LMFDB - Embedded newform 8001.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 22 x^{13} + 186 x^{11} - 763 x^{9} - 7 x^{8} + 1588 x^{7} + 64 x^{6} - 1625 x^{5} - 185 x^{4} + \cdots - 13 $ x^15 - 22x^13 + 186x^11 - 763x^9 - 7x^8 + 1588x^7 + 64x^6 - 1625x^5 - 185x^4 + 726x^3 + 145x^2 - 83*x - 13
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			1.2
Root			$-2.51564$ 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)

$f(q)$	$=$	$q-2.51564 q^{2} +4.32847 q^{4} +1.19101 q^{5} -1.00000 q^{7} -5.85760 q^{8} +O(q^{10})$	\(q-2.51564 q^{2} +4.32847 q^{4} +1.19101 q^{5} -1.00000 q^{7} -5.85760 q^{8} -2.99615 q^{10} +4.18912 q^{11} +1.43411 q^{13} +2.51564 q^{14} +6.07870 q^{16} -1.76349 q^{17} -4.92899 q^{19} +5.15524 q^{20} -10.5383 q^{22} +0.661413 q^{23} -3.58150 q^{25} -3.60772 q^{26} -4.32847 q^{28} +0.521062 q^{29} +3.67252 q^{31} -3.57666 q^{32} +4.43630 q^{34} -1.19101 q^{35} +1.30196 q^{37} +12.3996 q^{38} -6.97644 q^{40} -8.43934 q^{41} +4.56744 q^{43} +18.1325 q^{44} -1.66388 q^{46} -0.797594 q^{47} +1.00000 q^{49} +9.00978 q^{50} +6.20752 q^{52} +9.82487 q^{53} +4.98928 q^{55} +5.85760 q^{56} -1.31081 q^{58} -12.7579 q^{59} +0.0608610 q^{61} -9.23874 q^{62} -3.15981 q^{64} +1.70804 q^{65} -7.96005 q^{67} -7.63319 q^{68} +2.99615 q^{70} -6.48720 q^{71} -9.61580 q^{73} -3.27527 q^{74} -21.3350 q^{76} -4.18912 q^{77} +15.3521 q^{79} +7.23978 q^{80} +21.2304 q^{82} -5.82288 q^{83} -2.10033 q^{85} -11.4900 q^{86} -24.5382 q^{88} -12.1345 q^{89} -1.43411 q^{91} +2.86290 q^{92} +2.00646 q^{94} -5.87047 q^{95} +0.0612387 q^{97} -2.51564 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * q^97

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 $

$2$	−2.51564	−1.77883	−0.889415	−	0.457101i	$-0.848888\pi$
$2$	−2.51564	−1.77883	−0.889415	+	0.457101i	$0.848888\pi$
$3$	0	0
$3$	0	0
$4$	4.32847	2.16423
$5$	1.19101	0.532635	0.266317	−	0.963885i	$-0.414193\pi$
$5$	1.19101	0.532635	0.266317	+	0.963885i	$0.414193\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−5.85760	−2.07097
$9$	0	0
$10$	−2.99615	−0.947466
$11$	4.18912	1.26307	0.631534	−	0.775348i	$-0.282426\pi$
$11$	4.18912	1.26307	0.631534	+	0.775348i	$0.282426\pi$
$12$	0	0
$13$	1.43411	0.397752	0.198876	−	0.980025i	$-0.436271\pi$
$13$	1.43411	0.397752	0.198876	+	0.980025i	$0.436271\pi$
$14$	2.51564	0.672334
$15$	0	0
$16$	6.07870	1.51968
$17$	−1.76349	−0.427708	−0.213854	−	0.976866i	$-0.568602\pi$
$17$	−1.76349	−0.427708	−0.213854	+	0.976866i	$0.568602\pi$
$18$	0	0
$19$	−4.92899	−1.13079	−0.565394	−	0.824821i	$-0.691276\pi$
$19$	−4.92899	−1.13079	−0.565394	+	0.824821i	$0.691276\pi$
$20$	5.15524	1.15275
$21$	0	0
$22$	−10.5383	−2.24678
$23$	0.661413	0.137914	0.0689570	−	0.997620i	$-0.478033\pi$
$23$	0.661413	0.137914	0.0689570	+	0.997620i	$0.478033\pi$
$24$	0	0
$25$	−3.58150	−0.716300
$26$	−3.60772	−0.707532
$27$	0	0
$28$	−4.32847	−0.818004
$29$	0.521062	0.0967588	0.0483794	−	0.998829i	$-0.484594\pi$
$29$	0.521062	0.0967588	0.0483794	+	0.998829i	$0.484594\pi$
$30$	0	0
$31$	3.67252	0.659603	0.329802	−	0.944050i	$-0.393018\pi$
$31$	3.67252	0.659603	0.329802	+	0.944050i	$0.393018\pi$
$32$	−3.57666	−0.632270
$33$	0	0
$34$	4.43630	0.760820
$35$	−1.19101	−0.201317
$36$	0	0
$37$	1.30196	0.214041	0.107021	−	0.994257i	$-0.465869\pi$
