LMFDB - Embedded newform 5733.2.a.y.1.2

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:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ 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)

$f(q)$	$=$	$q+0.347296 q^{2} -1.87939 q^{4} -0.652704 q^{5} -1.34730 q^{8} +O(q^{10})$	$q+0.347296 q^{2} -1.87939 q^{4} -0.652704 q^{5} -1.34730 q^{8} -0.226682 q^{10} -0.532089 q^{11} +1.00000 q^{13} +3.29086 q^{16} -1.12061 q^{17} +0.305407 q^{19} +1.22668 q^{20} -0.184793 q^{22} +8.00774 q^{23} -4.57398 q^{25} +0.347296 q^{26} -7.78106 q^{29} +0.588526 q^{31} +3.83750 q^{32} -0.389185 q^{34} -2.87939 q^{37} +0.106067 q^{38} +0.879385 q^{40} +1.04189 q^{41} +2.94356 q^{43} +1.00000 q^{44} +2.78106 q^{46} +4.12836 q^{47} -1.58853 q^{50} -1.87939 q^{52} +2.53209 q^{53} +0.347296 q^{55} -2.70233 q^{58} -14.3824 q^{59} +7.85978 q^{61} +0.204393 q^{62} -5.24897 q^{64} -0.652704 q^{65} +4.66044 q^{67} +2.10607 q^{68} +3.92902 q^{71} +11.6236 q^{73} -1.00000 q^{74} -0.573978 q^{76} +2.92902 q^{79} -2.14796 q^{80} +0.361844 q^{82} +2.46791 q^{83} +0.731429 q^{85} +1.02229 q^{86} +0.716881 q^{88} -10.9855 q^{89} -15.0496 q^{92} +1.43376 q^{94} -0.199340 q^{95} +10.2686 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^5 - 3 * q^8	$ 3 q - 3 q^{5} - 3 q^{8} + 6 q^{10} + 3 q^{11} + 3 q^{13} - 6 q^{16} - 9 q^{17} + 3 q^{19} - 3 q^{20} + 3 q^{22} - 6 q^{25} - 6 q^{29} + 12 q^{31} + 9 q^{32} + 3 q^{34} - 3 q^{37} - 12 q^{38} - 3 q^{40} - 6 q^{43} + 3 q^{44} - 9 q^{46} - 6 q^{47} - 15 q^{50} + 3 q^{53} + 18 q^{58} + 3 q^{59} + 15 q^{61} - 3 q^{64} - 3 q^{65} - 9 q^{67} - 6 q^{68} - 21 q^{71} - 3 q^{74} + 6 q^{76} - 24 q^{79} + 9 q^{80} + 18 q^{82} + 12 q^{83} + 12 q^{85} - 3 q^{86} - 6 q^{88} - 15 q^{89} - 18 q^{92} - 12 q^{94} - 15 q^{95} + 21 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 - 3 * q^8 + 6 * q^10 + 3 * q^11 + 3 * q^13 - 6 * q^16 - 9 * q^17 + 3 * q^19 - 3 * q^20 + 3 * q^22 - 6 * q^25 - 6 * q^29 + 12 * q^31 + 9 * q^32 + 3 * q^34 - 3 * q^37 - 12 * q^38 - 3 * q^40 - 6 * q^43 + 3 * q^44 - 9 * q^46 - 6 * q^47 - 15 * q^50 + 3 * q^53 + 18 * q^58 + 3 * q^59 + 15 * q^61 - 3 * q^64 - 3 * q^65 - 9 * q^67 - 6 * q^68 - 21 * q^71 - 3 * q^74 + 6 * q^76 - 24 * q^79 + 9 * q^80 + 18 * q^82 + 12 * q^83 + 12 * q^85 - 3 * q^86 - 6 * q^88 - 15 * q^89 - 18 * q^92 - 12 * q^94 - 15 * q^95 + 21 * 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 $

$2$	0.347296	0.245576	0.122788	−	0.992433i	$-0.460817\pi$
$2$	0.347296	0.245576	0.122788	+	0.992433i	$0.460817\pi$
$3$	0	0
$3$	0	0
$4$	−1.87939	−0.939693
$5$	−0.652704	−0.291898	−0.145949	−	0.989292i	$-0.546624\pi$
$5$	−0.652704	−0.291898	−0.145949	+	0.989292i	$0.546624\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.34730	−0.476341
$9$	0	0
$10$	−0.226682	−0.0716830
$11$	−0.532089	−0.160431	−0.0802154	−	0.996778i	$-0.525561\pi$
$11$	−0.532089	−0.160431	−0.0802154	+	0.996778i	$0.525561\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	3.29086	0.822715
$17$	−1.12061	−0.271789	−0.135895	−	0.990723i	$-0.543391\pi$
$17$	−1.12061	−0.271789	−0.135895	+	0.990723i	$0.543391\pi$
$18$	0	0
$19$	0.305407	0.0700652	0.0350326	−	0.999386i	$-0.488846\pi$
$19$	0.305407	0.0700652	0.0350326	+	0.999386i	$0.488846\pi$
$20$	1.22668	0.274294
$21$	0	0
$22$	−0.184793	−0.0393979
$23$	8.00774	1.66973	0.834865	−	0.550455i	$-0.185546\pi$
$23$	8.00774	1.66973	0.834865	+	0.550455i	$0.185546\pi$
$24$	0	0
$25$	−4.57398	−0.914796
$26$	0.347296	0.0681104
$27$	0	0
$28$	0	0
$29$	−7.78106	−1.44491	−0.722453	−	0.691420i	$-0.756986\pi$
$29$	−7.78106	−1.44491	−0.722453	+	0.691420i	$0.756986\pi$
$30$	0	0
$31$	0.588526	0.105702	0.0528512	−	0.998602i	$-0.483169\pi$
$31$	0.588526	0.105702	0.0528512	+	0.998602i	$0.483169\pi$
$32$	3.83750	0.678380
$33$	0	0
$34$	−0.389185	−0.0667447
$35$	0	0
$36$	0	0
$37$	−2.87939	−0.473368	−0.236684	−	0.971587i	$-0.576061\pi$
$37$	−2.87939	−0.473368	−0.236684	+	0.971587i	$0.576061\pi$
$38$	0.106067	0.0172063
$39$	0	0
$40$	0.879385	0.139043
$41$	1.04189	0.162716	0.0813579	−	0.996685i	$-0.474074\pi$
$41$	1.04189	0.162716	0.0813579	+	0.996685i	$0.474074\pi$
$42$	0	0
$43$	2.94356	0.448889	0.224445	−	0.974487i	$-0.427943\pi$
$43$	2.94356	0.448889	0.224445	+	0.974487i	$0.427943\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	2.78106	0.410045
$47$	4.12836	0.602183	0.301091	−	0.953595i	$-0.402649\pi$
$47$	4.12836	0.602183	0.301091	+	0.953595i	$0.402649\pi$
$48$	0	0
$49$	0	0
$50$	−1.58853	−0.224651
$51$	0	0
$52$	−1.87939	−0.260624
$53$	2.53209	0.347809	0.173905	−	0.984763i	$-0.444362\pi$
$53$	2.53209	0.347809	0.173905	+	0.984763i	$0.444362\pi$
$54$	0	0
$55$	0.347296	0.0468294
$56$	0	0
$57$	0	0
$58$	−2.70233	−0.354834
$59$	−14.3824	−1.87243	−0.936213	−	0.351433i	$-0.885695\pi$
$59$	−14.3824	−1.87243	−0.936213	+	0.351433i	$0.885695\pi$
$60$	0	0
$61$	7.85978	1.00634	0.503171	−	0.864187i	$-0.332167\pi$
$61$	7.85978	1.00634	0.503171	+	0.864187i	$0.332167\pi$
