LMFDB - Embedded newform 4000.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,2,Mod(1,4000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, 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("4000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4000.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:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.26208800000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} + 34x^{4} - 30x^{2} + 5 $ x^8 - 11x^6 + 34x^4 - 30*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.464606$ of defining polynomial
Character	$\chi$	$=$	4000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.79703 q^{3} +2.30294 q^{7} +4.82335 q^{9} +O(q^{10})$	$q-2.79703 q^{3} +2.30294 q^{7} +4.82335 q^{9} -5.01977 q^{11} -1.36296 q^{13} +7.56828 q^{17} +4.96081 q^{19} -6.44139 q^{21} -0.225160 q^{23} -5.09997 q^{27} -8.18632 q^{29} -6.87819 q^{31} +14.0404 q^{33} +6.20532 q^{37} +3.81224 q^{39} +4.69646 q^{41} -4.30052 q^{43} +4.39592 q^{47} -1.69646 q^{49} -21.1687 q^{51} +10.4721 q^{53} -13.8755 q^{57} +0.0364368 q^{59} +9.99067 q^{61} +11.1079 q^{63} -14.6819 q^{67} +0.629779 q^{69} -1.24396 q^{71} -3.30147 q^{73} -11.5602 q^{77} -8.12215 q^{79} -0.205320 q^{81} +13.8824 q^{83} +22.8973 q^{87} -0.854102 q^{89} -3.13882 q^{91} +19.2385 q^{93} -5.51981 q^{97} -24.2121 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} + 8 q^{13} + 20 q^{17} - 12 q^{21} - 16 q^{29} + 36 q^{33} + 28 q^{37} + 8 q^{41} + 16 q^{49} + 48 q^{53} + 20 q^{57} - 28 q^{61} - 28 q^{69} + 44 q^{77} + 20 q^{81} + 20 q^{89} + 4 q^{93} + 16 q^{97}+O(q^{100}) $ 8 * q + 8 * q^9 + 8 * q^13 + 20 * q^17 - 12 * q^21 - 16 * q^29 + 36 * q^33 + 28 * q^37 + 8 * q^41 + 16 * q^49 + 48 * q^53 + 20 * q^57 - 28 * q^61 - 28 * q^69 + 44 * q^77 + 20 * q^81 + 20 * q^89 + 4 * q^93 + 16 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.79703	−1.61486	−0.807432	−	0.589961i	$-0.799143\pi$
$3$	−2.79703	−1.61486	−0.807432	+	0.589961i	$0.799143\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.30294	0.870430	0.435215	−	0.900327i	$-0.356672\pi$
$7$	2.30294	0.870430	0.435215	+	0.900327i	$0.356672\pi$
$8$	0	0
$9$	4.82335	1.60778
$10$	0	0
$11$	−5.01977	−1.51352	−0.756758	−	0.653695i	$-0.773218\pi$
$11$	−5.01977	−1.51352	−0.756758	+	0.653695i	$0.773218\pi$
$12$	0	0
$13$	−1.36296	−0.378018	−0.189009	−	0.981975i	$-0.560528\pi$
$13$	−1.36296	−0.378018	−0.189009	+	0.981975i	$0.560528\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.56828	1.83558	0.917789	−	0.397068i	$-0.129972\pi$
$17$	7.56828	1.83558	0.917789	+	0.397068i	$0.129972\pi$
$18$	0	0
$19$	4.96081	1.13809	0.569044	−	0.822307i	$-0.307313\pi$
$19$	4.96081	1.13809	0.569044	+	0.822307i	$0.307313\pi$
$20$	0	0
$21$	−6.44139	−1.40563
$22$	0	0
$23$	−0.225160	−0.0469491	−0.0234746	−	0.999724i	$-0.507473\pi$
$23$	−0.225160	−0.0469491	−0.0234746	+	0.999724i	$0.507473\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.09997	−0.981489
$28$	0	0
$29$	−8.18632	−1.52016	−0.760080	−	0.649829i	$-0.774840\pi$
$29$	−8.18632	−1.52016	−0.760080	+	0.649829i	$0.774840\pi$
$30$	0	0
$31$	−6.87819	−1.23536	−0.617680	−	0.786430i	$-0.711927\pi$
$31$	−6.87819	−1.23536	−0.617680	+	0.786430i	$0.711927\pi$
$32$	0	0
$33$	14.0404	2.44412
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.20532	1.02015	0.510074	−	0.860130i	$-0.329618\pi$
$37$	6.20532	1.02015	0.510074	+	0.860130i	$0.329618\pi$
$38$	0	0
$39$	3.81224	0.610447
$40$	0	0
$41$	4.69646	0.733464	0.366732	−	0.930327i	$-0.380477\pi$
$41$	4.69646	0.733464	0.366732	+	0.930327i	$0.380477\pi$
$42$	0	0
$43$	−4.30052	−0.655824	−0.327912	−	0.944708i	$-0.606345\pi$
$43$	−4.30052	−0.655824	−0.327912	+	0.944708i	$0.606345\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.39592	0.641210	0.320605	−	0.947213i	$-0.396114\pi$
$47$	4.39592	0.641210	0.320605	+	0.947213i	$0.396114\pi$
$48$	0	0
$49$	−1.69646	−0.242351
$50$	0	0
$51$	−21.1687	−2.96421
$52$	0	0
$53$	10.4721	1.43846	0.719229	−	0.694773i	$-0.244495\pi$
$53$	10.4721	1.43846	0.719229	+	0.694773i	$0.244495\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−13.8755	−1.83786
$58$	0	0
$59$	0.0364368	0.00474367	0.00237183	−	0.999997i	$-0.499245\pi$
$59$	0.0364368	0.00474367	0.00237183	+	0.999997i	$0.499245\pi$
$60$	0	0
$61$	9.99067	1.27917	0.639587	−	0.768719i	$-0.279105\pi$
$61$	9.99067	1.27917	0.639587	+	0.768719i	$0.279105\pi$
$62$	0	0
$63$	11.1079	1.39946
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−14.6819	−1.79367	−0.896837	−	0.442361i	$-0.854141\pi$
$67$	−14.6819	−1.79367	−0.896837	+	0.442361i	$0.854141\pi$
$68$	0	0
$69$	0.629779	0.0758165
$70$	0	0
