LMFDB - Embedded newform 6003.2.a.q.1.12

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			1.12
Root			$-1.28888$ 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)

$f(q)$	$=$	$q+1.28888 q^{2} -0.338793 q^{4} -4.18846 q^{5} -3.67888 q^{7} -3.01442 q^{8} +O(q^{10})$	\(q+1.28888 q^{2} -0.338793 q^{4} -4.18846 q^{5} -3.67888 q^{7} -3.01442 q^{8} -5.39841 q^{10} +5.13684 q^{11} +4.32617 q^{13} -4.74163 q^{14} -3.20763 q^{16} -0.381268 q^{17} +1.30789 q^{19} +1.41902 q^{20} +6.62076 q^{22} -1.00000 q^{23} +12.5432 q^{25} +5.57590 q^{26} +1.24638 q^{28} +1.00000 q^{29} +5.12340 q^{31} +1.89459 q^{32} -0.491408 q^{34} +15.4088 q^{35} -6.52227 q^{37} +1.68572 q^{38} +12.6258 q^{40} +5.19412 q^{41} -3.41912 q^{43} -1.74033 q^{44} -1.28888 q^{46} +4.78004 q^{47} +6.53417 q^{49} +16.1667 q^{50} -1.46568 q^{52} -1.50843 q^{53} -21.5154 q^{55} +11.0897 q^{56} +1.28888 q^{58} -5.31889 q^{59} -7.01094 q^{61} +6.60344 q^{62} +8.85716 q^{64} -18.1200 q^{65} -8.40996 q^{67} +0.129171 q^{68} +19.8601 q^{70} +2.40424 q^{71} +1.76206 q^{73} -8.40641 q^{74} -0.443106 q^{76} -18.8978 q^{77} -6.13455 q^{79} +13.4350 q^{80} +6.69459 q^{82} -3.80378 q^{83} +1.59692 q^{85} -4.40683 q^{86} -15.4846 q^{88} +1.90259 q^{89} -15.9155 q^{91} +0.338793 q^{92} +6.16089 q^{94} -5.47806 q^{95} -5.23059 q^{97} +8.42175 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.28888	0.911374	0.455687	−	0.890140i	$-0.349394\pi$
$2$	1.28888	0.911374	0.455687	+	0.890140i	$0.349394\pi$
$3$	0	0
$3$	0	0
$4$	−0.338793	−0.169397
$5$	−4.18846	−1.87314	−0.936568	−	0.350486i	$-0.886017\pi$
$5$	−4.18846	−1.87314	−0.936568	+	0.350486i	$0.886017\pi$
$6$	0	0
$7$	−3.67888	−1.39049	−0.695243	−	0.718774i	$-0.744703\pi$
$7$	−3.67888	−1.39049	−0.695243	+	0.718774i	$0.744703\pi$
$8$	−3.01442	−1.06576
$9$	0	0
$10$	−5.39841	−1.70713
$11$	5.13684	1.54881	0.774407	−	0.632687i	$-0.218048\pi$
$11$	5.13684	1.54881	0.774407	+	0.632687i	$0.218048\pi$
$12$	0	0
$13$	4.32617	1.19986	0.599932	−	0.800051i	$-0.295194\pi$
$13$	4.32617	1.19986	0.599932	+	0.800051i	$0.295194\pi$
$14$	−4.74163	−1.26725
$15$	0	0
$16$	−3.20763	−0.801908
$17$	−0.381268	−0.0924710	−0.0462355	−	0.998931i	$-0.514722\pi$
$17$	−0.381268	−0.0924710	−0.0462355	+	0.998931i	$0.514722\pi$
$18$	0	0
$19$	1.30789	0.300051	0.150026	−	0.988682i	$-0.452064\pi$
$19$	1.30789	0.300051	0.150026	+	0.988682i	$0.452064\pi$
$20$	1.41902	0.317303
$21$	0	0
$22$	6.62076	1.41155
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	12.5432	2.50864
$26$	5.57590	1.09352
$27$	0	0
$28$	1.24638	0.235544
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	5.12340	0.920190	0.460095	−	0.887870i	$-0.347815\pi$
$31$	5.12340	0.920190	0.460095	+	0.887870i	$0.347815\pi$
$32$	1.89459	0.334920
$33$	0	0
$34$	−0.491408	−0.0842757
$35$	15.4088	2.60457
$36$	0	0
$37$	−6.52227	−1.07225	−0.536127	−	0.844137i	$-0.680113\pi$
$37$	−6.52227	−1.07225	−0.536127	+	0.844137i	$0.680113\pi$
$38$	1.68572	0.273459
$39$	0	0
$40$	12.6258	1.99631
$41$	5.19412	0.811186	0.405593	−	0.914054i	$-0.367065\pi$
$41$	5.19412	0.811186	0.405593	+	0.914054i	$0.367065\pi$
$42$	0	0
$43$	−3.41912	−0.521411	−0.260706	−	0.965418i	$-0.583955\pi$
$43$	−3.41912	−0.521411	−0.260706	+	0.965418i	$0.583955\pi$
$44$	−1.74033	−0.262364
$45$	0	0
$46$	−1.28888	−0.190035
$47$	4.78004	0.697240	0.348620	−	0.937264i	$-0.386650\pi$
$47$	4.78004	0.697240	0.348620	+	0.937264i	$0.386650\pi$
$48$	0	0
$49$	6.53417	0.933453
$50$	16.1667	2.28631
$51$	0	0
$52$	−1.46568	−0.203253
$53$	−1.50843	−0.207199	−0.103599	−	0.994619i	$-0.533036\pi$
$53$	−1.50843	−0.207199	−0.103599	+	0.994619i	$0.533036\pi$
$54$	0	0
$55$	−21.5154	−2.90114
$56$	11.0897	1.48192
$57$	0	0
$58$	1.28888	0.169238
$59$	−5.31889	−0.692460	−0.346230	−	0.938150i	$-0.612538\pi$
$59$	−5.31889	−0.692460	−0.346230	+	0.938150i	$0.612538\pi$
$60$	0	0
$61$	−7.01094	−0.897659	−0.448829	−	0.893618i	$-0.648159\pi$
$61$	−7.01094	−0.897659	−0.448829	+	0.893618i	$0.648159\pi$
$62$	6.60344	0.838638
$63$	0	0
$64$	8.85716	1.10715
$65$	−18.1200	−2.24751
$66$	0	0
$67$	−8.40996	−1.02744	−0.513720	−	0.857958i	$-0.671733\pi$
$67$	−8.40996	−1.02744	−0.513720	+	0.857958i	$0.671733\pi$
$68$	0.129171	0.0156643
$69$	0	0
$70$	19.8601	2.37374
$71$	2.40424	0.285330	0.142665	−	0.989771i	$-0.454433\pi$
$71$	2.40424	0.285330	0.142665	+	0.989771i	$0.454433\pi$
$72$	0	0
$73$	1.76206	0.206234	0.103117	−	0.994669i	$-0.467118\pi$
$73$	1.76206	0.206234	0.103117	+	0.994669i	$0.467118\pi$
$74$	−8.40641	−0.977225
$75$	0	0
$76$	−0.443106	−0.0508277
$77$	−18.8978	−2.15361
$78$	0	0
$79$	−6.13455	−0.690191	−0.345095	−	0.938568i	$-0.612153\pi$
$79$	−6.13455	−0.690191	−0.345095	+	0.938568i	$0.612153\pi$
$80$	13.4350	1.50208
$81$	0	0
$82$	6.69459	0.739294
$83$	−3.80378	−0.417519	−0.208760	−	0.977967i	$-0.566943\pi$
