LMFDB - Embedded newform 207.4.a.g.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [207,4,Mod(1,207)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("207.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(207, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	207.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$12.2133953712$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 26x^{3} + 10x^{2} + 144x + 56 $ x^5 - x^4 - 26x^3 + 10x^2 + 144*x + 56
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.413197$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41320 q^{2} -6.00288 q^{4} +6.16513 q^{5} -0.219811 q^{7} +19.7888 q^{8} -8.71253 q^{10} -52.4758 q^{11} +86.8286 q^{13} +0.310636 q^{14} +20.0575 q^{16} -94.7709 q^{17} -41.3899 q^{19} -37.0085 q^{20} +74.1586 q^{22} +23.0000 q^{23} -86.9912 q^{25} -122.706 q^{26} +1.31950 q^{28} -221.765 q^{29} -86.5941 q^{31} -186.656 q^{32} +133.930 q^{34} -1.35516 q^{35} -363.291 q^{37} +58.4920 q^{38} +122.001 q^{40} -342.800 q^{41} +454.201 q^{43} +315.005 q^{44} -32.5035 q^{46} +442.861 q^{47} -342.952 q^{49} +122.936 q^{50} -521.221 q^{52} -456.142 q^{53} -323.520 q^{55} -4.34979 q^{56} +313.398 q^{58} -753.494 q^{59} -75.7151 q^{61} +122.374 q^{62} +103.321 q^{64} +535.309 q^{65} +551.758 q^{67} +568.898 q^{68} +1.91511 q^{70} -565.146 q^{71} +573.048 q^{73} +513.402 q^{74} +248.458 q^{76} +11.5347 q^{77} +37.6381 q^{79} +123.657 q^{80} +484.444 q^{82} +1163.49 q^{83} -584.274 q^{85} -641.875 q^{86} -1038.43 q^{88} +1015.64 q^{89} -19.0859 q^{91} -138.066 q^{92} -625.850 q^{94} -255.174 q^{95} -244.959 q^{97} +484.658 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{2} + 16 q^{4} - 20 q^{5} - 10 q^{7} - 48 q^{8} + 50 q^{10} - 46 q^{11} + 54 q^{13} - 164 q^{14} - 60 q^{16} - 250 q^{17} - 28 q^{19} - 242 q^{20} - 10 q^{22} + 115 q^{23} + 239 q^{25} - 368 q^{26}+ \cdots + 2400 q^{98}+O(q^{100}) $ 5 * q - 4 * q^2 + 16 * q^4 - 20 * q^5 - 10 * q^7 - 48 * q^8 + 50 * q^10 - 46 * q^11 + 54 * q^13 - 164 * q^14 - 60 * q^16 - 250 * q^17 - 28 * q^19 - 242 * q^20 - 10 * q^22 + 115 * q^23 + 239 * q^25 - 368 * q^26 + 224 * q^28 - 460 * q^29 - 360 * q^31 - 268 * q^32 - 20 * q^34 - 964 * q^35 - 92 * q^37 - 266 * q^38 - 278 * q^40 - 788 * q^41 + 108 * q^43 - 354 * q^44 - 92 * q^46 - 724 * q^47 + 149 * q^49 - 128 * q^50 - 288 * q^52 - 580 * q^53 - 672 * q^55 + 1172 * q^56 - 48 * q^58 + 408 * q^59 - 620 * q^61 + 1088 * q^62 + 696 * q^64 - 184 * q^65 + 100 * q^67 + 1584 * q^68 + 728 * q^70 + 124 * q^71 - 1946 * q^73 + 2866 * q^74 + 1362 * q^76 + 396 * q^77 - 198 * q^79 + 2398 * q^80 + 76 * q^82 + 578 * q^83 + 376 * q^85 + 3910 * q^86 + 1366 * q^88 - 930 * q^89 + 1972 * q^91 + 368 * q^92 + 976 * q^94 + 908 * q^95 + 2618 * q^97 + 2400 * q^98

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 $

$2$	−1.41320	−0.499640	−0.249820	−	0.968292i	$-0.580372\pi$
$2$	−1.41320	−0.499640	−0.249820	+	0.968292i	$0.580372\pi$
$3$	0	0
$3$	0	0
$4$	−6.00288	−0.750359
$5$	6.16513	0.551426	0.275713	−	0.961240i	$-0.411086\pi$
$5$	6.16513	0.551426	0.275713	+	0.961240i	$0.411086\pi$
$6$	0	0
$7$	−0.219811	−0.0118687	−0.00593433	−	0.999982i	$-0.501889\pi$
$7$	−0.219811	−0.0118687	−0.00593433	+	0.999982i	$0.501889\pi$
$8$	19.7888	0.874550
$9$	0	0
$10$	−8.71253	−0.275515
$11$	−52.4758	−1.43837	−0.719184	−	0.694820i	$-0.755484\pi$
$11$	−52.4758	−1.43837	−0.719184	+	0.694820i	$0.755484\pi$
$12$	0	0
$13$	86.8286	1.85246	0.926228	−	0.376965i	$-0.123032\pi$
$13$	86.8286	1.85246	0.926228	+	0.376965i	$0.123032\pi$
$14$	0.310636	0.00593006
$15$	0	0
$16$	20.0575	0.313399
$17$	−94.7709	−1.35208	−0.676038	−	0.736866i	$-0.736305\pi$
$17$	−94.7709	−1.35208	−0.676038	+	0.736866i	$0.736305\pi$
$18$	0	0
$19$	−41.3899	−0.499762	−0.249881	−	0.968277i	$-0.580392\pi$
$19$	−41.3899	−0.499762	−0.249881	+	0.968277i	$0.580392\pi$
$20$	−37.0085	−0.413767
$21$	0	0
$22$	74.1586	0.718666
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	−86.9912	−0.695930
$26$	−122.706	−0.925562
$27$	0	0
$28$	1.31950	0.00890576
$29$	−221.765	−1.42003	−0.710013	−	0.704189i	$-0.751311\pi$
$29$	−221.765	−1.42003	−0.710013	+	0.704189i	$0.751311\pi$
$30$	0	0
$31$	−86.5941	−0.501702	−0.250851	−	0.968026i	$-0.580710\pi$
$31$	−86.5941	−0.501702	−0.250851	+	0.968026i	$0.580710\pi$
$32$	−186.656	−1.03114
$33$	0	0
$34$	133.930	0.675552
$35$	−1.35516	−0.00654468
$36$	0	0
$37$	−363.291	−1.61418	−0.807091	−	0.590428i	$-0.798959\pi$
$37$	−363.291	−1.61418	−0.807091	+	0.590428i	$0.798959\pi$
$38$	58.4920	0.249701
$39$	0	0
$40$	122.001	0.482249
$41$	−342.800	−1.30577	−0.652883	−	0.757459i	$-0.726441\pi$
$41$	−342.800	−1.30577	−0.652883	+	0.757459i	$0.726441\pi$
$42$	0	0
$43$	454.201	1.61081	0.805406	−	0.592723i	$-0.201947\pi$
$43$	454.201	1.61081	0.805406	+	0.592723i	$0.201947\pi$
$44$	315.005	1.07929
$45$	0	0
$46$	−32.5035	−0.104182
$47$	442.861	1.37442	0.687212	−	0.726457i	$-0.258834\pi$
$47$	442.861	1.37442	0.687212	+	0.726457i	$0.258834\pi$
$48$	0	0
$49$	−342.952	−0.999859
$50$	122.936	0.347715
$51$	0	0
$52$	−521.221	−1.39001
$53$	−456.142	−1.18219	−0.591094	−	0.806603i	$-0.701304\pi$
