LMFDB - Embedded newform 4024.2.a.e.1.8

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:	$1$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q-1.85667 q^{3} +2.88879 q^{5} +1.11301 q^{7} +0.447227 q^{9} +O(q^{10})$	$q-1.85667 q^{3} +2.88879 q^{5} +1.11301 q^{7} +0.447227 q^{9} -4.87952 q^{11} +3.77695 q^{13} -5.36353 q^{15} -0.0587275 q^{17} -2.60223 q^{19} -2.06649 q^{21} -2.33470 q^{23} +3.34509 q^{25} +4.73966 q^{27} +6.17124 q^{29} -4.36717 q^{31} +9.05966 q^{33} +3.21524 q^{35} -4.47125 q^{37} -7.01255 q^{39} -2.97356 q^{41} -10.9407 q^{43} +1.29194 q^{45} +8.04535 q^{47} -5.76121 q^{49} +0.109038 q^{51} -11.7036 q^{53} -14.0959 q^{55} +4.83148 q^{57} -1.31653 q^{59} +7.26699 q^{61} +0.497768 q^{63} +10.9108 q^{65} -5.79019 q^{67} +4.33477 q^{69} -0.407049 q^{71} +1.44721 q^{73} -6.21073 q^{75} -5.43094 q^{77} +9.65376 q^{79} -10.1417 q^{81} -13.7915 q^{83} -0.169651 q^{85} -11.4580 q^{87} -16.0376 q^{89} +4.20378 q^{91} +8.10840 q^{93} -7.51728 q^{95} -4.02735 q^{97} -2.18225 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.85667	−1.07195	−0.535975	−	0.844234i	$-0.680056\pi$
$3$	−1.85667	−1.07195	−0.535975	+	0.844234i	$0.680056\pi$
$4$	0	0
$5$	2.88879	1.29190	0.645952	−	0.763378i	$-0.276460\pi$
$5$	2.88879	1.29190	0.645952	+	0.763378i	$0.276460\pi$
$6$	0	0
$7$	1.11301	0.420678	0.210339	−	0.977629i	$-0.432543\pi$
$7$	1.11301	0.420678	0.210339	+	0.977629i	$0.432543\pi$
$8$	0	0
$9$	0.447227	0.149076
$10$	0	0
$11$	−4.87952	−1.47123	−0.735615	−	0.677400i	$-0.763107\pi$
$11$	−4.87952	−1.47123	−0.735615	+	0.677400i	$0.763107\pi$
$12$	0	0
$13$	3.77695	1.04754	0.523769	−	0.851861i	$-0.324526\pi$
$13$	3.77695	1.04754	0.523769	+	0.851861i	$0.324526\pi$
$14$	0	0
$15$	−5.36353	−1.38486
$16$	0	0
$17$	−0.0587275	−0.0142435	−0.00712176	−	0.999975i	$-0.502267\pi$
$17$	−0.0587275	−0.0142435	−0.00712176	+	0.999975i	$0.502267\pi$
$18$	0	0
$19$	−2.60223	−0.596992	−0.298496	−	0.954411i	$-0.596485\pi$
$19$	−2.60223	−0.596992	−0.298496	+	0.954411i	$0.596485\pi$
$20$	0	0
$21$	−2.06649	−0.450945
$22$	0	0
$23$	−2.33470	−0.486819	−0.243409	−	0.969924i	$-0.578266\pi$
$23$	−2.33470	−0.486819	−0.243409	+	0.969924i	$0.578266\pi$
$24$	0	0
$25$	3.34509	0.669017
$26$	0	0
$27$	4.73966	0.912148
$28$	0	0
$29$	6.17124	1.14597	0.572985	−	0.819566i	$-0.305785\pi$
$29$	6.17124	1.14597	0.572985	+	0.819566i	$0.305785\pi$
$30$	0	0
$31$	−4.36717	−0.784367	−0.392184	−	0.919887i	$-0.628280\pi$
$31$	−4.36717	−0.784367	−0.392184	+	0.919887i	$0.628280\pi$
$32$	0	0
$33$	9.05966	1.57708
$34$	0	0
$35$	3.21524	0.543475
$36$	0	0
$37$	−4.47125	−0.735068	−0.367534	−	0.930010i	$-0.619798\pi$
$37$	−4.47125	−0.735068	−0.367534	+	0.930010i	$0.619798\pi$
$38$	0	0
$39$	−7.01255	−1.12291
$40$	0	0
$41$	−2.97356	−0.464393	−0.232196	−	0.972669i	$-0.574591\pi$
$41$	−2.97356	−0.464393	−0.232196	+	0.972669i	$0.574591\pi$
$42$	0	0
$43$	−10.9407	−1.66844	−0.834219	−	0.551433i	$-0.814081\pi$
$43$	−10.9407	−1.66844	−0.834219	+	0.551433i	$0.814081\pi$
$44$	0	0
$45$	1.29194	0.192592
$46$	0	0
$47$	8.04535	1.17353	0.586767	−	0.809756i	$-0.300400\pi$
$47$	8.04535	1.17353	0.586767	+	0.809756i	$0.300400\pi$
$48$	0	0
$49$	−5.76121	−0.823030
$50$	0	0
$51$	0.109038	0.0152683
$52$	0	0
$53$	−11.7036	−1.60762	−0.803809	−	0.594887i	$-0.797197\pi$
$53$	−11.7036	−1.60762	−0.803809	+	0.594887i	$0.797197\pi$
$54$	0	0
$55$	−14.0959	−1.90069
$56$	0	0
$57$	4.83148	0.639945
$58$	0	0
$59$	−1.31653	−0.171398	−0.0856991	−	0.996321i	$-0.527312\pi$
$59$	−1.31653	−0.171398	−0.0856991	+	0.996321i	$0.527312\pi$
$60$	0	0
$61$	7.26699	0.930443	0.465222	−	0.885194i	$-0.345975\pi$
$61$	7.26699	0.930443	0.465222	+	0.885194i	$0.345975\pi$
$62$	0	0
$63$	0.497768	0.0627128
$64$	0	0
$65$	10.9108	1.35332
$66$	0	0
$67$	−5.79019	−0.707384	−0.353692	−	0.935362i	$-0.615074\pi$
$67$	−5.79019	−0.707384	−0.353692	+	0.935362i	$0.615074\pi$
$68$	0	0
$69$	4.33477	0.521845
$70$	0	0
$71$	−0.407049	−0.0483078	−0.0241539	−	0.999708i	$-0.507689\pi$
$71$	−0.407049	−0.0483078	−0.0241539	+	0.999708i	$0.507689\pi$
$72$	0	0
$73$	1.44721	0.169383	0.0846913	−	0.996407i	$-0.473010\pi$
$73$	1.44721	0.169383	0.0846913	+	0.996407i	$0.473010\pi$
$74$	0	0
$75$	−6.21073	−0.717153
$76$	0	0
$77$	−5.43094	−0.618913
$78$	0	0
$79$	9.65376	1.08613	0.543067	−	0.839690i	$-0.317263\pi$
$79$	9.65376	1.08613	0.543067	+	0.839690i	$0.317263\pi$
$80$	0	0
$81$	−10.1417	−1.12685
$82$	0	0
$83$	−13.7915	−1.51382	−0.756910	−	0.653520i	$-0.773292\pi$
