LMFDB - Embedded newform 208.4.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,4,Mod(1,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, 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("208.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	208.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:	$12.2723972812$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	208.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.68466 q^{3} +0.561553 q^{5} -18.1771 q^{7} -13.4233 q^{9} +O(q^{10})$	$q+3.68466 q^{3} +0.561553 q^{5} -18.1771 q^{7} -13.4233 q^{9} -64.7386 q^{11} -13.0000 q^{13} +2.06913 q^{15} -25.5464 q^{17} +107.970 q^{19} -66.9763 q^{21} -73.2614 q^{23} -124.685 q^{25} -148.946 q^{27} +175.909 q^{29} +113.093 q^{31} -238.540 q^{33} -10.2074 q^{35} +114.808 q^{37} -47.9006 q^{39} -69.6458 q^{41} -438.302 q^{43} -7.53789 q^{45} +31.9479 q^{47} -12.5937 q^{49} -94.1298 q^{51} +2.84658 q^{53} -36.3542 q^{55} +397.831 q^{57} -71.6325 q^{59} -920.695 q^{61} +243.996 q^{63} -7.30019 q^{65} +444.280 q^{67} -269.943 q^{69} +541.719 q^{71} +764.004 q^{73} -459.420 q^{75} +1176.76 q^{77} +421.538 q^{79} -186.386 q^{81} -603.797 q^{83} -14.3457 q^{85} +648.165 q^{87} -1159.88 q^{89} +236.302 q^{91} +416.708 q^{93} +60.6307 q^{95} +583.269 q^{97} +869.006 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5 q^{3} - 3 q^{5} + 9 q^{7} + 35 q^{9}+O(q^{10}) $ 2 * q - 5 * q^3 - 3 * q^5 + 9 * q^7 + 35 * q^9	$ 2 q - 5 q^{3} - 3 q^{5} + 9 q^{7} + 35 q^{9} - 80 q^{11} - 26 q^{13} + 33 q^{15} + 19 q^{17} + 84 q^{19} - 303 q^{21} - 196 q^{23} - 237 q^{25} - 335 q^{27} - 44 q^{29} + 86 q^{31} - 106 q^{33} - 107 q^{35} + 209 q^{37} + 65 q^{39} - 230 q^{41} - 287 q^{43} - 180 q^{45} - 435 q^{47} + 383 q^{49} - 481 q^{51} - 118 q^{53} + 18 q^{55} + 606 q^{57} + 368 q^{59} - 1058 q^{61} + 1560 q^{63} + 39 q^{65} - 68 q^{67} + 796 q^{69} + 131 q^{71} + 456 q^{73} + 516 q^{75} + 762 q^{77} + 1008 q^{79} + 122 q^{81} - 1958 q^{83} - 173 q^{85} + 2558 q^{87} - 720 q^{89} - 117 q^{91} + 652 q^{93} + 146 q^{95} - 928 q^{97} + 130 q^{99}+O(q^{100}) $ 2 * q - 5 * q^3 - 3 * q^5 + 9 * q^7 + 35 * q^9 - 80 * q^11 - 26 * q^13 + 33 * q^15 + 19 * q^17 + 84 * q^19 - 303 * q^21 - 196 * q^23 - 237 * q^25 - 335 * q^27 - 44 * q^29 + 86 * q^31 - 106 * q^33 - 107 * q^35 + 209 * q^37 + 65 * q^39 - 230 * q^41 - 287 * q^43 - 180 * q^45 - 435 * q^47 + 383 * q^49 - 481 * q^51 - 118 * q^53 + 18 * q^55 + 606 * q^57 + 368 * q^59 - 1058 * q^61 + 1560 * q^63 + 39 * q^65 - 68 * q^67 + 796 * q^69 + 131 * q^71 + 456 * q^73 + 516 * q^75 + 762 * q^77 + 1008 * q^79 + 122 * q^81 - 1958 * q^83 - 173 * q^85 + 2558 * q^87 - 720 * q^89 - 117 * q^91 + 652 * q^93 + 146 * q^95 - 928 * q^97 + 130 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.68466	0.709113	0.354556	−	0.935035i	$-0.384632\pi$
$3$	3.68466	0.709113	0.354556	+	0.935035i	$0.384632\pi$
$4$	0	0
$5$	0.561553	0.0502268	0.0251134	−	0.999685i	$-0.492005\pi$
$5$	0.561553	0.0502268	0.0251134	+	0.999685i	$0.492005\pi$
$6$	0	0
$7$	−18.1771	−0.981470	−0.490735	−	0.871309i	$-0.663272\pi$
$7$	−18.1771	−0.981470	−0.490735	+	0.871309i	$0.663272\pi$
$8$	0	0
$9$	−13.4233	−0.497159
$10$	0	0
$11$	−64.7386	−1.77449	−0.887247	−	0.461295i	$-0.847385\pi$
$11$	−64.7386	−1.77449	−0.887247	+	0.461295i	$0.847385\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	2.06913	0.0356165
$16$	0	0
$17$	−25.5464	−0.364465	−0.182233	−	0.983255i	$-0.558332\pi$
$17$	−25.5464	−0.364465	−0.182233	+	0.983255i	$0.558332\pi$
$18$	0	0
$19$	107.970	1.30368	0.651841	−	0.758356i	$-0.273997\pi$
$19$	107.970	1.30368	0.651841	+	0.758356i	$0.273997\pi$
$20$	0	0
$21$	−66.9763	−0.695973
$22$	0	0
$23$	−73.2614	−0.664176	−0.332088	−	0.943248i	$-0.607753\pi$
$23$	−73.2614	−0.664176	−0.332088	+	0.943248i	$0.607753\pi$
$24$	0	0
$25$	−124.685	−0.997477
$26$	0	0
$27$	−148.946	−1.06165
$28$	0	0
$29$	175.909	1.12640	0.563198	−	0.826322i	$-0.309571\pi$
$29$	175.909	1.12640	0.563198	+	0.826322i	$0.309571\pi$
$30$	0	0
$31$	113.093	0.655228	0.327614	−	0.944812i	$-0.393755\pi$
$31$	113.093	0.655228	0.327614	+	0.944812i	$0.393755\pi$
$32$	0	0
$33$	−238.540	−1.25832
$34$	0	0
$35$	−10.2074	−0.0492961
$36$	0	0
$37$	114.808	0.510116	0.255058	−	0.966926i	$-0.417905\pi$
$37$	114.808	0.510116	0.255058	+	0.966926i	$0.417905\pi$
$38$	0	0
$39$	−47.9006	−0.196673
$40$	0	0
$41$	−69.6458	−0.265289	−0.132645	−	0.991164i	$-0.542347\pi$
$41$	−69.6458	−0.265289	−0.132645	+	0.991164i	$0.542347\pi$
$42$	0	0
$43$	−438.302	−1.55443	−0.777214	−	0.629236i	$-0.783368\pi$
$43$	−438.302	−1.55443	−0.777214	+	0.629236i	$0.783368\pi$
$44$	0	0
$45$	−7.53789	−0.0249707
$46$	0	0
$47$	31.9479	0.0991506	0.0495753	−	0.998770i	$-0.484213\pi$
$47$	31.9479	0.0991506	0.0495753	+	0.998770i	$0.484213\pi$
$48$	0	0
$49$	−12.5937	−0.0367164
$50$	0	0
$51$	−94.1298	−0.258447
$52$	0	0
$53$	2.84658	0.00737752	0.00368876	−	0.999993i	$-0.498826\pi$
$53$	2.84658	0.00737752	0.00368876	+	0.999993i	$0.498826\pi$
$54$	0	0
