LMFDB - Embedded newform 6008.2.a.e.1.36

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			1.36
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)

$f(q)$	$=$	$q+1.83858 q^{3} +4.41353 q^{5} +3.19108 q^{7} +0.380391 q^{9} +O(q^{10})$	$q+1.83858 q^{3} +4.41353 q^{5} +3.19108 q^{7} +0.380391 q^{9} -5.76845 q^{11} +6.18779 q^{13} +8.11464 q^{15} +2.08881 q^{17} +2.76564 q^{19} +5.86706 q^{21} +0.206321 q^{23} +14.4792 q^{25} -4.81637 q^{27} +3.05541 q^{29} +5.45060 q^{31} -10.6058 q^{33} +14.0839 q^{35} -5.94239 q^{37} +11.3768 q^{39} -4.72190 q^{41} -1.65760 q^{43} +1.67887 q^{45} -6.83514 q^{47} +3.18297 q^{49} +3.84045 q^{51} +8.67166 q^{53} -25.4592 q^{55} +5.08486 q^{57} -10.8182 q^{59} -6.58965 q^{61} +1.21386 q^{63} +27.3100 q^{65} -7.21819 q^{67} +0.379338 q^{69} -16.6430 q^{71} +11.4742 q^{73} +26.6213 q^{75} -18.4076 q^{77} -9.03959 q^{79} -9.99648 q^{81} +6.69813 q^{83} +9.21901 q^{85} +5.61763 q^{87} -7.60669 q^{89} +19.7457 q^{91} +10.0214 q^{93} +12.2062 q^{95} -0.349298 q^{97} -2.19426 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.83858	1.06151	0.530753	−	0.847526i	$-0.321909\pi$
$3$	1.83858	1.06151	0.530753	+	0.847526i	$0.321909\pi$
$4$	0	0
$5$	4.41353	1.97379	0.986895	−	0.161363i	$-0.0515889\pi$
$5$	4.41353	1.97379	0.986895	+	0.161363i	$0.0515889\pi$
$6$	0	0
$7$	3.19108	1.20611	0.603057	−	0.797698i	$-0.293949\pi$
$7$	3.19108	1.20611	0.603057	+	0.797698i	$0.293949\pi$
$8$	0	0
$9$	0.380391	0.126797
$10$	0	0
$11$	−5.76845	−1.73925	−0.869626	−	0.493711i	$-0.835640\pi$
$11$	−5.76845	−1.73925	−0.869626	+	0.493711i	$0.835640\pi$
$12$	0	0
$13$	6.18779	1.71618	0.858092	−	0.513495i	$-0.171650\pi$
$13$	6.18779	1.71618	0.858092	+	0.513495i	$0.171650\pi$
$14$	0	0
$15$	8.11464	2.09519
$16$	0	0
$17$	2.08881	0.506610	0.253305	−	0.967386i	$-0.418482\pi$
$17$	2.08881	0.506610	0.253305	+	0.967386i	$0.418482\pi$
$18$	0	0
$19$	2.76564	0.634481	0.317240	−	0.948345i	$-0.397244\pi$
$19$	2.76564	0.634481	0.317240	+	0.948345i	$0.397244\pi$
$20$	0	0
$21$	5.86706	1.28030
$22$	0	0
$23$	0.206321	0.0430209	0.0215104	−	0.999769i	$-0.493152\pi$
$23$	0.206321	0.0430209	0.0215104	+	0.999769i	$0.493152\pi$
$24$	0	0
$25$	14.4792	2.89585
$26$	0	0
$27$	−4.81637	−0.926911
$28$	0	0
$29$	3.05541	0.567376	0.283688	−	0.958917i	$-0.408442\pi$
$29$	3.05541	0.567376	0.283688	+	0.958917i	$0.408442\pi$
$30$	0	0
$31$	5.45060	0.978957	0.489478	−	0.872015i	$-0.337187\pi$
$31$	5.45060	0.978957	0.489478	+	0.872015i	$0.337187\pi$
$32$	0	0
$33$	−10.6058	−1.84623
$34$	0	0
$35$	14.0839	2.38062
$36$	0	0
$37$	−5.94239	−0.976923	−0.488461	−	0.872586i	$-0.662442\pi$
$37$	−5.94239	−0.976923	−0.488461	+	0.872586i	$0.662442\pi$
$38$	0	0
$39$	11.3768	1.82174
$40$	0	0
$41$	−4.72190	−0.737437	−0.368719	−	0.929541i	$-0.620203\pi$
$41$	−4.72190	−0.737437	−0.368719	+	0.929541i	$0.620203\pi$
$42$	0	0
$43$	−1.65760	−0.252781	−0.126391	−	0.991981i	$-0.540339\pi$
$43$	−1.65760	−0.252781	−0.126391	+	0.991981i	$0.540339\pi$
$44$	0	0
$45$	1.67887	0.250271
$46$	0	0
$47$	−6.83514	−0.997008	−0.498504	−	0.866888i	$-0.666117\pi$
$47$	−6.83514	−0.997008	−0.498504	+	0.866888i	$0.666117\pi$
$48$	0	0
$49$	3.18297	0.454710
$50$	0	0
$51$	3.84045	0.537770
$52$	0	0
$53$	8.67166	1.19114	0.595572	−	0.803302i	$-0.296925\pi$
$53$	8.67166	1.19114	0.595572	+	0.803302i	$0.296925\pi$
$54$	0	0
$55$	−25.4592	−3.43292
$56$	0	0
$57$	5.08486	0.673506
$58$	0	0
$59$	−10.8182	−1.40841	−0.704207	−	0.709995i	$-0.748697\pi$
$59$	−10.8182	−1.40841	−0.704207	+	0.709995i	$0.748697\pi$
$60$	0	0
$61$	−6.58965	−0.843718	−0.421859	−	0.906662i	$-0.638622\pi$
$61$	−6.58965	−0.843718	−0.421859	+	0.906662i	$0.638622\pi$
$62$	0	0
$63$	1.21386	0.152931
$64$	0	0
$65$	27.3100	3.38739
$66$	0	0
$67$	−7.21819	−0.881842	−0.440921	−	0.897546i	$-0.645348\pi$
$67$	−7.21819	−0.881842	−0.440921	+	0.897546i	$0.645348\pi$
$68$	0	0
$69$	0.379338	0.0456669
$70$	0	0
$71$	−16.6430	−1.97516	−0.987582	−	0.157104i	$-0.949784\pi$
$71$	−16.6430	−1.97516	−0.987582	+	0.157104i	$0.949784\pi$
$72$	0	0
$73$	11.4742	1.34295	0.671475	−	0.741028i	$-0.265661\pi$
$73$	11.4742	1.34295	0.671475	+	0.741028i	$0.265661\pi$
$74$	0	0
$75$	26.6213	3.07396
$76$	0	0
$77$	−18.4076	−2.09774
$78$	0	0
$79$	−9.03959	−1.01703	−0.508517	−	0.861052i	$-0.669806\pi$
$79$	−9.03959	−1.01703	−0.508517	+	0.861052i	$0.669806\pi$
$80$	0	0
$81$	−9.99648	−1.11072
$82$	0	0
$83$	6.69813	0.735215	0.367608	−	0.929981i	$-0.380177\pi$
