LMFDB - Embedded newform 200.6.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [200,6,Mod(1,200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("200.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	200.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:	$32.0767639626$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.1595208.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 20x^{2} + 33x - 3 $ x^4 - x^3 - 20x^2 + 33x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.64654$ of defining polynomial
Character	$\chi$	$=$	200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+28.9338 q^{3} +146.828 q^{7} +594.165 q^{9} +O(q^{10})$	$q+28.9338 q^{3} +146.828 q^{7} +594.165 q^{9} +191.129 q^{11} +83.9971 q^{13} -2000.23 q^{17} +677.481 q^{19} +4248.29 q^{21} +1296.23 q^{23} +10160.5 q^{27} -3266.00 q^{29} +6157.98 q^{31} +5530.08 q^{33} -11369.5 q^{37} +2430.36 q^{39} -10599.8 q^{41} +12926.2 q^{43} +9521.88 q^{47} +4751.45 q^{49} -57874.4 q^{51} +14786.8 q^{53} +19602.1 q^{57} -38226.9 q^{59} -3581.69 q^{61} +87240.0 q^{63} +21780.0 q^{67} +37504.9 q^{69} +51390.1 q^{71} -13305.9 q^{73} +28063.1 q^{77} +15944.3 q^{79} +149601. q^{81} -53305.0 q^{83} -94497.8 q^{87} -51330.4 q^{89} +12333.1 q^{91} +178174. q^{93} +80849.9 q^{97} +113562. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 148 q^{7} + 500 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 148 * q^7 + 500 * q^9	$ 4 q + 4 q^{3} + 148 q^{7} + 500 q^{9} - 368 q^{11} + 440 q^{13} - 672 q^{17} - 688 q^{19} + 992 q^{21} + 4492 q^{23} + 8152 q^{27} - 2936 q^{29} + 2112 q^{31} + 26864 q^{33} + 8792 q^{37} + 1504 q^{39} + 11800 q^{41} + 48276 q^{43} + 14724 q^{47} + 22500 q^{49} - 62400 q^{51} + 84296 q^{53} + 71024 q^{57} - 45840 q^{59} + 61928 q^{61} + 186292 q^{63} + 72700 q^{67} + 38368 q^{69} - 62816 q^{71} + 133072 q^{73} + 11440 q^{77} - 21632 q^{79} + 204836 q^{81} + 74660 q^{83} - 12472 q^{87} + 20952 q^{89} - 243808 q^{91} + 105600 q^{93} + 59456 q^{97} - 133424 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 148 * q^7 + 500 * q^9 - 368 * q^11 + 440 * q^13 - 672 * q^17 - 688 * q^19 + 992 * q^21 + 4492 * q^23 + 8152 * q^27 - 2936 * q^29 + 2112 * q^31 + 26864 * q^33 + 8792 * q^37 + 1504 * q^39 + 11800 * q^41 + 48276 * q^43 + 14724 * q^47 + 22500 * q^49 - 62400 * q^51 + 84296 * q^53 + 71024 * q^57 - 45840 * q^59 + 61928 * q^61 + 186292 * q^63 + 72700 * q^67 + 38368 * q^69 - 62816 * q^71 + 133072 * q^73 + 11440 * q^77 - 21632 * q^79 + 204836 * q^81 + 74660 * q^83 - 12472 * q^87 + 20952 * q^89 - 243808 * q^91 + 105600 * q^93 + 59456 * q^97 - 133424 * 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
$3$	28.9338	1.85610	0.928052	−	0.372451i	$-0.121482\pi$
$3$	28.9338	1.85610	0.928052	+	0.372451i	$0.121482\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	146.828	1.13257	0.566283	−	0.824211i	$-0.308381\pi$
$7$	146.828	1.13257	0.566283	+	0.824211i	$0.308381\pi$
$8$	0	0
$9$	594.165	2.44512
$10$	0	0
$11$	191.129	0.476260	0.238130	−	0.971233i	$-0.423466\pi$
$11$	191.129	0.476260	0.238130	+	0.971233i	$0.423466\pi$
$12$	0	0
$13$	83.9971	0.137850	0.0689249	−	0.997622i	$-0.478043\pi$
$13$	83.9971	0.137850	0.0689249	+	0.997622i	$0.478043\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2000.23	−1.67864	−0.839322	−	0.543635i	$-0.817048\pi$
$17$	−2000.23	−1.67864	−0.839322	+	0.543635i	$0.817048\pi$
$18$	0	0
$19$	677.481	0.430540	0.215270	−	0.976555i	$-0.430937\pi$
$19$	677.481	0.430540	0.215270	+	0.976555i	$0.430937\pi$
$20$	0	0
$21$	4248.29	2.10216
$22$	0	0
$23$	1296.23	0.510931	0.255466	−	0.966818i	$-0.417771\pi$
$23$	1296.23	0.510931	0.255466	+	0.966818i	$0.417771\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	10160.5	2.68230
$28$	0	0
$29$	−3266.00	−0.721143	−0.360571	−	0.932732i	$-0.617418\pi$
$29$	−3266.00	−0.721143	−0.360571	+	0.932732i	$0.617418\pi$
$30$	0	0
$31$	6157.98	1.15089	0.575445	−	0.817841i	$-0.304829\pi$
$31$	6157.98	1.15089	0.575445	+	0.817841i	$0.304829\pi$
$32$	0	0
$33$	5530.08	0.883989
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−11369.5	−1.36533	−0.682667	−	0.730730i	$-0.739180\pi$
$37$	−11369.5	−1.36533	−0.682667	+	0.730730i	$0.739180\pi$
$38$	0	0
$39$	2430.36	0.255863
$40$	0	0
$41$	−10599.8	−0.984774	−0.492387	−	0.870376i	$-0.663876\pi$
$41$	−10599.8	−0.984774	−0.492387	+	0.870376i	$0.663876\pi$
$42$	0	0
$43$	12926.2	1.06610	0.533052	−	0.846082i	$-0.321045\pi$
$43$	12926.2	1.06610	0.533052	+	0.846082i	$0.321045\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9521.88	0.628750	0.314375	−	0.949299i	$-0.398205\pi$
$47$	9521.88	0.628750	0.314375	+	0.949299i	$0.398205\pi$
$48$	0	0
$49$	4751.45	0.282706
$50$	0	0
$51$	−57874.4	−3.11574
$52$	0	0
$53$	14786.8	0.723078	0.361539	−	0.932357i	$-0.382251\pi$
$53$	14786.8	0.723078	0.361539	+	0.932357i	$0.382251\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	19602.1	0.799126
$58$	0	0
$59$	−38226.9	−1.42968	−0.714841	−	0.699287i	$-0.753501\pi$