$55$	−36.3542	−0.0891272
$56$	0	0
$57$	397.831	0.924457
$58$	0	0
$59$	−71.6325	−0.158064	−0.0790319	−	0.996872i	$-0.525183\pi$
$59$	−71.6325	−0.158064	−0.0790319	+	0.996872i	$0.525183\pi$
$60$	0	0
$61$	−920.695	−1.93251	−0.966253	−	0.257593i	$-0.917071\pi$
$61$	−920.695	−1.93251	−0.966253	+	0.257593i	$0.917071\pi$
$62$	0	0
$63$	243.996	0.487947
$64$	0	0
$65$	−7.30019	−0.0139304
$66$	0	0
$67$	444.280	0.810112	0.405056	−	0.914292i	$-0.367252\pi$
$67$	444.280	0.810112	0.405056	+	0.914292i	$0.367252\pi$
$68$	0	0
$69$	−269.943	−0.470976
$70$	0	0
$71$	541.719	0.905496	0.452748	−	0.891639i	$-0.350444\pi$
$71$	541.719	0.905496	0.452748	+	0.891639i	$0.350444\pi$
$72$	0	0
$73$	764.004	1.22493	0.612465	−	0.790498i	$-0.290178\pi$
$73$	764.004	1.22493	0.612465	+	0.790498i	$0.290178\pi$
$74$	0	0
$75$	−459.420	−0.707324
$76$	0	0
$77$	1176.76	1.74161
$78$	0	0
$79$	421.538	0.600338	0.300169	−	0.953886i	$-0.402957\pi$
$79$	421.538	0.600338	0.300169	+	0.953886i	$0.402957\pi$
$80$	0	0
$81$	−186.386	−0.255674
$82$	0	0
$83$	−603.797	−0.798498	−0.399249	−	0.916842i	$-0.630729\pi$
$83$	−603.797	−0.798498	−0.399249	+	0.916842i	$0.630729\pi$
$84$	0	0
$85$	−14.3457	−0.0183059
$86$	0	0
$87$	648.165	0.798742
$88$	0	0
$89$	−1159.88	−1.38143	−0.690715	−	0.723127i	$-0.742704\pi$
$89$	−1159.88	−1.38143	−0.690715	+	0.723127i	$0.742704\pi$
$90$	0	0
$91$	236.302	0.272211
$92$	0	0
$93$	416.708	0.464631
$94$	0	0
$95$	60.6307	0.0654798
$96$	0	0
$97$	583.269	0.610536	0.305268	−	0.952267i	$-0.401254\pi$
$97$	583.269	0.610536	0.305268	+	0.952267i	$0.401254\pi$
$98$	0	0
$99$	869.006	0.882206
$100$	0	0
$101$	921.740	0.908085	0.454043	−	0.890980i	$-0.349981\pi$
$101$	921.740	0.908085	0.454043	+	0.890980i	$0.349981\pi$
$102$	0	0
$103$	930.712	0.890347	0.445174	−	0.895444i	$-0.353142\pi$
$103$	930.712	0.890347	0.445174	+	0.895444i	$0.353142\pi$
$104$	0	0
$105$	−37.6107	−0.0349565
$106$	0	0
$107$	−857.383	−0.774638	−0.387319	−	0.921946i	$-0.626599\pi$
$107$	−857.383	−0.774638	−0.387319	+	0.921946i	$0.626599\pi$
$108$	0	0
$109$	671.853	0.590384	0.295192	−	0.955438i	$-0.404616\pi$
$109$	671.853	0.590384	0.295192	+	0.955438i	$0.404616\pi$
$110$	0	0
$111$	423.027	0.361730
$112$	0	0
$113$	641.474	0.534024	0.267012	−	0.963693i	$-0.413964\pi$
$113$	641.474	0.534024	0.267012	+	0.963693i	$0.413964\pi$
$114$	0	0
$115$	−41.1401	−0.0333594
$116$	0	0
$117$	174.503	0.137887
$118$	0	0
$119$	464.359	0.357712
$120$	0	0
$121$	2860.09	2.14883
$122$	0	0
$123$	−256.621	−0.188120
$124$	0	0
$125$	−140.211	−0.100327
$126$	0	0
$127$	553.174	0.386506	0.193253	−	0.981149i	$-0.438096\pi$
$127$	553.174	0.386506	0.193253	+	0.981149i	$0.438096\pi$
$128$	0	0
$129$	−1614.99	−1.10227
$130$	0	0
$131$	−2056.40	−1.37152	−0.685758	−	0.727830i	$-0.740529\pi$
$131$	−2056.40	−1.37152	−0.685758	+	0.727830i	$0.740529\pi$
$132$	0	0
$133$	−1962.57	−1.27952
$134$	0	0
$135$	−83.6411	−0.0533235
$136$	0	0
$137$	−1808.57	−1.12786	−0.563928	−	0.825824i	$-0.690710\pi$
$137$	−1808.57	−1.12786	−0.563928	+	0.825824i	$0.690710\pi$
$138$	0	0
$139$	−1493.64	−0.911428	−0.455714	−	0.890126i	$-0.650616\pi$
$139$	−1493.64	−0.911428	−0.455714	+	0.890126i	$0.650616\pi$
$140$	0	0
$141$	117.717	0.0703090
$142$	0	0
$143$	841.602	0.492156
$144$	0	0
$145$	98.7822	0.0565753
$146$	0	0
$147$	−46.4036	−0.0260361
$148$	0	0
$149$	−2759.02	−1.51696	−0.758482	−	0.651694i	$-0.774059\pi$
$149$	−2759.02	−1.51696	−0.758482	+	0.651694i	$0.774059\pi$
$150$	0	0
$151$	976.355	0.526190	0.263095	−	0.964770i	$-0.415257\pi$
$151$	976.355	0.526190	0.263095	+	0.964770i	$0.415257\pi$
$152$	0	0
$153$	342.917	0.181197
$154$	0	0
$155$	63.5076	0.0329100
$156$	0	0
$157$	−564.875	−0.287146	−0.143573	−	0.989640i	$-0.545859\pi$
$157$	−564.875	−0.287146	−0.143573	+	0.989640i	$0.545859\pi$
$158$	0	0
$159$	10.4887	0.00523149
$160$	0	0
$161$	1331.68	0.651869
$162$	0	0
$163$	−1508.53	−0.724892	−0.362446	−	0.932005i	$-0.618058\pi$
$163$	−1508.53	−0.724892	−0.362446	+	0.932005i	$0.618058\pi$
$164$	0	0
$165$	−133.953	−0.0632012
$166$	0	0
$167$	−592.521	−0.274555	−0.137277	−	0.990533i	$-0.543835\pi$
$167$	−592.521	−0.274555	−0.137277	+	0.990533i	$0.543835\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−1449.31	−0.648137
$172$	0	0
$173$	−4495.57	−1.97568	−0.987838	−	0.155488i	$-0.950305\pi$
$173$	−4495.57	−1.97568	−0.987838	+	0.155488i	$0.950305\pi$
$174$	0	0
$175$	2266.40	0.978994
$176$	0	0
$177$	−263.941	−0.112085
$178$	0	0
$179$	154.285	0.0644235	0.0322117	−	0.999481i	$-0.489745\pi$
$179$	154.285	0.0644235	0.0322117	+	0.999481i	$0.489745\pi$
$180$	0	0
$181$	1071.35	0.439959	0.219979	−	0.975505i	$-0.429401\pi$
$181$	1071.35	0.439959	0.219979	+	0.975505i	$0.429401\pi$
$182$	0	0
$183$	−3392.45	−1.37037
$184$	0	0
$185$	64.4706	0.0256215
$186$	0	0
$187$	1653.84	0.646742
$188$	0	0
$189$	2707.40	1.04198
$190$	0	0
$191$	−677.203	−0.256548	−0.128274	−	0.991739i	$-0.540944\pi$
$191$	−677.203	−0.256548	−0.128274	+	0.991739i	$0.540944\pi$
$192$	0	0
$193$	1321.68	0.492936	0.246468	−	0.969151i	$-0.420730\pi$
