LMFDB - Embedded newform 8048.2.a.u.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, 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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.69727 q^{3} +0.763844 q^{5} +0.178001 q^{7} -0.119277 q^{9} +O(q^{10})$	$q-1.69727 q^{3} +0.763844 q^{5} +0.178001 q^{7} -0.119277 q^{9} +3.15073 q^{11} -4.14465 q^{13} -1.29645 q^{15} -7.28992 q^{17} -7.21972 q^{19} -0.302115 q^{21} -1.98808 q^{23} -4.41654 q^{25} +5.29425 q^{27} +5.44144 q^{29} -9.61595 q^{31} -5.34764 q^{33} +0.135965 q^{35} +8.98278 q^{37} +7.03459 q^{39} +1.85423 q^{41} -8.91571 q^{43} -0.0911093 q^{45} -0.850791 q^{47} -6.96832 q^{49} +12.3730 q^{51} +6.82601 q^{53} +2.40667 q^{55} +12.2538 q^{57} -7.34734 q^{59} -3.45073 q^{61} -0.0212315 q^{63} -3.16587 q^{65} -8.65951 q^{67} +3.37431 q^{69} +12.7380 q^{71} +3.24032 q^{73} +7.49606 q^{75} +0.560833 q^{77} +14.1236 q^{79} -8.62794 q^{81} +6.24192 q^{83} -5.56836 q^{85} -9.23559 q^{87} +1.86316 q^{89} -0.737751 q^{91} +16.3209 q^{93} -5.51474 q^{95} +15.4553 q^{97} -0.375811 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) $ 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9	$ 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) $ 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9 + 17 * q^11 + 14 * q^13 - 18 * q^15 + 17 * q^17 + 22 * q^19 - 16 * q^21 - 27 * q^23 + 93 * q^25 - 31 * q^27 + 13 * q^29 - 26 * q^31 + 6 * q^33 + 22 * q^35 + 55 * q^37 + 15 * q^39 + 24 * q^41 - 20 * q^43 - 8 * q^45 + 25 * q^47 + 65 * q^49 - 7 * q^51 + 30 * q^53 - 25 * q^55 + 9 * q^57 + 26 * q^59 + 15 * q^61 + 19 * q^63 + 20 * q^65 + 20 * q^67 - 27 * q^69 + 35 * q^71 + 38 * q^73 - 2 * q^75 - 6 * q^77 - 21 * q^79 + 70 * q^81 + 48 * q^83 + 6 * q^85 + 9 * q^87 - 5 * q^89 + 24 * q^91 - 8 * q^93 - 43 * q^95 + 142 * q^97 + 50 * 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
$3$	−1.69727	−0.979919	−0.489959	−	0.871745i	$-0.662988\pi$
$3$	−1.69727	−0.979919	−0.489959	+	0.871745i	$0.662988\pi$
$4$	0	0
$5$	0.763844	0.341601	0.170801	−	0.985306i	$-0.445365\pi$
$5$	0.763844	0.341601	0.170801	+	0.985306i	$0.445365\pi$
$6$	0	0
$7$	0.178001	0.0672779	0.0336390	−	0.999434i	$-0.489290\pi$
$7$	0.178001	0.0672779	0.0336390	+	0.999434i	$0.489290\pi$
$8$	0	0
$9$	−0.119277	−0.0397591
$10$	0	0
$11$	3.15073	0.949982	0.474991	−	0.879991i	$-0.342451\pi$
$11$	3.15073	0.949982	0.474991	+	0.879991i	$0.342451\pi$
$12$	0	0
$13$	−4.14465	−1.14952	−0.574760	−	0.818322i	$-0.694905\pi$
$13$	−4.14465	−1.14952	−0.574760	+	0.818322i	$0.694905\pi$
$14$	0	0
$15$	−1.29645	−0.334742
$16$	0	0
$17$	−7.28992	−1.76807	−0.884033	−	0.467425i	$-0.845182\pi$
$17$	−7.28992	−1.76807	−0.884033	+	0.467425i	$0.845182\pi$
$18$	0	0
$19$	−7.21972	−1.65632	−0.828159	−	0.560493i	$-0.810612\pi$
$19$	−7.21972	−1.65632	−0.828159	+	0.560493i	$0.810612\pi$
$20$	0	0
$21$	−0.302115	−0.0659269
$22$	0	0
$23$	−1.98808	−0.414544	−0.207272	−	0.978283i	$-0.566458\pi$
$23$	−1.98808	−0.414544	−0.207272	+	0.978283i	$0.566458\pi$
$24$	0	0
$25$	−4.41654	−0.883308
$26$	0	0
$27$	5.29425	1.01888
$28$	0	0
$29$	5.44144	1.01045	0.505225	−	0.862988i	$-0.331409\pi$
$29$	5.44144	1.01045	0.505225	+	0.862988i	$0.331409\pi$
$30$	0	0
$31$	−9.61595	−1.72708	−0.863538	−	0.504284i	$-0.831757\pi$
$31$	−9.61595	−1.72708	−0.863538	+	0.504284i	$0.831757\pi$
$32$	0	0
$33$	−5.34764	−0.930905
$34$	0	0
$35$	0.135965	0.0229822
$36$	0	0
$37$	8.98278	1.47676	0.738380	−	0.674385i	$-0.235591\pi$
$37$	8.98278	1.47676	0.738380	+	0.674385i	$0.235591\pi$
$38$	0	0
$39$	7.03459	1.12644
$40$	0	0
$41$	1.85423	0.289582	0.144791	−	0.989462i	$-0.453749\pi$
$41$	1.85423	0.289582	0.144791	+	0.989462i	$0.453749\pi$
$42$	0	0
$43$	−8.91571	−1.35963	−0.679817	−	0.733382i	$-0.737941\pi$
$43$	−8.91571	−1.35963	−0.679817	+	0.733382i	$0.737941\pi$
$44$	0	0
$45$	−0.0911093	−0.0135818
$46$	0	0
$47$	−0.850791	−0.124101	−0.0620503	−	0.998073i	$-0.519764\pi$
$47$	−0.850791	−0.124101	−0.0620503	+	0.998073i	$0.519764\pi$
$48$	0	0
$49$	−6.96832	−0.995474
$50$	0	0
$51$	12.3730	1.73256
$52$	0	0
$53$	6.82601	0.937625	0.468813	−	0.883298i	$-0.344682\pi$
$53$	6.82601	0.937625	0.468813	+	0.883298i	$0.344682\pi$
$54$	0	0
$55$	2.40667	0.324515
$56$	0	0
$57$	12.2538	1.62306
$58$	0	0
$59$	−7.34734	−0.956543	−0.478271	−	0.878212i	$-0.658736\pi$
$59$	−7.34734	−0.956543	−0.478271	+	0.878212i	$0.658736\pi$
$60$	0	0
$61$	−3.45073	−0.441821	−0.220911	−	0.975294i	$-0.570903\pi$
$61$	−3.45073	−0.441821	−0.220911	+	0.975294i	$0.570903\pi$
$62$	0	0
$63$	−0.0212315	−0.00267491
$64$	0	0
$65$	−3.16587	−0.392678
$66$	0	0
$67$	−8.65951	−1.05793	−0.528964	−	0.848644i	$-0.677419\pi$
