LMFDB - Embedded newform 8024.2.a.z.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.64815 q^{3} -1.50738 q^{5} +4.54063 q^{7} -0.283613 q^{9} +O(q^{10})$	$q+1.64815 q^{3} -1.50738 q^{5} +4.54063 q^{7} -0.283613 q^{9} -2.48710 q^{11} +2.49245 q^{13} -2.48439 q^{15} +1.00000 q^{17} -2.40282 q^{19} +7.48363 q^{21} -8.83935 q^{23} -2.72779 q^{25} -5.41188 q^{27} +7.56615 q^{29} -11.0715 q^{31} -4.09911 q^{33} -6.84448 q^{35} +1.38592 q^{37} +4.10792 q^{39} -5.32311 q^{41} -8.07047 q^{43} +0.427513 q^{45} +8.91502 q^{47} +13.6173 q^{49} +1.64815 q^{51} +7.87420 q^{53} +3.74902 q^{55} -3.96020 q^{57} +1.00000 q^{59} -13.9530 q^{61} -1.28778 q^{63} -3.75707 q^{65} -5.77851 q^{67} -14.5685 q^{69} -5.35328 q^{71} -5.44696 q^{73} -4.49580 q^{75} -11.2930 q^{77} -4.89373 q^{79} -8.06873 q^{81} -10.3769 q^{83} -1.50738 q^{85} +12.4701 q^{87} +9.44373 q^{89} +11.3173 q^{91} -18.2474 q^{93} +3.62197 q^{95} -3.15283 q^{97} +0.705373 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9}+O(q^{10}) $ 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9	$ 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9} - 15 q^{11} - 17 q^{13} - 7 q^{15} + 24 q^{17} - q^{19} + 10 q^{21} - 21 q^{23} + 19 q^{25} - 26 q^{27} - 18 q^{29} - 23 q^{31} + 10 q^{33} - 5 q^{35} - 29 q^{37} - 44 q^{39} + 14 q^{41} - 23 q^{43} - 24 q^{45} - 17 q^{47} + 46 q^{49} - 5 q^{51} + 7 q^{53} - 33 q^{55} - 2 q^{57} + 24 q^{59} - 9 q^{61} - 52 q^{63} - 15 q^{65} + 2 q^{67} + 2 q^{69} - 51 q^{71} - 18 q^{73} + 7 q^{75} + 14 q^{77} - 36 q^{79} + 12 q^{81} - 39 q^{83} - 7 q^{85} - 5 q^{87} + 46 q^{89} - 52 q^{91} - 20 q^{93} - 52 q^{95} + 50 q^{97} - 53 q^{99}+O(q^{100}) $ 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9 - 15 * q^11 - 17 * q^13 - 7 * q^15 + 24 * q^17 - q^19 + 10 * q^21 - 21 * q^23 + 19 * q^25 - 26 * q^27 - 18 * q^29 - 23 * q^31 + 10 * q^33 - 5 * q^35 - 29 * q^37 - 44 * q^39 + 14 * q^41 - 23 * q^43 - 24 * q^45 - 17 * q^47 + 46 * q^49 - 5 * q^51 + 7 * q^53 - 33 * q^55 - 2 * q^57 + 24 * q^59 - 9 * q^61 - 52 * q^63 - 15 * q^65 + 2 * q^67 + 2 * q^69 - 51 * q^71 - 18 * q^73 + 7 * q^75 + 14 * q^77 - 36 * q^79 + 12 * q^81 - 39 * q^83 - 7 * q^85 - 5 * q^87 + 46 * q^89 - 52 * q^91 - 20 * q^93 - 52 * q^95 + 50 * q^97 - 53 * 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
$3$	1.64815	0.951558	0.475779	−	0.879565i	$-0.342166\pi$
$3$	1.64815	0.951558	0.475779	+	0.879565i	$0.342166\pi$
$4$	0	0
$5$	−1.50738	−0.674123	−0.337062	−	0.941483i	$-0.609433\pi$
$5$	−1.50738	−0.674123	−0.337062	+	0.941483i	$0.609433\pi$
$6$	0	0
$7$	4.54063	1.71620	0.858099	−	0.513484i	$-0.171646\pi$
$7$	4.54063	1.71620	0.858099	+	0.513484i	$0.171646\pi$
$8$	0	0
$9$	−0.283613	−0.0945375
$10$	0	0
$11$	−2.48710	−0.749889	−0.374945	−	0.927047i	$-0.622338\pi$
$11$	−2.48710	−0.749889	−0.374945	+	0.927047i	$0.622338\pi$
$12$	0	0
$13$	2.49245	0.691280	0.345640	−	0.938367i	$-0.387662\pi$
$13$	2.49245	0.691280	0.345640	+	0.938367i	$0.387662\pi$
$14$	0	0
$15$	−2.48439	−0.641467
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−2.40282	−0.551244	−0.275622	−	0.961266i	$-0.588884\pi$
$19$	−2.40282	−0.551244	−0.275622	+	0.961266i	$0.588884\pi$
$20$	0	0
$21$	7.48363	1.63306
$22$	0	0
$23$	−8.83935	−1.84313	−0.921566	−	0.388222i	$-0.873089\pi$
$23$	−8.83935	−1.84313	−0.921566	+	0.388222i	$0.873089\pi$
$24$	0	0
$25$	−2.72779	−0.545558
$26$	0	0
$27$	−5.41188	−1.04152
$28$	0	0
$29$	7.56615	1.40500	0.702499	−	0.711685i	$-0.252067\pi$
$29$	7.56615	1.40500	0.702499	+	0.711685i	$0.252067\pi$
$30$	0	0
$31$	−11.0715	−1.98849	−0.994246	−	0.107120i	$-0.965837\pi$
$31$	−11.0715	−1.98849	−0.994246	+	0.107120i	$0.965837\pi$
$32$	0	0
$33$	−4.09911	−0.713563
$34$	0	0
$35$	−6.84448	−1.15693
$36$	0	0
$37$	1.38592	0.227845	0.113922	−	0.993490i	$-0.463659\pi$
$37$	1.38592	0.227845	0.113922	+	0.993490i	$0.463659\pi$
$38$	0	0
$39$	4.10792	0.657793
$40$	0	0
$41$	−5.32311	−0.831329	−0.415665	−	0.909518i	$-0.636451\pi$
$41$	−5.32311	−0.831329	−0.415665	+	0.909518i	$0.636451\pi$
$42$	0	0
$43$	−8.07047	−1.23074	−0.615368	−	0.788240i	$-0.710992\pi$
$43$	−8.07047	−1.23074	−0.615368	+	0.788240i	$0.710992\pi$
$44$	0	0
$45$	0.427513	0.0637299
$46$	0	0
$47$	8.91502	1.30039	0.650194	−	0.759768i	$-0.274687\pi$
$47$	8.91502	1.30039	0.650194	+	0.759768i	$0.274687\pi$
$48$	0	0
$49$	13.6173	1.94533
$50$	0	0
$51$	1.64815	0.230787
$52$	0	0
$53$	7.87420	1.08160	0.540802	−	0.841150i	$-0.318121\pi$
$53$	7.87420	1.08160	0.540802	+	0.841150i	$0.318121\pi$
$54$	0	0
$55$	3.74902	0.505517
$56$	0	0
$57$	−3.96020	−0.524541
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−13.9530	−1.78650	−0.893249	−	0.449562i	$-0.851580\pi$
$61$	−13.9530	−1.78650	−0.893249	+	0.449562i	$0.851580\pi$
$62$	0	0
