LMFDB - Embedded newform 6003.2.a.t.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, 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("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.78641 q^{2} +1.19125 q^{4} -0.326188 q^{5} +1.17242 q^{7} +1.44475 q^{8} +O(q^{10})$	\(q-1.78641 q^{2} +1.19125 q^{4} -0.326188 q^{5} +1.17242 q^{7} +1.44475 q^{8} +0.582705 q^{10} +0.852209 q^{11} +3.22671 q^{13} -2.09442 q^{14} -4.96342 q^{16} +5.35744 q^{17} -1.49277 q^{19} -0.388572 q^{20} -1.52239 q^{22} +1.00000 q^{23} -4.89360 q^{25} -5.76422 q^{26} +1.39665 q^{28} +1.00000 q^{29} +0.352535 q^{31} +5.97719 q^{32} -9.57057 q^{34} -0.382430 q^{35} -10.9308 q^{37} +2.66670 q^{38} -0.471261 q^{40} -11.6880 q^{41} -3.16001 q^{43} +1.01520 q^{44} -1.78641 q^{46} +1.44410 q^{47} -5.62543 q^{49} +8.74197 q^{50} +3.84383 q^{52} +6.05996 q^{53} -0.277980 q^{55} +1.69386 q^{56} -1.78641 q^{58} +1.56212 q^{59} -6.48678 q^{61} -0.629771 q^{62} -0.750853 q^{64} -1.05251 q^{65} -6.67740 q^{67} +6.38206 q^{68} +0.683176 q^{70} -13.6764 q^{71} +12.6053 q^{73} +19.5269 q^{74} -1.77827 q^{76} +0.999148 q^{77} -15.9954 q^{79} +1.61901 q^{80} +20.8795 q^{82} -4.84681 q^{83} -1.74753 q^{85} +5.64507 q^{86} +1.23123 q^{88} -7.47777 q^{89} +3.78307 q^{91} +1.19125 q^{92} -2.57975 q^{94} +0.486925 q^{95} +3.68119 q^{97} +10.0493 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34} + 2 q^{35} - 28 q^{37} + 14 q^{38} - 30 q^{40} - 10 q^{41} - 14 q^{43} + 37 q^{44} - 3 q^{46} - 18 q^{47} + 2 q^{49} + 7 q^{50} - 57 q^{52} + 20 q^{53} - 42 q^{55} - 2 q^{56} - 3 q^{58} - 20 q^{59} - 38 q^{61} + 4 q^{62} - 24 q^{64} + 12 q^{65} - 50 q^{67} + 11 q^{68} - 48 q^{70} + 12 q^{71} - 46 q^{73} - 6 q^{74} - 16 q^{76} - 14 q^{77} - 20 q^{79} - 58 q^{80} - 42 q^{82} + 22 q^{83} - 66 q^{85} + 22 q^{86} - 68 q^{88} - 14 q^{89} - 16 q^{91} + 17 q^{92} - 27 q^{94} - 20 q^{95} - 48 q^{97} - 28 q^{98}+O(q^{100}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.78641	−1.26318	−0.631590	−	0.775302i	$-0.717598\pi$
$2$	−1.78641	−1.26318	−0.631590	+	0.775302i	$0.717598\pi$
$3$	0	0
$3$	0	0
$4$	1.19125	0.595626
$5$	−0.326188	−0.145876	−0.0729378	−	0.997336i	$-0.523237\pi$
$5$	−0.326188	−0.145876	−0.0729378	+	0.997336i	$0.523237\pi$
$6$	0	0
$7$	1.17242	0.443134	0.221567	−	0.975145i	$-0.428883\pi$
$7$	1.17242	0.443134	0.221567	+	0.975145i	$0.428883\pi$
$8$	1.44475	0.510797
$9$	0	0
$10$	0.582705	0.184267
$11$	0.852209	0.256951	0.128475	−	0.991713i	$-0.458992\pi$
$11$	0.852209	0.256951	0.128475	+	0.991713i	$0.458992\pi$
$12$	0	0
$13$	3.22671	0.894929	0.447464	−	0.894302i	$-0.352327\pi$
$13$	3.22671	0.894929	0.447464	+	0.894302i	$0.352327\pi$
$14$	−2.09442	−0.559758
$15$	0	0
$16$	−4.96342	−1.24086
$17$	5.35744	1.29937	0.649685	−	0.760203i	$-0.274901\pi$
$17$	5.35744	1.29937	0.649685	+	0.760203i	$0.274901\pi$
$18$	0	0
$19$	−1.49277	−0.342466	−0.171233	−	0.985231i	$-0.554775\pi$
$19$	−1.49277	−0.342466	−0.171233	+	0.985231i	$0.554775\pi$
$20$	−0.388572	−0.0868874
$21$	0	0
$22$	−1.52239	−0.324575
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.89360	−0.978720
$26$	−5.76422	−1.13046
$27$	0	0
$28$	1.39665	0.263942
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	0.352535	0.0633171	0.0316586	−	0.999499i	$-0.489921\pi$
$31$	0.352535	0.0633171	0.0316586	+	0.999499i	$0.489921\pi$
$32$	5.97719	1.05663
$33$	0	0
$34$	−9.57057	−1.64134
$35$	−0.382430	−0.0646424
$36$	0	0
$37$	−10.9308	−1.79702	−0.898509	−	0.438955i	$-0.855349\pi$
$37$	−10.9308	−1.79702	−0.898509	+	0.438955i	$0.855349\pi$
$38$	2.66670	0.432596
$39$	0	0
$40$	−0.471261	−0.0745129
$41$	−11.6880	−1.82535	−0.912677	−	0.408682i	$-0.865988\pi$
$41$	−11.6880	−1.82535	−0.912677	+	0.408682i	$0.865988\pi$
$42$	0	0
$43$	−3.16001	−0.481897	−0.240948	−	0.970538i	$-0.577458\pi$
$43$	−3.16001	−0.481897	−0.240948	+	0.970538i	$0.577458\pi$
$44$	1.01520	0.153047
$45$	0	0
$46$	−1.78641	−0.263391
$47$	1.44410	0.210643	0.105322	−	0.994438i	$-0.466413\pi$
$47$	1.44410	0.210643	0.105322	+	0.994438i	$0.466413\pi$
$48$	0	0
$49$	−5.62543	−0.803632
$50$	8.74197	1.23630
$51$	0	0
$52$	3.84383	0.533043
$53$	6.05996	0.832400	0.416200	−	0.909273i	$-0.363362\pi$
$53$	6.05996	0.832400	0.416200	+	0.909273i	$0.363362\pi$
$54$	0	0
$55$	−0.277980	−0.0374828
$56$	1.69386	0.226352
$57$	0	0
$58$	−1.78641	−0.234567
$59$	1.56212	0.203370	0.101685	−	0.994817i	$-0.467577\pi$
$59$	1.56212	0.203370	0.101685	+	0.994817i	$0.467577\pi$
$60$	0	0
$61$	−6.48678	−0.830546	−0.415273	−	0.909697i	$-0.636314\pi$
$61$	−6.48678	−0.830546	−0.415273	+	0.909697i	$0.636314\pi$
$62$	−0.629771	−0.0799810
$63$	0	0
$64$	−0.750853	−0.0938566
$65$	−1.05251	−0.130548
$66$	0	0
$67$	−6.67740	−0.815774	−0.407887	−	0.913032i	$-0.633734\pi$
$67$	−6.67740	−0.815774	−0.407887	+	0.913032i	$0.633734\pi$
$68$	6.38206	0.773939
$69$	0	0
$70$	0.683176	0.0816551
$71$	−13.6764	−1.62309	−0.811545	−	0.584291i	$-0.801373\pi$
$71$	−13.6764	−1.62309	−0.811545	+	0.584291i	$0.801373\pi$
$72$	0	0