$67$	−8.65951	−1.05793	−0.528964	+	0.848644i	$0.677419\pi$
$68$	0	0
$69$	3.37431	0.406219
$70$	0	0
$71$	12.7380	1.51172	0.755858	−	0.654735i	$-0.227220\pi$
$71$	12.7380	1.51172	0.755858	+	0.654735i	$0.227220\pi$
$72$	0	0
$73$	3.24032	0.379251	0.189626	−	0.981856i	$-0.439273\pi$
$73$	3.24032	0.379251	0.189626	+	0.981856i	$0.439273\pi$
$74$	0	0
$75$	7.49606	0.865571
$76$	0	0
$77$	0.560833	0.0639128
$78$	0	0
$79$	14.1236	1.58903	0.794513	−	0.607248i	$-0.207726\pi$
$79$	14.1236	1.58903	0.794513	+	0.607248i	$0.207726\pi$
$80$	0	0
$81$	−8.62794	−0.958660
$82$	0	0
$83$	6.24192	0.685139	0.342570	−	0.939492i	$-0.388703\pi$
$83$	6.24192	0.685139	0.342570	+	0.939492i	$0.388703\pi$
$84$	0	0
$85$	−5.56836	−0.603974
$86$	0	0
$87$	−9.23559	−0.990159
$88$	0	0
$89$	1.86316	0.197494	0.0987472	−	0.995113i	$-0.468516\pi$
$89$	1.86316	0.197494	0.0987472	+	0.995113i	$0.468516\pi$
$90$	0	0
$91$	−0.737751	−0.0773373
$92$	0	0
$93$	16.3209	1.69239
$94$	0	0
$95$	−5.51474	−0.565801
$96$	0	0
$97$	15.4553	1.56924	0.784622	−	0.619975i	$-0.212857\pi$
$97$	15.4553	1.56924	0.784622	+	0.619975i	$0.212857\pi$
$98$	0	0
$99$	−0.375811	−0.0377705
$100$	0	0
$101$	8.74903	0.870561	0.435280	−	0.900295i	$-0.356649\pi$
$101$	8.74903	0.870561	0.435280	+	0.900295i	$0.356649\pi$
$102$	0	0
$103$	14.0023	1.37968	0.689842	−	0.723960i	$-0.257680\pi$
$103$	14.0023	1.37968	0.689842	+	0.723960i	$0.257680\pi$
$104$	0	0
$105$	−0.230769	−0.0225207
$106$	0	0
$107$	11.6356	1.12486	0.562429	−	0.826845i	$-0.309867\pi$
$107$	11.6356	1.12486	0.562429	+	0.826845i	$0.309867\pi$
$108$	0	0
$109$	−10.4030	−0.996424	−0.498212	−	0.867055i	$-0.666010\pi$
$109$	−10.4030	−0.996424	−0.498212	+	0.867055i	$0.666010\pi$
$110$	0	0
$111$	−15.2462	−1.44711
$112$	0	0
$113$	−3.84149	−0.361377	−0.180689	−	0.983540i	$-0.557833\pi$
$113$	−3.84149	−0.361377	−0.180689	+	0.983540i	$0.557833\pi$
$114$	0	0
$115$	−1.51858	−0.141609
$116$	0	0
$117$	0.494363	0.0457039
$118$	0	0
$119$	−1.29761	−0.118952
$120$	0	0
$121$	−1.07288	−0.0975345
$122$	0	0
$123$	−3.14713	−0.283767
$124$	0	0
$125$	−7.19277	−0.643341
$126$	0	0
$127$	−9.22945	−0.818982	−0.409491	−	0.912314i	$-0.634294\pi$
$127$	−9.22945	−0.818982	−0.409491	+	0.912314i	$0.634294\pi$
$128$	0	0
$129$	15.1324	1.33233
$130$	0	0
$131$	16.7199	1.46082	0.730410	−	0.683009i	$-0.239329\pi$
$131$	16.7199	1.46082	0.730410	+	0.683009i	$0.239329\pi$
$132$	0	0
$133$	−1.28512	−0.111434
$134$	0	0
$135$	4.04398	0.348051
$136$	0	0
$137$	5.24892	0.448446	0.224223	−	0.974538i	$-0.428016\pi$
$137$	5.24892	0.448446	0.224223	+	0.974538i	$0.428016\pi$
$138$	0	0
$139$	15.8605	1.34527	0.672634	−	0.739976i	$-0.265163\pi$
$139$	15.8605	1.34527	0.672634	+	0.739976i	$0.265163\pi$
$140$	0	0
$141$	1.44402	0.121609
$142$	0	0
$143$	−13.0587	−1.09202
$144$	0	0
$145$	4.15641	0.345171
$146$	0	0
$147$	11.8271	0.975483
$148$	0	0
$149$	2.03830	0.166984	0.0834918	−	0.996508i	$-0.473393\pi$
$149$	2.03830	0.166984	0.0834918	+	0.996508i	$0.473393\pi$
$150$	0	0
$151$	−12.6466	−1.02917	−0.514583	−	0.857441i	$-0.672053\pi$
$151$	−12.6466	−1.02917	−0.514583	+	0.857441i	$0.672053\pi$
$152$	0	0
$153$	0.869523	0.0702968
$154$	0	0
$155$	−7.34509	−0.589972
$156$	0	0
$157$	−3.21459	−0.256553	−0.128276	−	0.991738i	$-0.540944\pi$
$157$	−3.21459	−0.256553	−0.128276	+	0.991738i	$0.540944\pi$
$158$	0	0
$159$	−11.5856	−0.918797
$160$	0	0
$161$	−0.353880	−0.0278897
$162$	0	0
$163$	−12.5440	−0.982520	−0.491260	−	0.871013i	$-0.663464\pi$
$163$	−12.5440	−0.982520	−0.491260	+	0.871013i	$0.663464\pi$
$164$	0	0
$165$	−4.08477	−0.317999
$166$	0	0
$167$	13.3877	1.03597	0.517987	−	0.855389i	$-0.326682\pi$
$167$	13.3877	1.03597	0.517987	+	0.855389i	$0.326682\pi$
$168$	0	0
$169$	4.17815	0.321396
$170$	0	0
$171$	0.861150	0.0658538
$172$	0	0
$173$	−7.07466	−0.537876	−0.268938	−	0.963157i	$-0.586673\pi$
$173$	−7.07466	−0.537876	−0.268938	+	0.963157i	$0.586673\pi$
$174$	0	0
$175$	−0.786148	−0.0594272
$176$	0	0
$177$	12.4704	0.937334
$178$	0	0
$179$	−20.8937	−1.56167	−0.780836	−	0.624736i	$-0.785207\pi$
$179$	−20.8937	−1.56167	−0.780836	+	0.624736i	$0.785207\pi$
$180$	0	0
$181$	−16.2179	−1.20547	−0.602733	−	0.797943i	$-0.705922\pi$
$181$	−16.2179	−1.20547	−0.602733	+	0.797943i	$0.705922\pi$
$182$	0	0
$183$	5.85682	0.432949
$184$	0	0
$185$	6.86144	0.504463
$186$	0	0
$187$	−22.9686	−1.67963
$188$	0	0
$189$	0.942381	0.0685481
$190$	0	0
$191$	20.6903	1.49710	0.748549	−	0.663080i	$-0.230751\pi$
$191$	20.6903	1.49710	0.748549	+	0.663080i	$0.230751\pi$
$192$	0	0
$193$	−12.5329	−0.902139	−0.451069	−	0.892489i	$-0.648957\pi$
$193$	−12.5329	−0.902139	−0.451069	+	0.892489i	$0.648957\pi$
$194$	0	0
