LMFDB - Embedded newform 6003.2.a.d.1.2

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:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{4} +3.56155 q^{5} +3.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} -1.00000 q^{4} +3.56155 q^{5} +3.00000 q^{8} -3.56155 q^{10} +3.56155 q^{11} -3.56155 q^{13} -1.00000 q^{16} -5.12311 q^{19} -3.56155 q^{20} -3.56155 q^{22} -1.00000 q^{23} +7.68466 q^{25} +3.56155 q^{26} +1.00000 q^{29} -2.43845 q^{31} -5.00000 q^{32} +2.43845 q^{37} +5.12311 q^{38} +10.6847 q^{40} -7.56155 q^{41} -4.24621 q^{43} -3.56155 q^{44} +1.00000 q^{46} -7.12311 q^{47} -7.00000 q^{49} -7.68466 q^{50} +3.56155 q^{52} +4.24621 q^{53} +12.6847 q^{55} -1.00000 q^{58} -9.56155 q^{59} -11.8078 q^{61} +2.43845 q^{62} +7.00000 q^{64} -12.6847 q^{65} +6.43845 q^{67} -10.4384 q^{71} -5.12311 q^{73} -2.43845 q^{74} +5.12311 q^{76} -2.00000 q^{79} -3.56155 q^{80} +7.56155 q^{82} +7.12311 q^{83} +4.24621 q^{86} +10.6847 q^{88} -17.3693 q^{89} +1.00000 q^{92} +7.12311 q^{94} -18.2462 q^{95} -13.3693 q^{97} +7.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{4} + 3 q^{5} + 6 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^4 + 3 * q^5 + 6 * q^8	$ 2 q - 2 q^{2} - 2 q^{4} + 3 q^{5} + 6 q^{8} - 3 q^{10} + 3 q^{11} - 3 q^{13} - 2 q^{16} - 2 q^{19} - 3 q^{20} - 3 q^{22} - 2 q^{23} + 3 q^{25} + 3 q^{26} + 2 q^{29} - 9 q^{31} - 10 q^{32} + 9 q^{37} + 2 q^{38} + 9 q^{40} - 11 q^{41} + 8 q^{43} - 3 q^{44} + 2 q^{46} - 6 q^{47} - 14 q^{49} - 3 q^{50} + 3 q^{52} - 8 q^{53} + 13 q^{55} - 2 q^{58} - 15 q^{59} - 3 q^{61} + 9 q^{62} + 14 q^{64} - 13 q^{65} + 17 q^{67} - 25 q^{71} - 2 q^{73} - 9 q^{74} + 2 q^{76} - 4 q^{79} - 3 q^{80} + 11 q^{82} + 6 q^{83} - 8 q^{86} + 9 q^{88} - 10 q^{89} + 2 q^{92} + 6 q^{94} - 20 q^{95} - 2 q^{97} + 14 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^4 + 3 * q^5 + 6 * q^8 - 3 * q^10 + 3 * q^11 - 3 * q^13 - 2 * q^16 - 2 * q^19 - 3 * q^20 - 3 * q^22 - 2 * q^23 + 3 * q^25 + 3 * q^26 + 2 * q^29 - 9 * q^31 - 10 * q^32 + 9 * q^37 + 2 * q^38 + 9 * q^40 - 11 * q^41 + 8 * q^43 - 3 * q^44 + 2 * q^46 - 6 * q^47 - 14 * q^49 - 3 * q^50 + 3 * q^52 - 8 * q^53 + 13 * q^55 - 2 * q^58 - 15 * q^59 - 3 * q^61 + 9 * q^62 + 14 * q^64 - 13 * q^65 + 17 * q^67 - 25 * q^71 - 2 * q^73 - 9 * q^74 + 2 * q^76 - 4 * q^79 - 3 * q^80 + 11 * q^82 + 6 * q^83 - 8 * q^86 + 9 * q^88 - 10 * q^89 + 2 * q^92 + 6 * q^94 - 20 * q^95 - 2 * q^97 + 14 * 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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	3.56155	1.59277	0.796387	−	0.604787i	$-0.206742\pi$
$5$	3.56155	1.59277	0.796387	+	0.604787i	$0.206742\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	3.00000	1.06066
$9$	0	0
$10$	−3.56155	−1.12626
$11$	3.56155	1.07385	0.536924	−	0.843630i	$-0.319586\pi$
$11$	3.56155	1.07385	0.536924	+	0.843630i	$0.319586\pi$
$12$	0	0
$13$	−3.56155	−0.987797	−0.493899	−	0.869520i	$-0.664429\pi$
$13$	−3.56155	−0.987797	−0.493899	+	0.869520i	$0.664429\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−5.12311	−1.17532	−0.587661	−	0.809108i	$-0.699951\pi$
$19$	−5.12311	−1.17532	−0.587661	+	0.809108i	$0.699951\pi$
$20$	−3.56155	−0.796387
$21$	0	0
$22$	−3.56155	−0.759326
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	7.68466	1.53693
$26$	3.56155	0.698478
$27$	0	0
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−2.43845	−0.437958	−0.218979	−	0.975730i	$-0.570273\pi$
$31$	−2.43845	−0.437958	−0.218979	+	0.975730i	$0.570273\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.43845	0.400878	0.200439	−	0.979706i	$-0.435763\pi$
$37$	2.43845	0.400878	0.200439	+	0.979706i	$0.435763\pi$
$38$	5.12311	0.831077
$39$	0	0
$40$	10.6847	1.68939
$41$	−7.56155	−1.18092	−0.590458	−	0.807068i	$-0.701053\pi$
$41$	−7.56155	−1.18092	−0.590458	+	0.807068i	$0.701053\pi$
$42$	0	0
$43$	−4.24621	−0.647541	−0.323771	−	0.946136i	$-0.604951\pi$
$43$	−4.24621	−0.647541	−0.323771	+	0.946136i	$0.604951\pi$
$44$	−3.56155	−0.536924
$45$	0	0
$46$	1.00000	0.147442
$47$	−7.12311	−1.03901	−0.519506	−	0.854467i	$-0.673884\pi$
$47$	−7.12311	−1.03901	−0.519506	+	0.854467i	$0.673884\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	−7.68466	−1.08677
$51$	0	0
$52$	3.56155	0.493899
$53$	4.24621	0.583262	0.291631	−	0.956531i	$-0.405802\pi$
$53$	4.24621	0.583262	0.291631	+	0.956531i	$0.405802\pi$
$54$	0	0
$55$	12.6847	1.71040
$56$	0	0
$57$	0	0
$58$	−1.00000	−0.131306
$59$	−9.56155	−1.24481	−0.622404	−	0.782696i	$-0.713844\pi$
$59$	−9.56155	−1.24481	−0.622404	+	0.782696i	$0.713844\pi$
$60$	0	0
$61$	−11.8078	−1.51183	−0.755915	−	0.654670i	$-0.772808\pi$
$61$	−11.8078	−1.51183	−0.755915	+	0.654670i	$0.772808\pi$
$62$	2.43845	0.309683
$63$	0	0
$64$	7.00000	0.875000