$73$	12.6053	1.47534	0.737669	−	0.675162i	$-0.235926\pi$
$73$	12.6053	1.47534	0.737669	+	0.675162i	$0.235926\pi$
$74$	19.5269	2.26996
$75$	0	0
$76$	−1.77827	−0.203982
$77$	0.999148	0.113864
$78$	0	0
$79$	−15.9954	−1.79962	−0.899809	−	0.436284i	$-0.856294\pi$
$79$	−15.9954	−1.79962	−0.899809	+	0.436284i	$0.856294\pi$
$80$	1.61901	0.181011
$81$	0	0
$82$	20.8795	2.30575
$83$	−4.84681	−0.532006	−0.266003	−	0.963972i	$-0.585703\pi$
$83$	−4.84681	−0.532006	−0.266003	+	0.963972i	$0.585703\pi$
$84$	0	0
$85$	−1.74753	−0.189546
$86$	5.64507	0.608723
$87$	0	0
$88$	1.23123	0.131250
$89$	−7.47777	−0.792642	−0.396321	−	0.918112i	$-0.629713\pi$
$89$	−7.47777	−0.792642	−0.396321	+	0.918112i	$0.629713\pi$
$90$	0	0
$91$	3.78307	0.396573
$92$	1.19125	0.124197
$93$	0	0
$94$	−2.57975	−0.266081
$95$	0.486925	0.0499574
$96$	0	0
$97$	3.68119	0.373769	0.186884	−	0.982382i	$-0.440161\pi$
$97$	3.68119	0.373769	0.186884	+	0.982382i	$0.440161\pi$
$98$	10.0493	1.01513
$99$	0	0
$100$	−5.82951	−0.582951
$101$	9.80422	0.975556	0.487778	−	0.872968i	$-0.337808\pi$
$101$	9.80422	0.975556	0.487778	+	0.872968i	$0.337808\pi$
$102$	0	0
$103$	−0.857878	−0.0845292	−0.0422646	−	0.999106i	$-0.513457\pi$
$103$	−0.857878	−0.0845292	−0.0422646	+	0.999106i	$0.513457\pi$
$104$	4.66180	0.457127
$105$	0	0
$106$	−10.8256	−1.05147
$107$	12.8882	1.24595	0.622977	−	0.782240i	$-0.285923\pi$
$107$	12.8882	1.24595	0.622977	+	0.782240i	$0.285923\pi$
$108$	0	0
$109$	−4.06867	−0.389708	−0.194854	−	0.980832i	$-0.562423\pi$
$109$	−4.06867	−0.389708	−0.194854	+	0.980832i	$0.562423\pi$
$110$	0.496586	0.0473476
$111$	0	0
$112$	−5.81923	−0.549865
$113$	−5.77823	−0.543570	−0.271785	−	0.962358i	$-0.587614\pi$
$113$	−5.77823	−0.543570	−0.271785	+	0.962358i	$0.587614\pi$
$114$	0	0
$115$	−0.326188	−0.0304172
$116$	1.19125	0.110605
$117$	0	0
$118$	−2.79057	−0.256893
$119$	6.28118	0.575795
$120$	0	0
$121$	−10.2737	−0.933976
$122$	11.5880	1.04913
$123$	0	0
$124$	0.419958	0.0377133
$125$	3.22717	0.288647
$126$	0	0
$127$	1.81622	0.161164	0.0805818	−	0.996748i	$-0.474322\pi$
$127$	1.81622	0.161164	0.0805818	+	0.996748i	$0.474322\pi$
$128$	−10.6131	−0.938070
$129$	0	0
$130$	1.88022	0.164906
$131$	−3.89276	−0.340112	−0.170056	−	0.985434i	$-0.554395\pi$
$131$	−3.89276	−0.340112	−0.170056	+	0.985434i	$0.554395\pi$
$132$	0	0
$133$	−1.75016	−0.151758
$134$	11.9286	1.03047
$135$	0	0
$136$	7.74018	0.663715
$137$	3.15056	0.269171	0.134585	−	0.990902i	$-0.457030\pi$
$137$	3.15056	0.269171	0.134585	+	0.990902i	$0.457030\pi$
$138$	0	0
$139$	−1.43304	−0.121549	−0.0607743	−	0.998152i	$-0.519357\pi$
$139$	−1.43304	−0.121549	−0.0607743	+	0.998152i	$0.519357\pi$
$140$	−0.455570	−0.0385027
$141$	0	0
$142$	24.4316	2.05026
$143$	2.74983	0.229952
$144$	0	0
$145$	−0.326188	−0.0270884
$146$	−22.5182	−1.86362
$147$	0	0
$148$	−13.0214	−1.07035
$149$	−0.913677	−0.0748513	−0.0374257	−	0.999299i	$-0.511916\pi$
$149$	−0.913677	−0.0748513	−0.0374257	+	0.999299i	$0.511916\pi$
$150$	0	0
$151$	−16.4830	−1.34137	−0.670684	−	0.741743i	$-0.733999\pi$
$151$	−16.4830	−1.34137	−0.670684	+	0.741743i	$0.733999\pi$
$152$	−2.15669	−0.174931
$153$	0	0
$154$	−1.78489	−0.143830
$155$	−0.114993	−0.00923643
$156$	0	0
$157$	3.94251	0.314647	0.157323	−	0.987547i	$-0.449713\pi$
$157$	3.94251	0.314647	0.157323	+	0.987547i	$0.449713\pi$
$158$	28.5742	2.27324
$159$	0	0
$160$	−1.94969	−0.154136
$161$	1.17242	0.0923998
$162$	0	0
$163$	−9.76256	−0.764663	−0.382331	−	0.924025i	$-0.624879\pi$
$163$	−9.76256	−0.764663	−0.382331	+	0.924025i	$0.624879\pi$
$164$	−13.9233	−1.08723
$165$	0	0
$166$	8.65837	0.672020
$167$	21.4326	1.65851	0.829254	−	0.558872i	$-0.188766\pi$
$167$	21.4326	1.65851	0.829254	+	0.558872i	$0.188766\pi$
$168$	0	0
$169$	−2.58834	−0.199103
$170$	3.12180	0.239432
$171$	0	0
$172$	−3.76437	−0.287030
$173$	−13.4151	−1.01993	−0.509967	−	0.860194i	$-0.670342\pi$
$173$	−13.4151	−1.01993	−0.509967	+	0.860194i	$0.670342\pi$
$174$	0	0
$175$	−5.73737	−0.433704
$176$	−4.22987	−0.318839
$177$	0	0
$178$	13.3583	1.00125
$179$	0.550115	0.0411175	0.0205588	−	0.999789i	$-0.493455\pi$
$179$	0.550115	0.0411175	0.0205588	+	0.999789i	$0.493455\pi$
$180$	0	0
$181$	7.93736	0.589979	0.294990	−	0.955500i	$-0.404684\pi$
$181$	7.93736	0.589979	0.294990	+	0.955500i	$0.404684\pi$
$182$	−6.75810	−0.500944
$183$	0	0
$184$	1.44475	0.106509
$185$	3.56551	0.262141
$186$	0	0
$187$	4.56566	0.333874
$188$	1.72029	0.125465
$189$	0	0
$190$	−0.869846	−0.0631052
$191$	1.69390	0.122566	0.0612831	−	0.998120i	$-0.480481\pi$
$191$	1.69390	0.122566	0.0612831	+	0.998120i	$0.480481\pi$
$192$	0	0
$193$	−2.47769	−0.178348	−0.0891741	−	0.996016i	$-0.528423\pi$
$193$	−2.47769	−0.178348	−0.0891741	+	0.996016i	$0.528423\pi$
$194$	−6.57611	−0.472138
$195$	0	0
$196$	−6.70130	−0.478664
$197$	0.544234	0.0387751	0.0193875	−	0.999812i	$-0.493828\pi$
$197$	0.544234	0.0387751	0.0193875	+	0.999812i	$0.493828\pi$
$198$	0	0
$199$	6.56260	0.465211	0.232605	−	0.972571i	$-0.425275\pi$
