LMFDB - Embedded newform 1500.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1500,2,Mod(1,1500)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$11.9775603032$
Analytic rank:	$1$
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:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -2.38197 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.38197 q^{7} +1.00000 q^{9} -3.47214 q^{11} -5.00000 q^{13} +4.85410 q^{17} +6.70820 q^{19} -2.38197 q^{21} -5.47214 q^{23} +1.00000 q^{27} -8.23607 q^{29} -6.85410 q^{31} -3.47214 q^{33} -3.00000 q^{37} -5.00000 q^{39} +5.32624 q^{41} -11.8541 q^{43} -9.00000 q^{47} -1.32624 q^{49} +4.85410 q^{51} -5.09017 q^{53} +6.70820 q^{57} +0.381966 q^{59} +1.14590 q^{61} -2.38197 q^{63} +4.94427 q^{67} -5.47214 q^{69} +0.326238 q^{71} +7.85410 q^{73} +8.27051 q^{77} +9.70820 q^{79} +1.00000 q^{81} -12.3820 q^{83} -8.23607 q^{87} +9.47214 q^{89} +11.9098 q^{91} -6.85410 q^{93} -0.0901699 q^{97} -3.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 7 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 7 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 7 q^{7} + 2 q^{9} + 2 q^{11} - 10 q^{13} + 3 q^{17} - 7 q^{21} - 2 q^{23} + 2 q^{27} - 12 q^{29} - 7 q^{31} + 2 q^{33} - 6 q^{37} - 10 q^{39} - 5 q^{41} - 17 q^{43} - 18 q^{47} + 13 q^{49} + 3 q^{51} + q^{53} + 3 q^{59} + 9 q^{61} - 7 q^{63} - 8 q^{67} - 2 q^{69} - 15 q^{71} + 9 q^{73} - 17 q^{77} + 6 q^{79} + 2 q^{81} - 27 q^{83} - 12 q^{87} + 10 q^{89} + 35 q^{91} - 7 q^{93} + 11 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 7 * q^7 + 2 * q^9 + 2 * q^11 - 10 * q^13 + 3 * q^17 - 7 * q^21 - 2 * q^23 + 2 * q^27 - 12 * q^29 - 7 * q^31 + 2 * q^33 - 6 * q^37 - 10 * q^39 - 5 * q^41 - 17 * q^43 - 18 * q^47 + 13 * q^49 + 3 * q^51 + q^53 + 3 * q^59 + 9 * q^61 - 7 * q^63 - 8 * q^67 - 2 * q^69 - 15 * q^71 + 9 * q^73 - 17 * q^77 + 6 * q^79 + 2 * q^81 - 27 * q^83 - 12 * q^87 + 10 * q^89 + 35 * q^91 - 7 * q^93 + 11 * q^97 + 2 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.38197	−0.900299	−0.450149	−	0.892953i	$-0.648629\pi$
$7$	−2.38197	−0.900299	−0.450149	+	0.892953i	$0.648629\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.47214	−1.04689	−0.523444	−	0.852060i	$-0.675353\pi$
$11$	−3.47214	−1.04689	−0.523444	+	0.852060i	$0.675353\pi$
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.85410	1.17729	0.588646	−	0.808391i	$-0.299661\pi$
$17$	4.85410	1.17729	0.588646	+	0.808391i	$0.299661\pi$
$18$	0	0
$19$	6.70820	1.53897	0.769484	−	0.638666i	$-0.220514\pi$
$19$	6.70820	1.53897	0.769484	+	0.638666i	$0.220514\pi$
$20$	0	0
$21$	−2.38197	−0.519788
$22$	0	0
$23$	−5.47214	−1.14102	−0.570510	−	0.821291i	$-0.693254\pi$
$23$	−5.47214	−1.14102	−0.570510	+	0.821291i	$0.693254\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.23607	−1.52940	−0.764700	−	0.644387i	$-0.777113\pi$
$29$	−8.23607	−1.52940	−0.764700	+	0.644387i	$0.777113\pi$
$30$	0	0
$31$	−6.85410	−1.23103	−0.615517	−	0.788124i	$-0.711053\pi$
$31$	−6.85410	−1.23103	−0.615517	+	0.788124i	$0.711053\pi$
$32$	0	0
$33$	−3.47214	−0.604421
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	0	0
$39$	−5.00000	−0.800641
$40$	0	0
$41$	5.32624	0.831819	0.415909	−	0.909406i	$-0.363463\pi$
$41$	5.32624	0.831819	0.415909	+	0.909406i	$0.363463\pi$
$42$	0	0
$43$	−11.8541	−1.80773	−0.903867	−	0.427814i	$-0.859284\pi$
$43$	−11.8541	−1.80773	−0.903867	+	0.427814i	$0.859284\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	0	0
$49$	−1.32624	−0.189463
$50$	0	0
$51$	4.85410	0.679710
$52$	0	0
$53$	−5.09017	−0.699189	−0.349594	−	0.936901i	$-0.613681\pi$
$53$	−5.09017	−0.699189	−0.349594	+	0.936901i	$0.613681\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.70820	0.888523
$58$	0	0
$59$	0.381966	0.0497277	0.0248639	−	0.999691i	$-0.492085\pi$
$59$	0.381966	0.0497277	0.0248639	+	0.999691i	$0.492085\pi$
$60$	0	0
$61$	1.14590	0.146717	0.0733586	−	0.997306i	$-0.476628\pi$
$61$	1.14590	0.146717	0.0733586	+	0.997306i	$0.476628\pi$
$62$	0	0
$63$	−2.38197	−0.300100
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.94427	0.604039	0.302019	−	0.953302i	$-0.402339\pi$
$67$	4.94427	0.604039	0.302019	+	0.953302i	$0.402339\pi$
$68$	0	0
$69$	−5.47214	−0.658768
$70$	0	0
$71$	0.326238	0.0387173	0.0193587	−	0.999813i	$-0.493838\pi$
$71$	0.326238	0.0387173	0.0193587	+	0.999813i	$0.493838\pi$
$72$	0	0
$73$	7.85410	0.919253	0.459627	−	0.888112i	$-0.347983\pi$
$73$	7.85410	0.919253	0.459627	+	0.888112i	$0.347983\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.27051	0.942512
$78$	0	0
$79$	9.70820	1.09226	0.546129	−	0.837701i	$-0.316101\pi$
