LMFDB - Embedded newform 4024.2.a.d.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, 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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.49100 q^{3} -1.09730 q^{5} -0.799854 q^{7} -0.776913 q^{9} +O(q^{10})$	$q+1.49100 q^{3} -1.09730 q^{5} -0.799854 q^{7} -0.776913 q^{9} -0.838888 q^{11} +2.03223 q^{13} -1.63608 q^{15} -1.82317 q^{17} +4.30282 q^{19} -1.19258 q^{21} -0.747572 q^{23} -3.79593 q^{25} -5.63138 q^{27} +2.79315 q^{29} -0.0226322 q^{31} -1.25078 q^{33} +0.877680 q^{35} +1.80488 q^{37} +3.03006 q^{39} +6.90939 q^{41} -8.84694 q^{43} +0.852507 q^{45} -6.50163 q^{47} -6.36023 q^{49} -2.71835 q^{51} -8.42159 q^{53} +0.920512 q^{55} +6.41551 q^{57} -1.81714 q^{59} -8.08359 q^{61} +0.621417 q^{63} -2.22997 q^{65} +11.6490 q^{67} -1.11463 q^{69} -1.18928 q^{71} +3.90124 q^{73} -5.65974 q^{75} +0.670988 q^{77} -8.22799 q^{79} -6.06567 q^{81} -5.12022 q^{83} +2.00056 q^{85} +4.16459 q^{87} -9.57753 q^{89} -1.62549 q^{91} -0.0337447 q^{93} -4.72149 q^{95} -0.843274 q^{97} +0.651742 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) $ 28 * q + 2 * q^3 - 12 * q^5 + 18 * q^9	$ 28 q + 2 q^{3} - 12 q^{5} + 18 q^{9} - 14 q^{11} - 31 q^{13} - 2 q^{15} - 9 q^{17} - 8 q^{19} - 28 q^{21} - 4 q^{23} + 22 q^{25} - 4 q^{27} - 47 q^{29} - 5 q^{31} - 26 q^{33} - 13 q^{35} - 67 q^{37} - 9 q^{39} - 28 q^{41} + 15 q^{43} - 57 q^{45} - 10 q^{47} + 20 q^{49} - 11 q^{51} - 58 q^{53} + 15 q^{55} - 31 q^{57} - 32 q^{59} - 55 q^{61} - 16 q^{63} - 44 q^{65} + 22 q^{67} - 44 q^{69} - 47 q^{71} - 5 q^{73} - 25 q^{75} - 50 q^{77} - 14 q^{79} - 28 q^{81} - 16 q^{83} - 78 q^{85} - 11 q^{87} - 20 q^{89} - 15 q^{91} - 83 q^{93} - 27 q^{95} - 8 q^{97} - 70 q^{99}+O(q^{100}) $ 28 * q + 2 * q^3 - 12 * q^5 + 18 * q^9 - 14 * q^11 - 31 * q^13 - 2 * q^15 - 9 * q^17 - 8 * q^19 - 28 * q^21 - 4 * q^23 + 22 * q^25 - 4 * q^27 - 47 * q^29 - 5 * q^31 - 26 * q^33 - 13 * q^35 - 67 * q^37 - 9 * q^39 - 28 * q^41 + 15 * q^43 - 57 * q^45 - 10 * q^47 + 20 * q^49 - 11 * q^51 - 58 * q^53 + 15 * q^55 - 31 * q^57 - 32 * q^59 - 55 * q^61 - 16 * q^63 - 44 * q^65 + 22 * q^67 - 44 * q^69 - 47 * q^71 - 5 * q^73 - 25 * q^75 - 50 * q^77 - 14 * q^79 - 28 * q^81 - 16 * q^83 - 78 * q^85 - 11 * q^87 - 20 * q^89 - 15 * q^91 - 83 * q^93 - 27 * q^95 - 8 * q^97 - 70 * 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
$3$	1.49100	0.860831	0.430415	−	0.902631i	$-0.358367\pi$
$3$	1.49100	0.860831	0.430415	+	0.902631i	$0.358367\pi$
$4$	0	0
$5$	−1.09730	−0.490728	−0.245364	−	0.969431i	$-0.578907\pi$
$5$	−1.09730	−0.490728	−0.245364	+	0.969431i	$0.578907\pi$
$6$	0	0
$7$	−0.799854	−0.302316	−0.151158	−	0.988510i	$-0.548300\pi$
$7$	−0.799854	−0.302316	−0.151158	+	0.988510i	$0.548300\pi$
$8$	0	0
$9$	−0.776913	−0.258971
$10$	0	0
$11$	−0.838888	−0.252934	−0.126467	−	0.991971i	$-0.540364\pi$
$11$	−0.838888	−0.252934	−0.126467	+	0.991971i	$0.540364\pi$
$12$	0	0
$13$	2.03223	0.563640	0.281820	−	0.959467i	$-0.409062\pi$
$13$	2.03223	0.563640	0.281820	+	0.959467i	$0.409062\pi$
$14$	0	0
$15$	−1.63608	−0.422433
$16$	0	0
$17$	−1.82317	−0.442183	−0.221091	−	0.975253i	$-0.570962\pi$
$17$	−1.82317	−0.442183	−0.221091	+	0.975253i	$0.570962\pi$
$18$	0	0
$19$	4.30282	0.987134	0.493567	−	0.869708i	$-0.335693\pi$
$19$	4.30282	0.987134	0.493567	+	0.869708i	$0.335693\pi$
$20$	0	0
$21$	−1.19258	−0.260243
$22$	0	0
$23$	−0.747572	−0.155880	−0.0779398	−	0.996958i	$-0.524834\pi$
$23$	−0.747572	−0.155880	−0.0779398	+	0.996958i	$0.524834\pi$
$24$	0	0
$25$	−3.79593	−0.759186
$26$	0	0
$27$	−5.63138	−1.08376
$28$	0	0
$29$	2.79315	0.518675	0.259337	−	0.965787i	$-0.416496\pi$
$29$	2.79315	0.518675	0.259337	+	0.965787i	$0.416496\pi$
$30$	0	0
$31$	−0.0226322	−0.00406487	−0.00203243	−	0.999998i	$-0.500647\pi$
$31$	−0.0226322	−0.00406487	−0.00203243	+	0.999998i	$0.500647\pi$
$32$	0	0
$33$	−1.25078	−0.217733
$34$	0	0
$35$	0.877680	0.148355
$36$	0	0
$37$	1.80488	0.296721	0.148360	−	0.988933i	$-0.452600\pi$
$37$	1.80488	0.296721	0.148360	+	0.988933i	$0.452600\pi$
$38$	0	0
$39$	3.03006	0.485198
$40$	0	0
$41$	6.90939	1.07907	0.539533	−	0.841965i	$-0.318601\pi$
$41$	6.90939	1.07907	0.539533	+	0.841965i	$0.318601\pi$
$42$	0	0
$43$	−8.84694	−1.34914	−0.674572	−	0.738209i	$-0.735672\pi$
$43$	−8.84694	−1.34914	−0.674572	+	0.738209i	$0.735672\pi$
$44$	0	0
$45$	0.852507	0.127084
$46$	0	0
$47$	−6.50163	−0.948360	−0.474180	−	0.880428i	$-0.657255\pi$
$47$	−6.50163	−0.948360	−0.474180	+	0.880428i	$0.657255\pi$
$48$	0	0
$49$	−6.36023	−0.908605
$50$	0	0
$51$	−2.71835	−0.380645
$52$	0	0
$53$	−8.42159	−1.15679	−0.578397	−	0.815755i	$-0.696321\pi$
$53$	−8.42159	−1.15679	−0.578397	+	0.815755i	$0.696321\pi$
$54$	0	0
$55$	0.920512	0.124122
$56$	0	0
$57$	6.41551	0.849755
$58$	0	0
