LMFDB - Embedded newform 800.4.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,Mod(1,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("800.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	800.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:	$47.2015280046$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{29}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 7 $ x^2 - x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.19258$ of defining polynomial
Character	$\chi$	$=$	800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{3} +4.00000 q^{7} -11.0000 q^{9} +O(q^{10})$	$q-4.00000 q^{3} +4.00000 q^{7} -11.0000 q^{9} -43.0813 q^{11} -21.5407 q^{13} -43.0813 q^{17} +129.244 q^{19} -16.0000 q^{21} -52.0000 q^{23} +152.000 q^{27} -158.000 q^{29} -172.325 q^{31} +172.325 q^{33} +280.029 q^{37} +86.1626 q^{39} -170.000 q^{41} +316.000 q^{43} +244.000 q^{47} -327.000 q^{49} +172.325 q^{51} -495.435 q^{53} -516.976 q^{57} -646.220 q^{59} +82.0000 q^{61} -44.0000 q^{63} +692.000 q^{67} +208.000 q^{69} +947.789 q^{71} +430.813 q^{73} -172.325 q^{77} -344.651 q^{79} -311.000 q^{81} +940.000 q^{83} +632.000 q^{87} +6.00000 q^{89} -86.1626 q^{91} +689.301 q^{93} +1077.03 q^{97} +473.895 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{3} + 8 q^{7} - 22 q^{9}+O(q^{10}) $ 2 * q - 8 * q^3 + 8 * q^7 - 22 * q^9	$ 2 q - 8 q^{3} + 8 q^{7} - 22 q^{9} - 32 q^{21} - 104 q^{23} + 304 q^{27} - 316 q^{29} - 340 q^{41} + 632 q^{43} + 488 q^{47} - 654 q^{49} + 164 q^{61} - 88 q^{63} + 1384 q^{67} + 416 q^{69} - 622 q^{81} + 1880 q^{83} + 1264 q^{87} + 12 q^{89}+O(q^{100}) $ 2 * q - 8 * q^3 + 8 * q^7 - 22 * q^9 - 32 * q^21 - 104 * q^23 + 304 * q^27 - 316 * q^29 - 340 * q^41 + 632 * q^43 + 488 * q^47 - 654 * q^49 + 164 * q^61 - 88 * q^63 + 1384 * q^67 + 416 * q^69 - 622 * q^81 + 1880 * q^83 + 1264 * q^87 + 12 * q^89

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
$3$	−4.00000	−0.769800	−0.384900	−	0.922958i	$-0.625764\pi$
$3$	−4.00000	−0.769800	−0.384900	+	0.922958i	$0.625764\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.00000	0.215980	0.107990	−	0.994152i	$-0.465559\pi$
$7$	4.00000	0.215980	0.107990	+	0.994152i	$0.465559\pi$
$8$	0	0
$9$	−11.0000	−0.407407
$10$	0	0
$11$	−43.0813	−1.18086	−0.590432	−	0.807087i	$-0.701043\pi$
$11$	−43.0813	−1.18086	−0.590432	+	0.807087i	$0.701043\pi$
$12$	0	0
$13$	−21.5407	−0.459562	−0.229781	−	0.973242i	$-0.573801\pi$
$13$	−21.5407	−0.459562	−0.229781	+	0.973242i	$0.573801\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−43.0813	−0.614633	−0.307316	−	0.951607i	$-0.599431\pi$
$17$	−43.0813	−0.614633	−0.307316	+	0.951607i	$0.599431\pi$
$18$	0	0
$19$	129.244	1.56056	0.780279	−	0.625432i	$-0.215077\pi$
$19$	129.244	1.56056	0.780279	+	0.625432i	$0.215077\pi$
$20$	0	0
$21$	−16.0000	−0.166261
$22$	0	0
$23$	−52.0000	−0.471424	−0.235712	−	0.971823i	$-0.575742\pi$
$23$	−52.0000	−0.471424	−0.235712	+	0.971823i	$0.575742\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	152.000	1.08342
$28$	0	0
$29$	−158.000	−1.01172	−0.505860	−	0.862616i	$-0.668825\pi$
$29$	−158.000	−1.01172	−0.505860	+	0.862616i	$0.668825\pi$
$30$	0	0
$31$	−172.325	−0.998404	−0.499202	−	0.866486i	$-0.666373\pi$
$31$	−172.325	−0.998404	−0.499202	+	0.866486i	$0.666373\pi$
$32$	0	0
$33$	172.325	0.909030
$34$	0	0
$35$	0	0
$36$	0	0
$37$	280.029	1.24423	0.622114	−	0.782927i	$-0.286274\pi$
$37$	280.029	1.24423	0.622114	+	0.782927i	$0.286274\pi$
$38$	0	0
$39$	86.1626	0.353771
$40$	0	0
$41$	−170.000	−0.647550	−0.323775	−	0.946134i	$-0.604952\pi$
$41$	−170.000	−0.647550	−0.323775	+	0.946134i	$0.604952\pi$
$42$	0	0
$43$	316.000	1.12069	0.560344	−	0.828260i	$-0.310669\pi$
$43$	316.000	1.12069	0.560344	+	0.828260i	$0.310669\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	244.000	0.757257	0.378628	−	0.925549i	$-0.376396\pi$
$47$	244.000	0.757257	0.378628	+	0.925549i	$0.376396\pi$
$48$	0	0
$49$	−327.000	−0.953353
$50$	0	0
$51$	172.325	0.473144
$52$	0	0
$53$	−495.435	−1.28402	−0.642012	−	0.766695i	$-0.721900\pi$
$53$	−495.435	−1.28402	−0.642012	+	0.766695i	$0.721900\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−516.976	−1.20132
$58$	0	0
$59$	−646.220	−1.42594	−0.712972	−	0.701193i	$-0.752651\pi$
$59$	−646.220	−1.42594	−0.712972	+	0.701193i	$0.752651\pi$
$60$	0	0
$61$	82.0000	0.172115	0.0860576	−	0.996290i	$-0.472573\pi$
$61$	82.0000	0.172115	0.0860576	+	0.996290i	$0.472573\pi$
$62$	0	0
$63$	−44.0000	−0.0879917
$64$	0	0
$65$	0	0
$66$	0	0
$67$	692.000	1.26181	0.630905	−	0.775860i	$-0.282684\pi$
$67$	692.000	1.26181	0.630905	+	0.775860i	$0.282684\pi$
$68$	0	0
$69$	208.000	0.362902
$70$	0	0
$71$	947.789	1.58425	0.792126	−	0.610358i	$-0.208974\pi$
