LMFDB - Embedded newform 8001.2.a.u.1.10

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:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 11 x^{16} + 123 x^{15} - 35 x^{14} - 982 x^{13} + 988 x^{12} + 3872 x^{11} + \cdots + 16 $ x^18 - 6x^17 - 11x^16 + 123x^15 - 35x^14 - 982x^13 + 988x^12 + 3872x^11 - 5421x^10 - 7882x^9 + 13376x^8 + 7948x^7 - 15795x^6 - 3858x^5 + 8199x^4 + 1453x^3 - 1610x^2 - 380*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$0.803961$ 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)

$f(q)$	$=$	$q+0.803961 q^{2} -1.35365 q^{4} -0.343335 q^{5} +1.00000 q^{7} -2.69620 q^{8} +O(q^{10})$	\(q+0.803961 q^{2} -1.35365 q^{4} -0.343335 q^{5} +1.00000 q^{7} -2.69620 q^{8} -0.276028 q^{10} -4.04620 q^{11} -5.07964 q^{13} +0.803961 q^{14} +0.539656 q^{16} +3.48245 q^{17} -2.23803 q^{19} +0.464755 q^{20} -3.25298 q^{22} +2.35865 q^{23} -4.88212 q^{25} -4.08383 q^{26} -1.35365 q^{28} -9.24369 q^{29} +5.54357 q^{31} +5.82626 q^{32} +2.79975 q^{34} -0.343335 q^{35} +5.87348 q^{37} -1.79929 q^{38} +0.925700 q^{40} +1.66919 q^{41} -10.3740 q^{43} +5.47712 q^{44} +1.89626 q^{46} +11.9067 q^{47} +1.00000 q^{49} -3.92503 q^{50} +6.87604 q^{52} -12.9408 q^{53} +1.38920 q^{55} -2.69620 q^{56} -7.43157 q^{58} -7.20507 q^{59} -10.3500 q^{61} +4.45681 q^{62} +3.60477 q^{64} +1.74402 q^{65} -4.97489 q^{67} -4.71400 q^{68} -0.276028 q^{70} -8.55324 q^{71} +10.9142 q^{73} +4.72205 q^{74} +3.02950 q^{76} -4.04620 q^{77} +15.8472 q^{79} -0.185283 q^{80} +1.34196 q^{82} +11.6255 q^{83} -1.19565 q^{85} -8.34032 q^{86} +10.9094 q^{88} +0.366424 q^{89} -5.07964 q^{91} -3.19278 q^{92} +9.57253 q^{94} +0.768394 q^{95} -7.75160 q^{97} +0.803961 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) $ 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8	$ 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) $ 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8 - 4 * q^10 + 9 * q^11 - 25 * q^13 + 6 * q^14 + 34 * q^16 + 17 * q^17 - 5 * q^19 + 21 * q^20 + 5 * q^22 + 14 * q^23 + 28 * q^25 + 8 * q^26 + 22 * q^28 + 17 * q^29 + 5 * q^31 + 53 * q^32 - 19 * q^34 + 10 * q^35 - 15 * q^37 + 22 * q^38 - q^40 + 17 * q^41 + q^43 + 33 * q^44 + 10 * q^46 + 31 * q^47 + 18 * q^49 + 35 * q^50 - 70 * q^52 + 35 * q^53 + 4 * q^55 + 21 * q^56 + 3 * q^58 + 46 * q^59 - 5 * q^61 + 10 * q^62 + 63 * q^64 + 12 * q^65 + 6 * q^67 + 56 * q^68 - 4 * q^70 + 22 * q^71 - 16 * q^73 - 18 * q^74 + 32 * q^76 + 9 * q^77 + 46 * q^79 + 30 * q^80 - 12 * q^82 + 46 * q^83 + 4 * q^85 - 18 * q^86 + 30 * q^88 + 42 * q^89 - 25 * q^91 + 48 * q^92 + 3 * q^94 + 2 * q^95 - 35 * q^97 + 6 * 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 $

$2$	0.803961	0.568486	0.284243	−	0.958752i	$-0.408258\pi$
$2$	0.803961	0.568486	0.284243	+	0.958752i	$0.408258\pi$
$3$	0	0
$3$	0	0
$4$	−1.35365	−0.676824
$5$	−0.343335	−0.153544	−0.0767720	−	0.997049i	$-0.524461\pi$
$5$	−0.343335	−0.153544	−0.0767720	+	0.997049i	$0.524461\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.69620	−0.953251
$9$	0	0
$10$	−0.276028	−0.0872876
$11$	−4.04620	−1.21997	−0.609987	−	0.792411i	$-0.708825\pi$
$11$	−4.04620	−1.21997	−0.609987	+	0.792411i	$0.708825\pi$
$12$	0	0
$13$	−5.07964	−1.40884	−0.704419	−	0.709784i	$-0.748792\pi$
$13$	−5.07964	−1.40884	−0.704419	+	0.709784i	$0.748792\pi$
$14$	0.803961	0.214867
$15$	0	0
$16$	0.539656	0.134914
$17$	3.48245	0.844617	0.422309	−	0.906452i	$-0.361220\pi$
$17$	3.48245	0.844617	0.422309	+	0.906452i	$0.361220\pi$
$18$	0	0
$19$	−2.23803	−0.513439	−0.256720	−	0.966486i	$-0.582642\pi$
$19$	−2.23803	−0.513439	−0.256720	+	0.966486i	$0.582642\pi$
$20$	0.464755	0.103922
$21$	0	0
$22$	−3.25298	−0.693538
$23$	2.35865	0.491813	0.245906	−	0.969294i	$-0.420914\pi$
$23$	2.35865	0.491813	0.245906	+	0.969294i	$0.420914\pi$
$24$	0	0
$25$	−4.88212	−0.976424
$26$	−4.08383	−0.800905
$27$	0	0
$28$	−1.35365	−0.255815
$29$	−9.24369	−1.71651	−0.858255	−	0.513223i	$-0.828451\pi$
$29$	−9.24369	−1.71651	−0.858255	+	0.513223i	$0.828451\pi$
$30$	0	0
$31$	5.54357	0.995654	0.497827	−	0.867276i	$-0.334131\pi$
$31$	5.54357	0.995654	0.497827	+	0.867276i	$0.334131\pi$
$32$	5.82626	1.02995
$33$	0	0
$34$	2.79975	0.480153
$35$	−0.343335	−0.0580342
$36$	0	0
$37$	5.87348	0.965594	0.482797	−	0.875732i	$-0.339621\pi$
$37$	5.87348	0.965594	0.482797	+	0.875732i	$0.339621\pi$
$38$	−1.79929	−0.291883
$39$	0	0
$40$	0.925700	0.146366
$41$	1.66919	0.260683	0.130342	−	0.991469i	$-0.458393\pi$
$41$	1.66919	0.260683	0.130342	+	0.991469i	$0.458393\pi$
$42$	0	0
$43$	−10.3740	−1.58203	−0.791013	−	0.611799i	$-0.790446\pi$
$43$	−10.3740	−1.58203	−0.791013	+	0.611799i	$0.790446\pi$
$44$	5.47712	0.825707
$45$	0	0
$46$	1.89626	0.279589
$47$	11.9067	1.73677	0.868387	−	0.495887i	$-0.165157\pi$
$47$	11.9067	1.73677	0.868387	+	0.495887i	$0.165157\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.92503	−0.555083
$51$	0	0
$52$	6.87604	0.953535
$53$	−12.9408	−1.77756	−0.888779	−	0.458336i	$-0.848445\pi$
$53$	−12.9408	−1.77756	−0.888779	+	0.458336i	$0.848445\pi$
$54$	0	0
$55$	1.38920	0.187320
