LMFDB - Embedded newform 6008.2.a.b.1.29

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:	$1$
Dimension:	$44$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.29
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)

$f(q)$	$=$	$q+0.663224 q^{3} -1.07871 q^{5} +2.50091 q^{7} -2.56013 q^{9} +O(q^{10})$	$q+0.663224 q^{3} -1.07871 q^{5} +2.50091 q^{7} -2.56013 q^{9} +2.33005 q^{11} +3.30037 q^{13} -0.715424 q^{15} -4.49872 q^{17} -1.81768 q^{19} +1.65866 q^{21} +1.64341 q^{23} -3.83639 q^{25} -3.68762 q^{27} -7.87508 q^{29} -4.59701 q^{31} +1.54534 q^{33} -2.69775 q^{35} -5.73720 q^{37} +2.18889 q^{39} +10.3035 q^{41} +9.47732 q^{43} +2.76163 q^{45} +10.0950 q^{47} -0.745444 q^{49} -2.98366 q^{51} -11.8738 q^{53} -2.51344 q^{55} -1.20553 q^{57} -13.4477 q^{59} -6.38387 q^{61} -6.40267 q^{63} -3.56013 q^{65} +4.16459 q^{67} +1.08995 q^{69} -2.34141 q^{71} -13.9134 q^{73} -2.54439 q^{75} +5.82724 q^{77} -15.9669 q^{79} +5.23469 q^{81} +13.0428 q^{83} +4.85280 q^{85} -5.22295 q^{87} -13.1072 q^{89} +8.25394 q^{91} -3.04885 q^{93} +1.96075 q^{95} -12.3368 q^{97} -5.96523 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * 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
$3$	0.663224	0.382913	0.191456	−	0.981501i	$-0.438679\pi$
$3$	0.663224	0.382913	0.191456	+	0.981501i	$0.438679\pi$
$4$	0	0
$5$	−1.07871	−0.482412	−0.241206	−	0.970474i	$-0.577543\pi$
$5$	−1.07871	−0.482412	−0.241206	+	0.970474i	$0.577543\pi$
$6$	0	0
$7$	2.50091	0.945255	0.472628	−	0.881262i	$-0.343306\pi$
$7$	2.50091	0.945255	0.472628	+	0.881262i	$0.343306\pi$
$8$	0	0
$9$	−2.56013	−0.853378
$10$	0	0
$11$	2.33005	0.702536	0.351268	−	0.936275i	$-0.385751\pi$
$11$	2.33005	0.702536	0.351268	+	0.936275i	$0.385751\pi$
$12$	0	0
$13$	3.30037	0.915358	0.457679	−	0.889117i	$-0.348681\pi$
$13$	3.30037	0.915358	0.457679	+	0.889117i	$0.348681\pi$
$14$	0	0
$15$	−0.715424	−0.184722
$16$	0	0
$17$	−4.49872	−1.09110	−0.545550	−	0.838078i	$-0.683679\pi$
$17$	−4.49872	−1.09110	−0.545550	+	0.838078i	$0.683679\pi$
$18$	0	0
$19$	−1.81768	−0.417005	−0.208503	−	0.978022i	$-0.566859\pi$
$19$	−1.81768	−0.417005	−0.208503	+	0.978022i	$0.566859\pi$
$20$	0	0
$21$	1.65866	0.361950
$22$	0	0
$23$	1.64341	0.342676	0.171338	−	0.985212i	$-0.445191\pi$
$23$	1.64341	0.342676	0.171338	+	0.985212i	$0.445191\pi$
$24$	0	0
$25$	−3.83639	−0.767279
$26$	0	0
$27$	−3.68762	−0.709682
$28$	0	0
$29$	−7.87508	−1.46237	−0.731183	−	0.682181i	$-0.761031\pi$
$29$	−7.87508	−1.46237	−0.731183	+	0.682181i	$0.761031\pi$
$30$	0	0
$31$	−4.59701	−0.825648	−0.412824	−	0.910811i	$-0.635458\pi$
$31$	−4.59701	−0.825648	−0.412824	+	0.910811i	$0.635458\pi$
$32$	0	0
$33$	1.54534	0.269010
$34$	0	0
$35$	−2.69775	−0.456003
$36$	0	0
$37$	−5.73720	−0.943190	−0.471595	−	0.881815i	$-0.656322\pi$
$37$	−5.73720	−0.943190	−0.471595	+	0.881815i	$0.656322\pi$
$38$	0	0
$39$	2.18889	0.350502
$40$	0	0
$41$	10.3035	1.60913	0.804566	−	0.593863i	$-0.202398\pi$
$41$	10.3035	1.60913	0.804566	+	0.593863i	$0.202398\pi$
$42$	0	0
$43$	9.47732	1.44528	0.722639	−	0.691225i	$-0.242929\pi$
$43$	9.47732	1.44528	0.722639	+	0.691225i	$0.242929\pi$
$44$	0	0
$45$	2.76163	0.411680
$46$	0	0
$47$	10.0950	1.47251	0.736256	−	0.676704i	$-0.236592\pi$
$47$	10.0950	1.47251	0.736256	+	0.676704i	$0.236592\pi$
$48$	0	0
$49$	−0.745444	−0.106492
$50$	0	0
$51$	−2.98366	−0.417796
$52$	0	0
$53$	−11.8738	−1.63099	−0.815496	−	0.578762i	$-0.803536\pi$
$53$	−11.8738	−1.63099	−0.815496	+	0.578762i	$0.803536\pi$
$54$	0	0
$55$	−2.51344	−0.338912
$56$	0	0
$57$	−1.20553	−0.159677
$58$	0	0
$59$	−13.4477	−1.75075	−0.875373	−	0.483449i	$-0.839384\pi$
$59$	−13.4477	−1.75075	−0.875373	+	0.483449i	$0.839384\pi$
$60$	0	0
$61$	−6.38387	−0.817371	−0.408685	−	0.912675i	$-0.634013\pi$
$61$	−6.38387	−0.817371	−0.408685	+	0.912675i	$0.634013\pi$
$62$	0	0
$63$	−6.40267	−0.806660
$64$	0	0
$65$	−3.56013	−0.441580
$66$	0	0
$67$	4.16459	0.508785	0.254393	−	0.967101i	$-0.418124\pi$
$67$	4.16459	0.508785	0.254393	+	0.967101i	$0.418124\pi$
$68$	0	0
$69$	1.08995	0.131215
$70$	0	0
$71$	−2.34141	−0.277874	−0.138937	−	0.990301i	$-0.544369\pi$
$71$	−2.34141	−0.277874	−0.138937	+	0.990301i	$0.544369\pi$
$72$	0	0
$73$	−13.9134	−1.62844	−0.814222	−	0.580554i	$-0.802836\pi$
$73$	−13.9134	−1.62844	−0.814222	+	0.580554i	$0.802836\pi$
$74$	0	0
$75$	−2.54439	−0.293801
$76$	0	0
$77$	5.82724	0.664076
$78$	0	0
$79$	−15.9669	−1.79642	−0.898209	−	0.439568i	$-0.855132\pi$