$71$	947.789	1.58425	0.792126	+	0.610358i	$0.208974\pi$
$72$	0	0
$73$	430.813	0.690724	0.345362	−	0.938470i	$-0.387756\pi$
$73$	430.813	0.690724	0.345362	+	0.938470i	$0.387756\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−172.325	−0.255043
$78$	0	0
$79$	−344.651	−0.490838	−0.245419	−	0.969417i	$-0.578926\pi$
$79$	−344.651	−0.490838	−0.245419	+	0.969417i	$0.578926\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	940.000	1.24311	0.621557	−	0.783369i	$-0.286501\pi$
$83$	940.000	1.24311	0.621557	+	0.783369i	$0.286501\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	632.000	0.778822
$88$	0	0
$89$	6.00000	0.00714605	0.00357303	−	0.999994i	$-0.498863\pi$
$89$	6.00000	0.00714605	0.00357303	+	0.999994i	$0.498863\pi$
$90$	0	0
$91$	−86.1626	−0.0992560
$92$	0	0
$93$	689.301	0.768572
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1077.03	1.12738	0.563691	−	0.825985i	$-0.309381\pi$
$97$	1077.03	1.12738	0.563691	+	0.825985i	$0.309381\pi$
$98$	0	0
$99$	473.895	0.481093
$100$	0	0
$101$	1014.00	0.998978	0.499489	−	0.866320i	$-0.333521\pi$
$101$	1014.00	0.998978	0.499489	+	0.866320i	$0.333521\pi$
$102$	0	0
$103$	44.0000	0.0420917	0.0210459	−	0.999779i	$-0.493300\pi$
$103$	44.0000	0.0420917	0.0210459	+	0.999779i	$0.493300\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1812.00	1.63713	0.818564	−	0.574416i	$-0.194771\pi$
$107$	1812.00	1.63713	0.818564	+	0.574416i	$0.194771\pi$
$108$	0	0
$109$	2014.00	1.76978	0.884891	−	0.465798i	$-0.154233\pi$
$109$	2014.00	1.76978	0.884891	+	0.465798i	$0.154233\pi$
$110$	0	0
$111$	−1120.11	−0.957807
$112$	0	0
$113$	−1637.09	−1.36287	−0.681436	−	0.731878i	$-0.738644\pi$
$113$	−1637.09	−1.36287	−0.681436	+	0.731878i	$0.738644\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	236.947	0.187229
$118$	0	0
$119$	−172.325	−0.132748
$120$	0	0
$121$	525.000	0.394440
$122$	0	0
$123$	680.000	0.498484
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2788.00	1.94799	0.973996	−	0.226565i	$-0.0727496\pi$
$127$	2788.00	1.94799	0.973996	+	0.226565i	$0.0727496\pi$
$128$	0	0
$129$	−1264.00	−0.862705
$130$	0	0
$131$	43.0813	0.0287331	0.0143665	−	0.999897i	$-0.495427\pi$
$131$	43.0813	0.0287331	0.0143665	+	0.999897i	$0.495427\pi$
$132$	0	0
$133$	516.976	0.337049
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−473.895	−0.295529	−0.147765	−	0.989023i	$-0.547208\pi$
$137$	−473.895	−0.295529	−0.147765	+	0.989023i	$0.547208\pi$
$138$	0	0
$139$	2110.98	1.28814	0.644070	−	0.764967i	$-0.277245\pi$
$139$	2110.98	1.28814	0.644070	+	0.764967i	$0.277245\pi$
$140$	0	0
$141$	−976.000	−0.582936
$142$	0	0
$143$	928.000	0.542680
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1308.00	0.733891
$148$	0	0
$149$	2550.00	1.40204	0.701021	−	0.713141i	$-0.252728\pi$
$149$	2550.00	1.40204	0.701021	+	0.713141i	$0.252728\pi$
$150$	0	0
$151$	−2154.07	−1.16090	−0.580448	−	0.814297i	$-0.697123\pi$
$151$	−2154.07	−1.16090	−0.580448	+	0.814297i	$0.697123\pi$
$152$	0	0
$153$	473.895	0.250406
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3554.21	−1.80673	−0.903365	−	0.428872i	$-0.858911\pi$
$157$	−3554.21	−1.80673	−0.903365	+	0.428872i	$0.858911\pi$
$158$	0	0
$159$	1981.74	0.988442
$160$	0	0
$161$	−208.000	−0.101818
$162$	0	0
$163$	−228.000	−0.109560	−0.0547802	−	0.998498i	$-0.517446\pi$
$163$	−228.000	−0.109560	−0.0547802	+	0.998498i	$0.517446\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	372.000	0.172373	0.0861863	−	0.996279i	$-0.472532\pi$
$167$	372.000	0.172373	0.0861863	+	0.996279i	$0.472532\pi$
$168$	0	0
$169$	−1733.00	−0.788803
$170$	0	0
$171$	−1421.68	−0.635783
$172$	0	0
$173$	1917.12	0.842519	0.421260	−	0.906940i	$-0.361588\pi$
$173$	1917.12	0.842519	0.421260	+	0.906940i	$0.361588\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2584.88	1.09769
$178$	0	0
$179$	−2800.29	−1.16929	−0.584646	−	0.811289i	$-0.698766\pi$
$179$	−2800.29	−1.16929	−0.584646	+	0.811289i	$0.698766\pi$
$180$	0	0
$181$	1542.00	0.633237	0.316619	−	0.948553i	$-0.397452\pi$
$181$	1542.00	0.633237	0.316619	+	0.948553i	$0.397452\pi$
$182$	0	0
$183$	−328.000	−0.132494
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1856.00	0.725798
$188$	0	0
$189$	608.000	0.233997
$190$	0	0
$191$	4480.46	1.69735	0.848677	−	0.528912i	$-0.177400\pi$
$191$	4480.46	1.69735	0.848677	+	0.528912i	$0.177400\pi$
$192$	0	0
$193$	−2541.80	−0.947993	−0.473996	−	0.880527i	$-0.657189\pi$
$193$	−2541.80	−0.947993	−0.473996	+	0.880527i	$0.657189\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2735.66	0.989381	0.494690	−	0.869069i	$-0.335282\pi$
$197$	2735.66	0.989381	0.494690	+	0.869069i	$0.335282\pi$
$198$	0	0
$199$	−258.488	−0.0920790	−0.0460395	−	0.998940i	$-0.514660\pi$
