LMFDB - Embedded newform 1500.2.a.d.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:	$0$
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:	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)

$f(q)$	$=$	$q-1.00000 q^{3} +4.61803 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.61803 q^{7} +1.00000 q^{9} +5.47214 q^{11} +5.00000 q^{13} +1.85410 q^{17} -6.70820 q^{19} -4.61803 q^{21} -3.47214 q^{23} -1.00000 q^{27} -3.76393 q^{29} -0.145898 q^{31} -5.47214 q^{33} +3.00000 q^{37} -5.00000 q^{39} -10.3262 q^{41} +5.14590 q^{43} +9.00000 q^{47} +14.3262 q^{49} -1.85410 q^{51} -6.09017 q^{53} +6.70820 q^{57} +2.61803 q^{59} +7.85410 q^{61} +4.61803 q^{63} +12.9443 q^{67} +3.47214 q^{69} -15.3262 q^{71} -1.14590 q^{73} +25.2705 q^{77} -3.70820 q^{79} +1.00000 q^{81} +14.6180 q^{83} +3.76393 q^{87} +0.527864 q^{89} +23.0902 q^{91} +0.145898 q^{93} -11.0902 q^{97} +5.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	4.61803	1.74545	0.872726	−	0.488210i	$-0.162350\pi$
$7$	4.61803	1.74545	0.872726	+	0.488210i	$0.162350\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.47214	1.64991	0.824956	−	0.565198i	$-0.191200\pi$
$11$	5.47214	1.64991	0.824956	+	0.565198i	$0.191200\pi$
$12$	0	0
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.85410	0.449686	0.224843	−	0.974395i	$-0.427813\pi$
$17$	1.85410	0.449686	0.224843	+	0.974395i	$0.427813\pi$
$18$	0	0
$19$	−6.70820	−1.53897	−0.769484	−	0.638666i	$-0.779486\pi$
$19$	−6.70820	−1.53897	−0.769484	+	0.638666i	$0.779486\pi$
$20$	0	0
$21$	−4.61803	−1.00774
$22$	0	0
$23$	−3.47214	−0.723990	−0.361995	−	0.932180i	$-0.617904\pi$
$23$	−3.47214	−0.723990	−0.361995	+	0.932180i	$0.617904\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.76393	−0.698945	−0.349472	−	0.936947i	$-0.613639\pi$
$29$	−3.76393	−0.698945	−0.349472	+	0.936947i	$0.613639\pi$
$30$	0	0
$31$	−0.145898	−0.0262041	−0.0131020	−	0.999914i	$-0.504171\pi$
$31$	−0.145898	−0.0262041	−0.0131020	+	0.999914i	$0.504171\pi$
$32$	0	0
$33$	−5.47214	−0.952577
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	0	0
$39$	−5.00000	−0.800641
$40$	0	0
$41$	−10.3262	−1.61269	−0.806344	−	0.591447i	$-0.798557\pi$
$41$	−10.3262	−1.61269	−0.806344	+	0.591447i	$0.798557\pi$
$42$	0	0
$43$	5.14590	0.784742	0.392371	−	0.919807i	$-0.371655\pi$
$43$	5.14590	0.784742	0.392371	+	0.919807i	$0.371655\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	0	0
$49$	14.3262	2.04661
$50$	0	0
$51$	−1.85410	−0.259626
$52$	0	0
$53$	−6.09017	−0.836549	−0.418275	−	0.908321i	$-0.637365\pi$
$53$	−6.09017	−0.836549	−0.418275	+	0.908321i	$0.637365\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.70820	0.888523
$58$	0	0
$59$	2.61803	0.340839	0.170419	−	0.985372i	$-0.445488\pi$
$59$	2.61803	0.340839	0.170419	+	0.985372i	$0.445488\pi$
$60$	0	0
$61$	7.85410	1.00561	0.502807	−	0.864398i	$-0.332301\pi$
$61$	7.85410	1.00561	0.502807	+	0.864398i	$0.332301\pi$
$62$	0	0
$63$	4.61803	0.581818
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.9443	1.58139	0.790697	−	0.612207i	$-0.209718\pi$
$67$	12.9443	1.58139	0.790697	+	0.612207i	$0.209718\pi$
$68$	0	0
$69$	3.47214	0.417996
$70$	0	0
$71$	−15.3262	−1.81889	−0.909445	−	0.415824i	$-0.863493\pi$
$71$	−15.3262	−1.81889	−0.909445	+	0.415824i	$0.863493\pi$
$72$	0	0
$73$	−1.14590	−0.134117	−0.0670586	−	0.997749i	$-0.521361\pi$
$73$	−1.14590	−0.134117	−0.0670586	+	0.997749i	$0.521361\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	25.2705	2.87984
$78$	0	0
$79$	−3.70820	−0.417206	−0.208603	−	0.978000i	$-0.566892\pi$
$79$	−3.70820	−0.417206	−0.208603	+	0.978000i	$0.566892\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	14.6180	1.60454	0.802269	−	0.596963i	$-0.203626\pi$
$83$	14.6180	1.60454	0.802269	+	0.596963i	$0.203626\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.76393	0.403536
$88$	0	0
$89$	0.527864	0.0559535	0.0279767	−	0.999609i	$-0.491094\pi$
$89$	0.527864	0.0559535	0.0279767	+	0.999609i	$0.491094\pi$
$90$	0	0
$91$	23.0902	2.42051
$92$	0	0
$93$	0.145898	0.0151289
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−11.0902	−1.12604	−0.563018	−	0.826445i	$-0.690360\pi$
$97$	−11.0902	−1.12604	−0.563018	+	0.826445i	$0.690360\pi$
$98$	0	0
$99$	5.47214	0.549970
$100$	0	0
$101$	2.47214	0.245987	0.122993	−	0.992407i	$-0.460751\pi$
