LMFDB - Embedded newform 8048.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, 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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$0$
Dimension:	$29$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.62135 q^{3} -1.76626 q^{5} -2.03701 q^{7} +3.87147 q^{9} +O(q^{10})$	$q-2.62135 q^{3} -1.76626 q^{5} -2.03701 q^{7} +3.87147 q^{9} +4.58536 q^{11} -0.991072 q^{13} +4.62999 q^{15} -5.18242 q^{17} -2.20838 q^{19} +5.33972 q^{21} +8.16377 q^{23} -1.88032 q^{25} -2.28443 q^{27} +0.471238 q^{29} -3.12170 q^{31} -12.0198 q^{33} +3.59789 q^{35} -4.70068 q^{37} +2.59794 q^{39} +4.39077 q^{41} +1.03048 q^{43} -6.83803 q^{45} +10.4741 q^{47} -2.85059 q^{49} +13.5849 q^{51} +8.99418 q^{53} -8.09894 q^{55} +5.78893 q^{57} -2.85288 q^{59} -8.50901 q^{61} -7.88623 q^{63} +1.75049 q^{65} +2.81969 q^{67} -21.4001 q^{69} -15.2859 q^{71} -10.2686 q^{73} +4.92898 q^{75} -9.34043 q^{77} +2.94631 q^{79} -5.62612 q^{81} +1.78724 q^{83} +9.15350 q^{85} -1.23528 q^{87} -8.79655 q^{89} +2.01882 q^{91} +8.18306 q^{93} +3.90057 q^{95} +10.3258 q^{97} +17.7521 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9	$ 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9} + 27 q^{11} + 16 q^{13} + 14 q^{15} - 15 q^{17} + 14 q^{19} + q^{21} + 25 q^{23} + 21 q^{25} + 25 q^{27} - 13 q^{29} + 27 q^{31} - 9 q^{33} + 29 q^{35} + 35 q^{37} + 38 q^{39} - 30 q^{41} + 38 q^{43} + q^{45} + 35 q^{47} + 14 q^{49} + 21 q^{51} + 2 q^{53} + 25 q^{55} - 25 q^{57} + 40 q^{59} + 10 q^{61} + 56 q^{63} - 50 q^{65} + 31 q^{67} + 11 q^{69} + 65 q^{71} - 23 q^{73} + 32 q^{75} + 13 q^{77} + 44 q^{79} - 7 q^{81} + 41 q^{83} + 26 q^{85} + 25 q^{87} - 48 q^{89} + 44 q^{91} + 25 q^{93} + 75 q^{95} - 18 q^{97} + 80 q^{99}+O(q^{100}) $ 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9 + 27 * q^11 + 16 * q^13 + 14 * q^15 - 15 * q^17 + 14 * q^19 + q^21 + 25 * q^23 + 21 * q^25 + 25 * q^27 - 13 * q^29 + 27 * q^31 - 9 * q^33 + 29 * q^35 + 35 * q^37 + 38 * q^39 - 30 * q^41 + 38 * q^43 + q^45 + 35 * q^47 + 14 * q^49 + 21 * q^51 + 2 * q^53 + 25 * q^55 - 25 * q^57 + 40 * q^59 + 10 * q^61 + 56 * q^63 - 50 * q^65 + 31 * q^67 + 11 * q^69 + 65 * q^71 - 23 * q^73 + 32 * q^75 + 13 * q^77 + 44 * q^79 - 7 * q^81 + 41 * q^83 + 26 * q^85 + 25 * q^87 - 48 * q^89 + 44 * q^91 + 25 * q^93 + 75 * q^95 - 18 * q^97 + 80 * 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
$3$	−2.62135	−1.51344	−0.756718	−	0.653741i	$-0.773199\pi$
$3$	−2.62135	−1.51344	−0.756718	+	0.653741i	$0.773199\pi$
$4$	0	0
$5$	−1.76626	−0.789896	−0.394948	−	0.918704i	$-0.629237\pi$
$5$	−1.76626	−0.789896	−0.394948	+	0.918704i	$0.629237\pi$
$6$	0	0
$7$	−2.03701	−0.769918	−0.384959	−	0.922934i	$-0.625784\pi$
$7$	−2.03701	−0.769918	−0.384959	+	0.922934i	$0.625784\pi$
$8$	0	0
$9$	3.87147	1.29049
$10$	0	0
$11$	4.58536	1.38254	0.691269	−	0.722598i	$-0.257052\pi$
$11$	4.58536	1.38254	0.691269	+	0.722598i	$0.257052\pi$
$12$	0	0
$13$	−0.991072	−0.274874	−0.137437	−	0.990511i	$-0.543886\pi$
$13$	−0.991072	−0.274874	−0.137437	+	0.990511i	$0.543886\pi$
$14$	0	0
$15$	4.62999	1.19546
$16$	0	0
$17$	−5.18242	−1.25692	−0.628460	−	0.777842i	$-0.716315\pi$
$17$	−5.18242	−1.25692	−0.628460	+	0.777842i	$0.716315\pi$
$18$	0	0
$19$	−2.20838	−0.506637	−0.253318	−	0.967383i	$-0.581522\pi$
$19$	−2.20838	−0.506637	−0.253318	+	0.967383i	$0.581522\pi$
$20$	0	0
$21$	5.33972	1.16522
$22$	0	0
$23$	8.16377	1.70226	0.851132	−	0.524952i	$-0.175917\pi$
$23$	8.16377	1.70226	0.851132	+	0.524952i	$0.175917\pi$
$24$	0	0
$25$	−1.88032	−0.376065
$26$	0	0
$27$	−2.28443	−0.439639
$28$	0	0
$29$	0.471238	0.0875068	0.0437534	−	0.999042i	$-0.486068\pi$
$29$	0.471238	0.0875068	0.0437534	+	0.999042i	$0.486068\pi$
$30$	0	0
$31$	−3.12170	−0.560674	−0.280337	−	0.959902i	$-0.590446\pi$
$31$	−3.12170	−0.560674	−0.280337	+	0.959902i	$0.590446\pi$
$32$	0	0
$33$	−12.0198	−2.09238
$34$	0	0
$35$	3.59789	0.608155
$36$	0	0
$37$	−4.70068	−0.772787	−0.386394	−	0.922334i	$-0.626279\pi$
$37$	−4.70068	−0.772787	−0.386394	+	0.922334i	$0.626279\pi$
$38$	0	0
$39$	2.59794	0.416004
$40$	0	0
$41$	4.39077	0.685723	0.342861	−	0.939386i	$-0.388604\pi$
$41$	4.39077	0.685723	0.342861	+	0.939386i	$0.388604\pi$
$42$	0	0
$43$	1.03048	0.157147	0.0785734	−	0.996908i	$-0.474964\pi$
$43$	1.03048	0.157147	0.0785734	+	0.996908i	$0.474964\pi$
$44$	0	0
$45$	−6.83803	−1.01935
$46$	0	0
$47$	10.4741	1.52781	0.763903	−	0.645332i	$-0.223281\pi$
$47$	10.4741	1.52781	0.763903	+	0.645332i	$0.223281\pi$
$48$	0	0
$49$	−2.85059	−0.407227
$50$	0	0
$51$	13.5849	1.90227
$52$	0	0
$53$	8.99418	1.23545	0.617723	−	0.786396i	$-0.288055\pi$
$53$	8.99418	1.23545	0.617723	+	0.786396i	$0.288055\pi$
$54$	0	0
$55$	−8.09894	−1.09206
$56$	0	0