$71$	−1.24396	−0.147631	−0.0738156	−	0.997272i	$-0.523518\pi$
$71$	−1.24396	−0.147631	−0.0738156	+	0.997272i	$0.523518\pi$
$72$	0	0
$73$	−3.30147	−0.386407	−0.193204	−	0.981159i	$-0.561888\pi$
$73$	−3.30147	−0.386407	−0.193204	+	0.981159i	$0.561888\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−11.5602	−1.31741
$78$	0	0
$79$	−8.12215	−0.913814	−0.456907	−	0.889515i	$-0.651043\pi$
$79$	−8.12215	−0.913814	−0.456907	+	0.889515i	$0.651043\pi$
$80$	0	0
$81$	−0.205320	−0.0228133
$82$	0	0
$83$	13.8824	1.52379	0.761896	−	0.647699i	$-0.224269\pi$
$83$	13.8824	1.52379	0.761896	+	0.647699i	$0.224269\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	22.8973	2.45485
$88$	0	0
$89$	−0.854102	−0.0905346	−0.0452673	−	0.998975i	$-0.514414\pi$
$89$	−0.854102	−0.0905346	−0.0452673	+	0.998975i	$0.514414\pi$
$90$	0	0
$91$	−3.13882	−0.329038
$92$	0	0
$93$	19.2385	1.99494
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−5.51981	−0.560452	−0.280226	−	0.959934i	$-0.590409\pi$
$97$	−5.51981	−0.560452	−0.280226	+	0.959934i	$0.590409\pi$
$98$	0	0
$99$	−24.2121	−2.43341
$100$	0	0
$101$	−19.4936	−1.93968	−0.969840	−	0.243741i	$-0.921625\pi$
$101$	−19.4936	−1.93968	−0.969840	+	0.243741i	$0.921625\pi$
$102$	0	0
$103$	−2.27231	−0.223897	−0.111949	−	0.993714i	$-0.535709\pi$
$103$	−2.27231	−0.223897	−0.111949	+	0.993714i	$0.535709\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.03615	−0.776884	−0.388442	−	0.921473i	$-0.626987\pi$
$107$	−8.03615	−0.776884	−0.388442	+	0.921473i	$0.626987\pi$
$108$	0	0
$109$	3.02867	0.290094	0.145047	−	0.989425i	$-0.453667\pi$
$109$	3.02867	0.290094	0.145047	+	0.989425i	$0.453667\pi$
$110$	0	0
$111$	−17.3564	−1.64740
$112$	0	0
$113$	−4.24574	−0.399405	−0.199703	−	0.979857i	$-0.563998\pi$
$113$	−4.24574	−0.399405	−0.199703	+	0.979857i	$0.563998\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−6.57405	−0.607771
$118$	0	0
$119$	17.4293	1.59774
$120$	0	0
$121$	14.1981	1.29073
$122$	0	0
$123$	−13.1361	−1.18444
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.19814	0.106317	0.0531587	−	0.998586i	$-0.483071\pi$
$127$	1.19814	0.106317	0.0531587	+	0.998586i	$0.483071\pi$
$128$	0	0
$129$	12.0287	1.05907
$130$	0	0
$131$	19.3103	1.68715	0.843573	−	0.537015i	$-0.180448\pi$
$131$	19.3103	1.68715	0.843573	+	0.537015i	$0.180448\pi$
$132$	0	0
$133$	11.4245	0.990626
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.3145	1.30841	0.654203	−	0.756319i	$-0.273004\pi$
$137$	15.3145	1.30841	0.654203	+	0.756319i	$0.273004\pi$
$138$	0	0
$139$	18.8715	1.60066	0.800332	−	0.599558i	$-0.204657\pi$
$139$	18.8715	1.60066	0.800332	+	0.599558i	$0.204657\pi$
$140$	0	0
$141$	−12.2955	−1.03547
$142$	0	0
$143$	6.84175	0.572136
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.74504	0.391364
$148$	0	0
$149$	19.4451	1.59300	0.796502	−	0.604636i	$-0.206682\pi$
$149$	19.4451	1.59300	0.796502	+	0.604636i	$0.206682\pi$
$150$	0	0
$151$	−18.0438	−1.46838	−0.734191	−	0.678943i	$-0.762438\pi$
$151$	−18.0438	−1.46838	−0.734191	+	0.678943i	$0.762438\pi$
$152$	0	0
$153$	36.5045	2.95121
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1.37599	−0.109816	−0.0549080	−	0.998491i	$-0.517487\pi$
$157$	−1.37599	−0.109816	−0.0549080	+	0.998491i	$0.517487\pi$
$158$	0	0
$159$	−29.2908	−2.32291
$160$	0	0
$161$	−0.518531	−0.0408659
$162$	0	0
$163$	4.77209	0.373779	0.186889	−	0.982381i	$-0.440159\pi$
$163$	4.77209	0.373779	0.186889	+	0.982381i	$0.440159\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.8140	0.991580	0.495790	−	0.868442i	$-0.334879\pi$
$167$	12.8140	0.991580	0.495790	+	0.868442i	$0.334879\pi$
$168$	0	0
$169$	−11.1423	−0.857103
$170$	0	0
$171$	23.9277	1.82980
$172$	0	0
$173$	13.7866	1.04818	0.524089	−	0.851664i	$-0.324406\pi$
$173$	13.7866	1.04818	0.524089	+	0.851664i	$0.324406\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.101915	−0.00766038
$178$	0	0
$179$	−8.92740	−0.667265	−0.333633	−	0.942703i	$-0.608275\pi$
$179$	−8.92740	−0.667265	−0.333633	+	0.942703i	$0.608275\pi$
$180$	0	0
$181$	−5.01174	−0.372520	−0.186260	−	0.982500i	$-0.559637\pi$
$181$	−5.01174	−0.372520	−0.186260	+	0.982500i	$0.559637\pi$
$182$	0	0
$183$	−27.9442	−2.06569
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−37.9910	−2.77818
$188$	0	0
$189$	−11.7449	−0.854318
$190$	0	0
$191$	22.2808	1.61218	0.806092	−	0.591791i	$-0.201579\pi$
$191$	22.2808	1.61218	0.806092	+	0.591791i	$0.201579\pi$
$192$	0	0
$193$	15.5602	1.12005	0.560025	−	0.828476i	$-0.310792\pi$
$193$	15.5602	1.12005	0.560025	+	0.828476i	$0.310792\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.5126	1.46146	0.730729	−	0.682667i	$-0.239180\pi$