$101$	2.47214	0.245987	0.122993	+	0.992407i	$0.460751\pi$
$102$	0	0
$103$	−6.85410	−0.675355	−0.337677	−	0.941262i	$-0.609641\pi$
$103$	−6.85410	−0.675355	−0.337677	+	0.941262i	$0.609641\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−19.0344	−1.84013	−0.920064	−	0.391767i	$-0.871864\pi$
$107$	−19.0344	−1.84013	−0.920064	+	0.391767i	$0.871864\pi$
$108$	0	0
$109$	−10.4164	−0.997711	−0.498855	−	0.866685i	$-0.666246\pi$
$109$	−10.4164	−0.997711	−0.498855	+	0.866685i	$0.666246\pi$
$110$	0	0
$111$	−3.00000	−0.284747
$112$	0	0
$113$	−12.6525	−1.19024	−0.595122	−	0.803635i	$-0.702896\pi$
$113$	−12.6525	−1.19024	−0.595122	+	0.803635i	$0.702896\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.00000	0.462250
$118$	0	0
$119$	8.56231	0.784905
$120$	0	0
$121$	18.9443	1.72221
$122$	0	0
$123$	10.3262	0.931086
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.7082	1.12767	0.563835	−	0.825887i	$-0.309325\pi$
$127$	12.7082	1.12767	0.563835	+	0.825887i	$0.309325\pi$
$128$	0	0
$129$	−5.14590	−0.453071
$130$	0	0
$131$	3.38197	0.295484	0.147742	−	0.989026i	$-0.452800\pi$
$131$	3.38197	0.295484	0.147742	+	0.989026i	$0.452800\pi$
$132$	0	0
$133$	−30.9787	−2.68620
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.23607	−0.361912	−0.180956	−	0.983491i	$-0.557919\pi$
$137$	−4.23607	−0.361912	−0.180956	+	0.983491i	$0.557919\pi$
$138$	0	0
$139$	6.52786	0.553686	0.276843	−	0.960915i	$-0.410712\pi$
$139$	6.52786	0.553686	0.276843	+	0.960915i	$0.410712\pi$
$140$	0	0
$141$	−9.00000	−0.757937
$142$	0	0
$143$	27.3607	2.28801
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−14.3262	−1.18161
$148$	0	0
$149$	−2.52786	−0.207091	−0.103545	−	0.994625i	$-0.533019\pi$
$149$	−2.52786	−0.207091	−0.103545	+	0.994625i	$0.533019\pi$
$150$	0	0
$151$	11.7082	0.952800	0.476400	−	0.879229i	$-0.341941\pi$
$151$	11.7082	0.952800	0.476400	+	0.879229i	$0.341941\pi$
$152$	0	0
$153$	1.85410	0.149895
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−0.437694	−0.0349318	−0.0174659	−	0.999847i	$-0.505560\pi$
$157$	−0.437694	−0.0349318	−0.0174659	+	0.999847i	$0.505560\pi$
$158$	0	0
$159$	6.09017	0.482982
$160$	0	0
$161$	−16.0344	−1.26369
$162$	0	0
$163$	3.29180	0.257833	0.128917	−	0.991655i	$-0.458850\pi$
$163$	3.29180	0.257833	0.128917	+	0.991655i	$0.458850\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.8885	1.22949	0.614746	−	0.788725i	$-0.289258\pi$
$167$	15.8885	1.22949	0.614746	+	0.788725i	$0.289258\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	−6.70820	−0.512989
$172$	0	0
$173$	4.32624	0.328918	0.164459	−	0.986384i	$-0.447412\pi$
$173$	4.32624	0.328918	0.164459	+	0.986384i	$0.447412\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.61803	−0.196783
$178$	0	0
$179$	22.4164	1.67548	0.837740	−	0.546069i	$-0.183876\pi$
$179$	22.4164	1.67548	0.837740	+	0.546069i	$0.183876\pi$
$180$	0	0
$181$	−13.4164	−0.997234	−0.498617	−	0.866822i	$-0.666159\pi$
$181$	−13.4164	−0.997234	−0.498617	+	0.866822i	$0.666159\pi$
$182$	0	0
$183$	−7.85410	−0.580592
$184$	0	0
$185$	0	0
$186$	0	0
$187$	10.1459	0.741942
$188$	0	0
$189$	−4.61803	−0.335913
$190$	0	0
$191$	−12.7082	−0.919533	−0.459767	−	0.888040i	$-0.652067\pi$
$191$	−12.7082	−0.919533	−0.459767	+	0.888040i	$0.652067\pi$
$192$	0	0
$193$	0.708204	0.0509776	0.0254888	−	0.999675i	$-0.491886\pi$
$193$	0.708204	0.0509776	0.0254888	+	0.999675i	$0.491886\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.6525	−1.54268	−0.771338	−	0.636426i	$-0.780412\pi$
$197$	−21.6525	−1.54268	−0.771338	+	0.636426i	$0.780412\pi$
$198$	0	0
$199$	−15.9443	−1.13026	−0.565130	−	0.825002i	$-0.691174\pi$
$199$	−15.9443	−1.13026	−0.565130	+	0.825002i	$0.691174\pi$
$200$	0	0
$201$	−12.9443	−0.913019
$202$	0	0
$203$	−17.3820	−1.21997
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3.47214	−0.241330
$208$	0	0
$209$	−36.7082	−2.53916
$210$	0	0
$211$	3.41641	0.235195	0.117598	−	0.993061i	$-0.462481\pi$
$211$	3.41641	0.235195	0.117598	+	0.993061i	$0.462481\pi$
$212$	0	0
$213$	15.3262	1.05014
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−0.673762	−0.0457380
$218$	0	0
$219$	1.14590	0.0774326
$220$	0	0
$221$	9.27051	0.623602
$222$	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
$227$	3.81966	0.253520	0.126760	−	0.991933i	$-0.459542\pi$