$195$	5.37333	0.384792
$196$	0	0
$197$	10.5062	0.748536	0.374268	−	0.927321i	$-0.377894\pi$
$197$	10.5062	0.748536	0.374268	+	0.927321i	$0.377894\pi$
$198$	0	0
$199$	5.58500	0.395910	0.197955	−	0.980211i	$-0.436570\pi$
$199$	5.58500	0.395910	0.197955	+	0.980211i	$0.436570\pi$
$200$	0	0
$201$	14.6975	1.03668
$202$	0	0
$203$	0.968580	0.0679810
$204$	0	0
$205$	1.41634	0.0989218
$206$	0	0
$207$	0.237133	0.0164819
$208$	0	0
$209$	−22.7474	−1.57347
$210$	0	0
$211$	−26.2800	−1.80919	−0.904596	−	0.426269i	$-0.859828\pi$
$211$	−26.2800	−1.80919	−0.904596	+	0.426269i	$0.859828\pi$
$212$	0	0
$213$	−21.6197	−1.48136
$214$	0	0
$215$	−6.81021	−0.464453
$216$	0	0
$217$	−1.71165	−0.116194
$218$	0	0
$219$	−5.49970	−0.371635
$220$	0	0
$221$	30.2142	2.03243
$222$	0	0
$223$	−4.26815	−0.285817	−0.142908	−	0.989736i	$-0.545645\pi$
$223$	−4.26815	−0.285817	−0.142908	+	0.989736i	$0.545645\pi$
$224$	0	0
$225$	0.526794	0.0351196
$226$	0	0
$227$	−5.61678	−0.372799	−0.186400	−	0.982474i	$-0.559682\pi$
$227$	−5.61678	−0.372799	−0.186400	+	0.982474i	$0.559682\pi$
$228$	0	0
$229$	−9.80988	−0.648255	−0.324128	−	0.946013i	$-0.605071\pi$
$229$	−9.80988	−0.648255	−0.324128	+	0.946013i	$0.605071\pi$
$230$	0	0
$231$	−0.951884	−0.0626294
$232$	0	0
$233$	23.5681	1.54400	0.771998	−	0.635625i	$-0.219258\pi$
$233$	23.5681	1.54400	0.771998	+	0.635625i	$0.219258\pi$
$234$	0	0
$235$	−0.649872	−0.0423930
$236$	0	0
$237$	−23.9715	−1.55712
$238$	0	0
$239$	6.56240	0.424486	0.212243	−	0.977217i	$-0.431923\pi$
$239$	6.56240	0.424486	0.212243	+	0.977217i	$0.431923\pi$
$240$	0	0
$241$	−12.2713	−0.790462	−0.395231	−	0.918582i	$-0.629335\pi$
$241$	−12.2713	−0.790462	−0.395231	+	0.918582i	$0.629335\pi$
$242$	0	0
$243$	−1.23882	−0.0794705
$244$	0	0
$245$	−5.32271	−0.340055
$246$	0	0
$247$	29.9233	1.90397
$248$	0	0
$249$	−10.5942	−0.671381
$250$	0	0
$251$	−3.95799	−0.249826	−0.124913	−	0.992168i	$-0.539865\pi$
$251$	−3.95799	−0.249826	−0.124913	+	0.992168i	$0.539865\pi$
$252$	0	0
$253$	−6.26392	−0.393809
$254$	0	0
$255$	9.45101	0.591845
$256$	0	0
$257$	−7.46970	−0.465947	−0.232974	−	0.972483i	$-0.574846\pi$
$257$	−7.46970	−0.465947	−0.232974	+	0.972483i	$0.574846\pi$
$258$	0	0
$259$	1.59894	0.0993534
$260$	0	0
$261$	−0.649041	−0.0401746
$262$	0	0
$263$	14.4773	0.892707	0.446354	−	0.894857i	$-0.352722\pi$
$263$	14.4773	0.892707	0.446354	+	0.894857i	$0.352722\pi$
$264$	0	0
$265$	5.21401	0.320294
$266$	0	0
$267$	−3.16228	−0.193529
$268$	0	0
$269$	−13.3061	−0.811286	−0.405643	−	0.914032i	$-0.632952\pi$
$269$	−13.3061	−0.811286	−0.405643	+	0.914032i	$0.632952\pi$
$270$	0	0
$271$	3.69517	0.224466	0.112233	−	0.993682i	$-0.464200\pi$
$271$	3.69517	0.224466	0.112233	+	0.993682i	$0.464200\pi$
$272$	0	0
$273$	1.25216	0.0757843
$274$	0	0
$275$	−13.9153	−0.839127
$276$	0	0
$277$	24.3011	1.46011	0.730055	−	0.683389i	$-0.239494\pi$
$277$	24.3011	1.46011	0.730055	+	0.683389i	$0.239494\pi$
$278$	0	0
$279$	1.14697	0.0686671
$280$	0	0
$281$	18.2514	1.08878	0.544392	−	0.838831i	$-0.316760\pi$
$281$	18.2514	1.08878	0.544392	+	0.838831i	$0.316760\pi$
$282$	0	0
$283$	6.09607	0.362374	0.181187	−	0.983449i	$-0.442006\pi$
$283$	6.09607	0.362374	0.181187	+	0.983449i	$0.442006\pi$
$284$	0	0
$285$	9.36000	0.554439
$286$	0	0
$287$	0.330055	0.0194825
$288$	0	0
$289$	36.1430	2.12606
$290$	0	0
$291$	−26.2317	−1.53773
$292$	0	0
$293$	−24.3564	−1.42291	−0.711457	−	0.702729i	$-0.751964\pi$
$293$	−24.3564	−1.42291	−0.711457	+	0.702729i	$0.751964\pi$
$294$	0	0
$295$	−5.61223	−0.326756
$296$	0	0
$297$	16.6808	0.967917
$298$	0	0
$299$	8.23991	0.476526
$300$	0	0
$301$	−1.58700	−0.0914733
$302$	0	0
$303$	−14.8495	−0.853079
$304$	0	0
$305$	−2.63582	−0.150927
$306$	0	0
$307$	−24.8534	−1.41846	−0.709231	−	0.704977i	$-0.750957\pi$
$307$	−24.8534	−1.41846	−0.709231	+	0.704977i	$0.750957\pi$
$308$	0	0
$309$	−23.7656	−1.35198
$310$	0	0
$311$	−19.2221	−1.08999	−0.544994	−	0.838440i	$-0.683468\pi$
$311$	−19.2221	−1.08999	−0.544994	+	0.838440i	$0.683468\pi$
$312$	0	0
$313$	14.2371	0.804730	0.402365	−	0.915479i	$-0.368188\pi$
$313$	14.2371	0.804730	0.402365	+	0.915479i	$0.368188\pi$
$314$	0	0
$315$	−0.0162175	−0.000913754 0
$316$	0	0
$317$	26.2691	1.47542	0.737710	−	0.675117i	$-0.235907\pi$
$317$	26.2691	1.47542	0.737710	+	0.675117i	$0.235907\pi$
$318$	0	0
$319$	17.1445	0.959909
$320$	0	0
$321$	−19.7488	−1.10227
$322$	0	0
$323$	52.6312	2.92848
$324$	0	0
$325$	18.3050	1.01538
$326$	0	0
$327$	17.6566	0.976415
$328$	0	0
$329$	−0.151441	−0.00834924
$330$	0	0
$331$	13.4210	0.737684	0.368842	−	0.929492i	$-0.379754\pi$
$331$	13.4210	0.737684	0.368842	+	0.929492i	$0.379754\pi$
$332$	0	0
$333$	−1.07144	−0.0587147