$53$	−456.142	−1.18219	−0.591094	+	0.806603i	$0.701304\pi$
$54$	0	0
$55$	−323.520	−0.793152
$56$	−4.34979	−0.0103797
$57$	0	0
$58$	313.398	0.709502
$59$	−753.494	−1.66265	−0.831327	−	0.555784i	$-0.812418\pi$
$59$	−753.494	−1.66265	−0.831327	+	0.555784i	$0.812418\pi$
$60$	0	0
$61$	−75.7151	−0.158923	−0.0794617	−	0.996838i	$-0.525320\pi$
$61$	−75.7151	−0.158923	−0.0794617	+	0.996838i	$0.525320\pi$
$62$	122.374	0.250671
$63$	0	0
$64$	103.321	0.201799
$65$	535.309	1.02149
$66$	0	0
$67$	551.758	1.00609	0.503044	−	0.864261i	$-0.332213\pi$
$67$	551.758	1.00609	0.503044	+	0.864261i	$0.332213\pi$
$68$	568.898	1.01454
$69$	0	0
$70$	1.91511	0.00326999
$71$	−565.146	−0.944655	−0.472328	−	0.881423i	$-0.656586\pi$
$71$	−565.146	−0.944655	−0.472328	+	0.881423i	$0.656586\pi$
$72$	0	0
$73$	573.048	0.918770	0.459385	−	0.888237i	$-0.348070\pi$
$73$	573.048	0.918770	0.459385	+	0.888237i	$0.348070\pi$
$74$	513.402	0.806510
$75$	0	0
$76$	248.458	0.375001
$77$	11.5347	0.0170715
$78$	0	0
$79$	37.6381	0.0536028	0.0268014	−	0.999641i	$-0.491468\pi$
$79$	37.6381	0.0536028	0.0268014	+	0.999641i	$0.491468\pi$
$80$	123.657	0.172816
$81$	0	0
$82$	484.444	0.652414
$83$	1163.49	1.53866	0.769332	−	0.638849i	$-0.220589\pi$
$83$	1163.49	1.53866	0.769332	+	0.638849i	$0.220589\pi$
$84$	0	0
$85$	−584.274	−0.745570
$86$	−641.875	−0.804827
$87$	0	0
$88$	−1038.43	−1.25792
$89$	1015.64	1.20964	0.604820	−	0.796362i	$-0.293245\pi$
$89$	1015.64	1.20964	0.604820	+	0.796362i	$0.293245\pi$
$90$	0	0
$91$	−19.0859	−0.0219862
$92$	−138.066	−0.156461
$93$	0	0
$94$	−625.850	−0.686718
$95$	−255.174	−0.275582
$96$	0	0
$97$	−244.959	−0.256411	−0.128205	−	0.991748i	$-0.540922\pi$
$97$	−244.959	−0.256411	−0.128205	+	0.991748i	$0.540922\pi$
$98$	484.658	0.499570
$99$	0	0
$100$	522.197	0.522197
$101$	−20.0731	−0.0197757	−0.00988786	−	0.999951i	$-0.503147\pi$
$101$	−20.0731	−0.0197757	−0.00988786	+	0.999951i	$0.503147\pi$
$102$	0	0
$103$	1101.13	1.05337	0.526686	−	0.850060i	$-0.323434\pi$
$103$	1101.13	1.05337	0.526686	+	0.850060i	$0.323434\pi$
$104$	1718.24	1.62007
$105$	0	0
$106$	644.619	0.590669
$107$	−17.2298	−0.0155670	−0.00778351	−	0.999970i	$-0.502478\pi$
$107$	−17.2298	−0.0155670	−0.00778351	+	0.999970i	$0.502478\pi$
$108$	0	0
$109$	−729.003	−0.640604	−0.320302	−	0.947315i	$-0.603784\pi$
$109$	−729.003	−0.640604	−0.320302	+	0.947315i	$0.603784\pi$
$110$	457.197	0.396291
$111$	0	0
$112$	−4.40886	−0.00371962
$113$	99.3322	0.0826937	0.0413468	−	0.999145i	$-0.486835\pi$
$113$	99.3322	0.0826937	0.0413468	+	0.999145i	$0.486835\pi$
$114$	0	0
$115$	141.798	0.114980
$116$	1331.23	1.06553
$117$	0	0
$118$	1064.83	0.830729
$119$	20.8316	0.0160473
$120$	0	0
$121$	1422.71	1.06890
$122$	107.000	0.0794046
$123$	0	0
$124$	519.814	0.376457
$125$	−1306.95	−0.935179
$126$	0	0
$127$	−1177.21	−0.822522	−0.411261	−	0.911518i	$-0.634912\pi$
$127$	−1177.21	−0.822522	−0.411261	+	0.911518i	$0.634912\pi$
$128$	1347.23	0.930310
$129$	0	0
$130$	−756.497	−0.510378
$131$	638.839	0.426074	0.213037	−	0.977044i	$-0.431665\pi$
$131$	638.839	0.426074	0.213037	+	0.977044i	$0.431665\pi$
$132$	0	0
$133$	9.09793	0.00593151
$134$	−779.742	−0.502682
$135$	0	0
$136$	−1875.40	−1.18246
$137$	282.549	0.176203	0.0881016	−	0.996111i	$-0.471920\pi$
$137$	282.549	0.176203	0.0881016	+	0.996111i	$0.471920\pi$
$138$	0	0
$139$	−1014.95	−0.619331	−0.309665	−	0.950846i	$-0.600217\pi$
$139$	−1014.95	−0.619331	−0.309665	+	0.950846i	$0.600217\pi$
$140$	8.13486	0.00491086
$141$	0	0
$142$	798.663	0.471988
$143$	−4556.40	−2.66451
$144$	0	0
$145$	−1367.21	−0.783038
$146$	−809.829	−0.459055
$147$	0	0
$148$	2180.79	1.21122
$149$	−2363.79	−1.29966	−0.649830	−	0.760079i	$-0.725160\pi$
$149$	−2363.79	−1.29966	−0.649830	+	0.760079i	$0.725160\pi$
$150$	0	0
$151$	−2721.03	−1.46645	−0.733227	−	0.679984i	$-0.761987\pi$
$151$	−2721.03	−1.46645	−0.733227	+	0.679984i	$0.761987\pi$
$152$	−819.056	−0.437067
$153$	0	0
$154$	−16.3008	−0.00852961
$155$	−533.863	−0.276651
$156$	0	0
$157$	783.921	0.398495	0.199248	−	0.979949i	$-0.436150\pi$
$157$	783.921	0.398495	0.199248	+	0.979949i	$0.436150\pi$
$158$	−53.1901	−0.0267821
$159$	0	0
$160$	−1150.76	−0.568595
$161$	−5.05565	−0.00247479
$162$	0	0
$163$	−1078.60	−0.518298	−0.259149	−	0.965837i	$-0.583442\pi$
$163$	−1078.60	−0.518298	−0.259149	+	0.965837i	$0.583442\pi$
$164$	2057.79	0.979794
$165$	0	0
$166$	−1644.23	−0.768779
$167$	2191.56	1.01550	0.507749	−	0.861505i	$-0.330478\pi$
$167$	2191.56	1.01550	0.507749	+	0.861505i	$0.330478\pi$
$168$	0	0
$169$	5342.20	2.43159
$170$	825.694	0.372517
$171$	0	0
$172$	−2726.51	−1.20869
$173$	−3254.30	−1.43017	−0.715086	−	0.699036i	$-0.753613\pi$
$173$	−3254.30	−1.43017	−0.715086	+	0.699036i	$0.753613\pi$
$174$	0	0
$175$	19.1216	0.00825975
$176$	−1052.53	−0.450782
$177$	0	0
$178$	−1435.30	−0.604385
$179$	−1969.15	−0.822241	−0.411121	−	0.911581i	$-0.634862\pi$
$179$	−1969.15	−0.822241	−0.411121	+	0.911581i	$0.634862\pi$
$180$	0	0
$181$	−57.4589	−0.0235961	−0.0117980	−	0.999930i	$-0.503756\pi$
$181$	−57.4589	−0.0235961	−0.0117980	+	0.999930i	$0.503756\pi$
$182$	26.9721	0.0109852
$183$	0	0
$184$	455.143	0.182356
$185$	−2239.74	−0.890101
$186$	0	0