$197$	20.5126	1.46146	0.730729	+	0.682667i	$0.239180\pi$
$198$	0	0
$199$	−1.83591	−0.130144	−0.0650719	−	0.997881i	$-0.520728\pi$
$199$	−1.83591	−0.130144	−0.0650719	+	0.997881i	$0.520728\pi$
$200$	0	0
$201$	41.0655	2.89654
$202$	0	0
$203$	−18.8526	−1.32319
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.08603	−0.0754841
$208$	0	0
$209$	−24.9021	−1.72252
$210$	0	0
$211$	4.99725	0.344025	0.172012	−	0.985095i	$-0.444973\pi$
$211$	4.99725	0.344025	0.172012	+	0.985095i	$0.444973\pi$
$212$	0	0
$213$	3.47939	0.238404
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−15.8401	−1.07529
$218$	0	0
$219$	9.23429	0.623995
$220$	0	0
$221$	−10.3153	−0.693881
$222$	0	0
$223$	−8.46769	−0.567039	−0.283519	−	0.958967i	$-0.591502\pi$
$223$	−8.46769	−0.567039	−0.283519	+	0.958967i	$0.591502\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.5737	0.768171	0.384086	−	0.923298i	$-0.374517\pi$
$227$	11.5737	0.768171	0.384086	+	0.923298i	$0.374517\pi$
$228$	0	0
$229$	4.61078	0.304689	0.152344	−	0.988327i	$-0.451318\pi$
$229$	4.61078	0.304689	0.152344	+	0.988327i	$0.451318\pi$
$230$	0	0
$231$	32.3343	2.12744
$232$	0	0
$233$	9.95958	0.652474	0.326237	−	0.945288i	$-0.394219\pi$
$233$	9.95958	0.652474	0.326237	+	0.945288i	$0.394219\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	22.7179	1.47568
$238$	0	0
$239$	27.3735	1.77064	0.885321	−	0.464981i	$-0.153939\pi$
$239$	27.3735	1.77064	0.885321	+	0.464981i	$0.153939\pi$
$240$	0	0
$241$	−6.23399	−0.401567	−0.200783	−	0.979636i	$-0.564349\pi$
$241$	−6.23399	−0.401567	−0.200783	+	0.979636i	$0.564349\pi$
$242$	0	0
$243$	15.8742	1.01833
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.76140	−0.430218
$248$	0	0
$249$	−38.8295	−2.46072
$250$	0	0
$251$	−8.89096	−0.561193	−0.280596	−	0.959826i	$-0.590532\pi$
$251$	−8.89096	−0.561193	−0.280596	+	0.959826i	$0.590532\pi$
$252$	0	0
$253$	1.13025	0.0710583
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.6644	0.789985	0.394993	−	0.918684i	$-0.370747\pi$
$257$	12.6644	0.789985	0.394993	+	0.918684i	$0.370747\pi$
$258$	0	0
$259$	14.2905	0.887968
$260$	0	0
$261$	−39.4855	−2.44409
$262$	0	0
$263$	−20.0945	−1.23908	−0.619540	−	0.784965i	$-0.712681\pi$
$263$	−20.0945	−1.23908	−0.619540	+	0.784965i	$0.712681\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.38895	0.146201
$268$	0	0
$269$	16.5530	1.00925	0.504626	−	0.863338i	$-0.331630\pi$
$269$	16.5530	1.00925	0.504626	+	0.863338i	$0.331630\pi$
$270$	0	0
$271$	24.3075	1.47658	0.738288	−	0.674486i	$-0.235635\pi$
$271$	24.3075	1.47658	0.738288	+	0.674486i	$0.235635\pi$
$272$	0	0
$273$	8.77937	0.531352
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.37022	0.142413	0.0712064	−	0.997462i	$-0.477315\pi$
$277$	2.37022	0.142413	0.0712064	+	0.997462i	$0.477315\pi$
$278$	0	0
$279$	−33.1760	−1.98619
$280$	0	0
$281$	10.2575	0.611910	0.305955	−	0.952046i	$-0.401024\pi$
$281$	10.2575	0.611910	0.305955	+	0.952046i	$0.401024\pi$
$282$	0	0
$283$	25.2405	1.50039	0.750195	−	0.661217i	$-0.229960\pi$
$283$	25.2405	1.50039	0.750195	+	0.661217i	$0.229960\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.8157	0.638429
$288$	0	0
$289$	40.2789	2.36935
$290$	0	0
$291$	15.4391	0.905054
$292$	0	0
$293$	−29.4438	−1.72013	−0.860063	−	0.510189i	$-0.829576\pi$
$293$	−29.4438	−1.72013	−0.860063	+	0.510189i	$0.829576\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	25.6006	1.48550
$298$	0	0
$299$	0.306885	0.0177476
$300$	0	0
$301$	−9.90385	−0.570849
$302$	0	0
$303$	54.5240	3.13232
$304$	0	0
$305$	0	0
$306$	0	0
$307$	19.3446	1.10406	0.552028	−	0.833825i	$-0.313854\pi$
$307$	19.3446	1.10406	0.552028	+	0.833825i	$0.313854\pi$
$308$	0	0
$309$	6.35570	0.361563
$310$	0	0
$311$	11.2245	0.636485	0.318243	−	0.948009i	$-0.396907\pi$
$311$	11.2245	0.636485	0.318243	+	0.948009i	$0.396907\pi$
$312$	0	0
$313$	13.8625	0.783554	0.391777	−	0.920060i	$-0.371860\pi$
$313$	13.8625	0.783554	0.391777	+	0.920060i	$0.371860\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.62173	−0.540410	−0.270205	−	0.962803i	$-0.587091\pi$
$317$	−9.62173	−0.540410	−0.270205	+	0.962803i	$0.587091\pi$
$318$	0	0
$319$	41.0934	2.30079
$320$	0	0
$321$	22.4773	1.25456
$322$	0	0
$323$	37.5448	2.08905
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−8.47128	−0.468463
$328$	0	0
$329$	10.1235	0.558129
$330$	0	0
$331$	28.6399	1.57419	0.787097	−	0.616830i	$-0.211583\pi$
$331$	28.6399	1.57419	0.787097	+	0.616830i	$0.211583\pi$
$332$	0	0
$333$	29.9305	1.64018
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.67169	−0.0910626	−0.0455313	−	0.998963i	$-0.514498\pi$
$337$	−1.67169	−0.0910626	−0.0455313	+	0.998963i	$0.514498\pi$