$59$	−38226.9	−1.42968	−0.714841	+	0.699287i	$0.753501\pi$
$60$	0	0
$61$	−3581.69	−0.123243	−0.0616217	−	0.998100i	$-0.519627\pi$
$61$	−3581.69	−0.123243	−0.0616217	+	0.998100i	$0.519627\pi$
$62$	0	0
$63$	87240.0	2.76926
$64$	0	0
$65$	0	0
$66$	0	0
$67$	21780.0	0.592748	0.296374	−	0.955072i	$-0.404223\pi$
$67$	21780.0	0.592748	0.296374	+	0.955072i	$0.404223\pi$
$68$	0	0
$69$	37504.9	0.948341
$70$	0	0
$71$	51390.1	1.20986	0.604928	−	0.796280i	$-0.293202\pi$
$71$	51390.1	1.20986	0.604928	+	0.796280i	$0.293202\pi$
$72$	0	0
$73$	−13305.9	−0.292238	−0.146119	−	0.989267i	$-0.546678\pi$
$73$	−13305.9	−0.292238	−0.146119	+	0.989267i	$0.546678\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	28063.1	0.539396
$78$	0	0
$79$	15944.3	0.287434	0.143717	−	0.989619i	$-0.454094\pi$
$79$	15944.3	0.287434	0.143717	+	0.989619i	$0.454094\pi$
$80$	0	0
$81$	149601.	2.53350
$82$	0	0
$83$	−53305.0	−0.849323	−0.424662	−	0.905352i	$-0.639607\pi$
$83$	−53305.0	−0.849323	−0.424662	+	0.905352i	$0.639607\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−94497.8	−1.33852
$88$	0	0
$89$	−51330.4	−0.686910	−0.343455	−	0.939169i	$-0.611597\pi$
$89$	−51330.4	−0.686910	−0.343455	+	0.939169i	$0.611597\pi$
$90$	0	0
$91$	12333.1	0.156124
$92$	0	0
$93$	178174.	2.13617
$94$	0	0
$95$	0	0
$96$	0	0
$97$	80849.9	0.872469	0.436234	−	0.899833i	$-0.356312\pi$
$97$	80849.9	0.872469	0.436234	+	0.899833i	$0.356312\pi$
$98$	0	0
$99$	113562.	1.16451
$100$	0	0
$101$	89471.2	0.872729	0.436365	−	0.899770i	$-0.356266\pi$
$101$	89471.2	0.872729	0.436365	+	0.899770i	$0.356266\pi$
$102$	0	0
$103$	−44184.6	−0.410372	−0.205186	−	0.978723i	$-0.565780\pi$
$103$	−44184.6	−0.410372	−0.205186	+	0.978723i	$0.565780\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	56446.0	0.476622	0.238311	−	0.971189i	$-0.423406\pi$
$107$	56446.0	0.476622	0.238311	+	0.971189i	$0.423406\pi$
$108$	0	0
$109$	52651.1	0.424465	0.212232	−	0.977219i	$-0.431927\pi$
$109$	52651.1	0.424465	0.212232	+	0.977219i	$0.431927\pi$
$110$	0	0
$111$	−328964.	−2.53420
$112$	0	0
$113$	−78399.6	−0.577587	−0.288794	−	0.957391i	$-0.593254\pi$
$113$	−78399.6	−0.577587	−0.288794	+	0.957391i	$0.593254\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	49908.1	0.337059
$118$	0	0
$119$	−293690.	−1.90118
$120$	0	0
$121$	−124521.	−0.773176
$122$	0	0
$123$	−306691.	−1.82784
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−296102.	−1.62904	−0.814522	−	0.580133i	$-0.803001\pi$
$127$	−296102.	−1.62904	−0.814522	+	0.580133i	$0.803001\pi$
$128$	0	0
$129$	374004.	1.97880
$130$	0	0
$131$	−199030.	−1.01331	−0.506653	−	0.862150i	$-0.669117\pi$
$131$	−199030.	−1.01331	−0.506653	+	0.862150i	$0.669117\pi$
$132$	0	0
$133$	99473.2	0.487615
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−361224.	−1.64428	−0.822138	−	0.569288i	$-0.807219\pi$
$137$	−361224.	−1.64428	−0.822138	+	0.569288i	$0.807219\pi$
$138$	0	0
$139$	−257315.	−1.12961	−0.564805	−	0.825224i	$-0.691049\pi$
$139$	−257315.	−1.12961	−0.564805	+	0.825224i	$0.691049\pi$
$140$	0	0
$141$	275504.	1.16702
$142$	0	0
$143$	16054.3	0.0656524
$144$	0	0
$145$	0	0
$146$	0	0
$147$	137477.	0.524733
$148$	0	0
$149$	−478333.	−1.76508	−0.882540	−	0.470238i	$-0.844168\pi$
$149$	−478333.	−1.76508	−0.882540	+	0.470238i	$0.844168\pi$
$150$	0	0
$151$	−193483.	−0.690558	−0.345279	−	0.938500i	$-0.612216\pi$
$151$	−193483.	−0.690558	−0.345279	+	0.938500i	$0.612216\pi$
$152$	0	0
$153$	−1.18847e6	−4.10449
$154$	0	0
$155$	0	0
$156$	0	0
$157$	460294.	1.49034	0.745172	−	0.666873i	$-0.232367\pi$
$157$	460294.	1.49034	0.745172	+	0.666873i	$0.232367\pi$
$158$	0	0
$159$	427839.	1.34211
$160$	0	0
$161$	190323.	0.578663
$162$	0	0
$163$	372211.	1.09729	0.548643	−	0.836057i	$-0.315145\pi$
$163$	372211.	1.09729	0.548643	+	0.836057i	$0.315145\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	423266.	1.17442	0.587208	−	0.809436i	$-0.300227\pi$
$167$	423266.	1.17442	0.587208	+	0.809436i	$0.300227\pi$
$168$	0	0
$169$	−364237.	−0.980997
$170$	0	0
$171$	402535.	1.05272
$172$	0	0
$173$	223915.	0.568810	0.284405	−	0.958704i	$-0.408204\pi$
$173$	223915.	0.568810	0.284405	+	0.958704i	$0.408204\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.10605e6	−2.65364
$178$	0	0
$179$	−29324.8	−0.0684072	−0.0342036	−	0.999415i	$-0.510889\pi$
$179$	−29324.8	−0.0684072	−0.0342036	+	0.999415i	$0.510889\pi$
$180$	0	0
$181$	−46594.3	−0.105715	−0.0528574	−	0.998602i	$-0.516833\pi$
$181$	−46594.3	−0.105715	−0.0528574	+	0.998602i	$0.516833\pi$
$182$	0	0
$183$	−103632.	−0.228752
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−382302.	−0.799472
$188$	0	0
$189$	1.49185e6	3.03788
$190$	0	0
$191$	327452.	0.649477	0.324739	−	0.945804i	$-0.394724\pi$
$191$	327452.	0.649477	0.324739	+	0.945804i	$0.394724\pi$
$192$	0	0
$193$	−406622.	−0.785773	−0.392887	−	0.919587i	$-0.628524\pi$
$193$	−406622.	−0.785773	−0.392887	+	0.919587i	$0.628524\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−364071.	−0.668376	−0.334188	−	0.942506i	$-0.608462\pi$