$199$	6.56260	0.465211	0.232605	+	0.972571i	$0.425275\pi$
$200$	−7.07005	−0.499928
$201$	0	0
$202$	−17.5143	−1.23230
$203$	1.17242	0.0822879
$204$	0	0
$205$	3.81247	0.266275
$206$	1.53252	0.106776
$207$	0	0
$208$	−16.0155	−1.11048
$209$	−1.27215	−0.0879968
$210$	0	0
$211$	−7.71720	−0.531274	−0.265637	−	0.964073i	$-0.585582\pi$
$211$	−7.71720	−0.531274	−0.265637	+	0.964073i	$0.585582\pi$
$212$	7.21894	0.495799
$213$	0	0
$214$	−23.0237	−1.57387
$215$	1.03076	0.0702970
$216$	0	0
$217$	0.413320	0.0280580
$218$	7.26830	0.492272
$219$	0	0
$220$	−0.331145	−0.0223258
$221$	17.2869	1.16284
$222$	0	0
$223$	11.6536	0.780386	0.390193	−	0.920733i	$-0.372408\pi$
$223$	11.6536	0.780386	0.390193	+	0.920733i	$0.372408\pi$
$224$	7.00779	0.468228
$225$	0	0
$226$	10.3223	0.686627
$227$	18.0449	1.19769	0.598843	−	0.800867i	$-0.295628\pi$
$227$	18.0449	1.19769	0.598843	+	0.800867i	$0.295628\pi$
$228$	0	0
$229$	7.11185	0.469964	0.234982	−	0.972000i	$-0.424497\pi$
$229$	7.11185	0.469964	0.234982	+	0.972000i	$0.424497\pi$
$230$	0.582705	0.0384224
$231$	0	0
$232$	1.44475	0.0948527
$233$	−5.44024	−0.356402	−0.178201	−	0.983994i	$-0.557028\pi$
$233$	−5.44024	−0.356402	−0.178201	+	0.983994i	$0.557028\pi$
$234$	0	0
$235$	−0.471048	−0.0307278
$236$	1.86087	0.121133
$237$	0	0
$238$	−11.2207	−0.727333
$239$	−5.34393	−0.345670	−0.172835	−	0.984951i	$-0.555293\pi$
$239$	−5.34393	−0.345670	−0.172835	+	0.984951i	$0.555293\pi$
$240$	0	0
$241$	4.78763	0.308398	0.154199	−	0.988040i	$-0.450720\pi$
$241$	4.78763	0.308398	0.154199	+	0.988040i	$0.450720\pi$
$242$	18.3531	1.17978
$243$	0	0
$244$	−7.72739	−0.494695
$245$	1.83495	0.117230
$246$	0	0
$247$	−4.81675	−0.306482
$248$	0.509326	0.0323422
$249$	0	0
$250$	−5.76505	−0.364614
$251$	−11.8178	−0.745934	−0.372967	−	0.927845i	$-0.621660\pi$
$251$	−11.8178	−0.745934	−0.372967	+	0.927845i	$0.621660\pi$
$252$	0	0
$253$	0.852209	0.0535779
$254$	−3.24451	−0.203579
$255$	0	0
$256$	20.4609	1.27881
$257$	−8.04832	−0.502041	−0.251020	−	0.967982i	$-0.580766\pi$
$257$	−8.04832	−0.502041	−0.251020	+	0.967982i	$0.580766\pi$
$258$	0	0
$259$	−12.8155	−0.796319
$260$	−1.25381	−0.0777580
$261$	0	0
$262$	6.95406	0.429623
$263$	−8.69157	−0.535945	−0.267973	−	0.963427i	$-0.586354\pi$
$263$	−8.69157	−0.535945	−0.267973	+	0.963427i	$0.586354\pi$
$264$	0	0
$265$	−1.97669	−0.121427
$266$	3.12650	0.191698
$267$	0	0
$268$	−7.95446	−0.485896
$269$	13.3150	0.811829	0.405915	−	0.913911i	$-0.366953\pi$
$269$	13.3150	0.811829	0.405915	+	0.913911i	$0.366953\pi$
$270$	0	0
$271$	15.6543	0.950932	0.475466	−	0.879734i	$-0.342279\pi$
$271$	15.6543	0.950932	0.475466	+	0.879734i	$0.342279\pi$
$272$	−26.5912	−1.61233
$273$	0	0
$274$	−5.62819	−0.340011
$275$	−4.17037	−0.251483
$276$	0	0
$277$	−17.8305	−1.07133	−0.535664	−	0.844431i	$-0.679939\pi$
$277$	−17.8305	−1.07133	−0.535664	+	0.844431i	$0.679939\pi$
$278$	2.55999	0.153538
$279$	0	0
$280$	−0.552517	−0.0330192
$281$	2.21112	0.131904	0.0659522	−	0.997823i	$-0.478992\pi$
$281$	2.21112	0.131904	0.0659522	+	0.997823i	$0.478992\pi$
$282$	0	0
$283$	7.97755	0.474216	0.237108	−	0.971483i	$-0.423800\pi$
$283$	7.97755	0.474216	0.237108	+	0.971483i	$0.423800\pi$
$284$	−16.2920	−0.966754
$285$	0	0
$286$	−4.91232	−0.290472
$287$	−13.7032	−0.808876
$288$	0	0
$289$	11.7022	0.688363
$290$	0.582705	0.0342176
$291$	0	0
$292$	15.0161	0.878750
$293$	10.3091	0.602267	0.301133	−	0.953582i	$-0.402635\pi$
$293$	10.3091	0.602267	0.301133	+	0.953582i	$0.402635\pi$
$294$	0	0
$295$	−0.509543	−0.0296667
$296$	−15.7924	−0.917912
$297$	0	0
$298$	1.63220	0.0945508
$299$	3.22671	0.186606
$300$	0	0
$301$	−3.70486	−0.213545
$302$	29.4454	1.69439
$303$	0	0
$304$	7.40926	0.424950
$305$	2.11591	0.121157
$306$	0	0
$307$	−3.75282	−0.214185	−0.107092	−	0.994249i	$-0.534154\pi$
$307$	−3.75282	−0.214185	−0.107092	+	0.994249i	$0.534154\pi$
$308$	1.19024	0.0678201
$309$	0	0
$310$	0.205424	0.0116673
$311$	−1.70068	−0.0964367	−0.0482184	−	0.998837i	$-0.515354\pi$
$311$	−1.70068	−0.0964367	−0.0482184	+	0.998837i	$0.515354\pi$
$312$	0	0
$313$	−28.4550	−1.60837	−0.804186	−	0.594378i	$-0.797398\pi$
$313$	−28.4550	−1.60837	−0.804186	+	0.594378i	$0.797398\pi$
$314$	−7.04294	−0.397456
$315$	0	0
$316$	−19.0545	−1.07190
$317$	4.71662	0.264912	0.132456	−	0.991189i	$-0.457714\pi$
$317$	4.71662	0.264912	0.132456	+	0.991189i	$0.457714\pi$
$318$	0	0
$319$	0.852209	0.0477145
$320$	0.244919	0.0136914
$321$	0	0
$322$	−2.09442	−0.116718
$323$	−7.99744	−0.444990
$324$	0	0
$325$	−15.7902	−0.875885
$326$	17.4399	0.965908
$327$	0	0
$328$	−16.8862	−0.932386
$329$	1.69309	0.0933432
$330$	0	0
$331$	−0.172157	−0.00946260	−0.00473130	−	0.999989i	$-0.501506\pi$
$331$	−0.172157	−0.00946260	−0.00473130	+	0.999989i	$0.501506\pi$
$332$	−5.77377	−0.316877
$333$	0	0
$334$	−38.2874	−2.09500
$335$	2.17809	0.119002
$336$	0	0
$337$	5.30625	0.289050	0.144525	−	0.989501i	$-0.453835\pi$
$337$	5.30625	0.289050	0.144525	+	0.989501i	$0.453835\pi$
$338$	4.62382	0.251503
$339$	0	0
$340$	−2.08175	−0.112899
