LMFDB - Embedded newform 6633.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6633,2,Mod(1,6633)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6633, 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("6633.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6633 = 3^{2} \cdot 11 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6633.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:	$52.9647716607$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - x^{16} - 26 x^{15} + 25 x^{14} + 272 x^{13} - 244 x^{12} - 1472 x^{11} + 1186 x^{10} + 4406 x^{9} + \cdots - 2 $ x^17 - x^16 - 26x^15 + 25x^14 + 272x^13 - 244x^12 - 1472x^11 + 1186x^10 + 4406x^9 - 3042x^8 - 7180x^7 + 4055x^6 + 5665x^5 - 2655x^4 - 1418x^3 + 810x^2 - 76*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 737)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.47482$ of defining polynomial
Character	$\chi$	$=$	6633.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.47482 q^{2} +4.12472 q^{4} -1.41072 q^{5} +1.07503 q^{7} -5.25828 q^{8} +O(q^{10})$	\(q-2.47482 q^{2} +4.12472 q^{4} -1.41072 q^{5} +1.07503 q^{7} -5.25828 q^{8} +3.49127 q^{10} -1.00000 q^{11} -5.36271 q^{13} -2.66051 q^{14} +4.76386 q^{16} -7.50614 q^{17} -2.87505 q^{19} -5.81882 q^{20} +2.47482 q^{22} +0.298348 q^{23} -3.00987 q^{25} +13.2717 q^{26} +4.43421 q^{28} -8.79227 q^{29} -6.51720 q^{31} -1.27310 q^{32} +18.5763 q^{34} -1.51657 q^{35} -9.04221 q^{37} +7.11522 q^{38} +7.41796 q^{40} +3.80931 q^{41} -0.489532 q^{43} -4.12472 q^{44} -0.738358 q^{46} -8.13263 q^{47} -5.84430 q^{49} +7.44888 q^{50} -22.1197 q^{52} -4.28080 q^{53} +1.41072 q^{55} -5.65283 q^{56} +21.7593 q^{58} -14.2678 q^{59} +1.03289 q^{61} +16.1289 q^{62} -6.37702 q^{64} +7.56528 q^{65} -1.00000 q^{67} -30.9607 q^{68} +3.75323 q^{70} +0.424042 q^{71} +16.1947 q^{73} +22.3778 q^{74} -11.8588 q^{76} -1.07503 q^{77} -1.90066 q^{79} -6.72046 q^{80} -9.42733 q^{82} +17.1012 q^{83} +10.5891 q^{85} +1.21150 q^{86} +5.25828 q^{88} -8.92771 q^{89} -5.76509 q^{91} +1.23060 q^{92} +20.1268 q^{94} +4.05588 q^{95} -16.0441 q^{97} +14.4636 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7}+O(q^{10}) $ 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7	$ 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7} + 10 q^{10} - 17 q^{11} + q^{13} + 11 q^{14} + 19 q^{16} - 2 q^{17} + 13 q^{19} - 3 q^{20} + q^{22} - 16 q^{23} + 33 q^{25} - 12 q^{26} + 44 q^{28} + 5 q^{29} + 16 q^{31} + 24 q^{32} + 4 q^{34} + 2 q^{35} + 29 q^{37} + 19 q^{38} + 31 q^{40} + 6 q^{41} + 19 q^{43} - 19 q^{44} - 33 q^{46} - 40 q^{47} + 23 q^{49} + 3 q^{50} - 28 q^{52} - 15 q^{53} + 10 q^{55} + 38 q^{56} - 12 q^{58} + 2 q^{59} - 6 q^{61} - 3 q^{62} - 4 q^{64} + 30 q^{65} - 17 q^{67} + 13 q^{68} + 71 q^{70} - 2 q^{71} + 41 q^{73} + 13 q^{74} + 21 q^{76} - 20 q^{77} + 41 q^{79} + 23 q^{80} - 8 q^{82} - 2 q^{83} - 36 q^{85} + 54 q^{86} - q^{89} + 16 q^{91} - 36 q^{92} + 12 q^{94} + 31 q^{95} + 3 q^{97} + 64 q^{98}+O(q^{100}) $ 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7 + 10 * q^10 - 17 * q^11 + q^13 + 11 * q^14 + 19 * q^16 - 2 * q^17 + 13 * q^19 - 3 * q^20 + q^22 - 16 * q^23 + 33 * q^25 - 12 * q^26 + 44 * q^28 + 5 * q^29 + 16 * q^31 + 24 * q^32 + 4 * q^34 + 2 * q^35 + 29 * q^37 + 19 * q^38 + 31 * q^40 + 6 * q^41 + 19 * q^43 - 19 * q^44 - 33 * q^46 - 40 * q^47 + 23 * q^49 + 3 * q^50 - 28 * q^52 - 15 * q^53 + 10 * q^55 + 38 * q^56 - 12 * q^58 + 2 * q^59 - 6 * q^61 - 3 * q^62 - 4 * q^64 + 30 * q^65 - 17 * q^67 + 13 * q^68 + 71 * q^70 - 2 * q^71 + 41 * q^73 + 13 * q^74 + 21 * q^76 - 20 * q^77 + 41 * q^79 + 23 * q^80 - 8 * q^82 - 2 * q^83 - 36 * q^85 + 54 * q^86 - q^89 + 16 * q^91 - 36 * q^92 + 12 * q^94 + 31 * q^95 + 3 * q^97 + 64 * 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 $

$2$	−2.47482	−1.74996	−0.874980	−	0.484159i	$-0.839125\pi$
$2$	−2.47482	−1.74996	−0.874980	+	0.484159i	$0.839125\pi$
$3$	0	0
$3$	0	0
$4$	4.12472	2.06236
$5$	−1.41072	−0.630893	−0.315446	−	0.948943i	$-0.602154\pi$
$5$	−1.41072	−0.630893	−0.315446	+	0.948943i	$0.602154\pi$
$6$	0	0
$7$	1.07503	0.406324	0.203162	−	0.979145i	$-0.434878\pi$
$7$	1.07503	0.406324	0.203162	+	0.979145i	$0.434878\pi$
$8$	−5.25828	−1.85908
$9$	0	0
$10$	3.49127	1.10404
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−5.36271	−1.48735	−0.743675	−	0.668542i	$-0.766919\pi$
$13$	−5.36271	−1.48735	−0.743675	+	0.668542i	$0.766919\pi$
$14$	−2.66051	−0.711051
$15$	0	0
$16$	4.76386	1.19096
$17$	−7.50614	−1.82051	−0.910253	−	0.414053i	$-0.864113\pi$
$17$	−7.50614	−1.82051	−0.910253	+	0.414053i	$0.864113\pi$
$18$	0	0
$19$	−2.87505	−0.659581	−0.329791	−	0.944054i	$-0.606978\pi$
$19$	−2.87505	−0.659581	−0.329791	+	0.944054i	$0.606978\pi$
$20$	−5.81882	−1.30113
$21$	0	0
$22$	2.47482	0.527633
$23$	0.298348	0.0622100	0.0311050	−	0.999516i	$-0.490097\pi$
$23$	0.298348	0.0622100	0.0311050	+	0.999516i	$0.490097\pi$
$24$	0	0
$25$	−3.00987	−0.601974
$26$	13.2717	2.60280
$27$	0	0
$28$	4.43421	0.837986
$29$	−8.79227	−1.63268	−0.816342	−	0.577569i	$-0.804001\pi$
$29$	−8.79227	−1.63268	−0.816342	+	0.577569i	$0.804001\pi$
$30$	0	0
$31$	−6.51720	−1.17052	−0.585261	−	0.810845i	$-0.699008\pi$
$31$	−6.51720	−1.17052	−0.585261	+	0.810845i	$0.699008\pi$
$32$	−1.27310	−0.225054
$33$	0	0
$34$	18.5763	3.18581
$35$	−1.51657	−0.256347
