LMFDB - Embedded newform 1500.2.a.b.1.1

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

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.23607 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.23607 q^{7} +1.00000 q^{9} -4.00000 q^{11} +4.47214 q^{13} +6.85410 q^{17} -8.61803 q^{19} +1.23607 q^{21} +2.38197 q^{23} -1.00000 q^{27} -7.70820 q^{29} +2.09017 q^{31} +4.00000 q^{33} -9.23607 q^{37} -4.47214 q^{39} +8.94427 q^{41} -4.00000 q^{43} -6.32624 q^{47} -5.47214 q^{49} -6.85410 q^{51} -1.09017 q^{53} +8.61803 q^{57} -6.00000 q^{59} -2.14590 q^{61} -1.23607 q^{63} -3.23607 q^{67} -2.38197 q^{69} -4.47214 q^{71} -13.7082 q^{73} +4.94427 q^{77} +8.32624 q^{79} +1.00000 q^{81} +3.56231 q^{83} +7.70820 q^{87} -11.7082 q^{89} -5.52786 q^{91} -2.09017 q^{93} -5.23607 q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} - 8 q^{11} + 7 q^{17} - 15 q^{19} - 2 q^{21} + 7 q^{23} - 2 q^{27} - 2 q^{29} - 7 q^{31} + 8 q^{33} - 14 q^{37} - 8 q^{43} + 3 q^{47} - 2 q^{49} - 7 q^{51} + 9 q^{53} + 15 q^{57} - 12 q^{59} - 11 q^{61} + 2 q^{63} - 2 q^{67} - 7 q^{69} - 14 q^{73} - 8 q^{77} + q^{79} + 2 q^{81} - 13 q^{83} + 2 q^{87} - 10 q^{89} - 20 q^{91} + 7 q^{93} - 6 q^{97} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 - 8 * q^11 + 7 * q^17 - 15 * q^19 - 2 * q^21 + 7 * q^23 - 2 * q^27 - 2 * q^29 - 7 * q^31 + 8 * q^33 - 14 * q^37 - 8 * q^43 + 3 * q^47 - 2 * q^49 - 7 * q^51 + 9 * q^53 + 15 * q^57 - 12 * q^59 - 11 * q^61 + 2 * q^63 - 2 * q^67 - 7 * q^69 - 14 * q^73 - 8 * q^77 + q^79 + 2 * q^81 - 13 * q^83 + 2 * q^87 - 10 * q^89 - 20 * q^91 + 7 * q^93 - 6 * q^97 - 8 * 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
$7$	−1.23607	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	4.47214	1.24035	0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214	1.24035	0.620174	+	0.784465i	$0.287062\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.85410	1.66236	0.831182	−	0.556001i	$-0.187665\pi$
$17$	6.85410	1.66236	0.831182	+	0.556001i	$0.187665\pi$
$18$	0	0
$19$	−8.61803	−1.97711	−0.988556	−	0.150852i	$-0.951798\pi$
$19$	−8.61803	−1.97711	−0.988556	+	0.150852i	$0.951798\pi$
$20$	0	0
$21$	1.23607	0.269732
$22$	0	0
$23$	2.38197	0.496674	0.248337	−	0.968674i	$-0.420116\pi$
$23$	2.38197	0.496674	0.248337	+	0.968674i	$0.420116\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.70820	−1.43138	−0.715689	−	0.698419i	$-0.753887\pi$
$29$	−7.70820	−1.43138	−0.715689	+	0.698419i	$0.753887\pi$
$30$	0	0
$31$	2.09017	0.375406	0.187703	−	0.982226i	$-0.439896\pi$
$31$	2.09017	0.375406	0.187703	+	0.982226i	$0.439896\pi$
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.23607	−1.51840	−0.759200	−	0.650857i	$-0.774410\pi$
$37$	−9.23607	−1.51840	−0.759200	+	0.650857i	$0.774410\pi$
$38$	0	0
$39$	−4.47214	−0.716115
$40$	0	0
$41$	8.94427	1.39686	0.698430	−	0.715678i	$-0.253882\pi$
$41$	8.94427	1.39686	0.698430	+	0.715678i	$0.253882\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.32624	−0.922777	−0.461388	−	0.887198i	$-0.652648\pi$
$47$	−6.32624	−0.922777	−0.461388	+	0.887198i	$0.652648\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	0	0
$51$	−6.85410	−0.959766
$52$	0	0
$53$	−1.09017	−0.149746	−0.0748732	−	0.997193i	$-0.523855\pi$
$53$	−1.09017	−0.149746	−0.0748732	+	0.997193i	$0.523855\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	8.61803	1.14149
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−2.14590	−0.274754	−0.137377	−	0.990519i	$-0.543867\pi$
$61$	−2.14590	−0.274754	−0.137377	+	0.990519i	$0.543867\pi$
$62$	0	0
$63$	−1.23607	−0.155730
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.23607	−0.395349	−0.197674	−	0.980268i	$-0.563339\pi$
$67$	−3.23607	−0.395349	−0.197674	+	0.980268i	$0.563339\pi$
$68$	0	0
$69$	−2.38197	−0.286755
$70$	0	0
$71$	−4.47214	−0.530745	−0.265372	−	0.964146i	$-0.585495\pi$
$71$	−4.47214	−0.530745	−0.265372	+	0.964146i	$0.585495\pi$
$72$	0	0
$73$	−13.7082	−1.60442	−0.802212	−	0.597039i	$-0.796344\pi$
$73$	−13.7082	−1.60442	−0.802212	+	0.597039i	$0.796344\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.94427	0.563452
$78$	0	0
$79$	8.32624	0.936775	0.468387	−	0.883523i	$-0.344835\pi$
$79$	8.32624	0.936775	0.468387	+	0.883523i	$0.344835\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.56231	0.391014	0.195507	−	0.980702i	$-0.437365\pi$
$83$	3.56231	0.391014	0.195507	+	0.980702i	$0.437365\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.70820	0.826406
