LMFDB - Embedded newform 6003.2.a.q.1.1

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 3 x^{15} - 22 x^{14} + 68 x^{13} + 187 x^{12} - 597 x^{11} - 795 x^{10} + 2592 x^{9} + \cdots + 64 $ x^16 - 3x^15 - 22x^14 + 68x^13 + 187x^12 - 597x^11 - 795x^10 + 2592x^9 + 1860x^8 - 5877x^7 - 2496x^6 + 6612x^5 + 1842x^4 - 3011x^3 - 505x^2 + 336*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.73896$ 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)

$f(q)$	$=$	$q-2.73896 q^{2} +5.50188 q^{4} +1.91750 q^{5} -0.472052 q^{7} -9.59151 q^{8} +O(q^{10})$	\(q-2.73896 q^{2} +5.50188 q^{4} +1.91750 q^{5} -0.472052 q^{7} -9.59151 q^{8} -5.25195 q^{10} +2.34056 q^{11} +0.319606 q^{13} +1.29293 q^{14} +15.2670 q^{16} +0.228054 q^{17} +3.36342 q^{19} +10.5499 q^{20} -6.41070 q^{22} -1.00000 q^{23} -1.32319 q^{25} -0.875388 q^{26} -2.59718 q^{28} +1.00000 q^{29} -2.04991 q^{31} -22.6325 q^{32} -0.624629 q^{34} -0.905161 q^{35} -5.23336 q^{37} -9.21226 q^{38} -18.3917 q^{40} -1.97999 q^{41} -11.8584 q^{43} +12.8775 q^{44} +2.73896 q^{46} -6.50673 q^{47} -6.77717 q^{49} +3.62415 q^{50} +1.75844 q^{52} +0.128482 q^{53} +4.48804 q^{55} +4.52769 q^{56} -2.73896 q^{58} -4.92843 q^{59} +6.57063 q^{61} +5.61460 q^{62} +31.4556 q^{64} +0.612846 q^{65} +7.39486 q^{67} +1.25472 q^{68} +2.47920 q^{70} -0.455585 q^{71} +9.53772 q^{73} +14.3339 q^{74} +18.5051 q^{76} -1.10487 q^{77} -10.7528 q^{79} +29.2744 q^{80} +5.42311 q^{82} -15.0544 q^{83} +0.437293 q^{85} +32.4798 q^{86} -22.4495 q^{88} +2.76934 q^{89} -0.150871 q^{91} -5.50188 q^{92} +17.8217 q^{94} +6.44936 q^{95} -7.87806 q^{97} +18.5624 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8}+O(q^{10}) $ 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8	$ 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8} - 14 q^{10} - 4 q^{11} + 15 q^{13} - 8 q^{14} + 23 q^{16} - 20 q^{17} - 4 q^{19} - 25 q^{20} + 13 q^{22} - 16 q^{23} + 30 q^{25} - 25 q^{26} - 13 q^{28} + 16 q^{29} + 19 q^{32} - 23 q^{34} - 5 q^{35} + 5 q^{37} - 38 q^{38} - 20 q^{40} - 7 q^{41} - 17 q^{43} + 21 q^{44} + 3 q^{46} - 29 q^{47} + 31 q^{49} + 44 q^{50} + 20 q^{52} - 63 q^{53} + q^{55} + 19 q^{56} - 3 q^{58} - 11 q^{59} - 33 q^{62} + 29 q^{64} - 53 q^{65} - 13 q^{67} - 63 q^{68} - 46 q^{70} + 23 q^{71} - 38 q^{73} + 47 q^{74} - 56 q^{76} - 97 q^{77} - 27 q^{79} - 8 q^{80} + 9 q^{82} - 36 q^{83} + 6 q^{85} + 11 q^{86} - 24 q^{88} + 16 q^{89} - 47 q^{91} - 21 q^{92} + 37 q^{94} + 12 q^{95} - 30 q^{97} + 27 q^{98}+O(q^{100}) $ 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8 - 14 * q^10 - 4 * q^11 + 15 * q^13 - 8 * q^14 + 23 * q^16 - 20 * q^17 - 4 * q^19 - 25 * q^20 + 13 * q^22 - 16 * q^23 + 30 * q^25 - 25 * q^26 - 13 * q^28 + 16 * q^29 + 19 * q^32 - 23 * q^34 - 5 * q^35 + 5 * q^37 - 38 * q^38 - 20 * q^40 - 7 * q^41 - 17 * q^43 + 21 * q^44 + 3 * q^46 - 29 * q^47 + 31 * q^49 + 44 * q^50 + 20 * q^52 - 63 * q^53 + q^55 + 19 * q^56 - 3 * q^58 - 11 * q^59 - 33 * q^62 + 29 * q^64 - 53 * q^65 - 13 * q^67 - 63 * q^68 - 46 * q^70 + 23 * q^71 - 38 * q^73 + 47 * q^74 - 56 * q^76 - 97 * q^77 - 27 * q^79 - 8 * q^80 + 9 * q^82 - 36 * q^83 + 6 * q^85 + 11 * q^86 - 24 * q^88 + 16 * q^89 - 47 * q^91 - 21 * q^92 + 37 * q^94 + 12 * q^95 - 30 * q^97 + 27 * 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 $

$2$	−2.73896	−1.93673	−0.968367	−	0.249529i	$-0.919724\pi$
$2$	−2.73896	−1.93673	−0.968367	+	0.249529i	$0.919724\pi$
$3$	0	0
$3$	0	0
$4$	5.50188	2.75094
$5$	1.91750	0.857533	0.428766	−	0.903415i	$-0.358948\pi$
$5$	1.91750	0.857533	0.428766	+	0.903415i	$0.358948\pi$
$6$	0	0
$7$	−0.472052	−0.178419	−0.0892095	−	0.996013i	$-0.528434\pi$
$7$	−0.472052	−0.178419	−0.0892095	+	0.996013i	$0.528434\pi$
$8$	−9.59151	−3.39111
$9$	0	0
$10$	−5.25195	−1.66081
$11$	2.34056	0.705707	0.352853	−	0.935679i	$-0.385211\pi$
$11$	2.34056	0.705707	0.352853	+	0.935679i	$0.385211\pi$
$12$	0	0
$13$	0.319606	0.0886428	0.0443214	−	0.999017i	$-0.485887\pi$
$13$	0.319606	0.0886428	0.0443214	+	0.999017i	$0.485887\pi$
$14$	1.29293	0.345550
$15$	0	0
$16$	15.2670	3.81674
$17$	0.228054	0.0553111	0.0276556	−	0.999618i	$-0.491196\pi$
$17$	0.228054	0.0553111	0.0276556	+	0.999618i	$0.491196\pi$
$18$	0	0
$19$	3.36342	0.771621	0.385811	−	0.922578i	$-0.373922\pi$
$19$	3.36342	0.771621	0.385811	+	0.922578i	$0.373922\pi$
$20$	10.5499	2.35902
$21$	0	0
$22$	−6.41070	−1.36677
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−1.32319	−0.264637
$26$	−0.875388	−0.171678
$27$	0	0
$28$	−2.59718	−0.490820
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−2.04991	−0.368174	−0.184087	−	0.982910i	$-0.558933\pi$
$31$	−2.04991	−0.368174	−0.184087	+	0.982910i	$0.558933\pi$
$32$	−22.6325	−4.00090
$33$	0	0
$34$	−0.624629	−0.107123
$35$	−0.905161	−0.153000
$36$	0	0
$37$	−5.23336	−0.860359	−0.430180	−	0.902743i	$-0.641550\pi$
$37$	−5.23336	−0.860359	−0.430180	+	0.902743i	$0.641550\pi$
$38$	−9.21226	−1.49443
$39$	0	0
$40$	−18.3917	−2.90799
$41$	−1.97999	−0.309222	−0.154611	−	0.987975i	$-0.549413\pi$
$41$	−1.97999	−0.309222	−0.154611	+	0.987975i	$0.549413\pi$
$42$	0	0
$43$	−11.8584	−1.80840	−0.904198	−	0.427114i	$-0.859530\pi$
$43$	−11.8584	−1.80840	−0.904198	+	0.427114i	$0.859530\pi$
$44$	12.8775	1.94136
$45$	0	0
$46$	2.73896	0.403837
