LMFDB - Embedded newform 208.4.a.l.1.3

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:	$3$
Coefficient field:	3.3.18257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 26x + 8 $ x^3 - x^2 - 26*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$5.47894$ 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)

$f(q)$	$=$	$q+8.74887 q^{3} -21.7068 q^{5} -20.7489 q^{7} +49.5428 q^{9} +O(q^{10})$	$q+8.74887 q^{3} -21.7068 q^{5} -20.7489 q^{7} +49.5428 q^{9} -28.6617 q^{11} +13.0000 q^{13} -189.910 q^{15} -111.210 q^{17} -17.6662 q^{19} -181.529 q^{21} -89.6721 q^{23} +346.183 q^{25} +197.224 q^{27} +27.0162 q^{29} -61.4919 q^{31} -250.758 q^{33} +450.391 q^{35} +55.5986 q^{37} +113.735 q^{39} +281.877 q^{41} -276.604 q^{43} -1075.41 q^{45} +89.4403 q^{47} +87.5158 q^{49} -972.966 q^{51} -127.603 q^{53} +622.153 q^{55} -154.559 q^{57} -132.300 q^{59} +674.731 q^{61} -1027.96 q^{63} -282.188 q^{65} +398.983 q^{67} -784.530 q^{69} -767.268 q^{71} -533.231 q^{73} +3028.71 q^{75} +594.698 q^{77} +579.596 q^{79} +387.833 q^{81} -1075.40 q^{83} +2414.02 q^{85} +236.362 q^{87} +586.903 q^{89} -269.735 q^{91} -537.985 q^{93} +383.476 q^{95} -6.09013 q^{97} -1419.98 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 8 q^{5} - 36 q^{7} + 73 q^{9}+O(q^{10}) $ 3 * q - 8 * q^5 - 36 * q^7 + 73 * q^9	$ 3 q - 8 q^{5} - 36 q^{7} + 73 q^{9} - 52 q^{11} + 39 q^{13} - 244 q^{15} - 116 q^{17} - 124 q^{19} - 154 q^{21} - 232 q^{23} + 191 q^{25} - 12 q^{27} - 30 q^{29} - 240 q^{31} - 564 q^{33} + 340 q^{35} - 264 q^{37} - 374 q^{41} - 248 q^{43} - 966 q^{45} + 412 q^{47} - 443 q^{49} - 92 q^{51} - 386 q^{53} + 400 q^{55} + 52 q^{57} + 940 q^{59} + 1206 q^{61} - 864 q^{63} - 104 q^{65} + 564 q^{67} + 512 q^{69} - 1260 q^{71} + 142 q^{73} + 3788 q^{75} + 1188 q^{77} - 1040 q^{79} + 1571 q^{81} - 756 q^{83} + 2510 q^{85} - 1536 q^{87} - 18 q^{89} - 468 q^{91} - 768 q^{93} - 384 q^{95} + 2174 q^{97} + 1940 q^{99}+O(q^{100}) $ 3 * q - 8 * q^5 - 36 * q^7 + 73 * q^9 - 52 * q^11 + 39 * q^13 - 244 * q^15 - 116 * q^17 - 124 * q^19 - 154 * q^21 - 232 * q^23 + 191 * q^25 - 12 * q^27 - 30 * q^29 - 240 * q^31 - 564 * q^33 + 340 * q^35 - 264 * q^37 - 374 * q^41 - 248 * q^43 - 966 * q^45 + 412 * q^47 - 443 * q^49 - 92 * q^51 - 386 * q^53 + 400 * q^55 + 52 * q^57 + 940 * q^59 + 1206 * q^61 - 864 * q^63 - 104 * q^65 + 564 * q^67 + 512 * q^69 - 1260 * q^71 + 142 * q^73 + 3788 * q^75 + 1188 * q^77 - 1040 * q^79 + 1571 * q^81 - 756 * q^83 + 2510 * q^85 - 1536 * q^87 - 18 * q^89 - 468 * q^91 - 768 * q^93 - 384 * q^95 + 2174 * q^97 + 1940 * 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
$3$	8.74887	1.68372	0.841861	−	0.539695i	$-0.181460\pi$
$3$	8.74887	1.68372	0.841861	+	0.539695i	$0.181460\pi$
$4$	0	0
$5$	−21.7068	−1.94151	−0.970756	−	0.240069i	$-0.922830\pi$
$5$	−21.7068	−1.94151	−0.970756	+	0.240069i	$0.922830\pi$
$6$	0	0
$7$	−20.7489	−1.12033	−0.560167	−	0.828380i	$-0.689263\pi$
$7$	−20.7489	−1.12033	−0.560167	+	0.828380i	$0.689263\pi$
$8$	0	0
$9$	49.5428	1.83492
$10$	0	0
$11$	−28.6617	−0.785621	−0.392810	−	0.919619i	$-0.628497\pi$
$11$	−28.6617	−0.785621	−0.392810	+	0.919619i	$0.628497\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−189.910	−3.26896
$16$	0	0
$17$	−111.210	−1.58662	−0.793308	−	0.608820i	$-0.791643\pi$
$17$	−111.210	−1.58662	−0.793308	+	0.608820i	$0.791643\pi$
$18$	0	0
$19$	−17.6662	−0.213311	−0.106655	−	0.994296i	$-0.534014\pi$
$19$	−17.6662	−0.213311	−0.106655	+	0.994296i	$0.534014\pi$
$20$	0	0
$21$	−181.529	−1.88633
$22$	0	0
$23$	−89.6721	−0.812953	−0.406477	−	0.913661i	$-0.633243\pi$
$23$	−89.6721	−0.812953	−0.406477	+	0.913661i	$0.633243\pi$
$24$	0	0
$25$	346.183	2.76947
$26$	0	0
$27$	197.224	1.40577
$28$	0	0
$29$	27.0162	0.172993	0.0864964	−	0.996252i	$-0.472433\pi$
$29$	27.0162	0.172993	0.0864964	+	0.996252i	$0.472433\pi$
$30$	0	0
$31$	−61.4919	−0.356267	−0.178133	−	0.984006i	$-0.557006\pi$
$31$	−61.4919	−0.356267	−0.178133	+	0.984006i	$0.557006\pi$
$32$	0	0
$33$	−250.758	−1.32277
$34$	0	0
$35$	450.391	2.17514
$36$	0	0
$37$	55.5986	0.247037	0.123518	−	0.992342i	$-0.460582\pi$
$37$	55.5986	0.247037	0.123518	+	0.992342i	$0.460582\pi$
$38$	0	0
$39$	113.735	0.466980
$40$	0	0
$41$	281.877	1.07370	0.536850	−	0.843677i	$-0.319614\pi$
$41$	281.877	1.07370	0.536850	+	0.843677i	$0.319614\pi$
$42$	0	0
$43$	−276.604	−0.980970	−0.490485	−	0.871450i	$-0.663180\pi$
$43$	−276.604	−0.980970	−0.490485	+	0.871450i	$0.663180\pi$
$44$	0	0
$45$	−1075.41	−3.56252
$46$	0	0
$47$	89.4403	0.277579	0.138790	−	0.990322i	$-0.455679\pi$
$47$	89.4403	0.277579	0.138790	+	0.990322i	$0.455679\pi$
$48$	0	0
$49$	87.5158	0.255148
$50$	0	0
$51$	−972.966	−2.67142
$52$	0	0
$53$	−127.603	−0.330709	−0.165355	−	0.986234i	$-0.552877\pi$
$53$	−127.603	−0.330709	−0.165355	+	0.986234i	$0.552877\pi$
$54$	0	0