$338$	0	0
$339$	11.8754	0.644985
$340$	0	0
$341$	34.5269	1.86974
$342$	0	0
$343$	−20.0274	−1.08138
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.4862	0.938709	0.469354	−	0.883010i	$-0.344487\pi$
$347$	17.4862	0.938709	0.469354	+	0.883010i	$0.344487\pi$
$348$	0	0
$349$	−8.43044	−0.451271	−0.225635	−	0.974212i	$-0.572446\pi$
$349$	−8.43044	−0.451271	−0.225635	+	0.974212i	$0.572446\pi$
$350$	0	0
$351$	6.95107	0.371020
$352$	0	0
$353$	4.54168	0.241729	0.120865	−	0.992669i	$-0.461433\pi$
$353$	4.54168	0.241729	0.120865	+	0.992669i	$0.461433\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−48.7502	−2.58014
$358$	0	0
$359$	26.1295	1.37906	0.689531	−	0.724256i	$-0.257817\pi$
$359$	26.1295	1.37906	0.689531	+	0.724256i	$0.257817\pi$
$360$	0	0
$361$	5.60965	0.295245
$362$	0	0
$363$	−39.7123	−2.08436
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5.72008	0.298586	0.149293	−	0.988793i	$-0.452300\pi$
$367$	5.72008	0.298586	0.149293	+	0.988793i	$0.452300\pi$
$368$	0	0
$369$	22.6527	1.17925
$370$	0	0
$371$	24.1167	1.25208
$372$	0	0
$373$	21.7026	1.12372	0.561858	−	0.827234i	$-0.310087\pi$
$373$	21.7026	1.12372	0.561858	+	0.827234i	$0.310087\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11.1576	0.574648
$378$	0	0
$379$	−24.8266	−1.27526	−0.637628	−	0.770345i	$-0.720084\pi$
$379$	−24.8266	−1.27526	−0.637628	+	0.770345i	$0.720084\pi$
$380$	0	0
$381$	−3.35122	−0.171688
$382$	0	0
$383$	−17.3245	−0.885242	−0.442621	−	0.896709i	$-0.645951\pi$
$383$	−17.3245	−0.885242	−0.442621	+	0.896709i	$0.645951\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−20.7429	−1.05442
$388$	0	0
$389$	−15.2445	−0.772925	−0.386462	−	0.922305i	$-0.626303\pi$
$389$	−15.2445	−0.772925	−0.386462	+	0.922305i	$0.626303\pi$
$390$	0	0
$391$	−1.70408	−0.0861788
$392$	0	0
$393$	−54.0113	−2.72451
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−23.2255	−1.16565	−0.582826	−	0.812597i	$-0.698053\pi$
$397$	−23.2255	−1.16565	−0.582826	+	0.812597i	$0.698053\pi$
$398$	0	0
$399$	−31.9545	−1.59973
$400$	0	0
$401$	6.75875	0.337516	0.168758	−	0.985658i	$-0.446024\pi$
$401$	6.75875	0.337516	0.168758	+	0.985658i	$0.446024\pi$
$402$	0	0
$403$	9.37472	0.466988
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−31.1493	−1.54401
$408$	0	0
$409$	10.7676	0.532425	0.266212	−	0.963914i	$-0.414228\pi$
$409$	10.7676	0.532425	0.266212	+	0.963914i	$0.414228\pi$
$410$	0	0
$411$	−42.8350	−2.11290
$412$	0	0
$413$	0.0839119	0.00412903
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−52.7842	−2.58485
$418$	0	0
$419$	0.900641	0.0439992	0.0219996	−	0.999758i	$-0.492997\pi$
$419$	0.900641	0.0439992	0.0219996	+	0.999758i	$0.492997\pi$
$420$	0	0
$421$	−29.9114	−1.45779	−0.728897	−	0.684623i	$-0.759967\pi$
$421$	−29.9114	−1.45779	−0.728897	+	0.684623i	$0.759967\pi$
$422$	0	0
$423$	21.2031	1.03093
$424$	0	0
$425$	0	0
$426$	0	0
$427$	23.0079	1.11343
$428$	0	0
$429$	−19.1366	−0.923922
$430$	0	0
$431$	29.2769	1.41022	0.705110	−	0.709098i	$-0.250898\pi$
$431$	29.2769	1.41022	0.705110	+	0.709098i	$0.250898\pi$
$432$	0	0
$433$	5.54089	0.266278	0.133139	−	0.991097i	$-0.457494\pi$
$433$	5.54089	0.266278	0.133139	+	0.991097i	$0.457494\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.11698	−0.0534323
$438$	0	0
$439$	3.81224	0.181948	0.0909742	−	0.995853i	$-0.471002\pi$
$439$	3.81224	0.181948	0.0909742	+	0.995853i	$0.471002\pi$
$440$	0	0
$441$	−8.18262	−0.389649
$442$	0	0
$443$	−23.0329	−1.09433	−0.547163	−	0.837026i	$-0.684292\pi$
$443$	−23.0329	−1.09433	−0.547163	+	0.837026i	$0.684292\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−54.3884	−2.57248
$448$	0	0
$449$	27.4857	1.29713	0.648565	−	0.761159i	$-0.275369\pi$
$449$	27.4857	1.29713	0.648565	+	0.761159i	$0.275369\pi$
$450$	0	0
$451$	−23.5751	−1.11011
$452$	0	0
$453$	50.4689	2.37124
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−26.4721	−1.23831	−0.619157	−	0.785267i	$-0.712526\pi$
$457$	−26.4721	−1.23831	−0.619157	+	0.785267i	$0.712526\pi$
$458$	0	0
$459$	−38.5980	−1.80160
$460$	0	0
$461$	−16.9406	−0.789001	−0.394501	−	0.918896i	$-0.629082\pi$
$461$	−16.9406	−0.789001	−0.394501	+	0.918896i	$0.629082\pi$
$462$	0	0
$463$	30.8062	1.43168	0.715842	−	0.698262i	$-0.246043\pi$
$463$	30.8062	1.43168	0.715842	+	0.698262i	$0.246043\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.7852	1.19320	0.596599	−	0.802539i	$-0.296518\pi$
$467$	25.7852	1.19320	0.596599	+	0.802539i	$0.296518\pi$
$468$	0	0
$469$	−33.8115	−1.56127
$470$	0	0
$471$	3.84868	0.177338
$472$	0	0
$473$	21.5876	0.992600
$474$	0	0
$475$	0	0
$476$	0	0
$477$	50.5108	2.31273
$478$	0	0
$479$	−4.80646	−0.219613	−0.109806	−	0.993953i	$-0.535023\pi$
