LMFDB - Embedded newform 208.4.a.i.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{321}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 80 $ x^2 - x - 80
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$9.45824$ 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)

$f(q)$	$=$	$q+9.45824 q^{3} +3.45824 q^{5} +26.3747 q^{7} +62.4582 q^{9} +O(q^{10})$	$q+9.45824 q^{3} +3.45824 q^{5} +26.3747 q^{7} +62.4582 q^{9} -58.0000 q^{11} -13.0000 q^{13} +32.7088 q^{15} +88.2077 q^{17} -99.8329 q^{19} +249.458 q^{21} -9.49884 q^{23} -113.041 q^{25} +335.372 q^{27} -6.00000 q^{29} -23.4176 q^{31} -548.578 q^{33} +91.2100 q^{35} +44.9571 q^{37} -122.957 q^{39} +58.0812 q^{41} -457.955 q^{43} +215.995 q^{45} +496.208 q^{47} +352.625 q^{49} +834.289 q^{51} +601.747 q^{53} -200.578 q^{55} -944.244 q^{57} -211.336 q^{59} +414.912 q^{61} +1647.32 q^{63} -44.9571 q^{65} -744.167 q^{67} -89.8422 q^{69} -882.537 q^{71} -319.499 q^{73} -1069.16 q^{75} -1529.73 q^{77} +490.664 q^{79} +1485.66 q^{81} +224.420 q^{83} +305.043 q^{85} -56.7494 q^{87} -580.501 q^{89} -342.871 q^{91} -221.490 q^{93} -345.246 q^{95} +403.499 q^{97} -3622.58 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 11 q^{5} - q^{7} + 107 q^{9}+O(q^{10}) $ 2 * q + q^3 - 11 * q^5 - q^7 + 107 * q^9	$ 2 q + q^{3} - 11 q^{5} - q^{7} + 107 q^{9} - 116 q^{11} - 26 q^{13} + 155 q^{15} + 51 q^{17} - 128 q^{19} + 481 q^{21} + 196 q^{23} - 29 q^{25} + 187 q^{27} - 12 q^{29} - 226 q^{31} - 58 q^{33} + 487 q^{35} - 143 q^{37} - 13 q^{39} - 278 q^{41} - 253 q^{43} - 428 q^{45} + 867 q^{47} + 759 q^{49} + 1149 q^{51} + 666 q^{53} + 638 q^{55} - 706 q^{57} - 996 q^{59} - 66 q^{61} + 428 q^{63} + 143 q^{65} - 1560 q^{67} - 1828 q^{69} - 923 q^{71} - 424 q^{73} - 1780 q^{75} + 58 q^{77} + 408 q^{79} + 1538 q^{81} + 1058 q^{83} + 843 q^{85} - 6 q^{87} - 1376 q^{89} + 13 q^{91} + 1492 q^{93} + 62 q^{95} + 592 q^{97} - 6206 q^{99}+O(q^{100}) $ 2 * q + q^3 - 11 * q^5 - q^7 + 107 * q^9 - 116 * q^11 - 26 * q^13 + 155 * q^15 + 51 * q^17 - 128 * q^19 + 481 * q^21 + 196 * q^23 - 29 * q^25 + 187 * q^27 - 12 * q^29 - 226 * q^31 - 58 * q^33 + 487 * q^35 - 143 * q^37 - 13 * q^39 - 278 * q^41 - 253 * q^43 - 428 * q^45 + 867 * q^47 + 759 * q^49 + 1149 * q^51 + 666 * q^53 + 638 * q^55 - 706 * q^57 - 996 * q^59 - 66 * q^61 + 428 * q^63 + 143 * q^65 - 1560 * q^67 - 1828 * q^69 - 923 * q^71 - 424 * q^73 - 1780 * q^75 + 58 * q^77 + 408 * q^79 + 1538 * q^81 + 1058 * q^83 + 843 * q^85 - 6 * q^87 - 1376 * q^89 + 13 * q^91 + 1492 * q^93 + 62 * q^95 + 592 * q^97 - 6206 * 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
$3$	9.45824	1.82024	0.910119	−	0.414346i	$-0.135990\pi$
$3$	9.45824	1.82024	0.910119	+	0.414346i	$0.135990\pi$
$4$	0	0
$5$	3.45824	0.309314	0.154657	−	0.987968i	$-0.450573\pi$
$5$	3.45824	0.309314	0.154657	+	0.987968i	$0.450573\pi$
$6$	0	0
$7$	26.3747	1.42410	0.712050	−	0.702129i	$-0.247767\pi$
$7$	26.3747	1.42410	0.712050	+	0.702129i	$0.247767\pi$
$8$	0	0
$9$	62.4582	2.31327
$10$	0	0
$11$	−58.0000	−1.58979	−0.794894	−	0.606749i	$-0.792473\pi$
$11$	−58.0000	−1.58979	−0.794894	+	0.606749i	$0.792473\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	32.7088	0.563025
$16$	0	0
$17$	88.2077	1.25844	0.629221	−	0.777227i	$-0.283374\pi$
$17$	88.2077	1.25844	0.629221	+	0.777227i	$0.283374\pi$
$18$	0	0
$19$	−99.8329	−1.20543	−0.602717	−	0.797955i	$-0.705915\pi$
$19$	−99.8329	−1.20543	−0.602717	+	0.797955i	$0.705915\pi$
$20$	0	0
$21$	249.458	2.59220
$22$	0	0
$23$	−9.49884	−0.0861150	−0.0430575	−	0.999073i	$-0.513710\pi$
$23$	−9.49884	−0.0861150	−0.0430575	+	0.999073i	$0.513710\pi$
$24$	0	0
$25$	−113.041	−0.904325
$26$	0	0
$27$	335.372	2.39046
$28$	0	0
$29$	−6.00000	−0.0384197	−0.0192099	−	0.999815i	$-0.506115\pi$
$29$	−6.00000	−0.0384197	−0.0192099	+	0.999815i	$0.506115\pi$
$30$	0	0
$31$	−23.4176	−0.135675	−0.0678376	−	0.997696i	$-0.521610\pi$
$31$	−23.4176	−0.135675	−0.0678376	+	0.997696i	$0.521610\pi$
$32$	0	0
$33$	−548.578	−2.89379
$34$	0	0
$35$	91.2100	0.440494
$36$	0	0
$37$	44.9571	0.199754	0.0998770	−	0.995000i	$-0.468155\pi$
$37$	44.9571	0.199754	0.0998770	+	0.995000i	$0.468155\pi$
$38$	0	0
$39$	−122.957	−0.504843
$40$	0	0
$41$	58.0812	0.221238	0.110619	−	0.993863i	$-0.464717\pi$
$41$	58.0812	0.221238	0.110619	+	0.993863i	$0.464717\pi$
$42$	0	0
$43$	−457.955	−1.62413	−0.812063	−	0.583569i	$-0.801656\pi$
$43$	−457.955	−1.62413	−0.812063	+	0.583569i	$0.801656\pi$
$44$	0	0
$45$	215.995	0.715526
$46$	0	0
$47$	496.208	1.53999	0.769993	−	0.638053i	$-0.220260\pi$
$47$	496.208	1.53999	0.769993	+	0.638053i	$0.220260\pi$
$48$	0	0
$49$	352.625	1.02806
$50$	0	0
$51$	834.289	2.29066
$52$	0	0
$53$	601.747	1.55955	0.779777	−	0.626058i	$-0.215333\pi$
$53$	601.747	1.55955	0.779777	+	0.626058i	$0.215333\pi$
$54$	0	0
$55$	−200.578	−0.491744