$47$	−6.50673	−0.949104	−0.474552	−	0.880227i	$-0.657390\pi$
$47$	−6.50673	−0.949104	−0.474552	+	0.880227i	$0.657390\pi$
$48$	0	0
$49$	−6.77717	−0.968167
$50$	3.62415	0.512532
$51$	0	0
$52$	1.75844	0.243851
$53$	0.128482	0.0176484	0.00882420	−	0.999961i	$-0.497191\pi$
$53$	0.128482	0.0176484	0.00882420	+	0.999961i	$0.497191\pi$
$54$	0	0
$55$	4.48804	0.605167
$56$	4.52769	0.605038
$57$	0	0
$58$	−2.73896	−0.359643
$59$	−4.92843	−0.641627	−0.320814	−	0.947142i	$-0.603956\pi$
$59$	−4.92843	−0.641627	−0.320814	+	0.947142i	$0.603956\pi$
$60$	0	0
$61$	6.57063	0.841283	0.420641	−	0.907227i	$-0.361805\pi$
$61$	6.57063	0.841283	0.420641	+	0.907227i	$0.361805\pi$
$62$	5.61460	0.713055
$63$	0	0
$64$	31.4556	3.93194
$65$	0.612846	0.0760141
$66$	0	0
$67$	7.39486	0.903426	0.451713	−	0.892163i	$-0.350813\pi$
$67$	7.39486	0.903426	0.451713	+	0.892163i	$0.350813\pi$
$68$	1.25472	0.152158
$69$	0	0
$70$	2.47920	0.296321
$71$	−0.455585	−0.0540680	−0.0270340	−	0.999635i	$-0.508606\pi$
$71$	−0.455585	−0.0540680	−0.0270340	+	0.999635i	$0.508606\pi$
$72$	0	0
$73$	9.53772	1.11631	0.558153	−	0.829738i	$-0.311510\pi$
$73$	9.53772	1.11631	0.558153	+	0.829738i	$0.311510\pi$
$74$	14.3339	1.66629
$75$	0	0
$76$	18.5051	2.12268
$77$	−1.10487	−0.125911
$78$	0	0
$79$	−10.7528	−1.20979	−0.604893	−	0.796307i	$-0.706784\pi$
$79$	−10.7528	−1.20979	−0.604893	+	0.796307i	$0.706784\pi$
$80$	29.2744	3.27298
$81$	0	0
$82$	5.42311	0.598882
$83$	−15.0544	−1.65243	−0.826215	−	0.563355i	$-0.809510\pi$
$83$	−15.0544	−1.65243	−0.826215	+	0.563355i	$0.809510\pi$
$84$	0	0
$85$	0.437293	0.0474311
$86$	32.4798	3.50238
$87$	0	0
$88$	−22.4495	−2.39313
$89$	2.76934	0.293550	0.146775	−	0.989170i	$-0.453111\pi$
$89$	2.76934	0.293550	0.146775	+	0.989170i	$0.453111\pi$
$90$	0	0
$91$	−0.150871	−0.0158156
$92$	−5.50188	−0.573611
$93$	0	0
$94$	17.8217	1.83816
$95$	6.44936	0.661690
$96$	0	0
$97$	−7.87806	−0.799896	−0.399948	−	0.916538i	$-0.630972\pi$
$97$	−7.87806	−0.799896	−0.399948	+	0.916538i	$0.630972\pi$
$98$	18.5624	1.87508
$99$	0	0
$100$	−7.28002	−0.728002
$101$	−12.8819	−1.28180	−0.640900	−	0.767625i	$-0.721438\pi$
$101$	−12.8819	−1.28180	−0.640900	+	0.767625i	$0.721438\pi$
$102$	0	0
$103$	3.80646	0.375062	0.187531	−	0.982259i	$-0.439951\pi$
$103$	3.80646	0.375062	0.187531	+	0.982259i	$0.439951\pi$
$104$	−3.06551	−0.300598
$105$	0	0
$106$	−0.351908	−0.0341803
$107$	−8.57347	−0.828829	−0.414414	−	0.910088i	$-0.636014\pi$
$107$	−8.57347	−0.828829	−0.414414	+	0.910088i	$0.636014\pi$
$108$	0	0
$109$	−9.57100	−0.916736	−0.458368	−	0.888763i	$-0.651566\pi$
$109$	−9.57100	−0.916736	−0.458368	+	0.888763i	$0.651566\pi$
$110$	−12.2925	−1.17205
$111$	0	0
$112$	−7.20680	−0.680979
$113$	−10.1083	−0.950912	−0.475456	−	0.879740i	$-0.657717\pi$
$113$	−10.1083	−0.950912	−0.475456	+	0.879740i	$0.657717\pi$
$114$	0	0
$115$	−1.91750	−0.178808
$116$	5.50188	0.510837
$117$	0	0
$118$	13.4988	1.24266
$119$	−0.107653	−0.00986856
$120$	0	0
$121$	−5.52176	−0.501978
$122$	−17.9967	−1.62934
$123$	0	0
$124$	−11.2783	−1.01283
$125$	−12.1247	−1.08447
$126$	0	0
$127$	11.6804	1.03646	0.518232	−	0.855240i	$-0.326590\pi$
$127$	11.6804	1.03646	0.518232	+	0.855240i	$0.326590\pi$
$128$	−40.8904	−3.61423
$129$	0	0
$130$	−1.67856	−0.147219
$131$	7.44339	0.650332	0.325166	−	0.945657i	$-0.394580\pi$
$131$	7.44339	0.650332	0.325166	+	0.945657i	$0.394580\pi$
$132$	0	0
$133$	−1.58771	−0.137672
$134$	−20.2542	−1.74970
$135$	0	0
$136$	−2.18738	−0.187566
$137$	−11.7505	−1.00392	−0.501958	−	0.864892i	$-0.667387\pi$
$137$	−11.7505	−1.00392	−0.501958	+	0.864892i	$0.667387\pi$
$138$	0	0
$139$	21.6918	1.83987	0.919935	−	0.392070i	$-0.128241\pi$
$139$	21.6918	1.83987	0.919935	+	0.392070i	$0.128241\pi$
$140$	−4.98009	−0.420895
$141$	0	0
$142$	1.24783	0.104715
$143$	0.748059	0.0625558
$144$	0	0
$145$	1.91750	0.159240
$146$	−26.1234	−2.16199
$147$	0	0
$148$	−28.7933	−2.36680
$149$	−8.78218	−0.719464	−0.359732	−	0.933056i	$-0.617132\pi$
$149$	−8.78218	−0.719464	−0.359732	+	0.933056i	$0.617132\pi$
$150$	0	0
$151$	16.0772	1.30835	0.654174	−	0.756344i	$-0.273017\pi$
$151$	16.0772	1.30835	0.654174	+	0.756344i	$0.273017\pi$
$152$	−32.2602	−2.61665
$153$	0	0
$154$	3.02619	0.243857
$155$	−3.93070	−0.315721
$156$	0	0
$157$	−16.9632	−1.35381	−0.676907	−	0.736069i	$-0.736680\pi$
$157$	−16.9632	−1.35381	−0.676907	+	0.736069i	$0.736680\pi$
$158$	29.4515	2.34303
$159$	0	0
$160$	−43.3979	−3.43090
$161$	0.472052	0.0372029
$162$	0	0
$163$	−20.2343	−1.58487	−0.792435	−	0.609957i	$-0.791187\pi$
$163$	−20.2343	−1.58487	−0.792435	+	0.609957i	$0.791187\pi$
$164$	−10.8937	−0.850653
$165$	0	0
$166$	41.2332	3.20032
$167$	17.3251	1.34065	0.670327	−	0.742066i	$-0.266154\pi$
$167$	17.3251	1.34065	0.670327	+	0.742066i	$0.266154\pi$
$168$	0	0
$169$	−12.8979	−0.992142
$170$	−1.19773	−0.0918615
$171$	0	0
$172$	−65.2438	−4.97479
$173$	17.0135	1.29351	0.646755	−	0.762698i	$-0.276126\pi$
$173$	17.0135	1.29351	0.646755	+	0.762698i	$0.276126\pi$
$174$	0	0
$175$	0.624613	0.0472163
$176$	35.7333	2.69350
$177$	0	0
$178$	−7.58511	−0.568528
$179$	7.06258	0.527882	0.263941	−	0.964539i	$-0.414978\pi$
$179$	7.06258	0.527882	0.263941	+	0.964539i	$0.414978\pi$
$180$	0	0
$181$	14.6788	1.09106	0.545532	−	0.838090i	$-0.316328\pi$