$63$	−1.28778	−0.162245
$64$	0	0
$65$	−3.75707	−0.466008
$66$	0	0
$67$	−5.77851	−0.705957	−0.352979	−	0.935631i	$-0.614831\pi$
$67$	−5.77851	−0.705957	−0.352979	+	0.935631i	$0.614831\pi$
$68$	0	0
$69$	−14.5685	−1.75385
$70$	0	0
$71$	−5.35328	−0.635317	−0.317659	−	0.948205i	$-0.602897\pi$
$71$	−5.35328	−0.635317	−0.317659	+	0.948205i	$0.602897\pi$
$72$	0	0
$73$	−5.44696	−0.637519	−0.318759	−	0.947836i	$-0.603266\pi$
$73$	−5.44696	−0.637519	−0.318759	+	0.947836i	$0.603266\pi$
$74$	0	0
$75$	−4.49580	−0.519130
$76$	0	0
$77$	−11.2930	−1.28696
$78$	0	0
$79$	−4.89373	−0.550588	−0.275294	−	0.961360i	$-0.588775\pi$
$79$	−4.89373	−0.550588	−0.275294	+	0.961360i	$0.588775\pi$
$80$	0	0
$81$	−8.06873	−0.896525
$82$	0	0
$83$	−10.3769	−1.13901	−0.569504	−	0.821989i	$-0.692865\pi$
$83$	−10.3769	−1.13901	−0.569504	+	0.821989i	$0.692865\pi$
$84$	0	0
$85$	−1.50738	−0.163499
$86$	0	0
$87$	12.4701	1.33694
$88$	0	0
$89$	9.44373	1.00103	0.500516	−	0.865727i	$-0.333143\pi$
$89$	9.44373	1.00103	0.500516	+	0.865727i	$0.333143\pi$
$90$	0	0
$91$	11.3173	1.18637
$92$	0	0
$93$	−18.2474	−1.89217
$94$	0	0
$95$	3.62197	0.371606
$96$	0	0
$97$	−3.15283	−0.320122	−0.160061	−	0.987107i	$-0.551169\pi$
$97$	−3.15283	−0.320122	−0.160061	+	0.987107i	$0.551169\pi$
$98$	0	0
$99$	0.705373	0.0708927
$100$	0	0
$101$	−11.5993	−1.15418	−0.577088	−	0.816682i	$-0.695811\pi$
$101$	−11.5993	−1.15418	−0.577088	+	0.816682i	$0.695811\pi$
$102$	0	0
$103$	−5.33946	−0.526113	−0.263056	−	0.964780i	$-0.584731\pi$
$103$	−5.33946	−0.526113	−0.263056	+	0.964780i	$0.584731\pi$
$104$	0	0
$105$	−11.2807	−1.10088
$106$	0	0
$107$	18.4024	1.77903	0.889515	−	0.456906i	$-0.151042\pi$
$107$	18.4024	1.77903	0.889515	+	0.456906i	$0.151042\pi$
$108$	0	0
$109$	−10.1884	−0.975869	−0.487935	−	0.872880i	$-0.662250\pi$
$109$	−10.1884	−0.975869	−0.487935	+	0.872880i	$0.662250\pi$
$110$	0	0
$111$	2.28421	0.216807
$112$	0	0
$113$	−13.0954	−1.23191	−0.615957	−	0.787779i	$-0.711231\pi$
$113$	−13.0954	−1.23191	−0.615957	+	0.787779i	$0.711231\pi$
$114$	0	0
$115$	13.3243	1.24250
$116$	0	0
$117$	−0.706889	−0.0653519
$118$	0	0
$119$	4.54063	0.416239
$120$	0	0
$121$	−4.81433	−0.437666
$122$	0	0
$123$	−8.77326	−0.791058
$124$	0	0
$125$	11.6488	1.04190
$126$	0	0
$127$	14.0980	1.25099	0.625497	−	0.780227i	$-0.284896\pi$
$127$	14.0980	1.25099	0.625497	+	0.780227i	$0.284896\pi$
$128$	0	0
$129$	−13.3013	−1.17112
$130$	0	0
$131$	−7.25204	−0.633614	−0.316807	−	0.948490i	$-0.602611\pi$
$131$	−7.25204	−0.633614	−0.316807	+	0.948490i	$0.602611\pi$
$132$	0	0
$133$	−10.9103	−0.946044
$134$	0	0
$135$	8.15778	0.702110
$136$	0	0
$137$	13.5628	1.15875	0.579376	−	0.815061i	$-0.303296\pi$
$137$	13.5628	1.15875	0.579376	+	0.815061i	$0.303296\pi$
$138$	0	0
$139$	−12.2231	−1.03675	−0.518373	−	0.855154i	$-0.673462\pi$
$139$	−12.2231	−1.03675	−0.518373	+	0.855154i	$0.673462\pi$
$140$	0	0
$141$	14.6933	1.23740
$142$	0	0
$143$	−6.19896	−0.518383
$144$	0	0
$145$	−11.4051	−0.947142
$146$	0	0
$147$	22.4434	1.85110
$148$	0	0
$149$	20.2058	1.65533	0.827663	−	0.561226i	$-0.189670\pi$
$149$	20.2058	1.65533	0.827663	+	0.561226i	$0.189670\pi$
$150$	0	0
$151$	−0.231934	−0.0188745	−0.00943724	−	0.999955i	$-0.503004\pi$
$151$	−0.231934	−0.0188745	−0.00943724	+	0.999955i	$0.503004\pi$
$152$	0	0
$153$	−0.283613	−0.0229287
$154$	0	0
$155$	16.6889	1.34049
$156$	0	0
$157$	20.1062	1.60465	0.802324	−	0.596888i	$-0.203596\pi$
$157$	20.1062	1.60465	0.802324	+	0.596888i	$0.203596\pi$
$158$	0	0
$159$	12.9778	1.02921
$160$	0	0
$161$	−40.1362	−3.16318
$162$	0	0
$163$	−16.3982	−1.28440	−0.642202	−	0.766535i	$-0.721979\pi$
$163$	−16.3982	−1.28440	−0.642202	+	0.766535i	$0.721979\pi$
$164$	0	0
$165$	6.17893	0.481029
$166$	0	0
$167$	7.95500	0.615576	0.307788	−	0.951455i	$-0.400411\pi$
$167$	7.95500	0.615576	0.307788	+	0.951455i	$0.400411\pi$
$168$	0	0
$169$	−6.78771	−0.522132
$170$	0	0
$171$	0.681469	0.0521133
$172$	0	0
$173$	14.2335	1.08215	0.541075	−	0.840974i	$-0.318018\pi$
$173$	14.2335	1.08215	0.541075	+	0.840974i	$0.318018\pi$
$174$	0	0
$175$	−12.3859	−0.936286
$176$	0	0
$177$	1.64815	0.123882
$178$	0	0
$179$	15.2176	1.13742	0.568709	−	0.822539i	$-0.307443\pi$
$179$	15.2176	1.13742	0.568709	+	0.822539i	$0.307443\pi$
$180$	0	0
$181$	14.1430	1.05124	0.525620	−	0.850719i	$-0.323833\pi$
$181$	14.1430	1.05124	0.525620	+	0.850719i	$0.323833\pi$
$182$	0	0
$183$	−22.9966	−1.69996
$184$	0	0
$185$	−2.08912	−0.153595
$186$	0	0
$187$	−2.48710	−0.181875
$188$	0	0
$189$	−24.5733	−1.78745
$190$	0	0
$191$	15.7674	1.14089	0.570444	−	0.821337i	$-0.306771\pi$
$191$	15.7674	1.14089	0.570444	+	0.821337i	$0.306771\pi$
$192$	0	0
$193$	4.23439	0.304798	0.152399	−	0.988319i	$-0.451300\pi$
$193$	4.23439	0.304798	0.152399	+	0.988319i	$0.451300\pi$
