LMFDB - Embedded newform 9000.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9000,2,Mod(1,9000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9000, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("9000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9000 = 2^{3} \cdot 3^{2} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9000.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:	$71.8653618192$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3000)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	9000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.38197 q^{7} +O(q^{10})$	$q+1.38197 q^{7} +1.00000 q^{11} +0.236068 q^{13} +3.85410 q^{17} -6.23607 q^{19} -8.70820 q^{23} +1.76393 q^{29} +1.09017 q^{31} -1.76393 q^{37} +9.09017 q^{41} +3.14590 q^{43} +2.23607 q^{47} -5.09017 q^{49} +5.61803 q^{53} +11.8541 q^{59} +3.85410 q^{61} +1.52786 q^{67} -10.5623 q^{71} -0.854102 q^{73} +1.38197 q^{77} +0.291796 q^{79} +0.909830 q^{83} +11.0000 q^{89} +0.326238 q^{91} +16.7984 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5 q^{7}+O(q^{10}) $ 2 * q + 5 * q^7	$ 2 q + 5 q^{7} + 2 q^{11} - 4 q^{13} + q^{17} - 8 q^{19} - 4 q^{23} + 8 q^{29} - 9 q^{31} - 8 q^{37} + 7 q^{41} + 13 q^{43} + q^{49} + 9 q^{53} + 17 q^{59} + q^{61} + 12 q^{67} - q^{71} + 5 q^{73} + 5 q^{77} + 14 q^{79} + 13 q^{83} + 22 q^{89} - 15 q^{91} + 9 q^{97}+O(q^{100}) $ 2 * q + 5 * q^7 + 2 * q^11 - 4 * q^13 + q^17 - 8 * q^19 - 4 * q^23 + 8 * q^29 - 9 * q^31 - 8 * q^37 + 7 * q^41 + 13 * q^43 + q^49 + 9 * q^53 + 17 * q^59 + q^61 + 12 * q^67 - q^71 + 5 * q^73 + 5 * q^77 + 14 * q^79 + 13 * q^83 + 22 * q^89 - 15 * q^91 + 9 * 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
$7$	1.38197	0.522334	0.261167	−	0.965294i	$-0.415893\pi$
$7$	1.38197	0.522334	0.261167	+	0.965294i	$0.415893\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	0.236068	0.0654735	0.0327367	−	0.999464i	$-0.489578\pi$
$13$	0.236068	0.0654735	0.0327367	+	0.999464i	$0.489578\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.85410	0.934757	0.467379	−	0.884057i	$-0.345199\pi$
$17$	3.85410	0.934757	0.467379	+	0.884057i	$0.345199\pi$
$18$	0	0
$19$	−6.23607	−1.43065	−0.715326	−	0.698791i	$-0.753722\pi$
$19$	−6.23607	−1.43065	−0.715326	+	0.698791i	$0.753722\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.70820	−1.81579	−0.907893	−	0.419202i	$-0.862310\pi$
$23$	−8.70820	−1.81579	−0.907893	+	0.419202i	$0.862310\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.76393	0.327554	0.163777	−	0.986497i	$-0.447632\pi$
$29$	1.76393	0.327554	0.163777	+	0.986497i	$0.447632\pi$
$30$	0	0
$31$	1.09017	0.195800	0.0979002	−	0.995196i	$-0.468787\pi$
$31$	1.09017	0.195800	0.0979002	+	0.995196i	$0.468787\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.76393	−0.289989	−0.144994	−	0.989432i	$-0.546316\pi$
$37$	−1.76393	−0.289989	−0.144994	+	0.989432i	$0.546316\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.09017	1.41965	0.709823	−	0.704380i	$-0.248775\pi$
$41$	9.09017	1.41965	0.709823	+	0.704380i	$0.248775\pi$
$42$	0	0
$43$	3.14590	0.479745	0.239872	−	0.970804i	$-0.422894\pi$
$43$	3.14590	0.479745	0.239872	+	0.970804i	$0.422894\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.23607	0.326164	0.163082	−	0.986613i	$-0.447856\pi$
$47$	2.23607	0.326164	0.163082	+	0.986613i	$0.447856\pi$
$48$	0	0
$49$	−5.09017	−0.727167
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.61803	0.771696	0.385848	−	0.922562i	$-0.373909\pi$
$53$	5.61803	0.771696	0.385848	+	0.922562i	$0.373909\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.8541	1.54327	0.771636	−	0.636064i	$-0.219439\pi$
$59$	11.8541	1.54327	0.771636	+	0.636064i	$0.219439\pi$
$60$	0	0
$61$	3.85410	0.493467	0.246734	−	0.969083i	$-0.420643\pi$
$61$	3.85410	0.493467	0.246734	+	0.969083i	$0.420643\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.52786	0.186658	0.0933292	−	0.995635i	$-0.470249\pi$
$67$	1.52786	0.186658	0.0933292	+	0.995635i	$0.470249\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.5623	−1.25352	−0.626758	−	0.779214i	$-0.715618\pi$
$71$	−10.5623	−1.25352	−0.626758	+	0.779214i	$0.715618\pi$
$72$	0	0
$73$	−0.854102	−0.0999651	−0.0499825	−	0.998750i	$-0.515917\pi$