$181$	14.6788	1.09106	0.545532	+	0.838090i	$0.316328\pi$
$182$	0.413229	0.0306306
$183$	0	0
$184$	9.59151	0.707095
$185$	−10.0350	−0.737786
$186$	0	0
$187$	0.533774	0.0390334
$188$	−35.7993	−2.61093
$189$	0	0
$190$	−17.6645	−1.28152
$191$	−20.2462	−1.46497	−0.732483	−	0.680785i	$-0.761639\pi$
$191$	−20.2462	−1.46497	−0.732483	+	0.680785i	$0.761639\pi$
$192$	0	0
$193$	−10.4593	−0.752880	−0.376440	−	0.926441i	$-0.622852\pi$
$193$	−10.4593	−0.752880	−0.376440	+	0.926441i	$0.622852\pi$
$194$	21.5777	1.54919
$195$	0	0
$196$	−37.2872	−2.66337
$197$	−1.06522	−0.0758938	−0.0379469	−	0.999280i	$-0.512082\pi$
$197$	−1.06522	−0.0758938	−0.0379469	+	0.999280i	$0.512082\pi$
$198$	0	0
$199$	10.6750	0.756728	0.378364	−	0.925657i	$-0.376487\pi$
$199$	10.6750	0.756728	0.378364	+	0.925657i	$0.376487\pi$
$200$	12.6914	0.897414
$201$	0	0
$202$	35.2830	2.48250
$203$	−0.472052	−0.0331316
$204$	0	0
$205$	−3.79663	−0.265168
$206$	−10.4257	−0.726396
$207$	0	0
$208$	4.87941	0.338326
$209$	7.87230	0.544538
$210$	0	0
$211$	−18.6423	−1.28339	−0.641695	−	0.766960i	$-0.721769\pi$
$211$	−18.6423	−1.28339	−0.641695	+	0.766960i	$0.721769\pi$
$212$	0.706895	0.0485497
$213$	0	0
$214$	23.4824	1.60522
$215$	−22.7386	−1.55076
$216$	0	0
$217$	0.967663	0.0656892
$218$	26.2146	1.77547
$219$	0	0
$220$	24.6926	1.66478
$221$	0.0728874	0.00490294
$222$	0	0
$223$	−19.7223	−1.32070	−0.660350	−	0.750958i	$-0.729592\pi$
$223$	−19.7223	−1.32070	−0.660350	+	0.750958i	$0.729592\pi$
$224$	10.6837	0.713837
$225$	0	0
$226$	27.6863	1.84166
$227$	3.83497	0.254536	0.127268	−	0.991868i	$-0.459379\pi$
$227$	3.83497	0.254536	0.127268	+	0.991868i	$0.459379\pi$
$228$	0	0
$229$	8.27021	0.546511	0.273255	−	0.961941i	$-0.411900\pi$
$229$	8.27021	0.546511	0.273255	+	0.961941i	$0.411900\pi$
$230$	5.25195	0.346304
$231$	0	0
$232$	−9.59151	−0.629713
$233$	−21.3381	−1.39790	−0.698951	−	0.715169i	$-0.746350\pi$
$233$	−21.3381	−1.39790	−0.698951	+	0.715169i	$0.746350\pi$
$234$	0	0
$235$	−12.4767	−0.813888
$236$	−27.1157	−1.76508
$237$	0	0
$238$	0.294858	0.0191128
$239$	2.68311	0.173556	0.0867781	−	0.996228i	$-0.472343\pi$
$239$	2.68311	0.173556	0.0867781	+	0.996228i	$0.472343\pi$
$240$	0	0
$241$	−17.1681	−1.10590	−0.552948	−	0.833216i	$-0.686497\pi$
$241$	−17.1681	−1.10590	−0.552948	+	0.833216i	$0.686497\pi$
$242$	15.1239	0.972199
$243$	0	0
$244$	36.1508	2.31432
$245$	−12.9952	−0.830235
$246$	0	0
$247$	1.07497	0.0683987
$248$	19.6617	1.24852
$249$	0	0
$250$	33.2091	2.10033
$251$	24.7244	1.56059	0.780295	−	0.625411i	$-0.215069\pi$
$251$	24.7244	1.56059	0.780295	+	0.625411i	$0.215069\pi$
$252$	0	0
$253$	−2.34056	−0.147150
$254$	−31.9920	−2.00736
$255$	0	0
$256$	49.0859	3.06787
$257$	25.0454	1.56229	0.781145	−	0.624350i	$-0.214636\pi$
$257$	25.0454	1.56229	0.781145	+	0.624350i	$0.214636\pi$
$258$	0	0
$259$	2.47042	0.153504
$260$	3.37180	0.209110
$261$	0	0
$262$	−20.3871	−1.25952
$263$	−15.6651	−0.965950	−0.482975	−	0.875634i	$-0.660444\pi$
$263$	−15.6651	−0.965950	−0.482975	+	0.875634i	$0.660444\pi$
$264$	0	0
$265$	0.246365	0.0151341
$266$	4.34867	0.266634
$267$	0	0
$268$	40.6857	2.48527
$269$	−13.0997	−0.798701	−0.399350	−	0.916798i	$-0.630764\pi$
$269$	−13.0997	−0.798701	−0.399350	+	0.916798i	$0.630764\pi$
$270$	0	0
$271$	16.3884	0.995525	0.497762	−	0.867313i	$-0.334155\pi$
$271$	16.3884	0.995525	0.497762	+	0.867313i	$0.334155\pi$
$272$	3.48168	0.211108
$273$	0	0
$274$	32.1842	1.94432
$275$	−3.09700	−0.186756
$276$	0	0
$277$	14.7704	0.887465	0.443732	−	0.896159i	$-0.353654\pi$
$277$	14.7704	0.887465	0.443732	+	0.896159i	$0.353654\pi$
$278$	−59.4128	−3.56334
$279$	0	0
$280$	8.68186	0.518840
$281$	32.4768	1.93740	0.968700	−	0.248233i	$-0.0798499\pi$
$281$	32.4768	1.93740	0.968700	+	0.248233i	$0.0798499\pi$
$282$	0	0
$283$	−9.21618	−0.547845	−0.273922	−	0.961752i	$-0.588321\pi$
$283$	−9.21618	−0.547845	−0.273922	+	0.961752i	$0.588321\pi$
$284$	−2.50658	−0.148738
$285$	0	0
$286$	−2.04890	−0.121154
$287$	0.934659	0.0551712
$288$	0	0
$289$	−16.9480	−0.996941
$290$	−5.25195	−0.308405
$291$	0	0
$292$	52.4754	3.07089
$293$	−11.9851	−0.700177	−0.350088	−	0.936717i	$-0.613848\pi$
$293$	−11.9851	−0.700177	−0.350088	+	0.936717i	$0.613848\pi$
$294$	0	0
$295$	−9.45028	−0.550217
$296$	50.1958	2.91757
$297$	0	0
$298$	24.0540	1.39341
$299$	−0.319606	−0.0184833
$300$	0	0
$301$	5.59781	0.322652
$302$	−44.0349	−2.53392
$303$	0	0
$304$	51.3491	2.94508
$305$	12.5992	0.721428
$306$	0	0
$307$	−15.0754	−0.860398	−0.430199	−	0.902734i	$-0.641557\pi$
$307$	−15.0754	−0.860398	−0.430199	+	0.902734i	$0.641557\pi$
$308$	−6.07886	−0.346375
$309$	0	0
$310$	10.7660	0.611468
$311$	10.6937	0.606384	0.303192	−	0.952929i	$-0.401948\pi$
$311$	10.6937	0.606384	0.303192	+	0.952929i	$0.401948\pi$
$312$	0	0
$313$	34.4367	1.94647	0.973237	−	0.229802i	$-0.0738080\pi$
$313$	34.4367	1.94647	0.973237	+	0.229802i	$0.0738080\pi$
$314$	46.4615	2.62198
$315$	0	0
$316$	−59.1607	−3.32805
$317$	29.2559	1.64317	0.821587	−	0.570084i	$-0.193089\pi$
$317$	29.2559	1.64317	0.821587	+	0.570084i	$0.193089\pi$
$318$	0	0
$319$	2.34056	0.131046
$320$	60.3161	3.37177
$321$	0	0
$322$	−1.29293	−0.0720522
$323$	0.767040	0.0426792
$324$	0	0
$325$	−0.422899	−0.0234582
$326$	55.4207	3.06947
$327$	0	0
$328$	18.9911	1.04861
