LMFDB - Embedded newform 4024.2.a.f.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.27275 q^{3} +2.37900 q^{5} +1.75584 q^{7} +2.16537 q^{9} +O(q^{10})$	$q-2.27275 q^{3} +2.37900 q^{5} +1.75584 q^{7} +2.16537 q^{9} +6.02465 q^{11} +5.29882 q^{13} -5.40685 q^{15} +7.04287 q^{17} -6.15087 q^{19} -3.99059 q^{21} +5.05348 q^{23} +0.659618 q^{25} +1.89689 q^{27} +2.49852 q^{29} +4.57755 q^{31} -13.6925 q^{33} +4.17715 q^{35} +8.89524 q^{37} -12.0429 q^{39} +10.6215 q^{41} +11.1940 q^{43} +5.15141 q^{45} -10.2780 q^{47} -3.91701 q^{49} -16.0067 q^{51} -2.73408 q^{53} +14.3326 q^{55} +13.9794 q^{57} -13.9984 q^{59} -8.80697 q^{61} +3.80206 q^{63} +12.6059 q^{65} -9.78880 q^{67} -11.4853 q^{69} -8.39309 q^{71} +7.86197 q^{73} -1.49914 q^{75} +10.5783 q^{77} -2.93993 q^{79} -10.8073 q^{81} -9.64967 q^{83} +16.7550 q^{85} -5.67851 q^{87} -18.6612 q^{89} +9.30390 q^{91} -10.4036 q^{93} -14.6329 q^{95} +6.29361 q^{97} +13.0456 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) $ 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9	$ 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9} + 22 q^{11} + 25 q^{13} - 4 q^{15} + 17 q^{17} + 6 q^{19} + 18 q^{21} + 16 q^{23} + 47 q^{25} - 20 q^{27} + 47 q^{29} - 7 q^{31} - 6 q^{33} + 19 q^{35} + 75 q^{37} + 21 q^{39} + 22 q^{41} - 5 q^{43} + 33 q^{45} + 10 q^{47} + 31 q^{49} + 9 q^{51} + 64 q^{53} - 3 q^{55} + 5 q^{57} + 28 q^{59} + 49 q^{61} - 10 q^{63} + 46 q^{65} - 14 q^{67} + 30 q^{69} + 35 q^{71} + 19 q^{73} - 33 q^{75} + 32 q^{77} - 12 q^{79} + 57 q^{81} + 82 q^{85} - 5 q^{87} + 42 q^{89} - 15 q^{91} + 55 q^{93} + 33 q^{95} + 4 q^{97} + 22 q^{99}+O(q^{100}) $ 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9 + 22 * q^11 + 25 * q^13 - 4 * q^15 + 17 * q^17 + 6 * q^19 + 18 * q^21 + 16 * q^23 + 47 * q^25 - 20 * q^27 + 47 * q^29 - 7 * q^31 - 6 * q^33 + 19 * q^35 + 75 * q^37 + 21 * q^39 + 22 * q^41 - 5 * q^43 + 33 * q^45 + 10 * q^47 + 31 * q^49 + 9 * q^51 + 64 * q^53 - 3 * q^55 + 5 * q^57 + 28 * q^59 + 49 * q^61 - 10 * q^63 + 46 * q^65 - 14 * q^67 + 30 * q^69 + 35 * q^71 + 19 * q^73 - 33 * q^75 + 32 * q^77 - 12 * q^79 + 57 * q^81 + 82 * q^85 - 5 * q^87 + 42 * q^89 - 15 * q^91 + 55 * q^93 + 33 * q^95 + 4 * q^97 + 22 * 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
$3$	−2.27275	−1.31217	−0.656085	−	0.754687i	$-0.727789\pi$
$3$	−2.27275	−1.31217	−0.656085	+	0.754687i	$0.727789\pi$
$4$	0	0
$5$	2.37900	1.06392	0.531959	−	0.846770i	$-0.321456\pi$
$5$	2.37900	1.06392	0.531959	+	0.846770i	$0.321456\pi$
$6$	0	0
$7$	1.75584	0.663647	0.331823	−	0.943342i	$-0.392336\pi$
$7$	1.75584	0.663647	0.331823	+	0.943342i	$0.392336\pi$
$8$	0	0
$9$	2.16537	0.721791
$10$	0	0
$11$	6.02465	1.81650	0.908250	−	0.418428i	$-0.137419\pi$
$11$	6.02465	1.81650	0.908250	+	0.418428i	$0.137419\pi$
$12$	0	0
$13$	5.29882	1.46963	0.734814	−	0.678269i	$-0.237270\pi$
$13$	5.29882	1.46963	0.734814	+	0.678269i	$0.237270\pi$
$14$	0	0
$15$	−5.40685	−1.39604
$16$	0	0
$17$	7.04287	1.70815	0.854074	−	0.520151i	$-0.174125\pi$
$17$	7.04287	1.70815	0.854074	+	0.520151i	$0.174125\pi$
$18$	0	0
$19$	−6.15087	−1.41111	−0.705553	−	0.708657i	$-0.749302\pi$
$19$	−6.15087	−1.41111	−0.705553	+	0.708657i	$0.749302\pi$
$20$	0	0
$21$	−3.99059	−0.870818
$22$	0	0
$23$	5.05348	1.05372	0.526862	−	0.849951i	$-0.323368\pi$
$23$	5.05348	1.05372	0.526862	+	0.849951i	$0.323368\pi$
$24$	0	0
$25$	0.659618	0.131924
$26$	0	0
$27$	1.89689	0.365057
$28$	0	0
$29$	2.49852	0.463964	0.231982	−	0.972720i	$-0.425479\pi$
$29$	2.49852	0.463964	0.231982	+	0.972720i	$0.425479\pi$
$30$	0	0
$31$	4.57755	0.822152	0.411076	−	0.911601i	$-0.365153\pi$
$31$	4.57755	0.822152	0.411076	+	0.911601i	$0.365153\pi$
$32$	0	0
$33$	−13.6925	−2.38356
$34$	0	0
$35$	4.17715	0.706066
$36$	0	0
$37$	8.89524	1.46237	0.731184	−	0.682180i	$-0.238968\pi$
$37$	8.89524	1.46237	0.731184	+	0.682180i	$0.238968\pi$
$38$	0	0
$39$	−12.0429	−1.92840
$40$	0	0
$41$	10.6215	1.65880	0.829401	−	0.558654i	$-0.188682\pi$
$41$	10.6215	1.65880	0.829401	+	0.558654i	$0.188682\pi$
$42$	0	0
$43$	11.1940	1.70707	0.853536	−	0.521034i	$-0.174454\pi$
$43$	11.1940	1.70707	0.853536	+	0.521034i	$0.174454\pi$
$44$	0	0
$45$	5.15141	0.767927
$46$	0	0
$47$	−10.2780	−1.49920	−0.749599	−	0.661892i	$-0.769754\pi$
$47$	−10.2780	−1.49920	−0.749599	+	0.661892i	$0.769754\pi$
$48$	0	0
$49$	−3.91701	−0.559573
$50$	0	0
$51$	−16.0067	−2.24138
$52$	0	0
$53$	−2.73408	−0.375555	−0.187777	−	0.982212i	$-0.560128\pi$
$53$	−2.73408	−0.375555	−0.187777	+	0.982212i	$0.560128\pi$
$54$	0	0
$55$	14.3326	1.93261
$56$	0	0
$57$	13.9794	1.85161
$58$	0	0
$59$	−13.9984	−1.82243	−0.911215	−	0.411931i	$-0.864855\pi$
$59$	−13.9984	−1.82243	−0.911215	+	0.411931i	$0.864855\pi$
$60$	0	0
