LMFDB - Embedded newform 208.4.a.l.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:	$3$
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			1.2
Root			$-4.78415$ 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)

$f(q)$	$=$	$q+0.0519499 q^{3} +7.51634 q^{5} -12.0519 q^{7} -26.9973 q^{9} +O(q^{10})$	$q+0.0519499 q^{3} +7.51634 q^{5} -12.0519 q^{7} -26.9973 q^{9} -58.5849 q^{11} +13.0000 q^{13} +0.390473 q^{15} +94.7514 q^{17} -82.3771 q^{19} -0.626097 q^{21} +4.96192 q^{23} -68.5046 q^{25} -2.80515 q^{27} -256.886 q^{29} -203.443 q^{31} -3.04348 q^{33} -90.5866 q^{35} +5.51726 q^{37} +0.675348 q^{39} -266.398 q^{41} +451.577 q^{43} -202.921 q^{45} -68.8724 q^{47} -197.751 q^{49} +4.92232 q^{51} +97.8531 q^{53} -440.344 q^{55} -4.27948 q^{57} +598.977 q^{59} +218.290 q^{61} +325.370 q^{63} +97.7124 q^{65} +65.7000 q^{67} +0.257771 q^{69} +319.255 q^{71} -285.588 q^{73} -3.55881 q^{75} +706.062 q^{77} -917.746 q^{79} +728.781 q^{81} -133.337 q^{83} +712.184 q^{85} -13.3452 q^{87} +244.626 q^{89} -156.675 q^{91} -10.5688 q^{93} -619.174 q^{95} +926.249 q^{97} +1581.63 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.0519499	0.00999776	0.00499888	−	0.999988i	$-0.498409\pi$
$3$	0.0519499	0.00999776	0.00499888	+	0.999988i	$0.498409\pi$
$4$	0	0
$5$	7.51634	0.672282	0.336141	−	0.941812i	$-0.390878\pi$
$5$	7.51634	0.672282	0.336141	+	0.941812i	$0.390878\pi$
$6$	0	0
$7$	−12.0519	−0.650744	−0.325372	−	0.945586i	$-0.605490\pi$
$7$	−12.0519	−0.650744	−0.325372	+	0.945586i	$0.605490\pi$
$8$	0	0
$9$	−26.9973	−0.999900
$10$	0	0
$11$	−58.5849	−1.60582	−0.802909	−	0.596101i	$-0.796716\pi$
$11$	−58.5849	−1.60582	−0.802909	+	0.596101i	$0.796716\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	0.390473	0.00672131
$16$	0	0
$17$	94.7514	1.35180	0.675900	−	0.736993i	$-0.263755\pi$
$17$	94.7514	1.35180	0.675900	+	0.736993i	$0.263755\pi$
$18$	0	0
$19$	−82.3771	−0.994663	−0.497331	−	0.867561i	$-0.665687\pi$
$19$	−82.3771	−0.994663	−0.497331	+	0.867561i	$0.665687\pi$
$20$	0	0
$21$	−0.626097	−0.00650598
$22$	0	0
$23$	4.96192	0.0449840	0.0224920	−	0.999747i	$-0.492840\pi$
$23$	4.96192	0.0449840	0.0224920	+	0.999747i	$0.492840\pi$
$24$	0	0
$25$	−68.5046	−0.548037
$26$	0	0
$27$	−2.80515	−0.0199945
$28$	0	0
$29$	−256.886	−1.64491	−0.822457	−	0.568828i	$-0.807397\pi$
$29$	−256.886	−1.64491	−0.822457	+	0.568828i	$0.807397\pi$
$30$	0	0
$31$	−203.443	−1.17869	−0.589345	−	0.807881i	$-0.700614\pi$
$31$	−203.443	−1.17869	−0.589345	+	0.807881i	$0.700614\pi$
$32$	0	0
$33$	−3.04348	−0.0160546
$34$	0	0
$35$	−90.5866	−0.437484
$36$	0	0
$37$	5.51726	0.0245144	0.0122572	−	0.999925i	$-0.496098\pi$
$37$	5.51726	0.0245144	0.0122572	+	0.999925i	$0.496098\pi$
$38$	0	0
$39$	0.675348	0.00277288
$40$	0	0
$41$	−266.398	−1.01474	−0.507370	−	0.861729i	$-0.669382\pi$
$41$	−266.398	−1.01474	−0.507370	+	0.861729i	$0.669382\pi$
$42$	0	0
$43$	451.577	1.60151	0.800754	−	0.598993i	$-0.204432\pi$
$43$	451.577	1.60151	0.800754	+	0.598993i	$0.204432\pi$
$44$	0	0
$45$	−202.921	−0.672215
$46$	0	0
$47$	−68.8724	−0.213746	−0.106873	−	0.994273i	$-0.534084\pi$
$47$	−68.8724	−0.213746	−0.106873	+	0.994273i	$0.534084\pi$
$48$	0	0
$49$	−197.751	−0.576532
$50$	0	0
$51$	4.92232	0.0135150
$52$	0	0
$53$	97.8531	0.253607	0.126803	−	0.991928i	$-0.459528\pi$
$53$	97.8531	0.253607	0.126803	+	0.991928i	$0.459528\pi$
$54$	0	0
$55$	−440.344	−1.07956
$56$	0	0
$57$	−4.27948	−0.00994440
$58$	0	0
$59$	598.977	1.32170	0.660849	−	0.750519i	$-0.270196\pi$
$59$	598.977	1.32170	0.660849	+	0.750519i	$0.270196\pi$
$60$	0	0
$61$	218.290	0.458184	0.229092	−	0.973405i	$-0.426424\pi$
$61$	218.290	0.458184	0.229092	+	0.973405i	$0.426424\pi$
$62$	0	0
$63$	325.370	0.650679
$64$	0	0
$65$	97.7124	0.186457
$66$	0	0
$67$	65.7000	0.119799	0.0598995	−	0.998204i	$-0.480922\pi$
$67$	65.7000	0.119799	0.0598995	+	0.998204i	$0.480922\pi$
$68$	0	0
$69$	0.257771	0.000449739 0
$70$	0	0
$71$	319.255	0.533642	0.266821	−	0.963746i	$-0.414027\pi$
$71$	319.255	0.533642	0.266821	+	0.963746i	$0.414027\pi$
$72$	0	0
$73$	−285.588	−0.457884	−0.228942	−	0.973440i	$-0.573527\pi$
$73$	−285.588	−0.457884	−0.228942	+	0.973440i	$0.573527\pi$
$74$	0	0
$75$	−3.55881	−0.00547914
$76$	0	0
$77$	706.062	1.04498
$78$	0	0
$79$	−917.746	−1.30702	−0.653510	−	0.756918i	$-0.726704\pi$
$79$	−917.746	−1.30702	−0.653510	+	0.756918i	$0.726704\pi$
$80$	0	0
$81$	728.781	0.999700
$82$	0	0
$83$	−133.337	−0.176332	−0.0881662	−	0.996106i	$-0.528101\pi$
$83$	−133.337	−0.176332	−0.0881662	+	0.996106i	$0.528101\pi$
