LMFDB - Embedded newform 5733.2.a.br.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5733,2,Mod(1,5733)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5733, 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("5733.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5733 = 3^{2} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5733.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:	$45.7782354788$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.4507648.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 $ x^6 - 2x^5 - 5x^4 + 8x^3 + 7x^2 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 637)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.20475$ of defining polynomial
Character	$\chi$	$=$	5733.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.656184 q^{2} -1.56942 q^{4} +1.35996 q^{5} -2.34220 q^{8} +O(q^{10})$	$q+0.656184 q^{2} -1.56942 q^{4} +1.35996 q^{5} -2.34220 q^{8} +0.892385 q^{10} +1.90551 q^{11} +1.00000 q^{13} +1.60193 q^{16} -3.56633 q^{17} +0.985255 q^{19} -2.13436 q^{20} +1.25037 q^{22} -1.69605 q^{23} -3.15050 q^{25} +0.656184 q^{26} -6.54835 q^{29} +7.69550 q^{31} +5.73556 q^{32} -2.34017 q^{34} -2.02005 q^{37} +0.646508 q^{38} -3.18530 q^{40} -9.88779 q^{41} -3.16639 q^{43} -2.99055 q^{44} -1.11292 q^{46} -7.76416 q^{47} -2.06731 q^{50} -1.56942 q^{52} -0.354194 q^{53} +2.59143 q^{55} -4.29692 q^{58} -2.16385 q^{59} +12.2002 q^{61} +5.04967 q^{62} +0.559715 q^{64} +1.35996 q^{65} +11.3134 q^{67} +5.59709 q^{68} +9.05268 q^{71} -7.13619 q^{73} -1.32553 q^{74} -1.54628 q^{76} -5.39629 q^{79} +2.17857 q^{80} -6.48821 q^{82} +2.03494 q^{83} -4.85008 q^{85} -2.07773 q^{86} -4.46309 q^{88} +6.89864 q^{89} +2.66182 q^{92} -5.09472 q^{94} +1.33991 q^{95} -14.6223 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{4} - 6 q^{5}+O(q^{10}) $ 6 * q + 4 * q^4 - 6 * q^5	$ 6 q + 4 q^{4} - 6 q^{5} + 4 q^{10} - 4 q^{11} + 6 q^{13} - 16 q^{17} + 2 q^{19} - 16 q^{20} - 12 q^{22} + 6 q^{23} - 4 q^{25} + 6 q^{29} + 6 q^{31} + 20 q^{32} - 8 q^{38} + 4 q^{40} + 8 q^{41} + 2 q^{43} + 4 q^{44} + 8 q^{46} - 30 q^{47} - 8 q^{50} + 4 q^{52} + 14 q^{53} - 8 q^{55} - 8 q^{58} - 24 q^{59} - 28 q^{62} - 20 q^{64} - 6 q^{65} + 16 q^{67} - 28 q^{68} - 8 q^{71} - 6 q^{73} + 12 q^{74} - 16 q^{76} - 22 q^{79} + 28 q^{80} - 40 q^{82} - 50 q^{83} - 8 q^{85} + 16 q^{86} - 44 q^{88} - 26 q^{89} - 20 q^{92} - 32 q^{94} + 6 q^{95} - 14 q^{97}+O(q^{100}) $ 6 * q + 4 * q^4 - 6 * q^5 + 4 * q^10 - 4 * q^11 + 6 * q^13 - 16 * q^17 + 2 * q^19 - 16 * q^20 - 12 * q^22 + 6 * q^23 - 4 * q^25 + 6 * q^29 + 6 * q^31 + 20 * q^32 - 8 * q^38 + 4 * q^40 + 8 * q^41 + 2 * q^43 + 4 * q^44 + 8 * q^46 - 30 * q^47 - 8 * q^50 + 4 * q^52 + 14 * q^53 - 8 * q^55 - 8 * q^58 - 24 * q^59 - 28 * q^62 - 20 * q^64 - 6 * q^65 + 16 * q^67 - 28 * q^68 - 8 * q^71 - 6 * q^73 + 12 * q^74 - 16 * q^76 - 22 * q^79 + 28 * q^80 - 40 * q^82 - 50 * q^83 - 8 * q^85 + 16 * q^86 - 44 * q^88 - 26 * q^89 - 20 * q^92 - 32 * q^94 + 6 * q^95 - 14 * q^97

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 $

$2$	0.656184	0.463992	0.231996	−	0.972717i	$-0.425474\pi$
$2$	0.656184	0.463992	0.231996	+	0.972717i	$0.425474\pi$
$3$	0	0
$3$	0	0
$4$	−1.56942	−0.784711
$5$	1.35996	0.608194	0.304097	−	0.952641i	$-0.401645\pi$
$5$	1.35996	0.608194	0.304097	+	0.952641i	$0.401645\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.34220	−0.828092
$9$	0	0
$10$	0.892385	0.282197
$11$	1.90551	0.574534	0.287267	−	0.957851i	$-0.407253\pi$
$11$	1.90551	0.574534	0.287267	+	0.957851i	$0.407253\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	1.60193	0.400483
$17$	−3.56633	−0.864963	−0.432482	−	0.901643i	$-0.642362\pi$
$17$	−3.56633	−0.864963	−0.432482	+	0.901643i	$0.642362\pi$
$18$	0	0
$19$	0.985255	0.226033	0.113016	−	0.993593i	$-0.463949\pi$
$19$	0.985255	0.226033	0.113016	+	0.993593i	$0.463949\pi$
$20$	−2.13436	−0.477256
$21$	0	0
$22$	1.25037	0.266579
$23$	−1.69605	−0.353651	−0.176826	−	0.984242i	$-0.556583\pi$
$23$	−1.69605	−0.353651	−0.176826	+	0.984242i	$0.556583\pi$
$24$	0	0
$25$	−3.15050	−0.630100
$26$	0.656184	0.128688
$27$	0	0
$28$	0	0
$29$	−6.54835	−1.21600	−0.607999	−	0.793938i	$-0.708028\pi$
$29$	−6.54835	−1.21600	−0.607999	+	0.793938i	$0.708028\pi$
$30$	0	0
$31$	7.69550	1.38215	0.691077	−	0.722781i	$-0.257137\pi$
$31$	7.69550	1.38215	0.691077	+	0.722781i	$0.257137\pi$
$32$	5.73556	1.01391
$33$	0	0
$34$	−2.34017	−0.401336
$35$	0	0
$36$	0	0
$37$	−2.02005	−0.332095	−0.166047	−	0.986118i	$-0.553100\pi$
$37$	−2.02005	−0.332095	−0.166047	+	0.986118i	$0.553100\pi$
$38$	0.646508	0.104877
$39$	0	0
$40$	−3.18530	−0.503640
$41$	−9.88779	−1.54421	−0.772107	−	0.635493i	$-0.780797\pi$
$41$	−9.88779	−1.54421	−0.772107	+	0.635493i	$0.780797\pi$
$42$	0	0
$43$	−3.16639	−0.482869	−0.241435	−	0.970417i	$-0.577618\pi$
$43$	−3.16639	−0.482869	−0.241435	+	0.970417i	$0.577618\pi$
$44$	−2.99055	−0.450843
$45$	0	0
$46$	−1.11292	−0.164091
$47$	−7.76416	−1.13252	−0.566260	−	0.824227i	$-0.691610\pi$
$47$	−7.76416	−1.13252	−0.566260	+	0.824227i	$0.691610\pi$
$48$	0	0
