LMFDB - Embedded newform 1800.4.a.br.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,4,Mod(1,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, 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("1800.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1800.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:	$106.203438010$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.121909.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 87x + 270 $ x^3 - x^2 - 87*x + 270
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-10.1679$ of defining polynomial
Character	$\chi$	$=$	1800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.96885 q^{7} +O(q^{10})$	$q-7.96885 q^{7} -44.6716 q^{11} +68.4055 q^{13} -90.1395 q^{17} +75.3744 q^{19} +11.0792 q^{23} +145.890 q^{29} +202.593 q^{31} -124.936 q^{37} -150.955 q^{41} -134.435 q^{43} +510.716 q^{47} -279.498 q^{49} +16.2978 q^{53} -399.215 q^{59} -486.344 q^{61} +353.433 q^{67} +76.9802 q^{71} -419.939 q^{73} +355.981 q^{77} +110.437 q^{79} -45.6005 q^{83} -724.874 q^{89} -545.113 q^{91} +539.123 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{7}+O(q^{10}) $ 3 * q - 9 * q^7	$ 3 q - 9 q^{7} - 8 q^{11} - 17 q^{13} + 48 q^{17} - 11 q^{19} + 40 q^{23} + 59 q^{31} + 10 q^{37} - 400 q^{41} + 159 q^{43} + 272 q^{47} + 110 q^{49} - 256 q^{53} - 176 q^{59} - 375 q^{61} + 143 q^{67} - 1288 q^{71} - 398 q^{73} + 736 q^{77} - 292 q^{79} - 1424 q^{83} - 928 q^{89} + 851 q^{91} - 697 q^{97}+O(q^{100}) $ 3 * q - 9 * q^7 - 8 * q^11 - 17 * q^13 + 48 * q^17 - 11 * q^19 + 40 * q^23 + 59 * q^31 + 10 * q^37 - 400 * q^41 + 159 * q^43 + 272 * q^47 + 110 * q^49 - 256 * q^53 - 176 * q^59 - 375 * q^61 + 143 * q^67 - 1288 * q^71 - 398 * q^73 + 736 * q^77 - 292 * q^79 - 1424 * q^83 - 928 * q^89 + 851 * q^91 - 697 * 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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.96885	−0.430277	−0.215139	−	0.976584i	$-0.569020\pi$
$7$	−7.96885	−0.430277	−0.215139	+	0.976584i	$0.569020\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−44.6716	−1.22445	−0.612227	−	0.790682i	$-0.709726\pi$
$11$	−44.6716	−1.22445	−0.612227	+	0.790682i	$0.709726\pi$
$12$	0	0
$13$	68.4055	1.45941	0.729703	−	0.683764i	$-0.239658\pi$
$13$	68.4055	1.45941	0.729703	+	0.683764i	$0.239658\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−90.1395	−1.28600	−0.643001	−	0.765865i	$-0.722311\pi$
$17$	−90.1395	−1.28600	−0.643001	+	0.765865i	$0.722311\pi$
$18$	0	0
$19$	75.3744	0.910109	0.455054	−	0.890464i	$-0.349620\pi$
$19$	75.3744	0.910109	0.455054	+	0.890464i	$0.349620\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	11.0792	0.100442	0.0502209	−	0.998738i	$-0.484007\pi$
$23$	11.0792	0.100442	0.0502209	+	0.998738i	$0.484007\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	145.890	0.934177	0.467088	−	0.884211i	$-0.345303\pi$
$29$	145.890	0.934177	0.467088	+	0.884211i	$0.345303\pi$
$30$	0	0
$31$	202.593	1.17377	0.586883	−	0.809671i	$-0.300355\pi$
$31$	202.593	1.17377	0.586883	+	0.809671i	$0.300355\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−124.936	−0.555116	−0.277558	−	0.960709i	$-0.589525\pi$
$37$	−124.936	−0.555116	−0.277558	+	0.960709i	$0.589525\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−150.955	−0.575003	−0.287502	−	0.957780i	$-0.592825\pi$
$41$	−150.955	−0.575003	−0.287502	+	0.957780i	$0.592825\pi$
$42$	0	0
$43$	−134.435	−0.476770	−0.238385	−	0.971171i	$-0.576618\pi$
$43$	−134.435	−0.476770	−0.238385	+	0.971171i	$0.576618\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	510.716	1.58501	0.792506	−	0.609864i	$-0.208776\pi$
$47$	510.716	1.58501	0.792506	+	0.609864i	$0.208776\pi$
$48$	0	0
$49$	−279.498	−0.814862
$50$	0	0
$51$	0	0
$52$	0	0
$53$	16.2978	0.0422390	0.0211195	−	0.999777i	$-0.493277\pi$
$53$	16.2978	0.0422390	0.0211195	+	0.999777i	$0.493277\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−399.215	−0.880904	−0.440452	−	0.897776i	$-0.645182\pi$
$59$	−399.215	−0.880904	−0.440452	+	0.897776i	$0.645182\pi$
$60$	0	0
$61$	−486.344	−1.02082	−0.510410	−	0.859931i	$-0.670506\pi$
$61$	−486.344	−1.02082	−0.510410	+	0.859931i	$0.670506\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	353.433	0.644458	0.322229	−	0.946662i	$-0.395568\pi$
