LMFDB - Embedded newform 8001.2.a.n.1.6

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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 3 x^{10} + 41 x^{9} - 11 x^{8} - 123 x^{7} + 44 x^{6} + 159 x^{5} - 39 x^{4} + \cdots - 1 $ x^12 - 5x^11 - 3x^10 + 41x^9 - 11x^8 - 123x^7 + 44x^6 + 159x^5 - 39x^4 - 71x^3 + 16x^2 + 7*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.482477$ 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)

$f(q)$	$=$	$q+0.517523 q^{2} -1.73217 q^{4} +3.15891 q^{5} +1.00000 q^{7} -1.93148 q^{8} +O(q^{10})$	\(q+0.517523 q^{2} -1.73217 q^{4} +3.15891 q^{5} +1.00000 q^{7} -1.93148 q^{8} +1.63481 q^{10} +1.34212 q^{11} +6.17634 q^{13} +0.517523 q^{14} +2.46476 q^{16} -0.162696 q^{17} +4.03864 q^{19} -5.47177 q^{20} +0.694579 q^{22} +9.14878 q^{23} +4.97873 q^{25} +3.19640 q^{26} -1.73217 q^{28} +3.08083 q^{29} -1.81945 q^{31} +5.13853 q^{32} -0.0841990 q^{34} +3.15891 q^{35} -1.51273 q^{37} +2.09008 q^{38} -6.10138 q^{40} -6.78579 q^{41} +3.59974 q^{43} -2.32479 q^{44} +4.73470 q^{46} +0.216887 q^{47} +1.00000 q^{49} +2.57660 q^{50} -10.6985 q^{52} -2.35396 q^{53} +4.23965 q^{55} -1.93148 q^{56} +1.59440 q^{58} -9.56132 q^{59} -3.42969 q^{61} -0.941605 q^{62} -2.27021 q^{64} +19.5105 q^{65} +3.15443 q^{67} +0.281818 q^{68} +1.63481 q^{70} +11.3492 q^{71} +7.71360 q^{73} -0.782869 q^{74} -6.99560 q^{76} +1.34212 q^{77} -12.1515 q^{79} +7.78595 q^{80} -3.51180 q^{82} +7.78225 q^{83} -0.513943 q^{85} +1.86295 q^{86} -2.59229 q^{88} -15.4859 q^{89} +6.17634 q^{91} -15.8473 q^{92} +0.112244 q^{94} +12.7577 q^{95} -16.3571 q^{97} +0.517523 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) $ 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8	$ 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8} - 2 q^{10} + 22 q^{11} + 7 q^{14} + 7 q^{16} + 6 q^{17} - 7 q^{19} + 8 q^{20} + 13 q^{22} + 29 q^{23} + 3 q^{25} + 9 q^{28} + 22 q^{29} - 16 q^{31} + 27 q^{32} - 5 q^{34} + 7 q^{35} - 4 q^{37} - 2 q^{38} + 16 q^{40} + 21 q^{41} + 11 q^{43} + 11 q^{44} + 31 q^{47} + 12 q^{49} + 21 q^{50} + 3 q^{52} + 38 q^{53} - 11 q^{55} + 15 q^{56} + 20 q^{58} + 15 q^{59} - 3 q^{61} + 4 q^{62} + 29 q^{64} + 32 q^{65} - q^{67} - 17 q^{68} - 2 q^{70} + 57 q^{71} - 7 q^{73} + 42 q^{74} - 44 q^{76} + 22 q^{77} - 18 q^{79} - q^{80} + 56 q^{82} + 21 q^{83} - 5 q^{85} + 32 q^{86} - 10 q^{88} - 6 q^{89} + 15 q^{92} + 35 q^{94} + 57 q^{95} + 4 q^{97} + 7 q^{98}+O(q^{100}) $ 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8 - 2 * q^10 + 22 * q^11 + 7 * q^14 + 7 * q^16 + 6 * q^17 - 7 * q^19 + 8 * q^20 + 13 * q^22 + 29 * q^23 + 3 * q^25 + 9 * q^28 + 22 * q^29 - 16 * q^31 + 27 * q^32 - 5 * q^34 + 7 * q^35 - 4 * q^37 - 2 * q^38 + 16 * q^40 + 21 * q^41 + 11 * q^43 + 11 * q^44 + 31 * q^47 + 12 * q^49 + 21 * q^50 + 3 * q^52 + 38 * q^53 - 11 * q^55 + 15 * q^56 + 20 * q^58 + 15 * q^59 - 3 * q^61 + 4 * q^62 + 29 * q^64 + 32 * q^65 - q^67 - 17 * q^68 - 2 * q^70 + 57 * q^71 - 7 * q^73 + 42 * q^74 - 44 * q^76 + 22 * q^77 - 18 * q^79 - q^80 + 56 * q^82 + 21 * q^83 - 5 * q^85 + 32 * q^86 - 10 * q^88 - 6 * q^89 + 15 * q^92 + 35 * q^94 + 57 * q^95 + 4 * q^97 + 7 * 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 $

$2$	0.517523	0.365944	0.182972	−	0.983118i	$-0.441428\pi$
$2$	0.517523	0.365944	0.182972	+	0.983118i	$0.441428\pi$
$3$	0	0
$3$	0	0
$4$	−1.73217	−0.866085
$5$	3.15891	1.41271	0.706354	−	0.707858i	$-0.250339\pi$
$5$	3.15891	1.41271	0.706354	+	0.707858i	$0.250339\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.93148	−0.682882
$9$	0	0
$10$	1.63481	0.516972
$11$	1.34212	0.404665	0.202333	−	0.979317i	$-0.435148\pi$
$11$	1.34212	0.404665	0.202333	+	0.979317i	$0.435148\pi$
$12$	0	0
$13$	6.17634	1.71301	0.856505	−	0.516139i	$-0.172631\pi$
$13$	6.17634	1.71301	0.856505	+	0.516139i	$0.172631\pi$
$14$	0.517523	0.138314
$15$	0	0
$16$	2.46476	0.616189
$17$	−0.162696	−0.0394596	−0.0197298	−	0.999805i	$-0.506281\pi$
$17$	−0.162696	−0.0394596	−0.0197298	+	0.999805i	$0.506281\pi$
$18$	0	0
$19$	4.03864	0.926526	0.463263	−	0.886221i	$-0.346678\pi$
$19$	4.03864	0.926526	0.463263	+	0.886221i	$0.346678\pi$
$20$	−5.47177	−1.22353
$21$	0	0
$22$	0.694579	0.148085
$23$	9.14878	1.90765	0.953827	−	0.300358i	$-0.0971061\pi$
$23$	9.14878	1.90765	0.953827	+	0.300358i	$0.0971061\pi$
$24$	0	0
$25$	4.97873	0.995745
$26$	3.19640	0.626865
$27$	0	0
$28$	−1.73217	−0.327349
$29$	3.08083	0.572097	0.286048	−	0.958215i	$-0.407658\pi$
$29$	3.08083	0.572097	0.286048	+	0.958215i	$0.407658\pi$
$30$	0	0
$31$	−1.81945	−0.326783	−0.163391	−	0.986561i	$-0.552243\pi$
$31$	−1.81945	−0.326783	−0.163391	+	0.986561i	$0.552243\pi$
$32$	5.13853	0.908372
$33$	0	0
$34$	−0.0841990	−0.0144400
$35$	3.15891	0.533954
$36$	0	0
$37$	−1.51273	−0.248690	−0.124345	−	0.992239i	$-0.539683\pi$
$37$	−1.51273	−0.248690	−0.124345	+	0.992239i	$0.539683\pi$
$38$	2.09008	0.339057
$39$	0	0
$40$	−6.10138	−0.964713
$41$	−6.78579	−1.05976	−0.529881	−	0.848072i	$-0.677764\pi$
$41$	−6.78579	−1.05976	−0.529881	+	0.848072i	$0.677764\pi$
$42$	0	0
$43$	3.59974	0.548956	0.274478	−	0.961593i	$-0.411495\pi$
$43$	3.59974	0.548956	0.274478	+	0.961593i	$0.411495\pi$
$44$	−2.32479	−0.350475
$45$	0	0
$46$	4.73470	0.698094
