LMFDB - Embedded newform 6008.2.a.e.1.50

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			1.50
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)

$f(q)$	$=$	$q+3.42580 q^{3} +1.19116 q^{5} -1.06018 q^{7} +8.73609 q^{9} +O(q^{10})$	$q+3.42580 q^{3} +1.19116 q^{5} -1.06018 q^{7} +8.73609 q^{9} +0.730687 q^{11} +5.49734 q^{13} +4.08066 q^{15} +0.933752 q^{17} +5.09856 q^{19} -3.63197 q^{21} -6.73974 q^{23} -3.58115 q^{25} +19.6507 q^{27} +5.95222 q^{29} +7.12567 q^{31} +2.50318 q^{33} -1.26284 q^{35} +7.45858 q^{37} +18.8328 q^{39} -11.8661 q^{41} -12.0255 q^{43} +10.4060 q^{45} -11.0394 q^{47} -5.87601 q^{49} +3.19885 q^{51} -7.95898 q^{53} +0.870361 q^{55} +17.4666 q^{57} -8.66266 q^{59} +11.6731 q^{61} -9.26185 q^{63} +6.54818 q^{65} +0.0992160 q^{67} -23.0890 q^{69} -0.301153 q^{71} +1.95916 q^{73} -12.2683 q^{75} -0.774662 q^{77} +0.440104 q^{79} +41.1110 q^{81} -10.9971 q^{83} +1.11224 q^{85} +20.3911 q^{87} +5.62519 q^{89} -5.82818 q^{91} +24.4111 q^{93} +6.07317 q^{95} +2.36384 q^{97} +6.38334 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.42580	1.97789	0.988943	−	0.148299i	$-0.0473798\pi$
$3$	3.42580	1.97789	0.988943	+	0.148299i	$0.0473798\pi$
$4$	0	0
$5$	1.19116	0.532701	0.266350	−	0.963876i	$-0.414182\pi$
$5$	1.19116	0.532701	0.266350	+	0.963876i	$0.414182\pi$
$6$	0	0
$7$	−1.06018	−0.400712	−0.200356	−	0.979723i	$-0.564210\pi$
$7$	−1.06018	−0.400712	−0.200356	+	0.979723i	$0.564210\pi$
$8$	0	0
$9$	8.73609	2.91203
$10$	0	0
$11$	0.730687	0.220310	0.110155	−	0.993914i	$-0.464865\pi$
$11$	0.730687	0.220310	0.110155	+	0.993914i	$0.464865\pi$
$12$	0	0
$13$	5.49734	1.52469	0.762343	−	0.647173i	$-0.224049\pi$
$13$	5.49734	1.52469	0.762343	+	0.647173i	$0.224049\pi$
$14$	0	0
$15$	4.08066	1.05362
$16$	0	0
$17$	0.933752	0.226468	0.113234	−	0.993568i	$-0.463879\pi$
$17$	0.933752	0.226468	0.113234	+	0.993568i	$0.463879\pi$
$18$	0	0
$19$	5.09856	1.16969	0.584844	−	0.811145i	$-0.301156\pi$
$19$	5.09856	1.16969	0.584844	+	0.811145i	$0.301156\pi$
$20$	0	0
$21$	−3.63197	−0.792561
$22$	0	0
$23$	−6.73974	−1.40533	−0.702666	−	0.711520i	$-0.748007\pi$
$23$	−6.73974	−1.40533	−0.702666	+	0.711520i	$0.748007\pi$
$24$	0	0
$25$	−3.58115	−0.716230
$26$	0	0
$27$	19.6507	3.78177
$28$	0	0
$29$	5.95222	1.10530	0.552650	−	0.833413i	$-0.313617\pi$
$29$	5.95222	1.10530	0.552650	+	0.833413i	$0.313617\pi$
$30$	0	0
$31$	7.12567	1.27981	0.639904	−	0.768455i	$-0.278974\pi$
$31$	7.12567	1.27981	0.639904	+	0.768455i	$0.278974\pi$
$32$	0	0
$33$	2.50318	0.435749
$34$	0	0
$35$	−1.26284	−0.213459
$36$	0	0
$37$	7.45858	1.22618	0.613092	−	0.790012i	$-0.289926\pi$
$37$	7.45858	1.22618	0.613092	+	0.790012i	$0.289926\pi$
$38$	0	0
$39$	18.8328	3.01565
$40$	0	0
$41$	−11.8661	−1.85318	−0.926589	−	0.376076i	$-0.877273\pi$
$41$	−11.8661	−1.85318	−0.926589	+	0.376076i	$0.877273\pi$
$42$	0	0
$43$	−12.0255	−1.83388	−0.916939	−	0.399028i	$-0.869348\pi$
$43$	−12.0255	−1.83388	−0.916939	+	0.399028i	$0.869348\pi$
$44$	0	0
$45$	10.4060	1.55124
$46$	0	0
$47$	−11.0394	−1.61026	−0.805131	−	0.593096i	$-0.797905\pi$
$47$	−11.0394	−1.61026	−0.805131	+	0.593096i	$0.797905\pi$
$48$	0	0
$49$	−5.87601	−0.839430
$50$	0	0
$51$	3.19885	0.447928
$52$	0	0
$53$	−7.95898	−1.09325	−0.546625	−	0.837378i	$-0.684088\pi$
$53$	−7.95898	−1.09325	−0.546625	+	0.837378i	$0.684088\pi$
$54$	0	0
$55$	0.870361	0.117359
$56$	0	0
$57$	17.4666	2.31351
$58$	0	0
$59$	−8.66266	−1.12778	−0.563891	−	0.825849i	$-0.690696\pi$
$59$	−8.66266	−1.12778	−0.563891	+	0.825849i	$0.690696\pi$
$60$	0	0
$61$	11.6731	1.49459	0.747297	−	0.664490i	$-0.231351\pi$
$61$	11.6731	1.49459	0.747297	+	0.664490i	$0.231351\pi$
$62$	0	0
$63$	−9.26185	−1.16688
$64$	0	0
$65$	6.54818	0.812202
$66$	0	0
$67$	0.0992160	0.0121212	0.00606058	−	0.999982i	$-0.498071\pi$
$67$	0.0992160	0.0121212	0.00606058	+	0.999982i	$0.498071\pi$
$68$	0	0
$69$	−23.0890	−2.77959
$70$	0	0
$71$	−0.301153	−0.0357403	−0.0178701	−	0.999840i	$-0.505689\pi$
$71$	−0.301153	−0.0357403	−0.0178701	+	0.999840i	$0.505689\pi$
$72$	0	0
$73$	1.95916	0.229302	0.114651	−	0.993406i	$-0.463425\pi$
$73$	1.95916	0.229302	0.114651	+	0.993406i	$0.463425\pi$
$74$	0	0
$75$	−12.2683	−1.41662
$76$	0	0
$77$	−0.774662	−0.0882809
$78$	0	0
$79$	0.440104	0.0495155	0.0247578	−	0.999693i	$-0.492119\pi$
$79$	0.440104	0.0495155	0.0247578	+	0.999693i	$0.492119\pi$
$80$	0	0
$81$	41.1110	4.56789
$82$	0	0
$83$	−10.9971	−1.20709	−0.603543	−	0.797330i	$-0.706245\pi$
