LMFDB - Embedded newform 8024.2.a.x.1.1

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:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-3.05613 q^{3} -1.88060 q^{5} -1.70270 q^{7} +6.33993 q^{9} +O(q^{10})$	$q-3.05613 q^{3} -1.88060 q^{5} -1.70270 q^{7} +6.33993 q^{9} +1.92982 q^{11} -4.25032 q^{13} +5.74735 q^{15} -1.00000 q^{17} -1.66467 q^{19} +5.20368 q^{21} +3.93909 q^{23} -1.46335 q^{25} -10.2073 q^{27} -1.25788 q^{29} -4.61517 q^{31} -5.89779 q^{33} +3.20210 q^{35} +7.09623 q^{37} +12.9895 q^{39} -7.76214 q^{41} +9.07127 q^{43} -11.9229 q^{45} -2.80294 q^{47} -4.10081 q^{49} +3.05613 q^{51} +2.34421 q^{53} -3.62922 q^{55} +5.08745 q^{57} +1.00000 q^{59} +9.05707 q^{61} -10.7950 q^{63} +7.99315 q^{65} -5.52443 q^{67} -12.0384 q^{69} -6.21118 q^{71} +5.85896 q^{73} +4.47219 q^{75} -3.28591 q^{77} +11.0709 q^{79} +12.1750 q^{81} +13.4284 q^{83} +1.88060 q^{85} +3.84424 q^{87} -16.6287 q^{89} +7.23703 q^{91} +14.1046 q^{93} +3.13058 q^{95} -1.64029 q^{97} +12.2350 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) $ 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9	$ 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9} + 2 q^{11} - 9 q^{13} - 7 q^{15} - 22 q^{17} - 10 q^{19} - 10 q^{21} + 14 q^{23} + 3 q^{25} + 6 q^{29} - 15 q^{31} - 52 q^{33} - 7 q^{35} + 9 q^{37} - 52 q^{41} - 7 q^{43} - 30 q^{45} - 7 q^{47} - 6 q^{49} + 3 q^{51} - 18 q^{53} - 39 q^{55} - 2 q^{57} + 22 q^{59} - 42 q^{61} - 35 q^{65} - 28 q^{67} - 10 q^{69} + 23 q^{71} - 33 q^{73} - 3 q^{75} - 28 q^{77} - 30 q^{79} - 2 q^{81} + 11 q^{83} - 3 q^{85} + 27 q^{87} - 34 q^{89} - 18 q^{91} - 28 q^{93} - 10 q^{95} - 3 q^{97} - 28 q^{99}+O(q^{100}) $ 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9 + 2 * q^11 - 9 * q^13 - 7 * q^15 - 22 * q^17 - 10 * q^19 - 10 * q^21 + 14 * q^23 + 3 * q^25 + 6 * q^29 - 15 * q^31 - 52 * q^33 - 7 * q^35 + 9 * q^37 - 52 * q^41 - 7 * q^43 - 30 * q^45 - 7 * q^47 - 6 * q^49 + 3 * q^51 - 18 * q^53 - 39 * q^55 - 2 * q^57 + 22 * q^59 - 42 * q^61 - 35 * q^65 - 28 * q^67 - 10 * q^69 + 23 * q^71 - 33 * q^73 - 3 * q^75 - 28 * q^77 - 30 * q^79 - 2 * q^81 + 11 * q^83 - 3 * q^85 + 27 * q^87 - 34 * q^89 - 18 * q^91 - 28 * q^93 - 10 * q^95 - 3 * q^97 - 28 * 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
$3$	−3.05613	−1.76446	−0.882229	−	0.470821i	$-0.843958\pi$
$3$	−3.05613	−1.76446	−0.882229	+	0.470821i	$0.843958\pi$
$4$	0	0
$5$	−1.88060	−0.841029	−0.420515	−	0.907286i	$-0.638150\pi$
$5$	−1.88060	−0.841029	−0.420515	+	0.907286i	$0.638150\pi$
$6$	0	0
$7$	−1.70270	−0.643560	−0.321780	−	0.946814i	$-0.604281\pi$
$7$	−1.70270	−0.643560	−0.321780	+	0.946814i	$0.604281\pi$
$8$	0	0
$9$	6.33993	2.11331
$10$	0	0
$11$	1.92982	0.581864	0.290932	−	0.956744i	$-0.406035\pi$
$11$	1.92982	0.581864	0.290932	+	0.956744i	$0.406035\pi$
$12$	0	0
$13$	−4.25032	−1.17883	−0.589414	−	0.807831i	$-0.700641\pi$
$13$	−4.25032	−1.17883	−0.589414	+	0.807831i	$0.700641\pi$
$14$	0	0
$15$	5.74735	1.48396
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−1.66467	−0.381902	−0.190951	−	0.981600i	$-0.561157\pi$
$19$	−1.66467	−0.381902	−0.190951	+	0.981600i	$0.561157\pi$
$20$	0	0
$21$	5.20368	1.13554
$22$	0	0
$23$	3.93909	0.821357	0.410679	−	0.911780i	$-0.365292\pi$
$23$	3.93909	0.821357	0.410679	+	0.911780i	$0.365292\pi$
$24$	0	0
$25$	−1.46335	−0.292670
$26$	0	0
$27$	−10.2073	−1.96439
$28$	0	0
$29$	−1.25788	−0.233582	−0.116791	−	0.993157i	$-0.537261\pi$
$29$	−1.25788	−0.233582	−0.116791	+	0.993157i	$0.537261\pi$
$30$	0	0
$31$	−4.61517	−0.828910	−0.414455	−	0.910070i	$-0.636028\pi$
$31$	−4.61517	−0.828910	−0.414455	+	0.910070i	$0.636028\pi$
$32$	0	0
$33$	−5.89779	−1.02667
$34$	0	0
$35$	3.20210	0.541253
$36$	0	0
$37$	7.09623	1.16661	0.583306	−	0.812252i	$-0.301759\pi$
$37$	7.09623	1.16661	0.583306	+	0.812252i	$0.301759\pi$
$38$	0	0
$39$	12.9895	2.07999
$40$	0	0
$41$	−7.76214	−1.21224	−0.606121	−	0.795372i	$-0.707275\pi$
$41$	−7.76214	−1.21224	−0.606121	+	0.795372i	$0.707275\pi$
$42$	0	0
$43$	9.07127	1.38336	0.691678	−	0.722206i	$-0.256872\pi$
$43$	9.07127	1.38336	0.691678	+	0.722206i	$0.256872\pi$
$44$	0	0
$45$	−11.9229	−1.77736
$46$	0	0
$47$	−2.80294	−0.408851	−0.204425	−	0.978882i	$-0.565533\pi$
$47$	−2.80294	−0.408851	−0.204425	+	0.978882i	$0.565533\pi$
$48$	0	0
$49$	−4.10081	−0.585830
$50$	0	0
$51$	3.05613	0.427944
$52$	0	0
$53$	2.34421	0.322002	0.161001	−	0.986954i	$-0.448528\pi$
$53$	2.34421	0.322002	0.161001	+	0.986954i	$0.448528\pi$
$54$	0	0
$55$	−3.62922	−0.489364
$56$	0	0
$57$	5.08745	0.673849
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	9.05707	1.15964	0.579819	−	0.814745i	$-0.303123\pi$
$61$	9.05707	1.15964	0.579819	+	0.814745i	$0.303123\pi$
$62$	0	0