$47$	0.216887	0.0316362	0.0158181	−	0.999875i	$-0.494965\pi$
$47$	0.216887	0.0316362	0.0158181	+	0.999875i	$0.494965\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.57660	0.364387
$51$	0	0
$52$	−10.6985	−1.48361
$53$	−2.35396	−0.323342	−0.161671	−	0.986845i	$-0.551688\pi$
$53$	−2.35396	−0.323342	−0.161671	+	0.986845i	$0.551688\pi$
$54$	0	0
$55$	4.23965	0.571674
$56$	−1.93148	−0.258105
$57$	0	0
$58$	1.59440	0.209355
$59$	−9.56132	−1.24478	−0.622389	−	0.782708i	$-0.713838\pi$
$59$	−9.56132	−1.24478	−0.622389	+	0.782708i	$0.713838\pi$
$60$	0	0
$61$	−3.42969	−0.439127	−0.219563	−	0.975598i	$-0.570463\pi$
$61$	−3.42969	−0.439127	−0.219563	+	0.975598i	$0.570463\pi$
$62$	−0.941605	−0.119584
$63$	0	0
$64$	−2.27021	−0.283776
$65$	19.5105	2.41998
$66$	0	0
$67$	3.15443	0.385375	0.192687	−	0.981260i	$-0.438280\pi$
$67$	3.15443	0.385375	0.192687	+	0.981260i	$0.438280\pi$
$68$	0.281818	0.0341754
$69$	0	0
$70$	1.63481	0.195397
$71$	11.3492	1.34690	0.673448	−	0.739234i	$-0.264812\pi$
$71$	11.3492	1.34690	0.673448	+	0.739234i	$0.264812\pi$
$72$	0	0
$73$	7.71360	0.902809	0.451404	−	0.892320i	$-0.350923\pi$
$73$	7.71360	0.902809	0.451404	+	0.892320i	$0.350923\pi$
$74$	−0.782869	−0.0910067
$75$	0	0
$76$	−6.99560	−0.802451
$77$	1.34212	0.152949
$78$	0	0
$79$	−12.1515	−1.36715	−0.683574	−	0.729881i	$-0.739575\pi$
$79$	−12.1515	−1.36715	−0.683574	+	0.729881i	$0.739575\pi$
$80$	7.78595	0.870495
$81$	0	0
$82$	−3.51180	−0.387813
$83$	7.78225	0.854213	0.427107	−	0.904201i	$-0.359533\pi$
$83$	7.78225	0.854213	0.427107	+	0.904201i	$0.359533\pi$
$84$	0	0
$85$	−0.513943	−0.0557450
$86$	1.86295	0.200887
$87$	0	0
$88$	−2.59229	−0.276339
$89$	−15.4859	−1.64150	−0.820750	−	0.571287i	$-0.806444\pi$
$89$	−15.4859	−1.64150	−0.820750	+	0.571287i	$0.806444\pi$
$90$	0	0
$91$	6.17634	0.647457
$92$	−15.8473	−1.65219
$93$	0	0
$94$	0.112244	0.0115771
$95$	12.7577	1.30891
$96$	0	0
$97$	−16.3571	−1.66081	−0.830406	−	0.557158i	$-0.811892\pi$
$97$	−16.3571	−1.66081	−0.830406	+	0.557158i	$0.811892\pi$
$98$	0.517523	0.0522777
$99$	0	0
$100$	−8.62400	−0.862400
$101$	16.3648	1.62835	0.814177	−	0.580617i	$-0.197189\pi$
$101$	16.3648	1.62835	0.814177	+	0.580617i	$0.197189\pi$
$102$	0	0
$103$	−7.61305	−0.750136	−0.375068	−	0.926997i	$-0.622381\pi$
$103$	−7.61305	−0.750136	−0.375068	+	0.926997i	$0.622381\pi$
$104$	−11.9295	−1.16978
$105$	0	0
$106$	−1.21823	−0.118325
$107$	−17.8980	−1.73027	−0.865134	−	0.501541i	$-0.832767\pi$
$107$	−17.8980	−1.73027	−0.865134	+	0.501541i	$0.832767\pi$
$108$	0	0
$109$	1.12996	0.108231	0.0541153	−	0.998535i	$-0.482766\pi$
$109$	1.12996	0.108231	0.0541153	+	0.998535i	$0.482766\pi$
$110$	2.19411	0.209201
$111$	0	0
$112$	2.46476	0.232897
$113$	0.790903	0.0744019	0.0372010	−	0.999308i	$-0.488156\pi$
$113$	0.790903	0.0744019	0.0372010	+	0.999308i	$0.488156\pi$
$114$	0	0
$115$	28.9002	2.69496
$116$	−5.33653	−0.495484
$117$	0	0
$118$	−4.94820	−0.455518
$119$	−0.162696	−0.0149143
$120$	0	0
$121$	−9.19871	−0.836246
$122$	−1.77494	−0.160696
$123$	0	0
$124$	3.15159	0.283022
$125$	−0.0671982	−0.00601039
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−11.4519	−1.01222
$129$	0	0
$130$	10.0971	0.885578
$131$	−8.87329	−0.775263	−0.387631	−	0.921814i	$-0.626707\pi$
$131$	−8.87329	−0.775263	−0.387631	+	0.921814i	$0.626707\pi$
$132$	0	0
$133$	4.03864	0.350194
$134$	1.63249	0.141026
$135$	0	0
$136$	0.314245	0.0269463
$137$	15.3202	1.30890	0.654448	−	0.756107i	$-0.272901\pi$
$137$	15.3202	1.30890	0.654448	+	0.756107i	$0.272901\pi$
$138$	0	0
$139$	−9.77425	−0.829041	−0.414521	−	0.910040i	$-0.636051\pi$
$139$	−9.77425	−0.829041	−0.414521	+	0.910040i	$0.636051\pi$
$140$	−5.47177	−0.462449
$141$	0	0
$142$	5.87344	0.492888
$143$	8.28941	0.693196
$144$	0	0
$145$	9.73208	0.808206
$146$	3.99196	0.330377
$147$	0	0
$148$	2.62030	0.215387
$149$	9.14286	0.749012	0.374506	−	0.927224i	$-0.377812\pi$
$149$	9.14286	0.749012	0.374506	+	0.927224i	$0.377812\pi$
$150$	0	0
$151$	−18.4447	−1.50101	−0.750504	−	0.660866i	$-0.770189\pi$
$151$	−18.4447	−1.50101	−0.750504	+	0.660866i	$0.770189\pi$
$152$	−7.80055	−0.632708
$153$	0	0
$154$	0.694579	0.0559708
$155$	−5.74748	−0.461648
$156$	0	0
$157$	16.6124	1.32581	0.662907	−	0.748702i	$-0.269323\pi$
$157$	16.6124	1.32581	0.662907	+	0.748702i	$0.269323\pi$
$158$	−6.28866	−0.500299
$159$	0	0
$160$	16.2322	1.28327
$161$	9.14878	0.721025
$162$	0	0
$163$	6.58131	0.515488	0.257744	−	0.966213i	$-0.417021\pi$
$163$	6.58131	0.515488	0.257744	+	0.966213i	$0.417021\pi$
$164$	11.7541	0.917845
$165$	0	0
$166$	4.02749	0.312594
$167$	21.7467	1.68281	0.841405	−	0.540406i	$-0.181729\pi$
$167$	21.7467	1.68281	0.841405	+	0.540406i	$0.181729\pi$
$168$	0	0
$169$	25.1472	1.93440
$170$	−0.265977	−0.0203995
$171$	0	0
$172$	−6.23537	−0.475442
$173$	15.0542	1.14455	0.572277	−	0.820061i	$-0.306060\pi$
$173$	15.0542	1.14455	0.572277	+	0.820061i	$0.306060\pi$
$174$	0	0
$175$	4.97873	0.376356
$176$	3.30800	0.249350
$177$	0	0
$178$	−8.01429	−0.600697
$179$	16.7529	1.25217	0.626086	−	0.779754i	$-0.284656\pi$
$179$	16.7529	1.25217	0.626086	+	0.779754i	$0.284656\pi$
$180$	0	0
$181$	−9.75592	−0.725152	−0.362576	−	0.931954i	$-0.618103\pi$
$181$	−9.75592	−0.725152	−0.362576	+	0.931954i	$0.618103\pi$
$182$	3.19640	0.236933
