LMFDB - Embedded newform 5733.2.a.bu.1.3

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:	$0$
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			1.3
Root			$1.90903$ 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)

$f(q)$	$=$	$q-0.264627 q^{2} -1.92997 q^{4} -1.43515 q^{5} +1.03998 q^{8} +O(q^{10})$	$q-0.264627 q^{2} -1.92997 q^{4} -1.43515 q^{5} +1.03998 q^{8} +0.379780 q^{10} -5.50474 q^{11} -1.00000 q^{13} +3.58474 q^{16} +4.83072 q^{17} -2.82036 q^{19} +2.76981 q^{20} +1.45670 q^{22} +5.99956 q^{23} -2.94033 q^{25} +0.264627 q^{26} -1.04188 q^{29} -9.20895 q^{31} -3.02857 q^{32} -1.27834 q^{34} +0.612497 q^{37} +0.746342 q^{38} -1.49252 q^{40} -10.6196 q^{41} -8.43685 q^{43} +10.6240 q^{44} -1.58764 q^{46} +2.40922 q^{47} +0.778091 q^{50} +1.92997 q^{52} +1.82959 q^{53} +7.90015 q^{55} +0.275709 q^{58} -0.870914 q^{59} +3.33253 q^{61} +2.43693 q^{62} -6.36804 q^{64} +1.43515 q^{65} -6.62741 q^{67} -9.32316 q^{68} +6.85856 q^{71} -3.14147 q^{73} -0.162083 q^{74} +5.44322 q^{76} -17.5723 q^{79} -5.14465 q^{80} +2.81022 q^{82} +11.4525 q^{83} -6.93283 q^{85} +2.23261 q^{86} -5.72479 q^{88} -0.995318 q^{89} -11.5790 q^{92} -0.637545 q^{94} +4.04765 q^{95} -13.5090 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.264627	−0.187119	−0.0935596	−	0.995614i	$-0.529825\pi$
$2$	−0.264627	−0.187119	−0.0935596	+	0.995614i	$0.529825\pi$
$3$	0	0
$3$	0	0
$4$	−1.92997	−0.964986
$5$	−1.43515	−0.641820	−0.320910	−	0.947110i	$-0.603989\pi$
$5$	−1.43515	−0.641820	−0.320910	+	0.947110i	$0.603989\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.03998	0.367687
$9$	0	0
$10$	0.379780	0.120097
$11$	−5.50474	−1.65974	−0.829871	−	0.557955i	$-0.811586\pi$
$11$	−5.50474	−1.65974	−0.829871	+	0.557955i	$0.811586\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	3.58474	0.896185
$17$	4.83072	1.17162	0.585811	−	0.810448i	$-0.300776\pi$
$17$	4.83072	1.17162	0.585811	+	0.810448i	$0.300776\pi$
$18$	0	0
$19$	−2.82036	−0.647035	−0.323518	−	0.946222i	$-0.604865\pi$
$19$	−2.82036	−0.647035	−0.323518	+	0.946222i	$0.604865\pi$
$20$	2.76981	0.619348
$21$	0	0
$22$	1.45670	0.310570
$23$	5.99956	1.25100	0.625498	−	0.780226i	$-0.284896\pi$
$23$	5.99956	1.25100	0.625498	+	0.780226i	$0.284896\pi$
$24$	0	0
$25$	−2.94033	−0.588067
$26$	0.264627	0.0518975
$27$	0	0
$28$	0	0
$29$	−1.04188	−0.193472	−0.0967361	−	0.995310i	$-0.530840\pi$
$29$	−1.04188	−0.193472	−0.0967361	+	0.995310i	$0.530840\pi$
$30$	0	0
$31$	−9.20895	−1.65398	−0.826988	−	0.562219i	$-0.809948\pi$
$31$	−9.20895	−1.65398	−0.826988	+	0.562219i	$0.809948\pi$
$32$	−3.02857	−0.535380
$33$	0	0
$34$	−1.27834	−0.219233
$35$	0	0
$36$	0	0
$37$	0.612497	0.100694	0.0503470	−	0.998732i	$-0.483967\pi$
$37$	0.612497	0.100694	0.0503470	+	0.998732i	$0.483967\pi$
$38$	0.746342	0.121073
$39$	0	0
$40$	−1.49252	−0.235989
$41$	−10.6196	−1.65850	−0.829249	−	0.558879i	$-0.811232\pi$
$41$	−10.6196	−1.65850	−0.829249	+	0.558879i	$0.811232\pi$
$42$	0	0
$43$	−8.43685	−1.28661	−0.643304	−	0.765611i	$-0.722437\pi$
$43$	−8.43685	−1.28661	−0.643304	+	0.765611i	$0.722437\pi$
$44$	10.6240	1.60163
$45$	0	0
$46$	−1.58764	−0.234085
$47$	2.40922	0.351422	0.175711	−	0.984442i	$-0.443778\pi$
$47$	2.40922	0.351422	0.175711	+	0.984442i	$0.443778\pi$
$48$	0	0
$49$	0	0
$50$	0.778091	0.110039
$51$	0	0
$52$	1.92997	0.267639
$53$	1.82959	0.251313	0.125657	−	0.992074i	$-0.459896\pi$
$53$	1.82959	0.251313	0.125657	+	0.992074i	$0.459896\pi$
$54$	0	0
$55$	7.90015	1.06526
$56$	0	0
$57$	0	0
$58$	0.275709	0.0362024
$59$	−0.870914	−0.113383	−0.0566917	−	0.998392i	$-0.518055\pi$
$59$	−0.870914	−0.113383	−0.0566917	+	0.998392i	$0.518055\pi$
$60$	0	0
$61$	3.33253	0.426686	0.213343	−	0.976977i	$-0.431565\pi$
$61$	3.33253	0.426686	0.213343	+	0.976977i	$0.431565\pi$
$62$	2.43693	0.309491
$63$	0	0
$64$	−6.36804	−0.796005
$65$	1.43515	0.178009
$66$	0	0
$67$	−6.62741	−0.809667	−0.404833	−	0.914390i	$-0.632671\pi$
$67$	−6.62741	−0.809667	−0.404833	+	0.914390i	$0.632671\pi$
$68$	−9.32316	−1.13060
$69$	0	0
$70$	0	0
$71$	6.85856	0.813961	0.406980	−	0.913437i	$-0.366582\pi$
$71$	6.85856	0.813961	0.406980	+	0.913437i	$0.366582\pi$
$72$	0	0
$73$	−3.14147	−0.367682	−0.183841	−	0.982956i	$-0.558853\pi$
$73$	−3.14147	−0.367682	−0.183841	+	0.982956i	$0.558853\pi$
$74$	−0.162083	−0.0188418
$75$	0	0
$76$	5.44322	0.624380
$77$	0	0
$78$	0	0
$79$	−17.5723	−1.97704	−0.988518	−	0.151101i	$-0.951718\pi$
$79$	−17.5723	−1.97704	−0.988518	+	0.151101i	$0.951718\pi$
$80$	−5.14465	−0.575190
$81$	0	0
$82$	2.81022	0.310337
$83$	11.4525	1.25708	0.628538	−	0.777779i	$-0.283654\pi$
$83$	11.4525	1.25708	0.628538	+	0.777779i	$0.283654\pi$
$84$	0	0
$85$	−6.93283	−0.751971
