LMFDB - Embedded newform 6003.2.a.r.1.8

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:	$0$
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} - x^{15} - 28 x^{14} + 27 x^{13} + 316 x^{12} - 295 x^{11} - 1835 x^{10} + 1665 x^{9} + \cdots - 192 $ x^16 - x^15 - 28x^14 + 27x^13 + 316x^12 - 295x^11 - 1835x^10 + 1665x^9 + 5778x^8 - 5124x^7 - 9405x^6 + 8288x^5 + 6405x^4 - 6032x^3 - 400x^2 + 1088x - 192
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.300211$ 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)

$f(q)$	$=$	$q+0.300211 q^{2} -1.90987 q^{4} +3.45593 q^{5} -1.15345 q^{7} -1.17379 q^{8} +O(q^{10})$	\(q+0.300211 q^{2} -1.90987 q^{4} +3.45593 q^{5} -1.15345 q^{7} -1.17379 q^{8} +1.03751 q^{10} -6.20470 q^{11} -0.943654 q^{13} -0.346278 q^{14} +3.46736 q^{16} +7.87703 q^{17} -8.23644 q^{19} -6.60039 q^{20} -1.86272 q^{22} +1.00000 q^{23} +6.94347 q^{25} -0.283296 q^{26} +2.20294 q^{28} +1.00000 q^{29} -3.76222 q^{31} +3.38852 q^{32} +2.36477 q^{34} -3.98623 q^{35} +7.78036 q^{37} -2.47267 q^{38} -4.05653 q^{40} +0.655932 q^{41} +10.5468 q^{43} +11.8502 q^{44} +0.300211 q^{46} -10.0563 q^{47} -5.66956 q^{49} +2.08451 q^{50} +1.80226 q^{52} -6.17512 q^{53} -21.4430 q^{55} +1.35390 q^{56} +0.300211 q^{58} +2.25531 q^{59} +3.80068 q^{61} -1.12946 q^{62} -5.91745 q^{64} -3.26121 q^{65} +4.88992 q^{67} -15.0441 q^{68} -1.19671 q^{70} -2.81082 q^{71} +8.34919 q^{73} +2.33575 q^{74} +15.7305 q^{76} +7.15678 q^{77} +17.5279 q^{79} +11.9830 q^{80} +0.196918 q^{82} -4.47839 q^{83} +27.2225 q^{85} +3.16628 q^{86} +7.28300 q^{88} +0.0918378 q^{89} +1.08845 q^{91} -1.90987 q^{92} -3.01902 q^{94} -28.4646 q^{95} +2.85148 q^{97} -1.70207 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7}+O(q^{10}) $ 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7	$ 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7} + 11 q^{10} - 8 q^{11} + 19 q^{13} - 16 q^{14} + 31 q^{16} + 4 q^{17} + 19 q^{19} - 16 q^{20} + 6 q^{22} + 16 q^{23} + 23 q^{25} + 15 q^{26} + 18 q^{28} + 16 q^{29} + 24 q^{31} + 21 q^{32} - 9 q^{34} + 13 q^{35} + 26 q^{37} - 22 q^{40} + 15 q^{41} + 33 q^{43} - 6 q^{44} + q^{46} - 13 q^{47} + 41 q^{49} - 13 q^{50} - 26 q^{52} - 5 q^{53} + 9 q^{55} - 40 q^{56} + q^{58} - 2 q^{59} + 29 q^{61} + 32 q^{62} + 28 q^{64} - 18 q^{65} + 32 q^{67} + 26 q^{68} + 18 q^{70} - 29 q^{71} + 19 q^{73} + 16 q^{74} + 64 q^{76} + 21 q^{77} + 56 q^{79} + 14 q^{82} - 5 q^{83} + 16 q^{85} + 20 q^{86} + q^{88} - 7 q^{89} - 6 q^{91} + 25 q^{92} - 11 q^{94} - 39 q^{95} + 35 q^{97} + 109 q^{98}+O(q^{100}) $ 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7 + 11 * q^10 - 8 * q^11 + 19 * q^13 - 16 * q^14 + 31 * q^16 + 4 * q^17 + 19 * q^19 - 16 * q^20 + 6 * q^22 + 16 * q^23 + 23 * q^25 + 15 * q^26 + 18 * q^28 + 16 * q^29 + 24 * q^31 + 21 * q^32 - 9 * q^34 + 13 * q^35 + 26 * q^37 - 22 * q^40 + 15 * q^41 + 33 * q^43 - 6 * q^44 + q^46 - 13 * q^47 + 41 * q^49 - 13 * q^50 - 26 * q^52 - 5 * q^53 + 9 * q^55 - 40 * q^56 + q^58 - 2 * q^59 + 29 * q^61 + 32 * q^62 + 28 * q^64 - 18 * q^65 + 32 * q^67 + 26 * q^68 + 18 * q^70 - 29 * q^71 + 19 * q^73 + 16 * q^74 + 64 * q^76 + 21 * q^77 + 56 * q^79 + 14 * q^82 - 5 * q^83 + 16 * q^85 + 20 * q^86 + q^88 - 7 * q^89 - 6 * q^91 + 25 * q^92 - 11 * q^94 - 39 * q^95 + 35 * q^97 + 109 * 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 $

$2$	0.300211	0.212281	0.106141	−	0.994351i	$-0.466151\pi$
$2$	0.300211	0.212281	0.106141	+	0.994351i	$0.466151\pi$
$3$	0	0
$3$	0	0
$4$	−1.90987	−0.954937
$5$	3.45593	1.54554	0.772770	−	0.634686i	$-0.218871\pi$
$5$	3.45593	1.54554	0.772770	+	0.634686i	$0.218871\pi$
$6$	0	0
$7$	−1.15345	−0.435962	−0.217981	−	0.975953i	$-0.569947\pi$
$7$	−1.15345	−0.435962	−0.217981	+	0.975953i	$0.569947\pi$
$8$	−1.17379	−0.414997
$9$	0	0
$10$	1.03751	0.328089
$11$	−6.20470	−1.87079	−0.935393	−	0.353609i	$-0.884954\pi$
$11$	−6.20470	−1.87079	−0.935393	+	0.353609i	$0.884954\pi$
$12$	0	0
$13$	−0.943654	−0.261723	−0.130861	−	0.991401i	$-0.541774\pi$
$13$	−0.943654	−0.261723	−0.130861	+	0.991401i	$0.541774\pi$
$14$	−0.346278	−0.0925466
$15$	0	0
$16$	3.46736	0.866840
$17$	7.87703	1.91046	0.955230	−	0.295865i	$-0.0956079\pi$
$17$	7.87703	1.91046	0.955230	+	0.295865i	$0.0956079\pi$
$18$	0	0
$19$	−8.23644	−1.88957	−0.944784	−	0.327694i	$-0.893729\pi$
$19$	−8.23644	−1.88957	−0.944784	+	0.327694i	$0.893729\pi$
$20$	−6.60039	−1.47589
$21$	0	0
$22$	−1.86272	−0.397133
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	6.94347	1.38869
$26$	−0.283296	−0.0555589
$27$	0	0
$28$	2.20294	0.416316
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−3.76222	−0.675715	−0.337858	−	0.941197i	$-0.609702\pi$
$31$	−3.76222	−0.675715	−0.337858	+	0.941197i	$0.609702\pi$
$32$	3.38852	0.599011
$33$	0	0
$34$	2.36477	0.405555
$35$	−3.98623	−0.673796
$36$	0	0
$37$	7.78036	1.27908	0.639541	−	0.768757i	$-0.279124\pi$
$37$	7.78036	1.27908	0.639541	+	0.768757i	$0.279124\pi$
$38$	−2.47267	−0.401120
$39$	0	0
$40$	−4.05653	−0.641394
$41$	0.655932	0.102439	0.0512197	−	0.998687i	$-0.483689\pi$
$41$	0.655932	0.102439	0.0512197	+	0.998687i	$0.483689\pi$
$42$	0	0
$43$	10.5468	1.60838	0.804188	−	0.594375i	$-0.202600\pi$
$43$	10.5468	1.60838	0.804188	+	0.594375i	$0.202600\pi$
$44$	11.8502	1.78648