$61$	−8.80697	−1.12762	−0.563809	−	0.825905i	$-0.690664\pi$
$61$	−8.80697	−1.12762	−0.563809	+	0.825905i	$0.690664\pi$
$62$	0	0
$63$	3.80206	0.479014
$64$	0	0
$65$	12.6059	1.56356
$66$	0	0
$67$	−9.78880	−1.19589	−0.597946	−	0.801536i	$-0.704016\pi$
$67$	−9.78880	−1.19589	−0.597946	+	0.801536i	$0.704016\pi$
$68$	0	0
$69$	−11.4853	−1.38267
$70$	0	0
$71$	−8.39309	−0.996076	−0.498038	−	0.867155i	$-0.665946\pi$
$71$	−8.39309	−0.996076	−0.498038	+	0.867155i	$0.665946\pi$
$72$	0	0
$73$	7.86197	0.920174	0.460087	−	0.887874i	$-0.347818\pi$
$73$	7.86197	0.920174	0.460087	+	0.887874i	$0.347818\pi$
$74$	0	0
$75$	−1.49914	−0.173106
$76$	0	0
$77$	10.5783	1.20551
$78$	0	0
$79$	−2.93993	−0.330768	−0.165384	−	0.986229i	$-0.552886\pi$
$79$	−2.93993	−0.330768	−0.165384	+	0.986229i	$0.552886\pi$
$80$	0	0
$81$	−10.8073	−1.20081
$82$	0	0
$83$	−9.64967	−1.05919	−0.529594	−	0.848251i	$-0.677656\pi$
$83$	−9.64967	−1.05919	−0.529594	+	0.848251i	$0.677656\pi$
$84$	0	0
$85$	16.7550	1.81733
$86$	0	0
$87$	−5.67851	−0.608800
$88$	0	0
$89$	−18.6612	−1.97808	−0.989039	−	0.147652i	$-0.952828\pi$
$89$	−18.6612	−1.97808	−0.989039	+	0.147652i	$0.952828\pi$
$90$	0	0
$91$	9.30390	0.975313
$92$	0	0
$93$	−10.4036	−1.07880
$94$	0	0
$95$	−14.6329	−1.50130
$96$	0	0
$97$	6.29361	0.639020	0.319510	−	0.947583i	$-0.396482\pi$
$97$	6.29361	0.639020	0.319510	+	0.947583i	$0.396482\pi$
$98$	0	0
$99$	13.0456	1.31113
$100$	0	0
$101$	−1.35205	−0.134534	−0.0672670	−	0.997735i	$-0.521428\pi$
$101$	−1.35205	−0.134534	−0.0672670	+	0.997735i	$0.521428\pi$
$102$	0	0
$103$	11.9678	1.17923	0.589613	−	0.807686i	$-0.299280\pi$
$103$	11.9678	1.17923	0.589613	+	0.807686i	$0.299280\pi$
$104$	0	0
$105$	−9.49359	−0.926479
$106$	0	0
$107$	−7.83355	−0.757298	−0.378649	−	0.925540i	$-0.623611\pi$
$107$	−7.83355	−0.757298	−0.378649	+	0.925540i	$0.623611\pi$
$108$	0	0
$109$	0.729328	0.0698569	0.0349285	−	0.999390i	$-0.488880\pi$
$109$	0.729328	0.0698569	0.0349285	+	0.999390i	$0.488880\pi$
$110$	0	0
$111$	−20.2166	−1.91888
$112$	0	0
$113$	−13.3996	−1.26053	−0.630263	−	0.776382i	$-0.717053\pi$
$113$	−13.3996	−1.26053	−0.630263	+	0.776382i	$0.717053\pi$
$114$	0	0
$115$	12.0222	1.12108
$116$	0	0
$117$	11.4739	1.06076
$118$	0	0
$119$	12.3662	1.13361
$120$	0	0
$121$	25.2964	2.29967
$122$	0	0
$123$	−24.1400	−2.17663
$124$	0	0
$125$	−10.3257	−0.923563
$126$	0	0
$127$	−6.44554	−0.571949	−0.285974	−	0.958237i	$-0.592317\pi$
$127$	−6.44554	−0.571949	−0.285974	+	0.958237i	$0.592317\pi$
$128$	0	0
$129$	−25.4412	−2.23997
$130$	0	0
$131$	8.51788	0.744211	0.372105	−	0.928190i	$-0.378636\pi$
$131$	8.51788	0.744211	0.372105	+	0.928190i	$0.378636\pi$
$132$	0	0
$133$	−10.8000	−0.936476
$134$	0	0
$135$	4.51270	0.388391
$136$	0	0
$137$	−0.810341	−0.0692321	−0.0346161	−	0.999401i	$-0.511021\pi$
$137$	−0.810341	−0.0692321	−0.0346161	+	0.999401i	$0.511021\pi$
$138$	0	0
$139$	2.11616	0.179490	0.0897451	−	0.995965i	$-0.471395\pi$
$139$	2.11616	0.179490	0.0897451	+	0.995965i	$0.471395\pi$
$140$	0	0
$141$	23.3592	1.96720
$142$	0	0
$143$	31.9235	2.66958
$144$	0	0
$145$	5.94397	0.493620
$146$	0	0
$147$	8.90237	0.734255
$148$	0	0
$149$	12.7700	1.04616	0.523081	−	0.852283i	$-0.324783\pi$
$149$	12.7700	1.04616	0.523081	+	0.852283i	$0.324783\pi$
$150$	0	0
$151$	−12.1692	−0.990312	−0.495156	−	0.868804i	$-0.664889\pi$
$151$	−12.1692	−0.990312	−0.495156	+	0.868804i	$0.664889\pi$
$152$	0	0
$153$	15.2505	1.23293
$154$	0	0
$155$	10.8900	0.874703
$156$	0	0
$157$	−10.6444	−0.849514	−0.424757	−	0.905307i	$-0.639641\pi$
$157$	−10.6444	−0.849514	−0.424757	+	0.905307i	$0.639641\pi$
$158$	0	0
$159$	6.21387	0.492792
$160$	0	0
$161$	8.87313	0.699301
$162$	0	0
$163$	−11.6842	−0.915178	−0.457589	−	0.889164i	$-0.651287\pi$
$163$	−11.6842	−0.915178	−0.457589	+	0.889164i	$0.651287\pi$
$164$	0	0
$165$	−32.5744	−2.53591
$166$	0	0
$167$	−21.9602	−1.69933	−0.849666	−	0.527321i	$-0.823197\pi$
$167$	−21.9602	−1.69933	−0.849666	+	0.527321i	$0.823197\pi$
$168$	0	0
$169$	15.0775	1.15980
$170$	0	0
$171$	−13.3189	−1.01852
$172$	0	0
$173$	−0.674020	−0.0512448	−0.0256224	−	0.999672i	$-0.508157\pi$
$173$	−0.674020	−0.0512448	−0.0256224	+	0.999672i	$0.508157\pi$
$174$	0	0
$175$	1.15819	0.0875507
$176$	0	0
$177$	31.8147	2.39134
$178$	0	0
$179$	−4.27939	−0.319857	−0.159928	−	0.987129i	$-0.551126\pi$
$179$	−4.27939	−0.319857	−0.159928	+	0.987129i	$0.551126\pi$
$180$	0	0
$181$	8.38251	0.623067	0.311534	−	0.950235i	$-0.399157\pi$
$181$	8.38251	0.623067	0.311534	+	0.950235i	$0.399157\pi$
$182$	0	0
$183$	20.0160	1.47963
$184$	0	0
$185$	21.1617	1.55584
$186$	0	0
$187$	42.4308	3.10285
$188$	0	0
$189$	3.33065	0.242269
$190$	0	0
$191$	7.72863	0.559224	0.279612	−	0.960113i	$-0.409794\pi$
$191$	7.72863	0.559224	0.279612	+	0.960113i	$0.409794\pi$
