LMFDB - Embedded newform 8048.2.a.s.1.18

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:	$1$
Dimension:	$21$
Twist minimal:	no (minimal twist has level 2012)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
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)

$f(q)$	$=$	$q+1.91007 q^{3} +1.16052 q^{5} -1.88100 q^{7} +0.648366 q^{9} +O(q^{10})$	$q+1.91007 q^{3} +1.16052 q^{5} -1.88100 q^{7} +0.648366 q^{9} +4.20980 q^{11} -6.01487 q^{13} +2.21667 q^{15} +2.39994 q^{17} -7.69389 q^{19} -3.59285 q^{21} -2.06742 q^{23} -3.65319 q^{25} -4.49178 q^{27} +4.33712 q^{29} +2.53030 q^{31} +8.04102 q^{33} -2.18294 q^{35} +8.55077 q^{37} -11.4888 q^{39} -1.28933 q^{41} +5.01401 q^{43} +0.752441 q^{45} +0.868781 q^{47} -3.46182 q^{49} +4.58405 q^{51} -3.75499 q^{53} +4.88556 q^{55} -14.6959 q^{57} -5.59396 q^{59} +7.36118 q^{61} -1.21958 q^{63} -6.98038 q^{65} -8.24073 q^{67} -3.94892 q^{69} +5.04019 q^{71} +6.47359 q^{73} -6.97785 q^{75} -7.91866 q^{77} -11.4394 q^{79} -10.5247 q^{81} -15.5955 q^{83} +2.78518 q^{85} +8.28420 q^{87} +2.71606 q^{89} +11.3140 q^{91} +4.83305 q^{93} -8.92891 q^{95} -11.1814 q^{97} +2.72949 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9}+O(q^{10}) $ 21 * q - 10 * q^3 + 3 * q^5 - 13 * q^7 + 21 * q^9	$ 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9} - 7 q^{11} + 12 q^{13} - 14 q^{15} + q^{17} - 14 q^{19} + 14 q^{21} - 26 q^{23} + 18 q^{25} - 37 q^{27} + 9 q^{29} - 28 q^{31} + 3 q^{33} - 20 q^{35} + 31 q^{37} - 29 q^{39} + 4 q^{41} - 38 q^{43} + 24 q^{45} - 9 q^{47} + 16 q^{49} - 15 q^{51} + 22 q^{53} - 35 q^{55} - q^{57} - 10 q^{59} + 22 q^{61} - 35 q^{63} - 14 q^{65} - 58 q^{67} + 15 q^{69} - 27 q^{71} - 6 q^{73} - 48 q^{75} + 16 q^{77} - 47 q^{79} + 29 q^{81} - 22 q^{83} + 14 q^{85} - 29 q^{87} + q^{89} - 51 q^{91} + 34 q^{93} - 23 q^{95} - 2 q^{97} - 22 q^{99}+O(q^{100}) $ 21 * q - 10 * q^3 + 3 * q^5 - 13 * q^7 + 21 * q^9 - 7 * q^11 + 12 * q^13 - 14 * q^15 + q^17 - 14 * q^19 + 14 * q^21 - 26 * q^23 + 18 * q^25 - 37 * q^27 + 9 * q^29 - 28 * q^31 + 3 * q^33 - 20 * q^35 + 31 * q^37 - 29 * q^39 + 4 * q^41 - 38 * q^43 + 24 * q^45 - 9 * q^47 + 16 * q^49 - 15 * q^51 + 22 * q^53 - 35 * q^55 - q^57 - 10 * q^59 + 22 * q^61 - 35 * q^63 - 14 * q^65 - 58 * q^67 + 15 * q^69 - 27 * q^71 - 6 * q^73 - 48 * q^75 + 16 * q^77 - 47 * q^79 + 29 * q^81 - 22 * q^83 + 14 * q^85 - 29 * q^87 + q^89 - 51 * q^91 + 34 * q^93 - 23 * q^95 - 2 * q^97 - 22 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.91007	1.10278	0.551390	−	0.834248i	$-0.314098\pi$
$3$	1.91007	1.10278	0.551390	+	0.834248i	$0.314098\pi$
$4$	0	0
$5$	1.16052	0.519000	0.259500	−	0.965743i	$-0.416442\pi$
$5$	1.16052	0.519000	0.259500	+	0.965743i	$0.416442\pi$
$6$	0	0
$7$	−1.88100	−0.710953	−0.355476	−	0.934685i	$-0.615681\pi$
$7$	−1.88100	−0.710953	−0.355476	+	0.934685i	$0.615681\pi$
$8$	0	0
$9$	0.648366	0.216122
$10$	0	0
$11$	4.20980	1.26930	0.634652	−	0.772798i	$-0.281143\pi$
$11$	4.20980	1.26930	0.634652	+	0.772798i	$0.281143\pi$
$12$	0	0
$13$	−6.01487	−1.66822	−0.834112	−	0.551594i	$-0.814020\pi$
$13$	−6.01487	−1.66822	−0.834112	+	0.551594i	$0.814020\pi$
$14$	0	0
$15$	2.21667	0.572343
$16$	0	0
$17$	2.39994	0.582071	0.291035	−	0.956712i	$-0.406000\pi$
$17$	2.39994	0.582071	0.291035	+	0.956712i	$0.406000\pi$
$18$	0	0
$19$	−7.69389	−1.76510	−0.882550	−	0.470219i	$-0.844175\pi$
$19$	−7.69389	−1.76510	−0.882550	+	0.470219i	$0.844175\pi$
$20$	0	0
$21$	−3.59285	−0.784024
$22$	0	0
$23$	−2.06742	−0.431087	−0.215543	−	0.976494i	$-0.569152\pi$
$23$	−2.06742	−0.431087	−0.215543	+	0.976494i	$0.569152\pi$
$24$	0	0
$25$	−3.65319	−0.730639
$26$	0	0
$27$	−4.49178	−0.864444
$28$	0	0
$29$	4.33712	0.805383	0.402691	−	0.915336i	$-0.368075\pi$
$29$	4.33712	0.805383	0.402691	+	0.915336i	$0.368075\pi$
$30$	0	0
$31$	2.53030	0.454456	0.227228	−	0.973842i	$-0.427034\pi$
$31$	2.53030	0.454456	0.227228	+	0.973842i	$0.427034\pi$
$32$	0	0
$33$	8.04102	1.39976
$34$	0	0
$35$	−2.18294	−0.368985
$36$	0	0
$37$	8.55077	1.40574	0.702869	−	0.711319i	$-0.251902\pi$
$37$	8.55077	1.40574	0.702869	+	0.711319i	$0.251902\pi$
$38$	0	0
$39$	−11.4888	−1.83968
$40$	0	0
$41$	−1.28933	−0.201360	−0.100680	−	0.994919i	$-0.532102\pi$
$41$	−1.28933	−0.201360	−0.100680	+	0.994919i	$0.532102\pi$
$42$	0	0
$43$	5.01401	0.764630	0.382315	−	0.924032i	$-0.375127\pi$
$43$	5.01401	0.764630	0.382315	+	0.924032i	$0.375127\pi$
$44$	0	0
$45$	0.752441	0.112167
$46$	0	0
$47$	0.868781	0.126725	0.0633623	−	0.997991i	$-0.479818\pi$
$47$	0.868781	0.126725	0.0633623	+	0.997991i	$0.479818\pi$
$48$	0	0
$49$	−3.46182	−0.494546
$50$	0	0
$51$	4.58405	0.641895
$52$	0	0
$53$	−3.75499	−0.515787	−0.257893	−	0.966173i	$-0.583028\pi$
$53$	−3.75499	−0.515787	−0.257893	+	0.966173i	$0.583028\pi$
$54$	0	0