$86$	2.23261	0.240749
$87$	0	0
$88$	−5.72479	−0.610265
$89$	−0.995318	−0.105503	−0.0527517	−	0.998608i	$-0.516799\pi$
$89$	−0.995318	−0.105503	−0.0527517	+	0.998608i	$0.516799\pi$
$90$	0	0
$91$	0	0
$92$	−11.5790	−1.20719
$93$	0	0
$94$	−0.637545	−0.0657577
$95$	4.04765	0.415280
$96$	0	0
$97$	−13.5090	−1.37163	−0.685817	−	0.727774i	$-0.740555\pi$
$97$	−13.5090	−1.37163	−0.685817	+	0.727774i	$0.740555\pi$
$98$	0	0
$99$	0	0
$100$	5.67477	0.567477
$101$	−1.00807	−0.100306	−0.0501532	−	0.998742i	$-0.515971\pi$
$101$	−1.00807	−0.100306	−0.0501532	+	0.998742i	$0.515971\pi$
$102$	0	0
$103$	12.7754	1.25880	0.629401	−	0.777081i	$-0.283300\pi$
$103$	12.7754	1.25880	0.629401	+	0.777081i	$0.283300\pi$
$104$	−1.03998	−0.101978
$105$	0	0
$106$	−0.484157	−0.0470255
$107$	0.685495	0.0662693	0.0331347	−	0.999451i	$-0.489451\pi$
$107$	0.685495	0.0662693	0.0331347	+	0.999451i	$0.489451\pi$
$108$	0	0
$109$	−2.90344	−0.278099	−0.139050	−	0.990285i	$-0.544405\pi$
$109$	−2.90344	−0.278099	−0.139050	+	0.990285i	$0.544405\pi$
$110$	−2.09059	−0.199330
$111$	0	0
$112$	0	0
$113$	−12.0315	−1.13183	−0.565915	−	0.824464i	$-0.691477\pi$
$113$	−12.0315	−1.13183	−0.565915	+	0.824464i	$0.691477\pi$
$114$	0	0
$115$	−8.61029	−0.802914
$116$	2.01080	0.186698
$117$	0	0
$118$	0.230467	0.0212162
$119$	0	0
$120$	0	0
$121$	19.3022	1.75474
$122$	−0.881875	−0.0798412
$123$	0	0
$124$	17.7730	1.59606
$125$	11.3956	1.01925
$126$	0	0
$127$	15.6659	1.39012	0.695062	−	0.718950i	$-0.255377\pi$
$127$	15.6659	1.39012	0.695062	+	0.718950i	$0.255377\pi$
$128$	7.74229	0.684328
$129$	0	0
$130$	−0.379780	−0.0333089
$131$	12.1273	1.05957	0.529784	−	0.848132i	$-0.322273\pi$
$131$	12.1273	1.05957	0.529784	+	0.848132i	$0.322273\pi$
$132$	0	0
$133$	0	0
$134$	1.75379	0.151504
$135$	0	0
$136$	5.02383	0.430790
$137$	−15.9375	−1.36163	−0.680815	−	0.732456i	$-0.738374\pi$
$137$	−15.9375	−1.36163	−0.680815	+	0.732456i	$0.738374\pi$
$138$	0	0
$139$	6.64088	0.563272	0.281636	−	0.959521i	$-0.409123\pi$
$139$	6.64088	0.563272	0.281636	+	0.959521i	$0.409123\pi$
$140$	0	0
$141$	0	0
$142$	−1.81496	−0.152308
$143$	5.50474	0.460330
$144$	0	0
$145$	1.49526	0.124174
$146$	0.831317	0.0688003
$147$	0	0
$148$	−1.18210	−0.0971683
$149$	19.5502	1.60162	0.800809	−	0.598920i	$-0.204403\pi$
$149$	19.5502	1.60162	0.800809	+	0.598920i	$0.204403\pi$
$150$	0	0
$151$	10.6880	0.869779	0.434890	−	0.900484i	$-0.356787\pi$
$151$	10.6880	0.869779	0.434890	+	0.900484i	$0.356787\pi$
$152$	−2.93311	−0.237906
$153$	0	0
$154$	0	0
$155$	13.2163	1.06156
$156$	0	0
$157$	15.0734	1.20299	0.601496	−	0.798876i	$-0.294572\pi$
$157$	15.0734	1.20299	0.601496	+	0.798876i	$0.294572\pi$
$158$	4.65009	0.369942
$159$	0	0
$160$	4.34646	0.343618
$161$	0	0
$162$	0	0
$163$	−23.8135	−1.86521	−0.932607	−	0.360894i	$-0.882472\pi$
$163$	−23.8135	−1.86521	−0.932607	+	0.360894i	$0.882472\pi$
$164$	20.4955	1.60043
$165$	0	0
$166$	−3.03064	−0.235223
$167$	−7.12371	−0.551249	−0.275625	−	0.961265i	$-0.588885\pi$
$167$	−7.12371	−0.551249	−0.275625	+	0.961265i	$0.588885\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	1.83461	0.140708
$171$	0	0
$172$	16.2829	1.24156
$173$	11.2367	0.854309	0.427155	−	0.904179i	$-0.359516\pi$
$173$	11.2367	0.854309	0.427155	+	0.904179i	$0.359516\pi$
$174$	0	0
$175$	0	0
$176$	−19.7331	−1.48744
$177$	0	0
$178$	0.263387	0.0197417
$179$	13.1945	0.986204	0.493102	−	0.869972i	$-0.335863\pi$
$179$	13.1945	0.986204	0.493102	+	0.869972i	$0.335863\pi$
$180$	0	0
$181$	13.7414	1.02139	0.510696	−	0.859761i	$-0.329388\pi$
$181$	13.7414	1.02139	0.510696	+	0.859761i	$0.329388\pi$
$182$	0	0
$183$	0	0
$184$	6.23939	0.459974
$185$	−0.879028	−0.0646274
$186$	0	0
$187$	−26.5919	−1.94459
$188$	−4.64974	−0.339117
$189$	0	0
$190$	−1.07112	−0.0777069
$191$	16.3307	1.18165	0.590824	−	0.806800i	$-0.298803\pi$
$191$	16.3307	1.18165	0.590824	+	0.806800i	$0.298803\pi$
$192$	0	0
$193$	−14.0533	−1.01158	−0.505790	−	0.862656i	$-0.668799\pi$
$193$	−14.0533	−1.01158	−0.505790	+	0.862656i	$0.668799\pi$
$194$	3.57485	0.256659
$195$	0	0
$196$	0	0
$197$	1.46898	0.104660	0.0523302	−	0.998630i	$-0.483335\pi$
$197$	1.46898	0.104660	0.0523302	+	0.998630i	$0.483335\pi$
$198$	0	0
$199$	13.3772	0.948285	0.474142	−	0.880448i	$-0.342758\pi$
$199$	13.3772	0.948285	0.474142	+	0.880448i	$0.342758\pi$
$200$	−3.05787	−0.216224
$201$	0	0
$202$	0.266761	0.0187692
$203$	0	0
$204$	0	0
$205$	15.2407	1.06446
$206$	−3.38072	−0.235546
$207$	0	0
$208$	−3.58474	−0.248557
$209$	15.5254	1.07391
$210$	0	0
$211$	3.47044	0.238915	0.119457	−	0.992839i	$-0.461885\pi$
$211$	3.47044	0.238915	0.119457	+	0.992839i	$0.461885\pi$
$212$	−3.53105	−0.242514
$213$	0	0
$214$	−0.181400	−0.0124003
$215$	12.1082	0.825771