$45$	0	0
$46$	0.300211	0.0442637
$47$	−10.0563	−1.46687	−0.733433	−	0.679762i	$-0.762083\pi$
$47$	−10.0563	−1.46687	−0.733433	+	0.679762i	$0.762083\pi$
$48$	0	0
$49$	−5.66956	−0.809937
$50$	2.08451	0.294794
$51$	0	0
$52$	1.80226	0.249929
$53$	−6.17512	−0.848218	−0.424109	−	0.905611i	$-0.639413\pi$
$53$	−6.17512	−0.848218	−0.424109	+	0.905611i	$0.639413\pi$
$54$	0	0
$55$	−21.4430	−2.89138
$56$	1.35390	0.180923
$57$	0	0
$58$	0.300211	0.0394197
$59$	2.25531	0.293617	0.146808	−	0.989165i	$-0.453100\pi$
$59$	2.25531	0.293617	0.146808	+	0.989165i	$0.453100\pi$
$60$	0	0
$61$	3.80068	0.486628	0.243314	−	0.969948i	$-0.421765\pi$
$61$	3.80068	0.486628	0.243314	+	0.969948i	$0.421765\pi$
$62$	−1.12946	−0.143442
$63$	0	0
$64$	−5.91745	−0.739682
$65$	−3.26121	−0.404503
$66$	0	0
$67$	4.88992	0.597399	0.298699	−	0.954347i	$-0.403447\pi$
$67$	4.88992	0.597399	0.298699	+	0.954347i	$0.403447\pi$
$68$	−15.0441	−1.82437
$69$	0	0
$70$	−1.19671	−0.143034
$71$	−2.81082	−0.333583	−0.166792	−	0.985992i	$-0.553341\pi$
$71$	−2.81082	−0.333583	−0.166792	+	0.985992i	$0.553341\pi$
$72$	0	0
$73$	8.34919	0.977198	0.488599	−	0.872508i	$-0.337508\pi$
$73$	8.34919	0.977198	0.488599	+	0.872508i	$0.337508\pi$
$74$	2.33575	0.271526
$75$	0	0
$76$	15.7305	1.80442
$77$	7.15678	0.815591
$78$	0	0
$79$	17.5279	1.97204	0.986021	−	0.166623i	$-0.0532862\pi$
$79$	17.5279	1.97204	0.986021	+	0.166623i	$0.0532862\pi$
$80$	11.9830	1.33974
$81$	0	0
$82$	0.196918	0.0217460
$83$	−4.47839	−0.491568	−0.245784	−	0.969325i	$-0.579045\pi$
$83$	−4.47839	−0.491568	−0.245784	+	0.969325i	$0.579045\pi$
$84$	0	0
$85$	27.2225	2.95269
$86$	3.16628	0.341429
$87$	0	0
$88$	7.28300	0.776370
$89$	0.0918378	0.00973479	0.00486739	−	0.999988i	$-0.498451\pi$
$89$	0.0918378	0.00973479	0.00486739	+	0.999988i	$0.498451\pi$
$90$	0	0
$91$	1.08845	0.114101
$92$	−1.90987	−0.199118
$93$	0	0
$94$	−3.01902	−0.311388
$95$	−28.4646	−2.92040
$96$	0	0
$97$	2.85148	0.289524	0.144762	−	0.989467i	$-0.453758\pi$
$97$	2.85148	0.289524	0.144762	+	0.989467i	$0.453758\pi$
$98$	−1.70207	−0.171935
$99$	0	0
$100$	−13.2611	−1.32611
$101$	10.9655	1.09111	0.545556	−	0.838075i	$-0.316319\pi$
$101$	10.9655	1.09111	0.545556	+	0.838075i	$0.316319\pi$
$102$	0	0
$103$	15.3519	1.51267	0.756336	−	0.654184i	$-0.226988\pi$
$103$	15.3519	1.51267	0.756336	+	0.654184i	$0.226988\pi$
$104$	1.10765	0.108614
$105$	0	0
$106$	−1.85384	−0.180061
$107$	17.6070	1.70213	0.851066	−	0.525059i	$-0.175957\pi$
$107$	17.6070	1.70213	0.851066	+	0.525059i	$0.175957\pi$
$108$	0	0
$109$	14.4058	1.37982	0.689912	−	0.723893i	$-0.257649\pi$
$109$	14.4058	1.37982	0.689912	+	0.723893i	$0.257649\pi$
$110$	−6.43743	−0.613785
$111$	0	0
$112$	−3.99941	−0.377909
$113$	5.02763	0.472960	0.236480	−	0.971636i	$-0.424006\pi$
$113$	5.02763	0.472960	0.236480	+	0.971636i	$0.424006\pi$
$114$	0	0
$115$	3.45593	0.322267
$116$	−1.90987	−0.177327
$117$	0	0
$118$	0.677071	0.0623294
$119$	−9.08572	−0.832887
$120$	0	0
$121$	27.4983	2.49984
$122$	1.14101	0.103302
$123$	0	0
$124$	7.18537	0.645265
$125$	6.71650	0.600742
$126$	0	0
$127$	−0.900765	−0.0799300	−0.0399650	−	0.999201i	$-0.512725\pi$
$127$	−0.900765	−0.0799300	−0.0399650	+	0.999201i	$0.512725\pi$
$128$	−8.55352	−0.756032
$129$	0	0
$130$	−0.979051	−0.0858685
$131$	−0.266107	−0.0232499	−0.0116250	−	0.999932i	$-0.503700\pi$
$131$	−0.266107	−0.0232499	−0.0116250	+	0.999932i	$0.503700\pi$
$132$	0	0
$133$	9.50028	0.823779
$134$	1.46801	0.126817
$135$	0	0
$136$	−9.24596	−0.792835
$137$	−8.85258	−0.756327	−0.378163	−	0.925739i	$-0.623444\pi$
$137$	−8.85258	−0.756327	−0.378163	+	0.925739i	$0.623444\pi$
$138$	0	0
$139$	−3.57896	−0.303563	−0.151782	−	0.988414i	$-0.548501\pi$
$139$	−3.57896	−0.303563	−0.151782	+	0.988414i	$0.548501\pi$
$140$	7.61320	0.643433
$141$	0	0
$142$	−0.843840	−0.0708135
$143$	5.85509	0.489627
$144$	0	0
$145$	3.45593	0.287000
$146$	2.50652	0.207441
$147$	0	0
$148$	−14.8595	−1.22144
$149$	−5.91048	−0.484205	−0.242103	−	0.970251i	$-0.577837\pi$
$149$	−5.91048	−0.484205	−0.242103	+	0.970251i	$0.577837\pi$
$150$	0	0
$151$	−17.8283	−1.45085	−0.725424	−	0.688302i	$-0.758356\pi$
$151$	−17.8283	−1.45085	−0.725424	+	0.688302i	$0.758356\pi$
$152$	9.66783	0.784165
$153$	0	0
$154$	2.14855	0.173135
$155$	−13.0020	−1.04434
$156$	0	0
$157$	8.09832	0.646316	0.323158	−	0.946345i	$-0.395255\pi$
$157$	8.09832	0.646316	0.323158	+	0.946345i	$0.395255\pi$
$158$	5.26207	0.418628
$159$	0	0
$160$	11.7105	0.925795
$161$	−1.15345	−0.0909043
$162$	0	0
$163$	−6.16803	−0.483117	−0.241559	−	0.970386i	$-0.577659\pi$
$163$	−6.16803	−0.483117	−0.241559	+	0.970386i	$0.577659\pi$
$164$	−1.25275	−0.0978232
$165$	0	0
$166$	−1.34446	−0.104351
$167$	19.3211	1.49511	0.747555	−	0.664200i	$-0.231228\pi$
$167$	19.3211	1.49511	0.747555	+	0.664200i	$0.231228\pi$
$168$	0	0
$169$	−12.1095	−0.931501
$170$	8.17249	0.626802
$171$	0	0
$172$	−20.1431	−1.53590
$173$	−24.1734	−1.83787	−0.918933	−	0.394413i	$-0.870948\pi$
$173$	−24.1734	−1.83787	−0.918933	+	0.394413i	$0.870948\pi$
$174$	0	0
$175$	−8.00892	−0.605417
$176$	−21.5139	−1.62167
$177$	0	0
$178$	0.0275708	0.00206652
$179$	10.3202	0.771365	0.385683	−	0.922632i	$-0.373966\pi$
$179$	10.3202	0.771365	0.385683	+	0.922632i	$0.373966\pi$
$180$	0	0