$63$	−10.7950	−1.36004
$64$	0	0
$65$	7.99315	0.991428
$66$	0	0
$67$	−5.52443	−0.674917	−0.337458	−	0.941340i	$-0.609567\pi$
$67$	−5.52443	−0.674917	−0.337458	+	0.941340i	$0.609567\pi$
$68$	0	0
$69$	−12.0384	−1.44925
$70$	0	0
$71$	−6.21118	−0.737131	−0.368566	−	0.929602i	$-0.620151\pi$
$71$	−6.21118	−0.737131	−0.368566	+	0.929602i	$0.620151\pi$
$72$	0	0
$73$	5.85896	0.685739	0.342870	−	0.939383i	$-0.388601\pi$
$73$	5.85896	0.685739	0.342870	+	0.939383i	$0.388601\pi$
$74$	0	0
$75$	4.47219	0.516404
$76$	0	0
$77$	−3.28591	−0.374465
$78$	0	0
$79$	11.0709	1.24557	0.622784	−	0.782394i	$-0.286001\pi$
$79$	11.0709	1.24557	0.622784	+	0.782394i	$0.286001\pi$
$80$	0	0
$81$	12.1750	1.35277
$82$	0	0
$83$	13.4284	1.47396	0.736979	−	0.675916i	$-0.236252\pi$
$83$	13.4284	1.47396	0.736979	+	0.675916i	$0.236252\pi$
$84$	0	0
$85$	1.88060	0.203980
$86$	0	0
$87$	3.84424	0.412146
$88$	0	0
$89$	−16.6287	−1.76264	−0.881319	−	0.472521i	$-0.843344\pi$
$89$	−16.6287	−1.76264	−0.881319	+	0.472521i	$0.843344\pi$
$90$	0	0
$91$	7.23703	0.758647
$92$	0	0
$93$	14.1046	1.46258
$94$	0	0
$95$	3.13058	0.321190
$96$	0	0
$97$	−1.64029	−0.166547	−0.0832733	−	0.996527i	$-0.526537\pi$
$97$	−1.64029	−0.166547	−0.0832733	+	0.996527i	$0.526537\pi$
$98$	0	0
$99$	12.2350	1.22966
$100$	0	0
$101$	6.84254	0.680858	0.340429	−	0.940270i	$-0.389428\pi$
$101$	6.84254	0.680858	0.340429	+	0.940270i	$0.389428\pi$
$102$	0	0
$103$	0.921416	0.0907898	0.0453949	−	0.998969i	$-0.485545\pi$
$103$	0.921416	0.0907898	0.0453949	+	0.998969i	$0.485545\pi$
$104$	0	0
$105$	−9.78602	−0.955018
$106$	0	0
$107$	17.7287	1.71389	0.856947	−	0.515404i	$-0.172358\pi$
$107$	17.7287	1.71389	0.856947	+	0.515404i	$0.172358\pi$
$108$	0	0
$109$	9.67133	0.926345	0.463173	−	0.886268i	$-0.346711\pi$
$109$	9.67133	0.926345	0.463173	+	0.886268i	$0.346711\pi$
$110$	0	0
$111$	−21.6870	−2.05844
$112$	0	0
$113$	−12.5107	−1.17691	−0.588454	−	0.808531i	$-0.700263\pi$
$113$	−12.5107	−1.17691	−0.588454	+	0.808531i	$0.700263\pi$
$114$	0	0
$115$	−7.40785	−0.690785
$116$	0	0
$117$	−26.9468	−2.49123
$118$	0	0
$119$	1.70270	0.156086
$120$	0	0
$121$	−7.27578	−0.661434
$122$	0	0
$123$	23.7221	2.13895
$124$	0	0
$125$	12.1550	1.08717
$126$	0	0
$127$	18.4936	1.64104	0.820521	−	0.571617i	$-0.193684\pi$
$127$	18.4936	1.64104	0.820521	+	0.571617i	$0.193684\pi$
$128$	0	0
$129$	−27.7230	−2.44087
$130$	0	0
$131$	15.0349	1.31361	0.656803	−	0.754062i	$-0.271908\pi$
$131$	15.0349	1.31361	0.656803	+	0.754062i	$0.271908\pi$
$132$	0	0
$133$	2.83444	0.245777
$134$	0	0
$135$	19.1958	1.65211
$136$	0	0
$137$	21.0598	1.79926	0.899632	−	0.436650i	$-0.143835\pi$
$137$	21.0598	1.79926	0.899632	+	0.436650i	$0.143835\pi$
$138$	0	0
$139$	−3.28912	−0.278979	−0.139490	−	0.990224i	$-0.544546\pi$
$139$	−3.28912	−0.278979	−0.139490	+	0.990224i	$0.544546\pi$
$140$	0	0
$141$	8.56615	0.721400
$142$	0	0
$143$	−8.20238	−0.685917
$144$	0	0
$145$	2.36556	0.196449
$146$	0	0
$147$	12.5326	1.03367
$148$	0	0
$149$	−11.8742	−0.972774	−0.486387	−	0.873744i	$-0.661685\pi$
$149$	−11.8742	−0.972774	−0.486387	+	0.873744i	$0.661685\pi$
$150$	0	0
$151$	−10.3661	−0.843578	−0.421789	−	0.906694i	$-0.638598\pi$
$151$	−10.3661	−0.843578	−0.421789	+	0.906694i	$0.638598\pi$
$152$	0	0
$153$	−6.33993	−0.512553
$154$	0	0
$155$	8.67929	0.697137
$156$	0	0
$157$	−3.63775	−0.290324	−0.145162	−	0.989408i	$-0.546370\pi$
$157$	−3.63775	−0.290324	−0.145162	+	0.989408i	$0.546370\pi$
$158$	0	0
$159$	−7.16422	−0.568160
$160$	0	0
$161$	−6.70709	−0.528593
$162$	0	0
$163$	10.4086	0.815266	0.407633	−	0.913146i	$-0.366354\pi$
$163$	10.4086	0.815266	0.407633	+	0.913146i	$0.366354\pi$
$164$	0	0
$165$	11.0914	0.863463
$166$	0	0
$167$	−1.91873	−0.148476	−0.0742381	−	0.997241i	$-0.523652\pi$
$167$	−1.91873	−0.148476	−0.0742381	+	0.997241i	$0.523652\pi$
$168$	0	0
$169$	5.06525	0.389635
$170$	0	0
$171$	−10.5539	−0.807077
$172$	0	0
$173$	7.52751	0.572306	0.286153	−	0.958184i	$-0.407623\pi$
$173$	7.52751	0.572306	0.286153	+	0.958184i	$0.407623\pi$
$174$	0	0
$175$	2.49165	0.188351
$176$	0	0
$177$	−3.05613	−0.229713
$178$	0	0
$179$	22.5219	1.68336	0.841682	−	0.539974i	$-0.181566\pi$
$179$	22.5219	1.68336	0.841682	+	0.539974i	$0.181566\pi$
$180$	0	0
$181$	−7.76847	−0.577426	−0.288713	−	0.957416i	$-0.593227\pi$
$181$	−7.76847	−0.577426	−0.288713	+	0.957416i	$0.593227\pi$
$182$	0	0
$183$	−27.6796	−2.04613
$184$	0	0
$185$	−13.3452	−0.981155
$186$	0	0
$187$	−1.92982	−0.141123
$188$	0	0
$189$	17.3799	1.26420
$190$	0	0
$191$	−3.88269	−0.280942	−0.140471	−	0.990085i	$-0.544862\pi$
$191$	−3.88269	−0.280942	−0.140471	+	0.990085i	$0.544862\pi$
$192$	0	0
$193$	12.1663	0.875752	0.437876	−	0.899035i	$-0.355731\pi$
$193$	12.1663	0.875752	0.437876	+	0.899035i	$0.355731\pi$