$37$	1.30196	0.214041	0.107021	+	0.994257i	$0.465869\pi$
$38$	12.3996	2.01148
$39$	0	0
$40$	−6.97644	−1.10307
$41$	−8.43934	−1.31800	−0.659002	−	0.752142i	$-0.729021\pi$
$41$	−8.43934	−1.31800	−0.659002	+	0.752142i	$0.729021\pi$
$42$	0	0
$43$	4.56744	0.696527	0.348264	−	0.937397i	$-0.386771\pi$
$43$	4.56744	0.696527	0.348264	+	0.937397i	$0.386771\pi$
$44$	18.1325	2.73358
$45$	0	0
$46$	−1.66388	−0.245326
$47$	−0.797594	−0.116341	−0.0581705	−	0.998307i	$-0.518527\pi$
$47$	−0.797594	−0.116341	−0.0581705	+	0.998307i	$0.518527\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	9.00978	1.27418
$51$	0	0
$52$	6.20752	0.860828
$53$	9.82487	1.34955	0.674775	−	0.738023i	$-0.264241\pi$
$53$	9.82487	1.34955	0.674775	+	0.738023i	$0.264241\pi$
$54$	0	0
$55$	4.98928	0.672754
$56$	5.85760	0.782755
$57$	0	0
$58$	−1.31081	−0.172117
$59$	−12.7579	−1.66093	−0.830466	−	0.557070i	$-0.811926\pi$
$59$	−12.7579	−1.66093	−0.830466	+	0.557070i	$0.811926\pi$
$60$	0	0
$61$	0.0608610	0.00779246	0.00389623	−	0.999992i	$-0.498760\pi$
$61$	0.0608610	0.00779246	0.00389623	+	0.999992i	$0.498760\pi$
$62$	−9.23874	−1.17332
$63$	0	0
$64$	−3.15981	−0.394976
$65$	1.70804	0.211856
$66$	0	0
$67$	−7.96005	−0.972474	−0.486237	−	0.873827i	$-0.661631\pi$
$67$	−7.96005	−0.972474	−0.486237	+	0.873827i	$0.661631\pi$
$68$	−7.63319	−0.925661
$69$	0	0
$70$	2.99615	0.358109
$71$	−6.48720	−0.769889	−0.384945	−	0.922940i	$-0.625779\pi$
$71$	−6.48720	−0.769889	−0.384945	+	0.922940i	$0.625779\pi$
$72$	0	0
$73$	−9.61580	−1.12544	−0.562722	−	0.826646i	$-0.690246\pi$
$73$	−9.61580	−1.12544	−0.562722	+	0.826646i	$0.690246\pi$
$74$	−3.27527	−0.380743
$75$	0	0
$76$	−21.3350	−2.44729
$77$	−4.18912	−0.477395
$78$	0	0
$79$	15.3521	1.72725	0.863625	−	0.504135i	$-0.168189\pi$
$79$	15.3521	1.72725	0.863625	+	0.504135i	$0.168189\pi$
$80$	7.23978	0.809432
$81$	0	0
$82$	21.2304	2.34450
$83$	−5.82288	−0.639144	−0.319572	−	0.947562i	$-0.603539\pi$
$83$	−5.82288	−0.639144	−0.319572	+	0.947562i	$0.603539\pi$
$84$	0	0
$85$	−2.10033	−0.227812
$86$	−11.4900	−1.23900
$87$	0	0
$88$	−24.5382	−2.61578
$89$	−12.1345	−1.28625	−0.643125	−	0.765761i	$-0.722362\pi$
$89$	−12.1345	−1.28625	−0.643125	+	0.765761i	$0.722362\pi$
$90$	0	0
$91$	−1.43411	−0.150336
$92$	2.86290	0.298478
$93$	0	0
$94$	2.00646	0.206951
$95$	−5.87047	−0.602297
$96$	0	0
$97$	0.0612387	0.00621785	0.00310893	−	0.999995i	$-0.499010\pi$
$97$	0.0612387	0.00621785	0.00310893	+	0.999995i	$0.499010\pi$
$98$	−2.51564	−0.254118
$99$	0	0
$100$	−15.5024	−1.55024
$101$	9.53969	0.949235	0.474617	−	0.880192i	$-0.342586\pi$
$101$	9.53969	0.949235	0.474617	+	0.880192i	$0.342586\pi$
$102$	0	0
$103$	−2.73565	−0.269552	−0.134776	−	0.990876i	$-0.543031\pi$
$103$	−2.73565	−0.269552	−0.134776	+	0.990876i	$0.543031\pi$
$104$	−8.40046	−0.823733
$105$	0	0
$106$	−24.7159	−2.40062
$107$	6.98917	0.675669	0.337834	−	0.941206i	$-0.390306\pi$
$107$	6.98917	0.675669	0.337834	+	0.941206i	$0.390306\pi$
$108$	0	0
$109$	−1.71721	−0.164479	−0.0822395	−	0.996613i	$-0.526207\pi$
$109$	−1.71721	−0.164479	−0.0822395	+	0.996613i	$0.526207\pi$
$110$	−12.5512	−1.19671
$111$	0	0
$112$	−6.07870	−0.574383
$113$	19.9992	1.88136	0.940682	−	0.339290i	$-0.110187\pi$
$113$	19.9992	1.88136	0.940682	+	0.339290i	$0.110187\pi$