$187$	4973.17	1.94478
$188$	−2658.44	−1.03131
$189$	0	0
$190$	360.611	0.137692
$191$	760.313	0.288033	0.144017	−	0.989575i	$-0.453998\pi$
$191$	760.313	0.288033	0.144017	+	0.989575i	$0.453998\pi$
$192$	0	0
$193$	−2971.55	−1.10827	−0.554137	−	0.832426i	$-0.686952\pi$
$193$	−2971.55	−1.10827	−0.554137	+	0.832426i	$0.686952\pi$
$194$	346.175	0.128113
$195$	0	0
$196$	2058.70	0.750254
$197$	496.563	0.179587	0.0897935	−	0.995960i	$-0.471379\pi$
$197$	496.563	0.179587	0.0897935	+	0.995960i	$0.471379\pi$
$198$	0	0
$199$	1626.81	0.579504	0.289752	−	0.957102i	$-0.406427\pi$
$199$	1626.81	0.579504	0.289752	+	0.957102i	$0.406427\pi$
$200$	−1721.45	−0.608626
$201$	0	0
$202$	28.3672	0.00988075
$203$	48.7463	0.0168538
$204$	0	0
$205$	−2113.41	−0.720033
$206$	−1556.11	−0.526307
$207$	0	0
$208$	1741.57	0.580557
$209$	2171.96	0.718842
$210$	0	0
$211$	1857.31	0.605984	0.302992	−	0.952993i	$-0.402014\pi$
$211$	1857.31	0.605984	0.302992	+	0.952993i	$0.402014\pi$
$212$	2738.17	0.887066
$213$	0	0
$214$	24.3491	0.00777791
$215$	2800.20	0.888243
$216$	0	0
$217$	19.0343	0.00595453
$218$	1030.22	0.320072
$219$	0	0
$220$	1942.05	0.595149
$221$	−8228.82	−2.50466
$222$	0	0
$223$	−3204.53	−0.962292	−0.481146	−	0.876640i	$-0.659779\pi$
$223$	−3204.53	−0.962292	−0.481146	+	0.876640i	$0.659779\pi$
$224$	41.0289	0.0122382
$225$	0	0
$226$	−140.376	−0.0413171
$227$	5402.34	1.57959	0.789793	−	0.613374i	$-0.210188\pi$
$227$	5402.34	1.57959	0.789793	+	0.613374i	$0.210188\pi$
$228$	0	0
$229$	1242.46	0.358533	0.179267	−	0.983801i	$-0.442628\pi$
$229$	1242.46	0.358533	0.179267	+	0.983801i	$0.442628\pi$
$230$	−200.388	−0.0574488
$231$	0	0
$232$	−4388.47	−1.24188
$233$	−4861.95	−1.36703	−0.683513	−	0.729939i	$-0.739549\pi$
$233$	−4861.95	−1.36703	−0.683513	+	0.729939i	$0.739549\pi$
$234$	0	0
$235$	2730.29	0.757892
$236$	4523.13	1.24759
$237$	0	0
$238$	−29.4392	−0.00801790
$239$	2370.98	0.641698	0.320849	−	0.947130i	$-0.396032\pi$
$239$	2370.98	0.641698	0.320849	+	0.947130i	$0.396032\pi$
$240$	0	0
$241$	5487.10	1.46662	0.733309	−	0.679895i	$-0.237975\pi$
$241$	5487.10	1.46662	0.733309	+	0.679895i	$0.237975\pi$
$242$	−2010.56	−0.534066
$243$	0	0
$244$	454.509	0.119250
$245$	−2114.34	−0.551348
$246$	0	0
$247$	−3593.82	−0.925787
$248$	−1713.59	−0.438764
$249$	0	0
$250$	1846.98	0.467253
$251$	−3003.63	−0.755328	−0.377664	−	0.925943i	$-0.623273\pi$
$251$	−3003.63	−0.755328	−0.377664	+	0.925943i	$0.623273\pi$
$252$	0	0
$253$	−1206.94	−0.299920
$254$	1663.63	0.410965
$255$	0	0
$256$	−2730.47	−0.666620
$257$	−329.844	−0.0800587	−0.0400293	−	0.999199i	$-0.512745\pi$
$257$	−329.844	−0.0800587	−0.0400293	+	0.999199i	$0.512745\pi$
$258$	0	0
$259$	79.8553	0.0191582
$260$	−3213.39	−0.766486
$261$	0	0
$262$	−902.805	−0.212884
$263$	4571.82	1.07190	0.535951	−	0.844249i	$-0.319953\pi$
$263$	4571.82	1.07190	0.535951	+	0.844249i	$0.319953\pi$
$264$	0	0
$265$	−2812.17	−0.651889
$266$	−12.8572	−0.00296362
$267$	0	0
$268$	−3312.13	−0.754928
$269$	7095.23	1.60819	0.804096	−	0.594500i	$-0.202650\pi$
$269$	7095.23	1.60819	0.804096	+	0.594500i	$0.202650\pi$
$270$	0	0
$271$	561.320	0.125822	0.0629110	−	0.998019i	$-0.479962\pi$
$271$	561.320	0.125822	0.0629110	+	0.998019i	$0.479962\pi$
$272$	−1900.87	−0.423739
$273$	0	0
$274$	−399.298	−0.0880382
$275$	4564.93	1.00100
$276$	0	0
$277$	−4627.68	−1.00379	−0.501895	−	0.864928i	$-0.667364\pi$
$277$	−4627.68	−1.00379	−0.501895	+	0.864928i	$0.667364\pi$
$278$	1434.32	0.309443
$279$	0	0
$280$	−26.8170	−0.00572366
$281$	1859.67	0.394800	0.197400	−	0.980323i	$-0.436750\pi$
$281$	1859.67	0.394800	0.197400	+	0.980323i	$0.436750\pi$
$282$	0	0
$283$	6419.56	1.34842	0.674210	−	0.738539i	$-0.264484\pi$
$283$	6419.56	1.34842	0.674210	+	0.738539i	$0.264484\pi$
$284$	3392.50	0.708831
$285$	0	0
$286$	6439.08	1.33130
$287$	75.3512	0.0154977
$288$	0	0
$289$	4068.51	0.828112
$290$	1932.14	0.391238
$291$	0	0
$292$	−3439.94	−0.689408
$293$	9042.97	1.80306	0.901529	−	0.432720i	$-0.142446\pi$
$293$	9042.97	1.80306	0.901529	+	0.432720i	$0.142446\pi$
$294$	0	0
$295$	−4645.38	−0.916830
$296$	−7189.10	−1.41168
$297$	0	0
$298$	3340.50	0.649363
$299$	1997.06	0.386264
$300$	0	0
$301$	−99.8381	−0.0191182
$302$	3845.36	0.732700
$303$	0	0
$304$	−830.178	−0.156625
$305$	−466.793	−0.0876345
$306$	0	0
$307$	−1068.09	−0.198563	−0.0992816	−	0.995059i	$-0.531654\pi$
$307$	−1068.09	−0.198563	−0.0992816	+	0.995059i	$0.531654\pi$
$308$	−69.2416	−0.0128098
$309$	0	0
$310$	754.454	0.138226
$311$	4627.28	0.843694	0.421847	−	0.906667i	$-0.361382\pi$
$311$	4627.28	0.843694	0.421847	+	0.906667i	$0.361382\pi$
$312$	0	0
$313$	−4404.79	−0.795443	−0.397722	−	0.917506i	$-0.630199\pi$
$313$	−4404.79	−0.795443	−0.397722	+	0.917506i	$0.630199\pi$
$314$	−1107.83	−0.199104
$315$	0	0
$316$	−225.937	−0.0402214
$317$	1869.08	0.331161	0.165581	−	0.986196i	$-0.447050\pi$
$317$	1869.08	0.331161	0.165581	+	0.986196i	$0.447050\pi$
$318$	0	0
$319$	11637.3	2.04252
$320$	636.988	0.111277
$321$	0	0
$322$	7.14462	0.00123650
$323$	3922.55	0.675717
$324$	0	0
$325$	−7553.33	−1.28918
$326$	1524.28	0.258963
$327$	0	0
$328$	−6783.61	−1.14196
$329$	−97.3456	−0.0163126
$330$	0	0
