LMFDB - Embedded newform 8024.2.a.x.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.49900 q^{3} -0.909441 q^{5} +1.38706 q^{7} -0.753011 q^{9} +O(q^{10})$	$q+1.49900 q^{3} -0.909441 q^{5} +1.38706 q^{7} -0.753011 q^{9} +5.30850 q^{11} -1.69618 q^{13} -1.36325 q^{15} -1.00000 q^{17} -4.39103 q^{19} +2.07919 q^{21} -0.533354 q^{23} -4.17292 q^{25} -5.62575 q^{27} +2.90827 q^{29} -1.91846 q^{31} +7.95742 q^{33} -1.26145 q^{35} +1.11796 q^{37} -2.54257 q^{39} -9.43844 q^{41} -0.548133 q^{43} +0.684818 q^{45} -9.09856 q^{47} -5.07607 q^{49} -1.49900 q^{51} -1.54119 q^{53} -4.82776 q^{55} -6.58214 q^{57} +1.00000 q^{59} -4.47346 q^{61} -1.04447 q^{63} +1.54258 q^{65} +0.954313 q^{67} -0.799496 q^{69} -9.26909 q^{71} +11.6937 q^{73} -6.25519 q^{75} +7.36319 q^{77} +15.5502 q^{79} -6.17394 q^{81} -10.4150 q^{83} +0.909441 q^{85} +4.35948 q^{87} +9.19589 q^{89} -2.35271 q^{91} -2.87577 q^{93} +3.99338 q^{95} +11.7161 q^{97} -3.99735 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.49900	0.865446	0.432723	−	0.901527i	$-0.357553\pi$
$3$	1.49900	0.865446	0.432723	+	0.901527i	$0.357553\pi$
$4$	0	0
$5$	−0.909441	−0.406714	−0.203357	−	0.979105i	$-0.565185\pi$
$5$	−0.909441	−0.406714	−0.203357	+	0.979105i	$0.565185\pi$
$6$	0	0
$7$	1.38706	0.524258	0.262129	−	0.965033i	$-0.415575\pi$
$7$	1.38706	0.524258	0.262129	+	0.965033i	$0.415575\pi$
$8$	0	0
$9$	−0.753011	−0.251004
$10$	0	0
$11$	5.30850	1.60057	0.800286	−	0.599619i	$-0.204681\pi$
$11$	5.30850	1.60057	0.800286	+	0.599619i	$0.204681\pi$
$12$	0	0
$13$	−1.69618	−0.470437	−0.235219	−	0.971943i	$-0.575581\pi$
$13$	−1.69618	−0.470437	−0.235219	+	0.971943i	$0.575581\pi$
$14$	0	0
$15$	−1.36325	−0.351989
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−4.39103	−1.00737	−0.503686	−	0.863887i	$-0.668023\pi$
$19$	−4.39103	−1.00737	−0.503686	+	0.863887i	$0.668023\pi$
$20$	0	0
$21$	2.07919	0.453717
$22$	0	0
$23$	−0.533354	−0.111212	−0.0556060	−	0.998453i	$-0.517709\pi$
$23$	−0.533354	−0.111212	−0.0556060	+	0.998453i	$0.517709\pi$
$24$	0	0
$25$	−4.17292	−0.834584
$26$	0	0
$27$	−5.62575	−1.08268
$28$	0	0
$29$	2.90827	0.540051	0.270026	−	0.962853i	$-0.412968\pi$
$29$	2.90827	0.540051	0.270026	+	0.962853i	$0.412968\pi$
$30$	0	0
$31$	−1.91846	−0.344566	−0.172283	−	0.985048i	$-0.555114\pi$
$31$	−1.91846	−0.344566	−0.172283	+	0.985048i	$0.555114\pi$
$32$	0	0
$33$	7.95742	1.38521
$34$	0	0
$35$	−1.26145	−0.213223
$36$	0	0
$37$	1.11796	0.183792	0.0918960	−	0.995769i	$-0.470707\pi$
$37$	1.11796	0.183792	0.0918960	+	0.995769i	$0.470707\pi$
$38$	0	0
$39$	−2.54257	−0.407138
$40$	0	0
$41$	−9.43844	−1.47404	−0.737018	−	0.675873i	$-0.763767\pi$
$41$	−9.43844	−1.47404	−0.737018	+	0.675873i	$0.763767\pi$
$42$	0	0
$43$	−0.548133	−0.0835895	−0.0417948	−	0.999126i	$-0.513308\pi$
$43$	−0.548133	−0.0835895	−0.0417948	+	0.999126i	$0.513308\pi$
$44$	0	0
$45$	0.684818	0.102087
$46$	0	0
$47$	−9.09856	−1.32716	−0.663581	−	0.748105i	$-0.730964\pi$
$47$	−9.09856	−1.32716	−0.663581	+	0.748105i	$0.730964\pi$
$48$	0	0
$49$	−5.07607	−0.725153
$50$	0	0
$51$	−1.49900	−0.209901
$52$	0	0
$53$	−1.54119	−0.211698	−0.105849	−	0.994382i	$-0.533756\pi$
$53$	−1.54119	−0.211698	−0.105849	+	0.994382i	$0.533756\pi$
$54$	0	0
$55$	−4.82776	−0.650975
$56$	0	0
$57$	−6.58214	−0.871826
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−4.47346	−0.572768	−0.286384	−	0.958115i	$-0.592453\pi$
$61$	−4.47346	−0.572768	−0.286384	+	0.958115i	$0.592453\pi$
$62$	0	0
$63$	−1.04447	−0.131591
$64$	0	0
$65$	1.54258	0.191333
$66$	0	0
$67$	0.954313	0.116588	0.0582939	−	0.998299i	$-0.481434\pi$
$67$	0.954313	0.116588	0.0582939	+	0.998299i	$0.481434\pi$
$68$	0	0
$69$	−0.799496	−0.0962480
$70$	0	0
$71$	−9.26909	−1.10004	−0.550020	−	0.835152i	$-0.685380\pi$
$71$	−9.26909	−1.10004	−0.550020	+	0.835152i	$0.685380\pi$
$72$	0	0
$73$	11.6937	1.36865	0.684324	−	0.729178i	$-0.260097\pi$
$73$	11.6937	1.36865	0.684324	+	0.729178i	$0.260097\pi$
$74$	0	0
$75$	−6.25519	−0.722287
$76$	0	0
$77$	7.36319	0.839113
$78$	0	0
$79$	15.5502	1.74953	0.874766	−	0.484546i	$-0.161015\pi$
$79$	15.5502	1.74953	0.874766	+	0.484546i	$0.161015\pi$
$80$	0	0
$81$	−6.17394	−0.685994
$82$	0	0
$83$	−10.4150	−1.14319	−0.571597	−	0.820534i	$-0.693676\pi$
$83$	−10.4150	−1.14319	−0.571597	+	0.820534i	$0.693676\pi$
$84$	0	0
$85$	0.909441	0.0986427
$86$	0	0
$87$	4.35948	0.467385
$88$	0	0
$89$	9.19589	0.974762	0.487381	−	0.873189i	$-0.337952\pi$