$79$	−15.9669	−1.79642	−0.898209	+	0.439568i	$0.855132\pi$
$80$	0	0
$81$	5.23469	0.581632
$82$	0	0
$83$	13.0428	1.43163	0.715816	−	0.698289i	$-0.246055\pi$
$83$	13.0428	1.43163	0.715816	+	0.698289i	$0.246055\pi$
$84$	0	0
$85$	4.85280	0.526360
$86$	0	0
$87$	−5.22295	−0.559959
$88$	0	0
$89$	−13.1072	−1.38937	−0.694683	−	0.719316i	$-0.744455\pi$
$89$	−13.1072	−1.38937	−0.694683	+	0.719316i	$0.744455\pi$
$90$	0	0
$91$	8.25394	0.865248
$92$	0	0
$93$	−3.04885	−0.316151
$94$	0	0
$95$	1.96075	0.201168
$96$	0	0
$97$	−12.3368	−1.25261	−0.626307	−	0.779576i	$-0.715435\pi$
$97$	−12.3368	−1.25261	−0.626307	+	0.779576i	$0.715435\pi$
$98$	0	0
$99$	−5.96523	−0.599529
$100$	0	0
$101$	4.20440	0.418354	0.209177	−	0.977878i	$-0.432922\pi$
$101$	4.20440	0.418354	0.209177	+	0.977878i	$0.432922\pi$
$102$	0	0
$103$	12.9174	1.27279	0.636393	−	0.771365i	$-0.280426\pi$
$103$	12.9174	1.27279	0.636393	+	0.771365i	$0.280426\pi$
$104$	0	0
$105$	−1.78921	−0.174609
$106$	0	0
$107$	12.1232	1.17199	0.585995	−	0.810315i	$-0.300704\pi$
$107$	12.1232	1.17199	0.585995	+	0.810315i	$0.300704\pi$
$108$	0	0
$109$	5.76783	0.552458	0.276229	−	0.961092i	$-0.410915\pi$
$109$	5.76783	0.552458	0.276229	+	0.961092i	$0.410915\pi$
$110$	0	0
$111$	−3.80505	−0.361160
$112$	0	0
$113$	17.4675	1.64320	0.821600	−	0.570064i	$-0.193082\pi$
$113$	17.4675	1.64320	0.821600	+	0.570064i	$0.193082\pi$
$114$	0	0
$115$	−1.77276	−0.165311
$116$	0	0
$117$	−8.44939	−0.781147
$118$	0	0
$119$	−11.2509	−1.03137
$120$	0	0
$121$	−5.57088	−0.506443
$122$	0	0
$123$	6.83351	0.616157
$124$	0	0
$125$	9.53187	0.852557
$126$	0	0
$127$	4.92480	0.437005	0.218503	−	0.975836i	$-0.429883\pi$
$127$	4.92480	0.437005	0.218503	+	0.975836i	$0.429883\pi$
$128$	0	0
$129$	6.28559	0.553415
$130$	0	0
$131$	−0.308622	−0.0269645	−0.0134822	−	0.999909i	$-0.504292\pi$
$131$	−0.308622	−0.0269645	−0.0134822	+	0.999909i	$0.504292\pi$
$132$	0	0
$133$	−4.54587	−0.394177
$134$	0	0
$135$	3.97785	0.342359
$136$	0	0
$137$	1.71206	0.146271	0.0731354	−	0.997322i	$-0.476699\pi$
$137$	1.71206	0.146271	0.0731354	+	0.997322i	$0.476699\pi$
$138$	0	0
$139$	−11.5652	−0.980949	−0.490474	−	0.871456i	$-0.663176\pi$
$139$	−11.5652	−0.980949	−0.490474	+	0.871456i	$0.663176\pi$
$140$	0	0
$141$	6.69527	0.563843
$142$	0	0
$143$	7.69002	0.643072
$144$	0	0
$145$	8.49490	0.705463
$146$	0	0
$147$	−0.494397	−0.0407772
$148$	0	0
$149$	−1.39289	−0.114110	−0.0570548	−	0.998371i	$-0.518171\pi$
$149$	−1.39289	−0.114110	−0.0570548	+	0.998371i	$0.518171\pi$
$150$	0	0
$151$	−4.24994	−0.345855	−0.172928	−	0.984935i	$-0.555323\pi$
$151$	−4.24994	−0.345855	−0.172928	+	0.984935i	$0.555323\pi$
$152$	0	0
$153$	11.5173	0.931121
$154$	0	0
$155$	4.95883	0.398303
$156$	0	0
$157$	0.895739	0.0714877	0.0357439	−	0.999361i	$-0.488620\pi$
$157$	0.895739	0.0714877	0.0357439	+	0.999361i	$0.488620\pi$
$158$	0	0
$159$	−7.87499	−0.624528
$160$	0	0
$161$	4.11003	0.323916
$162$	0	0
$163$	6.20923	0.486345	0.243172	−	0.969983i	$-0.421812\pi$
$163$	6.20923	0.486345	0.243172	+	0.969983i	$0.421812\pi$
$164$	0	0
$165$	−1.66697	−0.129774
$166$	0	0
$167$	12.2306	0.946430	0.473215	−	0.880947i	$-0.343093\pi$
$167$	12.2306	0.946430	0.473215	+	0.880947i	$0.343093\pi$
$168$	0	0
$169$	−2.10755	−0.162119
$170$	0	0
$171$	4.65352	0.355863
$172$	0	0
$173$	−23.9772	−1.82295	−0.911476	−	0.411354i	$-0.865056\pi$
$173$	−23.9772	−1.82295	−0.911476	+	0.411354i	$0.865056\pi$
$174$	0	0
$175$	−9.59448	−0.725274
$176$	0	0
$177$	−8.91886	−0.670383
$178$	0	0
$179$	−14.3798	−1.07480	−0.537399	−	0.843328i	$-0.680593\pi$
$179$	−14.3798	−1.07480	−0.537399	+	0.843328i	$0.680593\pi$
$180$	0	0
$181$	−19.3019	−1.43470	−0.717350	−	0.696713i	$-0.754645\pi$
$181$	−19.3019	−1.43470	−0.717350	+	0.696713i	$0.754645\pi$
$182$	0	0
$183$	−4.23394	−0.312982
$184$	0	0
$185$	6.18876	0.455006
$186$	0	0
$187$	−10.4822	−0.766537
$188$	0	0
$189$	−9.22240	−0.670831
$190$	0	0
$191$	−12.1365	−0.878163	−0.439082	−	0.898447i	$-0.644696\pi$
$191$	−12.1365	−0.878163	−0.439082	+	0.898447i	$0.644696\pi$
$192$	0	0
$193$	−27.0151	−1.94459	−0.972294	−	0.233759i	$-0.924897\pi$
$193$	−27.0151	−1.94459	−0.972294	+	0.233759i	$0.924897\pi$
$194$	0	0
$195$	−2.36117	−0.169087
$196$	0	0
$197$	17.5885	1.25313	0.626563	−	0.779371i	$-0.284461\pi$
$197$	17.5885	1.25313	0.626563	+	0.779371i	$0.284461\pi$
$198$	0	0
$199$	8.49827	0.602427	0.301213	−	0.953557i	$-0.402608\pi$
$199$	8.49827	0.602427	0.301213	+	0.953557i	$0.402608\pi$
$200$	0	0
$201$	2.76206	0.194820