$83$	−10.9971	−1.20709	−0.603543	+	0.797330i	$0.706245\pi$
$84$	0	0
$85$	1.11224	0.120640
$86$	0	0
$87$	20.3911	2.18616
$88$	0	0
$89$	5.62519	0.596269	0.298134	−	0.954524i	$-0.403636\pi$
$89$	5.62519	0.596269	0.298134	+	0.954524i	$0.403636\pi$
$90$	0	0
$91$	−5.82818	−0.610959
$92$	0	0
$93$	24.4111	2.53131
$94$	0	0
$95$	6.07317	0.623094
$96$	0	0
$97$	2.36384	0.240012	0.120006	−	0.992773i	$-0.461709\pi$
$97$	2.36384	0.240012	0.120006	+	0.992773i	$0.461709\pi$
$98$	0	0
$99$	6.38334	0.641550
$100$	0	0
$101$	0.990036	0.0985122	0.0492561	−	0.998786i	$-0.484315\pi$
$101$	0.990036	0.0985122	0.0492561	+	0.998786i	$0.484315\pi$
$102$	0	0
$103$	−7.92463	−0.780837	−0.390419	−	0.920637i	$-0.627670\pi$
$103$	−7.92463	−0.780837	−0.390419	+	0.920637i	$0.627670\pi$
$104$	0	0
$105$	−4.32624	−0.422198
$106$	0	0
$107$	2.72000	0.262952	0.131476	−	0.991319i	$-0.458028\pi$
$107$	2.72000	0.262952	0.131476	+	0.991319i	$0.458028\pi$
$108$	0	0
$109$	2.88417	0.276254	0.138127	−	0.990415i	$-0.455892\pi$
$109$	2.88417	0.276254	0.138127	+	0.990415i	$0.455892\pi$
$110$	0	0
$111$	25.5516	2.42525
$112$	0	0
$113$	3.59853	0.338521	0.169261	−	0.985571i	$-0.445862\pi$
$113$	3.59853	0.338521	0.169261	+	0.985571i	$0.445862\pi$
$114$	0	0
$115$	−8.02807	−0.748622
$116$	0	0
$117$	48.0252	4.43993
$118$	0	0
$119$	−0.989948	−0.0907484
$120$	0	0
$121$	−10.4661	−0.951463
$122$	0	0
$123$	−40.6510	−3.66537
$124$	0	0
$125$	−10.2215	−0.914237
$126$	0	0
$127$	19.5079	1.73104	0.865522	−	0.500871i	$-0.166987\pi$
$127$	19.5079	1.73104	0.865522	+	0.500871i	$0.166987\pi$
$128$	0	0
$129$	−41.1971	−3.62720
$130$	0	0
$131$	−6.57371	−0.574348	−0.287174	−	0.957878i	$-0.592716\pi$
$131$	−6.57371	−0.574348	−0.287174	+	0.957878i	$0.592716\pi$
$132$	0	0
$133$	−5.40540	−0.468708
$134$	0	0
$135$	23.4070	2.01455
$136$	0	0
$137$	−7.44055	−0.635689	−0.317845	−	0.948143i	$-0.602959\pi$
$137$	−7.44055	−0.635689	−0.317845	+	0.948143i	$0.602959\pi$
$138$	0	0
$139$	−9.31538	−0.790120	−0.395060	−	0.918655i	$-0.629276\pi$
$139$	−9.31538	−0.790120	−0.395060	+	0.918655i	$0.629276\pi$
$140$	0	0
$141$	−37.8188	−3.18491
$142$	0	0
$143$	4.01683	0.335904
$144$	0	0
$145$	7.09002	0.588794
$146$	0	0
$147$	−20.1300	−1.66030
$148$	0	0
$149$	−14.8200	−1.21410	−0.607050	−	0.794664i	$-0.707647\pi$
$149$	−14.8200	−1.21410	−0.607050	+	0.794664i	$0.707647\pi$
$150$	0	0
$151$	10.0357	0.816690	0.408345	−	0.912828i	$-0.366106\pi$
$151$	10.0357	0.816690	0.408345	+	0.912828i	$0.366106\pi$
$152$	0	0
$153$	8.15734	0.659482
$154$	0	0
$155$	8.48778	0.681755
$156$	0	0
$157$	13.3876	1.06845	0.534223	−	0.845344i	$-0.320604\pi$
$157$	13.3876	1.06845	0.534223	+	0.845344i	$0.320604\pi$
$158$	0	0
$159$	−27.2659	−2.16232
$160$	0	0
$161$	7.14535	0.563133
$162$	0	0
$163$	−18.0020	−1.41003	−0.705014	−	0.709194i	$-0.749059\pi$
$163$	−18.0020	−1.41003	−0.705014	+	0.709194i	$0.749059\pi$
$164$	0	0
$165$	2.98168	0.232124
$166$	0	0
$167$	10.5541	0.816702	0.408351	−	0.912825i	$-0.366104\pi$
$167$	10.5541	0.816702	0.408351	+	0.912825i	$0.366104\pi$
$168$	0	0
$169$	17.2207	1.32467
$170$	0	0
$171$	44.5414	3.40617
$172$	0	0
$173$	−7.00754	−0.532773	−0.266387	−	0.963866i	$-0.585830\pi$
$173$	−7.00754	−0.532773	−0.266387	+	0.963866i	$0.585830\pi$
$174$	0	0
$175$	3.79667	0.287002
$176$	0	0
$177$	−29.6765	−2.23062
$178$	0	0
$179$	19.4480	1.45362	0.726808	−	0.686841i	$-0.241003\pi$
$179$	19.4480	1.45362	0.726808	+	0.686841i	$0.241003\pi$
$180$	0	0
$181$	14.0724	1.04599	0.522996	−	0.852335i	$-0.324814\pi$
$181$	14.0724	1.04599	0.522996	+	0.852335i	$0.324814\pi$
$182$	0	0
$183$	39.9898	2.95613
$184$	0	0
$185$	8.88433	0.653189
$186$	0	0
$187$	0.682280	0.0498933
$188$	0	0
$189$	−20.8333	−1.51540
$190$	0	0
$191$	−23.1982	−1.67856	−0.839280	−	0.543700i	$-0.817023\pi$
$191$	−23.1982	−1.67856	−0.839280	+	0.543700i	$0.817023\pi$
$192$	0	0
$193$	12.5544	0.903688	0.451844	−	0.892097i	$-0.350766\pi$
$193$	12.5544	0.903688	0.451844	+	0.892097i	$0.350766\pi$
$194$	0	0
$195$	22.4327	1.60644
$196$	0	0
$197$	−8.14448	−0.580270	−0.290135	−	0.956986i	$-0.593700\pi$
$197$	−8.14448	−0.580270	−0.290135	+	0.956986i	$0.593700\pi$
$198$	0	0
$199$	1.53179	0.108586	0.0542929	−	0.998525i	$-0.482710\pi$
$199$	1.53179	0.108586	0.0542929	+	0.998525i	$0.482710\pi$
$200$	0	0
$201$	0.339894	0.0239743
$202$	0	0
$203$	−6.31044	−0.442906
$204$	0	0
$205$	−14.1344	−0.987189
$206$	0	0
$207$	−58.8789	−4.09237
$208$	0	0