$36$	0	0
$37$	−9.04221	−1.48653	−0.743265	−	0.668997i	$-0.766724\pi$
$37$	−9.04221	−1.48653	−0.743265	+	0.668997i	$0.766724\pi$
$38$	7.11522	1.15424
$39$	0	0
$40$	7.41796	1.17288
$41$	3.80931	0.594914	0.297457	−	0.954735i	$-0.403862\pi$
$41$	3.80931	0.594914	0.297457	+	0.954735i	$0.403862\pi$
$42$	0	0
$43$	−0.489532	−0.0746530	−0.0373265	−	0.999303i	$-0.511884\pi$
$43$	−0.489532	−0.0746530	−0.0373265	+	0.999303i	$0.511884\pi$
$44$	−4.12472	−0.621824
$45$	0	0
$46$	−0.738358	−0.108865
$47$	−8.13263	−1.18627	−0.593133	−	0.805105i	$-0.702109\pi$
$47$	−8.13263	−1.18627	−0.593133	+	0.805105i	$0.702109\pi$
$48$	0	0
$49$	−5.84430	−0.834901
$50$	7.44888	1.05343
$51$	0	0
$52$	−22.1197	−3.06745
$53$	−4.28080	−0.588014	−0.294007	−	0.955803i	$-0.594989\pi$
$53$	−4.28080	−0.588014	−0.294007	+	0.955803i	$0.594989\pi$
$54$	0	0
$55$	1.41072	0.190221
$56$	−5.65283	−0.755391
$57$	0	0
$58$	21.7593	2.85713
$59$	−14.2678	−1.85751	−0.928755	−	0.370693i	$-0.879120\pi$
$59$	−14.2678	−1.85751	−0.928755	+	0.370693i	$0.879120\pi$
$60$	0	0
$61$	1.03289	0.132248	0.0661242	−	0.997811i	$-0.478937\pi$
$61$	1.03289	0.132248	0.0661242	+	0.997811i	$0.478937\pi$
$62$	16.1289	2.04837
$63$	0	0
$64$	−6.37702	−0.797128
$65$	7.56528	0.938358
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	−30.9607	−3.75454
$69$	0	0
$70$	3.75323	0.448597
$71$	0.424042	0.0503245	0.0251622	−	0.999683i	$-0.491990\pi$
$71$	0.424042	0.0503245	0.0251622	+	0.999683i	$0.491990\pi$
$72$	0	0
$73$	16.1947	1.89545	0.947724	−	0.319091i	$-0.103378\pi$
$73$	16.1947	1.89545	0.947724	+	0.319091i	$0.103378\pi$
$74$	22.3778	2.60137
$75$	0	0
$76$	−11.8588	−1.36029
$77$	−1.07503	−0.122511
$78$	0	0
$79$	−1.90066	−0.213841	−0.106921	−	0.994268i	$-0.534099\pi$
$79$	−1.90066	−0.213841	−0.106921	+	0.994268i	$0.534099\pi$
$80$	−6.72046	−0.751370
$81$	0	0
$82$	−9.42733	−1.04107
$83$	17.1012	1.87710	0.938548	−	0.345149i	$-0.112172\pi$
$83$	17.1012	1.87710	0.938548	+	0.345149i	$0.112172\pi$
$84$	0	0
$85$	10.5891	1.14854
$86$	1.21150	0.130640
$87$	0	0
$88$	5.25828	0.560535
$89$	−8.92771	−0.946336	−0.473168	−	0.880972i	$-0.656890\pi$
$89$	−8.92771	−0.946336	−0.473168	+	0.880972i	$0.656890\pi$
$90$	0	0
$91$	−5.76509	−0.604346
$92$	1.23060	0.128299
$93$	0	0
$94$	20.1268	2.07592
$95$	4.05588	0.416125
$96$	0	0
$97$	−16.0441	−1.62903	−0.814516	−	0.580141i	$-0.802997\pi$
$97$	−16.0441	−1.62903	−0.814516	+	0.580141i	$0.802997\pi$
$98$	14.4636	1.46104
$99$	0	0
$100$	−12.4149	−1.24149
$101$	8.75222	0.870878	0.435439	−	0.900218i	$-0.356593\pi$
$101$	8.75222	0.870878	0.435439	+	0.900218i	$0.356593\pi$
$102$	0	0
$103$	17.1503	1.68987	0.844936	−	0.534868i	$-0.179639\pi$
$103$	17.1503	1.68987	0.844936	+	0.534868i	$0.179639\pi$
$104$	28.1987	2.76511
$105$	0	0
$106$	10.5942	1.02900
$107$	5.64497	0.545720	0.272860	−	0.962054i	$-0.412030\pi$
$107$	5.64497	0.545720	0.272860	+	0.962054i	$0.412030\pi$
$108$	0	0
$109$	8.41516	0.806026	0.403013	−	0.915194i	$-0.367963\pi$
$109$	8.41516	0.806026	0.403013	+	0.915194i	$0.367963\pi$
$110$	−3.49127	−0.332880
$111$	0	0
$112$	5.12130	0.483917
$113$	7.69947	0.724305	0.362153	−	0.932119i	$-0.382042\pi$
$113$	7.69947	0.724305	0.362153	+	0.932119i	$0.382042\pi$
$114$	0	0
$115$	−0.420886	−0.0392478
$116$	−36.2656	−3.36718
$117$	0	0
$118$	35.3102	3.25057
$119$	−8.06934	−0.739716
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.55622	−0.231429
$123$	0	0
$124$	−26.8816	−2.41404
$125$	11.2997	1.01067
$126$	0	0
$127$	−3.79179	−0.336467	−0.168234	−	0.985747i	$-0.553806\pi$
$127$	−3.79179	−0.336467	−0.168234	+	0.985747i	$0.553806\pi$
$128$	18.3282	1.62000
$129$	0	0
$130$	−18.7227	−1.64209
$131$	1.32670	0.115914	0.0579570	−	0.998319i	$-0.481541\pi$
$131$	1.32670	0.115914	0.0579570	+	0.998319i	$0.481541\pi$
$132$	0	0
$133$	−3.09077	−0.268004
$134$	2.47482	0.213792
$135$	0	0
$136$	39.4694	3.38447
$137$	−0.325743	−0.0278301	−0.0139151	−	0.999903i	$-0.504429\pi$
$137$	−0.325743	−0.0278301	−0.0139151	+	0.999903i	$0.504429\pi$
$138$	0	0
$139$	−13.8231	−1.17246	−0.586229	−	0.810146i	$-0.699388\pi$
$139$	−13.8231	−1.17246	−0.586229	+	0.810146i	$0.699388\pi$
$140$	−6.25542	−0.528679
$141$	0	0
$142$	−1.04943	−0.0880658
$143$	5.36271	0.448453
$144$	0	0
$145$	12.4034	1.03005
$146$	−40.0789	−3.31696
$147$	0	0
$148$	−37.2965	−3.06576
$149$	3.11999	0.255600	0.127800	−	0.991800i	$-0.459208\pi$
$149$	3.11999	0.255600	0.127800	+	0.991800i	$0.459208\pi$
$150$	0	0
$151$	−16.0278	−1.30433	−0.652163	−	0.758079i	$-0.726138\pi$
$151$	−16.0278	−1.30433	−0.652163	+	0.758079i	$0.726138\pi$
$152$	15.1178	1.22622
$153$	0	0
$154$	2.66051	0.214390
$155$	9.19393	0.738474
$156$	0	0
$157$	23.6203	1.88511	0.942553	−	0.334057i	$-0.108418\pi$
$157$	23.6203	1.88511	0.942553	+	0.334057i	$0.108418\pi$
$158$	4.70379	0.374213
$159$	0	0
$160$	1.79598	0.141985
$161$	0.320734	0.0252774
$162$	0	0
$163$	5.65546	0.442970	0.221485	−	0.975164i	$-0.428910\pi$
$163$	5.65546	0.442970	0.221485	+	0.975164i	$0.428910\pi$
$164$	15.7123	1.22693
$165$	0	0
$166$	−42.3222	−3.28484
$167$	−19.6173	−1.51803	−0.759014	−	0.651074i	$-0.774319\pi$
$167$	−19.6173	−1.51803	−0.759014	+	0.651074i	$0.774319\pi$
$168$	0	0
$169$	15.7587	1.21221
$170$	−26.2060	−2.00991
$171$	0	0
$172$	−2.01918	−0.153961
$173$	13.2966	1.01092	0.505461	−	0.862849i	$-0.331322\pi$