$202$	0	0
$203$	−19.6949	−1.38231
$204$	0	0
$205$	−11.1144	−0.776265
$206$	0	0
$207$	−4.20736	−0.292432
$208$	0	0
$209$	−4.23529	−0.292961
$210$	0	0
$211$	9.46151	0.651357	0.325679	−	0.945480i	$-0.394407\pi$
$211$	9.46151	0.651357	0.325679	+	0.945480i	$0.394407\pi$
$212$	0	0
$213$	−1.55288	−0.106402
$214$	0	0
$215$	−10.2232	−0.697220
$216$	0	0
$217$	−11.4967	−0.780448
$218$	0	0
$219$	−9.22772	−0.623552
$220$	0	0
$221$	−14.8475	−0.998748
$222$	0	0
$223$	−20.7227	−1.38769	−0.693847	−	0.720122i	$-0.744086\pi$
$223$	−20.7227	−1.38769	−0.693847	+	0.720122i	$0.744086\pi$
$224$	0	0
$225$	9.82168	0.654779
$226$	0	0
$227$	−10.3845	−0.689240	−0.344620	−	0.938742i	$-0.611992\pi$
$227$	−10.3845	−0.689240	−0.344620	+	0.938742i	$0.611992\pi$
$228$	0	0
$229$	−0.506045	−0.0334404	−0.0167202	−	0.999860i	$-0.505322\pi$
$229$	−0.506045	−0.0334404	−0.0167202	+	0.999860i	$0.505322\pi$
$230$	0	0
$231$	3.86477	0.254283
$232$	0	0
$233$	5.09021	0.333471	0.166735	−	0.986002i	$-0.446677\pi$
$233$	5.09021	0.333471	0.166735	+	0.986002i	$0.446677\pi$
$234$	0	0
$235$	−10.8896	−0.710357
$236$	0	0
$237$	−10.5896	−0.687872
$238$	0	0
$239$	0.711863	0.0460466	0.0230233	−	0.999735i	$-0.492671\pi$
$239$	0.711863	0.0460466	0.0230233	+	0.999735i	$0.492671\pi$
$240$	0	0
$241$	27.3259	1.76021	0.880107	−	0.474776i	$-0.157471\pi$
$241$	27.3259	1.76021	0.880107	+	0.474776i	$0.157471\pi$
$242$	0	0
$243$	14.5346	0.932396
$244$	0	0
$245$	0.804115	0.0513731
$246$	0	0
$247$	−5.99903	−0.381709
$248$	0	0
$249$	8.65030	0.548190
$250$	0	0
$251$	−18.7231	−1.18179	−0.590896	−	0.806748i	$-0.701226\pi$
$251$	−18.7231	−1.18179	−0.590896	+	0.806748i	$0.701226\pi$
$252$	0	0
$253$	3.82923	0.240742
$254$	0	0
$255$	3.21849	0.201550
$256$	0	0
$257$	14.5018	0.904600	0.452300	−	0.891866i	$-0.350604\pi$
$257$	14.5018	0.904600	0.452300	+	0.891866i	$0.350604\pi$
$258$	0	0
$259$	−14.3482	−0.891556
$260$	0	0
$261$	20.1613	1.24795
$262$	0	0
$263$	−26.7006	−1.64643	−0.823213	−	0.567732i	$-0.807821\pi$
$263$	−26.7006	−1.64643	−0.823213	+	0.567732i	$0.807821\pi$
$264$	0	0
$265$	12.8083	0.786810
$266$	0	0
$267$	−8.69304	−0.532006
$268$	0	0
$269$	−10.1497	−0.618837	−0.309418	−	0.950926i	$-0.600134\pi$
$269$	−10.1497	−0.618837	−0.309418	+	0.950926i	$0.600134\pi$
$270$	0	0
$271$	1.47199	0.0894171	0.0447085	−	0.999000i	$-0.485764\pi$
$271$	1.47199	0.0894171	0.0447085	+	0.999000i	$0.485764\pi$
$272$	0	0
$273$	5.47421	0.331314
$274$	0	0
$275$	−8.93898	−0.539041
$276$	0	0
$277$	−23.7554	−1.42732	−0.713661	−	0.700491i	$-0.752964\pi$
$277$	−23.7554	−1.42732	−0.713661	+	0.700491i	$0.752964\pi$
$278$	0	0
$279$	11.7690	0.704590
$280$	0	0
$281$	15.4337	0.920700	0.460350	−	0.887737i	$-0.347724\pi$
$281$	15.4337	0.920700	0.460350	+	0.887737i	$0.347724\pi$
$282$	0	0
$283$	−7.12244	−0.423385	−0.211693	−	0.977336i	$-0.567898\pi$
$283$	−7.12244	−0.423385	−0.211693	+	0.977336i	$0.567898\pi$
$284$	0	0
$285$	1.30042	0.0770299
$286$	0	0
$287$	25.7681	1.52104
$288$	0	0
$289$	3.23850	0.190500
$290$	0	0
$291$	−8.18208	−0.479642
$292$	0	0
$293$	−5.92650	−0.346230	−0.173115	−	0.984902i	$-0.555383\pi$
$293$	−5.92650	−0.346230	−0.173115	+	0.984902i	$0.555383\pi$
$294$	0	0
$295$	14.5061	0.844581
$296$	0	0
$297$	−8.59232	−0.498577
$298$	0	0
$299$	5.42388	0.313671
$300$	0	0
$301$	23.7019	1.36616
$302$	0	0
$303$	2.78846	0.160193
$304$	0	0
$305$	6.88632	0.394310
$306$	0	0
$307$	−3.75785	−0.214472	−0.107236	−	0.994234i	$-0.534200\pi$
$307$	−3.75785	−0.214472	−0.107236	+	0.994234i	$0.534200\pi$
$308$	0	0
$309$	8.56711	0.487366
$310$	0	0
$311$	11.5667	0.655885	0.327943	−	0.944698i	$-0.393645\pi$
$311$	11.5667	0.655885	0.327943	+	0.944698i	$0.393645\pi$
$312$	0	0
$313$	−18.0294	−1.01908	−0.509541	−	0.860446i	$-0.670185\pi$
$313$	−18.0294	−1.01908	−0.509541	+	0.860446i	$0.670185\pi$
$314$	0	0
$315$	6.90660	0.389143
$316$	0	0
$317$	7.77961	0.436946	0.218473	−	0.975843i	$-0.429892\pi$
$317$	7.77961	0.436946	0.218473	+	0.975843i	$0.429892\pi$
$318$	0	0
$319$	−18.3493	−1.02736
$320$	0	0
$321$	8.04038	0.448770
$322$	0	0
$323$	8.17726	0.454995
$324$	0	0
$325$	−12.6615	−0.702335
$326$	0	0
$327$	3.82537	0.211543
$328$	0	0
$329$	25.2468	1.39190
$330$	0	0
$331$	0.760125	0.0417803	0.0208901	−	0.999782i	$-0.493350\pi$
$331$	0.760125	0.0417803	0.0208901	+	0.999782i	$0.493350\pi$
$332$	0	0
$333$	14.6880	0.804898
$334$	0	0
$335$	−4.49237	−0.245444
$336$	0	0
$337$	1.93081	0.105178	0.0525889	−	0.998616i	$-0.483253\pi$
