LMFDB - Embedded newform 8048.2.a.o.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:	$5$
Coefficient field:	5.5.205225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 3 $ x^5 - x^4 - 6x^3 + 3x^2 + 7*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1006)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.17073$ of defining polynomial
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)

$f(q)$	$=$	$q-0.170728 q^{3} -0.411173 q^{5} +3.93028 q^{7} -2.97085 q^{9} +O(q^{10})$	$q-0.170728 q^{3} -0.411173 q^{5} +3.93028 q^{7} -2.97085 q^{9} -2.24904 q^{11} -7.09487 q^{13} +0.0701989 q^{15} +1.46831 q^{17} +1.42952 q^{19} -0.671011 q^{21} -5.89500 q^{23} -4.83094 q^{25} +1.01939 q^{27} -10.5941 q^{29} +5.37060 q^{31} +0.383974 q^{33} -1.61603 q^{35} +1.09137 q^{37} +1.21130 q^{39} +4.54661 q^{41} +5.73685 q^{43} +1.22153 q^{45} -8.31027 q^{47} +8.44713 q^{49} -0.250682 q^{51} +6.80693 q^{53} +0.924742 q^{55} -0.244059 q^{57} -2.69911 q^{59} +6.35288 q^{61} -11.6763 q^{63} +2.91722 q^{65} +10.1931 q^{67} +1.00644 q^{69} +4.76139 q^{71} +1.54133 q^{73} +0.824778 q^{75} -8.83935 q^{77} +13.3467 q^{79} +8.73852 q^{81} +13.9631 q^{83} -0.603728 q^{85} +1.80871 q^{87} +11.5045 q^{89} -27.8849 q^{91} -0.916915 q^{93} -0.587779 q^{95} +11.2102 q^{97} +6.68155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{3} - 3 q^{5} + 9 q^{7} + q^{9}+O(q^{10}) $ 5 * q + 4 * q^3 - 3 * q^5 + 9 * q^7 + q^9	$ 5 q + 4 q^{3} - 3 q^{5} + 9 q^{7} + q^{9} + 15 q^{11} - 5 q^{13} + 4 q^{15} - 6 q^{17} - 10 q^{19} - 12 q^{21} + 12 q^{23} + 6 q^{25} + 7 q^{27} - 16 q^{29} + 33 q^{31} + 21 q^{33} + 11 q^{35} + 2 q^{37} - 4 q^{39} - 12 q^{41} + 17 q^{43} + 3 q^{45} + 7 q^{47} + 36 q^{49} - 14 q^{51} - 2 q^{55} - 6 q^{57} - 18 q^{59} + q^{61} - 31 q^{63} - 14 q^{65} - 6 q^{67} + 15 q^{69} + 26 q^{71} + 9 q^{73} + 33 q^{75} + 16 q^{77} + 21 q^{79} - 3 q^{81} - 28 q^{85} - 34 q^{87} + 15 q^{89} - 26 q^{91} + 19 q^{93} + 18 q^{95} + 26 q^{97} + 44 q^{99}+O(q^{100}) $ 5 * q + 4 * q^3 - 3 * q^5 + 9 * q^7 + q^9 + 15 * q^11 - 5 * q^13 + 4 * q^15 - 6 * q^17 - 10 * q^19 - 12 * q^21 + 12 * q^23 + 6 * q^25 + 7 * q^27 - 16 * q^29 + 33 * q^31 + 21 * q^33 + 11 * q^35 + 2 * q^37 - 4 * q^39 - 12 * q^41 + 17 * q^43 + 3 * q^45 + 7 * q^47 + 36 * q^49 - 14 * q^51 - 2 * q^55 - 6 * q^57 - 18 * q^59 + q^61 - 31 * q^63 - 14 * q^65 - 6 * q^67 + 15 * q^69 + 26 * q^71 + 9 * q^73 + 33 * q^75 + 16 * q^77 + 21 * q^79 - 3 * q^81 - 28 * q^85 - 34 * q^87 + 15 * q^89 - 26 * q^91 + 19 * q^93 + 18 * q^95 + 26 * q^97 + 44 * 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
$3$	−0.170728	−0.0985701	−0.0492850	−	0.998785i	$-0.515694\pi$
$3$	−0.170728	−0.0985701	−0.0492850	+	0.998785i	$0.515694\pi$
$4$	0	0
$5$	−0.411173	−0.183882	−0.0919410	−	0.995764i	$-0.529307\pi$
$5$	−0.411173	−0.183882	−0.0919410	+	0.995764i	$0.529307\pi$
$6$	0	0
$7$	3.93028	1.48551	0.742754	−	0.669565i	$-0.233519\pi$
$7$	3.93028	1.48551	0.742754	+	0.669565i	$0.233519\pi$
$8$	0	0
$9$	−2.97085	−0.990284
$10$	0	0
$11$	−2.24904	−0.678110	−0.339055	−	0.940767i	$-0.610107\pi$
$11$	−2.24904	−0.678110	−0.339055	+	0.940767i	$0.610107\pi$
$12$	0	0
$13$	−7.09487	−1.96776	−0.983882	−	0.178820i	$-0.942772\pi$
$13$	−7.09487	−1.96776	−0.983882	+	0.178820i	$0.942772\pi$
$14$	0	0
$15$	0.0701989	0.0181253
$16$	0	0
$17$	1.46831	0.356117	0.178058	−	0.984020i	$-0.443018\pi$
$17$	1.46831	0.356117	0.178058	+	0.984020i	$0.443018\pi$
$18$	0	0
$19$	1.42952	0.327954	0.163977	−	0.986464i	$-0.447568\pi$
$19$	1.42952	0.327954	0.163977	+	0.986464i	$0.447568\pi$
$20$	0	0
$21$	−0.671011	−0.146427
$22$	0	0
$23$	−5.89500	−1.22919	−0.614596	−	0.788842i	$-0.710681\pi$
$23$	−5.89500	−1.22919	−0.614596	+	0.788842i	$0.710681\pi$
$24$	0	0
$25$	−4.83094	−0.966187
$26$	0	0
$27$	1.01939	0.196182
$28$	0	0
$29$	−10.5941	−1.96728	−0.983638	−	0.180155i	$-0.942340\pi$
$29$	−10.5941	−1.96728	−0.983638	+	0.180155i	$0.942340\pi$
$30$	0	0
$31$	5.37060	0.964589	0.482295	−	0.876009i	$-0.339803\pi$
$31$	5.37060	0.964589	0.482295	+	0.876009i	$0.339803\pi$
$32$	0	0
$33$	0.383974	0.0668413
$34$	0	0
$35$	−1.61603	−0.273158
$36$	0	0
$37$	1.09137	0.179421	0.0897103	−	0.995968i	$-0.471406\pi$
$37$	1.09137	0.179421	0.0897103	+	0.995968i	$0.471406\pi$
$38$	0	0
$39$	1.21130	0.193963
$40$	0	0
$41$	4.54661	0.710062	0.355031	−	0.934855i	$-0.384470\pi$
$41$	4.54661	0.710062	0.355031	+	0.934855i	$0.384470\pi$
$42$	0	0
$43$	5.73685	0.874862	0.437431	−	0.899252i	$-0.355889\pi$
$43$	5.73685	0.874862	0.437431	+	0.899252i	$0.355889\pi$
$44$	0	0
$45$	1.22153	0.182095
$46$	0	0
$47$	−8.31027	−1.21218	−0.606089	−	0.795397i	$-0.707262\pi$
$47$	−8.31027	−1.21218	−0.606089	+	0.795397i	$0.707262\pi$
$48$	0	0
$49$	8.44713	1.20673
$50$	0	0
$51$	−0.250682	−0.0351024
$52$	0	0
$53$	6.80693	0.935004	0.467502	−	0.883992i	$-0.345154\pi$
