LMFDB - Embedded newform 4024.2.a.f.1.5

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			1.5
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)

$f(q)$	$=$	$q-2.88194 q^{3} -1.00235 q^{5} -1.34540 q^{7} +5.30561 q^{9} +O(q^{10})$	$q-2.88194 q^{3} -1.00235 q^{5} -1.34540 q^{7} +5.30561 q^{9} -0.428384 q^{11} +2.65042 q^{13} +2.88873 q^{15} +8.02576 q^{17} +4.96196 q^{19} +3.87738 q^{21} -7.95187 q^{23} -3.99528 q^{25} -6.64463 q^{27} +7.89547 q^{29} -1.02881 q^{31} +1.23458 q^{33} +1.34857 q^{35} -6.45548 q^{37} -7.63835 q^{39} -0.415875 q^{41} -2.10498 q^{43} -5.31810 q^{45} -2.82498 q^{47} -5.18989 q^{49} -23.1298 q^{51} -10.7571 q^{53} +0.429393 q^{55} -14.3001 q^{57} -2.97252 q^{59} +9.70439 q^{61} -7.13818 q^{63} -2.65666 q^{65} -1.01210 q^{67} +22.9168 q^{69} +6.32535 q^{71} +4.36612 q^{73} +11.5142 q^{75} +0.576349 q^{77} +10.1132 q^{79} +3.23264 q^{81} +9.87602 q^{83} -8.04466 q^{85} -22.7543 q^{87} -6.59783 q^{89} -3.56588 q^{91} +2.96498 q^{93} -4.97365 q^{95} -2.55755 q^{97} -2.27284 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.88194	−1.66389	−0.831946	−	0.554857i	$-0.812773\pi$
$3$	−2.88194	−1.66389	−0.831946	+	0.554857i	$0.812773\pi$
$4$	0	0
$5$	−1.00235	−0.448267	−0.224133	−	0.974558i	$-0.571955\pi$
$5$	−1.00235	−0.448267	−0.224133	+	0.974558i	$0.571955\pi$
$6$	0	0
$7$	−1.34540	−0.508515	−0.254257	−	0.967137i	$-0.581831\pi$
$7$	−1.34540	−0.508515	−0.254257	+	0.967137i	$0.581831\pi$
$8$	0	0
$9$	5.30561	1.76854
$10$	0	0
$11$	−0.428384	−0.129163	−0.0645813	−	0.997912i	$-0.520571\pi$
$11$	−0.428384	−0.129163	−0.0645813	+	0.997912i	$0.520571\pi$
$12$	0	0
$13$	2.65042	0.735093	0.367547	−	0.930005i	$-0.380198\pi$
$13$	2.65042	0.735093	0.367547	+	0.930005i	$0.380198\pi$
$14$	0	0
$15$	2.88873	0.745867
$16$	0	0
$17$	8.02576	1.94653	0.973266	−	0.229682i	$-0.0737685\pi$
$17$	8.02576	1.94653	0.973266	+	0.229682i	$0.0737685\pi$
$18$	0	0
$19$	4.96196	1.13835	0.569176	−	0.822215i	$-0.307262\pi$
$19$	4.96196	1.13835	0.569176	+	0.822215i	$0.307262\pi$
$20$	0	0
$21$	3.87738	0.846113
$22$	0	0
$23$	−7.95187	−1.65808	−0.829040	−	0.559190i	$-0.811112\pi$
$23$	−7.95187	−1.65808	−0.829040	+	0.559190i	$0.811112\pi$
$24$	0	0
$25$	−3.99528	−0.799057
$26$	0	0
$27$	−6.64463	−1.27876
$28$	0	0
$29$	7.89547	1.46615	0.733076	−	0.680146i	$-0.238084\pi$
$29$	7.89547	1.46615	0.733076	+	0.680146i	$0.238084\pi$
$30$	0	0
$31$	−1.02881	−0.184780	−0.0923900	−	0.995723i	$-0.529451\pi$
$31$	−1.02881	−0.184780	−0.0923900	+	0.995723i	$0.529451\pi$
$32$	0	0
$33$	1.23458	0.214913
$34$	0	0
$35$	1.34857	0.227950
$36$	0	0
$37$	−6.45548	−1.06127	−0.530637	−	0.847599i	$-0.678047\pi$
$37$	−6.45548	−1.06127	−0.530637	+	0.847599i	$0.678047\pi$
$38$	0	0
$39$	−7.63835	−1.22312
$40$	0	0
$41$	−0.415875	−0.0649488	−0.0324744	−	0.999473i	$-0.510339\pi$
$41$	−0.415875	−0.0649488	−0.0324744	+	0.999473i	$0.510339\pi$
$42$	0	0
$43$	−2.10498	−0.321007	−0.160503	−	0.987035i	$-0.551312\pi$
$43$	−2.10498	−0.321007	−0.160503	+	0.987035i	$0.551312\pi$
$44$	0	0
$45$	−5.31810	−0.792776
$46$	0	0
$47$	−2.82498	−0.412066	−0.206033	−	0.978545i	$-0.566055\pi$
$47$	−2.82498	−0.412066	−0.206033	+	0.978545i	$0.566055\pi$
$48$	0	0
$49$	−5.18989	−0.741413
$50$	0	0
$51$	−23.1298	−3.23882
$52$	0	0
$53$	−10.7571	−1.47760	−0.738800	−	0.673925i	$-0.764607\pi$
$53$	−10.7571	−1.47760	−0.738800	+	0.673925i	$0.764607\pi$
$54$	0	0
$55$	0.429393	0.0578993
$56$	0	0
$57$	−14.3001	−1.89410
$58$	0	0
$59$	−2.97252	−0.386990	−0.193495	−	0.981101i	$-0.561982\pi$
$59$	−2.97252	−0.386990	−0.193495	+	0.981101i	$0.561982\pi$
$60$	0	0
$61$	9.70439	1.24252	0.621260	−	0.783605i	$-0.286621\pi$
$61$	9.70439	1.24252	0.621260	+	0.783605i	$0.286621\pi$
$62$	0	0
$63$	−7.13818	−0.899326
$64$	0	0
$65$	−2.65666	−0.329518
$66$	0	0
$67$	−1.01210	−0.123648	−0.0618241	−	0.998087i	$-0.519692\pi$
$67$	−1.01210	−0.123648	−0.0618241	+	0.998087i	$0.519692\pi$
$68$	0	0
$69$	22.9168	2.75886
$70$	0	0
$71$	6.32535	0.750680	0.375340	−	0.926887i	$-0.377526\pi$
$71$	6.32535	0.750680	0.375340	+	0.926887i	$0.377526\pi$
$72$	0	0
$73$	4.36612	0.511015	0.255508	−	0.966807i	$-0.417757\pi$
$73$	4.36612	0.511015	0.255508	+	0.966807i	$0.417757\pi$
$74$	0	0
$75$	11.5142	1.32954
$76$	0	0
$77$	0.576349	0.0656811
$78$	0	0
$79$	10.1132	1.13782	0.568910	−	0.822400i	$-0.307365\pi$
$79$	10.1132	1.13782	0.568910	+	0.822400i	$0.307365\pi$
$80$	0	0
$81$	3.23264	0.359183
$82$	0	0