$62$	0.204393	0.0259579
$63$	0	0
$64$	−5.24897	−0.656121
$65$	−0.652704	−0.0809579
$66$	0	0
$67$	4.66044	0.569364	0.284682	−	0.958622i	$-0.408112\pi$
$67$	4.66044	0.569364	0.284682	+	0.958622i	$0.408112\pi$
$68$	2.10607	0.255398
$69$	0	0
$70$	0	0
$71$	3.92902	0.466288	0.233144	−	0.972442i	$-0.425099\pi$
$71$	3.92902	0.466288	0.233144	+	0.972442i	$0.425099\pi$
$72$	0	0
$73$	11.6236	1.36044	0.680220	−	0.733008i	$-0.261884\pi$
$73$	11.6236	1.36044	0.680220	+	0.733008i	$0.261884\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−0.573978	−0.0658398
$77$	0	0
$78$	0	0
$79$	2.92902	0.329540	0.164770	−	0.986332i	$-0.447312\pi$
$79$	2.92902	0.329540	0.164770	+	0.986332i	$0.447312\pi$
$80$	−2.14796	−0.240149
$81$	0	0
$82$	0.361844	0.0399590
$83$	2.46791	0.270888	0.135444	−	0.990785i	$-0.456754\pi$
$83$	2.46791	0.270888	0.135444	+	0.990785i	$0.456754\pi$
$84$	0	0
$85$	0.731429	0.0793347
$86$	1.02229	0.110236
$87$	0	0
$88$	0.716881	0.0764198
$89$	−10.9855	−1.16446	−0.582228	−	0.813026i	$-0.697819\pi$
$89$	−10.9855	−1.16446	−0.582228	+	0.813026i	$0.697819\pi$
$90$	0	0
$91$	0	0
$92$	−15.0496	−1.56903
$93$	0	0
$94$	1.43376	0.147881
$95$	−0.199340	−0.0204519
$96$	0	0
$97$	10.2686	1.04262	0.521308	−	0.853369i	$-0.325444\pi$
$97$	10.2686	1.04262	0.521308	+	0.853369i	$0.325444\pi$
$98$	0	0
$99$	0	0
$100$	8.59627	0.859627
$101$	−13.3473	−1.32811	−0.664053	−	0.747686i	$-0.731165\pi$
$101$	−13.3473	−1.32811	−0.664053	+	0.747686i	$0.731165\pi$
$102$	0	0
$103$	7.24123	0.713500	0.356750	−	0.934200i	$-0.383885\pi$
$103$	7.24123	0.713500	0.356750	+	0.934200i	$0.383885\pi$
$104$	−1.34730	−0.132113
$105$	0	0
$106$	0.879385	0.0854134
$107$	7.18479	0.694580	0.347290	−	0.937758i	$-0.387102\pi$
$107$	7.18479	0.694580	0.347290	+	0.937758i	$0.387102\pi$
$108$	0	0
$109$	−17.2422	−1.65150	−0.825750	−	0.564037i	$-0.809248\pi$
$109$	−17.2422	−1.65150	−0.825750	+	0.564037i	$0.809248\pi$
$110$	0.120615	0.0115002
$111$	0	0
$112$	0	0
$113$	−11.3523	−1.06794	−0.533970	−	0.845504i	$-0.679300\pi$
$113$	−11.3523	−1.06794	−0.533970	+	0.845504i	$0.679300\pi$
$114$	0	0
$115$	−5.22668	−0.487391
$116$	14.6236	1.35777
$117$	0	0
$118$	−4.99495	−0.459822
$119$	0	0
$120$	0	0
$121$	−10.7169	−0.974262
$122$	2.72967	0.247133
$123$	0	0
$124$	−1.10607	−0.0993277
$125$	6.24897	0.558925
$126$	0	0
$127$	−3.55169	−0.315161	−0.157581	−	0.987506i	$-0.550369\pi$
$127$	−3.55169	−0.315161	−0.157581	+	0.987506i	$0.550369\pi$
$128$	−9.49794	−0.839507
$129$	0	0
$130$	−0.226682	−0.0198813
$131$	−15.5449	−1.35816	−0.679081	−	0.734063i	$-0.737622\pi$
$131$	−15.5449	−1.35816	−0.679081	+	0.734063i	$0.737622\pi$
$132$	0	0
$133$	0	0
$134$	1.61856	0.139822
$135$	0	0
$136$	1.50980	0.129464
$137$	−8.12061	−0.693791	−0.346895	−	0.937904i	$-0.612764\pi$
$137$	−8.12061	−0.693791	−0.346895	+	0.937904i	$0.612764\pi$
$138$	0	0
$139$	16.0223	1.35899	0.679496	−	0.733679i	$-0.262198\pi$
$139$	16.0223	1.35899	0.679496	+	0.733679i	$0.262198\pi$
$140$	0	0
$141$	0	0
$142$	1.36453	0.114509
$143$	−0.532089	−0.0444955
$144$	0	0
$145$	5.07873	0.421765
$146$	4.03684	0.334091
$147$	0	0
$148$	5.41147	0.444820
$149$	−23.8871	−1.95691	−0.978455	−	0.206461i	$-0.933805\pi$
$149$	−23.8871	−1.95691	−0.978455	+	0.206461i	$0.933805\pi$
$150$	0	0
$151$	−4.90673	−0.399304	−0.199652	−	0.979867i	$-0.563981\pi$
$151$	−4.90673	−0.399304	−0.199652	+	0.979867i	$0.563981\pi$
$152$	−0.411474	−0.0333750
$153$	0	0
$154$	0	0
$155$	−0.384133	−0.0308543
$156$	0	0
$157$	−6.90941	−0.551431	−0.275716	−	0.961239i	$-0.588915\pi$
$157$	−6.90941	−0.551431	−0.275716	+	0.961239i	$0.588915\pi$
$158$	1.01724	0.0809270
$159$	0	0
$160$	−2.50475	−0.198018
$161$	0	0
$162$	0	0
$163$	−14.3277	−1.12223	−0.561116	−	0.827737i	$-0.689628\pi$
$163$	−14.3277	−1.12223	−0.561116	+	0.827737i	$0.689628\pi$
$164$	−1.95811	−0.152903
$165$	0	0
$166$	0.857097	0.0665236
$167$	11.4979	0.889737	0.444869	−	0.895596i	$-0.353250\pi$
$167$	11.4979	0.889737	0.444869	+	0.895596i	$0.353250\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0.254023	0.0194827
$171$	0	0
$172$	−5.53209	−0.421818
$173$	−15.4243	−1.17269	−0.586343	−	0.810063i	$-0.699433\pi$
$173$	−15.4243	−1.17269	−0.586343	+	0.810063i	$0.699433\pi$
$174$	0	0
$175$	0	0
$176$	−1.75103	−0.131989
$177$	0	0
$178$	−3.81521	−0.285962
$179$	−25.1489	−1.87972	−0.939858	−	0.341565i	$-0.889043\pi$
$179$	−25.1489	−1.87972	−0.939858	+	0.341565i	$0.889043\pi$
$180$	0	0
$181$	13.4584	1.00036	0.500178	−	0.865923i	$-0.333268\pi$
$181$	13.4584	1.00036	0.500178	+	0.865923i	$0.333268\pi$
$182$	0	0
$183$	0	0
$184$	−10.7888	−0.795361
$185$	1.87939	0.138175
$186$	0	0
$187$	0.596267	0.0436033
$188$	−7.75877	−0.565866
$189$	0	0
$190$	−0.0692302	−0.00502249
$191$	−16.7074	−1.20890	−0.604452	−	0.796642i	$-0.706608\pi$
$191$	−16.7074	−1.20890	−0.604452	+	0.796642i	$0.706608\pi$
$192$	0	0
$193$	18.2618	1.31451	0.657255	−	0.753668i	$-0.271717\pi$
$193$	18.2618	1.31451	0.657255	+	0.753668i	$0.271717\pi$
$194$	3.56624	0.256041
