LMFDB - Embedded newform 2151.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.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:	$17.1758214748$
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} - 3 x^{11} - 15 x^{10} + 47 x^{9} + 75 x^{8} - 256 x^{7} - 134 x^{6} + 571 x^{5} + 23 x^{4} + \cdots - 31 $ x^12 - 3x^11 - 15x^10 + 47x^9 + 75x^8 - 256x^7 - 134x^6 + 571x^5 + 23x^4 - 479x^3 + 129x^2 + 88*x - 31
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.34937$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.34937 q^{2} +3.51954 q^{4} -1.00390 q^{5} -3.03107 q^{7} -3.56996 q^{8} +O(q^{10})$	\(q-2.34937 q^{2} +3.51954 q^{4} -1.00390 q^{5} -3.03107 q^{7} -3.56996 q^{8} +2.35853 q^{10} -5.41595 q^{11} +5.93326 q^{13} +7.12110 q^{14} +1.34808 q^{16} +1.60348 q^{17} +4.16355 q^{19} -3.53326 q^{20} +12.7241 q^{22} -8.35553 q^{23} -3.99219 q^{25} -13.9394 q^{26} -10.6680 q^{28} +4.52830 q^{29} -6.03940 q^{31} +3.97279 q^{32} -3.76718 q^{34} +3.04289 q^{35} -6.51496 q^{37} -9.78173 q^{38} +3.58388 q^{40} +3.85290 q^{41} -4.15662 q^{43} -19.0616 q^{44} +19.6302 q^{46} -4.57526 q^{47} +2.18738 q^{49} +9.37912 q^{50} +20.8823 q^{52} +3.88340 q^{53} +5.43707 q^{55} +10.8208 q^{56} -10.6387 q^{58} +13.2432 q^{59} -11.2500 q^{61} +14.1888 q^{62} -12.0297 q^{64} -5.95640 q^{65} +8.26031 q^{67} +5.64352 q^{68} -7.14887 q^{70} -12.7951 q^{71} +9.76782 q^{73} +15.3061 q^{74} +14.6538 q^{76} +16.4161 q^{77} +11.2884 q^{79} -1.35333 q^{80} -9.05189 q^{82} +3.07527 q^{83} -1.60974 q^{85} +9.76543 q^{86} +19.3347 q^{88} +10.8732 q^{89} -17.9841 q^{91} -29.4076 q^{92} +10.7490 q^{94} -4.17979 q^{95} -7.06760 q^{97} -5.13896 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 15 * q^4 + q^5 + 11 * q^7 - 9 * q^8	$ 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8} - 15 q^{11} + 7 q^{13} + 6 q^{14} + 21 q^{16} + 3 q^{17} + 10 q^{19} + 4 q^{20} + 23 q^{22} - 20 q^{23} + 19 q^{25} + 10 q^{26} + 34 q^{28} - 2 q^{29} + 10 q^{31} - 26 q^{32} + 12 q^{34} - 7 q^{35} + 30 q^{37} + 3 q^{38} + 25 q^{40} + 28 q^{41} + 48 q^{43} - 25 q^{44} + 22 q^{46} - 13 q^{47} + 19 q^{49} - 12 q^{50} + 24 q^{52} + 2 q^{53} + 8 q^{55} + 7 q^{56} + 42 q^{58} + 14 q^{59} + 14 q^{61} - 8 q^{62} + 9 q^{64} + 35 q^{65} + 52 q^{67} - 3 q^{68} - 33 q^{70} + 7 q^{71} + 14 q^{73} + 13 q^{74} - 12 q^{76} + 6 q^{77} + 15 q^{79} + 8 q^{80} - 61 q^{82} - 29 q^{83} + 8 q^{85} + 9 q^{86} + 11 q^{88} + 71 q^{89} + 13 q^{91} - 2 q^{92} - 22 q^{94} - 2 q^{95} - 2 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 15 * q^4 + q^5 + 11 * q^7 - 9 * q^8 - 15 * q^11 + 7 * q^13 + 6 * q^14 + 21 * q^16 + 3 * q^17 + 10 * q^19 + 4 * q^20 + 23 * q^22 - 20 * q^23 + 19 * q^25 + 10 * q^26 + 34 * q^28 - 2 * q^29 + 10 * q^31 - 26 * q^32 + 12 * q^34 - 7 * q^35 + 30 * q^37 + 3 * q^38 + 25 * q^40 + 28 * q^41 + 48 * q^43 - 25 * q^44 + 22 * q^46 - 13 * q^47 + 19 * q^49 - 12 * q^50 + 24 * q^52 + 2 * q^53 + 8 * q^55 + 7 * q^56 + 42 * q^58 + 14 * q^59 + 14 * q^61 - 8 * q^62 + 9 * q^64 + 35 * q^65 + 52 * q^67 - 3 * q^68 - 33 * q^70 + 7 * q^71 + 14 * q^73 + 13 * q^74 - 12 * q^76 + 6 * q^77 + 15 * q^79 + 8 * q^80 - 61 * q^82 - 29 * q^83 + 8 * q^85 + 9 * q^86 + 11 * q^88 + 71 * q^89 + 13 * q^91 - 2 * q^92 - 22 * q^94 - 2 * q^95 - 2 * 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 $

$2$	−2.34937	−1.66126	−0.830628	−	0.556828i	$-0.812018\pi$
$2$	−2.34937	−1.66126	−0.830628	+	0.556828i	$0.812018\pi$
$3$	0	0
$3$	0	0
$4$	3.51954	1.75977
$5$	−1.00390	−0.448958	−0.224479	−	0.974479i	$-0.572068\pi$
$5$	−1.00390	−0.448958	−0.224479	+	0.974479i	$0.572068\pi$
$6$	0	0
$7$	−3.03107	−1.14564	−0.572818	−	0.819682i	$-0.694150\pi$
$7$	−3.03107	−1.14564	−0.572818	+	0.819682i	$0.694150\pi$
$8$	−3.56996	−1.26217
$9$	0	0
$10$	2.35853	0.745833
$11$	−5.41595	−1.63297	−0.816485	−	0.577366i	$-0.804080\pi$
$11$	−5.41595	−1.63297	−0.816485	+	0.577366i	$0.804080\pi$
$12$	0	0
$13$	5.93326	1.64559	0.822795	−	0.568338i	$-0.192413\pi$
$13$	5.93326	1.64559	0.822795	+	0.568338i	$0.192413\pi$
$14$	7.12110	1.90319
$15$	0	0
$16$	1.34808	0.337019
$17$	1.60348	0.388902	0.194451	−	0.980912i	$-0.437707\pi$
$17$	1.60348	0.388902	0.194451	+	0.980912i	$0.437707\pi$
$18$	0	0
$19$	4.16355	0.955185	0.477592	−	0.878582i	$-0.341510\pi$
$19$	4.16355	0.955185	0.477592	+	0.878582i	$0.341510\pi$
$20$	−3.53326	−0.790062
$21$	0	0
$22$	12.7241	2.71278
$23$	−8.35553	−1.74225	−0.871124	−	0.491063i	$-0.836608\pi$
$23$	−8.35553	−1.74225	−0.871124	+	0.491063i	$0.836608\pi$
$24$	0	0
$25$	−3.99219	−0.798437
$26$	−13.9394	−2.73375
$27$	0	0
$28$	−10.6680	−2.01606
$29$	4.52830	0.840885	0.420443	−	0.907319i	$-0.361875\pi$
$29$	4.52830	0.840885	0.420443	+	0.907319i	$0.361875\pi$
$30$	0	0
$31$	−6.03940	−1.08471	−0.542354	−	0.840150i	$-0.682467\pi$
$31$	−6.03940	−1.08471	−0.542354	+	0.840150i	$0.682467\pi$
$32$	3.97279	0.702296
$33$	0	0
$34$	−3.76718	−0.646065
$35$	3.04289	0.514342
$36$	0	0
$37$	−6.51496	−1.07105	−0.535527	−	0.844518i	$-0.679887\pi$
$37$	−6.51496	−1.07105	−0.535527	+	0.844518i	$0.679887\pi$
$38$	−9.78173	−1.58681
$39$	0	0
$40$	3.58388	0.566661
$41$	3.85290	0.601722	0.300861	−	0.953668i	$-0.402726\pi$
$41$	3.85290	0.601722	0.300861	+	0.953668i	$0.402726\pi$