$56$	0	0
$57$	−944.244	−2.19418
$58$	0	0
$59$	−211.336	−0.466333	−0.233167	−	0.972437i	$-0.574909\pi$
$59$	−211.336	−0.466333	−0.233167	+	0.972437i	$0.574909\pi$
$60$	0	0
$61$	414.912	0.870885	0.435443	−	0.900216i	$-0.356592\pi$
$61$	414.912	0.870885	0.435443	+	0.900216i	$0.356592\pi$
$62$	0	0
$63$	1647.32	3.29433
$64$	0	0
$65$	−44.9571	−0.0857883
$66$	0	0
$67$	−744.167	−1.35693	−0.678466	−	0.734632i	$-0.737355\pi$
$67$	−744.167	−1.35693	−0.678466	+	0.734632i	$0.737355\pi$
$68$	0	0
$69$	−89.8422	−0.156750
$70$	0	0
$71$	−882.537	−1.47518	−0.737591	−	0.675248i	$-0.764037\pi$
$71$	−882.537	−1.47518	−0.737591	+	0.675248i	$0.764037\pi$
$72$	0	0
$73$	−319.499	−0.512254	−0.256127	−	0.966643i	$-0.582446\pi$
$73$	−319.499	−0.512254	−0.256127	+	0.966643i	$0.582446\pi$
$74$	0	0
$75$	−1069.16	−1.64609
$76$	0	0
$77$	−1529.73	−2.26402
$78$	0	0
$79$	490.664	0.698784	0.349392	−	0.936977i	$-0.386388\pi$
$79$	490.664	0.698784	0.349392	+	0.936977i	$0.386388\pi$
$80$	0	0
$81$	1485.66	2.03794
$82$	0	0
$83$	224.420	0.296787	0.148393	−	0.988928i	$-0.452590\pi$
$83$	224.420	0.296787	0.148393	+	0.988928i	$0.452590\pi$
$84$	0	0
$85$	305.043	0.389254
$86$	0	0
$87$	−56.7494	−0.0699331
$88$	0	0
$89$	−580.501	−0.691382	−0.345691	−	0.938348i	$-0.612355\pi$
$89$	−580.501	−0.691382	−0.345691	+	0.938348i	$0.612355\pi$
$90$	0	0
$91$	−342.871	−0.394974
$92$	0	0
$93$	−221.490	−0.246961
$94$	0	0
$95$	−345.246	−0.372858
$96$	0	0
$97$	403.499	0.422362	0.211181	−	0.977447i	$-0.432269\pi$
$97$	403.499	0.422362	0.211181	+	0.977447i	$0.432269\pi$
$98$	0	0
$99$	−3622.58	−3.67760
$100$	0	0
$101$	−698.244	−0.687899	−0.343950	−	0.938988i	$-0.611765\pi$
$101$	−698.244	−0.687899	−0.343950	+	0.938988i	$0.611765\pi$
$102$	0	0
$103$	−1285.33	−1.22958	−0.614791	−	0.788690i	$-0.710760\pi$
$103$	−1285.33	−1.22958	−0.614791	+	0.788690i	$0.710760\pi$
$104$	0	0
$105$	862.686	0.801805
$106$	0	0
$107$	−1329.82	−1.20149	−0.600743	−	0.799443i	$-0.705128\pi$
$107$	−1329.82	−1.20149	−0.600743	+	0.799443i	$0.705128\pi$
$108$	0	0
$109$	−951.034	−0.835711	−0.417856	−	0.908513i	$-0.637218\pi$
$109$	−951.034	−0.835711	−0.417856	+	0.908513i	$0.637218\pi$
$110$	0	0
$111$	425.215	0.363600
$112$	0	0
$113$	89.8283	0.0747817	0.0373909	−	0.999301i	$-0.488095\pi$
$113$	89.8283	0.0747817	0.0373909	+	0.999301i	$0.488095\pi$
$114$	0	0
$115$	−32.8492	−0.0266366
$116$	0	0
$117$	−811.957	−0.641585
$118$	0	0
$119$	2326.45	1.79215
$120$	0	0
$121$	2033.00	1.52742
$122$	0	0
$123$	549.346	0.402706
$124$	0	0
$125$	−823.201	−0.589034
$126$	0	0
$127$	−345.012	−0.241062	−0.120531	−	0.992710i	$-0.538460\pi$
$127$	−345.012	−0.241062	−0.120531	+	0.992710i	$0.538460\pi$
$128$	0	0
$129$	−4331.44	−2.95630
$130$	0	0
$131$	2052.03	1.36860	0.684301	−	0.729200i	$-0.260108\pi$
$131$	2052.03	1.36860	0.684301	+	0.729200i	$0.260108\pi$
$132$	0	0
$133$	−2633.06	−1.71666
$134$	0	0
$135$	1159.80	0.739403
$136$	0	0
$137$	1952.58	1.21767	0.608833	−	0.793298i	$-0.291638\pi$
$137$	1952.58	1.21767	0.608833	+	0.793298i	$0.291638\pi$
$138$	0	0
$139$	2238.69	1.36606	0.683032	−	0.730389i	$-0.260661\pi$
$139$	2238.69	1.36606	0.683032	+	0.730389i	$0.260661\pi$
$140$	0	0
$141$	4693.25	2.80314
$142$	0	0
$143$	754.000	0.440928
$144$	0	0
$145$	−20.7494	−0.0118838
$146$	0	0
$147$	3335.21	1.87132
$148$	0	0
$149$	−1097.81	−0.603598	−0.301799	−	0.953372i	$-0.597587\pi$
$149$	−1097.81	−0.603598	−0.301799	+	0.953372i	$0.597587\pi$
$150$	0	0
$151$	−3593.17	−1.93648	−0.968239	−	0.250026i	$-0.919561\pi$
$151$	−3593.17	−1.93648	−0.968239	+	0.250026i	$0.919561\pi$
$152$	0	0
$153$	5509.29	2.91111
$154$	0	0
$155$	−80.9837	−0.0419663
$156$	0	0
$157$	−2517.48	−1.27972	−0.639861	−	0.768490i	$-0.721008\pi$
$157$	−2517.48	−1.27972	−0.639861	+	0.768490i	$0.721008\pi$
$158$	0	0
$159$	5691.47	2.83876
$160$	0	0
$161$	−250.529	−0.122636
$162$	0	0
$163$	251.084	0.120653	0.0603263	−	0.998179i	$-0.480786\pi$
$163$	251.084	0.120653	0.0603263	+	0.998179i	$0.480786\pi$
$164$	0	0
$165$	−1897.11	−0.895090
$166$	0	0
$167$	450.596	0.208792	0.104396	−	0.994536i	$-0.466709\pi$
$167$	450.596	0.208792	0.104396	+	0.994536i	$0.466709\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−6235.39	−2.78849
$172$	0	0
$173$	−247.914	−0.108951	−0.0544756	−	0.998515i	$-0.517349\pi$
$173$	−247.914	−0.108951	−0.0544756	+	0.998515i	$0.517349\pi$
$174$	0	0
$175$	−2981.41	−1.28785
$176$	0	0
$177$	−1998.87	−0.848838
$178$	0	0
$179$	1266.87	0.528995	0.264497	−	0.964386i	$-0.414794\pi$
$179$	1266.87	0.528995	0.264497	+	0.964386i	$0.414794\pi$
$180$	0	0
$181$	4048.38	1.66251	0.831254	−	0.555893i	$-0.187624\pi$
$181$	4048.38	1.66251	0.831254	+	0.555893i	$0.187624\pi$
$182$	0	0
$183$	3924.33	1.58522
$184$	0	0
$185$	155.472	0.0617867
$186$	0	0
$187$	−5116.04	−2.00065
$188$	0	0
$189$	8845.35	3.40426
$190$	0	0
$191$	4021.38	1.52344	0.761720	−	0.647907i	$-0.224355\pi$