$199$	−258.488	−0.0920790	−0.0460395	+	0.998940i	$0.514660\pi$
$200$	0	0
$201$	−2768.00	−0.971342
$202$	0	0
$203$	−632.000	−0.218511
$204$	0	0
$205$	0	0
$206$	0	0
$207$	572.000	0.192062
$208$	0	0
$209$	−5568.00	−1.84281
$210$	0	0
$211$	904.708	0.295178	0.147589	−	0.989049i	$-0.452849\pi$
$211$	904.708	0.295178	0.147589	+	0.989049i	$0.452849\pi$
$212$	0	0
$213$	−3791.16	−1.21956
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−689.301	−0.215635
$218$	0	0
$219$	−1723.25	−0.531720
$220$	0	0
$221$	928.000	0.282462
$222$	0	0
$223$	4284.00	1.28645	0.643224	−	0.765678i	$-0.277596\pi$
$223$	4284.00	1.28645	0.643224	+	0.765678i	$0.277596\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4956.00	−1.44908	−0.724540	−	0.689232i	$-0.757948\pi$
$227$	−4956.00	−1.44908	−0.724540	+	0.689232i	$0.757948\pi$
$228$	0	0
$229$	1770.00	0.510764	0.255382	−	0.966840i	$-0.417799\pi$
$229$	1770.00	0.510764	0.255382	+	0.966840i	$0.417799\pi$
$230$	0	0
$231$	689.301	0.196332
$232$	0	0
$233$	5428.25	1.52625	0.763125	−	0.646251i	$-0.223664\pi$
$233$	5428.25	1.52625	0.763125	+	0.646251i	$0.223664\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1378.60	0.377847
$238$	0	0
$239$	−5514.41	−1.49246	−0.746229	−	0.665689i	$-0.768138\pi$
$239$	−5514.41	−1.49246	−0.746229	+	0.665689i	$0.768138\pi$
$240$	0	0
$241$	−1618.00	−0.432467	−0.216233	−	0.976342i	$-0.569377\pi$
$241$	−1618.00	−0.432467	−0.216233	+	0.976342i	$0.569377\pi$
$242$	0	0
$243$	−2860.00	−0.755017
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2784.00	−0.717173
$248$	0	0
$249$	−3760.00	−0.956949
$250$	0	0
$251$	4954.35	1.24588	0.622940	−	0.782270i	$-0.285938\pi$
$251$	4954.35	1.24588	0.622940	+	0.782270i	$0.285938\pi$
$252$	0	0
$253$	2240.23	0.556688
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4652.78	−1.12931	−0.564655	−	0.825327i	$-0.690991\pi$
$257$	−4652.78	−1.12931	−0.564655	+	0.825327i	$0.690991\pi$
$258$	0	0
$259$	1120.11	0.268728
$260$	0	0
$261$	1738.00	0.412182
$262$	0	0
$263$	604.000	0.141613	0.0708065	−	0.997490i	$-0.477443\pi$
$263$	604.000	0.141613	0.0708065	+	0.997490i	$0.477443\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−24.0000	−0.00550103
$268$	0	0
$269$	3262.00	0.739359	0.369680	−	0.929159i	$-0.379467\pi$
$269$	3262.00	0.739359	0.369680	+	0.929159i	$0.379467\pi$
$270$	0	0
$271$	−2067.90	−0.463528	−0.231764	−	0.972772i	$-0.574450\pi$
$271$	−2067.90	−0.463528	−0.231764	+	0.972772i	$0.574450\pi$
$272$	0	0
$273$	344.651	0.0764073
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1227.82	0.266326	0.133163	−	0.991094i	$-0.457487\pi$
$277$	1227.82	0.266326	0.133163	+	0.991094i	$0.457487\pi$
$278$	0	0
$279$	1895.58	0.406757
$280$	0	0
$281$	−1290.00	−0.273861	−0.136931	−	0.990581i	$-0.543724\pi$
$281$	−1290.00	−0.273861	−0.136931	+	0.990581i	$0.543724\pi$
$282$	0	0
$283$	−692.000	−0.145354	−0.0726769	−	0.997356i	$-0.523154\pi$
$283$	−692.000	−0.145354	−0.0726769	+	0.997356i	$0.523154\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−680.000	−0.139858
$288$	0	0
$289$	−3057.00	−0.622227
$290$	0	0
$291$	−4308.13	−0.867860
$292$	0	0
$293$	−5535.95	−1.10380	−0.551900	−	0.833910i	$-0.686097\pi$
$293$	−5535.95	−1.10380	−0.551900	+	0.833910i	$0.686097\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−6548.36	−1.27938
$298$	0	0
$299$	1120.11	0.216648
$300$	0	0
$301$	1264.00	0.242046
$302$	0	0
$303$	−4056.00	−0.769014
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−7436.00	−1.38239	−0.691197	−	0.722666i	$-0.742916\pi$
$307$	−7436.00	−1.38239	−0.691197	+	0.722666i	$0.742916\pi$
$308$	0	0
$309$	−176.000	−0.0324022
$310$	0	0
$311$	−2498.72	−0.455592	−0.227796	−	0.973709i	$-0.573152\pi$
$311$	−2498.72	−0.455592	−0.227796	+	0.973709i	$0.573152\pi$
$312$	0	0
$313$	−2714.12	−0.490132	−0.245066	−	0.969506i	$-0.578810\pi$
$313$	−2714.12	−0.490132	−0.245066	+	0.969506i	$0.578810\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3295.72	0.583931	0.291965	−	0.956429i	$-0.405691\pi$
$317$	3295.72	0.583931	0.291965	+	0.956429i	$0.405691\pi$
$318$	0	0
$319$	6806.85	1.19470
$320$	0	0
$321$	−7248.00	−1.26026
$322$	0	0
$323$	−5568.00	−0.959170
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−8056.00	−1.36238
$328$	0	0
$329$	976.000	0.163552
$330$	0	0
$331$	−2972.61	−0.493624	−0.246812	−	0.969063i	$-0.579383\pi$
$331$	−2972.61	−0.493624	−0.246812	+	0.969063i	$0.579383\pi$
$332$	0	0
$333$	−3080.31	−0.506907
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9133.24	1.47632	0.738159	−	0.674627i	$-0.235695\pi$
$337$	9133.24	1.47632	0.738159	+	0.674627i	$0.235695\pi$
$338$	0	0
$339$	6548.36	1.04914
$340$	0	0
$341$	7424.00	1.17898
$342$	0	0
$343$	−2680.00	−0.421885
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5916.00	−0.915238	−0.457619	−	0.889148i	$-0.651298\pi$
