LMFDB - Embedded newform 3006.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3006,2,Mod(1,3006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3006, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3006 = 2 \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3006.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:	$24.0030308476$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2777.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + x + 2 $ x^4 - x^3 - 4*x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1002)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.50848$ of defining polynomial
Character	$\chi$	$=$	3006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.69113 q^{5} -2.24216 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -3.69113 q^{5} -2.24216 q^{7} -1.00000 q^{8} +3.69113 q^{10} +5.56799 q^{13} +2.24216 q^{14} +1.00000 q^{16} +1.01696 q^{17} -2.65167 q^{19} -3.69113 q^{20} -2.65167 q^{23} +8.62441 q^{25} -5.56799 q^{26} -2.24216 q^{28} +1.38225 q^{29} +4.89383 q^{31} -1.00000 q^{32} -1.01696 q^{34} +8.27608 q^{35} -1.97751 q^{37} +2.65167 q^{38} +3.69113 q^{40} -4.28639 q^{41} +0.551031 q^{43} +2.65167 q^{46} +5.62441 q^{47} -1.97273 q^{49} -8.62441 q^{50} +5.56799 q^{52} -1.03945 q^{53} +2.24216 q^{56} -1.38225 q^{58} +2.42790 q^{59} +5.38225 q^{61} -4.89383 q^{62} +1.00000 q^{64} -20.5522 q^{65} +14.8610 q^{67} +1.01696 q^{68} -8.27608 q^{70} +8.68560 q^{71} -1.75374 q^{73} +1.97751 q^{74} -2.65167 q^{76} +3.74754 q^{79} -3.69113 q^{80} +4.28639 q^{82} +5.43867 q^{83} -3.75374 q^{85} -0.551031 q^{86} +6.27608 q^{89} -12.4843 q^{91} -2.65167 q^{92} -5.62441 q^{94} +9.78766 q^{95} -11.0067 q^{97} +1.97273 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 5 q^{5} + q^{7} - 4 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 - 5 * q^5 + q^7 - 4 * q^8	$ 4 q - 4 q^{2} + 4 q^{4} - 5 q^{5} + q^{7} - 4 q^{8} + 5 q^{10} + 8 q^{13} - q^{14} + 4 q^{16} - 10 q^{17} - 2 q^{19} - 5 q^{20} - 2 q^{23} + 5 q^{25} - 8 q^{26} + q^{28} - 14 q^{29} + q^{31} - 4 q^{32} + 10 q^{34} - 5 q^{35} + 5 q^{37} + 2 q^{38} + 5 q^{40} - 14 q^{41} + 2 q^{43} + 2 q^{46} - 7 q^{47} + 13 q^{49} - 5 q^{50} + 8 q^{52} - 3 q^{53} - q^{56} + 14 q^{58} + 5 q^{59} + 2 q^{61} - q^{62} + 4 q^{64} - 6 q^{65} - 7 q^{67} - 10 q^{68} + 5 q^{70} - 2 q^{71} + 2 q^{73} - 5 q^{74} - 2 q^{76} - 10 q^{79} - 5 q^{80} + 14 q^{82} - 13 q^{83} - 6 q^{85} - 2 q^{86} - 13 q^{89} - 30 q^{91} - 2 q^{92} + 7 q^{94} + 2 q^{95} + 5 q^{97} - 13 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 5 * q^5 + q^7 - 4 * q^8 + 5 * q^10 + 8 * q^13 - q^14 + 4 * q^16 - 10 * q^17 - 2 * q^19 - 5 * q^20 - 2 * q^23 + 5 * q^25 - 8 * q^26 + q^28 - 14 * q^29 + q^31 - 4 * q^32 + 10 * q^34 - 5 * q^35 + 5 * q^37 + 2 * q^38 + 5 * q^40 - 14 * q^41 + 2 * q^43 + 2 * q^46 - 7 * q^47 + 13 * q^49 - 5 * q^50 + 8 * q^52 - 3 * q^53 - q^56 + 14 * q^58 + 5 * q^59 + 2 * q^61 - q^62 + 4 * q^64 - 6 * q^65 - 7 * q^67 - 10 * q^68 + 5 * q^70 - 2 * q^71 + 2 * q^73 - 5 * q^74 - 2 * q^76 - 10 * q^79 - 5 * q^80 + 14 * q^82 - 13 * q^83 - 6 * q^85 - 2 * q^86 - 13 * q^89 - 30 * q^91 - 2 * q^92 + 7 * q^94 + 2 * q^95 + 5 * q^97 - 13 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.69113	−1.65072	−0.825361	−	0.564606i	$-0.809028\pi$
$5$	−3.69113	−1.65072	−0.825361	+	0.564606i	$0.809028\pi$
$6$	0	0
$7$	−2.24216	−0.847456	−0.423728	−	0.905790i	$-0.639279\pi$
$7$	−2.24216	−0.847456	−0.423728	+	0.905790i	$0.639279\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.69113	1.16724
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	5.56799	1.54428	0.772142	−	0.635450i	$-0.219186\pi$
$13$	5.56799	1.54428	0.772142	+	0.635450i	$0.219186\pi$
$14$	2.24216	0.599242
$15$	0	0
$16$	1.00000	0.250000
$17$	1.01696	0.246650	0.123325	−	0.992366i	$-0.460644\pi$
$17$	1.01696	0.246650	0.123325	+	0.992366i	$0.460644\pi$
$18$	0	0
$19$	−2.65167	−0.608336	−0.304168	−	0.952618i	$-0.598378\pi$
$19$	−2.65167	−0.608336	−0.304168	+	0.952618i	$0.598378\pi$
$20$	−3.69113	−0.825361
$21$	0	0
$22$	0	0
$23$	−2.65167	−0.552912	−0.276456	−	0.961027i	$-0.589160\pi$
$23$	−2.65167	−0.552912	−0.276456	+	0.961027i	$0.589160\pi$
$24$	0	0
$25$	8.62441	1.72488
$26$	−5.56799	−1.09197
$27$	0	0
$28$	−2.24216	−0.423728
$29$	1.38225	0.256678	0.128339	−	0.991730i	$-0.459036\pi$
$29$	1.38225	0.256678	0.128339	+	0.991730i	$0.459036\pi$
$30$	0	0
$31$	4.89383	0.878958	0.439479	−	0.898253i	$-0.355163\pi$
$31$	4.89383	0.878958	0.439479	+	0.898253i	$0.355163\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.01696	−0.174408
$35$	8.27608	1.39891
$36$	0	0
$37$	−1.97751	−0.325101	−0.162550	−	0.986700i	$-0.551972\pi$
$37$	−1.97751	−0.325101	−0.162550	+	0.986700i	$0.551972\pi$
$38$	2.65167	0.430158
$39$	0	0
$40$	3.69113	0.583618
$41$	−4.28639	−0.669421	−0.334710	−	0.942321i	$-0.608639\pi$
$41$	−4.28639	−0.669421	−0.334710	+	0.942321i	$0.608639\pi$
$42$	0	0
$43$	0.551031	0.0840314	0.0420157	−	0.999117i	$-0.486622\pi$
$43$	0.551031	0.0840314	0.0420157	+	0.999117i	$0.486622\pi$
$44$	0	0
$45$	0	0
$46$	2.65167	0.390968
$47$	5.62441	0.820404	0.410202	−	0.911995i	$-0.365458\pi$