$227$	3.81966	0.253520	0.126760	+	0.991933i	$0.459542\pi$
$228$	0	0
$229$	4.29180	0.283610	0.141805	−	0.989895i	$-0.454709\pi$
$229$	4.29180	0.283610	0.141805	+	0.989895i	$0.454709\pi$
$230$	0	0
$231$	−25.2705	−1.66268
$232$	0	0
$233$	−16.3820	−1.07322	−0.536609	−	0.843831i	$-0.680295\pi$
$233$	−16.3820	−1.07322	−0.536609	+	0.843831i	$0.680295\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3.70820	0.240874
$238$	0	0
$239$	−10.6180	−0.686824	−0.343412	−	0.939185i	$-0.611583\pi$
$239$	−10.6180	−0.686824	−0.343412	+	0.939185i	$0.611583\pi$
$240$	0	0
$241$	4.18034	0.269279	0.134640	−	0.990895i	$-0.457012\pi$
$241$	4.18034	0.269279	0.134640	+	0.990895i	$0.457012\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−33.5410	−2.13416
$248$	0	0
$249$	−14.6180	−0.926380
$250$	0	0
$251$	16.0902	1.01560	0.507801	−	0.861474i	$-0.330458\pi$
$251$	16.0902	1.01560	0.507801	+	0.861474i	$0.330458\pi$
$252$	0	0
$253$	−19.0000	−1.19452
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−9.76393	−0.609057	−0.304529	−	0.952503i	$-0.598499\pi$
$257$	−9.76393	−0.609057	−0.304529	+	0.952503i	$0.598499\pi$
$258$	0	0
$259$	13.8541	0.860852
$260$	0	0
$261$	−3.76393	−0.232982
$262$	0	0
$263$	10.1459	0.625623	0.312811	−	0.949815i	$-0.398729\pi$
$263$	10.1459	0.625623	0.312811	+	0.949815i	$0.398729\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−0.527864	−0.0323048
$268$	0	0
$269$	0.798374	0.0486777	0.0243389	−	0.999704i	$-0.492252\pi$
$269$	0.798374	0.0486777	0.0243389	+	0.999704i	$0.492252\pi$
$270$	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
$273$	−23.0902	−1.39748
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−8.20163	−0.492788	−0.246394	−	0.969170i	$-0.579246\pi$
$277$	−8.20163	−0.492788	−0.246394	+	0.969170i	$0.579246\pi$
$278$	0	0
$279$	−0.145898	−0.00873469
$280$	0	0
$281$	−4.20163	−0.250648	−0.125324	−	0.992116i	$-0.539997\pi$
$281$	−4.20163	−0.250648	−0.125324	+	0.992116i	$0.539997\pi$
$282$	0	0
$283$	19.1803	1.14015	0.570076	−	0.821592i	$-0.306914\pi$
$283$	19.1803	1.14015	0.570076	+	0.821592i	$0.306914\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−47.6869	−2.81487
$288$	0	0
$289$	−13.5623	−0.797783
$290$	0	0
$291$	11.0902	0.650117
$292$	0	0
$293$	13.5279	0.790306	0.395153	−	0.918615i	$-0.370692\pi$
$293$	13.5279	0.790306	0.395153	+	0.918615i	$0.370692\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.47214	−0.317526
$298$	0	0
$299$	−17.3607	−1.00399
$300$	0	0
$301$	23.7639	1.36973
$302$	0	0
$303$	−2.47214	−0.142020
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−29.3607	−1.67570	−0.837851	−	0.545899i	$-0.816188\pi$
$307$	−29.3607	−1.67570	−0.837851	+	0.545899i	$0.816188\pi$
$308$	0	0
$309$	6.85410	0.389916
$310$	0	0
$311$	−17.5066	−0.992707	−0.496353	−	0.868121i	$-0.665328\pi$
$311$	−17.5066	−0.992707	−0.496353	+	0.868121i	$0.665328\pi$
$312$	0	0
$313$	32.2148	1.82089	0.910444	−	0.413633i	$-0.135740\pi$
$313$	32.2148	1.82089	0.910444	+	0.413633i	$0.135740\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.41641	0.248050	0.124025	−	0.992279i	$-0.460420\pi$
$317$	4.41641	0.248050	0.124025	+	0.992279i	$0.460420\pi$
$318$	0	0
$319$	−20.5967	−1.15320
$320$	0	0
$321$	19.0344	1.06240
$322$	0	0
$323$	−12.4377	−0.692052
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.4164	0.576029
$328$	0	0
$329$	41.5623	2.29140
$330$	0	0
$331$	24.5623	1.35007	0.675033	−	0.737787i	$-0.264129\pi$
$331$	24.5623	1.35007	0.675033	+	0.737787i	$0.264129\pi$
$332$	0	0
$333$	3.00000	0.164399
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−24.3607	−1.32701	−0.663505	−	0.748172i	$-0.730932\pi$
$337$	−24.3607	−1.32701	−0.663505	+	0.748172i	$0.730932\pi$
$338$	0	0
$339$	12.6525	0.687188
$340$	0	0
$341$	−0.798374	−0.0432344
$342$	0	0
$343$	33.8328	1.82680
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.38197	0.235236	0.117618	−	0.993059i	$-0.462474\pi$
$347$	4.38197	0.235236	0.117618	+	0.993059i	$0.462474\pi$
$348$	0	0
$349$	3.43769	0.184016	0.0920078	−	0.995758i	$-0.470672\pi$
$349$	3.43769	0.184016	0.0920078	+	0.995758i	$0.470672\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	−33.3820	−1.77674	−0.888371	−	0.459126i	$-0.848163\pi$
$353$	−33.3820	−1.77674	−0.888371	+	0.459126i	$0.848163\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−8.56231	−0.453165
