LMFDB - Embedded newform 8001.2.a.r.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
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} - 5 x^{15} - 13 x^{14} + 98 x^{13} + 9 x^{12} - 712 x^{11} + 565 x^{10} + 2282 x^{9} - 3082 x^{8} + \cdots + 4 $ x^16 - 5x^15 - 13x^14 + 98x^13 + 9x^12 - 712x^11 + 565x^10 + 2282x^9 - 3082x^8 - 2747x^7 + 5821x^6 - 158x^5 - 3341x^4 + 1002x^3 + 416x^2 - 148*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.68977$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.68977 q^{2} +0.855315 q^{4} -2.75353 q^{5} -1.00000 q^{7} +1.93425 q^{8} +O(q^{10})$	\(q-1.68977 q^{2} +0.855315 q^{4} -2.75353 q^{5} -1.00000 q^{7} +1.93425 q^{8} +4.65282 q^{10} -3.55608 q^{11} +4.67022 q^{13} +1.68977 q^{14} -4.97907 q^{16} -4.02992 q^{17} -1.08856 q^{19} -2.35514 q^{20} +6.00896 q^{22} +7.04757 q^{23} +2.58192 q^{25} -7.89159 q^{26} -0.855315 q^{28} -9.79449 q^{29} -2.65900 q^{31} +4.54496 q^{32} +6.80962 q^{34} +2.75353 q^{35} +10.4203 q^{37} +1.83942 q^{38} -5.32602 q^{40} +9.96789 q^{41} -6.65929 q^{43} -3.04157 q^{44} -11.9088 q^{46} -6.75602 q^{47} +1.00000 q^{49} -4.36285 q^{50} +3.99451 q^{52} +3.41123 q^{53} +9.79178 q^{55} -1.93425 q^{56} +16.5504 q^{58} -8.15705 q^{59} +15.2219 q^{61} +4.49309 q^{62} +2.27820 q^{64} -12.8596 q^{65} -10.7563 q^{67} -3.44685 q^{68} -4.65282 q^{70} -5.93668 q^{71} +8.88205 q^{73} -17.6079 q^{74} -0.931064 q^{76} +3.55608 q^{77} +10.1644 q^{79} +13.7100 q^{80} -16.8434 q^{82} -12.6631 q^{83} +11.0965 q^{85} +11.2527 q^{86} -6.87836 q^{88} +6.83138 q^{89} -4.67022 q^{91} +6.02790 q^{92} +11.4161 q^{94} +2.99739 q^{95} +13.2112 q^{97} -1.68977 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) $ 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8	$ 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8} - 12 q^{10} - 11 q^{11} + 18 q^{13} + 5 q^{14} + 25 q^{16} + 5 q^{17} - 11 q^{19} + q^{20} + q^{22} - 13 q^{23} + 33 q^{25} - 8 q^{26} - 19 q^{28} - 24 q^{29} - 42 q^{31} - 42 q^{32} + 9 q^{34} - q^{35} + 40 q^{37} - 38 q^{38} - 61 q^{40} - 9 q^{41} + 7 q^{43} - 3 q^{44} + 24 q^{46} - 31 q^{47} + 16 q^{49} - 6 q^{50} + 52 q^{52} - 66 q^{53} - 36 q^{55} + 6 q^{56} + 19 q^{58} + 7 q^{59} + 6 q^{61} - 52 q^{62} + 10 q^{64} - 51 q^{65} + 16 q^{67} - 14 q^{68} + 12 q^{70} - 46 q^{71} + 39 q^{73} - 72 q^{74} + 24 q^{76} + 11 q^{77} + 4 q^{79} + 2 q^{80} - 18 q^{82} - 15 q^{83} - 4 q^{85} - 14 q^{86} + 58 q^{88} + q^{89} - 18 q^{91} - 26 q^{92} + 5 q^{94} - 44 q^{95} + 41 q^{97} - 5 q^{98}+O(q^{100}) $ 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8 - 12 * q^10 - 11 * q^11 + 18 * q^13 + 5 * q^14 + 25 * q^16 + 5 * q^17 - 11 * q^19 + q^20 + q^22 - 13 * q^23 + 33 * q^25 - 8 * q^26 - 19 * q^28 - 24 * q^29 - 42 * q^31 - 42 * q^32 + 9 * q^34 - q^35 + 40 * q^37 - 38 * q^38 - 61 * q^40 - 9 * q^41 + 7 * q^43 - 3 * q^44 + 24 * q^46 - 31 * q^47 + 16 * q^49 - 6 * q^50 + 52 * q^52 - 66 * q^53 - 36 * q^55 + 6 * q^56 + 19 * q^58 + 7 * q^59 + 6 * q^61 - 52 * q^62 + 10 * q^64 - 51 * q^65 + 16 * q^67 - 14 * q^68 + 12 * q^70 - 46 * q^71 + 39 * q^73 - 72 * q^74 + 24 * q^76 + 11 * q^77 + 4 * q^79 + 2 * q^80 - 18 * q^82 - 15 * q^83 - 4 * q^85 - 14 * q^86 + 58 * q^88 + q^89 - 18 * q^91 - 26 * q^92 + 5 * q^94 - 44 * q^95 + 41 * q^97 - 5 * 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 $

$2$	−1.68977	−1.19485	−0.597423	−	0.801926i	$-0.703809\pi$
$2$	−1.68977	−1.19485	−0.597423	+	0.801926i	$0.703809\pi$
$3$	0	0
$3$	0	0
$4$	0.855315	0.427658
$5$	−2.75353	−1.23142	−0.615708	−	0.787975i	$-0.711130\pi$
$5$	−2.75353	−1.23142	−0.615708	+	0.787975i	$0.711130\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.93425	0.683861
$9$	0	0
$10$	4.65282	1.47135
$11$	−3.55608	−1.07220	−0.536100	−	0.844154i	$-0.680103\pi$
$11$	−3.55608	−1.07220	−0.536100	+	0.844154i	$0.680103\pi$
$12$	0	0
$13$	4.67022	1.29529	0.647643	−	0.761944i	$-0.275755\pi$
$13$	4.67022	1.29529	0.647643	+	0.761944i	$0.275755\pi$
$14$	1.68977	0.451609
$15$	0	0
$16$	−4.97907	−1.24477
$17$	−4.02992	−0.977398	−0.488699	−	0.872452i	$-0.662528\pi$
$17$	−4.02992	−0.977398	−0.488699	+	0.872452i	$0.662528\pi$
$18$	0	0
$19$	−1.08856	−0.249733	−0.124867	−	0.992174i	$-0.539850\pi$
$19$	−1.08856	−0.249733	−0.124867	+	0.992174i	$0.539850\pi$
$20$	−2.35514	−0.526624
$21$	0	0
$22$	6.00896	1.28111
$23$	7.04757	1.46952	0.734760	−	0.678327i	$-0.237295\pi$
$23$	7.04757	1.46952	0.734760	+	0.678327i	$0.237295\pi$
$24$	0	0
$25$	2.58192	0.516385
$26$	−7.89159	−1.54767
$27$	0	0
$28$	−0.855315	−0.161639
$29$	−9.79449	−1.81879	−0.909395	−	0.415933i	$-0.863455\pi$
$29$	−9.79449	−1.81879	−0.909395	+	0.415933i	$0.863455\pi$
$30$	0	0
$31$	−2.65900	−0.477570	−0.238785	−	0.971072i	$-0.576749\pi$
$31$	−2.65900	−0.477570	−0.238785	+	0.971072i	$0.576749\pi$
$32$	4.54496	0.803443
$33$	0	0
$34$	6.80962	1.16784
$35$	2.75353	0.465431
$36$	0	0
$37$	10.4203	1.71309	0.856543	−	0.516076i	$-0.172608\pi$
$37$	10.4203	1.71309	0.856543	+	0.516076i	$0.172608\pi$
$38$	1.83942	0.298393
$39$	0	0
$40$	−5.32602	−0.842117
$41$	9.96789	1.55672	0.778362	−	0.627816i	$-0.216051\pi$
$41$	9.96789	1.55672	0.778362	+	0.627816i	$0.216051\pi$
$42$	0	0
$43$	−6.65929	−1.01553	−0.507766	−	0.861495i	$-0.669529\pi$
