LMFDB - Embedded newform 8024.2.a.z.1.16

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			1.16
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)

$f(q)$	$=$	$q+1.18884 q^{3} +2.32992 q^{5} -1.08746 q^{7} -1.58666 q^{9} +O(q^{10})$	$q+1.18884 q^{3} +2.32992 q^{5} -1.08746 q^{7} -1.58666 q^{9} -2.20244 q^{11} -2.34980 q^{13} +2.76990 q^{15} +1.00000 q^{17} +5.72648 q^{19} -1.29282 q^{21} -1.43649 q^{23} +0.428518 q^{25} -5.45281 q^{27} +9.96495 q^{29} -4.20479 q^{31} -2.61835 q^{33} -2.53370 q^{35} -8.24478 q^{37} -2.79354 q^{39} -1.83311 q^{41} -8.43087 q^{43} -3.69678 q^{45} -6.34454 q^{47} -5.81742 q^{49} +1.18884 q^{51} -6.76676 q^{53} -5.13151 q^{55} +6.80787 q^{57} +1.00000 q^{59} +11.2963 q^{61} +1.72543 q^{63} -5.47485 q^{65} +5.66963 q^{67} -1.70775 q^{69} -13.8681 q^{71} -4.79161 q^{73} +0.509439 q^{75} +2.39508 q^{77} +12.9169 q^{79} -1.72254 q^{81} +0.463991 q^{83} +2.32992 q^{85} +11.8467 q^{87} -3.64787 q^{89} +2.55533 q^{91} -4.99883 q^{93} +13.3422 q^{95} +3.88374 q^{97} +3.49452 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.18884	0.686378	0.343189	−	0.939266i	$-0.388493\pi$
$3$	1.18884	0.686378	0.343189	+	0.939266i	$0.388493\pi$
$4$	0	0
$5$	2.32992	1.04197	0.520985	−	0.853566i	$-0.325565\pi$
$5$	2.32992	1.04197	0.520985	+	0.853566i	$0.325565\pi$
$6$	0	0
$7$	−1.08746	−0.411023	−0.205511	−	0.978655i	$-0.565886\pi$
$7$	−1.08746	−0.411023	−0.205511	+	0.978655i	$0.565886\pi$
$8$	0	0
$9$	−1.58666	−0.528886
$10$	0	0
$11$	−2.20244	−0.664061	−0.332031	−	0.943269i	$-0.607734\pi$
$11$	−2.20244	−0.664061	−0.332031	+	0.943269i	$0.607734\pi$
$12$	0	0
$13$	−2.34980	−0.651718	−0.325859	−	0.945418i	$-0.605653\pi$
$13$	−2.34980	−0.651718	−0.325859	+	0.945418i	$0.605653\pi$
$14$	0	0
$15$	2.76990	0.715185
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	5.72648	1.31374	0.656872	−	0.754002i	$-0.271879\pi$
$19$	5.72648	1.31374	0.656872	+	0.754002i	$0.271879\pi$
$20$	0	0
$21$	−1.29282	−0.282117
$22$	0	0
$23$	−1.43649	−0.299528	−0.149764	−	0.988722i	$-0.547851\pi$
$23$	−1.43649	−0.299528	−0.149764	+	0.988722i	$0.547851\pi$
$24$	0	0
$25$	0.428518	0.0857036
$26$	0	0
$27$	−5.45281	−1.04939
$28$	0	0
$29$	9.96495	1.85044	0.925222	−	0.379426i	$-0.123878\pi$
$29$	9.96495	1.85044	0.925222	+	0.379426i	$0.123878\pi$
$30$	0	0
$31$	−4.20479	−0.755203	−0.377602	−	0.925968i	$-0.623251\pi$
$31$	−4.20479	−0.755203	−0.377602	+	0.925968i	$0.623251\pi$
$32$	0	0
$33$	−2.61835	−0.455797
$34$	0	0
$35$	−2.53370	−0.428274
$36$	0	0
$37$	−8.24478	−1.35543	−0.677717	−	0.735323i	$-0.737030\pi$
$37$	−8.24478	−1.35543	−0.677717	+	0.735323i	$0.737030\pi$
$38$	0	0
$39$	−2.79354	−0.447324
$40$	0	0
$41$	−1.83311	−0.286284	−0.143142	−	0.989702i	$-0.545721\pi$
$41$	−1.83311	−0.286284	−0.143142	+	0.989702i	$0.545721\pi$
$42$	0	0
$43$	−8.43087	−1.28570	−0.642848	−	0.765994i	$-0.722247\pi$
$43$	−8.43087	−1.28570	−0.642848	+	0.765994i	$0.722247\pi$
$44$	0	0
$45$	−3.69678	−0.551084
$46$	0	0
$47$	−6.34454	−0.925447	−0.462723	−	0.886503i	$-0.653128\pi$
$47$	−6.34454	−0.925447	−0.462723	+	0.886503i	$0.653128\pi$
$48$	0	0
$49$	−5.81742	−0.831060
$50$	0	0
$51$	1.18884	0.166471
$52$	0	0
$53$	−6.76676	−0.929486	−0.464743	−	0.885446i	$-0.653853\pi$
$53$	−6.76676	−0.929486	−0.464743	+	0.885446i	$0.653853\pi$
$54$	0	0
$55$	−5.13151	−0.691933
$56$	0	0
$57$	6.80787	0.901724
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	11.2963	1.44634	0.723171	−	0.690669i	$-0.242684\pi$
$61$	11.2963	1.44634	0.723171	+	0.690669i	$0.242684\pi$
$62$	0	0
$63$	1.72543	0.217384
$64$	0	0
$65$	−5.47485	−0.679071
$66$	0	0
$67$	5.66963	0.692656	0.346328	−	0.938113i	$-0.387428\pi$
$67$	5.66963	0.692656	0.346328	+	0.938113i	$0.387428\pi$
$68$	0	0
$69$	−1.70775	−0.205589
$70$	0	0
$71$	−13.8681	−1.64584	−0.822918	−	0.568161i	$-0.807655\pi$
$71$	−13.8681	−1.64584	−0.822918	+	0.568161i	$0.807655\pi$
$72$	0	0
$73$	−4.79161	−0.560815	−0.280408	−	0.959881i	$-0.590470\pi$
$73$	−4.79161	−0.560815	−0.280408	+	0.959881i	$0.590470\pi$
$74$	0	0
$75$	0.509439	0.0588250
$76$	0	0
$77$	2.39508	0.272944
$78$	0	0
$79$	12.9169	1.45326	0.726631	−	0.687028i	$-0.241085\pi$
$79$	12.9169	1.45326	0.726631	+	0.687028i	$0.241085\pi$
$80$	0	0
$81$	−1.72254	−0.191394
$82$	0	0
$83$	0.463991	0.0509296	0.0254648	−	0.999676i	$-0.491893\pi$
$83$	0.463991	0.0509296	0.0254648	+	0.999676i	$0.491893\pi$
$84$	0	0
$85$	2.32992	0.252715
$86$	0	0
$87$	11.8467	1.27010
$88$	0	0
