LMFDB - Embedded newform 2100.4.a.y.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,4,Mod(1,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2100.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2100.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:	$123.904011012$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.4281.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 12x + 15 $ x^3 - x^2 - 12*x + 15
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 420)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.29027$ of defining polynomial
Character	$\chi$	$=$	2100.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -7.00000 q^{7} +9.00000 q^{9} +1.01865 q^{11} -63.5205 q^{13} +33.8576 q^{17} +30.4646 q^{19} -21.0000 q^{21} +147.169 q^{23} +27.0000 q^{27} -275.745 q^{29} +39.5205 q^{31} +3.05596 q^{33} +289.101 q^{37} -190.562 q^{39} +173.476 q^{41} -132.981 q^{43} -202.360 q^{47} +49.0000 q^{49} +101.573 q^{51} +76.1889 q^{53} +91.3938 q^{57} -799.100 q^{59} +780.418 q^{61} -63.0000 q^{63} -465.687 q^{67} +441.506 q^{69} -280.728 q^{71} -840.569 q^{73} -7.13057 q^{77} -303.041 q^{79} +81.0000 q^{81} -785.695 q^{83} -827.235 q^{87} -706.075 q^{89} +444.644 q^{91} +118.562 q^{93} +1075.31 q^{97} +9.16787 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 9 q^{3} - 21 q^{7} + 27 q^{9} - 54 q^{11} - 42 q^{13} + 56 q^{17} + 114 q^{19} - 63 q^{21} + 64 q^{23} + 81 q^{27} - 130 q^{29} - 30 q^{31} - 162 q^{33} + 216 q^{37} - 126 q^{39} - 6 q^{41} - 456 q^{43}+ \cdots - 486 q^{99}+O(q^{100}) $ 3 * q + 9 * q^3 - 21 * q^7 + 27 * q^9 - 54 * q^11 - 42 * q^13 + 56 * q^17 + 114 * q^19 - 63 * q^21 + 64 * q^23 + 81 * q^27 - 130 * q^29 - 30 * q^31 - 162 * q^33 + 216 * q^37 - 126 * q^39 - 6 * q^41 - 456 * q^43 + 388 * q^47 + 147 * q^49 + 168 * q^51 + 66 * q^53 + 342 * q^57 - 616 * q^59 - 138 * q^61 - 189 * q^63 - 468 * q^67 + 192 * q^69 - 474 * q^71 - 1710 * q^73 + 378 * q^77 - 612 * q^79 + 243 * q^81 - 1920 * q^83 - 390 * q^87 - 2026 * q^89 + 294 * q^91 - 90 * q^93 - 1218 * q^97 - 486 * q^99

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	1.01865	0.0279214	0.0139607	−	0.999903i	$-0.495556\pi$
$11$	1.01865	0.0279214	0.0139607	+	0.999903i	$0.495556\pi$
$12$	0	0
$13$	−63.5205	−1.35519	−0.677593	−	0.735437i	$-0.736977\pi$
$13$	−63.5205	−1.35519	−0.677593	+	0.735437i	$0.736977\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	33.8576	0.483039	0.241520	−	0.970396i	$-0.422354\pi$
$17$	33.8576	0.483039	0.241520	+	0.970396i	$0.422354\pi$
$18$	0	0
$19$	30.4646	0.367845	0.183923	−	0.982941i	$-0.441120\pi$
$19$	30.4646	0.367845	0.183923	+	0.982941i	$0.441120\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	147.169	1.33421	0.667104	−	0.744965i	$-0.267534\pi$
$23$	147.169	1.33421	0.667104	+	0.744965i	$0.267534\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−275.745	−1.76568	−0.882838	−	0.469678i	$-0.844370\pi$
$29$	−275.745	−1.76568	−0.882838	+	0.469678i	$0.844370\pi$
$30$	0	0
$31$	39.5205	0.228971	0.114485	−	0.993425i	$-0.463478\pi$
$31$	39.5205	0.228971	0.114485	+	0.993425i	$0.463478\pi$
$32$	0	0
$33$	3.05596	0.0161204
$34$	0	0
$35$	0	0
$36$	0	0
$37$	289.101	1.28454	0.642269	−	0.766480i	$-0.277993\pi$
$37$	289.101	1.28454	0.642269	+	0.766480i	$0.277993\pi$
$38$	0	0
$39$	−190.562	−0.782418
$40$	0	0
$41$	173.476	0.660790	0.330395	−	0.943843i	$-0.392818\pi$
$41$	173.476	0.660790	0.330395	+	0.943843i	$0.392818\pi$
$42$	0	0
$43$	−132.981	−0.471615	−0.235808	−	0.971800i	$-0.575774\pi$
$43$	−132.981	−0.471615	−0.235808	+	0.971800i	$0.575774\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−202.360	−0.628026	−0.314013	−	0.949419i	$-0.601673\pi$
$47$	−202.360	−0.628026	−0.314013	+	0.949419i	$0.601673\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	101.573	0.278883
$52$	0	0
$53$	76.1889	0.197459	0.0987297	−	0.995114i	$-0.468522\pi$
$53$	76.1889	0.197459	0.0987297	+	0.995114i	$0.468522\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	91.3938	0.212375
$58$	0	0
$59$	−799.100	−1.76329	−0.881643	−	0.471917i	$-0.843562\pi$
$59$	−799.100	−1.76329	−0.881643	+	0.471917i	$0.843562\pi$
$60$	0	0
$61$	780.418	1.63807	0.819035	−	0.573743i	$-0.194509\pi$
$61$	780.418	1.63807	0.819035	+	0.573743i	$0.194509\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−465.687	−0.849146	−0.424573	−	0.905394i	$-0.639576\pi$
$67$	−465.687	−0.849146	−0.424573	+	0.905394i	$0.639576\pi$
$68$	0	0
$69$	441.506	0.770305
$70$	0	0
$71$	−280.728	−0.469244	−0.234622	−	0.972087i	$-0.575385\pi$
$71$	−280.728	−0.469244	−0.234622	+	0.972087i	$0.575385\pi$
$72$	0	0