$83$	6.69813	0.735215	0.367608	+	0.929981i	$0.380177\pi$
$84$	0	0
$85$	9.21901	0.999942
$86$	0	0
$87$	5.61763	0.602273
$88$	0	0
$89$	−7.60669	−0.806308	−0.403154	−	0.915132i	$-0.632086\pi$
$89$	−7.60669	−0.806308	−0.403154	+	0.915132i	$0.632086\pi$
$90$	0	0
$91$	19.7457	2.06991
$92$	0	0
$93$	10.0214	1.03917
$94$	0	0
$95$	12.2062	1.25233
$96$	0	0
$97$	−0.349298	−0.0354659	−0.0177329	−	0.999843i	$-0.505645\pi$
$97$	−0.349298	−0.0354659	−0.0177329	+	0.999843i	$0.505645\pi$
$98$	0	0
$99$	−2.19426	−0.220532
$100$	0	0
$101$	2.79592	0.278204	0.139102	−	0.990278i	$-0.455578\pi$
$101$	2.79592	0.278204	0.139102	+	0.990278i	$0.455578\pi$
$102$	0	0
$103$	11.0391	1.08772	0.543859	−	0.839176i	$-0.316962\pi$
$103$	11.0391	1.08772	0.543859	+	0.839176i	$0.316962\pi$
$104$	0	0
$105$	25.8945	2.52704
$106$	0	0
$107$	−3.67417	−0.355196	−0.177598	−	0.984103i	$-0.556833\pi$
$107$	−3.67417	−0.355196	−0.177598	+	0.984103i	$0.556833\pi$
$108$	0	0
$109$	−12.6464	−1.21130	−0.605651	−	0.795730i	$-0.707087\pi$
$109$	−12.6464	−1.21130	−0.605651	+	0.795730i	$0.707087\pi$
$110$	0	0
$111$	−10.9256	−1.03701
$112$	0	0
$113$	−12.5845	−1.18385	−0.591926	−	0.805993i	$-0.701632\pi$
$113$	−12.5845	−1.18385	−0.591926	+	0.805993i	$0.701632\pi$
$114$	0	0
$115$	0.910603	0.0849141
$116$	0	0
$117$	2.35378	0.217607
$118$	0	0
$119$	6.66554	0.611029
$120$	0	0
$121$	22.2750	2.02500
$122$	0	0
$123$	−8.68161	−0.782795
$124$	0	0
$125$	41.8369	3.74201
$126$	0	0
$127$	0.372511	0.0330550	0.0165275	−	0.999863i	$-0.494739\pi$
$127$	0.372511	0.0330550	0.0165275	+	0.999863i	$0.494739\pi$
$128$	0	0
$129$	−3.04763	−0.268329
$130$	0	0
$131$	−4.66689	−0.407748	−0.203874	−	0.978997i	$-0.565353\pi$
$131$	−4.66689	−0.407748	−0.203874	+	0.978997i	$0.565353\pi$
$132$	0	0
$133$	8.82536	0.765256
$134$	0	0
$135$	−21.2572	−1.82953
$136$	0	0
$137$	−4.51917	−0.386099	−0.193049	−	0.981189i	$-0.561838\pi$
$137$	−4.51917	−0.386099	−0.193049	+	0.981189i	$0.561838\pi$
$138$	0	0
$139$	15.4332	1.30902	0.654512	−	0.756051i	$-0.272874\pi$
$139$	15.4332	1.30902	0.654512	+	0.756051i	$0.272874\pi$
$140$	0	0
$141$	−12.5670	−1.05833
$142$	0	0
$143$	−35.6940	−2.98488
$144$	0	0
$145$	13.4851	1.11988
$146$	0	0
$147$	5.85216	0.482678
$148$	0	0
$149$	−18.5931	−1.52321	−0.761603	−	0.648044i	$-0.775587\pi$
$149$	−18.5931	−1.52321	−0.761603	+	0.648044i	$0.775587\pi$
$150$	0	0
$151$	−0.643894	−0.0523994	−0.0261997	−	0.999657i	$-0.508341\pi$
$151$	−0.643894	−0.0523994	−0.0261997	+	0.999657i	$0.508341\pi$
$152$	0	0
$153$	0.794563	0.0642366
$154$	0	0
$155$	24.0564	1.93226
$156$	0	0
$157$	3.85080	0.307327	0.153664	−	0.988123i	$-0.450893\pi$
$157$	3.85080	0.307327	0.153664	+	0.988123i	$0.450893\pi$
$158$	0	0
$159$	15.9436	1.26441
$160$	0	0
$161$	0.658385	0.0518880
$162$	0	0
$163$	19.1544	1.50029	0.750144	−	0.661275i	$-0.229984\pi$
$163$	19.1544	1.50029	0.750144	+	0.661275i	$0.229984\pi$
$164$	0	0
$165$	−46.8089	−3.64407
$166$	0	0
$167$	8.46701	0.655197	0.327598	−	0.944817i	$-0.393761\pi$
$167$	8.46701	0.655197	0.327598	+	0.944817i	$0.393761\pi$
$168$	0	0
$169$	25.2888	1.94529
$170$	0	0
$171$	1.05202	0.0804502
$172$	0	0
$173$	22.3964	1.70277	0.851384	−	0.524543i	$-0.175764\pi$
$173$	22.3964	1.70277	0.851384	+	0.524543i	$0.175764\pi$
$174$	0	0
$175$	46.2044	3.49272
$176$	0	0
$177$	−19.8902	−1.49504
$178$	0	0
$179$	−0.210884	−0.0157622	−0.00788110	−	0.999969i	$-0.502509\pi$
$179$	−0.210884	−0.0157622	−0.00788110	+	0.999969i	$0.502509\pi$
$180$	0	0
$181$	−3.71607	−0.276214	−0.138107	−	0.990417i	$-0.544102\pi$
$181$	−3.71607	−0.276214	−0.138107	+	0.990417i	$0.544102\pi$
$182$	0	0
$183$	−12.1156	−0.895612
$184$	0	0
$185$	−26.2269	−1.92824
$186$	0	0
$187$	−12.0492	−0.881123
$188$	0	0
$189$	−15.3694	−1.11796
$190$	0	0
$191$	6.88090	0.497884	0.248942	−	0.968518i	$-0.419917\pi$
$191$	6.88090	0.497884	0.248942	+	0.968518i	$0.419917\pi$
$192$	0	0
$193$	14.4359	1.03912	0.519558	−	0.854435i	$-0.326097\pi$
$193$	14.4359	1.03912	0.519558	+	0.854435i	$0.326097\pi$
$194$	0	0
$195$	50.2117	3.59574
$196$	0	0
$197$	−11.1143	−0.791860	−0.395930	−	0.918281i	$-0.629578\pi$
$197$	−11.1143	−0.791860	−0.395930	+	0.918281i	$0.629578\pi$
$198$	0	0
$199$	−19.9596	−1.41490	−0.707450	−	0.706764i	$-0.750154\pi$
$199$	−19.9596	−1.41490	−0.707450	+	0.706764i	$0.750154\pi$
$200$	0	0
$201$	−13.2712	−0.936081
$202$	0	0
$203$	9.75005	0.684320
$204$	0	0
$205$	−20.8403	−1.45555
$206$	0	0
$207$	0.0784825	0.00545491
$208$	0	0
$209$	−15.9534	−1.10352
