LMFDB - Embedded newform 6008.2.a.e.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.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:	$47.9741215344$
Analytic rank:	$0$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.685000 q^{3} +2.14563 q^{5} +4.94928 q^{7} -2.53078 q^{9} +O(q^{10})$	$q-0.685000 q^{3} +2.14563 q^{5} +4.94928 q^{7} -2.53078 q^{9} +5.84000 q^{11} -0.179952 q^{13} -1.46976 q^{15} +3.12097 q^{17} +1.14439 q^{19} -3.39026 q^{21} -6.64547 q^{23} -0.396256 q^{25} +3.78858 q^{27} +0.232956 q^{29} +7.29442 q^{31} -4.00040 q^{33} +10.6193 q^{35} -5.55468 q^{37} +0.123267 q^{39} +1.51030 q^{41} +2.08398 q^{43} -5.43012 q^{45} +9.61601 q^{47} +17.4954 q^{49} -2.13786 q^{51} +10.2717 q^{53} +12.5305 q^{55} -0.783904 q^{57} -7.92347 q^{59} -0.773842 q^{61} -12.5255 q^{63} -0.386112 q^{65} -7.77866 q^{67} +4.55215 q^{69} +1.12432 q^{71} -10.6802 q^{73} +0.271436 q^{75} +28.9038 q^{77} +3.02508 q^{79} +4.99715 q^{81} +13.8174 q^{83} +6.69646 q^{85} -0.159575 q^{87} -12.5658 q^{89} -0.890634 q^{91} -4.99668 q^{93} +2.45543 q^{95} +6.78459 q^{97} -14.7797 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * 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
$3$	−0.685000	−0.395485	−0.197742	−	0.980254i	$-0.563361\pi$
$3$	−0.685000	−0.395485	−0.197742	+	0.980254i	$0.563361\pi$
$4$	0	0
$5$	2.14563	0.959557	0.479778	−	0.877390i	$-0.340717\pi$
$5$	2.14563	0.959557	0.479778	+	0.877390i	$0.340717\pi$
$6$	0	0
$7$	4.94928	1.87065	0.935326	−	0.353787i	$-0.115106\pi$
$7$	4.94928	1.87065	0.935326	+	0.353787i	$0.115106\pi$
$8$	0	0
$9$	−2.53078	−0.843592
$10$	0	0
$11$	5.84000	1.76083	0.880413	−	0.474208i	$-0.157265\pi$
$11$	5.84000	1.76083	0.880413	+	0.474208i	$0.157265\pi$
$12$	0	0
$13$	−0.179952	−0.0499098	−0.0249549	−	0.999689i	$-0.507944\pi$
$13$	−0.179952	−0.0499098	−0.0249549	+	0.999689i	$0.507944\pi$
$14$	0	0
$15$	−1.46976	−0.379490
$16$	0	0
$17$	3.12097	0.756946	0.378473	−	0.925612i	$-0.376449\pi$
$17$	3.12097	0.756946	0.378473	+	0.925612i	$0.376449\pi$
$18$	0	0
$19$	1.14439	0.262540	0.131270	−	0.991347i	$-0.458095\pi$
$19$	1.14439	0.262540	0.131270	+	0.991347i	$0.458095\pi$
$20$	0	0
$21$	−3.39026	−0.739815
$22$	0	0
$23$	−6.64547	−1.38568	−0.692838	−	0.721093i	$-0.743640\pi$
$23$	−6.64547	−1.38568	−0.692838	+	0.721093i	$0.743640\pi$
$24$	0	0
$25$	−0.396256	−0.0792513
$26$	0	0
$27$	3.78858	0.729113
$28$	0	0
$29$	0.232956	0.0432588	0.0216294	−	0.999766i	$-0.493115\pi$
$29$	0.232956	0.0432588	0.0216294	+	0.999766i	$0.493115\pi$
$30$	0	0
$31$	7.29442	1.31012	0.655059	−	0.755578i	$-0.272644\pi$
$31$	7.29442	1.31012	0.655059	+	0.755578i	$0.272644\pi$
$32$	0	0
$33$	−4.00040	−0.696380
$34$	0	0
$35$	10.6193	1.79500
$36$	0	0
$37$	−5.55468	−0.913184	−0.456592	−	0.889676i	$-0.650930\pi$
$37$	−5.55468	−0.913184	−0.456592	+	0.889676i	$0.650930\pi$
$38$	0	0
$39$	0.123267	0.0197386
$40$	0	0
$41$	1.51030	0.235868	0.117934	−	0.993021i	$-0.462373\pi$
$41$	1.51030	0.235868	0.117934	+	0.993021i	$0.462373\pi$
$42$	0	0
$43$	2.08398	0.317804	0.158902	−	0.987294i	$-0.449205\pi$
$43$	2.08398	0.317804	0.158902	+	0.987294i	$0.449205\pi$
$44$	0	0
$45$	−5.43012	−0.809474
$46$	0	0
$47$	9.61601	1.40264	0.701320	−	0.712847i	$-0.252595\pi$
$47$	9.61601	1.40264	0.701320	+	0.712847i	$0.252595\pi$
$48$	0	0
$49$	17.4954	2.49934
$50$	0	0
$51$	−2.13786	−0.299361
$52$	0	0
$53$	10.2717	1.41093	0.705463	−	0.708747i	$-0.250739\pi$
$53$	10.2717	1.41093	0.705463	+	0.708747i	$0.250739\pi$
$54$	0	0
$55$	12.5305	1.68961
$56$	0	0
$57$	−0.783904	−0.103831
$58$	0	0
$59$	−7.92347	−1.03155	−0.515774	−	0.856725i	$-0.672496\pi$
$59$	−7.92347	−1.03155	−0.515774	+	0.856725i	$0.672496\pi$
$60$	0	0
$61$	−0.773842	−0.0990803	−0.0495401	−	0.998772i	$-0.515776\pi$
$61$	−0.773842	−0.0990803	−0.0495401	+	0.998772i	$0.515776\pi$
$62$	0	0
$63$	−12.5255	−1.57807
$64$	0	0
$65$	−0.386112	−0.0478913
$66$	0	0
$67$	−7.77866	−0.950314	−0.475157	−	0.879901i	$-0.657609\pi$
$67$	−7.77866	−0.950314	−0.475157	+	0.879901i	$0.657609\pi$
$68$	0	0
$69$	4.55215	0.548014
$70$	0	0
$71$	1.12432	0.133432	0.0667159	−	0.997772i	$-0.478748\pi$
$71$	1.12432	0.133432	0.0667159	+	0.997772i	$0.478748\pi$
$72$	0	0
$73$	−10.6802	−1.25003	−0.625013	−	0.780615i	$-0.714906\pi$
$73$	−10.6802	−1.25003	−0.625013	+	0.780615i	$0.714906\pi$
$74$	0	0
$75$	0.271436	0.0313427
$76$	0	0
$77$	28.9038	3.29389
$78$	0	0
$79$	3.02508	0.340348	0.170174	−	0.985414i	$-0.445567\pi$