$341$	0.300433	0.0162694
$342$	0	0
$343$	−14.8023	−0.799251
$344$	−4.56543	−0.246152
$345$	0	0
$346$	23.9649	1.28836
$347$	−6.97399	−0.374383	−0.187192	−	0.982323i	$-0.559939\pi$
$347$	−6.97399	−0.374383	−0.187192	+	0.982323i	$0.559939\pi$
$348$	0	0
$349$	−16.2958	−0.872291	−0.436146	−	0.899876i	$-0.643657\pi$
$349$	−16.2958	−0.872291	−0.436146	+	0.899876i	$0.643657\pi$
$350$	10.2493	0.547847
$351$	0	0
$352$	5.09381	0.271501
$353$	1.29005	0.0686623	0.0343311	−	0.999411i	$-0.489070\pi$
$353$	1.29005	0.0686623	0.0343311	+	0.999411i	$0.489070\pi$
$354$	0	0
$355$	4.46107	0.236769
$356$	−8.90791	−0.472118
$357$	0	0
$358$	−0.982729	−0.0519389
$359$	−30.4439	−1.60677	−0.803385	−	0.595460i	$-0.796970\pi$
$359$	−30.4439	−1.60677	−0.803385	+	0.595460i	$0.796970\pi$
$360$	0	0
$361$	−16.7716	−0.882717
$362$	−14.1794	−0.745250
$363$	0	0
$364$	4.50659	0.236209
$365$	−4.11170	−0.215216
$366$	0	0
$367$	−20.3547	−1.06251	−0.531253	−	0.847213i	$-0.678279\pi$
$367$	−20.3547	−1.06251	−0.531253	+	0.847213i	$0.678279\pi$
$368$	−4.96342	−0.258736
$369$	0	0
$370$	−6.36945	−0.331132
$371$	7.10483	0.368864
$372$	0	0
$373$	−23.7681	−1.23067	−0.615334	−	0.788266i	$-0.710979\pi$
$373$	−23.7681	−1.23067	−0.615334	+	0.788266i	$0.710979\pi$
$374$	−8.15613	−0.421743
$375$	0	0
$376$	2.08637	0.107596
$377$	3.22671	0.166184
$378$	0	0
$379$	5.60685	0.288005	0.144002	−	0.989577i	$-0.454003\pi$
$379$	5.60685	0.288005	0.144002	+	0.989577i	$0.454003\pi$
$380$	0.580050	0.0297559
$381$	0	0
$382$	−3.02600	−0.154823
$383$	34.5495	1.76540	0.882699	−	0.469938i	$-0.155724\pi$
$383$	34.5495	1.76540	0.882699	+	0.469938i	$0.155724\pi$
$384$	0	0
$385$	−0.325910	−0.0166099
$386$	4.42617	0.225286
$387$	0	0
$388$	4.38523	0.222626
$389$	−12.1605	−0.616560	−0.308280	−	0.951296i	$-0.599753\pi$
$389$	−12.1605	−0.616560	−0.308280	+	0.951296i	$0.599753\pi$
$390$	0	0
$391$	5.35744	0.270937
$392$	−8.12735	−0.410493
$393$	0	0
$394$	−0.972224	−0.0489800
$395$	5.21749	0.262520
$396$	0	0
$397$	14.8664	0.746122	0.373061	−	0.927807i	$-0.378308\pi$
$397$	14.8664	0.746122	0.373061	+	0.927807i	$0.378308\pi$
$398$	−11.7235	−0.587645
$399$	0	0
$400$	24.2890	1.21445
$401$	−11.9695	−0.597728	−0.298864	−	0.954296i	$-0.596608\pi$
$401$	−11.9695	−0.597728	−0.298864	+	0.954296i	$0.596608\pi$
$402$	0	0
$403$	1.13753	0.0566643
$404$	11.6793	0.581067
$405$	0	0
$406$	−2.09442	−0.103944
$407$	−9.31535	−0.461745
$408$	0	0
$409$	36.9672	1.82791	0.913954	−	0.405817i	$-0.133013\pi$
$409$	36.9672	1.82791	0.913954	+	0.405817i	$0.133013\pi$
$410$	−6.81063	−0.336353
$411$	0	0
$412$	−1.02195	−0.0503478
$413$	1.83146	0.0901202
$414$	0	0
$415$	1.58097	0.0776068
$416$	19.2867	0.945607
$417$	0	0
$418$	2.27259	0.111156
$419$	−18.3400	−0.895969	−0.447984	−	0.894041i	$-0.647858\pi$
$419$	−18.3400	−0.895969	−0.447984	+	0.894041i	$0.647858\pi$
$420$	0	0
$421$	−6.09440	−0.297023	−0.148512	−	0.988911i	$-0.547448\pi$
$421$	−6.09440	−0.297023	−0.148512	+	0.988911i	$0.547448\pi$
$422$	13.7861	0.671095
$423$	0	0
$424$	8.75515	0.425187
$425$	−26.2172	−1.27172
$426$	0	0
$427$	−7.60524	−0.368043
$428$	15.3532	0.742123
$429$	0	0
$430$	−1.84135	−0.0887979
$431$	−7.49900	−0.361214	−0.180607	−	0.983555i	$-0.557806\pi$
$431$	−7.49900	−0.361214	−0.180607	+	0.983555i	$0.557806\pi$
$432$	0	0
$433$	16.7396	0.804453	0.402226	−	0.915540i	$-0.368237\pi$
$433$	16.7396	0.804453	0.402226	+	0.915540i	$0.368237\pi$
$434$	−0.738357	−0.0354423
$435$	0	0
$436$	−4.84681	−0.232120
$437$	−1.49277	−0.0714090
$438$	0	0
$439$	29.9913	1.43141	0.715704	−	0.698404i	$-0.246106\pi$
$439$	29.9913	1.43141	0.715704	+	0.698404i	$0.246106\pi$
$440$	−0.401613	−0.0191461
$441$	0	0
$442$	−30.8815	−1.46888
$443$	27.1617	1.29049	0.645245	−	0.763976i	$-0.276755\pi$
$443$	27.1617	1.29049	0.645245	+	0.763976i	$0.276755\pi$
$444$	0	0
$445$	2.43916	0.115627
$446$	−20.8182	−0.985768
$447$	0	0
$448$	−0.880317	−0.0415910
$449$	21.8977	1.03342	0.516708	−	0.856161i	$-0.327157\pi$
$449$	21.8977	1.03342	0.516708	+	0.856161i	$0.327157\pi$
$450$	0	0
$451$	−9.96059	−0.469026
$452$	−6.88333	−0.323764
$453$	0	0
$454$	−32.2356	−1.51289
$455$	−1.23399	−0.0578504
$456$	0	0
$457$	21.7853	1.01908	0.509538	−	0.860448i	$-0.329817\pi$
$457$	21.7853	1.01908	0.509538	+	0.860448i	$0.329817\pi$
$458$	−12.7047	−0.593650
$459$	0	0
$460$	−0.388572	−0.0181173
$461$	−18.0738	−0.841781	−0.420890	−	0.907111i	$-0.638282\pi$
$461$	−18.0738	−0.841781	−0.420890	+	0.907111i	$0.638282\pi$
$462$	0	0
$463$	−8.22965	−0.382464	−0.191232	−	0.981545i	$-0.561248\pi$
$463$	−8.22965	−0.382464	−0.191232	+	0.981545i	$0.561248\pi$
$464$	−4.96342	−0.230421
$465$	0	0
$466$	9.71848	0.450200
$467$	25.0014	1.15693	0.578463	−	0.815708i	$-0.303653\pi$
$467$	25.0014	1.15693	0.578463	+	0.815708i	$0.303653\pi$
$468$	0	0
$469$	−7.82873	−0.361497
$470$	0.841483	0.0388147
$471$	0	0
$472$	2.25687	0.103881
$473$	−2.69299	−0.123824
$474$	0	0
$475$	7.30504	0.335178
$476$	7.48247	0.342959
$477$	0	0
$478$	9.54644	0.436644