$59$	−1.81714	−0.236572	−0.118286	−	0.992980i	$-0.537740\pi$
$59$	−1.81714	−0.236572	−0.118286	+	0.992980i	$0.537740\pi$
$60$	0	0
$61$	−8.08359	−1.03500	−0.517499	−	0.855684i	$-0.673137\pi$
$61$	−8.08359	−1.03500	−0.517499	+	0.855684i	$0.673137\pi$
$62$	0	0
$63$	0.621417	0.0782911
$64$	0	0
$65$	−2.22997	−0.276594
$66$	0	0
$67$	11.6490	1.42315	0.711577	−	0.702608i	$-0.247981\pi$
$67$	11.6490	1.42315	0.711577	+	0.702608i	$0.247981\pi$
$68$	0	0
$69$	−1.11463	−0.134186
$70$	0	0
$71$	−1.18928	−0.141142	−0.0705709	−	0.997507i	$-0.522482\pi$
$71$	−1.18928	−0.141142	−0.0705709	+	0.997507i	$0.522482\pi$
$72$	0	0
$73$	3.90124	0.456605	0.228303	−	0.973590i	$-0.426682\pi$
$73$	3.90124	0.456605	0.228303	+	0.973590i	$0.426682\pi$
$74$	0	0
$75$	−5.65974	−0.653531
$76$	0	0
$77$	0.670988	0.0764661
$78$	0	0
$79$	−8.22799	−0.925721	−0.462861	−	0.886431i	$-0.653177\pi$
$79$	−8.22799	−0.925721	−0.462861	+	0.886431i	$0.653177\pi$
$80$	0	0
$81$	−6.06567	−0.673963
$82$	0	0
$83$	−5.12022	−0.562017	−0.281009	−	0.959705i	$-0.590669\pi$
$83$	−5.12022	−0.562017	−0.281009	+	0.959705i	$0.590669\pi$
$84$	0	0
$85$	2.00056	0.216991
$86$	0	0
$87$	4.16459	0.446491
$88$	0	0
$89$	−9.57753	−1.01522	−0.507608	−	0.861588i	$-0.669470\pi$
$89$	−9.57753	−1.01522	−0.507608	+	0.861588i	$0.669470\pi$
$90$	0	0
$91$	−1.62549	−0.170398
$92$	0	0
$93$	−0.0337447	−0.00349916
$94$	0	0
$95$	−4.72149	−0.484414
$96$	0	0
$97$	−0.843274	−0.0856215	−0.0428108	−	0.999083i	$-0.513631\pi$
$97$	−0.843274	−0.0856215	−0.0428108	+	0.999083i	$0.513631\pi$
$98$	0	0
$99$	0.651742	0.0655026
$100$	0	0
$101$	−14.2484	−1.41777	−0.708885	−	0.705324i	$-0.750802\pi$
$101$	−14.2484	−1.41777	−0.708885	+	0.705324i	$0.750802\pi$
$102$	0	0
$103$	−17.0465	−1.67964	−0.839822	−	0.542862i	$-0.817340\pi$
$103$	−17.0465	−1.67964	−0.839822	+	0.542862i	$0.817340\pi$
$104$	0	0
$105$	1.30862	0.127709
$106$	0	0
$107$	0.481214	0.0465207	0.0232604	−	0.999729i	$-0.492595\pi$
$107$	0.481214	0.0465207	0.0232604	+	0.999729i	$0.492595\pi$
$108$	0	0
$109$	17.2858	1.65568	0.827840	−	0.560964i	$-0.189569\pi$
$109$	17.2858	1.65568	0.827840	+	0.560964i	$0.189569\pi$
$110$	0	0
$111$	2.69108	0.255426
$112$	0	0
$113$	−5.97071	−0.561677	−0.280839	−	0.959755i	$-0.590613\pi$
$113$	−5.97071	−0.561677	−0.280839	+	0.959755i	$0.590613\pi$
$114$	0	0
$115$	0.820311	0.0764944
$116$	0	0
$117$	−1.57887	−0.145966
$118$	0	0
$119$	1.45827	0.133679
$120$	0	0
$121$	−10.2963	−0.936024
$122$	0	0
$123$	10.3019	0.928892
$124$	0	0
$125$	9.65178	0.863282
$126$	0	0
$127$	0.211759	0.0187906	0.00939528	−	0.999956i	$-0.497009\pi$
$127$	0.211759	0.0187906	0.00939528	+	0.999956i	$0.497009\pi$
$128$	0	0
$129$	−13.1908	−1.16139
$130$	0	0
$131$	13.0638	1.14139	0.570696	−	0.821161i	$-0.306673\pi$
$131$	13.0638	1.14139	0.570696	+	0.821161i	$0.306673\pi$
$132$	0	0
$133$	−3.44163	−0.298427
$134$	0	0
$135$	6.17932	0.531831
$136$	0	0
$137$	−3.99672	−0.341463	−0.170731	−	0.985318i	$-0.554613\pi$
$137$	−3.99672	−0.341463	−0.170731	+	0.985318i	$0.554613\pi$
$138$	0	0
$139$	−0.0245475	−0.00208209	−0.00104104	−	0.999999i	$-0.500331\pi$
$139$	−0.0245475	−0.00208209	−0.00104104	+	0.999999i	$0.500331\pi$
$140$	0	0
$141$	−9.69394	−0.816377
$142$	0	0
$143$	−1.70481	−0.142564
$144$	0	0
$145$	−3.06493	−0.254528
$146$	0	0
$147$	−9.48312	−0.782155
$148$	0	0
$149$	−22.5192	−1.84485	−0.922423	−	0.386181i	$-0.873794\pi$
$149$	−22.5192	−1.84485	−0.922423	+	0.386181i	$0.873794\pi$
$150$	0	0
$151$	−18.9657	−1.54341	−0.771703	−	0.635983i	$-0.780595\pi$
$151$	−18.9657	−1.54341	−0.771703	+	0.635983i	$0.780595\pi$
$152$	0	0
$153$	1.41644	0.114512
$154$	0	0
$155$	0.0248343	0.00199474
$156$	0	0
$157$	20.2221	1.61390	0.806951	−	0.590619i	$-0.201116\pi$
$157$	20.2221	1.61390	0.806951	+	0.590619i	$0.201116\pi$
$158$	0	0
$159$	−12.5566	−0.995803
$160$	0	0
$161$	0.597949	0.0471249
$162$	0	0
$163$	2.22151	0.174002	0.0870011	−	0.996208i	$-0.472272\pi$
$163$	2.22151	0.174002	0.0870011	+	0.996208i	$0.472272\pi$
$164$	0	0
$165$	1.37249	0.106848
$166$	0	0
$167$	−17.3196	−1.34023	−0.670117	−	0.742256i	$-0.733756\pi$
$167$	−17.3196	−1.34023	−0.670117	+	0.742256i	$0.733756\pi$
$168$	0	0
$169$	−8.87003	−0.682310
$170$	0	0
$171$	−3.34291	−0.255639
$172$	0	0
$173$	3.64302	0.276974	0.138487	−	0.990364i	$-0.455776\pi$
$173$	3.64302	0.276974	0.138487	+	0.990364i	$0.455776\pi$
$174$	0	0
$175$	3.03619	0.229514
$176$	0	0
$177$	−2.70936	−0.203648
$178$	0	0
$179$	−13.8085	−1.03210	−0.516050	−	0.856559i	$-0.672598\pi$
$179$	−13.8085	−1.03210	−0.516050	+	0.856559i	$0.672598\pi$
$180$	0	0
$181$	−7.95829	−0.591535	−0.295768	−	0.955260i	$-0.595575\pi$
$181$	−7.95829	−0.591535	−0.295768	+	0.955260i	$0.595575\pi$
$182$	0	0
$183$	−12.0527	−0.890958
$184$	0	0
$185$	−1.98050	−0.145609
$186$	0	0
$187$	1.52943	0.111843
$188$	0	0
$189$	4.50429	0.327639