$79$	9.70820	1.09226	0.546129	+	0.837701i	$0.316101\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.3820	−1.35910	−0.679549	−	0.733630i	$-0.737824\pi$
$83$	−12.3820	−1.35910	−0.679549	+	0.733630i	$0.737824\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−8.23607	−0.882999
$88$	0	0
$89$	9.47214	1.00404	0.502022	−	0.864855i	$-0.332590\pi$
$89$	9.47214	1.00404	0.502022	+	0.864855i	$0.332590\pi$
$90$	0	0
$91$	11.9098	1.24849
$92$	0	0
$93$	−6.85410	−0.710737
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.0901699	−0.00915537	−0.00457769	−	0.999990i	$-0.501457\pi$
$97$	−0.0901699	−0.00915537	−0.00457769	+	0.999990i	$0.501457\pi$
$98$	0	0
$99$	−3.47214	−0.348963
$100$	0	0
$101$	−6.47214	−0.644002	−0.322001	−	0.946739i	$-0.604355\pi$
$101$	−6.47214	−0.644002	−0.322001	+	0.946739i	$0.604355\pi$
$102$	0	0
$103$	0.145898	0.0143758	0.00718788	−	0.999974i	$-0.497712\pi$
$103$	0.145898	0.0143758	0.00718788	+	0.999974i	$0.497712\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.0344	−0.970066	−0.485033	−	0.874496i	$-0.661192\pi$
$107$	−10.0344	−0.970066	−0.485033	+	0.874496i	$0.661192\pi$
$108$	0	0
$109$	16.4164	1.57241	0.786203	−	0.617968i	$-0.212044\pi$
$109$	16.4164	1.57241	0.786203	+	0.617968i	$0.212044\pi$
$110$	0	0
$111$	−3.00000	−0.284747
$112$	0	0
$113$	−18.6525	−1.75468	−0.877339	−	0.479872i	$-0.840683\pi$
$113$	−18.6525	−1.75468	−0.877339	+	0.479872i	$0.840683\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5.00000	−0.462250
$118$	0	0
$119$	−11.5623	−1.05991
$120$	0	0
$121$	1.05573	0.0959753
$122$	0	0
$123$	5.32624	0.480251
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0.708204	0.0628429	0.0314215	−	0.999506i	$-0.489997\pi$
$127$	0.708204	0.0628429	0.0314215	+	0.999506i	$0.489997\pi$
$128$	0	0
$129$	−11.8541	−1.04370
$130$	0	0
$131$	5.61803	0.490850	0.245425	−	0.969416i	$-0.421073\pi$
$131$	5.61803	0.490850	0.245425	+	0.969416i	$0.421073\pi$
$132$	0	0
$133$	−15.9787	−1.38553
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−0.236068	−0.0201686	−0.0100843	−	0.999949i	$-0.503210\pi$
$137$	−0.236068	−0.0201686	−0.0100843	+	0.999949i	$0.503210\pi$
$138$	0	0
$139$	15.4721	1.31233	0.656165	−	0.754618i	$-0.272178\pi$
$139$	15.4721	1.31233	0.656165	+	0.754618i	$0.272178\pi$
$140$	0	0
$141$	−9.00000	−0.757937
$142$	0	0
$143$	17.3607	1.45177
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.32624	−0.109386
$148$	0	0
$149$	−11.4721	−0.939834	−0.469917	−	0.882711i	$-0.655716\pi$
$149$	−11.4721	−0.939834	−0.469917	+	0.882711i	$0.655716\pi$
$150$	0	0
$151$	−1.70820	−0.139012	−0.0695058	−	0.997582i	$-0.522142\pi$
$151$	−1.70820	−0.139012	−0.0695058	+	0.997582i	$0.522142\pi$
$152$	0	0
$153$	4.85410	0.392431
$154$	0	0
$155$	0	0
$156$	0	0
$157$	20.5623	1.64105	0.820525	−	0.571610i	$-0.193681\pi$
$157$	20.5623	1.64105	0.820525	+	0.571610i	$0.193681\pi$
$158$	0	0
$159$	−5.09017	−0.403677
$160$	0	0
$161$	13.0344	1.02726
$162$	0	0
$163$	−16.7082	−1.30869	−0.654344	−	0.756197i	$-0.727055\pi$
$163$	−16.7082	−1.30869	−0.654344	+	0.756197i	$0.727055\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.8885	1.53902	0.769511	−	0.638634i	$-0.220500\pi$
$167$	19.8885	1.53902	0.769511	+	0.638634i	$0.220500\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	6.70820	0.512989
$172$	0	0
$173$	11.3262	0.861118	0.430559	−	0.902562i	$-0.358316\pi$
$173$	11.3262	0.861118	0.430559	+	0.902562i	$0.358316\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.381966	0.0287103
$178$	0	0
$179$	−4.41641	−0.330098	−0.165049	−	0.986285i	$-0.552778\pi$
$179$	−4.41641	−0.330098	−0.165049	+	0.986285i	$0.552778\pi$
$180$	0	0
$181$	13.4164	0.997234	0.498617	−	0.866822i	$-0.333841\pi$
$181$	13.4164	0.997234	0.498617	+	0.866822i	$0.333841\pi$
$182$	0	0
$183$	1.14590	0.0847072
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−16.8541	−1.23249
$188$	0	0
$189$	−2.38197	−0.173263
$190$	0	0
$191$	0.708204	0.0512438	0.0256219	−	0.999672i	$-0.491843\pi$
$191$	0.708204	0.0512438	0.0256219	+	0.999672i	$0.491843\pi$
$192$	0	0
$193$	12.7082	0.914757	0.457378	−	0.889272i	$-0.348789\pi$
$193$	12.7082	0.914757	0.457378	+	0.889272i	$0.348789\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.65248	−0.687710	−0.343855	−	0.939023i	$-0.611733\pi$
$197$	−9.65248	−0.687710	−0.343855	+	0.939023i	$0.611733\pi$
$198$	0	0
$199$	1.94427	0.137826	0.0689129	−	0.997623i	$-0.478047\pi$
$199$	1.94427	0.137826	0.0689129	+	0.997623i	$0.478047\pi$
$200$	0	0
$201$	4.94427	0.348742
$202$	0	0
$203$	19.6180	1.37692
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−5.47214	−0.380340
$208$	0	0
$209$	−23.2918	−1.61113
$210$	0	0
$211$	−23.4164	−1.61205	−0.806026	−	0.591880i	$-0.798386\pi$
