LMFDB - Embedded newform 8037.2.a.x.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, 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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.1757681045$
Analytic rank:	$0$
Dimension:	$34$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.625588 q^{2} -1.60864 q^{4} +3.29167 q^{5} -1.24186 q^{7} -2.25752 q^{8} +O(q^{10})$	\(q+0.625588 q^{2} -1.60864 q^{4} +3.29167 q^{5} -1.24186 q^{7} -2.25752 q^{8} +2.05923 q^{10} +1.56749 q^{11} +0.814958 q^{13} -0.776892 q^{14} +1.80500 q^{16} +5.32195 q^{17} +1.00000 q^{19} -5.29512 q^{20} +0.980603 q^{22} +0.971204 q^{23} +5.83512 q^{25} +0.509828 q^{26} +1.99770 q^{28} +7.80878 q^{29} +2.39295 q^{31} +5.64423 q^{32} +3.32935 q^{34} -4.08779 q^{35} -2.63282 q^{37} +0.625588 q^{38} -7.43103 q^{40} -11.7752 q^{41} +7.57546 q^{43} -2.52153 q^{44} +0.607574 q^{46} -1.00000 q^{47} -5.45779 q^{49} +3.65038 q^{50} -1.31097 q^{52} -3.52312 q^{53} +5.15967 q^{55} +2.80352 q^{56} +4.88508 q^{58} -11.9322 q^{59} +5.01448 q^{61} +1.49700 q^{62} -0.0790374 q^{64} +2.68258 q^{65} -2.61700 q^{67} -8.56109 q^{68} -2.55727 q^{70} +10.0628 q^{71} +4.61705 q^{73} -1.64706 q^{74} -1.60864 q^{76} -1.94660 q^{77} +2.99979 q^{79} +5.94147 q^{80} -7.36644 q^{82} +7.94848 q^{83} +17.5181 q^{85} +4.73912 q^{86} -3.53864 q^{88} -11.4278 q^{89} -1.01206 q^{91} -1.56232 q^{92} -0.625588 q^{94} +3.29167 q^{95} -11.0720 q^{97} -3.41433 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 5 q^{2} + 31 q^{4} + 14 q^{5} + 15 q^{8}+O(q^{10}) $ 34 * q + 5 * q^2 + 31 * q^4 + 14 * q^5 + 15 * q^8	$ 34 q + 5 q^{2} + 31 q^{4} + 14 q^{5} + 15 q^{8} + 18 q^{11} - 6 q^{13} + 12 q^{14} + 21 q^{16} + 36 q^{17} + 34 q^{19} + 40 q^{20} + 12 q^{22} + 38 q^{23} + 32 q^{25} + 15 q^{26} + 28 q^{28} + 14 q^{29} - 6 q^{31} + 35 q^{32} + 10 q^{34} + 46 q^{35} - 2 q^{37} + 5 q^{38} + 31 q^{40} + 18 q^{41} - 6 q^{43} + 42 q^{44} - 14 q^{46} - 34 q^{47} + 44 q^{49} + 9 q^{50} + 2 q^{52} + 32 q^{53} + 8 q^{55} - 4 q^{56} + 8 q^{58} + 62 q^{59} - 10 q^{61} + 30 q^{62} - 37 q^{64} + 8 q^{65} + 92 q^{68} - 62 q^{70} + 4 q^{71} - 8 q^{73} + 34 q^{74} + 31 q^{76} + 52 q^{77} + 40 q^{79} + 48 q^{80} - 2 q^{82} + 110 q^{83} - 12 q^{85} + 16 q^{86} - 44 q^{88} + 2 q^{89} - 28 q^{91} + 60 q^{92} - 5 q^{94} + 14 q^{95} + 2 q^{97} + 45 q^{98}+O(q^{100}) $ 34 * q + 5 * q^2 + 31 * q^4 + 14 * q^5 + 15 * q^8 + 18 * q^11 - 6 * q^13 + 12 * q^14 + 21 * q^16 + 36 * q^17 + 34 * q^19 + 40 * q^20 + 12 * q^22 + 38 * q^23 + 32 * q^25 + 15 * q^26 + 28 * q^28 + 14 * q^29 - 6 * q^31 + 35 * q^32 + 10 * q^34 + 46 * q^35 - 2 * q^37 + 5 * q^38 + 31 * q^40 + 18 * q^41 - 6 * q^43 + 42 * q^44 - 14 * q^46 - 34 * q^47 + 44 * q^49 + 9 * q^50 + 2 * q^52 + 32 * q^53 + 8 * q^55 - 4 * q^56 + 8 * q^58 + 62 * q^59 - 10 * q^61 + 30 * q^62 - 37 * q^64 + 8 * q^65 + 92 * q^68 - 62 * q^70 + 4 * q^71 - 8 * q^73 + 34 * q^74 + 31 * q^76 + 52 * q^77 + 40 * q^79 + 48 * q^80 - 2 * q^82 + 110 * q^83 - 12 * q^85 + 16 * q^86 - 44 * q^88 + 2 * q^89 - 28 * q^91 + 60 * q^92 - 5 * q^94 + 14 * q^95 + 2 * q^97 + 45 * 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 $

$2$	0.625588	0.442358	0.221179	−	0.975233i	$-0.429010\pi$
$2$	0.625588	0.442358	0.221179	+	0.975233i	$0.429010\pi$
$3$	0	0
$3$	0	0
$4$	−1.60864	−0.804320
$5$	3.29167	1.47208	0.736041	−	0.676937i	$-0.236693\pi$
$5$	3.29167	1.47208	0.736041	+	0.676937i	$0.236693\pi$
$6$	0	0
$7$	−1.24186	−0.469378	−0.234689	−	0.972070i	$-0.575407\pi$
$7$	−1.24186	−0.469378	−0.234689	+	0.972070i	$0.575407\pi$
$8$	−2.25752	−0.798155
$9$	0	0
$10$	2.05923	0.651186
$11$	1.56749	0.472616	0.236308	−	0.971678i	$-0.424063\pi$
$11$	1.56749	0.472616	0.236308	+	0.971678i	$0.424063\pi$
$12$	0	0
$13$	0.814958	0.226029	0.113014	−	0.993593i	$-0.463949\pi$
$13$	0.814958	0.226029	0.113014	+	0.993593i	$0.463949\pi$
$14$	−0.776892	−0.207633
$15$	0	0
$16$	1.80500	0.451250
$17$	5.32195	1.29076	0.645381	−	0.763861i	$-0.276699\pi$
$17$	5.32195	1.29076	0.645381	+	0.763861i	$0.276699\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−5.29512	−1.18402
$21$	0	0
$22$	0.980603	0.209065
$23$	0.971204	0.202510	0.101255	−	0.994861i	$-0.467714\pi$
$23$	0.971204	0.202510	0.101255	+	0.994861i	$0.467714\pi$
$24$	0	0
$25$	5.83512	1.16702
$26$	0.509828	0.0999855
$27$	0	0
$28$	1.99770	0.377530
$29$	7.80878	1.45005	0.725027	−	0.688720i	$-0.241827\pi$
$29$	7.80878	1.45005	0.725027	+	0.688720i	$0.241827\pi$
$30$	0	0
$31$	2.39295	0.429786	0.214893	−	0.976638i	$-0.431060\pi$
$31$	2.39295	0.429786	0.214893	+	0.976638i	$0.431060\pi$
$32$	5.64423	0.997768
$33$	0	0
$34$	3.32935	0.570978
$35$	−4.08779	−0.690963
$36$	0	0
$37$	−2.63282	−0.432833	−0.216417	−	0.976301i	$-0.569437\pi$
$37$	−2.63282	−0.432833	−0.216417	+	0.976301i	$0.569437\pi$
$38$	0.625588	0.101484
$39$	0	0
$40$	−7.43103	−1.17495
$41$	−11.7752	−1.83898	−0.919491	−	0.393112i	$-0.871399\pi$
$41$	−11.7752	−1.83898	−0.919491	+	0.393112i	$0.871399\pi$
$42$	0	0
$43$	7.57546	1.15525	0.577623	−	0.816303i	$-0.303980\pi$
$43$	7.57546	1.15525	0.577623	+	0.816303i	$0.303980\pi$
$44$	−2.52153	−0.380134
$45$	0	0
$46$	0.607574	0.0895819
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−5.45779	−0.779684
$50$	3.65038	0.516242
