LMFDB - Embedded newform 4023.2.a.e.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4023,2,Mod(1,4023)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4023, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4023.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4023 = 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4023.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:	$32.1238167332$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	4023.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.43306 q^{2} +0.0536673 q^{4} -2.81862 q^{5} -4.82450 q^{7} +2.78922 q^{8} +O(q^{10})$	\(q-1.43306 q^{2} +0.0536673 q^{4} -2.81862 q^{5} -4.82450 q^{7} +2.78922 q^{8} +4.03925 q^{10} -4.36775 q^{11} +4.43488 q^{13} +6.91381 q^{14} -4.10445 q^{16} -1.03366 q^{17} +3.14167 q^{19} -0.151268 q^{20} +6.25926 q^{22} +2.48378 q^{23} +2.94461 q^{25} -6.35546 q^{26} -0.258918 q^{28} -3.24413 q^{29} -7.20431 q^{31} +0.303507 q^{32} +1.48129 q^{34} +13.5984 q^{35} +2.29108 q^{37} -4.50220 q^{38} -7.86173 q^{40} -6.79046 q^{41} +9.90469 q^{43} -0.234405 q^{44} -3.55941 q^{46} +1.92072 q^{47} +16.2758 q^{49} -4.21980 q^{50} +0.238008 q^{52} +8.05990 q^{53} +12.3110 q^{55} -13.4566 q^{56} +4.64905 q^{58} +11.1428 q^{59} +2.44472 q^{61} +10.3242 q^{62} +7.77396 q^{64} -12.5002 q^{65} -0.733174 q^{67} -0.0554735 q^{68} -19.4874 q^{70} +5.15270 q^{71} +7.86030 q^{73} -3.28327 q^{74} +0.168605 q^{76} +21.0722 q^{77} +10.1300 q^{79} +11.5689 q^{80} +9.73115 q^{82} -6.30024 q^{83} +2.91348 q^{85} -14.1940 q^{86} -12.1826 q^{88} -14.0502 q^{89} -21.3961 q^{91} +0.133298 q^{92} -2.75251 q^{94} -8.85516 q^{95} +16.0458 q^{97} -23.3242 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8}+O(q^{10}) $ 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8	$ 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8} - 4 q^{10} - 12 q^{11} - 10 q^{14} + 35 q^{16} - 26 q^{17} - 30 q^{20} + 8 q^{22} - 26 q^{23} + 27 q^{25} - 16 q^{26} + 4 q^{28} - 20 q^{29} - 6 q^{31} - 49 q^{32} - 14 q^{34} - 16 q^{35} - 2 q^{37} - 27 q^{38} + 2 q^{40} - 35 q^{41} + 4 q^{43} - 22 q^{44} + 6 q^{46} - 38 q^{47} + 19 q^{49} - 22 q^{50} + 4 q^{52} - 36 q^{53} + 10 q^{55} - 79 q^{56} - 22 q^{58} - 15 q^{59} + 10 q^{61} + 14 q^{62} + 41 q^{64} - 80 q^{65} - 6 q^{67} - 33 q^{68} + 8 q^{70} - 26 q^{71} - 6 q^{73} - 75 q^{74} - 10 q^{76} - 47 q^{77} - 6 q^{79} - 66 q^{80} + 12 q^{82} - 40 q^{83} - 12 q^{85} - 4 q^{86} + 12 q^{88} - 30 q^{89} - 158 q^{92} + 18 q^{94} - 22 q^{95} - 20 q^{97} - 35 q^{98}+O(q^{100}) $ 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8 - 4 * q^10 - 12 * q^11 - 10 * q^14 + 35 * q^16 - 26 * q^17 - 30 * q^20 + 8 * q^22 - 26 * q^23 + 27 * q^25 - 16 * q^26 + 4 * q^28 - 20 * q^29 - 6 * q^31 - 49 * q^32 - 14 * q^34 - 16 * q^35 - 2 * q^37 - 27 * q^38 + 2 * q^40 - 35 * q^41 + 4 * q^43 - 22 * q^44 + 6 * q^46 - 38 * q^47 + 19 * q^49 - 22 * q^50 + 4 * q^52 - 36 * q^53 + 10 * q^55 - 79 * q^56 - 22 * q^58 - 15 * q^59 + 10 * q^61 + 14 * q^62 + 41 * q^64 - 80 * q^65 - 6 * q^67 - 33 * q^68 + 8 * q^70 - 26 * q^71 - 6 * q^73 - 75 * q^74 - 10 * q^76 - 47 * q^77 - 6 * q^79 - 66 * q^80 + 12 * q^82 - 40 * q^83 - 12 * q^85 - 4 * q^86 + 12 * q^88 - 30 * q^89 - 158 * q^92 + 18 * q^94 - 22 * q^95 - 20 * q^97 - 35 * q^98

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 $

$2$	−1.43306	−1.01333	−0.506664	−	0.862144i	$-0.669122\pi$
$2$	−1.43306	−1.01333	−0.506664	+	0.862144i	$0.669122\pi$
$3$	0	0
$3$	0	0
$4$	0.0536673	0.0268336
$5$	−2.81862	−1.26052	−0.630262	−	0.776382i	$-0.717053\pi$
$5$	−2.81862	−1.26052	−0.630262	+	0.776382i	$0.717053\pi$
$6$	0	0
$7$	−4.82450	−1.82349	−0.911745	−	0.410757i	$-0.865265\pi$
$7$	−4.82450	−1.82349	−0.911745	+	0.410757i	$0.865265\pi$
$8$	2.78922	0.986137
$9$	0	0
$10$	4.03925	1.27732
$11$	−4.36775	−1.31693	−0.658463	−	0.752613i	$-0.728793\pi$
$11$	−4.36775	−1.31693	−0.658463	+	0.752613i	$0.728793\pi$
$12$	0	0
$13$	4.43488	1.23002	0.615008	−	0.788521i	$-0.289153\pi$
$13$	4.43488	1.23002	0.615008	+	0.788521i	$0.289153\pi$
$14$	6.91381	1.84779
$15$	0	0
$16$	−4.10445	−1.02611
$17$	−1.03366	−0.250698	−0.125349	−	0.992113i	$-0.540005\pi$
$17$	−1.03366	−0.250698	−0.125349	+	0.992113i	$0.540005\pi$
$18$	0	0
$19$	3.14167	0.720748	0.360374	−	0.932808i	$-0.382649\pi$
$19$	3.14167	0.720748	0.360374	+	0.932808i	$0.382649\pi$
$20$	−0.151268	−0.0338245
$21$	0	0
$22$	6.25926	1.33448
$23$	2.48378	0.517904	0.258952	−	0.965890i	$-0.416623\pi$
$23$	2.48378	0.517904	0.258952	+	0.965890i	$0.416623\pi$
$24$	0	0
$25$	2.94461	0.588921
$26$	−6.35546	−1.24641
$27$	0	0
$28$	−0.258918	−0.0489309
$29$	−3.24413	−0.602420	−0.301210	−	0.953558i	$-0.597391\pi$
$29$	−3.24413	−0.602420	−0.301210	+	0.953558i	$0.597391\pi$
$30$	0	0
$31$	−7.20431	−1.29393	−0.646967	−	0.762518i	$-0.723963\pi$
$31$	−7.20431	−1.29393	−0.646967	+	0.762518i	$0.723963\pi$
$32$	0.303507	0.0536529
$33$	0	0
$34$	1.48129	0.254040
$35$	13.5984	2.29855
$36$	0	0
$37$	2.29108	0.376652	0.188326	−	0.982107i	$-0.439694\pi$
$37$	2.29108	0.376652	0.188326	+	0.982107i	$0.439694\pi$
$38$	−4.50220	−0.730354
$39$	0	0
$40$	−7.86173	−1.24305
$41$	−6.79046	−1.06049	−0.530246	−	0.847844i	$-0.677900\pi$
$41$	−6.79046	−1.06049	−0.530246	+	0.847844i	$0.677900\pi$
$42$	0	0