$57$	5.78893	0.766762
$58$	0	0
$59$	−2.85288	−0.371414	−0.185707	−	0.982605i	$-0.559457\pi$
$59$	−2.85288	−0.371414	−0.185707	+	0.982605i	$0.559457\pi$
$60$	0	0
$61$	−8.50901	−1.08947	−0.544734	−	0.838609i	$-0.683369\pi$
$61$	−8.50901	−1.08947	−0.544734	+	0.838609i	$0.683369\pi$
$62$	0	0
$63$	−7.88623	−0.993572
$64$	0	0
$65$	1.75049	0.217122
$66$	0	0
$67$	2.81969	0.344479	0.172240	−	0.985055i	$-0.444900\pi$
$67$	2.81969	0.344479	0.172240	+	0.985055i	$0.444900\pi$
$68$	0	0
$69$	−21.4001	−2.57627
$70$	0	0
$71$	−15.2859	−1.81411	−0.907053	−	0.421016i	$-0.861674\pi$
$71$	−15.2859	−1.81411	−0.907053	+	0.421016i	$0.861674\pi$
$72$	0	0
$73$	−10.2686	−1.20184	−0.600922	−	0.799307i	$-0.705200\pi$
$73$	−10.2686	−1.20184	−0.600922	+	0.799307i	$0.705200\pi$
$74$	0	0
$75$	4.92898	0.569150
$76$	0	0
$77$	−9.34043	−1.06444
$78$	0	0
$79$	2.94631	0.331485	0.165743	−	0.986169i	$-0.446998\pi$
$79$	2.94631	0.331485	0.165743	+	0.986169i	$0.446998\pi$
$80$	0	0
$81$	−5.62612	−0.625124
$82$	0	0
$83$	1.78724	0.196175	0.0980876	−	0.995178i	$-0.468727\pi$
$83$	1.78724	0.196175	0.0980876	+	0.995178i	$0.468727\pi$
$84$	0	0
$85$	9.15350	0.992836
$86$	0	0
$87$	−1.23528	−0.132436
$88$	0	0
$89$	−8.79655	−0.932432	−0.466216	−	0.884671i	$-0.654383\pi$
$89$	−8.79655	−0.932432	−0.466216	+	0.884671i	$0.654383\pi$
$90$	0	0
$91$	2.01882	0.211630
$92$	0	0
$93$	8.18306	0.848544
$94$	0	0
$95$	3.90057	0.400190
$96$	0	0
$97$	10.3258	1.04842	0.524211	−	0.851588i	$-0.324360\pi$
$97$	10.3258	1.04842	0.524211	+	0.851588i	$0.324360\pi$
$98$	0	0
$99$	17.7521	1.78415
$100$	0	0
$101$	−16.2418	−1.61612	−0.808060	−	0.589100i	$-0.799482\pi$
$101$	−16.2418	−1.61612	−0.808060	+	0.589100i	$0.799482\pi$
$102$	0	0
$103$	−13.3551	−1.31592	−0.657961	−	0.753052i	$-0.728581\pi$
$103$	−13.3551	−1.31592	−0.657961	+	0.753052i	$0.728581\pi$
$104$	0	0
$105$	−9.43133	−0.920404
$106$	0	0
$107$	17.1455	1.65752	0.828758	−	0.559607i	$-0.189048\pi$
$107$	17.1455	1.65752	0.828758	+	0.559607i	$0.189048\pi$
$108$	0	0
$109$	9.70773	0.929832	0.464916	−	0.885355i	$-0.346085\pi$
$109$	9.70773	0.929832	0.464916	+	0.885355i	$0.346085\pi$
$110$	0	0
$111$	12.3221	1.16956
$112$	0	0
$113$	−3.46064	−0.325549	−0.162775	−	0.986663i	$-0.552044\pi$
$113$	−3.46064	−0.325549	−0.162775	+	0.986663i	$0.552044\pi$
$114$	0	0
$115$	−14.4193	−1.34461
$116$	0	0
$117$	−3.83691	−0.354722
$118$	0	0
$119$	10.5566	0.967725
$120$	0	0
$121$	10.0255	0.911411
$122$	0	0
$123$	−11.5097	−1.03780
$124$	0	0
$125$	12.1524	1.08695
$126$	0	0
$127$	−10.7698	−0.955661	−0.477831	−	0.878452i	$-0.658577\pi$
$127$	−10.7698	−0.955661	−0.477831	+	0.878452i	$0.658577\pi$
$128$	0	0
$129$	−2.70125	−0.237832
$130$	0	0
$131$	−0.741434	−0.0647794	−0.0323897	−	0.999475i	$-0.510312\pi$
$131$	−0.741434	−0.0647794	−0.0323897	+	0.999475i	$0.510312\pi$
$132$	0	0
$133$	4.49849	0.390068
$134$	0	0
$135$	4.03491	0.347269
$136$	0	0
$137$	2.00704	0.171473	0.0857366	−	0.996318i	$-0.472676\pi$
$137$	2.00704	0.171473	0.0857366	+	0.996318i	$0.472676\pi$
$138$	0	0
$139$	−20.1764	−1.71134	−0.855669	−	0.517523i	$-0.826854\pi$
$139$	−20.1764	−1.71134	−0.855669	+	0.517523i	$0.826854\pi$
$140$	0	0
$141$	−27.4563	−2.31224
$142$	0	0
$143$	−4.54442	−0.380023
$144$	0	0
$145$	−0.832330	−0.0691212
$146$	0	0
$147$	7.47238	0.616312
$148$	0	0
$149$	−5.29717	−0.433961	−0.216981	−	0.976176i	$-0.569621\pi$
$149$	−5.29717	−0.433961	−0.216981	+	0.976176i	$0.569621\pi$
$150$	0	0
$151$	−22.1501	−1.80255	−0.901273	−	0.433252i	$-0.857366\pi$
$151$	−22.1501	−1.80255	−0.901273	+	0.433252i	$0.857366\pi$
$152$	0	0
$153$	−20.0636	−1.62204
$154$	0	0
$155$	5.51373	0.442874
$156$	0	0
$157$	11.1866	0.892786	0.446393	−	0.894837i	$-0.352708\pi$
$157$	11.1866	0.892786	0.446393	+	0.894837i	$0.352708\pi$
$158$	0	0
$159$	−23.5769	−1.86977
$160$	0	0
$161$	−16.6297	−1.31060
$162$	0	0
$163$	−9.80151	−0.767714	−0.383857	−	0.923393i	$-0.625404\pi$
$163$	−9.80151	−0.767714	−0.383857	+	0.923393i	$0.625404\pi$
$164$	0	0
$165$	21.2302	1.65277
$166$	0	0
$167$	−10.1293	−0.783832	−0.391916	−	0.920001i	$-0.628188\pi$
$167$	−10.1293	−0.783832	−0.391916	+	0.920001i	$0.628188\pi$
$168$	0	0
$169$	−12.0178	−0.924444
$170$	0	0
$171$	−8.54967	−0.653810
$172$	0	0
$173$	−15.8733	−1.20682	−0.603411	−	0.797431i	$-0.706192\pi$
$173$	−15.8733	−1.20682	−0.603411	+	0.797431i	$0.706192\pi$
$174$	0	0
$175$	3.83024	0.289539
$176$	0	0
$177$	7.47840	0.562111
$178$	0	0
$179$	21.7275	1.62399	0.811994	−	0.583666i	$-0.198382\pi$
$179$	21.7275	1.62399	0.811994	+	0.583666i	$0.198382\pi$
$180$	0	0
$181$	−3.46032	−0.257203	−0.128602	−	0.991696i	$-0.541049\pi$
$181$	−3.46032	−0.257203	−0.128602	+	0.991696i	$0.541049\pi$
$182$	0	0
$183$	22.3051	1.64884
$184$	0	0
$185$	8.30263	0.610421
$186$	0	0
$187$	−23.7632	−1.73774