$73$	−840.569	−1.34769	−0.673844	−	0.738874i	$-0.735358\pi$
$73$	−840.569	−1.34769	−0.673844	+	0.738874i	$0.735358\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−7.13057	−0.0105533
$78$	0	0
$79$	−303.041	−0.431580	−0.215790	−	0.976440i	$-0.569233\pi$
$79$	−303.041	−0.431580	−0.215790	+	0.976440i	$0.569233\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−785.695	−1.03905	−0.519525	−	0.854455i	$-0.673891\pi$
$83$	−785.695	−1.03905	−0.519525	+	0.854455i	$0.673891\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−827.235	−1.01941
$88$	0	0
$89$	−706.075	−0.840942	−0.420471	−	0.907306i	$-0.638135\pi$
$89$	−706.075	−0.840942	−0.420471	+	0.907306i	$0.638135\pi$
$90$	0	0
$91$	444.644	0.512212
$92$	0	0
$93$	118.562	0.132196
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1075.31	1.12558	0.562789	−	0.826601i	$-0.309728\pi$
$97$	1075.31	1.12558	0.562789	+	0.826601i	$0.309728\pi$
$98$	0	0
$99$	9.16787	0.00930713
$100$	0	0
$101$	−1370.39	−1.35009	−0.675044	−	0.737777i	$-0.735876\pi$
$101$	−1370.39	−1.35009	−0.675044	+	0.737777i	$0.735876\pi$
$102$	0	0
$103$	−346.003	−0.330997	−0.165498	−	0.986210i	$-0.552923\pi$
$103$	−346.003	−0.330997	−0.165498	+	0.986210i	$0.552923\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−163.450	−0.147676	−0.0738381	−	0.997270i	$-0.523525\pi$
$107$	−163.450	−0.147676	−0.0738381	+	0.997270i	$0.523525\pi$
$108$	0	0
$109$	−1069.81	−0.940082	−0.470041	−	0.882645i	$-0.655761\pi$
$109$	−1069.81	−0.940082	−0.470041	+	0.882645i	$0.655761\pi$
$110$	0	0
$111$	867.302	0.741628
$112$	0	0
$113$	−273.537	−0.227719	−0.113859	−	0.993497i	$-0.536321\pi$
$113$	−273.537	−0.227719	−0.113859	+	0.993497i	$0.536321\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−571.685	−0.451729
$118$	0	0
$119$	−237.003	−0.182572
$120$	0	0
$121$	−1329.96	−0.999220
$122$	0	0
$123$	520.428	0.381507
$124$	0	0
$125$	0	0
$126$	0	0
$127$	125.154	0.0874455	0.0437228	−	0.999044i	$-0.486078\pi$
$127$	125.154	0.0874455	0.0437228	+	0.999044i	$0.486078\pi$
$128$	0	0
$129$	−398.944	−0.272287
$130$	0	0
$131$	2425.29	1.61755	0.808773	−	0.588121i	$-0.200132\pi$
$131$	2425.29	1.61755	0.808773	+	0.588121i	$0.200132\pi$
$132$	0	0
$133$	−213.252	−0.139032
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−537.340	−0.335095	−0.167548	−	0.985864i	$-0.553585\pi$
$137$	−537.340	−0.335095	−0.167548	+	0.985864i	$0.553585\pi$
$138$	0	0
$139$	−668.910	−0.408174	−0.204087	−	0.978953i	$-0.565423\pi$
$139$	−668.910	−0.408174	−0.204087	+	0.978953i	$0.565423\pi$
$140$	0	0
$141$	−607.079	−0.362591
$142$	0	0
$143$	−64.7053	−0.0378387
$144$	0	0
$145$	0	0
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	396.285	0.217886	0.108943	−	0.994048i	$-0.465253\pi$
$149$	396.285	0.217886	0.108943	+	0.994048i	$0.465253\pi$
$150$	0	0
$151$	−1753.87	−0.945218	−0.472609	−	0.881272i	$-0.656688\pi$
$151$	−1753.87	−0.945218	−0.472609	+	0.881272i	$0.656688\pi$
$152$	0	0
$153$	304.718	0.161013
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1897.59	0.964610	0.482305	−	0.876003i	$-0.339800\pi$
$157$	1897.59	0.964610	0.482305	+	0.876003i	$0.339800\pi$
$158$	0	0
$159$	228.567	0.114003
$160$	0	0
$161$	−1030.18	−0.504283
$162$	0	0
$163$	−319.695	−0.153622	−0.0768111	−	0.997046i	$-0.524474\pi$
$163$	−319.695	−0.153622	−0.0768111	+	0.997046i	$0.524474\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	685.713	0.317737	0.158868	−	0.987300i	$-0.449215\pi$
$167$	685.713	0.317737	0.158868	+	0.987300i	$0.449215\pi$
$168$	0	0
$169$	1837.86	0.836531
$170$	0	0
$171$	274.181	0.122615
$172$	0	0
$173$	2310.88	1.01557	0.507783	−	0.861485i	$-0.330465\pi$
$173$	2310.88	1.01557	0.507783	+	0.861485i	$0.330465\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2397.30	−1.01803
$178$	0	0
$179$	479.478	0.200211	0.100106	−	0.994977i	$-0.468082\pi$
$179$	479.478	0.200211	0.100106	+	0.994977i	$0.468082\pi$
$180$	0	0
$181$	−4110.84	−1.68816	−0.844078	−	0.536220i	$-0.819852\pi$
$181$	−4110.84	−1.68816	−0.844078	+	0.536220i	$0.819852\pi$
$182$	0	0
$183$	2341.25	0.945741
$184$	0	0
$185$	0	0
$186$	0	0
$187$	34.4891	0.0134871
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	−4898.91	−1.85588	−0.927940	−	0.372730i	$-0.878422\pi$
$191$	−4898.91	−1.85588	−0.927940	+	0.372730i	$0.878422\pi$
$192$	0	0
$193$	−4253.09	−1.58624	−0.793119	−	0.609066i	$-0.791544\pi$
$193$	−4253.09	−1.58624	−0.793119	+	0.609066i	$0.791544\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4329.51	−1.56581	−0.782906	−	0.622140i	$-0.786264\pi$
$197$	−4329.51	−1.56581	−0.782906	+	0.622140i	$0.786264\pi$
$198$	0	0
$199$	−3387.04	−1.20654	−0.603269	−	0.797538i	$-0.706135\pi$