$47$	5.62441	0.820404	0.410202	+	0.911995i	$0.365458\pi$
$48$	0	0
$49$	−1.97273	−0.281819
$50$	−8.62441	−1.21968
$51$	0	0
$52$	5.56799	0.772142
$53$	−1.03945	−0.142780	−0.0713898	−	0.997448i	$-0.522743\pi$
$53$	−1.03945	−0.142780	−0.0713898	+	0.997448i	$0.522743\pi$
$54$	0	0
$55$	0	0
$56$	2.24216	0.299621
$57$	0	0
$58$	−1.38225	−0.181498
$59$	2.42790	0.316086	0.158043	−	0.987432i	$-0.449482\pi$
$59$	2.42790	0.316086	0.158043	+	0.987432i	$0.449482\pi$
$60$	0	0
$61$	5.38225	0.689127	0.344563	−	0.938763i	$-0.388027\pi$
$61$	5.38225	0.689127	0.344563	+	0.938763i	$0.388027\pi$
$62$	−4.89383	−0.621517
$63$	0	0
$64$	1.00000	0.125000
$65$	−20.5522	−2.54918
$66$	0	0
$67$	14.8610	1.81556	0.907782	−	0.419442i	$-0.137774\pi$
$67$	14.8610	1.81556	0.907782	+	0.419442i	$0.137774\pi$
$68$	1.01696	0.123325
$69$	0	0
$70$	−8.27608	−0.989181
$71$	8.68560	1.03079	0.515395	−	0.856952i	$-0.327645\pi$
$71$	8.68560	1.03079	0.515395	+	0.856952i	$0.327645\pi$
$72$	0	0
$73$	−1.75374	−0.205259	−0.102630	−	0.994720i	$-0.532726\pi$
$73$	−1.75374	−0.205259	−0.102630	+	0.994720i	$0.532726\pi$
$74$	1.97751	0.229881
$75$	0	0
$76$	−2.65167	−0.304168
$77$	0	0
$78$	0	0
$79$	3.74754	0.421631	0.210816	−	0.977526i	$-0.432388\pi$
$79$	3.74754	0.421631	0.210816	+	0.977526i	$0.432388\pi$
$80$	−3.69113	−0.412680
$81$	0	0
$82$	4.28639	0.473352
$83$	5.43867	0.596971	0.298486	−	0.954414i	$-0.403519\pi$
$83$	5.43867	0.596971	0.298486	+	0.954414i	$0.403519\pi$
$84$	0	0
$85$	−3.75374	−0.407150
$86$	−0.551031	−0.0594192
$87$	0	0
$88$	0	0
$89$	6.27608	0.665263	0.332632	−	0.943057i	$-0.392063\pi$
$89$	6.27608	0.665263	0.332632	+	0.943057i	$0.392063\pi$
$90$	0	0
$91$	−12.4843	−1.30871
$92$	−2.65167	−0.276456
$93$	0	0
$94$	−5.62441	−0.580113
$95$	9.78766	1.00419
$96$	0	0
$97$	−11.0067	−1.11756	−0.558778	−	0.829317i	$-0.688730\pi$
$97$	−11.0067	−1.11756	−0.558778	+	0.829317i	$0.688730\pi$
$98$	1.97273	0.199276
$99$	0	0
$100$	8.62441	0.862441
$101$	−15.3114	−1.52354	−0.761772	−	0.647845i	$-0.775670\pi$
$101$	−15.3114	−1.52354	−0.761772	+	0.647845i	$0.775670\pi$
$102$	0	0
$103$	−13.2972	−1.31021	−0.655104	−	0.755539i	$-0.727375\pi$
$103$	−13.2972	−1.31021	−0.655104	+	0.755539i	$0.727375\pi$
$104$	−5.56799	−0.545987
$105$	0	0
$106$	1.03945	0.100960
$107$	−18.3848	−1.77733	−0.888663	−	0.458561i	$-0.848365\pi$
$107$	−18.3848	−1.77733	−0.888663	+	0.458561i	$0.848365\pi$
$108$	0	0
$109$	−18.1651	−1.73990	−0.869952	−	0.493136i	$-0.835850\pi$
$109$	−18.1651	−1.73990	−0.869952	+	0.493136i	$0.835850\pi$
$110$	0	0
$111$	0	0
$112$	−2.24216	−0.211864
$113$	−14.4331	−1.35776	−0.678878	−	0.734251i	$-0.737533\pi$
$113$	−14.4331	−1.35776	−0.678878	+	0.734251i	$0.737533\pi$
$114$	0	0
$115$	9.78766	0.912704
$116$	1.38225	0.128339
$117$	0	0
$118$	−2.42790	−0.223506
$119$	−2.28019	−0.209025
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−5.38225	−0.487286
$123$	0	0
$124$	4.89383	0.439479
$125$	−13.3781	−1.19658
$126$	0	0
$127$	−14.4434	−1.28165	−0.640824	−	0.767688i	$-0.721407\pi$
$127$	−14.4434	−1.28165	−0.640824	+	0.767688i	$0.721407\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	20.5522	1.80254
$131$	−11.4346	−0.999042	−0.499521	−	0.866302i	$-0.666491\pi$
$131$	−11.4346	−0.999042	−0.499521	+	0.866302i	$0.666491\pi$
$132$	0	0
$133$	5.94547	0.515537
$134$	−14.8610	−1.28380
$135$	0	0
$136$	−1.01696	−0.0872038
$137$	17.6134	1.50481	0.752405	−	0.658701i	$-0.228894\pi$
$137$	17.6134	1.50481	0.752405	+	0.658701i	$0.228894\pi$
$138$	0	0
$139$	−14.9060	−1.26431	−0.632156	−	0.774841i	$-0.717830\pi$
$139$	−14.9060	−1.26431	−0.632156	+	0.774841i	$0.717830\pi$
$140$	8.27608	0.699457
$141$	0	0
$142$	−8.68560	−0.728879
$143$	0	0
$144$	0	0
$145$	−5.10206	−0.423703
$146$	1.75374	0.145140
$147$	0	0
$148$	−1.97751	−0.162550
$149$	−0.308874	−0.0253040	−0.0126520	−	0.999920i	$-0.504027\pi$
$149$	−0.308874	−0.0253040	−0.0126520	+	0.999920i	$0.504027\pi$
$150$	0	0
$151$	−4.61155	−0.375283	−0.187641	−	0.982238i	$-0.560084\pi$
$151$	−4.61155	−0.375283	−0.187641	+	0.982238i	$0.560084\pi$
$152$	2.65167	0.215079
$153$	0	0
$154$	0	0
$155$	−18.0637	−1.45091
$156$	0	0
$157$	−7.30335	−0.582871	−0.291435	−	0.956591i	$-0.594133\pi$
$157$	−7.30335	−0.582871	−0.291435	+	0.956591i	$0.594133\pi$
$158$	−3.74754	−0.298138
$159$	0	0
$160$	3.69113	0.291809
$161$	5.94547	0.468569
$162$	0	0
$163$	−1.73610	−0.135982	−0.0679911	−	0.997686i	$-0.521659\pi$
$163$	−1.73610	−0.135982	−0.0679911	+	0.997686i	$0.521659\pi$
$164$	−4.28639	−0.334710
$165$	0	0
$166$	−5.43867	−0.422122
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	18.0026	1.38481
$170$	3.75374	0.287898
$171$	0	0
$172$	0.551031	0.0420157
$173$	5.38225	0.409205	0.204602	−	0.978845i	$-0.434410\pi$
$173$	5.38225	0.409205	0.204602	+	0.978845i	$0.434410\pi$
$174$	0	0
$175$	−19.3373	−1.46176
$176$	0	0
$177$	0	0
$178$	−6.27608	−0.470412
$179$	−1.30335	−0.0974168	−0.0487084	−	0.998813i	$-0.515510\pi$
$179$	−1.30335	−0.0974168	−0.0487084	+	0.998813i	$0.515510\pi$
$180$	0	0
$181$	−16.5632	−1.23113	−0.615567	−	0.788084i	$-0.711073\pi$
$181$	−16.5632	−1.23113	−0.615567	+	0.788084i	$0.711073\pi$