$331$	−11426.4	−1.89743	−0.948715	−	0.316132i	$-0.897616\pi$
$331$	−11426.4	−1.89743	−0.948715	+	0.316132i	$0.897616\pi$
$332$	−6984.26	−1.15455
$333$	0	0
$334$	−3097.11	−0.507384
$335$	3401.66	0.554783
$336$	0	0
$337$	−3554.86	−0.574615	−0.287308	−	0.957838i	$-0.592760\pi$
$337$	−3554.86	−0.574615	−0.287308	+	0.957838i	$0.592760\pi$
$338$	−7549.59	−1.21492
$339$	0	0
$340$	3507.33	0.559445
$341$	4544.09	0.721631
$342$	0	0
$343$	150.780	0.0237357
$344$	8988.09	1.40874
$345$	0	0
$346$	4598.96	0.714572
$347$	−311.236	−0.0481499	−0.0240750	−	0.999710i	$-0.507664\pi$
$347$	−311.236	−0.0481499	−0.0240750	+	0.999710i	$0.507664\pi$
$348$	0	0
$349$	6597.31	1.01188	0.505940	−	0.862569i	$-0.331146\pi$
$349$	6597.31	1.01188	0.505940	+	0.862569i	$0.331146\pi$
$350$	−27.0226	−0.00412691
$351$	0	0
$352$	9794.90	1.48315
$353$	1485.90	0.224041	0.112020	−	0.993706i	$-0.464268\pi$
$353$	1485.90	0.224041	0.112020	+	0.993706i	$0.464268\pi$
$354$	0	0
$355$	−3484.20	−0.520907
$356$	−6096.78	−0.907665
$357$	0	0
$358$	2782.80	0.410825
$359$	5855.25	0.860802	0.430401	−	0.902638i	$-0.358372\pi$
$359$	5855.25	0.860802	0.430401	+	0.902638i	$0.358372\pi$
$360$	0	0
$361$	−5145.88	−0.750238
$362$	81.2007	0.0117895
$363$	0	0
$364$	114.570	0.0164975
$365$	3532.91	0.506633
$366$	0	0
$367$	2413.50	0.343280	0.171640	−	0.985160i	$-0.445093\pi$
$367$	2413.50	0.343280	0.171640	+	0.985160i	$0.445093\pi$
$368$	461.323	0.0653481
$369$	0	0
$370$	3165.19	0.444730
$371$	100.265	0.0140310
$372$	0	0
$373$	9146.62	1.26969	0.634844	−	0.772640i	$-0.281064\pi$
$373$	9146.62	1.26969	0.634844	+	0.772640i	$0.281064\pi$
$374$	−7028.07	−0.971692
$375$	0	0
$376$	8763.70	1.20200
$377$	−19255.6	−2.63053
$378$	0	0
$379$	−11710.4	−1.58714	−0.793568	−	0.608481i	$-0.791779\pi$
$379$	−11710.4	−1.58714	−0.793568	+	0.608481i	$0.791779\pi$
$380$	1531.78	0.206785
$381$	0	0
$382$	−1074.47	−0.143913
$383$	−11769.1	−1.57017	−0.785084	−	0.619389i	$-0.787380\pi$
$383$	−11769.1	−1.57017	−0.785084	+	0.619389i	$0.787380\pi$
$384$	0	0
$385$	71.1131	0.00941366
$386$	4199.38	0.553738
$387$	0	0
$388$	1470.46	0.192400
$389$	−5080.01	−0.662125	−0.331063	−	0.943609i	$-0.607407\pi$
$389$	−5080.01	−0.662125	−0.331063	+	0.943609i	$0.607407\pi$
$390$	0	0
$391$	−2179.73	−0.281928
$392$	−6786.61	−0.874427
$393$	0	0
$394$	−701.741	−0.0897289
$395$	232.044	0.0295579
$396$	0	0
$397$	4377.90	0.553452	0.276726	−	0.960949i	$-0.410751\pi$
$397$	4377.90	0.553452	0.276726	+	0.960949i	$0.410751\pi$
$398$	−2299.00	−0.289544
$399$	0	0
$400$	−1744.83	−0.218103
$401$	−14961.0	−1.86313	−0.931567	−	0.363569i	$-0.881558\pi$
$401$	−14961.0	−1.86313	−0.931567	+	0.363569i	$0.881558\pi$
$402$	0	0
$403$	−7518.84	−0.929380
$404$	120.496	0.0148389
$405$	0	0
$406$	−68.8882	−0.00842084
$407$	19064.0	2.32178
$408$	0	0
$409$	4236.25	0.512149	0.256075	−	0.966657i	$-0.417571\pi$
$409$	4236.25	0.512149	0.256075	+	0.966657i	$0.417571\pi$
$410$	2986.66	0.359758
$411$	0	0
$412$	−6609.93	−0.790408
$413$	165.626	0.0197335
$414$	0	0
$415$	7173.03	0.848459
$416$	−16207.1	−1.91014
$417$	0	0
$418$	−3069.41	−0.359162
$419$	1497.60	0.174612	0.0873059	−	0.996182i	$-0.472174\pi$
$419$	1497.60	0.174612	0.0873059	+	0.996182i	$0.472174\pi$
$420$	0	0
$421$	2464.85	0.285343	0.142671	−	0.989770i	$-0.454431\pi$
$421$	2464.85	0.285343	0.142671	+	0.989770i	$0.454431\pi$
$422$	−2624.75	−0.302774
$423$	0	0
$424$	−9026.52	−1.03388
$425$	8244.23	0.940951
$426$	0	0
$427$	16.6430	0.00188621
$428$	103.429	0.0116809
$429$	0	0
$430$	−3957.24	−0.443802
$431$	−5637.58	−0.630052	−0.315026	−	0.949083i	$-0.602013\pi$
$431$	−5637.58	−0.630052	−0.315026	+	0.949083i	$0.602013\pi$
$432$	0	0
$433$	14222.1	1.57845	0.789225	−	0.614104i	$-0.210482\pi$
$433$	14222.1	1.57845	0.789225	+	0.614104i	$0.210482\pi$
$434$	−26.8992	−0.00297512
$435$	0	0
$436$	4376.12	0.480683
$437$	−951.967	−0.104208
$438$	0	0
$439$	−17165.9	−1.86625	−0.933124	−	0.359555i	$-0.882929\pi$
$439$	−17165.9	−1.86625	−0.933124	+	0.359555i	$0.882929\pi$
$440$	−6402.07	−0.693652
$441$	0	0
$442$	11628.9	1.25143
$443$	4580.12	0.491214	0.245607	−	0.969369i	$-0.421013\pi$
$443$	4580.12	0.491214	0.245607	+	0.969369i	$0.421013\pi$
$444$	0	0
$445$	6261.57	0.667027
$446$	4528.63	0.480800
$447$	0	0
$448$	−22.7111	−0.00239508
$449$	1540.91	0.161960	0.0809802	−	0.996716i	$-0.474195\pi$
$449$	1540.91	0.161960	0.0809802	+	0.996716i	$0.474195\pi$
$450$	0	0
$451$	17988.7	1.87817
$452$	−596.279	−0.0620500
$453$	0	0
$454$	−7634.57	−0.789225
$455$	−117.667	−0.0121237
$456$	0	0
$457$	7134.74	0.730304	0.365152	−	0.930948i	$-0.381017\pi$
$457$	7134.74	0.730304	0.365152	+	0.930948i	$0.381017\pi$
$458$	−1755.84	−0.179138
$459$	0	0
$460$	−851.195	−0.0862765
$461$	3720.33	0.375864	0.187932	−	0.982182i	$-0.439822\pi$
$461$	3720.33	0.375864	0.187932	+	0.982182i	$0.439822\pi$
$462$	0	0
$463$	5612.05	0.563314	0.281657	−	0.959515i	$-0.409116\pi$
$463$	5612.05	0.563314	0.281657	+	0.959515i	$0.409116\pi$
$464$	−4448.06	−0.445034
$465$	0	0
$466$	6870.89	0.683021
$467$	−93.2112	−0.00923619	−0.00461810	−	0.999989i	$-0.501470\pi$
$467$	−93.2112	−0.00923619	−0.00461810	+	0.999989i	$0.501470\pi$
$468$	0	0
$469$	−121.282	−0.0119409