$49$	0	0
$50$	−2.06731	−0.292362
$51$	0	0
$52$	−1.56942	−0.217640
$53$	−0.354194	−0.0486522	−0.0243261	−	0.999704i	$-0.507744\pi$
$53$	−0.354194	−0.0486522	−0.0243261	+	0.999704i	$0.507744\pi$
$54$	0	0
$55$	2.59143	0.349428
$56$	0	0
$57$	0	0
$58$	−4.29692	−0.564214
$59$	−2.16385	−0.281709	−0.140854	−	0.990030i	$-0.544985\pi$
$59$	−2.16385	−0.281709	−0.140854	+	0.990030i	$0.544985\pi$
$60$	0	0
$61$	12.2002	1.56207	0.781037	−	0.624484i	$-0.214691\pi$
$61$	12.2002	1.56207	0.781037	+	0.624484i	$0.214691\pi$
$62$	5.04967	0.641308
$63$	0	0
$64$	0.559715	0.0699644
$65$	1.35996	0.168683
$66$	0	0
$67$	11.3134	1.38215	0.691076	−	0.722782i	$-0.257137\pi$
$67$	11.3134	1.38215	0.691076	+	0.722782i	$0.257137\pi$
$68$	5.59709	0.678746
$69$	0	0
$70$	0	0
$71$	9.05268	1.07436	0.537178	−	0.843469i	$-0.319490\pi$
$71$	9.05268	1.07436	0.537178	+	0.843469i	$0.319490\pi$
$72$	0	0
$73$	−7.13619	−0.835228	−0.417614	−	0.908625i	$-0.637134\pi$
$73$	−7.13619	−0.835228	−0.417614	+	0.908625i	$0.637134\pi$
$74$	−1.32553	−0.154089
$75$	0	0
$76$	−1.54628	−0.177371
$77$	0	0
$78$	0	0
$79$	−5.39629	−0.607130	−0.303565	−	0.952811i	$-0.598177\pi$
$79$	−5.39629	−0.607130	−0.303565	+	0.952811i	$0.598177\pi$
$80$	2.17857	0.243571
$81$	0	0
$82$	−6.48821	−0.716503
$83$	2.03494	0.223364	0.111682	−	0.993744i	$-0.464376\pi$
$83$	2.03494	0.223364	0.111682	+	0.993744i	$0.464376\pi$
$84$	0	0
$85$	−4.85008	−0.526065
$86$	−2.07773	−0.224048
$87$	0	0
$88$	−4.46309	−0.475767
$89$	6.89864	0.731254	0.365627	−	0.930761i	$-0.380855\pi$
$89$	6.89864	0.731254	0.365627	+	0.930761i	$0.380855\pi$
$90$	0	0
$91$	0	0
$92$	2.66182	0.277514
$93$	0	0
$94$	−5.09472	−0.525480
$95$	1.33991	0.137472
$96$	0	0
$97$	−14.6223	−1.48467	−0.742334	−	0.670030i	$-0.766281\pi$
$97$	−14.6223	−1.48467	−0.742334	+	0.670030i	$0.766281\pi$
$98$	0	0
$99$	0	0
$100$	4.94447	0.494447
$101$	−2.81368	−0.279972	−0.139986	−	0.990153i	$-0.544706\pi$
$101$	−2.81368	−0.279972	−0.139986	+	0.990153i	$0.544706\pi$
$102$	0	0
$103$	6.04988	0.596112	0.298056	−	0.954548i	$-0.403662\pi$
$103$	6.04988	0.596112	0.298056	+	0.954548i	$0.403662\pi$
$104$	−2.34220	−0.229671
$105$	0	0
$106$	−0.232416	−0.0225743
$107$	−17.3204	−1.67442	−0.837211	−	0.546880i	$-0.815815\pi$
$107$	−17.3204	−1.67442	−0.837211	+	0.546880i	$0.815815\pi$
$108$	0	0
$109$	10.5762	1.01302	0.506509	−	0.862234i	$-0.330936\pi$
$109$	10.5762	1.01302	0.506509	+	0.862234i	$0.330936\pi$
$110$	1.70045	0.162132
$111$	0	0
$112$	0	0
$113$	2.34665	0.220755	0.110377	−	0.993890i	$-0.464794\pi$
$113$	2.34665	0.220755	0.110377	+	0.993890i	$0.464794\pi$
$114$	0	0
$115$	−2.30657	−0.215089
$116$	10.2771	0.954208
$117$	0	0
$118$	−1.41988	−0.130711
$119$	0	0
$120$	0	0
$121$	−7.36902	−0.669911
$122$	8.00557	0.724790
$123$	0	0
$124$	−12.0775	−1.08459
$125$	−11.0844	−0.991417
$126$	0	0
$127$	−18.1639	−1.61179	−0.805894	−	0.592060i	$-0.798315\pi$
$127$	−18.1639	−1.61179	−0.805894	+	0.592060i	$0.798315\pi$
$128$	−11.1038	−0.981450
$129$	0	0
$130$	0.892385	0.0782674
$131$	−10.1014	−0.882560	−0.441280	−	0.897369i	$-0.645476\pi$
$131$	−10.1014	−0.882560	−0.441280	+	0.897369i	$0.645476\pi$
$132$	0	0
$133$	0	0
$134$	7.42367	0.641308
$135$	0	0
$136$	8.35306	0.716269
$137$	−6.55601	−0.560118	−0.280059	−	0.959983i	$-0.590354\pi$
$137$	−6.55601	−0.560118	−0.280059	+	0.959983i	$0.590354\pi$
$138$	0	0
$139$	−13.8243	−1.17256	−0.586281	−	0.810108i	$-0.699409\pi$
$139$	−13.8243	−1.17256	−0.586281	+	0.810108i	$0.699409\pi$
$140$	0	0
$141$	0	0
$142$	5.94023	0.498493
$143$	1.90551	0.159347
$144$	0	0
$145$	−8.90551	−0.739563
$146$	−4.68265	−0.387539
$147$	0	0
$148$	3.17032	0.260598
$149$	−8.09389	−0.663078	−0.331539	−	0.943442i	$-0.607568\pi$
$149$	−8.09389	−0.663078	−0.331539	+	0.943442i	$0.607568\pi$
$150$	0	0
$151$	12.0365	0.979520	0.489760	−	0.871857i	$-0.337084\pi$
$151$	12.0365	0.979520	0.489760	+	0.871857i	$0.337084\pi$
$152$	−2.30766	−0.187176
$153$	0	0
$154$	0	0
$155$	10.4656	0.840617
$156$	0	0
$157$	−18.3935	−1.46796	−0.733982	−	0.679169i	$-0.762340\pi$
$157$	−18.3935	−1.46796	−0.733982	+	0.679169i	$0.762340\pi$
$158$	−3.54096	−0.281704
$159$	0	0
$160$	7.80014	0.616655
$161$	0	0
$162$	0	0
$163$	2.51286	0.196823	0.0984114	−	0.995146i	$-0.468624\pi$
$163$	2.51286	0.196823	0.0984114	+	0.995146i	$0.468624\pi$
$164$	15.5181	1.21176
$165$	0	0
$166$	1.33530	0.103639
$167$	−17.9095	−1.38588	−0.692941	−	0.720995i	$-0.743685\pi$
$167$	−17.9095	−1.38588	−0.692941	+	0.720995i	$0.743685\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−3.18254	−0.244090
$171$	0	0
$172$	4.96940	0.378913
$173$	−6.51981	−0.495692	−0.247846	−	0.968799i	$-0.579723\pi$
$173$	−6.51981	−0.495692	−0.247846	+	0.968799i	$0.579723\pi$
$174$	0	0
$175$	0	0
$176$	3.05250	0.230091
$177$	0	0
$178$	4.52678	0.339296
$179$	26.6014	1.98828	0.994141	−	0.108091i	$-0.0344738\pi$
$179$	26.6014	1.98828	0.994141	+	0.108091i	$0.0344738\pi$
$180$	0	0
$181$	−24.6402	−1.83149	−0.915746	−	0.401758i	$-0.868399\pi$
$181$	−24.6402	−1.83149	−0.915746	+	0.401758i	$0.868399\pi$
$182$	0	0
$183$	0	0
$184$	3.97249	0.292856