$84$	0	0
$85$	712.184	0.908791
$86$	0	0
$87$	−13.3452	−0.0164454
$88$	0	0
$89$	244.626	0.291352	0.145676	−	0.989332i	$-0.453464\pi$
$89$	244.626	0.291352	0.145676	+	0.989332i	$0.453464\pi$
$90$	0	0
$91$	−156.675	−0.180484
$92$	0	0
$93$	−10.5688	−0.0117843
$94$	0	0
$95$	−619.174	−0.668694
$96$	0	0
$97$	926.249	0.969550	0.484775	−	0.874639i	$-0.338902\pi$
$97$	926.249	0.969550	0.484775	+	0.874639i	$0.338902\pi$
$98$	0	0
$99$	1581.63	1.60566
$100$	0	0
$101$	−1358.74	−1.33862	−0.669308	−	0.742985i	$-0.733409\pi$
$101$	−1358.74	−1.33862	−0.669308	+	0.742985i	$0.733409\pi$
$102$	0	0
$103$	−1144.49	−1.09486	−0.547429	−	0.836852i	$-0.684393\pi$
$103$	−1144.49	−1.09486	−0.547429	+	0.836852i	$0.684393\pi$
$104$	0	0
$105$	−4.70596	−0.00437385
$106$	0	0
$107$	1770.30	1.59945	0.799727	−	0.600364i	$-0.204978\pi$
$107$	1770.30	1.59945	0.799727	+	0.600364i	$0.204978\pi$
$108$	0	0
$109$	1250.33	1.09871	0.549355	−	0.835589i	$-0.314873\pi$
$109$	1250.33	1.09871	0.549355	+	0.835589i	$0.314873\pi$
$110$	0	0
$111$	0.286621	0.000245089 0
$112$	0	0
$113$	823.180	0.685294	0.342647	−	0.939464i	$-0.388676\pi$
$113$	823.180	0.685294	0.342647	+	0.939464i	$0.388676\pi$
$114$	0	0
$115$	37.2955	0.0302419
$116$	0	0
$117$	−350.965	−0.277322
$118$	0	0
$119$	−1141.94	−0.879676
$120$	0	0
$121$	2101.19	1.57865
$122$	0	0
$123$	−13.8393	−0.0101451
$124$	0	0
$125$	−1454.45	−1.04072
$126$	0	0
$127$	1962.83	1.37144	0.685722	−	0.727864i	$-0.259487\pi$
$127$	1962.83	1.37144	0.685722	+	0.727864i	$0.259487\pi$
$128$	0	0
$129$	23.4594	0.0160115
$130$	0	0
$131$	1166.78	0.778183	0.389092	−	0.921199i	$-0.372789\pi$
$131$	1166.78	0.778183	0.389092	+	0.921199i	$0.372789\pi$
$132$	0	0
$133$	992.804	0.647271
$134$	0	0
$135$	−21.0845	−0.0134420
$136$	0	0
$137$	−3104.18	−1.93582	−0.967912	−	0.251288i	$-0.919146\pi$
$137$	−3104.18	−1.93582	−0.967912	+	0.251288i	$0.919146\pi$
$138$	0	0
$139$	−528.389	−0.322427	−0.161213	−	0.986920i	$-0.551541\pi$
$139$	−528.389	−0.322427	−0.161213	+	0.986920i	$0.551541\pi$
$140$	0	0
$141$	−3.57791	−0.00213698
$142$	0	0
$143$	−761.603	−0.445374
$144$	0	0
$145$	−1930.84	−1.10585
$146$	0	0
$147$	−10.2731	−0.00576403
$148$	0	0
$149$	−1599.76	−0.879581	−0.439790	−	0.898100i	$-0.644947\pi$
$149$	−1599.76	−0.879581	−0.439790	+	0.898100i	$0.644947\pi$
$150$	0	0
$151$	195.543	0.105385	0.0526923	−	0.998611i	$-0.483220\pi$
$151$	195.543	0.105385	0.0526923	+	0.998611i	$0.483220\pi$
$152$	0	0
$153$	−2558.03	−1.35166
$154$	0	0
$155$	−1529.15	−0.792413
$156$	0	0
$157$	1414.57	0.719078	0.359539	−	0.933130i	$-0.382934\pi$
$157$	1414.57	0.719078	0.359539	+	0.933130i	$0.382934\pi$
$158$	0	0
$159$	5.08346	0.00253550
$160$	0	0
$161$	−59.8008	−0.0292731
$162$	0	0
$163$	3177.92	1.52708	0.763539	−	0.645761i	$-0.223460\pi$
$163$	3177.92	1.52708	0.763539	+	0.645761i	$0.223460\pi$
$164$	0	0
$165$	−22.8758	−0.0107932
$166$	0	0
$167$	−3356.99	−1.55552	−0.777760	−	0.628561i	$-0.783644\pi$
$167$	−3356.99	−1.55552	−0.777760	+	0.628561i	$0.783644\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	2223.96	0.994563
$172$	0	0
$173$	−3828.33	−1.68244	−0.841222	−	0.540690i	$-0.818163\pi$
$173$	−3828.33	−1.68244	−0.841222	+	0.540690i	$0.818163\pi$
$174$	0	0
$175$	825.614	0.356632
$176$	0	0
$177$	31.1168	0.0132140
$178$	0	0
$179$	−804.950	−0.336116	−0.168058	−	0.985777i	$-0.553750\pi$
$179$	−804.950	−0.336116	−0.168058	+	0.985777i	$0.553750\pi$
$180$	0	0
$181$	4274.16	1.75523	0.877614	−	0.479369i	$-0.159134\pi$
$181$	4274.16	1.75523	0.877614	+	0.479369i	$0.159134\pi$
$182$	0	0
$183$	11.3402	0.00458081
$184$	0	0
$185$	41.4696	0.0164806
$186$	0	0
$187$	−5551.00	−2.17074
$188$	0	0
$189$	33.8076	0.0130113
$190$	0	0
$191$	−3993.20	−1.51276	−0.756382	−	0.654130i	$-0.773035\pi$
$191$	−3993.20	−1.51276	−0.756382	+	0.654130i	$0.773035\pi$
$192$	0	0
$193$	−3346.26	−1.24803	−0.624014	−	0.781413i	$-0.714499\pi$
$193$	−3346.26	−1.24803	−0.624014	+	0.781413i	$0.714499\pi$
$194$	0	0
$195$	5.07615	0.00186416
$196$	0	0
$197$	97.2020	0.0351541	0.0175770	−	0.999846i	$-0.494405\pi$
$197$	97.2020	0.0351541	0.0175770	+	0.999846i	$0.494405\pi$
$198$	0	0
$199$	−640.175	−0.228044	−0.114022	−	0.993478i	$-0.536374\pi$
$199$	−640.175	−0.228044	−0.114022	+	0.993478i	$0.536374\pi$
$200$	0	0
$201$	3.41311	0.00119772
$202$	0	0
$203$	3095.97	1.07042
$204$	0	0
$205$	−2002.33	−0.682191
$206$	0	0
$207$	−133.958	−0.0449795
$208$	0	0
$209$	4826.05	1.59725
$210$	0	0
$211$	−1423.20	−0.464346	−0.232173	−	0.972674i	$-0.574584\pi$
$211$	−1423.20	−0.464346	−0.232173	+	0.972674i	$0.574584\pi$