$199$	−3387.04	−1.20654	−0.603269	+	0.797538i	$0.706135\pi$
$200$	0	0
$201$	−1397.06	−0.490254
$202$	0	0
$203$	1930.22	0.667363
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1324.52	0.444736
$208$	0	0
$209$	31.0328	0.0102707
$210$	0	0
$211$	3182.61	1.03839	0.519193	−	0.854657i	$-0.326232\pi$
$211$	3182.61	1.03839	0.519193	+	0.854657i	$0.326232\pi$
$212$	0	0
$213$	−842.184	−0.270918
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−276.644	−0.0865429
$218$	0	0
$219$	−2521.71	−0.778088
$220$	0	0
$221$	−2150.65	−0.654608
$222$	0	0
$223$	−6598.42	−1.98145	−0.990724	−	0.135889i	$-0.956611\pi$
$223$	−6598.42	−1.98145	−0.990724	+	0.135889i	$0.956611\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1362.55	0.398396	0.199198	−	0.979959i	$-0.436166\pi$
$227$	1362.55	0.398396	0.199198	+	0.979959i	$0.436166\pi$
$228$	0	0
$229$	440.818	0.127206	0.0636028	−	0.997975i	$-0.479741\pi$
$229$	440.818	0.127206	0.0636028	+	0.997975i	$0.479741\pi$
$230$	0	0
$231$	−21.3917	−0.00609295
$232$	0	0
$233$	4405.63	1.23872	0.619362	−	0.785106i	$-0.287391\pi$
$233$	4405.63	1.23872	0.619362	+	0.785106i	$0.287391\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−909.123	−0.249173
$238$	0	0
$239$	−2645.24	−0.715926	−0.357963	−	0.933736i	$-0.616529\pi$
$239$	−2645.24	−0.715926	−0.357963	+	0.933736i	$0.616529\pi$
$240$	0	0
$241$	−2914.75	−0.779070	−0.389535	−	0.921012i	$-0.627364\pi$
$241$	−2914.75	−0.779070	−0.389535	+	0.921012i	$0.627364\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1935.13	−0.498499
$248$	0	0
$249$	−2357.08	−0.599896
$250$	0	0
$251$	1789.69	0.450058	0.225029	−	0.974352i	$-0.427752\pi$
$251$	1789.69	0.450058	0.225029	+	0.974352i	$0.427752\pi$
$252$	0	0
$253$	149.914	0.0372529
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5468.14	1.32721	0.663606	−	0.748082i	$-0.269025\pi$
$257$	5468.14	1.32721	0.663606	+	0.748082i	$0.269025\pi$
$258$	0	0
$259$	−2023.71	−0.485510
$260$	0	0
$261$	−2481.71	−0.588559
$262$	0	0
$263$	−1558.82	−0.365478	−0.182739	−	0.983161i	$-0.558496\pi$
$263$	−1558.82	−0.365478	−0.182739	+	0.983161i	$0.558496\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2118.23	−0.485518
$268$	0	0
$269$	−518.723	−0.117573	−0.0587865	−	0.998271i	$-0.518723\pi$
$269$	−518.723	−0.117573	−0.0587865	+	0.998271i	$0.518723\pi$
$270$	0	0
$271$	−2487.65	−0.557616	−0.278808	−	0.960347i	$-0.589939\pi$
$271$	−2487.65	−0.557616	−0.278808	+	0.960347i	$0.589939\pi$
$272$	0	0
$273$	1333.93	0.295726
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7631.02	1.65525	0.827624	−	0.561283i	$-0.189692\pi$
$277$	7631.02	1.65525	0.827624	+	0.561283i	$0.189692\pi$
$278$	0	0
$279$	355.685	0.0763237
$280$	0	0
$281$	3696.97	0.784849	0.392425	−	0.919784i	$-0.371636\pi$
$281$	3696.97	0.784849	0.392425	+	0.919784i	$0.371636\pi$
$282$	0	0
$283$	8631.55	1.81305	0.906524	−	0.422155i	$-0.138726\pi$
$283$	8631.55	1.81305	0.906524	+	0.422155i	$0.138726\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1214.33	−0.249755
$288$	0	0
$289$	−3766.66	−0.766673
$290$	0	0
$291$	3225.92	0.649852
$292$	0	0
$293$	3002.34	0.598630	0.299315	−	0.954154i	$-0.403242\pi$
$293$	3002.34	0.598630	0.299315	+	0.954154i	$0.403242\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	27.5036	0.00537347
$298$	0	0
$299$	−9348.23	−1.80810
$300$	0	0
$301$	930.869	0.178254
$302$	0	0
$303$	−4111.17	−0.779474
$304$	0	0
$305$	0	0
$306$	0	0
$307$	7206.48	1.33972	0.669862	−	0.742485i	$-0.266353\pi$
$307$	7206.48	1.33972	0.669862	+	0.742485i	$0.266353\pi$
$308$	0	0
$309$	−1038.01	−0.191101
$310$	0	0
$311$	−2413.36	−0.440028	−0.220014	−	0.975497i	$-0.570610\pi$
$311$	−2413.36	−0.440028	−0.220014	+	0.975497i	$0.570610\pi$
$312$	0	0
$313$	1795.30	0.324206	0.162103	−	0.986774i	$-0.448172\pi$
$313$	1795.30	0.324206	0.162103	+	0.986774i	$0.448172\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4393.88	−0.778501	−0.389250	−	0.921132i	$-0.627266\pi$
$317$	−4393.88	−0.778501	−0.389250	+	0.921132i	$0.627266\pi$
$318$	0	0
$319$	−280.888	−0.0493001
$320$	0	0
$321$	−490.351	−0.0852608
$322$	0	0
$323$	1031.46	0.177684
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3209.42	−0.542757
$328$	0	0
$329$	1416.52	0.237371
$330$	0	0
$331$	6917.15	1.14864	0.574322	−	0.818630i	$-0.305266\pi$
$331$	6917.15	1.14864	0.574322	+	0.818630i	$0.305266\pi$
$332$	0	0
$333$	2601.91	0.428179
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−5560.89	−0.898876	−0.449438	−	0.893312i	$-0.648376\pi$
$337$	−5560.89	−0.898876	−0.449438	+	0.893312i	$0.648376\pi$
$338$	0	0
$339$	−820.611	−0.131473
$340$	0	0
