LMFDB - Embedded newform 8024.2.a.y.1.4

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:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-2.64010 q^{3} +3.09841 q^{5} -1.88185 q^{7} +3.97013 q^{9} +O(q^{10})$	$q-2.64010 q^{3} +3.09841 q^{5} -1.88185 q^{7} +3.97013 q^{9} +0.597371 q^{11} +2.98307 q^{13} -8.18011 q^{15} -1.00000 q^{17} +4.89604 q^{19} +4.96827 q^{21} -4.83192 q^{23} +4.60014 q^{25} -2.56124 q^{27} -2.75562 q^{29} -5.24822 q^{31} -1.57712 q^{33} -5.83074 q^{35} +1.62507 q^{37} -7.87561 q^{39} +0.567272 q^{41} -4.98079 q^{43} +12.3011 q^{45} -13.6250 q^{47} -3.45864 q^{49} +2.64010 q^{51} +12.5718 q^{53} +1.85090 q^{55} -12.9260 q^{57} -1.00000 q^{59} -3.62825 q^{61} -7.47118 q^{63} +9.24278 q^{65} +3.19537 q^{67} +12.7567 q^{69} -0.921306 q^{71} -6.23436 q^{73} -12.1448 q^{75} -1.12416 q^{77} +5.92392 q^{79} -5.14846 q^{81} +1.29606 q^{83} -3.09841 q^{85} +7.27512 q^{87} -11.8793 q^{89} -5.61369 q^{91} +13.8558 q^{93} +15.1699 q^{95} +18.6143 q^{97} +2.37164 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * 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
$3$	−2.64010	−1.52426	−0.762131	−	0.647423i	$-0.775847\pi$
$3$	−2.64010	−1.52426	−0.762131	+	0.647423i	$0.775847\pi$
$4$	0	0
$5$	3.09841	1.38565	0.692825	−	0.721105i	$-0.256366\pi$
$5$	3.09841	1.38565	0.692825	+	0.721105i	$0.256366\pi$
$6$	0	0
$7$	−1.88185	−0.711272	−0.355636	−	0.934625i	$-0.615736\pi$
$7$	−1.88185	−0.711272	−0.355636	+	0.934625i	$0.615736\pi$
$8$	0	0
$9$	3.97013	1.32338
$10$	0	0
$11$	0.597371	0.180114	0.0900571	−	0.995937i	$-0.471295\pi$
$11$	0.597371	0.180114	0.0900571	+	0.995937i	$0.471295\pi$
$12$	0	0
$13$	2.98307	0.827355	0.413678	−	0.910423i	$-0.364244\pi$
$13$	2.98307	0.827355	0.413678	+	0.910423i	$0.364244\pi$
$14$	0	0
$15$	−8.18011	−2.11210
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	4.89604	1.12323	0.561614	−	0.827399i	$-0.310181\pi$
$19$	4.89604	1.12323	0.561614	+	0.827399i	$0.310181\pi$
$20$	0	0
$21$	4.96827	1.08417
$22$	0	0
$23$	−4.83192	−1.00752	−0.503762	−	0.863843i	$-0.668051\pi$
$23$	−4.83192	−1.00752	−0.503762	+	0.863843i	$0.668051\pi$
$24$	0	0
$25$	4.60014	0.920028
$26$	0	0
$27$	−2.56124	−0.492910
$28$	0	0
$29$	−2.75562	−0.511706	−0.255853	−	0.966716i	$-0.582356\pi$
$29$	−2.75562	−0.511706	−0.255853	+	0.966716i	$0.582356\pi$
$30$	0	0
$31$	−5.24822	−0.942608	−0.471304	−	0.881971i	$-0.656217\pi$
$31$	−5.24822	−0.942608	−0.471304	+	0.881971i	$0.656217\pi$
$32$	0	0
$33$	−1.57712	−0.274541
$34$	0	0
$35$	−5.83074	−0.985575
$36$	0	0
$37$	1.62507	0.267160	0.133580	−	0.991038i	$-0.457353\pi$
$37$	1.62507	0.267160	0.133580	+	0.991038i	$0.457353\pi$
$38$	0	0
$39$	−7.87561	−1.26111
$40$	0	0
$41$	0.567272	0.0885930	0.0442965	−	0.999018i	$-0.485895\pi$
$41$	0.567272	0.0885930	0.0442965	+	0.999018i	$0.485895\pi$
$42$	0	0
$43$	−4.98079	−0.759563	−0.379782	−	0.925076i	$-0.624001\pi$
$43$	−4.98079	−0.759563	−0.379782	+	0.925076i	$0.624001\pi$
$44$	0	0
$45$	12.3011	1.83374
$46$	0	0
$47$	−13.6250	−1.98741	−0.993706	−	0.112024i	$-0.964267\pi$
$47$	−13.6250	−1.98741	−0.993706	+	0.112024i	$0.964267\pi$
$48$	0	0
$49$	−3.45864	−0.494092
$50$	0	0
$51$	2.64010	0.369688
$52$	0	0
$53$	12.5718	1.72686	0.863431	−	0.504466i	$-0.168311\pi$
$53$	12.5718	1.72686	0.863431	+	0.504466i	$0.168311\pi$
$54$	0	0
$55$	1.85090	0.249575
$56$	0	0
$57$	−12.9260	−1.71210
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−3.62825	−0.464550	−0.232275	−	0.972650i	$-0.574617\pi$
$61$	−3.62825	−0.464550	−0.232275	+	0.972650i	$0.574617\pi$
$62$	0	0
$63$	−7.47118	−0.941280
$64$	0	0
$65$	9.24278	1.14643
$66$	0	0
$67$	3.19537	0.390376	0.195188	−	0.980766i	$-0.437468\pi$
$67$	3.19537	0.390376	0.195188	+	0.980766i	$0.437468\pi$
$68$	0	0
$69$	12.7567	1.53573
$70$	0	0
$71$	−0.921306	−0.109339	−0.0546695	−	0.998505i	$-0.517411\pi$
$71$	−0.921306	−0.109339	−0.0546695	+	0.998505i	$0.517411\pi$
$72$	0	0
$73$	−6.23436	−0.729677	−0.364838	−	0.931071i	$-0.618876\pi$
$73$	−6.23436	−0.729677	−0.364838	+	0.931071i	$0.618876\pi$
$74$	0	0
$75$	−12.1448	−1.40236
$76$	0	0
$77$	−1.12416	−0.128110
$78$	0	0
$79$	5.92392	0.666493	0.333246	−	0.942840i	$-0.391856\pi$
$79$	5.92392	0.666493	0.333246	+	0.942840i	$0.391856\pi$
$80$	0	0
$81$	−5.14846	−0.572052
$82$	0	0
$83$	1.29606	0.142262	0.0711308	−	0.997467i	$-0.477339\pi$
$83$	1.29606	0.142262	0.0711308	+	0.997467i	$0.477339\pi$
$84$	0	0
$85$	−3.09841	−0.336070
$86$	0	0