$195$	0	0
$196$	0	0
$197$	−17.1702	−1.22333	−0.611665	−	0.791117i	$-0.709500\pi$
$197$	−17.1702	−1.22333	−0.611665	+	0.791117i	$0.709500\pi$
$198$	0	0
$199$	−5.07873	−0.360021	−0.180011	−	0.983665i	$-0.557613\pi$
$199$	−5.07873	−0.360021	−0.180011	+	0.983665i	$0.557613\pi$
$200$	6.16250	0.435755
$201$	0	0
$202$	−4.63547	−0.326150
$203$	0	0
$204$	0	0
$205$	−0.680045	−0.0474964
$206$	2.51485	0.175218
$207$	0	0
$208$	3.29086	0.228180
$209$	−0.162504	−0.0112406
$210$	0	0
$211$	−23.4243	−1.61259	−0.806297	−	0.591512i	$-0.798531\pi$
$211$	−23.4243	−1.61259	−0.806297	+	0.591512i	$0.798531\pi$
$212$	−4.75877	−0.326834
$213$	0	0
$214$	2.49525	0.170572
$215$	−1.92127	−0.131030
$216$	0	0
$217$	0	0
$218$	−5.98814	−0.405568
$219$	0	0
$220$	−0.652704	−0.0440053
$221$	−1.12061	−0.0753807
$222$	0	0
$223$	−12.5202	−0.838417	−0.419208	−	0.907890i	$-0.637692\pi$
$223$	−12.5202	−0.838417	−0.419208	+	0.907890i	$0.637692\pi$
$224$	0	0
$225$	0	0
$226$	−3.94263	−0.262260
$227$	−7.80335	−0.517926	−0.258963	−	0.965887i	$-0.583381\pi$
$227$	−7.80335	−0.517926	−0.258963	+	0.965887i	$0.583381\pi$
$228$	0	0
$229$	−20.6459	−1.36432	−0.682160	−	0.731203i	$-0.738959\pi$
$229$	−20.6459	−1.36432	−0.682160	+	0.731203i	$0.738959\pi$
$230$	−1.81521	−0.119691
$231$	0	0
$232$	10.4834	0.688268
$233$	−9.61856	−0.630133	−0.315066	−	0.949070i	$-0.602027\pi$
$233$	−9.61856	−0.630133	−0.315066	+	0.949070i	$0.602027\pi$
$234$	0	0
$235$	−2.69459	−0.175776
$236$	27.0300	1.75951
$237$	0	0
$238$	0	0
$239$	−2.35235	−0.152161	−0.0760804	−	0.997102i	$-0.524241\pi$
$239$	−2.35235	−0.152161	−0.0760804	+	0.997102i	$0.524241\pi$
$240$	0	0
$241$	−15.7324	−1.01341	−0.506705	−	0.862119i	$-0.669137\pi$
$241$	−15.7324	−1.01341	−0.506705	+	0.862119i	$0.669137\pi$
$242$	−3.72193	−0.239255
$243$	0	0
$244$	−14.7716	−0.945652
$245$	0	0
$246$	0	0
$247$	0.305407	0.0194326
$248$	−0.792919	−0.0503504
$249$	0	0
$250$	2.17024	0.137258
$251$	21.8161	1.37702	0.688511	−	0.725226i	$-0.258265\pi$
$251$	21.8161	1.37702	0.688511	+	0.725226i	$0.258265\pi$
$252$	0	0
$253$	−4.26083	−0.267876
$254$	−1.23349	−0.0773960
$255$	0	0
$256$	7.19934	0.449959
$257$	23.2772	1.45199	0.725997	−	0.687698i	$-0.241378\pi$
$257$	23.2772	1.45199	0.725997	+	0.687698i	$0.241378\pi$
$258$	0	0
$259$	0	0
$260$	1.22668	0.0760756
$261$	0	0
$262$	−5.39868	−0.333532
$263$	−2.75103	−0.169636	−0.0848179	−	0.996396i	$-0.527031\pi$
$263$	−2.75103	−0.169636	−0.0848179	+	0.996396i	$0.527031\pi$
$264$	0	0
$265$	−1.65270	−0.101525
$266$	0	0
$267$	0	0
$268$	−8.75877	−0.535027
$269$	−5.42333	−0.330666	−0.165333	−	0.986238i	$-0.552870\pi$
$269$	−5.42333	−0.330666	−0.165333	+	0.986238i	$0.552870\pi$
$270$	0	0
$271$	1.61587	0.0981569	0.0490785	−	0.998795i	$-0.484372\pi$
$271$	1.61587	0.0981569	0.0490785	+	0.998795i	$0.484372\pi$
$272$	−3.68779	−0.223605
$273$	0	0
$274$	−2.82026	−0.170378
$275$	2.43376	0.146761
$276$	0	0
$277$	7.85029	0.471678	0.235839	−	0.971792i	$-0.424216\pi$
$277$	7.85029	0.471678	0.235839	+	0.971792i	$0.424216\pi$
$278$	5.56448	0.333735
$279$	0	0
$280$	0	0
$281$	19.0428	1.13600	0.568000	−	0.823029i	$-0.307717\pi$
$281$	19.0428	1.13600	0.568000	+	0.823029i	$0.307717\pi$
$282$	0	0
$283$	23.2867	1.38425	0.692127	−	0.721776i	$-0.256674\pi$
$283$	23.2867	1.38425	0.692127	+	0.721776i	$0.256674\pi$
$284$	−7.38413	−0.438168
$285$	0	0
$286$	−0.184793	−0.0109270
$287$	0	0
$288$	0	0
$289$	−15.7442	−0.926131
$290$	1.76382	0.103575
$291$	0	0
$292$	−21.8452	−1.27840
$293$	−26.9736	−1.57581	−0.787907	−	0.615794i	$-0.788835\pi$
$293$	−26.9736	−1.57581	−0.787907	+	0.615794i	$0.788835\pi$
$294$	0	0
$295$	9.38743	0.546557
$296$	3.87939	0.225485
$297$	0	0
$298$	−8.29591	−0.480569
$299$	8.00774	0.463100
$300$	0	0
$301$	0	0
$302$	−1.70409	−0.0980593
$303$	0	0
$304$	1.00505	0.0576437
$305$	−5.13011	−0.293749
$306$	0	0
$307$	6.28405	0.358650	0.179325	−	0.983790i	$-0.442609\pi$
$307$	6.28405	0.358650	0.179325	+	0.983790i	$0.442609\pi$
$308$	0	0
$309$	0	0
$310$	−0.133408	−0.00757706
$311$	−2.22668	−0.126264	−0.0631318	−	0.998005i	$-0.520109\pi$
$311$	−2.22668	−0.126264	−0.0631318	+	0.998005i	$0.520109\pi$
$312$	0	0
$313$	19.8212	1.12036	0.560180	−	0.828371i	$-0.310732\pi$
$313$	19.8212	1.12036	0.560180	+	0.828371i	$0.310732\pi$
$314$	−2.39961	−0.135418
$315$	0	0
$316$	−5.50475	−0.309666
$317$	11.7023	0.657269	0.328634	−	0.944457i	$-0.393412\pi$
$317$	11.7023	0.657269	0.328634	+	0.944457i	$0.393412\pi$
$318$	0	0
$319$	4.14022	0.231808
$320$	3.42602	0.191520
$321$	0	0
$322$	0	0
$323$	−0.342244	−0.0190430
$324$	0	0
$325$	−4.57398	−0.253719
$326$	−4.97596	−0.275593
$327$	0	0
$328$	−1.40373	−0.0775082
$329$	0	0
$330$	0	0
$331$	20.9118	1.14942	0.574708	−	0.818359i	$-0.305116\pi$
$331$	20.9118	1.14942	0.574708	+	0.818359i	$0.305116\pi$
$332$	−4.63816	−0.254552
$333$	0	0
$334$	3.99319	0.218498
$335$	−3.04189	−0.166196
$336$	0	0
$337$	−16.0419	−0.873857	−0.436929	−	0.899496i	$-0.643934\pi$
$337$	−16.0419	−0.873857	−0.436929	+	0.899496i	$0.643934\pi$