$329$	3.07152	0.169338
$330$	0	0
$331$	−24.9937	−1.37378	−0.686889	−	0.726763i	$-0.741024\pi$
$331$	−24.9937	−1.37378	−0.686889	+	0.726763i	$0.741024\pi$
$332$	−82.8273	−4.54574
$333$	0	0
$334$	−47.4526	−2.59649
$335$	14.1797	0.774718
$336$	0	0
$337$	−1.08292	−0.0589902	−0.0294951	−	0.999565i	$-0.509390\pi$
$337$	−1.08292	−0.0589902	−0.0294951	+	0.999565i	$0.509390\pi$
$338$	35.3267	1.92152
$339$	0	0
$340$	2.40594	0.130480
$341$	−4.79794	−0.259823
$342$	0	0
$343$	6.50354	0.351158
$344$	113.740	6.13247
$345$	0	0
$346$	−46.5992	−2.50519
$347$	10.6720	0.572904	0.286452	−	0.958095i	$-0.407524\pi$
$347$	10.6720	0.572904	0.286452	+	0.958095i	$0.407524\pi$
$348$	0	0
$349$	35.6630	1.90900	0.954498	−	0.298216i	$-0.0963916\pi$
$349$	35.6630	1.90900	0.954498	+	0.298216i	$0.0963916\pi$
$350$	−1.71079	−0.0914455
$351$	0	0
$352$	−52.9728	−2.82346
$353$	2.01254	0.107117	0.0535584	−	0.998565i	$-0.482944\pi$
$353$	2.01254	0.107117	0.0535584	+	0.998565i	$0.482944\pi$
$354$	0	0
$355$	−0.873586	−0.0463651
$356$	15.2366	0.807538
$357$	0	0
$358$	−19.3441	−1.02237
$359$	1.80858	0.0954531	0.0477265	−	0.998860i	$-0.484802\pi$
$359$	1.80858	0.0954531	0.0477265	+	0.998860i	$0.484802\pi$
$360$	0	0
$361$	−7.68742	−0.404601
$362$	−40.2045	−2.11310
$363$	0	0
$364$	−0.830074	−0.0435077
$365$	18.2886	0.957269
$366$	0	0
$367$	−26.7995	−1.39892	−0.699460	−	0.714672i	$-0.746576\pi$
$367$	−26.7995	−1.39892	−0.699460	+	0.714672i	$0.746576\pi$
$368$	−15.2670	−0.795845
$369$	0	0
$370$	27.4854	1.42890
$371$	−0.0606504	−0.00314881
$372$	0	0
$373$	10.4556	0.541369	0.270685	−	0.962668i	$-0.412750\pi$
$373$	10.4556	0.541369	0.270685	+	0.962668i	$0.412750\pi$
$374$	−1.46198	−0.0755974
$375$	0	0
$376$	62.4094	3.21852
$377$	0.319606	0.0164606
$378$	0	0
$379$	36.1379	1.85628	0.928140	−	0.372231i	$-0.121407\pi$
$379$	36.1379	1.85628	0.928140	+	0.372231i	$0.121407\pi$
$380$	35.4836	1.82027
$381$	0	0
$382$	55.4535	2.83725
$383$	−5.70146	−0.291331	−0.145666	−	0.989334i	$-0.546532\pi$
$383$	−5.70146	−0.291331	−0.145666	+	0.989334i	$0.546532\pi$
$384$	0	0
$385$	−2.11859	−0.107973
$386$	28.6477	1.45813
$387$	0	0
$388$	−43.3442	−2.20047
$389$	−34.8012	−1.76449	−0.882245	−	0.470790i	$-0.843969\pi$
$389$	−34.8012	−1.76449	−0.882245	+	0.470790i	$0.843969\pi$
$390$	0	0
$391$	−0.228054	−0.0115332
$392$	65.0032	3.28316
$393$	0	0
$394$	2.91759	0.146986
$395$	−20.6185	−1.03743
$396$	0	0
$397$	9.47664	0.475619	0.237809	−	0.971312i	$-0.423571\pi$
$397$	9.47664	0.475619	0.237809	+	0.971312i	$0.423571\pi$
$398$	−29.2382	−1.46558
$399$	0	0
$400$	−20.2010	−1.01005
$401$	13.3593	0.667132	0.333566	−	0.942727i	$-0.391748\pi$
$401$	13.3593	0.667132	0.333566	+	0.942727i	$0.391748\pi$
$402$	0	0
$403$	−0.655163	−0.0326360
$404$	−70.8748	−3.52615
$405$	0	0
$406$	1.29293	0.0641671
$407$	−12.2490	−0.607161
$408$	0	0
$409$	11.0887	0.548302	0.274151	−	0.961687i	$-0.411603\pi$
$409$	11.0887	0.548302	0.274151	+	0.961687i	$0.411603\pi$
$410$	10.3988	0.513561
$411$	0	0
$412$	20.9427	1.03177
$413$	2.32648	0.114479
$414$	0	0
$415$	−28.8668	−1.41701
$416$	−7.23349	−0.354651
$417$	0	0
$418$	−21.5619	−1.05463
$419$	22.8227	1.11496	0.557480	−	0.830190i	$-0.311768\pi$
$419$	22.8227	1.11496	0.557480	+	0.830190i	$0.311768\pi$
$420$	0	0
$421$	−7.94252	−0.387095	−0.193547	−	0.981091i	$-0.561999\pi$
$421$	−7.94252	−0.387095	−0.193547	+	0.981091i	$0.561999\pi$
$422$	51.0605	2.48559
$423$	0	0
$424$	−1.23234	−0.0598477
$425$	−0.301758	−0.0146374
$426$	0	0
$427$	−3.10168	−0.150101
$428$	−47.1702	−2.28006
$429$	0	0
$430$	62.2800	3.00341
$431$	6.31249	0.304062	0.152031	−	0.988376i	$-0.451419\pi$
$431$	6.31249	0.304062	0.152031	+	0.988376i	$0.451419\pi$
$432$	0	0
$433$	15.9635	0.767155	0.383578	−	0.923509i	$-0.374692\pi$
$433$	15.9635	0.767155	0.383578	+	0.923509i	$0.374692\pi$
$434$	−2.65039	−0.127223
$435$	0	0
$436$	−52.6585	−2.52189
$437$	−3.36342	−0.160894
$438$	0	0
$439$	−10.3550	−0.494216	−0.247108	−	0.968988i	$-0.579480\pi$
$439$	−10.3550	−0.494216	−0.247108	+	0.968988i	$0.579480\pi$
$440$	−43.0470	−2.05219
$441$	0	0
$442$	−0.199635	−0.00949569
$443$	16.3370	0.776196	0.388098	−	0.921618i	$-0.373132\pi$
$443$	16.3370	0.776196	0.388098	+	0.921618i	$0.373132\pi$
$444$	0	0
$445$	5.31022	0.251729
$446$	54.0184	2.55785
$447$	0	0
$448$	−14.8487	−0.701534
$449$	−13.9838	−0.659938	−0.329969	−	0.943992i	$-0.607038\pi$
$449$	−13.9838	−0.659938	−0.329969	+	0.943992i	$0.607038\pi$
$450$	0	0
$451$	−4.63429	−0.218220
$452$	−55.6149	−2.61590
$453$	0	0
$454$	−10.5038	−0.492969
$455$	−0.289295	−0.0135624
$456$	0	0
$457$	10.2930	0.481485	0.240743	−	0.970589i	$-0.422609\pi$
$457$	10.2930	0.481485	0.240743	+	0.970589i	$0.422609\pi$
$458$	−22.6518	−1.05845
$459$	0	0
$460$	−10.5499	−0.491890
$461$	−31.6586	−1.47449	−0.737244	−	0.675626i	$-0.763873\pi$
$461$	−31.6586	−1.47449	−0.737244	+	0.675626i	$0.763873\pi$
$462$	0	0
$463$	9.21454	0.428236	0.214118	−	0.976808i	$-0.431312\pi$
$463$	9.21454	0.428236	0.214118	+	0.976808i	$0.431312\pi$
$464$	15.2670	0.708751
$465$	0	0
$466$	58.4440	2.70737
$467$	−15.9397	−0.737601	−0.368801	−	0.929509i	$-0.620232\pi$
$467$	−15.9397	−0.737601	−0.368801	+	0.929509i	$0.620232\pi$
$468$	0	0
$469$	−3.49076	−0.161188
$470$	34.1731	1.57629
$471$	0	0