$89$	9.19589	0.974762	0.487381	+	0.873189i	$0.337952\pi$
$90$	0	0
$91$	−2.35271	−0.246631
$92$	0	0
$93$	−2.87577	−0.298203
$94$	0	0
$95$	3.99338	0.409712
$96$	0	0
$97$	11.7161	1.18959	0.594797	−	0.803876i	$-0.297233\pi$
$97$	11.7161	1.18959	0.594797	+	0.803876i	$0.297233\pi$
$98$	0	0
$99$	−3.99735	−0.401749
$100$	0	0
$101$	−9.48685	−0.943977	−0.471988	−	0.881605i	$-0.656464\pi$
$101$	−9.48685	−0.943977	−0.471988	+	0.881605i	$0.656464\pi$
$102$	0	0
$103$	−15.0587	−1.48378	−0.741889	−	0.670523i	$-0.766070\pi$
$103$	−15.0587	−1.48378	−0.741889	+	0.670523i	$0.766070\pi$
$104$	0	0
$105$	−1.89090	−0.184533
$106$	0	0
$107$	−7.17296	−0.693436	−0.346718	−	0.937969i	$-0.612704\pi$
$107$	−7.17296	−0.693436	−0.346718	+	0.937969i	$0.612704\pi$
$108$	0	0
$109$	−15.8691	−1.51998	−0.759992	−	0.649932i	$-0.774797\pi$
$109$	−15.8691	−1.51998	−0.759992	+	0.649932i	$0.774797\pi$
$110$	0	0
$111$	1.67582	0.159062
$112$	0	0
$113$	5.59226	0.526076	0.263038	−	0.964785i	$-0.415276\pi$
$113$	5.59226	0.526076	0.263038	+	0.964785i	$0.415276\pi$
$114$	0	0
$115$	0.485054	0.0452315
$116$	0	0
$117$	1.27725	0.118081
$118$	0	0
$119$	−1.38706	−0.127151
$120$	0	0
$121$	17.1801	1.56183
$122$	0	0
$123$	−14.1482	−1.27570
$124$	0	0
$125$	8.34222	0.746151
$126$	0	0
$127$	−0.549155	−0.0487296	−0.0243648	−	0.999703i	$-0.507756\pi$
$127$	−0.549155	−0.0487296	−0.0243648	+	0.999703i	$0.507756\pi$
$128$	0	0
$129$	−0.821649	−0.0723422
$130$	0	0
$131$	−9.03459	−0.789356	−0.394678	−	0.918820i	$-0.629144\pi$
$131$	−9.03459	−0.789356	−0.394678	+	0.918820i	$0.629144\pi$
$132$	0	0
$133$	−6.09061	−0.528123
$134$	0	0
$135$	5.11628	0.440340
$136$	0	0
$137$	5.15276	0.440230	0.220115	−	0.975474i	$-0.429357\pi$
$137$	5.15276	0.440230	0.220115	+	0.975474i	$0.429357\pi$
$138$	0	0
$139$	−19.4410	−1.64896	−0.824480	−	0.565891i	$-0.808532\pi$
$139$	−19.4410	−1.64896	−0.824480	+	0.565891i	$0.808532\pi$
$140$	0	0
$141$	−13.6387	−1.14859
$142$	0	0
$143$	−9.00419	−0.752968
$144$	0	0
$145$	−2.64489	−0.219647
$146$	0	0
$147$	−7.60901	−0.627581
$148$	0	0
$149$	4.84269	0.396728	0.198364	−	0.980128i	$-0.436437\pi$
$149$	4.84269	0.396728	0.198364	+	0.980128i	$0.436437\pi$
$150$	0	0
$151$	17.1126	1.39260	0.696301	−	0.717750i	$-0.254828\pi$
$151$	17.1126	1.39260	0.696301	+	0.717750i	$0.254828\pi$
$152$	0	0
$153$	0.753011	0.0608773
$154$	0	0
$155$	1.74473	0.140140
$156$	0	0
$157$	−5.53069	−0.441397	−0.220698	−	0.975342i	$-0.570834\pi$
$157$	−5.53069	−0.441397	−0.220698	+	0.975342i	$0.570834\pi$
$158$	0	0
$159$	−2.31023	−0.183213
$160$	0	0
$161$	−0.739793	−0.0583038
$162$	0	0
$163$	−0.875538	−0.0685775	−0.0342887	−	0.999412i	$-0.510917\pi$
$163$	−0.875538	−0.0685775	−0.0342887	+	0.999412i	$0.510917\pi$
$164$	0	0
$165$	−7.23680	−0.563384
$166$	0	0
$167$	−17.4544	−1.35066	−0.675330	−	0.737515i	$-0.735999\pi$
$167$	−17.4544	−1.35066	−0.675330	+	0.737515i	$0.735999\pi$
$168$	0	0
$169$	−10.1230	−0.778689
$170$	0	0
$171$	3.30649	0.252854
$172$	0	0
$173$	−7.14081	−0.542906	−0.271453	−	0.962452i	$-0.587504\pi$
$173$	−7.14081	−0.542906	−0.271453	+	0.962452i	$0.587504\pi$
$174$	0	0
$175$	−5.78808	−0.437537
$176$	0	0
$177$	1.49900	0.112671
$178$	0	0
$179$	12.1119	0.905285	0.452642	−	0.891692i	$-0.350481\pi$
$179$	12.1119	0.905285	0.452642	+	0.891692i	$0.350481\pi$
$180$	0	0
$181$	23.8356	1.77169	0.885844	−	0.463984i	$-0.153580\pi$
$181$	23.8356	1.77169	0.885844	+	0.463984i	$0.153580\pi$
$182$	0	0
$183$	−6.70570	−0.495700
$184$	0	0
$185$	−1.01672	−0.0747508
$186$	0	0
$187$	−5.30850	−0.388196
$188$	0	0
$189$	−7.80324	−0.567602
$190$	0	0
$191$	23.0729	1.66950	0.834750	−	0.550629i	$-0.185612\pi$
$191$	23.0729	1.66950	0.834750	+	0.550629i	$0.185612\pi$
$192$	0	0
$193$	17.3540	1.24917	0.624584	−	0.780958i	$-0.285269\pi$
$193$	17.3540	1.24917	0.624584	+	0.780958i	$0.285269\pi$
$194$	0	0
$195$	2.31232	0.165589
$196$	0	0
$197$	2.44008	0.173849	0.0869243	−	0.996215i	$-0.472296\pi$
$197$	2.44008	0.173849	0.0869243	+	0.996215i	$0.472296\pi$
$198$	0	0
$199$	−4.95908	−0.351540	−0.175770	−	0.984431i	$-0.556241\pi$
$199$	−4.95908	−0.351540	−0.175770	+	0.984431i	$0.556241\pi$
$200$	0	0
$201$	1.43051	0.100901
$202$	0	0
$203$	4.03393	0.283126
$204$	0	0
$205$	8.58370	0.599511
$206$	0	0
$207$	0.401621	0.0279146
$208$	0	0
$209$	−23.3098	−1.61237
$210$	0	0
$211$	−4.52600	−0.311583	−0.155791	−	0.987790i	$-0.549793\pi$
$211$	−4.52600	−0.311583	−0.155791	+	0.987790i	$0.549793\pi$
$212$	0	0
$213$	−13.8943	−0.952024