$470$	−3858.44	−0.378674
$471$	0	0
$472$	−14910.8	−1.45407
$473$	−23834.5	−2.31694
$474$	0	0
$475$	3600.55	0.347799
$476$	−125.050	−0.0120413
$477$	0	0
$478$	−3350.66	−0.320618
$479$	114.073	0.0108813	0.00544063	−	0.999985i	$-0.498268\pi$
$479$	114.073	0.0108813	0.00544063	+	0.999985i	$0.498268\pi$
$480$	0	0
$481$	−31544.1	−2.99020
$482$	−7754.35	−0.732782
$483$	0	0
$484$	−8540.32	−0.802059
$485$	−1510.20	−0.141391
$486$	0	0
$487$	−10495.0	−0.976541	−0.488271	−	0.872692i	$-0.662372\pi$
$487$	−10495.0	−0.976541	−0.488271	+	0.872692i	$0.662372\pi$
$488$	−1498.31	−0.138987
$489$	0	0
$490$	2987.98	0.275476
$491$	8987.02	0.826025	0.413013	−	0.910725i	$-0.364477\pi$
$491$	8987.02	0.826025	0.413013	+	0.910725i	$0.364477\pi$
$492$	0	0
$493$	21016.9	1.91998
$494$	5078.78	0.462561
$495$	0	0
$496$	−1736.86	−0.157233
$497$	124.225	0.0112118
$498$	0	0
$499$	−8539.75	−0.766115	−0.383058	−	0.923724i	$-0.625129\pi$
$499$	−8539.75	−0.766115	−0.383058	+	0.923724i	$0.625129\pi$
$500$	7845.47	0.701720
$501$	0	0
$502$	4244.72	0.377392
$503$	−4858.42	−0.430668	−0.215334	−	0.976540i	$-0.569084\pi$
$503$	−4858.42	−0.430668	−0.215334	+	0.976540i	$0.569084\pi$
$504$	0	0
$505$	−123.753	−0.0109048
$506$	1705.65	0.149852
$507$	0	0
$508$	7066.63	0.617187
$509$	8906.97	0.775628	0.387814	−	0.921738i	$-0.373230\pi$
$509$	8906.97	0.775628	0.387814	+	0.921738i	$0.373230\pi$
$510$	0	0
$511$	−125.962	−0.0109046
$512$	−6919.17	−0.597240
$513$	0	0
$514$	466.134	0.0400006
$515$	6788.59	0.580857
$516$	0	0
$517$	−23239.5	−1.97693
$518$	−112.851	−0.00957220
$519$	0	0
$520$	10593.1	0.893346
$521$	6022.66	0.506444	0.253222	−	0.967408i	$-0.418510\pi$
$521$	6022.66	0.506444	0.253222	+	0.967408i	$0.418510\pi$
$522$	0	0
$523$	−12922.8	−1.08045	−0.540225	−	0.841521i	$-0.681661\pi$
$523$	−12922.8	−1.08045	−0.540225	+	0.841521i	$0.681661\pi$
$524$	−3834.87	−0.319708
$525$	0	0
$526$	−6460.88	−0.535566
$527$	8206.60	0.678340
$528$	0	0
$529$	529.000	0.0434783
$530$	3974.16	0.325710
$531$	0	0
$532$	−54.6137	−0.00445076
$533$	−29764.9	−2.41887
$534$	0	0
$535$	−106.224	−0.00858405
$536$	10918.6	0.879875
$537$	0	0
$538$	−10026.9	−0.803518
$539$	17996.7	1.43816
$540$	0	0
$541$	−20376.1	−1.61929	−0.809647	−	0.586917i	$-0.800342\pi$
$541$	−20376.1	−1.61929	−0.809647	+	0.586917i	$0.800342\pi$
$542$	−793.256	−0.0628658
$543$	0	0
$544$	17689.5	1.39418
$545$	−4494.40	−0.353245
$546$	0	0
$547$	2667.19	0.208484	0.104242	−	0.994552i	$-0.466758\pi$
$547$	2667.19	0.208484	0.104242	+	0.994552i	$0.466758\pi$
$548$	−1696.11	−0.132216
$549$	0	0
$550$	−6451.14	−0.500141
$551$	9178.82	0.709675
$552$	0	0
$553$	−8.27326	−0.000636193 0
$554$	6539.82	0.501535
$555$	0	0
$556$	6092.62	0.464721
$557$	−15757.0	−1.19864	−0.599321	−	0.800508i	$-0.704563\pi$
$557$	−15757.0	−1.19864	−0.599321	+	0.800508i	$0.704563\pi$
$558$	0	0
$559$	39437.6	2.98396
$560$	−27.1812	−0.00205110
$561$	0	0
$562$	−2628.08	−0.197258
$563$	−4829.45	−0.361522	−0.180761	−	0.983527i	$-0.557856\pi$
$563$	−4829.45	−0.361522	−0.180761	+	0.983527i	$0.557856\pi$
$564$	0	0
$565$	612.395	0.0455994
$566$	−9072.10	−0.673726
$567$	0	0
$568$	−11183.6	−0.826149
$569$	427.019	0.0314615	0.0157307	−	0.999876i	$-0.494993\pi$
$569$	427.019	0.0314615	0.0157307	+	0.999876i	$0.494993\pi$
$570$	0	0
$571$	24460.0	1.79268	0.896338	−	0.443372i	$-0.146218\pi$
$571$	24460.0	1.79268	0.896338	+	0.443372i	$0.146218\pi$
$572$	27351.5	1.99934
$573$	0	0
$574$	−106.486	−0.00774328
$575$	−2000.80	−0.145111
$576$	0	0
$577$	8363.36	0.603417	0.301708	−	0.953400i	$-0.402443\pi$
$577$	8363.36	0.603417	0.301708	+	0.953400i	$0.402443\pi$
$578$	−5749.61	−0.413758
$579$	0	0
$580$	8207.19	0.587560
$581$	−255.747	−0.0182619
$582$	0	0
$583$	23936.4	1.70042
$584$	11339.9	0.803510
$585$	0	0
$586$	−12779.5	−0.900880
$587$	−14654.5	−1.03042	−0.515208	−	0.857065i	$-0.672285\pi$
$587$	−14654.5	−1.03042	−0.515208	+	0.857065i	$0.672285\pi$
$588$	0	0
$589$	3584.12	0.250732
$590$	6564.84	0.458085
$591$	0	0
$592$	−7286.72	−0.505882
$593$	4385.56	0.303698	0.151849	−	0.988404i	$-0.451477\pi$
$593$	4385.56	0.303698	0.151849	+	0.988404i	$0.451477\pi$
$594$	0	0
$595$	128.430	0.00884892
$596$	14189.6	0.975213
$597$	0	0
$598$	−2822.24	−0.192993
$599$	21443.4	1.46269	0.731347	−	0.682006i	$-0.238892\pi$
$599$	21443.4	1.46269	0.731347	+	0.682006i	$0.238892\pi$
$600$	0	0
$601$	−1264.93	−0.0858532	−0.0429266	−	0.999078i	$-0.513668\pi$
$601$	−1264.93	−0.0858532	−0.0429266	+	0.999078i	$0.513668\pi$
$602$	141.091	0.00955222
$603$	0	0
$604$	16334.0	1.10037
$605$	8771.16	0.589419
$606$	0	0
$607$	−7414.17	−0.495769	−0.247884	−	0.968790i	$-0.579735\pi$
$607$	−7414.17	−0.495769	−0.247884	+	0.968790i	$0.579735\pi$
$608$	7725.65	0.515323
$609$	0	0
$610$	659.671	0.0437857
$611$	38453.0	2.54606
$612$	0	0
$613$	−21120.5	−1.39160	−0.695798	−	0.718238i	$-0.744949\pi$
$613$	−21120.5	−1.39160	−0.695798	+	0.718238i	$0.744949\pi$
$614$	1509.42	0.0992102
$615$	0	0
$616$	228.259	0.0149299
$617$	−3598.75	−0.234814	−0.117407	−	0.993084i	$-0.537458\pi$
$617$	−3598.75	−0.234814	−0.117407	+	0.993084i	$0.537458\pi$
$618$	0	0
$619$	−19636.2	−1.27503	−0.637516	−	0.770437i	$-0.720038\pi$