$212$	0	0
$213$	16.5852	0.00533522
$214$	0	0
$215$	3394.21	1.07667
$216$	0	0
$217$	2451.88	0.767026
$218$	0	0
$219$	−14.8363	−0.00457782
$220$	0	0
$221$	1231.77	0.374922
$222$	0	0
$223$	−1186.75	−0.356371	−0.178186	−	0.983997i	$-0.557023\pi$
$223$	−1186.75	−0.356371	−0.178186	+	0.983997i	$0.557023\pi$
$224$	0	0
$225$	1849.44	0.547982
$226$	0	0
$227$	−3539.02	−1.03477	−0.517386	−	0.855752i	$-0.673095\pi$
$227$	−3539.02	−1.03477	−0.517386	+	0.855752i	$0.673095\pi$
$228$	0	0
$229$	−4176.67	−1.20525	−0.602625	−	0.798025i	$-0.705878\pi$
$229$	−4176.67	−1.20525	−0.602625	+	0.798025i	$0.705878\pi$
$230$	0	0
$231$	36.6798	0.0104474
$232$	0	0
$233$	−198.372	−0.0557758	−0.0278879	−	0.999611i	$-0.508878\pi$
$233$	−198.372	−0.0557758	−0.0278879	+	0.999611i	$0.508878\pi$
$234$	0	0
$235$	−517.668	−0.143698
$236$	0	0
$237$	−47.6768	−0.0130673
$238$	0	0
$239$	−5193.18	−1.40552	−0.702760	−	0.711427i	$-0.748049\pi$
$239$	−5193.18	−1.40552	−0.702760	+	0.711427i	$0.748049\pi$
$240$	0	0
$241$	5435.97	1.45295	0.726476	−	0.687192i	$-0.241157\pi$
$241$	5435.97	1.45295	0.726476	+	0.687192i	$0.241157\pi$
$242$	0	0
$243$	113.599	0.0299893
$244$	0	0
$245$	−1486.36	−0.387592
$246$	0	0
$247$	−1070.90	−0.275870
$248$	0	0
$249$	−6.92681	−0.00176293
$250$	0	0
$251$	2269.27	0.570657	0.285328	−	0.958430i	$-0.407897\pi$
$251$	2269.27	0.570657	0.285328	+	0.958430i	$0.407897\pi$
$252$	0	0
$253$	−290.693	−0.0722361
$254$	0	0
$255$	36.9979	0.00908587
$256$	0	0
$257$	−6947.78	−1.68634	−0.843172	−	0.537643i	$-0.819315\pi$
$257$	−6947.78	−1.68634	−0.843172	+	0.537643i	$0.819315\pi$
$258$	0	0
$259$	−66.4937	−0.0159526
$260$	0	0
$261$	6935.22	1.64475
$262$	0	0
$263$	5220.32	1.22395	0.611974	−	0.790878i	$-0.290376\pi$
$263$	5220.32	1.22395	0.611974	+	0.790878i	$0.290376\pi$
$264$	0	0
$265$	735.497	0.170495
$266$	0	0
$267$	12.7083	0.00291287
$268$	0	0
$269$	−1641.66	−0.372095	−0.186048	−	0.982541i	$-0.559568\pi$
$269$	−1641.66	−0.372095	−0.186048	+	0.982541i	$0.559568\pi$
$270$	0	0
$271$	−1558.50	−0.349345	−0.174672	−	0.984627i	$-0.555887\pi$
$271$	−1558.50	−0.349345	−0.174672	+	0.984627i	$0.555887\pi$
$272$	0	0
$273$	−8.13926	−0.00180443
$274$	0	0
$275$	4013.33	0.880048
$276$	0	0
$277$	5192.23	1.12625	0.563124	−	0.826372i	$-0.309599\pi$
$277$	5192.23	1.12625	0.563124	+	0.826372i	$0.309599\pi$
$278$	0	0
$279$	5492.41	1.17857
$280$	0	0
$281$	−5602.55	−1.18939	−0.594697	−	0.803950i	$-0.702728\pi$
$281$	−5602.55	−1.18939	−0.594697	+	0.803950i	$0.702728\pi$
$282$	0	0
$283$	5928.63	1.24530	0.622651	−	0.782500i	$-0.286056\pi$
$283$	5928.63	1.24530	0.622651	+	0.782500i	$0.286056\pi$
$284$	0	0
$285$	−32.1660	−0.00668544
$286$	0	0
$287$	3210.61	0.660336
$288$	0	0
$289$	4064.83	0.827363
$290$	0	0
$291$	48.1185	0.00969332
$292$	0	0
$293$	3604.56	0.718705	0.359353	−	0.933202i	$-0.382997\pi$
$293$	3604.56	0.718705	0.359353	+	0.933202i	$0.382997\pi$
$294$	0	0
$295$	4502.12	0.888554
$296$	0	0
$297$	164.339	0.0321076
$298$	0	0
$299$	64.5050	0.0124763
$300$	0	0
$301$	−5442.39	−1.04217
$302$	0	0
$303$	−70.5866	−0.0133832
$304$	0	0
$305$	1640.74	0.308029
$306$	0	0
$307$	−1487.38	−0.276511	−0.138256	−	0.990397i	$-0.544150\pi$
$307$	−1487.38	−0.276511	−0.138256	+	0.990397i	$0.544150\pi$
$308$	0	0
$309$	−59.4563	−0.0109461
$310$	0	0
$311$	−3760.35	−0.685627	−0.342814	−	0.939403i	$-0.611380\pi$
$311$	−3760.35	−0.685627	−0.342814	+	0.939403i	$0.611380\pi$
$312$	0	0
$313$	−3546.69	−0.640481	−0.320241	−	0.947336i	$-0.603764\pi$
$313$	−3546.69	−0.640481	−0.320241	+	0.947336i	$0.603764\pi$
$314$	0	0
$315$	2445.59	0.437440
$316$	0	0
$317$	−2015.56	−0.357113	−0.178557	−	0.983930i	$-0.557143\pi$
$317$	−2015.56	−0.357113	−0.178557	+	0.983930i	$0.557143\pi$
$318$	0	0
$319$	15049.6	2.64143
$320$	0	0
$321$	91.9670	0.0159909
$322$	0	0
$323$	−7805.34	−1.34458
$324$	0	0
$325$	−890.560	−0.151998
$326$	0	0
$327$	64.9542	0.0109846
$328$	0	0
$329$	830.046	0.139094
$330$	0	0
$331$	−3112.38	−0.516834	−0.258417	−	0.966033i	$-0.583201\pi$
$331$	−3112.38	−0.516834	−0.258417	+	0.966033i	$0.583201\pi$
$332$	0	0
$333$	−148.951	−0.0245119
$334$	0	0
$335$	493.824	0.0805387
$336$	0	0
$337$	3594.16	0.580969	0.290484	−	0.956880i	$-0.406184\pi$
$337$	3594.16	0.580969	0.290484	+	0.956880i	$0.406184\pi$
$338$	0	0
$339$	42.7641	0.00685141
$340$	0	0
$341$	11918.7	1.89276
$342$	0	0
$343$	6517.10	1.02592
$344$	0	0
$345$	1.93750	0.000302352 0
$346$	0	0
$347$	−9718.13	−1.50345	−0.751724	−	0.659478i	$-0.770777\pi$
$347$	−9718.13	−1.50345	−0.751724	+	0.659478i	$0.770777\pi$
$348$	0	0