$67$	353.433	0.644458	0.322229	+	0.946662i	$0.395568\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	76.9802	0.128674	0.0643371	−	0.997928i	$-0.479507\pi$
$71$	76.9802	0.128674	0.0643371	+	0.997928i	$0.479507\pi$
$72$	0	0
$73$	−419.939	−0.673289	−0.336645	−	0.941632i	$-0.609292\pi$
$73$	−419.939	−0.673289	−0.336645	+	0.941632i	$0.609292\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	355.981	0.526855
$78$	0	0
$79$	110.437	0.157280	0.0786402	−	0.996903i	$-0.474942\pi$
$79$	110.437	0.157280	0.0786402	+	0.996903i	$0.474942\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−45.6005	−0.0603049	−0.0301524	−	0.999545i	$-0.509599\pi$
$83$	−45.6005	−0.0603049	−0.0301524	+	0.999545i	$0.509599\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−724.874	−0.863332	−0.431666	−	0.902034i	$-0.642074\pi$
$89$	−724.874	−0.863332	−0.431666	+	0.902034i	$0.642074\pi$
$90$	0	0
$91$	−545.113	−0.627949
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	539.123	0.564326	0.282163	−	0.959366i	$-0.408948\pi$
$97$	539.123	0.564326	0.282163	+	0.959366i	$0.408948\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1489.32	−1.46725	−0.733626	−	0.679553i	$-0.762174\pi$
$101$	−1489.32	−1.46725	−0.733626	+	0.679553i	$0.762174\pi$
$102$	0	0
$103$	1207.05	1.15470	0.577352	−	0.816495i	$-0.304086\pi$
$103$	1207.05	1.15470	0.577352	+	0.816495i	$0.304086\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−709.348	−0.640890	−0.320445	−	0.947267i	$-0.603832\pi$
$107$	−709.348	−0.640890	−0.320445	+	0.947267i	$0.603832\pi$
$108$	0	0
$109$	886.269	0.778800	0.389400	−	0.921069i	$-0.372682\pi$
$109$	886.269	0.778800	0.389400	+	0.921069i	$0.372682\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2174.62	−1.81037	−0.905183	−	0.425022i	$-0.860266\pi$
$113$	−2174.62	−1.81037	−0.905183	+	0.425022i	$0.860266\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	718.307	0.553337
$120$	0	0
$121$	664.553	0.499288
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1189.23	−0.830925	−0.415463	−	0.909610i	$-0.636380\pi$
$127$	−1189.23	−0.830925	−0.415463	+	0.909610i	$0.636380\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2711.13	−1.80819	−0.904094	−	0.427334i	$-0.859453\pi$
$131$	−2711.13	−1.80819	−0.904094	+	0.427334i	$0.859453\pi$
$132$	0	0
$133$	−600.647	−0.391599
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1504.60	0.938294	0.469147	−	0.883120i	$-0.344561\pi$
$137$	1504.60	0.938294	0.469147	+	0.883120i	$0.344561\pi$
$138$	0	0
$139$	−2156.67	−1.31602	−0.658008	−	0.753011i	$-0.728601\pi$
$139$	−2156.67	−1.31602	−0.658008	+	0.753011i	$0.728601\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3055.79	−1.78698
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−668.220	−0.367401	−0.183700	−	0.982982i	$-0.558808\pi$
$149$	−668.220	−0.367401	−0.183700	+	0.982982i	$0.558808\pi$
$150$	0	0
$151$	3206.57	1.72812	0.864062	−	0.503385i	$-0.167912\pi$
$151$	3206.57	1.72812	0.864062	+	0.503385i	$0.167912\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2290.10	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$157$	−2290.10	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−88.2880	−0.0432178
$162$	0	0
$163$	−629.302	−0.302397	−0.151199	−	0.988503i	$-0.548313\pi$
$163$	−629.302	−0.302397	−0.151199	+	0.988503i	$0.548313\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1860.85	0.862255	0.431127	−	0.902291i	$-0.358116\pi$
$167$	1860.85	0.862255	0.431127	+	0.902291i	$0.358116\pi$
$168$	0	0
$169$	2482.32	1.12987
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2030.58	0.892384	0.446192	−	0.894937i	$-0.352780\pi$
$173$	2030.58	0.892384	0.446192	+	0.894937i	$0.352780\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3025.16	−1.26319	−0.631595	−	0.775299i	$-0.717599\pi$
$179$	−3025.16	−1.26319	−0.631595	+	0.775299i	$0.717599\pi$
$180$	0	0
$181$	−1238.77	−0.508714	−0.254357	−	0.967110i	$-0.581864\pi$
$181$	−1238.77	−0.508714	−0.254357	+	0.967110i	$0.581864\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4026.67	1.57465
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4731.20	−1.79234	−0.896171	−	0.443708i	$-0.853663\pi$
$191$	−4731.20	−1.79234	−0.896171	+	0.443708i	$0.853663\pi$
$192$	0	0
$193$	1115.42	0.416008	0.208004	−	0.978128i	$-0.433303\pi$