$83$	9.87602	1.08403	0.542017	−	0.840367i	$-0.317661\pi$
$83$	9.87602	1.08403	0.542017	+	0.840367i	$0.317661\pi$
$84$	0	0
$85$	−8.04466	−0.872565
$86$	0	0
$87$	−22.7543	−2.43952
$88$	0	0
$89$	−6.59783	−0.699368	−0.349684	−	0.936868i	$-0.613711\pi$
$89$	−6.59783	−0.699368	−0.349684	+	0.936868i	$0.613711\pi$
$90$	0	0
$91$	−3.56588	−0.373806
$92$	0	0
$93$	2.96498	0.307454
$94$	0	0
$95$	−4.97365	−0.510286
$96$	0	0
$97$	−2.55755	−0.259680	−0.129840	−	0.991535i	$-0.541446\pi$
$97$	−2.55755	−0.259680	−0.129840	+	0.991535i	$0.541446\pi$
$98$	0	0
$99$	−2.27284	−0.228429
$100$	0	0
$101$	6.14129	0.611081	0.305541	−	0.952179i	$-0.401163\pi$
$101$	6.14129	0.611081	0.305541	+	0.952179i	$0.401163\pi$
$102$	0	0
$103$	6.44844	0.635384	0.317692	−	0.948194i	$-0.397092\pi$
$103$	6.44844	0.635384	0.317692	+	0.948194i	$0.397092\pi$
$104$	0	0
$105$	−3.88651	−0.379284
$106$	0	0
$107$	−6.51897	−0.630213	−0.315106	−	0.949056i	$-0.602040\pi$
$107$	−6.51897	−0.630213	−0.315106	+	0.949056i	$0.602040\pi$
$108$	0	0
$109$	13.2590	1.26998	0.634989	−	0.772522i	$-0.281005\pi$
$109$	13.2590	1.26998	0.634989	+	0.772522i	$0.281005\pi$
$110$	0	0
$111$	18.6043	1.76585
$112$	0	0
$113$	3.83592	0.360853	0.180427	−	0.983588i	$-0.442252\pi$
$113$	3.83592	0.360853	0.180427	+	0.983588i	$0.442252\pi$
$114$	0	0
$115$	7.97060	0.743262
$116$	0	0
$117$	14.0621	1.30004
$118$	0	0
$119$	−10.7979	−0.989840
$120$	0	0
$121$	−10.8165	−0.983317
$122$	0	0
$123$	1.19853	0.108068
$124$	0	0
$125$	9.01647	0.806457
$126$	0	0
$127$	−14.7890	−1.31231	−0.656155	−	0.754626i	$-0.727818\pi$
$127$	−14.7890	−1.31231	−0.656155	+	0.754626i	$0.727818\pi$
$128$	0	0
$129$	6.06644	0.534121
$130$	0	0
$131$	−15.3567	−1.34172	−0.670861	−	0.741583i	$-0.734075\pi$
$131$	−15.3567	−1.34172	−0.670861	+	0.741583i	$0.734075\pi$
$132$	0	0
$133$	−6.67584	−0.578869
$134$	0	0
$135$	6.66028	0.573226
$136$	0	0
$137$	−9.97870	−0.852538	−0.426269	−	0.904597i	$-0.640172\pi$
$137$	−9.97870	−0.852538	−0.426269	+	0.904597i	$0.640172\pi$
$138$	0	0
$139$	−3.71707	−0.315278	−0.157639	−	0.987497i	$-0.550388\pi$
$139$	−3.71707	−0.315278	−0.157639	+	0.987497i	$0.550388\pi$
$140$	0	0
$141$	8.14144	0.685633
$142$	0	0
$143$	−1.13540	−0.0949466
$144$	0	0
$145$	−7.91407	−0.657228
$146$	0	0
$147$	14.9570	1.23363
$148$	0	0
$149$	6.59431	0.540227	0.270114	−	0.962828i	$-0.412939\pi$
$149$	6.59431	0.540227	0.270114	+	0.962828i	$0.412939\pi$
$150$	0	0
$151$	4.43670	0.361053	0.180527	−	0.983570i	$-0.442220\pi$
$151$	4.43670	0.361053	0.180527	+	0.983570i	$0.442220\pi$
$152$	0	0
$153$	42.5815	3.44251
$154$	0	0
$155$	1.03123	0.0828307
$156$	0	0
$157$	8.40008	0.670399	0.335200	−	0.942147i	$-0.391196\pi$
$157$	8.40008	0.670399	0.335200	+	0.942147i	$0.391196\pi$
$158$	0	0
$159$	31.0013	2.45857
$160$	0	0
$161$	10.6985	0.843158
$162$	0	0
$163$	6.30867	0.494133	0.247067	−	0.968999i	$-0.420533\pi$
$163$	6.30867	0.494133	0.247067	+	0.968999i	$0.420533\pi$
$164$	0	0
$165$	−1.23749	−0.0963382
$166$	0	0
$167$	9.57931	0.741270	0.370635	−	0.928779i	$-0.379140\pi$
$167$	9.57931	0.741270	0.370635	+	0.928779i	$0.379140\pi$
$168$	0	0
$169$	−5.97529	−0.459638
$170$	0	0
$171$	26.3262	2.01322
$172$	0	0
$173$	15.9246	1.21072	0.605362	−	0.795950i	$-0.293028\pi$
$173$	15.9246	1.21072	0.605362	+	0.795950i	$0.293028\pi$
$174$	0	0
$175$	5.37527	0.406332
$176$	0	0
$177$	8.56665	0.643909
$178$	0	0
$179$	7.26099	0.542712	0.271356	−	0.962479i	$-0.412528\pi$
$179$	7.26099	0.542712	0.271356	+	0.962479i	$0.412528\pi$
$180$	0	0
$181$	6.80128	0.505535	0.252767	−	0.967527i	$-0.418659\pi$
$181$	6.80128	0.505535	0.252767	+	0.967527i	$0.418659\pi$
$182$	0	0
$183$	−27.9675	−2.06742
$184$	0	0
$185$	6.47068	0.475734
$186$	0	0
$187$	−3.43811	−0.251419
$188$	0	0
$189$	8.93971	0.650268
$190$	0	0
$191$	15.0187	1.08672	0.543359	−	0.839500i	$-0.317152\pi$
$191$	15.0187	1.08672	0.543359	+	0.839500i	$0.317152\pi$
$192$	0	0
$193$	19.4250	1.39824	0.699121	−	0.715003i	$-0.253575\pi$
$193$	19.4250	1.39824	0.699121	+	0.715003i	$0.253575\pi$
$194$	0	0
$195$	7.65634	0.548282
$196$	0	0
$197$	22.7280	1.61931	0.809653	−	0.586909i	$-0.199656\pi$
$197$	22.7280	1.61931	0.809653	+	0.586909i	$0.199656\pi$
$198$	0	0
$199$	−10.2553	−0.726981	−0.363490	−	0.931598i	$-0.618415\pi$
$199$	−10.2553	−0.726981	−0.363490	+	0.931598i	$0.618415\pi$
$200$	0	0
$201$	2.91683	0.205737
$202$	0	0
$203$	−10.6226	−0.745560
$204$	0	0
$205$	0.416855	0.0291144
$206$	0	0
$207$	−42.1895	−2.93237
$208$	0	0
$209$	−2.12563	−0.147033