$349$	−10264.7	−1.57438	−0.787188	−	0.616713i	$-0.788464\pi$
$349$	−10264.7	−1.57438	−0.787188	+	0.616713i	$0.788464\pi$
$350$	0	0
$351$	−36.4670	−0.00554548
$352$	0	0
$353$	−4486.65	−0.676489	−0.338244	−	0.941058i	$-0.609833\pi$
$353$	−4486.65	−0.676489	−0.338244	+	0.941058i	$0.609833\pi$
$354$	0	0
$355$	2399.63	0.358758
$356$	0	0
$357$	−59.3236	−0.00879478
$358$	0	0
$359$	−4448.49	−0.653990	−0.326995	−	0.945026i	$-0.606036\pi$
$359$	−4448.49	−0.653990	−0.326995	+	0.945026i	$0.606036\pi$
$360$	0	0
$361$	−73.0198	−0.0106458
$362$	0	0
$363$	109.156	0.0157830
$364$	0	0
$365$	−2146.58	−0.307827
$366$	0	0
$367$	12147.3	1.72775	0.863876	−	0.503704i	$-0.168030\pi$
$367$	12147.3	1.72775	0.863876	+	0.503704i	$0.168030\pi$
$368$	0	0
$369$	7192.02	1.01464
$370$	0	0
$371$	−1179.32	−0.165033
$372$	0	0
$373$	7131.62	0.989977	0.494989	−	0.868900i	$-0.335172\pi$
$373$	7131.62	0.989977	0.494989	+	0.868900i	$0.335172\pi$
$374$	0	0
$375$	−75.5583	−0.0104048
$376$	0	0
$377$	−3339.51	−0.456217
$378$	0	0
$379$	−11916.2	−1.61503	−0.807515	−	0.589847i	$-0.799188\pi$
$379$	−11916.2	−1.61503	−0.807515	+	0.589847i	$0.799188\pi$
$380$	0	0
$381$	101.969	0.0137114
$382$	0	0
$383$	−353.179	−0.0471191	−0.0235596	−	0.999722i	$-0.507500\pi$
$383$	−353.179	−0.0471191	−0.0235596	+	0.999722i	$0.507500\pi$
$384$	0	0
$385$	5307.00	0.702519
$386$	0	0
$387$	−12191.4	−1.60135
$388$	0	0
$389$	−808.659	−0.105400	−0.0527001	−	0.998610i	$-0.516783\pi$
$389$	−808.659	−0.105400	−0.0527001	+	0.998610i	$0.516783\pi$
$390$	0	0
$391$	470.149	0.0608094
$392$	0	0
$393$	60.6141	0.00778009
$394$	0	0
$395$	−6898.09	−0.878685
$396$	0	0
$397$	375.922	0.0475239	0.0237620	−	0.999718i	$-0.492436\pi$
$397$	375.922	0.0475239	0.0237620	+	0.999718i	$0.492436\pi$
$398$	0	0
$399$	51.5761	0.00647126
$400$	0	0
$401$	5039.29	0.627556	0.313778	−	0.949496i	$-0.398405\pi$
$401$	5039.29	0.627556	0.313778	+	0.949496i	$0.398405\pi$
$402$	0	0
$403$	−2644.76	−0.326910
$404$	0	0
$405$	5477.77	0.672080
$406$	0	0
$407$	−323.228	−0.0393656
$408$	0	0
$409$	−9366.18	−1.13234	−0.566171	−	0.824288i	$-0.691576\pi$
$409$	−9366.18	−1.13234	−0.566171	+	0.824288i	$0.691576\pi$
$410$	0	0
$411$	−161.262	−0.0193539
$412$	0	0
$413$	−7218.84	−0.860087
$414$	0	0
$415$	−1002.20	−0.118545
$416$	0	0
$417$	−27.4497	−0.00322355
$418$	0	0
$419$	−4560.60	−0.531743	−0.265871	−	0.964009i	$-0.585660\pi$
$419$	−4560.60	−0.531743	−0.265871	+	0.964009i	$0.585660\pi$
$420$	0	0
$421$	−9468.60	−1.09613	−0.548066	−	0.836435i	$-0.684636\pi$
$421$	−9468.60	−1.09613	−0.548066	+	0.836435i	$0.684636\pi$
$422$	0	0
$423$	1859.37	0.213725
$424$	0	0
$425$	−6490.91	−0.740836
$426$	0	0
$427$	−2630.82	−0.298160
$428$	0	0
$429$	−39.5652	−0.00445274
$430$	0	0
$431$	−15675.3	−1.75186	−0.875930	−	0.482439i	$-0.839751\pi$
$431$	−15675.3	−1.75186	−0.875930	+	0.482439i	$0.839751\pi$
$432$	0	0
$433$	−4434.92	−0.492214	−0.246107	−	0.969243i	$-0.579151\pi$
$433$	−4434.92	−0.492214	−0.246107	+	0.969243i	$0.579151\pi$
$434$	0	0
$435$	−100.307	−0.0110560
$436$	0	0
$437$	−408.749	−0.0447439
$438$	0	0
$439$	−16390.8	−1.78198	−0.890988	−	0.454026i	$-0.849987\pi$
$439$	−16390.8	−1.78198	−0.890988	+	0.454026i	$0.849987\pi$
$440$	0	0
$441$	5338.73	0.576474
$442$	0	0
$443$	3793.91	0.406894	0.203447	−	0.979086i	$-0.434785\pi$
$443$	3793.91	0.406894	0.203447	+	0.979086i	$0.434785\pi$
$444$	0	0
$445$	1838.69	0.195871
$446$	0	0
$447$	−83.1074	−0.00879383
$448$	0	0
$449$	16415.2	1.72535	0.862673	−	0.505762i	$-0.168789\pi$
$449$	16415.2	1.72535	0.862673	+	0.505762i	$0.168789\pi$
$450$	0	0
$451$	15606.9	1.62949
$452$	0	0
$453$	10.1584	0.00105361
$454$	0	0
$455$	−1177.63	−0.121336
$456$	0	0
$457$	−2312.29	−0.236683	−0.118342	−	0.992973i	$-0.537758\pi$
$457$	−2312.29	−0.236683	−0.118342	+	0.992973i	$0.537758\pi$
$458$	0	0
$459$	−265.792	−0.0270286
$460$	0	0
$461$	8067.63	0.815069	0.407535	−	0.913190i	$-0.366389\pi$
$461$	8067.63	0.815069	0.407535	+	0.913190i	$0.366389\pi$
$462$	0	0
$463$	12116.9	1.21625	0.608124	−	0.793842i	$-0.291923\pi$
$463$	12116.9	1.21625	0.608124	+	0.793842i	$0.291923\pi$
$464$	0	0
$465$	−79.4389	−0.00792235
$466$	0	0
$467$	11811.6	1.17039	0.585196	−	0.810892i	$-0.301017\pi$
$467$	11811.6	1.17039	0.585196	+	0.810892i	$0.301017\pi$
$468$	0	0
$469$	−791.813	−0.0779585
$470$	0	0
$471$	73.4869	0.00718917
$472$	0	0
$473$	−26455.6	−2.57173
$474$	0	0
$475$	5643.21	0.545112
$476$	0	0
$477$	−2641.77	−0.253581
$478$	0	0
$479$	6707.40	0.639810	0.319905	−	0.947450i	$-0.396349\pi$
$479$	6707.40	0.639810	0.319905	+	0.947450i	$0.396349\pi$