$193$	1115.42	0.416008	0.208004	+	0.978128i	$0.433303\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4571.46	1.65331	0.826657	−	0.562705i	$-0.190239\pi$
$197$	4571.46	1.65331	0.826657	+	0.562705i	$0.190239\pi$
$198$	0	0
$199$	294.940	0.105064	0.0525320	−	0.998619i	$-0.483271\pi$
$199$	294.940	0.105064	0.0525320	+	0.998619i	$0.483271\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1162.58	−0.401955
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3367.09	−1.11439
$210$	0	0
$211$	−3803.10	−1.24083	−0.620417	−	0.784272i	$-0.713037\pi$
$211$	−3803.10	−1.24083	−0.620417	+	0.784272i	$0.713037\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1614.43	−0.505045
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6166.04	−1.87680
$222$	0	0
$223$	−274.982	−0.0825747	−0.0412873	−	0.999147i	$-0.513146\pi$
$223$	−274.982	−0.0825747	−0.0412873	+	0.999147i	$0.513146\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6321.48	1.84833	0.924166	−	0.381991i	$-0.124762\pi$
$227$	6321.48	1.84833	0.924166	+	0.381991i	$0.124762\pi$
$228$	0	0
$229$	−3506.58	−1.01188	−0.505941	−	0.862568i	$-0.668855\pi$
$229$	−3506.58	−1.01188	−0.505941	+	0.862568i	$0.668855\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5448.54	−1.53195	−0.765977	−	0.642867i	$-0.777745\pi$
$233$	−5448.54	−1.53195	−0.765977	+	0.642867i	$0.777745\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1075.25	−0.291013	−0.145506	−	0.989357i	$-0.546481\pi$
$239$	−1075.25	−0.291013	−0.145506	+	0.989357i	$0.546481\pi$
$240$	0	0
$241$	−3812.76	−1.01909	−0.509547	−	0.860443i	$-0.670187\pi$
$241$	−3812.76	−1.01909	−0.509547	+	0.860443i	$0.670187\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5156.02	1.32822
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3217.74	−0.809171	−0.404585	−	0.914500i	$-0.632584\pi$
$251$	−3217.74	−0.809171	−0.404585	+	0.914500i	$0.632584\pi$
$252$	0	0
$253$	−494.923	−0.122986
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1772.39	−0.430189	−0.215095	−	0.976593i	$-0.569006\pi$
$257$	−1772.39	−0.430189	−0.215095	+	0.976593i	$0.569006\pi$
$258$	0	0
$259$	995.593	0.238854
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−483.991	−0.113476	−0.0567380	−	0.998389i	$-0.518070\pi$
$263$	−483.991	−0.113476	−0.0567380	+	0.998389i	$0.518070\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3801.35	0.861608	0.430804	−	0.902446i	$-0.358230\pi$
$269$	3801.35	0.861608	0.430804	+	0.902446i	$0.358230\pi$
$270$	0	0
$271$	3309.65	0.741871	0.370936	−	0.928659i	$-0.379037\pi$
$271$	3309.65	0.741871	0.370936	+	0.928659i	$0.379037\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−5762.05	−1.24985	−0.624924	−	0.780686i	$-0.714870\pi$
$277$	−5762.05	−1.24985	−0.624924	+	0.780686i	$0.714870\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−9106.13	−1.93319	−0.966595	−	0.256308i	$-0.917494\pi$
$281$	−9106.13	−1.93319	−0.966595	+	0.256308i	$0.917494\pi$
$282$	0	0
$283$	275.519	0.0578724	0.0289362	−	0.999581i	$-0.490788\pi$
$283$	275.519	0.0578724	0.0289362	+	0.999581i	$0.490788\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1202.93	0.247411
$288$	0	0
$289$	3212.12	0.653800
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6947.24	−1.38519	−0.692597	−	0.721325i	$-0.743534\pi$
$293$	−6947.24	−1.38519	−0.692597	+	0.721325i	$0.743534\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	757.875	0.146585
$300$	0	0
$301$	1071.29	0.205143
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4368.00	0.812036	0.406018	−	0.913865i	$-0.366917\pi$
$307$	4368.00	0.812036	0.406018	+	0.913865i	$0.366917\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4267.08	−0.778019	−0.389010	−	0.921234i	$-0.627183\pi$
$311$	−4267.08	−0.778019	−0.389010	+	0.921234i	$0.627183\pi$
$312$	0	0
$313$	−6854.48	−1.23782	−0.618910	−	0.785462i	$-0.712426\pi$
$313$	−6854.48	−1.23782	−0.618910	+	0.785462i	$0.712426\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1349.88	0.239170	0.119585	−	0.992824i	$-0.461844\pi$
$317$	1349.88	0.239170	0.119585	+	0.992824i	$0.461844\pi$
$318$	0	0
$319$	−6517.15	−1.14386
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6794.21	−1.17040
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4069.82	−0.681995
$330$	0	0
$331$	10112.3	1.67923	0.839613	−	0.543185i	$-0.182782\pi$
