LMFDB - Embedded newform 4023.2.a.e.1.12

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			1.12
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)

$f(q)$	$=$	$q-0.614681 q^{2} -1.62217 q^{4} -3.33074 q^{5} +3.77407 q^{7} +2.22648 q^{8} +O(q^{10})$	\(q-0.614681 q^{2} -1.62217 q^{4} -3.33074 q^{5} +3.77407 q^{7} +2.22648 q^{8} +2.04734 q^{10} -3.16692 q^{11} +5.82586 q^{13} -2.31985 q^{14} +1.87576 q^{16} -6.41335 q^{17} -5.07037 q^{19} +5.40302 q^{20} +1.94665 q^{22} -0.325802 q^{23} +6.09384 q^{25} -3.58104 q^{26} -6.12217 q^{28} +1.11212 q^{29} +0.683695 q^{31} -5.60595 q^{32} +3.94216 q^{34} -12.5704 q^{35} +4.69579 q^{37} +3.11666 q^{38} -7.41582 q^{40} +6.58556 q^{41} -4.10487 q^{43} +5.13728 q^{44} +0.200264 q^{46} +5.09923 q^{47} +7.24359 q^{49} -3.74577 q^{50} -9.45052 q^{52} +2.82713 q^{53} +10.5482 q^{55} +8.40287 q^{56} -0.683601 q^{58} +0.550183 q^{59} -0.115741 q^{61} -0.420254 q^{62} -0.305655 q^{64} -19.4044 q^{65} +13.1749 q^{67} +10.4035 q^{68} +7.72681 q^{70} +9.11323 q^{71} -16.5170 q^{73} -2.88641 q^{74} +8.22499 q^{76} -11.9522 q^{77} -17.6173 q^{79} -6.24768 q^{80} -4.04802 q^{82} +15.7830 q^{83} +21.3612 q^{85} +2.52319 q^{86} -7.05108 q^{88} +4.50599 q^{89} +21.9872 q^{91} +0.528505 q^{92} -3.13440 q^{94} +16.8881 q^{95} -2.68958 q^{97} -4.45249 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.614681	−0.434645	−0.217322	−	0.976100i	$-0.569732\pi$
$2$	−0.614681	−0.434645	−0.217322	+	0.976100i	$0.569732\pi$
$3$	0	0
$3$	0	0
$4$	−1.62217	−0.811084
$5$	−3.33074	−1.48955	−0.744777	−	0.667314i	$-0.767444\pi$
$5$	−3.33074	−1.48955	−0.744777	+	0.667314i	$0.767444\pi$
$6$	0	0
$7$	3.77407	1.42646	0.713232	−	0.700928i	$-0.247231\pi$
$7$	3.77407	1.42646	0.713232	+	0.700928i	$0.247231\pi$
$8$	2.22648	0.787178
$9$	0	0
$10$	2.04734	0.647427
$11$	−3.16692	−0.954863	−0.477432	−	0.878669i	$-0.658432\pi$
$11$	−3.16692	−0.954863	−0.477432	+	0.878669i	$0.658432\pi$
$12$	0	0
$13$	5.82586	1.61580	0.807901	−	0.589318i	$-0.200603\pi$
$13$	5.82586	1.61580	0.807901	+	0.589318i	$0.200603\pi$
$14$	−2.31985	−0.620005
$15$	0	0
$16$	1.87576	0.468941
$17$	−6.41335	−1.55547	−0.777733	−	0.628595i	$-0.783631\pi$
$17$	−6.41335	−1.55547	−0.777733	+	0.628595i	$0.783631\pi$
$18$	0	0
$19$	−5.07037	−1.16322	−0.581612	−	0.813467i	$-0.697578\pi$
$19$	−5.07037	−1.16322	−0.581612	+	0.813467i	$0.697578\pi$
$20$	5.40302	1.20815
$21$	0	0
$22$	1.94665	0.415026
$23$	−0.325802	−0.0679343	−0.0339672	−	0.999423i	$-0.510814\pi$
$23$	−0.325802	−0.0679343	−0.0339672	+	0.999423i	$0.510814\pi$
$24$	0	0
$25$	6.09384	1.21877
$26$	−3.58104	−0.702301
$27$	0	0
$28$	−6.12217	−1.15698
$29$	1.11212	0.206516	0.103258	−	0.994655i	$-0.467073\pi$
$29$	1.11212	0.206516	0.103258	+	0.994655i	$0.467073\pi$
$30$	0	0
$31$	0.683695	0.122795	0.0613976	−	0.998113i	$-0.480444\pi$
$31$	0.683695	0.122795	0.0613976	+	0.998113i	$0.480444\pi$
$32$	−5.60595	−0.991001
$33$	0	0
$34$	3.94216	0.676076
$35$	−12.5704	−2.12479
$36$	0	0
$37$	4.69579	0.771983	0.385992	−	0.922502i	$-0.373859\pi$
$37$	4.69579	0.771983	0.385992	+	0.922502i	$0.373859\pi$
$38$	3.11666	0.505589
$39$	0	0
$40$	−7.41582	−1.17254
$41$	6.58556	1.02849	0.514246	−	0.857643i	$-0.328072\pi$
$41$	6.58556	1.02849	0.514246	+	0.857643i	$0.328072\pi$
$42$	0	0
$43$	−4.10487	−0.625987	−0.312994	−	0.949755i	$-0.601332\pi$
$43$	−4.10487	−0.625987	−0.312994	+	0.949755i	$0.601332\pi$
$44$	5.13728	0.774474
$45$	0	0
$46$	0.200264	0.0295273
$47$	5.09923	0.743799	0.371899	−	0.928273i	$-0.378707\pi$
$47$	5.09923	0.743799	0.371899	+	0.928273i	$0.378707\pi$
$48$	0	0
$49$	7.24359	1.03480
$50$	−3.74577	−0.529732
$51$	0	0
$52$	−9.45052	−1.31055
$53$	2.82713	0.388336	0.194168	−	0.980968i	$-0.437799\pi$
$53$	2.82713	0.388336	0.194168	+	0.980968i	$0.437799\pi$
$54$	0	0
$55$	10.5482	1.42232
$56$	8.40287	1.12288
$57$	0	0
$58$	−0.683601	−0.0897612
$59$	0.550183	0.0716278	0.0358139	−	0.999358i	$-0.488598\pi$
$59$	0.550183	0.0716278	0.0358139	+	0.999358i	$0.488598\pi$
$60$	0	0
$61$	−0.115741	−0.0148191	−0.00740957	−	0.999973i	$-0.502359\pi$
$61$	−0.115741	−0.0148191	−0.00740957	+	0.999973i	$0.502359\pi$
$62$	−0.420254	−0.0533723
$63$	0	0
$64$	−0.305655	−0.0382069
$65$	−19.4044	−2.40682
$66$	0	0
$67$	13.1749	1.60957	0.804786	−	0.593565i	$-0.202280\pi$
$67$	13.1749	1.60957	0.804786	+	0.593565i	$0.202280\pi$
$68$	10.4035	1.26161
$69$	0	0
$70$	7.72681	0.923531
$71$	9.11323	1.08154	0.540771	−	0.841170i	$-0.318133\pi$
$71$	9.11323	1.08154	0.540771	+	0.841170i	$0.318133\pi$
$72$	0	0
$73$	−16.5170	−1.93317	−0.966586	−	0.256343i	$-0.917482\pi$
$73$	−16.5170	−1.93317	−0.966586	+	0.256343i	$0.917482\pi$
$74$	−2.88641	−0.335539
$75$	0	0
$76$	8.22499	0.943471
$77$	−11.9522	−1.36208
$78$	0	0
$79$	−17.6173	−1.98210	−0.991050	−	0.133490i	$-0.957382\pi$
$79$	−17.6173	−1.98210	−0.991050	+	0.133490i	$0.957382\pi$