$43$	−6.65929	−1.01553	−0.507766	+	0.861495i	$0.669529\pi$
$44$	−3.04157	−0.458534
$45$	0	0
$46$	−11.9088	−1.75585
$47$	−6.75602	−0.985467	−0.492734	−	0.870180i	$-0.664002\pi$
$47$	−6.75602	−0.985467	−0.492734	+	0.870180i	$0.664002\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−4.36285	−0.617000
$51$	0	0
$52$	3.99451	0.553939
$53$	3.41123	0.468569	0.234284	−	0.972168i	$-0.424725\pi$
$53$	3.41123	0.468569	0.234284	+	0.972168i	$0.424725\pi$
$54$	0	0
$55$	9.79178	1.32032
$56$	−1.93425	−0.258475
$57$	0	0
$58$	16.5504	2.17318
$59$	−8.15705	−1.06196	−0.530979	−	0.847385i	$-0.678176\pi$
$59$	−8.15705	−1.06196	−0.530979	+	0.847385i	$0.678176\pi$
$60$	0	0
$61$	15.2219	1.94896	0.974481	−	0.224471i	$-0.0720653\pi$
$61$	15.2219	1.94896	0.974481	+	0.224471i	$0.0720653\pi$
$62$	4.49309	0.570623
$63$	0	0
$64$	2.27820	0.284775
$65$	−12.8596	−1.59504
$66$	0	0
$67$	−10.7563	−1.31409	−0.657045	−	0.753851i	$-0.728194\pi$
$67$	−10.7563	−1.31409	−0.657045	+	0.753851i	$0.728194\pi$
$68$	−3.44685	−0.417992
$69$	0	0
$70$	−4.65282	−0.556119
$71$	−5.93668	−0.704554	−0.352277	−	0.935896i	$-0.614593\pi$
$71$	−5.93668	−0.704554	−0.352277	+	0.935896i	$0.614593\pi$
$72$	0	0
$73$	8.88205	1.03956	0.519782	−	0.854299i	$-0.326013\pi$
$73$	8.88205	1.03956	0.519782	+	0.854299i	$0.326013\pi$
$74$	−17.6079	−2.04687
$75$	0	0
$76$	−0.931064	−0.106800
$77$	3.55608	0.405253
$78$	0	0
$79$	10.1644	1.14358	0.571792	−	0.820398i	$-0.306248\pi$
$79$	10.1644	1.14358	0.571792	+	0.820398i	$0.306248\pi$
$80$	13.7100	1.53283
$81$	0	0
$82$	−16.8434	−1.86004
$83$	−12.6631	−1.38995	−0.694976	−	0.719033i	$-0.744585\pi$
$83$	−12.6631	−1.38995	−0.694976	+	0.719033i	$0.744585\pi$
$84$	0	0
$85$	11.0965	1.20358
$86$	11.2527	1.21340
$87$	0	0
$88$	−6.87836	−0.733236
$89$	6.83138	0.724124	0.362062	−	0.932154i	$-0.382073\pi$
$89$	6.83138	0.724124	0.362062	+	0.932154i	$0.382073\pi$
$90$	0	0
$91$	−4.67022	−0.489572
$92$	6.02790	0.628452
$93$	0	0
$94$	11.4161	1.17748
$95$	2.99739	0.307526
$96$	0	0
$97$	13.2112	1.34139	0.670696	−	0.741733i	$-0.265996\pi$
$97$	13.2112	1.34139	0.670696	+	0.741733i	$0.265996\pi$
$98$	−1.68977	−0.170692
$99$	0	0
$100$	2.20836	0.220836
$101$	15.2914	1.52156	0.760778	−	0.649013i	$-0.224818\pi$
$101$	15.2914	1.52156	0.760778	+	0.649013i	$0.224818\pi$
$102$	0	0
$103$	−10.3343	−1.01827	−0.509137	−	0.860686i	$-0.670035\pi$
$103$	−10.3343	−1.01827	−0.509137	+	0.860686i	$0.670035\pi$
$104$	9.03338	0.885796
$105$	0	0
$106$	−5.76419	−0.559868
$107$	−10.6111	−1.02582	−0.512909	−	0.858443i	$-0.671432\pi$
$107$	−10.6111	−1.02582	−0.512909	+	0.858443i	$0.671432\pi$
$108$	0	0
$109$	3.64845	0.349458	0.174729	−	0.984617i	$-0.444095\pi$
$109$	3.64845	0.349458	0.174729	+	0.984617i	$0.444095\pi$
$110$	−16.5458	−1.57758
$111$	0	0
$112$	4.97907	0.470478
$113$	−2.75509	−0.259177	−0.129588	−	0.991568i	$-0.541366\pi$
$113$	−2.75509	−0.259177	−0.129588	+	0.991568i	$0.541366\pi$
$114$	0	0
$115$	−19.4057	−1.80959
$116$	−8.37737	−0.777820
$117$	0	0
$118$	13.7835	1.26888
$119$	4.02992	0.369422
$120$	0	0
$121$	1.64574	0.149613
$122$	−25.7214	−2.32871
$123$	0	0
$124$	−2.27428	−0.204236
$125$	6.65825	0.595532
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−12.9396	−1.14371
$129$	0	0
$130$	21.7297	1.90582
$131$	13.2906	1.16120	0.580601	−	0.814188i	$-0.302817\pi$
$131$	13.2906	1.16120	0.580601	+	0.814188i	$0.302817\pi$
$132$	0	0
$133$	1.08856	0.0943903
$134$	18.1756	1.57014
$135$	0	0
$136$	−7.79487	−0.668405
$137$	−8.51182	−0.727214	−0.363607	−	0.931552i	$-0.618455\pi$
$137$	−8.51182	−0.727214	−0.363607	+	0.931552i	$0.618455\pi$
$138$	0	0
$139$	−16.0437	−1.36081	−0.680406	−	0.732836i	$-0.738196\pi$
$139$	−16.0437	−1.36081	−0.680406	+	0.732836i	$0.738196\pi$
$140$	2.35514	0.199045
$141$	0	0
$142$	10.0316	0.841834
$143$	−16.6077	−1.38881
$144$	0	0
$145$	26.9694	2.23969
$146$	−15.0086	−1.24212
$147$	0	0
$148$	8.91263	0.732614
$149$	10.5162	0.861523	0.430762	−	0.902466i	$-0.358245\pi$
$149$	10.5162	0.861523	0.430762	+	0.902466i	$0.358245\pi$
$150$	0	0
$151$	4.75409	0.386882	0.193441	−	0.981112i	$-0.438035\pi$
$151$	4.75409	0.386882	0.193441	+	0.981112i	$0.438035\pi$
$152$	−2.10555	−0.170783
$153$	0	0
$154$	−6.00896	−0.484216
$155$	7.32163	0.588087
$156$	0	0
$157$	−9.39212	−0.749573	−0.374786	−	0.927111i	$-0.622284\pi$
$157$	−9.39212	−0.749573	−0.374786	+	0.927111i	$0.622284\pi$
$158$	−17.1755	−1.36641
$159$	0	0
$160$	−12.5147	−0.989373
$161$	−7.04757	−0.555427
$162$	0	0
$163$	13.8305	1.08329	0.541643	−	0.840609i	$-0.317802\pi$
$163$	13.8305	1.08329	0.541643	+	0.840609i	$0.317802\pi$
$164$	8.52569	0.665744
$165$	0	0
$166$	21.3977	1.66078
$167$	11.9916	0.927938	0.463969	−	0.885851i	$-0.346425\pi$
$167$	11.9916	0.927938	0.463969	+	0.885851i	$0.346425\pi$
$168$	0	0
$169$	8.81097	0.677767
$170$	−18.7505	−1.43810
$171$	0	0
$172$	−5.69579	−0.434300
$173$	11.9449	0.908154	0.454077	−	0.890963i	$-0.349969\pi$
$173$	11.9449	0.908154	0.454077	+	0.890963i	$0.349969\pi$
$174$	0	0
$175$	−2.58192	−0.195175
$176$	17.7060	1.33464
$177$	0	0
$178$	−11.5434	−0.865217
$179$	14.0073	1.04695	0.523477	−	0.852040i	$-0.324634\pi$
$179$	14.0073	1.04695	0.523477	+	0.852040i	$0.324634\pi$
$180$	0	0
$181$	−11.1494	−0.828732	−0.414366	−	0.910110i	$-0.635997\pi$