$331$	10112.3	1.67923	0.839613	+	0.543185i	$0.182782\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−10697.0	−1.72909	−0.864544	−	0.502557i	$-0.832393\pi$
$337$	−10697.0	−1.72909	−0.864544	+	0.502557i	$0.832393\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−9050.15	−1.43722
$342$	0	0
$343$	4960.59	0.780894
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2110.60	0.326521	0.163261	−	0.986583i	$-0.447799\pi$
$347$	2110.60	0.326521	0.163261	+	0.986583i	$0.447799\pi$
$348$	0	0
$349$	7685.29	1.17875	0.589375	−	0.807859i	$-0.299374\pi$
$349$	7685.29	1.17875	0.589375	+	0.807859i	$0.299374\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3154.92	−0.475693	−0.237847	−	0.971303i	$-0.576442\pi$
$353$	−3154.92	−0.475693	−0.237847	+	0.971303i	$0.576442\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12017.3	−1.76670	−0.883352	−	0.468710i	$-0.844719\pi$
$359$	−12017.3	−1.76670	−0.883352	+	0.468710i	$0.844719\pi$
$360$	0	0
$361$	−1177.70	−0.171702
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1967.69	0.279870	0.139935	−	0.990161i	$-0.455311\pi$
$367$	1967.69	0.279870	0.139935	+	0.990161i	$0.455311\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−129.874	−0.0181745
$372$	0	0
$373$	5256.37	0.729664	0.364832	−	0.931073i	$-0.381126\pi$
$373$	5256.37	0.729664	0.364832	+	0.931073i	$0.381126\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9979.70	1.36334
$378$	0	0
$379$	−6935.16	−0.939934	−0.469967	−	0.882684i	$-0.655734\pi$
$379$	−6935.16	−0.939934	−0.469967	+	0.882684i	$0.655734\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	351.455	0.0468890	0.0234445	−	0.999725i	$-0.492537\pi$
$383$	351.455	0.0468890	0.0234445	+	0.999725i	$0.492537\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10147.5	−1.32262	−0.661310	−	0.750113i	$-0.729999\pi$
$389$	−10147.5	−1.32262	−0.661310	+	0.750113i	$0.729999\pi$
$390$	0	0
$391$	−998.669	−0.129168
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3265.56	0.412831	0.206416	−	0.978464i	$-0.433820\pi$
$397$	3265.56	0.412831	0.206416	+	0.978464i	$0.433820\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5713.85	0.711561	0.355781	−	0.934570i	$-0.384215\pi$
$401$	5713.85	0.711561	0.355781	+	0.934570i	$0.384215\pi$
$402$	0	0
$403$	13858.5	1.71300
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5581.08	0.679714
$408$	0	0
$409$	15470.5	1.87034	0.935168	−	0.354205i	$-0.115248\pi$
$409$	15470.5	1.87034	0.935168	+	0.354205i	$0.115248\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3181.28	0.379033
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4986.09	0.581352	0.290676	−	0.956822i	$-0.406120\pi$
$419$	4986.09	0.581352	0.290676	+	0.956822i	$0.406120\pi$
$420$	0	0
$421$	−14428.4	−1.67030	−0.835152	−	0.550019i	$-0.814620\pi$
$421$	−14428.4	−1.67030	−0.835152	+	0.550019i	$0.814620\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3875.60	0.439235
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7342.19	−0.820559	−0.410280	−	0.911960i	$-0.634569\pi$
$431$	−7342.19	−0.820559	−0.410280	+	0.911960i	$0.634569\pi$
$432$	0	0
$433$	−9721.35	−1.07893	−0.539467	−	0.842007i	$-0.681374\pi$
$433$	−9721.35	−1.07893	−0.539467	+	0.842007i	$0.681374\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	835.084	0.0914130
$438$	0	0
$439$	−11550.9	−1.25580	−0.627898	−	0.778296i	$-0.716084\pi$
$439$	−11550.9	−1.25580	−0.627898	+	0.778296i	$0.716084\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9079.23	−0.973741	−0.486870	−	0.873474i	$-0.661862\pi$
$443$	−9079.23	−0.973741	−0.486870	+	0.873474i	$0.661862\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8743.18	0.918967	0.459484	−	0.888186i	$-0.348035\pi$
$449$	8743.18	0.918967	0.459484	+	0.888186i	$0.348035\pi$
$450$	0	0
$451$	6743.38	0.704065
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−552.591	−0.0565627	−0.0282813	−	0.999600i	$-0.509003\pi$
$457$	−552.591	−0.0565627	−0.0282813	+	0.999600i	$0.509003\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15025.8	1.51805	0.759027	−	0.651059i	$-0.225675\pi$
$461$	15025.8	1.51805	0.759027	+	0.651059i	$0.225675\pi$
$462$	0	0
$463$	−15101.2	−1.51579	−0.757896	−	0.652375i	$-0.773773\pi$
$463$	−15101.2	−1.51579	−0.757896	+	0.652375i	$0.773773\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1294.15	0.128236	0.0641179	−	0.997942i	$-0.479577\pi$