$214$	0	0
$215$	0.498495	0.0339970
$216$	0	0
$217$	−2.66102	−0.180641
$218$	0	0
$219$	17.5289	1.18449
$220$	0	0
$221$	1.69618	0.114098
$222$	0	0
$223$	4.06890	0.272474	0.136237	−	0.990676i	$-0.456499\pi$
$223$	4.06890	0.272474	0.136237	+	0.990676i	$0.456499\pi$
$224$	0	0
$225$	3.14225	0.209483
$226$	0	0
$227$	−14.2751	−0.947472	−0.473736	−	0.880667i	$-0.657095\pi$
$227$	−14.2751	−0.947472	−0.473736	+	0.880667i	$0.657095\pi$
$228$	0	0
$229$	12.4125	0.820242	0.410121	−	0.912031i	$-0.365486\pi$
$229$	12.4125	0.820242	0.410121	+	0.912031i	$0.365486\pi$
$230$	0	0
$231$	11.0374	0.726207
$232$	0	0
$233$	−4.73436	−0.310158	−0.155079	−	0.987902i	$-0.549563\pi$
$233$	−4.73436	−0.310158	−0.155079	+	0.987902i	$0.549563\pi$
$234$	0	0
$235$	8.27460	0.539775
$236$	0	0
$237$	23.3097	1.51412
$238$	0	0
$239$	−20.8808	−1.35067	−0.675334	−	0.737512i	$-0.736000\pi$
$239$	−20.8808	−1.35067	−0.675334	+	0.737512i	$0.736000\pi$
$240$	0	0
$241$	−12.0592	−0.776801	−0.388401	−	0.921491i	$-0.626972\pi$
$241$	−12.0592	−0.776801	−0.388401	+	0.921491i	$0.626972\pi$
$242$	0	0
$243$	7.62253	0.488985
$244$	0	0
$245$	4.61639	0.294930
$246$	0	0
$247$	7.44800	0.473905
$248$	0	0
$249$	−15.6120	−0.989373
$250$	0	0
$251$	16.1551	1.01970	0.509849	−	0.860264i	$-0.329701\pi$
$251$	16.1551	1.01970	0.509849	+	0.860264i	$0.329701\pi$
$252$	0	0
$253$	−2.83131	−0.178003
$254$	0	0
$255$	1.36325	0.0853699
$256$	0	0
$257$	−28.8675	−1.80070	−0.900352	−	0.435163i	$-0.856691\pi$
$257$	−28.8675	−1.80070	−0.900352	+	0.435163i	$0.856691\pi$
$258$	0	0
$259$	1.55068	0.0963545
$260$	0	0
$261$	−2.18995	−0.135555
$262$	0	0
$263$	15.9106	0.981090	0.490545	−	0.871416i	$-0.336798\pi$
$263$	15.9106	0.981090	0.490545	+	0.871416i	$0.336798\pi$
$264$	0	0
$265$	1.40162	0.0861007
$266$	0	0
$267$	13.7846	0.843604
$268$	0	0
$269$	8.18184	0.498855	0.249428	−	0.968393i	$-0.419758\pi$
$269$	8.18184	0.498855	0.249428	+	0.968393i	$0.419758\pi$
$270$	0	0
$271$	13.6995	0.832184	0.416092	−	0.909322i	$-0.363399\pi$
$271$	13.6995	0.832184	0.416092	+	0.909322i	$0.363399\pi$
$272$	0	0
$273$	−3.52670	−0.213445
$274$	0	0
$275$	−22.1519	−1.33581
$276$	0	0
$277$	13.9582	0.838669	0.419335	−	0.907832i	$-0.362263\pi$
$277$	13.9582	0.838669	0.419335	+	0.907832i	$0.362263\pi$
$278$	0	0
$279$	1.44462	0.0864872
$280$	0	0
$281$	−25.6871	−1.53237	−0.766183	−	0.642622i	$-0.777847\pi$
$281$	−25.6871	−1.53237	−0.766183	+	0.642622i	$0.777847\pi$
$282$	0	0
$283$	8.32901	0.495108	0.247554	−	0.968874i	$-0.420373\pi$
$283$	8.32901	0.495108	0.247554	+	0.968874i	$0.420373\pi$
$284$	0	0
$285$	5.98607	0.354584
$286$	0	0
$287$	−13.0917	−0.772776
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	17.5624	1.02953
$292$	0	0
$293$	−9.77066	−0.570808	−0.285404	−	0.958407i	$-0.592128\pi$
$293$	−9.77066	−0.570808	−0.285404	+	0.958407i	$0.592128\pi$
$294$	0	0
$295$	−0.909441	−0.0529497
$296$	0	0
$297$	−29.8643	−1.73290
$298$	0	0
$299$	0.904667	0.0523182
$300$	0	0
$301$	−0.760292	−0.0438225
$302$	0	0
$303$	−14.2207	−0.816961
$304$	0	0
$305$	4.06835	0.232953
$306$	0	0
$307$	−32.3756	−1.84778	−0.923888	−	0.382663i	$-0.875007\pi$
$307$	−32.3756	−1.84778	−0.923888	+	0.382663i	$0.875007\pi$
$308$	0	0
$309$	−22.5729	−1.28413
$310$	0	0
$311$	−18.2680	−1.03589	−0.517943	−	0.855415i	$-0.673302\pi$
$311$	−18.2680	−1.03589	−0.517943	+	0.855415i	$0.673302\pi$
$312$	0	0
$313$	−19.0831	−1.07864	−0.539321	−	0.842100i	$-0.681319\pi$
$313$	−19.0831	−1.07864	−0.539321	+	0.842100i	$0.681319\pi$
$314$	0	0
$315$	0.949882	0.0535198
$316$	0	0
$317$	0.582778	0.0327321	0.0163660	−	0.999866i	$-0.494790\pi$
$317$	0.582778	0.0327321	0.0163660	+	0.999866i	$0.494790\pi$
$318$	0	0
$319$	15.4385	0.864391
$320$	0	0
$321$	−10.7522	−0.600131
$322$	0	0
$323$	4.39103	0.244324
$324$	0	0
$325$	7.07804	0.392619
$326$	0	0
$327$	−23.7877	−1.31546
$328$	0	0
$329$	−12.6202	−0.695775
$330$	0	0
$331$	−26.9937	−1.48371	−0.741855	−	0.670560i	$-0.766054\pi$
$331$	−26.9937	−1.48371	−0.741855	+	0.670560i	$0.766054\pi$
$332$	0	0
$333$	−0.841838	−0.0461324
$334$	0	0
$335$	−0.867891	−0.0474180
$336$	0	0
$337$	−4.32399	−0.235543	−0.117771	−	0.993041i	$-0.537575\pi$
$337$	−4.32399	−0.235543	−0.117771	+	0.993041i	$0.537575\pi$
$338$	0	0
$339$	8.38278	0.455290
$340$	0	0
$341$	−10.1841	−0.551502
$342$	0	0
$343$	−16.7502	−0.904426
$344$	0	0
$345$	0.727094	0.0391454
$346$	0	0
$347$	22.1397	1.18852	0.594260	−	0.804273i	$-0.297445\pi$