$194$	0	0
$195$	−6.19221	−0.443433
$196$	0	0
$197$	−12.6816	−0.903525	−0.451763	−	0.892138i	$-0.649205\pi$
$197$	−12.6816	−0.903525	−0.451763	+	0.892138i	$0.649205\pi$
$198$	0	0
$199$	0.143596	0.0101793	0.00508963	−	0.999987i	$-0.498380\pi$
$199$	0.143596	0.0101793	0.00508963	+	0.999987i	$0.498380\pi$
$200$	0	0
$201$	−9.52383	−0.671759
$202$	0	0
$203$	34.3551	2.41125
$204$	0	0
$205$	8.02397	0.560418
$206$	0	0
$207$	2.50695	0.174245
$208$	0	0
$209$	5.97605	0.413372
$210$	0	0
$211$	2.03079	0.139805	0.0699025	−	0.997554i	$-0.477731\pi$
$211$	2.03079	0.139805	0.0699025	+	0.997554i	$0.477731\pi$
$212$	0	0
$213$	−8.82299	−0.604541
$214$	0	0
$215$	12.1653	0.829667
$216$	0	0
$217$	−50.2714	−3.41265
$218$	0	0
$219$	−8.97739	−0.606636
$220$	0	0
$221$	2.49245	0.167660
$222$	0	0
$223$	−6.95921	−0.466023	−0.233012	−	0.972474i	$-0.574858\pi$
$223$	−6.95921	−0.466023	−0.233012	+	0.972474i	$0.574858\pi$
$224$	0	0
$225$	0.773636	0.0515757
$226$	0	0
$227$	−29.3054	−1.94507	−0.972535	−	0.232756i	$-0.925226\pi$
$227$	−29.3054	−1.94507	−0.972535	+	0.232756i	$0.925226\pi$
$228$	0	0
$229$	12.2485	0.809401	0.404701	−	0.914449i	$-0.367376\pi$
$229$	12.2485	0.809401	0.404701	+	0.914449i	$0.367376\pi$
$230$	0	0
$231$	−18.6125	−1.22461
$232$	0	0
$233$	−15.9929	−1.04773	−0.523866	−	0.851801i	$-0.675511\pi$
$233$	−15.9929	−1.04773	−0.523866	+	0.851801i	$0.675511\pi$
$234$	0	0
$235$	−13.4384	−0.876622
$236$	0	0
$237$	−8.06558	−0.523916
$238$	0	0
$239$	−1.11043	−0.0718281	−0.0359140	−	0.999355i	$-0.511434\pi$
$239$	−1.11043	−0.0718281	−0.0359140	+	0.999355i	$0.511434\pi$
$240$	0	0
$241$	−8.28678	−0.533798	−0.266899	−	0.963724i	$-0.585999\pi$
$241$	−8.28678	−0.533798	−0.266899	+	0.963724i	$0.585999\pi$
$242$	0	0
$243$	2.93718	0.188420
$244$	0	0
$245$	−20.5266	−1.31139
$246$	0	0
$247$	−5.98889	−0.381064
$248$	0	0
$249$	−17.1026	−1.08383
$250$	0	0
$251$	−25.5264	−1.61121	−0.805605	−	0.592453i	$-0.798159\pi$
$251$	−25.5264	−1.61121	−0.805605	+	0.592453i	$0.798159\pi$
$252$	0	0
$253$	21.9843	1.38214
$254$	0	0
$255$	−2.48439	−0.155579
$256$	0	0
$257$	13.7563	0.858096	0.429048	−	0.903282i	$-0.358849\pi$
$257$	13.7563	0.858096	0.429048	+	0.903282i	$0.358849\pi$
$258$	0	0
$259$	6.29297	0.391026
$260$	0	0
$261$	−2.14585	−0.132825
$262$	0	0
$263$	−1.46008	−0.0900323	−0.0450162	−	0.998986i	$-0.514334\pi$
$263$	−1.46008	−0.0900323	−0.0450162	+	0.998986i	$0.514334\pi$
$264$	0	0
$265$	−11.8694	−0.729134
$266$	0	0
$267$	15.5646	0.952541
$268$	0	0
$269$	−25.9679	−1.58329	−0.791645	−	0.610982i	$-0.790775\pi$
$269$	−25.9679	−1.58329	−0.791645	+	0.610982i	$0.790775\pi$
$270$	0	0
$271$	−0.461150	−0.0280129	−0.0140064	−	0.999902i	$-0.504459\pi$
$271$	−0.461150	−0.0280129	−0.0140064	+	0.999902i	$0.504459\pi$
$272$	0	0
$273$	18.6525	1.12890
$274$	0	0
$275$	6.78429	0.409108
$276$	0	0
$277$	14.9822	0.900192	0.450096	−	0.892980i	$-0.351390\pi$
$277$	14.9822	0.900192	0.450096	+	0.892980i	$0.351390\pi$
$278$	0	0
$279$	3.14000	0.187987
$280$	0	0
$281$	9.70373	0.578876	0.289438	−	0.957197i	$-0.406532\pi$
$281$	9.70373	0.578876	0.289438	+	0.957197i	$0.406532\pi$
$282$	0	0
$283$	−21.9112	−1.30249	−0.651244	−	0.758869i	$-0.725753\pi$
$283$	−21.9112	−1.30249	−0.651244	+	0.758869i	$0.725753\pi$
$284$	0	0
$285$	5.96954	0.353605
$286$	0	0
$287$	−24.1703	−1.42673
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−5.19633	−0.304614
$292$	0	0
$293$	3.76228	0.219795	0.109897	−	0.993943i	$-0.464948\pi$
$293$	3.76228	0.219795	0.109897	+	0.993943i	$0.464948\pi$
$294$	0	0
$295$	−1.50738	−0.0877633
$296$	0	0
$297$	13.4599	0.781021
$298$	0	0
$299$	−22.0316	−1.27412
$300$	0	0
$301$	−36.6450	−2.11219
$302$	0	0
$303$	−19.1174	−1.09826
$304$	0	0
$305$	21.0325	1.20432
$306$	0	0
$307$	24.3647	1.39057	0.695283	−	0.718736i	$-0.255279\pi$
$307$	24.3647	1.39057	0.695283	+	0.718736i	$0.255279\pi$
$308$	0	0
$309$	−8.80021	−0.500627
$310$	0	0
$311$	11.2729	0.639230	0.319615	−	0.947547i	$-0.396446\pi$
$311$	11.2729	0.639230	0.319615	+	0.947547i	$0.396446\pi$
$312$	0	0
$313$	19.7356	1.11552	0.557762	−	0.830001i	$-0.311660\pi$
$313$	19.7356	1.11552	0.557762	+	0.830001i	$0.311660\pi$
$314$	0	0
$315$	1.94118	0.109373
$316$	0	0
$317$	−22.8296	−1.28224	−0.641120	−	0.767441i	$-0.721530\pi$
$317$	−22.8296	−1.28224	−0.641120	+	0.767441i	$0.721530\pi$
$318$	0	0
$319$	−18.8178	−1.05359
$320$	0	0
$321$	30.3299	1.69285
$322$	0	0
$323$	−2.40282	−0.133696
$324$	0	0
$325$	−6.79887	−0.377133
$326$	0	0
$327$	−16.7919	−0.928596
$328$	0	0
$329$	40.4798	2.23172
$330$	0	0
$331$	20.2204	1.11141	0.555706	−	0.831379i	$-0.312448\pi$
$331$	20.2204	1.11141	0.555706	+	0.831379i	$0.312448\pi$
$332$	0	0
$333$	−0.393066	−0.0215399
$334$	0	0
$335$	8.71044	0.475902
$336$	0	0