$188$	0	0
$189$	4.65342	0.338486
$190$	0	0
$191$	13.1625	0.952406	0.476203	−	0.879335i	$-0.342013\pi$
$191$	13.1625	0.952406	0.476203	+	0.879335i	$0.342013\pi$
$192$	0	0
$193$	−18.1853	−1.30901	−0.654504	−	0.756058i	$-0.727123\pi$
$193$	−18.1853	−1.30901	−0.654504	+	0.756058i	$0.727123\pi$
$194$	0	0
$195$	−4.58865	−0.328600
$196$	0	0
$197$	16.4318	1.17072	0.585358	−	0.810775i	$-0.300954\pi$
$197$	16.4318	1.17072	0.585358	+	0.810775i	$0.300954\pi$
$198$	0	0
$199$	−9.68840	−0.686792	−0.343396	−	0.939191i	$-0.611577\pi$
$199$	−9.68840	−0.686792	−0.343396	+	0.939191i	$0.611577\pi$
$200$	0	0
$201$	−7.39138	−0.521348
$202$	0	0
$203$	−0.959918	−0.0673730
$204$	0	0
$205$	−7.75524	−0.541649
$206$	0	0
$207$	31.6058	2.19676
$208$	0	0
$209$	−10.1262	−0.700444
$210$	0	0
$211$	13.3989	0.922418	0.461209	−	0.887292i	$-0.347416\pi$
$211$	13.3989	0.922418	0.461209	+	0.887292i	$0.347416\pi$
$212$	0	0
$213$	40.0698	2.74553
$214$	0	0
$215$	−1.82010	−0.124130
$216$	0	0
$217$	6.35893	0.431672
$218$	0	0
$219$	26.9175	1.81892
$220$	0	0
$221$	5.13615	0.345495
$222$	0	0
$223$	22.9306	1.53554	0.767772	−	0.640723i	$-0.221365\pi$
$223$	22.9306	1.53554	0.767772	+	0.640723i	$0.221365\pi$
$224$	0	0
$225$	−7.27962	−0.485308
$226$	0	0
$227$	−8.04813	−0.534173	−0.267086	−	0.963673i	$-0.586061\pi$
$227$	−8.04813	−0.534173	−0.267086	+	0.963673i	$0.586061\pi$
$228$	0	0
$229$	−12.7599	−0.843200	−0.421600	−	0.906782i	$-0.638531\pi$
$229$	−12.7599	−0.843200	−0.421600	+	0.906782i	$0.638531\pi$
$230$	0	0
$231$	24.4845	1.61096
$232$	0	0
$233$	−9.52154	−0.623777	−0.311889	−	0.950119i	$-0.600962\pi$
$233$	−9.52154	−0.623777	−0.311889	+	0.950119i	$0.600962\pi$
$234$	0	0
$235$	−18.5000	−1.20681
$236$	0	0
$237$	−7.72330	−0.501682
$238$	0	0
$239$	27.3741	1.77069	0.885343	−	0.464939i	$-0.153924\pi$
$239$	27.3741	1.77069	0.885343	+	0.464939i	$0.153924\pi$
$240$	0	0
$241$	12.4055	0.799107	0.399553	−	0.916710i	$-0.369165\pi$
$241$	12.4055	0.799107	0.399553	+	0.916710i	$0.369165\pi$
$242$	0	0
$243$	21.6013	1.38573
$244$	0	0
$245$	5.03488	0.321667
$246$	0	0
$247$	2.18866	0.139261
$248$	0	0
$249$	−4.68498	−0.296899
$250$	0	0
$251$	3.39817	0.214491	0.107245	−	0.994233i	$-0.465797\pi$
$251$	3.39817	0.214491	0.107245	+	0.994233i	$0.465797\pi$
$252$	0	0
$253$	37.4338	2.35344
$254$	0	0
$255$	−23.9945	−1.50260
$256$	0	0
$257$	20.1542	1.25718	0.628592	−	0.777735i	$-0.283632\pi$
$257$	20.1542	1.25718	0.628592	+	0.777735i	$0.283632\pi$
$258$	0	0
$259$	9.57534	0.594983
$260$	0	0
$261$	1.82439	0.112927
$262$	0	0
$263$	−15.3265	−0.945074	−0.472537	−	0.881311i	$-0.656662\pi$
$263$	−15.3265	−0.945074	−0.472537	+	0.881311i	$0.656662\pi$
$264$	0	0
$265$	−15.8861	−0.975873
$266$	0	0
$267$	23.0588	1.41118
$268$	0	0
$269$	16.2807	0.992650	0.496325	−	0.868137i	$-0.334682\pi$
$269$	16.2807	0.992650	0.496325	+	0.868137i	$0.334682\pi$
$270$	0	0
$271$	31.0910	1.88865	0.944323	−	0.329019i	$-0.106718\pi$
$271$	31.0910	1.88865	0.944323	+	0.329019i	$0.106718\pi$
$272$	0	0
$273$	−5.29204	−0.320289
$274$	0	0
$275$	−8.62196	−0.519923
$276$	0	0
$277$	10.2952	0.618577	0.309289	−	0.950968i	$-0.399909\pi$
$277$	10.2952	0.618577	0.309289	+	0.950968i	$0.399909\pi$
$278$	0	0
$279$	−12.0856	−0.723544
$280$	0	0
$281$	−18.3867	−1.09686	−0.548430	−	0.836196i	$-0.684774\pi$
$281$	−18.3867	−1.09686	−0.548430	+	0.836196i	$0.684774\pi$
$282$	0	0
$283$	15.6523	0.930433	0.465216	−	0.885197i	$-0.345976\pi$
$283$	15.6523	0.930433	0.465216	+	0.885197i	$0.345976\pi$
$284$	0	0
$285$	−10.2248	−0.605662
$286$	0	0
$287$	−8.94404	−0.527950
$288$	0	0
$289$	9.85744	0.579849
$290$	0	0
$291$	−27.0674	−1.58672
$292$	0	0
$293$	30.2482	1.76712	0.883560	−	0.468318i	$-0.155140\pi$
$293$	30.2482	1.76712	0.883560	+	0.468318i	$0.155140\pi$
$294$	0	0
$295$	5.03893	0.293378
$296$	0	0
$297$	−10.4749	−0.607818
$298$	0	0
$299$	−8.09088	−0.467908
$300$	0	0
$301$	−2.09910	−0.120990
$302$	0	0
$303$	42.5754	2.44590
$304$	0	0
$305$	15.0291	0.860566
$306$	0	0
$307$	−16.6264	−0.948916	−0.474458	−	0.880278i	$-0.657356\pi$
$307$	−16.6264	−0.948916	−0.474458	+	0.880278i	$0.657356\pi$
$308$	0	0
$309$	35.0085	1.99156
$310$	0	0
$311$	−0.0759674	−0.00430772	−0.00215386	−	0.999998i	$-0.500686\pi$
$311$	−0.0759674	−0.00430772	−0.00215386	+	0.999998i	$0.500686\pi$
$312$	0	0
$313$	−9.51757	−0.537965	−0.268982	−	0.963145i	$-0.586687\pi$
$313$	−9.51757	−0.537965	−0.268982	+	0.963145i	$0.586687\pi$
$314$	0	0
$315$	13.9291	0.784818
$316$	0	0
$317$	7.92825	0.445295	0.222647	−	0.974899i	$-0.428530\pi$
$317$	7.92825	0.445295	0.222647	+	0.974899i	$0.428530\pi$
$318$	0	0
$319$	2.16080	0.120981
$320$	0	0
$321$	−44.9443	−2.50855
$322$	0	0
$323$	11.4447	0.636802
$324$	0	0
$325$	1.86353	0.103370
$326$	0	0
$327$	−25.4473	−1.40724
$328$	0	0