$83$	−3.80378	−0.417519	−0.208760	+	0.977967i	$0.566943\pi$
$84$	0	0
$85$	1.59692	0.173211
$86$	−4.40683	−0.475201
$87$	0	0
$88$	−15.4846	−1.65066
$89$	1.90259	0.201674	0.100837	−	0.994903i	$-0.467848\pi$
$89$	1.90259	0.201674	0.100837	+	0.994903i	$0.467848\pi$
$90$	0	0
$91$	−15.9155	−1.66839
$92$	0.338793	0.0353216
$93$	0	0
$94$	6.16089	0.635447
$95$	−5.47806	−0.562037
$96$	0	0
$97$	−5.23059	−0.531086	−0.265543	−	0.964099i	$-0.585551\pi$
$97$	−5.23059	−0.531086	−0.265543	+	0.964099i	$0.585551\pi$
$98$	8.42175	0.850725
$99$	0	0
$100$	−4.24955	−0.424955
$101$	−18.5662	−1.84741	−0.923703	−	0.383110i	$-0.874853\pi$
$101$	−18.5662	−1.84741	−0.923703	+	0.383110i	$0.874853\pi$
$102$	0	0
$103$	6.34875	0.625561	0.312780	−	0.949825i	$-0.398740\pi$
$103$	6.34875	0.625561	0.312780	+	0.949825i	$0.398740\pi$
$104$	−13.0409	−1.27876
$105$	0	0
$106$	−1.94418	−0.188835
$107$	−4.81598	−0.465579	−0.232789	−	0.972527i	$-0.574785\pi$
$107$	−4.81598	−0.465579	−0.232789	+	0.972527i	$0.574785\pi$
$108$	0	0
$109$	−3.37159	−0.322940	−0.161470	−	0.986878i	$-0.551624\pi$
$109$	−3.37159	−0.322940	−0.161470	+	0.986878i	$0.551624\pi$
$110$	−27.7308	−2.64403
$111$	0	0
$112$	11.8005	1.11504
$113$	1.63180	0.153507	0.0767534	−	0.997050i	$-0.475545\pi$
$113$	1.63180	0.153507	0.0767534	+	0.997050i	$0.475545\pi$
$114$	0	0
$115$	4.18846	0.390576
$116$	−0.338793	−0.0314562
$117$	0	0
$118$	−6.85540	−0.631091
$119$	1.40264	0.128580
$120$	0	0
$121$	15.3871	1.39883
$122$	−9.03624	−0.818103
$123$	0	0
$124$	−1.73577	−0.155877
$125$	−31.5944	−2.82589
$126$	0	0
$127$	−13.2651	−1.17709	−0.588545	−	0.808464i	$-0.700299\pi$
$127$	−13.2651	−1.17709	−0.588545	+	0.808464i	$0.700299\pi$
$128$	7.62662	0.674104
$129$	0	0
$130$	−23.3544	−2.04832
$131$	9.78850	0.855225	0.427613	−	0.903962i	$-0.359355\pi$
$131$	9.78850	0.855225	0.427613	+	0.903962i	$0.359355\pi$
$132$	0	0
$133$	−4.81159	−0.417217
$134$	−10.8394	−0.936382
$135$	0	0
$136$	1.14930	0.0985518
$137$	13.0750	1.11708	0.558538	−	0.829479i	$-0.311363\pi$
$137$	13.0750	1.11708	0.558538	+	0.829479i	$0.311363\pi$
$138$	0	0
$139$	−15.5582	−1.31963	−0.659813	−	0.751430i	$-0.729364\pi$
$139$	−15.5582	−1.31963	−0.659813	+	0.751430i	$0.729364\pi$
$140$	−5.22041	−0.441206
$141$	0	0
$142$	3.09877	0.260043
$143$	22.2228	1.85837
$144$	0	0
$145$	−4.18846	−0.347833
$146$	2.27108	0.187956
$147$	0	0
$148$	2.20970	0.181636
$149$	1.57713	0.129203	0.0646016	−	0.997911i	$-0.479422\pi$
$149$	1.57713	0.129203	0.0646016	+	0.997911i	$0.479422\pi$
$150$	0	0
$151$	−0.660885	−0.0537820	−0.0268910	−	0.999638i	$-0.508561\pi$
$151$	−0.660885	−0.0537820	−0.0268910	+	0.999638i	$0.508561\pi$
$152$	−3.94254	−0.319782
$153$	0	0
$154$	−24.3570	−1.96274
$155$	−21.4592	−1.72364
$156$	0	0
$157$	6.45683	0.515311	0.257656	−	0.966237i	$-0.417050\pi$
$157$	6.45683	0.515311	0.257656	+	0.966237i	$0.417050\pi$
$158$	−7.90668	−0.629022
$159$	0	0
$160$	−7.93542	−0.627350
$161$	3.67888	0.289936
$162$	0	0
$163$	7.40570	0.580059	0.290029	−	0.957018i	$-0.406335\pi$
$163$	7.40570	0.580059	0.290029	+	0.957018i	$0.406335\pi$
$164$	−1.75973	−0.137412
$165$	0	0
$166$	−4.90261	−0.380516
$167$	12.5137	0.968337	0.484168	−	0.874975i	$-0.339122\pi$
$167$	12.5137	0.968337	0.484168	+	0.874975i	$0.339122\pi$
$168$	0	0
$169$	5.71573	0.439672
$170$	2.05824	0.157860
$171$	0	0
$172$	1.15838	0.0883253
$173$	1.51830	0.115435	0.0577173	−	0.998333i	$-0.481618\pi$
$173$	1.51830	0.115435	0.0577173	+	0.998333i	$0.481618\pi$
$174$	0	0
$175$	−46.1449	−3.48823
$176$	−16.4771	−1.24201
$177$	0	0
$178$	2.45221	0.183801
$179$	20.0342	1.49742	0.748712	−	0.662896i	$-0.230673\pi$
$179$	20.0342	1.49742	0.748712	+	0.662896i	$0.230673\pi$
$180$	0	0
$181$	8.19072	0.608812	0.304406	−	0.952542i	$-0.401542\pi$
$181$	8.19072	0.608812	0.304406	+	0.952542i	$0.401542\pi$
$182$	−20.5131	−1.52053
$183$	0	0
$184$	3.01442	0.222226
$185$	27.3183	2.00848
$186$	0	0
$187$	−1.95851	−0.143220
$188$	−1.61944	−0.118110
$189$	0	0
$190$	−7.06055	−0.512226
$191$	1.37096	0.0991992	0.0495996	−	0.998769i	$-0.484205\pi$
$191$	1.37096	0.0991992	0.0495996	+	0.998769i	$0.484205\pi$
$192$	0	0
$193$	−18.5215	−1.33320	−0.666602	−	0.745414i	$-0.732252\pi$
$193$	−18.5215	−1.33320	−0.666602	+	0.745414i	$0.732252\pi$
$194$	−6.74159	−0.484018
$195$	0	0
$196$	−2.21373	−0.158124
$197$	20.7478	1.47822	0.739111	−	0.673583i	$-0.235246\pi$
$197$	20.7478	1.47822	0.739111	+	0.673583i	$0.235246\pi$
$198$	0	0
$199$	9.02496	0.639763	0.319881	−	0.947458i	$-0.396357\pi$
$199$	9.02496	0.639763	0.319881	+	0.947458i	$0.396357\pi$
$200$	−37.8105	−2.67360
$201$	0	0
$202$	−23.9296	−1.68368
$203$	−3.67888	−0.258207
$204$	0	0
$205$	−21.7554	−1.51946
$206$	8.18276	0.570120
$207$	0	0
$208$	−13.8768	−0.962180
$209$	6.71844	0.464724
$210$	0	0
$211$	11.9605	0.823396	0.411698	−	0.911320i	$-0.364936\pi$
