LMFDB - Embedded newform 200.4.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [200,4,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, 4, names="a")

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

chi := DirichletCharacter("200.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
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:	$11.8003820011$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ 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)

$f(q)$	$=$	$q-6.89898 q^{3} +12.6969 q^{7} +20.5959 q^{9} +59.1918 q^{11} -42.2020 q^{13} -126.384 q^{17} -19.1918 q^{19} -87.5959 q^{21} -78.3133 q^{23} +44.1816 q^{27} -148.384 q^{29} -139.151 q^{31} -408.363 q^{33} +66.5653 q^{37} +291.151 q^{39} -203.212 q^{41} -288.879 q^{43} -360.434 q^{47} -181.788 q^{49} +871.918 q^{51} +686.888 q^{53} +132.404 q^{57} -83.1102 q^{59} -208.829 q^{61} +261.505 q^{63} -192.293 q^{67} +540.282 q^{69} +500.767 q^{71} -122.706 q^{73} +751.555 q^{77} -289.616 q^{79} -860.898 q^{81} +573.950 q^{83} +1023.70 q^{87} -565.151 q^{89} -535.837 q^{91} +960.000 q^{93} -643.959 q^{97} +1219.11 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} - 4 q^{7} + 2 q^{9} + 40 q^{11} - 104 q^{13} - 96 q^{17} + 40 q^{19} - 136 q^{21} - 284 q^{23} - 88 q^{27} - 140 q^{29} + 192 q^{31} - 464 q^{33} - 200 q^{37} + 112 q^{39} - 524 q^{41} - 372 q^{43}+ \cdots + 1576 q^{99}+O(q^{100}) $ 2 * q - 4 * q^3 - 4 * q^7 + 2 * q^9 + 40 * q^11 - 104 * q^13 - 96 * q^17 + 40 * q^19 - 136 * q^21 - 284 * q^23 - 88 * q^27 - 140 * q^29 + 192 * q^31 - 464 * q^33 - 200 * q^37 + 112 * q^39 - 524 * q^41 - 372 * q^43 - 84 * q^47 - 246 * q^49 + 960 * q^51 + 296 * q^53 + 304 * q^57 + 696 * q^59 - 692 * q^61 + 572 * q^63 - 316 * q^67 - 56 * q^69 + 688 * q^71 + 656 * q^73 + 1072 * q^77 - 736 * q^79 - 742 * q^81 + 1628 * q^83 + 1048 * q^87 - 660 * q^89 + 496 * q^91 + 1920 * q^93 - 896 * q^97 + 1576 * q^99

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
$3$	−6.89898	−1.32771	−0.663855	−	0.747862i	$-0.731081\pi$
$3$	−6.89898	−1.32771	−0.663855	+	0.747862i	$0.731081\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	12.6969	0.685570	0.342785	−	0.939414i	$-0.388630\pi$
$7$	12.6969	0.685570	0.342785	+	0.939414i	$0.388630\pi$
$8$	0	0
$9$	20.5959	0.762812
$10$	0	0
$11$	59.1918	1.62246	0.811228	−	0.584730i	$-0.198800\pi$
$11$	59.1918	1.62246	0.811228	+	0.584730i	$0.198800\pi$
$12$	0	0
$13$	−42.2020	−0.900365	−0.450182	−	0.892937i	$-0.648641\pi$
$13$	−42.2020	−0.900365	−0.450182	+	0.892937i	$0.648641\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−126.384	−1.80309	−0.901545	−	0.432685i	$-0.857566\pi$
$17$	−126.384	−1.80309	−0.901545	+	0.432685i	$0.857566\pi$
$18$	0	0
$19$	−19.1918	−0.231732	−0.115866	−	0.993265i	$-0.536964\pi$
$19$	−19.1918	−0.231732	−0.115866	+	0.993265i	$0.536964\pi$
$20$	0	0
$21$	−87.5959	−0.910238
$22$	0	0
$23$	−78.3133	−0.709976	−0.354988	−	0.934871i	$-0.615515\pi$
$23$	−78.3133	−0.709976	−0.354988	+	0.934871i	$0.615515\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	44.1816	0.314917
$28$	0	0
$29$	−148.384	−0.950143	−0.475072	−	0.879947i	$-0.657578\pi$
$29$	−148.384	−0.950143	−0.475072	+	0.879947i	$0.657578\pi$
$30$	0	0
$31$	−139.151	−0.806202	−0.403101	−	0.915156i	$-0.632068\pi$
$31$	−139.151	−0.806202	−0.403101	+	0.915156i	$0.632068\pi$
$32$	0	0
$33$	−408.363	−2.15415
$34$	0	0
$35$	0	0
$36$	0	0
$37$	66.5653	0.295764	0.147882	−	0.989005i	$-0.452754\pi$
$37$	66.5653	0.295764	0.147882	+	0.989005i	$0.452754\pi$
$38$	0	0
$39$	291.151	1.19542
$40$	0	0
$41$	−203.212	−0.774059	−0.387030	−	0.922067i	$-0.626499\pi$
$41$	−203.212	−0.774059	−0.387030	+	0.922067i	$0.626499\pi$
$42$	0	0
$43$	−288.879	−1.02450	−0.512251	−	0.858836i	$-0.671188\pi$
$43$	−288.879	−1.02450	−0.512251	+	0.858836i	$0.671188\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−360.434	−1.11861	−0.559305	−	0.828962i	$-0.688932\pi$
$47$	−360.434	−1.11861	−0.559305	+	0.828962i	$0.688932\pi$
$48$	0	0
$49$	−181.788	−0.529993
$50$	0	0
$51$	871.918	2.39398
$52$	0	0
$53$	686.888	1.78021	0.890106	−	0.455753i	$-0.150630\pi$
$53$	686.888	1.78021	0.890106	+	0.455753i	$0.150630\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	132.404	0.307673
$58$	0	0
$59$	−83.1102	−0.183390	−0.0916951	−	0.995787i	$-0.529229\pi$
$59$	−83.1102	−0.183390	−0.0916951	+	0.995787i	$0.529229\pi$
$60$	0	0
$61$	−208.829	−0.438324	−0.219162	−	0.975688i	$-0.570332\pi$
$61$	−208.829	−0.438324	−0.219162	+	0.975688i	$0.570332\pi$
$62$	0	0
$63$	261.505	0.522961
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−192.293	−0.350632	−0.175316	−	0.984512i	$-0.556095\pi$
$67$	−192.293	−0.350632	−0.175316	+	0.984512i	$0.556095\pi$
$68$	0	0
$69$	540.282	0.942642
$70$	0	0