$192$	0	0
$193$	−10.7748	−0.775591	−0.387795	−	0.921746i	$-0.626763\pi$
$193$	−10.7748	−0.775591	−0.387795	+	0.921746i	$0.626763\pi$
$194$	0	0
$195$	−28.6499	−2.05166
$196$	0	0
$197$	−4.40483	−0.313831	−0.156915	−	0.987612i	$-0.550155\pi$
$197$	−4.40483	−0.313831	−0.156915	+	0.987612i	$0.550155\pi$
$198$	0	0
$199$	−20.6540	−1.46412	−0.732061	−	0.681239i	$-0.761442\pi$
$199$	−20.6540	−1.46412	−0.732061	+	0.681239i	$0.761442\pi$
$200$	0	0
$201$	22.2475	1.56921
$202$	0	0
$203$	4.38702	0.307908
$204$	0	0
$205$	25.2685	1.76483
$206$	0	0
$207$	10.9427	0.760569
$208$	0	0
$209$	−37.0568	−2.56328
$210$	0	0
$211$	4.39715	0.302712	0.151356	−	0.988479i	$-0.451636\pi$
$211$	4.39715	0.302712	0.151356	+	0.988479i	$0.451636\pi$
$212$	0	0
$213$	19.0753	1.30702
$214$	0	0
$215$	26.6305	1.81619
$216$	0	0
$217$	8.03746	0.545618
$218$	0	0
$219$	−17.8683	−1.20742
$220$	0	0
$221$	37.3189	2.51034
$222$	0	0
$223$	3.55237	0.237884	0.118942	−	0.992901i	$-0.462050\pi$
$223$	3.55237	0.237884	0.118942	+	0.992901i	$0.462050\pi$
$224$	0	0
$225$	1.42832	0.0952213
$226$	0	0
$227$	6.49131	0.430843	0.215422	−	0.976521i	$-0.430887\pi$
$227$	6.49131	0.430843	0.215422	+	0.976521i	$0.430887\pi$
$228$	0	0
$229$	8.19048	0.541242	0.270621	−	0.962686i	$-0.412771\pi$
$229$	8.19048	0.541242	0.270621	+	0.962686i	$0.412771\pi$
$230$	0	0
$231$	−24.0419	−1.58184
$232$	0	0
$233$	16.8456	1.10359	0.551796	−	0.833979i	$-0.313943\pi$
$233$	16.8456	1.10359	0.551796	+	0.833979i	$0.313943\pi$
$234$	0	0
$235$	−24.4513	−1.59503
$236$	0	0
$237$	6.68172	0.434024
$238$	0	0
$239$	−1.46758	−0.0949300	−0.0474650	−	0.998873i	$-0.515114\pi$
$239$	−1.46758	−0.0949300	−0.0474650	+	0.998873i	$0.515114\pi$
$240$	0	0
$241$	−20.5771	−1.32549	−0.662744	−	0.748846i	$-0.730608\pi$
$241$	−20.5771	−1.32549	−0.662744	+	0.748846i	$0.730608\pi$
$242$	0	0
$243$	18.8715	1.21061
$244$	0	0
$245$	−9.31855	−0.595340
$246$	0	0
$247$	−32.5923	−2.07380
$248$	0	0
$249$	21.9312	1.38984
$250$	0	0
$251$	19.4458	1.22740	0.613702	−	0.789537i	$-0.289679\pi$
$251$	19.4458	1.22740	0.613702	+	0.789537i	$0.289679\pi$
$252$	0	0
$253$	30.4455	1.91409
$254$	0	0
$255$	−38.0798	−2.38465
$256$	0	0
$257$	1.66313	0.103743	0.0518716	−	0.998654i	$-0.483481\pi$
$257$	1.66313	0.103743	0.0518716	+	0.998654i	$0.483481\pi$
$258$	0	0
$259$	15.6187	0.970496
$260$	0	0
$261$	5.41023	0.334885
$262$	0	0
$263$	−10.6220	−0.654978	−0.327489	−	0.944855i	$-0.606202\pi$
$263$	−10.6220	−0.654978	−0.327489	+	0.944855i	$0.606202\pi$
$264$	0	0
$265$	−6.50436	−0.399560
$266$	0	0
$267$	42.4121	2.59558
$268$	0	0
$269$	2.70089	0.164676	0.0823382	−	0.996604i	$-0.473761\pi$
$269$	2.70089	0.164676	0.0823382	+	0.996604i	$0.473761\pi$
$270$	0	0
$271$	−24.5184	−1.48939	−0.744693	−	0.667407i	$-0.767404\pi$
$271$	−24.5184	−1.48939	−0.744693	+	0.667407i	$0.767404\pi$
$272$	0	0
$273$	−21.1454	−1.27978
$274$	0	0
$275$	3.97397	0.239639
$276$	0	0
$277$	17.0212	1.02271	0.511353	−	0.859371i	$-0.329145\pi$
$277$	17.0212	1.02271	0.511353	+	0.859371i	$0.329145\pi$
$278$	0	0
$279$	9.91210	0.593422
$280$	0	0
$281$	−1.93716	−0.115561	−0.0577807	−	0.998329i	$-0.518402\pi$
$281$	−1.93716	−0.115561	−0.0577807	+	0.998329i	$0.518402\pi$
$282$	0	0
$283$	5.57886	0.331629	0.165814	−	0.986157i	$-0.446975\pi$
$283$	5.57886	0.331629	0.165814	+	0.986157i	$0.446975\pi$
$284$	0	0
$285$	33.2569	1.96997
$286$	0	0
$287$	18.6497	1.10086
$288$	0	0
$289$	32.6021	1.91777
$290$	0	0
$291$	−14.3038	−0.838503
$292$	0	0
$293$	3.49884	0.204404	0.102202	−	0.994764i	$-0.467411\pi$
$293$	3.49884	0.204404	0.102202	+	0.994764i	$0.467411\pi$
$294$	0	0
$295$	−33.3020	−1.93892
$296$	0	0
$297$	11.4281	0.663127
$298$	0	0
$299$	26.7775	1.54858
$300$	0	0
$301$	19.6549	1.13289
$302$	0	0
$303$	3.07287	0.176531
$304$	0	0
$305$	−20.9517	−1.19969
$306$	0	0
$307$	−28.0509	−1.60095	−0.800476	−	0.599365i	$-0.795420\pi$
$307$	−28.0509	−1.60095	−0.800476	+	0.599365i	$0.795420\pi$
$308$	0	0
$309$	−27.1998	−1.54734
$310$	0	0
$311$	6.26062	0.355007	0.177503	−	0.984120i	$-0.443198\pi$
$311$	6.26062	0.355007	0.177503	+	0.984120i	$0.443198\pi$
$312$	0	0
$313$	−5.27889	−0.298381	−0.149190	−	0.988808i	$-0.547667\pi$
$313$	−5.27889	−0.298381	−0.149190	+	0.988808i	$0.547667\pi$
$314$	0	0
$315$	9.04508	0.509633
$316$	0	0
$317$	5.64794	0.317220	0.158610	−	0.987341i	$-0.449299\pi$
$317$	5.64794	0.317220	0.158610	+	0.987341i	$0.449299\pi$
$318$	0	0
$319$	15.0527	0.842790
$320$	0	0
$321$	17.8037	0.993704
$322$	0	0
$323$	−43.3198	−2.41038
$324$	0	0
$325$	3.49520	0.193879
$326$	0	0
$327$	−1.65758	−0.0916642
$328$	0	0
$329$	−18.0465	−0.994938
$330$	0	0
$331$	22.5370	1.23874	0.619372	−	0.785098i	$-0.287387\pi$
$331$	22.5370	1.23874	0.619372	+	0.785098i	$0.287387\pi$
$332$	0	0
$333$	19.2615	1.05552
