LMFDB - Embedded newform 1500.2.a.c.1.2

Error: no document with id 234476386 found in table mf_hecke_traces.

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1500,2,Mod(1,1500)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1500.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1500, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1500 = 2^{2} \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1500.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,0,0,2,0,2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$11.9775603032$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	1500.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.23607 q^{7} +1.00000 q^{9} -1.23607 q^{11} -0.381966 q^{17} +5.85410 q^{19} -3.23607 q^{21} -4.85410 q^{23} -1.00000 q^{27} +8.47214 q^{29} +7.61803 q^{31} +1.23607 q^{33} -2.00000 q^{37} -4.47214 q^{41} -5.70820 q^{43} -9.09017 q^{47} +3.47214 q^{49} +0.381966 q^{51} +5.09017 q^{53} -5.85410 q^{57} +11.2361 q^{59} -5.56231 q^{61} +3.23607 q^{63} +15.7082 q^{67} +4.85410 q^{69} +5.52786 q^{71} +3.52786 q^{73} -4.00000 q^{77} +13.8541 q^{79} +1.00000 q^{81} +6.32624 q^{83} -8.47214 q^{87} -2.76393 q^{89} -7.61803 q^{93} +12.0000 q^{97} -1.23607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} + 2 q^{11} - 3 q^{17} + 5 q^{19} - 2 q^{21} - 3 q^{23} - 2 q^{27} + 8 q^{29} + 13 q^{31} - 2 q^{33} - 4 q^{37} + 2 q^{43} - 7 q^{47} - 2 q^{49} + 3 q^{51} - q^{53} - 5 q^{57}+ \cdots + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 + 2 * q^11 - 3 * q^17 + 5 * q^19 - 2 * q^21 - 3 * q^23 - 2 * q^27 + 8 * q^29 + 13 * q^31 - 2 * q^33 - 4 * q^37 + 2 * q^43 - 7 * q^47 - 2 * q^49 + 3 * q^51 - q^53 - 5 * q^57 + 18 * q^59 + 9 * q^61 + 2 * q^63 + 18 * q^67 + 3 * q^69 + 20 * q^71 + 16 * q^73 - 8 * q^77 + 21 * q^79 + 2 * q^81 - 3 * q^83 - 8 * q^87 - 10 * q^89 - 13 * q^93 + 24 * q^97 + 2 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.23607	1.22312	0.611559	−	0.791199i	$-0.290543\pi$
$7$	3.23607	1.22312	0.611559	+	0.791199i	$0.290543\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.23607	−0.372689	−0.186344	−	0.982485i	$-0.559664\pi$
$11$	−1.23607	−0.372689	−0.186344	+	0.982485i	$0.559664\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.381966	−0.0926404	−0.0463202	−	0.998927i	$-0.514749\pi$
$17$	−0.381966	−0.0926404	−0.0463202	+	0.998927i	$0.514749\pi$
$18$	0	0
$19$	5.85410	1.34302	0.671512	−	0.740994i	$-0.265645\pi$
$19$	5.85410	1.34302	0.671512	+	0.740994i	$0.265645\pi$
$20$	0	0
$21$	−3.23607	−0.706168
$22$	0	0
$23$	−4.85410	−1.01215	−0.506075	−	0.862489i	$-0.668904\pi$
$23$	−4.85410	−1.01215	−0.506075	+	0.862489i	$0.668904\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.47214	1.57324	0.786618	−	0.617440i	$-0.211830\pi$
$29$	8.47214	1.57324	0.786618	+	0.617440i	$0.211830\pi$
$30$	0	0
$31$	7.61803	1.36824	0.684120	−	0.729370i	$-0.260187\pi$
$31$	7.61803	1.36824	0.684120	+	0.729370i	$0.260187\pi$
$32$	0	0
$33$	1.23607	0.215172
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.47214	−0.698430	−0.349215	−	0.937043i	$-0.613552\pi$
$41$	−4.47214	−0.698430	−0.349215	+	0.937043i	$0.613552\pi$
$42$	0	0
$43$	−5.70820	−0.870493	−0.435246	−	0.900311i	$-0.643339\pi$
$43$	−5.70820	−0.870493	−0.435246	+	0.900311i	$0.643339\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.09017	−1.32594	−0.662969	−	0.748647i	$-0.730704\pi$
$47$	−9.09017	−1.32594	−0.662969	+	0.748647i	$0.730704\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	0	0
$51$	0.381966	0.0534859
$52$	0	0
$53$	5.09017	0.699189	0.349594	−	0.936901i	$-0.386319\pi$
$53$	5.09017	0.699189	0.349594	+	0.936901i	$0.386319\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.85410	−0.775395
$58$	0	0
$59$	11.2361	1.46281	0.731406	−	0.681943i	$-0.238865\pi$
$59$	11.2361	1.46281	0.731406	+	0.681943i	$0.238865\pi$
$60$	0	0
$61$	−5.56231	−0.712180	−0.356090	−	0.934452i	$-0.615890\pi$
$61$	−5.56231	−0.712180	−0.356090	+	0.934452i	$0.615890\pi$
$62$	0	0
$63$	3.23607	0.407706
$64$	0	0
$65$	0	0
$66$	0	0
$67$	15.7082	1.91906	0.959531	−	0.281602i	$-0.0908658\pi$
$67$	15.7082	1.91906	0.959531	+	0.281602i	$0.0908658\pi$
$68$	0	0
$69$	4.85410	0.584365
$70$	0	0
$71$	5.52786	0.656037	0.328018	−	0.944671i	$-0.393619\pi$
$71$	5.52786	0.656037	0.328018	+	0.944671i	$0.393619\pi$
$72$	0	0
$73$	3.52786	0.412905	0.206453	−	0.978457i	$-0.433808\pi$
$73$	3.52786	0.412905	0.206453	+	0.978457i	$0.433808\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	13.8541	1.55871	0.779354	−	0.626584i	$-0.215547\pi$