$467$	1294.15	0.128236	0.0641179	+	0.997942i	$0.479577\pi$
$468$	0	0
$469$	−2816.45	−0.277295
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6005.41	0.583783
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3911.63	0.373125	0.186562	−	0.982443i	$-0.440265\pi$
$479$	3911.63	0.373125	0.186562	+	0.982443i	$0.440265\pi$
$480$	0	0
$481$	−8546.29	−0.810140
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	16588.1	1.54349	0.771743	−	0.635935i	$-0.219385\pi$
$487$	16588.1	1.54349	0.771743	+	0.635935i	$0.219385\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−15969.8	−1.46784	−0.733918	−	0.679238i	$-0.762311\pi$
$491$	−15969.8	−1.46784	−0.733918	+	0.679238i	$0.762311\pi$
$492$	0	0
$493$	−13150.5	−1.20135
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−613.443	−0.0553656
$498$	0	0
$499$	4521.53	0.405634	0.202817	−	0.979217i	$-0.434990\pi$
$499$	4521.53	0.405634	0.202817	+	0.979217i	$0.434990\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−569.755	−0.0505052	−0.0252526	−	0.999681i	$-0.508039\pi$
$503$	−569.755	−0.0505052	−0.0252526	+	0.999681i	$0.508039\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5363.79	−0.467084	−0.233542	−	0.972347i	$-0.575032\pi$
$509$	−5363.79	−0.467084	−0.233542	+	0.972347i	$0.575032\pi$
$510$	0	0
$511$	3346.43	0.289701
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−22814.5	−1.94078
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−13591.9	−1.14294	−0.571469	−	0.820624i	$-0.693626\pi$
$521$	−13591.9	−1.14294	−0.571469	+	0.820624i	$0.693626\pi$
$522$	0	0
$523$	16083.5	1.34471	0.672355	−	0.740229i	$-0.265283\pi$
$523$	16083.5	1.34471	0.672355	+	0.740229i	$0.265283\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−18261.6	−1.50947
$528$	0	0
$529$	−12044.3	−0.989911
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−10326.1	−0.839163
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	12485.6	0.997761
$540$	0	0
$541$	−5699.53	−0.452942	−0.226471	−	0.974018i	$-0.572719\pi$
$541$	−5699.53	−0.452942	−0.226471	+	0.974018i	$0.572719\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	11499.3	0.898853	0.449426	−	0.893317i	$-0.351628\pi$
$547$	11499.3	0.898853	0.449426	+	0.893317i	$0.351628\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	10996.4	0.850203
$552$	0	0
$553$	−880.057	−0.0676742
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6223.25	−0.473407	−0.236703	−	0.971582i	$-0.576067\pi$
$557$	−6223.25	−0.473407	−0.236703	+	0.971582i	$0.576067\pi$
$558$	0	0
$559$	−9196.08	−0.695801
$560$	0	0
$561$	0	0
$562$	0	0
$563$	19671.8	1.47259	0.736294	−	0.676661i	$-0.236574\pi$
$563$	19671.8	1.47259	0.736294	+	0.676661i	$0.236574\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5800.40	0.427356	0.213678	−	0.976904i	$-0.431456\pi$
$569$	5800.40	0.427356	0.213678	+	0.976904i	$0.431456\pi$
$570$	0	0
$571$	18249.2	1.33749	0.668745	−	0.743492i	$-0.266832\pi$
$571$	18249.2	1.33749	0.668745	+	0.743492i	$0.266832\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−22705.9	−1.63823	−0.819115	−	0.573629i	$-0.805535\pi$
$577$	−22705.9	−1.63823	−0.819115	+	0.573629i	$0.805535\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	363.383	0.0259478
$582$	0	0
$583$	−728.047	−0.0517198
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15450.3	−1.08637	−0.543186	−	0.839612i	$-0.682782\pi$
$587$	−15450.3	−1.08637	−0.543186	+	0.839612i	$0.682782\pi$
$588$	0	0
$589$	15270.3	1.06826
$590$	0	0
$591$	0	0
$592$	0	0
$593$	25508.9	1.76649	0.883243	−	0.468916i	$-0.155355\pi$
$593$	25508.9	1.76649	0.883243	+	0.468916i	$0.155355\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1271.74	0.0867479	0.0433740	−	0.999059i	$-0.486189\pi$
$599$	1271.74	0.0867479	0.0433740	+	0.999059i	$0.486189\pi$
$600$	0	0
$601$	−3159.87	−0.214466	−0.107233	−	0.994234i	$-0.534199\pi$
$601$	−3159.87	−0.214466	−0.107233	+	0.994234i	$0.534199\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	26336.1	1.76104	0.880518	−	0.474013i	$-0.157195\pi$
$607$	26336.1	1.76104	0.880518	+	0.474013i	$0.157195\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	34935.8	2.31318
$612$	0	0
$613$	3079.40	0.202897	0.101448	−	0.994841i	$-0.467652\pi$
$613$	3079.40	0.202897	0.101448	+	0.994841i	$0.467652\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4069.83	−0.265551	−0.132776	−	0.991146i	$-0.542389\pi$