$334$	0	0
$335$	−23.2875	−1.27233
$336$	0	0
$337$	−6.91252	−0.376549	−0.188275	−	0.982116i	$-0.560289\pi$
$337$	−6.91252	−0.376549	−0.188275	+	0.982116i	$0.560289\pi$
$338$	0	0
$339$	30.4538	1.65402
$340$	0	0
$341$	27.5781	1.49344
$342$	0	0
$343$	−19.1686	−1.03501
$344$	0	0
$345$	−27.3234	−1.47104
$346$	0	0
$347$	3.34912	0.179790	0.0898952	−	0.995951i	$-0.471347\pi$
$347$	3.34912	0.179790	0.0898952	+	0.995951i	$0.471347\pi$
$348$	0	0
$349$	−29.1823	−1.56209	−0.781047	−	0.624472i	$-0.785314\pi$
$349$	−29.1823	−1.56209	−0.781047	+	0.624472i	$0.785314\pi$
$350$	0	0
$351$	10.0513	0.536498
$352$	0	0
$353$	−18.1208	−0.964475	−0.482237	−	0.876041i	$-0.660176\pi$
$353$	−18.1208	−0.964475	−0.482237	+	0.876041i	$0.660176\pi$
$354$	0	0
$355$	−19.9671	−1.05974
$356$	0	0
$357$	−28.1052	−1.48749
$358$	0	0
$359$	7.07034	0.373158	0.186579	−	0.982440i	$-0.440260\pi$
$359$	7.07034	0.373158	0.186579	+	0.982440i	$0.440260\pi$
$360$	0	0
$361$	18.8332	0.991223
$362$	0	0
$363$	−57.4922	−3.01756
$364$	0	0
$365$	18.7036	0.978990
$366$	0	0
$367$	−8.27997	−0.432211	−0.216105	−	0.976370i	$-0.569335\pi$
$367$	−8.27997	−0.432211	−0.216105	+	0.976370i	$0.569335\pi$
$368$	0	0
$369$	22.9995	1.19731
$370$	0	0
$371$	−4.80062	−0.249236
$372$	0	0
$373$	−31.4256	−1.62715	−0.813577	−	0.581458i	$-0.802483\pi$
$373$	−31.4256	−1.62715	−0.813577	+	0.581458i	$0.802483\pi$
$374$	0	0
$375$	23.4678	1.21187
$376$	0	0
$377$	13.2392	0.681854
$378$	0	0
$379$	24.5274	1.25989	0.629944	−	0.776641i	$-0.283078\pi$
$379$	24.5274	1.25989	0.629944	+	0.776641i	$0.283078\pi$
$380$	0	0
$381$	14.6491	0.750494
$382$	0	0
$383$	24.3767	1.24559	0.622796	−	0.782384i	$-0.285997\pi$
$383$	24.3767	1.24559	0.622796	+	0.782384i	$0.285997\pi$
$384$	0	0
$385$	25.1658	1.28257
$386$	0	0
$387$	24.2392	1.23215
$388$	0	0
$389$	−23.2403	−1.17833	−0.589165	−	0.808013i	$-0.700543\pi$
$389$	−23.2403	−1.17833	−0.589165	+	0.808013i	$0.700543\pi$
$390$	0	0
$391$	35.5910	1.79992
$392$	0	0
$393$	−19.3590	−0.976531
$394$	0	0
$395$	−6.99408	−0.351911
$396$	0	0
$397$	−20.0240	−1.00497	−0.502487	−	0.864585i	$-0.667582\pi$
$397$	−20.0240	−1.00497	−0.502487	+	0.864585i	$0.667582\pi$
$398$	0	0
$399$	24.5456	1.22882
$400$	0	0
$401$	−25.8131	−1.28904	−0.644522	−	0.764586i	$-0.722944\pi$
$401$	−25.8131	−1.28904	−0.644522	+	0.764586i	$0.722944\pi$
$402$	0	0
$403$	24.2556	1.20826
$404$	0	0
$405$	−25.7105	−1.27756
$406$	0	0
$407$	53.5907	2.65639
$408$	0	0
$409$	22.1162	1.09358	0.546788	−	0.837271i	$-0.315850\pi$
$409$	22.1162	1.09358	0.546788	+	0.837271i	$0.315850\pi$
$410$	0	0
$411$	1.84170	0.0908443
$412$	0	0
$413$	−24.5789	−1.20945
$414$	0	0
$415$	−22.9565	−1.12689
$416$	0	0
$417$	−4.80949	−0.235522
$418$	0	0
$419$	23.6545	1.15560	0.577798	−	0.816180i	$-0.303912\pi$
$419$	23.6545	1.15560	0.577798	+	0.816180i	$0.303912\pi$
$420$	0	0
$421$	26.5694	1.29491	0.647456	−	0.762103i	$-0.275833\pi$
$421$	26.5694	1.29491	0.647456	+	0.762103i	$0.275833\pi$
$422$	0	0
$423$	−22.2557	−1.08211
$424$	0	0
$425$	4.64561	0.225345
$426$	0	0
$427$	−15.4637	−0.748339
$428$	0	0
$429$	−72.5540	−3.50294
$430$	0	0
$431$	−6.76258	−0.325742	−0.162871	−	0.986647i	$-0.552075\pi$
$431$	−6.76258	−0.325742	−0.162871	+	0.986647i	$0.552075\pi$
$432$	0	0
$433$	8.03624	0.386197	0.193098	−	0.981179i	$-0.438146\pi$
$433$	8.03624	0.386197	0.193098	+	0.981179i	$0.438146\pi$
$434$	0	0
$435$	−13.5091	−0.647714
$436$	0	0
$437$	−31.0833	−1.48692
$438$	0	0
$439$	−31.0164	−1.48033	−0.740167	−	0.672424i	$-0.765253\pi$
$439$	−31.0164	−1.48033	−0.740167	+	0.672424i	$0.765253\pi$
$440$	0	0
$441$	−8.48179	−0.403895
$442$	0	0
$443$	39.5947	1.88120	0.940599	−	0.339518i	$-0.110264\pi$
$443$	39.5947	1.88120	0.940599	+	0.339518i	$0.110264\pi$
$444$	0	0
$445$	−44.3948	−2.10452
$446$	0	0
$447$	−29.0230	−1.37274
$448$	0	0
$449$	19.4746	0.919061	0.459531	−	0.888162i	$-0.348018\pi$
$449$	19.4746	0.919061	0.459531	+	0.888162i	$0.348018\pi$
$450$	0	0
$451$	63.9909	3.01321
$452$	0	0
$453$	27.6574	1.29946
$454$	0	0
$455$	22.1339	1.03765
$456$	0	0
$457$	−32.8643	−1.53733	−0.768663	−	0.639654i	$-0.779078\pi$
$457$	−32.8643	−1.53733	−0.768663	+	0.639654i	$0.779078\pi$
$458$	0	0
$459$	13.3596	0.623572
$460$	0	0
$461$	4.85196	0.225978	0.112989	−	0.993596i	$-0.463957\pi$
$461$	4.85196	0.225978	0.112989	+	0.993596i	$0.463957\pi$
$462$	0	0
$463$	28.2559	1.31316	0.656581	−	0.754255i	$-0.272002\pi$
$463$	28.2559	1.31316	0.656581	+	0.754255i	$0.272002\pi$
$464$	0	0
$465$	−24.7501	−1.14776
$466$	0	0
$467$	13.0184	0.602418	0.301209	−	0.953558i	$-0.402610\pi$
$467$	13.0184	0.602418	0.301209	+	0.953558i	$0.402610\pi$
$468$	0	0
$469$	−17.1876	−0.793650
$470$	0	0
$471$	24.1920	1.11471
$472$	0	0
$473$	67.4400	3.10089
$474$	0	0
$475$	−4.05723	−0.186158