$334$	0	0
$335$	−6.61452	−0.361390
$336$	0	0
$337$	19.4548	1.05977	0.529886	−	0.848069i	$-0.322235\pi$
$337$	19.4548	1.05977	0.529886	+	0.848069i	$0.322235\pi$
$338$	0	0
$339$	6.52005	0.354120
$340$	0	0
$341$	−30.2973	−1.64069
$342$	0	0
$343$	−2.48637	−0.134251
$344$	0	0
$345$	2.57745	0.138765
$346$	0	0
$347$	−22.0398	−1.18316	−0.591580	−	0.806246i	$-0.701496\pi$
$347$	−22.0398	−1.18316	−0.591580	+	0.806246i	$0.701496\pi$
$348$	0	0
$349$	7.40079	0.396155	0.198078	−	0.980186i	$-0.436530\pi$
$349$	7.40079	0.396155	0.198078	+	0.980186i	$0.436530\pi$
$350$	0	0
$351$	−21.9428	−1.17122
$352$	0	0
$353$	1.96245	0.104451	0.0522255	−	0.998635i	$-0.483369\pi$
$353$	1.96245	0.104451	0.0522255	+	0.998635i	$0.483369\pi$
$354$	0	0
$355$	9.72981	0.516405
$356$	0	0
$357$	2.20240	0.116563
$358$	0	0
$359$	−10.3304	−0.545216	−0.272608	−	0.962125i	$-0.587886\pi$
$359$	−10.3304	−0.545216	−0.272608	+	0.962125i	$0.587886\pi$
$360$	0	0
$361$	33.1244	1.74339
$362$	0	0
$363$	1.82096	0.0955759
$364$	0	0
$365$	2.47510	0.129553
$366$	0	0
$367$	12.5048	0.652746	0.326373	−	0.945241i	$-0.394173\pi$
$367$	12.5048	0.652746	0.326373	+	0.945241i	$0.394173\pi$
$368$	0	0
$369$	−0.221168	−0.0115135
$370$	0	0
$371$	1.21504	0.0630815
$372$	0	0
$373$	−18.9839	−0.982947	−0.491473	−	0.870893i	$-0.663541\pi$
$373$	−18.9839	−0.982947	−0.491473	+	0.870893i	$0.663541\pi$
$374$	0	0
$375$	12.2081	0.630422
$376$	0	0
$377$	−22.5529	−1.16153
$378$	0	0
$379$	−13.3520	−0.685845	−0.342922	−	0.939364i	$-0.611417\pi$
$379$	−13.3520	−0.685845	−0.342922	+	0.939364i	$0.611417\pi$
$380$	0	0
$381$	15.6649	0.802535
$382$	0	0
$383$	−8.76007	−0.447619	−0.223809	−	0.974633i	$-0.571849\pi$
$383$	−8.76007	−0.447619	−0.223809	+	0.974633i	$0.571849\pi$
$384$	0	0
$385$	0.428389	0.0218327
$386$	0	0
$387$	1.06344	0.0540578
$388$	0	0
$389$	−5.65881	−0.286913	−0.143457	−	0.989657i	$-0.545822\pi$
$389$	−5.65881	−0.286913	−0.143457	+	0.989657i	$0.545822\pi$
$390$	0	0
$391$	14.4930	0.732941
$392$	0	0
$393$	−28.3781	−1.43149
$394$	0	0
$395$	10.7882	0.542813
$396$	0	0
$397$	35.6924	1.79135	0.895675	−	0.444709i	$-0.146693\pi$
$397$	35.6924	1.79135	0.895675	+	0.444709i	$0.146693\pi$
$398$	0	0
$399$	2.18119	0.109196
$400$	0	0
$401$	19.8815	0.992835	0.496417	−	0.868084i	$-0.334649\pi$
$401$	19.8815	0.992835	0.496417	+	0.868084i	$0.334649\pi$
$402$	0	0
$403$	39.8548	1.98531
$404$	0	0
$405$	−6.59040	−0.327480
$406$	0	0
$407$	28.3024	1.40290
$408$	0	0
$409$	24.2109	1.19715	0.598576	−	0.801066i	$-0.295733\pi$
$409$	24.2109	1.19715	0.598576	+	0.801066i	$0.295733\pi$
$410$	0	0
$411$	−8.90883	−0.439440
$412$	0	0
$413$	−1.30783	−0.0643542
$414$	0	0
$415$	4.76785	0.234045
$416$	0	0
$417$	−26.9195	−1.31825
$418$	0	0
$419$	19.8580	0.970125	0.485062	−	0.874480i	$-0.338797\pi$
$419$	19.8580	0.970125	0.485062	+	0.874480i	$0.338797\pi$
$420$	0	0
$421$	17.6819	0.861762	0.430881	−	0.902409i	$-0.358203\pi$
$421$	17.6819	0.861762	0.430881	+	0.902409i	$0.358203\pi$
$422$	0	0
$423$	0.101480	0.00493413
$424$	0	0
$425$	32.1962	1.56175
$426$	0	0
$427$	−0.614233	−0.0297248
$428$	0	0
$429$	22.1641	1.07009
$430$	0	0
$431$	39.9460	1.92413	0.962065	−	0.272820i	$-0.0879565\pi$
$431$	39.9460	1.92413	0.962065	+	0.272820i	$0.0879565\pi$
$432$	0	0
$433$	−1.83609	−0.0882370	−0.0441185	−	0.999026i	$-0.514048\pi$
$433$	−1.83609	−0.0882370	−0.0441185	+	0.999026i	$0.514048\pi$
$434$	0	0
$435$	−7.05455	−0.338240
$436$	0	0
$437$	14.3534	0.686617
$438$	0	0
$439$	−5.95524	−0.284228	−0.142114	−	0.989850i	$-0.545390\pi$
$439$	−5.95524	−0.284228	−0.142114	+	0.989850i	$0.545390\pi$
$440$	0	0
$441$	0.831163	0.0395792
$442$	0	0
$443$	30.5541	1.45167	0.725835	−	0.687869i	$-0.241454\pi$
$443$	30.5541	1.45167	0.725835	+	0.687869i	$0.241454\pi$
$444$	0	0
$445$	1.42316	0.0674644
$446$	0	0
$447$	−3.45954	−0.163630
$448$	0	0
$449$	−12.9240	−0.609923	−0.304962	−	0.952365i	$-0.598644\pi$
$449$	−12.9240	−0.609923	−0.304962	+	0.952365i	$0.598644\pi$
$450$	0	0
$451$	5.84219	0.275098
$452$	0	0
$453$	21.4647	1.00850
$454$	0	0
$455$	−0.563527	−0.0264185
$456$	0	0
$457$	26.1772	1.22452	0.612259	−	0.790657i	$-0.290261\pi$
$457$	26.1772	1.22452	0.612259	+	0.790657i	$0.290261\pi$
$458$	0	0
$459$	−38.5947	−1.80145
$460$	0	0
$461$	−7.13865	−0.332480	−0.166240	−	0.986085i	$-0.553163\pi$
$461$	−7.13865	−0.332480	−0.166240	+	0.986085i	$0.553163\pi$
$462$	0	0
$463$	−12.3558	−0.574223	−0.287111	−	0.957897i	$-0.592695\pi$
$463$	−12.3558	−0.574223	−0.287111	+	0.957897i	$0.592695\pi$
$464$	0	0
$465$	12.4666	0.578124
$466$	0	0
$467$	26.7335	1.23708	0.618540	−	0.785753i	$-0.287725\pi$
$467$	26.7335	1.23708	0.618540	+	0.785753i	$0.287725\pi$