$216$	0	0
$217$	0	0
$218$	0.768328	0.0520378
$219$	0	0
$220$	−15.2471	−1.02796
$221$	−4.83072	−0.324950
$222$	0	0
$223$	9.91318	0.663836	0.331918	−	0.943308i	$-0.392304\pi$
$223$	9.91318	0.663836	0.331918	+	0.943308i	$0.392304\pi$
$224$	0	0
$225$	0	0
$226$	3.18386	0.211787
$227$	12.0727	0.801292	0.400646	−	0.916233i	$-0.368786\pi$
$227$	12.0727	0.801292	0.400646	+	0.916233i	$0.368786\pi$
$228$	0	0
$229$	−4.05171	−0.267745	−0.133872	−	0.990999i	$-0.542741\pi$
$229$	−4.05171	−0.267745	−0.133872	+	0.990999i	$0.542741\pi$
$230$	2.27851	0.150241
$231$	0	0
$232$	−1.08353	−0.0711372
$233$	−12.5450	−0.821850	−0.410925	−	0.911669i	$-0.634794\pi$
$233$	−12.5450	−0.821850	−0.410925	+	0.911669i	$0.634794\pi$
$234$	0	0
$235$	−3.45761	−0.225549
$236$	1.68084	0.109413
$237$	0	0
$238$	0	0
$239$	−13.3463	−0.863299	−0.431649	−	0.902042i	$-0.642068\pi$
$239$	−13.3463	−0.863299	−0.431649	+	0.902042i	$0.642068\pi$
$240$	0	0
$241$	−20.3854	−1.31314	−0.656568	−	0.754267i	$-0.727993\pi$
$241$	−20.3854	−1.31314	−0.656568	+	0.754267i	$0.727993\pi$
$242$	−5.10787	−0.328346
$243$	0	0
$244$	−6.43169	−0.411747
$245$	0	0
$246$	0	0
$247$	2.82036	0.179455
$248$	−9.57708	−0.608145
$249$	0	0
$250$	−3.01558	−0.190722
$251$	17.1921	1.08515	0.542577	−	0.840006i	$-0.317449\pi$
$251$	17.1921	1.08515	0.542577	+	0.840006i	$0.317449\pi$
$252$	0	0
$253$	−33.0260	−2.07633
$254$	−4.14561	−0.260119
$255$	0	0
$256$	10.6873	0.667954
$257$	−7.64695	−0.477004	−0.238502	−	0.971142i	$-0.576656\pi$
$257$	−7.64695	−0.477004	−0.238502	+	0.971142i	$0.576656\pi$
$258$	0	0
$259$	0	0
$260$	−2.76981	−0.171776
$261$	0	0
$262$	−3.20921	−0.198266
$263$	0.101037	0.00623022	0.00311511	−	0.999995i	$-0.499008\pi$
$263$	0.101037	0.00623022	0.00311511	+	0.999995i	$0.499008\pi$
$264$	0	0
$265$	−2.62574	−0.161298
$266$	0	0
$267$	0	0
$268$	12.7907	0.781318
$269$	−7.56852	−0.461461	−0.230730	−	0.973018i	$-0.574111\pi$
$269$	−7.56852	−0.461461	−0.230730	+	0.973018i	$0.574111\pi$
$270$	0	0
$271$	13.8554	0.841653	0.420826	−	0.907141i	$-0.361740\pi$
$271$	13.8554	0.841653	0.420826	+	0.907141i	$0.361740\pi$
$272$	17.3169	1.04999
$273$	0	0
$274$	4.21748	0.254787
$275$	16.1858	0.976039
$276$	0	0
$277$	0.552935	0.0332226	0.0166113	−	0.999862i	$-0.494712\pi$
$277$	0.552935	0.0332226	0.0166113	+	0.999862i	$0.494712\pi$
$278$	−1.75735	−0.105399
$279$	0	0
$280$	0	0
$281$	−1.14667	−0.0684043	−0.0342022	−	0.999415i	$-0.510889\pi$
$281$	−1.14667	−0.0684043	−0.0342022	+	0.999415i	$0.510889\pi$
$282$	0	0
$283$	4.05396	0.240983	0.120491	−	0.992714i	$-0.461553\pi$
$283$	4.05396	0.240983	0.120491	+	0.992714i	$0.461553\pi$
$284$	−13.2368	−0.785461
$285$	0	0
$286$	−1.45670	−0.0861365
$287$	0	0
$288$	0	0
$289$	6.33588	0.372699
$290$	−0.395685	−0.0232354
$291$	0	0
$292$	6.06296	0.354808
$293$	15.0649	0.880102	0.440051	−	0.897973i	$-0.354960\pi$
$293$	15.0649	0.880102	0.440051	+	0.897973i	$0.354960\pi$
$294$	0	0
$295$	1.24990	0.0727718
$296$	0.636982	0.0370238
$297$	0	0
$298$	−5.17351	−0.299694
$299$	−5.99956	−0.346964
$300$	0	0
$301$	0	0
$302$	−2.82834	−0.162752
$303$	0	0
$304$	−10.1103	−0.579863
$305$	−4.78269	−0.273856
$306$	0	0
$307$	19.9408	1.13808	0.569040	−	0.822310i	$-0.307315\pi$
$307$	19.9408	1.13808	0.569040	+	0.822310i	$0.307315\pi$
$308$	0	0
$309$	0	0
$310$	−3.49737	−0.198637
$311$	−10.8956	−0.617833	−0.308916	−	0.951089i	$-0.599966\pi$
$311$	−10.8956	−0.617833	−0.308916	+	0.951089i	$0.599966\pi$
$312$	0	0
$313$	0.0519190	0.00293464	0.00146732	−	0.999999i	$-0.499533\pi$
$313$	0.0519190	0.00293464	0.00146732	+	0.999999i	$0.499533\pi$
$314$	−3.98883	−0.225103
$315$	0	0
$316$	33.9140	1.90781
$317$	−16.1010	−0.904321	−0.452161	−	0.891937i	$-0.649347\pi$
$317$	−16.1010	−0.904321	−0.452161	+	0.891937i	$0.649347\pi$
$318$	0	0
$319$	5.73528	0.321114
$320$	9.13912	0.510892
$321$	0	0
$322$	0	0
$323$	−13.6244	−0.758081
$324$	0	0
$325$	2.94033	0.163100
$326$	6.30167	0.349017
$327$	0	0
$328$	−11.0441	−0.609808
$329$	0	0
$330$	0	0
$331$	30.7862	1.69216	0.846081	−	0.533054i	$-0.178956\pi$
$331$	30.7862	1.69216	0.846081	+	0.533054i	$0.178956\pi$
$332$	−22.1030	−1.21306
$333$	0	0
$334$	1.88512	0.103149
$335$	9.51135	0.519661
$336$	0	0
$337$	−2.41842	−0.131740	−0.0658700	−	0.997828i	$-0.520982\pi$
$337$	−2.41842	−0.131740	−0.0658700	+	0.997828i	$0.520982\pi$
$338$	−0.264627	−0.0143938
$339$	0	0
$340$	13.3802	0.725642
$341$	50.6929	2.74517
$342$	0	0
$343$	0	0
$344$	−8.77411	−0.473069
$345$	0	0
$346$	−2.97353	−0.159858
$347$	0.492527	0.0264403	0.0132201	−	0.999913i	$-0.495792\pi$
$347$	0.492527	0.0264403	0.0132201	+	0.999913i	$0.495792\pi$
$348$	0	0
$349$	11.9442	0.639356	0.319678	−	0.947526i	$-0.396425\pi$
$349$	11.9442	0.639356	0.319678	+	0.947526i	$0.396425\pi$