$43$	9.90469	1.51045	0.755225	−	0.655465i	$-0.227527\pi$
$43$	9.90469	1.51045	0.755225	+	0.655465i	$0.227527\pi$
$44$	−0.234405	−0.0353379
$45$	0	0
$46$	−3.55941	−0.524807
$47$	1.92072	0.280166	0.140083	−	0.990140i	$-0.455263\pi$
$47$	1.92072	0.280166	0.140083	+	0.990140i	$0.455263\pi$
$48$	0	0
$49$	16.2758	2.32512
$50$	−4.21980	−0.596770
$51$	0	0
$52$	0.238008	0.0330058
$53$	8.05990	1.10711	0.553556	−	0.832812i	$-0.313270\pi$
$53$	8.05990	1.10711	0.553556	+	0.832812i	$0.313270\pi$
$54$	0	0
$55$	12.3110	1.66002
$56$	−13.4566	−1.79821
$57$	0	0
$58$	4.64905	0.610450
$59$	11.1428	1.45067	0.725337	−	0.688394i	$-0.241684\pi$
$59$	11.1428	1.45067	0.725337	+	0.688394i	$0.241684\pi$
$60$	0	0
$61$	2.44472	0.313015	0.156507	−	0.987677i	$-0.449977\pi$
$61$	2.44472	0.313015	0.156507	+	0.987677i	$0.449977\pi$
$62$	10.3242	1.31118
$63$	0	0
$64$	7.77396	0.971746
$65$	−12.5002	−1.55046
$66$	0	0
$67$	−0.733174	−0.0895714	−0.0447857	−	0.998997i	$-0.514261\pi$
$67$	−0.733174	−0.0895714	−0.0447857	+	0.998997i	$0.514261\pi$
$68$	−0.0554735	−0.00672715
$69$	0	0
$70$	−19.4874	−2.32919
$71$	5.15270	0.611513	0.305756	−	0.952110i	$-0.401091\pi$
$71$	5.15270	0.611513	0.305756	+	0.952110i	$0.401091\pi$
$72$	0	0
$73$	7.86030	0.919978	0.459989	−	0.887925i	$-0.347853\pi$
$73$	7.86030	0.919978	0.459989	+	0.887925i	$0.347853\pi$
$74$	−3.28327	−0.381672
$75$	0	0
$76$	0.168605	0.0193403
$77$	21.0722	2.40140
$78$	0	0
$79$	10.1300	1.13971	0.569856	−	0.821745i	$-0.306999\pi$
$79$	10.1300	1.13971	0.569856	+	0.821745i	$0.306999\pi$
$80$	11.5689	1.29344
$81$	0	0
$82$	9.73115	1.07463
$83$	−6.30024	−0.691541	−0.345771	−	0.938319i	$-0.612382\pi$
$83$	−6.30024	−0.691541	−0.345771	+	0.938319i	$0.612382\pi$
$84$	0	0
$85$	2.91348	0.316011
$86$	−14.1940	−1.53058
$87$	0	0
$88$	−12.1826	−1.29867
$89$	−14.0502	−1.48932	−0.744660	−	0.667444i	$-0.767388\pi$
$89$	−14.0502	−1.48932	−0.744660	+	0.667444i	$0.767388\pi$
$90$	0	0
$91$	−21.3961	−2.24292
$92$	0.133298	0.0138973
$93$	0	0
$94$	−2.75251	−0.283900
$95$	−8.85516	−0.908520
$96$	0	0
$97$	16.0458	1.62920	0.814602	−	0.580020i	$-0.196955\pi$
$97$	16.0458	1.62920	0.814602	+	0.580020i	$0.196955\pi$
$98$	−23.3242	−2.35610
$99$	0	0
$100$	0.158029	0.0158029
$101$	−10.4253	−1.03736	−0.518679	−	0.854969i	$-0.673576\pi$
$101$	−10.4253	−1.03736	−0.518679	+	0.854969i	$0.673576\pi$
$102$	0	0
$103$	14.6142	1.43998	0.719988	−	0.693986i	$-0.244147\pi$
$103$	14.6142	1.43998	0.719988	+	0.693986i	$0.244147\pi$
$104$	12.3698	1.21296
$105$	0	0
$106$	−11.5503	−1.12187
$107$	−8.40162	−0.812216	−0.406108	−	0.913825i	$-0.633114\pi$
$107$	−8.40162	−0.812216	−0.406108	+	0.913825i	$0.633114\pi$
$108$	0	0
$109$	2.55795	0.245007	0.122503	−	0.992468i	$-0.460908\pi$
$109$	2.55795	0.245007	0.122503	+	0.992468i	$0.460908\pi$
$110$	−17.6425	−1.68214
$111$	0	0
$112$	19.8019	1.87111
$113$	−13.0546	−1.22807	−0.614035	−	0.789279i	$-0.710455\pi$
$113$	−13.0546	−1.22807	−0.614035	+	0.789279i	$0.710455\pi$
$114$	0	0
$115$	−7.00083	−0.652831
$116$	−0.174104	−0.0161651
$117$	0	0
$118$	−15.9684	−1.47001
$119$	4.98687	0.457146
$120$	0	0
$121$	8.07724	0.734294
$122$	−3.50344	−0.317186
$123$	0	0
$124$	−0.386636	−0.0347209
$125$	5.79337	0.518175
$126$	0	0
$127$	13.3636	1.18582	0.592912	−	0.805267i	$-0.297978\pi$
$127$	13.3636	1.18582	0.592912	+	0.805267i	$0.297978\pi$
$128$	−11.7476	−1.03835
$129$	0	0
$130$	17.9136	1.57113
$131$	4.34932	0.380002	0.190001	−	0.981784i	$-0.439151\pi$
$131$	4.34932	0.380002	0.190001	+	0.981784i	$0.439151\pi$
$132$	0	0
$133$	−15.1570	−1.31428
$134$	1.05068	0.0907652
$135$	0	0
$136$	−2.88309	−0.247223
$137$	−7.54148	−0.644312	−0.322156	−	0.946687i	$-0.604408\pi$
$137$	−7.54148	−0.644312	−0.322156	+	0.946687i	$0.604408\pi$
$138$	0	0
$139$	4.23393	0.359117	0.179559	−	0.983747i	$-0.442533\pi$
$139$	4.23393	0.359117	0.179559	+	0.983747i	$0.442533\pi$
$140$	0.729791	0.0616786
$141$	0	0
$142$	−7.38414	−0.619663
$143$	−19.3705	−1.61984
$144$	0	0
$145$	9.14397	0.759366
$146$	−11.2643	−0.932240
$147$	0	0
$148$	0.122956	0.0101069
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−12.1678	−0.990201	−0.495101	−	0.868836i	$-0.664869\pi$
$151$	−12.1678	−0.990201	−0.495101	+	0.868836i	$0.664869\pi$
$152$	8.76279	0.710756
$153$	0	0
$154$	−30.1978	−2.43341
$155$	20.3062	1.63103
$156$	0	0
$157$	−10.5652	−0.843197	−0.421598	−	0.906783i	$-0.638531\pi$
$157$	−10.5652	−0.843197	−0.421598	+	0.906783i	$0.638531\pi$
$158$	−14.5169	−1.15490
$159$	0	0
$160$	−0.855470	−0.0676308
$161$	−11.9830	−0.944393
$162$	0	0
$163$	−19.4385	−1.52254	−0.761271	−	0.648433i	$-0.775425\pi$
$163$	−19.4385	−1.52254	−0.761271	+	0.648433i	$0.775425\pi$
$164$	−0.364426	−0.0284569
$165$	0	0
$166$	9.02864	0.700758
$167$	0.580265	0.0449023	0.0224511	−	0.999748i	$-0.492853\pi$
$167$	0.580265	0.0449023	0.0224511	+	0.999748i	$0.492853\pi$
$168$	0	0
$169$	6.66818	0.512937
$170$	−4.17520	−0.320223
$171$	0	0
$172$	0.531558	0.0405309
$173$	−6.59700	−0.501561	−0.250780	−	0.968044i	$-0.580687\pi$
$173$	−6.59700	−0.501561	−0.250780	+	0.968044i	$0.580687\pi$
$174$	0	0
$175$	−14.2063	−1.07389
$176$	17.9272	1.35132
$177$	0	0
$178$	20.1348	1.50917
$179$	−18.2314	−1.36268	−0.681338	−	0.731969i	$-0.738602\pi$
$179$	−18.2314	−1.36268	−0.681338	+	0.731969i	$0.738602\pi$
$180$	0	0