$194$	0	0
$195$	−24.4281	−1.74933
$196$	0	0
$197$	−8.23142	−0.586464	−0.293232	−	0.956041i	$-0.594731\pi$
$197$	−8.23142	−0.586464	−0.293232	+	0.956041i	$0.594731\pi$
$198$	0	0
$199$	14.9934	1.06285	0.531427	−	0.847104i	$-0.321656\pi$
$199$	14.9934	1.06285	0.531427	+	0.847104i	$0.321656\pi$
$200$	0	0
$201$	16.8834	1.19086
$202$	0	0
$203$	2.14179	0.150324
$204$	0	0
$205$	14.5975	1.01953
$206$	0	0
$207$	24.9736	1.73578
$208$	0	0
$209$	−3.21252	−0.222215
$210$	0	0
$211$	6.38772	0.439749	0.219874	−	0.975528i	$-0.429435\pi$
$211$	6.38772	0.439749	0.219874	+	0.975528i	$0.429435\pi$
$212$	0	0
$213$	18.9822	1.30064
$214$	0	0
$215$	−17.0594	−1.16344
$216$	0	0
$217$	7.85826	0.533454
$218$	0	0
$219$	−17.9057	−1.20996
$220$	0	0
$221$	4.25032	0.285908
$222$	0	0
$223$	0.646260	0.0432768	0.0216384	−	0.999766i	$-0.493112\pi$
$223$	0.646260	0.0432768	0.0216384	+	0.999766i	$0.493112\pi$
$224$	0	0
$225$	−9.27754	−0.618503
$226$	0	0
$227$	23.8057	1.58004	0.790020	−	0.613081i	$-0.210070\pi$
$227$	23.8057	1.58004	0.790020	+	0.613081i	$0.210070\pi$
$228$	0	0
$229$	−13.4775	−0.890620	−0.445310	−	0.895376i	$-0.646907\pi$
$229$	−13.4775	−0.890620	−0.445310	+	0.895376i	$0.646907\pi$
$230$	0	0
$231$	10.0422	0.660727
$232$	0	0
$233$	−8.81810	−0.577693	−0.288847	−	0.957375i	$-0.593272\pi$
$233$	−8.81810	−0.577693	−0.288847	+	0.957375i	$0.593272\pi$
$234$	0	0
$235$	5.27121	0.343856
$236$	0	0
$237$	−33.8340	−2.19775
$238$	0	0
$239$	−25.7089	−1.66297	−0.831486	−	0.555545i	$-0.812510\pi$
$239$	−25.7089	−1.66297	−0.831486	+	0.555545i	$0.812510\pi$
$240$	0	0
$241$	11.6327	0.749325	0.374663	−	0.927161i	$-0.377759\pi$
$241$	11.6327	0.749325	0.374663	+	0.927161i	$0.377759\pi$
$242$	0	0
$243$	−6.58645	−0.422521
$244$	0	0
$245$	7.71198	0.492700
$246$	0	0
$247$	7.07539	0.450196
$248$	0	0
$249$	−41.0389	−2.60074
$250$	0	0
$251$	−16.3761	−1.03365	−0.516827	−	0.856090i	$-0.672887\pi$
$251$	−16.3761	−1.03365	−0.516827	+	0.856090i	$0.672887\pi$
$252$	0	0
$253$	7.60175	0.477918
$254$	0	0
$255$	−5.74735	−0.359913
$256$	0	0
$257$	−10.7530	−0.670757	−0.335378	−	0.942084i	$-0.608864\pi$
$257$	−10.7530	−0.670757	−0.335378	+	0.942084i	$0.608864\pi$
$258$	0	0
$259$	−12.0828	−0.750786
$260$	0	0
$261$	−7.97487	−0.493632
$262$	0	0
$263$	−8.41034	−0.518604	−0.259302	−	0.965796i	$-0.583493\pi$
$263$	−8.41034	−0.518604	−0.259302	+	0.965796i	$0.583493\pi$
$264$	0	0
$265$	−4.40852	−0.270813
$266$	0	0
$267$	50.8195	3.11010
$268$	0	0
$269$	−26.8586	−1.63760	−0.818798	−	0.574082i	$-0.805359\pi$
$269$	−26.8586	−1.63760	−0.818798	+	0.574082i	$0.805359\pi$
$270$	0	0
$271$	24.5681	1.49241	0.746204	−	0.665717i	$-0.231874\pi$
$271$	24.5681	1.49241	0.746204	+	0.665717i	$0.231874\pi$
$272$	0	0
$273$	−22.1173	−1.33860
$274$	0	0
$275$	−2.82401	−0.170294
$276$	0	0
$277$	−11.5162	−0.691939	−0.345969	−	0.938246i	$-0.612450\pi$
$277$	−11.5162	−0.691939	−0.345969	+	0.938246i	$0.612450\pi$
$278$	0	0
$279$	−29.2599	−1.75174
$280$	0	0
$281$	−7.26357	−0.433309	−0.216654	−	0.976248i	$-0.569514\pi$
$281$	−7.26357	−0.433309	−0.216654	+	0.976248i	$0.569514\pi$
$282$	0	0
$283$	−16.6558	−0.990083	−0.495042	−	0.868869i	$-0.664847\pi$
$283$	−16.6558	−0.990083	−0.495042	+	0.868869i	$0.664847\pi$
$284$	0	0
$285$	−9.56745	−0.566727
$286$	0	0
$287$	13.2166	0.780151
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	5.01295	0.293864
$292$	0	0
$293$	12.9992	0.759419	0.379710	−	0.925106i	$-0.376024\pi$
$293$	12.9992	0.759419	0.379710	+	0.925106i	$0.376024\pi$
$294$	0	0
$295$	−1.88060	−0.109493
$296$	0	0
$297$	−19.6982	−1.14301
$298$	0	0
$299$	−16.7424	−0.968239
$300$	0	0
$301$	−15.4457	−0.890273
$302$	0	0
$303$	−20.9117	−1.20135
$304$	0	0
$305$	−17.0327	−0.975290
$306$	0	0
$307$	−30.2396	−1.72587	−0.862933	−	0.505318i	$-0.831375\pi$
$307$	−30.2396	−1.72587	−0.862933	+	0.505318i	$0.831375\pi$
$308$	0	0
$309$	−2.81597	−0.160195
$310$	0	0
$311$	3.47875	0.197262	0.0986309	−	0.995124i	$-0.468554\pi$
$311$	3.47875	0.197262	0.0986309	+	0.995124i	$0.468554\pi$
$312$	0	0
$313$	−17.6365	−0.996873	−0.498437	−	0.866926i	$-0.666092\pi$
$313$	−17.6365	−0.996873	−0.498437	+	0.866926i	$0.666092\pi$
$314$	0	0
$315$	20.3011	1.14384
$316$	0	0
$317$	−13.9952	−0.786047	−0.393023	−	0.919529i	$-0.628571\pi$
$317$	−13.9952	−0.786047	−0.393023	+	0.919529i	$0.628571\pi$
$318$	0	0
$319$	−2.42748	−0.135913
$320$	0	0
$321$	−54.1811	−3.02410
$322$	0	0
$323$	1.66467	0.0926248
$324$	0	0
$325$	6.21971	0.345008
$326$	0	0
$327$	−29.5568	−1.63450
$328$	0	0
$329$	4.77257	0.263120
$330$	0	0
$331$	−26.8173	−1.47401	−0.737005	−	0.675887i	$-0.763761\pi$
$331$	−26.8173	−1.47401	−0.737005	+	0.675887i	$0.763761\pi$
$332$	0	0
$333$	44.9896	2.46542
$334$	0	0
$335$	10.3892	0.567625