$211$	11.9605	0.823396	0.411698	+	0.911320i	$0.364936\pi$
$212$	0.511045	0.0350987
$213$	0	0
$214$	−6.20722	−0.424317
$215$	14.3209	0.976674
$216$	0	0
$217$	−18.8484	−1.27951
$218$	−4.34557	−0.294319
$219$	0	0
$220$	7.28929	0.491444
$221$	−1.64943	−0.110953
$222$	0	0
$223$	−10.0296	−0.671632	−0.335816	−	0.941928i	$-0.609012\pi$
$223$	−10.0296	−0.671632	−0.335816	+	0.941928i	$0.609012\pi$
$224$	−6.96998	−0.465701
$225$	0	0
$226$	2.10319	0.139902
$227$	−12.8144	−0.850523	−0.425261	−	0.905071i	$-0.639818\pi$
$227$	−12.8144	−0.850523	−0.425261	+	0.905071i	$0.639818\pi$
$228$	0	0
$229$	3.19732	0.211285	0.105642	−	0.994404i	$-0.466310\pi$
$229$	3.19732	0.211285	0.105642	+	0.994404i	$0.466310\pi$
$230$	5.39841	0.355961
$231$	0	0
$232$	−3.01442	−0.197906
$233$	27.1798	1.78061	0.890305	−	0.455364i	$-0.150491\pi$
$233$	27.1798	1.78061	0.890305	+	0.455364i	$0.150491\pi$
$234$	0	0
$235$	−20.0210	−1.30603
$236$	1.80200	0.117300
$237$	0	0
$238$	1.80783	0.117184
$239$	4.34212	0.280869	0.140434	−	0.990090i	$-0.455150\pi$
$239$	4.34212	0.280869	0.140434	+	0.990090i	$0.455150\pi$
$240$	0	0
$241$	11.6814	0.752466	0.376233	−	0.926525i	$-0.377219\pi$
$241$	11.6814	0.752466	0.376233	+	0.926525i	$0.377219\pi$
$242$	19.8321	1.27486
$243$	0	0
$244$	2.37526	0.152060
$245$	−27.3681	−1.74848
$246$	0	0
$247$	5.65817	0.360021
$248$	−15.4441	−0.980700
$249$	0	0
$250$	−40.7213	−2.57544
$251$	3.07955	0.194380	0.0971898	−	0.995266i	$-0.469015\pi$
$251$	3.07955	0.194380	0.0971898	+	0.995266i	$0.469015\pi$
$252$	0	0
$253$	−5.13684	−0.322950
$254$	−17.0971	−1.07277
$255$	0	0
$256$	−7.88454	−0.492784
$257$	−29.1604	−1.81898	−0.909489	−	0.415729i	$-0.863527\pi$
$257$	−29.1604	−1.81898	−0.909489	+	0.415729i	$0.863527\pi$
$258$	0	0
$259$	23.9947	1.49096
$260$	6.13893	0.380720
$261$	0	0
$262$	12.6162	0.779430
$263$	−22.1485	−1.36574	−0.682868	−	0.730542i	$-0.739268\pi$
$263$	−22.1485	−1.36574	−0.682868	+	0.730542i	$0.739268\pi$
$264$	0	0
$265$	6.31799	0.388111
$266$	−6.20155	−0.380241
$267$	0	0
$268$	2.84924	0.174045
$269$	−13.7724	−0.839718	−0.419859	−	0.907589i	$-0.637921\pi$
$269$	−13.7724	−0.839718	−0.419859	+	0.907589i	$0.637921\pi$
$270$	0	0
$271$	6.71784	0.408080	0.204040	−	0.978963i	$-0.434593\pi$
$271$	6.71784	0.408080	0.204040	+	0.978963i	$0.434593\pi$
$272$	1.22297	0.0741533
$273$	0	0
$274$	16.8521	1.01807
$275$	64.4324	3.88542
$276$	0	0
$277$	−3.89335	−0.233929	−0.116964	−	0.993136i	$-0.537316\pi$
$277$	−3.89335	−0.233929	−0.116964	+	0.993136i	$0.537316\pi$
$278$	−20.0526	−1.20267
$279$	0	0
$280$	−46.4487	−2.77584
$281$	1.01407	0.0604942	0.0302471	−	0.999542i	$-0.490371\pi$
$281$	1.01407	0.0604942	0.0302471	+	0.999542i	$0.490371\pi$
$282$	0	0
$283$	−4.16457	−0.247558	−0.123779	−	0.992310i	$-0.539501\pi$
$283$	−4.16457	−0.247558	−0.123779	+	0.992310i	$0.539501\pi$
$284$	−0.814539	−0.0483340
$285$	0	0
$286$	28.6425	1.69367
$287$	−19.1086	−1.12794
$288$	0	0
$289$	−16.8546	−0.991449
$290$	−5.39841	−0.317006
$291$	0	0
$292$	−0.596974	−0.0349353
$293$	−32.2403	−1.88350	−0.941750	−	0.336313i	$-0.890820\pi$
$293$	−32.2403	−1.88350	−0.941750	+	0.336313i	$0.890820\pi$
$294$	0	0
$295$	22.2779	1.29707
$296$	19.6609	1.14276
$297$	0	0
$298$	2.03272	0.117752
$299$	−4.32617	−0.250189
$300$	0	0
$301$	12.5785	0.725015
$302$	−0.851800	−0.0490156
$303$	0	0
$304$	−4.19524	−0.240614
$305$	29.3650	1.68144
$306$	0	0
$307$	−30.7765	−1.75651	−0.878255	−	0.478193i	$-0.841292\pi$
$307$	−30.7765	−1.75651	−0.878255	+	0.478193i	$0.841292\pi$
$308$	6.40245	0.364814
$309$	0	0
$310$	−27.6583	−1.57088
$311$	−22.4954	−1.27560	−0.637799	−	0.770203i	$-0.720155\pi$
$311$	−22.4954	−1.27560	−0.637799	+	0.770203i	$0.720155\pi$
$312$	0	0
$313$	26.6925	1.50875	0.754375	−	0.656444i	$-0.227940\pi$
$313$	26.6925	1.50875	0.754375	+	0.656444i	$0.227940\pi$
$314$	8.32207	0.469642
$315$	0	0
$316$	2.07834	0.116916
$317$	−25.5022	−1.43235	−0.716173	−	0.697923i	$-0.754108\pi$
$317$	−25.5022	−1.43235	−0.716173	+	0.697923i	$0.754108\pi$
$318$	0	0
$319$	5.13684	0.287608
$320$	−37.0979	−2.07383
$321$	0	0
$322$	4.74163	0.264241
$323$	−0.498658	−0.0277461
$324$	0	0
$325$	54.2640	3.01002
$326$	9.54504	0.528651
$327$	0	0
$328$	−15.6573	−0.864528
$329$	−17.5852	−0.969503
$330$	0	0
$331$	4.67948	0.257208	0.128604	−	0.991696i	$-0.458950\pi$
$331$	4.67948	0.257208	0.128604	+	0.991696i	$0.458950\pi$
$332$	1.28870	0.0707263
$333$	0	0
$334$	16.1286	0.882517
$335$	35.2248	1.92453
$336$	0	0
$337$	−24.1609	−1.31613	−0.658065	−	0.752962i	$-0.728625\pi$
$337$	−24.1609	−1.31613	−0.658065	+	0.752962i	$0.728625\pi$
$338$	7.36688	0.400706
$339$	0	0
$340$	−0.541027	−0.0293413
$341$	26.3181	1.42520
$342$	0	0
$343$	1.71373	0.0925329
$344$	10.3067	0.555698
$345$	0	0
$346$	1.95691	0.105204
$347$	−21.5069	−1.15455	−0.577277	−	0.816549i	$-0.695885\pi$
$347$	−21.5069	−1.15455	−0.577277	+	0.816549i	$0.695885\pi$