$56$	−2.69620	−0.360295
$57$	0	0
$58$	−7.43157	−0.975812
$59$	−7.20507	−0.938020	−0.469010	−	0.883193i	$-0.655389\pi$
$59$	−7.20507	−0.938020	−0.469010	+	0.883193i	$0.655389\pi$
$60$	0	0
$61$	−10.3500	−1.32519	−0.662593	−	0.748980i	$-0.730544\pi$
$61$	−10.3500	−1.32519	−0.662593	+	0.748980i	$0.730544\pi$
$62$	4.45681	0.566015
$63$	0	0
$64$	3.60477	0.450597
$65$	1.74402	0.216319
$66$	0	0
$67$	−4.97489	−0.607779	−0.303890	−	0.952707i	$-0.598285\pi$
$67$	−4.97489	−0.607779	−0.303890	+	0.952707i	$0.598285\pi$
$68$	−4.71400	−0.571657
$69$	0	0
$70$	−0.276028	−0.0329916
$71$	−8.55324	−1.01508	−0.507541	−	0.861628i	$-0.669445\pi$
$71$	−8.55324	−1.01508	−0.507541	+	0.861628i	$0.669445\pi$
$72$	0	0
$73$	10.9142	1.27741	0.638703	−	0.769454i	$-0.279471\pi$
$73$	10.9142	1.27741	0.638703	+	0.769454i	$0.279471\pi$
$74$	4.72205	0.548927
$75$	0	0
$76$	3.02950	0.347508
$77$	−4.04620	−0.461107
$78$	0	0
$79$	15.8472	1.78295	0.891474	−	0.453072i	$-0.149672\pi$
$79$	15.8472	1.78295	0.891474	+	0.453072i	$0.149672\pi$
$80$	−0.185283	−0.0207153
$81$	0	0
$82$	1.34196	0.148195
$83$	11.6255	1.27606	0.638031	−	0.770010i	$-0.279749\pi$
$83$	11.6255	1.27606	0.638031	+	0.770010i	$0.279749\pi$
$84$	0	0
$85$	−1.19565	−0.129686
$86$	−8.34032	−0.899360
$87$	0	0
$88$	10.9094	1.16294
$89$	0.366424	0.0388408	0.0194204	−	0.999811i	$-0.493818\pi$
$89$	0.366424	0.0388408	0.0194204	+	0.999811i	$0.493818\pi$
$90$	0	0
$91$	−5.07964	−0.532491
$92$	−3.19278	−0.332871
$93$	0	0
$94$	9.57253	0.987332
$95$	0.768394	0.0788356
$96$	0	0
$97$	−7.75160	−0.787056	−0.393528	−	0.919313i	$-0.628746\pi$
$97$	−7.75160	−0.787056	−0.393528	+	0.919313i	$0.628746\pi$
$98$	0.803961	0.0812123
$99$	0	0
$100$	6.60867	0.660867
$101$	13.1619	1.30966	0.654831	−	0.755776i	$-0.272740\pi$
$101$	13.1619	1.30966	0.654831	+	0.755776i	$0.272740\pi$
$102$	0	0
$103$	−10.2020	−1.00523	−0.502616	−	0.864510i	$-0.667629\pi$
$103$	−10.2020	−1.00523	−0.502616	+	0.864510i	$0.667629\pi$
$104$	13.6957	1.34298
$105$	0	0
$106$	−10.4039	−1.01052
$107$	5.75880	0.556724	0.278362	−	0.960476i	$-0.410208\pi$
$107$	5.75880	0.556724	0.278362	+	0.960476i	$0.410208\pi$
$108$	0	0
$109$	10.3606	0.992362	0.496181	−	0.868219i	$-0.334735\pi$
$109$	10.3606	0.992362	0.496181	+	0.868219i	$0.334735\pi$
$110$	1.11686	0.106489
$111$	0	0
$112$	0.539656	0.0509927
$113$	12.1901	1.14675	0.573376	−	0.819292i	$-0.305633\pi$
$113$	12.1901	1.14675	0.573376	+	0.819292i	$0.305633\pi$
$114$	0	0
$115$	−0.809807	−0.0755149
$116$	12.5127	1.16178
$117$	0	0
$118$	−5.79259	−0.533251
$119$	3.48245	0.319235
$120$	0	0
$121$	5.37170	0.488337
$122$	−8.32101	−0.753349
$123$	0	0
$124$	−7.50403	−0.673882
$125$	3.39288	0.303468
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−8.75443	−0.773790
$129$	0	0
$130$	1.40212	0.122974
$131$	11.7624	1.02768	0.513842	−	0.857885i	$-0.328222\pi$
$131$	11.7624	1.02768	0.513842	+	0.857885i	$0.328222\pi$
$132$	0	0
$133$	−2.23803	−0.194062
$134$	−3.99961	−0.345514
$135$	0	0
$136$	−9.38937	−0.805132
$137$	22.2429	1.90034	0.950169	−	0.311737i	$-0.100911\pi$
$137$	22.2429	1.90034	0.950169	+	0.311737i	$0.100911\pi$
$138$	0	0
$139$	4.45596	0.377950	0.188975	−	0.981982i	$-0.439484\pi$
$139$	4.45596	0.377950	0.188975	+	0.981982i	$0.439484\pi$
$140$	0.464755	0.0392789
$141$	0	0
$142$	−6.87646	−0.577060
$143$	20.5532	1.71875
$144$	0	0
$145$	3.17368	0.263560
$146$	8.77455	0.726187
$147$	0	0
$148$	−7.95062	−0.653537
$149$	18.9721	1.55425	0.777126	−	0.629345i	$-0.216677\pi$
$149$	18.9721	1.55425	0.777126	+	0.629345i	$0.216677\pi$
$150$	0	0
$151$	−10.7144	−0.871925	−0.435962	−	0.899965i	$-0.643592\pi$
$151$	−10.7144	−0.871925	−0.435962	+	0.899965i	$0.643592\pi$
$152$	6.03418	0.489436
$153$	0	0
$154$	−3.25298	−0.262133
$155$	−1.90330	−0.152877
$156$	0	0
$157$	11.4000	0.909817	0.454909	−	0.890538i	$-0.349672\pi$
$157$	11.4000	0.909817	0.454909	+	0.890538i	$0.349672\pi$
$158$	12.7405	1.01358
$159$	0	0
$160$	−2.00036	−0.158142
$161$	2.35865	0.185888
$162$	0	0
$163$	−18.3009	−1.43344	−0.716720	−	0.697361i	$-0.754357\pi$
$163$	−18.3009	−1.43344	−0.716720	+	0.697361i	$0.754357\pi$
$164$	−2.25949	−0.176437
$165$	0	0
$166$	9.34643	0.725424
$167$	−13.7796	−1.06630	−0.533150	−	0.846020i	$-0.678992\pi$
$167$	−13.7796	−1.06630	−0.533150	+	0.846020i	$0.678992\pi$
$168$	0	0
$169$	12.8027	0.984825
$170$	−0.961252	−0.0737246
$171$	0	0
$172$	14.0428	1.07075
$173$	23.7433	1.80517	0.902586	−	0.430510i	$-0.141666\pi$
$173$	23.7433	1.80517	0.902586	+	0.430510i	$0.141666\pi$
$174$	0	0
$175$	−4.88212	−0.369054
$176$	−2.18356	−0.164592
$177$	0	0
$178$	0.294590	0.0220805
$179$	−6.69909	−0.500714	−0.250357	−	0.968154i	$-0.580548\pi$
$179$	−6.69909	−0.500714	−0.250357	+	0.968154i	$0.580548\pi$
$180$	0	0
$181$	5.85446	0.435159	0.217579	−	0.976043i	$-0.430184\pi$
$181$	5.85446	0.435159	0.217579	+	0.976043i	$0.430184\pi$
$182$	−4.08383	−0.302714
$183$	0	0
$184$	−6.35940	−0.468821
$185$	−2.01657	−0.148261
$186$	0	0
$187$	−14.0907	−1.03041
$188$	−16.1175	−1.17549
$189$	0	0
$190$	0.617759	0.0448169
$191$	−6.96386	−0.503887	−0.251944	−	0.967742i	$-0.581070\pi$