$182$	12.4843	0.925399
$183$	0	0
$184$	2.65167	0.195484
$185$	7.29924	0.536651
$186$	0	0
$187$	0	0
$188$	5.62441	0.410202
$189$	0	0
$190$	−9.78766	−0.710072
$191$	21.3332	1.54361	0.771807	−	0.635857i	$-0.219353\pi$
$191$	21.3332	1.54361	0.771807	+	0.635857i	$0.219353\pi$
$192$	0	0
$193$	−13.1360	−0.945549	−0.472775	−	0.881183i	$-0.656747\pi$
$193$	−13.1360	−0.945549	−0.472775	+	0.881183i	$0.656747\pi$
$194$	11.0067	0.790232
$195$	0	0
$196$	−1.97273	−0.140910
$197$	6.55103	0.466742	0.233371	−	0.972388i	$-0.425024\pi$
$197$	6.55103	0.466742	0.233371	+	0.972388i	$0.425024\pi$
$198$	0	0
$199$	7.13599	0.505857	0.252928	−	0.967485i	$-0.418606\pi$
$199$	7.13599	0.505857	0.252928	+	0.967485i	$0.418606\pi$
$200$	−8.62441	−0.609838
$201$	0	0
$202$	15.3114	1.07731
$203$	−3.09922	−0.217523
$204$	0	0
$205$	15.8216	1.10503
$206$	13.2972	0.926456
$207$	0	0
$208$	5.56799	0.386071
$209$	0	0
$210$	0	0
$211$	−18.6517	−1.28403	−0.642017	−	0.766690i	$-0.721902\pi$
$211$	−18.6517	−1.28403	−0.642017	+	0.766690i	$0.721902\pi$
$212$	−1.03945	−0.0713898
$213$	0	0
$214$	18.3848	1.25676
$215$	−2.03392	−0.138713
$216$	0	0
$217$	−10.9727	−0.744878
$218$	18.1651	1.23030
$219$	0	0
$220$	0	0
$221$	5.66244	0.380897
$222$	0	0
$223$	2.24216	0.150146	0.0750730	−	0.997178i	$-0.476081\pi$
$223$	2.24216	0.150146	0.0750730	+	0.997178i	$0.476081\pi$
$224$	2.24216	0.149810
$225$	0	0
$226$	14.4331	0.960078
$227$	18.3829	1.22012	0.610059	−	0.792356i	$-0.291146\pi$
$227$	18.3829	1.22012	0.610059	+	0.792356i	$0.291146\pi$
$228$	0	0
$229$	−11.2149	−0.741101	−0.370550	−	0.928812i	$-0.620831\pi$
$229$	−11.2149	−0.741101	−0.370550	+	0.928812i	$0.620831\pi$
$230$	−9.78766	−0.645379
$231$	0	0
$232$	−1.38225	−0.0907492
$233$	17.5249	1.14809	0.574047	−	0.818822i	$-0.305373\pi$
$233$	17.5249	1.14809	0.574047	+	0.818822i	$0.305373\pi$
$234$	0	0
$235$	−20.7604	−1.35426
$236$	2.42790	0.158043
$237$	0	0
$238$	2.28019	0.147803
$239$	7.13599	0.461589	0.230794	−	0.973003i	$-0.425868\pi$
$239$	7.13599	0.461589	0.230794	+	0.973003i	$0.425868\pi$
$240$	0	0
$241$	−5.86656	−0.377899	−0.188949	−	0.981987i	$-0.560508\pi$
$241$	−5.86656	−0.377899	−0.188949	+	0.981987i	$0.560508\pi$
$242$	11.0000	0.707107
$243$	0	0
$244$	5.38225	0.344563
$245$	7.28161	0.465205
$246$	0	0
$247$	−14.7645	−0.939443
$248$	−4.89383	−0.310759
$249$	0	0
$250$	13.3781	0.846108
$251$	−17.0815	−1.07817	−0.539086	−	0.842251i	$-0.681230\pi$
$251$	−17.0815	−1.07817	−0.539086	+	0.842251i	$0.681230\pi$
$252$	0	0
$253$	0	0
$254$	14.4434	0.906262
$255$	0	0
$256$	1.00000	0.0625000
$257$	−11.0959	−0.692141	−0.346071	−	0.938208i	$-0.612484\pi$
$257$	−11.0959	−0.692141	−0.346071	+	0.938208i	$0.612484\pi$
$258$	0	0
$259$	4.43389	0.275508
$260$	−20.5522	−1.27459
$261$	0	0
$262$	11.4346	0.706429
$263$	−23.7836	−1.46656	−0.733278	−	0.679929i	$-0.762011\pi$
$263$	−23.7836	−1.46656	−0.733278	+	0.679929i	$0.762011\pi$
$264$	0	0
$265$	3.83675	0.235689
$266$	−5.94547	−0.364540
$267$	0	0
$268$	14.8610	0.907782
$269$	24.8404	1.51455	0.757274	−	0.653097i	$-0.226531\pi$
$269$	24.8404	1.51455	0.757274	+	0.653097i	$0.226531\pi$
$270$	0	0
$271$	−16.6918	−1.01395	−0.506977	−	0.861959i	$-0.669237\pi$
$271$	−16.6918	−1.01395	−0.506977	+	0.861959i	$0.669237\pi$
$272$	1.01696	0.0616624
$273$	0	0
$274$	−17.6134	−1.06406
$275$	0	0
$276$	0	0
$277$	30.8888	1.85593	0.927963	−	0.372672i	$-0.121558\pi$
$277$	30.8888	1.85593	0.927963	+	0.372672i	$0.121558\pi$
$278$	14.9060	0.894003
$279$	0	0
$280$	−8.27608	−0.494590
$281$	−25.0406	−1.49380	−0.746898	−	0.664939i	$-0.768458\pi$
$281$	−25.0406	−1.49380	−0.746898	+	0.664939i	$0.768458\pi$
$282$	0	0
$283$	27.6964	1.64638	0.823189	−	0.567767i	$-0.192193\pi$
$283$	27.6964	1.64638	0.823189	+	0.567767i	$0.192193\pi$
$284$	8.68560	0.515395
$285$	0	0
$286$	0	0
$287$	9.61075	0.567304
$288$	0	0
$289$	−15.9658	−0.939164
$290$	5.10206	0.299603
$291$	0	0
$292$	−1.75374	−0.102630
$293$	−21.3823	−1.24916	−0.624582	−	0.780959i	$-0.714731\pi$
$293$	−21.3823	−1.24916	−0.624582	+	0.780959i	$0.714731\pi$
$294$	0	0
$295$	−8.96168	−0.521769
$296$	1.97751	0.114940
$297$	0	0
$298$	0.308874	0.0178926
$299$	−14.7645	−0.853853
$300$	0	0
$301$	−1.23550	−0.0712129
$302$	4.61155	0.265365
$303$	0	0
$304$	−2.65167	−0.152084
$305$	−19.8666	−1.13756
$306$	0	0
$307$	5.35640	0.305706	0.152853	−	0.988249i	$-0.451154\pi$
$307$	5.35640	0.305706	0.152853	+	0.988249i	$0.451154\pi$
$308$	0	0
$309$	0	0
$310$	18.0637	1.02595
$311$	−11.0980	−0.629307	−0.314654	−	0.949207i	$-0.601888\pi$
$311$	−11.0980	−0.629307	−0.314654	+	0.949207i	$0.601888\pi$
$312$	0	0
$313$	−20.4527	−1.15605	−0.578026	−	0.816018i	$-0.696177\pi$
$313$	−20.4527	−1.15605	−0.578026	+	0.816018i	$0.696177\pi$
$314$	7.30335	0.412152
$315$	0	0
$316$	3.74754	0.210816
$317$	−0.889723	−0.0499718	−0.0249859	−	0.999688i	$-0.507954\pi$
$317$	−0.889723	−0.0499718	−0.0249859	+	0.999688i	$0.507954\pi$
$318$	0	0
$319$	0	0
$320$	−3.69113	−0.206340
$321$	0	0
$322$	−5.94547	−0.331328
$323$	−2.69665	−0.150046
$324$	0	0
$325$	48.0206	2.66371
$326$	1.73610	0.0961539
$327$	0	0
$328$	4.28639	0.236676
$329$	−12.6108	−0.695256