$79$	13.8541	1.55871	0.779354	+	0.626584i	$0.215547\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.32624	0.694395	0.347197	−	0.937792i	$-0.387133\pi$
$83$	6.32624	0.694395	0.347197	+	0.937792i	$0.387133\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−8.47214	−0.908308
$88$	0	0
$89$	−2.76393	−0.292976	−0.146488	−	0.989212i	$-0.546797\pi$
$89$	−2.76393	−0.292976	−0.146488	+	0.989212i	$0.546797\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−7.61803	−0.789953
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	0	0
$99$	−1.23607	−0.124230
$100$	0	0
$101$	0.763932	0.0760141	0.0380070	−	0.999277i	$-0.487899\pi$
$101$	0.763932	0.0760141	0.0380070	+	0.999277i	$0.487899\pi$
$102$	0	0
$103$	2.29180	0.225817	0.112909	−	0.993605i	$-0.463983\pi$
$103$	2.29180	0.225817	0.112909	+	0.993605i	$0.463983\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.3820	1.39036	0.695179	−	0.718837i	$-0.255325\pi$
$107$	14.3820	1.39036	0.695179	+	0.718837i	$0.255325\pi$
$108$	0	0
$109$	1.61803	0.154980	0.0774898	−	0.996993i	$-0.475309\pi$
$109$	1.61803	0.154980	0.0774898	+	0.996993i	$0.475309\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	−1.67376	−0.157454	−0.0787271	−	0.996896i	$-0.525086\pi$
$113$	−1.67376	−0.157454	−0.0787271	+	0.996896i	$0.525086\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.23607	−0.113310
$120$	0	0
$121$	−9.47214	−0.861103
$122$	0	0
$123$	4.47214	0.403239
$124$	0	0
$125$	0	0
$126$	0	0
$127$	9.41641	0.835571	0.417786	−	0.908546i	$-0.362806\pi$
$127$	9.41641	0.835571	0.417786	+	0.908546i	$0.362806\pi$
$128$	0	0
$129$	5.70820	0.502579
$130$	0	0
$131$	10.9443	0.956205	0.478103	−	0.878304i	$-0.341325\pi$
$131$	10.9443	0.956205	0.478103	+	0.878304i	$0.341325\pi$
$132$	0	0
$133$	18.9443	1.64268
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.4721	−1.40731	−0.703655	−	0.710542i	$-0.748450\pi$
$137$	−16.4721	−1.40731	−0.703655	+	0.710542i	$0.748450\pi$
$138$	0	0
$139$	5.14590	0.436469	0.218235	−	0.975896i	$-0.429970\pi$
$139$	5.14590	0.436469	0.218235	+	0.975896i	$0.429970\pi$
$140$	0	0
$141$	9.09017	0.765530
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.47214	−0.286377
$148$	0	0
$149$	23.1246	1.89444	0.947221	−	0.320581i	$-0.103878\pi$
$149$	23.1246	1.89444	0.947221	+	0.320581i	$0.103878\pi$
$150$	0	0
$151$	−5.85410	−0.476400	−0.238200	−	0.971216i	$-0.576557\pi$
$151$	−5.85410	−0.476400	−0.238200	+	0.971216i	$0.576557\pi$
$152$	0	0
$153$	−0.381966	−0.0308801
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.7639	1.17829	0.589145	−	0.808027i	$-0.299465\pi$
$157$	14.7639	1.17829	0.589145	+	0.808027i	$0.299465\pi$
$158$	0	0
$159$	−5.09017	−0.403677
$160$	0	0
$161$	−15.7082	−1.23798
$162$	0	0
$163$	2.76393	0.216488	0.108244	−	0.994124i	$-0.465477\pi$
$163$	2.76393	0.216488	0.108244	+	0.994124i	$0.465477\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.85410	−0.607769	−0.303884	−	0.952709i	$-0.598284\pi$
$167$	−7.85410	−0.607769	−0.303884	+	0.952709i	$0.598284\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	5.85410	0.447674
$172$	0	0
$173$	−23.8885	−1.81621	−0.908106	−	0.418740i	$-0.862472\pi$
$173$	−23.8885	−1.81621	−0.908106	+	0.418740i	$0.862472\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−11.2361	−0.844555
$178$	0	0
$179$	−23.8885	−1.78551	−0.892757	−	0.450539i	$-0.851232\pi$
$179$	−23.8885	−1.78551	−0.892757	+	0.450539i	$0.851232\pi$
$180$	0	0
$181$	−17.5623	−1.30540	−0.652698	−	0.757618i	$-0.726363\pi$
$181$	−17.5623	−1.30540	−0.652698	+	0.757618i	$0.726363\pi$
$182$	0	0
$183$	5.56231	0.411177
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.472136	0.0345260
$188$	0	0
$189$	−3.23607	−0.235389
$190$	0	0
$191$	12.9443	0.936615	0.468307	−	0.883566i	$-0.344864\pi$
$191$	12.9443	0.936615	0.468307	+	0.883566i	$0.344864\pi$
$192$	0	0
$193$	−21.1246	−1.52058	−0.760291	−	0.649582i	$-0.774944\pi$
$193$	−21.1246	−1.52058	−0.760291	+	0.649582i	$0.774944\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−23.0344	−1.64114	−0.820568	−	0.571549i	$-0.806343\pi$
$197$	−23.0344	−1.64114	−0.820568	+	0.571549i	$0.806343\pi$
$198$	0	0
$199$	−11.7984	−0.836365	−0.418182	−	0.908363i	$-0.637333\pi$
$199$	−11.7984	−0.836365	−0.418182	+	0.908363i	$0.637333\pi$
$200$	0	0
$201$	−15.7082	−1.10797
$202$	0	0
$203$	27.4164	1.92425
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.85410	−0.337383
$208$	0	0
$209$	−7.23607	−0.500529
$210$	0	0