$193$	1321.68	0.492936	0.246468	+	0.969151i	$0.420730\pi$
$194$	0	0
$195$	−26.8987	−0.00987823
$196$	0	0
$197$	1267.37	0.458356	0.229178	−	0.973385i	$-0.426396\pi$
$197$	1267.37	0.458356	0.229178	+	0.973385i	$0.426396\pi$
$198$	0	0
$199$	−2396.24	−0.853593	−0.426796	−	0.904348i	$-0.640358\pi$
$199$	−2396.24	−0.853593	−0.426796	+	0.904348i	$0.640358\pi$
$200$	0	0
$201$	1637.02	0.574460
$202$	0	0
$203$	−3197.51	−1.10552
$204$	0	0
$205$	−39.1098	−0.0133246
$206$	0	0
$207$	983.409	0.330201
$208$	0	0
$209$	−6989.81	−2.31337
$210$	0	0
$211$	91.5539	0.0298712	0.0149356	−	0.999888i	$-0.495246\pi$
$211$	91.5539	0.0298712	0.0149356	+	0.999888i	$0.495246\pi$
$212$	0	0
$213$	1996.05	0.642099
$214$	0	0
$215$	−246.130	−0.0780740
$216$	0	0
$217$	−2055.70	−0.643087
$218$	0	0
$219$	2815.09	0.868613
$220$	0	0
$221$	332.103	0.101085
$222$	0	0
$223$	−1235.42	−0.370985	−0.185493	−	0.982646i	$-0.559388\pi$
$223$	−1235.42	−0.370985	−0.185493	+	0.982646i	$0.559388\pi$
$224$	0	0
$225$	1673.68	0.495905
$226$	0	0
$227$	−3301.66	−0.965370	−0.482685	−	0.875794i	$-0.660338\pi$
$227$	−3301.66	−0.965370	−0.482685	+	0.875794i	$0.660338\pi$
$228$	0	0
$229$	211.283	0.0609694	0.0304847	−	0.999535i	$-0.490295\pi$
$229$	211.283	0.0609694	0.0304847	+	0.999535i	$0.490295\pi$
$230$	0	0
$231$	4335.96	1.23500
$232$	0	0
$233$	−256.724	−0.0721827	−0.0360913	−	0.999348i	$-0.511491\pi$
$233$	−256.724	−0.0721827	−0.0360913	+	0.999348i	$0.511491\pi$
$234$	0	0
$235$	17.9404	0.00498002
$236$	0	0
$237$	1553.22	0.425708
$238$	0	0
$239$	3549.62	0.960694	0.480347	−	0.877078i	$-0.340511\pi$
$239$	3549.62	0.960694	0.480347	+	0.877078i	$0.340511\pi$
$240$	0	0
$241$	−5030.10	−1.34447	−0.672235	−	0.740338i	$-0.734665\pi$
$241$	−5030.10	−1.34447	−0.672235	+	0.740338i	$0.734665\pi$
$242$	0	0
$243$	3334.77	0.880353
$244$	0	0
$245$	−7.07204	−0.00184415
$246$	0	0
$247$	−1403.61	−0.361576
$248$	0	0
$249$	−2224.79	−0.566226
$250$	0	0
$251$	718.784	0.180754	0.0903770	−	0.995908i	$-0.471193\pi$
$251$	718.784	0.180754	0.0903770	+	0.995908i	$0.471193\pi$
$252$	0	0
$253$	4742.84	1.17858
$254$	0	0
$255$	−52.8588	−0.0129810
$256$	0	0
$257$	1280.79	0.310871	0.155435	−	0.987846i	$-0.450322\pi$
$257$	1280.79	0.310871	0.155435	+	0.987846i	$0.450322\pi$
$258$	0	0
$259$	−2086.87	−0.500663
$260$	0	0
$261$	−2361.28	−0.559998
$262$	0	0
$263$	−5225.55	−1.22517	−0.612587	−	0.790403i	$-0.709871\pi$
$263$	−5225.55	−1.22517	−0.612587	+	0.790403i	$0.709871\pi$
$264$	0	0
$265$	1.59851	0.000370549 0
$266$	0	0
$267$	−4273.77	−0.979590
$268$	0	0
$269$	6443.80	1.46054	0.730270	−	0.683158i	$-0.239394\pi$
$269$	6443.80	1.46054	0.730270	+	0.683158i	$0.239394\pi$
$270$	0	0
$271$	3929.93	0.880909	0.440455	−	0.897775i	$-0.354817\pi$
$271$	3929.93	0.880909	0.440455	+	0.897775i	$0.354817\pi$
$272$	0	0
$273$	870.692	0.193028
$274$	0	0
$275$	8071.91	1.77002
$276$	0	0
$277$	−5884.40	−1.27639	−0.638194	−	0.769876i	$-0.720318\pi$
$277$	−5884.40	−1.27639	−0.638194	+	0.769876i	$0.720318\pi$
$278$	0	0
$279$	−1518.08	−0.325752
$280$	0	0
$281$	3529.79	0.749358	0.374679	−	0.927155i	$-0.377753\pi$
$281$	3529.79	0.749358	0.374679	+	0.927155i	$0.377753\pi$
$282$	0	0
$283$	2611.00	0.548438	0.274219	−	0.961667i	$-0.411581\pi$
$283$	2611.00	0.548438	0.274219	+	0.961667i	$0.411581\pi$
$284$	0	0
$285$	223.403	0.0464325
$286$	0	0
$287$	1265.96	0.260373
$288$	0	0
$289$	−4260.38	−0.867165
$290$	0	0
$291$	2149.15	0.432939
$292$	0	0
$293$	−5491.03	−1.09484	−0.547422	−	0.836857i	$-0.684391\pi$
$293$	−5491.03	−1.09484	−0.547422	+	0.836857i	$0.684391\pi$
$294$	0	0
$295$	−40.2255	−0.00793904
$296$	0	0
$297$	9642.56	1.88390
$298$	0	0
$299$	952.398	0.184209
$300$	0	0
$301$	7967.05	1.52563
$302$	0	0
$303$	3396.30	0.643935
$304$	0	0
$305$	−517.019	−0.0970637
$306$	0	0
$307$	7307.59	1.35852	0.679261	−	0.733897i	$-0.262300\pi$
$307$	7307.59	1.35852	0.679261	+	0.733897i	$0.262300\pi$
$308$	0	0
$309$	3429.36	0.631357
$310$	0	0
$311$	−7904.92	−1.44131	−0.720654	−	0.693295i	$-0.756158\pi$
$311$	−7904.92	−1.44131	−0.720654	+	0.693295i	$0.756158\pi$
$312$	0	0
$313$	10002.4	1.80629	0.903145	−	0.429336i	$-0.141252\pi$
$313$	10002.4	1.80629	0.903145	+	0.429336i	$0.141252\pi$
$314$	0	0
$315$	137.017	0.0245080
$316$	0	0
$317$	−6230.81	−1.10397	−0.551983	−	0.833856i	$-0.686129\pi$
$317$	−6230.81	−1.10397	−0.551983	+	0.833856i	$0.686129\pi$
$318$	0	0
$319$	−11388.1	−1.99878
$320$	0	0
$321$	−3159.16	−0.549306
$322$	0	0
$323$	−2758.24	−0.475147
$324$	0	0
$325$	1620.90	0.276650
$326$	0	0
$327$	2475.55	0.418649
$328$	0	0
$329$	−580.719	−0.0973134
$330$	0	0
$331$	4634.51	0.769594	0.384797	−	0.923001i	$-0.374271\pi$
$331$	4634.51	0.769594	0.384797	+	0.923001i	$0.374271\pi$
$332$	0	0
$333$	−1541.10	−0.253609
$334$	0	0
$335$	249.487	0.0406893
$336$	0	0
$337$	3029.82	0.489747	0.244874	−	0.969555i	$-0.421254\pi$
$337$	3029.82	0.489747	0.244874	+	0.969555i	$0.421254\pi$
$338$	0	0
$339$	2363.61	0.378684
$340$	0	0
$341$	−7321.47	−1.16270
$342$	0	0
$343$	6463.66	1.01751
$344$	0	0
$345$	−151.587	−0.0236556
