	0	0
$229$	500.883	0.144538	0.0722692	−	0.997385i	$-0.476976\pi$
$229$	500.883	0.144538	0.0722692	+	0.997385i	$0.476976\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4249.83	1.19492	0.597459	−	0.801900i	$-0.296177\pi$
$233$	4249.83	1.19492	0.597459	+	0.801900i	$0.296177\pi$
$234$	0	0
$235$	648.373	0.179979
$236$	0	0
$237$	0	0
$238$	0	0
$239$	451.391	0.122168	0.0610838	−	0.998133i	$-0.480544\pi$
$239$	451.391	0.122168	0.0610838	+	0.998133i	$0.480544\pi$
$240$	0	0
$241$	3323.55	0.888335	0.444168	−	0.895944i	$-0.353499\pi$
$241$	3323.55	0.888335	0.444168	+	0.895944i	$0.353499\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	787.663	0.205396
$246$	0	0
$247$	9570.38	2.46538
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4266.89	1.07300	0.536501	−	0.843900i	$-0.319746\pi$
$251$	4266.89	1.07300	0.536501	+	0.843900i	$0.319746\pi$
$252$	0	0
$253$	−3670.05	−0.911991
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−588.967	−0.142952	−0.0714761	−	0.997442i	$-0.522771\pi$
$257$	−588.967	−0.142952	−0.0714761	+	0.997442i	$0.522771\pi$
$258$	0	0
$259$	3218.52	0.772160
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2758.21	0.646686	0.323343	−	0.946282i	$-0.395193\pi$
$263$	2758.21	0.646686	0.323343	+	0.946282i	$0.395193\pi$
$264$	0	0
$265$	−2460.62	−0.570395
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4510.91	−1.02243	−0.511217	−	0.859451i	$-0.670805\pi$
$269$	−4510.91	−1.02243	−0.511217	+	0.859451i	$0.670805\pi$
$270$	0	0
$271$	942.723	0.211315	0.105658	−	0.994403i	$-0.466305\pi$
$271$	942.723	0.211315	0.105658	+	0.994403i	$0.466305\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−695.499	−0.152510
$276$	0	0
$277$	7726.43	1.67594	0.837972	−	0.545713i	$-0.183741\pi$
$277$	7726.43	1.67594	0.837972	+	0.545713i	$0.183741\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6614.42	1.40421	0.702105	−	0.712073i	$-0.252244\pi$
$281$	6614.42	1.40421	0.702105	+	0.712073i	$0.252244\pi$
$282$	0	0
$283$	6864.27	1.44183	0.720916	−	0.693023i	$-0.243721\pi$
$283$	6864.27	1.44183	0.720916	+	0.693023i	$0.243721\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3087.36	−0.634987
$288$	0	0
$289$	2750.32	0.559804
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1020.61	−0.203498	−0.101749	−	0.994810i	$-0.532444\pi$
$293$	−1020.61	−0.203498	−0.101749	+	0.994810i	$0.532444\pi$
$294$	0	0
$295$	−4395.05	−0.867423
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7817.61	1.51205
$300$	0	0
$301$	−2071.36	−0.396649
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2170.07	0.407402
$306$	0	0
$307$	−648.350	−0.120532	−0.0602659	−	0.998182i	$-0.519195\pi$
$307$	−648.350	−0.120532	−0.0602659	+	0.998182i	$0.519195\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5275.73	−0.961927	−0.480963	−	0.876741i	$-0.659713\pi$
$311$	−5275.73	−0.961927	−0.480963	+	0.876741i	$0.659713\pi$
$312$	0	0
$313$	197.203	0.0356121	0.0178060	−	0.999841i	$-0.494332\pi$
$313$	197.203	0.0356121	0.0178060	+	0.999841i	$0.494332\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11195.2	−1.98355	−0.991777	−	0.127977i	$-0.959152\pi$
$317$	−11195.2	−1.98355	−0.991777	+	0.127977i	$0.959152\pi$
$318$	0	0
$319$	4210.06	0.738929
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−14137.7	−2.43543
$324$	0	0
$325$	1481.49	0.252856