$348$	0	0
$349$	−26.7629	−1.43259	−0.716293	−	0.697799i	$-0.754163\pi$
$349$	−26.7629	−1.43259	−0.716293	+	0.697799i	$0.754163\pi$
$350$	−59.4752	−3.17908
$351$	0	0
$352$	9.73221	0.518728
$353$	−31.2917	−1.66549	−0.832744	−	0.553659i	$-0.813231\pi$
$353$	−31.2917	−1.66549	−0.832744	+	0.553659i	$0.813231\pi$
$354$	0	0
$355$	−10.0700	−0.534462
$356$	−0.644584	−0.0341629
$357$	0	0
$358$	25.8216	1.36471
$359$	−24.0817	−1.27098	−0.635492	−	0.772108i	$-0.719202\pi$
$359$	−24.0817	−1.27098	−0.635492	+	0.772108i	$0.719202\pi$
$360$	0	0
$361$	−17.2894	−0.909969
$362$	10.5568	0.554855
$363$	0	0
$364$	5.39205	0.282620
$365$	−7.38032	−0.386303
$366$	0	0
$367$	−34.2762	−1.78920	−0.894601	−	0.446866i	$-0.852540\pi$
$367$	−34.2762	−1.78920	−0.894601	+	0.446866i	$0.852540\pi$
$368$	3.20763	0.167209
$369$	0	0
$370$	35.2099	1.83048
$371$	5.54933	0.288107
$372$	0	0
$373$	27.2701	1.41199	0.705996	−	0.708215i	$-0.250499\pi$
$373$	27.2701	1.41199	0.705996	+	0.708215i	$0.250499\pi$
$374$	−2.52428	−0.130527
$375$	0	0
$376$	−14.4090	−0.743090
$377$	4.32617	0.222809
$378$	0	0
$379$	−20.0842	−1.03165	−0.515827	−	0.856693i	$-0.672515\pi$
$379$	−20.0842	−1.03165	−0.515827	+	0.856693i	$0.672515\pi$
$380$	1.85593	0.0952072
$381$	0	0
$382$	1.76700	0.0904076
$383$	−1.34037	−0.0684897	−0.0342448	−	0.999413i	$-0.510903\pi$
$383$	−1.34037	−0.0684897	−0.0342448	+	0.999413i	$0.510903\pi$
$384$	0	0
$385$	79.1527	4.03400
$386$	−23.8719	−1.21505
$387$	0	0
$388$	1.77209	0.0899642
$389$	−30.1105	−1.52666	−0.763330	−	0.646008i	$-0.776437\pi$
$389$	−30.1105	−1.52666	−0.763330	+	0.646008i	$0.776437\pi$
$390$	0	0
$391$	0.381268	0.0192815
$392$	−19.6967	−0.994835
$393$	0	0
$394$	26.7414	1.34721
$395$	25.6943	1.29282
$396$	0	0
$397$	−21.1262	−1.06030	−0.530148	−	0.847905i	$-0.677864\pi$
$397$	−21.1262	−1.06030	−0.530148	+	0.847905i	$0.677864\pi$
$398$	11.6321	0.583063
$399$	0	0
$400$	−40.2340	−2.01170
$401$	29.6567	1.48099	0.740493	−	0.672064i	$-0.234592\pi$
$401$	29.6567	1.48099	0.740493	+	0.672064i	$0.234592\pi$
$402$	0	0
$403$	22.1647	1.10410
$404$	6.29010	0.312944
$405$	0	0
$406$	−4.74163	−0.235323
$407$	−33.5038	−1.66072
$408$	0	0
$409$	15.8519	0.783826	0.391913	−	0.920002i	$-0.371813\pi$
$409$	15.8519	0.783826	0.391913	+	0.920002i	$0.371813\pi$
$410$	−28.0400	−1.38480
$411$	0	0
$412$	−2.15091	−0.105968
$413$	19.5676	0.962857
$414$	0	0
$415$	15.9320	0.782070
$416$	8.19632	0.401858
$417$	0	0
$418$	8.65925	0.423538
$419$	−8.95208	−0.437338	−0.218669	−	0.975799i	$-0.570171\pi$
$419$	−8.95208	−0.437338	−0.218669	+	0.975799i	$0.570171\pi$
$420$	0	0
$421$	−37.9501	−1.84958	−0.924788	−	0.380484i	$-0.875757\pi$
$421$	−37.9501	−1.84958	−0.924788	+	0.380484i	$0.875757\pi$
$422$	15.4156	0.750422
$423$	0	0
$424$	4.54703	0.220824
$425$	−4.78232	−0.231976
$426$	0	0
$427$	25.7924	1.24818
$428$	1.63162	0.0788675
$429$	0	0
$430$	18.4578	0.890116
$431$	−35.4309	−1.70665	−0.853324	−	0.521380i	$-0.825417\pi$
$431$	−35.4309	−1.70665	−0.853324	+	0.521380i	$0.825417\pi$
$432$	0	0
$433$	−10.6178	−0.510261	−0.255131	−	0.966907i	$-0.582118\pi$
$433$	−10.6178	−0.510261	−0.255131	+	0.966907i	$0.582118\pi$
$434$	−24.2933	−1.16611
$435$	0	0
$436$	1.14227	0.0547050
$437$	−1.30789	−0.0625650
$438$	0	0
$439$	11.6557	0.556295	0.278147	−	0.960538i	$-0.410280\pi$
$439$	11.6557	0.556295	0.278147	+	0.960538i	$0.410280\pi$
$440$	64.8566	3.09191
$441$	0	0
$442$	−2.12591	−0.101119
$443$	23.0692	1.09605	0.548026	−	0.836461i	$-0.315379\pi$
$443$	23.0692	1.09605	0.548026	+	0.836461i	$0.315379\pi$
$444$	0	0
$445$	−7.96892	−0.377763
$446$	−12.9269	−0.612108
$447$	0	0
$448$	−32.5845	−1.53947
$449$	−21.0609	−0.993926	−0.496963	−	0.867772i	$-0.665552\pi$
$449$	−21.0609	−0.993926	−0.496963	+	0.867772i	$0.665552\pi$
$450$	0	0
$451$	26.6814	1.25638
$452$	−0.552842	−0.0260035
$453$	0	0
$454$	−16.5162	−0.775145
$455$	66.6613	3.12513
$456$	0	0
$457$	21.3393	0.998209	0.499105	−	0.866542i	$-0.333662\pi$
$457$	21.3393	0.998209	0.499105	+	0.866542i	$0.333662\pi$
$458$	4.12095	0.192559
$459$	0	0
$460$	−1.41902	−0.0661622
$461$	4.11175	0.191503	0.0957517	−	0.995405i	$-0.469475\pi$
$461$	4.11175	0.191503	0.0957517	+	0.995405i	$0.469475\pi$
$462$	0	0
$463$	−13.9335	−0.647543	−0.323771	−	0.946135i	$-0.604951\pi$
$463$	−13.9335	−0.647543	−0.323771	+	0.946135i	$0.604951\pi$
$464$	−3.20763	−0.148911
$465$	0	0
$466$	35.0315	1.62280
$467$	32.6447	1.51062	0.755308	−	0.655370i	$-0.227487\pi$
$467$	32.6447	1.51062	0.755308	+	0.655370i	$0.227487\pi$
$468$	0	0
$469$	30.9392	1.42864
$470$	−25.8046	−1.19028
$471$	0	0
$472$	16.0334	0.737995
$473$	−17.5635	−0.807569
$474$	0	0
$475$	16.4052	0.752721
$476$	−0.475205	−0.0217810
$477$	0	0
$478$	5.59647	0.255976
$479$	11.8247	0.540285	0.270143	−	0.962820i	$-0.412929\pi$
$479$	11.8247	0.540285	0.270143	+	0.962820i	$0.412929\pi$