$210$	0	0
$211$	4.57584	0.315014	0.157507	−	0.987518i	$-0.449654\pi$
$211$	4.57584	0.315014	0.157507	+	0.987518i	$0.449654\pi$
$212$	0	0
$213$	−18.2293	−1.24905
$214$	0	0
$215$	2.10994	0.143897
$216$	0	0
$217$	1.38417	0.0939633
$218$	0	0
$219$	−12.5829	−0.850274
$220$	0	0
$221$	21.2716	1.43088
$222$	0	0
$223$	−14.0123	−0.938332	−0.469166	−	0.883110i	$-0.655445\pi$
$223$	−14.0123	−0.938332	−0.469166	+	0.883110i	$0.655445\pi$
$224$	0	0
$225$	−21.1974	−1.41316
$226$	0	0
$227$	−22.4574	−1.49055	−0.745276	−	0.666756i	$-0.767682\pi$
$227$	−22.4574	−1.49055	−0.745276	+	0.666756i	$0.767682\pi$
$228$	0	0
$229$	−21.2008	−1.40099	−0.700493	−	0.713659i	$-0.747037\pi$
$229$	−21.2008	−1.40099	−0.700493	+	0.713659i	$0.747037\pi$
$230$	0	0
$231$	−1.66101	−0.109286
$232$	0	0
$233$	1.56951	0.102822	0.0514110	−	0.998678i	$-0.483628\pi$
$233$	1.56951	0.102822	0.0514110	+	0.998678i	$0.483628\pi$
$234$	0	0
$235$	2.83163	0.184715
$236$	0	0
$237$	−29.1456	−1.89321
$238$	0	0
$239$	10.1738	0.658086	0.329043	−	0.944315i	$-0.393274\pi$
$239$	10.1738	0.658086	0.329043	+	0.944315i	$0.393274\pi$
$240$	0	0
$241$	−19.2698	−1.24127	−0.620637	−	0.784098i	$-0.713126\pi$
$241$	−19.2698	−1.24127	−0.620637	+	0.784098i	$0.713126\pi$
$242$	0	0
$243$	10.6176	0.681119
$244$	0	0
$245$	5.20211	0.332351
$246$	0	0
$247$	13.1513	0.836795
$248$	0	0
$249$	−28.4622	−1.80372
$250$	0	0
$251$	22.6683	1.43081	0.715407	−	0.698708i	$-0.246241\pi$
$251$	22.6683	1.43081	0.715407	+	0.698708i	$0.246241\pi$
$252$	0	0
$253$	3.40645	0.214162
$254$	0	0
$255$	23.1843	1.45185
$256$	0	0
$257$	−5.59245	−0.348848	−0.174424	−	0.984671i	$-0.555806\pi$
$257$	−5.59245	−0.348848	−0.174424	+	0.984671i	$0.555806\pi$
$258$	0	0
$259$	8.68522	0.539674
$260$	0	0
$261$	41.8903	2.59294
$262$	0	0
$263$	3.06029	0.188706	0.0943529	−	0.995539i	$-0.469922\pi$
$263$	3.06029	0.188706	0.0943529	+	0.995539i	$0.469922\pi$
$264$	0	0
$265$	10.7824	0.662359
$266$	0	0
$267$	19.0146	1.16367
$268$	0	0
$269$	25.3323	1.54454	0.772268	−	0.635297i	$-0.219122\pi$
$269$	25.3323	1.54454	0.772268	+	0.635297i	$0.219122\pi$
$270$	0	0
$271$	2.11573	0.128521	0.0642607	−	0.997933i	$-0.479531\pi$
$271$	2.11573	0.128521	0.0642607	+	0.997933i	$0.479531\pi$
$272$	0	0
$273$	10.2767	0.621972
$274$	0	0
$275$	1.71152	0.103208
$276$	0	0
$277$	6.30888	0.379064	0.189532	−	0.981875i	$-0.439303\pi$
$277$	6.30888	0.379064	0.189532	+	0.981875i	$0.439303\pi$
$278$	0	0
$279$	−5.45847	−0.326790
$280$	0	0
$281$	−17.3431	−1.03460	−0.517301	−	0.855804i	$-0.673063\pi$
$281$	−17.3431	−1.03460	−0.517301	+	0.855804i	$0.673063\pi$
$282$	0	0
$283$	5.31115	0.315715	0.157858	−	0.987462i	$-0.449541\pi$
$283$	5.31115	0.315715	0.157858	+	0.987462i	$0.449541\pi$
$284$	0	0
$285$	14.3338	0.849060
$286$	0	0
$287$	0.559520	0.0330274
$288$	0	0
$289$	47.4127	2.78899
$290$	0	0
$291$	7.37071	0.432079
$292$	0	0
$293$	30.8180	1.80041	0.900203	−	0.435471i	$-0.143418\pi$
$293$	30.8180	1.80041	0.900203	+	0.435471i	$0.143418\pi$
$294$	0	0
$295$	2.97952	0.173475
$296$	0	0
$297$	2.84645	0.165168
$298$	0	0
$299$	−21.0758	−1.21884
$300$	0	0
$301$	2.83205	0.163237
$302$	0	0
$303$	−17.6989	−1.01677
$304$	0	0
$305$	−9.72724	−0.556980
$306$	0	0
$307$	0.251426	0.0143497	0.00717483	−	0.999974i	$-0.497716\pi$
$307$	0.251426	0.0143497	0.00717483	+	0.999974i	$0.497716\pi$
$308$	0	0
$309$	−18.5841	−1.05721
$310$	0	0
$311$	−8.04502	−0.456191	−0.228096	−	0.973639i	$-0.573250\pi$
$311$	−8.04502	−0.456191	−0.228096	+	0.973639i	$0.573250\pi$
$312$	0	0
$313$	−23.7939	−1.34491	−0.672457	−	0.740137i	$-0.734761\pi$
$313$	−23.7939	−1.34491	−0.672457	+	0.740137i	$0.734761\pi$
$314$	0	0
$315$	7.15499	0.403138
$316$	0	0
$317$	11.3848	0.639435	0.319717	−	0.947513i	$-0.396412\pi$
$317$	11.3848	0.639435	0.319717	+	0.947513i	$0.396412\pi$
$318$	0	0
$319$	−3.38230	−0.189372
$320$	0	0
$321$	18.7873	1.04861
$322$	0	0
$323$	39.8235	2.21584
$324$	0	0
$325$	−10.5892	−0.587381
$326$	0	0
$327$	−38.2116	−2.11310
$328$	0	0
$329$	3.80074	0.209541
$330$	0	0
$331$	16.7230	0.919181	0.459591	−	0.888131i	$-0.347996\pi$
$331$	16.7230	0.919181	0.459591	+	0.888131i	$0.347996\pi$
$332$	0	0
$333$	−34.2502	−1.87690
$334$	0	0
$335$	1.01449	0.0554273
$336$	0	0
$337$	1.88712	0.102798	0.0513990	−	0.998678i	$-0.483632\pi$
$337$	1.88712	0.102798	0.0513990	+	0.998678i	$0.483632\pi$
$338$	0	0
$339$	−11.0549	−0.600420
$340$	0	0
$341$	0.440726	0.0238667
$342$	0	0
$343$	16.4003	0.885534
$344$	0	0