$479$	−2.60344	−0.118954	−0.0594772	−	0.998230i	$-0.518943\pi$
$479$	−2.60344	−0.118954	−0.0594772	+	0.998230i	$0.518943\pi$
$480$	0	0
$481$	−35.2706	−1.60820
$482$	−8.55265	−0.389563
$483$	0	0
$484$	−12.2386	−0.556301
$485$	−1.20076	−0.0545238
$486$	0	0
$487$	5.77061	0.261491	0.130746	−	0.991416i	$-0.458263\pi$
$487$	5.77061	0.261491	0.130746	+	0.991416i	$0.458263\pi$
$488$	−9.37179	−0.424241
$489$	0	0
$490$	−3.27796	−0.148083
$491$	−26.5293	−1.19725	−0.598625	−	0.801030i	$-0.704286\pi$
$491$	−26.5293	−1.19725	−0.598625	+	0.801030i	$0.704286\pi$
$492$	0	0
$493$	5.35744	0.241287
$494$	8.60468	0.387143
$495$	0	0
$496$	−1.74978	−0.0785674
$497$	−16.0345	−0.719246
$498$	0	0
$499$	38.1318	1.70701	0.853507	−	0.521081i	$-0.174471\pi$
$499$	38.1318	1.70701	0.853507	+	0.521081i	$0.174471\pi$
$500$	3.84438	0.171926
$501$	0	0
$502$	21.1114	0.942250
$503$	−9.14152	−0.407600	−0.203800	−	0.979013i	$-0.565329\pi$
$503$	−9.14152	−0.407600	−0.203800	+	0.979013i	$0.565329\pi$
$504$	0	0
$505$	−3.19802	−0.142310
$506$	−1.52239	−0.0676786
$507$	0	0
$508$	2.16358	0.0959932
$509$	−36.4615	−1.61613	−0.808063	−	0.589096i	$-0.799484\pi$
$509$	−36.4615	−1.61613	−0.808063	+	0.589096i	$0.799484\pi$
$510$	0	0
$511$	14.7787	0.653773
$512$	−15.3255	−0.677297
$513$	0	0
$514$	14.3776	0.634168
$515$	0.279829	0.0123308
$516$	0	0
$517$	1.23067	0.0541250
$518$	22.8938	1.00590
$519$	0	0
$520$	−1.52062	−0.0666837
$521$	2.69985	0.118282	0.0591412	−	0.998250i	$-0.481164\pi$
$521$	2.69985	0.118282	0.0591412	+	0.998250i	$0.481164\pi$
$522$	0	0
$523$	32.0986	1.40358	0.701788	−	0.712386i	$-0.252386\pi$
$523$	32.0986	1.40358	0.701788	+	0.712386i	$0.252386\pi$
$524$	−4.63726	−0.202580
$525$	0	0
$526$	15.5267	0.676996
$527$	1.88868	0.0822724
$528$	0	0
$529$	1.00000	0.0434783
$530$	3.53117	0.153384
$531$	0	0
$532$	−2.08488	−0.0903911
$533$	−37.7137	−1.63356
$534$	0	0
$535$	−4.20399	−0.181754
$536$	−9.64719	−0.416695
$537$	0	0
$538$	−23.7860	−1.02549
$539$	−4.79404	−0.206494
$540$	0	0
$541$	−15.4359	−0.663641	−0.331821	−	0.943342i	$-0.607663\pi$
$541$	−15.4359	−0.663641	−0.331821	+	0.943342i	$0.607663\pi$
$542$	−27.9650	−1.20120
$543$	0	0
$544$	32.0224	1.37295
$545$	1.32715	0.0568489
$546$	0	0
$547$	−5.05171	−0.215996	−0.107998	−	0.994151i	$-0.534444\pi$
$547$	−5.05171	−0.215996	−0.107998	+	0.994151i	$0.534444\pi$
$548$	3.75312	0.160325
$549$	0	0
$550$	7.44998	0.317668
$551$	−1.49277	−0.0635943
$552$	0	0
$553$	−18.7533	−0.797472
$554$	31.8525	1.35328
$555$	0	0
$556$	−1.70711	−0.0723975
$557$	10.4714	0.443688	0.221844	−	0.975082i	$-0.428792\pi$
$557$	10.4714	0.443688	0.221844	+	0.975082i	$0.428792\pi$
$558$	0	0
$559$	−10.1964	−0.431263
$560$	1.89816	0.0802119
$561$	0	0
$562$	−3.94996	−0.166619
$563$	−15.1161	−0.637069	−0.318535	−	0.947911i	$-0.603191\pi$
$563$	−15.1161	−0.637069	−0.318535	+	0.947911i	$0.603191\pi$
$564$	0	0
$565$	1.88479	0.0792936
$566$	−14.2512	−0.599021
$567$	0	0
$568$	−19.7590	−0.829070
$569$	−34.2755	−1.43691	−0.718453	−	0.695576i	$-0.755149\pi$
$569$	−34.2755	−1.43691	−0.718453	+	0.695576i	$0.755149\pi$
$570$	0	0
$571$	−39.8728	−1.66862	−0.834312	−	0.551293i	$-0.814135\pi$
$571$	−39.8728	−1.66862	−0.834312	+	0.551293i	$0.814135\pi$
$572$	3.27574	0.136966
$573$	0	0
$574$	24.4796	1.02176
$575$	−4.89360	−0.204077
$576$	0	0
$577$	−30.0528	−1.25111	−0.625556	−	0.780179i	$-0.715128\pi$
$577$	−30.0528	−1.25111	−0.625556	+	0.780179i	$0.715128\pi$
$578$	−20.9048	−0.869527
$579$	0	0
$580$	−0.388572	−0.0161346
$581$	−5.68250	−0.235750
$582$	0	0
$583$	5.16435	0.213886
$584$	18.2115	0.753599
$585$	0	0
$586$	−18.4163	−0.760772
$587$	−11.6696	−0.481656	−0.240828	−	0.970568i	$-0.577419\pi$
$587$	−11.6696	−0.481656	−0.240828	+	0.970568i	$0.577419\pi$
$588$	0	0
$589$	−0.526255	−0.0216839
$590$	0.910252	0.0374745
$591$	0	0
$592$	54.2543	2.22984
$593$	−32.5160	−1.33527	−0.667635	−	0.744489i	$-0.732693\pi$
$593$	−32.5160	−1.33527	−0.667635	+	0.744489i	$0.732693\pi$
$594$	0	0
$595$	−2.04885	−0.0839945
$596$	−1.08842	−0.0445834
$597$	0	0
$598$	−5.76422	−0.235717
$599$	−30.3911	−1.24175	−0.620874	−	0.783911i	$-0.713222\pi$
$599$	−30.3911	−1.24175	−0.620874	+	0.783911i	$0.713222\pi$
$600$	0	0
$601$	−35.2479	−1.43779	−0.718896	−	0.695118i	$-0.755352\pi$
$601$	−35.2479	−1.43779	−0.718896	+	0.695118i	$0.755352\pi$
$602$	6.61840	0.269746
$603$	0	0
$604$	−19.6354	−0.798954
$605$	3.35117	0.136244
$606$	0	0
$607$	−2.10437	−0.0854139	−0.0427070	−	0.999088i	$-0.513598\pi$
$607$	−2.10437	−0.0854139	−0.0427070	+	0.999088i	$0.513598\pi$
$608$	−8.92259	−0.361859
$609$	0	0
$610$	−3.77987	−0.153043
$611$	4.65969	0.188511
$612$	0	0
$613$	−16.4130	−0.662914	−0.331457	−	0.943470i	$-0.607540\pi$
$613$	−16.4130	−0.662914	−0.331457	+	0.943470i	$0.607540\pi$
$614$	6.70406	0.270554
$615$	0	0
$616$	1.44352	0.0581612
$617$	7.81267	0.314526	0.157263	−	0.987557i	$-0.449733\pi$
$617$	7.81267	0.314526	0.157263	+	0.987557i	$0.449733\pi$
$618$	0	0
$619$	−27.8878	−1.12091	−0.560453	−	0.828186i	$-0.689373\pi$