$185$	−2.74720	−0.201978
$186$	0	0
$187$	−6.79569	−0.496950
$188$	12.1853	0.888701
$189$	0	0
$190$	0.879227	0.0637858
$191$	−3.02099	−0.218591	−0.109295	−	0.994009i	$-0.534859\pi$
$191$	−3.02099	−0.218591	−0.109295	+	0.994009i	$0.534859\pi$
$192$	0	0
$193$	21.2522	1.52977	0.764884	−	0.644168i	$-0.222796\pi$
$193$	21.2522	1.52977	0.764884	+	0.644168i	$0.222796\pi$
$194$	−9.59491	−0.688874
$195$	0	0
$196$	0	0
$197$	2.72191	0.193928	0.0969639	−	0.995288i	$-0.469087\pi$
$197$	2.72191	0.193928	0.0969639	+	0.995288i	$0.469087\pi$
$198$	0	0
$199$	−21.6670	−1.53593	−0.767967	−	0.640490i	$-0.778731\pi$
$199$	−21.6670	−1.53593	−0.767967	+	0.640490i	$0.778731\pi$
$200$	7.37910	0.521781
$201$	0	0
$202$	−1.84629	−0.129905
$203$	0	0
$204$	0	0
$205$	−13.4470	−0.939181
$206$	3.96983	0.276591
$207$	0	0
$208$	1.60193	0.111074
$209$	1.87741	0.129864
$210$	0	0
$211$	18.3885	1.26592	0.632959	−	0.774186i	$-0.281840\pi$
$211$	18.3885	1.26592	0.632959	+	0.774186i	$0.281840\pi$
$212$	0.555879	0.0381780
$213$	0	0
$214$	−11.3653	−0.776918
$215$	−4.30617	−0.293678
$216$	0	0
$217$	0	0
$218$	6.93995	0.470033
$219$	0	0
$220$	−4.06704	−0.274200
$221$	−3.56633	−0.239898
$222$	0	0
$223$	22.2216	1.48807	0.744035	−	0.668141i	$-0.232910\pi$
$223$	22.2216	1.48807	0.744035	+	0.668141i	$0.232910\pi$
$224$	0	0
$225$	0	0
$226$	1.53984	0.102428
$227$	−26.7374	−1.77463	−0.887313	−	0.461167i	$-0.847431\pi$
$227$	−26.7374	−1.77463	−0.887313	+	0.461167i	$0.847431\pi$
$228$	0	0
$229$	−16.7008	−1.10362	−0.551809	−	0.833970i	$-0.686062\pi$
$229$	−16.7008	−1.10362	−0.551809	+	0.833970i	$0.686062\pi$
$230$	−1.51353	−0.0997994
$231$	0	0
$232$	15.3375	1.00696
$233$	13.8776	0.909149	0.454575	−	0.890709i	$-0.349791\pi$
$233$	13.8776	0.909149	0.454575	+	0.890709i	$0.349791\pi$
$234$	0	0
$235$	−10.5590	−0.688791
$236$	3.39599	0.221060
$237$	0	0
$238$	0	0
$239$	−0.465845	−0.0301330	−0.0150665	−	0.999886i	$-0.504796\pi$
$239$	−0.465845	−0.0301330	−0.0150665	+	0.999886i	$0.504796\pi$
$240$	0	0
$241$	5.69314	0.366727	0.183364	−	0.983045i	$-0.441301\pi$
$241$	5.69314	0.366727	0.183364	+	0.983045i	$0.441301\pi$
$242$	−4.83543	−0.310833
$243$	0	0
$244$	−19.1473	−1.22578
$245$	0	0
$246$	0	0
$247$	0.985255	0.0626902
$248$	−18.0244	−1.14455
$249$	0	0
$250$	−7.27339	−0.460010
$251$	−17.9066	−1.13025	−0.565126	−	0.825005i	$-0.691172\pi$
$251$	−17.9066	−1.13025	−0.565126	+	0.825005i	$0.691172\pi$
$252$	0	0
$253$	−3.23185	−0.203185
$254$	−11.9189	−0.747857
$255$	0	0
$256$	−8.40559	−0.525350
$257$	−25.9756	−1.62031	−0.810156	−	0.586214i	$-0.800618\pi$
$257$	−25.9756	−1.62031	−0.810156	+	0.586214i	$0.800618\pi$
$258$	0	0
$259$	0	0
$260$	−2.13436	−0.132367
$261$	0	0
$262$	−6.62835	−0.409501
$263$	−27.8402	−1.71670	−0.858349	−	0.513067i	$-0.828509\pi$
$263$	−27.8402	−1.71670	−0.858349	+	0.513067i	$0.828509\pi$
$264$	0	0
$265$	−0.481690	−0.0295900
$266$	0	0
$267$	0	0
$268$	−17.7555	−1.08459
$269$	−5.90486	−0.360025	−0.180013	−	0.983664i	$-0.557614\pi$
$269$	−5.90486	−0.360025	−0.180013	+	0.983664i	$0.557614\pi$
$270$	0	0
$271$	−4.34932	−0.264203	−0.132101	−	0.991236i	$-0.542172\pi$
$271$	−4.34932	−0.264203	−0.132101	+	0.991236i	$0.542172\pi$
$272$	−5.71303	−0.346403
$273$	0	0
$274$	−4.30195	−0.259890
$275$	−6.00332	−0.362014
$276$	0	0
$277$	1.08051	0.0649217	0.0324609	−	0.999473i	$-0.489666\pi$
$277$	1.08051	0.0649217	0.0324609	+	0.999473i	$0.489666\pi$
$278$	−9.07129	−0.544060
$279$	0	0
$280$	0	0
$281$	−14.4912	−0.864473	−0.432236	−	0.901760i	$-0.642275\pi$
$281$	−14.4912	−0.864473	−0.432236	+	0.901760i	$0.642275\pi$
$282$	0	0
$283$	24.4979	1.45625	0.728123	−	0.685446i	$-0.240393\pi$
$283$	24.4979	1.45625	0.728123	+	0.685446i	$0.240393\pi$
$284$	−14.2075	−0.843059
$285$	0	0
$286$	1.25037	0.0739357
$287$	0	0
$288$	0	0
$289$	−4.28126	−0.251839
$290$	−5.84365	−0.343151
$291$	0	0
$292$	11.1997	0.655413
$293$	−1.16105	−0.0678296	−0.0339148	−	0.999425i	$-0.510797\pi$
$293$	−1.16105	−0.0678296	−0.0339148	+	0.999425i	$0.510797\pi$
$294$	0	0
$295$	−2.94275	−0.171334
$296$	4.73136	0.275005
$297$	0	0
$298$	−5.31108	−0.307663
$299$	−1.69605	−0.0980852
$300$	0	0
$301$	0	0
$302$	7.89818	0.454489
$303$	0	0
$304$	1.57831	0.0905224
$305$	16.5918	0.950044
$306$	0	0
$307$	25.3731	1.44812	0.724060	−	0.689737i	$-0.242274\pi$
$307$	25.3731	1.44812	0.724060	+	0.689737i	$0.242274\pi$
$308$	0	0
$309$	0	0
$310$	6.86736	0.390040
$311$	−24.4645	−1.38726	−0.693628	−	0.720333i	$-0.743989\pi$
$311$	−24.4645	−1.38726	−0.693628	+	0.720333i	$0.743989\pi$
$312$	0	0
$313$	6.61665	0.373995	0.186997	−	0.982360i	$-0.440124\pi$
$313$	6.61665	0.373995	0.186997	+	0.982360i	$0.440124\pi$
$314$	−12.0695	−0.681124
$315$	0	0
$316$	8.46906	0.476422
$317$	−8.19803	−0.460447	−0.230224	−	0.973138i	$-0.573946\pi$
$317$	−8.19803	−0.460447	−0.230224	+	0.973138i	$0.573946\pi$
$318$	0	0
$319$	−12.4780	−0.698632
$320$	0.761192	0.0425519
$321$	0	0
$322$	0	0
$323$	−3.51375	−0.195510
$324$	0	0
$325$	−3.15050	−0.174758
$326$	1.64890	0.0913242
$327$	0	0
$328$	23.1592	1.27875
$329$	0	0
$330$	0	0
$331$	−0.724715	−0.0398339	−0.0199170	−	0.999802i	$-0.506340\pi$