$345$	−22.9708	−1.23671
$346$	0	0
$347$	30.9677	1.66243	0.831217	−	0.555948i	$-0.187645\pi$
$347$	30.9677	1.66243	0.831217	+	0.555948i	$0.187645\pi$
$348$	0	0
$349$	11.7177	0.627234	0.313617	−	0.949550i	$-0.398459\pi$
$349$	11.7177	0.627234	0.313617	+	0.949550i	$0.398459\pi$
$350$	0	0
$351$	−17.6110	−0.940008
$352$	0	0
$353$	2.12777	0.113250	0.0566250	−	0.998396i	$-0.481966\pi$
$353$	2.12777	0.113250	0.0566250	+	0.998396i	$0.481966\pi$
$354$	0	0
$355$	−6.34024	−0.336505
$356$	0	0
$357$	31.1189	1.64699
$358$	0	0
$359$	29.8269	1.57420	0.787101	−	0.616824i	$-0.211581\pi$
$359$	29.8269	1.57420	0.787101	+	0.616824i	$0.211581\pi$
$360$	0	0
$361$	5.62109	0.295847
$362$	0	0
$363$	31.1725	1.63613
$364$	0	0
$365$	−4.37640	−0.229071
$366$	0	0
$367$	−4.31545	−0.225265	−0.112632	−	0.993637i	$-0.535928\pi$
$367$	−4.31545	−0.225265	−0.112632	+	0.993637i	$0.535928\pi$
$368$	0	0
$369$	−2.20647	−0.114864
$370$	0	0
$371$	14.4726	0.751381
$372$	0	0
$373$	10.1736	0.526769	0.263385	−	0.964691i	$-0.415161\pi$
$373$	10.1736	0.526769	0.263385	+	0.964691i	$0.415161\pi$
$374$	0	0
$375$	−25.9850	−1.34186
$376$	0	0
$377$	20.9263	1.07776
$378$	0	0
$379$	1.74845	0.0898118	0.0449059	−	0.998991i	$-0.485701\pi$
$379$	1.74845	0.0898118	0.0449059	+	0.998991i	$0.485701\pi$
$380$	0	0
$381$	42.6210	2.18354
$382$	0	0
$383$	−16.2351	−0.829575	−0.414787	−	0.909918i	$-0.636144\pi$
$383$	−16.2351	−0.829575	−0.414787	+	0.909918i	$0.636144\pi$
$384$	0	0
$385$	−0.577707	−0.0294427
$386$	0	0
$387$	−11.1682	−0.567712
$388$	0	0
$389$	4.78821	0.242772	0.121386	−	0.992605i	$-0.461266\pi$
$389$	4.78821	0.242772	0.121386	+	0.992605i	$0.461266\pi$
$390$	0	0
$391$	−63.8198	−3.22750
$392$	0	0
$393$	44.2572	2.23248
$394$	0	0
$395$	−10.1370	−0.510047
$396$	0	0
$397$	−7.84294	−0.393626	−0.196813	−	0.980441i	$-0.563059\pi$
$397$	−7.84294	−0.393626	−0.196813	+	0.980441i	$0.563059\pi$
$398$	0	0
$399$	19.2394	0.963175
$400$	0	0
$401$	−14.7160	−0.734884	−0.367442	−	0.930046i	$-0.619766\pi$
$401$	−14.7160	−0.734884	−0.367442	+	0.930046i	$0.619766\pi$
$402$	0	0
$403$	−2.72678	−0.135831
$404$	0	0
$405$	−3.24026	−0.161010
$406$	0	0
$407$	2.76542	0.137077
$408$	0	0
$409$	1.64048	0.0811165	0.0405582	−	0.999177i	$-0.487086\pi$
$409$	1.64048	0.0811165	0.0405582	+	0.999177i	$0.487086\pi$
$410$	0	0
$411$	28.7581	1.41853
$412$	0	0
$413$	3.99924	0.196790
$414$	0	0
$415$	−9.89928	−0.485937
$416$	0	0
$417$	10.7124	0.524589
$418$	0	0
$419$	−37.0308	−1.80907	−0.904536	−	0.426396i	$-0.859783\pi$
$419$	−37.0308	−1.80907	−0.904536	+	0.426396i	$0.859783\pi$
$420$	0	0
$421$	2.78280	0.135625	0.0678127	−	0.997698i	$-0.478398\pi$
$421$	2.78280	0.135625	0.0678127	+	0.997698i	$0.478398\pi$
$422$	0	0
$423$	−14.9882	−0.728753
$424$	0	0
$425$	−32.0652	−1.55539
$426$	0	0
$427$	−13.0563	−0.631840
$428$	0	0
$429$	3.27215	0.157981
$430$	0	0
$431$	24.1263	1.16212	0.581062	−	0.813859i	$-0.302637\pi$
$431$	24.1263	1.16212	0.581062	+	0.813859i	$0.302637\pi$
$432$	0	0
$433$	4.78708	0.230052	0.115026	−	0.993362i	$-0.463305\pi$
$433$	4.78708	0.230052	0.115026	+	0.993362i	$0.463305\pi$
$434$	0	0
$435$	22.8079	1.09356
$436$	0	0
$437$	−39.4569	−1.88748
$438$	0	0
$439$	17.6901	0.844301	0.422150	−	0.906526i	$-0.361275\pi$
$439$	17.6901	0.844301	0.422150	+	0.906526i	$0.361275\pi$
$440$	0	0
$441$	−27.5355	−1.31121
$442$	0	0
$443$	6.20424	0.294772	0.147386	−	0.989079i	$-0.452914\pi$
$443$	6.20424	0.294772	0.147386	+	0.989079i	$0.452914\pi$
$444$	0	0
$445$	6.61336	0.313504
$446$	0	0
$447$	−19.0044	−0.898880
$448$	0	0
$449$	24.4851	1.15552	0.577761	−	0.816206i	$-0.303927\pi$
$449$	24.4851	1.15552	0.577761	+	0.816206i	$0.303927\pi$
$450$	0	0
$451$	0.178154	0.00838896
$452$	0	0
$453$	−12.7863	−0.600754
$454$	0	0
$455$	3.57428	0.167565
$456$	0	0
$457$	−4.58682	−0.214563	−0.107281	−	0.994229i	$-0.534215\pi$
$457$	−4.58682	−0.214563	−0.107281	+	0.994229i	$0.534215\pi$
$458$	0	0
$459$	−53.3282	−2.48915
$460$	0	0
$461$	38.6550	1.80034	0.900171	−	0.435536i	$-0.143441\pi$
$461$	38.6550	1.80034	0.900171	+	0.435536i	$0.143441\pi$
$462$	0	0
$463$	−13.6467	−0.634215	−0.317107	−	0.948390i	$-0.602712\pi$
$463$	−13.6467	−0.634215	−0.317107	+	0.948390i	$0.602712\pi$
$464$	0	0
$465$	−2.97196	−0.137821
$466$	0	0
$467$	−16.7565	−0.775399	−0.387699	−	0.921786i	$-0.626730\pi$
$467$	−16.7565	−0.775399	−0.387699	+	0.921786i	$0.626730\pi$
$468$	0	0
$469$	1.36169	0.0628769
$470$	0	0
$471$	−24.2086	−1.11547
$472$	0	0
$473$	0.901741	0.0414621