$181$	−24.9222	−1.85245	−0.926227	−	0.376967i	$-0.876967\pi$
$181$	−24.9222	−1.85245	−0.926227	+	0.376967i	$0.876967\pi$
$182$	30.6619	2.27281
$183$	0	0
$184$	6.92780	0.510724
$185$	−6.45769	−0.474779
$186$	0	0
$187$	4.51475	0.330151
$188$	0.103080	0.00751788
$189$	0	0
$190$	12.6900	0.920629
$191$	21.2674	1.53886	0.769428	−	0.638733i	$-0.220541\pi$
$191$	21.2674	1.53886	0.769428	+	0.638733i	$0.220541\pi$
$192$	0	0
$193$	−15.8097	−1.13801	−0.569005	−	0.822334i	$-0.692671\pi$
$193$	−15.8097	−1.13801	−0.569005	+	0.822334i	$0.692671\pi$
$194$	−22.9946	−1.65092
$195$	0	0
$196$	0.873479	0.0623913
$197$	−20.9546	−1.49295	−0.746477	−	0.665411i	$-0.768256\pi$
$197$	−20.9546	−1.49295	−0.746477	+	0.665411i	$0.768256\pi$
$198$	0	0
$199$	−10.6122	−0.752278	−0.376139	−	0.926563i	$-0.622748\pi$
$199$	−10.6122	−0.752278	−0.376139	+	0.926563i	$0.622748\pi$
$200$	8.21314	0.580757
$201$	0	0
$202$	14.9401	1.05118
$203$	15.6513	1.09851
$204$	0	0
$205$	19.1397	1.33678
$206$	−20.9430	−1.45917
$207$	0	0
$208$	−18.2028	−1.26214
$209$	−13.7220	−0.949171
$210$	0	0
$211$	2.73631	0.188376	0.0941878	−	0.995554i	$-0.469975\pi$
$211$	2.73631	0.188376	0.0941878	+	0.995554i	$0.469975\pi$
$212$	0.432553	0.0297079
$213$	0	0
$214$	12.0400	0.823041
$215$	−27.9175	−1.90396
$216$	0	0
$217$	34.7572	2.35947
$218$	−3.66570	−0.248272
$219$	0	0
$220$	0.660699	0.0445443
$221$	−4.58414	−0.308363
$222$	0	0
$223$	−11.5994	−0.776752	−0.388376	−	0.921501i	$-0.626964\pi$
$223$	−11.5994	−0.776752	−0.388376	+	0.921501i	$0.626964\pi$
$224$	−1.46427	−0.0978356
$225$	0	0
$226$	18.7080	1.24444
$227$	0.930629	0.0617680	0.0308840	−	0.999523i	$-0.490168\pi$
$227$	0.930629	0.0617680	0.0308840	+	0.999523i	$0.490168\pi$
$228$	0	0
$229$	−18.4706	−1.22057	−0.610286	−	0.792181i	$-0.708945\pi$
$229$	−18.4706	−1.22057	−0.610286	+	0.792181i	$0.708945\pi$
$230$	10.0326	0.661532
$231$	0	0
$232$	−9.04859	−0.594069
$233$	22.1940	1.45398	0.726988	−	0.686650i	$-0.240920\pi$
$233$	22.1940	1.45398	0.726988	+	0.686650i	$0.240920\pi$
$234$	0	0
$235$	−5.41378	−0.353156
$236$	0.598006	0.0389269
$237$	0	0
$238$	−7.14650	−0.463239
$239$	17.8761	1.15631	0.578156	−	0.815927i	$-0.303773\pi$
$239$	17.8761	1.15631	0.578156	+	0.815927i	$0.303773\pi$
$240$	0	0
$241$	−11.2160	−0.722484	−0.361242	−	0.932472i	$-0.617647\pi$
$241$	−11.2160	−0.722484	−0.361242	+	0.932472i	$0.617647\pi$
$242$	−11.5752	−0.744081
$243$	0	0
$244$	0.131202	0.00839932
$245$	−45.8753	−2.93086
$246$	0	0
$247$	13.9329	0.886531
$248$	−20.0944	−1.27599
$249$	0	0
$250$	−8.30226	−0.525081
$251$	−3.42143	−0.215959	−0.107979	−	0.994153i	$-0.534438\pi$
$251$	−3.42143	−0.215959	−0.107979	+	0.994153i	$0.534438\pi$
$252$	0	0
$253$	−10.8485	−0.682041
$254$	−19.1508	−1.20163
$255$	0	0
$256$	1.28710	0.0804435
$257$	0.124118	0.00774224	0.00387112	−	0.999993i	$-0.498768\pi$
$257$	0.124118	0.00774224	0.00387112	+	0.999993i	$0.498768\pi$
$258$	0	0
$259$	−11.0533	−0.686821
$260$	−0.670854	−0.0416046
$261$	0	0
$262$	−6.23284	−0.385066
$263$	4.86327	0.299882	0.149941	−	0.988695i	$-0.452092\pi$
$263$	4.86327	0.299882	0.149941	+	0.988695i	$0.452092\pi$
$264$	0	0
$265$	−22.7178	−1.39554
$266$	21.7209	1.33179
$267$	0	0
$268$	−0.0393475	−0.00240353
$269$	10.0675	0.613826	0.306913	−	0.951738i	$-0.400704\pi$
$269$	10.0675	0.613826	0.306913	+	0.951738i	$0.400704\pi$
$270$	0	0
$271$	24.8753	1.51107	0.755534	−	0.655109i	$-0.227377\pi$
$271$	24.8753	1.51107	0.755534	+	0.655109i	$0.227377\pi$
$272$	4.24259	0.257245
$273$	0	0
$274$	10.8074	0.652900
$275$	−12.8613	−0.775566
$276$	0	0
$277$	−3.25130	−0.195351	−0.0976757	−	0.995218i	$-0.531141\pi$
$277$	−3.25130	−0.195351	−0.0976757	+	0.995218i	$0.531141\pi$
$278$	−6.06748	−0.363903
$279$	0	0
$280$	37.9289	2.26669
$281$	7.23900	0.431843	0.215921	−	0.976411i	$-0.430725\pi$
$281$	7.23900	0.431843	0.215921	+	0.976411i	$0.430725\pi$
$282$	0	0
$283$	7.31920	0.435081	0.217541	−	0.976051i	$-0.430197\pi$
$283$	7.31920	0.435081	0.217541	+	0.976051i	$0.430197\pi$
$284$	0.276531	0.0164091
$285$	0	0
$286$	27.7591	1.64143
$287$	32.7606	1.93380
$288$	0	0
$289$	−15.9316	−0.937150
$290$	−13.1039	−0.769486
$291$	0	0
$292$	0.421841	0.0246864
$293$	19.8777	1.16127	0.580633	−	0.814165i	$-0.302805\pi$
$293$	19.8777	1.16127	0.580633	+	0.814165i	$0.302805\pi$
$294$	0	0
$295$	−31.4074	−1.82861
$296$	6.39033	0.371430
$297$	0	0
$298$	1.43306	0.0830151
$299$	11.0153	0.637030
$300$	0	0
$301$	−47.7852	−2.75429
$302$	17.4372	1.00340
$303$	0	0
$304$	−12.8948	−0.739569
$305$	−6.89074	−0.394562
$306$	0	0
$307$	21.6884	1.23782	0.618912	−	0.785460i	$-0.287574\pi$
$307$	21.6884	1.23782	0.618912	+	0.785460i	$0.287574\pi$
$308$	1.13089	0.0644384
$309$	0	0
$310$	−29.1001	−1.65277
$311$	16.8860	0.957514	0.478757	−	0.877947i	$-0.341087\pi$
$311$	16.8860	0.957514	0.478757	+	0.877947i	$0.341087\pi$
$312$	0	0
$313$	7.71275	0.435950	0.217975	−	0.975954i	$-0.430055\pi$
$313$	7.71275	0.435950	0.217975	+	0.975954i	$0.430055\pi$
$314$	15.1406	0.854435
$315$	0	0
$316$	0.543649	0.0305826
$317$	−29.3019	−1.64576	−0.822878	−	0.568218i	$-0.807633\pi$
$317$	−29.3019	−1.64576	−0.822878	+	0.568218i	$0.807633\pi$
$318$	0	0
$319$	14.1696	0.793343
$320$	−21.9118	−1.22491
$321$	0	0
$322$	17.1724	0.956980
$323$	−3.24740	−0.180690
$324$	0	0
$325$	13.0590	0.724382
