LMFDB - Embedded newform 1800.4.a.bt.1.1

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			1.1
Root			$7.73485$ 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)

$f(q)$	$=$	$q-22.6681 q^{7} +O(q^{10})$	$q-22.6681 q^{7} +26.9394 q^{11} +13.5426 q^{13} -2.14580 q^{17} -37.2107 q^{19} +162.405 q^{23} -191.491 q^{29} -175.762 q^{31} +83.5872 q^{37} +145.078 q^{41} +103.049 q^{43} +205.394 q^{47} +170.843 q^{49} +422.957 q^{53} -173.296 q^{59} +617.335 q^{61} -341.972 q^{67} -608.867 q^{71} -601.792 q^{73} -610.665 q^{77} -421.102 q^{79} +67.7724 q^{83} +338.205 q^{89} -306.985 q^{91} +1219.39 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−22.6681	−1.22396	−0.611981	−	0.790872i	$-0.709627\pi$
$7$	−22.6681	−1.22396	−0.611981	+	0.790872i	$0.709627\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	26.9394	0.738412	0.369206	−	0.929348i	$-0.379630\pi$
$11$	26.9394	0.738412	0.369206	+	0.929348i	$0.379630\pi$
$12$	0	0
$13$	13.5426	0.288926	0.144463	−	0.989510i	$-0.453855\pi$
$13$	13.5426	0.288926	0.144463	+	0.989510i	$0.453855\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.14580	−0.0306136	−0.0153068	−	0.999883i	$-0.504873\pi$
$17$	−2.14580	−0.0306136	−0.0153068	+	0.999883i	$0.504873\pi$
$18$	0	0
$19$	−37.2107	−0.449301	−0.224651	−	0.974439i	$-0.572124\pi$
$19$	−37.2107	−0.449301	−0.224651	+	0.974439i	$0.572124\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	162.405	1.47234	0.736171	−	0.676796i	$-0.236632\pi$
$23$	162.405	1.47234	0.736171	+	0.676796i	$0.236632\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−191.491	−1.22617	−0.613085	−	0.790017i	$-0.710072\pi$
$29$	−191.491	−1.22617	−0.613085	+	0.790017i	$0.710072\pi$
$30$	0	0
$31$	−175.762	−1.01832	−0.509158	−	0.860673i	$-0.670043\pi$
$31$	−175.762	−1.01832	−0.509158	+	0.860673i	$0.670043\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	83.5872	0.371396	0.185698	−	0.982607i	$-0.440545\pi$
$37$	83.5872	0.371396	0.185698	+	0.982607i	$0.440545\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	145.078	0.552618	0.276309	−	0.961069i	$-0.410889\pi$
$41$	145.078	0.552618	0.276309	+	0.961069i	$0.410889\pi$
$42$	0	0
$43$	103.049	0.365461	0.182730	−	0.983163i	$-0.441506\pi$
$43$	103.049	0.365461	0.182730	+	0.983163i	$0.441506\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	205.394	0.637442	0.318721	−	0.947849i	$-0.396747\pi$
$47$	205.394	0.637442	0.318721	+	0.947849i	$0.396747\pi$
$48$	0	0
$49$	170.843	0.498084
$50$	0	0
$51$	0	0
$52$	0	0
$53$	422.957	1.09618	0.548090	−	0.836419i	$-0.315355\pi$
$53$	422.957	1.09618	0.548090	+	0.836419i	$0.315355\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−173.296	−0.382393	−0.191196	−	0.981552i	$-0.561237\pi$
$59$	−173.296	−0.382393	−0.191196	+	0.981552i	$0.561237\pi$
$60$	0	0
$61$	617.335	1.29576	0.647882	−	0.761741i	$-0.275655\pi$
$61$	617.335	1.29576	0.647882	+	0.761741i	$0.275655\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−341.972	−0.623561	−0.311780	−	0.950154i	$-0.600925\pi$
$67$	−341.972	−0.623561	−0.311780	+	0.950154i	$0.600925\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−608.867	−1.01774	−0.508868	−	0.860845i	$-0.669936\pi$
$71$	−608.867	−1.01774	−0.508868	+	0.860845i	$0.669936\pi$
$72$	0	0
$73$	−601.792	−0.964855	−0.482428	−	0.875936i	$-0.660245\pi$
$73$	−601.792	−0.964855	−0.482428	+	0.875936i	$0.660245\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−610.665	−0.903789
$78$	0	0
$79$	−421.102	−0.599718	−0.299859	−	0.953984i	$-0.596940\pi$
$79$	−421.102	−0.599718	−0.299859	+	0.953984i	$0.596940\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	67.7724	0.0896263	0.0448132	−	0.998995i	$-0.485731\pi$
$83$	67.7724	0.0896263	0.0448132	+	0.998995i	$0.485731\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	338.205	0.402805	0.201402	−	0.979509i	$-0.435450\pi$
$89$	338.205	0.402805	0.201402	+	0.979509i	$0.435450\pi$
$90$	0	0
$91$	−306.985	−0.353635
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1219.39	1.27640	0.638199	−	0.769872i	$-0.279680\pi$