$173$	13.2966	1.01092	0.505461	+	0.862849i	$0.331322\pi$
$174$	0	0
$175$	−3.23571	−0.244597
$176$	−4.76386	−0.359089
$177$	0	0
$178$	22.0944	1.65605
$179$	−22.5756	−1.68738	−0.843688	−	0.536834i	$-0.819620\pi$
$179$	−22.5756	−1.68738	−0.843688	+	0.536834i	$0.819620\pi$
$180$	0	0
$181$	6.30754	0.468836	0.234418	−	0.972136i	$-0.424682\pi$
$181$	6.30754	0.468836	0.234418	+	0.972136i	$0.424682\pi$
$182$	14.2675	1.05758
$183$	0	0
$184$	−1.56880	−0.115654
$185$	12.7560	0.937841
$186$	0	0
$187$	7.50614	0.548903
$188$	−33.5448	−2.44650
$189$	0	0
$190$	−10.0376	−0.728202
$191$	−4.60549	−0.333241	−0.166621	−	0.986021i	$-0.553286\pi$
$191$	−4.60549	−0.333241	−0.166621	+	0.986021i	$0.553286\pi$
$192$	0	0
$193$	−6.72013	−0.483726	−0.241863	−	0.970310i	$-0.577758\pi$
$193$	−6.72013	−0.483726	−0.241863	+	0.970310i	$0.577758\pi$
$194$	39.7062	2.85074
$195$	0	0
$196$	−24.1061	−1.72186
$197$	24.5462	1.74885	0.874424	−	0.485163i	$-0.161240\pi$
$197$	24.5462	1.74885	0.874424	+	0.485163i	$0.161240\pi$
$198$	0	0
$199$	4.55996	0.323247	0.161623	−	0.986852i	$-0.448327\pi$
$199$	4.55996	0.323247	0.161623	+	0.986852i	$0.448327\pi$
$200$	15.8268	1.11912
$201$	0	0
$202$	−21.6601	−1.52400
$203$	−9.45198	−0.663399
$204$	0	0
$205$	−5.37386	−0.375327
$206$	−42.4439	−2.95721
$207$	0	0
$208$	−25.5472	−1.77138
$209$	2.87505	0.198871
$210$	0	0
$211$	25.7632	1.77361	0.886805	−	0.462144i	$-0.152920\pi$
$211$	25.7632	1.77361	0.886805	+	0.462144i	$0.152920\pi$
$212$	−17.6571	−1.21269
$213$	0	0
$214$	−13.9703	−0.954987
$215$	0.690592	0.0470980
$216$	0	0
$217$	−7.00620	−0.475612
$218$	−20.8260	−1.41051
$219$	0	0
$220$	5.81882	0.392305
$221$	40.2533	2.70773
$222$	0	0
$223$	19.0210	1.27374	0.636872	−	0.770970i	$-0.280228\pi$
$223$	19.0210	1.27374	0.636872	+	0.770970i	$0.280228\pi$
$224$	−1.36862	−0.0914450
$225$	0	0
$226$	−19.0548	−1.26751
$227$	−1.01213	−0.0671772	−0.0335886	−	0.999436i	$-0.510694\pi$
$227$	−1.01213	−0.0671772	−0.0335886	+	0.999436i	$0.510694\pi$
$228$	0	0
$229$	−11.0060	−0.727295	−0.363647	−	0.931537i	$-0.618469\pi$
$229$	−11.0060	−0.727295	−0.363647	+	0.931537i	$0.618469\pi$
$230$	1.04162	0.0686821
$231$	0	0
$232$	46.2323	3.03530
$233$	−12.7703	−0.836611	−0.418305	−	0.908306i	$-0.637376\pi$
$233$	−12.7703	−0.836611	−0.418305	+	0.908306i	$0.637376\pi$
$234$	0	0
$235$	11.4728	0.748406
$236$	−58.8507	−3.83085
$237$	0	0
$238$	19.9701	1.29447
$239$	−27.1892	−1.75872	−0.879361	−	0.476156i	$-0.842030\pi$
$239$	−27.1892	−1.75872	−0.879361	+	0.476156i	$0.842030\pi$
$240$	0	0
$241$	7.84708	0.505475	0.252738	−	0.967535i	$-0.418669\pi$
$241$	7.84708	0.505475	0.252738	+	0.967535i	$0.418669\pi$
$242$	−2.47482	−0.159087
$243$	0	0
$244$	4.26039	0.272744
$245$	8.24467	0.526733
$246$	0	0
$247$	15.4181	0.981028
$248$	34.2693	2.17610
$249$	0	0
$250$	−27.9646	−1.76864
$251$	−16.4041	−1.03542	−0.517710	−	0.855556i	$-0.673215\pi$
$251$	−16.4041	−1.03542	−0.517710	+	0.855556i	$0.673215\pi$
$252$	0	0
$253$	−0.298348	−0.0187570
$254$	9.38400	0.588804
$255$	0	0
$256$	−32.6048	−2.03780
$257$	−11.6722	−0.728095	−0.364047	−	0.931380i	$-0.618605\pi$
$257$	−11.6722	−0.728095	−0.364047	+	0.931380i	$0.618605\pi$
$258$	0	0
$259$	−9.72067	−0.604013
$260$	31.2047	1.93523
$261$	0	0
$262$	−3.28333	−0.202845
$263$	−8.21520	−0.506571	−0.253285	−	0.967392i	$-0.581511\pi$
$263$	−8.21520	−0.506571	−0.253285	+	0.967392i	$0.581511\pi$
$264$	0	0
$265$	6.03901	0.370973
$266$	7.64909	0.468996
$267$	0	0
$268$	−4.12472	−0.251957
$269$	−29.1597	−1.77790	−0.888948	−	0.458008i	$-0.848563\pi$
$269$	−29.1597	−1.77790	−0.888948	+	0.458008i	$0.848563\pi$
$270$	0	0
$271$	9.85653	0.598741	0.299371	−	0.954137i	$-0.403223\pi$
$271$	9.85653	0.598741	0.299371	+	0.954137i	$0.403223\pi$
$272$	−35.7582	−2.16816
$273$	0	0
$274$	0.806155	0.0487016
$275$	3.00987	0.181502
$276$	0	0
$277$	−16.7523	−1.00655	−0.503273	−	0.864128i	$-0.667871\pi$
$277$	−16.7523	−1.00655	−0.503273	+	0.864128i	$0.667871\pi$
$278$	34.2096	2.05175
$279$	0	0
$280$	7.97455	0.476571
$281$	17.1935	1.02568	0.512838	−	0.858486i	$-0.328594\pi$
$281$	17.1935	1.02568	0.512838	+	0.858486i	$0.328594\pi$
$282$	0	0
$283$	−2.75862	−0.163983	−0.0819916	−	0.996633i	$-0.526128\pi$
$283$	−2.75862	−0.163983	−0.0819916	+	0.996633i	$0.526128\pi$
$284$	1.74905	0.103787
$285$	0	0
$286$	−13.2717	−0.784774
$287$	4.09513	0.241728
$288$	0	0
$289$	39.3421	2.31424
$290$	−30.6962	−1.80254
$291$	0	0
$292$	66.7986	3.90909
$293$	−11.8582	−0.692763	−0.346381	−	0.938094i	$-0.612590\pi$
$293$	−11.8582	−0.692763	−0.346381	+	0.938094i	$0.612590\pi$
$294$	0	0
$295$	20.1279	1.17189
$296$	47.5465	2.76358
$297$	0	0
$298$	−7.72141	−0.447289
$299$	−1.59996	−0.0925280
$300$	0	0
$301$	−0.526263	−0.0303333
$302$	39.6659	2.28252
$303$	0	0
$304$	−13.6963	−0.785537
$305$	−1.45712	−0.0834345
$306$	0	0
$307$	9.98141	0.569669	0.284834	−	0.958577i	$-0.408061\pi$
$307$	9.98141	0.569669	0.284834	+	0.958577i	$0.408061\pi$
$308$	−4.43421	−0.252662
$309$	0	0
$310$	−22.7533	−1.29230
$311$	8.47944	0.480825	0.240412	−	0.970671i	$-0.422717\pi$
$311$	8.47944	0.480825	0.240412	+	0.970671i	$0.422717\pi$
$312$	0	0
$313$	−14.7813	−0.835491	−0.417746	−	0.908564i	$-0.637180\pi$
$313$	−14.7813	−0.835491	−0.417746	+	0.908564i	$0.637180\pi$
$314$	−58.4559	−3.29886
$315$	0	0
$316$	−7.83969	−0.441017
$317$	−12.9553	−0.727642	−0.363821	−	0.931469i	$-0.618528\pi$