$190$	0	0
$191$	−4.49411	−0.325182	−0.162591	−	0.986694i	$-0.551985\pi$
$191$	−4.49411	−0.325182	−0.162591	+	0.986694i	$0.551985\pi$
$192$	0	0
$193$	26.8118	1.92996	0.964979	−	0.262328i	$-0.0844903\pi$
$193$	26.8118	1.92996	0.964979	+	0.262328i	$0.0844903\pi$
$194$	0	0
$195$	−3.32489	−0.238100
$196$	0	0
$197$	−7.67895	−0.547103	−0.273551	−	0.961857i	$-0.588198\pi$
$197$	−7.67895	−0.547103	−0.273551	+	0.961857i	$0.588198\pi$
$198$	0	0
$199$	−1.27465	−0.0903578	−0.0451789	−	0.998979i	$-0.514386\pi$
$199$	−1.27465	−0.0903578	−0.0451789	+	0.998979i	$0.514386\pi$
$200$	0	0
$201$	17.3687	1.22509
$202$	0	0
$203$	−2.23411	−0.156804
$204$	0	0
$205$	−7.58168	−0.529527
$206$	0	0
$207$	0.580798	0.0403683
$208$	0	0
$209$	−3.60958	−0.249680
$210$	0	0
$211$	16.0159	1.10258	0.551291	−	0.834313i	$-0.314135\pi$
$211$	16.0159	1.10258	0.551291	+	0.834313i	$0.314135\pi$
$212$	0	0
$213$	−1.77322	−0.121499
$214$	0	0
$215$	9.70775	0.662063
$216$	0	0
$217$	0.0181025	0.00122888
$218$	0	0
$219$	5.81675	0.393060
$220$	0	0
$221$	−3.70510	−0.249232
$222$	0	0
$223$	20.3323	1.36155	0.680775	−	0.732492i	$-0.261643\pi$
$223$	20.3323	1.36155	0.680775	+	0.732492i	$0.261643\pi$
$224$	0	0
$225$	2.94911	0.196607
$226$	0	0
$227$	2.56302	0.170113	0.0850567	−	0.996376i	$-0.472893\pi$
$227$	2.56302	0.170113	0.0850567	+	0.996376i	$0.472893\pi$
$228$	0	0
$229$	−10.5606	−0.697865	−0.348932	−	0.937148i	$-0.613456\pi$
$229$	−10.5606	−0.697865	−0.348932	+	0.937148i	$0.613456\pi$
$230$	0	0
$231$	1.00044	0.0658244
$232$	0	0
$233$	19.1168	1.25238	0.626192	−	0.779669i	$-0.284613\pi$
$233$	19.1168	1.25238	0.626192	+	0.779669i	$0.284613\pi$
$234$	0	0
$235$	7.13424	0.465387
$236$	0	0
$237$	−12.2680	−0.796889
$238$	0	0
$239$	−8.24459	−0.533298	−0.266649	−	0.963794i	$-0.585916\pi$
$239$	−8.24459	−0.533298	−0.266649	+	0.963794i	$0.585916\pi$
$240$	0	0
$241$	8.11563	0.522774	0.261387	−	0.965234i	$-0.415820\pi$
$241$	8.11563	0.522774	0.261387	+	0.965234i	$0.415820\pi$
$242$	0	0
$243$	7.85023	0.503592
$244$	0	0
$245$	6.97909	0.445878
$246$	0	0
$247$	8.74433	0.556388
$248$	0	0
$249$	−7.63426	−0.483802
$250$	0	0
$251$	3.53539	0.223152	0.111576	−	0.993756i	$-0.464410\pi$
$251$	3.53539	0.223152	0.111576	+	0.993756i	$0.464410\pi$
$252$	0	0
$253$	0.627129	0.0394273
$254$	0	0
$255$	2.98284	0.186793
$256$	0	0
$257$	−21.3707	−1.33307	−0.666535	−	0.745474i	$-0.732223\pi$
$257$	−21.3707	−1.33307	−0.666535	+	0.745474i	$0.732223\pi$
$258$	0	0
$259$	−1.44364	−0.0897036
$260$	0	0
$261$	−2.17003	−0.134322
$262$	0	0
$263$	2.15569	0.132925	0.0664627	−	0.997789i	$-0.478829\pi$
$263$	2.15569	0.132925	0.0664627	+	0.997789i	$0.478829\pi$
$264$	0	0
$265$	9.24101	0.567671
$266$	0	0
$267$	−14.2801	−0.873929
$268$	0	0
$269$	−5.55734	−0.338837	−0.169419	−	0.985544i	$-0.554189\pi$
$269$	−5.55734	−0.338837	−0.169419	+	0.985544i	$0.554189\pi$
$270$	0	0
$271$	20.2990	1.23307	0.616537	−	0.787326i	$-0.288535\pi$
$271$	20.2990	1.23307	0.616537	+	0.787326i	$0.288535\pi$
$272$	0	0
$273$	−2.42361	−0.146683
$274$	0	0
$275$	3.18436	0.192024
$276$	0	0
$277$	−24.2147	−1.45492	−0.727461	−	0.686149i	$-0.759300\pi$
$277$	−24.2147	−1.45492	−0.727461	+	0.686149i	$0.759300\pi$
$278$	0	0
$279$	0.0175833	0.00105268
$280$	0	0
$281$	32.0352	1.91106	0.955530	−	0.294894i	$-0.0952842\pi$
$281$	32.0352	1.91106	0.955530	+	0.294894i	$0.0952842\pi$
$282$	0	0
$283$	24.0778	1.43128	0.715638	−	0.698472i	$-0.246136\pi$
$283$	24.0778	1.43128	0.715638	+	0.698472i	$0.246136\pi$
$284$	0	0
$285$	−7.03975	−0.416999
$286$	0	0
$287$	−5.52650	−0.326219
$288$	0	0
$289$	−13.6761	−0.804474
$290$	0	0
$291$	−1.25732	−0.0737056
$292$	0	0
$293$	−0.373309	−0.0218089	−0.0109045	−	0.999941i	$-0.503471\pi$
$293$	−0.373309	−0.0218089	−0.0109045	+	0.999941i	$0.503471\pi$
$294$	0	0
$295$	1.99395	0.116092
$296$	0	0
$297$	4.72410	0.274120
$298$	0	0
$299$	−1.51924	−0.0878599
$300$	0	0
$301$	7.07626	0.407869
$302$	0	0
$303$	−21.2444	−1.22046
$304$	0	0
$305$	8.87013	0.507902
$306$	0	0
$307$	28.5239	1.62795	0.813974	−	0.580901i	$-0.197300\pi$
$307$	28.5239	1.62795	0.813974	+	0.580901i	$0.197300\pi$
$308$	0	0
$309$	−25.4164	−1.44589
$310$	0	0
$311$	−21.2504	−1.20500	−0.602500	−	0.798119i	$-0.705829\pi$
$311$	−21.2504	−1.20500	−0.602500	+	0.798119i	$0.705829\pi$
$312$	0	0
$313$	−14.0831	−0.796022	−0.398011	−	0.917381i	$-0.630299\pi$
$313$	−14.0831	−0.796022	−0.398011	+	0.917381i	$0.630299\pi$
$314$	0	0
$315$	−0.681881	−0.0384196
$316$	0	0
$317$	−27.0436	−1.51892	−0.759459	−	0.650555i	$-0.774536\pi$
$317$	−27.0436	−1.51892	−0.759459	+	0.650555i	$0.774536\pi$
$318$	0	0
$319$	−2.34314	−0.131191
$320$	0	0
$321$	0.717491	0.0400464
$322$	0	0
$323$	−7.84476	−0.436494
$324$	0	0
$325$	−7.71421	−0.427908
$326$	0	0
$327$	25.7732	1.42526
$328$	0	0
$329$	5.20035	0.286705
$330$	0	0