$338$	0.347296	0.0188904
$339$	0	0
$340$	−1.37464	−0.0745502
$341$	−0.313148	−0.0169579
$342$	0	0
$343$	0	0
$344$	−3.96585	−0.213824
$345$	0	0
$346$	−5.35679	−0.287983
$347$	−18.5868	−0.997790	−0.498895	−	0.866662i	$-0.666261\pi$
$347$	−18.5868	−0.997790	−0.498895	+	0.866662i	$0.666261\pi$
$348$	0	0
$349$	15.2267	0.815066	0.407533	−	0.913191i	$-0.366389\pi$
$349$	15.2267	0.815066	0.407533	+	0.913191i	$0.366389\pi$
$350$	0	0
$351$	0	0
$352$	−2.04189	−0.108833
$353$	5.90941	0.314526	0.157263	−	0.987557i	$-0.449733\pi$
$353$	5.90941	0.314526	0.157263	+	0.987557i	$0.449733\pi$
$354$	0	0
$355$	−2.56448	−0.136109
$356$	20.6459	1.09423
$357$	0	0
$358$	−8.73412	−0.461612
$359$	−10.3901	−0.548370	−0.274185	−	0.961677i	$-0.588408\pi$
$359$	−10.3901	−0.548370	−0.274185	+	0.961677i	$0.588408\pi$
$360$	0	0
$361$	−18.9067	−0.995091
$362$	4.67406	0.245663
$363$	0	0
$364$	0	0
$365$	−7.58677	−0.397110
$366$	0	0
$367$	−24.1803	−1.26220	−0.631102	−	0.775700i	$-0.717397\pi$
$367$	−24.1803	−1.26220	−0.631102	+	0.775700i	$0.717397\pi$
$368$	26.3523	1.37371
$369$	0	0
$370$	0.652704	0.0339324
$371$	0	0
$372$	0	0
$373$	6.50206	0.336664	0.168332	−	0.985730i	$-0.446162\pi$
$373$	6.50206	0.336664	0.168332	+	0.985730i	$0.446162\pi$
$374$	0.207081	0.0107079
$375$	0	0
$376$	−5.56212	−0.286844
$377$	−7.78106	−0.400745
$378$	0	0
$379$	−17.7915	−0.913887	−0.456944	−	0.889496i	$-0.651056\pi$
$379$	−17.7915	−0.913887	−0.456944	+	0.889496i	$0.651056\pi$
$380$	0.374638	0.0192185
$381$	0	0
$382$	−5.80241	−0.296877
$383$	9.96316	0.509094	0.254547	−	0.967060i	$-0.418074\pi$
$383$	9.96316	0.509094	0.254547	+	0.967060i	$0.418074\pi$
$384$	0	0
$385$	0	0
$386$	6.34224	0.322812
$387$	0	0
$388$	−19.2986	−0.979738
$389$	3.90167	0.197823	0.0989114	−	0.995096i	$-0.468464\pi$
$389$	3.90167	0.197823	0.0989114	+	0.995096i	$0.468464\pi$
$390$	0	0
$391$	−8.97359	−0.453814
$392$	0	0
$393$	0	0
$394$	−5.96316	−0.300420
$395$	−1.91178	−0.0961920
$396$	0	0
$397$	−33.8999	−1.70139	−0.850694	−	0.525661i	$-0.823818\pi$
$397$	−33.8999	−1.70139	−0.850694	+	0.525661i	$0.823818\pi$
$398$	−1.76382	−0.0884125
$399$	0	0
$400$	−15.0523	−0.752616
$401$	−32.8580	−1.64085	−0.820426	−	0.571753i	$-0.806264\pi$
$401$	−32.8580	−1.64085	−0.820426	+	0.571753i	$0.806264\pi$
$402$	0	0
$403$	0.588526	0.0293166
$404$	25.0847	1.24801
$405$	0	0
$406$	0	0
$407$	1.53209	0.0759428
$408$	0	0
$409$	3.63816	0.179895	0.0899476	−	0.995946i	$-0.471330\pi$
$409$	3.63816	0.179895	0.0899476	+	0.995946i	$0.471330\pi$
$410$	−0.236177	−0.0116640
$411$	0	0
$412$	−13.6091	−0.670470
$413$	0	0
$414$	0	0
$415$	−1.61081	−0.0790718
$416$	3.83750	0.188149
$417$	0	0
$418$	−0.0564370	−0.00276042
$419$	−16.6905	−0.815383	−0.407692	−	0.913120i	$-0.633666\pi$
$419$	−16.6905	−0.815383	−0.407692	+	0.913120i	$0.633666\pi$
$420$	0	0
$421$	−21.5425	−1.04992	−0.524959	−	0.851127i	$-0.675919\pi$
$421$	−21.5425	−1.04992	−0.524959	+	0.851127i	$0.675919\pi$
$422$	−8.13516	−0.396014
$423$	0	0
$424$	−3.41147	−0.165676
$425$	5.12567	0.248631
$426$	0	0
$427$	0	0
$428$	−13.5030	−0.652692
$429$	0	0
$430$	−0.667252	−0.0321777
$431$	−10.3473	−0.498412	−0.249206	−	0.968451i	$-0.580170\pi$
$431$	−10.3473	−0.498412	−0.249206	+	0.968451i	$0.580170\pi$
$432$	0	0
$433$	−3.22399	−0.154935	−0.0774676	−	0.996995i	$-0.524683\pi$
$433$	−3.22399	−0.154935	−0.0774676	+	0.996995i	$0.524683\pi$
$434$	0	0
$435$	0	0
$436$	32.4047	1.55190
$437$	2.44562	0.116990
$438$	0	0
$439$	28.4989	1.36018	0.680089	−	0.733130i	$-0.261941\pi$
$439$	28.4989	1.36018	0.680089	+	0.733130i	$0.261941\pi$
$440$	−0.467911	−0.0223068
$441$	0	0
$442$	−0.389185	−0.0185117
$443$	21.1334	1.00408	0.502039	−	0.864845i	$-0.332583\pi$
$443$	21.1334	1.00408	0.502039	+	0.864845i	$0.332583\pi$
$444$	0	0
$445$	7.17024	0.339902
$446$	−4.34823	−0.205895
$447$	0	0
$448$	0	0
$449$	−25.6468	−1.21035	−0.605174	−	0.796093i	$-0.706897\pi$
$449$	−25.6468	−1.21035	−0.605174	+	0.796093i	$0.706897\pi$
$450$	0	0
$451$	−0.554378	−0.0261046
$452$	21.3354	1.00353
$453$	0	0
$454$	−2.71007	−0.127190
$455$	0	0
$456$	0	0
$457$	33.6587	1.57449	0.787244	−	0.616642i	$-0.211507\pi$
$457$	33.6587	1.57449	0.787244	+	0.616642i	$0.211507\pi$
$458$	−7.17024	−0.335044
$459$	0	0
$460$	9.82295	0.457997
$461$	−9.85710	−0.459091	−0.229545	−	0.973298i	$-0.573724\pi$
$461$	−9.85710	−0.459091	−0.229545	+	0.973298i	$0.573724\pi$
$462$	0	0
$463$	2.39693	0.111395	0.0556973	−	0.998448i	$-0.482262\pi$
$463$	2.39693	0.111395	0.0556973	+	0.998448i	$0.482262\pi$
$464$	−25.6064	−1.18875
$465$	0	0
$466$	−3.34049	−0.154745
$467$	27.2790	1.26232	0.631161	−	0.775652i	$-0.282579\pi$
$467$	27.2790	1.26232	0.631161	+	0.775652i	$0.282579\pi$
$468$	0	0
$469$	0	0
$470$	−0.935822	−0.0431663
$471$	0	0
$472$	19.3773	0.891914
$473$	−1.56624	−0.0720157
$474$	0	0
$475$	−1.39693	−0.0640954
$476$	0	0
$477$	0	0
$478$	−0.816962	−0.0373670
$479$	40.8212	1.86517	0.932584	−	0.360953i	$-0.117549\pi$
