LMFDB - Embedded newform 4024.2.a.f.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.1318017734$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.368053 q^{3} +2.18272 q^{5} +4.72026 q^{7} -2.86454 q^{9} +O(q^{10})$	$q+0.368053 q^{3} +2.18272 q^{5} +4.72026 q^{7} -2.86454 q^{9} +3.40506 q^{11} -4.21858 q^{13} +0.803356 q^{15} -1.82044 q^{17} +3.78676 q^{19} +1.73731 q^{21} +7.58438 q^{23} -0.235736 q^{25} -2.15846 q^{27} -8.32141 q^{29} -1.94502 q^{31} +1.25324 q^{33} +10.3030 q^{35} +8.92717 q^{37} -1.55266 q^{39} -1.04154 q^{41} +4.41479 q^{43} -6.25248 q^{45} +9.78750 q^{47} +15.2809 q^{49} -0.670019 q^{51} +11.7070 q^{53} +7.43229 q^{55} +1.39373 q^{57} -3.47206 q^{59} -5.21182 q^{61} -13.5214 q^{63} -9.20797 q^{65} -0.364754 q^{67} +2.79145 q^{69} -6.01242 q^{71} +9.30781 q^{73} -0.0867632 q^{75} +16.0728 q^{77} -6.94128 q^{79} +7.79918 q^{81} +13.8604 q^{83} -3.97352 q^{85} -3.06272 q^{87} -8.66640 q^{89} -19.9128 q^{91} -0.715871 q^{93} +8.26544 q^{95} -8.05012 q^{97} -9.75392 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) $ 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9	$ 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9} + 22 q^{11} + 25 q^{13} - 4 q^{15} + 17 q^{17} + 6 q^{19} + 18 q^{21} + 16 q^{23} + 47 q^{25} - 20 q^{27} + 47 q^{29} - 7 q^{31} - 6 q^{33} + 19 q^{35} + 75 q^{37} + 21 q^{39} + 22 q^{41} - 5 q^{43} + 33 q^{45} + 10 q^{47} + 31 q^{49} + 9 q^{51} + 64 q^{53} - 3 q^{55} + 5 q^{57} + 28 q^{59} + 49 q^{61} - 10 q^{63} + 46 q^{65} - 14 q^{67} + 30 q^{69} + 35 q^{71} + 19 q^{73} - 33 q^{75} + 32 q^{77} - 12 q^{79} + 57 q^{81} + 82 q^{85} - 5 q^{87} + 42 q^{89} - 15 q^{91} + 55 q^{93} + 33 q^{95} + 4 q^{97} + 22 q^{99}+O(q^{100}) $ 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9 + 22 * q^11 + 25 * q^13 - 4 * q^15 + 17 * q^17 + 6 * q^19 + 18 * q^21 + 16 * q^23 + 47 * q^25 - 20 * q^27 + 47 * q^29 - 7 * q^31 - 6 * q^33 + 19 * q^35 + 75 * q^37 + 21 * q^39 + 22 * q^41 - 5 * q^43 + 33 * q^45 + 10 * q^47 + 31 * q^49 + 9 * q^51 + 64 * q^53 - 3 * q^55 + 5 * q^57 + 28 * q^59 + 49 * q^61 - 10 * q^63 + 46 * q^65 - 14 * q^67 + 30 * q^69 + 35 * q^71 + 19 * q^73 - 33 * q^75 + 32 * q^77 - 12 * q^79 + 57 * q^81 + 82 * q^85 - 5 * q^87 + 42 * q^89 - 15 * q^91 + 55 * q^93 + 33 * q^95 + 4 * q^97 + 22 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.368053	0.212495	0.106248	−	0.994340i	$-0.466116\pi$
$3$	0.368053	0.212495	0.106248	+	0.994340i	$0.466116\pi$
$4$	0	0
$5$	2.18272	0.976142	0.488071	−	0.872804i	$-0.337701\pi$
$5$	2.18272	0.976142	0.488071	+	0.872804i	$0.337701\pi$
$6$	0	0
$7$	4.72026	1.78409	0.892045	−	0.451946i	$-0.149270\pi$
$7$	4.72026	1.78409	0.892045	+	0.451946i	$0.149270\pi$
$8$	0	0
$9$	−2.86454	−0.954846
$10$	0	0
$11$	3.40506	1.02666	0.513332	−	0.858190i	$-0.328411\pi$
$11$	3.40506	1.02666	0.513332	+	0.858190i	$0.328411\pi$
$12$	0	0
$13$	−4.21858	−1.17002	−0.585011	−	0.811025i	$-0.698910\pi$
$13$	−4.21858	−1.17002	−0.585011	+	0.811025i	$0.698910\pi$
$14$	0	0
$15$	0.803356	0.207426
$16$	0	0
$17$	−1.82044	−0.441522	−0.220761	−	0.975328i	$-0.570854\pi$
$17$	−1.82044	−0.441522	−0.220761	+	0.975328i	$0.570854\pi$
$18$	0	0
$19$	3.78676	0.868743	0.434372	−	0.900734i	$-0.356970\pi$
$19$	3.78676	0.868743	0.434372	+	0.900734i	$0.356970\pi$
$20$	0	0
$21$	1.73731	0.379111
$22$	0	0
$23$	7.58438	1.58145	0.790726	−	0.612170i	$-0.209703\pi$
$23$	7.58438	1.58145	0.790726	+	0.612170i	$0.209703\pi$
$24$	0	0
$25$	−0.235736	−0.0471471
$26$	0	0
$27$	−2.15846	−0.415396
$28$	0	0
$29$	−8.32141	−1.54525	−0.772623	−	0.634865i	$-0.781056\pi$
$29$	−8.32141	−1.54525	−0.772623	+	0.634865i	$0.781056\pi$
$30$	0	0
$31$	−1.94502	−0.349336	−0.174668	−	0.984627i	$-0.555885\pi$
$31$	−1.94502	−0.349336	−0.174668	+	0.984627i	$0.555885\pi$
$32$	0	0
$33$	1.25324	0.218162
$34$	0	0
$35$	10.3030	1.74153
$36$	0	0
$37$	8.92717	1.46762	0.733809	−	0.679356i	$-0.237741\pi$
$37$	8.92717	1.46762	0.733809	+	0.679356i	$0.237741\pi$
$38$	0	0
$39$	−1.55266	−0.248624
$40$	0	0
$41$	−1.04154	−0.162661	−0.0813304	−	0.996687i	$-0.525917\pi$
$41$	−1.04154	−0.162661	−0.0813304	+	0.996687i	$0.525917\pi$
$42$	0	0
$43$	4.41479	0.673250	0.336625	−	0.941639i	$-0.390715\pi$
$43$	4.41479	0.673250	0.336625	+	0.941639i	$0.390715\pi$
$44$	0	0
$45$	−6.25248	−0.932065
$46$	0	0
$47$	9.78750	1.42765	0.713827	−	0.700322i	$-0.246960\pi$
$47$	9.78750	1.42765	0.713827	+	0.700322i	$0.246960\pi$
$48$	0	0
$49$	15.2809	2.18298
$50$	0	0
$51$	−0.670019	−0.0938215
$52$	0	0
$53$	11.7070	1.60808	0.804041	−	0.594574i	$-0.202679\pi$
$53$	11.7070	1.60808	0.804041	+	0.594574i	$0.202679\pi$
$54$	0	0
$55$	7.43229	1.00217
$56$	0	0
$57$	1.39373	0.184604
$58$	0	0
$59$	−3.47206	−0.452024	−0.226012	−	0.974125i	$-0.572569\pi$