$89$	−3.64787	−0.386673	−0.193337	−	0.981132i	$-0.561931\pi$
$89$	−3.64787	−0.386673	−0.193337	+	0.981132i	$0.561931\pi$
$90$	0	0
$91$	2.55533	0.267871
$92$	0	0
$93$	−4.99883	−0.518355
$94$	0	0
$95$	13.3422	1.36888
$96$	0	0
$97$	3.88374	0.394334	0.197167	−	0.980370i	$-0.436826\pi$
$97$	3.88374	0.394334	0.197167	+	0.980370i	$0.436826\pi$
$98$	0	0
$99$	3.49452	0.351213
$100$	0	0
$101$	−4.47470	−0.445249	−0.222624	−	0.974904i	$-0.571462\pi$
$101$	−4.47470	−0.445249	−0.222624	+	0.974904i	$0.571462\pi$
$102$	0	0
$103$	−9.41604	−0.927790	−0.463895	−	0.885890i	$-0.653548\pi$
$103$	−9.41604	−0.927790	−0.463895	+	0.885890i	$0.653548\pi$
$104$	0	0
$105$	−3.01217	−0.293958
$106$	0	0
$107$	11.3129	1.09366	0.546828	−	0.837245i	$-0.315835\pi$
$107$	11.3129	1.09366	0.546828	+	0.837245i	$0.315835\pi$
$108$	0	0
$109$	16.7663	1.60592	0.802959	−	0.596034i	$-0.203258\pi$
$109$	16.7663	1.60592	0.802959	+	0.596034i	$0.203258\pi$
$110$	0	0
$111$	−9.80173	−0.930339
$112$	0	0
$113$	−14.7545	−1.38798	−0.693991	−	0.719983i	$-0.744149\pi$
$113$	−14.7545	−1.38798	−0.693991	+	0.719983i	$0.744149\pi$
$114$	0	0
$115$	−3.34690	−0.312100
$116$	0	0
$117$	3.72833	0.344684
$118$	0	0
$119$	−1.08746	−0.0996877
$120$	0	0
$121$	−6.14925	−0.559023
$122$	0	0
$123$	−2.17928	−0.196499
$124$	0	0
$125$	−10.6512	−0.952670
$126$	0	0
$127$	−6.92686	−0.614660	−0.307330	−	0.951603i	$-0.599435\pi$
$127$	−6.92686	−0.614660	−0.307330	+	0.951603i	$0.599435\pi$
$128$	0	0
$129$	−10.0230	−0.882472
$130$	0	0
$131$	−2.20295	−0.192472	−0.0962361	−	0.995359i	$-0.530680\pi$
$131$	−2.20295	−0.192472	−0.0962361	+	0.995359i	$0.530680\pi$
$132$	0	0
$133$	−6.22734	−0.539979
$134$	0	0
$135$	−12.7046	−1.09344
$136$	0	0
$137$	17.1677	1.46674	0.733370	−	0.679830i	$-0.237946\pi$
$137$	17.1677	1.46674	0.733370	+	0.679830i	$0.237946\pi$
$138$	0	0
$139$	−19.0880	−1.61902	−0.809511	−	0.587105i	$-0.800267\pi$
$139$	−19.0880	−1.61902	−0.809511	+	0.587105i	$0.800267\pi$
$140$	0	0
$141$	−7.54265	−0.635206
$142$	0	0
$143$	5.17530	0.432781
$144$	0	0
$145$	23.2175	1.92811
$146$	0	0
$147$	−6.91599	−0.570421
$148$	0	0
$149$	−19.7308	−1.61641	−0.808204	−	0.588902i	$-0.799560\pi$
$149$	−19.7308	−1.61641	−0.808204	+	0.588902i	$0.799560\pi$
$150$	0	0
$151$	−14.1199	−1.14907	−0.574533	−	0.818482i	$-0.694816\pi$
$151$	−14.1199	−1.14907	−0.574533	+	0.818482i	$0.694816\pi$
$152$	0	0
$153$	−1.58666	−0.128274
$154$	0	0
$155$	−9.79682	−0.786900
$156$	0	0
$157$	15.8596	1.26574	0.632869	−	0.774259i	$-0.281877\pi$
$157$	15.8596	1.26574	0.632869	+	0.774259i	$0.281877\pi$
$158$	0	0
$159$	−8.04460	−0.637978
$160$	0	0
$161$	1.56213	0.123113
$162$	0	0
$163$	2.00954	0.157399	0.0786997	−	0.996898i	$-0.474923\pi$
$163$	2.00954	0.157399	0.0786997	+	0.996898i	$0.474923\pi$
$164$	0	0
$165$	−6.10055	−0.474927
$166$	0	0
$167$	−10.1618	−0.786341	−0.393171	−	0.919465i	$-0.628622\pi$
$167$	−10.1618	−0.786341	−0.393171	+	0.919465i	$0.628622\pi$
$168$	0	0
$169$	−7.47843	−0.575264
$170$	0	0
$171$	−9.08596	−0.694821
$172$	0	0
$173$	1.03603	0.0787682	0.0393841	−	0.999224i	$-0.487460\pi$
$173$	1.03603	0.0787682	0.0393841	+	0.999224i	$0.487460\pi$
$174$	0	0
$175$	−0.465998	−0.0352261
$176$	0	0
$177$	1.18884	0.0893587
$178$	0	0
$179$	−19.0090	−1.42080	−0.710398	−	0.703800i	$-0.751485\pi$
$179$	−19.0090	−1.42080	−0.710398	+	0.703800i	$0.751485\pi$
$180$	0	0
$181$	−25.6257	−1.90475	−0.952373	−	0.304934i	$-0.901365\pi$
$181$	−25.6257	−1.90475	−0.952373	+	0.304934i	$0.901365\pi$
$182$	0	0
$183$	13.4295	0.992737
$184$	0	0
$185$	−19.2097	−1.41232
$186$	0	0
$187$	−2.20244	−0.161059
$188$	0	0
$189$	5.92973	0.431324
$190$	0	0
$191$	10.6896	0.773473	0.386736	−	0.922190i	$-0.373602\pi$
$191$	10.6896	0.773473	0.386736	+	0.922190i	$0.373602\pi$
$192$	0	0
$193$	10.9295	0.786722	0.393361	−	0.919384i	$-0.371312\pi$
$193$	10.9295	0.786722	0.393361	+	0.919384i	$0.371312\pi$
$194$	0	0
$195$	−6.50872	−0.466099
$196$	0	0
$197$	18.7392	1.33511	0.667555	−	0.744561i	$-0.267341\pi$
$197$	18.7392	1.33511	0.667555	+	0.744561i	$0.267341\pi$
$198$	0	0
$199$	−13.7663	−0.975866	−0.487933	−	0.872881i	$-0.662249\pi$
$199$	−13.7663	−0.975866	−0.487933	+	0.872881i	$0.662249\pi$
$200$	0	0
$201$	6.74029	0.475424
$202$	0	0
$203$	−10.8365	−0.760575
$204$	0	0
$205$	−4.27100	−0.298299
$206$	0	0
$207$	2.27921	0.158416
$208$	0	0
$209$	−12.6122	−0.872406
$210$	0	0
$211$	−4.05421	−0.279103	−0.139552	−	0.990215i	$-0.544566\pi$
$211$	−4.05421	−0.279103	−0.139552	+	0.990215i	$0.544566\pi$