$183$	0	0
$184$	−17.6707	−1.30270
$185$	−4.77857	−0.351327
$186$	0	0
$187$	−0.218358	−0.0159679
$188$	−0.375685	−0.0273996
$189$	0	0
$190$	6.60239	0.478988
$191$	7.37930	0.533947	0.266974	−	0.963704i	$-0.413976\pi$
$191$	7.37930	0.533947	0.266974	+	0.963704i	$0.413976\pi$
$192$	0	0
$193$	21.5861	1.55380	0.776902	−	0.629622i	$-0.216790\pi$
$193$	21.5861	1.55380	0.776902	+	0.629622i	$0.216790\pi$
$194$	−8.46517	−0.607764
$195$	0	0
$196$	−1.73217	−0.123726
$197$	−26.7663	−1.90702	−0.953508	−	0.301367i	$-0.902557\pi$
$197$	−26.7663	−1.90702	−0.953508	+	0.301367i	$0.902557\pi$
$198$	0	0
$199$	−18.4869	−1.31050	−0.655250	−	0.755412i	$-0.727437\pi$
$199$	−18.4869	−1.31050	−0.655250	+	0.755412i	$0.727437\pi$
$200$	−9.61632	−0.679977
$201$	0	0
$202$	8.46913	0.595886
$203$	3.08083	0.216232
$204$	0	0
$205$	−21.4357	−1.49714
$206$	−3.93993	−0.274508
$207$	0	0
$208$	15.2232	1.05554
$209$	5.42035	0.374933
$210$	0	0
$211$	−15.2544	−1.05015	−0.525077	−	0.851055i	$-0.675963\pi$
$211$	−15.2544	−1.05015	−0.525077	+	0.851055i	$0.675963\pi$
$212$	4.07747	0.280042
$213$	0	0
$214$	−9.26263	−0.633181
$215$	11.3713	0.775514
$216$	0	0
$217$	−1.81945	−0.123512
$218$	0.584781	0.0396063
$219$	0	0
$220$	−7.34379	−0.495119
$221$	−1.00487	−0.0675947
$222$	0	0
$223$	−8.17242	−0.547266	−0.273633	−	0.961834i	$-0.588225\pi$
$223$	−8.17242	−0.547266	−0.273633	+	0.961834i	$0.588225\pi$
$224$	5.13853	0.343333
$225$	0	0
$226$	0.409310	0.0272269
$227$	15.8036	1.04892	0.524460	−	0.851435i	$-0.324267\pi$
$227$	15.8036	1.04892	0.524460	+	0.851435i	$0.324267\pi$
$228$	0	0
$229$	18.5377	1.22501	0.612503	−	0.790468i	$-0.290163\pi$
$229$	18.5377	1.22501	0.612503	+	0.790468i	$0.290163\pi$
$230$	14.9565	0.986203
$231$	0	0
$232$	−5.95058	−0.390674
$233$	2.38959	0.156547	0.0782737	−	0.996932i	$-0.475059\pi$
$233$	2.38959	0.156547	0.0782737	+	0.996932i	$0.475059\pi$
$234$	0	0
$235$	0.685126	0.0446927
$236$	16.5618	1.07808
$237$	0	0
$238$	−0.0841990	−0.00545781
$239$	−25.2323	−1.63214	−0.816071	−	0.577952i	$-0.803852\pi$
$239$	−25.2323	−1.63214	−0.816071	+	0.577952i	$0.803852\pi$
$240$	0	0
$241$	−2.20906	−0.142298	−0.0711490	−	0.997466i	$-0.522667\pi$
$241$	−2.20906	−0.142298	−0.0711490	+	0.997466i	$0.522667\pi$
$242$	−4.76054	−0.306019
$243$	0	0
$244$	5.94080	0.380321
$245$	3.15891	0.201816
$246$	0	0
$247$	24.9440	1.58715
$248$	3.51423	0.223154
$249$	0	0
$250$	−0.0347766	−0.00219946
$251$	16.7979	1.06027	0.530137	−	0.847912i	$-0.322140\pi$
$251$	16.7979	1.06027	0.530137	+	0.847912i	$0.322140\pi$
$252$	0	0
$253$	12.2788	0.771961
$254$	0.517523	0.0324723
$255$	0	0
$256$	−1.38623	−0.0866393
$257$	−5.08796	−0.317378	−0.158689	−	0.987329i	$-0.550727\pi$
$257$	−5.08796	−0.317378	−0.158689	+	0.987329i	$0.550727\pi$
$258$	0	0
$259$	−1.51273	−0.0939962
$260$	−33.7956	−2.09591
$261$	0	0
$262$	−4.59213	−0.283703
$263$	−0.638166	−0.0393510	−0.0196755	−	0.999806i	$-0.506263\pi$
$263$	−0.638166	−0.0393510	−0.0196755	+	0.999806i	$0.506263\pi$
$264$	0	0
$265$	−7.43597	−0.456788
$266$	2.09008	0.128151
$267$	0	0
$268$	−5.46401	−0.333768
$269$	−13.9106	−0.848142	−0.424071	−	0.905629i	$-0.639399\pi$
$269$	−13.9106	−0.848142	−0.424071	+	0.905629i	$0.639399\pi$
$270$	0	0
$271$	−7.33095	−0.445324	−0.222662	−	0.974896i	$-0.571475\pi$
$271$	−7.33095	−0.445324	−0.222662	+	0.974896i	$0.571475\pi$
$272$	−0.401006	−0.0243146
$273$	0	0
$274$	7.92856	0.478982
$275$	6.68206	0.402944
$276$	0	0
$277$	5.36407	0.322296	0.161148	−	0.986930i	$-0.448480\pi$
$277$	5.36407	0.322296	0.161148	+	0.986930i	$0.448480\pi$
$278$	−5.05840	−0.303382
$279$	0	0
$280$	−6.10138	−0.364627
$281$	6.45659	0.385168	0.192584	−	0.981281i	$-0.438313\pi$
$281$	6.45659	0.385168	0.192584	+	0.981281i	$0.438313\pi$
$282$	0	0
$283$	12.8655	0.764777	0.382388	−	0.924002i	$-0.375102\pi$
$283$	12.8655	0.764777	0.382388	+	0.924002i	$0.375102\pi$
$284$	−19.6587	−1.16653
$285$	0	0
$286$	4.28996	0.253671
$287$	−6.78579	−0.400553
$288$	0	0
$289$	−16.9735	−0.998443
$290$	5.03657	0.295758
$291$	0	0
$292$	−13.3613	−0.781909
$293$	6.06884	0.354546	0.177273	−	0.984162i	$-0.443273\pi$
$293$	6.06884	0.354546	0.177273	+	0.984162i	$0.443273\pi$
$294$	0	0
$295$	−30.2034	−1.75851
$296$	2.92180	0.169826
$297$	0	0
$298$	4.73163	0.274096
$299$	56.5060	3.26783
$300$	0	0
$301$	3.59974	0.207486
$302$	−9.54555	−0.549285
$303$	0	0
$304$	9.95425	0.570915
$305$	−10.8341	−0.620358
$306$	0	0
$307$	−29.1722	−1.66495	−0.832473	−	0.554066i	$-0.813075\pi$
$307$	−29.1722	−1.66495	−0.832473	+	0.554066i	$0.813075\pi$
$308$	−2.32479	−0.132467
$309$	0	0
$310$	−2.97445	−0.168937
$311$	21.1048	1.19675	0.598373	−	0.801218i	$-0.295814\pi$
$311$	21.1048	1.19675	0.598373	+	0.801218i	$0.295814\pi$
$312$	0	0
$313$	5.00576	0.282942	0.141471	−	0.989942i	$-0.454817\pi$
$313$	5.00576	0.282942	0.141471	+	0.989942i	$0.454817\pi$
$314$	8.59729	0.485173
$315$	0	0
$316$	21.0484	1.18407
$317$	−2.91530	−0.163739	−0.0818697	−	0.996643i	$-0.526089\pi$
$317$	−2.91530	−0.163739	−0.0818697	+	0.996643i	$0.526089\pi$
$318$	0	0
$319$	4.13486	0.231508
$320$	−7.17138	−0.400892
$321$	0	0
$322$	4.73470	0.263855
$323$	−0.657071	−0.0365604
$324$	0	0
$325$	30.7503	1.70572
$326$	3.40597	0.188639
$327$	0	0
$328$	13.1066	0.723693
$329$	0.216887	0.0119574
$330$	0	0
