LMFDB - Embedded newform 207.4.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,4,Mod(1,207)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	207.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:	$12.2133953712$
Analytic rank:	$0$
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	$-0.419628\pi$
$2$	1.41320	0.499640	0.249820	+	0.968292i	$0.419628\pi$
$3$	0	0
$3$	0	0
$4$	−6.00288	−0.750359
$5$	−6.16513	−0.551426	−0.275713	−	0.961240i	$-0.588914\pi$
$5$	−6.16513	−0.551426	−0.275713	+	0.961240i	$0.588914\pi$
$6$	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	$-0.244516\pi$
$11$	52.4758	1.43837	0.719184	+	0.694820i	$0.244516\pi$
$12$	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	$-0.263695\pi$
$17$	94.7709	1.35208	0.676038	+	0.736866i	$0.263695\pi$
$18$	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	$-0.248689\pi$
$29$	221.765	1.42003	0.710013	+	0.704189i	$0.248689\pi$
$30$	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	$-0.273559\pi$
$41$	342.800	1.30577	0.652883	+	0.757459i	$0.273559\pi$
$42$	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	$-0.741166\pi$
$47$	−442.861	−1.37442	−0.687212	+	0.726457i	$0.741166\pi$
$48$	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	$-0.298696\pi$
$53$	456.142	1.18219	0.591094	+	0.806603i	$0.298696\pi$
$54$	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	$-0.187582\pi$
$59$	753.494	1.66265	0.831327	+	0.555784i	$0.187582\pi$
$60$	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	$-0.343414\pi$
$71$	565.146	0.944655	0.472328	+	0.881423i	$0.343414\pi$
$72$	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	$-0.779411\pi$
$83$	−1163.49	−1.53866	−0.769332	+	0.638849i	$0.779411\pi$
$84$	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	$-0.706755\pi$
$89$	−1015.64	−1.20964	−0.604820	+	0.796362i	$0.706755\pi$
$90$	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	$-0.496853\pi$
$101$	20.0731	0.0197757	0.00988786	+	0.999951i	$0.496853\pi$
$102$	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	$-0.497522\pi$
$107$	17.2298	0.0155670	0.00778351	+	0.999970i	$0.497522\pi$
$108$	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	$-0.513165\pi$
$113$	−99.3322	−0.0826937	−0.0413468	+	0.999145i	$0.513165\pi$
$114$	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	$-0.568335\pi$
$131$	−638.839	−0.426074	−0.213037	+	0.977044i	$0.568335\pi$
$132$	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	$-0.528080\pi$
$137$	−282.549	−0.176203	−0.0881016	+	0.996111i	$0.528080\pi$
$138$	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	$-0.274840\pi$
$149$	2363.79	1.29966	0.649830	+	0.760079i	$0.274840\pi$
$150$	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	$-0.669522\pi$
$167$	−2191.56	−1.01550	−0.507749	+	0.861505i	$0.669522\pi$
$168$	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	$-0.246387\pi$
$173$	3254.30	1.43017	0.715086	+	0.699036i	$0.246387\pi$
$174$	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	$-0.365138\pi$
$179$	1969.15	0.822241	0.411121	+	0.911581i	$0.365138\pi$
$180$	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	$-0.546002\pi$
$191$	−760.313	−0.288033	−0.144017	+	0.989575i	$0.546002\pi$
$192$	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	$-0.528621\pi$
$197$	−496.563	−0.179587	−0.0897935	+	0.995960i	$0.528621\pi$
$198$	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	$-0.789812\pi$
$227$	−5402.34	−1.57959	−0.789793	+	0.613374i	$0.789812\pi$
$228$	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	$-0.260451\pi$
$233$	4861.95	1.36703	0.683513	+	0.729939i	$0.260451\pi$
$234$	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	$-0.603968\pi$
$239$	−2370.98	−0.641698	−0.320849	+	0.947130i	$0.603968\pi$
$240$	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	$-0.376727\pi$
$251$	3003.63	0.755328	0.377664	+	0.925943i	$0.376727\pi$
$252$	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	$-0.487255\pi$
$257$	329.844	0.0800587	0.0400293	+	0.999199i	$0.487255\pi$