$59$	−3.47206	−0.452024	−0.226012	+	0.974125i	$0.572569\pi$
$60$	0	0
$61$	−5.21182	−0.667305	−0.333653	−	0.942696i	$-0.608281\pi$
$61$	−5.21182	−0.667305	−0.333653	+	0.942696i	$0.608281\pi$
$62$	0	0
$63$	−13.5214	−1.70353
$64$	0	0
$65$	−9.20797	−1.14211
$66$	0	0
$67$	−0.364754	−0.0445618	−0.0222809	−	0.999752i	$-0.507093\pi$
$67$	−0.364754	−0.0445618	−0.0222809	+	0.999752i	$0.507093\pi$
$68$	0	0
$69$	2.79145	0.336052
$70$	0	0
$71$	−6.01242	−0.713543	−0.356771	−	0.934192i	$-0.616122\pi$
$71$	−6.01242	−0.713543	−0.356771	+	0.934192i	$0.616122\pi$
$72$	0	0
$73$	9.30781	1.08940	0.544698	−	0.838632i	$-0.316644\pi$
$73$	9.30781	1.08940	0.544698	+	0.838632i	$0.316644\pi$
$74$	0	0
$75$	−0.0867632	−0.0100185
$76$	0	0
$77$	16.0728	1.83166
$78$	0	0
$79$	−6.94128	−0.780955	−0.390478	−	0.920612i	$-0.627690\pi$
$79$	−6.94128	−0.780955	−0.390478	+	0.920612i	$0.627690\pi$
$80$	0	0
$81$	7.79918	0.866576
$82$	0	0
$83$	13.8604	1.52138	0.760690	−	0.649115i	$-0.224861\pi$
$83$	13.8604	1.52138	0.760690	+	0.649115i	$0.224861\pi$
$84$	0	0
$85$	−3.97352	−0.430988
$86$	0	0
$87$	−3.06272	−0.328358
$88$	0	0
$89$	−8.66640	−0.918637	−0.459318	−	0.888272i	$-0.651906\pi$
$89$	−8.66640	−0.918637	−0.459318	+	0.888272i	$0.651906\pi$
$90$	0	0
$91$	−19.9128	−2.08743
$92$	0	0
$93$	−0.715871	−0.0742324
$94$	0	0
$95$	8.26544	0.848017
$96$	0	0
$97$	−8.05012	−0.817366	−0.408683	−	0.912676i	$-0.634012\pi$
$97$	−8.05012	−0.817366	−0.408683	+	0.912676i	$0.634012\pi$
$98$	0	0
$99$	−9.75392	−0.980306
$100$	0	0
$101$	−2.99868	−0.298380	−0.149190	−	0.988809i	$-0.547667\pi$
$101$	−2.99868	−0.298380	−0.149190	+	0.988809i	$0.547667\pi$
$102$	0	0
$103$	10.4224	1.02695	0.513474	−	0.858105i	$-0.328358\pi$
$103$	10.4224	1.02695	0.513474	+	0.858105i	$0.328358\pi$
$104$	0	0
$105$	3.79205	0.370066
$106$	0	0
$107$	−0.825509	−0.0798050	−0.0399025	−	0.999204i	$-0.512705\pi$
$107$	−0.825509	−0.0798050	−0.0399025	+	0.999204i	$0.512705\pi$
$108$	0	0
$109$	16.7426	1.60365	0.801825	−	0.597559i	$-0.203863\pi$
$109$	16.7426	1.60365	0.801825	+	0.597559i	$0.203863\pi$
$110$	0	0
$111$	3.28567	0.311862
$112$	0	0
$113$	−2.42295	−0.227932	−0.113966	−	0.993485i	$-0.536355\pi$
$113$	−2.42295	−0.227932	−0.113966	+	0.993485i	$0.536355\pi$
$114$	0	0
$115$	16.5546	1.54372
$116$	0	0
$117$	12.0843	1.11719
$118$	0	0
$119$	−8.59296	−0.787716
$120$	0	0
$121$	0.594443	0.0540402
$122$	0	0
$123$	−0.383341	−0.0345647
$124$	0	0
$125$	−11.4281	−1.02216
$126$	0	0
$127$	−19.2656	−1.70954	−0.854772	−	0.519004i	$-0.826303\pi$
$127$	−19.2656	−1.70954	−0.854772	+	0.519004i	$0.826303\pi$
$128$	0	0
$129$	1.62488	0.143063
$130$	0	0
$131$	8.73513	0.763192	0.381596	−	0.924329i	$-0.375375\pi$
$131$	8.73513	0.763192	0.381596	+	0.924329i	$0.375375\pi$
$132$	0	0
$133$	17.8745	1.54992
$134$	0	0
$135$	−4.71131	−0.405485
$136$	0	0
$137$	22.3117	1.90622	0.953110	−	0.302623i	$-0.0978622\pi$
$137$	22.3117	1.90622	0.953110	+	0.302623i	$0.0978622\pi$
$138$	0	0
$139$	−5.86288	−0.497283	−0.248641	−	0.968596i	$-0.579984\pi$
$139$	−5.86288	−0.497283	−0.248641	+	0.968596i	$0.579984\pi$
$140$	0	0
$141$	3.60232	0.303370
$142$	0	0
$143$	−14.3645	−1.20122
$144$	0	0
$145$	−18.1633	−1.50838
$146$	0	0
$147$	5.62416	0.463873
$148$	0	0
$149$	−20.9130	−1.71326	−0.856631	−	0.515929i	$-0.827447\pi$
$149$	−20.9130	−1.71326	−0.856631	+	0.515929i	$0.827447\pi$
$150$	0	0
$151$	3.77929	0.307554	0.153777	−	0.988106i	$-0.450856\pi$
$151$	3.77929	0.307554	0.153777	+	0.988106i	$0.450856\pi$
$152$	0	0
$153$	5.21473	0.421586
$154$	0	0
$155$	−4.24544	−0.341002
$156$	0	0
$157$	3.17010	0.253002	0.126501	−	0.991966i	$-0.459625\pi$
$157$	3.17010	0.253002	0.126501	+	0.991966i	$0.459625\pi$
$158$	0	0
$159$	4.30880	0.341710
$160$	0	0
$161$	35.8002	2.82145
$162$	0	0
$163$	0.648951	0.0508298	0.0254149	−	0.999677i	$-0.491909\pi$
$163$	0.648951	0.0508298	0.0254149	+	0.999677i	$0.491909\pi$
$164$	0	0
$165$	2.73548	0.212957
$166$	0	0
$167$	−7.50595	−0.580828	−0.290414	−	0.956901i	$-0.593793\pi$
$167$	−7.50595	−0.580828	−0.290414	+	0.956901i	$0.593793\pi$
$168$	0	0
$169$	4.79638	0.368952
$170$	0	0
$171$	−10.8473	−0.829516
$172$	0	0
$173$	−23.0641	−1.75353	−0.876766	−	0.480917i	$-0.840304\pi$
$173$	−23.0641	−1.75353	−0.876766	+	0.480917i	$0.840304\pi$
$174$	0	0
$175$	−1.11273	−0.0841147
$176$	0	0
$177$	−1.27790	−0.0960531
$178$	0	0
$179$	−18.6835	−1.39647	−0.698236	−	0.715868i	$-0.746031\pi$
$179$	−18.6835	−1.39647	−0.698236	+	0.715868i	$0.746031\pi$
$180$	0	0
$181$	−24.5907	−1.82781	−0.913906	−	0.405927i	$-0.866949\pi$
$181$	−24.5907	−1.82781	−0.913906	+	0.405927i	$0.866949\pi$
$182$	0	0
$183$	−1.91823	−0.141799
$184$	0	0
$185$	19.4855	1.43260
$186$	0	0
$187$	−6.19872	−0.453295
$188$	0	0
$189$	−10.1885	−0.741104
$190$	0	0
$191$	−6.18341	−0.447416	−0.223708	−	0.974656i	$-0.571816\pi$