$71$	500.767	0.837044	0.418522	−	0.908207i	$-0.362548\pi$
$71$	500.767	0.837044	0.418522	+	0.908207i	$0.362548\pi$
$72$	0	0
$73$	−122.706	−0.196735	−0.0983676	−	0.995150i	$-0.531362\pi$
$73$	−122.706	−0.196735	−0.0983676	+	0.995150i	$0.531362\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	751.555	1.11231
$78$	0	0
$79$	−289.616	−0.412461	−0.206230	−	0.978503i	$-0.566120\pi$
$79$	−289.616	−0.412461	−0.206230	+	0.978503i	$0.566120\pi$
$80$	0	0
$81$	−860.898	−1.18093
$82$	0	0
$83$	573.950	0.759026	0.379513	−	0.925186i	$-0.376091\pi$
$83$	573.950	0.759026	0.379513	+	0.925186i	$0.376091\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1023.70	1.26151
$88$	0	0
$89$	−565.151	−0.673100	−0.336550	−	0.941666i	$-0.609260\pi$
$89$	−565.151	−0.673100	−0.336550	+	0.941666i	$0.609260\pi$
$90$	0	0
$91$	−535.837	−0.617263
$92$	0	0
$93$	960.000	1.07040
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−643.959	−0.674063	−0.337032	−	0.941493i	$-0.609423\pi$
$97$	−643.959	−0.674063	−0.337032	+	0.941493i	$0.609423\pi$
$98$	0	0
$99$	1219.11	1.23763
$100$	0	0
$101$	−745.918	−0.734868	−0.367434	−	0.930050i	$-0.619764\pi$
$101$	−745.918	−0.734868	−0.367434	+	0.930050i	$0.619764\pi$
$102$	0	0
$103$	738.413	0.706389	0.353194	−	0.935550i	$-0.385095\pi$
$103$	738.413	0.706389	0.353194	+	0.935550i	$0.385095\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	558.536	0.504633	0.252316	−	0.967645i	$-0.418808\pi$
$107$	558.536	0.504633	0.252316	+	0.967645i	$0.418808\pi$
$108$	0	0
$109$	−1523.69	−1.33893	−0.669465	−	0.742843i	$-0.733477\pi$
$109$	−1523.69	−1.33893	−0.669465	+	0.742843i	$0.733477\pi$
$110$	0	0
$111$	−459.233	−0.392689
$112$	0	0
$113$	58.4245	0.0486382	0.0243191	−	0.999704i	$-0.492258\pi$
$113$	58.4245	0.0486382	0.0243191	+	0.999704i	$0.492258\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−869.190	−0.686809
$118$	0	0
$119$	−1604.69	−1.23615
$120$	0	0
$121$	2172.67	1.63236
$122$	0	0
$123$	1401.96	1.02773
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−941.283	−0.657680	−0.328840	−	0.944386i	$-0.606658\pi$
$127$	−941.283	−0.657680	−0.328840	+	0.944386i	$0.606658\pi$
$128$	0	0
$129$	1992.97	1.36024
$130$	0	0
$131$	−214.343	−0.142956	−0.0714779	−	0.997442i	$-0.522772\pi$
$131$	−214.343	−0.142956	−0.0714779	+	0.997442i	$0.522772\pi$
$132$	0	0
$133$	−243.678	−0.158869
$134$	0	0
$135$	0	0
$136$	0	0
$137$	499.514	0.311506	0.155753	−	0.987796i	$-0.450220\pi$
$137$	499.514	0.311506	0.155753	+	0.987796i	$0.450220\pi$
$138$	0	0
$139$	2660.48	1.62345	0.811723	−	0.584042i	$-0.198530\pi$
$139$	2660.48	1.62345	0.811723	+	0.584042i	$0.198530\pi$
$140$	0	0
$141$	2486.62	1.48519
$142$	0	0
$143$	−2498.02	−1.46080
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1254.15	0.703677
$148$	0	0
$149$	1334.89	0.733947	0.366973	−	0.930231i	$-0.380394\pi$
$149$	1334.89	0.733947	0.366973	+	0.930231i	$0.380394\pi$
$150$	0	0
$151$	−259.478	−0.139841	−0.0699205	−	0.997553i	$-0.522275\pi$
$151$	−259.478	−0.139841	−0.0699205	+	0.997553i	$0.522275\pi$
$152$	0	0
$153$	−2602.99	−1.37542
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1063.58	−0.540654	−0.270327	−	0.962769i	$-0.587132\pi$
$157$	−1063.58	−0.540654	−0.270327	+	0.962769i	$0.587132\pi$
$158$	0	0
$159$	−4738.82	−2.36361
$160$	0	0
$161$	−994.339	−0.486738
$162$	0	0
$163$	−3139.67	−1.50870	−0.754349	−	0.656474i	$-0.772047\pi$
$163$	−3139.67	−1.50870	−0.754349	+	0.656474i	$0.772047\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	936.411	0.433902	0.216951	−	0.976182i	$-0.430389\pi$
$167$	936.411	0.433902	0.216951	+	0.976182i	$0.430389\pi$
$168$	0	0
$169$	−415.988	−0.189344
$170$	0	0
$171$	−395.273	−0.176768
$172$	0	0
$173$	−1439.78	−0.632741	−0.316371	−	0.948636i	$-0.602464\pi$
$173$	−1439.78	−0.632741	−0.316371	+	0.948636i	$0.602464\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	573.376	0.243489
$178$	0	0
$179$	2494.42	1.04158	0.520788	−	0.853686i	$-0.325638\pi$
$179$	2494.42	1.04158	0.520788	+	0.853686i	$0.325638\pi$
$180$	0	0
$181$	−1590.36	−0.653096	−0.326548	−	0.945181i	$-0.605886\pi$
$181$	−1590.36	−0.653096	−0.326548	+	0.945181i	$0.605886\pi$
$182$	0	0
$183$	1440.70	0.581967
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−7480.88	−2.92543
$188$	0	0
$189$	560.971	0.215898
$190$	0	0
$191$	−2652.77	−1.00496	−0.502480	−	0.864589i	$-0.667579\pi$
$191$	−2652.77	−1.00496	−0.502480	+	0.864589i	$0.667579\pi$
$192$	0	0
$193$	3001.61	1.11948	0.559742	−	0.828667i	$-0.310900\pi$
$193$	3001.61	1.11948	0.559742	+	0.828667i	$0.310900\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2176.06	0.786994	0.393497	−	0.919326i	$-0.371265\pi$
$197$	2176.06	0.786994	0.393497	+	0.919326i	$0.371265\pi$
$198$	0	0
$199$	−4270.30	−1.52117	−0.760587	−	0.649236i	$-0.775089\pi$