$197$	−364071.	−0.668376	−0.334188	+	0.942506i	$0.608462\pi$
$198$	0	0
$199$	55181.3	0.0987777	0.0493889	−	0.998780i	$-0.484273\pi$
$199$	55181.3	0.0987777	0.0493889	+	0.998780i	$0.484273\pi$
$200$	0	0
$201$	630177.	1.10020
$202$	0	0
$203$	−479540.	−0.816742
$204$	0	0
$205$	0	0
$206$	0	0
$207$	770174.	1.24929
$208$	0	0
$209$	129486.	0.205049
$210$	0	0
$211$	−831145.	−1.28520	−0.642600	−	0.766202i	$-0.722144\pi$
$211$	−831145.	−1.28520	−0.642600	+	0.766202i	$0.722144\pi$
$212$	0	0
$213$	1.48691e6	2.24562
$214$	0	0
$215$	0	0
$216$	0	0
$217$	904163.	1.30346
$218$	0	0
$219$	−384989.	−0.542423
$220$	0	0
$221$	−168014.	−0.231401
$222$	0	0
$223$	−458628.	−0.617587	−0.308793	−	0.951129i	$-0.599925\pi$
$223$	−458628.	−0.617587	−0.308793	+	0.951129i	$0.599925\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−22779.7	−0.0293416	−0.0146708	−	0.999892i	$-0.504670\pi$
$227$	−22779.7	−0.0293416	−0.0146708	+	0.999892i	$0.504670\pi$
$228$	0	0
$229$	−1.36560e6	−1.72082	−0.860408	−	0.509606i	$-0.829791\pi$
$229$	−1.36560e6	−1.72082	−0.860408	+	0.509606i	$0.829791\pi$
$230$	0	0
$231$	811971.	1.00118
$232$	0	0
$233$	673470.	0.812696	0.406348	−	0.913718i	$-0.366802\pi$
$233$	673470.	0.812696	0.406348	+	0.913718i	$0.366802\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	461330.	0.533508
$238$	0	0
$239$	461339.	0.522426	0.261213	−	0.965281i	$-0.415877\pi$
$239$	461339.	0.522426	0.261213	+	0.965281i	$0.415877\pi$
$240$	0	0
$241$	−842868.	−0.934796	−0.467398	−	0.884047i	$-0.654809\pi$
$241$	−842868.	−0.934796	−0.467398	+	0.884047i	$0.654809\pi$
$242$	0	0
$243$	1.85951e6	2.02014
$244$	0	0
$245$	0	0
$246$	0	0
$247$	56906.4	0.0593498
$248$	0	0
$249$	−1.54232e6	−1.57643
$250$	0	0
$251$	−1.34039e6	−1.34291	−0.671457	−	0.741044i	$-0.734331\pi$
$251$	−1.34039e6	−1.34291	−0.671457	+	0.741044i	$0.734331\pi$
$252$	0	0
$253$	247747.	0.243336
$254$	0	0
$255$	0	0
$256$	0	0
$257$	337841.	0.319066	0.159533	−	0.987193i	$-0.449001\pi$
$257$	337841.	0.319066	0.159533	+	0.987193i	$0.449001\pi$
$258$	0	0
$259$	−1.66937e6	−1.54633
$260$	0	0
$261$	−1.94054e6	−1.76328
$262$	0	0
$263$	−836410.	−0.745641	−0.372821	−	0.927903i	$-0.621609\pi$
$263$	−836410.	−0.745641	−0.372821	+	0.927903i	$0.621609\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.48518e6	−1.27498
$268$	0	0
$269$	−36624.0	−0.0308592	−0.0154296	−	0.999881i	$-0.504912\pi$
$269$	−36624.0	−0.0308592	−0.0154296	+	0.999881i	$0.504912\pi$
$270$	0	0
$271$	1.92834e6	1.59500	0.797500	−	0.603319i	$-0.206155\pi$
$271$	1.92834e6	1.59500	0.797500	+	0.603319i	$0.206155\pi$
$272$	0	0
$273$	356844.	0.289782
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.42694e6	1.11739	0.558696	−	0.829373i	$-0.311302\pi$
$277$	1.42694e6	1.11739	0.558696	+	0.829373i	$0.311302\pi$
$278$	0	0
$279$	3.65885e6	2.81407
$280$	0	0
$281$	−2.36724e6	−1.78845	−0.894223	−	0.447621i	$-0.852271\pi$
$281$	−2.36724e6	−1.78845	−0.894223	+	0.447621i	$0.852271\pi$
$282$	0	0
$283$	1.35021e6	1.00215	0.501077	−	0.865403i	$-0.332937\pi$
$283$	1.35021e6	1.00215	0.501077	+	0.865403i	$0.332937\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.55634e6	−1.11532
$288$	0	0
$289$	2.58108e6	1.81785
$290$	0	0
$291$	2.33929e6	1.61939
$292$	0	0
$293$	506336.	0.344564	0.172282	−	0.985048i	$-0.444886\pi$
$293$	506336.	0.344564	0.172282	+	0.985048i	$0.444886\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.94197e6	1.27747
$298$	0	0
$299$	108880.	0.0704317
$300$	0	0
$301$	1.89793e6	1.20743
$302$	0	0
$303$	2.58874e6	1.61988
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.15053e6	0.696712	0.348356	−	0.937362i	$-0.386740\pi$
$307$	1.15053e6	0.696712	0.348356	+	0.937362i	$0.386740\pi$
$308$	0	0
$309$	−1.27843e6	−0.761693
$310$	0	0
$311$	−742231.	−0.435149	−0.217574	−	0.976044i	$-0.569815\pi$
$311$	−742231.	−0.435149	−0.217574	+	0.976044i	$0.569815\pi$
$312$	0	0
$313$	1.67101e6	0.964090	0.482045	−	0.876146i	$-0.339894\pi$
$313$	1.67101e6	0.964090	0.482045	+	0.876146i	$0.339894\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−646531.	−0.361361	−0.180681	−	0.983542i	$-0.557830\pi$
$317$	−646531.	−0.361361	−0.180681	+	0.983542i	$0.557830\pi$
$318$	0	0
$319$	−624227.	−0.343452
$320$	0	0
$321$	1.63320e6	0.884660
$322$	0	0
$323$	−1.35512e6	−0.722723
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.52340e6	0.787851
$328$	0	0
$329$	1.39808e6	0.712101
$330$	0	0
$331$	2.73052e6	1.36986	0.684928	−	0.728610i	$-0.259834\pi$
$331$	2.73052e6	1.36986	0.684928	+	0.728610i	$0.259834\pi$
$332$	0	0
$333$	−6.75538e6	−3.33841
$334$	0	0
$335$	0	0
$336$	0	0
$337$	390252.	0.187185	0.0935923	−	0.995611i	$-0.470165\pi$
$337$	390252.	0.187185	0.0935923	+	0.995611i	$0.470165\pi$
$338$	0	0
$339$	−2.26840e6	−1.07206
$340$	0	0
$341$	1.17697e6	0.548123
$342$	0	0
$343$	−1.77009e6	−0.812382
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.23128e6	−0.994788	−0.497394	−	0.867525i	$-0.665710\pi$