$617$	−4069.83	−0.265551	−0.132776	+	0.991146i	$0.542389\pi$
$618$	0	0
$619$	9667.13	0.627714	0.313857	−	0.949470i	$-0.398379\pi$
$619$	9667.13	0.627714	0.313857	+	0.949470i	$0.398379\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5776.41	0.371472
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11261.6	0.713880
$630$	0	0
$631$	10789.2	0.680685	0.340343	−	0.940302i	$-0.389457\pi$
$631$	10789.2	0.680685	0.340343	+	0.940302i	$0.389457\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−19119.2	−1.18921
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6793.94	0.418634	0.209317	−	0.977848i	$-0.432876\pi$
$641$	6793.94	0.418634	0.209317	+	0.977848i	$0.432876\pi$
$642$	0	0
$643$	13709.4	0.840819	0.420410	−	0.907334i	$-0.361886\pi$
$643$	13709.4	0.840819	0.420410	+	0.907334i	$0.361886\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21264.3	1.29210	0.646048	−	0.763296i	$-0.276420\pi$
$647$	21264.3	1.29210	0.646048	+	0.763296i	$0.276420\pi$
$648$	0	0
$649$	17833.6	1.07863
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7595.42	−0.455179	−0.227589	−	0.973757i	$-0.573084\pi$
$653$	−7595.42	−0.455179	−0.227589	+	0.973757i	$0.573084\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−28633.4	−1.69256	−0.846281	−	0.532737i	$-0.821164\pi$
$659$	−28633.4	−1.69256	−0.846281	+	0.532737i	$0.821164\pi$
$660$	0	0
$661$	512.739	0.0301713	0.0150857	−	0.999886i	$-0.495198\pi$
$661$	512.739	0.0301713	0.0150857	+	0.999886i	$0.495198\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1616.34	0.0938305
$668$	0	0
$669$	0	0
$670$	0	0
$671$	21725.8	1.24995
$672$	0	0
$673$	29639.6	1.69765	0.848827	−	0.528671i	$-0.177309\pi$
$673$	29639.6	1.69765	0.848827	+	0.528671i	$0.177309\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10148.1	−0.576107	−0.288054	−	0.957614i	$-0.593008\pi$
$677$	−10148.1	−0.576107	−0.288054	+	0.957614i	$0.593008\pi$
$678$	0	0
$679$	−4296.19	−0.242817
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9437.38	0.528714	0.264357	−	0.964425i	$-0.414840\pi$
$683$	9437.38	0.528714	0.264357	+	0.964425i	$0.414840\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1114.86	0.0616439
$690$	0	0
$691$	−6931.84	−0.381620	−0.190810	−	0.981627i	$-0.561111\pi$
$691$	−6931.84	−0.381620	−0.190810	+	0.981627i	$0.561111\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	13607.0	0.739455
$698$	0	0
$699$	0	0
$700$	0	0
$701$	34139.5	1.83942	0.919708	−	0.392603i	$-0.128425\pi$
$701$	34139.5	1.83942	0.919708	+	0.392603i	$0.128425\pi$
$702$	0	0
$703$	−9416.95	−0.505216
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11868.1	0.631325
$708$	0	0
$709$	2693.16	0.142657	0.0713285	−	0.997453i	$-0.477276\pi$
$709$	2693.16	0.142657	0.0713285	+	0.997453i	$0.477276\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2244.56	0.117895
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	36927.3	1.91537	0.957687	−	0.287810i	$-0.0929273\pi$
$719$	36927.3	1.91537	0.957687	+	0.287810i	$0.0929273\pi$
$720$	0	0
$721$	−9618.83	−0.496843
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−17669.9	−0.901431	−0.450715	−	0.892668i	$-0.648831\pi$
$727$	−17669.9	−0.901431	−0.450715	+	0.892668i	$0.648831\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	12117.9	0.613127
$732$	0	0
$733$	−30218.9	−1.52273	−0.761364	−	0.648324i	$-0.775470\pi$
$733$	−30218.9	−1.52273	−0.761364	+	0.648324i	$0.775470\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−15788.4	−0.789109
$738$	0	0
$739$	−25857.5	−1.28712	−0.643562	−	0.765394i	$-0.722544\pi$
$739$	−25857.5	−1.28712	−0.643562	+	0.765394i	$0.722544\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10855.2	0.535988	0.267994	−	0.963421i	$-0.413639\pi$
$743$	10855.2	0.535988	0.267994	+	0.963421i	$0.413639\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5652.69	0.275761
$750$	0	0
$751$	13363.2	0.649308	0.324654	−	0.945833i	$-0.394752\pi$
$751$	13363.2	0.649308	0.324654	+	0.945833i	$0.394752\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−11032.0	−0.529675	−0.264838	−	0.964293i	$-0.585318\pi$
$757$	−11032.0	−0.529675	−0.264838	+	0.964293i	$0.585318\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	21916.5	1.04399	0.521993	−	0.852950i	$-0.325189\pi$
$761$	21916.5	1.04399	0.521993	+	0.852950i	$0.325189\pi$
$762$	0	0