$347$	22.1397	1.18852	0.594260	+	0.804273i	$0.297445\pi$
$348$	0	0
$349$	−14.5794	−0.780415	−0.390208	−	0.920727i	$-0.627597\pi$
$349$	−14.5794	−0.780415	−0.390208	+	0.920727i	$0.627597\pi$
$350$	0	0
$351$	9.54231	0.509331
$352$	0	0
$353$	7.88478	0.419665	0.209832	−	0.977737i	$-0.432708\pi$
$353$	7.88478	0.419665	0.209832	+	0.977737i	$0.432708\pi$
$354$	0	0
$355$	8.42969	0.447402
$356$	0	0
$357$	−2.07919	−0.110043
$358$	0	0
$359$	16.9897	0.896681	0.448341	−	0.893863i	$-0.352015\pi$
$359$	16.9897	0.896681	0.448341	+	0.893863i	$0.352015\pi$
$360$	0	0
$361$	0.281166	0.0147982
$362$	0	0
$363$	25.7530	1.35168
$364$	0	0
$365$	−10.6348	−0.556649
$366$	0	0
$367$	9.27832	0.484324	0.242162	−	0.970236i	$-0.422143\pi$
$367$	9.27832	0.484324	0.242162	+	0.970236i	$0.422143\pi$
$368$	0	0
$369$	7.10724	0.369988
$370$	0	0
$371$	−2.13771	−0.110985
$372$	0	0
$373$	−36.6176	−1.89599	−0.947993	−	0.318291i	$-0.896891\pi$
$373$	−36.6176	−1.89599	−0.947993	+	0.318291i	$0.896891\pi$
$374$	0	0
$375$	12.5050	0.645753
$376$	0	0
$377$	−4.93296	−0.254060
$378$	0	0
$379$	−23.5179	−1.20803	−0.604016	−	0.796972i	$-0.706434\pi$
$379$	−23.5179	−1.20803	−0.604016	+	0.796972i	$0.706434\pi$
$380$	0	0
$381$	−0.823181	−0.0421728
$382$	0	0
$383$	20.7227	1.05888	0.529441	−	0.848347i	$-0.322402\pi$
$383$	20.7227	1.05888	0.529441	+	0.848347i	$0.322402\pi$
$384$	0	0
$385$	−6.69638	−0.341279
$386$	0	0
$387$	0.412750	0.0209813
$388$	0	0
$389$	−24.6671	−1.25067	−0.625336	−	0.780356i	$-0.715038\pi$
$389$	−24.6671	−1.25067	−0.625336	+	0.780356i	$0.715038\pi$
$390$	0	0
$391$	0.533354	0.0269729
$392$	0	0
$393$	−13.5428	−0.683145
$394$	0	0
$395$	−14.1420	−0.711559
$396$	0	0
$397$	11.1413	0.559167	0.279583	−	0.960121i	$-0.409804\pi$
$397$	11.1413	0.559167	0.279583	+	0.960121i	$0.409804\pi$
$398$	0	0
$399$	−9.12981	−0.457062
$400$	0	0
$401$	−16.5261	−0.825274	−0.412637	−	0.910895i	$-0.635392\pi$
$401$	−16.5261	−0.825274	−0.412637	+	0.910895i	$0.635392\pi$
$402$	0	0
$403$	3.25406	0.162096
$404$	0	0
$405$	5.61483	0.279003
$406$	0	0
$407$	5.93470	0.294172
$408$	0	0
$409$	14.4313	0.713582	0.356791	−	0.934184i	$-0.383871\pi$
$409$	14.4313	0.713582	0.356791	+	0.934184i	$0.383871\pi$
$410$	0	0
$411$	7.72397	0.380995
$412$	0	0
$413$	1.38706	0.0682526
$414$	0	0
$415$	9.47182	0.464953
$416$	0	0
$417$	−29.1419	−1.42709
$418$	0	0
$419$	−17.6755	−0.863503	−0.431751	−	0.901993i	$-0.642104\pi$
$419$	−17.6755	−0.863503	−0.431751	+	0.901993i	$0.642104\pi$
$420$	0	0
$421$	16.3727	0.797957	0.398978	−	0.916960i	$-0.369365\pi$
$421$	16.3727	0.797957	0.398978	+	0.916960i	$0.369365\pi$
$422$	0	0
$423$	6.85131	0.333122
$424$	0	0
$425$	4.17292	0.202416
$426$	0	0
$427$	−6.20495	−0.300278
$428$	0	0
$429$	−13.4972	−0.651653
$430$	0	0
$431$	−1.66765	−0.0803279	−0.0401640	−	0.999193i	$-0.512788\pi$
$431$	−1.66765	−0.0803279	−0.0401640	+	0.999193i	$0.512788\pi$
$432$	0	0
$433$	11.2458	0.540439	0.270219	−	0.962799i	$-0.412904\pi$
$433$	11.2458	0.540439	0.270219	+	0.962799i	$0.412904\pi$
$434$	0	0
$435$	−3.96469	−0.190092
$436$	0	0
$437$	2.34197	0.112032
$438$	0	0
$439$	−24.9711	−1.19180	−0.595902	−	0.803057i	$-0.703205\pi$
$439$	−24.9711	−1.19180	−0.595902	+	0.803057i	$0.703205\pi$
$440$	0	0
$441$	3.82234	0.182016
$442$	0	0
$443$	15.8538	0.753235	0.376617	−	0.926369i	$-0.377087\pi$
$443$	15.8538	0.753235	0.376617	+	0.926369i	$0.377087\pi$
$444$	0	0
$445$	−8.36311	−0.396450
$446$	0	0
$447$	7.25917	0.343347
$448$	0	0
$449$	−9.16449	−0.432499	−0.216250	−	0.976338i	$-0.569382\pi$
$449$	−9.16449	−0.432499	−0.216250	+	0.976338i	$0.569382\pi$
$450$	0	0
$451$	−50.1039	−2.35930
$452$	0	0
$453$	25.6517	1.20522
$454$	0	0
$455$	2.13965	0.100308
$456$	0	0
$457$	−12.8295	−0.600138	−0.300069	−	0.953918i	$-0.597010\pi$
$457$	−12.8295	−0.600138	−0.300069	+	0.953918i	$0.597010\pi$
$458$	0	0
$459$	5.62575	0.262587
$460$	0	0
$461$	35.5041	1.65359	0.826796	−	0.562501i	$-0.190161\pi$
$461$	35.5041	1.65359	0.826796	+	0.562501i	$0.190161\pi$
$462$	0	0
$463$	−0.560282	−0.0260385	−0.0130193	−	0.999915i	$-0.504144\pi$
$463$	−0.560282	−0.0260385	−0.0130193	+	0.999915i	$0.504144\pi$
$464$	0	0
$465$	2.61534	0.121283
$466$	0	0
$467$	−12.3465	−0.571327	−0.285664	−	0.958330i	$-0.592214\pi$
$467$	−12.3465	−0.571327	−0.285664	+	0.958330i	$0.592214\pi$
$468$	0	0
$469$	1.32369	0.0611222
$470$	0	0
$471$	−8.29048	−0.382005
$472$	0	0
$473$	−2.90976	−0.133791
$474$	0	0
$475$	18.3234	0.840736
$476$	0	0
$477$	1.16053	0.0531370
$478$	0	0