$211$	−23.4164	−1.61205	−0.806026	+	0.591880i	$0.798386\pi$
$212$	0	0
$213$	0.326238	0.0223535
$214$	0	0
$215$	0	0
$216$	0	0
$217$	16.3262	1.10830
$218$	0	0
$219$	7.85410	0.530731
$220$	0	0
$221$	−24.2705	−1.63261
$222$	0	0
$223$	20.1246	1.34764	0.673822	−	0.738894i	$-0.264652\pi$
$223$	20.1246	1.34764	0.673822	+	0.738894i	$0.264652\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−26.1803	−1.73765	−0.868825	−	0.495119i	$-0.835124\pi$
$227$	−26.1803	−1.73765	−0.868825	+	0.495119i	$0.835124\pi$
$228$	0	0
$229$	17.7082	1.17019	0.585096	−	0.810964i	$-0.301057\pi$
$229$	17.7082	1.17019	0.585096	+	0.810964i	$0.301057\pi$
$230$	0	0
$231$	8.27051	0.544160
$232$	0	0
$233$	18.6180	1.21971	0.609854	−	0.792514i	$-0.291228\pi$
$233$	18.6180	1.21971	0.609854	+	0.792514i	$0.291228\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	9.70820	0.630616
$238$	0	0
$239$	−8.38197	−0.542184	−0.271092	−	0.962553i	$-0.587385\pi$
$239$	−8.38197	−0.542184	−0.271092	+	0.962553i	$0.587385\pi$
$240$	0	0
$241$	−18.1803	−1.17110	−0.585549	−	0.810637i	$-0.699121\pi$
$241$	−18.1803	−1.17110	−0.585549	+	0.810637i	$0.699121\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−33.5410	−2.13416
$248$	0	0
$249$	−12.3820	−0.784675
$250$	0	0
$251$	4.90983	0.309906	0.154953	−	0.987922i	$-0.450477\pi$
$251$	4.90983	0.309906	0.154953	+	0.987922i	$0.450477\pi$
$252$	0	0
$253$	19.0000	1.19452
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.2361	0.888022	0.444011	−	0.896021i	$-0.353555\pi$
$257$	14.2361	0.888022	0.444011	+	0.896021i	$0.353555\pi$
$258$	0	0
$259$	7.14590	0.444024
$260$	0	0
$261$	−8.23607	−0.509800
$262$	0	0
$263$	−16.8541	−1.03927	−0.519634	−	0.854389i	$-0.673932\pi$
$263$	−16.8541	−1.03927	−0.519634	+	0.854389i	$0.673932\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	9.47214	0.579685
$268$	0	0
$269$	−23.7984	−1.45101	−0.725506	−	0.688216i	$-0.758394\pi$
$269$	−23.7984	−1.45101	−0.725506	+	0.688216i	$0.758394\pi$
$270$	0	0
$271$	−15.2705	−0.927617	−0.463809	−	0.885935i	$-0.653518\pi$
$271$	−15.2705	−0.927617	−0.463809	+	0.885935i	$0.653518\pi$
$272$	0	0
$273$	11.9098	0.720816
$274$	0	0
$275$	0	0
$276$	0	0
$277$	32.7984	1.97066	0.985332	−	0.170650i	$-0.0545869\pi$
$277$	32.7984	1.97066	0.985332	+	0.170650i	$0.0545869\pi$
$278$	0	0
$279$	−6.85410	−0.410344
$280$	0	0
$281$	−28.7984	−1.71797	−0.858983	−	0.512003i	$-0.828904\pi$
$281$	−28.7984	−1.71797	−0.858983	+	0.512003i	$0.828904\pi$
$282$	0	0
$283$	3.18034	0.189052	0.0945258	−	0.995522i	$-0.469867\pi$
$283$	3.18034	0.189052	0.0945258	+	0.995522i	$0.469867\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.6869	−0.748885
$288$	0	0
$289$	6.56231	0.386018
$290$	0	0
$291$	−0.0901699	−0.00528586
$292$	0	0
$293$	−22.4721	−1.31284	−0.656418	−	0.754397i	$-0.727929\pi$
$293$	−22.4721	−1.31284	−0.656418	+	0.754397i	$0.727929\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.47214	−0.201474
$298$	0	0
$299$	27.3607	1.58231
$300$	0	0
$301$	28.2361	1.62750
$302$	0	0
$303$	−6.47214	−0.371814
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−15.3607	−0.876680	−0.438340	−	0.898809i	$-0.644433\pi$
$307$	−15.3607	−0.876680	−0.438340	+	0.898809i	$0.644433\pi$
$308$	0	0
$309$	0.145898	0.00829985
$310$	0	0
$311$	20.5066	1.16282	0.581411	−	0.813610i	$-0.302501\pi$
$311$	20.5066	1.16282	0.581411	+	0.813610i	$0.302501\pi$
$312$	0	0
$313$	19.2148	1.08608	0.543042	−	0.839706i	$-0.317272\pi$
$313$	19.2148	1.08608	0.543042	+	0.839706i	$0.317272\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.4164	1.25903	0.629515	−	0.776988i	$-0.283253\pi$
$317$	22.4164	1.25903	0.629515	+	0.776988i	$0.283253\pi$
$318$	0	0
$319$	28.5967	1.60111
$320$	0	0
$321$	−10.0344	−0.560068
$322$	0	0
$323$	32.5623	1.81182
$324$	0	0
$325$	0	0
$326$	0	0
$327$	16.4164	0.907829
$328$	0	0
$329$	21.4377	1.18190
$330$	0	0
$331$	4.43769	0.243918	0.121959	−	0.992535i	$-0.461082\pi$
$331$	4.43769	0.243918	0.121959	+	0.992535i	$0.461082\pi$
$332$	0	0
$333$	−3.00000	−0.164399
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−20.3607	−1.10912	−0.554558	−	0.832145i	$-0.687113\pi$
$337$	−20.3607	−1.10912	−0.554558	+	0.832145i	$0.687113\pi$
$338$	0	0
$339$	−18.6525	−1.01306
$340$	0	0
$341$	23.7984	1.28875
$342$	0	0
$343$	19.8328	1.07087
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.61803	−0.355275	−0.177637	−	0.984096i	$-0.556845\pi$
$347$	−6.61803	−0.355275	−0.177637	+	0.984096i	$0.556845\pi$
$348$	0	0
$349$	23.5623	1.26126	0.630631	−	0.776083i	$-0.282796\pi$
$349$	23.5623	1.26126	0.630631	+	0.776083i	$0.282796\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	35.6180	1.89576	0.947878	−	0.318632i	$-0.103224\pi$