$212$	0	0
$213$	−16.4869	−1.12966
$214$	0	0
$215$	−19.6432	−1.33966
$216$	0	0
$217$	4.57256	0.310406
$218$	0	0
$219$	−5.69646	−0.384931
$220$	0	0
$221$	−2.34980	−0.158065
$222$	0	0
$223$	−6.34834	−0.425117	−0.212558	−	0.977148i	$-0.568180\pi$
$223$	−6.34834	−0.425117	−0.212558	+	0.977148i	$0.568180\pi$
$224$	0	0
$225$	−0.679911	−0.0453274
$226$	0	0
$227$	−6.75204	−0.448149	−0.224074	−	0.974572i	$-0.571936\pi$
$227$	−6.75204	−0.448149	−0.224074	+	0.974572i	$0.571936\pi$
$228$	0	0
$229$	10.2316	0.676121	0.338060	−	0.941124i	$-0.390229\pi$
$229$	10.2316	0.676121	0.338060	+	0.941124i	$0.390229\pi$
$230$	0	0
$231$	2.84736	0.187343
$232$	0	0
$233$	14.3953	0.943067	0.471534	−	0.881848i	$-0.343701\pi$
$233$	14.3953	0.943067	0.471534	+	0.881848i	$0.343701\pi$
$234$	0	0
$235$	−14.7823	−0.964289
$236$	0	0
$237$	15.3561	0.997486
$238$	0	0
$239$	−13.2493	−0.857028	−0.428514	−	0.903535i	$-0.640963\pi$
$239$	−13.2493	−0.857028	−0.428514	+	0.903535i	$0.640963\pi$
$240$	0	0
$241$	5.42968	0.349757	0.174878	−	0.984590i	$-0.444047\pi$
$241$	5.42968	0.349757	0.174878	+	0.984590i	$0.444047\pi$
$242$	0	0
$243$	14.3106	0.918025
$244$	0	0
$245$	−13.5541	−0.865941
$246$	0	0
$247$	−13.4561	−0.856190
$248$	0	0
$249$	0.551611	0.0349569
$250$	0	0
$251$	17.9964	1.13592	0.567960	−	0.823056i	$-0.307733\pi$
$251$	17.9964	1.13592	0.567960	+	0.823056i	$0.307733\pi$
$252$	0	0
$253$	3.16378	0.198905
$254$	0	0
$255$	2.76990	0.173458
$256$	0	0
$257$	−10.3704	−0.646890	−0.323445	−	0.946247i	$-0.604841\pi$
$257$	−10.3704	−0.646890	−0.323445	+	0.946247i	$0.604841\pi$
$258$	0	0
$259$	8.96590	0.557114
$260$	0	0
$261$	−15.8110	−0.978674
$262$	0	0
$263$	−14.8099	−0.913217	−0.456608	−	0.889668i	$-0.650936\pi$
$263$	−14.8099	−0.913217	−0.456608	+	0.889668i	$0.650936\pi$
$264$	0	0
$265$	−15.7660	−0.968497
$266$	0	0
$267$	−4.33674	−0.265404
$268$	0	0
$269$	−23.1468	−1.41128	−0.705642	−	0.708569i	$-0.749341\pi$
$269$	−23.1468	−1.41128	−0.705642	+	0.708569i	$0.749341\pi$
$270$	0	0
$271$	14.2640	0.866478	0.433239	−	0.901279i	$-0.357371\pi$
$271$	14.2640	0.866478	0.433239	+	0.901279i	$0.357371\pi$
$272$	0	0
$273$	3.03788	0.183861
$274$	0	0
$275$	−0.943786	−0.0569124
$276$	0	0
$277$	26.6303	1.60006	0.800029	−	0.599962i	$-0.204818\pi$
$277$	26.6303	1.60006	0.800029	+	0.599962i	$0.204818\pi$
$278$	0	0
$279$	6.67157	0.399416
$280$	0	0
$281$	−17.3347	−1.03410	−0.517051	−	0.855955i	$-0.672970\pi$
$281$	−17.3347	−1.03410	−0.517051	+	0.855955i	$0.672970\pi$
$282$	0	0
$283$	8.17815	0.486141	0.243070	−	0.970009i	$-0.421845\pi$
$283$	8.17815	0.486141	0.243070	+	0.970009i	$0.421845\pi$
$284$	0	0
$285$	15.8618	0.939570
$286$	0	0
$287$	1.99344	0.117669
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	4.61715	0.270662
$292$	0	0
$293$	13.0369	0.761623	0.380811	−	0.924653i	$-0.375645\pi$
$293$	13.0369	0.761623	0.380811	+	0.924653i	$0.375645\pi$
$294$	0	0
$295$	2.32992	0.135653
$296$	0	0
$297$	12.0095	0.696861
$298$	0	0
$299$	3.37546	0.195208
$300$	0	0
$301$	9.16827	0.528450
$302$	0	0
$303$	−5.31970	−0.305609
$304$	0	0
$305$	26.3195	1.50705
$306$	0	0
$307$	−5.55562	−0.317076	−0.158538	−	0.987353i	$-0.550678\pi$
$307$	−5.55562	−0.317076	−0.158538	+	0.987353i	$0.550678\pi$
$308$	0	0
$309$	−11.1942	−0.636814
$310$	0	0
$311$	4.25119	0.241063	0.120531	−	0.992710i	$-0.461540\pi$
$311$	4.25119	0.241063	0.120531	+	0.992710i	$0.461540\pi$
$312$	0	0
$313$	−23.7197	−1.34072	−0.670358	−	0.742037i	$-0.733860\pi$
$313$	−23.7197	−1.34072	−0.670358	+	0.742037i	$0.733860\pi$
$314$	0	0
$315$	4.02012	0.226508
$316$	0	0
$317$	0.416890	0.0234149	0.0117074	−	0.999931i	$-0.496273\pi$
$317$	0.416890	0.0234149	0.0117074	+	0.999931i	$0.496273\pi$
$318$	0	0
$319$	−21.9472	−1.22881
$320$	0	0
$321$	13.4492	0.750661
$322$	0	0
$323$	5.72648	0.318630
$324$	0	0
$325$	−1.00693	−0.0558545
$326$	0	0
$327$	19.9324	1.10227
$328$	0	0
$329$	6.89946	0.380380
$330$	0	0
$331$	−21.7460	−1.19527	−0.597635	−	0.801768i	$-0.703893\pi$
$331$	−21.7460	−1.19527	−0.597635	+	0.801768i	$0.703893\pi$
$332$	0	0
$333$	13.0816	0.716870
$334$	0	0
$335$	13.2098	0.721728
$336$	0	0
$337$	27.3529	1.49001	0.745003	−	0.667061i	$-0.232448\pi$
$337$	27.3529	1.49001	0.745003	+	0.667061i	$0.232448\pi$
$338$	0	0
$339$	−17.5407	−0.952680
$340$	0	0
$341$	9.26081	0.501501
$342$	0	0
$343$	13.9385	0.752608
$344$	0	0
$345$	−3.97893	−0.214218
$346$	0	0
$347$	18.8775	1.01340	0.506699	−	0.862123i	$-0.330866\pi$