$88$	0	0
$89$	−11.7082	−1.24107	−0.620534	−	0.784180i	$-0.713084\pi$
$89$	−11.7082	−1.24107	−0.620534	+	0.784180i	$0.713084\pi$
$90$	0	0
$91$	−5.52786	−0.579478
$92$	0	0
$93$	−2.09017	−0.216741
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−5.23607	−0.531642	−0.265821	−	0.964022i	$-0.585643\pi$
$97$	−5.23607	−0.531642	−0.265821	+	0.964022i	$0.585643\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−0.291796	−0.0290348	−0.0145174	−	0.999895i	$-0.504621\pi$
$101$	−0.291796	−0.0290348	−0.0145174	+	0.999895i	$0.504621\pi$
$102$	0	0
$103$	14.6525	1.44375	0.721876	−	0.692023i	$-0.243280\pi$
$103$	14.6525	1.44375	0.721876	+	0.692023i	$0.243280\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.5623	1.02110	0.510548	−	0.859849i	$-0.329442\pi$
$107$	10.5623	1.02110	0.510548	+	0.859849i	$0.329442\pi$
$108$	0	0
$109$	−16.2705	−1.55843	−0.779216	−	0.626755i	$-0.784383\pi$
$109$	−16.2705	−1.55843	−0.779216	+	0.626755i	$0.784383\pi$
$110$	0	0
$111$	9.23607	0.876649
$112$	0	0
$113$	−7.85410	−0.738852	−0.369426	−	0.929260i	$-0.620446\pi$
$113$	−7.85410	−0.738852	−0.369426	+	0.929260i	$0.620446\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.47214	0.413449
$118$	0	0
$119$	−8.47214	−0.776639
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−8.94427	−0.806478
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−18.4721	−1.63914	−0.819569	−	0.572981i	$-0.805787\pi$
$127$	−18.4721	−1.63914	−0.819569	+	0.572981i	$0.805787\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	0.944272	0.0825014	0.0412507	−	0.999149i	$-0.486866\pi$
$131$	0.944272	0.0825014	0.0412507	+	0.999149i	$0.486866\pi$
$132$	0	0
$133$	10.6525	0.923687
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.9443	−0.935032	−0.467516	−	0.883985i	$-0.654851\pi$
$137$	−10.9443	−0.935032	−0.467516	+	0.883985i	$0.654851\pi$
$138$	0	0
$139$	−9.32624	−0.791041	−0.395521	−	0.918457i	$-0.629436\pi$
$139$	−9.32624	−0.791041	−0.395521	+	0.918457i	$0.629436\pi$
$140$	0	0
$141$	6.32624	0.532765
$142$	0	0
$143$	−17.8885	−1.49592
$144$	0	0
$145$	0	0
$146$	0	0
$147$	5.47214	0.451334
$148$	0	0
$149$	11.4164	0.935269	0.467634	−	0.883922i	$-0.345106\pi$
$149$	11.4164	0.935269	0.467634	+	0.883922i	$0.345106\pi$
$150$	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
$153$	6.85410	0.554121
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4.18034	−0.333627	−0.166814	−	0.985988i	$-0.553348\pi$
$157$	−4.18034	−0.333627	−0.166814	+	0.985988i	$0.553348\pi$
$158$	0	0
$159$	1.09017	0.0864561
$160$	0	0
$161$	−2.94427	−0.232041
$162$	0	0
$163$	−8.29180	−0.649464	−0.324732	−	0.945806i	$-0.605274\pi$
$163$	−8.29180	−0.649464	−0.324732	+	0.945806i	$0.605274\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.3262	1.41813	0.709063	−	0.705145i	$-0.249118\pi$
$167$	18.3262	1.41813	0.709063	+	0.705145i	$0.249118\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	−8.61803	−0.659038
$172$	0	0
$173$	5.05573	0.384380	0.192190	−	0.981358i	$-0.438441\pi$
$173$	5.05573	0.384380	0.192190	+	0.981358i	$0.438441\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.00000	0.450988
$178$	0	0
$179$	1.23607	0.0923881	0.0461940	−	0.998932i	$-0.485291\pi$
$179$	1.23607	0.0923881	0.0461940	+	0.998932i	$0.485291\pi$
$180$	0	0
$181$	11.3820	0.846015	0.423007	−	0.906126i	$-0.360974\pi$
$181$	11.3820	0.846015	0.423007	+	0.906126i	$0.360974\pi$
$182$	0	0
$183$	2.14590	0.158629
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−27.4164	−2.00489
$188$	0	0
$189$	1.23607	0.0899107
$190$	0	0
$191$	−21.1246	−1.52852	−0.764262	−	0.644906i	$-0.776896\pi$
$191$	−21.1246	−1.52852	−0.764262	+	0.644906i	$0.776896\pi$
$192$	0	0
$193$	−19.4164	−1.39762	−0.698812	−	0.715306i	$-0.746288\pi$
$193$	−19.4164	−1.39762	−0.698812	+	0.715306i	$0.746288\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.0902	1.57386	0.786930	−	0.617043i	$-0.211669\pi$
$197$	22.0902	1.57386	0.786930	+	0.617043i	$0.211669\pi$
$198$	0	0
$199$	6.09017	0.431721	0.215860	−	0.976424i	$-0.430744\pi$
$199$	6.09017	0.431721	0.215860	+	0.976424i	$0.430744\pi$
$200$	0	0
$201$	3.23607	0.228255
$202$	0	0
$203$	9.52786	0.668725
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.38197	0.165558
$208$	0	0
$209$	34.4721	2.38449
$210$	0	0
$211$	−14.2705	−0.982422	−0.491211	−	0.871040i	$-0.663446\pi$
$211$	−14.2705	−0.982422	−0.491211	+	0.871040i	$0.663446\pi$
$212$	0	0
$213$	4.47214	0.306426