$479$	21.5713	0.985618	0.492809	−	0.870138i	$-0.335970\pi$
$479$	21.5713	0.985618	0.492809	+	0.870138i	$0.335970\pi$
$480$	0	0
$481$	−1.89627	−0.0864625
$482$	0	0
$483$	−1.10895	−0.0504588
$484$	0	0
$485$	−10.6551	−0.483825
$486$	0	0
$487$	−32.1082	−1.45496	−0.727482	−	0.686127i	$-0.759309\pi$
$487$	−32.1082	−1.45496	−0.727482	+	0.686127i	$0.759309\pi$
$488$	0	0
$489$	−1.31243	−0.0593501
$490$	0	0
$491$	10.5518	0.476197	0.238098	−	0.971241i	$-0.423476\pi$
$491$	10.5518	0.476197	0.238098	+	0.971241i	$0.423476\pi$
$492$	0	0
$493$	−2.90827	−0.130982
$494$	0	0
$495$	3.63536	0.163397
$496$	0	0
$497$	−12.8568	−0.576705
$498$	0	0
$499$	−20.7042	−0.926847	−0.463423	−	0.886137i	$-0.653379\pi$
$499$	−20.7042	−0.926847	−0.463423	+	0.886137i	$0.653379\pi$
$500$	0	0
$501$	−26.1641	−1.16892
$502$	0	0
$503$	41.8932	1.86792	0.933962	−	0.357373i	$-0.116327\pi$
$503$	41.8932	1.86792	0.933962	+	0.357373i	$0.116327\pi$
$504$	0	0
$505$	8.62772	0.383929
$506$	0	0
$507$	−15.1743	−0.673913
$508$	0	0
$509$	31.9353	1.41551	0.707755	−	0.706458i	$-0.249708\pi$
$509$	31.9353	1.41551	0.707755	+	0.706458i	$0.249708\pi$
$510$	0	0
$511$	16.2199	0.717525
$512$	0	0
$513$	24.7028	1.09066
$514$	0	0
$515$	13.6950	0.603474
$516$	0	0
$517$	−48.2997	−2.12422
$518$	0	0
$519$	−10.7040	−0.469855
$520$	0	0
$521$	−39.1684	−1.71600	−0.857999	−	0.513651i	$-0.828293\pi$
$521$	−39.1684	−1.71600	−0.857999	+	0.513651i	$0.828293\pi$
$522$	0	0
$523$	22.3922	0.979142	0.489571	−	0.871963i	$-0.337153\pi$
$523$	22.3922	0.979142	0.489571	+	0.871963i	$0.337153\pi$
$524$	0	0
$525$	−8.67630	−0.378665
$526$	0	0
$527$	1.91846	0.0835695
$528$	0	0
$529$	−22.7155	−0.987632
$530$	0	0
$531$	−0.753011	−0.0326779
$532$	0	0
$533$	16.0093	0.693441
$534$	0	0
$535$	6.52338	0.282030
$536$	0	0
$537$	18.1557	0.783475
$538$	0	0
$539$	−26.9463	−1.16066
$540$	0	0
$541$	33.9881	1.46126	0.730632	−	0.682771i	$-0.239225\pi$
$541$	33.9881	1.46126	0.730632	+	0.682771i	$0.239225\pi$
$542$	0	0
$543$	35.7295	1.53330
$544$	0	0
$545$	14.4320	0.618199
$546$	0	0
$547$	4.87414	0.208403	0.104201	−	0.994556i	$-0.466771\pi$
$547$	4.87414	0.208403	0.104201	+	0.994556i	$0.466771\pi$
$548$	0	0
$549$	3.36856	0.143767
$550$	0	0
$551$	−12.7703	−0.544033
$552$	0	0
$553$	21.5690	0.917206
$554$	0	0
$555$	−1.52406	−0.0646928
$556$	0	0
$557$	2.47610	0.104916	0.0524579	−	0.998623i	$-0.483294\pi$
$557$	2.47610	0.104916	0.0524579	+	0.998623i	$0.483294\pi$
$558$	0	0
$559$	0.929735	0.0393236
$560$	0	0
$561$	−7.95742	−0.335962
$562$	0	0
$563$	−1.19799	−0.0504891	−0.0252445	−	0.999681i	$-0.508036\pi$
$563$	−1.19799	−0.0504891	−0.0252445	+	0.999681i	$0.508036\pi$
$564$	0	0
$565$	−5.08583	−0.213963
$566$	0	0
$567$	−8.56361	−0.359638
$568$	0	0
$569$	36.9421	1.54869	0.774346	−	0.632762i	$-0.218079\pi$
$569$	36.9421	1.54869	0.774346	+	0.632762i	$0.218079\pi$
$570$	0	0
$571$	−6.70986	−0.280799	−0.140400	−	0.990095i	$-0.544839\pi$
$571$	−6.70986	−0.280799	−0.140400	+	0.990095i	$0.544839\pi$
$572$	0	0
$573$	34.5863	1.44486
$574$	0	0
$575$	2.22564	0.0928157
$576$	0	0
$577$	−37.4578	−1.55939	−0.779694	−	0.626160i	$-0.784626\pi$
$577$	−37.4578	−1.55939	−0.779694	+	0.626160i	$0.784626\pi$
$578$	0	0
$579$	26.0136	1.08109
$580$	0	0
$581$	−14.4462	−0.599329
$582$	0	0
$583$	−8.18138	−0.338838
$584$	0	0
$585$	−1.16158	−0.0480254
$586$	0	0
$587$	1.61392	0.0666136	0.0333068	−	0.999445i	$-0.489396\pi$
$587$	1.61392	0.0666136	0.0333068	+	0.999445i	$0.489396\pi$
$588$	0	0
$589$	8.42402	0.347106
$590$	0	0
$591$	3.65767	0.150457
$592$	0	0
$593$	10.5794	0.434445	0.217222	−	0.976122i	$-0.430300\pi$
$593$	10.5794	0.434445	0.217222	+	0.976122i	$0.430300\pi$
$594$	0	0
$595$	1.26145	0.0517143
$596$	0	0
$597$	−7.43363	−0.304238
$598$	0	0
$599$	25.8115	1.05463	0.527315	−	0.849670i	$-0.323199\pi$
$599$	25.8115	1.05463	0.527315	+	0.849670i	$0.323199\pi$
$600$	0	0
$601$	−31.6675	−1.29174	−0.645872	−	0.763446i	$-0.723506\pi$
$601$	−31.6675	−1.29174	−0.645872	+	0.763446i	$0.723506\pi$
$602$	0	0
$603$	−0.718608	−0.0292640
$604$	0	0
$605$	−15.6243	−0.635219
$606$	0	0
$607$	30.4594	1.23631	0.618154	−	0.786057i	$-0.287881\pi$
$607$	30.4594	1.23631	0.618154	+	0.786057i	$0.287881\pi$
$608$	0	0
$609$	6.04685	0.245031
$610$	0	0
$611$	15.4328	0.624346
$612$	0	0
$613$	−0.661027	−0.0266986	−0.0133493	−	0.999911i	$-0.504249\pi$
$613$	−0.661027	−0.0266986	−0.0133493	+	0.999911i	$0.504249\pi$
$614$	0	0
$615$	12.8669	0.518845
$616$	0	0