$329$	−21.3359	−1.17628
$330$	0	0
$331$	−0.975116	−0.0535972	−0.0267986	−	0.999641i	$-0.508531\pi$
$331$	−0.975116	−0.0535972	−0.0267986	+	0.999641i	$0.508531\pi$
$332$	0	0
$333$	−18.1986	−0.997275
$334$	0	0
$335$	−4.98030	−0.272103
$336$	0	0
$337$	3.90206	0.212559	0.106280	−	0.994336i	$-0.466106\pi$
$337$	3.90206	0.212559	0.106280	+	0.994336i	$0.466106\pi$
$338$	0	0
$339$	9.07154	0.492699
$340$	0	0
$341$	−14.3141	−0.775152
$342$	0	0
$343$	20.0658	1.08345
$344$	0	0
$345$	37.7982	2.03498
$346$	0	0
$347$	−17.8069	−0.955923	−0.477961	−	0.878381i	$-0.658624\pi$
$347$	−17.8069	−0.955923	−0.477961	+	0.878381i	$0.658624\pi$
$348$	0	0
$349$	7.94010	0.425024	0.212512	−	0.977158i	$-0.431836\pi$
$349$	7.94010	0.425024	0.212512	+	0.977158i	$0.431836\pi$
$350$	0	0
$351$	2.26404	0.120845
$352$	0	0
$353$	−21.5408	−1.14650	−0.573251	−	0.819380i	$-0.694318\pi$
$353$	−21.5408	−1.14650	−0.573251	+	0.819380i	$0.694318\pi$
$354$	0	0
$355$	26.9989	1.43295
$356$	0	0
$357$	−27.6726	−1.46459
$358$	0	0
$359$	21.4643	1.13284	0.566421	−	0.824116i	$-0.308328\pi$
$359$	21.4643	1.13284	0.566421	+	0.824116i	$0.308328\pi$
$360$	0	0
$361$	−14.1231	−0.743319
$362$	0	0
$363$	−26.2804	−1.37936
$364$	0	0
$365$	18.1370	0.949332
$366$	0	0
$367$	26.3654	1.37626	0.688132	−	0.725585i	$-0.258431\pi$
$367$	26.3654	1.37626	0.688132	+	0.725585i	$0.258431\pi$
$368$	0	0
$369$	16.9987	0.884919
$370$	0	0
$371$	−18.3212	−0.951191
$372$	0	0
$373$	29.9955	1.55311	0.776554	−	0.630051i	$-0.216966\pi$
$373$	29.9955	1.55311	0.776554	+	0.630051i	$0.216966\pi$
$374$	0	0
$375$	−31.8558	−1.64503
$376$	0	0
$377$	−0.467031	−0.0240533
$378$	0	0
$379$	3.68565	0.189319	0.0946594	−	0.995510i	$-0.469824\pi$
$379$	3.68565	0.189319	0.0946594	+	0.995510i	$0.469824\pi$
$380$	0	0
$381$	28.2313	1.44633
$382$	0	0
$383$	−14.5257	−0.742228	−0.371114	−	0.928587i	$-0.621024\pi$
$383$	−14.5257	−0.742228	−0.371114	+	0.928587i	$0.621024\pi$
$384$	0	0
$385$	16.4976	0.840797
$386$	0	0
$387$	3.98947	0.202796
$388$	0	0
$389$	5.30912	0.269183	0.134592	−	0.990901i	$-0.457028\pi$
$389$	5.30912	0.269183	0.134592	+	0.990901i	$0.457028\pi$
$390$	0	0
$391$	−42.3081	−2.13961
$392$	0	0
$393$	1.94356	0.0980396
$394$	0	0
$395$	−5.20395	−0.261839
$396$	0	0
$397$	32.4886	1.63056	0.815279	−	0.579068i	$-0.196584\pi$
$397$	32.4886	1.63056	0.815279	+	0.579068i	$0.196584\pi$
$398$	0	0
$399$	−11.7921	−0.590344
$400$	0	0
$401$	11.4855	0.573558	0.286779	−	0.957997i	$-0.407415\pi$
$401$	11.4855	0.573558	0.286779	+	0.957997i	$0.407415\pi$
$402$	0	0
$403$	3.09383	0.154114
$404$	0	0
$405$	9.93719	0.493783
$406$	0	0
$407$	−21.5543	−1.06841
$408$	0	0
$409$	−25.9961	−1.28543	−0.642713	−	0.766107i	$-0.722191\pi$
$409$	−25.9961	−1.28543	−0.642713	+	0.766107i	$0.722191\pi$
$410$	0	0
$411$	−5.26116	−0.259514
$412$	0	0
$413$	5.81135	0.285958
$414$	0	0
$415$	−3.15673	−0.154958
$416$	0	0
$417$	52.8893	2.59000
$418$	0	0
$419$	17.3387	0.847049	0.423525	−	0.905885i	$-0.360793\pi$
$419$	17.3387	0.847049	0.423525	+	0.905885i	$0.360793\pi$
$420$	0	0
$421$	18.1269	0.883449	0.441724	−	0.897151i	$-0.354367\pi$
$421$	18.1269	0.883449	0.441724	+	0.897151i	$0.354367\pi$
$422$	0	0
$423$	40.5502	1.97162
$424$	0	0
$425$	9.74462	0.472683
$426$	0	0
$427$	17.3330	0.838800
$428$	0	0
$429$	11.9125	0.575141
$430$	0	0
$431$	−19.3984	−0.934387	−0.467193	−	0.884155i	$-0.654735\pi$
$431$	−19.3984	−0.934387	−0.467193	+	0.884155i	$0.654735\pi$
$432$	0	0
$433$	26.0787	1.25326	0.626631	−	0.779316i	$-0.284433\pi$
$433$	26.0787	1.25326	0.626631	+	0.779316i	$0.284433\pi$
$434$	0	0
$435$	2.18183	0.104611
$436$	0	0
$437$	−18.0287	−0.862429
$438$	0	0
$439$	11.0121	0.525579	0.262789	−	0.964853i	$-0.415358\pi$
$439$	11.0121	0.525579	0.262789	+	0.964853i	$0.415358\pi$
$440$	0	0
$441$	−11.0360	−0.525522
$442$	0	0
$443$	31.2335	1.48395	0.741973	−	0.670430i	$-0.233890\pi$
$443$	31.2335	1.48395	0.741973	+	0.670430i	$0.233890\pi$
$444$	0	0
$445$	15.5370	0.736524
$446$	0	0
$447$	13.8857	0.656773
$448$	0	0
$449$	−11.7127	−0.552758	−0.276379	−	0.961049i	$-0.589134\pi$
$449$	−11.7127	−0.552758	−0.276379	+	0.961049i	$0.589134\pi$
$450$	0	0
$451$	20.1332	0.948037
$452$	0	0
$453$	58.0630	2.72804
$454$	0	0
$455$	−3.56577	−0.167166
$456$	0	0
$457$	22.4100	1.04829	0.524147	−	0.851628i	$-0.324384\pi$
$457$	22.4100	1.04829	0.524147	+	0.851628i	$0.324384\pi$
$458$	0	0
$459$	11.8389	0.552592
$460$	0	0
$461$	−9.71946	−0.452680	−0.226340	−	0.974048i	$-0.572676\pi$
$461$	−9.71946	−0.452680	−0.226340	+	0.974048i	$0.572676\pi$
$462$	0	0
$463$	3.78757	0.176023	0.0880116	−	0.996119i	$-0.471949\pi$
$463$	3.78757	0.176023	0.0880116	+	0.996119i	$0.471949\pi$
$464$	0	0
$465$	−14.4534	−0.670261
$466$	0	0
$467$	32.8951	1.52220	0.761102	−	0.648632i	$-0.224659\pi$