$341$	40.2577	0.00639319
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8105.46	1.25396	0.626980	−	0.779035i	$-0.284291\pi$
$347$	8105.46	1.25396	0.626980	+	0.779035i	$0.284291\pi$
$348$	0	0
$349$	−8455.40	−1.29687	−0.648434	−	0.761271i	$-0.724576\pi$
$349$	−8455.40	−1.29687	−0.648434	+	0.761271i	$0.724576\pi$
$350$	0	0
$351$	−1715.05	−0.260806
$352$	0	0
$353$	−8078.23	−1.21802	−0.609010	−	0.793163i	$-0.708433\pi$
$353$	−8078.23	−1.21802	−0.609010	+	0.793163i	$0.708433\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−711.009	−0.105408
$358$	0	0
$359$	−7211.89	−1.06025	−0.530124	−	0.847920i	$-0.677855\pi$
$359$	−7211.89	−1.06025	−0.530124	+	0.847920i	$0.677855\pi$
$360$	0	0
$361$	−5930.91	−0.864690
$362$	0	0
$363$	−3989.89	−0.576900
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−5013.11	−0.713031	−0.356515	−	0.934289i	$-0.616035\pi$
$367$	−5013.11	−0.713031	−0.356515	+	0.934289i	$0.616035\pi$
$368$	0	0
$369$	1561.28	0.220263
$370$	0	0
$371$	−533.322	−0.0746326
$372$	0	0
$373$	8283.03	1.14981	0.574905	−	0.818220i	$-0.305039\pi$
$373$	8283.03	1.14981	0.574905	+	0.818220i	$0.305039\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	17515.5	2.39282
$378$	0	0
$379$	−11249.8	−1.52470	−0.762351	−	0.647163i	$-0.775955\pi$
$379$	−11249.8	−1.52470	−0.762351	+	0.647163i	$0.775955\pi$
$380$	0	0
$381$	375.461	0.0504867
$382$	0	0
$383$	6512.53	0.868864	0.434432	−	0.900705i	$-0.356949\pi$
$383$	6512.53	0.868864	0.434432	+	0.900705i	$0.356949\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1196.83	−0.157205
$388$	0	0
$389$	−7913.44	−1.03143	−0.515716	−	0.856759i	$-0.672474\pi$
$389$	−7913.44	−1.03143	−0.515716	+	0.856759i	$0.672474\pi$
$390$	0	0
$391$	4982.77	0.644475
$392$	0	0
$393$	7275.87	0.933891
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−10278.6	−1.29941	−0.649705	−	0.760186i	$-0.725108\pi$
$397$	−10278.6	−1.29941	−0.649705	+	0.760186i	$0.725108\pi$
$398$	0	0
$399$	−639.756	−0.0802704
$400$	0	0
$401$	3434.29	0.427682	0.213841	−	0.976868i	$-0.431403\pi$
$401$	3434.29	0.427682	0.213841	+	0.976868i	$0.431403\pi$
$402$	0	0
$403$	−2510.37	−0.310298
$404$	0	0
$405$	0	0
$406$	0	0
$407$	294.493	0.0358661
$408$	0	0
$409$	−6768.53	−0.818294	−0.409147	−	0.912468i	$-0.634174\pi$
$409$	−6768.53	−0.818294	−0.409147	+	0.912468i	$0.634174\pi$
$410$	0	0
$411$	−1612.02	−0.193467
$412$	0	0
$413$	5593.70	0.666460
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2006.73	−0.235659
$418$	0	0
$419$	−3996.73	−0.465998	−0.232999	−	0.972477i	$-0.574854\pi$
$419$	−3996.73	−0.465998	−0.232999	+	0.972477i	$0.574854\pi$
$420$	0	0
$421$	3039.08	0.351818	0.175909	−	0.984406i	$-0.443713\pi$
$421$	3039.08	0.351818	0.175909	+	0.984406i	$0.443713\pi$
$422$	0	0
$423$	−1821.24	−0.209342
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−5462.93	−0.619133
$428$	0	0
$429$	−194.116	−0.0218462
$430$	0	0
$431$	13547.8	1.51410	0.757050	−	0.653357i	$-0.226640\pi$
$431$	13547.8	1.51410	0.757050	+	0.653357i	$0.226640\pi$
$432$	0	0
$433$	−14052.0	−1.55957	−0.779787	−	0.626045i	$-0.784673\pi$
$433$	−14052.0	−1.55957	−0.779787	+	0.626045i	$0.784673\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4483.43	0.490782
$438$	0	0
$439$	246.377	0.0267858	0.0133929	−	0.999910i	$-0.495737\pi$
$439$	246.377	0.0267858	0.0133929	+	0.999910i	$0.495737\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	2217.44	0.237819	0.118909	−	0.992905i	$-0.462060\pi$
$443$	2217.44	0.237819	0.118909	+	0.992905i	$0.462060\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1188.86	0.125796
$448$	0	0
$449$	−8163.10	−0.857996	−0.428998	−	0.903305i	$-0.641133\pi$
$449$	−8163.10	−0.857996	−0.428998	+	0.903305i	$0.641133\pi$
$450$	0	0
$451$	176.712	0.0184502
$452$	0	0
$453$	−5261.61	−0.545722
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−354.013	−0.0362364	−0.0181182	−	0.999836i	$-0.505768\pi$
$457$	−354.013	−0.0362364	−0.0181182	+	0.999836i	$0.505768\pi$
$458$	0	0
$459$	914.154	0.0929609
$460$	0	0
$461$	−8981.87	−0.907434	−0.453717	−	0.891146i	$-0.649902\pi$
$461$	−8981.87	−0.907434	−0.453717	+	0.891146i	$0.649902\pi$
$462$	0	0
$463$	8101.65	0.813208	0.406604	−	0.913604i	$-0.366713\pi$
$463$	8101.65	0.813208	0.406604	+	0.913604i	$0.366713\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6010.43	−0.595567	−0.297783	−	0.954633i	$-0.596247\pi$
$467$	−6010.43	−0.595567	−0.297783	+	0.954633i	$0.596247\pi$
$468$	0	0
$469$	3259.81	0.320947
$470$	0	0
$471$	5692.76	0.556918
$472$	0	0
$473$	−135.462	−0.0131682
$474$	0	0
$475$	0	0
$476$	0	0
$477$	685.700	0.0658198
$478$	0	0