$358$	0	0
$359$	2.94427	0.155393	0.0776964	−	0.996977i	$-0.475244\pi$
$359$	2.94427	0.155393	0.0776964	+	0.996977i	$0.475244\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	0	0
$363$	−18.9443	−0.994316
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−16.5623	−0.864545	−0.432273	−	0.901743i	$-0.642288\pi$
$367$	−16.5623	−0.864545	−0.432273	+	0.901743i	$0.642288\pi$
$368$	0	0
$369$	−10.3262	−0.537562
$370$	0	0
$371$	−28.1246	−1.46016
$372$	0	0
$373$	−29.3262	−1.51846	−0.759228	−	0.650825i	$-0.774423\pi$
$373$	−29.3262	−1.51846	−0.759228	+	0.650825i	$0.774423\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−18.8197	−0.969262
$378$	0	0
$379$	−18.9098	−0.971333	−0.485666	−	0.874144i	$-0.661423\pi$
$379$	−18.9098	−0.971333	−0.485666	+	0.874144i	$0.661423\pi$
$380$	0	0
$381$	−12.7082	−0.651061
$382$	0	0
$383$	−16.7984	−0.858357	−0.429178	−	0.903220i	$-0.641197\pi$
$383$	−16.7984	−0.858357	−0.429178	+	0.903220i	$0.641197\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.14590	0.261581
$388$	0	0
$389$	25.1459	1.27495	0.637474	−	0.770472i	$-0.279979\pi$
$389$	25.1459	1.27495	0.637474	+	0.770472i	$0.279979\pi$
$390$	0	0
$391$	−6.43769	−0.325568
$392$	0	0
$393$	−3.38197	−0.170598
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−0.944272	−0.0473916	−0.0236958	−	0.999719i	$-0.507543\pi$
$397$	−0.944272	−0.0473916	−0.0236958	+	0.999719i	$0.507543\pi$
$398$	0	0
$399$	30.9787	1.55088
$400$	0	0
$401$	−12.8197	−0.640183	−0.320092	−	0.947387i	$-0.603714\pi$
$401$	−12.8197	−0.640183	−0.320092	+	0.947387i	$0.603714\pi$
$402$	0	0
$403$	−0.729490	−0.0363385
$404$	0	0
$405$	0	0
$406$	0	0
$407$	16.4164	0.813731
$408$	0	0
$409$	31.1246	1.53901	0.769507	−	0.638639i	$-0.220502\pi$
$409$	31.1246	1.53901	0.769507	+	0.638639i	$0.220502\pi$
$410$	0	0
$411$	4.23607	0.208950
$412$	0	0
$413$	12.0902	0.594918
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−6.52786	−0.319671
$418$	0	0
$419$	−19.9098	−0.972659	−0.486329	−	0.873776i	$-0.661664\pi$
$419$	−19.9098	−0.972659	−0.486329	+	0.873776i	$0.661664\pi$
$420$	0	0
$421$	−7.88854	−0.384464	−0.192232	−	0.981349i	$-0.561573\pi$
$421$	−7.88854	−0.384464	−0.192232	+	0.981349i	$0.561573\pi$
$422$	0	0
$423$	9.00000	0.437595
$424$	0	0
$425$	0	0
$426$	0	0
$427$	36.2705	1.75525
$428$	0	0
$429$	−27.3607	−1.32099
$430$	0	0
$431$	16.4164	0.790751	0.395375	−	0.918520i	$-0.370615\pi$
$431$	16.4164	0.790751	0.395375	+	0.918520i	$0.370615\pi$
$432$	0	0
$433$	−0.472136	−0.0226894	−0.0113447	−	0.999936i	$-0.503611\pi$
$433$	−0.472136	−0.0226894	−0.0113447	+	0.999936i	$0.503611\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	23.2918	1.11420
$438$	0	0
$439$	−25.6180	−1.22268	−0.611341	−	0.791367i	$-0.709370\pi$
$439$	−25.6180	−1.22268	−0.611341	+	0.791367i	$0.709370\pi$
$440$	0	0
$441$	14.3262	0.682202
$442$	0	0
$443$	30.5066	1.44941	0.724706	−	0.689059i	$-0.241976\pi$
$443$	30.5066	1.44941	0.724706	+	0.689059i	$0.241976\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.52786	0.119564
$448$	0	0
$449$	−10.0902	−0.476185	−0.238092	−	0.971243i	$-0.576522\pi$
$449$	−10.0902	−0.476185	−0.238092	+	0.971243i	$0.576522\pi$
$450$	0	0
$451$	−56.5066	−2.66079
$452$	0	0
$453$	−11.7082	−0.550099
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.38197	−0.0646456	−0.0323228	−	0.999477i	$-0.510290\pi$
$457$	−1.38197	−0.0646456	−0.0323228	+	0.999477i	$0.510290\pi$
$458$	0	0
$459$	−1.85410	−0.0865421
$460$	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
$463$	0.437694	0.0203414	0.0101707	−	0.999948i	$-0.496763\pi$
$463$	0.437694	0.0203414	0.0101707	+	0.999948i	$0.496763\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.14590	−0.0530258	−0.0265129	−	0.999648i	$-0.508440\pi$
$467$	−1.14590	−0.0530258	−0.0265129	+	0.999648i	$0.508440\pi$
$468$	0	0
$469$	59.7771	2.76025
$470$	0	0
$471$	0.437694	0.0201679
$472$	0	0
$473$	28.1591	1.29475
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.09017	−0.278850
$478$	0	0
$479$	−36.6525	−1.67469	−0.837347	−	0.546671i	$-0.815895\pi$
$479$	−36.6525	−1.67469	−0.837347	+	0.546671i	$0.815895\pi$
$480$	0	0
$481$	15.0000	0.683941
$482$	0	0
$483$	16.0344	0.729592
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−32.4721	−1.47145	−0.735726	−	0.677279i	$-0.763159\pi$