$80$	−6.24768	−0.698512
$81$	0	0
$82$	−4.04802	−0.447029
$83$	15.7830	1.73241	0.866204	−	0.499691i	$-0.166553\pi$
$83$	15.7830	1.73241	0.866204	+	0.499691i	$0.166553\pi$
$84$	0	0
$85$	21.3612	2.31695
$86$	2.52319	0.272082
$87$	0	0
$88$	−7.05108	−0.751648
$89$	4.50599	0.477634	0.238817	−	0.971065i	$-0.423240\pi$
$89$	4.50599	0.477634	0.238817	+	0.971065i	$0.423240\pi$
$90$	0	0
$91$	21.9872	2.30488
$92$	0.528505	0.0551004
$93$	0	0
$94$	−3.13440	−0.323288
$95$	16.8881	1.73268
$96$	0	0
$97$	−2.68958	−0.273086	−0.136543	−	0.990634i	$-0.543599\pi$
$97$	−2.68958	−0.273086	−0.136543	+	0.990634i	$0.543599\pi$
$98$	−4.45249	−0.449770
$99$	0	0
$100$	−9.88523	−0.988523
$101$	−13.7539	−1.36857	−0.684283	−	0.729216i	$-0.739885\pi$
$101$	−13.7539	−1.36857	−0.684283	+	0.729216i	$0.739885\pi$
$102$	0	0
$103$	−3.23880	−0.319129	−0.159564	−	0.987188i	$-0.551009\pi$
$103$	−3.23880	−0.319129	−0.159564	+	0.987188i	$0.551009\pi$
$104$	12.9711	1.27193
$105$	0	0
$106$	−1.73778	−0.168788
$107$	−6.13368	−0.592965	−0.296483	−	0.955038i	$-0.595814\pi$
$107$	−6.13368	−0.592965	−0.296483	+	0.955038i	$0.595814\pi$
$108$	0	0
$109$	0.934734	0.0895312	0.0447656	−	0.998998i	$-0.485746\pi$
$109$	0.934734	0.0895312	0.0447656	+	0.998998i	$0.485746\pi$
$110$	−6.48378	−0.618204
$111$	0	0
$112$	7.07925	0.668927
$113$	−5.78284	−0.544004	−0.272002	−	0.962297i	$-0.587686\pi$
$113$	−5.78284	−0.544004	−0.272002	+	0.962297i	$0.587686\pi$
$114$	0	0
$115$	1.08516	0.101192
$116$	−1.80405	−0.167502
$117$	0	0
$118$	−0.338187	−0.0311326
$119$	−24.2044	−2.21882
$120$	0	0
$121$	−0.970600	−0.0882363
$122$	0.0711439	0.00644106
$123$	0	0
$124$	−1.10907	−0.0995972
$125$	−3.64330	−0.325867
$126$	0	0
$127$	−4.75020	−0.421512	−0.210756	−	0.977539i	$-0.567593\pi$
$127$	−4.75020	−0.421512	−0.210756	+	0.977539i	$0.567593\pi$
$128$	11.3998	1.00761
$129$	0	0
$130$	11.9275	1.04611
$131$	8.89706	0.777339	0.388670	−	0.921377i	$-0.372935\pi$
$131$	8.89706	0.777339	0.388670	+	0.921377i	$0.372935\pi$
$132$	0	0
$133$	−19.1359	−1.65930
$134$	−8.09837	−0.699592
$135$	0	0
$136$	−14.2792	−1.22443
$137$	−3.24858	−0.277545	−0.138772	−	0.990324i	$-0.544316\pi$
$137$	−3.24858	−0.277545	−0.138772	+	0.990324i	$0.544316\pi$
$138$	0	0
$139$	−12.6131	−1.06983	−0.534913	−	0.844907i	$-0.679655\pi$
$139$	−12.6131	−1.06983	−0.534913	+	0.844907i	$0.679655\pi$
$140$	20.3914	1.72339
$141$	0	0
$142$	−5.60173	−0.470087
$143$	−18.4500	−1.54287
$144$	0	0
$145$	−3.70420	−0.307617
$146$	10.1527	0.840244
$147$	0	0
$148$	−7.61736	−0.626143
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−0.348932	−0.0283957	−0.0141978	−	0.999899i	$-0.504519\pi$
$151$	−0.348932	−0.0283957	−0.0141978	+	0.999899i	$0.504519\pi$
$152$	−11.2891	−0.915664
$153$	0	0
$154$	7.34678	0.592020
$155$	−2.27721	−0.182910
$156$	0	0
$157$	14.4226	1.15105	0.575523	−	0.817786i	$-0.304798\pi$
$157$	14.4226	1.15105	0.575523	+	0.817786i	$0.304798\pi$
$158$	10.8290	0.861510
$159$	0	0
$160$	18.6720	1.47615
$161$	−1.22960	−0.0969058
$162$	0	0
$163$	18.4114	1.44209	0.721045	−	0.692888i	$-0.243662\pi$
$163$	18.4114	1.44209	0.721045	+	0.692888i	$0.243662\pi$
$164$	−10.6829	−0.834193
$165$	0	0
$166$	−9.70150	−0.752982
$167$	13.6299	1.05472	0.527358	−	0.849643i	$-0.323183\pi$
$167$	13.6299	1.05472	0.527358	+	0.849643i	$0.323183\pi$
$168$	0	0
$169$	20.9406	1.61082
$170$	−13.1303	−1.00705
$171$	0	0
$172$	6.65879	0.507728
$173$	−23.1396	−1.75927	−0.879634	−	0.475651i	$-0.842213\pi$
$173$	−23.1396	−1.75927	−0.879634	+	0.475651i	$0.842213\pi$
$174$	0	0
$175$	22.9986	1.73853
$176$	−5.94039	−0.447774
$177$	0	0
$178$	−2.76975	−0.207601
$179$	−23.2956	−1.74120	−0.870599	−	0.491993i	$-0.836269\pi$
$179$	−23.2956	−1.74120	−0.870599	+	0.491993i	$0.836269\pi$
$180$	0	0
$181$	−9.27000	−0.689034	−0.344517	−	0.938780i	$-0.611957\pi$
$181$	−9.27000	−0.689034	−0.344517	+	0.938780i	$0.611957\pi$
$182$	−13.5151	−1.00181
$183$	0	0
$184$	−0.725390	−0.0534764
$185$	−15.6405	−1.14991
$186$	0	0
$187$	20.3106	1.48526
$188$	−8.27180	−0.603283
$189$	0	0
$190$	−10.3808	−0.753102
$191$	−24.4916	−1.77215	−0.886075	−	0.463542i	$-0.846578\pi$
$191$	−24.4916	−1.77215	−0.886075	+	0.463542i	$0.846578\pi$
$192$	0	0
$193$	−9.49923	−0.683769	−0.341885	−	0.939742i	$-0.611065\pi$
$193$	−9.49923	−0.683769	−0.341885	+	0.939742i	$0.611065\pi$
$194$	1.65323	0.118695
$195$	0	0
$196$	−11.7503	−0.839308
$197$	−4.46997	−0.318472	−0.159236	−	0.987241i	$-0.550903\pi$
$197$	−4.46997	−0.318472	−0.159236	+	0.987241i	$0.550903\pi$
$198$	0	0
$199$	−14.3416	−1.01665	−0.508325	−	0.861165i	$-0.669735\pi$
$199$	−14.3416	−1.01665	−0.508325	+	0.861165i	$0.669735\pi$
$200$	13.5678	0.959388
$201$	0	0
$202$	8.45427	0.594841
$203$	4.19723	0.294588
$204$	0	0
$205$	−21.9348	−1.53199
$206$	1.99083	0.138708
$207$	0	0
$208$	10.9279	0.757716
$209$	16.0575	1.11072
$210$	0	0