$350$	0	0
$351$	0	0
$352$	16.6715	0.888593
$353$	15.5299	0.826575	0.413288	−	0.910601i	$-0.364380\pi$
$353$	15.5299	0.826575	0.413288	+	0.910601i	$0.364380\pi$
$354$	0	0
$355$	−9.84308	−0.522417
$356$	1.92094	0.101809
$357$	0	0
$358$	−3.49162	−0.184538
$359$	8.50709	0.448987	0.224493	−	0.974476i	$-0.427927\pi$
$359$	8.50709	0.448987	0.224493	+	0.974476i	$0.427927\pi$
$360$	0	0
$361$	−11.0456	−0.581345
$362$	−3.63635	−0.191122
$363$	0	0
$364$	0	0
$365$	4.50850	0.235985
$366$	0	0
$367$	−5.19084	−0.270960	−0.135480	−	0.990780i	$-0.543258\pi$
$367$	−5.19084	−0.270960	−0.135480	+	0.990780i	$0.543258\pi$
$368$	21.5069	1.12112
$369$	0	0
$370$	0.232614	0.0120930
$371$	0	0
$372$	0	0
$373$	−10.1427	−0.525169	−0.262585	−	0.964909i	$-0.584575\pi$
$373$	−10.1427	−0.525169	−0.262585	+	0.964909i	$0.584575\pi$
$374$	7.03692	0.363870
$375$	0	0
$376$	2.50553	0.129213
$377$	1.04188	0.0536595
$378$	0	0
$379$	−3.63670	−0.186805	−0.0934024	−	0.995628i	$-0.529774\pi$
$379$	−3.63670	−0.186805	−0.0934024	+	0.995628i	$0.529774\pi$
$380$	−7.81186	−0.400740
$381$	0	0
$382$	−4.32154	−0.221109
$383$	4.60281	0.235192	0.117596	−	0.993061i	$-0.462481\pi$
$383$	4.60281	0.235192	0.117596	+	0.993061i	$0.462481\pi$
$384$	0	0
$385$	0	0
$386$	3.71888	0.189286
$387$	0	0
$388$	26.0721	1.32361
$389$	−19.6104	−0.994286	−0.497143	−	0.867669i	$-0.665618\pi$
$389$	−19.6104	−0.994286	−0.497143	+	0.867669i	$0.665618\pi$
$390$	0	0
$391$	28.9822	1.46569
$392$	0	0
$393$	0	0
$394$	−0.388731	−0.0195840
$395$	25.2189	1.26890
$396$	0	0
$397$	19.8635	0.996919	0.498459	−	0.866913i	$-0.333899\pi$
$397$	19.8635	0.996919	0.498459	+	0.866913i	$0.333899\pi$
$398$	−3.53996	−0.177442
$399$	0	0
$400$	−10.5403	−0.527017
$401$	15.1117	0.754644	0.377322	−	0.926082i	$-0.376845\pi$
$401$	15.1117	0.754644	0.377322	+	0.926082i	$0.376845\pi$
$402$	0	0
$403$	9.20895	0.458731
$404$	1.94554	0.0967943
$405$	0	0
$406$	0	0
$407$	−3.37164	−0.167126
$408$	0	0
$409$	35.2443	1.74272	0.871360	−	0.490644i	$-0.163238\pi$
$409$	35.2443	1.74272	0.871360	+	0.490644i	$0.163238\pi$
$410$	−4.03310	−0.199181
$411$	0	0
$412$	−24.6563	−1.21473
$413$	0	0
$414$	0	0
$415$	−16.4361	−0.806816
$416$	3.02857	0.148488
$417$	0	0
$418$	−4.10842	−0.200950
$419$	1.50468	0.0735084	0.0367542	−	0.999324i	$-0.488298\pi$
$419$	1.50468	0.0735084	0.0367542	+	0.999324i	$0.488298\pi$
$420$	0	0
$421$	−24.5079	−1.19444	−0.597221	−	0.802077i	$-0.703728\pi$
$421$	−24.5079	−1.19444	−0.597221	+	0.802077i	$0.703728\pi$
$422$	−0.918370	−0.0447056
$423$	0	0
$424$	1.90272	0.0924045
$425$	−14.2039	−0.688992
$426$	0	0
$427$	0	0
$428$	−1.32299	−0.0639490
$429$	0	0
$430$	−3.20414	−0.154518
$431$	−41.0655	−1.97805	−0.989027	−	0.147732i	$-0.952803\pi$
$431$	−41.0655	−1.97805	−0.989027	+	0.147732i	$0.952803\pi$
$432$	0	0
$433$	−6.65603	−0.319869	−0.159934	−	0.987128i	$-0.551128\pi$
$433$	−6.65603	−0.319869	−0.159934	+	0.987128i	$0.551128\pi$
$434$	0	0
$435$	0	0
$436$	5.60357	0.268362
$437$	−16.9209	−0.809438
$438$	0	0
$439$	−8.22990	−0.392792	−0.196396	−	0.980525i	$-0.562924\pi$
$439$	−8.22990	−0.392792	−0.196396	+	0.980525i	$0.562924\pi$
$440$	8.21596	0.391681
$441$	0	0
$442$	1.27834	0.0608043
$443$	−17.6856	−0.840266	−0.420133	−	0.907463i	$-0.638017\pi$
$443$	−17.6856	−0.840266	−0.420133	+	0.907463i	$0.638017\pi$
$444$	0	0
$445$	1.42843	0.0677143
$446$	−2.62329	−0.124216
$447$	0	0
$448$	0	0
$449$	14.5250	0.685477	0.342738	−	0.939431i	$-0.388646\pi$
$449$	14.5250	0.685477	0.342738	+	0.939431i	$0.388646\pi$
$450$	0	0
$451$	58.4580	2.75268
$452$	23.2205	1.09220
$453$	0	0
$454$	−3.19475	−0.149937
$455$	0	0
$456$	0	0
$457$	3.78919	0.177251	0.0886255	−	0.996065i	$-0.471753\pi$
$457$	3.78919	0.177251	0.0886255	+	0.996065i	$0.471753\pi$
$458$	1.07219	0.0501002
$459$	0	0
$460$	16.6176	0.774801
$461$	13.1107	0.610627	0.305314	−	0.952252i	$-0.401239\pi$
$461$	13.1107	0.610627	0.305314	+	0.952252i	$0.401239\pi$
$462$	0	0
$463$	15.3027	0.711176	0.355588	−	0.934643i	$-0.384281\pi$
$463$	15.3027	0.711176	0.355588	+	0.934643i	$0.384281\pi$
$464$	−3.73487	−0.173387
$465$	0	0
$466$	3.31974	0.153784
$467$	30.7738	1.42404	0.712022	−	0.702158i	$-0.247780\pi$
$467$	30.7738	1.42404	0.712022	+	0.702158i	$0.247780\pi$
$468$	0	0
$469$	0	0
$470$	0.914975	0.0422046
$471$	0	0
$472$	−0.905729	−0.0416896
$473$	46.4427	2.13544
$474$	0	0
$475$	8.29280	0.380500
$476$	0	0
$477$	0	0
$478$	3.53178	0.161540
$479$	−1.71740	−0.0784699	−0.0392350	−	0.999230i	$-0.512492\pi$
$479$	−1.71740	−0.0784699	−0.0392350	+	0.999230i	$0.512492\pi$
$480$	0	0
$481$	−0.612497	−0.0279275
$482$	5.39451	0.245713
$483$	0	0
$484$	−37.2527	−1.69330
$485$	19.3875	0.880342
$486$	0	0