$191$	−6.96386	−0.503887	−0.251944	+	0.967742i	$0.581070\pi$
$192$	0	0
$193$	−4.33351	−0.311933	−0.155966	−	0.987762i	$-0.549849\pi$
$193$	−4.33351	−0.311933	−0.155966	+	0.987762i	$0.549849\pi$
$194$	−6.23198	−0.447430
$195$	0	0
$196$	−1.35365	−0.0966891
$197$	−6.88472	−0.490516	−0.245258	−	0.969458i	$-0.578873\pi$
$197$	−6.88472	−0.490516	−0.245258	+	0.969458i	$0.578873\pi$
$198$	0	0
$199$	−6.03022	−0.427471	−0.213735	−	0.976892i	$-0.568563\pi$
$199$	−6.03022	−0.427471	−0.213735	+	0.976892i	$0.568563\pi$
$200$	13.1632	0.930777
$201$	0	0
$202$	10.5817	0.744524
$203$	−9.24369	−0.648780
$204$	0	0
$205$	−0.573091	−0.0400264
$206$	−8.20199	−0.571460
$207$	0	0
$208$	−2.74126	−0.190072
$209$	9.05551	0.626383
$210$	0	0
$211$	−18.7052	−1.28772	−0.643858	−	0.765145i	$-0.722667\pi$
$211$	−18.7052	−1.28772	−0.643858	+	0.765145i	$0.722667\pi$
$212$	17.5173	1.20309
$213$	0	0
$214$	4.62985	0.316490
$215$	3.56177	0.242911
$216$	0	0
$217$	5.54357	0.376322
$218$	8.32948	0.564144
$219$	0	0
$220$	−1.88049	−0.126782
$221$	−17.6896	−1.18993
$222$	0	0
$223$	−10.6279	−0.711696	−0.355848	−	0.934544i	$-0.615808\pi$
$223$	−10.6279	−0.711696	−0.355848	+	0.934544i	$0.615808\pi$
$224$	5.82626	0.389284
$225$	0	0
$226$	9.80039	0.651912
$227$	2.03230	0.134888	0.0674442	−	0.997723i	$-0.478516\pi$
$227$	2.03230	0.134888	0.0674442	+	0.997723i	$0.478516\pi$
$228$	0	0
$229$	14.7071	0.971873	0.485937	−	0.873994i	$-0.338479\pi$
$229$	14.7071	0.971873	0.485937	+	0.873994i	$0.338479\pi$
$230$	−0.651053	−0.0429292
$231$	0	0
$232$	24.9228	1.63627
$233$	18.8646	1.23586	0.617932	−	0.786232i	$-0.287971\pi$
$233$	18.8646	1.23586	0.617932	+	0.786232i	$0.287971\pi$
$234$	0	0
$235$	−4.08799	−0.266671
$236$	9.75313	0.634874
$237$	0	0
$238$	2.79975	0.181481
$239$	−3.80322	−0.246010	−0.123005	−	0.992406i	$-0.539253\pi$
$239$	−3.80322	−0.246010	−0.123005	+	0.992406i	$0.539253\pi$
$240$	0	0
$241$	−23.5196	−1.51503	−0.757515	−	0.652818i	$-0.773587\pi$
$241$	−23.5196	−1.51503	−0.757515	+	0.652818i	$0.773587\pi$
$242$	4.31864	0.277613
$243$	0	0
$244$	14.0103	0.896917
$245$	−0.343335	−0.0219349
$246$	0	0
$247$	11.3684	0.723353
$248$	−14.9466	−0.949108
$249$	0	0
$250$	2.72774	0.172517
$251$	24.4633	1.54411	0.772056	−	0.635555i	$-0.219228\pi$
$251$	24.4633	1.54411	0.772056	+	0.635555i	$0.219228\pi$
$252$	0	0
$253$	−9.54357	−0.599999
$254$	0.803961	0.0504450
$255$	0	0
$256$	−14.2478	−0.890485
$257$	1.68358	0.105019	0.0525096	−	0.998620i	$-0.483278\pi$
$257$	1.68358	0.105019	0.0525096	+	0.998620i	$0.483278\pi$
$258$	0	0
$259$	5.87348	0.364960
$260$	−2.36078	−0.146410
$261$	0	0
$262$	9.45649	0.584224
$263$	22.2555	1.37233	0.686166	−	0.727445i	$-0.259292\pi$
$263$	22.2555	1.37233	0.686166	+	0.727445i	$0.259292\pi$
$264$	0	0
$265$	4.44303	0.272933
$266$	−1.79929	−0.110321
$267$	0	0
$268$	6.73425	0.411360
$269$	−10.4180	−0.635195	−0.317597	−	0.948226i	$-0.602876\pi$
$269$	−10.4180	−0.635195	−0.317597	+	0.948226i	$0.602876\pi$
$270$	0	0
$271$	2.90437	0.176428	0.0882141	−	0.996102i	$-0.471884\pi$
$271$	2.90437	0.176428	0.0882141	+	0.996102i	$0.471884\pi$
$272$	1.87932	0.113951
$273$	0	0
$274$	17.8824	1.08031
$275$	19.7540	1.19121
$276$	0	0
$277$	15.7288	0.945054	0.472527	−	0.881316i	$-0.343342\pi$
$277$	15.7288	0.945054	0.472527	+	0.881316i	$0.343342\pi$
$278$	3.58242	0.214859
$279$	0	0
$280$	0.925700	0.0553211
$281$	27.1991	1.62256	0.811280	−	0.584657i	$-0.198771\pi$
$281$	27.1991	1.62256	0.811280	+	0.584657i	$0.198771\pi$
$282$	0	0
$283$	11.1363	0.661982	0.330991	−	0.943634i	$-0.392617\pi$
$283$	11.1363	0.661982	0.330991	+	0.943634i	$0.392617\pi$
$284$	11.5781	0.687032
$285$	0	0
$286$	16.5240	0.977083
$287$	1.66919	0.0985290
$288$	0	0
$289$	−4.87257	−0.286622
$290$	2.55152	0.149830
$291$	0	0
$292$	−14.7739	−0.864578
$293$	−22.3106	−1.30340	−0.651700	−	0.758477i	$-0.725944\pi$
$293$	−22.3106	−1.30340	−0.651700	+	0.758477i	$0.725944\pi$
$294$	0	0
$295$	2.47375	0.144027
$296$	−15.8361	−0.920454
$297$	0	0
$298$	15.2528	0.883570
$299$	−11.9811	−0.692885
$300$	0	0
$301$	−10.3740	−0.597950
$302$	−8.61395	−0.495677
$303$	0	0
$304$	−1.20777	−0.0692702
$305$	3.55353	0.203474
$306$	0	0
$307$	20.8361	1.18918	0.594588	−	0.804030i	$-0.297315\pi$
$307$	20.8361	1.18918	0.594588	+	0.804030i	$0.297315\pi$
$308$	5.47712	0.312088
$309$	0	0
$310$	−1.53018	−0.0869083
$311$	−7.05946	−0.400305	−0.200153	−	0.979765i	$-0.564144\pi$
$311$	−7.05946	−0.400305	−0.200153	+	0.979765i	$0.564144\pi$
$312$	0	0
$313$	8.81011	0.497977	0.248988	−	0.968506i	$-0.419902\pi$
$313$	8.81011	0.497977	0.248988	+	0.968506i	$0.419902\pi$
$314$	9.16513	0.517218
$315$	0	0
$316$	−21.4515	−1.20674
$317$	9.91942	0.557130	0.278565	−	0.960417i	$-0.410141\pi$
$317$	9.91942	0.557130	0.278565	+	0.960417i	$0.410141\pi$
$318$	0	0
$319$	37.4018	2.09410
$320$	−1.23764	−0.0691864
$321$	0	0
$322$	1.89626	0.105675
$323$	−7.79382	−0.433660
$324$	0	0
$325$	24.7994	1.37562
$326$	−14.7132	−0.814890
$327$	0	0
$328$	−4.50046	−0.248497
$329$	11.9067	0.656439
$330$	0	0
$331$	−4.64788	−0.255471	−0.127735	−	0.991808i	$-0.540771\pi$