$258$	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	$-0.680047\pi$
$263$	−4571.82	−1.07190	−0.535951	+	0.844249i	$0.680047\pi$
$264$	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	$-0.797350\pi$
$269$	−7095.23	−1.60819	−0.804096	+	0.594500i	$0.797350\pi$
$270$	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	$-0.563250\pi$
$281$	−1859.67	−0.394800	−0.197400	+	0.980323i	$0.563250\pi$
$282$	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	$-0.857554\pi$
$293$	−9042.97	−1.80306	−0.901529	+	0.432720i	$0.857554\pi$
$294$	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	$-0.638618\pi$
$311$	−4627.28	−0.843694	−0.421847	+	0.906667i	$0.638618\pi$
$312$	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	$-0.552950\pi$
$317$	−1869.08	−0.331161	−0.165581	+	0.986196i	$0.552950\pi$
$318$	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	$-0.492336\pi$
$347$	311.236	0.0481499	0.0240750	+	0.999710i	$0.492336\pi$
$348$	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	$-0.535732\pi$
$353$	−1485.90	−0.224041	−0.112020	+	0.993706i	$0.535732\pi$
$354$	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	$-0.641628\pi$
$359$	−5855.25	−0.860802	−0.430401	+	0.902638i	$0.641628\pi$
$360$	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	$-0.212620\pi$
$383$	11769.1	1.57017	0.785084	+	0.619389i	$0.212620\pi$
$384$	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	$-0.392593\pi$
$389$	5080.01	0.662125	0.331063	+	0.943609i	$0.392593\pi$
$390$	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	$-0.118442\pi$
$401$	14961.0	1.86313	0.931567	+	0.363569i	$0.118442\pi$
$402$	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	$-0.527826\pi$
$419$	−1497.60	−0.174612	−0.0873059	+	0.996182i	$0.527826\pi$
$420$	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	$-0.397987\pi$
$431$	5637.58	0.630052	0.315026	+	0.949083i	$0.397987\pi$
$432$	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	$-0.578987\pi$
$443$	−4580.12	−0.491214	−0.245607	+	0.969369i	$0.578987\pi$
$444$	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	$-0.525805\pi$
$449$	−1540.91	−0.161960	−0.0809802	+	0.996716i	$0.525805\pi$
$450$	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	$-0.560178\pi$
$461$	−3720.33	−0.375864	−0.187932	+	0.982182i	$0.560178\pi$
$462$	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	$-0.498530\pi$
$467$	93.2112	0.00923619	0.00461810	+	0.999989i	$0.498530\pi$
$468$	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	$-0.501732\pi$
$479$	−114.073	−0.0108813	−0.00544063	+	0.999985i	$0.501732\pi$
$480$	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	$-0.635523\pi$
$491$	−8987.02	−0.826025	−0.413013	+	0.910725i	$0.635523\pi$
$492$	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	$-0.430916\pi$
$503$	4858.42	0.430668	0.215334	+	0.976540i	$0.430916\pi$
$504$	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	$-0.626770\pi$
$509$	−8906.97	−0.775628	−0.387814	+	0.921738i	$0.626770\pi$
$510$	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	$-0.581490\pi$
$521$	−6022.66	−0.506444	−0.253222	+	0.967408i	$0.581490\pi$
$522$	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	$-0.295437\pi$
$557$	15757.0	1.19864	0.599321	+	0.800508i	$0.295437\pi$
$558$	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	$-0.442144\pi$
$563$	4829.45	0.361522	0.180761	+	0.983527i	$0.442144\pi$
$564$	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	$-0.505007\pi$
$569$	−427.019	−0.0314615	−0.0157307	+	0.999876i	$0.505007\pi$
$570$	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	$-0.327715\pi$
$587$	14654.5	1.03042	0.515208	+	0.857065i	$0.327715\pi$
$588$	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	$-0.548523\pi$
$593$	−4385.56	−0.303698	−0.151849	+	0.988404i	$0.548523\pi$
$594$	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	$-0.761108\pi$
$599$	−21443.4	−1.46269	−0.731347	+	0.682006i	$0.761108\pi$
$600$	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	$-0.462542\pi$
$617$	3598.75	0.234814	0.117407	+	0.993084i	$0.462542\pi$