$53$	6.80693	0.935004	0.467502	+	0.883992i	$0.345154\pi$
$54$	0	0
$55$	0.924742	0.124692
$56$	0	0
$57$	−0.244059	−0.0323265
$58$	0	0
$59$	−2.69911	−0.351394	−0.175697	−	0.984444i	$-0.556218\pi$
$59$	−2.69911	−0.351394	−0.175697	+	0.984444i	$0.556218\pi$
$60$	0	0
$61$	6.35288	0.813402	0.406701	−	0.913561i	$-0.366679\pi$
$61$	6.35288	0.813402	0.406701	+	0.913561i	$0.366679\pi$
$62$	0	0
$63$	−11.6763	−1.47107
$64$	0	0
$65$	2.91722	0.361836
$66$	0	0
$67$	10.1931	1.24528	0.622640	−	0.782508i	$-0.286060\pi$
$67$	10.1931	1.24528	0.622640	+	0.782508i	$0.286060\pi$
$68$	0	0
$69$	1.00644	0.121162
$70$	0	0
$71$	4.76139	0.565073	0.282537	−	0.959257i	$-0.408824\pi$
$71$	4.76139	0.565073	0.282537	+	0.959257i	$0.408824\pi$
$72$	0	0
$73$	1.54133	0.180399	0.0901997	−	0.995924i	$-0.471249\pi$
$73$	1.54133	0.180399	0.0901997	+	0.995924i	$0.471249\pi$
$74$	0	0
$75$	0.824778	0.0952372
$76$	0	0
$77$	−8.83935	−1.00734
$78$	0	0
$79$	13.3467	1.50163	0.750813	−	0.660515i	$-0.229662\pi$
$79$	13.3467	1.50163	0.750813	+	0.660515i	$0.229662\pi$
$80$	0	0
$81$	8.73852	0.970946
$82$	0	0
$83$	13.9631	1.53265	0.766323	−	0.642456i	$-0.222084\pi$
$83$	13.9631	1.53265	0.766323	+	0.642456i	$0.222084\pi$
$84$	0	0
$85$	−0.603728	−0.0654834
$86$	0	0
$87$	1.80871	0.193915
$88$	0	0
$89$	11.5045	1.21948	0.609738	−	0.792603i	$-0.291275\pi$
$89$	11.5045	1.21948	0.609738	+	0.792603i	$0.291275\pi$
$90$	0	0
$91$	−27.8849	−2.92313
$92$	0	0
$93$	−0.916915	−0.0950796
$94$	0	0
$95$	−0.587779	−0.0603049
$96$	0	0
$97$	11.2102	1.13823	0.569114	−	0.822259i	$-0.307286\pi$
$97$	11.2102	1.13823	0.569114	+	0.822259i	$0.307286\pi$
$98$	0	0
$99$	6.68155	0.671521
$100$	0	0
$101$	11.1533	1.10980	0.554898	−	0.831919i	$-0.312757\pi$
$101$	11.1533	1.10980	0.554898	+	0.831919i	$0.312757\pi$
$102$	0	0
$103$	16.9392	1.66907	0.834533	−	0.550957i	$-0.185737\pi$
$103$	16.9392	1.66907	0.834533	+	0.550957i	$0.185737\pi$
$104$	0	0
$105$	0.275901	0.0269252
$106$	0	0
$107$	−19.8223	−1.91630	−0.958149	−	0.286270i	$-0.907584\pi$
$107$	−19.8223	−1.91630	−0.958149	+	0.286270i	$0.907584\pi$
$108$	0	0
$109$	−12.3114	−1.17922	−0.589612	−	0.807687i	$-0.700719\pi$
$109$	−12.3114	−1.17922	−0.589612	+	0.807687i	$0.700719\pi$
$110$	0	0
$111$	−0.186328	−0.0176855
$112$	0	0
$113$	−5.50704	−0.518059	−0.259029	−	0.965869i	$-0.583403\pi$
$113$	−5.50704	−0.518059	−0.259029	+	0.965869i	$0.583403\pi$
$114$	0	0
$115$	2.42386	0.226026
$116$	0	0
$117$	21.0778	1.94864
$118$	0	0
$119$	5.77086	0.529014
$120$	0	0
$121$	−5.94184	−0.540167
$122$	0	0
$123$	−0.776236	−0.0699908
$124$	0	0
$125$	4.04221	0.361547
$126$	0	0
$127$	4.19012	0.371813	0.185907	−	0.982567i	$-0.440478\pi$
$127$	4.19012	0.371813	0.185907	+	0.982567i	$0.440478\pi$
$128$	0	0
$129$	−0.979443	−0.0862352
$130$	0	0
$131$	4.67972	0.408869	0.204434	−	0.978880i	$-0.434464\pi$
$131$	4.67972	0.408869	0.204434	+	0.978880i	$0.434464\pi$
$132$	0	0
$133$	5.61841	0.487178
$134$	0	0
$135$	−0.419147	−0.0360744
$136$	0	0
$137$	−0.408827	−0.0349285	−0.0174642	−	0.999847i	$-0.505559\pi$
$137$	−0.408827	−0.0349285	−0.0174642	+	0.999847i	$0.505559\pi$
$138$	0	0
$139$	−2.62314	−0.222492	−0.111246	−	0.993793i	$-0.535484\pi$
$139$	−2.62314	−0.222492	−0.111246	+	0.993793i	$0.535484\pi$
$140$	0	0
$141$	1.41880	0.119484
$142$	0	0
$143$	15.9566	1.33436
$144$	0	0
$145$	4.35601	0.361747
$146$	0	0
$147$	−1.44217	−0.118948
$148$	0	0
$149$	−22.3729	−1.83286	−0.916431	−	0.400192i	$-0.868944\pi$
$149$	−22.3729	−1.83286	−0.916431	+	0.400192i	$0.868944\pi$
$150$	0	0
$151$	7.06211	0.574706	0.287353	−	0.957825i	$-0.407225\pi$
$151$	7.06211	0.574706	0.287353	+	0.957825i	$0.407225\pi$
$152$	0	0
$153$	−4.36212	−0.352657
$154$	0	0
$155$	−2.20825	−0.177371
$156$	0	0
$157$	20.0175	1.59757	0.798785	−	0.601617i	$-0.205477\pi$
$157$	20.0175	1.59757	0.798785	+	0.601617i	$0.205477\pi$
$158$	0	0
$159$	−1.16214	−0.0921635
$160$	0	0
$161$	−23.1690	−1.82597
$162$	0	0
$163$	−12.4550	−0.975549	−0.487775	−	0.872970i	$-0.662191\pi$
$163$	−12.4550	−0.975549	−0.487775	+	0.872970i	$0.662191\pi$
$164$	0	0
$165$	−0.157880	−0.0122909
$166$	0	0
$167$	−0.265044	−0.0205097	−0.0102549	−	0.999947i	$-0.503264\pi$
$167$	−0.265044	−0.0205097	−0.0102549	+	0.999947i	$0.503264\pi$
$168$	0	0
$169$	37.3372	2.87209
$170$	0	0
$171$	−4.24689	−0.324768
$172$	0	0
$173$	10.6582	0.810325	0.405163	−	0.914245i	$-0.367215\pi$
$173$	10.6582	0.810325	0.405163	+	0.914245i	$0.367215\pi$
$174$	0	0
$175$	−18.9870	−1.43528
$176$	0	0
$177$	0.460815	0.0346370
$178$	0	0
$179$	11.3081	0.845208	0.422604	−	0.906314i	$-0.361116\pi$
$179$	11.3081	0.845208	0.422604	+	0.906314i	$0.361116\pi$
$180$	0	0
$181$	−1.18097	−0.0877805	−0.0438902	−	0.999036i	$-0.513975\pi$
$181$	−1.18097	−0.0877805	−0.0438902	+	0.999036i	$0.513975\pi$
$182$	0	0
$183$	−1.08462	−0.0801771