$617$	−22.4957	−0.905641	−0.452821	−	0.891602i	$-0.649582\pi$
$617$	−22.4957	−0.905641	−0.452821	+	0.891602i	$0.649582\pi$
$618$	0	0
$619$	−28.7325	−1.15486	−0.577428	−	0.816442i	$-0.695944\pi$
$619$	−28.7325	−1.15486	−0.577428	+	0.816442i	$0.695944\pi$
$620$	0	0
$621$	3.00052	0.120407
$622$	0	0
$623$	12.7552	0.511027
$624$	0	0
$625$	13.2778	0.531113
$626$	0	0
$627$	−34.9413	−1.39542
$628$	0	0
$629$	−1.11796	−0.0445761
$630$	0	0
$631$	8.11219	0.322941	0.161471	−	0.986878i	$-0.448376\pi$
$631$	8.11219	0.322941	0.161471	+	0.986878i	$0.448376\pi$
$632$	0	0
$633$	−6.78446	−0.269658
$634$	0	0
$635$	0.499424	0.0198190
$636$	0	0
$637$	8.60996	0.341139
$638$	0	0
$639$	6.97973	0.276114
$640$	0	0
$641$	−21.5345	−0.850560	−0.425280	−	0.905062i	$-0.639824\pi$
$641$	−21.5345	−0.850560	−0.425280	+	0.905062i	$0.639824\pi$
$642$	0	0
$643$	−31.2957	−1.23418	−0.617090	−	0.786892i	$-0.711689\pi$
$643$	−31.2957	−1.23418	−0.617090	+	0.786892i	$0.711689\pi$
$644$	0	0
$645$	0.747241	0.0294226
$646$	0	0
$647$	36.9626	1.45315	0.726575	−	0.687088i	$-0.241111\pi$
$647$	36.9626	1.45315	0.726575	+	0.687088i	$0.241111\pi$
$648$	0	0
$649$	5.30850	0.208377
$650$	0	0
$651$	−3.98885	−0.156335
$652$	0	0
$653$	12.7241	0.497933	0.248967	−	0.968512i	$-0.419909\pi$
$653$	12.7241	0.497933	0.248967	+	0.968512i	$0.419909\pi$
$654$	0	0
$655$	8.21642	0.321042
$656$	0	0
$657$	−8.80551	−0.343536
$658$	0	0
$659$	−26.3440	−1.02622	−0.513109	−	0.858323i	$-0.671506\pi$
$659$	−26.3440	−1.02622	−0.513109	+	0.858323i	$0.671506\pi$
$660$	0	0
$661$	−3.92121	−0.152518	−0.0762588	−	0.997088i	$-0.524298\pi$
$661$	−3.92121	−0.152518	−0.0762588	+	0.997088i	$0.524298\pi$
$662$	0	0
$663$	2.54257	0.0987454
$664$	0	0
$665$	5.53905	0.214795
$666$	0	0
$667$	−1.55113	−0.0600602
$668$	0	0
$669$	6.09927	0.235811
$670$	0	0
$671$	−23.7473	−0.916756
$672$	0	0
$673$	1.89292	0.0729667	0.0364834	−	0.999334i	$-0.488384\pi$
$673$	1.89292	0.0729667	0.0364834	+	0.999334i	$0.488384\pi$
$674$	0	0
$675$	23.4758	0.903583
$676$	0	0
$677$	−43.9258	−1.68821	−0.844103	−	0.536181i	$-0.819866\pi$
$677$	−43.9258	−1.68821	−0.844103	+	0.536181i	$0.819866\pi$
$678$	0	0
$679$	16.2510	0.623654
$680$	0	0
$681$	−21.3983	−0.819986
$682$	0	0
$683$	39.7295	1.52021	0.760103	−	0.649803i	$-0.225149\pi$
$683$	39.7295	1.52021	0.760103	+	0.649803i	$0.225149\pi$
$684$	0	0
$685$	−4.68613	−0.179048
$686$	0	0
$687$	18.6063	0.709875
$688$	0	0
$689$	2.61414	0.0995907
$690$	0	0
$691$	29.4323	1.11966	0.559829	−	0.828608i	$-0.310867\pi$
$691$	29.4323	1.11966	0.559829	+	0.828608i	$0.310867\pi$
$692$	0	0
$693$	−5.54456	−0.210620
$694$	0	0
$695$	17.6804	0.670655
$696$	0	0
$697$	9.43844	0.357506
$698$	0	0
$699$	−7.09679	−0.268425
$700$	0	0
$701$	30.4942	1.15175	0.575875	−	0.817538i	$-0.304661\pi$
$701$	30.4942	1.15175	0.575875	+	0.817538i	$0.304661\pi$
$702$	0	0
$703$	−4.90901	−0.185147
$704$	0	0
$705$	12.4036	0.467146
$706$	0	0
$707$	−13.1588	−0.494888
$708$	0	0
$709$	−27.6143	−1.03708	−0.518538	−	0.855054i	$-0.673524\pi$
$709$	−27.6143	−1.03708	−0.518538	+	0.855054i	$0.673524\pi$
$710$	0	0
$711$	−11.7094	−0.439138
$712$	0	0
$713$	1.02322	0.0383198
$714$	0	0
$715$	8.18878	0.306243
$716$	0	0
$717$	−31.3003	−1.16893
$718$	0	0
$719$	−4.32118	−0.161153	−0.0805764	−	0.996748i	$-0.525676\pi$
$719$	−4.32118	−0.161153	−0.0805764	+	0.996748i	$0.525676\pi$
$720$	0	0
$721$	−20.8873	−0.777883
$722$	0	0
$723$	−18.0767	−0.672279
$724$	0	0
$725$	−12.1360	−0.450718
$726$	0	0
$727$	−40.0759	−1.48633	−0.743167	−	0.669106i	$-0.766677\pi$
$727$	−40.0759	−1.48633	−0.743167	+	0.669106i	$0.766677\pi$
$728$	0	0
$729$	29.9480	1.10918
$730$	0	0
$731$	0.548133	0.0202734
$732$	0	0
$733$	−0.547067	−0.0202064	−0.0101032	−	0.999949i	$-0.503216\pi$
$733$	−0.547067	−0.0202064	−0.0101032	+	0.999949i	$0.503216\pi$
$734$	0	0
$735$	6.91994	0.255246
$736$	0	0
$737$	5.06597	0.186607
$738$	0	0
$739$	43.0740	1.58450	0.792252	−	0.610194i	$-0.208909\pi$
$739$	43.0740	1.58450	0.792252	+	0.610194i	$0.208909\pi$
$740$	0	0
$741$	11.1645	0.410139
$742$	0	0
$743$	−13.5341	−0.496518	−0.248259	−	0.968694i	$-0.579858\pi$
$743$	−13.5341	−0.496518	−0.248259	+	0.968694i	$0.579858\pi$
$744$	0	0
$745$	−4.40414	−0.161355
$746$	0	0
$747$	7.84260	0.286946
$748$	0	0
$749$	−9.94930	−0.363540
$750$	0	0
$751$	−18.4354	−0.672718	−0.336359	−	0.941734i	$-0.609195\pi$
$751$	−18.4354	−0.672718	−0.336359	+	0.941734i	$0.609195\pi$
$752$	0	0
$753$	24.2164	0.882494
$754$	0	0
$755$	−15.5629	−0.566391