$346$	0	0
$347$	−2841.60	−0.439611	−0.219805	−	0.975544i	$-0.570542\pi$
$347$	−2841.60	−0.439611	−0.219805	+	0.975544i	$0.570542\pi$
$348$	0	0
$349$	7565.68	1.16040	0.580202	−	0.814472i	$-0.302973\pi$
$349$	7565.68	1.16040	0.580202	+	0.814472i	$0.302973\pi$
$350$	0	0
$351$	1936.30	0.294450
$352$	0	0
$353$	−2339.44	−0.352736	−0.176368	−	0.984324i	$-0.556435\pi$
$353$	−2339.44	−0.352736	−0.176368	+	0.984324i	$0.556435\pi$
$354$	0	0
$355$	304.204	0.0454802
$356$	0	0
$357$	1711.00	0.253658
$358$	0	0
$359$	2531.68	0.372192	0.186096	−	0.982532i	$-0.440417\pi$
$359$	2531.68	0.372192	0.186096	+	0.982532i	$0.440417\pi$
$360$	0	0
$361$	4798.45	0.699585
$362$	0	0
$363$	10538.5	1.52376
$364$	0	0
$365$	429.028	0.0615243
$366$	0	0
$367$	−6577.81	−0.935583	−0.467792	−	0.883839i	$-0.654950\pi$
$367$	−6577.81	−0.935583	−0.467792	+	0.883839i	$0.654950\pi$
$368$	0	0
$369$	934.876	0.131891
$370$	0	0
$371$	−51.7426	−0.00724081
$372$	0	0
$373$	2902.72	0.402942	0.201471	−	0.979495i	$-0.435428\pi$
$373$	2902.72	0.402942	0.201471	+	0.979495i	$0.435428\pi$
$374$	0	0
$375$	−516.630	−0.0711431
$376$	0	0
$377$	−2286.82	−0.312406
$378$	0	0
$379$	1865.73	0.252866	0.126433	−	0.991975i	$-0.459647\pi$
$379$	1865.73	0.252866	0.126433	+	0.991975i	$0.459647\pi$
$380$	0	0
$381$	2038.26	0.274076
$382$	0	0
$383$	−10836.0	−1.44567	−0.722837	−	0.691019i	$-0.757162\pi$
$383$	−10836.0	−1.44567	−0.722837	+	0.691019i	$0.757162\pi$
$384$	0	0
$385$	660.813	0.0874757
$386$	0	0
$387$	5883.46	0.772798
$388$	0	0
$389$	−9520.34	−1.24088	−0.620438	−	0.784256i	$-0.713045\pi$
$389$	−9520.34	−1.24088	−0.620438	+	0.784256i	$0.713045\pi$
$390$	0	0
$391$	1871.56	0.242069
$392$	0	0
$393$	−7577.13	−0.972559
$394$	0	0
$395$	236.716	0.0301531
$396$	0	0
$397$	−10108.8	−1.27796	−0.638978	−	0.769225i	$-0.720642\pi$
$397$	−10108.8	−1.27796	−0.638978	+	0.769225i	$0.720642\pi$
$398$	0	0
$399$	−7231.41	−0.907327
$400$	0	0
$401$	2084.38	0.259573	0.129787	−	0.991542i	$-0.458571\pi$
$401$	2084.38	0.259573	0.129787	+	0.991542i	$0.458571\pi$
$402$	0	0
$403$	−1470.21	−0.181728
$404$	0	0
$405$	−104.666	−0.0128417
$406$	0	0
$407$	−7432.50	−0.905197
$408$	0	0
$409$	−9716.53	−1.17470	−0.587349	−	0.809334i	$-0.699828\pi$
$409$	−9716.53	−1.17470	−0.587349	+	0.809334i	$0.699828\pi$
$410$	0	0
$411$	−6663.95	−0.799777
$412$	0	0
$413$	1302.07	0.155135
$414$	0	0
$415$	−339.064	−0.0401060
$416$	0	0
$417$	−5503.54	−0.646305
$418$	0	0
$419$	−13381.9	−1.56026	−0.780129	−	0.625619i	$-0.784847\pi$
$419$	−13381.9	−1.56026	−0.780129	+	0.625619i	$0.784847\pi$
$420$	0	0
$421$	−9463.37	−1.09553	−0.547763	−	0.836633i	$-0.684521\pi$
$421$	−9463.37	−1.09553	−0.547763	+	0.836633i	$0.684521\pi$
$422$	0	0
$423$	−428.846	−0.0492936
$424$	0	0
$425$	3185.24	0.363546
$426$	0	0
$427$	16735.5	1.89670
$428$	0	0
$429$	3101.02	0.348994
$430$	0	0
$431$	−4852.28	−0.542288	−0.271144	−	0.962539i	$-0.587402\pi$
$431$	−4852.28	−0.542288	−0.271144	+	0.962539i	$0.587402\pi$
$432$	0	0
$433$	−8208.00	−0.910973	−0.455486	−	0.890243i	$-0.650535\pi$
$433$	−8208.00	−0.910973	−0.455486	+	0.890243i	$0.650535\pi$
$434$	0	0
$435$	363.979	0.0401183
$436$	0	0
$437$	−7910.01	−0.865874
$438$	0	0
$439$	2993.80	0.325481	0.162741	−	0.986669i	$-0.447967\pi$
$439$	2993.80	0.325481	0.162741	+	0.986669i	$0.447967\pi$
$440$	0	0
$441$	169.049	0.0182539
$442$	0	0
$443$	−9743.67	−1.04500	−0.522501	−	0.852639i	$-0.675001\pi$
$443$	−9743.67	−1.04500	−0.522501	+	0.852639i	$0.675001\pi$
$444$	0	0
$445$	−651.335	−0.0693848
$446$	0	0
$447$	−10166.0	−1.07570
$448$	0	0
$449$	−561.459	−0.0590131	−0.0295065	−	0.999565i	$-0.509394\pi$
$449$	−561.459	−0.0590131	−0.0295065	+	0.999565i	$0.509394\pi$
$450$	0	0
$451$	4508.78	0.470754
$452$	0	0
$453$	3597.54	0.373128
$454$	0	0
$455$	132.696	0.0136723
$456$	0	0
$457$	13758.4	1.40830	0.704148	−	0.710054i	$-0.251329\pi$
$457$	13758.4	1.40830	0.704148	+	0.710054i	$0.251329\pi$
$458$	0	0
$459$	3805.03	0.386936
$460$	0	0
$461$	12009.2	1.21329	0.606644	−	0.794974i	$-0.292515\pi$
$461$	12009.2	1.21329	0.606644	+	0.794974i	$0.292515\pi$
$462$	0	0
$463$	−13635.7	−1.36870	−0.684348	−	0.729156i	$-0.739913\pi$
$463$	−13635.7	−1.36870	−0.684348	+	0.729156i	$0.739913\pi$
$464$	0	0
$465$	234.004	0.0233369
$466$	0	0
$467$	−8821.95	−0.874157	−0.437079	−	0.899423i	$-0.643987\pi$
$467$	−8821.95	−0.874157	−0.437079	+	0.899423i	$0.643987\pi$
$468$	0	0
$469$	−8075.72	−0.795100
$470$	0	0
$471$	−2081.37	−0.203619
$472$	0	0
$473$	28375.1	2.75832
$474$	0	0
$475$	−13462.2	−1.30039
$476$	0	0
$477$	−38.2105	−0.00366780
$478$	0	0
$479$	14620.0	1.39459	0.697293	−	0.716786i	$-0.254388\pi$
$479$	14620.0	1.39459	0.697293	+	0.716786i	$0.254388\pi$
$480$	0	0
$481$	−1492.50	−0.141481
$482$	0	0
$483$	4906.78	0.462249
$484$	0	0
$485$	327.536	0.0306653
$486$	0	0
$487$	9798.86	0.911763	0.455882	−	0.890040i	$-0.349324\pi$
$487$	9798.86	0.911763	0.455882	+	0.890040i	$0.349324\pi$
$488$	0	0
$489$	−5558.43	−0.514030
$490$	0	0
$491$	10836.1	0.995977	0.497989	−	0.867184i	$-0.334072\pi$
