LMFDB - Embedded newform 208.4.a.j.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{73}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 18 $ x^2 - x - 18
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			$4.77200$ 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)

$f(q)$	$=$	$q+5.77200 q^{3} +11.3160 q^{5} +8.22800 q^{7} +6.31601 q^{9} +O(q^{10})$	$q+5.77200 q^{3} +11.3160 q^{5} +8.22800 q^{7} +6.31601 q^{9} +45.0880 q^{11} -13.0000 q^{13} +65.3160 q^{15} -87.6680 q^{17} +96.1760 q^{19} +47.4920 q^{21} +34.7360 q^{23} +3.05198 q^{25} -119.388 q^{27} +29.6480 q^{29} -106.248 q^{31} +260.248 q^{33} +93.1081 q^{35} +349.636 q^{37} -75.0360 q^{39} -216.216 q^{41} +321.252 q^{43} +71.4720 q^{45} +97.1400 q^{47} -275.300 q^{49} -506.020 q^{51} +375.720 q^{53} +510.216 q^{55} +555.128 q^{57} -666.080 q^{59} -115.720 q^{61} +51.9681 q^{63} -147.108 q^{65} -271.408 q^{67} +200.496 q^{69} +371.212 q^{71} -620.384 q^{73} +17.6161 q^{75} +370.984 q^{77} +550.528 q^{79} -859.640 q^{81} -976.632 q^{83} -992.052 q^{85} +171.128 q^{87} +906.368 q^{89} -106.964 q^{91} -613.264 q^{93} +1088.33 q^{95} -1883.14 q^{97} +284.776 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} - 3 q^{5} + 25 q^{7} - 13 q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 - 3 * q^5 + 25 * q^7 - 13 * q^9	$ 2 q + 3 q^{3} - 3 q^{5} + 25 q^{7} - 13 q^{9} + 56 q^{11} - 26 q^{13} + 105 q^{15} - 13 q^{17} + 124 q^{19} + q^{21} + 172 q^{23} + 83 q^{25} + 9 q^{27} + 196 q^{29} + 78 q^{31} + 230 q^{33} - 147 q^{35} + 161 q^{37} - 39 q^{39} + 234 q^{41} - 135 q^{43} + 348 q^{45} + 237 q^{47} - 337 q^{49} - 713 q^{51} + 666 q^{53} + 354 q^{55} + 478 q^{57} - 136 q^{59} - 146 q^{61} - 272 q^{63} + 39 q^{65} + 4 q^{67} - 180 q^{69} + 563 q^{71} - 1480 q^{73} - 204 q^{75} + 554 q^{77} + 896 q^{79} - 694 q^{81} - 1902 q^{83} - 2061 q^{85} - 290 q^{87} - 272 q^{89} - 325 q^{91} - 1124 q^{93} + 690 q^{95} - 2160 q^{97} + 74 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 - 3 * q^5 + 25 * q^7 - 13 * q^9 + 56 * q^11 - 26 * q^13 + 105 * q^15 - 13 * q^17 + 124 * q^19 + q^21 + 172 * q^23 + 83 * q^25 + 9 * q^27 + 196 * q^29 + 78 * q^31 + 230 * q^33 - 147 * q^35 + 161 * q^37 - 39 * q^39 + 234 * q^41 - 135 * q^43 + 348 * q^45 + 237 * q^47 - 337 * q^49 - 713 * q^51 + 666 * q^53 + 354 * q^55 + 478 * q^57 - 136 * q^59 - 146 * q^61 - 272 * q^63 + 39 * q^65 + 4 * q^67 - 180 * q^69 + 563 * q^71 - 1480 * q^73 - 204 * q^75 + 554 * q^77 + 896 * q^79 - 694 * q^81 - 1902 * q^83 - 2061 * q^85 - 290 * q^87 - 272 * q^89 - 325 * q^91 - 1124 * q^93 + 690 * q^95 - 2160 * q^97 + 74 * 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
$3$	5.77200	1.11082	0.555411	−	0.831576i	$-0.312561\pi$
$3$	5.77200	1.11082	0.555411	+	0.831576i	$0.312561\pi$
$4$	0	0
$5$	11.3160	1.01213	0.506067	−	0.862494i	$-0.331099\pi$
$5$	11.3160	1.01213	0.506067	+	0.862494i	$0.331099\pi$
$6$	0	0
$7$	8.22800	0.444270	0.222135	−	0.975016i	$-0.428697\pi$
$7$	8.22800	0.444270	0.222135	+	0.975016i	$0.428697\pi$
$8$	0	0
$9$	6.31601	0.233926
$10$	0	0
$11$	45.0880	1.23587	0.617934	−	0.786230i	$-0.287970\pi$
$11$	45.0880	1.23587	0.617934	+	0.786230i	$0.287970\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	65.3160	1.12430
$16$	0	0
$17$	−87.6680	−1.25074	−0.625371	−	0.780327i	$-0.715052\pi$
$17$	−87.6680	−1.25074	−0.625371	+	0.780327i	$0.715052\pi$
$18$	0	0
$19$	96.1760	1.16128	0.580639	−	0.814161i	$-0.302803\pi$
$19$	96.1760	1.16128	0.580639	+	0.814161i	$0.302803\pi$
$20$	0	0
$21$	47.4920	0.493505
$22$	0	0
$23$	34.7360	0.314911	0.157455	−	0.987526i	$-0.449671\pi$
$23$	34.7360	0.314911	0.157455	+	0.987526i	$0.449671\pi$
$24$	0	0
$25$	3.05198	0.0244159
$26$	0	0
$27$	−119.388	−0.850972
$28$	0	0
$29$	29.6480	0.189844	0.0949222	−	0.995485i	$-0.469740\pi$
$29$	29.6480	0.189844	0.0949222	+	0.995485i	$0.469740\pi$
$30$	0	0
$31$	−106.248	−0.615571	−0.307786	−	0.951456i	$-0.599588\pi$
$31$	−106.248	−0.615571	−0.307786	+	0.951456i	$0.599588\pi$
$32$	0	0
$33$	260.248	1.37283
$34$	0	0
$35$	93.1081	0.449661
$36$	0	0
$37$	349.636	1.55351	0.776754	−	0.629804i	$-0.216865\pi$
$37$	349.636	1.55351	0.776754	+	0.629804i	$0.216865\pi$
$38$	0	0
$39$	−75.0360	−0.308087
$40$	0	0
$41$	−216.216	−0.823592	−0.411796	−	0.911276i	$-0.635098\pi$
$41$	−216.216	−0.823592	−0.411796	+	0.911276i	$0.635098\pi$
$42$	0	0
$43$	321.252	1.13931	0.569657	−	0.821883i	$-0.307076\pi$
$43$	321.252	1.13931	0.569657	+	0.821883i	$0.307076\pi$
$44$	0	0
$45$	71.4720	0.236765
$46$	0	0
$47$	97.1400	0.301475	0.150737	−	0.988574i	$-0.451835\pi$
$47$	97.1400	0.301475	0.150737	+	0.988574i	$0.451835\pi$
$48$	0	0
$49$	−275.300	−0.802624
$50$	0	0
$51$	−506.020	−1.38935
$52$	0	0
$53$	375.720	0.973757	0.486878	−	0.873470i	$-0.338135\pi$
$53$	375.720	0.973757	0.486878	+	0.873470i	$0.338135\pi$
$54$	0	0