$83$	−13.7915	−1.51382	−0.756910	+	0.653520i	$0.773292\pi$
$84$	0	0
$85$	−0.169651	−0.0184013
$86$	0	0
$87$	−11.4580	−1.22842
$88$	0	0
$89$	−16.0376	−1.69999	−0.849994	−	0.526793i	$-0.823394\pi$
$89$	−16.0376	−1.69999	−0.849994	+	0.526793i	$0.823394\pi$
$90$	0	0
$91$	4.20378	0.440676
$92$	0	0
$93$	8.10840	0.840802
$94$	0	0
$95$	−7.51728	−0.771256
$96$	0	0
$97$	−4.02735	−0.408916	−0.204458	−	0.978875i	$-0.565543\pi$
$97$	−4.02735	−0.408916	−0.204458	+	0.978875i	$0.565543\pi$
$98$	0	0
$99$	−2.18225	−0.219325
$100$	0	0
$101$	4.99010	0.496533	0.248267	−	0.968692i	$-0.420139\pi$
$101$	4.99010	0.496533	0.248267	+	0.968692i	$0.420139\pi$
$102$	0	0
$103$	4.47999	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$103$	4.47999	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$104$	0	0
$105$	−5.96965	−0.582578
$106$	0	0
$107$	18.6239	1.80044	0.900220	−	0.435435i	$-0.143406\pi$
$107$	18.6239	1.80044	0.900220	+	0.435435i	$0.143406\pi$
$108$	0	0
$109$	13.7389	1.31594	0.657972	−	0.753042i	$-0.271415\pi$
$109$	13.7389	1.31594	0.657972	+	0.753042i	$0.271415\pi$
$110$	0	0
$111$	8.30163	0.787956
$112$	0	0
$113$	−5.02636	−0.472840	−0.236420	−	0.971651i	$-0.575974\pi$
$113$	−5.02636	−0.472840	−0.236420	+	0.971651i	$0.575974\pi$
$114$	0	0
$115$	−6.74445	−0.628924
$116$	0	0
$117$	1.68916	0.156162
$118$	0	0
$119$	−0.0653642	−0.00599193
$120$	0	0
$121$	12.8097	1.16452
$122$	0	0
$123$	5.52093	0.497805
$124$	0	0
$125$	−4.78069	−0.427598
$126$	0	0
$127$	−9.98671	−0.886177	−0.443089	−	0.896478i	$-0.646117\pi$
$127$	−9.98671	−0.886177	−0.443089	+	0.896478i	$0.646117\pi$
$128$	0	0
$129$	20.3132	1.78848
$130$	0	0
$131$	5.50533	0.481003	0.240501	−	0.970649i	$-0.422688\pi$
$131$	5.50533	0.481003	0.240501	+	0.970649i	$0.422688\pi$
$132$	0	0
$133$	−2.89630	−0.251141
$134$	0	0
$135$	13.6919	1.17841
$136$	0	0
$137$	22.7488	1.94356	0.971780	−	0.235890i	$-0.0758006\pi$
$137$	22.7488	1.94356	0.971780	+	0.235890i	$0.0758006\pi$
$138$	0	0
$139$	17.7634	1.50667	0.753335	−	0.657637i	$-0.228444\pi$
$139$	17.7634	1.50667	0.753335	+	0.657637i	$0.228444\pi$
$140$	0	0
$141$	−14.9376	−1.25797
$142$	0	0
$143$	−18.4297	−1.54117
$144$	0	0
$145$	17.8274	1.48048
$146$	0	0
$147$	10.6967	0.882247
$148$	0	0
$149$	−0.485953	−0.0398108	−0.0199054	−	0.999802i	$-0.506337\pi$
$149$	−0.485953	−0.0398108	−0.0199054	+	0.999802i	$0.506337\pi$
$150$	0	0
$151$	−6.38647	−0.519724	−0.259862	−	0.965646i	$-0.583677\pi$
$151$	−6.38647	−0.519724	−0.259862	+	0.965646i	$0.583677\pi$
$152$	0	0
$153$	−0.0262646	−0.00212336
$154$	0	0
$155$	−12.6158	−1.01333
$156$	0	0
$157$	−24.2468	−1.93510	−0.967552	−	0.252670i	$-0.918691\pi$
$157$	−24.2468	−1.93510	−0.967552	+	0.252670i	$0.918691\pi$
$158$	0	0
$159$	21.7298	1.72329
$160$	0	0
$161$	−2.59854	−0.204794
$162$	0	0
$163$	1.30653	0.102335	0.0511675	−	0.998690i	$-0.483706\pi$
$163$	1.30653	0.102335	0.0511675	+	0.998690i	$0.483706\pi$
$164$	0	0
$165$	26.1714	2.03744
$166$	0	0
$167$	−24.8231	−1.92087	−0.960434	−	0.278507i	$-0.910161\pi$
$167$	−24.8231	−1.92087	−0.960434	+	0.278507i	$0.910161\pi$
$168$	0	0
$169$	1.26535	0.0973347
$170$	0	0
$171$	−1.16379	−0.0889970
$172$	0	0
$173$	−11.8581	−0.901552	−0.450776	−	0.892637i	$-0.648853\pi$
$173$	−11.8581	−0.901552	−0.450776	+	0.892637i	$0.648853\pi$
$174$	0	0
$175$	3.72311	0.281441
$176$	0	0
$177$	2.44437	0.183730
$178$	0	0
$179$	−24.2933	−1.81577	−0.907883	−	0.419224i	$-0.862302\pi$
$179$	−24.2933	−1.81577	−0.907883	+	0.419224i	$0.862302\pi$
$180$	0	0
$181$	−7.51823	−0.558825	−0.279413	−	0.960171i	$-0.590140\pi$
$181$	−7.51823	−0.558825	−0.279413	+	0.960171i	$0.590140\pi$
$182$	0	0
$183$	−13.4924	−0.997388
$184$	0	0
$185$	−12.9165	−0.949638
$186$	0	0
$187$	0.286562	0.0209555
$188$	0	0
$189$	5.27528	0.383720
$190$	0	0
$191$	−10.0197	−0.725000	−0.362500	−	0.931984i	$-0.618077\pi$
$191$	−10.0197	−0.725000	−0.362500	+	0.931984i	$0.618077\pi$
$192$	0	0
$193$	−4.58241	−0.329849	−0.164925	−	0.986306i	$-0.552738\pi$
$193$	−4.58241	−0.329849	−0.164925	+	0.986306i	$0.552738\pi$
$194$	0	0
$195$	−20.2578	−1.45069
$196$	0	0
$197$	−13.4906	−0.961162	−0.480581	−	0.876950i	$-0.659574\pi$
$197$	−13.4906	−0.961162	−0.480581	+	0.876950i	$0.659574\pi$
$198$	0	0
$199$	−16.3235	−1.15714	−0.578570	−	0.815633i	$-0.696389\pi$
$199$	−16.3235	−1.15714	−0.578570	+	0.815633i	$0.696389\pi$
$200$	0	0
$201$	10.7505	0.758280
$202$	0	0
$203$	6.86864	0.482084
$204$	0	0
$205$	−8.58999	−0.599951
$206$	0	0
$207$	−1.04414	−0.0725729
$208$	0	0