$347$	−2.23128e6	−0.994788	−0.497394	+	0.867525i	$0.665710\pi$
$348$	0	0
$349$	−502679.	−0.220916	−0.110458	−	0.993881i	$-0.535232\pi$
$349$	−502679.	−0.220916	−0.110458	+	0.993881i	$0.535232\pi$
$350$	0	0
$351$	853455.	0.369754
$352$	0	0
$353$	4.57148e6	1.95263	0.976316	−	0.216350i	$-0.0694153\pi$
$353$	4.57148e6	1.95263	0.976316	+	0.216350i	$0.0694153\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−8.49758e6	−3.52878
$358$	0	0
$359$	−448308.	−0.183586	−0.0917931	−	0.995778i	$-0.529260\pi$
$359$	−448308.	−0.183586	−0.0917931	+	0.995778i	$0.529260\pi$
$360$	0	0
$361$	−2.01712e6	−0.814636
$362$	0	0
$363$	−3.60286e6	−1.43510
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2.56484e6	0.994018	0.497009	−	0.867745i	$-0.334432\pi$
$367$	2.56484e6	0.994018	0.497009	+	0.867745i	$0.334432\pi$
$368$	0	0
$369$	−6.29801e6	−2.40789
$370$	0	0
$371$	2.17112e6	0.818934
$372$	0	0
$373$	484728.	0.180396	0.0901978	−	0.995924i	$-0.471250\pi$
$373$	484728.	0.180396	0.0901978	+	0.995924i	$0.471250\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−274335.	−0.0994094
$378$	0	0
$379$	4.26934e6	1.52673	0.763366	−	0.645966i	$-0.223545\pi$
$379$	4.26934e6	1.52673	0.763366	+	0.645966i	$0.223545\pi$
$380$	0	0
$381$	−8.56737e6	−3.02367
$382$	0	0
$383$	−1.27791e6	−0.445147	−0.222573	−	0.974916i	$-0.571446\pi$
$383$	−1.27791e6	−0.445147	−0.222573	+	0.974916i	$0.571446\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.68029e6	2.60676
$388$	0	0
$389$	3.37341e6	1.13030	0.565151	−	0.824987i	$-0.308818\pi$
$389$	3.37341e6	1.13030	0.565151	+	0.824987i	$0.308818\pi$
$390$	0	0
$391$	−2.59276e6	−0.857672
$392$	0	0
$393$	−5.75870e6	−1.88080
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.82804e6	0.582117	0.291058	−	0.956705i	$-0.405993\pi$
$397$	1.82804e6	0.582117	0.291058	+	0.956705i	$0.405993\pi$
$398$	0	0
$399$	2.87814e6	0.905064
$400$	0	0
$401$	1.34118e6	0.416511	0.208256	−	0.978074i	$-0.433221\pi$
$401$	1.34118e6	0.416511	0.208256	+	0.978074i	$0.433221\pi$
$402$	0	0
$403$	517252.	0.158650
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.17305e6	−0.650254
$408$	0	0
$409$	−803937.	−0.237637	−0.118818	−	0.992916i	$-0.537911\pi$
$409$	−803937.	−0.237637	−0.118818	+	0.992916i	$0.537911\pi$
$410$	0	0
$411$	−1.04516e7	−3.05195
$412$	0	0
$413$	−5.61278e6	−1.61921
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−7.44511e6	−2.09667
$418$	0	0
$419$	−3.90295e6	−1.08607	−0.543035	−	0.839710i	$-0.682725\pi$
$419$	−3.90295e6	−1.08607	−0.543035	+	0.839710i	$0.682725\pi$
$420$	0	0
$421$	3.04306e6	0.836769	0.418384	−	0.908270i	$-0.362597\pi$
$421$	3.04306e6	0.836769	0.418384	+	0.908270i	$0.362597\pi$
$422$	0	0
$423$	5.65756e6	1.53737
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−525892.	−0.139581
$428$	0	0
$429$	464511.	0.121858
$430$	0	0
$431$	772580.	0.200332	0.100166	−	0.994971i	$-0.468063\pi$
$431$	772580.	0.200332	0.100166	+	0.994971i	$0.468063\pi$
$432$	0	0
$433$	4.83430e6	1.23912	0.619561	−	0.784948i	$-0.287311\pi$
$433$	4.83430e6	1.23912	0.619561	+	0.784948i	$0.287311\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	878171.	0.219976
$438$	0	0
$439$	4.98655e6	1.23492	0.617460	−	0.786602i	$-0.288162\pi$
$439$	4.98655e6	1.23492	0.617460	+	0.786602i	$0.288162\pi$
$440$	0	0
$441$	2.82314e6	0.691252
$442$	0	0
$443$	−7.63238e6	−1.84778	−0.923891	−	0.382655i	$-0.875010\pi$
$443$	−7.63238e6	−1.84778	−0.923891	+	0.382655i	$0.875010\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1.38400e7	−3.27617
$448$	0	0
$449$	1.92156e6	0.449819	0.224910	−	0.974380i	$-0.427791\pi$
$449$	1.92156e6	0.449819	0.224910	+	0.974380i	$0.427791\pi$
$450$	0	0
$451$	−2.02592e6	−0.469009
$452$	0	0
$453$	−5.59820e6	−1.28175
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.64057e6	1.48735	0.743677	−	0.668539i	$-0.233080\pi$
$457$	6.64057e6	1.48735	0.743677	+	0.668539i	$0.233080\pi$
$458$	0	0
$459$	−2.03234e7	−4.50262
$460$	0	0
$461$	5.78658e6	1.26815	0.634074	−	0.773272i	$-0.281381\pi$
$461$	5.78658e6	1.26815	0.634074	+	0.773272i	$0.281381\pi$
$462$	0	0
$463$	−4.55738e6	−0.988013	−0.494007	−	0.869458i	$-0.664468\pi$
$463$	−4.55738e6	−0.988013	−0.494007	+	0.869458i	$0.664468\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.02351e6	0.217171	0.108585	−	0.994087i	$-0.465368\pi$
$467$	1.02351e6	0.217171	0.108585	+	0.994087i	$0.465368\pi$
$468$	0	0
$469$	3.19791e6	0.671326
$470$	0	0
$471$	1.33181e7	2.76623
$472$	0	0
$473$	2.47057e6	0.507743
$474$	0	0
$475$	0	0
$476$	0	0
$477$	8.78581e6	1.76801
$478$	0	0
$479$	5.42421e6	1.08018	0.540092	−	0.841606i	$-0.318390\pi$
$479$	5.42421e6	1.08018	0.540092	+	0.841606i	$0.318390\pi$
$480$	0	0
$481$	−955009.	−0.188211
$482$	0	0
$483$	5.50676e6	1.07406
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8.76364e6	1.67441	0.837206	−	0.546888i	$-0.184188\pi$
$487$	8.76364e6	1.67441	0.837206	+	0.546888i	$0.184188\pi$
$488$	0	0
$489$	1.07695e7	2.03668
$490$	0	0