$347$	−5916.00	−0.915238	−0.457619	+	0.889148i	$0.651298\pi$
$348$	0	0
$349$	7522.00	1.15371	0.576853	−	0.816848i	$-0.304281\pi$
$349$	7522.00	1.15371	0.576853	+	0.816848i	$0.304281\pi$
$350$	0	0
$351$	−3274.18	−0.497900
$352$	0	0
$353$	8185.45	1.23419	0.617093	−	0.786890i	$-0.288310\pi$
$353$	8185.45	1.23419	0.617093	+	0.786890i	$0.288310\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	689.301	0.102190
$358$	0	0
$359$	−1464.76	−0.215341	−0.107670	−	0.994187i	$-0.534339\pi$
$359$	−1464.76	−0.215341	−0.107670	+	0.994187i	$0.534339\pi$
$360$	0	0
$361$	9845.00	1.43534
$362$	0	0
$363$	−2100.00	−0.303640
$364$	0	0
$365$	0	0
$366$	0	0
$367$	516.000	0.0733923	0.0366962	−	0.999326i	$-0.488317\pi$
$367$	516.000	0.0733923	0.0366962	+	0.999326i	$0.488317\pi$
$368$	0	0
$369$	1870.00	0.263817
$370$	0	0
$371$	−1981.74	−0.277323
$372$	0	0
$373$	8077.75	1.12131	0.560657	−	0.828048i	$-0.310549\pi$
$373$	8077.75	1.12131	0.560657	+	0.828048i	$0.310549\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3403.42	0.464948
$378$	0	0
$379$	−2197.15	−0.297783	−0.148892	−	0.988854i	$-0.547571\pi$
$379$	−2197.15	−0.297783	−0.148892	+	0.988854i	$0.547571\pi$
$380$	0	0
$381$	−11152.0	−1.49957
$382$	0	0
$383$	−5988.00	−0.798884	−0.399442	−	0.916759i	$-0.630796\pi$
$383$	−5988.00	−0.798884	−0.399442	+	0.916759i	$0.630796\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3476.00	−0.456576
$388$	0	0
$389$	7974.00	1.03933	0.519663	−	0.854371i	$-0.326057\pi$
$389$	7974.00	1.03933	0.519663	+	0.854371i	$0.326057\pi$
$390$	0	0
$391$	2240.23	0.289753
$392$	0	0
$393$	−172.325	−0.0221187
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−7991.58	−1.01029	−0.505146	−	0.863034i	$-0.668561\pi$
$397$	−7991.58	−1.01029	−0.505146	+	0.863034i	$0.668561\pi$
$398$	0	0
$399$	−2067.90	−0.259460
$400$	0	0
$401$	418.000	0.0520547	0.0260273	−	0.999661i	$-0.491714\pi$
$401$	418.000	0.0520547	0.0260273	+	0.999661i	$0.491714\pi$
$402$	0	0
$403$	3712.00	0.458829
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12064.0	−1.46926
$408$	0	0
$409$	−1414.00	−0.170948	−0.0854741	−	0.996340i	$-0.527240\pi$
$409$	−1414.00	−0.170948	−0.0854741	+	0.996340i	$0.527240\pi$
$410$	0	0
$411$	1895.58	0.227499
$412$	0	0
$413$	−2584.88	−0.307975
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−8443.94	−0.991610
$418$	0	0
$419$	−3661.91	−0.426960	−0.213480	−	0.976947i	$-0.568480\pi$
$419$	−3661.91	−0.426960	−0.213480	+	0.976947i	$0.568480\pi$
$420$	0	0
$421$	3290.00	0.380866	0.190433	−	0.981700i	$-0.439011\pi$
$421$	3290.00	0.380866	0.190433	+	0.981700i	$0.439011\pi$
$422$	0	0
$423$	−2684.00	−0.308512
$424$	0	0
$425$	0	0
$426$	0	0
$427$	328.000	0.0371734
$428$	0	0
$429$	−3712.00	−0.417755
$430$	0	0
$431$	1723.25	0.192590	0.0962949	−	0.995353i	$-0.469301\pi$
$431$	1723.25	0.192590	0.0962949	+	0.995353i	$0.469301\pi$
$432$	0	0
$433$	−16414.0	−1.82172	−0.910861	−	0.412713i	$-0.864581\pi$
$433$	−16414.0	−1.82172	−0.910861	+	0.412713i	$0.864581\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6720.69	−0.735684
$438$	0	0
$439$	−3704.99	−0.402801	−0.201401	−	0.979509i	$-0.564549\pi$
$439$	−3704.99	−0.402801	−0.201401	+	0.979509i	$0.564549\pi$
$440$	0	0
$441$	3597.00	0.388403
$442$	0	0
$443$	5484.00	0.588155	0.294078	−	0.955782i	$-0.404988\pi$
$443$	5484.00	0.588155	0.294078	+	0.955782i	$0.404988\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−10200.0	−1.07929
$448$	0	0
$449$	3458.00	0.363459	0.181730	−	0.983349i	$-0.441830\pi$
$449$	3458.00	0.363459	0.181730	+	0.983349i	$0.441830\pi$
$450$	0	0
$451$	7323.82	0.764668
$452$	0	0
$453$	8616.26	0.893659
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10468.8	1.07157	0.535786	−	0.844354i	$-0.320016\pi$
$457$	10468.8	1.07157	0.535786	+	0.844354i	$0.320016\pi$
$458$	0	0
$459$	−6548.36	−0.665907
$460$	0	0
$461$	5806.00	0.586578	0.293289	−	0.956024i	$-0.405250\pi$
$461$	5806.00	0.586578	0.293289	+	0.956024i	$0.405250\pi$
$462$	0	0
$463$	5004.00	0.502280	0.251140	−	0.967951i	$-0.419195\pi$
$463$	5004.00	0.502280	0.251140	+	0.967951i	$0.419195\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5148.00	−0.510109	−0.255055	−	0.966927i	$-0.582093\pi$
$467$	−5148.00	−0.510109	−0.255055	+	0.966927i	$0.582093\pi$
$468$	0	0
$469$	2768.00	0.272525
$470$	0	0
$471$	14216.8	1.39082
$472$	0	0
$473$	−13613.7	−1.32338
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5449.79	0.523121
$478$	0	0
$479$	4480.46	0.427385	0.213692	−	0.976901i	$-0.431451\pi$
$479$	4480.46	0.427385	0.213692	+	0.976901i	$0.431451\pi$
$480$	0	0
$481$	−6032.00	−0.571799
$482$	0	0
$483$	832.000	0.0783795
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3044.00	0.283238	0.141619	−	0.989921i	$-0.454769\pi$
