LMFDB - Embedded newform 1600.4.a.ce.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,4,Mod(1,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, 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("1600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1600.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:	$94.4030560092$
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	$\chi$	$=$	1600.1

$q$-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	$-0.611370\pi$
$7$	−12.6969	−0.685570	−0.342785	+	0.939414i	$0.611370\pi$
$8$	0	0
$9$	20.5959	0.762812
$10$	0	0
$11$	−59.1918	−1.62246	−0.811228	−	0.584730i	$-0.801200\pi$
$11$	−59.1918	−1.62246	−0.811228	+	0.584730i	$0.801200\pi$
$12$	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	$-0.142434\pi$
$17$	126.384	1.80309	0.901545	+	0.432685i	$0.142434\pi$
$18$	0	0
$19$	19.1918	0.231732	0.115866	−	0.993265i	$-0.463036\pi$
$19$	19.1918	0.231732	0.115866	+	0.993265i	$0.463036\pi$
$20$	0	0
$21$	87.5959	0.910238
$22$	0	0
$23$	78.3133	0.709976	0.354988	−	0.934871i	$-0.384485\pi$
$23$	78.3133	0.709976	0.354988	+	0.934871i	$0.384485\pi$
$24$	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	$-0.342422\pi$
$29$	148.384	0.950143	0.475072	+	0.879947i	$0.342422\pi$
$30$	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	$-0.311068\pi$
$47$	360.434	1.11861	0.559305	+	0.828962i	$0.311068\pi$
$48$	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	$-0.470771\pi$
$59$	83.1102	0.183390	0.0916951	+	0.995787i	$0.470771\pi$
$60$	0	0
$61$	208.829	0.438324	0.219162	−	0.975688i	$-0.429668\pi$
$61$	208.829	0.438324	0.219162	+	0.975688i	$0.429668\pi$
$62$	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	$-0.468638\pi$
$73$	122.706	0.196735	0.0983676	+	0.995150i	$0.468638\pi$
$74$	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	$-0.390577\pi$
$97$	643.959	0.674063	0.337032	+	0.941493i	$0.390577\pi$
$98$	0	0
$99$	−1219.11	−1.23763
$100$	0	0
$101$	745.918	0.734868	0.367434	−	0.930050i	$-0.380236\pi$
$101$	745.918	0.734868	0.367434	+	0.930050i	$0.380236\pi$
$102$	0	0
$103$	−738.413	−0.706389	−0.353194	−	0.935550i	$-0.614905\pi$
$103$	−738.413	−0.706389	−0.353194	+	0.935550i	$0.614905\pi$
$104$	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	$-0.266523\pi$
$109$	1523.69	1.33893	0.669465	+	0.742843i	$0.266523\pi$
$110$	0	0
$111$	−459.233	−0.392689
$112$	0	0
$113$	−58.4245	−0.0486382	−0.0243191	−	0.999704i	$-0.507742\pi$
$113$	−58.4245	−0.0486382	−0.0243191	+	0.999704i	$0.507742\pi$
$114$	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	$-0.393342\pi$
$127$	941.283	0.657680	0.328840	+	0.944386i	$0.393342\pi$
$128$	0	0
$129$	1992.97	1.36024
$130$	0	0
$131$	214.343	0.142956	0.0714779	−	0.997442i	$-0.477228\pi$
$131$	214.343	0.142956	0.0714779	+	0.997442i	$0.477228\pi$
$132$	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	$-0.549780\pi$
$137$	−499.514	−0.311506	−0.155753	+	0.987796i	$0.549780\pi$
$138$	0	0
$139$	−2660.48	−1.62345	−0.811723	−	0.584042i	$-0.801470\pi$
$139$	−2660.48	−1.62345	−0.811723	+	0.584042i	$0.801470\pi$
$140$	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	$-0.619606\pi$
$149$	−1334.89	−0.733947	−0.366973	+	0.930231i	$0.619606\pi$
$150$	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	$-0.569611\pi$
$167$	−936.411	−0.433902	−0.216951	+	0.976182i	$0.569611\pi$
$168$	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	$-0.674362\pi$