$55$	622.153	1.52529
$56$	0	0
$57$	−154.559	−0.359156
$58$	0	0
$59$	−132.300	−0.291933	−0.145967	−	0.989290i	$-0.546629\pi$
$59$	−132.300	−0.291933	−0.145967	+	0.989290i	$0.546629\pi$
$60$	0	0
$61$	674.731	1.41624	0.708118	−	0.706094i	$-0.249544\pi$
$61$	674.731	1.41624	0.708118	+	0.706094i	$0.249544\pi$
$62$	0	0
$63$	−1027.96	−2.05572
$64$	0	0
$65$	−282.188	−0.538478
$66$	0	0
$67$	398.983	0.727516	0.363758	−	0.931493i	$-0.381494\pi$
$67$	398.983	0.727516	0.363758	+	0.931493i	$0.381494\pi$
$68$	0	0
$69$	−784.530	−1.36879
$70$	0	0
$71$	−767.268	−1.28251	−0.641254	−	0.767329i	$-0.721585\pi$
$71$	−767.268	−1.28251	−0.641254	+	0.767329i	$0.721585\pi$
$72$	0	0
$73$	−533.231	−0.854930	−0.427465	−	0.904032i	$-0.640593\pi$
$73$	−533.231	−0.854930	−0.427465	+	0.904032i	$0.640593\pi$
$74$	0	0
$75$	3028.71	4.66301
$76$	0	0
$77$	594.698	0.880158
$78$	0	0
$79$	579.596	0.825439	0.412720	−	0.910858i	$-0.364579\pi$
$79$	579.596	0.825439	0.412720	+	0.910858i	$0.364579\pi$
$80$	0	0
$81$	387.833	0.532007
$82$	0	0
$83$	−1075.40	−1.42218	−0.711090	−	0.703101i	$-0.751798\pi$
$83$	−1075.40	−1.42218	−0.711090	+	0.703101i	$0.751798\pi$
$84$	0	0
$85$	2414.02	3.08043
$86$	0	0
$87$	236.362	0.291272
$88$	0	0
$89$	586.903	0.699006	0.349503	−	0.936935i	$-0.386350\pi$
$89$	586.903	0.699006	0.349503	+	0.936935i	$0.386350\pi$
$90$	0	0
$91$	−269.735	−0.310725
$92$	0	0
$93$	−537.985	−0.599854
$94$	0	0
$95$	383.476	0.414146
$96$	0	0
$97$	−6.09013	−0.00637483	−0.00318742	−	0.999995i	$-0.501015\pi$
$97$	−6.09013	−0.00637483	−0.00318742	+	0.999995i	$0.501015\pi$
$98$	0	0
$99$	−1419.98	−1.44155
$100$	0	0
$101$	1274.13	1.25526	0.627629	−	0.778512i	$-0.284025\pi$
$101$	1274.13	1.25526	0.627629	+	0.778512i	$0.284025\pi$
$102$	0	0
$103$	−1641.25	−1.57007	−0.785035	−	0.619451i	$-0.787355\pi$
$103$	−1641.25	−1.57007	−0.785035	+	0.619451i	$0.787355\pi$
$104$	0	0
$105$	3940.41	3.66233
$106$	0	0
$107$	901.794	0.814764	0.407382	−	0.913258i	$-0.366442\pi$
$107$	901.794	0.814764	0.407382	+	0.913258i	$0.366442\pi$
$108$	0	0
$109$	668.627	0.587549	0.293775	−	0.955875i	$-0.405088\pi$
$109$	668.627	0.587549	0.293775	+	0.955875i	$0.405088\pi$
$110$	0	0
$111$	486.426	0.415941
$112$	0	0
$113$	−963.665	−0.802248	−0.401124	−	0.916024i	$-0.631380\pi$
$113$	−963.665	−0.802248	−0.401124	+	0.916024i	$0.631380\pi$
$114$	0	0
$115$	1946.49	1.57836
$116$	0	0
$117$	644.056	0.508915
$118$	0	0
$119$	2307.49	1.77754
$120$	0	0
$121$	−509.506	−0.382800
$122$	0	0
$123$	2466.10	1.80781
$124$	0	0
$125$	−4801.17	−3.43544
$126$	0	0
$127$	−1726.42	−1.20626	−0.603130	−	0.797643i	$-0.706080\pi$
$127$	−1726.42	−1.20626	−0.603130	+	0.797643i	$0.706080\pi$
$128$	0	0
$129$	−2419.97	−1.65168
$130$	0	0
$131$	−656.621	−0.437933	−0.218967	−	0.975732i	$-0.570269\pi$
$131$	−656.621	−0.437933	−0.218967	+	0.975732i	$0.570269\pi$
$132$	0	0
$133$	366.554	0.238979
$134$	0	0
$135$	−4281.10	−2.72932
$136$	0	0
$137$	−2939.76	−1.83329	−0.916643	−	0.399706i	$-0.869112\pi$
$137$	−2939.76	−1.83329	−0.916643	+	0.399706i	$0.869112\pi$
$138$	0	0
$139$	−179.480	−0.109520	−0.0547599	−	0.998500i	$-0.517439\pi$
$139$	−179.480	−0.109520	−0.0547599	+	0.998500i	$0.517439\pi$
$140$	0	0
$141$	782.502	0.467366
$142$	0	0
$143$	−372.602	−0.217892
$144$	0	0
$145$	−586.435	−0.335867
$146$	0	0
$147$	765.665	0.429598
$148$	0	0
$149$	−725.796	−0.399057	−0.199529	−	0.979892i	$-0.563941\pi$
$149$	−725.796	−0.399057	−0.199529	+	0.979892i	$0.563941\pi$
$150$	0	0
$151$	−915.895	−0.493606	−0.246803	−	0.969066i	$-0.579380\pi$
$151$	−915.895	−0.493606	−0.246803	+	0.969066i	$0.579380\pi$
$152$	0	0
$153$	−5509.67	−2.91131
$154$	0	0
$155$	1334.79	0.691696
$156$	0	0
$157$	2338.51	1.18875	0.594374	−	0.804188i	$-0.297400\pi$
$157$	2338.51	1.18875	0.594374	+	0.804188i	$0.297400\pi$
$158$	0	0
$159$	−1116.38	−0.556822
$160$	0	0
$161$	1860.59	0.910779
$162$	0	0
$163$	3158.32	1.51766	0.758830	−	0.651289i	$-0.225771\pi$
$163$	3158.32	1.51766	0.758830	+	0.651289i	$0.225771\pi$
$164$	0	0
$165$	5443.14	2.56817
$166$	0	0
$167$	2564.13	1.18813	0.594067	−	0.804415i	$-0.297521\pi$
$167$	2564.13	1.18813	0.594067	+	0.804415i	$0.297521\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−875.234	−0.391408
$172$	0	0
$173$	510.758	0.224464	0.112232	−	0.993682i	$-0.464200\pi$
$173$	510.758	0.224464	0.112232	+	0.993682i	$0.464200\pi$
$174$	0	0
$175$	−7182.91	−3.10273
$176$	0	0
$177$	−1157.48	−0.491534
$178$	0	0
$179$	−1560.70	−0.651689	−0.325845	−	0.945423i	$-0.605649\pi$
$179$	−1560.70	−0.651689	−0.325845	+	0.945423i	$0.605649\pi$
$180$	0	0
$181$	728.404	0.299126	0.149563	−	0.988752i	$-0.452213\pi$
$181$	728.404	0.299126	0.149563	+	0.988752i	$0.452213\pi$
$182$	0	0
$183$	5903.13	2.38455
$184$	0	0
$185$	−1206.87	−0.479625
$186$	0	0
$187$	3187.48	1.24648
$188$	0	0
$189$	−4092.18	−1.57493
$190$	0	0
$191$	−3999.39	−1.51511	−0.757555	−	0.652772i	$-0.773606\pi$