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1765.99	−0.295934
$330$	0	0
$331$	−8156.78	−1.35449	−0.677247	−	0.735756i	$-0.736827\pi$
$331$	−8156.78	−1.35449	−0.677247	+	0.735756i	$0.736827\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2698.58	−0.440117
$336$	0	0
$337$	−7762.96	−1.25482	−0.627412	−	0.778688i	$-0.715886\pi$
$337$	−7762.96	−1.25482	−0.627412	+	0.778688i	$0.715886\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6346.70	1.00790
$342$	0	0
$343$	−6816.57	−1.07306
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6996.22	1.08235	0.541177	−	0.840909i	$-0.317979\pi$
$347$	6996.22	1.08235	0.541177	+	0.840909i	$0.317979\pi$
$348$	0	0
$349$	6579.70	1.00918	0.504589	−	0.863360i	$-0.331644\pi$
$349$	6579.70	1.00918	0.504589	+	0.863360i	$0.331644\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10319.7	−1.55598	−0.777989	−	0.628278i	$-0.783760\pi$
$353$	−10319.7	−1.55598	−0.777989	+	0.628278i	$0.783760\pi$
$354$	0	0
$355$	327.310	0.0489346
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5878.30	0.864192	0.432096	−	0.901828i	$-0.357774\pi$
$359$	5878.30	0.864192	0.432096	+	0.901828i	$0.357774\pi$
$360$	0	0
$361$	19223.0	2.80260
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1568.96	−0.224995
$366$	0	0
$367$	9953.87	1.41577	0.707885	−	0.706327i	$-0.249649\pi$
$367$	9953.87	1.41577	0.707885	+	0.706327i	$0.249649\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6702.06	0.937881
$372$	0	0
$373$	−3583.36	−0.497424	−0.248712	−	0.968577i	$-0.580007\pi$
$373$	−3583.36	−0.497424	−0.248712	+	0.968577i	$0.580007\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8967.91	−1.22512
$378$	0	0
$379$	−3724.30	−0.504761	−0.252380	−	0.967628i	$-0.581213\pi$
$379$	−3724.30	−0.504761	−0.252380	+	0.967628i	$0.581213\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3313.13	−0.442019	−0.221009	−	0.975272i	$-0.570935\pi$
$383$	−3313.13	−0.442019	−0.221009	+	0.975272i	$0.570935\pi$
$384$	0	0
$385$	1894.35	0.250766
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2541.56	0.331265	0.165632	−	0.986188i	$-0.447034\pi$
$389$	2541.56	0.331265	0.165632	+	0.986188i	$0.447034\pi$
$390$	0	0
$391$	−11548.5	−1.49368
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4428.79	−0.564144
$396$	0	0
$397$	−3595.24	−0.454508	−0.227254	−	0.973835i	$-0.572975\pi$
$397$	−3595.24	−0.454508	−0.227254	+	0.973835i	$0.572975\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4811.47	0.599185	0.299593	−	0.954067i	$-0.403149\pi$
$401$	4811.47	0.599185	0.299593	+	0.954067i	$0.403149\pi$
$402$	0	0
$403$	−13519.2	−1.67106
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6574.75	−0.800733
$408$	0	0
$409$	13355.7	1.61467	0.807333	−	0.590096i	$-0.200910\pi$
$409$	13355.7	1.61467	0.807333	+	0.590096i	$0.200910\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	11970.9	1.42627
$414$	0	0
$415$	3724.64	0.440568
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14486.3	1.68903	0.844513	−	0.535535i	$-0.179890\pi$
$419$	14486.3	1.68903	0.844513	+	0.535535i	$0.179890\pi$
$420$	0	0
$421$	−8027.60	−0.929314	−0.464657	−	0.885491i	$-0.653822\pi$
$421$	−8027.60	−0.929314	−0.464657	+	0.885491i	$0.653822\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2188.51	−0.249784
$426$	0	0
$427$	−5910.67	−0.669877
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8130.75	0.908688	0.454344	−	0.890826i	$-0.349874\pi$