$209$	12.6976	0.878312
$210$	0	0
$211$	27.6825	1.90574	0.952871	−	0.303376i	$-0.0981138\pi$
$211$	27.6825	1.90574	0.952871	+	0.303376i	$0.0981138\pi$
$212$	0	0
$213$	0.755756	0.0517835
$214$	0	0
$215$	−31.6053	−2.15546
$216$	0	0
$217$	−4.86070	−0.329966
$218$	0	0
$219$	−2.68698	−0.181570
$220$	0	0
$221$	−0.221811	−0.0149206
$222$	0	0
$223$	−12.5125	−0.837900	−0.418950	−	0.908009i	$-0.637602\pi$
$223$	−12.5125	−0.837900	−0.418950	+	0.908009i	$0.637602\pi$
$224$	0	0
$225$	1.49601	0.0997343
$226$	0	0
$227$	11.1524	0.740208	0.370104	−	0.928990i	$-0.379322\pi$
$227$	11.1524	0.740208	0.370104	+	0.928990i	$0.379322\pi$
$228$	0	0
$229$	−11.1364	−0.735912	−0.367956	−	0.929843i	$-0.619942\pi$
$229$	−11.1364	−0.735912	−0.367956	+	0.929843i	$0.619942\pi$
$230$	0	0
$231$	10.0835	0.663444
$232$	0	0
$233$	−1.18213	−0.0774440	−0.0387220	−	0.999250i	$-0.512329\pi$
$233$	−1.18213	−0.0774440	−0.0387220	+	0.999250i	$0.512329\pi$
$234$	0	0
$235$	23.2413	1.51610
$236$	0	0
$237$	−17.9239	−1.16428
$238$	0	0
$239$	−9.84806	−0.637018	−0.318509	−	0.947920i	$-0.603182\pi$
$239$	−9.84806	−0.637018	−0.318509	+	0.947920i	$0.603182\pi$
$240$	0	0
$241$	20.9461	1.34926	0.674630	−	0.738156i	$-0.264303\pi$
$241$	20.9461	1.34926	0.674630	+	0.738156i	$0.264303\pi$
$242$	0	0
$243$	4.61077	0.295781
$244$	0	0
$245$	−16.6429	−1.06328
$246$	0	0
$247$	−9.82848	−0.625371
$248$	0	0
$249$	25.6064	1.62274
$250$	0	0
$251$	−13.2545	−0.836618	−0.418309	−	0.908305i	$-0.637377\pi$
$251$	−13.2545	−0.836618	−0.418309	+	0.908305i	$0.637377\pi$
$252$	0	0
$253$	11.3922	0.716222
$254$	0	0
$255$	0.314987	0.0197252
$256$	0	0
$257$	−14.5008	−0.904535	−0.452267	−	0.891882i	$-0.649385\pi$
$257$	−14.5008	−0.904535	−0.452267	+	0.891882i	$0.649385\pi$
$258$	0	0
$259$	−4.97653	−0.309227
$260$	0	0
$261$	2.75995	0.170836
$262$	0	0
$263$	−6.36847	−0.392696	−0.196348	−	0.980534i	$-0.562908\pi$
$263$	−6.36847	−0.392696	−0.196348	+	0.980534i	$0.562908\pi$
$264$	0	0
$265$	−33.8093	−2.07689
$266$	0	0
$267$	29.7766	1.82230
$268$	0	0
$269$	−2.00443	−0.122212	−0.0611062	−	0.998131i	$-0.519463\pi$
$269$	−2.00443	−0.122212	−0.0611062	+	0.998131i	$0.519463\pi$
$270$	0	0
$271$	5.59194	0.339686	0.169843	−	0.985471i	$-0.445674\pi$
$271$	5.59194	0.339686	0.169843	+	0.985471i	$0.445674\pi$
$272$	0	0
$273$	−7.80503	−0.472382
$274$	0	0
$275$	−16.3224	−0.984278
$276$	0	0
$277$	18.7905	1.12901	0.564505	−	0.825430i	$-0.309067\pi$
$277$	18.7905	1.12901	0.564505	+	0.825430i	$0.309067\pi$
$278$	0	0
$279$	−1.95312	−0.116930
$280$	0	0
$281$	−15.6536	−0.933815	−0.466908	−	0.884306i	$-0.654632\pi$
$281$	−15.6536	−0.933815	−0.466908	+	0.884306i	$0.654632\pi$
$282$	0	0
$283$	−31.9192	−1.89740	−0.948700	−	0.316179i	$-0.897600\pi$
$283$	−31.9192	−1.89740	−0.948700	+	0.316179i	$0.897600\pi$
$284$	0	0
$285$	13.9571	0.826748
$286$	0	0
$287$	−3.30960	−0.195360
$288$	0	0
$289$	−16.9966	−0.999797
$290$	0	0
$291$	7.47747	0.438337
$292$	0	0
$293$	23.4215	1.36830	0.684149	−	0.729343i	$-0.260174\pi$
$293$	23.4215	1.36830	0.684149	+	0.729343i	$0.260174\pi$
$294$	0	0
$295$	−3.80319	−0.221430
$296$	0	0
$297$	−23.1272	−1.34198
$298$	0	0
$299$	−8.81805	−0.509961
$300$	0	0
$301$	−12.1771	−0.701875
$302$	0	0
$303$	−9.26497	−0.532259
$304$	0	0
$305$	20.9928	1.20204
$306$	0	0
$307$	12.2974	0.701851	0.350926	−	0.936403i	$-0.385867\pi$
$307$	12.2974	0.701851	0.350926	+	0.936403i	$0.385867\pi$
$308$	0	0
$309$	−8.31788	−0.473187
$310$	0	0
$311$	−10.8902	−0.617526	−0.308763	−	0.951139i	$-0.599915\pi$
$311$	−10.8902	−0.617526	−0.308763	+	0.951139i	$0.599915\pi$
$312$	0	0
$313$	17.3148	0.978692	0.489346	−	0.872090i	$-0.337236\pi$
$313$	17.3148	0.978692	0.489346	+	0.872090i	$0.337236\pi$
$314$	0	0
$315$	1.43794	0.0810190
$316$	0	0
$317$	13.7856	0.774274	0.387137	−	0.922022i	$-0.373464\pi$
$317$	13.7856	0.774274	0.387137	+	0.922022i	$0.373464\pi$
$318$	0	0
$319$	−30.1127	−1.68599
$320$	0	0
$321$	−34.5785	−1.92998
$322$	0	0
$323$	0.152822	0.00850326
$324$	0	0
$325$	12.6342	0.700821
$326$	0	0
$327$	−25.5085	−1.41063
$328$	0	0
$329$	8.95454	0.493680
$330$	0	0
$331$	9.84632	0.541203	0.270601	−	0.962691i	$-0.412777\pi$
$331$	9.84632	0.541203	0.270601	+	0.962691i	$0.412777\pi$
$332$	0	0
$333$	−1.99966	−0.109581
$334$	0	0
$335$	−16.7266	−0.913872
$336$	0	0
$337$	17.8600	0.972897	0.486448	−	0.873709i	$-0.338292\pi$
$337$	17.8600	0.972897	0.486448	+	0.873709i	$0.338292\pi$
$338$	0	0
$339$	9.33230	0.506861
$340$	0	0
$341$	21.3097	1.15398
$342$	0	0
$343$	−14.2033	−0.766908