$181$	−11.1494	−0.828732	−0.414366	+	0.910110i	$0.635997\pi$
$182$	7.89159	0.584964
$183$	0	0
$184$	13.6318	1.00495
$185$	−28.6926	−2.10952
$186$	0	0
$187$	14.3307	1.04797
$188$	−5.77853	−0.421442
$189$	0	0
$190$	−5.06489	−0.367446
$191$	−2.61439	−0.189170	−0.0945851	−	0.995517i	$-0.530152\pi$
$191$	−2.61439	−0.189170	−0.0945851	+	0.995517i	$0.530152\pi$
$192$	0	0
$193$	18.7102	1.34679	0.673393	−	0.739284i	$-0.264836\pi$
$193$	18.7102	1.34679	0.673393	+	0.739284i	$0.264836\pi$
$194$	−22.3238	−1.60276
$195$	0	0
$196$	0.855315	0.0610939
$197$	17.8976	1.27515	0.637574	−	0.770389i	$-0.279938\pi$
$197$	17.8976	1.27515	0.637574	+	0.770389i	$0.279938\pi$
$198$	0	0
$199$	9.53328	0.675796	0.337898	−	0.941183i	$-0.390284\pi$
$199$	9.53328	0.675796	0.337898	+	0.941183i	$0.390284\pi$
$200$	4.99409	0.353135
$201$	0	0
$202$	−25.8390	−1.81802
$203$	9.79449	0.687438
$204$	0	0
$205$	−27.4469	−1.91697
$206$	17.4626	1.21668
$207$	0	0
$208$	−23.2533	−1.61233
$209$	3.87102	0.267764
$210$	0	0
$211$	24.4748	1.68492	0.842459	−	0.538761i	$-0.181107\pi$
$211$	24.4748	1.68492	0.842459	+	0.538761i	$0.181107\pi$
$212$	2.91768	0.200387
$213$	0	0
$214$	17.9304	1.22570
$215$	18.3365	1.25054
$216$	0	0
$217$	2.65900	0.180504
$218$	−6.16504	−0.417549
$219$	0	0
$220$	8.37506	0.564647
$221$	−18.8206	−1.26601
$222$	0	0
$223$	−17.6622	−1.18275	−0.591375	−	0.806397i	$-0.701415\pi$
$223$	−17.6622	−1.18275	−0.591375	+	0.806397i	$0.701415\pi$
$224$	−4.54496	−0.303673
$225$	0	0
$226$	4.65546	0.309677
$227$	−3.42690	−0.227451	−0.113726	−	0.993512i	$-0.536278\pi$
$227$	−3.42690	−0.227451	−0.113726	+	0.993512i	$0.536278\pi$
$228$	0	0
$229$	12.2286	0.808089	0.404044	−	0.914739i	$-0.367604\pi$
$229$	12.2286	0.808089	0.404044	+	0.914739i	$0.367604\pi$
$230$	32.7911	2.16218
$231$	0	0
$232$	−18.9450	−1.24380
$233$	−15.1582	−0.993045	−0.496522	−	0.868024i	$-0.665390\pi$
$233$	−15.1582	−0.993045	−0.496522	+	0.868024i	$0.665390\pi$
$234$	0	0
$235$	18.6029	1.21352
$236$	−6.97685	−0.454154
$237$	0	0
$238$	−6.80962	−0.441402
$239$	18.7940	1.21568	0.607840	−	0.794059i	$-0.292036\pi$
$239$	18.7940	1.21568	0.607840	+	0.794059i	$0.292036\pi$
$240$	0	0
$241$	−13.2751	−0.855127	−0.427564	−	0.903985i	$-0.640628\pi$
$241$	−13.2751	−0.855127	−0.427564	+	0.903985i	$0.640628\pi$
$242$	−2.78092	−0.178764
$243$	0	0
$244$	13.0195	0.833488
$245$	−2.75353	−0.175917
$246$	0	0
$247$	−5.08383	−0.323476
$248$	−5.14317	−0.326592
$249$	0	0
$250$	−11.2509	−0.711569
$251$	11.5852	0.731254	0.365627	−	0.930762i	$-0.380855\pi$
$251$	11.5852	0.731254	0.365627	+	0.930762i	$0.380855\pi$
$252$	0	0
$253$	−25.0618	−1.57562
$254$	−1.68977	−0.106025
$255$	0	0
$256$	17.3084	1.08178
$257$	−9.60457	−0.599117	−0.299558	−	0.954078i	$-0.596839\pi$
$257$	−9.60457	−0.599117	−0.299558	+	0.954078i	$0.596839\pi$
$258$	0	0
$259$	−10.4203	−0.647485
$260$	−10.9990	−0.682129
$261$	0	0
$262$	−22.4580	−1.38746
$263$	−10.9340	−0.674217	−0.337109	−	0.941466i	$-0.609449\pi$
$263$	−10.9340	−0.674217	−0.337109	+	0.941466i	$0.609449\pi$
$264$	0	0
$265$	−9.39293	−0.577003
$266$	−1.83942	−0.112782
$267$	0	0
$268$	−9.20002	−0.561981
$269$	3.61584	0.220462	0.110231	−	0.993906i	$-0.464841\pi$
$269$	3.61584	0.220462	0.110231	+	0.993906i	$0.464841\pi$
$270$	0	0
$271$	−24.6308	−1.49621	−0.748106	−	0.663579i	$-0.769037\pi$
$271$	−24.6308	−1.49621	−0.748106	+	0.663579i	$0.769037\pi$
$272$	20.0652	1.21663
$273$	0	0
$274$	14.3830	0.868909
$275$	−9.18154	−0.553668
$276$	0	0
$277$	−26.9909	−1.62173	−0.810863	−	0.585236i	$-0.801002\pi$
$277$	−26.9909	−1.62173	−0.810863	+	0.585236i	$0.801002\pi$
$278$	27.1102	1.62596
$279$	0	0
$280$	5.32602	0.318290
$281$	−1.18138	−0.0704753	−0.0352376	−	0.999379i	$-0.511219\pi$
$281$	−1.18138	−0.0704753	−0.0352376	+	0.999379i	$0.511219\pi$
$282$	0	0
$283$	−11.6645	−0.693380	−0.346690	−	0.937980i	$-0.612694\pi$
$283$	−11.6645	−0.693380	−0.346690	+	0.937980i	$0.612694\pi$
$284$	−5.07773	−0.301308
$285$	0	0
$286$	28.0632	1.65941
$287$	−9.96789	−0.588386
$288$	0	0
$289$	−0.759770	−0.0446924
$290$	−45.5720	−2.67608
$291$	0	0
$292$	7.59695	0.444578
$293$	33.8712	1.97877	0.989387	−	0.145305i	$-0.0464163\pi$
$293$	33.8712	1.97877	0.989387	+	0.145305i	$0.0464163\pi$
$294$	0	0
$295$	22.4607	1.30771
$296$	20.1555	1.17151
$297$	0	0
$298$	−17.7700	−1.02939
$299$	32.9137	1.90345
$300$	0	0
$301$	6.65929	0.383835
$302$	−8.03330	−0.462264
$303$	0	0
$304$	5.42002	0.310860
$305$	−41.9139	−2.39998
$306$	0	0
$307$	−16.9237	−0.965888	−0.482944	−	0.875651i	$-0.660433\pi$
$307$	−16.9237	−0.965888	−0.482944	+	0.875651i	$0.660433\pi$
$308$	3.04157	0.173310
$309$	0	0
$310$	−12.3718	−0.702674
$311$	−26.5396	−1.50492	−0.752462	−	0.658636i	$-0.771134\pi$
$311$	−26.5396	−1.50492	−0.752462	+	0.658636i	$0.771134\pi$
$312$	0	0
$313$	−14.9423	−0.844590	−0.422295	−	0.906458i	$-0.638775\pi$
$313$	−14.9423	−0.844590	−0.422295	+	0.906458i	$0.638775\pi$
$314$	15.8705	0.895624
$315$	0	0
$316$	8.69377	0.489063
$317$	15.1302	0.849797	0.424899	−	0.905241i	$-0.360310\pi$
$317$	15.1302	0.849797	0.424899	+	0.905241i	$0.360310\pi$
$318$	0	0
$319$	34.8300	1.95011
$320$	−6.27309	−0.350677
$321$	0	0
$322$	11.9088	0.663649
$323$	4.38682	0.244089
$324$	0	0
$325$	12.0582	0.668866
$326$	−23.3703	−1.29436
$327$	0	0
$328$	19.2804	1.06458