$199$	−4270.30	−1.52117	−0.760587	+	0.649236i	$0.775089\pi$
$200$	0	0
$201$	1326.62	0.465537
$202$	0	0
$203$	−1884.02	−0.651390
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1612.93	−0.541578
$208$	0	0
$209$	−1136.00	−0.375975
$210$	0	0
$211$	−2978.78	−0.971886	−0.485943	−	0.873991i	$-0.661524\pi$
$211$	−2978.78	−0.971886	−0.485943	+	0.873991i	$0.661524\pi$
$212$	0	0
$213$	−3454.78	−1.11135
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1766.79	−0.552708
$218$	0	0
$219$	846.547	0.261207
$220$	0	0
$221$	5333.65	1.62344
$222$	0	0
$223$	−3183.16	−0.955875	−0.477937	−	0.878394i	$-0.658615\pi$
$223$	−3183.16	−0.955875	−0.477937	+	0.878394i	$0.658615\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5910.41	1.72814	0.864070	−	0.503372i	$-0.167908\pi$
$227$	5910.41	1.72814	0.864070	+	0.503372i	$0.167908\pi$
$228$	0	0
$229$	3465.59	1.00006	0.500028	−	0.866009i	$-0.333323\pi$
$229$	3465.59	1.00006	0.500028	+	0.866009i	$0.333323\pi$
$230$	0	0
$231$	−5184.96	−1.47682
$232$	0	0
$233$	4611.51	1.29661	0.648305	−	0.761380i	$-0.275478\pi$
$233$	4611.51	1.29661	0.648305	+	0.761380i	$0.275478\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1998.06	0.547628
$238$	0	0
$239$	4816.47	1.30356	0.651782	−	0.758406i	$-0.274022\pi$
$239$	4816.47	1.30356	0.651782	+	0.758406i	$0.274022\pi$
$240$	0	0
$241$	3057.47	0.817217	0.408608	−	0.912710i	$-0.366014\pi$
$241$	3057.47	0.817217	0.408608	+	0.912710i	$0.366014\pi$
$242$	0	0
$243$	4746.41	1.25301
$244$	0	0
$245$	0	0
$246$	0	0
$247$	809.935	0.208643
$248$	0	0
$249$	−3959.67	−1.00777
$250$	0	0
$251$	7498.54	1.88567	0.942836	−	0.333258i	$-0.108148\pi$
$251$	7498.54	1.88567	0.942836	+	0.333258i	$0.108148\pi$
$252$	0	0
$253$	−4635.51	−1.15190
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2272.33	0.551532	0.275766	−	0.961225i	$-0.411068\pi$
$257$	2272.33	0.551532	0.275766	+	0.961225i	$0.411068\pi$
$258$	0	0
$259$	845.176	0.202767
$260$	0	0
$261$	−3056.10	−0.724781
$262$	0	0
$263$	5271.34	1.23591	0.617955	−	0.786213i	$-0.287961\pi$
$263$	5271.34	1.23591	0.617955	+	0.786213i	$0.287961\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3898.97	0.893681
$268$	0	0
$269$	−4315.61	−0.978169	−0.489085	−	0.872236i	$-0.662669\pi$
$269$	−4315.61	−0.978169	−0.489085	+	0.872236i	$0.662669\pi$
$270$	0	0
$271$	4471.27	1.00225	0.501125	−	0.865375i	$-0.332919\pi$
$271$	4471.27	1.00225	0.501125	+	0.865375i	$0.332919\pi$
$272$	0	0
$273$	3696.73	0.819546
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−8112.09	−1.75960	−0.879798	−	0.475348i	$-0.842322\pi$
$277$	−8112.09	−1.75960	−0.879798	+	0.475348i	$0.842322\pi$
$278$	0	0
$279$	−2865.94	−0.614980
$280$	0	0
$281$	−3811.46	−0.809155	−0.404577	−	0.914504i	$-0.632581\pi$
$281$	−3811.46	−0.809155	−0.404577	+	0.914504i	$0.632581\pi$
$282$	0	0
$283$	2362.60	0.496261	0.248131	−	0.968727i	$-0.420184\pi$
$283$	2362.60	0.496261	0.248131	+	0.968727i	$0.420184\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2580.17	−0.530672
$288$	0	0
$289$	11059.8	2.25114
$290$	0	0
$291$	4442.66	0.894960
$292$	0	0
$293$	6337.79	1.26368	0.631839	−	0.775099i	$-0.282300\pi$
$293$	6337.79	1.26368	0.631839	+	0.775099i	$0.282300\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2615.19	0.510939
$298$	0	0
$299$	3304.98	0.639237
$300$	0	0
$301$	−3667.87	−0.702368
$302$	0	0
$303$	5146.08	0.975691
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4170.44	0.775308	0.387654	−	0.921805i	$-0.373285\pi$
$307$	4170.44	0.775308	0.387654	+	0.921805i	$0.373285\pi$
$308$	0	0
$309$	−5094.30	−0.937879
$310$	0	0
$311$	−5083.71	−0.926915	−0.463457	−	0.886119i	$-0.653391\pi$
$311$	−5083.71	−0.926915	−0.463457	+	0.886119i	$0.653391\pi$
$312$	0	0
$313$	1522.86	0.275007	0.137504	−	0.990501i	$-0.456092\pi$
$313$	1522.86	0.275007	0.137504	+	0.990501i	$0.456092\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4996.09	−0.885200	−0.442600	−	0.896719i	$-0.645944\pi$
$317$	−4996.09	−0.885200	−0.442600	+	0.896719i	$0.645944\pi$
$318$	0	0
$319$	−8783.10	−1.54157
$320$	0	0
$321$	−3853.33	−0.670005
$322$	0	0
$323$	2425.53	0.417834
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10511.9	1.77771
$328$	0	0
$329$	−4576.40	−0.766885
$330$	0	0
$331$	2832.04	0.470281	0.235141	−	0.971961i	$-0.424445\pi$
$331$	2832.04	0.470281	0.235141	+	0.971961i	$0.424445\pi$
$332$	0	0
$333$	1370.97	0.225612
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1684.43	0.272276	0.136138	−	0.990690i	$-0.456531\pi$
$337$	1684.43	0.272276	0.136138	+	0.990690i	$0.456531\pi$
$338$	0	0
$339$	−403.069	−0.0645774
$340$	0	0
$341$	−8236.60	−1.30803
$342$	0	0
$343$	−6663.20	−1.04892
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9350.30	−1.44654	−0.723271	−	0.690564i	$-0.757362\pi$