$211$	12.5623	0.864825	0.432412	−	0.901676i	$-0.357662\pi$
$211$	12.5623	0.864825	0.432412	+	0.901676i	$0.357662\pi$
$212$	0	0
$213$	−5.52786	−0.378763
$214$	0	0
$215$	0	0
$216$	0	0
$217$	24.6525	1.67352
$218$	0	0
$219$	−3.52786	−0.238391
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−26.8328	−1.79686	−0.898429	−	0.439119i	$-0.855291\pi$
$223$	−26.8328	−1.79686	−0.898429	+	0.439119i	$0.855291\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.9098	−0.790483	−0.395242	−	0.918577i	$-0.629339\pi$
$227$	−11.9098	−0.790483	−0.395242	+	0.918577i	$0.629339\pi$
$228$	0	0
$229$	−3.14590	−0.207887	−0.103943	−	0.994583i	$-0.533146\pi$
$229$	−3.14590	−0.207887	−0.103943	+	0.994583i	$0.533146\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	13.4164	0.878938	0.439469	−	0.898258i	$-0.355167\pi$
$233$	13.4164	0.878938	0.439469	+	0.898258i	$0.355167\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−13.8541	−0.899921
$238$	0	0
$239$	0.763932	0.0494147	0.0247073	−	0.999695i	$-0.492135\pi$
$239$	0.763932	0.0494147	0.0247073	+	0.999695i	$0.492135\pi$
$240$	0	0
$241$	12.2705	0.790413	0.395207	−	0.918592i	$-0.370673\pi$
$241$	12.2705	0.790413	0.395207	+	0.918592i	$0.370673\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−6.32624	−0.400909
$250$	0	0
$251$	14.1803	0.895055	0.447528	−	0.894270i	$-0.352305\pi$
$251$	14.1803	0.895055	0.447528	+	0.894270i	$0.352305\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.38197	0.585231	0.292615	−	0.956230i	$-0.405474\pi$
$257$	9.38197	0.585231	0.292615	+	0.956230i	$0.405474\pi$
$258$	0	0
$259$	−6.47214	−0.402159
$260$	0	0
$261$	8.47214	0.524412
$262$	0	0
$263$	−17.6180	−1.08637	−0.543187	−	0.839612i	$-0.682783\pi$
$263$	−17.6180	−1.08637	−0.543187	+	0.839612i	$0.682783\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.76393	0.169150
$268$	0	0
$269$	−29.1246	−1.77576	−0.887879	−	0.460076i	$-0.847822\pi$
$269$	−29.1246	−1.77576	−0.887879	+	0.460076i	$0.847822\pi$
$270$	0	0
$271$	18.2705	1.10985	0.554927	−	0.831899i	$-0.312746\pi$
$271$	18.2705	1.10985	0.554927	+	0.831899i	$0.312746\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	15.4164	0.926282	0.463141	−	0.886285i	$-0.346722\pi$
$277$	15.4164	0.926282	0.463141	+	0.886285i	$0.346722\pi$
$278$	0	0
$279$	7.61803	0.456080
$280$	0	0
$281$	−26.7639	−1.59660	−0.798301	−	0.602258i	$-0.794268\pi$
$281$	−26.7639	−1.59660	−0.798301	+	0.602258i	$0.794268\pi$
$282$	0	0
$283$	4.18034	0.248495	0.124248	−	0.992251i	$-0.460348\pi$
$283$	4.18034	0.248495	0.124248	+	0.992251i	$0.460348\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−14.4721	−0.854263
$288$	0	0
$289$	−16.8541	−0.991418
$290$	0	0
$291$	−12.0000	−0.703452
$292$	0	0
$293$	−19.0344	−1.11200	−0.556002	−	0.831181i	$-0.687665\pi$
$293$	−19.0344	−1.11200	−0.556002	+	0.831181i	$0.687665\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.23607	0.0717239
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−18.4721	−1.06472
$302$	0	0
$303$	−0.763932	−0.0438867
$304$	0	0
$305$	0	0
$306$	0	0
$307$	24.1803	1.38004	0.690022	−	0.723788i	$-0.257601\pi$
$307$	24.1803	1.38004	0.690022	+	0.723788i	$0.257601\pi$
$308$	0	0
$309$	−2.29180	−0.130376
$310$	0	0
$311$	−14.2918	−0.810413	−0.405207	−	0.914225i	$-0.632800\pi$
$311$	−14.2918	−0.810413	−0.405207	+	0.914225i	$0.632800\pi$
$312$	0	0
$313$	15.7082	0.887880	0.443940	−	0.896056i	$-0.353580\pi$
$313$	15.7082	0.887880	0.443940	+	0.896056i	$0.353580\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	32.8328	1.84407	0.922037	−	0.387101i	$-0.126524\pi$
$317$	32.8328	1.84407	0.922037	+	0.387101i	$0.126524\pi$
$318$	0	0
$319$	−10.4721	−0.586327
$320$	0	0
$321$	−14.3820	−0.802723
$322$	0	0
$323$	−2.23607	−0.124418
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.61803	−0.0894775
$328$	0	0
$329$	−29.4164	−1.62178
$330$	0	0
$331$	24.0344	1.32105	0.660526	−	0.750803i	$-0.270333\pi$
$331$	24.0344	1.32105	0.660526	+	0.750803i	$0.270333\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−18.1803	−0.990346	−0.495173	−	0.868794i	$-0.664895\pi$
$337$	−18.1803	−0.990346	−0.495173	+	0.868794i	$0.664895\pi$
$338$	0	0
$339$	1.67376	0.0909063
$340$	0	0
$341$	−9.41641	−0.509927
$342$	0	0
$343$	−11.4164	−0.616428
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8.50658	−0.456657	−0.228329	−	0.973584i	$-0.573326\pi$
$347$	−8.50658	−0.456657	−0.228329	+	0.973584i	$0.573326\pi$
$348$	0	0
$349$	−4.85410	−0.259834	−0.129917	−	0.991525i	$-0.541471\pi$