$336$	0	0
$337$	19.4850	1.06141	0.530707	−	0.847555i	$-0.321926\pi$
$337$	19.4850	1.06141	0.530707	+	0.847555i	$0.321926\pi$
$338$	0	0
$339$	38.2344	2.07661
$340$	0	0
$341$	−8.90648	−0.482313
$342$	0	0
$343$	18.9014	1.02058
$344$	0	0
$345$	22.6393	1.21886
$346$	0	0
$347$	−21.2345	−1.13993	−0.569963	−	0.821670i	$-0.693043\pi$
$347$	−21.2345	−1.13993	−0.569963	+	0.821670i	$0.693043\pi$
$348$	0	0
$349$	8.12523	0.434933	0.217467	−	0.976068i	$-0.430221\pi$
$349$	8.12523	0.434933	0.217467	+	0.976068i	$0.430221\pi$
$350$	0	0
$351$	43.3842	2.31568
$352$	0	0
$353$	−11.9022	−0.633488	−0.316744	−	0.948511i	$-0.602590\pi$
$353$	−11.9022	−0.633488	−0.316744	+	0.948511i	$0.602590\pi$
$354$	0	0
$355$	11.6807	0.619949
$356$	0	0
$357$	−5.20368	−0.275408
$358$	0	0
$359$	37.0893	1.95750	0.978749	−	0.205062i	$-0.0657395\pi$
$359$	37.0893	1.95750	0.978749	+	0.205062i	$0.0657395\pi$
$360$	0	0
$361$	−16.2289	−0.854151
$362$	0	0
$363$	22.2357	1.16707
$364$	0	0
$365$	−11.0183	−0.576727
$366$	0	0
$367$	20.8163	1.08660	0.543300	−	0.839539i	$-0.317175\pi$
$367$	20.8163	1.08660	0.543300	+	0.839539i	$0.317175\pi$
$368$	0	0
$369$	−49.2115	−2.56185
$370$	0	0
$371$	−3.99149	−0.207228
$372$	0	0
$373$	20.8948	1.08189	0.540947	−	0.841057i	$-0.318066\pi$
$373$	20.8948	1.08189	0.540947	+	0.841057i	$0.318066\pi$
$374$	0	0
$375$	−37.1472	−1.91827
$376$	0	0
$377$	5.34639	0.275353
$378$	0	0
$379$	7.10370	0.364892	0.182446	−	0.983216i	$-0.441598\pi$
$379$	7.10370	0.364892	0.182446	+	0.983216i	$0.441598\pi$
$380$	0	0
$381$	−56.5188	−2.89555
$382$	0	0
$383$	−14.1167	−0.721332	−0.360666	−	0.932695i	$-0.617451\pi$
$383$	−14.1167	−0.721332	−0.360666	+	0.932695i	$0.617451\pi$
$384$	0	0
$385$	6.17948	0.314936
$386$	0	0
$387$	57.5112	2.92346
$388$	0	0
$389$	−3.78292	−0.191802	−0.0959009	−	0.995391i	$-0.530573\pi$
$389$	−3.78292	−0.191802	−0.0959009	+	0.995391i	$0.530573\pi$
$390$	0	0
$391$	−3.93909	−0.199208
$392$	0	0
$393$	−45.9487	−2.31780
$394$	0	0
$395$	−20.8198	−1.04756
$396$	0	0
$397$	7.27117	0.364930	0.182465	−	0.983212i	$-0.441592\pi$
$397$	7.27117	0.364930	0.182465	+	0.983212i	$0.441592\pi$
$398$	0	0
$399$	−8.66241	−0.433663
$400$	0	0
$401$	−3.99524	−0.199513	−0.0997563	−	0.995012i	$-0.531806\pi$
$401$	−3.99524	−0.199513	−0.0997563	+	0.995012i	$0.531806\pi$
$402$	0	0
$403$	19.6160	0.977142
$404$	0	0
$405$	−22.8962	−1.13772
$406$	0	0
$407$	13.6945	0.678810
$408$	0	0
$409$	−19.7795	−0.978033	−0.489016	−	0.872275i	$-0.662644\pi$
$409$	−19.7795	−0.978033	−0.489016	+	0.872275i	$0.662644\pi$
$410$	0	0
$411$	−64.3616	−3.17472
$412$	0	0
$413$	−1.70270	−0.0837844
$414$	0	0
$415$	−25.2534	−1.23964
$416$	0	0
$417$	10.0520	0.492247
$418$	0	0
$419$	3.70189	0.180849	0.0904245	−	0.995903i	$-0.471178\pi$
$419$	3.70189	0.180849	0.0904245	+	0.995903i	$0.471178\pi$
$420$	0	0
$421$	−20.2401	−0.986441	−0.493221	−	0.869904i	$-0.664181\pi$
$421$	−20.2401	−0.986441	−0.493221	+	0.869904i	$0.664181\pi$
$422$	0	0
$423$	−17.7705	−0.864029
$424$	0	0
$425$	1.46335	0.0709829
$426$	0	0
$427$	−15.4215	−0.746298
$428$	0	0
$429$	25.0675	1.21027
$430$	0	0
$431$	20.6169	0.993083	0.496542	−	0.868013i	$-0.334603\pi$
$431$	20.6169	0.993083	0.496542	+	0.868013i	$0.334603\pi$
$432$	0	0
$433$	−6.51058	−0.312878	−0.156439	−	0.987688i	$-0.550002\pi$
$433$	−6.51058	−0.312878	−0.156439	+	0.987688i	$0.550002\pi$
$434$	0	0
$435$	−7.22947	−0.346627
$436$	0	0
$437$	−6.55729	−0.313678
$438$	0	0
$439$	−18.9600	−0.904912	−0.452456	−	0.891787i	$-0.649452\pi$
$439$	−18.9600	−0.904912	−0.452456	+	0.891787i	$0.649452\pi$
$440$	0	0
$441$	−25.9989	−1.23804
$442$	0	0
$443$	−22.1131	−1.05063	−0.525313	−	0.850909i	$-0.676052\pi$
$443$	−22.1131	−1.05063	−0.525313	+	0.850909i	$0.676052\pi$
$444$	0	0
$445$	31.2719	1.48243
$446$	0	0
$447$	36.2892	1.71642
$448$	0	0
$449$	−20.7452	−0.979026	−0.489513	−	0.871996i	$-0.662825\pi$
$449$	−20.7452	−0.979026	−0.489513	+	0.871996i	$0.662825\pi$
$450$	0	0
$451$	−14.9796	−0.705360
$452$	0	0
$453$	31.6800	1.48846
$454$	0	0
$455$	−13.6099	−0.638044
$456$	0	0
$457$	−19.1912	−0.897725	−0.448863	−	0.893601i	$-0.648171\pi$
$457$	−19.1912	−0.897725	−0.448863	+	0.893601i	$0.648171\pi$
$458$	0	0
$459$	10.2073	0.476435
$460$	0	0
$461$	39.3376	1.83214	0.916068	−	0.401023i	$-0.131345\pi$
$461$	39.3376	1.83214	0.916068	+	0.401023i	$0.131345\pi$
$462$	0	0
$463$	−27.1640	−1.26242	−0.631210	−	0.775612i	$-0.717441\pi$
$463$	−27.1640	−1.26242	−0.631210	+	0.775612i	$0.717441\pi$
$464$	0	0
$465$	−26.5250	−1.23007
$466$	0	0
$467$	−0.579634	−0.0268222	−0.0134111	−	0.999910i	$-0.504269\pi$
$467$	−0.579634	−0.0268222	−0.0134111	+	0.999910i	$0.504269\pi$
$468$	0	0
$469$	9.40645	0.434350
$470$	0	0
$471$	11.1174	0.512264
$472$	0	0
$473$	17.5060	0.804925