$619$	−19636.2	−1.27503	−0.637516	+	0.770437i	$0.720038\pi$
$620$	3204.72	0.207588
$621$	0	0
$622$	−6539.26	−0.421544
$623$	−223.249	−0.0143568
$624$	0	0
$625$	2816.38	0.180248
$626$	6224.84	0.397436
$627$	0	0
$628$	−4705.78	−0.299015
$629$	34429.4	2.18250
$630$	0	0
$631$	−1500.90	−0.0946907	−0.0473454	−	0.998879i	$-0.515076\pi$
$631$	−1500.90	−0.0946907	−0.0473454	+	0.998879i	$0.515076\pi$
$632$	744.814	0.0468783
$633$	0	0
$634$	−2641.38	−0.165462
$635$	−7257.64	−0.453560
$636$	0	0
$637$	−29778.0	−1.85219
$638$	−16445.8	−1.02052
$639$	0	0
$640$	8305.86	0.512997
$641$	−26926.8	−1.65920	−0.829599	−	0.558359i	$-0.811431\pi$
$641$	−26926.8	−1.65920	−0.829599	+	0.558359i	$0.811431\pi$
$642$	0	0
$643$	9023.36	0.553416	0.276708	−	0.960954i	$-0.410757\pi$
$643$	9023.36	0.553416	0.276708	+	0.960954i	$0.410757\pi$
$644$	30.3484	0.00185698
$645$	0	0
$646$	−5543.34	−0.337616
$647$	−1532.87	−0.0931427	−0.0465714	−	0.998915i	$-0.514829\pi$
$647$	−1532.87	−0.0931427	−0.0465714	+	0.998915i	$0.514829\pi$
$648$	0	0
$649$	39540.2	2.39151
$650$	10674.3	0.644126
$651$	0	0
$652$	6474.71	0.388910
$653$	−2699.48	−0.161774	−0.0808872	−	0.996723i	$-0.525775\pi$
$653$	−2699.48	−0.161774	−0.0808872	+	0.996723i	$0.525775\pi$
$654$	0	0
$655$	3938.52	0.234948
$656$	−6875.72	−0.409226
$657$	0	0
$658$	137.568	0.00815042
$659$	12669.5	0.748911	0.374456	−	0.927245i	$-0.377830\pi$
$659$	12669.5	0.748911	0.374456	+	0.927245i	$0.377830\pi$
$660$	0	0
$661$	645.043	0.0379565	0.0189783	−	0.999820i	$-0.493959\pi$
$661$	645.043	0.0379565	0.0189783	+	0.999820i	$0.493959\pi$
$662$	16147.7	0.948033
$663$	0	0
$664$	23024.0	1.34564
$665$	56.0899	0.00327079
$666$	0	0
$667$	−5100.60	−0.296096
$668$	−13155.7	−0.761989
$669$	0	0
$670$	−4807.21	−0.277192
$671$	3973.21	0.228590
$672$	0	0
$673$	−10045.7	−0.575384	−0.287692	−	0.957723i	$-0.592888\pi$
$673$	−10045.7	−0.575384	−0.287692	+	0.957723i	$0.592888\pi$
$674$	5023.71	0.287101
$675$	0	0
$676$	−32068.6	−1.82457
$677$	−9757.06	−0.553906	−0.276953	−	0.960883i	$-0.589325\pi$
$677$	−9757.06	−0.553906	−0.276953	+	0.960883i	$0.589325\pi$
$678$	0	0
$679$	53.8446	0.00304325
$680$	−11562.1	−0.652038
$681$	0	0
$682$	−6421.69	−0.360556
$683$	−7439.12	−0.416764	−0.208382	−	0.978047i	$-0.566820\pi$
$683$	−7439.12	−0.416764	−0.208382	+	0.978047i	$0.566820\pi$
$684$	0	0
$685$	1741.95	0.0971629
$686$	−213.081	−0.0118593
$687$	0	0
$688$	9110.13	0.504826
$689$	−39606.2	−2.18995
$690$	0	0
$691$	26441.7	1.45570	0.727850	−	0.685736i	$-0.240520\pi$
$691$	26441.7	1.45570	0.727850	+	0.685736i	$0.240520\pi$
$692$	19535.2	1.07314
$693$	0	0
$694$	439.838	0.0240576
$695$	−6257.30	−0.341515
$696$	0	0
$697$	32487.5	1.76550
$698$	−9323.29	−0.505576
$699$	0	0
$700$	−114.785	−0.00619779
$701$	−5020.78	−0.270517	−0.135258	−	0.990810i	$-0.543186\pi$
$701$	−5020.78	−0.270517	−0.135258	+	0.990810i	$0.543186\pi$
$702$	0	0
$703$	15036.6	0.806707
$704$	−5421.85	−0.290261
$705$	0	0
$706$	−2099.86	−0.111940
$707$	4.41228	0.000234711 0
$708$	0	0
$709$	−31105.5	−1.64766	−0.823830	−	0.566837i	$-0.808167\pi$
$709$	−31105.5	−1.64766	−0.823830	+	0.566837i	$0.808167\pi$
$710$	4923.86	0.260266
$711$	0	0
$712$	20098.4	1.05789
$713$	−1991.66	−0.104612
$714$	0	0
$715$	−28090.8	−1.46928
$716$	11820.6	0.616976
$717$	0	0
$718$	−8274.61	−0.430092
$719$	23922.6	1.24084	0.620419	−	0.784270i	$-0.286963\pi$
$719$	23922.6	1.24084	0.620419	+	0.784270i	$0.286963\pi$
$720$	0	0
$721$	−242.040	−0.0125021
$722$	7272.14	0.374849
$723$	0	0
$724$	344.919	0.0177055
$725$	19291.6	0.988238
$726$	0	0
$727$	−2445.05	−0.124735	−0.0623673	−	0.998053i	$-0.519865\pi$
$727$	−2445.05	−0.124735	−0.0623673	+	0.998053i	$0.519865\pi$
$728$	−377.686	−0.0192280
$729$	0	0
$730$	−4992.70	−0.253134
$731$	−43045.0	−2.17794
$732$	0	0
$733$	22351.6	1.12630	0.563148	−	0.826356i	$-0.309590\pi$
$733$	22351.6	1.12630	0.563148	+	0.826356i	$0.309590\pi$
$734$	−3410.76	−0.171517
$735$	0	0
$736$	−4293.08	−0.215007
$737$	−28953.9	−1.44712
$738$	0	0
$739$	−26118.7	−1.30013	−0.650063	−	0.759880i	$-0.725257\pi$
$739$	−26118.7	−1.30013	−0.650063	+	0.759880i	$0.725257\pi$
$740$	13444.9	0.667896
$741$	0	0
$742$	−141.694	−0.00701045
$743$	−22855.8	−1.12853	−0.564264	−	0.825594i	$-0.690840\pi$
$743$	−22855.8	−1.12853	−0.564264	+	0.825594i	$0.690840\pi$
$744$	0	0
$745$	−14573.1	−0.716666
$746$	−12926.0	−0.634388
$747$	0	0
$748$	−29853.3	−1.45929
$749$	3.78730	0.000184760 0
$750$	0	0
$751$	−21847.5	−1.06155	−0.530776	−	0.847512i	$-0.678099\pi$
$751$	−21847.5	−1.06155	−0.530776	+	0.847512i	$0.678099\pi$
$752$	8882.69	0.430743
$753$	0	0
$754$	27211.9	1.31432
$755$	−16775.5	−0.808641
$756$	0	0
$757$	−24521.8	−1.17736	−0.588679	−	0.808367i	$-0.700352\pi$
$757$	−24521.8	−1.17736	−0.588679	+	0.808367i	$0.700352\pi$
$758$	16549.2	0.792998
$759$	0	0
$760$	−5049.58	−0.241010
$761$	−11368.0	−0.541513	−0.270756	−	0.962648i	$-0.587274\pi$
$761$	−11368.0	−0.541513	−0.270756	+	0.962648i	$0.587274\pi$
$762$	0	0
$763$	160.243	0.00760311
$764$	−4564.06	−0.216128
$765$	0	0
$766$	16632.1	0.784519
$767$	−65424.8	−3.07999
$768$	0	0
$769$	24734.7	1.15989	0.579945	−	0.814655i	$-0.303074\pi$