$51$	0	0
$52$	−1.31097	−0.181799
$53$	−3.52312	−0.483937	−0.241969	−	0.970284i	$-0.577793\pi$
$53$	−3.52312	−0.483937	−0.241969	+	0.970284i	$0.577793\pi$
$54$	0	0
$55$	5.15967	0.695729
$56$	2.80352	0.374636
$57$	0	0
$58$	4.88508	0.641443
$59$	−11.9322	−1.55344	−0.776719	−	0.629848i	$-0.783117\pi$
$59$	−11.9322	−1.55344	−0.776719	+	0.629848i	$0.783117\pi$
$60$	0	0
$61$	5.01448	0.642038	0.321019	−	0.947073i	$-0.395975\pi$
$61$	5.01448	0.642038	0.321019	+	0.947073i	$0.395975\pi$
$62$	1.49700	0.190119
$63$	0	0
$64$	−0.0790374	−0.00987967
$65$	2.68258	0.332733
$66$	0	0
$67$	−2.61700	−0.319718	−0.159859	−	0.987140i	$-0.551104\pi$
$67$	−2.61700	−0.319718	−0.159859	+	0.987140i	$0.551104\pi$
$68$	−8.56109	−1.03819
$69$	0	0
$70$	−2.55727	−0.305653
$71$	10.0628	1.19423	0.597116	−	0.802155i	$-0.296313\pi$
$71$	10.0628	1.19423	0.597116	+	0.802155i	$0.296313\pi$
$72$	0	0
$73$	4.61705	0.540384	0.270192	−	0.962806i	$-0.412913\pi$
$73$	4.61705	0.540384	0.270192	+	0.962806i	$0.412913\pi$
$74$	−1.64706	−0.191467
$75$	0	0
$76$	−1.60864	−0.184524
$77$	−1.94660	−0.221836
$78$	0	0
$79$	2.99979	0.337503	0.168752	−	0.985659i	$-0.446026\pi$
$79$	2.99979	0.337503	0.168752	+	0.985659i	$0.446026\pi$
$80$	5.94147	0.664277
$81$	0	0
$82$	−7.36644	−0.813487
$83$	7.94848	0.872460	0.436230	−	0.899835i	$-0.356313\pi$
$83$	7.94848	0.872460	0.436230	+	0.899835i	$0.356313\pi$
$84$	0	0
$85$	17.5181	1.90011
$86$	4.73912	0.511032
$87$	0	0
$88$	−3.53864	−0.377221
$89$	−11.4278	−1.21134	−0.605671	−	0.795715i	$-0.707095\pi$
$89$	−11.4278	−1.21134	−0.605671	+	0.795715i	$0.707095\pi$
$90$	0	0
$91$	−1.01206	−0.106093
$92$	−1.56232	−0.162883
$93$	0	0
$94$	−0.625588	−0.0645245
$95$	3.29167	0.337719
$96$	0	0
$97$	−11.0720	−1.12419	−0.562096	−	0.827072i	$-0.690005\pi$
$97$	−11.0720	−1.12419	−0.562096	+	0.827072i	$0.690005\pi$
$98$	−3.41433	−0.344899
$99$	0	0
$100$	−9.38660	−0.938660
$101$	−1.76990	−0.176111	−0.0880556	−	0.996116i	$-0.528065\pi$
$101$	−1.76990	−0.176111	−0.0880556	+	0.996116i	$0.528065\pi$
$102$	0	0
$103$	7.51984	0.740952	0.370476	−	0.928842i	$-0.379195\pi$
$103$	7.51984	0.740952	0.370476	+	0.928842i	$0.379195\pi$
$104$	−1.83979	−0.180406
$105$	0	0
$106$	−2.20402	−0.214073
$107$	−8.40815	−0.812847	−0.406423	−	0.913685i	$-0.633224\pi$
$107$	−8.40815	−0.812847	−0.406423	+	0.913685i	$0.633224\pi$
$108$	0	0
$109$	9.67555	0.926749	0.463375	−	0.886162i	$-0.346638\pi$
$109$	9.67555	0.926749	0.463375	+	0.886162i	$0.346638\pi$
$110$	3.22783	0.307761
$111$	0	0
$112$	−2.24155	−0.211807
$113$	−5.52033	−0.519309	−0.259654	−	0.965702i	$-0.583609\pi$
$113$	−5.52033	−0.519309	−0.259654	+	0.965702i	$0.583609\pi$
$114$	0	0
$115$	3.19689	0.298111
$116$	−12.5615	−1.16631
$117$	0	0
$118$	−7.46463	−0.687175
$119$	−6.60910	−0.605855
$120$	0	0
$121$	−8.54298	−0.776634
$122$	3.13700	0.284010
$123$	0	0
$124$	−3.84939	−0.345686
$125$	2.74894	0.245872
$126$	0	0
$127$	15.2803	1.35590	0.677952	−	0.735106i	$-0.262868\pi$
$127$	15.2803	1.35590	0.677952	+	0.735106i	$0.262868\pi$
$128$	−11.3379	−1.00214
$129$	0	0
$130$	1.67819	0.147187
$131$	6.76473	0.591037	0.295518	−	0.955337i	$-0.404508\pi$
$131$	6.76473	0.591037	0.295518	+	0.955337i	$0.404508\pi$
$132$	0	0
$133$	−1.24186	−0.107683
$134$	−1.63717	−0.141430
$135$	0	0
$136$	−12.0144	−1.03023
$137$	13.6023	1.16212	0.581060	−	0.813861i	$-0.302638\pi$
$137$	13.6023	1.16212	0.581060	+	0.813861i	$0.302638\pi$
$138$	0	0
$139$	15.6957	1.33129	0.665646	−	0.746268i	$-0.268156\pi$
$139$	15.6957	1.33129	0.665646	+	0.746268i	$0.268156\pi$
$140$	6.57578	0.555755
$141$	0	0
$142$	6.29515	0.528277
$143$	1.27744	0.106825
$144$	0	0
$145$	25.7040	2.13460
$146$	2.88837	0.239043
$147$	0	0
$148$	4.23526	0.348136
$149$	16.1861	1.32601	0.663007	−	0.748613i	$-0.269280\pi$
$149$	16.1861	1.32601	0.663007	+	0.748613i	$0.269280\pi$
$150$	0	0
$151$	19.4276	1.58099	0.790497	−	0.612466i	$-0.209822\pi$
$151$	19.4276	1.58099	0.790497	+	0.612466i	$0.209822\pi$
$152$	−2.25752	−0.183109
$153$	0	0
$154$	−1.21777	−0.0981307
$155$	7.87681	0.632680
$156$	0	0
$157$	6.11360	0.487918	0.243959	−	0.969785i	$-0.421554\pi$
$157$	6.11360	0.487918	0.243959	+	0.969785i	$0.421554\pi$
$158$	1.87663	0.149297
$159$	0	0
$160$	18.5790	1.46880
$161$	−1.20610	−0.0950538
$162$	0	0
$163$	0.743444	0.0582310	0.0291155	−	0.999576i	$-0.490731\pi$
$163$	0.743444	0.0582310	0.0291155	+	0.999576i	$0.490731\pi$
$164$	18.9421	1.47913
$165$	0	0
$166$	4.97248	0.385939
$167$	−5.11951	−0.396159	−0.198080	−	0.980186i	$-0.563470\pi$
$167$	−5.11951	−0.396159	−0.198080	+	0.980186i	$0.563470\pi$
$168$	0	0
$169$	−12.3358	−0.948911
$170$	10.9591	0.840526
$171$	0	0
$172$	−12.1862	−0.929188
$173$	21.5243	1.63647	0.818233	−	0.574887i	$-0.194954\pi$
$173$	21.5243	1.63647	0.818233	+	0.574887i	$0.194954\pi$
$174$	0	0
$175$	−7.24639	−0.547776
$176$	2.82932	0.213268
$177$	0	0
$178$	−7.14908	−0.535846
$179$	−9.51848	−0.711444	−0.355722	−	0.934592i	$-0.615765\pi$
$179$	−9.51848	−0.711444	−0.355722	+	0.934592i	$0.615765\pi$
$180$	0	0
$181$	10.5753	0.786056	0.393028	−	0.919526i	$-0.371428\pi$
$181$	10.5753	0.786056	0.393028	+	0.919526i	$0.371428\pi$
$182$	−0.633134	−0.0469310
$183$	0	0
$184$	−2.19251	−0.161634
$185$	−8.66639	−0.637166
$186$	0	0
$187$	8.34210	0.610034