$317$	−12.9553	−0.727642	−0.363821	+	0.931469i	$0.618528\pi$
$318$	0	0
$319$	8.79227	0.492273
$320$	8.99619	0.502902
$321$	0	0
$322$	−0.793759	−0.0442344
$323$	21.5805	1.20077
$324$	0	0
$325$	16.1411	0.895346
$326$	−13.9962	−0.775179
$327$	0	0
$328$	−20.0304	−1.10599
$329$	−8.74284	−0.482008
$330$	0	0
$331$	−28.4300	−1.56265	−0.781326	−	0.624123i	$-0.785456\pi$
$331$	−28.4300	−1.56265	−0.781326	+	0.624123i	$0.785456\pi$
$332$	70.5374	3.87124
$333$	0	0
$334$	48.5491	2.65649
$335$	1.41072	0.0770758
$336$	0	0
$337$	−12.8361	−0.699225	−0.349613	−	0.936894i	$-0.613687\pi$
$337$	−12.8361	−0.699225	−0.349613	+	0.936894i	$0.613687\pi$
$338$	−38.9999	−2.12132
$339$	0	0
$340$	43.6768	2.36871
$341$	6.51720	0.352926
$342$	0	0
$343$	−13.8080	−0.745564
$344$	2.57410	0.138786
$345$	0	0
$346$	−32.9067	−1.76907
$347$	−4.04989	−0.217410	−0.108705	−	0.994074i	$-0.534670\pi$
$347$	−4.04989	−0.217410	−0.108705	+	0.994074i	$0.534670\pi$
$348$	0	0
$349$	−8.27152	−0.442764	−0.221382	−	0.975187i	$-0.571057\pi$
$349$	−8.27152	−0.442764	−0.221382	+	0.975187i	$0.571057\pi$
$350$	8.00779	0.428034
$351$	0	0
$352$	1.27310	0.0678564
$353$	14.6513	0.779810	0.389905	−	0.920855i	$-0.372508\pi$
$353$	14.6513	0.779810	0.389905	+	0.920855i	$0.372508\pi$
$354$	0	0
$355$	−0.598204	−0.0317494
$356$	−36.8243	−1.95168
$357$	0	0
$358$	55.8704	2.95284
$359$	16.3509	0.862969	0.431485	−	0.902120i	$-0.357990\pi$
$359$	16.3509	0.862969	0.431485	+	0.902120i	$0.357990\pi$
$360$	0	0
$361$	−10.7341	−0.564953
$362$	−15.6100	−0.820444
$363$	0	0
$364$	−23.7794	−1.24638
$365$	−22.8462	−1.19582
$366$	0	0
$367$	−2.81802	−0.147100	−0.0735498	−	0.997292i	$-0.523433\pi$
$367$	−2.81802	−0.147100	−0.0735498	+	0.997292i	$0.523433\pi$
$368$	1.42129	0.0740898
$369$	0	0
$370$	−31.5688	−1.64118
$371$	−4.60200	−0.238924
$372$	0	0
$373$	−27.0345	−1.39979	−0.699896	−	0.714245i	$-0.746770\pi$
$373$	−27.0345	−1.39979	−0.699896	+	0.714245i	$0.746770\pi$
$374$	−18.5763	−0.960558
$375$	0	0
$376$	42.7637	2.20537
$377$	47.1504	2.42837
$378$	0	0
$379$	−24.5068	−1.25883	−0.629414	−	0.777070i	$-0.716705\pi$
$379$	−24.5068	−1.25883	−0.629414	+	0.777070i	$0.716705\pi$
$380$	16.7294	0.858199
$381$	0	0
$382$	11.3977	0.583159
$383$	−13.1025	−0.669507	−0.334754	−	0.942306i	$-0.608653\pi$
$383$	−13.1025	−0.669507	−0.334754	+	0.942306i	$0.608653\pi$
$384$	0	0
$385$	1.51657	0.0772915
$386$	16.6311	0.846500
$387$	0	0
$388$	−66.1774	−3.35965
$389$	17.3039	0.877344	0.438672	−	0.898647i	$-0.355449\pi$
$389$	17.3039	0.877344	0.438672	+	0.898647i	$0.355449\pi$
$390$	0	0
$391$	−2.23944	−0.113254
$392$	30.7310	1.55215
$393$	0	0
$394$	−60.7475	−3.06041
$395$	2.68130	0.134911
$396$	0	0
$397$	1.60027	0.0803152	0.0401576	−	0.999193i	$-0.487214\pi$
$397$	1.60027	0.0803152	0.0401576	+	0.999193i	$0.487214\pi$
$398$	−11.2851	−0.565669
$399$	0	0
$400$	−14.3386	−0.716930
$401$	−11.1024	−0.554429	−0.277214	−	0.960808i	$-0.589411\pi$
$401$	−11.1024	−0.554429	−0.277214	+	0.960808i	$0.589411\pi$
$402$	0	0
$403$	34.9499	1.74098
$404$	36.1004	1.79606
$405$	0	0
$406$	23.3919	1.16092
$407$	9.04221	0.448206
$408$	0	0
$409$	−18.1330	−0.896617	−0.448308	−	0.893879i	$-0.647973\pi$
$409$	−18.1330	−0.896617	−0.448308	+	0.893879i	$0.647973\pi$
$410$	13.2993	0.656807
$411$	0	0
$412$	70.7402	3.48512
$413$	−15.3384	−0.754751
$414$	0	0
$415$	−24.1249	−1.18425
$416$	6.82727	0.334734
$417$	0	0
$418$	−7.11522	−0.348017
$419$	−31.5663	−1.54211	−0.771057	−	0.636766i	$-0.780272\pi$
$419$	−31.5663	−1.54211	−0.771057	+	0.636766i	$0.780272\pi$
$420$	0	0
$421$	−27.0593	−1.31879	−0.659395	−	0.751797i	$-0.729188\pi$
$421$	−27.0593	−1.31879	−0.659395	+	0.751797i	$0.729188\pi$
$422$	−63.7591	−3.10375
$423$	0	0
$424$	22.5097	1.09317
$425$	22.5925	1.09590
$426$	0	0
$427$	1.11039	0.0537357
$428$	23.2839	1.12547
$429$	0	0
$430$	−1.70909	−0.0824196
$431$	11.9693	0.576541	0.288271	−	0.957549i	$-0.406920\pi$
$431$	11.9693	0.576541	0.288271	+	0.957549i	$0.406920\pi$
$432$	0	0
$433$	11.6358	0.559182	0.279591	−	0.960119i	$-0.409801\pi$
$433$	11.6358	0.559182	0.279591	+	0.960119i	$0.409801\pi$
$434$	17.3391	0.832301
$435$	0	0
$436$	34.7101	1.66231
$437$	−0.857766	−0.0410325
$438$	0	0
$439$	3.98251	0.190075	0.0950374	−	0.995474i	$-0.469703\pi$
$439$	3.98251	0.190075	0.0950374	+	0.995474i	$0.469703\pi$
$440$	−7.41796	−0.353637
$441$	0	0
$442$	−99.6195	−4.73842
$443$	−0.516457	−0.0245376	−0.0122688	−	0.999925i	$-0.503905\pi$
$443$	−0.516457	−0.0245376	−0.0122688	+	0.999925i	$0.503905\pi$
$444$	0	0
$445$	12.5945	0.597036
$446$	−47.0736	−2.22900
$447$	0	0
$448$	−6.85551	−0.323892
$449$	20.3091	0.958445	0.479223	−	0.877693i	$-0.340919\pi$
$449$	20.3091	0.958445	0.479223	+	0.877693i	$0.340919\pi$
$450$	0	0
$451$	−3.80931	−0.179373
$452$	31.7581	1.49378
$453$	0	0
$454$	2.50483	0.117557
$455$	8.13293	0.381277
$456$	0	0
$457$	−9.77358	−0.457189	−0.228594	−	0.973522i	$-0.573413\pi$
$457$	−9.77358	−0.457189	−0.228594	+	0.973522i	$0.573413\pi$
$458$	27.2377	1.27274
$459$	0	0
$460$	−1.73604	−0.0809431
$461$	3.54561	0.165136	0.0825678	−	0.996585i	$-0.473688\pi$
$461$	3.54561	0.165136	0.0825678	+	0.996585i	$0.473688\pi$
$462$	0	0
$463$	−12.3969	−0.576132	−0.288066	−	0.957611i	$-0.593012\pi$
$463$	−12.3969	−0.576132	−0.288066	+	0.957611i	$0.593012\pi$
$464$	−41.8851	−1.94447
$465$	0	0
$466$	31.6042	1.46404