$87$	7.27512	0.779975
$88$	0	0
$89$	−11.8793	−1.25920	−0.629602	−	0.776918i	$-0.716782\pi$
$89$	−11.8793	−1.25920	−0.629602	+	0.776918i	$0.716782\pi$
$90$	0	0
$91$	−5.61369	−0.588475
$92$	0	0
$93$	13.8558	1.43678
$94$	0	0
$95$	15.1699	1.55640
$96$	0	0
$97$	18.6143	1.88999	0.944996	−	0.327082i	$-0.106065\pi$
$97$	18.6143	1.88999	0.944996	+	0.327082i	$0.106065\pi$
$98$	0	0
$99$	2.37164	0.238359
$100$	0	0
$101$	−16.5349	−1.64528	−0.822640	−	0.568562i	$-0.807500\pi$
$101$	−16.5349	−1.64528	−0.822640	+	0.568562i	$0.807500\pi$
$102$	0	0
$103$	−5.63356	−0.555091	−0.277545	−	0.960713i	$-0.589521\pi$
$103$	−5.63356	−0.555091	−0.277545	+	0.960713i	$0.589521\pi$
$104$	0	0
$105$	15.3937	1.50227
$106$	0	0
$107$	15.1779	1.46731	0.733653	−	0.679524i	$-0.237814\pi$
$107$	15.1779	1.46731	0.733653	+	0.679524i	$0.237814\pi$
$108$	0	0
$109$	−19.3868	−1.85692	−0.928458	−	0.371437i	$-0.878865\pi$
$109$	−19.3868	−1.85692	−0.928458	+	0.371437i	$0.878865\pi$
$110$	0	0
$111$	−4.29034	−0.407221
$112$	0	0
$113$	13.7574	1.29418	0.647092	−	0.762412i	$-0.275985\pi$
$113$	13.7574	1.29418	0.647092	+	0.762412i	$0.275985\pi$
$114$	0	0
$115$	−14.9713	−1.39608
$116$	0	0
$117$	11.8432	1.09490
$118$	0	0
$119$	1.88185	0.172509
$120$	0	0
$121$	−10.6431	−0.967559
$122$	0	0
$123$	−1.49765	−0.135039
$124$	0	0
$125$	−1.23893	−0.110813
$126$	0	0
$127$	−21.2558	−1.88615	−0.943074	−	0.332582i	$-0.892080\pi$
$127$	−21.2558	−1.88615	−0.943074	+	0.332582i	$0.892080\pi$
$128$	0	0
$129$	13.1498	1.15777
$130$	0	0
$131$	11.3956	0.995636	0.497818	−	0.867282i	$-0.334135\pi$
$131$	11.3956	0.995636	0.497818	+	0.867282i	$0.334135\pi$
$132$	0	0
$133$	−9.21361	−0.798921
$134$	0	0
$135$	−7.93576	−0.683001
$136$	0	0
$137$	−0.174089	−0.0148734	−0.00743671	−	0.999972i	$-0.502367\pi$
$137$	−0.174089	−0.0148734	−0.00743671	+	0.999972i	$0.502367\pi$
$138$	0	0
$139$	4.60245	0.390374	0.195187	−	0.980766i	$-0.437469\pi$
$139$	4.60245	0.390374	0.195187	+	0.980766i	$0.437469\pi$
$140$	0	0
$141$	35.9714	3.02934
$142$	0	0
$143$	1.78200	0.149018
$144$	0	0
$145$	−8.53805	−0.709046
$146$	0	0
$147$	9.13117	0.753126
$148$	0	0
$149$	10.1384	0.830567	0.415284	−	0.909692i	$-0.363682\pi$
$149$	10.1384	0.830567	0.415284	+	0.909692i	$0.363682\pi$
$150$	0	0
$151$	−20.0810	−1.63417	−0.817084	−	0.576519i	$-0.804411\pi$
$151$	−20.0810	−1.63417	−0.817084	+	0.576519i	$0.804411\pi$
$152$	0	0
$153$	−3.97013	−0.320966
$154$	0	0
$155$	−16.2611	−1.30613
$156$	0	0
$157$	14.3233	1.14312	0.571561	−	0.820560i	$-0.306338\pi$
$157$	14.3233	1.14312	0.571561	+	0.820560i	$0.306338\pi$
$158$	0	0
$159$	−33.1907	−2.63219
$160$	0	0
$161$	9.09293	0.716624
$162$	0	0
$163$	2.58984	0.202852	0.101426	−	0.994843i	$-0.467660\pi$
$163$	2.58984	0.202852	0.101426	+	0.994843i	$0.467660\pi$
$164$	0	0
$165$	−4.88656	−0.380418
$166$	0	0
$167$	13.7417	1.06336	0.531681	−	0.846944i	$-0.321560\pi$
$167$	13.7417	1.06336	0.531681	+	0.846944i	$0.321560\pi$
$168$	0	0
$169$	−4.10128	−0.315483
$170$	0	0
$171$	19.4379	1.48645
$172$	0	0
$173$	−12.4631	−0.947554	−0.473777	−	0.880645i	$-0.657110\pi$
$173$	−12.4631	−0.947554	−0.473777	+	0.880645i	$0.657110\pi$
$174$	0	0
$175$	−8.65677	−0.654390
$176$	0	0
$177$	2.64010	0.198442
$178$	0	0
$179$	6.02229	0.450127	0.225064	−	0.974344i	$-0.427741\pi$
$179$	6.02229	0.450127	0.225064	+	0.974344i	$0.427741\pi$
$180$	0	0
$181$	20.0689	1.49171	0.745856	−	0.666107i	$-0.232041\pi$
$181$	20.0689	1.49171	0.745856	+	0.666107i	$0.232041\pi$
$182$	0	0
$183$	9.57895	0.708097
$184$	0	0
$185$	5.03513	0.370190
$186$	0	0
$187$	−0.597371	−0.0436841
$188$	0	0
$189$	4.81986	0.350593
$190$	0	0
$191$	−7.88848	−0.570791	−0.285395	−	0.958410i	$-0.592125\pi$
$191$	−7.88848	−0.570791	−0.285395	+	0.958410i	$0.592125\pi$
$192$	0	0
$193$	13.3107	0.958127	0.479064	−	0.877780i	$-0.340976\pi$
$193$	13.3107	0.958127	0.479064	+	0.877780i	$0.340976\pi$
$194$	0	0
$195$	−24.4019	−1.74745
$196$	0	0
$197$	−2.95073	−0.210231	−0.105115	−	0.994460i	$-0.533521\pi$
$197$	−2.95073	−0.210231	−0.105115	+	0.994460i	$0.533521\pi$
$198$	0	0
$199$	−17.0945	−1.21180	−0.605898	−	0.795542i	$-0.707186\pi$
$199$	−17.0945	−1.21180	−0.605898	+	0.795542i	$0.707186\pi$
$200$	0	0
$201$	−8.43609	−0.595036
$202$	0	0
$203$	5.18567	0.363963
$204$	0	0
$205$	1.75764	0.122759
$206$	0	0
$207$	−19.1833	−1.33333
$208$	0	0
$209$	2.92475	0.202309
$210$	0	0
$211$	13.2152	0.909774	0.454887	−	0.890549i	$-0.349680\pi$
$211$	13.2152	0.909774	0.454887	+	0.890549i	$0.349680\pi$