$472$	47.2711	2.17583
$473$	−27.7554	−1.27620
$474$	0	0
$475$	−4.45043	−0.204200
$476$	−0.592296	−0.0271478
$477$	0	0
$478$	−7.34894	−0.336132
$479$	9.02738	0.412471	0.206236	−	0.978502i	$-0.433879\pi$
$479$	9.02738	0.412471	0.206236	+	0.978502i	$0.433879\pi$
$480$	0	0
$481$	−1.67261	−0.0762647
$482$	47.0227	2.14183
$483$	0	0
$484$	−30.3801	−1.38091
$485$	−15.1062	−0.685937
$486$	0	0
$487$	−34.8413	−1.57881	−0.789405	−	0.613873i	$-0.789611\pi$
$487$	−34.8413	−1.57881	−0.789405	+	0.613873i	$0.789611\pi$
$488$	−63.0222	−2.85288
$489$	0	0
$490$	35.5934	1.60794
$491$	2.35939	0.106478	0.0532388	−	0.998582i	$-0.483046\pi$
$491$	2.35939	0.106478	0.0532388	+	0.998582i	$0.483046\pi$
$492$	0	0
$493$	0.228054	0.0102710
$494$	−2.94429	−0.132470
$495$	0	0
$496$	−31.2958	−1.40522
$497$	0.215060	0.00964677
$498$	0	0
$499$	−20.0500	−0.897559	−0.448780	−	0.893643i	$-0.648141\pi$
$499$	−20.0500	−0.897559	−0.448780	+	0.893643i	$0.648141\pi$
$500$	−66.7088	−2.98331
$501$	0	0
$502$	−67.7191	−3.02245
$503$	18.8490	0.840434	0.420217	−	0.907424i	$-0.361954\pi$
$503$	18.8490	0.840434	0.420217	+	0.907424i	$0.361954\pi$
$504$	0	0
$505$	−24.7011	−1.09918
$506$	6.41070	0.284991
$507$	0	0
$508$	64.2640	2.85125
$509$	−9.07142	−0.402084	−0.201042	−	0.979583i	$-0.564433\pi$
$509$	−9.07142	−0.402084	−0.201042	+	0.979583i	$0.564433\pi$
$510$	0	0
$511$	−4.50230	−0.199170
$512$	−52.6633	−2.32741
$513$	0	0
$514$	−68.5983	−3.02574
$515$	7.29890	0.321628
$516$	0	0
$517$	−15.2294	−0.669789
$518$	−6.76637	−0.297297
$519$	0	0
$520$	−5.87811	−0.257772
$521$	−14.2415	−0.623933	−0.311967	−	0.950093i	$-0.600988\pi$
$521$	−14.2415	−0.623933	−0.311967	+	0.950093i	$0.600988\pi$
$522$	0	0
$523$	33.1353	1.44891	0.724454	−	0.689324i	$-0.242092\pi$
$523$	33.1353	1.44891	0.724454	+	0.689324i	$0.242092\pi$
$524$	40.9526	1.78902
$525$	0	0
$526$	42.9059	1.87079
$527$	−0.467489	−0.0203641
$528$	0	0
$529$	1.00000	0.0434783
$530$	−0.674783	−0.0293107
$531$	0	0
$532$	−8.73539	−0.378727
$533$	−0.632817	−0.0274104
$534$	0	0
$535$	−16.4396	−0.710748
$536$	−70.9279	−3.06362
$537$	0	0
$538$	35.8794	1.54687
$539$	−15.8624	−0.683242
$540$	0	0
$541$	28.2274	1.21359	0.606796	−	0.794858i	$-0.292455\pi$
$541$	28.2274	1.21359	0.606796	+	0.794858i	$0.292455\pi$
$542$	−44.8871	−1.92807
$543$	0	0
$544$	−5.16143	−0.221294
$545$	−18.3524	−0.786131
$546$	0	0
$547$	−19.2164	−0.821633	−0.410817	−	0.911718i	$-0.634756\pi$
$547$	−19.2164	−0.821633	−0.410817	+	0.911718i	$0.634756\pi$
$548$	−64.6501	−2.76171
$549$	0	0
$550$	8.48256	0.361697
$551$	3.36342	0.143286
$552$	0	0
$553$	5.07589	0.215849
$554$	−40.4554	−1.71878
$555$	0	0
$556$	119.346	5.06138
$557$	7.31679	0.310022	0.155011	−	0.987913i	$-0.450459\pi$
$557$	7.31679	0.310022	0.155011	+	0.987913i	$0.450459\pi$
$558$	0	0
$559$	−3.79003	−0.160301
$560$	−13.8191	−0.583962
$561$	0	0
$562$	−88.9524	−3.75223
$563$	23.8300	1.00431	0.502157	−	0.864776i	$-0.332540\pi$
$563$	23.8300	1.00431	0.502157	+	0.864776i	$0.332540\pi$
$564$	0	0
$565$	−19.3827	−0.815438
$566$	25.2427	1.06103
$567$	0	0
$568$	4.36975	0.183351
$569$	2.37588	0.0996022	0.0498011	−	0.998759i	$-0.484141\pi$
$569$	2.37588	0.0996022	0.0498011	+	0.998759i	$0.484141\pi$
$570$	0	0
$571$	−3.18919	−0.133463	−0.0667317	−	0.997771i	$-0.521257\pi$
$571$	−3.18919	−0.133463	−0.0667317	+	0.997771i	$0.521257\pi$
$572$	4.11573	0.172087
$573$	0	0
$574$	−2.55999	−0.106852
$575$	1.32319	0.0551807
$576$	0	0
$577$	−33.9831	−1.41473	−0.707367	−	0.706846i	$-0.750117\pi$
$577$	−33.9831	−1.41473	−0.707367	+	0.706846i	$0.750117\pi$
$578$	46.4198	1.93081
$579$	0	0
$580$	10.5499	0.438060
$581$	7.10644	0.294825
$582$	0	0
$583$	0.300721	0.0124546
$584$	−91.4811	−3.78552
$585$	0	0
$586$	32.8267	1.35606
$587$	−16.7339	−0.690682	−0.345341	−	0.938477i	$-0.612237\pi$
$587$	−16.7339	−0.690682	−0.345341	+	0.938477i	$0.612237\pi$
$588$	0	0
$589$	−6.89469	−0.284091
$590$	25.8839	1.06562
$591$	0	0
$592$	−79.8975	−3.28377
$593$	−32.3693	−1.32925	−0.664623	−	0.747178i	$-0.731408\pi$
$593$	−32.3693	−1.32925	−0.664623	+	0.747178i	$0.731408\pi$
$594$	0	0
$595$	−0.206425	−0.00846261
$596$	−48.3185	−1.97920
$597$	0	0
$598$	0.875388	0.0357973
$599$	−21.6112	−0.883010	−0.441505	−	0.897259i	$-0.645555\pi$
$599$	−21.6112	−0.883010	−0.441505	+	0.897259i	$0.645555\pi$
$600$	0	0
$601$	−12.1545	−0.495793	−0.247897	−	0.968786i	$-0.579739\pi$
$601$	−12.1545	−0.495793	−0.247897	+	0.968786i	$0.579739\pi$
$602$	−15.3321	−0.624892
$603$	0	0
$604$	88.4551	3.59919
$605$	−10.5880	−0.430463
$606$	0	0
$607$	−8.60603	−0.349308	−0.174654	−	0.984630i	$-0.555881\pi$
$607$	−8.60603	−0.349308	−0.174654	+	0.984630i	$0.555881\pi$
$608$	−76.1226	−3.08718
$609$	0	0
$610$	−34.5086	−1.39721
$611$	−2.07959	−0.0841313
$612$	0	0
$613$	−15.7991	−0.638120	−0.319060	−	0.947735i	$-0.603367\pi$
$613$	−15.7991	−0.638120	−0.319060	+	0.947735i	$0.603367\pi$
$614$	41.2909	1.66636
$615$	0	0
$616$	10.5974	0.426980
$617$	20.9455	0.843235	0.421617	−	0.906774i	$-0.361463\pi$
$617$	20.9455	0.843235	0.421617	+	0.906774i	$0.361463\pi$
$618$	0	0
$619$	−39.3700	−1.58242	−0.791208	−	0.611547i	$-0.790547\pi$
$619$	−39.3700	−1.58242	−0.791208	+	0.611547i	$0.790547\pi$
$620$	−21.6262	−0.868531
$621$	0	0
$622$	−29.2896	−1.17441
$623$	−1.30727	−0.0523749