$331$	−4.64788	−0.255471	−0.127735	+	0.991808i	$0.540771\pi$
$332$	−15.7368	−0.863669
$333$	0	0
$334$	−11.0783	−0.606177
$335$	1.70805	0.0933209
$336$	0	0
$337$	20.2867	1.10509	0.552545	−	0.833483i	$-0.313657\pi$
$337$	20.2867	1.10509	0.552545	+	0.833483i	$0.313657\pi$
$338$	10.2929	0.559859
$339$	0	0
$340$	1.61848	0.0877745
$341$	−22.4304	−1.21467
$342$	0	0
$343$	1.00000	0.0539949
$344$	27.9705	1.50807
$345$	0	0
$346$	19.0887	1.02621
$347$	7.49407	0.402303	0.201151	−	0.979560i	$-0.435532\pi$
$347$	7.49407	0.402303	0.201151	+	0.979560i	$0.435532\pi$
$348$	0	0
$349$	−32.3002	−1.72899	−0.864496	−	0.502640i	$-0.832362\pi$
$349$	−32.3002	−1.72899	−0.864496	+	0.502640i	$0.832362\pi$
$350$	−3.92503	−0.209802
$351$	0	0
$352$	−23.5742	−1.25651
$353$	−20.5000	−1.09110	−0.545552	−	0.838077i	$-0.683680\pi$
$353$	−20.5000	−1.09110	−0.545552	+	0.838077i	$0.683680\pi$
$354$	0	0
$355$	2.93662	0.155860
$356$	−0.496009	−0.0262884
$357$	0	0
$358$	−5.38581	−0.284649
$359$	24.6060	1.29866	0.649328	−	0.760508i	$-0.275050\pi$
$359$	24.6060	1.29866	0.649328	+	0.760508i	$0.275050\pi$
$360$	0	0
$361$	−13.9912	−0.736380
$362$	4.70675	0.247382
$363$	0	0
$364$	6.87604	0.360402
$365$	−3.74721	−0.196138
$366$	0	0
$367$	15.9470	0.832429	0.416214	−	0.909267i	$-0.363357\pi$
$367$	15.9470	0.832429	0.416214	+	0.909267i	$0.363357\pi$
$368$	1.27286	0.0663525
$369$	0	0
$370$	−1.62124	−0.0842845
$371$	−12.9408	−0.671854
$372$	0	0
$373$	−28.0230	−1.45098	−0.725488	−	0.688235i	$-0.758386\pi$
$373$	−28.0230	−1.45098	−0.725488	+	0.688235i	$0.758386\pi$
$374$	−11.3283	−0.585774
$375$	0	0
$376$	−32.1029	−1.65558
$377$	46.9546	2.41829
$378$	0	0
$379$	32.1526	1.65157	0.825783	−	0.563987i	$-0.190733\pi$
$379$	32.1526	1.65157	0.825783	+	0.563987i	$0.190733\pi$
$380$	−1.04013	−0.0533578
$381$	0	0
$382$	−5.59867	−0.286453
$383$	21.9187	1.11999	0.559996	−	0.828495i	$-0.310803\pi$
$383$	21.9187	1.11999	0.559996	+	0.828495i	$0.310803\pi$
$384$	0	0
$385$	1.38920	0.0708002
$386$	−3.48397	−0.177329
$387$	0	0
$388$	10.4929	0.532698
$389$	−0.259292	−0.0131466	−0.00657331	−	0.999978i	$-0.502092\pi$
$389$	−0.259292	−0.0131466	−0.00657331	+	0.999978i	$0.502092\pi$
$390$	0	0
$391$	8.21387	0.415394
$392$	−2.69620	−0.136179
$393$	0	0
$394$	−5.53504	−0.278852
$395$	−5.44089	−0.273761
$396$	0	0
$397$	8.76662	0.439984	0.219992	−	0.975502i	$-0.429397\pi$
$397$	8.76662	0.439984	0.219992	+	0.975502i	$0.429397\pi$
$398$	−4.84805	−0.243011
$399$	0	0
$400$	−2.63467	−0.131733
$401$	12.3572	0.617091	0.308546	−	0.951210i	$-0.400158\pi$
$401$	12.3572	0.617091	0.308546	+	0.951210i	$0.400158\pi$
$402$	0	0
$403$	−28.1593	−1.40272
$404$	−17.8166	−0.886410
$405$	0	0
$406$	−7.43157	−0.368822
$407$	−23.7653	−1.17800
$408$	0	0
$409$	10.7862	0.533346	0.266673	−	0.963787i	$-0.414076\pi$
$409$	10.7862	0.533346	0.266673	+	0.963787i	$0.414076\pi$
$410$	−0.460742	−0.0227544
$411$	0	0
$412$	13.8099	0.680365
$413$	−7.20507	−0.354538
$414$	0	0
$415$	−3.99143	−0.195932
$416$	−29.5953	−1.45103
$417$	0	0
$418$	7.28027	0.356090
$419$	17.4411	0.852054	0.426027	−	0.904710i	$-0.359913\pi$
$419$	17.4411	0.852054	0.426027	+	0.904710i	$0.359913\pi$
$420$	0	0
$421$	−40.2522	−1.96177	−0.980886	−	0.194583i	$-0.937665\pi$
$421$	−40.2522	−1.96177	−0.980886	+	0.194583i	$0.937665\pi$
$422$	−15.0382	−0.732048
$423$	0	0
$424$	34.8910	1.69446
$425$	−17.0017	−0.824705
$426$	0	0
$427$	−10.3500	−0.500873
$428$	−7.79538	−0.376804
$429$	0	0
$430$	2.86352	0.138091
$431$	−5.00922	−0.241286	−0.120643	−	0.992696i	$-0.538496\pi$
$431$	−5.00922	−0.241286	−0.120643	+	0.992696i	$0.538496\pi$
$432$	0	0
$433$	26.9224	1.29381	0.646904	−	0.762571i	$-0.276063\pi$
$433$	26.9224	1.29381	0.646904	+	0.762571i	$0.276063\pi$
$434$	4.45681	0.213934
$435$	0	0
$436$	−14.0245	−0.671654
$437$	−5.27873	−0.252516
$438$	0	0
$439$	−30.1663	−1.43976	−0.719879	−	0.694099i	$-0.755803\pi$
$439$	−30.1663	−1.43976	−0.719879	+	0.694099i	$0.755803\pi$
$440$	−3.74556	−0.178563
$441$	0	0
$442$	−14.2217	−0.676458
$443$	−30.6696	−1.45716	−0.728579	−	0.684962i	$-0.759819\pi$
$443$	−30.6696	−1.45716	−0.728579	+	0.684962i	$0.759819\pi$
$444$	0	0
$445$	−0.125806	−0.00596378
$446$	−8.54440	−0.404589
$447$	0	0
$448$	3.60477	0.170309
$449$	12.2072	0.576093	0.288047	−	0.957616i	$-0.406994\pi$
$449$	12.2072	0.576093	0.288047	+	0.957616i	$0.406994\pi$
$450$	0	0
$451$	−6.75386	−0.318027
$452$	−16.5012	−0.776149
$453$	0	0
$454$	1.63389	0.0766822
$455$	1.74402	0.0817608
$456$	0	0
$457$	−32.4700	−1.51888	−0.759442	−	0.650575i	$-0.774528\pi$
$457$	−32.4700	−1.51888	−0.759442	+	0.650575i	$0.774528\pi$
$458$	11.8239	0.552496
$459$	0	0
$460$	1.09619	0.0511103
$461$	12.1968	0.568060	0.284030	−	0.958815i	$-0.408329\pi$
$461$	12.1968	0.568060	0.284030	+	0.958815i	$0.408329\pi$
$462$	0	0
$463$	35.1369	1.63295	0.816475	−	0.577380i	$-0.195925\pi$
$463$	35.1369	1.63295	0.816475	+	0.577380i	$0.195925\pi$
$464$	−4.98842	−0.231582
$465$	0	0
$466$	15.1664	0.702571
$467$	17.0341	0.788246	0.394123	−	0.919058i	$-0.371048\pi$
$467$	17.0341	0.788246	0.394123	+	0.919058i	$0.371048\pi$
$468$	0	0