$487$	3044.00	0.283238	0.141619	+	0.989921i	$0.454769\pi$
$488$	0	0
$489$	912.000	0.0843396
$490$	0	0
$491$	−215.407	−0.0197987	−0.00989935	−	0.999951i	$-0.503151\pi$
$491$	−215.407	−0.0197987	−0.00989935	+	0.999951i	$0.503151\pi$
$492$	0	0
$493$	6806.85	0.621836
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3791.16	0.342166
$498$	0	0
$499$	−11416.5	−1.02420	−0.512099	−	0.858926i	$-0.671132\pi$
$499$	−11416.5	−1.02420	−0.512099	+	0.858926i	$0.671132\pi$
$500$	0	0
$501$	−1488.00	−0.132692
$502$	0	0
$503$	−15348.0	−1.36050	−0.680252	−	0.732978i	$-0.738130\pi$
$503$	−15348.0	−1.36050	−0.680252	+	0.732978i	$0.738130\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6932.00	0.607221
$508$	0	0
$509$	−17502.0	−1.52409	−0.762046	−	0.647523i	$-0.775805\pi$
$509$	−17502.0	−1.52409	−0.762046	+	0.647523i	$0.775805\pi$
$510$	0	0
$511$	1723.25	0.149182
$512$	0	0
$513$	19645.1	1.69074
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−10511.8	−0.894217
$518$	0	0
$519$	−7668.47	−0.648572
$520$	0	0
$521$	2874.00	0.241674	0.120837	−	0.992672i	$-0.461442\pi$
$521$	2874.00	0.241674	0.120837	+	0.992672i	$0.461442\pi$
$522$	0	0
$523$	14604.0	1.22101	0.610505	−	0.792012i	$-0.290966\pi$
$523$	14604.0	1.22101	0.610505	+	0.792012i	$0.290966\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7424.00	0.613652
$528$	0	0
$529$	−9463.00	−0.777760
$530$	0	0
$531$	7108.42	0.580940
$532$	0	0
$533$	3661.91	0.297589
$534$	0	0
$535$	0	0
$536$	0	0
$537$	11201.1	0.900121
$538$	0	0
$539$	14087.6	1.12578
$540$	0	0
$541$	−19330.0	−1.53616	−0.768079	−	0.640355i	$-0.778787\pi$
$541$	−19330.0	−1.53616	−0.768079	+	0.640355i	$0.778787\pi$
$542$	0	0
$543$	−6168.00	−0.487466
$544$	0	0
$545$	0	0
$546$	0	0
$547$	708.000	0.0553417	0.0276708	−	0.999617i	$-0.491191\pi$
$547$	708.000	0.0553417	0.0276708	+	0.999617i	$0.491191\pi$
$548$	0	0
$549$	−902.000	−0.0701210
$550$	0	0
$551$	−20420.5	−1.57885
$552$	0	0
$553$	−1378.60	−0.106011
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9284.02	0.706242	0.353121	−	0.935578i	$-0.385120\pi$
$557$	9284.02	0.706242	0.353121	+	0.935578i	$0.385120\pi$
$558$	0	0
$559$	−6806.85	−0.515025
$560$	0	0
$561$	−7424.00	−0.558719
$562$	0	0
$563$	−9220.00	−0.690189	−0.345095	−	0.938568i	$-0.612153\pi$
$563$	−9220.00	−0.690189	−0.345095	+	0.938568i	$0.612153\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1244.00	−0.0921395
$568$	0	0
$569$	10458.0	0.770513	0.385257	−	0.922809i	$-0.374113\pi$
$569$	10458.0	0.770513	0.385257	+	0.922809i	$0.374113\pi$
$570$	0	0
$571$	−13312.1	−0.975648	−0.487824	−	0.872942i	$-0.662209\pi$
$571$	−13312.1	−0.975648	−0.487824	+	0.872942i	$0.662209\pi$
$572$	0	0
$573$	−17921.8	−1.30662
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−17491.0	−1.26198	−0.630988	−	0.775792i	$-0.717350\pi$
$577$	−17491.0	−1.26198	−0.630988	+	0.775792i	$0.717350\pi$
$578$	0	0
$579$	10167.2	0.729765
$580$	0	0
$581$	3760.00	0.268487
$582$	0	0
$583$	21344.0	1.51626
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11460.0	0.805800	0.402900	−	0.915244i	$-0.368002\pi$
$587$	11460.0	0.805800	0.402900	+	0.915244i	$0.368002\pi$
$588$	0	0
$589$	−22272.0	−1.55807
$590$	0	0
$591$	−10942.7	−0.761626
$592$	0	0
$593$	2326.39	0.161102	0.0805510	−	0.996750i	$-0.474332\pi$
$593$	2326.39	0.161102	0.0805510	+	0.996750i	$0.474332\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1033.95	0.0708825
$598$	0	0
$599$	24384.0	1.66328	0.831640	−	0.555316i	$-0.187403\pi$
$599$	24384.0	1.66328	0.831640	+	0.555316i	$0.187403\pi$
$600$	0	0
$601$	24054.0	1.63258	0.816292	−	0.577639i	$-0.196026\pi$
$601$	24054.0	1.63258	0.816292	+	0.577639i	$0.196026\pi$
$602$	0	0
$603$	−7612.00	−0.514071
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17364.0	1.16109	0.580546	−	0.814227i	$-0.302839\pi$
$607$	17364.0	1.16109	0.580546	+	0.814227i	$0.302839\pi$
$608$	0	0
$609$	2528.00	0.168210
$610$	0	0
$611$	−5255.92	−0.348006
$612$	0	0
$613$	2434.09	0.160379	0.0801894	−	0.996780i	$-0.474447\pi$
$613$	2434.09	0.160379	0.0801894	+	0.996780i	$0.474447\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8616.26	−0.562201	−0.281100	−	0.959678i	$-0.590699\pi$
$617$	−8616.26	−0.562201	−0.281100	+	0.959678i	$0.590699\pi$
$618$	0	0
$619$	9176.32	0.595844	0.297922	−	0.954590i	$-0.403706\pi$
$619$	9176.32	0.595844	0.297922	+	0.954590i	$0.403706\pi$
$620$	0	0
$621$	−7904.00	−0.510751
$622$	0	0
$623$	24.0000	0.00154340
$624$	0	0
$625$	0	0
$626$	0	0
$627$	22272.0	1.41859
$628$	0	0
$629$	−12064.0	−0.764743
$630$	0	0
$631$	9219.40	0.581646	0.290823	−	0.956777i	$-0.406071\pi$
$631$	9219.40	0.581646	0.290823	+	0.956777i	$0.406071\pi$
$632$	0	0
$633$	−3618.83	−0.227228