$431$	8130.75	0.908688	0.454344	+	0.890826i	$0.349874\pi$
$432$	0	0
$433$	15235.0	1.69087	0.845435	−	0.534078i	$-0.179341\pi$
$433$	15235.0	1.69087	0.845435	+	0.534078i	$0.179341\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	21305.2	2.33219
$438$	0	0
$439$	−15660.5	−1.70258	−0.851290	−	0.524695i	$-0.824179\pi$
$439$	−15660.5	−1.70258	−0.851290	+	0.524695i	$0.824179\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−850.329	−0.0911972	−0.0455986	−	0.998960i	$-0.514520\pi$
$443$	−850.329	−0.0911972	−0.0455986	+	0.998960i	$0.514520\pi$
$444$	0	0
$445$	−2009.49	−0.214065
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6341.65	0.666550	0.333275	−	0.942830i	$-0.391846\pi$
$449$	6341.65	0.666550	0.333275	+	0.942830i	$0.391846\pi$
$450$	0	0
$451$	6306.82	0.658485
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4035.17	−0.415762
$456$	0	0
$457$	3907.04	0.399921	0.199960	−	0.979804i	$-0.435919\pi$
$457$	3907.04	0.399921	0.199960	+	0.979804i	$0.435919\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2168.52	0.219085	0.109543	−	0.993982i	$-0.465061\pi$
$461$	2168.52	0.219085	0.109543	+	0.993982i	$0.465061\pi$
$462$	0	0
$463$	−9111.18	−0.914541	−0.457271	−	0.889328i	$-0.651173\pi$
$463$	−9111.18	−0.914541	−0.457271	+	0.889328i	$0.651173\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3019.67	−0.299216	−0.149608	−	0.988745i	$-0.547801\pi$
$467$	−3019.67	−0.299216	−0.149608	+	0.988745i	$0.547801\pi$
$468$	0	0
$469$	7350.20	0.723669
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4231.35	0.411327
$474$	0	0
$475$	4037.48	0.390005
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3105.37	−0.296217	−0.148108	−	0.988971i	$-0.547318\pi$
$479$	−3105.37	−0.296217	−0.148108	+	0.988971i	$0.547318\pi$
$480$	0	0
$481$	14005.0	1.32759
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5385.73	0.504234
$486$	0	0
$487$	−13984.4	−1.30122	−0.650610	−	0.759412i	$-0.725487\pi$
$487$	−13984.4	−1.30122	−0.650610	+	0.759412i	$0.725487\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−18639.6	−1.71323	−0.856613	−	0.515960i	$-0.827435\pi$
$491$	−18639.6	−1.71323	−0.856613	+	0.515960i	$0.827435\pi$
$492$	0	0
$493$	13247.7	1.21024
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−891.502	−0.0804615
$498$	0	0
$499$	9501.38	0.852385	0.426192	−	0.904633i	$-0.359855\pi$
$499$	9501.38	0.852385	0.426192	+	0.904633i	$0.359855\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−11812.6	−1.04711	−0.523555	−	0.851992i	$-0.675395\pi$
$503$	−11812.6	−1.04711	−0.523555	+	0.851992i	$0.675395\pi$
$504$	0	0
$505$	4522.31	0.398496
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−11768.8	−1.02484	−0.512421	−	0.858734i	$-0.671251\pi$
$509$	−11768.8	−1.02484	−0.512421	+	0.858734i	$0.671251\pi$
$510$	0	0
$511$	4273.42	0.369951
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1196.66	0.102390
$516$	0	0
$517$	3607.54	0.306885
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10546.1	0.886823	0.443412	−	0.896318i	$-0.353768\pi$
$521$	10546.1	0.886823	0.443412	+	0.896318i	$0.353768\pi$
$522$	0	0
$523$	−2003.65	−0.167521	−0.0837605	−	0.996486i	$-0.526693\pi$
$523$	−2003.65	−0.167521	−0.0837605	+	0.996486i	$0.526693\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	19971.0	1.65076
$528$	0	0
$529$	5236.26	0.430366