$337$	21.7306	1.18374	0.591871	−	0.806033i	$-0.298390\pi$
$337$	21.7306	1.18374	0.591871	+	0.806033i	$0.298390\pi$
$338$	0	0
$339$	−21.5832	−1.17224
$340$	0	0
$341$	27.5358	1.49115
$342$	0	0
$343$	30.0469	1.62238
$344$	0	0
$345$	21.9604	1.18231
$346$	0	0
$347$	1.55357	0.0833998	0.0416999	−	0.999130i	$-0.486723\pi$
$347$	1.55357	0.0833998	0.0416999	+	0.999130i	$0.486723\pi$
$348$	0	0
$349$	−5.89380	−0.315488	−0.157744	−	0.987480i	$-0.550422\pi$
$349$	−5.89380	−0.315488	−0.157744	+	0.987480i	$0.550422\pi$
$350$	0	0
$351$	−13.4888	−0.719979
$352$	0	0
$353$	36.5438	1.94503	0.972514	−	0.232844i	$-0.0748031\pi$
$353$	36.5438	1.94503	0.972514	+	0.232844i	$0.0748031\pi$
$354$	0	0
$355$	8.06945	0.428282
$356$	0	0
$357$	7.48363	0.396076
$358$	0	0
$359$	29.9202	1.57913	0.789563	−	0.613669i	$-0.210307\pi$
$359$	29.9202	1.57913	0.789563	+	0.613669i	$0.210307\pi$
$360$	0	0
$361$	−13.2265	−0.696130
$362$	0	0
$363$	−7.93472	−0.416465
$364$	0	0
$365$	8.21067	0.429766
$366$	0	0
$367$	−37.0958	−1.93638	−0.968192	−	0.250210i	$-0.919500\pi$
$367$	−37.0958	−1.93638	−0.968192	+	0.250210i	$0.919500\pi$
$368$	0	0
$369$	1.50970	0.0785918
$370$	0	0
$371$	35.7538	1.85625
$372$	0	0
$373$	−18.0818	−0.936239	−0.468120	−	0.883665i	$-0.655068\pi$
$373$	−18.0818	−0.936239	−0.468120	+	0.883665i	$0.655068\pi$
$374$	0	0
$375$	19.1989	0.991425
$376$	0	0
$377$	18.8582	0.971247
$378$	0	0
$379$	−34.5628	−1.77537	−0.887686	−	0.460448i	$-0.847689\pi$
$379$	−34.5628	−1.77537	−0.887686	+	0.460448i	$0.847689\pi$
$380$	0	0
$381$	23.2355	1.19039
$382$	0	0
$383$	1.08724	0.0555552	0.0277776	−	0.999614i	$-0.491157\pi$
$383$	1.08724	0.0555552	0.0277776	+	0.999614i	$0.491157\pi$
$384$	0	0
$385$	17.0229	0.867568
$386$	0	0
$387$	2.28889	0.116351
$388$	0	0
$389$	−15.1357	−0.767413	−0.383706	−	0.923455i	$-0.625353\pi$
$389$	−15.1357	−0.767413	−0.383706	+	0.923455i	$0.625353\pi$
$390$	0	0
$391$	−8.83935	−0.447025
$392$	0	0
$393$	−11.9524	−0.602921
$394$	0	0
$395$	7.37673	0.371164
$396$	0	0
$397$	−32.6941	−1.64087	−0.820434	−	0.571741i	$-0.806268\pi$
$397$	−32.6941	−1.64087	−0.820434	+	0.571741i	$0.806268\pi$
$398$	0	0
$399$	−17.9818	−0.900216
$400$	0	0
$401$	11.5983	0.579189	0.289595	−	0.957149i	$-0.406480\pi$
$401$	11.5983	0.579189	0.289595	+	0.957149i	$0.406480\pi$
$402$	0	0
$403$	−27.5950	−1.37461
$404$	0	0
$405$	12.1627	0.604368
$406$	0	0
$407$	−3.44693	−0.170858
$408$	0	0
$409$	−28.3737	−1.40299	−0.701493	−	0.712676i	$-0.747483\pi$
$409$	−28.3737	−1.40299	−0.701493	+	0.712676i	$0.747483\pi$
$410$	0	0
$411$	22.3535	1.10262
$412$	0	0
$413$	4.54063	0.223430
$414$	0	0
$415$	15.6419	0.767831
$416$	0	0
$417$	−20.1454	−0.986525
$418$	0	0
$419$	−2.47509	−0.120916	−0.0604581	−	0.998171i	$-0.519256\pi$
$419$	−2.47509	−0.120916	−0.0604581	+	0.998171i	$0.519256\pi$
$420$	0	0
$421$	−36.2777	−1.76807	−0.884033	−	0.467424i	$-0.845182\pi$
$421$	−36.2777	−1.76807	−0.884033	+	0.467424i	$0.845182\pi$
$422$	0	0
$423$	−2.52841	−0.122936
$424$	0	0
$425$	−2.72779	−0.132317
$426$	0	0
$427$	−63.3554	−3.06598
$428$	0	0
$429$	−10.2168	−0.493272
$430$	0	0
$431$	−15.6127	−0.752037	−0.376019	−	0.926612i	$-0.622707\pi$
$431$	−15.6127	−0.752037	−0.376019	+	0.926612i	$0.622707\pi$
$432$	0	0
$433$	−22.9365	−1.10226	−0.551130	−	0.834419i	$-0.685803\pi$
$433$	−22.9365	−1.10226	−0.551130	+	0.834419i	$0.685803\pi$
$434$	0	0
$435$	−18.7973	−0.901260
$436$	0	0
$437$	21.2393	1.01602
$438$	0	0
$439$	−4.04204	−0.192916	−0.0964579	−	0.995337i	$-0.530751\pi$
$439$	−4.04204	−0.192916	−0.0964579	+	0.995337i	$0.530751\pi$
$440$	0	0
$441$	−3.86205	−0.183907
$442$	0	0
$443$	0.226670	0.0107694	0.00538471	−	0.999986i	$-0.498286\pi$
$443$	0.226670	0.0107694	0.00538471	+	0.999986i	$0.498286\pi$
$444$	0	0
$445$	−14.2353	−0.674819
$446$	0	0
$447$	33.3022	1.57514
$448$	0	0
$449$	34.5066	1.62847	0.814233	−	0.580539i	$-0.197158\pi$
$449$	34.5066	1.62847	0.814233	+	0.580539i	$0.197158\pi$
$450$	0	0
$451$	13.2391	0.623405
$452$	0	0
$453$	−0.382261	−0.0179602
$454$	0	0
$455$	−17.0595	−0.799762
$456$	0	0
$457$	20.8219	0.974009	0.487004	−	0.873400i	$-0.338090\pi$
$457$	20.8219	0.974009	0.487004	+	0.873400i	$0.338090\pi$
$458$	0	0
$459$	−5.41188	−0.252605
$460$	0	0
$461$	3.36213	0.156590	0.0782949	−	0.996930i	$-0.475052\pi$
$461$	3.36213	0.156590	0.0782949	+	0.996930i	$0.475052\pi$
$462$	0	0
$463$	−36.6539	−1.70345	−0.851726	−	0.523987i	$-0.824444\pi$
$463$	−36.6539	−1.70345	−0.851726	+	0.523987i	$0.824444\pi$
$464$	0	0
$465$	27.5058	1.27555
$466$	0	0
$467$	2.93926	0.136013	0.0680063	−	0.997685i	$-0.478336\pi$
$467$	2.93926	0.136013	0.0680063	+	0.997685i	$0.478336\pi$
$468$	0	0
$469$	−26.2381	−1.21156
$470$	0	0
$471$	33.1380	1.52692
$472$	0	0
$473$	20.0721	0.922915
$474$	0	0
$475$	6.55438	0.300736
$476$	0	0