$469$	−4.97489	−0.229719
$470$	−3.28659	−0.151599
$471$	0	0
$472$	19.4263	0.894169
$473$	41.9754	1.93003
$474$	0	0
$475$	10.9263	0.501335
$476$	−4.71400	−0.216066
$477$	0	0
$478$	−3.05764	−0.139853
$479$	−12.1072	−0.553190	−0.276595	−	0.960987i	$-0.589206\pi$
$479$	−12.1072	−0.553190	−0.276595	+	0.960987i	$0.589206\pi$
$480$	0	0
$481$	−29.8352	−1.36037
$482$	−18.9088	−0.861273
$483$	0	0
$484$	−7.27139	−0.330518
$485$	2.66140	0.120848
$486$	0	0
$487$	15.5980	0.706812	0.353406	−	0.935470i	$-0.385023\pi$
$487$	15.5980	0.706812	0.353406	+	0.935470i	$0.385023\pi$
$488$	27.9057	1.26323
$489$	0	0
$490$	−0.276028	−0.0124697
$491$	−8.87764	−0.400642	−0.200321	−	0.979730i	$-0.564199\pi$
$491$	−8.87764	−0.400642	−0.200321	+	0.979730i	$0.564199\pi$
$492$	0	0
$493$	−32.1907	−1.44979
$494$	9.13973	0.411216
$495$	0	0
$496$	2.99162	0.134328
$497$	−8.55324	−0.383665
$498$	0	0
$499$	15.3071	0.685241	0.342620	−	0.939474i	$-0.388686\pi$
$499$	15.3071	0.685241	0.342620	+	0.939474i	$0.388686\pi$
$500$	−4.59276	−0.205394
$501$	0	0
$502$	19.6676	0.877806
$503$	14.5352	0.648091	0.324045	−	0.946042i	$-0.394957\pi$
$503$	14.5352	0.648091	0.324045	+	0.946042i	$0.394957\pi$
$504$	0	0
$505$	−4.51895	−0.201091
$506$	−7.67265	−0.341091
$507$	0	0
$508$	−1.35365	−0.0600584
$509$	29.5667	1.31052	0.655260	−	0.755403i	$-0.272559\pi$
$509$	29.5667	1.31052	0.655260	+	0.755403i	$0.272559\pi$
$510$	0	0
$511$	10.9142	0.482814
$512$	6.05422	0.267561
$513$	0	0
$514$	1.35354	0.0597019
$515$	3.50270	0.154347
$516$	0	0
$517$	−48.1769	−2.11882
$518$	4.72205	0.207475
$519$	0	0
$520$	−4.70222	−0.206206
$521$	14.6876	0.643478	0.321739	−	0.946828i	$-0.395733\pi$
$521$	14.6876	0.643478	0.321739	+	0.946828i	$0.395733\pi$
$522$	0	0
$523$	16.6693	0.728897	0.364448	−	0.931224i	$-0.381258\pi$
$523$	16.6693	0.728897	0.364448	+	0.931224i	$0.381258\pi$
$524$	−15.9221	−0.695561
$525$	0	0
$526$	17.8925	0.780152
$527$	19.3052	0.840946
$528$	0	0
$529$	−17.4368	−0.758120
$530$	3.57202	0.155159
$531$	0	0
$532$	3.02950	0.131346
$533$	−8.47887	−0.367261
$534$	0	0
$535$	−1.97720	−0.0854817
$536$	13.4133	0.579366
$537$	0	0
$538$	−8.37563	−0.361099
$539$	−4.04620	−0.174282
$540$	0	0
$541$	−9.00282	−0.387061	−0.193531	−	0.981094i	$-0.561994\pi$
$541$	−9.00282	−0.387061	−0.193531	+	0.981094i	$0.561994\pi$
$542$	2.33500	0.100297
$543$	0	0
$544$	20.2896	0.869911
$545$	−3.55714	−0.152371
$546$	0	0
$547$	34.5632	1.47782	0.738909	−	0.673805i	$-0.235341\pi$
$547$	34.5632	1.47782	0.738909	+	0.673805i	$0.235341\pi$
$548$	−30.1090	−1.28619
$549$	0	0
$550$	15.8815	0.677187
$551$	20.6877	0.881324
$552$	0	0
$553$	15.8472	0.673891
$554$	12.6454	0.537250
$555$	0	0
$556$	−6.03180	−0.255805
$557$	−5.54071	−0.234767	−0.117384	−	0.993087i	$-0.537451\pi$
$557$	−5.54071	−0.234767	−0.117384	+	0.993087i	$0.537451\pi$
$558$	0	0
$559$	52.6964	2.22882
$560$	−0.185283	−0.00782963
$561$	0	0
$562$	21.8670	0.922403
$563$	32.8794	1.38570	0.692852	−	0.721080i	$-0.256354\pi$
$563$	32.8794	1.38570	0.692852	+	0.721080i	$0.256354\pi$
$564$	0	0
$565$	−4.18530	−0.176077
$566$	8.95311	0.376327
$567$	0	0
$568$	23.0612	0.967628
$569$	−39.4015	−1.65180	−0.825898	−	0.563819i	$-0.809331\pi$
$569$	−39.4015	−1.65180	−0.825898	+	0.563819i	$0.809331\pi$
$570$	0	0
$571$	−8.97776	−0.375707	−0.187854	−	0.982197i	$-0.560153\pi$
$571$	−8.97776	−0.375707	−0.187854	+	0.982197i	$0.560153\pi$
$572$	−27.8218	−1.16329
$573$	0	0
$574$	1.34196	0.0560124
$575$	−11.5152	−0.480218
$576$	0	0
$577$	29.6596	1.23474	0.617372	−	0.786671i	$-0.288197\pi$
$577$	29.6596	1.23474	0.617372	+	0.786671i	$0.288197\pi$
$578$	−3.91736	−0.162941
$579$	0	0
$580$	−4.29605	−0.178384
$581$	11.6255	0.482306
$582$	0	0
$583$	52.3611	2.16857
$584$	−29.4268	−1.21769
$585$	0	0
$586$	−17.9369	−0.740964
$587$	−24.2422	−1.00058	−0.500292	−	0.865857i	$-0.666774\pi$
$587$	−24.2422	−1.00058	−0.500292	+	0.865857i	$0.666774\pi$
$588$	0	0
$589$	−12.4067	−0.511208
$590$	1.98880	0.0818776
$591$	0	0
$592$	3.16966	0.130272
$593$	−4.03464	−0.165683	−0.0828413	−	0.996563i	$-0.526399\pi$
$593$	−4.03464	−0.165683	−0.0828413	+	0.996563i	$0.526399\pi$
$594$	0	0
$595$	−1.19565	−0.0490167
$596$	−25.6815	−1.05195
$597$	0	0
$598$	−9.63233	−0.393895
$599$	−13.9844	−0.571386	−0.285693	−	0.958321i	$-0.592224\pi$
$599$	−13.9844	−0.571386	−0.285693	+	0.958321i	$0.592224\pi$
$600$	0	0
$601$	2.49583	0.101807	0.0509034	−	0.998704i	$-0.483790\pi$
$601$	2.49583	0.101807	0.0509034	+	0.998704i	$0.483790\pi$
$602$	−8.34032	−0.339926
$603$	0	0
$604$	14.5035	0.590139
$605$	−1.84429	−0.0749812
$606$	0	0
$607$	−31.0017	−1.25832	−0.629160	−	0.777276i	$-0.716601\pi$
$607$	−31.0017	−1.25832	−0.629160	+	0.777276i	$0.716601\pi$
$608$	−13.0394	−0.528816
$609$	0	0
$610$	2.85689	0.115672
$611$	−60.4818	−2.44683
$612$	0	0
$613$	16.1533	0.652424	0.326212	−	0.945297i	$-0.394228\pi$
$613$	16.1533	0.652424	0.326212	+	0.945297i	$0.394228\pi$
$614$	16.7514	0.676030
$615$	0	0
$616$	10.9094	0.439550
$617$	28.2570	1.13759	0.568793	−	0.822481i	$-0.307411\pi$
$617$	28.2570	1.13759	0.568793	+	0.822481i	$0.307411\pi$
$618$	0	0