$349$	−4.85410	−0.259834	−0.129917	+	0.991525i	$0.541471\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−22.8541	−1.21640	−0.608201	−	0.793783i	$-0.708108\pi$
$353$	−22.8541	−1.21640	−0.608201	+	0.793783i	$0.708108\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.23607	0.0654197
$358$	0	0
$359$	−10.4721	−0.552698	−0.276349	−	0.961057i	$-0.589125\pi$
$359$	−10.4721	−0.552698	−0.276349	+	0.961057i	$0.589125\pi$
$360$	0	0
$361$	15.2705	0.803711
$362$	0	0
$363$	9.47214	0.497158
$364$	0	0
$365$	0	0
$366$	0	0
$367$	19.4164	1.01353	0.506764	−	0.862085i	$-0.330842\pi$
$367$	19.4164	1.01353	0.506764	+	0.862085i	$0.330842\pi$
$368$	0	0
$369$	−4.47214	−0.232810
$370$	0	0
$371$	16.4721	0.855191
$372$	0	0
$373$	21.5279	1.11467	0.557335	−	0.830288i	$-0.311824\pi$
$373$	21.5279	1.11467	0.557335	+	0.830288i	$0.311824\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	−9.41641	−0.482417
$382$	0	0
$383$	21.2148	1.08402	0.542012	−	0.840371i	$-0.317663\pi$
$383$	21.2148	1.08402	0.542012	+	0.840371i	$0.317663\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.70820	−0.290164
$388$	0	0
$389$	6.65248	0.337294	0.168647	−	0.985677i	$-0.446060\pi$
$389$	6.65248	0.337294	0.168647	+	0.985677i	$0.446060\pi$
$390$	0	0
$391$	1.85410	0.0937660
$392$	0	0
$393$	−10.9443	−0.552065
$394$	0	0
$395$	0	0
$396$	0	0
$397$	6.29180	0.315776	0.157888	−	0.987457i	$-0.449531\pi$
$397$	6.29180	0.315776	0.157888	+	0.987457i	$0.449531\pi$
$398$	0	0
$399$	−18.9443	−0.948400
$400$	0	0
$401$	18.3607	0.916889	0.458444	−	0.888723i	$-0.348407\pi$
$401$	18.3607	0.916889	0.458444	+	0.888723i	$0.348407\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.47214	0.122539
$408$	0	0
$409$	−21.5623	−1.06619	−0.533094	−	0.846056i	$-0.678971\pi$
$409$	−21.5623	−1.06619	−0.533094	+	0.846056i	$0.678971\pi$
$410$	0	0
$411$	16.4721	0.812511
$412$	0	0
$413$	36.3607	1.78919
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−5.14590	−0.251996
$418$	0	0
$419$	5.81966	0.284309	0.142155	−	0.989844i	$-0.454597\pi$
$419$	5.81966	0.284309	0.142155	+	0.989844i	$0.454597\pi$
$420$	0	0
$421$	−20.3262	−0.990640	−0.495320	−	0.868711i	$-0.664949\pi$
$421$	−20.3262	−0.990640	−0.495320	+	0.868711i	$0.664949\pi$
$422$	0	0
$423$	−9.09017	−0.441979
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−18.0000	−0.871081
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7.52786	−0.362604	−0.181302	−	0.983427i	$-0.558031\pi$
$431$	−7.52786	−0.362604	−0.181302	+	0.983427i	$0.558031\pi$
$432$	0	0
$433$	27.4164	1.31755	0.658774	−	0.752341i	$-0.271075\pi$
$433$	27.4164	1.31755	0.658774	+	0.752341i	$0.271075\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−28.4164	−1.35934
$438$	0	0
$439$	−15.4164	−0.735785	−0.367893	−	0.929868i	$-0.619921\pi$
$439$	−15.4164	−0.735785	−0.367893	+	0.929868i	$0.619921\pi$
$440$	0	0
$441$	3.47214	0.165340
$442$	0	0
$443$	−20.7984	−0.988161	−0.494080	−	0.869416i	$-0.664495\pi$
$443$	−20.7984	−0.988161	−0.494080	+	0.869416i	$0.664495\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−23.1246	−1.09376
$448$	0	0
$449$	−5.81966	−0.274647	−0.137323	−	0.990526i	$-0.543850\pi$
$449$	−5.81966	−0.274647	−0.137323	+	0.990526i	$0.543850\pi$
$450$	0	0
$451$	5.52786	0.260297
$452$	0	0
$453$	5.85410	0.275050
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.12461	−0.239719	−0.119860	−	0.992791i	$-0.538244\pi$
$457$	−5.12461	−0.239719	−0.119860	+	0.992791i	$0.538244\pi$
$458$	0	0
$459$	0.381966	0.0178286
$460$	0	0
$461$	−20.1803	−0.939892	−0.469946	−	0.882695i	$-0.655727\pi$
$461$	−20.1803	−0.939892	−0.469946	+	0.882695i	$0.655727\pi$
$462$	0	0
$463$	−30.9443	−1.43810	−0.719051	−	0.694957i	$-0.755423\pi$
$463$	−30.9443	−1.43810	−0.719051	+	0.694957i	$0.755423\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.74265	−0.265738	−0.132869	−	0.991134i	$-0.542419\pi$
$467$	−5.74265	−0.265738	−0.132869	+	0.991134i	$0.542419\pi$
$468$	0	0
$469$	50.8328	2.34724
$470$	0	0
$471$	−14.7639	−0.680286
$472$	0	0
$473$	7.05573	0.324423
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.09017	0.233063
$478$	0	0
$479$	−11.1246	−0.508296	−0.254148	−	0.967165i	$-0.581795\pi$
$479$	−11.1246	−0.508296	−0.254148	+	0.967165i	$0.581795\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	15.7082	0.714748
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−10.7639	−0.487760	−0.243880	−	0.969805i	$-0.578420\pi$
$487$	−10.7639	−0.487760	−0.243880	+	0.969805i	$0.578420\pi$
$488$	0	0
$489$	−2.76393	−0.124989
$490$	0	0