$326$	27.8566	1.54284
$327$	0	0
$328$	−18.9401	−1.04579
$329$	−9.26653	−0.510880
$330$	0	0
$331$	5.28966	0.290746	0.145373	−	0.989377i	$-0.453562\pi$
$331$	5.28966	0.290746	0.145373	+	0.989377i	$0.453562\pi$
$332$	−0.338117	−0.0185566
$333$	0	0
$334$	−0.831556	−0.0455007
$335$	2.06654	0.112907
$336$	0	0
$337$	−11.0266	−0.600656	−0.300328	−	0.953836i	$-0.597096\pi$
$337$	−11.0266	−0.600656	−0.300328	+	0.953836i	$0.597096\pi$
$338$	−9.55592	−0.519774
$339$	0	0
$340$	0.156359	0.00847974
$341$	31.4666	1.70401
$342$	0	0
$343$	−44.7512	−2.41633
$344$	27.6263	1.48951
$345$	0	0
$346$	9.45392	0.508246
$347$	−8.83844	−0.474472	−0.237236	−	0.971452i	$-0.576242\pi$
$347$	−8.83844	−0.474472	−0.237236	+	0.971452i	$0.576242\pi$
$348$	0	0
$349$	−8.85379	−0.473933	−0.236966	−	0.971518i	$-0.576153\pi$
$349$	−8.85379	−0.473933	−0.236966	+	0.971518i	$0.576153\pi$
$350$	20.3584	1.08820
$351$	0	0
$352$	−1.32564	−0.0706569
$353$	26.8833	1.43085	0.715427	−	0.698688i	$-0.246232\pi$
$353$	26.8833	1.43085	0.715427	+	0.698688i	$0.246232\pi$
$354$	0	0
$355$	−14.5235	−0.770827
$356$	−0.754037	−0.0399639
$357$	0	0
$358$	26.1267	1.38084
$359$	14.0915	0.743722	0.371861	−	0.928288i	$-0.378720\pi$
$359$	14.0915	0.743722	0.371861	+	0.928288i	$0.378720\pi$
$360$	0	0
$361$	−9.12993	−0.480523
$362$	35.7151	1.87714
$363$	0	0
$364$	−1.14827	−0.0601857
$365$	−22.1552	−1.15965
$366$	0	0
$367$	−20.6954	−1.08029	−0.540145	−	0.841572i	$-0.681631\pi$
$367$	−20.6954	−1.08029	−0.540145	+	0.841572i	$0.681631\pi$
$368$	−10.1946	−0.531428
$369$	0	0
$370$	9.25427	0.481107
$371$	−38.8850	−2.01881
$372$	0	0
$373$	−10.8281	−0.560657	−0.280328	−	0.959904i	$-0.590443\pi$
$373$	−10.8281	−0.560657	−0.280328	+	0.959904i	$0.590443\pi$
$374$	−6.46992	−0.334551
$375$	0	0
$376$	5.35731	0.276282
$377$	−14.3874	−0.740986
$378$	0	0
$379$	26.7747	1.37532	0.687662	−	0.726031i	$-0.258637\pi$
$379$	26.7747	1.37532	0.687662	+	0.726031i	$0.258637\pi$
$380$	−0.475232	−0.0243789
$381$	0	0
$382$	−30.4775	−1.55937
$383$	23.8219	1.21724	0.608621	−	0.793461i	$-0.291723\pi$
$383$	23.8219	1.21724	0.608621	+	0.793461i	$0.291723\pi$
$384$	0	0
$385$	−59.3945	−3.02702
$386$	22.6563	1.15318
$387$	0	0
$388$	0.861135	0.0437175
$389$	8.43975	0.427912	0.213956	−	0.976843i	$-0.431365\pi$
$389$	8.43975	0.427912	0.213956	+	0.976843i	$0.431365\pi$
$390$	0	0
$391$	−2.56737	−0.129838
$392$	45.3967	2.29288
$393$	0	0
$394$	30.0293	1.51285
$395$	−28.5525	−1.43663
$396$	0	0
$397$	−2.47735	−0.124335	−0.0621673	−	0.998066i	$-0.519801\pi$
$397$	−2.47735	−0.124335	−0.0621673	+	0.998066i	$0.519801\pi$
$398$	15.2079	0.762304
$399$	0	0
$400$	−12.0860	−0.604300
$401$	−13.3313	−0.665733	−0.332867	−	0.942974i	$-0.608016\pi$
$401$	−13.3313	−0.665733	−0.332867	+	0.942974i	$0.608016\pi$
$402$	0	0
$403$	−31.9503	−1.59156
$404$	−0.559499	−0.0278361
$405$	0	0
$406$	−22.4293	−1.11315
$407$	−10.0069	−0.496023
$408$	0	0
$409$	−3.00246	−0.148462	−0.0742310	−	0.997241i	$-0.523650\pi$
$409$	−3.00246	−0.148462	−0.0742310	+	0.997241i	$0.523650\pi$
$410$	−27.4284	−1.35459
$411$	0	0
$412$	0.784303	0.0386398
$413$	−53.7586	−2.64529
$414$	0	0
$415$	17.7580	0.871705
$416$	1.34602	0.0659939
$417$	0	0
$418$	19.6645	0.961822
$419$	21.0746	1.02956	0.514781	−	0.857322i	$-0.327873\pi$
$419$	21.0746	1.02956	0.514781	+	0.857322i	$0.327873\pi$
$420$	0	0
$421$	31.4197	1.53130	0.765650	−	0.643257i	$-0.222417\pi$
$421$	31.4197	1.53130	0.765650	+	0.643257i	$0.222417\pi$
$422$	−3.92131	−0.190886
$423$	0	0
$424$	22.4808	1.09176
$425$	−3.04371	−0.147642
$426$	0	0
$427$	−11.7946	−0.570779
$428$	−0.450892	−0.0217947
$429$	0	0
$430$	40.0076	1.92934
$431$	20.0343	0.965019	0.482510	−	0.875891i	$-0.339725\pi$
$431$	20.0343	0.965019	0.482510	+	0.875891i	$0.339725\pi$
$432$	0	0
$433$	8.42597	0.404926	0.202463	−	0.979290i	$-0.435105\pi$
$433$	8.42597	0.404926	0.202463	+	0.979290i	$0.435105\pi$
$434$	−49.8093	−2.39092
$435$	0	0
$436$	0.137278	0.00657443
$437$	7.80321	0.373278
$438$	0	0
$439$	39.1154	1.86687	0.933437	−	0.358741i	$-0.116794\pi$
$439$	39.1154	1.86687	0.933437	+	0.358741i	$0.116794\pi$
$440$	34.3381	1.63700
$441$	0	0
$442$	6.56936	0.312473
$443$	31.8555	1.51350	0.756749	−	0.653705i	$-0.226786\pi$
$443$	31.8555	1.51350	0.756749	+	0.653705i	$0.226786\pi$
$444$	0	0
$445$	39.6022	1.87732
$446$	16.6226	0.787104
$447$	0	0
$448$	−37.5055	−1.77197
$449$	2.56837	0.121209	0.0606043	−	0.998162i	$-0.480697\pi$
$449$	2.56837	0.121209	0.0606043	+	0.998162i	$0.480697\pi$
$450$	0	0
$451$	29.6590	1.39659
$452$	−0.700603	−0.0329536
$453$	0	0
$454$	−1.33365	−0.0625913
$455$	60.3074	2.82726
$456$	0	0
$457$	−31.1945	−1.45921	−0.729607	−	0.683866i	$-0.760297\pi$
$457$	−31.1945	−1.45921	−0.729607	+	0.683866i	$0.760297\pi$
$458$	26.4695	1.23684
$459$	0	0
$460$	−0.375716	−0.0175178
$461$	−6.27989	−0.292484	−0.146242	−	0.989249i	$-0.546718\pi$
$461$	−6.27989	−0.292484	−0.146242	+	0.989249i	$0.546718\pi$
$462$	0	0
$463$	−34.7917	−1.61691	−0.808454	−	0.588560i	$-0.799695\pi$
$463$	−34.7917	−1.61691	−0.808454	+	0.588560i	$0.799695\pi$
$464$	13.3154	0.618152
$465$	0	0
$466$	−31.8054	−1.47335
$467$	−38.3525	−1.77474	−0.887370	−	0.461058i	$-0.847470\pi$
$467$	−38.3525	−1.77474	−0.887370	+	0.461058i	$0.847470\pi$
$468$	0	0
$469$	3.53720	0.163333
$470$	7.75829	0.357863