$468$	0	0
$469$	−1.54140	−0.0711752
$470$	0	0
$471$	5.45603	0.251401
$472$	0	0
$473$	−28.0910	−1.29163
$474$	0	0
$475$	31.8862	1.46304
$476$	0	0
$477$	−0.814189	−0.0372792
$478$	0	0
$479$	31.2813	1.42928	0.714640	−	0.699492i	$-0.246590\pi$
$479$	31.2813	1.42928	0.714640	+	0.699492i	$0.246590\pi$
$480$	0	0
$481$	−37.2305	−1.69757
$482$	0	0
$483$	0.600630	0.0273296
$484$	0	0
$485$	11.8054	0.536056
$486$	0	0
$487$	−1.00551	−0.0455639	−0.0227819	−	0.999740i	$-0.507252\pi$
$487$	−1.00551	−0.0455639	−0.0227819	+	0.999740i	$0.507252\pi$
$488$	0	0
$489$	21.2905	0.962790
$490$	0	0
$491$	25.1974	1.13714	0.568572	−	0.822633i	$-0.307496\pi$
$491$	25.1974	1.13714	0.568572	+	0.822633i	$0.307496\pi$
$492$	0	0
$493$	−39.6677	−1.78654
$494$	0	0
$495$	−0.287061	−0.0129024
$496$	0	0
$497$	2.26736	0.101705
$498$	0	0
$499$	−15.0752	−0.674860	−0.337430	−	0.941351i	$-0.609558\pi$
$499$	−15.0752	−0.674860	−0.337430	+	0.941351i	$0.609558\pi$
$500$	0	0
$501$	−22.7226	−1.01517
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	6.68289	0.297385
$506$	0	0
$507$	−7.09144	−0.314942
$508$	0	0
$509$	−3.23616	−0.143440	−0.0717201	−	0.997425i	$-0.522849\pi$
$509$	−3.23616	−0.143440	−0.0717201	+	0.997425i	$0.522849\pi$
$510$	0	0
$511$	0.576780	0.0255152
$512$	0	0
$513$	−38.2231	−1.68759
$514$	0	0
$515$	10.6955	0.471302
$516$	0	0
$517$	−2.68062	−0.117893
$518$	0	0
$519$	12.0076	0.527075
$520$	0	0
$521$	−37.3233	−1.63517	−0.817583	−	0.575811i	$-0.804686\pi$
$521$	−37.3233	−1.63517	−0.817583	+	0.575811i	$0.804686\pi$
$522$	0	0
$523$	21.8970	0.957490	0.478745	−	0.877954i	$-0.341092\pi$
$523$	21.8970	0.957490	0.478745	+	0.877954i	$0.341092\pi$
$524$	0	0
$525$	1.33430	0.0582338
$526$	0	0
$527$	70.0996	3.05358
$528$	0	0
$529$	−19.0475	−0.828153
$530$	0	0
$531$	0.876372	0.0380313
$532$	0	0
$533$	−7.68515	−0.332881
$534$	0	0
$535$	8.88781	0.384253
$536$	0	0
$537$	35.4623	1.53031
$538$	0	0
$539$	−21.9553	−0.945682
$540$	0	0
$541$	−8.09760	−0.348143	−0.174072	−	0.984733i	$-0.555692\pi$
$541$	−8.09760	−0.348143	−0.174072	+	0.984733i	$0.555692\pi$
$542$	0	0
$543$	27.5261	1.18126
$544$	0	0
$545$	−7.94625	−0.340380
$546$	0	0
$547$	4.88452	0.208847	0.104424	−	0.994533i	$-0.466700\pi$
$547$	4.88452	0.208847	0.104424	+	0.994533i	$0.466700\pi$
$548$	0	0
$549$	0.411595	0.0175664
$550$	0	0
$551$	−39.2857	−1.67363
$552$	0	0
$553$	2.51400	0.106906
$554$	0	0
$555$	−11.6457	−0.494333
$556$	0	0
$557$	20.7330	0.878483	0.439242	−	0.898369i	$-0.355247\pi$
$557$	20.7330	0.878483	0.439242	+	0.898369i	$0.355247\pi$
$558$	0	0
$559$	36.9525	1.56293
$560$	0	0
$561$	38.9839	1.64590
$562$	0	0
$563$	32.8686	1.38525	0.692624	−	0.721299i	$-0.256455\pi$
$563$	32.8686	1.38525	0.692624	+	0.721299i	$0.256455\pi$
$564$	0	0
$565$	−2.93430	−0.123447
$566$	0	0
$567$	−1.53578	−0.0644967
$568$	0	0
$569$	−30.1141	−1.26245	−0.631224	−	0.775601i	$-0.717447\pi$
$569$	−30.1141	−1.26245	−0.631224	+	0.775601i	$0.717447\pi$
$570$	0	0
$571$	−4.76186	−0.199277	−0.0996387	−	0.995024i	$-0.531769\pi$
$571$	−4.76186	−0.199277	−0.0996387	+	0.995024i	$0.531769\pi$
$572$	0	0
$573$	−35.1170	−1.46703
$574$	0	0
$575$	8.78045	0.366170
$576$	0	0
$577$	−16.6419	−0.692812	−0.346406	−	0.938085i	$-0.612598\pi$
$577$	−16.6419	−0.692812	−0.346406	+	0.938085i	$0.612598\pi$
$578$	0	0
$579$	21.2717	0.884023
$580$	0	0
$581$	1.11107	0.0460948
$582$	0	0
$583$	21.5070	0.890727
$584$	0	0
$585$	0.377617	0.0156125
$586$	0	0
$587$	−24.1899	−0.998422	−0.499211	−	0.866480i	$-0.666377\pi$
$587$	−24.1899	−0.998422	−0.499211	+	0.866480i	$0.666377\pi$
$588$	0	0
$589$	69.4245	2.86059
$590$	0	0
$591$	−17.8319	−0.733504
$592$	0	0
$593$	−32.5555	−1.33689	−0.668447	−	0.743760i	$-0.733041\pi$
$593$	−32.5555	−1.33689	−0.668447	+	0.743760i	$0.733041\pi$
$594$	0	0
$595$	−0.991173	−0.0406341
$596$	0	0
$597$	−9.47925	−0.387960
$598$	0	0
$599$	6.73100	0.275021	0.137511	−	0.990500i	$-0.456090\pi$
$599$	6.73100	0.275021	0.137511	+	0.990500i	$0.456090\pi$
$600$	0	0
$601$	0.824904	0.0336485	0.0168243	−	0.999858i	$-0.494644\pi$
$601$	0.824904	0.0336485	0.0168243	+	0.999858i	$0.494644\pi$
$602$	0	0
$603$	1.03288	0.0420623
$604$	0	0
$605$	−0.819513	−0.0333179
$606$	0	0
$607$	17.1831	0.697442	0.348721	−	0.937227i	$-0.386616\pi$
$607$	17.1831	0.697442	0.348721	+	0.937227i	$0.386616\pi$
$608$	0	0
$609$	−1.64394	−0.0666159
$610$	0	0
$611$	3.52623	0.142656
$612$	0	0
$613$	8.47269	0.342209	0.171104	−	0.985253i	$-0.445266\pi$
$613$	8.47269	0.342209	0.171104	+	0.985253i	$0.445266\pi$
$614$	0	0
$615$	−2.40392	−0.0969353
$616$	0	0
$617$	13.7189	0.552300	0.276150	−	0.961115i	$-0.410941\pi$
$617$	13.7189	0.552300	0.276150	+	0.961115i	$0.410941\pi$