$55$	510.216	1.25086
$56$	0	0
$57$	555.128	1.28997
$58$	0	0
$59$	−666.080	−1.46977	−0.734884	−	0.678193i	$-0.762763\pi$
$59$	−666.080	−1.46977	−0.734884	+	0.678193i	$0.762763\pi$
$60$	0	0
$61$	−115.720	−0.242892	−0.121446	−	0.992598i	$-0.538753\pi$
$61$	−115.720	−0.242892	−0.121446	+	0.992598i	$0.538753\pi$
$62$	0	0
$63$	51.9681	0.103926
$64$	0	0
$65$	−147.108	−0.280716
$66$	0	0
$67$	−271.408	−0.494892	−0.247446	−	0.968902i	$-0.579591\pi$
$67$	−271.408	−0.494892	−0.247446	+	0.968902i	$0.579591\pi$
$68$	0	0
$69$	200.496	0.349810
$70$	0	0
$71$	371.212	0.620490	0.310245	−	0.950657i	$-0.399589\pi$
$71$	371.212	0.620490	0.310245	+	0.950657i	$0.399589\pi$
$72$	0	0
$73$	−620.384	−0.994664	−0.497332	−	0.867560i	$-0.665687\pi$
$73$	−620.384	−0.994664	−0.497332	+	0.867560i	$0.665687\pi$
$74$	0	0
$75$	17.6161	0.0271217
$76$	0	0
$77$	370.984	0.549059
$78$	0	0
$79$	550.528	0.784041	0.392021	−	0.919956i	$-0.371776\pi$
$79$	550.528	0.784041	0.392021	+	0.919956i	$0.371776\pi$
$80$	0	0
$81$	−859.640	−1.17920
$82$	0	0
$83$	−976.632	−1.29156	−0.645779	−	0.763525i	$-0.723467\pi$
$83$	−976.632	−1.29156	−0.645779	+	0.763525i	$0.723467\pi$
$84$	0	0
$85$	−992.052	−1.26592
$86$	0	0
$87$	171.128	0.210883
$88$	0	0
$89$	906.368	1.07949	0.539746	−	0.841828i	$-0.318520\pi$
$89$	906.368	1.07949	0.539746	+	0.841828i	$0.318520\pi$
$90$	0	0
$91$	−106.964	−0.123218
$92$	0	0
$93$	−613.264	−0.683791
$94$	0	0
$95$	1088.33	1.17537
$96$	0	0
$97$	−1883.14	−1.97117	−0.985585	−	0.169180i	$-0.945888\pi$
$97$	−1883.14	−1.97117	−0.985585	+	0.169180i	$0.945888\pi$
$98$	0	0
$99$	284.776	0.289102
$100$	0	0
$101$	−1722.39	−1.69688	−0.848438	−	0.529295i	$-0.822456\pi$
$101$	−1722.39	−1.69688	−0.848438	+	0.529295i	$0.822456\pi$
$102$	0	0
$103$	−1908.10	−1.82534	−0.912672	−	0.408694i	$-0.865984\pi$
$103$	−1908.10	−1.82534	−0.912672	+	0.408694i	$0.865984\pi$
$104$	0	0
$105$	537.420	0.499494
$106$	0	0
$107$	−1713.36	−1.54801	−0.774004	−	0.633181i	$-0.781749\pi$
$107$	−1713.36	−1.54801	−0.774004	+	0.633181i	$0.781749\pi$
$108$	0	0
$109$	−1293.76	−1.13688	−0.568438	−	0.822726i	$-0.692452\pi$
$109$	−1293.76	−1.13688	−0.568438	+	0.822726i	$0.692452\pi$
$110$	0	0
$111$	2018.10	1.72567
$112$	0	0
$113$	1345.02	1.11973	0.559864	−	0.828585i	$-0.310853\pi$
$113$	1345.02	1.11973	0.559864	+	0.828585i	$0.310853\pi$
$114$	0	0
$115$	393.073	0.318732
$116$	0	0
$117$	−82.1081	−0.0648794
$118$	0	0
$119$	−721.332	−0.555668
$120$	0	0
$121$	701.928	0.527369
$122$	0	0
$123$	−1248.00	−0.914865
$124$	0	0
$125$	−1379.96	−0.987422
$126$	0	0
$127$	757.600	0.529339	0.264670	−	0.964339i	$-0.414737\pi$
$127$	757.600	0.529339	0.264670	+	0.964339i	$0.414737\pi$
$128$	0	0
$129$	1854.27	1.26558
$130$	0	0
$131$	685.413	0.457136	0.228568	−	0.973528i	$-0.426596\pi$
$131$	685.413	0.457136	0.228568	+	0.973528i	$0.426596\pi$
$132$	0	0
$133$	791.336	0.515921
$134$	0	0
$135$	−1351.00	−0.861298
$136$	0	0
$137$	1057.40	0.659414	0.329707	−	0.944083i	$-0.393050\pi$
$137$	1057.40	0.659414	0.329707	+	0.944083i	$0.393050\pi$
$138$	0	0
$139$	−738.748	−0.450790	−0.225395	−	0.974267i	$-0.572367\pi$
$139$	−738.748	−0.450790	−0.225395	+	0.974267i	$0.572367\pi$
$140$	0	0
$141$	560.692	0.334885
$142$	0	0
$143$	−586.144	−0.342768
$144$	0	0
$145$	335.497	0.192148
$146$	0	0
$147$	−1589.03	−0.891573
$148$	0	0
$149$	1999.42	1.09932	0.549662	−	0.835387i	$-0.314757\pi$
$149$	1999.42	1.09932	0.549662	+	0.835387i	$0.314757\pi$
$150$	0	0
$151$	1410.28	0.760048	0.380024	−	0.924977i	$-0.375916\pi$
$151$	1410.28	0.760048	0.380024	+	0.924977i	$0.375916\pi$
$152$	0	0
$153$	−553.712	−0.292581
$154$	0	0
$155$	−1202.30	−0.623041
$156$	0	0
$157$	2558.43	1.30054	0.650271	−	0.759702i	$-0.274655\pi$
$157$	2558.43	1.30054	0.650271	+	0.759702i	$0.274655\pi$
$158$	0	0
$159$	2168.66	1.08167
$160$	0	0
$161$	285.808	0.139906
$162$	0	0
$163$	3069.77	1.47511	0.737555	−	0.675287i	$-0.235980\pi$
$163$	3069.77	1.47511	0.737555	+	0.675287i	$0.235980\pi$
$164$	0	0
$165$	2944.97	1.38949
$166$	0	0
$167$	2268.02	1.05093	0.525464	−	0.850816i	$-0.323892\pi$
$167$	2268.02	1.05093	0.525464	+	0.850816i	$0.323892\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	607.448	0.271653
$172$	0	0
$173$	2648.97	1.16415	0.582073	−	0.813136i	$-0.302242\pi$
$173$	2648.97	1.16415	0.582073	+	0.813136i	$0.302242\pi$
$174$	0	0
$175$	25.1117	0.0108472
$176$	0	0
$177$	−3844.62	−1.63265
$178$	0	0
$179$	3931.03	1.64145	0.820723	−	0.571326i	$-0.193571\pi$
$179$	3931.03	1.64145	0.820723	+	0.571326i	$0.193571\pi$
$180$	0	0
$181$	−3708.47	−1.52292	−0.761460	−	0.648211i	$-0.775517\pi$
$181$	−3708.47	−1.52292	−0.761460	+	0.648211i	$0.775517\pi$
$182$	0	0
$183$	−667.936	−0.269810
$184$	0	0
$185$	3956.48	1.57236
$186$	0	0
$187$	−3952.78	−1.54575
$188$	0	0
$189$	−982.325	−0.378061
$190$	0	0
$191$	−162.233	−0.0614594	−0.0307297	−	0.999528i	$-0.509783\pi$