$211$	−8.45619	−0.582148	−0.291074	−	0.956701i	$-0.594013\pi$
$211$	−8.45619	−0.582148	−0.291074	+	0.956701i	$0.594013\pi$
$212$	−4.58608	−0.314973
$213$	0	0
$214$	3.77025	0.257729
$215$	13.6723	0.932441
$216$	0	0
$217$	2.58031	0.175163
$218$	−0.574563	−0.0389143
$219$	0	0
$220$	−17.1110	−1.15362
$221$	−37.3633	−2.51333
$222$	0	0
$223$	−10.6424	−0.712671	−0.356335	−	0.934358i	$-0.615974\pi$
$223$	−10.6424	−0.712671	−0.356335	+	0.934358i	$0.615974\pi$
$224$	−21.1572	−1.41363
$225$	0	0
$226$	3.55460	0.236448
$227$	14.4438	0.958668	0.479334	−	0.877633i	$-0.340878\pi$
$227$	14.4438	0.958668	0.479334	+	0.877633i	$0.340878\pi$
$228$	0	0
$229$	−26.8393	−1.77359	−0.886796	−	0.462162i	$-0.847074\pi$
$229$	−26.8393	−1.77359	−0.886796	+	0.462162i	$0.847074\pi$
$230$	−0.667028	−0.0439825
$231$	0	0
$232$	2.47612	0.162565
$233$	−19.0274	−1.24653	−0.623263	−	0.782013i	$-0.714193\pi$
$233$	−19.0274	−1.24653	−0.623263	+	0.782013i	$0.714193\pi$
$234$	0	0
$235$	−16.9842	−1.10793
$236$	−0.892489	−0.0580961
$237$	0	0
$238$	14.8780	0.964397
$239$	−14.3891	−0.930753	−0.465377	−	0.885113i	$-0.654081\pi$
$239$	−14.3891	−0.930753	−0.465377	+	0.885113i	$0.654081\pi$
$240$	0	0
$241$	17.2643	1.11209	0.556046	−	0.831151i	$-0.312318\pi$
$241$	17.2643	1.11209	0.556046	+	0.831151i	$0.312318\pi$
$242$	0.596609	0.0383515
$243$	0	0
$244$	0.187751	0.0120196
$245$	−24.1265	−1.54139
$246$	0	0
$247$	−29.5393	−1.87954
$248$	1.52223	0.0966618
$249$	0	0
$250$	2.23947	0.141636
$251$	−22.1153	−1.39590	−0.697951	−	0.716145i	$-0.745905\pi$
$251$	−22.1153	−1.39590	−0.697951	+	0.716145i	$0.745905\pi$
$252$	0	0
$253$	1.03179	0.0648680
$254$	2.91986	0.183208
$255$	0	0
$256$	−6.39591	−0.399745
$257$	21.3975	1.33474	0.667368	−	0.744728i	$-0.267421\pi$
$257$	21.3975	1.33474	0.667368	+	0.744728i	$0.267421\pi$
$258$	0	0
$259$	17.7222	1.10121
$260$	31.4772	1.95214
$261$	0	0
$262$	−5.46885	−0.337867
$263$	5.32111	0.328114	0.164057	−	0.986451i	$-0.447542\pi$
$263$	5.32111	0.328114	0.164057	+	0.986451i	$0.447542\pi$
$264$	0	0
$265$	−9.41643	−0.578447
$266$	11.7625	0.721204
$267$	0	0
$268$	−21.3719	−1.30550
$269$	−14.3517	−0.875037	−0.437519	−	0.899209i	$-0.644143\pi$
$269$	−14.3517	−0.875037	−0.437519	+	0.899209i	$0.644143\pi$
$270$	0	0
$271$	−30.0122	−1.82311	−0.911556	−	0.411175i	$-0.865118\pi$
$271$	−30.0122	−1.82311	−0.911556	+	0.411175i	$0.865118\pi$
$272$	−12.0299	−0.729421
$273$	0	0
$274$	1.99684	0.120633
$275$	−19.2987	−1.16376
$276$	0	0
$277$	−19.8405	−1.19210	−0.596050	−	0.802947i	$-0.703264\pi$
$277$	−19.8405	−1.19210	−0.596050	+	0.802947i	$0.703264\pi$
$278$	7.75300	0.464994
$279$	0	0
$280$	−27.9878	−1.67259
$281$	−1.90641	−0.113727	−0.0568633	−	0.998382i	$-0.518110\pi$
$281$	−1.90641	−0.113727	−0.0568633	+	0.998382i	$0.518110\pi$
$282$	0	0
$283$	−3.65834	−0.217466	−0.108733	−	0.994071i	$-0.534679\pi$
$283$	−3.65834	−0.217466	−0.108733	+	0.994071i	$0.534679\pi$
$284$	−14.7832	−0.877221
$285$	0	0
$286$	11.3409	0.670601
$287$	24.8544	1.46711
$288$	0	0
$289$	24.1311	1.41948
$290$	2.27690	0.133704
$291$	0	0
$292$	26.7934	1.56796
$293$	13.3936	0.782465	0.391232	−	0.920292i	$-0.372049\pi$
$293$	13.3936	0.782465	0.391232	+	0.920292i	$0.372049\pi$
$294$	0	0
$295$	−1.83252	−0.106693
$296$	10.4551	0.607689
$297$	0	0
$298$	0.614681	0.0356075
$299$	−1.89807	−0.109768
$300$	0	0
$301$	−15.4921	−0.892948
$302$	0.214482	0.0123420
$303$	0	0
$304$	−9.51081	−0.545482
$305$	0.385504	0.0220739
$306$	0	0
$307$	−15.3049	−0.873494	−0.436747	−	0.899584i	$-0.643869\pi$
$307$	−15.3049	−0.873494	−0.436747	+	0.899584i	$0.643869\pi$
$308$	19.3884	1.10476
$309$	0	0
$310$	1.39976	0.0795009
$311$	−30.7291	−1.74249	−0.871244	−	0.490850i	$-0.836686\pi$
$311$	−30.7291	−1.74249	−0.871244	+	0.490850i	$0.836686\pi$
$312$	0	0
$313$	−5.53987	−0.313132	−0.156566	−	0.987668i	$-0.550042\pi$
$313$	−5.53987	−0.313132	−0.156566	+	0.987668i	$0.550042\pi$
$314$	−8.86528	−0.500296
$315$	0	0
$316$	28.5782	1.60765
$317$	−1.30203	−0.0731292	−0.0365646	−	0.999331i	$-0.511641\pi$
$317$	−1.30203	−0.0731292	−0.0365646	+	0.999331i	$0.511641\pi$
$318$	0	0
$319$	−3.52201	−0.197195
$320$	1.01806	0.0569112
$321$	0	0
$322$	0.755810	0.0421196
$323$	32.5181	1.80935
$324$	0	0
$325$	35.5019	1.96929
$326$	−11.3171	−0.626797
$327$	0	0
$328$	14.6626	0.809607
$329$	19.2448	1.06100
$330$	0	0
$331$	−8.52115	−0.468365	−0.234183	−	0.972193i	$-0.575241\pi$
$331$	−8.52115	−0.468365	−0.234183	+	0.972193i	$0.575241\pi$
$332$	−25.6026	−1.40513
$333$	0	0
$334$	−8.37806	−0.458427
$335$	−43.8822	−2.39754
$336$	0	0
$337$	−14.9022	−0.811773	−0.405886	−	0.913924i	$-0.633037\pi$
$337$	−14.9022	−0.811773	−0.405886	+	0.913924i	$0.633037\pi$
$338$	−12.8718	−0.700134
$339$	0	0
$340$	−34.6515	−1.87924
$341$	−2.16521	−0.117253
$342$	0	0
$343$	0.919310	0.0496381
$344$	−9.13940	−0.492764
$345$	0	0
$346$	14.2234	0.764657