$471$	0	0
$472$	31.0798	1.43056
$473$	−43.2612	−1.98915
$474$	0	0
$475$	9.25097	0.424464
$476$	0.267632	0.0122669
$477$	0	0
$478$	−25.6176	−1.17172
$479$	−25.7578	−1.17690	−0.588452	−	0.808532i	$-0.700262\pi$
$479$	−25.7578	−1.17690	−0.588452	+	0.808532i	$0.700262\pi$
$480$	0	0
$481$	10.1607	0.463288
$482$	16.0732	0.732113
$483$	0	0
$484$	0.433484	0.0197038
$485$	−45.2270	−2.05365
$486$	0	0
$487$	9.13475	0.413935	0.206967	−	0.978348i	$-0.433641\pi$
$487$	9.13475	0.413935	0.206967	+	0.978348i	$0.433641\pi$
$488$	6.81886	0.308675
$489$	0	0
$490$	65.7421	2.96993
$491$	−20.3194	−0.917000	−0.458500	−	0.888694i	$-0.651613\pi$
$491$	−20.3194	−0.917000	−0.458500	+	0.888694i	$0.651613\pi$
$492$	0	0
$493$	3.35332	0.151026
$494$	−19.9667	−0.898346
$495$	0	0
$496$	29.5698	1.32772
$497$	−24.8592	−1.11509
$498$	0	0
$499$	−27.7623	−1.24281	−0.621405	−	0.783490i	$-0.713438\pi$
$499$	−27.7623	−1.24281	−0.621405	+	0.783490i	$0.713438\pi$
$500$	0.310914	0.0139045
$501$	0	0
$502$	4.90312	0.218837
$503$	−30.8611	−1.37603	−0.688015	−	0.725697i	$-0.741518\pi$
$503$	−30.8611	−1.37603	−0.688015	+	0.725697i	$0.741518\pi$
$504$	0	0
$505$	29.3850	1.30762
$506$	15.5466	0.691132
$507$	0	0
$508$	0.717186	0.0318200
$509$	−16.6592	−0.738406	−0.369203	−	0.929349i	$-0.620369\pi$
$509$	−16.6592	−0.738406	−0.369203	+	0.929349i	$0.620369\pi$
$510$	0	0
$511$	−37.9220	−1.67757
$512$	21.6507	0.956834
$513$	0	0
$514$	−0.177868	−0.00784543
$515$	−41.1917	−1.81513
$516$	0	0
$517$	−8.38923	−0.368958
$518$	15.8401	0.695975
$519$	0	0
$520$	−34.8659	−1.52897
$521$	−15.3670	−0.673241	−0.336621	−	0.941640i	$-0.609284\pi$
$521$	−15.3670	−0.673241	−0.336621	+	0.941640i	$0.609284\pi$
$522$	0	0
$523$	28.7771	1.25834	0.629168	−	0.777269i	$-0.283396\pi$
$523$	28.7771	1.25834	0.629168	+	0.777269i	$0.283396\pi$
$524$	0.233416	0.0101968
$525$	0	0
$526$	−6.96937	−0.303879
$527$	7.44678	0.324387
$528$	0	0
$529$	−16.8308	−0.731775
$530$	32.5560	1.41414
$531$	0	0
$532$	−0.813434	−0.0352668
$533$	−30.1149	−1.30442
$534$	0	0
$535$	23.6810	1.02382
$536$	−2.04498	−0.0883297
$537$	0	0
$538$	−14.4274	−0.622008
$539$	−71.0887	−3.06201
$540$	0	0
$541$	−32.3614	−1.39133	−0.695663	−	0.718368i	$-0.744889\pi$
$541$	−32.3614	−1.39133	−0.695663	+	0.718368i	$0.744889\pi$
$542$	−35.6479	−1.53121
$543$	0	0
$544$	−0.313722	−0.0134507
$545$	−7.20988	−0.308837
$546$	0	0
$547$	−23.7876	−1.01708	−0.508542	−	0.861037i	$-0.669816\pi$
$547$	−23.7876	−1.01708	−0.508542	+	0.861037i	$0.669816\pi$
$548$	−0.404731	−0.0172893
$549$	0	0
$550$	18.4310	0.785903
$551$	−10.1920	−0.434193
$552$	0	0
$553$	−48.8721	−2.07825
$554$	4.65931	0.197955
$555$	0	0
$556$	0.227223	0.00963642
$557$	−38.9638	−1.65095	−0.825474	−	0.564439i	$-0.809092\pi$
$557$	−38.9638	−1.65095	−0.825474	+	0.564439i	$0.809092\pi$
$558$	0	0
$559$	43.9261	1.85788
$560$	−55.8141	−2.35858
$561$	0	0
$562$	−10.3739	−0.437598
$563$	−7.49806	−0.316005	−0.158003	−	0.987439i	$-0.550505\pi$
$563$	−7.49806	−0.316005	−0.158003	+	0.987439i	$0.550505\pi$
$564$	0	0
$565$	36.7958	1.54801
$566$	−10.4889	−0.440880
$567$	0	0
$568$	14.3720	0.603035
$569$	22.1830	0.929961	0.464980	−	0.885321i	$-0.346061\pi$
$569$	22.1830	0.929961	0.464980	+	0.885321i	$0.346061\pi$
$570$	0	0
$571$	−4.47888	−0.187435	−0.0937177	−	0.995599i	$-0.529875\pi$
$571$	−4.47888	−0.187435	−0.0937177	+	0.995599i	$0.529875\pi$
$572$	−1.03956	−0.0434662
$573$	0	0
$574$	−46.9480	−1.95957
$575$	7.31376	0.305005
$576$	0	0
$577$	38.0885	1.58565	0.792823	−	0.609452i	$-0.208610\pi$
$577$	38.0885	1.58565	0.792823	+	0.609452i	$0.208610\pi$
$578$	22.8309	0.949641
$579$	0	0
$580$	0.490732	0.0203765
$581$	30.3955	1.26102
$582$	0	0
$583$	−35.2036	−1.45798
$584$	21.9241	0.907224
$585$	0	0
$586$	−28.4860	−1.17674
$587$	−6.28458	−0.259393	−0.129696	−	0.991554i	$-0.541400\pi$
$587$	−6.28458	−0.259393	−0.129696	+	0.991554i	$0.541400\pi$
$588$	0	0
$589$	−22.6336	−0.932599
$590$	45.0088	1.85298
$591$	0	0
$592$	−9.40365	−0.386488
$593$	−25.8680	−1.06227	−0.531136	−	0.847287i	$-0.678235\pi$
$593$	−25.8680	−1.06227	−0.531136	+	0.847287i	$0.678235\pi$
$594$	0	0
$595$	−14.0561	−0.576243
$596$	−0.0536673	−0.00219830
$597$	0	0
$598$	−15.7856	−0.645520
$599$	41.4256	1.69260	0.846302	−	0.532704i	$-0.178824\pi$
$599$	41.4256	1.69260	0.846302	+	0.532704i	$0.178824\pi$
$600$	0	0
$601$	0.255365	0.0104166	0.00520829	−	0.999986i	$-0.498342\pi$
$601$	0.255365	0.0104166	0.00520829	+	0.999986i	$0.498342\pi$
$602$	68.4791	2.79100
$603$	0	0
$604$	−0.653013	−0.0265707
$605$	−22.7666	−0.925596
$606$	0	0
$607$	−14.3968	−0.584350	−0.292175	−	0.956365i	$-0.594379\pi$
$607$	−14.3968	−0.584350	−0.292175	+	0.956365i	$0.594379\pi$
$608$	0.953517	0.0386702
$609$	0	0
$610$	9.87485	0.399821
$611$	8.51818	0.344609
$612$	0	0
$613$	23.3649	0.943700	0.471850	−	0.881679i	$-0.343586\pi$
$613$	23.3649	0.943700	0.471850	+	0.881679i	$0.343586\pi$
$614$	−31.0809	−1.25432
$615$	0	0
$616$	58.7749	2.36811
$617$	−17.3511	−0.698529	−0.349264	−	0.937024i	$-0.613569\pi$
$617$	−17.3511	−0.698529	−0.349264	+	0.937024i	$0.613569\pi$
$618$	0	0
$619$	24.2434	0.974423	0.487211	−	0.873284i	$-0.338014\pi$
$619$	24.2434	0.974423	0.487211	+	0.873284i	$0.338014\pi$
$620$	1.08978	0.0437666
$621$	0	0
$622$	−24.1986	−0.970276
$623$	67.7853	2.71576