$487$	−32.4721	−1.47145	−0.735726	+	0.677279i	$0.763159\pi$
$488$	0	0
$489$	−3.29180	−0.148860
$490$	0	0
$491$	−39.8673	−1.79918	−0.899592	−	0.436731i	$-0.856136\pi$
$491$	−39.8673	−1.79918	−0.899592	+	0.436731i	$0.856136\pi$
$492$	0	0
$493$	−6.97871	−0.314305
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−70.7771	−3.17479
$498$	0	0
$499$	3.47214	0.155434	0.0777171	−	0.996975i	$-0.475237\pi$
$499$	3.47214	0.155434	0.0777171	+	0.996975i	$0.475237\pi$
$500$	0	0
$501$	−15.8885	−0.709848
$502$	0	0
$503$	0.381966	0.0170310	0.00851551	−	0.999964i	$-0.497289\pi$
$503$	0.381966	0.0170310	0.00851551	+	0.999964i	$0.497289\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	−5.23607	−0.232085	−0.116042	−	0.993244i	$-0.537021\pi$
$509$	−5.23607	−0.232085	−0.116042	+	0.993244i	$0.537021\pi$
$510$	0	0
$511$	−5.29180	−0.234095
$512$	0	0
$513$	6.70820	0.296174
$514$	0	0
$515$	0	0
$516$	0	0
$517$	49.2492	2.16598
$518$	0	0
$519$	−4.32624	−0.189901
$520$	0	0
$521$	0.493422	0.0216172	0.0108086	−	0.999942i	$-0.496559\pi$
$521$	0.493422	0.0216172	0.0108086	+	0.999942i	$0.496559\pi$
$522$	0	0
$523$	−0.944272	−0.0412901	−0.0206451	−	0.999787i	$-0.506572\pi$
$523$	−0.944272	−0.0412901	−0.0206451	+	0.999787i	$0.506572\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.270510	−0.0117836
$528$	0	0
$529$	−10.9443	−0.475838
$530$	0	0
$531$	2.61803	0.113613
$532$	0	0
$533$	−51.6312	−2.23640
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−22.4164	−0.967339
$538$	0	0
$539$	78.3951	3.37672
$540$	0	0
$541$	17.2705	0.742517	0.371259	−	0.928530i	$-0.378926\pi$
$541$	17.2705	0.742517	0.371259	+	0.928530i	$0.378926\pi$
$542$	0	0
$543$	13.4164	0.575753
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−2.14590	−0.0917520	−0.0458760	−	0.998947i	$-0.514608\pi$
$547$	−2.14590	−0.0917520	−0.0458760	+	0.998947i	$0.514608\pi$
$548$	0	0
$549$	7.85410	0.335205
$550$	0	0
$551$	25.2492	1.07565
$552$	0	0
$553$	−17.1246	−0.728213
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.0902	0.427534	0.213767	−	0.976885i	$-0.431427\pi$
$557$	10.0902	0.427534	0.213767	+	0.976885i	$0.431427\pi$
$558$	0	0
$559$	25.7295	1.08824
$560$	0	0
$561$	−10.1459	−0.428360
$562$	0	0
$563$	45.3262	1.91027	0.955137	−	0.296166i	$-0.0957080\pi$
$563$	45.3262	1.91027	0.955137	+	0.296166i	$0.0957080\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.61803	0.193939
$568$	0	0
$569$	−19.6525	−0.823875	−0.411937	−	0.911212i	$-0.635148\pi$
$569$	−19.6525	−0.823875	−0.411937	+	0.911212i	$0.635148\pi$
$570$	0	0
$571$	−43.2148	−1.80848	−0.904241	−	0.427022i	$-0.859563\pi$
$571$	−43.2148	−1.80848	−0.904241	+	0.427022i	$0.859563\pi$
$572$	0	0
$573$	12.7082	0.530893
$574$	0	0
$575$	0	0
$576$	0	0
$577$	12.5279	0.521542	0.260771	−	0.965401i	$-0.416023\pi$
$577$	12.5279	0.521542	0.260771	+	0.965401i	$0.416023\pi$
$578$	0	0
$579$	−0.708204	−0.0294320
$580$	0	0
$581$	67.5066	2.80064
$582$	0	0
$583$	−33.3262	−1.38023
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.6525	0.893693	0.446847	−	0.894611i	$-0.352547\pi$
$587$	21.6525	0.893693	0.446847	+	0.894611i	$0.352547\pi$
$588$	0	0
$589$	0.978714	0.0403272
$590$	0	0
$591$	21.6525	0.890664
$592$	0	0
$593$	−26.8885	−1.10418	−0.552090	−	0.833784i	$-0.686170\pi$
$593$	−26.8885	−1.10418	−0.552090	+	0.833784i	$0.686170\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	15.9443	0.652556
$598$	0	0
$599$	16.3262	0.667072	0.333536	−	0.942737i	$-0.391758\pi$
$599$	16.3262	0.667072	0.333536	+	0.942737i	$0.391758\pi$
$600$	0	0
$601$	−23.8328	−0.972161	−0.486080	−	0.873914i	$-0.661574\pi$
$601$	−23.8328	−0.972161	−0.486080	+	0.873914i	$0.661574\pi$
$602$	0	0
$603$	12.9443	0.527132
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−15.4164	−0.625733	−0.312866	−	0.949797i	$-0.601289\pi$
$607$	−15.4164	−0.625733	−0.312866	+	0.949797i	$0.601289\pi$
$608$	0	0
$609$	17.3820	0.704353
$610$	0	0
$611$	45.0000	1.82051
$612$	0	0
$613$	−40.0132	−1.61612	−0.808058	−	0.589103i	$-0.799481\pi$
$613$	−40.0132	−1.61612	−0.808058	+	0.589103i	$0.799481\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.8197	0.516100	0.258050	−	0.966132i	$-0.416920\pi$
$617$	12.8197	0.516100	0.258050	+	0.966132i	$0.416920\pi$
$618$	0	0
$619$	−39.4164	−1.58428	−0.792140	−	0.610340i	$-0.791033\pi$