$353$	35.6180	1.89576	0.947878	+	0.318632i	$0.103224\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−11.5623	−0.611942
$358$	0	0
$359$	−14.9443	−0.788729	−0.394364	−	0.918954i	$-0.629035\pi$
$359$	−14.9443	−0.788729	−0.394364	+	0.918954i	$0.629035\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	0	0
$363$	1.05573	0.0554114
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−3.56231	−0.185951	−0.0929754	−	0.995668i	$-0.529638\pi$
$367$	−3.56231	−0.185951	−0.0929754	+	0.995668i	$0.529638\pi$
$368$	0	0
$369$	5.32624	0.277273
$370$	0	0
$371$	12.1246	0.629478
$372$	0	0
$373$	13.6738	0.708001	0.354000	−	0.935245i	$-0.384821\pi$
$373$	13.6738	0.708001	0.354000	+	0.935245i	$0.384821\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	41.1803	2.12090
$378$	0	0
$379$	−30.0902	−1.54563	−0.772814	−	0.634632i	$-0.781151\pi$
$379$	−30.0902	−1.54563	−0.772814	+	0.634632i	$0.781151\pi$
$380$	0	0
$381$	0.708204	0.0362824
$382$	0	0
$383$	−7.79837	−0.398478	−0.199239	−	0.979951i	$-0.563847\pi$
$383$	−7.79837	−0.398478	−0.199239	+	0.979951i	$0.563847\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−11.8541	−0.602578
$388$	0	0
$389$	31.8541	1.61507	0.807534	−	0.589822i	$-0.200802\pi$
$389$	31.8541	1.61507	0.807534	+	0.589822i	$0.200802\pi$
$390$	0	0
$391$	−26.5623	−1.34331
$392$	0	0
$393$	5.61803	0.283392
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−16.9443	−0.850409	−0.425204	−	0.905097i	$-0.639798\pi$
$397$	−16.9443	−0.850409	−0.425204	+	0.905097i	$0.639798\pi$
$398$	0	0
$399$	−15.9787	−0.799936
$400$	0	0
$401$	−35.1803	−1.75682	−0.878411	−	0.477906i	$-0.841396\pi$
$401$	−35.1803	−1.75682	−0.878411	+	0.477906i	$0.841396\pi$
$402$	0	0
$403$	34.2705	1.70714
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.4164	0.516322
$408$	0	0
$409$	−9.12461	−0.451183	−0.225592	−	0.974222i	$-0.572431\pi$
$409$	−9.12461	−0.451183	−0.225592	+	0.974222i	$0.572431\pi$
$410$	0	0
$411$	−0.236068	−0.0116444
$412$	0	0
$413$	−0.909830	−0.0447698
$414$	0	0
$415$	0	0
$416$	0	0
$417$	15.4721	0.757674
$418$	0	0
$419$	−31.0902	−1.51885	−0.759427	−	0.650592i	$-0.774521\pi$
$419$	−31.0902	−1.51885	−0.759427	+	0.650592i	$0.774521\pi$
$420$	0	0
$421$	27.8885	1.35920	0.679602	−	0.733581i	$-0.262152\pi$
$421$	27.8885	1.35920	0.679602	+	0.733581i	$0.262152\pi$
$422$	0	0
$423$	−9.00000	−0.437595
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.72949	−0.132089
$428$	0	0
$429$	17.3607	0.838182
$430$	0	0
$431$	−10.4164	−0.501741	−0.250870	−	0.968021i	$-0.580717\pi$
$431$	−10.4164	−0.501741	−0.250870	+	0.968021i	$0.580717\pi$
$432$	0	0
$433$	−8.47214	−0.407145	−0.203572	−	0.979060i	$-0.565255\pi$
$433$	−8.47214	−0.407145	−0.203572	+	0.979060i	$0.565255\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−36.7082	−1.75599
$438$	0	0
$439$	−23.3820	−1.11596	−0.557980	−	0.829854i	$-0.688423\pi$
$439$	−23.3820	−1.11596	−0.557980	+	0.829854i	$0.688423\pi$
$440$	0	0
$441$	−1.32624	−0.0631542
$442$	0	0
$443$	7.50658	0.356648	0.178324	−	0.983972i	$-0.442932\pi$
$443$	7.50658	0.356648	0.178324	+	0.983972i	$0.442932\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−11.4721	−0.542613
$448$	0	0
$449$	1.09017	0.0514483	0.0257242	−	0.999669i	$-0.491811\pi$
$449$	1.09017	0.0514483	0.0257242	+	0.999669i	$0.491811\pi$
$450$	0	0
$451$	−18.4934	−0.870821
$452$	0	0
$453$	−1.70820	−0.0802584
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.61803	0.169244	0.0846222	−	0.996413i	$-0.473032\pi$
$457$	3.61803	0.169244	0.0846222	+	0.996413i	$0.473032\pi$
$458$	0	0
$459$	4.85410	0.226570
$460$	0	0
$461$	−9.00000	−0.419172	−0.209586	−	0.977790i	$-0.567212\pi$
$461$	−9.00000	−0.419172	−0.209586	+	0.977790i	$0.567212\pi$
$462$	0	0
$463$	−20.5623	−0.955611	−0.477806	−	0.878466i	$-0.658568\pi$
$463$	−20.5623	−0.955611	−0.477806	+	0.878466i	$0.658568\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.85410	0.363444	0.181722	−	0.983350i	$-0.441833\pi$
$467$	7.85410	0.363444	0.181722	+	0.983350i	$0.441833\pi$
$468$	0	0
$469$	−11.7771	−0.543815
$470$	0	0
$471$	20.5623	0.947461
$472$	0	0
$473$	41.1591	1.89250
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−5.09017	−0.233063
$478$	0	0
$479$	−5.34752	−0.244335	−0.122167	−	0.992510i	$-0.538984\pi$
$479$	−5.34752	−0.244335	−0.122167	+	0.992510i	$0.538984\pi$
$480$	0	0
$481$	15.0000	0.683941
$482$	0	0
$483$	13.0344	0.593088
$484$	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
$489$	−16.7082	−0.755571
$490$	0	0
$491$	42.8673	1.93457	0.967286	−	0.253688i	$-0.0816436\pi$
$491$	42.8673	1.93457	0.967286	+	0.253688i	$0.0816436\pi$
$492$	0	0
$493$	−39.9787	−1.80055