$209$	3.72545	0.257695
$210$	0	0
$211$	−1.59340	−0.109694	−0.0548471	−	0.998495i	$-0.517467\pi$
$211$	−1.59340	−0.109694	−0.0548471	+	0.998495i	$0.517467\pi$
$212$	0	0
$213$	−1.03169	−0.0706902
$214$	0	0
$215$	−14.3243	−0.976908
$216$	0	0
$217$	−7.55451	−0.512834
$218$	0	0
$219$	6.71167	0.453533
$220$	0	0
$221$	5.13315	0.345293
$222$	0	0
$223$	15.2605	1.02192	0.510959	−	0.859605i	$-0.329290\pi$
$223$	15.2605	1.02192	0.510959	+	0.859605i	$0.329290\pi$
$224$	0	0
$225$	−31.2852	−2.08568
$226$	0	0
$227$	28.8322	1.91366	0.956832	−	0.290643i	$-0.0938692\pi$
$227$	28.8322	1.91366	0.956832	+	0.290643i	$0.0938692\pi$
$228$	0	0
$229$	16.6559	1.10065	0.550327	−	0.834949i	$-0.314503\pi$
$229$	16.6559	1.10065	0.550327	+	0.834949i	$0.314503\pi$
$230$	0	0
$231$	−2.65383	−0.174609
$232$	0	0
$233$	4.30821	0.282240	0.141120	−	0.989992i	$-0.454930\pi$
$233$	4.30821	0.282240	0.141120	+	0.989992i	$0.454930\pi$
$234$	0	0
$235$	−13.1496	−0.857788
$236$	0	0
$237$	1.50771	0.0979360
$238$	0	0
$239$	15.4629	1.00021	0.500105	−	0.865965i	$-0.333295\pi$
$239$	15.4629	1.00021	0.500105	+	0.865965i	$0.333295\pi$
$240$	0	0
$241$	−9.26533	−0.596832	−0.298416	−	0.954436i	$-0.596458\pi$
$241$	−9.26533	−0.596832	−0.298416	+	0.954436i	$0.596458\pi$
$242$	0	0
$243$	81.8859	5.25298
$244$	0	0
$245$	−6.99924	−0.447165
$246$	0	0
$247$	28.0285	1.78341
$248$	0	0
$249$	−37.6738	−2.38748
$250$	0	0
$251$	−2.87265	−0.181320	−0.0906601	−	0.995882i	$-0.528898\pi$
$251$	−2.87265	−0.181320	−0.0906601	+	0.995882i	$0.528898\pi$
$252$	0	0
$253$	−4.92464	−0.309609
$254$	0	0
$255$	3.81032	0.238612
$256$	0	0
$257$	25.1631	1.56963	0.784817	−	0.619728i	$-0.212757\pi$
$257$	25.1631	1.56963	0.784817	+	0.619728i	$0.212757\pi$
$258$	0	0
$259$	−7.90746	−0.491346
$260$	0	0
$261$	51.9991	3.21867
$262$	0	0
$263$	21.7659	1.34214	0.671072	−	0.741392i	$-0.265834\pi$
$263$	21.7659	1.34214	0.671072	+	0.741392i	$0.265834\pi$
$264$	0	0
$265$	−9.48038	−0.582375
$266$	0	0
$267$	19.2707	1.17935
$268$	0	0
$269$	16.4541	1.00322	0.501611	−	0.865093i	$-0.332741\pi$
$269$	16.4541	1.00322	0.501611	+	0.865093i	$0.332741\pi$
$270$	0	0
$271$	−22.7471	−1.38179	−0.690895	−	0.722956i	$-0.742783\pi$
$271$	−22.7471	−1.38179	−0.690895	+	0.722956i	$0.742783\pi$
$272$	0	0
$273$	−19.9662	−1.20841
$274$	0	0
$275$	−2.61670	−0.157793
$276$	0	0
$277$	−11.8828	−0.713967	−0.356983	−	0.934111i	$-0.616195\pi$
$277$	−11.8828	−0.713967	−0.356983	+	0.934111i	$0.616195\pi$
$278$	0	0
$279$	62.2505	3.72684
$280$	0	0
$281$	−12.3086	−0.734268	−0.367134	−	0.930168i	$-0.619661\pi$
$281$	−12.3086	−0.734268	−0.367134	+	0.930168i	$0.619661\pi$
$282$	0	0
$283$	−16.2708	−0.967199	−0.483599	−	0.875289i	$-0.660671\pi$
$283$	−16.2708	−0.967199	−0.483599	+	0.875289i	$0.660671\pi$
$284$	0	0
$285$	20.8055	1.23241
$286$	0	0
$287$	12.5803	0.742590
$288$	0	0
$289$	−16.1281	−0.948712
$290$	0	0
$291$	8.09804	0.474716
$292$	0	0
$293$	30.3002	1.77016	0.885079	−	0.465441i	$-0.154104\pi$
$293$	30.3002	1.77016	0.885079	+	0.465441i	$0.154104\pi$
$294$	0	0
$295$	−10.3186	−0.600770
$296$	0	0
$297$	14.3585	0.833164
$298$	0	0
$299$	−37.0506	−2.14269
$300$	0	0
$301$	12.7493	0.734856
$302$	0	0
$303$	3.39166	0.194846
$304$	0	0
$305$	13.9045	0.796171
$306$	0	0
$307$	−8.56308	−0.488721	−0.244360	−	0.969684i	$-0.578578\pi$
$307$	−8.56308	−0.488721	−0.244360	+	0.969684i	$0.578578\pi$
$308$	0	0
$309$	−27.1482	−1.54441
$310$	0	0
$311$	2.09759	0.118944	0.0594718	−	0.998230i	$-0.481058\pi$
$311$	2.09759	0.118944	0.0594718	+	0.998230i	$0.481058\pi$
$312$	0	0
$313$	25.9664	1.46771	0.733854	−	0.679307i	$-0.237720\pi$
$313$	25.9664	1.46771	0.733854	+	0.679307i	$0.237720\pi$
$314$	0	0
$315$	−11.0323	−0.621600
$316$	0	0
$317$	−26.8548	−1.50832	−0.754158	−	0.656693i	$-0.771955\pi$
$317$	−26.8548	−1.50832	−0.754158	+	0.656693i	$0.771955\pi$
$318$	0	0
$319$	4.34921	0.243509
$320$	0	0
$321$	9.31817	0.520090
$322$	0	0
$323$	4.76079	0.264897
$324$	0	0
$325$	−19.6868	−1.09203
$326$	0	0
$327$	9.88059	0.546398
$328$	0	0
$329$	11.7038	0.645251
$330$	0	0
$331$	−4.73548	−0.260285	−0.130143	−	0.991495i	$-0.541544\pi$
$331$	−4.73548	−0.260285	−0.130143	+	0.991495i	$0.541544\pi$
$332$	0	0
$333$	65.1588	3.57068
$334$	0	0
$335$	0.118182	0.00645695
$336$	0	0
$337$	−22.8801	−1.24636	−0.623179	−	0.782079i	$-0.714159\pi$
$337$	−22.8801	−1.24636	−0.623179	+	0.782079i	$0.714159\pi$
$338$	0	0
$339$	12.3278	0.669556
$340$	0	0
$341$	5.20663	0.281955
$342$	0	0