$619$	−39.4164	−1.58428	−0.792140	+	0.610340i	$0.791033\pi$
$620$	0	0
$621$	3.47214	0.139332
$622$	0	0
$623$	2.43769	0.0976642
$624$	0	0
$625$	0	0
$626$	0	0
$627$	36.7082	1.46598
$628$	0	0
$629$	5.56231	0.221784
$630$	0	0
$631$	24.8885	0.990797	0.495399	−	0.868666i	$-0.335022\pi$
$631$	24.8885	0.990797	0.495399	+	0.868666i	$0.335022\pi$
$632$	0	0
$633$	−3.41641	−0.135790
$634$	0	0
$635$	0	0
$636$	0	0
$637$	71.6312	2.83813
$638$	0	0
$639$	−15.3262	−0.606297
$640$	0	0
$641$	−5.34752	−0.211215	−0.105607	−	0.994408i	$-0.533679\pi$
$641$	−5.34752	−0.211215	−0.105607	+	0.994408i	$0.533679\pi$
$642$	0	0
$643$	−2.58359	−0.101887	−0.0509435	−	0.998702i	$-0.516223\pi$
$643$	−2.58359	−0.101887	−0.0509435	+	0.998702i	$0.516223\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.6180	0.653322	0.326661	−	0.945142i	$-0.394076\pi$
$647$	16.6180	0.653322	0.326661	+	0.945142i	$0.394076\pi$
$648$	0	0
$649$	14.3262	0.562354
$650$	0	0
$651$	0.673762	0.0264068
$652$	0	0
$653$	46.6525	1.82565	0.912826	−	0.408348i	$-0.133895\pi$
$653$	46.6525	1.82565	0.912826	+	0.408348i	$0.133895\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.14590	−0.0447057
$658$	0	0
$659$	−40.4164	−1.57440	−0.787200	−	0.616698i	$-0.788470\pi$
$659$	−40.4164	−1.57440	−0.787200	+	0.616698i	$0.788470\pi$
$660$	0	0
$661$	−17.8541	−0.694444	−0.347222	−	0.937783i	$-0.612875\pi$
$661$	−17.8541	−0.694444	−0.347222	+	0.937783i	$0.612875\pi$
$662$	0	0
$663$	−9.27051	−0.360037
$664$	0	0
$665$	0	0
$666$	0	0
$667$	13.0689	0.506029
$668$	0	0
$669$	−20.1246	−0.778062
$670$	0	0
$671$	42.9787	1.65917
$672$	0	0
$673$	21.3050	0.821246	0.410623	−	0.911805i	$-0.365311\pi$
$673$	21.3050	0.821246	0.410623	+	0.911805i	$0.365311\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	43.4853	1.67128	0.835638	−	0.549281i	$-0.185098\pi$
$677$	43.4853	1.67128	0.835638	+	0.549281i	$0.185098\pi$
$678$	0	0
$679$	−51.2148	−1.96544
$680$	0	0
$681$	−3.81966	−0.146370
$682$	0	0
$683$	−1.65248	−0.0632302	−0.0316151	−	0.999500i	$-0.510065\pi$
$683$	−1.65248	−0.0632302	−0.0316151	+	0.999500i	$0.510065\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−4.29180	−0.163742
$688$	0	0
$689$	−30.4508	−1.16008
$690$	0	0
$691$	1.94427	0.0739636	0.0369818	−	0.999316i	$-0.488226\pi$
$691$	1.94427	0.0739636	0.0369818	+	0.999316i	$0.488226\pi$
$692$	0	0
$693$	25.2705	0.959947
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−19.1459	−0.725203
$698$	0	0
$699$	16.3820	0.619623
$700$	0	0
$701$	15.5967	0.589081	0.294541	−	0.955639i	$-0.404833\pi$
$701$	15.5967	0.589081	0.294541	+	0.955639i	$0.404833\pi$
$702$	0	0
$703$	−20.1246	−0.759014
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11.4164	0.429358
$708$	0	0
$709$	14.1246	0.530461	0.265230	−	0.964185i	$-0.414552\pi$
$709$	14.1246	0.530461	0.265230	+	0.964185i	$0.414552\pi$
$710$	0	0
$711$	−3.70820	−0.139069
$712$	0	0
$713$	0.506578	0.0189715
$714$	0	0
$715$	0	0
$716$	0	0
$717$	10.6180	0.396538
$718$	0	0
$719$	43.6525	1.62796	0.813981	−	0.580891i	$-0.197296\pi$
$719$	43.6525	1.62796	0.813981	+	0.580891i	$0.197296\pi$
$720$	0	0
$721$	−31.6525	−1.17880
$722$	0	0
$723$	−4.18034	−0.155469
$724$	0	0
$725$	0	0
$726$	0	0
$727$	4.23607	0.157107	0.0785535	−	0.996910i	$-0.474970\pi$
$727$	4.23607	0.157107	0.0785535	+	0.996910i	$0.474970\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	9.54102	0.352887
$732$	0	0
$733$	36.0000	1.32969	0.664845	−	0.746981i	$-0.268498\pi$
$733$	36.0000	1.32969	0.664845	+	0.746981i	$0.268498\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	70.8328	2.60916
$738$	0	0
$739$	−17.9656	−0.660874	−0.330437	−	0.943828i	$-0.607196\pi$
$739$	−17.9656	−0.660874	−0.330437	+	0.943828i	$0.607196\pi$
$740$	0	0
$741$	33.5410	1.23216
$742$	0	0
$743$	−26.6525	−0.977785	−0.488892	−	0.872344i	$-0.662599\pi$
$743$	−26.6525	−0.977785	−0.488892	+	0.872344i	$0.662599\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	14.6180	0.534846
$748$	0	0
$749$	−87.9017	−3.21186
$750$	0	0
$751$	2.72949	0.0996005	0.0498003	−	0.998759i	$-0.484142\pi$
$751$	2.72949	0.0996005	0.0498003	+	0.998759i	$0.484142\pi$
$752$	0	0
$753$	−16.0902	−0.586358
$754$	0	0
$755$	0	0
$756$	0	0
$757$	20.7082	0.752652	0.376326	−	0.926487i	$-0.377187\pi$
$757$	20.7082	0.752652	0.376326	+	0.926487i	$0.377187\pi$
$758$	0	0
$759$	19.0000	0.689656
$760$	0	0