$55$	4.88556	0.658769
$56$	0	0
$57$	−14.6959	−1.94652
$58$	0	0
$59$	−5.59396	−0.728272	−0.364136	−	0.931346i	$-0.618636\pi$
$59$	−5.59396	−0.728272	−0.364136	+	0.931346i	$0.618636\pi$
$60$	0	0
$61$	7.36118	0.942502	0.471251	−	0.881999i	$-0.343803\pi$
$61$	7.36118	0.942502	0.471251	+	0.881999i	$0.343803\pi$
$62$	0	0
$63$	−1.21958	−0.153653
$64$	0	0
$65$	−6.98038	−0.865809
$66$	0	0
$67$	−8.24073	−1.00677	−0.503383	−	0.864064i	$-0.667911\pi$
$67$	−8.24073	−1.00677	−0.503383	+	0.864064i	$0.667911\pi$
$68$	0	0
$69$	−3.94892	−0.475394
$70$	0	0
$71$	5.04019	0.598160	0.299080	−	0.954228i	$-0.403320\pi$
$71$	5.04019	0.598160	0.299080	+	0.954228i	$0.403320\pi$
$72$	0	0
$73$	6.47359	0.757677	0.378838	−	0.925463i	$-0.376324\pi$
$73$	6.47359	0.757677	0.378838	+	0.925463i	$0.376324\pi$
$74$	0	0
$75$	−6.97785	−0.805733
$76$	0	0
$77$	−7.91866	−0.902415
$78$	0	0
$79$	−11.4394	−1.28703	−0.643515	−	0.765433i	$-0.722525\pi$
$79$	−11.4394	−1.28703	−0.643515	+	0.765433i	$0.722525\pi$
$80$	0	0
$81$	−10.5247	−1.16941
$82$	0	0
$83$	−15.5955	−1.71183	−0.855915	−	0.517116i	$-0.827006\pi$
$83$	−15.5955	−1.71183	−0.855915	+	0.517116i	$0.827006\pi$
$84$	0	0
$85$	2.78518	0.302095
$86$	0	0
$87$	8.28420	0.888160
$88$	0	0
$89$	2.71606	0.287901	0.143951	−	0.989585i	$-0.454019\pi$
$89$	2.71606	0.287901	0.143951	+	0.989585i	$0.454019\pi$
$90$	0	0
$91$	11.3140	1.18603
$92$	0	0
$93$	4.83305	0.501164
$94$	0	0
$95$	−8.92891	−0.916087
$96$	0	0
$97$	−11.1814	−1.13530	−0.567652	−	0.823269i	$-0.692148\pi$
$97$	−11.1814	−1.13530	−0.567652	+	0.823269i	$0.692148\pi$
$98$	0	0
$99$	2.72949	0.274324
$100$	0	0
$101$	8.96289	0.891841	0.445920	−	0.895073i	$-0.352876\pi$
$101$	8.96289	0.891841	0.445920	+	0.895073i	$0.352876\pi$
$102$	0	0
$103$	−12.9005	−1.27112	−0.635560	−	0.772051i	$-0.719231\pi$
$103$	−12.9005	−1.27112	−0.635560	+	0.772051i	$0.719231\pi$
$104$	0	0
$105$	−4.16957	−0.406909
$106$	0	0
$107$	−19.8633	−1.92026	−0.960130	−	0.279553i	$-0.909814\pi$
$107$	−19.8633	−1.92026	−0.960130	+	0.279553i	$0.909814\pi$
$108$	0	0
$109$	−5.71694	−0.547583	−0.273792	−	0.961789i	$-0.588278\pi$
$109$	−5.71694	−0.547583	−0.273792	+	0.961789i	$0.588278\pi$
$110$	0	0
$111$	16.3326	1.55022
$112$	0	0
$113$	−16.5754	−1.55928	−0.779642	−	0.626225i	$-0.784599\pi$
$113$	−16.5754	−1.55928	−0.779642	+	0.626225i	$0.784599\pi$
$114$	0	0
$115$	−2.39928	−0.223734
$116$	0	0
$117$	−3.89984	−0.360540
$118$	0	0
$119$	−4.51430	−0.413825
$120$	0	0
$121$	6.72245	0.611132
$122$	0	0
$123$	−2.46272	−0.222056
$124$	0	0
$125$	−10.0422	−0.898202
$126$	0	0
$127$	−9.60710	−0.852492	−0.426246	−	0.904607i	$-0.640164\pi$
$127$	−9.60710	−0.852492	−0.426246	+	0.904607i	$0.640164\pi$
$128$	0	0
$129$	9.57712	0.843218
$130$	0	0
$131$	−0.817283	−0.0714064	−0.0357032	−	0.999362i	$-0.511367\pi$
$131$	−0.817283	−0.0714064	−0.0357032	+	0.999362i	$0.511367\pi$
$132$	0	0
$133$	14.4722	1.25490
$134$	0	0
$135$	−5.21281	−0.448647
$136$	0	0
$137$	−14.9738	−1.27930	−0.639650	−	0.768666i	$-0.720921\pi$
$137$	−14.9738	−1.27930	−0.639650	+	0.768666i	$0.720921\pi$
$138$	0	0
$139$	−22.9772	−1.94890	−0.974452	−	0.224595i	$-0.927894\pi$
$139$	−22.9772	−1.94890	−0.974452	+	0.224595i	$0.927894\pi$
$140$	0	0
$141$	1.65943	0.139749
$142$	0	0
$143$	−25.3214	−2.11748
$144$	0	0
$145$	5.03331	0.417994
$146$	0	0
$147$	−6.61232	−0.545375
$148$	0	0
$149$	17.0560	1.39728	0.698640	−	0.715473i	$-0.253789\pi$
$149$	17.0560	1.39728	0.698640	+	0.715473i	$0.253789\pi$
$150$	0	0
$151$	3.08245	0.250846	0.125423	−	0.992103i	$-0.459971\pi$
$151$	3.08245	0.250846	0.125423	+	0.992103i	$0.459971\pi$
$152$	0	0
$153$	1.55604	0.125798
$154$	0	0
$155$	2.93647	0.235863
$156$	0	0
$157$	21.1467	1.68769	0.843845	−	0.536588i	$-0.180287\pi$
$157$	21.1467	1.68769	0.843845	+	0.536588i	$0.180287\pi$
$158$	0	0
$159$	−7.17228	−0.568799
$160$	0	0
$161$	3.88883	0.306482
$162$	0	0
$163$	−17.4723	−1.36854	−0.684270	−	0.729229i	$-0.739879\pi$
$163$	−17.4723	−1.36854	−0.684270	+	0.729229i	$0.739879\pi$
$164$	0	0
$165$	9.33176	0.726477
$166$	0	0
$167$	−0.820654	−0.0635041	−0.0317520	−	0.999496i	$-0.510109\pi$
$167$	−0.820654	−0.0635041	−0.0317520	+	0.999496i	$0.510109\pi$
$168$	0	0
$169$	23.1787	1.78297
$170$	0	0
$171$	−4.98846	−0.381477
$172$	0	0
$173$	−17.4855	−1.32940	−0.664698	−	0.747112i	$-0.731440\pi$
$173$	−17.4855	−1.32940	−0.664698	+	0.747112i	$0.731440\pi$
$174$	0	0
$175$	6.87167	0.519450
$176$	0	0
$177$	−10.6849	−0.803123
$178$	0	0
$179$	−5.24836	−0.392281	−0.196140	−	0.980576i	$-0.562841\pi$
$179$	−5.24836	−0.392281	−0.196140	+	0.980576i	$0.562841\pi$
$180$	0	0
$181$	7.31807	0.543948	0.271974	−	0.962305i	$-0.412323\pi$
$181$	7.31807	0.543948	0.271974	+	0.962305i	$0.412323\pi$
$182$	0	0
$183$	14.0604	1.03937
$184$	0	0
$185$	9.92334	0.729579
$186$	0	0