$210$	0	0
$211$	20.1947	1.39026	0.695131	−	0.718883i	$-0.255346\pi$
$211$	20.1947	1.39026	0.695131	+	0.718883i	$0.255346\pi$
$212$	0	0
$213$	−30.5996	−2.09665
$214$	0	0
$215$	−7.31585	−0.498937
$216$	0	0
$217$	17.3933	1.18073
$218$	0	0
$219$	21.0962	1.42555
$220$	0	0
$221$	12.9251	0.869437
$222$	0	0
$223$	0.171091	0.0114571	0.00572856	−	0.999984i	$-0.498177\pi$
$223$	0.171091	0.0114571	0.00572856	+	0.999984i	$0.498177\pi$
$224$	0	0
$225$	5.50777	0.367185
$226$	0	0
$227$	11.5019	0.763406	0.381703	−	0.924285i	$-0.375338\pi$
$227$	11.5019	0.763406	0.381703	+	0.924285i	$0.375338\pi$
$228$	0	0
$229$	−6.76731	−0.447196	−0.223598	−	0.974681i	$-0.571780\pi$
$229$	−6.76731	−0.447196	−0.223598	+	0.974681i	$0.571780\pi$
$230$	0	0
$231$	−33.8438	−2.22676
$232$	0	0
$233$	5.74422	0.376316	0.188158	−	0.982139i	$-0.439748\pi$
$233$	5.74422	0.376316	0.188158	+	0.982139i	$0.439748\pi$
$234$	0	0
$235$	−30.1671	−1.96788
$236$	0	0
$237$	−16.6200	−1.07959
$238$	0	0
$239$	−20.0459	−1.29666	−0.648330	−	0.761360i	$-0.724532\pi$
$239$	−20.0459	−1.29666	−0.648330	+	0.761360i	$0.724532\pi$
$240$	0	0
$241$	18.4927	1.19122	0.595611	−	0.803273i	$-0.296910\pi$
$241$	18.4927	1.19122	0.595611	+	0.803273i	$0.296910\pi$
$242$	0	0
$243$	−3.93024	−0.252125
$244$	0	0
$245$	14.0481	0.897502
$246$	0	0
$247$	17.1132	1.08889
$248$	0	0
$249$	12.3151	0.780436
$250$	0	0
$251$	−11.6549	−0.735652	−0.367826	−	0.929895i	$-0.619898\pi$
$251$	−11.6549	−0.735652	−0.367826	+	0.929895i	$0.619898\pi$
$252$	0	0
$253$	−1.19015	−0.0748241
$254$	0	0
$255$	16.9499	1.06145
$256$	0	0
$257$	−14.4909	−0.903915	−0.451957	−	0.892039i	$-0.649274\pi$
$257$	−14.4909	−0.903915	−0.451957	+	0.892039i	$0.649274\pi$
$258$	0	0
$259$	−18.9626	−1.17828
$260$	0	0
$261$	1.16225	0.0719415
$262$	0	0
$263$	−24.2691	−1.49650	−0.748248	−	0.663419i	$-0.769105\pi$
$263$	−24.2691	−1.49650	−0.748248	+	0.663419i	$0.769105\pi$
$264$	0	0
$265$	38.2726	2.35107
$266$	0	0
$267$	−13.9855	−0.855902
$268$	0	0
$269$	24.1915	1.47498	0.737492	−	0.675356i	$-0.236010\pi$
$269$	24.1915	1.47498	0.737492	+	0.675356i	$0.236010\pi$
$270$	0	0
$271$	3.85814	0.234365	0.117183	−	0.993110i	$-0.462614\pi$
$271$	3.85814	0.234365	0.117183	+	0.993110i	$0.462614\pi$
$272$	0	0
$273$	36.3042	2.19723
$274$	0	0
$275$	−83.5228	−5.03661
$276$	0	0
$277$	−9.92750	−0.596486	−0.298243	−	0.954490i	$-0.596401\pi$
$277$	−9.92750	−0.596486	−0.298243	+	0.954490i	$0.596401\pi$
$278$	0	0
$279$	2.07336	0.124129
$280$	0	0
$281$	3.73454	0.222784	0.111392	−	0.993777i	$-0.464469\pi$
$281$	3.73454	0.222784	0.111392	+	0.993777i	$0.464469\pi$
$282$	0	0
$283$	15.0927	0.897167	0.448584	−	0.893741i	$-0.351929\pi$
$283$	15.0927	0.897167	0.448584	+	0.893741i	$0.351929\pi$
$284$	0	0
$285$	22.4422	1.32936
$286$	0	0
$287$	−15.0680	−0.889433
$288$	0	0
$289$	−12.6369	−0.743346
$290$	0	0
$291$	−0.642214	−0.0376473
$292$	0	0
$293$	−9.71078	−0.567310	−0.283655	−	0.958926i	$-0.591547\pi$
$293$	−9.71078	−0.567310	−0.283655	+	0.958926i	$0.591547\pi$
$294$	0	0
$295$	−47.7466	−2.77991
$296$	0	0
$297$	27.7830	1.61213
$298$	0	0
$299$	1.27667	0.0738317
$300$	0	0
$301$	−5.28952	−0.304883
$302$	0	0
$303$	5.14052	0.295315
$304$	0	0
$305$	−29.0836	−1.66532
$306$	0	0
$307$	−19.8412	−1.13240	−0.566198	−	0.824269i	$-0.691586\pi$
$307$	−19.8412	−1.13240	−0.566198	+	0.824269i	$0.691586\pi$
$308$	0	0
$309$	20.2964	1.15462
$310$	0	0
$311$	24.3097	1.37848	0.689238	−	0.724535i	$-0.257945\pi$
$311$	24.3097	1.37848	0.689238	+	0.724535i	$0.257945\pi$
$312$	0	0
$313$	−32.1551	−1.81751	−0.908757	−	0.417327i	$-0.862967\pi$
$313$	−32.1551	−1.81751	−0.908757	+	0.417327i	$0.862967\pi$
$314$	0	0
$315$	5.35739	0.301855
$316$	0	0
$317$	−22.4920	−1.26328	−0.631639	−	0.775263i	$-0.717617\pi$
$317$	−22.4920	−1.26328	−0.631639	+	0.775263i	$0.717617\pi$
$318$	0	0
$319$	−17.6250	−0.986810
$320$	0	0
$321$	−6.75528	−0.377043
$322$	0	0
$323$	5.77688	0.321434
$324$	0	0
$325$	89.5945	4.96981
$326$	0	0
$327$	−23.2514	−1.28581
$328$	0	0
$329$	−21.8115	−1.20250
$330$	0	0
$331$	15.1564	0.833070	0.416535	−	0.909120i	$-0.363244\pi$
$331$	15.1564	0.833070	0.416535	+	0.909120i	$0.363244\pi$
$332$	0	0
$333$	−2.26043	−0.123871
$334$	0	0
$335$	−31.8577	−1.74057
$336$	0	0
$337$	−7.26897	−0.395966	−0.197983	−	0.980205i	$-0.563439\pi$
$337$	−7.26897	−0.395966	−0.197983	+	0.980205i	$0.563439\pi$
$338$	0	0
$339$	−23.1377	−1.25667
$340$	0	0
$341$	−31.4415	−1.70265
$342$	0	0
$343$	−12.1804	−0.657682