$761$	−5.76393	−0.208942	−0.104471	−	0.994528i	$-0.533315\pi$
$761$	−5.76393	−0.208942	−0.104471	+	0.994528i	$0.533315\pi$
$762$	0	0
$763$	−48.1033	−1.74146
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.0902	0.472659
$768$	0	0
$769$	−45.8885	−1.65478	−0.827392	−	0.561625i	$-0.810176\pi$
$769$	−45.8885	−1.65478	−0.827392	+	0.561625i	$0.810176\pi$
$770$	0	0
$771$	9.76393	0.351639
$772$	0	0
$773$	28.0557	1.00909	0.504547	−	0.863384i	$-0.331659\pi$
$773$	28.0557	1.00909	0.504547	+	0.863384i	$0.331659\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−13.8541	−0.497013
$778$	0	0
$779$	69.2705	2.48187
$780$	0	0
$781$	−83.8673	−3.00101
$782$	0	0
$783$	3.76393	0.134512
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−47.3951	−1.68945	−0.844727	−	0.535198i	$-0.820237\pi$
$787$	−47.3951	−1.68945	−0.844727	+	0.535198i	$0.820237\pi$
$788$	0	0
$789$	−10.1459	−0.361204
$790$	0	0
$791$	−58.4296	−2.07752
$792$	0	0
$793$	39.2705	1.39454
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25.0344	−0.886765	−0.443383	−	0.896332i	$-0.646222\pi$
$797$	−25.0344	−0.886765	−0.443383	+	0.896332i	$0.646222\pi$
$798$	0	0
$799$	16.6869	0.590341
$800$	0	0
$801$	0.527864	0.0186512
$802$	0	0
$803$	−6.27051	−0.221281
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.798374	−0.0281041
$808$	0	0
$809$	0.798374	0.0280693	0.0140347	−	0.999902i	$-0.495532\pi$
$809$	0.798374	0.0280693	0.0140347	+	0.999902i	$0.495532\pi$
$810$	0	0
$811$	−29.1459	−1.02345	−0.511725	−	0.859149i	$-0.670993\pi$
$811$	−29.1459	−1.02345	−0.511725	+	0.859149i	$0.670993\pi$
$812$	0	0
$813$	−18.2705	−0.640775
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−34.5197	−1.20769
$818$	0	0
$819$	23.0902	0.806836
$820$	0	0
$821$	−56.2361	−1.96265	−0.981326	−	0.192351i	$-0.938389\pi$
$821$	−56.2361	−1.96265	−0.981326	+	0.192351i	$0.938389\pi$
$822$	0	0
$823$	−28.8328	−1.00505	−0.502524	−	0.864563i	$-0.667595\pi$
$823$	−28.8328	−1.00505	−0.502524	+	0.864563i	$0.667595\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.9787	0.451314	0.225657	−	0.974207i	$-0.427547\pi$
$827$	12.9787	0.451314	0.225657	+	0.974207i	$0.427547\pi$
$828$	0	0
$829$	32.6525	1.13407	0.567034	−	0.823695i	$-0.308091\pi$
$829$	32.6525	1.13407	0.567034	+	0.823695i	$0.308091\pi$
$830$	0	0
$831$	8.20163	0.284511
$832$	0	0
$833$	26.5623	0.920329
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0.145898	0.00504297
$838$	0	0
$839$	3.87539	0.133793	0.0668966	−	0.997760i	$-0.478690\pi$
$839$	3.87539	0.133793	0.0668966	+	0.997760i	$0.478690\pi$
$840$	0	0
$841$	−14.8328	−0.511476
$842$	0	0
$843$	4.20163	0.144712
$844$	0	0
$845$	0	0
$846$	0	0
$847$	87.4853	3.00603
$848$	0	0
$849$	−19.1803	−0.658268
$850$	0	0
$851$	−10.4164	−0.357070
$852$	0	0
$853$	19.7639	0.676704	0.338352	−	0.941020i	$-0.390131\pi$
$853$	19.7639	0.676704	0.338352	+	0.941020i	$0.390131\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−16.0902	−0.549630	−0.274815	−	0.961497i	$-0.588617\pi$
$857$	−16.0902	−0.549630	−0.274815	+	0.961497i	$0.588617\pi$
$858$	0	0
$859$	−31.5836	−1.07762	−0.538809	−	0.842428i	$-0.681126\pi$
$859$	−31.5836	−1.07762	−0.538809	+	0.842428i	$0.681126\pi$
$860$	0	0
$861$	47.6869	1.62517
$862$	0	0
$863$	−7.14590	−0.243249	−0.121625	−	0.992576i	$-0.538810\pi$
$863$	−7.14590	−0.243249	−0.121625	+	0.992576i	$0.538810\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.5623	0.460600
$868$	0	0
$869$	−20.2918	−0.688352
$870$	0	0
$871$	64.7214	2.19300
$872$	0	0
$873$	−11.0902	−0.375345
$874$	0	0
$875$	0	0
$876$	0	0
$877$	9.00000	0.303908	0.151954	−	0.988388i	$-0.451443\pi$
$877$	9.00000	0.303908	0.151954	+	0.988388i	$0.451443\pi$
$878$	0	0
$879$	−13.5279	−0.456284
$880$	0	0
$881$	9.34752	0.314926	0.157463	−	0.987525i	$-0.449668\pi$
$881$	9.34752	0.314926	0.157463	+	0.987525i	$0.449668\pi$
$882$	0	0
$883$	22.1246	0.744552	0.372276	−	0.928122i	$-0.378577\pi$
$883$	22.1246	0.744552	0.372276	+	0.928122i	$0.378577\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.81966	−0.0946749	−0.0473375	−	0.998879i	$-0.515074\pi$
$887$	−2.81966	−0.0946749	−0.0473375	+	0.998879i	$0.515074\pi$
$888$	0	0
$889$	58.6869	1.96830
$890$	0	0
$891$	5.47214	0.183323
$892$	0	0
$893$	−60.3738	−2.02033
$894$	0	0
$895$	0	0
$896$	0	0
$897$	17.3607	0.579656
$898$	0	0