$97$	1219.39	1.27640	0.638199	+	0.769872i	$0.279680\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1680.48	−1.65558	−0.827792	−	0.561035i	$-0.810403\pi$
$101$	−1680.48	−1.65558	−0.827792	+	0.561035i	$0.810403\pi$
$102$	0	0
$103$	552.958	0.528977	0.264488	−	0.964389i	$-0.414797\pi$
$103$	552.958	0.528977	0.264488	+	0.964389i	$0.414797\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1450.07	−1.31013	−0.655063	−	0.755575i	$-0.727358\pi$
$107$	−1450.07	−1.31013	−0.655063	+	0.755575i	$0.727358\pi$
$108$	0	0
$109$	801.325	0.704157	0.352078	−	0.935971i	$-0.385475\pi$
$109$	801.325	0.704157	0.352078	+	0.935971i	$0.385475\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1014.61	−0.844663	−0.422331	−	0.906441i	$-0.638788\pi$
$113$	−1014.61	−0.844663	−0.422331	+	0.906441i	$0.638788\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	48.6411	0.0374699
$120$	0	0
$121$	−605.269	−0.454747
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1267.71	−0.885758	−0.442879	−	0.896581i	$-0.646043\pi$
$127$	−1267.71	−0.885758	−0.442879	+	0.896581i	$0.646043\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1758.02	−1.17251	−0.586254	−	0.810127i	$-0.699398\pi$
$131$	−1758.02	−1.17251	−0.586254	+	0.810127i	$0.699398\pi$
$132$	0	0
$133$	843.496	0.549928
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−626.087	−0.390439	−0.195220	−	0.980760i	$-0.562542\pi$
$137$	−626.087	−0.390439	−0.195220	+	0.980760i	$0.562542\pi$
$138$	0	0
$139$	−148.565	−0.0906555	−0.0453277	−	0.998972i	$-0.514433\pi$
$139$	−148.565	−0.0906555	−0.0453277	+	0.998972i	$0.514433\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	364.829	0.213347
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1342.98	−0.738398	−0.369199	−	0.929350i	$-0.620368\pi$
$149$	−1342.98	−0.738398	−0.369199	+	0.929350i	$0.620368\pi$
$150$	0	0
$151$	225.798	0.121690	0.0608451	−	0.998147i	$-0.480620\pi$
$151$	225.798	0.121690	0.0608451	+	0.998147i	$0.480620\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−979.591	−0.497961	−0.248980	−	0.968508i	$-0.580096\pi$
$157$	−979.591	−0.497961	−0.248980	+	0.968508i	$0.580096\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3681.42	−1.80209
$162$	0	0
$163$	−1345.92	−0.646751	−0.323376	−	0.946271i	$-0.604818\pi$
$163$	−1345.92	−0.646751	−0.323376	+	0.946271i	$0.604818\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3053.07	−1.41469	−0.707345	−	0.706868i	$-0.750107\pi$
$167$	−3053.07	−1.41469	−0.707345	+	0.706868i	$0.750107\pi$
$168$	0	0
$169$	−2013.60	−0.916522
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4506.69	1.98056	0.990282	−	0.139077i	$-0.0444135\pi$
$173$	4506.69	1.98056	0.990282	+	0.139077i	$0.0444135\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1586.95	−0.662648	−0.331324	−	0.943517i	$-0.607495\pi$
$179$	−1586.95	−0.662648	−0.331324	+	0.943517i	$0.607495\pi$
$180$	0	0
$181$	−1604.17	−0.658767	−0.329384	−	0.944196i	$-0.606841\pi$
$181$	−1604.17	−0.658767	−0.329384	+	0.944196i	$0.606841\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−57.8064	−0.0226055
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2839.05	−1.07553	−0.537765	−	0.843095i	$-0.680731\pi$
$191$	−2839.05	−1.07553	−0.537765	+	0.843095i	$0.680731\pi$
$192$	0	0
$193$	1137.78	0.424348	0.212174	−	0.977232i	$-0.431946\pi$
$193$	1137.78	0.424348	0.212174	+	0.977232i	$0.431946\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−828.511	−0.299639	−0.149820	−	0.988713i	$-0.547869\pi$
$197$	−828.511	−0.299639	−0.149820	+	0.988713i	$0.547869\pi$
$198$	0	0
$199$	−397.273	−0.141517	−0.0707586	−	0.997493i	$-0.522542\pi$
$199$	−397.273	−0.141517	−0.0707586	+	0.997493i	$0.522542\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4340.73	1.50079
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1002.43	−0.331769
$210$	0	0
$211$	4840.04	1.57916	0.789578	−	0.613650i	$-0.210300\pi$
$211$	4840.04	1.57916	0.789578	+	0.613650i	$0.210300\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3984.19	1.24638
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−29.0596	−0.00884508
$222$	0	0