$619$	−27.8878	−1.12091	−0.560453	+	0.828186i	$0.689373\pi$
$620$	−0.136985	−0.00550146
$621$	0	0
$622$	3.03811	0.121817
$623$	−8.76710	−0.351246
$624$	0	0
$625$	23.4153	0.936614
$626$	50.8322	2.03166
$627$	0	0
$628$	4.69653	0.187412
$629$	−58.5613	−2.33499
$630$	0	0
$631$	19.5765	0.779330	0.389665	−	0.920957i	$-0.372591\pi$
$631$	19.5765	0.779330	0.389665	+	0.920957i	$0.372591\pi$
$632$	−23.1093	−0.919240
$633$	0	0
$634$	−8.42581	−0.334632
$635$	−0.592429	−0.0235098
$636$	0	0
$637$	−18.1516	−0.719194
$638$	−1.52239	−0.0602721
$639$	0	0
$640$	3.46185	0.136842
$641$	−9.47844	−0.374376	−0.187188	−	0.982324i	$-0.559937\pi$
$641$	−9.47844	−0.374376	−0.187188	+	0.982324i	$0.559937\pi$
$642$	0	0
$643$	−4.98714	−0.196674	−0.0983368	−	0.995153i	$-0.531352\pi$
$643$	−4.98714	−0.196674	−0.0983368	+	0.995153i	$0.531352\pi$
$644$	1.39665	0.0550357
$645$	0	0
$646$	14.2867	0.562103
$647$	−10.2381	−0.402501	−0.201251	−	0.979540i	$-0.564501\pi$
$647$	−10.2381	−0.402501	−0.201251	+	0.979540i	$0.564501\pi$
$648$	0	0
$649$	1.33125	0.0522561
$650$	28.2078	1.10640
$651$	0	0
$652$	−11.6297	−0.455453
$653$	29.3913	1.15017	0.575086	−	0.818093i	$-0.304969\pi$
$653$	29.3913	1.15017	0.575086	+	0.818093i	$0.304969\pi$
$654$	0	0
$655$	1.26977	0.0496141
$656$	58.0123	2.26500
$657$	0	0
$658$	−3.02455	−0.117909
$659$	−37.1953	−1.44892	−0.724462	−	0.689315i	$-0.757912\pi$
$659$	−37.1953	−1.44892	−0.724462	+	0.689315i	$0.757912\pi$
$660$	0	0
$661$	7.87718	0.306387	0.153193	−	0.988196i	$-0.451044\pi$
$661$	7.87718	0.306387	0.153193	+	0.988196i	$0.451044\pi$
$662$	0.307542	0.0119530
$663$	0	0
$664$	−7.00244	−0.271747
$665$	0.570881	0.0221378
$666$	0	0
$667$	1.00000	0.0387202
$668$	25.5317	0.987850
$669$	0	0
$670$	−3.89095	−0.150321
$671$	−5.52809	−0.213409
$672$	0	0
$673$	8.46069	0.326136	0.163068	−	0.986615i	$-0.447861\pi$
$673$	8.46069	0.326136	0.163068	+	0.986615i	$0.447861\pi$
$674$	−9.47913	−0.365122
$675$	0	0
$676$	−3.08336	−0.118591
$677$	−0.648694	−0.0249313	−0.0124657	−	0.999922i	$-0.503968\pi$
$677$	−0.648694	−0.0249313	−0.0124657	+	0.999922i	$0.503968\pi$
$678$	0	0
$679$	4.31591	0.165630
$680$	−2.52475	−0.0968198
$681$	0	0
$682$	−0.536696	−0.0205512
$683$	15.5457	0.594838	0.297419	−	0.954747i	$-0.403874\pi$
$683$	15.5457	0.594838	0.297419	+	0.954747i	$0.403874\pi$
$684$	0	0
$685$	−1.02768	−0.0392655
$686$	26.4430	1.00960
$687$	0	0
$688$	15.6845	0.597965
$689$	19.5537	0.744938
$690$	0	0
$691$	−17.1134	−0.651023	−0.325511	−	0.945538i	$-0.605536\pi$
$691$	−17.1134	−0.651023	−0.325511	+	0.945538i	$0.605536\pi$
$692$	−15.9808	−0.607499
$693$	0	0
$694$	12.4584	0.472914
$695$	0.467439	0.0177310
$696$	0	0
$697$	−62.6176	−2.37181
$698$	29.1109	1.10186
$699$	0	0
$700$	−6.83465	−0.258326
$701$	16.8345	0.635829	0.317915	−	0.948119i	$-0.397017\pi$
$701$	16.8345	0.635829	0.317915	+	0.948119i	$0.397017\pi$
$702$	0	0
$703$	16.3173	0.615417
$704$	−0.639884	−0.0241165
$705$	0	0
$706$	−2.30455	−0.0867329
$707$	11.4947	0.432302
$708$	0	0
$709$	−30.6600	−1.15146	−0.575730	−	0.817640i	$-0.695282\pi$
$709$	−30.6600	−1.15146	−0.575730	+	0.817640i	$0.695282\pi$
$710$	−7.96930	−0.299082
$711$	0	0
$712$	−10.8035	−0.404879
$713$	0.352535	0.0132025
$714$	0	0
$715$	−0.896962	−0.0335445
$716$	0.655326	0.0244907
$717$	0	0
$718$	54.3853	2.02964
$719$	26.6976	0.995654	0.497827	−	0.867276i	$-0.334132\pi$
$719$	26.6976	0.995654	0.497827	+	0.867276i	$0.334132\pi$
$720$	0	0
$721$	−1.00579	−0.0374577
$722$	29.9610	1.11503
$723$	0	0
$724$	9.45539	0.351407
$725$	−4.89360	−0.181744
$726$	0	0
$727$	45.0018	1.66903	0.834513	−	0.550989i	$-0.185749\pi$
$727$	45.0018	1.66903	0.834513	+	0.550989i	$0.185749\pi$
$728$	5.46560	0.202569
$729$	0	0
$730$	7.34517	0.271857
$731$	−16.9296	−0.626163
$732$	0	0
$733$	−20.8424	−0.769833	−0.384917	−	0.922951i	$-0.625770\pi$
$733$	−20.8424	−0.769833	−0.384917	+	0.922951i	$0.625770\pi$
$734$	36.3618	1.34214
$735$	0	0
$736$	5.97719	0.220322
$737$	−5.69054	−0.209614
$738$	0	0
$739$	28.5125	1.04885	0.524425	−	0.851457i	$-0.324280\pi$
$739$	28.5125	1.04885	0.524425	+	0.851457i	$0.324280\pi$
$740$	4.24742	0.156138
$741$	0	0
$742$	−12.6921	−0.465943
$743$	−22.4337	−0.823013	−0.411506	−	0.911407i	$-0.634997\pi$
$743$	−22.4337	−0.823013	−0.411506	+	0.911407i	$0.634997\pi$
$744$	0	0
$745$	0.298030	0.0109190
$746$	42.4596	1.55456
$747$	0	0
$748$	5.43885	0.198864
$749$	15.1105	0.552124
$750$	0	0
$751$	8.21818	0.299886	0.149943	−	0.988695i	$-0.452091\pi$
$751$	8.21818	0.299886	0.149943	+	0.988695i	$0.452091\pi$
$752$	−7.16767	−0.261378
$753$	0	0
$754$	−5.76422	−0.209921
$755$	5.37656	0.195673
$756$	0	0
$757$	−15.6868	−0.570145	−0.285073	−	0.958506i	$-0.592018\pi$
$757$	−15.6868	−0.570145	−0.285073	+	0.958506i	$0.592018\pi$
$758$	−10.0161	−0.363802
$759$	0	0
$760$	0.703486	0.0255181
$761$	−16.0488	−0.581768	−0.290884	−	0.956758i	$-0.593949\pi$
$761$	−16.0488	−0.581768	−0.290884	+	0.956758i	$0.593949\pi$
$762$	0	0
$763$	−4.77020	−0.172693
$764$	2.01786	0.0730037
$765$	0	0