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.58359	−0.175386
$218$	0	0
$219$	13.7082	0.926315
$220$	0	0
$221$	30.6525	2.06191
$222$	0	0
$223$	7.23607	0.484563	0.242281	−	0.970206i	$-0.422104\pi$
$223$	7.23607	0.484563	0.242281	+	0.970206i	$0.422104\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.79837	0.650341	0.325170	−	0.945655i	$-0.394578\pi$
$227$	9.79837	0.650341	0.325170	+	0.945655i	$0.394578\pi$
$228$	0	0
$229$	2.38197	0.157405	0.0787024	−	0.996898i	$-0.474922\pi$
$229$	2.38197	0.157405	0.0787024	+	0.996898i	$0.474922\pi$
$230$	0	0
$231$	−4.94427	−0.325309
$232$	0	0
$233$	27.8885	1.82704	0.913520	−	0.406795i	$-0.133354\pi$
$233$	27.8885	1.82704	0.913520	+	0.406795i	$0.133354\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.32624	−0.540847
$238$	0	0
$239$	8.65248	0.559682	0.279841	−	0.960046i	$-0.409718\pi$
$239$	8.65248	0.559682	0.279841	+	0.960046i	$0.409718\pi$
$240$	0	0
$241$	14.3820	0.926424	0.463212	−	0.886248i	$-0.346697\pi$
$241$	14.3820	0.926424	0.463212	+	0.886248i	$0.346697\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−38.5410	−2.45231
$248$	0	0
$249$	−3.56231	−0.225752
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	−9.52786	−0.599012
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.20163	0.199712	0.0998560	−	0.995002i	$-0.468162\pi$
$257$	3.20163	0.199712	0.0998560	+	0.995002i	$0.468162\pi$
$258$	0	0
$259$	11.4164	0.709381
$260$	0	0
$261$	−7.70820	−0.477126
$262$	0	0
$263$	24.0902	1.48546	0.742732	−	0.669589i	$-0.233530\pi$
$263$	24.0902	1.48546	0.742732	+	0.669589i	$0.233530\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	11.7082	0.716530
$268$	0	0
$269$	−19.1246	−1.16605	−0.583024	−	0.812455i	$-0.698131\pi$
$269$	−19.1246	−1.16605	−0.583024	+	0.812455i	$0.698131\pi$
$270$	0	0
$271$	−28.5623	−1.73504	−0.867518	−	0.497405i	$-0.834286\pi$
$271$	−28.5623	−1.73504	−0.867518	+	0.497405i	$0.834286\pi$
$272$	0	0
$273$	5.52786	0.334562
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−10.7639	−0.646742	−0.323371	−	0.946272i	$-0.604816\pi$
$277$	−10.7639	−0.646742	−0.323371	+	0.946272i	$0.604816\pi$
$278$	0	0
$279$	2.09017	0.125135
$280$	0	0
$281$	3.23607	0.193048	0.0965238	−	0.995331i	$-0.469228\pi$
$281$	3.23607	0.193048	0.0965238	+	0.995331i	$0.469228\pi$
$282$	0	0
$283$	32.0689	1.90630	0.953149	−	0.302502i	$-0.0978220\pi$
$283$	32.0689	1.90630	0.953149	+	0.302502i	$0.0978220\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−11.0557	−0.652599
$288$	0	0
$289$	29.9787	1.76345
$290$	0	0
$291$	5.23607	0.306944
$292$	0	0
$293$	−17.3262	−1.01221	−0.506105	−	0.862472i	$-0.668915\pi$
$293$	−17.3262	−1.01221	−0.506105	+	0.862472i	$0.668915\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	10.6525	0.616049
$300$	0	0
$301$	4.94427	0.284983
$302$	0	0
$303$	0.291796	0.0167632
$304$	0	0
$305$	0	0
$306$	0	0
$307$	32.4721	1.85328	0.926641	−	0.375947i	$-0.122682\pi$
$307$	32.4721	1.85328	0.926641	+	0.375947i	$0.122682\pi$
$308$	0	0
$309$	−14.6525	−0.833550
$310$	0	0
$311$	−32.1803	−1.82478	−0.912390	−	0.409322i	$-0.865765\pi$
$311$	−32.1803	−1.82478	−0.912390	+	0.409322i	$0.865765\pi$
$312$	0	0
$313$	−2.58359	−0.146033	−0.0730166	−	0.997331i	$-0.523263\pi$
$313$	−2.58359	−0.146033	−0.0730166	+	0.997331i	$0.523263\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.0000	1.46031	0.730153	−	0.683284i	$-0.239449\pi$
$317$	26.0000	1.46031	0.730153	+	0.683284i	$0.239449\pi$
$318$	0	0
$319$	30.8328	1.72631
$320$	0	0
$321$	−10.5623	−0.589530
$322$	0	0
$323$	−59.0689	−3.28668
$324$	0	0
$325$	0	0
$326$	0	0
$327$	16.2705	0.899761
$328$	0	0
$329$	7.81966	0.431112
$330$	0	0
$331$	−2.79837	−0.153813	−0.0769063	−	0.997038i	$-0.524504\pi$
$331$	−2.79837	−0.153813	−0.0769063	+	0.997038i	$0.524504\pi$
$332$	0	0
$333$	−9.23607	−0.506133
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−12.0000	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$338$	0	0
$339$	7.85410	0.426576
$340$	0	0
$341$	−8.36068	−0.452756
$342$	0	0
$343$	15.4164	0.832408
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.7984	−0.901784	−0.450892	−	0.892579i	$-0.648894\pi$
$347$	−16.7984	−0.901784	−0.450892	+	0.892579i	$0.648894\pi$
$348$	0	0
$349$	−8.27051	−0.442710	−0.221355	−	0.975193i	$-0.571048\pi$
$349$	−8.27051	−0.442710	−0.221355	+	0.975193i	$0.571048\pi$
$350$	0	0
$351$	−4.47214	−0.238705
$352$	0	0
$353$	22.2705	1.18534	0.592670	−	0.805446i	$-0.298074\pi$