$756$	0	0
$757$	28.2855	1.02806	0.514028	−	0.857774i	$-0.328153\pi$
$757$	28.2855	1.02806	0.514028	+	0.857774i	$0.328153\pi$
$758$	0	0
$759$	−4.24412	−0.154052
$760$	0	0
$761$	13.3251	0.483034	0.241517	−	0.970397i	$-0.422355\pi$
$761$	13.3251	0.483034	0.241517	+	0.970397i	$0.422355\pi$
$762$	0	0
$763$	−22.0114	−0.796865
$764$	0	0
$765$	−0.684818	−0.0247597
$766$	0	0
$767$	−1.69618	−0.0612457
$768$	0	0
$769$	−18.7718	−0.676928	−0.338464	−	0.940979i	$-0.609907\pi$
$769$	−18.7718	−0.676928	−0.338464	+	0.940979i	$0.609907\pi$
$770$	0	0
$771$	−43.2722	−1.55841
$772$	0	0
$773$	26.6737	0.959388	0.479694	−	0.877436i	$-0.340748\pi$
$773$	26.6737	0.959388	0.479694	+	0.877436i	$0.340748\pi$
$774$	0	0
$775$	8.00558	0.287569
$776$	0	0
$777$	2.32446	0.0833896
$778$	0	0
$779$	41.4445	1.48490
$780$	0	0
$781$	−49.2050	−1.76069
$782$	0	0
$783$	−16.3612	−0.584700
$784$	0	0
$785$	5.02983	0.179522
$786$	0	0
$787$	−17.5866	−0.626893	−0.313447	−	0.949606i	$-0.601484\pi$
$787$	−17.5866	−0.626893	−0.313447	+	0.949606i	$0.601484\pi$
$788$	0	0
$789$	23.8499	0.849080
$790$	0	0
$791$	7.75679	0.275800
$792$	0	0
$793$	7.58781	0.269451
$794$	0	0
$795$	2.10102	0.0745155
$796$	0	0
$797$	−6.50821	−0.230533	−0.115266	−	0.993335i	$-0.536772\pi$
$797$	−6.50821	−0.230533	−0.115266	+	0.993335i	$0.536772\pi$
$798$	0	0
$799$	9.09856	0.321884
$800$	0	0
$801$	−6.92460	−0.244669
$802$	0	0
$803$	62.0762	2.19062
$804$	0	0
$805$	0.672797	0.0237130
$806$	0	0
$807$	12.2645	0.431732
$808$	0	0
$809$	30.5933	1.07560	0.537801	−	0.843072i	$-0.319255\pi$
$809$	30.5933	1.07560	0.537801	+	0.843072i	$0.319255\pi$
$810$	0	0
$811$	−2.56468	−0.0900580	−0.0450290	−	0.998986i	$-0.514338\pi$
$811$	−2.56468	−0.0900580	−0.0450290	+	0.998986i	$0.514338\pi$
$812$	0	0
$813$	20.5355	0.720210
$814$	0	0
$815$	0.796250	0.0278914
$816$	0	0
$817$	2.40687	0.0842057
$818$	0	0
$819$	1.77161	0.0619051
$820$	0	0
$821$	5.48274	0.191349	0.0956744	−	0.995413i	$-0.469499\pi$
$821$	5.48274	0.191349	0.0956744	+	0.995413i	$0.469499\pi$
$822$	0	0
$823$	47.8997	1.66968	0.834839	−	0.550495i	$-0.185561\pi$
$823$	47.8997	1.66968	0.834839	+	0.550495i	$0.185561\pi$
$824$	0	0
$825$	−33.2056	−1.15607
$826$	0	0
$827$	40.7577	1.41728	0.708642	−	0.705568i	$-0.249308\pi$
$827$	40.7577	1.41728	0.708642	+	0.705568i	$0.249308\pi$
$828$	0	0
$829$	35.6577	1.23844	0.619222	−	0.785216i	$-0.287448\pi$
$829$	35.6577	1.23844	0.619222	+	0.785216i	$0.287448\pi$
$830$	0	0
$831$	20.9233	0.725823
$832$	0	0
$833$	5.07607	0.175875
$834$	0	0
$835$	15.8737	0.549333
$836$	0	0
$837$	10.7928	0.373053
$838$	0	0
$839$	−41.9639	−1.44876	−0.724378	−	0.689403i	$-0.757873\pi$
$839$	−41.9639	−1.44876	−0.724378	+	0.689403i	$0.757873\pi$
$840$	0	0
$841$	−20.5420	−0.708345
$842$	0	0
$843$	−38.5049	−1.32618
$844$	0	0
$845$	9.20623	0.316704
$846$	0	0
$847$	23.8298	0.818803
$848$	0	0
$849$	12.4851	0.428489
$850$	0	0
$851$	−0.596270	−0.0204399
$852$	0	0
$853$	−11.5052	−0.393932	−0.196966	−	0.980410i	$-0.563109\pi$
$853$	−11.5052	−0.393932	−0.196966	+	0.980410i	$0.563109\pi$
$854$	0	0
$855$	−3.00706	−0.102839
$856$	0	0
$857$	9.90046	0.338193	0.169097	−	0.985599i	$-0.445915\pi$
$857$	9.90046	0.338193	0.169097	+	0.985599i	$0.445915\pi$
$858$	0	0
$859$	−19.4959	−0.665191	−0.332596	−	0.943070i	$-0.607924\pi$
$859$	−19.4959	−0.665191	−0.332596	+	0.943070i	$0.607924\pi$
$860$	0	0
$861$	−19.6243	−0.668796
$862$	0	0
$863$	38.8038	1.32090	0.660449	−	0.750871i	$-0.270366\pi$
$863$	38.8038	1.32090	0.660449	+	0.750871i	$0.270366\pi$
$864$	0	0
$865$	6.49414	0.220807
$866$	0	0
$867$	1.49900	0.0509086
$868$	0	0
$869$	82.5480	2.80025
$870$	0	0
$871$	−1.61869	−0.0548473
$872$	0	0
$873$	−8.82238	−0.298592
$874$	0	0
$875$	11.5711	0.391176
$876$	0	0
$877$	10.9660	0.370297	0.185148	−	0.982711i	$-0.440723\pi$
$877$	10.9660	0.370297	0.185148	+	0.982711i	$0.440723\pi$
$878$	0	0
$879$	−14.6462	−0.494004
$880$	0	0
$881$	−33.9790	−1.14478	−0.572391	−	0.819980i	$-0.693984\pi$
$881$	−33.9790	−1.14478	−0.572391	+	0.819980i	$0.693984\pi$
$882$	0	0
$883$	−4.30032	−0.144717	−0.0723586	−	0.997379i	$-0.523053\pi$
$883$	−4.30032	−0.144717	−0.0723586	+	0.997379i	$0.523053\pi$
$884$	0	0
$885$	−1.36325	−0.0458251
$886$	0	0
$887$	−22.1724	−0.744475	−0.372237	−	0.928138i	$-0.621409\pi$
$887$	−22.1724	−0.744475	−0.372237	+	0.928138i	$0.621409\pi$
$888$	0	0
$889$	−0.761709	−0.0255469
$890$	0	0
$891$	−32.7744	−1.09798
$892$	0	0
$893$	39.9521	1.33694
$894$	0	0