$480$	0	0
$481$	−28.2164	−1.28656
$482$	15.0559	0.685778
$483$	0	0
$484$	−5.21305	−0.236957
$485$	21.9081	0.994796
$486$	0	0
$487$	33.8627	1.53447	0.767233	−	0.641369i	$-0.221633\pi$
$487$	33.8627	1.53447	0.767233	+	0.641369i	$0.221633\pi$
$488$	21.1339	0.956687
$489$	0	0
$490$	−35.2742	−1.59352
$491$	7.01774	0.316706	0.158353	−	0.987383i	$-0.449382\pi$
$491$	7.01774	0.316706	0.158353	+	0.987383i	$0.449382\pi$
$492$	0	0
$493$	−0.381268	−0.0171714
$494$	7.29269	0.328114
$495$	0	0
$496$	−16.4340	−0.737908
$497$	−8.84490	−0.396748
$498$	0	0
$499$	−17.8635	−0.799681	−0.399841	−	0.916585i	$-0.630935\pi$
$499$	−17.8635	−0.799681	−0.399841	+	0.916585i	$0.630935\pi$
$500$	10.7040	0.478696
$501$	0	0
$502$	3.96917	0.177153
$503$	−21.7670	−0.970544	−0.485272	−	0.874363i	$-0.661279\pi$
$503$	−21.7670	−0.970544	−0.485272	+	0.874363i	$0.661279\pi$
$504$	0	0
$505$	77.7638	3.46044
$506$	−6.62076	−0.294329
$507$	0	0
$508$	4.49414	0.199395
$509$	21.6851	0.961175	0.480587	−	0.876947i	$-0.340424\pi$
$509$	21.6851	0.961175	0.480587	+	0.876947i	$0.340424\pi$
$510$	0	0
$511$	−6.48241	−0.286765
$512$	−25.4155	−1.12321
$513$	0	0
$514$	−37.5842	−1.65777
$515$	−26.5915	−1.17176
$516$	0	0
$517$	24.5543	1.07990
$518$	30.9262	1.35882
$519$	0	0
$520$	54.6212	2.39530
$521$	23.2799	1.01991	0.509955	−	0.860201i	$-0.329662\pi$
$521$	23.2799	1.01991	0.509955	+	0.860201i	$0.329662\pi$
$522$	0	0
$523$	5.23245	0.228799	0.114400	−	0.993435i	$-0.463506\pi$
$523$	5.23245	0.228799	0.114400	+	0.993435i	$0.463506\pi$
$524$	−3.31628	−0.144872
$525$	0	0
$526$	−28.5467	−1.24470
$527$	−1.95339	−0.0850910
$528$	0	0
$529$	1.00000	0.0434783
$530$	8.14312	0.353714
$531$	0	0
$532$	1.63013	0.0706752
$533$	22.4707	0.973312
$534$	0	0
$535$	20.1716	0.872093
$536$	25.3511	1.09500
$537$	0	0
$538$	−17.7509	−0.765298
$539$	33.5650	1.44575
$540$	0	0
$541$	42.6054	1.83175	0.915875	−	0.401465i	$-0.131499\pi$
$541$	42.6054	1.83175	0.915875	+	0.401465i	$0.131499\pi$
$542$	8.65848	0.371913
$543$	0	0
$544$	−0.722347	−0.0309704
$545$	14.1218	0.604911
$546$	0	0
$547$	9.76176	0.417383	0.208691	−	0.977982i	$-0.433080\pi$
$547$	9.76176	0.417383	0.208691	+	0.977982i	$0.433080\pi$
$548$	−4.42974	−0.189229
$549$	0	0
$550$	83.0455	3.54107
$551$	1.30789	0.0557181
$552$	0	0
$553$	22.5683	0.959701
$554$	−5.01806	−0.213197
$555$	0	0
$556$	5.27100	0.223540
$557$	−22.0226	−0.933126	−0.466563	−	0.884488i	$-0.654508\pi$
$557$	−22.0226	−0.933126	−0.466563	+	0.884488i	$0.654508\pi$
$558$	0	0
$559$	−14.7917	−0.625622
$560$	−49.4259	−2.08863
$561$	0	0
$562$	1.30701	0.0551329
$563$	−1.66816	−0.0703045	−0.0351523	−	0.999382i	$-0.511192\pi$
$563$	−1.66816	−0.0703045	−0.0351523	+	0.999382i	$0.511192\pi$
$564$	0	0
$565$	−6.83472	−0.287539
$566$	−5.36762	−0.225618
$567$	0	0
$568$	−7.24737	−0.304093
$569$	7.92623	0.332285	0.166142	−	0.986102i	$-0.446869\pi$
$569$	7.92623	0.332285	0.166142	+	0.986102i	$0.446869\pi$
$570$	0	0
$571$	22.2556	0.931370	0.465685	−	0.884951i	$-0.345808\pi$
$571$	22.2556	0.931370	0.465685	+	0.884951i	$0.345808\pi$
$572$	−7.52894	−0.314801
$573$	0	0
$574$	−24.6286	−1.02798
$575$	−12.5432	−0.523087
$576$	0	0
$577$	1.76754	0.0735835	0.0367918	−	0.999323i	$-0.488286\pi$
$577$	1.76754	0.0735835	0.0367918	+	0.999323i	$0.488286\pi$
$578$	−21.7236	−0.903581
$579$	0	0
$580$	1.41902	0.0589217
$581$	13.9937	0.580555
$582$	0	0
$583$	−7.74855	−0.320912
$584$	−5.31159	−0.219795
$585$	0	0
$586$	−41.5539	−1.71657
$587$	−31.6969	−1.30827	−0.654135	−	0.756377i	$-0.726967\pi$
$587$	−31.6969	−1.30827	−0.654135	+	0.756377i	$0.726967\pi$
$588$	0	0
$589$	6.70087	0.276104
$590$	28.7136	1.18212
$591$	0	0
$592$	20.9210	0.859850
$593$	−11.5949	−0.476146	−0.238073	−	0.971247i	$-0.576516\pi$
$593$	−11.5949	−0.476146	−0.238073	+	0.971247i	$0.576516\pi$
$594$	0	0
$595$	−5.87490	−0.240847
$596$	−0.534320	−0.0218866
$597$	0	0
$598$	−5.57590	−0.228016
$599$	−44.2328	−1.80730	−0.903652	−	0.428268i	$-0.859124\pi$
$599$	−44.2328	−1.80730	−0.903652	+	0.428268i	$0.859124\pi$
$600$	0	0
$601$	−17.7972	−0.725964	−0.362982	−	0.931796i	$-0.618241\pi$
$601$	−17.7972	−0.725964	−0.362982	+	0.931796i	$0.618241\pi$
$602$	16.2122	0.660760
$603$	0	0
$604$	0.223903	0.00911050
$605$	−64.4482	−2.62019
$606$	0	0
$607$	16.6439	0.675556	0.337778	−	0.941226i	$-0.390325\pi$
$607$	16.6439	0.675556	0.337778	+	0.941226i	$0.390325\pi$
$608$	2.47792	0.100493
$609$	0	0
$610$	37.8479	1.53242
$611$	20.6793	0.836593
$612$	0	0
$613$	47.0438	1.90008	0.950040	−	0.312129i	$-0.101042\pi$
$613$	47.0438	1.90008	0.950040	+	0.312129i	$0.101042\pi$
$614$	−39.6672	−1.60084
$615$	0	0
$616$	56.9659	2.29522
$617$	−2.14822	−0.0864842	−0.0432421	−	0.999065i	$-0.513769\pi$
$617$	−2.14822	−0.0864842	−0.0432421	+	0.999065i	$0.513769\pi$
$618$	0	0
$619$	−42.3549	−1.70239	−0.851193	−	0.524853i	$-0.824120\pi$