$467$	−4.52161	−0.209235	−0.104618	−	0.994513i	$-0.533362\pi$
$467$	−4.52161	−0.209235	−0.104618	+	0.994513i	$0.533362\pi$
$468$	0	0
$469$	−1.07503	−0.0496404
$470$	−28.3932	−1.30968
$471$	0	0
$472$	75.0242	3.45327
$473$	0.489532	0.0225087
$474$	0	0
$475$	8.65353	0.397051
$476$	−33.2838	−1.52556
$477$	0	0
$478$	67.2882	3.07769
$479$	−20.2911	−0.927124	−0.463562	−	0.886064i	$-0.653429\pi$
$479$	−20.2911	−0.927124	−0.463562	+	0.886064i	$0.653429\pi$
$480$	0	0
$481$	48.4908	2.21099
$482$	−19.4201	−0.884561
$483$	0	0
$484$	4.12472	0.187487
$485$	22.6337	1.02774
$486$	0	0
$487$	18.1440	0.822181	0.411091	−	0.911595i	$-0.365148\pi$
$487$	18.1440	0.822181	0.411091	+	0.911595i	$0.365148\pi$
$488$	−5.43124	−0.245861
$489$	0	0
$490$	−20.4040	−0.921761
$491$	−36.1591	−1.63184	−0.815918	−	0.578168i	$-0.803768\pi$
$491$	−36.1591	−1.63184	−0.815918	+	0.578168i	$0.803768\pi$
$492$	0	0
$493$	65.9960	2.97231
$494$	−38.1569	−1.71676
$495$	0	0
$496$	−31.0470	−1.39405
$497$	0.455859	0.0204481
$498$	0	0
$499$	0.616756	0.0276098	0.0138049	−	0.999905i	$-0.495606\pi$
$499$	0.616756	0.0276098	0.0138049	+	0.999905i	$0.495606\pi$
$500$	46.6080	2.08437
$501$	0	0
$502$	40.5972	1.81194
$503$	−27.6342	−1.23215	−0.616073	−	0.787689i	$-0.711277\pi$
$503$	−27.6342	−1.23215	−0.616073	+	0.787689i	$0.711277\pi$
$504$	0	0
$505$	−12.3469	−0.549431
$506$	0.738358	0.0328240
$507$	0	0
$508$	−15.6401	−0.693916
$509$	7.49366	0.332151	0.166075	−	0.986113i	$-0.446890\pi$
$509$	7.49366	0.332151	0.166075	+	0.986113i	$0.446890\pi$
$510$	0	0
$511$	17.4098	0.770166
$512$	44.0346	1.94607
$513$	0	0
$514$	28.8867	1.27414
$515$	−24.1943	−1.06613
$516$	0	0
$517$	8.13263	0.357672
$518$	24.0569	1.05700
$519$	0	0
$520$	−39.7804	−1.74449
$521$	2.77252	0.121466	0.0607331	−	0.998154i	$-0.480656\pi$
$521$	2.77252	0.121466	0.0607331	+	0.998154i	$0.480656\pi$
$522$	0	0
$523$	2.66495	0.116530	0.0582652	−	0.998301i	$-0.481443\pi$
$523$	2.66495	0.116530	0.0582652	+	0.998301i	$0.481443\pi$
$524$	5.47225	0.239056
$525$	0	0
$526$	20.3311	0.886478
$527$	48.9190	2.13094
$528$	0	0
$529$	−22.9110	−0.996130
$530$	−14.9454	−0.649189
$531$	0	0
$532$	−12.7486	−0.552720
$533$	−20.4282	−0.884844
$534$	0	0
$535$	−7.96346	−0.344290
$536$	5.25828	0.227123
$537$	0	0
$538$	72.1648	3.11125
$539$	5.84430	0.251732
$540$	0	0
$541$	13.2897	0.571370	0.285685	−	0.958324i	$-0.407779\pi$
$541$	13.2897	0.571370	0.285685	+	0.958324i	$0.407779\pi$
$542$	−24.3931	−1.04777
$543$	0	0
$544$	9.55606	0.409713
$545$	−11.8714	−0.508516
$546$	0	0
$547$	−31.3904	−1.34216	−0.671078	−	0.741387i	$-0.734168\pi$
$547$	−31.3904	−1.34216	−0.671078	+	0.741387i	$0.734168\pi$
$548$	−1.34360	−0.0573957
$549$	0	0
$550$	−7.44888	−0.317621
$551$	25.2782	1.07689
$552$	0	0
$553$	−2.04327	−0.0868888
$554$	41.4588	1.76141
$555$	0	0
$556$	−57.0163	−2.41803
$557$	−8.62963	−0.365649	−0.182825	−	0.983146i	$-0.558524\pi$
$557$	−8.62963	−0.365649	−0.182825	+	0.983146i	$0.558524\pi$
$558$	0	0
$559$	2.62522	0.111035
$560$	−7.22472	−0.305300
$561$	0	0
$562$	−42.5506	−1.79489
$563$	−22.7799	−0.960059	−0.480029	−	0.877252i	$-0.659374\pi$
$563$	−22.7799	−0.960059	−0.480029	+	0.877252i	$0.659374\pi$
$564$	0	0
$565$	−10.8618	−0.456959
$566$	6.82709	0.286964
$567$	0	0
$568$	−2.22973	−0.0935575
$569$	19.6605	0.824211	0.412105	−	0.911136i	$-0.364794\pi$
$569$	19.6605	0.824211	0.412105	+	0.911136i	$0.364794\pi$
$570$	0	0
$571$	3.44149	0.144022	0.0720110	−	0.997404i	$-0.477058\pi$
$571$	3.44149	0.144022	0.0720110	+	0.997404i	$0.477058\pi$
$572$	22.1197	0.924870
$573$	0	0
$574$	−10.1347	−0.423014
$575$	−0.897991	−0.0374488
$576$	0	0
$577$	35.9511	1.49666	0.748332	−	0.663324i	$-0.230855\pi$
$577$	35.9511	1.49666	0.748332	+	0.663324i	$0.230855\pi$
$578$	−97.3645	−4.04983
$579$	0	0
$580$	51.1606	2.12433
$581$	18.3843	0.762709
$582$	0	0
$583$	4.28080	0.177293
$584$	−85.1564	−3.52380
$585$	0	0
$586$	29.3468	1.21231
$587$	−32.9177	−1.35866	−0.679330	−	0.733833i	$-0.737729\pi$
$587$	−32.9177	−1.35866	−0.679330	+	0.733833i	$0.737729\pi$
$588$	0	0
$589$	18.7372	0.772055
$590$	−49.8128	−2.05076
$591$	0	0
$592$	−43.0758	−1.77040
$593$	33.5406	1.37735	0.688673	−	0.725072i	$-0.258193\pi$
$593$	33.5406	1.37735	0.688673	+	0.725072i	$0.258193\pi$
$594$	0	0
$595$	11.3836	0.466681
$596$	12.8691	0.527138
$597$	0	0
$598$	3.95960	0.161920
$599$	18.2356	0.745087	0.372544	−	0.928015i	$-0.378486\pi$
$599$	18.2356	0.745087	0.372544	+	0.928015i	$0.378486\pi$
$600$	0	0
$601$	−31.0173	−1.26522	−0.632611	−	0.774469i	$-0.718017\pi$
$601$	−31.0173	−1.26522	−0.632611	+	0.774469i	$0.718017\pi$
$602$	1.30240	0.0530820
$603$	0	0
$604$	−66.1102	−2.68999
$605$	−1.41072	−0.0573539
$606$	0	0
$607$	25.2349	1.02425	0.512127	−	0.858910i	$-0.328858\pi$
$607$	25.2349	1.02425	0.512127	+	0.858910i	$0.328858\pi$
$608$	3.66022	0.148442
$609$	0	0
$610$	3.60611	0.146007
$611$	43.6130	1.76439
$612$	0	0
$613$	−1.27527	−0.0515076	−0.0257538	−	0.999668i	$-0.508199\pi$
$613$	−1.27527	−0.0515076	−0.0257538	+	0.999668i	$0.508199\pi$
$614$	−24.7021	−0.996898
$615$	0	0
$616$	5.65283	0.227759
$617$	17.0407	0.686033	0.343016	−	0.939329i	$-0.388551\pi$
$617$	17.0407	0.686033	0.343016	+	0.939329i	$0.388551\pi$
$618$	0	0
$619$	0.465701	0.0187181	0.00935904	−	0.999956i	$-0.497021\pi$
$619$	0.465701	0.0187181	0.00935904	+	0.999956i	$0.497021\pi$