$895$	−11.0150	−0.368192
$896$	0	0
$897$	1.35609	0.0452786
$898$	0	0
$899$	−5.57939	−0.186083
$900$	0	0
$901$	1.54119	0.0513444
$902$	0	0
$903$	−1.13967	−0.0379260
$904$	0	0
$905$	−21.6771	−0.720570
$906$	0	0
$907$	−19.7896	−0.657103	−0.328552	−	0.944486i	$-0.606560\pi$
$907$	−19.7896	−0.657103	−0.328552	+	0.944486i	$0.606560\pi$
$908$	0	0
$909$	7.14370	0.236941
$910$	0	0
$911$	16.2218	0.537451	0.268726	−	0.963217i	$-0.413397\pi$
$911$	16.2218	0.537451	0.268726	+	0.963217i	$0.413397\pi$
$912$	0	0
$913$	−55.2880	−1.82976
$914$	0	0
$915$	6.09844	0.201608
$916$	0	0
$917$	−12.5315	−0.413826
$918$	0	0
$919$	41.1678	1.35800	0.679001	−	0.734137i	$-0.262413\pi$
$919$	41.1678	1.35800	0.679001	+	0.734137i	$0.262413\pi$
$920$	0	0
$921$	−48.5310	−1.59915
$922$	0	0
$923$	15.7221	0.517499
$924$	0	0
$925$	−4.66517	−0.153390
$926$	0	0
$927$	11.3394	0.372433
$928$	0	0
$929$	−23.5656	−0.773161	−0.386580	−	0.922256i	$-0.626344\pi$
$929$	−23.5656	−0.773161	−0.386580	+	0.922256i	$0.626344\pi$
$930$	0	0
$931$	22.2892	0.730499
$932$	0	0
$933$	−27.3837	−0.896503
$934$	0	0
$935$	4.82776	0.157885
$936$	0	0
$937$	30.2453	0.988071	0.494036	−	0.869442i	$-0.335521\pi$
$937$	30.2453	0.988071	0.494036	+	0.869442i	$0.335521\pi$
$938$	0	0
$939$	−28.6056	−0.933507
$940$	0	0
$941$	2.28373	0.0744475	0.0372237	−	0.999307i	$-0.488149\pi$
$941$	2.28373	0.0744475	0.0372237	+	0.999307i	$0.488149\pi$
$942$	0	0
$943$	5.03403	0.163931
$944$	0	0
$945$	7.09658	0.230852
$946$	0	0
$947$	49.8704	1.62057	0.810286	−	0.586035i	$-0.199312\pi$
$947$	49.8704	1.62057	0.810286	+	0.586035i	$0.199312\pi$
$948$	0	0
$949$	−19.8347	−0.643863
$950$	0	0
$951$	0.873582	0.0283278
$952$	0	0
$953$	−0.559837	−0.0181349	−0.00906746	−	0.999959i	$-0.502886\pi$
$953$	−0.559837	−0.0181349	−0.00906746	+	0.999959i	$0.502886\pi$
$954$	0	0
$955$	−20.9835	−0.679009
$956$	0	0
$957$	23.1423	0.748084
$958$	0	0
$959$	7.14717	0.230794
$960$	0	0
$961$	−27.3195	−0.881274
$962$	0	0
$963$	5.40131	0.174055
$964$	0	0
$965$	−15.7824	−0.508054
$966$	0	0
$967$	−33.5072	−1.07752	−0.538759	−	0.842460i	$-0.681107\pi$
$967$	−33.5072	−1.07752	−0.538759	+	0.842460i	$0.681107\pi$
$968$	0	0
$969$	6.58214	0.211449
$970$	0	0
$971$	−11.3815	−0.365250	−0.182625	−	0.983183i	$-0.558459\pi$
$971$	−11.3815	−0.365250	−0.182625	+	0.983183i	$0.558459\pi$
$972$	0	0
$973$	−26.9657	−0.864481
$974$	0	0
$975$	10.6100	0.339790
$976$	0	0
$977$	−23.7865	−0.760997	−0.380499	−	0.924781i	$-0.624248\pi$
$977$	−23.7865	−0.760997	−0.380499	+	0.924781i	$0.624248\pi$
$978$	0	0
$979$	48.8163	1.56018
$980$	0	0
$981$	11.9496	0.381521
$982$	0	0
$983$	6.14485	0.195990	0.0979952	−	0.995187i	$-0.468757\pi$
$983$	6.14485	0.195990	0.0979952	+	0.995187i	$0.468757\pi$
$984$	0	0
$985$	−2.21911	−0.0707067
$986$	0	0
$987$	−18.9177	−0.602156
$988$	0	0
$989$	0.292349	0.00929616
$990$	0	0
$991$	14.3267	0.455102	0.227551	−	0.973766i	$-0.426928\pi$
$991$	14.3267	0.455102	0.227551	+	0.973766i	$0.426928\pi$
$992$	0	0
$993$	−40.4635	−1.28407
$994$	0	0
$995$	4.50998	0.142976
$996$	0	0
$997$	42.0695	1.33235	0.666177	−	0.745793i	$-0.267929\pi$
$997$	42.0695	1.33235	0.666177	+	0.745793i	$0.267929\pi$
$998$	0	0
$999$	−6.28938	−0.198987

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.x.1.18	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.x.1.18	✓	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)

\(f(q)\)	\(=\)	\(q+1.49900 q^{3} -0.909441 q^{5} +1.38706 q^{7} -0.753011 q^{9} +O(q^{10})\)	\(q+1.49900 q^{3} -0.909441 q^{5} +1.38706 q^{7} -0.753011 q^{9} +5.30850 q^{11} -1.69618 q^{13} -1.36325 q^{15} -1.00000 q^{17} -4.39103 q^{19} +2.07919 q^{21} -0.533354 q^{23} -4.17292 q^{25} -5.62575 q^{27} +2.90827 q^{29} -1.91846 q^{31} +7.95742 q^{33} -1.26145 q^{35} +1.11796 q^{37} -2.54257 q^{39} -9.43844 q^{41} -0.548133 q^{43} +0.684818 q^{45} -9.09856 q^{47} -5.07607 q^{49} -1.49900 q^{51} -1.54119 q^{53} -4.82776 q^{55} -6.58214 q^{57} +1.00000 q^{59} -4.47346 q^{61} -1.04447 q^{63} +1.54258 q^{65} +0.954313 q^{67} -0.799496 q^{69} -9.26909 q^{71} +11.6937 q^{73} -6.25519 q^{75} +7.36319 q^{77} +15.5502 q^{79} -6.17394 q^{81} -10.4150 q^{83} +0.909441 q^{85} +4.35948 q^{87} +9.19589 q^{89} -2.35271 q^{91} -2.87577 q^{93} +3.99338 q^{95} +11.7161 q^{97} -3.99735 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) \) 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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