$476$	0	0
$477$	−5.92030	−0.271072
$478$	0	0
$479$	1.20261	0.0549487	0.0274743	−	0.999623i	$-0.491254\pi$
$479$	1.20261	0.0549487	0.0274743	+	0.999623i	$0.491254\pi$
$480$	0	0
$481$	47.1342	2.14914
$482$	0	0
$483$	−20.1664	−0.917601
$484$	0	0
$485$	14.9725	0.679865
$486$	0	0
$487$	−37.7297	−1.70970	−0.854848	−	0.518878i	$-0.826350\pi$
$487$	−37.7297	−1.70970	−0.854848	+	0.518878i	$0.826350\pi$
$488$	0	0
$489$	26.5552	1.20087
$490$	0	0
$491$	15.4692	0.698117	0.349059	−	0.937101i	$-0.386501\pi$
$491$	15.4692	0.698117	0.349059	+	0.937101i	$0.386501\pi$
$492$	0	0
$493$	17.5968	0.792519
$494$	0	0
$495$	31.0355	1.39494
$496$	0	0
$497$	−14.7370	−0.661043
$498$	0	0
$499$	17.9852	0.805128	0.402564	−	0.915392i	$-0.368119\pi$
$499$	17.9852	0.805128	0.402564	+	0.915392i	$0.368119\pi$
$500$	0	0
$501$	49.9100	2.22981
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−3.21652	−0.143133
$506$	0	0
$507$	−34.2672	−1.52186
$508$	0	0
$509$	8.04342	0.356518	0.178259	−	0.983984i	$-0.442953\pi$
$509$	8.04342	0.356518	0.178259	+	0.983984i	$0.442953\pi$
$510$	0	0
$511$	13.8044	0.610670
$512$	0	0
$513$	−11.6676	−0.515135
$514$	0	0
$515$	28.4714	1.25460
$516$	0	0
$517$	−61.9212	−2.72329
$518$	0	0
$519$	1.53188	0.0672419
$520$	0	0
$521$	18.1571	0.795476	0.397738	−	0.917499i	$-0.369795\pi$
$521$	18.1571	0.795476	0.397738	+	0.917499i	$0.369795\pi$
$522$	0	0
$523$	−19.0883	−0.834674	−0.417337	−	0.908752i	$-0.637037\pi$
$523$	−19.0883	−0.834674	−0.417337	+	0.908752i	$0.637037\pi$
$524$	0	0
$525$	−2.63226	−0.114881
$526$	0	0
$527$	32.2391	1.40436
$528$	0	0
$529$	2.53769	0.110334
$530$	0	0
$531$	−30.3117	−1.31541
$532$	0	0
$533$	56.2814	2.43782
$534$	0	0
$535$	−18.6360	−0.805704
$536$	0	0
$537$	9.72597	0.419707
$538$	0	0
$539$	−23.5986	−1.01646
$540$	0	0
$541$	−41.9750	−1.80465	−0.902323	−	0.431060i	$-0.858140\pi$
$541$	−41.9750	−1.80465	−0.902323	+	0.431060i	$0.858140\pi$
$542$	0	0
$543$	−19.0513	−0.817571
$544$	0	0
$545$	1.73507	0.0743221
$546$	0	0
$547$	−11.5492	−0.493808	−0.246904	−	0.969040i	$-0.579413\pi$
$547$	−11.5492	−0.493808	−0.246904	+	0.969040i	$0.579413\pi$
$548$	0	0
$549$	−19.0704	−0.813904
$550$	0	0
$551$	−15.3681	−0.654703
$552$	0	0
$553$	−5.16206	−0.219513
$554$	0	0
$555$	−48.0952	−2.04153
$556$	0	0
$557$	−2.01946	−0.0855671	−0.0427836	−	0.999084i	$-0.513623\pi$
$557$	−2.01946	−0.0855671	−0.0427836	+	0.999084i	$0.513623\pi$
$558$	0	0
$559$	59.3150	2.50876
$560$	0	0
$561$	−96.4345	−4.07147
$562$	0	0
$563$	−6.20952	−0.261700	−0.130850	−	0.991402i	$-0.541771\pi$
$563$	−6.20952	−0.261700	−0.130850	+	0.991402i	$0.541771\pi$
$564$	0	0
$565$	−31.8775	−1.34110
$566$	0	0
$567$	−18.9759	−0.796913
$568$	0	0
$569$	11.0151	0.461776	0.230888	−	0.972980i	$-0.425837\pi$
$569$	11.0151	0.461776	0.230888	+	0.972980i	$0.425837\pi$
$570$	0	0
$571$	−19.9176	−0.833526	−0.416763	−	0.909015i	$-0.636836\pi$
$571$	−19.9176	−0.833526	−0.416763	+	0.909015i	$0.636836\pi$
$572$	0	0
$573$	−17.5652	−0.733797
$574$	0	0
$575$	3.33337	0.139011
$576$	0	0
$577$	−10.6638	−0.443939	−0.221969	−	0.975054i	$-0.571248\pi$
$577$	−10.6638	−0.443939	−0.221969	+	0.975054i	$0.571248\pi$
$578$	0	0
$579$	24.4885	1.01771
$580$	0	0
$581$	−16.9433	−0.702927
$582$	0	0
$583$	−16.4719	−0.682195
$584$	0	0
$585$	27.2964	1.12857
$586$	0	0
$587$	18.1164	0.747744	0.373872	−	0.927480i	$-0.378030\pi$
$587$	18.1164	0.747744	0.373872	+	0.927480i	$0.378030\pi$
$588$	0	0
$589$	−28.1559	−1.16014
$590$	0	0
$591$	10.0111	0.411800
$592$	0	0
$593$	−8.73140	−0.358556	−0.179278	−	0.983798i	$-0.557376\pi$
$593$	−8.73140	−0.358556	−0.179278	+	0.983798i	$0.557376\pi$
$594$	0	0
$595$	29.4191	1.20607
$596$	0	0
$597$	46.9413	1.92118
$598$	0	0
$599$	−32.2888	−1.31929	−0.659643	−	0.751579i	$-0.729293\pi$
$599$	−32.2888	−1.31929	−0.659643	+	0.751579i	$0.729293\pi$
$600$	0	0
$601$	−5.91504	−0.241279	−0.120640	−	0.992696i	$-0.538495\pi$
$601$	−5.91504	−0.241279	−0.120640	+	0.992696i	$0.538495\pi$
$602$	0	0
$603$	−21.1964	−0.863184
$604$	0	0
$605$	60.1800	2.44666
$606$	0	0
$607$	−35.5739	−1.44390	−0.721951	−	0.691944i	$-0.756754\pi$
$607$	−35.5739	−1.44390	−0.721951	+	0.691944i	$0.756754\pi$
$608$	0	0
$609$	−9.97057	−0.404028
$610$	0	0
$611$	−54.4612	−2.20326
$612$	0	0
$613$	13.1329	0.530434	0.265217	−	0.964189i	$-0.414556\pi$
$613$	13.1329	0.530434	0.265217	+	0.964189i	$0.414556\pi$
$614$	0	0
$615$	−57.4289	−2.31576
$616$	0	0
$617$	−12.4250	−0.500213	−0.250107	−	0.968218i	$-0.580466\pi$
$617$	−12.4250	−0.500213	−0.250107	+	0.968218i	$0.580466\pi$
$618$	0	0
$619$	16.7477	0.673149	0.336574	−	0.941657i	$-0.390732\pi$
$619$	16.7477	0.673149	0.336574	+	0.941657i	$0.390732\pi$
$620$	0	0
$621$	9.58592	0.384670
$622$	0	0
$623$	−32.7661	−1.31275
$624$	0	0
$625$	−27.8630	−1.11452