$212$	0	0
$213$	2.43234	0.166661
$214$	0	0
$215$	−15.4325	−1.05249
$216$	0	0
$217$	9.87636	0.670451
$218$	0	0
$219$	16.4593	1.11222
$220$	0	0
$221$	−2.98307	−0.200663
$222$	0	0
$223$	−16.0443	−1.07440	−0.537202	−	0.843454i	$-0.680519\pi$
$223$	−16.0443	−1.07440	−0.537202	+	0.843454i	$0.680519\pi$
$224$	0	0
$225$	18.2631	1.21754
$226$	0	0
$227$	13.8961	0.922318	0.461159	−	0.887317i	$-0.347434\pi$
$227$	13.8961	0.922318	0.461159	+	0.887317i	$0.347434\pi$
$228$	0	0
$229$	−0.397114	−0.0262421	−0.0131210	−	0.999914i	$-0.504177\pi$
$229$	−0.397114	−0.0262421	−0.0131210	+	0.999914i	$0.504177\pi$
$230$	0	0
$231$	2.96790	0.195273
$232$	0	0
$233$	−22.3412	−1.46362	−0.731810	−	0.681509i	$-0.761324\pi$
$233$	−22.3412	−1.46362	−0.731810	+	0.681509i	$0.761324\pi$
$234$	0	0
$235$	−42.2158	−2.75386
$236$	0	0
$237$	−15.6397	−1.01591
$238$	0	0
$239$	−24.6079	−1.59175	−0.795877	−	0.605459i	$-0.792990\pi$
$239$	−24.6079	−1.59175	−0.795877	+	0.605459i	$0.792990\pi$
$240$	0	0
$241$	−8.68646	−0.559544	−0.279772	−	0.960066i	$-0.590259\pi$
$241$	−8.68646	−0.559544	−0.279772	+	0.960066i	$0.590259\pi$
$242$	0	0
$243$	21.2762	1.36487
$244$	0	0
$245$	−10.7163	−0.684639
$246$	0	0
$247$	14.6052	0.929309
$248$	0	0
$249$	−3.42174	−0.216844
$250$	0	0
$251$	−21.0928	−1.33137	−0.665683	−	0.746235i	$-0.731860\pi$
$251$	−21.0928	−1.33137	−0.665683	+	0.746235i	$0.731860\pi$
$252$	0	0
$253$	−2.88645	−0.181469
$254$	0	0
$255$	8.18011	0.512258
$256$	0	0
$257$	−11.6927	−0.729373	−0.364686	−	0.931130i	$-0.618824\pi$
$257$	−11.6927	−0.729373	−0.364686	+	0.931130i	$0.618824\pi$
$258$	0	0
$259$	−3.05813	−0.190023
$260$	0	0
$261$	−10.9402	−0.677180
$262$	0	0
$263$	10.1676	0.626958	0.313479	−	0.949595i	$-0.398505\pi$
$263$	10.1676	0.626958	0.313479	+	0.949595i	$0.398505\pi$
$264$	0	0
$265$	38.9524	2.39283
$266$	0	0
$267$	31.3626	1.91936
$268$	0	0
$269$	27.8493	1.69800	0.849001	−	0.528391i	$-0.177205\pi$
$269$	27.8493	1.69800	0.849001	+	0.528391i	$0.177205\pi$
$270$	0	0
$271$	23.9963	1.45767	0.728834	−	0.684690i	$-0.240062\pi$
$271$	23.9963	1.45767	0.728834	+	0.684690i	$0.240062\pi$
$272$	0	0
$273$	14.8207	0.896990
$274$	0	0
$275$	2.74799	0.165710
$276$	0	0
$277$	−6.70010	−0.402570	−0.201285	−	0.979533i	$-0.564512\pi$
$277$	−6.70010	−0.402570	−0.201285	+	0.979533i	$0.564512\pi$
$278$	0	0
$279$	−20.8361	−1.24743
$280$	0	0
$281$	31.4486	1.87607	0.938033	−	0.346546i	$-0.112646\pi$
$281$	31.4486	1.87607	0.938033	+	0.346546i	$0.112646\pi$
$282$	0	0
$283$	22.1776	1.31832	0.659161	−	0.752002i	$-0.270912\pi$
$283$	22.1776	1.31832	0.659161	+	0.752002i	$0.270912\pi$
$284$	0	0
$285$	−40.0501	−2.37237
$286$	0	0
$287$	−1.06752	−0.0630137
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−49.1435	−2.88084
$292$	0	0
$293$	−16.5035	−0.964143	−0.482072	−	0.876132i	$-0.660115\pi$
$293$	−16.5035	−0.964143	−0.482072	+	0.876132i	$0.660115\pi$
$294$	0	0
$295$	−3.09841	−0.180396
$296$	0	0
$297$	−1.53001	−0.0887801
$298$	0	0
$299$	−14.4139	−0.833580
$300$	0	0
$301$	9.37309	0.540256
$302$	0	0
$303$	43.6537	2.50784
$304$	0	0
$305$	−11.2418	−0.643705
$306$	0	0
$307$	−23.2501	−1.32696	−0.663478	−	0.748196i	$-0.730920\pi$
$307$	−23.2501	−1.32696	−0.663478	+	0.748196i	$0.730920\pi$
$308$	0	0
$309$	14.8732	0.846104
$310$	0	0
$311$	−0.529911	−0.0300485	−0.0150242	−	0.999887i	$-0.504783\pi$
$311$	−0.529911	−0.0300485	−0.0150242	+	0.999887i	$0.504783\pi$
$312$	0	0
$313$	−7.27055	−0.410956	−0.205478	−	0.978662i	$-0.565875\pi$
$313$	−7.27055	−0.410956	−0.205478	+	0.978662i	$0.565875\pi$
$314$	0	0
$315$	−23.1488	−1.30429
$316$	0	0
$317$	−15.4420	−0.867309	−0.433654	−	0.901079i	$-0.642776\pi$
$317$	−15.4420	−0.867309	−0.433654	+	0.901079i	$0.642776\pi$
$318$	0	0
$319$	−1.64613	−0.0921656
$320$	0	0
$321$	−40.0713	−2.23656
$322$	0	0
$323$	−4.89604	−0.272423
$324$	0	0
$325$	13.7225	0.761190
$326$	0	0
$327$	51.1830	2.83043
$328$	0	0
$329$	25.6402	1.41359
$330$	0	0
$331$	24.7132	1.35836	0.679180	−	0.733971i	$-0.262335\pi$
$331$	24.7132	1.35836	0.679180	+	0.733971i	$0.262335\pi$
$332$	0	0
$333$	6.45173	0.353553
$334$	0	0
$335$	9.90056	0.540925
$336$	0	0
$337$	3.17452	0.172927	0.0864636	−	0.996255i	$-0.472443\pi$
$337$	3.17452	0.172927	0.0864636	+	0.996255i	$0.472443\pi$
$338$	0	0
$339$	−36.3208	−1.97268
$340$	0	0
$341$	−3.13513	−0.169777
$342$	0	0
$343$	19.6816	1.06271
$344$	0	0
$345$	39.5256	2.12799
$346$	0	0
$347$	6.12945	0.329046	0.164523	−	0.986373i	$-0.447391\pi$