$337$	1.93081	0.105178	0.0525889	+	0.998616i	$0.483253\pi$
$338$	0	0
$339$	11.5848	0.629202
$340$	0	0
$341$	−10.7113	−0.580047
$342$	0	0
$343$	−19.3707	−1.04592
$344$	0	0
$345$	−1.17574	−0.0632996
$346$	0	0
$347$	−3.95219	−0.212165	−0.106082	−	0.994357i	$-0.533831\pi$
$347$	−3.95219	−0.212165	−0.106082	+	0.994357i	$0.533831\pi$
$348$	0	0
$349$	−19.2682	−1.03140	−0.515702	−	0.856768i	$-0.672469\pi$
$349$	−19.2682	−1.03140	−0.515702	+	0.856768i	$0.672469\pi$
$350$	0	0
$351$	−12.1705	−0.649613
$352$	0	0
$353$	36.0745	1.92005	0.960025	−	0.279914i	$-0.0903061\pi$
$353$	36.0745	1.92005	0.960025	+	0.279914i	$0.0903061\pi$
$354$	0	0
$355$	2.52569	0.134050
$356$	0	0
$357$	−7.46187	−0.394924
$358$	0	0
$359$	−12.0312	−0.634984	−0.317492	−	0.948261i	$-0.602841\pi$
$359$	−12.0312	−0.634984	−0.317492	+	0.948261i	$0.602841\pi$
$360$	0	0
$361$	−15.6960	−0.826106
$362$	0	0
$363$	−3.69474	−0.193924
$364$	0	0
$365$	15.0085	0.785581
$366$	0	0
$367$	7.08143	0.369648	0.184824	−	0.982772i	$-0.440829\pi$
$367$	7.08143	0.369648	0.184824	+	0.982772i	$0.440829\pi$
$368$	0	0
$369$	−26.3783	−1.37320
$370$	0	0
$371$	−29.6953	−1.54170
$372$	0	0
$373$	−2.94072	−0.152265	−0.0761324	−	0.997098i	$-0.524257\pi$
$373$	−2.94072	−0.152265	−0.0761324	+	0.997098i	$0.524257\pi$
$374$	0	0
$375$	6.32177	0.326455
$376$	0	0
$377$	−25.9907	−1.33859
$378$	0	0
$379$	−7.12347	−0.365908	−0.182954	−	0.983121i	$-0.558566\pi$
$379$	−7.12347	−0.365908	−0.182954	+	0.983121i	$0.558566\pi$
$380$	0	0
$381$	3.26625	0.167335
$382$	0	0
$383$	20.2810	1.03631	0.518155	−	0.855287i	$-0.326619\pi$
$383$	20.2810	1.03631	0.518155	+	0.855287i	$0.326619\pi$
$384$	0	0
$385$	−6.28588	−0.320358
$386$	0	0
$387$	−24.2632	−1.23337
$388$	0	0
$389$	34.2986	1.73901	0.869504	−	0.493925i	$-0.164438\pi$
$389$	34.2986	1.73901	0.869504	+	0.493925i	$0.164438\pi$
$390$	0	0
$391$	−7.39326	−0.373893
$392$	0	0
$393$	−0.204686	−0.0103250
$394$	0	0
$395$	17.2236	0.866614
$396$	0	0
$397$	2.31254	0.116063	0.0580315	−	0.998315i	$-0.481518\pi$
$397$	2.31254	0.116063	0.0580315	+	0.998315i	$0.481518\pi$
$398$	0	0
$399$	−3.01493	−0.150935
$400$	0	0
$401$	−16.3015	−0.814056	−0.407028	−	0.913416i	$-0.633435\pi$
$401$	−16.3015	−0.814056	−0.407028	+	0.913416i	$0.633435\pi$
$402$	0	0
$403$	−15.1719	−0.755764
$404$	0	0
$405$	−5.64669	−0.280586
$406$	0	0
$407$	−13.3680	−0.662625
$408$	0	0
$409$	−17.4264	−0.861680	−0.430840	−	0.902428i	$-0.641783\pi$
$409$	−17.4264	−0.861680	−0.430840	+	0.902428i	$0.641783\pi$
$410$	0	0
$411$	1.13548	0.0560090
$412$	0	0
$413$	−33.6316	−1.65490
$414$	0	0
$415$	−14.0693	−0.690637
$416$	0	0
$417$	−7.67033	−0.375618
$418$	0	0
$419$	−32.0218	−1.56437	−0.782183	−	0.623049i	$-0.785894\pi$
$419$	−32.0218	−1.56437	−0.782183	+	0.623049i	$0.785894\pi$
$420$	0	0
$421$	−5.63190	−0.274482	−0.137241	−	0.990538i	$-0.543824\pi$
$421$	−5.63190	−0.274482	−0.137241	+	0.990538i	$0.543824\pi$
$422$	0	0
$423$	−25.8446	−1.25661
$424$	0	0
$425$	17.2589	0.837178
$426$	0	0
$427$	−15.9655	−0.772624
$428$	0	0
$429$	5.10021	0.246240
$430$	0	0
$431$	−0.666942	−0.0321255	−0.0160627	−	0.999871i	$-0.505113\pi$
$431$	−0.666942	−0.0321255	−0.0160627	+	0.999871i	$0.505113\pi$
$432$	0	0
$433$	18.2033	0.874793	0.437397	−	0.899269i	$-0.355901\pi$
$433$	18.2033	0.874793	0.437397	+	0.899269i	$0.355901\pi$
$434$	0	0
$435$	5.63402	0.270131
$436$	0	0
$437$	−2.98721	−0.142898
$438$	0	0
$439$	−11.4257	−0.545320	−0.272660	−	0.962110i	$-0.587903\pi$
$439$	−11.4257	−0.545320	−0.272660	+	0.962110i	$0.587903\pi$
$440$	0	0
$441$	1.90844	0.0908780
$442$	0	0
$443$	−31.3327	−1.48866	−0.744331	−	0.667811i	$-0.767231\pi$
$443$	−31.3327	−1.48866	−0.744331	+	0.667811i	$0.767231\pi$
$444$	0	0
$445$	14.1389	0.670247
$446$	0	0
$447$	−0.923795	−0.0436940
$448$	0	0
$449$	−15.1648	−0.715671	−0.357835	−	0.933785i	$-0.616485\pi$
$449$	−15.1648	−0.715671	−0.357835	+	0.933785i	$0.616485\pi$
$450$	0	0
$451$	24.0076	1.13047
$452$	0	0
$453$	−2.81867	−0.132432
$454$	0	0
$455$	−8.90357	−0.417406
$456$	0	0
$457$	−12.6347	−0.591025	−0.295513	−	0.955339i	$-0.595490\pi$
$457$	−12.6347	−0.591025	−0.295513	+	0.955339i	$0.595490\pi$
$458$	0	0
$459$	16.5896	0.774334
$460$	0	0
$461$	33.4631	1.55853	0.779265	−	0.626694i	$-0.215592\pi$
$461$	33.4631	1.55853	0.779265	+	0.626694i	$0.215592\pi$
$462$	0	0
$463$	5.07551	0.235879	0.117940	−	0.993021i	$-0.462371\pi$
$463$	5.07551	0.235879	0.117940	+	0.993021i	$0.462371\pi$
$464$	0	0
$465$	3.28881	0.152515
$466$	0	0