$73$	−0.854102	−0.0999651	−0.0499825	+	0.998750i	$0.515917\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.38197	0.157490
$78$	0	0
$79$	0.291796	0.0328296	0.0164148	−	0.999865i	$-0.494775\pi$
$79$	0.291796	0.0328296	0.0164148	+	0.999865i	$0.494775\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.909830	0.0998668	0.0499334	−	0.998753i	$-0.484099\pi$
$83$	0.909830	0.0998668	0.0499334	+	0.998753i	$0.484099\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.0000	1.16600	0.582999	−	0.812473i	$-0.301879\pi$
$89$	11.0000	1.16600	0.582999	+	0.812473i	$0.301879\pi$
$90$	0	0
$91$	0.326238	0.0341990
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.7984	1.70562	0.852808	−	0.522224i	$-0.174898\pi$
$97$	16.7984	1.70562	0.852808	+	0.522224i	$0.174898\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−0.944272	−0.0939586	−0.0469793	−	0.998896i	$-0.514959\pi$
$101$	−0.944272	−0.0939586	−0.0469793	+	0.998896i	$0.514959\pi$
$102$	0	0
$103$	9.61803	0.947693	0.473847	−	0.880607i	$-0.342865\pi$
$103$	9.61803	0.947693	0.473847	+	0.880607i	$0.342865\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.32624	0.708254	0.354127	−	0.935197i	$-0.384778\pi$
$107$	7.32624	0.708254	0.354127	+	0.935197i	$0.384778\pi$
$108$	0	0
$109$	1.94427	0.186227	0.0931137	−	0.995655i	$-0.470318\pi$
$109$	1.94427	0.186227	0.0931137	+	0.995655i	$0.470318\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.70820	0.348838	0.174419	−	0.984671i	$-0.444195\pi$
$113$	3.70820	0.348838	0.174419	+	0.984671i	$0.444195\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.32624	0.488255
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.4721	1.01799	0.508994	−	0.860770i	$-0.330018\pi$
$127$	11.4721	1.01799	0.508994	+	0.860770i	$0.330018\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.6180	1.10244	0.551221	−	0.834359i	$-0.314162\pi$
$131$	12.6180	1.10244	0.551221	+	0.834359i	$0.314162\pi$
$132$	0	0
$133$	−8.61803	−0.747278
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.0000	0.939793	0.469897	−	0.882721i	$-0.344291\pi$
$137$	11.0000	0.939793	0.469897	+	0.882721i	$0.344291\pi$
$138$	0	0
$139$	−20.4164	−1.73170	−0.865849	−	0.500306i	$-0.833221\pi$
$139$	−20.4164	−1.73170	−0.865849	+	0.500306i	$0.833221\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.236068	0.0197410
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.4164	1.18104	0.590519	−	0.807024i	$-0.298923\pi$
$149$	14.4164	1.18104	0.590519	+	0.807024i	$0.298923\pi$
$150$	0	0
$151$	−2.18034	−0.177434	−0.0887168	−	0.996057i	$-0.528277\pi$
$151$	−2.18034	−0.177434	−0.0887168	+	0.996057i	$0.528277\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−21.2705	−1.69757	−0.848786	−	0.528737i	$-0.822666\pi$
$157$	−21.2705	−1.69757	−0.848786	+	0.528737i	$0.822666\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−12.0344	−0.948447
$162$	0	0
$163$	7.00000	0.548282	0.274141	−	0.961689i	$-0.411606\pi$
$163$	7.00000	0.548282	0.274141	+	0.961689i	$0.411606\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.9443	1.15642	0.578211	−	0.815887i	$-0.303751\pi$
$167$	14.9443	1.15642	0.578211	+	0.815887i	$0.303751\pi$
$168$	0	0
$169$	−12.9443	−0.995713
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.9098	0.829459	0.414730	−	0.909945i	$-0.363876\pi$
$173$	10.9098	0.829459	0.414730	+	0.909945i	$0.363876\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−10.4164	−0.778559	−0.389279	−	0.921120i	$-0.627276\pi$
$179$	−10.4164	−0.778559	−0.389279	+	0.921120i	$0.627276\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.85410	0.281840
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.23607	0.161796	0.0808981	−	0.996722i	$-0.474221\pi$
$191$	2.23607	0.161796	0.0808981	+	0.996722i	$0.474221\pi$
$192$	0	0
$193$	−21.9443	−1.57958	−0.789792	−	0.613375i	$-0.789811\pi$
$193$	−21.9443	−1.57958	−0.789792	+	0.613375i	$0.789811\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.05573	0.288959	0.144479	−	0.989508i	$-0.453849\pi$
$197$	4.05573	0.288959	0.144479	+	0.989508i	$0.453849\pi$
$198$	0	0
$199$	−23.4721	−1.66390	−0.831948	−	0.554854i	$-0.812774\pi$
$199$	−23.4721	−1.66390	−0.831948	+	0.554854i	$0.812774\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.43769	0.171093