$65$	−12.6847	−1.57334
$66$	0	0
$67$	6.43845	0.786582	0.393291	−	0.919414i	$-0.371337\pi$
$67$	6.43845	0.786582	0.393291	+	0.919414i	$0.371337\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.4384	−1.23882	−0.619408	−	0.785069i	$-0.712627\pi$
$71$	−10.4384	−1.23882	−0.619408	+	0.785069i	$0.712627\pi$
$72$	0	0
$73$	−5.12311	−0.599614	−0.299807	−	0.954000i	$-0.596922\pi$
$73$	−5.12311	−0.599614	−0.299807	+	0.954000i	$0.596922\pi$
$74$	−2.43845	−0.283464
$75$	0	0
$76$	5.12311	0.587661
$77$	0	0
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	−3.56155	−0.398194
$81$	0	0
$82$	7.56155	0.835034
$83$	7.12311	0.781862	0.390931	−	0.920420i	$-0.372153\pi$
$83$	7.12311	0.781862	0.390931	+	0.920420i	$0.372153\pi$
$84$	0	0
$85$	0	0
$86$	4.24621	0.457881
$87$	0	0
$88$	10.6847	1.13899
$89$	−17.3693	−1.84114	−0.920572	−	0.390573i	$-0.872277\pi$
$89$	−17.3693	−1.84114	−0.920572	+	0.390573i	$0.872277\pi$
$90$	0	0
$91$	0	0
$92$	1.00000	0.104257
$93$	0	0
$94$	7.12311	0.734692
$95$	−18.2462	−1.87202
$96$	0	0
$97$	−13.3693	−1.35745	−0.678724	−	0.734393i	$-0.737467\pi$
$97$	−13.3693	−1.35745	−0.678724	+	0.734393i	$0.737467\pi$
$98$	7.00000	0.707107
$99$	0	0
$100$	−7.68466	−0.768466
$101$	7.56155	0.752403	0.376201	−	0.926538i	$-0.377230\pi$
$101$	7.56155	0.752403	0.376201	+	0.926538i	$0.377230\pi$
$102$	0	0
$103$	1.56155	0.153864	0.0769322	−	0.997036i	$-0.475488\pi$
$103$	1.56155	0.153864	0.0769322	+	0.997036i	$0.475488\pi$
$104$	−10.6847	−1.04772
$105$	0	0
$106$	−4.24621	−0.412428
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−12.6847	−1.20943
$111$	0	0
$112$	0	0
$113$	−13.3693	−1.25768	−0.628840	−	0.777535i	$-0.716470\pi$
$113$	−13.3693	−1.25768	−0.628840	+	0.777535i	$0.716470\pi$
$114$	0	0
$115$	−3.56155	−0.332117
$116$	−1.00000	−0.0928477
$117$	0	0
$118$	9.56155	0.880212
$119$	0	0
$120$	0	0
$121$	1.68466	0.153151
$122$	11.8078	1.06902
$123$	0	0
$124$	2.43845	0.218979
$125$	9.56155	0.855211
$126$	0	0
$127$	11.8078	1.04777	0.523885	−	0.851789i	$-0.324482\pi$
$127$	11.8078	1.04777	0.523885	+	0.851789i	$0.324482\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	12.6847	1.11252
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	0	0
$134$	−6.43845	−0.556197
$135$	0	0
$136$	0	0
$137$	−1.75379	−0.149836	−0.0749181	−	0.997190i	$-0.523870\pi$
$137$	−1.75379	−0.149836	−0.0749181	+	0.997190i	$0.523870\pi$
$138$	0	0
$139$	−19.6155	−1.66377	−0.831884	−	0.554950i	$-0.812737\pi$
$139$	−19.6155	−1.66377	−0.831884	+	0.554950i	$0.812737\pi$
$140$	0	0
$141$	0	0
$142$	10.4384	0.875975
$143$	−12.6847	−1.06074
$144$	0	0
$145$	3.56155	0.295771
$146$	5.12311	0.423991
$147$	0	0
$148$	−2.43845	−0.200439
$149$	16.9309	1.38703	0.693515	−	0.720442i	$-0.256061\pi$
$149$	16.9309	1.38703	0.693515	+	0.720442i	$0.256061\pi$
$150$	0	0
$151$	6.24621	0.508309	0.254155	−	0.967164i	$-0.418203\pi$
$151$	6.24621	0.508309	0.254155	+	0.967164i	$0.418203\pi$
$152$	−15.3693	−1.24662
$153$	0	0
$154$	0	0
$155$	−8.68466	−0.697569
$156$	0	0
$157$	19.1231	1.52619	0.763095	−	0.646286i	$-0.223679\pi$
$157$	19.1231	1.52619	0.763095	+	0.646286i	$0.223679\pi$
$158$	2.00000	0.159111
$159$	0	0
$160$	−17.8078	−1.40783
$161$	0	0
$162$	0	0
$163$	5.56155	0.435614	0.217807	−	0.975992i	$-0.430110\pi$
$163$	5.56155	0.435614	0.217807	+	0.975992i	$0.430110\pi$
$164$	7.56155	0.590458
$165$	0	0
$166$	−7.12311	−0.552860
$167$	6.93087	0.536327	0.268163	−	0.963373i	$-0.413583\pi$
$167$	6.93087	0.536327	0.268163	+	0.963373i	$0.413583\pi$
$168$	0	0
$169$	−0.315342	−0.0242570
$170$	0	0
$171$	0	0
$172$	4.24621	0.323771
$173$	−10.4924	−0.797724	−0.398862	−	0.917011i	$-0.630595\pi$
$173$	−10.4924	−0.797724	−0.398862	+	0.917011i	$0.630595\pi$
$174$	0	0
$175$	0	0
$176$	−3.56155	−0.268462
$177$	0	0
$178$	17.3693	1.30189
$179$	16.4924	1.23270	0.616351	−	0.787472i	$-0.288610\pi$
$179$	16.4924	1.23270	0.616351	+	0.787472i	$0.288610\pi$
$180$	0	0
$181$	18.4924	1.37453	0.687265	−	0.726406i	$-0.258811\pi$
$181$	18.4924	1.37453	0.687265	+	0.726406i	$0.258811\pi$
$182$	0	0
$183$	0	0
$184$	−3.00000	−0.221163
$185$	8.68466	0.638509
$186$	0	0
$187$	0	0
$188$	7.12311	0.519506
$189$	0	0
$190$	18.2462	1.32372
$191$	19.5616	1.41542	0.707712	−	0.706501i	$-0.249727\pi$
$191$	19.5616	1.41542	0.707712	+	0.706501i	$0.249727\pi$
$192$	0	0
$193$	−6.87689	−0.495010	−0.247505	−	0.968887i	$-0.579611\pi$
$193$	−6.87689	−0.495010	−0.247505	+	0.968887i	$0.579611\pi$
$194$	13.3693	0.959861
$195$	0	0
$196$	7.00000	0.500000
$197$	−12.2462	−0.872506	−0.436253	−	0.899824i	$-0.643695\pi$
$197$	−12.2462	−0.872506	−0.436253	+	0.899824i	$0.643695\pi$
$198$	0	0
$199$	3.31534	0.235018	0.117509	−	0.993072i	$-0.462509\pi$
$199$	3.31534	0.235018	0.117509	+	0.993072i	$0.462509\pi$
$200$	23.0540	1.63016
$201$	0	0