$491$	−32.8328	−1.48172	−0.740862	−	0.671657i	$-0.765583\pi$
$491$	−32.8328	−1.48172	−0.740862	+	0.671657i	$0.765583\pi$
$492$	0	0
$493$	−3.23607	−0.145745
$494$	0	0
$495$	0	0
$496$	0	0
$497$	17.8885	0.802411
$498$	0	0
$499$	−28.6869	−1.28420	−0.642101	−	0.766620i	$-0.721937\pi$
$499$	−28.6869	−1.28420	−0.642101	+	0.766620i	$0.721937\pi$
$500$	0	0
$501$	7.85410	0.350895
$502$	0	0
$503$	32.9443	1.46891	0.734456	−	0.678656i	$-0.237437\pi$
$503$	32.9443	1.46891	0.734456	+	0.678656i	$0.237437\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	13.0000	0.577350
$508$	0	0
$509$	29.8885	1.32479	0.662393	−	0.749156i	$-0.269541\pi$
$509$	29.8885	1.32479	0.662393	+	0.749156i	$0.269541\pi$
$510$	0	0
$511$	11.4164	0.505032
$512$	0	0
$513$	−5.85410	−0.258465
$514$	0	0
$515$	0	0
$516$	0	0
$517$	11.2361	0.494162
$518$	0	0
$519$	23.8885	1.04859
$520$	0	0
$521$	29.8885	1.30944	0.654720	−	0.755871i	$-0.272786\pi$
$521$	29.8885	1.30944	0.654720	+	0.755871i	$0.272786\pi$
$522$	0	0
$523$	−39.8885	−1.74420	−0.872102	−	0.489324i	$-0.837244\pi$
$523$	−39.8885	−1.74420	−0.872102	+	0.489324i	$0.837244\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.90983	−0.126754
$528$	0	0
$529$	0.562306	0.0244481
$530$	0	0
$531$	11.2361	0.487604
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	23.8885	1.03087
$538$	0	0
$539$	−4.29180	−0.184861
$540$	0	0
$541$	−16.2705	−0.699524	−0.349762	−	0.936839i	$-0.613738\pi$
$541$	−16.2705	−0.699524	−0.349762	+	0.936839i	$0.613738\pi$
$542$	0	0
$543$	17.5623	0.753671
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−44.8328	−1.91691	−0.958456	−	0.285239i	$-0.907927\pi$
$547$	−44.8328	−1.91691	−0.958456	+	0.285239i	$0.907927\pi$
$548$	0	0
$549$	−5.56231	−0.237393
$550$	0	0
$551$	49.5967	2.11289
$552$	0	0
$553$	44.8328	1.90649
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10.3607	−0.438996	−0.219498	−	0.975613i	$-0.570442\pi$
$557$	−10.3607	−0.438996	−0.219498	+	0.975613i	$0.570442\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−0.472136	−0.0199336
$562$	0	0
$563$	6.38197	0.268968	0.134484	−	0.990916i	$-0.457062\pi$
$563$	6.38197	0.268968	0.134484	+	0.990916i	$0.457062\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.23607	0.135902
$568$	0	0
$569$	−25.3050	−1.06084	−0.530419	−	0.847735i	$-0.677966\pi$
$569$	−25.3050	−1.06084	−0.530419	+	0.847735i	$0.677966\pi$
$570$	0	0
$571$	−37.8885	−1.58559	−0.792793	−	0.609491i	$-0.791374\pi$
$571$	−37.8885	−1.58559	−0.792793	+	0.609491i	$0.791374\pi$
$572$	0	0
$573$	−12.9443	−0.540755
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−23.1246	−0.962690	−0.481345	−	0.876531i	$-0.659852\pi$
$577$	−23.1246	−0.962690	−0.481345	+	0.876531i	$0.659852\pi$
$578$	0	0
$579$	21.1246	0.877909
$580$	0	0
$581$	20.4721	0.849327
$582$	0	0
$583$	−6.29180	−0.260580
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.0344	0.537989	0.268994	−	0.963142i	$-0.413309\pi$
$587$	13.0344	0.537989	0.268994	+	0.963142i	$0.413309\pi$
$588$	0	0
$589$	44.5967	1.83758
$590$	0	0
$591$	23.0344	0.947510
$592$	0	0
$593$	−20.5066	−0.842104	−0.421052	−	0.907036i	$-0.638339\pi$
$593$	−20.5066	−0.842104	−0.421052	+	0.907036i	$0.638339\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	11.7984	0.482875
$598$	0	0
$599$	32.8328	1.34151	0.670756	−	0.741678i	$-0.265970\pi$
$599$	32.8328	1.34151	0.670756	+	0.741678i	$0.265970\pi$
$600$	0	0
$601$	−12.8541	−0.524330	−0.262165	−	0.965023i	$-0.584436\pi$
$601$	−12.8541	−0.524330	−0.262165	+	0.965023i	$0.584436\pi$
$602$	0	0
$603$	15.7082	0.639688
$604$	0	0
$605$	0	0
$606$	0	0
$607$	48.2492	1.95838	0.979188	−	0.202956i	$-0.0650550\pi$
$607$	48.2492	1.95838	0.979188	+	0.202956i	$0.0650550\pi$
$608$	0	0
$609$	−27.4164	−1.11097
$610$	0	0
$611$	0	0
$612$	0	0
$613$	7.59675	0.306830	0.153415	−	0.988162i	$-0.450973\pi$
$613$	7.59675	0.306830	0.153415	+	0.988162i	$0.450973\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	34.4508	1.38694	0.693469	−	0.720486i	$-0.256081\pi$
$617$	34.4508	1.38694	0.693469	+	0.720486i	$0.256081\pi$
$618$	0	0
$619$	23.7984	0.956537	0.478269	−	0.878214i	$-0.341265\pi$
$619$	23.7984	0.956537	0.478269	+	0.878214i	$0.341265\pi$
$620$	0	0
$621$	4.85410	0.194788
$622$	0	0
$623$	−8.94427	−0.358345
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7.23607	0.288981
$628$	0	0
$629$	0.763932	0.0304600
$630$	0	0
$631$	−29.3050	−1.16661	−0.583306	−	0.812253i	$-0.698241\pi$
$631$	−29.3050	−1.16661	−0.583306	+	0.812253i	$0.698241\pi$
$632$	0	0
$633$	−12.5623	−0.499307