$491$	10836.1	0.995977	0.497989	+	0.867184i	$0.334072\pi$
$492$	0	0
$493$	−4493.84	−0.410532
$494$	0	0
$495$	487.993	0.0443104
$496$	0	0
$497$	−9846.86	−0.888717
$498$	0	0
$499$	−2589.96	−0.232349	−0.116175	−	0.993229i	$-0.537063\pi$
$499$	−2589.96	−0.232349	−0.116175	+	0.993229i	$0.537063\pi$
$500$	0	0
$501$	−2183.24	−0.194690
$502$	0	0
$503$	17067.5	1.51292	0.756462	−	0.654038i	$-0.226926\pi$
$503$	17067.5	1.51292	0.756462	+	0.654038i	$0.226926\pi$
$504$	0	0
$505$	517.606	0.0456102
$506$	0	0
$507$	622.707	0.0545471
$508$	0	0
$509$	−1012.89	−0.0882038	−0.0441019	−	0.999027i	$-0.514043\pi$
$509$	−1012.89	−0.0882038	−0.0441019	+	0.999027i	$0.514043\pi$
$510$	0	0
$511$	−13887.4	−1.20223
$512$	0	0
$513$	−16081.7	−1.38406
$514$	0	0
$515$	522.644	0.0447193
$516$	0	0
$517$	−2068.26	−0.175942
$518$	0	0
$519$	−16564.6	−1.40098
$520$	0	0
$521$	−14367.7	−1.20818	−0.604089	−	0.796917i	$-0.706463\pi$
$521$	−14367.7	−1.20818	−0.604089	+	0.796917i	$0.706463\pi$
$522$	0	0
$523$	16219.9	1.35611	0.678057	−	0.735010i	$-0.262822\pi$
$523$	16219.9	1.35611	0.678057	+	0.735010i	$0.262822\pi$
$524$	0	0
$525$	8350.92	0.694217
$526$	0	0
$527$	−2889.11	−0.238808
$528$	0	0
$529$	−6799.77	−0.558870
$530$	0	0
$531$	961.545	0.0785828
$532$	0	0
$533$	905.396	0.0735780
$534$	0	0
$535$	−481.466	−0.0389076
$536$	0	0
$537$	568.488	0.0456835
$538$	0	0
$539$	815.301	0.0651530
$540$	0	0
$541$	17592.2	1.39806	0.699029	−	0.715094i	$-0.253616\pi$
$541$	17592.2	1.39806	0.699029	+	0.715094i	$0.253616\pi$
$542$	0	0
$543$	3947.55	0.311980
$544$	0	0
$545$	377.281	0.0296531
$546$	0	0
$547$	−10504.6	−0.821103	−0.410552	−	0.911837i	$-0.634664\pi$
$547$	−10504.6	−0.821103	−0.410552	+	0.911837i	$0.634664\pi$
$548$	0	0
$549$	12358.8	0.960763
$550$	0	0
$551$	18992.8	1.46846
$552$	0	0
$553$	−7662.33	−0.589214
$554$	0	0
$555$	237.552	0.0181685
$556$	0	0
$557$	−507.558	−0.0386102	−0.0193051	−	0.999814i	$-0.506145\pi$
$557$	−507.558	−0.0386102	−0.0193051	+	0.999814i	$0.506145\pi$
$558$	0	0
$559$	5697.93	0.431121
$560$	0	0
$561$	6093.83	0.458613
$562$	0	0
$563$	3443.14	0.257746	0.128873	−	0.991661i	$-0.458864\pi$
$563$	3443.14	0.257746	0.128873	+	0.991661i	$0.458864\pi$
$564$	0	0
$565$	360.221	0.0268223
$566$	0	0
$567$	3387.96	0.250936
$568$	0	0
$569$	23972.2	1.76620	0.883098	−	0.469189i	$-0.155454\pi$
$569$	23972.2	1.76620	0.883098	+	0.469189i	$0.155454\pi$
$570$	0	0
$571$	7458.32	0.546622	0.273311	−	0.961926i	$-0.411881\pi$
$571$	7458.32	0.546622	0.273311	+	0.961926i	$0.411881\pi$
$572$	0	0
$573$	−2495.26	−0.181922
$574$	0	0
$575$	9134.57	0.662501
$576$	0	0
$577$	5669.57	0.409059	0.204530	−	0.978860i	$-0.434434\pi$
$577$	5669.57	0.409059	0.204530	+	0.978860i	$0.434434\pi$
$578$	0	0
$579$	4869.94	0.349547
$580$	0	0
$581$	10975.3	0.783702
$582$	0	0
$583$	−184.284	−0.0130914
$584$	0	0
$585$	97.9925	0.00692563
$586$	0	0
$587$	−1017.39	−0.0715371	−0.0357685	−	0.999360i	$-0.511388\pi$
$587$	−1017.39	−0.0715371	−0.0357685	+	0.999360i	$0.511388\pi$
$588$	0	0
$589$	12210.6	0.854208
$590$	0	0
$591$	4669.81	0.325026
$592$	0	0
$593$	−10198.2	−0.706221	−0.353111	−	0.935582i	$-0.614876\pi$
$593$	−10198.2	−0.706221	−0.353111	+	0.935582i	$0.614876\pi$
$594$	0	0
$595$	260.762	0.0179667
$596$	0	0
$597$	−8829.33	−0.605294
$598$	0	0
$599$	−12516.3	−0.853763	−0.426881	−	0.904308i	$-0.640388\pi$
$599$	−12516.3	−0.853763	−0.426881	+	0.904308i	$0.640388\pi$
$600$	0	0
$601$	9627.46	0.653431	0.326716	−	0.945123i	$-0.394058\pi$
$601$	9627.46	0.653431	0.326716	+	0.945123i	$0.394058\pi$
$602$	0	0
$603$	−5963.70	−0.402754
$604$	0	0
$605$	1606.09	0.107929
$606$	0	0
$607$	−6667.20	−0.445821	−0.222910	−	0.974839i	$-0.571556\pi$
$607$	−6667.20	−0.445821	−0.222910	+	0.974839i	$0.571556\pi$
$608$	0	0
$609$	−11781.7	−0.783942
$610$	0	0
$611$	−415.323	−0.0274994
$612$	0	0
$613$	−23085.4	−1.52106	−0.760530	−	0.649302i	$-0.775061\pi$
$613$	−23085.4	−1.52106	−0.760530	+	0.649302i	$0.775061\pi$
$614$	0	0
$615$	−144.106	−0.00944866
$616$	0	0
$617$	3049.24	0.198959	0.0994796	−	0.995040i	$-0.468282\pi$
$617$	3049.24	0.198959	0.0994796	+	0.995040i	$0.468282\pi$
$618$	0	0
$619$	−7296.58	−0.473787	−0.236894	−	0.971536i	$-0.576129\pi$
$619$	−7296.58	−0.473787	−0.236894	+	0.971536i	$0.576129\pi$
$620$	0	0
$621$	10912.0	0.705126
$622$	0	0
$623$	21083.3	1.35583
$624$	0	0
$625$	15506.8	0.992438
$626$	0	0
$627$	−25755.1	−1.64044
$628$	0	0
$629$	−2932.92	−0.185920
$630$	0	0
$631$	23829.5	1.50339	0.751694	−	0.659512i	$-0.229237\pi$
$631$	23829.5	1.50339	0.751694	+	0.659512i	$0.229237\pi$
$632$	0	0
$633$	337.345	0.0211821
$634$	0	0
$635$	310.637	0.0194130
$636$	0	0
$637$	163.718	0.0101833
$638$	0	0
$639$	−7271.65	−0.450175
$640$	0	0
$641$	13405.3	0.826016	0.413008	−	0.910727i	$-0.364478\pi$
$641$	13405.3	0.826016	0.413008	+	0.910727i	$0.364478\pi$
$642$	0	0
$643$	−5251.51	−0.322083	−0.161042	−	0.986948i	$-0.551485\pi$
$643$	−5251.51	−0.322083	−0.161042	+	0.986948i	$0.551485\pi$
$644$	0	0
$645$	−906.904	−0.0553633
$646$	0	0
$647$	−21611.4	−1.31319	−0.656595	−	0.754244i	$-0.728004\pi$