$347$	6.12945	0.329046	0.164523	+	0.986373i	$0.447391\pi$
$348$	0	0
$349$	−16.9644	−0.908081	−0.454040	−	0.890981i	$-0.650018\pi$
$349$	−16.9644	−0.908081	−0.454040	+	0.890981i	$0.650018\pi$
$350$	0	0
$351$	−7.64035	−0.407812
$352$	0	0
$353$	21.9799	1.16987	0.584936	−	0.811080i	$-0.301120\pi$
$353$	21.9799	1.16987	0.584936	+	0.811080i	$0.301120\pi$
$354$	0	0
$355$	−2.85458	−0.151506
$356$	0	0
$357$	−4.96827	−0.262949
$358$	0	0
$359$	−37.1601	−1.96124	−0.980618	−	0.195930i	$-0.937227\pi$
$359$	−37.1601	−1.96124	−0.980618	+	0.195930i	$0.937227\pi$
$360$	0	0
$361$	4.97120	0.261642
$362$	0	0
$363$	28.0990	1.47481
$364$	0	0
$365$	−19.3166	−1.01108
$366$	0	0
$367$	15.9881	0.834569	0.417285	−	0.908776i	$-0.362982\pi$
$367$	15.9881	0.834569	0.417285	+	0.908776i	$0.362982\pi$
$368$	0	0
$369$	2.25214	0.117242
$370$	0	0
$371$	−23.6581	−1.22827
$372$	0	0
$373$	4.89983	0.253703	0.126852	−	0.991922i	$-0.459513\pi$
$373$	4.89983	0.253703	0.126852	+	0.991922i	$0.459513\pi$
$374$	0	0
$375$	3.27090	0.168909
$376$	0	0
$377$	−8.22022	−0.423363
$378$	0	0
$379$	−26.7490	−1.37400	−0.687002	−	0.726656i	$-0.741073\pi$
$379$	−26.7490	−1.37400	−0.687002	+	0.726656i	$0.741073\pi$
$380$	0	0
$381$	56.1175	2.87499
$382$	0	0
$383$	−20.9419	−1.07008	−0.535041	−	0.844826i	$-0.679704\pi$
$383$	−20.9419	−1.07008	−0.535041	+	0.844826i	$0.679704\pi$
$384$	0	0
$385$	−3.48311	−0.177516
$386$	0	0
$387$	−19.7744	−1.00519
$388$	0	0
$389$	−11.3189	−0.573893	−0.286946	−	0.957947i	$-0.592640\pi$
$389$	−11.3189	−0.573893	−0.286946	+	0.957947i	$0.592640\pi$
$390$	0	0
$391$	4.83192	0.244360
$392$	0	0
$393$	−30.0855	−1.51761
$394$	0	0
$395$	18.3547	0.923526
$396$	0	0
$397$	−20.2420	−1.01592	−0.507958	−	0.861382i	$-0.669599\pi$
$397$	−20.2420	−1.01592	−0.507958	+	0.861382i	$0.669599\pi$
$398$	0	0
$399$	24.3248	1.21777
$400$	0	0
$401$	4.91967	0.245677	0.122838	−	0.992427i	$-0.460800\pi$
$401$	4.91967	0.245677	0.122838	+	0.992427i	$0.460800\pi$
$402$	0	0
$403$	−15.6558	−0.779872
$404$	0	0
$405$	−15.9521	−0.792664
$406$	0	0
$407$	0.970769	0.0481192
$408$	0	0
$409$	1.70945	0.0845270	0.0422635	−	0.999106i	$-0.486543\pi$
$409$	1.70945	0.0845270	0.0422635	+	0.999106i	$0.486543\pi$
$410$	0	0
$411$	0.459612	0.0226710
$412$	0	0
$413$	1.88185	0.0925997
$414$	0	0
$415$	4.01574	0.197125
$416$	0	0
$417$	−12.1509	−0.595033
$418$	0	0
$419$	−40.6629	−1.98651	−0.993256	−	0.115939i	$-0.963012\pi$
$419$	−40.6629	−1.98651	−0.993256	+	0.115939i	$0.963012\pi$
$420$	0	0
$421$	−30.8825	−1.50512	−0.752561	−	0.658522i	$-0.771182\pi$
$421$	−30.8825	−1.50512	−0.752561	+	0.658522i	$0.771182\pi$
$422$	0	0
$423$	−54.0930	−2.63009
$424$	0	0
$425$	−4.60014	−0.223140
$426$	0	0
$427$	6.82783	0.330422
$428$	0	0
$429$	−4.70466	−0.227143
$430$	0	0
$431$	32.0926	1.54585	0.772924	−	0.634499i	$-0.218794\pi$
$431$	32.0926	1.54585	0.772924	+	0.634499i	$0.218794\pi$
$432$	0	0
$433$	16.2642	0.781610	0.390805	−	0.920474i	$-0.372197\pi$
$433$	16.2642	0.781610	0.390805	+	0.920474i	$0.372197\pi$
$434$	0	0
$435$	22.5413	1.08077
$436$	0	0
$437$	−23.6572	−1.13168
$438$	0	0
$439$	−18.0452	−0.861249	−0.430624	−	0.902531i	$-0.641707\pi$
$439$	−18.0452	−0.861249	−0.430624	+	0.902531i	$0.641707\pi$
$440$	0	0
$441$	−13.7313	−0.653870
$442$	0	0
$443$	6.71841	0.319202	0.159601	−	0.987182i	$-0.448979\pi$
$443$	6.71841	0.319202	0.159601	+	0.987182i	$0.448979\pi$
$444$	0	0
$445$	−36.8070	−1.74482
$446$	0	0
$447$	−26.7663	−1.26600
$448$	0	0
$449$	−39.0190	−1.84142	−0.920709	−	0.390250i	$-0.872389\pi$
$449$	−39.0190	−1.84142	−0.920709	+	0.390250i	$0.872389\pi$
$450$	0	0
$451$	0.338872	0.0159568
$452$	0	0
$453$	53.0158	2.49090
$454$	0	0
$455$	−17.3935	−0.815420
$456$	0	0
$457$	−18.1464	−0.848853	−0.424427	−	0.905462i	$-0.639524\pi$
$457$	−18.1464	−0.848853	−0.424427	+	0.905462i	$0.639524\pi$
$458$	0	0
$459$	2.56124	0.119548
$460$	0	0
$461$	−32.3716	−1.50769	−0.753847	−	0.657049i	$-0.771804\pi$
$461$	−32.3716	−1.50769	−0.753847	+	0.657049i	$0.771804\pi$
$462$	0	0
$463$	8.09908	0.376396	0.188198	−	0.982131i	$-0.439735\pi$
$463$	8.09908	0.376396	0.188198	+	0.982131i	$0.439735\pi$
$464$	0	0
$465$	42.9310	1.99088
$466$	0	0
$467$	14.0590	0.650572	0.325286	−	0.945616i	$-0.394539\pi$
$467$	14.0590	0.650572	0.325286	+	0.945616i	$0.394539\pi$
$468$	0	0
$469$	−6.01320	−0.277664
$470$	0	0
$471$	−37.8149	−1.74242
$472$	0	0
$473$	−2.97538	−0.136808
$474$	0	0
$475$	22.5225	1.03340