$467$	32.8951	1.52220	0.761102	+	0.648632i	$0.224659\pi$
$468$	0	0
$469$	−5.74373	−0.265221
$470$	0	0
$471$	−29.3239	−1.35118
$472$	0	0
$473$	4.72512	0.217261
$474$	0	0
$475$	4.15246	0.190528
$476$	0	0
$477$	34.8207	1.59433
$478$	0	0
$479$	−16.8932	−0.771870	−0.385935	−	0.922526i	$-0.626121\pi$
$479$	−16.8932	−0.771870	−0.385935	+	0.922526i	$0.626121\pi$
$480$	0	0
$481$	4.65871	0.212419
$482$	0	0
$483$	43.5922	1.98351
$484$	0	0
$485$	−18.2380	−0.828145
$486$	0	0
$487$	19.9778	0.905279	0.452639	−	0.891694i	$-0.350482\pi$
$487$	19.9778	0.905279	0.452639	+	0.891694i	$0.350482\pi$
$488$	0	0
$489$	25.6932	1.16189
$490$	0	0
$491$	−12.7314	−0.574561	−0.287280	−	0.957847i	$-0.592751\pi$
$491$	−12.7314	−0.574561	−0.287280	+	0.957847i	$0.592751\pi$
$492$	0	0
$493$	−2.44215	−0.109989
$494$	0	0
$495$	−31.3548	−1.40929
$496$	0	0
$497$	31.1376	1.39671
$498$	0	0
$499$	−5.49664	−0.246063	−0.123032	−	0.992403i	$-0.539262\pi$
$499$	−5.49664	−0.246063	−0.123032	+	0.992403i	$0.539262\pi$
$500$	0	0
$501$	26.5526	1.18628
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	28.6873	1.27657
$506$	0	0
$507$	31.5028	1.39909
$508$	0	0
$509$	−35.3509	−1.56690	−0.783451	−	0.621453i	$-0.786543\pi$
$509$	−35.3509	−1.56690	−0.783451	+	0.621453i	$0.786543\pi$
$510$	0	0
$511$	20.9172	0.925322
$512$	0	0
$513$	5.04489	0.222737
$514$	0	0
$515$	23.5887	1.03944
$516$	0	0
$517$	48.0275	2.11225
$518$	0	0
$519$	41.6093	1.82645
$520$	0	0
$521$	4.84756	0.212375	0.106188	−	0.994346i	$-0.466136\pi$
$521$	4.84756	0.212375	0.106188	+	0.994346i	$0.466136\pi$
$522$	0	0
$523$	−2.70375	−0.118227	−0.0591133	−	0.998251i	$-0.518827\pi$
$523$	−2.70375	−0.118227	−0.0591133	+	0.998251i	$0.518827\pi$
$524$	0	0
$525$	−10.0404	−0.438199
$526$	0	0
$527$	16.1779	0.704722
$528$	0	0
$529$	43.6472	1.89770
$530$	0	0
$531$	−11.0449	−0.479306
$532$	0	0
$533$	−4.35156	−0.188487
$534$	0	0
$535$	−30.2834	−1.30927
$536$	0	0
$537$	−56.9553	−2.45780
$538$	0	0
$539$	−13.0710	−0.563006
$540$	0	0
$541$	−38.8418	−1.66994	−0.834970	−	0.550296i	$-0.814515\pi$
$541$	−38.8418	−1.66994	−0.834970	+	0.550296i	$0.814515\pi$
$542$	0	0
$543$	9.07070	0.389261
$544$	0	0
$545$	−17.1464	−0.734470
$546$	0	0
$547$	−21.4560	−0.917394	−0.458697	−	0.888593i	$-0.651684\pi$
$547$	−21.4560	−0.917394	−0.458697	+	0.888593i	$0.651684\pi$
$548$	0	0
$549$	−32.9424	−1.40595
$550$	0	0
$551$	−1.04067	−0.0443341
$552$	0	0
$553$	−6.00166	−0.255216
$554$	0	0
$555$	−21.7641	−0.923834
$556$	0	0
$557$	−9.89982	−0.419469	−0.209734	−	0.977758i	$-0.567260\pi$
$557$	−9.89982	−0.419469	−0.209734	+	0.977758i	$0.567260\pi$
$558$	0	0
$559$	−1.02128	−0.0431955
$560$	0	0
$561$	62.2918	2.62996
$562$	0	0
$563$	−11.6652	−0.491631	−0.245815	−	0.969317i	$-0.579056\pi$
$563$	−11.6652	−0.491631	−0.245815	+	0.969317i	$0.579056\pi$
$564$	0	0
$565$	6.11239	0.257150
$566$	0	0
$567$	11.4605	0.481294
$568$	0	0
$569$	39.4870	1.65538	0.827691	−	0.561183i	$-0.189654\pi$
$569$	39.4870	1.65538	0.827691	+	0.561183i	$0.189654\pi$
$570$	0	0
$571$	−32.7418	−1.37020	−0.685101	−	0.728448i	$-0.740242\pi$
$571$	−32.7418	−1.37020	−0.685101	+	0.728448i	$0.740242\pi$
$572$	0	0
$573$	−34.5035	−1.44141
$574$	0	0
$575$	−15.3505	−0.640161
$576$	0	0
$577$	−41.9623	−1.74691	−0.873456	−	0.486903i	$-0.838126\pi$
$577$	−41.9623	−1.74691	−0.873456	+	0.486903i	$0.838126\pi$
$578$	0	0
$579$	47.6701	1.98110
$580$	0	0
$581$	−3.64063	−0.151039
$582$	0	0
$583$	41.2415	1.70805
$584$	0	0
$585$	6.77698	0.280194
$586$	0	0
$587$	22.8079	0.941384	0.470692	−	0.882298i	$-0.344004\pi$
$587$	22.8079	0.941384	0.470692	+	0.882298i	$0.344004\pi$
$588$	0	0
$589$	6.89389	0.284058
$590$	0	0
$591$	−43.0734	−1.77180
$592$	0	0
$593$	13.5716	0.557319	0.278659	−	0.960390i	$-0.410110\pi$
$593$	13.5716	0.557319	0.278659	+	0.960390i	$0.410110\pi$
$594$	0	0
$595$	−18.6458	−0.764402
$596$	0	0
$597$	25.3967	1.03942
$598$	0	0
$599$	−8.87846	−0.362764	−0.181382	−	0.983413i	$-0.558057\pi$
$599$	−8.87846	−0.362764	−0.181382	+	0.983413i	$0.558057\pi$
$600$	0	0
$601$	1.95603	0.0797880	0.0398940	−	0.999204i	$-0.487298\pi$
$601$	1.95603	0.0797880	0.0398940	+	0.999204i	$0.487298\pi$
$602$	0	0
$603$	10.9163	0.444548
$604$	0	0
$605$	−17.7077	−0.719920
$606$	0	0
$607$	−37.6937	−1.52994	−0.764970	−	0.644066i	$-0.777246\pi$
$607$	−37.6937	−1.52994	−0.764970	+	0.644066i	$0.777246\pi$
$608$	0	0
$609$	2.51628	0.101965
$610$	0	0
$611$	−10.3806	−0.419954
$612$	0	0
$613$	20.8575	0.842425	0.421212	−	0.906962i	$-0.361605\pi$
$613$	20.8575	0.842425	0.421212	+	0.906962i	$0.361605\pi$
$614$	0	0
$615$	20.3292	0.819752
$616$	0	0
$617$	−39.9935	−1.61008	−0.805039	−	0.593223i	$-0.797855\pi$
$617$	−39.9935	−1.61008	−0.805039	+	0.593223i	$0.797855\pi$
$618$	0	0
$619$	27.7558	1.11560	0.557800	−	0.829975i	$-0.311646\pi$