$331$	8.83232	0.485468	0.242734	−	0.970093i	$-0.421956\pi$
$331$	8.83232	0.485468	0.242734	+	0.970093i	$0.421956\pi$
$332$	0	0
$333$	−1.40224	−0.0768421
$334$	0	0
$335$	−12.7825	−0.698381
$336$	0	0
$337$	5.01026	0.272926	0.136463	−	0.990645i	$-0.456426\pi$
$337$	5.01026	0.272926	0.136463	+	0.990645i	$0.456426\pi$
$338$	0	0
$339$	−8.90234	−0.483509
$340$	0	0
$341$	0.0189859	0.00102814
$342$	0	0
$343$	10.6862	0.577003
$344$	0	0
$345$	1.22309	0.0658487
$346$	0	0
$347$	18.1935	0.976679	0.488339	−	0.872654i	$-0.337603\pi$
$347$	18.1935	0.976679	0.488339	+	0.872654i	$0.337603\pi$
$348$	0	0
$349$	−19.8229	−1.06109	−0.530547	−	0.847656i	$-0.678013\pi$
$349$	−19.8229	−1.06109	−0.530547	+	0.847656i	$0.678013\pi$
$350$	0	0
$351$	−11.4443	−0.610851
$352$	0	0
$353$	−25.7268	−1.36930	−0.684650	−	0.728872i	$-0.740045\pi$
$353$	−25.7268	−1.36930	−0.684650	+	0.728872i	$0.740045\pi$
$354$	0	0
$355$	1.30500	0.0692622
$356$	0	0
$357$	2.17428	0.115075
$358$	0	0
$359$	32.6690	1.72421	0.862103	−	0.506733i	$-0.169147\pi$
$359$	32.6690	1.72421	0.862103	+	0.506733i	$0.169147\pi$
$360$	0	0
$361$	−0.485748	−0.0255657
$362$	0	0
$363$	−15.3518	−0.805758
$364$	0	0
$365$	−4.28083	−0.224069
$366$	0	0
$367$	6.33622	0.330748	0.165374	−	0.986231i	$-0.447117\pi$
$367$	6.33622	0.330748	0.165374	+	0.986231i	$0.447117\pi$
$368$	0	0
$369$	−5.36799	−0.279446
$370$	0	0
$371$	6.73604	0.349718
$372$	0	0
$373$	2.77574	0.143722	0.0718612	−	0.997415i	$-0.477106\pi$
$373$	2.77574	0.143722	0.0718612	+	0.997415i	$0.477106\pi$
$374$	0	0
$375$	14.3908	0.743139
$376$	0	0
$377$	5.67633	0.292346
$378$	0	0
$379$	1.36981	0.0703624	0.0351812	−	0.999381i	$-0.488799\pi$
$379$	1.36981	0.0703624	0.0351812	+	0.999381i	$0.488799\pi$
$380$	0	0
$381$	0.315733	0.0161755
$382$	0	0
$383$	21.9702	1.12262	0.561312	−	0.827604i	$-0.310297\pi$
$383$	21.9702	1.12262	0.561312	+	0.827604i	$0.310297\pi$
$384$	0	0
$385$	−0.736275	−0.0375241
$386$	0	0
$387$	6.87329	0.349389
$388$	0	0
$389$	8.99985	0.456310	0.228155	−	0.973625i	$-0.426731\pi$
$389$	8.99985	0.456310	0.228155	+	0.973625i	$0.426731\pi$
$390$	0	0
$391$	1.36295	0.0689273
$392$	0	0
$393$	19.4782	0.982545
$394$	0	0
$395$	9.02858	0.454277
$396$	0	0
$397$	−6.02280	−0.302276	−0.151138	−	0.988513i	$-0.548294\pi$
$397$	−6.02280	−0.302276	−0.151138	+	0.988513i	$0.548294\pi$
$398$	0	0
$399$	−5.13147	−0.256895
$400$	0	0
$401$	21.1265	1.05501	0.527503	−	0.849553i	$-0.323128\pi$
$401$	21.1265	1.05501	0.527503	+	0.849553i	$0.323128\pi$
$402$	0	0
$403$	−0.0459939	−0.00229112
$404$	0	0
$405$	6.65586	0.330733
$406$	0	0
$407$	−1.51409	−0.0750508
$408$	0	0
$409$	−31.2674	−1.54607	−0.773036	−	0.634363i	$-0.781263\pi$
$409$	−31.2674	−1.54607	−0.773036	+	0.634363i	$0.781263\pi$
$410$	0	0
$411$	−5.95911	−0.293941
$412$	0	0
$413$	1.45345	0.0715195
$414$	0	0
$415$	5.61842	0.275798
$416$	0	0
$417$	−0.0366003	−0.00179233
$418$	0	0
$419$	−3.24693	−0.158623	−0.0793115	−	0.996850i	$-0.525272\pi$
$419$	−3.24693	−0.158623	−0.0793115	+	0.996850i	$0.525272\pi$
$420$	0	0
$421$	17.2288	0.839680	0.419840	−	0.907598i	$-0.362086\pi$
$421$	17.2288	0.839680	0.419840	+	0.907598i	$0.362086\pi$
$422$	0	0
$423$	5.05120	0.245598
$424$	0	0
$425$	6.92062	0.335699
$426$	0	0
$427$	6.46569	0.312897
$428$	0	0
$429$	−2.54188	−0.122723
$430$	0	0
$431$	16.4443	0.792095	0.396047	−	0.918230i	$-0.370382\pi$
$431$	16.4443	0.792095	0.396047	+	0.918230i	$0.370382\pi$
$432$	0	0
$433$	−13.2198	−0.635305	−0.317652	−	0.948207i	$-0.602895\pi$
$433$	−13.2198	−0.635305	−0.317652	+	0.948207i	$0.602895\pi$
$434$	0	0
$435$	−4.56981	−0.219106
$436$	0	0
$437$	−3.21667	−0.153874
$438$	0	0
$439$	20.5145	0.979106	0.489553	−	0.871974i	$-0.337160\pi$
$439$	20.5145	0.979106	0.489553	+	0.871974i	$0.337160\pi$
$440$	0	0
$441$	4.94135	0.235302
$442$	0	0
$443$	34.5081	1.63953	0.819764	−	0.572702i	$-0.194105\pi$
$443$	34.5081	1.63953	0.819764	+	0.572702i	$0.194105\pi$
$444$	0	0
$445$	10.5094	0.498195
$446$	0	0
$447$	−33.5762	−1.58810
$448$	0	0
$449$	27.1657	1.28203	0.641014	−	0.767529i	$-0.278514\pi$
$449$	27.1657	1.28203	0.641014	+	0.767529i	$0.278514\pi$
$450$	0	0
$451$	−5.79620	−0.272932
$452$	0	0
$453$	−28.2779	−1.32861
$454$	0	0
$455$	1.78365	0.0836188
$456$	0	0
$457$	−18.1314	−0.848153	−0.424077	−	0.905626i	$-0.639401\pi$
$457$	−18.1314	−0.848153	−0.424077	+	0.905626i	$0.639401\pi$
$458$	0	0
$459$	10.2670	0.479220
$460$	0	0
$461$	−24.1606	−1.12527	−0.562636	−	0.826705i	$-0.690213\pi$
$461$	−24.1606	−1.12527	−0.562636	+	0.826705i	$0.690213\pi$
$462$	0	0
$463$	−25.8126	−1.19961	−0.599806	−	0.800146i	$-0.704755\pi$
$463$	−25.8126	−1.19961	−0.599806	+	0.800146i	$0.704755\pi$
$464$	0	0
$465$	0.0370281	0.00171714
$466$	0	0
$467$	−23.9669	−1.10906	−0.554528	−	0.832165i	$-0.687101\pi$
$467$	−23.9669	−1.10906	−0.554528	+	0.832165i	$0.687101\pi$