$114$	0	0
$115$	0.787747	0.0734578
$116$	2.25540	0.209409
$117$	0	0
$118$	32.0942	2.95451
$119$	1.76349	0.161659
$120$	0	0
$121$	6.54875	0.595341
$122$	−0.153105	−0.0138615
$123$	0	0
$124$	15.8964	1.42754
$125$	−10.2206	−0.914161
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	15.1023	1.33486
$129$	0	0
$130$	−4.29682	−0.376856
$131$	−10.3792	−0.906832	−0.453416	−	0.891299i	$-0.649795\pi$
$131$	−10.3792	−0.906832	−0.453416	+	0.891299i	$0.649795\pi$
$132$	0	0
$133$	4.92899	0.427398
$134$	20.0246	1.72987
$135$	0	0
$136$	10.3298	0.885773
$137$	−13.4955	−1.15300	−0.576500	−	0.817097i	$-0.695582\pi$
$137$	−13.4955	−1.15300	−0.576500	+	0.817097i	$0.695582\pi$
$138$	0	0
$139$	−13.9654	−1.18453	−0.592263	−	0.805745i	$-0.701765\pi$
$139$	−13.9654	−1.18453	−0.592263	+	0.805745i	$0.701765\pi$
$140$	−5.15524	−0.435697
$141$	0	0
$142$	16.3195	1.36950
$143$	6.00768	0.502387
$144$	0	0
$145$	0.620589	0.0515371
$146$	24.1899	2.00197
$147$	0	0
$148$	5.63550	0.463235
$149$	14.4117	1.18065	0.590325	−	0.807166i	$-0.299000\pi$
$149$	14.4117	1.18065	0.590325	+	0.807166i	$0.299000\pi$
$150$	0	0
$151$	−9.13744	−0.743595	−0.371797	−	0.928314i	$-0.621258\pi$
$151$	−9.13744	−0.743595	−0.371797	+	0.928314i	$0.621258\pi$
$152$	28.8721	2.34183
$153$	0	0
$154$	10.5383	0.849204
$155$	4.37399	0.351328
$156$	0	0
$157$	9.89040	0.789340	0.394670	−	0.918823i	$-0.370859\pi$
$157$	9.89040	0.789340	0.394670	+	0.918823i	$0.370859\pi$
$158$	−38.6205	−3.07248
$159$	0	0
$160$	−4.25983	−0.336769
$161$	−0.661413	−0.0521266
$162$	0	0
$163$	12.7487	0.998553	0.499276	−	0.866443i	$-0.333599\pi$
$163$	12.7487	0.998553	0.499276	+	0.866443i	$0.333599\pi$
$164$	−36.5294	−2.85247
$165$	0	0
$166$	14.6483	1.13693
$167$	−20.4473	−1.58226	−0.791129	−	0.611649i	$-0.790506\pi$
$167$	−20.4473	−1.58226	−0.791129	+	0.611649i	$0.790506\pi$
$168$	0	0
$169$	−10.9433	−0.841794
$170$	5.28367	0.405239
$171$	0	0
$172$	19.7700	1.50745
$173$	−21.7709	−1.65521	−0.827606	−	0.561309i	$-0.810298\pi$
$173$	−21.7709	−1.65521	−0.827606	+	0.561309i	$0.810298\pi$
$174$	0	0
$175$	3.58150	0.270736
$176$	25.4644	1.91945
$177$	0	0
$178$	30.5260	2.28802
$179$	−3.27383	−0.244698	−0.122349	−	0.992487i	$-0.539043\pi$
$179$	−3.27383	−0.244698	−0.122349	+	0.992487i	$0.539043\pi$
$180$	0	0
$181$	−16.5329	−1.22888	−0.614440	−	0.788964i	$-0.710618\pi$
$181$	−16.5329	−1.22888	−0.614440	+	0.788964i	$0.710618\pi$
$182$	3.60772	0.267422
$183$	0	0
$184$	−3.87429	−0.285616
$185$	1.55065	0.114006
$186$	0	0
$187$	−7.38746	−0.540225
$188$	−3.45236	−0.251789
$189$	0	0
$190$	14.7680	1.07138
$191$	−23.8442	−1.72530	−0.862651	−	0.505799i	$-0.831198\pi$
$191$	−23.8442	−1.72530	−0.862651	+	0.505799i	$0.831198\pi$
$192$	0	0
$193$	2.23275	0.160717	0.0803586	−	0.996766i	$-0.474393\pi$
$193$	2.23275	0.160717	0.0803586	+	0.996766i	$0.474393\pi$
$194$	−0.154055	−0.0110605
$195$	0	0
$196$	4.32847	0.309176
$197$	−20.4153	−1.45453	−0.727266	−	0.686355i	$-0.759209\pi$
$197$	−20.4153	−1.45453	−0.727266	+	0.686355i	$0.759209\pi$
$198$	0	0
$199$	9.51842	0.674743	0.337372	−	0.941372i	$-0.390462\pi$
$199$	9.51842	0.674743	0.337372	+	0.941372i	$0.390462\pi$
$200$	20.9790	1.48344
$201$	0	0
$202$	−23.9985	−1.68853
$203$	−0.521062	−0.0365714
$204$	0	0
$205$	−10.0513	−0.702014