$181$	15.5976	1.15936	0.579680	−	0.814844i	$-0.303177\pi$
$181$	15.5976	1.15936	0.579680	+	0.814844i	$0.303177\pi$
$182$	0.326766	0.0242215
$183$	0	0
$184$	−1.17379	−0.0865328
$185$	26.8884	1.97687
$186$	0	0
$187$	−48.8746	−3.57406
$188$	19.2063	1.40076
$189$	0	0
$190$	−8.54538	−0.619947
$191$	3.26029	0.235906	0.117953	−	0.993019i	$-0.462367\pi$
$191$	3.26029	0.235906	0.117953	+	0.993019i	$0.462367\pi$
$192$	0	0
$193$	−20.6694	−1.48781	−0.743907	−	0.668283i	$-0.767029\pi$
$193$	−20.6694	−1.48781	−0.743907	+	0.668283i	$0.767029\pi$
$194$	0.856047	0.0614606
$195$	0	0
$196$	10.8281	0.773439
$197$	−5.83911	−0.416019	−0.208010	−	0.978127i	$-0.566699\pi$
$197$	−5.83911	−0.416019	−0.208010	+	0.978127i	$0.566699\pi$
$198$	0	0
$199$	13.6581	0.968199	0.484100	−	0.875013i	$-0.339147\pi$
$199$	13.6581	0.968199	0.484100	+	0.875013i	$0.339147\pi$
$200$	−8.15016	−0.576304
$201$	0	0
$202$	3.29198	0.231623
$203$	−1.15345	−0.0809560
$204$	0	0
$205$	2.26686	0.158324
$206$	4.60883	0.321112
$207$	0	0
$208$	−3.27199	−0.226872
$209$	51.1046	3.53498
$210$	0	0
$211$	11.0985	0.764055	0.382027	−	0.924151i	$-0.375226\pi$
$211$	11.0985	0.764055	0.382027	+	0.924151i	$0.375226\pi$
$212$	11.7937	0.809995
$213$	0	0
$214$	5.28582	0.361331
$215$	36.4491	2.48581
$216$	0	0
$217$	4.33952	0.294586
$218$	4.32478	0.292911
$219$	0	0
$220$	40.9534	2.76108
$221$	−7.43319	−0.500011
$222$	0	0
$223$	−7.33605	−0.491258	−0.245629	−	0.969364i	$-0.578994\pi$
$223$	−7.33605	−0.491258	−0.245629	+	0.969364i	$0.578994\pi$
$224$	−3.90847	−0.261146
$225$	0	0
$226$	1.50935	0.100401
$227$	−4.98508	−0.330871	−0.165436	−	0.986221i	$-0.552903\pi$
$227$	−4.98508	−0.330871	−0.165436	+	0.986221i	$0.552903\pi$
$228$	0	0
$229$	15.6936	1.03706	0.518531	−	0.855059i	$-0.326479\pi$
$229$	15.6936	1.03706	0.518531	+	0.855059i	$0.326479\pi$
$230$	1.03751	0.0684114
$231$	0	0
$232$	−1.17379	−0.0770630
$233$	−0.556241	−0.0364405	−0.0182203	−	0.999834i	$-0.505800\pi$
$233$	−0.556241	−0.0364405	−0.0182203	+	0.999834i	$0.505800\pi$
$234$	0	0
$235$	−34.7540	−2.26710
$236$	−4.30736	−0.280386
$237$	0	0
$238$	−2.72764	−0.176806
$239$	1.56997	0.101553	0.0507765	−	0.998710i	$-0.483830\pi$
$239$	1.56997	0.101553	0.0507765	+	0.998710i	$0.483830\pi$
$240$	0	0
$241$	12.6190	0.812860	0.406430	−	0.913682i	$-0.366773\pi$
$241$	12.6190	0.812860	0.406430	+	0.913682i	$0.366773\pi$
$242$	8.25529	0.530670
$243$	0	0
$244$	−7.25882	−0.464699
$245$	−19.5936	−1.25179
$246$	0	0
$247$	7.77235	0.494543
$248$	4.41605	0.280420
$249$	0	0
$250$	2.01637	0.127526
$251$	12.4570	0.786281	0.393140	−	0.919478i	$-0.371389\pi$
$251$	12.4570	0.786281	0.393140	+	0.919478i	$0.371389\pi$
$252$	0	0
$253$	−6.20470	−0.390086
$254$	−0.270420	−0.0169677
$255$	0	0
$256$	9.26704	0.579190
$257$	18.1076	1.12952	0.564759	−	0.825256i	$-0.308969\pi$
$257$	18.1076	1.12952	0.564759	+	0.825256i	$0.308969\pi$
$258$	0	0
$259$	−8.97422	−0.557631
$260$	6.22849	0.386275
$261$	0	0
$262$	−0.0798884	−0.00493552
$263$	9.93615	0.612689	0.306345	−	0.951921i	$-0.400894\pi$
$263$	9.93615	0.612689	0.306345	+	0.951921i	$0.400894\pi$
$264$	0	0
$265$	−21.3408	−1.31096
$266$	2.85209	0.174873
$267$	0	0
$268$	−9.33913	−0.570478
$269$	−18.4197	−1.12307	−0.561535	−	0.827453i	$-0.689789\pi$
$269$	−18.4197	−1.12307	−0.561535	+	0.827453i	$0.689789\pi$
$270$	0	0
$271$	17.9509	1.09044	0.545219	−	0.838293i	$-0.316446\pi$
$271$	17.9509	1.09044	0.545219	+	0.838293i	$0.316446\pi$
$272$	27.3125	1.65606
$273$	0	0
$274$	−2.65764	−0.160554
$275$	−43.0821	−2.59795
$276$	0	0
$277$	−7.55490	−0.453930	−0.226965	−	0.973903i	$-0.572880\pi$
$277$	−7.55490	−0.453930	−0.226965	+	0.973903i	$0.572880\pi$
$278$	−1.07444	−0.0644409
$279$	0	0
$280$	4.67899	0.279623
$281$	−10.1815	−0.607376	−0.303688	−	0.952772i	$-0.598218\pi$
$281$	−10.1815	−0.607376	−0.303688	+	0.952772i	$0.598218\pi$
$282$	0	0
$283$	−24.4121	−1.45115	−0.725574	−	0.688144i	$-0.758426\pi$
$283$	−24.4121	−1.45115	−0.725574	+	0.688144i	$0.758426\pi$
$284$	5.36831	0.318551
$285$	0	0
$286$	1.75776	0.103939
$287$	−0.756583	−0.0446597
$288$	0	0
$289$	45.0475	2.64986
$290$	1.03751	0.0609247
$291$	0	0
$292$	−15.9459	−0.933162
$293$	11.4637	0.669716	0.334858	−	0.942269i	$-0.391312\pi$
$293$	11.4637	0.669716	0.334858	+	0.942269i	$0.391312\pi$
$294$	0	0
$295$	7.79422	0.453797
$296$	−9.13249	−0.530815
$297$	0	0
$298$	−1.77439	−0.102788
$299$	−0.943654	−0.0545729
$300$	0	0
$301$	−12.1652	−0.701191
$302$	−5.35226	−0.307988
$303$	0	0
$304$	−28.5587	−1.63795
$305$	13.1349	0.752103
$306$	0	0
$307$	19.3330	1.10339	0.551695	−	0.834046i	$-0.313981\pi$
$307$	19.3330	1.10339	0.551695	+	0.834046i	$0.313981\pi$
$308$	−13.6685	−0.778838
$309$	0	0
$310$	−3.90334	−0.221695
$311$	−1.32217	−0.0749731	−0.0374866	−	0.999297i	$-0.511935\pi$
$311$	−1.32217	−0.0749731	−0.0374866	+	0.999297i	$0.511935\pi$
$312$	0	0
$313$	13.8895	0.785082	0.392541	−	0.919734i	$-0.371596\pi$
$313$	13.8895	0.785082	0.392541	+	0.919734i	$0.371596\pi$
$314$	2.43121	0.137201
$315$	0	0
$316$	−33.4760	−1.88317
$317$	8.23874	0.462734	0.231367	−	0.972867i	$-0.425680\pi$
$317$	8.23874	0.462734	0.231367	+	0.972867i	$0.425680\pi$
$318$	0	0
$319$	−6.20470	−0.347396
$320$	−20.4503	−1.14321
$321$	0	0
$322$	−0.346278	−0.0192973
$323$	−64.8786	−3.60994
$324$	0	0
$325$	−6.55224	−0.363453