$331$	−0.724715	−0.0398339	−0.0199170	+	0.999802i	$0.506340\pi$
$332$	−3.19369	−0.175276
$333$	0	0
$334$	−11.7519	−0.643038
$335$	15.3858	0.840616
$336$	0	0
$337$	−8.58299	−0.467545	−0.233773	−	0.972291i	$-0.575107\pi$
$337$	−8.58299	−0.467545	−0.233773	+	0.972291i	$0.575107\pi$
$338$	0.656184	0.0356917
$339$	0	0
$340$	7.61183	0.412809
$341$	14.6639	0.794094
$342$	0	0
$343$	0	0
$344$	7.41630	0.399860
$345$	0	0
$346$	−4.27820	−0.229997
$347$	25.9250	1.39173	0.695864	−	0.718173i	$-0.255021\pi$
$347$	25.9250	1.39173	0.695864	+	0.718173i	$0.255021\pi$
$348$	0	0
$349$	17.1403	0.917501	0.458750	−	0.888565i	$-0.348297\pi$
$349$	17.1403	0.917501	0.458750	+	0.888565i	$0.348297\pi$
$350$	0	0
$351$	0	0
$352$	10.9292	0.582527
$353$	18.9101	1.00648	0.503242	−	0.864146i	$-0.332140\pi$
$353$	18.9101	1.00648	0.503242	+	0.864146i	$0.332140\pi$
$354$	0	0
$355$	12.3113	0.653417
$356$	−10.8269	−0.573823
$357$	0	0
$358$	17.4554	0.922547
$359$	−14.1049	−0.744431	−0.372215	−	0.928146i	$-0.621402\pi$
$359$	−14.1049	−0.744431	−0.372215	+	0.928146i	$0.621402\pi$
$360$	0	0
$361$	−18.0293	−0.948909
$362$	−16.1685	−0.849798
$363$	0	0
$364$	0	0
$365$	−9.70495	−0.507980
$366$	0	0
$367$	0.360258	0.0188053	0.00940266	−	0.999956i	$-0.497007\pi$
$367$	0.360258	0.0188053	0.00940266	+	0.999956i	$0.497007\pi$
$368$	−2.71696	−0.141631
$369$	0	0
$370$	−1.80267	−0.0937162
$371$	0	0
$372$	0	0
$373$	−25.1680	−1.30315	−0.651575	−	0.758584i	$-0.725892\pi$
$373$	−25.1680	−1.30315	−0.651575	+	0.758584i	$0.725892\pi$
$374$	−4.45923	−0.230581
$375$	0	0
$376$	18.1852	0.937830
$377$	−6.54835	−0.337257
$378$	0	0
$379$	31.8947	1.63832	0.819161	−	0.573564i	$-0.194440\pi$
$379$	31.8947	1.63832	0.819161	+	0.573564i	$0.194440\pi$
$380$	−2.10288	−0.107876
$381$	0	0
$382$	−1.98232	−0.101424
$383$	−32.8172	−1.67688	−0.838440	−	0.544994i	$-0.816532\pi$
$383$	−32.8172	−1.67688	−0.838440	+	0.544994i	$0.816532\pi$
$384$	0	0
$385$	0	0
$386$	13.9454	0.709800
$387$	0	0
$388$	22.9486	1.16504
$389$	25.3087	1.28320	0.641600	−	0.767040i	$-0.278271\pi$
$389$	25.3087	1.28320	0.641600	+	0.767040i	$0.278271\pi$
$390$	0	0
$391$	6.04869	0.305895
$392$	0	0
$393$	0	0
$394$	1.78607	0.0899810
$395$	−7.33875	−0.369253
$396$	0	0
$397$	−5.29395	−0.265696	−0.132848	−	0.991136i	$-0.542412\pi$
$397$	−5.29395	−0.265696	−0.132848	+	0.991136i	$0.542412\pi$
$398$	−14.2175	−0.712661
$399$	0	0
$400$	−5.04689	−0.252345
$401$	24.6335	1.23014	0.615070	−	0.788473i	$-0.289128\pi$
$401$	24.6335	1.23014	0.615070	+	0.788473i	$0.289128\pi$
$402$	0	0
$403$	7.69550	0.383340
$404$	4.41586	0.219697
$405$	0	0
$406$	0	0
$407$	−3.84924	−0.190800
$408$	0	0
$409$	10.2223	0.505458	0.252729	−	0.967537i	$-0.418672\pi$
$409$	10.2223	0.505458	0.252729	+	0.967537i	$0.418672\pi$
$410$	−8.82372	−0.435773
$411$	0	0
$412$	−9.49481	−0.467776
$413$	0	0
$414$	0	0
$415$	2.76745	0.135849
$416$	5.73556	0.281209
$417$	0	0
$418$	1.23193	0.0602556
$419$	−30.6503	−1.49736	−0.748682	−	0.662929i	$-0.769313\pi$
$419$	−30.6503	−1.49736	−0.748682	+	0.662929i	$0.769313\pi$
$420$	0	0
$421$	25.9764	1.26601	0.633007	−	0.774146i	$-0.281820\pi$
$421$	25.9764	1.26601	0.633007	+	0.774146i	$0.281820\pi$
$422$	12.0662	0.587376
$423$	0	0
$424$	0.829592	0.0402885
$425$	11.2357	0.545014
$426$	0	0
$427$	0	0
$428$	27.1830	1.31394
$429$	0	0
$430$	−2.82564	−0.136264
$431$	−14.8818	−0.716829	−0.358415	−	0.933563i	$-0.616683\pi$
$431$	−14.8818	−0.716829	−0.358415	+	0.933563i	$0.616683\pi$
$432$	0	0
$433$	−40.0871	−1.92647	−0.963233	−	0.268669i	$-0.913416\pi$
$433$	−40.0871	−1.92647	−0.963233	+	0.268669i	$0.913416\pi$
$434$	0	0
$435$	0	0
$436$	−16.5986	−0.794927
$437$	−1.67104	−0.0799368
$438$	0	0
$439$	−3.00243	−0.143298	−0.0716491	−	0.997430i	$-0.522826\pi$
$439$	−3.00243	−0.143298	−0.0716491	+	0.997430i	$0.522826\pi$
$440$	−6.06963	−0.289358
$441$	0	0
$442$	−2.34017	−0.111311
$443$	−1.43566	−0.0682101	−0.0341051	−	0.999418i	$-0.510858\pi$
$443$	−1.43566	−0.0682101	−0.0341051	+	0.999418i	$0.510858\pi$
$444$	0	0
$445$	9.38189	0.444744
$446$	14.5815	0.690453
$447$	0	0
$448$	0	0
$449$	−19.6313	−0.926460	−0.463230	−	0.886238i	$-0.653310\pi$
$449$	−19.6313	−0.926460	−0.463230	+	0.886238i	$0.653310\pi$
$450$	0	0
$451$	−18.8413	−0.887203
$452$	−3.68289	−0.173229
$453$	0	0
$454$	−17.5447	−0.823413
$455$	0	0
$456$	0	0
$457$	10.6421	0.497815	0.248907	−	0.968527i	$-0.419929\pi$
$457$	10.6421	0.497815	0.248907	+	0.968527i	$0.419929\pi$
$458$	−10.9588	−0.512070
$459$	0	0
$460$	3.61998	0.168782
$461$	38.5930	1.79745	0.898727	−	0.438509i	$-0.144493\pi$
$461$	38.5930	1.79745	0.898727	+	0.438509i	$0.144493\pi$
$462$	0	0
$463$	−27.9993	−1.30124	−0.650618	−	0.759405i	$-0.725490\pi$
$463$	−27.9993	−1.30124	−0.650618	+	0.759405i	$0.725490\pi$
$464$	−10.4900	−0.486987
$465$	0	0
$466$	9.10623	0.421838
$467$	25.8199	1.19480	0.597401	−	0.801943i	$-0.296200\pi$
$467$	25.8199	1.19480	0.597401	+	0.801943i	$0.296200\pi$
$468$	0	0
$469$	0	0
$470$	−6.92862	−0.319594
$471$	0	0
$472$	5.06816	0.233281
$473$	−6.03359	−0.277425
$474$	0	0
$475$	−3.10405	−0.142423
$476$	0	0