$626$	0	0
$627$	84.2208	3.36345
$628$	0	0
$629$	62.6481	2.49794
$630$	0	0
$631$	−12.5128	−0.498126	−0.249063	−	0.968487i	$-0.580123\pi$
$631$	−12.5128	−0.498126	−0.249063	+	0.968487i	$0.580123\pi$
$632$	0	0
$633$	−9.99360	−0.397210
$634$	0	0
$635$	−15.3339	−0.608507
$636$	0	0
$637$	−20.7555	−0.822364
$638$	0	0
$639$	−18.1742	−0.718959
$640$	0	0
$641$	−14.6249	−0.577649	−0.288824	−	0.957382i	$-0.593264\pi$
$641$	−14.6249	−0.577649	−0.288824	+	0.957382i	$0.593264\pi$
$642$	0	0
$643$	15.6194	0.615968	0.307984	−	0.951391i	$-0.400346\pi$
$643$	15.6194	0.615968	0.307984	+	0.951391i	$0.400346\pi$
$644$	0	0
$645$	−60.5244	−2.38315
$646$	0	0
$647$	30.0056	1.17964	0.589821	−	0.807534i	$-0.299198\pi$
$647$	30.0056	1.17964	0.589821	+	0.807534i	$0.299198\pi$
$648$	0	0
$649$	−84.3352	−3.31044
$650$	0	0
$651$	−18.2671	−0.715944
$652$	0	0
$653$	4.16483	0.162982	0.0814911	−	0.996674i	$-0.474032\pi$
$653$	4.16483	0.162982	0.0814911	+	0.996674i	$0.474032\pi$
$654$	0	0
$655$	20.2640	0.791780
$656$	0	0
$657$	17.0241	0.664173
$658$	0	0
$659$	−16.4275	−0.639923	−0.319962	−	0.947430i	$-0.603670\pi$
$659$	−16.4275	−0.639923	−0.319962	+	0.947430i	$0.603670\pi$
$660$	0	0
$661$	−21.3368	−0.829904	−0.414952	−	0.909843i	$-0.636202\pi$
$661$	−21.3368	−0.829904	−0.414952	+	0.909843i	$0.636202\pi$
$662$	0	0
$663$	−84.8164	−3.29400
$664$	0	0
$665$	−25.6931	−0.996335
$666$	0	0
$667$	12.6262	0.488890
$668$	0	0
$669$	−8.07363	−0.312144
$670$	0	0
$671$	−53.0589	−2.04832
$672$	0	0
$673$	−5.26167	−0.202823	−0.101411	−	0.994845i	$-0.532336\pi$
$673$	−5.26167	−0.202823	−0.101411	+	0.994845i	$0.532336\pi$
$674$	0	0
$675$	1.25123	0.0481597
$676$	0	0
$677$	−37.7776	−1.45191	−0.725955	−	0.687742i	$-0.758602\pi$
$677$	−37.7776	−1.45191	−0.725955	+	0.687742i	$0.758602\pi$
$678$	0	0
$679$	11.0506	0.424083
$680$	0	0
$681$	−14.7531	−0.565340
$682$	0	0
$683$	−7.27905	−0.278525	−0.139263	−	0.990255i	$-0.544473\pi$
$683$	−7.27905	−0.278525	−0.139263	+	0.990255i	$0.544473\pi$
$684$	0	0
$685$	−1.92780	−0.0736574
$686$	0	0
$687$	−18.6149	−0.710202
$688$	0	0
$689$	−14.4874	−0.551925
$690$	0	0
$691$	−10.6128	−0.403730	−0.201865	−	0.979413i	$-0.564700\pi$
$691$	−10.6128	−0.403730	−0.201865	+	0.979413i	$0.564700\pi$
$692$	0	0
$693$	22.9061	0.870129
$694$	0	0
$695$	5.03433	0.190963
$696$	0	0
$697$	74.8060	2.83348
$698$	0	0
$699$	−38.2858	−1.44810
$700$	0	0
$701$	13.5094	0.510242	0.255121	−	0.966909i	$-0.417885\pi$
$701$	13.5094	0.510242	0.255121	+	0.966909i	$0.417885\pi$
$702$	0	0
$703$	−54.7135	−2.06356
$704$	0	0
$705$	55.5715	2.09295
$706$	0	0
$707$	−2.37399	−0.0892830
$708$	0	0
$709$	7.09711	0.266538	0.133269	−	0.991080i	$-0.457453\pi$
$709$	7.09711	0.266538	0.133269	+	0.991080i	$0.457453\pi$
$710$	0	0
$711$	−6.36605	−0.238746
$712$	0	0
$713$	23.1326	0.866321
$714$	0	0
$715$	75.9459	2.84021
$716$	0	0
$717$	3.33544	0.124564
$718$	0	0
$719$	−20.0618	−0.748180	−0.374090	−	0.927392i	$-0.622045\pi$
$719$	−20.0618	−0.748180	−0.374090	+	0.927392i	$0.622045\pi$
$720$	0	0
$721$	21.0137	0.782589
$722$	0	0
$723$	46.7666	1.73927
$724$	0	0
$725$	1.64807	0.0612078
$726$	0	0
$727$	13.3637	0.495634	0.247817	−	0.968807i	$-0.420287\pi$
$727$	13.3637	0.495634	0.247817	+	0.968807i	$0.420287\pi$
$728$	0	0
$729$	−10.4683	−0.387716
$730$	0	0
$731$	78.8381	2.91593
$732$	0	0
$733$	14.3289	0.529251	0.264626	−	0.964351i	$-0.414752\pi$
$733$	14.3289	0.529251	0.264626	+	0.964351i	$0.414752\pi$
$734$	0	0
$735$	21.1787	0.781188
$736$	0	0
$737$	−58.9741	−2.17234
$738$	0	0
$739$	13.7342	0.505222	0.252611	−	0.967568i	$-0.418711\pi$
$739$	13.7342	0.505222	0.252611	+	0.967568i	$0.418711\pi$
$740$	0	0
$741$	74.0741	2.72118
$742$	0	0
$743$	−6.51304	−0.238940	−0.119470	−	0.992838i	$-0.538120\pi$
$743$	−6.51304	−0.238940	−0.119470	+	0.992838i	$0.538120\pi$
$744$	0	0
$745$	30.3798	1.11303
$746$	0	0
$747$	−20.8951	−0.764513
$748$	0	0
$749$	−13.7545	−0.502578
$750$	0	0
$751$	10.9515	0.399625	0.199813	−	0.979834i	$-0.435967\pi$
$751$	10.9515	0.399625	0.199813	+	0.979834i	$0.435967\pi$
$752$	0	0
$753$	−44.1953	−1.61056
$754$	0	0
$755$	−28.9504	−1.05361
$756$	0	0
$757$	49.1895	1.78782	0.893912	−	0.448243i	$-0.147950\pi$
$757$	49.1895	1.78782	0.893912	+	0.448243i	$0.147950\pi$
$758$	0	0
$759$	−69.1948	−2.51161
$760$	0	0
$761$	13.3168	0.482734	0.241367	−	0.970434i	$-0.422404\pi$
$761$	13.3168	0.482734	0.241367	+	0.970434i	$0.422404\pi$
$762$	0	0
$763$	1.28059	0.0463603
$764$	0	0
$765$	36.2808	1.31173
$766$	0	0
$767$	−74.1747	−2.67829
$768$	0	0
$769$	30.4056	1.09645	0.548226	−	0.836330i	$-0.315303\pi$
$769$	30.4056	1.09645	0.548226	+	0.836330i	$0.315303\pi$
$770$	0	0
$771$	−3.77987	−0.136129
$772$	0	0
$773$	−47.3326	−1.70244	−0.851219	−	0.524811i	$-0.824136\pi$