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−13434.2	−1.09175
$534$	0	0
$535$	−4666.34	−0.377090
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4382.55	0.350222
$540$	0	0
$541$	9845.65	0.782436	0.391218	−	0.920298i	$-0.372054\pi$
$541$	9845.65	0.782436	0.391218	+	0.920298i	$0.372054\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5914.43	−0.464856
$546$	0	0
$547$	−9883.20	−0.772532	−0.386266	−	0.922387i	$-0.626235\pi$
$547$	−9883.20	−0.772532	−0.386266	+	0.922387i	$0.626235\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−24440.1	−1.88963
$552$	0	0
$553$	12062.8	0.927602
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19026.9	1.44739	0.723694	−	0.690121i	$-0.242443\pi$
$557$	19026.9	1.44739	0.723694	+	0.690121i	$0.242443\pi$
$558$	0	0
$559$	−9013.25	−0.681967
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−15704.9	−1.17563	−0.587817	−	0.808994i	$-0.700012\pi$
$563$	−15704.9	−1.17563	−0.587817	+	0.808994i	$0.700012\pi$
$564$	0	0
$565$	−7997.30	−0.595485
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8049.12	0.593035	0.296517	−	0.955027i	$-0.404175\pi$
$569$	8049.12	0.593035	0.296517	+	0.955027i	$0.404175\pi$
$570$	0	0
$571$	−17218.1	−1.26192	−0.630959	−	0.775816i	$-0.717338\pi$
$571$	−17218.1	−1.26192	−0.630959	+	0.775816i	$0.717338\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3298.04	0.239196
$576$	0	0
$577$	5463.08	0.394161	0.197081	−	0.980387i	$-0.436854\pi$
$577$	5463.08	0.394161	0.197081	+	0.980387i	$0.436854\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−10144.9	−0.724410
$582$	0	0
$583$	−13690.9	−0.972586
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3729.26	−0.262220	−0.131110	−	0.991368i	$-0.541854\pi$
$587$	−3729.26	−0.262220	−0.131110	+	0.991368i	$0.541854\pi$
$588$	0	0
$589$	−36843.7	−2.57745
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−19960.4	−1.38225	−0.691127	−	0.722734i	$-0.742885\pi$
$593$	−19960.4	−1.38225	−0.691127	+	0.722734i	$0.742885\pi$
$594$	0	0
$595$	5960.90	0.410711
$596$	0	0
$597$	0	0
$598$	0	0
$599$	19375.8	1.32166	0.660830	−	0.750535i	$-0.270204\pi$
$599$	19375.8	1.32166	0.660830	+	0.750535i	$0.270204\pi$
$600$	0	0
$601$	−23674.5	−1.60682	−0.803412	−	0.595423i	$-0.796985\pi$
$601$	−23674.5	−1.60682	−0.803412	+	0.595423i	$0.796985\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2785.25	0.187168
$606$	0	0
$607$	252.857	0.0169080	0.00845401	−	0.999964i	$-0.497309\pi$
$607$	252.857	0.0169080	0.00845401	+	0.999964i	$0.497309\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7684.46	−0.508805
$612$	0	0
$613$	−28573.8	−1.88268	−0.941340	−	0.337459i	$-0.890432\pi$
$613$	−28573.8	−1.88268	−0.941340	+	0.337459i	$0.890432\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3522.00	0.229806	0.114903	−	0.993377i	$-0.463344\pi$
$617$	3522.00	0.229806	0.114903	+	0.993377i	$0.463344\pi$
$618$	0	0
$619$	−4711.05	−0.305901	−0.152951	−	0.988234i	$-0.548878\pi$
$619$	−4711.05	−0.305901	−0.152951	+	0.988234i	$0.548878\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5473.31	0.351980
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−20688.6	−1.31146
$630$	0	0
$631$	−18288.5	−1.15381	−0.576903	−	0.816812i	$-0.695739\pi$
$631$	−18288.5	−1.15381	−0.576903	+	0.816812i	$0.695739\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3372.45	0.210758