$344$	0	0
$345$	12.5222	0.674174
$346$	0	0
$347$	−1.14166	−0.0612877	−0.0306439	−	0.999530i	$-0.509756\pi$
$347$	−1.14166	−0.0612877	−0.0306439	+	0.999530i	$0.509756\pi$
$348$	0	0
$349$	−21.8294	−1.16850	−0.584250	−	0.811574i	$-0.698611\pi$
$349$	−21.8294	−1.16850	−0.584250	+	0.811574i	$0.698611\pi$
$350$	0	0
$351$	17.9015	0.955509
$352$	0	0
$353$	3.45503	0.183893	0.0919465	−	0.995764i	$-0.470691\pi$
$353$	3.45503	0.183893	0.0919465	+	0.995764i	$0.470691\pi$
$354$	0	0
$355$	−1.17588	−0.0624091
$356$	0	0
$357$	0.121360	0.00642305
$358$	0	0
$359$	0.521451	0.0275211	0.0137606	−	0.999905i	$-0.495620\pi$
$359$	0.521451	0.0275211	0.0137606	+	0.999905i	$0.495620\pi$
$360$	0	0
$361$	−12.2284	−0.643601
$362$	0	0
$363$	−23.7834	−1.24830
$364$	0	0
$365$	4.18067	0.218826
$366$	0	0
$367$	17.2271	0.899249	0.449625	−	0.893218i	$-0.351558\pi$
$367$	17.2271	0.899249	0.449625	+	0.893218i	$0.351558\pi$
$368$	0	0
$369$	−1.32986	−0.0692297
$370$	0	0
$371$	−13.0263	−0.676289
$372$	0	0
$373$	−11.7368	−0.607711	−0.303855	−	0.952718i	$-0.598274\pi$
$373$	−11.7368	−0.607711	−0.303855	+	0.952718i	$0.598274\pi$
$374$	0	0
$375$	8.87617	0.458363
$376$	0	0
$377$	23.3085	1.20045
$378$	0	0
$379$	−22.7612	−1.16917	−0.584583	−	0.811334i	$-0.698742\pi$
$379$	−22.7612	−1.16917	−0.584583	+	0.811334i	$0.698742\pi$
$380$	0	0
$381$	18.5420	0.949937
$382$	0	0
$383$	−35.6044	−1.81930	−0.909651	−	0.415374i	$-0.863651\pi$
$383$	−35.6044	−1.81930	−0.909651	+	0.415374i	$0.863651\pi$
$384$	0	0
$385$	−15.6888	−0.799577
$386$	0	0
$387$	−4.89297	−0.248724
$388$	0	0
$389$	11.3339	0.574652	0.287326	−	0.957833i	$-0.407234\pi$
$389$	11.3339	0.574652	0.287326	+	0.957833i	$0.407234\pi$
$390$	0	0
$391$	0.137111	0.00693401
$392$	0	0
$393$	−10.2216	−0.515611
$394$	0	0
$395$	27.8877	1.40318
$396$	0	0
$397$	−6.87670	−0.345132	−0.172566	−	0.984998i	$-0.555206\pi$
$397$	−6.87670	−0.345132	−0.172566	+	0.984998i	$0.555206\pi$
$398$	0	0
$399$	5.37748	0.269211
$400$	0	0
$401$	−22.4857	−1.12288	−0.561441	−	0.827517i	$-0.689753\pi$
$401$	−22.4857	−1.12288	−0.561441	+	0.827517i	$0.689753\pi$
$402$	0	0
$403$	−16.4946	−0.821654
$404$	0	0
$405$	−29.2971	−1.45579
$406$	0	0
$407$	21.8175	1.08145
$408$	0	0
$409$	−15.4836	−0.765614	−0.382807	−	0.923828i	$-0.625043\pi$
$409$	−15.4836	−0.765614	−0.382807	+	0.923828i	$0.625043\pi$
$410$	0	0
$411$	−42.2370	−2.08340
$412$	0	0
$413$	−1.46531	−0.0721034
$414$	0	0
$415$	−39.8408	−1.95571
$416$	0	0
$417$	−32.9808	−1.61507
$418$	0	0
$419$	26.4400	1.29168	0.645839	−	0.763473i	$-0.276508\pi$
$419$	26.4400	1.29168	0.645839	+	0.763473i	$0.276508\pi$
$420$	0	0
$421$	32.6596	1.59173	0.795866	−	0.605473i	$-0.207016\pi$
$421$	32.6596	1.59173	0.795866	+	0.605473i	$0.207016\pi$
$422$	0	0
$423$	3.59810	0.174946
$424$	0	0
$425$	−0.196449	−0.00952916
$426$	0	0
$427$	8.08822	0.391417
$428$	0	0
$429$	34.2179	1.65205
$430$	0	0
$431$	−5.54207	−0.266952	−0.133476	−	0.991052i	$-0.542614\pi$
$431$	−5.54207	−0.266952	−0.133476	+	0.991052i	$0.542614\pi$
$432$	0	0
$433$	−15.9421	−0.766128	−0.383064	−	0.923722i	$-0.625131\pi$
$433$	−15.9421	−0.766128	−0.383064	+	0.923722i	$0.625131\pi$
$434$	0	0
$435$	−33.0996	−1.58700
$436$	0	0
$437$	6.07542	0.290627
$438$	0	0
$439$	−8.27398	−0.394896	−0.197448	−	0.980313i	$-0.563265\pi$
$439$	−8.27398	−0.394896	−0.197448	+	0.980313i	$0.563265\pi$
$440$	0	0
$441$	−2.57657	−0.122694
$442$	0	0
$443$	−21.5514	−1.02394	−0.511968	−	0.859005i	$-0.671083\pi$
$443$	−21.5514	−1.02394	−0.511968	+	0.859005i	$0.671083\pi$
$444$	0	0
$445$	−46.3293	−2.19622
$446$	0	0
$447$	0.902255	0.0426752
$448$	0	0
$449$	−21.7423	−1.02608	−0.513041	−	0.858364i	$-0.671481\pi$
$449$	−21.7423	−1.02608	−0.513041	+	0.858364i	$0.671481\pi$
$450$	0	0
$451$	14.5096	0.683228
$452$	0	0
$453$	11.8576	0.557118
$454$	0	0
$455$	12.1438	0.569311
$456$	0	0
$457$	8.63515	0.403935	0.201968	−	0.979392i	$-0.435266\pi$
$457$	8.63515	0.403935	0.201968	+	0.979392i	$0.435266\pi$
$458$	0	0
$459$	−0.278348	−0.0129922
$460$	0	0
$461$	−30.3731	−1.41462	−0.707309	−	0.706904i	$-0.750091\pi$
$461$	−30.3731	−1.41462	−0.707309	+	0.706904i	$0.750091\pi$
$462$	0	0
$463$	13.3348	0.619722	0.309861	−	0.950782i	$-0.399717\pi$
$463$	13.3348	0.619722	0.309861	+	0.950782i	$0.399717\pi$
$464$	0	0
$465$	23.4234	1.08624
$466$	0	0
$467$	15.1550	0.701292	0.350646	−	0.936508i	$-0.385962\pi$
$467$	15.1550	0.701292	0.350646	+	0.936508i	$0.385962\pi$
$468$	0	0
$469$	−6.44452	−0.297581
$470$	0	0
$471$	45.0183	2.07433