$773$	−47.3326	−1.70244	−0.851219	+	0.524811i	$0.824136\pi$
$774$	0	0
$775$	3.01943	0.108461
$776$	0	0
$777$	−35.4972	−1.27346
$778$	0	0
$779$	−65.3316	−2.34075
$780$	0	0
$781$	−50.5654	−1.80937
$782$	0	0
$783$	4.73943	0.169373
$784$	0	0
$785$	−25.3229	−0.903814
$786$	0	0
$787$	49.4887	1.76408	0.882041	−	0.471173i	$-0.156169\pi$
$787$	49.4887	1.76408	0.882041	+	0.471173i	$0.156169\pi$
$788$	0	0
$789$	24.1410	0.859442
$790$	0	0
$791$	−23.5276	−0.836544
$792$	0	0
$793$	−46.6665	−1.65718
$794$	0	0
$795$	14.7828	0.524291
$796$	0	0
$797$	−24.9387	−0.883374	−0.441687	−	0.897169i	$-0.645620\pi$
$797$	−24.9387	−0.883374	−0.441687	+	0.897169i	$0.645620\pi$
$798$	0	0
$799$	−72.3866	−2.56085
$800$	0	0
$801$	−40.4084	−1.42776
$802$	0	0
$803$	47.3656	1.67150
$804$	0	0
$805$	21.1091	0.743999
$806$	0	0
$807$	−6.13844	−0.216083
$808$	0	0
$809$	−49.3622	−1.73548	−0.867741	−	0.497016i	$-0.834429\pi$
$809$	−49.3622	−1.73548	−0.867741	+	0.497016i	$0.834429\pi$
$810$	0	0
$811$	40.6724	1.42820	0.714101	−	0.700042i	$-0.246836\pi$
$811$	40.6724	1.42820	0.714101	+	0.700042i	$0.246836\pi$
$812$	0	0
$813$	55.7241	1.95433
$814$	0	0
$815$	−27.7967	−0.973675
$816$	0	0
$817$	−68.8530	−2.40886
$818$	0	0
$819$	20.1464	0.703973
$820$	0	0
$821$	−16.4017	−0.572422	−0.286211	−	0.958167i	$-0.592396\pi$
$821$	−16.4017	−0.572422	−0.286211	+	0.958167i	$0.592396\pi$
$822$	0	0
$823$	−39.9047	−1.39099	−0.695495	−	0.718531i	$-0.744815\pi$
$823$	−39.9047	−1.39099	−0.695495	+	0.718531i	$0.744815\pi$
$824$	0	0
$825$	−9.03182	−0.314447
$826$	0	0
$827$	−14.8884	−0.517720	−0.258860	−	0.965915i	$-0.583347\pi$
$827$	−14.8884	−0.517720	−0.258860	+	0.965915i	$0.583347\pi$
$828$	0	0
$829$	21.8735	0.759699	0.379850	−	0.925048i	$-0.375976\pi$
$829$	21.8735	0.759699	0.379850	+	0.925048i	$0.375976\pi$
$830$	0	0
$831$	−38.6849	−1.34196
$832$	0	0
$833$	−27.5870	−0.955833
$834$	0	0
$835$	−52.2433	−1.80795
$836$	0	0
$837$	8.68312	0.300133
$838$	0	0
$839$	−7.04644	−0.243270	−0.121635	−	0.992575i	$-0.538814\pi$
$839$	−7.04644	−0.243270	−0.121635	+	0.992575i	$0.538814\pi$
$840$	0	0
$841$	−22.7574	−0.784738
$842$	0	0
$843$	4.40267	0.151636
$844$	0	0
$845$	35.8692	1.23394
$846$	0	0
$847$	44.4165	1.52617
$848$	0	0
$849$	−12.6793	−0.435153
$850$	0	0
$851$	44.9519	1.54093
$852$	0	0
$853$	−6.19194	−0.212008	−0.106004	−	0.994366i	$-0.533806\pi$
$853$	−6.19194	−0.212008	−0.106004	+	0.994366i	$0.533806\pi$
$854$	0	0
$855$	−31.6857	−1.08363
$856$	0	0
$857$	−29.8629	−1.02010	−0.510049	−	0.860146i	$-0.670373\pi$
$857$	−29.8629	−1.02010	−0.510049	+	0.860146i	$0.670373\pi$
$858$	0	0
$859$	−9.19222	−0.313635	−0.156817	−	0.987628i	$-0.550123\pi$
$859$	−9.19222	−0.313635	−0.156817	+	0.987628i	$0.550123\pi$
$860$	0	0
$861$	−42.3861	−1.44451
$862$	0	0
$863$	15.4314	0.525292	0.262646	−	0.964892i	$-0.415405\pi$
$863$	15.4314	0.525292	0.262646	+	0.964892i	$0.415405\pi$
$864$	0	0
$865$	−1.60349	−0.0545203
$866$	0	0
$867$	−74.0963	−2.51644
$868$	0	0
$869$	−17.7121	−0.600840
$870$	0	0
$871$	−51.8690	−1.75752
$872$	0	0
$873$	13.6280	0.461239
$874$	0	0
$875$	−18.1304	−0.612920
$876$	0	0
$877$	25.0182	0.844805	0.422402	−	0.906408i	$-0.361187\pi$
$877$	25.0182	0.844805	0.422402	+	0.906408i	$0.361187\pi$
$878$	0	0
$879$	−7.95197	−0.268213
$880$	0	0
$881$	−10.9920	−0.370330	−0.185165	−	0.982707i	$-0.559282\pi$
$881$	−10.9920	−0.370330	−0.185165	+	0.982707i	$0.559282\pi$
$882$	0	0
$883$	49.8934	1.67905	0.839523	−	0.543324i	$-0.182834\pi$
$883$	49.8934	1.67905	0.839523	+	0.543324i	$0.182834\pi$
$884$	0	0
$885$	75.6870	2.54419
$886$	0	0
$887$	24.5933	0.825761	0.412881	−	0.910785i	$-0.364523\pi$
$887$	24.5933	0.825761	0.412881	+	0.910785i	$0.364523\pi$
$888$	0	0
$889$	−11.3174	−0.379572
$890$	0	0
$891$	−65.1100	−2.18127
$892$	0	0
$893$	63.2186	2.11553
$894$	0	0
$895$	−10.1807	−0.340302
$896$	0	0
$897$	−60.8584	−2.03200
$898$	0	0
$899$	11.4371	0.381449
$900$	0	0
$901$	−19.2558	−0.641503
$902$	0	0
$903$	−44.6707	−1.48655
$904$	0	0
$905$	19.9420	0.662893
$906$	0	0
$907$	−28.1785	−0.935652	−0.467826	−	0.883821i	$-0.654963\pi$
$907$	−28.1785	−0.935652	−0.467826	+	0.883821i	$0.654963\pi$
$908$	0	0
$909$	−2.92769	−0.0971054
$910$	0	0
$911$	16.1510	0.535107	0.267553	−	0.963543i	$-0.413785\pi$
$911$	16.1510	0.535107	0.267553	+	0.963543i	$0.413785\pi$
$912$	0	0
$913$	−58.1358	−1.92402
$914$	0	0
$915$	47.6180	1.57420
$916$	0	0
$917$	14.9561	0.493893
$918$	0	0
$919$	3.35825	0.110778	0.0553892	−	0.998465i	$-0.482360\pi$
$919$	3.35825	0.110778	0.0553892	+	0.998465i	$0.482360\pi$
$920$	0	0
$921$	63.7527	2.10072
$922$	0	0
$923$	−44.4734	−1.46386
$924$	0	0
$925$	5.86746	0.192921
$926$	0	0
$927$	25.9148	0.851155
$928$	0	0
$929$	16.5377	0.542584	0.271292	−	0.962497i	$-0.412549\pi$
$929$	16.5377	0.542584	0.271292	+	0.962497i	$0.412549\pi$