$206$	6.88193	0.479487
$207$	0	0
$208$	8.71755	0.604453
$209$	−20.6482	−1.42826
$210$	0	0
$211$	−12.1258	−0.834773	−0.417386	−	0.908729i	$-0.637054\pi$
$211$	−12.1258	−0.834773	−0.417386	+	0.908729i	$0.637054\pi$
$212$	42.5267	2.92074
$213$	0	0
$214$	−17.5823	−1.20190
$215$	5.43985	0.370995
$216$	0	0
$217$	−3.67252	−0.249307
$218$	4.31989	0.292580
$219$	0	0
$220$	21.5959	1.45600
$221$	−2.52904	−0.170122
$222$	0	0
$223$	6.01349	0.402693	0.201347	−	0.979520i	$-0.435468\pi$
$223$	6.01349	0.402693	0.201347	+	0.979520i	$0.435468\pi$
$224$	3.57666	0.238975
$225$	0	0
$226$	−50.3108	−3.34662
$227$	20.1839	1.33965	0.669825	−	0.742519i	$-0.266369\pi$
$227$	20.1839	1.33965	0.669825	+	0.742519i	$0.266369\pi$
$228$	0	0
$229$	10.8054	0.714043	0.357021	−	0.934096i	$-0.383792\pi$
$229$	10.8054	0.714043	0.357021	+	0.934096i	$0.383792\pi$
$230$	−1.98169	−0.130669
$231$	0	0
$232$	−3.05217	−0.200385
$233$	17.7238	1.16112	0.580561	−	0.814217i	$-0.302833\pi$
$233$	17.7238	1.16112	0.580561	+	0.814217i	$0.302833\pi$
$234$	0	0
$235$	−0.949941	−0.0619673
$236$	−55.2220	−3.59465
$237$	0	0
$238$	−4.43630	−0.287563
$239$	−16.3003	−1.05438	−0.527189	−	0.849748i	$-0.676754\pi$
$239$	−16.3003	−1.05438	−0.527189	+	0.849748i	$0.676754\pi$
$240$	0	0
$241$	−16.4252	−1.05804	−0.529020	−	0.848609i	$-0.677440\pi$
$241$	−16.4252	−1.05804	−0.529020	+	0.848609i	$0.677440\pi$
$242$	−16.4743	−1.05901
$243$	0	0
$244$	0.263435	0.0168647
$245$	1.19101	0.0760907
$246$	0	0
$247$	−7.06874	−0.449773
$248$	−21.5121	−1.36602
$249$	0	0
$250$	25.7115	1.62614
$251$	10.0670	0.635421	0.317711	−	0.948188i	$-0.397086\pi$
$251$	10.0670	0.635421	0.317711	+	0.948188i	$0.397086\pi$
$252$	0	0
$253$	2.77074	0.174195
$254$	−2.51564	−0.157846
$255$	0	0
$256$	−31.6723	−1.97952
$257$	−12.9478	−0.807664	−0.403832	−	0.914833i	$-0.632322\pi$
$257$	−12.9478	−0.807664	−0.403832	+	0.914833i	$0.632322\pi$
$258$	0	0
$259$	−1.30196	−0.0808999
$260$	7.39320	0.458507
$261$	0	0
$262$	26.1103	1.61310
$263$	26.8400	1.65503	0.827513	−	0.561446i	$-0.189755\pi$
$263$	26.8400	1.65503	0.827513	+	0.561446i	$0.189755\pi$
$264$	0	0
$265$	11.7015	0.718817
$266$	−12.3996	−0.760268
$267$	0	0
$268$	−34.4548	−2.10466
$269$	17.1313	1.04451	0.522256	−	0.852789i	$-0.325090\pi$
$269$	17.1313	1.04451	0.522256	+	0.852789i	$0.325090\pi$
$270$	0	0
$271$	25.5813	1.55395	0.776975	−	0.629531i	$-0.216753\pi$
$271$	25.5813	1.55395	0.776975	+	0.629531i	$0.216753\pi$
$272$	−10.7197	−0.649978
$273$	0	0
$274$	33.9499	2.05099
$275$	−15.0033	−0.904736
$276$	0	0
$277$	−14.5241	−0.872669	−0.436334	−	0.899785i	$-0.643724\pi$
$277$	−14.5241	−0.872669	−0.436334	+	0.899785i	$0.643724\pi$
$278$	35.1319	2.10707
$279$	0	0
$280$	6.97644	0.416922
$281$	−10.3069	−0.614857	−0.307429	−	0.951571i	$-0.599469\pi$
$281$	−10.3069	−0.614857	−0.307429	+	0.951571i	$0.599469\pi$
$282$	0	0
$283$	−19.8952	−1.18264	−0.591322	−	0.806435i	$-0.701394\pi$
$283$	−19.8952	−1.18264	−0.591322	+	0.806435i	$0.701394\pi$
$284$	−28.0796	−1.66622
$285$	0	0
$286$	−15.1132	−0.893661
$287$	8.43934	0.498158
$288$	0	0
$289$	−13.8901	−0.817066
$290$	−1.56118	−0.0916757
$291$	0	0
$292$	−41.6217	−2.43572
$293$	−8.29939	−0.484855	−0.242428	−	0.970169i	$-0.577944\pi$
$293$	−8.29939	−0.484855	−0.242428	+	0.970169i	$0.577944\pi$