$353$	22.2705	1.18534	0.592670	+	0.805446i	$0.298074\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	8.47214	0.448393
$358$	0	0
$359$	5.70820	0.301267	0.150634	−	0.988590i	$-0.451869\pi$
$359$	5.70820	0.301267	0.150634	+	0.988590i	$0.451869\pi$
$360$	0	0
$361$	55.2705	2.90897
$362$	0	0
$363$	−5.00000	−0.262432
$364$	0	0
$365$	0	0
$366$	0	0
$367$	24.2918	1.26802	0.634011	−	0.773324i	$-0.281408\pi$
$367$	24.2918	1.26802	0.634011	+	0.773324i	$0.281408\pi$
$368$	0	0
$369$	8.94427	0.465620
$370$	0	0
$371$	1.34752	0.0699600
$372$	0	0
$373$	−14.6525	−0.758676	−0.379338	−	0.925258i	$-0.623848\pi$
$373$	−14.6525	−0.758676	−0.379338	+	0.925258i	$0.623848\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−34.4721	−1.77541
$378$	0	0
$379$	16.9443	0.870369	0.435184	−	0.900341i	$-0.356683\pi$
$379$	16.9443	0.870369	0.435184	+	0.900341i	$0.356683\pi$
$380$	0	0
$381$	18.4721	0.946356
$382$	0	0
$383$	−3.90983	−0.199783	−0.0998915	−	0.994998i	$-0.531850\pi$
$383$	−3.90983	−0.199783	−0.0998915	+	0.994998i	$0.531850\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	17.7082	0.897842	0.448921	−	0.893572i	$-0.351809\pi$
$389$	17.7082	0.897842	0.448921	+	0.893572i	$0.351809\pi$
$390$	0	0
$391$	16.3262	0.825653
$392$	0	0
$393$	−0.944272	−0.0476322
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−28.1803	−1.41433	−0.707165	−	0.707048i	$-0.750026\pi$
$397$	−28.1803	−1.41433	−0.707165	+	0.707048i	$0.750026\pi$
$398$	0	0
$399$	−10.6525	−0.533291
$400$	0	0
$401$	−0.180340	−0.00900574	−0.00450287	−	0.999990i	$-0.501433\pi$
$401$	−0.180340	−0.00900574	−0.00450287	+	0.999990i	$0.501433\pi$
$402$	0	0
$403$	9.34752	0.465633
$404$	0	0
$405$	0	0
$406$	0	0
$407$	36.9443	1.83126
$408$	0	0
$409$	28.6869	1.41848	0.709238	−	0.704969i	$-0.249039\pi$
$409$	28.6869	1.41848	0.709238	+	0.704969i	$0.249039\pi$
$410$	0	0
$411$	10.9443	0.539841
$412$	0	0
$413$	7.41641	0.364938
$414$	0	0
$415$	0	0
$416$	0	0
$417$	9.32624	0.456708
$418$	0	0
$419$	24.7639	1.20980	0.604899	−	0.796302i	$-0.293214\pi$
$419$	24.7639	1.20980	0.604899	+	0.796302i	$0.293214\pi$
$420$	0	0
$421$	−2.43769	−0.118806	−0.0594030	−	0.998234i	$-0.518920\pi$
$421$	−2.43769	−0.118806	−0.0594030	+	0.998234i	$0.518920\pi$
$422$	0	0
$423$	−6.32624	−0.307592
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.65248	0.128362
$428$	0	0
$429$	17.8885	0.863667
$430$	0	0
$431$	−16.4721	−0.793435	−0.396717	−	0.917941i	$-0.629851\pi$
$431$	−16.4721	−0.793435	−0.396717	+	0.917941i	$0.629851\pi$
$432$	0	0
$433$	−3.23607	−0.155516	−0.0777578	−	0.996972i	$-0.524776\pi$
$433$	−3.23607	−0.155516	−0.0777578	+	0.996972i	$0.524776\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−20.5279	−0.981981
$438$	0	0
$439$	−24.3607	−1.16267	−0.581336	−	0.813664i	$-0.697470\pi$
$439$	−24.3607	−1.16267	−0.581336	+	0.813664i	$0.697470\pi$
$440$	0	0
$441$	−5.47214	−0.260578
$442$	0	0
$443$	−3.56231	−0.169250	−0.0846251	−	0.996413i	$-0.526969\pi$
$443$	−3.56231	−0.169250	−0.0846251	+	0.996413i	$0.526969\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−11.4164	−0.539978
$448$	0	0
$449$	34.8328	1.64386	0.821931	−	0.569587i	$-0.192897\pi$
$449$	34.8328	1.64386	0.821931	+	0.569587i	$0.192897\pi$
$450$	0	0
$451$	−35.7771	−1.68468
$452$	0	0
$453$	5.85410	0.275050
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.7639	−1.06485	−0.532426	−	0.846477i	$-0.678720\pi$
$457$	−22.7639	−1.06485	−0.532426	+	0.846477i	$0.678720\pi$
$458$	0	0
$459$	−6.85410	−0.319922
$460$	0	0
$461$	3.88854	0.181108	0.0905538	−	0.995892i	$-0.471136\pi$
$461$	3.88854	0.181108	0.0905538	+	0.995892i	$0.471136\pi$
$462$	0	0
$463$	0.763932	0.0355029	0.0177515	−	0.999842i	$-0.494349\pi$
$463$	0.763932	0.0355029	0.0177515	+	0.999842i	$0.494349\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−21.6738	−1.00294	−0.501471	−	0.865174i	$-0.667208\pi$
$467$	−21.6738	−1.00294	−0.501471	+	0.865174i	$0.667208\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	4.18034	0.192620
$472$	0	0
$473$	16.0000	0.735681
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.09017	−0.0499155
$478$	0	0
$479$	−37.7082	−1.72293	−0.861466	−	0.507815i	$-0.830453\pi$
$479$	−37.7082	−1.72293	−0.861466	+	0.507815i	$0.830453\pi$
$480$	0	0
$481$	−41.3050	−1.88334
$482$	0	0
$483$	2.94427	0.133969
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−2.47214	−0.112023	−0.0560116	−	0.998430i	$-0.517838\pi$
$487$	−2.47214	−0.112023	−0.0560116	+	0.998430i	$0.517838\pi$
$488$	0	0