$347$	−9350.30	−1.44654	−0.723271	+	0.690564i	$0.757362\pi$
$348$	0	0
$349$	4174.20	0.640228	0.320114	−	0.947379i	$-0.396279\pi$
$349$	4174.20	0.640228	0.320114	+	0.947379i	$0.396279\pi$
$350$	0	0
$351$	−1864.55	−0.283540
$352$	0	0
$353$	−6217.04	−0.937394	−0.468697	−	0.883359i	$-0.655276\pi$
$353$	−6217.04	−0.937394	−0.468697	+	0.883359i	$0.655276\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	11070.7	1.64124
$358$	0	0
$359$	8149.73	1.19812	0.599062	−	0.800703i	$-0.295540\pi$
$359$	8149.73	1.19812	0.599062	+	0.800703i	$0.295540\pi$
$360$	0	0
$361$	−6490.67	−0.946300
$362$	0	0
$363$	−14989.2	−2.16730
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−11868.2	−1.68806	−0.844028	−	0.536299i	$-0.819822\pi$
$367$	−11868.2	−1.68806	−0.844028	+	0.536299i	$0.819822\pi$
$368$	0	0
$369$	−4185.34	−0.590461
$370$	0	0
$371$	8721.37	1.22046
$372$	0	0
$373$	−3286.85	−0.456265	−0.228132	−	0.973630i	$-0.573262\pi$
$373$	−3286.85	−0.456265	−0.228132	+	0.973630i	$0.573262\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6262.09	0.855475
$378$	0	0
$379$	8326.24	1.12847	0.564235	−	0.825614i	$-0.309171\pi$
$379$	8326.24	1.12847	0.564235	+	0.825614i	$0.309171\pi$
$380$	0	0
$381$	6493.89	0.873207
$382$	0	0
$383$	4249.23	0.566908	0.283454	−	0.958986i	$-0.408520\pi$
$383$	4249.23	0.566908	0.283454	+	0.958986i	$0.408520\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5949.72	−0.781502
$388$	0	0
$389$	−7841.83	−1.02210	−0.511050	−	0.859551i	$-0.670743\pi$
$389$	−7841.83	−1.02210	−0.511050	+	0.859551i	$0.670743\pi$
$390$	0	0
$391$	9897.52	1.28015
$392$	0	0
$393$	1478.75	0.189804
$394$	0	0
$395$	0	0
$396$	0	0
$397$	401.794	0.0507946	0.0253973	−	0.999677i	$-0.491915\pi$
$397$	401.794	0.0507946	0.0253973	+	0.999677i	$0.491915\pi$
$398$	0	0
$399$	1681.13	0.210931
$400$	0	0
$401$	9732.89	1.21206	0.606032	−	0.795441i	$-0.292761\pi$
$401$	9732.89	1.21206	0.606032	+	0.795441i	$0.292761\pi$
$402$	0	0
$403$	5872.46	0.725876
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3940.12	0.479864
$408$	0	0
$409$	−5629.67	−0.680609	−0.340305	−	0.940315i	$-0.610530\pi$
$409$	−5629.67	−0.680609	−0.340305	+	0.940315i	$0.610530\pi$
$410$	0	0
$411$	−3446.14	−0.413590
$412$	0	0
$413$	−1055.25	−0.125727
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−18354.6	−2.15547
$418$	0	0
$419$	9773.77	1.13957	0.569785	−	0.821794i	$-0.307026\pi$
$419$	9773.77	1.13957	0.569785	+	0.821794i	$0.307026\pi$
$420$	0	0
$421$	3037.95	0.351688	0.175844	−	0.984418i	$-0.443735\pi$
$421$	3037.95	0.351688	0.175844	+	0.984418i	$0.443735\pi$
$422$	0	0
$423$	−7423.46	−0.853289
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2651.48	−0.300502
$428$	0	0
$429$	17233.8	1.93952
$430$	0	0
$431$	1873.45	0.209376	0.104688	−	0.994505i	$-0.466616\pi$
$431$	1873.45	0.209376	0.104688	+	0.994505i	$0.466616\pi$
$432$	0	0
$433$	−11855.8	−1.31583	−0.657915	−	0.753092i	$-0.728561\pi$
$433$	−11855.8	−1.31583	−0.657915	+	0.753092i	$0.728561\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1502.98	0.164524
$438$	0	0
$439$	−16243.8	−1.76600	−0.882999	−	0.469375i	$-0.844479\pi$
$439$	−16243.8	−1.76600	−0.882999	+	0.469375i	$0.844479\pi$
$440$	0	0
$441$	−3744.09	−0.404285
$442$	0	0
$443$	−3905.52	−0.418864	−0.209432	−	0.977823i	$-0.567161\pi$
$443$	−3905.52	−0.418864	−0.209432	+	0.977823i	$0.567161\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−9209.35	−0.974468
$448$	0	0
$449$	−6852.71	−0.720265	−0.360133	−	0.932901i	$-0.617269\pi$
$449$	−6852.71	−0.720265	−0.360133	+	0.932901i	$0.617269\pi$
$450$	0	0
$451$	−12028.5	−1.25588
$452$	0	0
$453$	1790.13	0.185668
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5800.75	−0.593758	−0.296879	−	0.954915i	$-0.595946\pi$
$457$	−5800.75	−0.593758	−0.296879	+	0.954915i	$0.595946\pi$
$458$	0	0
$459$	−5583.84	−0.567824
$460$	0	0
$461$	−5263.63	−0.531783	−0.265891	−	0.964003i	$-0.585666\pi$
$461$	−5263.63	−0.531783	−0.265891	+	0.964003i	$0.585666\pi$
$462$	0	0
$463$	−13636.6	−1.36879	−0.684394	−	0.729113i	$-0.739933\pi$
$463$	−13636.6	−1.36879	−0.684394	+	0.729113i	$0.739933\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13158.3	1.30384	0.651920	−	0.758287i	$-0.273964\pi$
$467$	13158.3	1.30384	0.651920	+	0.758287i	$0.273964\pi$
$468$	0	0
$469$	−2441.53	−0.240383
$470$	0	0
$471$	7337.60	0.717832
$472$	0	0
$473$	−17099.3	−1.66221
$474$	0	0
$475$	0	0
$476$	0	0
$477$	14147.1	1.35797
$478$	0	0
$479$	−6775.90	−0.646344	−0.323172	−	0.946340i	$-0.604749\pi$
$479$	−6775.90	−0.646344	−0.323172	+	0.946340i	$0.604749\pi$
$480$	0	0
$481$	−2809.19	−0.266295
$482$	0	0
$483$	6859.92	0.646247
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5933.64	0.552113	0.276057	−	0.961141i	$-0.410972\pi$
$487$	5933.64	0.552113	0.276057	+	0.961141i	$0.410972\pi$