$330$	0	0
$331$	7.31553	0.402098	0.201049	−	0.979581i	$-0.435565\pi$
$331$	7.31553	0.402098	0.201049	+	0.979581i	$0.435565\pi$
$332$	5.43867	0.298486
$333$	0	0
$334$	−1.00000	−0.0547176
$335$	−54.8540	−2.99699
$336$	0	0
$337$	6.11827	0.333284	0.166642	−	0.986017i	$-0.446708\pi$
$337$	6.11827	0.333284	0.166642	+	0.986017i	$0.446708\pi$
$338$	−18.0026	−0.979210
$339$	0	0
$340$	−3.75374	−0.203575
$341$	0	0
$342$	0	0
$343$	20.1183	1.08628
$344$	−0.551031	−0.0297096
$345$	0	0
$346$	−5.38225	−0.289352
$347$	−17.6400	−0.946962	−0.473481	−	0.880804i	$-0.657003\pi$
$347$	−17.6400	−0.946962	−0.473481	+	0.880804i	$0.657003\pi$
$348$	0	0
$349$	12.7420	0.682064	0.341032	−	0.940052i	$-0.389223\pi$
$349$	12.7420	0.682064	0.341032	+	0.940052i	$0.389223\pi$
$350$	19.3373	1.03362
$351$	0	0
$352$	0	0
$353$	−33.0406	−1.75857	−0.879286	−	0.476293i	$-0.841980\pi$
$353$	−33.0406	−1.75857	−0.879286	+	0.476293i	$0.841980\pi$
$354$	0	0
$355$	−32.0596	−1.70155
$356$	6.27608	0.332632
$357$	0	0
$358$	1.30335	0.0688841
$359$	10.4054	0.549177	0.274588	−	0.961562i	$-0.411458\pi$
$359$	10.4054	0.549177	0.274588	+	0.961562i	$0.411458\pi$
$360$	0	0
$361$	−11.9686	−0.629928
$362$	16.5632	0.870543
$363$	0	0
$364$	−12.4843	−0.654356
$365$	6.47326	0.338826
$366$	0	0
$367$	6.89794	0.360069	0.180035	−	0.983660i	$-0.442379\pi$
$367$	6.89794	0.360069	0.180035	+	0.983660i	$0.442379\pi$
$368$	−2.65167	−0.138228
$369$	0	0
$370$	−7.29924	−0.379469
$371$	2.33061	0.120999
$372$	0	0
$373$	9.15026	0.473783	0.236891	−	0.971536i	$-0.423871\pi$
$373$	9.15026	0.473783	0.236891	+	0.971536i	$0.423871\pi$
$374$	0	0
$375$	0	0
$376$	−5.62441	−0.290057
$377$	7.69636	0.396383
$378$	0	0
$379$	0.229971	0.0118128	0.00590641	−	0.999983i	$-0.498120\pi$
$379$	0.229971	0.0118128	0.00590641	+	0.999983i	$0.498120\pi$
$380$	9.78766	0.502096
$381$	0	0
$382$	−21.3332	−1.09150
$383$	−21.6992	−1.10878	−0.554389	−	0.832258i	$-0.687048\pi$
$383$	−21.6992	−1.10878	−0.554389	+	0.832258i	$0.687048\pi$
$384$	0	0
$385$	0	0
$386$	13.1360	0.668604
$387$	0	0
$388$	−11.0067	−0.558778
$389$	15.1917	0.770247	0.385124	−	0.922865i	$-0.374159\pi$
$389$	15.1917	0.770247	0.385124	+	0.922865i	$0.374159\pi$
$390$	0	0
$391$	−2.69665	−0.136376
$392$	1.97273	0.0996381
$393$	0	0
$394$	−6.55103	−0.330036
$395$	−13.8326	−0.695996
$396$	0	0
$397$	8.45039	0.424113	0.212056	−	0.977257i	$-0.431984\pi$
$397$	8.45039	0.424113	0.212056	+	0.977257i	$0.431984\pi$
$398$	−7.13599	−0.357695
$399$	0	0
$400$	8.62441	0.431220
$401$	2.56657	0.128169	0.0640843	−	0.997944i	$-0.479587\pi$
$401$	2.56657	0.128169	0.0640843	+	0.997944i	$0.479587\pi$
$402$	0	0
$403$	27.2488	1.35736
$404$	−15.3114	−0.761772
$405$	0	0
$406$	3.09922	0.153812
$407$	0	0
$408$	0	0
$409$	18.6650	0.922924	0.461462	−	0.887160i	$-0.347325\pi$
$409$	18.6650	0.922924	0.461462	+	0.887160i	$0.347325\pi$
$410$	−15.8216	−0.781372
$411$	0	0
$412$	−13.2972	−0.655104
$413$	−5.44373	−0.267868
$414$	0	0
$415$	−20.0748	−0.985433
$416$	−5.56799	−0.272993
$417$	0	0
$418$	0	0
$419$	1.83264	0.0895303	0.0447651	−	0.998998i	$-0.485746\pi$
$419$	1.83264	0.0895303	0.0447651	+	0.998998i	$0.485746\pi$
$420$	0	0
$421$	−38.2174	−1.86260	−0.931302	−	0.364248i	$-0.881326\pi$
$421$	−38.2174	−1.86260	−0.931302	+	0.364248i	$0.881326\pi$
$422$	18.6517	0.907949
$423$	0	0
$424$	1.03945	0.0504802
$425$	8.77070	0.425441
$426$	0	0
$427$	−12.0678	−0.584004
$428$	−18.3848	−0.888663
$429$	0	0
$430$	2.03392	0.0980846
$431$	1.31029	0.0631146	0.0315573	−	0.999502i	$-0.489953\pi$
$431$	1.31029	0.0631146	0.0315573	+	0.999502i	$0.489953\pi$
$432$	0	0
$433$	23.0434	1.10740	0.553698	−	0.832717i	$-0.313216\pi$
$433$	23.0434	1.10740	0.553698	+	0.832717i	$0.313216\pi$
$434$	10.9727	0.526708
$435$	0	0
$436$	−18.1651	−0.869952
$437$	7.03137	0.336356
$438$	0	0
$439$	38.4154	1.83347	0.916733	−	0.399501i	$-0.130817\pi$
$439$	38.4154	1.83347	0.916733	+	0.399501i	$0.130817\pi$
$440$	0	0
$441$	0	0
$442$	−5.66244	−0.269335
$443$	36.8127	1.74902	0.874512	−	0.485004i	$-0.161182\pi$
$443$	36.8127	1.74902	0.874512	+	0.485004i	$0.161182\pi$
$444$	0	0
$445$	−23.1658	−1.09816
$446$	−2.24216	−0.106169
$447$	0	0
$448$	−2.24216	−0.105932
$449$	−19.8257	−0.935632	−0.467816	−	0.883826i	$-0.654959\pi$
$449$	−19.8257	−0.935632	−0.467816	+	0.883826i	$0.654959\pi$
$450$	0	0
$451$	0	0
$452$	−14.4331	−0.678878
$453$	0	0
$454$	−18.3829	−0.862753
$455$	46.0812	2.16032
$456$	0	0
$457$	−3.25987	−0.152490	−0.0762451	−	0.997089i	$-0.524293\pi$
$457$	−3.25987	−0.152490	−0.0762451	+	0.997089i	$0.524293\pi$
$458$	11.2149	0.524037
$459$	0	0
$460$	9.78766	0.456352
$461$	6.97684	0.324944	0.162472	−	0.986713i	$-0.448053\pi$
$461$	6.97684	0.324944	0.162472	+	0.986713i	$0.448053\pi$
$462$	0	0
$463$	37.8943	1.76110	0.880549	−	0.473956i	$-0.157174\pi$
$463$	37.8943	1.76110	0.880549	+	0.473956i	$0.157174\pi$
$464$	1.38225	0.0641694
$465$	0	0
$466$	−17.5249	−0.811825
$467$	−4.32651	−0.200207	−0.100103	−	0.994977i	$-0.531917\pi$
$467$	−4.32651	−0.200207	−0.100103	+	0.994977i	$0.531917\pi$
$468$	0	0
$469$	−33.3208	−1.53861
$470$	20.7604	0.957606
$471$	0	0
$472$	−2.42790	−0.111753