$347$	15.4782	0.830916	0.415458	−	0.909612i	$-0.363621\pi$
$347$	15.4782	0.830916	0.415458	+	0.909612i	$0.363621\pi$
$348$	0	0
$349$	20.3591	1.08980	0.544900	−	0.838501i	$-0.316568\pi$
$349$	20.3591	1.08980	0.544900	+	0.838501i	$0.316568\pi$
$350$	−14.1368	−0.755643
$351$	0	0
$352$	17.7536	0.946270
$353$	−9.81594	−0.522450	−0.261225	−	0.965278i	$-0.584126\pi$
$353$	−9.81594	−0.522450	−0.261225	+	0.965278i	$0.584126\pi$
$354$	0	0
$355$	−30.3538	−1.61101
$356$	−7.30947	−0.387401
$357$	0	0
$358$	14.3194	0.756803
$359$	−20.0037	−1.05575	−0.527877	−	0.849321i	$-0.677012\pi$
$359$	−20.0037	−1.05575	−0.527877	+	0.849321i	$0.677012\pi$
$360$	0	0
$361$	6.70867	0.353088
$362$	5.69809	0.299485
$363$	0	0
$364$	−35.6669	−1.86945
$365$	55.0140	2.87956
$366$	0	0
$367$	32.3408	1.68818	0.844088	−	0.536204i	$-0.180142\pi$
$367$	32.3408	1.68818	0.844088	+	0.536204i	$0.180142\pi$
$368$	−0.611126	−0.0318572
$369$	0	0
$370$	9.61390	0.499803
$371$	10.6698	0.553947
$372$	0	0
$373$	16.8825	0.874142	0.437071	−	0.899427i	$-0.356016\pi$
$373$	16.8825	0.874142	0.437071	+	0.899427i	$0.356016\pi$
$374$	−12.4845	−0.645560
$375$	0	0
$376$	11.3533	0.585502
$377$	6.47908	0.333689
$378$	0	0
$379$	32.3144	1.65988	0.829939	−	0.557855i	$-0.188375\pi$
$379$	32.3144	1.65988	0.829939	+	0.557855i	$0.188375\pi$
$380$	−27.3953	−1.40535
$381$	0	0
$382$	15.0545	0.770256
$383$	23.7465	1.21339	0.606694	−	0.794935i	$-0.292495\pi$
$383$	23.7465	1.21339	0.606694	+	0.794935i	$0.292495\pi$
$384$	0	0
$385$	39.8096	2.02889
$386$	5.83899	0.297197
$387$	0	0
$388$	4.36295	0.221495
$389$	−18.2883	−0.927254	−0.463627	−	0.886030i	$-0.653452\pi$
$389$	−18.2883	−0.927254	−0.463627	+	0.886030i	$0.653452\pi$
$390$	0	0
$391$	2.08948	0.105670
$392$	16.1277	0.814571
$393$	0	0
$394$	2.74760	0.138422
$395$	58.6787	2.95244
$396$	0	0
$397$	7.37455	0.370118	0.185059	−	0.982727i	$-0.440752\pi$
$397$	7.37455	0.370118	0.185059	+	0.982727i	$0.440752\pi$
$398$	8.81552	0.441882
$399$	0	0
$400$	11.4306	0.571530
$401$	12.3488	0.616670	0.308335	−	0.951278i	$-0.400228\pi$
$401$	12.3488	0.616670	0.308335	+	0.951278i	$0.400228\pi$
$402$	0	0
$403$	3.98311	0.198413
$404$	22.3112	1.11002
$405$	0	0
$406$	−2.57996	−0.128041
$407$	−14.8712	−0.737139
$408$	0	0
$409$	9.84930	0.487016	0.243508	−	0.969899i	$-0.421702\pi$
$409$	9.84930	0.487016	0.243508	+	0.969899i	$0.421702\pi$
$410$	13.4829	0.665873
$411$	0	0
$412$	5.25388	0.258840
$413$	2.07643	0.102174
$414$	0	0
$415$	−52.5691	−2.58051
$416$	−32.6595	−1.60126
$417$	0	0
$418$	−9.87022	−0.482768
$419$	−6.99677	−0.341815	−0.170907	−	0.985287i	$-0.554670\pi$
$419$	−6.99677	−0.341815	−0.170907	+	0.985287i	$0.554670\pi$
$420$	0	0
$421$	25.4338	1.23957	0.619784	−	0.784773i	$-0.287220\pi$
$421$	25.4338	1.23957	0.619784	+	0.784773i	$0.287220\pi$
$422$	5.19786	0.253028
$423$	0	0
$424$	6.29454	0.305690
$425$	−39.0820	−1.89575
$426$	0	0
$427$	−0.436815	−0.0211390
$428$	9.94985	0.480944
$429$	0	0
$430$	−8.40408	−0.405281
$431$	25.9494	1.24994	0.624969	−	0.780650i	$-0.285112\pi$
$431$	25.9494	1.24994	0.624969	+	0.780650i	$0.285112\pi$
$432$	0	0
$433$	−9.23489	−0.443801	−0.221900	−	0.975069i	$-0.571226\pi$
$433$	−9.23489	−0.443801	−0.221900	+	0.975069i	$0.571226\pi$
$434$	−1.58607	−0.0761337
$435$	0	0
$436$	−1.51629	−0.0726173
$437$	1.65194	0.0790228
$438$	0	0
$439$	−33.0268	−1.57628	−0.788142	−	0.615493i	$-0.788957\pi$
$439$	−33.0268	−1.57628	−0.788142	+	0.615493i	$0.788957\pi$
$440$	23.4853	1.11962
$441$	0	0
$442$	22.9665	1.09240
$443$	−3.97217	−0.188724	−0.0943618	−	0.995538i	$-0.530081\pi$
$443$	−3.97217	−0.188724	−0.0943618	+	0.995538i	$0.530081\pi$
$444$	0	0
$445$	−15.0083	−0.711462
$446$	6.54170	0.309759
$447$	0	0
$448$	−1.15356	−0.0545008
$449$	−35.7041	−1.68498	−0.842491	−	0.538711i	$-0.818911\pi$
$449$	−35.7041	−1.68498	−0.842491	+	0.538711i	$0.818911\pi$
$450$	0	0
$451$	−20.8560	−0.982069
$452$	9.38073	0.441232
$453$	0	0
$454$	−8.87832	−0.416680
$455$	−73.2337	−3.43325
$456$	0	0
$457$	33.7502	1.57877	0.789385	−	0.613899i	$-0.210400\pi$
$457$	33.7502	1.57877	0.789385	+	0.613899i	$0.210400\pi$
$458$	16.4976	0.770883
$459$	0	0
$460$	−1.76031	−0.0820750
$461$	1.50442	0.0700677	0.0350338	−	0.999386i	$-0.488846\pi$
$461$	1.50442	0.0700677	0.0350338	+	0.999386i	$0.488846\pi$
$462$	0	0
$463$	−19.5029	−0.906375	−0.453188	−	0.891415i	$-0.649713\pi$
$463$	−19.5029	−0.906375	−0.453188	+	0.891415i	$0.649713\pi$
$464$	2.08608	0.0968438
$465$	0	0
$466$	11.6958	0.541796
$467$	−25.4609	−1.17819	−0.589096	−	0.808063i	$-0.700516\pi$
$467$	−25.4609	−1.17819	−0.589096	+	0.808063i	$0.700516\pi$
$468$	0	0
$469$	49.7230	2.29600
$470$	10.4399	0.481555
$471$	0	0
$472$	1.22497	0.0563838
$473$	12.9998	0.597732
$474$	0	0
$475$	−30.8980	−1.41770
$476$	39.2636	1.79965
$477$	0	0
$478$	8.84470	0.404547
$479$	−26.6932	−1.21965	−0.609823	−	0.792538i	$-0.708759\pi$