$619$	27.7558	1.11560	0.557800	+	0.829975i	$0.311646\pi$
$620$	0	0
$621$	−18.6496	−0.748382
$622$	0	0
$623$	17.9187	0.717896
$624$	0	0
$625$	−12.0628	−0.482511
$626$	0	0
$627$	26.5443	1.06008
$628$	0	0
$629$	24.3609	0.971332
$630$	0	0
$631$	18.5828	0.739769	0.369884	−	0.929078i	$-0.379397\pi$
$631$	18.5828	0.739769	0.369884	+	0.929078i	$0.379397\pi$
$632$	0	0
$633$	−35.1232	−1.39602
$634$	0	0
$635$	19.0222	0.754873
$636$	0	0
$637$	2.82514	0.111936
$638$	0	0
$639$	−59.1791	−2.34109
$640$	0	0
$641$	26.7543	1.05673	0.528366	−	0.849016i	$-0.322805\pi$
$641$	26.7543	1.05673	0.528366	+	0.849016i	$0.322805\pi$
$642$	0	0
$643$	−22.4012	−0.883416	−0.441708	−	0.897159i	$-0.645627\pi$
$643$	−22.4012	−0.883416	−0.441708	+	0.897159i	$0.645627\pi$
$644$	0	0
$645$	4.77111	0.187862
$646$	0	0
$647$	29.0005	1.14013	0.570064	−	0.821600i	$-0.306918\pi$
$647$	29.0005	1.14013	0.570064	+	0.821600i	$0.306918\pi$
$648$	0	0
$649$	−13.0815	−0.513493
$650$	0	0
$651$	−16.6690	−0.653309
$652$	0	0
$653$	41.0976	1.60827	0.804136	−	0.594445i	$-0.202628\pi$
$653$	41.0976	1.60827	0.804136	+	0.594445i	$0.202628\pi$
$654$	0	0
$655$	1.30957	0.0511690
$656$	0	0
$657$	−39.7545	−1.55097
$658$	0	0
$659$	−14.9877	−0.583837	−0.291919	−	0.956443i	$-0.594294\pi$
$659$	−14.9877	−0.583837	−0.291919	+	0.956443i	$0.594294\pi$
$660$	0	0
$661$	10.1877	0.396254	0.198127	−	0.980176i	$-0.436514\pi$
$661$	10.1877	0.396254	0.198127	+	0.980176i	$0.436514\pi$
$662$	0	0
$663$	−13.4636	−0.522884
$664$	0	0
$665$	−7.94550	−0.308113
$666$	0	0
$667$	3.84708	0.148960
$668$	0	0
$669$	−60.1090	−2.32395
$670$	0	0
$671$	−39.0169	−1.50623
$672$	0	0
$673$	−20.8006	−0.801802	−0.400901	−	0.916121i	$-0.631303\pi$
$673$	−20.8006	−0.801802	−0.400901	+	0.916121i	$0.631303\pi$
$674$	0	0
$675$	4.29547	0.165333
$676$	0	0
$677$	−3.74942	−0.144102	−0.0720509	−	0.997401i	$-0.522954\pi$
$677$	−3.74942	−0.144102	−0.0720509	+	0.997401i	$0.522954\pi$
$678$	0	0
$679$	−21.0337	−0.807199
$680$	0	0
$681$	21.0969	0.808437
$682$	0	0
$683$	6.45841	0.247124	0.123562	−	0.992337i	$-0.460568\pi$
$683$	6.45841	0.247124	0.123562	+	0.992337i	$0.460568\pi$
$684$	0	0
$685$	−3.54496	−0.135446
$686$	0	0
$687$	33.4482	1.27613
$688$	0	0
$689$	−8.91387	−0.339592
$690$	0	0
$691$	42.4181	1.61366	0.806830	−	0.590783i	$-0.201181\pi$
$691$	42.4181	1.61366	0.806830	+	0.590783i	$0.201181\pi$
$692$	0	0
$693$	−36.1612	−1.37365
$694$	0	0
$695$	35.6368	1.35178
$696$	0	0
$697$	−22.7548	−0.861899
$698$	0	0
$699$	24.9593	0.944047
$700$	0	0
$701$	−4.09328	−0.154601	−0.0773006	−	0.997008i	$-0.524630\pi$
$701$	−4.09328	−0.154601	−0.0773006	+	0.997008i	$0.524630\pi$
$702$	0	0
$703$	10.3809	0.391522
$704$	0	0
$705$	48.4950	1.82643
$706$	0	0
$707$	33.0847	1.24428
$708$	0	0
$709$	5.43271	0.204030	0.102015	−	0.994783i	$-0.467471\pi$
$709$	5.43271	0.204030	0.102015	+	0.994783i	$0.467471\pi$
$710$	0	0
$711$	11.4065	0.427779
$712$	0	0
$713$	−25.4848	−0.954414
$714$	0	0
$715$	8.02663	0.300179
$716$	0	0
$717$	−71.7571	−2.67982
$718$	0	0
$719$	−12.8926	−0.480812	−0.240406	−	0.970672i	$-0.577281\pi$
$719$	−12.8926	−0.480812	−0.240406	+	0.970672i	$0.577281\pi$
$720$	0	0
$721$	27.2046	1.01315
$722$	0	0
$723$	−32.5191	−1.20940
$724$	0	0
$725$	−0.886080	−0.0329082
$726$	0	0
$727$	5.26383	0.195225	0.0976124	−	0.995225i	$-0.468879\pi$
$727$	5.26383	0.195225	0.0976124	+	0.995225i	$0.468879\pi$
$728$	0	0
$729$	−39.7463	−1.47208
$730$	0	0
$731$	−5.34038	−0.197521
$732$	0	0
$733$	1.36026	0.0502424	0.0251212	−	0.999684i	$-0.492003\pi$
$733$	1.36026	0.0502424	0.0251212	+	0.999684i	$0.492003\pi$
$734$	0	0
$735$	−13.1982	−0.486822
$736$	0	0
$737$	12.9293	0.476256
$738$	0	0
$739$	48.3281	1.77778	0.888890	−	0.458121i	$-0.151478\pi$
$739$	48.3281	1.77778	0.888890	+	0.458121i	$0.151478\pi$
$740$	0	0
$741$	−5.73724	−0.210763
$742$	0	0
$743$	−13.4347	−0.492871	−0.246436	−	0.969159i	$-0.579259\pi$
$743$	−13.4347	−0.492871	−0.246436	+	0.969159i	$0.579259\pi$
$744$	0	0
$745$	9.35619	0.342784
$746$	0	0
$747$	6.91926	0.253162
$748$	0	0
$749$	−34.9255	−1.27615
$750$	0	0
$751$	−28.6678	−1.04610	−0.523052	−	0.852301i	$-0.675206\pi$
$751$	−28.6678	−1.04610	−0.523052	+	0.852301i	$0.675206\pi$
$752$	0	0
$753$	−8.90780	−0.324618
$754$	0	0
$755$	39.1228	1.42382
$756$	0	0
$757$	10.3286	0.375401	0.187701	−	0.982226i	$-0.439897\pi$
$757$	10.3286	0.375401	0.187701	+	0.982226i	$0.439897\pi$
$758$	0	0
$759$	−98.1271	−3.56179
$760$	0	0
$761$	−6.74657	−0.244563	−0.122281	−	0.992495i	$-0.539021\pi$
$761$	−6.74657	−0.244563	−0.122281	+	0.992495i	$0.539021\pi$
$762$	0	0
$763$	−19.7747	−0.715894
$764$	0	0
$765$	35.4375	1.28125
$766$	0	0
$767$	2.82741	0.102092
$768$	0	0
$769$	−29.9844	−1.08126	−0.540632	−	0.841259i	$-0.681815\pi$