$474$	0	0
$475$	−19.8245	−0.909609
$476$	0	0
$477$	−57.0729	−2.61319
$478$	0	0
$479$	−5.89416	−0.269311	−0.134655	−	0.990892i	$-0.542993\pi$
$479$	−5.89416	−0.269311	−0.134655	+	0.990892i	$0.542993\pi$
$480$	0	0
$481$	−17.1097	−0.780136
$482$	0	0
$483$	−30.8324	−1.40292
$484$	0	0
$485$	2.56357	0.116406
$486$	0	0
$487$	34.1947	1.54951	0.774755	−	0.632261i	$-0.217873\pi$
$487$	34.1947	1.54951	0.774755	+	0.632261i	$0.217873\pi$
$488$	0	0
$489$	−18.1812	−0.822184
$490$	0	0
$491$	24.8750	1.12259	0.561296	−	0.827615i	$-0.310303\pi$
$491$	24.8750	1.12259	0.561296	+	0.827615i	$0.310303\pi$
$492$	0	0
$493$	63.3671	2.85391
$494$	0	0
$495$	2.27819	0.102397
$496$	0	0
$497$	−8.51014	−0.381732
$498$	0	0
$499$	8.01241	0.358685	0.179342	−	0.983787i	$-0.442603\pi$
$499$	8.01241	0.358685	0.179342	+	0.983787i	$0.442603\pi$
$500$	0	0
$501$	−27.6071	−1.23339
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−6.15575	−0.273927
$506$	0	0
$507$	17.2205	0.764788
$508$	0	0
$509$	−37.2605	−1.65154	−0.825770	−	0.564007i	$-0.809259\pi$
$509$	−37.2605	−1.65154	−0.825770	+	0.564007i	$0.809259\pi$
$510$	0	0
$511$	−5.87419	−0.259859
$512$	0	0
$513$	−32.9704	−1.45568
$514$	0	0
$515$	−6.46363	−0.284822
$516$	0	0
$517$	1.21018	0.0532235
$518$	0	0
$519$	−45.8938	−2.01452
$520$	0	0
$521$	−36.7570	−1.61035	−0.805177	−	0.593034i	$-0.797930\pi$
$521$	−36.7570	−1.61035	−0.805177	+	0.593034i	$0.797930\pi$
$522$	0	0
$523$	−32.8403	−1.43601	−0.718003	−	0.696040i	$-0.754944\pi$
$523$	−32.8403	−1.43601	−0.718003	+	0.696040i	$0.754944\pi$
$524$	0	0
$525$	−15.4912	−0.676093
$526$	0	0
$527$	−8.25699	−0.359680
$528$	0	0
$529$	40.2322	1.74923
$530$	0	0
$531$	−15.7710	−0.684405
$532$	0	0
$533$	−1.10224	−0.0477434
$534$	0	0
$535$	6.53432	0.282503
$536$	0	0
$537$	−20.9258	−0.903014
$538$	0	0
$539$	2.22327	0.0957628
$540$	0	0
$541$	25.5512	1.09853	0.549266	−	0.835647i	$-0.314907\pi$
$541$	25.5512	1.09853	0.549266	+	0.835647i	$0.314907\pi$
$542$	0	0
$543$	−19.6009	−0.841155
$544$	0	0
$545$	−13.2902	−0.569289
$546$	0	0
$547$	−13.6482	−0.583557	−0.291778	−	0.956486i	$-0.594247\pi$
$547$	−13.6482	−0.583557	−0.291778	+	0.956486i	$0.594247\pi$
$548$	0	0
$549$	51.4877	2.19744
$550$	0	0
$551$	39.1771	1.66900
$552$	0	0
$553$	−13.6063	−0.578598
$554$	0	0
$555$	−18.6482	−0.791570
$556$	0	0
$557$	−28.9038	−1.22469	−0.612347	−	0.790589i	$-0.709775\pi$
$557$	−28.9038	−1.22469	−0.612347	+	0.790589i	$0.709775\pi$
$558$	0	0
$559$	−5.57908	−0.235970
$560$	0	0
$561$	9.90843	0.418334
$562$	0	0
$563$	43.7154	1.84238	0.921192	−	0.389109i	$-0.127217\pi$
$563$	43.7154	1.84238	0.921192	+	0.389109i	$0.127217\pi$
$564$	0	0
$565$	−3.84495	−0.161758
$566$	0	0
$567$	−4.34921	−0.182650
$568$	0	0
$569$	37.3991	1.56785	0.783927	−	0.620853i	$-0.213214\pi$
$569$	37.3991	1.56785	0.783927	+	0.620853i	$0.213214\pi$
$570$	0	0
$571$	−9.01866	−0.377419	−0.188710	−	0.982033i	$-0.560431\pi$
$571$	−9.01866	−0.377419	−0.188710	+	0.982033i	$0.560431\pi$
$572$	0	0
$573$	−43.2832	−1.80818
$574$	0	0
$575$	31.7700	1.32490
$576$	0	0
$577$	−15.8133	−0.658318	−0.329159	−	0.944275i	$-0.606765\pi$
$577$	−15.8133	−0.658318	−0.329159	+	0.944275i	$0.606765\pi$
$578$	0	0
$579$	−55.9818	−2.32652
$580$	0	0
$581$	−13.2872	−0.551247
$582$	0	0
$583$	4.60817	0.190851
$584$	0	0
$585$	−14.0952	−0.582764
$586$	0	0
$587$	33.2924	1.37413	0.687063	−	0.726598i	$-0.258900\pi$
$587$	33.2924	1.37413	0.687063	+	0.726598i	$0.258900\pi$
$588$	0	0
$589$	−5.10493	−0.210345
$590$	0	0
$591$	−65.5009	−2.69435
$592$	0	0
$593$	−2.62974	−0.107990	−0.0539951	−	0.998541i	$-0.517196\pi$
$593$	−2.62974	−0.107990	−0.0539951	+	0.998541i	$0.517196\pi$
$594$	0	0
$595$	10.8233	0.443712
$596$	0	0
$597$	29.5553	1.20962
$598$	0	0
$599$	21.6138	0.883117	0.441558	−	0.897232i	$-0.354426\pi$
$599$	21.6138	0.883117	0.441558	+	0.897232i	$0.354426\pi$
$600$	0	0
$601$	47.9324	1.95520	0.977601	−	0.210466i	$-0.0674980\pi$
$601$	47.9324	1.95520	0.977601	+	0.210466i	$0.0674980\pi$
$602$	0	0
$603$	−5.36982	−0.218676
$604$	0	0
$605$	10.8420	0.440788
$606$	0	0
$607$	36.5533	1.48365	0.741827	−	0.670591i	$-0.233959\pi$
$607$	36.5533	1.48365	0.741827	+	0.670591i	$0.233959\pi$
$608$	0	0
$609$	30.6137	1.24053
$610$	0	0
$611$	−7.48737	−0.302907
$612$	0	0
$613$	−30.7798	−1.24318	−0.621592	−	0.783341i	$-0.713514\pi$
$613$	−30.7798	−1.24318	−0.621592	+	0.783341i	$0.713514\pi$
$614$	0	0
$615$	−1.20135	−0.0484432
$616$	0	0
$617$	−48.5258	−1.95357	−0.976787	−	0.214212i	$-0.931282\pi$