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6.23607	−0.431358
$210$	0	0
$211$	17.8885	1.23150	0.615749	−	0.787942i	$-0.288854\pi$
$211$	17.8885	1.23150	0.615749	+	0.787942i	$0.288854\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.50658	0.102273
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.909830	0.0612018
$222$	0	0
$223$	−3.00000	−0.200895	−0.100447	−	0.994942i	$-0.532027\pi$
$223$	−3.00000	−0.200895	−0.100447	+	0.994942i	$0.532027\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.6525	1.23801	0.619004	−	0.785388i	$-0.287536\pi$
$227$	18.6525	1.23801	0.619004	+	0.785388i	$0.287536\pi$
$228$	0	0
$229$	9.23607	0.610337	0.305168	−	0.952298i	$-0.401287\pi$
$229$	9.23607	0.610337	0.305168	+	0.952298i	$0.401287\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.56231	−0.167862	−0.0839311	−	0.996472i	$-0.526748\pi$
$233$	−2.56231	−0.167862	−0.0839311	+	0.996472i	$0.526748\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.56231	0.230426	0.115213	−	0.993341i	$-0.463245\pi$
$239$	3.56231	0.230426	0.115213	+	0.993341i	$0.463245\pi$
$240$	0	0
$241$	6.29180	0.405290	0.202645	−	0.979252i	$-0.435046\pi$
$241$	6.29180	0.405290	0.202645	+	0.979252i	$0.435046\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.47214	−0.0936698
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−13.6180	−0.859563	−0.429781	−	0.902933i	$-0.641409\pi$
$251$	−13.6180	−0.859563	−0.429781	+	0.902933i	$0.641409\pi$
$252$	0	0
$253$	−8.70820	−0.547480
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.00000	0.187135	0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000	0.187135	0.0935674	+	0.995613i	$0.470173\pi$
$258$	0	0
$259$	−2.43769	−0.151471
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−22.9787	−1.41693	−0.708464	−	0.705747i	$-0.750612\pi$
$263$	−22.9787	−1.41693	−0.708464	+	0.705747i	$0.750612\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	28.2148	1.72029	0.860143	−	0.510053i	$-0.170374\pi$
$269$	28.2148	1.72029	0.860143	+	0.510053i	$0.170374\pi$
$270$	0	0
$271$	9.50658	0.577483	0.288742	−	0.957407i	$-0.406763\pi$
$271$	9.50658	0.577483	0.288742	+	0.957407i	$0.406763\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	21.0344	1.26384	0.631919	−	0.775035i	$-0.282268\pi$
$277$	21.0344	1.26384	0.631919	+	0.775035i	$0.282268\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.90983	0.352551	0.176275	−	0.984341i	$-0.443595\pi$
$281$	5.90983	0.352551	0.176275	+	0.984341i	$0.443595\pi$
$282$	0	0
$283$	25.9443	1.54223	0.771113	−	0.636698i	$-0.219700\pi$
$283$	25.9443	1.54223	0.771113	+	0.636698i	$0.219700\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.5623	0.741529
$288$	0	0
$289$	−2.14590	−0.126229
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.58359	0.267776	0.133888	−	0.990996i	$-0.457254\pi$
$293$	4.58359	0.267776	0.133888	+	0.990996i	$0.457254\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.05573	−0.118886
$300$	0	0
$301$	4.34752	0.250587
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	18.7082	1.06773	0.533867	−	0.845569i	$-0.320738\pi$
$307$	18.7082	1.06773	0.533867	+	0.845569i	$0.320738\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3.43769	−0.194934	−0.0974669	−	0.995239i	$-0.531074\pi$
$311$	−3.43769	−0.194934	−0.0974669	+	0.995239i	$0.531074\pi$
$312$	0	0
$313$	−22.5066	−1.27215	−0.636073	−	0.771628i	$-0.719442\pi$
$313$	−22.5066	−1.27215	−0.636073	+	0.771628i	$0.719442\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.7639	1.22238	0.611192	−	0.791482i	$-0.290690\pi$
$317$	21.7639	1.22238	0.611192	+	0.791482i	$0.290690\pi$
$318$	0	0
$319$	1.76393	0.0987612
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−24.0344	−1.33731
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.09017	0.170367
$330$	0	0
$331$	−23.6180	−1.29816	−0.649082	−	0.760718i	$-0.724847\pi$
$331$	−23.6180	−1.29816	−0.649082	+	0.760718i	$0.724847\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	16.9443	0.923013	0.461507	−	0.887137i	$-0.347309\pi$
$337$	16.9443	0.923013	0.461507	+	0.887137i	$0.347309\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.09017	0.0590360
$342$	0	0
$343$	−16.7082	−0.902158
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.2705	0.873447	0.436723	−	0.899596i	$-0.356139\pi$