$331$	−9.23683	−0.507702	−0.253851	−	0.967243i	$-0.581697\pi$
$331$	−9.23683	−0.507702	−0.253851	+	0.967243i	$0.581697\pi$
$332$	−13.4802	−0.739822
$333$	0	0
$334$	11.2544	0.615813
$335$	9.96457	0.544422
$336$	0	0
$337$	−25.8417	−1.40769	−0.703843	−	0.710356i	$-0.748534\pi$
$337$	−25.8417	−1.40769	−0.703843	+	0.710356i	$0.748534\pi$
$338$	13.0143	0.707882
$339$	0	0
$340$	0.890237	0.0482799
$341$	−2.44192	−0.132238
$342$	0	0
$343$	1.00000	0.0539949
$344$	−6.95284	−0.374872
$345$	0	0
$346$	7.79091	0.418842
$347$	18.5438	0.995484	0.497742	−	0.867325i	$-0.334163\pi$
$347$	18.5438	0.995484	0.497742	+	0.867325i	$0.334163\pi$
$348$	0	0
$349$	15.3366	0.820948	0.410474	−	0.911872i	$-0.365363\pi$
$349$	15.3366	0.820948	0.410474	+	0.911872i	$0.365363\pi$
$350$	2.57660	0.137725
$351$	0	0
$352$	6.89654	0.367587
$353$	−13.1579	−0.700324	−0.350162	−	0.936689i	$-0.613873\pi$
$353$	−13.1579	−0.700324	−0.350162	+	0.936689i	$0.613873\pi$
$354$	0	0
$355$	35.8510	1.90277
$356$	26.8242	1.42168
$357$	0	0
$358$	8.67001	0.458224
$359$	6.16565	0.325411	0.162705	−	0.986675i	$-0.447978\pi$
$359$	6.16565	0.325411	0.162705	+	0.986675i	$0.447978\pi$
$360$	0	0
$361$	−2.68942	−0.141549
$362$	−5.04891	−0.265365
$363$	0	0
$364$	−10.6985	−0.560753
$365$	24.3666	1.27541
$366$	0	0
$367$	26.4469	1.38052	0.690259	−	0.723562i	$-0.257496\pi$
$367$	26.4469	1.38052	0.690259	+	0.723562i	$0.257496\pi$
$368$	22.5495	1.17547
$369$	0	0
$370$	−2.47302	−0.128566
$371$	−2.35396	−0.122212
$372$	0	0
$373$	−6.96126	−0.360440	−0.180220	−	0.983626i	$-0.557681\pi$
$373$	−6.96126	−0.360440	−0.180220	+	0.983626i	$0.557681\pi$
$374$	−0.113005	−0.00584337
$375$	0	0
$376$	−0.418913	−0.0216038
$377$	19.0283	0.980007
$378$	0	0
$379$	1.76089	0.0904507	0.0452253	−	0.998977i	$-0.485599\pi$
$379$	1.76089	0.0904507	0.0452253	+	0.998977i	$0.485599\pi$
$380$	−22.0985	−1.13363
$381$	0	0
$382$	3.81895	0.195395
$383$	−27.7917	−1.42009	−0.710046	−	0.704156i	$-0.751326\pi$
$383$	−27.7917	−1.42009	−0.710046	+	0.704156i	$0.751326\pi$
$384$	0	0
$385$	4.23965	0.216073
$386$	11.1713	0.568605
$387$	0	0
$388$	28.3333	1.43841
$389$	12.5483	0.636226	0.318113	−	0.948053i	$-0.396951\pi$
$389$	12.5483	0.636226	0.318113	+	0.948053i	$0.396951\pi$
$390$	0	0
$391$	−1.48847	−0.0752753
$392$	−1.93148	−0.0975546
$393$	0	0
$394$	−13.8521	−0.697861
$395$	−38.3855	−1.93138
$396$	0	0
$397$	−19.2558	−0.966420	−0.483210	−	0.875504i	$-0.660529\pi$
$397$	−19.2558	−0.966420	−0.483210	+	0.875504i	$0.660529\pi$
$398$	−9.56738	−0.479569
$399$	0	0
$400$	12.2713	0.613567
$401$	−11.1377	−0.556191	−0.278096	−	0.960553i	$-0.589703\pi$
$401$	−11.1377	−0.556191	−0.278096	+	0.960553i	$0.589703\pi$
$402$	0	0
$403$	−11.2375	−0.559782
$404$	−28.3465	−1.41029
$405$	0	0
$406$	1.59440	0.0791288
$407$	−2.03026	−0.100636
$408$	0	0
$409$	−29.0313	−1.43550	−0.717752	−	0.696299i	$-0.754829\pi$
$409$	−29.0313	−1.43550	−0.717752	+	0.696299i	$0.754829\pi$
$410$	−11.0935	−0.547867
$411$	0	0
$412$	13.1871	0.649682
$413$	−9.56132	−0.470482
$414$	0	0
$415$	24.5835	1.20675
$416$	31.7373	1.55605
$417$	0	0
$418$	2.80515	0.137204
$419$	−10.1605	−0.496373	−0.248187	−	0.968712i	$-0.579835\pi$
$419$	−10.1605	−0.496373	−0.248187	+	0.968712i	$0.579835\pi$
$420$	0	0
$421$	−26.0457	−1.26939	−0.634696	−	0.772762i	$-0.718874\pi$
$421$	−26.0457	−1.26939	−0.634696	+	0.772762i	$0.718874\pi$
$422$	−7.89447	−0.384297
$423$	0	0
$424$	4.54664	0.220804
$425$	−0.810020	−0.0392918
$426$	0	0
$427$	−3.42969	−0.165974
$428$	31.0024	1.49856
$429$	0	0
$430$	5.88489	0.283795
$431$	15.7374	0.758043	0.379021	−	0.925388i	$-0.376261\pi$
$431$	15.7374	0.758043	0.379021	+	0.925388i	$0.376261\pi$
$432$	0	0
$433$	9.03622	0.434253	0.217127	−	0.976143i	$-0.430332\pi$
$433$	9.03622	0.434253	0.217127	+	0.976143i	$0.430332\pi$
$434$	−0.941605	−0.0451985
$435$	0	0
$436$	−1.95729	−0.0937370
$437$	36.9486	1.76749
$438$	0	0
$439$	−21.3707	−1.01997	−0.509983	−	0.860185i	$-0.670348\pi$
$439$	−21.3707	−1.01997	−0.509983	+	0.860185i	$0.670348\pi$
$440$	−8.18881	−0.390386
$441$	0	0
$442$	−0.520042	−0.0247359
$443$	5.94429	0.282422	0.141211	−	0.989980i	$-0.454900\pi$
$443$	5.94429	0.282422	0.141211	+	0.989980i	$0.454900\pi$
$444$	0	0
$445$	−48.9186	−2.31896
$446$	−4.22941	−0.200268
$447$	0	0
$448$	−2.27021	−0.107257
$449$	−13.8992	−0.655942	−0.327971	−	0.944688i	$-0.606365\pi$
$449$	−13.8992	−0.655942	−0.327971	+	0.944688i	$0.606365\pi$
$450$	0	0
$451$	−9.10737	−0.428849
$452$	−1.36998	−0.0644384
$453$	0	0
$454$	8.17871	0.383846
$455$	19.5105	0.914668
$456$	0	0
$457$	30.4265	1.42329	0.711646	−	0.702538i	$-0.247950\pi$
$457$	30.4265	1.42329	0.711646	+	0.702538i	$0.247950\pi$
$458$	9.59368	0.448283
$459$	0	0
$460$	−50.0601	−2.33406
$461$	−18.9867	−0.884301	−0.442150	−	0.896941i	$-0.645784\pi$
$461$	−18.9867	−0.884301	−0.442150	+	0.896941i	$0.645784\pi$
$462$	0	0
$463$	1.17036	0.0543913	0.0271956	−	0.999630i	$-0.491342\pi$
$463$	1.17036	0.0543913	0.0271956	+	0.999630i	$0.491342\pi$
$464$	7.59350	0.352519
$465$	0	0
$466$	1.23667	0.0572875
$467$	13.9204	0.644159	0.322079	−	0.946713i	$-0.395618\pi$
$467$	13.9204	0.644159	0.322079	+	0.946713i	$0.395618\pi$
$468$	0	0
$469$	3.15443	0.145658
$470$	0.354568	0.0163550
$471$	0	0
$472$	18.4675	0.850036