$479$	40.8212	1.86517	0.932584	+	0.360953i	$0.117549\pi$
$480$	0	0
$481$	−2.87939	−0.131289
$482$	−5.46379	−0.248869
$483$	0	0
$484$	20.1411	0.915507
$485$	−6.70233	−0.304337
$486$	0	0
$487$	−19.2276	−0.871286	−0.435643	−	0.900119i	$-0.643479\pi$
$487$	−19.2276	−0.871286	−0.435643	+	0.900119i	$0.643479\pi$
$488$	−10.5895	−0.479362
$489$	0	0
$490$	0	0
$491$	7.54219	0.340374	0.170187	−	0.985412i	$-0.445563\pi$
$491$	7.54219	0.340374	0.170187	+	0.985412i	$0.445563\pi$
$492$	0	0
$493$	8.71957	0.392710
$494$	0.106067	0.00477217
$495$	0	0
$496$	1.93676	0.0869629
$497$	0	0
$498$	0	0
$499$	−30.9959	−1.38757	−0.693783	−	0.720184i	$-0.744057\pi$
$499$	−30.9959	−1.38757	−0.693783	+	0.720184i	$0.744057\pi$
$500$	−11.7442	−0.525218
$501$	0	0
$502$	7.57667	0.338163
$503$	−18.4861	−0.824254	−0.412127	−	0.911126i	$-0.635214\pi$
$503$	−18.4861	−0.824254	−0.412127	+	0.911126i	$0.635214\pi$
$504$	0	0
$505$	8.71183	0.387671
$506$	−1.47977	−0.0657838
$507$	0	0
$508$	6.67499	0.296155
$509$	36.1735	1.60336	0.801682	−	0.597751i	$-0.203939\pi$
$509$	36.1735	1.60336	0.801682	+	0.597751i	$0.203939\pi$
$510$	0	0
$511$	0	0
$512$	21.4962	0.950006
$513$	0	0
$514$	8.08410	0.356574
$515$	−4.72638	−0.208269
$516$	0	0
$517$	−2.19665	−0.0966086
$518$	0	0
$519$	0	0
$520$	0.879385	0.0385636
$521$	−26.4861	−1.16038	−0.580188	−	0.814482i	$-0.697021\pi$
$521$	−26.4861	−1.16038	−0.580188	+	0.814482i	$0.697021\pi$
$522$	0	0
$523$	−10.9959	−0.480816	−0.240408	−	0.970672i	$-0.577281\pi$
$523$	−10.9959	−0.480816	−0.240408	+	0.970672i	$0.577281\pi$
$524$	29.2148	1.27626
$525$	0	0
$526$	−0.955423	−0.0416584
$527$	−0.659511	−0.0287287
$528$	0	0
$529$	41.1239	1.78800
$530$	−0.573978	−0.0249320
$531$	0	0
$532$	0	0
$533$	1.04189	0.0451292
$534$	0	0
$535$	−4.68954	−0.202747
$536$	−6.27900	−0.271211
$537$	0	0
$538$	−1.88350	−0.0812036
$539$	0	0
$540$	0	0
$541$	−3.38507	−0.145535	−0.0727677	−	0.997349i	$-0.523183\pi$
$541$	−3.38507	−0.145535	−0.0727677	+	0.997349i	$0.523183\pi$
$542$	0.561185	0.0241049
$543$	0	0
$544$	−4.30035	−0.184376
$545$	11.2540	0.482069
$546$	0	0
$547$	−8.37464	−0.358074	−0.179037	−	0.983842i	$-0.557298\pi$
$547$	−8.37464	−0.358074	−0.179037	+	0.983842i	$0.557298\pi$
$548$	15.2618	0.651950
$549$	0	0
$550$	0.845237	0.0360410
$551$	−2.37639	−0.101238
$552$	0	0
$553$	0	0
$554$	2.72638	0.115833
$555$	0	0
$556$	−30.1121	−1.27704
$557$	−36.3655	−1.54085	−0.770427	−	0.637528i	$-0.779957\pi$
$557$	−36.3655	−1.54085	−0.770427	+	0.637528i	$0.779957\pi$
$558$	0	0
$559$	2.94356	0.124499
$560$	0	0
$561$	0	0
$562$	6.61350	0.278974
$563$	−5.24628	−0.221104	−0.110552	−	0.993870i	$-0.535262\pi$
$563$	−5.24628	−0.221104	−0.110552	+	0.993870i	$0.535262\pi$
$564$	0	0
$565$	7.40972	0.311729
$566$	8.08740	0.339939
$567$	0	0
$568$	−5.29355	−0.222112
$569$	21.4858	0.900730	0.450365	−	0.892845i	$-0.351294\pi$
$569$	21.4858	0.900730	0.450365	+	0.892845i	$0.351294\pi$
$570$	0	0
$571$	33.1607	1.38773	0.693867	−	0.720103i	$-0.255906\pi$
$571$	33.1607	1.38773	0.693867	+	0.720103i	$0.255906\pi$
$572$	1.00000	0.0418121
$573$	0	0
$574$	0	0
$575$	−36.6272	−1.52746
$576$	0	0
$577$	27.1780	1.13143	0.565717	−	0.824599i	$-0.308599\pi$
$577$	27.1780	1.13143	0.565717	+	0.824599i	$0.308599\pi$
$578$	−5.46791	−0.227435
$579$	0	0
$580$	−9.54488	−0.396330
$581$	0	0
$582$	0	0
$583$	−1.34730	−0.0557993
$584$	−15.6604	−0.648034
$585$	0	0
$586$	−9.36783	−0.386982
$587$	30.8280	1.27241	0.636204	−	0.771521i	$-0.280504\pi$
$587$	30.8280	1.27241	0.636204	+	0.771521i	$0.280504\pi$
$588$	0	0
$589$	0.179740	0.00740606
$590$	3.26022	0.134221
$591$	0	0
$592$	−9.47565	−0.389447
$593$	−40.0729	−1.64559	−0.822797	−	0.568335i	$-0.807588\pi$
$593$	−40.0729	−1.64559	−0.822797	+	0.568335i	$0.807588\pi$
$594$	0	0
$595$	0	0
$596$	44.8931	1.83889
$597$	0	0
$598$	2.78106	0.113726
$599$	−30.8452	−1.26030	−0.630151	−	0.776473i	$-0.717007\pi$
$599$	−30.8452	−1.26030	−0.630151	+	0.776473i	$0.717007\pi$
$600$	0	0
$601$	−9.85803	−0.402117	−0.201059	−	0.979579i	$-0.564438\pi$
$601$	−9.85803	−0.402117	−0.201059	+	0.979579i	$0.564438\pi$
$602$	0	0
$603$	0	0
$604$	9.22163	0.375223
$605$	6.99495	0.284385
$606$	0	0
$607$	−31.3286	−1.27159	−0.635795	−	0.771858i	$-0.719328\pi$
$607$	−31.3286	−1.27159	−0.635795	+	0.771858i	$0.719328\pi$
$608$	1.17200	0.0475308
$609$	0	0
$610$	−1.78167	−0.0721377
$611$	4.12836	0.167015
$612$	0	0
$613$	−4.61856	−0.186542	−0.0932708	−	0.995641i	$-0.529732\pi$
$613$	−4.61856	−0.186542	−0.0932708	+	0.995641i	$0.529732\pi$
$614$	2.18243	0.0880756
$615$	0	0
$616$	0	0
$617$	14.5648	0.586357	0.293179	−	0.956058i	$-0.405287\pi$
$617$	14.5648	0.586357	0.293179	+	0.956058i	$0.405287\pi$
$618$	0	0
$619$	−49.4133	−1.98609	−0.993045	−	0.117736i	$-0.962436\pi$
$619$	−49.4133	−1.98609	−0.993045	+	0.117736i	$0.962436\pi$
$620$	0.721934	0.0289936
$621$	0	0
$622$	−0.773318	−0.0310072
$623$	0	0
$624$	0	0
$625$	18.7912	0.751647
$626$	6.88383	0.275133
$627$	0	0
$628$	12.9855	0.518176
$629$	3.22668	0.128656
$630$	0	0