$344$	0	0
$345$	1.67422	0.0901370
$346$	0	0
$347$	11.7570	0.631149	0.315575	−	0.948901i	$-0.397803\pi$
$347$	11.7570	0.631149	0.315575	+	0.948901i	$0.397803\pi$
$348$	0	0
$349$	25.0460	1.34068	0.670341	−	0.742054i	$-0.266148\pi$
$349$	25.0460	1.34068	0.670341	+	0.742054i	$0.266148\pi$
$350$	0	0
$351$	−29.8027	−1.59075
$352$	0	0
$353$	22.8400	1.21565	0.607827	−	0.794070i	$-0.292042\pi$
$353$	22.8400	1.21565	0.607827	+	0.794070i	$0.292042\pi$
$354$	0	0
$355$	−73.4545	−3.89856
$356$	0	0
$357$	12.2552	0.648612
$358$	0	0
$359$	1.60511	0.0847142	0.0423571	−	0.999103i	$-0.486513\pi$
$359$	1.60511	0.0847142	0.0423571	+	0.999103i	$0.486513\pi$
$360$	0	0
$361$	−11.3512	−0.597434
$362$	0	0
$363$	40.9544	2.14955
$364$	0	0
$365$	50.6416	2.65070
$366$	0	0
$367$	15.9548	0.832832	0.416416	−	0.909174i	$-0.363286\pi$
$367$	15.9548	0.832832	0.416416	+	0.909174i	$0.363286\pi$
$368$	0	0
$369$	−1.79617	−0.0935048
$370$	0	0
$371$	27.6719	1.43666
$372$	0	0
$373$	−20.9920	−1.08693	−0.543463	−	0.839433i	$-0.682887\pi$
$373$	−20.9920	−1.08693	−0.543463	+	0.839433i	$0.682887\pi$
$374$	0	0
$375$	76.9207	3.97217
$376$	0	0
$377$	18.9062	0.973721
$378$	0	0
$379$	−30.8432	−1.58431	−0.792154	−	0.610321i	$-0.791040\pi$
$379$	−30.8432	−1.58431	−0.792154	+	0.610321i	$0.791040\pi$
$380$	0	0
$381$	0.684892	0.0350881
$382$	0	0
$383$	32.8983	1.68102	0.840511	−	0.541794i	$-0.182255\pi$
$383$	32.8983	1.68102	0.840511	+	0.541794i	$0.182255\pi$
$384$	0	0
$385$	−81.2423	−4.14049
$386$	0	0
$387$	−0.630534	−0.0320519
$388$	0	0
$389$	−32.6721	−1.65654	−0.828270	−	0.560329i	$-0.810675\pi$
$389$	−32.6721	−1.65654	−0.828270	+	0.560329i	$0.810675\pi$
$390$	0	0
$391$	0.430964	0.0217948
$392$	0	0
$393$	−8.58047	−0.432827
$394$	0	0
$395$	−39.8965	−2.00741
$396$	0	0
$397$	19.0413	0.955654	0.477827	−	0.878454i	$-0.341425\pi$
$397$	19.0413	0.955654	0.477827	+	0.878454i	$0.341425\pi$
$398$	0	0
$399$	16.2262	0.812324
$400$	0	0
$401$	−27.5683	−1.37670	−0.688348	−	0.725380i	$-0.741664\pi$
$401$	−27.5683	−1.37670	−0.688348	+	0.725380i	$0.741664\pi$
$402$	0	0
$403$	33.7272	1.68007
$404$	0	0
$405$	−44.1197	−2.19233
$406$	0	0
$407$	34.2784	1.69912
$408$	0	0
$409$	18.0509	0.892560	0.446280	−	0.894893i	$-0.352749\pi$
$409$	18.0509	0.892560	0.446280	+	0.894893i	$0.352749\pi$
$410$	0	0
$411$	−8.30887	−0.409847
$412$	0	0
$413$	−34.5218	−1.69871
$414$	0	0
$415$	29.5624	1.45116
$416$	0	0
$417$	28.3752	1.38954
$418$	0	0
$419$	12.1865	0.595349	0.297675	−	0.954667i	$-0.403789\pi$
$419$	12.1865	0.595349	0.297675	+	0.954667i	$0.403789\pi$
$420$	0	0
$421$	14.6178	0.712427	0.356214	−	0.934405i	$-0.384068\pi$
$421$	14.6178	0.712427	0.356214	+	0.934405i	$0.384068\pi$
$422$	0	0
$423$	−2.60002	−0.126417
$424$	0	0
$425$	30.2444	1.46707
$426$	0	0
$427$	−21.0281	−1.01762
$428$	0	0
$429$	−65.6263	−3.16847
$430$	0	0
$431$	32.5109	1.56599	0.782997	−	0.622026i	$-0.213690\pi$
$431$	32.5109	1.56599	0.782997	+	0.622026i	$0.213690\pi$
$432$	0	0
$433$	−16.1939	−0.778229	−0.389115	−	0.921189i	$-0.627219\pi$
$433$	−16.1939	−0.778229	−0.389115	+	0.921189i	$0.627219\pi$
$434$	0	0
$435$	24.7936	1.18876
$436$	0	0
$437$	0.570608	0.0272959
$438$	0	0
$439$	−10.7643	−0.513750	−0.256875	−	0.966445i	$-0.582693\pi$
$439$	−10.7643	−0.513750	−0.256875	+	0.966445i	$0.582693\pi$
$440$	0	0
$441$	1.21077	0.0576558
$442$	0	0
$443$	−35.1136	−1.66830	−0.834148	−	0.551541i	$-0.814040\pi$
$443$	−35.1136	−1.66830	−0.834148	+	0.551541i	$0.814040\pi$
$444$	0	0
$445$	−33.5724	−1.59148
$446$	0	0
$447$	−34.1850	−1.61689
$448$	0	0
$449$	28.0647	1.32445	0.662227	−	0.749303i	$-0.269612\pi$
$449$	28.0647	1.32445	0.662227	+	0.749303i	$0.269612\pi$
$450$	0	0
$451$	27.2380	1.28259
$452$	0	0
$453$	−1.18385	−0.0556223
$454$	0	0
$455$	87.1483	4.08558
$456$	0	0
$457$	30.1216	1.40903	0.704515	−	0.709689i	$-0.251164\pi$
$457$	30.1216	1.40903	0.704515	+	0.709689i	$0.251164\pi$
$458$	0	0
$459$	−10.0605	−0.469583
$460$	0	0
$461$	25.2291	1.17504	0.587518	−	0.809211i	$-0.300105\pi$
$461$	25.2291	1.17504	0.587518	+	0.809211i	$0.300105\pi$
$462$	0	0
$463$	4.67366	0.217203	0.108602	−	0.994085i	$-0.465363\pi$
$463$	4.67366	0.217203	0.108602	+	0.994085i	$0.465363\pi$
$464$	0	0
$465$	44.2297	2.05110
$466$	0	0
$467$	−6.59519	−0.305189	−0.152595	−	0.988289i	$-0.548763\pi$
$467$	−6.59519	−0.305189	−0.152595	+	0.988289i	$0.548763\pi$
$468$	0	0
$469$	−23.0338	−1.06360
$470$	0	0
$471$	7.08002	0.326230
$472$	0	0
$473$	9.56176	0.439650