$191$	−6.18341	−0.447416	−0.223708	+	0.974656i	$0.571816\pi$
$192$	0	0
$193$	19.1868	1.38110	0.690548	−	0.723287i	$-0.257369\pi$
$193$	19.1868	1.38110	0.690548	+	0.723287i	$0.257369\pi$
$194$	0	0
$195$	−3.38902	−0.242693
$196$	0	0
$197$	9.95404	0.709196	0.354598	−	0.935019i	$-0.384618\pi$
$197$	9.95404	0.709196	0.354598	+	0.935019i	$0.384618\pi$
$198$	0	0
$199$	5.24500	0.371809	0.185904	−	0.982568i	$-0.440479\pi$
$199$	5.24500	0.371809	0.185904	+	0.982568i	$0.440479\pi$
$200$	0	0
$201$	−0.134249	−0.00946919
$202$	0	0
$203$	−39.2792	−2.75686
$204$	0	0
$205$	−2.27338	−0.158780
$206$	0	0
$207$	−21.7257	−1.51004
$208$	0	0
$209$	12.8942	0.891908
$210$	0	0
$211$	8.52487	0.586876	0.293438	−	0.955978i	$-0.405201\pi$
$211$	8.52487	0.586876	0.293438	+	0.955978i	$0.405201\pi$
$212$	0	0
$213$	−2.21289	−0.151625
$214$	0	0
$215$	9.63626	0.657187
$216$	0	0
$217$	−9.18101	−0.623247
$218$	0	0
$219$	3.42577	0.231492
$220$	0	0
$221$	7.67968	0.516591
$222$	0	0
$223$	−18.8074	−1.25943	−0.629717	−	0.776825i	$-0.716829\pi$
$223$	−18.8074	−1.25943	−0.629717	+	0.776825i	$0.716829\pi$
$224$	0	0
$225$	0.675273	0.0450182
$226$	0	0
$227$	16.4088	1.08909	0.544544	−	0.838732i	$-0.316703\pi$
$227$	16.4088	1.08909	0.544544	+	0.838732i	$0.316703\pi$
$228$	0	0
$229$	−0.416235	−0.0275056	−0.0137528	−	0.999905i	$-0.504378\pi$
$229$	−0.416235	−0.0275056	−0.0137528	+	0.999905i	$0.504378\pi$
$230$	0	0
$231$	5.91563	0.389220
$232$	0	0
$233$	−20.4309	−1.33848	−0.669238	−	0.743048i	$-0.733379\pi$
$233$	−20.4309	−1.33848	−0.669238	+	0.743048i	$0.733379\pi$
$234$	0	0
$235$	21.3634	1.39359
$236$	0	0
$237$	−2.55476	−0.165949
$238$	0	0
$239$	15.1071	0.977195	0.488597	−	0.872509i	$-0.337509\pi$
$239$	15.1071	0.977195	0.488597	+	0.872509i	$0.337509\pi$
$240$	0	0
$241$	−12.9140	−0.831864	−0.415932	−	0.909396i	$-0.636545\pi$
$241$	−12.9140	−0.831864	−0.415932	+	0.909396i	$0.636545\pi$
$242$	0	0
$243$	9.34589	0.599539
$244$	0	0
$245$	33.3538	2.13090
$246$	0	0
$247$	−15.9747	−1.01645
$248$	0	0
$249$	5.10137	0.323286
$250$	0	0
$251$	17.6571	1.11451	0.557253	−	0.830343i	$-0.311856\pi$
$251$	17.6571	1.11451	0.557253	+	0.830343i	$0.311856\pi$
$252$	0	0
$253$	25.8253	1.62362
$254$	0	0
$255$	−1.46246	−0.0915831
$256$	0	0
$257$	−11.0672	−0.690350	−0.345175	−	0.938538i	$-0.612181\pi$
$257$	−11.0672	−0.690350	−0.345175	+	0.938538i	$0.612181\pi$
$258$	0	0
$259$	42.1386	2.61836
$260$	0	0
$261$	23.8370	1.47547
$262$	0	0
$263$	4.80105	0.296045	0.148023	−	0.988984i	$-0.452709\pi$
$263$	4.80105	0.296045	0.148023	+	0.988984i	$0.452709\pi$
$264$	0	0
$265$	25.5531	1.56972
$266$	0	0
$267$	−3.18970	−0.195206
$268$	0	0
$269$	−18.6066	−1.13446	−0.567231	−	0.823559i	$-0.691985\pi$
$269$	−18.6066	−1.13446	−0.567231	+	0.823559i	$0.691985\pi$
$270$	0	0
$271$	14.4221	0.876079	0.438039	−	0.898956i	$-0.355673\pi$
$271$	14.4221	0.876079	0.438039	+	0.898956i	$0.355673\pi$
$272$	0	0
$273$	−7.32896	−0.443569
$274$	0	0
$275$	−0.802694	−0.0484043
$276$	0	0
$277$	19.3137	1.16045	0.580224	−	0.814457i	$-0.302965\pi$
$277$	19.3137	1.16045	0.580224	+	0.814457i	$0.302965\pi$
$278$	0	0
$279$	5.57159	0.333562
$280$	0	0
$281$	27.8638	1.66222	0.831108	−	0.556111i	$-0.187707\pi$
$281$	27.8638	1.66222	0.831108	+	0.556111i	$0.187707\pi$
$282$	0	0
$283$	−5.47167	−0.325257	−0.162629	−	0.986687i	$-0.551997\pi$
$283$	−5.47167	−0.325257	−0.162629	+	0.986687i	$0.551997\pi$
$284$	0	0
$285$	3.04212	0.180200
$286$	0	0
$287$	−4.91632	−0.290201
$288$	0	0
$289$	−13.6860	−0.805058
$290$	0	0
$291$	−2.96287	−0.173687
$292$	0	0
$293$	16.9301	0.989067	0.494534	−	0.869158i	$-0.335339\pi$
$293$	16.9301	0.989067	0.494534	+	0.869158i	$0.335339\pi$
$294$	0	0
$295$	−7.57854	−0.441240
$296$	0	0
$297$	−7.34969	−0.426472
$298$	0	0
$299$	−31.9953	−1.85034
$300$	0	0
$301$	20.8390	1.20114
$302$	0	0
$303$	−1.10367	−0.0634044
$304$	0	0
$305$	−11.3759	−0.651385
$306$	0	0
$307$	32.3514	1.84639	0.923195	−	0.384331i	$-0.125568\pi$
$307$	32.3514	1.84639	0.923195	+	0.384331i	$0.125568\pi$
$308$	0	0
$309$	3.83599	0.218222
$310$	0	0
$311$	9.54389	0.541184	0.270592	−	0.962694i	$-0.412780\pi$
$311$	9.54389	0.541184	0.270592	+	0.962694i	$0.412780\pi$
$312$	0	0
$313$	−23.0962	−1.30547	−0.652737	−	0.757585i	$-0.726379\pi$
$313$	−23.0962	−1.30547	−0.652737	+	0.757585i	$0.726379\pi$
$314$	0	0
$315$	−29.5133	−1.66289
$316$	0	0
$317$	3.90535	0.219346	0.109673	−	0.993968i	$-0.465020\pi$
$317$	3.90535	0.219346	0.109673	+	0.993968i	$0.465020\pi$
$318$	0	0
$319$	−28.3349	−1.58645
$320$	0	0
$321$	−0.303831	−0.0169582
$322$	0	0
$323$	−6.89359	−0.383569
$324$	0	0
$325$	0.994468	0.0551632
$326$	0	0
$327$	6.16216	0.340768
$328$	0	0
$329$	46.1995	2.54706
$330$	0	0
$331$	−11.8048	−0.648852	−0.324426	−	0.945911i	$-0.605171\pi$