$42$	0	0
$43$	−4.15662	−0.633878	−0.316939	−	0.948446i	$-0.602655\pi$
$43$	−4.15662	−0.633878	−0.316939	+	0.948446i	$0.602655\pi$
$44$	−19.0616	−2.87365
$45$	0	0
$46$	19.6302	2.89432
$47$	−4.57526	−0.667371	−0.333685	−	0.942684i	$-0.608292\pi$
$47$	−4.57526	−0.667371	−0.333685	+	0.942684i	$0.608292\pi$
$48$	0	0
$49$	2.18738	0.312482
$50$	9.37912	1.32641
$51$	0	0
$52$	20.8823	2.89586
$53$	3.88340	0.533426	0.266713	−	0.963776i	$-0.414062\pi$
$53$	3.88340	0.533426	0.266713	+	0.963776i	$0.414062\pi$
$54$	0	0
$55$	5.43707	0.733135
$56$	10.8208	1.44599
$57$	0	0
$58$	−10.6387	−1.39692
$59$	13.2432	1.72412	0.862059	−	0.506808i	$-0.169175\pi$
$59$	13.2432	1.72412	0.862059	+	0.506808i	$0.169175\pi$
$60$	0	0
$61$	−11.2500	−1.44042	−0.720208	−	0.693758i	$-0.755954\pi$
$61$	−11.2500	−1.44042	−0.720208	+	0.693758i	$0.755954\pi$
$62$	14.1888	1.80198
$63$	0	0
$64$	−12.0297	−1.50371
$65$	−5.95640	−0.738801
$66$	0	0
$67$	8.26031	1.00916	0.504579	−	0.863366i	$-0.331648\pi$
$67$	8.26031	1.00916	0.504579	+	0.863366i	$0.331648\pi$
$68$	5.64352	0.684378
$69$	0	0
$70$	−7.14887	−0.854454
$71$	−12.7951	−1.51850	−0.759252	−	0.650797i	$-0.774435\pi$
$71$	−12.7951	−1.51850	−0.759252	+	0.650797i	$0.774435\pi$
$72$	0	0
$73$	9.76782	1.14324	0.571619	−	0.820520i	$-0.306316\pi$
$73$	9.76782	1.14324	0.571619	+	0.820520i	$0.306316\pi$
$74$	15.3061	1.77929
$75$	0	0
$76$	14.6538	1.68091
$77$	16.4161	1.87079
$78$	0	0
$79$	11.2884	1.27005	0.635025	−	0.772492i	$-0.280990\pi$
$79$	11.2884	1.27005	0.635025	+	0.772492i	$0.280990\pi$
$80$	−1.35333	−0.151307
$81$	0	0
$82$	−9.05189	−0.999614
$83$	3.07527	0.337555	0.168777	−	0.985654i	$-0.446018\pi$
$83$	3.07527	0.337555	0.168777	+	0.985654i	$0.446018\pi$
$84$	0	0
$85$	−1.60974	−0.174601
$86$	9.76543	1.05303
$87$	0	0
$88$	19.3347	2.06109
$89$	10.8732	1.15255	0.576276	−	0.817255i	$-0.304505\pi$
$89$	10.8732	1.15255	0.576276	+	0.817255i	$0.304505\pi$
$90$	0	0
$91$	−17.9841	−1.88525
$92$	−29.4076	−3.06595
$93$	0	0
$94$	10.7490	1.10867
$95$	−4.17979	−0.428838
$96$	0	0
$97$	−7.06760	−0.717606	−0.358803	−	0.933413i	$-0.616815\pi$
$97$	−7.06760	−0.717606	−0.358803	+	0.933413i	$0.616815\pi$
$98$	−5.13896	−0.519113
$99$	0	0
$100$	−14.0507	−1.40507
$101$	0.531292	0.0528655	0.0264328	−	0.999651i	$-0.491585\pi$
$101$	0.531292	0.0528655	0.0264328	+	0.999651i	$0.491585\pi$
$102$	0	0
$103$	−13.7020	−1.35010	−0.675048	−	0.737774i	$-0.735877\pi$
$103$	−13.7020	−1.35010	−0.675048	+	0.737774i	$0.735877\pi$
$104$	−21.1815	−2.07702
$105$	0	0
$106$	−9.12354	−0.886157
$107$	−19.0002	−1.83682	−0.918408	−	0.395636i	$-0.870524\pi$
$107$	−19.0002	−1.83682	−0.918408	+	0.395636i	$0.870524\pi$
$108$	0	0
$109$	15.6383	1.49787	0.748937	−	0.662642i	$-0.230565\pi$
$109$	15.6383	1.49787	0.748937	+	0.662642i	$0.230565\pi$
$110$	−12.7737	−1.21792
$111$	0	0
$112$	−4.08611	−0.386101
$113$	−4.44730	−0.418367	−0.209183	−	0.977876i	$-0.567081\pi$
$113$	−4.44730	−0.418367	−0.209183	+	0.977876i	$0.567081\pi$
$114$	0	0
$115$	8.38811	0.782195
$116$	15.9375	1.47976
$117$	0	0
$118$	−31.1132	−2.86420
$119$	−4.86027	−0.445540
$120$	0	0
$121$	18.3325	1.66659
$122$	26.4304	2.39290
$123$	0	0
$124$	−21.2559	−1.90884
$125$	9.02725	0.807422
$126$	0	0
$127$	13.6473	1.21100	0.605500	−	0.795845i	$-0.292973\pi$
$127$	13.6473	1.21100	0.605500	+	0.795845i	$0.292973\pi$
$128$	20.3166	1.79575
$129$	0	0
$130$	13.9938	1.22734
$131$	2.12000	0.185225	0.0926126	−	0.995702i	$-0.470478\pi$
$131$	2.12000	0.185225	0.0926126	+	0.995702i	$0.470478\pi$
$132$	0	0
$133$	−12.6200	−1.09429
$134$	−19.4065	−1.67647
$135$	0	0
$136$	−5.72437	−0.490861
$137$	12.9691	1.10802	0.554010	−	0.832510i	$-0.313097\pi$
$137$	12.9691	1.10802	0.554010	+	0.832510i	$0.313097\pi$
$138$	0	0
$139$	−6.15312	−0.521901	−0.260950	−	0.965352i	$-0.584036\pi$
$139$	−6.15312	−0.521901	−0.260950	+	0.965352i	$0.584036\pi$
$140$	10.7096	0.905124
$141$	0	0
$142$	30.0605	2.52262
$143$	−32.1342	−2.68720
$144$	0	0
$145$	−4.54596	−0.377522
$146$	−22.9482	−1.89921
$147$	0	0
$148$	−22.9297	−1.88481
$149$	13.0828	1.07179	0.535894	−	0.844285i	$-0.319975\pi$
$149$	13.0828	1.07179	0.535894	+	0.844285i	$0.319975\pi$
$150$	0	0
$151$	−14.7720	−1.20213	−0.601064	−	0.799201i	$-0.705256\pi$
$151$	−14.7720	−1.20213	−0.601064	+	0.799201i	$0.705256\pi$
$152$	−14.8637	−1.20561
$153$	0	0
$154$	−38.5675	−3.10786
$155$	6.06296	0.486988
$156$	0	0
$157$	−9.47462	−0.756157	−0.378078	−	0.925774i	$-0.623415\pi$
$157$	−9.47462	−0.756157	−0.378078	+	0.925774i	$0.623415\pi$
$158$	−26.5207	−2.10988
$159$	0	0
$160$	−3.98828	−0.315301
$161$	25.3262	1.99598
$162$	0	0
$163$	18.0089	1.41056	0.705282	−	0.708927i	$-0.250820\pi$
$163$	18.0089	1.41056	0.705282	+	0.708927i	$0.250820\pi$
$164$	13.5604	1.05889
$165$	0	0
$166$	−7.22495	−0.560765
$167$	−1.38898	−0.107483	−0.0537413	−	0.998555i	$-0.517115\pi$
$167$	−1.38898	−0.107483	−0.0537413	+	0.998555i	$0.517115\pi$
$168$	0	0
$169$	22.2036	1.70797
$170$	3.78187	0.290056
$171$	0	0
$172$	−14.6294	−1.11548
$173$	13.1807	1.00211	0.501055	−	0.865415i	$-0.332945\pi$
$173$	13.1807	1.00211	0.501055	+	0.865415i	$0.332945\pi$
$174$	0	0
$175$	12.1006	0.914718
$176$	−7.30112	−0.550342
$177$	0	0
$178$	−25.5451	−1.91468
$179$	14.9009	1.11375	0.556873	−	0.830598i	$-0.312001\pi$