$899$	0.549150	0.0183152
$900$	0	0
$901$	−11.2918	−0.376184
$902$	0	0
$903$	−23.7639	−0.790814
$904$	0	0
$905$	0	0
$906$	0	0
$907$	6.41641	0.213053	0.106527	−	0.994310i	$-0.466027\pi$
$907$	6.41641	0.213053	0.106527	+	0.994310i	$0.466027\pi$
$908$	0	0
$909$	2.47214	0.0819956
$910$	0	0
$911$	−0.214782	−0.00711604	−0.00355802	−	0.999994i	$-0.501133\pi$
$911$	−0.214782	−0.00711604	−0.00355802	+	0.999994i	$0.501133\pi$
$912$	0	0
$913$	79.9919	2.64734
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15.6180	0.515753
$918$	0	0
$919$	39.5066	1.30320	0.651601	−	0.758562i	$-0.274098\pi$
$919$	39.5066	1.30320	0.651601	+	0.758562i	$0.274098\pi$
$920$	0	0
$921$	29.3607	0.967467
$922$	0	0
$923$	−76.6312	−2.52235
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−6.85410	−0.225118
$928$	0	0
$929$	18.9787	0.622671	0.311336	−	0.950300i	$-0.399224\pi$
$929$	18.9787	0.622671	0.311336	+	0.950300i	$0.399224\pi$
$930$	0	0
$931$	−96.1033	−3.14966
$932$	0	0
$933$	17.5066	0.573140
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−13.7082	−0.447828	−0.223914	−	0.974609i	$-0.571883\pi$
$937$	−13.7082	−0.447828	−0.223914	+	0.974609i	$0.571883\pi$
$938$	0	0
$939$	−32.2148	−1.05129
$940$	0	0
$941$	−8.21478	−0.267794	−0.133897	−	0.990995i	$-0.542749\pi$
$941$	−8.21478	−0.267794	−0.133897	+	0.990995i	$0.542749\pi$
$942$	0	0
$943$	35.8541	1.16757
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33.6525	1.09356	0.546779	−	0.837277i	$-0.315854\pi$
$947$	33.6525	1.09356	0.546779	+	0.837277i	$0.315854\pi$
$948$	0	0
$949$	−5.72949	−0.185987
$950$	0	0
$951$	−4.41641	−0.143212
$952$	0	0
$953$	16.9656	0.549568	0.274784	−	0.961506i	$-0.411394\pi$
$953$	16.9656	0.549568	0.274784	+	0.961506i	$0.411394\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	20.5967	0.665798
$958$	0	0
$959$	−19.5623	−0.631700
$960$	0	0
$961$	−30.9787	−0.999313
$962$	0	0
$963$	−19.0344	−0.613376
$964$	0	0
$965$	0	0
$966$	0	0
$967$	41.4296	1.33228	0.666142	−	0.745825i	$-0.267944\pi$
$967$	41.4296	1.33228	0.666142	+	0.745825i	$0.267944\pi$
$968$	0	0
$969$	12.4377	0.399556
$970$	0	0
$971$	14.1803	0.455069	0.227534	−	0.973770i	$-0.426934\pi$
$971$	14.1803	0.455069	0.227534	+	0.973770i	$0.426934\pi$
$972$	0	0
$973$	30.1459	0.966433
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−58.4508	−1.87001	−0.935004	−	0.354637i	$-0.884605\pi$
$977$	−58.4508	−1.87001	−0.935004	+	0.354637i	$0.884605\pi$
$978$	0	0
$979$	2.88854	0.0923183
$980$	0	0
$981$	−10.4164	−0.332570
$982$	0	0
$983$	39.1033	1.24720	0.623601	−	0.781743i	$-0.285669\pi$
$983$	39.1033	1.24720	0.623601	+	0.781743i	$0.285669\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−41.5623	−1.32294
$988$	0	0
$989$	−17.8673	−0.568146
$990$	0	0
$991$	15.6869	0.498311	0.249156	−	0.968463i	$-0.419847\pi$
$991$	15.6869	0.498311	0.249156	+	0.968463i	$0.419847\pi$
$992$	0	0
$993$	−24.5623	−0.779461
$994$	0	0
$995$	0	0
$996$	0	0
$997$	29.0000	0.918439	0.459220	−	0.888323i	$-0.348129\pi$
$997$	29.0000	0.918439	0.459220	+	0.888323i	$0.348129\pi$
$998$	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.d.1.2	✓	2
3.2	odd	2		4500.2.a.n.1.2		2
4.3	odd	2		6000.2.a.o.1.1		2
5.2	odd	4		1500.2.d.c.1249.4		4
5.3	odd	4		1500.2.d.c.1249.1		4
5.4	even	2		1500.2.a.e.1.1	yes	2
15.2	even	4		4500.2.d.b.4249.4		4
15.8	even	4		4500.2.d.b.4249.1		4
15.14	odd	2		4500.2.a.a.1.1		2
20.3	even	4		6000.2.f.d.1249.4		4
20.7	even	4		6000.2.f.d.1249.1		4
20.19	odd	2		6000.2.a.n.1.2		2

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.61803 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.61803 q^{7} +1.00000 q^{9} +5.47214 q^{11} +5.00000 q^{13} +1.85410 q^{17} -6.70820 q^{19} -4.61803 q^{21} -3.47214 q^{23} -1.00000 q^{27} -3.76393 q^{29} -0.145898 q^{31} -5.47214 q^{33} +3.00000 q^{37} -5.00000 q^{39} -10.3262 q^{41} +5.14590 q^{43} +9.00000 q^{47} +14.3262 q^{49} -1.85410 q^{51} -6.09017 q^{53} +6.70820 q^{57} +2.61803 q^{59} +7.85410 q^{61} +4.61803 q^{63} +12.9443 q^{67} +3.47214 q^{69} -15.3262 q^{71} -1.14590 q^{73} +25.2705 q^{77} -3.70820 q^{79} +1.00000 q^{81} +14.6180 q^{83} +3.76393 q^{87} +0.527864 q^{89} +23.0902 q^{91} +0.145898 q^{93} -11.0902 q^{97} +5.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 7 q^{7} + 2 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more