$489$	8.29180	0.374968
$490$	0	0
$491$	−13.2361	−0.597335	−0.298668	−	0.954357i	$-0.596542\pi$
$491$	−13.2361	−0.597335	−0.298668	+	0.954357i	$0.596542\pi$
$492$	0	0
$493$	−52.8328	−2.37947
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.52786	0.247959
$498$	0	0
$499$	−30.7984	−1.37872	−0.689362	−	0.724417i	$-0.742109\pi$
$499$	−30.7984	−1.37872	−0.689362	+	0.724417i	$0.742109\pi$
$500$	0	0
$501$	−18.3262	−0.818756
$502$	0	0
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−7.00000	−0.310881
$508$	0	0
$509$	−29.7082	−1.31679	−0.658396	−	0.752671i	$-0.728765\pi$
$509$	−29.7082	−1.31679	−0.658396	+	0.752671i	$0.728765\pi$
$510$	0	0
$511$	16.9443	0.749570
$512$	0	0
$513$	8.61803	0.380495
$514$	0	0
$515$	0	0
$516$	0	0
$517$	25.3050	1.11291
$518$	0	0
$519$	−5.05573	−0.221922
$520$	0	0
$521$	15.8197	0.693072	0.346536	−	0.938037i	$-0.387358\pi$
$521$	15.8197	0.693072	0.346536	+	0.938037i	$0.387358\pi$
$522$	0	0
$523$	−14.3607	−0.627949	−0.313974	−	0.949431i	$-0.601661\pi$
$523$	−14.3607	−0.627949	−0.313974	+	0.949431i	$0.601661\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.3262	0.624061
$528$	0	0
$529$	−17.3262	−0.753315
$530$	0	0
$531$	−6.00000	−0.260378
$532$	0	0
$533$	40.0000	1.73259
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.23607	−0.0533403
$538$	0	0
$539$	21.8885	0.942806
$540$	0	0
$541$	−9.43769	−0.405758	−0.202879	−	0.979204i	$-0.565030\pi$
$541$	−9.43769	−0.405758	−0.202879	+	0.979204i	$0.565030\pi$
$542$	0	0
$543$	−11.3820	−0.488447
$544$	0	0
$545$	0	0
$546$	0	0
$547$	38.1803	1.63247	0.816237	−	0.577718i	$-0.196056\pi$
$547$	38.1803	1.63247	0.816237	+	0.577718i	$0.196056\pi$
$548$	0	0
$549$	−2.14590	−0.0915847
$550$	0	0
$551$	66.4296	2.82999
$552$	0	0
$553$	−10.2918	−0.437652
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.11146	0.174208	0.0871040	−	0.996199i	$-0.472239\pi$
$557$	4.11146	0.174208	0.0871040	+	0.996199i	$0.472239\pi$
$558$	0	0
$559$	−17.8885	−0.756605
$560$	0	0
$561$	27.4164	1.15752
$562$	0	0
$563$	−29.7984	−1.25585	−0.627926	−	0.778273i	$-0.716096\pi$
$563$	−29.7984	−1.25585	−0.627926	+	0.778273i	$0.716096\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.23607	−0.0519100
$568$	0	0
$569$	−33.5967	−1.40845	−0.704224	−	0.709977i	$-0.748705\pi$
$569$	−33.5967	−1.40845	−0.704224	+	0.709977i	$0.748705\pi$
$570$	0	0
$571$	23.4164	0.979946	0.489973	−	0.871738i	$-0.337007\pi$
$571$	23.4164	0.979946	0.489973	+	0.871738i	$0.337007\pi$
$572$	0	0
$573$	21.1246	0.882493
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14.7639	0.614631	0.307315	−	0.951608i	$-0.400569\pi$
$577$	14.7639	0.614631	0.307315	+	0.951608i	$0.400569\pi$
$578$	0	0
$579$	19.4164	0.806918
$580$	0	0
$581$	−4.40325	−0.182678
$582$	0	0
$583$	4.36068	0.180601
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.49342	0.185463	0.0927317	−	0.995691i	$-0.470440\pi$
$587$	4.49342	0.185463	0.0927317	+	0.995691i	$0.470440\pi$
$588$	0	0
$589$	−18.0132	−0.742219
$590$	0	0
$591$	−22.0902	−0.908668
$592$	0	0
$593$	−37.7426	−1.54990	−0.774952	−	0.632020i	$-0.782226\pi$
$593$	−37.7426	−1.54990	−0.774952	+	0.632020i	$0.782226\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.09017	−0.249254
$598$	0	0
$599$	31.5279	1.28819	0.644097	−	0.764944i	$-0.277233\pi$
$599$	31.5279	1.28819	0.644097	+	0.764944i	$0.277233\pi$
$600$	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
$603$	−3.23607	−0.131783
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−3.70820	−0.150511	−0.0752557	−	0.997164i	$-0.523977\pi$
$607$	−3.70820	−0.150511	−0.0752557	+	0.997164i	$0.523977\pi$
$608$	0	0
$609$	−9.52786	−0.386089
$610$	0	0
$611$	−28.2918	−1.14456
$612$	0	0
$613$	14.1803	0.572739	0.286369	−	0.958119i	$-0.407552\pi$
$613$	14.1803	0.572739	0.286369	+	0.958119i	$0.407552\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.7426	0.513000	0.256500	−	0.966544i	$-0.417431\pi$
$617$	12.7426	0.513000	0.256500	+	0.966544i	$0.417431\pi$
$618$	0	0
$619$	−5.14590	−0.206831	−0.103416	−	0.994638i	$-0.532977\pi$
$619$	−5.14590	−0.206831	−0.103416	+	0.994638i	$0.532977\pi$
$620$	0	0
$621$	−2.38197	−0.0955850
$622$	0	0
$623$	14.4721	0.579814
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−34.4721	−1.37668
$628$	0	0
$629$	−63.3050	−2.52413
$630$	0	0
$631$	12.0000	0.477712	0.238856	−	0.971055i	$-0.423228\pi$
$631$	12.0000	0.477712	0.238856	+	0.971055i	$0.423228\pi$
$632$	0	0
$633$	14.2705	0.567202