$468$	0	0
$469$	−9.31751	−0.430243
$470$	0	0
$471$	30.1512	1.38930
$472$	0	0
$473$	7.42158	0.341245
$474$	0	0
$475$	−16.3332	−0.749419
$476$	0	0
$477$	6.54284	0.299576
$478$	0	0
$479$	−2.97853	−0.136092	−0.0680462	−	0.997682i	$-0.521677\pi$
$479$	−2.97853	−0.136092	−0.0680462	+	0.997682i	$0.521677\pi$
$480$	0	0
$481$	3.66794	0.167244
$482$	0	0
$483$	0.891543	0.0405666
$484$	0	0
$485$	0.925325	0.0420169
$486$	0	0
$487$	31.0404	1.40657	0.703287	−	0.710906i	$-0.251715\pi$
$487$	31.0404	1.40657	0.703287	+	0.710906i	$0.251715\pi$
$488$	0	0
$489$	3.31228	0.149786
$490$	0	0
$491$	−6.52070	−0.294275	−0.147138	−	0.989116i	$-0.547006\pi$
$491$	−6.52070	−0.294275	−0.147138	+	0.989116i	$0.547006\pi$
$492$	0	0
$493$	−5.09238	−0.229349
$494$	0	0
$495$	−0.715157	−0.0321439
$496$	0	0
$497$	0.951252	0.0426695
$498$	0	0
$499$	34.8087	1.55825	0.779125	−	0.626869i	$-0.215664\pi$
$499$	34.8087	1.55825	0.779125	+	0.626869i	$0.215664\pi$
$500$	0	0
$501$	−25.8236	−1.15371
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	15.6348	0.695739
$506$	0	0
$507$	−13.2252	−0.587353
$508$	0	0
$509$	8.27277	0.366684	0.183342	−	0.983049i	$-0.441308\pi$
$509$	8.27277	0.366684	0.183342	+	0.983049i	$0.441308\pi$
$510$	0	0
$511$	−3.12042	−0.138039
$512$	0	0
$513$	−24.2308	−1.06982
$514$	0	0
$515$	18.7052	0.824248
$516$	0	0
$517$	5.45413	0.239873
$518$	0	0
$519$	5.43175	0.238427
$520$	0	0
$521$	17.4742	0.765559	0.382779	−	0.923840i	$-0.374967\pi$
$521$	17.4742	0.765559	0.382779	+	0.923840i	$0.374967\pi$
$522$	0	0
$523$	3.26300	0.142681	0.0713404	−	0.997452i	$-0.477272\pi$
$523$	3.26300	0.142681	0.0713404	+	0.997452i	$0.477272\pi$
$524$	0	0
$525$	4.52697	0.197573
$526$	0	0
$527$	0.0412623	0.00179741
$528$	0	0
$529$	−22.4411	−0.975702
$530$	0	0
$531$	1.41176	0.0612651
$532$	0	0
$533$	14.0415	0.608204
$534$	0	0
$535$	−0.528036	−0.0228290
$536$	0	0
$537$	−20.5886	−0.888462
$538$	0	0
$539$	5.33552	0.229817
$540$	0	0
$541$	−36.8792	−1.58556	−0.792781	−	0.609507i	$-0.791367\pi$
$541$	−36.8792	−1.58556	−0.792781	+	0.609507i	$0.791367\pi$
$542$	0	0
$543$	−11.8658	−0.509212
$544$	0	0
$545$	−18.9677	−0.812489
$546$	0	0
$547$	3.57057	0.152667	0.0763334	−	0.997082i	$-0.475679\pi$
$547$	3.57057	0.152667	0.0763334	+	0.997082i	$0.475679\pi$
$548$	0	0
$549$	6.28024	0.268034
$550$	0	0
$551$	12.0184	0.512002
$552$	0	0
$553$	6.58119	0.279861
$554$	0	0
$555$	−2.95293	−0.125345
$556$	0	0
$557$	−30.2578	−1.28206	−0.641032	−	0.767514i	$-0.721494\pi$
$557$	−30.2578	−1.28206	−0.641032	+	0.767514i	$0.721494\pi$
$558$	0	0
$559$	−17.9790	−0.760432
$560$	0	0
$561$	2.28039	0.0962780
$562$	0	0
$563$	−22.4953	−0.948062	−0.474031	−	0.880508i	$-0.657202\pi$
$563$	−22.4953	−0.948062	−0.474031	+	0.880508i	$0.657202\pi$
$564$	0	0
$565$	6.55166	0.275631
$566$	0	0
$567$	4.85165	0.203750
$568$	0	0
$569$	−36.4374	−1.52753	−0.763767	−	0.645492i	$-0.776652\pi$
$569$	−36.4374	−1.52753	−0.763767	+	0.645492i	$0.776652\pi$
$570$	0	0
$571$	23.2768	0.974104	0.487052	−	0.873373i	$-0.338072\pi$
$571$	23.2768	0.974104	0.487052	+	0.873373i	$0.338072\pi$
$572$	0	0
$573$	−6.70072	−0.279927
$574$	0	0
$575$	2.83773	0.118342
$576$	0	0
$577$	15.7394	0.655241	0.327621	−	0.944809i	$-0.393753\pi$
$577$	15.7394	0.655241	0.327621	+	0.944809i	$0.393753\pi$
$578$	0	0
$579$	39.9765	1.66137
$580$	0	0
$581$	4.09543	0.169907
$582$	0	0
$583$	7.06476	0.292593
$584$	0	0
$585$	1.73249	0.0716297
$586$	0	0
$587$	19.7718	0.816068	0.408034	−	0.912967i	$-0.366214\pi$
$587$	19.7718	0.816068	0.408034	+	0.912967i	$0.366214\pi$
$588$	0	0
$589$	−0.0973823	−0.00401257
$590$	0	0
$591$	−11.4493	−0.470963
$592$	0	0
$593$	26.9384	1.10623	0.553114	−	0.833106i	$-0.313439\pi$
$593$	26.9384	1.10623	0.553114	+	0.833106i	$0.313439\pi$
$594$	0	0
$595$	−1.60016	−0.0656001
$596$	0	0
$597$	−1.90051	−0.0777827
$598$	0	0
$599$	−42.4760	−1.73552	−0.867761	−	0.496982i	$-0.834442\pi$
$599$	−42.4760	−1.73552	−0.867761	+	0.496982i	$0.834442\pi$
$600$	0	0
$601$	33.6101	1.37099	0.685493	−	0.728080i	$-0.259587\pi$
$601$	33.6101	1.37099	0.685493	+	0.728080i	$0.259587\pi$
$602$	0	0
$603$	−9.05027	−0.368555
$604$	0	0
$605$	11.2981	0.459333
$606$	0	0
$607$	11.3897	0.462295	0.231147	−	0.972919i	$-0.425752\pi$
$607$	11.3897	0.462295	0.231147	+	0.972919i	$0.425752\pi$
$608$	0	0
$609$	−3.33107	−0.134982
$610$	0	0
$611$	−13.2128	−0.534533
$612$	0	0
$613$	1.30427	0.0526789	0.0263395	−	0.999653i	$-0.491615\pi$
$613$	1.30427	0.0526789	0.0263395	+	0.999653i	$0.491615\pi$
$614$	0	0
$615$	−11.3043	−0.455833
$616$	0	0
$617$	−18.4802	−0.743985	−0.371992	−	0.928236i	$-0.621325\pi$
$617$	−18.4802	−0.743985	−0.371992	+	0.928236i	$0.621325\pi$
$618$	0	0
$619$	−31.5751	−1.26911	−0.634554	−	0.772878i	$-0.718816\pi$
$619$	−31.5751	−1.26911	−0.634554	+	0.772878i	$0.718816\pi$