$202$	−7.56155	−0.532029
$203$	0	0
$204$	0	0
$205$	−26.9309	−1.88093
$206$	−1.56155	−0.108799
$207$	0	0
$208$	3.56155	0.246949
$209$	−18.2462	−1.26212
$210$	0	0
$211$	21.1771	1.45789	0.728945	−	0.684572i	$-0.240011\pi$
$211$	21.1771	1.45789	0.728945	+	0.684572i	$0.240011\pi$
$212$	−4.24621	−0.291631
$213$	0	0
$214$	0	0
$215$	−15.1231	−1.03139
$216$	0	0
$217$	0	0
$218$	−2.00000	−0.135457
$219$	0	0
$220$	−12.6847	−0.855199
$221$	0	0
$222$	0	0
$223$	−28.4924	−1.90799	−0.953997	−	0.299817i	$-0.903075\pi$
$223$	−28.4924	−1.90799	−0.953997	+	0.299817i	$0.903075\pi$
$224$	0	0
$225$	0	0
$226$	13.3693	0.889314
$227$	29.3693	1.94931	0.974655	−	0.223713i	$-0.0718179\pi$
$227$	29.3693	1.94931	0.974655	+	0.223713i	$0.0718179\pi$
$228$	0	0
$229$	0.192236	0.0127033	0.00635165	−	0.999980i	$-0.497978\pi$
$229$	0.192236	0.0127033	0.00635165	+	0.999980i	$0.497978\pi$
$230$	3.56155	0.234842
$231$	0	0
$232$	3.00000	0.196960
$233$	−9.12311	−0.597675	−0.298837	−	0.954304i	$-0.596599\pi$
$233$	−9.12311	−0.597675	−0.298837	+	0.954304i	$0.596599\pi$
$234$	0	0
$235$	−25.3693	−1.65491
$236$	9.56155	0.622404
$237$	0	0
$238$	0	0
$239$	−10.4384	−0.675207	−0.337604	−	0.941288i	$-0.609616\pi$
$239$	−10.4384	−0.675207	−0.337604	+	0.941288i	$0.609616\pi$
$240$	0	0
$241$	−17.6155	−1.13472	−0.567358	−	0.823471i	$-0.692034\pi$
$241$	−17.6155	−1.13472	−0.567358	+	0.823471i	$0.692034\pi$
$242$	−1.68466	−0.108294
$243$	0	0
$244$	11.8078	0.755915
$245$	−24.9309	−1.59277
$246$	0	0
$247$	18.2462	1.16098
$248$	−7.31534	−0.464525
$249$	0	0
$250$	−9.56155	−0.604726
$251$	−10.6847	−0.674410	−0.337205	−	0.941431i	$-0.609481\pi$
$251$	−10.6847	−0.674410	−0.337205	+	0.941431i	$0.609481\pi$
$252$	0	0
$253$	−3.56155	−0.223913
$254$	−11.8078	−0.740885
$255$	0	0
$256$	−17.0000	−1.06250
$257$	4.24621	0.264871	0.132436	−	0.991192i	$-0.457720\pi$
$257$	4.24621	0.264871	0.132436	+	0.991192i	$0.457720\pi$
$258$	0	0
$259$	0	0
$260$	12.6847	0.786669
$261$	0	0
$262$	−4.00000	−0.247121
$263$	23.3693	1.44101	0.720507	−	0.693448i	$-0.243909\pi$
$263$	23.3693	1.44101	0.720507	+	0.693448i	$0.243909\pi$
$264$	0	0
$265$	15.1231	0.929005
$266$	0	0
$267$	0	0
$268$	−6.43845	−0.393291
$269$	4.93087	0.300640	0.150320	−	0.988637i	$-0.451970\pi$
$269$	4.93087	0.300640	0.150320	+	0.988637i	$0.451970\pi$
$270$	0	0
$271$	−13.5616	−0.823806	−0.411903	−	0.911228i	$-0.635136\pi$
$271$	−13.5616	−0.823806	−0.411903	+	0.911228i	$0.635136\pi$
$272$	0	0
$273$	0	0
$274$	1.75379	0.105950
$275$	27.3693	1.65043
$276$	0	0
$277$	12.9309	0.776941	0.388470	−	0.921461i	$-0.373004\pi$
$277$	12.9309	0.776941	0.388470	+	0.921461i	$0.373004\pi$
$278$	19.6155	1.17646
$279$	0	0
$280$	0	0
$281$	4.24621	0.253308	0.126654	−	0.991947i	$-0.459576\pi$
$281$	4.24621	0.253308	0.126654	+	0.991947i	$0.459576\pi$
$282$	0	0
$283$	0.684658	0.0406987	0.0203494	−	0.999793i	$-0.493522\pi$
$283$	0.684658	0.0406987	0.0203494	+	0.999793i	$0.493522\pi$
$284$	10.4384	0.619408
$285$	0	0
$286$	12.6847	0.750060
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	−3.56155	−0.209142
$291$	0	0
$292$	5.12311	0.299807
$293$	16.0000	0.934730	0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000	0.934730	0.467365	+	0.884064i	$0.345203\pi$
$294$	0	0
$295$	−34.0540	−1.98270
$296$	7.31534	0.425196
$297$	0	0
$298$	−16.9309	−0.980779
$299$	3.56155	0.205970
$300$	0	0
$301$	0	0
$302$	−6.24621	−0.359429
$303$	0	0
$304$	5.12311	0.293830
$305$	−42.0540	−2.40800
$306$	0	0
$307$	−7.31534	−0.417509	−0.208754	−	0.977968i	$-0.566941\pi$
$307$	−7.31534	−0.417509	−0.208754	+	0.977968i	$0.566941\pi$
$308$	0	0
$309$	0	0
$310$	8.68466	0.493255
$311$	−16.8769	−0.957001	−0.478500	−	0.878087i	$-0.658819\pi$
$311$	−16.8769	−0.957001	−0.478500	+	0.878087i	$0.658819\pi$
$312$	0	0
$313$	−3.75379	−0.212177	−0.106088	−	0.994357i	$-0.533833\pi$
$313$	−3.75379	−0.212177	−0.106088	+	0.994357i	$0.533833\pi$
$314$	−19.1231	−1.07918
$315$	0	0
$316$	2.00000	0.112509
$317$	−8.93087	−0.501608	−0.250804	−	0.968038i	$-0.580695\pi$
$317$	−8.93087	−0.501608	−0.250804	+	0.968038i	$0.580695\pi$
$318$	0	0
$319$	3.56155	0.199409
$320$	24.9309	1.39368
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−27.3693	−1.51818
$326$	−5.56155	−0.308026
$327$	0	0
$328$	−22.6847	−1.25255
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−7.12311	−0.390931
$333$	0	0
$334$	−6.93087	−0.379240
$335$	22.9309	1.25285
$336$	0	0
$337$	−23.8078	−1.29689	−0.648446	−	0.761261i	$-0.724581\pi$
$337$	−23.8078	−1.29689	−0.648446	+	0.761261i	$0.724581\pi$
$338$	0.315342	0.0171523
$339$	0	0
$340$	0	0
$341$	−8.68466	−0.470301
$342$	0	0
$343$	0	0
$344$	−12.7386	−0.686821
$345$	0	0
$346$	10.4924	0.564076
$347$	−24.4924	−1.31482	−0.657411	−	0.753532i	$-0.728348\pi$