$479$	3488.64	0.332777	0.166388	−	0.986060i	$-0.446789\pi$
$479$	3488.64	0.332777	0.166388	+	0.986060i	$0.446789\pi$
$480$	0	0
$481$	−18363.8	−1.74079
$482$	0	0
$483$	−3090.54	−0.291148
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−19351.4	−1.80061	−0.900304	−	0.435261i	$-0.856656\pi$
$487$	−19351.4	−1.80061	−0.900304	+	0.435261i	$0.856656\pi$
$488$	0	0
$489$	−959.084	−0.0886939
$490$	0	0
$491$	−7657.24	−0.703802	−0.351901	−	0.936037i	$-0.614465\pi$
$491$	−7657.24	−0.703802	−0.351901	+	0.936037i	$0.614465\pi$
$492$	0	0
$493$	−9336.06	−0.852891
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1965.10	0.177357
$498$	0	0
$499$	−882.964	−0.0792122	−0.0396061	−	0.999215i	$-0.512610\pi$
$499$	−882.964	−0.0792122	−0.0396061	+	0.999215i	$0.512610\pi$
$500$	0	0
$501$	2057.14	0.183445
$502$	0	0
$503$	−16298.4	−1.44475	−0.722374	−	0.691503i	$-0.756949\pi$
$503$	−16298.4	−1.44475	−0.722374	+	0.691503i	$0.756949\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	5513.58	0.482972
$508$	0	0
$509$	−2429.50	−0.211563	−0.105781	−	0.994389i	$-0.533734\pi$
$509$	−2429.50	−0.211563	−0.105781	+	0.994389i	$0.533734\pi$
$510$	0	0
$511$	5883.98	0.509378
$512$	0	0
$513$	822.544	0.0707918
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−206.134	−0.0175353
$518$	0	0
$519$	6932.64	0.586337
$520$	0	0
$521$	−16784.9	−1.41144	−0.705720	−	0.708491i	$-0.749376\pi$
$521$	−16784.9	−1.41144	−0.705720	+	0.708491i	$0.749376\pi$
$522$	0	0
$523$	11261.6	0.941560	0.470780	−	0.882251i	$-0.343973\pi$
$523$	11261.6	0.941560	0.470780	+	0.882251i	$0.343973\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1338.07	0.110602
$528$	0	0
$529$	9491.61	0.780111
$530$	0	0
$531$	−7191.90	−0.587762
$532$	0	0
$533$	−11019.3	−0.895494
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1438.43	0.115592
$538$	0	0
$539$	49.9140	0.00398877
$540$	0	0
$541$	−11072.4	−0.879922	−0.439961	−	0.898017i	$-0.645008\pi$
$541$	−11072.4	−0.879922	−0.439961	+	0.898017i	$0.645008\pi$
$542$	0	0
$543$	−12332.5	−0.974658
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−11800.9	−0.922434	−0.461217	−	0.887287i	$-0.652587\pi$
$547$	−11800.9	−0.922434	−0.461217	+	0.887287i	$0.652587\pi$
$548$	0	0
$549$	7023.76	0.546024
$550$	0	0
$551$	−8400.46	−0.649495
$552$	0	0
$553$	2121.29	0.163122
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8777.51	0.667711	0.333855	−	0.942624i	$-0.391650\pi$
$557$	8777.51	0.667711	0.333855	+	0.942624i	$0.391650\pi$
$558$	0	0
$559$	8447.05	0.639127
$560$	0	0
$561$	103.467	0.00778680
$562$	0	0
$563$	3225.26	0.241436	0.120718	−	0.992687i	$-0.461480\pi$
$563$	3225.26	0.241436	0.120718	+	0.992687i	$0.461480\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−2312.75	−0.170396	−0.0851981	−	0.996364i	$-0.527152\pi$
$569$	−2312.75	−0.170396	−0.0851981	+	0.996364i	$0.527152\pi$
$570$	0	0
$571$	2045.95	0.149948	0.0749740	−	0.997185i	$-0.476113\pi$
$571$	2045.95	0.149948	0.0749740	+	0.997185i	$0.476113\pi$
$572$	0	0
$573$	−14696.7	−1.07149
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16418.3	1.18458	0.592291	−	0.805724i	$-0.298224\pi$
$577$	16418.3	1.18458	0.592291	+	0.805724i	$0.298224\pi$
$578$	0	0
$579$	−12759.3	−0.915815
$580$	0	0
$581$	5499.86	0.392724
$582$	0	0
$583$	77.6100	0.00551334
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26289.2	1.84850	0.924251	−	0.381785i	$-0.124691\pi$
$587$	26289.2	1.84850	0.924251	+	0.381785i	$0.124691\pi$
$588$	0	0
$589$	1203.98	0.0842258
$590$	0	0
$591$	−12988.5	−0.904022
$592$	0	0
$593$	24597.7	1.70338	0.851690	−	0.524046i	$-0.175578\pi$
$593$	24597.7	1.70338	0.851690	+	0.524046i	$0.175578\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−10161.1	−0.696595
$598$	0	0
$599$	19963.1	1.36172	0.680859	−	0.732415i	$-0.261607\pi$
$599$	19963.1	1.36172	0.680859	+	0.732415i	$0.261607\pi$
$600$	0	0
$601$	17328.1	1.17608	0.588042	−	0.808830i	$-0.299899\pi$
$601$	17328.1	1.17608	0.588042	+	0.808830i	$0.299899\pi$
$602$	0	0
$603$	−4191.19	−0.283049
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−9765.74	−0.653013	−0.326507	−	0.945195i	$-0.605872\pi$
$607$	−9765.74	−0.653013	−0.326507	+	0.945195i	$0.605872\pi$
$608$	0	0
$609$	5790.65	0.385302
$610$	0	0
$611$	12854.0	0.851092
$612$	0	0
$613$	16655.7	1.09742	0.548710	−	0.836013i	$-0.315119\pi$
$613$	16655.7	1.09742	0.548710	+	0.836013i	$0.315119\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6655.24	0.434246	0.217123	−	0.976144i	$-0.430333\pi$
$617$	6655.24	0.434246	0.217123	+	0.976144i	$0.430333\pi$
$618$	0	0
$619$	4892.75	0.317700	0.158850	−	0.987303i	$-0.449221\pi$
$619$	4892.75	0.317700	0.158850	+	0.987303i	$0.449221\pi$