$624$	0	0
$625$	−16.6332	−0.665330
$626$	−94.3205	−3.76981
$627$	0	0
$628$	−93.3297	−3.72426
$629$	−1.19349	−0.0475874
$630$	0	0
$631$	−30.9614	−1.23256	−0.616278	−	0.787529i	$-0.711360\pi$
$631$	−30.9614	−1.23256	−0.616278	+	0.787529i	$0.711360\pi$
$632$	103.136	4.10252
$633$	0	0
$634$	−80.1305	−3.18239
$635$	22.3971	0.888802
$636$	0	0
$637$	−2.16602	−0.0858210
$638$	−6.41070	−0.253802
$639$	0	0
$640$	−78.4074	−3.09932
$641$	−11.3484	−0.448236	−0.224118	−	0.974562i	$-0.571950\pi$
$641$	−11.3484	−0.448236	−0.224118	+	0.974562i	$0.571950\pi$
$642$	0	0
$643$	−7.98456	−0.314880	−0.157440	−	0.987529i	$-0.550324\pi$
$643$	−7.98456	−0.314880	−0.157440	+	0.987529i	$0.550324\pi$
$644$	2.59718	0.102343
$645$	0	0
$646$	−2.10089	−0.0826584
$647$	−7.76293	−0.305192	−0.152596	−	0.988289i	$-0.548763\pi$
$647$	−7.76293	−0.305192	−0.152596	+	0.988289i	$0.548763\pi$
$648$	0	0
$649$	−11.5353	−0.452801
$650$	1.15830	0.0454323
$651$	0	0
$652$	−111.327	−4.35988
$653$	−41.7182	−1.63256	−0.816279	−	0.577658i	$-0.803967\pi$
$653$	−41.7182	−1.63256	−0.816279	+	0.577658i	$0.803967\pi$
$654$	0	0
$655$	14.2727	0.557681
$656$	−30.2284	−1.18022
$657$	0	0
$658$	−8.41276	−0.327963
$659$	50.7906	1.97852	0.989261	−	0.146158i	$-0.0466909\pi$
$659$	50.7906	1.97852	0.989261	+	0.146158i	$0.0466909\pi$
$660$	0	0
$661$	−4.51998	−0.175807	−0.0879035	−	0.996129i	$-0.528017\pi$
$661$	−4.51998	−0.175807	−0.0879035	+	0.996129i	$0.528017\pi$
$662$	68.4566	2.66064
$663$	0	0
$664$	144.394	5.60357
$665$	−3.04444	−0.118058
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	95.3205	3.68806
$669$	0	0
$670$	−38.8375	−1.50042
$671$	15.3790	0.593699
$672$	0	0
$673$	−11.8316	−0.456075	−0.228038	−	0.973652i	$-0.573231\pi$
$673$	−11.8316	−0.456075	−0.228038	+	0.973652i	$0.573231\pi$
$674$	2.96606	0.114248
$675$	0	0
$676$	−70.9625	−2.72933
$677$	−16.4707	−0.633022	−0.316511	−	0.948589i	$-0.602511\pi$
$677$	−16.4707	−0.633022	−0.316511	+	0.948589i	$0.602511\pi$
$678$	0	0
$679$	3.71886	0.142717
$680$	−4.19430	−0.160844
$681$	0	0
$682$	13.1413	0.503208
$683$	22.6484	0.866615	0.433308	−	0.901246i	$-0.357346\pi$
$683$	22.6484	0.866615	0.433308	+	0.901246i	$0.357346\pi$
$684$	0	0
$685$	−22.5317	−0.860891
$686$	−17.8129	−0.680101
$687$	0	0
$688$	−181.042	−6.90217
$689$	0.0410638	0.00156440
$690$	0	0
$691$	15.3540	0.584095	0.292048	−	0.956404i	$-0.405663\pi$
$691$	15.3540	0.584095	0.292048	+	0.956404i	$0.405663\pi$
$692$	93.6061	3.55837
$693$	0	0
$694$	−29.2302	−1.10956
$695$	41.5940	1.57775
$696$	0	0
$697$	−0.451544	−0.0171034
$698$	−97.6794	−3.69722
$699$	0	0
$700$	3.43655	0.129889
$701$	−33.3510	−1.25965	−0.629824	−	0.776737i	$-0.716873\pi$
$701$	−33.3510	−1.25965	−0.629824	+	0.776737i	$0.716873\pi$
$702$	0	0
$703$	−17.6020	−0.663871
$704$	73.6237	2.77480
$705$	0	0
$706$	−5.51227	−0.207457
$707$	6.08094	0.228697
$708$	0	0
$709$	−26.9569	−1.01239	−0.506194	−	0.862420i	$-0.668948\pi$
$709$	−26.9569	−1.01239	−0.506194	+	0.862420i	$0.668948\pi$
$710$	2.39271	0.0897970
$711$	0	0
$712$	−26.5622	−0.995459
$713$	2.04991	0.0767696
$714$	0	0
$715$	1.43440	0.0536437
$716$	38.8575	1.45217
$717$	0	0
$718$	−4.95361	−0.184867
$719$	15.0402	0.560905	0.280453	−	0.959868i	$-0.409515\pi$
$719$	15.0402	0.560905	0.280453	+	0.959868i	$0.409515\pi$
$720$	0	0
$721$	−1.79685	−0.0669182
$722$	21.0555	0.783605
$723$	0	0
$724$	80.7608	3.00145
$725$	−1.32319	−0.0491419
$726$	0	0
$727$	−40.9075	−1.51718	−0.758588	−	0.651571i	$-0.774110\pi$
$727$	−40.9075	−1.51718	−0.758588	+	0.651571i	$0.774110\pi$
$728$	1.44708	0.0536323
$729$	0	0
$730$	−50.0917	−1.85398
$731$	−2.70436	−0.100024
$732$	0	0
$733$	4.31515	0.159384	0.0796918	−	0.996820i	$-0.474606\pi$
$733$	4.31515	0.159384	0.0796918	+	0.996820i	$0.474606\pi$
$734$	73.4026	2.70934
$735$	0	0
$736$	22.6325	0.834245
$737$	17.3081	0.637554
$738$	0	0
$739$	14.2442	0.523980	0.261990	−	0.965071i	$-0.415621\pi$
$739$	14.2442	0.523980	0.261990	+	0.965071i	$0.415621\pi$
$740$	−55.2113	−2.02961
$741$	0	0
$742$	0.166119	0.00609841
$743$	24.3012	0.891523	0.445762	−	0.895152i	$-0.352933\pi$
$743$	24.3012	0.891523	0.445762	+	0.895152i	$0.352933\pi$
$744$	0	0
$745$	−16.8398	−0.616964
$746$	−28.6374	−1.04849
$747$	0	0
$748$	2.93676	0.107379
$749$	4.04713	0.147879
$750$	0	0
$751$	2.73017	0.0996254	0.0498127	−	0.998759i	$-0.484138\pi$
$751$	2.73017	0.0996254	0.0498127	+	0.998759i	$0.484138\pi$
$752$	−99.3380	−3.62248
$753$	0	0
$754$	−0.875388	−0.0318797
$755$	30.8281	1.12195
$756$	0	0
$757$	12.1676	0.442241	0.221120	−	0.975247i	$-0.429029\pi$
$757$	12.1676	0.442241	0.221120	+	0.975247i	$0.429029\pi$
$758$	−98.9802	−3.59512
$759$	0	0
$760$	−61.8591	−2.24386
$761$	11.9905	0.434655	0.217328	−	0.976099i	$-0.430266\pi$
$761$	11.9905	0.434655	0.217328	+	0.976099i	$0.430266\pi$
$762$	0	0
$763$	4.51801	0.163563
$764$	−111.392	−4.03004
$765$	0	0
$766$	15.6161	0.564231
$767$	−1.57516	−0.0568757
$768$	0	0
$769$	−13.6001	−0.490434	−0.245217	−	0.969468i	$-0.578859\pi$
$769$	−13.6001	−0.490434	−0.245217	+	0.969468i	$0.578859\pi$
$770$	5.80272	0.209116
$771$	0	0
$772$	−57.5461	−2.07113
$773$	25.6026	0.920860	0.460430	−	0.887696i	$-0.347695\pi$
$773$	25.6026	0.920860	0.460430	+	0.887696i	$0.347695\pi$
$774$	0	0
$775$	2.71241	0.0974326
$776$	75.5625	2.71253