$179$	−2494.42	−1.04158	−0.520788	+	0.853686i	$0.674362\pi$
$180$	0	0
$181$	1590.36	0.653096	0.326548	−	0.945181i	$-0.394114\pi$
$181$	1590.36	0.653096	0.326548	+	0.945181i	$0.394114\pi$
$182$	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	$-0.689100\pi$
$193$	−3001.61	−1.11948	−0.559742	+	0.828667i	$0.689100\pi$
$194$	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	$-0.338476\pi$
$211$	2978.78	0.971886	0.485943	+	0.873991i	$0.338476\pi$
$212$	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	$-0.341385\pi$
$223$	3183.16	0.955875	0.477937	+	0.878394i	$0.341385\pi$
$224$	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	$-0.666677\pi$
$229$	−3465.59	−1.00006	−0.500028	+	0.866009i	$0.666677\pi$
$230$	0	0
$231$	−5184.96	−1.47682
$232$	0	0
$233$	−4611.51	−1.29661	−0.648305	−	0.761380i	$-0.724522\pi$
$233$	−4611.51	−1.29661	−0.648305	+	0.761380i	$0.724522\pi$
$234$	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	$-0.891852\pi$
$251$	−7498.54	−1.88567	−0.942836	+	0.333258i	$0.891852\pi$
$252$	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	$-0.588932\pi$
$257$	−2272.33	−0.551532	−0.275766	+	0.961225i	$0.588932\pi$
$258$	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	$-0.712039\pi$
$263$	−5271.34	−1.23591	−0.617955	+	0.786213i	$0.712039\pi$
$264$	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	$-0.337331\pi$
$269$	4315.61	0.978169	0.489085	+	0.872236i	$0.337331\pi$
$270$	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	$-0.543908\pi$
$313$	−1522.86	−0.275007	−0.137504	+	0.990501i	$0.543908\pi$
$314$	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	$-0.575555\pi$
$331$	−2832.04	−0.470281	−0.235141	+	0.971961i	$0.575555\pi$
$332$	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	$-0.543469\pi$
$337$	−1684.43	−0.272276	−0.136138	+	0.990690i	$0.543469\pi$
$338$	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	$-0.603721\pi$
$349$	−4174.20	−0.640228	−0.320114	+	0.947379i	$0.603721\pi$
$350$	0	0
$351$	−1864.55	−0.283540
$352$	0	0
$353$	6217.04	0.937394	0.468697	−	0.883359i	$-0.344724\pi$
$353$	6217.04	0.937394	0.468697	+	0.883359i	$0.344724\pi$
$354$	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	$-0.180178\pi$
$367$	11868.2	1.68806	0.844028	+	0.536299i	$0.180178\pi$
$368$	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	$-0.690829\pi$
$379$	−8326.24	−1.12847	−0.564235	+	0.825614i	$0.690829\pi$
$380$	0	0
$381$	−6493.89	−0.873207
$382$	0	0
$383$	−4249.23	−0.566908	−0.283454	−	0.958986i	$-0.591480\pi$
$383$	−4249.23	−0.566908	−0.283454	+	0.958986i	$0.591480\pi$
$384$	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	$-0.329257\pi$
$389$	7841.83	1.02210	0.511050	+	0.859551i	$0.329257\pi$
$390$	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	$-0.692974\pi$
$419$	−9773.77	−1.13957	−0.569785	+	0.821794i	$0.692974\pi$
$420$	0	0
$421$	−3037.95	−0.351688	−0.175844	−	0.984418i	$-0.556265\pi$
$421$	−3037.95	−0.351688	−0.175844	+	0.984418i	$0.556265\pi$
$422$	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	$-0.271439\pi$
$433$	11855.8	1.31583	0.657915	+	0.753092i	$0.271439\pi$
$434$	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	$-0.404054\pi$
$457$	5800.75	0.593758	0.296879	+	0.954915i	$0.404054\pi$
$458$	0	0
$459$	5583.84	0.567824
$460$	0	0
$461$	5263.63	0.531783	0.265891	−	0.964003i	$-0.414334\pi$
$461$	5263.63	0.531783	0.265891	+	0.964003i	$0.414334\pi$
$462$	0	0