$477$	−2.23322	−0.102252
$478$	0	0
$479$	−9.10997	−0.416245	−0.208122	−	0.978103i	$-0.566735\pi$
$479$	−9.10997	−0.416245	−0.208122	+	0.978103i	$0.566735\pi$
$480$	0	0
$481$	3.45434	0.157504
$482$	0	0
$483$	−66.1504	−3.00995
$484$	0	0
$485$	4.75253	0.215801
$486$	0	0
$487$	36.8091	1.66798	0.833990	−	0.551780i	$-0.186051\pi$
$487$	36.8091	1.66798	0.833990	+	0.551780i	$0.186051\pi$
$488$	0	0
$489$	−27.0266	−1.22219
$490$	0	0
$491$	−12.1944	−0.550326	−0.275163	−	0.961398i	$-0.588732\pi$
$491$	−12.1944	−0.550326	−0.275163	+	0.961398i	$0.588732\pi$
$492$	0	0
$493$	7.56615	0.340762
$494$	0	0
$495$	−1.06327	−0.0477904
$496$	0	0
$497$	−24.3073	−1.09033
$498$	0	0
$499$	−13.1461	−0.588500	−0.294250	−	0.955729i	$-0.595070\pi$
$499$	−13.1461	−0.588500	−0.294250	+	0.955729i	$0.595070\pi$
$500$	0	0
$501$	13.1110	0.585756
$502$	0	0
$503$	−22.7479	−1.01428	−0.507139	−	0.861865i	$-0.669297\pi$
$503$	−22.7479	−1.01428	−0.507139	+	0.861865i	$0.669297\pi$
$504$	0	0
$505$	17.4846	0.778056
$506$	0	0
$507$	−11.1871	−0.496839
$508$	0	0
$509$	−39.3871	−1.74580	−0.872902	−	0.487896i	$-0.837764\pi$
$509$	−39.3871	−1.74580	−0.872902	+	0.487896i	$0.837764\pi$
$510$	0	0
$511$	−24.7327	−1.09411
$512$	0	0
$513$	13.0037	0.574129
$514$	0	0
$515$	8.04862	0.354665
$516$	0	0
$517$	−22.1725	−0.975147
$518$	0	0
$519$	23.4588	1.02973
$520$	0	0
$521$	3.40789	0.149302	0.0746511	−	0.997210i	$-0.476216\pi$
$521$	3.40789	0.149302	0.0746511	+	0.997210i	$0.476216\pi$
$522$	0	0
$523$	10.6469	0.465558	0.232779	−	0.972530i	$-0.425218\pi$
$523$	10.6469	0.465558	0.232779	+	0.972530i	$0.425218\pi$
$524$	0	0
$525$	−20.4138	−0.890930
$526$	0	0
$527$	−11.0715	−0.482280
$528$	0	0
$529$	55.1341	2.39713
$530$	0	0
$531$	−0.283613	−0.0123077
$532$	0	0
$533$	−13.2676	−0.574682
$534$	0	0
$535$	−27.7395	−1.19929
$536$	0	0
$537$	25.0808	1.08232
$538$	0	0
$539$	−33.8677	−1.45879
$540$	0	0
$541$	43.4839	1.86952	0.934759	−	0.355282i	$-0.115615\pi$
$541$	43.4839	1.86952	0.934759	+	0.355282i	$0.115615\pi$
$542$	0	0
$543$	23.3097	1.00032
$544$	0	0
$545$	15.3578	0.657856
$546$	0	0
$547$	19.3431	0.827049	0.413525	−	0.910493i	$-0.364298\pi$
$547$	19.3431	0.827049	0.413525	+	0.910493i	$0.364298\pi$
$548$	0	0
$549$	3.95725	0.168891
$550$	0	0
$551$	−18.1801	−0.774497
$552$	0	0
$553$	−22.2206	−0.944917
$554$	0	0
$555$	−3.44318	−0.146155
$556$	0	0
$557$	−39.2266	−1.66209	−0.831043	−	0.556209i	$-0.812256\pi$
$557$	−39.2266	−1.66209	−0.831043	+	0.556209i	$0.812256\pi$
$558$	0	0
$559$	−20.1152	−0.850783
$560$	0	0
$561$	−4.09911	−0.173064
$562$	0	0
$563$	29.1368	1.22797	0.613985	−	0.789317i	$-0.289565\pi$
$563$	29.1368	1.22797	0.613985	+	0.789317i	$0.289565\pi$
$564$	0	0
$565$	19.7399	0.830462
$566$	0	0
$567$	−36.6371	−1.53861
$568$	0	0
$569$	−12.9636	−0.543462	−0.271731	−	0.962373i	$-0.587596\pi$
$569$	−12.9636	−0.543462	−0.271731	+	0.962373i	$0.587596\pi$
$570$	0	0
$571$	−34.3662	−1.43818	−0.719089	−	0.694918i	$-0.755441\pi$
$571$	−34.3662	−1.43818	−0.719089	+	0.694918i	$0.755441\pi$
$572$	0	0
$573$	25.9870	1.08562
$574$	0	0
$575$	24.1119	1.00554
$576$	0	0
$577$	18.4852	0.769548	0.384774	−	0.923011i	$-0.374279\pi$
$577$	18.4852	0.769548	0.384774	+	0.923011i	$0.374279\pi$
$578$	0	0
$579$	6.97890	0.290033
$580$	0	0
$581$	−47.1175	−1.95476
$582$	0	0
$583$	−19.5839	−0.811083
$584$	0	0
$585$	1.06555	0.0440552
$586$	0	0
$587$	31.1451	1.28550	0.642748	−	0.766077i	$-0.277794\pi$
$587$	31.1451	1.28550	0.642748	+	0.766077i	$0.277794\pi$
$588$	0	0
$589$	26.6027	1.09614
$590$	0	0
$591$	−20.9011	−0.859756
$592$	0	0
$593$	−10.2490	−0.420876	−0.210438	−	0.977607i	$-0.567489\pi$
$593$	−10.2490	−0.420876	−0.210438	+	0.977607i	$0.567489\pi$
$594$	0	0
$595$	−6.84448	−0.280596
$596$	0	0
$597$	0.236668	0.00968616
$598$	0	0
$599$	24.6550	1.00738	0.503688	−	0.863885i	$-0.331976\pi$
$599$	24.6550	1.00738	0.503688	+	0.863885i	$0.331976\pi$
$600$	0	0
$601$	−2.27234	−0.0926905	−0.0463453	−	0.998925i	$-0.514757\pi$
$601$	−2.27234	−0.0926905	−0.0463453	+	0.998925i	$0.514757\pi$
$602$	0	0
$603$	1.63886	0.0667395
$604$	0	0
$605$	7.25705	0.295041
$606$	0	0
$607$	−23.4618	−0.952286	−0.476143	−	0.879368i	$-0.657966\pi$
$607$	−23.4618	−0.952286	−0.476143	+	0.879368i	$0.657966\pi$
$608$	0	0
$609$	56.6222	2.29445
$610$	0	0
$611$	22.2202	0.898933
$612$	0	0
$613$	−8.41166	−0.339744	−0.169872	−	0.985466i	$-0.554335\pi$
$613$	−8.41166	−0.339744	−0.169872	+	0.985466i	$0.554335\pi$
$614$	0	0
$615$	13.2247	0.533271
$616$	0	0
$617$	16.0634	0.646687	0.323343	−	0.946282i	$-0.395193\pi$
$617$	16.0634	0.646687	0.323343	+	0.946282i	$0.395193\pi$
$618$	0	0
$619$	27.0402	1.08684	0.543420	−	0.839461i	$-0.317129\pi$
$619$	27.0402	1.08684	0.543420	+	0.839461i	$0.317129\pi$
$620$	0	0
$621$	47.8374	1.91965
$622$	0	0