$188$	1.60864	0.117322
$189$	0	0
$190$	2.05923	0.149392
$191$	15.5421	1.12458	0.562292	−	0.826939i	$-0.309920\pi$
$191$	15.5421	1.12458	0.562292	+	0.826939i	$0.309920\pi$
$192$	0	0
$193$	1.08271	0.0779348	0.0389674	−	0.999240i	$-0.487593\pi$
$193$	1.08271	0.0779348	0.0389674	+	0.999240i	$0.487593\pi$
$194$	−6.92651	−0.497295
$195$	0	0
$196$	8.77961	0.627115
$197$	−4.20561	−0.299637	−0.149819	−	0.988714i	$-0.547869\pi$
$197$	−4.20561	−0.299637	−0.149819	+	0.988714i	$0.547869\pi$
$198$	0	0
$199$	19.6772	1.39488	0.697441	−	0.716642i	$-0.254322\pi$
$199$	19.6772	1.39488	0.697441	+	0.716642i	$0.254322\pi$
$200$	−13.1729	−0.931465
$201$	0	0
$202$	−1.10723	−0.0779042
$203$	−9.69740	−0.680624
$204$	0	0
$205$	−38.7602	−2.70713
$206$	4.70432	0.327766
$207$	0	0
$208$	1.47100	0.101995
$209$	1.56749	0.108426
$210$	0	0
$211$	−24.4496	−1.68318	−0.841591	−	0.540115i	$-0.818381\pi$
$211$	−24.4496	−1.68318	−0.841591	+	0.540115i	$0.818381\pi$
$212$	5.66743	0.389240
$213$	0	0
$214$	−5.26004	−0.359569
$215$	24.9359	1.70062
$216$	0	0
$217$	−2.97170	−0.201732
$218$	6.05291	0.409955
$219$	0	0
$220$	−8.30004	−0.559589
$221$	4.33716	0.291749
$222$	0	0
$223$	4.94233	0.330963	0.165482	−	0.986213i	$-0.447082\pi$
$223$	4.94233	0.330963	0.165482	+	0.986213i	$0.447082\pi$
$224$	−7.00933	−0.468331
$225$	0	0
$226$	−3.45345	−0.229720
$227$	5.30066	0.351817	0.175908	−	0.984407i	$-0.443714\pi$
$227$	5.30066	0.351817	0.175908	+	0.984407i	$0.443714\pi$
$228$	0	0
$229$	20.9677	1.38558	0.692791	−	0.721138i	$-0.256381\pi$
$229$	20.9677	1.38558	0.692791	+	0.721138i	$0.256381\pi$
$230$	1.99993	0.131872
$231$	0	0
$232$	−17.6285	−1.15737
$233$	−12.7018	−0.832121	−0.416060	−	0.909337i	$-0.636589\pi$
$233$	−12.7018	−0.832121	−0.416060	+	0.909337i	$0.636589\pi$
$234$	0	0
$235$	−3.29167	−0.214725
$236$	19.1946	1.24946
$237$	0	0
$238$	−4.13458	−0.268005
$239$	17.2243	1.11415	0.557073	−	0.830464i	$-0.311924\pi$
$239$	17.2243	1.11415	0.557073	+	0.830464i	$0.311924\pi$
$240$	0	0
$241$	1.07741	0.0694022	0.0347011	−	0.999398i	$-0.488952\pi$
$241$	1.07741	0.0694022	0.0347011	+	0.999398i	$0.488952\pi$
$242$	−5.34438	−0.343550
$243$	0	0
$244$	−8.06649	−0.516404
$245$	−17.9653	−1.14776
$246$	0	0
$247$	0.814958	0.0518545
$248$	−5.40213	−0.343036
$249$	0	0
$250$	1.71970	0.108764
$251$	16.3240	1.03036	0.515181	−	0.857082i	$-0.327725\pi$
$251$	16.3240	1.03036	0.515181	+	0.857082i	$0.327725\pi$
$252$	0	0
$253$	1.52235	0.0957095
$254$	9.55915	0.599794
$255$	0	0
$256$	−6.93478	−0.433424
$257$	30.9269	1.92917	0.964585	−	0.263772i	$-0.0849667\pi$
$257$	30.9269	1.92917	0.964585	+	0.263772i	$0.0849667\pi$
$258$	0	0
$259$	3.26959	0.203163
$260$	−4.31530	−0.267623
$261$	0	0
$262$	4.23193	0.261450
$263$	−23.0909	−1.42384	−0.711922	−	0.702258i	$-0.752175\pi$
$263$	−23.0909	−1.42384	−0.711922	+	0.702258i	$0.752175\pi$
$264$	0	0
$265$	−11.5970	−0.712395
$266$	−0.776892	−0.0476343
$267$	0	0
$268$	4.20981	0.257155
$269$	−27.9588	−1.70468	−0.852340	−	0.522987i	$-0.824817\pi$
$269$	−27.9588	−1.70468	−0.852340	+	0.522987i	$0.824817\pi$
$270$	0	0
$271$	21.8594	1.32786	0.663931	−	0.747794i	$-0.268887\pi$
$271$	21.8594	1.32786	0.663931	+	0.747794i	$0.268887\pi$
$272$	9.60611	0.582456
$273$	0	0
$274$	8.50942	0.514073
$275$	9.14649	0.551554
$276$	0	0
$277$	9.42621	0.566366	0.283183	−	0.959066i	$-0.408610\pi$
$277$	9.42621	0.566366	0.283183	+	0.959066i	$0.408610\pi$
$278$	9.81904	0.588907
$279$	0	0
$280$	9.22828	0.551495
$281$	−7.56605	−0.451353	−0.225676	−	0.974202i	$-0.572459\pi$
$281$	−7.56605	−0.451353	−0.225676	+	0.974202i	$0.572459\pi$
$282$	0	0
$283$	−9.18619	−0.546062	−0.273031	−	0.962005i	$-0.588026\pi$
$283$	−9.18619	−0.546062	−0.273031	+	0.962005i	$0.588026\pi$
$284$	−16.1874	−0.960544
$285$	0	0
$286$	0.799150	0.0472547
$287$	14.6232	0.863178
$288$	0	0
$289$	11.3231	0.666065
$290$	16.0801	0.944256
$291$	0	0
$292$	−7.42716	−0.434642
$293$	20.0799	1.17308	0.586540	−	0.809921i	$-0.300490\pi$
$293$	20.0799	1.17308	0.586540	+	0.809921i	$0.300490\pi$
$294$	0	0
$295$	−39.2768	−2.28679
$296$	5.94365	0.345468
$297$	0	0
$298$	10.1258	0.586572
$299$	0.791491	0.0457731
$300$	0	0
$301$	−9.40765	−0.542248
$302$	12.1537	0.699365
$303$	0	0
$304$	1.80500	0.103524
$305$	16.5060	0.945132
$306$	0	0
$307$	−30.2375	−1.72575	−0.862873	−	0.505422i	$-0.831337\pi$
$307$	−30.2375	−1.72575	−0.862873	+	0.505422i	$0.831337\pi$
$308$	3.13138	0.178427
$309$	0	0
$310$	4.92764	0.279871
$311$	−5.92245	−0.335831	−0.167916	−	0.985801i	$-0.553704\pi$
$311$	−5.92245	−0.335831	−0.167916	+	0.985801i	$0.553704\pi$
$312$	0	0
$313$	−7.26786	−0.410804	−0.205402	−	0.978678i	$-0.565850\pi$
$313$	−7.26786	−0.410804	−0.205402	+	0.978678i	$0.565850\pi$
$314$	3.82460	0.215834
$315$	0	0
$316$	−4.82558	−0.271460
$317$	−8.03230	−0.451139	−0.225569	−	0.974227i	$-0.572424\pi$
$317$	−8.03230	−0.451139	−0.225569	+	0.974227i	$0.572424\pi$
$318$	0	0
$319$	12.2402	0.685319
$320$	−0.260165	−0.0145437
$321$	0	0
$322$	−0.754520	−0.0420478
$323$	5.32195	0.296121
$324$	0	0
$325$	4.75538	0.263781
$326$	0.465090	0.0257589
$327$	0	0
$328$	26.5828	1.46779
$329$	1.24186	0.0684659
$330$	0	0
$331$	−5.16830	−0.284075	−0.142038	−	0.989861i	$-0.545365\pi$
$331$	−5.16830	−0.284075	−0.142038	+	0.989861i	$0.545365\pi$