$488$	0	0
$489$	21660.5	2.00311
$490$	0	0
$491$	−11934.5	−1.09694	−0.548468	−	0.836171i	$-0.684789\pi$
$491$	−11934.5	−1.09694	−0.548468	+	0.836171i	$0.684789\pi$
$492$	0	0
$493$	18753.3	1.71319
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6358.21	0.573853
$498$	0	0
$499$	−1567.65	−0.140636	−0.0703182	−	0.997525i	$-0.522401\pi$
$499$	−1567.65	−0.140636	−0.0703182	+	0.997525i	$0.522401\pi$
$500$	0	0
$501$	−6460.28	−0.576096
$502$	0	0
$503$	−450.509	−0.0399348	−0.0199674	−	0.999801i	$-0.506356\pi$
$503$	−450.509	−0.0399348	−0.0199674	+	0.999801i	$0.506356\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2869.89	0.251393
$508$	0	0
$509$	−7016.11	−0.610969	−0.305485	−	0.952197i	$-0.598818\pi$
$509$	−7016.11	−0.610969	−0.305485	+	0.952197i	$0.598818\pi$
$510$	0	0
$511$	−1557.99	−0.134876
$512$	0	0
$513$	−847.927	−0.0729764
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−21334.7	−1.81489
$518$	0	0
$519$	9933.00	0.840097
$520$	0	0
$521$	−16356.0	−1.37538	−0.687688	−	0.726006i	$-0.741374\pi$
$521$	−16356.0	−1.37538	−0.687688	+	0.726006i	$0.741374\pi$
$522$	0	0
$523$	13190.5	1.10283	0.551414	−	0.834232i	$-0.314089\pi$
$523$	13190.5	1.10283	0.551414	+	0.834232i	$0.314089\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	17586.4	1.45366
$528$	0	0
$529$	−6034.03	−0.495934
$530$	0	0
$531$	−1711.73	−0.139892
$532$	0	0
$533$	8575.97	0.696935
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−17209.0	−1.38291
$538$	0	0
$539$	−10760.4	−0.859891
$540$	0	0
$541$	21997.6	1.74815	0.874076	−	0.485789i	$-0.161468\pi$
$541$	21997.6	1.74815	0.874076	+	0.485789i	$0.161468\pi$
$542$	0	0
$543$	10971.9	0.867122
$544$	0	0
$545$	0	0
$546$	0	0
$547$	212.663	0.0166230	0.00831151	−	0.999965i	$-0.497354\pi$
$547$	212.663	0.0166230	0.00831151	+	0.999965i	$0.497354\pi$
$548$	0	0
$549$	−4301.02	−0.334359
$550$	0	0
$551$	2847.76	0.220179
$552$	0	0
$553$	−3677.24	−0.282771
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6641.66	−0.505235	−0.252618	−	0.967566i	$-0.581291\pi$
$557$	−6641.66	−0.505235	−0.252618	+	0.967566i	$0.581291\pi$
$558$	0	0
$559$	12191.3	0.922425
$560$	0	0
$561$	51610.4	3.88413
$562$	0	0
$563$	−7122.94	−0.533208	−0.266604	−	0.963806i	$-0.585902\pi$
$563$	−7122.94	−0.533208	−0.266604	+	0.963806i	$0.585902\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−10930.8	−0.809610
$568$	0	0
$569$	5951.39	0.438480	0.219240	−	0.975671i	$-0.429642\pi$
$569$	5951.39	0.438480	0.219240	+	0.975671i	$0.429642\pi$
$570$	0	0
$571$	−23612.2	−1.73054	−0.865272	−	0.501303i	$-0.832854\pi$
$571$	−23612.2	−1.73054	−0.865272	+	0.501303i	$0.832854\pi$
$572$	0	0
$573$	18301.4	1.33430
$574$	0	0
$575$	0	0
$576$	0	0
$577$	25398.1	1.83247	0.916237	−	0.400636i	$-0.131211\pi$
$577$	25398.1	1.83247	0.916237	+	0.400636i	$0.131211\pi$
$578$	0	0
$579$	−20708.0	−1.48635
$580$	0	0
$581$	7287.41	0.520366
$582$	0	0
$583$	40658.1	2.88832
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−17994.6	−1.26528	−0.632638	−	0.774448i	$-0.718028\pi$
$587$	−17994.6	−1.26528	−0.632638	+	0.774448i	$0.718028\pi$
$588$	0	0
$589$	2670.56	0.186823
$590$	0	0
$591$	−15012.6	−1.04490
$592$	0	0
$593$	14842.0	1.02780	0.513900	−	0.857850i	$-0.328200\pi$
$593$	14842.0	1.02780	0.513900	+	0.857850i	$0.328200\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	29460.7	2.01968
$598$	0	0
$599$	25071.6	1.71018	0.855089	−	0.518481i	$-0.173502\pi$
$599$	25071.6	1.71018	0.855089	+	0.518481i	$0.173502\pi$
$600$	0	0
$601$	−2772.27	−0.188159	−0.0940794	−	0.995565i	$-0.529991\pi$
$601$	−2772.27	−0.188159	−0.0940794	+	0.995565i	$0.529991\pi$
$602$	0	0
$603$	−3960.45	−0.267466
$604$	0	0
$605$	0	0
$606$	0	0
$607$	19585.8	1.30966	0.654830	−	0.755776i	$-0.272740\pi$
$607$	19585.8	1.30966	0.654830	+	0.755776i	$0.272740\pi$
$608$	0	0
$609$	12997.8	0.864856
$610$	0	0
$611$	15211.0	1.00716
$612$	0	0
$613$	−24247.5	−1.59763	−0.798814	−	0.601579i	$-0.794539\pi$
$613$	−24247.5	−1.59763	−0.798814	+	0.601579i	$0.794539\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1225.83	−0.0799841	−0.0399920	−	0.999200i	$-0.512733\pi$
$617$	−1225.83	−0.0799841	−0.0399920	+	0.999200i	$0.512733\pi$
$618$	0	0
$619$	−12430.1	−0.807122	−0.403561	−	0.914953i	$-0.632228\pi$
$619$	−12430.1	−0.807122	−0.403561	+	0.914953i	$0.632228\pi$
$620$	0	0
$621$	−3460.01	−0.223583
$622$	0	0
$623$	−7175.69	−0.461457
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7837.24	0.499185
$628$	0	0
$629$	−8412.77	−0.533289
$630$	0	0
$631$	−6215.00	−0.392100	−0.196050	−	0.980594i	$-0.562812\pi$
$631$	−6215.00	−0.392100	−0.196050	+	0.980594i	$0.562812\pi$
$632$	0	0
$633$	20550.6	1.29038
$634$	0	0
$635$	0	0
$636$	0	0
$637$	7671.81	0.477187
$638$	0	0
$639$	10313.8	0.638507
$640$	0	0
$641$	5059.76	0.311776	0.155888	−	0.987775i	$-0.450176\pi$