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	5.52786	0.218679
$640$	0	0
$641$	−41.7771	−1.65010	−0.825048	−	0.565063i	$-0.808852\pi$
$641$	−41.7771	−1.65010	−0.825048	+	0.565063i	$0.808852\pi$
$642$	0	0
$643$	4.65248	0.183476	0.0917379	−	0.995783i	$-0.470758\pi$
$643$	4.65248	0.183476	0.0917379	+	0.995783i	$0.470758\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19.0557	0.749158	0.374579	−	0.927195i	$-0.377787\pi$
$647$	19.0557	0.749158	0.374579	+	0.927195i	$0.377787\pi$
$648$	0	0
$649$	−13.8885	−0.545173
$650$	0	0
$651$	−24.6525	−0.966207
$652$	0	0
$653$	−46.6869	−1.82700	−0.913500	−	0.406838i	$-0.866631\pi$
$653$	−46.6869	−1.82700	−0.913500	+	0.406838i	$0.866631\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.52786	0.137635
$658$	0	0
$659$	32.0689	1.24923	0.624613	−	0.780934i	$-0.285257\pi$
$659$	32.0689	1.24923	0.624613	+	0.780934i	$0.285257\pi$
$660$	0	0
$661$	18.2016	0.707961	0.353981	−	0.935253i	$-0.384828\pi$
$661$	18.2016	0.707961	0.353981	+	0.935253i	$0.384828\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−41.1246	−1.59235
$668$	0	0
$669$	26.8328	1.03742
$670$	0	0
$671$	6.87539	0.265421
$672$	0	0
$673$	7.88854	0.304081	0.152041	−	0.988374i	$-0.451416\pi$
$673$	7.88854	0.304081	0.152041	+	0.988374i	$0.451416\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.97871	0.268214	0.134107	−	0.990967i	$-0.457183\pi$
$677$	6.97871	0.268214	0.134107	+	0.990967i	$0.457183\pi$
$678$	0	0
$679$	38.8328	1.49027
$680$	0	0
$681$	11.9098	0.456386
$682$	0	0
$683$	19.3262	0.739498	0.369749	−	0.929132i	$-0.379444\pi$
$683$	19.3262	0.739498	0.369749	+	0.929132i	$0.379444\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	3.14590	0.120023
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−4.56231	−0.173558	−0.0867791	−	0.996228i	$-0.527657\pi$
$691$	−4.56231	−0.173558	−0.0867791	+	0.996228i	$0.527657\pi$
$692$	0	0
$693$	−4.00000	−0.151947
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.70820	0.0647028
$698$	0	0
$699$	−13.4164	−0.507455
$700$	0	0
$701$	−1.88854	−0.0713293	−0.0356647	−	0.999364i	$-0.511355\pi$
$701$	−1.88854	−0.0713293	−0.0356647	+	0.999364i	$0.511355\pi$
$702$	0	0
$703$	−11.7082	−0.441583
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.47214	0.0929742
$708$	0	0
$709$	12.6180	0.473880	0.236940	−	0.971524i	$-0.423855\pi$
$709$	12.6180	0.473880	0.236940	+	0.971524i	$0.423855\pi$
$710$	0	0
$711$	13.8541	0.519569
$712$	0	0
$713$	−36.9787	−1.38486
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−0.763932	−0.0285296
$718$	0	0
$719$	20.7639	0.774364	0.387182	−	0.922003i	$-0.373449\pi$
$719$	20.7639	0.774364	0.387182	+	0.922003i	$0.373449\pi$
$720$	0	0
$721$	7.41641	0.276201
$722$	0	0
$723$	−12.2705	−0.456345
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18.8328	0.698470	0.349235	−	0.937035i	$-0.386441\pi$
$727$	18.8328	0.698470	0.349235	+	0.937035i	$0.386441\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.18034	0.0806428
$732$	0	0
$733$	−10.1803	−0.376019	−0.188010	−	0.982167i	$-0.560204\pi$
$733$	−10.1803	−0.376019	−0.188010	+	0.982167i	$0.560204\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−19.4164	−0.715213
$738$	0	0
$739$	−12.5623	−0.462112	−0.231056	−	0.972940i	$-0.574218\pi$
$739$	−12.5623	−0.462112	−0.231056	+	0.972940i	$0.574218\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7.41641	0.272082	0.136041	−	0.990703i	$-0.456562\pi$
$743$	7.41641	0.272082	0.136041	+	0.990703i	$0.456562\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.32624	0.231465
$748$	0	0
$749$	46.5410	1.70057
$750$	0	0
$751$	8.90983	0.325124	0.162562	−	0.986698i	$-0.448024\pi$
$751$	8.90983	0.325124	0.162562	+	0.986698i	$0.448024\pi$
$752$	0	0
$753$	−14.1803	−0.516760
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−43.4853	−1.58050	−0.790250	−	0.612785i	$-0.790049\pi$
$757$	−43.4853	−1.58050	−0.790250	+	0.612785i	$0.790049\pi$
$758$	0	0
$759$	−6.00000	−0.217786
$760$	0	0
$761$	−13.5279	−0.490385	−0.245192	−	0.969474i	$-0.578851\pi$
$761$	−13.5279	−0.490385	−0.245192	+	0.969474i	$0.578851\pi$
$762$	0	0
$763$	5.23607	0.189558
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−2.14590	−0.0773831	−0.0386915	−	0.999251i	$-0.512319\pi$
$769$	−2.14590	−0.0773831	−0.0386915	+	0.999251i	$0.512319\pi$
$770$	0	0
$771$	−9.38197	−0.337883
$772$	0	0
$773$	51.9230	1.86754	0.933770	−	0.357874i	$-0.116498\pi$
$773$	51.9230	1.86754	0.933770	+	0.357874i	$0.116498\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	6.47214	0.232187