$487$	22.6805	1.02775	0.513877	−	0.857864i	$-0.328209\pi$
$487$	22.6805	1.02775	0.513877	+	0.857864i	$0.328209\pi$
$488$	3.46575	0.156887
$489$	0	0
$490$	0	0
$491$	13.1366	0.592846	0.296423	−	0.955057i	$-0.404206\pi$
$491$	13.1366	0.592846	0.296423	+	0.955057i	$0.404206\pi$
$492$	0	0
$493$	−5.03303	−0.226676
$494$	−0.746342	−0.0335795
$495$	0	0
$496$	−33.0117	−1.48227
$497$	0	0
$498$	0	0
$499$	14.0395	0.628495	0.314248	−	0.949341i	$-0.398248\pi$
$499$	14.0395	0.628495	0.314248	+	0.949341i	$0.398248\pi$
$500$	−21.9932	−0.983566
$501$	0	0
$502$	−4.54948	−0.203053
$503$	0.367865	0.0164023	0.00820114	−	0.999966i	$-0.497389\pi$
$503$	0.367865	0.0164023	0.00820114	+	0.999966i	$0.497389\pi$
$504$	0	0
$505$	1.44673	0.0643786
$506$	8.73957	0.388521
$507$	0	0
$508$	−30.2348	−1.34145
$509$	41.2319	1.82757	0.913787	−	0.406194i	$-0.133144\pi$
$509$	41.2319	1.82757	0.913787	+	0.406194i	$0.133144\pi$
$510$	0	0
$511$	0	0
$512$	−18.3127	−0.809315
$513$	0	0
$514$	2.02359	0.0892566
$515$	−18.3347	−0.807925
$516$	0	0
$517$	−13.2622	−0.583269
$518$	0	0
$519$	0	0
$520$	1.49252	0.0654515
$521$	1.04099	0.0456065	0.0228032	−	0.999740i	$-0.492741\pi$
$521$	1.04099	0.0456065	0.0228032	+	0.999740i	$0.492741\pi$
$522$	0	0
$523$	20.0209	0.875451	0.437726	−	0.899109i	$-0.355784\pi$
$523$	20.0209	0.875451	0.437726	+	0.899109i	$0.355784\pi$
$524$	−23.4054	−1.02247
$525$	0	0
$526$	−0.0267371	−0.00116579
$527$	−44.4859	−1.93784
$528$	0	0
$529$	12.9947	0.564989
$530$	0.694840	0.0301819
$531$	0	0
$532$	0	0
$533$	10.6196	0.459985
$534$	0	0
$535$	−0.983791	−0.0425330
$536$	−6.89234	−0.297704
$537$	0	0
$538$	2.00283	0.0863481
$539$	0	0
$540$	0	0
$541$	−9.78749	−0.420797	−0.210399	−	0.977616i	$-0.567476\pi$
$541$	−9.78749	−0.420797	−0.210399	+	0.977616i	$0.567476\pi$
$542$	−3.66649	−0.157489
$543$	0	0
$544$	−14.6302	−0.627263
$545$	4.16689	0.178490
$546$	0	0
$547$	−2.56174	−0.109532	−0.0547660	−	0.998499i	$-0.517441\pi$
$547$	−2.56174	−0.109532	−0.0547660	+	0.998499i	$0.517441\pi$
$548$	30.7589	1.31395
$549$	0	0
$550$	−4.28319	−0.182636
$551$	2.93848	0.125183
$552$	0	0
$553$	0	0
$554$	−0.146321	−0.00621660
$555$	0	0
$556$	−12.8167	−0.543550
$557$	27.4442	1.16285	0.581424	−	0.813601i	$-0.302496\pi$
$557$	27.4442	1.16285	0.581424	+	0.813601i	$0.302496\pi$
$558$	0	0
$559$	8.43685	0.356841
$560$	0	0
$561$	0	0
$562$	0.303438	0.0127998
$563$	0.162708	0.00685734	0.00342867	−	0.999994i	$-0.498909\pi$
$563$	0.162708	0.00685734	0.00342867	+	0.999994i	$0.498909\pi$
$564$	0	0
$565$	17.2671	0.726431
$566$	−1.07278	−0.0450925
$567$	0	0
$568$	7.13273	0.299283
$569$	12.3901	0.519419	0.259709	−	0.965687i	$-0.416373\pi$
$569$	12.3901	0.519419	0.259709	+	0.965687i	$0.416373\pi$
$570$	0	0
$571$	−21.8122	−0.912810	−0.456405	−	0.889772i	$-0.650863\pi$
$571$	−21.8122	−0.912810	−0.456405	+	0.889772i	$0.650863\pi$
$572$	−10.6240	−0.444212
$573$	0	0
$574$	0	0
$575$	−17.6407	−0.735669
$576$	0	0
$577$	−6.06583	−0.252524	−0.126262	−	0.991997i	$-0.540298\pi$
$577$	−6.06583	−0.252524	−0.126262	+	0.991997i	$0.540298\pi$
$578$	−1.67664	−0.0697391
$579$	0	0
$580$	−2.88581	−0.119827
$581$	0	0
$582$	0	0
$583$	−10.0714	−0.417115
$584$	−3.26705	−0.135192
$585$	0	0
$586$	−3.98658	−0.164684
$587$	20.5820	0.849510	0.424755	−	0.905308i	$-0.360360\pi$
$587$	20.5820	0.849510	0.424755	+	0.905308i	$0.360360\pi$
$588$	0	0
$589$	25.9726	1.07018
$590$	−0.330756	−0.0136170
$591$	0	0
$592$	2.19564	0.0902404
$593$	−24.0397	−0.987190	−0.493595	−	0.869692i	$-0.664318\pi$
$593$	−24.0397	−0.987190	−0.493595	+	0.869692i	$0.664318\pi$
$594$	0	0
$595$	0	0
$596$	−37.7314	−1.54554
$597$	0	0
$598$	1.58764	0.0649236
$599$	−32.2523	−1.31779	−0.658896	−	0.752234i	$-0.728976\pi$
$599$	−32.2523	−1.31779	−0.658896	+	0.752234i	$0.728976\pi$
$600$	0	0
$601$	−5.21454	−0.212705	−0.106353	−	0.994328i	$-0.533917\pi$
$601$	−5.21454	−0.212705	−0.106353	+	0.994328i	$0.533917\pi$
$602$	0	0
$603$	0	0
$604$	−20.6276	−0.839325
$605$	−27.7016	−1.12623
$606$	0	0
$607$	−9.07048	−0.368160	−0.184080	−	0.982911i	$-0.558930\pi$
$607$	−9.07048	−0.368160	−0.184080	+	0.982911i	$0.558930\pi$
$608$	8.54165	0.346410
$609$	0	0
$610$	1.26563	0.0512437
$611$	−2.40922	−0.0974668
$612$	0	0
$613$	20.0920	0.811507	0.405754	−	0.913983i	$-0.367009\pi$
$613$	20.0920	0.811507	0.405754	+	0.913983i	$0.367009\pi$
$614$	−5.27686	−0.212957
$615$	0	0
$616$	0	0
$617$	12.9556	0.521572	0.260786	−	0.965397i	$-0.416018\pi$
$617$	12.9556	0.521572	0.260786	+	0.965397i	$0.416018\pi$
$618$	0	0
$619$	44.3644	1.78316	0.891578	−	0.452866i	$-0.149598\pi$
$619$	44.3644	1.78316	0.891578	+	0.452866i	$0.149598\pi$
$620$	−25.5070	−1.02439
$621$	0	0
$622$	2.88327	0.115608
$623$	0	0
$624$	0	0
$625$	−1.65276	−0.0661105