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.777088	−0.0348571
$498$	0	0
$499$	−5.47214	−0.244966	−0.122483	−	0.992471i	$-0.539086\pi$
$499$	−5.47214	−0.244966	−0.122483	+	0.992471i	$0.539086\pi$
$500$	0	0
$501$	19.8885	0.888555
$502$	0	0
$503$	−2.61803	−0.116732	−0.0583662	−	0.998295i	$-0.518589\pi$
$503$	−2.61803	−0.116732	−0.0583662	+	0.998295i	$0.518589\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.0000	0.532939
$508$	0	0
$509$	−0.763932	−0.0338607	−0.0169303	−	0.999857i	$-0.505389\pi$
$509$	−0.763932	−0.0338607	−0.0169303	+	0.999857i	$0.505389\pi$
$510$	0	0
$511$	−18.7082	−0.827602
$512$	0	0
$513$	6.70820	0.296174
$514$	0	0
$515$	0	0
$516$	0	0
$517$	31.2492	1.37434
$518$	0	0
$519$	11.3262	0.497167
$520$	0	0
$521$	38.5066	1.68700	0.843502	−	0.537126i	$-0.180490\pi$
$521$	38.5066	1.68700	0.843502	+	0.537126i	$0.180490\pi$
$522$	0	0
$523$	−16.9443	−0.740921	−0.370461	−	0.928848i	$-0.620800\pi$
$523$	−16.9443	−0.740921	−0.370461	+	0.928848i	$0.620800\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−33.2705	−1.44929
$528$	0	0
$529$	6.94427	0.301925
$530$	0	0
$531$	0.381966	0.0165759
$532$	0	0
$533$	−26.6312	−1.15352
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4.41641	−0.190582
$538$	0	0
$539$	4.60488	0.198346
$540$	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
$543$	13.4164	0.575753
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.85410	0.378574	0.189287	−	0.981922i	$-0.439382\pi$
$547$	8.85410	0.378574	0.189287	+	0.981922i	$0.439382\pi$
$548$	0	0
$549$	1.14590	0.0489057
$550$	0	0
$551$	−55.2492	−2.35370
$552$	0	0
$553$	−23.1246	−0.983359
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.09017	0.0461920	0.0230960	−	0.999733i	$-0.492648\pi$
$557$	1.09017	0.0461920	0.0230960	+	0.999733i	$0.492648\pi$
$558$	0	0
$559$	59.2705	2.50688
$560$	0	0
$561$	−16.8541	−0.711581
$562$	0	0
$563$	−29.6738	−1.25060	−0.625300	−	0.780384i	$-0.715023\pi$
$563$	−29.6738	−1.25060	−0.625300	+	0.780384i	$0.715023\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.38197	−0.100033
$568$	0	0
$569$	11.6525	0.488497	0.244249	−	0.969713i	$-0.421459\pi$
$569$	11.6525	0.488497	0.244249	+	0.969713i	$0.421459\pi$
$570$	0	0
$571$	8.21478	0.343778	0.171889	−	0.985116i	$-0.445013\pi$
$571$	8.21478	0.343778	0.171889	+	0.985116i	$0.445013\pi$
$572$	0	0
$573$	0.708204	0.0295856
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−21.4721	−0.893897	−0.446948	−	0.894560i	$-0.647489\pi$
$577$	−21.4721	−0.893897	−0.446948	+	0.894560i	$0.647489\pi$
$578$	0	0
$579$	12.7082	0.528135
$580$	0	0
$581$	29.4934	1.22359
$582$	0	0
$583$	17.6738	0.731972
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.65248	0.398400	0.199200	−	0.979959i	$-0.436166\pi$
$587$	9.65248	0.398400	0.199200	+	0.979959i	$0.436166\pi$
$588$	0	0
$589$	−45.9787	−1.89452
$590$	0	0
$591$	−9.65248	−0.397050
$592$	0	0
$593$	−8.88854	−0.365009	−0.182504	−	0.983205i	$-0.558420\pi$
$593$	−8.88854	−0.365009	−0.182504	+	0.983205i	$0.558420\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.94427	0.0795738
$598$	0	0
$599$	0.673762	0.0275292	0.0137646	−	0.999905i	$-0.495618\pi$
$599$	0.673762	0.0275292	0.0137646	+	0.999905i	$0.495618\pi$
$600$	0	0
$601$	29.8328	1.21691	0.608453	−	0.793590i	$-0.291790\pi$
$601$	29.8328	1.21691	0.608453	+	0.793590i	$0.291790\pi$
$602$	0	0
$603$	4.94427	0.201346
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−11.4164	−0.463378	−0.231689	−	0.972790i	$-0.574425\pi$
$607$	−11.4164	−0.463378	−0.231689	+	0.972790i	$0.574425\pi$
$608$	0	0
$609$	19.6180	0.794963
$610$	0	0
$611$	45.0000	1.82051
$612$	0	0
$613$	−36.0132	−1.45456	−0.727279	−	0.686342i	$-0.759215\pi$
$613$	−36.0132	−1.45456	−0.727279	+	0.686342i	$0.759215\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−35.1803	−1.41631	−0.708154	−	0.706058i	$-0.750472\pi$
$617$	−35.1803	−1.41631	−0.708154	+	0.706058i	$0.750472\pi$
$618$	0	0
$619$	−12.5836	−0.505777	−0.252889	−	0.967495i	$-0.581381\pi$
$619$	−12.5836	−0.505777	−0.252889	+	0.967495i	$0.581381\pi$
$620$	0	0
$621$	−5.47214	−0.219589
$622$	0	0
$623$	−22.5623	−0.903940
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−23.2918	−0.930185
$628$	0	0
$629$	−14.5623	−0.580637
$630$	0	0
$631$	−10.8885	−0.433466	−0.216733	−	0.976231i	$-0.569540\pi$
$631$	−10.8885	−0.433466	−0.216733	+	0.976231i	$0.569540\pi$
$632$	0	0
$633$	−23.4164	−0.930719
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.63119	0.262737
$638$	0	0
$639$	0.326238	0.0129058
$640$	0	0
$641$	−36.6525	−1.44769	−0.723843	−	0.689965i	$-0.757626\pi$
$641$	−36.6525	−1.44769	−0.723843	+	0.689965i	$0.757626\pi$
$642$	0	0
$643$	29.4164	1.16007	0.580035	−	0.814592i	$-0.303039\pi$