$476$	0	0
$477$	49.9115	2.28529
$478$	0	0
$479$	13.4726	0.615581	0.307791	−	0.951454i	$-0.400410\pi$
$479$	13.4726	0.615581	0.307791	+	0.951454i	$0.400410\pi$
$480$	0	0
$481$	4.84770	0.221036
$482$	0	0
$483$	−24.0063	−1.09232
$484$	0	0
$485$	57.6746	2.61887
$486$	0	0
$487$	10.1883	0.461674	0.230837	−	0.972992i	$-0.425854\pi$
$487$	10.1883	0.461674	0.230837	+	0.972992i	$0.425854\pi$
$488$	0	0
$489$	−6.83743	−0.309199
$490$	0	0
$491$	−34.9968	−1.57939	−0.789693	−	0.613502i	$-0.789760\pi$
$491$	−34.9968	−1.57939	−0.789693	+	0.613502i	$0.789760\pi$
$492$	0	0
$493$	2.75562	0.124107
$494$	0	0
$495$	7.34831	0.330282
$496$	0	0
$497$	1.73376	0.0777697
$498$	0	0
$499$	−21.5376	−0.964156	−0.482078	−	0.876128i	$-0.660118\pi$
$499$	−21.5376	−0.964156	−0.482078	+	0.876128i	$0.660118\pi$
$500$	0	0
$501$	−36.2794	−1.62084
$502$	0	0
$503$	−11.4976	−0.512653	−0.256327	−	0.966590i	$-0.582512\pi$
$503$	−11.4976	−0.512653	−0.256327	+	0.966590i	$0.582512\pi$
$504$	0	0
$505$	−51.2318	−2.27978
$506$	0	0
$507$	10.8278	0.480879
$508$	0	0
$509$	−19.9714	−0.885218	−0.442609	−	0.896715i	$-0.645947\pi$
$509$	−19.9714	−0.885218	−0.442609	+	0.896715i	$0.645947\pi$
$510$	0	0
$511$	11.7321	0.518999
$512$	0	0
$513$	−12.5399	−0.553651
$514$	0	0
$515$	−17.4551	−0.769162
$516$	0	0
$517$	−8.13918	−0.357961
$518$	0	0
$519$	32.9039	1.44432
$520$	0	0
$521$	−23.4369	−1.02679	−0.513395	−	0.858153i	$-0.671612\pi$
$521$	−23.4369	−1.02679	−0.513395	+	0.858153i	$0.671612\pi$
$522$	0	0
$523$	−12.6568	−0.553442	−0.276721	−	0.960950i	$-0.589248\pi$
$523$	−12.6568	−0.553442	−0.276721	+	0.960950i	$0.589248\pi$
$524$	0	0
$525$	22.8547	0.997462
$526$	0	0
$527$	5.24822	0.228616
$528$	0	0
$529$	0.347403	0.0151045
$530$	0	0
$531$	−3.97013	−0.172289
$532$	0	0
$533$	1.69221	0.0732979
$534$	0	0
$535$	47.0275	2.03317
$536$	0	0
$537$	−15.8994	−0.686112
$538$	0	0
$539$	−2.06609	−0.0889930
$540$	0	0
$541$	−34.5836	−1.48686	−0.743432	−	0.668812i	$-0.766803\pi$
$541$	−34.5836	−1.48686	−0.743432	+	0.668812i	$0.766803\pi$
$542$	0	0
$543$	−52.9840	−2.27376
$544$	0	0
$545$	−60.0682	−2.57304
$546$	0	0
$547$	−19.9236	−0.851871	−0.425936	−	0.904754i	$-0.640055\pi$
$547$	−19.9236	−0.851871	−0.425936	+	0.904754i	$0.640055\pi$
$548$	0	0
$549$	−14.4046	−0.614775
$550$	0	0
$551$	−13.4916	−0.574763
$552$	0	0
$553$	−11.1479	−0.474058
$554$	0	0
$555$	−13.2932	−0.564267
$556$	0	0
$557$	32.0465	1.35785	0.678927	−	0.734206i	$-0.262445\pi$
$557$	32.0465	1.35785	0.678927	+	0.734206i	$0.262445\pi$
$558$	0	0
$559$	−14.8580	−0.628429
$560$	0	0
$561$	1.57712	0.0665860
$562$	0	0
$563$	−5.95563	−0.251000	−0.125500	−	0.992094i	$-0.540053\pi$
$563$	−5.95563	−0.251000	−0.125500	+	0.992094i	$0.540053\pi$
$564$	0	0
$565$	42.6260	1.79329
$566$	0	0
$567$	9.68863	0.406884
$568$	0	0
$569$	40.2198	1.68610	0.843051	−	0.537833i	$-0.180757\pi$
$569$	40.2198	1.68610	0.843051	+	0.537833i	$0.180757\pi$
$570$	0	0
$571$	−33.5014	−1.40199	−0.700995	−	0.713166i	$-0.747261\pi$
$571$	−33.5014	−1.40199	−0.700995	+	0.713166i	$0.747261\pi$
$572$	0	0
$573$	20.8264	0.870035
$574$	0	0
$575$	−22.2275	−0.926950
$576$	0	0
$577$	−31.3005	−1.30306	−0.651528	−	0.758625i	$-0.725872\pi$
$577$	−31.3005	−1.30306	−0.651528	+	0.758625i	$0.725872\pi$
$578$	0	0
$579$	−35.1417	−1.46044
$580$	0	0
$581$	−2.43900	−0.101187
$582$	0	0
$583$	7.51000	0.311032
$584$	0	0
$585$	36.6950	1.51715
$586$	0	0
$587$	−7.49924	−0.309527	−0.154763	−	0.987952i	$-0.549461\pi$
$587$	−7.49924	−0.309527	−0.154763	+	0.987952i	$0.549461\pi$
$588$	0	0
$589$	−25.6955	−1.05876
$590$	0	0
$591$	7.79023	0.320447
$592$	0	0
$593$	12.9774	0.532917	0.266459	−	0.963846i	$-0.414146\pi$
$593$	12.9774	0.532917	0.266459	+	0.963846i	$0.414146\pi$
$594$	0	0
$595$	5.83074	0.239037
$596$	0	0
$597$	45.1311	1.84709
$598$	0	0
$599$	−19.2832	−0.787891	−0.393946	−	0.919134i	$-0.628890\pi$
$599$	−19.2832	−0.787891	−0.393946	+	0.919134i	$0.628890\pi$
$600$	0	0
$601$	−6.83449	−0.278785	−0.139392	−	0.990237i	$-0.544515\pi$
$601$	−6.83449	−0.278785	−0.139392	+	0.990237i	$0.544515\pi$
$602$	0	0
$603$	12.6860	0.516615
$604$	0	0
$605$	−32.9768	−1.34070
$606$	0	0
$607$	−2.33426	−0.0947445	−0.0473723	−	0.998877i	$-0.515085\pi$
$607$	−2.33426	−0.0947445	−0.0473723	+	0.998877i	$0.515085\pi$
$608$	0	0
$609$	−13.6907	−0.554774
$610$	0	0
$611$	−40.6444	−1.64429
$612$	0	0
$613$	22.3782	0.903846	0.451923	−	0.892057i	$-0.350738\pi$
$613$	22.3782	0.903846	0.451923	+	0.892057i	$0.350738\pi$