$187$	10.1033	0.738824
$188$	0	0
$189$	8.44907	0.614579
$190$	0	0
$191$	11.7620	0.851067	0.425534	−	0.904943i	$-0.360086\pi$
$191$	11.7620	0.851067	0.425534	+	0.904943i	$0.360086\pi$
$192$	0	0
$193$	19.2131	1.38299	0.691493	−	0.722383i	$-0.256953\pi$
$193$	19.2131	1.38299	0.691493	+	0.722383i	$0.256953\pi$
$194$	0	0
$195$	−13.3330	−0.954796
$196$	0	0
$197$	7.76453	0.553200	0.276600	−	0.960985i	$-0.410792\pi$
$197$	7.76453	0.553200	0.276600	+	0.960985i	$0.410792\pi$
$198$	0	0
$199$	7.84200	0.555905	0.277952	−	0.960595i	$-0.410344\pi$
$199$	7.84200	0.555905	0.277952	+	0.960595i	$0.410344\pi$
$200$	0	0
$201$	−15.7404	−1.11024
$202$	0	0
$203$	−8.15814	−0.572589
$204$	0	0
$205$	−1.49630	−0.104506
$206$	0	0
$207$	−1.34044	−0.0931673
$208$	0	0
$209$	−32.3898	−2.24045
$210$	0	0
$211$	14.1719	0.975632	0.487816	−	0.872946i	$-0.337794\pi$
$211$	14.1719	0.975632	0.487816	+	0.872946i	$0.337794\pi$
$212$	0	0
$213$	9.62711	0.659638
$214$	0	0
$215$	5.81886	0.396843
$216$	0	0
$217$	−4.75951	−0.323097
$218$	0	0
$219$	12.3650	0.835550
$220$	0	0
$221$	−14.4353	−0.971025
$222$	0	0
$223$	5.23022	0.350242	0.175121	−	0.984547i	$-0.443968\pi$
$223$	5.23022	0.350242	0.175121	+	0.984547i	$0.443968\pi$
$224$	0	0
$225$	−2.36861	−0.157907
$226$	0	0
$227$	1.99379	0.132332	0.0661662	−	0.997809i	$-0.478923\pi$
$227$	1.99379	0.132332	0.0661662	+	0.997809i	$0.478923\pi$
$228$	0	0
$229$	−10.5816	−0.699252	−0.349626	−	0.936889i	$-0.613691\pi$
$229$	−10.5816	−0.699252	−0.349626	+	0.936889i	$0.613691\pi$
$230$	0	0
$231$	−15.1252	−0.995165
$232$	0	0
$233$	6.43327	0.421458	0.210729	−	0.977545i	$-0.432416\pi$
$233$	6.43327	0.421458	0.210729	+	0.977545i	$0.432416\pi$
$234$	0	0
$235$	1.00824	0.0657701
$236$	0	0
$237$	−21.8500	−1.41931
$238$	0	0
$239$	−28.1942	−1.82373	−0.911867	−	0.410486i	$-0.865359\pi$
$239$	−28.1942	−1.82373	−0.911867	+	0.410486i	$0.865359\pi$
$240$	0	0
$241$	−6.57882	−0.423779	−0.211890	−	0.977294i	$-0.567962\pi$
$241$	−6.57882	−0.423779	−0.211890	+	0.977294i	$0.567962\pi$
$242$	0	0
$243$	−6.62759	−0.425160
$244$	0	0
$245$	−4.01751	−0.256669
$246$	0	0
$247$	46.2778	2.94458
$248$	0	0
$249$	−29.7885	−1.88777
$250$	0	0
$251$	18.1381	1.14487	0.572434	−	0.819951i	$-0.305999\pi$
$251$	18.1381	1.14487	0.572434	+	0.819951i	$0.305999\pi$
$252$	0	0
$253$	−8.70343	−0.547180
$254$	0	0
$255$	5.31988	0.333144
$256$	0	0
$257$	16.5957	1.03521	0.517604	−	0.855620i	$-0.326824\pi$
$257$	16.5957	1.03521	0.517604	+	0.855620i	$0.326824\pi$
$258$	0	0
$259$	−16.0840	−0.999414
$260$	0	0
$261$	2.81204	0.174061
$262$	0	0
$263$	17.0672	1.05241	0.526205	−	0.850358i	$-0.323615\pi$
$263$	17.0672	1.05241	0.526205	+	0.850358i	$0.323615\pi$
$264$	0	0
$265$	−4.35774	−0.267694
$266$	0	0
$267$	5.18785	0.317492
$268$	0	0
$269$	−10.4772	−0.638809	−0.319404	−	0.947618i	$-0.603483\pi$
$269$	−10.4772	−0.638809	−0.319404	+	0.947618i	$0.603483\pi$
$270$	0	0
$271$	−1.01899	−0.0618992	−0.0309496	−	0.999521i	$-0.509853\pi$
$271$	−1.01899	−0.0618992	−0.0309496	+	0.999521i	$0.509853\pi$
$272$	0	0
$273$	21.6105	1.30793
$274$	0	0
$275$	−15.3792	−0.927402
$276$	0	0
$277$	10.4975	0.630735	0.315367	−	0.948970i	$-0.397872\pi$
$277$	10.4975	0.630735	0.315367	+	0.948970i	$0.397872\pi$
$278$	0	0
$279$	1.64056	0.0982178
$280$	0	0
$281$	3.73288	0.222685	0.111342	−	0.993782i	$-0.464485\pi$
$281$	3.73288	0.222685	0.111342	+	0.993782i	$0.464485\pi$
$282$	0	0
$283$	17.7483	1.05503	0.527514	−	0.849546i	$-0.323124\pi$
$283$	17.7483	1.05503	0.527514	+	0.849546i	$0.323124\pi$
$284$	0	0
$285$	−17.0548	−1.01024
$286$	0	0
$287$	2.42524	0.143158
$288$	0	0
$289$	−11.2403	−0.661194
$290$	0	0
$291$	−21.3573	−1.25199
$292$	0	0
$293$	−2.19902	−0.128468	−0.0642341	−	0.997935i	$-0.520460\pi$
$293$	−2.19902	−0.128468	−0.0642341	+	0.997935i	$0.520460\pi$
$294$	0	0
$295$	−6.49191	−0.377973
$296$	0	0
$297$	−18.9095	−1.09724
$298$	0	0
$299$	12.4353	0.719150
$300$	0	0
$301$	−9.43138	−0.543616
$302$	0	0
$303$	17.1197	0.983503
$304$	0	0
$305$	8.54279	0.489159
$306$	0	0
$307$	−9.70573	−0.553935	−0.276968	−	0.960879i	$-0.589329\pi$
$307$	−9.70573	−0.553935	−0.276968	+	0.960879i	$0.589329\pi$
$308$	0	0
$309$	−24.6408	−1.40176
$310$	0	0
$311$	−6.81900	−0.386670	−0.193335	−	0.981133i	$-0.561930\pi$
$311$	−6.81900	−0.386670	−0.193335	+	0.981133i	$0.561930\pi$
$312$	0	0
$313$	−3.17898	−0.179686	−0.0898432	−	0.995956i	$-0.528637\pi$
$313$	−3.17898	−0.179686	−0.0898432	+	0.995956i	$0.528637\pi$
$314$	0	0
$315$	−1.41535	−0.0797457
$316$	0	0
$317$	−26.1728	−1.47001	−0.735004	−	0.678063i	$-0.762820\pi$
$317$	−26.1728	−1.47001	−0.735004	+	0.678063i	$0.762820\pi$
$318$	0	0
$319$	18.2584	1.02228
$320$	0	0
$321$	−37.9403	−2.11762
$322$	0	0
$323$	−18.4649	−1.02741
$324$	0	0
$325$	21.9735	1.21887
$326$	0	0
$327$	−10.9197	−0.603863
$328$	0	0