$643$	29.4164	1.16007	0.580035	+	0.814592i	$0.303039\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−14.3820	−0.565413	−0.282707	−	0.959206i	$-0.591232\pi$
$647$	−14.3820	−0.565413	−0.282707	+	0.959206i	$0.591232\pi$
$648$	0	0
$649$	−1.32624	−0.0520594
$650$	0	0
$651$	16.3262	0.639876
$652$	0	0
$653$	−15.3475	−0.600595	−0.300298	−	0.953846i	$-0.597086\pi$
$653$	−15.3475	−0.600595	−0.300298	+	0.953846i	$0.597086\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	7.85410	0.306418
$658$	0	0
$659$	−13.5836	−0.529142	−0.264571	−	0.964366i	$-0.585230\pi$
$659$	−13.5836	−0.529142	−0.264571	+	0.964366i	$0.585230\pi$
$660$	0	0
$661$	−11.1459	−0.433525	−0.216763	−	0.976224i	$-0.569550\pi$
$661$	−11.1459	−0.433525	−0.216763	+	0.976224i	$0.569550\pi$
$662$	0	0
$663$	−24.2705	−0.942588
$664$	0	0
$665$	0	0
$666$	0	0
$667$	45.0689	1.74507
$668$	0	0
$669$	20.1246	0.778062
$670$	0	0
$671$	−3.97871	−0.153597
$672$	0	0
$673$	41.3050	1.59219	0.796094	−	0.605172i	$-0.206896\pi$
$673$	41.3050	1.59219	0.796094	+	0.605172i	$0.206896\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	41.4853	1.59441	0.797205	−	0.603709i	$-0.206311\pi$
$677$	41.4853	1.59441	0.797205	+	0.603709i	$0.206311\pi$
$678$	0	0
$679$	0.214782	0.00824257
$680$	0	0
$681$	−26.1803	−1.00323
$682$	0	0
$683$	−29.6525	−1.13462	−0.567310	−	0.823504i	$-0.692016\pi$
$683$	−29.6525	−1.13462	−0.567310	+	0.823504i	$0.692016\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	17.7082	0.675610
$688$	0	0
$689$	25.4508	0.969600
$690$	0	0
$691$	−15.9443	−0.606549	−0.303274	−	0.952903i	$-0.598080\pi$
$691$	−15.9443	−0.606549	−0.303274	+	0.952903i	$0.598080\pi$
$692$	0	0
$693$	8.27051	0.314171
$694$	0	0
$695$	0	0
$696$	0	0
$697$	25.8541	0.979294
$698$	0	0
$699$	18.6180	0.704199
$700$	0	0
$701$	−33.5967	−1.26893	−0.634466	−	0.772951i	$-0.718780\pi$
$701$	−33.5967	−1.26893	−0.634466	+	0.772951i	$0.718780\pi$
$702$	0	0
$703$	−20.1246	−0.759014
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.4164	0.579794
$708$	0	0
$709$	−26.1246	−0.981130	−0.490565	−	0.871404i	$-0.663210\pi$
$709$	−26.1246	−0.981130	−0.490565	+	0.871404i	$0.663210\pi$
$710$	0	0
$711$	9.70820	0.364086
$712$	0	0
$713$	37.5066	1.40463
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.38197	−0.313030
$718$	0	0
$719$	12.3475	0.460485	0.230242	−	0.973133i	$-0.426048\pi$
$719$	12.3475	0.460485	0.230242	+	0.973133i	$0.426048\pi$
$720$	0	0
$721$	−0.347524	−0.0129425
$722$	0	0
$723$	−18.1803	−0.676134
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0.236068	0.00875528	0.00437764	−	0.999990i	$-0.498607\pi$
$727$	0.236068	0.00875528	0.00437764	+	0.999990i	$0.498607\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−57.5410	−2.12823
$732$	0	0
$733$	−36.0000	−1.32969	−0.664845	−	0.746981i	$-0.731502\pi$
$733$	−36.0000	−1.32969	−0.664845	+	0.746981i	$0.731502\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−17.1672	−0.632361
$738$	0	0
$739$	−47.0344	−1.73019	−0.865095	−	0.501608i	$-0.832742\pi$
$739$	−47.0344	−1.73019	−0.865095	+	0.501608i	$0.832742\pi$
$740$	0	0
$741$	−33.5410	−1.23216
$742$	0	0
$743$	−4.65248	−0.170683	−0.0853414	−	0.996352i	$-0.527198\pi$
$743$	−4.65248	−0.170683	−0.0853414	+	0.996352i	$0.527198\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−12.3820	−0.453032
$748$	0	0
$749$	23.9017	0.873349
$750$	0	0
$751$	36.2705	1.32353	0.661765	−	0.749711i	$-0.269808\pi$
$751$	36.2705	1.32353	0.661765	+	0.749711i	$0.269808\pi$
$752$	0	0
$753$	4.90983	0.178924
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−7.29180	−0.265025	−0.132512	−	0.991181i	$-0.542304\pi$
$757$	−7.29180	−0.265025	−0.132512	+	0.991181i	$0.542304\pi$
$758$	0	0
$759$	19.0000	0.689656
$760$	0	0
$761$	−10.2361	−0.371057	−0.185529	−	0.982639i	$-0.559400\pi$
$761$	−10.2361	−0.371057	−0.185529	+	0.982639i	$0.559400\pi$
$762$	0	0
$763$	−39.1033	−1.41564
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.90983	−0.0689600
$768$	0	0
$769$	−10.1115	−0.364628	−0.182314	−	0.983240i	$-0.558359\pi$
$769$	−10.1115	−0.364628	−0.182314	+	0.983240i	$0.558359\pi$
$770$	0	0
$771$	14.2361	0.512699
$772$	0	0
$773$	−45.9443	−1.65250	−0.826250	−	0.563303i	$-0.809530\pi$
$773$	−45.9443	−1.65250	−0.826250	+	0.563303i	$0.809530\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	7.14590	0.256358
$778$	0	0
$779$	35.7295	1.28014
$780$	0	0
$781$	−1.13274	−0.0405327
$782$	0	0
$783$	−8.23607	−0.294333
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−26.3951	−0.940884	−0.470442	−	0.882431i	$-0.655906\pi$
$787$	−26.3951	−0.940884	−0.470442	+	0.882431i	$0.655906\pi$
$788$	0	0
$789$	−16.8541	−0.600022
$790$	0	0