$473$	0	0
$474$	0	0
$475$	−22.8691	−1.04931
$476$	−2.28019	−0.104512
$477$	0	0
$478$	−7.13599	−0.326392
$479$	41.4323	1.89309	0.946546	−	0.322569i	$-0.104547\pi$
$479$	41.4323	1.89309	0.946546	+	0.322569i	$0.104547\pi$
$480$	0	0
$481$	−11.0108	−0.502048
$482$	5.86656	0.267215
$483$	0	0
$484$	−11.0000	−0.500000
$485$	40.6270	1.84478
$486$	0	0
$487$	−16.2319	−0.735535	−0.367768	−	0.929918i	$-0.619878\pi$
$487$	−16.2319	−0.735535	−0.367768	+	0.929918i	$0.619878\pi$
$488$	−5.38225	−0.243643
$489$	0	0
$490$	−7.28161	−0.328950
$491$	−17.6868	−0.798195	−0.399097	−	0.916909i	$-0.630676\pi$
$491$	−17.6868	−0.798195	−0.399097	+	0.916909i	$0.630676\pi$
$492$	0	0
$493$	1.40570	0.0633094
$494$	14.7645	0.664286
$495$	0	0
$496$	4.89383	0.219739
$497$	−19.4745	−0.873549
$498$	0	0
$499$	−4.88575	−0.218716	−0.109358	−	0.994002i	$-0.534880\pi$
$499$	−4.88575	−0.218716	−0.109358	+	0.994002i	$0.534880\pi$
$500$	−13.3781	−0.598289
$501$	0	0
$502$	17.0815	0.762383
$503$	−36.4637	−1.62584	−0.812918	−	0.582378i	$-0.802122\pi$
$503$	−36.4637	−1.62584	−0.812918	+	0.582378i	$0.802122\pi$
$504$	0	0
$505$	56.5164	2.51495
$506$	0	0
$507$	0	0
$508$	−14.4434	−0.640824
$509$	7.71981	0.342175	0.171087	−	0.985256i	$-0.445272\pi$
$509$	7.71981	0.342175	0.171087	+	0.985256i	$0.445272\pi$
$510$	0	0
$511$	3.93215	0.173948
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	11.0959	0.489418
$515$	49.0815	2.16279
$516$	0	0
$517$	0	0
$518$	−4.43389	−0.194814
$519$	0	0
$520$	20.5522	0.901272
$521$	−32.2658	−1.41359	−0.706795	−	0.707419i	$-0.749860\pi$
$521$	−32.2658	−1.41359	−0.706795	+	0.707419i	$0.749860\pi$
$522$	0	0
$523$	18.9782	0.829858	0.414929	−	0.909854i	$-0.363806\pi$
$523$	18.9782	0.829858	0.414929	+	0.909854i	$0.363806\pi$
$524$	−11.4346	−0.499521
$525$	0	0
$526$	23.7836	1.03701
$527$	4.97684	0.216795
$528$	0	0
$529$	−15.9686	−0.694288
$530$	−3.83675	−0.166658
$531$	0	0
$532$	5.94547	0.257769
$533$	−23.8666	−1.03378
$534$	0	0
$535$	67.8606	2.93387
$536$	−14.8610	−0.641899
$537$	0	0
$538$	−24.8404	−1.07095
$539$	0	0
$540$	0	0
$541$	−41.9037	−1.80158	−0.900791	−	0.434254i	$-0.857012\pi$
$541$	−41.9037	−1.80158	−0.900791	+	0.434254i	$0.857012\pi$
$542$	16.6918	0.716974
$543$	0	0
$544$	−1.01696	−0.0436019
$545$	67.0498	2.87210
$546$	0	0
$547$	2.27211	0.0971484	0.0485742	−	0.998820i	$-0.484532\pi$
$547$	2.27211	0.0971484	0.0485742	+	0.998820i	$0.484532\pi$
$548$	17.6134	0.752405
$549$	0	0
$550$	0	0
$551$	−3.66528	−0.156146
$552$	0	0
$553$	−8.40257	−0.357314
$554$	−30.8888	−1.31234
$555$	0	0
$556$	−14.9060	−0.632156
$557$	−1.58354	−0.0670966	−0.0335483	−	0.999437i	$-0.510681\pi$
$557$	−1.58354	−0.0670966	−0.0335483	+	0.999437i	$0.510681\pi$
$558$	0	0
$559$	3.06814	0.129768
$560$	8.27608	0.349728
$561$	0	0
$562$	25.0406	1.05627
$563$	−8.85580	−0.373227	−0.186614	−	0.982433i	$-0.559751\pi$
$563$	−8.85580	−0.373227	−0.186614	+	0.982433i	$0.559751\pi$
$564$	0	0
$565$	53.2745	2.24128
$566$	−27.6964	−1.16417
$567$	0	0
$568$	−8.68560	−0.364440
$569$	−44.1552	−1.85108	−0.925541	−	0.378646i	$-0.876390\pi$
$569$	−44.1552	−1.85108	−0.925541	+	0.378646i	$0.876390\pi$
$570$	0	0
$571$	−29.3114	−1.22664	−0.613322	−	0.789833i	$-0.710167\pi$
$571$	−29.3114	−1.22664	−0.613322	+	0.789833i	$0.710167\pi$
$572$	0	0
$573$	0	0
$574$	−9.61075	−0.401145
$575$	−22.8691	−0.953708
$576$	0	0
$577$	7.78221	0.323978	0.161989	−	0.986793i	$-0.448209\pi$
$577$	7.78221	0.323978	0.161989	+	0.986793i	$0.448209\pi$
$578$	15.9658	0.664089
$579$	0	0
$580$	−5.10206	−0.211852
$581$	−12.1943	−0.505906
$582$	0	0
$583$	0	0
$584$	1.75374	0.0725701
$585$	0	0
$586$	21.3823	0.883293
$587$	25.7895	1.06445	0.532224	−	0.846603i	$-0.321356\pi$
$587$	25.7895	1.06445	0.532224	+	0.846603i	$0.321356\pi$
$588$	0	0
$589$	−12.9768	−0.534701
$590$	8.96168	0.368947
$591$	0	0
$592$	−1.97751	−0.0812752
$593$	−12.9720	−0.532696	−0.266348	−	0.963877i	$-0.585817\pi$
$593$	−12.9720	−0.532696	−0.266348	+	0.963877i	$0.585817\pi$
$594$	0	0
$595$	8.41646	0.345041
$596$	−0.308874	−0.0126520
$597$	0	0
$598$	14.7645	0.603765
$599$	−23.0406	−0.941413	−0.470706	−	0.882290i	$-0.656001\pi$
$599$	−23.0406	−0.941413	−0.470706	+	0.882290i	$0.656001\pi$
$600$	0	0
$601$	19.0884	0.778632	0.389316	−	0.921104i	$-0.372711\pi$
$601$	19.0884	0.778632	0.389316	+	0.921104i	$0.372711\pi$
$602$	1.23550	0.0503551
$603$	0	0
$604$	−4.61155	−0.187641
$605$	40.6024	1.65072
$606$	0	0
$607$	−30.3704	−1.23270	−0.616348	−	0.787474i	$-0.711389\pi$
$607$	−30.3704	−1.23270	−0.616348	+	0.787474i	$0.711389\pi$
$608$	2.65167	0.107540
$609$	0	0
$610$	19.8666	0.804374
$611$	31.3167	1.26694
$612$	0	0
$613$	36.6678	1.48100	0.740500	−	0.672057i	$-0.234589\pi$
$613$	36.6678	1.48100	0.740500	+	0.672057i	$0.234589\pi$
$614$	−5.35640	−0.216167
$615$	0	0
$616$	0	0
$617$	−27.1331	−1.09234	−0.546170	−	0.837675i	$-0.683915\pi$
$617$	−27.1331	−1.09234	−0.546170	+	0.837675i	$0.683915\pi$
$618$	0	0
$619$	−31.2867	−1.25752	−0.628760	−	0.777600i	$-0.716437\pi$
$619$	−31.2867	−1.25752	−0.628760	+	0.777600i	$0.716437\pi$
$620$	−18.0637	−0.725457
$621$	0	0
$622$	11.0980	0.444987
$623$	−14.0720	−0.563781