$479$	−4.80646	−0.219613	−0.109806	+	0.993953i	$0.535023\pi$
$480$	0	0
$481$	−8.45762	−0.385634
$482$	0	0
$483$	1.45034	0.0659929
$484$	0	0
$485$	0	0
$486$	0	0
$487$	22.0258	0.998085	0.499042	−	0.866578i	$-0.333685\pi$
$487$	22.0258	0.998085	0.499042	+	0.866578i	$0.333685\pi$
$488$	0	0
$489$	−13.3477	−0.603602
$490$	0	0
$491$	−6.81924	−0.307748	−0.153874	−	0.988090i	$-0.549175\pi$
$491$	−6.81924	−0.307748	−0.153874	+	0.988090i	$0.549175\pi$
$492$	0	0
$493$	−61.9564	−2.79037
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.86477	−0.128503
$498$	0	0
$499$	33.5418	1.50154	0.750768	−	0.660566i	$-0.229683\pi$
$499$	33.5418	1.50154	0.750768	+	0.660566i	$0.229683\pi$
$500$	0	0
$501$	−35.8412	−1.60127
$502$	0	0
$503$	−18.4744	−0.823732	−0.411866	−	0.911244i	$-0.635123\pi$
$503$	−18.4744	−0.823732	−0.411866	+	0.911244i	$0.635123\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	31.1654	1.38410
$508$	0	0
$509$	−30.7713	−1.36391	−0.681957	−	0.731392i	$-0.738871\pi$
$509$	−30.7713	−1.36391	−0.681957	+	0.731392i	$0.738871\pi$
$510$	0	0
$511$	−7.60308	−0.336341
$512$	0	0
$513$	−25.3000	−1.11702
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−22.0665	−0.970482
$518$	0	0
$519$	−38.5616	−1.69266
$520$	0	0
$521$	6.42574	0.281517	0.140758	−	0.990044i	$-0.455046\pi$
$521$	6.42574	0.281517	0.140758	+	0.990044i	$0.455046\pi$
$522$	0	0
$523$	−14.8516	−0.649417	−0.324709	−	0.945814i	$-0.605266\pi$
$523$	−14.8516	−0.649417	−0.324709	+	0.945814i	$0.605266\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−52.0561	−2.26760
$528$	0	0
$529$	−22.9493	−0.997796
$530$	0	0
$531$	0.175748	0.00762680
$532$	0	0
$533$	−6.40110	−0.277262
$534$	0	0
$535$	0	0
$536$	0	0
$537$	24.9702	1.07754
$538$	0	0
$539$	8.51583	0.366803
$540$	0	0
$541$	−30.4975	−1.31119	−0.655594	−	0.755114i	$-0.727582\pi$
$541$	−30.4975	−1.31119	−0.655594	+	0.755114i	$0.727582\pi$
$542$	0	0
$543$	14.0180	0.601569
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−5.36403	−0.229349	−0.114675	−	0.993403i	$-0.536583\pi$
$547$	−5.36403	−0.229349	−0.114675	+	0.993403i	$0.536583\pi$
$548$	0	0
$549$	48.1885	2.05664
$550$	0	0
$551$	−40.6108	−1.73008
$552$	0	0
$553$	−18.7048	−0.795411
$554$	0	0
$555$	0	0
$556$	0	0
$557$	40.2166	1.70403	0.852016	−	0.523515i	$-0.175380\pi$
$557$	40.2166	1.70403	0.852016	+	0.523515i	$0.175380\pi$
$558$	0	0
$559$	5.86145	0.247913
$560$	0	0
$561$	106.262	4.48638
$562$	0	0
$563$	−12.3847	−0.521954	−0.260977	−	0.965345i	$-0.584045\pi$
$563$	−12.3847	−0.521954	−0.260977	+	0.965345i	$0.584045\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−0.472839	−0.0198574
$568$	0	0
$569$	−32.1129	−1.34624	−0.673121	−	0.739533i	$-0.735047\pi$
$569$	−32.1129	−1.34624	−0.673121	+	0.739533i	$0.735047\pi$
$570$	0	0
$571$	−29.5620	−1.23713	−0.618565	−	0.785734i	$-0.712286\pi$
$571$	−29.5620	−1.23713	−0.618565	+	0.785734i	$0.712286\pi$
$572$	0	0
$573$	−62.3200	−2.60346
$574$	0	0
$575$	0	0
$576$	0	0
$577$	21.6087	0.899582	0.449791	−	0.893134i	$-0.351498\pi$
$577$	21.6087	0.899582	0.449791	+	0.893134i	$0.351498\pi$
$578$	0	0
$579$	−43.5224	−1.80873
$580$	0	0
$581$	31.9704	1.32635
$582$	0	0
$583$	−52.5677	−2.17713
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.61166	−0.149069	−0.0745346	−	0.997218i	$-0.523747\pi$
$587$	−3.61166	−0.149069	−0.0745346	+	0.997218i	$0.523747\pi$
$588$	0	0
$589$	−34.1214	−1.40595
$590$	0	0
$591$	−57.3741	−2.36006
$592$	0	0
$593$	−29.9985	−1.23189	−0.615946	−	0.787789i	$-0.711226\pi$
$593$	−29.9985	−1.23189	−0.615946	+	0.787789i	$0.711226\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.13507	0.210165
$598$	0	0
$599$	16.8952	0.690319	0.345160	−	0.938544i	$-0.387825\pi$
$599$	16.8952	0.690319	0.345160	+	0.938544i	$0.387825\pi$
$600$	0	0
$601$	−10.6939	−0.436213	−0.218107	−	0.975925i	$-0.569988\pi$
$601$	−10.6939	−0.436213	−0.218107	+	0.975925i	$0.569988\pi$
$602$	0	0
$603$	−70.8158	−2.88384
$604$	0	0
$605$	0	0
$606$	0	0
$607$	35.7187	1.44978	0.724889	−	0.688866i	$-0.241891\pi$
$607$	35.7187	1.44978	0.724889	+	0.688866i	$0.241891\pi$
$608$	0	0
$609$	52.7312	2.13678
$610$	0	0
$611$	−5.99147	−0.242389
$612$	0	0
$613$	−9.55392	−0.385879	−0.192940	−	0.981211i	$-0.561802\pi$
$613$	−9.55392	−0.385879	−0.192940	+	0.981211i	$0.561802\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3.72513	−0.149968	−0.0749841	−	0.997185i	$-0.523891\pi$
$617$	−3.72513	−0.149968	−0.0749841	+	0.997185i	$0.523891\pi$
$618$	0	0
$619$	19.5450	0.785578	0.392789	−	0.919629i	$-0.371510\pi$
$619$	19.5450	0.785578	0.392789	+	0.919629i	$0.371510\pi$
$620$	0	0
$621$	1.14831	0.0460801
$622$	0	0