$647$	−21611.4	−1.31319	−0.656595	+	0.754244i	$0.728004\pi$
$648$	0	0
$649$	4637.39	0.280483
$650$	0	0
$651$	−7574.54	−0.456021
$652$	0	0
$653$	−21595.8	−1.29420	−0.647099	−	0.762406i	$-0.724018\pi$
$653$	−21595.8	−1.29420	−0.647099	+	0.762406i	$0.724018\pi$
$654$	0	0
$655$	−1154.78	−0.0688869
$656$	0	0
$657$	−10255.4	−0.608985
$658$	0	0
$659$	16642.6	0.983768	0.491884	−	0.870661i	$-0.336308\pi$
$659$	16642.6	0.983768	0.491884	+	0.870661i	$0.336308\pi$
$660$	0	0
$661$	26981.1	1.58766	0.793831	−	0.608139i	$-0.208084\pi$
$661$	26981.1	1.58766	0.793831	+	0.608139i	$0.208084\pi$
$662$	0	0
$663$	1223.69	0.0716803
$664$	0	0
$665$	−1102.09	−0.0642664
$666$	0	0
$667$	−12887.3	−0.748126
$668$	0	0
$669$	−4552.09	−0.263070
$670$	0	0
$671$	59604.5	3.42922
$672$	0	0
$673$	11149.2	0.638591	0.319296	−	0.947655i	$-0.396554\pi$
$673$	11149.2	0.638591	0.319296	+	0.947655i	$0.396554\pi$
$674$	0	0
$675$	18571.3	1.05898
$676$	0	0
$677$	3314.33	0.188154	0.0940769	−	0.995565i	$-0.470010\pi$
$677$	3314.33	0.188154	0.0940769	+	0.995565i	$0.470010\pi$
$678$	0	0
$679$	−10602.1	−0.599223
$680$	0	0
$681$	−12165.5	−0.684556
$682$	0	0
$683$	−24505.2	−1.37287	−0.686433	−	0.727193i	$-0.740824\pi$
$683$	−24505.2	−1.37287	−0.686433	+	0.727193i	$0.740824\pi$
$684$	0	0
$685$	−1015.61	−0.0566486
$686$	0	0
$687$	778.506	0.0432342
$688$	0	0
$689$	−37.0056	−0.00204616
$690$	0	0
$691$	21752.8	1.19756	0.598782	−	0.800912i	$-0.295652\pi$
$691$	21752.8	1.19756	0.598782	+	0.800912i	$0.295652\pi$
$692$	0	0
$693$	−15796.0	−0.865858
$694$	0	0
$695$	−838.755	−0.0457781
$696$	0	0
$697$	1779.20	0.0966887
$698$	0	0
$699$	−945.941	−0.0511857
$700$	0	0
$701$	34250.9	1.84542	0.922709	−	0.385496i	$-0.125970\pi$
$701$	34250.9	1.84542	0.922709	+	0.385496i	$0.125970\pi$
$702$	0	0
$703$	12395.8	0.665028
$704$	0	0
$705$	66.1043	0.00353140
$706$	0	0
$707$	−16754.6	−0.891259
$708$	0	0
$709$	−5527.11	−0.292771	−0.146386	−	0.989228i	$-0.546764\pi$
$709$	−5527.11	−0.292771	−0.146386	+	0.989228i	$0.546764\pi$
$710$	0	0
$711$	−5658.43	−0.298464
$712$	0	0
$713$	−8285.33	−0.435187
$714$	0	0
$715$	472.604	0.0247194
$716$	0	0
$717$	13079.1	0.681241
$718$	0	0
$719$	3777.78	0.195949	0.0979745	−	0.995189i	$-0.468764\pi$
$719$	3777.78	0.195949	0.0979745	+	0.995189i	$0.468764\pi$
$720$	0	0
$721$	−16917.6	−0.873849
$722$	0	0
$723$	−18534.2	−0.953380
$724$	0	0
$725$	−21933.2	−1.12355
$726$	0	0
$727$	−19076.8	−0.973204	−0.486602	−	0.873624i	$-0.661764\pi$
$727$	−19076.8	−0.973204	−0.486602	+	0.873624i	$0.661764\pi$
$728$	0	0
$729$	17319.9	0.879944
$730$	0	0
$731$	11197.0	0.566535
$732$	0	0
$733$	7997.30	0.402984	0.201492	−	0.979490i	$-0.435421\pi$
$733$	7997.30	0.402984	0.201492	+	0.979490i	$0.435421\pi$
$734$	0	0
$735$	−26.0581	−0.00130771
$736$	0	0
$737$	−28762.1	−1.43754
$738$	0	0
$739$	−28983.6	−1.44273	−0.721367	−	0.692553i	$-0.756486\pi$
$739$	−28983.6	−1.44273	−0.721367	+	0.692553i	$0.756486\pi$
$740$	0	0
$741$	−5171.81	−0.256398
$742$	0	0
$743$	19145.4	0.945324	0.472662	−	0.881244i	$-0.343293\pi$
$743$	19145.4	0.945324	0.472662	+	0.881244i	$0.343293\pi$
$744$	0	0
$745$	−1549.33	−0.0761923
$746$	0	0
$747$	8104.95	0.396981
$748$	0	0
$749$	15584.7	0.760284
$750$	0	0
$751$	25516.9	1.23985	0.619923	−	0.784663i	$-0.287164\pi$
$751$	25516.9	1.23985	0.619923	+	0.784663i	$0.287164\pi$
$752$	0	0
$753$	2648.47	0.128175
$754$	0	0
$755$	548.275	0.0264288
$756$	0	0
$757$	−17230.6	−0.827289	−0.413645	−	0.910438i	$-0.635744\pi$
$757$	−17230.6	−0.827289	−0.413645	+	0.910438i	$0.635744\pi$
$758$	0	0
$759$	17475.7	0.835744
$760$	0	0
$761$	−2343.06	−0.111611	−0.0558053	−	0.998442i	$-0.517773\pi$
$761$	−2343.06	−0.111611	−0.0558053	+	0.998442i	$0.517773\pi$
$762$	0	0
$763$	−12212.3	−0.579444
$764$	0	0
$765$	192.566	0.00910096
$766$	0	0
$767$	931.223	0.0438390
$768$	0	0
$769$	−7100.18	−0.332950	−0.166475	−	0.986046i	$-0.553239\pi$
$769$	−7100.18	−0.332950	−0.166475	+	0.986046i	$0.553239\pi$
$770$	0	0
$771$	4719.29	0.220442
$772$	0	0
$773$	12270.4	0.570940	0.285470	−	0.958388i	$-0.407850\pi$
$773$	12270.4	0.570940	0.285470	+	0.958388i	$0.407850\pi$
$774$	0	0
$775$	−14100.9	−0.653575
$776$	0	0
$777$	−7689.40	−0.355027
$778$	0	0
$779$	−7519.64	−0.345852
$780$	0	0
$781$	−35070.1	−1.60680
$782$	0	0
$783$	−26201.0	−1.19584
$784$	0	0
$785$	−317.207	−0.0144224
$786$	0	0
$787$	−3425.04	−0.155133	−0.0775663	−	0.996987i	$-0.524715\pi$
$787$	−3425.04	−0.155133	−0.0775663	+	0.996987i	$0.524715\pi$
$788$	0	0
$789$	−19254.3	−0.868787
$790$	0	0
$791$	−11660.1	−0.524129
$792$	0	0
$793$	11969.0	0.535981
$794$	0	0
$795$	5.88995	0.000262761 0
$796$	0	0
$797$	−11781.1	−0.523600	−0.261800	−	0.965122i	$-0.584316\pi$
$797$	−11781.1	−0.523600	−0.261800	+	0.965122i	$0.584316\pi$
$798$	0	0
$799$	−816.154	−0.0361370
$800$	0	0
$801$	15569.4	0.686790
$802$	0	0
$803$	−49460.6	−2.17363
$804$	0	0
$805$	747.807	0.0327413
$806$	0	0
$807$	23743.2	1.03569
$808$	0	0
$809$	18910.1	0.821810	0.410905	−	0.911678i	$-0.365213\pi$