$769$	24734.7	1.15989	0.579945	+	0.814655i	$0.303074\pi$
$770$	−100.497	−0.00470344
$771$	0	0
$772$	17837.8	0.831604
$773$	11383.5	0.529674	0.264837	−	0.964293i	$-0.414682\pi$
$773$	11383.5	0.529674	0.264837	+	0.964293i	$0.414682\pi$
$774$	0	0
$775$	7532.93	0.349149
$776$	−4847.45	−0.224244
$777$	0	0
$778$	7179.05	0.330824
$779$	14188.5	0.652573
$780$	0	0
$781$	29656.5	1.35876
$782$	3080.39	0.140862
$783$	0	0
$784$	−6878.76	−0.313355
$785$	4832.97	0.219740
$786$	0	0
$787$	6825.16	0.309137	0.154568	−	0.987982i	$-0.450601\pi$
$787$	6825.16	0.309137	0.154568	+	0.987982i	$0.450601\pi$
$788$	−2980.81	−0.134755
$789$	0	0
$790$	−327.923	−0.0147683
$791$	−21.8343	−0.000981463 0
$792$	0	0
$793$	−6574.24	−0.294399
$794$	−6186.83	−0.276527
$795$	0	0
$796$	−9765.52	−0.434836
$797$	−29398.7	−1.30659	−0.653297	−	0.757101i	$-0.726615\pi$
$797$	−29398.7	−1.30659	−0.653297	+	0.757101i	$0.726615\pi$
$798$	0	0
$799$	−41970.3	−1.85833
$800$	16237.4	0.717599
$801$	0	0
$802$	21142.8	0.930897
$803$	−30071.1	−1.32153
$804$	0	0
$805$	−31.1687	−0.00136466
$806$	10625.6	0.464356
$807$	0	0
$808$	−397.223	−0.0172949
$809$	29739.6	1.29245	0.646223	−	0.763149i	$-0.276348\pi$
$809$	29739.6	1.29245	0.646223	+	0.763149i	$0.276348\pi$
$810$	0	0
$811$	−31225.8	−1.35202	−0.676010	−	0.736893i	$-0.736292\pi$
$811$	−31225.8	−1.35202	−0.676010	+	0.736893i	$0.736292\pi$
$812$	−292.618	−0.0126464
$813$	0	0
$814$	−26941.2	−1.16006
$815$	−6649.71	−0.285803
$816$	0	0
$817$	−18799.3	−0.805023
$818$	−5986.65	−0.255890
$819$	0	0
$820$	12686.5	0.540284
$821$	−29267.2	−1.24413	−0.622066	−	0.782965i	$-0.713706\pi$
$821$	−29267.2	−1.24413	−0.622066	+	0.782965i	$0.713706\pi$
$822$	0	0
$823$	−2408.64	−0.102017	−0.0510084	−	0.998698i	$-0.516244\pi$
$823$	−2408.64	−0.102017	−0.0510084	+	0.998698i	$0.516244\pi$
$824$	21790.0	0.921227
$825$	0	0
$826$	−234.062	−0.00985964
$827$	21947.4	0.922837	0.461418	−	0.887183i	$-0.347341\pi$
$827$	21947.4	0.922837	0.461418	+	0.887183i	$0.347341\pi$
$828$	0	0
$829$	−3831.73	−0.160533	−0.0802663	−	0.996773i	$-0.525577\pi$
$829$	−3831.73	−0.160533	−0.0802663	+	0.996773i	$0.525577\pi$
$830$	−10136.9	−0.423924
$831$	0	0
$832$	8971.23	0.373824
$833$	32501.8	1.35189
$834$	0	0
$835$	13511.3	0.559972
$836$	−13038.0	−0.539390
$837$	0	0
$838$	−2116.40	−0.0872432
$839$	−2232.98	−0.0918843	−0.0459421	−	0.998944i	$-0.514629\pi$
$839$	−2232.98	−0.0918843	−0.0459421	+	0.998944i	$0.514629\pi$
$840$	0	0
$841$	24790.8	1.01647
$842$	−3483.32	−0.142569
$843$	0	0
$844$	−11149.2	−0.454706
$845$	32935.4	1.34084
$846$	0	0
$847$	−312.726	−0.0126864
$848$	−9149.08	−0.370496
$849$	0	0
$850$	−11650.7	−0.470137
$851$	−8355.70	−0.336580
$852$	0	0
$853$	−7431.74	−0.298309	−0.149155	−	0.988814i	$-0.547655\pi$
$853$	−7431.74	−0.298309	−0.149155	+	0.988814i	$0.547655\pi$
$854$	−23.5198	−0.000942426 0
$855$	0	0
$856$	−340.958	−0.0136141
$857$	−28373.0	−1.13092	−0.565462	−	0.824774i	$-0.691302\pi$
$857$	−28373.0	−1.13092	−0.565462	+	0.824774i	$0.691302\pi$
$858$	0	0
$859$	24814.5	0.985633	0.492816	−	0.870133i	$-0.335967\pi$
$859$	24814.5	0.985633	0.492816	+	0.870133i	$0.335967\pi$
$860$	−16809.3	−0.666502
$861$	0	0
$862$	7967.00	0.314800
$863$	5427.03	0.214065	0.107033	−	0.994256i	$-0.465865\pi$
$863$	5427.03	0.214065	0.107033	+	0.994256i	$0.465865\pi$
$864$	0	0
$865$	−20063.2	−0.788634
$866$	−20098.6	−0.788658
$867$	0	0
$868$	−114.261	−0.00446804
$869$	−1975.09	−0.0771005
$870$	0	0
$871$	47908.3	1.86373
$872$	−14426.1	−0.560240
$873$	0	0
$874$	1345.32	0.0520663
$875$	287.282	0.0110993
$876$	0	0
$877$	19222.6	0.740139	0.370069	−	0.929004i	$-0.379334\pi$
$877$	19222.6	0.740139	0.370069	+	0.929004i	$0.379334\pi$
$878$	24258.8	0.932453
$879$	0	0
$880$	−6489.00	−0.248573
$881$	−28542.6	−1.09152	−0.545758	−	0.837943i	$-0.683758\pi$
$881$	−28542.6	−1.09152	−0.545758	+	0.837943i	$0.683758\pi$
$882$	0	0
$883$	−8556.51	−0.326104	−0.163052	−	0.986617i	$-0.552134\pi$
$883$	−8556.51	−0.326104	−0.163052	+	0.986617i	$0.552134\pi$
$884$	49396.6	1.87940
$885$	0	0
$886$	−6472.61	−0.245431
$887$	5901.02	0.223379	0.111689	−	0.993743i	$-0.464374\pi$
$887$	5901.02	0.223379	0.111689	+	0.993743i	$0.464374\pi$
$888$	0	0
$889$	258.763	0.00976224
$890$	−8848.83	−0.333274
$891$	0	0
$892$	19236.4	0.722065
$893$	−18330.0	−0.686885
$894$	0	0
$895$	−12140.1	−0.453405
$896$	−296.136	−0.0110415
$897$	0	0
$898$	−2177.62	−0.0809220
$899$	19203.5	0.712430
$900$	0	0
$901$	43229.0	1.59841
$902$	−25421.6	−0.938410
$903$	0	0
$904$	1965.67	0.0723198
$905$	−354.241	−0.0130115
$906$	0	0
$907$	25144.5	0.920519	0.460259	−	0.887784i	$-0.347756\pi$
$907$	25144.5	0.920519	0.460259	+	0.887784i	$0.347756\pi$
$908$	−32429.6	−1.18526
$909$	0	0
$910$	166.286	0.00605751
$911$	47767.5	1.73722	0.868609	−	0.495497i	$-0.165014\pi$
$911$	47767.5	1.73722	0.868609	+	0.495497i	$0.165014\pi$
$912$	0	0
$913$	−61054.8	−2.21316
$914$	−10082.8	−0.364889
$915$	0	0
$916$	−7458.34	−0.269029
$917$	−140.424	−0.00505692
$918$	0	0
$919$	−42390.6	−1.52159	−0.760793	−	0.648995i	$-0.775190\pi$
$919$	−42390.6	−1.52159	−0.760793	+	0.648995i	$0.775190\pi$
$920$	2806.01	0.100556
$921$	0	0
$922$	−5257.56	−0.187797