$624$	0	0
$625$	−31.0523	−1.24209
$626$	−11.0529	−0.441761
$627$	0	0
$628$	−0.567007	−0.0226260
$629$	−2.36819	−0.0944260
$630$	0	0
$631$	9.42492	0.375200	0.187600	−	0.982245i	$-0.439929\pi$
$631$	9.42492	0.375200	0.187600	+	0.982245i	$0.439929\pi$
$632$	28.2547	1.12391
$633$	0	0
$634$	41.9914	1.66769
$635$	−37.6668	−1.49476
$636$	0	0
$637$	72.1813	2.85993
$638$	−20.3059	−0.803917
$639$	0	0
$640$	33.1120	1.30887
$641$	24.9521	0.985548	0.492774	−	0.870157i	$-0.335983\pi$
$641$	24.9521	0.985548	0.492774	+	0.870157i	$0.335983\pi$
$642$	0	0
$643$	25.6817	1.01279	0.506393	−	0.862303i	$-0.330979\pi$
$643$	25.6817	1.01279	0.506393	+	0.862303i	$0.330979\pi$
$644$	−0.643095	−0.0253415
$645$	0	0
$646$	4.65373	0.183098
$647$	−26.0980	−1.02602	−0.513010	−	0.858383i	$-0.671469\pi$
$647$	−26.0980	−1.02602	−0.513010	+	0.858383i	$0.671469\pi$
$648$	0	0
$649$	−48.6691	−1.91043
$650$	−18.7143	−0.734037
$651$	0	0
$652$	−1.04321	−0.0408554
$653$	41.5419	1.62566	0.812830	−	0.582501i	$-0.197926\pi$
$653$	41.5419	1.62566	0.812830	+	0.582501i	$0.197926\pi$
$654$	0	0
$655$	−12.2591	−0.479001
$656$	27.8711	1.08819
$657$	0	0
$658$	13.2795	0.517689
$659$	−5.91929	−0.230583	−0.115291	−	0.993332i	$-0.536780\pi$
$659$	−5.91929	−0.230583	−0.115291	+	0.993332i	$0.536780\pi$
$660$	0	0
$661$	14.0230	0.545430	0.272715	−	0.962095i	$-0.412078\pi$
$661$	14.0230	0.545430	0.272715	+	0.962095i	$0.412078\pi$
$662$	−7.58040	−0.294621
$663$	0	0
$664$	−17.5727	−0.681954
$665$	42.7217	1.65668
$666$	0	0
$667$	−8.05772	−0.311996
$668$	0.0311413	0.00120489
$669$	0	0
$670$	−2.96148	−0.114412
$671$	−10.6779	−0.412217
$672$	0	0
$673$	35.2881	1.36026	0.680129	−	0.733092i	$-0.261924\pi$
$673$	35.2881	1.36026	0.680129	+	0.733092i	$0.261924\pi$
$674$	15.8018	0.608662
$675$	0	0
$676$	0.357863	0.0137640
$677$	−40.5733	−1.55936	−0.779679	−	0.626179i	$-0.784618\pi$
$677$	−40.5733	−1.55936	−0.779679	+	0.626179i	$0.784618\pi$
$678$	0	0
$679$	−77.4130	−2.97084
$680$	8.12633	0.311630
$681$	0	0
$682$	−45.0937	−1.72673
$683$	−37.8000	−1.44638	−0.723188	−	0.690652i	$-0.757324\pi$
$683$	−37.8000	−1.44638	−0.723188	+	0.690652i	$0.757324\pi$
$684$	0	0
$685$	21.2566	0.812171
$686$	64.1312	2.44854
$687$	0	0
$688$	−40.6533	−1.54989
$689$	35.7447	1.36176
$690$	0	0
$691$	−31.8795	−1.21276	−0.606378	−	0.795177i	$-0.707378\pi$
$691$	−31.8795	−1.21276	−0.606378	+	0.795177i	$0.707378\pi$
$692$	−0.354043	−0.0134587
$693$	0	0
$694$	12.6660	0.480796
$695$	−11.9338	−0.452676
$696$	0	0
$697$	7.01900	0.265864
$698$	12.6880	0.480249
$699$	0	0
$700$	−0.762411	−0.0288164
$701$	31.9238	1.20574	0.602872	−	0.797838i	$-0.294023\pi$
$701$	31.9238	1.20574	0.602872	+	0.797838i	$0.294023\pi$
$702$	0	0
$703$	7.19782	0.271471
$704$	−33.9547	−1.27972
$705$	0	0
$706$	−38.5254	−1.44992
$707$	50.2970	1.89161
$708$	0	0
$709$	35.9065	1.34850	0.674249	−	0.738504i	$-0.264468\pi$
$709$	35.9065	1.34850	0.674249	+	0.738504i	$0.264468\pi$
$710$	20.8131	0.781100
$711$	0	0
$712$	−39.1891	−1.46867
$713$	−17.8939	−0.670133
$714$	0	0
$715$	54.5979	2.04185
$716$	−0.978428	−0.0365656
$717$	0	0
$718$	−20.1940	−0.753634
$719$	48.6493	1.81431	0.907157	−	0.420792i	$-0.138248\pi$
$719$	48.6493	1.81431	0.907157	+	0.420792i	$0.138248\pi$
$720$	0	0
$721$	−70.5061	−2.62578
$722$	13.0838	0.486927
$723$	0	0
$724$	−1.33751	−0.0497081
$725$	−9.55270	−0.354778
$726$	0	0
$727$	5.81485	0.215661	0.107830	−	0.994169i	$-0.465610\pi$
$727$	5.81485	0.215661	0.107830	+	0.994169i	$0.465610\pi$
$728$	−59.6783	−2.21183
$729$	0	0
$730$	31.7497	1.17511
$731$	−10.2380	−0.378667
$732$	0	0
$733$	6.97072	0.257469	0.128735	−	0.991679i	$-0.458908\pi$
$733$	6.97072	0.257469	0.128735	+	0.991679i	$0.458908\pi$
$734$	29.6578	1.09469
$735$	0	0
$736$	0.753844	0.0277871
$737$	3.20232	0.117959
$738$	0	0
$739$	−42.4412	−1.56123	−0.780613	−	0.625014i	$-0.785093\pi$
$739$	−42.4412	−1.56123	−0.780613	+	0.625014i	$0.785093\pi$
$740$	−0.346567	−0.0127401
$741$	0	0
$742$	55.7246	2.04571
$743$	−26.5640	−0.974538	−0.487269	−	0.873252i	$-0.662007\pi$
$743$	−26.5640	−0.974538	−0.487269	+	0.873252i	$0.662007\pi$
$744$	0	0
$745$	2.81862	0.103266
$746$	15.5173	0.568129
$747$	0	0
$748$	0.242294	0.00885916
$749$	40.5336	1.48107
$750$	0	0
$751$	47.1325	1.71989	0.859944	−	0.510389i	$-0.170498\pi$
$751$	47.1325	1.71989	0.859944	+	0.510389i	$0.170498\pi$
$752$	−7.88352	−0.287482
$753$	0	0
$754$	20.6180	0.750862
$755$	34.2964	1.24817
$756$	0	0
$757$	49.3912	1.79515	0.897577	−	0.440858i	$-0.145326\pi$
$757$	49.3912	1.79515	0.897577	+	0.440858i	$0.145326\pi$
$758$	−38.3698	−1.39366
$759$	0	0
$760$	−24.6989	−0.895925
$761$	−41.3621	−1.49938	−0.749688	−	0.661791i	$-0.769796\pi$
$761$	−41.3621	−1.49938	−0.749688	+	0.661791i	$0.769796\pi$
$762$	0	0
$763$	−12.3408	−0.446768
$764$	1.14137	0.0412931
$765$	0	0
$766$	−34.1382	−1.23346
$767$	49.4172	1.78435
$768$	0	0
$769$	−27.9986	−1.00966	−0.504828	−	0.863220i	$-0.668444\pi$
$769$	−27.9986	−1.00966	−0.504828	+	0.863220i	$0.668444\pi$
$770$	85.1160	3.06737
$771$	0	0
$772$	−0.848465	−0.0305369
$773$	42.4794	1.52788	0.763940	−	0.645288i	$-0.223263\pi$
$773$	42.4794	1.52788	0.763940	+	0.645288i	$0.223263\pi$
$774$	0	0
$775$	−21.2139	−0.762025
$776$	44.7552	1.60662
$777$	0	0
$778$	−12.0947	−0.433615
$779$	−21.3334	−0.764347