$619$	−42.3549	−1.70239	−0.851193	+	0.524853i	$0.824120\pi$
$620$	7.27022	0.291979
$621$	0	0
$622$	−28.9939	−1.16255
$623$	−6.99940	−0.280425
$624$	0	0
$625$	69.6158	2.78463
$626$	34.4034	1.37504
$627$	0	0
$628$	−2.18753	−0.0872920
$629$	2.48673	0.0991525
$630$	0	0
$631$	−24.6603	−0.981711	−0.490855	−	0.871241i	$-0.663316\pi$
$631$	−24.6603	−0.981711	−0.490855	+	0.871241i	$0.663316\pi$
$632$	18.4921	0.735576
$633$	0	0
$634$	−32.8692	−1.30540
$635$	55.5605	2.20485
$636$	0	0
$637$	28.2679	1.12002
$638$	6.62076	0.262118
$639$	0	0
$640$	−31.9438	−1.26269
$641$	−16.0011	−0.632004	−0.316002	−	0.948759i	$-0.602341\pi$
$641$	−16.0011	−0.632004	−0.316002	+	0.948759i	$0.602341\pi$
$642$	0	0
$643$	37.7703	1.48952	0.744759	−	0.667334i	$-0.232565\pi$
$643$	37.7703	1.48952	0.744759	+	0.667334i	$0.232565\pi$
$644$	−1.24638	−0.0491143
$645$	0	0
$646$	−0.642709	−0.0252871
$647$	−34.5123	−1.35682	−0.678408	−	0.734685i	$-0.737330\pi$
$647$	−34.5123	−1.35682	−0.678408	+	0.734685i	$0.737330\pi$
$648$	0	0
$649$	−27.3223	−1.07249
$650$	69.9397	2.74326
$651$	0	0
$652$	−2.50900	−0.0982600
$653$	9.67983	0.378801	0.189400	−	0.981900i	$-0.439346\pi$
$653$	9.67983	0.378801	0.189400	+	0.981900i	$0.439346\pi$
$654$	0	0
$655$	−40.9987	−1.60195
$656$	−16.6608	−0.650497
$657$	0	0
$658$	−22.6652	−0.883580
$659$	43.3472	1.68857	0.844283	−	0.535897i	$-0.180026\pi$
$659$	43.3472	1.68857	0.844283	+	0.535897i	$0.180026\pi$
$660$	0	0
$661$	−0.814945	−0.0316977	−0.0158489	−	0.999874i	$-0.505045\pi$
$661$	−0.814945	−0.0316977	−0.0158489	+	0.999874i	$0.505045\pi$
$662$	6.03128	0.234412
$663$	0	0
$664$	11.4662	0.444974
$665$	20.1531	0.781505
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−4.23955	−0.164033
$669$	0	0
$670$	45.4004	1.75397
$671$	−36.0140	−1.39031
$672$	0	0
$673$	42.2226	1.62756	0.813781	−	0.581172i	$-0.197406\pi$
$673$	42.2226	1.62756	0.813781	+	0.581172i	$0.197406\pi$
$674$	−31.1405	−1.19949
$675$	0	0
$676$	−1.93645	−0.0744789
$677$	−40.8174	−1.56874	−0.784370	−	0.620293i	$-0.787014\pi$
$677$	−40.8174	−1.56874	−0.784370	+	0.620293i	$0.787014\pi$
$678$	0	0
$679$	19.2427	0.738468
$680$	−4.81380	−0.184601
$681$	0	0
$682$	33.9208	1.29889
$683$	−32.0002	−1.22445	−0.612227	−	0.790682i	$-0.709726\pi$
$683$	−32.0002	−1.22445	−0.612227	+	0.790682i	$0.709726\pi$
$684$	0	0
$685$	−54.7643	−2.09244
$686$	2.20879	0.0843321
$687$	0	0
$688$	10.9673	0.418124
$689$	−6.52571	−0.248610
$690$	0	0
$691$	−17.0658	−0.649215	−0.324607	−	0.945849i	$-0.605232\pi$
$691$	−17.0658	−0.649215	−0.324607	+	0.945849i	$0.605232\pi$
$692$	−0.514391	−0.0195542
$693$	0	0
$694$	−27.7198	−1.05223
$695$	65.1647	2.47184
$696$	0	0
$697$	−1.98035	−0.0750112
$698$	−34.4942	−1.30562
$699$	0	0
$700$	15.6336	0.590894
$701$	8.92226	0.336989	0.168495	−	0.985703i	$-0.446109\pi$
$701$	8.92226	0.336989	0.168495	+	0.985703i	$0.446109\pi$
$702$	0	0
$703$	−8.53043	−0.321731
$704$	45.4978	1.71476
$705$	0	0
$706$	−40.3311	−1.51788
$707$	68.3028	2.56879
$708$	0	0
$709$	−0.494674	−0.0185779	−0.00928894	−	0.999957i	$-0.502957\pi$
$709$	−0.494674	−0.0185779	−0.00928894	+	0.999957i	$0.502957\pi$
$710$	−12.9791	−0.487095
$711$	0	0
$712$	−5.73520	−0.214936
$713$	−5.12340	−0.191873
$714$	0	0
$715$	−93.0794	−3.48097
$716$	−6.78744	−0.253658
$717$	0	0
$718$	−31.0384	−1.15834
$719$	3.19568	0.119179	0.0595895	−	0.998223i	$-0.481021\pi$
$719$	3.19568	0.119179	0.0595895	+	0.998223i	$0.481021\pi$
$720$	0	0
$721$	−23.3563	−0.869834
$722$	−22.2839	−0.829323
$723$	0	0
$724$	−2.77496	−0.103131
$725$	12.5432	0.465843
$726$	0	0
$727$	5.82605	0.216076	0.108038	−	0.994147i	$-0.465543\pi$
$727$	5.82605	0.216076	0.108038	+	0.994147i	$0.465543\pi$
$728$	47.9759	1.77810
$729$	0	0
$730$	−9.51233	−0.352067
$731$	1.30360	0.0482154
$732$	0	0
$733$	−37.5460	−1.38679	−0.693396	−	0.720557i	$-0.743886\pi$
$733$	−37.5460	−1.38679	−0.693396	+	0.720557i	$0.743886\pi$
$734$	−44.1778	−1.63063
$735$	0	0
$736$	−1.89459	−0.0698356
$737$	−43.2006	−1.59131
$738$	0	0
$739$	21.9698	0.808173	0.404086	−	0.914721i	$-0.367590\pi$
$739$	21.9698	0.808173	0.404086	+	0.914721i	$0.367590\pi$
$740$	−9.25524	−0.340230
$741$	0	0
$742$	7.15241	0.262573
$743$	−28.5154	−1.04613	−0.523065	−	0.852293i	$-0.675211\pi$
$743$	−28.5154	−1.04613	−0.523065	+	0.852293i	$0.675211\pi$
$744$	0	0
$745$	−6.60573	−0.242015
$746$	35.1478	1.28685
$747$	0	0
$748$	0.663530	0.0242611
$749$	17.7174	0.647381
$750$	0	0
$751$	44.0311	1.60672	0.803359	−	0.595495i	$-0.203044\pi$
$751$	44.0311	1.60672	0.803359	+	0.595495i	$0.203044\pi$
$752$	−15.3326	−0.559123
$753$	0	0
$754$	5.57590	0.203062
$755$	2.76809	0.100741
$756$	0	0
$757$	42.9767	1.56202	0.781008	−	0.624521i	$-0.214706\pi$
$757$	42.9767	1.56202	0.781008	+	0.624521i	$0.214706\pi$
$758$	−25.8860	−0.940223
$759$	0	0
$760$	16.5132	0.598996