$636$	0	0
$637$	−9335.32	−0.580657
$638$	0	0
$639$	0	0
$640$	0	0
$641$	29803.9	1.83648	0.918240	−	0.396023i	$-0.129610\pi$
$641$	29803.9	1.83648	0.918240	+	0.396023i	$0.129610\pi$
$642$	0	0
$643$	−15029.4	−0.921774	−0.460887	−	0.887459i	$-0.652469\pi$
$643$	−15029.4	−0.921774	−0.460887	+	0.887459i	$0.652469\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12528.7	0.761291	0.380646	−	0.924721i	$-0.375702\pi$
$647$	12528.7	0.761291	0.380646	+	0.924721i	$0.375702\pi$
$648$	0	0
$649$	−24454.0	−1.47905
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−24272.3	−1.45459	−0.727297	−	0.686323i	$-0.759224\pi$
$653$	−24272.3	−1.45459	−0.727297	+	0.686323i	$0.759224\pi$
$654$	0	0
$655$	−2234.43	−0.133292
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−19946.8	−1.17908	−0.589542	−	0.807738i	$-0.700692\pi$
$659$	−19946.8	−1.17908	−0.589542	+	0.807738i	$0.700692\pi$
$660$	0	0
$661$	−14580.6	−0.857972	−0.428986	−	0.903311i	$-0.641129\pi$
$661$	−14580.6	−0.857972	−0.428986	+	0.903311i	$0.641129\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−10997.0	−0.641272
$666$	0	0
$667$	−19964.0	−1.15894
$668$	0	0
$669$	0	0
$670$	0	0
$671$	12074.2	0.694665
$672$	0	0
$673$	8335.04	0.477403	0.238702	−	0.971093i	$-0.423278\pi$
$673$	8335.04	0.477403	0.238702	+	0.971093i	$0.423278\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29395.0	1.66875	0.834373	−	0.551200i	$-0.185830\pi$
$677$	29395.0	1.66875	0.834373	+	0.551200i	$0.185830\pi$
$678$	0	0
$679$	−14669.3	−0.829093
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−11333.3	−0.634930	−0.317465	−	0.948270i	$-0.602832\pi$
$683$	−11333.3	−0.634930	−0.317465	+	0.948270i	$0.602832\pi$
$684$	0	0
$685$	−9177.93	−0.511928
$686$	0	0
$687$	0	0
$688$	0	0
$689$	29163.1	1.61252
$690$	0	0
$691$	−8590.67	−0.472944	−0.236472	−	0.971638i	$-0.575991\pi$
$691$	−8590.67	−0.472944	−0.236472	+	0.971638i	$0.575991\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−658.548	−0.0359427
$696$	0	0
$697$	19845.5	1.07848
$698$	0	0
$699$	0	0
$700$	0	0
$701$	10609.3	0.571621	0.285810	−	0.958286i	$-0.407737\pi$
$701$	10609.3	0.571621	0.285810	+	0.958286i	$0.407737\pi$
$702$	0	0
$703$	38167.5	2.04767
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−12317.6	−0.655232
$708$	0	0
$709$	3733.08	0.197742	0.0988708	−	0.995100i	$-0.468477\pi$
$709$	3733.08	0.197742	0.0988708	+	0.995100i	$0.468477\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−30095.9	−1.58079
$714$	0	0
$715$	8242.99	0.431147
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−882.767	−0.0457881	−0.0228940	−	0.999738i	$-0.507288\pi$
$719$	−882.767	−0.0457881	−0.0228940	+	0.999738i	$0.507288\pi$
$720$	0	0
$721$	−3259.37	−0.168357
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3783.32	−0.193805
$726$	0	0
$727$	13239.5	0.675412	0.337706	−	0.941252i	$-0.390349\pi$
$727$	13239.5	0.675412	0.337706	+	0.941252i	$0.390349\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	13314.7	0.673682
$732$	0	0
$733$	15988.2	0.805643	0.402822	−	0.915278i	$-0.368029\pi$
$733$	15988.2	0.805643	0.402822	+	0.915278i	$0.368029\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−15014.9	−0.750447
$738$	0	0
$739$	−11634.4	−0.579134	−0.289567	−	0.957158i	$-0.593511\pi$
$739$	−11634.4	−0.579134	−0.289567	+	0.957158i	$0.593511\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−9024.76	−0.445607	−0.222804	−	0.974863i	$-0.571521\pi$