$809$	18910.1	0.821810	0.410905	+	0.911678i	$0.365213\pi$
$810$	0	0
$811$	−12803.3	−0.554359	−0.277180	−	0.960818i	$-0.589400\pi$
$811$	−12803.3	−0.554359	−0.277180	+	0.960818i	$0.589400\pi$
$812$	0	0
$813$	14480.5	0.624664
$814$	0	0
$815$	−847.121	−0.0364090
$816$	0	0
$817$	−47323.3	−2.02648
$818$	0	0
$819$	−3171.95	−0.135332
$820$	0	0
$821$	19335.1	0.821923	0.410962	−	0.911653i	$-0.365193\pi$
$821$	19335.1	0.821923	0.410962	+	0.911653i	$0.365193\pi$
$822$	0	0
$823$	2125.90	0.0900417	0.0450209	−	0.998986i	$-0.485665\pi$
$823$	2125.90	0.0900417	0.0450209	+	0.998986i	$0.485665\pi$
$824$	0	0
$825$	29742.2	1.25514
$826$	0	0
$827$	6989.24	0.293881	0.146941	−	0.989145i	$-0.453057\pi$
$827$	6989.24	0.293881	0.146941	+	0.989145i	$0.453057\pi$
$828$	0	0
$829$	−32649.7	−1.36788	−0.683938	−	0.729540i	$-0.739734\pi$
$829$	−32649.7	−1.36788	−0.683938	+	0.729540i	$0.739734\pi$
$830$	0	0
$831$	−21682.0	−0.905103
$832$	0	0
$833$	321.724	0.0133819
$834$	0	0
$835$	−332.732	−0.0137900
$836$	0	0
$837$	−16844.7	−0.695626
$838$	0	0
$839$	4038.23	0.166168	0.0830841	−	0.996543i	$-0.473523\pi$
$839$	4038.23	0.166168	0.0830841	+	0.996543i	$0.473523\pi$
$840$	0	0
$841$	6555.00	0.268769
$842$	0	0
$843$	13006.1	0.531380
$844$	0	0
$845$	94.9024	0.00386360
$846$	0	0
$847$	−51988.1	−2.10901
$848$	0	0
$849$	9620.64	0.388904
$850$	0	0
$851$	−8410.97	−0.338807
$852$	0	0
$853$	8114.12	0.325700	0.162850	−	0.986651i	$-0.447931\pi$
$853$	8114.12	0.325700	0.162850	+	0.986651i	$0.447931\pi$
$854$	0	0
$855$	−813.863	−0.0325538
$856$	0	0
$857$	−22298.1	−0.888786	−0.444393	−	0.895832i	$-0.646581\pi$
$857$	−22298.1	−0.888786	−0.444393	+	0.895832i	$0.646581\pi$
$858$	0	0
$859$	−33550.5	−1.33263	−0.666315	−	0.745670i	$-0.732130\pi$
$859$	−33550.5	−1.33263	−0.666315	+	0.745670i	$0.732130\pi$
$860$	0	0
$861$	4664.62	0.184634
$862$	0	0
$863$	14120.5	0.556972	0.278486	−	0.960440i	$-0.410167\pi$
$863$	14120.5	0.556972	0.278486	+	0.960440i	$0.410167\pi$
$864$	0	0
$865$	−2524.50	−0.0992319
$866$	0	0
$867$	−15698.1	−0.614918
$868$	0	0
$869$	−27289.8	−1.06530
$870$	0	0
$871$	−5775.64	−0.224685
$872$	0	0
$873$	−7829.39	−0.303533
$874$	0	0
$875$	2548.63	0.0984679
$876$	0	0
$877$	−1941.69	−0.0747619	−0.0373809	−	0.999301i	$-0.511901\pi$
$877$	−1941.69	−0.0747619	−0.0373809	+	0.999301i	$0.511901\pi$
$878$	0	0
$879$	−20232.6	−0.776368
$880$	0	0
$881$	−790.231	−0.0302197	−0.0151099	−	0.999886i	$-0.504810\pi$
$881$	−790.231	−0.0302197	−0.0151099	+	0.999886i	$0.504810\pi$
$882$	0	0
$883$	36638.6	1.39636	0.698180	−	0.715922i	$-0.253993\pi$
$883$	36638.6	1.39636	0.698180	+	0.715922i	$0.253993\pi$
$884$	0	0
$885$	−148.217	−0.00562968
$886$	0	0
$887$	40686.3	1.54015	0.770075	−	0.637954i	$-0.220219\pi$
$887$	40686.3	1.54015	0.770075	+	0.637954i	$0.220219\pi$
$888$	0	0
$889$	−10055.1	−0.379344
$890$	0	0
$891$	12066.4	0.453692
$892$	0	0
$893$	3449.40	0.129261
$894$	0	0
$895$	86.6392	0.00323579
$896$	0	0
$897$	3509.26	0.130625
$898$	0	0
$899$	19894.0	0.738046
$900$	0	0
$901$	−72.7200	−0.00268885
$902$	0	0
$903$	29355.9	1.08184
$904$	0	0
$905$	601.618	0.0220977
$906$	0	0
$907$	10464.4	0.383093	0.191547	−	0.981484i	$-0.438650\pi$
$907$	10464.4	0.383093	0.191547	+	0.981484i	$0.438650\pi$
$908$	0	0
$909$	−12372.8	−0.451463
$910$	0	0
$911$	35611.5	1.29513	0.647563	−	0.762011i	$-0.275788\pi$
$911$	35611.5	1.29513	0.647563	+	0.762011i	$0.275788\pi$
$912$	0	0
$913$	39089.0	1.41693
$914$	0	0
$915$	−1905.04	−0.0688291
$916$	0	0
$917$	37379.4	1.34610
$918$	0	0
$919$	−1077.25	−0.0386674	−0.0193337	−	0.999813i	$-0.506154\pi$
$919$	−1077.25	−0.0386674	−0.0193337	+	0.999813i	$0.506154\pi$
$920$	0	0
$921$	26926.0	0.963346
$922$	0	0
$923$	−7042.34	−0.251139
$924$	0	0
$925$	−14314.8	−0.508829
$926$	0	0
$927$	−12493.2	−0.442644
$928$	0	0
$929$	55733.8	1.96832	0.984159	−	0.177290i	$-0.0567330\pi$
$929$	55733.8	1.96832	0.984159	+	0.177290i	$0.0567330\pi$
$930$	0	0
$931$	−1359.74	−0.0478665
$932$	0	0
$933$	−29126.9	−1.02205
$934$	0	0
$935$	928.718	0.0324838
$936$	0	0
$937$	−3198.60	−0.111519	−0.0557596	−	0.998444i	$-0.517758\pi$
$937$	−3198.60	−0.111519	−0.0557596	+	0.998444i	$0.517758\pi$
$938$	0	0
$939$	36855.4	1.28086
$940$	0	0
$941$	8823.35	0.305667	0.152834	−	0.988252i	$-0.451160\pi$
$941$	8823.35	0.305667	0.152834	+	0.988252i	$0.451160\pi$
$942$	0	0
$943$	5102.35	0.176199
$944$	0	0
$945$	1520.35	0.0523354
$946$	0	0
$947$	−28290.4	−0.970766	−0.485383	−	0.874301i	$-0.661320\pi$
$947$	−28290.4	−0.970766	−0.485383	+	0.874301i	$0.661320\pi$
$948$	0	0
$949$	−9932.05	−0.339734
$950$	0	0
$951$	−22958.4	−0.782836
$952$	0	0
$953$	−12399.0	−0.421452	−0.210726	−	0.977545i	$-0.567583\pi$
$953$	−12399.0	−0.421452	−0.210726	+	0.977545i	$0.567583\pi$
$954$	0	0
$955$	−380.285	−0.0128856
$956$	0	0
$957$	−41961.3	−1.41736
$958$	0	0
$959$	32874.5	1.10696
$960$	0	0
$961$	−17001.0	−0.570676
$962$	0	0
$963$	11508.9	0.385118
$964$	0	0
$965$	742.193	0.0247586
$966$	0	0
$967$	26667.1	0.886820	0.443410	−	0.896319i	$-0.353769\pi$
$967$	26667.1	0.886820	0.443410	+	0.896319i	$0.353769\pi$