$343$	13.6509	0.737081
$344$	0	0
$345$	−27.5025	−1.48069
$346$	0	0
$347$	−32.3176	−1.73490	−0.867450	−	0.497525i	$-0.834242\pi$
$347$	−32.3176	−1.73490	−0.867450	+	0.497525i	$0.834242\pi$
$348$	0	0
$349$	−16.7908	−0.898792	−0.449396	−	0.893333i	$-0.648361\pi$
$349$	−16.7908	−0.898792	−0.449396	+	0.893333i	$0.648361\pi$
$350$	0	0
$351$	108.026	5.76602
$352$	0	0
$353$	−15.7745	−0.839591	−0.419795	−	0.907619i	$-0.637898\pi$
$353$	−15.7745	−0.839591	−0.419795	+	0.907619i	$0.637898\pi$
$354$	0	0
$355$	−0.358720	−0.0190389
$356$	0	0
$357$	−3.39136	−0.179490
$358$	0	0
$359$	−2.54133	−0.134126	−0.0670630	−	0.997749i	$-0.521363\pi$
$359$	−2.54133	−0.134126	−0.0670630	+	0.997749i	$0.521363\pi$
$360$	0	0
$361$	6.99527	0.368172
$362$	0	0
$363$	−35.8547	−1.88189
$364$	0	0
$365$	2.33366	0.122149
$366$	0	0
$367$	−1.27115	−0.0663537	−0.0331768	−	0.999449i	$-0.510562\pi$
$367$	−1.27115	−0.0663537	−0.0331768	+	0.999449i	$0.510562\pi$
$368$	0	0
$369$	−103.664	−5.39651
$370$	0	0
$371$	8.43797	0.438078
$372$	0	0
$373$	12.2315	0.633323	0.316661	−	0.948539i	$-0.397438\pi$
$373$	12.2315	0.633323	0.316661	+	0.948539i	$0.397438\pi$
$374$	0	0
$375$	−35.0167	−1.80826
$376$	0	0
$377$	32.7214	1.68524
$378$	0	0
$379$	7.81196	0.401273	0.200637	−	0.979666i	$-0.435699\pi$
$379$	7.81196	0.401273	0.200637	+	0.979666i	$0.435699\pi$
$380$	0	0
$381$	66.8300	3.42381
$382$	0	0
$383$	−7.92385	−0.404890	−0.202445	−	0.979294i	$-0.564889\pi$
$383$	−7.92385	−0.404890	−0.202445	+	0.979294i	$0.564889\pi$
$384$	0	0
$385$	−0.922742	−0.0470273
$386$	0	0
$387$	−105.056	−5.34031
$388$	0	0
$389$	−17.3213	−0.878223	−0.439111	−	0.898433i	$-0.644707\pi$
$389$	−17.3213	−0.878223	−0.439111	+	0.898433i	$0.644707\pi$
$390$	0	0
$391$	−6.29324	−0.318263
$392$	0	0
$393$	−22.5202	−1.13599
$394$	0	0
$395$	0.524232	0.0263770
$396$	0	0
$397$	1.16479	0.0584594	0.0292297	−	0.999573i	$-0.490695\pi$
$397$	1.16479	0.0584594	0.0292297	+	0.999573i	$0.490695\pi$
$398$	0	0
$399$	−18.5178	−0.927050
$400$	0	0
$401$	−37.1501	−1.85519	−0.927594	−	0.373590i	$-0.878127\pi$
$401$	−37.1501	−1.85519	−0.927594	+	0.373590i	$0.878127\pi$
$402$	0	0
$403$	39.1722	1.95131
$404$	0	0
$405$	48.9696	2.43332
$406$	0	0
$407$	5.44989	0.270141
$408$	0	0
$409$	−6.15989	−0.304587	−0.152293	−	0.988335i	$-0.548666\pi$
$409$	−6.15989	−0.304587	−0.152293	+	0.988335i	$0.548666\pi$
$410$	0	0
$411$	−25.4898	−1.25732
$412$	0	0
$413$	9.18400	0.451915
$414$	0	0
$415$	−13.0992	−0.643016
$416$	0	0
$417$	−31.9126	−1.56277
$418$	0	0
$419$	−5.25531	−0.256739	−0.128369	−	0.991726i	$-0.540974\pi$
$419$	−5.25531	−0.256739	−0.128369	+	0.991726i	$0.540974\pi$
$420$	0	0
$421$	−24.2976	−1.18419	−0.592096	−	0.805867i	$-0.701699\pi$
$421$	−24.2976	−1.18419	−0.592096	+	0.805867i	$0.701699\pi$
$422$	0	0
$423$	−96.4412	−4.68913
$424$	0	0
$425$	−3.34391	−0.162203
$426$	0	0
$427$	−12.3757	−0.598901
$428$	0	0
$429$	13.7608	0.664380
$430$	0	0
$431$	4.09597	0.197296	0.0986481	−	0.995122i	$-0.468548\pi$
$431$	4.09597	0.197296	0.0986481	+	0.995122i	$0.468548\pi$
$432$	0	0
$433$	14.4549	0.694657	0.347329	−	0.937743i	$-0.387089\pi$
$433$	14.4549	0.694657	0.347329	+	0.937743i	$0.387089\pi$
$434$	0	0
$435$	24.2890	1.16457
$436$	0	0
$437$	−34.3629	−1.64380
$438$	0	0
$439$	−37.0021	−1.76602	−0.883008	−	0.469358i	$-0.844485\pi$
$439$	−37.0021	−1.76602	−0.883008	+	0.469358i	$0.844485\pi$
$440$	0	0
$441$	−51.3334	−2.44445
$442$	0	0
$443$	17.9905	0.854757	0.427378	−	0.904073i	$-0.359437\pi$
$443$	17.9905	0.854757	0.427378	+	0.904073i	$0.359437\pi$
$444$	0	0
$445$	6.70047	0.317633
$446$	0	0
$447$	−50.7702	−2.40135
$448$	0	0
$449$	25.7971	1.21744	0.608721	−	0.793385i	$-0.291683\pi$
$449$	25.7971	1.21744	0.608721	+	0.793385i	$0.291683\pi$
$450$	0	0
$451$	−8.67042	−0.408274
$452$	0	0
$453$	34.3801	1.61532
$454$	0	0
$455$	−6.94227	−0.325459
$456$	0	0
$457$	−38.1478	−1.78448	−0.892239	−	0.451564i	$-0.850866\pi$
$457$	−38.1478	−1.78448	−0.892239	+	0.451564i	$0.850866\pi$
$458$	0	0
$459$	18.3489	0.856452
$460$	0	0
$461$	14.7727	0.688034	0.344017	−	0.938963i	$-0.388212\pi$
$461$	14.7727	0.688034	0.344017	+	0.938963i	$0.388212\pi$
$462$	0	0
$463$	40.3241	1.87402	0.937011	−	0.349300i	$-0.113581\pi$
$463$	40.3241	1.87402	0.937011	+	0.349300i	$0.113581\pi$
$464$	0	0
$465$	29.0774	1.34843
$466$	0	0
$467$	3.59372	0.166298	0.0831488	−	0.996537i	$-0.473502\pi$
$467$	3.59372	0.166298	0.0831488	+	0.996537i	$0.473502\pi$
$468$	0	0
$469$	−0.105187	−0.00485709