$191$	4021.38	1.52344	0.761720	+	0.647907i	$0.224355\pi$
$192$	0	0
$193$	−1529.58	−0.570476	−0.285238	−	0.958457i	$-0.592073\pi$
$193$	−1529.58	−0.570476	−0.285238	+	0.958457i	$0.592073\pi$
$194$	0	0
$195$	−425.215	−0.156155
$196$	0	0
$197$	1146.04	0.414477	0.207239	−	0.978290i	$-0.433552\pi$
$197$	1146.04	0.414477	0.207239	+	0.978290i	$0.433552\pi$
$198$	0	0
$199$	5190.08	1.84882	0.924409	−	0.381403i	$-0.124559\pi$
$199$	5190.08	1.84882	0.924409	+	0.381403i	$0.124559\pi$
$200$	0	0
$201$	−7038.51	−2.46994
$202$	0	0
$203$	−158.248	−0.0547135
$204$	0	0
$205$	200.859	0.0684320
$206$	0	0
$207$	−593.281	−0.199207
$208$	0	0
$209$	5790.31	1.91638
$210$	0	0
$211$	−1272.63	−0.415219	−0.207610	−	0.978212i	$-0.566568\pi$
$211$	−1272.63	−0.415219	−0.207610	+	0.978212i	$0.566568\pi$
$212$	0	0
$213$	−8347.24	−2.68518
$214$	0	0
$215$	−1583.72	−0.502365
$216$	0	0
$217$	−617.633	−0.193215
$218$	0	0
$219$	−3021.90	−0.932424
$220$	0	0
$221$	−1146.70	−0.349029
$222$	0	0
$223$	−3340.87	−1.00323	−0.501617	−	0.865090i	$-0.667261\pi$
$223$	−3340.87	−1.00323	−0.501617	+	0.865090i	$0.667261\pi$
$224$	0	0
$225$	−7060.32	−2.09195
$226$	0	0
$227$	3577.07	1.04590	0.522948	−	0.852365i	$-0.324832\pi$
$227$	3577.07	1.04590	0.522948	+	0.852365i	$0.324832\pi$
$228$	0	0
$229$	6215.35	1.79355	0.896773	−	0.442492i	$-0.145905\pi$
$229$	6215.35	1.79355	0.896773	+	0.442492i	$0.145905\pi$
$230$	0	0
$231$	−14468.6	−4.12105
$232$	0	0
$233$	2163.02	0.608171	0.304085	−	0.952645i	$-0.401649\pi$
$233$	2163.02	0.608171	0.304085	+	0.952645i	$0.401649\pi$
$234$	0	0
$235$	1716.00	0.476339
$236$	0	0
$237$	4640.81	1.27195
$238$	0	0
$239$	2908.51	0.787180	0.393590	−	0.919286i	$-0.371233\pi$
$239$	2908.51	0.787180	0.393590	+	0.919286i	$0.371233\pi$
$240$	0	0
$241$	1391.63	0.371963	0.185981	−	0.982553i	$-0.440454\pi$
$241$	1391.63	0.371963	0.185981	+	0.982553i	$0.440454\pi$
$242$	0	0
$243$	4996.66	1.31908
$244$	0	0
$245$	1219.46	0.317994
$246$	0	0
$247$	1297.83	0.334327
$248$	0	0
$249$	2122.62	0.540222
$250$	0	0
$251$	−2634.12	−0.662407	−0.331204	−	0.943559i	$-0.607455\pi$
$251$	−2634.12	−0.662407	−0.331204	+	0.943559i	$0.607455\pi$
$252$	0	0
$253$	550.933	0.136904
$254$	0	0
$255$	2885.17	0.708534
$256$	0	0
$257$	441.841	0.107242	0.0536212	−	0.998561i	$-0.482924\pi$
$257$	441.841	0.107242	0.0536212	+	0.998561i	$0.482924\pi$
$258$	0	0
$259$	1185.73	0.284470
$260$	0	0
$261$	−374.749	−0.0888751
$262$	0	0
$263$	2100.32	0.492439	0.246220	−	0.969214i	$-0.420812\pi$
$263$	2100.32	0.492439	0.246220	+	0.969214i	$0.420812\pi$
$264$	0	0
$265$	2080.98	0.482392
$266$	0	0
$267$	−5490.52	−1.25848
$268$	0	0
$269$	6432.40	1.45796	0.728978	−	0.684537i	$-0.239995\pi$
$269$	6432.40	1.45796	0.728978	+	0.684537i	$0.239995\pi$
$270$	0	0
$271$	−3093.14	−0.693339	−0.346669	−	0.937987i	$-0.612687\pi$
$271$	−3093.14	−0.693339	−0.346669	+	0.937987i	$0.612687\pi$
$272$	0	0
$273$	−3242.96	−0.718948
$274$	0	0
$275$	6556.35	1.43768
$276$	0	0
$277$	−1082.57	−0.234820	−0.117410	−	0.993084i	$-0.537459\pi$
$277$	−1082.57	−0.234820	−0.117410	+	0.993084i	$0.537459\pi$
$278$	0	0
$279$	−1462.62	−0.313853
$280$	0	0
$281$	4885.85	1.03724	0.518621	−	0.855004i	$-0.326445\pi$
$281$	4885.85	1.03724	0.518621	+	0.855004i	$0.326445\pi$
$282$	0	0
$283$	8190.58	1.72042	0.860211	−	0.509938i	$-0.170332\pi$
$283$	8190.58	1.72042	0.860211	+	0.509938i	$0.170332\pi$
$284$	0	0
$285$	−3265.42	−0.678690
$286$	0	0
$287$	1531.87	0.315065
$288$	0	0
$289$	2867.59	0.583674
$290$	0	0
$291$	3816.39	0.768799
$292$	0	0
$293$	−1167.54	−0.232793	−0.116397	−	0.993203i	$-0.537134\pi$
$293$	−1167.54	−0.232793	−0.116397	+	0.993203i	$0.537134\pi$
$294$	0	0
$295$	−730.851	−0.144243
$296$	0	0
$297$	−19451.6	−3.80032
$298$	0	0
$299$	123.485	0.0238840
$300$	0	0
$301$	−12078.4	−2.31292
$302$	0	0
$303$	−6604.15	−1.25214
$304$	0	0
$305$	1434.86	0.269377
$306$	0	0
$307$	−3928.01	−0.730240	−0.365120	−	0.930961i	$-0.618972\pi$
$307$	−3928.01	−0.730240	−0.365120	+	0.930961i	$0.618972\pi$
$308$	0	0
$309$	−12156.9	−2.23813
$310$	0	0
$311$	2848.86	0.519433	0.259717	−	0.965685i	$-0.416371\pi$
$311$	2848.86	0.519433	0.259717	+	0.965685i	$0.416371\pi$
$312$	0	0
$313$	−5990.54	−1.08181	−0.540903	−	0.841085i	$-0.681917\pi$
$313$	−5990.54	−1.08181	−0.540903	+	0.841085i	$0.681917\pi$
$314$	0	0
$315$	5696.81	1.01898
$316$	0	0
$317$	−3521.83	−0.623992	−0.311996	−	0.950083i	$-0.600998\pi$
$317$	−3521.83	−0.623992	−0.311996	+	0.950083i	$0.600998\pi$
$318$	0	0
$319$	348.000	0.0610792
$320$	0	0
$321$	−12577.8	−2.18699
$322$	0	0
$323$	−8806.03	−1.51697
$324$	0	0
$325$	1469.53	0.250815
$326$	0	0
$327$	−8995.10	−1.52119
$328$	0	0
$329$	13087.3	2.19309
$330$	0	0
$331$	−1487.38	−0.246990	−0.123495	−	0.992345i	$-0.539410\pi$
$331$	−1487.38	−0.246990	−0.123495	+	0.992345i	$0.539410\pi$
$332$	0	0
$333$	2807.94	0.462085