$619$	8.17631	0.328634	0.164317	−	0.986408i	$-0.447458\pi$
$619$	8.17631	0.328634	0.164317	+	0.986408i	$0.447458\pi$
$620$	2.57640	0.103471
$621$	0	0
$622$	−5.67553	−0.227568
$623$	0.366424	0.0146805
$624$	0	0
$625$	23.2457	0.929828
$626$	7.08298	0.283093
$627$	0	0
$628$	−15.4315	−0.615786
$629$	20.4541	0.815558
$630$	0	0
$631$	−4.89122	−0.194716	−0.0973582	−	0.995249i	$-0.531039\pi$
$631$	−4.89122	−0.194716	−0.0973582	+	0.995249i	$0.531039\pi$
$632$	−42.7272	−1.69960
$633$	0	0
$634$	7.97483	0.316721
$635$	−0.343335	−0.0136248
$636$	0	0
$637$	−5.07964	−0.201263
$638$	30.0696	1.19047
$639$	0	0
$640$	3.00570	0.118811
$641$	25.5390	1.00873	0.504364	−	0.863491i	$-0.331727\pi$
$641$	25.5390	1.00873	0.504364	+	0.863491i	$0.331727\pi$
$642$	0	0
$643$	−1.19858	−0.0472675	−0.0236338	−	0.999721i	$-0.507524\pi$
$643$	−1.19858	−0.0472675	−0.0236338	+	0.999721i	$0.507524\pi$
$644$	−3.19278	−0.125813
$645$	0	0
$646$	−6.26592	−0.246529
$647$	18.5349	0.728684	0.364342	−	0.931265i	$-0.381294\pi$
$647$	18.5349	0.728684	0.364342	+	0.931265i	$0.381294\pi$
$648$	0	0
$649$	29.1531	1.14436
$650$	19.9377	0.782023
$651$	0	0
$652$	24.7730	0.970186
$653$	17.4806	0.684070	0.342035	−	0.939687i	$-0.388884\pi$
$653$	17.4806	0.684070	0.342035	+	0.939687i	$0.388884\pi$
$654$	0	0
$655$	−4.03844	−0.157795
$656$	0.900788	0.0351699
$657$	0	0
$658$	9.57253	0.373176
$659$	−14.2329	−0.554433	−0.277217	−	0.960807i	$-0.589412\pi$
$659$	−14.2329	−0.554433	−0.277217	+	0.960807i	$0.589412\pi$
$660$	0	0
$661$	41.6188	1.61878	0.809392	−	0.587269i	$-0.199797\pi$
$661$	41.6188	1.61878	0.809392	+	0.587269i	$0.199797\pi$
$662$	−3.73671	−0.145231
$663$	0	0
$664$	−31.3446	−1.21641
$665$	0.768394	0.0297970
$666$	0	0
$667$	−21.8027	−0.844202
$668$	18.6528	0.721698
$669$	0	0
$670$	1.37321	0.0530516
$671$	41.8782	1.61669
$672$	0	0
$673$	−7.84024	−0.302219	−0.151110	−	0.988517i	$-0.548285\pi$
$673$	−7.84024	−0.302219	−0.151110	+	0.988517i	$0.548285\pi$
$674$	16.3097	0.628228
$675$	0	0
$676$	−17.3304	−0.666553
$677$	−10.4000	−0.399704	−0.199852	−	0.979826i	$-0.564046\pi$
$677$	−10.4000	−0.399704	−0.199852	+	0.979826i	$0.564046\pi$
$678$	0	0
$679$	−7.75160	−0.297479
$680$	3.22370	0.123623
$681$	0	0
$682$	−18.0331	−0.690524
$683$	−30.2573	−1.15776	−0.578882	−	0.815412i	$-0.696511\pi$
$683$	−30.2573	−1.15776	−0.578882	+	0.815412i	$0.696511\pi$
$684$	0	0
$685$	−7.63676	−0.291786
$686$	0.803961	0.0306954
$687$	0	0
$688$	−5.59842	−0.213438
$689$	65.7347	2.50429
$690$	0	0
$691$	−23.7198	−0.902343	−0.451172	−	0.892437i	$-0.648994\pi$
$691$	−23.7198	−0.902343	−0.451172	+	0.892437i	$0.648994\pi$
$692$	−32.1401	−1.22178
$693$	0	0
$694$	6.02494	0.228704
$695$	−1.52989	−0.0580319
$696$	0	0
$697$	5.81286	0.220178
$698$	−25.9681	−0.982907
$699$	0	0
$700$	6.60867	0.249784
$701$	33.5878	1.26859	0.634297	−	0.773090i	$-0.281290\pi$
$701$	33.5878	1.26859	0.634297	+	0.773090i	$0.281290\pi$
$702$	0	0
$703$	−13.1450	−0.495774
$704$	−14.5856	−0.549716
$705$	0	0
$706$	−16.4812	−0.620277
$707$	13.1619	0.495005
$708$	0	0
$709$	−27.9016	−1.04787	−0.523933	−	0.851759i	$-0.675536\pi$
$709$	−27.9016	−1.04787	−0.523933	+	0.851759i	$0.675536\pi$
$710$	2.36093	0.0886041
$711$	0	0
$712$	−0.987952	−0.0370251
$713$	13.0753	0.489675
$714$	0	0
$715$	−7.05664	−0.263903
$716$	9.06821	0.338895
$717$	0	0
$718$	19.7823	0.738268
$719$	8.25535	0.307873	0.153936	−	0.988081i	$-0.450805\pi$
$719$	8.25535	0.307873	0.153936	+	0.988081i	$0.450805\pi$
$720$	0	0
$721$	−10.2020	−0.379942
$722$	−11.2484	−0.418622
$723$	0	0
$724$	−7.92488	−0.294526
$725$	45.1288	1.67604
$726$	0	0
$727$	25.7683	0.955694	0.477847	−	0.878443i	$-0.341417\pi$
$727$	25.7683	0.955694	0.477847	+	0.878443i	$0.341417\pi$
$728$	13.6957	0.507597
$729$	0	0
$730$	−3.01261	−0.111502
$731$	−36.1270	−1.33621
$732$	0	0
$733$	−41.2704	−1.52436	−0.762179	−	0.647366i	$-0.775870\pi$
$733$	−41.2704	−1.52436	−0.762179	+	0.647366i	$0.775870\pi$
$734$	12.8208	0.473224
$735$	0	0
$736$	13.7421	0.506541
$737$	20.1294	0.741475
$738$	0	0
$739$	−12.2673	−0.451260	−0.225630	−	0.974213i	$-0.572444\pi$
$739$	−12.2673	−0.451260	−0.225630	+	0.974213i	$0.572444\pi$
$740$	2.72973	0.100347
$741$	0	0
$742$	−10.4039	−0.381939
$743$	−30.3770	−1.11443	−0.557213	−	0.830370i	$-0.688129\pi$
$743$	−30.3770	−1.11443	−0.557213	+	0.830370i	$0.688129\pi$
$744$	0	0
$745$	−6.51377	−0.238646
$746$	−22.5294	−0.824859
$747$	0	0
$748$	19.0738	0.697407
$749$	5.75880	0.210422
$750$	0	0
$751$	38.9352	1.42076	0.710382	−	0.703816i	$-0.248522\pi$
$751$	38.9352	1.42076	0.710382	+	0.703816i	$0.248522\pi$
$752$	6.42554	0.234315
$753$	0	0
$754$	37.7497	1.37476
$755$	3.67862	0.133879
$756$	0	0
$757$	0.998200	0.0362802	0.0181401	−	0.999835i	$-0.494226\pi$
$757$	0.998200	0.0362802	0.0181401	+	0.999835i	$0.494226\pi$
$758$	25.8494	0.938892
$759$	0	0
$760$	−2.07174	−0.0751501
$761$	−49.1530	−1.78180	−0.890898	−	0.454204i	$-0.849924\pi$
$761$	−49.1530	−1.78180	−0.890898	+	0.454204i	$0.849924\pi$
$762$	0	0
$763$	10.3606	0.375077
$764$	9.42662	0.341043
$765$	0	0
$766$	17.6218	0.636700
$767$	36.5992	1.32152
$768$	0	0