$332$	−12.7862	−0.701736
$333$	0	0
$334$	−3.20270	−0.175244
$335$	−8.61432	−0.470651
$336$	0	0
$337$	−32.5893	−1.77525	−0.887627	−	0.460562i	$-0.847648\pi$
$337$	−32.5893	−1.77525	−0.887627	+	0.460562i	$0.847648\pi$
$338$	−7.71716	−0.419758
$339$	0	0
$340$	−28.1803	−1.52829
$341$	3.75092	0.203124
$342$	0	0
$343$	15.4708	0.835345
$344$	−17.1018	−0.922066
$345$	0	0
$346$	13.4654	0.723903
$347$	30.6414	1.64492	0.822458	−	0.568826i	$-0.192602\pi$
$347$	30.6414	1.64492	0.822458	+	0.568826i	$0.192602\pi$
$348$	0	0
$349$	−30.8311	−1.65035	−0.825177	−	0.564875i	$-0.808924\pi$
$349$	−30.8311	−1.65035	−0.825177	+	0.564875i	$0.808924\pi$
$350$	−4.53325	−0.242313
$351$	0	0
$352$	8.84727	0.471561
$353$	−18.0043	−0.958271	−0.479136	−	0.877741i	$-0.659050\pi$
$353$	−18.0043	−0.958271	−0.479136	+	0.877741i	$0.659050\pi$
$354$	0	0
$355$	33.1234	1.75801
$356$	18.3832	0.974307
$357$	0	0
$358$	−5.95465	−0.314713
$359$	6.60845	0.348781	0.174390	−	0.984677i	$-0.444205\pi$
$359$	6.60845	0.348781	0.174390	+	0.984677i	$0.444205\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	6.61578	0.347718
$363$	0	0
$364$	1.62804	0.0853327
$365$	15.1978	0.795490
$366$	0	0
$367$	−11.7837	−0.615104	−0.307552	−	0.951531i	$-0.599510\pi$
$367$	−11.7837	−0.615104	−0.307552	+	0.951531i	$0.599510\pi$
$368$	1.75302	0.0913827
$369$	0	0
$370$	−5.42159	−0.281855
$371$	4.37521	0.227150
$372$	0	0
$373$	−1.23012	−0.0636932	−0.0318466	−	0.999493i	$-0.510139\pi$
$373$	−1.23012	−0.0636932	−0.0318466	+	0.999493i	$0.510139\pi$
$374$	5.21872	0.269853
$375$	0	0
$376$	2.25752	0.116423
$377$	6.36383	0.327754
$378$	0	0
$379$	22.2193	1.14133	0.570665	−	0.821183i	$-0.306685\pi$
$379$	22.2193	1.14133	0.570665	+	0.821183i	$0.306685\pi$
$380$	−5.29512	−0.271634
$381$	0	0
$382$	9.72292	0.497468
$383$	30.6513	1.56621	0.783104	−	0.621891i	$-0.213635\pi$
$383$	30.6513	1.56621	0.783104	+	0.621891i	$0.213635\pi$
$384$	0	0
$385$	−6.40757	−0.326560
$386$	0.677328	0.0344751
$387$	0	0
$388$	17.8109	0.904210
$389$	−3.74794	−0.190028	−0.0950141	−	0.995476i	$-0.530290\pi$
$389$	−3.74794	−0.190028	−0.0950141	+	0.995476i	$0.530290\pi$
$390$	0	0
$391$	5.16870	0.261392
$392$	12.3211	0.622308
$393$	0	0
$394$	−2.63098	−0.132547
$395$	9.87434	0.496832
$396$	0	0
$397$	−11.6962	−0.587016	−0.293508	−	0.955957i	$-0.594823\pi$
$397$	−11.6962	−0.587016	−0.293508	+	0.955957i	$0.594823\pi$
$398$	12.3098	0.617036
$399$	0	0
$400$	10.5324	0.526620
$401$	21.8833	1.09280	0.546400	−	0.837525i	$-0.315998\pi$
$401$	21.8833	1.09280	0.546400	+	0.837525i	$0.315998\pi$
$402$	0	0
$403$	1.95015	0.0971440
$404$	2.84713	0.141650
$405$	0	0
$406$	−6.06658	−0.301079
$407$	−4.12692	−0.204564
$408$	0	0
$409$	−23.4927	−1.16164	−0.580821	−	0.814032i	$-0.697268\pi$
$409$	−23.4927	−1.16164	−0.580821	+	0.814032i	$0.697268\pi$
$410$	−24.2479	−1.19752
$411$	0	0
$412$	−12.0967	−0.595962
$413$	14.8181	0.729150
$414$	0	0
$415$	26.1638	1.28433
$416$	4.59981	0.225524
$417$	0	0
$418$	0.980603	0.0479629
$419$	5.99341	0.292797	0.146399	−	0.989226i	$-0.453232\pi$
$419$	5.99341	0.292797	0.146399	+	0.989226i	$0.453232\pi$
$420$	0	0
$421$	−6.16198	−0.300317	−0.150158	−	0.988662i	$-0.547978\pi$
$421$	−6.16198	−0.300317	−0.150158	+	0.988662i	$0.547978\pi$
$422$	−15.2954	−0.744568
$423$	0	0
$424$	7.95352	0.386257
$425$	31.0542	1.50635
$426$	0	0
$427$	−6.22727	−0.301359
$428$	13.5257	0.653789
$429$	0	0
$430$	15.5996	0.752281
$431$	34.9437	1.68318	0.841590	−	0.540116i	$-0.181620\pi$
$431$	34.9437	1.68318	0.841590	+	0.540116i	$0.181620\pi$
$432$	0	0
$433$	−36.5278	−1.75541	−0.877706	−	0.479199i	$-0.840927\pi$
$433$	−36.5278	−1.75541	−0.877706	+	0.479199i	$0.840927\pi$
$434$	−1.85906	−0.0892378
$435$	0	0
$436$	−15.5645	−0.745403
$437$	0.971204	0.0464590
$438$	0	0
$439$	33.1212	1.58079	0.790394	−	0.612599i	$-0.209876\pi$
$439$	33.1212	1.58079	0.790394	+	0.612599i	$0.209876\pi$
$440$	−11.6481	−0.555299
$441$	0	0
$442$	2.71328	0.129057
$443$	6.57760	0.312511	0.156256	−	0.987717i	$-0.450058\pi$
$443$	6.57760	0.312511	0.156256	+	0.987717i	$0.450058\pi$
$444$	0	0
$445$	−37.6165	−1.78319
$446$	3.09187	0.146404
$447$	0	0
$448$	0.0981532	0.00463730
$449$	2.09866	0.0990418	0.0495209	−	0.998773i	$-0.484231\pi$
$449$	2.09866	0.0990418	0.0495209	+	0.998773i	$0.484231\pi$
$450$	0	0
$451$	−18.4575	−0.869132
$452$	8.88021	0.417690
$453$	0	0
$454$	3.31603	0.155629
$455$	−3.33138	−0.156177
$456$	0	0
$457$	−7.00120	−0.327502	−0.163751	−	0.986502i	$-0.552359\pi$
$457$	−7.00120	−0.327502	−0.163751	+	0.986502i	$0.552359\pi$
$458$	13.1171	0.612923
$459$	0	0
$460$	−5.14264	−0.239777
$461$	32.7531	1.52546	0.762731	−	0.646715i	$-0.223858\pi$
$461$	32.7531	1.52546	0.762731	+	0.646715i	$0.223858\pi$
$462$	0	0
$463$	20.1661	0.937200	0.468600	−	0.883410i	$-0.344759\pi$
$463$	20.1661	0.937200	0.468600	+	0.883410i	$0.344759\pi$
$464$	14.0949	0.654337
$465$	0	0
$466$	−7.94608	−0.368095
$467$	−29.4466	−1.36263	−0.681313	−	0.731992i	$-0.738591\pi$
$467$	−29.4466	−1.36263	−0.681313	+	0.731992i	$0.738591\pi$
$468$	0	0
$469$	3.24995	0.150069
$470$	−2.05923	−0.0949853
$471$	0	0
$472$	26.9372	1.23988
$473$	11.8745	0.545988
$474$	0	0
$475$	5.83512	0.267734
$476$	10.6317	0.487302
$477$	0	0
$478$	10.7753	0.492851