$480$	0	0
$481$	71.7244	0.00679906
$482$	0	0
$483$	−3.10665	−0.000292665 0
$484$	0	0
$485$	6962.00	0.651811
$486$	0	0
$487$	−8141.26	−0.757527	−0.378763	−	0.925494i	$-0.623651\pi$
$487$	−8141.26	−0.757527	−0.378763	+	0.925494i	$0.623651\pi$
$488$	0	0
$489$	165.092	0.0152674
$490$	0	0
$491$	−10112.7	−0.929490	−0.464745	−	0.885445i	$-0.653854\pi$
$491$	−10112.7	−0.929490	−0.464745	+	0.885445i	$0.653854\pi$
$492$	0	0
$493$	−24340.3	−2.22359
$494$	0	0
$495$	11888.1	1.07945
$496$	0	0
$497$	−3847.64	−0.347264
$498$	0	0
$499$	−14174.0	−1.27157	−0.635785	−	0.771866i	$-0.719324\pi$
$499$	−14174.0	−1.27157	−0.635785	+	0.771866i	$0.719324\pi$
$500$	0	0
$501$	−174.395	−0.0155517
$502$	0	0
$503$	4190.17	0.371432	0.185716	−	0.982603i	$-0.440540\pi$
$503$	4190.17	0.371432	0.185716	+	0.982603i	$0.440540\pi$
$504$	0	0
$505$	−10212.8	−0.899927
$506$	0	0
$507$	8.77953	0.000769058 0
$508$	0	0
$509$	−10761.4	−0.937111	−0.468555	−	0.883434i	$-0.655225\pi$
$509$	−10761.4	−0.937111	−0.468555	+	0.883434i	$0.655225\pi$
$510$	0	0
$511$	3441.89	0.297966
$512$	0	0
$513$	231.080	0.0198878
$514$	0	0
$515$	−8602.41	−0.736053
$516$	0	0
$517$	4034.88	0.343237
$518$	0	0
$519$	−198.881	−0.0168207
$520$	0	0
$521$	15561.6	1.30858	0.654288	−	0.756246i	$-0.272969\pi$
$521$	15561.6	1.30858	0.654288	+	0.756246i	$0.272969\pi$
$522$	0	0
$523$	7111.00	0.594536	0.297268	−	0.954794i	$-0.403924\pi$
$523$	7111.00	0.594536	0.297268	+	0.954794i	$0.403924\pi$
$524$	0	0
$525$	42.8906	0.00356552
$526$	0	0
$527$	−19276.5	−1.59335
$528$	0	0
$529$	−12142.4	−0.997976
$530$	0	0
$531$	−16170.8	−1.32157
$532$	0	0
$533$	−3463.17	−0.281438
$534$	0	0
$535$	13306.2	1.07528
$536$	0	0
$537$	−41.8170	−0.00336041
$538$	0	0
$539$	11585.2	0.925806
$540$	0	0
$541$	22162.3	1.76124	0.880620	−	0.473824i	$-0.157127\pi$
$541$	22162.3	1.76124	0.880620	+	0.473824i	$0.157127\pi$
$542$	0	0
$543$	222.042	0.0175483
$544$	0	0
$545$	9397.87	0.738643
$546$	0	0
$547$	9401.44	0.734875	0.367437	−	0.930048i	$-0.380235\pi$
$547$	9401.44	0.734875	0.367437	+	0.930048i	$0.380235\pi$
$548$	0	0
$549$	−5893.25	−0.458138
$550$	0	0
$551$	21161.5	1.63613
$552$	0	0
$553$	11060.6	0.850535
$554$	0	0
$555$	2.15434	0.000164769 0
$556$	0	0
$557$	−12244.7	−0.931463	−0.465731	−	0.884926i	$-0.654209\pi$
$557$	−12244.7	−0.931463	−0.465731	+	0.884926i	$0.654209\pi$
$558$	0	0
$559$	5870.50	0.444179
$560$	0	0
$561$	−288.374	−0.0217026
$562$	0	0
$563$	21958.0	1.64373	0.821865	−	0.569682i	$-0.192934\pi$
$563$	21958.0	1.64373	0.821865	+	0.569682i	$0.192934\pi$
$564$	0	0
$565$	6187.30	0.460711
$566$	0	0
$567$	−8783.24	−0.650549
$568$	0	0
$569$	501.601	0.0369564	0.0184782	−	0.999829i	$-0.494118\pi$
$569$	501.601	0.0369564	0.0184782	+	0.999829i	$0.494118\pi$
$570$	0	0
$571$	24616.5	1.80415	0.902076	−	0.431578i	$-0.142043\pi$
$571$	24616.5	1.80415	0.902076	+	0.431578i	$0.142043\pi$
$572$	0	0
$573$	−207.446	−0.0151242
$574$	0	0
$575$	−339.915	−0.0246529
$576$	0	0
$577$	2346.78	0.169320	0.0846600	−	0.996410i	$-0.473020\pi$
$577$	2346.78	0.169320	0.0846600	+	0.996410i	$0.473020\pi$
$578$	0	0
$579$	−173.838	−0.0124775
$580$	0	0
$581$	1606.96	0.114747
$582$	0	0
$583$	−5732.71	−0.407246
$584$	0	0
$585$	−2637.97	−0.186439
$586$	0	0
$587$	−1566.32	−0.110135	−0.0550673	−	0.998483i	$-0.517537\pi$
$587$	−1566.32	−0.110135	−0.0550673	+	0.998483i	$0.517537\pi$
$588$	0	0
$589$	16759.0	1.17240
$590$	0	0
$591$	5.04963	0.000351462 0
$592$	0	0
$593$	19887.0	1.37717	0.688583	−	0.725157i	$-0.258233\pi$
$593$	19887.0	1.37717	0.688583	+	0.725157i	$0.258233\pi$
$594$	0	0
$595$	−8583.21	−0.591390
$596$	0	0
$597$	−33.2570	−0.00227993
$598$	0	0
$599$	9073.88	0.618946	0.309473	−	0.950908i	$-0.399847\pi$
$599$	9073.88	0.618946	0.309473	+	0.950908i	$0.399847\pi$
$600$	0	0
$601$	16604.5	1.12697	0.563487	−	0.826125i	$-0.309459\pi$
$601$	16604.5	1.12697	0.563487	+	0.826125i	$0.309459\pi$
$602$	0	0
$603$	−1773.72	−0.119787
$604$	0	0
$605$	15793.2	1.06130
$606$	0	0
$607$	−26692.4	−1.78486	−0.892431	−	0.451183i	$-0.851002\pi$
$607$	−26692.4	−1.78486	−0.892431	+	0.451183i	$0.851002\pi$
$608$	0	0
$609$	160.835	0.0107018
$610$	0	0
$611$	−895.341	−0.0592825
$612$	0	0
$613$	8549.18	0.563292	0.281646	−	0.959518i	$-0.409120\pi$
$613$	8549.18	0.563292	0.281646	+	0.959518i	$0.409120\pi$
$614$	0	0
$615$	−104.021	−0.00682038
$616$	0	0
$617$	−776.812	−0.0506860	−0.0253430	−	0.999679i	$-0.508068\pi$
$617$	−776.812	−0.0506860	−0.0253430	+	0.999679i	$0.508068\pi$
$618$	0	0
$619$	−26846.2	−1.74320	−0.871599	−	0.490219i	$-0.836917\pi$