$347$	18.8775	1.01340	0.506699	+	0.862123i	$0.330866\pi$
$348$	0	0
$349$	29.0191	1.55335	0.776677	−	0.629899i	$-0.216904\pi$
$349$	29.0191	1.55335	0.776677	+	0.629899i	$0.216904\pi$
$350$	0	0
$351$	12.8130	0.683908
$352$	0	0
$353$	−7.80349	−0.415338	−0.207669	−	0.978199i	$-0.566588\pi$
$353$	−7.80349	−0.415338	−0.207669	+	0.978199i	$0.566588\pi$
$354$	0	0
$355$	−32.3114	−1.71491
$356$	0	0
$357$	−1.29282	−0.0684234
$358$	0	0
$359$	−20.8698	−1.10147	−0.550734	−	0.834681i	$-0.685652\pi$
$359$	−20.8698	−1.10147	−0.550734	+	0.834681i	$0.685652\pi$
$360$	0	0
$361$	13.7925	0.725923
$362$	0	0
$363$	−7.31048	−0.383701
$364$	0	0
$365$	−11.1641	−0.584353
$366$	0	0
$367$	−27.9214	−1.45748	−0.728742	−	0.684789i	$-0.759895\pi$
$367$	−27.9214	−1.45748	−0.728742	+	0.684789i	$0.759895\pi$
$368$	0	0
$369$	2.90852	0.151411
$370$	0	0
$371$	7.35861	0.382040
$372$	0	0
$373$	37.5556	1.94456	0.972278	−	0.233829i	$-0.0751256\pi$
$373$	37.5556	1.94456	0.972278	+	0.233829i	$0.0751256\pi$
$374$	0	0
$375$	−12.6626	−0.653892
$376$	0	0
$377$	−23.4157	−1.20597
$378$	0	0
$379$	−26.2292	−1.34730	−0.673652	−	0.739049i	$-0.735275\pi$
$379$	−26.2292	−1.34730	−0.673652	+	0.739049i	$0.735275\pi$
$380$	0	0
$381$	−8.23493	−0.421888
$382$	0	0
$383$	5.05411	0.258253	0.129127	−	0.991628i	$-0.458783\pi$
$383$	5.05411	0.258253	0.129127	+	0.991628i	$0.458783\pi$
$384$	0	0
$385$	5.58033	0.284400
$386$	0	0
$387$	13.3769	0.679986
$388$	0	0
$389$	15.2074	0.771046	0.385523	−	0.922698i	$-0.374021\pi$
$389$	15.2074	0.771046	0.385523	+	0.922698i	$0.374021\pi$
$390$	0	0
$391$	−1.43649	−0.0726463
$392$	0	0
$393$	−2.61895	−0.132109
$394$	0	0
$395$	30.0952	1.51426
$396$	0	0
$397$	−6.26572	−0.314468	−0.157234	−	0.987561i	$-0.550258\pi$
$397$	−6.26572	−0.314468	−0.157234	+	0.987561i	$0.550258\pi$
$398$	0	0
$399$	−7.40331	−0.370629
$400$	0	0
$401$	−33.0181	−1.64884	−0.824421	−	0.565976i	$-0.808499\pi$
$401$	−33.0181	−1.64884	−0.824421	+	0.565976i	$0.808499\pi$
$402$	0	0
$403$	9.88043	0.492179
$404$	0	0
$405$	−4.01339	−0.199427
$406$	0	0
$407$	18.1586	0.900091
$408$	0	0
$409$	−14.0203	−0.693261	−0.346631	−	0.938002i	$-0.612674\pi$
$409$	−14.0203	−0.693261	−0.346631	+	0.938002i	$0.612674\pi$
$410$	0	0
$411$	20.4097	1.00674
$412$	0	0
$413$	−1.08746	−0.0535106
$414$	0	0
$415$	1.08106	0.0530672
$416$	0	0
$417$	−22.6926	−1.11126
$418$	0	0
$419$	8.79712	0.429768	0.214884	−	0.976640i	$-0.431063\pi$
$419$	8.79712	0.429768	0.214884	+	0.976640i	$0.431063\pi$
$420$	0	0
$421$	−24.2198	−1.18040	−0.590200	−	0.807257i	$-0.700951\pi$
$421$	−24.2198	−1.18040	−0.590200	+	0.807257i	$0.700951\pi$
$422$	0	0
$423$	10.0666	0.489456
$424$	0	0
$425$	0.428518	0.0207862
$426$	0	0
$427$	−12.2843	−0.594480
$428$	0	0
$429$	6.15261	0.297051
$430$	0	0
$431$	−16.2962	−0.784961	−0.392480	−	0.919760i	$-0.628383\pi$
$431$	−16.2962	−0.784961	−0.392480	+	0.919760i	$0.628383\pi$
$432$	0	0
$433$	34.4816	1.65708	0.828539	−	0.559931i	$-0.189172\pi$
$433$	34.4816	1.65708	0.828539	+	0.559931i	$0.189172\pi$
$434$	0	0
$435$	27.6019	1.32341
$436$	0	0
$437$	−8.22601	−0.393503
$438$	0	0
$439$	−35.8039	−1.70883	−0.854413	−	0.519595i	$-0.826083\pi$
$439$	−35.8039	−1.70883	−0.854413	+	0.519595i	$0.826083\pi$
$440$	0	0
$441$	9.23026	0.439536
$442$	0	0
$443$	27.4934	1.30625	0.653127	−	0.757249i	$-0.273457\pi$
$443$	27.4934	1.30625	0.653127	+	0.757249i	$0.273457\pi$
$444$	0	0
$445$	−8.49924	−0.402902
$446$	0	0
$447$	−23.4568	−1.10947
$448$	0	0
$449$	1.44772	0.0683221	0.0341611	−	0.999416i	$-0.489124\pi$
$449$	1.44772	0.0683221	0.0341611	+	0.999416i	$0.489124\pi$
$450$	0	0
$451$	4.03732	0.190110
$452$	0	0
$453$	−16.7864	−0.788693
$454$	0	0
$455$	5.95370	0.279114
$456$	0	0
$457$	−30.1328	−1.40955	−0.704777	−	0.709429i	$-0.748953\pi$
$457$	−30.1328	−1.40955	−0.704777	+	0.709429i	$0.748953\pi$
$458$	0	0
$459$	−5.45281	−0.254515
$460$	0	0
$461$	3.19471	0.148792	0.0743962	−	0.997229i	$-0.476297\pi$
$461$	3.19471	0.148792	0.0743962	+	0.997229i	$0.476297\pi$
$462$	0	0
$463$	−5.62130	−0.261244	−0.130622	−	0.991432i	$-0.541697\pi$
$463$	−5.62130	−0.261244	−0.130622	+	0.991432i	$0.541697\pi$
$464$	0	0
$465$	−11.6469	−0.540110
$466$	0	0
$467$	−1.22711	−0.0567841	−0.0283921	−	0.999597i	$-0.509039\pi$
$467$	−1.22711	−0.0567841	−0.0283921	+	0.999597i	$0.509039\pi$
$468$	0	0
$469$	−6.16553	−0.284698
$470$	0	0
$471$	18.8546	0.868774
$472$	0	0
$473$	18.5685	0.853781
$474$	0	0
$475$	2.45390	0.112593
$476$	0	0