$617$	−48.5258	−1.95357	−0.976787	+	0.214212i	$0.931282\pi$
$618$	0	0
$619$	12.6866	0.509919	0.254960	−	0.966952i	$-0.417938\pi$
$619$	12.6866	0.509919	0.254960	+	0.966952i	$0.417938\pi$
$620$	0	0
$621$	52.8372	2.12029
$622$	0	0
$623$	8.87674	0.355639
$624$	0	0
$625$	10.9387	0.437549
$626$	0	0
$627$	6.12594	0.244646
$628$	0	0
$629$	−51.8101	−2.06580
$630$	0	0
$631$	−36.7385	−1.46254	−0.731268	−	0.682090i	$-0.761071\pi$
$631$	−36.7385	−1.46254	−0.731268	+	0.682090i	$0.761071\pi$
$632$	0	0
$633$	−13.1873	−0.524149
$634$	0	0
$635$	14.8238	0.588265
$636$	0	0
$637$	−13.7554	−0.545008
$638$	0	0
$639$	33.5598	1.32761
$640$	0	0
$641$	−9.54397	−0.376964	−0.188482	−	0.982077i	$-0.560357\pi$
$641$	−9.54397	−0.376964	−0.188482	+	0.982077i	$0.560357\pi$
$642$	0	0
$643$	16.2340	0.640207	0.320103	−	0.947383i	$-0.396282\pi$
$643$	16.2340	0.640207	0.320103	+	0.947383i	$0.396282\pi$
$644$	0	0
$645$	−6.08073	−0.239429
$646$	0	0
$647$	21.2630	0.835937	0.417968	−	0.908462i	$-0.362742\pi$
$647$	21.2630	0.835937	0.417968	+	0.908462i	$0.362742\pi$
$648$	0	0
$649$	1.27338	0.0499846
$650$	0	0
$651$	−3.98909	−0.156345
$652$	0	0
$653$	−20.1379	−0.788057	−0.394029	−	0.919098i	$-0.628919\pi$
$653$	−20.1379	−0.788057	−0.394029	+	0.919098i	$0.628919\pi$
$654$	0	0
$655$	15.3929	0.601449
$656$	0	0
$657$	23.1649	0.903749
$658$	0	0
$659$	41.2595	1.60724	0.803620	−	0.595142i	$-0.202904\pi$
$659$	41.2595	1.60724	0.803620	+	0.595142i	$0.202904\pi$
$660$	0	0
$661$	5.25508	0.204399	0.102199	−	0.994764i	$-0.467412\pi$
$661$	5.25508	0.204399	0.102199	+	0.994764i	$0.467412\pi$
$662$	0	0
$663$	−61.3036	−2.38083
$664$	0	0
$665$	6.69156	0.259488
$666$	0	0
$667$	−62.7838	−2.43100
$668$	0	0
$669$	40.3826	1.56128
$670$	0	0
$671$	−4.15721	−0.160487
$672$	0	0
$673$	3.27535	0.126256	0.0631278	−	0.998005i	$-0.479892\pi$
$673$	3.27535	0.126256	0.0631278	+	0.998005i	$0.479892\pi$
$674$	0	0
$675$	26.5472	1.02180
$676$	0	0
$677$	−45.1502	−1.73526	−0.867631	−	0.497208i	$-0.834359\pi$
$677$	−45.1502	−1.73526	−0.867631	+	0.497208i	$0.834359\pi$
$678$	0	0
$679$	3.44093	0.132051
$680$	0	0
$681$	64.7211	2.48012
$682$	0	0
$683$	1.24561	0.0476619	0.0238309	−	0.999716i	$-0.492414\pi$
$683$	1.24561	0.0476619	0.0238309	+	0.999716i	$0.492414\pi$
$684$	0	0
$685$	10.0022	0.382164
$686$	0	0
$687$	61.0995	2.33109
$688$	0	0
$689$	−28.5108	−1.08617
$690$	0	0
$691$	18.3525	0.698161	0.349080	−	0.937093i	$-0.386494\pi$
$691$	18.3525	0.698161	0.349080	+	0.937093i	$0.386494\pi$
$692$	0	0
$693$	3.05788	0.116159
$694$	0	0
$695$	3.72583	0.141329
$696$	0	0
$697$	−3.33771	−0.126425
$698$	0	0
$699$	−4.52324	−0.171085
$700$	0	0
$701$	13.5978	0.513582	0.256791	−	0.966467i	$-0.417335\pi$
$701$	13.5978	0.513582	0.256791	+	0.966467i	$0.417335\pi$
$702$	0	0
$703$	−32.0319	−1.20810
$704$	0	0
$705$	−8.16061	−0.307346
$706$	0	0
$707$	−8.26251	−0.310744
$708$	0	0
$709$	16.7216	0.627993	0.313997	−	0.949424i	$-0.398332\pi$
$709$	16.7216	0.627993	0.313997	+	0.949424i	$0.398332\pi$
$710$	0	0
$711$	53.6565	2.01228
$712$	0	0
$713$	8.18097	0.306380
$714$	0	0
$715$	1.13807	0.0425614
$716$	0	0
$717$	−29.3202	−1.09498
$718$	0	0
$719$	−13.6650	−0.509617	−0.254809	−	0.966991i	$-0.582013\pi$
$719$	−13.6650	−0.509617	−0.254809	+	0.966991i	$0.582013\pi$
$720$	0	0
$721$	−8.67576	−0.323102
$722$	0	0
$723$	55.5344	2.06535
$724$	0	0
$725$	−31.5447	−1.17154
$726$	0	0
$727$	12.0042	0.445212	0.222606	−	0.974908i	$-0.428544\pi$
$727$	12.0042	0.445212	0.222606	+	0.974908i	$0.428544\pi$
$728$	0	0
$729$	−40.2973	−1.49249
$730$	0	0
$731$	−16.8941	−0.624850
$732$	0	0
$733$	−43.3956	−1.60285	−0.801427	−	0.598092i	$-0.795926\pi$
$733$	−43.3956	−1.60285	−0.801427	+	0.598092i	$0.795926\pi$
$734$	0	0
$735$	−14.9922	−0.552996
$736$	0	0
$737$	0.433569	0.0159707
$738$	0	0
$739$	−3.83318	−0.141006	−0.0705029	−	0.997512i	$-0.522460\pi$
$739$	−3.83318	−0.141006	−0.0705029	+	0.997512i	$0.522460\pi$
$740$	0	0
$741$	−37.9012	−1.39234
$742$	0	0
$743$	8.65248	0.317429	0.158714	−	0.987325i	$-0.449265\pi$
$743$	8.65248	0.317429	0.158714	+	0.987325i	$0.449265\pi$
$744$	0	0
$745$	−6.60984	−0.242166
$746$	0	0
$747$	52.3983	1.91715
$748$	0	0
$749$	8.77065	0.320472
$750$	0	0
$751$	−2.02550	−0.0739114	−0.0369557	−	0.999317i	$-0.511766\pi$
$751$	−2.02550	−0.0739114	−0.0369557	+	0.999317i	$0.511766\pi$
$752$	0	0
$753$	−65.3289	−2.38072
$754$	0	0
$755$	−4.44715	−0.161848
$756$	0	0