$191$	−3999.39	−1.51511	−0.757555	+	0.652772i	$0.773606\pi$
$192$	0	0
$193$	547.659	0.204256	0.102128	−	0.994771i	$-0.467435\pi$
$193$	547.659	0.204256	0.102128	+	0.994771i	$0.467435\pi$
$194$	0	0
$195$	−2468.83	−0.906648
$196$	0	0
$197$	−593.202	−0.214538	−0.107269	−	0.994230i	$-0.534211\pi$
$197$	−593.202	−0.214538	−0.107269	+	0.994230i	$0.534211\pi$
$198$	0	0
$199$	−450.315	−0.160412	−0.0802060	−	0.996778i	$-0.525558\pi$
$199$	−450.315	−0.160412	−0.0802060	+	0.996778i	$0.525558\pi$
$200$	0	0
$201$	3490.66	1.22493
$202$	0	0
$203$	−560.557	−0.193810
$204$	0	0
$205$	−6118.63	−2.08460
$206$	0	0
$207$	−4442.61	−1.49170
$208$	0	0
$209$	506.344	0.167581
$210$	0	0
$211$	−1325.92	−0.432606	−0.216303	−	0.976326i	$-0.569400\pi$
$211$	−1325.92	−0.432606	−0.216303	+	0.976326i	$0.569400\pi$
$212$	0	0
$213$	−6712.73	−2.15938
$214$	0	0
$215$	6004.17	1.90456
$216$	0	0
$217$	1275.89	0.399138
$218$	0	0
$219$	−4665.17	−1.43946
$220$	0	0
$221$	−1445.73	−0.440048
$222$	0	0
$223$	−3473.90	−1.04318	−0.521591	−	0.853196i	$-0.674661\pi$
$223$	−3473.90	−1.04318	−0.521591	+	0.853196i	$0.674661\pi$
$224$	0	0
$225$	17150.9	5.08175
$226$	0	0
$227$	5280.97	1.54410	0.772049	−	0.635563i	$-0.219232\pi$
$227$	5280.97	1.54410	0.772049	+	0.635563i	$0.219232\pi$
$228$	0	0
$229$	1989.76	0.574180	0.287090	−	0.957904i	$-0.407312\pi$
$229$	1989.76	0.574180	0.287090	+	0.957904i	$0.407312\pi$
$230$	0	0
$231$	5202.94	1.48194
$232$	0	0
$233$	551.879	0.155171	0.0775854	−	0.996986i	$-0.475279\pi$
$233$	551.879	0.155171	0.0775854	+	0.996986i	$0.475279\pi$
$234$	0	0
$235$	−1941.46	−0.538923
$236$	0	0
$237$	5070.82	1.38981
$238$	0	0
$239$	−6913.26	−1.87105	−0.935527	−	0.353256i	$-0.885074\pi$
$239$	−6913.26	−1.87105	−0.935527	+	0.353256i	$0.885074\pi$
$240$	0	0
$241$	−1989.57	−0.531782	−0.265891	−	0.964003i	$-0.585666\pi$
$241$	−1989.57	−0.531782	−0.265891	+	0.964003i	$0.585666\pi$
$242$	0	0
$243$	−1931.95	−0.510018
$244$	0	0
$245$	−1899.68	−0.495373
$246$	0	0
$247$	−229.661	−0.0591618
$248$	0	0
$249$	−9408.58	−2.39456
$250$	0	0
$251$	−7720.98	−1.94161	−0.970804	−	0.239873i	$-0.922894\pi$
$251$	−7720.98	−1.94161	−0.970804	+	0.239873i	$0.922894\pi$
$252$	0	0
$253$	2570.15	0.638673
$254$	0	0
$255$	21119.9	5.18659
$256$	0	0
$257$	4365.83	1.05966	0.529831	−	0.848103i	$-0.322255\pi$
$257$	4365.83	1.05966	0.529831	+	0.848103i	$0.322255\pi$
$258$	0	0
$259$	−1153.61	−0.276764
$260$	0	0
$261$	1338.46	0.317428
$262$	0	0
$263$	−5014.63	−1.17572	−0.587861	−	0.808962i	$-0.700030\pi$
$263$	−5014.63	−1.17572	−0.587861	+	0.808962i	$0.700030\pi$
$264$	0	0
$265$	2769.84	0.642076
$266$	0	0
$267$	5134.74	1.17693
$268$	0	0
$269$	−4524.84	−1.02559	−0.512796	−	0.858511i	$-0.671390\pi$
$269$	−4524.84	−1.02559	−0.512796	+	0.858511i	$0.671390\pi$
$270$	0	0
$271$	7700.90	1.72619	0.863093	−	0.505045i	$-0.168524\pi$
$271$	7700.90	1.72619	0.863093	+	0.505045i	$0.168524\pi$
$272$	0	0
$273$	−2359.88	−0.523174
$274$	0	0
$275$	−9922.21	−2.17575
$276$	0	0
$277$	−6931.33	−1.50348	−0.751739	−	0.659461i	$-0.770785\pi$
$277$	−6931.33	−1.50348	−0.751739	+	0.659461i	$0.770785\pi$
$278$	0	0
$279$	−3046.48	−0.653720
$280$	0	0
$281$	2694.30	0.571988	0.285994	−	0.958231i	$-0.407676\pi$
$281$	2694.30	0.571988	0.285994	+	0.958231i	$0.407676\pi$
$282$	0	0
$283$	81.0681	0.0170283	0.00851413	−	0.999964i	$-0.497290\pi$
$283$	81.0681	0.0170283	0.00851413	+	0.999964i	$0.497290\pi$
$284$	0	0
$285$	3354.98	0.697306
$286$	0	0
$287$	−5848.62	−1.20290
$288$	0	0
$289$	7454.75	1.51735
$290$	0	0
$291$	−53.2818	−0.0107334
$292$	0	0
$293$	4870.60	0.971138	0.485569	−	0.874198i	$-0.338613\pi$
$293$	4870.60	0.971138	0.485569	+	0.874198i	$0.338613\pi$
$294$	0	0
$295$	2871.81	0.566792
$296$	0	0
$297$	−5652.78	−1.10440
$298$	0	0
$299$	−1165.74	−0.225473
$300$	0	0
$301$	5739.22	1.09901
$302$	0	0
$303$	11147.2	2.11351
$304$	0	0
$305$	−14646.2	−2.74964
$306$	0	0
$307$	−3209.22	−0.596612	−0.298306	−	0.954470i	$-0.596422\pi$
$307$	−3209.22	−0.596612	−0.298306	+	0.954470i	$0.596422\pi$
$308$	0	0
$309$	−14359.1	−2.64356
$310$	0	0
$311$	3095.46	0.564397	0.282198	−	0.959356i	$-0.408936\pi$
$311$	3095.46	0.564397	0.282198	+	0.959356i	$0.408936\pi$
$312$	0	0
$313$	−8894.23	−1.60617	−0.803085	−	0.595864i	$-0.796810\pi$
$313$	−8894.23	−1.60617	−0.803085	+	0.595864i	$0.796810\pi$
$314$	0	0
$315$	22313.6	3.99121
$316$	0	0
$317$	−2811.69	−0.498171	−0.249085	−	0.968482i	$-0.580130\pi$
$317$	−2811.69	−0.498171	−0.249085	+	0.968482i	$0.580130\pi$
$318$	0	0
$319$	−774.332	−0.135907
$320$	0	0
$321$	7889.69	1.37184
$322$	0	0
$323$	1964.67	0.338443
$324$	0	0
$325$	4500.38	0.768112
$326$	0	0
$327$	5849.74	0.989270
$328$	0	0
$329$	−1855.79	−0.310981
$330$	0	0
$331$	2935.22	0.487414	0.243707	−	0.969849i	$-0.421636\pi$
$331$	2935.22	0.487414	0.243707	+	0.969849i	$0.421636\pi$
$332$	0	0
$333$	2754.51	0.453292