$474$	0	0
$475$	40.0443	1.83736
$476$	0	0
$477$	3.29862	0.151033
$478$	0	0
$479$	−9.33595	−0.426571	−0.213285	−	0.976990i	$-0.568416\pi$
$479$	−9.33595	−0.426571	−0.213285	+	0.976990i	$0.568416\pi$
$480$	0	0
$481$	−36.7703	−1.67658
$482$	0	0
$483$	1.21050	0.0550795
$484$	0	0
$485$	−1.54164	−0.0700022
$486$	0	0
$487$	−15.7194	−0.712316	−0.356158	−	0.934426i	$-0.615913\pi$
$487$	−15.7194	−0.712316	−0.356158	+	0.934426i	$0.615913\pi$
$488$	0	0
$489$	35.2170	1.59257
$490$	0	0
$491$	−15.8117	−0.713571	−0.356786	−	0.934186i	$-0.616127\pi$
$491$	−15.8117	−0.713571	−0.356786	+	0.934186i	$0.616127\pi$
$492$	0	0
$493$	6.38217	0.287438
$494$	0	0
$495$	−9.68445	−0.435284
$496$	0	0
$497$	−53.1092	−2.38227
$498$	0	0
$499$	−15.3430	−0.686847	−0.343424	−	0.939181i	$-0.611587\pi$
$499$	−15.3430	−0.686847	−0.343424	+	0.939181i	$0.611587\pi$
$500$	0	0
$501$	15.5673	0.695496
$502$	0	0
$503$	−13.8434	−0.617248	−0.308624	−	0.951184i	$-0.599868\pi$
$503$	−13.8434	−0.617248	−0.308624	+	0.951184i	$0.599868\pi$
$504$	0	0
$505$	12.3399	0.549116
$506$	0	0
$507$	46.4955	2.06494
$508$	0	0
$509$	21.7069	0.962143	0.481072	−	0.876681i	$-0.340248\pi$
$509$	21.7069	0.962143	0.481072	+	0.876681i	$0.340248\pi$
$510$	0	0
$511$	36.6149	1.61975
$512$	0	0
$513$	−13.3203	−0.588107
$514$	0	0
$515$	48.7216	2.14693
$516$	0	0
$517$	39.4281	1.73405
$518$	0	0
$519$	41.1777	1.80750
$520$	0	0
$521$	21.0525	0.922327	0.461163	−	0.887315i	$-0.347432\pi$
$521$	21.0525	0.922327	0.461163	+	0.887315i	$0.347432\pi$
$522$	0	0
$523$	−22.5116	−0.984362	−0.492181	−	0.870493i	$-0.663800\pi$
$523$	−22.5116	−0.984362	−0.492181	+	0.870493i	$0.663800\pi$
$524$	0	0
$525$	84.9506	3.70755
$526$	0	0
$527$	11.3853	0.495950
$528$	0	0
$529$	−22.9574	−0.998149
$530$	0	0
$531$	−4.11515	−0.178582
$532$	0	0
$533$	−29.2181	−1.26558
$534$	0	0
$535$	−16.2161	−0.701082
$536$	0	0
$537$	−0.387728	−0.0167317
$538$	0	0
$539$	−18.3608	−0.790855
$540$	0	0
$541$	36.3683	1.56360	0.781798	−	0.623531i	$-0.214303\pi$
$541$	36.3683	1.56360	0.781798	+	0.623531i	$0.214303\pi$
$542$	0	0
$543$	−6.83232	−0.293203
$544$	0	0
$545$	−55.8151	−2.39086
$546$	0	0
$547$	21.5180	0.920042	0.460021	−	0.887908i	$-0.347842\pi$
$547$	21.5180	0.920042	0.460021	+	0.887908i	$0.347842\pi$
$548$	0	0
$549$	−2.50664	−0.106981
$550$	0	0
$551$	8.45016	0.359989
$552$	0	0
$553$	−28.8460	−1.22666
$554$	0	0
$555$	−48.2204	−2.04684
$556$	0	0
$557$	−40.1886	−1.70285	−0.851423	−	0.524480i	$-0.824260\pi$
$557$	−40.1886	−1.70285	−0.851423	+	0.524480i	$0.824260\pi$
$558$	0	0
$559$	−10.2569	−0.433819
$560$	0	0
$561$	−22.1534	−0.935318
$562$	0	0
$563$	18.0394	0.760269	0.380135	−	0.924931i	$-0.375878\pi$
$563$	18.0394	0.760269	0.380135	+	0.924931i	$0.375878\pi$
$564$	0	0
$565$	−55.5421	−2.33667
$566$	0	0
$567$	−31.8995	−1.33965
$568$	0	0
$569$	34.7862	1.45832	0.729158	−	0.684346i	$-0.239912\pi$
$569$	34.7862	1.45832	0.729158	+	0.684346i	$0.239912\pi$
$570$	0	0
$571$	−37.7914	−1.58152	−0.790760	−	0.612127i	$-0.790314\pi$
$571$	−37.7914	−1.58152	−0.790760	+	0.612127i	$0.790314\pi$
$572$	0	0
$573$	12.6511	0.528508
$574$	0	0
$575$	2.98737	0.124582
$576$	0	0
$577$	−23.5036	−0.978468	−0.489234	−	0.872152i	$-0.662724\pi$
$577$	−23.5036	−0.978468	−0.489234	+	0.872152i	$0.662724\pi$
$578$	0	0
$579$	26.5416	1.10303
$580$	0	0
$581$	21.3742	0.886753
$582$	0	0
$583$	−50.0220	−2.07170
$584$	0	0
$585$	10.3885	0.429510
$586$	0	0
$587$	24.9338	1.02913	0.514564	−	0.857452i	$-0.327954\pi$
$587$	24.9338	1.02913	0.514564	+	0.857452i	$0.327954\pi$
$588$	0	0
$589$	15.0744	0.621129
$590$	0	0
$591$	−20.4345	−0.840564
$592$	0	0
$593$	−9.20816	−0.378134	−0.189067	−	0.981964i	$-0.560546\pi$
$593$	−9.20816	−0.378134	−0.189067	+	0.981964i	$0.560546\pi$
$594$	0	0
$595$	29.4186	1.20604
$596$	0	0
$597$	−36.6974	−1.50193
$598$	0	0
$599$	45.7527	1.86941	0.934703	−	0.355430i	$-0.115665\pi$
$599$	45.7527	1.86941	0.934703	+	0.355430i	$0.115665\pi$
$600$	0	0
$601$	12.2404	0.499298	0.249649	−	0.968336i	$-0.419685\pi$
$601$	12.2404	0.499298	0.249649	+	0.968336i	$0.419685\pi$
$602$	0	0
$603$	−2.74573	−0.111815
$604$	0	0
$605$	98.3113	3.99692
$606$	0	0
$607$	30.9651	1.25684	0.628418	−	0.777876i	$-0.283703\pi$
$607$	30.9651	1.25684	0.628418	+	0.777876i	$0.283703\pi$
$608$	0	0
$609$	17.9263	0.726410
$610$	0	0
$611$	−42.2944	−1.71105
$612$	0	0
$613$	11.7066	0.472825	0.236412	−	0.971653i	$-0.424028\pi$
$613$	11.7066	0.472825	0.236412	+	0.971653i	$0.424028\pi$
$614$	0	0