$619$	−26846.2	−1.74320	−0.871599	+	0.490219i	$0.836917\pi$
$620$	0	0
$621$	−13.9189	−0.000899433 0
$622$	0	0
$623$	−2948.22	−0.189596
$624$	0	0
$625$	−2369.04	−0.151618
$626$	0	0
$627$	250.713	0.0159689
$628$	0	0
$629$	522.768	0.0331385
$630$	0	0
$631$	13226.6	0.834458	0.417229	−	0.908801i	$-0.363001\pi$
$631$	13226.6	0.834458	0.417229	+	0.908801i	$0.363001\pi$
$632$	0	0
$633$	−73.9350	−0.00464242
$634$	0	0
$635$	14753.3	0.921996
$636$	0	0
$637$	−2570.76	−0.159901
$638$	0	0
$639$	−8619.01	−0.533588
$640$	0	0
$641$	−2242.71	−0.138193	−0.0690964	−	0.997610i	$-0.522012\pi$
$641$	−2242.71	−0.138193	−0.0690964	+	0.997610i	$0.522012\pi$
$642$	0	0
$643$	−1337.91	−0.0820563	−0.0410282	−	0.999158i	$-0.513063\pi$
$643$	−1337.91	−0.0820563	−0.0410282	+	0.999158i	$0.513063\pi$
$644$	0	0
$645$	176.329	0.0107642
$646$	0	0
$647$	−17870.6	−1.08588	−0.542941	−	0.839771i	$-0.682689\pi$
$647$	−17870.6	−1.08588	−0.542941	+	0.839771i	$0.682689\pi$
$648$	0	0
$649$	−35091.0	−2.12241
$650$	0	0
$651$	127.375	0.00766854
$652$	0	0
$653$	6267.11	0.375576	0.187788	−	0.982210i	$-0.439868\pi$
$653$	6267.11	0.375576	0.187788	+	0.982210i	$0.439868\pi$
$654$	0	0
$655$	8769.91	0.523159
$656$	0	0
$657$	7710.11	0.457839
$658$	0	0
$659$	17737.7	1.04850	0.524252	−	0.851563i	$-0.324345\pi$
$659$	17737.7	1.04850	0.524252	+	0.851563i	$0.324345\pi$
$660$	0	0
$661$	21640.1	1.27338	0.636689	−	0.771121i	$-0.280304\pi$
$661$	21640.1	1.27338	0.636689	+	0.771121i	$0.280304\pi$
$662$	0	0
$663$	63.9902	0.00374838
$664$	0	0
$665$	7462.25	0.435149
$666$	0	0
$667$	−1274.65	−0.0739948
$668$	0	0
$669$	−61.6516	−0.00356291
$670$	0	0
$671$	−12788.5	−0.735760
$672$	0	0
$673$	26852.2	1.53800	0.769002	−	0.639246i	$-0.220753\pi$
$673$	26852.2	1.53800	0.769002	+	0.639246i	$0.220753\pi$
$674$	0	0
$675$	192.166	0.0109577
$676$	0	0
$677$	1645.08	0.0933910	0.0466955	−	0.998909i	$-0.485131\pi$
$677$	1645.08	0.0933910	0.0466955	+	0.998909i	$0.485131\pi$
$678$	0	0
$679$	−11163.1	−0.630929
$680$	0	0
$681$	−183.852	−0.0103454
$682$	0	0
$683$	−3172.63	−0.177742	−0.0888708	−	0.996043i	$-0.528326\pi$
$683$	−3172.63	−0.177742	−0.0888708	+	0.996043i	$0.528326\pi$
$684$	0	0
$685$	−23332.1	−1.30142
$686$	0	0
$687$	−216.978	−0.0120498
$688$	0	0
$689$	1272.09	0.0703378
$690$	0	0
$691$	15820.0	0.870943	0.435472	−	0.900203i	$-0.356582\pi$
$691$	15820.0	0.870943	0.435472	+	0.900203i	$0.356582\pi$
$692$	0	0
$693$	−19061.8	−1.04487
$694$	0	0
$695$	−3971.55	−0.216762
$696$	0	0
$697$	−25241.5	−1.37172
$698$	0	0
$699$	−10.3054	−0.000557633 0
$700$	0	0
$701$	12729.3	0.685848	0.342924	−	0.939363i	$-0.388583\pi$
$701$	12729.3	0.685848	0.342924	+	0.939363i	$0.388583\pi$
$702$	0	0
$703$	−454.496	−0.0243835
$704$	0	0
$705$	−26.8928	−0.00143665
$706$	0	0
$707$	16375.5	0.871096
$708$	0	0
$709$	15175.0	0.803821	0.401911	−	0.915679i	$-0.368346\pi$
$709$	15175.0	0.803821	0.401911	+	0.915679i	$0.368346\pi$
$710$	0	0
$711$	24776.7	1.30689
$712$	0	0
$713$	−1009.47	−0.0530222
$714$	0	0
$715$	−5724.47	−0.299417
$716$	0	0
$717$	−269.785	−0.0140520
$718$	0	0
$719$	−15055.8	−0.780926	−0.390463	−	0.920619i	$-0.627685\pi$
$719$	−15055.8	−0.780926	−0.390463	+	0.920619i	$0.627685\pi$
$720$	0	0
$721$	13793.4	0.712473
$722$	0	0
$723$	282.398	0.0145263
$724$	0	0
$725$	17597.9	0.901473
$726$	0	0
$727$	−12070.5	−0.615778	−0.307889	−	0.951422i	$-0.599622\pi$
$727$	−12070.5	−0.615778	−0.307889	+	0.951422i	$0.599622\pi$
$728$	0	0
$729$	−19671.2	−0.999400
$730$	0	0
$731$	42787.6	2.16492
$732$	0	0
$733$	−26625.2	−1.34165	−0.670823	−	0.741618i	$-0.734059\pi$
$733$	−26625.2	−1.34165	−0.670823	+	0.741618i	$0.734059\pi$
$734$	0	0
$735$	−77.2162	−0.00387505
$736$	0	0
$737$	−3849.03	−0.192375
$738$	0	0
$739$	−1063.52	−0.0529395	−0.0264697	−	0.999650i	$-0.508427\pi$
$739$	−1063.52	−0.0529395	−0.0264697	+	0.999650i	$0.508427\pi$
$740$	0	0
$741$	−55.6332	−0.00275808
$742$	0	0
$743$	−33166.8	−1.63765	−0.818825	−	0.574044i	$-0.805374\pi$
$743$	−33166.8	−1.63765	−0.818825	+	0.574044i	$0.805374\pi$
$744$	0	0
$745$	−12024.3	−0.591326
$746$	0	0
$747$	3599.73	0.176315
$748$	0	0
$749$	−21335.6	−1.04084
$750$	0	0
$751$	−18934.9	−0.920032	−0.460016	−	0.887911i	$-0.652156\pi$
$751$	−18934.9	−0.920032	−0.460016	+	0.887911i	$0.652156\pi$
$752$	0	0
$753$	117.888	0.00570529
$754$	0	0
$755$	1469.77	0.0708481
$756$	0	0
$757$	−9329.09	−0.447915	−0.223957	−	0.974599i	$-0.571898\pi$
$757$	−9329.09	−0.447915	−0.223957	+	0.974599i	$0.571898\pi$
$758$	0	0
$759$	−15.1015	−0.000722199 0
$760$	0	0
$761$	−17120.0	−0.815506	−0.407753	−	0.913092i	$-0.633687\pi$