$334$	0	0
$335$	−8660.64	−1.41248
$336$	0	0
$337$	10527.9	1.70175	0.850874	−	0.525369i	$-0.176073\pi$
$337$	10527.9	1.70175	0.850874	+	0.525369i	$0.176073\pi$
$338$	0	0
$339$	−8430.98	−1.35076
$340$	0	0
$341$	1762.46	0.279891
$342$	0	0
$343$	5301.01	0.834483
$344$	0	0
$345$	17029.6	2.65752
$346$	0	0
$347$	−395.789	−0.0612307	−0.0306153	−	0.999531i	$-0.509747\pi$
$347$	−395.789	−0.0612307	−0.0306153	+	0.999531i	$0.509747\pi$
$348$	0	0
$349$	−1103.52	−0.169255	−0.0846275	−	0.996413i	$-0.526970\pi$
$349$	−1103.52	−0.169255	−0.0846275	+	0.996413i	$0.526970\pi$
$350$	0	0
$351$	2563.91	0.389891
$352$	0	0
$353$	−8352.29	−1.25934	−0.629671	−	0.776862i	$-0.716810\pi$
$353$	−8352.29	−1.25934	−0.629671	+	0.776862i	$0.716810\pi$
$354$	0	0
$355$	16654.9	2.49000
$356$	0	0
$357$	20187.9	2.99288
$358$	0	0
$359$	−3982.54	−0.585488	−0.292744	−	0.956191i	$-0.594568\pi$
$359$	−3982.54	−0.585488	−0.292744	+	0.956191i	$0.594568\pi$
$360$	0	0
$361$	−6546.90	−0.954498
$362$	0	0
$363$	−4457.61	−0.644528
$364$	0	0
$365$	11574.7	1.65986
$366$	0	0
$367$	−6288.34	−0.894411	−0.447205	−	0.894431i	$-0.647581\pi$
$367$	−6288.34	−0.894411	−0.447205	+	0.894431i	$0.647581\pi$
$368$	0	0
$369$	13965.0	1.97015
$370$	0	0
$371$	2647.61	0.370505
$372$	0	0
$373$	4049.92	0.562190	0.281095	−	0.959680i	$-0.409302\pi$
$373$	4049.92	0.562190	0.281095	+	0.959680i	$0.409302\pi$
$374$	0	0
$375$	−42004.9	−5.78432
$376$	0	0
$377$	351.211	0.0479796
$378$	0	0
$379$	6827.64	0.925363	0.462681	−	0.886525i	$-0.346887\pi$
$379$	6827.64	0.925363	0.462681	+	0.886525i	$0.346887\pi$
$380$	0	0
$381$	−15104.2	−2.03101
$382$	0	0
$383$	14692.5	1.96019	0.980093	−	0.198539i	$-0.0636198\pi$
$383$	14692.5	1.96019	0.980093	+	0.198539i	$0.0636198\pi$
$384$	0	0
$385$	−12909.0	−1.70884
$386$	0	0
$387$	−13703.7	−1.80000
$388$	0	0
$389$	3129.28	0.407868	0.203934	−	0.978985i	$-0.434627\pi$
$389$	3129.28	0.407868	0.203934	+	0.978985i	$0.434627\pi$
$390$	0	0
$391$	9972.47	1.28984
$392$	0	0
$393$	−5744.70	−0.737358
$394$	0	0
$395$	−12581.2	−1.60260
$396$	0	0
$397$	10584.0	1.33803	0.669015	−	0.743249i	$-0.266716\pi$
$397$	10584.0	1.33803	0.669015	+	0.743249i	$0.266716\pi$
$398$	0	0
$399$	3206.93	0.402375
$400$	0	0
$401$	−8631.02	−1.07484	−0.537422	−	0.843313i	$-0.680602\pi$
$401$	−8631.02	−1.07484	−0.537422	+	0.843313i	$0.680602\pi$
$402$	0	0
$403$	−799.394	−0.0988106
$404$	0	0
$405$	−8418.60	−1.03290
$406$	0	0
$407$	−1593.55	−0.194077
$408$	0	0
$409$	−14224.5	−1.71969	−0.859847	−	0.510552i	$-0.829441\pi$
$409$	−14224.5	−1.71969	−0.859847	+	0.510552i	$0.829441\pi$
$410$	0	0
$411$	−25719.6	−3.08674
$412$	0	0
$413$	2745.09	0.327063
$414$	0	0
$415$	23343.5	2.76118
$416$	0	0
$417$	−1570.24	−0.184401
$418$	0	0
$419$	9631.13	1.12294	0.561470	−	0.827497i	$-0.310236\pi$
$419$	9631.13	1.12294	0.561470	+	0.827497i	$0.310236\pi$
$420$	0	0
$421$	6333.63	0.733212	0.366606	−	0.930376i	$-0.380520\pi$
$421$	6333.63	0.733212	0.366606	+	0.930376i	$0.380520\pi$
$422$	0	0
$423$	4431.12	0.509335
$424$	0	0
$425$	−38499.2	−4.39408
$426$	0	0
$427$	−13999.9	−1.58666
$428$	0	0
$429$	−3259.85	−0.366870
$430$	0	0
$431$	−9588.48	−1.07160	−0.535801	−	0.844344i	$-0.679990\pi$
$431$	−9588.48	−1.07160	−0.535801	+	0.844344i	$0.679990\pi$
$432$	0	0
$433$	7879.92	0.874561	0.437280	−	0.899325i	$-0.355942\pi$
$433$	7879.92	0.874561	0.437280	+	0.899325i	$0.355942\pi$
$434$	0	0
$435$	−5130.65	−0.565507
$436$	0	0
$437$	1584.17	0.173412
$438$	0	0
$439$	−1159.40	−0.126048	−0.0630240	−	0.998012i	$-0.520074\pi$
$439$	−1159.40	−0.126048	−0.0630240	+	0.998012i	$0.520074\pi$
$440$	0	0
$441$	4335.78	0.468176
$442$	0	0
$443$	−7975.52	−0.855369	−0.427684	−	0.903928i	$-0.640671\pi$
$443$	−7975.52	−0.855369	−0.427684	+	0.903928i	$0.640671\pi$
$444$	0	0
$445$	−12739.8	−1.35713
$446$	0	0
$447$	−6349.90	−0.671902
$448$	0	0
$449$	3074.87	0.323190	0.161595	−	0.986857i	$-0.448336\pi$
$449$	3074.87	0.323190	0.161595	+	0.986857i	$0.448336\pi$
$450$	0	0
$451$	−8079.06	−0.843522
$452$	0	0
$453$	−8013.05	−0.831095
$454$	0	0
$455$	5855.08	0.603276
$456$	0	0
$457$	5669.28	0.580302	0.290151	−	0.956981i	$-0.406295\pi$
$457$	5669.28	0.580302	0.290151	+	0.956981i	$0.406295\pi$
$458$	0	0
$459$	−21933.4	−2.23042
$460$	0	0
$461$	−719.799	−0.0727210	−0.0363605	−	0.999339i	$-0.511576\pi$
$461$	−719.799	−0.0727210	−0.0363605	+	0.999339i	$0.511576\pi$
$462$	0	0
$463$	−4645.80	−0.466325	−0.233163	−	0.972438i	$-0.574907\pi$
$463$	−4645.80	−0.466325	−0.233163	+	0.972438i	$0.574907\pi$
$464$	0	0
$465$	11677.9	1.16462
$466$	0	0
$467$	−13902.7	−1.37760	−0.688801	−	0.724951i	$-0.741862\pi$
$467$	−13902.7	−1.37760	−0.688801	+	0.724951i	$0.741862\pi$
$468$	0	0
$469$	−8278.46	−0.815061
$470$	0	0
$471$	20459.3	2.00152
$472$	0	0
$473$	7927.94	0.770670
$474$	0	0
$475$	−6115.75	−0.590757
$476$	0	0
$477$	−6321.80	−0.606824
$478$	0	0