$791$	44.4296	1.57973
$792$	0	0
$793$	−5.72949	−0.203460
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−4.03444	−0.142907	−0.0714536	−	0.997444i	$-0.522764\pi$
$797$	−4.03444	−0.142907	−0.0714536	+	0.997444i	$0.522764\pi$
$798$	0	0
$799$	−43.6869	−1.54553
$800$	0	0
$801$	9.47214	0.334681
$802$	0	0
$803$	−27.2705	−0.962355
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−23.7984	−0.837742
$808$	0	0
$809$	−23.7984	−0.836706	−0.418353	−	0.908284i	$-0.637393\pi$
$809$	−23.7984	−0.836706	−0.418353	+	0.908284i	$0.637393\pi$
$810$	0	0
$811$	−35.8541	−1.25901	−0.629504	−	0.776997i	$-0.716742\pi$
$811$	−35.8541	−1.25901	−0.629504	+	0.776997i	$0.716742\pi$
$812$	0	0
$813$	−15.2705	−0.535560
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−79.5197	−2.78204
$818$	0	0
$819$	11.9098	0.416163
$820$	0	0
$821$	−51.7639	−1.80657	−0.903287	−	0.429037i	$-0.858853\pi$
$821$	−51.7639	−1.80657	−0.903287	+	0.429037i	$0.858853\pi$
$822$	0	0
$823$	−24.8328	−0.865618	−0.432809	−	0.901486i	$-0.642477\pi$
$823$	−24.8328	−0.865618	−0.432809	+	0.901486i	$0.642477\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33.9787	1.18156	0.590778	−	0.806834i	$-0.298821\pi$
$827$	33.9787	1.18156	0.590778	+	0.806834i	$0.298821\pi$
$828$	0	0
$829$	1.34752	0.0468014	0.0234007	−	0.999726i	$-0.492551\pi$
$829$	1.34752	0.0468014	0.0234007	+	0.999726i	$0.492551\pi$
$830$	0	0
$831$	32.7984	1.13776
$832$	0	0
$833$	−6.43769	−0.223053
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−6.85410	−0.236912
$838$	0	0
$839$	44.1246	1.52335	0.761675	−	0.647959i	$-0.224377\pi$
$839$	44.1246	1.52335	0.761675	+	0.647959i	$0.224377\pi$
$840$	0	0
$841$	38.8328	1.33906
$842$	0	0
$843$	−28.7984	−0.991869
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.51471	−0.0864064
$848$	0	0
$849$	3.18034	0.109149
$850$	0	0
$851$	16.4164	0.562747
$852$	0	0
$853$	−24.2361	−0.829827	−0.414914	−	0.909861i	$-0.636188\pi$
$853$	−24.2361	−0.829827	−0.414914	+	0.909861i	$0.636188\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.90983	0.167717	0.0838583	−	0.996478i	$-0.473276\pi$
$857$	4.90983	0.167717	0.0838583	+	0.996478i	$0.473276\pi$
$858$	0	0
$859$	−58.4164	−1.99314	−0.996571	−	0.0827413i	$-0.973632\pi$
$859$	−58.4164	−1.99314	−0.996571	+	0.0827413i	$0.973632\pi$
$860$	0	0
$861$	−12.6869	−0.432369
$862$	0	0
$863$	13.8541	0.471599	0.235800	−	0.971802i	$-0.424229\pi$
$863$	13.8541	0.471599	0.235800	+	0.971802i	$0.424229\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.56231	0.222868
$868$	0	0
$869$	−33.7082	−1.14347
$870$	0	0
$871$	−24.7214	−0.837651
$872$	0	0
$873$	−0.0901699	−0.00305179
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−9.00000	−0.303908	−0.151954	−	0.988388i	$-0.548557\pi$
$877$	−9.00000	−0.303908	−0.151954	+	0.988388i	$0.548557\pi$
$878$	0	0
$879$	−22.4721	−0.757966
$880$	0	0
$881$	40.6525	1.36962	0.684808	−	0.728723i	$-0.259886\pi$
$881$	40.6525	1.36962	0.684808	+	0.728723i	$0.259886\pi$
$882$	0	0
$883$	18.1246	0.609942	0.304971	−	0.952362i	$-0.401353\pi$
$883$	18.1246	0.609942	0.304971	+	0.952362i	$0.401353\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.1803	0.845473	0.422737	−	0.906253i	$-0.361070\pi$
$887$	25.1803	0.845473	0.422737	+	0.906253i	$0.361070\pi$
$888$	0	0
$889$	−1.68692	−0.0565774
$890$	0	0
$891$	−3.47214	−0.116321
$892$	0	0
$893$	−60.3738	−2.02033
$894$	0	0
$895$	0	0
$896$	0	0
$897$	27.3607	0.913547
$898$	0	0
$899$	56.4508	1.88274
$900$	0	0
$901$	−24.7082	−0.823150
$902$	0	0
$903$	28.2361	0.939638
$904$	0	0
$905$	0	0
$906$	0	0
$907$	20.4164	0.677916	0.338958	−	0.940802i	$-0.389926\pi$
$907$	20.4164	0.677916	0.338958	+	0.940802i	$0.389926\pi$
$908$	0	0
$909$	−6.47214	−0.214667
$910$	0	0
$911$	51.2148	1.69682	0.848411	−	0.529339i	$-0.177560\pi$
$911$	51.2148	1.69682	0.848411	+	0.529339i	$0.177560\pi$
$912$	0	0
$913$	42.9919	1.42282
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−13.3820	−0.441911
$918$	0	0
$919$	1.49342	0.0492635	0.0246317	−	0.999697i	$-0.492159\pi$
$919$	1.49342	0.0492635	0.0246317	+	0.999697i	$0.492159\pi$
$920$	0	0
$921$	−15.3607	−0.506152
$922$	0	0
$923$	−1.63119	−0.0536913
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.145898	0.00479192
$928$	0	0
$929$	−27.9787	−0.917952	−0.458976	−	0.888449i	$-0.651784\pi$
$929$	−27.9787	−0.917952	−0.458976	+	0.888449i	$0.651784\pi$
$930$	0	0
$931$	−8.89667	−0.291577
$932$	0	0
$933$	20.5066	0.671355
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0.291796	0.00953256	0.00476628	−	0.999989i	$-0.498483\pi$
$937$	0.291796	0.00953256	0.00476628	+	0.999989i	$0.498483\pi$
$938$	0	0
$939$	19.2148	0.627051
$940$	0	0