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−24.4721	−0.969621
$638$	0	0
$639$	−4.47214	−0.176915
$640$	0	0
$641$	0.583592	0.0230505	0.0115253	−	0.999934i	$-0.496331\pi$
$641$	0.583592	0.0230505	0.0115253	+	0.999934i	$0.496331\pi$
$642$	0	0
$643$	15.0557	0.593740	0.296870	−	0.954918i	$-0.404057\pi$
$643$	15.0557	0.593740	0.296870	+	0.954918i	$0.404057\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8.58359	−0.337456	−0.168728	−	0.985663i	$-0.553966\pi$
$647$	−8.58359	−0.337456	−0.168728	+	0.985663i	$0.553966\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	0	0
$651$	2.58359	0.101259
$652$	0	0
$653$	−4.72949	−0.185079	−0.0925396	−	0.995709i	$-0.529498\pi$
$653$	−4.72949	−0.185079	−0.0925396	+	0.995709i	$0.529498\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−13.7082	−0.534808
$658$	0	0
$659$	10.3607	0.403595	0.201797	−	0.979427i	$-0.435322\pi$
$659$	10.3607	0.403595	0.201797	+	0.979427i	$0.435322\pi$
$660$	0	0
$661$	−9.43769	−0.367084	−0.183542	−	0.983012i	$-0.558756\pi$
$661$	−9.43769	−0.367084	−0.183542	+	0.983012i	$0.558756\pi$
$662$	0	0
$663$	−30.6525	−1.19044
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−18.3607	−0.710928
$668$	0	0
$669$	−7.23607	−0.279763
$670$	0	0
$671$	8.58359	0.331366
$672$	0	0
$673$	28.9443	1.11572	0.557860	−	0.829935i	$-0.311623\pi$
$673$	28.9443	1.11572	0.557860	+	0.829935i	$0.311623\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−0.506578	−0.0194694	−0.00973468	−	0.999953i	$-0.503099\pi$
$677$	−0.506578	−0.0194694	−0.00973468	+	0.999953i	$0.503099\pi$
$678$	0	0
$679$	6.47214	0.248378
$680$	0	0
$681$	−9.79837	−0.375475
$682$	0	0
$683$	1.03444	0.0395818	0.0197909	−	0.999804i	$-0.493700\pi$
$683$	1.03444	0.0395818	0.0197909	+	0.999804i	$0.493700\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.38197	−0.0908777
$688$	0	0
$689$	−4.87539	−0.185737
$690$	0	0
$691$	−15.6180	−0.594138	−0.297069	−	0.954856i	$-0.596009\pi$
$691$	−15.6180	−0.594138	−0.297069	+	0.954856i	$0.596009\pi$
$692$	0	0
$693$	4.94427	0.187817
$694$	0	0
$695$	0	0
$696$	0	0
$697$	61.3050	2.32209
$698$	0	0
$699$	−27.8885	−1.05484
$700$	0	0
$701$	34.5410	1.30460	0.652298	−	0.757962i	$-0.273805\pi$
$701$	34.5410	1.30460	0.652298	+	0.757962i	$0.273805\pi$
$702$	0	0
$703$	79.5967	3.00205
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.360680	0.0135648
$708$	0	0
$709$	−36.3262	−1.36426	−0.682130	−	0.731231i	$-0.738946\pi$
$709$	−36.3262	−1.36426	−0.682130	+	0.731231i	$0.738946\pi$
$710$	0	0
$711$	8.32624	0.312258
$712$	0	0
$713$	4.97871	0.186454
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.65248	−0.323133
$718$	0	0
$719$	−8.18034	−0.305075	−0.152538	−	0.988298i	$-0.548745\pi$
$719$	−8.18034	−0.305075	−0.152538	+	0.988298i	$0.548745\pi$
$720$	0	0
$721$	−18.1115	−0.674506
$722$	0	0
$723$	−14.3820	−0.534871
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9.23607	0.342547	0.171273	−	0.985224i	$-0.445212\pi$
$727$	9.23607	0.342547	0.171273	+	0.985224i	$0.445212\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−27.4164	−1.01403
$732$	0	0
$733$	−28.0689	−1.03675	−0.518374	−	0.855154i	$-0.673462\pi$
$733$	−28.0689	−1.03675	−0.518374	+	0.855154i	$0.673462\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.9443	0.476808
$738$	0	0
$739$	5.32624	0.195929	0.0979644	−	0.995190i	$-0.468767\pi$
$739$	5.32624	0.195929	0.0979644	+	0.995190i	$0.468767\pi$
$740$	0	0
$741$	38.5410	1.41584
$742$	0	0
$743$	47.4164	1.73954	0.869770	−	0.493458i	$-0.164267\pi$
$743$	47.4164	1.73954	0.869770	+	0.493458i	$0.164267\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.56231	0.130338
$748$	0	0
$749$	−13.0557	−0.477046
$750$	0	0
$751$	−7.67376	−0.280020	−0.140010	−	0.990150i	$-0.544713\pi$
$751$	−7.67376	−0.280020	−0.140010	+	0.990150i	$0.544713\pi$
$752$	0	0
$753$	−18.0000	−0.655956
$754$	0	0
$755$	0	0
$756$	0	0
$757$	28.4721	1.03484	0.517419	−	0.855732i	$-0.326893\pi$
$757$	28.4721	1.03484	0.517419	+	0.855732i	$0.326893\pi$
$758$	0	0
$759$	9.52786	0.345840
$760$	0	0
$761$	−53.7771	−1.94942	−0.974709	−	0.223478i	$-0.928259\pi$
$761$	−53.7771	−1.94942	−0.974709	+	0.223478i	$0.928259\pi$
$762$	0	0
$763$	20.1115	0.728084
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−26.8328	−0.968877
$768$	0	0
$769$	3.38197	0.121957	0.0609784	−	0.998139i	$-0.480578\pi$
$769$	3.38197	0.121957	0.0609784	+	0.998139i	$0.480578\pi$
$770$	0	0
$771$	−3.20163	−0.115304
$772$	0	0
$773$	−10.0344	−0.360914	−0.180457	−	0.983583i	$-0.557758\pi$