$491$	2.44420e6	0.457545	0.228772	−	0.973480i	$-0.426529\pi$
$491$	2.44420e6	0.457545	0.228772	+	0.973480i	$0.426529\pi$
$492$	0	0
$493$	6.53277e6	1.21054
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.54550e6	1.37024
$498$	0	0
$499$	5.77888e6	1.03894	0.519472	−	0.854487i	$-0.326129\pi$
$499$	5.77888e6	1.03894	0.519472	+	0.854487i	$0.326129\pi$
$500$	0	0
$501$	1.22467e7	2.17984
$502$	0	0
$503$	5.58777e6	0.984733	0.492367	−	0.870388i	$-0.336132\pi$
$503$	5.58777e6	0.984733	0.492367	+	0.870388i	$0.336132\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.05388e7	−1.82083
$508$	0	0
$509$	−9.32211e6	−1.59485	−0.797425	−	0.603418i	$-0.793805\pi$
$509$	−9.32211e6	−1.59485	−0.797425	+	0.603418i	$0.793805\pi$
$510$	0	0
$511$	−1.95367e6	−0.330978
$512$	0	0
$513$	6.88357e6	1.15484
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.81990e6	0.299449
$518$	0	0
$519$	6.47871e6	1.05577
$520$	0	0
$521$	7.51696e6	1.21324	0.606622	−	0.794991i	$-0.292524\pi$
$521$	7.51696e6	1.21324	0.606622	+	0.794991i	$0.292524\pi$
$522$	0	0
$523$	−618460.	−0.0988683	−0.0494342	−	0.998777i	$-0.515742\pi$
$523$	−618460.	−0.0988683	−0.0494342	+	0.998777i	$0.515742\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.23174e7	−1.93193
$528$	0	0
$529$	−4.75613e6	−0.738949
$530$	0	0
$531$	−2.27131e7	−3.49575
$532$	0	0
$533$	−890350.	−0.135751
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−848477.	−0.126971
$538$	0	0
$539$	908138.	0.134642
$540$	0	0
$541$	−7.78044e6	−1.14291	−0.571454	−	0.820634i	$-0.693620\pi$
$541$	−7.78044e6	−1.14291	−0.571454	+	0.820634i	$0.693620\pi$
$542$	0	0
$543$	−1.34815e6	−0.196218
$544$	0	0
$545$	0	0
$546$	0	0
$547$	6.38809e6	0.912857	0.456428	−	0.889760i	$-0.349128\pi$
$547$	6.38809e6	0.912857	0.456428	+	0.889760i	$0.349128\pi$
$548$	0	0
$549$	−2.12811e6	−0.301345
$550$	0	0
$551$	−2.21265e6	−0.310481
$552$	0	0
$553$	2.34107e6	0.325538
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8.07422e6	−1.10271	−0.551356	−	0.834270i	$-0.685890\pi$
$557$	−8.07422e6	−1.10271	−0.551356	+	0.834270i	$0.685890\pi$
$558$	0	0
$559$	1.08576e6	0.146962
$560$	0	0
$561$	−1.10615e7	−1.48390
$562$	0	0
$563$	5.02445e6	0.668064	0.334032	−	0.942562i	$-0.391591\pi$
$563$	5.02445e6	0.668064	0.334032	+	0.942562i	$0.391591\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.19656e7	2.86936
$568$	0	0
$569$	8.36647e6	1.08333	0.541666	−	0.840594i	$-0.317794\pi$
$569$	8.36647e6	1.08333	0.541666	+	0.840594i	$0.317794\pi$
$570$	0	0
$571$	−831558.	−0.106734	−0.0533670	−	0.998575i	$-0.516995\pi$
$571$	−831558.	−0.106734	−0.0533670	+	0.998575i	$0.516995\pi$
$572$	0	0
$573$	9.47443e6	1.20550
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−5.63868e6	−0.705080	−0.352540	−	0.935797i	$-0.614682\pi$
$577$	−5.63868e6	−0.705080	−0.352540	+	0.935797i	$0.614682\pi$
$578$	0	0
$579$	−1.17651e7	−1.45848
$580$	0	0
$581$	−7.82667e6	−0.961915
$582$	0	0
$583$	2.82619e6	0.344373
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.13074e6	−0.494803	−0.247402	−	0.968913i	$-0.579577\pi$
$587$	−4.13074e6	−0.494803	−0.247402	+	0.968913i	$0.579577\pi$
$588$	0	0
$589$	4.17191e6	0.495504
$590$	0	0
$591$	−1.05340e7	−1.24057
$592$	0	0
$593$	−2.07710e6	−0.242561	−0.121280	−	0.992618i	$-0.538700\pi$
$593$	−2.07710e6	−0.242561	−0.121280	+	0.992618i	$0.538700\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.59660e6	0.183342
$598$	0	0
$599$	−1.44039e7	−1.64026	−0.820128	−	0.572180i	$-0.806098\pi$
$599$	−1.44039e7	−1.64026	−0.820128	+	0.572180i	$0.806098\pi$
$600$	0	0
$601$	1.16092e7	1.31104	0.655522	−	0.755176i	$-0.272449\pi$
$601$	1.16092e7	1.31104	0.655522	+	0.755176i	$0.272449\pi$
$602$	0	0
$603$	1.29409e7	1.44934
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.75662e7	1.93511	0.967557	−	0.252654i	$-0.0813035\pi$
$607$	1.75662e7	1.93511	0.967557	+	0.252654i	$0.0813035\pi$
$608$	0	0
$609$	−1.38749e7	−1.51596
$610$	0	0
$611$	799810.	0.0866730
$612$	0	0
$613$	1.76294e7	1.89490	0.947449	−	0.319906i	$-0.103651\pi$
$613$	1.76294e7	1.89490	0.947449	+	0.319906i	$0.103651\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.44774e6	−0.681859	−0.340930	−	0.940089i	$-0.610742\pi$
$617$	−6.44774e6	−0.681859	−0.340930	+	0.940089i	$0.610742\pi$
$618$	0	0
$619$	−1.48717e7	−1.56003	−0.780017	−	0.625758i	$-0.784790\pi$
$619$	−1.48717e7	−1.56003	−0.780017	+	0.625758i	$0.784790\pi$
$620$	0	0
$621$	1.31704e7	1.37047
$622$	0	0
$623$	−7.53674e6	−0.777971
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.74653e6	0.380592
$628$	0	0
$629$	2.27418e7	2.29191
$630$	0	0
$631$	−5.05549e6	−0.505464	−0.252732	−	0.967536i	$-0.581329\pi$
$631$	−5.05549e6	−0.505464	−0.252732	+	0.967536i	$0.581329\pi$
$632$	0	0
$633$	−2.40482e7	−2.38547
$634$	0	0
$635$	0	0
$636$	0	0
$637$	399108.	0.0389710
$638$	0	0
$639$	3.05342e7	2.95824
$640$	0	0
$641$	6.66243e6	0.640453	0.320226	−	0.947341i	$-0.396241\pi$
$641$	6.66243e6	0.640453	0.320226	+	0.947341i	$0.396241\pi$
$642$	0	0