$334$	0	0
$335$	−2573.51	−0.419718
$336$	0	0
$337$	−2559.21	−0.413676	−0.206838	−	0.978375i	$-0.566317\pi$
$337$	−2559.21	−0.413676	−0.206838	+	0.978375i	$0.566317\pi$
$338$	0	0
$339$	849.617	0.136121
$340$	0	0
$341$	1358.22	0.215695
$342$	0	0
$343$	253.864	0.0399632
$344$	0	0
$345$	−310.696	−0.0484849
$346$	0	0
$347$	−11229.6	−1.73728	−0.868639	−	0.495446i	$-0.835004\pi$
$347$	−11229.6	−1.73728	−0.868639	+	0.495446i	$0.835004\pi$
$348$	0	0
$349$	2570.45	0.394249	0.197124	−	0.980378i	$-0.436840\pi$
$349$	2570.45	0.394249	0.197124	+	0.980378i	$0.436840\pi$
$350$	0	0
$351$	−4359.84	−0.662995
$352$	0	0
$353$	5401.58	0.814439	0.407219	−	0.913330i	$-0.366498\pi$
$353$	5401.58	0.814439	0.407219	+	0.913330i	$0.366498\pi$
$354$	0	0
$355$	−3052.02	−0.456294
$356$	0	0
$357$	22004.1	3.26213
$358$	0	0
$359$	−2650.71	−0.389691	−0.194846	−	0.980834i	$-0.562421\pi$
$359$	−2650.71	−0.389691	−0.194846	+	0.980834i	$0.562421\pi$
$360$	0	0
$361$	3107.62	0.453071
$362$	0	0
$363$	19228.6	2.78027
$364$	0	0
$365$	−1104.90	−0.158447
$366$	0	0
$367$	7298.53	1.03809	0.519046	−	0.854746i	$-0.326287\pi$
$367$	7298.53	1.03809	0.519046	+	0.854746i	$0.326287\pi$
$368$	0	0
$369$	3627.65	0.511783
$370$	0	0
$371$	15870.9	2.22096
$372$	0	0
$373$	−8960.03	−1.24379	−0.621894	−	0.783102i	$-0.713636\pi$
$373$	−8960.03	−1.24379	−0.621894	+	0.783102i	$0.713636\pi$
$374$	0	0
$375$	−7786.03	−1.07218
$376$	0	0
$377$	78.0000	0.0106557
$378$	0	0
$379$	3470.36	0.470344	0.235172	−	0.971954i	$-0.424435\pi$
$379$	3470.36	0.470344	0.235172	+	0.971954i	$0.424435\pi$
$380$	0	0
$381$	−3263.20	−0.438790
$382$	0	0
$383$	7611.42	1.01547	0.507736	−	0.861513i	$-0.330483\pi$
$383$	7611.42	1.01547	0.507736	+	0.861513i	$0.330483\pi$
$384$	0	0
$385$	−5290.18	−0.700292
$386$	0	0
$387$	−28603.0	−3.75704
$388$	0	0
$389$	−9394.26	−1.22444	−0.612221	−	0.790687i	$-0.709724\pi$
$389$	−9394.26	−1.22444	−0.612221	+	0.790687i	$0.709724\pi$
$390$	0	0
$391$	−837.870	−0.108371
$392$	0	0
$393$	19408.6	2.49118
$394$	0	0
$395$	1696.83	0.216144
$396$	0	0
$397$	8377.68	1.05910	0.529551	−	0.848278i	$-0.322360\pi$
$397$	8377.68	1.05910	0.529551	+	0.848278i	$0.322360\pi$
$398$	0	0
$399$	−24904.2	−3.12473
$400$	0	0
$401$	−1171.50	−0.145890	−0.0729448	−	0.997336i	$-0.523240\pi$
$401$	−1171.50	−0.145890	−0.0729448	+	0.997336i	$0.523240\pi$
$402$	0	0
$403$	304.429	0.0376295
$404$	0	0
$405$	5137.76	0.630364
$406$	0	0
$407$	−2607.51	−0.317566
$408$	0	0
$409$	1320.85	0.159687	0.0798434	−	0.996807i	$-0.474558\pi$
$409$	1320.85	0.159687	0.0798434	+	0.996807i	$0.474558\pi$
$410$	0	0
$411$	18468.0	2.21644
$412$	0	0
$413$	−5573.94	−0.664105
$414$	0	0
$415$	776.097	0.0918003
$416$	0	0
$417$	21174.0	2.48656
$418$	0	0
$419$	1553.50	0.181129	0.0905647	−	0.995891i	$-0.471133\pi$
$419$	1553.50	0.181129	0.0905647	+	0.995891i	$0.471133\pi$
$420$	0	0
$421$	−6746.63	−0.781022	−0.390511	−	0.920598i	$-0.627702\pi$
$421$	−6746.63	−0.781022	−0.390511	+	0.920598i	$0.627702\pi$
$422$	0	0
$423$	30992.3	3.56240
$424$	0	0
$425$	−9971.05	−1.13804
$426$	0	0
$427$	10943.2	1.24023
$428$	0	0
$429$	7131.51	0.802593
$430$	0	0
$431$	2646.81	0.295805	0.147903	−	0.989002i	$-0.452748\pi$
$431$	2646.81	0.295805	0.147903	+	0.989002i	$0.452748\pi$
$432$	0	0
$433$	9197.82	1.02083	0.510415	−	0.859928i	$-0.329492\pi$
$433$	9197.82	1.02083	0.510415	+	0.859928i	$0.329492\pi$
$434$	0	0
$435$	−196.253	−0.0216313
$436$	0	0
$437$	948.297	0.103806
$438$	0	0
$439$	9360.53	1.01766	0.508831	−	0.860866i	$-0.330078\pi$
$439$	9360.53	1.01766	0.508831	+	0.860866i	$0.330078\pi$
$440$	0	0
$441$	22024.4	2.37818
$442$	0	0
$443$	−3196.32	−0.342803	−0.171401	−	0.985201i	$-0.554830\pi$
$443$	−3196.32	−0.342803	−0.171401	+	0.985201i	$0.554830\pi$
$444$	0	0
$445$	−2007.51	−0.213854
$446$	0	0
$447$	−10383.3	−1.09869
$448$	0	0
$449$	−2844.45	−0.298970	−0.149485	−	0.988764i	$-0.547762\pi$
$449$	−2844.45	−0.298970	−0.149485	+	0.988764i	$0.547762\pi$
$450$	0	0
$451$	−3368.71	−0.351721
$452$	0	0
$453$	−33985.1	−3.52485
$454$	0	0
$455$	−1185.73	−0.122171
$456$	0	0
$457$	−4468.90	−0.457431	−0.228716	−	0.973493i	$-0.573453\pi$
$457$	−4468.90	−0.457431	−0.228716	+	0.973493i	$0.573453\pi$
$458$	0	0
$459$	29582.4	3.00825
$460$	0	0
$461$	15006.8	1.51613	0.758067	−	0.652177i	$-0.226144\pi$
$461$	15006.8	1.51613	0.758067	+	0.652177i	$0.226144\pi$
$462$	0	0
$463$	−6247.37	−0.627084	−0.313542	−	0.949574i	$-0.601516\pi$
$463$	−6247.37	−0.627084	−0.313542	+	0.949574i	$0.601516\pi$
$464$	0	0
$465$	−765.963	−0.0763886
$466$	0	0
$467$	977.327	0.0968422	0.0484211	−	0.998827i	$-0.484581\pi$
$467$	977.327	0.0968422	0.0484211	+	0.998827i	$0.484581\pi$
$468$	0	0
$469$	−19627.2	−1.93241
$470$	0	0
$471$	−23810.9	−2.32940
$472$	0	0
$473$	26561.4	2.58202
$474$	0	0
$475$	11285.2	1.09010
$476$	0	0
$477$	37584.1	3.60766
$478$	0	0