$778$	0	0
$779$	−26.1803	−0.938008
$780$	0	0
$781$	−6.83282	−0.244497
$782$	0	0
$783$	−8.47214	−0.302769
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−39.7082	−1.41544	−0.707722	−	0.706491i	$-0.750277\pi$
$787$	−39.7082	−1.41544	−0.707722	+	0.706491i	$0.750277\pi$
$788$	0	0
$789$	17.6180	0.627219
$790$	0	0
$791$	−5.41641	−0.192585
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20.7426	0.734742	0.367371	−	0.930074i	$-0.380258\pi$
$797$	20.7426	0.734742	0.367371	+	0.930074i	$0.380258\pi$
$798$	0	0
$799$	3.47214	0.122835
$800$	0	0
$801$	−2.76393	−0.0976587
$802$	0	0
$803$	−4.36068	−0.153885
$804$	0	0
$805$	0	0
$806$	0	0
$807$	29.1246	1.02523
$808$	0	0
$809$	49.4164	1.73739	0.868694	−	0.495349i	$-0.164960\pi$
$809$	49.4164	1.73739	0.868694	+	0.495349i	$0.164960\pi$
$810$	0	0
$811$	2.43769	0.0855990	0.0427995	−	0.999084i	$-0.486372\pi$
$811$	2.43769	0.0855990	0.0427995	+	0.999084i	$0.486372\pi$
$812$	0	0
$813$	−18.2705	−0.640775
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−33.4164	−1.16909
$818$	0	0
$819$	0	0
$820$	0	0
$821$	41.5279	1.44933	0.724666	−	0.689100i	$-0.241994\pi$
$821$	41.5279	1.44933	0.724666	+	0.689100i	$0.241994\pi$
$822$	0	0
$823$	−14.7639	−0.514638	−0.257319	−	0.966326i	$-0.582839\pi$
$823$	−14.7639	−0.514638	−0.257319	+	0.966326i	$0.582839\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16.1459	0.561448	0.280724	−	0.959789i	$-0.409425\pi$
$827$	16.1459	0.561448	0.280724	+	0.959789i	$0.409425\pi$
$828$	0	0
$829$	37.8541	1.31473	0.657364	−	0.753574i	$-0.271672\pi$
$829$	37.8541	1.31473	0.657364	+	0.753574i	$0.271672\pi$
$830$	0	0
$831$	−15.4164	−0.534789
$832$	0	0
$833$	−1.32624	−0.0459514
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−7.61803	−0.263318
$838$	0	0
$839$	27.0132	0.932598	0.466299	−	0.884627i	$-0.345587\pi$
$839$	27.0132	0.932598	0.466299	+	0.884627i	$0.345587\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	0	0
$843$	26.7639	0.921799
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−30.6525	−1.05323
$848$	0	0
$849$	−4.18034	−0.143469
$850$	0	0
$851$	9.70820	0.332793
$852$	0	0
$853$	−26.2918	−0.900214	−0.450107	−	0.892975i	$-0.648614\pi$
$853$	−26.2918	−0.900214	−0.450107	+	0.892975i	$0.648614\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−34.9098	−1.19250	−0.596249	−	0.802800i	$-0.703343\pi$
$857$	−34.9098	−1.19250	−0.596249	+	0.802800i	$0.703343\pi$
$858$	0	0
$859$	26.8328	0.915524	0.457762	−	0.889075i	$-0.348651\pi$
$859$	26.8328	0.915524	0.457762	+	0.889075i	$0.348651\pi$
$860$	0	0
$861$	14.4721	0.493209
$862$	0	0
$863$	17.5279	0.596655	0.298328	−	0.954464i	$-0.403571\pi$
$863$	17.5279	0.596655	0.298328	+	0.954464i	$0.403571\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	16.8541	0.572395
$868$	0	0
$869$	−17.1246	−0.580913
$870$	0	0
$871$	0	0
$872$	0	0
$873$	12.0000	0.406138
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−34.5410	−1.16637	−0.583184	−	0.812340i	$-0.698193\pi$
$877$	−34.5410	−1.16637	−0.583184	+	0.812340i	$0.698193\pi$
$878$	0	0
$879$	19.0344	0.642016
$880$	0	0
$881$	−16.5836	−0.558715	−0.279358	−	0.960187i	$-0.590122\pi$
$881$	−16.5836	−0.558715	−0.279358	+	0.960187i	$0.590122\pi$
$882$	0	0
$883$	−10.1115	−0.340278	−0.170139	−	0.985420i	$-0.554422\pi$
$883$	−10.1115	−0.340278	−0.170139	+	0.985420i	$0.554422\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22.9098	0.769237	0.384618	−	0.923076i	$-0.374333\pi$
$887$	22.9098	0.769237	0.384618	+	0.923076i	$0.374333\pi$
$888$	0	0
$889$	30.4721	1.02200
$890$	0	0
$891$	−1.23607	−0.0414098
$892$	0	0
$893$	−53.2148	−1.78076
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	64.5410	2.15256
$900$	0	0
$901$	−1.94427	−0.0647731
$902$	0	0
$903$	18.4721	0.614714
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−53.7082	−1.78335	−0.891676	−	0.452675i	$-0.850470\pi$
$907$	−53.7082	−1.78335	−0.891676	+	0.452675i	$0.850470\pi$
$908$	0	0
$909$	0.763932	0.0253380
$910$	0	0
$911$	−56.0689	−1.85764	−0.928822	−	0.370525i	$-0.879178\pi$
$911$	−56.0689	−1.85764	−0.928822	+	0.370525i	$0.879178\pi$
$912$	0	0
$913$	−7.81966	−0.258793
$914$	0	0
$915$	0	0
$916$	0	0
$917$	35.4164	1.16955
$918$	0	0
$919$	−3.25735	−0.107450	−0.0537251	−	0.998556i	$-0.517109\pi$
$919$	−3.25735	−0.107450	−0.0537251	+	0.998556i	$0.517109\pi$
$920$	0	0
$921$	−24.1803	−0.796769
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.29180	0.0752725
$928$	0	0
$929$	−49.4853	−1.62356	−0.811780	−	0.583964i	$-0.801501\pi$