$743$	−9024.76	−0.445607	−0.222804	+	0.974863i	$0.571521\pi$
$744$	0	0
$745$	2133.78	0.104934
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12709.8	0.620036
$750$	0	0
$751$	4775.68	0.232047	0.116023	−	0.993246i	$-0.462985\pi$
$751$	4775.68	0.232047	0.116023	+	0.993246i	$0.462985\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−81.8102	−0.00394355
$756$	0	0
$757$	−14333.5	−0.688188	−0.344094	−	0.938935i	$-0.611814\pi$
$757$	−14333.5	−0.688188	−0.344094	+	0.938935i	$0.611814\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	20806.4	0.991105	0.495553	−	0.868578i	$-0.334966\pi$
$761$	20806.4	0.991105	0.495553	+	0.868578i	$0.334966\pi$
$762$	0	0
$763$	16109.3	0.764346
$764$	0	0
$765$	0	0
$766$	0	0
$767$	52089.8	2.45222
$768$	0	0
$769$	2390.32	0.112090	0.0560449	−	0.998428i	$-0.482151\pi$
$769$	2390.32	0.112090	0.0560449	+	0.998428i	$0.482151\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−12562.6	−0.584534	−0.292267	−	0.956337i	$-0.594410\pi$
$773$	−12562.6	−0.584534	−0.292267	+	0.956337i	$0.594410\pi$
$774$	0	0
$775$	−5703.38	−0.264350
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−36612.1	−1.68391
$780$	0	0
$781$	1821.15	0.0834389
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7251.03	−0.329682
$786$	0	0
$787$	−32883.4	−1.48941	−0.744706	−	0.667392i	$-0.767410\pi$
$787$	−32883.4	−1.48941	−0.744706	+	0.667392i	$0.767410\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	21782.5	0.979135
$792$	0	0
$793$	−25719.4	−1.15173
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−36420.9	−1.61869	−0.809344	−	0.587335i	$-0.800177\pi$
$797$	−36420.9	−1.61869	−0.809344	+	0.587335i	$0.800177\pi$
$798$	0	0
$799$	11351.8	0.502624
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−8729.67	−0.383641
$804$	0	0
$805$	−8982.95	−0.393301
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8400.94	0.365094	0.182547	−	0.983197i	$-0.441566\pi$
$809$	8400.94	0.365094	0.182547	+	0.983197i	$0.441566\pi$
$810$	0	0
$811$	−23278.5	−1.00792	−0.503958	−	0.863728i	$-0.668123\pi$
$811$	−23278.5	−1.00792	−0.503958	+	0.863728i	$0.668123\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−14954.7	−0.642749
$816$	0	0
$817$	−24563.7	−1.05187
$818$	0	0
$819$	0	0
$820$	0	0
$821$	42994.1	1.82765	0.913827	−	0.406105i	$-0.133113\pi$
$821$	42994.1	1.82765	0.913827	+	0.406105i	$0.133113\pi$
$822$	0	0
$823$	−16806.8	−0.711843	−0.355921	−	0.934516i	$-0.615833\pi$
$823$	−16806.8	−0.711843	−0.355921	+	0.934516i	$0.615833\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−21733.5	−0.913843	−0.456922	−	0.889507i	$-0.651048\pi$
$827$	−21733.5	−0.913843	−0.456922	+	0.889507i	$0.651048\pi$
$828$	0	0
$829$	25186.5	1.05520	0.527601	−	0.849492i	$-0.323092\pi$
$829$	25186.5	1.05520	0.527601	+	0.849492i	$0.323092\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	13790.5	0.573603
$834$	0	0
$835$	−14046.8	−0.582165
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−5183.60	−0.213299	−0.106649	−	0.994297i	$-0.534012\pi$
$839$	−5183.60	−0.213299	−0.106649	+	0.994297i	$0.534012\pi$
$840$	0	0
$841$	−1487.44	−0.0609881
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−6573.48	−0.267615
$846$	0	0
$847$	−7586.28	−0.307754
$848$	0	0
$849$	0	0
$850$	0	0
$851$	31177.3	1.25587
$852$	0	0