$191$	−162.233	−0.0614594	−0.0307297	+	0.999528i	$0.509783\pi$
$192$	0	0
$193$	−2761.91	−1.03009	−0.515044	−	0.857164i	$-0.672224\pi$
$193$	−2761.91	−1.03009	−0.515044	+	0.857164i	$0.672224\pi$
$194$	0	0
$195$	−849.108	−0.311825
$196$	0	0
$197$	−2283.46	−0.825837	−0.412918	−	0.910768i	$-0.635490\pi$
$197$	−2283.46	−0.825837	−0.412918	+	0.910768i	$0.635490\pi$
$198$	0	0
$199$	−2409.21	−0.858213	−0.429106	−	0.903254i	$-0.641171\pi$
$199$	−2409.21	−0.858213	−0.429106	+	0.903254i	$0.641171\pi$
$200$	0	0
$201$	−1566.57	−0.549737
$202$	0	0
$203$	243.943	0.0843422
$204$	0	0
$205$	−2446.70	−0.833586
$206$	0	0
$207$	219.393	0.0736659
$208$	0	0
$209$	4336.38	1.43519
$210$	0	0
$211$	1747.68	0.570216	0.285108	−	0.958495i	$-0.407971\pi$
$211$	1747.68	0.570216	0.285108	+	0.958495i	$0.407971\pi$
$212$	0	0
$213$	2142.64	0.689254
$214$	0	0
$215$	3635.29	1.15314
$216$	0	0
$217$	−874.209	−0.273480
$218$	0	0
$219$	−3580.86	−1.10489
$220$	0	0
$221$	1139.68	0.346894
$222$	0	0
$223$	3682.45	1.10581	0.552903	−	0.833245i	$-0.313520\pi$
$223$	3682.45	1.10581	0.552903	+	0.833245i	$0.313520\pi$
$224$	0	0
$225$	19.2763	0.00571151
$226$	0	0
$227$	−2860.35	−0.836334	−0.418167	−	0.908370i	$-0.637327\pi$
$227$	−2860.35	−0.836334	−0.418167	+	0.908370i	$0.637327\pi$
$228$	0	0
$229$	−2670.24	−0.770544	−0.385272	−	0.922803i	$-0.625892\pi$
$229$	−2670.24	−0.770544	−0.385272	+	0.922803i	$0.625892\pi$
$230$	0	0
$231$	2141.32	0.609907
$232$	0	0
$233$	1017.98	0.286223	0.143112	−	0.989707i	$-0.454289\pi$
$233$	1017.98	0.286223	0.143112	+	0.989707i	$0.454289\pi$
$234$	0	0
$235$	1099.24	0.305133
$236$	0	0
$237$	3177.65	0.870930
$238$	0	0
$239$	3609.01	0.976769	0.488384	−	0.872629i	$-0.337586\pi$
$239$	3609.01	0.976769	0.488384	+	0.872629i	$0.337586\pi$
$240$	0	0
$241$	−4125.46	−1.10267	−0.551336	−	0.834283i	$-0.685882\pi$
$241$	−4125.46	−1.10267	−0.551336	+	0.834283i	$0.685882\pi$
$242$	0	0
$243$	−1738.37	−0.458915
$244$	0	0
$245$	−3115.30	−0.812363
$246$	0	0
$247$	−1250.29	−0.322081
$248$	0	0
$249$	−5637.12	−1.43469
$250$	0	0
$251$	5464.47	1.37416	0.687080	−	0.726582i	$-0.258892\pi$
$251$	5464.47	1.37416	0.687080	+	0.726582i	$0.258892\pi$
$252$	0	0
$253$	1566.18	0.389188
$254$	0	0
$255$	−5726.13	−1.40621
$256$	0	0
$257$	−2417.40	−0.586743	−0.293372	−	0.955999i	$-0.594777\pi$
$257$	−2417.40	−0.586743	−0.293372	+	0.955999i	$0.594777\pi$
$258$	0	0
$259$	2876.81	0.690178
$260$	0	0
$261$	187.257	0.0444096
$262$	0	0
$263$	4303.01	1.00888	0.504439	−	0.863447i	$-0.331699\pi$
$263$	4303.01	1.00888	0.504439	+	0.863447i	$0.331699\pi$
$264$	0	0
$265$	4251.65	0.985573
$266$	0	0
$267$	5231.56	1.19912
$268$	0	0
$269$	7912.01	1.79332	0.896661	−	0.442717i	$-0.145985\pi$
$269$	7912.01	1.79332	0.896661	+	0.442717i	$0.145985\pi$
$270$	0	0
$271$	−3745.36	−0.839536	−0.419768	−	0.907632i	$-0.637888\pi$
$271$	−3745.36	−0.839536	−0.419768	+	0.907632i	$0.637888\pi$
$272$	0	0
$273$	−617.396	−0.136874
$274$	0	0
$275$	137.608	0.0301748
$276$	0	0
$277$	−5074.49	−1.10071	−0.550355	−	0.834931i	$-0.685508\pi$
$277$	−5074.49	−1.10071	−0.550355	+	0.834931i	$0.685508\pi$
$278$	0	0
$279$	−671.063	−0.143998
$280$	0	0
$281$	−1905.44	−0.404516	−0.202258	−	0.979332i	$-0.564828\pi$
$281$	−1905.44	−0.404516	−0.202258	+	0.979332i	$0.564828\pi$
$282$	0	0
$283$	−6107.17	−1.28280	−0.641402	−	0.767205i	$-0.721647\pi$
$283$	−6107.17	−1.28280	−0.641402	+	0.767205i	$0.721647\pi$
$284$	0	0
$285$	6281.83	1.30563
$286$	0	0
$287$	−1779.03	−0.365898
$288$	0	0
$289$	2772.68	0.564357
$290$	0	0
$291$	−10869.5	−2.18962
$292$	0	0
$293$	7843.25	1.56385	0.781925	−	0.623373i	$-0.214238\pi$
$293$	7843.25	1.56385	0.781925	+	0.623373i	$0.214238\pi$
$294$	0	0
$295$	−7537.37	−1.48760
$296$	0	0
$297$	−5382.97	−1.05169
$298$	0	0
$299$	−451.568	−0.0873406
$300$	0	0
$301$	2643.26	0.506163
$302$	0	0
$303$	−9941.65	−1.88493
$304$	0	0
$305$	−1309.49	−0.245840
$306$	0	0
$307$	8009.00	1.48892	0.744458	−	0.667669i	$-0.232708\pi$
$307$	8009.00	1.48892	0.744458	+	0.667669i	$0.232708\pi$
$308$	0	0
$309$	−11013.5	−2.02763
$310$	0	0
$311$	−5071.14	−0.924625	−0.462312	−	0.886717i	$-0.652980\pi$
$311$	−5071.14	−0.924625	−0.462312	+	0.886717i	$0.652980\pi$
$312$	0	0
$313$	8580.50	1.54952	0.774758	−	0.632258i	$-0.217872\pi$
$313$	8580.50	1.54952	0.774758	+	0.632258i	$0.217872\pi$
$314$	0	0
$315$	588.071	0.105187
$316$	0	0
$317$	−7383.75	−1.30824	−0.654121	−	0.756390i	$-0.726961\pi$
$317$	−7383.75	−1.30824	−0.654121	+	0.756390i	$0.726961\pi$
$318$	0	0
$319$	1336.77	0.234623
$320$	0	0
$321$	−9889.52	−1.71956
$322$	0	0
$323$	−8431.56	−1.45246
$324$	0	0
$325$	−39.6758	−0.00677174
$326$	0	0
$327$	−7467.57	−1.26287
$328$	0	0
$329$	799.268	0.133936
$330$	0	0
$331$	4056.60	0.673628	0.336814	−	0.941571i	$-0.390651\pi$
$331$	4056.60	0.673628	0.336814	+	0.941571i	$0.390651\pi$
$332$	0	0
$333$	2208.30	0.363406