$479$	−8832.44	−0.842515	−0.421257	−	0.906941i	$-0.638411\pi$
$479$	−8832.44	−0.842515	−0.421257	+	0.906941i	$0.638411\pi$
$480$	0	0
$481$	−584.442	−0.0554018
$482$	0	0
$483$	−2369.56	−0.223227
$484$	0	0
$485$	1395.39	0.130642
$486$	0	0
$487$	9906.44	0.921774	0.460887	−	0.887459i	$-0.347531\pi$
$487$	9906.44	0.921774	0.460887	+	0.887459i	$0.347531\pi$
$488$	0	0
$489$	2374.81	0.219617
$490$	0	0
$491$	−11073.6	−1.01781	−0.508906	−	0.860822i	$-0.669950\pi$
$491$	−11073.6	−1.01781	−0.508906	+	0.860822i	$0.669950\pi$
$492$	0	0
$493$	−529.246	−0.0483490
$494$	0	0
$495$	−12527.7	−1.13753
$496$	0	0
$497$	−23276.7	−2.10081
$498$	0	0
$499$	3268.09	0.293186	0.146593	−	0.989197i	$-0.453169\pi$
$499$	3268.09	0.293186	0.146593	+	0.989197i	$0.453169\pi$
$500$	0	0
$501$	4261.85	0.380050
$502$	0	0
$503$	18758.6	1.66283	0.831416	−	0.555651i	$-0.187531\pi$
$503$	18758.6	1.66283	0.831416	+	0.555651i	$0.187531\pi$
$504$	0	0
$505$	−2414.69	−0.212777
$506$	0	0
$507$	1598.44	0.140018
$508$	0	0
$509$	6121.28	0.533047	0.266523	−	0.963828i	$-0.414125\pi$
$509$	6121.28	0.533047	0.266523	+	0.963828i	$0.414125\pi$
$510$	0	0
$511$	−8426.69	−0.729501
$512$	0	0
$513$	−33481.2	−2.88154
$514$	0	0
$515$	−4444.97	−0.380327
$516$	0	0
$517$	−28780.0	−2.44825
$518$	0	0
$519$	−2344.83	−0.198317
$520$	0	0
$521$	−12976.1	−1.09116	−0.545578	−	0.838060i	$-0.683690\pi$
$521$	−12976.1	−1.09116	−0.545578	+	0.838060i	$0.683690\pi$
$522$	0	0
$523$	11563.9	0.966832	0.483416	−	0.875391i	$-0.339396\pi$
$523$	11563.9	0.966832	0.483416	+	0.875391i	$0.339396\pi$
$524$	0	0
$525$	−28198.9	−2.34419
$526$	0	0
$527$	−2065.61	−0.170739
$528$	0	0
$529$	−12076.8	−0.992584
$530$	0	0
$531$	−13199.7	−1.07875
$532$	0	0
$533$	−755.056	−0.0613604
$534$	0	0
$535$	−4598.84	−0.371636
$536$	0	0
$537$	11982.3	0.962897
$538$	0	0
$539$	−20452.3	−1.63440
$540$	0	0
$541$	−11231.9	−0.892603	−0.446302	−	0.894883i	$-0.647259\pi$
$541$	−11231.9	−0.892603	−0.446302	+	0.894883i	$0.647259\pi$
$542$	0	0
$543$	38290.6	3.02616
$544$	0	0
$545$	−3288.90	−0.258497
$546$	0	0
$547$	−5604.16	−0.438056	−0.219028	−	0.975719i	$-0.570289\pi$
$547$	−5604.16	−0.438056	−0.219028	+	0.975719i	$0.570289\pi$
$548$	0	0
$549$	25914.7	2.01459
$550$	0	0
$551$	598.998	0.0463124
$552$	0	0
$553$	12941.1	0.995139
$554$	0	0
$555$	1470.49	0.112467
$556$	0	0
$557$	13863.4	1.05460	0.527299	−	0.849680i	$-0.323205\pi$
$557$	13863.4	1.05460	0.527299	+	0.849680i	$0.323205\pi$
$558$	0	0
$559$	5953.41	0.450452
$560$	0	0
$561$	−48388.8	−3.64167
$562$	0	0
$563$	−9602.64	−0.718833	−0.359417	−	0.933177i	$-0.617024\pi$
$563$	−9602.64	−0.718833	−0.359417	+	0.933177i	$0.617024\pi$
$564$	0	0
$565$	310.647	0.0231310
$566$	0	0
$567$	39183.8	2.90223
$568$	0	0
$569$	−21102.9	−1.55480	−0.777399	−	0.629008i	$-0.783461\pi$
$569$	−21102.9	−1.55480	−0.777399	+	0.629008i	$0.783461\pi$
$570$	0	0
$571$	−20570.0	−1.50758	−0.753790	−	0.657115i	$-0.771776\pi$
$571$	−20570.0	−1.50758	−0.753790	+	0.657115i	$0.771776\pi$
$572$	0	0
$573$	38035.2	2.77302
$574$	0	0
$575$	1073.75	0.0778759
$576$	0	0
$577$	7896.95	0.569765	0.284882	−	0.958562i	$-0.408045\pi$
$577$	7896.95	0.569765	0.284882	+	0.958562i	$0.408045\pi$
$578$	0	0
$579$	−14467.2	−1.03840
$580$	0	0
$581$	5919.01	0.422654
$582$	0	0
$583$	−34901.3	−2.47936
$584$	0	0
$585$	−2807.94	−0.198451
$586$	0	0
$587$	15872.7	1.11608	0.558039	−	0.829815i	$-0.311554\pi$
$587$	15872.7	1.11608	0.558039	+	0.829815i	$0.311554\pi$
$588$	0	0
$589$	2337.85	0.163548
$590$	0	0
$591$	10839.5	0.754447
$592$	0	0
$593$	−4037.35	−0.279585	−0.139793	−	0.990181i	$-0.544644\pi$
$593$	−4037.35	−0.279585	−0.139793	+	0.990181i	$0.544644\pi$
$594$	0	0
$595$	8045.42	0.554336
$596$	0	0
$597$	49089.0	3.36529
$598$	0	0
$599$	18225.7	1.24321	0.621606	−	0.783330i	$-0.286481\pi$
$599$	18225.7	1.24321	0.621606	+	0.783330i	$0.286481\pi$
$600$	0	0
$601$	5555.22	0.377042	0.188521	−	0.982069i	$-0.439631\pi$
$601$	5555.22	0.377042	0.188521	+	0.982069i	$0.439631\pi$
$602$	0	0
$603$	−46479.4	−3.13895
$604$	0	0
$605$	7030.59	0.472453
$606$	0	0
$607$	−4083.66	−0.273066	−0.136533	−	0.990636i	$-0.543596\pi$
$607$	−4083.66	−0.273066	−0.136533	+	0.990636i	$0.543596\pi$
$608$	0	0
$609$	−1496.75	−0.0995917
$610$	0	0
$611$	−6450.70	−0.427115
$612$	0	0
$613$	1065.87	0.0702285	0.0351143	−	0.999383i	$-0.488820\pi$
$613$	1065.87	0.0702285	0.0351143	+	0.999383i	$0.488820\pi$
$614$	0	0
$615$	1899.77	0.124563
$616$	0	0
$617$	−3183.51	−0.207720	−0.103860	−	0.994592i	$-0.533119\pi$
$617$	−3183.51	−0.207720	−0.103860	+	0.994592i	$0.533119\pi$
$618$	0	0
$619$	3361.17	0.218250	0.109125	−	0.994028i	$-0.465195\pi$
$619$	3361.17	0.218250	0.109125	+	0.994028i	$0.465195\pi$
$620$	0	0
$621$	−3185.65	−0.205854
$622$	0	0
$623$	−15310.5	−0.984597
$624$	0	0
$625$	11283.3	0.722128
$626$	0	0
$627$	54766.1	3.48828
$628$	0	0
$629$	3965.56	0.251379
$630$	0	0