$620$	37.9224	1.52300
$621$	0	0
$622$	−20.9851	−0.841424
$623$	−9.59758	−0.384519
$624$	0	0
$625$	−0.891311	−0.0356524
$626$	36.5811	1.46208
$627$	0	0
$628$	97.4271	3.88776
$629$	67.8721	2.70624
$630$	0	0
$631$	−4.71536	−0.187716	−0.0938578	−	0.995586i	$-0.529920\pi$
$631$	−4.71536	−0.187716	−0.0938578	+	0.995586i	$0.529920\pi$
$632$	9.99422	0.397549
$633$	0	0
$634$	32.0620	1.27334
$635$	5.34916	0.212275
$636$	0	0
$637$	31.3413	1.24179
$638$	−21.7593	−0.861457
$639$	0	0
$640$	−25.8559	−1.02204
$641$	−33.1712	−1.31018	−0.655092	−	0.755549i	$-0.727370\pi$
$641$	−33.1712	−1.31018	−0.655092	+	0.755549i	$0.727370\pi$
$642$	0	0
$643$	−25.8672	−1.02010	−0.510051	−	0.860144i	$-0.670373\pi$
$643$	−25.8672	−1.02010	−0.510051	+	0.860144i	$0.670373\pi$
$644$	1.32294	0.0521311
$645$	0	0
$646$	−53.4078	−2.10130
$647$	44.4593	1.74787	0.873937	−	0.486039i	$-0.161559\pi$
$647$	44.4593	1.74787	0.873937	+	0.486039i	$0.161559\pi$
$648$	0	0
$649$	14.2678	0.560061
$650$	−39.9462	−1.56682
$651$	0	0
$652$	23.3272	0.913563
$653$	22.8862	0.895608	0.447804	−	0.894132i	$-0.352206\pi$
$653$	22.8862	0.895608	0.447804	+	0.894132i	$0.352206\pi$
$654$	0	0
$655$	−1.87160	−0.0731293
$656$	18.1470	0.708521
$657$	0	0
$658$	21.6369	0.843495
$659$	20.6361	0.803867	0.401933	−	0.915669i	$-0.368338\pi$
$659$	20.6361	0.803867	0.401933	+	0.915669i	$0.368338\pi$
$660$	0	0
$661$	−30.1991	−1.17461	−0.587304	−	0.809366i	$-0.699811\pi$
$661$	−30.1991	−1.17461	−0.587304	+	0.809366i	$0.699811\pi$
$662$	70.3590	2.73458
$663$	0	0
$664$	−89.9228	−3.48968
$665$	4.36021	0.169082
$666$	0	0
$667$	−2.62316	−0.101569
$668$	−80.9156	−3.13072
$669$	0	0
$670$	−3.49127	−0.134880
$671$	−1.03289	−0.0398744
$672$	0	0
$673$	−27.4308	−1.05738	−0.528690	−	0.848815i	$-0.677317\pi$
$673$	−27.4308	−1.05738	−0.528690	+	0.848815i	$0.677317\pi$
$674$	31.7669	1.22362
$675$	0	0
$676$	65.0002	2.50001
$677$	−6.48875	−0.249383	−0.124691	−	0.992196i	$-0.539794\pi$
$677$	−6.48875	−0.249383	−0.124691	+	0.992196i	$0.539794\pi$
$678$	0	0
$679$	−17.2479	−0.661915
$680$	−55.6802	−2.13524
$681$	0	0
$682$	−16.1289	−0.617606
$683$	−41.5048	−1.58814	−0.794069	−	0.607828i	$-0.792041\pi$
$683$	−41.5048	−1.58814	−0.794069	+	0.607828i	$0.792041\pi$
$684$	0	0
$685$	0.459532	0.0175578
$686$	34.1724	1.30471
$687$	0	0
$688$	−2.33206	−0.0889090
$689$	22.9567	0.874582
$690$	0	0
$691$	−15.0247	−0.571566	−0.285783	−	0.958294i	$-0.592254\pi$
$691$	−15.0247	−0.571566	−0.285783	+	0.958294i	$0.592254\pi$
$692$	54.8447	2.08488
$693$	0	0
$694$	10.0227	0.380458
$695$	19.5005	0.739695
$696$	0	0
$697$	−28.5932	−1.08304
$698$	20.4705	0.774820
$699$	0	0
$700$	−13.3464	−0.504446
$701$	−14.2097	−0.536693	−0.268346	−	0.963323i	$-0.586477\pi$
$701$	−14.2097	−0.536693	−0.268346	+	0.963323i	$0.586477\pi$
$702$	0	0
$703$	25.9968	0.980487
$704$	6.37702	0.240343
$705$	0	0
$706$	−36.2593	−1.36464
$707$	9.40892	0.353859
$708$	0	0
$709$	−10.7534	−0.403852	−0.201926	−	0.979401i	$-0.564720\pi$
$709$	−10.7534	−0.403852	−0.201926	+	0.979401i	$0.564720\pi$
$710$	1.48044	0.0555601
$711$	0	0
$712$	46.9444	1.75932
$713$	−1.94440	−0.0728182
$714$	0	0
$715$	−7.56528	−0.282926
$716$	−93.1178	−3.47997
$717$	0	0
$718$	−40.4656	−1.51016
$719$	10.5747	0.394370	0.197185	−	0.980366i	$-0.436820\pi$
$719$	10.5747	0.394370	0.197185	+	0.980366i	$0.436820\pi$
$720$	0	0
$721$	18.4372	0.686636
$722$	26.5649	0.988644
$723$	0	0
$724$	26.0168	0.966908
$725$	26.4636	0.982834
$726$	0	0
$727$	10.6516	0.395044	0.197522	−	0.980298i	$-0.436711\pi$
$727$	10.6516	0.395044	0.197522	+	0.980298i	$0.436711\pi$
$728$	30.3145	1.12353
$729$	0	0
$730$	56.5401	2.09264
$731$	3.67450	0.135906
$732$	0	0
$733$	−30.1892	−1.11506	−0.557532	−	0.830155i	$-0.688252\pi$
$733$	−30.1892	−1.11506	−0.557532	+	0.830155i	$0.688252\pi$
$734$	6.97409	0.257418
$735$	0	0
$736$	−0.379827	−0.0140006
$737$	1.00000	0.0368355
$738$	0	0
$739$	−18.3060	−0.673397	−0.336698	−	0.941613i	$-0.609310\pi$
$739$	−18.3060	−0.673397	−0.336698	+	0.941613i	$0.609310\pi$
$740$	52.6149	1.93416
$741$	0	0
$742$	11.3891	0.418108
$743$	13.8457	0.507948	0.253974	−	0.967211i	$-0.418262\pi$
$743$	13.8457	0.507948	0.253974	+	0.967211i	$0.418262\pi$
$744$	0	0
$745$	−4.40143	−0.161256
$746$	66.9054	2.44958
$747$	0	0
$748$	30.9607	1.13204
$749$	6.06852	0.221739
$750$	0	0
$751$	3.94350	0.143901	0.0719503	−	0.997408i	$-0.477078\pi$
$751$	3.94350	0.143901	0.0719503	+	0.997408i	$0.477078\pi$
$752$	−38.7427	−1.41280
$753$	0	0
$754$	−116.689	−4.24955
$755$	22.6107	0.822889
$756$	0	0
$757$	−34.1274	−1.24038	−0.620190	−	0.784452i	$-0.712944\pi$
$757$	−34.1274	−1.24038	−0.620190	+	0.784452i	$0.712944\pi$
$758$	60.6497	2.20290
$759$	0	0
$760$	−21.3270	−0.773611
$761$	21.9753	0.796604	0.398302	−	0.917254i	$-0.369600\pi$
$761$	21.9753	0.796604	0.398302	+	0.917254i	$0.369600\pi$
$762$	0	0
$763$	9.04657	0.327508
$764$	−18.9963	−0.687263
$765$	0	0
$766$	32.4263	1.17161
$767$	76.5142	2.76277
$768$	0	0
$769$	51.0865	1.84223	0.921113	−	0.389295i	$-0.127281\pi$
$769$	51.0865	1.84223	0.921113	+	0.389295i	$0.127281\pi$
$770$	−3.75323	−0.135257
$771$	0	0
$772$	−27.7186	−0.997616
$773$	15.2348	0.547957	0.273978	−	0.961736i	$-0.411660\pi$
$773$	15.2348	0.547957	0.273978	+	0.961736i	$0.411660\pi$
$774$	0	0
$775$	19.6159	0.704625
$776$	84.3644	3.02851
$777$	0	0
$778$	−42.8240	−1.53532