$347$	16.2705	0.873447	0.436723	+	0.899596i	$0.356139\pi$
$348$	0	0
$349$	34.2705	1.83446	0.917229	−	0.398360i	$-0.130421\pi$
$349$	34.2705	1.83446	0.917229	+	0.398360i	$0.130421\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	28.9787	1.54238	0.771191	−	0.636604i	$-0.219662\pi$
$353$	28.9787	1.54238	0.771191	+	0.636604i	$0.219662\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−29.4164	−1.55254	−0.776269	−	0.630401i	$-0.782890\pi$
$359$	−29.4164	−1.55254	−0.776269	+	0.630401i	$0.782890\pi$
$360$	0	0
$361$	19.8885	1.04677
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.61803	−0.0844607	−0.0422303	−	0.999108i	$-0.513446\pi$
$367$	−1.61803	−0.0844607	−0.0422303	+	0.999108i	$0.513446\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.76393	0.403083
$372$	0	0
$373$	11.9098	0.616668	0.308334	−	0.951278i	$-0.400229\pi$
$373$	11.9098	0.616668	0.308334	+	0.951278i	$0.400229\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.416408	0.0214461
$378$	0	0
$379$	1.38197	0.0709868	0.0354934	−	0.999370i	$-0.488700\pi$
$379$	1.38197	0.0709868	0.0354934	+	0.999370i	$0.488700\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	14.7984	0.756162	0.378081	−	0.925773i	$-0.376584\pi$
$383$	14.7984	0.756162	0.378081	+	0.925773i	$0.376584\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7.79837	−0.395393	−0.197697	−	0.980263i	$-0.563346\pi$
$389$	−7.79837	−0.395393	−0.197697	+	0.980263i	$0.563346\pi$
$390$	0	0
$391$	−33.5623	−1.69732
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−8.36068	−0.419610	−0.209805	−	0.977743i	$-0.567283\pi$
$397$	−8.36068	−0.419610	−0.209805	+	0.977743i	$0.567283\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−23.2918	−1.16314	−0.581568	−	0.813498i	$-0.697561\pi$
$401$	−23.2918	−1.16314	−0.581568	+	0.813498i	$0.697561\pi$
$402$	0	0
$403$	0.257354	0.0128197
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.76393	−0.0874349
$408$	0	0
$409$	33.5967	1.66125	0.830626	−	0.556831i	$-0.187983\pi$
$409$	33.5967	1.66125	0.830626	+	0.556831i	$0.187983\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	16.3820	0.806104
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	23.3262	1.13956	0.569781	−	0.821797i	$-0.307028\pi$
$419$	23.3262	1.13956	0.569781	+	0.821797i	$0.307028\pi$
$420$	0	0
$421$	6.58359	0.320865	0.160432	−	0.987047i	$-0.448711\pi$
$421$	6.58359	0.320865	0.160432	+	0.987047i	$0.448711\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.32624	0.257755
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.3607	0.739898	0.369949	−	0.929052i	$-0.379375\pi$
$431$	15.3607	0.739898	0.369949	+	0.929052i	$0.379375\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	54.3050	2.59776
$438$	0	0
$439$	2.56231	0.122292	0.0611461	−	0.998129i	$-0.480524\pi$
$439$	2.56231	0.122292	0.0611461	+	0.998129i	$0.480524\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.14590	−0.196978	−0.0984888	−	0.995138i	$-0.531401\pi$
$443$	−4.14590	−0.196978	−0.0984888	+	0.995138i	$0.531401\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5.14590	−0.242850	−0.121425	−	0.992601i	$-0.538746\pi$
$449$	−5.14590	−0.242850	−0.121425	+	0.992601i	$0.538746\pi$
$450$	0	0
$451$	9.09017	0.428039
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	20.0344	0.937172	0.468586	−	0.883418i	$-0.344764\pi$
$457$	20.0344	0.937172	0.468586	+	0.883418i	$0.344764\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−20.0557	−0.934088	−0.467044	−	0.884234i	$-0.654681\pi$
$461$	−20.0557	−0.934088	−0.467044	+	0.884234i	$0.654681\pi$
$462$	0	0
$463$	37.9230	1.76243	0.881215	−	0.472715i	$-0.156726\pi$
$463$	37.9230	1.76243	0.881215	+	0.472715i	$0.156726\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.56231	−0.303667	−0.151834	−	0.988406i	$-0.548518\pi$
$467$	−6.56231	−0.303667	−0.151834	+	0.988406i	$0.548518\pi$
$468$	0	0
$469$	2.11146	0.0974980
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.14590	0.144649
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−33.5967	−1.53507	−0.767537	−	0.641004i	$-0.778518\pi$
$479$	−33.5967	−1.53507	−0.767537	+	0.641004i	$0.778518\pi$
$480$	0	0
$481$	−0.416408	−0.0189866