$477$	10.7365	0.491592
$478$	0	0
$479$	26.5903	1.21494	0.607470	−	0.794342i	$-0.292184\pi$
$479$	26.5903	1.21494	0.607470	+	0.794342i	$0.292184\pi$
$480$	0	0
$481$	19.3736	0.883360
$482$	0	0
$483$	1.85712	0.0845020
$484$	0	0
$485$	9.04879	0.410885
$486$	0	0
$487$	37.1379	1.68288	0.841440	−	0.540350i	$-0.181708\pi$
$487$	37.1379	1.68288	0.841440	+	0.540350i	$0.181708\pi$
$488$	0	0
$489$	2.38902	0.108035
$490$	0	0
$491$	6.41342	0.289434	0.144717	−	0.989473i	$-0.453773\pi$
$491$	6.41342	0.289434	0.144717	+	0.989473i	$0.453773\pi$
$492$	0	0
$493$	9.96495	0.448799
$494$	0	0
$495$	8.14195	0.365953
$496$	0	0
$497$	15.0810	0.676476
$498$	0	0
$499$	21.7809	0.975049	0.487524	−	0.873109i	$-0.337900\pi$
$499$	21.7809	0.975049	0.487524	+	0.873109i	$0.337900\pi$
$500$	0	0
$501$	−12.0807	−0.539727
$502$	0	0
$503$	−28.7676	−1.28268	−0.641341	−	0.767256i	$-0.721622\pi$
$503$	−28.7676	−1.28268	−0.641341	+	0.767256i	$0.721622\pi$
$504$	0	0
$505$	−10.4257	−0.463936
$506$	0	0
$507$	−8.89066	−0.394848
$508$	0	0
$509$	15.8718	0.703503	0.351751	−	0.936093i	$-0.385586\pi$
$509$	15.8718	0.703503	0.351751	+	0.936093i	$0.385586\pi$
$510$	0	0
$511$	5.21070	0.230508
$512$	0	0
$513$	−31.2254	−1.37863
$514$	0	0
$515$	−21.9386	−0.966730
$516$	0	0
$517$	13.9735	0.614553
$518$	0	0
$519$	1.23168	0.0540647
$520$	0	0
$521$	34.5446	1.51343	0.756714	−	0.653746i	$-0.226804\pi$
$521$	34.5446	1.51343	0.756714	+	0.653746i	$0.226804\pi$
$522$	0	0
$523$	20.5939	0.900509	0.450254	−	0.892900i	$-0.351333\pi$
$523$	20.5939	0.900509	0.450254	+	0.892900i	$0.351333\pi$
$524$	0	0
$525$	−0.553997	−0.0241784
$526$	0	0
$527$	−4.20479	−0.183164
$528$	0	0
$529$	−20.9365	−0.910283
$530$	0	0
$531$	−1.58666	−0.0688551
$532$	0	0
$533$	4.30745	0.186576
$534$	0	0
$535$	26.3580	1.13956
$536$	0	0
$537$	−22.5986	−0.975203
$538$	0	0
$539$	12.8125	0.551875
$540$	0	0
$541$	−25.5084	−1.09669	−0.548346	−	0.836252i	$-0.684742\pi$
$541$	−25.5084	−1.09669	−0.548346	+	0.836252i	$0.684742\pi$
$542$	0	0
$543$	−30.4649	−1.30738
$544$	0	0
$545$	39.0641	1.67332
$546$	0	0
$547$	13.8432	0.591891	0.295945	−	0.955205i	$-0.404365\pi$
$547$	13.8432	0.591891	0.295945	+	0.955205i	$0.404365\pi$
$548$	0	0
$549$	−17.9234	−0.764950
$550$	0	0
$551$	57.0640	2.43101
$552$	0	0
$553$	−14.0466	−0.597324
$554$	0	0
$555$	−22.8372	−0.969386
$556$	0	0
$557$	43.9503	1.86224	0.931118	−	0.364718i	$-0.118835\pi$
$557$	43.9503	1.86224	0.931118	+	0.364718i	$0.118835\pi$
$558$	0	0
$559$	19.8109	0.837911
$560$	0	0
$561$	−2.61835	−0.110547
$562$	0	0
$563$	−39.8890	−1.68112	−0.840560	−	0.541719i	$-0.817774\pi$
$563$	−39.8890	−1.68112	−0.840560	+	0.541719i	$0.817774\pi$
$564$	0	0
$565$	−34.3767	−1.44624
$566$	0	0
$567$	1.87320	0.0786672
$568$	0	0
$569$	−9.94567	−0.416944	−0.208472	−	0.978028i	$-0.566849\pi$
$569$	−9.94567	−0.416944	−0.208472	+	0.978028i	$0.566849\pi$
$570$	0	0
$571$	−6.23397	−0.260883	−0.130442	−	0.991456i	$-0.541640\pi$
$571$	−6.23397	−0.260883	−0.130442	+	0.991456i	$0.541640\pi$
$572$	0	0
$573$	12.7082	0.530894
$574$	0	0
$575$	−0.615560	−0.0256706
$576$	0	0
$577$	11.3023	0.470519	0.235260	−	0.971933i	$-0.424406\pi$
$577$	11.3023	0.470519	0.235260	+	0.971933i	$0.424406\pi$
$578$	0	0
$579$	12.9934	0.539988
$580$	0	0
$581$	−0.504573	−0.0209332
$582$	0	0
$583$	14.9034	0.617235
$584$	0	0
$585$	8.68671	0.359151
$586$	0	0
$587$	−23.0600	−0.951789	−0.475894	−	0.879502i	$-0.657876\pi$
$587$	−23.0600	−0.951789	−0.475894	+	0.879502i	$0.657876\pi$
$588$	0	0
$589$	−24.0786	−0.992143
$590$	0	0
$591$	22.2779	0.916389
$592$	0	0
$593$	17.5153	0.719268	0.359634	−	0.933094i	$-0.382902\pi$
$593$	17.5153	0.719268	0.359634	+	0.933094i	$0.382902\pi$
$594$	0	0
$595$	−2.53370	−0.103872
$596$	0	0
$597$	−16.3659	−0.669813
$598$	0	0
$599$	−4.12394	−0.168500	−0.0842499	−	0.996445i	$-0.526849\pi$
$599$	−4.12394	−0.168500	−0.0842499	+	0.996445i	$0.526849\pi$
$600$	0	0
$601$	−6.09857	−0.248766	−0.124383	−	0.992234i	$-0.539695\pi$
$601$	−6.09857	−0.248766	−0.124383	+	0.992234i	$0.539695\pi$
$602$	0	0
$603$	−8.99577	−0.366336
$604$	0	0
$605$	−14.3272	−0.582485
$606$	0	0
$607$	3.89530	0.158105	0.0790527	−	0.996870i	$-0.474810\pi$
$607$	3.89530	0.158105	0.0790527	+	0.996870i	$0.474810\pi$
$608$	0	0
$609$	−12.8829	−0.522042
$610$	0	0
$611$	14.9084	0.603130
$612$	0	0
$613$	2.53718	0.102476	0.0512379	−	0.998686i	$-0.483683\pi$
$613$	2.53718	0.102476	0.0512379	+	0.998686i	$0.483683\pi$
$614$	0	0
$615$	−5.07753	−0.204746
$616$	0	0