$618$	0	0
$619$	−23.9875	−0.964137	−0.482069	−	0.876133i	$-0.660114\pi$
$619$	−23.9875	−0.964137	−0.482069	+	0.876133i	$0.660114\pi$
$620$	0	0
$621$	−10.5254	−0.422370
$622$	0	0
$623$	0.331644	0.0132870
$624$	0	0
$625$	16.5886	0.663542
$626$	0	0
$627$	38.6085	1.54188
$628$	0	0
$629$	−65.4838	−2.61101
$630$	0	0
$631$	−24.5121	−0.975810	−0.487905	−	0.872897i	$-0.662239\pi$
$631$	−24.5121	−0.975810	−0.487905	+	0.872897i	$0.662239\pi$
$632$	0	0
$633$	44.6043	1.77286
$634$	0	0
$635$	−7.04986	−0.279765
$636$	0	0
$637$	28.8813	1.14432
$638$	0	0
$639$	−1.51935	−0.0601046
$640$	0	0
$641$	16.8124	0.664049	0.332024	−	0.943271i	$-0.392268\pi$
$641$	16.8124	0.664049	0.332024	+	0.943271i	$0.392268\pi$
$642$	0	0
$643$	−32.1501	−1.26788	−0.633938	−	0.773383i	$-0.718563\pi$
$643$	−32.1501	−1.26788	−0.633938	+	0.773383i	$0.718563\pi$
$644$	0	0
$645$	11.5588	0.455126
$646$	0	0
$647$	5.93065	0.233158	0.116579	−	0.993181i	$-0.462807\pi$
$647$	5.93065	0.233158	0.116579	+	0.993181i	$0.462807\pi$
$648$	0	0
$649$	−23.1495	−0.908698
$650$	0	0
$651$	2.90512	0.113861
$652$	0	0
$653$	17.7681	0.695321	0.347660	−	0.937621i	$-0.386976\pi$
$653$	17.7681	0.695321	0.347660	+	0.937621i	$0.386976\pi$
$654$	0	0
$655$	12.7714	0.499018
$656$	0	0
$657$	−0.386497	−0.0150787
$658$	0	0
$659$	2.56871	0.100063	0.0500313	−	0.998748i	$-0.484068\pi$
$659$	2.56871	0.100063	0.0500313	+	0.998748i	$0.484068\pi$
$660$	0	0
$661$	21.0135	0.817329	0.408664	−	0.912685i	$-0.365995\pi$
$661$	21.0135	0.817329	0.408664	+	0.912685i	$0.365995\pi$
$662$	0	0
$663$	−51.2816	−1.99161
$664$	0	0
$665$	−0.981628	−0.0380659
$666$	0	0
$667$	−10.8180	−0.418876
$668$	0	0
$669$	7.24421	0.280077
$670$	0	0
$671$	−10.8723	−0.419722
$672$	0	0
$673$	26.9992	1.04074	0.520371	−	0.853940i	$-0.325794\pi$
$673$	26.9992	1.04074	0.520371	+	0.853940i	$0.325794\pi$
$674$	0	0
$675$	−23.3823	−0.899985
$676$	0	0
$677$	−15.0981	−0.580265	−0.290133	−	0.956986i	$-0.593699\pi$
$677$	−15.0981	−0.580265	−0.290133	+	0.956986i	$0.593699\pi$
$678$	0	0
$679$	2.75105	0.105575
$680$	0	0
$681$	9.53319	0.365313
$682$	0	0
$683$	10.7833	0.412613	0.206306	−	0.978487i	$-0.433856\pi$
$683$	10.7833	0.412613	0.206306	+	0.978487i	$0.433856\pi$
$684$	0	0
$685$	4.00936	0.153190
$686$	0	0
$687$	16.6500	0.635237
$688$	0	0
$689$	−28.2915	−1.07782
$690$	0	0
$691$	−30.5909	−1.16373	−0.581866	−	0.813285i	$-0.697677\pi$
$691$	−30.5909	−1.16373	−0.581866	+	0.813285i	$0.697677\pi$
$692$	0	0
$693$	−0.0668947	−0.00254112
$694$	0	0
$695$	12.1149	0.459545
$696$	0	0
$697$	−13.5172	−0.512001
$698$	0	0
$699$	−40.0014	−1.51299
$700$	0	0
$701$	8.04569	0.303882	0.151941	−	0.988390i	$-0.451448\pi$
$701$	8.04569	0.303882	0.151941	+	0.988390i	$0.451448\pi$
$702$	0	0
$703$	−64.8532	−2.44599
$704$	0	0
$705$	1.10301	0.0415417
$706$	0	0
$707$	1.55733	0.0585695
$708$	0	0
$709$	−38.3346	−1.43968	−0.719842	−	0.694138i	$-0.755786\pi$
$709$	−38.3346	−1.43968	−0.719842	+	0.694138i	$0.755786\pi$
$710$	0	0
$711$	−1.68462	−0.0631783
$712$	0	0
$713$	19.1173	0.715949
$714$	0	0
$715$	−9.97481	−0.373037
$716$	0	0
$717$	−11.1382	−0.415962
$718$	0	0
$719$	−51.7385	−1.92952	−0.964760	−	0.263132i	$-0.915245\pi$
$719$	−51.7385	−1.92952	−0.964760	+	0.263132i	$0.915245\pi$
$720$	0	0
$721$	2.49241	0.0928223
$722$	0	0
$723$	20.8276	0.774588
$724$	0	0
$725$	−24.0323	−0.892539
$726$	0	0
$727$	16.6380	0.617068	0.308534	−	0.951213i	$-0.400162\pi$
$727$	16.6380	0.617068	0.308534	+	0.951213i	$0.400162\pi$
$728$	0	0
$729$	27.9864	1.03653
$730$	0	0
$731$	64.9948	2.40392
$732$	0	0
$733$	4.67080	0.172520	0.0862600	−	0.996273i	$-0.472508\pi$
$733$	4.67080	0.172520	0.0862600	+	0.996273i	$0.472508\pi$
$734$	0	0
$735$	9.03407	0.333227
$736$	0	0
$737$	−27.2838	−1.00501
$738$	0	0
$739$	22.8042	0.838865	0.419432	−	0.907787i	$-0.362229\pi$
$739$	22.8042	0.838865	0.419432	+	0.907787i	$0.362229\pi$
$740$	0	0
$741$	−50.7878	−1.86574
$742$	0	0
$743$	−2.20048	−0.0807276	−0.0403638	−	0.999185i	$-0.512852\pi$
$743$	−2.20048	−0.0807276	−0.0403638	+	0.999185i	$0.512852\pi$
$744$	0	0
$745$	1.55694	0.0570419
$746$	0	0
$747$	−0.744520	−0.0272405
$748$	0	0
$749$	2.07115	0.0756782
$750$	0	0
$751$	−37.3032	−1.36121	−0.680607	−	0.732648i	$-0.738284\pi$
$751$	−37.3032	−1.36121	−0.680607	+	0.732648i	$0.738284\pi$
$752$	0	0
$753$	6.71777	0.244809
$754$	0	0
$755$	−9.66004	−0.351565
$756$	0	0
$757$	46.1609	1.67775	0.838873	−	0.544327i	$-0.183215\pi$
$757$	46.1609	1.67775	0.838873	+	0.544327i	$0.183215\pi$
$758$	0	0
$759$	10.6316	0.385901
$760$	0	0
$761$	−46.6735	−1.69191	−0.845956	−	0.533253i	$-0.820969\pi$
$761$	−46.6735	−1.69191	−0.845956	+	0.533253i	$0.820969\pi$
$762$	0	0