$634$	0	0
$635$	0	0
$636$	0	0
$637$	7043.80	0.438125
$638$	0	0
$639$	−10425.7	−0.645436
$640$	0	0
$641$	−25538.0	−1.57362	−0.786810	−	0.617195i	$-0.788269\pi$
$641$	−25538.0	−1.57362	−0.786810	+	0.617195i	$0.788269\pi$
$642$	0	0
$643$	22060.0	1.35297	0.676486	−	0.736455i	$-0.263502\pi$
$643$	22060.0	1.35297	0.676486	+	0.736455i	$0.263502\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	29444.0	1.78912	0.894562	−	0.446944i	$-0.147488\pi$
$647$	29444.0	1.78912	0.894562	+	0.446944i	$0.147488\pi$
$648$	0	0
$649$	27840.0	1.68385
$650$	0	0
$651$	2757.20	0.165996
$652$	0	0
$653$	−20786.7	−1.24571	−0.622854	−	0.782338i	$-0.714027\pi$
$653$	−20786.7	−1.24571	−0.622854	+	0.782338i	$0.714027\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−4738.95	−0.281406
$658$	0	0
$659$	13742.9	0.812366	0.406183	−	0.913792i	$-0.366860\pi$
$659$	13742.9	0.812366	0.406183	+	0.913792i	$0.366860\pi$
$660$	0	0
$661$	11530.0	0.678464	0.339232	−	0.940703i	$-0.389833\pi$
$661$	11530.0	0.678464	0.339232	+	0.940703i	$0.389833\pi$
$662$	0	0
$663$	−3712.00	−0.217439
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8216.00	0.476949
$668$	0	0
$669$	−17136.0	−0.990308
$670$	0	0
$671$	−3532.67	−0.203245
$672$	0	0
$673$	23910.1	1.36949	0.684746	−	0.728782i	$-0.259913\pi$
$673$	23910.1	1.36949	0.684746	+	0.728782i	$0.259913\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2175.61	−0.123509	−0.0617543	−	0.998091i	$-0.519670\pi$
$677$	−2175.61	−0.123509	−0.0617543	+	0.998091i	$0.519670\pi$
$678$	0	0
$679$	4308.13	0.243492
$680$	0	0
$681$	19824.0	1.11550
$682$	0	0
$683$	−20708.0	−1.16013	−0.580066	−	0.814570i	$-0.696973\pi$
$683$	−20708.0	−1.16013	−0.580066	+	0.814570i	$0.696973\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−7080.00	−0.393186
$688$	0	0
$689$	10672.0	0.590088
$690$	0	0
$691$	−4609.70	−0.253779	−0.126890	−	0.991917i	$-0.540499\pi$
$691$	−4609.70	−0.253779	−0.126890	+	0.991917i	$0.540499\pi$
$692$	0	0
$693$	1895.58	0.103906
$694$	0	0
$695$	0	0
$696$	0	0
$697$	7323.82	0.398005
$698$	0	0
$699$	−21713.0	−1.17491
$700$	0	0
$701$	−8942.00	−0.481790	−0.240895	−	0.970551i	$-0.577441\pi$
$701$	−8942.00	−0.481790	−0.240895	+	0.970551i	$0.577441\pi$
$702$	0	0
$703$	36192.0	1.94169
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4056.00	0.215759
$708$	0	0
$709$	−3670.00	−0.194400	−0.0972001	−	0.995265i	$-0.530989\pi$
$709$	−3670.00	−0.194400	−0.0972001	+	0.995265i	$0.530989\pi$
$710$	0	0
$711$	3791.16	0.199971
$712$	0	0
$713$	8960.91	0.470672
$714$	0	0
$715$	0	0
$716$	0	0
$717$	22057.6	1.14889
$718$	0	0
$719$	8616.26	0.446916	0.223458	−	0.974714i	$-0.428265\pi$
$719$	8616.26	0.446916	0.223458	+	0.974714i	$0.428265\pi$
$720$	0	0
$721$	176.000	0.00909096
$722$	0	0
$723$	6472.00	0.332913
$724$	0	0
$725$	0	0
$726$	0	0
$727$	32004.0	1.63269	0.816343	−	0.577567i	$-0.195998\pi$
$727$	32004.0	1.63269	0.816343	+	0.577567i	$0.195998\pi$
$728$	0	0
$729$	19837.0	1.00782
$730$	0	0
$731$	−13613.7	−0.688811
$732$	0	0
$733$	5363.62	0.270273	0.135136	−	0.990827i	$-0.456853\pi$
$733$	5363.62	0.270273	0.135136	+	0.990827i	$0.456853\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−29812.3	−1.49003
$738$	0	0
$739$	−25374.9	−1.26310	−0.631550	−	0.775335i	$-0.717581\pi$
$739$	−25374.9	−1.26310	−0.631550	+	0.775335i	$0.717581\pi$
$740$	0	0
$741$	11136.0	0.552080
$742$	0	0
$743$	17404.0	0.859342	0.429671	−	0.902986i	$-0.358630\pi$
$743$	17404.0	0.859342	0.429671	+	0.902986i	$0.358630\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−10340.0	−0.506454
$748$	0	0
$749$	7248.00	0.353586
$750$	0	0
$751$	32569.5	1.58253	0.791263	−	0.611476i	$-0.209424\pi$
$751$	32569.5	1.58253	0.791263	+	0.611476i	$0.209424\pi$
$752$	0	0
$753$	−19817.4	−0.959079
$754$	0	0
$755$	0	0
$756$	0	0
$757$	6139.09	0.294754	0.147377	−	0.989080i	$-0.452917\pi$
$757$	6139.09	0.294754	0.147377	+	0.989080i	$0.452917\pi$
$758$	0	0
$759$	−8960.91	−0.428538
$760$	0	0
$761$	27850.0	1.32663	0.663313	−	0.748342i	$-0.269150\pi$
$761$	27850.0	1.32663	0.663313	+	0.748342i	$0.269150\pi$
$762$	0	0
$763$	8056.00	0.382237
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13920.0	0.655309
$768$	0	0
$769$	33550.0	1.57327	0.786635	−	0.617419i	$-0.211822\pi$
$769$	33550.0	1.57327	0.786635	+	0.617419i	$0.211822\pi$
$770$	0	0
$771$	18611.1	0.869343
$772$	0	0
$773$	1658.63	0.0771757	0.0385878	−	0.999255i	$-0.487714\pi$
$773$	1658.63	0.0771757	0.0385878	+	0.999255i	$0.487714\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−4480.46	−0.206867
$778$	0	0
$779$	−21971.5	−1.01054
$780$	0	0
$781$	−40832.0	−1.87079
$782$	0	0
$783$	−24016.0	−1.09612
$784$	0	0
$785$	0	0
$786$	0	0
$787$	27028.0	1.22420	0.612099	−	0.790781i	$-0.290325\pi$