$347$	−24.4924	−1.31482	−0.657411	+	0.753532i	$0.728348\pi$
$348$	0	0
$349$	22.3002	1.19370	0.596851	−	0.802352i	$-0.296418\pi$
$349$	22.3002	1.19370	0.596851	+	0.802352i	$0.296418\pi$
$350$	0	0
$351$	0	0
$352$	−17.8078	−0.949157
$353$	−10.4924	−0.558455	−0.279228	−	0.960225i	$-0.590078\pi$
$353$	−10.4924	−0.558455	−0.279228	+	0.960225i	$0.590078\pi$
$354$	0	0
$355$	−37.1771	−1.97315
$356$	17.3693	0.920572
$357$	0	0
$358$	−16.4924	−0.871652
$359$	5.12311	0.270387	0.135194	−	0.990819i	$-0.456834\pi$
$359$	5.12311	0.270387	0.135194	+	0.990819i	$0.456834\pi$
$360$	0	0
$361$	7.24621	0.381380
$362$	−18.4924	−0.971940
$363$	0	0
$364$	0	0
$365$	−18.2462	−0.955050
$366$	0	0
$367$	24.7386	1.29135	0.645673	−	0.763614i	$-0.276577\pi$
$367$	24.7386	1.29135	0.645673	+	0.763614i	$0.276577\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	−8.68466	−0.451494
$371$	0	0
$372$	0	0
$373$	2.49242	0.129053	0.0645264	−	0.997916i	$-0.479446\pi$
$373$	2.49242	0.129053	0.0645264	+	0.997916i	$0.479446\pi$
$374$	0	0
$375$	0	0
$376$	−21.3693	−1.10204
$377$	−3.56155	−0.183429
$378$	0	0
$379$	30.0000	1.54100	0.770498	−	0.637442i	$-0.220007\pi$
$379$	30.0000	1.54100	0.770498	+	0.637442i	$0.220007\pi$
$380$	18.2462	0.936011
$381$	0	0
$382$	−19.5616	−1.00086
$383$	−8.87689	−0.453588	−0.226794	−	0.973943i	$-0.572824\pi$
$383$	−8.87689	−0.453588	−0.226794	+	0.973943i	$0.572824\pi$
$384$	0	0
$385$	0	0
$386$	6.87689	0.350025
$387$	0	0
$388$	13.3693	0.678724
$389$	−38.2462	−1.93916	−0.969580	−	0.244775i	$-0.921286\pi$
$389$	−38.2462	−1.93916	−0.969580	+	0.244775i	$0.921286\pi$
$390$	0	0
$391$	0	0
$392$	−21.0000	−1.06066
$393$	0	0
$394$	12.2462	0.616955
$395$	−7.12311	−0.358402
$396$	0	0
$397$	−15.7538	−0.790660	−0.395330	−	0.918539i	$-0.629370\pi$
$397$	−15.7538	−0.790660	−0.395330	+	0.918539i	$0.629370\pi$
$398$	−3.31534	−0.166183
$399$	0	0
$400$	−7.68466	−0.384233
$401$	2.19224	0.109475	0.0547375	−	0.998501i	$-0.482568\pi$
$401$	2.19224	0.109475	0.0547375	+	0.998501i	$0.482568\pi$
$402$	0	0
$403$	8.68466	0.432614
$404$	−7.56155	−0.376201
$405$	0	0
$406$	0	0
$407$	8.68466	0.430483
$408$	0	0
$409$	20.2462	1.00111	0.500555	−	0.865705i	$-0.333129\pi$
$409$	20.2462	1.00111	0.500555	+	0.865705i	$0.333129\pi$
$410$	26.9309	1.33002
$411$	0	0
$412$	−1.56155	−0.0769322
$413$	0	0
$414$	0	0
$415$	25.3693	1.24533
$416$	17.8078	0.873097
$417$	0	0
$418$	18.2462	0.892451
$419$	−23.1231	−1.12964	−0.564819	−	0.825215i	$-0.691054\pi$
$419$	−23.1231	−1.12964	−0.564819	+	0.825215i	$0.691054\pi$
$420$	0	0
$421$	19.8078	0.965371	0.482686	−	0.875794i	$-0.339661\pi$
$421$	19.8078	0.965371	0.482686	+	0.875794i	$0.339661\pi$
$422$	−21.1771	−1.03088
$423$	0	0
$424$	12.7386	0.618643
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	15.1231	0.729301
$431$	−20.8769	−1.00560	−0.502802	−	0.864401i	$-0.667698\pi$
$431$	−20.8769	−1.00560	−0.502802	+	0.864401i	$0.667698\pi$
$432$	0	0
$433$	−20.8769	−1.00328	−0.501640	−	0.865077i	$-0.667270\pi$
$433$	−20.8769	−1.00328	−0.501640	+	0.865077i	$0.667270\pi$
$434$	0	0
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	5.12311	0.245071
$438$	0	0
$439$	29.8617	1.42522	0.712612	−	0.701559i	$-0.247512\pi$
$439$	29.8617	1.42522	0.712612	+	0.701559i	$0.247512\pi$
$440$	38.0540	1.81415
$441$	0	0
$442$	0	0
$443$	−4.87689	−0.231708	−0.115854	−	0.993266i	$-0.536961\pi$
$443$	−4.87689	−0.231708	−0.115854	+	0.993266i	$0.536961\pi$
$444$	0	0
$445$	−61.8617	−2.93253
$446$	28.4924	1.34916
$447$	0	0
$448$	0	0
$449$	−17.3153	−0.817161	−0.408581	−	0.912722i	$-0.633976\pi$
$449$	−17.3153	−0.817161	−0.408581	+	0.912722i	$0.633976\pi$
$450$	0	0
$451$	−26.9309	−1.26813
$452$	13.3693	0.628840
$453$	0	0
$454$	−29.3693	−1.37837
$455$	0	0
$456$	0	0
$457$	−1.12311	−0.0525367	−0.0262683	−	0.999655i	$-0.508362\pi$
$457$	−1.12311	−0.0525367	−0.0262683	+	0.999655i	$0.508362\pi$
$458$	−0.192236	−0.00898260
$459$	0	0
$460$	3.56155	0.166058
$461$	38.3002	1.78382	0.891909	−	0.452215i	$-0.149366\pi$
$461$	38.3002	1.78382	0.891909	+	0.452215i	$0.149366\pi$
$462$	0	0
$463$	20.4924	0.952364	0.476182	−	0.879347i	$-0.342020\pi$
$463$	20.4924	0.952364	0.476182	+	0.879347i	$0.342020\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	9.12311	0.422620
$467$	−25.4233	−1.17645	−0.588225	−	0.808697i	$-0.700173\pi$
$467$	−25.4233	−1.17645	−0.588225	+	0.808697i	$0.700173\pi$
$468$	0	0
$469$	0	0
$470$	25.3693	1.17020
$471$	0	0
$472$	−28.6847	−1.32032
$473$	−15.1231	−0.695361
$474$	0	0
$475$	−39.3693	−1.80639
$476$	0	0
$477$	0	0
$478$	10.4384	0.477443
$479$	−21.3153	−0.973923	−0.486961	−	0.873423i	$-0.661895\pi$
$479$	−21.3153	−0.973923	−0.486961	+	0.873423i	$0.661895\pi$
$480$	0	0
$481$	−8.68466	−0.395986
$482$	17.6155	0.802365
$483$	0	0
$484$	−1.68466	−0.0765754
$485$	−47.6155	−2.16211