$329$	6.75602	0.372472
$330$	0	0
$331$	6.77045	0.372138	0.186069	−	0.982537i	$-0.440425\pi$
$331$	6.77045	0.372138	0.186069	+	0.982537i	$0.440425\pi$
$332$	−10.8309	−0.594424
$333$	0	0
$334$	−20.2630	−1.10874
$335$	29.6178	1.61819
$336$	0	0
$337$	−6.04309	−0.329188	−0.164594	−	0.986361i	$-0.552631\pi$
$337$	−6.04309	−0.329188	−0.164594	+	0.986361i	$0.552631\pi$
$338$	−14.8885	−0.809827
$339$	0	0
$340$	9.49100	0.514722
$341$	9.45562	0.512050
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−12.8807	−0.694483
$345$	0	0
$346$	−20.1841	−1.08510
$347$	−1.21237	−0.0650834	−0.0325417	−	0.999470i	$-0.510360\pi$
$347$	−1.21237	−0.0650834	−0.0325417	+	0.999470i	$0.510360\pi$
$348$	0	0
$349$	−12.8685	−0.688838	−0.344419	−	0.938816i	$-0.611924\pi$
$349$	−12.8685	−0.688838	−0.344419	+	0.938816i	$0.611924\pi$
$350$	4.36285	0.233204
$351$	0	0
$352$	−16.1623	−0.861452
$353$	−10.7175	−0.570433	−0.285216	−	0.958463i	$-0.592065\pi$
$353$	−10.7175	−0.570433	−0.285216	+	0.958463i	$0.592065\pi$
$354$	0	0
$355$	16.3468	0.867599
$356$	5.84298	0.309677
$357$	0	0
$358$	−23.6691	−1.25095
$359$	−16.7933	−0.886315	−0.443158	−	0.896444i	$-0.646142\pi$
$359$	−16.7933	−0.886315	−0.443158	+	0.896444i	$0.646142\pi$
$360$	0	0
$361$	−17.8150	−0.937633
$362$	18.8400	0.990207
$363$	0	0
$364$	−3.99451	−0.209369
$365$	−24.4570	−1.28014
$366$	0	0
$367$	17.5414	0.915655	0.457828	−	0.889041i	$-0.348628\pi$
$367$	17.5414	0.915655	0.457828	+	0.889041i	$0.348628\pi$
$368$	−35.0903	−1.82921
$369$	0	0
$370$	48.4838	2.52055
$371$	−3.41123	−0.177102
$372$	0	0
$373$	33.3744	1.72806	0.864030	−	0.503440i	$-0.167932\pi$
$373$	33.3744	1.72806	0.864030	+	0.503440i	$0.167932\pi$
$374$	−24.2156	−1.25216
$375$	0	0
$376$	−13.0678	−0.673923
$377$	−45.7424	−2.35585
$378$	0	0
$379$	26.3319	1.35258	0.676289	−	0.736637i	$-0.263587\pi$
$379$	26.3319	1.35258	0.676289	+	0.736637i	$0.263587\pi$
$380$	2.56371	0.131516
$381$	0	0
$382$	4.41770	0.226029
$383$	11.7876	0.602320	0.301160	−	0.953574i	$-0.402626\pi$
$383$	11.7876	0.602320	0.301160	+	0.953574i	$0.402626\pi$
$384$	0	0
$385$	−9.79178	−0.499035
$386$	−31.6158	−1.60920
$387$	0	0
$388$	11.2997	0.573656
$389$	9.32061	0.472573	0.236287	−	0.971683i	$-0.424070\pi$
$389$	9.32061	0.472573	0.236287	+	0.971683i	$0.424070\pi$
$390$	0	0
$391$	−28.4011	−1.43631
$392$	1.93425	0.0976945
$393$	0	0
$394$	−30.2427	−1.52361
$395$	−27.9880	−1.40823
$396$	0	0
$397$	17.5856	0.882594	0.441297	−	0.897361i	$-0.354519\pi$
$397$	17.5856	0.882594	0.441297	+	0.897361i	$0.354519\pi$
$398$	−16.1090	−0.807473
$399$	0	0
$400$	−12.8556	−0.642778
$401$	−7.80332	−0.389679	−0.194840	−	0.980835i	$-0.562419\pi$
$401$	−7.80332	−0.389679	−0.194840	+	0.980835i	$0.562419\pi$
$402$	0	0
$403$	−12.4181	−0.618590
$404$	13.0790	0.650705
$405$	0	0
$406$	−16.5504	−0.821383
$407$	−37.0554	−1.83677
$408$	0	0
$409$	−34.3626	−1.69912	−0.849562	−	0.527489i	$-0.823134\pi$
$409$	−34.3626	−1.69912	−0.849562	+	0.527489i	$0.823134\pi$
$410$	46.3789	2.29049
$411$	0	0
$412$	−8.83912	−0.435472
$413$	8.15705	0.401382
$414$	0	0
$415$	34.8681	1.71161
$416$	21.2260	1.04069
$417$	0	0
$418$	−6.54113	−0.319937
$419$	15.4176	0.753200	0.376600	−	0.926376i	$-0.377093\pi$
$419$	15.4176	0.753200	0.376600	+	0.926376i	$0.377093\pi$
$420$	0	0
$421$	−24.4173	−1.19003	−0.595014	−	0.803716i	$-0.702853\pi$
$421$	−24.4173	−1.19003	−0.595014	+	0.803716i	$0.702853\pi$
$422$	−41.3568	−2.01322
$423$	0	0
$424$	6.59818	0.320436
$425$	−10.4049	−0.504713
$426$	0	0
$427$	−15.2219	−0.736638
$428$	−9.07587	−0.438699
$429$	0	0
$430$	−30.9845	−1.49421
$431$	7.47016	0.359825	0.179913	−	0.983683i	$-0.442419\pi$
$431$	7.47016	0.359825	0.179913	+	0.983683i	$0.442419\pi$
$432$	0	0
$433$	−32.3778	−1.55598	−0.777989	−	0.628278i	$-0.783760\pi$
$433$	−32.3778	−1.55598	−0.777989	+	0.628278i	$0.783760\pi$
$434$	−4.49309	−0.215675
$435$	0	0
$436$	3.12058	0.149448
$437$	−7.67172	−0.366988
$438$	0	0
$439$	−1.21369	−0.0579261	−0.0289630	−	0.999580i	$-0.509221\pi$
$439$	−1.21369	−0.0579261	−0.0289630	+	0.999580i	$0.509221\pi$
$440$	18.9398	0.902918
$441$	0	0
$442$	31.8024	1.51269
$443$	−13.3878	−0.636075	−0.318038	−	0.948078i	$-0.603024\pi$
$443$	−13.3878	−0.636075	−0.318038	+	0.948078i	$0.603024\pi$
$444$	0	0
$445$	−18.8104	−0.891698
$446$	29.8451	1.41320
$447$	0	0
$448$	−2.27820	−0.107635
$449$	9.25603	0.436819	0.218410	−	0.975857i	$-0.429913\pi$
$449$	9.25603	0.436819	0.218410	+	0.975857i	$0.429913\pi$
$450$	0	0
$451$	−35.4467	−1.66912
$452$	−2.35647	−0.110839
$453$	0	0
$454$	5.79066	0.271769
$455$	12.8596	0.602867
$456$	0	0
$457$	−30.3276	−1.41866	−0.709332	−	0.704875i	$-0.751003\pi$
$457$	−30.3276	−1.41866	−0.709332	+	0.704875i	$0.751003\pi$
$458$	−20.6635	−0.965542
$459$	0	0
$460$	−16.5980	−0.773885
$461$	4.21066	0.196110	0.0980549	−	0.995181i	$-0.468738\pi$
$461$	4.21066	0.196110	0.0980549	+	0.995181i	$0.468738\pi$
$462$	0	0
$463$	5.86507	0.272573	0.136286	−	0.990669i	$-0.456483\pi$
$463$	5.86507	0.272573	0.136286	+	0.990669i	$0.456483\pi$
$464$	48.7674	2.26397
$465$	0	0
$466$	25.6138	1.18654
$467$	−23.3865	−1.08220	−0.541100	−	0.840958i	$-0.681992\pi$
$467$	−23.3865	−1.08220	−0.541100	+	0.840958i	$0.681992\pi$
$468$	0	0
$469$	10.7563	0.496679
$470$	−31.4346	−1.44997