$615$	−38.3166	−1.54507
$616$	0	0
$617$	34.7155	1.39759	0.698797	−	0.715320i	$-0.253719\pi$
$617$	34.7155	1.39759	0.698797	+	0.715320i	$0.253719\pi$
$618$	0	0
$619$	−32.9516	−1.32444	−0.662219	−	0.749311i	$-0.730385\pi$
$619$	−32.9516	−1.32444	−0.662219	+	0.749311i	$0.730385\pi$
$620$	0	0
$621$	−0.993717	−0.0398765
$622$	0	0
$623$	−24.2735	−0.972499
$624$	0	0
$625$	112.252	4.49009
$626$	0	0
$627$	−29.3317	−1.17140
$628$	0	0
$629$	−12.4125	−0.494919
$630$	0	0
$631$	−9.20156	−0.366308	−0.183154	−	0.983084i	$-0.558631\pi$
$631$	−9.20156	−0.366308	−0.183154	+	0.983084i	$0.558631\pi$
$632$	0	0
$633$	37.1297	1.47577
$634$	0	0
$635$	1.64409	0.0652436
$636$	0	0
$637$	19.6956	0.780366
$638$	0	0
$639$	−6.33085	−0.250445
$640$	0	0
$641$	−2.14001	−0.0845253	−0.0422627	−	0.999107i	$-0.513457\pi$
$641$	−2.14001	−0.0845253	−0.0422627	+	0.999107i	$0.513457\pi$
$642$	0	0
$643$	4.33293	0.170874	0.0854370	−	0.996344i	$-0.472771\pi$
$643$	4.33293	0.170874	0.0854370	+	0.996344i	$0.472771\pi$
$644$	0	0
$645$	−13.4508	−0.529625
$646$	0	0
$647$	20.7194	0.814562	0.407281	−	0.913303i	$-0.366477\pi$
$647$	20.7194	0.814562	0.407281	+	0.913303i	$0.366477\pi$
$648$	0	0
$649$	62.4044	2.44959
$650$	0	0
$651$	31.9790	1.25336
$652$	0	0
$653$	−2.77847	−0.108730	−0.0543649	−	0.998521i	$-0.517313\pi$
$653$	−2.77847	−0.108730	−0.0543649	+	0.998521i	$0.517313\pi$
$654$	0	0
$655$	−20.5975	−0.804809
$656$	0	0
$657$	4.36467	0.170282
$658$	0	0
$659$	13.1545	0.512427	0.256213	−	0.966620i	$-0.417525\pi$
$659$	13.1545	0.512427	0.256213	+	0.966620i	$0.417525\pi$
$660$	0	0
$661$	6.97313	0.271223	0.135612	−	0.990762i	$-0.456700\pi$
$661$	6.97313	0.271223	0.135612	+	0.990762i	$0.456700\pi$
$662$	0	0
$663$	23.7639	0.922913
$664$	0	0
$665$	38.9510	1.51045
$666$	0	0
$667$	0.630395	0.0244090
$668$	0	0
$669$	0.314566	0.0121618
$670$	0	0
$671$	38.0120	1.46744
$672$	0	0
$673$	40.1282	1.54683	0.773415	−	0.633900i	$-0.218547\pi$
$673$	40.1282	1.54683	0.773415	+	0.633900i	$0.218547\pi$
$674$	0	0
$675$	−69.7374	−2.68419
$676$	0	0
$677$	−35.2764	−1.35578	−0.677892	−	0.735162i	$-0.737106\pi$
$677$	−35.2764	−1.35578	−0.677892	+	0.735162i	$0.737106\pi$
$678$	0	0
$679$	−1.11464	−0.0427759
$680$	0	0
$681$	21.1472	0.810361
$682$	0	0
$683$	35.4466	1.35633	0.678164	−	0.734910i	$-0.262776\pi$
$683$	35.4466	1.35633	0.678164	+	0.734910i	$0.262776\pi$
$684$	0	0
$685$	−19.9455	−0.762078
$686$	0	0
$687$	−12.4423	−0.474702
$688$	0	0
$689$	53.6584	2.04422
$690$	0	0
$691$	−4.21985	−0.160531	−0.0802653	−	0.996774i	$-0.525577\pi$
$691$	−4.21985	−0.160531	−0.0802653	+	0.996774i	$0.525577\pi$
$692$	0	0
$693$	−7.00207	−0.265986
$694$	0	0
$695$	68.1147	2.58374
$696$	0	0
$697$	−9.86314	−0.373593
$698$	0	0
$699$	10.5612	0.399462
$700$	0	0
$701$	−13.8064	−0.521462	−0.260731	−	0.965411i	$-0.583964\pi$
$701$	−13.8064	−0.521462	−0.260731	+	0.965411i	$0.583964\pi$
$702$	0	0
$703$	−16.4345	−0.619839
$704$	0	0
$705$	−55.4647	−2.08892
$706$	0	0
$707$	8.92198	0.335546
$708$	0	0
$709$	15.0247	0.564263	0.282132	−	0.959376i	$-0.408958\pi$
$709$	15.0247	0.564263	0.282132	+	0.959376i	$0.408958\pi$
$710$	0	0
$711$	−3.43858	−0.128957
$712$	0	0
$713$	1.12457	0.0421156
$714$	0	0
$715$	−157.536	−5.89152
$716$	0	0
$717$	−36.8560	−1.37641
$718$	0	0
$719$	−25.3960	−0.947109	−0.473555	−	0.880764i	$-0.657029\pi$
$719$	−25.3960	−0.947109	−0.473555	+	0.880764i	$0.657029\pi$
$720$	0	0
$721$	35.2267	1.31191
$722$	0	0
$723$	34.0005	1.26449
$724$	0	0
$725$	44.2400	1.64303
$726$	0	0
$727$	−31.9476	−1.18487	−0.592436	−	0.805618i	$-0.701834\pi$
$727$	−31.9476	−1.18487	−0.592436	+	0.805618i	$0.701834\pi$
$728$	0	0
$729$	22.7633	0.843087
$730$	0	0
$731$	−3.46240	−0.128061
$732$	0	0
$733$	8.98593	0.331903	0.165952	−	0.986134i	$-0.446930\pi$
$733$	8.98593	0.331903	0.165952	+	0.986134i	$0.446930\pi$
$734$	0	0
$735$	25.8287	0.952705
$736$	0	0
$737$	41.6377	1.53375
$738$	0	0
$739$	−30.3398	−1.11607	−0.558033	−	0.829819i	$-0.688444\pi$
$739$	−30.3398	−1.11607	−0.558033	+	0.829819i	$0.688444\pi$
$740$	0	0
$741$	31.4640	1.15586
$742$	0	0
$743$	−47.5979	−1.74620	−0.873098	−	0.487545i	$-0.837893\pi$
$743$	−47.5979	−1.74620	−0.873098	+	0.487545i	$0.837893\pi$
$744$	0	0
$745$	−82.0612	−3.00649
$746$	0	0
$747$	2.54791	0.0932230
$748$	0	0
$749$	−11.7246	−0.428407
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−21.4286	−0.780900
$754$	0	0
$755$	−2.84185	−0.103425
$756$	0	0
$757$	−16.9916	−0.617571	−0.308786	−	0.951132i	$-0.599923\pi$