$623$	−1.96695	−0.0788041
$624$	0	0
$625$	0	0
$626$	0	0
$627$	69.6519	2.78163
$628$	0	0
$629$	46.9636	1.87256
$630$	0	0
$631$	−4.38279	−0.174476	−0.0872380	−	0.996188i	$-0.527804\pi$
$631$	−4.38279	−0.174476	−0.0872380	+	0.996188i	$0.527804\pi$
$632$	0	0
$633$	−13.9774	−0.555553
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.31221	0.0916131
$638$	0	0
$639$	−6.00007	−0.237359
$640$	0	0
$641$	3.83302	0.151395	0.0756977	−	0.997131i	$-0.475882\pi$
$641$	3.83302	0.151395	0.0756977	+	0.997131i	$0.475882\pi$
$642$	0	0
$643$	26.6081	1.04932	0.524661	−	0.851311i	$-0.324192\pi$
$643$	26.6081	1.04932	0.524661	+	0.851311i	$0.324192\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−33.7874	−1.32832	−0.664160	−	0.747590i	$-0.731211\pi$
$647$	−33.7874	−1.32832	−0.664160	+	0.747590i	$0.731211\pi$
$648$	0	0
$649$	−0.182904	−0.00717962
$650$	0	0
$651$	44.3051	1.73645
$652$	0	0
$653$	30.7728	1.20423	0.602117	−	0.798408i	$-0.294324\pi$
$653$	30.7728	1.20423	0.602117	+	0.798408i	$0.294324\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−15.9241	−0.621260
$658$	0	0
$659$	27.6082	1.07546	0.537731	−	0.843117i	$-0.319282\pi$
$659$	27.6082	1.07546	0.537731	+	0.843117i	$0.319282\pi$
$660$	0	0
$661$	−10.7410	−0.417778	−0.208889	−	0.977939i	$-0.566985\pi$
$661$	−10.7410	−0.417778	−0.208889	+	0.977939i	$0.566985\pi$
$662$	0	0
$663$	28.8521	1.12052
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.84323	0.0713702
$668$	0	0
$669$	23.6844	0.915690
$670$	0	0
$671$	−50.1508	−1.93605
$672$	0	0
$673$	−3.59741	−0.138670	−0.0693350	−	0.997593i	$-0.522088\pi$
$673$	−3.59741	−0.138670	−0.0693350	+	0.997593i	$0.522088\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.8667	1.49377	0.746884	−	0.664955i	$-0.231549\pi$
$677$	38.8667	1.49377	0.746884	+	0.664955i	$0.231549\pi$
$678$	0	0
$679$	−12.7118	−0.487834
$680$	0	0
$681$	−32.3718	−1.24049
$682$	0	0
$683$	46.3746	1.77448	0.887238	−	0.461313i	$-0.152621\pi$
$683$	46.3746	1.77448	0.887238	+	0.461313i	$0.152621\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−12.8965	−0.492031
$688$	0	0
$689$	−14.2731	−0.543763
$690$	0	0
$691$	−31.0174	−1.17996	−0.589979	−	0.807418i	$-0.700864\pi$
$691$	−31.0174	−1.17996	−0.589979	+	0.807418i	$0.700864\pi$
$692$	0	0
$693$	−55.7591	−2.11811
$694$	0	0
$695$	0	0
$696$	0	0
$697$	35.5441	1.34633
$698$	0	0
$699$	−27.8572	−1.05366
$700$	0	0
$701$	−14.8473	−0.560776	−0.280388	−	0.959887i	$-0.590463\pi$
$701$	−14.8473	−0.560776	−0.280388	+	0.959887i	$0.590463\pi$
$702$	0	0
$703$	30.7834	1.16102
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−44.8925	−1.68836
$708$	0	0
$709$	7.85721	0.295084	0.147542	−	0.989056i	$-0.452864\pi$
$709$	7.85721	0.295084	0.147542	+	0.989056i	$0.452864\pi$
$710$	0	0
$711$	−39.1760	−1.46922
$712$	0	0
$713$	1.54869	0.0579991
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−76.5643	−2.85934
$718$	0	0
$719$	−18.4825	−0.689281	−0.344640	−	0.938735i	$-0.611999\pi$
$719$	−18.4825	−0.689281	−0.344640	+	0.938735i	$0.611999\pi$
$720$	0	0
$721$	−5.23299	−0.194887
$722$	0	0
$723$	17.4366	0.648476
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−31.7754	−1.17849	−0.589243	−	0.807956i	$-0.700574\pi$
$727$	−31.7754	−1.17849	−0.589243	+	0.807956i	$0.700574\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	−32.5476	−1.20382
$732$	0	0
$733$	13.5198	0.499366	0.249683	−	0.968328i	$-0.419674\pi$
$733$	13.5198	0.499366	0.249683	+	0.968328i	$0.419674\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	73.6995	2.71476
$738$	0	0
$739$	24.1757	0.889317	0.444658	−	0.895700i	$-0.353325\pi$
$739$	24.1757	0.889317	0.444658	+	0.895700i	$0.353325\pi$
$740$	0	0
$741$	18.9118	0.694743
$742$	0	0
$743$	−9.03371	−0.331415	−0.165707	−	0.986175i	$-0.552991\pi$
$743$	−9.03371	−0.331415	−0.165707	+	0.986175i	$0.552991\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	66.9598	2.44993
$748$	0	0
$749$	−18.5068	−0.676223
$750$	0	0
$751$	−9.73265	−0.355149	−0.177575	−	0.984107i	$-0.556825\pi$
$751$	−9.73265	−0.355149	−0.177575	+	0.984107i	$0.556825\pi$
$752$	0	0
$753$	24.8683	0.906250
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−3.75426	−0.136451	−0.0682255	−	0.997670i	$-0.521734\pi$
$757$	−3.75426	−0.136451	−0.0682255	+	0.997670i	$0.521734\pi$
$758$	0	0
$759$	−3.16134	−0.114749
$760$	0	0
$761$	−30.5962	−1.10911	−0.554555	−	0.832147i	$-0.687112\pi$
$761$	−30.5962	−1.10911	−0.554555	+	0.832147i	$0.687112\pi$
$762$	0	0
$763$	6.97486	0.252507
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−0.0496620	−0.00179319
$768$	0	0
$769$	33.3674	1.20326	0.601630	−	0.798775i	$-0.294518\pi$
$769$	33.3674	1.20326	0.601630	+	0.798775i	$0.294518\pi$