$631$	−17306.6	−1.09186	−0.545932	−	0.837829i	$-0.683824\pi$
$631$	−17306.6	−1.09186	−0.545932	+	0.837829i	$0.683824\pi$
$632$	0	0
$633$	−12036.8	−0.755798
$634$	0	0
$635$	−1193.13	−0.0745638
$636$	0	0
$637$	−4584.13	−0.285133
$638$	0	0
$639$	−55121.7	−3.41249
$640$	0	0
$641$	20526.1	1.26479	0.632397	−	0.774644i	$-0.282071\pi$
$641$	20526.1	1.26479	0.632397	+	0.774644i	$0.282071\pi$
$642$	0	0
$643$	−17395.8	−1.06691	−0.533456	−	0.845828i	$-0.679107\pi$
$643$	−17395.8	−1.06691	−0.533456	+	0.845828i	$0.679107\pi$
$644$	0	0
$645$	−14979.2	−0.914424
$646$	0	0
$647$	−1475.37	−0.0896486	−0.0448243	−	0.998995i	$-0.514273\pi$
$647$	−1475.37	−0.0896486	−0.0448243	+	0.998995i	$0.514273\pi$
$648$	0	0
$649$	12257.5	0.741371
$650$	0	0
$651$	−5841.72	−0.351698
$652$	0	0
$653$	−4918.78	−0.294773	−0.147387	−	0.989079i	$-0.547086\pi$
$653$	−4918.78	−0.294773	−0.147387	+	0.989079i	$0.547086\pi$
$654$	0	0
$655$	7096.41	0.423328
$656$	0	0
$657$	−19955.3	−1.18498
$658$	0	0
$659$	3938.97	0.232839	0.116419	−	0.993200i	$-0.462858\pi$
$659$	3938.97	0.232839	0.116419	+	0.993200i	$0.462858\pi$
$660$	0	0
$661$	31950.2	1.88006	0.940028	−	0.341097i	$-0.110798\pi$
$661$	31950.2	1.88006	0.940028	+	0.341097i	$0.110798\pi$
$662$	0	0
$663$	−10845.8	−0.635316
$664$	0	0
$665$	−9105.76	−0.530987
$666$	0	0
$667$	56.9930	0.00330851
$668$	0	0
$669$	−31598.7	−1.82612
$670$	0	0
$671$	−24064.9	−1.38452
$672$	0	0
$673$	14912.6	0.854141	0.427070	−	0.904218i	$-0.359546\pi$
$673$	14912.6	0.854141	0.427070	+	0.904218i	$0.359546\pi$
$674$	0	0
$675$	−37910.7	−2.16175
$676$	0	0
$677$	−24876.2	−1.41221	−0.706107	−	0.708105i	$-0.749550\pi$
$677$	−24876.2	−1.41221	−0.706107	+	0.708105i	$0.749550\pi$
$678$	0	0
$679$	10642.2	0.601486
$680$	0	0
$681$	33832.8	1.90378
$682$	0	0
$683$	5819.10	0.326005	0.163003	−	0.986626i	$-0.447882\pi$
$683$	5819.10	0.326005	0.163003	+	0.986626i	$0.447882\pi$
$684$	0	0
$685$	6752.49	0.376642
$686$	0	0
$687$	58786.2	3.26468
$688$	0	0
$689$	−7822.71	−0.432542
$690$	0	0
$691$	23926.4	1.31722	0.658612	−	0.752483i	$-0.271144\pi$
$691$	23926.4	1.31722	0.658612	+	0.752483i	$0.271144\pi$
$692$	0	0
$693$	−95544.4	−5.23728
$694$	0	0
$695$	7741.90	0.422543
$696$	0	0
$697$	5123.21	0.278415
$698$	0	0
$699$	20458.3	1.10702
$700$	0	0
$701$	−11493.3	−0.619251	−0.309626	−	0.950859i	$-0.600204\pi$
$701$	−11493.3	−0.619251	−0.309626	+	0.950859i	$0.600204\pi$
$702$	0	0
$703$	−4488.20	−0.240790
$704$	0	0
$705$	16230.4	0.867051
$706$	0	0
$707$	−18416.0	−0.979638
$708$	0	0
$709$	11011.4	0.583276	0.291638	−	0.956529i	$-0.405800\pi$
$709$	11011.4	0.583276	0.291638	+	0.956529i	$0.405800\pi$
$710$	0	0
$711$	30646.0	1.61648
$712$	0	0
$713$	222.440	0.0116837
$714$	0	0
$715$	2607.51	0.136385
$716$	0	0
$717$	27509.4	1.43286
$718$	0	0
$719$	14546.9	0.754529	0.377265	−	0.926106i	$-0.376865\pi$
$719$	14546.9	0.754529	0.377265	+	0.926106i	$0.376865\pi$
$720$	0	0
$721$	−33900.1	−1.75105
$722$	0	0
$723$	13162.4	0.677061
$724$	0	0
$725$	678.244	0.0347439
$726$	0	0
$727$	−7971.12	−0.406647	−0.203323	−	0.979112i	$-0.565174\pi$
$727$	−7971.12	−0.406647	−0.203323	+	0.979112i	$0.565174\pi$
$728$	0	0
$729$	7146.79	0.363095
$730$	0	0
$731$	−40395.1	−2.04387
$732$	0	0
$733$	−4191.29	−0.211199	−0.105599	−	0.994409i	$-0.533676\pi$
$733$	−4191.29	−0.211199	−0.105599	+	0.994409i	$0.533676\pi$
$734$	0	0
$735$	11534.0	0.578825
$736$	0	0
$737$	43161.7	2.15723
$738$	0	0
$739$	−15062.1	−0.749756	−0.374878	−	0.927074i	$-0.622315\pi$
$739$	−15062.1	−0.749756	−0.374878	+	0.927074i	$0.622315\pi$
$740$	0	0
$741$	12275.2	0.608555
$742$	0	0
$743$	8785.14	0.433776	0.216888	−	0.976197i	$-0.430409\pi$
$743$	8785.14	0.433776	0.216888	+	0.976197i	$0.430409\pi$
$744$	0	0
$745$	−3796.49	−0.186701
$746$	0	0
$747$	14016.9	0.686547
$748$	0	0
$749$	−35073.7	−1.71104
$750$	0	0
$751$	−18437.0	−0.895841	−0.447920	−	0.894073i	$-0.647835\pi$
$751$	−18437.0	−0.895841	−0.447920	+	0.894073i	$0.647835\pi$
$752$	0	0
$753$	−24914.1	−1.20574
$754$	0	0
$755$	−12426.0	−0.598980
$756$	0	0
$757$	−26856.4	−1.28945	−0.644725	−	0.764415i	$-0.723028\pi$
$757$	−26856.4	−1.28945	−0.644725	+	0.764415i	$0.723028\pi$
$758$	0	0
$759$	5210.85	0.249199
$760$	0	0
$761$	−21550.1	−1.02653	−0.513265	−	0.858230i	$-0.671564\pi$
$761$	−21550.1	−1.02653	−0.513265	+	0.858230i	$0.671564\pi$
$762$	0	0
$763$	−25083.2	−1.19014
$764$	0	0
$765$	19052.4	0.900448
$766$	0	0
$767$	2747.37	0.129338
$768$	0	0
$769$	35567.3	1.66787	0.833934	−	0.551864i	$-0.186083\pi$
$769$	35567.3	1.66787	0.833934	+	0.551864i	$0.186083\pi$
$770$	0	0
$771$	4179.04	0.195207
$772$	0	0
$773$	20762.0	0.966053	0.483026	−	0.875606i	$-0.339538\pi$
$773$	20762.0	0.966053	0.483026	+	0.875606i	$0.339538\pi$
$774$	0	0
$775$	2647.14	0.122694
$776$	0	0
$777$	11214.9	0.517803
$778$	0	0
$779$	−5798.42	−0.266688
$780$	0	0
$781$	51187.2	2.34522