$486$	0	0
$487$	−10.7386	−0.486614	−0.243307	−	0.969949i	$-0.578232\pi$
$487$	−10.7386	−0.486614	−0.243307	+	0.969949i	$0.578232\pi$
$488$	−35.4233	−1.60354
$489$	0	0
$490$	24.9309	1.12626
$491$	2.63068	0.118721	0.0593605	−	0.998237i	$-0.481094\pi$
$491$	2.63068	0.118721	0.0593605	+	0.998237i	$0.481094\pi$
$492$	0	0
$493$	0	0
$494$	−18.2462	−0.820936
$495$	0	0
$496$	2.43845	0.109490
$497$	0	0
$498$	0	0
$499$	5.36932	0.240364	0.120182	−	0.992752i	$-0.461652\pi$
$499$	5.36932	0.240364	0.120182	+	0.992752i	$0.461652\pi$
$500$	−9.56155	−0.427606
$501$	0	0
$502$	10.6847	0.476880
$503$	−16.6307	−0.741526	−0.370763	−	0.928728i	$-0.620904\pi$
$503$	−16.6307	−0.741526	−0.370763	+	0.928728i	$0.620904\pi$
$504$	0	0
$505$	26.9309	1.19841
$506$	3.56155	0.158330
$507$	0	0
$508$	−11.8078	−0.523885
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	0	0
$512$	11.0000	0.486136
$513$	0	0
$514$	−4.24621	−0.187292
$515$	5.56155	0.245071
$516$	0	0
$517$	−25.3693	−1.11574
$518$	0	0
$519$	0	0
$520$	−38.0540	−1.66878
$521$	28.2462	1.23749	0.618744	−	0.785592i	$-0.287642\pi$
$521$	28.2462	1.23749	0.618744	+	0.785592i	$0.287642\pi$
$522$	0	0
$523$	−7.80776	−0.341410	−0.170705	−	0.985322i	$-0.554604\pi$
$523$	−7.80776	−0.341410	−0.170705	+	0.985322i	$0.554604\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−23.3693	−1.01895
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	−15.1231	−0.656906
$531$	0	0
$532$	0	0
$533$	26.9309	1.16651
$534$	0	0
$535$	0	0
$536$	19.3153	0.834296
$537$	0	0
$538$	−4.93087	−0.212585
$539$	−24.9309	−1.07385
$540$	0	0
$541$	−4.24621	−0.182559	−0.0912794	−	0.995825i	$-0.529096\pi$
$541$	−4.24621	−0.182559	−0.0912794	+	0.995825i	$0.529096\pi$
$542$	13.5616	0.582519
$543$	0	0
$544$	0	0
$545$	7.12311	0.305120
$546$	0	0
$547$	42.2462	1.80632	0.903159	−	0.429307i	$-0.141242\pi$
$547$	42.2462	1.80632	0.903159	+	0.429307i	$0.141242\pi$
$548$	1.75379	0.0749181
$549$	0	0
$550$	−27.3693	−1.16703
$551$	−5.12311	−0.218252
$552$	0	0
$553$	0	0
$554$	−12.9309	−0.549380
$555$	0	0
$556$	19.6155	0.831884
$557$	3.17708	0.134617	0.0673086	−	0.997732i	$-0.478559\pi$
$557$	3.17708	0.134617	0.0673086	+	0.997732i	$0.478559\pi$
$558$	0	0
$559$	15.1231	0.639639
$560$	0	0
$561$	0	0
$562$	−4.24621	−0.179116
$563$	−15.0691	−0.635088	−0.317544	−	0.948244i	$-0.602858\pi$
$563$	−15.0691	−0.635088	−0.317544	+	0.948244i	$0.602858\pi$
$564$	0	0
$565$	−47.6155	−2.00320
$566$	−0.684658	−0.0287783
$567$	0	0
$568$	−31.3153	−1.31396
$569$	30.7386	1.28863	0.644315	−	0.764760i	$-0.277142\pi$
$569$	30.7386	1.28863	0.644315	+	0.764760i	$0.277142\pi$
$570$	0	0
$571$	−13.1771	−0.551444	−0.275722	−	0.961237i	$-0.588917\pi$
$571$	−13.1771	−0.551444	−0.275722	+	0.961237i	$0.588917\pi$
$572$	12.6847	0.530372
$573$	0	0
$574$	0	0
$575$	−7.68466	−0.320472
$576$	0	0
$577$	−5.12311	−0.213278	−0.106639	−	0.994298i	$-0.534009\pi$
$577$	−5.12311	−0.213278	−0.106639	+	0.994298i	$0.534009\pi$
$578$	17.0000	0.707107
$579$	0	0
$580$	−3.56155	−0.147885
$581$	0	0
$582$	0	0
$583$	15.1231	0.626335
$584$	−15.3693	−0.635987
$585$	0	0
$586$	−16.0000	−0.660954
$587$	−35.2311	−1.45414	−0.727071	−	0.686563i	$-0.759119\pi$
$587$	−35.2311	−1.45414	−0.727071	+	0.686563i	$0.759119\pi$
$588$	0	0
$589$	12.4924	0.514741
$590$	34.0540	1.40198
$591$	0	0
$592$	−2.43845	−0.100220
$593$	−4.63068	−0.190159	−0.0950797	−	0.995470i	$-0.530311\pi$
$593$	−4.63068	−0.190159	−0.0950797	+	0.995470i	$0.530311\pi$
$594$	0	0
$595$	0	0
$596$	−16.9309	−0.693515
$597$	0	0
$598$	−3.56155	−0.145643
$599$	−20.9848	−0.857418	−0.428709	−	0.903443i	$-0.641031\pi$
$599$	−20.9848	−0.857418	−0.428709	+	0.903443i	$0.641031\pi$
$600$	0	0
$601$	12.2462	0.499533	0.249767	−	0.968306i	$-0.419646\pi$
$601$	12.2462	0.499533	0.249767	+	0.968306i	$0.419646\pi$
$602$	0	0
$603$	0	0
$604$	−6.24621	−0.254155
$605$	6.00000	0.243935
$606$	0	0
$607$	−35.2311	−1.42998	−0.714992	−	0.699132i	$-0.753570\pi$
$607$	−35.2311	−1.42998	−0.714992	+	0.699132i	$0.753570\pi$
$608$	25.6155	1.03885
$609$	0	0
$610$	42.0540	1.70272
$611$	25.3693	1.02633
$612$	0	0
$613$	−10.4924	−0.423785	−0.211892	−	0.977293i	$-0.567963\pi$
$613$	−10.4924	−0.423785	−0.211892	+	0.977293i	$0.567963\pi$
$614$	7.31534	0.295223
$615$	0	0
$616$	0	0
$617$	−43.6155	−1.75590	−0.877948	−	0.478757i	$-0.841088\pi$
$617$	−43.6155	−1.75590	−0.877948	+	0.478757i	$0.841088\pi$
$618$	0	0
$619$	−8.24621	−0.331443	−0.165722	−	0.986173i	$-0.552995\pi$
$619$	−8.24621	−0.331443	−0.165722	+	0.986173i	$0.552995\pi$
$620$	8.68466	0.348784
$621$	0	0
$622$	16.8769	0.676702
$623$	0	0
$624$	0	0
$625$	−4.36932	−0.174773
$626$	3.75379	0.150032
$627$	0	0
$628$	−19.1231	−0.763095
$629$	0	0
$630$	0	0
$631$	30.4384	1.21174	0.605868	−	0.795565i	$-0.292826\pi$