$472$	0	0
$473$	53.3852	2.45466
$474$	0	0
$475$	−8.70467	−0.399398
$476$	0	0
$477$	−5.23419	−0.239657
$478$	0	0
$479$	26.5167	1.21158	0.605789	−	0.795626i	$-0.292858\pi$
$479$	26.5167	1.21158	0.605789	+	0.795626i	$0.292858\pi$
$480$	0	0
$481$	−16.8877	−0.770012
$482$	0	0
$483$	4.82464	0.219529
$484$	0	0
$485$	−11.6342	−0.528280
$486$	0	0
$487$	−12.3528	−0.559758	−0.279879	−	0.960035i	$-0.590294\pi$
$487$	−12.3528	−0.559758	−0.279879	+	0.960035i	$0.590294\pi$
$488$	0	0
$489$	−2.42579	−0.109698
$490$	0	0
$491$	31.0101	1.39947	0.699733	−	0.714405i	$-0.253303\pi$
$491$	31.0101	1.39947	0.699733	+	0.714405i	$0.253303\pi$
$492$	0	0
$493$	−0.362421	−0.0163226
$494$	0	0
$495$	−6.30406	−0.283347
$496$	0	0
$497$	−0.453049	−0.0203220
$498$	0	0
$499$	29.5327	1.32206	0.661032	−	0.750358i	$-0.270119\pi$
$499$	29.5327	1.32206	0.661032	+	0.750358i	$0.270119\pi$
$500$	0	0
$501$	46.0883	2.05907
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	14.4153	0.641474
$506$	0	0
$507$	−2.34934	−0.104338
$508$	0	0
$509$	−18.0736	−0.801100	−0.400550	−	0.916275i	$-0.631181\pi$
$509$	−18.0736	−0.801100	−0.400550	+	0.916275i	$0.631181\pi$
$510$	0	0
$511$	1.61075	0.0712555
$512$	0	0
$513$	−12.3337	−0.544545
$514$	0	0
$515$	12.9417	0.570282
$516$	0	0
$517$	−39.2574	−1.72654
$518$	0	0
$519$	22.0165	0.966418
$520$	0	0
$521$	−28.7707	−1.26047	−0.630233	−	0.776406i	$-0.717041\pi$
$521$	−28.7707	−1.26047	−0.630233	+	0.776406i	$0.717041\pi$
$522$	0	0
$523$	−12.8646	−0.562531	−0.281266	−	0.959630i	$-0.590754\pi$
$523$	−12.8646	−0.562531	−0.281266	+	0.959630i	$0.590754\pi$
$524$	0	0
$525$	−6.91259	−0.301690
$526$	0	0
$527$	0.256473	0.0111721
$528$	0	0
$529$	−17.5492	−0.763007
$530$	0	0
$531$	−0.588790	−0.0255513
$532$	0	0
$533$	−11.2310	−0.486469
$534$	0	0
$535$	53.8005	2.32600
$536$	0	0
$537$	45.1046	1.94641
$538$	0	0
$539$	28.1119	1.21087
$540$	0	0
$541$	1.99848	0.0859216	0.0429608	−	0.999077i	$-0.486321\pi$
$541$	1.99848	0.0859216	0.0429608	+	0.999077i	$0.486321\pi$
$542$	0	0
$543$	13.9589	0.599033
$544$	0	0
$545$	39.6886	1.70007
$546$	0	0
$547$	11.0938	0.474335	0.237167	−	0.971469i	$-0.423781\pi$
$547$	11.0938	0.474335	0.237167	+	0.971469i	$0.423781\pi$
$548$	0	0
$549$	3.25000	0.138707
$550$	0	0
$551$	−16.0590	−0.684135
$552$	0	0
$553$	10.7447	0.456912
$554$	0	0
$555$	23.9816	1.01796
$556$	0	0
$557$	−28.1225	−1.19159	−0.595795	−	0.803137i	$-0.703163\pi$
$557$	−28.1225	−1.19159	−0.595795	+	0.803137i	$0.703163\pi$
$558$	0	0
$559$	−41.3224	−1.74775
$560$	0	0
$561$	−0.532051	−0.0224632
$562$	0	0
$563$	12.9634	0.546341	0.273170	−	0.961966i	$-0.411928\pi$
$563$	12.9634	0.546341	0.273170	+	0.961966i	$0.411928\pi$
$564$	0	0
$565$	−14.5201	−0.610864
$566$	0	0
$567$	−11.2878	−0.474041
$568$	0	0
$569$	17.4452	0.731341	0.365670	−	0.930744i	$-0.380840\pi$
$569$	17.4452	0.731341	0.365670	+	0.930744i	$0.380840\pi$
$570$	0	0
$571$	29.2284	1.22317	0.611586	−	0.791178i	$-0.290532\pi$
$571$	29.2284	1.22317	0.611586	+	0.791178i	$0.290532\pi$
$572$	0	0
$573$	18.6033	0.777163
$574$	0	0
$575$	−7.80978	−0.325690
$576$	0	0
$577$	−23.6731	−0.985525	−0.492762	−	0.870164i	$-0.664013\pi$
$577$	−23.6731	−0.985525	−0.492762	+	0.870164i	$0.664013\pi$
$578$	0	0
$579$	8.50803	0.353582
$580$	0	0
$581$	−15.3501	−0.636830
$582$	0	0
$583$	57.1081	2.36518
$584$	0	0
$585$	4.87961	0.201747
$586$	0	0
$587$	−3.74869	−0.154725	−0.0773626	−	0.997003i	$-0.524650\pi$
$587$	−3.74869	−0.154725	−0.0773626	+	0.997003i	$0.524650\pi$
$588$	0	0
$589$	11.3644	0.468261
$590$	0	0
$591$	25.0475	1.03032
$592$	0	0
$593$	39.3373	1.61539	0.807695	−	0.589600i	$-0.200715\pi$
$593$	39.3373	1.61539	0.807695	+	0.589600i	$0.200715\pi$
$594$	0	0
$595$	−0.188823	−0.00774100
$596$	0	0
$597$	30.3073	1.24040
$598$	0	0
$599$	−29.6814	−1.21275	−0.606375	−	0.795179i	$-0.707377\pi$
$599$	−29.6814	−1.21275	−0.606375	+	0.795179i	$0.707377\pi$
$600$	0	0
$601$	17.4468	0.711669	0.355835	−	0.934549i	$-0.384197\pi$
$601$	17.4468	0.711669	0.355835	+	0.934549i	$0.384197\pi$
$602$	0	0
$603$	−2.58953	−0.105454
$604$	0	0
$605$	37.0044	1.50444
$606$	0	0
$607$	27.8459	1.13023	0.565114	−	0.825013i	$-0.308832\pi$
$607$	27.8459	1.13023	0.565114	+	0.825013i	$0.308832\pi$
$608$	0	0
$609$	−12.7528	−0.516770
$610$	0	0
$611$	30.3869	1.22932
$612$	0	0
$613$	15.2998	0.617953	0.308977	−	0.951070i	$-0.400014\pi$
$613$	15.2998	0.617953	0.308977	+	0.951070i	$0.400014\pi$
$614$	0	0
$615$	15.9488	0.643117
$616$	0	0
$617$	30.4672	1.22656	0.613281	−	0.789865i	$-0.289849\pi$