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	23.5279	1.06615	0.533075	−	0.846068i	$-0.321036\pi$
$487$	23.5279	1.06615	0.533075	+	0.846068i	$0.321036\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0.0901699	0.00406931	0.00203466	−	0.999998i	$-0.499352\pi$
$491$	0.0901699	0.00406931	0.00203466	+	0.999998i	$0.499352\pi$
$492$	0	0
$493$	6.79837	0.306183
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−14.5967	−0.654754
$498$	0	0
$499$	−28.7771	−1.28824	−0.644120	−	0.764925i	$-0.722776\pi$
$499$	−28.7771	−1.28824	−0.644120	+	0.764925i	$0.722776\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−29.9787	−1.33668	−0.668342	−	0.743854i	$-0.732996\pi$
$503$	−29.9787	−1.33668	−0.668342	+	0.743854i	$0.732996\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−40.0689	−1.77602	−0.888011	−	0.459822i	$-0.847913\pi$
$509$	−40.0689	−1.77602	−0.888011	+	0.459822i	$0.847913\pi$
$510$	0	0
$511$	−1.18034	−0.0522152
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.23607	0.0983422
$518$	0	0
$519$	0	0
$520$	0	0
$521$	30.3820	1.33106	0.665529	−	0.746372i	$-0.268206\pi$
$521$	30.3820	1.33106	0.665529	+	0.746372i	$0.268206\pi$
$522$	0	0
$523$	−7.05573	−0.308525	−0.154263	−	0.988030i	$-0.549300\pi$
$523$	−7.05573	−0.308525	−0.154263	+	0.988030i	$0.549300\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.20163	0.183026
$528$	0	0
$529$	52.8328	2.29708
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.14590	0.0929492
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.09017	−0.219249
$540$	0	0
$541$	−14.5066	−0.623686	−0.311843	−	0.950134i	$-0.600946\pi$
$541$	−14.5066	−0.623686	−0.311843	+	0.950134i	$0.600946\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−20.7984	−0.889274	−0.444637	−	0.895711i	$-0.646667\pi$
$547$	−20.7984	−0.889274	−0.444637	+	0.895711i	$0.646667\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−11.0000	−0.468616
$552$	0	0
$553$	0.403252	0.0171480
$554$	0	0
$555$	0	0
$556$	0	0
$557$	40.2705	1.70632	0.853158	−	0.521652i	$-0.174684\pi$
$557$	40.2705	1.70632	0.853158	+	0.521652i	$0.174684\pi$
$558$	0	0
$559$	0.742646	0.0314106
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.03444	−0.212176	−0.106088	−	0.994357i	$-0.533833\pi$
$563$	−5.03444	−0.212176	−0.106088	+	0.994357i	$0.533833\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	16.7082	0.700444	0.350222	−	0.936667i	$-0.386106\pi$
$569$	16.7082	0.700444	0.350222	+	0.936667i	$0.386106\pi$
$570$	0	0
$571$	6.85410	0.286835	0.143418	−	0.989662i	$-0.454191\pi$
$571$	6.85410	0.286835	0.143418	+	0.989662i	$0.454191\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	26.5967	1.10724	0.553619	−	0.832770i	$-0.313247\pi$
$577$	26.5967	1.10724	0.553619	+	0.832770i	$0.313247\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.25735	0.0521638
$582$	0	0
$583$	5.61803	0.232675
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.36068	0.0561613	0.0280806	−	0.999606i	$-0.491060\pi$
$587$	1.36068	0.0561613	0.0280806	+	0.999606i	$0.491060\pi$
$588$	0	0
$589$	−6.79837	−0.280122
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−29.5410	−1.21310	−0.606552	−	0.795044i	$-0.707448\pi$
$593$	−29.5410	−1.21310	−0.606552	+	0.795044i	$0.707448\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−31.1591	−1.27312	−0.636562	−	0.771225i	$-0.719644\pi$
$599$	−31.1591	−1.27312	−0.636562	+	0.771225i	$0.719644\pi$
$600$	0	0
$601$	−5.36068	−0.218667	−0.109333	−	0.994005i	$-0.534872\pi$
$601$	−5.36068	−0.218667	−0.109333	+	0.994005i	$0.534872\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−17.8885	−0.726074	−0.363037	−	0.931775i	$-0.618260\pi$
$607$	−17.8885	−0.726074	−0.363037	+	0.931775i	$0.618260\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.527864	0.0213551
$612$	0	0
$613$	−17.0000	−0.686624	−0.343312	−	0.939222i	$-0.611549\pi$
$613$	−17.0000	−0.686624	−0.343312	+	0.939222i	$0.611549\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−32.5279	−1.30952	−0.654761	−	0.755836i	$-0.727231\pi$
$617$	−32.5279	−1.30952	−0.654761	+	0.755836i	$0.727231\pi$
$618$	0	0
$619$	−30.8328	−1.23928	−0.619638	−	0.784888i	$-0.712720\pi$
$619$	−30.8328	−1.23928	−0.619638	+	0.784888i	$0.712720\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	15.2016	0.609040