$79$	3.02508	0.340348	0.170174	+	0.985414i	$0.445567\pi$
$80$	0	0
$81$	4.99715	0.555239
$82$	0	0
$83$	13.8174	1.51665	0.758326	−	0.651875i	$-0.226017\pi$
$83$	13.8174	1.51665	0.758326	+	0.651875i	$0.226017\pi$
$84$	0	0
$85$	6.69646	0.726333
$86$	0	0
$87$	−0.159575	−0.0171082
$88$	0	0
$89$	−12.5658	−1.33197	−0.665987	−	0.745963i	$-0.731989\pi$
$89$	−12.5658	−1.33197	−0.665987	+	0.745963i	$0.731989\pi$
$90$	0	0
$91$	−0.890634	−0.0933639
$92$	0	0
$93$	−4.99668	−0.518132
$94$	0	0
$95$	2.45543	0.251922
$96$	0	0
$97$	6.78459	0.688871	0.344436	−	0.938810i	$-0.388070\pi$
$97$	6.78459	0.688871	0.344436	+	0.938810i	$0.388070\pi$
$98$	0	0
$99$	−14.7797	−1.48542
$100$	0	0
$101$	−2.15273	−0.214204	−0.107102	−	0.994248i	$-0.534157\pi$
$101$	−2.15273	−0.214204	−0.107102	+	0.994248i	$0.534157\pi$
$102$	0	0
$103$	2.33703	0.230274	0.115137	−	0.993350i	$-0.463269\pi$
$103$	2.33703	0.230274	0.115137	+	0.993350i	$0.463269\pi$
$104$	0	0
$105$	−7.27425	−0.709894
$106$	0	0
$107$	−1.61371	−0.156003	−0.0780014	−	0.996953i	$-0.524854\pi$
$107$	−1.61371	−0.156003	−0.0780014	+	0.996953i	$0.524854\pi$
$108$	0	0
$109$	−3.15647	−0.302335	−0.151167	−	0.988508i	$-0.548303\pi$
$109$	−3.15647	−0.302335	−0.151167	+	0.988508i	$0.548303\pi$
$110$	0	0
$111$	3.80496	0.361150
$112$	0	0
$113$	−8.40627	−0.790795	−0.395398	−	0.918510i	$-0.629393\pi$
$113$	−8.40627	−0.790795	−0.395398	+	0.918510i	$0.629393\pi$
$114$	0	0
$115$	−14.2587	−1.32963
$116$	0	0
$117$	0.455419	0.0421035
$118$	0	0
$119$	15.4466	1.41598
$120$	0	0
$121$	23.1056	2.10051
$122$	0	0
$123$	−1.03455	−0.0932824
$124$	0	0
$125$	−11.5784	−1.03560
$126$	0	0
$127$	−20.1082	−1.78431	−0.892156	−	0.451728i	$-0.850808\pi$
$127$	−20.1082	−1.78431	−0.892156	+	0.451728i	$0.850808\pi$
$128$	0	0
$129$	−1.42753	−0.125687
$130$	0	0
$131$	−19.5208	−1.70554	−0.852769	−	0.522289i	$-0.825078\pi$
$131$	−19.5208	−1.70554	−0.852769	+	0.522289i	$0.825078\pi$
$132$	0	0
$133$	5.66388	0.491121
$134$	0	0
$135$	8.12891	0.699625
$136$	0	0
$137$	10.9285	0.933681	0.466840	−	0.884342i	$-0.345392\pi$
$137$	10.9285	0.933681	0.466840	+	0.884342i	$0.345392\pi$
$138$	0	0
$139$	−8.62610	−0.731657	−0.365828	−	0.930682i	$-0.619214\pi$
$139$	−8.62610	−0.731657	−0.365828	+	0.930682i	$0.619214\pi$
$140$	0	0
$141$	−6.58697	−0.554723
$142$	0	0
$143$	−1.05092	−0.0878825
$144$	0	0
$145$	0.499838	0.0415093
$146$	0	0
$147$	−11.9843	−0.988450
$148$	0	0
$149$	−2.74722	−0.225061	−0.112531	−	0.993648i	$-0.535896\pi$
$149$	−2.74722	−0.225061	−0.112531	+	0.993648i	$0.535896\pi$
$150$	0	0
$151$	17.6361	1.43520	0.717601	−	0.696455i	$-0.245240\pi$
$151$	17.6361	1.43520	0.717601	+	0.696455i	$0.245240\pi$
$152$	0	0
$153$	−7.89847	−0.638554
$154$	0	0
$155$	15.6512	1.25713
$156$	0	0
$157$	18.2560	1.45699	0.728495	−	0.685051i	$-0.240220\pi$
$157$	18.2560	1.45699	0.728495	+	0.685051i	$0.240220\pi$
$158$	0	0
$159$	−7.03611	−0.558000
$160$	0	0
$161$	−32.8903	−2.59212
$162$	0	0
$163$	−9.78433	−0.766368	−0.383184	−	0.923672i	$-0.625172\pi$
$163$	−9.78433	−0.766368	−0.383184	+	0.923672i	$0.625172\pi$
$164$	0	0
$165$	−8.58339	−0.668216
$166$	0	0
$167$	−7.30879	−0.565571	−0.282786	−	0.959183i	$-0.591259\pi$
$167$	−7.30879	−0.565571	−0.282786	+	0.959183i	$0.591259\pi$
$168$	0	0
$169$	−12.9676	−0.997509
$170$	0	0
$171$	−2.89618	−0.221477
$172$	0	0
$173$	−2.22954	−0.169508	−0.0847542	−	0.996402i	$-0.527011\pi$
$173$	−2.22954	−0.169508	−0.0847542	+	0.996402i	$0.527011\pi$
$174$	0	0
$175$	−1.96118	−0.148252
$176$	0	0
$177$	5.42758	0.407962
$178$	0	0
$179$	25.6296	1.91565	0.957825	−	0.287353i	$-0.0927753\pi$
$179$	25.6296	1.91565	0.957825	+	0.287353i	$0.0927753\pi$
$180$	0	0
$181$	0.997348	0.0741323	0.0370662	−	0.999313i	$-0.488199\pi$
$181$	0.997348	0.0741323	0.0370662	+	0.999313i	$0.488199\pi$
$182$	0	0
$183$	0.530081	0.0391847
$184$	0	0
$185$	−11.9183	−0.876252
$186$	0	0
$187$	18.2265	1.33285
$188$	0	0
$189$	18.7507	1.36392
$190$	0	0
$191$	−20.3884	−1.47525	−0.737626	−	0.675210i	$-0.764053\pi$
$191$	−20.3884	−1.47525	−0.737626	+	0.675210i	$0.764053\pi$
$192$	0	0
$193$	24.1537	1.73862	0.869312	−	0.494264i	$-0.164562\pi$
$193$	24.1537	1.73862	0.869312	+	0.494264i	$0.164562\pi$
$194$	0	0
$195$	0.264487	0.0189403
$196$	0	0
$197$	−22.8243	−1.62617	−0.813083	−	0.582148i	$-0.802213\pi$
$197$	−22.8243	−1.62617	−0.813083	+	0.582148i	$0.802213\pi$
$198$	0	0
$199$	−3.18881	−0.226049	−0.113024	−	0.993592i	$-0.536054\pi$
$199$	−3.18881	−0.226049	−0.113024	+	0.993592i	$0.536054\pi$
$200$	0	0