$631$	0.529401	0.0210751	0.0105376	−	0.999944i	$-0.496646\pi$
$631$	0.529401	0.0210751	0.0105376	+	0.999944i	$0.496646\pi$
$632$	−3.94625	−0.156973
$633$	0	0
$634$	4.06418	0.161409
$635$	2.31820	0.0919950
$636$	0	0
$637$	0	0
$638$	1.43788	0.0569263
$639$	0	0
$640$	6.19934	0.245050
$641$	32.1111	1.26831	0.634156	−	0.773205i	$-0.281347\pi$
$641$	32.1111	1.26831	0.634156	+	0.773205i	$0.281347\pi$
$642$	0	0
$643$	23.5740	0.929667	0.464833	−	0.885398i	$-0.346114\pi$
$643$	23.5740	0.929667	0.464833	+	0.885398i	$0.346114\pi$
$644$	0	0
$645$	0	0
$646$	−0.118860	−0.00467649
$647$	15.7520	0.619274	0.309637	−	0.950855i	$-0.399793\pi$
$647$	15.7520	0.619274	0.309637	+	0.950855i	$0.399793\pi$
$648$	0	0
$649$	7.65270	0.300395
$650$	−1.58853	−0.0623071
$651$	0	0
$652$	26.9273	1.05455
$653$	31.0675	1.21576	0.607882	−	0.794027i	$-0.292019\pi$
$653$	31.0675	1.21576	0.607882	+	0.794027i	$0.292019\pi$
$654$	0	0
$655$	10.1462	0.396445
$656$	3.42871	0.133869
$657$	0	0
$658$	0	0
$659$	−19.9162	−0.775826	−0.387913	−	0.921696i	$-0.626804\pi$
$659$	−19.9162	−0.775826	−0.387913	+	0.921696i	$0.626804\pi$
$660$	0	0
$661$	24.2722	0.944079	0.472039	−	0.881577i	$-0.343518\pi$
$661$	24.2722	0.944079	0.472039	+	0.881577i	$0.343518\pi$
$662$	7.26258	0.282268
$663$	0	0
$664$	−3.32501	−0.129035
$665$	0	0
$666$	0	0
$667$	−62.3087	−2.41260
$668$	−21.6091	−0.836080
$669$	0	0
$670$	−1.05644	−0.0408137
$671$	−4.18210	−0.161448
$672$	0	0
$673$	23.2668	0.896870	0.448435	−	0.893815i	$-0.351982\pi$
$673$	23.2668	0.896870	0.448435	+	0.893815i	$0.351982\pi$
$674$	−5.57129	−0.214598
$675$	0	0
$676$	−1.87939	−0.0722840
$677$	−22.0496	−0.847436	−0.423718	−	0.905794i	$-0.639275\pi$
$677$	−22.0496	−0.847436	−0.423718	+	0.905794i	$0.639275\pi$
$678$	0	0
$679$	0	0
$680$	−0.985452	−0.0377904
$681$	0	0
$682$	−0.108755	−0.00416445
$683$	7.47659	0.286084	0.143042	−	0.989717i	$-0.454312\pi$
$683$	7.47659	0.286084	0.143042	+	0.989717i	$0.454312\pi$
$684$	0	0
$685$	5.30035	0.202516
$686$	0	0
$687$	0	0
$688$	9.68685	0.369308
$689$	2.53209	0.0964649
$690$	0	0
$691$	11.3327	0.431118	0.215559	−	0.976491i	$-0.430843\pi$
$691$	11.3327	0.431118	0.215559	+	0.976491i	$0.430843\pi$
$692$	28.9881	1.10196
$693$	0	0
$694$	−6.45512	−0.245033
$695$	−10.4578	−0.396687
$696$	0	0
$697$	−1.16756	−0.0442243
$698$	5.28817	0.200160
$699$	0	0
$700$	0	0
$701$	15.8135	0.597266	0.298633	−	0.954368i	$-0.403469\pi$
$701$	15.8135	0.597266	0.298633	+	0.954368i	$0.403469\pi$
$702$	0	0
$703$	−0.879385	−0.0331666
$704$	2.79292	0.105262
$705$	0	0
$706$	2.05232	0.0772400
$707$	0	0
$708$	0	0
$709$	42.3236	1.58950	0.794748	−	0.606940i	$-0.207603\pi$
$709$	42.3236	1.58950	0.794748	+	0.606940i	$0.207603\pi$
$710$	−0.890635	−0.0334250
$711$	0	0
$712$	14.8007	0.554678
$713$	4.71276	0.176494
$714$	0	0
$715$	0.347296	0.0129881
$716$	47.2645	1.76636
$717$	0	0
$718$	−3.60845	−0.134666
$719$	−27.5740	−1.02834	−0.514168	−	0.857690i	$-0.671899\pi$
$719$	−27.5740	−1.02834	−0.514168	+	0.857690i	$0.671899\pi$
$720$	0	0
$721$	0	0
$722$	−6.56624	−0.244370
$723$	0	0
$724$	−25.2935	−0.940027
$725$	35.5904	1.32179
$726$	0	0
$727$	20.5844	0.763433	0.381717	−	0.924279i	$-0.375333\pi$
$727$	20.5844	0.763433	0.381717	+	0.924279i	$0.375333\pi$
$728$	0	0
$729$	0	0
$730$	−2.63486	−0.0975205
$731$	−3.29860	−0.122003
$732$	0	0
$733$	−11.6162	−0.429054	−0.214527	−	0.976718i	$-0.568821\pi$
$733$	−11.6162	−0.429054	−0.214527	+	0.976718i	$0.568821\pi$
$734$	−8.39775	−0.309967
$735$	0	0
$736$	30.7297	1.13271
$737$	−2.47977	−0.0913435
$738$	0	0
$739$	−8.61350	−0.316853	−0.158426	−	0.987371i	$-0.550642\pi$
$739$	−8.61350	−0.316853	−0.158426	+	0.987371i	$0.550642\pi$
$740$	−3.53209	−0.129842
$741$	0	0
$742$	0	0
$743$	−23.9828	−0.879842	−0.439921	−	0.898036i	$-0.644994\pi$
$743$	−23.9828	−0.879842	−0.439921	+	0.898036i	$0.644994\pi$
$744$	0	0
$745$	15.5912	0.571218
$746$	2.25814	0.0826764
$747$	0	0
$748$	−1.12061	−0.0409737
$749$	0	0
$750$	0	0
$751$	−9.12298	−0.332902	−0.166451	−	0.986050i	$-0.553231\pi$
$751$	−9.12298	−0.332902	−0.166451	+	0.986050i	$0.553231\pi$
$752$	13.5858	0.495425
$753$	0	0
$754$	−2.70233	−0.0984132
$755$	3.20264	0.116556
$756$	0	0
$757$	−26.4979	−0.963084	−0.481542	−	0.876423i	$-0.659923\pi$
$757$	−26.4979	−0.963084	−0.481542	+	0.876423i	$0.659923\pi$
$758$	−6.17892	−0.224428
$759$	0	0
$760$	0.268571	0.00974208
$761$	−52.2995	−1.89586	−0.947928	−	0.318484i	$-0.896826\pi$
$761$	−52.2995	−1.89586	−0.947928	+	0.318484i	$0.896826\pi$
$762$	0	0
$763$	0	0
$764$	31.3996	1.13600
$765$	0	0
$766$	3.46017	0.125021
$767$	−14.3824	−0.519318
$768$	0	0
$769$	44.9436	1.62071	0.810353	−	0.585942i	$-0.199275\pi$
$769$	44.9436	1.62071	0.810353	+	0.585942i	$0.199275\pi$
$770$	0	0
$771$	0	0
$772$	−34.3209	−1.23524
$773$	43.2559	1.55581	0.777903	−	0.628384i	$-0.216283\pi$
$773$	43.2559	1.55581	0.777903	+	0.628384i	$0.216283\pi$
$774$	0	0
$775$	−2.69190	−0.0966961
$776$	−13.8348	−0.496641
$777$	0	0
$778$	1.35504	0.0485804