$761$	−42.3746	−1.53608	−0.768039	−	0.640403i	$-0.778767\pi$
$761$	−42.3746	−1.53608	−0.768039	+	0.640403i	$0.778767\pi$
$762$	0	0
$763$	12.4037	0.449044
$764$	−0.464472	−0.0168040
$765$	0	0
$766$	−1.72757	−0.0624197
$767$	−23.0104	−0.830858
$768$	0	0
$769$	−44.7656	−1.61429	−0.807145	−	0.590354i	$-0.798988\pi$
$769$	−44.7656	−1.61429	−0.807145	+	0.590354i	$0.798988\pi$
$770$	102.018	3.67648
$771$	0	0
$772$	6.27495	0.225840
$773$	−27.3930	−0.985257	−0.492628	−	0.870240i	$-0.663964\pi$
$773$	−27.3930	−0.985257	−0.492628	+	0.870240i	$0.663964\pi$
$774$	0	0
$775$	64.2639	2.30843
$776$	15.7672	0.566009
$777$	0	0
$778$	−38.8087	−1.39136
$779$	6.79336	0.243397
$780$	0	0
$781$	12.3502	0.441924
$782$	0.491408	0.0175727
$783$	0	0
$784$	−20.9592	−0.748543
$785$	−27.0442	−0.965248
$786$	0	0
$787$	−20.4857	−0.730237	−0.365118	−	0.930961i	$-0.618971\pi$
$787$	−20.4857	−0.730237	−0.365118	+	0.930961i	$0.618971\pi$
$788$	−7.02923	−0.250406
$789$	0	0
$790$	33.1168	1.17824
$791$	−6.00319	−0.213449
$792$	0	0
$793$	−30.3305	−1.07707
$794$	−27.2292	−0.966327
$795$	0	0
$796$	−3.05760	−0.108374
$797$	−25.8146	−0.914402	−0.457201	−	0.889364i	$-0.651148\pi$
$797$	−25.8146	−0.914402	−0.457201	+	0.889364i	$0.651148\pi$
$798$	0	0
$799$	−1.82247	−0.0644745
$800$	23.7642	0.840192
$801$	0	0
$802$	38.2239	1.34973
$803$	9.05142	0.319418
$804$	0	0
$805$	−15.4088	−0.543091
$806$	28.5676	1.00625
$807$	0	0
$808$	55.9663	1.96889
$809$	47.5765	1.67270	0.836351	−	0.548195i	$-0.184685\pi$
$809$	47.5765	1.67270	0.836351	+	0.548195i	$0.184685\pi$
$810$	0	0
$811$	20.6697	0.725810	0.362905	−	0.931826i	$-0.381785\pi$
$811$	20.6697	0.725810	0.362905	+	0.931826i	$0.381785\pi$
$812$	1.24638	0.0437394
$813$	0	0
$814$	−43.1824	−1.51354
$815$	−31.0185	−1.08653
$816$	0	0
$817$	−4.47185	−0.156450
$818$	20.4312	0.714359
$819$	0	0
$820$	7.37058	0.257392
$821$	5.38616	0.187978	0.0939892	−	0.995573i	$-0.470038\pi$
$821$	5.38616	0.187978	0.0939892	+	0.995573i	$0.470038\pi$
$822$	0	0
$823$	−26.6246	−0.928075	−0.464038	−	0.885816i	$-0.653600\pi$
$823$	−26.6246	−0.928075	−0.464038	+	0.885816i	$0.653600\pi$
$824$	−19.1378	−0.666696
$825$	0	0
$826$	25.2202	0.877523
$827$	−23.3930	−0.813455	−0.406728	−	0.913549i	$-0.633330\pi$
$827$	−23.3930	−0.813455	−0.406728	+	0.913549i	$0.633330\pi$
$828$	0	0
$829$	−29.7744	−1.03411	−0.517054	−	0.855953i	$-0.672971\pi$
$829$	−29.7744	−1.03411	−0.517054	+	0.855953i	$0.672971\pi$
$830$	20.5344	0.712759
$831$	0	0
$832$	38.3176	1.32842
$833$	−2.49127	−0.0863173
$834$	0	0
$835$	−52.4130	−1.81383
$836$	−2.27616	−0.0787227
$837$	0	0
$838$	−11.5381	−0.398578
$839$	38.4282	1.32669	0.663344	−	0.748314i	$-0.269137\pi$
$839$	38.4282	1.32669	0.663344	+	0.748314i	$0.269137\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−48.9131	−1.68566
$843$	0	0
$844$	−4.05214	−0.139480
$845$	−23.9401	−0.823565
$846$	0	0
$847$	−56.6073	−1.94505
$848$	4.83848	0.166154
$849$	0	0
$850$	−6.16382	−0.211417
$851$	6.52227	0.223580
$852$	0	0
$853$	−46.2140	−1.58234	−0.791169	−	0.611597i	$-0.790527\pi$
$853$	−46.2140	−1.58234	−0.791169	+	0.611597i	$0.790527\pi$
$854$	33.2433	1.13756
$855$	0	0
$856$	14.5174	0.496194
$857$	10.4809	0.358019	0.179010	−	0.983847i	$-0.442711\pi$
$857$	10.4809	0.358019	0.179010	+	0.983847i	$0.442711\pi$
$858$	0	0
$859$	7.73093	0.263776	0.131888	−	0.991265i	$-0.457896\pi$
$859$	7.73093	0.263776	0.131888	+	0.991265i	$0.457896\pi$
$860$	−4.85181	−0.165445
$861$	0	0
$862$	−45.6662	−1.55540
$863$	30.3638	1.03360	0.516798	−	0.856107i	$-0.327124\pi$
$863$	30.3638	1.03360	0.516798	+	0.856107i	$0.327124\pi$
$864$	0	0
$865$	−6.35936	−0.216225
$866$	−13.6851	−0.465039
$867$	0	0
$868$	6.38571	0.216745
$869$	−31.5122	−1.06898
$870$	0	0
$871$	−36.3829	−1.23279
$872$	10.1634	0.344176
$873$	0	0
$874$	−1.68572	−0.0570202
$875$	116.232	3.92936
$876$	0	0
$877$	−28.6725	−0.968202	−0.484101	−	0.875012i	$-0.660853\pi$
$877$	−28.6725	−0.968202	−0.484101	+	0.875012i	$0.660853\pi$
$878$	15.0227	0.506993
$879$	0	0
$880$	69.0136	2.32645
$881$	27.1297	0.914024	0.457012	−	0.889461i	$-0.348920\pi$
$881$	27.1297	0.914024	0.457012	+	0.889461i	$0.348920\pi$
$882$	0	0
$883$	26.2496	0.883368	0.441684	−	0.897171i	$-0.354381\pi$
$883$	26.2496	0.883368	0.441684	+	0.897171i	$0.354381\pi$
$884$	0.558815	0.0187950
$885$	0	0
$886$	29.7334	0.998914
$887$	−38.2089	−1.28293	−0.641465	−	0.767153i	$-0.721673\pi$
$887$	−38.2089	−1.28293	−0.641465	+	0.767153i	$0.721673\pi$
$888$	0	0
$889$	48.8009	1.63673
$890$	−10.2710	−0.344283
$891$	0	0
$892$	3.39796	0.113772
$893$	6.25178	0.209208
$894$	0	0
$895$	−83.9123	−2.80488
$896$	−28.0574	−0.937333
$897$	0	0
$898$	−27.1450	−0.905839
$899$	5.12340	0.170875
$900$	0	0
$901$	0.575115	0.0191599
$902$	34.3890	1.14503
$903$	0	0
$904$	−4.91893	−0.163601
$905$	−34.3065	−1.14039
$906$	0	0
$907$	2.94859	0.0979063	0.0489531	−	0.998801i	$-0.484412\pi$