$479$	825.368	0.0787307	0.0393653	−	0.999225i	$-0.487466\pi$
$479$	825.368	0.0787307	0.0393653	+	0.999225i	$0.487466\pi$
$480$	0	0
$481$	722.782	0.0685157
$482$	0	0
$483$	16278.1	1.53350
$484$	0	0
$485$	132.197	0.0123768
$486$	0	0
$487$	−585.438	−0.0544738	−0.0272369	−	0.999629i	$-0.508671\pi$
$487$	−585.438	−0.0544738	−0.0272369	+	0.999629i	$0.508671\pi$
$488$	0	0
$489$	27631.7	2.55532
$490$	0	0
$491$	11112.8	1.02141	0.510705	−	0.859756i	$-0.329384\pi$
$491$	11112.8	1.02141	0.510705	+	0.859756i	$0.329384\pi$
$492$	0	0
$493$	−3004.49	−0.274473
$494$	0	0
$495$	30823.2	2.79879
$496$	0	0
$497$	15920.0	1.43684
$498$	0	0
$499$	18283.9	1.64028	0.820141	−	0.572161i	$-0.193895\pi$
$499$	18283.9	1.64028	0.820141	+	0.572161i	$0.193895\pi$
$500$	0	0
$501$	22433.3	2.00049
$502$	0	0
$503$	−10044.8	−0.890406	−0.445203	−	0.895430i	$-0.646868\pi$
$503$	−10044.8	−0.890406	−0.445203	+	0.895430i	$0.646868\pi$
$504$	0	0
$505$	−27657.3	−2.43710
$506$	0	0
$507$	1478.56	0.129517
$508$	0	0
$509$	19001.5	1.65467	0.827336	−	0.561707i	$-0.189855\pi$
$509$	19001.5	1.65467	0.827336	+	0.561707i	$0.189855\pi$
$510$	0	0
$511$	11063.9	0.957808
$512$	0	0
$513$	−3484.20	−0.299866
$514$	0	0
$515$	35626.2	3.04831
$516$	0	0
$517$	−2563.51	−0.218072
$518$	0	0
$519$	4468.56	0.377934
$520$	0	0
$521$	17562.6	1.47684	0.738418	−	0.674344i	$-0.235573\pi$
$521$	17562.6	1.47684	0.738418	+	0.674344i	$0.235573\pi$
$522$	0	0
$523$	284.358	0.0237746	0.0118873	−	0.999929i	$-0.496216\pi$
$523$	284.358	0.0237746	0.0118873	+	0.999929i	$0.496216\pi$
$524$	0	0
$525$	−62842.4	−5.22413
$526$	0	0
$527$	6838.54	0.565259
$528$	0	0
$529$	−4125.92	−0.339107
$530$	0	0
$531$	−6554.54	−0.535674
$532$	0	0
$533$	3664.40	0.297791
$534$	0	0
$535$	−19575.0	−1.58187
$536$	0	0
$537$	−13654.4	−1.09726
$538$	0	0
$539$	−2508.35	−0.200450
$540$	0	0
$541$	−13947.0	−1.10837	−0.554183	−	0.832395i	$-0.686969\pi$
$541$	−13947.0	−1.10837	−0.554183	+	0.832395i	$0.686969\pi$
$542$	0	0
$543$	6372.72	0.503645
$544$	0	0
$545$	−14513.7	−1.14073
$546$	0	0
$547$	−729.848	−0.0570495	−0.0285247	−	0.999593i	$-0.509081\pi$
$547$	−729.848	−0.0570495	−0.0285247	+	0.999593i	$0.509081\pi$
$548$	0	0
$549$	33428.0	2.59868
$550$	0	0
$551$	−477.275	−0.0369012
$552$	0	0
$553$	−12026.0	−0.924768
$554$	0	0
$555$	−10558.7	−0.807555
$556$	0	0
$557$	1073.95	0.0816964	0.0408482	−	0.999165i	$-0.486994\pi$
$557$	1073.95	0.0816964	0.0408482	+	0.999165i	$0.486994\pi$
$558$	0	0
$559$	−3595.85	−0.272072
$560$	0	0
$561$	27886.9	2.09872
$562$	0	0
$563$	9622.42	0.720313	0.360157	−	0.932892i	$-0.382723\pi$
$563$	9622.42	0.720313	0.360157	+	0.932892i	$0.382723\pi$
$564$	0	0
$565$	20918.0	1.55757
$566$	0	0
$567$	−8047.10	−0.596026
$568$	0	0
$569$	26893.8	1.98146	0.990728	−	0.135863i	$-0.0433808\pi$
$569$	26893.8	1.98146	0.990728	+	0.135863i	$0.0433808\pi$
$570$	0	0
$571$	14214.0	1.04175	0.520874	−	0.853633i	$-0.325606\pi$
$571$	14214.0	1.04175	0.520874	+	0.853633i	$0.325606\pi$
$572$	0	0
$573$	−34990.2	−2.55102
$574$	0	0
$575$	−31043.0	−2.25145
$576$	0	0
$577$	−21834.6	−1.57536	−0.787682	−	0.616082i	$-0.788719\pi$
$577$	−21834.6	−1.57536	−0.787682	+	0.616082i	$0.788719\pi$
$578$	0	0
$579$	4791.40	0.343910
$580$	0	0
$581$	22313.4	1.59332
$582$	0	0
$583$	3657.31	0.259812
$584$	0	0
$585$	−13980.4	−0.988064
$586$	0	0
$587$	14151.0	0.995018	0.497509	−	0.867459i	$-0.334248\pi$
$587$	14151.0	0.995018	0.497509	+	0.867459i	$0.334248\pi$
$588$	0	0
$589$	1086.33	0.0759956
$590$	0	0
$591$	−5189.85	−0.361222
$592$	0	0
$593$	−1195.81	−0.0828096	−0.0414048	−	0.999142i	$-0.513183\pi$
$593$	−1195.81	−0.0828096	−0.0414048	+	0.999142i	$0.513183\pi$
$594$	0	0
$595$	−50088.1	−3.45111
$596$	0	0
$597$	−3939.75	−0.270089
$598$	0	0
$599$	15744.3	1.07395	0.536973	−	0.843600i	$-0.319568\pi$
$599$	15744.3	1.07395	0.536973	+	0.843600i	$0.319568\pi$
$600$	0	0
$601$	12760.3	0.866063	0.433031	−	0.901379i	$-0.357444\pi$
$601$	12760.3	0.866063	0.433031	+	0.901379i	$0.357444\pi$
$602$	0	0
$603$	19766.8	1.33493
$604$	0	0
$605$	11059.7	0.743210
$606$	0	0
$607$	22994.0	1.53756	0.768779	−	0.639514i	$-0.220864\pi$
$607$	22994.0	1.53756	0.768779	+	0.639514i	$0.220864\pi$
$608$	0	0
$609$	−4904.24	−0.326321
$610$	0	0
$611$	1162.72	0.0769866
$612$	0	0
$613$	−6903.59	−0.454867	−0.227434	−	0.973794i	$-0.573033\pi$
$613$	−6903.59	−0.454867	−0.227434	+	0.973794i	$0.573033\pi$
$614$	0	0
$615$	−53531.1	−3.50989
$616$	0	0
$617$	13209.2	0.861884	0.430942	−	0.902380i	$-0.358181\pi$
$617$	13209.2	0.861884	0.430942	+	0.902380i	$0.358181\pi$
$618$	0	0
$619$	−13350.0	−0.866851	−0.433426	−	0.901189i	$-0.642695\pi$
$619$	−13350.0	−0.866851	−0.433426	+	0.901189i	$0.642695\pi$
$620$	0	0
$621$	−17685.5	−1.14283
$622$	0	0
$623$	−12177.6	−0.783120
$624$	0	0
$625$	60945.0	3.90048
$626$	0	0
$627$	4429.94	0.282161
$628$	0	0
$629$	−6183.15	−0.391953
$630$	0	0