$479$	−2.92930	−0.133843	−0.0669215	−	0.997758i	$-0.521318\pi$
$479$	−2.92930	−0.133843	−0.0669215	+	0.997758i	$0.521318\pi$
$480$	0	0
$481$	−2.14564	−0.0978327
$482$	0.674016	0.0307006
$483$	0	0
$484$	13.7426	0.624662
$485$	−36.4454	−1.65490
$486$	0	0
$487$	−13.6029	−0.616407	−0.308204	−	0.951320i	$-0.599728\pi$
$487$	−13.6029	−0.616407	−0.308204	+	0.951320i	$0.599728\pi$
$488$	−11.3203	−0.512446
$489$	0	0
$490$	−11.2389	−0.507720
$491$	−43.2731	−1.95289	−0.976445	−	0.215767i	$-0.930775\pi$
$491$	−43.2731	−1.95289	−0.976445	+	0.215767i	$0.930775\pi$
$492$	0	0
$493$	41.5579	1.87167
$494$	0.509828	0.0229382
$495$	0	0
$496$	4.31927	0.193941
$497$	−12.4965	−0.560546
$498$	0	0
$499$	−19.4997	−0.872925	−0.436462	−	0.899723i	$-0.643769\pi$
$499$	−19.4997	−0.872925	−0.436462	+	0.899723i	$0.643769\pi$
$500$	−4.42205	−0.197760
$501$	0	0
$502$	10.2121	0.455788
$503$	−1.52623	−0.0680513	−0.0340256	−	0.999421i	$-0.510833\pi$
$503$	−1.52623	−0.0680513	−0.0340256	+	0.999421i	$0.510833\pi$
$504$	0	0
$505$	−5.82592	−0.259250
$506$	0.952366	0.0423378
$507$	0	0
$508$	−24.5804	−1.09058
$509$	35.8997	1.59123	0.795613	−	0.605805i	$-0.207149\pi$
$509$	35.8997	1.59123	0.795613	+	0.605805i	$0.207149\pi$
$510$	0	0
$511$	−5.73372	−0.253645
$512$	18.3375	0.810410
$513$	0	0
$514$	19.3475	0.853383
$515$	24.7529	1.09074
$516$	0	0
$517$	−1.56749	−0.0689381
$518$	2.04542	0.0898705
$519$	0	0
$520$	−6.05597	−0.265572
$521$	−8.55671	−0.374876	−0.187438	−	0.982276i	$-0.560018\pi$
$521$	−8.55671	−0.374876	−0.187438	+	0.982276i	$0.560018\pi$
$522$	0	0
$523$	−28.9935	−1.26780	−0.633898	−	0.773416i	$-0.718546\pi$
$523$	−28.9935	−1.26780	−0.633898	+	0.773416i	$0.718546\pi$
$524$	−10.8820	−0.475383
$525$	0	0
$526$	−14.4454	−0.629848
$527$	12.7351	0.554752
$528$	0	0
$529$	−22.0568	−0.958990
$530$	−7.25492	−0.315133
$531$	0	0
$532$	1.99770	0.0866114
$533$	−9.59632	−0.415663
$534$	0	0
$535$	−27.6769	−1.19658
$536$	5.90794	0.255184
$537$	0	0
$538$	−17.4907	−0.754078
$539$	−8.55503	−0.368491
$540$	0	0
$541$	−41.4158	−1.78060	−0.890302	−	0.455371i	$-0.849506\pi$
$541$	−41.4158	−1.78060	−0.890302	+	0.455371i	$0.849506\pi$
$542$	13.6750	0.587390
$543$	0	0
$544$	30.0383	1.28788
$545$	31.8488	1.36425
$546$	0	0
$547$	−3.27430	−0.139999	−0.0699995	−	0.997547i	$-0.522300\pi$
$547$	−3.27430	−0.139999	−0.0699995	+	0.997547i	$0.522300\pi$
$548$	−21.8811	−0.934716
$549$	0	0
$550$	5.72193	0.243984
$551$	7.80878	0.332665
$552$	0	0
$553$	−3.72532	−0.158417
$554$	5.89693	0.250536
$555$	0	0
$556$	−25.2487	−1.07078
$557$	7.84945	0.332592	0.166296	−	0.986076i	$-0.446819\pi$
$557$	7.84945	0.332592	0.166296	+	0.986076i	$0.446819\pi$
$558$	0	0
$559$	6.17368	0.261119
$560$	−7.37847	−0.311797
$561$	0	0
$562$	−4.73323	−0.199659
$563$	28.8145	1.21439	0.607193	−	0.794554i	$-0.292295\pi$
$563$	28.8145	1.21439	0.607193	+	0.794554i	$0.292295\pi$
$564$	0	0
$565$	−18.1711	−0.764464
$566$	−5.74677	−0.241555
$567$	0	0
$568$	−22.7169	−0.953181
$569$	5.28058	0.221374	0.110687	−	0.993855i	$-0.464695\pi$
$569$	5.28058	0.221374	0.110687	+	0.993855i	$0.464695\pi$
$570$	0	0
$571$	15.8249	0.662251	0.331125	−	0.943587i	$-0.392572\pi$
$571$	15.8249	0.662251	0.331125	+	0.943587i	$0.392572\pi$
$572$	−2.05494	−0.0859213
$573$	0	0
$574$	9.14808	0.381833
$575$	5.66709	0.236334
$576$	0	0
$577$	41.2182	1.71594	0.857969	−	0.513702i	$-0.171726\pi$
$577$	41.2182	1.71594	0.857969	+	0.513702i	$0.171726\pi$
$578$	7.08360	0.294639
$579$	0	0
$580$	−41.3484	−1.71690
$581$	−9.87089	−0.409514
$582$	0	0
$583$	−5.52245	−0.228717
$584$	−10.4231	−0.431310
$585$	0	0
$586$	12.5617	0.518921
$587$	−7.91951	−0.326873	−0.163437	−	0.986554i	$-0.552258\pi$
$587$	−7.91951	−0.326873	−0.163437	+	0.986554i	$0.552258\pi$
$588$	0	0
$589$	2.39295	0.0985997
$590$	−24.5711	−1.01158
$591$	0	0
$592$	−4.75224	−0.195316
$593$	31.9289	1.31116	0.655582	−	0.755124i	$-0.272424\pi$
$593$	31.9289	1.31116	0.655582	+	0.755124i	$0.272424\pi$
$594$	0	0
$595$	−21.7550	−0.891869
$596$	−26.0375	−1.06654
$597$	0	0
$598$	0.495147	0.0202481
$599$	15.8085	0.645916	0.322958	−	0.946413i	$-0.395323\pi$
$599$	15.8085	0.645916	0.322958	+	0.946413i	$0.395323\pi$
$600$	0	0
$601$	13.7007	0.558865	0.279432	−	0.960165i	$-0.409854\pi$
$601$	13.7007	0.558865	0.279432	+	0.960165i	$0.409854\pi$
$602$	−5.88531	−0.239867
$603$	0	0
$604$	−31.2520	−1.27162
$605$	−28.1207	−1.14327
$606$	0	0
$607$	−1.18740	−0.0481953	−0.0240976	−	0.999710i	$-0.507671\pi$
$607$	−1.18740	−0.0481953	−0.0240976	+	0.999710i	$0.507671\pi$
$608$	5.64423	0.228904
$609$	0	0
$610$	10.3260	0.418086
$611$	−0.814958	−0.0329697
$612$	0	0
$613$	24.9129	1.00622	0.503111	−	0.864222i	$-0.332189\pi$
$613$	24.9129	1.00622	0.503111	+	0.864222i	$0.332189\pi$
$614$	−18.9162	−0.763396
$615$	0	0
$616$	4.39449	0.177059
$617$	−24.8801	−1.00163	−0.500817	−	0.865553i	$-0.666967\pi$
$617$	−24.8801	−1.00163	−0.500817	+	0.865553i	$0.666967\pi$
$618$	0	0
$619$	16.5989	0.667168	0.333584	−	0.942720i	$-0.391742\pi$
$619$	16.5989	0.667168	0.333584	+	0.942720i	$0.391742\pi$
$620$	−12.6709	−0.508877
$621$	0	0
$622$	−3.70501	−0.148558
$623$	14.1917	0.568578
$624$	0	0
$625$	−20.1270	−0.805079
$626$	−4.54669	−0.181722
$627$	0	0
$628$	−9.83458	−0.392442
$629$	−14.0117	−0.558685
$630$	0	0