$766$	−61.7196	−2.23002
$767$	5.04049	0.182002
$768$	0	0
$769$	−3.81368	−0.137525	−0.0687624	−	0.997633i	$-0.521905\pi$
$769$	−3.81368	−0.137525	−0.0687624	+	0.997633i	$0.521905\pi$
$770$	0.582208	0.0209813
$771$	0	0
$772$	−2.95156	−0.106229
$773$	9.76179	0.351107	0.175554	−	0.984470i	$-0.443828\pi$
$773$	9.76179	0.351107	0.175554	+	0.984470i	$0.443828\pi$
$774$	0	0
$775$	−1.72517	−0.0619698
$776$	5.31842	0.190920
$777$	0	0
$778$	21.7236	0.778827
$779$	17.4475	0.625121
$780$	0	0
$781$	−11.6551	−0.417054
$782$	−9.57057	−0.342243
$783$	0	0
$784$	27.9214	0.997192
$785$	−1.28600	−0.0458993
$786$	0	0
$787$	−9.60016	−0.342209	−0.171104	−	0.985253i	$-0.554734\pi$
$787$	−9.60016	−0.342209	−0.171104	+	0.985253i	$0.554734\pi$
$788$	0.648320	0.0230955
$789$	0	0
$790$	−9.32057	−0.331611
$791$	−6.77452	−0.240874
$792$	0	0
$793$	−20.9309	−0.743280
$794$	−26.5574	−0.942487
$795$	0	0
$796$	7.81772	0.277092
$797$	−44.4356	−1.57399	−0.786994	−	0.616960i	$-0.788364\pi$
$797$	−44.4356	−1.57399	−0.786994	+	0.616960i	$0.788364\pi$
$798$	0	0
$799$	7.73667	0.273704
$800$	−29.2500	−1.03414
$801$	0	0
$802$	21.3824	0.755039
$803$	10.7423	0.379089
$804$	0	0
$805$	−0.382430	−0.0134789
$806$	−2.03209	−0.0715773
$807$	0	0
$808$	14.1647	0.498311
$809$	14.6055	0.513501	0.256751	−	0.966478i	$-0.417348\pi$
$809$	14.6055	0.513501	0.256751	+	0.966478i	$0.417348\pi$
$810$	0	0
$811$	19.6349	0.689474	0.344737	−	0.938699i	$-0.387968\pi$
$811$	19.6349	0.689474	0.344737	+	0.938699i	$0.387968\pi$
$812$	1.39665	0.0490128
$813$	0	0
$814$	16.6410	0.583267
$815$	3.18443	0.111546
$816$	0	0
$817$	4.71718	0.165033
$818$	−66.0384	−2.30898
$819$	0	0
$820$	4.54162	0.158600
$821$	32.1373	1.12160	0.560800	−	0.827952i	$-0.310494\pi$
$821$	32.1373	1.12160	0.560800	+	0.827952i	$0.310494\pi$
$822$	0	0
$823$	30.5635	1.06538	0.532689	−	0.846311i	$-0.321182\pi$
$823$	30.5635	1.06538	0.532689	+	0.846311i	$0.321182\pi$
$824$	−1.23942	−0.0431773
$825$	0	0
$826$	−3.27173	−0.113838
$827$	−45.4056	−1.57891	−0.789453	−	0.613811i	$-0.789636\pi$
$827$	−45.4056	−1.57891	−0.789453	+	0.613811i	$0.789636\pi$
$828$	0	0
$829$	−27.3167	−0.948747	−0.474374	−	0.880324i	$-0.657325\pi$
$829$	−27.3167	−0.948747	−0.474374	+	0.880324i	$0.657325\pi$
$830$	−2.82426	−0.0980314
$831$	0	0
$832$	−2.42279	−0.0839950
$833$	−30.1379	−1.04422
$834$	0	0
$835$	−6.99107	−0.241936
$836$	−1.51546	−0.0524132
$837$	0	0
$838$	32.7628	1.13177
$839$	−5.77471	−0.199365	−0.0996826	−	0.995019i	$-0.531783\pi$
$839$	−5.77471	−0.199365	−0.0996826	+	0.995019i	$0.531783\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	10.8871	0.375194
$843$	0	0
$844$	−9.19313	−0.316441
$845$	0.844284	0.0290443
$846$	0	0
$847$	−12.0452	−0.413877
$848$	−30.0781	−1.03289
$849$	0	0
$850$	46.8346	1.60641
$851$	−10.9308	−0.374704
$852$	0	0
$853$	−2.09579	−0.0717585	−0.0358792	−	0.999356i	$-0.511423\pi$
$853$	−2.09579	−0.0717585	−0.0358792	+	0.999356i	$0.511423\pi$
$854$	13.5861	0.464905
$855$	0	0
$856$	18.6203	0.636430
$857$	−52.3317	−1.78762	−0.893809	−	0.448449i	$-0.851977\pi$
$857$	−52.3317	−1.78762	−0.893809	+	0.448449i	$0.851977\pi$
$858$	0	0
$859$	−1.34901	−0.0460277	−0.0230139	−	0.999735i	$-0.507326\pi$
$859$	−1.34901	−0.0460277	−0.0230139	+	0.999735i	$0.507326\pi$
$860$	1.22789	0.0418708
$861$	0	0
$862$	13.3963	0.456279
$863$	0.664000	0.0226028	0.0113014	−	0.999936i	$-0.496403\pi$
$863$	0.664000	0.0226028	0.0113014	+	0.999936i	$0.496403\pi$
$864$	0	0
$865$	4.37585	0.148783
$866$	−29.9037	−1.01617
$867$	0	0
$868$	0.492368	0.0167121
$869$	−13.6314	−0.462413
$870$	0	0
$871$	−21.5460	−0.730059
$872$	−5.87822	−0.199062
$873$	0	0
$874$	2.66670	0.0902025
$875$	3.78361	0.127909
$876$	0	0
$877$	14.2825	0.482285	0.241143	−	0.970490i	$-0.422478\pi$
$877$	14.2825	0.482285	0.241143	+	0.970490i	$0.422478\pi$
$878$	−53.5767	−1.80813
$879$	0	0
$880$	1.37973	0.0465108
$881$	45.7708	1.54206	0.771028	−	0.636801i	$-0.219743\pi$
$881$	45.7708	1.54206	0.771028	+	0.636801i	$0.219743\pi$
$882$	0	0
$883$	−37.0494	−1.24681	−0.623406	−	0.781898i	$-0.714252\pi$
$883$	−37.0494	−1.24681	−0.623406	+	0.781898i	$0.714252\pi$
$884$	20.5931	0.692620
$885$	0	0
$886$	−48.5218	−1.63012
$887$	−0.840415	−0.0282184	−0.0141092	−	0.999900i	$-0.504491\pi$
$887$	−0.840415	−0.0282184	−0.0141092	+	0.999900i	$0.504491\pi$
$888$	0	0
$889$	2.12938	0.0714170
$890$	−4.35733	−0.146058
$891$	0	0
$892$	13.8824	0.464818
$893$	−2.15571	−0.0721381
$894$	0	0
$895$	−0.179441	−0.00599804
$896$	−12.4430	−0.415691
$897$	0	0
$898$	−39.1182	−1.30539
$899$	0.352535	0.0117577
$900$	0	0
$901$	32.4659	1.08160
$902$	17.7937	0.592464
$903$	0	0
$904$	−8.34811	−0.277654
$905$	−2.58907	−0.0860636
$906$	0	0
$907$	−12.7632	−0.423794	−0.211897	−	0.977292i	$-0.567964\pi$
$907$	−12.7632	−0.423794	−0.211897	+	0.977292i	$0.567964\pi$
$908$	21.4961	0.713373
$909$	0	0
$910$	2.20441	0.0730755
$911$	0.499026	0.0165335	0.00826674	−	0.999966i	$-0.497369\pi$
$911$	0.499026	0.0165335	0.00826674	+	0.999966i	$0.497369\pi$
$912$	0	0