$941$	43.2148	1.40876	0.704381	−	0.709822i	$-0.251225\pi$
$941$	43.2148	1.40876	0.704381	+	0.709822i	$0.251225\pi$
$942$	0	0
$943$	−29.1459	−0.949121
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.34752	−0.0762843	−0.0381421	−	0.999272i	$-0.512144\pi$
$947$	−2.34752	−0.0762843	−0.0381421	+	0.999272i	$0.512144\pi$
$948$	0	0
$949$	−39.2705	−1.27477
$950$	0	0
$951$	22.4164	0.726902
$952$	0	0
$953$	−46.0344	−1.49120	−0.745601	−	0.666393i	$-0.767837\pi$
$953$	−46.0344	−1.49120	−0.745601	+	0.666393i	$0.767837\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	28.5967	0.924402
$958$	0	0
$959$	0.562306	0.0181578
$960$	0	0
$961$	15.9787	0.515442
$962$	0	0
$963$	−10.0344	−0.323355
$964$	0	0
$965$	0	0
$966$	0	0
$967$	61.4296	1.97544	0.987721	−	0.156229i	$-0.0499339\pi$
$967$	61.4296	1.97544	0.987721	+	0.156229i	$0.0499339\pi$
$968$	0	0
$969$	32.5623	1.04605
$970$	0	0
$971$	−8.18034	−0.262520	−0.131260	−	0.991348i	$-0.541902\pi$
$971$	−8.18034	−0.262520	−0.131260	+	0.991348i	$0.541902\pi$
$972$	0	0
$973$	−36.8541	−1.18149
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.54915	0.0815545	0.0407773	−	0.999168i	$-0.487017\pi$
$977$	2.54915	0.0815545	0.0407773	+	0.999168i	$0.487017\pi$
$978$	0	0
$979$	−32.8885	−1.05112
$980$	0	0
$981$	16.4164	0.524136
$982$	0	0
$983$	48.1033	1.53426	0.767129	−	0.641493i	$-0.221685\pi$
$983$	48.1033	1.53426	0.767129	+	0.641493i	$0.221685\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	21.4377	0.682369
$988$	0	0
$989$	64.8673	2.06266
$990$	0	0
$991$	−44.6869	−1.41953	−0.709763	−	0.704440i	$-0.751198\pi$
$991$	−44.6869	−1.41953	−0.709763	+	0.704440i	$0.751198\pi$
$992$	0	0
$993$	4.43769	0.140826
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−29.0000	−0.918439	−0.459220	−	0.888323i	$-0.651871\pi$
$997$	−29.0000	−0.918439	−0.459220	+	0.888323i	$0.651871\pi$
$998$	0	0
$999$	−3.00000	−0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1500.2.a.e.1.2	yes	2
3.2	odd	2		4500.2.a.a.1.2		2
4.3	odd	2		6000.2.a.n.1.1		2
5.2	odd	4		1500.2.d.c.1249.2		4
5.3	odd	4		1500.2.d.c.1249.3		4
5.4	even	2		1500.2.a.d.1.1	✓	2
15.2	even	4		4500.2.d.b.4249.2		4
15.8	even	4		4500.2.d.b.4249.3		4
15.14	odd	2		4500.2.a.n.1.1		2
20.3	even	4		6000.2.f.d.1249.2		4
20.7	even	4		6000.2.f.d.1249.3		4
20.19	odd	2		6000.2.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1500.2.a.d.1.1	✓	2	5.4	even	2
1500.2.a.e.1.2	yes	2	1.1	even	1	trivial
1500.2.d.c.1249.2		4	5.2	odd	4
1500.2.d.c.1249.3		4	5.3	odd	4
4500.2.a.a.1.2		2	3.2	odd	2
4500.2.a.n.1.1		2	15.14	odd	2
4500.2.d.b.4249.2		4	15.2	even	4
4500.2.d.b.4249.3		4	15.8	even	4
6000.2.a.n.1.1		2	4.3	odd	2
6000.2.a.o.1.2		2	20.19	odd	2
6000.2.f.d.1249.2		4	20.3	even	4
6000.2.f.d.1249.3		4	20.7	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:	\( N \)	\(=\)	\( 1500 = 2^{2} \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1500.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.38197 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.38197 q^{7} +1.00000 q^{9} -3.47214 q^{11} -5.00000 q^{13} +4.85410 q^{17} +6.70820 q^{19} -2.38197 q^{21} -5.47214 q^{23} +1.00000 q^{27} -8.23607 q^{29} -6.85410 q^{31} -3.47214 q^{33} -3.00000 q^{37} -5.00000 q^{39} +5.32624 q^{41} -11.8541 q^{43} -9.00000 q^{47} -1.32624 q^{49} +4.85410 q^{51} -5.09017 q^{53} +6.70820 q^{57} +0.381966 q^{59} +1.14590 q^{61} -2.38197 q^{63} +4.94427 q^{67} -5.47214 q^{69} +0.326238 q^{71} +7.85410 q^{73} +8.27051 q^{77} +9.70820 q^{79} +1.00000 q^{81} -12.3820 q^{83} -8.23607 q^{87} +9.47214 q^{89} +11.9098 q^{91} -6.85410 q^{93} -0.0901699 q^{97} -3.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 7 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 7 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 7 q^{7} + 2 q^{9} + 2 q^{11} - 10 q^{13} + 3 q^{17} - 7 q^{21} - 2 q^{23} + 2 q^{27} - 12 q^{29} - 7 q^{31} + 2 q^{33} - 6 q^{37} - 10 q^{39} - 5 q^{41} - 17 q^{43} - 18 q^{47} + 13 q^{49} + 3 q^{51} + q^{53} + 3 q^{59} + 9 q^{61} - 7 q^{63} - 8 q^{67} - 2 q^{69} - 15 q^{71} + 9 q^{73} - 17 q^{77} + 6 q^{79} + 2 q^{81} - 27 q^{83} - 12 q^{87} + 10 q^{89} + 35 q^{91} - 7 q^{93} + 11 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 7 * q^7 + 2 * q^9 + 2 * q^11 - 10 * q^13 + 3 * q^17 - 7 * q^21 - 2 * q^23 + 2 * q^27 - 12 * q^29 - 7 * q^31 + 2 * q^33 - 6 * q^37 - 10 * q^39 - 5 * q^41 - 17 * q^43 - 18 * q^47 + 13 * q^49 + 3 * q^51 + q^53 + 3 * q^59 + 9 * q^61 - 7 * q^63 - 8 * q^67 - 2 * q^69 - 15 * q^71 + 9 * q^73 - 17 * q^77 + 6 * q^79 + 2 * q^81 - 27 * q^83 - 12 * q^87 + 10 * q^89 + 35 * q^91 - 7 * q^93 + 11 * q^97 + 2 * q^99

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