$463$	13636.6	1.36879	0.684394	−	0.729113i	$-0.260067\pi$
$463$	13636.6	1.36879	0.684394	+	0.729113i	$0.260067\pi$
$464$	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	$-0.589028\pi$
$487$	−5933.64	−0.552113	−0.276057	+	0.961141i	$0.589028\pi$
$488$	0	0
$489$	21660.5	2.00311
$490$	0	0
$491$	11934.5	1.09694	0.548468	−	0.836171i	$-0.315211\pi$
$491$	11934.5	1.09694	0.548468	+	0.836171i	$0.315211\pi$
$492$	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	$-0.477599\pi$
$499$	1567.65	0.140636	0.0703182	+	0.997525i	$0.477599\pi$
$500$	0	0
$501$	6460.28	0.576096
$502$	0	0
$503$	450.509	0.0399348	0.0199674	−	0.999801i	$-0.493644\pi$
$503$	450.509	0.0399348	0.0199674	+	0.999801i	$0.493644\pi$
$504$	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	$-0.401182\pi$
$509$	7016.11	0.610969	0.305485	+	0.952197i	$0.401182\pi$
$510$	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	$-0.838532\pi$
$541$	−21997.6	−1.74815	−0.874076	+	0.485789i	$0.838532\pi$
$542$	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	$-0.167146\pi$
$571$	23612.2	1.73054	0.865272	+	0.501303i	$0.167146\pi$
$572$	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	$-0.868789\pi$
$577$	−25398.1	−1.83247	−0.916237	+	0.400636i	$0.868789\pi$
$578$	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	$-0.671800\pi$
$593$	−14842.0	−1.02780	−0.513900	+	0.857850i	$0.671800\pi$
$594$	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	$-0.727260\pi$
$607$	−19585.8	−1.30966	−0.654830	+	0.755776i	$0.727260\pi$
$608$	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	$-0.487267\pi$
$617$	1225.83	0.0799841	0.0399920	+	0.999200i	$0.487267\pi$
$618$	0	0
$619$	12430.1	0.807122	0.403561	−	0.914953i	$-0.367772\pi$
$619$	12430.1	0.807122	0.403561	+	0.914953i	$0.367772\pi$
$620$	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	$-0.256587\pi$
$647$	22787.4	1.38465	0.692323	+	0.721587i	$0.256587\pi$
$648$	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	$-0.481778\pi$
$659$	1935.78	0.114427	0.0572134	+	0.998362i	$0.481778\pi$
$660$	0	0
$661$	−3371.84	−0.198411	−0.0992053	−	0.995067i	$-0.531630\pi$
$661$	−3371.84	−0.198411	−0.0992053	+	0.995067i	$0.531630\pi$
$662$	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	$-0.717683\pi$
$673$	−22061.3	−1.26360	−0.631798	+	0.775133i	$0.717683\pi$
$674$	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	$-0.356259\pi$
$691$	15853.1	0.872767	0.436383	+	0.899761i	$0.356259\pi$
$692$	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	$-0.336020\pi$
$701$	18288.0	0.985347	0.492673	+	0.870214i	$0.336020\pi$
$702$	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	$-0.602821\pi$
$709$	−11985.3	−0.634864	−0.317432	+	0.948281i	$0.602821\pi$
$710$	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	$-0.490094\pi$
$727$	1219.85	0.0622307	0.0311154	+	0.999516i	$0.490094\pi$
$728$	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	$-0.277788\pi$
$739$	25825.4	1.28553	0.642763	+	0.766065i	$0.277788\pi$
$740$	0	0
$741$	5587.72	0.277018
$742$	0	0
$743$	−11549.8	−0.570282	−0.285141	−	0.958486i	$-0.592041\pi$
$743$	−11549.8	−0.570282	−0.285141	+	0.958486i	$0.592041\pi$
$744$	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	$-0.540535\pi$
$811$	−5866.38	−0.254003	−0.127001	+	0.991903i	$0.540535\pi$
$812$	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	$-0.292252\pi$
$821$	28572.6	1.21460	0.607301	+	0.794472i	$0.292252\pi$
$822$	0	0