$631$	16439.8	1.03718	0.518588	−	0.855024i	$-0.326458\pi$
$631$	16439.8	1.03718	0.518588	+	0.855024i	$0.326458\pi$
$632$	0	0
$633$	−11600.3	−0.728389
$634$	0	0
$635$	37475.0	2.34197
$636$	0	0
$637$	1137.71	0.0707653
$638$	0	0
$639$	−38012.6	−2.35330
$640$	0	0
$641$	17216.4	1.06086	0.530428	−	0.847730i	$-0.322031\pi$
$641$	17216.4	1.06086	0.530428	+	0.847730i	$0.322031\pi$
$642$	0	0
$643$	−1049.43	−0.0643634	−0.0321817	−	0.999482i	$-0.510246\pi$
$643$	−1049.43	−0.0643634	−0.0321817	+	0.999482i	$0.510246\pi$
$644$	0	0
$645$	52529.8	3.20676
$646$	0	0
$647$	−22455.8	−1.36450	−0.682248	−	0.731121i	$-0.738997\pi$
$647$	−22455.8	−1.36450	−0.682248	+	0.731121i	$0.738997\pi$
$648$	0	0
$649$	3791.96	0.229349
$650$	0	0
$651$	11162.6	0.672037
$652$	0	0
$653$	−1247.82	−0.0747797	−0.0373899	−	0.999301i	$-0.511904\pi$
$653$	−1247.82	−0.0747797	−0.0373899	+	0.999301i	$0.511904\pi$
$654$	0	0
$655$	14253.1	0.850252
$656$	0	0
$657$	−26417.7	−1.56873
$658$	0	0
$659$	29752.4	1.75871	0.879356	−	0.476165i	$-0.157974\pi$
$659$	29752.4	1.75871	0.879356	+	0.476165i	$0.157974\pi$
$660$	0	0
$661$	−11425.7	−0.672324	−0.336162	−	0.941804i	$-0.609129\pi$
$661$	−11425.7	−0.672324	−0.336162	+	0.941804i	$0.609129\pi$
$662$	0	0
$663$	−12648.6	−0.740919
$664$	0	0
$665$	−7956.70	−0.463981
$666$	0	0
$667$	−2422.60	−0.140635
$668$	0	0
$669$	−30392.7	−1.75643
$670$	0	0
$671$	−19338.9	−1.11262
$672$	0	0
$673$	8206.20	0.470024	0.235012	−	0.971993i	$-0.424487\pi$
$673$	8206.20	0.470024	0.235012	+	0.971993i	$0.424487\pi$
$674$	0	0
$675$	68275.7	3.89323
$676$	0	0
$677$	−29254.5	−1.66077	−0.830385	−	0.557190i	$-0.811880\pi$
$677$	−29254.5	−1.66077	−0.830385	+	0.557190i	$0.811880\pi$
$678$	0	0
$679$	126.363	0.00714194
$680$	0	0
$681$	46202.5	2.59983
$682$	0	0
$683$	−23500.9	−1.31660	−0.658298	−	0.752757i	$-0.728723\pi$
$683$	−23500.9	−1.31660	−0.658298	+	0.752757i	$0.728723\pi$
$684$	0	0
$685$	63812.6	3.55935
$686$	0	0
$687$	17408.2	0.966759
$688$	0	0
$689$	−1658.84	−0.0917222
$690$	0	0
$691$	−15576.6	−0.857541	−0.428771	−	0.903413i	$-0.641053\pi$
$691$	−15576.6	−0.857541	−0.428771	+	0.903413i	$0.641053\pi$
$692$	0	0
$693$	29463.0	1.61502
$694$	0	0
$695$	3895.92	0.212634
$696$	0	0
$697$	−31347.6	−1.70355
$698$	0	0
$699$	4828.32	0.261264
$700$	0	0
$701$	−21651.4	−1.16657	−0.583283	−	0.812269i	$-0.698232\pi$
$701$	−21651.4	−1.16657	−0.583283	+	0.812269i	$0.698232\pi$
$702$	0	0
$703$	−982.217	−0.0526956
$704$	0	0
$705$	−16985.6	−0.907396
$706$	0	0
$707$	−26436.9	−1.40631
$708$	0	0
$709$	−28594.2	−1.51463	−0.757317	−	0.653047i	$-0.773490\pi$
$709$	−28594.2	−1.51463	−0.757317	+	0.653047i	$0.773490\pi$
$710$	0	0
$711$	28714.8	1.51461
$712$	0	0
$713$	5514.10	0.289628
$714$	0	0
$715$	8087.99	0.423040
$716$	0	0
$717$	−60483.3	−3.15033
$718$	0	0
$719$	30854.5	1.60039	0.800193	−	0.599742i	$-0.204730\pi$
$719$	30854.5	1.60039	0.800193	+	0.599742i	$0.204730\pi$
$720$	0	0
$721$	34054.1	1.75900
$722$	0	0
$723$	−17406.5	−0.895373
$724$	0	0
$725$	9352.57	0.479098
$726$	0	0
$727$	−7122.13	−0.363336	−0.181668	−	0.983360i	$-0.558150\pi$
$727$	−7122.13	−0.363336	−0.181668	+	0.983360i	$0.558150\pi$
$728$	0	0
$729$	−27373.9	−1.39074
$730$	0	0
$731$	30761.2	1.55642
$732$	0	0
$733$	12449.8	0.627347	0.313674	−	0.949531i	$-0.398440\pi$
$733$	12449.8	0.627347	0.313674	+	0.949531i	$0.398440\pi$
$734$	0	0
$735$	−16620.1	−0.834070
$736$	0	0
$737$	−11435.5	−0.571552
$738$	0	0
$739$	−5608.43	−0.279174	−0.139587	−	0.990210i	$-0.544577\pi$
$739$	−5608.43	−0.279174	−0.139587	+	0.990210i	$0.544577\pi$
$740$	0	0
$741$	−2009.27	−0.0996120
$742$	0	0
$743$	848.085	0.0418751	0.0209376	−	0.999781i	$-0.493335\pi$
$743$	848.085	0.0418751	0.0209376	+	0.999781i	$0.493335\pi$
$744$	0	0
$745$	15754.7	0.774775
$746$	0	0
$747$	−53278.6	−2.60959
$748$	0	0
$749$	−18711.2	−0.912808
$750$	0	0
$751$	−9668.44	−0.469782	−0.234891	−	0.972022i	$-0.575473\pi$
$751$	−9668.44	−0.469782	−0.234891	+	0.972022i	$0.575473\pi$
$752$	0	0
$753$	−67549.8	−3.26913
$754$	0	0
$755$	19881.1	0.958342
$756$	0	0
$757$	−22002.9	−1.05642	−0.528210	−	0.849114i	$-0.677137\pi$
$757$	−22002.9	−1.05642	−0.528210	+	0.849114i	$0.677137\pi$
$758$	0	0
$759$	22486.0	1.07535
$760$	0	0
$761$	9084.21	0.432723	0.216362	−	0.976313i	$-0.430581\pi$
$761$	9084.21	0.432723	0.216362	+	0.976313i	$0.430581\pi$
$762$	0	0
$763$	−13873.3	−0.658252
$764$	0	0
$765$	119597.	5.65234
$766$	0	0
$767$	−1719.91	−0.0809677
$768$	0	0
$769$	−13443.2	−0.630395	−0.315197	−	0.949026i	$-0.602071\pi$
$769$	−13443.2	−0.630395	−0.315197	+	0.949026i	$0.602071\pi$
$770$	0	0
$771$	38196.1	1.78418
$772$	0	0
$773$	13215.7	0.614923	0.307461	−	0.951561i	$-0.400521\pi$
$773$	13215.7	0.614923	0.307461	+	0.951561i	$0.400521\pi$
$774$	0	0
$775$	−21287.5	−0.986669
$776$	0	0
$777$	−10092.8	−0.465993
$778$	0	0
$779$	−4979.69	−0.229032
$780$	0	0
$781$	21991.2	1.00756