$331$	−11.8048	−0.648852	−0.324426	+	0.945911i	$0.605171\pi$
$332$	0	0
$333$	−25.5722	−1.40135
$334$	0	0
$335$	−0.796157	−0.0434987
$336$	0	0
$337$	−19.9679	−1.08772	−0.543860	−	0.839176i	$-0.683038\pi$
$337$	−19.9679	−1.08772	−0.543860	+	0.839176i	$0.683038\pi$
$338$	0	0
$339$	−0.891773	−0.0484345
$340$	0	0
$341$	−6.62292	−0.358651
$342$	0	0
$343$	39.0878	2.11054
$344$	0	0
$345$	6.09296	0.328034
$346$	0	0
$347$	26.5872	1.42728	0.713638	−	0.700514i	$-0.247046\pi$
$347$	26.5872	1.42728	0.713638	+	0.700514i	$0.247046\pi$
$348$	0	0
$349$	−2.63696	−0.141153	−0.0705767	−	0.997506i	$-0.522484\pi$
$349$	−2.63696	−0.141153	−0.0705767	+	0.997506i	$0.522484\pi$
$350$	0	0
$351$	9.10563	0.486022
$352$	0	0
$353$	19.0117	1.01189	0.505947	−	0.862565i	$-0.331143\pi$
$353$	19.0117	1.01189	0.505947	+	0.862565i	$0.331143\pi$
$354$	0	0
$355$	−13.1234	−0.696519
$356$	0	0
$357$	−3.16267	−0.167386
$358$	0	0
$359$	15.5125	0.818721	0.409360	−	0.912373i	$-0.365752\pi$
$359$	15.5125	0.818721	0.409360	+	0.912373i	$0.365752\pi$
$360$	0	0
$361$	−4.66042	−0.245285
$362$	0	0
$363$	0.218786	0.0114833
$364$	0	0
$365$	20.3163	1.06341
$366$	0	0
$367$	21.4008	1.11712	0.558558	−	0.829466i	$-0.311355\pi$
$367$	21.4008	1.11712	0.558558	+	0.829466i	$0.311355\pi$
$368$	0	0
$369$	2.98352	0.155316
$370$	0	0
$371$	55.2602	2.86896
$372$	0	0
$373$	−20.3217	−1.05222	−0.526109	−	0.850417i	$-0.676349\pi$
$373$	−20.3217	−1.05222	−0.526109	+	0.850417i	$0.676349\pi$
$374$	0	0
$375$	−4.20616	−0.217205
$376$	0	0
$377$	35.1045	1.80797
$378$	0	0
$379$	−8.07775	−0.414926	−0.207463	−	0.978243i	$-0.566521\pi$
$379$	−8.07775	−0.414926	−0.207463	+	0.978243i	$0.566521\pi$
$380$	0	0
$381$	−7.09075	−0.363270
$382$	0	0
$383$	−24.1209	−1.23252	−0.616261	−	0.787542i	$-0.711353\pi$
$383$	−24.1209	−1.23252	−0.616261	+	0.787542i	$0.711353\pi$
$384$	0	0
$385$	35.0824	1.78796
$386$	0	0
$387$	−12.6463	−0.642850
$388$	0	0
$389$	−35.0398	−1.77659	−0.888295	−	0.459273i	$-0.848110\pi$
$389$	−35.0398	−1.77659	−0.888295	+	0.459273i	$0.848110\pi$
$390$	0	0
$391$	−13.8069	−0.698247
$392$	0	0
$393$	3.21499	0.162175
$394$	0	0
$395$	−15.1509	−0.762323
$396$	0	0
$397$	23.4028	1.17455	0.587275	−	0.809387i	$-0.300201\pi$
$397$	23.4028	1.17455	0.587275	+	0.809387i	$0.300201\pi$
$398$	0	0
$399$	6.57877	0.329350
$400$	0	0
$401$	−3.48869	−0.174217	−0.0871085	−	0.996199i	$-0.527763\pi$
$401$	−3.48869	−0.174217	−0.0871085	+	0.996199i	$0.527763\pi$
$402$	0	0
$403$	8.20522	0.408731
$404$	0	0
$405$	17.0234	0.845901
$406$	0	0
$407$	30.3976	1.50675
$408$	0	0
$409$	−31.4904	−1.55710	−0.778549	−	0.627584i	$-0.784044\pi$
$409$	−31.4904	−1.55710	−0.778549	+	0.627584i	$0.784044\pi$
$410$	0	0
$411$	8.21190	0.405063
$412$	0	0
$413$	−16.3890	−0.806452
$414$	0	0
$415$	30.2534	1.48508
$416$	0	0
$417$	−2.15785	−0.105670
$418$	0	0
$419$	−30.5786	−1.49386	−0.746932	−	0.664900i	$-0.768474\pi$
$419$	−30.5786	−1.49386	−0.746932	+	0.664900i	$0.768474\pi$
$420$	0	0
$421$	−24.7610	−1.20678	−0.603389	−	0.797447i	$-0.706184\pi$
$421$	−24.7610	−1.20678	−0.603389	+	0.797447i	$0.706184\pi$
$422$	0	0
$423$	−28.0367	−1.36319
$424$	0	0
$425$	0.429143	0.0208165
$426$	0	0
$427$	−24.6011	−1.19053
$428$	0	0
$429$	−5.28690	−0.255254
$430$	0	0
$431$	34.0981	1.64245	0.821224	−	0.570607i	$-0.193292\pi$
$431$	34.0981	1.64245	0.821224	+	0.570607i	$0.193292\pi$
$432$	0	0
$433$	−33.8717	−1.62777	−0.813885	−	0.581026i	$-0.802652\pi$
$433$	−33.8717	−1.62777	−0.813885	+	0.581026i	$0.802652\pi$
$434$	0	0
$435$	−6.68505	−0.320524
$436$	0	0
$437$	28.7203	1.37388
$438$	0	0
$439$	−36.2383	−1.72956	−0.864780	−	0.502151i	$-0.832542\pi$
$439$	−36.2383	−1.72956	−0.864780	+	0.502151i	$0.832542\pi$
$440$	0	0
$441$	−43.7726	−2.08441
$442$	0	0
$443$	23.4350	1.11343	0.556716	−	0.830703i	$-0.312061\pi$
$443$	23.4350	1.11343	0.556716	+	0.830703i	$0.312061\pi$
$444$	0	0
$445$	−18.9163	−0.896720
$446$	0	0
$447$	−7.69710	−0.364060
$448$	0	0
$449$	−1.51491	−0.0714929	−0.0357464	−	0.999361i	$-0.511381\pi$
$449$	−1.51491	−0.0714929	−0.0357464	+	0.999361i	$0.511381\pi$
$450$	0	0
$451$	−3.54650	−0.166998
$452$	0	0
$453$	1.39098	0.0653539
$454$	0	0
$455$	−43.4640	−2.03762
$456$	0	0
$457$	15.9404	0.745660	0.372830	−	0.927900i	$-0.378388\pi$
$457$	15.9404	0.745660	0.372830	+	0.927900i	$0.378388\pi$
$458$	0	0
$459$	3.92935	0.183407
$460$	0	0
$461$	14.7969	0.689161	0.344581	−	0.938757i	$-0.388021\pi$
$461$	14.7969	0.689161	0.344581	+	0.938757i	$0.388021\pi$
$462$	0	0
$463$	−17.6809	−0.821701	−0.410850	−	0.911703i	$-0.634768\pi$
$463$	−17.6809	−0.821701	−0.410850	+	0.911703i	$0.634768\pi$
$464$	0	0
$465$	−1.56255	−0.0724613
$466$	0	0
$467$	−25.4520	−1.17778	−0.588889	−	0.808214i	$-0.700435\pi$
$467$	−25.4520	−1.17778	−0.588889	+	0.808214i	$0.700435\pi$
$468$	0	0
$469$	−1.72174	−0.0795024