$782$	0	0
$783$	−2012.23	−0.0918409
$784$	0	0
$785$	−8706.03	−0.395836
$786$	0	0
$787$	19483.4	0.882475	0.441237	−	0.897390i	$-0.354540\pi$
$787$	19483.4	0.882475	0.441237	+	0.897390i	$0.354540\pi$
$788$	0	0
$789$	19865.4	0.896357
$790$	0	0
$791$	2369.20	0.106497
$792$	0	0
$793$	−5393.85	−0.241540
$794$	0	0
$795$	19682.4	0.878068
$796$	0	0
$797$	−11672.8	−0.518783	−0.259392	−	0.965772i	$-0.583522\pi$
$797$	−11672.8	−0.518783	−0.259392	+	0.965772i	$0.583522\pi$
$798$	0	0
$799$	43769.3	1.93798
$800$	0	0
$801$	−36257.1	−1.59935
$802$	0	0
$803$	18530.9	0.814374
$804$	0	0
$805$	−866.389	−0.0379332
$806$	0	0
$807$	60839.2	2.65383
$808$	0	0
$809$	−18423.5	−0.800663	−0.400331	−	0.916370i	$-0.631105\pi$
$809$	−18423.5	−0.800663	−0.400331	+	0.916370i	$0.631105\pi$
$810$	0	0
$811$	−16287.6	−0.705222	−0.352611	−	0.935770i	$-0.614706\pi$
$811$	−16287.6	−0.705222	−0.352611	+	0.935770i	$0.614706\pi$
$812$	0	0
$813$	−29255.6	−1.26204
$814$	0	0
$815$	868.306	0.0373196
$816$	0	0
$817$	45719.0	1.95778
$818$	0	0
$819$	−21415.1	−0.913682
$820$	0	0
$821$	−26118.8	−1.11030	−0.555148	−	0.831751i	$-0.687339\pi$
$821$	−26118.8	−1.11030	−0.555148	+	0.831751i	$0.687339\pi$
$822$	0	0
$823$	13571.1	0.574798	0.287399	−	0.957811i	$-0.407209\pi$
$823$	13571.1	0.574798	0.287399	+	0.957811i	$0.407209\pi$
$824$	0	0
$825$	62011.6	2.61693
$826$	0	0
$827$	−45842.8	−1.92758	−0.963790	−	0.266662i	$-0.914079\pi$
$827$	−45842.8	−1.92758	−0.963790	+	0.266662i	$0.914079\pi$
$828$	0	0
$829$	2847.06	0.119279	0.0596397	−	0.998220i	$-0.481005\pi$
$829$	2847.06	0.119279	0.0596397	+	0.998220i	$0.481005\pi$
$830$	0	0
$831$	−10239.2	−0.427429
$832$	0	0
$833$	31104.2	1.29376
$834$	0	0
$835$	1558.27	0.0645822
$836$	0	0
$837$	−7853.63	−0.324326
$838$	0	0
$839$	28995.5	1.19313	0.596564	−	0.802565i	$-0.296532\pi$
$839$	28995.5	1.19313	0.596564	+	0.802565i	$0.296532\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	0	0
$843$	46211.5	1.88803
$844$	0	0
$845$	584.442	0.0237934
$846$	0	0
$847$	53619.8	2.17520
$848$	0	0
$849$	77468.4	3.13158
$850$	0	0
$851$	−427.040	−0.0172018
$852$	0	0
$853$	14179.1	0.569148	0.284574	−	0.958654i	$-0.408148\pi$
$853$	14179.1	0.569148	0.284574	+	0.958654i	$0.408148\pi$
$854$	0	0
$855$	−21563.5	−0.862520
$856$	0	0
$857$	−28026.0	−1.11709	−0.558547	−	0.829473i	$-0.688641\pi$
$857$	−28026.0	−1.11709	−0.558547	+	0.829473i	$0.688641\pi$
$858$	0	0
$859$	20995.6	0.833946	0.416973	−	0.908919i	$-0.363091\pi$
$859$	20995.6	0.833946	0.416973	+	0.908919i	$0.363091\pi$
$860$	0	0
$861$	14488.8	0.573494
$862$	0	0
$863$	25139.2	0.991596	0.495798	−	0.868438i	$-0.334876\pi$
$863$	25139.2	0.991596	0.495798	+	0.868438i	$0.334876\pi$
$864$	0	0
$865$	−857.346	−0.0337001
$866$	0	0
$867$	27122.3	1.06243
$868$	0	0
$869$	−28458.5	−1.11092
$870$	0	0
$871$	9674.17	0.376345
$872$	0	0
$873$	25201.8	0.977036
$874$	0	0
$875$	−21711.7	−0.838844
$876$	0	0
$877$	34676.4	1.33516	0.667582	−	0.744537i	$-0.267330\pi$
$877$	34676.4	1.33516	0.667582	+	0.744537i	$0.267330\pi$
$878$	0	0
$879$	−11042.9	−0.423739
$880$	0	0
$881$	−25864.8	−0.989110	−0.494555	−	0.869146i	$-0.664669\pi$
$881$	−25864.8	−0.989110	−0.494555	+	0.869146i	$0.664669\pi$
$882$	0	0
$883$	29347.0	1.11847	0.559233	−	0.829011i	$-0.311096\pi$
$883$	29347.0	1.11847	0.559233	+	0.829011i	$0.311096\pi$
$884$	0	0
$885$	−6912.56	−0.262557
$886$	0	0
$887$	−14292.1	−0.541015	−0.270508	−	0.962718i	$-0.587192\pi$
$887$	−14292.1	−0.541015	−0.270508	+	0.962718i	$0.587192\pi$
$888$	0	0
$889$	−9099.58	−0.343296
$890$	0	0
$891$	−86168.2	−3.23989
$892$	0	0
$893$	−49537.9	−1.85635
$894$	0	0
$895$	4381.12	0.163626
$896$	0	0
$897$	1167.95	0.0434746
$898$	0	0
$899$	140.506	0.00521260
$900$	0	0
$901$	53078.7	1.96261
$902$	0	0
$903$	−114241.	−4.21006
$904$	0	0
$905$	14000.3	0.514237
$906$	0	0
$907$	19319.4	0.707264	0.353632	−	0.935385i	$-0.384946\pi$
$907$	19319.4	0.707264	0.353632	+	0.935385i	$0.384946\pi$
$908$	0	0
$909$	−43611.1	−1.59130
$910$	0	0
$911$	47558.1	1.72961	0.864803	−	0.502112i	$-0.167443\pi$
$911$	47558.1	1.72961	0.864803	+	0.502112i	$0.167443\pi$
$912$	0	0
$913$	−13016.4	−0.471828
$914$	0	0
$915$	13571.3	0.490331
$916$	0	0
$917$	54121.7	1.94903
$918$	0	0
$919$	−49795.2	−1.78737	−0.893684	−	0.448696i	$-0.851889\pi$
$919$	−49795.2	−1.78737	−0.893684	+	0.448696i	$0.851889\pi$
$920$	0	0
$921$	−37152.1	−1.32921
$922$	0	0
$923$	11473.0	0.409142
$924$	0	0
$925$	−5081.97	−0.180642
$926$	0	0
$927$	−80279.3	−2.84435
$928$	0	0
$929$	−18163.8	−0.641479	−0.320739	−	0.947167i	$-0.603931\pi$
$929$	−18163.8	−0.641479	−0.320739	+	0.947167i	$0.603931\pi$
$930$	0	0
$931$	−35203.6	−1.23926
$932$	0	0
$933$	26945.2	0.945493
$934$	0	0
$935$	−17692.5	−0.618830
$936$	0	0
$937$	3915.72	0.136522	0.0682610	−	0.997667i	$-0.478255\pi$
$937$	3915.72	0.136522	0.0682610	+	0.997667i	$0.478255\pi$
$938$	0	0