$617$	−3.03670	−0.122253	−0.0611264	−	0.998130i	$-0.519469\pi$
$617$	−3.03670	−0.122253	−0.0611264	+	0.998130i	$0.519469\pi$
$618$	0	0
$619$	−16.5908	−0.666842	−0.333421	−	0.942778i	$-0.608203\pi$
$619$	−16.5908	−0.666842	−0.333421	+	0.942778i	$0.608203\pi$
$620$	0	0
$621$	7.83289	0.314323
$622$	0	0
$623$	3.96693	0.158932
$624$	0	0
$625$	−26.9590	−1.07836
$626$	0	0
$627$	−14.9939	−0.598800
$628$	0	0
$629$	−8.24478	−0.328741
$630$	0	0
$631$	−41.4328	−1.64941	−0.824706	−	0.565562i	$-0.808659\pi$
$631$	−41.4328	−1.64941	−0.824706	+	0.565562i	$0.808659\pi$
$632$	0	0
$633$	−4.81981	−0.191570
$634$	0	0
$635$	−16.1390	−0.640457
$636$	0	0
$637$	13.6698	0.541617
$638$	0	0
$639$	22.0039	0.870459
$640$	0	0
$641$	10.4873	0.414222	0.207111	−	0.978318i	$-0.433594\pi$
$641$	10.4873	0.414222	0.207111	+	0.978318i	$0.433594\pi$
$642$	0	0
$643$	41.8648	1.65099	0.825493	−	0.564413i	$-0.190897\pi$
$643$	41.8648	1.65099	0.825493	+	0.564413i	$0.190897\pi$
$644$	0	0
$645$	−23.3527	−0.919511
$646$	0	0
$647$	−27.4395	−1.07876	−0.539379	−	0.842063i	$-0.681341\pi$
$647$	−27.4395	−1.07876	−0.539379	+	0.842063i	$0.681341\pi$
$648$	0	0
$649$	−2.20244	−0.0864534
$650$	0	0
$651$	5.43605	0.213056
$652$	0	0
$653$	−35.3932	−1.38504	−0.692522	−	0.721397i	$-0.743500\pi$
$653$	−35.3932	−1.38504	−0.692522	+	0.721397i	$0.743500\pi$
$654$	0	0
$655$	−5.13268	−0.200550
$656$	0	0
$657$	7.60264	0.296607
$658$	0	0
$659$	14.9347	0.581774	0.290887	−	0.956757i	$-0.406050\pi$
$659$	14.9347	0.581774	0.290887	+	0.956757i	$0.406050\pi$
$660$	0	0
$661$	−5.91962	−0.230247	−0.115123	−	0.993351i	$-0.536726\pi$
$661$	−5.91962	−0.230247	−0.115123	+	0.993351i	$0.536726\pi$
$662$	0	0
$663$	−2.79354	−0.108492
$664$	0	0
$665$	−14.5092	−0.562642
$666$	0	0
$667$	−14.3145	−0.554261
$668$	0	0
$669$	−7.54717	−0.291790
$670$	0	0
$671$	−24.8794	−0.960460
$672$	0	0
$673$	36.3638	1.40172	0.700861	−	0.713298i	$-0.252799\pi$
$673$	36.3638	1.40172	0.700861	+	0.713298i	$0.252799\pi$
$674$	0	0
$675$	−2.33662	−0.0899367
$676$	0	0
$677$	−41.7943	−1.60628	−0.803142	−	0.595787i	$-0.796840\pi$
$677$	−41.7943	−1.60628	−0.803142	+	0.595787i	$0.796840\pi$
$678$	0	0
$679$	−4.22343	−0.162080
$680$	0	0
$681$	−8.02711	−0.307599
$682$	0	0
$683$	31.2059	1.19406	0.597030	−	0.802219i	$-0.296347\pi$
$683$	31.2059	1.19406	0.597030	+	0.802219i	$0.296347\pi$
$684$	0	0
$685$	39.9994	1.52830
$686$	0	0
$687$	12.1637	0.464074
$688$	0	0
$689$	15.9005	0.605762
$690$	0	0
$691$	34.7565	1.32220	0.661099	−	0.750298i	$-0.270090\pi$
$691$	34.7565	1.32220	0.661099	+	0.750298i	$0.270090\pi$
$692$	0	0
$693$	−3.80017	−0.144356
$694$	0	0
$695$	−44.4734	−1.68697
$696$	0	0
$697$	−1.83311	−0.0694340
$698$	0	0
$699$	17.1137	0.647300
$700$	0	0
$701$	24.6008	0.929160	0.464580	−	0.885531i	$-0.346205\pi$
$701$	24.6008	0.929160	0.464580	+	0.885531i	$0.346205\pi$
$702$	0	0
$703$	−47.2135	−1.78069
$704$	0	0
$705$	−17.5738	−0.661866
$706$	0	0
$707$	4.86607	0.183007
$708$	0	0
$709$	−9.44473	−0.354704	−0.177352	−	0.984147i	$-0.556753\pi$
$709$	−9.44473	−0.354704	−0.177352	+	0.984147i	$0.556753\pi$
$710$	0	0
$711$	−20.4946	−0.768609
$712$	0	0
$713$	6.04013	0.226205
$714$	0	0
$715$	12.0580	0.450945
$716$	0	0
$717$	−15.7513	−0.588245
$718$	0	0
$719$	−37.0147	−1.38042	−0.690208	−	0.723611i	$-0.742481\pi$
$719$	−37.0147	−1.38042	−0.690208	+	0.723611i	$0.742481\pi$
$720$	0	0
$721$	10.2396	0.381343
$722$	0	0
$723$	6.45503	0.240065
$724$	0	0
$725$	4.27016	0.158590
$726$	0	0
$727$	−16.3970	−0.608130	−0.304065	−	0.952651i	$-0.598344\pi$
$727$	−16.3970	−0.608130	−0.304065	+	0.952651i	$0.598344\pi$
$728$	0	0
$729$	22.1806	0.821505
$730$	0	0
$731$	−8.43087	−0.311827
$732$	0	0
$733$	−20.0082	−0.739022	−0.369511	−	0.929226i	$-0.620475\pi$
$733$	−20.0082	−0.739022	−0.369511	+	0.929226i	$0.620475\pi$
$734$	0	0
$735$	−16.1137	−0.594362
$736$	0	0
$737$	−12.4870	−0.459966
$738$	0	0
$739$	−37.4325	−1.37698	−0.688488	−	0.725248i	$-0.741725\pi$
$739$	−37.4325	−1.37698	−0.688488	+	0.725248i	$0.741725\pi$
$740$	0	0
$741$	−15.9971	−0.587670
$742$	0	0
$743$	−21.5791	−0.791662	−0.395831	−	0.918323i	$-0.629543\pi$
$743$	−21.5791	−0.791662	−0.395831	+	0.918323i	$0.629543\pi$
$744$	0	0
$745$	−45.9711	−1.68425
$746$	0	0
$747$	−0.736195	−0.0269360
$748$	0	0
$749$	−12.3023	−0.449517
$750$	0	0
$751$	−40.5381	−1.47926	−0.739629	−	0.673015i	$-0.764999\pi$
$751$	−40.5381	−1.47926	−0.739629	+	0.673015i	$0.764999\pi$
$752$	0	0
$753$	21.3948	0.779670