$326$	−1.85171	−0.102557
$327$	0	0
$328$	−0.769926	−0.0425120
$329$	11.5994	0.639497
$330$	0	0
$331$	21.1426	1.16210	0.581052	−	0.813867i	$-0.302641\pi$
$331$	21.1426	1.16210	0.581052	+	0.813867i	$0.302641\pi$
$332$	8.55317	0.469416
$333$	0	0
$334$	5.80041	0.317384
$335$	16.8992	0.923304
$336$	0	0
$337$	−13.2115	−0.719675	−0.359837	−	0.933015i	$-0.617168\pi$
$337$	−13.2115	−0.719675	−0.359837	+	0.933015i	$0.617168\pi$
$338$	−3.63541	−0.197740
$339$	0	0
$340$	−51.9915	−2.81963
$341$	23.3434	1.26412
$342$	0	0
$343$	14.6137	0.789063
$344$	−12.3797	−0.667471
$345$	0	0
$346$	−7.25712	−0.390145
$347$	28.6201	1.53641	0.768204	−	0.640205i	$-0.221151\pi$
$347$	28.6201	1.53641	0.768204	+	0.640205i	$0.221151\pi$
$348$	0	0
$349$	−5.72184	−0.306283	−0.153142	−	0.988204i	$-0.548939\pi$
$349$	−5.72184	−0.306283	−0.153142	+	0.988204i	$0.548939\pi$
$350$	−2.40437	−0.128519
$351$	0	0
$352$	−21.0247	−1.12062
$353$	2.98281	0.158759	0.0793794	−	0.996844i	$-0.474706\pi$
$353$	2.98281	0.158759	0.0793794	+	0.996844i	$0.474706\pi$
$354$	0	0
$355$	−9.71401	−0.515566
$356$	−0.175399	−0.00929611
$357$	0	0
$358$	3.09823	0.163747
$359$	−10.1497	−0.535679	−0.267840	−	0.963464i	$-0.586310\pi$
$359$	−10.1497	−0.535679	−0.267840	+	0.963464i	$0.586310\pi$
$360$	0	0
$361$	48.8389	2.57047
$362$	4.68258	0.246111
$363$	0	0
$364$	−2.07881	−0.108959
$365$	28.8542	1.51030
$366$	0	0
$367$	−23.4739	−1.22533	−0.612663	−	0.790344i	$-0.709902\pi$
$367$	−23.4739	−1.22533	−0.612663	+	0.790344i	$0.709902\pi$
$368$	3.46736	0.180749
$369$	0	0
$370$	8.07220	0.419654
$371$	7.12267	0.369791
$372$	0	0
$373$	28.9502	1.49899	0.749493	−	0.662012i	$-0.230297\pi$
$373$	28.9502	1.49899	0.749493	+	0.662012i	$0.230297\pi$
$374$	−14.6727	−0.758707
$375$	0	0
$376$	11.8040	0.608745
$377$	−0.943654	−0.0486007
$378$	0	0
$379$	−5.64153	−0.289786	−0.144893	−	0.989447i	$-0.546284\pi$
$379$	−5.64153	−0.289786	−0.144893	+	0.989447i	$0.546284\pi$
$380$	54.3637	2.78880
$381$	0	0
$382$	0.978776	0.0500785
$383$	14.8674	0.759688	0.379844	−	0.925051i	$-0.375978\pi$
$383$	14.8674	0.759688	0.379844	+	0.925051i	$0.375978\pi$
$384$	0	0
$385$	24.7334	1.26053
$386$	−6.20518	−0.315835
$387$	0	0
$388$	−5.44597	−0.276477
$389$	−7.58542	−0.384596	−0.192298	−	0.981337i	$-0.561594\pi$
$389$	−7.58542	−0.384596	−0.192298	+	0.981337i	$0.561594\pi$
$390$	0	0
$391$	7.87703	0.398358
$392$	6.65486	0.336121
$393$	0	0
$394$	−1.75297	−0.0883132
$395$	60.5752	3.04787
$396$	0	0
$397$	24.4819	1.22871	0.614356	−	0.789029i	$-0.289416\pi$
$397$	24.4819	1.22871	0.614356	+	0.789029i	$0.289416\pi$
$398$	4.10033	0.205531
$399$	0	0
$400$	24.0755	1.20378
$401$	1.53449	0.0766287	0.0383143	−	0.999266i	$-0.487801\pi$
$401$	1.53449	0.0766287	0.0383143	+	0.999266i	$0.487801\pi$
$402$	0	0
$403$	3.55024	0.176850
$404$	−20.9428	−1.04194
$405$	0	0
$406$	−0.346278	−0.0171855
$407$	−48.2747	−2.39289
$408$	0	0
$409$	−18.6325	−0.921319	−0.460659	−	0.887577i	$-0.652387\pi$
$409$	−18.6325	−0.921319	−0.460659	+	0.887577i	$0.652387\pi$
$410$	0.680536	0.0336093
$411$	0	0
$412$	−29.3203	−1.44451
$413$	−2.60138	−0.128006
$414$	0	0
$415$	−15.4770	−0.759738
$416$	−3.19759	−0.156775
$417$	0	0
$418$	15.3422	0.750410
$419$	−27.2482	−1.33116	−0.665581	−	0.746326i	$-0.731816\pi$
$419$	−27.2482	−1.33116	−0.665581	+	0.746326i	$0.731816\pi$
$420$	0	0
$421$	16.3824	0.798428	0.399214	−	0.916858i	$-0.369283\pi$
$421$	16.3824	0.798428	0.399214	+	0.916858i	$0.369283\pi$
$422$	3.33191	0.162195
$423$	0	0
$424$	7.24828	0.352008
$425$	54.6939	2.65304
$426$	0	0
$427$	−4.38388	−0.212151
$428$	−33.6271	−1.62543
$429$	0	0
$430$	10.9424	0.527692
$431$	−30.8380	−1.48541	−0.742707	−	0.669616i	$-0.766459\pi$
$431$	−30.8380	−1.48541	−0.742707	+	0.669616i	$0.766459\pi$
$432$	0	0
$433$	−2.42962	−0.116760	−0.0583800	−	0.998294i	$-0.518594\pi$
$433$	−2.42962	−0.116760	−0.0583800	+	0.998294i	$0.518594\pi$
$434$	1.30277	0.0625351
$435$	0	0
$436$	−27.5132	−1.31765
$437$	−8.23644	−0.394002
$438$	0	0
$439$	17.0616	0.814306	0.407153	−	0.913360i	$-0.366522\pi$
$439$	17.0616	0.814306	0.407153	+	0.913360i	$0.366522\pi$
$440$	25.1696	1.19991
$441$	0	0
$442$	−2.23153	−0.106143
$443$	−20.3650	−0.967570	−0.483785	−	0.875187i	$-0.660738\pi$
$443$	−20.3650	−0.967570	−0.483785	+	0.875187i	$0.660738\pi$
$444$	0	0
$445$	0.317385	0.0150455
$446$	−2.20236	−0.104285
$447$	0	0
$448$	6.82546	0.322473
$449$	27.0498	1.27656	0.638281	−	0.769804i	$-0.279646\pi$
$449$	27.0498	1.27656	0.638281	+	0.769804i	$0.279646\pi$
$450$	0	0
$451$	−4.06986	−0.191642
$452$	−9.60214	−0.451647
$453$	0	0
$454$	−1.49658	−0.0702378
$455$	3.76163	0.176348
$456$	0	0
$457$	1.28469	0.0600950	0.0300475	−	0.999548i	$-0.490434\pi$
$457$	1.28469	0.0600950	0.0300475	+	0.999548i	$0.490434\pi$
$458$	4.71139	0.220149
$459$	0	0
$460$	−6.60039	−0.307745
$461$	−8.89199	−0.414141	−0.207071	−	0.978326i	$-0.566393\pi$
$461$	−8.89199	−0.414141	−0.207071	+	0.978326i	$0.566393\pi$
$462$	0	0
$463$	5.62247	0.261298	0.130649	−	0.991429i	$-0.458294\pi$
$463$	5.62247	0.261298	0.130649	+	0.991429i	$0.458294\pi$
$464$	3.46736	0.160968
$465$	0	0
$466$	−0.166990	−0.00773565
$467$	−17.0927	−0.790956	−0.395478	−	0.918475i	$-0.629421\pi$
$467$	−17.0927	−0.790956	−0.395478	+	0.918475i	$0.629421\pi$
$468$	0	0
$469$	−5.64026	−0.260443