$467$	33.0914	1.53129	0.765644	−	0.643265i	$-0.222421\pi$
$467$	33.0914	1.53129	0.765644	+	0.643265i	$0.222421\pi$
$468$	0	0
$469$	10.4153	0.480932
$470$	0	0
$471$	0.594076	0.0273736
$472$	0	0
$473$	22.0826	1.01536
$474$	0	0
$475$	6.97335	0.319959
$476$	0	0
$477$	30.3985	1.39185
$478$	0	0
$479$	−11.2745	−0.515147	−0.257573	−	0.966259i	$-0.582923\pi$
$479$	−11.2745	−0.515147	−0.257573	+	0.966259i	$0.582923\pi$
$480$	0	0
$481$	−18.9349	−0.863357
$482$	0	0
$483$	2.72587	0.124032
$484$	0	0
$485$	13.3078	0.604277
$486$	0	0
$487$	10.3991	0.471226	0.235613	−	0.971847i	$-0.424290\pi$
$487$	10.3991	0.471226	0.235613	+	0.971847i	$0.424290\pi$
$488$	0	0
$489$	4.11811	0.186228
$490$	0	0
$491$	−18.9188	−0.853794	−0.426897	−	0.904300i	$-0.640393\pi$
$491$	−18.9188	−0.853794	−0.426897	+	0.904300i	$0.640393\pi$
$492$	0	0
$493$	35.4278	1.59559
$494$	0	0
$495$	6.43474	0.289220
$496$	0	0
$497$	−5.85566	−0.262662
$498$	0	0
$499$	−7.07362	−0.316659	−0.158329	−	0.987386i	$-0.550611\pi$
$499$	−7.07362	−0.316659	−0.158329	+	0.987386i	$0.550611\pi$
$500$	0	0
$501$	8.11161	0.362400
$502$	0	0
$503$	−8.88782	−0.396288	−0.198144	−	0.980173i	$-0.563491\pi$
$503$	−8.88782	−0.396288	−0.198144	+	0.980173i	$0.563491\pi$
$504$	0	0
$505$	−4.53531	−0.201819
$506$	0	0
$507$	−1.39778	−0.0620774
$508$	0	0
$509$	27.4046	1.21469	0.607344	−	0.794439i	$-0.292235\pi$
$509$	27.4046	1.21469	0.607344	+	0.794439i	$0.292235\pi$
$510$	0	0
$511$	−34.7962	−1.53930
$512$	0	0
$513$	6.70292	0.295941
$514$	0	0
$515$	−13.9340	−0.614007
$516$	0	0
$517$	23.5219	1.03449
$518$	0	0
$519$	−15.9022	−0.698031
$520$	0	0
$521$	6.10425	0.267432	0.133716	−	0.991020i	$-0.457309\pi$
$521$	6.10425	0.267432	0.133716	+	0.991020i	$0.457309\pi$
$522$	0	0
$523$	12.5988	0.550908	0.275454	−	0.961314i	$-0.411172\pi$
$523$	12.5988	0.550908	0.275454	+	0.961314i	$0.411172\pi$
$524$	0	0
$525$	−6.36329	−0.277717
$526$	0	0
$527$	20.6807	0.900865
$528$	0	0
$529$	−20.2992	−0.882573
$530$	0	0
$531$	34.4280	1.49405
$532$	0	0
$533$	34.0053	1.47293
$534$	0	0
$535$	−13.0773	−0.565382
$536$	0	0
$537$	−9.53704	−0.411553
$538$	0	0
$539$	−1.73692	−0.0748145
$540$	0	0
$541$	40.8577	1.75661	0.878305	−	0.478102i	$-0.158675\pi$
$541$	40.8577	1.75661	0.878305	+	0.478102i	$0.158675\pi$
$542$	0	0
$543$	−12.8015	−0.549365
$544$	0	0
$545$	−6.22180	−0.266512
$546$	0	0
$547$	18.3384	0.784094	0.392047	−	0.919945i	$-0.371767\pi$
$547$	18.3384	0.784094	0.392047	+	0.919945i	$0.371767\pi$
$548$	0	0
$549$	16.3436	0.697526
$550$	0	0
$551$	14.3144	0.609815
$552$	0	0
$553$	−39.9318	−1.69807
$554$	0	0
$555$	4.10453	0.174228
$556$	0	0
$557$	14.3809	0.609338	0.304669	−	0.952458i	$-0.401454\pi$
$557$	14.3809	0.609338	0.304669	+	0.952458i	$0.401454\pi$
$558$	0	0
$559$	31.2787	1.32295
$560$	0	0
$561$	−6.95207	−0.293517
$562$	0	0
$563$	−23.8206	−1.00392	−0.501958	−	0.864892i	$-0.667387\pi$
$563$	−23.8206	−1.00392	−0.501958	+	0.864892i	$0.667387\pi$
$564$	0	0
$565$	−18.8423	−0.792700
$566$	0	0
$567$	13.0915	0.549791
$568$	0	0
$569$	−39.6960	−1.66414	−0.832072	−	0.554667i	$-0.812846\pi$
$569$	−39.6960	−1.66414	−0.832072	+	0.554667i	$0.812846\pi$
$570$	0	0
$571$	16.7390	0.700505	0.350253	−	0.936655i	$-0.386096\pi$
$571$	16.7390	0.700505	0.350253	+	0.936655i	$0.386096\pi$
$572$	0	0
$573$	−8.04919	−0.336260
$574$	0	0
$575$	−6.30478	−0.262928
$576$	0	0
$577$	9.71337	0.404373	0.202187	−	0.979347i	$-0.435195\pi$
$577$	9.71337	0.404373	0.202187	+	0.979347i	$0.435195\pi$
$578$	0	0
$579$	−17.9171	−0.744608
$580$	0	0
$581$	32.6189	1.35326
$582$	0	0
$583$	−27.6665	−1.14583
$584$	0	0
$585$	9.11441	0.376835
$586$	0	0
$587$	43.1562	1.78125	0.890623	−	0.454743i	$-0.150269\pi$
$587$	43.1562	1.78125	0.890623	+	0.454743i	$0.150269\pi$
$588$	0	0
$589$	8.35592	0.344300
$590$	0	0
$591$	11.6651	0.479838
$592$	0	0
$593$	25.8999	1.06358	0.531791	−	0.846875i	$-0.321519\pi$
$593$	25.8999	1.06358	0.531791	+	0.846875i	$0.321519\pi$
$594$	0	0
$595$	12.1364	0.497545
$596$	0	0
$597$	5.63626	0.230677
$598$	0	0
$599$	−3.99357	−0.163173	−0.0815864	−	0.996666i	$-0.525999\pi$
$599$	−3.99357	−0.163173	−0.0815864	+	0.996666i	$0.525999\pi$
$600$	0	0
$601$	−42.9892	−1.75357	−0.876784	−	0.480885i	$-0.840315\pi$
$601$	−42.9892	−1.75357	−0.876784	+	0.480885i	$0.840315\pi$
$602$	0	0
$603$	−10.6619	−0.434186
$604$	0	0
$605$	6.00934	0.244314
$606$	0	0
$607$	9.71218	0.394205	0.197103	−	0.980383i	$-0.436847\pi$
$607$	9.71218	0.394205	0.197103	+	0.980383i	$0.436847\pi$