$631$	30.4384	1.21174	0.605868	+	0.795565i	$0.292826\pi$
$632$	−6.00000	−0.238667
$633$	0	0
$634$	8.93087	0.354690
$635$	42.0540	1.66886
$636$	0	0
$637$	24.9309	0.987797
$638$	−3.56155	−0.141003
$639$	0	0
$640$	10.6847	0.422348
$641$	38.7386	1.53008	0.765042	−	0.643980i	$-0.222718\pi$
$641$	38.7386	1.53008	0.765042	+	0.643980i	$0.222718\pi$
$642$	0	0
$643$	16.4924	0.650398	0.325199	−	0.945646i	$-0.394569\pi$
$643$	16.4924	0.650398	0.325199	+	0.945646i	$0.394569\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24.6847	−0.970454	−0.485227	−	0.874388i	$-0.661263\pi$
$647$	−24.6847	−0.970454	−0.485227	+	0.874388i	$0.661263\pi$
$648$	0	0
$649$	−34.0540	−1.33674
$650$	27.3693	1.07351
$651$	0	0
$652$	−5.56155	−0.217807
$653$	−43.5616	−1.70470	−0.852348	−	0.522976i	$-0.824822\pi$
$653$	−43.5616	−1.70470	−0.852348	+	0.522976i	$0.824822\pi$
$654$	0	0
$655$	14.2462	0.556646
$656$	7.56155	0.295229
$657$	0	0
$658$	0	0
$659$	−33.1231	−1.29029	−0.645147	−	0.764059i	$-0.723204\pi$
$659$	−33.1231	−1.29029	−0.645147	+	0.764059i	$0.723204\pi$
$660$	0	0
$661$	17.6155	0.685165	0.342582	−	0.939488i	$-0.388698\pi$
$661$	17.6155	0.685165	0.342582	+	0.939488i	$0.388698\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	21.3693	0.829290
$665$	0	0
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−6.93087	−0.268163
$669$	0	0
$670$	−22.9309	−0.885897
$671$	−42.0540	−1.62348
$672$	0	0
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	23.8078	0.917041
$675$	0	0
$676$	0.315342	0.0121285
$677$	46.2462	1.77739	0.888693	−	0.458502i	$-0.151614\pi$
$677$	46.2462	1.77739	0.888693	+	0.458502i	$0.151614\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	8.68466	0.332553
$683$	−9.56155	−0.365863	−0.182931	−	0.983126i	$-0.558559\pi$
$683$	−9.56155	−0.365863	−0.182931	+	0.983126i	$0.558559\pi$
$684$	0	0
$685$	−6.24621	−0.238655
$686$	0	0
$687$	0	0
$688$	4.24621	0.161885
$689$	−15.1231	−0.576144
$690$	0	0
$691$	24.1080	0.917110	0.458555	−	0.888666i	$-0.348367\pi$
$691$	24.1080	0.917110	0.458555	+	0.888666i	$0.348367\pi$
$692$	10.4924	0.398862
$693$	0	0
$694$	24.4924	0.929720
$695$	−69.8617	−2.65001
$696$	0	0
$697$	0	0
$698$	−22.3002	−0.844075
$699$	0	0
$700$	0	0
$701$	−16.0540	−0.606350	−0.303175	−	0.952935i	$-0.598047\pi$
$701$	−16.0540	−0.606350	−0.303175	+	0.952935i	$0.598047\pi$
$702$	0	0
$703$	−12.4924	−0.471161
$704$	24.9309	0.939618
$705$	0	0
$706$	10.4924	0.394888
$707$	0	0
$708$	0	0
$709$	1.12311	0.0421791	0.0210896	−	0.999778i	$-0.493286\pi$
$709$	1.12311	0.0421791	0.0210896	+	0.999778i	$0.493286\pi$
$710$	37.1771	1.39523
$711$	0	0
$712$	−52.1080	−1.95283
$713$	2.43845	0.0913206
$714$	0	0
$715$	−45.1771	−1.68953
$716$	−16.4924	−0.616351
$717$	0	0
$718$	−5.12311	−0.191193
$719$	−40.3002	−1.50294	−0.751472	−	0.659765i	$-0.770656\pi$
$719$	−40.3002	−1.50294	−0.751472	+	0.659765i	$0.770656\pi$
$720$	0	0
$721$	0	0
$722$	−7.24621	−0.269676
$723$	0	0
$724$	−18.4924	−0.687265
$725$	7.68466	0.285401
$726$	0	0
$727$	−48.7386	−1.80762	−0.903808	−	0.427938i	$-0.859240\pi$
$727$	−48.7386	−1.80762	−0.903808	+	0.427938i	$0.859240\pi$
$728$	0	0
$729$	0	0
$730$	18.2462	0.675323
$731$	0	0
$732$	0	0
$733$	−33.5616	−1.23962	−0.619812	−	0.784750i	$-0.712791\pi$
$733$	−33.5616	−1.23962	−0.619812	+	0.784750i	$0.712791\pi$
$734$	−24.7386	−0.913120
$735$	0	0
$736$	5.00000	0.184302
$737$	22.9309	0.844669
$738$	0	0
$739$	−30.5464	−1.12367	−0.561834	−	0.827250i	$-0.689904\pi$
$739$	−30.5464	−1.12367	−0.561834	+	0.827250i	$0.689904\pi$
$740$	−8.68466	−0.319254
$741$	0	0
$742$	0	0
$743$	22.3002	0.818114	0.409057	−	0.912509i	$-0.365858\pi$
$743$	22.3002	0.818114	0.409057	+	0.912509i	$0.365858\pi$
$744$	0	0
$745$	60.3002	2.20923
$746$	−2.49242	−0.0912541
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	19.8617	0.724765	0.362383	−	0.932029i	$-0.381963\pi$
$751$	19.8617	0.724765	0.362383	+	0.932029i	$0.381963\pi$
$752$	7.12311	0.259753
$753$	0	0
$754$	3.56155	0.129704
$755$	22.2462	0.809623
$756$	0	0
$757$	20.6847	0.751797	0.375898	−	0.926661i	$-0.377334\pi$
$757$	20.6847	0.751797	0.375898	+	0.926661i	$0.377334\pi$
$758$	−30.0000	−1.08965
$759$	0	0
$760$	−54.7386	−1.98558
$761$	18.9848	0.688200	0.344100	−	0.938933i	$-0.388184\pi$
$761$	18.9848	0.688200	0.344100	+	0.938933i	$0.388184\pi$
$762$	0	0
$763$	0	0
$764$	−19.5616	−0.707712
$765$	0	0
$766$	8.87689	0.320735
$767$	34.0540	1.22962
$768$	0	0
$769$	−5.56155	−0.200555	−0.100277	−	0.994960i	$-0.531973\pi$
$769$	−5.56155	−0.200555	−0.100277	+	0.994960i	$0.531973\pi$
$770$	0	0
$771$	0	0
$772$	6.87689	0.247505
$773$	44.0000	1.58257	0.791285	−	0.611448i	$-0.209412\pi$
$773$	44.0000	1.58257	0.791285	+	0.611448i	$0.209412\pi$
$774$	0	0
$775$	−18.7386	−0.673112
$776$	−40.1080	−1.43979