$641$	5059.76	0.311776	0.155888	+	0.987775i	$0.450176\pi$
$642$	0	0
$643$	−8676.69	−0.532154	−0.266077	−	0.963952i	$-0.585728\pi$
$643$	−8676.69	−0.532154	−0.266077	+	0.963952i	$0.585728\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−22787.4	−1.38465	−0.692323	−	0.721587i	$-0.743413\pi$
$647$	−22787.4	−1.38465	−0.692323	+	0.721587i	$0.743413\pi$
$648$	0	0
$649$	−4919.45	−0.297543
$650$	0	0
$651$	12189.1	0.733836
$652$	0	0
$653$	−26813.1	−1.60686	−0.803428	−	0.595402i	$-0.796993\pi$
$653$	−26813.1	−1.60686	−0.803428	+	0.595402i	$0.796993\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2527.25	−0.150072
$658$	0	0
$659$	−1935.78	−0.114427	−0.0572134	−	0.998362i	$-0.518222\pi$
$659$	−1935.78	−0.114427	−0.0572134	+	0.998362i	$0.518222\pi$
$660$	0	0
$661$	3371.84	0.198411	0.0992053	−	0.995067i	$-0.468370\pi$
$661$	3371.84	0.198411	0.0992053	+	0.995067i	$0.468370\pi$
$662$	0	0
$663$	−36796.7	−2.15546
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11620.4	0.674579
$668$	0	0
$669$	21960.5	1.26912
$670$	0	0
$671$	−12360.9	−0.711161
$672$	0	0
$673$	22061.3	1.26360	0.631798	−	0.775133i	$-0.282317\pi$
$673$	22061.3	1.26360	0.631798	+	0.775133i	$0.282317\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−12747.5	−0.723674	−0.361837	−	0.932241i	$-0.617850\pi$
$677$	−12747.5	−0.723674	−0.361837	+	0.932241i	$0.617850\pi$
$678$	0	0
$679$	−8176.31	−0.462118
$680$	0	0
$681$	−40775.8	−2.29447
$682$	0	0
$683$	24562.0	1.37605	0.688023	−	0.725689i	$-0.258479\pi$
$683$	24562.0	1.37605	0.688023	+	0.725689i	$0.258479\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−23909.0	−1.32778
$688$	0	0
$689$	−28988.1	−1.60284
$690$	0	0
$691$	−15853.1	−0.872767	−0.436383	−	0.899761i	$-0.643741\pi$
$691$	−15853.1	−0.872767	−0.436383	+	0.899761i	$0.643741\pi$
$692$	0	0
$693$	15479.0	0.848481
$694$	0	0
$695$	0	0
$696$	0	0
$697$	25682.7	1.39570
$698$	0	0
$699$	−31814.7	−1.72152
$700$	0	0
$701$	−18288.0	−0.985347	−0.492673	−	0.870214i	$-0.663980\pi$
$701$	−18288.0	−0.985347	−0.492673	+	0.870214i	$0.663980\pi$
$702$	0	0
$703$	−1277.51	−0.0685380
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9470.88	−0.503804
$708$	0	0
$709$	11985.3	0.634864	0.317432	−	0.948281i	$-0.397179\pi$
$709$	11985.3	0.634864	0.317432	+	0.948281i	$0.397179\pi$
$710$	0	0
$711$	−5964.91	−0.314630
$712$	0	0
$713$	10897.4	0.572384
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−33228.8	−1.73075
$718$	0	0
$719$	17036.3	0.883653	0.441827	−	0.897101i	$-0.354331\pi$
$719$	17036.3	0.883653	0.441827	+	0.897101i	$0.354331\pi$
$720$	0	0
$721$	9375.59	0.484279
$722$	0	0
$723$	−21093.4	−1.08503
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1219.85	−0.0622307	−0.0311154	−	0.999516i	$-0.509906\pi$
$727$	−1219.85	−0.0622307	−0.0311154	+	0.999516i	$0.509906\pi$
$728$	0	0
$729$	−9501.16	−0.482709
$730$	0	0
$731$	36509.5	1.84727
$732$	0	0
$733$	−17750.6	−0.894453	−0.447226	−	0.894421i	$-0.647588\pi$
$733$	−17750.6	−0.894453	−0.447226	+	0.894421i	$0.647588\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−11382.2	−0.568884
$738$	0	0
$739$	−25825.4	−1.28553	−0.642763	−	0.766065i	$-0.722212\pi$
$739$	−25825.4	−1.28553	−0.642763	+	0.766065i	$0.722212\pi$
$740$	0	0
$741$	−5587.72	−0.277018
$742$	0	0
$743$	11549.8	0.570282	0.285141	−	0.958486i	$-0.407959\pi$
$743$	11549.8	0.570282	0.285141	+	0.958486i	$0.407959\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	11821.0	0.578994
$748$	0	0
$749$	7091.69	0.345961
$750$	0	0
$751$	29468.3	1.43184	0.715921	−	0.698182i	$-0.246007\pi$
$751$	29468.3	1.43184	0.715921	+	0.698182i	$0.246007\pi$
$752$	0	0
$753$	−51732.3	−2.50362
$754$	0	0
$755$	0	0
$756$	0	0
$757$	13820.2	0.663545	0.331772	−	0.943359i	$-0.392353\pi$
$757$	13820.2	0.663545	0.331772	+	0.943359i	$0.392353\pi$
$758$	0	0
$759$	31980.3	1.52939
$760$	0	0
$761$	−1188.09	−0.0565943	−0.0282971	−	0.999600i	$-0.509008\pi$
$761$	−1188.09	−0.0565943	−0.0282971	+	0.999600i	$0.509008\pi$
$762$	0	0
$763$	−19346.2	−0.917931
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3507.42	0.165118
$768$	0	0
$769$	29907.3	1.40245	0.701225	−	0.712940i	$-0.252637\pi$
$769$	29907.3	1.40245	0.701225	+	0.712940i	$0.252637\pi$
$770$	0	0
$771$	−15676.7	−0.732275
$772$	0	0
$773$	−27528.3	−1.28088	−0.640442	−	0.768007i	$-0.721249\pi$
$773$	−27528.3	−1.28088	−0.640442	+	0.768007i	$0.721249\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−5830.85	−0.269216
$778$	0	0
$779$	3900.02	0.179374
$780$	0	0
$781$	29641.3	1.35807
$782$	0	0
$783$	−6555.83	−0.299216
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−10038.2	−0.454666	−0.227333	−	0.973817i	$-0.573001\pi$
$787$	−10038.2	−0.454666	−0.227333	+	0.973817i	$0.573001\pi$
$788$	0	0
$789$	−36366.8	−1.64093