$773$	−10.0344	−0.360914	−0.180457	+	0.983583i	$0.557758\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−11.4164	−0.409561
$778$	0	0
$779$	−77.0820	−2.76175
$780$	0	0
$781$	17.8885	0.640102
$782$	0	0
$783$	7.70820	0.275469
$784$	0	0
$785$	0	0
$786$	0	0
$787$	33.7082	1.20157	0.600784	−	0.799412i	$-0.294855\pi$
$787$	33.7082	1.20157	0.600784	+	0.799412i	$0.294855\pi$
$788$	0	0
$789$	−24.0902	−0.857633
$790$	0	0
$791$	9.70820	0.345184
$792$	0	0
$793$	−9.59675	−0.340791
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−51.2148	−1.81412	−0.907060	−	0.421001i	$-0.861679\pi$
$797$	−51.2148	−1.81412	−0.907060	+	0.421001i	$0.861679\pi$
$798$	0	0
$799$	−43.3607	−1.53399
$800$	0	0
$801$	−11.7082	−0.413689
$802$	0	0
$803$	54.8328	1.93501
$804$	0	0
$805$	0	0
$806$	0	0
$807$	19.1246	0.673218
$808$	0	0
$809$	28.7639	1.01129	0.505643	−	0.862743i	$-0.331255\pi$
$809$	28.7639	1.01129	0.505643	+	0.862743i	$0.331255\pi$
$810$	0	0
$811$	36.1033	1.26776	0.633880	−	0.773432i	$-0.281461\pi$
$811$	36.1033	1.26776	0.633880	+	0.773432i	$0.281461\pi$
$812$	0	0
$813$	28.5623	1.00172
$814$	0	0
$815$	0	0
$816$	0	0
$817$	34.4721	1.20603
$818$	0	0
$819$	−5.52786	−0.193159
$820$	0	0
$821$	−1.63932	−0.0572127	−0.0286063	−	0.999591i	$-0.509107\pi$
$821$	−1.63932	−0.0572127	−0.0286063	+	0.999591i	$0.509107\pi$
$822$	0	0
$823$	−44.7639	−1.56037	−0.780186	−	0.625547i	$-0.784876\pi$
$823$	−44.7639	−1.56037	−0.780186	+	0.625547i	$0.784876\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−21.0902	−0.733377	−0.366689	−	0.930344i	$-0.619509\pi$
$827$	−21.0902	−0.733377	−0.366689	+	0.930344i	$0.619509\pi$
$828$	0	0
$829$	−7.67376	−0.266521	−0.133260	−	0.991081i	$-0.542545\pi$
$829$	−7.67376	−0.266521	−0.133260	+	0.991081i	$0.542545\pi$
$830$	0	0
$831$	10.7639	0.373397
$832$	0	0
$833$	−37.5066	−1.29953
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.09017	−0.0722468
$838$	0	0
$839$	−36.6525	−1.26538	−0.632692	−	0.774404i	$-0.718050\pi$
$839$	−36.6525	−1.26538	−0.632692	+	0.774404i	$0.718050\pi$
$840$	0	0
$841$	30.4164	1.04884
$842$	0	0
$843$	−3.23607	−0.111456
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−6.18034	−0.212359
$848$	0	0
$849$	−32.0689	−1.10060
$850$	0	0
$851$	−22.0000	−0.754150
$852$	0	0
$853$	−49.9574	−1.71051	−0.855255	−	0.518208i	$-0.826599\pi$
$853$	−49.9574	−1.71051	−0.855255	+	0.518208i	$0.826599\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−45.8115	−1.56489	−0.782446	−	0.622718i	$-0.786028\pi$
$857$	−45.8115	−1.56489	−0.782446	+	0.622718i	$0.786028\pi$
$858$	0	0
$859$	−17.8885	−0.610349	−0.305175	−	0.952296i	$-0.598715\pi$
$859$	−17.8885	−0.610349	−0.305175	+	0.952296i	$0.598715\pi$
$860$	0	0
$861$	11.0557	0.376778
$862$	0	0
$863$	−45.8885	−1.56206	−0.781032	−	0.624491i	$-0.785307\pi$
$863$	−45.8885	−1.56206	−0.781032	+	0.624491i	$0.785307\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−29.9787	−1.01813
$868$	0	0
$869$	−33.3050	−1.12979
$870$	0	0
$871$	−14.4721	−0.490370
$872$	0	0
$873$	−5.23607	−0.177214
$874$	0	0
$875$	0	0
$876$	0	0
$877$	24.2492	0.818838	0.409419	−	0.912346i	$-0.365731\pi$
$877$	24.2492	0.818838	0.409419	+	0.912346i	$0.365731\pi$
$878$	0	0
$879$	17.3262	0.584400
$880$	0	0
$881$	4.06888	0.137084	0.0685421	−	0.997648i	$-0.478165\pi$
$881$	4.06888	0.137084	0.0685421	+	0.997648i	$0.478165\pi$
$882$	0	0
$883$	20.2918	0.682873	0.341437	−	0.939905i	$-0.389087\pi$
$883$	20.2918	0.682873	0.341437	+	0.939905i	$0.389087\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.79837	−0.295420	−0.147710	−	0.989031i	$-0.547190\pi$
$887$	−8.79837	−0.295420	−0.147710	+	0.989031i	$0.547190\pi$
$888$	0	0
$889$	22.8328	0.765788
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	54.5197	1.82443
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−10.6525	−0.355676
$898$	0	0
$899$	−16.1115	−0.537347
$900$	0	0
$901$	−7.47214	−0.248933
$902$	0	0
$903$	−4.94427	−0.164535
$904$	0	0
$905$	0	0
$906$	0	0
$907$	33.7771	1.12155	0.560775	−	0.827968i	$-0.310503\pi$
$907$	33.7771	1.12155	0.560775	+	0.827968i	$0.310503\pi$
$908$	0	0
$909$	−0.291796	−0.00967826
$910$	0	0
$911$	6.94427	0.230074	0.115037	−	0.993361i	$-0.463301\pi$
$911$	6.94427	0.230074	0.115037	+	0.993361i	$0.463301\pi$
$912$	0	0
$913$	−14.2492	−0.471580
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.16718	−0.0385438
$918$	0	0
$919$	−26.6738	−0.879886	−0.439943	−	0.898026i	$-0.645001\pi$