$479$	−26.6932	−1.21965	−0.609823	+	0.792538i	$0.708759\pi$
$480$	0	0
$481$	27.3570	1.24737
$482$	−10.6120	−0.483366
$483$	0	0
$484$	1.57448	0.0715671
$485$	8.95830	0.406775
$486$	0	0
$487$	11.2238	0.508600	0.254300	−	0.967125i	$-0.418155\pi$
$487$	11.2238	0.508600	0.254300	+	0.967125i	$0.418155\pi$
$488$	−0.257695	−0.0116653
$489$	0	0
$490$	14.8301	0.669956
$491$	9.40237	0.424323	0.212162	−	0.977235i	$-0.431950\pi$
$491$	9.40237	0.424323	0.212162	+	0.977235i	$0.431950\pi$
$492$	0	0
$493$	−7.13244	−0.321229
$494$	18.1572	0.816932
$495$	0	0
$496$	1.28245	0.0575837
$497$	34.3940	1.54278
$498$	0	0
$499$	−23.7118	−1.06149	−0.530743	−	0.847533i	$-0.678087\pi$
$499$	−23.7118	−1.06149	−0.530743	+	0.847533i	$0.678087\pi$
$500$	5.91005	0.264305
$501$	0	0
$502$	13.5938	0.606722
$503$	−18.9108	−0.843189	−0.421594	−	0.906784i	$-0.638529\pi$
$503$	−18.9108	−0.843189	−0.421594	+	0.906784i	$0.638529\pi$
$504$	0	0
$505$	45.8108	2.03855
$506$	−0.634221	−0.0281945
$507$	0	0
$508$	7.70562	0.341881
$509$	19.7934	0.877328	0.438664	−	0.898651i	$-0.355452\pi$
$509$	19.7934	0.877328	0.438664	+	0.898651i	$0.355452\pi$
$510$	0	0
$511$	−62.3364	−2.75760
$512$	−18.8681	−0.833861
$513$	0	0
$514$	−13.1526	−0.580137
$515$	10.7876	0.475359
$516$	0	0
$517$	−16.1489	−0.710226
$518$	−10.8935	−0.478634
$519$	0	0
$520$	−43.2035	−1.89460
$521$	39.0038	1.70879	0.854395	−	0.519624i	$-0.173928\pi$
$521$	39.0038	1.70879	0.854395	+	0.519624i	$0.173928\pi$
$522$	0	0
$523$	38.6801	1.69136	0.845682	−	0.533687i	$-0.179194\pi$
$523$	38.6801	1.69136	0.845682	+	0.533687i	$0.179194\pi$
$524$	−14.4325	−0.630487
$525$	0	0
$526$	−3.27078	−0.142613
$527$	−4.38478	−0.191004
$528$	0	0
$529$	−22.8939	−0.995385
$530$	5.78810	0.251419
$531$	0	0
$532$	31.0417	1.34583
$533$	38.3666	1.66184
$534$	0	0
$535$	20.4297	0.883253
$536$	29.3336	1.26702
$537$	0	0
$538$	8.82170	0.380330
$539$	−22.9399	−0.988090
$540$	0	0
$541$	−26.3266	−1.13187	−0.565935	−	0.824450i	$-0.691485\pi$
$541$	−26.3266	−1.13187	−0.565935	+	0.824450i	$0.691485\pi$
$542$	18.4479	0.792407
$543$	0	0
$544$	35.9529	1.54147
$545$	−3.11336	−0.133362
$546$	0	0
$547$	−26.1105	−1.11640	−0.558202	−	0.829705i	$-0.688509\pi$
$547$	−26.1105	−1.11640	−0.558202	+	0.829705i	$0.688509\pi$
$548$	5.26974	0.225112
$549$	0	0
$550$	11.8626	0.505821
$551$	−5.63888	−0.240224
$552$	0	0
$553$	−66.4889	−2.82739
$554$	12.1956	0.518140
$555$	0	0
$556$	20.4605	0.867718
$557$	−1.75916	−0.0745378	−0.0372689	−	0.999305i	$-0.511866\pi$
$557$	−1.75916	−0.0745378	−0.0372689	+	0.999305i	$0.511866\pi$
$558$	0	0
$559$	−23.9144	−1.01147
$560$	−23.5792	−0.996402
$561$	0	0
$562$	1.17183	0.0494307
$563$	17.1135	0.721246	0.360623	−	0.932712i	$-0.382564\pi$
$563$	17.1135	0.721246	0.360623	+	0.932712i	$0.382564\pi$
$564$	0	0
$565$	19.2611	0.810322
$566$	2.24871	0.0945205
$567$	0	0
$568$	20.2904	0.851367
$569$	14.8826	0.623911	0.311956	−	0.950097i	$-0.399016\pi$
$569$	14.8826	0.623911	0.311956	+	0.950097i	$0.399016\pi$
$570$	0	0
$571$	26.7560	1.11970	0.559852	−	0.828593i	$-0.310858\pi$
$571$	26.7560	1.11970	0.559852	+	0.828593i	$0.310858\pi$
$572$	29.9291	1.25140
$573$	0	0
$574$	−15.2775	−0.637670
$575$	−1.98538	−0.0827962
$576$	0	0
$577$	−28.8391	−1.20059	−0.600294	−	0.799779i	$-0.704950\pi$
$577$	−28.8391	−1.20059	−0.600294	+	0.799779i	$0.704950\pi$
$578$	−14.8329	−0.616968
$579$	0	0
$580$	6.00883	0.249503
$581$	59.5661	2.47122
$582$	0	0
$583$	−8.95330	−0.370808
$584$	−36.7748	−1.52175
$585$	0	0
$586$	−8.23281	−0.340094
$587$	4.35948	0.179935	0.0899674	−	0.995945i	$-0.471324\pi$
$587$	4.35948	0.179935	0.0899674	+	0.995945i	$0.471324\pi$
$588$	0	0
$589$	−3.46659	−0.142838
$590$	1.12641	0.0463737
$591$	0	0
$592$	8.80819	0.362014
$593$	9.89117	0.406182	0.203091	−	0.979160i	$-0.434901\pi$
$593$	9.89117	0.406182	0.203091	+	0.979160i	$0.434901\pi$
$594$	0	0
$595$	80.6187	3.30504
$596$	1.62217	0.0664466
$597$	0	0
$598$	1.16671	0.0477103
$599$	−21.2684	−0.869005	−0.434503	−	0.900671i	$-0.643076\pi$
$599$	−21.2684	−0.869005	−0.434503	+	0.900671i	$0.643076\pi$
$600$	0	0
$601$	−2.55447	−0.104199	−0.0520995	−	0.998642i	$-0.516591\pi$
$601$	−2.55447	−0.104199	−0.0520995	+	0.998642i	$0.516591\pi$
$602$	9.52267	0.388115
$603$	0	0
$604$	0.566026	0.0230313
$605$	3.23282	0.131433
$606$	0	0
$607$	−4.09456	−0.166193	−0.0830965	−	0.996542i	$-0.526481\pi$
$607$	−4.09456	−0.166193	−0.0830965	+	0.996542i	$0.526481\pi$
$608$	28.4242	1.15276
$609$	0	0
$610$	−0.236962	−0.00959430
$611$	29.7074	1.20183
$612$	0	0
$613$	−23.8632	−0.963826	−0.481913	−	0.876219i	$-0.660058\pi$
$613$	−23.8632	−0.963826	−0.481913	+	0.876219i	$0.660058\pi$
$614$	9.40760	0.379660
$615$	0	0
$616$	−26.6113	−1.07220
$617$	−42.0102	−1.69127	−0.845633	−	0.533765i	$-0.820777\pi$
$617$	−42.0102	−1.69127	−0.845633	+	0.533765i	$0.820777\pi$
$618$	0	0