$471$	0	0
$472$	−15.7778	−0.726231
$473$	23.6810	1.08885
$474$	0	0
$475$	−2.81058	−0.128958
$476$	3.44685	0.157986
$477$	0	0
$478$	−31.7574	−1.45255
$479$	−34.8129	−1.59064	−0.795322	−	0.606188i	$-0.792698\pi$
$479$	−34.8129	−1.59064	−0.795322	+	0.606188i	$0.792698\pi$
$480$	0	0
$481$	48.6651	2.21894
$482$	22.4319	1.02175
$483$	0	0
$484$	1.40763	0.0639830
$485$	−36.3773	−1.65181
$486$	0	0
$487$	−38.0221	−1.72295	−0.861473	−	0.507804i	$-0.830457\pi$
$487$	−38.0221	−1.72295	−0.861473	+	0.507804i	$0.830457\pi$
$488$	29.4429	1.33282
$489$	0	0
$490$	4.65282	0.210193
$491$	−19.5076	−0.880367	−0.440183	−	0.897908i	$-0.645087\pi$
$491$	−19.5076	−0.880367	−0.440183	+	0.897908i	$0.645087\pi$
$492$	0	0
$493$	39.4710	1.77768
$494$	8.59049	0.386504
$495$	0	0
$496$	13.2393	0.594463
$497$	5.93668	0.266297
$498$	0	0
$499$	−36.0759	−1.61498	−0.807489	−	0.589883i	$-0.799174\pi$
$499$	−36.0759	−1.61498	−0.807489	+	0.589883i	$0.799174\pi$
$500$	5.69490	0.254684
$501$	0	0
$502$	−19.5764	−0.873736
$503$	9.82521	0.438085	0.219042	−	0.975715i	$-0.429707\pi$
$503$	9.82521	0.438085	0.219042	+	0.975715i	$0.429707\pi$
$504$	0	0
$505$	−42.1054	−1.87367
$506$	42.3486	1.88262
$507$	0	0
$508$	0.855315	0.0379485
$509$	−15.0794	−0.668381	−0.334191	−	0.942505i	$-0.608463\pi$
$509$	−15.0794	−0.668381	−0.334191	+	0.942505i	$0.608463\pi$
$510$	0	0
$511$	−8.88205	−0.392919
$512$	−3.36814	−0.148852
$513$	0	0
$514$	16.2295	0.715852
$515$	28.4559	1.25392
$516$	0	0
$517$	24.0250	1.05662
$518$	17.6079	0.773646
$519$	0	0
$520$	−24.8737	−1.09078
$521$	20.2909	0.888961	0.444481	−	0.895789i	$-0.353388\pi$
$521$	20.2909	0.888961	0.444481	+	0.895789i	$0.353388\pi$
$522$	0	0
$523$	−5.13571	−0.224569	−0.112285	−	0.993676i	$-0.535817\pi$
$523$	−5.13571	−0.224569	−0.112285	+	0.993676i	$0.535817\pi$
$524$	11.3676	0.496597
$525$	0	0
$526$	18.4759	0.805586
$527$	10.7155	0.466776
$528$	0	0
$529$	26.6683	1.15949
$530$	15.8719	0.689430
$531$	0	0
$532$	0.931064	0.0403667
$533$	46.5523	2.01640
$534$	0	0
$535$	29.2181	1.26321
$536$	−20.8054	−0.898655
$537$	0	0
$538$	−6.10994	−0.263418
$539$	−3.55608	−0.153171
$540$	0	0
$541$	−39.4878	−1.69771	−0.848856	−	0.528625i	$-0.822708\pi$
$541$	−39.4878	−1.69771	−0.848856	+	0.528625i	$0.822708\pi$
$542$	41.6203	1.78774
$543$	0	0
$544$	−18.3158	−0.785284
$545$	−10.0461	−0.430328
$546$	0	0
$547$	18.7996	0.803813	0.401907	−	0.915681i	$-0.368348\pi$
$547$	18.7996	0.803813	0.401907	+	0.915681i	$0.368348\pi$
$548$	−7.28029	−0.310999
$549$	0	0
$550$	15.5147	0.661548
$551$	10.6619	0.454213
$552$	0	0
$553$	−10.1644	−0.432234
$554$	45.6083	1.93771
$555$	0	0
$556$	−13.7224	−0.581961
$557$	−10.6884	−0.452881	−0.226440	−	0.974025i	$-0.572709\pi$
$557$	−10.6884	−0.452881	−0.226440	+	0.974025i	$0.572709\pi$
$558$	0	0
$559$	−31.1003	−1.31540
$560$	−13.7100	−0.579353
$561$	0	0
$562$	1.99626	0.0842071
$563$	27.2454	1.14826	0.574129	−	0.818765i	$-0.305341\pi$
$563$	27.2454	1.14826	0.574129	+	0.818765i	$0.305341\pi$
$564$	0	0
$565$	7.58622	0.319155
$566$	19.7102	0.828482
$567$	0	0
$568$	−11.4830	−0.481817
$569$	20.3136	0.851592	0.425796	−	0.904819i	$-0.359994\pi$
$569$	20.3136	0.851592	0.425796	+	0.904819i	$0.359994\pi$
$570$	0	0
$571$	−1.89715	−0.0793931	−0.0396966	−	0.999212i	$-0.512639\pi$
$571$	−1.89715	−0.0793931	−0.0396966	+	0.999212i	$0.512639\pi$
$572$	−14.2048	−0.593933
$573$	0	0
$574$	16.8434	0.703031
$575$	18.1963	0.758838
$576$	0	0
$577$	7.62020	0.317233	0.158617	−	0.987340i	$-0.449297\pi$
$577$	7.62020	0.317233	0.158617	+	0.987340i	$0.449297\pi$
$578$	1.28384	0.0534005
$579$	0	0
$580$	23.0673	0.957819
$581$	12.6631	0.525353
$582$	0	0
$583$	−12.1306	−0.502399
$584$	17.1801	0.710918
$585$	0	0
$586$	−57.2344	−2.36433
$587$	−19.7884	−0.816754	−0.408377	−	0.912813i	$-0.633905\pi$
$587$	−19.7884	−0.816754	−0.408377	+	0.912813i	$0.633905\pi$
$588$	0	0
$589$	2.89448	0.119265
$590$	−37.9533	−1.56251
$591$	0	0
$592$	−51.8833	−2.13239
$593$	−31.9458	−1.31186	−0.655928	−	0.754823i	$-0.727723\pi$
$593$	−31.9458	−1.31186	−0.655928	+	0.754823i	$0.727723\pi$
$594$	0	0
$595$	−11.0965	−0.454912
$596$	8.99469	0.368437
$597$	0	0
$598$	−55.6166	−2.27433
$599$	−3.32792	−0.135975	−0.0679876	−	0.997686i	$-0.521658\pi$
$599$	−3.32792	−0.135975	−0.0679876	+	0.997686i	$0.521658\pi$
$600$	0	0
$601$	−44.7853	−1.82683	−0.913415	−	0.407030i	$-0.866565\pi$
$601$	−44.7853	−1.82683	−0.913415	+	0.407030i	$0.866565\pi$
$602$	−11.2527	−0.458624
$603$	0	0
$604$	4.06624	0.165453
$605$	−4.53159	−0.184235
$606$	0	0
$607$	5.47481	0.222216	0.111108	−	0.993808i	$-0.464560\pi$
$607$	5.47481	0.222216	0.111108	+	0.993808i	$0.464560\pi$
$608$	−4.94748	−0.200647
$609$	0	0
$610$	70.8247	2.86761
$611$	−31.5521	−1.27646
$612$	0	0
$613$	20.5636	0.830556	0.415278	−	0.909695i	$-0.363684\pi$
$613$	20.5636	0.830556	0.415278	+	0.909695i	$0.363684\pi$
$614$	28.5972	1.15409
$615$	0	0
$616$	6.87836	0.277137
$617$	−14.8686	−0.598586	−0.299293	−	0.954161i	$-0.596751\pi$
$617$	−14.8686	−0.598586	−0.299293	+	0.954161i	$0.596751\pi$
$618$	0	0
$619$	−11.5606	−0.464658	−0.232329	−	0.972637i	$-0.574635\pi$
$619$	−11.5606	−0.464658	−0.232329	+	0.972637i	$0.574635\pi$
$620$	6.26230	0.251500
$621$	0	0
$622$	44.8458	1.79815
$623$	−6.83138	−0.273693
$624$	0	0