$179$	14.9009	1.11375	0.556873	+	0.830598i	$0.312001\pi$
$180$	0	0
$181$	8.25708	0.613744	0.306872	−	0.951751i	$-0.400718\pi$
$181$	8.25708	0.613744	0.306872	+	0.951751i	$0.400718\pi$
$182$	42.2514	3.13188
$183$	0	0
$184$	29.8289	2.19901
$185$	6.54037	0.480858
$186$	0	0
$187$	−8.68439	−0.635065
$188$	−16.1028	−1.17442
$189$	0	0
$190$	9.81988	0.712409
$191$	16.8935	1.22237	0.611186	−	0.791487i	$-0.290693\pi$
$191$	16.8935	1.22237	0.611186	+	0.791487i	$0.290693\pi$
$192$	0	0
$193$	9.60760	0.691570	0.345785	−	0.938314i	$-0.387613\pi$
$193$	9.60760	0.691570	0.345785	+	0.938314i	$0.387613\pi$
$194$	16.6044	1.19213
$195$	0	0
$196$	7.69856	0.549897
$197$	10.3337	0.736249	0.368125	−	0.929776i	$-0.380000\pi$
$197$	10.3337	0.736249	0.368125	+	0.929776i	$0.380000\pi$
$198$	0	0
$199$	3.72015	0.263715	0.131857	−	0.991269i	$-0.457906\pi$
$199$	3.72015	0.263715	0.131857	+	0.991269i	$0.457906\pi$
$200$	14.2519	1.00776
$201$	0	0
$202$	−1.24820	−0.0878231
$203$	−13.7256	−0.963348
$204$	0	0
$205$	−3.86793	−0.270148
$206$	32.1910	2.24285
$207$	0	0
$208$	7.99849	0.554596
$209$	−22.5496	−1.55979
$210$	0	0
$211$	−4.61156	−0.317473	−0.158736	−	0.987321i	$-0.550742\pi$
$211$	−4.61156	−0.317473	−0.158736	+	0.987321i	$0.550742\pi$
$212$	13.6678	0.938707
$213$	0	0
$214$	44.6384	3.05142
$215$	4.17283	0.284584
$216$	0	0
$217$	18.3058	1.24268
$218$	−36.7400	−2.48835
$219$	0	0
$220$	19.1360	1.29015
$221$	9.51389	0.639973
$222$	0	0
$223$	−6.06888	−0.406402	−0.203201	−	0.979137i	$-0.565134\pi$
$223$	−6.06888	−0.406402	−0.203201	+	0.979137i	$0.565134\pi$
$224$	−12.0418	−0.804576
$225$	0	0
$226$	10.4484	0.695014
$227$	22.1809	1.47220	0.736099	−	0.676874i	$-0.236666\pi$
$227$	22.1809	1.47220	0.736099	+	0.676874i	$0.236666\pi$
$228$	0	0
$229$	18.4944	1.22215	0.611073	−	0.791574i	$-0.290738\pi$
$229$	18.4944	1.22215	0.611073	+	0.791574i	$0.290738\pi$
$230$	−19.7068	−1.29943
$231$	0	0
$232$	−16.1659	−1.06134
$233$	−23.6569	−1.54981	−0.774907	−	0.632075i	$-0.782203\pi$
$233$	−23.6569	−1.54981	−0.774907	+	0.632075i	$0.782203\pi$
$234$	0	0
$235$	4.59311	0.299621
$236$	46.6100	3.03405
$237$	0	0
$238$	11.4186	0.740156
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	−6.39331	−0.411830	−0.205915	−	0.978570i	$-0.566017\pi$
$241$	−6.39331	−0.411830	−0.205915	+	0.978570i	$0.566017\pi$
$242$	−43.0699	−2.76864
$243$	0	0
$244$	−39.5949	−2.53480
$245$	−2.19591	−0.140291
$246$	0	0
$247$	24.7035	1.57184
$248$	21.5604	1.36909
$249$	0	0
$250$	−21.2084	−1.34133
$251$	14.7436	0.930605	0.465303	−	0.885152i	$-0.345946\pi$
$251$	14.7436	0.930605	0.465303	+	0.885152i	$0.345946\pi$
$252$	0	0
$253$	45.2531	2.84504
$254$	−32.0625	−2.01178
$255$	0	0
$256$	−23.6719	−1.47949
$257$	−30.1862	−1.88296	−0.941482	−	0.337063i	$-0.890566\pi$
$257$	−30.1862	−1.88296	−0.941482	+	0.337063i	$0.890566\pi$
$258$	0	0
$259$	19.7473	1.22704
$260$	−20.9638	−1.30012
$261$	0	0
$262$	−4.98066	−0.307706
$263$	3.47807	0.214467	0.107233	−	0.994234i	$-0.465801\pi$
$263$	3.47807	0.214467	0.107233	+	0.994234i	$0.465801\pi$
$264$	0	0
$265$	−3.89854	−0.239486
$266$	29.6491	1.81790
$267$	0	0
$268$	29.0725	1.77588
$269$	7.19245	0.438531	0.219266	−	0.975665i	$-0.429634\pi$
$269$	7.19245	0.438531	0.219266	+	0.975665i	$0.429634\pi$
$270$	0	0
$271$	1.10087	0.0668733	0.0334367	−	0.999441i	$-0.489355\pi$
$271$	1.10087	0.0668733	0.0334367	+	0.999441i	$0.489355\pi$
$272$	2.16162	0.131067
$273$	0	0
$274$	−30.4691	−1.84071
$275$	21.6215	1.30382
$276$	0	0
$277$	1.27890	0.0768419	0.0384209	−	0.999262i	$-0.487767\pi$
$277$	1.27890	0.0768419	0.0384209	+	0.999262i	$0.487767\pi$
$278$	14.4559	0.867010
$279$	0	0
$280$	−10.8630	−0.649188
$281$	17.2198	1.02725	0.513624	−	0.858015i	$-0.328303\pi$
$281$	17.2198	1.02725	0.513624	+	0.858015i	$0.328303\pi$
$282$	0	0
$283$	30.3875	1.80635	0.903176	−	0.429270i	$-0.141229\pi$
$283$	30.3875	1.80635	0.903176	+	0.429270i	$0.141229\pi$
$284$	−45.0330	−2.67222
$285$	0	0
$286$	75.4952	4.46413
$287$	−11.6784	−0.689354
$288$	0	0
$289$	−14.4288	−0.848755
$290$	10.6802	0.627160
$291$	0	0
$292$	34.3782	2.01183
$293$	−7.18081	−0.419507	−0.209754	−	0.977754i	$-0.567266\pi$
$293$	−7.18081	−0.419507	−0.209754	+	0.977754i	$0.567266\pi$
$294$	0	0
$295$	−13.2948	−0.774056
$296$	23.2582	1.35185
$297$	0	0
$298$	−30.7364	−1.78051
$299$	−49.5755	−2.86703
$300$	0	0
$301$	12.5990	0.726194
$302$	34.7049	1.99704
$303$	0	0
$304$	5.61279	0.321916
$305$	11.2939	0.646686
$306$	0	0
$307$	9.38748	0.535772	0.267886	−	0.963451i	$-0.413675\pi$
$307$	9.38748	0.535772	0.267886	+	0.963451i	$0.413675\pi$
$308$	57.7772	3.29216
$309$	0	0
$310$	−14.2441	−0.809012
$311$	14.3990	0.816493	0.408247	−	0.912872i	$-0.366140\pi$
$311$	14.3990	0.816493	0.408247	+	0.912872i	$0.366140\pi$
$312$	0	0
$313$	23.4709	1.32666	0.663328	−	0.748329i	$-0.269144\pi$
$313$	23.4709	1.32666	0.663328	+	0.748329i	$0.269144\pi$
$314$	22.2594	1.25617
$315$	0	0
$316$	39.7301	2.23499
$317$	−11.5355	−0.647896	−0.323948	−	0.946075i	$-0.605010\pi$
$317$	−11.5355	−0.647896	−0.323948	+	0.946075i	$0.605010\pi$
$318$	0	0
$319$	−24.5251	−1.37314
$320$	12.0766	0.675103
$321$	0	0
$322$	−59.5006	−3.31584
$323$	6.67619	0.371473
$324$	0	0
$325$	−23.6867	−1.31390