$770$	0	0
$771$	−35.4227	−1.27572
$772$	0	0
$773$	−33.0251	−1.18783	−0.593915	−	0.804528i	$-0.702419\pi$
$773$	−33.0251	−1.18783	−0.593915	+	0.804528i	$0.702419\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−39.9709	−1.43395
$778$	0	0
$779$	23.2982	0.834746
$780$	0	0
$781$	6.24440	0.223442
$782$	0	0
$783$	41.7499	1.49202
$784$	0	0
$785$	0	0
$786$	0	0
$787$	35.6747	1.27167	0.635833	−	0.771827i	$-0.280657\pi$
$787$	35.6747	1.27167	0.635833	+	0.771827i	$0.280657\pi$
$788$	0	0
$789$	56.2049	2.00095
$790$	0	0
$791$	−9.77769	−0.347655
$792$	0	0
$793$	−13.6169	−0.483550
$794$	0	0
$795$	0	0
$796$	0	0
$797$	37.5507	1.33011	0.665057	−	0.746793i	$-0.268407\pi$
$797$	37.5507	1.33011	0.665057	+	0.746793i	$0.268407\pi$
$798$	0	0
$799$	33.2695	1.17699
$800$	0	0
$801$	−4.11964	−0.145560
$802$	0	0
$803$	16.5726	0.584834
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−46.2991	−1.62981
$808$	0	0
$809$	38.1903	1.34270	0.671351	−	0.741140i	$-0.265715\pi$
$809$	38.1903	1.34270	0.671351	+	0.741140i	$0.265715\pi$
$810$	0	0
$811$	32.2614	1.13285	0.566425	−	0.824113i	$-0.308326\pi$
$811$	32.2614	1.13285	0.566425	+	0.824113i	$0.308326\pi$
$812$	0	0
$813$	−67.9887	−2.38447
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−21.3341	−0.746385
$818$	0	0
$819$	−15.1397	−0.529022
$820$	0	0
$821$	14.2705	0.498044	0.249022	−	0.968498i	$-0.419891\pi$
$821$	14.2705	0.498044	0.249022	+	0.968498i	$0.419891\pi$
$822$	0	0
$823$	−41.8784	−1.45979	−0.729895	−	0.683559i	$-0.760431\pi$
$823$	−41.8784	−1.45979	−0.729895	+	0.683559i	$0.760431\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−18.3183	−0.636991	−0.318496	−	0.947924i	$-0.603178\pi$
$827$	−18.3183	−0.636991	−0.318496	+	0.947924i	$0.603178\pi$
$828$	0	0
$829$	−17.5858	−0.610780	−0.305390	−	0.952227i	$-0.598787\pi$
$829$	−17.5858	−0.610780	−0.305390	+	0.952227i	$0.598787\pi$
$830$	0	0
$831$	−6.62957	−0.229977
$832$	0	0
$833$	−12.8393	−0.444855
$834$	0	0
$835$	0	0
$836$	0	0
$837$	35.0786	1.21249
$838$	0	0
$839$	−7.75631	−0.267778	−0.133889	−	0.990996i	$-0.542747\pi$
$839$	−7.75631	−0.267778	−0.133889	+	0.990996i	$0.542747\pi$
$840$	0	0
$841$	38.0158	1.31089
$842$	0	0
$843$	−28.6904	−0.988151
$844$	0	0
$845$	0	0
$846$	0	0
$847$	32.6973	1.12349
$848$	0	0
$849$	−70.5982	−2.42292
$850$	0	0
$851$	−1.39719	−0.0478951
$852$	0	0
$853$	−36.2377	−1.24075	−0.620377	−	0.784303i	$-0.713021\pi$
$853$	−36.2377	−1.24075	−0.620377	+	0.784303i	$0.713021\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4.98618	−0.170325	−0.0851623	−	0.996367i	$-0.527141\pi$
$857$	−4.98618	−0.170325	−0.0851623	+	0.996367i	$0.527141\pi$
$858$	0	0
$859$	−23.2768	−0.794196	−0.397098	−	0.917776i	$-0.629983\pi$
$859$	−23.2768	−0.794196	−0.397098	+	0.917776i	$0.629983\pi$
$860$	0	0
$861$	−30.2517	−1.03098
$862$	0	0
$863$	32.8930	1.11969	0.559845	−	0.828598i	$-0.310861\pi$
$863$	32.8930	1.11969	0.559845	+	0.828598i	$0.310861\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−112.661	−3.82617
$868$	0	0
$869$	40.7713	1.38307
$870$	0	0
$871$	20.0108	0.678041
$872$	0	0
$873$	−26.6240	−0.901086
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−56.8730	−1.92046	−0.960232	−	0.279203i	$-0.909930\pi$
$877$	−56.8730	−1.92046	−0.960232	+	0.279203i	$0.909930\pi$
$878$	0	0
$879$	82.3551	2.77777
$880$	0	0
$881$	−12.2955	−0.414246	−0.207123	−	0.978315i	$-0.566410\pi$
$881$	−12.2955	−0.414246	−0.207123	+	0.978315i	$0.566410\pi$
$882$	0	0
$883$	15.2668	0.513769	0.256884	−	0.966442i	$-0.417304\pi$
$883$	15.2668	0.513769	0.256884	+	0.966442i	$0.417304\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28.4403	0.954932	0.477466	−	0.878650i	$-0.341555\pi$
$887$	28.4403	0.954932	0.477466	+	0.878650i	$0.341555\pi$
$888$	0	0
$889$	2.75924	0.0925419
$890$	0	0
$891$	1.03066	0.0345283
$892$	0	0
$893$	21.8073	0.729754
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−0.858365	−0.0286600
$898$	0	0
$899$	56.3071	1.87795
$900$	0	0
$901$	79.2561	2.64040
$902$	0	0
$903$	27.7013	0.921843
$904$	0	0
$905$	0	0
$906$	0	0
$907$	16.1108	0.534950	0.267475	−	0.963565i	$-0.413811\pi$
$907$	16.1108	0.534950	0.267475	+	0.963565i	$0.413811\pi$
$908$	0	0
$909$	−94.0243	−3.11859
$910$	0	0
$911$	−0.482632	−0.0159903	−0.00799516	−	0.999968i	$-0.502545\pi$
$911$	−0.482632	−0.0159903	−0.00799516	+	0.999968i	$0.502545\pi$
$912$	0	0
$913$	−69.6865	−2.30629
$914$	0	0
$915$	0	0
$916$	0	0
$917$	44.4704	1.46854
$918$	0	0
$919$	−5.39953	−0.178114	−0.0890570	−	0.996027i	$-0.528385\pi$
$919$	−5.39953	−0.178114	−0.0890570	+	0.996027i	$0.528385\pi$
$920$	0	0
$921$	−54.1074	−1.78290
$922$	0	0
$923$	1.69547	0.0558072