$294$	0	0
$295$	−15.1947	−0.884670
$296$	−7.62637	−0.443274
$297$	0	0
$298$	−36.2547	−2.10018
$299$	0.948541	0.0548555
$300$	0	0
$301$	−4.56744	−0.263263
$302$	22.9866	1.32273
$303$	0	0
$304$	−29.9619	−1.71843
$305$	0.0724859	0.00415053
$306$	0	0
$307$	−16.4163	−0.936927	−0.468464	−	0.883483i	$-0.655192\pi$
$307$	−16.4163	−0.936927	−0.468464	+	0.883483i	$0.655192\pi$
$308$	−18.1325	−1.03319
$309$	0	0
$310$	−11.0034	−0.624952
$311$	9.12922	0.517671	0.258835	−	0.965921i	$-0.416661\pi$
$311$	9.12922	0.517671	0.258835	+	0.965921i	$0.416661\pi$
$312$	0	0
$313$	−3.23344	−0.182765	−0.0913824	−	0.995816i	$-0.529129\pi$
$313$	−3.23344	−0.182765	−0.0913824	+	0.995816i	$0.529129\pi$
$314$	−24.8807	−1.40410
$315$	0	0
$316$	66.4512	3.73817
$317$	12.6938	0.712953	0.356477	−	0.934304i	$-0.383978\pi$
$317$	12.6938	0.712953	0.356477	+	0.934304i	$0.383978\pi$
$318$	0	0
$319$	2.18279	0.122213
$320$	−3.76335	−0.210378
$321$	0	0
$322$	1.66388	0.0927243
$323$	8.69221	0.483647
$324$	0	0
$325$	−5.13628	−0.284910
$326$	−32.0711	−1.77625
$327$	0	0
$328$	49.4343	2.72955
$329$	0.797594	0.0439728
$330$	0	0
$331$	2.70814	0.148853	0.0744265	−	0.997227i	$-0.476287\pi$
$331$	2.70814	0.148853	0.0744265	+	0.997227i	$0.476287\pi$
$332$	−25.2041	−1.38326
$333$	0	0
$334$	51.4381	2.81457
$335$	−9.48047	−0.517974
$336$	0	0
$337$	28.3780	1.54585	0.772924	−	0.634499i	$-0.218793\pi$
$337$	28.3780	1.54585	0.772924	+	0.634499i	$0.218793\pi$
$338$	27.5295	1.49741
$339$	0	0
$340$	−9.09119	−0.493039
$341$	15.3846	0.833124
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−26.7542	−1.44249
$345$	0	0
$346$	54.7679	2.94434
$347$	1.19640	0.0642261	0.0321131	−	0.999484i	$-0.489776\pi$
$347$	1.19640	0.0642261	0.0321131	+	0.999484i	$0.489776\pi$
$348$	0	0
$349$	−6.91527	−0.370166	−0.185083	−	0.982723i	$-0.559255\pi$
$349$	−6.91527	−0.370166	−0.185083	+	0.982723i	$0.559255\pi$
$350$	−9.00978	−0.481593
$351$	0	0
$352$	−14.9831	−0.798600
$353$	−21.1758	−1.12707	−0.563537	−	0.826091i	$-0.690560\pi$
$353$	−21.1758	−1.12707	−0.563537	+	0.826091i	$0.690560\pi$
$354$	0	0
$355$	−7.72630	−0.410070
$356$	−52.5236	−2.78375
$357$	0	0
$358$	8.23580	0.435276
$359$	22.4862	1.18678	0.593388	−	0.804916i	$-0.297790\pi$
$359$	22.4862	1.18678	0.593388	+	0.804916i	$0.297790\pi$
$360$	0	0
$361$	5.29497	0.278682
$362$	41.5908	2.18597
$363$	0	0
$364$	−6.20752	−0.325362
$365$	−11.4525	−0.599450
$366$	0	0
$367$	17.1938	0.897512	0.448756	−	0.893654i	$-0.351867\pi$
$367$	17.1938	0.897512	0.448756	+	0.893654i	$0.351867\pi$
$368$	4.02053	0.209585
$369$	0	0
$370$	−3.90087	−0.202797
$371$	−9.82487	−0.510082
$372$	0	0
$373$	8.57902	0.444205	0.222102	−	0.975023i	$-0.428708\pi$
$373$	8.57902	0.444205	0.222102	+	0.975023i	$0.428708\pi$
$374$	18.5842	0.960967
$375$	0	0
$376$	4.67199	0.240939
$377$	0.747262	0.0384860
$378$	0	0
$379$	−4.60673	−0.236632	−0.118316	−	0.992976i	$-0.537750\pi$
$379$	−4.60673	−0.236632	−0.118316	+	0.992976i	$0.537750\pi$
$380$	−25.4101	−1.30351
$381$	0	0
$382$	59.9834	3.06902
$383$	21.3698	1.09195	0.545973	−	0.837803i	$-0.316160\pi$
$383$	21.3698	1.09195	0.545973	+	0.837803i	$0.316160\pi$
$384$	0	0
$385$	−4.98928	−0.254277
$386$	−5.61682	−0.285888
$387$	0	0
$388$	0.265070	0.0134569
$389$	24.6810	1.25138	0.625689	−	0.780073i	$-0.284818\pi$