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6.79837	−0.271069
$630$	0	0
$631$	−36.8885	−1.46851	−0.734255	−	0.678874i	$-0.762468\pi$
$631$	−36.8885	−1.46851	−0.734255	+	0.678874i	$0.762468\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.20163	−0.0476102
$638$	0	0
$639$	0	0
$640$	0	0
$641$	29.1246	1.15035	0.575177	−	0.818029i	$-0.304933\pi$
$641$	29.1246	1.15035	0.575177	+	0.818029i	$0.304933\pi$
$642$	0	0
$643$	−26.9443	−1.06258	−0.531289	−	0.847191i	$-0.678292\pi$
$643$	−26.9443	−1.06258	−0.531289	+	0.847191i	$0.678292\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	43.6312	1.71532	0.857660	−	0.514218i	$-0.171918\pi$
$647$	43.6312	1.71532	0.857660	+	0.514218i	$0.171918\pi$
$648$	0	0
$649$	11.8541	0.465314
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−24.5410	−0.960364	−0.480182	−	0.877169i	$-0.659429\pi$
$653$	−24.5410	−0.960364	−0.480182	+	0.877169i	$0.659429\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−28.3050	−1.10260	−0.551302	−	0.834306i	$-0.685869\pi$
$659$	−28.3050	−1.10260	−0.551302	+	0.834306i	$0.685869\pi$
$660$	0	0
$661$	32.0344	1.24600	0.622998	−	0.782224i	$-0.285915\pi$
$661$	32.0344	1.24600	0.622998	+	0.782224i	$0.285915\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−15.3607	−0.594768
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.85410	0.148786
$672$	0	0
$673$	−46.2492	−1.78278	−0.891388	−	0.453240i	$-0.850268\pi$
$673$	−46.2492	−1.78278	−0.891388	+	0.453240i	$0.850268\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	45.0132	1.73000	0.864998	−	0.501775i	$-0.167320\pi$
$677$	45.0132	1.73000	0.864998	+	0.501775i	$0.167320\pi$
$678$	0	0
$679$	23.2148	0.890902
$680$	0	0
$681$	0	0
$682$	0	0
$683$	25.4721	0.974664	0.487332	−	0.873217i	$-0.337970\pi$
$683$	25.4721	0.974664	0.487332	+	0.873217i	$0.337970\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.32624	0.0505256
$690$	0	0
$691$	36.5279	1.38959	0.694793	−	0.719210i	$-0.255496\pi$
$691$	36.5279	1.38959	0.694793	+	0.719210i	$0.255496\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	35.0344	1.32702
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18.0689	0.682452	0.341226	−	0.939981i	$-0.389158\pi$
$701$	18.0689	0.682452	0.341226	+	0.939981i	$0.389158\pi$
$702$	0	0
$703$	11.0000	0.414873
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.30495	−0.0490778
$708$	0	0
$709$	21.1803	0.795444	0.397722	−	0.917506i	$-0.369801\pi$
$709$	21.1803	0.795444	0.397722	+	0.917506i	$0.369801\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−9.49342	−0.355531
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−16.8197	−0.627267	−0.313634	−	0.949544i	$-0.601546\pi$
$719$	−16.8197	−0.627267	−0.313634	+	0.949544i	$0.601546\pi$
$720$	0	0
$721$	13.2918	0.495012
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−4.63932	−0.172063	−0.0860314	−	0.996292i	$-0.527419\pi$
$727$	−4.63932	−0.172063	−0.0860314	+	0.996292i	$0.527419\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	12.1246	0.448445
$732$	0	0
$733$	48.7214	1.79956	0.899782	−	0.436339i	$-0.143725\pi$
$733$	48.7214	1.79956	0.899782	+	0.436339i	$0.143725\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.52786	0.0562796
$738$	0	0
$739$	13.9656	0.513731	0.256866	−	0.966447i	$-0.417310\pi$
$739$	13.9656	0.513731	0.256866	+	0.966447i	$0.417310\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.34752	−0.0494359	−0.0247179	−	0.999694i	$-0.507869\pi$
$743$	−1.34752	−0.0494359	−0.0247179	+	0.999694i	$0.507869\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	10.1246	0.369945
$750$	0	0
$751$	9.85410	0.359581	0.179791	−	0.983705i	$-0.442458\pi$
$751$	9.85410	0.359581	0.179791	+	0.983705i	$0.442458\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−47.2492	−1.71730	−0.858651	−	0.512560i	$-0.828697\pi$
$757$	−47.2492	−1.71730	−0.858651	+	0.512560i	$0.828697\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−52.5967	−1.90663	−0.953315	−	0.301977i	$-0.902354\pi$
$761$	−52.5967	−1.90663	−0.953315	+	0.301977i	$0.902354\pi$
$762$	0	0
$763$	2.68692	0.0972730
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.79837	0.101043
$768$	0	0
$769$	35.4164	1.27715	0.638574	−	0.769560i	$-0.279525\pi$