$184$	0	0
$185$	−0.448743	−0.0329922
$186$	0	0
$187$	−3.30227	−0.241486
$188$	0	0
$189$	4.00651	0.291431
$190$	0	0
$191$	16.8778	1.22124	0.610619	−	0.791925i	$-0.290921\pi$
$191$	16.8778	1.22124	0.610619	+	0.791925i	$0.290921\pi$
$192$	0	0
$193$	3.32673	0.239463	0.119731	−	0.992806i	$-0.461797\pi$
$193$	3.32673	0.239463	0.119731	+	0.992806i	$0.461797\pi$
$194$	0	0
$195$	−0.498052	−0.0356662
$196$	0	0
$197$	−11.4195	−0.813606	−0.406803	−	0.913516i	$-0.633356\pi$
$197$	−11.4195	−0.813606	−0.406803	+	0.913516i	$0.633356\pi$
$198$	0	0
$199$	0.800238	0.0567274	0.0283637	−	0.999598i	$-0.490970\pi$
$199$	0.800238	0.0567274	0.0283637	+	0.999598i	$0.490970\pi$
$200$	0	0
$201$	−1.74024	−0.122747
$202$	0	0
$203$	−41.6379	−2.92240
$204$	0	0
$205$	−1.86944	−0.130568
$206$	0	0
$207$	17.5132	1.21725
$208$	0	0
$209$	−3.21504	−0.222389
$210$	0	0
$211$	9.81940	0.675996	0.337998	−	0.941147i	$-0.390250\pi$
$211$	9.81940	0.675996	0.337998	+	0.941147i	$0.390250\pi$
$212$	0	0
$213$	−0.812905	−0.0556993
$214$	0	0
$215$	−2.35884	−0.160871
$216$	0	0
$217$	21.1080	1.43290
$218$	0	0
$219$	−0.263149	−0.0177820
$220$	0	0
$221$	−10.4174	−0.700753
$222$	0	0
$223$	6.90132	0.462147	0.231073	−	0.972936i	$-0.425776\pi$
$223$	6.90132	0.462147	0.231073	+	0.972936i	$0.425776\pi$
$224$	0	0
$225$	14.3520	0.956800
$226$	0	0
$227$	−5.72972	−0.380295	−0.190147	−	0.981756i	$-0.560897\pi$
$227$	−5.72972	−0.380295	−0.190147	+	0.981756i	$0.560897\pi$
$228$	0	0
$229$	−23.1686	−1.53102	−0.765512	−	0.643422i	$-0.777514\pi$
$229$	−23.1686	−1.53102	−0.765512	+	0.643422i	$0.777514\pi$
$230$	0	0
$231$	1.50913	0.0992933
$232$	0	0
$233$	16.3858	1.07347	0.536734	−	0.843752i	$-0.319658\pi$
$233$	16.3858	1.07347	0.536734	+	0.843752i	$0.319658\pi$
$234$	0	0
$235$	3.41696	0.222898
$236$	0	0
$237$	−2.27867	−0.148015
$238$	0	0
$239$	−14.2792	−0.923646	−0.461823	−	0.886972i	$-0.652805\pi$
$239$	−14.2792	−0.923646	−0.461823	+	0.886972i	$0.652805\pi$
$240$	0	0
$241$	−2.91817	−0.187976	−0.0939881	−	0.995573i	$-0.529962\pi$
$241$	−2.91817	−0.187976	−0.0939881	+	0.995573i	$0.529962\pi$
$242$	0	0
$243$	−4.55009	−0.291889
$244$	0	0
$245$	−3.47323	−0.221897
$246$	0	0
$247$	−10.1423	−0.645336
$248$	0	0
$249$	−2.38389	−0.151073
$250$	0	0
$251$	15.8846	1.00263	0.501315	−	0.865265i	$-0.332850\pi$
$251$	15.8846	1.00263	0.501315	+	0.865265i	$0.332850\pi$
$252$	0	0
$253$	13.2581	0.833527
$254$	0	0
$255$	0.103073	0.00645471
$256$	0	0
$257$	28.5913	1.78348	0.891738	−	0.452552i	$-0.149486\pi$
$257$	28.5913	1.78348	0.891738	+	0.452552i	$0.149486\pi$
$258$	0	0
$259$	4.28941	0.266531
$260$	0	0
$261$	31.4735	1.94816
$262$	0	0
$263$	6.39509	0.394338	0.197169	−	0.980370i	$-0.436825\pi$
$263$	6.39509	0.394338	0.197169	+	0.980370i	$0.436825\pi$
$264$	0	0
$265$	−2.79883	−0.171931
$266$	0	0
$267$	−1.96415	−0.120204
$268$	0	0
$269$	3.68144	0.224461	0.112231	−	0.993682i	$-0.464200\pi$
$269$	3.68144	0.224461	0.112231	+	0.993682i	$0.464200\pi$
$270$	0	0
$271$	−5.93769	−0.360689	−0.180345	−	0.983603i	$-0.557721\pi$
$271$	−5.93769	−0.360689	−0.180345	+	0.983603i	$0.557721\pi$
$272$	0	0
$273$	4.76074	0.288133
$274$	0	0
$275$	10.8650	0.655181
$276$	0	0
$277$	−8.64858	−0.519643	−0.259821	−	0.965657i	$-0.583664\pi$
$277$	−8.64858	−0.519643	−0.259821	+	0.965657i	$0.583664\pi$
$278$	0	0
$279$	−15.9553	−0.955217
$280$	0	0
$281$	−28.3543	−1.69148	−0.845739	−	0.533597i	$-0.820840\pi$
$281$	−28.3543	−1.69148	−0.845739	+	0.533597i	$0.820840\pi$
$282$	0	0
$283$	−22.4481	−1.33440	−0.667201	−	0.744878i	$-0.732508\pi$
$283$	−22.4481	−1.33440	−0.667201	+	0.744878i	$0.732508\pi$
$284$	0	0
$285$	0.100351	0.00594425
$286$	0	0
$287$	17.8695	1.05480
$288$	0	0
$289$	−14.8441	−0.873181
$290$	0	0
$291$	−1.91391	−0.112195
$292$	0	0
$293$	−16.7042	−0.975873	−0.487936	−	0.872879i	$-0.662250\pi$
$293$	−16.7042	−0.975873	−0.487936	+	0.872879i	$0.662250\pi$
$294$	0	0
$295$	1.10980	0.0646151
$296$	0	0
$297$	−2.29265	−0.133033
$298$	0	0
$299$	41.8243	2.41876
$300$	0	0
$301$	22.5475	1.29961
$302$	0	0
$303$	−1.90419	−0.109393
$304$	0	0
$305$	−2.61213	−0.149570
$306$	0	0
$307$	11.6140	0.662845	0.331423	−	0.943482i	$-0.392471\pi$
$307$	11.6140	0.662845	0.331423	+	0.943482i	$0.392471\pi$
$308$	0	0
$309$	−2.89200	−0.164520
$310$	0	0
$311$	8.10150	0.459394	0.229697	−	0.973262i	$-0.426227\pi$
$311$	8.10150	0.459394	0.229697	+	0.973262i	$0.426227\pi$
$312$	0	0
$313$	0.924406	0.0522505	0.0261253	−	0.999659i	$-0.491683\pi$
$313$	0.924406	0.0522505	0.0261253	+	0.999659i	$0.491683\pi$
$314$	0	0
$315$	4.80097	0.270504
$316$	0	0
$317$	12.8528	0.721887	0.360944	−	0.932588i	$-0.382455\pi$
$317$	12.8528	0.721887	0.360944	+	0.932588i	$0.382455\pi$
$318$	0	0
$319$	23.8265	1.33403
$320$	0	0
$321$	3.38424	0.188890
$322$	0	0
$323$	2.09897	0.116790
$324$	0	0
$325$	34.2749	1.90123
$326$	0	0