$777$	0	0
$778$	95.3190	3.41735
$779$	−6.65953	−0.238603
$780$	0	0
$781$	−1.06633	−0.0381562
$782$	0.624629	0.0223367
$783$	0	0
$784$	−103.467	−3.69524
$785$	−32.5270	−1.16094
$786$	0	0
$787$	−5.97177	−0.212871	−0.106435	−	0.994320i	$-0.533944\pi$
$787$	−5.97177	−0.212871	−0.106435	+	0.994320i	$0.533944\pi$
$788$	−5.86071	−0.208779
$789$	0	0
$790$	56.4733	2.00923
$791$	4.77166	0.169661
$792$	0	0
$793$	2.10001	0.0745737
$794$	−25.9561	−0.921148
$795$	0	0
$796$	58.7324	2.08171
$797$	36.2551	1.28422	0.642111	−	0.766612i	$-0.278059\pi$
$797$	36.2551	1.28422	0.642111	+	0.766612i	$0.278059\pi$
$798$	0	0
$799$	−1.48388	−0.0524960
$800$	29.9470	1.05879
$801$	0	0
$802$	−36.5906	−1.29206
$803$	22.3236	0.787784
$804$	0	0
$805$	0.905161	0.0319027
$806$	1.79446	0.0632072
$807$	0	0
$808$	123.557	4.34672
$809$	−32.3249	−1.13648	−0.568242	−	0.822862i	$-0.692376\pi$
$809$	−32.3249	−1.13648	−0.568242	+	0.822862i	$0.692376\pi$
$810$	0	0
$811$	−23.8825	−0.838628	−0.419314	−	0.907841i	$-0.637729\pi$
$811$	−23.8825	−0.838628	−0.419314	+	0.907841i	$0.637729\pi$
$812$	−2.59718	−0.0911430
$813$	0	0
$814$	33.5495	1.17591
$815$	−38.7992	−1.35908
$816$	0	0
$817$	−39.8849	−1.39540
$818$	−30.3715	−1.06192
$819$	0	0
$820$	−20.8886	−0.729463
$821$	26.6795	0.931121	0.465560	−	0.885016i	$-0.345853\pi$
$821$	26.6795	0.931121	0.465560	+	0.885016i	$0.345853\pi$
$822$	0	0
$823$	50.7284	1.76828	0.884141	−	0.467221i	$-0.154745\pi$
$823$	50.7284	1.76828	0.884141	+	0.467221i	$0.154745\pi$
$824$	−36.5097	−1.27188
$825$	0	0
$826$	−6.37212	−0.221715
$827$	19.1478	0.665833	0.332917	−	0.942956i	$-0.391967\pi$
$827$	19.1478	0.665833	0.332917	+	0.942956i	$0.391967\pi$
$828$	0	0
$829$	2.03854	0.0708015	0.0354007	−	0.999373i	$-0.488729\pi$
$829$	2.03854	0.0708015	0.0354007	+	0.999373i	$0.488729\pi$
$830$	79.0648	2.74438
$831$	0	0
$832$	10.0534	0.348539
$833$	−1.54556	−0.0535504
$834$	0	0
$835$	33.2208	1.14965
$836$	43.3125	1.49799
$837$	0	0
$838$	−62.5103	−2.15938
$839$	14.4289	0.498142	0.249071	−	0.968485i	$-0.419875\pi$
$839$	14.4289	0.498142	0.249071	+	0.968485i	$0.419875\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	21.7542	0.749700
$843$	0	0
$844$	−102.568	−3.53053
$845$	−24.7317	−0.850795
$846$	0	0
$847$	2.60656	0.0895625
$848$	1.96153	0.0673594
$849$	0	0
$850$	0.826501	0.0283487
$851$	5.23336	0.179397
$852$	0	0
$853$	15.7135	0.538021	0.269011	−	0.963137i	$-0.413303\pi$
$853$	15.7135	0.538021	0.269011	+	0.963137i	$0.413303\pi$
$854$	8.49537	0.290706
$855$	0	0
$856$	82.2325	2.81065
$857$	20.1784	0.689281	0.344640	−	0.938735i	$-0.388001\pi$
$857$	20.1784	0.689281	0.344640	+	0.938735i	$0.388001\pi$
$858$	0	0
$859$	35.7174	1.21866	0.609331	−	0.792916i	$-0.291438\pi$
$859$	35.7174	1.21866	0.609331	+	0.792916i	$0.291438\pi$
$860$	−125.105	−4.26605
$861$	0	0
$862$	−17.2896	−0.588887
$863$	7.63686	0.259962	0.129981	−	0.991516i	$-0.458508\pi$
$863$	7.63686	0.259962	0.129981	+	0.991516i	$0.458508\pi$
$864$	0	0
$865$	32.6234	1.10923
$866$	−43.7232	−1.48578
$867$	0	0
$868$	5.32397	0.180707
$869$	−25.1676	−0.853754
$870$	0	0
$871$	2.36344	0.0800822
$872$	91.8003	3.10875
$873$	0	0
$874$	9.21226	0.311609
$875$	5.72350	0.193490
$876$	0	0
$877$	−39.3917	−1.33016	−0.665082	−	0.746771i	$-0.731603\pi$
$877$	−39.3917	−1.33016	−0.665082	+	0.746771i	$0.731603\pi$
$878$	28.3618	0.957165
$879$	0	0
$880$	68.5186	2.30976
$881$	−14.0836	−0.474487	−0.237243	−	0.971450i	$-0.576244\pi$
$881$	−14.0836	−0.474487	−0.237243	+	0.971450i	$0.576244\pi$
$882$	0	0
$883$	20.7640	0.698763	0.349381	−	0.936981i	$-0.386392\pi$
$883$	20.7640	0.698763	0.349381	+	0.936981i	$0.386392\pi$
$884$	0.401018	0.0134877
$885$	0	0
$886$	−44.7464	−1.50329
$887$	−55.5503	−1.86520	−0.932598	−	0.360918i	$-0.882464\pi$
$887$	−55.5503	−1.86520	−0.932598	+	0.360918i	$0.882464\pi$
$888$	0	0
$889$	−5.51374	−0.184925
$890$	−14.5445	−0.487531
$891$	0	0
$892$	−108.510	−3.63317
$893$	−21.8849	−0.732349
$894$	0	0
$895$	13.5425	0.452676
$896$	19.3024	0.644848
$897$	0	0
$898$	38.3011	1.27813
$899$	−2.04991	−0.0683682
$900$	0	0
$901$	0.0293009	0.000976153 0
$902$	12.6931	0.422635
$903$	0	0
$904$	96.9541	3.22465
$905$	28.1466	0.935623
$906$	0	0
$907$	−4.42952	−0.147080	−0.0735399	−	0.997292i	$-0.523430\pi$
$907$	−4.42952	−0.147080	−0.0735399	+	0.997292i	$0.523430\pi$
$908$	21.0996	0.700214
$909$	0	0
$910$	0.792367	0.0262667
$911$	1.22958	0.0407377	0.0203689	−	0.999793i	$-0.493516\pi$
$911$	1.22958	0.0407377	0.0203689	+	0.999793i	$0.493516\pi$
$912$	0	0
$913$	−35.2357	−1.16613
$914$	−28.1920	−0.932510
$915$	0	0
$916$	45.5017	1.50342
$917$	−3.51367	−0.116032
$918$	0	0
$919$	−8.29213	−0.273532	−0.136766	−	0.990603i	$-0.543671\pi$
$919$	−8.29213	−0.273532	−0.136766	+	0.990603i	$0.543671\pi$
$920$	18.3917	0.606357
$921$	0	0
$922$	86.7116	2.85569
$923$	−0.145608	−0.00479274
$924$	0	0
$925$	6.92471	0.227683
$926$	−25.2382	−0.829380
$927$	0	0
$928$	−22.6325	−0.742949
$929$	−15.6905	−0.514790	−0.257395	−	0.966306i	$-0.582864\pi$
$929$	−15.6905	−0.514790	−0.257395	+	0.966306i	$0.582864\pi$
$930$	0	0
$931$	−22.7944	−0.747058
$932$	−117.399	−3.84555
$933$	0	0
$934$	43.6581	1.42854
$935$	1.02351	0.0334725
$936$	0	0
$937$	−5.91998	−0.193397	−0.0966986	−	0.995314i	$-0.530828\pi$