$201$	5.32838	0.375835
$202$	0	0
$203$	1.15296	0.0809221
$204$	0	0
$205$	3.24054	0.226329
$206$	0	0
$207$	16.8182	1.16894
$208$	0	0
$209$	6.68321	0.462287
$210$	0	0
$211$	−1.22492	−0.0843270	−0.0421635	−	0.999111i	$-0.513425\pi$
$211$	−1.22492	−0.0843270	−0.0421635	+	0.999111i	$0.513425\pi$
$212$	0	0
$213$	−0.770157	−0.0527703
$214$	0	0
$215$	4.47146	0.304951
$216$	0	0
$217$	36.1021	2.45077
$218$	0	0
$219$	7.31595	0.494366
$220$	0	0
$221$	−0.561626	−0.0377791
$222$	0	0
$223$	−11.9491	−0.800171	−0.400085	−	0.916478i	$-0.631020\pi$
$223$	−11.9491	−0.800171	−0.400085	+	0.916478i	$0.631020\pi$
$224$	0	0
$225$	1.00284	0.0668557
$226$	0	0
$227$	6.68335	0.443590	0.221795	−	0.975093i	$-0.428808\pi$
$227$	6.68335	0.443590	0.221795	+	0.975093i	$0.428808\pi$
$228$	0	0
$229$	−3.68476	−0.243496	−0.121748	−	0.992561i	$-0.538850\pi$
$229$	−3.68476	−0.243496	−0.121748	+	0.992561i	$0.538850\pi$
$230$	0	0
$231$	−19.7991	−1.30268
$232$	0	0
$233$	12.3673	0.810209	0.405105	−	0.914270i	$-0.367235\pi$
$233$	12.3673	0.810209	0.405105	+	0.914270i	$0.367235\pi$
$234$	0	0
$235$	20.6324	1.34591
$236$	0	0
$237$	−2.07218	−0.134602
$238$	0	0
$239$	−4.19593	−0.271412	−0.135706	−	0.990749i	$-0.543330\pi$
$239$	−4.19593	−0.271412	−0.135706	+	0.990749i	$0.543330\pi$
$240$	0	0
$241$	12.5319	0.807252	0.403626	−	0.914924i	$-0.367750\pi$
$241$	12.5319	0.807252	0.403626	+	0.914924i	$0.367750\pi$
$242$	0	0
$243$	−14.7888	−0.948701
$244$	0	0
$245$	37.5386	2.39826
$246$	0	0
$247$	−0.205935	−0.0131033
$248$	0	0
$249$	−9.46489	−0.599813
$250$	0	0
$251$	−3.97786	−0.251080	−0.125540	−	0.992089i	$-0.540066\pi$
$251$	−3.97786	−0.251080	−0.125540	+	0.992089i	$0.540066\pi$
$252$	0	0
$253$	−38.8095	−2.43993
$254$	0	0
$255$	−4.58707	−0.287254
$256$	0	0
$257$	7.02618	0.438281	0.219141	−	0.975693i	$-0.429675\pi$
$257$	7.02618	0.438281	0.219141	+	0.975693i	$0.429675\pi$
$258$	0	0
$259$	−27.4917	−1.70825
$260$	0	0
$261$	−0.589559	−0.0364928
$262$	0	0
$263$	−14.0263	−0.864899	−0.432449	−	0.901658i	$-0.642351\pi$
$263$	−14.0263	−0.864899	−0.432449	+	0.901658i	$0.642351\pi$
$264$	0	0
$265$	22.0393	1.35386
$266$	0	0
$267$	8.60759	0.526776
$268$	0	0
$269$	10.2401	0.624348	0.312174	−	0.950025i	$-0.398943\pi$
$269$	10.2401	0.624348	0.312174	+	0.950025i	$0.398943\pi$
$270$	0	0
$271$	−22.7598	−1.38256	−0.691279	−	0.722588i	$-0.742952\pi$
$271$	−22.7598	−1.38256	−0.691279	+	0.722588i	$0.742952\pi$
$272$	0	0
$273$	0.610085	0.0369240
$274$	0	0
$275$	−2.31414	−0.139548
$276$	0	0
$277$	16.9060	1.01578	0.507891	−	0.861421i	$-0.330425\pi$
$277$	16.9060	1.01578	0.507891	+	0.861421i	$0.330425\pi$
$278$	0	0
$279$	−18.4605	−1.10520
$280$	0	0
$281$	33.1150	1.97548	0.987738	−	0.156119i	$-0.0498983\pi$
$281$	33.1150	1.97548	0.987738	+	0.156119i	$0.0498983\pi$
$282$	0	0
$283$	−19.3002	−1.14728	−0.573638	−	0.819109i	$-0.694468\pi$
$283$	−19.3002	−1.14728	−0.573638	+	0.819109i	$0.694468\pi$
$284$	0	0
$285$	−1.68197	−0.0996313
$286$	0	0
$287$	7.47487	0.441228
$288$	0	0
$289$	−7.25954	−0.427032
$290$	0	0
$291$	−4.64745	−0.272438
$292$	0	0
$293$	9.49110	0.554476	0.277238	−	0.960801i	$-0.410581\pi$
$293$	9.49110	0.554476	0.277238	+	0.960801i	$0.410581\pi$
$294$	0	0
$295$	−17.0009	−0.989829
$296$	0	0
$297$	22.1253	1.28384
$298$	0	0
$299$	1.19587	0.0691588
$300$	0	0
$301$	10.3142	0.594501
$302$	0	0
$303$	1.47462	0.0847146
$304$	0	0
$305$	−1.66038	−0.0950731
$306$	0	0
$307$	−22.8314	−1.30305	−0.651527	−	0.758625i	$-0.725871\pi$
$307$	−22.8314	−1.30305	−0.651527	+	0.758625i	$0.725871\pi$
$308$	0	0
$309$	−1.60086	−0.0910699
$310$	0	0
$311$	−11.0405	−0.626050	−0.313025	−	0.949745i	$-0.601342\pi$
$311$	−11.0405	−0.626050	−0.313025	+	0.949745i	$0.601342\pi$
$312$	0	0
$313$	1.37876	0.0779323	0.0389662	−	0.999241i	$-0.487594\pi$
$313$	1.37876	0.0779323	0.0389662	+	0.999241i	$0.487594\pi$
$314$	0	0
$315$	−26.8752	−1.51424
$316$	0	0
$317$	24.0543	1.35103	0.675513	−	0.737348i	$-0.263922\pi$
$317$	24.0543	1.35103	0.675513	+	0.737348i	$0.263922\pi$
$318$	0	0
$319$	1.36046	0.0761712
$320$	0	0
$321$	1.10539	0.0616968
$322$	0	0
$323$	3.57159	0.198729
$324$	0	0
$325$	0.0713073	0.00395542
$326$	0	0
$327$	2.16218	0.119569
$328$	0	0
$329$	47.5923	2.62385
$330$	0	0
$331$	16.9664	0.932557	0.466279	−	0.884638i	$-0.345594\pi$
$331$	16.9664	0.932557	0.466279	+	0.884638i	$0.345594\pi$
$332$	0	0
$333$	14.0576	0.770354
$334$	0	0
$335$	−16.6902	−0.911880
$336$	0	0
$337$	9.84136	0.536093	0.268047	−	0.963406i	$-0.413622\pi$