$763$	−1.85174	−0.0670374
$764$	0	0
$765$	0.664180	0.0240135
$766$	0	0
$767$	30.4522	1.09956
$768$	0	0
$769$	18.1370	0.654037	0.327019	−	0.945018i	$-0.393956\pi$
$769$	18.1370	0.654037	0.327019	+	0.945018i	$0.393956\pi$
$770$	0	0
$771$	12.6781	0.456590
$772$	0	0
$773$	5.52823	0.198837	0.0994183	−	0.995046i	$-0.468302\pi$
$773$	5.52823	0.198837	0.0994183	+	0.995046i	$0.468302\pi$
$774$	0	0
$775$	42.4693	1.52554
$776$	0	0
$777$	−2.71383	−0.0973583
$778$	0	0
$779$	−13.3870	−0.479641
$780$	0	0
$781$	40.1339	1.43610
$782$	0	0
$783$	28.8084	1.02953
$784$	0	0
$785$	−2.45545	−0.0876387
$786$	0	0
$787$	−35.1593	−1.25329	−0.626647	−	0.779304i	$-0.715573\pi$
$787$	−35.1593	−1.25329	−0.626647	+	0.779304i	$0.715573\pi$
$788$	0	0
$789$	−24.5718	−0.874780
$790$	0	0
$791$	−0.683788	−0.0243127
$792$	0	0
$793$	14.3021	0.507882
$794$	0	0
$795$	−8.84958	−0.313862
$796$	0	0
$797$	22.6479	0.802228	0.401114	−	0.916028i	$-0.368623\pi$
$797$	22.6479	0.802228	0.401114	+	0.916028i	$0.368623\pi$
$798$	0	0
$799$	6.20220	0.219418
$800$	0	0
$801$	−0.222233	−0.00785221
$802$	0	0
$803$	10.2094	0.360282
$804$	0	0
$805$	−0.270309	−0.00952715
$806$	0	0
$807$	22.5840	0.794995
$808$	0	0
$809$	32.2040	1.13223	0.566117	−	0.824325i	$-0.308445\pi$
$809$	32.2040	1.13223	0.566117	+	0.824325i	$0.308445\pi$
$810$	0	0
$811$	−14.9318	−0.524327	−0.262163	−	0.965023i	$-0.584436\pi$
$811$	−14.9318	−0.524327	−0.262163	+	0.965023i	$0.584436\pi$
$812$	0	0
$813$	−6.27170	−0.219958
$814$	0	0
$815$	−9.58164	−0.335630
$816$	0	0
$817$	64.3690	2.25199
$818$	0	0
$819$	0.0879970	0.00307487
$820$	0	0
$821$	21.4585	0.748907	0.374454	−	0.927246i	$-0.377830\pi$
$821$	21.4585	0.748907	0.374454	+	0.927246i	$0.377830\pi$
$822$	0	0
$823$	−30.1071	−1.04947	−0.524734	−	0.851266i	$-0.675835\pi$
$823$	−30.1071	−1.04947	−0.524734	+	0.851266i	$0.675835\pi$
$824$	0	0
$825$	23.6181	0.822276
$826$	0	0
$827$	−45.5737	−1.58475	−0.792376	−	0.610033i	$-0.791156\pi$
$827$	−45.5737	−1.58475	−0.792376	+	0.610033i	$0.791156\pi$
$828$	0	0
$829$	32.4221	1.12607	0.563033	−	0.826435i	$-0.309635\pi$
$829$	32.4221	1.12607	0.563033	+	0.826435i	$0.309635\pi$
$830$	0	0
$831$	−41.2455	−1.43079
$832$	0	0
$833$	50.7985	1.76006
$834$	0	0
$835$	10.2261	0.353890
$836$	0	0
$837$	−50.9093	−1.75968
$838$	0	0
$839$	10.0488	0.346922	0.173461	−	0.984841i	$-0.444505\pi$
$839$	10.0488	0.346922	0.173461	+	0.984841i	$0.444505\pi$
$840$	0	0
$841$	0.609268	0.0210093
$842$	0	0
$843$	−30.9775	−1.06692
$844$	0	0
$845$	3.19145	0.109789
$846$	0	0
$847$	−0.190973	−0.00656192
$848$	0	0
$849$	−10.3467	−0.355097
$850$	0	0
$851$	−17.8585	−0.612182
$852$	0	0
$853$	−42.8684	−1.46779	−0.733893	−	0.679265i	$-0.762299\pi$
$853$	−42.8684	−1.46779	−0.733893	+	0.679265i	$0.762299\pi$
$854$	0	0
$855$	0.657784	0.0224957
$856$	0	0
$857$	26.8378	0.916760	0.458380	−	0.888756i	$-0.348430\pi$
$857$	26.8378	0.916760	0.458380	+	0.888756i	$0.348430\pi$
$858$	0	0
$859$	31.3928	1.07111	0.535554	−	0.844501i	$-0.320103\pi$
$859$	31.3928	1.07111	0.535554	+	0.844501i	$0.320103\pi$
$860$	0	0
$861$	−0.560191	−0.0190913
$862$	0	0
$863$	45.5609	1.55091	0.775456	−	0.631402i	$-0.217520\pi$
$863$	45.5609	1.55091	0.775456	+	0.631402i	$0.217520\pi$
$864$	0	0
$865$	−5.40394	−0.183739
$866$	0	0
$867$	−61.3443	−2.08336
$868$	0	0
$869$	44.4996	1.50955
$870$	0	0
$871$	35.8907	1.21611
$872$	0	0
$873$	−1.84346	−0.0623918
$874$	0	0
$875$	−1.28032	−0.0432826
$876$	0	0
$877$	−34.7371	−1.17299	−0.586494	−	0.809954i	$-0.699492\pi$
$877$	−34.7371	−1.17299	−0.586494	+	0.809954i	$0.699492\pi$
$878$	0	0
$879$	41.3393	1.39434
$880$	0	0
$881$	−40.4516	−1.36285	−0.681425	−	0.731888i	$-0.738639\pi$
$881$	−40.4516	−1.36285	−0.681425	+	0.731888i	$0.738639\pi$
$882$	0	0
$883$	−12.1561	−0.409087	−0.204543	−	0.978858i	$-0.565571\pi$
$883$	−12.1561	−0.409087	−0.204543	+	0.978858i	$0.565571\pi$
$884$	0	0
$885$	9.52546	0.320195
$886$	0	0
$887$	−0.586469	−0.0196917	−0.00984586	−	0.999952i	$-0.503134\pi$
$887$	−0.586469	−0.0196917	−0.00984586	+	0.999952i	$0.503134\pi$
$888$	0	0
$889$	−1.64285	−0.0550994
$890$	0	0
$891$	−27.1843	−0.910710
$892$	0	0
$893$	6.14248	0.205550
$894$	0	0
$895$	−15.9596	−0.533469
$896$	0	0
$897$	−13.9853	−0.466957
$898$	0	0
$899$	−52.3246	−1.74512
$900$	0	0
$901$	−49.7611	−1.65778
$902$	0	0
$903$	2.69357	0.0896364
$904$	0	0
$905$	−12.3879	−0.411789
$906$	0	0
$907$	58.1545	1.93099	0.965494	−	0.260425i	$-0.0838627\pi$
$907$	58.1545	1.93099	0.965494	+	0.260425i	$0.0838627\pi$
$908$	0	0
$909$	−1.04356	−0.0346127
$910$	0	0
$911$	5.52664	0.183106	0.0915529	−	0.995800i	$-0.470817\pi$
$911$	5.52664	0.183106	0.0915529	+	0.995800i	$0.470817\pi$