$334$	0	0
$335$	−3071.26	−0.500897
$336$	0	0
$337$	2310.55	0.373483	0.186741	−	0.982409i	$-0.440207\pi$
$337$	2310.55	0.373483	0.186741	+	0.982409i	$0.440207\pi$
$338$	0	0
$339$	7763.48	1.24382
$340$	0	0
$341$	−4790.51	−0.760765
$342$	0	0
$343$	−5087.37	−0.800852
$344$	0	0
$345$	2268.82	0.354055
$346$	0	0
$347$	4720.04	0.730216	0.365108	−	0.930965i	$-0.381032\pi$
$347$	4720.04	0.730216	0.365108	+	0.930965i	$0.381032\pi$
$348$	0	0
$349$	−1796.40	−0.275527	−0.137763	−	0.990465i	$-0.543991\pi$
$349$	−1796.40	−0.275527	−0.137763	+	0.990465i	$0.543991\pi$
$350$	0	0
$351$	1552.04	0.236017
$352$	0	0
$353$	5060.70	0.763042	0.381521	−	0.924360i	$-0.375400\pi$
$353$	5060.70	0.763042	0.381521	+	0.924360i	$0.375400\pi$
$354$	0	0
$355$	4200.64	0.628019
$356$	0	0
$357$	−4163.53	−0.617248
$358$	0	0
$359$	7262.97	1.06776	0.533879	−	0.845561i	$-0.320734\pi$
$359$	7262.97	1.06776	0.533879	+	0.845561i	$0.320734\pi$
$360$	0	0
$361$	2390.83	0.348568
$362$	0	0
$363$	4051.53	0.585813
$364$	0	0
$365$	−7020.27	−1.00673
$366$	0	0
$367$	114.488	0.0162840	0.00814199	−	0.999967i	$-0.497408\pi$
$367$	114.488	0.0162840	0.00814199	+	0.999967i	$0.497408\pi$
$368$	0	0
$369$	−1365.62	−0.192660
$370$	0	0
$371$	3091.42	0.432611
$372$	0	0
$373$	−8702.99	−1.20811	−0.604053	−	0.796944i	$-0.706449\pi$
$373$	−8702.99	−1.20811	−0.604053	+	0.796944i	$0.706449\pi$
$374$	0	0
$375$	−7965.16	−1.09685
$376$	0	0
$377$	−385.424	−0.0526534
$378$	0	0
$379$	673.287	0.0912518	0.0456259	−	0.998959i	$-0.485472\pi$
$379$	673.287	0.0912518	0.0456259	+	0.998959i	$0.485472\pi$
$380$	0	0
$381$	4372.87	0.588002
$382$	0	0
$383$	8663.53	1.15584	0.577919	−	0.816094i	$-0.303865\pi$
$383$	8663.53	1.15584	0.577919	+	0.816094i	$0.303865\pi$
$384$	0	0
$385$	4198.06	0.555722
$386$	0	0
$387$	2029.03	0.266515
$388$	0	0
$389$	12935.7	1.68603	0.843013	−	0.537892i	$-0.180779\pi$
$389$	12935.7	1.68603	0.843013	+	0.537892i	$0.180779\pi$
$390$	0	0
$391$	−3045.23	−0.393873
$392$	0	0
$393$	3956.20	0.507797
$394$	0	0
$395$	6229.78	0.793555
$396$	0	0
$397$	13916.7	1.75935	0.879673	−	0.475579i	$-0.157762\pi$
$397$	13916.7	1.75935	0.879673	+	0.475579i	$0.157762\pi$
$398$	0	0
$399$	4567.59	0.573097
$400$	0	0
$401$	−8639.03	−1.07584	−0.537921	−	0.842995i	$-0.680790\pi$
$401$	−8639.03	−1.07584	−0.537921	+	0.842995i	$0.680790\pi$
$402$	0	0
$403$	1381.22	0.170729
$404$	0	0
$405$	−9727.69	−1.19351
$406$	0	0
$407$	15764.4	1.91993
$408$	0	0
$409$	5635.48	0.681312	0.340656	−	0.940188i	$-0.389351\pi$
$409$	5635.48	0.681312	0.340656	+	0.940188i	$0.389351\pi$
$410$	0	0
$411$	6103.31	0.732492
$412$	0	0
$413$	−5480.51	−0.652974
$414$	0	0
$415$	−11051.6	−1.30723
$416$	0	0
$417$	−4264.05	−0.500747
$418$	0	0
$419$	8549.58	0.996836	0.498418	−	0.866937i	$-0.333915\pi$
$419$	8549.58	0.996836	0.498418	+	0.866937i	$0.333915\pi$
$420$	0	0
$421$	−11008.1	−1.27435	−0.637177	−	0.770717i	$-0.719898\pi$
$421$	−11008.1	−1.27435	−0.637177	+	0.770717i	$0.719898\pi$
$422$	0	0
$423$	613.537	0.0705229
$424$	0	0
$425$	−267.561	−0.0305380
$426$	0	0
$427$	−952.144	−0.107910
$428$	0	0
$429$	−3383.22	−0.380754
$430$	0	0
$431$	−9253.94	−1.03421	−0.517107	−	0.855921i	$-0.672991\pi$
$431$	−9253.94	−1.03421	−0.517107	+	0.855921i	$0.672991\pi$
$432$	0	0
$433$	15204.0	1.68743	0.843714	−	0.536793i	$-0.180364\pi$
$433$	15204.0	1.68743	0.843714	+	0.536793i	$0.180364\pi$
$434$	0	0
$435$	1936.49	0.213442
$436$	0	0
$437$	3340.77	0.365699
$438$	0	0
$439$	−9892.87	−1.07554	−0.537769	−	0.843092i	$-0.680733\pi$
$439$	−9892.87	−1.07554	−0.537769	+	0.843092i	$0.680733\pi$
$440$	0	0
$441$	−1738.80	−0.187755
$442$	0	0
$443$	4718.14	0.506017	0.253009	−	0.967464i	$-0.418580\pi$
$443$	4718.14	0.506017	0.253009	+	0.967464i	$0.418580\pi$
$444$	0	0
$445$	10256.5	1.09259
$446$	0	0
$447$	11540.7	1.22115
$448$	0	0
$449$	5425.60	0.570267	0.285133	−	0.958488i	$-0.407962\pi$
$449$	5425.60	0.570267	0.285133	+	0.958488i	$0.407962\pi$
$450$	0	0
$451$	−9748.76	−1.01785
$452$	0	0
$453$	8140.16	0.844278
$454$	0	0
$455$	−1210.40	−0.124714
$456$	0	0
$457$	894.192	0.0915285	0.0457642	−	0.998952i	$-0.485428\pi$
$457$	894.192	0.0915285	0.0457642	+	0.998952i	$0.485428\pi$
$458$	0	0
$459$	10466.5	1.06435
$460$	0	0
$461$	7137.99	0.721148	0.360574	−	0.932731i	$-0.382581\pi$
$461$	7137.99	0.721148	0.360574	+	0.932731i	$0.382581\pi$
$462$	0	0
$463$	−5797.64	−0.581942	−0.290971	−	0.956732i	$-0.593978\pi$
$463$	−5797.64	−0.581942	−0.290971	+	0.956732i	$0.593978\pi$
$464$	0	0
$465$	−6939.70	−0.692088
$466$	0	0
$467$	10104.0	1.00119	0.500595	−	0.865681i	$-0.333115\pi$
$467$	10104.0	1.00119	0.500595	+	0.865681i	$0.333115\pi$
$468$	0	0
$469$	−2233.15	−0.219866
$470$	0	0
$471$	14767.3	1.44467
$472$	0	0
$473$	14484.6	1.40804
$474$	0	0
$475$	293.528	0.0283536
$476$	0	0
$477$	2373.05	0.227787
$478$	0	0
$479$	−9256.72	−0.882986	−0.441493	−	0.897265i	$-0.645551\pi$