$631$	−31.6017	−1.25804	−0.629022	−	0.777387i	$-0.716545\pi$
$631$	−31.6017	−1.25804	−0.629022	+	0.777387i	$0.716545\pi$
$632$	−6.77210	−0.269380
$633$	0	0
$634$	−5.02491	−0.199565
$635$	50.2976	1.99600
$636$	0	0
$637$	−4.44787	−0.176231
$638$	7.65732	0.303156
$639$	0	0
$640$	−37.3207	−1.47523
$641$	−24.6788	−0.974754	−0.487377	−	0.873192i	$-0.662046\pi$
$641$	−24.6788	−0.974754	−0.487377	+	0.873192i	$0.662046\pi$
$642$	0	0
$643$	23.8271	0.939651	0.469826	−	0.882759i	$-0.344317\pi$
$643$	23.8271	0.939651	0.469826	+	0.882759i	$0.344317\pi$
$644$	1.94018	0.0764537
$645$	0	0
$646$	3.32935	0.130991
$647$	−16.0565	−0.631248	−0.315624	−	0.948884i	$-0.602214\pi$
$647$	−16.0565	−0.631248	−0.315624	+	0.948884i	$0.602214\pi$
$648$	0	0
$649$	−18.7036	−0.734179
$650$	2.97491	0.116685
$651$	0	0
$652$	−1.19593	−0.0468364
$653$	−2.15810	−0.0844532	−0.0422266	−	0.999108i	$-0.513445\pi$
$653$	−2.15810	−0.0844532	−0.0422266	+	0.999108i	$0.513445\pi$
$654$	0	0
$655$	22.2673	0.870054
$656$	−21.2543	−0.829841
$657$	0	0
$658$	0.776892	0.0302864
$659$	40.7852	1.58877	0.794384	−	0.607416i	$-0.207794\pi$
$659$	40.7852	1.58877	0.794384	+	0.607416i	$0.207794\pi$
$660$	0	0
$661$	−2.98510	−0.116107	−0.0580534	−	0.998313i	$-0.518489\pi$
$661$	−2.98510	−0.116107	−0.0580534	+	0.998313i	$0.518489\pi$
$662$	−3.23323	−0.125663
$663$	0	0
$664$	−17.9439	−0.696358
$665$	−4.08779	−0.158518
$666$	0	0
$667$	7.58392	0.293651
$668$	8.23544	0.318639
$669$	0	0
$670$	−5.38902	−0.208196
$671$	7.86014	0.303437
$672$	0	0
$673$	14.6580	0.565023	0.282511	−	0.959264i	$-0.408832\pi$
$673$	14.6580	0.565023	0.282511	+	0.959264i	$0.408832\pi$
$674$	−20.3875	−0.785297
$675$	0	0
$676$	19.8439	0.763228
$677$	−24.9476	−0.958816	−0.479408	−	0.877592i	$-0.659149\pi$
$677$	−24.9476	−0.958816	−0.479408	+	0.877592i	$0.659149\pi$
$678$	0	0
$679$	13.7499	0.527671
$680$	−39.5475	−1.51658
$681$	0	0
$682$	2.34653	0.0898534
$683$	−3.61079	−0.138163	−0.0690815	−	0.997611i	$-0.522007\pi$
$683$	−3.61079	−0.138163	−0.0690815	+	0.997611i	$0.522007\pi$
$684$	0	0
$685$	44.7742	1.71074
$686$	9.67835	0.369521
$687$	0	0
$688$	13.6737	0.521305
$689$	−2.87119	−0.109384
$690$	0	0
$691$	−13.4656	−0.512256	−0.256128	−	0.966643i	$-0.582447\pi$
$691$	−13.4656	−0.512256	−0.256128	+	0.966643i	$0.582447\pi$
$692$	−34.6249	−1.31624
$693$	0	0
$694$	19.1689	0.727641
$695$	51.6651	1.95977
$696$	0	0
$697$	−62.6671	−2.37369
$698$	−19.2876	−0.730046
$699$	0	0
$700$	11.6568	0.440587
$701$	−23.4300	−0.884939	−0.442470	−	0.896783i	$-0.645898\pi$
$701$	−23.4300	−0.884939	−0.442470	+	0.896783i	$0.645898\pi$
$702$	0	0
$703$	−2.63282	−0.0992988
$704$	−0.123890	−0.00466929
$705$	0	0
$706$	−11.2633	−0.423899
$707$	2.19796	0.0826628
$708$	0	0
$709$	−5.07131	−0.190457	−0.0952286	−	0.995455i	$-0.530358\pi$
$709$	−5.07131	−0.190457	−0.0952286	+	0.995455i	$0.530358\pi$
$710$	20.7216	0.777667
$711$	0	0
$712$	25.7985	0.966838
$713$	2.32404	0.0870360
$714$	0	0
$715$	4.20491	0.157255
$716$	15.3118	0.572229
$717$	0	0
$718$	4.13417	0.154286
$719$	−13.6634	−0.509559	−0.254779	−	0.966999i	$-0.582003\pi$
$719$	−13.6634	−0.509559	−0.254779	+	0.966999i	$0.582003\pi$
$720$	0	0
$721$	−9.33858	−0.347787
$722$	0.625588	0.0232820
$723$	0	0
$724$	−17.0119	−0.632241
$725$	45.5652	1.69225
$726$	0	0
$727$	10.8201	0.401295	0.200647	−	0.979664i	$-0.435695\pi$
$727$	10.8201	0.401295	0.200647	+	0.979664i	$0.435695\pi$
$728$	2.28475	0.0846786
$729$	0	0
$730$	9.50757	0.351891
$731$	40.3162	1.49115
$732$	0	0
$733$	43.3040	1.59947	0.799735	−	0.600353i	$-0.204973\pi$
$733$	43.3040	1.59947	0.799735	+	0.600353i	$0.204973\pi$
$734$	−7.37174	−0.272096
$735$	0	0
$736$	5.48170	0.202058
$737$	−4.10213	−0.151104
$738$	0	0
$739$	38.4302	1.41368	0.706839	−	0.707375i	$-0.250121\pi$
$739$	38.4302	1.41368	0.706839	+	0.707375i	$0.250121\pi$
$740$	13.9411	0.512485
$741$	0	0
$742$	2.73708	0.100481
$743$	−0.139780	−0.00512804	−0.00256402	−	0.999997i	$-0.500816\pi$
$743$	−0.139780	−0.00512804	−0.00256402	+	0.999997i	$0.500816\pi$
$744$	0	0
$745$	53.2792	1.95200
$746$	−0.769548	−0.0281752
$747$	0	0
$748$	−13.4194	−0.490663
$749$	10.4417	0.381533
$750$	0	0
$751$	−17.1416	−0.625506	−0.312753	−	0.949834i	$-0.601251\pi$
$751$	−17.1416	−0.625506	−0.312753	+	0.949834i	$0.601251\pi$
$752$	−1.80500	−0.0658216
$753$	0	0
$754$	3.98114	0.144984
$755$	63.9493	2.32735
$756$	0	0
$757$	30.4639	1.10723	0.553615	−	0.832773i	$-0.313248\pi$
$757$	30.4639	1.10723	0.553615	+	0.832773i	$0.313248\pi$
$758$	13.9002	0.504876
$759$	0	0
$760$	−7.43103	−0.269552
$761$	−51.2727	−1.85863	−0.929317	−	0.369282i	$-0.879604\pi$
$761$	−51.2727	−1.85863	−0.929317	+	0.369282i	$0.879604\pi$
$762$	0	0
$763$	−12.0157	−0.434996
$764$	−25.0016	−0.904525
$765$	0	0
$766$	19.1751	0.692824
$767$	−9.72423	−0.351121
$768$	0	0
$769$	11.2463	0.405551	0.202776	−	0.979225i	$-0.435004\pi$
$769$	11.2463	0.405551	0.202776	+	0.979225i	$0.435004\pi$
$770$	−4.00850	−0.144456
$771$	0	0
$772$	−1.74168	−0.0626845
$773$	−43.0087	−1.54691	−0.773457	−	0.633848i	$-0.781474\pi$
$773$	−43.0087	−1.54691	−0.773457	+	0.633848i	$0.781474\pi$
$774$	0	0
$775$	13.9631	0.501571
$776$	24.9953	0.897279
$777$	0	0
$778$	−2.34467	−0.0840605
$779$	−11.7752	−0.421891
$780$	0	0
$781$	15.7733	0.564413
$782$	3.23347	0.115629