$643$	3.29414e6	0.314206	0.157103	−	0.987582i	$-0.449784\pi$
$643$	3.29414e6	0.314206	0.157103	+	0.987582i	$0.449784\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.79219e6	0.450063	0.225032	−	0.974351i	$-0.427751\pi$
$647$	4.79219e6	0.450063	0.225032	+	0.974351i	$0.427751\pi$
$648$	0	0
$649$	−7.30627e6	−0.680901
$650$	0	0
$651$	2.61609e7	2.41936
$652$	0	0
$653$	−1.27962e7	−1.17435	−0.587177	−	0.809459i	$-0.699761\pi$
$653$	−1.27962e7	−1.17435	−0.587177	+	0.809459i	$0.699761\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−7.90588e6	−0.714556
$658$	0	0
$659$	−3.06887e6	−0.275274	−0.137637	−	0.990483i	$-0.543951\pi$
$659$	−3.06887e6	−0.275274	−0.137637	+	0.990483i	$0.543951\pi$
$660$	0	0
$661$	−1.79240e6	−0.159563	−0.0797814	−	0.996812i	$-0.525422\pi$
$661$	−1.79240e6	−0.159563	−0.0797814	+	0.996812i	$0.525422\pi$
$662$	0	0
$663$	−4.86128e6	−0.429504
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.23349e6	−0.368454
$668$	0	0
$669$	−1.32698e7	−1.14631
$670$	0	0
$671$	−684564.	−0.0586959
$672$	0	0
$673$	−1.37614e7	−1.17119	−0.585593	−	0.810605i	$-0.699138\pi$
$673$	−1.37614e7	−1.17119	−0.585593	+	0.810605i	$0.699138\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.01097e7	−1.68629	−0.843147	−	0.537684i	$-0.819300\pi$
$677$	−2.01097e7	−1.68629	−0.843147	+	0.537684i	$0.819300\pi$
$678$	0	0
$679$	1.18710e7	0.988129
$680$	0	0
$681$	−659104.	−0.0544611
$682$	0	0
$683$	6.08331e6	0.498986	0.249493	−	0.968377i	$-0.419736\pi$
$683$	6.08331e6	0.498986	0.249493	+	0.968377i	$0.419736\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−3.95120e7	−3.19401
$688$	0	0
$689$	1.24205e6	0.0996761
$690$	0	0
$691$	2.23063e7	1.77719	0.888593	−	0.458697i	$-0.151684\pi$
$691$	2.23063e7	1.77719	0.888593	+	0.458697i	$0.151684\pi$
$692$	0	0
$693$	1.66741e7	1.31889
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.12020e7	1.65308
$698$	0	0
$699$	1.94860e7	1.50845
$700$	0	0
$701$	−2.01349e7	−1.54758	−0.773792	−	0.633439i	$-0.781643\pi$
$701$	−2.01349e7	−1.54758	−0.773792	+	0.633439i	$0.781643\pi$
$702$	0	0
$703$	−7.70265e6	−0.587830
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.31369e7	0.988424
$708$	0	0
$709$	−7.09436e6	−0.530027	−0.265013	−	0.964245i	$-0.585376\pi$
$709$	−7.09436e6	−0.530027	−0.265013	+	0.964245i	$0.585376\pi$
$710$	0	0
$711$	9.47356e6	0.702812
$712$	0	0
$713$	7.98215e6	0.588026
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.33483e7	0.969677
$718$	0	0
$719$	−1.57591e7	−1.13686	−0.568432	−	0.822730i	$-0.692450\pi$
$719$	−1.57591e7	−1.13686	−0.568432	+	0.822730i	$0.692450\pi$
$720$	0	0
$721$	−6.48753e6	−0.464774
$722$	0	0
$723$	−2.43874e7	−1.73508
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.93222e7	1.35588	0.677939	−	0.735118i	$-0.262873\pi$
$727$	1.93222e7	1.35588	0.677939	+	0.735118i	$0.262873\pi$
$728$	0	0
$729$	1.74496e7	1.21610
$730$	0	0
$731$	−2.58554e7	−1.78961
$732$	0	0
$733$	−7.53614e6	−0.518071	−0.259035	−	0.965868i	$-0.583405\pi$
$733$	−7.53614e6	−0.518071	−0.259035	+	0.965868i	$0.583405\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.16278e6	0.282302
$738$	0	0
$739$	−6.08996e6	−0.410207	−0.205104	−	0.978740i	$-0.565753\pi$
$739$	−6.08996e6	−0.410207	−0.205104	+	0.978740i	$0.565753\pi$
$740$	0	0
$741$	1.64652e6	0.110159
$742$	0	0
$743$	5.29284e6	0.351736	0.175868	−	0.984414i	$-0.443727\pi$
$743$	5.29284e6	0.351736	0.175868	+	0.984414i	$0.443727\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−3.16720e7	−2.07670
$748$	0	0
$749$	8.28785e6	0.539806
$750$	0	0
$751$	1.58109e7	1.02296	0.511478	−	0.859296i	$-0.329098\pi$
$751$	1.58109e7	1.02296	0.511478	+	0.859296i	$0.329098\pi$
$752$	0	0
$753$	−3.87827e7	−2.49259
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.01450e7	1.27770	0.638849	−	0.769332i	$-0.279411\pi$
$757$	2.01450e7	1.27770	0.638849	+	0.769332i	$0.279411\pi$
$758$	0	0
$759$	7.16826e6	0.451657
$760$	0	0
$761$	−2.10320e7	−1.31650	−0.658248	−	0.752801i	$-0.728702\pi$
$761$	−2.10320e7	−1.31650	−0.658248	+	0.752801i	$0.728702\pi$
$762$	0	0
$763$	7.73066e6	0.480734
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.21095e6	−0.197081
$768$	0	0
$769$	−4.09332e6	−0.249609	−0.124804	−	0.992181i	$-0.539830\pi$
$769$	−4.09332e6	−0.249609	−0.124804	+	0.992181i	$0.539830\pi$
$770$	0	0
$771$	9.77503e6	0.592219
$772$	0	0
$773$	−1.09247e7	−0.657599	−0.328799	−	0.944400i	$-0.606644\pi$
$773$	−1.09247e7	−0.657599	−0.328799	+	0.944400i	$0.606644\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−4.83011e7	−2.87015
$778$	0	0
$779$	−7.18114e6	−0.423984
$780$	0	0
$781$	9.82213e6	0.576206
$782$	0	0
$783$	−3.31843e7	−1.93432
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.38257e7	−0.795700	−0.397850	−	0.917450i	$-0.630244\pi$
$787$	−1.38257e7	−0.795700	−0.397850	+	0.917450i	$0.630244\pi$
$788$	0	0
$789$	−2.42005e7	−1.38399
$790$	0	0
$791$	−1.15113e7	−0.654156
$792$	0	0
$793$	−300852.	−0.0169891