$769$	35.4164	1.27715	0.638574	+	0.769560i	$0.279525\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−8.70820	−0.313212	−0.156606	−	0.987661i	$-0.550055\pi$
$773$	−8.70820	−0.313212	−0.156606	+	0.987661i	$0.550055\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−56.6869	−2.03102
$780$	0	0
$781$	−10.5623	−0.377949
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	55.6869	1.98502	0.992512	−	0.122146i	$-0.0389777\pi$
$787$	55.6869	1.98502	0.992512	+	0.122146i	$0.0389777\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.12461	0.182210
$792$	0	0
$793$	0.909830	0.0323090
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−8.38197	−0.296904	−0.148452	−	0.988920i	$-0.547429\pi$
$797$	−8.38197	−0.296904	−0.148452	+	0.988920i	$0.547429\pi$
$798$	0	0
$799$	8.61803	0.304884
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−0.854102	−0.0301406
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7.49342	0.263455	0.131727	−	0.991286i	$-0.457948\pi$
$809$	7.49342	0.263455	0.131727	+	0.991286i	$0.457948\pi$
$810$	0	0
$811$	28.9230	1.01562	0.507812	−	0.861468i	$-0.330455\pi$
$811$	28.9230	1.01562	0.507812	+	0.861468i	$0.330455\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−19.6180	−0.686348
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−39.0689	−1.36351	−0.681757	−	0.731579i	$-0.738784\pi$
$821$	−39.0689	−1.36351	−0.681757	+	0.731579i	$0.738784\pi$
$822$	0	0
$823$	16.1115	0.561610	0.280805	−	0.959765i	$-0.409399\pi$
$823$	16.1115	0.561610	0.280805	+	0.959765i	$0.409399\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0.437694	0.0152201	0.00761006	−	0.999971i	$-0.497578\pi$
$827$	0.437694	0.0152201	0.00761006	+	0.999971i	$0.497578\pi$
$828$	0	0
$829$	27.4853	0.954604	0.477302	−	0.878739i	$-0.341615\pi$
$829$	27.4853	0.954604	0.477302	+	0.878739i	$0.341615\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−19.6180	−0.679725
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	35.1803	1.21456	0.607280	−	0.794488i	$-0.292261\pi$
$839$	35.1803	1.21456	0.607280	+	0.794488i	$0.292261\pi$
$840$	0	0
$841$	−25.8885	−0.892708
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−13.8197	−0.474849
$848$	0	0
$849$	0	0
$850$	0	0
$851$	15.3607	0.526557
$852$	0	0
$853$	−1.58359	−0.0542212	−0.0271106	−	0.999632i	$-0.508631\pi$
$853$	−1.58359	−0.0542212	−0.0271106	+	0.999632i	$0.508631\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−53.1033	−1.81398	−0.906988	−	0.421157i	$-0.861624\pi$
$857$	−53.1033	−1.81398	−0.906988	+	0.421157i	$0.861624\pi$
$858$	0	0
$859$	26.4164	0.901316	0.450658	−	0.892697i	$-0.351189\pi$
$859$	26.4164	0.901316	0.450658	+	0.892697i	$0.351189\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	44.1591	1.50319	0.751596	−	0.659624i	$-0.229284\pi$
$863$	44.1591	1.50319	0.751596	+	0.659624i	$0.229284\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0.291796	0.00989850
$870$	0	0
$871$	0.360680	0.0122212
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	29.7639	1.00506	0.502528	−	0.864561i	$-0.332403\pi$
$877$	29.7639	1.00506	0.502528	+	0.864561i	$0.332403\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	50.6525	1.70653	0.853263	−	0.521481i	$-0.174620\pi$
$881$	50.6525	1.70653	0.853263	+	0.521481i	$0.174620\pi$
$882$	0	0
$883$	−39.2492	−1.32084	−0.660421	−	0.750896i	$-0.729622\pi$
$883$	−39.2492	−1.32084	−0.660421	+	0.750896i	$0.729622\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−34.3050	−1.15185	−0.575924	−	0.817503i	$-0.695358\pi$
$887$	−34.3050	−1.15185	−0.575924	+	0.817503i	$0.695358\pi$
$888$	0	0
$889$	15.8541	0.531730
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−13.9443	−0.466627
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.92299	0.0641352
$900$	0	0
$901$	21.6525	0.721349
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−27.1803	−0.902508	−0.451254	−	0.892395i	$-0.649023\pi$
$907$	−27.1803	−0.902508	−0.451254	+	0.892395i	$0.649023\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−20.1459	−0.667463	−0.333732	−	0.942668i	$-0.608308\pi$
$911$	−20.1459	−0.667463	−0.333732	+	0.942668i	$0.608308\pi$
$912$	0	0
$913$	0.909830	0.0301110
$914$	0	0
$915$	0	0
$916$	0	0
$917$	17.4377	0.575843
$918$	0	0
$919$	−35.5066	−1.17125	−0.585627	−	0.810581i	$-0.699152\pi$