$620$	0	0
$621$	4.20987	0.168936
$622$	0	0
$623$	7.66063	0.306917
$624$	0	0
$625$	8.38875	0.335550
$626$	0	0
$627$	−5.38189	−0.214932
$628$	0	0
$629$	−3.29060	−0.131205
$630$	0	0
$631$	42.9916	1.71147	0.855734	−	0.517416i	$-0.173106\pi$
$631$	42.9916	1.71147	0.855734	+	0.517416i	$0.173106\pi$
$632$	0	0
$633$	23.8798	0.949137
$634$	0	0
$635$	−0.232363	−0.00922105
$636$	0	0
$637$	−12.9255	−0.512126
$638$	0	0
$639$	0.923968	0.0365516
$640$	0	0
$641$	−20.0209	−0.790779	−0.395390	−	0.918513i	$-0.629390\pi$
$641$	−20.0209	−0.790779	−0.395390	+	0.918513i	$0.629390\pi$
$642$	0	0
$643$	−28.3527	−1.11812	−0.559060	−	0.829127i	$-0.688838\pi$
$643$	−28.3527	−1.11812	−0.559060	+	0.829127i	$0.688838\pi$
$644$	0	0
$645$	14.4743	0.569924
$646$	0	0
$647$	−25.7284	−1.01149	−0.505744	−	0.862683i	$-0.668782\pi$
$647$	−25.7284	−1.01149	−0.505744	+	0.862683i	$0.668782\pi$
$648$	0	0
$649$	1.52438	0.0598370
$650$	0	0
$651$	0.0269908	0.00105785
$652$	0	0
$653$	37.4989	1.46744	0.733722	−	0.679449i	$-0.237781\pi$
$653$	37.4989	1.46744	0.733722	+	0.679449i	$0.237781\pi$
$654$	0	0
$655$	−14.3349	−0.560113
$656$	0	0
$657$	−3.03092	−0.118247
$658$	0	0
$659$	−21.3066	−0.829988	−0.414994	−	0.909824i	$-0.636216\pi$
$659$	−21.3066	−0.829988	−0.414994	+	0.909824i	$0.636216\pi$
$660$	0	0
$661$	−23.3872	−0.909656	−0.454828	−	0.890579i	$-0.650299\pi$
$661$	−23.3872	−0.909656	−0.454828	+	0.890579i	$0.650299\pi$
$662$	0	0
$663$	−5.52431	−0.214546
$664$	0	0
$665$	3.77650	0.146446
$666$	0	0
$667$	−2.08808	−0.0808508
$668$	0	0
$669$	30.3155	1.17206
$670$	0	0
$671$	6.78123	0.261786
$672$	0	0
$673$	20.6684	0.796706	0.398353	−	0.917232i	$-0.369582\pi$
$673$	20.6684	0.796706	0.398353	+	0.917232i	$0.369582\pi$
$674$	0	0
$675$	21.3763	0.822776
$676$	0	0
$677$	26.0320	1.00049	0.500246	−	0.865883i	$-0.333243\pi$
$677$	26.0320	1.00049	0.500246	+	0.865883i	$0.333243\pi$
$678$	0	0
$679$	0.674496	0.0258848
$680$	0	0
$681$	3.82147	0.146439
$682$	0	0
$683$	−13.3035	−0.509043	−0.254522	−	0.967067i	$-0.581918\pi$
$683$	−13.3035	−0.509043	−0.254522	+	0.967067i	$0.581918\pi$
$684$	0	0
$685$	4.38560	0.167565
$686$	0	0
$687$	−15.7459	−0.600743
$688$	0	0
$689$	−17.1146	−0.652015
$690$	0	0
$691$	−11.2943	−0.429657	−0.214828	−	0.976652i	$-0.568919\pi$
$691$	−11.2943	−0.429657	−0.214828	+	0.976652i	$0.568919\pi$
$692$	0	0
$693$	−0.521299	−0.0198025
$694$	0	0
$695$	0.0269359	0.00102174
$696$	0	0
$697$	−12.5970	−0.477144
$698$	0	0
$699$	28.5032	1.07809
$700$	0	0
$701$	−6.64761	−0.251077	−0.125538	−	0.992089i	$-0.540066\pi$
$701$	−6.64761	−0.251077	−0.125538	+	0.992089i	$0.540066\pi$
$702$	0	0
$703$	7.76608	0.292903
$704$	0	0
$705$	10.6372	0.400619
$706$	0	0
$707$	11.3966	0.428615
$708$	0	0
$709$	−25.8987	−0.972648	−0.486324	−	0.873779i	$-0.661662\pi$
$709$	−25.8987	−0.972648	−0.486324	+	0.873779i	$0.661662\pi$
$710$	0	0
$711$	6.39243	0.239735
$712$	0	0
$713$	0.0169192	0.000633630 0
$714$	0	0
$715$	1.87069	0.0699600
$716$	0	0
$717$	−12.2927	−0.459079
$718$	0	0
$719$	−16.3355	−0.609210	−0.304605	−	0.952479i	$-0.598525\pi$
$719$	−16.3355	−0.609210	−0.304605	+	0.952479i	$0.598525\pi$
$720$	0	0
$721$	13.6347	0.507784
$722$	0	0
$723$	12.1004	0.450020
$724$	0	0
$725$	−10.6026	−0.393771
$726$	0	0
$727$	−19.6052	−0.727117	−0.363559	−	0.931571i	$-0.618438\pi$
$727$	−19.6052	−0.727117	−0.363559	+	0.931571i	$0.618438\pi$
$728$	0	0
$729$	29.9017	1.10747
$730$	0	0
$731$	16.1294	0.596569
$732$	0	0
$733$	−34.6268	−1.27897	−0.639484	−	0.768804i	$-0.720852\pi$
$733$	−34.6268	−1.27897	−0.639484	+	0.768804i	$0.720852\pi$
$734$	0	0
$735$	10.4058	0.383825
$736$	0	0
$737$	−9.77221	−0.359964
$738$	0	0
$739$	51.1192	1.88045	0.940225	−	0.340554i	$-0.110615\pi$
$739$	51.1192	1.88045	0.940225	+	0.340554i	$0.110615\pi$
$740$	0	0
$741$	13.0378	0.478956
$742$	0	0
$743$	18.3318	0.672530	0.336265	−	0.941767i	$-0.390836\pi$
$743$	18.3318	0.672530	0.336265	+	0.941767i	$0.390836\pi$
$744$	0	0
$745$	24.7104	0.905317
$746$	0	0
$747$	3.97797	0.145546
$748$	0	0
$749$	−0.384901	−0.0140640
$750$	0	0
$751$	19.2142	0.701136	0.350568	−	0.936537i	$-0.385989\pi$
$751$	19.2142	0.701136	0.350568	+	0.936537i	$0.385989\pi$
$752$	0	0
$753$	5.27127	0.192096
$754$	0	0
$755$	20.8111	0.757392
$756$	0	0
$757$	−25.5974	−0.930352	−0.465176	−	0.885218i	$-0.654009\pi$
$757$	−25.5974	−0.930352	−0.465176	+	0.885218i	$0.654009\pi$
$758$	0	0
$759$	0.935051	0.0339402
$760$	0	0
$761$	28.0281	1.01602	0.508009	−	0.861352i	$-0.330382\pi$
$761$	28.0281	1.01602	0.508009	+	0.861352i	$0.330382\pi$
$762$	0	0
$763$	−13.8261	−0.500539
$764$	0	0
$765$	−1.55426	−0.0561945
$766$	0	0
$767$	−3.69285	−0.133341
$768$	0	0
$769$	50.7564	1.83032	0.915162	−	0.403087i	$-0.132063\pi$
$769$	50.7564	1.83032	0.915162	+	0.403087i	$0.132063\pi$
$770$	0	0
$771$	−31.8638	−1.14755
$772$	0	0