$626$	−0.0137392	−0.000549127 0
$627$	0	0
$628$	−29.0913	−1.16087
$629$	2.95880	0.117975
$630$	0	0
$631$	6.61717	0.263426	0.131713	−	0.991288i	$-0.457952\pi$
$631$	6.61717	0.263426	0.131713	+	0.991288i	$0.457952\pi$
$632$	−18.2747	−0.726930
$633$	0	0
$634$	4.26075	0.169216
$635$	−22.4830	−0.892210
$636$	0	0
$637$	0	0
$638$	−1.51771	−0.0600866
$639$	0	0
$640$	−11.1114	−0.439216
$641$	−18.9567	−0.748744	−0.374372	−	0.927279i	$-0.622142\pi$
$641$	−18.9567	−0.748744	−0.374372	+	0.927279i	$0.622142\pi$
$642$	0	0
$643$	−13.4019	−0.528517	−0.264259	−	0.964452i	$-0.585127\pi$
$643$	−13.4019	−0.528517	−0.264259	+	0.964452i	$0.585127\pi$
$644$	0	0
$645$	0	0
$646$	3.60537	0.141851
$647$	42.7588	1.68102	0.840511	−	0.541794i	$-0.182255\pi$
$647$	42.7588	1.68102	0.840511	+	0.541794i	$0.182255\pi$
$648$	0	0
$649$	4.79416	0.188187
$650$	−0.778091	−0.0305192
$651$	0	0
$652$	45.9593	1.79991
$653$	−10.9852	−0.429884	−0.214942	−	0.976627i	$-0.568956\pi$
$653$	−10.9852	−0.429884	−0.214942	+	0.976627i	$0.568956\pi$
$654$	0	0
$655$	−17.4046	−0.680052
$656$	−38.0684	−1.48632
$657$	0	0
$658$	0	0
$659$	17.7614	0.691884	0.345942	−	0.938256i	$-0.387559\pi$
$659$	17.7614	0.691884	0.345942	+	0.938256i	$0.387559\pi$
$660$	0	0
$661$	−8.18255	−0.318264	−0.159132	−	0.987257i	$-0.550870\pi$
$661$	−8.18255	−0.318264	−0.159132	+	0.987257i	$0.550870\pi$
$662$	−8.14685	−0.316636
$663$	0	0
$664$	11.9103	0.462210
$665$	0	0
$666$	0	0
$667$	−6.25082	−0.242033
$668$	13.7486	0.531948
$669$	0	0
$670$	−2.51696	−0.0972385
$671$	−18.3447	−0.708189
$672$	0	0
$673$	9.30129	0.358539	0.179269	−	0.983800i	$-0.442627\pi$
$673$	9.30129	0.358539	0.179269	+	0.983800i	$0.442627\pi$
$674$	0.639979	0.0246511
$675$	0	0
$676$	−1.92997	−0.0742297
$677$	−41.1552	−1.58172	−0.790862	−	0.611994i	$-0.790367\pi$
$677$	−41.1552	−1.58172	−0.790862	+	0.611994i	$0.790367\pi$
$678$	0	0
$679$	0	0
$680$	−7.20997	−0.276490
$681$	0	0
$682$	−13.4147	−0.513675
$683$	−39.2842	−1.50317	−0.751583	−	0.659638i	$-0.770709\pi$
$683$	−39.2842	−1.50317	−0.751583	+	0.659638i	$0.770709\pi$
$684$	0	0
$685$	22.8727	0.873921
$686$	0	0
$687$	0	0
$688$	−30.2439	−1.15304
$689$	−1.82959	−0.0697017
$690$	0	0
$691$	3.03355	0.115402	0.0577009	−	0.998334i	$-0.481623\pi$
$691$	3.03355	0.115402	0.0577009	+	0.998334i	$0.481623\pi$
$692$	−21.6865	−0.824397
$693$	0	0
$694$	−0.130336	−0.00494748
$695$	−9.53069	−0.361520
$696$	0	0
$697$	−51.3002	−1.94313
$698$	−3.16074	−0.119636
$699$	0	0
$700$	0	0
$701$	26.2320	0.990767	0.495384	−	0.868674i	$-0.335028\pi$
$701$	26.2320	0.990767	0.495384	+	0.868674i	$0.335028\pi$
$702$	0	0
$703$	−1.72746	−0.0651525
$704$	35.0544	1.32116
$705$	0	0
$706$	−4.10963	−0.154668
$707$	0	0
$708$	0	0
$709$	−7.87770	−0.295853	−0.147927	−	0.988998i	$-0.547260\pi$
$709$	−7.87770	−0.295853	−0.147927	+	0.988998i	$0.547260\pi$
$710$	2.60474	0.0977542
$711$	0	0
$712$	−1.03511	−0.0387922
$713$	−55.2497	−2.06912
$714$	0	0
$715$	−7.90015	−0.295449
$716$	−25.4650	−0.951673
$717$	0	0
$718$	−2.25120	−0.0840141
$719$	45.6656	1.70304	0.851519	−	0.524323i	$-0.175682\pi$
$719$	45.6656	1.70304	0.851519	+	0.524323i	$0.175682\pi$
$720$	0	0
$721$	0	0
$722$	2.92295	0.108781
$723$	0	0
$724$	−26.5206	−0.985630
$725$	3.06348	0.113775
$726$	0	0
$727$	−37.5947	−1.39431	−0.697155	−	0.716921i	$-0.745551\pi$
$727$	−37.5947	−1.39431	−0.697155	+	0.716921i	$0.745551\pi$
$728$	0	0
$729$	0	0
$730$	−1.19307	−0.0441574
$731$	−40.7561	−1.50742
$732$	0	0
$733$	−53.1810	−1.96429	−0.982143	−	0.188138i	$-0.939755\pi$
$733$	−53.1810	−1.96429	−0.982143	+	0.188138i	$0.939755\pi$
$734$	1.37363	0.0507018
$735$	0	0
$736$	−18.1701	−0.669758
$737$	36.4822	1.34384
$738$	0	0
$739$	41.9633	1.54364	0.771822	−	0.635839i	$-0.219346\pi$
$739$	41.9633	1.54364	0.771822	+	0.635839i	$0.219346\pi$
$740$	1.69650	0.0623646
$741$	0	0
$742$	0	0
$743$	38.5424	1.41398	0.706991	−	0.707222i	$-0.250052\pi$
$743$	38.5424	1.41398	0.706991	+	0.707222i	$0.250052\pi$
$744$	0	0
$745$	−28.0576	−1.02795
$746$	2.68403	0.0982693
$747$	0	0
$748$	51.3216	1.87650
$749$	0	0
$750$	0	0
$751$	36.2434	1.32254	0.661270	−	0.750148i	$-0.270018\pi$
$751$	36.2434	1.32254	0.661270	+	0.750148i	$0.270018\pi$
$752$	8.63644	0.314939
$753$	0	0
$754$	−0.275709	−0.0100407
$755$	−15.3390	−0.558242
$756$	0	0
$757$	19.4752	0.707837	0.353919	−	0.935276i	$-0.384849\pi$
$757$	19.4752	0.707837	0.353919	+	0.935276i	$0.384849\pi$
$758$	0.962368	0.0349548
$759$	0	0
$760$	4.20946	0.152693
$761$	−51.9059	−1.88159	−0.940793	−	0.338981i	$-0.889918\pi$
$761$	−51.9059	−1.88159	−0.940793	+	0.338981i	$0.889918\pi$
$762$	0	0
$763$	0	0
$764$	−31.5178	−1.14028
$765$	0	0
$766$	−1.21803	−0.0440090
$767$	0.870914	0.0314469
$768$	0	0