$779$	−10.9519	−0.392394
$780$	0	0
$781$	−0.424042	−0.0151734
$782$	5.54222	0.198189
$783$	0	0
$784$	−27.8414	−0.994336
$785$	−33.3216	−1.18930
$786$	0	0
$787$	−24.9183	−0.888241	−0.444120	−	0.895967i	$-0.646484\pi$
$787$	−24.9183	−0.888241	−0.444120	+	0.895967i	$0.646484\pi$
$788$	101.246	3.60675
$789$	0	0
$790$	−6.63572	−0.236088
$791$	8.27718	0.294303
$792$	0	0
$793$	−5.53911	−0.196700
$794$	−3.96037	−0.140548
$795$	0	0
$796$	18.8085	0.666651
$797$	−15.3440	−0.543513	−0.271757	−	0.962366i	$-0.587605\pi$
$797$	−15.3440	−0.543513	−0.271757	+	0.962366i	$0.587605\pi$
$798$	0	0
$799$	61.0446	2.15960
$800$	3.83187	0.135477
$801$	0	0
$802$	27.4765	0.970228
$803$	−16.1947	−0.571499
$804$	0	0
$805$	−0.452466	−0.0159473
$806$	−86.4945	−3.04664
$807$	0	0
$808$	−46.0217	−1.61904
$809$	31.4869	1.10702	0.553511	−	0.832842i	$-0.313288\pi$
$809$	31.4869	1.10702	0.553511	+	0.832842i	$0.313288\pi$
$810$	0	0
$811$	−9.30604	−0.326779	−0.163390	−	0.986562i	$-0.552243\pi$
$811$	−9.30604	−0.326779	−0.163390	+	0.986562i	$0.552243\pi$
$812$	−38.9867	−1.36817
$813$	0	0
$814$	−22.3778	−0.784342
$815$	−7.97827	−0.279466
$816$	0	0
$817$	1.40743	0.0492397
$818$	44.8757	1.56904
$819$	0	0
$820$	−22.1657	−0.774058
$821$	−5.55715	−0.193946	−0.0969730	−	0.995287i	$-0.530916\pi$
$821$	−5.55715	−0.193946	−0.0969730	+	0.995287i	$0.530916\pi$
$822$	0	0
$823$	33.9641	1.18391	0.591957	−	0.805970i	$-0.298356\pi$
$823$	33.9641	1.18391	0.591957	+	0.805970i	$0.298356\pi$
$824$	−90.1813	−3.14161
$825$	0	0
$826$	37.9596	1.32078
$827$	40.2741	1.40047	0.700233	−	0.713914i	$-0.253079\pi$
$827$	40.2741	1.40047	0.700233	+	0.713914i	$0.253079\pi$
$828$	0	0
$829$	21.6426	0.751678	0.375839	−	0.926685i	$-0.377355\pi$
$829$	21.6426	0.751678	0.375839	+	0.926685i	$0.377355\pi$
$830$	59.7048	2.07238
$831$	0	0
$832$	34.1982	1.18561
$833$	43.8682	1.51994
$834$	0	0
$835$	27.6744	0.957713
$836$	11.8588	0.410144
$837$	0	0
$838$	78.1208	2.69864
$839$	−19.5438	−0.674726	−0.337363	−	0.941375i	$-0.609535\pi$
$839$	−19.5438	−0.674726	−0.337363	+	0.941375i	$0.609535\pi$
$840$	0	0
$841$	48.3040	1.66566
$842$	66.9668	2.30783
$843$	0	0
$844$	106.266	3.65782
$845$	−22.2311	−0.764773
$846$	0	0
$847$	1.07503	0.0369386
$848$	−20.3931	−0.700303
$849$	0	0
$850$	−55.9123	−1.91778
$851$	−2.69773	−0.0924770
$852$	0	0
$853$	−16.5059	−0.565153	−0.282576	−	0.959245i	$-0.591189\pi$
$853$	−16.5059	−0.565153	−0.282576	+	0.959245i	$0.591189\pi$
$854$	−2.74802	−0.0940353
$855$	0	0
$856$	−29.6828	−1.01454
$857$	−13.5931	−0.464330	−0.232165	−	0.972676i	$-0.574581\pi$
$857$	−13.5931	−0.464330	−0.232165	+	0.972676i	$0.574581\pi$
$858$	0	0
$859$	−26.8799	−0.917131	−0.458566	−	0.888661i	$-0.651637\pi$
$859$	−26.8799	−0.917131	−0.458566	+	0.888661i	$0.651637\pi$
$860$	2.84850	0.0971330
$861$	0	0
$862$	−29.6219	−1.00892
$863$	21.0071	0.715092	0.357546	−	0.933896i	$-0.383614\pi$
$863$	21.0071	0.715092	0.357546	+	0.933896i	$0.383614\pi$
$864$	0	0
$865$	−18.7578	−0.637783
$866$	−28.7965	−0.978546
$867$	0	0
$868$	−28.8986	−0.980882
$869$	1.90066	0.0644755
$870$	0	0
$871$	5.36271	0.181709
$872$	−44.2493	−1.49847
$873$	0	0
$874$	2.12281	0.0718053
$875$	12.1475	0.410661
$876$	0	0
$877$	−6.81574	−0.230151	−0.115076	−	0.993357i	$-0.536711\pi$
$877$	−6.81574	−0.230151	−0.115076	+	0.993357i	$0.536711\pi$
$878$	−9.85598	−0.332623
$879$	0	0
$880$	6.72046	0.226547
$881$	58.6675	1.97656	0.988279	−	0.152658i	$-0.0487834\pi$
$881$	58.6675	1.97656	0.988279	+	0.152658i	$0.0487834\pi$
$882$	0	0
$883$	33.6937	1.13388	0.566942	−	0.823758i	$-0.308126\pi$
$883$	33.6937	1.13388	0.566942	+	0.823758i	$0.308126\pi$
$884$	166.033	5.58431
$885$	0	0
$886$	1.27814	0.0429398
$887$	51.2454	1.72065	0.860326	−	0.509744i	$-0.170260\pi$
$887$	51.2454	1.72065	0.860326	+	0.509744i	$0.170260\pi$
$888$	0	0
$889$	−4.07630	−0.136715
$890$	−31.1691	−1.04479
$891$	0	0
$892$	78.4564	2.62692
$893$	23.3817	0.782438
$894$	0	0
$895$	31.8478	1.06455
$896$	19.7034	0.658244
$897$	0	0
$898$	−50.2613	−1.67724
$899$	57.3009	1.91109
$900$	0	0
$901$	32.1323	1.07048
$902$	9.42733	0.313896
$903$	0	0
$904$	−40.4860	−1.34654
$905$	−8.89817	−0.295785
$906$	0	0
$907$	−37.4568	−1.24373	−0.621867	−	0.783123i	$-0.713625\pi$
$907$	−37.4568	−1.24373	−0.621867	+	0.783123i	$0.713625\pi$
$908$	−4.17474	−0.138544
$909$	0	0
$910$	−20.1275	−0.667220
$911$	−1.80820	−0.0599084	−0.0299542	−	0.999551i	$-0.509536\pi$
$911$	−1.80820	−0.0599084	−0.0299542	+	0.999551i	$0.509536\pi$
$912$	0	0
$913$	−17.1012	−0.565966
$914$	24.1878	0.800062
$915$	0	0
$916$	−45.3965	−1.49994
$917$	1.42624	0.0470987
$918$	0	0
$919$	38.3165	1.26394	0.631972	−	0.774991i	$-0.282246\pi$
$919$	38.3165	1.26394	0.631972	+	0.774991i	$0.282246\pi$
$920$	2.21314	0.0729650
$921$	0	0
$922$	−8.77474	−0.288981
$923$	−2.27401	−0.0748501
$924$	0	0
$925$	27.2159	0.894853
$926$	30.6800	1.00821
$927$	0	0
$928$	11.1934	0.367442
$929$	−13.7067	−0.449701	−0.224850	−	0.974393i	$-0.572189\pi$
$929$	−13.7067	−0.449701	−0.224850	+	0.974393i	$0.572189\pi$
$930$	0	0
$931$	16.8027	0.550685
$932$	−52.6739	−1.72539
$933$	0	0
$934$	11.1902	0.366153
$935$	−10.5891	−0.346299
$936$	0	0
$937$	13.4141	0.438219	0.219110	−	0.975700i	$-0.429685\pi$
$937$	13.4141	0.438219	0.219110	+	0.975700i	$0.429685\pi$
$938$	2.66051	0.0868687
$939$	0	0
$940$	47.3223	1.54348