$470$	0	0
$471$	1.16677	0.0537618
$472$	0	0
$473$	15.0326	0.691202
$474$	0	0
$475$	−0.892675	−0.0409587
$476$	0	0
$477$	−33.5352	−1.53547
$478$	0	0
$479$	−27.1897	−1.24233	−0.621164	−	0.783680i	$-0.713340\pi$
$479$	−27.1897	−1.24233	−0.621164	+	0.783680i	$0.713340\pi$
$480$	0	0
$481$	−37.6599	−1.71715
$482$	0	0
$483$	13.1764	0.599546
$484$	0	0
$485$	−17.5712	−0.797865
$486$	0	0
$487$	5.16678	0.234129	0.117064	−	0.993124i	$-0.462652\pi$
$487$	5.16678	0.234129	0.117064	+	0.993124i	$0.462652\pi$
$488$	0	0
$489$	0.238848	0.0108011
$490$	0	0
$491$	10.4798	0.472947	0.236474	−	0.971638i	$-0.424008\pi$
$491$	10.4798	0.472947	0.236474	+	0.971638i	$0.424008\pi$
$492$	0	0
$493$	15.1486	0.682261
$494$	0	0
$495$	−21.2901	−0.956918
$496$	0	0
$497$	−28.3802	−1.27302
$498$	0	0
$499$	2.76601	0.123823	0.0619117	−	0.998082i	$-0.480280\pi$
$499$	2.76601	0.123823	0.0619117	+	0.998082i	$0.480280\pi$
$500$	0	0
$501$	−2.76259	−0.123423
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−6.54528	−0.291261
$506$	0	0
$507$	1.76532	0.0784007
$508$	0	0
$509$	21.7053	0.962071	0.481036	−	0.876701i	$-0.340261\pi$
$509$	21.7053	0.962071	0.481036	+	0.876701i	$0.340261\pi$
$510$	0	0
$511$	43.9353	1.94358
$512$	0	0
$513$	−8.17358	−0.360872
$514$	0	0
$515$	22.7491	1.00245
$516$	0	0
$517$	33.3270	1.46572
$518$	0	0
$519$	−8.48882	−0.372618
$520$	0	0
$521$	21.7134	0.951282	0.475641	−	0.879640i	$-0.342216\pi$
$521$	21.7134	0.951282	0.475641	+	0.879640i	$0.342216\pi$
$522$	0	0
$523$	9.98639	0.436674	0.218337	−	0.975873i	$-0.429937\pi$
$523$	9.98639	0.436674	0.218337	+	0.975873i	$0.429937\pi$
$524$	0	0
$525$	−0.409545	−0.0178740
$526$	0	0
$527$	3.54080	0.154240
$528$	0	0
$529$	34.5228	1.50099
$530$	0	0
$531$	9.94585	0.431613
$532$	0	0
$533$	4.39380	0.190317
$534$	0	0
$535$	−1.80185	−0.0779010
$536$	0	0
$537$	−6.87653	−0.296744
$538$	0	0
$539$	52.0322	2.24119
$540$	0	0
$541$	17.8396	0.766983	0.383491	−	0.923544i	$-0.374722\pi$
$541$	17.8396	0.766983	0.383491	+	0.923544i	$0.374722\pi$
$542$	0	0
$543$	−9.05067	−0.388402
$544$	0	0
$545$	36.5444	1.56539
$546$	0	0
$547$	13.0303	0.557137	0.278568	−	0.960416i	$-0.410140\pi$
$547$	13.0303	0.557137	0.278568	+	0.960416i	$0.410140\pi$
$548$	0	0
$549$	14.9295	0.637174
$550$	0	0
$551$	−31.5112	−1.34242
$552$	0	0
$553$	−32.7647	−1.39329
$554$	0	0
$555$	7.17170	0.304422
$556$	0	0
$557$	−11.3407	−0.480519	−0.240260	−	0.970709i	$-0.577233\pi$
$557$	−11.3407	−0.480519	−0.240260	+	0.970709i	$0.577233\pi$
$558$	0	0
$559$	−18.6241	−0.787717
$560$	0	0
$561$	−2.28146	−0.0963232
$562$	0	0
$563$	−26.0919	−1.09964	−0.549822	−	0.835282i	$-0.685305\pi$
$563$	−26.0919	−1.09964	−0.549822	+	0.835282i	$0.685305\pi$
$564$	0	0
$565$	−5.28862	−0.222494
$566$	0	0
$567$	36.8142	1.54605
$568$	0	0
$569$	36.9836	1.55043	0.775217	−	0.631695i	$-0.217640\pi$
$569$	36.9836	1.55043	0.775217	+	0.631695i	$0.217640\pi$
$570$	0	0
$571$	6.34744	0.265632	0.132816	−	0.991141i	$-0.457598\pi$
$571$	6.34744	0.265632	0.132816	+	0.991141i	$0.457598\pi$
$572$	0	0
$573$	−2.27582	−0.0950739
$574$	0	0
$575$	−1.78791	−0.0745609
$576$	0	0
$577$	−1.45666	−0.0606414	−0.0303207	−	0.999540i	$-0.509653\pi$
$577$	−1.45666	−0.0606414	−0.0303207	+	0.999540i	$0.509653\pi$
$578$	0	0
$579$	7.06176	0.293477
$580$	0	0
$581$	65.4248	2.71428
$582$	0	0
$583$	39.8631	1.65096
$584$	0	0
$585$	26.3766	1.09054
$586$	0	0
$587$	−25.4382	−1.04995	−0.524973	−	0.851119i	$-0.675925\pi$
$587$	−25.4382	−1.04995	−0.524973	+	0.851119i	$0.675925\pi$
$588$	0	0
$589$	−7.36534	−0.303483
$590$	0	0
$591$	3.66361	0.150701
$592$	0	0
$593$	−38.6709	−1.58802	−0.794012	−	0.607903i	$-0.792011\pi$
$593$	−38.6709	−1.58802	−0.794012	+	0.607903i	$0.792011\pi$
$594$	0	0
$595$	−18.7560	−0.768922
$596$	0	0
$597$	1.93044	0.0790076
$598$	0	0
$599$	−3.28351	−0.134161	−0.0670804	−	0.997748i	$-0.521368\pi$
$599$	−3.28351	−0.134161	−0.0670804	+	0.997748i	$0.521368\pi$
$600$	0	0
$601$	−47.5714	−1.94048	−0.970239	−	0.242147i	$-0.922148\pi$
$601$	−47.5714	−1.94048	−0.970239	+	0.242147i	$0.922148\pi$
$602$	0	0
$603$	1.04485	0.0425497
$604$	0	0
$605$	1.29750	0.0527509
$606$	0	0
$607$	−0.0789447	−0.00320427	−0.00160213	−	0.999999i	$-0.500510\pi$
$607$	−0.0789447	−0.00320427	−0.00160213	+	0.999999i	$0.500510\pi$
$608$	0	0
$609$	−14.4568	−0.585820
$610$	0	0
$611$	−41.2893	−1.67039
$612$	0	0
$613$	−12.7233	−0.513888	−0.256944	−	0.966426i	$-0.582716\pi$
$613$	−12.7233	−0.513888	−0.256944	+	0.966426i	$0.582716\pi$
$614$	0	0
$615$	−0.836725	−0.0337400
$616$	0	0
$617$	−23.4260	−0.943097	−0.471549	−	0.881840i	$-0.656305\pi$
$617$	−23.4260	−0.943097	−0.471549	+	0.881840i	$0.656305\pi$
$618$	0	0
$619$	20.5068	0.824240	0.412120	−	0.911130i	$-0.364788\pi$
$619$	20.5068	0.824240	0.412120	+	0.911130i	$0.364788\pi$
$620$	0	0