$223$	−2736.65	−0.821793	−0.410896	−	0.911682i	$-0.634784\pi$
$223$	−2736.65	−0.821793	−0.410896	+	0.911682i	$0.634784\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1478.13	0.432189	0.216095	−	0.976372i	$-0.430668\pi$
$227$	1478.13	0.432189	0.216095	+	0.976372i	$0.430668\pi$
$228$	0	0
$229$	6677.62	1.92694	0.963471	−	0.267814i	$-0.0863012\pi$
$229$	6677.62	1.92694	0.963471	+	0.267814i	$0.0863012\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5223.22	1.46860	0.734302	−	0.678823i	$-0.237510\pi$
$233$	5223.22	1.46860	0.734302	+	0.678823i	$0.237510\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6007.42	−1.62589	−0.812945	−	0.582340i	$-0.802137\pi$
$239$	−6007.42	−1.62589	−0.812945	+	0.582340i	$0.802137\pi$
$240$	0	0
$241$	−5067.61	−1.35450	−0.677248	−	0.735755i	$-0.736827\pi$
$241$	−5067.61	−1.35450	−0.677248	+	0.735755i	$0.736827\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−503.929	−0.129815
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5123.97	−1.28853	−0.644267	−	0.764801i	$-0.722837\pi$
$251$	−5123.97	−1.28853	−0.644267	+	0.764801i	$0.722837\pi$
$252$	0	0
$253$	4375.10	1.08720
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7655.10	−1.85802	−0.929011	−	0.370051i	$-0.879340\pi$
$257$	−7655.10	−1.85802	−0.929011	+	0.370051i	$0.879340\pi$
$258$	0	0
$259$	−1894.76	−0.454575
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6342.01	1.48694	0.743470	−	0.668769i	$-0.233179\pi$
$263$	6342.01	1.48694	0.743470	+	0.668769i	$0.233179\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4231.66	−0.959142	−0.479571	−	0.877503i	$-0.659208\pi$
$269$	−4231.66	−0.959142	−0.479571	+	0.877503i	$0.659208\pi$
$270$	0	0
$271$	6203.46	1.39053	0.695265	−	0.718754i	$-0.255287\pi$
$271$	6203.46	1.39053	0.695265	+	0.718754i	$0.255287\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−529.988	−0.114960	−0.0574800	−	0.998347i	$-0.518307\pi$
$277$	−529.988	−0.114960	−0.0574800	+	0.998347i	$0.518307\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−8792.13	−1.86653	−0.933264	−	0.359190i	$-0.883053\pi$
$281$	−8792.13	−1.86653	−0.933264	+	0.359190i	$0.883053\pi$
$282$	0	0
$283$	−3888.20	−0.816713	−0.408356	−	0.912823i	$-0.633898\pi$
$283$	−3888.20	−0.816713	−0.408356	+	0.912823i	$0.633898\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3288.64	−0.676384
$288$	0	0
$289$	−4908.40	−0.999063
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1232.60	−0.245765	−0.122883	−	0.992421i	$-0.539214\pi$
$293$	−1232.60	−0.245765	−0.122883	+	0.992421i	$0.539214\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2199.39	0.425398
$300$	0	0
$301$	−2335.92	−0.447310
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−6709.37	−1.24731	−0.623655	−	0.781700i	$-0.714353\pi$
$307$	−6709.37	−1.24731	−0.623655	+	0.781700i	$0.714353\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6030.09	−1.09947	−0.549735	−	0.835339i	$-0.685271\pi$
$311$	−6030.09	−1.09947	−0.549735	+	0.835339i	$0.685271\pi$
$312$	0	0
$313$	−2881.63	−0.520381	−0.260190	−	0.965557i	$-0.583785\pi$
$313$	−2881.63	−0.520381	−0.260190	+	0.965557i	$0.583785\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1298.34	−0.230038	−0.115019	−	0.993363i	$-0.536693\pi$
$317$	−1298.34	−0.230038	−0.115019	+	0.993363i	$0.536693\pi$
$318$	0	0
$319$	−5158.64	−0.905418
$320$	0	0
$321$	0	0
$322$	0	0
$323$	79.8465	0.0137547
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4655.89	−0.780206
$330$	0	0
$331$	−2408.51	−0.399950	−0.199975	−	0.979801i	$-0.564086\pi$
$331$	−2408.51	−0.399950	−0.199975	+	0.979801i	$0.564086\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	6593.61	1.06581	0.532903	−	0.846176i	$-0.321101\pi$
$337$	6593.61	1.06581	0.532903	+	0.846176i	$0.321101\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4734.92	−0.751936
$342$	0	0
$343$	3902.48	0.614326
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10415.7	−1.61136	−0.805682	−	0.592348i	$-0.798201\pi$
$347$	−10415.7	−1.61136	−0.805682	+	0.592348i	$0.798201\pi$
$348$	0	0
$349$	9870.18	1.51386	0.756932	−	0.653494i	$-0.226697\pi$