$326$	−42.3095	−2.34331
$327$	0	0
$328$	−13.7547	−0.759476
$329$	13.8679	0.764564
$330$	0	0
$331$	30.7530	1.69034	0.845168	−	0.534501i	$-0.179500\pi$
$331$	30.7530	1.69034	0.845168	+	0.534501i	$0.179500\pi$
$332$	10.8235	0.594019
$333$	0	0
$334$	3.26323	0.178556
$335$	−8.29252	−0.453069
$336$	0	0
$337$	−13.1335	−0.715427	−0.357713	−	0.933831i	$-0.616443\pi$
$337$	−13.1335	−0.715427	−0.357713	+	0.933831i	$0.616443\pi$
$338$	−52.1645	−2.83737
$339$	0	0
$340$	−5.66553	−0.307257
$341$	32.7091	1.77130
$342$	0	0
$343$	14.5874	0.787645
$344$	14.8390	0.800063
$345$	0	0
$346$	−30.9664	−1.66476
$347$	14.4065	0.773382	0.386691	−	0.922209i	$-0.373618\pi$
$347$	14.4065	0.773382	0.386691	+	0.922209i	$0.373618\pi$
$348$	0	0
$349$	29.1445	1.56007	0.780035	−	0.625736i	$-0.215201\pi$
$349$	29.1445	1.56007	0.780035	+	0.625736i	$0.215201\pi$
$350$	−28.4288	−1.51958
$351$	0	0
$352$	−21.5164	−1.14683
$353$	−21.7818	−1.15933	−0.579665	−	0.814855i	$-0.696816\pi$
$353$	−21.7818	−1.15933	−0.579665	+	0.814855i	$0.696816\pi$
$354$	0	0
$355$	12.8450	0.681744
$356$	38.2685	2.02823
$357$	0	0
$358$	−35.0078	−1.85022
$359$	6.22536	0.328562	0.164281	−	0.986414i	$-0.447470\pi$
$359$	6.22536	0.328562	0.164281	+	0.986414i	$0.447470\pi$
$360$	0	0
$361$	−1.66482	−0.0876220
$362$	−19.3989	−1.01959
$363$	0	0
$364$	−63.2958	−3.31760
$365$	−9.80591	−0.513265
$366$	0	0
$367$	−8.97848	−0.468673	−0.234336	−	0.972156i	$-0.575292\pi$
$367$	−8.97848	−0.468673	−0.234336	+	0.972156i	$0.575292\pi$
$368$	−11.2639	−0.587171
$369$	0	0
$370$	−15.3658	−0.798827
$371$	−11.7709	−0.611112
$372$	0	0
$373$	9.88717	0.511938	0.255969	−	0.966685i	$-0.417605\pi$
$373$	9.88717	0.511938	0.255969	+	0.966685i	$0.417605\pi$
$374$	20.4028	1.05501
$375$	0	0
$376$	16.3335	0.842336
$377$	26.8676	1.38375
$378$	0	0
$379$	23.2972	1.19670	0.598348	−	0.801237i	$-0.295824\pi$
$379$	23.2972	1.19670	0.598348	+	0.801237i	$0.295824\pi$
$380$	−14.7109	−0.754655
$381$	0	0
$382$	−39.6891	−2.03067
$383$	1.03131	0.0526973	0.0263486	−	0.999653i	$-0.491612\pi$
$383$	1.03131	0.0526973	0.0263486	+	0.999653i	$0.491612\pi$
$384$	0	0
$385$	−16.4801	−0.839906
$386$	−22.5718	−1.14887
$387$	0	0
$388$	−24.8747	−1.26282
$389$	8.30103	0.420879	0.210439	−	0.977607i	$-0.432511\pi$
$389$	8.30103	0.420879	0.210439	+	0.977607i	$0.432511\pi$
$390$	0	0
$391$	−13.3980	−0.677564
$392$	−7.80885	−0.394406
$393$	0	0
$394$	−24.2778	−1.22310
$395$	−11.3325	−0.570198
$396$	0	0
$397$	−20.6816	−1.03798	−0.518989	−	0.854781i	$-0.673691\pi$
$397$	−20.6816	−1.03798	−0.518989	+	0.854781i	$0.673691\pi$
$398$	−8.74001	−0.438097
$399$	0	0
$400$	−5.38177	−0.269089
$401$	−8.72249	−0.435580	−0.217790	−	0.975996i	$-0.569885\pi$
$401$	−8.72249	−0.435580	−0.217790	+	0.975996i	$0.569885\pi$
$402$	0	0
$403$	−35.8334	−1.78499
$404$	1.86990	0.0930311
$405$	0	0
$406$	32.2465	1.60037
$407$	35.2847	1.74900
$408$	0	0
$409$	10.5870	0.523496	0.261748	−	0.965136i	$-0.415701\pi$
$409$	10.5870	0.523496	0.261748	+	0.965136i	$0.415701\pi$
$410$	9.08719	0.448784
$411$	0	0
$412$	−48.2246	−2.37586
$413$	−40.1410	−1.97521
$414$	0	0
$415$	−3.08726	−0.151548
$416$	23.5716	1.15569
$417$	0	0
$418$	52.9774	2.59121
$419$	0.521081	0.0254565	0.0127282	−	0.999919i	$-0.495948\pi$
$419$	0.521081	0.0254565	0.0127282	+	0.999919i	$0.495948\pi$
$420$	0	0
$421$	−5.47847	−0.267004	−0.133502	−	0.991049i	$-0.542622\pi$
$421$	−5.47847	−0.267004	−0.133502	+	0.991049i	$0.542622\pi$
$422$	10.8343	0.527403
$423$	0	0
$424$	−13.8636	−0.673275
$425$	−6.40140	−0.310514
$426$	0	0
$427$	34.0996	1.65019
$428$	−66.8718	−3.23237
$429$	0	0
$430$	−9.80352	−0.472767
$431$	31.5388	1.51917	0.759585	−	0.650408i	$-0.225402\pi$
$431$	31.5388	1.51917	0.759585	+	0.650408i	$0.225402\pi$
$432$	0	0
$433$	0.702315	0.0337511	0.0168756	−	0.999858i	$-0.494628\pi$
$433$	0.702315	0.0337511	0.0168756	+	0.999858i	$0.494628\pi$
$434$	−43.0072	−2.06441
$435$	0	0
$436$	55.0394	2.63591
$437$	−34.7887	−1.66417
$438$	0	0
$439$	13.3330	0.636349	0.318174	−	0.948032i	$-0.396930\pi$
$439$	13.3330	0.636349	0.318174	+	0.948032i	$0.396930\pi$
$440$	−19.4101	−0.925341
$441$	0	0
$442$	−22.3516	−1.06316
$443$	−32.0255	−1.52158	−0.760788	−	0.649001i	$-0.775187\pi$
$443$	−32.0255	−1.52158	−0.760788	+	0.649001i	$0.775187\pi$
$444$	0	0
$445$	−10.9156	−0.517447
$446$	14.2580	0.675138
$447$	0	0
$448$	36.4628	1.72271
$449$	−7.64853	−0.360957	−0.180478	−	0.983579i	$-0.557765\pi$
$449$	−7.64853	−0.360957	−0.180478	+	0.983579i	$0.557765\pi$
$450$	0	0
$451$	−20.8671	−0.982594
$452$	−15.6525	−0.736229
$453$	0	0
$454$	−52.1111	−2.44570
$455$	18.0543	0.846397
$456$	0	0
$457$	14.7324	0.689152	0.344576	−	0.938758i	$-0.388023\pi$
$457$	14.7324	0.689152	0.344576	+	0.938758i	$0.388023\pi$
$458$	−43.4503	−2.03030
$459$	0	0
$460$	29.5223	1.37648
$461$	17.4109	0.810904	0.405452	−	0.914116i	$-0.367114\pi$
$461$	17.4109	0.810904	0.405452	+	0.914116i	$0.367114\pi$
$462$	0	0
$463$	−16.4261	−0.763387	−0.381694	−	0.924289i	$-0.624659\pi$
$463$	−16.4261	−0.763387	−0.381694	+	0.924289i	$0.624659\pi$
$464$	6.10450	0.283394
$465$	0	0
$466$	55.5788	2.57464
$467$	−29.0498	−1.34426	−0.672131	−	0.740432i	$-0.734621\pi$
$467$	−29.0498	−1.34426	−0.672131	+	0.740432i	$0.734621\pi$
$468$	0	0
$469$	−25.0376	−1.15613