$608$	0	0
$609$	−13.0621	−0.529304
$610$	0	0
$611$	33.3173	1.34788
$612$	0	0
$613$	19.9490	0.805731	0.402865	−	0.915259i	$-0.368014\pi$
$613$	19.9490	0.805731	0.402865	+	0.915259i	$0.368014\pi$
$614$	0	0
$615$	−7.37135	−0.297242
$616$	0	0
$617$	26.5224	1.06775	0.533876	−	0.845563i	$-0.320735\pi$
$617$	26.5224	1.06775	0.533876	+	0.845563i	$0.320735\pi$
$618$	0	0
$619$	−20.8690	−0.838798	−0.419399	−	0.907802i	$-0.637759\pi$
$619$	−20.8690	−0.838798	−0.419399	+	0.907802i	$0.637759\pi$
$620$	0	0
$621$	−6.06028	−0.243191
$622$	0	0
$623$	−32.7801	−1.31331
$624$	0	0
$625$	8.89988	0.355995
$626$	0	0
$627$	−2.80895	−0.112179
$628$	0	0
$629$	25.8101	1.02912
$630$	0	0
$631$	−11.5300	−0.459002	−0.229501	−	0.973308i	$-0.573709\pi$
$631$	−11.5300	−0.459002	−0.229501	+	0.973308i	$0.573709\pi$
$632$	0	0
$633$	6.27510	0.249413
$634$	0	0
$635$	−5.31241	−0.210817
$636$	0	0
$637$	−2.46024	−0.0974784
$638$	0	0
$639$	5.99432	0.237132
$640$	0	0
$641$	−33.8084	−1.33535	−0.667676	−	0.744452i	$-0.732711\pi$
$641$	−33.8084	−1.33535	−0.667676	+	0.744452i	$0.732711\pi$
$642$	0	0
$643$	−20.9985	−0.828098	−0.414049	−	0.910255i	$-0.635886\pi$
$643$	−20.9985	−0.828098	−0.414049	+	0.910255i	$0.635886\pi$
$644$	0	0
$645$	−6.78031	−0.266974
$646$	0	0
$647$	38.1804	1.50103	0.750514	−	0.660855i	$-0.229806\pi$
$647$	38.1804	1.50103	0.750514	+	0.660855i	$0.229806\pi$
$648$	0	0
$649$	−31.3339	−1.22996
$650$	0	0
$651$	−7.62491	−0.298844
$652$	0	0
$653$	8.44328	0.330411	0.165206	−	0.986259i	$-0.447171\pi$
$653$	8.44328	0.330411	0.165206	+	0.986259i	$0.447171\pi$
$654$	0	0
$655$	0.332913	0.0130080
$656$	0	0
$657$	35.6202	1.38968
$658$	0	0
$659$	−29.1024	−1.13367	−0.566834	−	0.823832i	$-0.691832\pi$
$659$	−29.1024	−1.13367	−0.566834	+	0.823832i	$0.691832\pi$
$660$	0	0
$661$	15.2277	0.592291	0.296145	−	0.955143i	$-0.404299\pi$
$661$	15.2277	0.592291	0.296145	+	0.955143i	$0.404299\pi$
$662$	0	0
$663$	−9.84719	−0.382433
$664$	0	0
$665$	4.90365	0.190156
$666$	0	0
$667$	−12.9420	−0.501117
$668$	0	0
$669$	−13.7438	−0.531366
$670$	0	0
$671$	−14.8747	−0.574232
$672$	0	0
$673$	12.9620	0.499649	0.249825	−	0.968291i	$-0.419627\pi$
$673$	12.9620	0.499649	0.249825	+	0.968291i	$0.419627\pi$
$674$	0	0
$675$	14.1471	0.544524
$676$	0	0
$677$	−38.8478	−1.49304	−0.746521	−	0.665362i	$-0.768277\pi$
$677$	−38.8478	−1.49304	−0.746521	+	0.665362i	$0.768277\pi$
$678$	0	0
$679$	−30.8533	−1.18404
$680$	0	0
$681$	−6.88722	−0.263919
$682$	0	0
$683$	11.8915	0.455015	0.227507	−	0.973776i	$-0.426942\pi$
$683$	11.8915	0.455015	0.227507	+	0.973776i	$0.426942\pi$
$684$	0	0
$685$	−1.84681	−0.0705628
$686$	0	0
$687$	−0.335621	−0.0128047
$688$	0	0
$689$	−39.1880	−1.49294
$690$	0	0
$691$	−25.6778	−0.976829	−0.488415	−	0.872612i	$-0.662425\pi$
$691$	−25.6778	−0.976829	−0.488415	+	0.872612i	$0.662425\pi$
$692$	0	0
$693$	−14.9185	−0.566708
$694$	0	0
$695$	12.4755	0.473221
$696$	0	0
$697$	−46.3525	−1.75573
$698$	0	0
$699$	3.37595	0.127690
$700$	0	0
$701$	−41.1163	−1.55294	−0.776470	−	0.630154i	$-0.782992\pi$
$701$	−41.1163	−1.55294	−0.776470	+	0.630154i	$0.782992\pi$
$702$	0	0
$703$	10.4284	0.393316
$704$	0	0
$705$	−7.22223	−0.272005
$706$	0	0
$707$	10.5148	0.395451
$708$	0	0
$709$	−9.65505	−0.362603	−0.181302	−	0.983428i	$-0.558031\pi$
$709$	−9.65505	−0.362603	−0.181302	+	0.983428i	$0.558031\pi$
$710$	0	0
$711$	40.8775	1.53302
$712$	0	0
$713$	−7.55480	−0.282929
$714$	0	0
$715$	−8.29528	−0.310226
$716$	0	0
$717$	0.472125	0.0176318
$718$	0	0
$719$	24.1834	0.901888	0.450944	−	0.892552i	$-0.351087\pi$
$719$	24.1834	0.901888	0.450944	+	0.892552i	$0.351087\pi$
$720$	0	0
$721$	32.3052	1.20311
$722$	0	0
$723$	18.1232	0.674008
$724$	0	0
$725$	30.2119	1.12204
$726$	0	0
$727$	19.4358	0.720832	0.360416	−	0.932792i	$-0.382635\pi$
$727$	19.4358	0.720832	0.360416	+	0.932792i	$0.382635\pi$
$728$	0	0
$729$	−6.06435	−0.224605
$730$	0	0
$731$	−42.6359	−1.57694
$732$	0	0
$733$	37.5471	1.38683	0.693416	−	0.720537i	$-0.256105\pi$
$733$	37.5471	1.38683	0.693416	+	0.720537i	$0.256105\pi$
$734$	0	0
$735$	0.533309	0.0196714
$736$	0	0
$737$	9.70369	0.357440
$738$	0	0
$739$	42.8033	1.57455	0.787273	−	0.616605i	$-0.211493\pi$
$739$	42.8033	1.57455	0.787273	+	0.616605i	$0.211493\pi$
$740$	0	0
$741$	−3.97870	−0.146161
$742$	0	0
$743$	−24.4878	−0.898369	−0.449184	−	0.893439i	$-0.648285\pi$
$743$	−24.4878	−0.898369	−0.449184	+	0.893439i	$0.648285\pi$
$744$	0	0
$745$	1.50251	0.0550478
$746$	0	0
$747$	−33.3913	−1.22172