$787$	27028.0	1.22420	0.612099	+	0.790781i	$0.290325\pi$
$788$	0	0
$789$	−2416.00	−0.109014
$790$	0	0
$791$	−6548.36	−0.294353
$792$	0	0
$793$	−1766.33	−0.0790976
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2692.58	0.119669	0.0598345	−	0.998208i	$-0.480943\pi$
$797$	2692.58	0.119669	0.0598345	+	0.998208i	$0.480943\pi$
$798$	0	0
$799$	−10511.8	−0.465435
$800$	0	0
$801$	−66.0000	−0.00291135
$802$	0	0
$803$	−18560.0	−0.815652
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−13048.0	−0.569159
$808$	0	0
$809$	−1434.00	−0.0623198	−0.0311599	−	0.999514i	$-0.509920\pi$
$809$	−1434.00	−0.0623198	−0.0311599	+	0.999514i	$0.509920\pi$
$810$	0	0
$811$	42004.3	1.81871	0.909353	−	0.416026i	$-0.136578\pi$
$811$	42004.3	1.81871	0.909353	+	0.416026i	$0.136578\pi$
$812$	0	0
$813$	8271.61	0.356824
$814$	0	0
$815$	0	0
$816$	0	0
$817$	40841.1	1.74890
$818$	0	0
$819$	947.789	0.0404376
$820$	0	0
$821$	21322.0	0.906386	0.453193	−	0.891412i	$-0.350285\pi$
$821$	21322.0	0.906386	0.453193	+	0.891412i	$0.350285\pi$
$822$	0	0
$823$	21548.0	0.912656	0.456328	−	0.889812i	$-0.349164\pi$
$823$	21548.0	0.912656	0.456328	+	0.889812i	$0.349164\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9660.00	−0.406180	−0.203090	−	0.979160i	$-0.565098\pi$
$827$	−9660.00	−0.406180	−0.203090	+	0.979160i	$0.565098\pi$
$828$	0	0
$829$	−24082.0	−1.00893	−0.504465	−	0.863432i	$-0.668310\pi$
$829$	−24082.0	−1.00893	−0.504465	+	0.863432i	$0.668310\pi$
$830$	0	0
$831$	−4911.27	−0.205018
$832$	0	0
$833$	14087.6	0.585962
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−26193.4	−1.08169
$838$	0	0
$839$	−14906.1	−0.613369	−0.306685	−	0.951811i	$-0.599220\pi$
$839$	−14906.1	−0.613369	−0.306685	+	0.951811i	$0.599220\pi$
$840$	0	0
$841$	575.000	0.0235762
$842$	0	0
$843$	5160.00	0.210818
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2100.00	0.0851911
$848$	0	0
$849$	2768.00	0.111893
$850$	0	0
$851$	−14561.5	−0.586559
$852$	0	0
$853$	−21002.1	−0.843024	−0.421512	−	0.906823i	$-0.638501\pi$
$853$	−21002.1	−0.843024	−0.421512	+	0.906823i	$0.638501\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19860.5	0.791624	0.395812	−	0.918332i	$-0.370463\pi$
$857$	19860.5	0.791624	0.395812	+	0.918332i	$0.370463\pi$
$858$	0	0
$859$	34852.8	1.38436	0.692178	−	0.721727i	$-0.256651\pi$
$859$	34852.8	1.38436	0.692178	+	0.721727i	$0.256651\pi$
$860$	0	0
$861$	2720.00	0.107662
$862$	0	0
$863$	11468.0	0.452347	0.226173	−	0.974087i	$-0.427378\pi$
$863$	11468.0	0.452347	0.226173	+	0.974087i	$0.427378\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	12228.0	0.478990
$868$	0	0
$869$	14848.0	0.579613
$870$	0	0
$871$	−14906.1	−0.579880
$872$	0	0
$873$	−11847.4	−0.459304
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−34572.8	−1.33117	−0.665587	−	0.746321i	$-0.731819\pi$
$877$	−34572.8	−1.33117	−0.665587	+	0.746321i	$0.731819\pi$
$878$	0	0
$879$	22143.8	0.849706
$880$	0	0
$881$	−33186.0	−1.26909	−0.634543	−	0.772888i	$-0.718812\pi$
$881$	−33186.0	−1.26909	−0.634543	+	0.772888i	$0.718812\pi$
$882$	0	0
$883$	15196.0	0.579146	0.289573	−	0.957156i	$-0.406487\pi$
$883$	15196.0	0.579146	0.289573	+	0.957156i	$0.406487\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3036.00	−0.114925	−0.0574627	−	0.998348i	$-0.518301\pi$
$887$	−3036.00	−0.114925	−0.0574627	+	0.998348i	$0.518301\pi$
$888$	0	0
$889$	11152.0	0.420727
$890$	0	0
$891$	13398.3	0.503771
$892$	0	0
$893$	31535.5	1.18174
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4480.46	−0.166776
$898$	0	0
$899$	27227.4	1.01011
$900$	0	0
$901$	21344.0	0.789203
$902$	0	0
$903$	−5056.00	−0.186327
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−7404.00	−0.271054	−0.135527	−	0.990774i	$-0.543273\pi$
$907$	−7404.00	−0.271054	−0.135527	+	0.990774i	$0.543273\pi$
$908$	0	0
$909$	−11154.0	−0.406991
$910$	0	0
$911$	−6548.36	−0.238152	−0.119076	−	0.992885i	$-0.537993\pi$
$911$	−6548.36	−0.238152	−0.119076	+	0.992885i	$0.537993\pi$
$912$	0	0
$913$	−40496.4	−1.46795
$914$	0	0
$915$	0	0
$916$	0	0
$917$	172.325	0.00620576
$918$	0	0
$919$	−34206.6	−1.22782	−0.613912	−	0.789374i	$-0.710405\pi$
$919$	−34206.6	−1.22782	−0.613912	+	0.789374i	$0.710405\pi$
$920$	0	0
$921$	29744.0	1.06417
$922$	0	0
$923$	−20416.0	−0.728062
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−484.000	−0.0171485
$928$	0	0
$929$	−24286.0	−0.857694	−0.428847	−	0.903377i	$-0.641080\pi$
$929$	−24286.0	−0.857694	−0.428847	+	0.903377i	$0.641080\pi$
$930$	0	0
$931$	−42262.8	−1.48776
$932$	0	0
$933$	9994.87	0.350715
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−37997.7	−1.32479	−0.662397	−	0.749153i	$-0.730461\pi$
$937$	−37997.7	−1.32479	−0.662397	+	0.749153i	$0.730461\pi$
$938$	0	0
$939$	10856.5	0.377304
$940$	0	0