$763$	−7062.54	−0.335100
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−27308.5	−1.28560
$768$	0	0
$769$	−5678.81	−0.266298	−0.133149	−	0.991096i	$-0.542509\pi$
$769$	−5678.81	−0.266298	−0.133149	+	0.991096i	$0.542509\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7095.51	0.330152	0.165076	−	0.986281i	$-0.447213\pi$
$773$	7095.51	0.330152	0.165076	+	0.986281i	$0.447213\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−11378.1	−0.523316
$780$	0	0
$781$	−3438.83	−0.157556
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	40797.4	1.84787	0.923934	−	0.382553i	$-0.124955\pi$
$787$	40797.4	1.84787	0.923934	+	0.382553i	$0.124955\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	17329.2	0.778959
$792$	0	0
$793$	−33268.6	−1.48979
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−40111.4	−1.78271	−0.891355	−	0.453306i	$-0.850244\pi$
$797$	−40111.4	−1.78271	−0.891355	+	0.453306i	$0.850244\pi$
$798$	0	0
$799$	−46035.7	−2.03833
$800$	0	0
$801$	0	0
$802$	0	0
$803$	18759.3	0.824412
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	18512.8	0.804544	0.402272	−	0.915520i	$-0.368221\pi$
$809$	18512.8	0.804544	0.402272	+	0.915520i	$0.368221\pi$
$810$	0	0
$811$	−34802.7	−1.50689	−0.753446	−	0.657510i	$-0.771610\pi$
$811$	−34802.7	−1.50689	−0.753446	+	0.657510i	$0.771610\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−10132.9	−0.433912
$818$	0	0
$819$	0	0
$820$	0	0
$821$	11737.8	0.498967	0.249484	−	0.968379i	$-0.419739\pi$
$821$	11737.8	0.498967	0.249484	+	0.968379i	$0.419739\pi$
$822$	0	0
$823$	−1225.95	−0.0519247	−0.0259624	−	0.999663i	$-0.508265\pi$
$823$	−1225.95	−0.0519247	−0.0259624	+	0.999663i	$0.508265\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−40735.7	−1.71284	−0.856420	−	0.516279i	$-0.827317\pi$
$827$	−40735.7	−1.71284	−0.856420	+	0.516279i	$0.827317\pi$
$828$	0	0
$829$	42348.2	1.77420	0.887101	−	0.461576i	$-0.152716\pi$
$829$	42348.2	1.77420	0.887101	+	0.461576i	$0.152716\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	25193.8	1.04791
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	7909.91	0.325483	0.162742	−	0.986669i	$-0.447966\pi$
$839$	7909.91	0.325483	0.162742	+	0.986669i	$0.447966\pi$
$840$	0	0
$841$	−3105.05	−0.127313
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−5295.72	−0.214832
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1384.18	−0.0557569
$852$	0	0
$853$	9376.70	0.376380	0.188190	−	0.982133i	$-0.439738\pi$
$853$	9376.70	0.376380	0.188190	+	0.982133i	$0.439738\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−19600.1	−0.781245	−0.390623	−	0.920551i	$-0.627740\pi$
$857$	−19600.1	−0.781245	−0.390623	+	0.920551i	$0.627740\pi$
$858$	0	0
$859$	10498.5	0.417003	0.208502	−	0.978022i	$-0.433141\pi$
$859$	10498.5	0.417003	0.208502	+	0.978022i	$0.433141\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24079.4	−0.949796	−0.474898	−	0.880041i	$-0.657515\pi$
$863$	−24079.4	−0.949796	−0.474898	+	0.880041i	$0.657515\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4933.41	−0.192583
$870$	0	0
$871$	24176.7	0.940526
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	16335.5	0.628974	0.314487	−	0.949262i	$-0.398168\pi$
$877$	16335.5	0.628974	0.314487	+	0.949262i	$0.398168\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	34411.0	1.31593	0.657966	−	0.753048i	$-0.271417\pi$
$881$	34411.0	1.31593	0.657966	+	0.753048i	$0.271417\pi$
$882$	0	0
$883$	33088.6	1.26106	0.630532	−	0.776163i	$-0.282837\pi$
$883$	33088.6	1.26106	0.630532	+	0.776163i	$0.282837\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9019.58	−0.341430	−0.170715	−	0.985320i	$-0.554608\pi$
$887$	−9019.58	−0.341430	−0.170715	+	0.985320i	$0.554608\pi$
$888$	0	0
$889$	9476.82	0.357528
$890$	0	0
$891$	0	0
$892$	0	0
$893$	38494.9	1.44253
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	29556.3	1.09651
$900$	0	0
$901$	−1469.07	−0.0543195
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	22228.1	0.813751	0.406876	−	0.913484i	$-0.366618\pi$
$907$	22228.1	0.813751	0.406876	+	0.913484i	$0.366618\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−4413.93	−0.160527	−0.0802635	−	0.996774i	$-0.525576\pi$
$911$	−4413.93	−0.160527	−0.0802635	+	0.996774i	$0.525576\pi$
$912$	0	0
$913$	2037.05	0.0738406
$914$	0	0
$915$	0	0
$916$	0	0