$470$	0	0
$471$	45.8631	2.11326
$472$	0	0
$473$	−8.78690	−0.404022
$474$	0	0
$475$	−18.2587	−0.837766
$476$	0	0
$477$	−69.5303	−3.18358
$478$	0	0
$479$	−28.6332	−1.30828	−0.654142	−	0.756372i	$-0.726970\pi$
$479$	−28.6332	−1.30828	−0.654142	+	0.756372i	$0.726970\pi$
$480$	0	0
$481$	41.0023	1.86955
$482$	0	0
$483$	24.4785	1.11381
$484$	0	0
$485$	2.81570	0.127854
$486$	0	0
$487$	6.67522	0.302483	0.151241	−	0.988497i	$-0.451673\pi$
$487$	6.67522	0.302483	0.151241	+	0.988497i	$0.451673\pi$
$488$	0	0
$489$	−61.6713	−2.78887
$490$	0	0
$491$	−37.0220	−1.67078	−0.835391	−	0.549656i	$-0.814759\pi$
$491$	−37.0220	−1.67078	−0.835391	+	0.549656i	$0.814759\pi$
$492$	0	0
$493$	5.55790	0.250315
$494$	0	0
$495$	7.60355	0.341754
$496$	0	0
$497$	0.319277	0.0143215
$498$	0	0
$499$	26.7752	1.19862	0.599310	−	0.800517i	$-0.295441\pi$
$499$	26.7752	1.19862	0.599310	+	0.800517i	$0.295441\pi$
$500$	0	0
$501$	36.1563	1.61534
$502$	0	0
$503$	−26.4446	−1.17911	−0.589554	−	0.807729i	$-0.700697\pi$
$503$	−26.4446	−1.17911	−0.589554	+	0.807729i	$0.700697\pi$
$504$	0	0
$505$	1.17929	0.0524775
$506$	0	0
$507$	58.9946	2.62004
$508$	0	0
$509$	−14.1788	−0.628466	−0.314233	−	0.949346i	$-0.601747\pi$
$509$	−14.1788	−0.628466	−0.314233	+	0.949346i	$0.601747\pi$
$510$	0	0
$511$	−2.07706	−0.0918839
$512$	0	0
$513$	100.190	4.42350
$514$	0	0
$515$	−9.43946	−0.415952
$516$	0	0
$517$	−8.06635	−0.354758
$518$	0	0
$519$	−24.0064	−1.05376
$520$	0	0
$521$	−20.5358	−0.899692	−0.449846	−	0.893106i	$-0.648521\pi$
$521$	−20.5358	−0.899692	−0.449846	+	0.893106i	$0.648521\pi$
$522$	0	0
$523$	19.0514	0.833060	0.416530	−	0.909122i	$-0.363246\pi$
$523$	19.0514	0.833060	0.416530	+	0.909122i	$0.363246\pi$
$524$	0	0
$525$	13.0066	0.567656
$526$	0	0
$527$	6.65361	0.289836
$528$	0	0
$529$	22.4240	0.974959
$530$	0	0
$531$	−75.6777	−3.28413
$532$	0	0
$533$	−65.2321	−2.82552
$534$	0	0
$535$	3.23994	0.140075
$536$	0	0
$537$	66.6251	2.87508
$538$	0	0
$539$	−4.29352	−0.184935
$540$	0	0
$541$	−22.7414	−0.977730	−0.488865	−	0.872359i	$-0.662589\pi$
$541$	−22.7414	−0.977730	−0.488865	+	0.872359i	$0.662589\pi$
$542$	0	0
$543$	48.2091	2.06885
$544$	0	0
$545$	3.43550	0.147161
$546$	0	0
$547$	−29.9724	−1.28153	−0.640763	−	0.767739i	$-0.721382\pi$
$547$	−29.9724	−1.28153	−0.640763	+	0.767739i	$0.721382\pi$
$548$	0	0
$549$	101.978	4.35230
$550$	0	0
$551$	30.3477	1.29286
$552$	0	0
$553$	−0.466590	−0.0198414
$554$	0	0
$555$	30.4359	1.29193
$556$	0	0
$557$	6.85807	0.290586	0.145293	−	0.989389i	$-0.453588\pi$
$557$	6.85807	0.290586	0.145293	+	0.989389i	$0.453588\pi$
$558$	0	0
$559$	−66.1084	−2.79609
$560$	0	0
$561$	2.33735	0.0986832
$562$	0	0
$563$	−32.1397	−1.35453	−0.677263	−	0.735741i	$-0.736834\pi$
$563$	−32.1397	−1.35453	−0.677263	+	0.735741i	$0.736834\pi$
$564$	0	0
$565$	4.28641	0.180331
$566$	0	0
$567$	−43.5852	−1.83040
$568$	0	0
$569$	40.0343	1.67832	0.839162	−	0.543881i	$-0.183046\pi$
$569$	40.0343	1.67832	0.839162	+	0.543881i	$0.183046\pi$
$570$	0	0
$571$	−16.8499	−0.705145	−0.352572	−	0.935785i	$-0.614693\pi$
$571$	−16.8499	−0.705145	−0.352572	+	0.935785i	$0.614693\pi$
$572$	0	0
$573$	−79.4722	−3.32000
$574$	0	0
$575$	24.1360	1.00654
$576$	0	0
$577$	13.0112	0.541662	0.270831	−	0.962627i	$-0.412702\pi$
$577$	13.0112	0.541662	0.270831	+	0.962627i	$0.412702\pi$
$578$	0	0
$579$	43.0090	1.78739
$580$	0	0
$581$	11.6589	0.483694
$582$	0	0
$583$	−5.81552	−0.240854
$584$	0	0
$585$	57.2055	2.36516
$586$	0	0
$587$	28.2923	1.16775	0.583874	−	0.811844i	$-0.301536\pi$
$587$	28.2923	1.16775	0.583874	+	0.811844i	$0.301536\pi$
$588$	0	0
$589$	36.3306	1.49698
$590$	0	0
$591$	−27.9013	−1.14771
$592$	0	0
$593$	43.0379	1.76735	0.883677	−	0.468098i	$-0.155060\pi$
$593$	43.0379	1.76735	0.883677	+	0.468098i	$0.155060\pi$
$594$	0	0
$595$	−1.17918	−0.0483417
$596$	0	0
$597$	5.24761	0.214770
$598$	0	0
$599$	−26.1921	−1.07018	−0.535090	−	0.844795i	$-0.679722\pi$
$599$	−26.1921	−1.07018	−0.535090	+	0.844795i	$0.679722\pi$
$600$	0	0
$601$	23.2334	0.947712	0.473856	−	0.880602i	$-0.342862\pi$
$601$	23.2334	0.947712	0.473856	+	0.880602i	$0.342862\pi$
$602$	0	0
$603$	0.866759	0.0352972
$604$	0	0
$605$	−12.4667	−0.506845
$606$	0	0
$607$	32.3906	1.31469	0.657347	−	0.753588i	$-0.271678\pi$
$607$	32.3906	1.31469	0.657347	+	0.753588i	$0.271678\pi$
$608$	0	0
$609$	−21.6183	−0.876018
$610$	0	0
$611$	−60.6873	−2.45515
$612$	0	0
$613$	−23.8610	−0.963737	−0.481869	−	0.876243i	$-0.660042\pi$