$757$	−16.9916	−0.617571	−0.308786	+	0.951132i	$0.599923\pi$
$758$	0	0
$759$	−2.18819	−0.0794263
$760$	0	0
$761$	37.1372	1.34622	0.673112	−	0.739541i	$-0.264957\pi$
$761$	37.1372	1.34622	0.673112	+	0.739541i	$0.264957\pi$
$762$	0	0
$763$	−40.3555	−1.46097
$764$	0	0
$765$	3.50683	0.126790
$766$	0	0
$767$	−66.9409	−2.41710
$768$	0	0
$769$	−19.6996	−0.710384	−0.355192	−	0.934793i	$-0.615585\pi$
$769$	−19.6996	−0.710384	−0.355192	+	0.934793i	$0.615585\pi$
$770$	0	0
$771$	−26.6427	−0.959512
$772$	0	0
$773$	32.4048	1.16552	0.582760	−	0.812644i	$-0.301973\pi$
$773$	32.4048	1.16552	0.582760	+	0.812644i	$0.301973\pi$
$774$	0	0
$775$	78.9206	2.83491
$776$	0	0
$777$	−34.8644	−1.25075
$778$	0	0
$779$	−13.0591	−0.467890
$780$	0	0
$781$	96.0044	3.43531
$782$	0	0
$783$	−14.7160	−0.525907
$784$	0	0
$785$	16.9956	0.606600
$786$	0	0
$787$	−5.04972	−0.180003	−0.0900016	−	0.995942i	$-0.528687\pi$
$787$	−5.04972	−0.180003	−0.0900016	+	0.995942i	$0.528687\pi$
$788$	0	0
$789$	−44.6208	−1.58854
$790$	0	0
$791$	−40.1581	−1.42786
$792$	0	0
$793$	−40.7754	−1.44798
$794$	0	0
$795$	70.3674	2.49568
$796$	0	0
$797$	−53.3119	−1.88840	−0.944202	−	0.329368i	$-0.893164\pi$
$797$	−53.3119	−1.88840	−0.944202	+	0.329368i	$0.893164\pi$
$798$	0	0
$799$	−14.2773	−0.505094
$800$	0	0
$801$	−2.89352	−0.102237
$802$	0	0
$803$	−66.1881	−2.33573
$804$	0	0
$805$	2.90580	0.102416
$806$	0	0
$807$	44.4782	1.56571
$808$	0	0
$809$	−27.8865	−0.980437	−0.490219	−	0.871599i	$-0.663083\pi$
$809$	−27.8865	−0.980437	−0.490219	+	0.871599i	$0.663083\pi$
$810$	0	0
$811$	−38.5146	−1.35243	−0.676214	−	0.736705i	$-0.736381\pi$
$811$	−38.5146	−1.35243	−0.676214	+	0.736705i	$0.736381\pi$
$812$	0	0
$813$	7.09351	0.248780
$814$	0	0
$815$	84.5385	2.96125
$816$	0	0
$817$	−4.58431	−0.160385
$818$	0	0
$819$	7.51109	0.262459
$820$	0	0
$821$	−45.0145	−1.57102	−0.785508	−	0.618851i	$-0.787598\pi$
$821$	−45.0145	−1.57102	−0.785508	+	0.618851i	$0.787598\pi$
$822$	0	0
$823$	−48.1363	−1.67792	−0.838962	−	0.544189i	$-0.816837\pi$
$823$	−48.1363	−1.67792	−0.838962	+	0.544189i	$0.816837\pi$
$824$	0	0
$825$	−153.564	−5.34640
$826$	0	0
$827$	−30.4803	−1.05990	−0.529951	−	0.848028i	$-0.677790\pi$
$827$	−30.4803	−1.05990	−0.529951	+	0.848028i	$0.677790\pi$
$828$	0	0
$829$	13.7029	0.475920	0.237960	−	0.971275i	$-0.423521\pi$
$829$	13.7029	0.475920	0.237960	+	0.971275i	$0.423521\pi$
$830$	0	0
$831$	−18.2525	−0.633174
$832$	0	0
$833$	6.64861	0.230361
$834$	0	0
$835$	37.3694	1.29322
$836$	0	0
$837$	−26.2521	−0.907406
$838$	0	0
$839$	−23.1845	−0.800417	−0.400209	−	0.916424i	$-0.631062\pi$
$839$	−23.1845	−0.800417	−0.400209	+	0.916424i	$0.631062\pi$
$840$	0	0
$841$	−19.6645	−0.678085
$842$	0	0
$843$	6.86627	0.236487
$844$	0	0
$845$	111.613	3.83959
$846$	0	0
$847$	71.0812	2.44238
$848$	0	0
$849$	27.7492	0.952349
$850$	0	0
$851$	−1.22604	−0.0420280
$852$	0	0
$853$	35.5179	1.21611	0.608055	−	0.793895i	$-0.291950\pi$
$853$	35.5179	1.21611	0.608055	+	0.793895i	$0.291950\pi$
$854$	0	0
$855$	4.64313	0.158792
$856$	0	0
$857$	−21.7468	−0.742857	−0.371428	−	0.928462i	$-0.621132\pi$
$857$	−21.7468	−0.742857	−0.371428	+	0.928462i	$0.621132\pi$
$858$	0	0
$859$	24.1470	0.823885	0.411943	−	0.911210i	$-0.364850\pi$
$859$	24.1470	0.823885	0.411943	+	0.911210i	$0.364850\pi$
$860$	0	0
$861$	−27.7037	−0.944139
$862$	0	0
$863$	−10.1227	−0.344582	−0.172291	−	0.985046i	$-0.555117\pi$
$863$	−10.1227	−0.344582	−0.172291	+	0.985046i	$0.555117\pi$
$864$	0	0
$865$	98.8473	3.36091
$866$	0	0
$867$	−23.2340	−0.789067
$868$	0	0
$869$	52.1444	1.76888
$870$	0	0
$871$	−44.6646	−1.51340
$872$	0	0
$873$	−0.132870	−0.00449696
$874$	0	0
$875$	133.505	4.51329
$876$	0	0
$877$	−3.96635	−0.133934	−0.0669671	−	0.997755i	$-0.521332\pi$
$877$	−3.96635	−0.133934	−0.0669671	+	0.997755i	$0.521332\pi$
$878$	0	0
$879$	−17.8541	−0.602204
$880$	0	0
$881$	−22.3579	−0.753256	−0.376628	−	0.926365i	$-0.622916\pi$
$881$	−22.3579	−0.753256	−0.376628	+	0.926365i	$0.622916\pi$
$882$	0	0
$883$	−9.09669	−0.306128	−0.153064	−	0.988216i	$-0.548914\pi$
$883$	−9.09669	−0.306128	−0.153064	+	0.988216i	$0.548914\pi$
$884$	0	0
$885$	−87.7861	−2.95090
$886$	0	0
$887$	−28.3378	−0.951490	−0.475745	−	0.879583i	$-0.657821\pi$
$887$	−28.3378	−0.951490	−0.475745	+	0.879583i	$0.657821\pi$
$888$	0	0
$889$	1.18871	0.0398680
$890$	0	0
$891$	57.6641	1.93182
$892$	0	0
$893$	−18.9035	−0.632582
$894$	0	0
$895$	−0.930742	−0.0311113
$896$	0	0
$897$	2.34726	0.0783729