$777$	0	0
$778$	38.2462	1.37119
$779$	38.7386	1.38796
$780$	0	0
$781$	−37.1771	−1.33030
$782$	0	0
$783$	0	0
$784$	7.00000	0.250000
$785$	68.1080	2.43088
$786$	0	0
$787$	−17.9460	−0.639707	−0.319853	−	0.947467i	$-0.603634\pi$
$787$	−17.9460	−0.639707	−0.319853	+	0.947467i	$0.603634\pi$
$788$	12.2462	0.436253
$789$	0	0
$790$	7.12311	0.253429
$791$	0	0
$792$	0	0
$793$	42.0540	1.49338
$794$	15.7538	0.559081
$795$	0	0
$796$	−3.31534	−0.117509
$797$	−32.1080	−1.13732	−0.568661	−	0.822572i	$-0.692538\pi$
$797$	−32.1080	−1.13732	−0.568661	+	0.822572i	$0.692538\pi$
$798$	0	0
$799$	0	0
$800$	−38.4233	−1.35847
$801$	0	0
$802$	−2.19224	−0.0774105
$803$	−18.2462	−0.643895
$804$	0	0
$805$	0	0
$806$	−8.68466	−0.305904
$807$	0	0
$808$	22.6847	0.798043
$809$	2.30019	0.0808703	0.0404351	−	0.999182i	$-0.487126\pi$
$809$	2.30019	0.0808703	0.0404351	+	0.999182i	$0.487126\pi$
$810$	0	0
$811$	−18.2462	−0.640711	−0.320356	−	0.947297i	$-0.603802\pi$
$811$	−18.2462	−0.640711	−0.320356	+	0.947297i	$0.603802\pi$
$812$	0	0
$813$	0	0
$814$	−8.68466	−0.304397
$815$	19.8078	0.693836
$816$	0	0
$817$	21.7538	0.761069
$818$	−20.2462	−0.707892
$819$	0	0
$820$	26.9309	0.940467
$821$	−3.36932	−0.117590	−0.0587950	−	0.998270i	$-0.518726\pi$
$821$	−3.36932	−0.117590	−0.0587950	+	0.998270i	$0.518726\pi$
$822$	0	0
$823$	28.3002	0.986482	0.493241	−	0.869893i	$-0.335812\pi$
$823$	28.3002	0.986482	0.493241	+	0.869893i	$0.335812\pi$
$824$	4.68466	0.163198
$825$	0	0
$826$	0	0
$827$	24.9309	0.866931	0.433466	−	0.901170i	$-0.357291\pi$
$827$	24.9309	0.866931	0.433466	+	0.901170i	$0.357291\pi$
$828$	0	0
$829$	−40.3542	−1.40156	−0.700779	−	0.713378i	$-0.747164\pi$
$829$	−40.3542	−1.40156	−0.700779	+	0.713378i	$0.747164\pi$
$830$	−25.3693	−0.880582
$831$	0	0
$832$	−24.9309	−0.864322
$833$	0	0
$834$	0	0
$835$	24.6847	0.854248
$836$	18.2462	0.631058
$837$	0	0
$838$	23.1231	0.798774
$839$	−49.8078	−1.71955	−0.859777	−	0.510669i	$-0.829398\pi$
$839$	−49.8078	−1.71955	−0.859777	+	0.510669i	$0.829398\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−19.8078	−0.682621
$843$	0	0
$844$	−21.1771	−0.728945
$845$	−1.12311	−0.0386360
$846$	0	0
$847$	0	0
$848$	−4.24621	−0.145815
$849$	0	0
$850$	0	0
$851$	−2.43845	−0.0835889
$852$	0	0
$853$	−19.3693	−0.663193	−0.331596	−	0.943421i	$-0.607587\pi$
$853$	−19.3693	−0.663193	−0.331596	+	0.943421i	$0.607587\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−15.7538	−0.538139	−0.269070	−	0.963121i	$-0.586716\pi$
$857$	−15.7538	−0.538139	−0.269070	+	0.963121i	$0.586716\pi$
$858$	0	0
$859$	12.9848	0.443037	0.221519	−	0.975156i	$-0.428899\pi$
$859$	12.9848	0.443037	0.221519	+	0.975156i	$0.428899\pi$
$860$	15.1231	0.515694
$861$	0	0
$862$	20.8769	0.711070
$863$	22.9309	0.780576	0.390288	−	0.920693i	$-0.372375\pi$
$863$	22.9309	0.780576	0.390288	+	0.920693i	$0.372375\pi$
$864$	0	0
$865$	−37.3693	−1.27059
$866$	20.8769	0.709426
$867$	0	0
$868$	0	0
$869$	−7.12311	−0.241635
$870$	0	0
$871$	−22.9309	−0.776983
$872$	6.00000	0.203186
$873$	0	0
$874$	−5.12311	−0.173292
$875$	0	0
$876$	0	0
$877$	33.2311	1.12213	0.561067	−	0.827771i	$-0.310391\pi$
$877$	33.2311	1.12213	0.561067	+	0.827771i	$0.310391\pi$
$878$	−29.8617	−1.00778
$879$	0	0
$880$	−12.6847	−0.427600
$881$	−36.1080	−1.21651	−0.608254	−	0.793743i	$-0.708130\pi$
$881$	−36.1080	−1.21651	−0.608254	+	0.793743i	$0.708130\pi$
$882$	0	0
$883$	54.7386	1.84210	0.921051	−	0.389442i	$-0.127332\pi$
$883$	54.7386	1.84210	0.921051	+	0.389442i	$0.127332\pi$
$884$	0	0
$885$	0	0
$886$	4.87689	0.163842
$887$	−4.49242	−0.150841	−0.0754204	−	0.997152i	$-0.524030\pi$
$887$	−4.49242	−0.150841	−0.0754204	+	0.997152i	$0.524030\pi$
$888$	0	0
$889$	0	0
$890$	61.8617	2.07361
$891$	0	0
$892$	28.4924	0.953997
$893$	36.4924	1.22117
$894$	0	0
$895$	58.7386	1.96342
$896$	0	0
$897$	0	0
$898$	17.3153	0.577820
$899$	−2.43845	−0.0813268
$900$	0	0
$901$	0	0
$902$	26.9309	0.896700
$903$	0	0
$904$	−40.1080	−1.33397
$905$	65.8617	2.18932
$906$	0	0
$907$	2.00000	0.0664089	0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000	0.0664089	0.0332045	+	0.999449i	$0.489429\pi$
$908$	−29.3693	−0.974655
$909$	0	0
$910$	0	0
$911$	29.3153	0.971261	0.485630	−	0.874164i	$-0.338590\pi$
$911$	29.3153	0.971261	0.485630	+	0.874164i	$0.338590\pi$
$912$	0	0
$913$	25.3693	0.839602
$914$	1.12311	0.0371490
$915$	0	0
$916$	−0.192236	−0.00635165
$917$	0	0
$918$	0	0
$919$	11.8078	0.389502	0.194751	−	0.980853i	$-0.437610\pi$
$919$	11.8078	0.389502	0.194751	+	0.980853i	$0.437610\pi$
$920$	−10.6847	−0.352263
$921$	0	0
$922$	−38.3002	−1.26135
$923$	37.1771	1.22370
$924$	0	0
$925$	18.7386	0.616122
$926$	−20.4924	−0.673423
$927$	0	0
$928$	−5.00000	−0.164133
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	35.8617	1.17532