$327$	2.10191	0.116236
$328$	0	0
$329$	−32.6617	−1.80070
$330$	0	0
$331$	−34.8858	−1.91750	−0.958749	−	0.284253i	$-0.908255\pi$
$331$	−34.8858	−1.91750	−0.958749	+	0.284253i	$0.908255\pi$
$332$	0	0
$333$	−3.24231	−0.177677
$334$	0	0
$335$	−4.19111	−0.228985
$336$	0	0
$337$	−15.6351	−0.851698	−0.425849	−	0.904794i	$-0.640025\pi$
$337$	−15.6351	−0.851698	−0.425849	+	0.904794i	$0.640025\pi$
$338$	0	0
$339$	0.940208	0.0510651
$340$	0	0
$341$	−12.0787	−0.654097
$342$	0	0
$343$	5.68764	0.307104
$344$	0	0
$345$	−0.413822	−0.0222794
$346$	0	0
$347$	23.8266	1.27908	0.639538	−	0.768759i	$-0.279126\pi$
$347$	23.8266	1.27908	0.639538	+	0.768759i	$0.279126\pi$
$348$	0	0
$349$	21.9405	1.17445	0.587224	−	0.809424i	$-0.300221\pi$
$349$	21.9405	1.17445	0.587224	+	0.809424i	$0.300221\pi$
$350$	0	0
$351$	−7.23247	−0.386041
$352$	0	0
$353$	−21.5923	−1.14924	−0.574622	−	0.818419i	$-0.694851\pi$
$353$	−21.5923	−1.14924	−0.574622	+	0.818419i	$0.694851\pi$
$354$	0	0
$355$	−1.95775	−0.103907
$356$	0	0
$357$	−0.985250	−0.0521449
$358$	0	0
$359$	−11.4829	−0.606042	−0.303021	−	0.952984i	$-0.597995\pi$
$359$	−11.4829	−0.606042	−0.303021	+	0.952984i	$0.597995\pi$
$360$	0	0
$361$	−16.9565	−0.892446
$362$	0	0
$363$	1.01444	0.0532443
$364$	0	0
$365$	−0.633754	−0.0331722
$366$	0	0
$367$	3.88096	0.202584	0.101292	−	0.994857i	$-0.467702\pi$
$367$	3.88096	0.202584	0.101292	+	0.994857i	$0.467702\pi$
$368$	0	0
$369$	−13.5073	−0.703163
$370$	0	0
$371$	26.7532	1.38896
$372$	0	0
$373$	−29.6185	−1.53359	−0.766795	−	0.641892i	$-0.778150\pi$
$373$	−29.6185	−1.53359	−0.766795	+	0.641892i	$0.778150\pi$
$374$	0	0
$375$	−0.690121	−0.0356377
$376$	0	0
$377$	75.1638	3.87114
$378$	0	0
$379$	−17.9581	−0.922445	−0.461222	−	0.887285i	$-0.652589\pi$
$379$	−17.9581	−0.922445	−0.461222	+	0.887285i	$0.652589\pi$
$380$	0	0
$381$	−0.715373	−0.0366497
$382$	0	0
$383$	−21.3569	−1.09129	−0.545644	−	0.838017i	$-0.683715\pi$
$383$	−21.3569	−1.09129	−0.545644	+	0.838017i	$0.683715\pi$
$384$	0	0
$385$	3.63450	0.185231
$386$	0	0
$387$	−17.0433	−0.866361
$388$	0	0
$389$	−14.7126	−0.745957	−0.372979	−	0.927840i	$-0.621663\pi$
$389$	−14.7126	−0.745957	−0.372979	+	0.927840i	$0.621663\pi$
$390$	0	0
$391$	−8.65566	−0.437736
$392$	0	0
$393$	−0.798961	−0.0403022
$394$	0	0
$395$	−5.48781	−0.276122
$396$	0	0
$397$	−26.7008	−1.34007	−0.670037	−	0.742327i	$-0.733722\pi$
$397$	−26.7008	−1.34007	−0.670037	+	0.742327i	$0.733722\pi$
$398$	0	0
$399$	−0.959223	−0.0480212
$400$	0	0
$401$	1.30956	0.0653964	0.0326982	−	0.999465i	$-0.489590\pi$
$401$	1.30956	0.0653964	0.0326982	+	0.999465i	$0.489590\pi$
$402$	0	0
$403$	−38.1038	−1.89808
$404$	0	0
$405$	−3.59304	−0.178540
$406$	0	0
$407$	−2.45454	−0.121667
$408$	0	0
$409$	−4.26983	−0.211130	−0.105565	−	0.994412i	$-0.533665\pi$
$409$	−4.26983	−0.211130	−0.105565	+	0.994412i	$0.533665\pi$
$410$	0	0
$411$	0.0697984	0.00344290
$412$	0	0
$413$	−10.6083	−0.521999
$414$	0	0
$415$	−5.74123	−0.281826
$416$	0	0
$417$	0.447845	0.0219310
$418$	0	0
$419$	13.4588	0.657508	0.328754	−	0.944416i	$-0.393371\pi$
$419$	13.4588	0.657508	0.328754	+	0.944416i	$0.393371\pi$
$420$	0	0
$421$	5.75702	0.280580	0.140290	−	0.990110i	$-0.455197\pi$
$421$	5.75702	0.280580	0.140290	+	0.990110i	$0.455197\pi$
$422$	0	0
$423$	24.6886	1.20040
$424$	0	0
$425$	−7.09329	−0.344075
$426$	0	0
$427$	24.9686	1.20832
$428$	0	0
$429$	−2.72425	−0.131528
$430$	0	0
$431$	3.00936	0.144956	0.0724780	−	0.997370i	$-0.476909\pi$
$431$	3.00936	0.144956	0.0724780	+	0.997370i	$0.476909\pi$
$432$	0	0
$433$	−20.1108	−0.966462	−0.483231	−	0.875493i	$-0.660537\pi$
$433$	−20.1108	−0.966462	−0.483231	+	0.875493i	$0.660537\pi$
$434$	0	0
$435$	−0.743694	−0.0356574
$436$	0	0
$437$	−8.42701	−0.403118
$438$	0	0
$439$	34.4627	1.64481	0.822407	−	0.568900i	$-0.192631\pi$
$439$	34.4627	1.64481	0.822407	+	0.568900i	$0.192631\pi$
$440$	0	0
$441$	−25.0952	−1.19501
$442$	0	0
$443$	5.13471	0.243957	0.121979	−	0.992533i	$-0.461076\pi$
$443$	5.13471	0.243957	0.121979	+	0.992533i	$0.461076\pi$
$444$	0	0
$445$	−4.73034	−0.224240
$446$	0	0
$447$	3.81970	0.180665
$448$	0	0
$449$	28.1953	1.33062	0.665310	−	0.746567i	$-0.268299\pi$
$449$	28.1953	1.33062	0.665310	+	0.746567i	$0.268299\pi$
$450$	0	0
$451$	−10.2255	−0.481500
$452$	0	0
$453$	−1.20570	−0.0566488
$454$	0	0
$455$	11.4655	0.537511
$456$	0	0
$457$	4.02044	0.188068	0.0940342	−	0.995569i	$-0.470024\pi$
$457$	4.02044	0.188068	0.0940342	+	0.995569i	$0.470024\pi$
$458$	0	0
$459$	1.49678	0.0698638
$460$	0	0
$461$	−28.5766	−1.33095	−0.665473	−	0.746422i	$-0.731770\pi$
$461$	−28.5766	−1.33095	−0.665473	+	0.746422i	$0.731770\pi$
$462$	0	0
$463$	20.8967	0.971150	0.485575	−	0.874195i	$-0.338610\pi$
$463$	20.8967	0.971150	0.485575	+	0.874195i	$0.338610\pi$
$464$	0	0
$465$	0.377010	0.0174834
$466$	0	0