$337$	9.84136	0.536093	0.268047	+	0.963406i	$0.413622\pi$
$338$	0	0
$339$	5.75829	0.312748
$340$	0	0
$341$	42.5994	2.30689
$342$	0	0
$343$	51.9445	2.80474
$344$	0	0
$345$	9.76724	0.525850
$346$	0	0
$347$	2.42800	0.130342	0.0651708	−	0.997874i	$-0.479241\pi$
$347$	2.42800	0.130342	0.0651708	+	0.997874i	$0.479241\pi$
$348$	0	0
$349$	−19.9933	−1.07022	−0.535108	−	0.844784i	$-0.679729\pi$
$349$	−19.9933	−1.07022	−0.535108	+	0.844784i	$0.679729\pi$
$350$	0	0
$351$	−0.681764	−0.0363899
$352$	0	0
$353$	−18.8138	−1.00136	−0.500678	−	0.865633i	$-0.666916\pi$
$353$	−18.8138	−1.00136	−0.500678	+	0.865633i	$0.666916\pi$
$354$	0	0
$355$	2.41237	0.128035
$356$	0	0
$357$	−10.5809	−0.560000
$358$	0	0
$359$	13.6608	0.720987	0.360494	−	0.932762i	$-0.382608\pi$
$359$	13.6608	0.720987	0.360494	+	0.932762i	$0.382608\pi$
$360$	0	0
$361$	−17.6904	−0.931073
$362$	0	0
$363$	−15.8273	−0.830719
$364$	0	0
$365$	−22.9158	−1.19947
$366$	0	0
$367$	8.47228	0.442250	0.221125	−	0.975246i	$-0.429027\pi$
$367$	8.47228	0.442250	0.221125	+	0.975246i	$0.429027\pi$
$368$	0	0
$369$	−3.82222	−0.198977
$370$	0	0
$371$	50.8375	2.63935
$372$	0	0
$373$	−0.228118	−0.0118115	−0.00590576	−	0.999983i	$-0.501880\pi$
$373$	−0.228118	−0.0118115	−0.00590576	+	0.999983i	$0.501880\pi$
$374$	0	0
$375$	7.93120	0.409565
$376$	0	0
$377$	−0.0419209	−0.00215904
$378$	0	0
$379$	24.5799	1.26258	0.631292	−	0.775545i	$-0.282525\pi$
$379$	24.5799	1.26258	0.631292	+	0.775545i	$0.282525\pi$
$380$	0	0
$381$	13.7741	0.705668
$382$	0	0
$383$	−24.6978	−1.26200	−0.631000	−	0.775783i	$-0.717355\pi$
$383$	−24.6978	−1.26200	−0.631000	+	0.775783i	$0.717355\pi$
$384$	0	0
$385$	62.0169	3.16068
$386$	0	0
$387$	−5.27409	−0.268097
$388$	0	0
$389$	−29.4000	−1.49064	−0.745321	−	0.666706i	$-0.767704\pi$
$389$	−29.4000	−1.49064	−0.745321	+	0.666706i	$0.767704\pi$
$390$	0	0
$391$	−20.7403	−1.04888
$392$	0	0
$393$	13.3717	0.674514
$394$	0	0
$395$	6.49071	0.326583
$396$	0	0
$397$	19.2696	0.967116	0.483558	−	0.875312i	$-0.339344\pi$
$397$	19.2696	0.967116	0.483558	+	0.875312i	$0.339344\pi$
$398$	0	0
$399$	−3.87976	−0.194231
$400$	0	0
$401$	29.3236	1.46435	0.732176	−	0.681116i	$-0.238505\pi$
$401$	29.3236	1.46435	0.732176	+	0.681116i	$0.238505\pi$
$402$	0	0
$403$	−1.31265	−0.0653877
$404$	0	0
$405$	10.7220	0.532783
$406$	0	0
$407$	−32.4393	−1.60796
$408$	0	0
$409$	13.3658	0.660899	0.330449	−	0.943824i	$-0.392800\pi$
$409$	13.3658	0.660899	0.330449	+	0.943824i	$0.392800\pi$
$410$	0	0
$411$	−7.48599	−0.369257
$412$	0	0
$413$	−39.2155	−1.92967
$414$	0	0
$415$	29.6470	1.45531
$416$	0	0
$417$	5.90888	0.289359
$418$	0	0
$419$	−13.8349	−0.675878	−0.337939	−	0.941168i	$-0.609730\pi$
$419$	−13.8349	−0.675878	−0.337939	+	0.941168i	$0.609730\pi$
$420$	0	0
$421$	24.1254	1.17580	0.587900	−	0.808934i	$-0.299955\pi$
$421$	24.1254	1.17580	0.587900	+	0.808934i	$0.299955\pi$
$422$	0	0
$423$	−24.3360	−1.18325
$424$	0	0
$425$	−1.23670	−0.0599890
$426$	0	0
$427$	−3.82996	−0.185345
$428$	0	0
$429$	0.719881	0.0347562
$430$	0	0
$431$	−18.4962	−0.890932	−0.445466	−	0.895299i	$-0.646962\pi$
$431$	−18.4962	−0.890932	−0.445466	+	0.895299i	$0.646962\pi$
$432$	0	0
$433$	9.86211	0.473943	0.236971	−	0.971517i	$-0.423845\pi$
$433$	9.86211	0.473943	0.236971	+	0.971517i	$0.423845\pi$
$434$	0	0
$435$	−0.342389	−0.0164163
$436$	0	0
$437$	−7.60498	−0.363795
$438$	0	0
$439$	−30.0086	−1.43223	−0.716117	−	0.697980i	$-0.754082\pi$
$439$	−30.0086	−1.43223	−0.716117	+	0.697980i	$0.754082\pi$
$440$	0	0
$441$	−44.2768	−2.10842
$442$	0	0
$443$	−4.16482	−0.197877	−0.0989384	−	0.995094i	$-0.531545\pi$
$443$	−4.16482	−0.197877	−0.0989384	+	0.995094i	$0.531545\pi$
$444$	0	0
$445$	−26.9616	−1.27810
$446$	0	0
$447$	1.88185	0.0890083
$448$	0	0
$449$	−1.26226	−0.0595697	−0.0297848	−	0.999556i	$-0.509482\pi$
$449$	−1.26226	−0.0595697	−0.0297848	+	0.999556i	$0.509482\pi$
$450$	0	0
$451$	8.82012	0.415323
$452$	0	0
$453$	−12.0807	−0.567601
$454$	0	0
$455$	−1.91098	−0.0895879
$456$	0	0
$457$	−35.2343	−1.64819	−0.824095	−	0.566452i	$-0.808316\pi$
$457$	−35.2343	−1.64819	−0.824095	+	0.566452i	$0.808316\pi$
$458$	0	0
$459$	11.8240	0.551899
$460$	0	0
$461$	31.8895	1.48524	0.742621	−	0.669712i	$-0.233582\pi$
$461$	31.8895	1.48524	0.742621	+	0.669712i	$0.233582\pi$
$462$	0	0
$463$	24.0028	1.11551	0.557753	−	0.830007i	$-0.311664\pi$
$463$	24.0028	1.11551	0.557753	+	0.830007i	$0.311664\pi$
$464$	0	0
$465$	−10.7210	−0.497177
$466$	0	0
$467$	3.44249	0.159299	0.0796497	−	0.996823i	$-0.474620\pi$