$761$	−17120.0	−0.815506	−0.407753	+	0.913092i	$0.633687\pi$
$762$	0	0
$763$	−15068.9	−0.714979
$764$	0	0
$765$	−19227.0	−0.908700
$766$	0	0
$767$	7786.70	0.366573
$768$	0	0
$769$	34301.4	1.60850	0.804252	−	0.594288i	$-0.202566\pi$
$769$	34301.4	1.60850	0.804252	+	0.594288i	$0.202566\pi$
$770$	0	0
$771$	−360.936	−0.0168597
$772$	0	0
$773$	−21732.1	−1.01119	−0.505595	−	0.862771i	$-0.668727\pi$
$773$	−21732.1	−1.01119	−0.505595	+	0.862771i	$0.668727\pi$
$774$	0	0
$775$	13936.8	0.645966
$776$	0	0
$777$	−3.45434	−0.000159490 0
$778$	0	0
$779$	21945.0	1.00932
$780$	0	0
$781$	−18703.5	−0.856931
$782$	0	0
$783$	720.604	0.0328892
$784$	0	0
$785$	10632.4	0.483423
$786$	0	0
$787$	−11556.4	−0.523434	−0.261717	−	0.965145i	$-0.584289\pi$
$787$	−11556.4	−0.523434	−0.261717	+	0.965145i	$0.584289\pi$
$788$	0	0
$789$	271.195	0.0122367
$790$	0	0
$791$	−9920.93	−0.445951
$792$	0	0
$793$	2837.77	0.127077
$794$	0	0
$795$	38.2090	0.00170457
$796$	0	0
$797$	−12158.2	−0.540357	−0.270178	−	0.962810i	$-0.587083\pi$
$797$	−12158.2	−0.540357	−0.270178	+	0.962810i	$0.587083\pi$
$798$	0	0
$799$	−6525.75	−0.288942
$800$	0	0
$801$	−6604.25	−0.291323
$802$	0	0
$803$	16731.1	0.735279
$804$	0	0
$805$	−449.483	−0.0196798
$806$	0	0
$807$	−85.2839	−0.00372012
$808$	0	0
$809$	1954.04	0.0849203	0.0424602	−	0.999098i	$-0.486480\pi$
$809$	1954.04	0.0849203	0.0424602	+	0.999098i	$0.486480\pi$
$810$	0	0
$811$	13701.0	0.593228	0.296614	−	0.954997i	$-0.404143\pi$
$811$	13701.0	0.593228	0.296614	+	0.954997i	$0.404143\pi$
$812$	0	0
$813$	−80.9641	−0.00349266
$814$	0	0
$815$	23886.3	1.02663
$816$	0	0
$817$	−37199.6	−1.59296
$818$	0	0
$819$	4229.81	0.180466
$820$	0	0
$821$	−11806.6	−0.501891	−0.250945	−	0.968001i	$-0.580741\pi$
$821$	−11806.6	−0.501891	−0.250945	+	0.968001i	$0.580741\pi$
$822$	0	0
$823$	−44947.6	−1.90374	−0.951868	−	0.306509i	$-0.900839\pi$
$823$	−44947.6	−1.90374	−0.951868	+	0.306509i	$0.900839\pi$
$824$	0	0
$825$	208.492	0.00879850
$826$	0	0
$827$	−11776.9	−0.495190	−0.247595	−	0.968864i	$-0.579640\pi$
$827$	−11776.9	−0.495190	−0.247595	+	0.968864i	$0.579640\pi$
$828$	0	0
$829$	−13479.2	−0.564720	−0.282360	−	0.959309i	$-0.591117\pi$
$829$	−13479.2	−0.564720	−0.282360	+	0.959309i	$0.591117\pi$
$830$	0	0
$831$	269.736	0.0112600
$832$	0	0
$833$	−18737.1	−0.779356
$834$	0	0
$835$	−25232.3	−1.04575
$836$	0	0
$837$	570.688	0.0235674
$838$	0	0
$839$	15662.9	0.644509	0.322254	−	0.946653i	$-0.395559\pi$
$839$	15662.9	0.644509	0.322254	+	0.946653i	$0.395559\pi$
$840$	0	0
$841$	41601.3	1.70574
$842$	0	0
$843$	−291.052	−0.0118913
$844$	0	0
$845$	1270.26	0.0517140
$846$	0	0
$847$	−25323.4	−1.02730
$848$	0	0
$849$	307.992	0.0124502
$850$	0	0
$851$	27.3762	0.00110275
$852$	0	0
$853$	39238.1	1.57501	0.787507	−	0.616305i	$-0.211371\pi$
$853$	39238.1	1.57501	0.787507	+	0.616305i	$0.211371\pi$
$854$	0	0
$855$	16716.0	0.668627
$856$	0	0
$857$	−17154.7	−0.683774	−0.341887	−	0.939741i	$-0.611066\pi$
$857$	−17154.7	−0.683774	−0.341887	+	0.939741i	$0.611066\pi$
$858$	0	0
$859$	−37572.5	−1.49238	−0.746192	−	0.665731i	$-0.768120\pi$
$859$	−37572.5	−1.49238	−0.746192	+	0.665731i	$0.768120\pi$
$860$	0	0
$861$	166.791	0.00660188
$862$	0	0
$863$	−37677.0	−1.48614	−0.743071	−	0.669213i	$-0.766631\pi$
$863$	−37677.0	−1.48614	−0.743071	+	0.669213i	$0.766631\pi$
$864$	0	0
$865$	−28775.1	−1.13108
$866$	0	0
$867$	211.168	0.00827177
$868$	0	0
$869$	53766.0	2.09884
$870$	0	0
$871$	854.100	0.0332263
$872$	0	0
$873$	−25006.2	−0.969453
$874$	0	0
$875$	17528.9	0.677241
$876$	0	0
$877$	3534.78	0.136102	0.0680508	−	0.997682i	$-0.478322\pi$
$877$	3534.78	0.136102	0.0680508	+	0.997682i	$0.478322\pi$
$878$	0	0
$879$	187.256	0.00718544
$880$	0	0
$881$	11107.5	0.424770	0.212385	−	0.977186i	$-0.431877\pi$
$881$	11107.5	0.424770	0.212385	+	0.977186i	$0.431877\pi$
$882$	0	0
$883$	−5892.96	−0.224591	−0.112296	−	0.993675i	$-0.535820\pi$
$883$	−5892.96	−0.224591	−0.112296	+	0.993675i	$0.535820\pi$
$884$	0	0
$885$	233.884	0.00888354
$886$	0	0
$887$	−21476.4	−0.812972	−0.406486	−	0.913657i	$-0.633246\pi$
$887$	−21476.4	−0.812972	−0.406486	+	0.913657i	$0.633246\pi$
$888$	0	0
$889$	−23656.0	−0.892459
$890$	0	0
$891$	−42695.6	−1.60534
$892$	0	0
$893$	5673.50	0.212605
$894$	0	0
$895$	−6050.28	−0.225965
$896$	0	0
$897$	3.35103	0.000124735 0
$898$	0	0
$899$	52261.6	1.93884
$900$	0	0
$901$	9271.72	0.342825
$902$	0	0
$903$	−282.731	−0.0104194
$904$	0	0
$905$	32126.1	1.18001
$906$	0	0
$907$	16593.2	0.607461	0.303731	−	0.952758i	$-0.401768\pi$
$907$	16593.2	0.607461	0.303731	+	0.952758i	$0.401768\pi$