$757$	−12.0324	−0.437324	−0.218662	−	0.975801i	$-0.570169\pi$
$757$	−12.0324	−0.437324	−0.218662	+	0.975801i	$0.570169\pi$
$758$	0	0
$759$	−9.81721	−0.356342
$760$	0	0
$761$	44.6969	1.62026	0.810130	−	0.586250i	$-0.199397\pi$
$761$	44.6969	1.62026	0.810130	+	0.586250i	$0.199397\pi$
$762$	0	0
$763$	−17.8386	−0.645802
$764$	0	0
$765$	−42.6818	−1.54316
$766$	0	0
$767$	−7.87843	−0.284474
$768$	0	0
$769$	8.32799	0.300315	0.150157	−	0.988662i	$-0.452022\pi$
$769$	8.32799	0.300315	0.150157	+	0.988662i	$0.452022\pi$
$770$	0	0
$771$	16.1171	0.580445
$772$	0	0
$773$	33.9520	1.22117	0.610585	−	0.791951i	$-0.290935\pi$
$773$	33.9520	1.22117	0.610585	+	0.791951i	$0.290935\pi$
$774$	0	0
$775$	4.11039	0.147650
$776$	0	0
$777$	−25.0303	−0.897958
$778$	0	0
$779$	−2.06356	−0.0739346
$780$	0	0
$781$	−2.70968	−0.0969599
$782$	0	0
$783$	−52.4625	−1.87486
$784$	0	0
$785$	−8.41986	−0.300518
$786$	0	0
$787$	−20.8424	−0.742951	−0.371475	−	0.928443i	$-0.621148\pi$
$787$	−20.8424	−0.742951	−0.371475	+	0.928443i	$0.621148\pi$
$788$	0	0
$789$	−8.81959	−0.313986
$790$	0	0
$791$	−5.16086	−0.183499
$792$	0	0
$793$	25.7207	0.913368
$794$	0	0
$795$	−31.0743	−1.10209
$796$	0	0
$797$	−29.5214	−1.04570	−0.522851	−	0.852424i	$-0.675132\pi$
$797$	−29.5214	−1.04570	−0.522851	+	0.852424i	$0.675132\pi$
$798$	0	0
$799$	−22.6726	−0.802099
$800$	0	0
$801$	−35.0055	−1.23686
$802$	0	0
$803$	−1.87037	−0.0660041
$804$	0	0
$805$	−10.7237	−0.377960
$806$	0	0
$807$	−73.0062	−2.56994
$808$	0	0
$809$	15.5688	0.547370	0.273685	−	0.961819i	$-0.411757\pi$
$809$	15.5688	0.547370	0.273685	+	0.961819i	$0.411757\pi$
$810$	0	0
$811$	−44.9735	−1.57923	−0.789616	−	0.613601i	$-0.789720\pi$
$811$	−44.9735	−1.57923	−0.789616	+	0.613601i	$0.789720\pi$
$812$	0	0
$813$	−6.09742	−0.213846
$814$	0	0
$815$	−6.32353	−0.221503
$816$	0	0
$817$	−10.4448	−0.365419
$818$	0	0
$819$	−18.9192	−0.661089
$820$	0	0
$821$	−25.1289	−0.877003	−0.438502	−	0.898730i	$-0.644491\pi$
$821$	−25.1289	−0.877003	−0.438502	+	0.898730i	$0.644491\pi$
$822$	0	0
$823$	54.6166	1.90381	0.951907	−	0.306386i	$-0.0991198\pi$
$823$	54.6166	1.90381	0.951907	+	0.306386i	$0.0991198\pi$
$824$	0	0
$825$	−4.93250	−0.171727
$826$	0	0
$827$	32.2349	1.12092	0.560458	−	0.828183i	$-0.310625\pi$
$827$	32.2349	1.12092	0.560458	+	0.828183i	$0.310625\pi$
$828$	0	0
$829$	−34.0786	−1.18360	−0.591799	−	0.806085i	$-0.701582\pi$
$829$	−34.0786	−1.18360	−0.591799	+	0.806085i	$0.701582\pi$
$830$	0	0
$831$	−18.1819	−0.630721
$832$	0	0
$833$	−41.6528	−1.44318
$834$	0	0
$835$	−9.60187	−0.332287
$836$	0	0
$837$	6.83607	0.236289
$838$	0	0
$839$	43.2105	1.49179	0.745896	−	0.666063i	$-0.232022\pi$
$839$	43.2105	1.49179	0.745896	+	0.666063i	$0.232022\pi$
$840$	0	0
$841$	33.3385	1.14960
$842$	0	0
$843$	49.9818	1.72147
$844$	0	0
$845$	5.98937	0.206040
$846$	0	0
$847$	14.5525	0.500031
$848$	0	0
$849$	−15.3064	−0.525316
$850$	0	0
$851$	51.3331	1.75968
$852$	0	0
$853$	−18.8505	−0.645429	−0.322714	−	0.946496i	$-0.604595\pi$
$853$	−18.8505	−0.645429	−0.322714	+	0.946496i	$0.604595\pi$
$854$	0	0
$855$	−26.3882	−0.902458
$856$	0	0
$857$	−39.9542	−1.36481	−0.682404	−	0.730975i	$-0.739065\pi$
$857$	−39.9542	−1.36481	−0.682404	+	0.730975i	$0.739065\pi$
$858$	0	0
$859$	5.79062	0.197573	0.0987867	−	0.995109i	$-0.468504\pi$
$859$	5.79062	0.197573	0.0987867	+	0.995109i	$0.468504\pi$
$860$	0	0
$861$	−1.61251	−0.0549540
$862$	0	0
$863$	−10.6197	−0.361498	−0.180749	−	0.983529i	$-0.557852\pi$
$863$	−10.6197	−0.361498	−0.180749	+	0.983529i	$0.557852\pi$
$864$	0	0
$865$	−15.9621	−0.542728
$866$	0	0
$867$	−136.641	−4.64057
$868$	0	0
$869$	−4.33232	−0.146964
$870$	0	0
$871$	−2.68250	−0.0908929
$872$	0	0
$873$	−13.5693	−0.459253
$874$	0	0
$875$	−12.1308	−0.410095
$876$	0	0
$877$	−23.8021	−0.803739	−0.401870	−	0.915697i	$-0.631640\pi$
$877$	−23.8021	−0.803739	−0.401870	+	0.915697i	$0.631640\pi$
$878$	0	0
$879$	−88.8157	−2.99568
$880$	0	0
$881$	41.5621	1.40026	0.700131	−	0.714015i	$-0.253125\pi$
$881$	41.5621	1.40026	0.700131	+	0.714015i	$0.253125\pi$
$882$	0	0
$883$	−2.47137	−0.0831683	−0.0415842	−	0.999135i	$-0.513240\pi$
$883$	−2.47137	−0.0831683	−0.0415842	+	0.999135i	$0.513240\pi$
$884$	0	0
$885$	−8.58683	−0.288643
$886$	0	0
$887$	−9.69064	−0.325380	−0.162690	−	0.986677i	$-0.552017\pi$
$887$	−9.69064	−0.325380	−0.162690	+	0.986677i	$0.552017\pi$
$888$	0	0
$889$	19.8971	0.667329
$890$	0	0
$891$	−1.38481	−0.0463930
$892$	0	0
$893$	−14.0174	−0.469076
$894$	0	0
$895$	−7.27809	−0.243280