$932$	9.12311	0.298837
$933$	0	0
$934$	25.4233	0.831876
$935$	0	0
$936$	0	0
$937$	18.9848	0.620208	0.310104	−	0.950703i	$-0.399636\pi$
$937$	18.9848	0.620208	0.310104	+	0.950703i	$0.399636\pi$
$938$	0	0
$939$	0	0
$940$	25.3693	0.827456
$941$	44.5464	1.45217	0.726086	−	0.687604i	$-0.241338\pi$
$941$	44.5464	1.45217	0.726086	+	0.687604i	$0.241338\pi$
$942$	0	0
$943$	7.56155	0.246238
$944$	9.56155	0.311202
$945$	0	0
$946$	15.1231	0.491695
$947$	49.4773	1.60780	0.803898	−	0.594768i	$-0.202756\pi$
$947$	49.4773	1.60780	0.803898	+	0.594768i	$0.202756\pi$
$948$	0	0
$949$	18.2462	0.592297
$950$	39.3693	1.27731
$951$	0	0
$952$	0	0
$953$	−26.9848	−0.874125	−0.437062	−	0.899431i	$-0.643981\pi$
$953$	−26.9848	−0.874125	−0.437062	+	0.899431i	$0.643981\pi$
$954$	0	0
$955$	69.6695	2.25445
$956$	10.4384	0.337604
$957$	0	0
$958$	21.3153	0.688667
$959$	0	0
$960$	0	0
$961$	−25.0540	−0.808193
$962$	8.68466	0.280005
$963$	0	0
$964$	17.6155	0.567358
$965$	−24.4924	−0.788439
$966$	0	0
$967$	37.7538	1.21408	0.607040	−	0.794671i	$-0.292357\pi$
$967$	37.7538	1.21408	0.607040	+	0.794671i	$0.292357\pi$
$968$	5.05398	0.162441
$969$	0	0
$970$	47.6155	1.52884
$971$	−17.8078	−0.571478	−0.285739	−	0.958307i	$-0.592239\pi$
$971$	−17.8078	−0.571478	−0.285739	+	0.958307i	$0.592239\pi$
$972$	0	0
$973$	0	0
$974$	10.7386	0.344088
$975$	0	0
$976$	11.8078	0.377957
$977$	15.1771	0.485558	0.242779	−	0.970082i	$-0.421941\pi$
$977$	15.1771	0.485558	0.242779	+	0.970082i	$0.421941\pi$
$978$	0	0
$979$	−61.8617	−1.97711
$980$	24.9309	0.796387
$981$	0	0
$982$	−2.63068	−0.0839485
$983$	7.56155	0.241176	0.120588	−	0.992703i	$-0.461522\pi$
$983$	7.56155	0.241176	0.120588	+	0.992703i	$0.461522\pi$
$984$	0	0
$985$	−43.6155	−1.38971
$986$	0	0
$987$	0	0
$988$	−18.2462	−0.580489
$989$	4.24621	0.135022
$990$	0	0
$991$	−11.5076	−0.365550	−0.182775	−	0.983155i	$-0.558508\pi$
$991$	−11.5076	−0.365550	−0.182775	+	0.983155i	$0.558508\pi$
$992$	12.1922	0.387104
$993$	0	0
$994$	0	0
$995$	11.8078	0.374331
$996$	0	0
$997$	−24.7386	−0.783480	−0.391740	−	0.920076i	$-0.628127\pi$
$997$	−24.7386	−0.783480	−0.391740	+	0.920076i	$0.628127\pi$
$998$	−5.36932	−0.169963
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.d.1.2		2
3.2	odd	2		2001.2.a.f.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.f.1.1	✓	2	3.2	odd	2
6003.2.a.d.1.2		2	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{4} +3.56155 q^{5} +3.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{4} +3.56155 q^{5} +3.00000 q^{8} -3.56155 q^{10} +3.56155 q^{11} -3.56155 q^{13} -1.00000 q^{16} -5.12311 q^{19} -3.56155 q^{20} -3.56155 q^{22} -1.00000 q^{23} +7.68466 q^{25} +3.56155 q^{26} +1.00000 q^{29} -2.43845 q^{31} -5.00000 q^{32} +2.43845 q^{37} +5.12311 q^{38} +10.6847 q^{40} -7.56155 q^{41} -4.24621 q^{43} -3.56155 q^{44} +1.00000 q^{46} -7.12311 q^{47} -7.00000 q^{49} -7.68466 q^{50} +3.56155 q^{52} +4.24621 q^{53} +12.6847 q^{55} -1.00000 q^{58} -9.56155 q^{59} -11.8078 q^{61} +2.43845 q^{62} +7.00000 q^{64} -12.6847 q^{65} +6.43845 q^{67} -10.4384 q^{71} -5.12311 q^{73} -2.43845 q^{74} +5.12311 q^{76} -2.00000 q^{79} -3.56155 q^{80} +7.56155 q^{82} +7.12311 q^{83} +4.24621 q^{86} +10.6847 q^{88} -17.3693 q^{89} +1.00000 q^{92} +7.12311 q^{94} -18.2462 q^{95} -13.3693 q^{97} +7.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{4} + 3 q^{5} + 6 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^4 + 3 * q^5 + 6 * q^8	\( 2 q - 2 q^{2} - 2 q^{4} + 3 q^{5} + 6 q^{8} - 3 q^{10} + 3 q^{11} - 3 q^{13} - 2 q^{16} - 2 q^{19} - 3 q^{20} - 3 q^{22} - 2 q^{23} + 3 q^{25} + 3 q^{26} + 2 q^{29} - 9 q^{31} - 10 q^{32} + 9 q^{37} + 2 q^{38} + 9 q^{40} - 11 q^{41} + 8 q^{43} - 3 q^{44} + 2 q^{46} - 6 q^{47} - 14 q^{49} - 3 q^{50} + 3 q^{52} - 8 q^{53} + 13 q^{55} - 2 q^{58} - 15 q^{59} - 3 q^{61} + 9 q^{62} + 14 q^{64} - 13 q^{65} + 17 q^{67} - 25 q^{71} - 2 q^{73} - 9 q^{74} + 2 q^{76} - 4 q^{79} - 3 q^{80} + 11 q^{82} + 6 q^{83} - 8 q^{86} + 9 q^{88} - 10 q^{89} + 2 q^{92} + 6 q^{94} - 20 q^{95} - 2 q^{97} + 14 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^4 + 3 * q^5 + 6 * q^8 - 3 * q^10 + 3 * q^11 - 3 * q^13 - 2 * q^16 - 2 * q^19 - 3 * q^20 - 3 * q^22 - 2 * q^23 + 3 * q^25 + 3 * q^26 + 2 * q^29 - 9 * q^31 - 10 * q^32 + 9 * q^37 + 2 * q^38 + 9 * q^40 - 11 * q^41 + 8 * q^43 - 3 * q^44 + 2 * q^46 - 6 * q^47 - 14 * q^49 - 3 * q^50 + 3 * q^52 - 8 * q^53 + 13 * q^55 - 2 * q^58 - 15 * q^59 - 3 * q^61 + 9 * q^62 + 14 * q^64 - 13 * q^65 + 17 * q^67 - 25 * q^71 - 2 * q^73 - 9 * q^74 + 2 * q^76 - 4 * q^79 - 3 * q^80 + 11 * q^82 + 6 * q^83 - 8 * q^86 + 9 * q^88 - 10 * q^89 + 2 * q^92 + 6 * q^94 - 20 * q^95 - 2 * q^97 + 14 * q^98

Introduction

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