$919$	−35.5066	−1.17125	−0.585627	+	0.810581i	$0.699152\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2.49342	−0.0820720
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	7.56231	0.248111	0.124056	−	0.992275i	$-0.460410\pi$
$929$	7.56231	0.248111	0.124056	+	0.992275i	$0.460410\pi$
$930$	0	0
$931$	31.7426	1.04032
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	11.5967	0.378849	0.189425	−	0.981895i	$-0.439338\pi$
$937$	11.5967	0.378849	0.189425	+	0.981895i	$0.439338\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	46.0344	1.50068	0.750340	−	0.661052i	$-0.229890\pi$
$941$	46.0344	1.50068	0.750340	+	0.661052i	$0.229890\pi$
$942$	0	0
$943$	−79.1591	−2.57777
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9.58359	0.311425	0.155712	−	0.987802i	$-0.450233\pi$
$947$	9.58359	0.311425	0.155712	+	0.987802i	$0.450233\pi$
$948$	0	0
$949$	−0.201626	−0.00654506
$950$	0	0
$951$	0	0
$952$	0	0
$953$	41.0344	1.32924	0.664618	−	0.747183i	$-0.268594\pi$
$953$	41.0344	1.32924	0.664618	+	0.747183i	$0.268594\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15.2016	0.490886
$960$	0	0
$961$	−29.8115	−0.961662
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	20.0557	0.644949	0.322474	−	0.946578i	$-0.395485\pi$
$967$	20.0557	0.644949	0.322474	+	0.946578i	$0.395485\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	19.1246	0.613738	0.306869	−	0.951752i	$-0.400719\pi$
$971$	19.1246	0.613738	0.306869	+	0.951752i	$0.400719\pi$
$972$	0	0
$973$	−28.2148	−0.904524
$974$	0	0
$975$	0	0
$976$	0	0
$977$	14.9230	0.477429	0.238714	−	0.971090i	$-0.423274\pi$
$977$	14.9230	0.477429	0.238714	+	0.971090i	$0.423274\pi$
$978$	0	0
$979$	11.0000	0.351562
$980$	0	0
$981$	0	0
$982$	0	0
$983$	20.3820	0.650084	0.325042	−	0.945700i	$-0.394622\pi$
$983$	20.3820	0.650084	0.325042	+	0.945700i	$0.394622\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−27.3951	−0.871114
$990$	0	0
$991$	−44.2705	−1.40630	−0.703150	−	0.711042i	$-0.748224\pi$
$991$	−44.2705	−1.40630	−0.703150	+	0.711042i	$0.748224\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−45.0689	−1.42735	−0.713673	−	0.700479i	$-0.752970\pi$
$997$	−45.0689	−1.42735	−0.713673	+	0.700479i	$0.752970\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9000.2.a.p.1.1		2
3.2	odd	2		3000.2.a.d.1.1	✓	2
5.4	even	2		9000.2.a.a.1.2		2
12.11	even	2		6000.2.a.p.1.2		2
15.2	even	4		3000.2.f.c.1249.3		4
15.8	even	4		3000.2.f.c.1249.2		4
15.14	odd	2		3000.2.a.e.1.2	yes	2
60.23	odd	4		6000.2.f.i.1249.3		4
60.47	odd	4		6000.2.f.i.1249.2		4
60.59	even	2		6000.2.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3000.2.a.d.1.1	✓	2	3.2	odd	2
3000.2.a.e.1.2	yes	2	15.14	odd	2
3000.2.f.c.1249.2		4	15.8	even	4
3000.2.f.c.1249.3		4	15.2	even	4
6000.2.a.l.1.1		2	60.59	even	2
6000.2.a.p.1.2		2	12.11	even	2
6000.2.f.i.1249.2		4	60.47	odd	4
6000.2.f.i.1249.3		4	60.23	odd	4
9000.2.a.a.1.2		2	5.4	even	2
9000.2.a.p.1.1		2	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:	\( N \)	\(=\)	\( 9000 = 2^{3} \cdot 3^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9000.a (trivial)

\(f(q)\)	\(=\)	\(q+1.38197 q^{7} +O(q^{10})\)	\(q+1.38197 q^{7} +1.00000 q^{11} +0.236068 q^{13} +3.85410 q^{17} -6.23607 q^{19} -8.70820 q^{23} +1.76393 q^{29} +1.09017 q^{31} -1.76393 q^{37} +9.09017 q^{41} +3.14590 q^{43} +2.23607 q^{47} -5.09017 q^{49} +5.61803 q^{53} +11.8541 q^{59} +3.85410 q^{61} +1.52786 q^{67} -10.5623 q^{71} -0.854102 q^{73} +1.38197 q^{77} +0.291796 q^{79} +0.909830 q^{83} +11.0000 q^{89} +0.326238 q^{91} +16.7984 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{7}+O(q^{10}) \) 2 * q + 5 * q^7	\( 2 q + 5 q^{7} + 2 q^{11} - 4 q^{13} + q^{17} - 8 q^{19} - 4 q^{23} + 8 q^{29} - 9 q^{31} - 8 q^{37} + 7 q^{41} + 13 q^{43} + q^{49} + 9 q^{53} + 17 q^{59} + q^{61} + 12 q^{67} - q^{71} + 5 q^{73} + 5 q^{77} + 14 q^{79} + 13 q^{83} + 22 q^{89} - 15 q^{91} + 9 q^{97}+O(q^{100}) \) 2 * q + 5 * q^7 + 2 * q^11 - 4 * q^13 + q^17 - 8 * q^19 - 4 * q^23 + 8 * q^29 - 9 * q^31 - 8 * q^37 + 7 * q^41 + 13 * q^43 + q^49 + 9 * q^53 + 17 * q^59 + q^61 + 12 * q^67 - q^71 + 5 * q^73 + 5 * q^77 + 14 * q^79 + 13 * q^83 + 22 * q^89 - 15 * q^91 + 9 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more