$619$	−30.8684	−1.24071	−0.620353	−	0.784323i	$-0.713011\pi$
$619$	−30.8684	−1.24071	−0.620353	+	0.784323i	$0.713011\pi$
$620$	3.69402	0.148355
$621$	0	0
$622$	18.8886	0.757364
$623$	17.0059	0.681328
$624$	0	0
$625$	−18.3343	−0.733372
$626$	3.40525	0.136101
$627$	0	0
$628$	−23.3958	−0.933595
$629$	−30.1158	−1.20079
$630$	0	0
$631$	23.1351	0.920992	0.460496	−	0.887662i	$-0.347672\pi$
$631$	23.1351	0.920992	0.460496	+	0.887662i	$0.347672\pi$
$632$	−39.2245	−1.56027
$633$	0	0
$634$	0.800332	0.0317853
$635$	15.8217	0.627864
$636$	0	0
$637$	42.2001	1.67203
$638$	2.16491	0.0857097
$639$	0	0
$640$	−37.9697	−1.50088
$641$	−43.6335	−1.72342	−0.861709	−	0.507402i	$-0.830606\pi$
$641$	−43.6335	−1.72342	−0.861709	+	0.507402i	$0.830606\pi$
$642$	0	0
$643$	34.2853	1.35208	0.676040	−	0.736865i	$-0.263695\pi$
$643$	34.2853	1.35208	0.676040	+	0.736865i	$0.263695\pi$
$644$	1.99461	0.0785987
$645$	0	0
$646$	−19.9882	−0.786427
$647$	29.4619	1.15827	0.579134	−	0.815232i	$-0.303391\pi$
$647$	29.4619	1.15827	0.579134	+	0.815232i	$0.303391\pi$
$648$	0	0
$649$	−1.74239	−0.0683947
$650$	−21.8223	−0.855942
$651$	0	0
$652$	−29.8663	−1.16966
$653$	47.0017	1.83932	0.919659	−	0.392717i	$-0.128465\pi$
$653$	47.0017	1.83932	0.919659	+	0.392717i	$0.128465\pi$
$654$	0	0
$655$	−29.6338	−1.15789
$656$	12.3529	0.482302
$657$	0	0
$658$	−11.8294	−0.461159
$659$	31.9305	1.24383	0.621917	−	0.783083i	$-0.286354\pi$
$659$	31.9305	1.24383	0.621917	+	0.783083i	$0.286354\pi$
$660$	0	0
$661$	−2.77704	−0.108014	−0.0540071	−	0.998541i	$-0.517199\pi$
$661$	−2.77704	−0.108014	−0.0540071	+	0.998541i	$0.517199\pi$
$662$	5.23779	0.203573
$663$	0	0
$664$	35.1405	1.36371
$665$	63.7368	2.47161
$666$	0	0
$667$	−0.362332	−0.0140295
$668$	−22.1100	−0.855463
$669$	0	0
$670$	26.9736	1.04208
$671$	0.366543	0.0141502
$672$	0	0
$673$	−18.1424	−0.699336	−0.349668	−	0.936874i	$-0.613706\pi$
$673$	−18.1424	−0.699336	−0.349668	+	0.936874i	$0.613706\pi$
$674$	9.16008	0.352833
$675$	0	0
$676$	−33.9692	−1.30651
$677$	42.1626	1.62044	0.810220	−	0.586125i	$-0.199347\pi$
$677$	42.1626	1.62044	0.810220	+	0.586125i	$0.199347\pi$
$678$	0	0
$679$	−10.1507	−0.389547
$680$	47.5603	1.82385
$681$	0	0
$682$	1.33091	0.0509633
$683$	6.79582	0.260035	0.130017	−	0.991512i	$-0.458497\pi$
$683$	6.79582	0.260035	0.130017	+	0.991512i	$0.458497\pi$
$684$	0	0
$685$	10.8202	0.413417
$686$	−0.565082	−0.0215749
$687$	0	0
$688$	−7.69976	−0.293551
$689$	16.4705	0.627474
$690$	0	0
$691$	−5.72339	−0.217728	−0.108864	−	0.994057i	$-0.534721\pi$
$691$	−5.72339	−0.217728	−0.108864	+	0.994057i	$0.534721\pi$
$692$	37.5362	1.42691
$693$	0	0
$694$	−9.51418	−0.361153
$695$	42.0108	1.59356
$696$	0	0
$697$	−42.2355	−1.59978
$698$	−12.5144	−0.473676
$699$	0	0
$700$	−37.3075	−1.41009
$701$	−19.9606	−0.753901	−0.376951	−	0.926233i	$-0.623027\pi$
$701$	−19.9606	−0.753901	−0.376951	+	0.926233i	$0.623027\pi$
$702$	0	0
$703$	−23.8094	−0.897989
$704$	0.967987	0.0364824
$705$	0	0
$706$	6.03367	0.227080
$707$	−51.9082	−1.95221
$708$	0	0
$709$	−30.7698	−1.15558	−0.577792	−	0.816184i	$-0.696086\pi$
$709$	−30.7698	−1.15558	−0.577792	+	0.816184i	$0.696086\pi$
$710$	18.6579	0.700219
$711$	0	0
$712$	10.0325	0.375983
$713$	−0.222749	−0.00834201
$714$	0	0
$715$	61.4524	2.29819
$716$	37.7894	1.41226
$717$	0	0
$718$	12.2959	0.458878
$719$	−37.2093	−1.38767	−0.693836	−	0.720133i	$-0.744081\pi$
$719$	−37.2093	−1.38767	−0.693836	+	0.720133i	$0.744081\pi$
$720$	0	0
$721$	−12.2235	−0.455226
$722$	−4.12369	−0.153468
$723$	0	0
$724$	15.0375	0.558864
$725$	6.77710	0.251695
$726$	0	0
$727$	48.3823	1.79440	0.897200	−	0.441625i	$-0.145598\pi$
$727$	48.3823	1.79440	0.897200	+	0.441625i	$0.145598\pi$
$728$	48.9540	1.81435
$729$	0	0
$730$	−33.8160	−1.25159
$731$	26.3260	0.973702
$732$	0	0
$733$	6.14112	0.226827	0.113414	−	0.993548i	$-0.463821\pi$
$733$	6.14112	0.226827	0.113414	+	0.993548i	$0.463821\pi$
$734$	−19.8793	−0.733758
$735$	0	0
$736$	1.82643	0.0673230
$737$	−41.7239	−1.53692
$738$	0	0
$739$	−23.6004	−0.868155	−0.434078	−	0.900876i	$-0.642926\pi$
$739$	−23.6004	−0.868155	−0.434078	+	0.900876i	$0.642926\pi$
$740$	25.3715	0.932674
$741$	0	0
$742$	−6.55850	−0.240770
$743$	40.0517	1.46935	0.734677	−	0.678417i	$-0.237334\pi$
$743$	40.0517	1.46935	0.734677	+	0.678417i	$0.237334\pi$
$744$	0	0
$745$	3.33074	0.122029
$746$	−10.3773	−0.379942
$747$	0	0
$748$	−32.9472	−1.20467
$749$	−23.1489	−0.845843
$750$	0	0
$751$	−16.0856	−0.586973	−0.293487	−	0.955963i	$-0.594816\pi$
$751$	−16.0856	−0.586973	−0.293487	+	0.955963i	$0.594816\pi$
$752$	9.56494	0.348797
$753$	0	0
$754$	−3.98256	−0.145036
$755$	1.16220	0.0422968
$756$	0	0
$757$	−24.2078	−0.879848	−0.439924	−	0.898035i	$-0.644995\pi$
$757$	−24.2078	−0.879848	−0.439924	+	0.898035i	$0.644995\pi$
$758$	−19.8630	−0.721457
$759$	0	0
$760$	37.6010	1.36393