$479$	−9256.72	−0.882986	−0.441493	+	0.897265i	$0.645551\pi$
$480$	0	0
$481$	−4545.27	−0.430866
$482$	0	0
$483$	1649.68	0.155410
$484$	0	0
$485$	−21309.6	−1.99509
$486$	0	0
$487$	9003.50	0.837757	0.418878	−	0.908042i	$-0.362423\pi$
$487$	9003.50	0.837757	0.418878	+	0.908042i	$0.362423\pi$
$488$	0	0
$489$	17718.7	1.63858
$490$	0	0
$491$	−18696.4	−1.71844	−0.859222	−	0.511603i	$-0.829052\pi$
$491$	−18696.4	−1.71844	−0.859222	+	0.511603i	$0.829052\pi$
$492$	0	0
$493$	−2599.18	−0.237447
$494$	0	0
$495$	3222.53	0.292610
$496$	0	0
$497$	3054.33	0.275665
$498$	0	0
$499$	−2279.45	−0.204493	−0.102247	−	0.994759i	$-0.532603\pi$
$499$	−2279.45	−0.204493	−0.102247	+	0.994759i	$0.532603\pi$
$500$	0	0
$501$	13091.0	1.16739
$502$	0	0
$503$	−5053.76	−0.447984	−0.223992	−	0.974591i	$-0.571909\pi$
$503$	−5053.76	−0.447984	−0.223992	+	0.974591i	$0.571909\pi$
$504$	0	0
$505$	−19490.6	−1.71747
$506$	0	0
$507$	975.468	0.0854479
$508$	0	0
$509$	3556.05	0.309664	0.154832	−	0.987941i	$-0.450516\pi$
$509$	3556.05	0.309664	0.154832	+	0.987941i	$0.450516\pi$
$510$	0	0
$511$	−5104.52	−0.441899
$512$	0	0
$513$	−11482.3	−0.988215
$514$	0	0
$515$	−21592.0	−1.84749
$516$	0	0
$517$	4379.85	0.372583
$518$	0	0
$519$	15289.8	1.29316
$520$	0	0
$521$	1262.58	0.106170	0.0530852	−	0.998590i	$-0.483095\pi$
$521$	1262.58	0.106170	0.0530852	+	0.998590i	$0.483095\pi$
$522$	0	0
$523$	−10799.5	−0.902927	−0.451464	−	0.892290i	$-0.649098\pi$
$523$	−10799.5	−0.902927	−0.451464	+	0.892290i	$0.649098\pi$
$524$	0	0
$525$	144.945	0.0120494
$526$	0	0
$527$	9314.56	0.769921
$528$	0	0
$529$	−10960.4	−0.900831
$530$	0	0
$531$	−4206.97	−0.343817
$532$	0	0
$533$	2810.81	0.228423
$534$	0	0
$535$	−19388.4	−1.56679
$536$	0	0
$537$	22689.9	1.82336
$538$	0	0
$539$	−12412.7	−0.991937
$540$	0	0
$541$	3182.97	0.252951	0.126476	−	0.991970i	$-0.459633\pi$
$541$	3182.97	0.252951	0.126476	+	0.991970i	$0.459633\pi$
$542$	0	0
$543$	−21405.3	−1.69169
$544$	0	0
$545$	−14640.2	−1.15067
$546$	0	0
$547$	−15282.8	−1.19460	−0.597298	−	0.802019i	$-0.703759\pi$
$547$	−15282.8	−1.19460	−0.597298	+	0.802019i	$0.703759\pi$
$548$	0	0
$549$	−730.888	−0.0568189
$550$	0	0
$551$	2851.42	0.220462
$552$	0	0
$553$	4529.74	0.348326
$554$	0	0
$555$	22836.8	1.74661
$556$	0	0
$557$	8777.73	0.667728	0.333864	−	0.942621i	$-0.391647\pi$
$557$	8777.73	0.667728	0.333864	+	0.942621i	$0.391647\pi$
$558$	0	0
$559$	−4176.28	−0.315989
$560$	0	0
$561$	−22815.4	−1.71706
$562$	0	0
$563$	3861.83	0.289088	0.144544	−	0.989498i	$-0.453828\pi$
$563$	3861.83	0.289088	0.144544	+	0.989498i	$0.453828\pi$
$564$	0	0
$565$	15220.3	1.13331
$566$	0	0
$567$	−7073.12	−0.523885
$568$	0	0
$569$	12459.0	0.917942	0.458971	−	0.888451i	$-0.348218\pi$
$569$	12459.0	0.917942	0.458971	+	0.888451i	$0.348218\pi$
$570$	0	0
$571$	19165.2	1.40462	0.702309	−	0.711872i	$-0.252152\pi$
$571$	19165.2	1.40462	0.702309	+	0.711872i	$0.252152\pi$
$572$	0	0
$573$	−936.407	−0.0682704
$574$	0	0
$575$	106.014	0.00768882
$576$	0	0
$577$	−9842.51	−0.710137	−0.355069	−	0.934840i	$-0.615542\pi$
$577$	−9842.51	−0.710137	−0.355069	+	0.934840i	$0.615542\pi$
$578$	0	0
$579$	−15941.8	−1.14424
$580$	0	0
$581$	−8035.73	−0.573801
$582$	0	0
$583$	16940.5	1.20343
$584$	0	0
$585$	−929.135	−0.0656667
$586$	0	0
$587$	5201.28	0.365723	0.182862	−	0.983139i	$-0.441464\pi$
$587$	5201.28	0.365723	0.182862	+	0.983139i	$0.441464\pi$
$588$	0	0
$589$	−10218.5	−0.714850
$590$	0	0
$591$	−13180.1	−0.917358
$592$	0	0
$593$	−19852.7	−1.37480	−0.687398	−	0.726281i	$-0.741247\pi$
$593$	−19852.7	−1.37480	−0.687398	+	0.726281i	$0.741247\pi$
$594$	0	0
$595$	−8162.60	−0.562410
$596$	0	0
$597$	−13906.0	−0.953322
$598$	0	0
$599$	18170.5	1.23944	0.619721	−	0.784822i	$-0.287246\pi$
$599$	18170.5	1.23944	0.619721	+	0.784822i	$0.287246\pi$
$600$	0	0
$601$	1129.65	0.0766709	0.0383354	−	0.999265i	$-0.487794\pi$
$601$	1129.65	0.0766709	0.0383354	+	0.999265i	$0.487794\pi$
$602$	0	0
$603$	−1714.22	−0.115768
$604$	0	0
$605$	7943.03	0.533768
$606$	0	0
$607$	7900.63	0.528298	0.264149	−	0.964482i	$-0.414909\pi$
$607$	7900.63	0.528298	0.264149	+	0.964482i	$0.414909\pi$
$608$	0	0
$609$	1408.04	0.0936892
$610$	0	0
$611$	−1262.82	−0.0836141
$612$	0	0
$613$	60.3675	0.00397752	0.00198876	−	0.999998i	$-0.499367\pi$
$613$	60.3675	0.00397752	0.00198876	+	0.999998i	$0.499367\pi$
$614$	0	0
$615$	−14122.4	−0.925966
$616$	0	0
$617$	−6755.06	−0.440759	−0.220380	−	0.975414i	$-0.570730\pi$
$617$	−6755.06	−0.440759	−0.220380	+	0.975414i	$0.570730\pi$
$618$	0	0
$619$	694.732	0.0451109	0.0225554	−	0.999746i	$-0.492820\pi$
$619$	694.732	0.0451109	0.0225554	+	0.999746i	$0.492820\pi$
$620$	0	0
$621$	−4147.06	−0.267980
$622$	0	0
$623$	7457.60	0.479586
$624$	0	0
$625$	−15997.2	−1.02382
$626$	0	0
$627$	25029.6	1.59424
$628$	0	0
$629$	−30651.9	−1.94304
$630$	0	0
$631$	27282.1	1.72121	0.860604	−	0.509274i	$-0.170086\pi$