$349$	9870.18	1.51386	0.756932	+	0.653494i	$0.226697\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−362.709	−0.0546886	−0.0273443	−	0.999626i	$-0.508705\pi$
$353$	−362.709	−0.0546886	−0.0273443	+	0.999626i	$0.508705\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8124.61	1.19443	0.597215	−	0.802081i	$-0.296274\pi$
$359$	8124.61	1.19443	0.597215	+	0.802081i	$0.296274\pi$
$360$	0	0
$361$	−5474.36	−0.798129
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−7995.22	−1.13719	−0.568593	−	0.822619i	$-0.692512\pi$
$367$	−7995.22	−1.13719	−0.568593	+	0.822619i	$0.692512\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−9587.62	−1.34168
$372$	0	0
$373$	9674.83	1.34301	0.671506	−	0.740999i	$-0.265648\pi$
$373$	9674.83	1.34301	0.671506	+	0.740999i	$0.265648\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2593.28	−0.354272
$378$	0	0
$379$	−274.211	−0.0371644	−0.0185822	−	0.999827i	$-0.505915\pi$
$379$	−274.211	−0.0371644	−0.0185822	+	0.999827i	$0.505915\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3990.74	−0.532421	−0.266211	−	0.963915i	$-0.585772\pi$
$383$	−3990.74	−0.532421	−0.266211	+	0.963915i	$0.585772\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5836.43	0.760716	0.380358	−	0.924839i	$-0.375801\pi$
$389$	5836.43	0.760716	0.380358	+	0.924839i	$0.375801\pi$
$390$	0	0
$391$	−348.489	−0.0450737
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	13724.1	1.73499	0.867495	−	0.497445i	$-0.165728\pi$
$397$	13724.1	1.73499	0.867495	+	0.497445i	$0.165728\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5892.52	−0.733812	−0.366906	−	0.930258i	$-0.619583\pi$
$401$	−5892.52	−0.733812	−0.366906	+	0.930258i	$0.619583\pi$
$402$	0	0
$403$	−2380.27	−0.294218
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2251.79	0.274243
$408$	0	0
$409$	2971.55	0.359251	0.179625	−	0.983735i	$-0.442511\pi$
$409$	2971.55	0.359251	0.179625	+	0.983735i	$0.442511\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3928.28	0.468034
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−11566.2	−1.34856	−0.674282	−	0.738474i	$-0.735547\pi$
$419$	−11566.2	−1.34856	−0.674282	+	0.738474i	$0.735547\pi$
$420$	0	0
$421$	4913.75	0.568840	0.284420	−	0.958700i	$-0.408199\pi$
$421$	4913.75	0.568840	0.284420	+	0.958700i	$0.408199\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−13993.8	−1.58597
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4137.08	−0.462358	−0.231179	−	0.972911i	$-0.574258\pi$
$431$	−4137.08	−0.462358	−0.231179	+	0.972911i	$0.574258\pi$
$432$	0	0
$433$	−8420.21	−0.934526	−0.467263	−	0.884118i	$-0.654760\pi$
$433$	−8420.21	−0.934526	−0.467263	+	0.884118i	$0.654760\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6043.22	−0.661525
$438$	0	0
$439$	−11257.2	−1.22386	−0.611932	−	0.790910i	$-0.709608\pi$
$439$	−11257.2	−1.22386	−0.611932	+	0.790910i	$0.709608\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	566.814	0.0607904	0.0303952	−	0.999538i	$-0.490323\pi$
$443$	566.814	0.0607904	0.0303952	+	0.999538i	$0.490323\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4772.52	0.501624	0.250812	−	0.968036i	$-0.419302\pi$
$449$	4772.52	0.501624	0.250812	+	0.968036i	$0.419302\pi$
$450$	0	0
$451$	3908.31	0.408060
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16953.6	1.73535	0.867677	−	0.497129i	$-0.165612\pi$
$457$	16953.6	1.73535	0.867677	+	0.497129i	$0.165612\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7787.22	−0.786739	−0.393370	−	0.919380i	$-0.628691\pi$
$461$	−7787.22	−0.786739	−0.393370	+	0.919380i	$0.628691\pi$
$462$	0	0
$463$	−6149.00	−0.617210	−0.308605	−	0.951190i	$-0.599862\pi$
$463$	−6149.00	−0.617210	−0.308605	+	0.951190i	$0.599862\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	798.751	0.0791473	0.0395737	−	0.999217i	$-0.487400\pi$
$467$	798.751	0.0791473	0.0395737	+	0.999217i	$0.487400\pi$
$468$	0	0
$469$	7751.86	0.763215
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2776.08	0.269861
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9632.15	−0.918798	−0.459399	−	0.888230i	$-0.651935\pi$