$618$	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	$-0.188569\pi$
$641$	26926.8	1.65920	0.829599	+	0.558359i	$0.188569\pi$
$642$	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	$-0.485171\pi$
$647$	1532.87	0.0931427	0.0465714	+	0.998915i	$0.485171\pi$
$648$	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	$-0.474225\pi$
$653$	2699.48	0.161774	0.0808872	+	0.996723i	$0.474225\pi$
$654$	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	$-0.622170\pi$
$659$	−12669.5	−0.748911	−0.374456	+	0.927245i	$0.622170\pi$
$660$	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	$-0.410675\pi$
$677$	9757.06	0.553906	0.276953	+	0.960883i	$0.410675\pi$
$678$	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	$-0.433180\pi$
$683$	7439.12	0.416764	0.208382	+	0.978047i	$0.433180\pi$
$684$	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	$-0.456814\pi$
$701$	5020.78	0.270517	0.135258	+	0.990810i	$0.456814\pi$
$702$	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	$-0.713037\pi$
$719$	−23922.6	−1.24084	−0.620419	+	0.784270i	$0.713037\pi$
$720$	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	$-0.309160\pi$
$743$	22855.8	1.12853	0.564264	+	0.825594i	$0.309160\pi$
$744$	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	$-0.412726\pi$
$761$	11368.0	0.541513	0.270756	+	0.962648i	$0.412726\pi$
$762$	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	$-0.585318\pi$
$773$	−11383.5	−0.529674	−0.264837	+	0.964293i	$0.585318\pi$
$774$	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	$-0.273385\pi$
$797$	29398.7	1.30659	0.653297	+	0.757101i	$0.273385\pi$
$798$	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	$-0.723652\pi$
$809$	−29739.6	−1.29245	−0.646223	+	0.763149i	$0.723652\pi$
$810$	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	$-0.286294\pi$
$821$	29267.2	1.24413	0.622066	+	0.782965i	$0.286294\pi$
$822$	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	$-0.652659\pi$
$827$	−21947.4	−0.922837	−0.461418	+	0.887183i	$0.652659\pi$
$828$	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	$-0.485371\pi$
$839$	2232.98	0.0918843	0.0459421	+	0.998944i	$0.485371\pi$
$840$	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	$-0.308698\pi$
$857$	28373.0	1.13092	0.565462	+	0.824774i	$0.308698\pi$
$858$	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	$-0.534135\pi$
$863$	−5427.03	−0.214065	−0.107033	+	0.994256i	$0.534135\pi$
$864$	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	$-0.316242\pi$
$881$	28542.6	1.09152	0.545758	+	0.837943i	$0.316242\pi$
$882$	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	$-0.535626\pi$
$887$	−5901.02	−0.223379	−0.111689	+	0.993743i	$0.535626\pi$
$888$	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	$-0.834986\pi$
$911$	−47767.5	−1.73722	−0.868609	+	0.495497i	$0.834986\pi$
$912$	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	$-0.414933\pi$
$929$	14954.8	0.528151	0.264075	+	0.964502i	$0.414933\pi$
$930$	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	$-0.525023\pi$
$941$	−4533.73	−0.157062	−0.0785311	+	0.996912i	$0.525023\pi$
$942$	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	$-0.937365\pi$
$947$	−57160.0	−1.96141	−0.980703	+	0.195505i	$0.937365\pi$
$948$	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	$-0.573601\pi$
$953$	−13484.3	−0.458340	−0.229170	+	0.973386i	$0.573601\pi$
$954$	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	$-0.720974\pi$
$971$	−38715.8	−1.27956	−0.639778	+	0.768560i	$0.720974\pi$
$972$	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	$-0.598586\pi$
$977$	−18615.4	−0.609579	−0.304789	+	0.952420i	$0.598586\pi$
$978$	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	$-0.357629\pi$
$983$	26659.7	0.865017	0.432508	+	0.901630i	$0.357629\pi$
$984$	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.h.1.3	yes	5
3.2	odd	2		207.4.a.g.1.3	✓	5

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

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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