$470$	−10.4335	−0.481263
$471$	0	0
$472$	−2.64726	−0.121850
$473$	−65.4399	−3.00893
$474$	0	0
$475$	−57.1894	−2.62403
$476$	17.3526	0.795354
$477$	0	0
$478$	0.471323	0.0215578
$479$	37.2224	1.70073	0.850367	−	0.526190i	$-0.176380\pi$
$479$	37.2224	1.70073	0.850367	+	0.526190i	$0.176380\pi$
$480$	0	0
$481$	−7.34197	−0.334765
$482$	3.78836	0.172555
$483$	0	0
$484$	−52.5182	−2.38719
$485$	9.85453	0.447471
$486$	0	0
$487$	−23.6747	−1.07280	−0.536402	−	0.843962i	$-0.680217\pi$
$487$	−23.6747	−1.07280	−0.536402	+	0.843962i	$0.680217\pi$
$488$	−4.46120	−0.201949
$489$	0	0
$490$	−5.88223	−0.265732
$491$	12.0799	0.545158	0.272579	−	0.962133i	$-0.412123\pi$
$491$	12.0799	0.545158	0.272579	+	0.962133i	$0.412123\pi$
$492$	0	0
$493$	7.87703	0.354763
$494$	2.33335	0.104982
$495$	0	0
$496$	−13.0450	−0.585737
$497$	3.24213	0.145429
$498$	0	0
$499$	40.4616	1.81131	0.905655	−	0.424014i	$-0.139379\pi$
$499$	40.4616	1.81131	0.905655	+	0.424014i	$0.139379\pi$
$500$	−12.8277	−0.573671
$501$	0	0
$502$	3.73974	0.166913
$503$	34.9076	1.55645	0.778227	−	0.627982i	$-0.216119\pi$
$503$	34.9076	1.55645	0.778227	+	0.627982i	$0.216119\pi$
$504$	0	0
$505$	37.8961	1.68636
$506$	−1.86272	−0.0828080
$507$	0	0
$508$	1.72035	0.0763281
$509$	14.9170	0.661184	0.330592	−	0.943774i	$-0.392752\pi$
$509$	14.9170	0.661184	0.330592	+	0.943774i	$0.392752\pi$
$510$	0	0
$511$	−9.63034	−0.426021
$512$	19.8891	0.878983
$513$	0	0
$514$	5.43609	0.239776
$515$	53.0553	2.33789
$516$	0	0
$517$	62.3964	2.74419
$518$	−2.69416	−0.118375
$519$	0	0
$520$	3.82797	0.167867
$521$	7.65620	0.335424	0.167712	−	0.985836i	$-0.446362\pi$
$521$	7.65620	0.335424	0.167712	+	0.985836i	$0.446362\pi$
$522$	0	0
$523$	−19.7492	−0.863573	−0.431786	−	0.901976i	$-0.642117\pi$
$523$	−19.7492	−0.863573	−0.431786	+	0.901976i	$0.642117\pi$
$524$	0.508231	0.0222022
$525$	0	0
$526$	2.98295	0.130063
$527$	−29.6351	−1.29093
$528$	0	0
$529$	1.00000	0.0434783
$530$	−6.40675	−0.278291
$531$	0	0
$532$	−18.1443	−0.786657
$533$	−0.618973	−0.0268107
$534$	0	0
$535$	60.8486	2.63071
$536$	−5.73973	−0.247919
$537$	0	0
$538$	−5.52980	−0.238407
$539$	35.1779	1.51522
$540$	0	0
$541$	−2.64097	−0.113544	−0.0567722	−	0.998387i	$-0.518081\pi$
$541$	−2.64097	−0.113544	−0.0567722	+	0.998387i	$0.518081\pi$
$542$	5.38906	0.231480
$543$	0	0
$544$	26.6914	1.14439
$545$	49.7855	2.13257
$546$	0	0
$547$	−26.2387	−1.12188	−0.560942	−	0.827855i	$-0.689561\pi$
$547$	−26.2387	−1.12188	−0.560942	+	0.827855i	$0.689561\pi$
$548$	16.9073	0.722244
$549$	0	0
$550$	−12.9337	−0.551497
$551$	−8.23644	−0.350884
$552$	0	0
$553$	−20.2175	−0.859734
$554$	−2.26806	−0.0963608
$555$	0	0
$556$	6.83536	0.289884
$557$	−30.5092	−1.29272	−0.646359	−	0.763034i	$-0.723709\pi$
$557$	−30.5092	−1.29272	−0.646359	+	0.763034i	$0.723709\pi$
$558$	0	0
$559$	−9.95256	−0.420949
$560$	−13.8217	−0.584074
$561$	0	0
$562$	−3.05659	−0.128935
$563$	−0.196690	−0.00828950	−0.00414475	−	0.999991i	$-0.501319\pi$
$563$	−0.196690	−0.00828950	−0.00414475	+	0.999991i	$0.501319\pi$
$564$	0	0
$565$	17.3752	0.730978
$566$	−7.32878	−0.308052
$567$	0	0
$568$	3.29931	0.138436
$569$	−38.7359	−1.62389	−0.811946	−	0.583732i	$-0.801592\pi$
$569$	−38.7359	−1.62389	−0.811946	+	0.583732i	$0.801592\pi$
$570$	0	0
$571$	22.2899	0.932803	0.466402	−	0.884573i	$-0.345550\pi$
$571$	22.2899	0.932803	0.466402	+	0.884573i	$0.345550\pi$
$572$	−11.1825	−0.467563
$573$	0	0
$574$	−0.227135	−0.00948042
$575$	6.94347	0.289563
$576$	0	0
$577$	5.78631	0.240887	0.120444	−	0.992720i	$-0.461568\pi$
$577$	5.78631	0.240887	0.120444	+	0.992720i	$0.461568\pi$
$578$	13.5238	0.562515
$579$	0	0
$580$	−6.60039	−0.274066
$581$	5.16559	0.214305
$582$	0	0
$583$	38.3148	1.58683
$584$	−9.80018	−0.405534
$585$	0	0
$586$	3.44153	0.142168
$587$	−24.3999	−1.00709	−0.503546	−	0.863968i	$-0.667972\pi$
$587$	−24.3999	−1.00709	−0.503546	+	0.863968i	$0.667972\pi$
$588$	0	0
$589$	30.9873	1.27681
$590$	2.33991	0.0963326
$591$	0	0
$592$	26.9773	1.10876
$593$	−26.0685	−1.07050	−0.535252	−	0.844692i	$-0.679783\pi$
$593$	−26.0685	−1.07050	−0.535252	+	0.844692i	$0.679783\pi$
$594$	0	0
$595$	−31.3997	−1.28726
$596$	11.2883	0.462385
$597$	0	0
$598$	−0.283296	−0.0115848
$599$	−16.9462	−0.692403	−0.346201	−	0.938160i	$-0.612529\pi$
$599$	−16.9462	−0.692403	−0.346201	+	0.938160i	$0.612529\pi$
$600$	0	0
$601$	−25.0008	−1.01980	−0.509901	−	0.860233i	$-0.670318\pi$
$601$	−25.0008	−1.01980	−0.509901	+	0.860233i	$0.670318\pi$
$602$	−3.65213	−0.148850
$603$	0	0
$604$	34.0498	1.38547
$605$	95.0321	3.86361
$606$	0	0
$607$	39.6979	1.61129	0.805643	−	0.592401i	$-0.201820\pi$
$607$	39.6979	1.61129	0.805643	+	0.592401i	$0.201820\pi$
$608$	−27.9093	−1.13187
$609$	0	0
$610$	3.94325	0.159657
$611$	9.48970	0.383912
$612$	0	0
$613$	−43.9646	−1.77571	−0.887857	−	0.460120i	$-0.847806\pi$
$613$	−43.9646	−1.77571	−0.887857	+	0.460120i	$0.847806\pi$
$614$	5.80397	0.234229
$615$	0	0
$616$	−8.40055	−0.338468
$617$	24.2946	0.978064	0.489032	−	0.872266i	$-0.337350\pi$
$617$	24.2946	0.978064	0.489032	+	0.872266i	$0.337350\pi$
$618$	0	0
$619$	22.5005	0.904371	0.452185	−	0.891924i	$-0.350645\pi$
$619$	22.5005	0.904371	0.452185	+	0.891924i	$0.350645\pi$
$620$	24.8321	0.997283
$621$	0	0
$622$	−0.396929	−0.0159154
$623$	−0.105930	−0.00424399