$631$	27282.1	1.72121	0.860604	+	0.509274i	$0.170086\pi$
$632$	0	0
$633$	10087.6	0.633408
$634$	0	0
$635$	8573.00	0.535763
$636$	0	0
$637$	3578.90	0.222608
$638$	0	0
$639$	2344.58	0.145149
$640$	0	0
$641$	−6762.80	−0.416716	−0.208358	−	0.978053i	$-0.566812\pi$
$641$	−6762.80	−0.416716	−0.208358	+	0.978053i	$0.566812\pi$
$642$	0	0
$643$	−5013.52	−0.307487	−0.153743	−	0.988111i	$-0.549133\pi$
$643$	−5013.52	−0.307487	−0.153743	+	0.988111i	$0.549133\pi$
$644$	0	0
$645$	20982.9	1.28093
$646$	0	0
$647$	−5841.03	−0.354922	−0.177461	−	0.984128i	$-0.556788\pi$
$647$	−5841.03	−0.354922	−0.177461	+	0.984128i	$0.556788\pi$
$648$	0	0
$649$	−30032.2	−1.81644
$650$	0	0
$651$	−5045.94	−0.303788
$652$	0	0
$653$	1916.88	0.114875	0.0574373	−	0.998349i	$-0.481707\pi$
$653$	1916.88	0.114875	0.0574373	+	0.998349i	$0.481707\pi$
$654$	0	0
$655$	7756.13	0.462683
$656$	0	0
$657$	−3918.35	−0.232678
$658$	0	0
$659$	−8744.88	−0.516923	−0.258462	−	0.966022i	$-0.583216\pi$
$659$	−8744.88	−0.516923	−0.258462	+	0.966022i	$0.583216\pi$
$660$	0	0
$661$	−20146.4	−1.18548	−0.592741	−	0.805393i	$-0.701954\pi$
$661$	−20146.4	−1.18548	−0.592741	+	0.805393i	$0.701954\pi$
$662$	0	0
$663$	6578.26	0.385337
$664$	0	0
$665$	8954.76	0.522182
$666$	0	0
$667$	1029.85	0.0597841
$668$	0	0
$669$	21255.1	1.22835
$670$	0	0
$671$	−5217.59	−0.300183
$672$	0	0
$673$	30825.8	1.76560	0.882798	−	0.469752i	$-0.155657\pi$
$673$	30825.8	1.76560	0.882798	+	0.469752i	$0.155657\pi$
$674$	0	0
$675$	−364.370	−0.0207772
$676$	0	0
$677$	16590.6	0.941843	0.470921	−	0.882175i	$-0.343922\pi$
$677$	16590.6	0.941843	0.470921	+	0.882175i	$0.343922\pi$
$678$	0	0
$679$	−15494.4	−0.875732
$680$	0	0
$681$	−16509.9	−0.929018
$682$	0	0
$683$	−20959.6	−1.17423	−0.587114	−	0.809505i	$-0.699736\pi$
$683$	−20959.6	−1.17423	−0.587114	+	0.809505i	$0.699736\pi$
$684$	0	0
$685$	11965.5	0.667416
$686$	0	0
$687$	−15412.7	−0.855938
$688$	0	0
$689$	−4884.36	−0.270072
$690$	0	0
$691$	−4178.88	−0.230061	−0.115030	−	0.993362i	$-0.536697\pi$
$691$	−4178.88	−0.230061	−0.115030	+	0.993362i	$0.536697\pi$
$692$	0	0
$693$	2343.14	0.128439
$694$	0	0
$695$	−8359.67	−0.456260
$696$	0	0
$697$	18955.2	1.03010
$698$	0	0
$699$	5875.78	0.317943
$700$	0	0
$701$	−25180.2	−1.35669	−0.678347	−	0.734742i	$-0.737303\pi$
$701$	−25180.2	−1.35669	−0.678347	+	0.734742i	$0.737303\pi$
$702$	0	0
$703$	33626.6	1.80406
$704$	0	0
$705$	6344.80	0.338949
$706$	0	0
$707$	−14171.8	−0.753871
$708$	0	0
$709$	−1917.57	−0.101574	−0.0507868	−	0.998710i	$-0.516173\pi$
$709$	−1917.57	−0.101574	−0.0507868	+	0.998710i	$0.516173\pi$
$710$	0	0
$711$	3477.14	0.183408
$712$	0	0
$713$	−3690.63	−0.193850
$714$	0	0
$715$	−6632.81	−0.346927
$716$	0	0
$717$	20831.2	1.08502
$718$	0	0
$719$	539.927	0.0280054	0.0140027	−	0.999902i	$-0.495543\pi$
$719$	539.927	0.0280054	0.0140027	+	0.999902i	$0.495543\pi$
$720$	0	0
$721$	−15699.8	−0.810945
$722$	0	0
$723$	−23812.1	−1.22487
$724$	0	0
$725$	90.4851	0.00463522
$726$	0	0
$727$	8866.96	0.452349	0.226174	−	0.974087i	$-0.427378\pi$
$727$	8866.96	0.452349	0.226174	+	0.974087i	$0.427378\pi$
$728$	0	0
$729$	13176.4	0.669432
$730$	0	0
$731$	−28163.5	−1.42499
$732$	0	0
$733$	−31431.1	−1.58381	−0.791905	−	0.610644i	$-0.790911\pi$
$733$	−31431.1	−1.58381	−0.791905	+	0.610644i	$0.790911\pi$
$734$	0	0
$735$	−17981.5	−0.902391
$736$	0	0
$737$	−12237.3	−0.611621
$738$	0	0
$739$	10299.6	0.512688	0.256344	−	0.966586i	$-0.417482\pi$
$739$	10299.6	0.512688	0.256344	+	0.966586i	$0.417482\pi$
$740$	0	0
$741$	−7216.67	−0.357774
$742$	0	0
$743$	−6335.15	−0.312805	−0.156403	−	0.987693i	$-0.549990\pi$
$743$	−6335.15	−0.312805	−0.156403	+	0.987693i	$0.549990\pi$
$744$	0	0
$745$	22625.5	1.11266
$746$	0	0
$747$	−6168.41	−0.302129
$748$	0	0
$749$	−14097.5	−0.687734
$750$	0	0
$751$	34964.8	1.69892	0.849458	−	0.527657i	$-0.176929\pi$
$751$	34964.8	1.69892	0.849458	+	0.527657i	$0.176929\pi$
$752$	0	0
$753$	31540.9	1.52645
$754$	0	0
$755$	15958.8	0.769271
$756$	0	0
$757$	8904.00	0.427505	0.213753	−	0.976888i	$-0.431431\pi$
$757$	8904.00	0.427505	0.213753	+	0.976888i	$0.431431\pi$
$758$	0	0
$759$	9039.97	0.432319
$760$	0	0
$761$	−3870.31	−0.184361	−0.0921804	−	0.995742i	$-0.529384\pi$
$761$	−3870.31	−0.184361	−0.0921804	+	0.995742i	$0.529384\pi$
$762$	0	0
$763$	−10645.0	−0.505080
$764$	0	0
$765$	−6265.81	−0.296132
$766$	0	0
$767$	8659.04	0.407640
$768$	0	0
$769$	10431.1	0.489148	0.244574	−	0.969631i	$-0.421352\pi$
$769$	10431.1	0.489148	0.244574	+	0.969631i	$0.421352\pi$
$770$	0	0
$771$	−13953.2	−0.651767
$772$	0	0
$773$	−32950.1	−1.53316	−0.766580	−	0.642148i	$-0.778043\pi$
$773$	−32950.1	−1.53316	−0.766580	+	0.642148i	$0.778043\pi$
$774$	0	0
$775$	−324.267	−0.0150297
$776$	0	0
$777$	16604.9	0.766665
$778$	0	0
$779$	−20794.8	−0.956420
$780$	0	0
$781$	16737.2	0.766843
$782$	0	0