$470$	−10.7909	−0.497747
$471$	0	0
$472$	−47.2777	−2.17613
$473$	22.5120	1.03510
$474$	0	0
$475$	−16.6217	−0.762655
$476$	−17.1059	−0.784048
$477$	0	0
$478$	2.34937	0.107458
$479$	−13.4958	−0.616637	−0.308319	−	0.951283i	$-0.599766\pi$
$479$	−13.4958	−0.616637	−0.308319	+	0.951283i	$0.599766\pi$
$480$	0	0
$481$	−38.6550	−1.76252
$482$	15.0203	0.684154
$483$	0	0
$484$	64.5220	2.93282
$485$	7.09516	0.322175
$486$	0	0
$487$	−11.1204	−0.503911	−0.251956	−	0.967739i	$-0.581074\pi$
$487$	−11.1204	−0.503911	−0.251956	+	0.967739i	$0.581074\pi$
$488$	40.1621	1.81805
$489$	0	0
$490$	5.15900	0.233060
$491$	−1.02832	−0.0464075	−0.0232037	−	0.999731i	$-0.507387\pi$
$491$	−1.02832	−0.0464075	−0.0232037	+	0.999731i	$0.507387\pi$
$492$	0	0
$493$	7.26106	0.327022
$494$	−58.0376	−2.61123
$495$	0	0
$496$	−8.14158	−0.365568
$497$	38.7829	1.73965
$498$	0	0
$499$	−2.02434	−0.0906219	−0.0453109	−	0.998973i	$-0.514428\pi$
$499$	−2.02434	−0.0906219	−0.0453109	+	0.998973i	$0.514428\pi$
$500$	31.7718	1.42088
$501$	0	0
$502$	−34.6381	−1.54597
$503$	−20.3575	−0.907694	−0.453847	−	0.891080i	$-0.649949\pi$
$503$	−20.3575	−0.907694	−0.453847	+	0.891080i	$0.649949\pi$
$504$	0	0
$505$	−0.533364	−0.0237344
$506$	−106.316	−4.72634
$507$	0	0
$508$	48.0322	2.13108
$509$	−6.11257	−0.270935	−0.135468	−	0.990782i	$-0.543254\pi$
$509$	−6.11257	−0.270935	−0.135468	+	0.990782i	$0.543254\pi$
$510$	0	0
$511$	−29.6069	−1.30973
$512$	14.9808	0.662063
$513$	0	0
$514$	70.9186	3.12808
$515$	13.7554	0.606136
$516$	0	0
$517$	24.7794	1.08980
$518$	−46.3937	−2.03842
$519$	0	0
$520$	21.2641	0.932493
$521$	3.89974	0.170851	0.0854253	−	0.996345i	$-0.472775\pi$
$521$	3.89974	0.170851	0.0854253	+	0.996345i	$0.472775\pi$
$522$	0	0
$523$	−41.6398	−1.82078	−0.910391	−	0.413750i	$-0.864219\pi$
$523$	−41.6398	−1.82078	−0.910391	+	0.413750i	$0.864219\pi$
$524$	7.46142	0.325954
$525$	0	0
$526$	−8.17127	−0.356284
$527$	−9.68409	−0.421845
$528$	0	0
$529$	46.8148	2.03543
$530$	9.15912	0.397847
$531$	0	0
$532$	−44.4166	−1.92571
$533$	22.8603	0.990188
$534$	0	0
$535$	19.0743	0.824652
$536$	−29.4890	−1.27373
$537$	0	0
$538$	−16.8977	−0.728513
$539$	−11.8467	−0.510275
$540$	0	0
$541$	−39.3222	−1.69059	−0.845297	−	0.534297i	$-0.820576\pi$
$541$	−39.3222	−1.69059	−0.845297	+	0.534297i	$0.820576\pi$
$542$	−2.58636	−0.111094
$543$	0	0
$544$	6.37030	0.273124
$545$	−15.6992	−0.672482
$546$	0	0
$547$	−7.29284	−0.311819	−0.155910	−	0.987771i	$-0.549831\pi$
$547$	−7.29284	−0.311819	−0.155910	+	0.987771i	$0.549831\pi$
$548$	45.6451	1.94986
$549$	0	0
$550$	−50.7968	−2.16598
$551$	18.8538	0.803201
$552$	0	0
$553$	−34.2161	−1.45501
$554$	−3.00462	−0.127654
$555$	0	0
$556$	−21.6561	−0.918425
$557$	−4.22304	−0.178936	−0.0894680	−	0.995990i	$-0.528517\pi$
$557$	−4.22304	−0.178936	−0.0894680	+	0.995990i	$0.528517\pi$
$558$	0	0
$559$	−24.6623	−1.04310
$560$	4.10205	0.173343
$561$	0	0
$562$	−40.4557	−1.70652
$563$	−25.8421	−1.08912	−0.544558	−	0.838723i	$-0.683302\pi$
$563$	−25.8421	−1.08912	−0.544558	+	0.838723i	$0.683302\pi$
$564$	0	0
$565$	4.46464	0.187829
$566$	−71.3916	−3.00081
$567$	0	0
$568$	45.6781	1.91661
$569$	5.65454	0.237051	0.118525	−	0.992951i	$-0.462183\pi$
$569$	5.65454	0.237051	0.118525	+	0.992951i	$0.462183\pi$
$570$	0	0
$571$	25.4427	1.06474	0.532372	−	0.846511i	$-0.321301\pi$
$571$	25.4427	1.06474	0.532372	+	0.846511i	$0.321301\pi$
$572$	−113.098	−4.72885
$573$	0	0
$574$	27.4369	1.14519
$575$	33.3568	1.39108
$576$	0	0
$577$	−42.2095	−1.75720	−0.878602	−	0.477555i	$-0.841523\pi$
$577$	−42.2095	−1.75720	−0.878602	+	0.477555i	$0.841523\pi$
$578$	33.8987	1.41000
$579$	0	0
$580$	−15.9997	−0.664351
$581$	−9.32136	−0.386715
$582$	0	0
$583$	−21.0323	−0.871069
$584$	−34.8707	−1.44296
$585$	0	0
$586$	16.8704	0.696909
$587$	12.3293	0.508886	0.254443	−	0.967088i	$-0.418108\pi$
$587$	12.3293	0.508886	0.254443	+	0.967088i	$0.418108\pi$
$588$	0	0
$589$	−25.1454	−1.03610
$590$	31.2345	1.28590
$591$	0	0
$592$	−8.78267	−0.360966
$593$	21.5755	0.886001	0.443000	−	0.896521i	$-0.353914\pi$
$593$	21.5755	0.886001	0.443000	+	0.896521i	$0.353914\pi$
$594$	0	0
$595$	4.87922	0.200029
$596$	46.0455	1.88610
$597$	0	0
$598$	116.471	4.76286
$599$	−11.5114	−0.470345	−0.235172	−	0.971954i	$-0.575565\pi$
$599$	−11.5114	−0.470345	−0.235172	+	0.971954i	$0.575565\pi$
$600$	0	0
$601$	0.244395	0.00996906	0.00498453	−	0.999988i	$-0.498413\pi$
$601$	0.244395	0.00996906	0.00498453	+	0.999988i	$0.498413\pi$
$602$	−29.5997	−1.20639
$603$	0	0
$604$	−51.9906	−2.11547
$605$	−18.4040	−0.748229
$606$	0	0
$607$	−31.4574	−1.27682	−0.638408	−	0.769698i	$-0.720407\pi$
$607$	−31.4574	−1.27682	−0.638408	+	0.769698i	$0.720407\pi$
$608$	16.5409	0.670823
$609$	0	0
$610$	−26.5335	−1.07431
$611$	−27.1462	−1.09822
$612$	0	0
$613$	23.1581	0.935348	0.467674	−	0.883901i	$-0.345092\pi$
$613$	23.1581	0.935348	0.467674	+	0.883901i	$0.345092\pi$
$614$	−22.0547	−0.890053
$615$	0	0
$616$	−58.6049	−2.36126
$617$	−41.0781	−1.65374	−0.826871	−	0.562392i	$-0.809881\pi$
$617$	−41.0781	−1.65374	−0.826871	+	0.562392i	$0.809881\pi$
$618$	0	0
$619$	36.5576	1.46938	0.734688	−	0.678405i	$-0.237329\pi$
$619$	36.5576	1.46938	0.734688	+	0.678405i	$0.237329\pi$
$620$	21.3388	0.856987
$621$	0	0