$477$	0	0
$478$	−0.305680	−0.0139815
$479$	39.4071	1.80055	0.900277	−	0.435317i	$-0.143364\pi$
$479$	39.4071	1.80055	0.900277	+	0.435317i	$0.143364\pi$
$480$	0	0
$481$	−2.02005	−0.0921065
$482$	3.73575	0.170159
$483$	0	0
$484$	11.5651	0.525687
$485$	−19.8858	−0.902966
$486$	0	0
$487$	−34.0959	−1.54503	−0.772517	−	0.634994i	$-0.781003\pi$
$487$	−34.0959	−1.54503	−0.772517	+	0.634994i	$0.781003\pi$
$488$	−28.5753	−1.29354
$489$	0	0
$490$	0	0
$491$	9.70414	0.437942	0.218971	−	0.975731i	$-0.429730\pi$
$491$	9.70414	0.437942	0.218971	+	0.975731i	$0.429730\pi$
$492$	0	0
$493$	23.3536	1.05179
$494$	0.646508	0.0290878
$495$	0	0
$496$	12.3277	0.553529
$497$	0	0
$498$	0	0
$499$	18.0463	0.807862	0.403931	−	0.914789i	$-0.367644\pi$
$499$	18.0463	0.807862	0.403931	+	0.914789i	$0.367644\pi$
$500$	17.3961	0.777976
$501$	0	0
$502$	−11.7500	−0.524428
$503$	−16.4922	−0.735352	−0.367676	−	0.929954i	$-0.619846\pi$
$503$	−16.4922	−0.735352	−0.367676	+	0.929954i	$0.619846\pi$
$504$	0	0
$505$	−3.82650	−0.170277
$506$	−2.12069	−0.0942761
$507$	0	0
$508$	28.5069	1.26479
$509$	19.1977	0.850921	0.425460	−	0.904977i	$-0.360112\pi$
$509$	19.1977	0.850921	0.425460	+	0.904977i	$0.360112\pi$
$510$	0	0
$511$	0	0
$512$	16.6921	0.737692
$513$	0	0
$514$	−17.0448	−0.751812
$515$	8.22760	0.362552
$516$	0	0
$517$	−14.7947	−0.650670
$518$	0	0
$519$	0	0
$520$	−3.18530	−0.139685
$521$	8.76034	0.383797	0.191899	−	0.981415i	$-0.438535\pi$
$521$	8.76034	0.383797	0.191899	+	0.981415i	$0.438535\pi$
$522$	0	0
$523$	−11.7406	−0.513381	−0.256690	−	0.966494i	$-0.582632\pi$
$523$	−11.7406	−0.513381	−0.256690	+	0.966494i	$0.582632\pi$
$524$	15.8533	0.692555
$525$	0	0
$526$	−18.2683	−0.796534
$527$	−27.4447	−1.19551
$528$	0	0
$529$	−20.1234	−0.874931
$530$	−0.316077	−0.0137295
$531$	0	0
$532$	0	0
$533$	−9.88779	−0.428288
$534$	0	0
$535$	−23.5550	−1.01837
$536$	−26.4982	−1.14455
$537$	0	0
$538$	−3.87467	−0.167049
$539$	0	0
$540$	0	0
$541$	16.4587	0.707614	0.353807	−	0.935318i	$-0.384887\pi$
$541$	16.4587	0.707614	0.353807	+	0.935318i	$0.384887\pi$
$542$	−2.85396	−0.122588
$543$	0	0
$544$	−20.4549	−0.876997
$545$	14.3833	0.616112
$546$	0	0
$547$	−8.19375	−0.350339	−0.175170	−	0.984538i	$-0.556047\pi$
$547$	−8.19375	−0.350339	−0.175170	+	0.984538i	$0.556047\pi$
$548$	10.2891	0.439531
$549$	0	0
$550$	−3.93928	−0.167972
$551$	−6.45179	−0.274856
$552$	0	0
$553$	0	0
$554$	0.709015	0.0301232
$555$	0	0
$556$	21.6962	0.920123
$557$	13.8144	0.585335	0.292667	−	0.956214i	$-0.405457\pi$
$557$	13.8144	0.585335	0.292667	+	0.956214i	$0.405457\pi$
$558$	0	0
$559$	−3.16639	−0.133924
$560$	0	0
$561$	0	0
$562$	−9.50889	−0.401108
$563$	−9.75559	−0.411149	−0.205575	−	0.978641i	$-0.565906\pi$
$563$	−9.75559	−0.411149	−0.205575	+	0.978641i	$0.565906\pi$
$564$	0	0
$565$	3.19136	0.134262
$566$	16.0751	0.675687
$567$	0	0
$568$	−21.2032	−0.889666
$569$	1.43264	0.0600593	0.0300296	−	0.999549i	$-0.490440\pi$
$569$	1.43264	0.0600593	0.0300296	+	0.999549i	$0.490440\pi$
$570$	0	0
$571$	16.2872	0.681598	0.340799	−	0.940136i	$-0.389303\pi$
$571$	16.2872	0.681598	0.340799	+	0.940136i	$0.389303\pi$
$572$	−2.99055	−0.125041
$573$	0	0
$574$	0	0
$575$	5.34342	0.222836
$576$	0	0
$577$	−4.81846	−0.200595	−0.100297	−	0.994957i	$-0.531979\pi$
$577$	−4.81846	−0.200595	−0.100297	+	0.994957i	$0.531979\pi$
$578$	−2.80929	−0.116851
$579$	0	0
$580$	13.9765	0.580343
$581$	0	0
$582$	0	0
$583$	−0.674920	−0.0279523
$584$	16.7144	0.691645
$585$	0	0
$586$	−0.761866	−0.0314724
$587$	−36.6215	−1.51153	−0.755765	−	0.654843i	$-0.772735\pi$
$587$	−36.6215	−1.51153	−0.755765	+	0.654843i	$0.772735\pi$
$588$	0	0
$589$	7.58203	0.312412
$590$	−1.93099	−0.0794974
$591$	0	0
$592$	−3.23599	−0.132998
$593$	18.7081	0.768251	0.384125	−	0.923281i	$-0.374503\pi$
$593$	18.7081	0.768251	0.384125	+	0.923281i	$0.374503\pi$
$594$	0	0
$595$	0	0
$596$	12.7027	0.520325
$597$	0	0
$598$	−1.11292	−0.0455108
$599$	25.4194	1.03861	0.519305	−	0.854589i	$-0.326191\pi$
$599$	25.4194	1.03861	0.519305	+	0.854589i	$0.326191\pi$
$600$	0	0
$601$	17.7790	0.725219	0.362609	−	0.931941i	$-0.381886\pi$
$601$	17.7790	0.725219	0.362609	+	0.931941i	$0.381886\pi$
$602$	0	0
$603$	0	0
$604$	−18.8904	−0.768640
$605$	−10.0216	−0.407436
$606$	0	0
$607$	26.0047	1.05550	0.527750	−	0.849400i	$-0.323036\pi$
$607$	26.0047	1.05550	0.527750	+	0.849400i	$0.323036\pi$
$608$	5.65098	0.229178
$609$	0	0
$610$	10.8873	0.440813
$611$	−7.76416	−0.314104
$612$	0	0
$613$	26.8230	1.08337	0.541685	−	0.840582i	$-0.317787\pi$
$613$	26.8230	1.08337	0.541685	+	0.840582i	$0.317787\pi$
$614$	16.6494	0.671916
$615$	0	0
$616$	0	0
$617$	18.6491	0.750784	0.375392	−	0.926866i	$-0.377508\pi$
$617$	18.6491	0.750784	0.375392	+	0.926866i	$0.377508\pi$
$618$	0	0
$619$	13.0685	0.525269	0.262634	−	0.964895i	$-0.415409\pi$
$619$	13.0685	0.525269	0.262634	+	0.964895i	$0.415409\pi$
$620$	−16.4249	−0.659642
$621$	0	0
$622$	−16.0532	−0.643676
$623$	0	0
$624$	0	0
$625$	0.678175	0.0271270
$626$	4.34174	0.173531
$627$	0	0
$628$	28.8672	1.15193
$629$	7.20418	0.287250
$630$	0	0