$625$	−31.2433	−1.24973
$626$	25.2491	1.00916
$627$	0	0
$628$	−8.03322	−0.320561
$629$	−41.9929	−1.67437
$630$	0	0
$631$	−30.9792	−1.23326	−0.616632	−	0.787252i	$-0.711503\pi$
$631$	−30.9792	−1.23326	−0.616632	+	0.787252i	$0.711503\pi$
$632$	19.6605	0.782053
$633$	0	0
$634$	−25.5665	−1.01538
$635$	−2.75353	−0.109270
$636$	0	0
$637$	4.67022	0.185041
$638$	−58.8547	−2.33008
$639$	0	0
$640$	35.6294	1.40838
$641$	−18.0424	−0.712630	−0.356315	−	0.934366i	$-0.615967\pi$
$641$	−18.0424	−0.712630	−0.356315	+	0.934366i	$0.615967\pi$
$642$	0	0
$643$	1.58037	0.0623236	0.0311618	−	0.999514i	$-0.490079\pi$
$643$	1.58037	0.0623236	0.0311618	+	0.999514i	$0.490079\pi$
$644$	−6.02790	−0.237532
$645$	0	0
$646$	−7.41270	−0.291649
$647$	7.03625	0.276624	0.138312	−	0.990389i	$-0.455832\pi$
$647$	7.03625	0.276624	0.138312	+	0.990389i	$0.455832\pi$
$648$	0	0
$649$	29.0072	1.13863
$650$	−20.3755	−0.799192
$651$	0	0
$652$	11.8294	0.463276
$653$	40.7136	1.59325	0.796624	−	0.604475i	$-0.206617\pi$
$653$	40.7136	1.59325	0.796624	+	0.604475i	$0.206617\pi$
$654$	0	0
$655$	−36.5960	−1.42992
$656$	−49.6308	−1.93776
$657$	0	0
$658$	−11.4161	−0.445046
$659$	24.3058	0.946819	0.473409	−	0.880843i	$-0.343023\pi$
$659$	24.3058	0.946819	0.473409	+	0.880843i	$0.343023\pi$
$660$	0	0
$661$	−33.1412	−1.28904	−0.644522	−	0.764586i	$-0.722943\pi$
$661$	−33.1412	−1.28904	−0.644522	+	0.764586i	$0.722943\pi$
$662$	−11.4405	−0.444648
$663$	0	0
$664$	−24.4936	−0.950535
$665$	−2.99739	−0.116234
$666$	0	0
$667$	−69.0274	−2.67275
$668$	10.2566	0.396840
$669$	0	0
$670$	−50.0472	−1.93349
$671$	−54.1303	−2.08968
$672$	0	0
$673$	24.6627	0.950675	0.475338	−	0.879803i	$-0.342326\pi$
$673$	24.6627	0.950675	0.475338	+	0.879803i	$0.342326\pi$
$674$	10.2114	0.393329
$675$	0	0
$676$	7.53615	0.289852
$677$	0.0186713	0.000717598 0	0.000358799	−	1.00000i	$-0.499886\pi$
$677$	0.0186713	0.000717598 0	0.000358799		1.00000i	$0.499886\pi$
$678$	0	0
$679$	−13.2112	−0.506998
$680$	21.4634	0.823084
$681$	0	0
$682$	−15.9778	−0.611822
$683$	−47.4850	−1.81696	−0.908482	−	0.417924i	$-0.862758\pi$
$683$	−47.4850	−1.81696	−0.908482	+	0.417924i	$0.862758\pi$
$684$	0	0
$685$	23.4376	0.895503
$686$	1.68977	0.0645156
$687$	0	0
$688$	33.1570	1.26410
$689$	15.9312	0.606931
$690$	0	0
$691$	32.4728	1.23532	0.617662	−	0.786444i	$-0.288080\pi$
$691$	32.4728	1.23532	0.617662	+	0.786444i	$0.288080\pi$
$692$	10.2166	0.388379
$693$	0	0
$694$	2.04862	0.0777646
$695$	44.1769	1.67572
$696$	0	0
$697$	−40.1698	−1.52154
$698$	21.7449	0.823055
$699$	0	0
$700$	−2.20836	−0.0834681
$701$	−27.9667	−1.05629	−0.528143	−	0.849155i	$-0.677112\pi$
$701$	−27.9667	−1.05629	−0.528143	+	0.849155i	$0.677112\pi$
$702$	0	0
$703$	−11.3431	−0.427815
$704$	−8.10148	−0.305336
$705$	0	0
$706$	18.1100	0.681579
$707$	−15.2914	−0.575094
$708$	0	0
$709$	43.0859	1.61813	0.809063	−	0.587721i	$-0.199975\pi$
$709$	43.0859	1.61813	0.809063	+	0.587721i	$0.199975\pi$
$710$	−27.6223	−1.03665
$711$	0	0
$712$	13.2136	0.495201
$713$	−18.7395	−0.701799
$714$	0	0
$715$	45.7298	1.71020
$716$	11.9807	0.447738
$717$	0	0
$718$	28.3767	1.05901
$719$	9.46438	0.352962	0.176481	−	0.984304i	$-0.443529\pi$
$719$	9.46438	0.352962	0.176481	+	0.984304i	$0.443529\pi$
$720$	0	0
$721$	10.3343	0.384871
$722$	30.1033	1.12033
$723$	0	0
$724$	−9.53629	−0.354413
$725$	−25.2886	−0.939195
$726$	0	0
$727$	18.8916	0.700650	0.350325	−	0.936628i	$-0.386071\pi$
$727$	18.8916	0.700650	0.350325	+	0.936628i	$0.386071\pi$
$728$	−9.03338	−0.334799
$729$	0	0
$730$	41.3266	1.52957
$731$	26.8364	0.992579
$732$	0	0
$733$	23.0268	0.850516	0.425258	−	0.905072i	$-0.360183\pi$
$733$	23.0268	0.850516	0.425258	+	0.905072i	$0.360183\pi$
$734$	−29.6409	−1.09407
$735$	0	0
$736$	32.0310	1.18068
$737$	38.2503	1.40897
$738$	0	0
$739$	−9.25786	−0.340556	−0.170278	−	0.985396i	$-0.554467\pi$
$739$	−9.25786	−0.340556	−0.170278	+	0.985396i	$0.554467\pi$
$740$	−24.5412	−0.902152
$741$	0	0
$742$	5.76419	0.211610
$743$	−11.8065	−0.433139	−0.216569	−	0.976267i	$-0.569487\pi$
$743$	−11.8065	−0.433139	−0.216569	+	0.976267i	$0.569487\pi$
$744$	0	0
$745$	−28.9568	−1.06089
$746$	−56.3950	−2.06477
$747$	0	0
$748$	12.2573	0.448171
$749$	10.6111	0.387723
$750$	0	0
$751$	28.4680	1.03881	0.519406	−	0.854528i	$-0.326153\pi$
$751$	28.4680	1.03881	0.519406	+	0.854528i	$0.326153\pi$
$752$	33.6387	1.22668
$753$	0	0
$754$	77.2941	2.81488
$755$	−13.0905	−0.476413
$756$	0	0
$757$	−29.7893	−1.08271	−0.541356	−	0.840794i	$-0.682089\pi$
$757$	−29.7893	−1.08271	−0.541356	+	0.840794i	$0.682089\pi$
$758$	−44.4948	−1.61612
$759$	0	0
$760$	5.79770	0.210305
$761$	−22.1044	−0.801285	−0.400642	−	0.916235i	$-0.631213\pi$
$761$	−22.1044	−0.801285	−0.400642	+	0.916235i	$0.631213\pi$
$762$	0	0
$763$	−3.64845	−0.132083
$764$	−2.23612	−0.0809001
$765$	0	0
$766$	−19.9184	−0.719680
$767$	−38.0952	−1.37554
$768$	0	0
$769$	2.55777	0.0922357	0.0461179	−	0.998936i	$-0.485315\pi$
$769$	2.55777	0.0922357	0.0461179	+	0.998936i	$0.485315\pi$
$770$	16.5458	0.596271
$771$	0	0
$772$	16.0031	0.575964
$773$	10.8724	0.391052	0.195526	−	0.980699i	$-0.437359\pi$
$773$	10.8724	0.391052	0.195526	+	0.980699i	$0.437359\pi$
$774$	0	0
$775$	−6.86533	−0.246610
$776$	25.5537	0.917325
$777$	0	0
$778$	−15.7497	−0.564653