$329$	−1.63418	−0.0900953
$330$	0	0
$331$	−21.0659	−1.15789	−0.578943	−	0.815368i	$-0.696535\pi$
$331$	−21.0659	−1.15789	−0.578943	+	0.815368i	$0.696535\pi$
$332$	0	0
$333$	5.54403	0.303811
$334$	0	0
$335$	−9.56353	−0.522511
$336$	0	0
$337$	28.7732	1.56737	0.783687	−	0.621156i	$-0.213337\pi$
$337$	28.7732	1.56737	0.783687	+	0.621156i	$0.213337\pi$
$338$	0	0
$339$	−31.6602	−1.71955
$340$	0	0
$341$	10.6521	0.576842
$342$	0	0
$343$	19.6787	1.06255
$344$	0	0
$345$	−4.58280	−0.246729
$346$	0	0
$347$	−4.91378	−0.263786	−0.131893	−	0.991264i	$-0.542105\pi$
$347$	−4.91378	−0.263786	−0.131893	+	0.991264i	$0.542105\pi$
$348$	0	0
$349$	−0.267076	−0.0142962	−0.00714812	−	0.999974i	$-0.502275\pi$
$349$	−0.267076	−0.0142962	−0.00714812	+	0.999974i	$0.502275\pi$
$350$	0	0
$351$	27.0175	1.44209
$352$	0	0
$353$	−21.1615	−1.12631	−0.563157	−	0.826350i	$-0.690413\pi$
$353$	−21.1615	−1.12631	−0.563157	+	0.826350i	$0.690413\pi$
$354$	0	0
$355$	5.84924	0.310445
$356$	0	0
$357$	−8.62262	−0.456357
$358$	0	0
$359$	3.50502	0.184988	0.0924938	−	0.995713i	$-0.470516\pi$
$359$	3.50502	0.184988	0.0924938	+	0.995713i	$0.470516\pi$
$360$	0	0
$361$	40.1959	2.11558
$362$	0	0
$363$	12.8404	0.673944
$364$	0	0
$365$	7.51274	0.393235
$366$	0	0
$367$	7.42218	0.387434	0.193717	−	0.981057i	$-0.437946\pi$
$367$	7.42218	0.387434	0.193717	+	0.981057i	$0.437946\pi$
$368$	0	0
$369$	−0.835960	−0.0435183
$370$	0	0
$371$	7.06315	0.366700
$372$	0	0
$373$	−14.2628	−0.738500	−0.369250	−	0.929330i	$-0.620385\pi$
$373$	−14.2628	−0.738500	−0.369250	+	0.929330i	$0.620385\pi$
$374$	0	0
$375$	−19.1813	−0.990518
$376$	0	0
$377$	−26.0872	−1.34356
$378$	0	0
$379$	16.5344	0.849316	0.424658	−	0.905354i	$-0.360394\pi$
$379$	16.5344	0.849316	0.424658	+	0.905354i	$0.360394\pi$
$380$	0	0
$381$	−18.3502	−0.940111
$382$	0	0
$383$	−19.3370	−0.988073	−0.494037	−	0.869441i	$-0.664479\pi$
$383$	−19.3370	−0.988073	−0.494037	+	0.869441i	$0.664479\pi$
$384$	0	0
$385$	−9.18977	−0.468354
$386$	0	0
$387$	3.25092	0.165253
$388$	0	0
$389$	−32.3230	−1.63884	−0.819419	−	0.573194i	$-0.805704\pi$
$389$	−32.3230	−1.63884	−0.819419	+	0.573194i	$0.805704\pi$
$390$	0	0
$391$	−4.96168	−0.250923
$392$	0	0
$393$	−1.56107	−0.0787455
$394$	0	0
$395$	−13.2756	−0.667969
$396$	0	0
$397$	18.0754	0.907178	0.453589	−	0.891211i	$-0.350143\pi$
$397$	18.0754	0.907178	0.453589	+	0.891211i	$0.350143\pi$
$398$	0	0
$399$	27.6430	1.38388
$400$	0	0
$401$	1.48738	0.0742761	0.0371381	−	0.999310i	$-0.488176\pi$
$401$	1.48738	0.0742761	0.0371381	+	0.999310i	$0.488176\pi$
$402$	0	0
$403$	−15.2194	−0.758134
$404$	0	0
$405$	−12.2141	−0.606926
$406$	0	0
$407$	35.9971	1.78431
$408$	0	0
$409$	10.6262	0.525431	0.262716	−	0.964873i	$-0.415382\pi$
$409$	10.6262	0.525431	0.262716	+	0.964873i	$0.415382\pi$
$410$	0	0
$411$	−28.6010	−1.41079
$412$	0	0
$413$	10.5223	0.517767
$414$	0	0
$415$	−18.0989	−0.888441
$416$	0	0
$417$	−43.8881	−2.14921
$418$	0	0
$419$	11.9620	0.584381	0.292191	−	0.956360i	$-0.405616\pi$
$419$	11.9620	0.584381	0.292191	+	0.956360i	$0.405616\pi$
$420$	0	0
$421$	−16.0661	−0.783012	−0.391506	−	0.920176i	$-0.628046\pi$
$421$	−16.0661	−0.783012	−0.391506	+	0.920176i	$0.628046\pi$
$422$	0	0
$423$	0.563288	0.0273880
$424$	0	0
$425$	−8.76744	−0.425283
$426$	0	0
$427$	−13.8464	−0.670075
$428$	0	0
$429$	−48.3657	−2.33512
$430$	0	0
$431$	27.9765	1.34758	0.673791	−	0.738922i	$-0.264665\pi$
$431$	27.9765	1.34758	0.673791	+	0.738922i	$0.264665\pi$
$432$	0	0
$433$	21.1268	1.01529	0.507644	−	0.861567i	$-0.330517\pi$
$433$	21.1268	1.01529	0.507644	+	0.861567i	$0.330517\pi$
$434$	0	0
$435$	9.61398	0.460955
$436$	0	0
$437$	15.9065	0.760911
$438$	0	0
$439$	15.0062	0.716208	0.358104	−	0.933682i	$-0.383423\pi$
$439$	15.0062	0.716208	0.358104	+	0.933682i	$0.383423\pi$
$440$	0	0
$441$	−2.24453	−0.106882
$442$	0	0
$443$	−32.5571	−1.54683	−0.773416	−	0.633898i	$-0.781454\pi$
$443$	−32.5571	−1.54683	−0.773416	+	0.633898i	$0.781454\pi$
$444$	0	0
$445$	3.15204	0.149421
$446$	0	0
$447$	32.5781	1.54089
$448$	0	0
$449$	6.40545	0.302292	0.151146	−	0.988511i	$-0.451704\pi$
$449$	6.40545	0.302292	0.151146	+	0.988511i	$0.451704\pi$
$450$	0	0
$451$	−5.42784	−0.255587
$452$	0	0
$453$	5.88770	0.276628
$454$	0	0
$455$	13.1301	0.615550
$456$	0	0
$457$	31.9707	1.49553	0.747764	−	0.663965i	$-0.231128\pi$
$457$	31.9707	1.49553	0.747764	+	0.663965i	$0.231128\pi$
$458$	0	0
$459$	−10.7800	−0.503168
$460$	0	0
$461$	−33.6280	−1.56621	−0.783107	−	0.621887i	$-0.786366\pi$
$461$	−33.6280	−1.56621	−0.783107	+	0.621887i	$0.786366\pi$
$462$	0	0
$463$	1.86049	0.0864643	0.0432322	−	0.999065i	$-0.486234\pi$
$463$	1.86049	0.0864643	0.0432322	+	0.999065i	$0.486234\pi$
$464$	0	0
$465$	5.60885	0.260104
$466$	0	0
$467$	−20.1263	−0.931335	−0.465667	−	0.884960i	$-0.654186\pi$