$923$	−49070.9	−1.74993
$924$	0	0
$925$	31603.1	1.12336
$926$	−7930.93	−0.281454
$927$	0	0
$928$	41393.7	1.46424
$929$	−14954.8	−0.528151	−0.264075	−	0.964502i	$-0.585067\pi$
$929$	−14954.8	−0.528151	−0.264075	+	0.964502i	$0.585067\pi$
$930$	0	0
$931$	14194.7	0.499692
$932$	29185.7	1.02576
$933$	0	0
$934$	131.726	0.00461477
$935$	30660.2	1.07240
$936$	0	0
$937$	3106.76	0.108318	0.0541588	−	0.998532i	$-0.482752\pi$
$937$	3106.76	0.108318	0.0541588	+	0.998532i	$0.482752\pi$
$938$	171.396	0.00596617
$939$	0	0
$940$	−16389.6	−0.568692
$941$	4533.73	0.157062	0.0785311	−	0.996912i	$-0.474977\pi$
$941$	4533.73	0.157062	0.0785311	+	0.996912i	$0.474977\pi$
$942$	0	0
$943$	−7884.41	−0.272271
$944$	−15113.2	−0.521073
$945$	0	0
$946$	33682.9	1.15764
$947$	57160.0	1.96141	0.980703	−	0.195505i	$-0.0626347\pi$
$947$	57160.0	1.96141	0.980703	+	0.195505i	$0.0626347\pi$
$948$	0	0
$949$	49756.9	1.70198
$950$	−5088.29	−0.173775
$951$	0	0
$952$	412.234	0.0140342
$953$	13484.3	0.458340	0.229170	−	0.973386i	$-0.426399\pi$
$953$	13484.3	0.458340	0.229170	+	0.973386i	$0.426399\pi$
$954$	0	0
$955$	4687.42	0.158829
$956$	−14232.7	−0.481504
$957$	0	0
$958$	−161.207	−0.00543672
$959$	−62.1074	−0.00209130
$960$	0	0
$961$	−22292.5	−0.748295
$962$	44578.0	1.49402
$963$	0	0
$964$	−32938.4	−1.10049
$965$	−18320.0	−0.611131
$966$	0	0
$967$	−3323.25	−0.110516	−0.0552578	−	0.998472i	$-0.517598\pi$
$967$	−3323.25	−0.110516	−0.0552578	+	0.998472i	$0.517598\pi$
$968$	28153.7	0.934807
$969$	0	0
$970$	2134.21	0.0706449
$971$	38715.8	1.27956	0.639778	−	0.768560i	$-0.279026\pi$
$971$	38715.8	1.27956	0.639778	+	0.768560i	$0.279026\pi$
$972$	0	0
$973$	223.097	0.00735063
$974$	14831.6	0.487920
$975$	0	0
$976$	−1518.66	−0.0498064
$977$	18615.4	0.609579	0.304789	−	0.952420i	$-0.401414\pi$
$977$	18615.4	0.609579	0.304789	+	0.952420i	$0.401414\pi$
$978$	0	0
$979$	−53296.7	−1.73991
$980$	12692.1	0.413709
$981$	0	0
$982$	−12700.4	−0.412716
$983$	−26659.7	−0.865017	−0.432508	−	0.901630i	$-0.642371\pi$
$983$	−26659.7	−0.865017	−0.432508	+	0.901630i	$0.642371\pi$
$984$	0	0
$985$	3061.37	0.0990289
$986$	−29701.0	−0.959302
$987$	0	0
$988$	21573.3	0.694673
$989$	10446.6	0.335878
$990$	0	0
$991$	−53383.3	−1.71118	−0.855589	−	0.517656i	$-0.826805\pi$
$991$	−53383.3	−1.71118	−0.855589	+	0.517656i	$0.826805\pi$
$992$	16163.3	0.517323
$993$	0	0
$994$	−175.555	−0.00560187
$995$	10029.5	0.319553
$996$	0	0
$997$	27221.4	0.864705	0.432353	−	0.901705i	$-0.357684\pi$
$997$	27221.4	0.864705	0.432353	+	0.901705i	$0.357684\pi$
$998$	12068.3	0.382782
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.a.g.1.3	✓	5
3.2	odd	2		207.4.a.h.1.3	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.4.a.g.1.3	✓	5	1.1	even	1	trivial
207.4.a.h.1.3	yes	5	3.2	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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41320 q^{2} -6.00288 q^{4} +6.16513 q^{5} -0.219811 q^{7} +19.7888 q^{8} -8.71253 q^{10} -52.4758 q^{11} +86.8286 q^{13} +0.310636 q^{14} +20.0575 q^{16} -94.7709 q^{17} -41.3899 q^{19} -37.0085 q^{20} +74.1586 q^{22} +23.0000 q^{23} -86.9912 q^{25} -122.706 q^{26} +1.31950 q^{28} -221.765 q^{29} -86.5941 q^{31} -186.656 q^{32} +133.930 q^{34} -1.35516 q^{35} -363.291 q^{37} +58.4920 q^{38} +122.001 q^{40} -342.800 q^{41} +454.201 q^{43} +315.005 q^{44} -32.5035 q^{46} +442.861 q^{47} -342.952 q^{49} +122.936 q^{50} -521.221 q^{52} -456.142 q^{53} -323.520 q^{55} -4.34979 q^{56} +313.398 q^{58} -753.494 q^{59} -75.7151 q^{61} +122.374 q^{62} +103.321 q^{64} +535.309 q^{65} +551.758 q^{67} +568.898 q^{68} +1.91511 q^{70} -565.146 q^{71} +573.048 q^{73} +513.402 q^{74} +248.458 q^{76} +11.5347 q^{77} +37.6381 q^{79} +123.657 q^{80} +484.444 q^{82} +1163.49 q^{83} -584.274 q^{85} -641.875 q^{86} -1038.43 q^{88} +1015.64 q^{89} -19.0859 q^{91} -138.066 q^{92} -625.850 q^{94} -255.174 q^{95} -244.959 q^{97} +484.658 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{2} + 16 q^{4} - 20 q^{5} - 10 q^{7} - 48 q^{8} + 50 q^{10} - 46 q^{11} + 54 q^{13} - 164 q^{14} - 60 q^{16} - 250 q^{17} - 28 q^{19} - 242 q^{20} - 10 q^{22} + 115 q^{23} + 239 q^{25} - 368 q^{26}+ \cdots + 2400 q^{98}+O(q^{100}) \) 5 * q - 4 * q^2 + 16 * q^4 - 20 * q^5 - 10 * q^7 - 48 * q^8 + 50 * q^10 - 46 * q^11 + 54 * q^13 - 164 * q^14 - 60 * q^16 - 250 * q^17 - 28 * q^19 - 242 * q^20 - 10 * q^22 + 115 * q^23 + 239 * q^25 - 368 * q^26 + 224 * q^28 - 460 * q^29 - 360 * q^31 - 268 * q^32 - 20 * q^34 - 964 * q^35 - 92 * q^37 - 266 * q^38 - 278 * q^40 - 788 * q^41 + 108 * q^43 - 354 * q^44 - 92 * q^46 - 724 * q^47 + 149 * q^49 - 128 * q^50 - 288 * q^52 - 580 * q^53 - 672 * q^55 + 1172 * q^56 - 48 * q^58 + 408 * q^59 - 620 * q^61 + 1088 * q^62 + 696 * q^64 - 184 * q^65 + 100 * q^67 + 1584 * q^68 + 728 * q^70 + 124 * q^71 - 1946 * q^73 + 2866 * q^74 + 1362 * q^76 + 396 * q^77 - 198 * q^79 + 2398 * q^80 + 76 * q^82 + 578 * q^83 + 376 * q^85 + 3910 * q^86 + 1366 * q^88 - 930 * q^89 + 1972 * q^91 + 368 * q^92 + 976 * q^94 + 908 * q^95 + 2618 * q^97 + 2400 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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