$853$	3426.60	0.137544	0.0687718	−	0.997632i	$-0.478092\pi$
$853$	3426.60	0.137544	0.0687718	+	0.997632i	$0.478092\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−33348.7	−1.32925	−0.664627	−	0.747175i	$-0.731409\pi$
$857$	−33348.7	−1.32925	−0.664627	+	0.747175i	$0.731409\pi$
$858$	0	0
$859$	2848.63	0.113148	0.0565740	−	0.998398i	$-0.481982\pi$
$859$	2848.63	0.113148	0.0565740	+	0.998398i	$0.481982\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	13389.8	0.528151	0.264075	−	0.964502i	$-0.414933\pi$
$863$	13389.8	0.528151	0.264075	+	0.964502i	$0.414933\pi$
$864$	0	0
$865$	−5782.10	−0.227280
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−24641.8	−0.961927
$870$	0	0
$871$	31983.3	1.24422
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1702.33	−0.0657706
$876$	0	0
$877$	28580.1	1.10044	0.550218	−	0.835021i	$-0.314545\pi$
$877$	28580.1	1.10044	0.550218	+	0.835021i	$0.314545\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−9302.28	−0.355734	−0.177867	−	0.984055i	$-0.556920\pi$
$881$	−9302.28	−0.355734	−0.177867	+	0.984055i	$0.556920\pi$
$882$	0	0
$883$	−38674.2	−1.47394	−0.736970	−	0.675925i	$-0.763744\pi$
$883$	−38674.2	−1.47394	−0.736970	+	0.675925i	$0.763744\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−22829.4	−0.864188	−0.432094	−	0.901829i	$-0.642225\pi$
$887$	−22829.4	−0.864188	−0.432094	+	0.901829i	$0.642225\pi$
$888$	0	0
$889$	−9185.62	−0.346542
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−20942.4	−0.784781
$894$	0	0
$895$	16357.5	0.610916
$896$	0	0
$897$	0	0
$898$	0	0
$899$	34524.3	1.28081
$900$	0	0
$901$	−43080.7	−1.59293
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−19599.5	−0.719899
$906$	0	0
$907$	−43396.7	−1.58872	−0.794358	−	0.607450i	$-0.792192\pi$
$907$	−43396.7	−1.58872	−0.794358	+	0.607450i	$0.792192\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	10206.6	0.371197	0.185599	−	0.982626i	$-0.440578\pi$
$911$	10206.6	0.371197	0.185599	+	0.982626i	$0.440578\pi$
$912$	0	0
$913$	20723.9	0.751216
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6085.98	0.219168
$918$	0	0
$919$	−34124.4	−1.22488	−0.612438	−	0.790519i	$-0.709811\pi$
$919$	−34124.4	−1.22488	−0.612438	+	0.790519i	$0.709811\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3879.25	−0.138339
$924$	0	0
$925$	5908.31	0.210015
$926$	0	0
$927$	0	0
$928$	0	0
$929$	30487.7	1.07672	0.538359	−	0.842716i	$-0.319045\pi$
$929$	30487.7	1.07672	0.538359	+	0.842716i	$0.319045\pi$
$930$	0	0
$931$	−25441.4	−0.895606
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−12176.8	−0.425909
$936$	0	0
$937$	23053.8	0.803774	0.401887	−	0.915689i	$-0.368355\pi$
$937$	23053.8	0.803774	0.401887	+	0.915689i	$0.368355\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	44284.4	1.53414	0.767072	−	0.641561i	$-0.221713\pi$
$941$	44284.4	1.53414	0.767072	+	0.641561i	$0.221713\pi$
$942$	0	0
$943$	−29906.8	−1.03277
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16100.7	−0.552483	−0.276241	−	0.961088i	$-0.589089\pi$
$947$	−16100.7	−0.552483	−0.276241	+	0.961088i	$0.589089\pi$
$948$	0	0
$949$	18595.2	0.636064
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−35933.7	−1.22141	−0.610706	−	0.791857i	$-0.709114\pi$
$953$	−35933.7	−1.22141	−0.610706	+	0.791857i	$0.709114\pi$
$954$	0	0
$955$	−16711.3	−0.566244
$956$	0	0
$957$	0	0