$769$	−49.4560	−1.78343	−0.891715	−	0.452598i	$-0.850497\pi$
$769$	−49.4560	−1.78343	−0.891715	+	0.452598i	$0.850497\pi$
$770$	1.11686	0.0402489
$771$	0	0
$772$	5.86604	0.211124
$773$	18.8474	0.677893	0.338946	−	0.940806i	$-0.389929\pi$
$773$	18.8474	0.677893	0.338946	+	0.940806i	$0.389929\pi$
$774$	0	0
$775$	−27.0644	−0.972181
$776$	20.8999	0.750261
$777$	0	0
$778$	−0.208460	−0.00747367
$779$	−3.73569	−0.133845
$780$	0	0
$781$	34.6081	1.23837
$782$	6.60363	0.236145
$783$	0	0
$784$	0.539656	0.0192734
$785$	−3.91401	−0.139697
$786$	0	0
$787$	32.2531	1.14970	0.574849	−	0.818259i	$-0.305061\pi$
$787$	32.2531	1.14970	0.574849	+	0.818259i	$0.305061\pi$
$788$	9.31949	0.331993
$789$	0	0
$790$	−4.37426	−0.155629
$791$	12.1901	0.433431
$792$	0	0
$793$	52.5744	1.86697
$794$	7.04802	0.250125
$795$	0	0
$796$	8.16279	0.289322
$797$	−21.3937	−0.757805	−0.378902	−	0.925437i	$-0.623698\pi$
$797$	−21.3937	−0.757805	−0.378902	+	0.925437i	$0.623698\pi$
$798$	0	0
$799$	41.4645	1.46691
$800$	−28.4445	−1.00567
$801$	0	0
$802$	9.93474	0.350808
$803$	−44.1608	−1.55840
$804$	0	0
$805$	−0.809807	−0.0285420
$806$	−22.6390	−0.797424
$807$	0	0
$808$	−35.4872	−1.24844
$809$	10.7150	0.376721	0.188360	−	0.982100i	$-0.439683\pi$
$809$	10.7150	0.376721	0.188360	+	0.982100i	$0.439683\pi$
$810$	0	0
$811$	8.31984	0.292149	0.146075	−	0.989274i	$-0.453336\pi$
$811$	8.31984	0.292149	0.146075	+	0.989274i	$0.453336\pi$
$812$	12.5127	0.439110
$813$	0	0
$814$	−19.1063	−0.669677
$815$	6.28335	0.220096
$816$	0	0
$817$	23.2174	0.812275
$818$	8.67172	0.303200
$819$	0	0
$820$	0.775763	0.0270908
$821$	5.64944	0.197167	0.0985834	−	0.995129i	$-0.468569\pi$
$821$	5.64944	0.197167	0.0985834	+	0.995129i	$0.468569\pi$
$822$	0	0
$823$	−17.7394	−0.618356	−0.309178	−	0.951004i	$-0.600054\pi$
$823$	−17.7394	−0.618356	−0.309178	+	0.951004i	$0.600054\pi$
$824$	27.5066	0.958238
$825$	0	0
$826$	−5.79259	−0.201550
$827$	−19.4735	−0.677160	−0.338580	−	0.940938i	$-0.609947\pi$
$827$	−19.4735	−0.677160	−0.338580	+	0.940938i	$0.609947\pi$
$828$	0	0
$829$	−7.64454	−0.265506	−0.132753	−	0.991149i	$-0.542382\pi$
$829$	−7.64454	−0.265506	−0.132753	+	0.991149i	$0.542382\pi$
$830$	−3.20896	−0.111384
$831$	0	0
$832$	−18.3109	−0.634818
$833$	3.48245	0.120660
$834$	0	0
$835$	4.73103	0.163724
$836$	−12.2580	−0.423951
$837$	0	0
$838$	14.0220	0.484381
$839$	−36.5165	−1.26069	−0.630345	−	0.776315i	$-0.717087\pi$
$839$	−36.5165	−1.26069	−0.630345	+	0.776315i	$0.717087\pi$
$840$	0	0
$841$	56.4459	1.94641
$842$	−32.3612	−1.11524
$843$	0	0
$844$	25.3202	0.871557
$845$	−4.39562	−0.151214
$846$	0	0
$847$	5.37170	0.184574
$848$	−6.98359	−0.239818
$849$	0	0
$850$	−13.6687	−0.468833
$851$	13.8535	0.474892
$852$	0	0
$853$	−17.2119	−0.589325	−0.294663	−	0.955601i	$-0.595207\pi$
$853$	−17.2119	−0.589325	−0.294663	+	0.955601i	$0.595207\pi$
$854$	−8.32101	−0.284739
$855$	0	0
$856$	−15.5269	−0.530698
$857$	−36.8613	−1.25916	−0.629579	−	0.776937i	$-0.716773\pi$
$857$	−36.8613	−1.25916	−0.629579	+	0.776937i	$0.716773\pi$
$858$	0	0
$859$	−39.9500	−1.36308	−0.681538	−	0.731783i	$-0.738688\pi$
$859$	−39.9500	−1.36308	−0.681538	+	0.731783i	$0.738688\pi$
$860$	−4.82138	−0.164408
$861$	0	0
$862$	−4.02722	−0.137168
$863$	4.30869	0.146670	0.0733348	−	0.997307i	$-0.476636\pi$
$863$	4.30869	0.146670	0.0733348	+	0.997307i	$0.476636\pi$
$864$	0	0
$865$	−8.15191	−0.277173
$866$	21.6446	0.735512
$867$	0	0
$868$	−7.50403	−0.254704
$869$	−64.1208	−2.17515
$870$	0	0
$871$	25.2706	0.856263
$872$	−27.9341	−0.945970
$873$	0	0
$874$	−4.24389	−0.143552
$875$	3.39288	0.114700
$876$	0	0
$877$	−41.7064	−1.40832	−0.704162	−	0.710039i	$-0.748677\pi$
$877$	−41.7064	−1.40832	−0.704162	+	0.710039i	$0.748677\pi$
$878$	−24.2525	−0.818483
$879$	0	0
$880$	0.749691	0.0252721
$881$	−14.6304	−0.492910	−0.246455	−	0.969154i	$-0.579266\pi$
$881$	−14.6304	−0.492910	−0.246455	+	0.969154i	$0.579266\pi$
$882$	0	0
$883$	20.2903	0.682821	0.341411	−	0.939914i	$-0.389095\pi$
$883$	20.2903	0.682821	0.341411	+	0.939914i	$0.389095\pi$
$884$	23.9454	0.805372
$885$	0	0
$886$	−24.6572	−0.828374
$887$	14.1451	0.474945	0.237473	−	0.971394i	$-0.423681\pi$
$887$	14.1451	0.474945	0.237473	+	0.971394i	$0.423681\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−0.101143	−0.00339033
$891$	0	0
$892$	14.3864	0.481692
$893$	−26.6476	−0.891728
$894$	0	0
$895$	2.30003	0.0768816
$896$	−8.75443	−0.292465
$897$	0	0
$898$	9.81410	0.327501
$899$	−51.2430	−1.70905
$900$	0	0
$901$	−45.0657	−1.50136
$902$	−5.42984	−0.180794
$903$	0	0
$904$	−32.8671	−1.09314
$905$	−2.01004	−0.0668160
$906$	0	0
$907$	−53.2224	−1.76722	−0.883611	−	0.468223i	$-0.844895\pi$
$907$	−53.2224	−1.76722	−0.883611	+	0.468223i	$0.844895\pi$
$908$	−2.75102	−0.0912957
$909$	0	0
$910$	1.40212	0.0464799
$911$	−35.5787	−1.17877	−0.589387	−	0.807851i	$-0.700631\pi$
$911$	−35.5787	−1.17877	−0.589387	+	0.807851i	$0.700631\pi$
$912$	0	0
$913$	−47.0390	−1.55676
$914$	−26.1046	−0.863464
$915$	0	0
$916$	−19.9082	−0.657787
$917$	11.7624	0.388428
$918$	0	0
$919$	55.8133	1.84111	0.920556	−	0.390610i	$-0.127736\pi$