$389$	24.6810	1.25138	0.625689	+	0.780073i	$0.284818\pi$
$390$	0	0
$391$	−1.16639	−0.0589870
$392$	−5.85760	−0.295853
$393$	0	0
$394$	51.3577	2.58737
$395$	18.2845	0.919993
$396$	0	0
$397$	21.0436	1.05615	0.528073	−	0.849199i	$-0.322915\pi$
$397$	21.0436	1.05615	0.528073	+	0.849199i	$0.322915\pi$
$398$	−23.9450	−1.20025
$399$	0	0
$400$	−21.7709	−1.08854
$401$	−12.7428	−0.636346	−0.318173	−	0.948033i	$-0.603069\pi$
$401$	−12.7428	−0.636346	−0.318173	+	0.948033i	$0.603069\pi$
$402$	0	0
$403$	5.26681	0.262358
$404$	41.2923	2.05437
$405$	0	0
$406$	1.31081	0.0650542
$407$	5.45407	0.270348
$408$	0	0
$409$	−10.4497	−0.516703	−0.258351	−	0.966051i	$-0.583179\pi$
$409$	−10.4497	−0.516703	−0.258351	+	0.966051i	$0.583179\pi$
$410$	25.2855	1.24876
$411$	0	0
$412$	−11.8412	−0.583374
$413$	12.7579	0.627773
$414$	0	0
$415$	−6.93509	−0.340430
$416$	−5.12933	−0.251486
$417$	0	0
$418$	51.9434	2.54064
$419$	−35.8726	−1.75249	−0.876246	−	0.481864i	$-0.839960\pi$
$419$	−35.8726	−1.75249	−0.876246	+	0.481864i	$0.839960\pi$
$420$	0	0
$421$	2.32495	0.113311	0.0566555	−	0.998394i	$-0.481956\pi$
$421$	2.32495	0.113311	0.0566555	+	0.998394i	$0.481956\pi$
$422$	30.5041	1.48492
$423$	0	0
$424$	−57.5502	−2.79488
$425$	6.31593	0.306368
$426$	0	0
$427$	−0.0608610	−0.00294527
$428$	30.2524	1.46231
$429$	0	0
$430$	−13.6847	−0.659936
$431$	−34.8419	−1.67828	−0.839138	−	0.543919i	$-0.816940\pi$
$431$	−34.8419	−1.67828	−0.839138	+	0.543919i	$0.816940\pi$
$432$	0	0
$433$	−2.46087	−0.118262	−0.0591309	−	0.998250i	$-0.518833\pi$
$433$	−2.46087	−0.118262	−0.0591309	+	0.998250i	$0.518833\pi$
$434$	9.23874	0.443474
$435$	0	0
$436$	−7.43289	−0.355971
$437$	−3.26010	−0.155952
$438$	0	0
$439$	4.86694	0.232286	0.116143	−	0.993232i	$-0.462947\pi$
$439$	4.86694	0.232286	0.116143	+	0.993232i	$0.462947\pi$
$440$	−29.2252	−1.39326
$441$	0	0
$442$	6.36217	0.302617
$443$	−5.94755	−0.282577	−0.141288	−	0.989968i	$-0.545124\pi$
$443$	−5.94755	−0.282577	−0.141288	+	0.989968i	$0.545124\pi$
$444$	0	0
$445$	−14.4522	−0.685101
$446$	−15.1278	−0.716323
$447$	0	0
$448$	3.15981	0.149287
$449$	36.7039	1.73216	0.866082	−	0.499902i	$-0.166631\pi$
$449$	36.7039	1.73216	0.866082	+	0.499902i	$0.166631\pi$
$450$	0	0
$451$	−35.3534	−1.66473
$452$	86.5658	4.07171
$453$	0	0
$454$	−50.7754	−2.38301
$455$	−1.70804	−0.0800742
$456$	0	0
$457$	−30.5030	−1.42687	−0.713436	−	0.700721i	$-0.752862\pi$
$457$	−30.5030	−1.42687	−0.713436	+	0.700721i	$0.752862\pi$
$458$	−27.1826	−1.27016
$459$	0	0
$460$	3.40974	0.158980
$461$	−26.1775	−1.21921	−0.609604	−	0.792706i	$-0.708671\pi$
$461$	−26.1775	−1.21921	−0.609604	+	0.792706i	$0.708671\pi$
$462$	0	0
$463$	29.8321	1.38642	0.693208	−	0.720738i	$-0.256197\pi$
$463$	29.8321	1.38642	0.693208	+	0.720738i	$0.256197\pi$
$464$	3.16738	0.147042
$465$	0	0
$466$	−44.5867	−2.06544
$467$	26.4147	1.22233	0.611163	−	0.791505i	$-0.290702\pi$
$467$	26.4147	1.22233	0.611163	+	0.791505i	$0.290702\pi$
$468$	0	0
$469$	7.96005	0.367561
$470$	2.38971	0.110229
$471$	0	0
$472$	74.7304	3.43975
$473$	19.1336	0.879762
$474$	0	0
$475$	17.6532	0.809984
$476$	7.63319	0.349867
$477$	0	0
$478$	41.0057	1.87556
$479$	−29.8315	−1.36304	−0.681519	−	0.731801i	$-0.738680\pi$
$479$	−29.8315	−1.36304	−0.681519	+	0.731801i	$0.738680\pi$