$613$	−23.8610	−0.963737	−0.481869	+	0.876243i	$0.660042\pi$
$614$	0	0
$615$	−48.4216	−1.95255
$616$	0	0
$617$	37.2777	1.50074	0.750371	−	0.661017i	$-0.229875\pi$
$617$	37.2777	1.50074	0.750371	+	0.661017i	$0.229875\pi$
$618$	0	0
$619$	−22.0151	−0.884862	−0.442431	−	0.896802i	$-0.645884\pi$
$619$	−22.0151	−0.884862	−0.442431	+	0.896802i	$0.645884\pi$
$620$	0	0
$621$	−132.440	−5.31465
$622$	0	0
$623$	−5.96373	−0.238932
$624$	0	0
$625$	5.73038	0.229215
$626$	0	0
$627$	12.7626	0.509690
$628$	0	0
$629$	6.96447	0.277692
$630$	0	0
$631$	−21.9633	−0.874347	−0.437173	−	0.899377i	$-0.644020\pi$
$631$	−21.9633	−0.874347	−0.437173	+	0.899377i	$0.644020\pi$
$632$	0	0
$633$	−5.45867	−0.216963
$634$	0	0
$635$	23.2369	0.922129
$636$	0	0
$637$	−32.3024	−1.27987
$638$	0	0
$639$	−2.63090	−0.104077
$640$	0	0
$641$	23.0944	0.912174	0.456087	−	0.889935i	$-0.349251\pi$
$641$	23.0944	0.912174	0.456087	+	0.889935i	$0.349251\pi$
$642$	0	0
$643$	−11.1967	−0.441554	−0.220777	−	0.975324i	$-0.570859\pi$
$643$	−11.1967	−0.441554	−0.220777	+	0.975324i	$0.570859\pi$
$644$	0	0
$645$	−49.0721	−1.93221
$646$	0	0
$647$	−24.8744	−0.977914	−0.488957	−	0.872308i	$-0.662623\pi$
$647$	−24.8744	−0.977914	−0.488957	+	0.872308i	$0.662623\pi$
$648$	0	0
$649$	−6.32969	−0.248462
$650$	0	0
$651$	−25.8802	−1.01433
$652$	0	0
$653$	19.3616	0.757678	0.378839	−	0.925463i	$-0.376323\pi$
$653$	19.3616	0.757678	0.378839	+	0.925463i	$0.376323\pi$
$654$	0	0
$655$	−7.83031	−0.305955
$656$	0	0
$657$	17.1154	0.667734
$658$	0	0
$659$	−4.91374	−0.191412	−0.0957060	−	0.995410i	$-0.530511\pi$
$659$	−4.91374	−0.191412	−0.0957060	+	0.995410i	$0.530511\pi$
$660$	0	0
$661$	−18.4510	−0.717662	−0.358831	−	0.933402i	$-0.616825\pi$
$661$	−18.4510	−0.717662	−0.358831	+	0.933402i	$0.616825\pi$
$662$	0	0
$663$	17.5851	0.682950
$664$	0	0
$665$	−6.43867	−0.249681
$666$	0	0
$667$	−40.1164	−1.55331
$668$	0	0
$669$	52.2794	2.02124
$670$	0	0
$671$	8.52941	0.329274
$672$	0	0
$673$	−3.71530	−0.143214	−0.0716072	−	0.997433i	$-0.522813\pi$
$673$	−3.71530	−0.143214	−0.0716072	+	0.997433i	$0.522813\pi$
$674$	0	0
$675$	−70.3720	−2.70862
$676$	0	0
$677$	−11.7177	−0.450349	−0.225174	−	0.974318i	$-0.572295\pi$
$677$	−11.7177	−0.450349	−0.225174	+	0.974318i	$0.572295\pi$
$678$	0	0
$679$	−2.50610	−0.0961754
$680$	0	0
$681$	98.7734	3.78501
$682$	0	0
$683$	33.8630	1.29573	0.647867	−	0.761754i	$-0.275661\pi$
$683$	33.8630	1.29573	0.647867	+	0.761754i	$0.275661\pi$
$684$	0	0
$685$	−8.86285	−0.338632
$686$	0	0
$687$	57.0598	2.17697
$688$	0	0
$689$	−43.7532	−1.66686
$690$	0	0
$691$	39.8390	1.51555	0.757773	−	0.652518i	$-0.226287\pi$
$691$	39.8390	1.51555	0.757773	+	0.652518i	$0.226287\pi$
$692$	0	0
$693$	−6.76751	−0.257077
$694$	0	0
$695$	−11.0961	−0.420898
$696$	0	0
$697$	−11.0800	−0.419686
$698$	0	0
$699$	14.7591	0.558239
$700$	0	0
$701$	36.3350	1.37235	0.686177	−	0.727435i	$-0.259288\pi$
$701$	36.3350	1.37235	0.686177	+	0.727435i	$0.259288\pi$
$702$	0	0
$703$	38.0280	1.43425
$704$	0	0
$705$	−45.0480	−1.69661
$706$	0	0
$707$	−1.04962	−0.0394750
$708$	0	0
$709$	8.77864	0.329689	0.164844	−	0.986320i	$-0.447288\pi$
$709$	8.77864	0.329689	0.164844	+	0.986320i	$0.447288\pi$
$710$	0	0
$711$	3.84478	0.144191
$712$	0	0
$713$	−48.0251	−1.79856
$714$	0	0
$715$	4.78467	0.178936
$716$	0	0
$717$	52.9727	1.97830
$718$	0	0
$719$	10.4227	0.388702	0.194351	−	0.980932i	$-0.437740\pi$
$719$	10.4227	0.388702	0.194351	+	0.980932i	$0.437740\pi$
$720$	0	0
$721$	8.40156	0.312890
$722$	0	0
$723$	−31.7411	−1.18047
$724$	0	0
$725$	−21.3158	−0.791649
$726$	0	0
$727$	−33.4171	−1.23937	−0.619686	−	0.784850i	$-0.712740\pi$
$727$	−33.4171	−1.23937	−0.619686	+	0.784850i	$0.712740\pi$
$728$	0	0
$729$	157.191	5.82190
$730$	0	0
$731$	−11.2289	−0.415315
$732$	0	0
$733$	−19.6742	−0.726684	−0.363342	−	0.931656i	$-0.618364\pi$
$733$	−19.6742	−0.726684	−0.363342	+	0.931656i	$0.618364\pi$
$734$	0	0
$735$	−23.9780	−0.884441
$736$	0	0
$737$	0.0724958	0.00267042
$738$	0	0
$739$	25.9998	0.956420	0.478210	−	0.878246i	$-0.341286\pi$
$739$	25.9998	0.956420	0.478210	+	0.878246i	$0.341286\pi$
$740$	0	0
$741$	96.0199	3.52738
$742$	0	0
$743$	−6.01446	−0.220649	−0.110325	−	0.993896i	$-0.535189\pi$
$743$	−6.01446	−0.220649	−0.110325	+	0.993896i	$0.535189\pi$
$744$	0	0
$745$	−17.6529	−0.646752
$746$	0	0
$747$	−96.0715	−3.51507
$748$	0	0
$749$	−2.88370	−0.105368
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−9.84113	−0.358631