$939$	−56659.9	−1.96914
$940$	0	0
$941$	−38184.8	−1.32284	−0.661418	−	0.750017i	$-0.730045\pi$
$941$	−38184.8	−1.32284	−0.661418	+	0.750017i	$0.730045\pi$
$942$	0	0
$943$	−551.704	−0.0190519
$944$	0	0
$945$	30589.3	1.05298
$946$	0	0
$947$	35496.7	1.21804	0.609022	−	0.793154i	$-0.291562\pi$
$947$	35496.7	1.21804	0.609022	+	0.793154i	$0.291562\pi$
$948$	0	0
$949$	4153.48	0.142074
$950$	0	0
$951$	−33310.3	−1.13581
$952$	0	0
$953$	51103.0	1.73703	0.868515	−	0.495663i	$-0.165075\pi$
$953$	51103.0	1.73703	0.868515	+	0.495663i	$0.165075\pi$
$954$	0	0
$955$	13906.9	0.471221
$956$	0	0
$957$	3291.47	0.111179
$958$	0	0
$959$	51498.8	1.73408
$960$	0	0
$961$	−29242.6	−0.981592
$962$	0	0
$963$	−83058.4	−2.77936
$964$	0	0
$965$	−5289.67	−0.176456
$966$	0	0
$967$	−52681.0	−1.75192	−0.875961	−	0.482383i	$-0.839771\pi$
$967$	−52681.0	−1.75192	−0.875961	+	0.482383i	$0.839771\pi$
$968$	0	0
$969$	−83289.5	−2.76124
$970$	0	0
$971$	−28878.6	−0.954437	−0.477219	−	0.878785i	$-0.658355\pi$
$971$	−28878.6	−0.954437	−0.477219	+	0.878785i	$0.658355\pi$
$972$	0	0
$973$	59044.7	1.94541
$974$	0	0
$975$	13899.1	0.456542
$976$	0	0
$977$	−20043.8	−0.656355	−0.328178	−	0.944616i	$-0.606434\pi$
$977$	−20043.8	−0.656355	−0.328178	+	0.944616i	$0.606434\pi$
$978$	0	0
$979$	33669.1	1.09915
$980$	0	0
$981$	−59399.9	−1.93322
$982$	0	0
$983$	14937.1	0.484659	0.242330	−	0.970194i	$-0.422088\pi$
$983$	14937.1	0.484659	0.242330	+	0.970194i	$0.422088\pi$
$984$	0	0
$985$	3963.28	0.128204
$986$	0	0
$987$	123783.	3.99195
$988$	0	0
$989$	4350.04	0.139862
$990$	0	0
$991$	−24226.9	−0.776582	−0.388291	−	0.921537i	$-0.626934\pi$
$991$	−24226.9	−0.776582	−0.388291	+	0.921537i	$0.626934\pi$
$992$	0	0
$993$	−14068.0	−0.449581
$994$	0	0
$995$	17948.5	0.571865
$996$	0	0
$997$	−38.6217	−0.00122684	−0.000613421	−	1.00000i	$-0.500195\pi$
$997$	−38.6217	−0.00122684	−0.000613421		1.00000i	$0.500195\pi$
$998$	0	0
$999$	15077.4	0.477504

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.4.a.i.1.2		2
3.2	odd	2		1872.4.a.bh.1.1		2
4.3	odd	2		104.4.a.d.1.1	✓	2
8.3	odd	2		832.4.a.w.1.2		2
8.5	even	2		832.4.a.v.1.1		2
12.11	even	2		936.4.a.i.1.1		2
52.51	odd	2		1352.4.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.4.a.d.1.1	✓	2	4.3	odd	2
208.4.a.i.1.2		2	1.1	even	1	trivial
832.4.a.v.1.1		2	8.5	even	2
832.4.a.w.1.2		2	8.3	odd	2
936.4.a.i.1.1		2	12.11	even	2
1352.4.a.g.1.1		2	52.51	odd	2
1872.4.a.bh.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)

\(f(q)\)	\(=\)	\(q+9.45824 q^{3} +3.45824 q^{5} +26.3747 q^{7} +62.4582 q^{9} +O(q^{10})\)	\(q+9.45824 q^{3} +3.45824 q^{5} +26.3747 q^{7} +62.4582 q^{9} -58.0000 q^{11} -13.0000 q^{13} +32.7088 q^{15} +88.2077 q^{17} -99.8329 q^{19} +249.458 q^{21} -9.49884 q^{23} -113.041 q^{25} +335.372 q^{27} -6.00000 q^{29} -23.4176 q^{31} -548.578 q^{33} +91.2100 q^{35} +44.9571 q^{37} -122.957 q^{39} +58.0812 q^{41} -457.955 q^{43} +215.995 q^{45} +496.208 q^{47} +352.625 q^{49} +834.289 q^{51} +601.747 q^{53} -200.578 q^{55} -944.244 q^{57} -211.336 q^{59} +414.912 q^{61} +1647.32 q^{63} -44.9571 q^{65} -744.167 q^{67} -89.8422 q^{69} -882.537 q^{71} -319.499 q^{73} -1069.16 q^{75} -1529.73 q^{77} +490.664 q^{79} +1485.66 q^{81} +224.420 q^{83} +305.043 q^{85} -56.7494 q^{87} -580.501 q^{89} -342.871 q^{91} -221.490 q^{93} -345.246 q^{95} +403.499 q^{97} -3622.58 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 11 q^{5} - q^{7} + 107 q^{9}+O(q^{10}) \) 2 * q + q^3 - 11 * q^5 - q^7 + 107 * q^9	\( 2 q + q^{3} - 11 q^{5} - q^{7} + 107 q^{9} - 116 q^{11} - 26 q^{13} + 155 q^{15} + 51 q^{17} - 128 q^{19} + 481 q^{21} + 196 q^{23} - 29 q^{25} + 187 q^{27} - 12 q^{29} - 226 q^{31} - 58 q^{33} + 487 q^{35} - 143 q^{37} - 13 q^{39} - 278 q^{41} - 253 q^{43} - 428 q^{45} + 867 q^{47} + 759 q^{49} + 1149 q^{51} + 666 q^{53} + 638 q^{55} - 706 q^{57} - 996 q^{59} - 66 q^{61} + 428 q^{63} + 143 q^{65} - 1560 q^{67} - 1828 q^{69} - 923 q^{71} - 424 q^{73} - 1780 q^{75} + 58 q^{77} + 408 q^{79} + 1538 q^{81} + 1058 q^{83} + 843 q^{85} - 6 q^{87} - 1376 q^{89} + 13 q^{91} + 1492 q^{93} + 62 q^{95} + 592 q^{97} - 6206 q^{99}+O(q^{100}) \) 2 * q + q^3 - 11 * q^5 - q^7 + 107 * q^9 - 116 * q^11 - 26 * q^13 + 155 * q^15 + 51 * q^17 - 128 * q^19 + 481 * q^21 + 196 * q^23 - 29 * q^25 + 187 * q^27 - 12 * q^29 - 226 * q^31 - 58 * q^33 + 487 * q^35 - 143 * q^37 - 13 * q^39 - 278 * q^41 - 253 * q^43 - 428 * q^45 + 867 * q^47 + 759 * q^49 + 1149 * q^51 + 666 * q^53 + 638 * q^55 - 706 * q^57 - 996 * q^59 - 66 * q^61 + 428 * q^63 + 143 * q^65 - 1560 * q^67 - 1828 * q^69 - 923 * q^71 - 424 * q^73 - 1780 * q^75 + 58 * q^77 + 408 * q^79 + 1538 * q^81 + 1058 * q^83 + 843 * q^85 - 6 * q^87 - 1376 * q^89 + 13 * q^91 + 1492 * q^93 + 62 * q^95 + 592 * q^97 - 6206 * q^99

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