$779$	0.318201	0.0114007
$780$	0	0
$781$	−2.09059	−0.0748070
$782$	−3.11650	−0.111446
$783$	0	0
$784$	0	0
$785$	4.50980	0.160962
$786$	0	0
$787$	−37.3523	−1.33147	−0.665734	−	0.746189i	$-0.731881\pi$
$787$	−37.3523	−1.33147	−0.665734	+	0.746189i	$0.731881\pi$
$788$	32.2695	1.14955
$789$	0	0
$790$	−0.663954	−0.0236224
$791$	0	0
$792$	0	0
$793$	7.85978	0.279109
$794$	−11.7733	−0.417819
$795$	0	0
$796$	9.54488	0.338309
$797$	50.5740	1.79142	0.895711	−	0.444636i	$-0.146667\pi$
$797$	50.5740	1.79142	0.895711	+	0.444636i	$0.146667\pi$
$798$	0	0
$799$	−4.62630	−0.163667
$800$	−17.5526	−0.620579
$801$	0	0
$802$	−11.4115	−0.402953
$803$	−6.18479	−0.218257
$804$	0	0
$805$	0	0
$806$	0.204393	0.00719943
$807$	0	0
$808$	17.9828	0.632631
$809$	50.9600	1.79166	0.895829	−	0.444400i	$-0.146583\pi$
$809$	50.9600	1.79166	0.895829	+	0.444400i	$0.146583\pi$
$810$	0	0
$811$	−42.1334	−1.47950	−0.739752	−	0.672880i	$-0.765057\pi$
$811$	−42.1334	−1.47950	−0.739752	+	0.672880i	$0.765057\pi$
$812$	0	0
$813$	0	0
$814$	0.532089	0.0186497
$815$	9.35174	0.327577
$816$	0	0
$817$	0.898986	0.0314515
$818$	1.26352	0.0441779
$819$	0	0
$820$	1.27807	0.0446320
$821$	−20.8794	−0.728696	−0.364348	−	0.931263i	$-0.618708\pi$
$821$	−20.8794	−0.728696	−0.364348	+	0.931263i	$0.618708\pi$
$822$	0	0
$823$	−11.3928	−0.397128	−0.198564	−	0.980088i	$-0.563628\pi$
$823$	−11.3928	−0.397128	−0.198564	+	0.980088i	$0.563628\pi$
$824$	−9.75608	−0.339869
$825$	0	0
$826$	0	0
$827$	−3.45842	−0.120261	−0.0601304	−	0.998191i	$-0.519152\pi$
$827$	−3.45842	−0.120261	−0.0601304	+	0.998191i	$0.519152\pi$
$828$	0	0
$829$	15.4970	0.538233	0.269117	−	0.963108i	$-0.413268\pi$
$829$	15.4970	0.538233	0.269117	+	0.963108i	$0.413268\pi$
$830$	−0.559430	−0.0194181
$831$	0	0
$832$	−5.24897	−0.181975
$833$	0	0
$834$	0	0
$835$	−7.50475	−0.259713
$836$	0.305407	0.0105627
$837$	0	0
$838$	−5.79654	−0.200238
$839$	−42.1385	−1.45478	−0.727391	−	0.686224i	$-0.759267\pi$
$839$	−42.1385	−1.45478	−0.727391	+	0.686224i	$0.759267\pi$
$840$	0	0
$841$	31.5449	1.08775
$842$	−7.48164	−0.257834
$843$	0	0
$844$	44.0232	1.51534
$845$	−0.652704	−0.0224537
$846$	0	0
$847$	0	0
$848$	8.33275	0.286148
$849$	0	0
$850$	1.78013	0.0610578
$851$	−23.0574	−0.790396
$852$	0	0
$853$	−28.6736	−0.981764	−0.490882	−	0.871226i	$-0.663325\pi$
$853$	−28.6736	−0.981764	−0.490882	+	0.871226i	$0.663325\pi$
$854$	0	0
$855$	0	0
$856$	−9.68004	−0.330857
$857$	−1.06181	−0.0362709	−0.0181354	−	0.999836i	$-0.505773\pi$
$857$	−1.06181	−0.0362709	−0.0181354	+	0.999836i	$0.505773\pi$
$858$	0	0
$859$	11.4970	0.392273	0.196136	−	0.980577i	$-0.437160\pi$
$859$	11.4970	0.392273	0.196136	+	0.980577i	$0.437160\pi$
$860$	3.61081	0.123128
$861$	0	0
$862$	−3.59358	−0.122398
$863$	52.2026	1.77700	0.888499	−	0.458878i	$-0.151749\pi$
$863$	52.2026	1.77700	0.888499	+	0.458878i	$0.151749\pi$
$864$	0	0
$865$	10.0675	0.342304
$866$	−1.11968	−0.0380483
$867$	0	0
$868$	0	0
$869$	−1.55850	−0.0528684
$870$	0	0
$871$	4.66044	0.157913
$872$	23.2303	0.786677
$873$	0	0
$874$	0.849356	0.0287299
$875$	0	0
$876$	0	0
$877$	46.6509	1.57529	0.787645	−	0.616129i	$-0.211300\pi$
$877$	46.6509	1.57529	0.787645	+	0.616129i	$0.211300\pi$
$878$	9.89756	0.334026
$879$	0	0
$880$	1.14290	0.0385273
$881$	−18.9691	−0.639087	−0.319543	−	0.947572i	$-0.603530\pi$
$881$	−18.9691	−0.639087	−0.319543	+	0.947572i	$0.603530\pi$
$882$	0	0
$883$	0.744223	0.0250451	0.0125225	−	0.999922i	$-0.496014\pi$
$883$	0.744223	0.0250451	0.0125225	+	0.999922i	$0.496014\pi$
$884$	2.10607	0.0708347
$885$	0	0
$886$	7.33956	0.246577
$887$	−32.3432	−1.08598	−0.542989	−	0.839740i	$-0.682707\pi$
$887$	−32.3432	−1.08598	−0.542989	+	0.839740i	$0.682707\pi$
$888$	0	0
$889$	0	0
$890$	2.49020	0.0834717
$891$	0	0
$892$	23.5303	0.787854
$893$	1.26083	0.0421921
$894$	0	0
$895$	16.4148	0.548685
$896$	0	0
$897$	0	0
$898$	−8.90705	−0.297232
$899$	−4.57935	−0.152730
$900$	0	0
$901$	−2.83750	−0.0945307
$902$	−0.192533	−0.00641066
$903$	0	0
$904$	15.2950	0.508703
$905$	−8.78436	−0.292002
$906$	0	0
$907$	−51.9050	−1.72348	−0.861738	−	0.507353i	$-0.830624\pi$
$907$	−51.9050	−1.72348	−0.861738	+	0.507353i	$0.830624\pi$
$908$	14.6655	0.486692
$909$	0	0
$910$	0	0
$911$	31.0324	1.02815	0.514075	−	0.857746i	$-0.328135\pi$
$911$	31.0324	1.02815	0.514075	+	0.857746i	$0.328135\pi$
$912$	0	0
$913$	−1.31315	−0.0434589
$914$	11.6895	0.386656
$915$	0	0
$916$	38.8016	1.28204
$917$	0	0
$918$	0	0
$919$	3.39187	0.111888	0.0559438	−	0.998434i	$-0.482183\pi$
$919$	3.39187	0.111888	0.0559438	+	0.998434i	$0.482183\pi$
$920$	7.04189	0.232164
$921$	0	0
$922$	−3.42333	−0.112741
$923$	3.92902	0.129325
$924$	0	0
$925$	13.1702	0.433035
$926$	0.832444	0.0273558
$927$	0	0
$928$	−29.8598	−0.980195
$929$	−48.9026	−1.60444	−0.802221	−	0.597027i	$-0.796349\pi$
$929$	−48.9026	−1.60444	−0.802221	+	0.597027i	$0.796349\pi$
$930$	0	0
$931$	0	0
$932$	18.0770	0.592131
$933$	0	0
$934$	9.47390	0.309995
$935$	−0.389185	−0.0127277
$936$	0	0