$754$	0	0
$755$	−32.8983	−1.19729
$756$	0	0
$757$	49.8307	1.81113	0.905563	−	0.424211i	$-0.139449\pi$
$757$	49.8307	1.81113	0.905563	+	0.424211i	$0.139449\pi$
$758$	0	0
$759$	3.76123	0.136524
$760$	0	0
$761$	17.5864	0.637505	0.318752	−	0.947838i	$-0.396736\pi$
$761$	17.5864	0.637505	0.318752	+	0.947838i	$0.396736\pi$
$762$	0	0
$763$	−18.2327	−0.660069
$764$	0	0
$765$	−3.69678	−0.133657
$766$	0	0
$767$	−2.34980	−0.0848464
$768$	0	0
$769$	3.81974	0.137743	0.0688717	−	0.997626i	$-0.478060\pi$
$769$	3.81974	0.137743	0.0688717	+	0.997626i	$0.478060\pi$
$770$	0	0
$771$	−12.3288	−0.444010
$772$	0	0
$773$	−11.9263	−0.428958	−0.214479	−	0.976729i	$-0.568805\pi$
$773$	−11.9263	−0.428958	−0.214479	+	0.976729i	$0.568805\pi$
$774$	0	0
$775$	−1.80183	−0.0647236
$776$	0	0
$777$	10.6590	0.382391
$778$	0	0
$779$	−10.4973	−0.376103
$780$	0	0
$781$	30.5436	1.09294
$782$	0	0
$783$	−54.3369	−1.94184
$784$	0	0
$785$	36.9517	1.31886
$786$	0	0
$787$	9.46289	0.337316	0.168658	−	0.985675i	$-0.446057\pi$
$787$	9.46289	0.337316	0.168658	+	0.985675i	$0.446057\pi$
$788$	0	0
$789$	−17.6066	−0.626812
$790$	0	0
$791$	16.0449	0.570493
$792$	0	0
$793$	−26.5441	−0.942607
$794$	0	0
$795$	−18.7433	−0.664755
$796$	0	0
$797$	1.23943	0.0439029	0.0219515	−	0.999759i	$-0.493012\pi$
$797$	1.23943	0.0439029	0.0219515	+	0.999759i	$0.493012\pi$
$798$	0	0
$799$	−6.34454	−0.224454
$800$	0	0
$801$	5.78792	0.204506
$802$	0	0
$803$	10.5532	0.372416
$804$	0	0
$805$	3.63963	0.128280
$806$	0	0
$807$	−27.5178	−0.968673
$808$	0	0
$809$	50.1559	1.76339	0.881693	−	0.471823i	$-0.156404\pi$
$809$	50.1559	1.76339	0.881693	+	0.471823i	$0.156404\pi$
$810$	0	0
$811$	29.4695	1.03481	0.517407	−	0.855740i	$-0.326897\pi$
$811$	29.4695	1.03481	0.517407	+	0.855740i	$0.326897\pi$
$812$	0	0
$813$	16.9577	0.594731
$814$	0	0
$815$	4.68206	0.164006
$816$	0	0
$817$	−48.2792	−1.68907
$818$	0	0
$819$	−4.05443	−0.141673
$820$	0	0
$821$	52.5180	1.83289	0.916445	−	0.400161i	$-0.131046\pi$
$821$	52.5180	1.83289	0.916445	+	0.400161i	$0.131046\pi$
$822$	0	0
$823$	42.6230	1.48574	0.742872	−	0.669433i	$-0.233463\pi$
$823$	42.6230	1.48574	0.742872	+	0.669433i	$0.233463\pi$
$824$	0	0
$825$	−1.12201	−0.0390634
$826$	0	0
$827$	−11.3976	−0.396333	−0.198167	−	0.980168i	$-0.563499\pi$
$827$	−11.3976	−0.396333	−0.198167	+	0.980168i	$0.563499\pi$
$828$	0	0
$829$	37.5868	1.30544	0.652721	−	0.757598i	$-0.273627\pi$
$829$	37.5868	1.30544	0.652721	+	0.757598i	$0.273627\pi$
$830$	0	0
$831$	31.6591	1.09824
$832$	0	0
$833$	−5.81742	−0.201562
$834$	0	0
$835$	−23.6761	−0.819345
$836$	0	0
$837$	22.9279	0.792505
$838$	0	0
$839$	53.5728	1.84954	0.924769	−	0.380528i	$-0.124258\pi$
$839$	53.5728	1.84954	0.924769	+	0.380528i	$0.124258\pi$
$840$	0	0
$841$	70.3002	2.42415
$842$	0	0
$843$	−20.6082	−0.709784
$844$	0	0
$845$	−17.4241	−0.599408
$846$	0	0
$847$	6.68709	0.229771
$848$	0	0
$849$	9.72252	0.333676
$850$	0	0
$851$	11.8435	0.405991
$852$	0	0
$853$	18.8807	0.646461	0.323231	−	0.946320i	$-0.395231\pi$
$853$	18.8807	0.646461	0.323231	+	0.946320i	$0.395231\pi$
$854$	0	0
$855$	−21.1695	−0.723983
$856$	0	0
$857$	−45.0601	−1.53923	−0.769613	−	0.638511i	$-0.779551\pi$
$857$	−45.0601	−1.53923	−0.769613	+	0.638511i	$0.779551\pi$
$858$	0	0
$859$	−4.21291	−0.143743	−0.0718713	−	0.997414i	$-0.522897\pi$
$859$	−4.21291	−0.143743	−0.0718713	+	0.997414i	$0.522897\pi$
$860$	0	0
$861$	2.36988	0.0807655
$862$	0	0
$863$	−7.47064	−0.254303	−0.127152	−	0.991883i	$-0.540584\pi$
$863$	−7.47064	−0.254303	−0.127152	+	0.991883i	$0.540584\pi$
$864$	0	0
$865$	2.41387	0.0820742
$866$	0	0
$867$	1.18884	0.0403751
$868$	0	0
$869$	−28.4487	−0.965054
$870$	0	0
$871$	−13.3225	−0.451416
$872$	0	0
$873$	−6.16216	−0.208558
$874$	0	0
$875$	11.5828	0.391569
$876$	0	0
$877$	32.1004	1.08395	0.541977	−	0.840393i	$-0.317676\pi$
$877$	32.1004	1.08395	0.541977	+	0.840393i	$0.317676\pi$
$878$	0	0
$879$	15.4988	0.522761
$880$	0	0
$881$	−25.8699	−0.871579	−0.435789	−	0.900049i	$-0.643531\pi$
$881$	−25.8699	−0.871579	−0.435789	+	0.900049i	$0.643531\pi$
$882$	0	0
$883$	20.2618	0.681865	0.340933	−	0.940088i	$-0.389257\pi$
$883$	20.2618	0.681865	0.340933	+	0.940088i	$0.389257\pi$
$884$	0	0
$885$	2.76990	0.0931092
$886$	0	0
$887$	17.6855	0.593820	0.296910	−	0.954906i	$-0.404044\pi$
$887$	17.6855	0.593820	0.296910	+	0.954906i	$0.404044\pi$
$888$	0	0
$889$	7.53271	0.252639
$890$	0	0
$891$	3.79380	0.127097
$892$	0	0
$893$	−36.3319	−1.21580
$894$	0	0