$614$	0	0
$615$	−4.64035	−0.187117
$616$	0	0
$617$	−15.2812	−0.615198	−0.307599	−	0.951516i	$-0.599525\pi$
$617$	−15.2812	−0.615198	−0.307599	+	0.951516i	$0.599525\pi$
$618$	0	0
$619$	43.8678	1.76320	0.881598	−	0.472000i	$-0.156468\pi$
$619$	43.8678	1.76320	0.881598	+	0.472000i	$0.156468\pi$
$620$	0	0
$621$	12.3757	0.496619
$622$	0	0
$623$	22.3551	0.895637
$624$	0	0
$625$	−26.8394	−1.07358
$626$	0	0
$627$	−7.72164	−0.308372
$628$	0	0
$629$	−1.62507	−0.0647957
$630$	0	0
$631$	−29.3235	−1.16735	−0.583675	−	0.811987i	$-0.698386\pi$
$631$	−29.3235	−1.16735	−0.583675	+	0.811987i	$0.698386\pi$
$632$	0	0
$633$	−34.8895	−1.38673
$634$	0	0
$635$	−65.8592	−2.61354
$636$	0	0
$637$	−10.3174	−0.408790
$638$	0	0
$639$	−3.65770	−0.144697
$640$	0	0
$641$	0.379717	0.0149979	0.00749897	−	0.999972i	$-0.497613\pi$
$641$	0.379717	0.0149979	0.00749897	+	0.999972i	$0.497613\pi$
$642$	0	0
$643$	−12.1713	−0.479991	−0.239995	−	0.970774i	$-0.577146\pi$
$643$	−12.1713	−0.479991	−0.239995	+	0.970774i	$0.577146\pi$
$644$	0	0
$645$	40.7434	1.60427
$646$	0	0
$647$	15.9328	0.626382	0.313191	−	0.949690i	$-0.398602\pi$
$647$	15.9328	0.626382	0.313191	+	0.949690i	$0.398602\pi$
$648$	0	0
$649$	−0.597371	−0.0234489
$650$	0	0
$651$	−26.0746	−1.02194
$652$	0	0
$653$	−17.0340	−0.666592	−0.333296	−	0.942822i	$-0.608161\pi$
$653$	−17.0340	−0.666592	−0.333296	+	0.942822i	$0.608161\pi$
$654$	0	0
$655$	35.3082	1.37960
$656$	0	0
$657$	−24.7512	−0.965637
$658$	0	0
$659$	−6.93126	−0.270004	−0.135002	−	0.990845i	$-0.543104\pi$
$659$	−6.93126	−0.270004	−0.135002	+	0.990845i	$0.543104\pi$
$660$	0	0
$661$	−13.1041	−0.509690	−0.254845	−	0.966982i	$-0.582024\pi$
$661$	−13.1041	−0.509690	−0.254845	+	0.966982i	$0.582024\pi$
$662$	0	0
$663$	7.87561	0.305863
$664$	0	0
$665$	−28.5475	−1.10703
$666$	0	0
$667$	13.3149	0.515557
$668$	0	0
$669$	42.3585	1.63767
$670$	0	0
$671$	−2.16741	−0.0836721
$672$	0	0
$673$	−25.8628	−0.996938	−0.498469	−	0.866907i	$-0.666104\pi$
$673$	−25.8628	−0.996938	−0.498469	+	0.866907i	$0.666104\pi$
$674$	0	0
$675$	−11.7820	−0.453491
$676$	0	0
$677$	−50.0947	−1.92530	−0.962648	−	0.270757i	$-0.912726\pi$
$677$	−50.0947	−1.92530	−0.962648	+	0.270757i	$0.912726\pi$
$678$	0	0
$679$	−35.0292	−1.34430
$680$	0	0
$681$	−36.6872	−1.40586
$682$	0	0
$683$	29.8403	1.14181	0.570903	−	0.821017i	$-0.306593\pi$
$683$	29.8403	1.14181	0.570903	+	0.821017i	$0.306593\pi$
$684$	0	0
$685$	−0.539398	−0.0206094
$686$	0	0
$687$	1.04842	0.0399998
$688$	0	0
$689$	37.5024	1.42873
$690$	0	0
$691$	30.9584	1.17771	0.588857	−	0.808237i	$-0.299578\pi$
$691$	30.9584	1.17771	0.588857	+	0.808237i	$0.299578\pi$
$692$	0	0
$693$	−4.46307	−0.169538
$694$	0	0
$695$	14.2603	0.540922
$696$	0	0
$697$	−0.567272	−0.0214870
$698$	0	0
$699$	58.9830	2.23094
$700$	0	0
$701$	−21.2361	−0.802078	−0.401039	−	0.916061i	$-0.631351\pi$
$701$	−21.2361	−0.802078	−0.401039	+	0.916061i	$0.631351\pi$
$702$	0	0
$703$	7.95640	0.300081
$704$	0	0
$705$	111.454	4.19760
$706$	0	0
$707$	31.1161	1.17024
$708$	0	0
$709$	−29.2617	−1.09895	−0.549473	−	0.835511i	$-0.685172\pi$
$709$	−29.2617	−1.09895	−0.549473	+	0.835511i	$0.685172\pi$
$710$	0	0
$711$	23.5187	0.882020
$712$	0	0
$713$	25.3590	0.949700
$714$	0	0
$715$	5.52137	0.206487
$716$	0	0
$717$	64.9673	2.42625
$718$	0	0
$719$	−2.65987	−0.0991965	−0.0495982	−	0.998769i	$-0.515794\pi$
$719$	−2.65987	−0.0991965	−0.0495982	+	0.998769i	$0.515794\pi$
$720$	0	0
$721$	10.6015	0.394821
$722$	0	0
$723$	22.9331	0.852892
$724$	0	0
$725$	−12.6763	−0.470784
$726$	0	0
$727$	−7.80268	−0.289386	−0.144693	−	0.989477i	$-0.546219\pi$
$727$	−7.80268	−0.289386	−0.144693	+	0.989477i	$0.546219\pi$
$728$	0	0
$729$	−40.7258	−1.50836
$730$	0	0
$731$	4.98079	0.184221
$732$	0	0
$733$	39.1888	1.44747	0.723735	−	0.690078i	$-0.242424\pi$
$733$	39.1888	1.44747	0.723735	+	0.690078i	$0.242424\pi$
$734$	0	0
$735$	28.2921	1.04357
$736$	0	0
$737$	1.90882	0.0703123
$738$	0	0
$739$	9.29964	0.342093	0.171046	−	0.985263i	$-0.445285\pi$
$739$	9.29964	0.342093	0.171046	+	0.985263i	$0.445285\pi$
$740$	0	0
$741$	−38.5593	−1.41651
$742$	0	0
$743$	−12.4069	−0.455166	−0.227583	−	0.973759i	$-0.573082\pi$
$743$	−12.4069	−0.455166	−0.227583	+	0.973759i	$0.573082\pi$
$744$	0	0
$745$	31.4128	1.15088
$746$	0	0
$747$	5.14554	0.188266
$748$	0	0
$749$	−28.5626	−1.04365
$750$	0	0
$751$	−10.2605	−0.374412	−0.187206	−	0.982321i	$-0.559943\pi$
$751$	−10.2605	−0.374412	−0.187206	+	0.982321i	$0.559943\pi$