$620$	0	0
$621$	3973.55	0.256768
$622$	0	0
$623$	4942.53	0.317846
$624$	0	0
$625$	0	0
$626$	0	0
$627$	93.0985	0.00592982
$628$	0	0
$629$	9788.25	0.620482
$630$	0	0
$631$	−13790.8	−0.870055	−0.435027	−	0.900417i	$-0.643261\pi$
$631$	−13790.8	−0.870055	−0.435027	+	0.900417i	$0.643261\pi$
$632$	0	0
$633$	9547.82	0.599513
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−3112.51	−0.193598
$638$	0	0
$639$	−2526.55	−0.156415
$640$	0	0
$641$	2380.10	0.146659	0.0733295	−	0.997308i	$-0.476638\pi$
$641$	2380.10	0.146659	0.0733295	+	0.997308i	$0.476638\pi$
$642$	0	0
$643$	−6098.34	−0.374021	−0.187010	−	0.982358i	$-0.559880\pi$
$643$	−6098.34	−0.374021	−0.187010	+	0.982358i	$0.559880\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2069.63	0.125758	0.0628791	−	0.998021i	$-0.479972\pi$
$647$	2069.63	0.125758	0.0628791	+	0.998021i	$0.479972\pi$
$648$	0	0
$649$	−814.005	−0.0492334
$650$	0	0
$651$	−829.931	−0.0499656
$652$	0	0
$653$	−9525.57	−0.570849	−0.285424	−	0.958401i	$-0.592135\pi$
$653$	−9525.57	−0.570849	−0.285424	+	0.958401i	$0.592135\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−7565.12	−0.449229
$658$	0	0
$659$	28827.7	1.70405	0.852023	−	0.523504i	$-0.175376\pi$
$659$	28827.7	1.70405	0.852023	+	0.523504i	$0.175376\pi$
$660$	0	0
$661$	−1696.86	−0.0998488	−0.0499244	−	0.998753i	$-0.515898\pi$
$661$	−1696.86	−0.0998488	−0.0499244	+	0.998753i	$0.515898\pi$
$662$	0	0
$663$	−6451.95	−0.377938
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−40581.0	−2.35578
$668$	0	0
$669$	−19795.3	−1.14399
$670$	0	0
$671$	794.975	0.0457372
$672$	0	0
$673$	25637.6	1.46843	0.734217	−	0.678915i	$-0.237550\pi$
$673$	25637.6	1.46843	0.734217	+	0.678915i	$0.237550\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21279.1	1.20801	0.604005	−	0.796980i	$-0.293571\pi$
$677$	21279.1	1.20801	0.604005	+	0.796980i	$0.293571\pi$
$678$	0	0
$679$	−7527.16	−0.425428
$680$	0	0
$681$	4087.66	0.230014
$682$	0	0
$683$	−6828.09	−0.382533	−0.191266	−	0.981538i	$-0.561259\pi$
$683$	−6828.09	−0.382533	−0.191266	+	0.981538i	$0.561259\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1322.45	0.0734422
$688$	0	0
$689$	−4839.56	−0.267594
$690$	0	0
$691$	7710.08	0.424465	0.212232	−	0.977219i	$-0.431927\pi$
$691$	7710.08	0.424465	0.212232	+	0.977219i	$0.431927\pi$
$692$	0	0
$693$	−64.1751	−0.00351776
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5873.47	0.319188
$698$	0	0
$699$	13216.9	0.715178
$700$	0	0
$701$	4712.20	0.253891	0.126945	−	0.991910i	$-0.459483\pi$
$701$	4712.20	0.253891	0.126945	+	0.991910i	$0.459483\pi$
$702$	0	0
$703$	8807.34	0.472511
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9592.73	0.510286
$708$	0	0
$709$	10005.7	0.530001	0.265001	−	0.964248i	$-0.414628\pi$
$709$	10005.7	0.530001	0.265001	+	0.964248i	$0.414628\pi$
$710$	0	0
$711$	−2727.37	−0.143860
$712$	0	0
$713$	5816.19	0.305495
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−7935.71	−0.413340
$718$	0	0
$719$	24420.6	1.26667	0.633333	−	0.773879i	$-0.281686\pi$
$719$	24420.6	1.26667	0.633333	+	0.773879i	$0.281686\pi$
$720$	0	0
$721$	2422.02	0.125105
$722$	0	0
$723$	−8744.26	−0.449796
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−18875.2	−0.962918	−0.481459	−	0.876469i	$-0.659893\pi$
$727$	−18875.2	−0.962918	−0.481459	+	0.876469i	$0.659893\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−4502.43	−0.227809
$732$	0	0
$733$	−19504.7	−0.982841	−0.491420	−	0.870923i	$-0.663522\pi$
$733$	−19504.7	−0.982841	−0.491420	+	0.870923i	$0.663522\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−474.373	−0.0237093
$738$	0	0
$739$	35927.8	1.78840	0.894199	−	0.447670i	$-0.147746\pi$
$739$	35927.8	1.78840	0.894199	+	0.447670i	$0.147746\pi$
$740$	0	0
$741$	−5805.38	−0.287808
$742$	0	0
$743$	−21322.5	−1.05282	−0.526412	−	0.850230i	$-0.676463\pi$
$743$	−21322.5	−1.05282	−0.526412	+	0.850230i	$0.676463\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−7071.25	−0.346350
$748$	0	0
$749$	1144.15	0.0558163
$750$	0	0
$751$	−30538.1	−1.48382	−0.741912	−	0.670497i	$-0.766081\pi$
$751$	−30538.1	−1.48382	−0.741912	+	0.670497i	$0.766081\pi$
$752$	0	0
$753$	5369.08	0.259841
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−581.620	−0.0279251	−0.0139626	−	0.999903i	$-0.504445\pi$
$757$	−581.620	−0.0279251	−0.0139626	+	0.999903i	$0.504445\pi$
$758$	0	0
$759$	449.741	0.0215080
$760$	0	0
$761$	−5358.99	−0.255274	−0.127637	−	0.991821i	$-0.540739\pi$
$761$	−5358.99	−0.255274	−0.127637	+	0.991821i	$0.540739\pi$
$762$	0	0
$763$	7488.65	0.355318
$764$	0	0
$765$	0	0
$766$	0	0
$767$	50759.2	2.38958