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−10.9601	−0.359978
$928$	0	0
$929$	−21.9776	−0.721063	−0.360531	−	0.932747i	$-0.617405\pi$
$929$	−21.9776	−0.721063	−0.360531	+	0.932747i	$0.617405\pi$
$930$	0	0
$931$	−8.41581	−0.275817
$932$	0	0
$933$	−31.3953	−1.02784
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−46.5269	−1.51997	−0.759984	−	0.649942i	$-0.774793\pi$
$937$	−46.5269	−1.51997	−0.759984	+	0.649942i	$0.774793\pi$
$938$	0	0
$939$	−38.7737	−1.26533
$940$	0	0
$941$	−24.3540	−0.793917	−0.396958	−	0.917837i	$-0.629934\pi$
$941$	−24.3540	−0.793917	−0.396958	+	0.917837i	$0.629934\pi$
$942$	0	0
$943$	−1.05746	−0.0344355
$944$	0	0
$945$	0	0
$946$	0	0
$947$	38.6134	1.25477	0.627384	−	0.778710i	$-0.284126\pi$
$947$	38.6134	1.25477	0.627384	+	0.778710i	$0.284126\pi$
$948$	0	0
$949$	4.49977	0.146069
$950$	0	0
$951$	26.9122	0.872689
$952$	0	0
$953$	−14.8068	−0.479638	−0.239819	−	0.970818i	$-0.577088\pi$
$953$	−14.8068	−0.479638	−0.239819	+	0.970818i	$0.577088\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−114.939	−3.71546
$958$	0	0
$959$	35.2684	1.13888
$960$	0	0
$961$	16.3095	0.526113
$962$	0	0
$963$	−38.7612	−1.24906
$964$	0	0
$965$	0	0
$966$	0	0
$967$	6.41848	0.206404	0.103202	−	0.994660i	$-0.467091\pi$
$967$	6.41848	0.206404	0.103202	+	0.994660i	$0.467091\pi$
$968$	0	0
$969$	−105.014	−3.37353
$970$	0	0
$971$	−22.3173	−0.716195	−0.358097	−	0.933684i	$-0.616574\pi$
$971$	−22.3173	−0.716195	−0.358097	+	0.933684i	$0.616574\pi$
$972$	0	0
$973$	43.4601	1.39327
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−8.48650	−0.271507	−0.135754	−	0.990743i	$-0.543346\pi$
$977$	−8.48650	−0.271507	−0.135754	+	0.990743i	$0.543346\pi$
$978$	0	0
$979$	4.28739	0.137026
$980$	0	0
$981$	14.6084	0.466409
$982$	0	0
$983$	−18.3183	−0.584265	−0.292132	−	0.956378i	$-0.594365\pi$
$983$	−18.3183	−0.584265	−0.292132	+	0.956378i	$0.594365\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−28.3158	−0.901302
$988$	0	0
$989$	0.968306	0.0307904
$990$	0	0
$991$	10.4286	0.331275	0.165637	−	0.986187i	$-0.447032\pi$
$991$	10.4286	0.331275	0.165637	+	0.986187i	$0.447032\pi$
$992$	0	0
$993$	−80.1066	−2.54211
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.10768	0.0984213	0.0492107	−	0.998788i	$-0.484329\pi$
$997$	3.10768	0.0984213	0.0492107	+	0.998788i	$0.484329\pi$
$998$	0	0
$999$	−31.6469	−1.00126

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4000.2.a.p.1.1	yes	8
4.3	odd	2	inner	4000.2.a.p.1.8	yes	8
5.2	odd	4		4000.2.c.i.1249.15		16
5.3	odd	4		4000.2.c.i.1249.1		16
5.4	even	2		4000.2.a.o.1.8	yes	8
8.3	odd	2		8000.2.a.by.1.1		8
8.5	even	2		8000.2.a.by.1.8		8
20.3	even	4		4000.2.c.i.1249.16		16
20.7	even	4		4000.2.c.i.1249.2		16
20.19	odd	2		4000.2.a.o.1.1	✓	8
40.19	odd	2		8000.2.a.bz.1.8		8
40.29	even	2		8000.2.a.bz.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4000.2.a.o.1.1	✓	8	20.19	odd	2
4000.2.a.o.1.8	yes	8	5.4	even	2
4000.2.a.p.1.1	yes	8	1.1	even	1	trivial
4000.2.a.p.1.8	yes	8	4.3	odd	2	inner
4000.2.c.i.1249.1		16	5.3	odd	4
4000.2.c.i.1249.2		16	20.7	even	4
4000.2.c.i.1249.15		16	5.2	odd	4
4000.2.c.i.1249.16		16	20.3	even	4
8000.2.a.by.1.1		8	8.3	odd	2
8000.2.a.by.1.8		8	8.5	even	2
8000.2.a.bz.1.1		8	40.29	even	2
8000.2.a.bz.1.8		8	40.19	odd	2

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:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4000.a (trivial)

\(f(q)\)	\(=\)	\(q-2.79703 q^{3} +2.30294 q^{7} +4.82335 q^{9} +O(q^{10})\)	\(q-2.79703 q^{3} +2.30294 q^{7} +4.82335 q^{9} -5.01977 q^{11} -1.36296 q^{13} +7.56828 q^{17} +4.96081 q^{19} -6.44139 q^{21} -0.225160 q^{23} -5.09997 q^{27} -8.18632 q^{29} -6.87819 q^{31} +14.0404 q^{33} +6.20532 q^{37} +3.81224 q^{39} +4.69646 q^{41} -4.30052 q^{43} +4.39592 q^{47} -1.69646 q^{49} -21.1687 q^{51} +10.4721 q^{53} -13.8755 q^{57} +0.0364368 q^{59} +9.99067 q^{61} +11.1079 q^{63} -14.6819 q^{67} +0.629779 q^{69} -1.24396 q^{71} -3.30147 q^{73} -11.5602 q^{77} -8.12215 q^{79} -0.205320 q^{81} +13.8824 q^{83} +22.8973 q^{87} -0.854102 q^{89} -3.13882 q^{91} +19.2385 q^{93} -5.51981 q^{97} -24.2121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} + 8 q^{13} + 20 q^{17} - 12 q^{21} - 16 q^{29} + 36 q^{33} + 28 q^{37} + 8 q^{41} + 16 q^{49} + 48 q^{53} + 20 q^{57} - 28 q^{61} - 28 q^{69} + 44 q^{77} + 20 q^{81} + 20 q^{89} + 4 q^{93} + 16 q^{97}+O(q^{100}) \) 8 * q + 8 * q^9 + 8 * q^13 + 20 * q^17 - 12 * q^21 - 16 * q^29 + 36 * q^33 + 28 * q^37 + 8 * q^41 + 16 * q^49 + 48 * q^53 + 20 * q^57 - 28 * q^61 - 28 * q^69 + 44 * q^77 + 20 * q^81 + 20 * q^89 + 4 * q^93 + 16 * q^97

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