$621$	−16.3706	−0.656929
$622$	0	0
$623$	−40.9077	−1.63893
$624$	0	0
$625$	−23.7658	−0.950630
$626$	0	0
$627$	4.74573	0.189526
$628$	0	0
$629$	−16.2514	−0.647986
$630$	0	0
$631$	1.09126	0.0434422	0.0217211	−	0.999764i	$-0.493085\pi$
$631$	1.09126	0.0434422	0.0217211	+	0.999764i	$0.493085\pi$
$632$	0	0
$633$	3.13760	0.124709
$634$	0	0
$635$	−42.0513	−1.66876
$636$	0	0
$637$	−64.4634	−2.55413
$638$	0	0
$639$	17.2228	0.681323
$640$	0	0
$641$	−1.25724	−0.0496580	−0.0248290	−	0.999692i	$-0.507904\pi$
$641$	−1.25724	−0.0496580	−0.0248290	+	0.999692i	$0.507904\pi$
$642$	0	0
$643$	−7.83042	−0.308802	−0.154401	−	0.988008i	$-0.549345\pi$
$643$	−7.83042	−0.308802	−0.154401	+	0.988008i	$0.549345\pi$
$644$	0	0
$645$	3.54665	0.139649
$646$	0	0
$647$	26.7267	1.05074	0.525368	−	0.850875i	$-0.323928\pi$
$647$	26.7267	1.05074	0.525368	+	0.850875i	$0.323928\pi$
$648$	0	0
$649$	−11.8226	−0.464077
$650$	0	0
$651$	−3.37910	−0.132437
$652$	0	0
$653$	20.2977	0.794310	0.397155	−	0.917752i	$-0.369998\pi$
$653$	20.2977	0.794310	0.397155	+	0.917752i	$0.369998\pi$
$654$	0	0
$655$	19.0663	0.744984
$656$	0	0
$657$	−26.6626	−1.04021
$658$	0	0
$659$	−29.8997	−1.16473	−0.582364	−	0.812928i	$-0.697872\pi$
$659$	−29.8997	−1.16473	−0.582364	+	0.812928i	$0.697872\pi$
$660$	0	0
$661$	10.8301	0.421243	0.210622	−	0.977568i	$-0.432451\pi$
$661$	10.8301	0.421243	0.210622	+	0.977568i	$0.432451\pi$
$662$	0	0
$663$	2.82653	0.109773
$664$	0	0
$665$	39.0150	1.51294
$666$	0	0
$667$	−63.1127	−2.44373
$668$	0	0
$669$	−6.92211	−0.267624
$670$	0	0
$671$	−17.7466	−0.685099
$672$	0	0
$673$	20.9536	0.807702	0.403851	−	0.914825i	$-0.367671\pi$
$673$	20.9536	0.807702	0.403851	+	0.914825i	$0.367671\pi$
$674$	0	0
$675$	0.508826	0.0195847
$676$	0	0
$677$	47.0107	1.80677	0.903383	−	0.428834i	$-0.141076\pi$
$677$	47.0107	1.80677	0.903383	+	0.428834i	$0.141076\pi$
$678$	0	0
$679$	−37.9987	−1.45826
$680$	0	0
$681$	6.03930	0.231426
$682$	0	0
$683$	5.61727	0.214939	0.107469	−	0.994208i	$-0.465725\pi$
$683$	5.61727	0.214939	0.107469	+	0.994208i	$0.465725\pi$
$684$	0	0
$685$	48.7003	1.86074
$686$	0	0
$687$	−0.153197	−0.00584482
$688$	0	0
$689$	−49.3869	−1.88149
$690$	0	0
$691$	−40.6949	−1.54811	−0.774054	−	0.633119i	$-0.781774\pi$
$691$	−40.6949	−1.54811	−0.774054	+	0.633119i	$0.781774\pi$
$692$	0	0
$693$	−46.0411	−1.74896
$694$	0	0
$695$	−12.7970	−0.485419
$696$	0	0
$697$	1.89606	0.0718183
$698$	0	0
$699$	−7.51967	−0.284420
$700$	0	0
$701$	−40.1273	−1.51559	−0.757794	−	0.652494i	$-0.773723\pi$
$701$	−40.1273	−1.51559	−0.757794	+	0.652494i	$0.773723\pi$
$702$	0	0
$703$	33.8051	1.27498
$704$	0	0
$705$	7.86285	0.296132
$706$	0	0
$707$	−14.1546	−0.532337
$708$	0	0
$709$	−28.7463	−1.07959	−0.539794	−	0.841797i	$-0.681498\pi$
$709$	−28.7463	−1.07959	−0.539794	+	0.841797i	$0.681498\pi$
$710$	0	0
$711$	19.8836	0.745692
$712$	0	0
$713$	−14.7518	−0.552459
$714$	0	0
$715$	−31.3537	−1.17256
$716$	0	0
$717$	5.56020	0.207649
$718$	0	0
$719$	−37.6856	−1.40544	−0.702718	−	0.711468i	$-0.748030\pi$
$719$	−37.6856	−1.40544	−0.702718	+	0.711468i	$0.748030\pi$
$720$	0	0
$721$	49.1963	1.83217
$722$	0	0
$723$	−4.75304	−0.176767
$724$	0	0
$725$	1.96165	0.0728539
$726$	0	0
$727$	2.68673	0.0996454	0.0498227	−	0.998758i	$-0.484134\pi$
$727$	2.68673	0.0996454	0.0498227	+	0.998758i	$0.484134\pi$
$728$	0	0
$729$	−19.9578	−0.739177
$730$	0	0
$731$	−8.03688	−0.297255
$732$	0	0
$733$	−1.62506	−0.0600229	−0.0300115	−	0.999550i	$-0.509554\pi$
$733$	−1.62506	−0.0600229	−0.0300115	+	0.999550i	$0.509554\pi$
$734$	0	0
$735$	12.2760	0.452806
$736$	0	0
$737$	−1.24201	−0.0457501
$738$	0	0
$739$	−50.5150	−1.85822	−0.929112	−	0.369798i	$-0.879427\pi$
$739$	−50.5150	−1.85822	−0.929112	+	0.369798i	$0.879427\pi$
$740$	0	0
$741$	−5.87955	−0.215991
$742$	0	0
$743$	−41.4117	−1.51925	−0.759623	−	0.650364i	$-0.774617\pi$
$743$	−41.4117	−1.51925	−0.759623	+	0.650364i	$0.774617\pi$
$744$	0	0
$745$	−45.6473	−1.67239
$746$	0	0
$747$	−39.7037	−1.45268
$748$	0	0
$749$	−3.89662	−0.142379
$750$	0	0
$751$	−31.3331	−1.14336	−0.571680	−	0.820477i	$-0.693708\pi$
$751$	−31.3331	−1.14336	−0.571680	+	0.820477i	$0.693708\pi$
$752$	0	0
$753$	6.49874	0.236827
$754$	0	0
$755$	8.24913	0.300217
$756$	0	0
$757$	−3.17434	−0.115373	−0.0576866	−	0.998335i	$-0.518372\pi$
$757$	−3.17434	−0.115373	−0.0576866	+	0.998335i	$0.518372\pi$
$758$	0	0
$759$	9.50507	0.345012
$760$	0	0
$761$	0.354677	0.0128570	0.00642852	−	0.999979i	$-0.497954\pi$
$761$	0.354677	0.0128570	0.00642852	+	0.999979i	$0.497954\pi$
$762$	0	0
$763$	79.0294	2.86106
$764$	0	0
$765$	11.3823	0.411527
$766$	0	0
$767$	14.6472	0.528878
$768$	0	0
$769$	−42.8871	−1.54655	−0.773274	−	0.634072i	$-0.781382\pi$
$769$	−42.8871	−1.54655	−0.773274	+	0.634072i	$0.781382\pi$
$770$	0	0
$771$	−4.07330	−0.146696