$624$	0	0
$625$	−11.5056	−0.460223
$626$	4.16979	0.166658
$627$	0	0
$628$	−15.4668	−0.617191
$629$	61.2861	2.44364
$630$	0	0
$631$	16.9323	0.674064	0.337032	−	0.941493i	$-0.390577\pi$
$631$	16.9323	0.674064	0.337032	+	0.941493i	$0.390577\pi$
$632$	−20.5740	−0.818391
$633$	0	0
$634$	2.47336	0.0982298
$635$	−3.11298	−0.123535
$636$	0	0
$637$	5.35011	0.211979
$638$	−1.86272	−0.0737458
$639$	0	0
$640$	−29.5604	−1.16848
$641$	−22.4355	−0.886147	−0.443074	−	0.896485i	$-0.646112\pi$
$641$	−22.4355	−0.886147	−0.443074	+	0.896485i	$0.646112\pi$
$642$	0	0
$643$	−22.5247	−0.888288	−0.444144	−	0.895955i	$-0.646492\pi$
$643$	−22.5247	−0.888288	−0.444144	+	0.895955i	$0.646492\pi$
$644$	2.20294	0.0868078
$645$	0	0
$646$	−19.4773	−0.766324
$647$	−49.1069	−1.93059	−0.965296	−	0.261157i	$-0.915896\pi$
$647$	−49.1069	−1.93059	−0.965296	+	0.261157i	$0.915896\pi$
$648$	0	0
$649$	−13.9935	−0.549295
$650$	−1.96706	−0.0771543
$651$	0	0
$652$	11.7801	0.461346
$653$	22.6444	0.886142	0.443071	−	0.896486i	$-0.353889\pi$
$653$	22.6444	0.886142	0.443071	+	0.896486i	$0.353889\pi$
$654$	0	0
$655$	−0.919649	−0.0359337
$656$	2.27435	0.0887986
$657$	0	0
$658$	3.48228	0.135753
$659$	43.5423	1.69617	0.848083	−	0.529863i	$-0.177757\pi$
$659$	43.5423	1.69617	0.848083	+	0.529863i	$0.177757\pi$
$660$	0	0
$661$	−32.5400	−1.26566	−0.632829	−	0.774292i	$-0.718106\pi$
$661$	−32.5400	−1.26566	−0.632829	+	0.774292i	$0.718106\pi$
$662$	6.34725	0.246693
$663$	0	0
$664$	5.25669	0.203999
$665$	32.8323	1.27318
$666$	0	0
$667$	1.00000	0.0387202
$668$	−36.9008	−1.42774
$669$	0	0
$670$	5.07334	0.196000
$671$	−23.5821	−0.910377
$672$	0	0
$673$	−14.4530	−0.557122	−0.278561	−	0.960419i	$-0.589857\pi$
$673$	−14.4530	−0.557122	−0.278561	+	0.960419i	$0.589857\pi$
$674$	−3.96623	−0.152774
$675$	0	0
$676$	23.1276	0.889525
$677$	38.0074	1.46074	0.730372	−	0.683050i	$-0.239347\pi$
$677$	38.0074	1.46074	0.730372	+	0.683050i	$0.239347\pi$
$678$	0	0
$679$	−3.28903	−0.126221
$680$	−31.9534	−1.22536
$681$	0	0
$682$	7.00797	0.268349
$683$	−2.20850	−0.0845057	−0.0422529	−	0.999107i	$-0.513454\pi$
$683$	−2.20850	−0.0845057	−0.0422529	+	0.999107i	$0.513454\pi$
$684$	0	0
$685$	−30.5939	−1.16893
$686$	4.38718	0.167503
$687$	0	0
$688$	36.5697	1.39421
$689$	5.82718	0.221998
$690$	0	0
$691$	−9.14713	−0.347973	−0.173987	−	0.984748i	$-0.555665\pi$
$691$	−9.14713	−0.347973	−0.173987	+	0.984748i	$0.555665\pi$
$692$	46.1680	1.75505
$693$	0	0
$694$	8.59208	0.326151
$695$	−12.3686	−0.469169
$696$	0	0
$697$	5.16680	0.195706
$698$	−1.71776	−0.0650182
$699$	0	0
$700$	15.2960	0.578135
$701$	−1.65508	−0.0625114	−0.0312557	−	0.999511i	$-0.509951\pi$
$701$	−1.65508	−0.0625114	−0.0312557	+	0.999511i	$0.509951\pi$
$702$	0	0
$703$	−64.0824	−2.41691
$704$	36.7160	1.38379
$705$	0	0
$706$	0.895473	0.0337016
$707$	−12.6482	−0.475683
$708$	0	0
$709$	−29.4769	−1.10703	−0.553514	−	0.832840i	$-0.686713\pi$
$709$	−29.4769	−1.10703	−0.553514	+	0.832840i	$0.686713\pi$
$710$	−2.91625	−0.109445
$711$	0	0
$712$	−0.107798	−0.00403991
$713$	−3.76222	−0.140896
$714$	0	0
$715$	20.2348	0.756738
$716$	−19.7102	−0.736605
$717$	0	0
$718$	−3.04705	−0.113715
$719$	9.10427	0.339532	0.169766	−	0.985484i	$-0.445699\pi$
$719$	9.10427	0.339532	0.169766	+	0.985484i	$0.445699\pi$
$720$	0	0
$721$	−17.7076	−0.659467
$722$	14.6620	0.545662
$723$	0	0
$724$	−29.7894	−1.10712
$725$	6.94347	0.257874
$726$	0	0
$727$	−39.3164	−1.45816	−0.729082	−	0.684426i	$-0.760053\pi$
$727$	−39.3164	−1.45816	−0.729082	+	0.684426i	$0.760053\pi$
$728$	−1.27761	−0.0473516
$729$	0	0
$730$	8.66237	0.320609
$731$	83.0777	3.07274
$732$	0	0
$733$	−9.03002	−0.333531	−0.166766	−	0.985997i	$-0.553332\pi$
$733$	−9.03002	−0.333531	−0.166766	+	0.985997i	$0.553332\pi$
$734$	−7.04712	−0.260114
$735$	0	0
$736$	3.38852	0.124902
$737$	−30.3405	−1.11761
$738$	0	0
$739$	13.6453	0.501950	0.250975	−	0.967994i	$-0.419249\pi$
$739$	13.6453	0.501950	0.250975	+	0.967994i	$0.419249\pi$
$740$	−51.3534	−1.88779
$741$	0	0
$742$	2.13831	0.0784997
$743$	13.5959	0.498785	0.249392	−	0.968403i	$-0.419769\pi$
$743$	13.5959	0.498785	0.249392	+	0.968403i	$0.419769\pi$
$744$	0	0
$745$	−20.4262	−0.748359
$746$	8.69119	0.318207
$747$	0	0
$748$	93.3442	3.41300
$749$	−20.3087	−0.742064
$750$	0	0
$751$	31.5969	1.15299	0.576493	−	0.817102i	$-0.304421\pi$
$751$	31.5969	1.15299	0.576493	+	0.817102i	$0.304421\pi$
$752$	−34.8689	−1.27154
$753$	0	0
$754$	−0.283296	−0.0103170
$755$	−61.6135	−2.24234
$756$	0	0
$757$	26.9038	0.977835	0.488918	−	0.872330i	$-0.337392\pi$
$757$	26.9038	0.977835	0.488918	+	0.872330i	$0.337392\pi$
$758$	−1.69365	−0.0615162
$759$	0	0
$760$	33.4114	1.21196
$761$	−3.60966	−0.130850	−0.0654250	−	0.997857i	$-0.520840\pi$
$761$	−3.60966	−0.130850	−0.0654250	+	0.997857i	$0.520840\pi$
$762$	0	0
$763$	−16.6163	−0.601551
$764$	−6.22674	−0.225276
$765$	0	0
$766$	4.46336	0.161268
$767$	−2.12824	−0.0768462
$768$	0	0
$769$	13.1300	0.473480	0.236740	−	0.971573i	$-0.423921\pi$
$769$	13.1300	0.473480	0.236740	+	0.971573i	$0.423921\pi$
$770$	7.42523	0.267587
$771$	0	0
$772$	39.4759	1.42077
$773$	−5.91443	−0.212727	−0.106364	−	0.994327i	$-0.533921\pi$
$773$	−5.91443	−0.212727	−0.106364	+	0.994327i	$0.533921\pi$
$774$	0	0
$775$	−26.1229	−0.938362
$776$	−3.34703	−0.120152