$782$	0	0
$783$	5328.25	0.243188
$784$	0	0
$785$	−50761.5	−2.30797
$786$	0	0
$787$	−2927.52	−0.132598	−0.0662992	−	0.997800i	$-0.521119\pi$
$787$	−2927.52	−0.132598	−0.0662992	+	0.997800i	$0.521119\pi$
$788$	0	0
$789$	−43872.3	−1.97959
$790$	0	0
$791$	19995.0	0.898785
$792$	0	0
$793$	8771.50	0.392793
$794$	0	0
$795$	24233.0	1.08108
$796$	0	0
$797$	−28075.9	−1.24780	−0.623902	−	0.781502i	$-0.714454\pi$
$797$	−28075.9	−1.24780	−0.623902	+	0.781502i	$0.714454\pi$
$798$	0	0
$799$	−9946.69	−0.440411
$800$	0	0
$801$	29076.8	1.28262
$802$	0	0
$803$	15283.3	0.671651
$804$	0	0
$805$	−40387.5	−1.76829
$806$	0	0
$807$	−39587.2	−1.72681
$808$	0	0
$809$	−5522.90	−0.240019	−0.120009	−	0.992773i	$-0.538292\pi$
$809$	−5522.90	−0.240019	−0.120009	+	0.992773i	$0.538292\pi$
$810$	0	0
$811$	−10829.1	−0.468880	−0.234440	−	0.972131i	$-0.575326\pi$
$811$	−10829.1	−0.468880	−0.234440	+	0.972131i	$0.575326\pi$
$812$	0	0
$813$	67374.2	2.90642
$814$	0	0
$815$	−68556.8	−2.94655
$816$	0	0
$817$	4886.54	0.209252
$818$	0	0
$819$	−13363.4	−0.570155
$820$	0	0
$821$	15170.3	0.644881	0.322440	−	0.946590i	$-0.395497\pi$
$821$	15170.3	0.644881	0.322440	+	0.946590i	$0.395497\pi$
$822$	0	0
$823$	15082.0	0.638793	0.319396	−	0.947621i	$-0.396520\pi$
$823$	15082.0	0.638793	0.319396	+	0.947621i	$0.396520\pi$
$824$	0	0
$825$	−86808.1	−3.66336
$826$	0	0
$827$	−27386.4	−1.15153	−0.575766	−	0.817614i	$-0.695296\pi$
$827$	−27386.4	−1.15153	−0.575766	+	0.817614i	$0.695296\pi$
$828$	0	0
$829$	−6199.23	−0.259720	−0.129860	−	0.991532i	$-0.541453\pi$
$829$	−6199.23	−0.259720	−0.129860	+	0.991532i	$0.541453\pi$
$830$	0	0
$831$	−60641.4	−2.53144
$832$	0	0
$833$	−9732.66	−0.404822
$834$	0	0
$835$	−55659.0	−2.30678
$836$	0	0
$837$	−12127.7	−0.500829
$838$	0	0
$839$	10377.1	0.427007	0.213503	−	0.976942i	$-0.431513\pi$
$839$	10377.1	0.427007	0.213503	+	0.976942i	$0.431513\pi$
$840$	0	0
$841$	−23659.1	−0.970074
$842$	0	0
$843$	23572.1	0.963068
$844$	0	0
$845$	−3668.44	−0.149347
$846$	0	0
$847$	10571.7	0.428864
$848$	0	0
$849$	709.255	0.0286709
$850$	0	0
$851$	−4985.65	−0.200829
$852$	0	0
$853$	−10825.2	−0.434523	−0.217262	−	0.976113i	$-0.569712\pi$
$853$	−10825.2	−0.434523	−0.217262	+	0.976113i	$0.569712\pi$
$854$	0	0
$855$	18998.5	0.759923
$856$	0	0
$857$	32614.4	1.29999	0.649993	−	0.759940i	$-0.274772\pi$
$857$	32614.4	1.29999	0.649993	+	0.759940i	$0.274772\pi$
$858$	0	0
$859$	−17790.1	−0.706626	−0.353313	−	0.935505i	$-0.614945\pi$
$859$	−17790.1	−0.706626	−0.353313	+	0.935505i	$0.614945\pi$
$860$	0	0
$861$	−51168.9	−2.02535
$862$	0	0
$863$	−3003.67	−0.118477	−0.0592387	−	0.998244i	$-0.518867\pi$
$863$	−3003.67	−0.118477	−0.0592387	+	0.998244i	$0.518867\pi$
$864$	0	0
$865$	−11086.9	−0.435799
$866$	0	0
$867$	65220.6	2.55480
$868$	0	0
$869$	−16612.2	−0.648482
$870$	0	0
$871$	5186.78	0.201777
$872$	0	0
$873$	−301.722	−0.0116973
$874$	0	0
$875$	99618.9	3.84884
$876$	0	0
$877$	13966.5	0.537759	0.268880	−	0.963174i	$-0.413347\pi$
$877$	13966.5	0.537759	0.268880	+	0.963174i	$0.413347\pi$
$878$	0	0
$879$	42612.2	1.63513
$880$	0	0
$881$	37487.3	1.43357	0.716787	−	0.697293i	$-0.245612\pi$
$881$	37487.3	1.43357	0.716787	+	0.697293i	$0.245612\pi$
$882$	0	0
$883$	−16040.1	−0.611317	−0.305659	−	0.952141i	$-0.598877\pi$
$883$	−16040.1	−0.611317	−0.305659	+	0.952141i	$0.598877\pi$
$884$	0	0
$885$	25125.1	0.954319
$886$	0	0
$887$	30070.1	1.13828	0.569140	−	0.822240i	$-0.307276\pi$
$887$	30070.1	1.13828	0.569140	+	0.822240i	$0.307276\pi$
$888$	0	0
$889$	35821.3	1.35141
$890$	0	0
$891$	−11116.0	−0.417956
$892$	0	0
$893$	−1580.07	−0.0592106
$894$	0	0
$895$	33877.8	1.26526
$896$	0	0
$897$	−10198.9	−0.379633
$898$	0	0
$899$	−1661.28	−0.0616316
$900$	0	0
$901$	14190.7	0.524708
$902$	0	0
$903$	50211.7	1.85043
$904$	0	0
$905$	−15811.3	−0.580757
$906$	0	0
$907$	−22228.6	−0.813769	−0.406885	−	0.913480i	$-0.633385\pi$
$907$	−22228.6	−0.813769	−0.406885	+	0.913480i	$0.633385\pi$
$908$	0	0
$909$	63124.2	2.30330
$910$	0	0
$911$	−38083.0	−1.38501	−0.692506	−	0.721412i	$-0.743493\pi$
$911$	−38083.0	−1.38501	−0.692506	+	0.721412i	$0.743493\pi$
$912$	0	0
$913$	30822.9	1.11729
$914$	0	0
$915$	−128138.	−4.62963
$916$	0	0
$917$	13624.1	0.490631
$918$	0	0
$919$	1240.35	0.0445217	0.0222609	−	0.999752i	$-0.492914\pi$
$919$	1240.35	0.0445217	0.0222609	+	0.999752i	$0.492914\pi$
$920$	0	0
$921$	−28077.1	−1.00453
$922$	0	0
$923$	−9974.49	−0.355703
$924$	0	0
$925$	19247.3	0.684160
$926$	0	0
$927$	−81312.2	−2.88095
$928$	0	0
$929$	−6303.84	−0.222629	−0.111314	−	0.993785i	$-0.535506\pi$
$929$	−6303.84	−0.222629	−0.111314	+	0.993785i	$0.535506\pi$
$930$	0	0
$931$	−1546.07	−0.0544259
$932$	0	0
$933$	27081.8	0.950287
$934$	0	0
$935$	−69189.8	−2.42005
$936$	0	0
$937$	−22170.5	−0.772977	−0.386488	−	0.922294i	$-0.626312\pi$
$937$	−22170.5	−0.772977	−0.386488	+	0.922294i	$0.626312\pi$