$474$	0	0
$475$	2.43600	0.111771
$476$	0	0
$477$	14.8622	0.680491
$478$	0	0
$479$	−5.42824	−0.248023	−0.124011	−	0.992281i	$-0.539576\pi$
$479$	−5.42824	−0.248023	−0.124011	+	0.992281i	$0.539576\pi$
$480$	0	0
$481$	−30.1613	−1.37524
$482$	0	0
$483$	20.4978	0.932680
$484$	0	0
$485$	3.08473	0.140070
$486$	0	0
$487$	−1.26685	−0.0574066	−0.0287033	−	0.999588i	$-0.509138\pi$
$487$	−1.26685	−0.0574066	−0.0287033	+	0.999588i	$0.509138\pi$
$488$	0	0
$489$	−31.8101	−1.43850
$490$	0	0
$491$	−28.6885	−1.29469	−0.647347	−	0.762196i	$-0.724121\pi$
$491$	−28.6885	−1.29469	−0.647347	+	0.762196i	$0.724121\pi$
$492$	0	0
$493$	1.25788	0.0566520
$494$	0	0
$495$	−23.0090	−1.03418
$496$	0	0
$497$	10.5758	0.474389
$498$	0	0
$499$	−0.690179	−0.0308966	−0.0154483	−	0.999881i	$-0.504918\pi$
$499$	−0.690179	−0.0308966	−0.0154483	+	0.999881i	$0.504918\pi$
$500$	0	0
$501$	5.86390	0.261980
$502$	0	0
$503$	2.32868	0.103831	0.0519154	−	0.998651i	$-0.483467\pi$
$503$	2.32868	0.103831	0.0519154	+	0.998651i	$0.483467\pi$
$504$	0	0
$505$	−12.8681	−0.572621
$506$	0	0
$507$	−15.4801	−0.687494
$508$	0	0
$509$	−11.2589	−0.499043	−0.249522	−	0.968369i	$-0.580273\pi$
$509$	−11.2589	−0.499043	−0.249522	+	0.968369i	$0.580273\pi$
$510$	0	0
$511$	−9.97605	−0.441315
$512$	0	0
$513$	16.9918	0.750204
$514$	0	0
$515$	−1.73281	−0.0763569
$516$	0	0
$517$	−5.40918	−0.237896
$518$	0	0
$519$	−23.0050	−1.00981
$520$	0	0
$521$	29.0370	1.27214	0.636068	−	0.771633i	$-0.280560\pi$
$521$	29.0370	1.27214	0.636068	+	0.771633i	$0.280560\pi$
$522$	0	0
$523$	−24.6200	−1.07656	−0.538278	−	0.842767i	$-0.680925\pi$
$523$	−24.6200	−1.07656	−0.538278	+	0.842767i	$0.680925\pi$
$524$	0	0
$525$	−7.61480	−0.332337
$526$	0	0
$527$	4.61517	0.201040
$528$	0	0
$529$	−7.48357	−0.325372
$530$	0	0
$531$	6.33993	0.275130
$532$	0	0
$533$	32.9916	1.42903
$534$	0	0
$535$	−33.3405	−1.44144
$536$	0	0
$537$	−68.8298	−2.97022
$538$	0	0
$539$	−7.91384	−0.340873
$540$	0	0
$541$	1.23680	0.0531741	0.0265871	−	0.999647i	$-0.491536\pi$
$541$	1.23680	0.0531741	0.0265871	+	0.999647i	$0.491536\pi$
$542$	0	0
$543$	23.7415	1.01884
$544$	0	0
$545$	−18.1879	−0.779083
$546$	0	0
$547$	19.6076	0.838360	0.419180	−	0.907903i	$-0.362318\pi$
$547$	19.6076	0.838360	0.419180	+	0.907903i	$0.362318\pi$
$548$	0	0
$549$	57.4212	2.45068
$550$	0	0
$551$	2.09395	0.0892054
$552$	0	0
$553$	−18.8504	−0.801599
$554$	0	0
$555$	40.7845	1.73121
$556$	0	0
$557$	22.5220	0.954290	0.477145	−	0.878825i	$-0.341672\pi$
$557$	22.5220	0.954290	0.477145	+	0.878825i	$0.341672\pi$
$558$	0	0
$559$	−38.5558	−1.63074
$560$	0	0
$561$	5.89779	0.249005
$562$	0	0
$563$	−42.7084	−1.79994	−0.899972	−	0.435948i	$-0.856413\pi$
$563$	−42.7084	−1.79994	−0.899972	+	0.435948i	$0.856413\pi$
$564$	0	0
$565$	23.5276	0.989814
$566$	0	0
$567$	−20.7303	−0.870592
$568$	0	0
$569$	2.88967	0.121141	0.0605707	−	0.998164i	$-0.480708\pi$
$569$	2.88967	0.121141	0.0605707	+	0.998164i	$0.480708\pi$
$570$	0	0
$571$	−45.4857	−1.90352	−0.951759	−	0.306847i	$-0.900726\pi$
$571$	−45.4857	−1.90352	−0.951759	+	0.306847i	$0.900726\pi$
$572$	0	0
$573$	11.8660	0.495710
$574$	0	0
$575$	−5.76427	−0.240387
$576$	0	0
$577$	13.0584	0.543628	0.271814	−	0.962350i	$-0.412377\pi$
$577$	13.0584	0.543628	0.271814	+	0.962350i	$0.412377\pi$
$578$	0	0
$579$	−37.1819	−1.54523
$580$	0	0
$581$	−22.8645	−0.948581
$582$	0	0
$583$	4.52392	0.187362
$584$	0	0
$585$	50.6761	2.09520
$586$	0	0
$587$	4.12069	0.170079	0.0850396	−	0.996378i	$-0.472898\pi$
$587$	4.12069	0.170079	0.0850396	+	0.996378i	$0.472898\pi$
$588$	0	0
$589$	7.68275	0.316562
$590$	0	0
$591$	25.1563	1.03479
$592$	0	0
$593$	28.4585	1.16865	0.584325	−	0.811520i	$-0.301359\pi$
$593$	28.4585	1.16865	0.584325	+	0.811520i	$0.301359\pi$
$594$	0	0
$595$	−3.20210	−0.131273
$596$	0	0
$597$	−45.8218	−1.87536
$598$	0	0
$599$	5.28412	0.215903	0.107952	−	0.994156i	$-0.465571\pi$
$599$	5.28412	0.215903	0.107952	+	0.994156i	$0.465571\pi$
$600$	0	0
$601$	−5.42716	−0.221378	−0.110689	−	0.993855i	$-0.535306\pi$
$601$	−5.42716	−0.221378	−0.110689	+	0.993855i	$0.535306\pi$
$602$	0	0
$603$	−35.0245	−1.42631
$604$	0	0
$605$	13.6828	0.556286
$606$	0	0
$607$	−13.9468	−0.566082	−0.283041	−	0.959108i	$-0.591343\pi$
$607$	−13.9468	−0.566082	−0.283041	+	0.959108i	$0.591343\pi$
$608$	0	0
$609$	−6.54559	−0.265241
$610$	0	0
$611$	11.9134	0.481965
$612$	0	0
$613$	21.1636	0.854790	0.427395	−	0.904065i	$-0.359431\pi$
$613$	21.1636	0.854790	0.427395	+	0.904065i	$0.359431\pi$
$614$	0	0
$615$	−44.6118	−1.79892
$616$	0	0
$617$	−32.1428	−1.29402	−0.647011	−	0.762481i	$-0.723981\pi$
$617$	−32.1428	−1.29402	−0.647011	+	0.762481i	$0.723981\pi$
$618$	0	0
$619$	−40.1469	−1.61364	−0.806819	−	0.590798i	$-0.798813\pi$