$467$	3.44249	0.159299	0.0796497	+	0.996823i	$0.474620\pi$
$468$	0	0
$469$	−38.4988	−1.77771
$470$	0	0
$471$	−12.5054	−0.576218
$472$	0	0
$473$	12.1705	0.559598
$474$	0	0
$475$	−0.453470	−0.0208066
$476$	0	0
$477$	−25.9953	−1.19025
$478$	0	0
$479$	19.1306	0.874102	0.437051	−	0.899437i	$-0.356023\pi$
$479$	19.1306	0.874102	0.437051	+	0.899437i	$0.356023\pi$
$480$	0	0
$481$	0.999578	0.0455768
$482$	0	0
$483$	22.5298	1.02514
$484$	0	0
$485$	14.5573	0.661011
$486$	0	0
$487$	27.6357	1.25229	0.626147	−	0.779705i	$-0.284631\pi$
$487$	27.6357	1.25229	0.626147	+	0.779705i	$0.284631\pi$
$488$	0	0
$489$	6.70227	0.303087
$490$	0	0
$491$	−8.91431	−0.402297	−0.201149	−	0.979561i	$-0.564467\pi$
$491$	−8.91431	−0.402297	−0.201149	+	0.979561i	$0.564467\pi$
$492$	0	0
$493$	0.727048	0.0327446
$494$	0	0
$495$	−31.7119	−1.42534
$496$	0	0
$497$	5.56456	0.249604
$498$	0	0
$499$	37.4955	1.67853	0.839265	−	0.543723i	$-0.182986\pi$
$499$	37.4955	1.67853	0.839265	+	0.543723i	$0.182986\pi$
$500$	0	0
$501$	5.00652	0.223675
$502$	0	0
$503$	−30.7058	−1.36911	−0.684553	−	0.728963i	$-0.740003\pi$
$503$	−30.7058	−1.36911	−0.684553	+	0.728963i	$0.740003\pi$
$504$	0	0
$505$	−4.61896	−0.205541
$506$	0	0
$507$	8.88282	0.394500
$508$	0	0
$509$	−2.52251	−0.111808	−0.0559042	−	0.998436i	$-0.517804\pi$
$509$	−2.52251	−0.111808	−0.0559042	+	0.998436i	$0.517804\pi$
$510$	0	0
$511$	−52.8594	−2.33836
$512$	0	0
$513$	4.33560	0.191421
$514$	0	0
$515$	5.01440	0.220961
$516$	0	0
$517$	56.1575	2.46980
$518$	0	0
$519$	1.52723	0.0670380
$520$	0	0
$521$	−24.9034	−1.09104	−0.545519	−	0.838098i	$-0.683667\pi$
$521$	−24.9034	−1.09104	−0.545519	+	0.838098i	$0.683667\pi$
$522$	0	0
$523$	0.118779	0.00519382	0.00259691	−	0.999997i	$-0.499173\pi$
$523$	0.118779	0.00519382	0.00259691	+	0.999997i	$0.499173\pi$
$524$	0	0
$525$	1.34341	0.0586313
$526$	0	0
$527$	22.7657	0.991689
$528$	0	0
$529$	21.1623	0.920098
$530$	0	0
$531$	20.0525	0.870206
$532$	0	0
$533$	−0.271781	−0.0117721
$534$	0	0
$535$	−3.46242	−0.149694
$536$	0	0
$537$	−17.5563	−0.757610
$538$	0	0
$539$	102.173	4.40090
$540$	0	0
$541$	13.6990	0.588965	0.294483	−	0.955657i	$-0.404853\pi$
$541$	13.6990	0.588965	0.294483	+	0.955657i	$0.404853\pi$
$542$	0	0
$543$	−0.683183	−0.0293182
$544$	0	0
$545$	−6.77263	−0.290107
$546$	0	0
$547$	11.0972	0.474481	0.237241	−	0.971451i	$-0.423757\pi$
$547$	11.0972	0.474481	0.237241	+	0.971451i	$0.423757\pi$
$548$	0	0
$549$	1.95842	0.0835833
$550$	0	0
$551$	0.266591	0.0113572
$552$	0	0
$553$	14.9720	0.636673
$554$	0	0
$555$	8.16404	0.346544
$556$	0	0
$557$	9.91757	0.420221	0.210110	−	0.977678i	$-0.432618\pi$
$557$	9.91757	0.420221	0.210110	+	0.977678i	$0.432618\pi$
$558$	0	0
$559$	−0.375017	−0.0158615
$560$	0	0
$561$	−12.4851	−0.527122
$562$	0	0
$563$	40.0932	1.68973	0.844864	−	0.534982i	$-0.179682\pi$
$563$	40.0932	1.68973	0.844864	+	0.534982i	$0.179682\pi$
$564$	0	0
$565$	−18.0368	−0.758813
$566$	0	0
$567$	24.7323	1.03866
$568$	0	0
$569$	−9.64889	−0.404503	−0.202251	−	0.979334i	$-0.564826\pi$
$569$	−9.64889	−0.404503	−0.202251	+	0.979334i	$0.564826\pi$
$570$	0	0
$571$	−5.11116	−0.213896	−0.106948	−	0.994265i	$-0.534108\pi$
$571$	−5.11116	−0.213896	−0.106948	+	0.994265i	$0.534108\pi$
$572$	0	0
$573$	13.9660	0.583440
$574$	0	0
$575$	2.63331	0.109817
$576$	0	0
$577$	13.2909	0.553309	0.276654	−	0.960969i	$-0.410774\pi$
$577$	13.2909	0.553309	0.276654	+	0.960969i	$0.410774\pi$
$578$	0	0
$579$	−16.5453	−0.687600
$580$	0	0
$581$	68.3860	2.83713
$582$	0	0
$583$	59.9867	2.48439
$584$	0	0
$585$	0.977162	0.0404007
$586$	0	0
$587$	42.8016	1.76661	0.883305	−	0.468798i	$-0.155313\pi$
$587$	42.8016	1.76661	0.883305	+	0.468798i	$0.155313\pi$
$588$	0	0
$589$	8.34763	0.343958
$590$	0	0
$591$	15.6347	0.643124
$592$	0	0
$593$	−33.1865	−1.36281	−0.681403	−	0.731909i	$-0.738630\pi$
$593$	−33.1865	−1.36281	−0.681403	+	0.731909i	$0.738630\pi$
$594$	0	0
$595$	33.1426	1.35872
$596$	0	0
$597$	2.18434	0.0893990
$598$	0	0
$599$	15.0989	0.616924	0.308462	−	0.951237i	$-0.400186\pi$
$599$	15.0989	0.616924	0.308462	+	0.951237i	$0.400186\pi$
$600$	0	0
$601$	−21.3445	−0.870662	−0.435331	−	0.900270i	$-0.643369\pi$
$601$	−21.3445	−0.870662	−0.435331	+	0.900270i	$0.643369\pi$
$602$	0	0
$603$	19.6860	0.801677
$604$	0	0
$605$	49.5761	2.01556
$606$	0	0
$607$	31.1557	1.26457	0.632286	−	0.774735i	$-0.282117\pi$
$607$	31.1557	1.26457	0.632286	+	0.774735i	$0.282117\pi$
$608$	0	0
$609$	−0.789780	−0.0320035