$769$	−29.9844	−1.08126	−0.540632	+	0.841259i	$0.681815\pi$
$770$	0	0
$771$	−52.8312	−1.90267
$772$	0	0
$773$	14.9499	0.537711	0.268856	−	0.963180i	$-0.413355\pi$
$773$	14.9499	0.537711	0.268856	+	0.963180i	$0.413355\pi$
$774$	0	0
$775$	5.86980	0.210849
$776$	0	0
$777$	−25.1003	−0.900468
$778$	0	0
$779$	−9.69647	−0.347412
$780$	0	0
$781$	−70.0915	−2.50807
$782$	0	0
$783$	−1.07651	−0.0384714
$784$	0	0
$785$	−19.7584	−0.705208
$786$	0	0
$787$	45.5072	1.62216	0.811078	−	0.584939i	$-0.198881\pi$
$787$	45.5072	1.62216	0.811078	+	0.584939i	$0.198881\pi$
$788$	0	0
$789$	40.1762	1.43031
$790$	0	0
$791$	7.04936	0.250646
$792$	0	0
$793$	8.43304	0.299466
$794$	0	0
$795$	41.6429	1.47692
$796$	0	0
$797$	1.90100	0.0673368	0.0336684	−	0.999433i	$-0.489281\pi$
$797$	1.90100	0.0673368	0.0336684	+	0.999433i	$0.489281\pi$
$798$	0	0
$799$	−54.2812	−1.92033
$800$	0	0
$801$	−34.0556	−1.20330
$802$	0	0
$803$	−47.0851	−1.66160
$804$	0	0
$805$	29.3724	1.03524
$806$	0	0
$807$	−42.6773	−1.50231
$808$	0	0
$809$	19.0465	0.669639	0.334819	−	0.942282i	$-0.391325\pi$
$809$	19.0465	0.669639	0.334819	+	0.942282i	$0.391325\pi$
$810$	0	0
$811$	31.6715	1.11214	0.556068	−	0.831137i	$-0.312309\pi$
$811$	31.6715	1.11214	0.556068	+	0.831137i	$0.312309\pi$
$812$	0	0
$813$	−81.5005	−2.85835
$814$	0	0
$815$	17.3120	0.606414
$816$	0	0
$817$	−2.27569	−0.0796163
$818$	0	0
$819$	7.81582	0.273107
$820$	0	0
$821$	40.9312	1.42851	0.714255	−	0.699886i	$-0.246766\pi$
$821$	40.9312	1.42851	0.714255	+	0.699886i	$0.246766\pi$
$822$	0	0
$823$	−2.41349	−0.0841290	−0.0420645	−	0.999115i	$-0.513394\pi$
$823$	−2.41349	−0.0841290	−0.0420645	+	0.999115i	$0.513394\pi$
$824$	0	0
$825$	22.6012	0.786871
$826$	0	0
$827$	27.3475	0.950965	0.475483	−	0.879725i	$-0.342273\pi$
$827$	27.3475	0.950965	0.475483	+	0.879725i	$0.342273\pi$
$828$	0	0
$829$	4.34056	0.150754	0.0753770	−	0.997155i	$-0.475984\pi$
$829$	4.34056	0.150754	0.0753770	+	0.997155i	$0.475984\pi$
$830$	0	0
$831$	−26.9873	−0.936178
$832$	0	0
$833$	14.7729	0.511852
$834$	0	0
$835$	17.8911	0.619146
$836$	0	0
$837$	7.13131	0.246494
$838$	0	0
$839$	−8.67338	−0.299438	−0.149719	−	0.988729i	$-0.547837\pi$
$839$	−8.67338	−0.299438	−0.149719	+	0.988729i	$0.547837\pi$
$840$	0	0
$841$	−28.7779	−0.992343
$842$	0	0
$843$	48.1980	1.66003
$844$	0	0
$845$	21.2265	0.730215
$846$	0	0
$847$	−20.4221	−0.701711
$848$	0	0
$849$	−41.0302	−1.40815
$850$	0	0
$851$	−38.3753	−1.31549
$852$	0	0
$853$	31.6291	1.08296	0.541479	−	0.840714i	$-0.317865\pi$
$853$	31.6291	1.08296	0.541479	+	0.840714i	$0.317865\pi$
$854$	0	0
$855$	15.1010	0.516442
$856$	0	0
$857$	47.9630	1.63838	0.819192	−	0.573519i	$-0.194422\pi$
$857$	47.9630	1.63838	0.819192	+	0.573519i	$0.194422\pi$
$858$	0	0
$859$	−32.8601	−1.12117	−0.560587	−	0.828096i	$-0.689424\pi$
$859$	−32.8601	−1.12117	−0.560587	+	0.828096i	$0.689424\pi$
$860$	0	0
$861$	23.4455	0.799019
$862$	0	0
$863$	2.09676	0.0713745	0.0356873	−	0.999363i	$-0.488638\pi$
$863$	2.09676	0.0713745	0.0356873	+	0.999363i	$0.488638\pi$
$864$	0	0
$865$	28.0363	0.953263
$866$	0	0
$867$	−25.8398	−0.877565
$868$	0	0
$869$	13.5099	0.458291
$870$	0	0
$871$	−2.79451	−0.0946884
$872$	0	0
$873$	39.9759	1.35298
$874$	0	0
$875$	−24.7547	−0.836860
$876$	0	0
$877$	17.9179	0.605044	0.302522	−	0.953142i	$-0.402171\pi$
$877$	17.9179	0.605044	0.302522	+	0.953142i	$0.402171\pi$
$878$	0	0
$879$	−79.2912	−2.67442
$880$	0	0
$881$	3.13995	0.105788	0.0528939	−	0.998600i	$-0.483155\pi$
$881$	3.13995	0.105788	0.0528939	+	0.998600i	$0.483155\pi$
$882$	0	0
$883$	19.2263	0.647016	0.323508	−	0.946225i	$-0.395138\pi$
$883$	19.2263	0.647016	0.323508	+	0.946225i	$0.395138\pi$
$884$	0	0
$885$	−13.2088	−0.444009
$886$	0	0
$887$	23.2848	0.781828	0.390914	−	0.920427i	$-0.372159\pi$
$887$	23.2848	0.781828	0.390914	+	0.920427i	$0.372159\pi$
$888$	0	0
$889$	21.9381	0.735781
$890$	0	0
$891$	−25.7978	−0.864258
$892$	0	0
$893$	−23.1308	−0.774042
$894$	0	0
$895$	−38.3764	−1.28278
$896$	0	0
$897$	21.2090	0.708149
$898$	0	0
$899$	−1.47106	−0.0490627
$900$	0	0
$901$	−46.6116	−1.55286
$902$	0	0
$903$	5.50247	0.183111
$904$	0	0
$905$	6.11182	0.203164
$906$	0	0
$907$	−26.6815	−0.885946	−0.442973	−	0.896535i	$-0.646076\pi$
$907$	−26.6815	−0.885946	−0.442973	+	0.896535i	$0.646076\pi$
$908$	0	0
$909$	−62.8797	−2.08559
$910$	0	0
$911$	17.5867	0.582672	0.291336	−	0.956621i	$-0.405900\pi$
$911$	17.5867	0.582672	0.291336	+	0.956621i	$0.405900\pi$
$912$	0	0
$913$	8.19514	0.271220
$914$	0	0
$915$	−39.3966	−1.30241
$916$	0	0
$917$	1.51031	0.0498748
$918$	0	0
$919$	30.7032	1.01280	0.506402	−	0.862297i	$-0.330975\pi$
$919$	30.7032	1.01280	0.506402	+	0.862297i	$0.330975\pi$
$920$	0	0
$921$	43.5835	1.43612
$922$	0	0
$923$	15.1495	0.498650
$924$	0	0
$925$	8.83880	0.290618
$926$	0	0
$927$	−51.7041	−1.69818