$624$	0	0
$625$	6.25837	0.250335
$626$	20.4527	0.817452
$627$	0	0
$628$	−7.30335	−0.291435
$629$	−2.01105	−0.0801860
$630$	0	0
$631$	2.37559	0.0945708	0.0472854	−	0.998881i	$-0.484943\pi$
$631$	2.37559	0.0945708	0.0472854	+	0.998881i	$0.484943\pi$
$632$	−3.74754	−0.149069
$633$	0	0
$634$	0.889723	0.0353354
$635$	53.3126	2.11564
$636$	0	0
$637$	−10.9842	−0.435209
$638$	0	0
$639$	0	0
$640$	3.69113	0.145905
$641$	1.27144	0.0502189	0.0251095	−	0.999685i	$-0.492007\pi$
$641$	1.27144	0.0502189	0.0251095	+	0.999685i	$0.492007\pi$
$642$	0	0
$643$	−0.789080	−0.0311183	−0.0155591	−	0.999879i	$-0.504953\pi$
$643$	−0.789080	−0.0311183	−0.0155591	+	0.999879i	$0.504953\pi$
$644$	5.94547	0.234284
$645$	0	0
$646$	2.69665	0.106098
$647$	20.6774	0.812912	0.406456	−	0.913670i	$-0.366764\pi$
$647$	20.6774	0.812912	0.406456	+	0.913670i	$0.366764\pi$
$648$	0	0
$649$	0	0
$650$	−48.0206	−1.88352
$651$	0	0
$652$	−1.73610	−0.0679911
$653$	22.8434	0.893932	0.446966	−	0.894551i	$-0.352505\pi$
$653$	22.8434	0.893932	0.446966	+	0.894551i	$0.352505\pi$
$654$	0	0
$655$	42.2064	1.64914
$656$	−4.28639	−0.167355
$657$	0	0
$658$	12.6108	0.491620
$659$	10.5040	0.409176	0.204588	−	0.978848i	$-0.434414\pi$
$659$	10.5040	0.409176	0.204588	+	0.978848i	$0.434414\pi$
$660$	0	0
$661$	15.0168	0.584084	0.292042	−	0.956405i	$-0.405665\pi$
$661$	15.0168	0.584084	0.292042	+	0.956405i	$0.405665\pi$
$662$	−7.31553	−0.284326
$663$	0	0
$664$	−5.43867	−0.211061
$665$	−21.9455	−0.851009
$666$	0	0
$667$	−3.66528	−0.141920
$668$	1.00000	0.0386912
$669$	0	0
$670$	54.8540	2.11919
$671$	0	0
$672$	0	0
$673$	−16.9370	−0.652872	−0.326436	−	0.945219i	$-0.605848\pi$
$673$	−16.9370	−0.652872	−0.326436	+	0.945219i	$0.605848\pi$
$674$	−6.11827	−0.235667
$675$	0	0
$676$	18.0026	0.692406
$677$	34.3509	1.32021	0.660106	−	0.751173i	$-0.270511\pi$
$677$	34.3509	1.32021	0.660106	+	0.751173i	$0.270511\pi$
$678$	0	0
$679$	24.6787	0.947080
$680$	3.75374	0.143949
$681$	0	0
$682$	0	0
$683$	−3.52452	−0.134862	−0.0674309	−	0.997724i	$-0.521480\pi$
$683$	−3.52452	−0.134862	−0.0674309	+	0.997724i	$0.521480\pi$
$684$	0	0
$685$	−65.0131	−2.48402
$686$	−20.1183	−0.768119
$687$	0	0
$688$	0.551031	0.0210079
$689$	−5.78766	−0.220492
$690$	0	0
$691$	2.91968	0.111070	0.0555349	−	0.998457i	$-0.482314\pi$
$691$	2.91968	0.111070	0.0555349	+	0.998457i	$0.482314\pi$
$692$	5.38225	0.204602
$693$	0	0
$694$	17.6400	0.669603
$695$	55.0200	2.08703
$696$	0	0
$697$	−4.35909	−0.165112
$698$	−12.7420	−0.482292
$699$	0	0
$700$	−19.3373	−0.730880
$701$	−42.3944	−1.60121	−0.800606	−	0.599191i	$-0.795489\pi$
$701$	−42.3944	−1.60121	−0.800606	+	0.599191i	$0.795489\pi$
$702$	0	0
$703$	5.24371	0.197770
$704$	0	0
$705$	0	0
$706$	33.0406	1.24350
$707$	34.3306	1.29114
$708$	0	0
$709$	−48.2207	−1.81096	−0.905482	−	0.424384i	$-0.860491\pi$
$709$	−48.2207	−1.81096	−0.905482	+	0.424384i	$0.860491\pi$
$710$	32.0596	1.20318
$711$	0	0
$712$	−6.27608	−0.235206
$713$	−12.9768	−0.485987
$714$	0	0
$715$	0	0
$716$	−1.30335	−0.0487084
$717$	0	0
$718$	−10.4054	−0.388326
$719$	−27.9344	−1.04178	−0.520889	−	0.853624i	$-0.674399\pi$
$719$	−27.9344	−1.04178	−0.520889	+	0.853624i	$0.674399\pi$
$720$	0	0
$721$	29.8143	1.11034
$722$	11.9686	0.445426
$723$	0	0
$724$	−16.5632	−0.615567
$725$	11.9211	0.442738
$726$	0	0
$727$	4.65653	0.172701	0.0863506	−	0.996265i	$-0.472479\pi$
$727$	4.65653	0.172701	0.0863506	+	0.996265i	$0.472479\pi$
$728$	12.4843	0.462699
$729$	0	0
$730$	−6.47326	−0.239586
$731$	0.560378	0.0207263
$732$	0	0
$733$	3.63957	0.134431	0.0672153	−	0.997738i	$-0.478589\pi$
$733$	3.63957	0.134431	0.0672153	+	0.997738i	$0.478589\pi$
$734$	−6.89794	−0.254608
$735$	0	0
$736$	2.65167	0.0977420
$737$	0	0
$738$	0	0
$739$	40.1562	1.47717	0.738584	−	0.674161i	$-0.235495\pi$
$739$	40.1562	1.47717	0.738584	+	0.674161i	$0.235495\pi$
$740$	7.29924	0.268325
$741$	0	0
$742$	−2.33061	−0.0855595
$743$	23.5167	0.862743	0.431372	−	0.902174i	$-0.358030\pi$
$743$	23.5167	0.862743	0.431372	+	0.902174i	$0.358030\pi$
$744$	0	0
$745$	1.14009	0.0417698
$746$	−9.15026	−0.335015
$747$	0	0
$748$	0	0
$749$	41.2216	1.50620
$750$	0	0
$751$	−9.49844	−0.346603	−0.173301	−	0.984869i	$-0.555443\pi$
$751$	−9.49844	−0.346603	−0.173301	+	0.984869i	$0.555443\pi$
$752$	5.62441	0.205101
$753$	0	0
$754$	−7.69636	−0.280285
$755$	17.0218	0.619487
$756$	0	0
$757$	40.6663	1.47804	0.739021	−	0.673682i	$-0.235288\pi$
$757$	40.6663	1.47804	0.739021	+	0.673682i	$0.235288\pi$
$758$	−0.229971	−0.00835292
$759$	0	0
$760$	−9.78766	−0.355036
$761$	16.3018	0.590939	0.295470	−	0.955352i	$-0.404524\pi$
$761$	16.3018	0.590939	0.295470	+	0.955352i	$0.404524\pi$
$762$	0	0
$763$	40.7291	1.47449
$764$	21.3332	0.771807
$765$	0	0
$766$	21.6992	0.784024
$767$	13.5185	0.488126
$768$	0	0
$769$	−28.9452	−1.04379	−0.521895	−	0.853010i	$-0.674775\pi$
$769$	−28.9452	−1.04379	−0.521895	+	0.853010i	$0.674775\pi$
$770$	0	0
$771$	0	0
$772$	−13.1360	−0.472775
$773$	−25.6351	−0.922030	−0.461015	−	0.887392i	$-0.652515\pi$
$773$	−25.6351	−0.922030	−0.461015	+	0.887392i	$0.652515\pi$
$774$	0	0
$775$	42.2064	1.51610
$776$	11.0067	0.395116
$777$	0	0