$610$	0	0
$611$	−1.73042	−0.0700054
$612$	0	0
$613$	34.8608	1.40802	0.704008	−	0.710193i	$-0.251392\pi$
$613$	34.8608	1.40802	0.704008	+	0.710193i	$0.251392\pi$
$614$	0	0
$615$	−2.21977	−0.0895097
$616$	0	0
$617$	−32.1336	−1.29365	−0.646825	−	0.762638i	$-0.723904\pi$
$617$	−32.1336	−1.29365	−0.646825	+	0.762638i	$0.723904\pi$
$618$	0	0
$619$	18.2548	0.733723	0.366861	−	0.930276i	$-0.380432\pi$
$619$	18.2548	0.733723	0.366861	+	0.930276i	$0.380432\pi$
$620$	0	0
$621$	−25.1769	−1.01031
$622$	0	0
$623$	−62.1918	−2.49166
$624$	0	0
$625$	−22.8617	−0.914468
$626$	0	0
$627$	−4.57800	−0.182828
$628$	0	0
$629$	−17.3360	−0.691231
$630$	0	0
$631$	−27.8618	−1.10916	−0.554581	−	0.832130i	$-0.687122\pi$
$631$	−27.8618	−1.10916	−0.554581	+	0.832130i	$0.687122\pi$
$632$	0	0
$633$	0.839070	0.0333500
$634$	0	0
$635$	−43.1448	−1.71215
$636$	0	0
$637$	−3.14833	−0.124741
$638$	0	0
$639$	−2.84539	−0.112562
$640$	0	0
$641$	−30.3218	−1.19764	−0.598820	−	0.800884i	$-0.704364\pi$
$641$	−30.3218	−1.19764	−0.598820	+	0.800884i	$0.704364\pi$
$642$	0	0
$643$	−29.6129	−1.16782	−0.583909	−	0.811819i	$-0.698477\pi$
$643$	−29.6129	−1.16782	−0.583909	+	0.811819i	$0.698477\pi$
$644$	0	0
$645$	−3.06295	−0.120604
$646$	0	0
$647$	−2.57974	−0.101420	−0.0507100	−	0.998713i	$-0.516148\pi$
$647$	−2.57974	−0.101420	−0.0507100	+	0.998713i	$0.516148\pi$
$648$	0	0
$649$	−46.2731	−1.81638
$650$	0	0
$651$	−24.7300	−0.969244
$652$	0	0
$653$	−25.6116	−1.00226	−0.501129	−	0.865373i	$-0.667082\pi$
$653$	−25.6116	−1.00226	−0.501129	+	0.865373i	$0.667082\pi$
$654$	0	0
$655$	−41.8844	−1.63656
$656$	0	0
$657$	27.0292	1.05451
$658$	0	0
$659$	1.91152	0.0744621	0.0372311	−	0.999307i	$-0.488146\pi$
$659$	1.91152	0.0744621	0.0372311	+	0.999307i	$0.488146\pi$
$660$	0	0
$661$	29.3213	1.14047	0.570233	−	0.821483i	$-0.306853\pi$
$661$	29.3213	1.14047	0.570233	+	0.821483i	$0.306853\pi$
$662$	0	0
$663$	0.384714	0.0149410
$664$	0	0
$665$	12.1526	0.471258
$666$	0	0
$667$	−1.54810	−0.0599427
$668$	0	0
$669$	8.18513	0.316455
$670$	0	0
$671$	−4.51923	−0.174463
$672$	0	0
$673$	32.3544	1.24717	0.623584	−	0.781756i	$-0.285676\pi$
$673$	32.3544	1.24717	0.623584	+	0.781756i	$0.285676\pi$
$674$	0	0
$675$	−1.50125	−0.0577831
$676$	0	0
$677$	13.4155	0.515598	0.257799	−	0.966199i	$-0.417003\pi$
$677$	13.4155	0.515598	0.257799	+	0.966199i	$0.417003\pi$
$678$	0	0
$679$	33.5788	1.28864
$680$	0	0
$681$	−4.57810	−0.175433
$682$	0	0
$683$	18.8229	0.720237	0.360118	−	0.932907i	$-0.382736\pi$
$683$	18.8229	0.720237	0.360118	+	0.932907i	$0.382736\pi$
$684$	0	0
$685$	23.4485	0.895920
$686$	0	0
$687$	2.52406	0.0962988
$688$	0	0
$689$	−1.84842	−0.0704190
$690$	0	0
$691$	−21.0021	−0.798959	−0.399479	−	0.916742i	$-0.630809\pi$
$691$	−21.0021	−0.798959	−0.399479	+	0.916742i	$0.630809\pi$
$692$	0	0
$693$	−73.1490	−2.77870
$694$	0	0
$695$	−18.5085	−0.702066
$696$	0	0
$697$	4.71359	0.178540
$698$	0	0
$699$	−8.47161	−0.320426
$700$	0	0
$701$	−8.66385	−0.327229	−0.163615	−	0.986524i	$-0.552315\pi$
$701$	−8.66385	−0.327229	−0.163615	+	0.986524i	$0.552315\pi$
$702$	0	0
$703$	−6.35670	−0.239747
$704$	0	0
$705$	−14.1332	−0.532288
$706$	0	0
$707$	−10.6544	−0.400702
$708$	0	0
$709$	5.20369	0.195429	0.0977144	−	0.995215i	$-0.468847\pi$
$709$	5.20369	0.195429	0.0977144	+	0.995215i	$0.468847\pi$
$710$	0	0
$711$	−7.65579	−0.287115
$712$	0	0
$713$	−48.4749	−1.81540
$714$	0	0
$715$	−2.25489	−0.0843282
$716$	0	0
$717$	2.87421	0.107340
$718$	0	0
$719$	32.4074	1.20859	0.604296	−	0.796760i	$-0.293454\pi$
$719$	32.4074	1.20859	0.604296	+	0.796760i	$0.293454\pi$
$720$	0	0
$721$	11.5666	0.430762
$722$	0	0
$723$	−8.58436	−0.319256
$724$	0	0
$725$	−0.0923102	−0.00342832
$726$	0	0
$727$	9.10375	0.337640	0.168820	−	0.985647i	$-0.446004\pi$
$727$	9.10375	0.337640	0.168820	+	0.985647i	$0.446004\pi$
$728$	0	0
$729$	−4.86112	−0.180042
$730$	0	0
$731$	6.50405	0.240561
$732$	0	0
$733$	−5.09445	−0.188168	−0.0940839	−	0.995564i	$-0.529992\pi$
$733$	−5.09445	−0.188168	−0.0940839	+	0.995564i	$0.529992\pi$
$734$	0	0
$735$	−25.7140	−0.948474
$736$	0	0
$737$	−45.4274	−1.67334
$738$	0	0
$739$	33.4122	1.22909	0.614544	−	0.788883i	$-0.289340\pi$
$739$	33.4122	1.22909	0.614544	+	0.788883i	$0.289340\pi$
$740$	0	0
$741$	0.141065	0.00518217
$742$	0	0
$743$	−16.9682	−0.622504	−0.311252	−	0.950327i	$-0.600748\pi$
$743$	−16.9682	−0.622504	−0.311252	+	0.950327i	$0.600748\pi$
$744$	0	0
$745$	−5.89453	−0.215959
$746$	0	0
$747$	−34.9686	−1.27944
$748$	0	0