$619$	−40.1469	−1.61364	−0.806819	+	0.590798i	$0.798813\pi$
$620$	0	0
$621$	−40.2074	−1.61347
$622$	0	0
$623$	28.3137	1.13436
$624$	0	0
$625$	−15.5419	−0.621674
$626$	0	0
$627$	9.81789	0.392089
$628$	0	0
$629$	−7.09623	−0.282945
$630$	0	0
$631$	−20.8458	−0.829856	−0.414928	−	0.909854i	$-0.636193\pi$
$631$	−20.8458	−0.829856	−0.414928	+	0.909854i	$0.636193\pi$
$632$	0	0
$633$	−19.5217	−0.775918
$634$	0	0
$635$	−34.7790	−1.38016
$636$	0	0
$637$	17.4298	0.690593
$638$	0	0
$639$	−39.3785	−1.55779
$640$	0	0
$641$	−31.1240	−1.22932	−0.614661	−	0.788791i	$-0.710707\pi$
$641$	−31.1240	−1.22932	−0.614661	+	0.788791i	$0.710707\pi$
$642$	0	0
$643$	−14.2143	−0.560558	−0.280279	−	0.959919i	$-0.590427\pi$
$643$	−14.2143	−0.560558	−0.280279	+	0.959919i	$0.590427\pi$
$644$	0	0
$645$	52.1358	2.05284
$646$	0	0
$647$	0.861015	0.0338500	0.0169250	−	0.999857i	$-0.494612\pi$
$647$	0.861015	0.0338500	0.0169250	+	0.999857i	$0.494612\pi$
$648$	0	0
$649$	1.92982	0.0757522
$650$	0	0
$651$	−24.0159	−0.941256
$652$	0	0
$653$	−30.4426	−1.19131	−0.595656	−	0.803240i	$-0.703108\pi$
$653$	−30.4426	−1.19131	−0.595656	+	0.803240i	$0.703108\pi$
$654$	0	0
$655$	−28.2746	−1.10478
$656$	0	0
$657$	37.1454	1.44918
$658$	0	0
$659$	−13.7327	−0.534950	−0.267475	−	0.963565i	$-0.586189\pi$
$659$	−13.7327	−0.534950	−0.267475	+	0.963565i	$0.586189\pi$
$660$	0	0
$661$	−26.1507	−1.01715	−0.508573	−	0.861019i	$-0.669827\pi$
$661$	−26.1507	−1.01715	−0.508573	+	0.861019i	$0.669827\pi$
$662$	0	0
$663$	−12.9895	−0.504472
$664$	0	0
$665$	−5.33044	−0.206705
$666$	0	0
$667$	−4.95490	−0.191854
$668$	0	0
$669$	−1.97505	−0.0763600
$670$	0	0
$671$	17.4785	0.674752
$672$	0	0
$673$	35.5344	1.36975	0.684875	−	0.728660i	$-0.259857\pi$
$673$	35.5344	1.36975	0.684875	+	0.728660i	$0.259857\pi$
$674$	0	0
$675$	14.9368	0.574918
$676$	0	0
$677$	−4.80949	−0.184844	−0.0924220	−	0.995720i	$-0.529461\pi$
$677$	−4.80949	−0.184844	−0.0924220	+	0.995720i	$0.529461\pi$
$678$	0	0
$679$	2.79293	0.107183
$680$	0	0
$681$	−72.7534	−2.78791
$682$	0	0
$683$	18.8545	0.721446	0.360723	−	0.932673i	$-0.382530\pi$
$683$	18.8545	0.721446	0.360723	+	0.932673i	$0.382530\pi$
$684$	0	0
$685$	−39.6051	−1.51323
$686$	0	0
$687$	41.1891	1.57146
$688$	0	0
$689$	−9.96366	−0.379585
$690$	0	0
$691$	16.4902	0.627315	0.313658	−	0.949536i	$-0.398446\pi$
$691$	16.4902	0.627315	0.313658	+	0.949536i	$0.398446\pi$
$692$	0	0
$693$	−20.8325	−0.791360
$694$	0	0
$695$	6.18551	0.234630
$696$	0	0
$697$	7.76214	0.294012
$698$	0	0
$699$	26.9493	1.01932
$700$	0	0
$701$	13.2354	0.499892	0.249946	−	0.968260i	$-0.419587\pi$
$701$	13.2354	0.499892	0.249946	+	0.968260i	$0.419587\pi$
$702$	0	0
$703$	−11.8129	−0.445531
$704$	0	0
$705$	−16.1095	−0.606719
$706$	0	0
$707$	−11.6508	−0.438173
$708$	0	0
$709$	10.8651	0.408048	0.204024	−	0.978966i	$-0.434598\pi$
$709$	10.8651	0.408048	0.204024	+	0.978966i	$0.434598\pi$
$710$	0	0
$711$	70.1885	2.63227
$712$	0	0
$713$	−18.1796	−0.680831
$714$	0	0
$715$	15.4254	0.576876
$716$	0	0
$717$	78.5698	2.93424
$718$	0	0
$719$	46.6007	1.73791	0.868956	−	0.494890i	$-0.164792\pi$
$719$	46.6007	1.73791	0.868956	+	0.494890i	$0.164792\pi$
$720$	0	0
$721$	−1.56890	−0.0584287
$722$	0	0
$723$	−35.5509	−1.32215
$724$	0	0
$725$	1.84072	0.0683625
$726$	0	0
$727$	23.5026	0.871664	0.435832	−	0.900028i	$-0.356454\pi$
$727$	23.5026	0.871664	0.435832	+	0.900028i	$0.356454\pi$
$728$	0	0
$729$	−16.3958	−0.607253
$730$	0	0
$731$	−9.07127	−0.335513
$732$	0	0
$733$	8.17025	0.301775	0.150888	−	0.988551i	$-0.451787\pi$
$733$	8.17025	0.301775	0.150888	+	0.988551i	$0.451787\pi$
$734$	0	0
$735$	−23.5688	−0.869348
$736$	0	0
$737$	−10.6612	−0.392710
$738$	0	0
$739$	34.3679	1.26424	0.632122	−	0.774869i	$-0.282184\pi$
$739$	34.3679	1.26424	0.632122	+	0.774869i	$0.282184\pi$
$740$	0	0
$741$	−21.6233	−0.794352
$742$	0	0
$743$	−4.67424	−0.171481	−0.0857407	−	0.996317i	$-0.527326\pi$
$743$	−4.67424	−0.171481	−0.0857407	+	0.996317i	$0.527326\pi$
$744$	0	0
$745$	22.3306	0.818131
$746$	0	0
$747$	85.1352	3.11493
$748$	0	0
$749$	−30.1866	−1.10299
$750$	0	0
$751$	−8.87430	−0.323828	−0.161914	−	0.986805i	$-0.551767\pi$
$751$	−8.87430	−0.323828	−0.161914	+	0.986805i	$0.551767\pi$
$752$	0	0
$753$	50.0476	1.82384
$754$	0	0
$755$	19.4944	0.709473
$756$	0	0
$757$	−20.3256	−0.738746	−0.369373	−	0.929281i	$-0.620428\pi$
$757$	−20.3256	−0.738746	−0.369373	+	0.929281i	$0.620428\pi$
$758$	0	0
$759$	−23.2319	−0.843266
$760$	0	0
$761$	51.4007	1.86327	0.931636	−	0.363392i	$-0.118382\pi$
$761$	51.4007	1.86327	0.931636	+	0.363392i	$0.118382\pi$
$762$	0	0
$763$	−16.4674	−0.596159
$764$	0	0
$765$	11.9229	0.431072
$766$	0	0
$767$	−4.25032	−0.153470
$768$	0	0
$769$	11.1901	0.403524	0.201762	−	0.979435i	$-0.435333\pi$