$783$	−3539.61	−0.161552
$784$	0	0
$785$	28951.2	1.31632
$786$	0	0
$787$	32299.0	1.46294	0.731471	−	0.681872i	$-0.238834\pi$
$787$	32299.0	1.46294	0.731471	+	0.681872i	$0.238834\pi$
$788$	0	0
$789$	24837.0	1.12068
$790$	0	0
$791$	11066.9	0.497462
$792$	0	0
$793$	1504.36	0.0673662
$794$	0	0
$795$	24540.5	1.09480
$796$	0	0
$797$	11682.9	0.519235	0.259617	−	0.965712i	$-0.416404\pi$
$797$	11682.9	0.519235	0.259617	+	0.965712i	$0.416404\pi$
$798$	0	0
$799$	−8516.07	−0.377068
$800$	0	0
$801$	5724.63	0.252522
$802$	0	0
$803$	−27971.9	−1.22927
$804$	0	0
$805$	3234.20	0.141603
$806$	0	0
$807$	45668.1	1.99206
$808$	0	0
$809$	7651.77	0.332536	0.166268	−	0.986081i	$-0.446828\pi$
$809$	7651.77	0.332536	0.166268	+	0.986081i	$0.446828\pi$
$810$	0	0
$811$	10113.4	0.437891	0.218946	−	0.975737i	$-0.429738\pi$
$811$	10113.4	0.437891	0.218946	+	0.975737i	$0.429738\pi$
$812$	0	0
$813$	−21618.2	−0.932575
$814$	0	0
$815$	34737.5	1.49301
$816$	0	0
$817$	30896.8	1.32306
$818$	0	0
$819$	−675.585	−0.0288240
$820$	0	0
$821$	25159.0	1.06949	0.534747	−	0.845012i	$-0.320407\pi$
$821$	25159.0	1.06949	0.534747	+	0.845012i	$0.320407\pi$
$822$	0	0
$823$	4679.66	0.198205	0.0991025	−	0.995077i	$-0.468403\pi$
$823$	4679.66	0.198205	0.0991025	+	0.995077i	$0.468403\pi$
$824$	0	0
$825$	794.273	0.0335188
$826$	0	0
$827$	−29404.0	−1.23637	−0.618184	−	0.786033i	$-0.712131\pi$
$827$	−29404.0	−1.23637	−0.618184	+	0.786033i	$0.712131\pi$
$828$	0	0
$829$	33722.7	1.41283	0.706416	−	0.707797i	$-0.250311\pi$
$829$	33722.7	1.41283	0.706416	+	0.707797i	$0.250311\pi$
$830$	0	0
$831$	−29290.0	−1.22269
$832$	0	0
$833$	24135.0	1.00388
$834$	0	0
$835$	25665.0	1.06368
$836$	0	0
$837$	12684.7	0.523834
$838$	0	0
$839$	−14439.6	−0.594172	−0.297086	−	0.954851i	$-0.596015\pi$
$839$	−14439.6	−0.594172	−0.297086	+	0.954851i	$0.596015\pi$
$840$	0	0
$841$	−23510.0	−0.963959
$842$	0	0
$843$	−10998.2	−0.449345
$844$	0	0
$845$	1912.40	0.0778565
$846$	0	0
$847$	5775.47	0.234294
$848$	0	0
$849$	−35250.6	−1.42497
$850$	0	0
$851$	12145.0	0.489217
$852$	0	0
$853$	−40881.3	−1.64097	−0.820486	−	0.571667i	$-0.806297\pi$
$853$	−40881.3	−1.64097	−0.820486	+	0.571667i	$0.806297\pi$
$854$	0	0
$855$	6873.89	0.274950
$856$	0	0
$857$	22406.1	0.893090	0.446545	−	0.894761i	$-0.352654\pi$
$857$	22406.1	0.893090	0.446545	+	0.894761i	$0.352654\pi$
$858$	0	0
$859$	−26451.9	−1.05067	−0.525337	−	0.850894i	$-0.676061\pi$
$859$	−26451.9	−1.05067	−0.525337	+	0.850894i	$0.676061\pi$
$860$	0	0
$861$	−10268.5	−0.406447
$862$	0	0
$863$	29947.3	1.18125	0.590624	−	0.806947i	$-0.298882\pi$
$863$	29947.3	1.18125	0.590624	+	0.806947i	$0.298882\pi$
$864$	0	0
$865$	29975.7	1.17827
$866$	0	0
$867$	16003.9	0.626900
$868$	0	0
$869$	24822.2	0.968971
$870$	0	0
$871$	3528.31	0.137258
$872$	0	0
$873$	−11893.9	−0.461108
$874$	0	0
$875$	−11354.3	−0.438682
$876$	0	0
$877$	25860.8	0.995734	0.497867	−	0.867254i	$-0.334117\pi$
$877$	25860.8	0.995734	0.497867	+	0.867254i	$0.334117\pi$
$878$	0	0
$879$	45271.3	1.73716
$880$	0	0
$881$	−16951.9	−0.648268	−0.324134	−	0.946011i	$-0.605073\pi$
$881$	−16951.9	−0.648268	−0.324134	+	0.946011i	$0.605073\pi$
$882$	0	0
$883$	25183.8	0.959799	0.479900	−	0.877323i	$-0.340673\pi$
$883$	25183.8	0.959799	0.479900	+	0.877323i	$0.340673\pi$
$884$	0	0
$885$	−43505.7	−1.65246
$886$	0	0
$887$	−12276.6	−0.464720	−0.232360	−	0.972630i	$-0.574645\pi$
$887$	−12276.6	−0.464720	−0.232360	+	0.972630i	$0.574645\pi$
$888$	0	0
$889$	6233.53	0.235170
$890$	0	0
$891$	−38759.5	−1.45734
$892$	0	0
$893$	9342.54	0.350096
$894$	0	0
$895$	44483.5	1.66136
$896$	0	0
$897$	−2606.45	−0.0970199
$898$	0	0
$899$	−3150.04	−0.116863
$900$	0	0
$901$	−32938.6	−1.21792
$902$	0	0
$903$	15256.9	0.562257
$904$	0	0
$905$	−41965.1	−1.54140
$906$	0	0
$907$	36313.2	1.32939	0.664697	−	0.747113i	$-0.268561\pi$
$907$	36313.2	1.32939	0.664697	+	0.747113i	$0.268561\pi$
$908$	0	0
$909$	−10878.6	−0.396944
$910$	0	0
$911$	−30504.4	−1.10939	−0.554696	−	0.832053i	$-0.687165\pi$
$911$	−30504.4	−1.10939	−0.554696	+	0.832053i	$0.687165\pi$
$912$	0	0
$913$	−44034.4	−1.59619
$914$	0	0
$915$	−7558.37	−0.273084
$916$	0	0
$917$	5639.57	0.203092
$918$	0	0
$919$	−11713.9	−0.420463	−0.210231	−	0.977652i	$-0.567422\pi$
$919$	−11713.9	−0.420463	−0.210231	+	0.977652i	$0.567422\pi$
$920$	0	0
$921$	46227.9	1.65392
$922$	0	0
$923$	−4825.76	−0.172093
$924$	0	0
$925$	1067.08	0.0379303
$926$	0	0
$927$	−12051.5	−0.426995
$928$	0	0
$929$	−25458.6	−0.899104	−0.449552	−	0.893254i	$-0.648416\pi$
$929$	−25458.6	−0.899104	−0.449552	+	0.893254i	$0.648416\pi$
$930$	0	0
$931$	−26477.3	−0.932070
$932$	0	0
$933$	−29270.7	−1.02709
$934$	0	0
$935$	−44729.6	−1.56451
$936$	0	0
$937$	−18146.0	−0.632661	−0.316331	−	0.948649i	$-0.602451\pi$
$937$	−18146.0	−0.632661	−0.316331	+	0.948649i	$0.602451\pi$
$938$	0	0
$939$	49526.7	1.72124
$940$	0	0