$778$	−15.1917	−0.544647
$779$	11.3661	0.407233
$780$	0	0
$781$	0	0
$782$	2.69665	0.0964321
$783$	0	0
$784$	−1.97273	−0.0704548
$785$	26.9576	0.962157
$786$	0	0
$787$	12.1428	0.432843	0.216422	−	0.976300i	$-0.430561\pi$
$787$	12.1428	0.432843	0.216422	+	0.976300i	$0.430561\pi$
$788$	6.55103	0.233371
$789$	0	0
$790$	13.8326	0.492143
$791$	32.3614	1.15064
$792$	0	0
$793$	29.9683	1.06421
$794$	−8.45039	−0.299893
$795$	0	0
$796$	7.13599	0.252928
$797$	−15.6163	−0.553159	−0.276579	−	0.960991i	$-0.589201\pi$
$797$	−15.6163	−0.553159	−0.276579	+	0.960991i	$0.589201\pi$
$798$	0	0
$799$	5.71981	0.202352
$800$	−8.62441	−0.304919
$801$	0	0
$802$	−2.56657	−0.0906289
$803$	0	0
$804$	0	0
$805$	−21.9455	−0.773476
$806$	−27.2488	−0.959799
$807$	0	0
$808$	15.3114	0.538654
$809$	2.93881	0.103323	0.0516615	−	0.998665i	$-0.483548\pi$
$809$	2.93881	0.103323	0.0516615	+	0.998665i	$0.483548\pi$
$810$	0	0
$811$	−38.3561	−1.34687	−0.673433	−	0.739249i	$-0.735181\pi$
$811$	−38.3561	−1.34687	−0.673433	+	0.739249i	$0.735181\pi$
$812$	−3.09922	−0.108761
$813$	0	0
$814$	0	0
$815$	6.40818	0.224469
$816$	0	0
$817$	−1.46115	−0.0511193
$818$	−18.6650	−0.652606
$819$	0	0
$820$	15.8216	0.552514
$821$	16.7552	0.584759	0.292379	−	0.956302i	$-0.405553\pi$
$821$	16.7552	0.584759	0.292379	+	0.956302i	$0.405553\pi$
$822$	0	0
$823$	−13.9827	−0.487408	−0.243704	−	0.969850i	$-0.578363\pi$
$823$	−13.9827	−0.487408	−0.243704	+	0.969850i	$0.578363\pi$
$824$	13.2972	0.463228
$825$	0	0
$826$	5.44373	0.189412
$827$	−0.468770	−0.0163007	−0.00815037	−	0.999967i	$-0.502594\pi$
$827$	−0.468770	−0.0163007	−0.00815037	+	0.999967i	$0.502594\pi$
$828$	0	0
$829$	54.6669	1.89866	0.949329	−	0.314283i	$-0.101764\pi$
$829$	54.6669	1.89866	0.949329	+	0.314283i	$0.101764\pi$
$830$	20.0748	0.696806
$831$	0	0
$832$	5.56799	0.193035
$833$	−2.00620	−0.0695106
$834$	0	0
$835$	−3.69113	−0.127737
$836$	0	0
$837$	0	0
$838$	−1.83264	−0.0633075
$839$	31.9399	1.10269	0.551343	−	0.834279i	$-0.314115\pi$
$839$	31.9399	1.10269	0.551343	+	0.834279i	$0.314115\pi$
$840$	0	0
$841$	−27.0894	−0.934117
$842$	38.2174	1.31706
$843$	0	0
$844$	−18.6517	−0.642017
$845$	−66.4497	−2.28594
$846$	0	0
$847$	24.6637	0.847456
$848$	−1.03945	−0.0356949
$849$	0	0
$850$	−8.77070	−0.300832
$851$	5.24371	0.179752
$852$	0	0
$853$	13.0108	0.445480	0.222740	−	0.974878i	$-0.428500\pi$
$853$	13.0108	0.445480	0.222740	+	0.974878i	$0.428500\pi$
$854$	12.0678	0.412953
$855$	0	0
$856$	18.3848	0.628380
$857$	−16.1331	−0.551096	−0.275548	−	0.961287i	$-0.588859\pi$
$857$	−16.1331	−0.551096	−0.275548	+	0.961287i	$0.588859\pi$
$858$	0	0
$859$	−6.77689	−0.231225	−0.115612	−	0.993294i	$-0.536883\pi$
$859$	−6.77689	−0.231225	−0.115612	+	0.993294i	$0.536883\pi$
$860$	−2.03392	−0.0693563
$861$	0	0
$862$	−1.31029	−0.0446288
$863$	16.4517	0.560023	0.280012	−	0.959997i	$-0.409662\pi$
$863$	16.4517	0.560023	0.280012	+	0.959997i	$0.409662\pi$
$864$	0	0
$865$	−19.8666	−0.675483
$866$	−23.0434	−0.783047
$867$	0	0
$868$	−10.9727	−0.372439
$869$	0	0
$870$	0	0
$871$	82.7462	2.80375
$872$	18.1651	0.615149
$873$	0	0
$874$	−7.03137	−0.237840
$875$	29.9959	1.01405
$876$	0	0
$877$	3.08145	0.104053	0.0520267	−	0.998646i	$-0.483432\pi$
$877$	3.08145	0.104053	0.0520267	+	0.998646i	$0.483432\pi$
$878$	−38.4154	−1.29646
$879$	0	0
$880$	0	0
$881$	−26.5909	−0.895872	−0.447936	−	0.894066i	$-0.647841\pi$
$881$	−26.5909	−0.895872	−0.447936	+	0.894066i	$0.647841\pi$
$882$	0	0
$883$	−40.9997	−1.37975	−0.689875	−	0.723928i	$-0.742335\pi$
$883$	−40.9997	−1.37975	−0.689875	+	0.723928i	$0.742335\pi$
$884$	5.66244	0.190448
$885$	0	0
$886$	−36.8127	−1.23675
$887$	−24.5233	−0.823413	−0.411707	−	0.911316i	$-0.635067\pi$
$887$	−24.5233	−0.823413	−0.411707	+	0.911316i	$0.635067\pi$
$888$	0	0
$889$	32.3845	1.08614
$890$	23.1658	0.776520
$891$	0	0
$892$	2.24216	0.0750730
$893$	−14.9141	−0.499081
$894$	0	0
$895$	4.81082	0.160808
$896$	2.24216	0.0749052
$897$	0	0
$898$	19.8257	0.661592
$899$	6.76450	0.225609
$900$	0	0
$901$	−1.05708	−0.0352166
$902$	0	0
$903$	0	0
$904$	14.4331	0.480039
$905$	61.1369	2.03226
$906$	0	0
$907$	8.73058	0.289894	0.144947	−	0.989439i	$-0.453699\pi$
$907$	8.73058	0.289894	0.144947	+	0.989439i	$0.453699\pi$
$908$	18.3829	0.610059
$909$	0	0
$910$	−46.0812	−1.52758
$911$	−43.0841	−1.42744	−0.713719	−	0.700432i	$-0.752991\pi$
$911$	−43.0841	−1.42744	−0.713719	+	0.700432i	$0.752991\pi$
$912$	0	0
$913$	0	0
$914$	3.25987	0.107827
$915$	0	0
$916$	−11.2149	−0.370550
$917$	25.6381	0.846644
$918$	0	0
$919$	−40.3233	−1.33014	−0.665072	−	0.746779i	$-0.731599\pi$
$919$	−40.3233	−1.33014	−0.665072	+	0.746779i	$0.731599\pi$
$920$	−9.78766	−0.322690
$921$	0	0
$922$	−6.97684	−0.229770
$923$	48.3614	1.59183
$924$	0	0
$925$	−17.0549	−0.560760
$926$	−37.8943	−1.24528
$927$	0	0
$928$	−1.38225	−0.0453746
$929$	6.18762	0.203009	0.101505	−	0.994835i	$-0.467634\pi$
$929$	6.18762	0.203009	0.101505	+	0.994835i	$0.467634\pi$
$930$	0	0
$931$	5.23105	0.171441
$932$	17.5249	0.574047
$933$	0	0
$934$	4.32651	0.141568
$935$	0	0
$936$	0	0
$937$	6.88688	0.224985	0.112492	−	0.993653i	$-0.464117\pi$