$473$	4.83130	0.222143
$474$	0	0
$475$	20.1073	0.922585
$476$	0.281818	0.0129171
$477$	0	0
$478$	−13.0583	−0.597272
$479$	24.7214	1.12955	0.564776	−	0.825244i	$-0.308963\pi$
$479$	24.7214	1.12955	0.564776	+	0.825244i	$0.308963\pi$
$480$	0	0
$481$	−9.34311	−0.426009
$482$	−1.14324	−0.0520731
$483$	0	0
$484$	15.9337	0.724260
$485$	−51.6707	−2.34624
$486$	0	0
$487$	33.9569	1.53873	0.769367	−	0.638807i	$-0.220572\pi$
$487$	33.9569	1.53873	0.769367	+	0.638807i	$0.220572\pi$
$488$	6.62438	0.299872
$489$	0	0
$490$	1.63481	0.0738531
$491$	30.6917	1.38510	0.692548	−	0.721372i	$-0.256488\pi$
$491$	30.6917	1.38510	0.692548	+	0.721372i	$0.256488\pi$
$492$	0	0
$493$	−0.501240	−0.0225747
$494$	12.9091	0.580807
$495$	0	0
$496$	−4.48449	−0.201360
$497$	11.3492	0.509079
$498$	0	0
$499$	−37.6012	−1.68326	−0.841630	−	0.540055i	$-0.818404\pi$
$499$	−37.6012	−1.68326	−0.841630	+	0.540055i	$0.818404\pi$
$500$	0.116399	0.00520551
$501$	0	0
$502$	8.69329	0.388001
$503$	35.2163	1.57022	0.785108	−	0.619359i	$-0.212607\pi$
$503$	35.2163	1.57022	0.785108	+	0.619359i	$0.212607\pi$
$504$	0	0
$505$	51.6948	2.30039
$506$	6.35455	0.282494
$507$	0	0
$508$	−1.73217	−0.0768526
$509$	4.26090	0.188861	0.0944305	−	0.995531i	$-0.469897\pi$
$509$	4.26090	0.188861	0.0944305	+	0.995531i	$0.469897\pi$
$510$	0	0
$511$	7.71360	0.341230
$512$	22.1865	0.980513
$513$	0	0
$514$	−2.63314	−0.116143
$515$	−24.0490	−1.05972
$516$	0	0
$517$	0.291089	0.0128021
$518$	−0.782869	−0.0343973
$519$	0	0
$520$	−37.6842	−1.65256
$521$	4.39856	0.192704	0.0963522	−	0.995347i	$-0.469282\pi$
$521$	4.39856	0.192704	0.0963522	+	0.995347i	$0.469282\pi$
$522$	0	0
$523$	18.8425	0.823923	0.411962	−	0.911201i	$-0.364844\pi$
$523$	18.8425	0.823923	0.411962	+	0.911201i	$0.364844\pi$
$524$	15.3701	0.671444
$525$	0	0
$526$	−0.330265	−0.0144002
$527$	0.296017	0.0128947
$528$	0	0
$529$	60.7002	2.63914
$530$	−3.84828	−0.167159
$531$	0	0
$532$	−6.99560	−0.303298
$533$	−41.9114	−1.81538
$534$	0	0
$535$	−56.5383	−2.44436
$536$	−6.09272	−0.263166
$537$	0	0
$538$	−7.19903	−0.310372
$539$	1.34212	0.0578093
$540$	0	0
$541$	−21.2325	−0.912859	−0.456429	−	0.889760i	$-0.650872\pi$
$541$	−21.2325	−0.912859	−0.456429	+	0.889760i	$0.650872\pi$
$542$	−3.79393	−0.162963
$543$	0	0
$544$	−0.836020	−0.0358440
$545$	3.56945	0.152898
$546$	0	0
$547$	−12.5316	−0.535814	−0.267907	−	0.963445i	$-0.586332\pi$
$547$	−12.5316	−0.535814	−0.267907	+	0.963445i	$0.586332\pi$
$548$	−26.5372	−1.13361
$549$	0	0
$550$	3.45812	0.147455
$551$	12.4424	0.530063
$552$	0	0
$553$	−12.1515	−0.516733
$554$	2.77603	0.117942
$555$	0	0
$556$	16.9307	0.718020
$557$	2.14647	0.0909487	0.0454744	−	0.998966i	$-0.485520\pi$
$557$	2.14647	0.0909487	0.0454744	+	0.998966i	$0.485520\pi$
$558$	0	0
$559$	22.2333	0.940367
$560$	7.78595	0.329016
$561$	0	0
$562$	3.34143	0.140950
$563$	−37.3240	−1.57302	−0.786509	−	0.617578i	$-0.788114\pi$
$563$	−37.3240	−1.57302	−0.786509	+	0.617578i	$0.788114\pi$
$564$	0	0
$565$	2.49839	0.105108
$566$	6.65821	0.279865
$567$	0	0
$568$	−21.9207	−0.919771
$569$	12.5848	0.527584	0.263792	−	0.964580i	$-0.415027\pi$
$569$	12.5848	0.527584	0.263792	+	0.964580i	$0.415027\pi$
$570$	0	0
$571$	−4.20681	−0.176049	−0.0880247	−	0.996118i	$-0.528055\pi$
$571$	−4.20681	−0.176049	−0.0880247	+	0.996118i	$0.528055\pi$
$572$	−14.3587	−0.600367
$573$	0	0
$574$	−3.51180	−0.146580
$575$	45.5493	1.89954
$576$	0	0
$577$	14.2143	0.591751	0.295875	−	0.955227i	$-0.404389\pi$
$577$	14.2143	0.591751	0.295875	+	0.955227i	$0.404389\pi$
$578$	−8.78418	−0.365374
$579$	0	0
$580$	−16.8576	−0.699975
$581$	7.78225	0.322862
$582$	0	0
$583$	−3.15931	−0.130845
$584$	−14.8987	−0.616512
$585$	0	0
$586$	3.14076	0.129744
$587$	22.8702	0.943954	0.471977	−	0.881611i	$-0.343541\pi$
$587$	22.8702	0.943954	0.471977	+	0.881611i	$0.343541\pi$
$588$	0	0
$589$	−7.34809	−0.302773
$590$	−15.6309	−0.643515
$591$	0	0
$592$	−3.72850	−0.153240
$593$	−29.1075	−1.19530	−0.597651	−	0.801756i	$-0.703899\pi$
$593$	−29.1075	−1.19530	−0.597651	+	0.801756i	$0.703899\pi$
$594$	0	0
$595$	−0.513943	−0.0210696
$596$	−15.8370	−0.648708
$597$	0	0
$598$	29.2431	1.19584
$599$	3.05934	0.125001	0.0625007	−	0.998045i	$-0.480092\pi$
$599$	3.05934	0.125001	0.0625007	+	0.998045i	$0.480092\pi$
$600$	0	0
$601$	−33.3234	−1.35929	−0.679645	−	0.733541i	$-0.737866\pi$
$601$	−33.3234	−1.35929	−0.679645	+	0.733541i	$0.737866\pi$
$602$	1.86295	0.0759281
$603$	0	0
$604$	31.9494	1.30000
$605$	−29.0579	−1.18137
$606$	0	0
$607$	32.8226	1.33223	0.666114	−	0.745850i	$-0.267956\pi$
$607$	32.8226	1.33223	0.666114	+	0.745850i	$0.267956\pi$
$608$	20.7527	0.841631
$609$	0	0
$610$	−5.60688	−0.227016
$611$	1.33957	0.0541931
$612$	0	0
$613$	29.3076	1.18372	0.591862	−	0.806039i	$-0.298393\pi$
$613$	29.3076	1.18372	0.591862	+	0.806039i	$0.298393\pi$
$614$	−15.0973	−0.609276
$615$	0	0
$616$	−2.59229	−0.104446
$617$	10.0422	0.404285	0.202142	−	0.979356i	$-0.435210\pi$
$617$	10.0422	0.404285	0.202142	+	0.979356i	$0.435210\pi$
$618$	0	0
$619$	44.1986	1.77649	0.888245	−	0.459370i	$-0.151925\pi$
$619$	44.1986	1.77649	0.888245	+	0.459370i	$0.151925\pi$
$620$	9.95561	0.399827
$621$	0	0
$622$	10.9222	0.437942
$623$	−15.4859	−0.620429
$624$	0	0
$625$	−25.1059	−1.00424
$626$	2.59059	0.103541