$479$	−9632.15	−0.918798	−0.459399	+	0.888230i	$0.651935\pi$
$480$	0	0
$481$	1131.99	0.107306
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−14395.3	−1.33946	−0.669728	−	0.742606i	$-0.733589\pi$
$487$	−14395.3	−1.33946	−0.669728	+	0.742606i	$0.733589\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17812.6	1.63721	0.818607	−	0.574355i	$-0.194747\pi$
$491$	17812.6	1.63721	0.818607	+	0.574355i	$0.194747\pi$
$492$	0	0
$493$	410.900	0.0375375
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13801.9	1.24567
$498$	0	0
$499$	−11256.8	−1.00987	−0.504933	−	0.863159i	$-0.668483\pi$
$499$	−11256.8	−1.00987	−0.504933	+	0.863159i	$0.668483\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7682.62	0.681016	0.340508	−	0.940242i	$-0.389401\pi$
$503$	7682.62	0.681016	0.340508	+	0.940242i	$0.389401\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−4307.91	−0.375137	−0.187568	−	0.982252i	$-0.560061\pi$
$509$	−4307.91	−0.375137	−0.187568	+	0.982252i	$0.560061\pi$
$510$	0	0
$511$	13641.5	1.18095
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5533.19	0.470695
$518$	0	0
$519$	0	0
$520$	0	0
$521$	22866.8	1.92287	0.961434	−	0.275036i	$-0.0886898\pi$
$521$	22866.8	1.92287	0.961434	+	0.275036i	$0.0886898\pi$
$522$	0	0
$523$	−726.169	−0.0607135	−0.0303567	−	0.999539i	$-0.509664\pi$
$523$	−726.169	−0.0607135	−0.0303567	+	0.999539i	$0.509664\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	377.149	0.0311743
$528$	0	0
$529$	14208.5	1.16779
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1964.73	0.159666
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4602.40	0.367791
$540$	0	0
$541$	−8641.42	−0.686735	−0.343368	−	0.939201i	$-0.611568\pi$
$541$	−8641.42	−0.686735	−0.343368	+	0.939201i	$0.611568\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−7223.56	−0.564638	−0.282319	−	0.959321i	$-0.591104\pi$
$547$	−7223.56	−0.564638	−0.282319	+	0.959321i	$0.591104\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7125.50	0.550919
$552$	0	0
$553$	9545.59	0.734032
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1424.42	0.108356	0.0541782	−	0.998531i	$-0.482746\pi$
$557$	1424.42	0.108356	0.0541782	+	0.998531i	$0.482746\pi$
$558$	0	0
$559$	1395.55	0.105591
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2450.01	0.183403	0.0917013	−	0.995787i	$-0.470770\pi$
$563$	2450.01	0.183403	0.0917013	+	0.995787i	$0.470770\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7342.47	−0.540971	−0.270486	−	0.962724i	$-0.587184\pi$
$569$	−7342.47	−0.540971	−0.270486	+	0.962724i	$0.587184\pi$
$570$	0	0
$571$	−10662.7	−0.781468	−0.390734	−	0.920504i	$-0.627779\pi$
$571$	−10662.7	−0.781468	−0.390734	+	0.920504i	$0.627779\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	11435.4	0.825064	0.412532	−	0.910943i	$-0.364644\pi$
$577$	11435.4	0.825064	0.412532	+	0.910943i	$0.364644\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1536.27	−0.109699
$582$	0	0
$583$	11394.2	0.809433
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3781.86	0.265918	0.132959	−	0.991122i	$-0.457552\pi$
$587$	3781.86	0.265918	0.132959	+	0.991122i	$0.457552\pi$
$588$	0	0
$589$	6540.22	0.457530
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7288.32	0.504714	0.252357	−	0.967634i	$-0.418794\pi$
$593$	7288.32	0.504714	0.252357	+	0.967634i	$0.418794\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8298.02	0.566023	0.283012	−	0.959116i	$-0.408666\pi$
$599$	8298.02	0.566023	0.283012	+	0.959116i	$0.408666\pi$
$600$	0	0
$601$	−13502.7	−0.916452	−0.458226	−	0.888836i	$-0.651515\pi$
$601$	−13502.7	−0.916452	−0.458226	+	0.888836i	$0.651515\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5781.00	0.386563	0.193281	−	0.981143i	$-0.438087\pi$
$607$	5781.00	0.386563	0.193281	+	0.981143i	$0.438087\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2781.57	0.184174
$612$	0	0
$613$	−1442.85	−0.0950669	−0.0475334	−	0.998870i	$-0.515136\pi$
$613$	−1442.85	−0.0950669	−0.0475334	+	0.998870i	$0.515136\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12608.0	0.822657	0.411329	−	0.911487i	$-0.365065\pi$