$780$	0	0
$781$	−22.5057	−0.805317
$782$	3.67921	0.131568
$783$	0	0
$784$	−66.8033	−2.38583
$785$	29.7793	1.06287
$786$	0	0
$787$	−32.8529	−1.17108	−0.585540	−	0.810643i	$-0.699118\pi$
$787$	−32.8529	−1.17108	−0.585540	+	0.810643i	$0.699118\pi$
$788$	−1.12458	−0.0400614
$789$	0	0
$790$	40.9176	1.45578
$791$	62.9817	2.23937
$792$	0	0
$793$	10.8421	0.385013
$794$	3.55020	0.125992
$795$	0	0
$796$	−0.569527	−0.0201864
$797$	6.12549	0.216976	0.108488	−	0.994098i	$-0.465399\pi$
$797$	6.12549	0.216976	0.108488	+	0.994098i	$0.465399\pi$
$798$	0	0
$799$	−1.98537	−0.0702372
$800$	0.893708	0.0315974
$801$	0	0
$802$	19.1046	0.674606
$803$	−34.3318	−1.21154
$804$	0	0
$805$	33.7755	1.19043
$806$	45.7867	1.61277
$807$	0	0
$808$	−29.0785	−1.02298
$809$	−4.50692	−0.158455	−0.0792274	−	0.996857i	$-0.525245\pi$
$809$	−4.50692	−0.158455	−0.0792274	+	0.996857i	$0.525245\pi$
$810$	0	0
$811$	−38.6490	−1.35715	−0.678574	−	0.734532i	$-0.737402\pi$
$811$	−38.6490	−1.35715	−0.678574	+	0.734532i	$0.737402\pi$
$812$	0.839964	0.0294770
$813$	0	0
$814$	14.3405	0.502634
$815$	54.7898	1.91920
$816$	0	0
$817$	31.1172	1.08865
$818$	4.30271	0.150441
$819$	0	0
$820$	1.02718	0.0358706
$821$	51.1455	1.78499	0.892495	−	0.451057i	$-0.148953\pi$
$821$	51.1455	1.78499	0.892495	+	0.451057i	$0.148953\pi$
$822$	0	0
$823$	−36.4309	−1.26990	−0.634951	−	0.772553i	$-0.718980\pi$
$823$	−36.4309	−1.26990	−0.634951	+	0.772553i	$0.718980\pi$
$824$	40.7621	1.42001
$825$	0	0
$826$	77.0395	2.68055
$827$	41.4802	1.44241	0.721204	−	0.692723i	$-0.243589\pi$
$827$	41.4802	1.44241	0.721204	+	0.692723i	$0.243589\pi$
$828$	0	0
$829$	−14.4574	−0.502128	−0.251064	−	0.967971i	$-0.580780\pi$
$829$	−14.4574	−0.502128	−0.251064	+	0.967971i	$0.580780\pi$
$830$	−25.4483	−0.883323
$831$	0	0
$832$	34.4766	1.19526
$833$	−16.8236	−0.582903
$834$	0	0
$835$	−1.63555	−0.0566004
$836$	−0.736423	−0.0254697
$837$	0	0
$838$	−30.2012	−1.04328
$839$	−39.2269	−1.35426	−0.677131	−	0.735862i	$-0.736777\pi$
$839$	−39.2269	−1.35426	−0.677131	+	0.735862i	$0.736777\pi$
$840$	0	0
$841$	−18.4756	−0.637090
$842$	−45.0263	−1.55171
$843$	0	0
$844$	0.146851	0.00505480
$845$	−18.7951	−0.646570
$846$	0	0
$847$	−38.9686	−1.33898
$848$	−33.0815	−1.13602
$849$	0	0
$850$	4.36182	0.149609
$851$	5.69055	0.195070
$852$	0	0
$853$	10.1766	0.348440	0.174220	−	0.984707i	$-0.444260\pi$
$853$	10.1766	0.348440	0.174220	+	0.984707i	$0.444260\pi$
$854$	16.9023	0.578386
$855$	0	0
$856$	−23.4339	−0.800956
$857$	30.5675	1.04416	0.522082	−	0.852895i	$-0.325155\pi$
$857$	30.5675	1.04416	0.522082	+	0.852895i	$0.325155\pi$
$858$	0	0
$859$	−41.7313	−1.42386	−0.711928	−	0.702253i	$-0.752178\pi$
$859$	−41.7313	−1.42386	−0.711928	+	0.702253i	$0.752178\pi$
$860$	−1.49826	−0.0510902
$861$	0	0
$862$	−28.7104	−0.977881
$863$	0.540652	0.0184040	0.00920201	−	0.999958i	$-0.497071\pi$
$863$	0.540652	0.0184040	0.00920201	+	0.999958i	$0.497071\pi$
$864$	0	0
$865$	18.5944	0.632230
$866$	−12.0749	−0.410323
$867$	0	0
$868$	1.86533	0.0633133
$869$	−44.2452	−1.50092
$870$	0	0
$871$	−3.25154	−0.110174
$872$	7.13467	0.241610
$873$	0	0
$874$	−11.1825	−0.378253
$875$	−27.9501	−0.944886
$876$	0	0
$877$	−48.9716	−1.65365	−0.826827	−	0.562456i	$-0.809857\pi$
$877$	−48.9716	−1.65365	−0.826827	+	0.562456i	$0.809857\pi$
$878$	−56.0547	−1.89176
$879$	0	0
$880$	−50.5300	−1.70337
$881$	−4.71393	−0.158816	−0.0794081	−	0.996842i	$-0.525303\pi$
$881$	−4.71393	−0.158816	−0.0794081	+	0.996842i	$0.525303\pi$
$882$	0	0
$883$	27.6528	0.930591	0.465296	−	0.885155i	$-0.345948\pi$
$883$	27.6528	0.930591	0.465296	+	0.885155i	$0.345948\pi$
$884$	−0.246018	−0.00827450
$885$	0	0
$886$	−45.6509	−1.53367
$887$	46.0763	1.54709	0.773545	−	0.633742i	$-0.218482\pi$
$887$	46.0763	1.54709	0.773545	+	0.633742i	$0.218482\pi$
$888$	0	0
$889$	−64.4725	−2.16234
$890$	−56.7524	−1.90234
$891$	0	0
$892$	−0.622507	−0.0208431
$893$	6.03427	0.201929
$894$	0	0
$895$	51.3873	1.71769
$896$	56.6763	1.89342
$897$	0	0
$898$	−3.68063	−0.122824
$899$	23.3718	0.779492
$900$	0	0
$901$	−8.33116	−0.277551
$902$	−42.5032	−1.41520
$903$	0	0
$904$	−36.4120	−1.21104
$905$	70.2462	2.33506
$906$	0	0
$907$	−19.4669	−0.646389	−0.323195	−	0.946332i	$-0.604757\pi$
$907$	−19.4669	−0.646389	−0.323195	+	0.946332i	$0.604757\pi$
$908$	0.0499444	0.00165746
$909$	0	0
$910$	−86.4243	−2.86494
$911$	4.13455	0.136984	0.0684919	−	0.997652i	$-0.478181\pi$
$911$	4.13455	0.136984	0.0684919	+	0.997652i	$0.478181\pi$
$912$	0	0
$913$	27.5179	0.910709
$914$	44.7036	1.47866
$915$	0	0
$916$	−0.991267	−0.0327524
$917$	−20.9833	−0.692929
$918$	0	0
$919$	11.6964	0.385827	0.192914	−	0.981216i	$-0.438206\pi$
$919$	11.6964	0.385827	0.192914	+	0.981216i	$0.438206\pi$
$920$	−19.5268	−0.643780
$921$	0	0
$922$	8.99947	0.296382
$923$	22.8516	0.752170
$924$	0	0
$925$	6.74634	0.221818
$926$	49.8587	1.63846
$927$	0	0
$928$	−0.984617	−0.0323216
$929$	−44.1711	−1.44921	−0.724603	−	0.689166i	$-0.757977\pi$
$929$	−44.1711	−1.44921	−0.724603	+	0.689166i	$0.757977\pi$
$930$	0	0
$931$	51.1332	1.67582
$932$	1.19109	0.0390155
$933$	0	0
$934$	54.9615	1.79839
$935$	−12.7254	−0.416164
$936$	0	0
$937$	−36.9817	−1.20814	−0.604069	−	0.796932i	$-0.706455\pi$
$937$	−36.9817	−1.20814	−0.604069	+	0.796932i	$0.706455\pi$
$938$	−5.06902	−0.165510
$939$	0	0