$623$	42.8805	1.71797
$624$	0	0
$625$	−3.92020	−0.156808
$626$	0	0
$627$	9.84940	0.393347
$628$	0	0
$629$	1.38592	0.0552604
$630$	0	0
$631$	−8.84312	−0.352039	−0.176019	−	0.984387i	$-0.556322\pi$
$631$	−8.84312	−0.352039	−0.176019	+	0.984387i	$0.556322\pi$
$632$	0	0
$633$	3.34703	0.133033
$634$	0	0
$635$	−21.2511	−0.843324
$636$	0	0
$637$	33.9405	1.34477
$638$	0	0
$639$	1.51826	0.0600613
$640$	0	0
$641$	4.77767	0.188707	0.0943534	−	0.995539i	$-0.469922\pi$
$641$	4.77767	0.188707	0.0943534	+	0.995539i	$0.469922\pi$
$642$	0	0
$643$	−27.8185	−1.09706	−0.548528	−	0.836132i	$-0.684811\pi$
$643$	−27.8185	−1.09706	−0.548528	+	0.836132i	$0.684811\pi$
$644$	0	0
$645$	20.0502	0.789476
$646$	0	0
$647$	−27.3653	−1.07584	−0.537921	−	0.842995i	$-0.680790\pi$
$647$	−27.3653	−1.07584	−0.537921	+	0.842995i	$0.680790\pi$
$648$	0	0
$649$	−2.48710	−0.0976272
$650$	0	0
$651$	−82.8547	−3.24733
$652$	0	0
$653$	−27.7960	−1.08774	−0.543872	−	0.839168i	$-0.683042\pi$
$653$	−27.7960	−1.08774	−0.543872	+	0.839168i	$0.683042\pi$
$654$	0	0
$655$	10.9316	0.427134
$656$	0	0
$657$	1.54483	0.0602694
$658$	0	0
$659$	13.8244	0.538521	0.269261	−	0.963067i	$-0.413221\pi$
$659$	13.8244	0.538521	0.269261	+	0.963067i	$0.413221\pi$
$660$	0	0
$661$	−30.3555	−1.18069	−0.590346	−	0.807150i	$-0.701009\pi$
$661$	−30.3555	−1.18069	−0.590346	+	0.807150i	$0.701009\pi$
$662$	0	0
$663$	4.10792	0.159538
$664$	0	0
$665$	16.4460	0.637750
$666$	0	0
$667$	−66.8798	−2.58960
$668$	0	0
$669$	−11.4698	−0.443448
$670$	0	0
$671$	34.7025	1.33968
$672$	0	0
$673$	4.64346	0.178992	0.0894961	−	0.995987i	$-0.471474\pi$
$673$	4.64346	0.178992	0.0894961	+	0.995987i	$0.471474\pi$
$674$	0	0
$675$	14.7625	0.568207
$676$	0	0
$677$	−15.9778	−0.614078	−0.307039	−	0.951697i	$-0.599338\pi$
$677$	−15.9778	−0.614078	−0.307039	+	0.951697i	$0.599338\pi$
$678$	0	0
$679$	−14.3159	−0.549392
$680$	0	0
$681$	−48.2997	−1.85085
$682$	0	0
$683$	−18.0850	−0.692004	−0.346002	−	0.938234i	$-0.612461\pi$
$683$	−18.0850	−0.692004	−0.346002	+	0.938234i	$0.612461\pi$
$684$	0	0
$685$	−20.4444	−0.781141
$686$	0	0
$687$	20.1873	0.770192
$688$	0	0
$689$	19.6260	0.747691
$690$	0	0
$691$	21.2281	0.807557	0.403778	−	0.914857i	$-0.367697\pi$
$691$	21.2281	0.807557	0.403778	+	0.914857i	$0.367697\pi$
$692$	0	0
$693$	3.20284	0.121666
$694$	0	0
$695$	18.4249	0.698895
$696$	0	0
$697$	−5.32311	−0.201627
$698$	0	0
$699$	−26.3587	−0.996977
$700$	0	0
$701$	−40.7156	−1.53781	−0.768904	−	0.639364i	$-0.779198\pi$
$701$	−40.7156	−1.53781	−0.768904	+	0.639364i	$0.779198\pi$
$702$	0	0
$703$	−3.33012	−0.125598
$704$	0	0
$705$	−22.1484	−0.834157
$706$	0	0
$707$	−52.6683	−1.98079
$708$	0	0
$709$	3.07309	0.115412	0.0577061	−	0.998334i	$-0.481621\pi$
$709$	3.07309	0.115412	0.0577061	+	0.998334i	$0.481621\pi$
$710$	0	0
$711$	1.38792	0.0520512
$712$	0	0
$713$	97.8645	3.66505
$714$	0	0
$715$	9.34422	0.349454
$716$	0	0
$717$	−1.83016	−0.0683486
$718$	0	0
$719$	6.04559	0.225463	0.112731	−	0.993626i	$-0.464040\pi$
$719$	6.04559	0.225463	0.112731	+	0.993626i	$0.464040\pi$
$720$	0	0
$721$	−24.2445	−0.902913
$722$	0	0
$723$	−13.6578	−0.507940
$724$	0	0
$725$	−20.6389	−0.766508
$726$	0	0
$727$	−4.87930	−0.180963	−0.0904816	−	0.995898i	$-0.528841\pi$
$727$	−4.87930	−0.180963	−0.0904816	+	0.995898i	$0.528841\pi$
$728$	0	0
$729$	29.0471	1.07582
$730$	0	0
$731$	−8.07047	−0.298497
$732$	0	0
$733$	11.6155	0.429027	0.214514	−	0.976721i	$-0.431183\pi$
$733$	11.6155	0.429027	0.214514	+	0.976721i	$0.431183\pi$
$734$	0	0
$735$	−33.8308	−1.24787
$736$	0	0
$737$	14.3717	0.529390
$738$	0	0
$739$	30.6202	1.12638	0.563192	−	0.826326i	$-0.309573\pi$
$739$	30.6202	1.12638	0.563192	+	0.826326i	$0.309573\pi$
$740$	0	0
$741$	−9.87057	−0.362605
$742$	0	0
$743$	44.5439	1.63416	0.817079	−	0.576526i	$-0.195592\pi$
$743$	44.5439	1.63416	0.817079	+	0.576526i	$0.195592\pi$
$744$	0	0
$745$	−30.4579	−1.11589
$746$	0	0
$747$	2.94301	0.107679
$748$	0	0
$749$	83.5587	3.05317
$750$	0	0
$751$	3.00269	0.109570	0.0547848	−	0.998498i	$-0.482553\pi$
$751$	3.00269	0.109570	0.0547848	+	0.998498i	$0.482553\pi$
$752$	0	0
$753$	−42.0712	−1.53316
$754$	0	0
$755$	0.349613	0.0127237
$756$	0	0
$757$	−32.2131	−1.17081	−0.585403	−	0.810743i	$-0.699064\pi$
$757$	−32.2131	−1.17081	−0.585403	+	0.810743i	$0.699064\pi$
$758$	0	0
$759$	36.2334	1.31519
$760$	0	0
$761$	−42.2368	−1.53108	−0.765541	−	0.643387i	$-0.777529\pi$
$761$	−42.2368	−1.53108	−0.765541	+	0.643387i	$0.777529\pi$
$762$	0	0
$763$	−46.2617	−1.67478
$764$	0	0
$765$	0.427513	0.0154568
$766$	0	0
$767$	2.49245	0.0899970
$768$	0	0
$769$	10.2782	0.370641	0.185321	−	0.982678i	$-0.440668\pi$
$769$	10.2782	0.370641	0.185321	+	0.982678i	$0.440668\pi$
$770$	0	0
$771$	22.6724	0.816528
$772$	0	0