$938$	0	0
$939$	−77814.5	−2.70434
$940$	0	0
$941$	279.026	0.00966630	0.00483315	−	0.999988i	$-0.498462\pi$
$941$	279.026	0.00966630	0.00483315	+	0.999988i	$0.498462\pi$
$942$	0	0
$943$	−25276.5	−0.872868
$944$	0	0
$945$	88827.9	3.05775
$946$	0	0
$947$	−32168.3	−1.10383	−0.551916	−	0.833900i	$-0.686103\pi$
$947$	−32168.3	−1.10383	−0.551916	+	0.833900i	$0.686103\pi$
$948$	0	0
$949$	−6932.00	−0.237115
$950$	0	0
$951$	−24599.1	−0.838781
$952$	0	0
$953$	4387.33	0.149129	0.0745643	−	0.997216i	$-0.476243\pi$
$953$	4387.33	0.149129	0.0745643	+	0.997216i	$0.476243\pi$
$954$	0	0
$955$	86813.9	2.94160
$956$	0	0
$957$	−6774.53	−0.228829
$958$	0	0
$959$	60996.6	2.05389
$960$	0	0
$961$	−26009.7	−0.873074
$962$	0	0
$963$	44677.4	1.49503
$964$	0	0
$965$	−11887.9	−0.396565
$966$	0	0
$967$	55979.6	1.86162	0.930808	−	0.365509i	$-0.119105\pi$
$967$	55979.6	1.86162	0.930808	+	0.365509i	$0.119105\pi$
$968$	0	0
$969$	17188.6	0.569843
$970$	0	0
$971$	45097.6	1.49048	0.745238	−	0.666798i	$-0.232336\pi$
$971$	45097.6	1.49048	0.745238	+	0.666798i	$0.232336\pi$
$972$	0	0
$973$	3724.00	0.122699
$974$	0	0
$975$	39373.3	1.29329
$976$	0	0
$977$	39725.4	1.30085	0.650424	−	0.759571i	$-0.274591\pi$
$977$	39725.4	1.30085	0.650424	+	0.759571i	$0.274591\pi$
$978$	0	0
$979$	−16821.6	−0.549154
$980$	0	0
$981$	33125.7	1.07811
$982$	0	0
$983$	1112.35	0.0360919	0.0180460	−	0.999837i	$-0.494255\pi$
$983$	1112.35	0.0360919	0.0180460	+	0.999837i	$0.494255\pi$
$984$	0	0
$985$	12876.5	0.416527
$986$	0	0
$987$	−16236.0	−0.523606
$988$	0	0
$989$	24803.6	0.797482
$990$	0	0
$991$	−18494.5	−0.592831	−0.296416	−	0.955059i	$-0.595791\pi$
$991$	−18494.5	−0.592831	−0.296416	+	0.955059i	$0.595791\pi$
$992$	0	0
$993$	25679.9	0.820670
$994$	0	0
$995$	9774.88	0.311442
$996$	0	0
$997$	−21205.1	−0.673594	−0.336797	−	0.941577i	$-0.609344\pi$
$997$	−21205.1	−0.673594	−0.336797	+	0.941577i	$0.609344\pi$
$998$	0	0
$999$	10965.4	0.347277

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.4.a.l.1.3		3
3.2	odd	2		1872.4.a.bm.1.3		3
4.3	odd	2		104.4.a.e.1.1	✓	3
8.3	odd	2		832.4.a.bc.1.3		3
8.5	even	2		832.4.a.bb.1.1		3
12.11	even	2		936.4.a.m.1.3		3
52.51	odd	2		1352.4.a.h.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.4.a.e.1.1	✓	3	4.3	odd	2
208.4.a.l.1.3		3	1.1	even	1	trivial
832.4.a.bb.1.1		3	8.5	even	2
832.4.a.bc.1.3		3	8.3	odd	2
936.4.a.m.1.3		3	12.11	even	2
1352.4.a.h.1.1		3	52.51	odd	2
1872.4.a.bm.1.3		3	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)

\(f(q)\)	\(=\)	\(q+8.74887 q^{3} -21.7068 q^{5} -20.7489 q^{7} +49.5428 q^{9} +O(q^{10})\)	\(q+8.74887 q^{3} -21.7068 q^{5} -20.7489 q^{7} +49.5428 q^{9} -28.6617 q^{11} +13.0000 q^{13} -189.910 q^{15} -111.210 q^{17} -17.6662 q^{19} -181.529 q^{21} -89.6721 q^{23} +346.183 q^{25} +197.224 q^{27} +27.0162 q^{29} -61.4919 q^{31} -250.758 q^{33} +450.391 q^{35} +55.5986 q^{37} +113.735 q^{39} +281.877 q^{41} -276.604 q^{43} -1075.41 q^{45} +89.4403 q^{47} +87.5158 q^{49} -972.966 q^{51} -127.603 q^{53} +622.153 q^{55} -154.559 q^{57} -132.300 q^{59} +674.731 q^{61} -1027.96 q^{63} -282.188 q^{65} +398.983 q^{67} -784.530 q^{69} -767.268 q^{71} -533.231 q^{73} +3028.71 q^{75} +594.698 q^{77} +579.596 q^{79} +387.833 q^{81} -1075.40 q^{83} +2414.02 q^{85} +236.362 q^{87} +586.903 q^{89} -269.735 q^{91} -537.985 q^{93} +383.476 q^{95} -6.09013 q^{97} -1419.98 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 8 q^{5} - 36 q^{7} + 73 q^{9}+O(q^{10}) \) 3 * q - 8 * q^5 - 36 * q^7 + 73 * q^9	\( 3 q - 8 q^{5} - 36 q^{7} + 73 q^{9} - 52 q^{11} + 39 q^{13} - 244 q^{15} - 116 q^{17} - 124 q^{19} - 154 q^{21} - 232 q^{23} + 191 q^{25} - 12 q^{27} - 30 q^{29} - 240 q^{31} - 564 q^{33} + 340 q^{35} - 264 q^{37} - 374 q^{41} - 248 q^{43} - 966 q^{45} + 412 q^{47} - 443 q^{49} - 92 q^{51} - 386 q^{53} + 400 q^{55} + 52 q^{57} + 940 q^{59} + 1206 q^{61} - 864 q^{63} - 104 q^{65} + 564 q^{67} + 512 q^{69} - 1260 q^{71} + 142 q^{73} + 3788 q^{75} + 1188 q^{77} - 1040 q^{79} + 1571 q^{81} - 756 q^{83} + 2510 q^{85} - 1536 q^{87} - 18 q^{89} - 468 q^{91} - 768 q^{93} - 384 q^{95} + 2174 q^{97} + 1940 q^{99}+O(q^{100}) \) 3 * q - 8 * q^5 - 36 * q^7 + 73 * q^9 - 52 * q^11 + 39 * q^13 - 244 * q^15 - 116 * q^17 - 124 * q^19 - 154 * q^21 - 232 * q^23 + 191 * q^25 - 12 * q^27 - 30 * q^29 - 240 * q^31 - 564 * q^33 + 340 * q^35 - 264 * q^37 - 374 * q^41 - 248 * q^43 - 966 * q^45 + 412 * q^47 - 443 * q^49 - 92 * q^51 - 386 * q^53 + 400 * q^55 + 52 * q^57 + 940 * q^59 + 1206 * q^61 - 864 * q^63 - 104 * q^65 + 564 * q^67 + 512 * q^69 - 1260 * q^71 + 142 * q^73 + 3788 * q^75 + 1188 * q^77 - 1040 * q^79 + 1571 * q^81 - 756 * q^83 + 2510 * q^85 - 1536 * q^87 - 18 * q^89 - 468 * q^91 - 768 * q^93 - 384 * q^95 + 2174 * q^97 + 1940 * q^99

Introduction

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