$917$	21604.6	0.778022
$918$	0	0
$919$	16833.6	0.604233	0.302116	−	0.953271i	$-0.402307\pi$
$919$	16833.6	0.604233	0.302116	+	0.953271i	$0.402307\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5265.87	0.187788
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	24485.1	0.864724	0.432362	−	0.901700i	$-0.357680\pi$
$929$	24485.1	0.864724	0.432362	+	0.901700i	$0.357680\pi$
$930$	0	0
$931$	−21067.0	−0.741613
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−29815.7	−1.03953	−0.519763	−	0.854310i	$-0.673980\pi$
$937$	−29815.7	−1.03953	−0.519763	+	0.854310i	$0.673980\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1739.37	0.0602571	0.0301286	−	0.999546i	$-0.490408\pi$
$941$	1739.37	0.0602571	0.0301286	+	0.999546i	$0.490408\pi$
$942$	0	0
$943$	−1672.45	−0.0577544
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7911.11	−0.271464	−0.135732	−	0.990746i	$-0.543339\pi$
$947$	−7911.11	−0.271464	−0.135732	+	0.990746i	$0.543339\pi$
$948$	0	0
$949$	−28726.1	−0.982602
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−25369.3	−0.862321	−0.431160	−	0.902275i	$-0.641896\pi$
$953$	−25369.3	−0.862321	−0.431160	+	0.902275i	$0.641896\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−11989.9	−0.403726
$960$	0	0
$961$	11252.9	0.377729
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−46797.7	−1.55627	−0.778135	−	0.628097i	$-0.783834\pi$
$967$	−46797.7	−1.55627	−0.778135	+	0.628097i	$0.783834\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	24387.3	0.806001	0.403001	−	0.915200i	$-0.367967\pi$
$971$	24387.3	0.806001	0.403001	+	0.915200i	$0.367967\pi$
$972$	0	0
$973$	17186.2	0.566252
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24999.5	0.818635	0.409317	−	0.912392i	$-0.365767\pi$
$977$	24999.5	0.818635	0.409317	+	0.912392i	$0.365767\pi$
$978$	0	0
$979$	32381.3	1.05711
$980$	0	0
$981$	0	0
$982$	0	0
$983$	49558.1	1.60800	0.803998	−	0.594632i	$-0.202702\pi$
$983$	49558.1	1.60800	0.803998	+	0.594632i	$0.202702\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1489.42	−0.0478876
$990$	0	0
$991$	47258.6	1.51485	0.757427	−	0.652920i	$-0.226456\pi$
$991$	47258.6	1.51485	0.757427	+	0.652920i	$0.226456\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	20040.5	0.636599	0.318299	−	0.947990i	$-0.396888\pi$
$997$	20040.5	0.636599	0.318299	+	0.947990i	$0.396888\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.4.a.br.1.2	✓	3
3.2	odd	2		1800.4.a.bs.1.2	yes	3
5.2	odd	4		1800.4.f.ba.649.3		6
5.3	odd	4		1800.4.f.ba.649.4		6
5.4	even	2		1800.4.a.bt.1.2	yes	3
15.2	even	4		1800.4.f.bb.649.3		6
15.8	even	4		1800.4.f.bb.649.4		6
15.14	odd	2		1800.4.a.bu.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1800.4.a.br.1.2	✓	3	1.1	even	1	trivial
1800.4.a.bs.1.2	yes	3	3.2	odd	2
1800.4.a.bt.1.2	yes	3	5.4	even	2
1800.4.a.bu.1.2	yes	3	15.14	odd	2
1800.4.f.ba.649.3		6	5.2	odd	4
1800.4.f.ba.649.4		6	5.3	odd	4
1800.4.f.bb.649.3		6	15.2	even	4
1800.4.f.bb.649.4		6	15.8	even	4

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:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1800.a (trivial)

\(f(q)\)	\(=\)	\(q-7.96885 q^{7} +O(q^{10})\)	\(q-7.96885 q^{7} -44.6716 q^{11} +68.4055 q^{13} -90.1395 q^{17} +75.3744 q^{19} +11.0792 q^{23} +145.890 q^{29} +202.593 q^{31} -124.936 q^{37} -150.955 q^{41} -134.435 q^{43} +510.716 q^{47} -279.498 q^{49} +16.2978 q^{53} -399.215 q^{59} -486.344 q^{61} +353.433 q^{67} +76.9802 q^{71} -419.939 q^{73} +355.981 q^{77} +110.437 q^{79} -45.6005 q^{83} -724.874 q^{89} -545.113 q^{91} +539.123 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{7}+O(q^{10}) \) 3 * q - 9 * q^7	\( 3 q - 9 q^{7} - 8 q^{11} - 17 q^{13} + 48 q^{17} - 11 q^{19} + 40 q^{23} + 59 q^{31} + 10 q^{37} - 400 q^{41} + 159 q^{43} + 272 q^{47} + 110 q^{49} - 256 q^{53} - 176 q^{59} - 375 q^{61} + 143 q^{67} - 1288 q^{71} - 398 q^{73} + 736 q^{77} - 292 q^{79} - 1424 q^{83} - 928 q^{89} + 851 q^{91} - 697 q^{97}+O(q^{100}) \) 3 * q - 9 * q^7 - 8 * q^11 - 17 * q^13 + 48 * q^17 - 11 * q^19 + 40 * q^23 + 59 * q^31 + 10 * q^37 - 400 * q^41 + 159 * q^43 + 272 * q^47 + 110 * q^49 - 256 * q^53 - 176 * q^59 - 375 * q^61 + 143 * q^67 - 1288 * q^71 - 398 * q^73 + 736 * q^77 - 292 * q^79 - 1424 * q^83 - 928 * q^89 + 851 * q^91 - 697 * q^97

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