$749$	−7.98668	−0.291827
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	2.72483	0.0992985
$754$	0	0
$755$	37.8405	1.37716
$756$	0	0
$757$	32.3558	1.17599	0.587995	−	0.808864i	$-0.299917\pi$
$757$	32.3558	1.17599	0.587995	+	0.808864i	$0.299917\pi$
$758$	0	0
$759$	26.5845	0.964957
$760$	0	0
$761$	−29.7123	−1.07707	−0.538535	−	0.842603i	$-0.681022\pi$
$761$	−29.7123	−1.07707	−0.538535	+	0.842603i	$0.681022\pi$
$762$	0	0
$763$	−15.6222	−0.565563
$764$	0	0
$765$	−16.9472	−0.612728
$766$	0	0
$767$	1.42585	0.0514844
$768$	0	0
$769$	0.187541	0.00676290	0.00338145	−	0.999994i	$-0.498924\pi$
$769$	0.187541	0.00676290	0.00338145	+	0.999994i	$0.498924\pi$
$770$	0	0
$771$	−4.81294	−0.173334
$772$	0	0
$773$	−53.6009	−1.92789	−0.963946	−	0.266099i	$-0.914265\pi$
$773$	−53.6009	−1.92789	−0.963946	+	0.266099i	$0.914265\pi$
$774$	0	0
$775$	−2.89046	−0.103829
$776$	0	0
$777$	18.8318	0.675587
$778$	0	0
$779$	1.72836	0.0619249
$780$	0	0
$781$	6.56601	0.234950
$782$	0	0
$783$	0.882572	0.0315405
$784$	0	0
$785$	39.1708	1.39806
$786$	0	0
$787$	−27.5297	−0.981327	−0.490664	−	0.871349i	$-0.663246\pi$
$787$	−27.5297	−0.981327	−0.490664	+	0.871349i	$0.663246\pi$
$788$	0	0
$789$	9.60801	0.342054
$790$	0	0
$791$	−41.6050	−1.47930
$792$	0	0
$793$	0.139255	0.00494508
$794$	0	0
$795$	−15.0969	−0.535432
$796$	0	0
$797$	−16.1756	−0.572970	−0.286485	−	0.958085i	$-0.592487\pi$
$797$	−16.1756	−0.572970	−0.286485	+	0.958085i	$0.592487\pi$
$798$	0	0
$799$	30.0113	1.06172
$800$	0	0
$801$	31.8013	1.12364
$802$	0	0
$803$	−62.3725	−2.20108
$804$	0	0
$805$	−70.5705	−2.48728
$806$	0	0
$807$	−7.01445	−0.246920
$808$	0	0
$809$	40.8674	1.43682	0.718410	−	0.695620i	$-0.244870\pi$
$809$	40.8674	1.43682	0.718410	+	0.695620i	$0.244870\pi$
$810$	0	0
$811$	−1.35801	−0.0476863	−0.0238432	−	0.999716i	$-0.507590\pi$
$811$	−1.35801	−0.0476863	−0.0238432	+	0.999716i	$0.507590\pi$
$812$	0	0
$813$	15.5904	0.546781
$814$	0	0
$815$	−20.9936	−0.735373
$816$	0	0
$817$	2.38488	0.0834363
$818$	0	0
$819$	2.25400	0.0787610
$820$	0	0
$821$	−38.4273	−1.34112	−0.670560	−	0.741855i	$-0.733946\pi$
$821$	−38.4273	−1.34112	−0.670560	+	0.741855i	$0.733946\pi$
$822$	0	0
$823$	−37.2785	−1.29945	−0.649723	−	0.760171i	$-0.725115\pi$
$823$	−37.2785	−1.29945	−0.649723	+	0.760171i	$0.725115\pi$
$824$	0	0
$825$	1.58518	0.0551890
$826$	0	0
$827$	35.6293	1.23895	0.619477	−	0.785015i	$-0.287345\pi$
$827$	35.6293	1.23895	0.619477	+	0.785015i	$0.287345\pi$
$828$	0	0
$829$	−1.40717	−0.0488730	−0.0244365	−	0.999701i	$-0.507779\pi$
$829$	−1.40717	−0.0488730	−0.0244365	+	0.999701i	$0.507779\pi$
$830$	0	0
$831$	−11.5806	−0.401727
$832$	0	0
$833$	54.6025	1.89186
$834$	0	0
$835$	−15.6820	−0.542698
$836$	0	0
$837$	27.6355	0.955223
$838$	0	0
$839$	33.1429	1.14422	0.572110	−	0.820177i	$-0.306125\pi$
$839$	33.1429	1.14422	0.572110	+	0.820177i	$0.306125\pi$
$840$	0	0
$841$	−28.9457	−0.998129
$842$	0	0
$843$	−22.6838	−0.781271
$844$	0	0
$845$	−27.8238	−0.957166
$846$	0	0
$847$	114.356	3.92932
$848$	0	0
$849$	13.2206	0.453731
$850$	0	0
$851$	36.9135	1.26538
$852$	0	0
$853$	−24.1624	−0.827306	−0.413653	−	0.910435i	$-0.635747\pi$
$853$	−24.1624	−0.827306	−0.413653	+	0.910435i	$0.635747\pi$
$854$	0	0
$855$	−6.21415	−0.212519
$856$	0	0
$857$	−11.3547	−0.387869	−0.193935	−	0.981014i	$-0.562125\pi$
$857$	−11.3547	−0.387869	−0.193935	+	0.981014i	$0.562125\pi$
$858$	0	0
$859$	3.48115	0.118775	0.0593877	−	0.998235i	$-0.481085\pi$
$859$	3.48115	0.118775	0.0593877	+	0.998235i	$0.481085\pi$
$860$	0	0
$861$	−5.12029	−0.174499
$862$	0	0
$863$	−31.9669	−1.08816	−0.544082	−	0.839032i	$-0.683122\pi$
$863$	−31.9669	−1.08816	−0.544082	+	0.839032i	$0.683122\pi$
$864$	0	0
$865$	−4.78377	−0.162653
$866$	0	0
$867$	4.97279	0.168885
$868$	0	0
$869$	17.6665	0.599294
$870$	0	0
$871$	1.39979	0.0474300
$872$	0	0
$873$	−17.1703	−0.581126
$874$	0	0
$875$	−57.3047	−1.93725
$876$	0	0
$877$	−2.97280	−0.100384	−0.0501922	−	0.998740i	$-0.515983\pi$
$877$	−2.97280	−0.100384	−0.0501922	+	0.998740i	$0.515983\pi$
$878$	0	0
$879$	−6.50141	−0.219287
$880$	0	0
$881$	16.2874	0.548736	0.274368	−	0.961625i	$-0.411531\pi$
$881$	16.2874	0.548736	0.274368	+	0.961625i	$0.411531\pi$
$882$	0	0
$883$	−52.6411	−1.77151	−0.885757	−	0.464149i	$-0.846360\pi$
$883$	−52.6411	−1.77151	−0.885757	+	0.464149i	$0.846360\pi$
$884$	0	0
$885$	11.6456	0.391462
$886$	0	0
$887$	−7.12621	−0.239275	−0.119637	−	0.992818i	$-0.538173\pi$
$887$	−7.12621	−0.239275	−0.119637	+	0.992818i	$0.538173\pi$