$779$	−10.8507	−0.388766
$780$	0	0
$781$	21.1113	0.755423
$782$	47.9913	1.71617
$783$	0	0
$784$	−4.97907	−0.177824
$785$	25.8615	0.923036
$786$	0	0
$787$	−45.1429	−1.60917	−0.804586	−	0.593837i	$-0.797613\pi$
$787$	−45.1429	−1.60917	−0.804586	+	0.593837i	$0.797613\pi$
$788$	15.3081	0.545327
$789$	0	0
$790$	47.2932	1.68262
$791$	2.75509	0.0979597
$792$	0	0
$793$	71.0895	2.52446
$794$	−29.7155	−1.05456
$795$	0	0
$796$	8.15396	0.289009
$797$	26.6371	0.943534	0.471767	−	0.881723i	$-0.343616\pi$
$797$	26.6371	0.943534	0.471767	+	0.881723i	$0.343616\pi$
$798$	0	0
$799$	27.2262	0.963194
$800$	11.7347	0.414886
$801$	0	0
$802$	13.1858	0.465607
$803$	−31.5853	−1.11462
$804$	0	0
$805$	19.4057	0.683961
$806$	20.9837	0.739120
$807$	0	0
$808$	29.5775	1.04053
$809$	−37.2858	−1.31090	−0.655449	−	0.755239i	$-0.727521\pi$
$809$	−37.2858	−1.31090	−0.655449	+	0.755239i	$0.727521\pi$
$810$	0	0
$811$	−46.7113	−1.64025	−0.820127	−	0.572181i	$-0.806097\pi$
$811$	−46.7113	−1.64025	−0.820127	+	0.572181i	$0.806097\pi$
$812$	8.37737	0.293988
$813$	0	0
$814$	62.6151	2.19466
$815$	−38.0826	−1.33398
$816$	0	0
$817$	7.24905	0.253612
$818$	58.0649	2.03019
$819$	0	0
$820$	−23.4757	−0.819808
$821$	22.2468	0.776418	0.388209	−	0.921571i	$-0.373094\pi$
$821$	22.2468	0.776418	0.388209	+	0.921571i	$0.373094\pi$
$822$	0	0
$823$	25.9321	0.903936	0.451968	−	0.892034i	$-0.350722\pi$
$823$	25.9321	0.903936	0.451968	+	0.892034i	$0.350722\pi$
$824$	−19.9892	−0.696358
$825$	0	0
$826$	−13.7835	−0.479590
$827$	−13.2579	−0.461021	−0.230511	−	0.973070i	$-0.574040\pi$
$827$	−13.2579	−0.461021	−0.230511	+	0.973070i	$0.574040\pi$
$828$	0	0
$829$	28.7272	0.997737	0.498868	−	0.866678i	$-0.333749\pi$
$829$	28.7272	0.997737	0.498868	+	0.866678i	$0.333749\pi$
$830$	−58.9191	−2.04511
$831$	0	0
$832$	10.6397	0.368865
$833$	−4.02992	−0.139628
$834$	0	0
$835$	−33.0192	−1.14268
$836$	3.31094	0.114511
$837$	0	0
$838$	−26.0522	−0.899958
$839$	−0.285401	−0.00985315	−0.00492657	−	0.999988i	$-0.501568\pi$
$839$	−0.285401	−0.00985315	−0.00492657	+	0.999988i	$0.501568\pi$
$840$	0	0
$841$	66.9320	2.30800
$842$	41.2596	1.42190
$843$	0	0
$844$	20.9337	0.720568
$845$	−24.2613	−0.834612
$846$	0	0
$847$	−1.64574	−0.0565483
$848$	−16.9847	−0.583259
$849$	0	0
$850$	17.5819	0.603055
$851$	73.4378	2.51741
$852$	0	0
$853$	−13.2862	−0.454910	−0.227455	−	0.973789i	$-0.573040\pi$
$853$	−13.2862	−0.454910	−0.227455	+	0.973789i	$0.573040\pi$
$854$	25.7214	0.880170
$855$	0	0
$856$	−20.5246	−0.701517
$857$	−14.2129	−0.485504	−0.242752	−	0.970088i	$-0.578050\pi$
$857$	−14.2129	−0.485504	−0.242752	+	0.970088i	$0.578050\pi$
$858$	0	0
$859$	33.9814	1.15943	0.579715	−	0.814820i	$-0.303164\pi$
$859$	33.9814	1.15943	0.579715	+	0.814820i	$0.303164\pi$
$860$	15.6835	0.534804
$861$	0	0
$862$	−12.6228	−0.429936
$863$	52.2183	1.77753	0.888765	−	0.458363i	$-0.151564\pi$
$863$	52.2183	1.77753	0.888765	+	0.458363i	$0.151564\pi$
$864$	0	0
$865$	−32.8906	−1.11831
$866$	54.7110	1.85915
$867$	0	0
$868$	2.27428	0.0771941
$869$	−36.1455	−1.22615
$870$	0	0
$871$	−50.2343	−1.70212
$872$	7.05702	0.238981
$873$	0	0
$874$	12.9634	0.438495
$875$	−6.65825	−0.225090
$876$	0	0
$877$	1.77493	0.0599352	0.0299676	−	0.999551i	$-0.490460\pi$
$877$	1.77493	0.0599352	0.0299676	+	0.999551i	$0.490460\pi$
$878$	2.05085	0.0692128
$879$	0	0
$880$	−48.7539	−1.64349
$881$	−18.3924	−0.619656	−0.309828	−	0.950793i	$-0.600271\pi$
$881$	−18.3924	−0.619656	−0.309828	+	0.950793i	$0.600271\pi$
$882$	0	0
$883$	12.2614	0.412628	0.206314	−	0.978486i	$-0.433853\pi$
$883$	12.2614	0.412628	0.206314	+	0.978486i	$0.433853\pi$
$884$	−16.0975	−0.541419
$885$	0	0
$886$	22.6223	0.760012
$887$	45.4442	1.52587	0.762933	−	0.646478i	$-0.223759\pi$
$887$	45.4442	1.52587	0.762933	+	0.646478i	$0.223759\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	31.7852	1.06544
$891$	0	0
$892$	−15.1068	−0.505812
$893$	7.35435	0.246104
$894$	0	0
$895$	−38.5695	−1.28924
$896$	12.9396	0.432280
$897$	0	0
$898$	−15.6405	−0.521932
$899$	26.0435	0.868600
$900$	0	0
$901$	−13.7470	−0.457978
$902$	59.8966	1.99434
$903$	0	0
$904$	−5.32903	−0.177241
$905$	30.7003	1.02051
$906$	0	0
$907$	−25.9990	−0.863283	−0.431642	−	0.902045i	$-0.642065\pi$
$907$	−25.9990	−0.863283	−0.431642	+	0.902045i	$0.642065\pi$
$908$	−2.93108	−0.0972712
$909$	0	0
$910$	−21.7297	−0.720333
$911$	42.6190	1.41203	0.706015	−	0.708197i	$-0.250491\pi$
$911$	42.6190	1.41203	0.706015	+	0.708197i	$0.250491\pi$
$912$	0	0
$913$	45.0310	1.49031
$914$	51.2466	1.69509
$915$	0	0
$916$	10.4593	0.345585
$917$	−13.2906	−0.438893
$918$	0	0
$919$	−15.7378	−0.519142	−0.259571	−	0.965724i	$-0.583581\pi$
$919$	−15.7378	−0.519142	−0.259571	+	0.965724i	$0.583581\pi$
$920$	−37.5355	−1.23751
$921$	0	0
$922$	−7.11503	−0.234321
$923$	−27.7256	−0.912600
$924$	0	0
$925$	26.9044	0.884611
$926$	−9.91060	−0.325683
$927$	0	0
$928$	−44.5156	−1.46130
$929$	−20.9070	−0.685935	−0.342968	−	0.939347i	$-0.611432\pi$
$929$	−20.9070	−0.685935	−0.342968	+	0.939347i	$0.611432\pi$
$930$	0	0
$931$	−1.08856	−0.0356762
$932$	−12.9650	−0.424683
$933$	0	0
$934$	39.5178	1.29306
$935$	−39.4601	−1.29048
$936$	0	0
$937$	−55.9490	−1.82778	−0.913888	−	0.405967i	$-0.866935\pi$
$937$	−55.9490	−1.82778	−0.913888	+	0.405967i	$0.866935\pi$