$480$	0	0
$481$	1.86716	0.0851352
$482$	41.3200	1.88207
$483$	0	0
$484$	28.3461	1.28846
$485$	0.0729358	0.00331184
$486$	0	0
$487$	30.4878	1.38153	0.690767	−	0.723077i	$-0.257273\pi$
$487$	30.4878	1.38153	0.690767	+	0.723077i	$0.257273\pi$
$488$	−0.356500	−0.0161380
$489$	0	0
$490$	−2.99615	−0.135352
$491$	−12.6414	−0.570496	−0.285248	−	0.958454i	$-0.592076\pi$
$491$	−12.6414	−0.570496	−0.285248	+	0.958454i	$0.592076\pi$
$492$	0	0
$493$	−0.918885	−0.0413845
$494$	17.7824	0.800069
$495$	0	0
$496$	22.3241	1.00238
$497$	6.48720	0.290991
$498$	0	0
$499$	−32.9525	−1.47516	−0.737578	−	0.675262i	$-0.764031\pi$
$499$	−32.9525	−1.47516	−0.737578	+	0.675262i	$0.764031\pi$
$500$	−44.2397	−1.97846
$501$	0	0
$502$	−25.3249	−1.13031
$503$	−6.65795	−0.296863	−0.148432	−	0.988923i	$-0.547423\pi$
$503$	−6.65795	−0.296863	−0.148432	+	0.988923i	$0.547423\pi$
$504$	0	0
$505$	11.3618	0.505595
$506$	−6.97019	−0.309863
$507$	0	0
$508$	4.32847	0.192045
$509$	30.1687	1.33721	0.668603	−	0.743619i	$-0.266893\pi$
$509$	30.1687	1.33721	0.668603	+	0.743619i	$0.266893\pi$
$510$	0	0
$511$	9.61580	0.425378
$512$	49.4718	2.18636
$513$	0	0
$514$	32.5722	1.43670
$515$	−3.25818	−0.143573
$516$	0	0
$517$	−3.34122	−0.146947
$518$	3.27527	0.143907
$519$	0	0
$520$	−10.0050	−0.438749
$521$	45.2001	1.98025	0.990126	−	0.140184i	$-0.0447693\pi$
$521$	45.2001	1.98025	0.990126	+	0.140184i	$0.0447693\pi$
$522$	0	0
$523$	−31.0371	−1.35716	−0.678578	−	0.734528i	$-0.737404\pi$
$523$	−31.0371	−1.35716	−0.678578	+	0.734528i	$0.737404\pi$
$524$	−44.9259	−1.96260
$525$	0	0
$526$	−67.5199	−2.94401
$527$	−6.47643	−0.282118
$528$	0	0
$529$	−22.5625	−0.980980
$530$	−29.4368	−1.27865
$531$	0	0
$532$	21.3350	0.924989
$533$	−12.1030	−0.524238
$534$	0	0
$535$	8.32415	0.359885
$536$	46.6268	2.01397
$537$	0	0
$538$	−43.0962	−1.85801
$539$	4.18912	0.180438
$540$	0	0
$541$	28.5580	1.22781	0.613903	−	0.789381i	$-0.289599\pi$
$541$	28.5580	1.22781	0.613903	+	0.789381i	$0.289599\pi$
$542$	−64.3533	−2.76421
$543$	0	0
$544$	6.30739	0.270427
$545$	−2.04521	−0.0876072
$546$	0	0
$547$	44.5596	1.90523	0.952615	−	0.304179i	$-0.0983822\pi$
$547$	44.5596	1.90523	0.952615	+	0.304179i	$0.0983822\pi$
$548$	−58.4149	−2.49536
$549$	0	0
$550$	37.7431	1.60937
$551$	−2.56831	−0.109414
$552$	0	0
$553$	−15.3521	−0.652839
$554$	36.5375	1.55233
$555$	0	0
$556$	−60.4486	−2.56359
$557$	4.97061	0.210612	0.105306	−	0.994440i	$-0.466418\pi$
$557$	4.97061	0.210612	0.105306	+	0.994440i	$0.466418\pi$
$558$	0	0
$559$	6.55022	0.277045
$560$	−7.23978	−0.305937
$561$	0	0
$562$	25.9285	1.09373
$563$	−35.6218	−1.50128	−0.750641	−	0.660710i	$-0.770255\pi$
$563$	−35.6218	−1.50128	−0.750641	+	0.660710i	$0.770255\pi$
$564$	0	0
$565$	23.8192	1.00208
$566$	50.0492	2.10372
$567$	0	0
$568$	37.9994	1.59442
$569$	3.49593	0.146557	0.0732785	−	0.997312i	$-0.476654\pi$
$569$	3.49593	0.146557	0.0732785	+	0.997312i	$0.476654\pi$
$570$	0	0
$571$	−8.82565	−0.369342	−0.184671	−	0.982800i	$-0.559122\pi$
$571$	−8.82565	−0.369342	−0.184671	+	0.982800i	$0.559122\pi$
$572$	26.0040	1.08728
$573$	0	0
$574$	−21.2304	−0.886139
$575$	−2.36885	−0.0987879
$576$	0	0
$577$	−23.9481	−0.996972	−0.498486	−	0.866898i	$-0.666110\pi$
$577$	−23.9481	−0.996972	−0.498486	+	0.866898i	$0.666110\pi$