$752$	0	0
$753$	55.6871	2.02935
$754$	0	0
$755$	−62.2191	−2.26439
$756$	0	0
$757$	53.8023	1.95548	0.977740	−	0.209822i	$-0.0672884\pi$
$757$	53.8023	1.95548	0.977740	+	0.209822i	$0.0672884\pi$
$758$	0	0
$759$	7.62051	0.276607
$760$	0	0
$761$	−18.7034	−0.677999	−0.339000	−	0.940786i	$-0.610089\pi$
$761$	−18.7034	−0.677999	−0.339000	+	0.940786i	$0.610089\pi$
$762$	0	0
$763$	36.4830	1.32077
$764$	0	0
$765$	−12.3011	−0.444747
$766$	0	0
$767$	−2.98307	−0.107712
$768$	0	0
$769$	20.7356	0.747746	0.373873	−	0.927480i	$-0.378030\pi$
$769$	20.7356	0.747746	0.373873	+	0.927480i	$0.378030\pi$
$770$	0	0
$771$	30.8700	1.11176
$772$	0	0
$773$	10.6824	0.384219	0.192109	−	0.981374i	$-0.438467\pi$
$773$	10.6824	0.384219	0.192109	+	0.981374i	$0.438467\pi$
$774$	0	0
$775$	−24.1425	−0.867226
$776$	0	0
$777$	8.07378	0.289645
$778$	0	0
$779$	2.77739	0.0995102
$780$	0	0
$781$	−0.550361	−0.0196935
$782$	0	0
$783$	7.05780	0.252225
$784$	0	0
$785$	44.3794	1.58397
$786$	0	0
$787$	−46.1603	−1.64544	−0.822718	−	0.568449i	$-0.807544\pi$
$787$	−46.1603	−1.64544	−0.822718	+	0.568449i	$0.807544\pi$
$788$	0	0
$789$	−26.8434	−0.955649
$790$	0	0
$791$	−25.8893	−0.920517
$792$	0	0
$793$	−10.8233	−0.384348
$794$	0	0
$795$	−102.838	−3.64730
$796$	0	0
$797$	−49.8192	−1.76469	−0.882343	−	0.470608i	$-0.844035\pi$
$797$	−49.8192	−1.76469	−0.882343	+	0.470608i	$0.844035\pi$
$798$	0	0
$799$	13.6250	0.482018
$800$	0	0
$801$	−47.1624	−1.66640
$802$	0	0
$803$	−3.72423	−0.131425
$804$	0	0
$805$	28.1736	0.992990
$806$	0	0
$807$	−73.5249	−2.58820
$808$	0	0
$809$	−33.6142	−1.18181	−0.590907	−	0.806740i	$-0.701230\pi$
$809$	−33.6142	−1.18181	−0.590907	+	0.806740i	$0.701230\pi$
$810$	0	0
$811$	45.8223	1.60904	0.804518	−	0.593928i	$-0.202423\pi$
$811$	45.8223	1.60904	0.804518	+	0.593928i	$0.202423\pi$
$812$	0	0
$813$	−63.3525	−2.22187
$814$	0	0
$815$	8.02437	0.281081
$816$	0	0
$817$	−24.3861	−0.853163
$818$	0	0
$819$	−22.2871	−0.778773
$820$	0	0
$821$	39.8311	1.39012	0.695058	−	0.718954i	$-0.255379\pi$
$821$	39.8311	1.39012	0.695058	+	0.718954i	$0.255379\pi$
$822$	0	0
$823$	47.9620	1.67185	0.835925	−	0.548844i	$-0.184932\pi$
$823$	47.9620	1.67185	0.835925	+	0.548844i	$0.184932\pi$
$824$	0	0
$825$	−7.25497	−0.252586
$826$	0	0
$827$	17.7749	0.618093	0.309046	−	0.951047i	$-0.399990\pi$
$827$	17.7749	0.618093	0.309046	+	0.951047i	$0.399990\pi$
$828$	0	0
$829$	24.4615	0.849582	0.424791	−	0.905292i	$-0.360348\pi$
$829$	24.4615	0.849582	0.424791	+	0.905292i	$0.360348\pi$
$830$	0	0
$831$	17.6889	0.613622
$832$	0	0
$833$	3.45864	0.119835
$834$	0	0
$835$	42.5773	1.47345
$836$	0	0
$837$	13.4419	0.464621
$838$	0	0
$839$	41.1306	1.41998	0.709992	−	0.704209i	$-0.248698\pi$
$839$	41.1306	1.41998	0.709992	+	0.704209i	$0.248698\pi$
$840$	0	0
$841$	−21.4065	−0.738156
$842$	0	0
$843$	−83.0274	−2.85962
$844$	0	0
$845$	−12.7075	−0.437150
$846$	0	0
$847$	20.0288	0.688198
$848$	0	0
$849$	−58.5511	−2.00947
$850$	0	0
$851$	−7.85219	−0.269170
$852$	0	0
$853$	−17.5746	−0.601743	−0.300871	−	0.953665i	$-0.597277\pi$
$853$	−17.5746	−0.601743	−0.300871	+	0.953665i	$0.597277\pi$
$854$	0	0
$855$	60.2266	2.05971
$856$	0	0
$857$	21.8219	0.745421	0.372710	−	0.927948i	$-0.378428\pi$
$857$	21.8219	0.745421	0.372710	+	0.927948i	$0.378428\pi$
$858$	0	0
$859$	13.8327	0.471964	0.235982	−	0.971757i	$-0.424169\pi$
$859$	13.8327	0.471964	0.235982	+	0.971757i	$0.424169\pi$
$860$	0	0
$861$	2.81836	0.0960494
$862$	0	0
$863$	−44.7815	−1.52438	−0.762190	−	0.647353i	$-0.775876\pi$
$863$	−44.7815	−1.52438	−0.762190	+	0.647353i	$0.775876\pi$
$864$	0	0
$865$	−38.6159	−1.31298
$866$	0	0
$867$	−2.64010	−0.0896625
$868$	0	0
$869$	3.53878	0.120045
$870$	0	0
$871$	9.53201	0.322980
$872$	0	0
$873$	73.9010	2.50117
$874$	0	0
$875$	2.33148	0.0788184
$876$	0	0
$877$	−2.87835	−0.0971948	−0.0485974	−	0.998818i	$-0.515475\pi$
$877$	−2.87835	−0.0971948	−0.0485974	+	0.998818i	$0.515475\pi$
$878$	0	0
$879$	43.5708	1.46961
$880$	0	0
$881$	−18.2968	−0.616433	−0.308217	−	0.951316i	$-0.599732\pi$
$881$	−18.2968	−0.616433	−0.308217	+	0.951316i	$0.599732\pi$
$882$	0	0
$883$	−8.24553	−0.277484	−0.138742	−	0.990329i	$-0.544306\pi$
$883$	−8.24553	−0.277484	−0.138742	+	0.990329i	$0.544306\pi$
$884$	0	0
$885$	8.18011	0.274971
$886$	0	0
$887$	49.9377	1.67674	0.838372	−	0.545098i	$-0.183508\pi$
$887$	49.9377	1.67674	0.838372	+	0.545098i	$0.183508\pi$
$888$	0	0
$889$	40.0002	1.34156
$890$	0	0