$908$	0	0
$909$	36682.4	1.33848
$910$	0	0
$911$	48547.9	1.76560	0.882802	−	0.469745i	$-0.155654\pi$
$911$	48547.9	1.76560	0.882802	+	0.469745i	$0.155654\pi$
$912$	0	0
$913$	7811.50	0.283158
$914$	0	0
$915$	85.2364	0.00307960
$916$	0	0
$917$	−14062.0	−0.506398
$918$	0	0
$919$	−34369.8	−1.23368	−0.616842	−	0.787087i	$-0.711588\pi$
$919$	−34369.8	−1.23368	−0.616842	+	0.787087i	$0.711588\pi$
$920$	0	0
$921$	−77.2690	−0.00276449
$922$	0	0
$923$	4150.31	0.148006
$924$	0	0
$925$	−377.958	−0.0134348
$926$	0	0
$927$	30898.3	1.09475
$928$	0	0
$929$	−19337.2	−0.682920	−0.341460	−	0.939896i	$-0.610921\pi$
$929$	−19337.2	−0.682920	−0.341460	+	0.939896i	$0.610921\pi$
$930$	0	0
$931$	16290.1	0.573455
$932$	0	0
$933$	−195.350	−0.00685474
$934$	0	0
$935$	−41723.2	−1.45935
$936$	0	0
$937$	53408.0	1.86207	0.931037	−	0.364924i	$-0.118905\pi$
$937$	53408.0	1.86207	0.931037	+	0.364924i	$0.118905\pi$
$938$	0	0
$939$	−184.250	−0.00640338
$940$	0	0
$941$	46543.7	1.61241	0.806207	−	0.591634i	$-0.201517\pi$
$941$	46543.7	1.61241	0.806207	+	0.591634i	$0.201517\pi$
$942$	0	0
$943$	−1321.84	−0.0456470
$944$	0	0
$945$	254.109	0.00874727
$946$	0	0
$947$	14885.8	0.510794	0.255397	−	0.966836i	$-0.417794\pi$
$947$	14885.8	0.510794	0.255397	+	0.966836i	$0.417794\pi$
$948$	0	0
$949$	−3712.65	−0.126994
$950$	0	0
$951$	−104.708	−0.00357033
$952$	0	0
$953$	31363.2	1.06606	0.533028	−	0.846097i	$-0.321054\pi$
$953$	31363.2	1.06606	0.533028	+	0.846097i	$0.321054\pi$
$954$	0	0
$955$	−30014.3	−1.01700
$956$	0	0
$957$	781.826	0.0264084
$958$	0	0
$959$	37411.4	1.25973
$960$	0	0
$961$	11598.0	0.389312
$962$	0	0
$963$	−47793.4	−1.59929
$964$	0	0
$965$	−25151.7	−0.839027
$966$	0	0
$967$	34784.3	1.15676	0.578380	−	0.815767i	$-0.303685\pi$
$967$	34784.3	1.15676	0.578380	+	0.815767i	$0.303685\pi$
$968$	0	0
$969$	−405.487	−0.0134428
$970$	0	0
$971$	21889.3	0.723439	0.361720	−	0.932287i	$-0.382190\pi$
$971$	21889.3	0.723439	0.361720	+	0.932287i	$0.382190\pi$
$972$	0	0
$973$	6368.11	0.209817
$974$	0	0
$975$	−46.2645	−0.00151964
$976$	0	0
$977$	−2790.44	−0.0913757	−0.0456878	−	0.998956i	$-0.514548\pi$
$977$	−2790.44	−0.0913757	−0.0456878	+	0.998956i	$0.514548\pi$
$978$	0	0
$979$	−14331.4	−0.467858
$980$	0	0
$981$	−33755.4	−1.09860
$982$	0	0
$983$	30018.4	0.973996	0.486998	−	0.873403i	$-0.338092\pi$
$983$	30018.4	0.973996	0.486998	+	0.873403i	$0.338092\pi$
$984$	0	0
$985$	730.603	0.0236334
$986$	0	0
$987$	43.1208	0.00139063
$988$	0	0
$989$	2240.69	0.0720423
$990$	0	0
$991$	−46961.9	−1.50534	−0.752670	−	0.658398i	$-0.771235\pi$
$991$	−46961.9	−1.50534	−0.752670	+	0.658398i	$0.771235\pi$
$992$	0	0
$993$	−161.688	−0.00516718
$994$	0	0
$995$	−4811.78	−0.153310
$996$	0	0
$997$	−6426.34	−0.204137	−0.102068	−	0.994777i	$-0.532546\pi$
$997$	−6426.34	−0.204137	−0.102068	+	0.994777i	$0.532546\pi$
$998$	0	0
$999$	−15.4768	−0.000490153 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.4.a.l.1.2		3
3.2	odd	2		1872.4.a.bm.1.1		3
4.3	odd	2		104.4.a.e.1.2	✓	3
8.3	odd	2		832.4.a.bc.1.2		3
8.5	even	2		832.4.a.bb.1.2		3
12.11	even	2		936.4.a.m.1.1		3
52.51	odd	2		1352.4.a.h.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.4.a.e.1.2	✓	3	4.3	odd	2
208.4.a.l.1.2		3	1.1	even	1	trivial
832.4.a.bb.1.2		3	8.5	even	2
832.4.a.bc.1.2		3	8.3	odd	2
936.4.a.m.1.1		3	12.11	even	2
1352.4.a.h.1.2		3	52.51	odd	2
1872.4.a.bm.1.1		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)

\(f(q)\)	\(=\)	\(q+0.0519499 q^{3} +7.51634 q^{5} -12.0519 q^{7} -26.9973 q^{9} +O(q^{10})\)	\(q+0.0519499 q^{3} +7.51634 q^{5} -12.0519 q^{7} -26.9973 q^{9} -58.5849 q^{11} +13.0000 q^{13} +0.390473 q^{15} +94.7514 q^{17} -82.3771 q^{19} -0.626097 q^{21} +4.96192 q^{23} -68.5046 q^{25} -2.80515 q^{27} -256.886 q^{29} -203.443 q^{31} -3.04348 q^{33} -90.5866 q^{35} +5.51726 q^{37} +0.675348 q^{39} -266.398 q^{41} +451.577 q^{43} -202.921 q^{45} -68.8724 q^{47} -197.751 q^{49} +4.92232 q^{51} +97.8531 q^{53} -440.344 q^{55} -4.27948 q^{57} +598.977 q^{59} +218.290 q^{61} +325.370 q^{63} +97.7124 q^{65} +65.7000 q^{67} +0.257771 q^{69} +319.255 q^{71} -285.588 q^{73} -3.55881 q^{75} +706.062 q^{77} -917.746 q^{79} +728.781 q^{81} -133.337 q^{83} +712.184 q^{85} -13.3452 q^{87} +244.626 q^{89} -156.675 q^{91} -10.5688 q^{93} -619.174 q^{95} +926.249 q^{97} +1581.63 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 8 q^{5} - 36 q^{7} + 73 q^{9}+O(q^{10}) \) 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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