$896$	0	0
$897$	60.7392	2.02802
$898$	0	0
$899$	−8.12295	−0.270916
$900$	0	0
$901$	−86.3338	−2.87619
$902$	0	0
$903$	−8.16181	−0.271608
$904$	0	0
$905$	−6.81729	−0.226614
$906$	0	0
$907$	16.7900	0.557504	0.278752	−	0.960363i	$-0.410079\pi$
$907$	16.7900	0.557504	0.278752	+	0.960363i	$0.410079\pi$
$908$	0	0
$909$	32.5833	1.08072
$910$	0	0
$911$	5.07623	0.168183	0.0840915	−	0.996458i	$-0.473201\pi$
$911$	5.07623	0.168183	0.0840915	+	0.996458i	$0.473201\pi$
$912$	0	0
$913$	−4.23073	−0.140017
$914$	0	0
$915$	28.0334	0.926755
$916$	0	0
$917$	20.6610	0.682285
$918$	0	0
$919$	−41.3656	−1.36453	−0.682263	−	0.731107i	$-0.739004\pi$
$919$	−41.3656	−1.36453	−0.682263	+	0.731107i	$0.739004\pi$
$920$	0	0
$921$	−0.724597	−0.0238763
$922$	0	0
$923$	16.7648	0.551820
$924$	0	0
$925$	25.7915	0.848019
$926$	0	0
$927$	34.2129	1.12370
$928$	0	0
$929$	12.8057	0.420140	0.210070	−	0.977686i	$-0.432631\pi$
$929$	12.8057	0.420140	0.210070	+	0.977686i	$0.432631\pi$
$930$	0	0
$931$	−25.7520	−0.843989
$932$	0	0
$933$	23.1853	0.759053
$934$	0	0
$935$	3.44620	0.112703
$936$	0	0
$937$	35.6143	1.16347	0.581735	−	0.813378i	$-0.302374\pi$
$937$	35.6143	1.16347	0.581735	+	0.813378i	$0.302374\pi$
$938$	0	0
$939$	68.5728	2.23779
$940$	0	0
$941$	51.4683	1.67782	0.838909	−	0.544272i	$-0.183194\pi$
$941$	51.4683	1.67782	0.838909	+	0.544272i	$0.183194\pi$
$942$	0	0
$943$	3.30698	0.107690
$944$	0	0
$945$	−8.96076	−0.291494
$946$	0	0
$947$	36.5976	1.18926	0.594631	−	0.803999i	$-0.297298\pi$
$947$	36.5976	1.18926	0.594631	+	0.803999i	$0.297298\pi$
$948$	0	0
$949$	11.5720	0.375644
$950$	0	0
$951$	−32.8104	−1.06395
$952$	0	0
$953$	49.7603	1.61190	0.805948	−	0.591987i	$-0.201656\pi$
$953$	49.7603	1.61190	0.805948	+	0.591987i	$0.201656\pi$
$954$	0	0
$955$	−15.0541	−0.487140
$956$	0	0
$957$	9.74759	0.315095
$958$	0	0
$959$	13.4254	0.433528
$960$	0	0
$961$	−29.9415	−0.965856
$962$	0	0
$963$	−34.5871	−1.11455
$964$	0	0
$965$	−19.4708	−0.626786
$966$	0	0
$967$	−61.8099	−1.98767	−0.993837	−	0.110855i	$-0.964641\pi$
$967$	−61.8099	−1.98767	−0.993837	+	0.110855i	$0.964641\pi$
$968$	0	0
$969$	−114.769	−3.68692
$970$	0	0
$971$	30.4268	0.976443	0.488222	−	0.872720i	$-0.337646\pi$
$971$	30.4268	0.976443	0.488222	+	0.872720i	$0.337646\pi$
$972$	0	0
$973$	5.00096	0.160324
$974$	0	0
$975$	30.5174	0.977339
$976$	0	0
$977$	−9.19272	−0.294101	−0.147051	−	0.989129i	$-0.546978\pi$
$977$	−9.19272	−0.294101	−0.147051	+	0.989129i	$0.546978\pi$
$978$	0	0
$979$	2.82640	0.0903323
$980$	0	0
$981$	70.3468	2.24600
$982$	0	0
$983$	55.1555	1.75919	0.879594	−	0.475724i	$-0.157814\pi$
$983$	55.1555	1.75919	0.879594	+	0.475724i	$0.157814\pi$
$984$	0	0
$985$	−22.7816	−0.725881
$986$	0	0
$987$	−10.9535	−0.348654
$988$	0	0
$989$	16.7385	0.532255
$990$	0	0
$991$	−5.89220	−0.187172	−0.0935859	−	0.995611i	$-0.529833\pi$
$991$	−5.89220	−0.187172	−0.0935859	+	0.995611i	$0.529833\pi$
$992$	0	0
$993$	−48.1949	−1.52942
$994$	0	0
$995$	10.2795	0.325881
$996$	0	0
$997$	−30.3605	−0.961528	−0.480764	−	0.876850i	$-0.659641\pi$
$997$	−30.3605	−0.961528	−0.480764	+	0.876850i	$0.659641\pi$
$998$	0	0
$999$	42.8943	1.35712

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.f.1.5	✓	33
4.3	odd	2		8048.2.a.y.1.29		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.f.1.5	✓	33	1.1	even	1	trivial
8048.2.a.y.1.29		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)

\(f(q)\)	\(=\)	\(q-2.88194 q^{3} -1.00235 q^{5} -1.34540 q^{7} +5.30561 q^{9} +O(q^{10})\)	\(q-2.88194 q^{3} -1.00235 q^{5} -1.34540 q^{7} +5.30561 q^{9} -0.428384 q^{11} +2.65042 q^{13} +2.88873 q^{15} +8.02576 q^{17} +4.96196 q^{19} +3.87738 q^{21} -7.95187 q^{23} -3.99528 q^{25} -6.64463 q^{27} +7.89547 q^{29} -1.02881 q^{31} +1.23458 q^{33} +1.34857 q^{35} -6.45548 q^{37} -7.63835 q^{39} -0.415875 q^{41} -2.10498 q^{43} -5.31810 q^{45} -2.82498 q^{47} -5.18989 q^{49} -23.1298 q^{51} -10.7571 q^{53} +0.429393 q^{55} -14.3001 q^{57} -2.97252 q^{59} +9.70439 q^{61} -7.13818 q^{63} -2.65666 q^{65} -1.01210 q^{67} +22.9168 q^{69} +6.32535 q^{71} +4.36612 q^{73} +11.5142 q^{75} +0.576349 q^{77} +10.1132 q^{79} +3.23264 q^{81} +9.87602 q^{83} -8.04466 q^{85} -22.7543 q^{87} -6.59783 q^{89} -3.56588 q^{91} +2.96498 q^{93} -4.97365 q^{95} -2.55755 q^{97} -2.27284 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) \) 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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