$790$	0	0
$791$	741.812	0.0333449
$792$	0	0
$793$	8812.99	0.394651
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−41075.3	−1.82555	−0.912773	−	0.408466i	$-0.866064\pi$
$797$	−41075.3	−1.82555	−0.912773	+	0.408466i	$0.866064\pi$
$798$	0	0
$799$	45552.9	2.01695
$800$	0	0
$801$	−11639.8	−0.513449
$802$	0	0
$803$	−7263.20	−0.319194
$804$	0	0
$805$	0	0
$806$	0	0
$807$	29773.3	1.29872
$808$	0	0
$809$	−2134.47	−0.0927616	−0.0463808	−	0.998924i	$-0.514769\pi$
$809$	−2134.47	−0.0927616	−0.0463808	+	0.998924i	$0.514769\pi$
$810$	0	0
$811$	5866.38	0.254003	0.127001	−	0.991903i	$-0.459465\pi$
$811$	5866.38	0.254003	0.127001	+	0.991903i	$0.459465\pi$
$812$	0	0
$813$	−30847.2	−1.33070
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5544.11	0.237410
$818$	0	0
$819$	−11036.0	−0.470856
$820$	0	0
$821$	−28572.6	−1.21460	−0.607301	−	0.794472i	$-0.707748\pi$
$821$	−28572.6	−1.21460	−0.607301	+	0.794472i	$0.707748\pi$
$822$	0	0
$823$	−15900.5	−0.673460	−0.336730	−	0.941601i	$-0.609321\pi$
$823$	−15900.5	−0.673460	−0.336730	+	0.941601i	$0.609321\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34717.6	1.45979	0.729896	−	0.683558i	$-0.239569\pi$
$827$	34717.6	1.45979	0.729896	+	0.683558i	$0.239569\pi$
$828$	0	0
$829$	−812.559	−0.0340426	−0.0170213	−	0.999855i	$-0.505418\pi$
$829$	−812.559	−0.0340426	−0.0170213	+	0.999855i	$0.505418\pi$
$830$	0	0
$831$	55965.1	2.33623
$832$	0	0
$833$	22975.0	0.955626
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−6147.92	−0.253887
$838$	0	0
$839$	−852.588	−0.0350829	−0.0175415	−	0.999846i	$-0.505584\pi$
$839$	−852.588	−0.0350829	−0.0175415	+	0.999846i	$0.505584\pi$
$840$	0	0
$841$	−2371.29	−0.0972277
$842$	0	0
$843$	26295.2	1.07432
$844$	0	0
$845$	0	0
$846$	0	0
$847$	27586.3	1.11910
$848$	0	0
$849$	−16299.5	−0.658891
$850$	0	0
$851$	−5212.95	−0.209985
$852$	0	0
$853$	8851.54	0.355300	0.177650	−	0.984094i	$-0.443150\pi$
$853$	8851.54	0.355300	0.177650	+	0.984094i	$0.443150\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1712.28	−0.0682502	−0.0341251	−	0.999418i	$-0.510864\pi$
$857$	−1712.28	−0.0682502	−0.0341251	+	0.999418i	$0.510864\pi$
$858$	0	0
$859$	33276.7	1.32175	0.660877	−	0.750495i	$-0.270185\pi$
$859$	33276.7	1.32175	0.660877	+	0.750495i	$0.270185\pi$
$860$	0	0
$861$	17800.6	0.704578
$862$	0	0
$863$	5913.49	0.233253	0.116627	−	0.993176i	$-0.462792\pi$
$863$	5913.49	0.233253	0.116627	+	0.993176i	$0.462792\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−76301.6	−2.98885
$868$	0	0
$869$	−17142.9	−0.669199
$870$	0	0
$871$	8115.15	0.315696
$872$	0	0
$873$	−13262.9	−0.514183
$874$	0	0
$875$	0	0
$876$	0	0
$877$	38661.1	1.48859	0.744295	−	0.667851i	$-0.232786\pi$
$877$	38661.1	1.48859	0.744295	+	0.667851i	$0.232786\pi$
$878$	0	0
$879$	−43724.3	−1.67780
$880$	0	0
$881$	20876.4	0.798348	0.399174	−	0.916875i	$-0.369297\pi$
$881$	20876.4	0.798348	0.399174	+	0.916875i	$0.369297\pi$
$882$	0	0
$883$	14713.8	0.560770	0.280385	−	0.959888i	$-0.409538\pi$
$883$	14713.8	0.560770	0.280385	+	0.959888i	$0.409538\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−38213.1	−1.44653	−0.723264	−	0.690571i	$-0.757359\pi$
$887$	−38213.1	−1.44653	−0.723264	+	0.690571i	$0.757359\pi$
$888$	0	0
$889$	−11951.4	−0.450886
$890$	0	0
$891$	−50958.1	−1.91601
$892$	0	0
$893$	6917.38	0.259218
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−22801.0	−0.848721
$898$	0	0
$899$	20647.7	0.766007
$900$	0	0
$901$	−86811.4	−3.20989
$902$	0	0
$903$	25304.6	0.932540
$904$	0	0
$905$	0	0
$906$	0	0
$907$	30912.4	1.13168	0.565838	−	0.824516i	$-0.308553\pi$
$907$	30912.4	1.13168	0.565838	+	0.824516i	$0.308553\pi$
$908$	0	0
$909$	−15362.9	−0.560566
$910$	0	0
$911$	−1390.58	−0.0505730	−0.0252865	−	0.999680i	$-0.508050\pi$
$911$	−1390.58	−0.0505730	−0.0252865	+	0.999680i	$0.508050\pi$
$912$	0	0
$913$	33973.2	1.23149
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2721.50	−0.0980063
$918$	0	0
$919$	−14064.7	−0.504843	−0.252422	−	0.967617i	$-0.581227\pi$
$919$	−14064.7	−0.504843	−0.252422	+	0.967617i	$0.581227\pi$
$920$	0	0
$921$	−28771.8	−1.02938
$922$	0	0
$923$	−21133.4	−0.753645
$924$	0	0
$925$	0	0
$926$	0	0
$927$	15208.3	0.538841
$928$	0	0
$929$	51692.7	1.82560	0.912799	−	0.408408i	$-0.133916\pi$
$929$	51692.7	1.82560	0.912799	+	0.408408i	$0.133916\pi$
$930$	0	0
$931$	3488.84	0.122816
$932$	0	0
$933$	35072.4	1.23067
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−12479.3	−0.435092	−0.217546	−	0.976050i	$-0.569805\pi$
$937$	−12479.3	−0.435092	−0.217546	+	0.976050i	$0.569805\pi$
$938$	0	0
$939$	−10506.2	−0.365129
$940$	0	0
$941$	41800.9	1.44811	0.724054	−	0.689743i	$-0.242277\pi$
$941$	41800.9	1.44811	0.724054	+	0.689743i	$0.242277\pi$
$942$	0	0
$943$	15914.2	0.549563