$773$	18.5080	0.665685	0.332843	−	0.942982i	$-0.391992\pi$
$773$	18.5080	0.665685	0.332843	+	0.942982i	$0.391992\pi$
$774$	0	0
$775$	30.2006	1.08484
$776$	0	0
$777$	10.3717	0.372084
$778$	0	0
$779$	12.7905	0.458265
$780$	0	0
$781$	13.3141	0.476417
$782$	0	0
$783$	−40.9470	−1.46333
$784$	0	0
$785$	−30.3078	−1.08173
$786$	0	0
$787$	4.85120	0.172926	0.0864632	−	0.996255i	$-0.472443\pi$
$787$	4.85120	0.172926	0.0864632	+	0.996255i	$0.472443\pi$
$788$	0	0
$789$	−2.40642	−0.0856710
$790$	0	0
$791$	−59.4615	−2.11421
$792$	0	0
$793$	−34.7771	−1.23497
$794$	0	0
$795$	−19.5626	−0.693813
$796$	0	0
$797$	−1.18480	−0.0419678	−0.0209839	−	0.999780i	$-0.506680\pi$
$797$	−1.18480	−0.0419678	−0.0209839	+	0.999780i	$0.506680\pi$
$798$	0	0
$799$	8.91502	0.315391
$800$	0	0
$801$	−2.67836	−0.0946352
$802$	0	0
$803$	13.5471	0.478068
$804$	0	0
$805$	60.5007	2.13237
$806$	0	0
$807$	−42.7989	−1.50659
$808$	0	0
$809$	−5.06137	−0.177948	−0.0889741	−	0.996034i	$-0.528359\pi$
$809$	−5.06137	−0.177948	−0.0889741	+	0.996034i	$0.528359\pi$
$810$	0	0
$811$	14.5159	0.509721	0.254860	−	0.966978i	$-0.417970\pi$
$811$	14.5159	0.509721	0.254860	+	0.966978i	$0.417970\pi$
$812$	0	0
$813$	−0.760043	−0.0266559
$814$	0	0
$815$	24.7184	0.865847
$816$	0	0
$817$	19.3919	0.678436
$818$	0	0
$819$	−3.20972	−0.112157
$820$	0	0
$821$	−10.8095	−0.377255	−0.188627	−	0.982049i	$-0.560404\pi$
$821$	−10.8095	−0.377255	−0.188627	+	0.982049i	$0.560404\pi$
$822$	0	0
$823$	6.15204	0.214447	0.107223	−	0.994235i	$-0.465804\pi$
$823$	6.15204	0.214447	0.107223	+	0.994235i	$0.465804\pi$
$824$	0	0
$825$	11.1815	0.389290
$826$	0	0
$827$	44.2378	1.53830	0.769150	−	0.639069i	$-0.220680\pi$
$827$	44.2378	1.53830	0.769150	+	0.639069i	$0.220680\pi$
$828$	0	0
$829$	6.65432	0.231114	0.115557	−	0.993301i	$-0.463135\pi$
$829$	6.65432	0.231114	0.115557	+	0.993301i	$0.463135\pi$
$830$	0	0
$831$	24.6928	0.856584
$832$	0	0
$833$	13.6173	0.471813
$834$	0	0
$835$	−11.9912	−0.414974
$836$	0	0
$837$	59.9173	2.07105
$838$	0	0
$839$	−16.0475	−0.554021	−0.277011	−	0.960867i	$-0.589344\pi$
$839$	−16.0475	−0.554021	−0.277011	+	0.960867i	$0.589344\pi$
$840$	0	0
$841$	28.2466	0.974020
$842$	0	0
$843$	15.9932	0.550834
$844$	0	0
$845$	10.2317	0.351981
$846$	0	0
$847$	−21.8601	−0.751122
$848$	0	0
$849$	−36.1129	−1.23939
$850$	0	0
$851$	−12.2507	−0.419947
$852$	0	0
$853$	45.4734	1.55698	0.778490	−	0.627657i	$-0.215986\pi$
$853$	45.4734	1.55698	0.778490	+	0.627657i	$0.215986\pi$
$854$	0	0
$855$	−1.02724	−0.0351307
$856$	0	0
$857$	19.1008	0.652470	0.326235	−	0.945289i	$-0.394220\pi$
$857$	19.1008	0.652470	0.326235	+	0.945289i	$0.394220\pi$
$858$	0	0
$859$	25.8806	0.883034	0.441517	−	0.897253i	$-0.354440\pi$
$859$	25.8806	0.883034	0.441517	+	0.897253i	$0.354440\pi$
$860$	0	0
$861$	−39.8361	−1.35761
$862$	0	0
$863$	11.8078	0.401942	0.200971	−	0.979597i	$-0.435590\pi$
$863$	11.8078	0.401942	0.200971	+	0.979597i	$0.435590\pi$
$864$	0	0
$865$	−21.4553	−0.729502
$866$	0	0
$867$	1.64815	0.0559740
$868$	0	0
$869$	12.1712	0.412880
$870$	0	0
$871$	−14.4026	−0.488014
$872$	0	0
$873$	0.894183	0.0302635
$874$	0	0
$875$	52.8927	1.78810
$876$	0	0
$877$	−5.02125	−0.169556	−0.0847778	−	0.996400i	$-0.527018\pi$
$877$	−5.02125	−0.169556	−0.0847778	+	0.996400i	$0.527018\pi$
$878$	0	0
$879$	6.20079	0.209147
$880$	0	0
$881$	43.6317	1.46999	0.734995	−	0.678072i	$-0.237184\pi$
$881$	43.6317	1.46999	0.734995	+	0.678072i	$0.237184\pi$
$882$	0	0
$883$	−11.8608	−0.399147	−0.199573	−	0.979883i	$-0.563956\pi$
$883$	−11.8608	−0.399147	−0.199573	+	0.979883i	$0.563956\pi$
$884$	0	0
$885$	−2.48439	−0.0835119
$886$	0	0
$887$	−28.0771	−0.942737	−0.471369	−	0.881936i	$-0.656240\pi$
$887$	−28.0771	−0.942737	−0.471369	+	0.881936i	$0.656240\pi$
$888$	0	0
$889$	64.0138	2.14695
$890$	0	0
$891$	20.0677	0.672294
$892$	0	0
$893$	−21.4212	−0.716832
$894$	0	0
$895$	−22.9388	−0.766759
$896$	0	0
$897$	−36.3113	−1.21240
$898$	0	0
$899$	−83.7683	−2.79383
$900$	0	0
$901$	7.87420	0.262328
$902$	0	0
$903$	−60.3964	−2.00987
$904$	0	0
$905$	−21.3189	−0.708666
$906$	0	0
$907$	−15.9388	−0.529238	−0.264619	−	0.964353i	$-0.585246\pi$
$907$	−15.9388	−0.529238	−0.264619	+	0.964353i	$0.585246\pi$
$908$	0	0
$909$	3.28971	0.109113
$910$	0	0
$911$	46.5078	1.54087	0.770436	−	0.637517i	$-0.220038\pi$
$911$	46.5078	1.54087	0.770436	+	0.637517i	$0.220038\pi$
$912$	0	0
$913$	25.8083	0.854130
$914$	0	0
$915$	34.6647	1.14598
$916$	0	0
$917$	−32.9289	−1.08741
$918$	0	0
$919$	−1.88068	−0.0620379	−0.0310190	−	0.999519i	$-0.509875\pi$
$919$	−1.88068	−0.0620379	−0.0310190	+	0.999519i	$0.509875\pi$
$920$	0	0
$921$	40.1566	1.32320
$922$	0	0
$923$	−13.3428	−0.439182
$924$	0	0
$925$	−3.78051	−0.124302
$926$	0	0
$927$	1.51434	0.0497374
$928$	0	0
$929$	54.9565	1.80306	0.901532	−	0.432713i	$-0.142444\pi$