$772$	0	0
$773$	−16.8575	−0.606321	−0.303161	−	0.952939i	$-0.598042\pi$
$773$	−16.8575	−0.606321	−0.303161	+	0.952939i	$0.598042\pi$
$774$	0	0
$775$	0.458511	0.0164702
$776$	0	0
$777$	15.5092	0.556390
$778$	0	0
$779$	−3.94405	−0.141310
$780$	0	0
$781$	−20.4726	−0.732569
$782$	0	0
$783$	17.9614	0.641889
$784$	0	0
$785$	6.91945	0.246966
$786$	0	0
$787$	−36.9562	−1.31735	−0.658673	−	0.752429i	$-0.728882\pi$
$787$	−36.9562	−1.31735	−0.658673	+	0.752429i	$0.728882\pi$
$788$	0	0
$789$	1.76704	0.0629083
$790$	0	0
$791$	−11.4369	−0.406651
$792$	0	0
$793$	21.9865	0.780762
$794$	0	0
$795$	9.40491	0.333558
$796$	0	0
$797$	13.3540	0.473024	0.236512	−	0.971629i	$-0.423996\pi$
$797$	13.3540	0.473024	0.236512	+	0.971629i	$0.423996\pi$
$798$	0	0
$799$	−17.8176	−0.630341
$800$	0	0
$801$	24.8252	0.877156
$802$	0	0
$803$	31.6937	1.11845
$804$	0	0
$805$	78.1419	2.75414
$806$	0	0
$807$	−6.84820	−0.241068
$808$	0	0
$809$	44.2379	1.55532	0.777662	−	0.628683i	$-0.216406\pi$
$809$	44.2379	1.55532	0.777662	+	0.628683i	$0.216406\pi$
$810$	0	0
$811$	45.9262	1.61269	0.806344	−	0.591447i	$-0.201443\pi$
$811$	45.9262	1.61269	0.806344	+	0.591447i	$0.201443\pi$
$812$	0	0
$813$	5.30809	0.186163
$814$	0	0
$815$	1.41648	0.0496171
$816$	0	0
$817$	16.7178	0.584881
$818$	0	0
$819$	57.0409	1.99317
$820$	0	0
$821$	19.3447	0.675136	0.337568	−	0.941301i	$-0.390396\pi$
$821$	19.3447	0.675136	0.337568	+	0.941301i	$0.390396\pi$
$822$	0	0
$823$	−44.4395	−1.54906	−0.774532	−	0.632535i	$-0.782014\pi$
$823$	−44.4395	−1.54906	−0.774532	+	0.632535i	$0.782014\pi$
$824$	0	0
$825$	−0.295434	−0.0102857
$826$	0	0
$827$	−41.6379	−1.44789	−0.723945	−	0.689857i	$-0.757673\pi$
$827$	−41.6379	−1.44789	−0.723945	+	0.689857i	$0.757673\pi$
$828$	0	0
$829$	47.9391	1.66499	0.832497	−	0.554030i	$-0.186911\pi$
$829$	47.9391	1.66499	0.832497	+	0.554030i	$0.186911\pi$
$830$	0	0
$831$	7.10846	0.246590
$832$	0	0
$833$	−27.8179	−0.963834
$834$	0	0
$835$	−16.3834	−0.566970
$836$	0	0
$837$	4.19825	0.145113
$838$	0	0
$839$	6.36476	0.219736	0.109868	−	0.993946i	$-0.464957\pi$
$839$	6.36476	0.219736	0.109868	+	0.993946i	$0.464957\pi$
$840$	0	0
$841$	40.2458	1.38779
$842$	0	0
$843$	10.2554	0.353213
$844$	0	0
$845$	10.4692	0.360150
$846$	0	0
$847$	2.80592	0.0964127
$848$	0	0
$849$	−2.01386	−0.0691157
$850$	0	0
$851$	67.7071	2.32097
$852$	0	0
$853$	−7.16165	−0.245210	−0.122605	−	0.992456i	$-0.539125\pi$
$853$	−7.16165	−0.245210	−0.122605	+	0.992456i	$0.539125\pi$
$854$	0	0
$855$	−23.6767	−0.809725
$856$	0	0
$857$	−24.5418	−0.838330	−0.419165	−	0.907910i	$-0.637677\pi$
$857$	−24.5418	−0.838330	−0.419165	+	0.907910i	$0.637677\pi$
$858$	0	0
$859$	2.61539	0.0892359	0.0446179	−	0.999004i	$-0.485793\pi$
$859$	2.61539	0.0892359	0.0446179	+	0.999004i	$0.485793\pi$
$860$	0	0
$861$	−1.80947	−0.0616665
$862$	0	0
$863$	20.5423	0.699269	0.349635	−	0.936886i	$-0.386306\pi$
$863$	20.5423	0.699269	0.349635	+	0.936886i	$0.386306\pi$
$864$	0	0
$865$	−50.3425	−1.71170
$866$	0	0
$867$	−5.03717	−0.171071
$868$	0	0
$869$	−23.6355	−0.801779
$870$	0	0
$871$	1.53874	0.0521384
$872$	0	0
$873$	23.0599	0.780459
$874$	0	0
$875$	−53.9438	−1.82363
$876$	0	0
$877$	49.0431	1.65607	0.828034	−	0.560678i	$-0.189459\pi$
$877$	49.0431	1.65607	0.828034	+	0.560678i	$0.189459\pi$
$878$	0	0
$879$	6.23118	0.210172
$880$	0	0
$881$	−26.1098	−0.879661	−0.439831	−	0.898081i	$-0.644962\pi$
$881$	−26.1098	−0.879661	−0.439831	+	0.898081i	$0.644962\pi$
$882$	0	0
$883$	−33.2381	−1.11855	−0.559276	−	0.828982i	$-0.688921\pi$
$883$	−33.2381	−1.11855	−0.559276	+	0.828982i	$0.688921\pi$
$884$	0	0
$885$	−2.78930	−0.0937614
$886$	0	0
$887$	1.82213	0.0611813	0.0305906	−	0.999532i	$-0.490261\pi$
$887$	1.82213	0.0611813	0.0305906	+	0.999532i	$0.490261\pi$
$888$	0	0
$889$	−90.9385	−3.04998
$890$	0	0
$891$	26.5567	0.889683
$892$	0	0
$893$	37.0630	1.24026
$894$	0	0
$895$	−40.7809	−1.36315
$896$	0	0
$897$	−11.7760	−0.393188
$898$	0	0
$899$	16.1853	0.539810
$900$	0	0
$901$	−21.3120	−0.710004
$902$	0	0
$903$	7.66985	0.255236
$904$	0	0
$905$	−53.6746	−1.78420
$906$	0	0
$907$	−52.0234	−1.72741	−0.863704	−	0.504000i	$-0.831861\pi$
$907$	−52.0234	−1.72741	−0.863704	+	0.504000i	$0.831861\pi$
$908$	0	0
$909$	8.58984	0.284907
$910$	0	0
$911$	16.3660	0.542230	0.271115	−	0.962547i	$-0.412608\pi$
$911$	16.3660	0.542230	0.271115	+	0.962547i	$0.412608\pi$
$912$	0	0
$913$	47.1956	1.56195
$914$	0	0
$915$	−4.18695	−0.138416
$916$	0	0
$917$	41.2321	1.36160
$918$	0	0
$919$	−45.3912	−1.49732	−0.748659	−	0.662955i	$-0.769302\pi$
$919$	−45.3912	−1.49732	−0.748659	+	0.662955i	$0.769302\pi$
$920$	0	0
$921$	11.9070	0.392350
$922$	0	0
$923$	25.3638	0.834861
$924$	0	0
$925$	−2.10445	−0.0691940
$926$	0	0
$927$	−29.8553	−0.980577
$928$	0	0