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.38429e6	−0.0771937	−0.0385969	−	0.999255i	$-0.512289\pi$
$797$	−1.38429e6	−0.0771937	−0.0385969	+	0.999255i	$0.512289\pi$
$798$	0	0
$799$	−1.90460e7	−1.05545
$800$	0	0
$801$	−3.04987e7	−1.67958
$802$	0	0
$803$	−2.54313e6	−0.139181
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.05967e6	−0.0572780
$808$	0	0
$809$	−2.08074e7	−1.11776	−0.558878	−	0.829250i	$-0.688768\pi$
$809$	−2.08074e7	−1.11776	−0.558878	+	0.829250i	$0.688768\pi$
$810$	0	0
$811$	−2.24363e7	−1.19784	−0.598919	−	0.800810i	$-0.704403\pi$
$811$	−2.24363e7	−1.19784	−0.598919	+	0.800810i	$0.704403\pi$
$812$	0	0
$813$	5.57942e7	2.96049
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.75726e6	0.459000
$818$	0	0
$819$	7.32791e6	0.381742
$820$	0	0
$821$	1.30693e7	0.676698	0.338349	−	0.941021i	$-0.390131\pi$
$821$	1.30693e7	0.676698	0.338349	+	0.941021i	$0.390131\pi$
$822$	0	0
$823$	2.09593e7	1.07864	0.539321	−	0.842100i	$-0.318681\pi$
$823$	2.09593e7	1.07864	0.539321	+	0.842100i	$0.318681\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−4.54670e6	−0.231171	−0.115585	−	0.993298i	$-0.536874\pi$
$827$	−4.54670e6	−0.231171	−0.115585	+	0.993298i	$0.536874\pi$
$828$	0	0
$829$	−2.05898e7	−1.04055	−0.520277	−	0.853997i	$-0.674171\pi$
$829$	−2.05898e7	−1.04055	−0.520277	+	0.853997i	$0.674171\pi$
$830$	0	0
$831$	4.12867e7	2.07399
$832$	0	0
$833$	−9.50401e6	−0.474564
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.25683e7	3.08703
$838$	0	0
$839$	1.09532e7	0.537200	0.268600	−	0.963252i	$-0.413439\pi$
$839$	1.09532e7	0.537200	0.268600	+	0.963252i	$0.413439\pi$
$840$	0	0
$841$	−9.84438e6	−0.479953
$842$	0	0
$843$	−6.84932e7	−3.31954
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.82831e7	−0.875673
$848$	0	0
$849$	3.90666e7	1.86010
$850$	0	0
$851$	−1.47375e7	−0.697592
$852$	0	0
$853$	−1.27066e7	−0.597939	−0.298970	−	0.954263i	$-0.596643\pi$
$853$	−1.27066e7	−0.597939	−0.298970	+	0.954263i	$0.596643\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.46258e6	−0.347086	−0.173543	−	0.984826i	$-0.555522\pi$
$857$	−7.46258e6	−0.347086	−0.173543	+	0.984826i	$0.555522\pi$
$858$	0	0
$859$	1.13390e7	0.524315	0.262158	−	0.965025i	$-0.415566\pi$
$859$	1.13390e7	0.524315	0.262158	+	0.965025i	$0.415566\pi$
$860$	0	0
$861$	−4.50309e7	−2.07015
$862$	0	0
$863$	1.61275e7	0.737121	0.368561	−	0.929604i	$-0.379851\pi$
$863$	1.61275e7	0.737121	0.368561	+	0.929604i	$0.379851\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	7.46805e7	3.37411
$868$	0	0
$869$	3.04742e6	0.136894
$870$	0	0
$871$	1.82945e6	0.0817101
$872$	0	0
$873$	4.80381e7	2.13329
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−3.89336e7	−1.70933	−0.854663	−	0.519182i	$-0.826237\pi$
$877$	−3.89336e7	−1.70933	−0.854663	+	0.519182i	$0.826237\pi$
$878$	0	0
$879$	1.46502e7	0.639546
$880$	0	0
$881$	3.57060e7	1.54989	0.774946	−	0.632027i	$-0.217777\pi$
$881$	3.57060e7	1.54989	0.774946	+	0.632027i	$0.217777\pi$
$882$	0	0
$883$	3.48264e7	1.50317	0.751583	−	0.659638i	$-0.229291\pi$
$883$	3.48264e7	1.50317	0.751583	+	0.659638i	$0.229291\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.97196e6	−0.425571	−0.212785	−	0.977099i	$-0.568253\pi$
$887$	−9.97196e6	−0.425571	−0.212785	+	0.977099i	$0.568253\pi$
$888$	0	0
$889$	−4.34761e7	−1.84500
$890$	0	0
$891$	2.85930e7	1.20661
$892$	0	0
$893$	6.45089e6	0.270702
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.15030e6	0.130729
$898$	0	0
$899$	−2.01120e7	−0.829956
$900$	0	0
$901$	−2.95771e7	−1.21379
$902$	0	0
$903$	5.49143e7	2.24112
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−2.17790e7	−0.879064	−0.439532	−	0.898227i	$-0.644856\pi$
$907$	−2.17790e7	−0.879064	−0.439532	+	0.898227i	$0.644856\pi$
$908$	0	0
$909$	5.31606e7	2.13393
$910$	0	0
$911$	4.45704e7	1.77931	0.889653	−	0.456637i	$-0.150946\pi$
$911$	4.45704e7	1.77931	0.889653	+	0.456637i	$0.150946\pi$
$912$	0	0
$913$	−1.01881e7	−0.404499
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.92232e7	−1.14764
$918$	0	0
$919$	5.05848e7	1.97575	0.987873	−	0.155264i	$-0.0496227\pi$
$919$	5.05848e7	1.97575	0.987873	+	0.155264i	$0.0496227\pi$
$920$	0	0
$921$	3.32893e7	1.29317
$922$	0	0
$923$	4.31662e6	0.166778
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2.62529e7	−1.00341
$928$	0	0
$929$	9.61614e6	0.365563	0.182781	−	0.983154i	$-0.441490\pi$
$929$	9.61614e6	0.365563	0.182781	+	0.983154i	$0.441490\pi$
$930$	0	0
$931$	3.21902e6	0.121716
$932$	0	0
$933$	−2.14756e7	−0.807682
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.07884e7	1.14562	0.572808	−	0.819690i	$-0.305854\pi$
$937$	3.07884e7	1.14562	0.572808	+	0.819690i	$0.305854\pi$
$938$	0	0
$939$	4.83486e7	1.78945
$940$	0	0
$941$	3.94367e7	1.45186	0.725932	−	0.687766i	$-0.241409\pi$
$941$	3.94367e7	1.45186	0.725932	+	0.687766i	$0.241409\pi$
$942$	0	0
$943$	−1.37397e7	−0.503152
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.02857e7	0.735045	0.367523	−	0.930015i	$-0.380206\pi$
$947$	2.02857e7	0.735045	0.367523	+	0.930015i	$0.380206\pi$