$928$	0	0
$929$	−43.1658	−1.41622	−0.708112	−	0.706100i	$-0.750453\pi$
$929$	−43.1658	−1.41622	−0.708112	+	0.706100i	$0.750453\pi$
$930$	0	0
$931$	6.29517	0.206316
$932$	0	0
$933$	0.199137	0.00651946
$934$	0	0
$935$	41.9721	1.37263
$936$	0	0
$937$	18.3753	0.600294	0.300147	−	0.953893i	$-0.402964\pi$
$937$	18.3753	0.600294	0.300147	+	0.953893i	$0.402964\pi$
$938$	0	0
$939$	24.9489	0.814176
$940$	0	0
$941$	−17.5137	−0.570930	−0.285465	−	0.958389i	$-0.592148\pi$
$941$	−17.5137	−0.570930	−0.285465	+	0.958389i	$0.592148\pi$
$942$	0	0
$943$	35.8452	1.16728
$944$	0	0
$945$	−8.21915	−0.267369
$946$	0	0
$947$	−9.62328	−0.312715	−0.156357	−	0.987701i	$-0.549975\pi$
$947$	−9.62328	−0.312715	−0.156357	+	0.987701i	$0.549975\pi$
$948$	0	0
$949$	10.1769	0.330356
$950$	0	0
$951$	−20.7827	−0.673925
$952$	0	0
$953$	−4.06279	−0.131607	−0.0658033	−	0.997833i	$-0.520961\pi$
$953$	−4.06279	−0.131607	−0.0658033	+	0.997833i	$0.520961\pi$
$954$	0	0
$955$	−23.2484	−0.752301
$956$	0	0
$957$	−5.66420	−0.183098
$958$	0	0
$959$	−4.08837	−0.132020
$960$	0	0
$961$	−21.2550	−0.685645
$962$	0	0
$963$	66.3783	2.13901
$964$	0	0
$965$	32.1200	1.03398
$966$	0	0
$967$	13.2971	0.427607	0.213804	−	0.976877i	$-0.431415\pi$
$967$	13.2971	0.427607	0.213804	+	0.976877i	$0.431415\pi$
$968$	0	0
$969$	−30.0006	−0.963759
$970$	0	0
$971$	−55.1313	−1.76925	−0.884624	−	0.466305i	$-0.845585\pi$
$971$	−55.1313	−1.76925	−0.884624	+	0.466305i	$0.845585\pi$
$972$	0	0
$973$	41.0995	1.31759
$974$	0	0
$975$	−4.88497	−0.156444
$976$	0	0
$977$	−1.98310	−0.0634448	−0.0317224	−	0.999497i	$-0.510099\pi$
$977$	−1.98310	−0.0634448	−0.0317224	+	0.999497i	$0.510099\pi$
$978$	0	0
$979$	−40.3353	−1.28912
$980$	0	0
$981$	37.5832	1.19994
$982$	0	0
$983$	35.3606	1.12783	0.563913	−	0.825834i	$-0.309295\pi$
$983$	35.3606	1.12783	0.563913	+	0.825834i	$0.309295\pi$
$984$	0	0
$985$	−29.0228	−0.924744
$986$	0	0
$987$	55.9287	1.78023
$988$	0	0
$989$	8.41260	0.267505
$990$	0	0
$991$	44.0460	1.39917	0.699584	−	0.714551i	$-0.253369\pi$
$991$	44.0460	1.39917	0.699584	+	0.714551i	$0.253369\pi$
$992$	0	0
$993$	2.55612	0.0811160
$994$	0	0
$995$	17.1122	0.542494
$996$	0	0
$997$	−9.78987	−0.310048	−0.155024	−	0.987911i	$-0.549546\pi$
$997$	−9.78987	−0.310048	−0.155024	+	0.987911i	$0.549546\pi$
$998$	0	0
$999$	10.7384	0.339748

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.w.1.2		29
4.3	odd	2		4024.2.a.e.1.28	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.28	✓	29	4.3	odd	2
8048.2.a.w.1.2		29	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-2.62135 q^{3} -1.76626 q^{5} -2.03701 q^{7} +3.87147 q^{9} +O(q^{10})\)	\(q-2.62135 q^{3} -1.76626 q^{5} -2.03701 q^{7} +3.87147 q^{9} +4.58536 q^{11} -0.991072 q^{13} +4.62999 q^{15} -5.18242 q^{17} -2.20838 q^{19} +5.33972 q^{21} +8.16377 q^{23} -1.88032 q^{25} -2.28443 q^{27} +0.471238 q^{29} -3.12170 q^{31} -12.0198 q^{33} +3.59789 q^{35} -4.70068 q^{37} +2.59794 q^{39} +4.39077 q^{41} +1.03048 q^{43} -6.83803 q^{45} +10.4741 q^{47} -2.85059 q^{49} +13.5849 q^{51} +8.99418 q^{53} -8.09894 q^{55} +5.78893 q^{57} -2.85288 q^{59} -8.50901 q^{61} -7.88623 q^{63} +1.75049 q^{65} +2.81969 q^{67} -21.4001 q^{69} -15.2859 q^{71} -10.2686 q^{73} +4.92898 q^{75} -9.34043 q^{77} +2.94631 q^{79} -5.62612 q^{81} +1.78724 q^{83} +9.15350 q^{85} -1.23528 q^{87} -8.79655 q^{89} +2.01882 q^{91} +8.18306 q^{93} +3.90057 q^{95} +10.3258 q^{97} +17.7521 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9	\( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9} + 27 q^{11} + 16 q^{13} + 14 q^{15} - 15 q^{17} + 14 q^{19} + q^{21} + 25 q^{23} + 21 q^{25} + 25 q^{27} - 13 q^{29} + 27 q^{31} - 9 q^{33} + 29 q^{35} + 35 q^{37} + 38 q^{39} - 30 q^{41} + 38 q^{43} + q^{45} + 35 q^{47} + 14 q^{49} + 21 q^{51} + 2 q^{53} + 25 q^{55} - 25 q^{57} + 40 q^{59} + 10 q^{61} + 56 q^{63} - 50 q^{65} + 31 q^{67} + 11 q^{69} + 65 q^{71} - 23 q^{73} + 32 q^{75} + 13 q^{77} + 44 q^{79} - 7 q^{81} + 41 q^{83} + 26 q^{85} + 25 q^{87} - 48 q^{89} + 44 q^{91} + 25 q^{93} + 75 q^{95} - 18 q^{97} + 80 q^{99}+O(q^{100}) \) 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9 + 27 * q^11 + 16 * q^13 + 14 * q^15 - 15 * q^17 + 14 * q^19 + q^21 + 25 * q^23 + 21 * q^25 + 25 * q^27 - 13 * q^29 + 27 * q^31 - 9 * q^33 + 29 * q^35 + 35 * q^37 + 38 * q^39 - 30 * q^41 + 38 * q^43 + q^45 + 35 * q^47 + 14 * q^49 + 21 * q^51 + 2 * q^53 + 25 * q^55 - 25 * q^57 + 40 * q^59 + 10 * q^61 + 56 * q^63 - 50 * q^65 + 31 * q^67 + 11 * q^69 + 65 * q^71 - 23 * q^73 + 32 * q^75 + 13 * q^77 + 44 * q^79 - 7 * q^81 + 41 * q^83 + 26 * q^85 + 25 * q^87 - 48 * q^89 + 44 * q^91 + 25 * q^93 + 75 * q^95 - 18 * q^97 + 80 * q^99

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