$968$	0	0
$969$	−10163.2	−0.336933
$970$	0	0
$971$	−49420.7	−1.63335	−0.816676	−	0.577096i	$-0.804186\pi$
$971$	−49420.7	−1.63335	−0.816676	+	0.577096i	$0.804186\pi$
$972$	0	0
$973$	27149.9	0.894539
$974$	0	0
$975$	5972.46	0.196176
$976$	0	0
$977$	778.759	0.0255012	0.0127506	−	0.999919i	$-0.495941\pi$
$977$	778.759	0.0255012	0.0127506	+	0.999919i	$0.495941\pi$
$978$	0	0
$979$	75089.2	2.45134
$980$	0	0
$981$	−9018.48	−0.293515
$982$	0	0
$983$	−5997.90	−0.194612	−0.0973059	−	0.995255i	$-0.531023\pi$
$983$	−5997.90	−0.194612	−0.0973059	+	0.995255i	$0.531023\pi$
$984$	0	0
$985$	711.693	0.0230218
$986$	0	0
$987$	−2139.75	−0.0690062
$988$	0	0
$989$	32110.6	1.03241
$990$	0	0
$991$	−8974.94	−0.287688	−0.143844	−	0.989600i	$-0.545946\pi$
$991$	−8974.94	−0.287688	−0.143844	+	0.989600i	$0.545946\pi$
$992$	0	0
$993$	17076.6	0.545729
$994$	0	0
$995$	−1345.62	−0.0428732
$996$	0	0
$997$	28530.2	0.906280	0.453140	−	0.891439i	$-0.350304\pi$
$997$	28530.2	0.906280	0.453140	+	0.891439i	$0.350304\pi$
$998$	0	0
$999$	−17100.2	−0.541567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.4.a.h.1.2		2
3.2	odd	2		1872.4.a.bb.1.1		2
4.3	odd	2		13.4.a.b.1.2	✓	2
8.3	odd	2		832.4.a.s.1.2		2
8.5	even	2		832.4.a.z.1.1		2
12.11	even	2		117.4.a.d.1.1		2
20.3	even	4		325.4.b.e.274.1		4
20.7	even	4		325.4.b.e.274.4		4
20.19	odd	2		325.4.a.f.1.1		2
28.27	even	2		637.4.a.b.1.2		2
44.43	even	2		1573.4.a.b.1.1		2
52.3	odd	6		169.4.c.g.22.1		4
52.7	even	12		169.4.e.f.23.4		8
52.11	even	12		169.4.e.f.147.1		8
52.15	even	12		169.4.e.f.147.4		8
52.19	even	12		169.4.e.f.23.1		8
52.23	odd	6		169.4.c.j.22.2		4
52.31	even	4		169.4.b.f.168.1		4
52.35	odd	6		169.4.c.g.146.1		4
52.43	odd	6		169.4.c.j.146.2		4
52.47	even	4		169.4.b.f.168.4		4
52.51	odd	2		169.4.a.g.1.1		2
156.155	even	2		1521.4.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.a.b.1.2	✓	2	4.3	odd	2
117.4.a.d.1.1		2	12.11	even	2
169.4.a.g.1.1		2	52.51	odd	2
169.4.b.f.168.1		4	52.31	even	4
169.4.b.f.168.4		4	52.47	even	4
169.4.c.g.22.1		4	52.3	odd	6
169.4.c.g.146.1		4	52.35	odd	6
169.4.c.j.22.2		4	52.23	odd	6
169.4.c.j.146.2		4	52.43	odd	6
169.4.e.f.23.1		8	52.19	even	12
169.4.e.f.23.4		8	52.7	even	12
169.4.e.f.147.1		8	52.11	even	12
169.4.e.f.147.4		8	52.15	even	12
208.4.a.h.1.2		2	1.1	even	1	trivial
325.4.a.f.1.1		2	20.19	odd	2
325.4.b.e.274.1		4	20.3	even	4
325.4.b.e.274.4		4	20.7	even	4
637.4.a.b.1.2		2	28.27	even	2
832.4.a.s.1.2		2	8.3	odd	2
832.4.a.z.1.1		2	8.5	even	2
1521.4.a.r.1.2		2	156.155	even	2
1573.4.a.b.1.1		2	44.43	even	2
1872.4.a.bb.1.1		2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

\(f(q)\)	\(=\)	\(q+3.68466 q^{3} +0.561553 q^{5} -18.1771 q^{7} -13.4233 q^{9} +O(q^{10})\)	\(q+3.68466 q^{3} +0.561553 q^{5} -18.1771 q^{7} -13.4233 q^{9} -64.7386 q^{11} -13.0000 q^{13} +2.06913 q^{15} -25.5464 q^{17} +107.970 q^{19} -66.9763 q^{21} -73.2614 q^{23} -124.685 q^{25} -148.946 q^{27} +175.909 q^{29} +113.093 q^{31} -238.540 q^{33} -10.2074 q^{35} +114.808 q^{37} -47.9006 q^{39} -69.6458 q^{41} -438.302 q^{43} -7.53789 q^{45} +31.9479 q^{47} -12.5937 q^{49} -94.1298 q^{51} +2.84658 q^{53} -36.3542 q^{55} +397.831 q^{57} -71.6325 q^{59} -920.695 q^{61} +243.996 q^{63} -7.30019 q^{65} +444.280 q^{67} -269.943 q^{69} +541.719 q^{71} +764.004 q^{73} -459.420 q^{75} +1176.76 q^{77} +421.538 q^{79} -186.386 q^{81} -603.797 q^{83} -14.3457 q^{85} +648.165 q^{87} -1159.88 q^{89} +236.302 q^{91} +416.708 q^{93} +60.6307 q^{95} +583.269 q^{97} +869.006 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{3} - 3 q^{5} + 9 q^{7} + 35 q^{9}+O(q^{10}) \) 2 * q - 5 * q^3 - 3 * q^5 + 9 * q^7 + 35 * q^9	\( 2 q - 5 q^{3} - 3 q^{5} + 9 q^{7} + 35 q^{9} - 80 q^{11} - 26 q^{13} + 33 q^{15} + 19 q^{17} + 84 q^{19} - 303 q^{21} - 196 q^{23} - 237 q^{25} - 335 q^{27} - 44 q^{29} + 86 q^{31} - 106 q^{33} - 107 q^{35} + 209 q^{37} + 65 q^{39} - 230 q^{41} - 287 q^{43} - 180 q^{45} - 435 q^{47} + 383 q^{49} - 481 q^{51} - 118 q^{53} + 18 q^{55} + 606 q^{57} + 368 q^{59} - 1058 q^{61} + 1560 q^{63} + 39 q^{65} - 68 q^{67} + 796 q^{69} + 131 q^{71} + 456 q^{73} + 516 q^{75} + 762 q^{77} + 1008 q^{79} + 122 q^{81} - 1958 q^{83} - 173 q^{85} + 2558 q^{87} - 720 q^{89} - 117 q^{91} + 652 q^{93} + 146 q^{95} - 928 q^{97} + 130 q^{99}+O(q^{100}) \) 2 * q - 5 * q^3 - 3 * q^5 + 9 * q^7 + 35 * q^9 - 80 * q^11 - 26 * q^13 + 33 * q^15 + 19 * q^17 + 84 * q^19 - 303 * q^21 - 196 * q^23 - 237 * q^25 - 335 * q^27 - 44 * q^29 + 86 * q^31 - 106 * q^33 - 107 * q^35 + 209 * q^37 + 65 * q^39 - 230 * q^41 - 287 * q^43 - 180 * q^45 - 435 * q^47 + 383 * q^49 - 481 * q^51 - 118 * q^53 + 18 * q^55 + 606 * q^57 + 368 * q^59 - 1058 * q^61 + 1560 * q^63 + 39 * q^65 - 68 * q^67 + 796 * q^69 + 131 * q^71 + 456 * q^73 + 516 * q^75 + 762 * q^77 + 1008 * q^79 + 122 * q^81 - 1958 * q^83 - 173 * q^85 + 2558 * q^87 - 720 * q^89 - 117 * q^91 + 652 * q^93 + 146 * q^95 - 928 * q^97 + 130 * q^99

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