$958$	0	0
$959$	24998.2	0.841745
$960$	0	0
$961$	22254.6	0.747025
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−5990.35	−0.199830
$966$	0	0
$967$	−4887.24	−0.162527	−0.0812633	−	0.996693i	$-0.525895\pi$
$967$	−4887.24	−0.162527	−0.0812633	+	0.996693i	$0.525895\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	31305.2	1.03463	0.517317	−	0.855794i	$-0.326931\pi$
$971$	31305.2	1.03463	0.517317	+	0.855794i	$0.326931\pi$
$972$	0	0
$973$	1793.71	0.0590993
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−22083.0	−0.723130	−0.361565	−	0.932347i	$-0.617757\pi$
$977$	−22083.0	−0.723130	−0.361565	+	0.932347i	$0.617757\pi$
$978$	0	0
$979$	−11180.8	−0.365005
$980$	0	0
$981$	0	0
$982$	0	0
$983$	26669.4	0.865332	0.432666	−	0.901554i	$-0.357573\pi$
$983$	26669.4	0.865332	0.432666	+	0.901554i	$0.357573\pi$
$984$	0	0
$985$	10761.7	0.348117
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−20065.0	−0.645125
$990$	0	0
$991$	40357.4	1.29364	0.646819	−	0.762644i	$-0.276099\pi$
$991$	40357.4	1.29364	0.646819	+	0.762644i	$0.276099\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	21383.8	0.681319
$996$	0	0
$997$	23774.0	0.755194	0.377597	−	0.925970i	$-0.376750\pi$
$997$	23774.0	0.755194	0.377597	+	0.925970i	$0.376750\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.a.k.1.5		7
3.2	odd	2		1620.4.a.l.1.5		7
9.2	odd	6		180.4.i.c.121.2	yes	14
9.4	even	3		540.4.i.c.181.3		14
9.5	odd	6		180.4.i.c.61.2	✓	14
9.7	even	3		540.4.i.c.361.3		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.4.i.c.61.2	✓	14	9.5	odd	6
180.4.i.c.121.2	yes	14	9.2	odd	6
540.4.i.c.181.3		14	9.4	even	3
540.4.i.c.361.3		14	9.7	even	3
1620.4.a.k.1.5		7	1.1	even	1	trivial
1620.4.a.l.1.5		7	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} +13.6186 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} +13.6186 q^{7} -27.8199 q^{11} +59.2596 q^{13} -87.5404 q^{17} +161.499 q^{19} +131.921 q^{23} +25.0000 q^{25} -151.333 q^{29} -228.135 q^{31} -68.0932 q^{35} +236.332 q^{37} -226.701 q^{41} -152.098 q^{43} -129.675 q^{47} -157.533 q^{49} +492.124 q^{53} +139.100 q^{55} +879.010 q^{59} -434.013 q^{61} -296.298 q^{65} +539.716 q^{67} -65.4619 q^{71} +313.792 q^{73} -378.870 q^{77} +885.759 q^{79} -744.929 q^{83} +437.702 q^{85} +401.899 q^{89} +807.035 q^{91} -807.497 q^{95} -1077.15 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 35 q^{5} + 8 q^{7}+O(q^{10}) \) 7 * q - 35 * q^5 + 8 * q^7	\( 7 q - 35 q^{5} + 8 q^{7} - 27 q^{11} + 32 q^{13} - 123 q^{17} - 67 q^{19} - 42 q^{23} + 175 q^{25} - 324 q^{29} + 98 q^{31} - 40 q^{35} + 356 q^{37} - 339 q^{41} + 119 q^{43} + 96 q^{47} + 813 q^{49} - 858 q^{53} + 135 q^{55} - 549 q^{59} + 260 q^{61} - 160 q^{65} + 881 q^{67} - 342 q^{71} + 737 q^{73} + 456 q^{77} + 1886 q^{79} + 132 q^{83} + 615 q^{85} - 396 q^{89} + 1264 q^{91} + 335 q^{95} + 1991 q^{97}+O(q^{100}) \) 7 * q - 35 * q^5 + 8 * q^7 - 27 * q^11 + 32 * q^13 - 123 * q^17 - 67 * q^19 - 42 * q^23 + 175 * q^25 - 324 * q^29 + 98 * q^31 - 40 * q^35 + 356 * q^37 - 339 * q^41 + 119 * q^43 + 96 * q^47 + 813 * q^49 - 858 * q^53 + 135 * q^55 - 549 * q^59 + 260 * q^61 - 160 * q^65 + 881 * q^67 - 342 * q^71 + 737 * q^73 + 456 * q^77 + 1886 * q^79 + 132 * q^83 + 615 * q^85 - 396 * q^89 + 1264 * q^91 + 335 * q^95 + 1991 * q^97

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