$769$	11.1901	0.403524	0.201762	+	0.979435i	$0.435333\pi$
$770$	0	0
$771$	32.8627	1.18352
$772$	0	0
$773$	−48.4051	−1.74101	−0.870505	−	0.492160i	$-0.836207\pi$
$773$	−48.4051	−1.74101	−0.870505	+	0.492160i	$0.836207\pi$
$774$	0	0
$775$	6.75362	0.242597
$776$	0	0
$777$	36.9265	1.32473
$778$	0	0
$779$	12.9214	0.462958
$780$	0	0
$781$	−11.9865	−0.428910
$782$	0	0
$783$	12.8395	0.458847
$784$	0	0
$785$	6.84114	0.244171
$786$	0	0
$787$	18.0618	0.643832	0.321916	−	0.946768i	$-0.395673\pi$
$787$	18.0618	0.643832	0.321916	+	0.946768i	$0.395673\pi$
$788$	0	0
$789$	25.7031	0.915055
$790$	0	0
$791$	21.3020	0.757412
$792$	0	0
$793$	−38.4955	−1.36701
$794$	0	0
$795$	13.4730	0.477839
$796$	0	0
$797$	19.7910	0.701034	0.350517	−	0.936556i	$-0.386006\pi$
$797$	19.7910	0.701034	0.350517	+	0.936556i	$0.386006\pi$
$798$	0	0
$799$	2.80294	0.0991609
$800$	0	0
$801$	−105.425	−3.72500
$802$	0	0
$803$	11.3068	0.399007
$804$	0	0
$805$	12.6133	0.444562
$806$	0	0
$807$	82.0833	2.88947
$808$	0	0
$809$	20.6308	0.725341	0.362671	−	0.931917i	$-0.381865\pi$
$809$	20.6308	0.725341	0.362671	+	0.931917i	$0.381865\pi$
$810$	0	0
$811$	33.4860	1.17585	0.587927	−	0.808914i	$-0.299944\pi$
$811$	33.4860	1.17585	0.587927	+	0.808914i	$0.299944\pi$
$812$	0	0
$813$	−75.0835	−2.63329
$814$	0	0
$815$	−19.5744	−0.685663
$816$	0	0
$817$	−15.1007	−0.528306
$818$	0	0
$819$	45.8823	1.60326
$820$	0	0
$821$	27.3859	0.955773	0.477887	−	0.878422i	$-0.341403\pi$
$821$	27.3859	0.955773	0.477887	+	0.878422i	$0.341403\pi$
$822$	0	0
$823$	3.30954	0.115363	0.0576817	−	0.998335i	$-0.481629\pi$
$823$	3.30954	0.115363	0.0576817	+	0.998335i	$0.481629\pi$
$824$	0	0
$825$	8.63054	0.300477
$826$	0	0
$827$	−34.1279	−1.18674	−0.593371	−	0.804929i	$-0.702203\pi$
$827$	−34.1279	−1.18674	−0.593371	+	0.804929i	$0.702203\pi$
$828$	0	0
$829$	−21.9366	−0.761891	−0.380945	−	0.924598i	$-0.624401\pi$
$829$	−21.9366	−0.761891	−0.380945	+	0.924598i	$0.624401\pi$
$830$	0	0
$831$	35.1949	1.22090
$832$	0	0
$833$	4.10081	0.142085
$834$	0	0
$835$	3.60837	0.124873
$836$	0	0
$837$	47.1084	1.62830
$838$	0	0
$839$	10.9586	0.378331	0.189166	−	0.981945i	$-0.439422\pi$
$839$	10.9586	0.378331	0.189166	+	0.981945i	$0.439422\pi$
$840$	0	0
$841$	−27.4177	−0.945439
$842$	0	0
$843$	22.1984	0.764555
$844$	0	0
$845$	−9.52571	−0.327694
$846$	0	0
$847$	12.3885	0.425673
$848$	0	0
$849$	50.9022	1.74696
$850$	0	0
$851$	27.9527	0.958205
$852$	0	0
$853$	1.04015	0.0356142	0.0178071	−	0.999841i	$-0.494332\pi$
$853$	1.04015	0.0356142	0.0178071	+	0.999841i	$0.494332\pi$
$854$	0	0
$855$	19.8477	0.678775
$856$	0	0
$857$	29.8776	1.02060	0.510299	−	0.859997i	$-0.329535\pi$
$857$	29.8776	1.02060	0.510299	+	0.859997i	$0.329535\pi$
$858$	0	0
$859$	3.40971	0.116338	0.0581689	−	0.998307i	$-0.481474\pi$
$859$	3.40971	0.116338	0.0581689	+	0.998307i	$0.481474\pi$
$860$	0	0
$861$	−40.3917	−1.37654
$862$	0	0
$863$	48.7079	1.65803	0.829017	−	0.559223i	$-0.188900\pi$
$863$	48.7079	1.65803	0.829017	+	0.559223i	$0.188900\pi$
$864$	0	0
$865$	−14.1562	−0.481326
$866$	0	0
$867$	−3.05613	−0.103792
$868$	0	0
$869$	21.3648	0.724752
$870$	0	0
$871$	23.4806	0.795611
$872$	0	0
$873$	−10.3994	−0.351965
$874$	0	0
$875$	−20.6963	−0.699662
$876$	0	0
$877$	−5.78020	−0.195183	−0.0975917	−	0.995227i	$-0.531114\pi$
$877$	−5.78020	−0.195183	−0.0975917	+	0.995227i	$0.531114\pi$
$878$	0	0
$879$	−39.7271	−1.33996
$880$	0	0
$881$	−45.8887	−1.54603	−0.773015	−	0.634388i	$-0.781252\pi$
$881$	−45.8887	−1.54603	−0.773015	+	0.634388i	$0.781252\pi$
$882$	0	0
$883$	10.1758	0.342442	0.171221	−	0.985233i	$-0.445229\pi$
$883$	10.1758	0.342442	0.171221	+	0.985233i	$0.445229\pi$
$884$	0	0
$885$	5.74735	0.193195
$886$	0	0
$887$	39.4098	1.32325	0.661626	−	0.749834i	$-0.269867\pi$
$887$	39.4098	1.32325	0.661626	+	0.749834i	$0.269867\pi$
$888$	0	0
$889$	−31.4891	−1.05611
$890$	0	0
$891$	23.4955	0.787130
$892$	0	0
$893$	4.66597	0.156141
$894$	0	0
$895$	−42.3546	−1.41576
$896$	0	0
$897$	51.1670	1.70842
$898$	0	0
$899$	5.80533	0.193619
$900$	0	0
$901$	−2.34421	−0.0780971
$902$	0	0
$903$	47.2039	1.57085
$904$	0	0
$905$	14.6094	0.485632
$906$	0	0
$907$	−47.8976	−1.59041	−0.795207	−	0.606338i	$-0.792638\pi$
$907$	−47.8976	−1.59041	−0.795207	+	0.606338i	$0.792638\pi$
$908$	0	0
$909$	43.3812	1.43887
$910$	0	0
$911$	−48.5816	−1.60958	−0.804791	−	0.593559i	$-0.797722\pi$
$911$	−48.5816	−1.60958	−0.804791	+	0.593559i	$0.797722\pi$
$912$	0	0
$913$	25.9144	0.857643
$914$	0	0
$915$	52.0542	1.72086
$916$	0	0
$917$	−25.6000	−0.845385
$918$	0	0
$919$	−26.9500	−0.888999	−0.444500	−	0.895779i	$-0.646618\pi$
$919$	−26.9500	−0.888999	−0.444500	+	0.895779i	$0.646618\pi$
$920$	0	0
$921$	92.4163	3.04522
$922$	0	0
$923$	26.3995	0.868951
$924$	0	0