$929$	−49.4853	−1.62356	−0.811780	+	0.583964i	$0.801501\pi$
$930$	0	0
$931$	20.3262	0.666166
$932$	0	0
$933$	14.2918	0.467892
$934$	0	0
$935$	0	0
$936$	0	0
$937$	14.5836	0.476425	0.238213	−	0.971213i	$-0.423438\pi$
$937$	14.5836	0.476425	0.238213	+	0.971213i	$0.423438\pi$
$938$	0	0
$939$	−15.7082	−0.512618
$940$	0	0
$941$	38.2918	1.24828	0.624138	−	0.781314i	$-0.285450\pi$
$941$	38.2918	1.24828	0.624138	+	0.781314i	$0.285450\pi$
$942$	0	0
$943$	21.7082	0.706916
$944$	0	0
$945$	0	0
$946$	0	0
$947$	21.2148	0.689388	0.344694	−	0.938715i	$-0.387983\pi$
$947$	21.2148	0.689388	0.344694	+	0.938715i	$0.387983\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−32.8328	−1.06468
$952$	0	0
$953$	12.7426	0.412775	0.206387	−	0.978470i	$-0.433829\pi$
$953$	12.7426	0.412775	0.206387	+	0.978470i	$0.433829\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	10.4721	0.338516
$958$	0	0
$959$	−53.3050	−1.72131
$960$	0	0
$961$	27.0344	0.872079
$962$	0	0
$963$	14.3820	0.463452
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−18.9443	−0.609207	−0.304603	−	0.952479i	$-0.598524\pi$
$967$	−18.9443	−0.609207	−0.304603	+	0.952479i	$0.598524\pi$
$968$	0	0
$969$	2.23607	0.0718329
$970$	0	0
$971$	−30.5410	−0.980108	−0.490054	−	0.871692i	$-0.663023\pi$
$971$	−30.5410	−0.980108	−0.490054	+	0.871692i	$0.663023\pi$
$972$	0	0
$973$	16.6525	0.533854
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17.7295	0.567217	0.283608	−	0.958940i	$-0.408468\pi$
$977$	17.7295	0.567217	0.283608	+	0.958940i	$0.408468\pi$
$978$	0	0
$979$	3.41641	0.109189
$980$	0	0
$981$	1.61803	0.0516598
$982$	0	0
$983$	−48.5066	−1.54712	−0.773560	−	0.633723i	$-0.781526\pi$
$983$	−48.5066	−1.54712	−0.773560	+	0.633723i	$0.781526\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	29.4164	0.936335
$988$	0	0
$989$	27.7082	0.881070
$990$	0	0
$991$	5.56231	0.176692	0.0883462	−	0.996090i	$-0.471842\pi$
$991$	5.56231	0.176692	0.0883462	+	0.996090i	$0.471842\pi$
$992$	0	0
$993$	−24.0344	−0.762710
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−6.00000	−0.190022	−0.0950110	−	0.995476i	$-0.530289\pi$
$997$	−6.00000	−0.190022	−0.0950110	+	0.995476i	$0.530289\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1500.2.a.c.1.2	✓	2
3.2	odd	2		4500.2.a.h.1.2		2
4.3	odd	2		6000.2.a.s.1.1		2
5.2	odd	4		1500.2.d.b.1249.4		4
5.3	odd	4		1500.2.d.b.1249.1		4
5.4	even	2		1500.2.a.g.1.1	yes	2
15.2	even	4		4500.2.d.a.4249.4		4
15.8	even	4		4500.2.d.a.4249.1		4
15.14	odd	2		4500.2.a.d.1.1		2
20.3	even	4		6000.2.f.f.1249.4		4
20.7	even	4		6000.2.f.f.1249.1		4
20.19	odd	2		6000.2.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1500.2.a.c.1.2	✓	2	1.1	even	1	trivial
1500.2.a.g.1.1	yes	2	5.4	even	2
1500.2.d.b.1249.1		4	5.3	odd	4
1500.2.d.b.1249.4		4	5.2	odd	4
4500.2.a.d.1.1		2	15.14	odd	2
4500.2.a.h.1.2		2	3.2	odd	2
4500.2.d.a.4249.1		4	15.8	even	4
4500.2.d.a.4249.4		4	15.2	even	4
6000.2.a.i.1.2		2	20.19	odd	2
6000.2.a.s.1.1		2	4.3	odd	2
6000.2.f.f.1249.1		4	20.7	even	4
6000.2.f.f.1249.4		4	20.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1500 = 2^{2} \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1500.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.23607 q^{7} +1.00000 q^{9} -1.23607 q^{11} -0.381966 q^{17} +5.85410 q^{19} -3.23607 q^{21} -4.85410 q^{23} -1.00000 q^{27} +8.47214 q^{29} +7.61803 q^{31} +1.23607 q^{33} -2.00000 q^{37} -4.47214 q^{41} -5.70820 q^{43} -9.09017 q^{47} +3.47214 q^{49} +0.381966 q^{51} +5.09017 q^{53} -5.85410 q^{57} +11.2361 q^{59} -5.56231 q^{61} +3.23607 q^{63} +15.7082 q^{67} +4.85410 q^{69} +5.52786 q^{71} +3.52786 q^{73} -4.00000 q^{77} +13.8541 q^{79} +1.00000 q^{81} +6.32624 q^{83} -8.47214 q^{87} -2.76393 q^{89} -7.61803 q^{93} +12.0000 q^{97} -1.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} + 2 q^{11} - 3 q^{17} + 5 q^{19} - 2 q^{21} - 3 q^{23} - 2 q^{27} + 8 q^{29} + 13 q^{31} - 2 q^{33} - 4 q^{37} + 2 q^{43} - 7 q^{47} - 2 q^{49} + 3 q^{51} - q^{53} - 5 q^{57}+ \cdots + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 + 2 * q^11 - 3 * q^17 + 5 * q^19 - 2 * q^21 - 3 * q^23 - 2 * q^27 + 8 * q^29 + 13 * q^31 - 2 * q^33 - 4 * q^37 + 2 * q^43 - 7 * q^47 - 2 * q^49 + 3 * q^51 - q^53 - 5 * q^57 + 18 * q^59 + 9 * q^61 + 2 * q^63 + 18 * q^67 + 3 * q^69 + 20 * q^71 + 16 * q^73 - 8 * q^77 + 21 * q^79 + 2 * q^81 - 3 * q^83 - 8 * q^87 - 10 * q^89 - 13 * q^93 + 24 * q^97 + 2 * q^99

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