$888$	0	0
$889$	−99.5209	−3.33783
$890$	0	0
$891$	29.1833	0.977679
$892$	0	0
$893$	11.0044	0.368249
$894$	0	0
$895$	54.9918	1.83817
$896$	0	0
$897$	−0.819169	−0.0273513
$898$	0	0
$899$	1.69928	0.0566741
$900$	0	0
$901$	32.0577	1.06800
$902$	0	0
$903$	−7.06523	−0.235116
$904$	0	0
$905$	2.13994	0.0711341
$906$	0	0
$907$	−12.9951	−0.431496	−0.215748	−	0.976449i	$-0.569219\pi$
$907$	−12.9951	−0.431496	−0.215748	+	0.976449i	$0.569219\pi$
$908$	0	0
$909$	5.44807	0.180701
$910$	0	0
$911$	−17.0255	−0.564082	−0.282041	−	0.959402i	$-0.591011\pi$
$911$	−17.0255	−0.564082	−0.282041	+	0.959402i	$0.591011\pi$
$912$	0	0
$913$	80.6934	2.67056
$914$	0	0
$915$	1.13736	0.0376000
$916$	0	0
$917$	−96.6137	−3.19047
$918$	0	0
$919$	−34.2565	−1.13002	−0.565008	−	0.825085i	$-0.691127\pi$
$919$	−34.2565	−1.13002	−0.565008	+	0.825085i	$0.691127\pi$
$920$	0	0
$921$	15.6395	0.515339
$922$	0	0
$923$	−0.202323	−0.00665956
$924$	0	0
$925$	2.20108	0.0723710
$926$	0	0
$927$	−5.91449	−0.194257
$928$	0	0
$929$	−3.68750	−0.120983	−0.0604915	−	0.998169i	$-0.519267\pi$
$929$	−3.68750	−0.120983	−0.0604915	+	0.998169i	$0.519267\pi$
$930$	0	0
$931$	20.0214	0.656176
$932$	0	0
$933$	7.56276	0.247594
$934$	0	0
$935$	39.1073	1.27895
$936$	0	0
$937$	50.1654	1.63883	0.819417	−	0.573198i	$-0.194298\pi$
$937$	50.1654	1.63883	0.819417	+	0.573198i	$0.194298\pi$
$938$	0	0
$939$	−0.944453	−0.0308211
$940$	0	0
$941$	−13.0686	−0.426023	−0.213011	−	0.977050i	$-0.568327\pi$
$941$	−13.0686	−0.426023	−0.213011	+	0.977050i	$0.568327\pi$
$942$	0	0
$943$	−10.0366	−0.326837
$944$	0	0
$945$	40.2322	1.30875
$946$	0	0
$947$	−9.54515	−0.310176	−0.155088	−	0.987901i	$-0.549566\pi$
$947$	−9.54515	−0.310176	−0.155088	+	0.987901i	$0.549566\pi$
$948$	0	0
$949$	1.92193	0.0623885
$950$	0	0
$951$	−16.4772	−0.534311
$952$	0	0
$953$	−54.5055	−1.76561	−0.882803	−	0.469744i	$-0.844346\pi$
$953$	−54.5055	−1.76561	−0.882803	+	0.469744i	$0.844346\pi$
$954$	0	0
$955$	−43.7460	−1.41559
$956$	0	0
$957$	−0.931916	−0.0301246
$958$	0	0
$959$	54.0880	1.74659
$960$	0	0
$961$	22.2086	0.716408
$962$	0	0
$963$	4.08393	0.131603
$964$	0	0
$965$	51.8251	1.66831
$966$	0	0
$967$	4.73060	0.152126	0.0760628	−	0.997103i	$-0.475765\pi$
$967$	4.73060	0.152126	0.0760628	+	0.997103i	$0.475765\pi$
$968$	0	0
$969$	−2.44654	−0.0785942
$970$	0	0
$971$	25.9431	0.832553	0.416277	−	0.909238i	$-0.363335\pi$
$971$	25.9431	0.832553	0.416277	+	0.909238i	$0.363335\pi$
$972$	0	0
$973$	−42.6930	−1.36867
$974$	0	0
$975$	−0.0488455	−0.00156431
$976$	0	0
$977$	−43.4569	−1.39031	−0.695155	−	0.718859i	$-0.744664\pi$
$977$	−43.4569	−1.39031	−0.695155	+	0.718859i	$0.744664\pi$
$978$	0	0
$979$	−73.3844	−2.34538
$980$	0	0
$981$	7.98831	0.255047
$982$	0	0
$983$	−13.7376	−0.438161	−0.219081	−	0.975707i	$-0.570306\pi$
$983$	−13.7376	−0.438161	−0.219081	+	0.975707i	$0.570306\pi$
$984$	0	0
$985$	−48.9726	−1.56040
$986$	0	0
$987$	−32.6007	−1.03769
$988$	0	0
$989$	−13.8490	−0.440374
$990$	0	0
$991$	−45.9927	−1.46101	−0.730504	−	0.682909i	$-0.760715\pi$
$991$	−45.9927	−1.46101	−0.730504	+	0.682909i	$0.760715\pi$
$992$	0	0
$993$	−11.6220	−0.368812
$994$	0	0
$995$	−6.84203	−0.216907
$996$	0	0
$997$	−23.1200	−0.732219	−0.366109	−	0.930572i	$-0.619310\pi$
$997$	−23.1200	−0.732219	−0.366109	+	0.930572i	$0.619310\pi$
$998$	0	0
$999$	−21.0444	−0.665814

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.19	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.19	✓	50	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q-0.685000 q^{3} +2.14563 q^{5} +4.94928 q^{7} -2.53078 q^{9} +O(q^{10})\)	\(q-0.685000 q^{3} +2.14563 q^{5} +4.94928 q^{7} -2.53078 q^{9} +5.84000 q^{11} -0.179952 q^{13} -1.46976 q^{15} +3.12097 q^{17} +1.14439 q^{19} -3.39026 q^{21} -6.64547 q^{23} -0.396256 q^{25} +3.78858 q^{27} +0.232956 q^{29} +7.29442 q^{31} -4.00040 q^{33} +10.6193 q^{35} -5.55468 q^{37} +0.123267 q^{39} +1.51030 q^{41} +2.08398 q^{43} -5.43012 q^{45} +9.61601 q^{47} +17.4954 q^{49} -2.13786 q^{51} +10.2717 q^{53} +12.5305 q^{55} -0.783904 q^{57} -7.92347 q^{59} -0.773842 q^{61} -12.5255 q^{63} -0.386112 q^{65} -7.77866 q^{67} +4.55215 q^{69} +1.12432 q^{71} -10.6802 q^{73} +0.271436 q^{75} +28.9038 q^{77} +3.02508 q^{79} +4.99715 q^{81} +13.8174 q^{83} +6.69646 q^{85} -0.159575 q^{87} -12.5658 q^{89} -0.890634 q^{91} -4.99668 q^{93} +2.45543 q^{95} +6.78459 q^{97} -14.7797 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more