$768$	0	0
$769$	−12576.2	−0.589741	−0.294871	−	0.955537i	$-0.595277\pi$
$769$	−12576.2	−0.589741	−0.294871	+	0.955537i	$0.595277\pi$
$770$	0	0
$771$	16404.4	0.766266
$772$	0	0
$773$	8574.25	0.398958	0.199479	−	0.979902i	$-0.436075\pi$
$773$	8574.25	0.398958	0.199479	+	0.979902i	$0.436075\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−6071.12	−0.280309
$778$	0	0
$779$	5284.87	0.243068
$780$	0	0
$781$	−285.964	−0.0131019
$782$	0	0
$783$	−7445.12	−0.339804
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−24471.6	−1.10841	−0.554206	−	0.832380i	$-0.686978\pi$
$787$	−24471.6	−1.10841	−0.554206	+	0.832380i	$0.686978\pi$
$788$	0	0
$789$	−4676.45	−0.211009
$790$	0	0
$791$	1914.76	0.0860695
$792$	0	0
$793$	−49572.6	−2.21989
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−31962.3	−1.42053	−0.710266	−	0.703934i	$-0.751425\pi$
$797$	−31962.3	−1.42053	−0.710266	+	0.703934i	$0.751425\pi$
$798$	0	0
$799$	−6851.41	−0.303361
$800$	0	0
$801$	−6354.68	−0.280314
$802$	0	0
$803$	−856.248	−0.0376293
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1556.17	−0.0678807
$808$	0	0
$809$	−31743.1	−1.37952	−0.689758	−	0.724040i	$-0.742283\pi$
$809$	−31743.1	−1.37952	−0.689758	+	0.724040i	$0.742283\pi$
$810$	0	0
$811$	40594.6	1.75767	0.878834	−	0.477128i	$-0.158322\pi$
$811$	40594.6	1.75767	0.878834	+	0.477128i	$0.158322\pi$
$812$	0	0
$813$	−7462.95	−0.321940
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−4051.22	−0.173481
$818$	0	0
$819$	4001.79	0.170737
$820$	0	0
$821$	−4840.20	−0.205754	−0.102877	−	0.994694i	$-0.532805\pi$
$821$	−4840.20	−0.205754	−0.102877	+	0.994694i	$0.532805\pi$
$822$	0	0
$823$	−3706.56	−0.156990	−0.0784948	−	0.996915i	$-0.525011\pi$
$823$	−3706.56	−0.156990	−0.0784948	+	0.996915i	$0.525011\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3280.08	−0.137920	−0.0689599	−	0.997619i	$-0.521968\pi$
$827$	−3280.08	−0.137920	−0.0689599	+	0.997619i	$0.521968\pi$
$828$	0	0
$829$	−42382.8	−1.77565	−0.887825	−	0.460181i	$-0.847785\pi$
$829$	−42382.8	−1.77565	−0.887825	+	0.460181i	$0.847785\pi$
$830$	0	0
$831$	22893.1	0.955658
$832$	0	0
$833$	1659.02	0.0690056
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1067.05	0.0440655
$838$	0	0
$839$	40306.9	1.65858	0.829290	−	0.558818i	$-0.188745\pi$
$839$	40306.9	1.65858	0.829290	+	0.558818i	$0.188745\pi$
$840$	0	0
$841$	51646.4	2.11761
$842$	0	0
$843$	11090.9	0.453133
$844$	0	0
$845$	0	0
$846$	0	0
$847$	9309.74	0.377670
$848$	0	0
$849$	25894.6	1.04676
$850$	0	0
$851$	42546.6	1.71384
$852$	0	0
$853$	−22158.9	−0.889456	−0.444728	−	0.895666i	$-0.646700\pi$
$853$	−22158.9	−0.889456	−0.444728	+	0.895666i	$0.646700\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	45815.6	1.82617	0.913087	−	0.407764i	$-0.133691\pi$
$857$	45815.6	1.82617	0.913087	+	0.407764i	$0.133691\pi$
$858$	0	0
$859$	14724.2	0.584846	0.292423	−	0.956289i	$-0.405538\pi$
$859$	14724.2	0.584846	0.292423	+	0.956289i	$0.405538\pi$
$860$	0	0
$861$	−3642.99	−0.144196
$862$	0	0
$863$	27168.6	1.07165	0.535824	−	0.844330i	$-0.320001\pi$
$863$	27168.6	1.07165	0.535824	+	0.844330i	$0.320001\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−11300.0	−0.442639
$868$	0	0
$869$	−308.694	−0.0120503
$870$	0	0
$871$	29580.7	1.15075
$872$	0	0
$873$	9677.77	0.375192
$874$	0	0
$875$	0	0
$876$	0	0
$877$	5839.59	0.224845	0.112422	−	0.993661i	$-0.464139\pi$
$877$	5839.59	0.224845	0.112422	+	0.993661i	$0.464139\pi$
$878$	0	0
$879$	9007.02	0.345619
$880$	0	0
$881$	−28684.0	−1.09692	−0.548461	−	0.836176i	$-0.684786\pi$
$881$	−28684.0	−1.09692	−0.548461	+	0.836176i	$0.684786\pi$
$882$	0	0
$883$	−17221.1	−0.656326	−0.328163	−	0.944621i	$-0.606430\pi$
$883$	−17221.1	−0.656326	−0.328163	+	0.944621i	$0.606430\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32495.5	1.23009	0.615046	−	0.788491i	$-0.289137\pi$
$887$	32495.5	1.23009	0.615046	+	0.788491i	$0.289137\pi$
$888$	0	0
$889$	−876.075	−0.0330513
$890$	0	0
$891$	82.5108	0.00310238
$892$	0	0
$893$	−6164.81	−0.231016
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−28044.7	−1.04391
$898$	0	0
$899$	−10897.6	−0.404288
$900$	0	0
$901$	2579.57	0.0953806
$902$	0	0
$903$	2792.61	0.102915
$904$	0	0
$905$	0	0
$906$	0	0
$907$	9095.47	0.332977	0.166488	−	0.986043i	$-0.446757\pi$
$907$	9095.47	0.332977	0.166488	+	0.986043i	$0.446757\pi$
$908$	0	0
$909$	−12333.5	−0.450030
$910$	0	0
$911$	25214.0	0.916989	0.458495	−	0.888697i	$-0.348389\pi$
$911$	25214.0	0.916989	0.458495	+	0.888697i	$0.348389\pi$
$912$	0	0
$913$	−800.350	−0.0290117
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−16977.0	−0.611375