$467$	8.76726	0.405700	0.202850	−	0.979210i	$-0.434980\pi$
$467$	8.76726	0.405700	0.202850	+	0.979210i	$0.434980\pi$
$468$	0	0
$469$	40.0616	1.84987
$470$	0	0
$471$	−3.41755	−0.157473
$472$	0	0
$473$	−12.9024	−0.593252
$474$	0	0
$475$	−6.90591	−0.316865
$476$	0	0
$477$	−20.2224	−0.925920
$478$	0	0
$479$	29.1667	1.33266	0.666329	−	0.745658i	$-0.267865\pi$
$479$	29.1667	1.33266	0.666329	+	0.745658i	$0.267865\pi$
$480$	0	0
$481$	−7.74315	−0.353057
$482$	0	0
$483$	3.95561	0.179986
$484$	0	0
$485$	−4.60935	−0.209300
$486$	0	0
$487$	25.5737	1.15885	0.579427	−	0.815024i	$-0.303276\pi$
$487$	25.5737	1.15885	0.579427	+	0.815024i	$0.303276\pi$
$488$	0	0
$489$	2.12642	0.0961600
$490$	0	0
$491$	−11.7038	−0.528184	−0.264092	−	0.964497i	$-0.585072\pi$
$491$	−11.7038	−0.528184	−0.264092	+	0.964497i	$0.585072\pi$
$492$	0	0
$493$	−15.5554	−0.700580
$494$	0	0
$495$	−2.74727	−0.123481
$496$	0	0
$497$	18.7136	0.839420
$498$	0	0
$499$	26.9301	1.20556	0.602778	−	0.797909i	$-0.294060\pi$
$499$	26.9301	1.20556	0.602778	+	0.797909i	$0.294060\pi$
$500$	0	0
$501$	0.0452506	0.00202165
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−4.58593	−0.204071
$506$	0	0
$507$	−6.37452	−0.283103
$508$	0	0
$509$	4.55748	0.202007	0.101003	−	0.994886i	$-0.467795\pi$
$509$	4.55748	0.202007	0.101003	+	0.994886i	$0.467795\pi$
$510$	0	0
$511$	6.05788	0.267985
$512$	0	0
$513$	1.45724	0.0643388
$514$	0	0
$515$	−6.96493	−0.306911
$516$	0	0
$517$	18.6901	0.821989
$518$	0	0
$519$	−1.81965	−0.0798738
$520$	0	0
$521$	7.72281	0.338342	0.169171	−	0.985587i	$-0.445891\pi$
$521$	7.72281	0.338342	0.169171	+	0.985587i	$0.445891\pi$
$522$	0	0
$523$	−25.1581	−1.10009	−0.550044	−	0.835136i	$-0.685389\pi$
$523$	−25.1581	−1.10009	−0.550044	+	0.835136i	$0.685389\pi$
$524$	0	0
$525$	3.24161	0.141476
$526$	0	0
$527$	7.88569	0.343506
$528$	0	0
$529$	11.7510	0.510912
$530$	0	0
$531$	8.01866	0.347980
$532$	0	0
$533$	−32.2576	−1.39723
$534$	0	0
$535$	8.15041	0.352373
$536$	0	0
$537$	−1.93062	−0.0833123
$538$	0	0
$539$	−18.9979	−0.818298
$540$	0	0
$541$	16.0052	0.688115	0.344058	−	0.938948i	$-0.388198\pi$
$541$	16.0052	0.688115	0.344058	+	0.938948i	$0.388198\pi$
$542$	0	0
$543$	0.201624	0.00865253
$544$	0	0
$545$	5.06213	0.216838
$546$	0	0
$547$	13.9550	0.596673	0.298336	−	0.954461i	$-0.403568\pi$
$547$	13.9550	0.596673	0.298336	+	0.954461i	$0.403568\pi$
$548$	0	0
$549$	−18.8735	−0.805499
$550$	0	0
$551$	−15.1445	−0.645176
$552$	0	0
$553$	52.4565	2.23068
$554$	0	0
$555$	0.0766131	0.00325205
$556$	0	0
$557$	18.7466	0.794320	0.397160	−	0.917749i	$-0.369996\pi$
$557$	18.7466	0.794320	0.397160	+	0.917749i	$0.369996\pi$
$558$	0	0
$559$	−40.7022	−1.72152
$560$	0	0
$561$	0.563792	0.0238033
$562$	0	0
$563$	7.91309	0.333497	0.166749	−	0.985999i	$-0.446673\pi$
$563$	7.91309	0.333497	0.166749	+	0.985999i	$0.446673\pi$
$564$	0	0
$565$	2.26434	0.0952617
$566$	0	0
$567$	34.3448	1.44235
$568$	0	0
$569$	5.69925	0.238925	0.119462	−	0.992839i	$-0.461883\pi$
$569$	5.69925	0.238925	0.119462	+	0.992839i	$0.461883\pi$
$570$	0	0
$571$	42.7467	1.78889	0.894447	−	0.447174i	$-0.147570\pi$
$571$	42.7467	1.78889	0.894447	+	0.447174i	$0.147570\pi$
$572$	0	0
$573$	−2.88153	−0.120377
$574$	0	0
$575$	28.4784	1.18763
$576$	0	0
$577$	30.3256	1.26247	0.631236	−	0.775591i	$-0.282548\pi$
$577$	30.3256	1.26247	0.631236	+	0.775591i	$0.282548\pi$
$578$	0	0
$579$	−0.567967	−0.0236039
$580$	0	0
$581$	54.8788	2.27676
$582$	0	0
$583$	−15.3090	−0.634036
$584$	0	0
$585$	−8.66662	−0.358321
$586$	0	0
$587$	8.92059	0.368192	0.184096	−	0.982908i	$-0.441064\pi$
$587$	8.92059	0.368192	0.184096	+	0.982908i	$0.441064\pi$
$588$	0	0
$589$	7.67738	0.316341
$590$	0	0
$591$	1.94963	0.0801972
$592$	0	0
$593$	−11.1031	−0.455951	−0.227975	−	0.973667i	$-0.573211\pi$
$593$	−11.1031	−0.455951	−0.227975	+	0.973667i	$0.573211\pi$
$594$	0	0
$595$	−2.37282	−0.0972762
$596$	0	0
$597$	−0.136623	−0.00559162
$598$	0	0
$599$	−41.8120	−1.70839	−0.854195	−	0.519953i	$-0.825950\pi$
$599$	−41.8120	−1.70839	−0.854195	+	0.519953i	$0.825950\pi$
$600$	0	0
$601$	3.19617	0.130374	0.0651872	−	0.997873i	$-0.479236\pi$
$601$	3.19617	0.130374	0.0651872	+	0.997873i	$0.479236\pi$
$602$	0	0
$603$	−30.2821	−1.23318
$604$	0	0
$605$	2.44312	0.0993270
$606$	0	0
$607$	−12.9602	−0.526038	−0.263019	−	0.964791i	$-0.584718\pi$
$607$	−12.9602	−0.526038	−0.263019	+	0.964791i	$0.584718\pi$
$608$	0	0
$609$	7.10876	0.288062
$610$	0	0
$611$	58.9603	2.38528
$612$	0	0
$613$	−16.9644	−0.685184	−0.342592	−	0.939484i	$-0.611305\pi$
$613$	−16.9644	−0.685184	−0.342592	+	0.939484i	$0.611305\pi$
$614$	0	0
$615$	0.319167	0.0128701
$616$	0	0
$617$	24.3974	0.982201	0.491101	−	0.871103i	$-0.336595\pi$
$617$	24.3974	0.982201	0.491101	+	0.871103i	$0.336595\pi$
$618$	0	0
$619$	36.7157	1.47573	0.737865	−	0.674948i	$-0.235834\pi$
$619$	36.7157	1.47573	0.737865	+	0.674948i	$0.235834\pi$