$940$	−0.290543	−0.00947647
$941$	−52.4501	−1.70983	−0.854913	−	0.518772i	$-0.826389\pi$
$941$	−52.4501	−1.70983	−0.854913	+	0.518772i	$0.826389\pi$
$942$	0	0
$943$	−16.8660	−0.549233
$944$	−45.7353	−1.48856
$945$	0	0
$946$	61.9960	2.01566
$947$	42.9602	1.39602	0.698009	−	0.716089i	$-0.254070\pi$
$947$	42.9602	1.39602	0.698009	+	0.716089i	$0.254070\pi$
$948$	0	0
$949$	34.8595	1.13159
$950$	−13.2572	−0.430121
$951$	0	0
$952$	13.9095	0.450808
$953$	−29.5707	−0.957890	−0.478945	−	0.877845i	$-0.658981\pi$
$953$	−29.5707	−0.957890	−0.478945	+	0.877845i	$0.658981\pi$
$954$	0	0
$955$	−59.9447	−1.93977
$956$	0.959364	0.0310280
$957$	0	0
$958$	36.9125	1.19259
$959$	36.3839	1.17490
$960$	0	0
$961$	20.9021	0.674263
$962$	−14.5609	−0.469462
$963$	0	0
$964$	−0.601931	−0.0193869
$965$	44.5616	1.43449
$966$	0	0
$967$	−23.2815	−0.748683	−0.374341	−	0.927291i	$-0.622131\pi$
$967$	−23.2815	−0.748683	−0.374341	+	0.927291i	$0.622131\pi$
$968$	22.5292	0.724115
$969$	0	0
$970$	64.8131	2.08102
$971$	32.8511	1.05424	0.527121	−	0.849790i	$-0.323272\pi$
$971$	32.8511	1.05424	0.527121	+	0.849790i	$0.323272\pi$
$972$	0	0
$973$	−20.4266	−0.654846
$974$	−13.0907	−0.419452
$975$	0	0
$976$	−10.0343	−0.321189
$977$	−41.5580	−1.32956	−0.664780	−	0.747039i	$-0.731475\pi$
$977$	−41.5580	−1.32956	−0.664780	+	0.747039i	$0.731475\pi$
$978$	0	0
$979$	61.3678	1.96132
$980$	−2.46200	−0.0786458
$981$	0	0
$982$	29.1189	0.929222
$983$	−7.62816	−0.243300	−0.121650	−	0.992573i	$-0.538819\pi$
$983$	−7.62816	−0.243300	−0.121650	+	0.992573i	$0.538819\pi$
$984$	0	0
$985$	59.0630	1.88190
$986$	−4.80551	−0.153039
$987$	0	0
$988$	0.747742	0.0237888
$989$	24.6011	0.782268
$990$	0	0
$991$	−48.7780	−1.54948	−0.774742	−	0.632278i	$-0.782120\pi$
$991$	−48.7780	−1.54948	−0.774742	+	0.632278i	$0.782120\pi$
$992$	−2.18656	−0.0694233
$993$	0	0
$994$	35.6248	1.12995
$995$	29.9117	0.948264
$996$	0	0
$997$	10.8459	0.343493	0.171747	−	0.985141i	$-0.445059\pi$
$997$	10.8459	0.343493	0.171747	+	0.985141i	$0.445059\pi$
$998$	39.7850	1.25937
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4023.2.a.e.1.9	✓	25
3.2	odd	2		4023.2.a.f.1.17	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4023.2.a.e.1.9	✓	25	1.1	even	1	trivial
4023.2.a.f.1.17	yes	25	3.2	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 4023 = 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4023.a (trivial)

\(f(q)\)	\(=\)	\(q-1.43306 q^{2} +0.0536673 q^{4} -2.81862 q^{5} -4.82450 q^{7} +2.78922 q^{8} +O(q^{10})\)	\(q-1.43306 q^{2} +0.0536673 q^{4} -2.81862 q^{5} -4.82450 q^{7} +2.78922 q^{8} +4.03925 q^{10} -4.36775 q^{11} +4.43488 q^{13} +6.91381 q^{14} -4.10445 q^{16} -1.03366 q^{17} +3.14167 q^{19} -0.151268 q^{20} +6.25926 q^{22} +2.48378 q^{23} +2.94461 q^{25} -6.35546 q^{26} -0.258918 q^{28} -3.24413 q^{29} -7.20431 q^{31} +0.303507 q^{32} +1.48129 q^{34} +13.5984 q^{35} +2.29108 q^{37} -4.50220 q^{38} -7.86173 q^{40} -6.79046 q^{41} +9.90469 q^{43} -0.234405 q^{44} -3.55941 q^{46} +1.92072 q^{47} +16.2758 q^{49} -4.21980 q^{50} +0.238008 q^{52} +8.05990 q^{53} +12.3110 q^{55} -13.4566 q^{56} +4.64905 q^{58} +11.1428 q^{59} +2.44472 q^{61} +10.3242 q^{62} +7.77396 q^{64} -12.5002 q^{65} -0.733174 q^{67} -0.0554735 q^{68} -19.4874 q^{70} +5.15270 q^{71} +7.86030 q^{73} -3.28327 q^{74} +0.168605 q^{76} +21.0722 q^{77} +10.1300 q^{79} +11.5689 q^{80} +9.73115 q^{82} -6.30024 q^{83} +2.91348 q^{85} -14.1940 q^{86} -12.1826 q^{88} -14.0502 q^{89} -21.3961 q^{91} +0.133298 q^{92} -2.75251 q^{94} -8.85516 q^{95} +16.0458 q^{97} -23.3242 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8}+O(q^{10}) \) 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8	\( 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8} - 4 q^{10} - 12 q^{11} - 10 q^{14} + 35 q^{16} - 26 q^{17} - 30 q^{20} + 8 q^{22} - 26 q^{23} + 27 q^{25} - 16 q^{26} + 4 q^{28} - 20 q^{29} - 6 q^{31} - 49 q^{32} - 14 q^{34} - 16 q^{35} - 2 q^{37} - 27 q^{38} + 2 q^{40} - 35 q^{41} + 4 q^{43} - 22 q^{44} + 6 q^{46} - 38 q^{47} + 19 q^{49} - 22 q^{50} + 4 q^{52} - 36 q^{53} + 10 q^{55} - 79 q^{56} - 22 q^{58} - 15 q^{59} + 10 q^{61} + 14 q^{62} + 41 q^{64} - 80 q^{65} - 6 q^{67} - 33 q^{68} + 8 q^{70} - 26 q^{71} - 6 q^{73} - 75 q^{74} - 10 q^{76} - 47 q^{77} - 6 q^{79} - 66 q^{80} + 12 q^{82} - 40 q^{83} - 12 q^{85} - 4 q^{86} + 12 q^{88} - 30 q^{89} - 158 q^{92} + 18 q^{94} - 22 q^{95} - 20 q^{97} - 35 q^{98}+O(q^{100}) \) 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8 - 4 * q^10 - 12 * q^11 - 10 * q^14 + 35 * q^16 - 26 * q^17 - 30 * q^20 + 8 * q^22 - 26 * q^23 + 27 * q^25 - 16 * q^26 + 4 * q^28 - 20 * q^29 - 6 * q^31 - 49 * q^32 - 14 * q^34 - 16 * q^35 - 2 * q^37 - 27 * q^38 + 2 * q^40 - 35 * q^41 + 4 * q^43 - 22 * q^44 + 6 * q^46 - 38 * q^47 + 19 * q^49 - 22 * q^50 + 4 * q^52 - 36 * q^53 + 10 * q^55 - 79 * q^56 - 22 * q^58 - 15 * q^59 + 10 * q^61 + 14 * q^62 + 41 * q^64 - 80 * q^65 - 6 * q^67 - 33 * q^68 + 8 * q^70 - 26 * q^71 - 6 * q^73 - 75 * q^74 - 10 * q^76 - 47 * q^77 - 6 * q^79 - 66 * q^80 + 12 * q^82 - 40 * q^83 - 12 * q^85 - 4 * q^86 + 12 * q^88 - 30 * q^89 - 158 * q^92 + 18 * q^94 - 22 * q^95 - 20 * q^97 - 35 * q^98

Introduction

L-functions

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