$918$	0	0
$919$	11204.3	0.402171	0.201085	−	0.979574i	$-0.435553\pi$
$919$	11204.3	0.402171	0.201085	+	0.979574i	$0.435553\pi$
$920$	0	0
$921$	21619.4	0.773491
$922$	0	0
$923$	17832.0	0.635913
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−3114.03	−0.110332
$928$	0	0
$929$	−4395.54	−0.155235	−0.0776173	−	0.996983i	$-0.524731\pi$
$929$	−4395.54	−0.155235	−0.0776173	+	0.996983i	$0.524731\pi$
$930$	0	0
$931$	1492.76	0.0525493
$932$	0	0
$933$	−7240.07	−0.254051
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1389.66	0.0484504	0.0242252	−	0.999707i	$-0.492288\pi$
$937$	1389.66	0.0484504	0.0242252	+	0.999707i	$0.492288\pi$
$938$	0	0
$939$	5385.90	0.187180
$940$	0	0
$941$	2342.75	0.0811600	0.0405800	−	0.999176i	$-0.487079\pi$
$941$	2342.75	0.0811600	0.0405800	+	0.999176i	$0.487079\pi$
$942$	0	0
$943$	25530.2	0.881631
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39418.7	1.35263	0.676313	−	0.736614i	$-0.263577\pi$
$947$	39418.7	1.35263	0.676313	+	0.736614i	$0.263577\pi$
$948$	0	0
$949$	53393.4	1.82637
$950$	0	0
$951$	−13181.6	−0.449468
$952$	0	0
$953$	52564.0	1.78669	0.893345	−	0.449371i	$-0.148352\pi$
$953$	52564.0	1.78669	0.893345	+	0.449371i	$0.148352\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−842.665	−0.0284634
$958$	0	0
$959$	3761.38	0.126654
$960$	0	0
$961$	−28229.1	−0.947572
$962$	0	0
$963$	−1471.05	−0.0492254
$964$	0	0
$965$	0	0
$966$	0	0
$967$	34020.4	1.13136	0.565679	−	0.824626i	$-0.308614\pi$
$967$	34020.4	1.13136	0.565679	+	0.824626i	$0.308614\pi$
$968$	0	0
$969$	3094.37	0.102586
$970$	0	0
$971$	23646.8	0.781526	0.390763	−	0.920491i	$-0.372211\pi$
$971$	23646.8	0.781526	0.390763	+	0.920491i	$0.372211\pi$
$972$	0	0
$973$	4682.37	0.154275
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−56300.6	−1.84362	−0.921809	−	0.387645i	$-0.873289\pi$
$977$	−56300.6	−1.84362	−0.921809	+	0.387645i	$0.873289\pi$
$978$	0	0
$979$	−719.245	−0.0234803
$980$	0	0
$981$	−9628.26	−0.313361
$982$	0	0
$983$	50882.8	1.65098	0.825488	−	0.564420i	$-0.190900\pi$
$983$	50882.8	1.65098	0.825488	+	0.564420i	$0.190900\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4249.55	0.137046
$988$	0	0
$989$	−19570.7	−0.629233
$990$	0	0
$991$	−56050.7	−1.79668	−0.898339	−	0.439302i	$-0.855226\pi$
$991$	−56050.7	−1.79668	−0.898339	+	0.439302i	$0.855226\pi$
$992$	0	0
$993$	20751.4	0.663169
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1487.65	−0.0472560	−0.0236280	−	0.999721i	$-0.507522\pi$
$997$	−1487.65	−0.0472560	−0.0236280	+	0.999721i	$0.507522\pi$
$998$	0	0
$999$	7805.72	0.247209

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.4.a.y.1.3		3
5.2	odd	4		420.4.k.b.169.1	✓	6
5.3	odd	4		420.4.k.b.169.4	yes	6
5.4	even	2		2100.4.a.x.1.3		3
15.2	even	4		1260.4.k.e.1009.5		6
15.8	even	4		1260.4.k.e.1009.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.4.k.b.169.1	✓	6	5.2	odd	4
420.4.k.b.169.4	yes	6	5.3	odd	4
1260.4.k.e.1009.5		6	15.2	even	4
1260.4.k.e.1009.6		6	15.8	even	4
2100.4.a.x.1.3		3	5.4	even	2
2100.4.a.y.1.3		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2100.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -7.00000 q^{7} +9.00000 q^{9} +1.01865 q^{11} -63.5205 q^{13} +33.8576 q^{17} +30.4646 q^{19} -21.0000 q^{21} +147.169 q^{23} +27.0000 q^{27} -275.745 q^{29} +39.5205 q^{31} +3.05596 q^{33} +289.101 q^{37} -190.562 q^{39} +173.476 q^{41} -132.981 q^{43} -202.360 q^{47} +49.0000 q^{49} +101.573 q^{51} +76.1889 q^{53} +91.3938 q^{57} -799.100 q^{59} +780.418 q^{61} -63.0000 q^{63} -465.687 q^{67} +441.506 q^{69} -280.728 q^{71} -840.569 q^{73} -7.13057 q^{77} -303.041 q^{79} +81.0000 q^{81} -785.695 q^{83} -827.235 q^{87} -706.075 q^{89} +444.644 q^{91} +118.562 q^{93} +1075.31 q^{97} +9.16787 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 9 q^{3} - 21 q^{7} + 27 q^{9} - 54 q^{11} - 42 q^{13} + 56 q^{17} + 114 q^{19} - 63 q^{21} + 64 q^{23} + 81 q^{27} - 130 q^{29} - 30 q^{31} - 162 q^{33} + 216 q^{37} - 126 q^{39} - 6 q^{41} - 456 q^{43}+ \cdots - 486 q^{99}+O(q^{100}) \) 3 * q + 9 * q^3 - 21 * q^7 + 27 * q^9 - 54 * q^11 - 42 * q^13 + 56 * q^17 + 114 * q^19 - 63 * q^21 + 64 * q^23 + 81 * q^27 - 130 * q^29 - 30 * q^31 - 162 * q^33 + 216 * q^37 - 126 * q^39 - 6 * q^41 - 456 * q^43 + 388 * q^47 + 147 * q^49 + 168 * q^51 + 66 * q^53 + 342 * q^57 - 616 * q^59 - 138 * q^61 - 189 * q^63 - 468 * q^67 + 192 * q^69 - 474 * q^71 - 1710 * q^73 + 378 * q^77 - 612 * q^79 + 243 * q^81 - 1920 * q^83 - 390 * q^87 - 2026 * q^89 + 294 * q^91 - 90 * q^93 - 1218 * q^97 - 486 * q^99

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