$769$	7.31376	0.263741	0.131870	−	0.991267i	$-0.457902\pi$
$769$	7.31376	0.263741	0.131870	+	0.991267i	$0.457902\pi$
$770$	0	0
$771$	0	0
$772$	27.1225	0.976161
$773$	−14.1844	−0.510178	−0.255089	−	0.966918i	$-0.582105\pi$
$773$	−14.1844	−0.510178	−0.255089	+	0.966918i	$0.582105\pi$
$774$	0	0
$775$	27.0774	0.972649
$776$	−14.0491	−0.504332
$777$	0	0
$778$	5.18943	0.186050
$779$	29.9510	1.07311
$780$	0	0
$781$	−37.7546	−1.35097
$782$	−7.66946	−0.274259
$783$	0	0
$784$	0	0
$785$	−21.6327	−0.772104
$786$	0	0
$787$	−31.2777	−1.11493	−0.557465	−	0.830201i	$-0.688226\pi$
$787$	−31.2777	−1.11493	−0.557465	+	0.830201i	$0.688226\pi$
$788$	−2.83509	−0.100996
$789$	0	0
$790$	−6.67360	−0.237436
$791$	0	0
$792$	0	0
$793$	−3.33253	−0.118342
$794$	−5.25640	−0.186543
$795$	0	0
$796$	−25.8176	−0.915082
$797$	20.2422	0.717017	0.358509	−	0.933526i	$-0.383285\pi$
$797$	20.2422	0.717017	0.358509	+	0.933526i	$0.383285\pi$
$798$	0	0
$799$	11.6383	0.411733
$800$	8.90500	0.314839
$801$	0	0
$802$	−3.99897	−0.141208
$803$	17.2930	0.610257
$804$	0	0
$805$	0	0
$806$	−2.43693	−0.0858373
$807$	0	0
$808$	−1.04836	−0.0368813
$809$	−7.88265	−0.277139	−0.138570	−	0.990353i	$-0.544250\pi$
$809$	−7.88265	−0.277139	−0.138570	+	0.990353i	$0.544250\pi$
$810$	0	0
$811$	5.99962	0.210675	0.105338	−	0.994437i	$-0.466408\pi$
$811$	5.99962	0.210675	0.105338	+	0.994437i	$0.466408\pi$
$812$	0	0
$813$	0	0
$814$	0.892226	0.0312725
$815$	34.1760	1.19713
$816$	0	0
$817$	23.7950	0.832480
$818$	−9.32659	−0.326096
$819$	0	0
$820$	−29.4142	−1.02719
$821$	19.1692	0.669011	0.334505	−	0.942394i	$-0.391431\pi$
$821$	19.1692	0.669011	0.334505	+	0.942394i	$0.391431\pi$
$822$	0	0
$823$	−30.3735	−1.05875	−0.529376	−	0.848387i	$-0.677574\pi$
$823$	−30.3735	−1.05875	−0.529376	+	0.848387i	$0.677574\pi$
$824$	13.2861	0.462845
$825$	0	0
$826$	0	0
$827$	14.6870	0.510717	0.255359	−	0.966846i	$-0.417807\pi$
$827$	14.6870	0.510717	0.255359	+	0.966846i	$0.417807\pi$
$828$	0	0
$829$	34.9985	1.21555	0.607774	−	0.794110i	$-0.292062\pi$
$829$	34.9985	1.21555	0.607774	+	0.794110i	$0.292062\pi$
$830$	4.34943	0.150971
$831$	0	0
$832$	6.36804	0.220772
$833$	0	0
$834$	0	0
$835$	10.2236	0.353803
$836$	−29.9635	−1.03631
$837$	0	0
$838$	−0.398178	−0.0137548
$839$	27.6333	0.954008	0.477004	−	0.878901i	$-0.341723\pi$
$839$	27.6333	0.954008	0.477004	+	0.878901i	$0.341723\pi$
$840$	0	0
$841$	−27.9145	−0.962568
$842$	6.48544	0.223503
$843$	0	0
$844$	−6.69785	−0.230550
$845$	−1.43515	−0.0493708
$846$	0	0
$847$	0	0
$848$	6.55859	0.225223
$849$	0	0
$850$	3.75874	0.128924
$851$	3.67472	0.125968
$852$	0	0
$853$	−32.6336	−1.11735	−0.558676	−	0.829386i	$-0.688690\pi$
$853$	−32.6336	−1.11735	−0.558676	+	0.829386i	$0.688690\pi$
$854$	0	0
$855$	0	0
$856$	0.712898	0.0243664
$857$	−18.8742	−0.644730	−0.322365	−	0.946616i	$-0.604478\pi$
$857$	−18.8742	−0.644730	−0.322365	+	0.946616i	$0.604478\pi$
$858$	0	0
$859$	−15.8242	−0.539915	−0.269957	−	0.962872i	$-0.587010\pi$
$859$	−15.8242	−0.539915	−0.269957	+	0.962872i	$0.587010\pi$
$860$	−23.3684	−0.796857
$861$	0	0
$862$	10.8670	0.370132
$863$	−52.3212	−1.78104	−0.890518	−	0.454948i	$-0.849658\pi$
$863$	−52.3212	−1.78104	−0.890518	+	0.454948i	$0.849658\pi$
$864$	0	0
$865$	−16.1264	−0.548313
$866$	1.76136	0.0598536
$867$	0	0
$868$	0	0
$869$	96.7309	3.28137
$870$	0	0
$871$	6.62741	0.224561
$872$	−3.01951	−0.102253
$873$	0	0
$874$	4.47773	0.151461
$875$	0	0
$876$	0	0
$877$	−54.0162	−1.82400	−0.911999	−	0.410193i	$-0.865461\pi$
$877$	−54.0162	−1.82400	−0.911999	+	0.410193i	$0.865461\pi$
$878$	2.17785	0.0734989
$879$	0	0
$880$	28.3200	0.954667
$881$	−42.0823	−1.41779	−0.708895	−	0.705314i	$-0.750806\pi$
$881$	−42.0823	−1.41779	−0.708895	+	0.705314i	$0.750806\pi$
$882$	0	0
$883$	36.5314	1.22938	0.614689	−	0.788769i	$-0.289281\pi$
$883$	36.5314	1.22938	0.614689	+	0.788769i	$0.289281\pi$
$884$	9.32316	0.313572
$885$	0	0
$886$	4.68007	0.157230
$887$	−11.4648	−0.384950	−0.192475	−	0.981302i	$-0.561651\pi$
$887$	−11.4648	−0.384950	−0.192475	+	0.981302i	$0.561651\pi$
$888$	0	0
$889$	0	0
$890$	−0.378001	−0.0126706
$891$	0	0
$892$	−19.1322	−0.640592
$893$	−6.79488	−0.227382
$894$	0	0
$895$	−18.9361	−0.632965
$896$	0	0
$897$	0	0
$898$	−3.84370	−0.128266
$899$	9.59462	0.319999
$900$	0	0
$901$	8.83823	0.294444
$902$	−15.4695	−0.515079
$903$	0	0
$904$	−12.5125	−0.416159
$905$	−19.7211	−0.655550
$906$	0	0
$907$	−5.04665	−0.167571	−0.0837856	−	0.996484i	$-0.526701\pi$
$907$	−5.04665	−0.167571	−0.0837856	+	0.996484i	$0.526701\pi$
$908$	−23.2999	−0.773236
$909$	0	0
$910$	0	0
$911$	−47.5236	−1.57453	−0.787263	−	0.616618i	$-0.788503\pi$
$911$	−47.5236	−1.57453	−0.787263	+	0.616618i	$0.788503\pi$
$912$	0	0
$913$	−63.0431	−2.08642