$622$	−33.8286	−1.35640
$623$	−32.9573	−1.32040
$624$	0	0
$625$	10.8985	0.435939
$626$	−55.1419	−2.20391
$627$	0	0
$628$	−33.3463	−1.33066
$629$	−10.4466	−0.416535
$630$	0	0
$631$	−11.0220	−0.438777	−0.219389	−	0.975638i	$-0.570406\pi$
$631$	−11.0220	−0.438777	−0.219389	+	0.975638i	$0.570406\pi$
$632$	−40.2993	−1.60302
$633$	0	0
$634$	27.1011	1.07632
$635$	−13.7005	−0.543688
$636$	0	0
$637$	12.9783	0.514218
$638$	57.6185	2.28114
$639$	0	0
$640$	−20.3959	−0.806218
$641$	−1.20590	−0.0476300	−0.0238150	−	0.999716i	$-0.507581\pi$
$641$	−1.20590	−0.0476300	−0.0238150	+	0.999716i	$0.507581\pi$
$642$	0	0
$643$	4.55025	0.179444	0.0897221	−	0.995967i	$-0.471402\pi$
$643$	4.55025	0.179444	0.0897221	+	0.995967i	$0.471402\pi$
$644$	89.1365	3.51247
$645$	0	0
$646$	−15.6848	−0.617112
$647$	−45.2956	−1.78075	−0.890377	−	0.455224i	$-0.849559\pi$
$647$	−45.2956	−1.78075	−0.890377	+	0.455224i	$0.849559\pi$
$648$	0	0
$649$	−71.7245	−2.81543
$650$	55.6488	2.18272
$651$	0	0
$652$	63.3829	2.48227
$653$	9.08772	0.355630	0.177815	−	0.984064i	$-0.443097\pi$
$653$	9.08772	0.355630	0.177815	+	0.984064i	$0.443097\pi$
$654$	0	0
$655$	−2.12827	−0.0831583
$656$	5.19401	0.202792
$657$	0	0
$658$	−32.5809	−1.27014
$659$	41.5123	1.61709	0.808544	−	0.588435i	$-0.200256\pi$
$659$	41.5123	1.61709	0.808544	+	0.588435i	$0.200256\pi$
$660$	0	0
$661$	−4.86598	−0.189264	−0.0946322	−	0.995512i	$-0.530168\pi$
$661$	−4.86598	−0.189264	−0.0946322	+	0.995512i	$0.530168\pi$
$662$	−72.2501	−2.80808
$663$	0	0
$664$	−10.9786	−0.426052
$665$	12.6692	0.491292
$666$	0	0
$667$	−37.8364	−1.46503
$668$	−4.88858	−0.189145
$669$	0	0
$670$	19.4822	0.752663
$671$	60.9295	2.35216
$672$	0	0
$673$	7.06998	0.272528	0.136264	−	0.990673i	$-0.456491\pi$
$673$	7.06998	0.272528	0.136264	+	0.990673i	$0.456491\pi$
$674$	30.8554	1.18851
$675$	0	0
$676$	78.1464	3.00563
$677$	5.51786	0.212068	0.106034	−	0.994362i	$-0.466185\pi$
$677$	5.51786	0.212068	0.106034	+	0.994362i	$0.466185\pi$
$678$	0	0
$679$	21.4224	0.822116
$680$	5.74670	0.220376
$681$	0	0
$682$	−76.8458	−2.94258
$683$	23.7902	0.910306	0.455153	−	0.890413i	$-0.349584\pi$
$683$	23.7902	0.910306	0.455153	+	0.890413i	$0.349584\pi$
$684$	0	0
$685$	−13.0196	−0.497454
$686$	−34.2712	−1.30848
$687$	0	0
$688$	−5.60344	−0.213629
$689$	23.0412	0.877801
$690$	0	0
$691$	5.96615	0.226963	0.113481	−	0.993540i	$-0.463800\pi$
$691$	5.96615	0.226963	0.113481	+	0.993540i	$0.463800\pi$
$692$	46.3900	1.76348
$693$	0	0
$694$	−33.8462	−1.28478
$695$	6.17711	0.234311
$696$	0	0
$697$	6.17806	0.234011
$698$	−68.4712	−2.59167
$699$	0	0
$700$	42.5885	1.60969
$701$	−24.5759	−0.928219	−0.464110	−	0.885778i	$-0.653626\pi$
$701$	−24.5759	−0.928219	−0.464110	+	0.885778i	$0.653626\pi$
$702$	0	0
$703$	−27.1254	−1.02305
$704$	65.1523	2.45552
$705$	0	0
$706$	51.1736	1.92594
$707$	−1.61038	−0.0605646
$708$	0	0
$709$	−11.9824	−0.450010	−0.225005	−	0.974358i	$-0.572240\pi$
$709$	−11.9824	−0.450010	−0.225005	+	0.974358i	$0.572240\pi$
$710$	−30.1777	−1.13255
$711$	0	0
$712$	−38.8167	−1.45472
$713$	50.4624	1.88983
$714$	0	0
$715$	32.2596	1.20644
$716$	52.4443	1.95994
$717$	0	0
$718$	−14.6257	−0.545825
$719$	14.5964	0.544354	0.272177	−	0.962247i	$-0.412256\pi$
$719$	14.5964	0.544354	0.272177	+	0.962247i	$0.412256\pi$
$720$	0	0
$721$	41.5316	1.54672
$722$	3.91127	0.145563
$723$	0	0
$724$	29.0611	1.08005
$725$	−18.0778	−0.671394
$726$	0	0
$727$	10.5557	0.391487	0.195744	−	0.980655i	$-0.437288\pi$
$727$	10.5557	0.391487	0.195744	+	0.980655i	$0.437288\pi$
$728$	64.2026	2.37951
$729$	0	0
$730$	23.0377	0.852664
$731$	−6.66507	−0.246516
$732$	0	0
$733$	9.78662	0.361477	0.180739	−	0.983531i	$-0.442151\pi$
$733$	9.78662	0.361477	0.180739	+	0.983531i	$0.442151\pi$
$734$	21.0938	0.778585
$735$	0	0
$736$	−33.1947	−1.22357
$737$	−44.7374	−1.64792
$738$	0	0
$739$	34.0495	1.25253	0.626265	−	0.779610i	$-0.284583\pi$
$739$	34.0495	1.25253	0.626265	+	0.779610i	$0.284583\pi$
$740$	23.0191	0.846199
$741$	0	0
$742$	27.6541	1.01521
$743$	−19.5931	−0.718802	−0.359401	−	0.933183i	$-0.617019\pi$
$743$	−19.5931	−0.718802	−0.359401	+	0.933183i	$0.617019\pi$
$744$	0	0
$745$	−13.1339	−0.481187
$746$	−23.2286	−0.850461
$747$	0	0
$748$	−30.5650	−1.11757
$749$	57.5908	2.10432
$750$	0	0
$751$	52.5008	1.91578	0.957891	−	0.287131i	$-0.0927014\pi$
$751$	52.5008	1.91578	0.957891	+	0.287131i	$0.0927014\pi$
$752$	−6.16781	−0.224917
$753$	0	0
$754$	−63.1220	−2.29877
$755$	14.8296	0.539705
$756$	0	0
$757$	26.6845	0.969864	0.484932	−	0.874552i	$-0.338844\pi$
$757$	26.6845	0.969864	0.484932	+	0.874552i	$0.338844\pi$
$758$	−54.7337	−1.98802
$759$	0	0
$760$	14.9217	0.541266
$761$	−23.9937	−0.869772	−0.434886	−	0.900485i	$-0.643211\pi$
$761$	−23.9937	−0.869772	−0.434886	+	0.900485i	$0.643211\pi$
$762$	0	0
$763$	−47.4006	−1.71602
$764$	59.4574	2.15109
$765$	0	0
$766$	−2.42292	−0.0875436
$767$	78.5754	2.83719
$768$	0	0
$769$	10.8966	0.392940	0.196470	−	0.980510i	$-0.437052\pi$
$769$	10.8966	0.392940	0.196470	+	0.980510i	$0.437052\pi$
$770$	38.7179	1.39530
$771$	0	0
$772$	33.8143	1.21700
$773$	16.7110	0.601052	0.300526	−	0.953774i	$-0.402838\pi$
$773$	16.7110	0.601052	0.300526	+	0.953774i	$0.402838\pi$
$774$	0	0
$775$	24.1104	0.866072