$937$	−12.9676	−0.423633	−0.211817	−	0.977309i	$-0.567938\pi$
$937$	−12.9676	−0.423633	−0.211817	+	0.977309i	$0.567938\pi$
$938$	0	0
$939$	0	0
$940$	5.06418	0.165175
$941$	−27.0906	−0.883128	−0.441564	−	0.897230i	$-0.645576\pi$
$941$	−27.0906	−0.883128	−0.441564	+	0.897230i	$0.645576\pi$
$942$	0	0
$943$	8.34318	0.271691
$944$	−47.3304	−1.54047
$945$	0	0
$946$	−0.543948	−0.0176853
$947$	−21.4587	−0.697315	−0.348658	−	0.937250i	$-0.613362\pi$
$947$	−21.4587	−0.697315	−0.348658	+	0.937250i	$0.613362\pi$
$948$	0	0
$949$	11.6236	0.377318
$950$	−0.485147	−0.0157403
$951$	0	0
$952$	0	0
$953$	28.1147	0.910726	0.455363	−	0.890306i	$-0.349510\pi$
$953$	28.1147	0.910726	0.455363	+	0.890306i	$0.349510\pi$
$954$	0	0
$955$	10.9050	0.352877
$956$	4.42097	0.142984
$957$	0	0
$958$	14.1771	0.458040
$959$	0	0
$960$	0	0
$961$	−30.6536	−0.988827
$962$	−1.00000	−0.0322413
$963$	0	0
$964$	29.5672	0.952294
$965$	−11.9195	−0.383703
$966$	0	0
$967$	43.7006	1.40532	0.702658	−	0.711528i	$-0.251996\pi$
$967$	43.7006	1.40532	0.702658	+	0.711528i	$0.251996\pi$
$968$	14.4388	0.464081
$969$	0	0
$970$	−2.32770	−0.0747378
$971$	37.1884	1.19343	0.596717	−	0.802452i	$-0.296472\pi$
$971$	37.1884	1.19343	0.596717	+	0.802452i	$0.296472\pi$
$972$	0	0
$973$	0	0
$974$	−6.67768	−0.213967
$975$	0	0
$976$	25.8654	0.827933
$977$	22.4406	0.717937	0.358969	−	0.933350i	$-0.383128\pi$
$977$	22.4406	0.717937	0.358969	+	0.933350i	$0.383128\pi$
$978$	0	0
$979$	5.84524	0.186815
$980$	0	0
$981$	0	0
$982$	2.61938	0.0835877
$983$	23.3122	0.743544	0.371772	−	0.928324i	$-0.378750\pi$
$983$	23.3122	0.743544	0.371772	+	0.928324i	$0.378750\pi$
$984$	0	0
$985$	11.2071	0.357087
$986$	3.02827	0.0964399
$987$	0	0
$988$	−0.573978	−0.0182607
$989$	23.5713	0.749523
$990$	0	0
$991$	5.28169	0.167778	0.0838892	−	0.996475i	$-0.473266\pi$
$991$	5.28169	0.167778	0.0838892	+	0.996475i	$0.473266\pi$
$992$	2.25847	0.0717064
$993$	0	0
$994$	0	0
$995$	3.31490	0.105089
$996$	0	0
$997$	28.9695	0.917472	0.458736	−	0.888572i	$-0.348302\pi$
$997$	28.9695	0.917472	0.458736	+	0.888572i	$0.348302\pi$
$998$	−10.7648	−0.340752
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.y.1.2		3
3.2	odd	2		1911.2.a.o.1.2		3
7.2	even	3		819.2.j.e.235.2		6
7.4	even	3		819.2.j.e.352.2		6
7.6	odd	2		5733.2.a.z.1.2		3
21.2	odd	6		273.2.i.b.235.2	yes	6
21.11	odd	6		273.2.i.b.79.2	✓	6
21.20	even	2		1911.2.a.p.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.i.b.79.2	✓	6	21.11	odd	6
273.2.i.b.235.2	yes	6	21.2	odd	6
819.2.j.e.235.2		6	7.2	even	3
819.2.j.e.352.2		6	7.4	even	3
1911.2.a.o.1.2		3	3.2	odd	2
1911.2.a.p.1.2		3	21.20	even	2
5733.2.a.y.1.2		3	1.1	even	1	trivial
5733.2.a.z.1.2		3	7.6	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)

\(f(q)\)	\(=\)	\(q+0.347296 q^{2} -1.87939 q^{4} -0.652704 q^{5} -1.34730 q^{8} +O(q^{10})\)	\(q+0.347296 q^{2} -1.87939 q^{4} -0.652704 q^{5} -1.34730 q^{8} -0.226682 q^{10} -0.532089 q^{11} +1.00000 q^{13} +3.29086 q^{16} -1.12061 q^{17} +0.305407 q^{19} +1.22668 q^{20} -0.184793 q^{22} +8.00774 q^{23} -4.57398 q^{25} +0.347296 q^{26} -7.78106 q^{29} +0.588526 q^{31} +3.83750 q^{32} -0.389185 q^{34} -2.87939 q^{37} +0.106067 q^{38} +0.879385 q^{40} +1.04189 q^{41} +2.94356 q^{43} +1.00000 q^{44} +2.78106 q^{46} +4.12836 q^{47} -1.58853 q^{50} -1.87939 q^{52} +2.53209 q^{53} +0.347296 q^{55} -2.70233 q^{58} -14.3824 q^{59} +7.85978 q^{61} +0.204393 q^{62} -5.24897 q^{64} -0.652704 q^{65} +4.66044 q^{67} +2.10607 q^{68} +3.92902 q^{71} +11.6236 q^{73} -1.00000 q^{74} -0.573978 q^{76} +2.92902 q^{79} -2.14796 q^{80} +0.361844 q^{82} +2.46791 q^{83} +0.731429 q^{85} +1.02229 q^{86} +0.716881 q^{88} -10.9855 q^{89} -15.0496 q^{92} +1.43376 q^{94} -0.199340 q^{95} +10.2686 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^5 - 3 * q^8	\( 3 q - 3 q^{5} - 3 q^{8} + 6 q^{10} + 3 q^{11} + 3 q^{13} - 6 q^{16} - 9 q^{17} + 3 q^{19} - 3 q^{20} + 3 q^{22} - 6 q^{25} - 6 q^{29} + 12 q^{31} + 9 q^{32} + 3 q^{34} - 3 q^{37} - 12 q^{38} - 3 q^{40} - 6 q^{43} + 3 q^{44} - 9 q^{46} - 6 q^{47} - 15 q^{50} + 3 q^{53} + 18 q^{58} + 3 q^{59} + 15 q^{61} - 3 q^{64} - 3 q^{65} - 9 q^{67} - 6 q^{68} - 21 q^{71} - 3 q^{74} + 6 q^{76} - 24 q^{79} + 9 q^{80} + 18 q^{82} + 12 q^{83} + 12 q^{85} - 3 q^{86} - 6 q^{88} - 15 q^{89} - 18 q^{92} - 12 q^{94} - 15 q^{95} + 21 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 - 3 * q^8 + 6 * q^10 + 3 * q^11 + 3 * q^13 - 6 * q^16 - 9 * q^17 + 3 * q^19 - 3 * q^20 + 3 * q^22 - 6 * q^25 - 6 * q^29 + 12 * q^31 + 9 * q^32 + 3 * q^34 - 3 * q^37 - 12 * q^38 - 3 * q^40 - 6 * q^43 + 3 * q^44 - 9 * q^46 - 6 * q^47 - 15 * q^50 + 3 * q^53 + 18 * q^58 + 3 * q^59 + 15 * q^61 - 3 * q^64 - 3 * q^65 - 9 * q^67 - 6 * q^68 - 21 * q^71 - 3 * q^74 + 6 * q^76 - 24 * q^79 + 9 * q^80 + 18 * q^82 + 12 * q^83 + 12 * q^85 - 3 * q^86 - 6 * q^88 - 15 * q^89 - 18 * q^92 - 12 * q^94 - 15 * q^95 + 21 * q^97

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