$761$	14.4822	0.524978	0.262489	−	0.964935i	$-0.415457\pi$
$761$	14.4822	0.524978	0.262489	+	0.964935i	$0.415457\pi$
$762$	0	0
$763$	3.52775	0.127713
$764$	39.7295	1.43736
$765$	0	0
$766$	−14.5965	−0.527393
$767$	3.20529	0.115736
$768$	0	0
$769$	−5.80489	−0.209330	−0.104665	−	0.994508i	$-0.533377\pi$
$769$	−5.80489	−0.209330	−0.104665	+	0.994508i	$0.533377\pi$
$770$	−24.4702	−0.881845
$771$	0	0
$772$	15.4093	0.554594
$773$	24.1955	0.870253	0.435127	−	0.900369i	$-0.356704\pi$
$773$	24.1955	0.870253	0.435127	+	0.900369i	$0.356704\pi$
$774$	0	0
$775$	4.16633	0.149659
$776$	−5.98829	−0.214967
$777$	0	0
$778$	11.2415	0.403026
$779$	−33.3912	−1.19637
$780$	0	0
$781$	−28.8609	−1.03272
$782$	−1.28436	−0.0459287
$783$	0	0
$784$	13.5872	0.485259
$785$	−48.0379	−1.71454
$786$	0	0
$787$	−32.6210	−1.16281	−0.581407	−	0.813613i	$-0.697498\pi$
$787$	−32.6210	−1.16281	−0.581407	+	0.813613i	$0.697498\pi$
$788$	7.25104	0.258307
$789$	0	0
$790$	−36.0686	−1.28326
$791$	−21.8248	−0.776001
$792$	0	0
$793$	−0.674292	−0.0239448
$794$	−4.53300	−0.160870
$795$	0	0
$796$	23.2645	0.824589
$797$	−2.98511	−0.105738	−0.0528689	−	0.998601i	$-0.516837\pi$
$797$	−2.98511	−0.105738	−0.0528689	+	0.998601i	$0.516837\pi$
$798$	0	0
$799$	−32.7031	−1.15695
$800$	−34.1618	−1.20780
$801$	0	0
$802$	−7.59057	−0.268032
$803$	52.3082	1.84591
$804$	0	0
$805$	4.09547	0.144346
$806$	−2.44834	−0.0862392
$807$	0	0
$808$	−30.6228	−1.07731
$809$	−37.3160	−1.31196	−0.655980	−	0.754778i	$-0.727745\pi$
$809$	−37.3160	−1.31196	−0.655980	+	0.754778i	$0.727745\pi$
$810$	0	0
$811$	10.8265	0.380171	0.190086	−	0.981768i	$-0.439123\pi$
$811$	10.8265	0.380171	0.190086	+	0.981768i	$0.439123\pi$
$812$	−6.80861	−0.238935
$813$	0	0
$814$	9.14105	0.320394
$815$	−61.3235	−2.14807
$816$	0	0
$817$	20.8132	0.728163
$818$	−6.05417	−0.211679
$819$	0	0
$820$	35.5819	1.24257
$821$	−31.8704	−1.11228	−0.556142	−	0.831087i	$-0.687719\pi$
$821$	−31.8704	−1.11228	−0.556142	+	0.831087i	$0.687719\pi$
$822$	0	0
$823$	13.0962	0.456506	0.228253	−	0.973602i	$-0.426699\pi$
$823$	13.0962	0.456506	0.228253	+	0.973602i	$0.426699\pi$
$824$	−7.21112	−0.251211
$825$	0	0
$826$	−1.27634	−0.0444096
$827$	−1.69446	−0.0589222	−0.0294611	−	0.999566i	$-0.509379\pi$
$827$	−1.69446	−0.0589222	−0.0294611	+	0.999566i	$0.509379\pi$
$828$	0	0
$829$	12.1194	0.420925	0.210463	−	0.977602i	$-0.432503\pi$
$829$	12.1194	0.420925	0.210463	+	0.977602i	$0.432503\pi$
$830$	32.3132	1.12161
$831$	0	0
$832$	−1.78071	−0.0617348
$833$	−46.4557	−1.60959
$834$	0	0
$835$	−45.3978	−1.57105
$836$	−26.0479	−0.900886
$837$	0	0
$838$	4.30078	0.148568
$839$	−43.2576	−1.49342	−0.746710	−	0.665150i	$-0.768368\pi$
$839$	−43.2576	−1.49342	−0.746710	+	0.665150i	$0.768368\pi$
$840$	0	0
$841$	−27.7632	−0.957351
$842$	−15.6337	−0.538772
$843$	0	0
$844$	13.7173	0.472171
$845$	−69.7479	−2.39940
$846$	0	0
$847$	−3.66311	−0.125866
$848$	5.30302	0.182106
$849$	0	0
$850$	24.0229	0.823980
$851$	−1.52990	−0.0524442
$852$	0	0
$853$	19.7436	0.676007	0.338003	−	0.941145i	$-0.390248\pi$
$853$	19.7436	0.676007	0.338003	+	0.941145i	$0.390248\pi$
$854$	0.268502	0.00918794
$855$	0	0
$856$	−13.6565	−0.466769
$857$	27.1626	0.927855	0.463928	−	0.885873i	$-0.346440\pi$
$857$	27.1626	0.927855	0.463928	+	0.885873i	$0.346440\pi$
$858$	0	0
$859$	−22.7652	−0.776740	−0.388370	−	0.921504i	$-0.626962\pi$
$859$	−22.7652	−0.776740	−0.388370	+	0.921504i	$0.626962\pi$
$860$	−22.1787	−0.756288
$861$	0	0
$862$	−15.9506	−0.543279
$863$	−52.7312	−1.79499	−0.897496	−	0.441023i	$-0.854616\pi$
$863$	−52.7312	−1.79499	−0.897496	+	0.441023i	$0.854616\pi$
$864$	0	0
$865$	77.0719	2.62052
$866$	5.67651	0.192896
$867$	0	0
$868$	−4.18570	−0.142072
$869$	55.7926	1.89263
$870$	0	0
$871$	76.7552	2.60075
$872$	2.08116	0.0704771
$873$	0	0
$874$	−1.01541	−0.0343469
$875$	−13.7501	−0.464837
$876$	0	0
$877$	−46.6117	−1.57396	−0.786982	−	0.616976i	$-0.788358\pi$
$877$	−46.6117	−1.57396	−0.786982	+	0.616976i	$0.788358\pi$
$878$	20.3010	0.685124
$879$	0	0
$880$	19.7859	0.666983
$881$	−40.8308	−1.37563	−0.687813	−	0.725888i	$-0.741429\pi$
$881$	−40.8308	−1.37563	−0.687813	+	0.725888i	$0.741429\pi$
$882$	0	0
$883$	−27.5992	−0.928786	−0.464393	−	0.885629i	$-0.653727\pi$
$883$	−27.5992	−0.928786	−0.464393	+	0.885629i	$0.653727\pi$
$884$	60.6095	2.03852
$885$	0	0
$886$	2.44162	0.0820278
$887$	−53.5299	−1.79736	−0.898680	−	0.438605i	$-0.855473\pi$
$887$	−53.5299	−1.79736	−0.898680	+	0.438605i	$0.855473\pi$
$888$	0	0
$889$	−17.9276	−0.601271
$890$	9.22531	0.309233
$891$	0	0
$892$	17.2638	0.578036
$893$	−25.8550	−0.865204
$894$	0	0
$895$	77.5918	2.59361
$896$	43.0235	1.43732
$897$	0	0
$898$	21.9466	0.732369
$899$	0.760353	0.0253592
$900$	0	0
$901$	−18.1314	−0.604043
$902$	12.8198	0.426851
$903$	0	0
$904$	−12.8754	−0.428228
$905$	30.8760	1.02635
$906$	0	0
$907$	−24.6130	−0.817262	−0.408631	−	0.912700i	$-0.633994\pi$