$617$	12608.0	0.822657	0.411329	+	0.911487i	$0.365065\pi$
$618$	0	0
$619$	−8422.21	−0.546877	−0.273439	−	0.961889i	$-0.588161\pi$
$619$	−8422.21	−0.546877	−0.273439	+	0.961889i	$0.588161\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−7666.46	−0.493018
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−179.361	−0.0113698
$630$	0	0
$631$	12825.0	0.809119	0.404559	−	0.914512i	$-0.367425\pi$
$631$	12825.0	0.809119	0.404559	+	0.914512i	$0.367425\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2313.66	0.143910
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24631.4	1.51776	0.758878	−	0.651233i	$-0.225748\pi$
$641$	24631.4	1.51776	0.758878	+	0.651233i	$0.225748\pi$
$642$	0	0
$643$	−30883.5	−1.89413	−0.947065	−	0.321041i	$-0.895967\pi$
$643$	−30883.5	−1.89413	−0.947065	+	0.321041i	$0.895967\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7129.59	0.433219	0.216610	−	0.976258i	$-0.430500\pi$
$647$	7129.59	0.433219	0.216610	+	0.976258i	$0.430500\pi$
$648$	0	0
$649$	−4668.48	−0.282363
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−26275.7	−1.57465	−0.787326	−	0.616537i	$-0.788535\pi$
$653$	−26275.7	−1.57465	−0.787326	+	0.616537i	$0.788535\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	20785.1	1.22864	0.614319	−	0.789058i	$-0.289431\pi$
$659$	20785.1	1.22864	0.614319	+	0.789058i	$0.289431\pi$
$660$	0	0
$661$	−19301.6	−1.13577	−0.567887	−	0.823106i	$-0.692239\pi$
$661$	−19301.6	−1.13577	−0.567887	+	0.823106i	$0.692239\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−31099.1	−1.80534
$668$	0	0
$669$	0	0
$670$	0	0
$671$	16630.6	0.956808
$672$	0	0
$673$	31104.0	1.78153	0.890765	−	0.454464i	$-0.150169\pi$
$673$	31104.0	1.78153	0.890765	+	0.454464i	$0.150169\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1093.98	0.0621048	0.0310524	−	0.999518i	$-0.490114\pi$
$677$	1093.98	0.0621048	0.0310524	+	0.999518i	$0.490114\pi$
$678$	0	0
$679$	−27641.3	−1.56226
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−15564.2	−0.871958	−0.435979	−	0.899957i	$-0.643598\pi$
$683$	−15564.2	−0.871958	−0.435979	+	0.899957i	$0.643598\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5727.93	0.316715
$690$	0	0
$691$	24605.8	1.35463	0.677314	−	0.735694i	$-0.263144\pi$
$691$	24605.8	1.35463	0.677314	+	0.735694i	$0.263144\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−311.307	−0.0169177
$698$	0	0
$699$	0	0
$700$	0	0
$701$	10600.1	0.571130	0.285565	−	0.958359i	$-0.407819\pi$
$701$	10600.1	0.571130	0.285565	+	0.958359i	$0.407819\pi$
$702$	0	0
$703$	−3110.34	−0.166869
$704$	0	0
$705$	0	0
$706$	0	0
$707$	38093.3	2.02637
$708$	0	0
$709$	−3814.01	−0.202029	−0.101014	−	0.994885i	$-0.532209\pi$
$709$	−3814.01	−0.202029	−0.101014	+	0.994885i	$0.532209\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−28544.7	−1.49931
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6697.54	0.347394	0.173697	−	0.984799i	$-0.444429\pi$
$719$	6697.54	0.347394	0.173697	+	0.984799i	$0.444429\pi$
$720$	0	0
$721$	−12534.5	−0.647448
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−10701.3	−0.545928	−0.272964	−	0.962024i	$-0.588004\pi$
$727$	−10701.3	−0.545928	−0.272964	+	0.962024i	$0.588004\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−221.122	−0.0111881
$732$	0	0
$733$	−2465.69	−0.124246	−0.0621229	−	0.998069i	$-0.519787\pi$
$733$	−2465.69	−0.124246	−0.0621229	+	0.998069i	$0.519787\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9212.53	−0.460445
$738$	0	0
$739$	34265.8	1.70567	0.852833	−	0.522184i	$-0.174883\pi$
$739$	34265.8	1.70567	0.852833	+	0.522184i	$0.174883\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−29710.2	−1.46697	−0.733487	−	0.679704i	$-0.762108\pi$
$743$	−29710.2	−1.46697	−0.733487	+	0.679704i	$0.762108\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	32870.3	1.60354
$750$	0	0
$751$	31553.0	1.53314	0.766569	−	0.642162i	$-0.221963\pi$
$751$	31553.0	1.53314	0.766569	+	0.642162i	$0.221963\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	23864.3	1.14579	0.572895	−	0.819629i	$-0.305820\pi$