$898$	0	0
$899$	16.6538	0.555436
$900$	0	0
$901$	18.1134	0.603446
$902$	0	0
$903$	−9.72522	−0.323635
$904$	0	0
$905$	−16.4010	−0.545188
$906$	0	0
$907$	−53.8586	−1.78835	−0.894173	−	0.447721i	$-0.852236\pi$
$907$	−53.8586	−1.78835	−0.894173	+	0.447721i	$0.852236\pi$
$908$	0	0
$909$	1.06354	0.0352754
$910$	0	0
$911$	−2.36505	−0.0783575	−0.0391788	−	0.999232i	$-0.512474\pi$
$911$	−2.36505	−0.0783575	−0.0391788	+	0.999232i	$0.512474\pi$
$912$	0	0
$913$	−38.6378	−1.27872
$914$	0	0
$915$	−53.4726	−1.76775
$916$	0	0
$917$	−14.8924	−0.491790
$918$	0	0
$919$	51.2506	1.69060	0.845300	−	0.534292i	$-0.179422\pi$
$919$	51.2506	1.69060	0.845300	+	0.534292i	$0.179422\pi$
$920$	0	0
$921$	−36.4797	−1.20205
$922$	0	0
$923$	−102.984	−3.38975
$924$	0	0
$925$	−86.0413	−2.82902
$926$	0	0
$927$	4.19919	0.137919
$928$	0	0
$929$	−4.90182	−0.160823	−0.0804117	−	0.996762i	$-0.525624\pi$
$929$	−4.90182	−0.160823	−0.0804117	+	0.996762i	$0.525624\pi$
$930$	0	0
$931$	8.80294	0.288505
$932$	0	0
$933$	44.6954	1.46326
$934$	0	0
$935$	−53.1794	−1.73915
$936$	0	0
$937$	51.2289	1.67357	0.836787	−	0.547528i	$-0.184431\pi$
$937$	51.2289	1.67357	0.836787	+	0.547528i	$0.184431\pi$
$938$	0	0
$939$	−59.1198	−1.92930
$940$	0	0
$941$	1.95788	0.0638252	0.0319126	−	0.999491i	$-0.489840\pi$
$941$	1.95788	0.0638252	0.0319126	+	0.999491i	$0.489840\pi$
$942$	0	0
$943$	−0.974227	−0.0317252
$944$	0	0
$945$	−67.8333	−2.20662
$946$	0	0
$947$	1.20853	0.0392718	0.0196359	−	0.999807i	$-0.493749\pi$
$947$	1.20853	0.0392718	0.0196359	+	0.999807i	$0.493749\pi$
$948$	0	0
$949$	70.9997	2.30475
$950$	0	0
$951$	−41.3535	−1.34098
$952$	0	0
$953$	−30.1358	−0.976195	−0.488097	−	0.872789i	$-0.662309\pi$
$953$	−30.1358	−0.976195	−0.488097	+	0.872789i	$0.662309\pi$
$954$	0	0
$955$	30.3690	0.982719
$956$	0	0
$957$	−32.4050	−1.04751
$958$	0	0
$959$	−14.4210	−0.465679
$960$	0	0
$961$	−1.29095	−0.0416437
$962$	0	0
$963$	−1.39762	−0.0450377
$964$	0	0
$965$	63.7131	2.05100
$966$	0	0
$967$	−25.3222	−0.814307	−0.407153	−	0.913360i	$-0.633479\pi$
$967$	−25.3222	−0.814307	−0.407153	+	0.913360i	$0.633479\pi$
$968$	0	0
$969$	10.6213	0.341205
$970$	0	0
$971$	−7.78253	−0.249753	−0.124877	−	0.992172i	$-0.539854\pi$
$971$	−7.78253	−0.249753	−0.124877	+	0.992172i	$0.539854\pi$
$972$	0	0
$973$	49.2484	1.57883
$974$	0	0
$975$	164.727	5.27549
$976$	0	0
$977$	−47.6249	−1.52365	−0.761827	−	0.647780i	$-0.775698\pi$
$977$	−47.6249	−1.52365	−0.761827	+	0.647780i	$0.775698\pi$
$978$	0	0
$979$	43.8788	1.40237
$980$	0	0
$981$	−4.81056	−0.153589
$982$	0	0
$983$	−19.8158	−0.632027	−0.316013	−	0.948755i	$-0.602344\pi$
$983$	−19.8158	−0.632027	−0.316013	+	0.948755i	$0.602344\pi$
$984$	0	0
$985$	−49.0532	−1.56296
$986$	0	0
$987$	−40.1022	−1.27647
$988$	0	0
$989$	−0.341997	−0.0108749
$990$	0	0
$991$	19.3883	0.615890	0.307945	−	0.951404i	$-0.400359\pi$
$991$	19.3883	0.615890	0.307945	+	0.951404i	$0.400359\pi$
$992$	0	0
$993$	27.8663	0.884310
$994$	0	0
$995$	−88.0924	−2.79272
$996$	0	0
$997$	−36.4954	−1.15582	−0.577910	−	0.816100i	$-0.696132\pi$
$997$	−36.4954	−1.15582	−0.577910	+	0.816100i	$0.696132\pi$
$998$	0	0
$999$	28.6208	0.905520

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.36	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.36	✓	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)

\(f(q)\)	\(=\)	\(q+1.83858 q^{3} +4.41353 q^{5} +3.19108 q^{7} +0.380391 q^{9} +O(q^{10})\)	\(q+1.83858 q^{3} +4.41353 q^{5} +3.19108 q^{7} +0.380391 q^{9} -5.76845 q^{11} +6.18779 q^{13} +8.11464 q^{15} +2.08881 q^{17} +2.76564 q^{19} +5.86706 q^{21} +0.206321 q^{23} +14.4792 q^{25} -4.81637 q^{27} +3.05541 q^{29} +5.45060 q^{31} -10.6058 q^{33} +14.0839 q^{35} -5.94239 q^{37} +11.3768 q^{39} -4.72190 q^{41} -1.65760 q^{43} +1.67887 q^{45} -6.83514 q^{47} +3.18297 q^{49} +3.84045 q^{51} +8.67166 q^{53} -25.4592 q^{55} +5.08486 q^{57} -10.8182 q^{59} -6.58965 q^{61} +1.21386 q^{63} +27.3100 q^{65} -7.21819 q^{67} +0.379338 q^{69} -16.6430 q^{71} +11.4742 q^{73} +26.6213 q^{75} -18.4076 q^{77} -9.03959 q^{79} -9.99648 q^{81} +6.69813 q^{83} +9.21901 q^{85} +5.61763 q^{87} -7.60669 q^{89} +19.7457 q^{91} +10.0214 q^{93} +12.2062 q^{95} -0.349298 q^{97} -2.19426 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 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

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Database

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