$617$	30.4672	1.22656	0.613281	+	0.789865i	$0.289849\pi$
$618$	0	0
$619$	−5.17771	−0.208110	−0.104055	−	0.994572i	$-0.533182\pi$
$619$	−5.17771	−0.208110	−0.104055	+	0.994572i	$0.533182\pi$
$620$	0	0
$621$	−11.0657	−0.444051
$622$	0	0
$623$	−17.8500	−0.715147
$624$	0	0
$625$	−30.5358	−1.22143
$626$	0	0
$627$	−23.5753	−0.941506
$628$	0	0
$629$	0.262585	0.0104700
$630$	0	0
$631$	−42.3192	−1.68470	−0.842351	−	0.538929i	$-0.818829\pi$
$631$	−42.3192	−1.68470	−0.842351	+	0.538929i	$0.818829\pi$
$632$	0	0
$633$	−51.3973	−2.04286
$634$	0	0
$635$	−28.8495	−1.14486
$636$	0	0
$637$	−21.7598	−0.862155
$638$	0	0
$639$	−0.182043	−0.00720152
$640$	0	0
$641$	−46.1804	−1.82402	−0.912008	−	0.410172i	$-0.865469\pi$
$641$	−46.1804	−1.82402	−0.912008	+	0.410172i	$0.865469\pi$
$642$	0	0
$643$	44.3225	1.74791	0.873954	−	0.486008i	$-0.161547\pi$
$643$	44.3225	1.74791	0.873954	+	0.486008i	$0.161547\pi$
$644$	0	0
$645$	58.6806	2.31055
$646$	0	0
$647$	−4.36792	−0.171721	−0.0858604	−	0.996307i	$-0.527364\pi$
$647$	−4.36792	−0.171721	−0.0858604	+	0.996307i	$0.527364\pi$
$648$	0	0
$649$	6.42405	0.252166
$650$	0	0
$651$	9.02472	0.353707
$652$	0	0
$653$	36.1824	1.41593	0.707964	−	0.706249i	$-0.249614\pi$
$653$	36.1824	1.41593	0.707964	+	0.706249i	$0.249614\pi$
$654$	0	0
$655$	15.9037	0.621410
$656$	0	0
$657$	0.647230	0.0252508
$658$	0	0
$659$	−18.5190	−0.721398	−0.360699	−	0.932682i	$-0.617462\pi$
$659$	−18.5190	−0.721398	−0.360699	+	0.932682i	$0.617462\pi$
$660$	0	0
$661$	30.7736	1.19695	0.598477	−	0.801140i	$-0.295773\pi$
$661$	30.7736	1.19695	0.598477	+	0.801140i	$0.295773\pi$
$662$	0	0
$663$	0.411830	0.0159941
$664$	0	0
$665$	−8.36679	−0.324450
$666$	0	0
$667$	−14.4080	−0.557880
$668$	0	0
$669$	23.2316	0.898187
$670$	0	0
$671$	−35.4594	−1.36890
$672$	0	0
$673$	1.67082	0.0644054	0.0322027	−	0.999481i	$-0.489748\pi$
$673$	1.67082	0.0644054	0.0322027	+	0.999481i	$0.489748\pi$
$674$	0	0
$675$	15.8546	0.610243
$676$	0	0
$677$	−1.01155	−0.0388770	−0.0194385	−	0.999811i	$-0.506188\pi$
$677$	−1.01155	−0.0388770	−0.0194385	+	0.999811i	$0.506188\pi$
$678$	0	0
$679$	−4.48248	−0.172022
$680$	0	0
$681$	−20.7063	−0.793466
$682$	0	0
$683$	2.31115	0.0884336	0.0442168	−	0.999022i	$-0.485921\pi$
$683$	2.31115	0.0884336	0.0442168	+	0.999022i	$0.485921\pi$
$684$	0	0
$685$	65.7164	2.51089
$686$	0	0
$687$	20.6766	0.788861
$688$	0	0
$689$	−44.2041	−1.68404
$690$	0	0
$691$	−9.36349	−0.356204	−0.178102	−	0.984012i	$-0.556996\pi$
$691$	−9.36349	−0.356204	−0.178102	+	0.984012i	$0.556996\pi$
$692$	0	0
$693$	−2.42887	−0.0922650
$694$	0	0
$695$	51.3146	1.94647
$696$	0	0
$697$	0.174630	0.00661458
$698$	0	0
$699$	2.19483	0.0830160
$700$	0	0
$701$	−5.30552	−0.200387	−0.100193	−	0.994968i	$-0.531946\pi$
$701$	−5.30552	−0.200387	−0.100193	+	0.994968i	$0.531946\pi$
$702$	0	0
$703$	11.6352	0.438830
$704$	0	0
$705$	−43.1514	−1.62518
$706$	0	0
$707$	5.55402	0.208880
$708$	0	0
$709$	31.6295	1.18787	0.593935	−	0.804513i	$-0.297574\pi$
$709$	31.6295	1.18787	0.593935	+	0.804513i	$0.297574\pi$
$710$	0	0
$711$	4.31743	0.161916
$712$	0	0
$713$	10.1960	0.381845
$714$	0	0
$715$	−53.2394	−1.99104
$716$	0	0
$717$	18.2846	0.682851
$718$	0	0
$719$	40.4332	1.50790	0.753952	−	0.656929i	$-0.228145\pi$
$719$	40.4332	1.50790	0.753952	+	0.656929i	$0.228145\pi$
$720$	0	0
$721$	4.98627	0.185698
$722$	0	0
$723$	−38.8901	−1.44634
$724$	0	0
$725$	20.6433	0.766674
$726$	0	0
$727$	36.0124	1.33563	0.667814	−	0.744329i	$-0.267230\pi$
$727$	36.0124	1.33563	0.667814	+	0.744329i	$0.267230\pi$
$728$	0	0
$729$	21.8643	0.809790
$730$	0	0
$731$	0.642519	0.0237644
$732$	0	0
$733$	16.4883	0.609008	0.304504	−	0.952511i	$-0.401509\pi$
$733$	16.4883	0.609008	0.304504	+	0.952511i	$0.401509\pi$
$734$	0	0
$735$	30.9004	1.13978
$736$	0	0
$737$	28.2533	1.04072
$738$	0	0
$739$	41.2515	1.51746	0.758730	−	0.651405i	$-0.225820\pi$
$739$	41.2515	1.51746	0.758730	+	0.651405i	$0.225820\pi$
$740$	0	0
$741$	18.2483	0.670366
$742$	0	0
$743$	43.5627	1.59816	0.799081	−	0.601224i	$-0.205320\pi$
$743$	43.5627	1.59816	0.799081	+	0.601224i	$0.205320\pi$
$744$	0	0
$745$	−1.40381	−0.0514318
$746$	0	0
$747$	−6.16796	−0.225674
$748$	0	0
$749$	20.7286	0.757405
$750$	0	0
$751$	17.2147	0.628174	0.314087	−	0.949394i	$-0.398302\pi$
$751$	17.2147	0.628174	0.314087	+	0.949394i	$0.398302\pi$
$752$	0	0
$753$	24.6093	0.896812
$754$	0	0
$755$	−18.4492	−0.671433
$756$	0	0
$757$	9.54239	0.346824	0.173412	−	0.984849i	$-0.444521\pi$