$941$	−36722.0	−1.27216	−0.636080	−	0.771623i	$-0.719445\pi$
$941$	−36722.0	−1.27216	−0.636080	+	0.771623i	$0.719445\pi$
$942$	0	0
$943$	8840.00	0.305270
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16364.0	−0.561519	−0.280760	−	0.959778i	$-0.590586\pi$
$947$	−16364.0	−0.561519	−0.280760	+	0.959778i	$0.590586\pi$
$948$	0	0
$949$	−9280.00	−0.317431
$950$	0	0
$951$	−13182.9	−0.449510
$952$	0	0
$953$	−7797.72	−0.265050	−0.132525	−	0.991180i	$-0.542309\pi$
$953$	−7797.72	−0.265050	−0.132525	+	0.991180i	$0.542309\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−27227.4	−0.919683
$958$	0	0
$959$	−1895.58	−0.0638284
$960$	0	0
$961$	−95.0000	−0.00318888
$962$	0	0
$963$	−19932.0	−0.666978
$964$	0	0
$965$	0	0
$966$	0	0
$967$	16756.0	0.557225	0.278613	−	0.960404i	$-0.410125\pi$
$967$	16756.0	0.557225	0.278613	+	0.960404i	$0.410125\pi$
$968$	0	0
$969$	22272.0	0.738369
$970$	0	0
$971$	−16414.0	−0.542482	−0.271241	−	0.962512i	$-0.587434\pi$
$971$	−16414.0	−0.542482	−0.271241	+	0.962512i	$0.587434\pi$
$972$	0	0
$973$	8443.94	0.278212
$974$	0	0
$975$	0	0
$976$	0	0
$977$	50362.1	1.64916	0.824578	−	0.565749i	$-0.191413\pi$
$977$	50362.1	1.64916	0.824578	+	0.565749i	$0.191413\pi$
$978$	0	0
$979$	−258.488	−0.00843852
$980$	0	0
$981$	−22154.0	−0.721022
$982$	0	0
$983$	−60612.0	−1.96666	−0.983328	−	0.181841i	$-0.941794\pi$
$983$	−60612.0	−1.96666	−0.983328	+	0.181841i	$0.941794\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−3904.00	−0.125902
$988$	0	0
$989$	−16432.0	−0.528319
$990$	0	0
$991$	−16198.6	−0.519238	−0.259619	−	0.965711i	$-0.583597\pi$
$991$	−16198.6	−0.519238	−0.259619	+	0.965711i	$0.583597\pi$
$992$	0	0
$993$	11890.4	0.379992
$994$	0	0
$995$	0	0
$996$	0	0
$997$	54002.4	1.71542	0.857710	−	0.514133i	$-0.171886\pi$
$997$	54002.4	1.71542	0.857710	+	0.514133i	$0.171886\pi$
$998$	0	0
$999$	42564.3	1.34802

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.4.a.n.1.1		2
4.3	odd	2		800.4.a.r.1.2		2
5.2	odd	4		160.4.c.b.129.4	yes	4
5.3	odd	4		160.4.c.b.129.1	✓	4
5.4	even	2		800.4.a.r.1.1		2
8.3	odd	2		1600.4.a.cc.1.1		2
8.5	even	2		1600.4.a.co.1.2		2
15.2	even	4		1440.4.f.h.289.2		4
15.8	even	4		1440.4.f.h.289.3		4
20.3	even	4		160.4.c.b.129.3	yes	4
20.7	even	4		160.4.c.b.129.2	yes	4
20.19	odd	2	inner	800.4.a.n.1.2		2
40.3	even	4		320.4.c.i.129.2		4
40.13	odd	4		320.4.c.i.129.4		4
40.19	odd	2		1600.4.a.co.1.1		2
40.27	even	4		320.4.c.i.129.3		4
40.29	even	2		1600.4.a.cc.1.2		2
40.37	odd	4		320.4.c.i.129.1		4
60.23	odd	4		1440.4.f.h.289.4		4
60.47	odd	4		1440.4.f.h.289.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.c.b.129.1	✓	4	5.3	odd	4
160.4.c.b.129.2	yes	4	20.7	even	4
160.4.c.b.129.3	yes	4	20.3	even	4
160.4.c.b.129.4	yes	4	5.2	odd	4
320.4.c.i.129.1		4	40.37	odd	4
320.4.c.i.129.2		4	40.3	even	4
320.4.c.i.129.3		4	40.27	even	4
320.4.c.i.129.4		4	40.13	odd	4
800.4.a.n.1.1		2	1.1	even	1	trivial
800.4.a.n.1.2		2	20.19	odd	2	inner
800.4.a.r.1.1		2	5.4	even	2
800.4.a.r.1.2		2	4.3	odd	2
1440.4.f.h.289.1		4	60.47	odd	4
1440.4.f.h.289.2		4	15.2	even	4
1440.4.f.h.289.3		4	15.8	even	4
1440.4.f.h.289.4		4	60.23	odd	4
1600.4.a.cc.1.1		2	8.3	odd	2
1600.4.a.cc.1.2		2	40.29	even	2
1600.4.a.co.1.1		2	40.19	odd	2
1600.4.a.co.1.2		2	8.5	even	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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{3} +4.00000 q^{7} -11.0000 q^{9} +O(q^{10})\)	\(q-4.00000 q^{3} +4.00000 q^{7} -11.0000 q^{9} -43.0813 q^{11} -21.5407 q^{13} -43.0813 q^{17} +129.244 q^{19} -16.0000 q^{21} -52.0000 q^{23} +152.000 q^{27} -158.000 q^{29} -172.325 q^{31} +172.325 q^{33} +280.029 q^{37} +86.1626 q^{39} -170.000 q^{41} +316.000 q^{43} +244.000 q^{47} -327.000 q^{49} +172.325 q^{51} -495.435 q^{53} -516.976 q^{57} -646.220 q^{59} +82.0000 q^{61} -44.0000 q^{63} +692.000 q^{67} +208.000 q^{69} +947.789 q^{71} +430.813 q^{73} -172.325 q^{77} -344.651 q^{79} -311.000 q^{81} +940.000 q^{83} +632.000 q^{87} +6.00000 q^{89} -86.1626 q^{91} +689.301 q^{93} +1077.03 q^{97} +473.895 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{3} + 8 q^{7} - 22 q^{9}+O(q^{10}) \) 2 * q - 8 * q^3 + 8 * q^7 - 22 * q^9	\( 2 q - 8 q^{3} + 8 q^{7} - 22 q^{9} - 32 q^{21} - 104 q^{23} + 304 q^{27} - 316 q^{29} - 340 q^{41} + 632 q^{43} + 488 q^{47} - 654 q^{49} + 164 q^{61} - 88 q^{63} + 1384 q^{67} + 416 q^{69} - 622 q^{81} + 1880 q^{83} + 1264 q^{87} + 12 q^{89}+O(q^{100}) \) 2 * q - 8 * q^3 + 8 * q^7 - 22 * q^9 - 32 * q^21 - 104 * q^23 + 304 * q^27 - 316 * q^29 - 340 * q^41 + 632 * q^43 + 488 * q^47 - 654 * q^49 + 164 * q^61 - 88 * q^63 + 1384 * q^67 + 416 * q^69 - 622 * q^81 + 1880 * q^83 + 1264 * q^87 + 12 * q^89

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