$757$	23864.3	1.14579	0.572895	+	0.819629i	$0.305820\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−4184.66	−0.199335	−0.0996673	−	0.995021i	$-0.531778\pi$
$761$	−4184.66	−0.199335	−0.0996673	+	0.995021i	$0.531778\pi$
$762$	0	0
$763$	−18164.5	−0.861861
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2346.87	−0.110483
$768$	0	0
$769$	31262.9	1.46602	0.733010	−	0.680218i	$-0.238115\pi$
$769$	31262.9	1.46602	0.733010	+	0.680218i	$0.238115\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−17353.0	−0.807430	−0.403715	−	0.914885i	$-0.632281\pi$
$773$	−17353.0	−0.807430	−0.403715	+	0.914885i	$0.632281\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5398.45	−0.248292
$780$	0	0
$781$	−16402.5	−0.751509
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	22887.2	1.03664	0.518322	−	0.855185i	$-0.326557\pi$
$787$	22887.2	1.03664	0.518322	+	0.855185i	$0.326557\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	22999.4	1.03384
$792$	0	0
$793$	8360.31	0.374380
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16671.0	0.740923	0.370462	−	0.928848i	$-0.379199\pi$
$797$	16671.0	0.740923	0.370462	+	0.928848i	$0.379199\pi$
$798$	0	0
$799$	−440.733	−0.0195144
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−16211.9	−0.712461
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	29389.3	1.27722	0.638611	−	0.769529i	$-0.279509\pi$
$809$	29389.3	1.27722	0.638611	+	0.769529i	$0.279509\pi$
$810$	0	0
$811$	−5775.63	−0.250074	−0.125037	−	0.992152i	$-0.539905\pi$
$811$	−5775.63	−0.250074	−0.125037	+	0.992152i	$0.539905\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−3834.52	−0.164202
$818$	0	0
$819$	0	0
$820$	0	0
$821$	46248.9	1.96602	0.983008	−	0.183563i	$-0.0587630\pi$
$821$	46248.9	1.96602	0.983008	+	0.183563i	$0.0587630\pi$
$822$	0	0
$823$	11664.3	0.494036	0.247018	−	0.969011i	$-0.420549\pi$
$823$	11664.3	0.494036	0.247018	+	0.969011i	$0.420549\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11448.3	−0.481376	−0.240688	−	0.970603i	$-0.577373\pi$
$827$	−11448.3	−0.481376	−0.240688	+	0.970603i	$0.577373\pi$
$828$	0	0
$829$	−38593.9	−1.61691	−0.808456	−	0.588557i	$-0.799696\pi$
$829$	−38593.9	−1.61691	−0.808456	+	0.588557i	$0.799696\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−366.594	−0.0152482
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2111.74	−0.0868956	−0.0434478	−	0.999056i	$-0.513834\pi$
$839$	−2111.74	−0.0868956	−0.0434478	+	0.999056i	$0.513834\pi$
$840$	0	0
$841$	12279.7	0.503491
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	13720.3	0.556594
$848$	0	0
$849$	0	0
$850$	0	0
$851$	13575.0	0.546822
$852$	0	0
$853$	−13960.0	−0.560352	−0.280176	−	0.959949i	$-0.590393\pi$
$853$	−13960.0	−0.560352	−0.280176	+	0.959949i	$0.590393\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2864.10	0.114161	0.0570805	−	0.998370i	$-0.481821\pi$
$857$	2864.10	0.114161	0.0570805	+	0.998370i	$0.481821\pi$
$858$	0	0
$859$	−9817.43	−0.389949	−0.194975	−	0.980808i	$-0.562462\pi$
$859$	−9817.43	−0.389949	−0.194975	+	0.980808i	$0.562462\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3583.97	−0.141367	−0.0706834	−	0.997499i	$-0.522518\pi$
$863$	−3583.97	−0.141367	−0.0706834	+	0.997499i	$0.522518\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−11344.2	−0.442839
$870$	0	0
$871$	−4631.19	−0.180163
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	11564.5	0.445273	0.222637	−	0.974901i	$-0.428534\pi$
$877$	11564.5	0.445273	0.222637	+	0.974901i	$0.428534\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	7126.88	0.272543	0.136272	−	0.990672i	$-0.456488\pi$
$881$	7126.88	0.272543	0.136272	+	0.990672i	$0.456488\pi$
$882$	0	0
$883$	−14390.6	−0.548450	−0.274225	−	0.961666i	$-0.588421\pi$
$883$	−14390.6	−0.548450	−0.274225	+	0.961666i	$0.588421\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37434.5	1.41705	0.708527	−	0.705684i	$-0.249360\pi$
$887$	37434.5	1.41705	0.708527	+	0.705684i	$0.249360\pi$
$888$	0	0
$889$	28736.6	1.08413
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−7642.85	−0.286404