$757$	9.54239	0.346824	0.173412	+	0.984849i	$0.444521\pi$
$758$	0	0
$759$	−21.1516	−0.767754
$760$	0	0
$761$	−40.3944	−1.46430	−0.732149	−	0.681145i	$-0.761482\pi$
$761$	−40.3944	−1.46430	−0.732149	+	0.681145i	$0.761482\pi$
$762$	0	0
$763$	15.2915	0.553588
$764$	0	0
$765$	−0.0758727	−0.00274318
$766$	0	0
$767$	−4.97249	−0.179546
$768$	0	0
$769$	−39.7566	−1.43366	−0.716829	−	0.697249i	$-0.754407\pi$
$769$	−39.7566	−1.43366	−0.716829	+	0.697249i	$0.754407\pi$
$770$	0	0
$771$	26.9232	0.969615
$772$	0	0
$773$	15.6639	0.563390	0.281695	−	0.959504i	$-0.409103\pi$
$773$	15.6639	0.563390	0.281695	+	0.959504i	$0.409103\pi$
$774$	0	0
$775$	−14.6086	−0.524755
$776$	0	0
$777$	9.23979	0.331476
$778$	0	0
$779$	7.73789	0.277239
$780$	0	0
$781$	1.98620	0.0710719
$782$	0	0
$783$	29.2496	1.04529
$784$	0	0
$785$	−70.0438	−2.49997
$786$	0	0
$787$	19.1080	0.681125	0.340563	−	0.940222i	$-0.389382\pi$
$787$	19.1080	0.681125	0.340563	+	0.940222i	$0.389382\pi$
$788$	0	0
$789$	11.8241	0.420951
$790$	0	0
$791$	−5.59438	−0.198913
$792$	0	0
$793$	27.4471	0.974674
$794$	0	0
$795$	62.7728	2.22632
$796$	0	0
$797$	−25.6636	−0.909050	−0.454525	−	0.890734i	$-0.650191\pi$
$797$	−25.6636	−0.909050	−0.454525	+	0.890734i	$0.650191\pi$
$798$	0	0
$799$	−0.472483	−0.0167153
$800$	0	0
$801$	−7.17247	−0.253427
$802$	0	0
$803$	−7.06166	−0.249201
$804$	0	0
$805$	−7.50663	−0.264574
$806$	0	0
$807$	3.72157	0.131006
$808$	0	0
$809$	32.3932	1.13888	0.569442	−	0.822031i	$-0.307159\pi$
$809$	32.3932	1.13888	0.569442	+	0.822031i	$0.307159\pi$
$810$	0	0
$811$	−26.5460	−0.932156	−0.466078	−	0.884744i	$-0.654333\pi$
$811$	−26.5460	−0.932156	−0.466078	+	0.884744i	$0.654333\pi$
$812$	0	0
$813$	−10.3824	−0.364127
$814$	0	0
$815$	3.77427	0.132207
$816$	0	0
$817$	28.4701	0.996044
$818$	0	0
$819$	1.88004	0.0656940
$820$	0	0
$821$	2.34239	0.0817500	0.0408750	−	0.999164i	$-0.486985\pi$
$821$	2.34239	0.0817500	0.0408750	+	0.999164i	$0.486985\pi$
$822$	0	0
$823$	−29.3882	−1.02441	−0.512204	−	0.858864i	$-0.671171\pi$
$823$	−29.3882	−1.02441	−0.512204	+	0.858864i	$0.671171\pi$
$824$	0	0
$825$	30.3053	1.05510
$826$	0	0
$827$	−8.16844	−0.284044	−0.142022	−	0.989863i	$-0.545360\pi$
$827$	−8.16844	−0.284044	−0.142022	+	0.989863i	$0.545360\pi$
$828$	0	0
$829$	52.7126	1.83078	0.915392	−	0.402564i	$-0.131881\pi$
$829$	52.7126	1.83078	0.915392	+	0.402564i	$0.131881\pi$
$830$	0	0
$831$	−34.8877	−1.21024
$832$	0	0
$833$	0.338342	0.0117228
$834$	0	0
$835$	−71.7086	−2.48158
$836$	0	0
$837$	−20.6989	−0.715459
$838$	0	0
$839$	21.0536	0.726852	0.363426	−	0.931623i	$-0.381607\pi$
$839$	21.0536	0.726852	0.363426	+	0.931623i	$0.381607\pi$
$840$	0	0
$841$	9.08417	0.313247
$842$	0	0
$843$	29.0636	1.00100
$844$	0	0
$845$	3.65533	0.125747
$846$	0	0
$847$	14.2573	0.489886
$848$	0	0
$849$	59.2634	2.03392
$850$	0	0
$851$	10.4390	0.357845
$852$	0	0
$853$	−33.6427	−1.15191	−0.575953	−	0.817483i	$-0.695369\pi$
$853$	−33.6427	−1.15191	−0.575953	+	0.817483i	$0.695369\pi$
$854$	0	0
$855$	−3.36193	−0.114976
$856$	0	0
$857$	18.1320	0.619376	0.309688	−	0.950838i	$-0.399775\pi$
$857$	18.1320	0.619376	0.309688	+	0.950838i	$0.399775\pi$
$858$	0	0
$859$	−37.4621	−1.27819	−0.639095	−	0.769128i	$-0.720691\pi$
$859$	−37.4621	−1.27819	−0.639095	+	0.769128i	$0.720691\pi$
$860$	0	0
$861$	6.14484	0.209416
$862$	0	0
$863$	3.90183	0.132820	0.0664099	−	0.997792i	$-0.478846\pi$
$863$	3.90183	0.132820	0.0664099	+	0.997792i	$0.478846\pi$
$864$	0	0
$865$	−34.2554	−1.16472
$866$	0	0
$867$	31.5570	1.07173
$868$	0	0
$869$	−47.1057	−1.59795
$870$	0	0
$871$	−21.8692	−0.741011
$872$	0	0
$873$	−1.80114	−0.0609594
$874$	0	0
$875$	−5.32095	−0.179881
$876$	0	0
$877$	13.8299	0.467004	0.233502	−	0.972356i	$-0.424981\pi$
$877$	13.8299	0.467004	0.233502	+	0.972356i	$0.424981\pi$
$878$	0	0
$879$	−43.4860	−1.46675
$880$	0	0
$881$	26.7736	0.902025	0.451012	−	0.892518i	$-0.351063\pi$
$881$	26.7736	0.902025	0.451012	+	0.892518i	$0.351063\pi$
$882$	0	0
$883$	26.9929	0.908384	0.454192	−	0.890904i	$-0.349928\pi$
$883$	26.9929	0.908384	0.454192	+	0.890904i	$0.349928\pi$
$884$	0	0
$885$	7.06127	0.237362
$886$	0	0
$887$	−5.21969	−0.175260	−0.0876300	−	0.996153i	$-0.527929\pi$
$887$	−5.21969	−0.175260	−0.0876300	+	0.996153i	$0.527929\pi$
$888$	0	0
$889$	−11.1153	−0.372795
$890$	0	0
$891$	49.4864	1.65786
$892$	0	0
$893$	−20.9358	−0.700591
$894$	0	0
$895$	−70.1781	−2.34580
$896$	0	0
$897$	16.3722	0.546652
$898$	0	0
$899$	−26.9509	−0.898861
$900$	0	0
$901$	0.687326	0.0228981
$902$	0	0