$467$	−20.1263	−0.931335	−0.465667	+	0.884960i	$0.654186\pi$
$468$	0	0
$469$	15.5008	0.715763
$470$	0	0
$471$	40.3916	1.86115
$472$	0	0
$473$	21.1080	0.970548
$474$	0	0
$475$	28.1073	1.28965
$476$	0	0
$477$	−2.43460	−0.111473
$478$	0	0
$479$	20.3663	0.930560	0.465280	−	0.885164i	$-0.345954\pi$
$479$	20.3663	0.930560	0.465280	+	0.885164i	$0.345954\pi$
$480$	0	0
$481$	−51.4318	−2.34509
$482$	0	0
$483$	7.42793	0.337983
$484$	0	0
$485$	−12.9763	−0.589223
$486$	0	0
$487$	−30.5363	−1.38373	−0.691866	−	0.722026i	$-0.743211\pi$
$487$	−30.5363	−1.38373	−0.691866	+	0.722026i	$0.743211\pi$
$488$	0	0
$489$	−33.3734	−1.50920
$490$	0	0
$491$	7.79487	0.351778	0.175889	−	0.984410i	$-0.443720\pi$
$491$	7.79487	0.351778	0.175889	+	0.984410i	$0.443720\pi$
$492$	0	0
$493$	10.4088	0.468790
$494$	0	0
$495$	3.16763	0.142374
$496$	0	0
$497$	−9.48061	−0.425264
$498$	0	0
$499$	26.6309	1.19216	0.596081	−	0.802925i	$-0.296724\pi$
$499$	26.6309	1.19216	0.596081	+	0.802925i	$0.296724\pi$
$500$	0	0
$501$	−1.56751	−0.0700310
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	10.4016	0.462866
$506$	0	0
$507$	44.2729	1.96623
$508$	0	0
$509$	9.19305	0.407475	0.203737	−	0.979026i	$-0.434691\pi$
$509$	9.19305	0.407475	0.203737	+	0.979026i	$0.434691\pi$
$510$	0	0
$511$	−12.1769	−0.538673
$512$	0	0
$513$	34.5593	1.52583
$514$	0	0
$515$	−14.9712	−0.659712
$516$	0	0
$517$	3.65740	0.160852
$518$	0	0
$519$	−33.3985	−1.46603
$520$	0	0
$521$	42.5048	1.86217	0.931084	−	0.364804i	$-0.118864\pi$
$521$	42.5048	1.86217	0.931084	+	0.364804i	$0.118864\pi$
$522$	0	0
$523$	36.3376	1.58893	0.794465	−	0.607309i	$-0.207751\pi$
$523$	36.3376	1.58893	0.794465	+	0.607309i	$0.207751\pi$
$524$	0	0
$525$	13.1254	0.572838
$526$	0	0
$527$	6.07257	0.264525
$528$	0	0
$529$	−18.7258	−0.814164
$530$	0	0
$531$	−3.62694	−0.157396
$532$	0	0
$533$	7.75517	0.335914
$534$	0	0
$535$	−23.0518	−0.996616
$536$	0	0
$537$	−10.0247	−0.432599
$538$	0	0
$539$	−14.5736	−0.627729
$540$	0	0
$541$	8.72722	0.375213	0.187606	−	0.982244i	$-0.439927\pi$
$541$	8.72722	0.375213	0.187606	+	0.982244i	$0.439927\pi$
$542$	0	0
$543$	13.9780	0.599855
$544$	0	0
$545$	−6.63462	−0.284196
$546$	0	0
$547$	−1.00179	−0.0428334	−0.0214167	−	0.999771i	$-0.506818\pi$
$547$	−1.00179	−0.0428334	−0.0214167	+	0.999771i	$0.506818\pi$
$548$	0	0
$549$	4.77274	0.203695
$550$	0	0
$551$	−33.3693	−1.42158
$552$	0	0
$553$	21.5175	0.915018
$554$	0	0
$555$	18.9543	0.804564
$556$	0	0
$557$	23.6427	1.00177	0.500886	−	0.865513i	$-0.333008\pi$
$557$	23.6427	1.00177	0.500886	+	0.865513i	$0.333008\pi$
$558$	0	0
$559$	−30.1586	−1.27557
$560$	0	0
$561$	19.2980	0.814760
$562$	0	0
$563$	−14.2325	−0.599829	−0.299915	−	0.953966i	$-0.596958\pi$
$563$	−14.2325	−0.599829	−0.299915	+	0.953966i	$0.596958\pi$
$564$	0	0
$565$	−19.2361	−0.809269
$566$	0	0
$567$	19.7970	0.831398
$568$	0	0
$569$	21.2672	0.891566	0.445783	−	0.895141i	$-0.352925\pi$
$569$	21.2672	0.891566	0.445783	+	0.895141i	$0.352925\pi$
$570$	0	0
$571$	−11.7723	−0.492655	−0.246328	−	0.969187i	$-0.579224\pi$
$571$	−11.7723	−0.492655	−0.246328	+	0.969187i	$0.579224\pi$
$572$	0	0
$573$	22.4662	0.938539
$574$	0	0
$575$	7.55269	0.314969
$576$	0	0
$577$	19.8916	0.828099	0.414050	−	0.910254i	$-0.364114\pi$
$577$	19.8916	0.828099	0.414050	+	0.910254i	$0.364114\pi$
$578$	0	0
$579$	36.6983	1.52513
$580$	0	0
$581$	29.3352	1.21703
$582$	0	0
$583$	−15.8078	−0.654690
$584$	0	0
$585$	−4.52584	−0.187120
$586$	0	0
$587$	21.4587	0.885697	0.442849	−	0.896596i	$-0.353968\pi$
$587$	21.4587	0.885697	0.442849	+	0.896596i	$0.353968\pi$
$588$	0	0
$589$	−19.4679	−0.802159
$590$	0	0
$591$	14.8308	0.610057
$592$	0	0
$593$	32.8072	1.34723	0.673615	−	0.739083i	$-0.264741\pi$
$593$	32.8072	1.34723	0.673615	+	0.739083i	$0.264741\pi$
$594$	0	0
$595$	−5.23893	−0.214775
$596$	0	0
$597$	14.9788	0.613040
$598$	0	0
$599$	26.5366	1.08426	0.542128	−	0.840296i	$-0.317619\pi$
$599$	26.5366	1.08426	0.542128	+	0.840296i	$0.317619\pi$
$600$	0	0
$601$	−36.4085	−1.48513	−0.742566	−	0.669773i	$-0.766392\pi$
$601$	−36.4085	−1.48513	−0.742566	+	0.669773i	$0.766392\pi$
$602$	0	0
$603$	−5.34301	−0.217584
$604$	0	0
$605$	7.80154	0.317178
$606$	0	0
$607$	9.31565	0.378111	0.189055	−	0.981966i	$-0.439457\pi$
$607$	9.31565	0.378111	0.189055	+	0.981966i	$0.439457\pi$
$608$	0	0
$609$	−15.5826	−0.631440
$610$	0	0
$611$	−5.22560	−0.211405
$612$	0	0
$613$	−3.67875	−0.148583	−0.0742916	−	0.997237i	$-0.523670\pi$
$613$	−3.67875	−0.148583	−0.0742916	+	0.997237i	$0.523670\pi$
$614$	0	0
$615$	−2.85803	−0.115247
$616$	0	0
$617$	−45.8034	−1.84398	−0.921988	−	0.387218i	$-0.873436\pi$
$617$	−45.8034	−1.84398	−0.921988	+	0.387218i	$0.873436\pi$
$618$	0	0
$619$	−1.11337	−0.0447502	−0.0223751	−	0.999750i	$-0.507123\pi$
$619$	−1.11337	−0.0447502	−0.0223751	+	0.999750i	$0.507123\pi$