$919$	55.8133	1.84111	0.920556	+	0.390610i	$0.127736\pi$
$920$	2.18340	0.0719847
$921$	0	0
$922$	9.80571	0.322934
$923$	43.4473	1.43009
$924$	0	0
$925$	−28.6751	−0.942830
$926$	28.2487	0.928310
$927$	0	0
$928$	−53.8562	−1.76792
$929$	−18.8585	−0.618726	−0.309363	−	0.950944i	$-0.600116\pi$
$929$	−18.8585	−0.618726	−0.309363	+	0.950944i	$0.600116\pi$
$930$	0	0
$931$	−2.23803	−0.0733485
$932$	−25.5361	−0.836462
$933$	0	0
$934$	13.6948	0.448107
$935$	4.83782	0.158213
$936$	0	0
$937$	−36.7884	−1.20183	−0.600913	−	0.799315i	$-0.705196\pi$
$937$	−36.7884	−1.20183	−0.600913	+	0.799315i	$0.705196\pi$
$938$	−3.99961	−0.130592
$939$	0	0
$940$	5.53370	0.180489
$941$	16.0773	0.524105	0.262053	−	0.965054i	$-0.415601\pi$
$941$	16.0773	0.524105	0.262053	+	0.965054i	$0.415601\pi$
$942$	0	0
$943$	3.93703	0.128207
$944$	−3.88826	−0.126552
$945$	0	0
$946$	33.7466	1.09720
$947$	40.8039	1.32595	0.662974	−	0.748642i	$-0.269294\pi$
$947$	40.8039	1.32595	0.662974	+	0.748642i	$0.269294\pi$
$948$	0	0
$949$	−55.4400	−1.79966
$950$	8.78434	0.285002
$951$	0	0
$952$	−9.38937	−0.304311
$953$	−2.02328	−0.0655406	−0.0327703	−	0.999463i	$-0.510433\pi$
$953$	−2.02328	−0.0655406	−0.0327703	+	0.999463i	$0.510433\pi$
$954$	0	0
$955$	2.39094	0.0773689
$956$	5.14823	0.166506
$957$	0	0
$958$	−9.73368	−0.314481
$959$	22.2429	0.718260
$960$	0	0
$961$	−0.268875	−0.00867340
$962$	−23.9863	−0.773349
$963$	0	0
$964$	31.8372	1.02541
$965$	1.48785	0.0478954
$966$	0	0
$967$	54.7174	1.75959	0.879796	−	0.475351i	$-0.157679\pi$
$967$	54.7174	1.75959	0.879796	+	0.475351i	$0.157679\pi$
$968$	−14.4832	−0.465507
$969$	0	0
$970$	2.13966	0.0687002
$971$	−13.3453	−0.428271	−0.214136	−	0.976804i	$-0.568693\pi$
$971$	−13.3453	−0.428271	−0.214136	+	0.976804i	$0.568693\pi$
$972$	0	0
$973$	4.45596	0.142852
$974$	12.5402	0.401812
$975$	0	0
$976$	−5.58546	−0.178786
$977$	−30.3067	−0.969596	−0.484798	−	0.874626i	$-0.661107\pi$
$977$	−30.3067	−0.969596	−0.484798	+	0.874626i	$0.661107\pi$
$978$	0	0
$979$	−1.48262	−0.0473848
$980$	0.464755	0.0148460
$981$	0	0
$982$	−7.13727	−0.227759
$983$	−25.8700	−0.825124	−0.412562	−	0.910930i	$-0.635366\pi$
$983$	−25.8700	−0.825124	−0.412562	+	0.910930i	$0.635366\pi$
$984$	0	0
$985$	2.36377	0.0753158
$986$	−25.8800	−0.824188
$987$	0	0
$988$	−15.3888	−0.489582
$989$	−24.4688	−0.778061
$990$	0	0
$991$	−19.5822	−0.622050	−0.311025	−	0.950402i	$-0.600672\pi$
$991$	−19.5822	−0.622050	−0.311025	+	0.950402i	$0.600672\pi$
$992$	32.2983	1.02547
$993$	0	0
$994$	−6.87646	−0.218108
$995$	2.07038	0.0656356
$996$	0	0
$997$	−51.3683	−1.62685	−0.813425	−	0.581670i	$-0.802399\pi$
$997$	−51.3683	−1.62685	−0.813425	+	0.581670i	$0.802399\pi$
$998$	12.3063	0.389550
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.u.1.10		18
3.2	odd	2		2667.2.a.p.1.9	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.p.1.9	✓	18	3.2	odd	2
8001.2.a.u.1.10		18	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)

\(f(q)\)	\(=\)	\(q+0.803961 q^{2} -1.35365 q^{4} -0.343335 q^{5} +1.00000 q^{7} -2.69620 q^{8} +O(q^{10})\)	\(q+0.803961 q^{2} -1.35365 q^{4} -0.343335 q^{5} +1.00000 q^{7} -2.69620 q^{8} -0.276028 q^{10} -4.04620 q^{11} -5.07964 q^{13} +0.803961 q^{14} +0.539656 q^{16} +3.48245 q^{17} -2.23803 q^{19} +0.464755 q^{20} -3.25298 q^{22} +2.35865 q^{23} -4.88212 q^{25} -4.08383 q^{26} -1.35365 q^{28} -9.24369 q^{29} +5.54357 q^{31} +5.82626 q^{32} +2.79975 q^{34} -0.343335 q^{35} +5.87348 q^{37} -1.79929 q^{38} +0.925700 q^{40} +1.66919 q^{41} -10.3740 q^{43} +5.47712 q^{44} +1.89626 q^{46} +11.9067 q^{47} +1.00000 q^{49} -3.92503 q^{50} +6.87604 q^{52} -12.9408 q^{53} +1.38920 q^{55} -2.69620 q^{56} -7.43157 q^{58} -7.20507 q^{59} -10.3500 q^{61} +4.45681 q^{62} +3.60477 q^{64} +1.74402 q^{65} -4.97489 q^{67} -4.71400 q^{68} -0.276028 q^{70} -8.55324 q^{71} +10.9142 q^{73} +4.72205 q^{74} +3.02950 q^{76} -4.04620 q^{77} +15.8472 q^{79} -0.185283 q^{80} +1.34196 q^{82} +11.6255 q^{83} -1.19565 q^{85} -8.34032 q^{86} +10.9094 q^{88} +0.366424 q^{89} -5.07964 q^{91} -3.19278 q^{92} +9.57253 q^{94} +0.768394 q^{95} -7.75160 q^{97} +0.803961 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) \) 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) \) 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8 - 4 * q^10 + 9 * q^11 - 25 * q^13 + 6 * q^14 + 34 * q^16 + 17 * q^17 - 5 * q^19 + 21 * q^20 + 5 * q^22 + 14 * q^23 + 28 * q^25 + 8 * q^26 + 22 * q^28 + 17 * q^29 + 5 * q^31 + 53 * q^32 - 19 * q^34 + 10 * q^35 - 15 * q^37 + 22 * q^38 - q^40 + 17 * q^41 + q^43 + 33 * q^44 + 10 * q^46 + 31 * q^47 + 18 * q^49 + 35 * q^50 - 70 * q^52 + 35 * q^53 + 4 * q^55 + 21 * q^56 + 3 * q^58 + 46 * q^59 - 5 * q^61 + 10 * q^62 + 63 * q^64 + 12 * q^65 + 6 * q^67 + 56 * q^68 - 4 * q^70 + 22 * q^71 - 16 * q^73 - 18 * q^74 + 32 * q^76 + 9 * q^77 + 46 * q^79 + 30 * q^80 - 12 * q^82 + 46 * q^83 + 4 * q^85 - 18 * q^86 + 30 * q^88 + 42 * q^89 - 25 * q^91 + 48 * q^92 + 3 * q^94 + 2 * q^95 - 35 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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