$620$	0	0
$621$	−6.00932	−0.241146
$622$	0	0
$623$	45.2160	1.81154
$624$	0	0
$625$	22.4926	0.899705
$626$	0	0
$627$	0.548898	0.0219209
$628$	0	0
$629$	1.60247	0.0638946
$630$	0	0
$631$	3.84265	0.152973	0.0764867	−	0.997071i	$-0.475630\pi$
$631$	3.84265	0.152973	0.0764867	+	0.997071i	$0.475630\pi$
$632$	0	0
$633$	−1.67645	−0.0666329
$634$	0	0
$635$	−1.72286	−0.0683698
$636$	0	0
$637$	−59.9313	−2.37457
$638$	0	0
$639$	−14.1454	−0.559583
$640$	0	0
$641$	18.6599	0.737023	0.368511	−	0.929623i	$-0.379868\pi$
$641$	18.6599	0.737023	0.368511	+	0.929623i	$0.379868\pi$
$642$	0	0
$643$	43.8543	1.72945	0.864723	−	0.502250i	$-0.167494\pi$
$643$	43.8543	1.72945	0.864723	+	0.502250i	$0.167494\pi$
$644$	0	0
$645$	0.402720	0.0158571
$646$	0	0
$647$	12.9599	0.509505	0.254753	−	0.967006i	$-0.418006\pi$
$647$	12.9599	0.509505	0.254753	+	0.967006i	$0.418006\pi$
$648$	0	0
$649$	6.07040	0.238284
$650$	0	0
$651$	−3.60374	−0.141242
$652$	0	0
$653$	−45.3451	−1.77449	−0.887245	−	0.461299i	$-0.847384\pi$
$653$	−45.3451	−1.77449	−0.887245	+	0.461299i	$0.847384\pi$
$654$	0	0
$655$	−1.92417	−0.0751836
$656$	0	0
$657$	−4.57907	−0.178647
$658$	0	0
$659$	−12.3560	−0.481320	−0.240660	−	0.970610i	$-0.577364\pi$
$659$	−12.3560	−0.481320	−0.240660	+	0.970610i	$0.577364\pi$
$660$	0	0
$661$	20.8801	0.812141	0.406070	−	0.913842i	$-0.366899\pi$
$661$	20.8801	0.812141	0.406070	+	0.913842i	$0.366899\pi$
$662$	0	0
$663$	1.77855	0.0690733
$664$	0	0
$665$	−2.31014	−0.0895833
$666$	0	0
$667$	62.4522	2.41816
$668$	0	0
$669$	−1.17825	−0.0455539
$670$	0	0
$671$	−14.2878	−0.551576
$672$	0	0
$673$	46.6034	1.79643	0.898215	−	0.439556i	$-0.144864\pi$
$673$	46.6034	1.79643	0.898215	+	0.439556i	$0.144864\pi$
$674$	0	0
$675$	−4.92463	−0.189549
$676$	0	0
$677$	−8.54908	−0.328568	−0.164284	−	0.986413i	$-0.552531\pi$
$677$	−8.54908	−0.328568	−0.164284	+	0.986413i	$0.552531\pi$
$678$	0	0
$679$	44.0595	1.69085
$680$	0	0
$681$	0.978225	0.0374857
$682$	0	0
$683$	−32.0271	−1.22548	−0.612741	−	0.790284i	$-0.709933\pi$
$683$	−32.0271	−1.22548	−0.612741	+	0.790284i	$0.709933\pi$
$684$	0	0
$685$	0.168099	0.00642272
$686$	0	0
$687$	3.95554	0.150913
$688$	0	0
$689$	−48.2943	−1.83987
$690$	0	0
$691$	42.9204	1.63277	0.816385	−	0.577508i	$-0.195975\pi$
$691$	42.9204	1.63277	0.816385	+	0.577508i	$0.195975\pi$
$692$	0	0
$693$	26.2604	0.997550
$694$	0	0
$695$	1.07856	0.0409123
$696$	0	0
$697$	6.67582	0.252865
$698$	0	0
$699$	−2.79752	−0.105812
$700$	0	0
$701$	16.2342	0.613157	0.306578	−	0.951845i	$-0.400816\pi$
$701$	16.2342	0.613157	0.306578	+	0.951845i	$0.400816\pi$
$702$	0	0
$703$	1.56014	0.0588417
$704$	0	0
$705$	−0.583371	−0.0219710
$706$	0	0
$707$	43.8357	1.64861
$708$	0	0
$709$	7.11609	0.267250	0.133625	−	0.991032i	$-0.457338\pi$
$709$	7.11609	0.267250	0.133625	+	0.991032i	$0.457338\pi$
$710$	0	0
$711$	−39.6512	−1.48704
$712$	0	0
$713$	−31.6597	−1.18566
$714$	0	0
$715$	−6.56093	−0.245365
$716$	0	0
$717$	2.43787	0.0910439
$718$	0	0
$719$	26.8896	1.00281	0.501406	−	0.865212i	$-0.332816\pi$
$719$	26.8896	1.00281	0.501406	+	0.865212i	$0.332816\pi$
$720$	0	0
$721$	66.5758	2.47941
$722$	0	0
$723$	0.498215	0.0185288
$724$	0	0
$725$	51.1795	1.90076
$726$	0	0
$727$	−10.5295	−0.390519	−0.195259	−	0.980752i	$-0.562555\pi$
$727$	−10.5295	−0.390519	−0.195259	+	0.980752i	$0.562555\pi$
$728$	0	0
$729$	−25.4387	−0.942175
$730$	0	0
$731$	8.42345	0.311553
$732$	0	0
$733$	8.80160	0.325094	0.162547	−	0.986701i	$-0.448029\pi$
$733$	8.80160	0.325094	0.162547	+	0.986701i	$0.448029\pi$
$734$	0	0
$735$	0.592979	0.0218724
$736$	0	0
$737$	−22.9246	−0.844437
$738$	0	0
$739$	53.9591	1.98492	0.992459	−	0.122579i	$-0.0391164\pi$
$739$	53.9591	1.98492	0.992459	+	0.122579i	$0.0391164\pi$
$740$	0	0
$741$	1.73157	0.0636108
$742$	0	0
$743$	−28.1867	−1.03407	−0.517035	−	0.855964i	$-0.672964\pi$
$743$	−28.1867	−1.03407	−0.517035	+	0.855964i	$0.672964\pi$
$744$	0	0
$745$	9.19914	0.337031
$746$	0	0
$747$	−41.4822	−1.51775
$748$	0	0
$749$	−77.9074	−2.84668
$750$	0	0
$751$	−4.05056	−0.147807	−0.0739034	−	0.997265i	$-0.523546\pi$
$751$	−4.05056	−0.147807	−0.0739034	+	0.997265i	$0.523546\pi$
$752$	0	0
$753$	−2.71196	−0.0988293
$754$	0	0
$755$	−2.90375	−0.105678
$756$	0	0
$757$	20.0644	0.729255	0.364627	−	0.931154i	$-0.381196\pi$
$757$	20.0644	0.729255	0.364627	+	0.931154i	$0.381196\pi$
$758$	0	0
$759$	−2.26353	−0.0821608
$760$	0	0
$761$	40.9388	1.48403	0.742015	−	0.670383i	$-0.233870\pi$
$761$	40.9388	1.48403	0.742015	+	0.670383i	$0.233870\pi$
$762$	0	0
$763$	−48.3875	−1.75174
$764$	0	0
$765$	1.79359	0.0648472
$766$	0	0
$767$	19.1499	0.691461
$768$	0	0
$769$	7.58748	0.273611	0.136806	−	0.990598i	$-0.456316\pi$
$769$	7.58748	0.273611	0.136806	+	0.990598i	$0.456316\pi$
$770$	0	0
$771$	−4.88135	−0.175797
$772$	0	0