$925$	−10.3843	−0.341433
$926$	0	0
$927$	5.84172	0.191867
$928$	0	0
$929$	24.7572	0.812258	0.406129	−	0.913816i	$-0.366878\pi$
$929$	24.7572	0.812258	0.406129	+	0.913816i	$0.366878\pi$
$930$	0	0
$931$	6.82650	0.223729
$932$	0	0
$933$	−10.6315	−0.348060
$934$	0	0
$935$	3.62922	0.118688
$936$	0	0
$937$	23.2854	0.760699	0.380350	−	0.924843i	$-0.375804\pi$
$937$	23.2854	0.760699	0.380350	+	0.924843i	$0.375804\pi$
$938$	0	0
$939$	53.8994	1.75894
$940$	0	0
$941$	−25.1290	−0.819183	−0.409591	−	0.912269i	$-0.634329\pi$
$941$	−25.1290	−0.819183	−0.409591	+	0.912269i	$0.634329\pi$
$942$	0	0
$943$	−30.5758	−0.995684
$944$	0	0
$945$	−32.6847	−1.06323
$946$	0	0
$947$	−24.1541	−0.784904	−0.392452	−	0.919773i	$-0.628373\pi$
$947$	−24.1541	−0.784904	−0.392452	+	0.919773i	$0.628373\pi$
$948$	0	0
$949$	−24.9025	−0.808368
$950$	0	0
$951$	42.7710	1.38695
$952$	0	0
$953$	21.8877	0.709011	0.354506	−	0.935054i	$-0.384649\pi$
$953$	21.8877	0.709011	0.354506	+	0.935054i	$0.384649\pi$
$954$	0	0
$955$	7.30178	0.236280
$956$	0	0
$957$	7.41871	0.239813
$958$	0	0
$959$	−35.8586	−1.15793
$960$	0	0
$961$	−9.70016	−0.312908
$962$	0	0
$963$	112.399	3.62199
$964$	0	0
$965$	−22.8800	−0.736533
$966$	0	0
$967$	28.4191	0.913898	0.456949	−	0.889493i	$-0.348942\pi$
$967$	28.4191	0.913898	0.456949	+	0.889493i	$0.348942\pi$
$968$	0	0
$969$	−5.08745	−0.163433
$970$	0	0
$971$	12.2191	0.392130	0.196065	−	0.980591i	$-0.437184\pi$
$971$	12.2191	0.392130	0.196065	+	0.980591i	$0.437184\pi$
$972$	0	0
$973$	5.60038	0.179540
$974$	0	0
$975$	−19.0083	−0.608751
$976$	0	0
$977$	26.9217	0.861301	0.430650	−	0.902519i	$-0.358284\pi$
$977$	26.9217	0.861301	0.430650	+	0.902519i	$0.358284\pi$
$978$	0	0
$979$	−32.0905	−1.02562
$980$	0	0
$981$	61.3156	1.95766
$982$	0	0
$983$	52.1676	1.66389	0.831944	−	0.554860i	$-0.187228\pi$
$983$	52.1676	1.66389	0.831944	+	0.554860i	$0.187228\pi$
$984$	0	0
$985$	15.4800	0.493234
$986$	0	0
$987$	−14.5856	−0.464265
$988$	0	0
$989$	35.7325	1.13623
$990$	0	0
$991$	−33.2577	−1.05646	−0.528232	−	0.849100i	$-0.677145\pi$
$991$	−33.2577	−1.05646	−0.528232	+	0.849100i	$0.677145\pi$
$992$	0	0
$993$	81.9571	2.60083
$994$	0	0
$995$	−28.1965	−0.893891
$996$	0	0
$997$	−52.4536	−1.66122	−0.830611	−	0.556853i	$-0.812009\pi$
$997$	−52.4536	−1.66122	−0.830611	+	0.556853i	$0.812009\pi$
$998$	0	0
$999$	−72.4331	−2.29168

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.x.1.1	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.x.1.1	✓	22	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-3.05613 q^{3} -1.88060 q^{5} -1.70270 q^{7} +6.33993 q^{9} +O(q^{10})\)	\(q-3.05613 q^{3} -1.88060 q^{5} -1.70270 q^{7} +6.33993 q^{9} +1.92982 q^{11} -4.25032 q^{13} +5.74735 q^{15} -1.00000 q^{17} -1.66467 q^{19} +5.20368 q^{21} +3.93909 q^{23} -1.46335 q^{25} -10.2073 q^{27} -1.25788 q^{29} -4.61517 q^{31} -5.89779 q^{33} +3.20210 q^{35} +7.09623 q^{37} +12.9895 q^{39} -7.76214 q^{41} +9.07127 q^{43} -11.9229 q^{45} -2.80294 q^{47} -4.10081 q^{49} +3.05613 q^{51} +2.34421 q^{53} -3.62922 q^{55} +5.08745 q^{57} +1.00000 q^{59} +9.05707 q^{61} -10.7950 q^{63} +7.99315 q^{65} -5.52443 q^{67} -12.0384 q^{69} -6.21118 q^{71} +5.85896 q^{73} +4.47219 q^{75} -3.28591 q^{77} +11.0709 q^{79} +12.1750 q^{81} +13.4284 q^{83} +1.88060 q^{85} +3.84424 q^{87} -16.6287 q^{89} +7.23703 q^{91} +14.1046 q^{93} +3.13058 q^{95} -1.64029 q^{97} +12.2350 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) \) 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9	\( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9} + 2 q^{11} - 9 q^{13} - 7 q^{15} - 22 q^{17} - 10 q^{19} - 10 q^{21} + 14 q^{23} + 3 q^{25} + 6 q^{29} - 15 q^{31} - 52 q^{33} - 7 q^{35} + 9 q^{37} - 52 q^{41} - 7 q^{43} - 30 q^{45} - 7 q^{47} - 6 q^{49} + 3 q^{51} - 18 q^{53} - 39 q^{55} - 2 q^{57} + 22 q^{59} - 42 q^{61} - 35 q^{65} - 28 q^{67} - 10 q^{69} + 23 q^{71} - 33 q^{73} - 3 q^{75} - 28 q^{77} - 30 q^{79} - 2 q^{81} + 11 q^{83} - 3 q^{85} + 27 q^{87} - 34 q^{89} - 18 q^{91} - 28 q^{93} - 10 q^{95} - 3 q^{97} - 28 q^{99}+O(q^{100}) \) 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9 + 2 * q^11 - 9 * q^13 - 7 * q^15 - 22 * q^17 - 10 * q^19 - 10 * q^21 + 14 * q^23 + 3 * q^25 + 6 * q^29 - 15 * q^31 - 52 * q^33 - 7 * q^35 + 9 * q^37 - 52 * q^41 - 7 * q^43 - 30 * q^45 - 7 * q^47 - 6 * q^49 + 3 * q^51 - 18 * q^53 - 39 * q^55 - 2 * q^57 + 22 * q^59 - 42 * q^61 - 35 * q^65 - 28 * q^67 - 10 * q^69 + 23 * q^71 - 33 * q^73 - 3 * q^75 - 28 * q^77 - 30 * q^79 - 2 * q^81 + 11 * q^83 - 3 * q^85 + 27 * q^87 - 34 * q^89 - 18 * q^91 - 28 * q^93 - 10 * q^95 - 3 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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