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−6482.77	−0.222452	−0.111226	−	0.993795i	$-0.535478\pi$
$947$	−6482.77	−0.222452	−0.111226	+	0.993795i	$0.535478\pi$
$948$	0	0
$949$	5178.45	0.177133
$950$	0	0
$951$	34467.9	1.17529
$952$	0	0
$953$	28635.8	0.973351	0.486675	−	0.873583i	$-0.338209\pi$
$953$	28635.8	0.973351	0.486675	+	0.873583i	$0.338209\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	60594.4	2.04675
$958$	0	0
$959$	6342.30	0.213560
$960$	0	0
$961$	−10428.0	−0.350038
$962$	0	0
$963$	11503.6	0.384940
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−3086.04	−0.102627	−0.0513135	−	0.998683i	$-0.516341\pi$
$967$	−3086.04	−0.102627	−0.0513135	+	0.998683i	$0.516341\pi$
$968$	0	0
$969$	−16733.7	−0.554762
$970$	0	0
$971$	11625.0	0.384205	0.192102	−	0.981375i	$-0.438469\pi$
$971$	11625.0	0.384205	0.192102	+	0.981375i	$0.438469\pi$
$972$	0	0
$973$	33780.0	1.11299
$974$	0	0
$975$	0	0
$976$	0	0
$977$	38232.0	1.25194	0.625972	−	0.779845i	$-0.284702\pi$
$977$	38232.0	1.25194	0.625972	+	0.779845i	$0.284702\pi$
$978$	0	0
$979$	−33452.3	−1.09207
$980$	0	0
$981$	−31381.9	−1.02135
$982$	0	0
$983$	−54290.3	−1.76154	−0.880769	−	0.473546i	$-0.842974\pi$
$983$	−54290.3	−1.76154	−0.880769	+	0.473546i	$0.842974\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	31572.5	1.01820
$988$	0	0
$989$	22623.0	0.727371
$990$	0	0
$991$	13673.7	0.438303	0.219152	−	0.975691i	$-0.429671\pi$
$991$	13673.7	0.438303	0.219152	+	0.975691i	$0.429671\pi$
$992$	0	0
$993$	−19538.2	−0.624397
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−3392.56	−0.107767	−0.0538834	−	0.998547i	$-0.517160\pi$
$997$	−3392.56	−0.107767	−0.0538834	+	0.998547i	$0.517160\pi$
$998$	0	0
$999$	2940.96	0.0931411

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.4.a.k.1.1		2
3.2	odd	2		1800.4.a.bk.1.2		2
4.3	odd	2		400.4.a.x.1.2		2
5.2	odd	4		40.4.c.a.9.4	yes	4
5.3	odd	4		40.4.c.a.9.1	✓	4
5.4	even	2		200.4.a.l.1.2		2
8.3	odd	2		1600.4.a.cf.1.1		2
8.5	even	2		1600.4.a.cl.1.2		2
15.2	even	4		360.4.f.e.289.3		4
15.8	even	4		360.4.f.e.289.4		4
15.14	odd	2		1800.4.a.bp.1.1		2
20.3	even	4		80.4.c.c.49.4		4
20.7	even	4		80.4.c.c.49.1		4
20.19	odd	2		400.4.a.v.1.1		2
40.3	even	4		320.4.c.h.129.1		4
40.13	odd	4		320.4.c.g.129.4		4
40.19	odd	2		1600.4.a.cm.1.2		2
40.27	even	4		320.4.c.h.129.4		4
40.29	even	2		1600.4.a.ce.1.1		2
40.37	odd	4		320.4.c.g.129.1		4
60.23	odd	4		720.4.f.m.289.4		4
60.47	odd	4		720.4.f.m.289.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.c.a.9.1	✓	4	5.3	odd	4
40.4.c.a.9.4	yes	4	5.2	odd	4
80.4.c.c.49.1		4	20.7	even	4
80.4.c.c.49.4		4	20.3	even	4
200.4.a.k.1.1		2	1.1	even	1	trivial
200.4.a.l.1.2		2	5.4	even	2
320.4.c.g.129.1		4	40.37	odd	4
320.4.c.g.129.4		4	40.13	odd	4
320.4.c.h.129.1		4	40.3	even	4
320.4.c.h.129.4		4	40.27	even	4
360.4.f.e.289.3		4	15.2	even	4
360.4.f.e.289.4		4	15.8	even	4
400.4.a.v.1.1		2	20.19	odd	2
400.4.a.x.1.2		2	4.3	odd	2
720.4.f.m.289.3		4	60.47	odd	4
720.4.f.m.289.4		4	60.23	odd	4
1600.4.a.ce.1.1		2	40.29	even	2
1600.4.a.cf.1.1		2	8.3	odd	2
1600.4.a.cl.1.2		2	8.5	even	2
1600.4.a.cm.1.2		2	40.19	odd	2
1800.4.a.bk.1.2		2	3.2	odd	2
1800.4.a.bp.1.1		2	15.14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	200.a (trivial)

\(f(q)\)	\(=\)	\(q-6.89898 q^{3} +12.6969 q^{7} +20.5959 q^{9} +59.1918 q^{11} -42.2020 q^{13} -126.384 q^{17} -19.1918 q^{19} -87.5959 q^{21} -78.3133 q^{23} +44.1816 q^{27} -148.384 q^{29} -139.151 q^{31} -408.363 q^{33} +66.5653 q^{37} +291.151 q^{39} -203.212 q^{41} -288.879 q^{43} -360.434 q^{47} -181.788 q^{49} +871.918 q^{51} +686.888 q^{53} +132.404 q^{57} -83.1102 q^{59} -208.829 q^{61} +261.505 q^{63} -192.293 q^{67} +540.282 q^{69} +500.767 q^{71} -122.706 q^{73} +751.555 q^{77} -289.616 q^{79} -860.898 q^{81} +573.950 q^{83} +1023.70 q^{87} -565.151 q^{89} -535.837 q^{91} +960.000 q^{93} -643.959 q^{97} +1219.11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} - 4 q^{7} + 2 q^{9} + 40 q^{11} - 104 q^{13} - 96 q^{17} + 40 q^{19} - 136 q^{21} - 284 q^{23} - 88 q^{27} - 140 q^{29} + 192 q^{31} - 464 q^{33} - 200 q^{37} + 112 q^{39} - 524 q^{41} - 372 q^{43}+ \cdots + 1576 q^{99}+O(q^{100}) \) 2 * q - 4 * q^3 - 4 * q^7 + 2 * q^9 + 40 * q^11 - 104 * q^13 - 96 * q^17 + 40 * q^19 - 136 * q^21 - 284 * q^23 - 88 * q^27 - 140 * q^29 + 192 * q^31 - 464 * q^33 - 200 * q^37 + 112 * q^39 - 524 * q^41 - 372 * q^43 - 84 * q^47 - 246 * q^49 + 960 * q^51 + 296 * q^53 + 304 * q^57 + 696 * q^59 - 692 * q^61 + 572 * q^63 - 316 * q^67 - 56 * q^69 + 688 * q^71 + 656 * q^73 + 1072 * q^77 - 736 * q^79 - 742 * q^81 + 1628 * q^83 + 1048 * q^87 - 660 * q^89 + 496 * q^91 + 1920 * q^93 - 896 * q^97 + 1576 * q^99

Introduction

L-functions

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