$919$	−26.6738	−0.879886	−0.439943	+	0.898026i	$0.645001\pi$
$920$	0	0
$921$	−32.4721	−1.06999
$922$	0	0
$923$	−20.0000	−0.658308
$924$	0	0
$925$	0	0
$926$	0	0
$927$	14.6525	0.481250
$928$	0	0
$929$	56.5410	1.85505	0.927525	−	0.373760i	$-0.121932\pi$
$929$	56.5410	1.85505	0.927525	+	0.373760i	$0.121932\pi$
$930$	0	0
$931$	47.1591	1.54558
$932$	0	0
$933$	32.1803	1.05354
$934$	0	0
$935$	0	0
$936$	0	0
$937$	14.1803	0.463252	0.231626	−	0.972805i	$-0.425596\pi$
$937$	14.1803	0.463252	0.231626	+	0.972805i	$0.425596\pi$
$938$	0	0
$939$	2.58359	0.0843123
$940$	0	0
$941$	−41.3050	−1.34650	−0.673251	−	0.739414i	$-0.735103\pi$
$941$	−41.3050	−1.34650	−0.673251	+	0.739414i	$0.735103\pi$
$942$	0	0
$943$	21.3050	0.693785
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−57.5755	−1.87095	−0.935476	−	0.353391i	$-0.885028\pi$
$947$	−57.5755	−1.87095	−0.935476	+	0.353391i	$0.885028\pi$
$948$	0	0
$949$	−61.3050	−1.99004
$950$	0	0
$951$	−26.0000	−0.843108
$952$	0	0
$953$	−33.6869	−1.09123	−0.545613	−	0.838037i	$-0.683703\pi$
$953$	−33.6869	−1.09123	−0.545613	+	0.838037i	$0.683703\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−30.8328	−0.996683
$958$	0	0
$959$	13.5279	0.436838
$960$	0	0
$961$	−26.6312	−0.859071
$962$	0	0
$963$	10.5623	0.340366
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−21.7082	−0.698089	−0.349044	−	0.937106i	$-0.613494\pi$
$967$	−21.7082	−0.698089	−0.349044	+	0.937106i	$0.613494\pi$
$968$	0	0
$969$	59.0689	1.89757
$970$	0	0
$971$	34.8328	1.11784	0.558919	−	0.829222i	$-0.311216\pi$
$971$	34.8328	1.11784	0.558919	+	0.829222i	$0.311216\pi$
$972$	0	0
$973$	11.5279	0.369566
$974$	0	0
$975$	0	0
$976$	0	0
$977$	28.3820	0.908020	0.454010	−	0.890997i	$-0.349993\pi$
$977$	28.3820	0.908020	0.454010	+	0.890997i	$0.349993\pi$
$978$	0	0
$979$	46.8328	1.49678
$980$	0	0
$981$	−16.2705	−0.519477
$982$	0	0
$983$	−18.9098	−0.603130	−0.301565	−	0.953446i	$-0.597509\pi$
$983$	−18.9098	−0.603130	−0.301565	+	0.953446i	$0.597509\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−7.81966	−0.248903
$988$	0	0
$989$	−9.52786	−0.302968
$990$	0	0
$991$	−15.7426	−0.500082	−0.250041	−	0.968235i	$-0.580444\pi$
$991$	−15.7426	−0.500082	−0.250041	+	0.968235i	$0.580444\pi$
$992$	0	0
$993$	2.79837	0.0888037
$994$	0	0
$995$	0	0
$996$	0	0
$997$	38.7214	1.22632	0.613159	−	0.789960i	$-0.289899\pi$
$997$	38.7214	1.22632	0.613159	+	0.789960i	$0.289899\pi$
$998$	0	0
$999$	9.23607	0.292216

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1500.2.a.b.1.1	✓	2	1.1	even	1	trivial
1500.2.a.f.1.2	yes	2	5.4	even	2
1500.2.d.a.1249.2		4	5.3	odd	4
1500.2.d.a.1249.3		4	5.2	odd	4
4500.2.a.g.1.2		2	15.14	odd	2
4500.2.a.k.1.1		2	3.2	odd	2
4500.2.d.j.4249.2		4	15.2	even	4
4500.2.d.j.4249.3		4	15.8	even	4
6000.2.a.j.1.1		2	20.19	odd	2
6000.2.a.t.1.2		2	4.3	odd	2
6000.2.f.m.1249.2		4	20.7	even	4
6000.2.f.m.1249.3		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.23607 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.23607 q^{7} +1.00000 q^{9} -4.00000 q^{11} +4.47214 q^{13} +6.85410 q^{17} -8.61803 q^{19} +1.23607 q^{21} +2.38197 q^{23} -1.00000 q^{27} -7.70820 q^{29} +2.09017 q^{31} +4.00000 q^{33} -9.23607 q^{37} -4.47214 q^{39} +8.94427 q^{41} -4.00000 q^{43} -6.32624 q^{47} -5.47214 q^{49} -6.85410 q^{51} -1.09017 q^{53} +8.61803 q^{57} -6.00000 q^{59} -2.14590 q^{61} -1.23607 q^{63} -3.23607 q^{67} -2.38197 q^{69} -4.47214 q^{71} -13.7082 q^{73} +4.94427 q^{77} +8.32624 q^{79} +1.00000 q^{81} +3.56231 q^{83} +7.70820 q^{87} -11.7082 q^{89} -5.52786 q^{91} -2.09017 q^{93} -5.23607 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} - 8 q^{11} + 7 q^{17} - 15 q^{19} - 2 q^{21} + 7 q^{23} - 2 q^{27} - 2 q^{29} - 7 q^{31} + 8 q^{33} - 14 q^{37} - 8 q^{43} + 3 q^{47} - 2 q^{49} - 7 q^{51} + 9 q^{53} + 15 q^{57} - 12 q^{59} - 11 q^{61} + 2 q^{63} - 2 q^{67} - 7 q^{69} - 14 q^{73} - 8 q^{77} + q^{79} + 2 q^{81} - 13 q^{83} + 2 q^{87} - 10 q^{89} - 20 q^{91} + 7 q^{93} - 6 q^{97} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 - 8 * q^11 + 7 * q^17 - 15 * q^19 - 2 * q^21 + 7 * q^23 - 2 * q^27 - 2 * q^29 - 7 * q^31 + 8 * q^33 - 14 * q^37 - 8 * q^43 + 3 * q^47 - 2 * q^49 - 7 * q^51 + 9 * q^53 + 15 * q^57 - 12 * q^59 - 11 * q^61 + 2 * q^63 - 2 * q^67 - 7 * q^69 - 14 * q^73 - 8 * q^77 + q^79 + 2 * q^81 - 13 * q^83 + 2 * q^87 - 10 * q^89 - 20 * q^91 + 7 * q^93 - 6 * q^97 - 8 * q^99

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