$773$	−55.0059	−1.97842	−0.989212	−	0.146494i	$-0.953201\pi$
$773$	−55.0059	−1.97842	−0.989212	+	0.146494i	$0.953201\pi$
$774$	0	0
$775$	0.0859103	0.00308599
$776$	0	0
$777$	−2.15247	−0.0772196
$778$	0	0
$779$	29.7299	1.06518
$780$	0	0
$781$	0.997674	0.0356996
$782$	0	0
$783$	−15.7293	−0.562119
$784$	0	0
$785$	−22.1898	−0.791986
$786$	0	0
$787$	6.78536	0.241872	0.120936	−	0.992660i	$-0.461410\pi$
$787$	6.78536	0.241872	0.120936	+	0.992660i	$0.461410\pi$
$788$	0	0
$789$	3.21414	0.114426
$790$	0	0
$791$	4.77570	0.169804
$792$	0	0
$793$	−16.4277	−0.583366
$794$	0	0
$795$	13.7784	0.488668
$796$	0	0
$797$	−42.3449	−1.49993	−0.749967	−	0.661475i	$-0.769931\pi$
$797$	−42.3449	−1.49993	−0.749967	+	0.661475i	$0.769931\pi$
$798$	0	0
$799$	11.8536	0.419349
$800$	0	0
$801$	7.44091	0.262911
$802$	0	0
$803$	−3.27270	−0.115491
$804$	0	0
$805$	−0.656129	−0.0231255
$806$	0	0
$807$	−8.28601	−0.291681
$808$	0	0
$809$	−25.3991	−0.892984	−0.446492	−	0.894788i	$-0.647327\pi$
$809$	−25.3991	−0.892984	−0.446492	+	0.894788i	$0.647327\pi$
$810$	0	0
$811$	20.0288	0.703306	0.351653	−	0.936130i	$-0.385620\pi$
$811$	20.0288	0.703306	0.351653	+	0.936130i	$0.385620\pi$
$812$	0	0
$813$	30.2658	1.06147
$814$	0	0
$815$	−2.43767	−0.0853877
$816$	0	0
$817$	−38.0668	−1.33179
$818$	0	0
$819$	1.26286	0.0441280
$820$	0	0
$821$	12.1634	0.424506	0.212253	−	0.977215i	$-0.431920\pi$
$821$	12.1634	0.424506	0.212253	+	0.977215i	$0.431920\pi$
$822$	0	0
$823$	−40.1250	−1.39867	−0.699335	−	0.714794i	$-0.746520\pi$
$823$	−40.1250	−1.39867	−0.699335	+	0.714794i	$0.746520\pi$
$824$	0	0
$825$	4.74789	0.165300
$826$	0	0
$827$	−37.2441	−1.29510	−0.647552	−	0.762022i	$-0.724207\pi$
$827$	−37.2441	−1.29510	−0.647552	+	0.762022i	$0.724207\pi$
$828$	0	0
$829$	25.9066	0.899774	0.449887	−	0.893086i	$-0.351464\pi$
$829$	25.9066	0.899774	0.449887	+	0.893086i	$0.351464\pi$
$830$	0	0
$831$	−36.1042	−1.25244
$832$	0	0
$833$	11.5958	0.401770
$834$	0	0
$835$	19.0048	0.657690
$836$	0	0
$837$	0.127451	0.00440534
$838$	0	0
$839$	5.02724	0.173560	0.0867798	−	0.996228i	$-0.472342\pi$
$839$	5.02724	0.173560	0.0867798	+	0.996228i	$0.472342\pi$
$840$	0	0
$841$	−21.1983	−0.730976
$842$	0	0
$843$	47.7646	1.64510
$844$	0	0
$845$	9.73309	0.334829
$846$	0	0
$847$	8.23551	0.282976
$848$	0	0
$849$	35.9000	1.23209
$850$	0	0
$851$	−1.34928	−0.0462527
$852$	0	0
$853$	26.7681	0.916524	0.458262	−	0.888817i	$-0.348472\pi$
$853$	26.7681	0.916524	0.458262	+	0.888817i	$0.348472\pi$
$854$	0	0
$855$	3.66818	0.125449
$856$	0	0
$857$	48.8760	1.66957	0.834787	−	0.550574i	$-0.185591\pi$
$857$	48.8760	1.66957	0.834787	+	0.550574i	$0.185591\pi$
$858$	0	0
$859$	1.54729	0.0527928	0.0263964	−	0.999652i	$-0.491597\pi$
$859$	1.54729	0.0527928	0.0263964	+	0.999652i	$0.491597\pi$
$860$	0	0
$861$	−8.24003	−0.280819
$862$	0	0
$863$	−3.76724	−0.128238	−0.0641192	−	0.997942i	$-0.520424\pi$
$863$	−3.76724	−0.128238	−0.0641192	+	0.997942i	$0.520424\pi$
$864$	0	0
$865$	−3.99749	−0.135919
$866$	0	0
$867$	−20.3910	−0.692516
$868$	0	0
$869$	6.90236	0.234146
$870$	0	0
$871$	23.6735	0.802146
$872$	0	0
$873$	0.655150	0.0221735
$874$	0	0
$875$	−7.72002	−0.260984
$876$	0	0
$877$	−4.29648	−0.145082	−0.0725410	−	0.997365i	$-0.523111\pi$
$877$	−4.29648	−0.145082	−0.0725410	+	0.997365i	$0.523111\pi$
$878$	0	0
$879$	−0.556604	−0.0187738
$880$	0	0
$881$	−27.7084	−0.933519	−0.466760	−	0.884384i	$-0.654579\pi$
$881$	−27.7084	−0.933519	−0.466760	+	0.884384i	$0.654579\pi$
$882$	0	0
$883$	37.4492	1.26027	0.630134	−	0.776487i	$-0.283000\pi$
$883$	37.4492	1.26027	0.630134	+	0.776487i	$0.283000\pi$
$884$	0	0
$885$	2.97298	0.0999357
$886$	0	0
$887$	0.219360	0.00736539	0.00368270	−	0.999993i	$-0.498828\pi$
$887$	0.219360	0.00736539	0.00368270	+	0.999993i	$0.498828\pi$
$888$	0	0
$889$	−0.169376	−0.00568070
$890$	0	0
$891$	5.08841	0.170468
$892$	0	0
$893$	−27.9753	−0.936159
$894$	0	0
$895$	15.1521	0.506480
$896$	0	0
$897$	−2.26519	−0.0756325
$898$	0	0
$899$	−0.0632152	−0.00210834
$900$	0	0
$901$	15.3540	0.511515
$902$	0	0
$903$	10.5507	0.351106
$904$	0	0
$905$	8.73264	0.290283
$906$	0	0
$907$	52.0680	1.72889	0.864444	−	0.502729i	$-0.167670\pi$
$907$	52.0680	1.72889	0.864444	+	0.502729i	$0.167670\pi$
$908$	0	0
$909$	11.0698	0.367161
$910$	0	0
$911$	41.6078	1.37853	0.689263	−	0.724511i	$-0.257934\pi$
$911$	41.6078	1.37853	0.689263	+	0.724511i	$0.257934\pi$
$912$	0	0
$913$	4.29529	0.142153
$914$	0	0
$915$	13.2254	0.437218
$916$	0	0
$917$	−10.4492	−0.345062
$918$	0	0
$919$	54.3382	1.79245	0.896226	−	0.443597i	$-0.146298\pi$
$919$	54.3382	1.79245	0.896226	+	0.443597i	$0.146298\pi$
$920$	0	0
$921$	42.5293	1.40139
$922$	0	0
$923$	−2.41690	−0.0795532
$924$	0	0
$925$	−6.85121	−0.225266
$926$	0	0
$927$	13.2437	0.434979
$928$	0	0
$929$	31.9579	1.04851	0.524253	−	0.851563i	$-0.324345\pi$