$773$	31.2763	1.12493	0.562466	−	0.826820i	$-0.309853\pi$
$773$	31.2763	1.12493	0.562466	+	0.826820i	$0.309853\pi$
$774$	0	0
$775$	−25.9451	−0.931974
$776$	0	0
$777$	−0.732323	−0.0262719
$778$	0	0
$779$	6.49947	0.232868
$780$	0	0
$781$	−10.7085	−0.383182
$782$	0	0
$783$	−10.7996	−0.385945
$784$	0	0
$785$	−8.23065	−0.293764
$786$	0	0
$787$	−28.2527	−1.00710	−0.503551	−	0.863966i	$-0.667973\pi$
$787$	−28.2527	−1.00710	−0.503551	+	0.863966i	$0.667973\pi$
$788$	0	0
$789$	−1.09182	−0.0388699
$790$	0	0
$791$	−21.6442	−0.769580
$792$	0	0
$793$	−45.0729	−1.60058
$794$	0	0
$795$	0.477839	0.0169472
$796$	0	0
$797$	54.1437	1.91787	0.958934	−	0.283631i	$-0.0915390\pi$
$797$	54.1437	1.91787	0.958934	+	0.283631i	$0.0915390\pi$
$798$	0	0
$799$	−12.2020	−0.431676
$800$	0	0
$801$	−34.1782	−1.20763
$802$	0	0
$803$	−3.46651	−0.122331
$804$	0	0
$805$	9.52647	0.335764
$806$	0	0
$807$	−0.628526	−0.0221252
$808$	0	0
$809$	−45.1417	−1.58710	−0.793548	−	0.608507i	$-0.791769\pi$
$809$	−45.1417	−1.58710	−0.793548	+	0.608507i	$0.791769\pi$
$810$	0	0
$811$	27.5987	0.969121	0.484561	−	0.874758i	$-0.338979\pi$
$811$	27.5987	0.969121	0.484561	+	0.874758i	$0.338979\pi$
$812$	0	0
$813$	1.01373	0.0355532
$814$	0	0
$815$	5.12115	0.179386
$816$	0	0
$817$	8.20093	0.286914
$818$	0	0
$819$	82.8418	2.89473
$820$	0	0
$821$	−30.3794	−1.06025	−0.530125	−	0.847920i	$-0.677855\pi$
$821$	−30.3794	−1.06025	−0.530125	+	0.847920i	$0.677855\pi$
$822$	0	0
$823$	22.3809	0.780149	0.390075	−	0.920783i	$-0.372449\pi$
$823$	22.3809	0.780149	0.390075	+	0.920783i	$0.372449\pi$
$824$	0	0
$825$	−1.85496	−0.0645813
$826$	0	0
$827$	−5.04169	−0.175317	−0.0876584	−	0.996151i	$-0.527938\pi$
$827$	−5.04169	−0.175317	−0.0876584	+	0.996151i	$0.527938\pi$
$828$	0	0
$829$	8.45313	0.293589	0.146795	−	0.989167i	$-0.453104\pi$
$829$	8.45313	0.293589	0.146795	+	0.989167i	$0.453104\pi$
$830$	0	0
$831$	1.47656	0.0512212
$832$	0	0
$833$	12.4030	0.429738
$834$	0	0
$835$	0.108979	0.00377137
$836$	0	0
$837$	5.47476	0.189235
$838$	0	0
$839$	19.8482	0.685237	0.342618	−	0.939475i	$-0.388686\pi$
$839$	19.8482	0.685237	0.342618	+	0.939475i	$0.388686\pi$
$840$	0	0
$841$	83.2351	2.87018
$842$	0	0
$843$	4.84089	0.166729
$844$	0	0
$845$	−15.3520	−0.528127
$846$	0	0
$847$	−23.3531	−0.802422
$848$	0	0
$849$	3.83253	0.131532
$850$	0	0
$851$	−6.43364	−0.220542
$852$	0	0
$853$	17.0929	0.585249	0.292625	−	0.956227i	$-0.405471\pi$
$853$	17.0929	0.585249	0.292625	+	0.956227i	$0.405471\pi$
$854$	0	0
$855$	1.74620	0.0597189
$856$	0	0
$857$	29.7049	1.01470	0.507350	−	0.861740i	$-0.330625\pi$
$857$	29.7049	1.01470	0.507350	+	0.861740i	$0.330625\pi$
$858$	0	0
$859$	46.2790	1.57902	0.789509	−	0.613739i	$-0.210335\pi$
$859$	46.2790	1.57902	0.789509	+	0.613739i	$0.210335\pi$
$860$	0	0
$861$	−3.05083	−0.103972
$862$	0	0
$863$	−20.0702	−0.683196	−0.341598	−	0.939846i	$-0.610968\pi$
$863$	−20.0702	−0.683196	−0.341598	+	0.939846i	$0.610968\pi$
$864$	0	0
$865$	−4.38235	−0.149004
$866$	0	0
$867$	2.53431	0.0860695
$868$	0	0
$869$	−30.0173	−1.01827
$870$	0	0
$871$	−72.3184	−2.45042
$872$	0	0
$873$	−33.3040	−1.12717
$874$	0	0
$875$	15.8870	0.537080
$876$	0	0
$877$	−20.1494	−0.680397	−0.340199	−	0.940354i	$-0.610494\pi$
$877$	−20.1494	−0.680397	−0.340199	+	0.940354i	$0.610494\pi$
$878$	0	0
$879$	2.85189	0.0961918
$880$	0	0
$881$	44.9490	1.51437	0.757184	−	0.653201i	$-0.226574\pi$
$881$	44.9490	1.51437	0.757184	+	0.653201i	$0.226574\pi$
$882$	0	0
$883$	−52.4437	−1.76487	−0.882435	−	0.470435i	$-0.844097\pi$
$883$	−52.4437	−1.76487	−0.882435	+	0.470435i	$0.844097\pi$
$884$	0	0
$885$	−0.189475	−0.00636912
$886$	0	0
$887$	−40.6273	−1.36413	−0.682066	−	0.731291i	$-0.738918\pi$
$887$	−40.6273	−1.36413	−0.682066	+	0.731291i	$0.738918\pi$
$888$	0	0
$889$	16.4684	0.552331
$890$	0	0
$891$	−19.6532	−0.658408
$892$	0	0
$893$	−11.8797	−0.397538
$894$	0	0
$895$	−4.64959	−0.155419
$896$	0	0
$897$	−7.14059	−0.238417
$898$	0	0
$899$	−56.8968	−1.89761
$900$	0	0
$901$	9.99466	0.332971
$902$	0	0
$903$	−3.84949	−0.128103
$904$	0	0
$905$	0.485581	0.0161413
$906$	0	0
$907$	−30.4460	−1.01094	−0.505471	−	0.862843i	$-0.668681\pi$
$907$	−30.4460	−1.01094	−0.505471	+	0.862843i	$0.668681\pi$
$908$	0	0
$909$	−33.1348	−1.09901
$910$	0	0
$911$	−11.9611	−0.396289	−0.198144	−	0.980173i	$-0.563492\pi$
$911$	−11.9611	−0.396289	−0.198144	+	0.980173i	$0.563492\pi$
$912$	0	0
$913$	−31.4034	−1.03930
$914$	0	0
$915$	0.445965	0.0147431
$916$	0	0
$917$	18.3926	0.607378
$918$	0	0
$919$	−21.0296	−0.693702	−0.346851	−	0.937920i	$-0.612749\pi$
$919$	−21.0296	−0.693702	−0.346851	+	0.937920i	$0.612749\pi$
$920$	0	0
$921$	−1.98284	−0.0653367
$922$	0	0
$923$	−33.7815	−1.11193
$924$	0	0
$925$	−5.27235	−0.173354
$926$	0	0
$927$	−50.3238	−1.65285
$928$	0	0
$929$	−32.2070	−1.05668	−0.528338	−	0.849034i	$-0.677185\pi$