$907$	−24.6130	−0.817262	−0.408631	+	0.912700i	$0.633994\pi$
$908$	−23.4302	−0.777560
$909$	0	0
$910$	45.0153	1.49224
$911$	23.7889	0.788161	0.394081	−	0.919076i	$-0.371063\pi$
$911$	23.7889	0.788161	0.394081	+	0.919076i	$0.371063\pi$
$912$	0	0
$913$	−49.9835	−1.65421
$914$	−20.7456	−0.686204
$915$	0	0
$916$	43.5378	1.43853
$917$	33.5781	1.10885
$918$	0	0
$919$	12.5034	0.412449	0.206224	−	0.978505i	$-0.433882\pi$
$919$	12.5034	0.412449	0.206224	+	0.978505i	$0.433882\pi$
$920$	2.41609	0.0796560
$921$	0	0
$922$	−0.924736	−0.0304546
$923$	53.0924	1.74756
$924$	0	0
$925$	28.6154	0.940869
$926$	11.9880	0.393951
$927$	0	0
$928$	−6.23451	−0.204658
$929$	10.8042	0.354474	0.177237	−	0.984168i	$-0.443284\pi$
$929$	10.8042	0.354474	0.177237	+	0.984168i	$0.443284\pi$
$930$	0	0
$931$	−36.7277	−1.20370
$932$	30.8656	1.01104
$933$	0	0
$934$	15.6503	0.512095
$935$	−67.6493	−2.21237
$936$	0	0
$937$	−24.4052	−0.797283	−0.398642	−	0.917107i	$-0.630518\pi$
$937$	−24.4052	−0.797283	−0.398642	+	0.917107i	$0.630518\pi$
$938$	−30.5638	−0.997943
$939$	0	0
$940$	27.5512	0.898622
$941$	−51.5189	−1.67947	−0.839734	−	0.542998i	$-0.817289\pi$
$941$	−51.5189	−1.67947	−0.839734	+	0.542998i	$0.817289\pi$
$942$	0	0
$943$	−2.14559	−0.0698699
$944$	1.03201	0.0335892
$945$	0	0
$946$	−7.99073	−0.259801
$947$	31.0735	1.00975	0.504877	−	0.863191i	$-0.331538\pi$
$947$	31.0735	1.00975	0.504877	+	0.863191i	$0.331538\pi$
$948$	0	0
$949$	−96.2259	−3.12362
$950$	18.9924	0.616196
$951$	0	0
$952$	−53.8906	−1.74660
$953$	14.9924	0.485651	0.242825	−	0.970070i	$-0.421926\pi$
$953$	14.9924	0.485651	0.242825	+	0.970070i	$0.421926\pi$
$954$	0	0
$955$	81.5752	2.63971
$956$	23.3415	0.754919
$957$	0	0
$958$	16.4078	0.530113
$959$	−12.2603	−0.395907
$960$	0	0
$961$	−30.5326	−0.984921
$962$	−16.8158	−0.542164
$963$	0	0
$964$	−28.0056	−0.902001
$965$	31.6395	1.01851
$966$	0	0
$967$	16.9269	0.544332	0.272166	−	0.962250i	$-0.412260\pi$
$967$	16.9269	0.544332	0.272166	+	0.962250i	$0.412260\pi$
$968$	−2.16102	−0.0694577
$969$	0	0
$970$	−5.50650	−0.176803
$971$	−11.6172	−0.372813	−0.186407	−	0.982473i	$-0.559684\pi$
$971$	−11.6172	−0.372813	−0.186407	+	0.982473i	$0.559684\pi$
$972$	0	0
$973$	−47.6025	−1.52607
$974$	−6.89907	−0.221060
$975$	0	0
$976$	−0.217103	−0.00694929
$977$	−4.74070	−0.151669	−0.0758343	−	0.997120i	$-0.524162\pi$
$977$	−4.74070	−0.151669	−0.0758343	+	0.997120i	$0.524162\pi$
$978$	0	0
$979$	−14.2701	−0.456075
$980$	39.1372	1.25019
$981$	0	0
$982$	−5.77946	−0.184430
$983$	9.59968	0.306182	0.153091	−	0.988212i	$-0.451077\pi$
$983$	9.59968	0.306182	0.153091	+	0.988212i	$0.451077\pi$
$984$	0	0
$985$	14.8883	0.474381
$986$	4.38417	0.139621
$987$	0	0
$988$	47.9177	1.52446
$989$	1.33737	0.0425260
$990$	0	0
$991$	19.1373	0.607917	0.303958	−	0.952685i	$-0.401692\pi$
$991$	19.1373	0.607917	0.303958	+	0.952685i	$0.401692\pi$
$992$	−3.83276	−0.121690
$993$	0	0
$994$	−21.1413	−0.670562
$995$	47.7682	1.51436
$996$	0	0
$997$	48.7582	1.54419	0.772093	−	0.635509i	$-0.219210\pi$
$997$	48.7582	1.54419	0.772093	+	0.635509i	$0.219210\pi$
$998$	14.5752	0.461369
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4023.2.a.e.1.12	✓	25
3.2	odd	2		4023.2.a.f.1.14	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4023.2.a.e.1.12	✓	25	1.1	even	1	trivial
4023.2.a.f.1.14	yes	25	3.2	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:	\( N \)	\(=\)	\( 4023 = 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4023.a (trivial)

\(f(q)\)	\(=\)	\(q-0.614681 q^{2} -1.62217 q^{4} -3.33074 q^{5} +3.77407 q^{7} +2.22648 q^{8} +O(q^{10})\)	\(q-0.614681 q^{2} -1.62217 q^{4} -3.33074 q^{5} +3.77407 q^{7} +2.22648 q^{8} +2.04734 q^{10} -3.16692 q^{11} +5.82586 q^{13} -2.31985 q^{14} +1.87576 q^{16} -6.41335 q^{17} -5.07037 q^{19} +5.40302 q^{20} +1.94665 q^{22} -0.325802 q^{23} +6.09384 q^{25} -3.58104 q^{26} -6.12217 q^{28} +1.11212 q^{29} +0.683695 q^{31} -5.60595 q^{32} +3.94216 q^{34} -12.5704 q^{35} +4.69579 q^{37} +3.11666 q^{38} -7.41582 q^{40} +6.58556 q^{41} -4.10487 q^{43} +5.13728 q^{44} +0.200264 q^{46} +5.09923 q^{47} +7.24359 q^{49} -3.74577 q^{50} -9.45052 q^{52} +2.82713 q^{53} +10.5482 q^{55} +8.40287 q^{56} -0.683601 q^{58} +0.550183 q^{59} -0.115741 q^{61} -0.420254 q^{62} -0.305655 q^{64} -19.4044 q^{65} +13.1749 q^{67} +10.4035 q^{68} +7.72681 q^{70} +9.11323 q^{71} -16.5170 q^{73} -2.88641 q^{74} +8.22499 q^{76} -11.9522 q^{77} -17.6173 q^{79} -6.24768 q^{80} -4.04802 q^{82} +15.7830 q^{83} +21.3612 q^{85} +2.52319 q^{86} -7.05108 q^{88} +4.50599 q^{89} +21.9872 q^{91} +0.528505 q^{92} -3.13440 q^{94} +16.8881 q^{95} -2.68958 q^{97} -4.45249 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8}+O(q^{10}) \) 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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