$907$	2.94859	0.0979063	0.0489531	+	0.998801i	$0.484412\pi$
$908$	4.34144	0.144076
$909$	0	0
$910$	85.9183	2.84816
$911$	−42.3040	−1.40159	−0.700797	−	0.713361i	$-0.747172\pi$
$911$	−42.3040	−1.40159	−0.700797	+	0.713361i	$0.747172\pi$
$912$	0	0
$913$	−19.5394	−0.646660
$914$	27.5037	0.909742
$915$	0	0
$916$	−1.08323	−0.0357909
$917$	−36.0107	−1.18918
$918$	0	0
$919$	−21.7786	−0.718411	−0.359205	−	0.933258i	$-0.616952\pi$
$919$	−21.7786	−0.718411	−0.359205	+	0.933258i	$0.616952\pi$
$920$	−12.6258	−0.416259
$921$	0	0
$922$	5.29955	0.174531
$923$	10.4011	0.342357
$924$	0	0
$925$	−81.8101	−2.68990
$926$	−17.9585	−0.590154
$927$	0	0
$928$	1.89459	0.0621930
$929$	12.7455	0.418168	0.209084	−	0.977898i	$-0.432952\pi$
$929$	12.7455	0.418168	0.209084	+	0.977898i	$0.432952\pi$
$930$	0	0
$931$	8.54600	0.280084
$932$	−9.20835	−0.301630
$933$	0	0
$934$	42.0750	1.37674
$935$	8.20314	0.268271
$936$	0	0
$937$	−7.64509	−0.249754	−0.124877	−	0.992172i	$-0.539854\pi$
$937$	−7.64509	−0.249754	−0.124877	+	0.992172i	$0.539854\pi$
$938$	39.8769	1.30203
$939$	0	0
$940$	6.78298	0.221236
$941$	−15.7444	−0.513254	−0.256627	−	0.966511i	$-0.582611\pi$
$941$	−15.7444	−0.513254	−0.256627	+	0.966511i	$0.582611\pi$
$942$	0	0
$943$	−5.19412	−0.169144
$944$	17.0610	0.555289
$945$	0	0
$946$	−22.6372	−0.735998
$947$	11.3219	0.367911	0.183956	−	0.982935i	$-0.441110\pi$
$947$	11.3219	0.367911	0.183956	+	0.982935i	$0.441110\pi$
$948$	0	0
$949$	7.62297	0.247452
$950$	21.1443	0.686010
$951$	0	0
$952$	−4.22814	−0.137035
$953$	−43.7432	−1.41698	−0.708491	−	0.705720i	$-0.750624\pi$
$953$	−43.7432	−1.41698	−0.708491	+	0.705720i	$0.750624\pi$
$954$	0	0
$955$	−5.74221	−0.185814
$956$	−1.47108	−0.0475782
$957$	0	0
$958$	15.2406	0.492402
$959$	−48.1015	−1.55328
$960$	0	0
$961$	−4.75074	−0.153250
$962$	−36.3675	−1.17254
$963$	0	0
$964$	−3.95758	−0.127465
$965$	77.5764	2.49727
$966$	0	0
$967$	27.7940	0.893794	0.446897	−	0.894586i	$-0.352529\pi$
$967$	27.7940	0.893794	0.446897	+	0.894586i	$0.352529\pi$
$968$	−46.3832	−1.49081
$969$	0	0
$970$	28.2369	0.906632
$971$	−14.8469	−0.476460	−0.238230	−	0.971209i	$-0.576567\pi$
$971$	−14.8469	−0.476460	−0.238230	+	0.971209i	$0.576567\pi$
$972$	0	0
$973$	57.2366	1.83492
$974$	43.6449	1.39847
$975$	0	0
$976$	22.4885	0.719840
$977$	11.2742	0.360693	0.180346	−	0.983603i	$-0.442278\pi$
$977$	11.2742	0.360693	0.180346	+	0.983603i	$0.442278\pi$
$978$	0	0
$979$	9.77329	0.312356
$980$	9.27213	0.296187
$981$	0	0
$982$	9.04502	0.288638
$983$	7.29224	0.232586	0.116293	−	0.993215i	$-0.462899\pi$
$983$	7.29224	0.232586	0.116293	+	0.993215i	$0.462899\pi$
$984$	0	0
$985$	−86.9015	−2.76891
$986$	−0.491408	−0.0156496
$987$	0	0
$988$	−1.91695	−0.0609863
$989$	3.41912	0.108722
$990$	0	0
$991$	49.5478	1.57394	0.786969	−	0.616993i	$-0.211649\pi$
$991$	49.5478	1.57394	0.786969	+	0.616993i	$0.211649\pi$
$992$	9.70676	0.308190
$993$	0	0
$994$	−11.4000	−0.361586
$995$	−37.8007	−1.19836
$996$	0	0
$997$	24.4536	0.774454	0.387227	−	0.921985i	$-0.373433\pi$
$997$	24.4536	0.774454	0.387227	+	0.921985i	$0.373433\pi$
$998$	−23.0239	−0.728809
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.q.1.12		16
3.2	odd	2		667.2.a.d.1.5	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.d.1.5	✓	16	3.2	odd	2
6003.2.a.q.1.12		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)

\(f(q)\)	\(=\)	\(q+1.28888 q^{2} -0.338793 q^{4} -4.18846 q^{5} -3.67888 q^{7} -3.01442 q^{8} +O(q^{10})\)	\(q+1.28888 q^{2} -0.338793 q^{4} -4.18846 q^{5} -3.67888 q^{7} -3.01442 q^{8} -5.39841 q^{10} +5.13684 q^{11} +4.32617 q^{13} -4.74163 q^{14} -3.20763 q^{16} -0.381268 q^{17} +1.30789 q^{19} +1.41902 q^{20} +6.62076 q^{22} -1.00000 q^{23} +12.5432 q^{25} +5.57590 q^{26} +1.24638 q^{28} +1.00000 q^{29} +5.12340 q^{31} +1.89459 q^{32} -0.491408 q^{34} +15.4088 q^{35} -6.52227 q^{37} +1.68572 q^{38} +12.6258 q^{40} +5.19412 q^{41} -3.41912 q^{43} -1.74033 q^{44} -1.28888 q^{46} +4.78004 q^{47} +6.53417 q^{49} +16.1667 q^{50} -1.46568 q^{52} -1.50843 q^{53} -21.5154 q^{55} +11.0897 q^{56} +1.28888 q^{58} -5.31889 q^{59} -7.01094 q^{61} +6.60344 q^{62} +8.85716 q^{64} -18.1200 q^{65} -8.40996 q^{67} +0.129171 q^{68} +19.8601 q^{70} +2.40424 q^{71} +1.76206 q^{73} -8.40641 q^{74} -0.443106 q^{76} -18.8978 q^{77} -6.13455 q^{79} +13.4350 q^{80} +6.69459 q^{82} -3.80378 q^{83} +1.59692 q^{85} -4.40683 q^{86} -15.4846 q^{88} +1.90259 q^{89} -15.9155 q^{91} +0.338793 q^{92} +6.16089 q^{94} -5.47806 q^{95} -5.23059 q^{97} +8.42175 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8}+O(q^{10}) \) 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

Modular forms

Varieties

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Groups

Database

Properties

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