$891$	−3.07554	−0.103035
$892$	0	0
$893$	−66.7086	−2.23232
$894$	0	0
$895$	18.6595	0.623719
$896$	0	0
$897$	38.0543	1.27059
$898$	0	0
$899$	14.4621	0.482339
$900$	0	0
$901$	−12.5718	−0.418826
$902$	0	0
$903$	−24.7459	−0.823492
$904$	0	0
$905$	62.1818	2.06699
$906$	0	0
$907$	10.8602	0.360606	0.180303	−	0.983611i	$-0.442292\pi$
$907$	10.8602	0.360606	0.180303	+	0.983611i	$0.442292\pi$
$908$	0	0
$909$	−65.6455	−2.17733
$910$	0	0
$911$	−19.6137	−0.649831	−0.324916	−	0.945743i	$-0.605336\pi$
$911$	−19.6137	−0.649831	−0.324916	+	0.945743i	$0.605336\pi$
$912$	0	0
$913$	0.774231	0.0256233
$914$	0	0
$915$	29.6795	0.981175
$916$	0	0
$917$	−21.4447	−0.708168
$918$	0	0
$919$	40.3847	1.33217	0.666085	−	0.745876i	$-0.267969\pi$
$919$	40.3847	1.33217	0.666085	+	0.745876i	$0.267969\pi$
$920$	0	0
$921$	61.3827	2.02263
$922$	0	0
$923$	−2.74832	−0.0904621
$924$	0	0
$925$	7.47554	0.245794
$926$	0	0
$927$	−22.3659	−0.734594
$928$	0	0
$929$	−0.151599	−0.00497379	−0.00248689	−	0.999997i	$-0.500792\pi$
$929$	−0.151599	−0.00497379	−0.00248689	+	0.999997i	$0.500792\pi$
$930$	0	0
$931$	−16.9337	−0.554978
$932$	0	0
$933$	1.39902	0.0458018
$934$	0	0
$935$	−1.85090	−0.0605309
$936$	0	0
$937$	8.52978	0.278656	0.139328	−	0.990246i	$-0.455506\pi$
$937$	8.52978	0.278656	0.139328	+	0.990246i	$0.455506\pi$
$938$	0	0
$939$	19.1950	0.626405
$940$	0	0
$941$	10.9745	0.357759	0.178880	−	0.983871i	$-0.442753\pi$
$941$	10.9745	0.357759	0.178880	+	0.983871i	$0.442753\pi$
$942$	0	0
$943$	−2.74101	−0.0892596
$944$	0	0
$945$	14.9339	0.485800
$946$	0	0
$947$	−24.3297	−0.790609	−0.395305	−	0.918550i	$-0.629361\pi$
$947$	−24.3297	−0.790609	−0.395305	+	0.918550i	$0.629361\pi$
$948$	0	0
$949$	−18.5975	−0.603702
$950$	0	0
$951$	40.7684	1.32201
$952$	0	0
$953$	−36.0423	−1.16752	−0.583762	−	0.811925i	$-0.698420\pi$
$953$	−36.0423	−1.16752	−0.583762	+	0.811925i	$0.698420\pi$
$954$	0	0
$955$	−24.4418	−0.790917
$956$	0	0
$957$	4.34595	0.140485
$958$	0	0
$959$	0.327609	0.0105790
$960$	0	0
$961$	−3.45618	−0.111490
$962$	0	0
$963$	60.2583	1.94180
$964$	0	0
$965$	41.2421	1.32763
$966$	0	0
$967$	−19.4758	−0.626298	−0.313149	−	0.949704i	$-0.601384\pi$
$967$	−19.4758	−0.626298	−0.313149	+	0.949704i	$0.601384\pi$
$968$	0	0
$969$	12.9260	0.415244
$970$	0	0
$971$	−11.1700	−0.358462	−0.179231	−	0.983807i	$-0.557361\pi$
$971$	−11.1700	−0.358462	−0.179231	+	0.983807i	$0.557361\pi$
$972$	0	0
$973$	−8.66111	−0.277662
$974$	0	0
$975$	−36.2289	−1.16025
$976$	0	0
$977$	0.289206	0.00925253	0.00462627	−	0.999989i	$-0.498527\pi$
$977$	0.289206	0.00925253	0.00462627	+	0.999989i	$0.498527\pi$
$978$	0	0
$979$	−7.09635	−0.226800
$980$	0	0
$981$	−76.9680	−2.45740
$982$	0	0
$983$	−16.1293	−0.514446	−0.257223	−	0.966352i	$-0.582807\pi$
$983$	−16.1293	−0.514446	−0.257223	+	0.966352i	$0.582807\pi$
$984$	0	0
$985$	−9.14257	−0.291307
$986$	0	0
$987$	−67.6927	−2.15468
$988$	0	0
$989$	24.0667	0.765278
$990$	0	0
$991$	30.0622	0.954958	0.477479	−	0.878643i	$-0.341551\pi$
$991$	30.0622	0.954958	0.477479	+	0.878643i	$0.341551\pi$
$992$	0	0
$993$	−65.2454	−2.07050
$994$	0	0
$995$	−52.9657	−1.67913
$996$	0	0
$997$	−55.7586	−1.76589	−0.882946	−	0.469474i	$-0.844443\pi$
$997$	−55.7586	−1.76589	−0.882946	+	0.469474i	$0.844443\pi$
$998$	0	0
$999$	−4.16219	−0.131686

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.y.1.4	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.y.1.4	✓	23	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-2.64010 q^{3} +3.09841 q^{5} -1.88185 q^{7} +3.97013 q^{9} +O(q^{10})\)	\(q-2.64010 q^{3} +3.09841 q^{5} -1.88185 q^{7} +3.97013 q^{9} +0.597371 q^{11} +2.98307 q^{13} -8.18011 q^{15} -1.00000 q^{17} +4.89604 q^{19} +4.96827 q^{21} -4.83192 q^{23} +4.60014 q^{25} -2.56124 q^{27} -2.75562 q^{29} -5.24822 q^{31} -1.57712 q^{33} -5.83074 q^{35} +1.62507 q^{37} -7.87561 q^{39} +0.567272 q^{41} -4.98079 q^{43} +12.3011 q^{45} -13.6250 q^{47} -3.45864 q^{49} +2.64010 q^{51} +12.5718 q^{53} +1.85090 q^{55} -12.9260 q^{57} -1.00000 q^{59} -3.62825 q^{61} -7.47118 q^{63} +9.24278 q^{65} +3.19537 q^{67} +12.7567 q^{69} -0.921306 q^{71} -6.23436 q^{73} -12.1448 q^{75} -1.12416 q^{77} +5.92392 q^{79} -5.14846 q^{81} +1.29606 q^{83} -3.09841 q^{85} +7.27512 q^{87} -11.8793 q^{89} -5.61369 q^{91} +13.8558 q^{93} +15.1699 q^{95} +18.6143 q^{97} +2.37164 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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