$748$	0	0
$749$	30.3190	1.10783
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−12.4176	−0.452523
$754$	0	0
$755$	4.58444	0.166845
$756$	0	0
$757$	9.33425	0.339259	0.169630	−	0.985508i	$-0.445743\pi$
$757$	9.33425	0.339259	0.169630	+	0.985508i	$0.445743\pi$
$758$	0	0
$759$	2.53964	0.0921831
$760$	0	0
$761$	−17.6711	−0.640577	−0.320289	−	0.947320i	$-0.603780\pi$
$761$	−17.6711	−0.640577	−0.320289	+	0.947320i	$0.603780\pi$
$762$	0	0
$763$	14.4248	0.522214
$764$	0	0
$765$	−12.4238	−0.449184
$766$	0	0
$767$	−44.3825	−1.60256
$768$	0	0
$769$	−23.1060	−0.833224	−0.416612	−	0.909084i	$-0.636783\pi$
$769$	−23.1060	−0.833224	−0.416612	+	0.909084i	$0.636783\pi$
$770$	0	0
$771$	9.61797	0.346383
$772$	0	0
$773$	−3.82912	−0.137724	−0.0688619	−	0.997626i	$-0.521937\pi$
$773$	−3.82912	−0.137724	−0.0688619	+	0.997626i	$0.521937\pi$
$774$	0	0
$775$	17.6360	0.633502
$776$	0	0
$777$	−9.51610	−0.341388
$778$	0	0
$779$	−18.7285	−0.671017
$780$	0	0
$781$	−5.45560	−0.195217
$782$	0	0
$783$	29.0403	1.03781
$784$	0	0
$785$	−0.966239	−0.0344865
$786$	0	0
$787$	−34.0734	−1.21458	−0.607292	−	0.794478i	$-0.707744\pi$
$787$	−34.0734	−1.21458	−0.607292	+	0.794478i	$0.707744\pi$
$788$	0	0
$789$	−17.7085	−0.630438
$790$	0	0
$791$	43.6846	1.55324
$792$	0	0
$793$	−21.0691	−0.748187
$794$	0	0
$795$	8.49480	0.301280
$796$	0	0
$797$	24.8759	0.881148	0.440574	−	0.897716i	$-0.354775\pi$
$797$	24.8759	0.881148	0.440574	+	0.897716i	$0.354775\pi$
$798$	0	0
$799$	−45.4147	−1.60666
$800$	0	0
$801$	33.5563	1.18565
$802$	0	0
$803$	−32.4190	−1.14404
$804$	0	0
$805$	−4.43352	−0.156261
$806$	0	0
$807$	−6.73151	−0.236960
$808$	0	0
$809$	16.0430	0.564042	0.282021	−	0.959408i	$-0.408995\pi$
$809$	16.0430	0.564042	0.282021	+	0.959408i	$0.408995\pi$
$810$	0	0
$811$	18.5400	0.651026	0.325513	−	0.945538i	$-0.394463\pi$
$811$	18.5400	0.651026	0.325513	+	0.945538i	$0.394463\pi$
$812$	0	0
$813$	0.976260	0.0342389
$814$	0	0
$815$	−6.69794	−0.234619
$816$	0	0
$817$	−17.2268	−0.602689
$818$	0	0
$819$	−21.1312	−0.738383
$820$	0	0
$821$	36.8643	1.28657	0.643287	−	0.765625i	$-0.277570\pi$
$821$	36.8643	1.28657	0.643287	+	0.765625i	$0.277570\pi$
$822$	0	0
$823$	16.3958	0.571522	0.285761	−	0.958301i	$-0.407754\pi$
$823$	16.3958	0.571522	0.285761	+	0.958301i	$0.407754\pi$
$824$	0	0
$825$	−5.92855	−0.206406
$826$	0	0
$827$	18.1533	0.631252	0.315626	−	0.948884i	$-0.397785\pi$
$827$	18.1533	0.631252	0.315626	+	0.948884i	$0.397785\pi$
$828$	0	0
$829$	−32.2184	−1.11899	−0.559495	−	0.828834i	$-0.689005\pi$
$829$	−32.2184	−1.11899	−0.559495	+	0.828834i	$0.689005\pi$
$830$	0	0
$831$	−15.7551	−0.546540
$832$	0	0
$833$	3.35355	0.116194
$834$	0	0
$835$	−13.1932	−0.456569
$836$	0	0
$837$	16.9520	0.585948
$838$	0	0
$839$	40.7231	1.40592	0.702959	−	0.711230i	$-0.251862\pi$
$839$	40.7231	1.40592	0.702959	+	0.711230i	$0.251862\pi$
$840$	0	0
$841$	33.0169	1.13851
$842$	0	0
$843$	10.2360	0.352548
$844$	0	0
$845$	2.27342	0.0782082
$846$	0	0
$847$	−13.9323	−0.478718
$848$	0	0
$849$	−4.72378	−0.162120
$850$	0	0
$851$	−9.42860	−0.323208
$852$	0	0
$853$	35.1478	1.20344	0.601719	−	0.798708i	$-0.294483\pi$
$853$	35.1478	1.20344	0.601719	+	0.798708i	$0.294483\pi$
$854$	0	0
$855$	−5.01978	−0.171673
$856$	0	0
$857$	−34.5643	−1.18069	−0.590347	−	0.807149i	$-0.701009\pi$
$857$	−34.5643	−1.18069	−0.590347	+	0.807149i	$0.701009\pi$
$858$	0	0
$859$	39.3824	1.34371	0.671855	−	0.740682i	$-0.265498\pi$
$859$	39.3824	1.34371	0.671855	+	0.740682i	$0.265498\pi$
$860$	0	0
$861$	17.0900	0.582426
$862$	0	0
$863$	−25.8229	−0.879021	−0.439510	−	0.898237i	$-0.644848\pi$
$863$	−25.8229	−0.879021	−0.439510	+	0.898237i	$0.644848\pi$
$864$	0	0
$865$	25.8643	0.879414
$866$	0	0
$867$	2.14785	0.0729450
$868$	0	0
$869$	−37.2037	−1.26205
$870$	0	0
$871$	13.7447	0.465721
$872$	0	0
$873$	31.5839	1.06895
$874$	0	0
$875$	23.8384	0.805884
$876$	0	0
$877$	20.0866	0.678278	0.339139	−	0.940736i	$-0.389864\pi$
$877$	20.0866	0.678278	0.339139	+	0.940736i	$0.389864\pi$
$878$	0	0
$879$	−3.93060	−0.132576
$880$	0	0
$881$	−37.8109	−1.27388	−0.636941	−	0.770913i	$-0.719800\pi$
$881$	−37.8109	−1.27388	−0.636941	+	0.770913i	$0.719800\pi$
$882$	0	0
$883$	19.6487	0.661231	0.330615	−	0.943766i	$-0.392744\pi$
$883$	19.6487	0.661231	0.330615	+	0.943766i	$0.392744\pi$
$884$	0	0
$885$	9.62083	0.323401
$886$	0	0
$887$	52.3896	1.75907	0.879536	−	0.475833i	$-0.157853\pi$
$887$	52.3896	1.75907	0.879536	+	0.475833i	$0.157853\pi$
$888$	0	0