$938$	−18.1756	−0.593456
$939$	0	0
$940$	15.9113	0.518971
$941$	−50.9901	−1.66223	−0.831115	−	0.556101i	$-0.812297\pi$
$941$	−50.9901	−1.66223	−0.831115	+	0.556101i	$0.812297\pi$
$942$	0	0
$943$	70.2495	2.28764
$944$	40.6145	1.32189
$945$	0	0
$946$	−40.0154	−1.30101
$947$	6.27706	0.203977	0.101989	−	0.994786i	$-0.467479\pi$
$947$	6.27706	0.203977	0.101989	+	0.994786i	$0.467479\pi$
$948$	0	0
$949$	41.4811	1.34653
$950$	4.74923	0.154086
$951$	0	0
$952$	7.79487	0.252633
$953$	−22.9897	−0.744710	−0.372355	−	0.928090i	$-0.621450\pi$
$953$	−22.9897	−0.744710	−0.372355	+	0.928090i	$0.621450\pi$
$954$	0	0
$955$	7.19879	0.232947
$956$	16.0748	0.519895
$957$	0	0
$958$	58.8258	1.90057
$959$	8.51182	0.274861
$960$	0	0
$961$	−23.9297	−0.771927
$962$	−82.2327	−2.65129
$963$	0	0
$964$	−11.3544	−0.365702
$965$	−51.5190	−1.65845
$966$	0	0
$967$	34.8738	1.12147	0.560733	−	0.827997i	$-0.310520\pi$
$967$	34.8738	1.12147	0.560733	+	0.827997i	$0.310520\pi$
$968$	3.18328	0.102314
$969$	0	0
$970$	61.4693	1.97366
$971$	−44.1849	−1.41796	−0.708980	−	0.705228i	$-0.750845\pi$
$971$	−44.1849	−1.41796	−0.708980	+	0.705228i	$0.750845\pi$
$972$	0	0
$973$	16.0437	0.514338
$974$	64.2485	2.05865
$975$	0	0
$976$	−75.7907	−2.42600
$977$	−57.8785	−1.85170	−0.925849	−	0.377893i	$-0.876649\pi$
$977$	−57.8785	−1.85170	−0.925849	+	0.377893i	$0.876649\pi$
$978$	0	0
$979$	−24.2930	−0.776406
$980$	−2.35514	−0.0752320
$981$	0	0
$982$	32.9634	1.05190
$983$	−41.0411	−1.30901	−0.654504	−	0.756059i	$-0.727122\pi$
$983$	−41.0411	−1.30901	−0.654504	+	0.756059i	$0.727122\pi$
$984$	0	0
$985$	−49.2815	−1.57024
$986$	−66.6968	−2.12406
$987$	0	0
$988$	−4.34827	−0.138337
$989$	−46.9318	−1.49235
$990$	0	0
$991$	−45.2928	−1.43877	−0.719387	−	0.694609i	$-0.755577\pi$
$991$	−45.2928	−1.43877	−0.719387	+	0.694609i	$0.755577\pi$
$992$	−12.0850	−0.383700
$993$	0	0
$994$	−10.0316	−0.318183
$995$	−26.2502	−0.832186
$996$	0	0
$997$	0.745112	0.0235979	0.0117990	−	0.999930i	$-0.496244\pi$
$997$	0.745112	0.0235979	0.0117990	+	0.999930i	$0.496244\pi$
$998$	60.9598	1.92965
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.r.1.5		16
3.2	odd	2		2667.2.a.o.1.12	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.o.1.12	✓	16	3.2	odd	2
8001.2.a.r.1.5		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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-1.68977 q^{2} +0.855315 q^{4} -2.75353 q^{5} -1.00000 q^{7} +1.93425 q^{8} +O(q^{10})\)	\(q-1.68977 q^{2} +0.855315 q^{4} -2.75353 q^{5} -1.00000 q^{7} +1.93425 q^{8} +4.65282 q^{10} -3.55608 q^{11} +4.67022 q^{13} +1.68977 q^{14} -4.97907 q^{16} -4.02992 q^{17} -1.08856 q^{19} -2.35514 q^{20} +6.00896 q^{22} +7.04757 q^{23} +2.58192 q^{25} -7.89159 q^{26} -0.855315 q^{28} -9.79449 q^{29} -2.65900 q^{31} +4.54496 q^{32} +6.80962 q^{34} +2.75353 q^{35} +10.4203 q^{37} +1.83942 q^{38} -5.32602 q^{40} +9.96789 q^{41} -6.65929 q^{43} -3.04157 q^{44} -11.9088 q^{46} -6.75602 q^{47} +1.00000 q^{49} -4.36285 q^{50} +3.99451 q^{52} +3.41123 q^{53} +9.79178 q^{55} -1.93425 q^{56} +16.5504 q^{58} -8.15705 q^{59} +15.2219 q^{61} +4.49309 q^{62} +2.27820 q^{64} -12.8596 q^{65} -10.7563 q^{67} -3.44685 q^{68} -4.65282 q^{70} -5.93668 q^{71} +8.88205 q^{73} -17.6079 q^{74} -0.931064 q^{76} +3.55608 q^{77} +10.1644 q^{79} +13.7100 q^{80} -16.8434 q^{82} -12.6631 q^{83} +11.0965 q^{85} +11.2527 q^{86} -6.87836 q^{88} +6.83138 q^{89} -4.67022 q^{91} +6.02790 q^{92} +11.4161 q^{94} +2.99739 q^{95} +13.2112 q^{97} -1.68977 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) \) 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8	\( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8} - 12 q^{10} - 11 q^{11} + 18 q^{13} + 5 q^{14} + 25 q^{16} + 5 q^{17} - 11 q^{19} + q^{20} + q^{22} - 13 q^{23} + 33 q^{25} - 8 q^{26} - 19 q^{28} - 24 q^{29} - 42 q^{31} - 42 q^{32} + 9 q^{34} - q^{35} + 40 q^{37} - 38 q^{38} - 61 q^{40} - 9 q^{41} + 7 q^{43} - 3 q^{44} + 24 q^{46} - 31 q^{47} + 16 q^{49} - 6 q^{50} + 52 q^{52} - 66 q^{53} - 36 q^{55} + 6 q^{56} + 19 q^{58} + 7 q^{59} + 6 q^{61} - 52 q^{62} + 10 q^{64} - 51 q^{65} + 16 q^{67} - 14 q^{68} + 12 q^{70} - 46 q^{71} + 39 q^{73} - 72 q^{74} + 24 q^{76} + 11 q^{77} + 4 q^{79} + 2 q^{80} - 18 q^{82} - 15 q^{83} - 4 q^{85} - 14 q^{86} + 58 q^{88} + q^{89} - 18 q^{91} - 26 q^{92} + 5 q^{94} - 44 q^{95} + 41 q^{97} - 5 q^{98}+O(q^{100}) \) 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8 - 12 * q^10 - 11 * q^11 + 18 * q^13 + 5 * q^14 + 25 * q^16 + 5 * q^17 - 11 * q^19 + q^20 + q^22 - 13 * q^23 + 33 * q^25 - 8 * q^26 - 19 * q^28 - 24 * q^29 - 42 * q^31 - 42 * q^32 + 9 * q^34 - q^35 + 40 * q^37 - 38 * q^38 - 61 * q^40 - 9 * q^41 + 7 * q^43 - 3 * q^44 + 24 * q^46 - 31 * q^47 + 16 * q^49 - 6 * q^50 + 52 * q^52 - 66 * q^53 - 36 * q^55 + 6 * q^56 + 19 * q^58 + 7 * q^59 + 6 * q^61 - 52 * q^62 + 10 * q^64 - 51 * q^65 + 16 * q^67 - 14 * q^68 + 12 * q^70 - 46 * q^71 + 39 * q^73 - 72 * q^74 + 24 * q^76 + 11 * q^77 + 4 * q^79 + 2 * q^80 - 18 * q^82 - 15 * q^83 - 4 * q^85 - 14 * q^86 + 58 * q^88 + q^89 - 18 * q^91 - 26 * q^92 + 5 * q^94 - 44 * q^95 + 41 * q^97 - 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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