$941$	7124.16	0.246802	0.123401	−	0.992357i	$-0.460620\pi$
$941$	7124.16	0.246802	0.123401	+	0.992357i	$0.460620\pi$
$942$	0	0
$943$	−7510.48	−0.259358
$944$	0	0
$945$	−11116.0	−0.382649
$946$	0	0
$947$	11813.7	0.405379	0.202689	−	0.979243i	$-0.435032\pi$
$947$	11813.7	0.405379	0.202689	+	0.979243i	$0.435032\pi$
$948$	0	0
$949$	8064.99	0.275870
$950$	0	0
$951$	−42619.0	−1.45322
$952$	0	0
$953$	−18341.8	−0.623451	−0.311725	−	0.950172i	$-0.600907\pi$
$953$	−18341.8	−0.623451	−0.311725	+	0.950172i	$0.600907\pi$
$954$	0	0
$955$	−1835.82	−0.0622051
$956$	0	0
$957$	7715.83	0.260624
$958$	0	0
$959$	8700.28	0.292958
$960$	0	0
$961$	−18502.3	−0.621072
$962$	0	0
$963$	−10821.6	−0.362119
$964$	0	0
$965$	−31253.8	−1.04259
$966$	0	0
$967$	−31077.4	−1.03349	−0.516743	−	0.856140i	$-0.672856\pi$
$967$	−31077.4	−1.03349	−0.516743	+	0.856140i	$0.672856\pi$
$968$	0	0
$969$	−48667.0	−1.61343
$970$	0	0
$971$	2493.06	0.0823955	0.0411977	−	0.999151i	$-0.486883\pi$
$971$	2493.06	0.0823955	0.0411977	+	0.999151i	$0.486883\pi$
$972$	0	0
$973$	−6078.42	−0.200272
$974$	0	0
$975$	−229.009	−0.00752220
$976$	0	0
$977$	2659.66	0.0870931	0.0435465	−	0.999051i	$-0.486134\pi$
$977$	2659.66	0.0870931	0.0435465	+	0.999051i	$0.486134\pi$
$978$	0	0
$979$	40866.3	1.33411
$980$	0	0
$981$	−8171.37	−0.265945
$982$	0	0
$983$	6061.61	0.196679	0.0983394	−	0.995153i	$-0.468647\pi$
$983$	6061.61	0.196679	0.0983394	+	0.995153i	$0.468647\pi$
$984$	0	0
$985$	−25839.6	−0.835857
$986$	0	0
$987$	4613.37	0.148779
$988$	0	0
$989$	11159.0	0.358782
$990$	0	0
$991$	−53791.1	−1.72425	−0.862125	−	0.506696i	$-0.830866\pi$
$991$	−53791.1	−1.72425	−0.862125	+	0.506696i	$0.830866\pi$
$992$	0	0
$993$	23414.7	0.748281
$994$	0	0
$995$	−27262.6	−0.868626
$996$	0	0
$997$	18039.9	0.573049	0.286524	−	0.958073i	$-0.407500\pi$
$997$	18039.9	0.573049	0.286524	+	0.958073i	$0.407500\pi$
$998$	0	0
$999$	−41742.4	−1.32199

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.4.a.j.1.2		2
3.2	odd	2		1872.4.a.bc.1.1		2
4.3	odd	2		104.4.a.c.1.1	✓	2
8.3	odd	2		832.4.a.y.1.2		2
8.5	even	2		832.4.a.u.1.1		2
12.11	even	2		936.4.a.e.1.1		2
52.51	odd	2		1352.4.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.4.a.c.1.1	✓	2	4.3	odd	2
208.4.a.j.1.2		2	1.1	even	1	trivial
832.4.a.u.1.1		2	8.5	even	2
832.4.a.y.1.2		2	8.3	odd	2
936.4.a.e.1.1		2	12.11	even	2
1352.4.a.f.1.1		2	52.51	odd	2
1872.4.a.bc.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)

\(f(q)\)	\(=\)	\(q+5.77200 q^{3} +11.3160 q^{5} +8.22800 q^{7} +6.31601 q^{9} +O(q^{10})\)	\(q+5.77200 q^{3} +11.3160 q^{5} +8.22800 q^{7} +6.31601 q^{9} +45.0880 q^{11} -13.0000 q^{13} +65.3160 q^{15} -87.6680 q^{17} +96.1760 q^{19} +47.4920 q^{21} +34.7360 q^{23} +3.05198 q^{25} -119.388 q^{27} +29.6480 q^{29} -106.248 q^{31} +260.248 q^{33} +93.1081 q^{35} +349.636 q^{37} -75.0360 q^{39} -216.216 q^{41} +321.252 q^{43} +71.4720 q^{45} +97.1400 q^{47} -275.300 q^{49} -506.020 q^{51} +375.720 q^{53} +510.216 q^{55} +555.128 q^{57} -666.080 q^{59} -115.720 q^{61} +51.9681 q^{63} -147.108 q^{65} -271.408 q^{67} +200.496 q^{69} +371.212 q^{71} -620.384 q^{73} +17.6161 q^{75} +370.984 q^{77} +550.528 q^{79} -859.640 q^{81} -976.632 q^{83} -992.052 q^{85} +171.128 q^{87} +906.368 q^{89} -106.964 q^{91} -613.264 q^{93} +1088.33 q^{95} -1883.14 q^{97} +284.776 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} - 3 q^{5} + 25 q^{7} - 13 q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 - 3 * q^5 + 25 * q^7 - 13 * q^9	\( 2 q + 3 q^{3} - 3 q^{5} + 25 q^{7} - 13 q^{9} + 56 q^{11} - 26 q^{13} + 105 q^{15} - 13 q^{17} + 124 q^{19} + q^{21} + 172 q^{23} + 83 q^{25} + 9 q^{27} + 196 q^{29} + 78 q^{31} + 230 q^{33} - 147 q^{35} + 161 q^{37} - 39 q^{39} + 234 q^{41} - 135 q^{43} + 348 q^{45} + 237 q^{47} - 337 q^{49} - 713 q^{51} + 666 q^{53} + 354 q^{55} + 478 q^{57} - 136 q^{59} - 146 q^{61} - 272 q^{63} + 39 q^{65} + 4 q^{67} - 180 q^{69} + 563 q^{71} - 1480 q^{73} - 204 q^{75} + 554 q^{77} + 896 q^{79} - 694 q^{81} - 1902 q^{83} - 2061 q^{85} - 290 q^{87} - 272 q^{89} - 325 q^{91} - 1124 q^{93} + 690 q^{95} - 2160 q^{97} + 74 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 - 3 * q^5 + 25 * q^7 - 13 * q^9 + 56 * q^11 - 26 * q^13 + 105 * q^15 - 13 * q^17 + 124 * q^19 + q^21 + 172 * q^23 + 83 * q^25 + 9 * q^27 + 196 * q^29 + 78 * q^31 + 230 * q^33 - 147 * q^35 + 161 * q^37 - 39 * q^39 + 234 * q^41 - 135 * q^43 + 348 * q^45 + 237 * q^47 - 337 * q^49 - 713 * q^51 + 666 * q^53 + 354 * q^55 + 478 * q^57 - 136 * q^59 - 146 * q^61 - 272 * q^63 + 39 * q^65 + 4 * q^67 - 180 * q^69 + 563 * q^71 - 1480 * q^73 - 204 * q^75 + 554 * q^77 + 896 * q^79 - 694 * q^81 - 1902 * q^83 - 2061 * q^85 - 290 * q^87 - 272 * q^89 - 325 * q^91 - 1124 * q^93 + 690 * q^95 - 2160 * q^97 + 74 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more