$941$	−5.99982	−0.195588	−0.0977942	−	0.995207i	$-0.531179\pi$
$941$	−5.99982	−0.195588	−0.0977942	+	0.995207i	$0.531179\pi$
$942$	0	0
$943$	1.13650	0.0370096
$944$	−67.9698	−2.21223
$945$	0	0
$946$	−1.21150	−0.0393893
$947$	25.0566	0.814230	0.407115	−	0.913377i	$-0.366535\pi$
$947$	25.0566	0.814230	0.407115	+	0.913377i	$0.366535\pi$
$948$	0	0
$949$	−86.8476	−2.81919
$950$	−21.4159	−0.694823
$951$	0	0
$952$	42.4309	1.37519
$953$	22.2391	0.720395	0.360197	−	0.932876i	$-0.382709\pi$
$953$	22.2391	0.720395	0.360197	+	0.932876i	$0.382709\pi$
$954$	0	0
$955$	6.49705	0.210240
$956$	−112.148	−3.62711
$957$	0	0
$958$	50.2167	1.62243
$959$	−0.350185	−0.0113081
$960$	0	0
$961$	11.4738	0.370124
$962$	−120.006	−3.86914
$963$	0	0
$964$	32.3670	1.04247
$965$	9.48022	0.305179
$966$	0	0
$967$	2.58503	0.0831289	0.0415645	−	0.999136i	$-0.486766\pi$
$967$	2.58503	0.0831289	0.0415645	+	0.999136i	$0.486766\pi$
$968$	−5.25828	−0.169008
$969$	0	0
$970$	−56.0143	−1.79851
$971$	32.6654	1.04828	0.524141	−	0.851631i	$-0.324386\pi$
$971$	32.6654	1.04828	0.524141	+	0.851631i	$0.324386\pi$
$972$	0	0
$973$	−14.8603	−0.476398
$974$	−44.9030	−1.43878
$975$	0	0
$976$	4.92055	0.157503
$977$	−23.1907	−0.741937	−0.370968	−	0.928645i	$-0.620974\pi$
$977$	−23.1907	−0.741937	−0.370968	+	0.928645i	$0.620974\pi$
$978$	0	0
$979$	8.92771	0.285331
$980$	34.0069	1.08631
$981$	0	0
$982$	89.4871	2.85565
$983$	−24.9304	−0.795155	−0.397577	−	0.917569i	$-0.630149\pi$
$983$	−24.9304	−0.795155	−0.397577	+	0.917569i	$0.630149\pi$
$984$	0	0
$985$	−34.6279	−1.10334
$986$	−163.328	−5.20142
$987$	0	0
$988$	63.5951	2.02323
$989$	−0.146051	−0.00464416
$990$	0	0
$991$	−6.61193	−0.210035	−0.105018	−	0.994470i	$-0.533490\pi$
$991$	−6.61193	−0.210035	−0.105018	+	0.994470i	$0.533490\pi$
$992$	8.29704	0.263431
$993$	0	0
$994$	−1.12817	−0.0357833
$995$	−6.43282	−0.203934
$996$	0	0
$997$	−28.1569	−0.891737	−0.445868	−	0.895099i	$-0.647105\pi$
$997$	−28.1569	−0.891737	−0.445868	+	0.895099i	$0.647105\pi$
$998$	−1.52636	−0.0483160
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6633.2.a.w.1.2		17
3.2	odd	2		737.2.a.f.1.16	✓	17
33.32	even	2		8107.2.a.o.1.2		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
737.2.a.f.1.16	✓	17	3.2	odd	2
6633.2.a.w.1.2		17	1.1	even	1	trivial
8107.2.a.o.1.2		17	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6633 = 3^{2} \cdot 11 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6633.a (trivial)

\(f(q)\)	\(=\)	\(q-2.47482 q^{2} +4.12472 q^{4} -1.41072 q^{5} +1.07503 q^{7} -5.25828 q^{8} +O(q^{10})\)	\(q-2.47482 q^{2} +4.12472 q^{4} -1.41072 q^{5} +1.07503 q^{7} -5.25828 q^{8} +3.49127 q^{10} -1.00000 q^{11} -5.36271 q^{13} -2.66051 q^{14} +4.76386 q^{16} -7.50614 q^{17} -2.87505 q^{19} -5.81882 q^{20} +2.47482 q^{22} +0.298348 q^{23} -3.00987 q^{25} +13.2717 q^{26} +4.43421 q^{28} -8.79227 q^{29} -6.51720 q^{31} -1.27310 q^{32} +18.5763 q^{34} -1.51657 q^{35} -9.04221 q^{37} +7.11522 q^{38} +7.41796 q^{40} +3.80931 q^{41} -0.489532 q^{43} -4.12472 q^{44} -0.738358 q^{46} -8.13263 q^{47} -5.84430 q^{49} +7.44888 q^{50} -22.1197 q^{52} -4.28080 q^{53} +1.41072 q^{55} -5.65283 q^{56} +21.7593 q^{58} -14.2678 q^{59} +1.03289 q^{61} +16.1289 q^{62} -6.37702 q^{64} +7.56528 q^{65} -1.00000 q^{67} -30.9607 q^{68} +3.75323 q^{70} +0.424042 q^{71} +16.1947 q^{73} +22.3778 q^{74} -11.8588 q^{76} -1.07503 q^{77} -1.90066 q^{79} -6.72046 q^{80} -9.42733 q^{82} +17.1012 q^{83} +10.5891 q^{85} +1.21150 q^{86} +5.25828 q^{88} -8.92771 q^{89} -5.76509 q^{91} +1.23060 q^{92} +20.1268 q^{94} +4.05588 q^{95} -16.0441 q^{97} +14.4636 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7}+O(q^{10}) \) 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7	\( 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7} + 10 q^{10} - 17 q^{11} + q^{13} + 11 q^{14} + 19 q^{16} - 2 q^{17} + 13 q^{19} - 3 q^{20} + q^{22} - 16 q^{23} + 33 q^{25} - 12 q^{26} + 44 q^{28} + 5 q^{29} + 16 q^{31} + 24 q^{32} + 4 q^{34} + 2 q^{35} + 29 q^{37} + 19 q^{38} + 31 q^{40} + 6 q^{41} + 19 q^{43} - 19 q^{44} - 33 q^{46} - 40 q^{47} + 23 q^{49} + 3 q^{50} - 28 q^{52} - 15 q^{53} + 10 q^{55} + 38 q^{56} - 12 q^{58} + 2 q^{59} - 6 q^{61} - 3 q^{62} - 4 q^{64} + 30 q^{65} - 17 q^{67} + 13 q^{68} + 71 q^{70} - 2 q^{71} + 41 q^{73} + 13 q^{74} + 21 q^{76} - 20 q^{77} + 41 q^{79} + 23 q^{80} - 8 q^{82} - 2 q^{83} - 36 q^{85} + 54 q^{86} - q^{89} + 16 q^{91} - 36 q^{92} + 12 q^{94} + 31 q^{95} + 3 q^{97} + 64 q^{98}+O(q^{100}) \) 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7 + 10 * q^10 - 17 * q^11 + q^13 + 11 * q^14 + 19 * q^16 - 2 * q^17 + 13 * q^19 - 3 * q^20 + q^22 - 16 * q^23 + 33 * q^25 - 12 * q^26 + 44 * q^28 + 5 * q^29 + 16 * q^31 + 24 * q^32 + 4 * q^34 + 2 * q^35 + 29 * q^37 + 19 * q^38 + 31 * q^40 + 6 * q^41 + 19 * q^43 - 19 * q^44 - 33 * q^46 - 40 * q^47 + 23 * q^49 + 3 * q^50 - 28 * q^52 - 15 * q^53 + 10 * q^55 + 38 * q^56 - 12 * q^58 + 2 * q^59 - 6 * q^61 - 3 * q^62 - 4 * q^64 + 30 * q^65 - 17 * q^67 + 13 * q^68 + 71 * q^70 - 2 * q^71 + 41 * q^73 + 13 * q^74 + 21 * q^76 - 20 * q^77 + 41 * q^79 + 23 * q^80 - 8 * q^82 - 2 * q^83 - 36 * q^85 + 54 * q^86 - q^89 + 16 * q^91 - 36 * q^92 + 12 * q^94 + 31 * q^95 + 3 * q^97 + 64 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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