$627$	0	0
$628$	−28.7755	−1.14827
$629$	0.246115	0.00981324
$630$	0	0
$631$	−32.6757	−1.30080	−0.650399	−	0.759592i	$-0.725398\pi$
$631$	−32.6757	−1.30080	−0.650399	+	0.759592i	$0.725398\pi$
$632$	23.4704	0.933601
$633$	0	0
$634$	−1.50873	−0.0599194
$635$	3.15891	0.125358
$636$	0	0
$637$	6.17634	0.244716
$638$	2.13988	0.0847187
$639$	0	0
$640$	−36.1757	−1.42997
$641$	23.1010	0.912434	0.456217	−	0.889869i	$-0.349204\pi$
$641$	23.1010	0.912434	0.456217	+	0.889869i	$0.349204\pi$
$642$	0	0
$643$	1.96242	0.0773901	0.0386951	−	0.999251i	$-0.487680\pi$
$643$	1.96242	0.0773901	0.0386951	+	0.999251i	$0.487680\pi$
$644$	−15.8473	−0.624469
$645$	0	0
$646$	−0.340049	−0.0133790
$647$	13.7497	0.540556	0.270278	−	0.962782i	$-0.412884\pi$
$647$	13.7497	0.540556	0.270278	+	0.962782i	$0.412884\pi$
$648$	0	0
$649$	−12.8325	−0.503718
$650$	15.9140	0.624198
$651$	0	0
$652$	−11.3999	−0.446456
$653$	35.4718	1.38812	0.694060	−	0.719918i	$-0.255820\pi$
$653$	35.4718	1.38812	0.694060	+	0.719918i	$0.255820\pi$
$654$	0	0
$655$	−28.0299	−1.09522
$656$	−16.7253	−0.653014
$657$	0	0
$658$	0.112244	0.00437572
$659$	−20.3449	−0.792524	−0.396262	−	0.918137i	$-0.629693\pi$
$659$	−20.3449	−0.792524	−0.396262	+	0.918137i	$0.629693\pi$
$660$	0	0
$661$	−35.2983	−1.37295	−0.686473	−	0.727155i	$-0.740842\pi$
$661$	−35.2983	−1.37295	−0.686473	+	0.727155i	$0.740842\pi$
$662$	−4.78027	−0.185790
$663$	0	0
$664$	−15.0313	−0.583327
$665$	12.7577	0.494722
$666$	0	0
$667$	28.1859	1.09136
$668$	−37.6690	−1.45746
$669$	0	0
$670$	5.15689	0.199228
$671$	−4.60306	−0.177699
$672$	0	0
$673$	8.25013	0.318019	0.159010	−	0.987277i	$-0.449170\pi$
$673$	8.25013	0.318019	0.159010	+	0.987277i	$0.449170\pi$
$674$	−13.3736	−0.515133
$675$	0	0
$676$	−43.5593	−1.67536
$677$	−9.47419	−0.364123	−0.182061	−	0.983287i	$-0.558277\pi$
$677$	−9.47419	−0.364123	−0.182061	+	0.983287i	$0.558277\pi$
$678$	0	0
$679$	−16.3571	−0.627728
$680$	0.992672	0.0380672
$681$	0	0
$682$	−1.26375	−0.0483915
$683$	47.0626	1.80080	0.900401	−	0.435062i	$-0.143273\pi$
$683$	47.0626	1.80080	0.900401	+	0.435062i	$0.143273\pi$
$684$	0	0
$685$	48.3953	1.84909
$686$	0.517523	0.0197591
$687$	0	0
$688$	8.87249	0.338260
$689$	−14.5389	−0.553888
$690$	0	0
$691$	0.867701	0.0330089	0.0165045	−	0.999864i	$-0.494746\pi$
$691$	0.867701	0.0330089	0.0165045	+	0.999864i	$0.494746\pi$
$692$	−26.0765	−0.991281
$693$	0	0
$694$	9.59684	0.364291
$695$	−30.8760	−1.17119
$696$	0	0
$697$	1.10402	0.0418178
$698$	7.93703	0.300421
$699$	0	0
$700$	−8.62400	−0.325957
$701$	−10.5160	−0.397184	−0.198592	−	0.980082i	$-0.563637\pi$
$701$	−10.5160	−0.397184	−0.198592	+	0.980082i	$0.563637\pi$
$702$	0	0
$703$	−6.10935	−0.230418
$704$	−3.04689	−0.114834
$705$	0	0
$706$	−6.80950	−0.256279
$707$	16.3648	0.615460
$708$	0	0
$709$	−50.2781	−1.88823	−0.944117	−	0.329610i	$-0.893083\pi$
$709$	−50.2781	−1.88823	−0.944117	+	0.329610i	$0.893083\pi$
$710$	18.5537	0.696307
$711$	0	0
$712$	29.9107	1.12095
$713$	−16.6457	−0.623388
$714$	0	0
$715$	26.1855	0.979283
$716$	−29.0189	−1.08449
$717$	0	0
$718$	3.19086	0.119082
$719$	15.0270	0.560413	0.280207	−	0.959940i	$-0.409597\pi$
$719$	15.0270	0.560413	0.280207	+	0.959940i	$0.409597\pi$
$720$	0	0
$721$	−7.61305	−0.283525
$722$	−1.39184	−0.0517988
$723$	0	0
$724$	16.8989	0.628044
$725$	15.3386	0.569663
$726$	0	0
$727$	−44.3097	−1.64335	−0.821677	−	0.569953i	$-0.806961\pi$
$727$	−44.3097	−1.64335	−0.821677	+	0.569953i	$0.806961\pi$
$728$	−11.9295	−0.442137
$729$	0	0
$730$	12.6103	0.466727
$731$	−0.585665	−0.0216616
$732$	0	0
$733$	34.9040	1.28921	0.644604	−	0.764517i	$-0.277022\pi$
$733$	34.9040	1.28921	0.644604	+	0.764517i	$0.277022\pi$
$734$	13.6869	0.505192
$735$	0	0
$736$	47.0113	1.73286
$737$	4.23363	0.155948
$738$	0	0
$739$	36.9096	1.35774	0.678871	−	0.734257i	$-0.262469\pi$
$739$	36.9096	1.35774	0.678871	+	0.734257i	$0.262469\pi$
$740$	8.27729	0.304279
$741$	0	0
$742$	−1.21823	−0.0447226
$743$	−27.2179	−0.998529	−0.499265	−	0.866450i	$-0.666396\pi$
$743$	−27.2179	−0.998529	−0.499265	+	0.866450i	$0.666396\pi$
$744$	0	0
$745$	28.8815	1.05814
$746$	−3.60261	−0.131901
$747$	0	0
$748$	0.378234	0.0138296
$749$	−17.8980	−0.653980
$750$	0	0
$751$	21.0710	0.768892	0.384446	−	0.923148i	$-0.374393\pi$
$751$	21.0710	0.768892	0.384446	+	0.923148i	$0.374393\pi$
$752$	0.534573	0.0194939
$753$	0	0
$754$	9.84757	0.358627
$755$	−58.2652	−2.12049
$756$	0	0
$757$	−5.60314	−0.203650	−0.101825	−	0.994802i	$-0.532468\pi$
$757$	−5.60314	−0.203650	−0.101825	+	0.994802i	$0.532468\pi$
$758$	0.911298	0.0330998
$759$	0	0
$760$	−24.6413	−0.893833
$761$	33.9425	1.23042	0.615208	−	0.788365i	$-0.289072\pi$
$761$	33.9425	1.23042	0.615208	+	0.788365i	$0.289072\pi$
$762$	0	0
$763$	1.12996	0.0409074
$764$	−12.7822	−0.462444
$765$	0	0
$766$	−14.3828	−0.519673
$767$	−59.0540	−2.13232
$768$	0	0
$769$	6.93112	0.249943	0.124971	−	0.992160i	$-0.460116\pi$
$769$	6.93112	0.249943	0.124971	+	0.992160i	$0.460116\pi$
$770$	2.19411	0.0790704
$771$	0	0
$772$	−37.3909	−1.34573
$773$	−44.4533	−1.59887	−0.799437	−	0.600750i	$-0.794869\pi$
$773$	−44.4533	−1.59887	−0.799437	+	0.600750i	$0.794869\pi$
$774$	0	0
$775$	−9.05854	−0.325392
$776$	31.5935	1.13414
$777$	0	0
$778$	6.49405	0.232823