$889$	12.3165	0.413082
$890$	0	0
$891$	12.1971	0.408617
$892$	0	0
$893$	−18.3496	−0.614045
$894$	0	0
$895$	15.5116	0.518495
$896$	0	0
$897$	3.59725	0.120109
$898$	0	0
$899$	36.2019	1.20740
$900$	0	0
$901$	53.4169	1.77958
$902$	0	0
$903$	15.7197	0.523119
$904$	0	0
$905$	20.8211	0.692117
$906$	0	0
$907$	−37.0499	−1.23022	−0.615111	−	0.788440i	$-0.710889\pi$
$907$	−37.0499	−1.23022	−0.615111	+	0.788440i	$0.710889\pi$
$908$	0	0
$909$	−10.7638	−0.357014
$910$	0	0
$911$	−6.35015	−0.210390	−0.105195	−	0.994452i	$-0.533547\pi$
$911$	−6.35015	−0.210390	−0.105195	+	0.994452i	$0.533547\pi$
$912$	0	0
$913$	30.3903	1.00577
$914$	0	0
$915$	4.56717	0.150986
$916$	0	0
$917$	−0.771837	−0.0254883
$918$	0	0
$919$	19.5575	0.645143	0.322571	−	0.946545i	$-0.395453\pi$
$919$	19.5575	0.645143	0.322571	+	0.946545i	$0.395453\pi$
$920$	0	0
$921$	−2.49230	−0.0821240
$922$	0	0
$923$	−7.72752	−0.254354
$924$	0	0
$925$	22.0102	0.723690
$926$	0	0
$927$	−33.0702	−1.08617
$928$	0	0
$929$	−17.1791	−0.563629	−0.281815	−	0.959469i	$-0.590936\pi$
$929$	−17.1791	−0.563629	−0.281815	+	0.959469i	$0.590936\pi$
$930$	0	0
$931$	1.35498	0.0444078
$932$	0	0
$933$	7.67129	0.251147
$934$	0	0
$935$	11.3073	0.369787
$936$	0	0
$937$	−60.0003	−1.96013	−0.980063	−	0.198689i	$-0.936332\pi$
$937$	−60.0003	−1.96013	−0.980063	+	0.198689i	$0.936332\pi$
$938$	0	0
$939$	−11.9575	−0.390220
$940$	0	0
$941$	−6.70401	−0.218544	−0.109272	−	0.994012i	$-0.534852\pi$
$941$	−6.70401	−0.218544	−0.109272	+	0.994012i	$0.534852\pi$
$942$	0	0
$943$	16.9329	0.551410
$944$	0	0
$945$	9.94826	0.323617
$946$	0	0
$947$	−8.72414	−0.283496	−0.141748	−	0.989903i	$-0.545272\pi$
$947$	−8.72414	−0.283496	−0.141748	+	0.989903i	$0.545272\pi$
$948$	0	0
$949$	−45.9195	−1.49061
$950$	0	0
$951$	5.15962	0.167312
$952$	0	0
$953$	−7.65381	−0.247931	−0.123966	−	0.992287i	$-0.539561\pi$
$953$	−7.65381	−0.247931	−0.123966	+	0.992287i	$0.539561\pi$
$954$	0	0
$955$	13.0917	0.423637
$956$	0	0
$957$	−12.1697	−0.393391
$958$	0	0
$959$	4.28170	0.138263
$960$	0	0
$961$	−9.86746	−0.318305
$962$	0	0
$963$	−31.0369	−1.00015
$964$	0	0
$965$	29.1413	0.938093
$966$	0	0
$967$	27.6620	0.889549	0.444774	−	0.895643i	$-0.353284\pi$
$967$	27.6620	0.889549	0.444774	+	0.895643i	$0.353284\pi$
$968$	0	0
$969$	5.42336	0.174223
$970$	0	0
$971$	25.3859	0.814672	0.407336	−	0.913278i	$-0.366458\pi$
$971$	25.3859	0.814672	0.407336	+	0.913278i	$0.366458\pi$
$972$	0	0
$973$	−28.9236	−0.927247
$974$	0	0
$975$	−8.39743	−0.268933
$976$	0	0
$977$	27.0512	0.865444	0.432722	−	0.901527i	$-0.357553\pi$
$977$	27.0512	0.865444	0.432722	+	0.901527i	$0.357553\pi$
$978$	0	0
$979$	−30.5405	−0.976079
$980$	0	0
$981$	−14.7664	−0.471456
$982$	0	0
$983$	−55.0126	−1.75463	−0.877315	−	0.479915i	$-0.840667\pi$
$983$	−55.0126	−1.75463	−0.877315	+	0.479915i	$0.840667\pi$
$984$	0	0
$985$	−18.9728	−0.604523
$986$	0	0
$987$	16.7443	0.532976
$988$	0	0
$989$	15.5752	0.495262
$990$	0	0
$991$	−30.0813	−0.955564	−0.477782	−	0.878478i	$-0.658559\pi$
$991$	−30.0813	−0.955564	−0.477782	+	0.878478i	$0.658559\pi$
$992$	0	0
$993$	0.504133	0.0159982
$994$	0	0
$995$	−9.16714	−0.290618
$996$	0	0
$997$	20.4955	0.649098	0.324549	−	0.945869i	$-0.394787\pi$
$997$	20.4955	0.649098	0.324549	+	0.945869i	$0.394787\pi$
$998$	0	0
$999$	21.1566	0.669365

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.29	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.29	✓	44	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)

\(f(q)\)	\(=\)	\(q+0.663224 q^{3} -1.07871 q^{5} +2.50091 q^{7} -2.56013 q^{9} +O(q^{10})\)	\(q+0.663224 q^{3} -1.07871 q^{5} +2.50091 q^{7} -2.56013 q^{9} +2.33005 q^{11} +3.30037 q^{13} -0.715424 q^{15} -4.49872 q^{17} -1.81768 q^{19} +1.65866 q^{21} +1.64341 q^{23} -3.83639 q^{25} -3.68762 q^{27} -7.87508 q^{29} -4.59701 q^{31} +1.54534 q^{33} -2.69775 q^{35} -5.73720 q^{37} +2.18889 q^{39} +10.3035 q^{41} +9.47732 q^{43} +2.76163 q^{45} +10.0950 q^{47} -0.745444 q^{49} -2.98366 q^{51} -11.8738 q^{53} -2.51344 q^{55} -1.20553 q^{57} -13.4477 q^{59} -6.38387 q^{61} -6.40267 q^{63} -3.56013 q^{65} +4.16459 q^{67} +1.08995 q^{69} -2.34141 q^{71} -13.9134 q^{73} -2.54439 q^{75} +5.82724 q^{77} -15.9669 q^{79} +5.23469 q^{81} +13.0428 q^{83} +4.85280 q^{85} -5.22295 q^{87} -13.1072 q^{89} +8.25394 q^{91} -3.04885 q^{93} +1.96075 q^{95} -12.3368 q^{97} -5.96523 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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