$895$	−44.2893	−1.48043
$896$	0	0
$897$	4.01289	0.133986
$898$	0	0
$899$	−41.9006	−1.39746
$900$	0	0
$901$	−6.76676	−0.225433
$902$	0	0
$903$	10.8996	0.362716
$904$	0	0
$905$	−59.7059	−1.98469
$906$	0	0
$907$	−2.85017	−0.0946384	−0.0473192	−	0.998880i	$-0.515068\pi$
$907$	−2.85017	−0.0946384	−0.0473192	+	0.998880i	$0.515068\pi$
$908$	0	0
$909$	7.09981	0.235486
$910$	0	0
$911$	−25.3727	−0.840635	−0.420318	−	0.907377i	$-0.638081\pi$
$911$	−25.3727	−0.840635	−0.420318	+	0.907377i	$0.638081\pi$
$912$	0	0
$913$	−1.02191	−0.0338204
$914$	0	0
$915$	31.2896	1.03440
$916$	0	0
$917$	2.39562	0.0791105
$918$	0	0
$919$	−21.1358	−0.697204	−0.348602	−	0.937271i	$-0.613344\pi$
$919$	−21.1358	−0.697204	−0.348602	+	0.937271i	$0.613344\pi$
$920$	0	0
$921$	−6.60475	−0.217634
$922$	0	0
$923$	32.5872	1.07262
$924$	0	0
$925$	−3.53303	−0.116165
$926$	0	0
$927$	14.9400	0.490695
$928$	0	0
$929$	40.8927	1.34165	0.670823	−	0.741618i	$-0.265941\pi$
$929$	40.8927	1.34165	0.670823	+	0.741618i	$0.265941\pi$
$930$	0	0
$931$	−33.3133	−1.09180
$932$	0	0
$933$	5.05399	0.165460
$934$	0	0
$935$	−5.13151	−0.167818
$936$	0	0
$937$	11.2235	0.366654	0.183327	−	0.983052i	$-0.441313\pi$
$937$	11.2235	0.366654	0.183327	+	0.983052i	$0.441313\pi$
$938$	0	0
$939$	−28.1989	−0.920238
$940$	0	0
$941$	−0.679848	−0.0221624	−0.0110812	−	0.999939i	$-0.503527\pi$
$941$	−0.679848	−0.0221624	−0.0110812	+	0.999939i	$0.503527\pi$
$942$	0	0
$943$	2.63324	0.0857501
$944$	0	0
$945$	13.8158	0.449428
$946$	0	0
$947$	−23.8139	−0.773846	−0.386923	−	0.922112i	$-0.626462\pi$
$947$	−23.8139	−0.773846	−0.386923	+	0.922112i	$0.626462\pi$
$948$	0	0
$949$	11.2593	0.365493
$950$	0	0
$951$	0.495616	0.0160715
$952$	0	0
$953$	−40.3136	−1.30589	−0.652943	−	0.757407i	$-0.726466\pi$
$953$	−40.3136	−1.30589	−0.652943	+	0.757407i	$0.726466\pi$
$954$	0	0
$955$	24.9059	0.805936
$956$	0	0
$957$	−26.0917	−0.843426
$958$	0	0
$959$	−18.6693	−0.602863
$960$	0	0
$961$	−13.3197	−0.429668
$962$	0	0
$963$	−17.9496	−0.578419
$964$	0	0
$965$	25.4648	0.819742
$966$	0	0
$967$	48.2296	1.55096	0.775480	−	0.631372i	$-0.217508\pi$
$967$	48.2296	1.55096	0.775480	+	0.631372i	$0.217508\pi$
$968$	0	0
$969$	6.80787	0.218700
$970$	0	0
$971$	−46.5145	−1.49272	−0.746360	−	0.665542i	$-0.768200\pi$
$971$	−46.5145	−1.49272	−0.746360	+	0.665542i	$0.768200\pi$
$972$	0	0
$973$	20.7575	0.665455
$974$	0	0
$975$	−1.19708	−0.0383373
$976$	0	0
$977$	43.0316	1.37670	0.688352	−	0.725377i	$-0.258334\pi$
$977$	43.0316	1.37670	0.688352	+	0.725377i	$0.258334\pi$
$978$	0	0
$979$	8.03422	0.256775
$980$	0	0
$981$	−26.6024	−0.849348
$982$	0	0
$983$	−11.1855	−0.356762	−0.178381	−	0.983961i	$-0.557086\pi$
$983$	−11.1855	−0.356762	−0.178381	+	0.983961i	$0.557086\pi$
$984$	0	0
$985$	43.6607	1.39115
$986$	0	0
$987$	8.20236	0.261084
$988$	0	0
$989$	12.1108	0.385102
$990$	0	0
$991$	−11.3163	−0.359473	−0.179737	−	0.983715i	$-0.557525\pi$
$991$	−11.3163	−0.359473	−0.179737	+	0.983715i	$0.557525\pi$
$992$	0	0
$993$	−25.8526	−0.820407
$994$	0	0
$995$	−32.0743	−1.01682
$996$	0	0
$997$	48.7349	1.54345	0.771725	−	0.635957i	$-0.219394\pi$
$997$	48.7349	1.54345	0.771725	+	0.635957i	$0.219394\pi$
$998$	0	0
$999$	44.9572	1.42238

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.z.1.16	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.z.1.16	✓	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)

\(f(q)\)	\(=\)	\(q+1.18884 q^{3} +2.32992 q^{5} -1.08746 q^{7} -1.58666 q^{9} +O(q^{10})\)	\(q+1.18884 q^{3} +2.32992 q^{5} -1.08746 q^{7} -1.58666 q^{9} -2.20244 q^{11} -2.34980 q^{13} +2.76990 q^{15} +1.00000 q^{17} +5.72648 q^{19} -1.29282 q^{21} -1.43649 q^{23} +0.428518 q^{25} -5.45281 q^{27} +9.96495 q^{29} -4.20479 q^{31} -2.61835 q^{33} -2.53370 q^{35} -8.24478 q^{37} -2.79354 q^{39} -1.83311 q^{41} -8.43087 q^{43} -3.69678 q^{45} -6.34454 q^{47} -5.81742 q^{49} +1.18884 q^{51} -6.76676 q^{53} -5.13151 q^{55} +6.80787 q^{57} +1.00000 q^{59} +11.2963 q^{61} +1.72543 q^{63} -5.47485 q^{65} +5.66963 q^{67} -1.70775 q^{69} -13.8681 q^{71} -4.79161 q^{73} +0.509439 q^{75} +2.39508 q^{77} +12.9169 q^{79} -1.72254 q^{81} +0.463991 q^{83} +2.32992 q^{85} +11.8467 q^{87} -3.64787 q^{89} +2.55533 q^{91} -4.99883 q^{93} +13.3422 q^{95} +3.88374 q^{97} +3.49452 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9}+O(q^{10}) \) 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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