$937$	−5.91998	−0.193397	−0.0966986	+	0.995314i	$0.530828\pi$
$938$	9.56105	0.312179
$939$	0	0
$940$	−68.6452	−2.23896
$941$	7.94486	0.258995	0.129498	−	0.991580i	$-0.458664\pi$
$941$	7.94486	0.258995	0.129498	+	0.991580i	$0.458664\pi$
$942$	0	0
$943$	1.97999	0.0644773
$944$	−75.2422	−2.44892
$945$	0	0
$946$	76.0210	2.47165
$947$	−48.2636	−1.56836	−0.784179	−	0.620535i	$-0.786915\pi$
$947$	−48.2636	−1.56836	−0.784179	+	0.620535i	$0.786915\pi$
$948$	0	0
$949$	3.04832	0.0989525
$950$	12.1895	0.395481
$951$	0	0
$952$	1.03256	0.0334654
$953$	−58.0127	−1.87922	−0.939608	−	0.342252i	$-0.888810\pi$
$953$	−58.0127	−1.87922	−0.939608	+	0.342252i	$0.888810\pi$
$954$	0	0
$955$	−38.8222	−1.25626
$956$	14.7622	0.477443
$957$	0	0
$958$	−24.7256	−0.798848
$959$	5.54687	0.179118
$960$	0	0
$961$	−26.7979	−0.864448
$962$	4.58122	0.147704
$963$	0	0
$964$	−94.4570	−3.04226
$965$	−20.0558	−0.645619
$966$	0	0
$967$	42.8336	1.37744	0.688718	−	0.725030i	$-0.258174\pi$
$967$	42.8336	1.37744	0.688718	+	0.725030i	$0.258174\pi$
$968$	52.9620	1.70226
$969$	0	0
$970$	41.3752	1.32848
$971$	−54.8657	−1.76072	−0.880362	−	0.474303i	$-0.842700\pi$
$971$	−54.8657	−1.76072	−0.880362	+	0.474303i	$0.842700\pi$
$972$	0	0
$973$	−10.2396	−0.328268
$974$	95.4288	3.05774
$975$	0	0
$976$	100.313	3.21096
$977$	0.842056	0.0269398	0.0134699	−	0.999909i	$-0.495712\pi$
$977$	0.842056	0.0269398	0.0134699	+	0.999909i	$0.495712\pi$
$978$	0	0
$979$	6.48182	0.207160
$980$	−71.4982	−2.28393
$981$	0	0
$982$	−6.46226	−0.206219
$983$	26.6803	0.850969	0.425485	−	0.904966i	$-0.360104\pi$
$983$	26.6803	0.850969	0.425485	+	0.904966i	$0.360104\pi$
$984$	0	0
$985$	−2.04256	−0.0650814
$986$	−0.624629	−0.0198922
$987$	0	0
$988$	5.91436	0.188161
$989$	11.8584	0.377077
$990$	0	0
$991$	−59.7070	−1.89665	−0.948327	−	0.317294i	$-0.897226\pi$
$991$	−59.7070	−1.89665	−0.948327	+	0.317294i	$0.897226\pi$
$992$	46.3945	1.47303
$993$	0	0
$994$	−0.589041	−0.0186832
$995$	20.4693	0.648919
$996$	0	0
$997$	54.7306	1.73333	0.866667	−	0.498887i	$-0.166258\pi$
$997$	54.7306	1.73333	0.866667	+	0.498887i	$0.166258\pi$
$998$	54.9159	1.73833
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.q.1.1		16
3.2	odd	2		667.2.a.d.1.16	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.d.1.16	✓	16	3.2	odd	2
6003.2.a.q.1.1		16	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)

\(f(q)\)	\(=\)	\(q-2.73896 q^{2} +5.50188 q^{4} +1.91750 q^{5} -0.472052 q^{7} -9.59151 q^{8} +O(q^{10})\)	\(q-2.73896 q^{2} +5.50188 q^{4} +1.91750 q^{5} -0.472052 q^{7} -9.59151 q^{8} -5.25195 q^{10} +2.34056 q^{11} +0.319606 q^{13} +1.29293 q^{14} +15.2670 q^{16} +0.228054 q^{17} +3.36342 q^{19} +10.5499 q^{20} -6.41070 q^{22} -1.00000 q^{23} -1.32319 q^{25} -0.875388 q^{26} -2.59718 q^{28} +1.00000 q^{29} -2.04991 q^{31} -22.6325 q^{32} -0.624629 q^{34} -0.905161 q^{35} -5.23336 q^{37} -9.21226 q^{38} -18.3917 q^{40} -1.97999 q^{41} -11.8584 q^{43} +12.8775 q^{44} +2.73896 q^{46} -6.50673 q^{47} -6.77717 q^{49} +3.62415 q^{50} +1.75844 q^{52} +0.128482 q^{53} +4.48804 q^{55} +4.52769 q^{56} -2.73896 q^{58} -4.92843 q^{59} +6.57063 q^{61} +5.61460 q^{62} +31.4556 q^{64} +0.612846 q^{65} +7.39486 q^{67} +1.25472 q^{68} +2.47920 q^{70} -0.455585 q^{71} +9.53772 q^{73} +14.3339 q^{74} +18.5051 q^{76} -1.10487 q^{77} -10.7528 q^{79} +29.2744 q^{80} +5.42311 q^{82} -15.0544 q^{83} +0.437293 q^{85} +32.4798 q^{86} -22.4495 q^{88} +2.76934 q^{89} -0.150871 q^{91} -5.50188 q^{92} +17.8217 q^{94} +6.44936 q^{95} -7.87806 q^{97} +18.5624 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8}+O(q^{10}) \) 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8	\( 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8} - 14 q^{10} - 4 q^{11} + 15 q^{13} - 8 q^{14} + 23 q^{16} - 20 q^{17} - 4 q^{19} - 25 q^{20} + 13 q^{22} - 16 q^{23} + 30 q^{25} - 25 q^{26} - 13 q^{28} + 16 q^{29} + 19 q^{32} - 23 q^{34} - 5 q^{35} + 5 q^{37} - 38 q^{38} - 20 q^{40} - 7 q^{41} - 17 q^{43} + 21 q^{44} + 3 q^{46} - 29 q^{47} + 31 q^{49} + 44 q^{50} + 20 q^{52} - 63 q^{53} + q^{55} + 19 q^{56} - 3 q^{58} - 11 q^{59} - 33 q^{62} + 29 q^{64} - 53 q^{65} - 13 q^{67} - 63 q^{68} - 46 q^{70} + 23 q^{71} - 38 q^{73} + 47 q^{74} - 56 q^{76} - 97 q^{77} - 27 q^{79} - 8 q^{80} + 9 q^{82} - 36 q^{83} + 6 q^{85} + 11 q^{86} - 24 q^{88} + 16 q^{89} - 47 q^{91} - 21 q^{92} + 37 q^{94} + 12 q^{95} - 30 q^{97} + 27 q^{98}+O(q^{100}) \) 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8 - 14 * q^10 - 4 * q^11 + 15 * q^13 - 8 * q^14 + 23 * q^16 - 20 * q^17 - 4 * q^19 - 25 * q^20 + 13 * q^22 - 16 * q^23 + 30 * q^25 - 25 * q^26 - 13 * q^28 + 16 * q^29 + 19 * q^32 - 23 * q^34 - 5 * q^35 + 5 * q^37 - 38 * q^38 - 20 * q^40 - 7 * q^41 - 17 * q^43 + 21 * q^44 + 3 * q^46 - 29 * q^47 + 31 * q^49 + 44 * q^50 + 20 * q^52 - 63 * q^53 + q^55 + 19 * q^56 - 3 * q^58 - 11 * q^59 - 33 * q^62 + 29 * q^64 - 53 * q^65 - 13 * q^67 - 63 * q^68 - 46 * q^70 + 23 * q^71 - 38 * q^73 + 47 * q^74 - 56 * q^76 - 97 * q^77 - 27 * q^79 - 8 * q^80 + 9 * q^82 - 36 * q^83 + 6 * q^85 + 11 * q^86 - 24 * q^88 + 16 * q^89 - 47 * q^91 - 21 * q^92 + 37 * q^94 + 12 * q^95 - 30 * q^97 + 27 * q^98

Introduction

L-functions

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