$783$	0	0
$784$	−9.85131	−0.351832
$785$	20.1240	0.718256
$786$	0	0
$787$	26.7432	0.953293	0.476646	−	0.879095i	$-0.341852\pi$
$787$	26.7432	0.953293	0.476646	+	0.879095i	$0.341852\pi$
$788$	6.76531	0.241004
$789$	0	0
$790$	6.17727	0.219777
$791$	6.85546	0.243752
$792$	0	0
$793$	4.08659	0.145119
$794$	−7.31701	−0.259671
$795$	0	0
$796$	−31.6536	−1.12193
$797$	−5.96325	−0.211229	−0.105614	−	0.994407i	$-0.533681\pi$
$797$	−5.96325	−0.211229	−0.105614	+	0.994407i	$0.533681\pi$
$798$	0	0
$799$	−5.32195	−0.188277
$800$	32.9348	1.16442
$801$	0	0
$802$	13.6899	0.483408
$803$	7.23717	0.255394
$804$	0	0
$805$	−3.97008	−0.139927
$806$	1.21999	0.0429724
$807$	0	0
$808$	3.99558	0.140564
$809$	−15.3076	−0.538187	−0.269094	−	0.963114i	$-0.586724\pi$
$809$	−15.3076	−0.538187	−0.269094	+	0.963114i	$0.586724\pi$
$810$	0	0
$811$	35.5399	1.24797	0.623987	−	0.781435i	$-0.285512\pi$
$811$	35.5399	1.24797	0.623987	+	0.781435i	$0.285512\pi$
$812$	15.5996	0.547439
$813$	0	0
$814$	−2.58175	−0.0904904
$815$	2.44718	0.0857208
$816$	0	0
$817$	7.57546	0.265032
$818$	−14.6968	−0.513861
$819$	0	0
$820$	62.3512	2.17740
$821$	15.2074	0.530742	0.265371	−	0.964146i	$-0.414505\pi$
$821$	15.2074	0.530742	0.265371	+	0.964146i	$0.414505\pi$
$822$	0	0
$823$	40.1253	1.39868	0.699340	−	0.714790i	$-0.253478\pi$
$823$	40.1253	1.39868	0.699340	+	0.714790i	$0.253478\pi$
$824$	−16.9762	−0.591394
$825$	0	0
$826$	9.27001	0.322545
$827$	−26.7739	−0.931019	−0.465509	−	0.885043i	$-0.654129\pi$
$827$	−26.7739	−0.931019	−0.465509	+	0.885043i	$0.654129\pi$
$828$	0	0
$829$	−25.4297	−0.883211	−0.441606	−	0.897209i	$-0.645591\pi$
$829$	−25.4297	−0.883211	−0.441606	+	0.897209i	$0.645591\pi$
$830$	16.3678	0.568134
$831$	0	0
$832$	−0.0644121	−0.00223309
$833$	−29.0461	−1.00639
$834$	0	0
$835$	−16.8518	−0.583179
$836$	−2.52153	−0.0872088
$837$	0	0
$838$	3.74941	0.129521
$839$	−25.0844	−0.866011	−0.433005	−	0.901391i	$-0.642547\pi$
$839$	−25.0844	−0.866011	−0.433005	+	0.901391i	$0.642547\pi$
$840$	0	0
$841$	31.9771	1.10266
$842$	−3.85486	−0.132847
$843$	0	0
$844$	39.3306	1.35382
$845$	−40.6056	−1.39687
$846$	0	0
$847$	10.6092	0.364535
$848$	−6.35923	−0.218377
$849$	0	0
$850$	19.4271	0.666345
$851$	−2.55701	−0.0876531
$852$	0	0
$853$	−31.7906	−1.08849	−0.544246	−	0.838926i	$-0.683184\pi$
$853$	−31.7906	−1.08849	−0.544246	+	0.838926i	$0.683184\pi$
$854$	−3.89571	−0.133308
$855$	0	0
$856$	18.9816	0.648777
$857$	9.66593	0.330182	0.165091	−	0.986278i	$-0.447208\pi$
$857$	9.66593	0.330182	0.165091	+	0.986278i	$0.447208\pi$
$858$	0	0
$859$	−50.7739	−1.73238	−0.866192	−	0.499711i	$-0.833439\pi$
$859$	−50.7739	−1.73238	−0.866192	+	0.499711i	$0.833439\pi$
$860$	−40.1129	−1.36784
$861$	0	0
$862$	21.8604	0.744568
$863$	−46.8392	−1.59443	−0.797213	−	0.603698i	$-0.793693\pi$
$863$	−46.8392	−1.59443	−0.797213	+	0.603698i	$0.793693\pi$
$864$	0	0
$865$	70.8511	2.40901
$866$	−22.8513	−0.776520
$867$	0	0
$868$	4.78040	0.162257
$869$	4.70214	0.159509
$870$	0	0
$871$	−2.13275	−0.0722654
$872$	−21.8428	−0.739689
$873$	0	0
$874$	0.607574	0.0205515
$875$	−3.41379	−0.115407
$876$	0	0
$877$	−8.88567	−0.300048	−0.150024	−	0.988682i	$-0.547935\pi$
$877$	−8.88567	−0.300048	−0.150024	+	0.988682i	$0.547935\pi$
$878$	20.7202	0.699274
$879$	0	0
$880$	9.31320	0.313948
$881$	23.3855	0.787876	0.393938	−	0.919137i	$-0.371112\pi$
$881$	23.3855	0.787876	0.393938	+	0.919137i	$0.371112\pi$
$882$	0	0
$883$	−30.3815	−1.02242	−0.511210	−	0.859456i	$-0.670803\pi$
$883$	−30.3815	−1.02242	−0.511210	+	0.859456i	$0.670803\pi$
$884$	−6.97693	−0.234660
$885$	0	0
$886$	4.11487	0.138242
$887$	−14.0914	−0.473142	−0.236571	−	0.971614i	$-0.576024\pi$
$887$	−14.0914	−0.473142	−0.236571	+	0.971614i	$0.576024\pi$
$888$	0	0
$889$	−18.9759	−0.636432
$890$	−23.5325	−0.788810
$891$	0	0
$892$	−7.95043	−0.266200
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	−31.3317	−1.04730
$896$	14.0801	0.470382
$897$	0	0
$898$	1.31290	0.0438119
$899$	18.6860	0.623214
$900$	0	0
$901$	−18.7498	−0.624648
$902$	−11.5468	−0.384467
$903$	0	0
$904$	12.4623	0.414488
$905$	34.8105	1.15714
$906$	0	0
$907$	−53.0851	−1.76266	−0.881332	−	0.472498i	$-0.843352\pi$
$907$	−53.0851	−1.76266	−0.881332	+	0.472498i	$0.843352\pi$
$908$	−8.52684	−0.282973
$909$	0	0
$910$	−2.08407	−0.0690863
$911$	−10.1886	−0.337564	−0.168782	−	0.985653i	$-0.553983\pi$
$911$	−10.1886	−0.337564	−0.168782	+	0.985653i	$0.553983\pi$
$912$	0	0
$913$	12.4592	0.412338
$914$	−4.37987	−0.144873
$915$	0	0
$916$	−33.7294	−1.11445
$917$	−8.40083	−0.277420
$918$	0	0
$919$	48.1854	1.58949	0.794745	−	0.606943i	$-0.207604\pi$
$919$	48.1854	1.58949	0.794745	+	0.606943i	$0.207604\pi$
$920$	−7.21704	−0.237939
$921$	0	0
$922$	20.4899	0.674800
$923$	8.20074	0.269931
$924$	0	0
$925$	−15.3628	−0.505127
$926$	12.6157	0.414578
$927$	0	0
$928$	44.0746	1.44682
$929$	26.9731	0.884960	0.442480	−	0.896778i	$-0.354099\pi$
$929$	26.9731	0.884960	0.442480	+	0.896778i	$0.354099\pi$
$930$	0	0
$931$	−5.45779	−0.178872
$932$	20.4326	0.669291
$933$	0	0
$934$	−18.4214	−0.602768
$935$	27.4595	0.898020
$936$	0	0
$937$	−48.1541	−1.57313	−0.786563	−	0.617510i	$-0.788141\pi$
$937$	−48.1541	−1.57313	−0.786563	+	0.617510i	$0.788141\pi$
$938$	2.03313	0.0663840
$939$	0	0