$929$	31.9579	1.04851	0.524253	+	0.851563i	$0.324345\pi$
$930$	0	0
$931$	−27.3669	−0.896915
$932$	0	0
$933$	−31.6844	−1.03730
$934$	0	0
$935$	−1.67825	−0.0548845
$936$	0	0
$937$	−52.6379	−1.71960	−0.859802	−	0.510627i	$-0.829413\pi$
$937$	−52.6379	−1.71960	−0.859802	+	0.510627i	$0.829413\pi$
$938$	0	0
$939$	−20.9979	−0.685240
$940$	0	0
$941$	7.74283	0.252409	0.126205	−	0.992004i	$-0.459720\pi$
$941$	7.74283	0.252409	0.126205	+	0.992004i	$0.459720\pi$
$942$	0	0
$943$	−5.16527	−0.168204
$944$	0	0
$945$	−4.94256	−0.160781
$946$	0	0
$947$	−54.5052	−1.77118	−0.885591	−	0.464467i	$-0.846246\pi$
$947$	−54.5052	−1.77118	−0.885591	+	0.464467i	$0.846246\pi$
$948$	0	0
$949$	7.92822	0.257361
$950$	0	0
$951$	−40.3220	−1.30753
$952$	0	0
$953$	19.8770	0.643880	0.321940	−	0.946760i	$-0.395665\pi$
$953$	19.8770	0.643880	0.321940	+	0.946760i	$0.395665\pi$
$954$	0	0
$955$	4.93139	0.159576
$956$	0	0
$957$	−3.49362	−0.112933
$958$	0	0
$959$	3.19679	0.103230
$960$	0	0
$961$	−30.9995	−0.999983
$962$	0	0
$963$	−0.373861	−0.0120475
$964$	0	0
$965$	−29.4206	−0.947084
$966$	0	0
$967$	18.0432	0.580230	0.290115	−	0.956992i	$-0.406306\pi$
$967$	18.0432	0.580230	0.290115	+	0.956992i	$0.406306\pi$
$968$	0	0
$969$	−11.6966	−0.375747
$970$	0	0
$971$	−30.4956	−0.978649	−0.489324	−	0.872102i	$-0.662757\pi$
$971$	−30.4956	−0.978649	−0.489324	+	0.872102i	$0.662757\pi$
$972$	0	0
$973$	0.0196344	0.000629450 0
$974$	0	0
$975$	−11.5019	−0.368356
$976$	0	0
$977$	37.3960	1.19640	0.598201	−	0.801346i	$-0.295882\pi$
$977$	37.3960	1.19640	0.598201	+	0.801346i	$0.295882\pi$
$978$	0	0
$979$	8.03447	0.256783
$980$	0	0
$981$	−13.4296	−0.428773
$982$	0	0
$983$	−17.8473	−0.569239	−0.284620	−	0.958641i	$-0.591867\pi$
$983$	−17.8473	−0.569239	−0.284620	+	0.958641i	$0.591867\pi$
$984$	0	0
$985$	8.42612	0.268479
$986$	0	0
$987$	7.75374	0.246804
$988$	0	0
$989$	6.61372	0.210304
$990$	0	0
$991$	−21.9819	−0.698277	−0.349138	−	0.937071i	$-0.613526\pi$
$991$	−21.9819	−0.698277	−0.349138	+	0.937071i	$0.613526\pi$
$992$	0	0
$993$	13.1690	0.417906
$994$	0	0
$995$	1.39868	0.0443411
$996$	0	0
$997$	31.0250	0.982571	0.491286	−	0.870999i	$-0.336527\pi$
$997$	31.0250	0.982571	0.491286	+	0.870999i	$0.336527\pi$
$998$	0	0
$999$	−10.1640	−0.321574

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.d.1.19	✓	28
4.3	odd	2		8048.2.a.v.1.10		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.19	✓	28	1.1	even	1	trivial
8048.2.a.v.1.10		28	4.3	odd	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:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q+1.49100 q^{3} -1.09730 q^{5} -0.799854 q^{7} -0.776913 q^{9} +O(q^{10})\)	\(q+1.49100 q^{3} -1.09730 q^{5} -0.799854 q^{7} -0.776913 q^{9} -0.838888 q^{11} +2.03223 q^{13} -1.63608 q^{15} -1.82317 q^{17} +4.30282 q^{19} -1.19258 q^{21} -0.747572 q^{23} -3.79593 q^{25} -5.63138 q^{27} +2.79315 q^{29} -0.0226322 q^{31} -1.25078 q^{33} +0.877680 q^{35} +1.80488 q^{37} +3.03006 q^{39} +6.90939 q^{41} -8.84694 q^{43} +0.852507 q^{45} -6.50163 q^{47} -6.36023 q^{49} -2.71835 q^{51} -8.42159 q^{53} +0.920512 q^{55} +6.41551 q^{57} -1.81714 q^{59} -8.08359 q^{61} +0.621417 q^{63} -2.22997 q^{65} +11.6490 q^{67} -1.11463 q^{69} -1.18928 q^{71} +3.90124 q^{73} -5.65974 q^{75} +0.670988 q^{77} -8.22799 q^{79} -6.06567 q^{81} -5.12022 q^{83} +2.00056 q^{85} +4.16459 q^{87} -9.57753 q^{89} -1.62549 q^{91} -0.0337447 q^{93} -4.72149 q^{95} -0.843274 q^{97} +0.651742 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 28 * q + 2 * q^3 - 12 * q^5 + 18 * q^9	\( 28 q + 2 q^{3} - 12 q^{5} + 18 q^{9} - 14 q^{11} - 31 q^{13} - 2 q^{15} - 9 q^{17} - 8 q^{19} - 28 q^{21} - 4 q^{23} + 22 q^{25} - 4 q^{27} - 47 q^{29} - 5 q^{31} - 26 q^{33} - 13 q^{35} - 67 q^{37} - 9 q^{39} - 28 q^{41} + 15 q^{43} - 57 q^{45} - 10 q^{47} + 20 q^{49} - 11 q^{51} - 58 q^{53} + 15 q^{55} - 31 q^{57} - 32 q^{59} - 55 q^{61} - 16 q^{63} - 44 q^{65} + 22 q^{67} - 44 q^{69} - 47 q^{71} - 5 q^{73} - 25 q^{75} - 50 q^{77} - 14 q^{79} - 28 q^{81} - 16 q^{83} - 78 q^{85} - 11 q^{87} - 20 q^{89} - 15 q^{91} - 83 q^{93} - 27 q^{95} - 8 q^{97} - 70 q^{99}+O(q^{100}) \) 28 * q + 2 * q^3 - 12 * q^5 + 18 * q^9 - 14 * q^11 - 31 * q^13 - 2 * q^15 - 9 * q^17 - 8 * q^19 - 28 * q^21 - 4 * q^23 + 22 * q^25 - 4 * q^27 - 47 * q^29 - 5 * q^31 - 26 * q^33 - 13 * q^35 - 67 * q^37 - 9 * q^39 - 28 * q^41 + 15 * q^43 - 57 * q^45 - 10 * q^47 + 20 * q^49 - 11 * q^51 - 58 * q^53 + 15 * q^55 - 31 * q^57 - 32 * q^59 - 55 * q^61 - 16 * q^63 - 44 * q^65 + 22 * q^67 - 44 * q^69 - 47 * q^71 - 5 * q^73 - 25 * q^75 - 50 * q^77 - 14 * q^79 - 28 * q^81 - 16 * q^83 - 78 * q^85 - 11 * q^87 - 20 * q^89 - 15 * q^91 - 83 * q^93 - 27 * q^95 - 8 * q^97 - 70 * q^99

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