$903$	22.6088	0.752374
$904$	0	0
$905$	−21.7185	−0.721949
$906$	0	0
$907$	−41.3415	−1.37272	−0.686361	−	0.727261i	$-0.740793\pi$
$907$	−41.3415	−1.37272	−0.686361	+	0.727261i	$0.740793\pi$
$908$	0	0
$909$	2.23171	0.0740211
$910$	0	0
$911$	−44.0700	−1.46010	−0.730052	−	0.683392i	$-0.760504\pi$
$911$	−44.0700	−1.46010	−0.730052	+	0.683392i	$0.760504\pi$
$912$	0	0
$913$	67.2961	2.22718
$914$	0	0
$915$	−38.9767	−1.28853
$916$	0	0
$917$	6.12748	0.202347
$918$	0	0
$919$	33.0024	1.08865	0.544324	−	0.838875i	$-0.316786\pi$
$919$	33.0024	1.08865	0.544324	+	0.838875i	$0.316786\pi$
$920$	0	0
$921$	−22.8323	−0.752349
$922$	0	0
$923$	−1.53740	−0.0506042
$924$	0	0
$925$	−14.9567	−0.491774
$926$	0	0
$927$	2.00358	0.0658061
$928$	0	0
$929$	54.5554	1.78990	0.894952	−	0.446162i	$-0.147210\pi$
$929$	54.5554	1.78990	0.894952	+	0.446162i	$0.147210\pi$
$930$	0	0
$931$	14.9920	0.491342
$932$	0	0
$933$	20.2195	0.661957
$934$	0	0
$935$	0.827816	0.0270725
$936$	0	0
$937$	−26.5026	−0.865802	−0.432901	−	0.901442i	$-0.642510\pi$
$937$	−26.5026	−0.865802	−0.432901	+	0.901442i	$0.642510\pi$
$938$	0	0
$939$	−32.1479	−1.04911
$940$	0	0
$941$	−7.77446	−0.253440	−0.126720	−	0.991939i	$-0.540445\pi$
$941$	−7.77446	−0.253440	−0.126720	+	0.991939i	$0.540445\pi$
$942$	0	0
$943$	6.94238	0.226075
$944$	0	0
$945$	15.2392	0.495730
$946$	0	0
$947$	38.7989	1.26079	0.630397	−	0.776273i	$-0.282892\pi$
$947$	38.7989	1.26079	0.630397	+	0.776273i	$0.282892\pi$
$948$	0	0
$949$	5.46602	0.177435
$950$	0	0
$951$	−25.5952	−0.829983
$952$	0	0
$953$	0.896106	0.0290277	0.0145139	−	0.999895i	$-0.495380\pi$
$953$	0.896106	0.0290277	0.0145139	+	0.999895i	$0.495380\pi$
$954$	0	0
$955$	−28.9448	−0.936631
$956$	0	0
$957$	55.9093	1.80729
$958$	0	0
$959$	25.3196	0.817612
$960$	0	0
$961$	−11.9278	−0.384768
$962$	0	0
$963$	8.32912	0.268402
$964$	0	0
$965$	−13.2376	−0.426134
$966$	0	0
$967$	−15.1730	−0.487931	−0.243965	−	0.969784i	$-0.578448\pi$
$967$	−15.1730	−0.487931	−0.243965	+	0.969784i	$0.578448\pi$
$968$	0	0
$969$	−0.283741	−0.00911507
$970$	0	0
$971$	26.4609	0.849172	0.424586	−	0.905388i	$-0.360420\pi$
$971$	26.4609	0.849172	0.424586	+	0.905388i	$0.360420\pi$
$972$	0	0
$973$	19.7708	0.633823
$974$	0	0
$975$	−23.4576	−0.751244
$976$	0	0
$977$	0.971157	0.0310701	0.0155350	−	0.999879i	$-0.495055\pi$
$977$	0.971157	0.0310701	0.0155350	+	0.999879i	$0.495055\pi$
$978$	0	0
$979$	78.2560	2.50107
$980$	0	0
$981$	6.14439	0.196175
$982$	0	0
$983$	−15.4561	−0.492973	−0.246486	−	0.969146i	$-0.579276\pi$
$983$	−15.4561	−0.492973	−0.246486	+	0.969146i	$0.579276\pi$
$984$	0	0
$985$	−38.9713	−1.24173
$986$	0	0
$987$	−16.6256	−0.529200
$988$	0	0
$989$	25.5432	0.812227
$990$	0	0
$991$	−8.77045	−0.278603	−0.139301	−	0.990250i	$-0.544486\pi$
$991$	−8.77045	−0.278603	−0.139301	+	0.990250i	$0.544486\pi$
$992$	0	0
$993$	−18.2814	−0.580142
$994$	0	0
$995$	−47.1550	−1.49491
$996$	0	0
$997$	23.6414	0.748731	0.374365	−	0.927281i	$-0.377861\pi$
$997$	23.6414	0.748731	0.374365	+	0.927281i	$0.377861\pi$
$998$	0	0
$999$	−21.1922	−0.670491

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.e.1.8	✓	29
4.3	odd	2		8048.2.a.w.1.22		29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.8	✓	29	1.1	even	1	trivial
8048.2.a.w.1.22		29	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)

\(f(q)\)	\(=\)	\(q-1.85667 q^{3} +2.88879 q^{5} +1.11301 q^{7} +0.447227 q^{9} +O(q^{10})\)	\(q-1.85667 q^{3} +2.88879 q^{5} +1.11301 q^{7} +0.447227 q^{9} -4.87952 q^{11} +3.77695 q^{13} -5.36353 q^{15} -0.0587275 q^{17} -2.60223 q^{19} -2.06649 q^{21} -2.33470 q^{23} +3.34509 q^{25} +4.73966 q^{27} +6.17124 q^{29} -4.36717 q^{31} +9.05966 q^{33} +3.21524 q^{35} -4.47125 q^{37} -7.01255 q^{39} -2.97356 q^{41} -10.9407 q^{43} +1.29194 q^{45} +8.04535 q^{47} -5.76121 q^{49} +0.109038 q^{51} -11.7036 q^{53} -14.0959 q^{55} +4.83148 q^{57} -1.31653 q^{59} +7.26699 q^{61} +0.497768 q^{63} +10.9108 q^{65} -5.79019 q^{67} +4.33477 q^{69} -0.407049 q^{71} +1.44721 q^{73} -6.21073 q^{75} -5.43094 q^{77} +9.65376 q^{79} -10.1417 q^{81} -13.7915 q^{83} -0.169651 q^{85} -11.4580 q^{87} -16.0376 q^{89} +4.20378 q^{91} +8.10840 q^{93} -7.51728 q^{95} -4.02735 q^{97} -2.18225 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * q^99

Introduction

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Modular forms

Varieties

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Database

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