$754$	0	0
$755$	11.9540	0.435051
$756$	0	0
$757$	28.1700	1.02386	0.511928	−	0.859029i	$-0.328932\pi$
$757$	28.1700	1.02386	0.511928	+	0.859029i	$0.328932\pi$
$758$	0	0
$759$	−16.8708	−0.612371
$760$	0	0
$761$	4.22909	0.153304	0.0766521	−	0.997058i	$-0.475577\pi$
$761$	4.22909	0.153304	0.0766521	+	0.997058i	$0.475577\pi$
$762$	0	0
$763$	−3.05775	−0.110698
$764$	0	0
$765$	9.71666	0.351307
$766$	0	0
$767$	−47.6215	−1.71951
$768$	0	0
$769$	−23.3601	−0.842386	−0.421193	−	0.906971i	$-0.638389\pi$
$769$	−23.3601	−0.842386	−0.421193	+	0.906971i	$0.638389\pi$
$770$	0	0
$771$	86.2038	3.10455
$772$	0	0
$773$	15.8859	0.571375	0.285687	−	0.958323i	$-0.407778\pi$
$773$	15.8859	0.571375	0.285687	+	0.958323i	$0.407778\pi$
$774$	0	0
$775$	−25.5181	−0.916637
$776$	0	0
$777$	−27.0894	−0.971825
$778$	0	0
$779$	−60.5001	−2.16764
$780$	0	0
$781$	−0.220048	−0.00787395
$782$	0	0
$783$	116.965	4.18000
$784$	0	0
$785$	15.9467	0.569162
$786$	0	0
$787$	−3.77666	−0.134624	−0.0673118	−	0.997732i	$-0.521442\pi$
$787$	−3.77666	−0.134624	−0.0673118	+	0.997732i	$0.521442\pi$
$788$	0	0
$789$	74.5656	2.65461
$790$	0	0
$791$	−3.81510	−0.135649
$792$	0	0
$793$	64.1712	2.27879
$794$	0	0
$795$	−32.4779	−1.15187
$796$	0	0
$797$	−10.8272	−0.383518	−0.191759	−	0.981442i	$-0.561419\pi$
$797$	−10.8272	−0.383518	−0.191759	+	0.981442i	$0.561419\pi$
$798$	0	0
$799$	−10.3081	−0.364673
$800$	0	0
$801$	49.1421	1.73635
$802$	0	0
$803$	1.43153	0.0505176
$804$	0	0
$805$	8.51122	0.299981
$806$	0	0
$807$	56.3683	1.98426
$808$	0	0
$809$	−42.2346	−1.48489	−0.742444	−	0.669908i	$-0.766333\pi$
$809$	−42.2346	−1.48489	−0.742444	+	0.669908i	$0.766333\pi$
$810$	0	0
$811$	−54.3277	−1.90770	−0.953852	−	0.300278i	$-0.902920\pi$
$811$	−54.3277	−1.90770	−0.953852	+	0.300278i	$0.902920\pi$
$812$	0	0
$813$	−77.9270	−2.73302
$814$	0	0
$815$	−21.4432	−0.751123
$816$	0	0
$817$	−61.3129	−2.14507
$818$	0	0
$819$	−50.9155	−1.77913
$820$	0	0
$821$	7.15332	0.249653	0.124826	−	0.992179i	$-0.460163\pi$
$821$	7.15332	0.249653	0.124826	+	0.992179i	$0.460163\pi$
$822$	0	0
$823$	−27.2663	−0.950443	−0.475221	−	0.879866i	$-0.657632\pi$
$823$	−27.2663	−0.950443	−0.475221	+	0.879866i	$0.657632\pi$
$824$	0	0
$825$	−8.96428	−0.312096
$826$	0	0
$827$	−40.2047	−1.39806	−0.699028	−	0.715095i	$-0.746384\pi$
$827$	−40.2047	−1.39806	−0.699028	+	0.715095i	$0.746384\pi$
$828$	0	0
$829$	−39.6423	−1.37683	−0.688417	−	0.725315i	$-0.741694\pi$
$829$	−39.6423	−1.37683	−0.688417	+	0.725315i	$0.741694\pi$
$830$	0	0
$831$	−40.7080	−1.41214
$832$	0	0
$833$	−5.48674	−0.190104
$834$	0	0
$835$	12.5716	0.435058
$836$	0	0
$837$	140.024	4.83995
$838$	0	0
$839$	−33.5662	−1.15883	−0.579417	−	0.815031i	$-0.696720\pi$
$839$	−33.5662	−1.15883	−0.579417	+	0.815031i	$0.696720\pi$
$840$	0	0
$841$	6.42895	0.221688
$842$	0	0
$843$	−42.1667	−1.45230
$844$	0	0
$845$	20.5125	0.705652
$846$	0	0
$847$	11.0960	0.381262
$848$	0	0
$849$	−55.7405	−1.91301
$850$	0	0
$851$	−50.2689	−1.72319
$852$	0	0
$853$	−38.7464	−1.32665	−0.663325	−	0.748331i	$-0.730855\pi$
$853$	−38.7464	−1.32665	−0.663325	+	0.748331i	$0.730855\pi$
$854$	0	0
$855$	53.0558	1.81447
$856$	0	0
$857$	21.4319	0.732099	0.366049	−	0.930595i	$-0.380710\pi$
$857$	21.4319	0.732099	0.366049	+	0.930595i	$0.380710\pi$
$858$	0	0
$859$	−24.6465	−0.840929	−0.420464	−	0.907309i	$-0.638133\pi$
$859$	−24.6465	−0.840929	−0.420464	+	0.907309i	$0.638133\pi$
$860$	0	0
$861$	43.0974	1.46876
$862$	0	0
$863$	3.84033	0.130726	0.0653632	−	0.997862i	$-0.479179\pi$
$863$	3.84033	0.130726	0.0653632	+	0.997862i	$0.479179\pi$
$864$	0	0
$865$	−8.34706	−0.283809
$866$	0	0
$867$	−55.2516	−1.87644
$868$	0	0
$869$	0.321578	0.0109088
$870$	0	0
$871$	0.545423	0.0184810
$872$	0	0
$873$	20.6507	0.698921
$874$	0	0
$875$	10.8366	0.366345
$876$	0	0
$877$	10.9204	0.368756	0.184378	−	0.982855i	$-0.440973\pi$
$877$	10.9204	0.368756	0.184378	+	0.982855i	$0.440973\pi$
$878$	0	0
$879$	103.802	3.50117
$880$	0	0
$881$	14.9811	0.504727	0.252364	−	0.967632i	$-0.418792\pi$
$881$	14.9811	0.504727	0.252364	+	0.967632i	$0.418792\pi$
$882$	0	0
$883$	8.44415	0.284168	0.142084	−	0.989855i	$-0.454620\pi$
$883$	8.44415	0.284168	0.142084	+	0.989855i	$0.454620\pi$
$884$	0	0
$885$	−35.3493	−1.18825
$886$	0	0
$887$	−2.76937	−0.0929864	−0.0464932	−	0.998919i	$-0.514805\pi$
$887$	−2.76937	−0.0929864	−0.0464932	+	0.998919i	$0.514805\pi$
$888$	0	0
$889$	−20.6819	−0.693649
$890$	0	0
$891$	30.0392	1.00635