$929$	−32.2070	−1.05668	−0.528338	+	0.849034i	$0.677185\pi$
$930$	0	0
$931$	12.0753	0.395753
$932$	0	0
$933$	−1.38316	−0.0452825
$934$	0	0
$935$	1.35780	0.0444050
$936$	0	0
$937$	−22.5855	−0.737835	−0.368917	−	0.929462i	$-0.620271\pi$
$937$	−22.5855	−0.737835	−0.368917	+	0.929462i	$0.620271\pi$
$938$	0	0
$939$	−0.157822	−0.00515034
$940$	0	0
$941$	−45.2634	−1.47554	−0.737772	−	0.675050i	$-0.764122\pi$
$941$	−45.2634	−1.47554	−0.737772	+	0.675050i	$0.764122\pi$
$942$	0	0
$943$	−26.8023	−0.872802
$944$	0	0
$945$	−1.64737	−0.0535888
$946$	0	0
$947$	16.8133	0.546359	0.273179	−	0.961963i	$-0.411925\pi$
$947$	16.8133	0.546359	0.273179	+	0.961963i	$0.411925\pi$
$948$	0	0
$949$	−10.9356	−0.354983
$950$	0	0
$951$	−2.19434	−0.0711565
$952$	0	0
$953$	−27.3758	−0.886790	−0.443395	−	0.896326i	$-0.646226\pi$
$953$	−27.3758	−0.886790	−0.443395	+	0.896326i	$0.646226\pi$
$954$	0	0
$955$	−6.93971	−0.224564
$956$	0	0
$957$	−4.06786	−0.131495
$958$	0	0
$959$	−1.60681	−0.0518865
$960$	0	0
$961$	−2.15660	−0.0695678
$962$	0	0
$963$	58.8892	1.89768
$964$	0	0
$965$	−1.36786	−0.0440329
$966$	0	0
$967$	−12.4210	−0.399433	−0.199717	−	0.979854i	$-0.564002\pi$
$967$	−12.4210	−0.399433	−0.199717	+	0.979854i	$0.564002\pi$
$968$	0	0
$969$	−0.358354	−0.0115120
$970$	0	0
$971$	−1.89323	−0.0607568	−0.0303784	−	0.999538i	$-0.509671\pi$
$971$	−1.89323	−0.0607568	−0.0303784	+	0.999538i	$0.509671\pi$
$972$	0	0
$973$	−10.3097	−0.330513
$974$	0	0
$975$	−5.85170	−0.187404
$976$	0	0
$977$	−3.07793	−0.0984717	−0.0492359	−	0.998787i	$-0.515679\pi$
$977$	−3.07793	−0.0984717	−0.0492359	+	0.998787i	$0.515679\pi$
$978$	0	0
$979$	−25.8741	−0.826939
$980$	0	0
$981$	36.5755	1.16777
$982$	0	0
$983$	−28.4420	−0.907160	−0.453580	−	0.891216i	$-0.649853\pi$
$983$	−28.4420	−0.907160	−0.453580	+	0.891216i	$0.649853\pi$
$984$	0	0
$985$	4.69539	0.149608
$986$	0	0
$987$	5.57628	0.177495
$988$	0	0
$989$	−33.8187	−1.07537
$990$	0	0
$991$	−43.9358	−1.39567	−0.697834	−	0.716260i	$-0.745853\pi$
$991$	−43.9358	−1.39567	−0.697834	+	0.716260i	$0.745853\pi$
$992$	0	0
$993$	5.95600	0.189008
$994$	0	0
$995$	−0.329036	−0.0104311
$996$	0	0
$997$	52.8092	1.67248	0.836241	−	0.548361i	$-0.184748\pi$
$997$	52.8092	1.67248	0.836241	+	0.548361i	$0.184748\pi$
$998$	0	0
$999$	1.11254	0.0351992

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.o.1.2		5
4.3	odd	2		1006.2.a.h.1.4	✓	5
12.11	even	2		9054.2.a.ba.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.h.1.4	✓	5	4.3	odd	2
8048.2.a.o.1.2		5	1.1	even	1	trivial
9054.2.a.ba.1.2		5	12.11	even	2

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)

\(f(q)\)	\(=\)	\(q-0.170728 q^{3} -0.411173 q^{5} +3.93028 q^{7} -2.97085 q^{9} +O(q^{10})\)	\(q-0.170728 q^{3} -0.411173 q^{5} +3.93028 q^{7} -2.97085 q^{9} -2.24904 q^{11} -7.09487 q^{13} +0.0701989 q^{15} +1.46831 q^{17} +1.42952 q^{19} -0.671011 q^{21} -5.89500 q^{23} -4.83094 q^{25} +1.01939 q^{27} -10.5941 q^{29} +5.37060 q^{31} +0.383974 q^{33} -1.61603 q^{35} +1.09137 q^{37} +1.21130 q^{39} +4.54661 q^{41} +5.73685 q^{43} +1.22153 q^{45} -8.31027 q^{47} +8.44713 q^{49} -0.250682 q^{51} +6.80693 q^{53} +0.924742 q^{55} -0.244059 q^{57} -2.69911 q^{59} +6.35288 q^{61} -11.6763 q^{63} +2.91722 q^{65} +10.1931 q^{67} +1.00644 q^{69} +4.76139 q^{71} +1.54133 q^{73} +0.824778 q^{75} -8.83935 q^{77} +13.3467 q^{79} +8.73852 q^{81} +13.9631 q^{83} -0.603728 q^{85} +1.80871 q^{87} +11.5045 q^{89} -27.8849 q^{91} -0.916915 q^{93} -0.587779 q^{95} +11.2102 q^{97} +6.68155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{3} - 3 q^{5} + 9 q^{7} + q^{9}+O(q^{10}) \) 5 * q + 4 * q^3 - 3 * q^5 + 9 * q^7 + q^9	\( 5 q + 4 q^{3} - 3 q^{5} + 9 q^{7} + q^{9} + 15 q^{11} - 5 q^{13} + 4 q^{15} - 6 q^{17} - 10 q^{19} - 12 q^{21} + 12 q^{23} + 6 q^{25} + 7 q^{27} - 16 q^{29} + 33 q^{31} + 21 q^{33} + 11 q^{35} + 2 q^{37} - 4 q^{39} - 12 q^{41} + 17 q^{43} + 3 q^{45} + 7 q^{47} + 36 q^{49} - 14 q^{51} - 2 q^{55} - 6 q^{57} - 18 q^{59} + q^{61} - 31 q^{63} - 14 q^{65} - 6 q^{67} + 15 q^{69} + 26 q^{71} + 9 q^{73} + 33 q^{75} + 16 q^{77} + 21 q^{79} - 3 q^{81} - 28 q^{85} - 34 q^{87} + 15 q^{89} - 26 q^{91} + 19 q^{93} + 18 q^{95} + 26 q^{97} + 44 q^{99}+O(q^{100}) \) 5 * q + 4 * q^3 - 3 * q^5 + 9 * q^7 + q^9 + 15 * q^11 - 5 * q^13 + 4 * q^15 - 6 * q^17 - 10 * q^19 - 12 * q^21 + 12 * q^23 + 6 * q^25 + 7 * q^27 - 16 * q^29 + 33 * q^31 + 21 * q^33 + 11 * q^35 + 2 * q^37 - 4 * q^39 - 12 * q^41 + 17 * q^43 + 3 * q^45 + 7 * q^47 + 36 * q^49 - 14 * q^51 - 2 * q^55 - 6 * q^57 - 18 * q^59 + q^61 - 31 * q^63 - 14 * q^65 - 6 * q^67 + 15 * q^69 + 26 * q^71 + 9 * q^73 + 33 * q^75 + 16 * q^77 + 21 * q^79 - 3 * q^81 - 28 * q^85 - 34 * q^87 + 15 * q^89 - 26 * q^91 + 19 * q^93 + 18 * q^95 + 26 * q^97 + 44 * q^99

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