$631$	−49.1745	−1.95761	−0.978804	−	0.204801i	$-0.934345\pi$
$631$	−49.1745	−1.95761	−0.978804	+	0.204801i	$0.934345\pi$
$632$	12.6392	0.502760
$633$	0	0
$634$	−5.37941	−0.213644
$635$	−24.7023	−0.980279
$636$	0	0
$637$	0	0
$638$	−8.18784	−0.324160
$639$	0	0
$640$	−15.1008	−0.596912
$641$	−4.37503	−0.172803	−0.0864016	−	0.996260i	$-0.527537\pi$
$641$	−4.37503	−0.172803	−0.0864016	+	0.996260i	$0.527537\pi$
$642$	0	0
$643$	−5.34651	−0.210846	−0.105423	−	0.994427i	$-0.533620\pi$
$643$	−5.34651	−0.210846	−0.105423	+	0.994427i	$0.533620\pi$
$644$	0	0
$645$	0	0
$646$	−2.30566	−0.0907151
$647$	29.7346	1.16899	0.584493	−	0.811398i	$-0.301293\pi$
$647$	29.7346	1.16899	0.584493	+	0.811398i	$0.301293\pi$
$648$	0	0
$649$	−4.12324	−0.161851
$650$	−2.06731	−0.0810865
$651$	0	0
$652$	−3.94375	−0.154449
$653$	2.54262	0.0995005	0.0497503	−	0.998762i	$-0.484157\pi$
$653$	2.54262	0.0995005	0.0497503	+	0.998762i	$0.484157\pi$
$654$	0	0
$655$	−13.7375	−0.536768
$656$	−15.8396	−0.618432
$657$	0	0
$658$	0	0
$659$	−1.87019	−0.0728523	−0.0364261	−	0.999336i	$-0.511597\pi$
$659$	−1.87019	−0.0728523	−0.0364261	+	0.999336i	$0.511597\pi$
$660$	0	0
$661$	−12.0763	−0.469714	−0.234857	−	0.972030i	$-0.575462\pi$
$661$	−12.0763	−0.469714	−0.234857	+	0.972030i	$0.575462\pi$
$662$	−0.475546	−0.0184826
$663$	0	0
$664$	−4.76624	−0.184966
$665$	0	0
$666$	0	0
$667$	11.1063	0.430040
$668$	28.1076	1.08752
$669$	0	0
$670$	10.0959	0.390039
$671$	23.2476	0.897464
$672$	0	0
$673$	−7.81691	−0.301320	−0.150660	−	0.988586i	$-0.548140\pi$
$673$	−7.81691	−0.301320	−0.150660	+	0.988586i	$0.548140\pi$
$674$	−5.63202	−0.216937
$675$	0	0
$676$	−1.56942	−0.0603624
$677$	−6.25628	−0.240448	−0.120224	−	0.992747i	$-0.538361\pi$
$677$	−6.25628	−0.240448	−0.120224	+	0.992747i	$0.538361\pi$
$678$	0	0
$679$	0	0
$680$	11.3598	0.435630
$681$	0	0
$682$	9.62220	0.368453
$683$	−22.3656	−0.855795	−0.427898	−	0.903827i	$-0.640746\pi$
$683$	−22.3656	−0.855795	−0.427898	+	0.903827i	$0.640746\pi$
$684$	0	0
$685$	−8.91593	−0.340660
$686$	0	0
$687$	0	0
$688$	−5.07234	−0.193381
$689$	−0.354194	−0.0134937
$690$	0	0
$691$	−3.80771	−0.144852	−0.0724260	−	0.997374i	$-0.523074\pi$
$691$	−3.80771	−0.144852	−0.0724260	+	0.997374i	$0.523074\pi$
$692$	10.2323	0.388975
$693$	0	0
$694$	17.0116	0.645751
$695$	−18.8005	−0.713145
$696$	0	0
$697$	35.2632	1.33569
$698$	11.2472	0.425713
$699$	0	0
$700$	0	0
$701$	8.14218	0.307526	0.153763	−	0.988108i	$-0.450861\pi$
$701$	8.14218	0.307526	0.153763	+	0.988108i	$0.450861\pi$
$702$	0	0
$703$	−1.99027	−0.0750643
$704$	1.06654	0.0401969
$705$	0	0
$706$	12.4085	0.467001
$707$	0	0
$708$	0	0
$709$	−24.0319	−0.902536	−0.451268	−	0.892388i	$-0.649028\pi$
$709$	−24.0319	−0.902536	−0.451268	+	0.892388i	$0.649028\pi$
$710$	8.07848	0.303180
$711$	0	0
$712$	−16.1580	−0.605546
$713$	−13.0520	−0.488800
$714$	0	0
$715$	2.59143	0.0969138
$716$	−41.7488	−1.56023
$717$	0	0
$718$	−9.25544	−0.345410
$719$	32.6960	1.21936	0.609678	−	0.792649i	$-0.291299\pi$
$719$	32.6960	1.21936	0.609678	+	0.792649i	$0.291299\pi$
$720$	0	0
$721$	0	0
$722$	−11.8305	−0.440286
$723$	0	0
$724$	38.6709	1.43719
$725$	20.6306	0.766201
$726$	0	0
$727$	−15.7712	−0.584921	−0.292460	−	0.956278i	$-0.594474\pi$
$727$	−15.7712	−0.584921	−0.292460	+	0.956278i	$0.594474\pi$
$728$	0	0
$729$	0	0
$730$	−6.36823	−0.235699
$731$	11.2924	0.417664
$732$	0	0
$733$	23.5094	0.868340	0.434170	−	0.900831i	$-0.357042\pi$
$733$	23.5094	0.868340	0.434170	+	0.900831i	$0.357042\pi$
$734$	0.236396	0.00872552
$735$	0	0
$736$	−9.72781	−0.358572
$737$	21.5578	0.794093
$738$	0	0
$739$	−22.5267	−0.828658	−0.414329	−	0.910127i	$-0.635984\pi$
$739$	−22.5267	−0.828658	−0.414329	+	0.910127i	$0.635984\pi$
$740$	4.31151	0.158494
$741$	0	0
$742$	0	0
$743$	13.9795	0.512857	0.256429	−	0.966563i	$-0.417454\pi$
$743$	13.9795	0.512857	0.256429	+	0.966563i	$0.417454\pi$
$744$	0	0
$745$	−11.0074	−0.403280
$746$	−16.5149	−0.604652
$747$	0	0
$748$	10.6653	0.389963
$749$	0	0
$750$	0	0
$751$	2.54591	0.0929017	0.0464509	−	0.998921i	$-0.485209\pi$
$751$	2.54591	0.0929017	0.0464509	+	0.998921i	$0.485209\pi$
$752$	−12.4377	−0.453555
$753$	0	0
$754$	−4.29692	−0.156485
$755$	16.3692	0.595738
$756$	0	0
$757$	19.6921	0.715721	0.357861	−	0.933775i	$-0.383506\pi$
$757$	19.6921	0.715721	0.357861	+	0.933775i	$0.383506\pi$
$758$	20.9288	0.760168
$759$	0	0
$760$	−3.13833	−0.113839
$761$	39.0905	1.41703	0.708515	−	0.705696i	$-0.249365\pi$
$761$	39.0905	1.41703	0.708515	+	0.705696i	$0.249365\pi$
$762$	0	0
$763$	0	0
$764$	4.74120	0.171531
$765$	0	0
$766$	−21.5341	−0.778059
$767$	−2.16385	−0.0781320
$768$	0	0
$769$	−21.7389	−0.783926	−0.391963	−	0.919981i	$-0.628204\pi$
$769$	−21.7389	−0.783926	−0.391963	+	0.919981i	$0.628204\pi$
$770$	0	0
$771$	0	0
$772$	−33.3537	−1.20043
$773$	−4.12806	−0.148476	−0.0742381	−	0.997241i	$-0.523652\pi$
$773$	−4.12806	−0.148476	−0.0742381	+	0.997241i	$0.523652\pi$
$774$	0	0
$775$	−24.2447	−0.870895
$776$	34.2483	1.22944
$777$	0	0
$778$	16.6071	0.595395
$779$	−9.74199	−0.349043
$780$	0	0
$781$	17.2500	0.617254
$782$	3.96905	0.141933
$783$	0	0
$784$	0	0