$777$	0	0
$778$	−2.27723	−0.0816427
$779$	−5.40254	−0.193566
$780$	0	0
$781$	17.4403	0.624063
$782$	2.36477	0.0845641
$783$	0	0
$784$	−19.6584	−0.702087
$785$	27.9872	0.998908
$786$	0	0
$787$	29.3448	1.04603	0.523015	−	0.852323i	$-0.324807\pi$
$787$	29.3448	1.04603	0.523015	+	0.852323i	$0.324807\pi$
$788$	11.1520	0.397272
$789$	0	0
$790$	18.1854	0.647006
$791$	−5.79910	−0.206192
$792$	0	0
$793$	−3.58653	−0.127362
$794$	7.34975	0.260833
$795$	0	0
$796$	−26.0853	−0.924569
$797$	34.3360	1.21624	0.608121	−	0.793844i	$-0.291923\pi$
$797$	34.3360	1.21624	0.608121	+	0.793844i	$0.291923\pi$
$798$	0	0
$799$	−79.2139	−2.80239
$800$	23.5281	0.831843
$801$	0	0
$802$	0.460671	0.0162668
$803$	−51.8042	−1.82813
$804$	0	0
$805$	−3.98623	−0.140496
$806$	1.06582	0.0375420
$807$	0	0
$808$	−12.8712	−0.452808
$809$	17.1828	0.604114	0.302057	−	0.953290i	$-0.402327\pi$
$809$	17.1828	0.604114	0.302057	+	0.953290i	$0.402327\pi$
$810$	0	0
$811$	25.2548	0.886817	0.443408	−	0.896320i	$-0.353769\pi$
$811$	25.2548	0.886817	0.443408	+	0.896320i	$0.353769\pi$
$812$	2.20294	0.0773079
$813$	0	0
$814$	−14.4926	−0.507966
$815$	−21.3163	−0.746677
$816$	0	0
$817$	−86.8683	−3.03914
$818$	−5.59369	−0.195579
$819$	0	0
$820$	−4.32941	−0.151190
$821$	22.9175	0.799825	0.399913	−	0.916553i	$-0.369040\pi$
$821$	22.9175	0.799825	0.399913	+	0.916553i	$0.369040\pi$
$822$	0	0
$823$	−30.0041	−1.04588	−0.522938	−	0.852371i	$-0.675164\pi$
$823$	−30.0041	−1.04588	−0.522938	+	0.852371i	$0.675164\pi$
$824$	−18.0199	−0.627754
$825$	0	0
$826$	−0.780965	−0.0271732
$827$	23.3972	0.813600	0.406800	−	0.913517i	$-0.366645\pi$
$827$	23.3972	0.813600	0.406800	+	0.913517i	$0.366645\pi$
$828$	0	0
$829$	23.1743	0.804875	0.402438	−	0.915447i	$-0.368163\pi$
$829$	23.1743	0.804875	0.402438	+	0.915447i	$0.368163\pi$
$830$	−4.64638	−0.161278
$831$	0	0
$832$	5.58403	0.193591
$833$	−44.6593	−1.54735
$834$	0	0
$835$	66.7724	2.31075
$836$	−97.6033	−3.37568
$837$	0	0
$838$	−8.18022	−0.282581
$839$	9.05018	0.312447	0.156223	−	0.987722i	$-0.450068\pi$
$839$	9.05018	0.312447	0.156223	+	0.987722i	$0.450068\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	4.91817	0.169491
$843$	0	0
$844$	−21.1968	−0.729624
$845$	−41.8497	−1.43967
$846$	0	0
$847$	−31.7178	−1.08983
$848$	−21.4114	−0.735270
$849$	0	0
$850$	16.4197	0.563192
$851$	7.78036	0.266707
$852$	0	0
$853$	−41.0322	−1.40492	−0.702459	−	0.711724i	$-0.747914\pi$
$853$	−41.0322	−1.40492	−0.702459	+	0.711724i	$0.747914\pi$
$854$	−1.31609	−0.0450357
$855$	0	0
$856$	−20.6669	−0.706379
$857$	−49.2805	−1.68339	−0.841695	−	0.539953i	$-0.818442\pi$
$857$	−49.2805	−1.68339	−0.841695	+	0.539953i	$0.818442\pi$
$858$	0	0
$859$	52.0011	1.77426	0.887128	−	0.461524i	$-0.152697\pi$
$859$	52.0011	1.77426	0.887128	+	0.461524i	$0.152697\pi$
$860$	−69.6132	−2.37379
$861$	0	0
$862$	−9.25792	−0.315326
$863$	−25.3554	−0.863109	−0.431555	−	0.902087i	$-0.642035\pi$
$863$	−25.3554	−0.863109	−0.431555	+	0.902087i	$0.642035\pi$
$864$	0	0
$865$	−83.5415	−2.84050
$866$	−0.729399	−0.0247860
$867$	0	0
$868$	−8.28793	−0.281311
$869$	−108.755	−3.68927
$870$	0	0
$871$	−4.61440	−0.156353
$872$	−16.9093	−0.572623
$873$	0	0
$874$	−2.47267	−0.0836393
$875$	−7.74713	−0.261901
$876$	0	0
$877$	11.2745	0.380713	0.190356	−	0.981715i	$-0.439036\pi$
$877$	11.2745	0.380713	0.190356	+	0.981715i	$0.439036\pi$
$878$	5.12208	0.172862
$879$	0	0
$880$	−74.3507	−2.50636
$881$	−46.6317	−1.57106	−0.785531	−	0.618822i	$-0.787610\pi$
$881$	−46.6317	−1.57106	−0.785531	+	0.618822i	$0.787610\pi$
$882$	0	0
$883$	−52.3625	−1.76214	−0.881069	−	0.472987i	$-0.843176\pi$
$883$	−52.3625	−1.76214	−0.881069	+	0.472987i	$0.843176\pi$
$884$	14.1965	0.477478
$885$	0	0
$886$	−6.11380	−0.205397
$887$	−4.46305	−0.149854	−0.0749272	−	0.997189i	$-0.523872\pi$
$887$	−4.46305	−0.149854	−0.0749272	+	0.997189i	$0.523872\pi$
$888$	0	0
$889$	1.03898	0.0348464
$890$	0.0952827	0.00319388
$891$	0	0
$892$	14.0109	0.469120
$893$	82.8283	2.77174
$894$	0	0
$895$	35.6658	1.19218
$896$	9.86602	0.329601
$897$	0	0
$898$	8.12067	0.270990
$899$	−3.76222	−0.125477
$900$	0	0
$901$	−48.6416	−1.62049
$902$	−1.22182	−0.0406821
$903$	0	0
$904$	−5.90137	−0.196277
$905$	53.9043	1.79184
$906$	0	0
$907$	15.2771	0.507269	0.253635	−	0.967300i	$-0.418374\pi$
$907$	15.2771	0.507269	0.253635	+	0.967300i	$0.418374\pi$
$908$	9.52087	0.315961
$909$	0	0
$910$	1.12928	0.0374353
$911$	−44.0988	−1.46106	−0.730529	−	0.682882i	$-0.760726\pi$
$911$	−44.0988	−1.46106	−0.730529	+	0.682882i	$0.760726\pi$
$912$	0	0
$913$	27.7871	0.919618
$914$	0.385677	0.0127571
$915$	0	0
$916$	−29.9728	−0.990328
$917$	0.306940	0.0101361
$918$	0	0
$919$	7.09251	0.233960	0.116980	−	0.993134i	$-0.462679\pi$
$919$	7.09251	0.233960	0.116980	+	0.993134i	$0.462679\pi$
$920$	−4.05653	−0.133740
$921$	0	0
$922$	−2.66947	−0.0879145
$923$	2.65244	0.0873062
$924$	0	0
$925$	54.0227	1.77625
$926$	1.68793	0.0554688
$927$	0	0
$928$	3.38852	0.111234
$929$	−36.2161	−1.18821	−0.594106	−	0.804387i	$-0.702494\pi$
$929$	−36.2161	−1.18821	−0.594106	+	0.804387i	$0.702494\pi$
$930$	0	0
$931$	46.6970	1.53043
$932$	1.06235	0.0347984
$933$	0	0
$934$	−5.13142	−0.167905
$935$	−168.907	−5.52386
$936$	0	0
$937$	3.79839	0.124088	0.0620441	−	0.998073i	$-0.480238\pi$