$929$	54.9565	1.80306	0.901532	+	0.432713i	$0.142444\pi$
$930$	0	0
$931$	−32.7200	−1.07235
$932$	0	0
$933$	18.5795	0.608265
$934$	0	0
$935$	3.74902	0.122606
$936$	0	0
$937$	26.4126	0.862863	0.431432	−	0.902146i	$-0.358009\pi$
$937$	26.4126	0.862863	0.431432	+	0.902146i	$0.358009\pi$
$938$	0	0
$939$	32.5272	1.06149
$940$	0	0
$941$	−14.4289	−0.470369	−0.235184	−	0.971951i	$-0.575569\pi$
$941$	−14.4289	−0.470369	−0.235184	+	0.971951i	$0.575569\pi$
$942$	0	0
$943$	47.0528	1.53225
$944$	0	0
$945$	37.0415	1.20496
$946$	0	0
$947$	−38.4884	−1.25070	−0.625352	−	0.780343i	$-0.715045\pi$
$947$	−38.4884	−1.25070	−0.625352	+	0.780343i	$0.715045\pi$
$948$	0	0
$949$	−13.5763	−0.440704
$950$	0	0
$951$	−37.6266	−1.22013
$952$	0	0
$953$	16.4086	0.531526	0.265763	−	0.964038i	$-0.414376\pi$
$953$	16.4086	0.531526	0.265763	+	0.964038i	$0.414376\pi$
$954$	0	0
$955$	−23.7675	−0.769099
$956$	0	0
$957$	−31.0144	−1.00255
$958$	0	0
$959$	61.5838	1.98865
$960$	0	0
$961$	91.5771	2.95410
$962$	0	0
$963$	−5.21916	−0.168185
$964$	0	0
$965$	−6.38286	−0.205472
$966$	0	0
$967$	−5.44108	−0.174973	−0.0874867	−	0.996166i	$-0.527884\pi$
$967$	−5.44108	−0.174973	−0.0874867	+	0.996166i	$0.527884\pi$
$968$	0	0
$969$	−3.96020	−0.127220
$970$	0	0
$971$	−7.71930	−0.247724	−0.123862	−	0.992299i	$-0.539528\pi$
$971$	−7.71930	−0.247724	−0.123862	+	0.992299i	$0.539528\pi$
$972$	0	0
$973$	−55.5004	−1.77926
$974$	0	0
$975$	−11.2055	−0.358864
$976$	0	0
$977$	21.8944	0.700465	0.350233	−	0.936663i	$-0.386103\pi$
$977$	21.8944	0.700465	0.350233	+	0.936663i	$0.386103\pi$
$978$	0	0
$979$	−23.4875	−0.750664
$980$	0	0
$981$	2.88955	0.0922563
$982$	0	0
$983$	−52.9214	−1.68793	−0.843965	−	0.536398i	$-0.819785\pi$
$983$	−52.9214	−1.68793	−0.843965	+	0.536398i	$0.819785\pi$
$984$	0	0
$985$	19.1160	0.609087
$986$	0	0
$987$	66.7167	2.12361
$988$	0	0
$989$	71.3377	2.26841
$990$	0	0
$991$	−9.58748	−0.304556	−0.152278	−	0.988338i	$-0.548661\pi$
$991$	−9.58748	−0.304556	−0.152278	+	0.988338i	$0.548661\pi$
$992$	0	0
$993$	33.3261	1.05757
$994$	0	0
$995$	−0.216455	−0.00686208
$996$	0	0
$997$	−15.3000	−0.484557	−0.242279	−	0.970207i	$-0.577895\pi$
$997$	−15.3000	−0.484557	−0.242279	+	0.970207i	$0.577895\pi$
$998$	0	0
$999$	−7.50045	−0.237304

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.z.1.18	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.z.1.18	✓	24	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q+1.64815 q^{3} -1.50738 q^{5} +4.54063 q^{7} -0.283613 q^{9} +O(q^{10})\)	\(q+1.64815 q^{3} -1.50738 q^{5} +4.54063 q^{7} -0.283613 q^{9} -2.48710 q^{11} +2.49245 q^{13} -2.48439 q^{15} +1.00000 q^{17} -2.40282 q^{19} +7.48363 q^{21} -8.83935 q^{23} -2.72779 q^{25} -5.41188 q^{27} +7.56615 q^{29} -11.0715 q^{31} -4.09911 q^{33} -6.84448 q^{35} +1.38592 q^{37} +4.10792 q^{39} -5.32311 q^{41} -8.07047 q^{43} +0.427513 q^{45} +8.91502 q^{47} +13.6173 q^{49} +1.64815 q^{51} +7.87420 q^{53} +3.74902 q^{55} -3.96020 q^{57} +1.00000 q^{59} -13.9530 q^{61} -1.28778 q^{63} -3.75707 q^{65} -5.77851 q^{67} -14.5685 q^{69} -5.35328 q^{71} -5.44696 q^{73} -4.49580 q^{75} -11.2930 q^{77} -4.89373 q^{79} -8.06873 q^{81} -10.3769 q^{83} -1.50738 q^{85} +12.4701 q^{87} +9.44373 q^{89} +11.3173 q^{91} -18.2474 q^{93} +3.62197 q^{95} -3.15283 q^{97} +0.705373 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9}+O(q^{10}) \) 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9	\( 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9} - 15 q^{11} - 17 q^{13} - 7 q^{15} + 24 q^{17} - q^{19} + 10 q^{21} - 21 q^{23} + 19 q^{25} - 26 q^{27} - 18 q^{29} - 23 q^{31} + 10 q^{33} - 5 q^{35} - 29 q^{37} - 44 q^{39} + 14 q^{41} - 23 q^{43} - 24 q^{45} - 17 q^{47} + 46 q^{49} - 5 q^{51} + 7 q^{53} - 33 q^{55} - 2 q^{57} + 24 q^{59} - 9 q^{61} - 52 q^{63} - 15 q^{65} + 2 q^{67} + 2 q^{69} - 51 q^{71} - 18 q^{73} + 7 q^{75} + 14 q^{77} - 36 q^{79} + 12 q^{81} - 39 q^{83} - 7 q^{85} - 5 q^{87} + 46 q^{89} - 52 q^{91} - 20 q^{93} - 52 q^{95} + 50 q^{97} - 53 q^{99}+O(q^{100}) \) 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9 - 15 * q^11 - 17 * q^13 - 7 * q^15 + 24 * q^17 - q^19 + 10 * q^21 - 21 * q^23 + 19 * q^25 - 26 * q^27 - 18 * q^29 - 23 * q^31 + 10 * q^33 - 5 * q^35 - 29 * q^37 - 44 * q^39 + 14 * q^41 - 23 * q^43 - 24 * q^45 - 17 * q^47 + 46 * q^49 - 5 * q^51 + 7 * q^53 - 33 * q^55 - 2 * q^57 + 24 * q^59 - 9 * q^61 - 52 * q^63 - 15 * q^65 + 2 * q^67 + 2 * q^69 - 51 * q^71 - 18 * q^73 + 7 * q^75 + 14 * q^77 - 36 * q^79 + 12 * q^81 - 39 * q^83 - 7 * q^85 - 5 * q^87 + 46 * q^89 - 52 * q^91 - 20 * q^93 - 52 * q^95 + 50 * q^97 - 53 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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