$785$	−25.0145	−0.892807
$786$	0	0
$787$	38.7863	1.38258	0.691291	−	0.722576i	$-0.257042\pi$
$787$	38.7863	1.38258	0.691291	+	0.722576i	$0.257042\pi$
$788$	−4.27182	−0.152177
$789$	0	0
$790$	−4.81557	−0.171330
$791$	0	0
$792$	0	0
$793$	12.2002	0.433242
$794$	−3.47381	−0.123281
$795$	0	0
$796$	34.0047	1.20526
$797$	−19.9651	−0.707201	−0.353601	−	0.935397i	$-0.615043\pi$
$797$	−19.9651	−0.707201	−0.353601	+	0.935397i	$0.615043\pi$
$798$	0	0
$799$	27.6896	0.979587
$800$	−18.0699	−0.638867
$801$	0	0
$802$	16.1641	0.570775
$803$	−13.5981	−0.479866
$804$	0	0
$805$	0	0
$806$	5.04967	0.177867
$807$	0	0
$808$	6.59020	0.231842
$809$	30.2441	1.06332	0.531662	−	0.846956i	$-0.321568\pi$
$809$	30.2441	1.06332	0.531662	+	0.846956i	$0.321568\pi$
$810$	0	0
$811$	23.4099	0.822034	0.411017	−	0.911628i	$-0.365174\pi$
$811$	23.4099	0.822034	0.411017	+	0.911628i	$0.365174\pi$
$812$	0	0
$813$	0	0
$814$	−2.52581	−0.0885295
$815$	3.41740	0.119706
$816$	0	0
$817$	−3.11970	−0.109144
$818$	6.70768	0.234528
$819$	0	0
$820$	21.1041	0.736986
$821$	46.5536	1.62473	0.812366	−	0.583147i	$-0.198179\pi$
$821$	46.5536	1.62473	0.812366	+	0.583147i	$0.198179\pi$
$822$	0	0
$823$	−5.68031	−0.198003	−0.0990017	−	0.995087i	$-0.531565\pi$
$823$	−5.68031	−0.198003	−0.0990017	+	0.995087i	$0.531565\pi$
$824$	−14.1700	−0.493636
$825$	0	0
$826$	0	0
$827$	−48.3198	−1.68024	−0.840122	−	0.542398i	$-0.817517\pi$
$827$	−48.3198	−1.68024	−0.840122	+	0.542398i	$0.817517\pi$
$828$	0	0
$829$	−11.8545	−0.411722	−0.205861	−	0.978581i	$-0.566000\pi$
$829$	−11.8545	−0.411722	−0.205861	+	0.978581i	$0.566000\pi$
$830$	1.81595	0.0630327
$831$	0	0
$832$	0.559715	0.0194046
$833$	0	0
$834$	0	0
$835$	−24.3563	−0.842884
$836$	−2.94646	−0.101905
$837$	0	0
$838$	−20.1122	−0.694765
$839$	2.69386	0.0930025	0.0465013	−	0.998918i	$-0.485193\pi$
$839$	2.69386	0.0930025	0.0465013	+	0.998918i	$0.485193\pi$
$840$	0	0
$841$	13.8809	0.478652
$842$	17.0453	0.587420
$843$	0	0
$844$	−28.8593	−0.993380
$845$	1.35996	0.0467841
$846$	0	0
$847$	0	0
$848$	−0.567394	−0.0194844
$849$	0	0
$850$	7.37271	0.252882
$851$	3.42612	0.117446
$852$	0	0
$853$	−2.48965	−0.0852440	−0.0426220	−	0.999091i	$-0.513571\pi$
$853$	−2.48965	−0.0852440	−0.0426220	+	0.999091i	$0.513571\pi$
$854$	0	0
$855$	0	0
$856$	40.5677	1.38658
$857$	−46.2110	−1.57854	−0.789270	−	0.614047i	$-0.789541\pi$
$857$	−46.2110	−1.57854	−0.789270	+	0.614047i	$0.789541\pi$
$858$	0	0
$859$	−10.9170	−0.372482	−0.186241	−	0.982504i	$-0.559631\pi$
$859$	−10.9170	−0.372482	−0.186241	+	0.982504i	$0.559631\pi$
$860$	6.75820	0.230453
$861$	0	0
$862$	−9.76517	−0.332603
$863$	29.2330	0.995104	0.497552	−	0.867434i	$-0.334232\pi$
$863$	29.2330	0.995104	0.497552	+	0.867434i	$0.334232\pi$
$864$	0	0
$865$	−8.86670	−0.301477
$866$	−26.3045	−0.893865
$867$	0	0
$868$	0	0
$869$	−10.2827	−0.348817
$870$	0	0
$871$	11.3134	0.383340
$872$	−24.7716	−0.838873
$873$	0	0
$874$	−1.09651	−0.0370901
$875$	0	0
$876$	0	0
$877$	−7.38110	−0.249242	−0.124621	−	0.992204i	$-0.539772\pi$
$877$	−7.38110	−0.249242	−0.124621	+	0.992204i	$0.539772\pi$
$878$	−1.97015	−0.0664892
$879$	0	0
$880$	4.15129	0.139940
$881$	−16.4854	−0.555407	−0.277703	−	0.960667i	$-0.589573\pi$
$881$	−16.4854	−0.555407	−0.277703	+	0.960667i	$0.589573\pi$
$882$	0	0
$883$	12.4427	0.418732	0.209366	−	0.977837i	$-0.432860\pi$
$883$	12.4427	0.418732	0.209366	+	0.977837i	$0.432860\pi$
$884$	5.59709	0.188250
$885$	0	0
$886$	−0.942055	−0.0316490
$887$	8.11331	0.272418	0.136209	−	0.990680i	$-0.456508\pi$
$887$	8.11331	0.272418	0.136209	+	0.990680i	$0.456508\pi$
$888$	0	0
$889$	0	0
$890$	6.15624	0.206358
$891$	0	0
$892$	−34.8751	−1.16771
$893$	−7.64968	−0.255987
$894$	0	0
$895$	36.1769	1.20926
$896$	0	0
$897$	0	0
$898$	−12.8818	−0.429870
$899$	−50.3929	−1.68070
$900$	0	0
$901$	1.26317	0.0420824
$902$	−12.3634	−0.411655
$903$	0	0
$904$	−5.49633	−0.182805
$905$	−33.5098	−1.11390
$906$	0	0
$907$	−18.4796	−0.613605	−0.306803	−	0.951773i	$-0.599259\pi$
$907$	−18.4796	−0.613605	−0.306803	+	0.951773i	$0.599259\pi$
$908$	41.9623	1.39257
$909$	0	0
$910$	0	0
$911$	−15.8660	−0.525663	−0.262831	−	0.964842i	$-0.584656\pi$
$911$	−15.8660	−0.525663	−0.262831	+	0.964842i	$0.584656\pi$
$912$	0	0
$913$	3.87761	0.128330
$914$	6.98315	0.230982
$915$	0	0
$916$	26.2106	0.866022
$917$	0	0
$918$	0	0
$919$	43.4568	1.43351	0.716753	−	0.697327i	$-0.245627\pi$
$919$	43.4568	1.43351	0.716753	+	0.697327i	$0.245627\pi$
$920$	5.40244	0.178113
$921$	0	0
$922$	25.3241	0.834004
$923$	9.05268	0.297973
$924$	0	0
$925$	6.36418	0.209253
$926$	−18.3727	−0.603763
$927$	0	0
$928$	−37.5585	−1.23292
$929$	25.6312	0.840931	0.420465	−	0.907309i	$-0.361867\pi$
$929$	25.6312	0.840931	0.420465	+	0.907309i	$0.361867\pi$
$930$	0	0
$931$	0	0
$932$	−21.7798	−0.713420
$933$	0	0
$934$	16.9426	0.554379
$935$	−9.24189	−0.302242
$936$	0	0
$937$	23.9639	0.782867	0.391433	−	0.920206i	$-0.371979\pi$
$937$	23.9639	0.782867	0.391433	+	0.920206i	$0.371979\pi$
$938$	0	0
$939$	0	0
$940$	16.5715	0.540502
$941$	42.2934	1.37872	0.689362	−	0.724417i	$-0.257891\pi$