$937$	6.88688	0.224985	0.112492	+	0.993653i	$0.464117\pi$
$938$	33.3208	1.08796
$939$	0	0
$940$	−20.7604	−0.677129
$941$	−48.3203	−1.57520	−0.787599	−	0.616188i	$-0.788676\pi$
$941$	−48.3203	−1.57520	−0.787599	+	0.616188i	$0.788676\pi$
$942$	0	0
$943$	11.3661	0.370131
$944$	2.42790	0.0790214
$945$	0	0
$946$	0	0
$947$	13.9867	0.454506	0.227253	−	0.973836i	$-0.427026\pi$
$947$	13.9867	0.454506	0.227253	+	0.973836i	$0.427026\pi$
$948$	0	0
$949$	−9.76479	−0.316978
$950$	22.8691	0.741972
$951$	0	0
$952$	2.28019	0.0739013
$953$	−49.2220	−1.59446	−0.797229	−	0.603677i	$-0.793701\pi$
$953$	−49.2220	−1.59446	−0.797229	+	0.603677i	$0.793701\pi$
$954$	0	0
$955$	−78.7434	−2.54808
$956$	7.13599	0.230794
$957$	0	0
$958$	−41.4323	−1.33862
$959$	−39.4919	−1.27526
$960$	0	0
$961$	−7.05042	−0.227433
$962$	11.0108	0.355001
$963$	0	0
$964$	−5.86656	−0.188949
$965$	48.4866	1.56084
$966$	0	0
$967$	5.91692	0.190275	0.0951376	−	0.995464i	$-0.469671\pi$
$967$	5.91692	0.190275	0.0951376	+	0.995464i	$0.469671\pi$
$968$	11.0000	0.353553
$969$	0	0
$970$	−40.6270	−1.30445
$971$	27.8006	0.892164	0.446082	−	0.894992i	$-0.352819\pi$
$971$	27.8006	0.892164	0.446082	+	0.894992i	$0.352819\pi$
$972$	0	0
$973$	33.4216	1.07145
$974$	16.2319	0.520102
$975$	0	0
$976$	5.38225	0.172282
$977$	34.4493	1.10213	0.551065	−	0.834462i	$-0.314222\pi$
$977$	34.4493	1.10213	0.551065	+	0.834462i	$0.314222\pi$
$978$	0	0
$979$	0	0
$980$	7.28161	0.232602
$981$	0	0
$982$	17.6868	0.564409
$983$	7.58504	0.241925	0.120963	−	0.992657i	$-0.461402\pi$
$983$	7.58504	0.241925	0.120963	+	0.992657i	$0.461402\pi$
$984$	0	0
$985$	−24.1807	−0.770460
$986$	−1.40570	−0.0447665
$987$	0	0
$988$	−14.7645	−0.469721
$989$	−1.46115	−0.0464620
$990$	0	0
$991$	−30.1199	−0.956792	−0.478396	−	0.878144i	$-0.658782\pi$
$991$	−30.1199	−0.956792	−0.478396	+	0.878144i	$0.658782\pi$
$992$	−4.89383	−0.155379
$993$	0	0
$994$	19.4745	0.617693
$995$	−26.3398	−0.835029
$996$	0	0
$997$	−17.5318	−0.555239	−0.277620	−	0.960691i	$-0.589545\pi$
$997$	−17.5318	−0.555239	−0.277620	+	0.960691i	$0.589545\pi$
$998$	4.88575	0.154656
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3006.2.a.s.1.1		4
3.2	odd	2		1002.2.a.i.1.4	✓	4
12.11	even	2		8016.2.a.o.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.i.1.4	✓	4	3.2	odd	2
3006.2.a.s.1.1		4	1.1	even	1	trivial
8016.2.a.o.1.4		4	12.11	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:	\( N \)	\(=\)	\( 3006 = 2 \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3006.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.69113 q^{5} -2.24216 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.69113 q^{5} -2.24216 q^{7} -1.00000 q^{8} +3.69113 q^{10} +5.56799 q^{13} +2.24216 q^{14} +1.00000 q^{16} +1.01696 q^{17} -2.65167 q^{19} -3.69113 q^{20} -2.65167 q^{23} +8.62441 q^{25} -5.56799 q^{26} -2.24216 q^{28} +1.38225 q^{29} +4.89383 q^{31} -1.00000 q^{32} -1.01696 q^{34} +8.27608 q^{35} -1.97751 q^{37} +2.65167 q^{38} +3.69113 q^{40} -4.28639 q^{41} +0.551031 q^{43} +2.65167 q^{46} +5.62441 q^{47} -1.97273 q^{49} -8.62441 q^{50} +5.56799 q^{52} -1.03945 q^{53} +2.24216 q^{56} -1.38225 q^{58} +2.42790 q^{59} +5.38225 q^{61} -4.89383 q^{62} +1.00000 q^{64} -20.5522 q^{65} +14.8610 q^{67} +1.01696 q^{68} -8.27608 q^{70} +8.68560 q^{71} -1.75374 q^{73} +1.97751 q^{74} -2.65167 q^{76} +3.74754 q^{79} -3.69113 q^{80} +4.28639 q^{82} +5.43867 q^{83} -3.75374 q^{85} -0.551031 q^{86} +6.27608 q^{89} -12.4843 q^{91} -2.65167 q^{92} -5.62441 q^{94} +9.78766 q^{95} -11.0067 q^{97} +1.97273 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 5 q^{5} + q^{7} - 4 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 - 5 * q^5 + q^7 - 4 * q^8	\( 4 q - 4 q^{2} + 4 q^{4} - 5 q^{5} + q^{7} - 4 q^{8} + 5 q^{10} + 8 q^{13} - q^{14} + 4 q^{16} - 10 q^{17} - 2 q^{19} - 5 q^{20} - 2 q^{23} + 5 q^{25} - 8 q^{26} + q^{28} - 14 q^{29} + q^{31} - 4 q^{32} + 10 q^{34} - 5 q^{35} + 5 q^{37} + 2 q^{38} + 5 q^{40} - 14 q^{41} + 2 q^{43} + 2 q^{46} - 7 q^{47} + 13 q^{49} - 5 q^{50} + 8 q^{52} - 3 q^{53} - q^{56} + 14 q^{58} + 5 q^{59} + 2 q^{61} - q^{62} + 4 q^{64} - 6 q^{65} - 7 q^{67} - 10 q^{68} + 5 q^{70} - 2 q^{71} + 2 q^{73} - 5 q^{74} - 2 q^{76} - 10 q^{79} - 5 q^{80} + 14 q^{82} - 13 q^{83} - 6 q^{85} - 2 q^{86} - 13 q^{89} - 30 q^{91} - 2 q^{92} + 7 q^{94} + 2 q^{95} + 5 q^{97} - 13 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 5 * q^5 + q^7 - 4 * q^8 + 5 * q^10 + 8 * q^13 - q^14 + 4 * q^16 - 10 * q^17 - 2 * q^19 - 5 * q^20 - 2 * q^23 + 5 * q^25 - 8 * q^26 + q^28 - 14 * q^29 + q^31 - 4 * q^32 + 10 * q^34 - 5 * q^35 + 5 * q^37 + 2 * q^38 + 5 * q^40 - 14 * q^41 + 2 * q^43 + 2 * q^46 - 7 * q^47 + 13 * q^49 - 5 * q^50 + 8 * q^52 - 3 * q^53 - q^56 + 14 * q^58 + 5 * q^59 + 2 * q^61 - q^62 + 4 * q^64 - 6 * q^65 - 7 * q^67 - 10 * q^68 + 5 * q^70 - 2 * q^71 + 2 * q^73 - 5 * q^74 - 2 * q^76 - 10 * q^79 - 5 * q^80 + 14 * q^82 - 13 * q^83 - 6 * q^85 - 2 * q^86 - 13 * q^89 - 30 * q^91 - 2 * q^92 + 7 * q^94 + 2 * q^95 + 5 * q^97 - 13 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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