$620$	0	0
$621$	9.28641	0.372651
$622$	0	0
$623$	−5.10891	−0.204684
$624$	0	0
$625$	6.61179	0.264472
$626$	0	0
$627$	−61.8667	−2.47072
$628$	0	0
$629$	20.5213	0.818239
$630$	0	0
$631$	−35.2716	−1.40414	−0.702069	−	0.712109i	$-0.747740\pi$
$631$	−35.2716	−1.40414	−0.702069	+	0.712109i	$0.747740\pi$
$632$	0	0
$633$	27.0693	1.07591
$634$	0	0
$635$	−11.1492	−0.442444
$636$	0	0
$637$	20.8224	0.825014
$638$	0	0
$639$	3.26788	0.129276
$640$	0	0
$641$	−34.8629	−1.37700	−0.688500	−	0.725236i	$-0.741731\pi$
$641$	−34.8629	−1.37700	−0.688500	+	0.725236i	$0.741731\pi$
$642$	0	0
$643$	28.0247	1.10519	0.552594	−	0.833451i	$-0.313638\pi$
$643$	28.0247	1.10519	0.552594	+	0.833451i	$0.313638\pi$
$644$	0	0
$645$	11.1144	0.437630
$646$	0	0
$647$	42.6463	1.67660	0.838299	−	0.545210i	$-0.183550\pi$
$647$	42.6463	1.67660	0.838299	+	0.545210i	$0.183550\pi$
$648$	0	0
$649$	−23.5495	−0.924399
$650$	0	0
$651$	−9.09100	−0.356304
$652$	0	0
$653$	−35.6968	−1.39693	−0.698463	−	0.715647i	$-0.746132\pi$
$653$	−35.6968	−1.39693	−0.698463	+	0.715647i	$0.746132\pi$
$654$	0	0
$655$	−0.948474	−0.0370599
$656$	0	0
$657$	4.19726	0.163751
$658$	0	0
$659$	23.7441	0.924941	0.462470	−	0.886635i	$-0.346963\pi$
$659$	23.7441	0.924941	0.462470	+	0.886635i	$0.346963\pi$
$660$	0	0
$661$	18.6705	0.726198	0.363099	−	0.931751i	$-0.381719\pi$
$661$	18.6705	0.726198	0.363099	+	0.931751i	$0.381719\pi$
$662$	0	0
$663$	−27.5725	−1.07083
$664$	0	0
$665$	16.7953	0.651295
$666$	0	0
$667$	−8.96665	−0.347190
$668$	0	0
$669$	9.99009	0.386239
$670$	0	0
$671$	30.9891	1.19632
$672$	0	0
$673$	−19.6875	−0.758897	−0.379448	−	0.925213i	$-0.623886\pi$
$673$	−19.6875	−0.758897	−0.379448	+	0.925213i	$0.623886\pi$
$674$	0	0
$675$	16.4094	0.631597
$676$	0	0
$677$	−20.9840	−0.806482	−0.403241	−	0.915094i	$-0.632116\pi$
$677$	−20.9840	−0.806482	−0.403241	+	0.915094i	$0.632116\pi$
$678$	0	0
$679$	21.0323	0.807147
$680$	0	0
$681$	3.80827	0.145933
$682$	0	0
$683$	42.7462	1.63564	0.817818	−	0.575477i	$-0.195184\pi$
$683$	42.7462	1.63564	0.817818	+	0.575477i	$0.195184\pi$
$684$	0	0
$685$	−17.3774	−0.663957
$686$	0	0
$687$	−20.2116	−0.771121
$688$	0	0
$689$	22.5858	0.860449
$690$	0	0
$691$	1.35769	0.0516488	0.0258244	−	0.999666i	$-0.491779\pi$
$691$	1.35769	0.0516488	0.0258244	+	0.999666i	$0.491779\pi$
$692$	0	0
$693$	−5.13419	−0.195032
$694$	0	0
$695$	−26.6656	−1.01148
$696$	0	0
$697$	−3.09432	−0.117206
$698$	0	0
$699$	12.2880	0.464775
$700$	0	0
$701$	−4.62550	−0.174703	−0.0873514	−	0.996178i	$-0.527840\pi$
$701$	−4.62550	−0.174703	−0.0873514	+	0.996178i	$0.527840\pi$
$702$	0	0
$703$	−65.7887	−2.48127
$704$	0	0
$705$	1.92580	0.0725299
$706$	0	0
$707$	−16.8592	−0.634057
$708$	0	0
$709$	7.24083	0.271935	0.135968	−	0.990713i	$-0.456586\pi$
$709$	7.24083	0.271935	0.135968	+	0.990713i	$0.456586\pi$
$710$	0	0
$711$	−7.41690	−0.278155
$712$	0	0
$713$	−5.23120	−0.195910
$714$	0	0
$715$	−29.3860	−1.09898
$716$	0	0
$717$	−53.8530	−2.01118
$718$	0	0
$719$	29.0836	1.08463	0.542317	−	0.840174i	$-0.317547\pi$
$719$	29.0836	1.08463	0.542317	+	0.840174i	$0.317547\pi$
$720$	0	0
$721$	24.2658	0.903706
$722$	0	0
$723$	−12.5660	−0.467335
$724$	0	0
$725$	−15.8443	−0.588444
$726$	0	0
$727$	−44.2104	−1.63967	−0.819837	−	0.572597i	$-0.805936\pi$
$727$	−44.2104	−1.63967	−0.819837	+	0.572597i	$0.805936\pi$
$728$	0	0
$729$	18.9150	0.700555
$730$	0	0
$731$	12.0333	0.445069
$732$	0	0
$733$	17.8407	0.658959	0.329480	−	0.944163i	$-0.393127\pi$
$733$	17.8407	0.658959	0.329480	+	0.944163i	$0.393127\pi$
$734$	0	0
$735$	−7.67373	−0.283050
$736$	0	0
$737$	−34.6919	−1.27789
$738$	0	0
$739$	37.3837	1.37518	0.687590	−	0.726099i	$-0.258669\pi$
$739$	37.3837	1.37518	0.687590	+	0.726099i	$0.258669\pi$
$740$	0	0
$741$	88.3937	3.24722
$742$	0	0
$743$	−42.0535	−1.54279	−0.771397	−	0.636355i	$-0.780442\pi$
$743$	−42.0535	−1.54279	−0.771397	+	0.636355i	$0.780442\pi$
$744$	0	0
$745$	19.7938	0.725189
$746$	0	0
$747$	−10.1116	−0.369964
$748$	0	0
$749$	37.3630	1.36521
$750$	0	0
$751$	−23.4385	−0.855284	−0.427642	−	0.903948i	$-0.640656\pi$
$751$	−23.4385	−0.855284	−0.427642	+	0.903948i	$0.640656\pi$
$752$	0	0
$753$	34.6451	1.26254
$754$	0	0
$755$	3.57725	0.130189
$756$	0	0
$757$	−24.4852	−0.889930	−0.444965	−	0.895548i	$-0.646784\pi$
$757$	−24.4852	−0.889930	−0.444965	+	0.895548i	$0.646784\pi$
$758$	0	0
$759$	−16.6242	−0.603419
$760$	0	0
$761$	−13.4993	−0.489350	−0.244675	−	0.969605i	$-0.578681\pi$
$761$	−13.4993	−0.489350	−0.244675	+	0.969605i	$0.578681\pi$
$762$	0	0
$763$	10.7536	0.389306
$764$	0	0
$765$	1.80581	0.0652893
$766$	0	0
$767$	33.6470	1.21492
$768$	0	0
$769$	21.4735	0.774353	0.387177	−	0.922006i	$-0.373450\pi$
$769$	21.4735	0.774353	0.387177	+	0.922006i	$0.373450\pi$
$770$	0	0
$771$	31.6989	1.14161