$779$	−27.4053	−0.981898
$780$	0	0
$781$	15.2320	0.545042
$782$	−0.770318	−0.0275465
$783$	0	0
$784$	2.46476	0.0880270
$785$	52.4771	1.87299
$786$	0	0
$787$	−14.4097	−0.513652	−0.256826	−	0.966458i	$-0.582677\pi$
$787$	−14.4097	−0.513652	−0.256826	+	0.966458i	$0.582677\pi$
$788$	46.3637	1.65164
$789$	0	0
$790$	−19.8653	−0.706777
$791$	0.790903	0.0281213
$792$	0	0
$793$	−21.1829	−0.752228
$794$	−9.96530	−0.353655
$795$	0	0
$796$	32.0224	1.13500
$797$	23.0841	0.817681	0.408841	−	0.912606i	$-0.365933\pi$
$797$	23.0841	0.817681	0.408841	+	0.912606i	$0.365933\pi$
$798$	0	0
$799$	−0.0352867	−0.00124835
$800$	25.5833	0.904508
$801$	0	0
$802$	−5.76402	−0.203535
$803$	10.3526	0.365335
$804$	0	0
$805$	28.9002	1.01860
$806$	−5.81568	−0.204849
$807$	0	0
$808$	−31.6082	−1.11197
$809$	−43.1548	−1.51724	−0.758622	−	0.651531i	$-0.774127\pi$
$809$	−43.1548	−1.51724	−0.758622	+	0.651531i	$0.774127\pi$
$810$	0	0
$811$	44.9870	1.57971	0.789854	−	0.613295i	$-0.210156\pi$
$811$	44.9870	1.57971	0.789854	+	0.613295i	$0.210156\pi$
$812$	−5.33653	−0.187275
$813$	0	0
$814$	−1.05071	−0.0368273
$815$	20.7898	0.728234
$816$	0	0
$817$	14.5381	0.508622
$818$	−15.0243	−0.525314
$819$	0	0
$820$	37.1303	1.29665
$821$	−48.9521	−1.70844	−0.854221	−	0.519910i	$-0.825965\pi$
$821$	−48.9521	−1.70844	−0.854221	+	0.519910i	$0.825965\pi$
$822$	0	0
$823$	5.62288	0.196001	0.0980006	−	0.995186i	$-0.468755\pi$
$823$	5.62288	0.196001	0.0980006	+	0.995186i	$0.468755\pi$
$824$	14.7045	0.512255
$825$	0	0
$826$	−4.94820	−0.172170
$827$	22.3849	0.778399	0.389199	−	0.921154i	$-0.372752\pi$
$827$	22.3849	0.778399	0.389199	+	0.921154i	$0.372752\pi$
$828$	0	0
$829$	24.3326	0.845105	0.422552	−	0.906339i	$-0.361134\pi$
$829$	24.3326	0.845105	0.422552	+	0.906339i	$0.361134\pi$
$830$	12.7225	0.441604
$831$	0	0
$832$	−14.0216	−0.486110
$833$	−0.162696	−0.00563709
$834$	0	0
$835$	68.6959	2.37732
$836$	−9.38896	−0.324724
$837$	0	0
$838$	−5.25829	−0.181645
$839$	−15.8504	−0.547217	−0.273608	−	0.961841i	$-0.588217\pi$
$839$	−15.8504	−0.547217	−0.273608	+	0.961841i	$0.588217\pi$
$840$	0	0
$841$	−19.5085	−0.672706
$842$	−13.4792	−0.464526
$843$	0	0
$844$	26.4231	0.909522
$845$	79.4379	2.73275
$846$	0	0
$847$	−9.19871	−0.316071
$848$	−5.80195	−0.199240
$849$	0	0
$850$	−0.419204	−0.0143786
$851$	−13.8396	−0.474415
$852$	0	0
$853$	15.6740	0.536669	0.268335	−	0.963326i	$-0.413527\pi$
$853$	15.6740	0.536669	0.268335	+	0.963326i	$0.413527\pi$
$854$	−1.77494	−0.0607372
$855$	0	0
$856$	34.5697	1.18157
$857$	−47.0971	−1.60881	−0.804403	−	0.594084i	$-0.797515\pi$
$857$	−47.0971	−1.60881	−0.804403	+	0.594084i	$0.797515\pi$
$858$	0	0
$859$	54.5622	1.86164	0.930819	−	0.365480i	$-0.119095\pi$
$859$	54.5622	1.86164	0.930819	+	0.365480i	$0.119095\pi$
$860$	−19.6970	−0.671662
$861$	0	0
$862$	8.14445	0.277401
$863$	56.1087	1.90996	0.954981	−	0.296668i	$-0.0958755\pi$
$863$	56.1087	1.90996	0.954981	+	0.296668i	$0.0958755\pi$
$864$	0	0
$865$	47.5550	1.61692
$866$	4.67645	0.158912
$867$	0	0
$868$	3.15159	0.106972
$869$	−16.3088	−0.553238
$870$	0	0
$871$	19.4828	0.660151
$872$	−2.18250	−0.0739088
$873$	0	0
$874$	19.1217	0.646802
$875$	−0.0671982	−0.00227171
$876$	0	0
$877$	−18.8442	−0.636322	−0.318161	−	0.948037i	$-0.603065\pi$
$877$	−18.8442	−0.636322	−0.318161	+	0.948037i	$0.603065\pi$
$878$	−11.0598	−0.373250
$879$	0	0
$880$	10.4497	0.352259
$881$	−44.4858	−1.49876	−0.749382	−	0.662138i	$-0.769649\pi$
$881$	−44.4858	−1.49876	−0.749382	+	0.662138i	$0.769649\pi$
$882$	0	0
$883$	9.94328	0.334618	0.167309	−	0.985905i	$-0.446492\pi$
$883$	9.94328	0.334618	0.167309	+	0.985905i	$0.446492\pi$
$884$	1.74060	0.0585428
$885$	0	0
$886$	3.07631	0.103351
$887$	−7.61358	−0.255639	−0.127819	−	0.991797i	$-0.540798\pi$
$887$	−7.61358	−0.255639	−0.127819	+	0.991797i	$0.540798\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−25.3165	−0.848609
$891$	0	0
$892$	14.1560	0.473979
$893$	0.875926	0.0293118
$894$	0	0
$895$	52.9210	1.76895
$896$	−11.4519	−0.382583
$897$	0	0
$898$	−7.19313	−0.240038
$899$	−5.60542	−0.186951
$900$	0	0
$901$	0.382981	0.0127590
$902$	−4.71327	−0.156935
$903$	0	0
$904$	−1.52762	−0.0508078
$905$	−30.8181	−1.02443
$906$	0	0
$907$	−2.49691	−0.0829086	−0.0414543	−	0.999140i	$-0.513199\pi$
$907$	−2.49691	−0.0829086	−0.0414543	+	0.999140i	$0.513199\pi$
$908$	−27.3745	−0.908455
$909$	0	0
$910$	10.0971	0.334717
$911$	−18.6159	−0.616772	−0.308386	−	0.951261i	$-0.599789\pi$
$911$	−18.6159	−0.616772	−0.308386	+	0.951261i	$0.599789\pi$
$912$	0	0
$913$	10.4447	0.345671
$914$	15.7464	0.520845
$915$	0	0
$916$	−32.1105	−1.06096
$917$	−8.87329	−0.293022
$918$	0	0
$919$	5.14844	0.169831	0.0849157	−	0.996388i	$-0.472938\pi$
$919$	5.14844	0.169831	0.0849157	+	0.996388i	$0.472938\pi$
$920$	−55.8202	−1.84034
$921$	0	0
$922$	−9.82607	−0.323604
$923$	70.0963	2.30725
$924$	0	0
$925$	−7.53145	−0.247632
$926$	0.605688	0.0199041
$927$	0	0
$928$	15.8310	0.519677
$929$	10.6717	0.350126	0.175063	−	0.984557i	$-0.443987\pi$
$929$	10.6717	0.350126	0.175063	+	0.984557i	$0.443987\pi$
$930$	0	0
$931$	4.03864	0.132361
$932$	−4.13918	−0.135583
$933$	0	0
$934$	7.20411	0.235726
$935$	−0.689775	−0.0225581
$936$	0	0
$937$	−18.5179	−0.604953	−0.302476	−	0.953157i	$-0.597813\pi$