$930$	0	0
$931$	24.0930	0.789617
$932$	0	0
$933$	−14.2288	−0.465830
$934$	0	0
$935$	100.943	3.30118
$936$	0	0
$937$	−35.4939	−1.15954	−0.579768	−	0.814782i	$-0.696857\pi$
$937$	−35.4939	−1.15954	−0.579768	+	0.814782i	$0.696857\pi$
$938$	0	0
$939$	11.9976	0.391526
$940$	0	0
$941$	24.4477	0.796971	0.398486	−	0.917175i	$-0.369536\pi$
$941$	24.4477	0.796971	0.398486	+	0.917175i	$0.369536\pi$
$942$	0	0
$943$	53.6756	1.74792
$944$	0	0
$945$	7.92360	0.257755
$946$	0	0
$947$	−15.6688	−0.509168	−0.254584	−	0.967051i	$-0.581938\pi$
$947$	−15.6688	−0.509168	−0.254584	+	0.967051i	$0.581938\pi$
$948$	0	0
$949$	41.6591	1.35231
$950$	0	0
$951$	−12.8363	−0.416246
$952$	0	0
$953$	−48.5550	−1.57285	−0.786426	−	0.617685i	$-0.788071\pi$
$953$	−48.5550	−1.57285	−0.786426	+	0.617685i	$0.788071\pi$
$954$	0	0
$955$	18.3864	0.594969
$956$	0	0
$957$	−34.2110	−1.10588
$958$	0	0
$959$	−1.42283	−0.0459457
$960$	0	0
$961$	−10.0461	−0.324066
$962$	0	0
$963$	−16.9626	−0.546611
$964$	0	0
$965$	−25.6333	−0.825166
$966$	0	0
$967$	38.7728	1.24685	0.623425	−	0.781883i	$-0.285741\pi$
$967$	38.7728	1.24685	0.623425	+	0.781883i	$0.285741\pi$
$968$	0	0
$969$	98.4549	3.16283
$970$	0	0
$971$	−27.9588	−0.897241	−0.448620	−	0.893722i	$-0.648084\pi$
$971$	−27.9588	−0.897241	−0.448620	+	0.893722i	$0.648084\pi$
$972$	0	0
$973$	3.71564	0.119118
$974$	0	0
$975$	−7.94369	−0.254402
$976$	0	0
$977$	−27.0689	−0.866010	−0.433005	−	0.901392i	$-0.642547\pi$
$977$	−27.0689	−0.866010	−0.433005	+	0.901392i	$0.642547\pi$
$978$	0	0
$979$	−112.427	−3.59318
$980$	0	0
$981$	1.57927	0.0504221
$982$	0	0
$983$	−10.8320	−0.345487	−0.172743	−	0.984967i	$-0.555263\pi$
$983$	−10.8320	−0.345487	−0.172743	+	0.984967i	$0.555263\pi$
$984$	0	0
$985$	−10.4791	−0.333891
$986$	0	0
$987$	41.0152	1.30553
$988$	0	0
$989$	56.5688	1.79878
$990$	0	0
$991$	39.0517	1.24052	0.620259	−	0.784397i	$-0.287027\pi$
$991$	39.0517	1.24052	0.620259	+	0.784397i	$0.287027\pi$
$992$	0	0
$993$	−51.2208	−1.62544
$994$	0	0
$995$	−49.1358	−1.55771
$996$	0	0
$997$	12.8638	0.407400	0.203700	−	0.979033i	$-0.434703\pi$
$997$	12.8638	0.407400	0.203700	+	0.979033i	$0.434703\pi$
$998$	0	0
$999$	16.8733	0.533848

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.f.1.7	✓	33
4.3	odd	2		8048.2.a.y.1.27		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.f.1.7	✓	33	1.1	even	1	trivial
8048.2.a.y.1.27		33	4.3	odd	2

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:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.27275 q^{3} +2.37900 q^{5} +1.75584 q^{7} +2.16537 q^{9} +O(q^{10})\)	\(q-2.27275 q^{3} +2.37900 q^{5} +1.75584 q^{7} +2.16537 q^{9} +6.02465 q^{11} +5.29882 q^{13} -5.40685 q^{15} +7.04287 q^{17} -6.15087 q^{19} -3.99059 q^{21} +5.05348 q^{23} +0.659618 q^{25} +1.89689 q^{27} +2.49852 q^{29} +4.57755 q^{31} -13.6925 q^{33} +4.17715 q^{35} +8.89524 q^{37} -12.0429 q^{39} +10.6215 q^{41} +11.1940 q^{43} +5.15141 q^{45} -10.2780 q^{47} -3.91701 q^{49} -16.0067 q^{51} -2.73408 q^{53} +14.3326 q^{55} +13.9794 q^{57} -13.9984 q^{59} -8.80697 q^{61} +3.80206 q^{63} +12.6059 q^{65} -9.78880 q^{67} -11.4853 q^{69} -8.39309 q^{71} +7.86197 q^{73} -1.49914 q^{75} +10.5783 q^{77} -2.93993 q^{79} -10.8073 q^{81} -9.64967 q^{83} +16.7550 q^{85} -5.67851 q^{87} -18.6612 q^{89} +9.30390 q^{91} -10.4036 q^{93} -14.6329 q^{95} +6.29361 q^{97} +13.0456 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) \) 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9	\( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9} + 22 q^{11} + 25 q^{13} - 4 q^{15} + 17 q^{17} + 6 q^{19} + 18 q^{21} + 16 q^{23} + 47 q^{25} - 20 q^{27} + 47 q^{29} - 7 q^{31} - 6 q^{33} + 19 q^{35} + 75 q^{37} + 21 q^{39} + 22 q^{41} - 5 q^{43} + 33 q^{45} + 10 q^{47} + 31 q^{49} + 9 q^{51} + 64 q^{53} - 3 q^{55} + 5 q^{57} + 28 q^{59} + 49 q^{61} - 10 q^{63} + 46 q^{65} - 14 q^{67} + 30 q^{69} + 35 q^{71} + 19 q^{73} - 33 q^{75} + 32 q^{77} - 12 q^{79} + 57 q^{81} + 82 q^{85} - 5 q^{87} + 42 q^{89} - 15 q^{91} + 55 q^{93} + 33 q^{95} + 4 q^{97} + 22 q^{99}+O(q^{100}) \) 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9 + 22 * q^11 + 25 * q^13 - 4 * q^15 + 17 * q^17 + 6 * q^19 + 18 * q^21 + 16 * q^23 + 47 * q^25 - 20 * q^27 + 47 * q^29 - 7 * q^31 - 6 * q^33 + 19 * q^35 + 75 * q^37 + 21 * q^39 + 22 * q^41 - 5 * q^43 + 33 * q^45 + 10 * q^47 + 31 * q^49 + 9 * q^51 + 64 * q^53 - 3 * q^55 + 5 * q^57 + 28 * q^59 + 49 * q^61 - 10 * q^63 + 46 * q^65 - 14 * q^67 + 30 * q^69 + 35 * q^71 + 19 * q^73 - 33 * q^75 + 32 * q^77 - 12 * q^79 + 57 * q^81 + 82 * q^85 - 5 * q^87 + 42 * q^89 - 15 * q^91 + 55 * q^93 + 33 * q^95 + 4 * q^97 + 22 * q^99

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