$892$	0	0
$893$	−56.2850	−1.88351
$894$	0	0
$895$	23.1656	0.774342
$896$	0	0
$897$	−126.928	−4.23800
$898$	0	0
$899$	42.4136	1.41457
$900$	0	0
$901$	−7.43172	−0.247586
$902$	0	0
$903$	43.6764	1.45346
$904$	0	0
$905$	16.7624	0.557201
$906$	0	0
$907$	3.53360	0.117331	0.0586657	−	0.998278i	$-0.481315\pi$
$907$	3.53360	0.117331	0.0586657	+	0.998278i	$0.481315\pi$
$908$	0	0
$909$	8.64904	0.286871
$910$	0	0
$911$	36.2467	1.20091	0.600454	−	0.799659i	$-0.294987\pi$
$911$	36.2467	1.20091	0.600454	+	0.799659i	$0.294987\pi$
$912$	0	0
$913$	−8.03542	−0.265934
$914$	0	0
$915$	47.6341	1.57473
$916$	0	0
$917$	6.96934	0.230148
$918$	0	0
$919$	0.698427	0.0230390	0.0115195	−	0.999934i	$-0.496333\pi$
$919$	0.698427	0.0230390	0.0115195	+	0.999934i	$0.496333\pi$
$920$	0	0
$921$	−29.3354	−0.966633
$922$	0	0
$923$	−1.65554	−0.0544927
$924$	0	0
$925$	−26.7103	−0.878229
$926$	0	0
$927$	−69.2303	−2.27382
$928$	0	0
$929$	37.2289	1.22144	0.610720	−	0.791846i	$-0.290880\pi$
$929$	37.2289	1.22144	0.610720	+	0.791846i	$0.290880\pi$
$930$	0	0
$931$	−29.9592	−0.981872
$932$	0	0
$933$	7.18593	0.235257
$934$	0	0
$935$	0.812702	0.0265782
$936$	0	0
$937$	28.1562	0.919822	0.459911	−	0.887965i	$-0.347881\pi$
$937$	28.1562	0.919822	0.459911	+	0.887965i	$0.347881\pi$
$938$	0	0
$939$	88.9557	2.90296
$940$	0	0
$941$	−28.9329	−0.943186	−0.471593	−	0.881816i	$-0.656321\pi$
$941$	−28.9329	−0.943186	−0.471593	+	0.881816i	$0.656321\pi$
$942$	0	0
$943$	79.9746	2.60433
$944$	0	0
$945$	−24.8157	−0.807255
$946$	0	0
$947$	38.6193	1.25496	0.627479	−	0.778633i	$-0.284087\pi$
$947$	38.6193	1.25496	0.627479	+	0.778633i	$0.284087\pi$
$948$	0	0
$949$	10.7701	0.349613
$950$	0	0
$951$	−91.9991	−2.98328
$952$	0	0
$953$	40.2597	1.30414	0.652069	−	0.758160i	$-0.273901\pi$
$953$	40.2597	1.30414	0.652069	+	0.758160i	$0.273901\pi$
$954$	0	0
$955$	−27.6326	−0.894170
$956$	0	0
$957$	14.8995	0.481633
$958$	0	0
$959$	7.88835	0.254728
$960$	0	0
$961$	19.7752	0.637909
$962$	0	0
$963$	23.7622	0.765725
$964$	0	0
$965$	14.9543	0.481395
$966$	0	0
$967$	16.6932	0.536818	0.268409	−	0.963305i	$-0.413502\pi$
$967$	16.6932	0.536818	0.268409	+	0.963305i	$0.413502\pi$
$968$	0	0
$969$	16.3095	0.523937
$970$	0	0
$971$	−58.3243	−1.87172	−0.935858	−	0.352376i	$-0.885374\pi$
$971$	−58.3243	−1.87172	−0.935858	+	0.352376i	$0.885374\pi$
$972$	0	0
$973$	9.87601	0.316610
$974$	0	0
$975$	−67.4429	−2.15990
$976$	0	0
$977$	29.0123	0.928185	0.464093	−	0.885787i	$-0.346381\pi$
$977$	29.0123	0.928185	0.464093	+	0.885787i	$0.346381\pi$
$978$	0	0
$979$	4.11025	0.131364
$980$	0	0
$981$	25.1964	0.804459
$982$	0	0
$983$	29.1448	0.929574	0.464787	−	0.885422i	$-0.346131\pi$
$983$	29.1448	0.929574	0.464787	+	0.885422i	$0.346131\pi$
$984$	0	0
$985$	−9.70134	−0.309110
$986$	0	0
$987$	40.0948	1.27623
$988$	0	0
$989$	81.0490	2.57721
$990$	0	0
$991$	−8.95498	−0.284464	−0.142232	−	0.989833i	$-0.545428\pi$
$991$	−8.95498	−0.284464	−0.142232	+	0.989833i	$0.545428\pi$
$992$	0	0
$993$	−16.2228	−0.514814
$994$	0	0
$995$	1.82460	0.0578437
$996$	0	0
$997$	−23.4520	−0.742732	−0.371366	−	0.928487i	$-0.621111\pi$
$997$	−23.4520	−0.742732	−0.371366	+	0.928487i	$0.621111\pi$
$998$	0	0
$999$	146.566	4.63715

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.50	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.50	✓	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)

\(f(q)\)	\(=\)	\(q+3.42580 q^{3} +1.19116 q^{5} -1.06018 q^{7} +8.73609 q^{9} +O(q^{10})\)	\(q+3.42580 q^{3} +1.19116 q^{5} -1.06018 q^{7} +8.73609 q^{9} +0.730687 q^{11} +5.49734 q^{13} +4.08066 q^{15} +0.933752 q^{17} +5.09856 q^{19} -3.63197 q^{21} -6.73974 q^{23} -3.58115 q^{25} +19.6507 q^{27} +5.95222 q^{29} +7.12567 q^{31} +2.50318 q^{33} -1.26284 q^{35} +7.45858 q^{37} +18.8328 q^{39} -11.8661 q^{41} -12.0255 q^{43} +10.4060 q^{45} -11.0394 q^{47} -5.87601 q^{49} +3.19885 q^{51} -7.95898 q^{53} +0.870361 q^{55} +17.4666 q^{57} -8.66266 q^{59} +11.6731 q^{61} -9.26185 q^{63} +6.54818 q^{65} +0.0992160 q^{67} -23.0890 q^{69} -0.301153 q^{71} +1.95916 q^{73} -12.2683 q^{75} -0.774662 q^{77} +0.440104 q^{79} +41.1110 q^{81} -10.9971 q^{83} +1.11224 q^{85} +20.3911 q^{87} +5.62519 q^{89} -5.82818 q^{91} +24.4111 q^{93} +6.07317 q^{95} +2.36384 q^{97} +6.38334 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 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

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