$772$	0	0
$773$	−46.1813	−1.66103	−0.830513	−	0.556999i	$-0.811953\pi$
$773$	−46.1813	−1.66103	−0.830513	+	0.556999i	$0.811953\pi$
$774$	0	0
$775$	−9.24368	−0.332043
$776$	0	0
$777$	−30.7216	−1.10213
$778$	0	0
$779$	9.91999	0.355421
$780$	0	0
$781$	21.2182	0.759247
$782$	0	0
$783$	−19.4814	−0.696209
$784$	0	0
$785$	24.5411	0.875911
$786$	0	0
$787$	−23.7314	−0.845932	−0.422966	−	0.906146i	$-0.639011\pi$
$787$	−23.7314	−0.845932	−0.422966	+	0.906146i	$0.639011\pi$
$788$	0	0
$789$	32.5996	1.16058
$790$	0	0
$791$	31.1784	1.10858
$792$	0	0
$793$	−44.2765	−1.57231
$794$	0	0
$795$	−8.32358	−0.295207
$796$	0	0
$797$	4.75292	0.168357	0.0841786	−	0.996451i	$-0.473173\pi$
$797$	4.75292	0.168357	0.0841786	+	0.996451i	$0.473173\pi$
$798$	0	0
$799$	2.08502	0.0737627
$800$	0	0
$801$	1.76100	0.0622218
$802$	0	0
$803$	27.2526	0.961722
$804$	0	0
$805$	4.51306	0.159065
$806$	0	0
$807$	−20.0123	−0.704465
$808$	0	0
$809$	−4.74474	−0.166816	−0.0834081	−	0.996515i	$-0.526581\pi$
$809$	−4.74474	−0.166816	−0.0834081	+	0.996515i	$0.526581\pi$
$810$	0	0
$811$	12.2717	0.430919	0.215460	−	0.976513i	$-0.430875\pi$
$811$	12.2717	0.430919	0.215460	+	0.976513i	$0.430875\pi$
$812$	0	0
$813$	−1.94634	−0.0682612
$814$	0	0
$815$	−20.2770	−0.710272
$816$	0	0
$817$	−38.5773	−1.34965
$818$	0	0
$819$	7.33561	0.256327
$820$	0	0
$821$	14.0916	0.491801	0.245901	−	0.969295i	$-0.420916\pi$
$821$	14.0916	0.491801	0.245901	+	0.969295i	$0.420916\pi$
$822$	0	0
$823$	−31.8453	−1.11006	−0.555028	−	0.831831i	$-0.687293\pi$
$823$	−31.8453	−1.11006	−0.555028	+	0.831831i	$0.687293\pi$
$824$	0	0
$825$	−29.3754	−1.02272
$826$	0	0
$827$	12.8188	0.445754	0.222877	−	0.974847i	$-0.428455\pi$
$827$	12.8188	0.445754	0.222877	+	0.974847i	$0.428455\pi$
$828$	0	0
$829$	27.4123	0.952069	0.476034	−	0.879427i	$-0.342074\pi$
$829$	27.4123	0.952069	0.476034	+	0.879427i	$0.342074\pi$
$830$	0	0
$831$	20.0510	0.695561
$832$	0	0
$833$	−8.30816	−0.287861
$834$	0	0
$835$	−0.952385	−0.0329586
$836$	0	0
$837$	−11.3656	−0.392852
$838$	0	0
$839$	−28.0642	−0.968882	−0.484441	−	0.874824i	$-0.660977\pi$
$839$	−28.0642	−0.968882	−0.484441	+	0.874824i	$0.660977\pi$
$840$	0	0
$841$	−10.1894	−0.351358
$842$	0	0
$843$	7.13006	0.245572
$844$	0	0
$845$	26.8993	0.925364
$846$	0	0
$847$	−12.6450	−0.434486
$848$	0	0
$849$	33.9005	1.16346
$850$	0	0
$851$	−17.6780	−0.605995
$852$	0	0
$853$	−23.7741	−0.814011	−0.407005	−	0.913426i	$-0.633427\pi$
$853$	−23.7741	−0.814011	−0.407005	+	0.913426i	$0.633427\pi$
$854$	0	0
$855$	−5.78920	−0.197987
$856$	0	0
$857$	−23.9040	−0.816546	−0.408273	−	0.912860i	$-0.633869\pi$
$857$	−23.9040	−0.816546	−0.408273	+	0.912860i	$0.633869\pi$
$858$	0	0
$859$	37.0038	1.26255	0.631277	−	0.775558i	$-0.282531\pi$
$859$	37.0038	1.26255	0.631277	+	0.775558i	$0.282531\pi$
$860$	0	0
$861$	4.63238	0.157871
$862$	0	0
$863$	23.9065	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$863$	23.9065	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$864$	0	0
$865$	−20.2923	−0.689957
$866$	0	0
$867$	−21.4697	−0.729151
$868$	0	0
$869$	−48.1575	−1.63363
$870$	0	0
$871$	49.5669	1.67951
$872$	0	0
$873$	−7.24966	−0.245364
$874$	0	0
$875$	18.8894	0.638579
$876$	0	0
$877$	40.2307	1.35849	0.679247	−	0.733910i	$-0.262307\pi$
$877$	40.2307	1.35849	0.679247	+	0.733910i	$0.262307\pi$
$878$	0	0
$879$	−4.20029	−0.141672
$880$	0	0
$881$	43.8485	1.47729	0.738646	−	0.674093i	$-0.235466\pi$
$881$	43.8485	1.47729	0.738646	+	0.674093i	$0.235466\pi$
$882$	0	0
$883$	−37.5869	−1.26490	−0.632451	−	0.774601i	$-0.717951\pi$
$883$	−37.5869	−1.26490	−0.632451	+	0.774601i	$0.717951\pi$
$884$	0	0
$885$	−12.4000	−0.416821
$886$	0	0
$887$	−14.9205	−0.500982	−0.250491	−	0.968119i	$-0.580592\pi$
$887$	−14.9205	−0.500982	−0.250491	+	0.968119i	$0.580592\pi$
$888$	0	0
$889$	18.0710	0.606082
$890$	0	0
$891$	−44.3070	−1.48434
$892$	0	0
$893$	−6.68430	−0.223682
$894$	0	0
$895$	−6.09083	−0.203594
$896$	0	0
$897$	23.7522	0.793064
$898$	0	0
$899$	10.9742	0.366011
$900$	0	0
$901$	−9.01174	−0.300224
$902$	0	0
$903$	−18.0146	−0.599488
$904$	0	0
$905$	8.49277	0.282309
$906$	0	0
$907$	−38.4045	−1.27520	−0.637600	−	0.770368i	$-0.720073\pi$
$907$	−38.4045	−1.27520	−0.637600	+	0.770368i	$0.720073\pi$
$908$	0	0
$909$	5.81123	0.192746
$910$	0	0
$911$	−39.0130	−1.29256	−0.646279	−	0.763101i	$-0.723676\pi$
$911$	−39.0130	−1.29256	−0.646279	+	0.763101i	$0.723676\pi$
$912$	0	0
$913$	−65.6541	−2.17283
$914$	0	0
$915$	16.3173	0.539434
$916$	0	0
$917$	1.53731	0.0507666
$918$	0	0
$919$	29.8587	0.984948	0.492474	−	0.870327i	$-0.336093\pi$
$919$	29.8587	0.984948	0.492474	+	0.870327i	$0.336093\pi$
$920$	0	0
$921$	−18.5386	−0.610868
$922$	0	0
$923$	−30.3161	−0.997866
$924$	0	0
$925$	−31.2376	−1.02709
$926$	0	0
$927$	−8.36422	−0.274717
$928$	0	0