$937$	3.79839	0.124088	0.0620441	+	0.998073i	$0.480238\pi$
$938$	−1.69327	−0.0552872
$939$	0	0
$940$	66.3757	2.16494
$941$	−20.0500	−0.653612	−0.326806	−	0.945091i	$-0.605972\pi$
$941$	−20.0500	−0.653612	−0.326806	+	0.945091i	$0.605972\pi$
$942$	0	0
$943$	0.655932	0.0213601
$944$	7.81999	0.254519
$945$	0	0
$946$	−19.6458	−0.638740
$947$	18.9623	0.616191	0.308096	−	0.951355i	$-0.400308\pi$
$947$	18.9623	0.616191	0.308096	+	0.951355i	$0.400308\pi$
$948$	0	0
$949$	−7.87875	−0.255755
$950$	−17.1689	−0.557033
$951$	0	0
$952$	10.6647	0.345645
$953$	4.29444	0.139110	0.0695552	−	0.997578i	$-0.477842\pi$
$953$	4.29444	0.139110	0.0695552	+	0.997578i	$0.477842\pi$
$954$	0	0
$955$	11.2673	0.364603
$956$	−2.99845	−0.0969767
$957$	0	0
$958$	11.1746	0.361034
$959$	10.2110	0.329729
$960$	0	0
$961$	−16.8457	−0.543409
$962$	−2.20414	−0.0710644
$963$	0	0
$964$	−24.1007	−0.776230
$965$	−71.4320	−2.29948
$966$	0	0
$967$	−26.5197	−0.852816	−0.426408	−	0.904531i	$-0.640221\pi$
$967$	−26.5197	−0.852816	−0.426408	+	0.904531i	$0.640221\pi$
$968$	−32.2771	−1.03743
$969$	0	0
$970$	2.95844	0.0949898
$971$	−9.41900	−0.302270	−0.151135	−	0.988513i	$-0.548293\pi$
$971$	−9.41900	−0.302270	−0.151135	+	0.988513i	$0.548293\pi$
$972$	0	0
$973$	4.12814	0.132342
$974$	−7.10742	−0.227737
$975$	0	0
$976$	13.1783	0.421829
$977$	−27.4172	−0.877153	−0.438576	−	0.898694i	$-0.644517\pi$
$977$	−27.4172	−0.877153	−0.438576	+	0.898694i	$0.644517\pi$
$978$	0	0
$979$	−0.569826	−0.0182117
$980$	37.4213	1.19538
$981$	0	0
$982$	3.62652	0.115727
$983$	2.38222	0.0759811	0.0379906	−	0.999278i	$-0.487904\pi$
$983$	2.38222	0.0759811	0.0379906	+	0.999278i	$0.487904\pi$
$984$	0	0
$985$	−20.1796	−0.642975
$986$	2.36477	0.0753097
$987$	0	0
$988$	−14.8442	−0.472257
$989$	10.5468	0.335370
$990$	0	0
$991$	15.7910	0.501617	0.250809	−	0.968037i	$-0.419303\pi$
$991$	15.7910	0.501617	0.250809	+	0.968037i	$0.419303\pi$
$992$	−12.7484	−0.404761
$993$	0	0
$994$	0.973324	0.0308720
$995$	47.2016	1.49639
$996$	0	0
$997$	34.2551	1.08487	0.542435	−	0.840098i	$-0.317503\pi$
$997$	34.2551	1.08487	0.542435	+	0.840098i	$0.317503\pi$
$998$	12.1470	0.384508
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.r.1.8		16
3.2	odd	2		2001.2.a.n.1.9	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.n.1.9	✓	16	3.2	odd	2
6003.2.a.r.1.8		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)

\(f(q)\)	\(=\)	\(q+0.300211 q^{2} -1.90987 q^{4} +3.45593 q^{5} -1.15345 q^{7} -1.17379 q^{8} +O(q^{10})\)	\(q+0.300211 q^{2} -1.90987 q^{4} +3.45593 q^{5} -1.15345 q^{7} -1.17379 q^{8} +1.03751 q^{10} -6.20470 q^{11} -0.943654 q^{13} -0.346278 q^{14} +3.46736 q^{16} +7.87703 q^{17} -8.23644 q^{19} -6.60039 q^{20} -1.86272 q^{22} +1.00000 q^{23} +6.94347 q^{25} -0.283296 q^{26} +2.20294 q^{28} +1.00000 q^{29} -3.76222 q^{31} +3.38852 q^{32} +2.36477 q^{34} -3.98623 q^{35} +7.78036 q^{37} -2.47267 q^{38} -4.05653 q^{40} +0.655932 q^{41} +10.5468 q^{43} +11.8502 q^{44} +0.300211 q^{46} -10.0563 q^{47} -5.66956 q^{49} +2.08451 q^{50} +1.80226 q^{52} -6.17512 q^{53} -21.4430 q^{55} +1.35390 q^{56} +0.300211 q^{58} +2.25531 q^{59} +3.80068 q^{61} -1.12946 q^{62} -5.91745 q^{64} -3.26121 q^{65} +4.88992 q^{67} -15.0441 q^{68} -1.19671 q^{70} -2.81082 q^{71} +8.34919 q^{73} +2.33575 q^{74} +15.7305 q^{76} +7.15678 q^{77} +17.5279 q^{79} +11.9830 q^{80} +0.196918 q^{82} -4.47839 q^{83} +27.2225 q^{85} +3.16628 q^{86} +7.28300 q^{88} +0.0918378 q^{89} +1.08845 q^{91} -1.90987 q^{92} -3.01902 q^{94} -28.4646 q^{95} +2.85148 q^{97} -1.70207 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7}+O(q^{10}) \) 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7	\( 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7} + 11 q^{10} - 8 q^{11} + 19 q^{13} - 16 q^{14} + 31 q^{16} + 4 q^{17} + 19 q^{19} - 16 q^{20} + 6 q^{22} + 16 q^{23} + 23 q^{25} + 15 q^{26} + 18 q^{28} + 16 q^{29} + 24 q^{31} + 21 q^{32} - 9 q^{34} + 13 q^{35} + 26 q^{37} - 22 q^{40} + 15 q^{41} + 33 q^{43} - 6 q^{44} + q^{46} - 13 q^{47} + 41 q^{49} - 13 q^{50} - 26 q^{52} - 5 q^{53} + 9 q^{55} - 40 q^{56} + q^{58} - 2 q^{59} + 29 q^{61} + 32 q^{62} + 28 q^{64} - 18 q^{65} + 32 q^{67} + 26 q^{68} + 18 q^{70} - 29 q^{71} + 19 q^{73} + 16 q^{74} + 64 q^{76} + 21 q^{77} + 56 q^{79} + 14 q^{82} - 5 q^{83} + 16 q^{85} + 20 q^{86} + q^{88} - 7 q^{89} - 6 q^{91} + 25 q^{92} - 11 q^{94} - 39 q^{95} + 35 q^{97} + 109 q^{98}+O(q^{100}) \) 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7 + 11 * q^10 - 8 * q^11 + 19 * q^13 - 16 * q^14 + 31 * q^16 + 4 * q^17 + 19 * q^19 - 16 * q^20 + 6 * q^22 + 16 * q^23 + 23 * q^25 + 15 * q^26 + 18 * q^28 + 16 * q^29 + 24 * q^31 + 21 * q^32 - 9 * q^34 + 13 * q^35 + 26 * q^37 - 22 * q^40 + 15 * q^41 + 33 * q^43 - 6 * q^44 + q^46 - 13 * q^47 + 41 * q^49 - 13 * q^50 - 26 * q^52 - 5 * q^53 + 9 * q^55 - 40 * q^56 + q^58 - 2 * q^59 + 29 * q^61 + 32 * q^62 + 28 * q^64 - 18 * q^65 + 32 * q^67 + 26 * q^68 + 18 * q^70 - 29 * q^71 + 19 * q^73 + 16 * q^74 + 64 * q^76 + 21 * q^77 + 56 * q^79 + 14 * q^82 - 5 * q^83 + 16 * q^85 + 20 * q^86 + q^88 - 7 * q^89 - 6 * q^91 + 25 * q^92 - 11 * q^94 - 39 * q^95 + 35 * q^97 + 109 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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