$929$	−8.06938	−0.264748	−0.132374	−	0.991200i	$-0.542260\pi$
$929$	−8.06938	−0.264748	−0.132374	+	0.991200i	$0.542260\pi$
$930$	0	0
$931$	57.8650	1.89645
$932$	0	0
$933$	3.51266	0.114999
$934$	0	0
$935$	−13.5301	−0.442480
$936$	0	0
$937$	14.9527	0.488483	0.244242	−	0.969714i	$-0.421461\pi$
$937$	14.9527	0.488483	0.244242	+	0.969714i	$0.421461\pi$
$938$	0	0
$939$	−8.50062	−0.277407
$940$	0	0
$941$	16.8225	0.548399	0.274199	−	0.961673i	$-0.411587\pi$
$941$	16.8225	0.548399	0.274199	+	0.961673i	$0.411587\pi$
$942$	0	0
$943$	−7.89941	−0.257240
$944$	0	0
$945$	−22.2386	−0.723422
$946$	0	0
$947$	44.8247	1.45661	0.728303	−	0.685255i	$-0.240309\pi$
$947$	44.8247	1.45661	0.728303	+	0.685255i	$0.240309\pi$
$948$	0	0
$949$	−39.2657	−1.27462
$950$	0	0
$951$	1.43738	0.0466101
$952$	0	0
$953$	23.5939	0.764281	0.382141	−	0.924104i	$-0.375187\pi$
$953$	23.5939	0.764281	0.382141	+	0.924104i	$0.375187\pi$
$954$	0	0
$955$	−13.4967	−0.436742
$956$	0	0
$957$	−10.4287	−0.337113
$958$	0	0
$959$	105.317	3.40087
$960$	0	0
$961$	−27.2169	−0.877964
$962$	0	0
$963$	2.36470	0.0762014
$964$	0	0
$965$	41.8794	1.34815
$966$	0	0
$967$	14.3917	0.462807	0.231404	−	0.972858i	$-0.425668\pi$
$967$	14.3917	0.462807	0.231404	+	0.972858i	$0.425668\pi$
$968$	0	0
$969$	−2.53720	−0.0815068
$970$	0	0
$971$	49.4033	1.58543	0.792714	−	0.609594i	$-0.208668\pi$
$971$	49.4033	1.58543	0.792714	+	0.609594i	$0.208668\pi$
$972$	0	0
$973$	−27.6743	−0.887198
$974$	0	0
$975$	0.366017	0.0117219
$976$	0	0
$977$	−33.3026	−1.06545	−0.532723	−	0.846290i	$-0.678831\pi$
$977$	−33.3026	−1.06545	−0.532723	+	0.846290i	$0.678831\pi$
$978$	0	0
$979$	−29.5096	−0.943132
$980$	0	0
$981$	−47.9598	−1.53124
$982$	0	0
$983$	−24.2402	−0.773142	−0.386571	−	0.922260i	$-0.626341\pi$
$983$	−24.2402	−0.773142	−0.386571	+	0.922260i	$0.626341\pi$
$984$	0	0
$985$	21.7269	0.692276
$986$	0	0
$987$	17.0039	0.541239
$988$	0	0
$989$	33.4835	1.06471
$990$	0	0
$991$	21.4967	0.682865	0.341432	−	0.939906i	$-0.389088\pi$
$991$	21.4967	0.682865	0.341432	+	0.939906i	$0.389088\pi$
$992$	0	0
$993$	−4.34480	−0.137878
$994$	0	0
$995$	11.4484	0.362938
$996$	0	0
$997$	−19.9113	−0.630599	−0.315299	−	0.948992i	$-0.602105\pi$
$997$	−19.9113	−0.630599	−0.315299	+	0.948992i	$0.602105\pi$
$998$	0	0
$999$	−19.2689	−0.609642

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.f.1.20	✓	33
4.3	odd	2		8048.2.a.y.1.14		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.f.1.20	✓	33	1.1	even	1	trivial
8048.2.a.y.1.14		33	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.368053 q^{3} +2.18272 q^{5} +4.72026 q^{7} -2.86454 q^{9} +O(q^{10})\)	\(q+0.368053 q^{3} +2.18272 q^{5} +4.72026 q^{7} -2.86454 q^{9} +3.40506 q^{11} -4.21858 q^{13} +0.803356 q^{15} -1.82044 q^{17} +3.78676 q^{19} +1.73731 q^{21} +7.58438 q^{23} -0.235736 q^{25} -2.15846 q^{27} -8.32141 q^{29} -1.94502 q^{31} +1.25324 q^{33} +10.3030 q^{35} +8.92717 q^{37} -1.55266 q^{39} -1.04154 q^{41} +4.41479 q^{43} -6.25248 q^{45} +9.78750 q^{47} +15.2809 q^{49} -0.670019 q^{51} +11.7070 q^{53} +7.43229 q^{55} +1.39373 q^{57} -3.47206 q^{59} -5.21182 q^{61} -13.5214 q^{63} -9.20797 q^{65} -0.364754 q^{67} +2.79145 q^{69} -6.01242 q^{71} +9.30781 q^{73} -0.0867632 q^{75} +16.0728 q^{77} -6.94128 q^{79} +7.79918 q^{81} +13.8604 q^{83} -3.97352 q^{85} -3.06272 q^{87} -8.66640 q^{89} -19.9128 q^{91} -0.715871 q^{93} +8.26544 q^{95} -8.05012 q^{97} -9.75392 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) \) 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9	\( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9} + 22 q^{11} + 25 q^{13} - 4 q^{15} + 17 q^{17} + 6 q^{19} + 18 q^{21} + 16 q^{23} + 47 q^{25} - 20 q^{27} + 47 q^{29} - 7 q^{31} - 6 q^{33} + 19 q^{35} + 75 q^{37} + 21 q^{39} + 22 q^{41} - 5 q^{43} + 33 q^{45} + 10 q^{47} + 31 q^{49} + 9 q^{51} + 64 q^{53} - 3 q^{55} + 5 q^{57} + 28 q^{59} + 49 q^{61} - 10 q^{63} + 46 q^{65} - 14 q^{67} + 30 q^{69} + 35 q^{71} + 19 q^{73} - 33 q^{75} + 32 q^{77} - 12 q^{79} + 57 q^{81} + 82 q^{85} - 5 q^{87} + 42 q^{89} - 15 q^{91} + 55 q^{93} + 33 q^{95} + 4 q^{97} + 22 q^{99}+O(q^{100}) \) 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9 + 22 * q^11 + 25 * q^13 - 4 * q^15 + 17 * q^17 + 6 * q^19 + 18 * q^21 + 16 * q^23 + 47 * q^25 - 20 * q^27 + 47 * q^29 - 7 * q^31 - 6 * q^33 + 19 * q^35 + 75 * q^37 + 21 * q^39 + 22 * q^41 - 5 * q^43 + 33 * q^45 + 10 * q^47 + 31 * q^49 + 9 * q^51 + 64 * q^53 - 3 * q^55 + 5 * q^57 + 28 * q^59 + 49 * q^61 - 10 * q^63 + 46 * q^65 - 14 * q^67 + 30 * q^69 + 35 * q^71 + 19 * q^73 - 33 * q^75 + 32 * q^77 - 12 * q^79 + 57 * q^81 + 82 * q^85 - 5 * q^87 + 42 * q^89 - 15 * q^91 + 55 * q^93 + 33 * q^95 + 4 * q^97 + 22 * q^99

Introduction

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