$948$	0	0
$949$	−1.11765e6	−0.0402849
$950$	0	0
$951$	−1.87066e7	−0.670724
$952$	0	0
$953$	7.23933e6	0.258206	0.129103	−	0.991631i	$-0.458790\pi$
$953$	7.23933e6	0.258206	0.129103	+	0.991631i	$0.458790\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.80613e7	−0.637482
$958$	0	0
$959$	−5.30377e7	−1.86225
$960$	0	0
$961$	9.29152e6	0.324548
$962$	0	0
$963$	3.35382e7	1.16540
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.74626e7	1.63224	0.816122	−	0.577880i	$-0.196120\pi$
$967$	4.74626e7	1.63224	0.816122	+	0.577880i	$0.196120\pi$
$968$	0	0
$969$	−3.92088e7	−1.34145
$970$	0	0
$971$	−1.13308e7	−0.385668	−0.192834	−	0.981231i	$-0.561768\pi$
$971$	−1.13308e7	−0.385668	−0.192834	+	0.981231i	$0.561768\pi$
$972$	0	0
$973$	−3.77811e7	−1.27936
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.09471e7	1.37242	0.686210	−	0.727404i	$-0.259273\pi$
$977$	4.09471e7	1.37242	0.686210	+	0.727404i	$0.259273\pi$
$978$	0	0
$979$	−9.81072e6	−0.327148
$980$	0	0
$981$	3.12835e7	1.03787
$982$	0	0
$983$	1.17862e7	0.389036	0.194518	−	0.980899i	$-0.437686\pi$
$983$	1.17862e7	0.389036	0.194518	+	0.980899i	$0.437686\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.04517e7	1.32173
$988$	0	0
$989$	1.67553e7	0.544706
$990$	0	0
$991$	−4.78243e7	−1.54691	−0.773454	−	0.633853i	$-0.781473\pi$
$991$	−4.78243e7	−1.54691	−0.773454	+	0.633853i	$0.781473\pi$
$992$	0	0
$993$	7.90043e7	2.54260
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−5.93264e6	−0.189021	−0.0945105	−	0.995524i	$-0.530129\pi$
$997$	−5.93264e6	−0.189021	−0.0945105	+	0.995524i	$0.530129\pi$
$998$	0	0
$999$	−1.15521e8	−3.66223

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.6.a.k.1.4		4
4.3	odd	2		400.6.a.z.1.1		4
5.2	odd	4		40.6.c.a.9.1	✓	8
5.3	odd	4		40.6.c.a.9.8	yes	8
5.4	even	2		200.6.a.j.1.1		4
15.2	even	4		360.6.f.b.289.6		8
15.8	even	4		360.6.f.b.289.5		8
20.3	even	4		80.6.c.d.49.1		8
20.7	even	4		80.6.c.d.49.8		8
20.19	odd	2		400.6.a.ba.1.4		4
40.3	even	4		320.6.c.i.129.8		8
40.13	odd	4		320.6.c.j.129.1		8
40.27	even	4		320.6.c.i.129.1		8
40.37	odd	4		320.6.c.j.129.8		8
60.23	odd	4		720.6.f.n.289.5		8
60.47	odd	4		720.6.f.n.289.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.c.a.9.1	✓	8	5.2	odd	4
40.6.c.a.9.8	yes	8	5.3	odd	4
80.6.c.d.49.1		8	20.3	even	4
80.6.c.d.49.8		8	20.7	even	4
200.6.a.j.1.1		4	5.4	even	2
200.6.a.k.1.4		4	1.1	even	1	trivial
320.6.c.i.129.1		8	40.27	even	4
320.6.c.i.129.8		8	40.3	even	4
320.6.c.j.129.1		8	40.13	odd	4
320.6.c.j.129.8		8	40.37	odd	4
360.6.f.b.289.5		8	15.8	even	4
360.6.f.b.289.6		8	15.2	even	4
400.6.a.z.1.1		4	4.3	odd	2
400.6.a.ba.1.4		4	20.19	odd	2
720.6.f.n.289.5		8	60.23	odd	4
720.6.f.n.289.6		8	60.47	odd	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:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	200.a (trivial)

\(f(q)\)	\(=\)	\(q+28.9338 q^{3} +146.828 q^{7} +594.165 q^{9} +O(q^{10})\)	\(q+28.9338 q^{3} +146.828 q^{7} +594.165 q^{9} +191.129 q^{11} +83.9971 q^{13} -2000.23 q^{17} +677.481 q^{19} +4248.29 q^{21} +1296.23 q^{23} +10160.5 q^{27} -3266.00 q^{29} +6157.98 q^{31} +5530.08 q^{33} -11369.5 q^{37} +2430.36 q^{39} -10599.8 q^{41} +12926.2 q^{43} +9521.88 q^{47} +4751.45 q^{49} -57874.4 q^{51} +14786.8 q^{53} +19602.1 q^{57} -38226.9 q^{59} -3581.69 q^{61} +87240.0 q^{63} +21780.0 q^{67} +37504.9 q^{69} +51390.1 q^{71} -13305.9 q^{73} +28063.1 q^{77} +15944.3 q^{79} +149601. q^{81} -53305.0 q^{83} -94497.8 q^{87} -51330.4 q^{89} +12333.1 q^{91} +178174. q^{93} +80849.9 q^{97} +113562. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 148 q^{7} + 500 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 148 * q^7 + 500 * q^9	\( 4 q + 4 q^{3} + 148 q^{7} + 500 q^{9} - 368 q^{11} + 440 q^{13} - 672 q^{17} - 688 q^{19} + 992 q^{21} + 4492 q^{23} + 8152 q^{27} - 2936 q^{29} + 2112 q^{31} + 26864 q^{33} + 8792 q^{37} + 1504 q^{39} + 11800 q^{41} + 48276 q^{43} + 14724 q^{47} + 22500 q^{49} - 62400 q^{51} + 84296 q^{53} + 71024 q^{57} - 45840 q^{59} + 61928 q^{61} + 186292 q^{63} + 72700 q^{67} + 38368 q^{69} - 62816 q^{71} + 133072 q^{73} + 11440 q^{77} - 21632 q^{79} + 204836 q^{81} + 74660 q^{83} - 12472 q^{87} + 20952 q^{89} - 243808 q^{91} + 105600 q^{93} + 59456 q^{97} - 133424 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 148 * q^7 + 500 * q^9 - 368 * q^11 + 440 * q^13 - 672 * q^17 - 688 * q^19 + 992 * q^21 + 4492 * q^23 + 8152 * q^27 - 2936 * q^29 + 2112 * q^31 + 26864 * q^33 + 8792 * q^37 + 1504 * q^39 + 11800 * q^41 + 48276 * q^43 + 14724 * q^47 + 22500 * q^49 - 62400 * q^51 + 84296 * q^53 + 71024 * q^57 - 45840 * q^59 + 61928 * q^61 + 186292 * q^63 + 72700 * q^67 + 38368 * q^69 - 62816 * q^71 + 133072 * q^73 + 11440 * q^77 - 21632 * q^79 + 204836 * q^81 + 74660 * q^83 - 12472 * q^87 + 20952 * q^89 - 243808 * q^91 + 105600 * q^93 + 59456 * q^97 - 133424 * q^99

Introduction

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