$937$	−18.5179	−0.604953	−0.302476	+	0.953157i	$0.597813\pi$
$938$	1.63249	0.0533026
$939$	0	0
$940$	−1.18676	−0.0387077
$941$	−35.6456	−1.16201	−0.581006	−	0.813899i	$-0.697341\pi$
$941$	−35.6456	−1.16201	−0.581006	+	0.813899i	$0.697341\pi$
$942$	0	0
$943$	−62.0817	−2.02166
$944$	−23.5663	−0.767018
$945$	0	0
$946$	2.50031	0.0812920
$947$	0.303541	0.00986376	0.00493188	−	0.999988i	$-0.498430\pi$
$947$	0.303541	0.00986376	0.00493188	+	0.999988i	$0.498430\pi$
$948$	0	0
$949$	47.6419	1.54652
$950$	10.4060	0.337614
$951$	0	0
$952$	0.314245	0.0101847
$953$	11.1478	0.361114	0.180557	−	0.983565i	$-0.442210\pi$
$953$	11.1478	0.361114	0.180557	+	0.983565i	$0.442210\pi$
$954$	0	0
$955$	23.3106	0.754312
$956$	43.7067	1.41357
$957$	0	0
$958$	12.7939	0.413352
$959$	15.3202	0.494716
$960$	0	0
$961$	−27.6896	−0.893213
$962$	−4.83527	−0.155895
$963$	0	0
$964$	3.82647	0.123242
$965$	68.1887	2.19507
$966$	0	0
$967$	8.30747	0.267150	0.133575	−	0.991039i	$-0.457354\pi$
$967$	8.30747	0.267150	0.133575	+	0.991039i	$0.457354\pi$
$968$	17.7671	0.571057
$969$	0	0
$970$	−26.7407	−0.858593
$971$	31.7444	1.01873	0.509363	−	0.860552i	$-0.329881\pi$
$971$	31.7444	1.01873	0.509363	+	0.860552i	$0.329881\pi$
$972$	0	0
$973$	−9.77425	−0.313348
$974$	17.5735	0.563090
$975$	0	0
$976$	−8.45334	−0.270585
$977$	17.7631	0.568292	0.284146	−	0.958781i	$-0.408290\pi$
$977$	17.7631	0.568292	0.284146	+	0.958781i	$0.408290\pi$
$978$	0	0
$979$	−20.7840	−0.664258
$980$	−5.47177	−0.174789
$981$	0	0
$982$	15.8836	0.506867
$983$	8.89430	0.283684	0.141842	−	0.989889i	$-0.454697\pi$
$983$	8.89430	0.283684	0.141842	+	0.989889i	$0.454697\pi$
$984$	0	0
$985$	−84.5523	−2.69406
$986$	−0.259403	−0.00826108
$987$	0	0
$988$	−43.2073	−1.37461
$989$	32.9333	1.04722
$990$	0	0
$991$	26.5270	0.842657	0.421329	−	0.906908i	$-0.361564\pi$
$991$	26.5270	0.842657	0.421329	+	0.906908i	$0.361564\pi$
$992$	−9.34929	−0.296840
$993$	0	0
$994$	5.87344	0.186294
$995$	−58.3984	−1.85135
$996$	0	0
$997$	7.05253	0.223356	0.111678	−	0.993744i	$-0.464378\pi$
$997$	7.05253	0.223356	0.111678	+	0.993744i	$0.464378\pi$
$998$	−19.4595	−0.615978
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.n.1.6		12
3.2	odd	2		889.2.a.a.1.7	✓	12
21.20	even	2		6223.2.a.i.1.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.a.1.7	✓	12	3.2	odd	2
6223.2.a.i.1.7		12	21.20	even	2
8001.2.a.n.1.6		12	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)

\(f(q)\)	\(=\)	\(q+0.517523 q^{2} -1.73217 q^{4} +3.15891 q^{5} +1.00000 q^{7} -1.93148 q^{8} +O(q^{10})\)	\(q+0.517523 q^{2} -1.73217 q^{4} +3.15891 q^{5} +1.00000 q^{7} -1.93148 q^{8} +1.63481 q^{10} +1.34212 q^{11} +6.17634 q^{13} +0.517523 q^{14} +2.46476 q^{16} -0.162696 q^{17} +4.03864 q^{19} -5.47177 q^{20} +0.694579 q^{22} +9.14878 q^{23} +4.97873 q^{25} +3.19640 q^{26} -1.73217 q^{28} +3.08083 q^{29} -1.81945 q^{31} +5.13853 q^{32} -0.0841990 q^{34} +3.15891 q^{35} -1.51273 q^{37} +2.09008 q^{38} -6.10138 q^{40} -6.78579 q^{41} +3.59974 q^{43} -2.32479 q^{44} +4.73470 q^{46} +0.216887 q^{47} +1.00000 q^{49} +2.57660 q^{50} -10.6985 q^{52} -2.35396 q^{53} +4.23965 q^{55} -1.93148 q^{56} +1.59440 q^{58} -9.56132 q^{59} -3.42969 q^{61} -0.941605 q^{62} -2.27021 q^{64} +19.5105 q^{65} +3.15443 q^{67} +0.281818 q^{68} +1.63481 q^{70} +11.3492 q^{71} +7.71360 q^{73} -0.782869 q^{74} -6.99560 q^{76} +1.34212 q^{77} -12.1515 q^{79} +7.78595 q^{80} -3.51180 q^{82} +7.78225 q^{83} -0.513943 q^{85} +1.86295 q^{86} -2.59229 q^{88} -15.4859 q^{89} +6.17634 q^{91} -15.8473 q^{92} +0.112244 q^{94} +12.7577 q^{95} -16.3571 q^{97} +0.517523 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) \) 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8	\( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8} - 2 q^{10} + 22 q^{11} + 7 q^{14} + 7 q^{16} + 6 q^{17} - 7 q^{19} + 8 q^{20} + 13 q^{22} + 29 q^{23} + 3 q^{25} + 9 q^{28} + 22 q^{29} - 16 q^{31} + 27 q^{32} - 5 q^{34} + 7 q^{35} - 4 q^{37} - 2 q^{38} + 16 q^{40} + 21 q^{41} + 11 q^{43} + 11 q^{44} + 31 q^{47} + 12 q^{49} + 21 q^{50} + 3 q^{52} + 38 q^{53} - 11 q^{55} + 15 q^{56} + 20 q^{58} + 15 q^{59} - 3 q^{61} + 4 q^{62} + 29 q^{64} + 32 q^{65} - q^{67} - 17 q^{68} - 2 q^{70} + 57 q^{71} - 7 q^{73} + 42 q^{74} - 44 q^{76} + 22 q^{77} - 18 q^{79} - q^{80} + 56 q^{82} + 21 q^{83} - 5 q^{85} + 32 q^{86} - 10 q^{88} - 6 q^{89} + 15 q^{92} + 35 q^{94} + 57 q^{95} + 4 q^{97} + 7 q^{98}+O(q^{100}) \) 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8 - 2 * q^10 + 22 * q^11 + 7 * q^14 + 7 * q^16 + 6 * q^17 - 7 * q^19 + 8 * q^20 + 13 * q^22 + 29 * q^23 + 3 * q^25 + 9 * q^28 + 22 * q^29 - 16 * q^31 + 27 * q^32 - 5 * q^34 + 7 * q^35 - 4 * q^37 - 2 * q^38 + 16 * q^40 + 21 * q^41 + 11 * q^43 + 11 * q^44 + 31 * q^47 + 12 * q^49 + 21 * q^50 + 3 * q^52 + 38 * q^53 - 11 * q^55 + 15 * q^56 + 20 * q^58 + 15 * q^59 - 3 * q^61 + 4 * q^62 + 29 * q^64 + 32 * q^65 - q^67 - 17 * q^68 - 2 * q^70 + 57 * q^71 - 7 * q^73 + 42 * q^74 - 44 * q^76 + 22 * q^77 - 18 * q^79 - q^80 + 56 * q^82 + 21 * q^83 - 5 * q^85 + 32 * q^86 - 10 * q^88 - 6 * q^89 + 15 * q^92 + 35 * q^94 + 57 * q^95 + 4 * q^97 + 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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