$913$	−4.13049	−0.136699
$914$	−38.9175	−1.28728
$915$	0	0
$916$	8.47201	0.279923
$917$	−4.56396	−0.150715
$918$	0	0
$919$	−5.62310	−0.185489	−0.0927445	−	0.995690i	$-0.529564\pi$
$919$	−5.62310	−0.185489	−0.0927445	+	0.995690i	$0.529564\pi$
$920$	−0.471261	−0.0155370
$921$	0	0
$922$	32.2872	1.06332
$923$	−44.1298	−1.45255
$924$	0	0
$925$	53.4911	1.75878
$926$	14.7015	0.483121
$927$	0	0
$928$	5.97719	0.196211
$929$	3.39932	0.111528	0.0557641	−	0.998444i	$-0.482241\pi$
$929$	3.39932	0.111528	0.0557641	+	0.998444i	$0.482241\pi$
$930$	0	0
$931$	8.39749	0.275217
$932$	−6.48069	−0.212282
$933$	0	0
$934$	−44.6627	−1.46141
$935$	−1.48926	−0.0487041
$936$	0	0
$937$	34.5379	1.12830	0.564152	−	0.825671i	$-0.309203\pi$
$937$	34.5379	1.12830	0.564152	+	0.825671i	$0.309203\pi$
$938$	13.9853	0.456636
$939$	0	0
$940$	−0.561136	−0.0183023
$941$	0.850674	0.0277312	0.0138656	−	0.999904i	$-0.495586\pi$
$941$	0.850674	0.0277312	0.0138656	+	0.999904i	$0.495586\pi$
$942$	0	0
$943$	−11.6880	−0.380613
$944$	−7.75344	−0.252353
$945$	0	0
$946$	4.81077	0.156412
$947$	−48.6038	−1.57941	−0.789706	−	0.613485i	$-0.789767\pi$
$947$	−48.6038	−1.57941	−0.789706	+	0.613485i	$0.789767\pi$
$948$	0	0
$949$	40.6737	1.32032
$950$	−13.0498	−0.423391
$951$	0	0
$952$	9.07476	0.294115
$953$	45.2493	1.46577	0.732885	−	0.680353i	$-0.238173\pi$
$953$	45.2493	1.46577	0.732885	+	0.680353i	$0.238173\pi$
$954$	0	0
$955$	−0.552530	−0.0178794
$956$	−6.36597	−0.205890
$957$	0	0
$958$	4.65081	0.150261
$959$	3.69379	0.119279
$960$	0	0
$961$	−30.8757	−0.995991
$962$	63.0077	2.03145
$963$	0	0
$964$	5.70327	0.183690
$965$	0.808193	0.0260167
$966$	0	0
$967$	−27.1176	−0.872043	−0.436021	−	0.899936i	$-0.643613\pi$
$967$	−27.1176	−0.872043	−0.436021	+	0.899936i	$0.643613\pi$
$968$	−14.8430	−0.477073
$969$	0	0
$970$	2.14505	0.0688734
$971$	40.2983	1.29323	0.646617	−	0.762815i	$-0.276183\pi$
$971$	40.2983	1.29323	0.646617	+	0.762815i	$0.276183\pi$
$972$	0	0
$973$	−1.68012	−0.0538623
$974$	−10.3087	−0.330311
$975$	0	0
$976$	32.1966	1.03059
$977$	43.0726	1.37801	0.689007	−	0.724755i	$-0.258047\pi$
$977$	43.0726	1.37801	0.689007	+	0.724755i	$0.258047\pi$
$978$	0	0
$979$	−6.37262	−0.203670
$980$	2.18588	0.0698255
$981$	0	0
$982$	47.3921	1.51234
$983$	−24.0627	−0.767481	−0.383740	−	0.923441i	$-0.625364\pi$
$983$	−24.0627	−0.767481	−0.383740	+	0.923441i	$0.625364\pi$
$984$	0	0
$985$	−0.177523	−0.00565634
$986$	−9.57057	−0.304789
$987$	0	0
$988$	−5.73796	−0.182549
$989$	−3.16001	−0.100482
$990$	0	0
$991$	0.202445	0.00643088	0.00321544	−	0.999995i	$-0.498976\pi$
$991$	0.202445	0.00643088	0.00321544	+	0.999995i	$0.498976\pi$
$992$	2.10717	0.0669026
$993$	0	0
$994$	28.6442	0.908537
$995$	−2.14064	−0.0678629
$996$	0	0
$997$	−15.0857	−0.477770	−0.238885	−	0.971048i	$-0.576782\pi$
$997$	−15.0857	−0.477770	−0.238885	+	0.971048i	$0.576782\pi$
$998$	−68.1190	−2.15627
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.t.1.6	✓	22
3.2	odd	2		6003.2.a.u.1.17	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.6	✓	22	1.1	even	1	trivial
6003.2.a.u.1.17	yes	22	3.2	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q-1.78641 q^{2} +1.19125 q^{4} -0.326188 q^{5} +1.17242 q^{7} +1.44475 q^{8} +O(q^{10})\)	\(q-1.78641 q^{2} +1.19125 q^{4} -0.326188 q^{5} +1.17242 q^{7} +1.44475 q^{8} +0.582705 q^{10} +0.852209 q^{11} +3.22671 q^{13} -2.09442 q^{14} -4.96342 q^{16} +5.35744 q^{17} -1.49277 q^{19} -0.388572 q^{20} -1.52239 q^{22} +1.00000 q^{23} -4.89360 q^{25} -5.76422 q^{26} +1.39665 q^{28} +1.00000 q^{29} +0.352535 q^{31} +5.97719 q^{32} -9.57057 q^{34} -0.382430 q^{35} -10.9308 q^{37} +2.66670 q^{38} -0.471261 q^{40} -11.6880 q^{41} -3.16001 q^{43} +1.01520 q^{44} -1.78641 q^{46} +1.44410 q^{47} -5.62543 q^{49} +8.74197 q^{50} +3.84383 q^{52} +6.05996 q^{53} -0.277980 q^{55} +1.69386 q^{56} -1.78641 q^{58} +1.56212 q^{59} -6.48678 q^{61} -0.629771 q^{62} -0.750853 q^{64} -1.05251 q^{65} -6.67740 q^{67} +6.38206 q^{68} +0.683176 q^{70} -13.6764 q^{71} +12.6053 q^{73} +19.5269 q^{74} -1.77827 q^{76} +0.999148 q^{77} -15.9954 q^{79} +1.61901 q^{80} +20.8795 q^{82} -4.84681 q^{83} -1.74753 q^{85} +5.64507 q^{86} +1.23123 q^{88} -7.47777 q^{89} +3.78307 q^{91} +1.19125 q^{92} -2.57975 q^{94} +0.486925 q^{95} +3.68119 q^{97} +10.0493 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34} + 2 q^{35} - 28 q^{37} + 14 q^{38} - 30 q^{40} - 10 q^{41} - 14 q^{43} + 37 q^{44} - 3 q^{46} - 18 q^{47} + 2 q^{49} + 7 q^{50} - 57 q^{52} + 20 q^{53} - 42 q^{55} - 2 q^{56} - 3 q^{58} - 20 q^{59} - 38 q^{61} + 4 q^{62} - 24 q^{64} + 12 q^{65} - 50 q^{67} + 11 q^{68} - 48 q^{70} + 12 q^{71} - 46 q^{73} - 6 q^{74} - 16 q^{76} - 14 q^{77} - 20 q^{79} - 58 q^{80} - 42 q^{82} + 22 q^{83} - 66 q^{85} + 22 q^{86} - 68 q^{88} - 14 q^{89} - 16 q^{91} + 17 q^{92} - 27 q^{94} - 20 q^{95} - 48 q^{97} - 28 q^{98}+O(q^{100}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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