$823$	15900.5	0.673460	0.336730	−	0.941601i	$-0.390679\pi$
$823$	15900.5	0.673460	0.336730	+	0.941601i	$0.390679\pi$
$824$	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	$-0.494582\pi$
$829$	812.559	0.0340426	0.0170213	+	0.999855i	$0.494582\pi$
$830$	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	$-0.489136\pi$
$857$	1712.28	0.0682502	0.0341251	+	0.999418i	$0.489136\pi$
$858$	0	0
$859$	−33276.7	−1.32175	−0.660877	−	0.750495i	$-0.729815\pi$
$859$	−33276.7	−1.32175	−0.660877	+	0.750495i	$0.729815\pi$
$860$	0	0
$861$	−17800.6	−0.704578
$862$	0	0
$863$	−5913.49	−0.233253	−0.116627	−	0.993176i	$-0.537208\pi$
$863$	−5913.49	−0.233253	−0.116627	+	0.993176i	$0.537208\pi$
$864$	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	$-0.242641\pi$
$887$	38213.1	1.44653	0.723264	+	0.690571i	$0.242641\pi$
$888$	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	$-0.430195\pi$
$937$	12479.3	0.435092	0.217546	+	0.976050i	$0.430195\pi$
$938$	0	0
$939$	10506.2	0.365129
$940$	0	0
$941$	−41800.9	−1.44811	−0.724054	−	0.689743i	$-0.757723\pi$
$941$	−41800.9	−1.44811	−0.724054	+	0.689743i	$0.757723\pi$
$942$	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	$-0.661791\pi$
$953$	−28635.8	−0.973351	−0.486675	+	0.873583i	$0.661791\pi$
$954$	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	$-0.483659\pi$
$967$	3086.04	0.102627	0.0513135	+	0.998683i	$0.483659\pi$
$968$	0	0
$969$	−16733.7	−0.554762
$970$	0	0
$971$	−11625.0	−0.384205	−0.192102	−	0.981375i	$-0.561531\pi$
$971$	−11625.0	−0.384205	−0.192102	+	0.981375i	$0.561531\pi$
$972$	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	$-0.715298\pi$
$977$	−38232.0	−1.25194	−0.625972	+	0.779845i	$0.715298\pi$
$978$	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	$-0.157026\pi$
$983$	54290.3	1.76154	0.880769	+	0.473546i	$0.157026\pi$
$984$	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	1600.4.a.ce.1.1		2
4.3	odd	2		1600.4.a.cm.1.2		2
5.2	odd	4		320.4.c.g.129.4		4
5.3	odd	4		320.4.c.g.129.1		4
5.4	even	2		1600.4.a.cl.1.2		2
8.3	odd	2		400.4.a.v.1.1		2
8.5	even	2		200.4.a.l.1.2		2
20.3	even	4		320.4.c.h.129.4		4
20.7	even	4		320.4.c.h.129.1		4
20.19	odd	2		1600.4.a.cf.1.1		2
24.5	odd	2		1800.4.a.bp.1.1		2
40.3	even	4		80.4.c.c.49.1		4
40.13	odd	4		40.4.c.a.9.4	yes	4
40.19	odd	2		400.4.a.x.1.2		2
40.27	even	4		80.4.c.c.49.4		4
40.29	even	2		200.4.a.k.1.1		2
40.37	odd	4		40.4.c.a.9.1	✓	4
120.29	odd	2		1800.4.a.bk.1.2		2
120.53	even	4		360.4.f.e.289.3		4
120.77	even	4		360.4.f.e.289.4		4
120.83	odd	4		720.4.f.m.289.3		4
120.107	odd	4		720.4.f.m.289.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.c.a.9.1	✓	4	40.37	odd	4
40.4.c.a.9.4	yes	4	40.13	odd	4
80.4.c.c.49.1		4	40.3	even	4
80.4.c.c.49.4		4	40.27	even	4
200.4.a.k.1.1		2	40.29	even	2
200.4.a.l.1.2		2	8.5	even	2
320.4.c.g.129.1		4	5.3	odd	4
320.4.c.g.129.4		4	5.2	odd	4
320.4.c.h.129.1		4	20.7	even	4
320.4.c.h.129.4		4	20.3	even	4
360.4.f.e.289.3		4	120.53	even	4
360.4.f.e.289.4		4	120.77	even	4
400.4.a.v.1.1		2	8.3	odd	2
400.4.a.x.1.2		2	40.19	odd	2
720.4.f.m.289.3		4	120.83	odd	4
720.4.f.m.289.4		4	120.107	odd	4
1600.4.a.ce.1.1		2	1.1	even	1	trivial
1600.4.a.cf.1.1		2	20.19	odd	2
1600.4.a.cl.1.2		2	5.4	even	2
1600.4.a.cm.1.2		2	4.3	odd	2
1800.4.a.bk.1.2		2	120.29	odd	2
1800.4.a.bp.1.1		2	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1600.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

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