$912$	0	0
$913$	19.6666	0.650870
$914$	0	0
$915$	4.47370	0.147896
$916$	0	0
$917$	2.97615	0.0982810
$918$	0	0
$919$	15.1749	0.500575	0.250288	−	0.968172i	$-0.419475\pi$
$919$	15.1749	0.500575	0.250288	+	0.968172i	$0.419475\pi$
$920$	0	0
$921$	42.1830	1.38998
$922$	0	0
$923$	−52.7944	−1.73775
$924$	0	0
$925$	−39.6728	−1.30443
$926$	0	0
$927$	−1.67015	−0.0548550
$928$	0	0
$929$	−16.7667	−0.550097	−0.275049	−	0.961430i	$-0.588694\pi$
$929$	−16.7667	−0.550097	−0.275049	+	0.961430i	$0.588694\pi$
$930$	0	0
$931$	50.3093	1.64882
$932$	0	0
$933$	32.6252	1.06810
$934$	0	0
$935$	−17.5444	−0.573764
$936$	0	0
$937$	6.28444	0.205304	0.102652	−	0.994717i	$-0.467267\pi$
$937$	6.28444	0.205304	0.102652	+	0.994717i	$0.467267\pi$
$938$	0	0
$939$	−24.1642	−0.788570
$940$	0	0
$941$	1.10359	0.0359759	0.0179879	−	0.999838i	$-0.494274\pi$
$941$	1.10359	0.0359759	0.0179879	+	0.999838i	$0.494274\pi$
$942$	0	0
$943$	−3.68637	−0.120045
$944$	0	0
$945$	0.719832	0.0234161
$946$	0	0
$947$	−21.9848	−0.714411	−0.357205	−	0.934026i	$-0.616270\pi$
$947$	−21.9848	−0.714411	−0.357205	+	0.934026i	$0.616270\pi$
$948$	0	0
$949$	−13.4300	−0.435957
$950$	0	0
$951$	−44.5858	−1.44579
$952$	0	0
$953$	37.2811	1.20765	0.603826	−	0.797116i	$-0.293642\pi$
$953$	37.2811	1.20765	0.603826	+	0.797116i	$0.293642\pi$
$954$	0	0
$955$	15.8042	0.511411
$956$	0	0
$957$	−29.0989	−0.940633
$958$	0	0
$959$	0.934312	0.0301705
$960$	0	0
$961$	61.4666	1.98279
$962$	0	0
$963$	−1.38787	−0.0447234
$964$	0	0
$965$	−9.57319	−0.308172
$966$	0	0
$967$	−28.7115	−0.923299	−0.461650	−	0.887062i	$-0.652742\pi$
$967$	−28.7115	−0.923299	−0.461650	+	0.887062i	$0.652742\pi$
$968$	0	0
$969$	−89.3294	−2.86967
$970$	0	0
$971$	−5.60323	−0.179816	−0.0899081	−	0.995950i	$-0.528657\pi$
$971$	−5.60323	−0.179816	−0.0899081	+	0.995950i	$0.528657\pi$
$972$	0	0
$973$	2.82317	0.0905068
$974$	0	0
$975$	−31.0686	−0.994991
$976$	0	0
$977$	28.7456	0.919652	0.459826	−	0.888009i	$-0.347912\pi$
$977$	28.7456	0.919652	0.459826	+	0.888009i	$0.347912\pi$
$978$	0	0
$979$	5.87032	0.187616
$980$	0	0
$981$	1.24084	0.0396170
$982$	0	0
$983$	4.79868	0.153054	0.0765271	−	0.997068i	$-0.475617\pi$
$983$	4.79868	0.153054	0.0765271	+	0.997068i	$0.475617\pi$
$984$	0	0
$985$	8.02510	0.255701
$986$	0	0
$987$	0.257037	0.00818157
$988$	0	0
$989$	17.7252	0.563627
$990$	0	0
$991$	11.0797	0.351958	0.175979	−	0.984394i	$-0.443691\pi$
$991$	11.0797	0.351958	0.175979	+	0.984394i	$0.443691\pi$
$992$	0	0
$993$	−22.7790	−0.722870
$994$	0	0
$995$	4.26607	0.135243
$996$	0	0
$997$	15.5322	0.491909	0.245954	−	0.969281i	$-0.420899\pi$
$997$	15.5322	0.491909	0.245954	+	0.969281i	$0.420899\pi$
$998$	0	0
$999$	47.5571	1.50464

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.u.1.9		26
4.3	odd	2		503.2.a.f.1.22	✓	26
12.11	even	2		4527.2.a.o.1.5		26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.f.1.22	✓	26	4.3	odd	2
4527.2.a.o.1.5		26	12.11	even	2
8048.2.a.u.1.9		26	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-1.69727 q^{3} +0.763844 q^{5} +0.178001 q^{7} -0.119277 q^{9} +O(q^{10})\)	\(q-1.69727 q^{3} +0.763844 q^{5} +0.178001 q^{7} -0.119277 q^{9} +3.15073 q^{11} -4.14465 q^{13} -1.29645 q^{15} -7.28992 q^{17} -7.21972 q^{19} -0.302115 q^{21} -1.98808 q^{23} -4.41654 q^{25} +5.29425 q^{27} +5.44144 q^{29} -9.61595 q^{31} -5.34764 q^{33} +0.135965 q^{35} +8.98278 q^{37} +7.03459 q^{39} +1.85423 q^{41} -8.91571 q^{43} -0.0911093 q^{45} -0.850791 q^{47} -6.96832 q^{49} +12.3730 q^{51} +6.82601 q^{53} +2.40667 q^{55} +12.2538 q^{57} -7.34734 q^{59} -3.45073 q^{61} -0.0212315 q^{63} -3.16587 q^{65} -8.65951 q^{67} +3.37431 q^{69} +12.7380 q^{71} +3.24032 q^{73} +7.49606 q^{75} +0.560833 q^{77} +14.1236 q^{79} -8.62794 q^{81} +6.24192 q^{83} -5.56836 q^{85} -9.23559 q^{87} +1.86316 q^{89} -0.737751 q^{91} +16.3209 q^{93} -5.51474 q^{95} +15.4553 q^{97} -0.375811 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) \) 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) \) 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9 + 17 * q^11 + 14 * q^13 - 18 * q^15 + 17 * q^17 + 22 * q^19 - 16 * q^21 - 27 * q^23 + 93 * q^25 - 31 * q^27 + 13 * q^29 - 26 * q^31 + 6 * q^33 + 22 * q^35 + 55 * q^37 + 15 * q^39 + 24 * q^41 - 20 * q^43 - 8 * q^45 + 25 * q^47 + 65 * q^49 - 7 * q^51 + 30 * q^53 - 25 * q^55 + 9 * q^57 + 26 * q^59 + 15 * q^61 + 19 * q^63 + 20 * q^65 + 20 * q^67 - 27 * q^69 + 35 * q^71 + 38 * q^73 - 2 * q^75 - 6 * q^77 - 21 * q^79 + 70 * q^81 + 48 * q^83 + 6 * q^85 + 9 * q^87 - 5 * q^89 + 24 * q^91 - 8 * q^93 - 43 * q^95 + 142 * q^97 + 50 * q^99

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