$940$	5.29512	0.172708
$941$	−13.0046	−0.423939	−0.211969	−	0.977276i	$-0.567988\pi$
$941$	−13.0046	−0.423939	−0.211969	+	0.977276i	$0.567988\pi$
$942$	0	0
$943$	−11.4361	−0.372412
$944$	−21.5376	−0.700989
$945$	0	0
$946$	7.42852	0.241522
$947$	42.8964	1.39394	0.696972	−	0.717098i	$-0.254530\pi$
$947$	42.8964	1.39394	0.696972	+	0.717098i	$0.254530\pi$
$948$	0	0
$949$	3.76270	0.122142
$950$	3.65038	0.118434
$951$	0	0
$952$	14.9202	0.483566
$953$	34.4364	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$953$	34.4364	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$954$	0	0
$955$	51.1594	1.65548
$956$	−27.7076	−0.896129
$957$	0	0
$958$	−1.83253	−0.0592064
$959$	−16.8921	−0.545474
$960$	0	0
$961$	−25.2738	−0.815284
$962$	−1.34229	−0.0432771
$963$	0	0
$964$	−1.73317	−0.0558215
$965$	3.56391	0.114726
$966$	0	0
$967$	−45.0585	−1.44898	−0.724492	−	0.689283i	$-0.757925\pi$
$967$	−45.0585	−1.44898	−0.724492	+	0.689283i	$0.757925\pi$
$968$	19.2860	0.619874
$969$	0	0
$970$	−22.7998	−0.732058
$971$	−28.0378	−0.899776	−0.449888	−	0.893085i	$-0.648536\pi$
$971$	−28.0378	−0.899776	−0.449888	+	0.893085i	$0.648536\pi$
$972$	0	0
$973$	−19.4918	−0.624879
$974$	−8.50983	−0.272672
$975$	0	0
$976$	9.05113	0.289720
$977$	11.6350	0.372236	0.186118	−	0.982527i	$-0.440409\pi$
$977$	11.6350	0.372236	0.186118	+	0.982527i	$0.440409\pi$
$978$	0	0
$979$	−17.9129	−0.572500
$980$	28.8996	0.923165
$981$	0	0
$982$	−27.0712	−0.863876
$983$	43.7920	1.39675	0.698374	−	0.715733i	$-0.253907\pi$
$983$	43.7920	1.39675	0.698374	+	0.715733i	$0.253907\pi$
$984$	0	0
$985$	−13.8435	−0.441090
$986$	25.9981	0.827950
$987$	0	0
$988$	−1.31097	−0.0417076
$989$	7.35732	0.233949
$990$	0	0
$991$	−21.3358	−0.677753	−0.338876	−	0.940831i	$-0.610047\pi$
$991$	−21.3358	−0.677753	−0.338876	+	0.940831i	$0.610047\pi$
$992$	13.5064	0.428827
$993$	0	0
$994$	−7.81769	−0.247962
$995$	64.7710	2.05338
$996$	0	0
$997$	0.0367939	0.00116528	0.000582638	−	1.00000i	$-0.499815\pi$
$997$	0.0367939	0.00116528	0.000582638		1.00000i	$0.499815\pi$
$998$	−12.1988	−0.386145
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.x.1.20	yes	34
3.2	odd	2		8037.2.a.u.1.15	✓	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8037.2.a.u.1.15	✓	34	3.2	odd	2
8037.2.a.x.1.20	yes	34	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q+0.625588 q^{2} -1.60864 q^{4} +3.29167 q^{5} -1.24186 q^{7} -2.25752 q^{8} +O(q^{10})\)	\(q+0.625588 q^{2} -1.60864 q^{4} +3.29167 q^{5} -1.24186 q^{7} -2.25752 q^{8} +2.05923 q^{10} +1.56749 q^{11} +0.814958 q^{13} -0.776892 q^{14} +1.80500 q^{16} +5.32195 q^{17} +1.00000 q^{19} -5.29512 q^{20} +0.980603 q^{22} +0.971204 q^{23} +5.83512 q^{25} +0.509828 q^{26} +1.99770 q^{28} +7.80878 q^{29} +2.39295 q^{31} +5.64423 q^{32} +3.32935 q^{34} -4.08779 q^{35} -2.63282 q^{37} +0.625588 q^{38} -7.43103 q^{40} -11.7752 q^{41} +7.57546 q^{43} -2.52153 q^{44} +0.607574 q^{46} -1.00000 q^{47} -5.45779 q^{49} +3.65038 q^{50} -1.31097 q^{52} -3.52312 q^{53} +5.15967 q^{55} +2.80352 q^{56} +4.88508 q^{58} -11.9322 q^{59} +5.01448 q^{61} +1.49700 q^{62} -0.0790374 q^{64} +2.68258 q^{65} -2.61700 q^{67} -8.56109 q^{68} -2.55727 q^{70} +10.0628 q^{71} +4.61705 q^{73} -1.64706 q^{74} -1.60864 q^{76} -1.94660 q^{77} +2.99979 q^{79} +5.94147 q^{80} -7.36644 q^{82} +7.94848 q^{83} +17.5181 q^{85} +4.73912 q^{86} -3.53864 q^{88} -11.4278 q^{89} -1.01206 q^{91} -1.56232 q^{92} -0.625588 q^{94} +3.29167 q^{95} -11.0720 q^{97} -3.41433 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 5 q^{2} + 31 q^{4} + 14 q^{5} + 15 q^{8}+O(q^{10}) \) 34 * q + 5 * q^2 + 31 * q^4 + 14 * q^5 + 15 * q^8	\( 34 q + 5 q^{2} + 31 q^{4} + 14 q^{5} + 15 q^{8} + 18 q^{11} - 6 q^{13} + 12 q^{14} + 21 q^{16} + 36 q^{17} + 34 q^{19} + 40 q^{20} + 12 q^{22} + 38 q^{23} + 32 q^{25} + 15 q^{26} + 28 q^{28} + 14 q^{29} - 6 q^{31} + 35 q^{32} + 10 q^{34} + 46 q^{35} - 2 q^{37} + 5 q^{38} + 31 q^{40} + 18 q^{41} - 6 q^{43} + 42 q^{44} - 14 q^{46} - 34 q^{47} + 44 q^{49} + 9 q^{50} + 2 q^{52} + 32 q^{53} + 8 q^{55} - 4 q^{56} + 8 q^{58} + 62 q^{59} - 10 q^{61} + 30 q^{62} - 37 q^{64} + 8 q^{65} + 92 q^{68} - 62 q^{70} + 4 q^{71} - 8 q^{73} + 34 q^{74} + 31 q^{76} + 52 q^{77} + 40 q^{79} + 48 q^{80} - 2 q^{82} + 110 q^{83} - 12 q^{85} + 16 q^{86} - 44 q^{88} + 2 q^{89} - 28 q^{91} + 60 q^{92} - 5 q^{94} + 14 q^{95} + 2 q^{97} + 45 q^{98}+O(q^{100}) \) 34 * q + 5 * q^2 + 31 * q^4 + 14 * q^5 + 15 * q^8 + 18 * q^11 - 6 * q^13 + 12 * q^14 + 21 * q^16 + 36 * q^17 + 34 * q^19 + 40 * q^20 + 12 * q^22 + 38 * q^23 + 32 * q^25 + 15 * q^26 + 28 * q^28 + 14 * q^29 - 6 * q^31 + 35 * q^32 + 10 * q^34 + 46 * q^35 - 2 * q^37 + 5 * q^38 + 31 * q^40 + 18 * q^41 - 6 * q^43 + 42 * q^44 - 14 * q^46 - 34 * q^47 + 44 * q^49 + 9 * q^50 + 2 * q^52 + 32 * q^53 + 8 * q^55 - 4 * q^56 + 8 * q^58 + 62 * q^59 - 10 * q^61 + 30 * q^62 - 37 * q^64 + 8 * q^65 + 92 * q^68 - 62 * q^70 + 4 * q^71 - 8 * q^73 + 34 * q^74 + 31 * q^76 + 52 * q^77 + 40 * q^79 + 48 * q^80 - 2 * q^82 + 110 * q^83 - 12 * q^85 + 16 * q^86 - 44 * q^88 + 2 * q^89 - 28 * q^91 + 60 * q^92 - 5 * q^94 + 14 * q^95 + 2 * q^97 + 45 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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