$929$	−53.2891	−1.74836	−0.874180	−	0.485602i	$-0.838600\pi$
$929$	−53.2891	−1.74836	−0.874180	+	0.485602i	$0.838600\pi$
$930$	0	0
$931$	26.6349	0.872923
$932$	0	0
$933$	−13.0248	−0.426411
$934$	0	0
$935$	11.7250	0.383450
$936$	0	0
$937$	11.9106	0.389103	0.194551	−	0.980892i	$-0.437675\pi$
$937$	11.9106	0.389103	0.194551	+	0.980892i	$0.437675\pi$
$938$	0	0
$939$	−6.07207	−0.198154
$940$	0	0
$941$	52.0049	1.69531	0.847656	−	0.530547i	$-0.178013\pi$
$941$	52.0049	1.69531	0.847656	+	0.530547i	$0.178013\pi$
$942$	0	0
$943$	2.66559	0.0868037
$944$	0	0
$945$	9.80531	0.318967
$946$	0	0
$947$	34.0555	1.10666	0.553328	−	0.832963i	$-0.313358\pi$
$947$	34.0555	1.10666	0.553328	+	0.832963i	$0.313358\pi$
$948$	0	0
$949$	−38.9378	−1.26398
$950$	0	0
$951$	−49.9918	−1.62109
$952$	0	0
$953$	39.7524	1.28771	0.643854	−	0.765149i	$-0.277335\pi$
$953$	39.7524	1.28771	0.643854	+	0.765149i	$0.277335\pi$
$954$	0	0
$955$	13.6500	0.441704
$956$	0	0
$957$	34.8749	1.12734
$958$	0	0
$959$	28.1658	0.909522
$960$	0	0
$961$	−24.5976	−0.793470
$962$	0	0
$963$	−12.8787	−0.415010
$964$	0	0
$965$	22.2971	0.717770
$966$	0	0
$967$	−56.1622	−1.80605	−0.903027	−	0.429584i	$-0.858660\pi$
$967$	−56.1622	−1.80605	−0.903027	+	0.429584i	$0.858660\pi$
$968$	0	0
$969$	−35.2692	−1.13301
$970$	0	0
$971$	47.4231	1.52188	0.760939	−	0.648823i	$-0.224738\pi$
$971$	47.4231	1.52188	0.760939	+	0.648823i	$0.224738\pi$
$972$	0	0
$973$	43.2203	1.38558
$974$	0	0
$975$	41.9709	1.34414
$976$	0	0
$977$	33.0025	1.05584	0.527921	−	0.849293i	$-0.322972\pi$
$977$	33.0025	1.05584	0.527921	+	0.849293i	$0.322972\pi$
$978$	0	0
$979$	11.4341	0.365434
$980$	0	0
$981$	−3.70667	−0.118345
$982$	0	0
$983$	30.2653	0.965312	0.482656	−	0.875810i	$-0.339672\pi$
$983$	30.2653	0.965312	0.482656	+	0.875810i	$0.339672\pi$
$984$	0	0
$985$	9.01089	0.287111
$986$	0	0
$987$	−3.12140	−0.0993552
$988$	0	0
$989$	−10.3661	−0.329622
$990$	0	0
$991$	−4.33315	−0.137647	−0.0688235	−	0.997629i	$-0.521925\pi$
$991$	−4.33315	−0.137647	−0.0688235	+	0.997629i	$0.521925\pi$
$992$	0	0
$993$	−40.2373	−1.27689
$994$	0	0
$995$	9.10080	0.288515
$996$	0	0
$997$	56.7614	1.79765	0.898825	−	0.438307i	$-0.144422\pi$
$997$	56.7614	1.79765	0.898825	+	0.438307i	$0.144422\pi$
$998$	0	0
$999$	−38.4082	−1.21518

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.s.1.18		21
4.3	odd	2		2012.2.a.b.1.4	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2012.2.a.b.1.4	✓	21	4.3	odd	2
8048.2.a.s.1.18		21	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)

\(f(q)\)	\(=\)	\(q+1.91007 q^{3} +1.16052 q^{5} -1.88100 q^{7} +0.648366 q^{9} +O(q^{10})\)	\(q+1.91007 q^{3} +1.16052 q^{5} -1.88100 q^{7} +0.648366 q^{9} +4.20980 q^{11} -6.01487 q^{13} +2.21667 q^{15} +2.39994 q^{17} -7.69389 q^{19} -3.59285 q^{21} -2.06742 q^{23} -3.65319 q^{25} -4.49178 q^{27} +4.33712 q^{29} +2.53030 q^{31} +8.04102 q^{33} -2.18294 q^{35} +8.55077 q^{37} -11.4888 q^{39} -1.28933 q^{41} +5.01401 q^{43} +0.752441 q^{45} +0.868781 q^{47} -3.46182 q^{49} +4.58405 q^{51} -3.75499 q^{53} +4.88556 q^{55} -14.6959 q^{57} -5.59396 q^{59} +7.36118 q^{61} -1.21958 q^{63} -6.98038 q^{65} -8.24073 q^{67} -3.94892 q^{69} +5.04019 q^{71} +6.47359 q^{73} -6.97785 q^{75} -7.91866 q^{77} -11.4394 q^{79} -10.5247 q^{81} -15.5955 q^{83} +2.78518 q^{85} +8.28420 q^{87} +2.71606 q^{89} +11.3140 q^{91} +4.83305 q^{93} -8.92891 q^{95} -11.1814 q^{97} +2.72949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9}+O(q^{10}) \) 21 * q - 10 * q^3 + 3 * q^5 - 13 * q^7 + 21 * q^9	\( 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9} - 7 q^{11} + 12 q^{13} - 14 q^{15} + q^{17} - 14 q^{19} + 14 q^{21} - 26 q^{23} + 18 q^{25} - 37 q^{27} + 9 q^{29} - 28 q^{31} + 3 q^{33} - 20 q^{35} + 31 q^{37} - 29 q^{39} + 4 q^{41} - 38 q^{43} + 24 q^{45} - 9 q^{47} + 16 q^{49} - 15 q^{51} + 22 q^{53} - 35 q^{55} - q^{57} - 10 q^{59} + 22 q^{61} - 35 q^{63} - 14 q^{65} - 58 q^{67} + 15 q^{69} - 27 q^{71} - 6 q^{73} - 48 q^{75} + 16 q^{77} - 47 q^{79} + 29 q^{81} - 22 q^{83} + 14 q^{85} - 29 q^{87} + q^{89} - 51 q^{91} + 34 q^{93} - 23 q^{95} - 2 q^{97} - 22 q^{99}+O(q^{100}) \) 21 * q - 10 * q^3 + 3 * q^5 - 13 * q^7 + 21 * q^9 - 7 * q^11 + 12 * q^13 - 14 * q^15 + q^17 - 14 * q^19 + 14 * q^21 - 26 * q^23 + 18 * q^25 - 37 * q^27 + 9 * q^29 - 28 * q^31 + 3 * q^33 - 20 * q^35 + 31 * q^37 - 29 * q^39 + 4 * q^41 - 38 * q^43 + 24 * q^45 - 9 * q^47 + 16 * q^49 - 15 * q^51 + 22 * q^53 - 35 * q^55 - q^57 - 10 * q^59 + 22 * q^61 - 35 * q^63 - 14 * q^65 - 58 * q^67 + 15 * q^69 - 27 * q^71 - 6 * q^73 - 48 * q^75 + 16 * q^77 - 47 * q^79 + 29 * q^81 - 22 * q^83 + 14 * q^85 - 29 * q^87 + q^89 - 51 * q^91 + 34 * q^93 - 23 * q^95 - 2 * q^97 - 22 * q^99

Introduction

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