$776$	25.2310	0.905742
$777$	0	0
$778$	−19.5022	−0.699187
$779$	16.0418	0.574756
$780$	0	0
$781$	69.2978	2.47967
$782$	31.4767	1.12561
$783$	0	0
$784$	2.94875	0.105313
$785$	9.51157	0.339482
$786$	0	0
$787$	16.7898	0.598493	0.299246	−	0.954176i	$-0.403265\pi$
$787$	16.7898	0.598493	0.299246	+	0.954176i	$0.403265\pi$
$788$	36.3700	1.29563
$789$	0	0
$790$	26.6242	0.947245
$791$	13.4801	0.479296
$792$	0	0
$793$	−66.7493	−2.37034
$794$	48.5886	1.72435
$795$	0	0
$796$	13.0932	0.464077
$797$	7.73287	0.273912	0.136956	−	0.990577i	$-0.456268\pi$
$797$	7.73287	0.273912	0.136956	+	0.990577i	$0.456268\pi$
$798$	0	0
$799$	−7.33636	−0.259542
$800$	−15.8601	−0.560739
$801$	0	0
$802$	20.4924	0.723610
$803$	−52.9020	−1.86687
$804$	0	0
$805$	−25.4249	−0.896111
$806$	84.1858	2.96532
$807$	0	0
$808$	−1.89669	−0.0667253
$809$	−29.2906	−1.02980	−0.514901	−	0.857250i	$-0.672171\pi$
$809$	−29.2906	−1.02980	−0.514901	+	0.857250i	$0.672171\pi$
$810$	0	0
$811$	−46.3133	−1.62628	−0.813140	−	0.582069i	$-0.802243\pi$
$811$	−46.3133	−1.62628	−0.813140	+	0.582069i	$0.802243\pi$
$812$	−48.3078	−1.69527
$813$	0	0
$814$	−82.8969	−2.90553
$815$	−18.0791	−0.633284
$816$	0	0
$817$	−17.3063	−0.605471
$818$	−24.8729	−0.869660
$819$	0	0
$820$	−13.6133	−0.475398
$821$	35.7976	1.24934	0.624672	−	0.780887i	$-0.285233\pi$
$821$	35.7976	1.24934	0.624672	+	0.780887i	$0.285233\pi$
$822$	0	0
$823$	13.2733	0.462679	0.231339	−	0.972873i	$-0.425689\pi$
$823$	13.2733	0.462679	0.231339	+	0.972873i	$0.425689\pi$
$824$	48.9155	1.70405
$825$	0	0
$826$	94.3062	3.28133
$827$	−18.4893	−0.642936	−0.321468	−	0.946920i	$-0.604176\pi$
$827$	−18.4893	−0.642936	−0.321468	+	0.946920i	$0.604176\pi$
$828$	0	0
$829$	3.79338	0.131750	0.0658748	−	0.997828i	$-0.479016\pi$
$829$	3.79338	0.131750	0.0658748	+	0.997828i	$0.479016\pi$
$830$	7.25312	0.251760
$831$	0	0
$832$	−71.3754	−2.47450
$833$	3.50742	0.121525
$834$	0	0
$835$	1.39440	0.0482552
$836$	−79.3642	−2.74487
$837$	0	0
$838$	−1.22421	−0.0422897
$839$	2.93323	0.101266	0.0506332	−	0.998717i	$-0.483876\pi$
$839$	2.93323	0.101266	0.0506332	+	0.998717i	$0.483876\pi$
$840$	0	0
$841$	−8.49446	−0.292912
$842$	12.8709	0.443562
$843$	0	0
$844$	−16.2306	−0.558679
$845$	−22.2902	−0.766806
$846$	0	0
$847$	−55.5671	−1.90931
$848$	5.23512	0.179775
$849$	0	0
$850$	15.0393	0.515843
$851$	54.4360	1.86604
$852$	0	0
$853$	−50.9904	−1.74588	−0.872940	−	0.487828i	$-0.837789\pi$
$853$	−50.9904	−1.74588	−0.872940	+	0.487828i	$0.837789\pi$
$854$	−80.1125	−2.74139
$855$	0	0
$856$	67.8298	2.31838
$857$	9.42608	0.321989	0.160994	−	0.986955i	$-0.448530\pi$
$857$	9.42608	0.321989	0.160994	+	0.986955i	$0.448530\pi$
$858$	0	0
$859$	16.6140	0.566862	0.283431	−	0.958993i	$-0.408527\pi$
$859$	16.6140	0.566862	0.283431	+	0.958993i	$0.408527\pi$
$860$	14.6864	0.500803
$861$	0	0
$862$	−74.0963	−2.52373
$863$	38.2003	1.30035	0.650176	−	0.759784i	$-0.274695\pi$
$863$	38.2003	1.30035	0.650176	+	0.759784i	$0.274695\pi$
$864$	0	0
$865$	−13.2321	−0.449905
$866$	−1.65000	−0.0560692
$867$	0	0
$868$	64.4281	2.18683
$869$	−61.1376	−2.07395
$870$	0	0
$871$	49.0106	1.66066
$872$	−55.8279	−1.89057
$873$	0	0
$874$	81.7315	2.76461
$875$	−27.3622	−0.925012
$876$	0	0
$877$	8.21545	0.277416	0.138708	−	0.990333i	$-0.455705\pi$
$877$	8.21545	0.277416	0.138708	+	0.990333i	$0.455705\pi$
$878$	−31.3241	−1.05714
$879$	0	0
$880$	7.32959	0.247080
$881$	12.6333	0.425627	0.212814	−	0.977093i	$-0.431737\pi$
$881$	12.6333	0.425627	0.212814	+	0.977093i	$0.431737\pi$
$882$	0	0
$883$	18.5155	0.623096	0.311548	−	0.950230i	$-0.399153\pi$
$883$	18.5155	0.623096	0.311548	+	0.950230i	$0.399153\pi$
$884$	33.4845	1.12621
$885$	0	0
$886$	75.2397	2.52773
$887$	5.17355	0.173711	0.0868555	−	0.996221i	$-0.472318\pi$
$887$	5.17355	0.173711	0.0868555	+	0.996221i	$0.472318\pi$
$888$	0	0
$889$	−41.3659	−1.38737
$890$	25.6447	0.859611
$891$	0	0
$892$	−21.3596	−0.715174
$893$	−19.0494	−0.637463
$894$	0	0
$895$	−14.9590	−0.500025
$896$	−61.5811	−2.05728
$897$	0	0
$898$	17.9692	0.599641
$899$	−27.3483	−0.912115
$900$	0	0
$901$	6.22697	0.207450
$902$	49.0246	1.63234
$903$	0	0
$904$	15.8767	0.528051
$905$	−8.28928	−0.275545
$906$	0	0
$907$	37.4786	1.24446	0.622228	−	0.782836i	$-0.286228\pi$
$907$	37.4786	1.24446	0.622228	+	0.782836i	$0.286228\pi$
$908$	78.0665	2.59073
$909$	0	0
$910$	−42.4161	−1.40608
$911$	1.45955	0.0483571	0.0241786	−	0.999708i	$-0.492303\pi$
$911$	1.45955	0.0483571	0.0241786	+	0.999708i	$0.492303\pi$
$912$	0	0
$913$	−16.6555	−0.551217
$914$	−34.6118	−1.14486
$915$	0	0
$916$	65.0919	2.15070
$917$	−6.42586	−0.212201
$918$	0	0
$919$	34.5584	1.13998	0.569989	−	0.821652i	$-0.306947\pi$
$919$	34.5584	1.13998	0.569989	+	0.821652i	$0.306947\pi$
$920$	−29.9452	−0.987265
$921$	0	0
$922$	−40.9045	−1.34712
$923$	−75.9169	−2.49883
$924$	0	0
$925$	26.0089	0.855169
$926$	38.5911	1.26818
$927$	0	0
$928$	17.9900	0.590550
$929$	8.07174	0.264825	0.132413	−	0.991195i	$-0.457728\pi$
$929$	8.07174	0.264825	0.132413	+	0.991195i	$0.457728\pi$
$930$	0	0
$931$	9.10726	0.298479
$932$	−83.2613	−2.72732
$933$	0	0
$934$	68.2486	2.23316
$935$	8.71826	0.285117
$936$	0	0
$937$	32.7210	1.06895	0.534474	−	0.845185i	$-0.320510\pi$
$937$	32.7210	1.06895	0.534474	+	0.845185i	$0.320510\pi$