$941$	42.2934	1.37872	0.689362	+	0.724417i	$0.257891\pi$
$942$	0	0
$943$	16.7702	0.546113
$944$	−3.46634	−0.112820
$945$	0	0
$946$	−3.95914	−0.128723
$947$	49.5957	1.61165	0.805823	−	0.592157i	$-0.201723\pi$
$947$	49.5957	1.61165	0.805823	+	0.592157i	$0.201723\pi$
$948$	0	0
$949$	−7.13619	−0.231650
$950$	−2.03683	−0.0660833
$951$	0	0
$952$	0	0
$953$	−18.7156	−0.606259	−0.303129	−	0.952949i	$-0.598031\pi$
$953$	−18.7156	−0.606259	−0.303129	+	0.952949i	$0.598031\pi$
$954$	0	0
$955$	−4.10843	−0.132946
$956$	0.731107	0.0236457
$957$	0	0
$958$	25.8583	0.835443
$959$	0	0
$960$	0	0
$961$	28.2208	0.910348
$962$	−1.32553	−0.0427367
$963$	0	0
$964$	−8.93494	−0.287775
$965$	28.9022	0.930395
$966$	0	0
$967$	−34.3284	−1.10393	−0.551964	−	0.833868i	$-0.686121\pi$
$967$	−34.3284	−1.10393	−0.551964	+	0.833868i	$0.686121\pi$
$968$	17.2597	0.554748
$969$	0	0
$970$	−13.0487	−0.418969
$971$	−56.7118	−1.81997	−0.909984	−	0.414642i	$-0.863907\pi$
$971$	−56.7118	−1.81997	−0.909984	+	0.414642i	$0.863907\pi$
$972$	0	0
$973$	0	0
$974$	−22.3732	−0.716884
$975$	0	0
$976$	19.5439	0.625585
$977$	39.8809	1.27590	0.637951	−	0.770077i	$-0.279782\pi$
$977$	39.8809	1.27590	0.637951	+	0.770077i	$0.279782\pi$
$978$	0	0
$979$	13.1454	0.420130
$980$	0	0
$981$	0	0
$982$	6.36770	0.203201
$983$	−41.2678	−1.31624	−0.658120	−	0.752913i	$-0.728648\pi$
$983$	−41.2678	−1.31624	−0.658120	+	0.752913i	$0.728648\pi$
$984$	0	0
$985$	3.70169	0.117946
$986$	15.3243	0.488024
$987$	0	0
$988$	−1.54628	−0.0491937
$989$	5.37036	0.170767
$990$	0	0
$991$	−26.1784	−0.831583	−0.415792	−	0.909460i	$-0.636495\pi$
$991$	−26.1784	−0.831583	−0.415792	+	0.909460i	$0.636495\pi$
$992$	44.1380	1.40138
$993$	0	0
$994$	0	0
$995$	−29.4663	−0.934145
$996$	0	0
$997$	8.62009	0.273001	0.136501	−	0.990640i	$-0.456414\pi$
$997$	8.62009	0.273001	0.136501	+	0.990640i	$0.456414\pi$
$998$	11.8417	0.374842
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.br.1.4		6
3.2	odd	2		637.2.a.n.1.3	yes	6
7.6	odd	2		5733.2.a.bu.1.4		6
21.2	odd	6		637.2.e.n.508.4		12
21.5	even	6		637.2.e.o.508.4		12
21.11	odd	6		637.2.e.n.79.4		12
21.17	even	6		637.2.e.o.79.4		12
21.20	even	2		637.2.a.m.1.3	✓	6
39.38	odd	2		8281.2.a.cd.1.4		6
273.272	even	2		8281.2.a.cc.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.a.m.1.3	✓	6	21.20	even	2
637.2.a.n.1.3	yes	6	3.2	odd	2
637.2.e.n.79.4		12	21.11	odd	6
637.2.e.n.508.4		12	21.2	odd	6
637.2.e.o.79.4		12	21.17	even	6
637.2.e.o.508.4		12	21.5	even	6
5733.2.a.br.1.4		6	1.1	even	1	trivial
5733.2.a.bu.1.4		6	7.6	odd	2
8281.2.a.cc.1.4		6	273.272	even	2
8281.2.a.cd.1.4		6	39.38	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:	\( N \)	\(=\)	\( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5733.a (trivial)

\(f(q)\)	\(=\)	\(q+0.656184 q^{2} -1.56942 q^{4} +1.35996 q^{5} -2.34220 q^{8} +O(q^{10})\)	\(q+0.656184 q^{2} -1.56942 q^{4} +1.35996 q^{5} -2.34220 q^{8} +0.892385 q^{10} +1.90551 q^{11} +1.00000 q^{13} +1.60193 q^{16} -3.56633 q^{17} +0.985255 q^{19} -2.13436 q^{20} +1.25037 q^{22} -1.69605 q^{23} -3.15050 q^{25} +0.656184 q^{26} -6.54835 q^{29} +7.69550 q^{31} +5.73556 q^{32} -2.34017 q^{34} -2.02005 q^{37} +0.646508 q^{38} -3.18530 q^{40} -9.88779 q^{41} -3.16639 q^{43} -2.99055 q^{44} -1.11292 q^{46} -7.76416 q^{47} -2.06731 q^{50} -1.56942 q^{52} -0.354194 q^{53} +2.59143 q^{55} -4.29692 q^{58} -2.16385 q^{59} +12.2002 q^{61} +5.04967 q^{62} +0.559715 q^{64} +1.35996 q^{65} +11.3134 q^{67} +5.59709 q^{68} +9.05268 q^{71} -7.13619 q^{73} -1.32553 q^{74} -1.54628 q^{76} -5.39629 q^{79} +2.17857 q^{80} -6.48821 q^{82} +2.03494 q^{83} -4.85008 q^{85} -2.07773 q^{86} -4.46309 q^{88} +6.89864 q^{89} +2.66182 q^{92} -5.09472 q^{94} +1.33991 q^{95} -14.6223 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{4} - 6 q^{5}+O(q^{10}) \) 6 * q + 4 * q^4 - 6 * q^5	\( 6 q + 4 q^{4} - 6 q^{5} + 4 q^{10} - 4 q^{11} + 6 q^{13} - 16 q^{17} + 2 q^{19} - 16 q^{20} - 12 q^{22} + 6 q^{23} - 4 q^{25} + 6 q^{29} + 6 q^{31} + 20 q^{32} - 8 q^{38} + 4 q^{40} + 8 q^{41} + 2 q^{43} + 4 q^{44} + 8 q^{46} - 30 q^{47} - 8 q^{50} + 4 q^{52} + 14 q^{53} - 8 q^{55} - 8 q^{58} - 24 q^{59} - 28 q^{62} - 20 q^{64} - 6 q^{65} + 16 q^{67} - 28 q^{68} - 8 q^{71} - 6 q^{73} + 12 q^{74} - 16 q^{76} - 22 q^{79} + 28 q^{80} - 40 q^{82} - 50 q^{83} - 8 q^{85} + 16 q^{86} - 44 q^{88} - 26 q^{89} - 20 q^{92} - 32 q^{94} + 6 q^{95} - 14 q^{97}+O(q^{100}) \) 6 * q + 4 * q^4 - 6 * q^5 + 4 * q^10 - 4 * q^11 + 6 * q^13 - 16 * q^17 + 2 * q^19 - 16 * q^20 - 12 * q^22 + 6 * q^23 - 4 * q^25 + 6 * q^29 + 6 * q^31 + 20 * q^32 - 8 * q^38 + 4 * q^40 + 8 * q^41 + 2 * q^43 + 4 * q^44 + 8 * q^46 - 30 * q^47 - 8 * q^50 + 4 * q^52 + 14 * q^53 - 8 * q^55 - 8 * q^58 - 24 * q^59 - 28 * q^62 - 20 * q^64 - 6 * q^65 + 16 * q^67 - 28 * q^68 - 8 * q^71 - 6 * q^73 + 12 * q^74 - 16 * q^76 - 22 * q^79 + 28 * q^80 - 40 * q^82 - 50 * q^83 - 8 * q^85 + 16 * q^86 - 44 * q^88 - 26 * q^89 - 20 * q^92 - 32 * q^94 + 6 * q^95 - 14 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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