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	33656.8	1.24863
$900$	0	0
$901$	−907.578	−0.0335581
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−30306.1	−1.10948	−0.554740	−	0.832024i	$-0.687182\pi$
$907$	−30306.1	−1.10948	−0.554740	+	0.832024i	$0.687182\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	38500.5	1.40020	0.700099	−	0.714046i	$-0.253139\pi$
$911$	38500.5	1.40020	0.700099	+	0.714046i	$0.253139\pi$
$912$	0	0
$913$	1825.75	0.0661812
$914$	0	0
$915$	0	0
$916$	0	0
$917$	39850.9	1.43511
$918$	0	0
$919$	−19369.3	−0.695248	−0.347624	−	0.937634i	$-0.613011\pi$
$919$	−19369.3	−0.695248	−0.347624	+	0.937634i	$0.613011\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−8245.65	−0.294051
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−47243.4	−1.66847	−0.834234	−	0.551411i	$-0.814090\pi$
$929$	−47243.4	−1.66847	−0.834234	+	0.551411i	$0.814090\pi$
$930$	0	0
$931$	−6357.18	−0.223790
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−24443.3	−0.852217	−0.426109	−	0.904672i	$-0.640116\pi$
$937$	−24443.3	−0.852217	−0.426109	+	0.904672i	$0.640116\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	29399.8	1.01850	0.509249	−	0.860619i	$-0.329923\pi$
$941$	29399.8	1.01850	0.509249	+	0.860619i	$0.329923\pi$
$942$	0	0
$943$	23561.4	0.813643
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−11138.9	−0.382225	−0.191112	−	0.981568i	$-0.561209\pi$
$947$	−11138.9	−0.382225	−0.191112	+	0.981568i	$0.561209\pi$
$948$	0	0
$949$	−8149.83	−0.278772
$950$	0	0
$951$	0	0
$952$	0	0
$953$	13482.5	0.458279	0.229140	−	0.973394i	$-0.426409\pi$
$953$	13482.5	0.458279	0.229140	+	0.973394i	$0.426409\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14192.2	0.477883
$960$	0	0
$961$	1101.25	0.0369657
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−27747.8	−0.922761	−0.461381	−	0.887202i	$-0.652646\pi$
$967$	−27747.8	−0.922761	−0.461381	+	0.887202i	$0.652646\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	58622.8	1.93748	0.968742	−	0.248071i	$-0.0797967\pi$
$971$	58622.8	1.93748	0.968742	+	0.248071i	$0.0797967\pi$
$972$	0	0
$973$	3367.68	0.110959
$974$	0	0
$975$	0	0
$976$	0	0
$977$	53085.8	1.73835	0.869174	−	0.494506i	$-0.164651\pi$
$977$	53085.8	1.73835	0.869174	+	0.494506i	$0.164651\pi$
$978$	0	0
$979$	9111.03	0.297436
$980$	0	0
$981$	0	0
$982$	0	0
$983$	9458.90	0.306909	0.153455	−	0.988156i	$-0.450960\pi$
$983$	9458.90	0.306909	0.153455	+	0.988156i	$0.450960\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16735.7	0.538083
$990$	0	0
$991$	13893.7	0.445356	0.222678	−	0.974892i	$-0.428520\pi$
$991$	13893.7	0.445356	0.222678	+	0.974892i	$0.428520\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	15005.2	0.476649	0.238324	−	0.971186i	$-0.423402\pi$
$997$	15005.2	0.476649	0.238324	+	0.971186i	$0.423402\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.4.a.bt.1.1	yes	3
3.2	odd	2		1800.4.a.bu.1.1	yes	3
5.2	odd	4		1800.4.f.ba.649.2		6
5.3	odd	4		1800.4.f.ba.649.5		6
5.4	even	2		1800.4.a.br.1.3	✓	3
15.2	even	4		1800.4.f.bb.649.2		6
15.8	even	4		1800.4.f.bb.649.5		6
15.14	odd	2		1800.4.a.bs.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1800.4.a.br.1.3	✓	3	5.4	even	2
1800.4.a.bs.1.3	yes	3	15.14	odd	2
1800.4.a.bt.1.1	yes	3	1.1	even	1	trivial
1800.4.a.bu.1.1	yes	3	3.2	odd	2
1800.4.f.ba.649.2		6	5.2	odd	4
1800.4.f.ba.649.5		6	5.3	odd	4
1800.4.f.bb.649.2		6	15.2	even	4
1800.4.f.bb.649.5		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)

\(f(q)\)	\(=\)	\(q-22.6681 q^{7} +O(q^{10})\)	\(q-22.6681 q^{7} +26.9394 q^{11} +13.5426 q^{13} -2.14580 q^{17} -37.2107 q^{19} +162.405 q^{23} -191.491 q^{29} -175.762 q^{31} +83.5872 q^{37} +145.078 q^{41} +103.049 q^{43} +205.394 q^{47} +170.843 q^{49} +422.957 q^{53} -173.296 q^{59} +617.335 q^{61} -341.972 q^{67} -608.867 q^{71} -601.792 q^{73} -610.665 q^{77} -421.102 q^{79} +67.7724 q^{83} +338.205 q^{89} -306.985 q^{91} +1219.39 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 9 q^{7}+O(q^{10}) \) 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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