$938$	58.8225	1.92062
$939$	0	0
$940$	16.1656	0.527264
$941$	−37.3568	−1.21780	−0.608899	−	0.793248i	$-0.708388\pi$
$941$	−37.3568	−1.21780	−0.608899	+	0.793248i	$0.708388\pi$
$942$	0	0
$943$	−32.1930	−1.04835
$944$	17.8529	0.581061
$945$	0	0
$946$	−52.8891	−1.71957
$947$	−30.4423	−0.989243	−0.494622	−	0.869108i	$-0.664693\pi$
$947$	−30.4423	−0.989243	−0.494622	+	0.869108i	$0.664693\pi$
$948$	0	0
$949$	57.9550	1.88130
$950$	39.0505	1.26696
$951$	0	0
$952$	17.3510	0.562348
$953$	55.3075	1.79158	0.895792	−	0.444473i	$-0.146609\pi$
$953$	55.3075	1.79158	0.895792	+	0.444473i	$0.146609\pi$
$954$	0	0
$955$	−16.9594	−0.548793
$956$	−3.51954	−0.113830
$957$	0	0
$958$	31.7066	1.02439
$959$	−39.3101	−1.26939
$960$	0	0
$961$	5.47439	0.176593
$962$	90.8149	2.92799
$963$	0	0
$964$	−22.5015	−0.724725
$965$	−9.64507	−0.310486
$966$	0	0
$967$	30.1750	0.970364	0.485182	−	0.874413i	$-0.338753\pi$
$967$	30.1750	0.970364	0.485182	+	0.874413i	$0.338753\pi$
$968$	−65.4463	−2.10352
$969$	0	0
$970$	−16.6692	−0.535215
$971$	−7.15915	−0.229748	−0.114874	−	0.993380i	$-0.536646\pi$
$971$	−7.15915	−0.229748	−0.114874	+	0.993380i	$0.536646\pi$
$972$	0	0
$973$	18.6505	0.597908
$974$	26.1258	0.837125
$975$	0	0
$976$	−15.1659	−0.485448
$977$	−5.80624	−0.185758	−0.0928790	−	0.995677i	$-0.529607\pi$
$977$	−5.80624	−0.185758	−0.0928790	+	0.995677i	$0.529607\pi$
$978$	0	0
$979$	−58.8884	−1.88208
$980$	−7.72858	−0.246881
$981$	0	0
$982$	2.41591	0.0770947
$983$	34.1351	1.08874	0.544370	−	0.838845i	$-0.316769\pi$
$983$	34.1351	1.08874	0.544370	+	0.838845i	$0.316769\pi$
$984$	0	0
$985$	−10.3740	−0.330545
$986$	−17.0589	−0.543267
$987$	0	0
$988$	86.9448	2.76608
$989$	34.7307	1.10437
$990$	0	0
$991$	5.72087	0.181729	0.0908647	−	0.995863i	$-0.471037\pi$
$991$	5.72087	0.181729	0.0908647	+	0.995863i	$0.471037\pi$
$992$	−23.9933	−0.761787
$993$	0	0
$994$	−91.1155	−2.89001
$995$	−3.73466	−0.118397
$996$	0	0
$997$	−13.8469	−0.438535	−0.219267	−	0.975665i	$-0.570367\pi$
$997$	−13.8469	−0.438535	−0.219267	+	0.975665i	$0.570367\pi$
$998$	4.75592	0.150546
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.h.1.2		12
3.2	odd	2		717.2.a.g.1.11	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.2.a.g.1.11	✓	12	3.2	odd	2
2151.2.a.h.1.2		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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-2.34937 q^{2} +3.51954 q^{4} -1.00390 q^{5} -3.03107 q^{7} -3.56996 q^{8} +O(q^{10})\)	\(q-2.34937 q^{2} +3.51954 q^{4} -1.00390 q^{5} -3.03107 q^{7} -3.56996 q^{8} +2.35853 q^{10} -5.41595 q^{11} +5.93326 q^{13} +7.12110 q^{14} +1.34808 q^{16} +1.60348 q^{17} +4.16355 q^{19} -3.53326 q^{20} +12.7241 q^{22} -8.35553 q^{23} -3.99219 q^{25} -13.9394 q^{26} -10.6680 q^{28} +4.52830 q^{29} -6.03940 q^{31} +3.97279 q^{32} -3.76718 q^{34} +3.04289 q^{35} -6.51496 q^{37} -9.78173 q^{38} +3.58388 q^{40} +3.85290 q^{41} -4.15662 q^{43} -19.0616 q^{44} +19.6302 q^{46} -4.57526 q^{47} +2.18738 q^{49} +9.37912 q^{50} +20.8823 q^{52} +3.88340 q^{53} +5.43707 q^{55} +10.8208 q^{56} -10.6387 q^{58} +13.2432 q^{59} -11.2500 q^{61} +14.1888 q^{62} -12.0297 q^{64} -5.95640 q^{65} +8.26031 q^{67} +5.64352 q^{68} -7.14887 q^{70} -12.7951 q^{71} +9.76782 q^{73} +15.3061 q^{74} +14.6538 q^{76} +16.4161 q^{77} +11.2884 q^{79} -1.35333 q^{80} -9.05189 q^{82} +3.07527 q^{83} -1.60974 q^{85} +9.76543 q^{86} +19.3347 q^{88} +10.8732 q^{89} -17.9841 q^{91} -29.4076 q^{92} +10.7490 q^{94} -4.17979 q^{95} -7.06760 q^{97} -5.13896 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 15 * q^4 + q^5 + 11 * q^7 - 9 * q^8	\( 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8} - 15 q^{11} + 7 q^{13} + 6 q^{14} + 21 q^{16} + 3 q^{17} + 10 q^{19} + 4 q^{20} + 23 q^{22} - 20 q^{23} + 19 q^{25} + 10 q^{26} + 34 q^{28} - 2 q^{29} + 10 q^{31} - 26 q^{32} + 12 q^{34} - 7 q^{35} + 30 q^{37} + 3 q^{38} + 25 q^{40} + 28 q^{41} + 48 q^{43} - 25 q^{44} + 22 q^{46} - 13 q^{47} + 19 q^{49} - 12 q^{50} + 24 q^{52} + 2 q^{53} + 8 q^{55} + 7 q^{56} + 42 q^{58} + 14 q^{59} + 14 q^{61} - 8 q^{62} + 9 q^{64} + 35 q^{65} + 52 q^{67} - 3 q^{68} - 33 q^{70} + 7 q^{71} + 14 q^{73} + 13 q^{74} - 12 q^{76} + 6 q^{77} + 15 q^{79} + 8 q^{80} - 61 q^{82} - 29 q^{83} + 8 q^{85} + 9 q^{86} + 11 q^{88} + 71 q^{89} + 13 q^{91} - 2 q^{92} - 22 q^{94} - 2 q^{95} - 2 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 15 * q^4 + q^5 + 11 * q^7 - 9 * q^8 - 15 * q^11 + 7 * q^13 + 6 * q^14 + 21 * q^16 + 3 * q^17 + 10 * q^19 + 4 * q^20 + 23 * q^22 - 20 * q^23 + 19 * q^25 + 10 * q^26 + 34 * q^28 - 2 * q^29 + 10 * q^31 - 26 * q^32 + 12 * q^34 - 7 * q^35 + 30 * q^37 + 3 * q^38 + 25 * q^40 + 28 * q^41 + 48 * q^43 - 25 * q^44 + 22 * q^46 - 13 * q^47 + 19 * q^49 - 12 * q^50 + 24 * q^52 + 2 * q^53 + 8 * q^55 + 7 * q^56 + 42 * q^58 + 14 * q^59 + 14 * q^61 - 8 * q^62 + 9 * q^64 + 35 * q^65 + 52 * q^67 - 3 * q^68 - 33 * q^70 + 7 * q^71 + 14 * q^73 + 13 * q^74 - 12 * q^76 + 6 * q^77 + 15 * q^79 + 8 * q^80 - 61 * q^82 - 29 * q^83 + 8 * q^85 + 9 * q^86 + 11 * q^88 + 71 * q^89 + 13 * q^91 - 2 * q^92 - 22 * q^94 - 2 * q^95 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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