$914$	−1.00272	−0.0331671
$915$	0	0
$916$	7.81969	0.258370
$917$	0	0
$918$	0	0
$919$	−45.6698	−1.50651	−0.753254	−	0.657730i	$-0.771517\pi$
$919$	−45.6698	−1.50651	−0.753254	+	0.657730i	$0.771517\pi$
$920$	−8.95449	−0.295221
$921$	0	0
$922$	−3.46945	−0.114260
$923$	−6.85856	−0.225752
$924$	0	0
$925$	−1.80095	−0.0592148
$926$	−4.04950	−0.133075
$927$	0	0
$928$	3.15540	0.103581
$929$	−44.4449	−1.45819	−0.729095	−	0.684412i	$-0.760059\pi$
$929$	−44.4449	−1.45819	−0.729095	+	0.684412i	$0.760059\pi$
$930$	0	0
$931$	0	0
$932$	24.2115	0.793074
$933$	0	0
$934$	−8.14357	−0.266466
$935$	38.1634	1.24808
$936$	0	0
$937$	−46.9796	−1.53476	−0.767379	−	0.641194i	$-0.778439\pi$
$937$	−46.9796	−1.53476	−0.767379	+	0.641194i	$0.778439\pi$
$938$	0	0
$939$	0	0
$940$	6.67309	0.217652
$941$	35.5654	1.15940	0.579699	−	0.814830i	$-0.303170\pi$
$941$	35.5654	1.15940	0.579699	+	0.814830i	$0.303170\pi$
$942$	0	0
$943$	−63.7128	−2.07477
$944$	−3.12200	−0.101613
$945$	0	0
$946$	−12.2900	−0.399581
$947$	22.3592	0.726576	0.363288	−	0.931677i	$-0.381654\pi$
$947$	22.3592	0.726576	0.363288	+	0.931677i	$0.381654\pi$
$948$	0	0
$949$	3.14147	0.101977
$950$	−2.19450	−0.0711989
$951$	0	0
$952$	0	0
$953$	46.7684	1.51498	0.757488	−	0.652849i	$-0.226426\pi$
$953$	46.7684	1.51498	0.757488	+	0.652849i	$0.226426\pi$
$954$	0	0
$955$	−23.4371	−0.758406
$956$	25.7579	0.833071
$957$	0	0
$958$	0.454469	0.0146832
$959$	0	0
$960$	0	0
$961$	53.8048	1.73564
$962$	0.162083	0.00522577
$963$	0	0
$964$	39.3432	1.26716
$965$	20.1687	0.649253
$966$	0	0
$967$	8.22976	0.264651	0.132326	−	0.991206i	$-0.457756\pi$
$967$	8.22976	0.264651	0.132326	+	0.991206i	$0.457756\pi$
$968$	20.0738	0.645196
$969$	0	0
$970$	−5.13046	−0.164729
$971$	−23.6326	−0.758407	−0.379204	−	0.925313i	$-0.623802\pi$
$971$	−23.6326	−0.758407	−0.379204	+	0.925313i	$0.623802\pi$
$972$	0	0
$973$	0	0
$974$	−6.00187	−0.192312
$975$	0	0
$976$	11.9462	0.382390
$977$	26.6428	0.852379	0.426190	−	0.904634i	$-0.359856\pi$
$977$	26.6428	0.852379	0.426190	+	0.904634i	$0.359856\pi$
$978$	0	0
$979$	5.47897	0.175109
$980$	0	0
$981$	0	0
$982$	−3.47629	−0.110933
$983$	43.6302	1.39159	0.695793	−	0.718242i	$-0.255053\pi$
$983$	43.6302	1.39159	0.695793	+	0.718242i	$0.255053\pi$
$984$	0	0
$985$	−2.10821	−0.0671732
$986$	1.33187	0.0424155
$987$	0	0
$988$	−5.44322	−0.173172
$989$	−50.6174	−1.60954
$990$	0	0
$991$	−7.60816	−0.241681	−0.120841	−	0.992672i	$-0.538559\pi$
$991$	−7.60816	−0.241681	−0.120841	+	0.992672i	$0.538559\pi$
$992$	27.8899	0.885506
$993$	0	0
$994$	0	0
$995$	−19.1983	−0.608628
$996$	0	0
$997$	−19.4356	−0.615532	−0.307766	−	0.951462i	$-0.599581\pi$
$997$	−19.4356	−0.615532	−0.307766	+	0.951462i	$0.599581\pi$
$998$	−3.71523	−0.117604
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.bu.1.3		6
3.2	odd	2		637.2.a.m.1.4	✓	6
7.6	odd	2		5733.2.a.br.1.3		6
21.2	odd	6		637.2.e.o.508.3		12
21.5	even	6		637.2.e.n.508.3		12
21.11	odd	6		637.2.e.o.79.3		12
21.17	even	6		637.2.e.n.79.3		12
21.20	even	2		637.2.a.n.1.4	yes	6
39.38	odd	2		8281.2.a.cc.1.3		6
273.272	even	2		8281.2.a.cd.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.a.m.1.4	✓	6	3.2	odd	2
637.2.a.n.1.4	yes	6	21.20	even	2
637.2.e.n.79.3		12	21.17	even	6
637.2.e.n.508.3		12	21.5	even	6
637.2.e.o.79.3		12	21.11	odd	6
637.2.e.o.508.3		12	21.2	odd	6
5733.2.a.br.1.3		6	7.6	odd	2
5733.2.a.bu.1.3		6	1.1	even	1	trivial
8281.2.a.cc.1.3		6	39.38	odd	2
8281.2.a.cd.1.3		6	273.272	even	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)

\(f(q)\)	\(=\)	\(q-0.264627 q^{2} -1.92997 q^{4} -1.43515 q^{5} +1.03998 q^{8} +O(q^{10})\)	\(q-0.264627 q^{2} -1.92997 q^{4} -1.43515 q^{5} +1.03998 q^{8} +0.379780 q^{10} -5.50474 q^{11} -1.00000 q^{13} +3.58474 q^{16} +4.83072 q^{17} -2.82036 q^{19} +2.76981 q^{20} +1.45670 q^{22} +5.99956 q^{23} -2.94033 q^{25} +0.264627 q^{26} -1.04188 q^{29} -9.20895 q^{31} -3.02857 q^{32} -1.27834 q^{34} +0.612497 q^{37} +0.746342 q^{38} -1.49252 q^{40} -10.6196 q^{41} -8.43685 q^{43} +10.6240 q^{44} -1.58764 q^{46} +2.40922 q^{47} +0.778091 q^{50} +1.92997 q^{52} +1.82959 q^{53} +7.90015 q^{55} +0.275709 q^{58} -0.870914 q^{59} +3.33253 q^{61} +2.43693 q^{62} -6.36804 q^{64} +1.43515 q^{65} -6.62741 q^{67} -9.32316 q^{68} +6.85856 q^{71} -3.14147 q^{73} -0.162083 q^{74} +5.44322 q^{76} -17.5723 q^{79} -5.14465 q^{80} +2.81022 q^{82} +11.4525 q^{83} -6.93283 q^{85} +2.23261 q^{86} -5.72479 q^{88} -0.995318 q^{89} -11.5790 q^{92} -0.637545 q^{94} +4.04765 q^{95} -13.5090 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{4} + 6 q^{5}+O(q^{10}) \) 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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