$578$	34.9426	1.45342
$579$	0	0
$580$	2.68620	0.111538
$581$	5.82288	0.241574
$582$	0	0
$583$	41.1576	1.70457
$584$	56.3255	2.33077
$585$	0	0
$586$	20.8783	0.862475
$587$	23.7662	0.980935	0.490468	−	0.871459i	$-0.336826\pi$
$587$	23.7662	0.980935	0.490468	+	0.871459i	$0.336826\pi$
$588$	0	0
$589$	−18.1018	−0.745872
$590$	38.2245	1.57368
$591$	0	0
$592$	7.91423	0.325273
$593$	6.53535	0.268375	0.134187	−	0.990956i	$-0.457158\pi$
$593$	6.53535	0.268375	0.134187	+	0.990956i	$0.457158\pi$
$594$	0	0
$595$	2.10033	0.0861049
$596$	62.3805	2.55520
$597$	0	0
$598$	−2.38619	−0.0975786
$599$	−9.00587	−0.367970	−0.183985	−	0.982929i	$-0.558900\pi$
$599$	−9.00587	−0.367970	−0.183985	+	0.982929i	$0.558900\pi$
$600$	0	0
$601$	16.6060	0.677375	0.338687	−	0.940899i	$-0.390017\pi$
$601$	16.6060	0.677375	0.338687	+	0.940899i	$0.390017\pi$
$602$	11.4900	0.468299
$603$	0	0
$604$	−39.5511	−1.60931
$605$	7.79961	0.317099
$606$	0	0
$607$	−26.6853	−1.08312	−0.541561	−	0.840661i	$-0.682167\pi$
$607$	−26.6853	−1.08312	−0.541561	+	0.840661i	$0.682167\pi$
$608$	17.6293	0.714963
$609$	0	0
$610$	−0.182349	−0.00738309
$611$	−1.14384	−0.0462749
$612$	0	0
$613$	−7.72787	−0.312126	−0.156063	−	0.987747i	$-0.549880\pi$
$613$	−7.72787	−0.312126	−0.156063	+	0.987747i	$0.549880\pi$
$614$	41.2976	1.66663
$615$	0	0
$616$	24.5382	0.988672
$617$	5.07794	0.204430	0.102215	−	0.994762i	$-0.467407\pi$
$617$	5.07794	0.204430	0.102215	+	0.994762i	$0.467407\pi$
$618$	0	0
$619$	−27.0052	−1.08543	−0.542715	−	0.839917i	$-0.682604\pi$
$619$	−27.0052	−1.08543	−0.542715	+	0.839917i	$0.682604\pi$
$620$	18.9327	0.760355
$621$	0	0
$622$	−22.9659	−0.920848
$623$	12.1345	0.486157
$624$	0	0
$625$	5.73466	0.229386
$626$	8.13418	0.325107
$627$	0	0
$628$	42.8103	1.70832
$629$	−2.29599	−0.0915471
$630$	0	0
$631$	−9.84678	−0.391994	−0.195997	−	0.980604i	$-0.562794\pi$
$631$	−9.84678	−0.391994	−0.195997	+	0.980604i	$0.562794\pi$
$632$	−89.9266	−3.57709
$633$	0	0
$634$	−31.9330	−1.26822
$635$	1.19101	0.0472637
$636$	0	0
$637$	1.43411	0.0568217
$638$	−5.49113	−0.217396
$639$	0	0
$640$	17.9869	0.710995
$641$	−37.7457	−1.49087	−0.745433	−	0.666580i	$-0.767757\pi$
$641$	−37.7457	−1.49087	−0.745433	+	0.666580i	$0.767757\pi$
$642$	0	0
$643$	47.5016	1.87328	0.936640	−	0.350293i	$-0.113918\pi$
$643$	47.5016	1.87328	0.936640	+	0.350293i	$0.113918\pi$
$644$	−2.86290	−0.112814
$645$	0	0
$646$	−21.8665	−0.860326
$647$	−34.5662	−1.35894	−0.679469	−	0.733704i	$-0.737790\pi$
$647$	−34.5662	−1.35894	−0.679469	+	0.733704i	$0.737790\pi$
$648$	0	0
$649$	−53.4442	−2.09787
$650$	12.9211	0.506806
$651$	0	0
$652$	55.1822	2.16110
$653$	4.89201	0.191439	0.0957196	−	0.995408i	$-0.469485\pi$
$653$	4.89201	0.191439	0.0957196	+	0.995408i	$0.469485\pi$
$654$	0	0
$655$	−12.3617	−0.483010
$656$	−51.3002	−2.00294
$657$	0	0
$658$	−2.00646	−0.0782201
$659$	−22.2731	−0.867635	−0.433817	−	0.901001i	$-0.642834\pi$
$659$	−22.2731	−0.867635	−0.433817	+	0.901001i	$0.642834\pi$
$660$	0	0
$661$	24.9111	0.968929	0.484464	−	0.874811i	$-0.339015\pi$
$661$	24.9111	0.968929	0.484464	+	0.874811i	$0.339015\pi$
$662$	−6.81273	−0.264784
$663$	0	0
$664$	34.1081	1.32365
$665$	5.87047	0.227647
$666$	0	0
$667$	0.344637	0.0133444
$668$	−88.5054	−3.42438
$669$	0	0
$670$	23.8495	0.921387
$671$	0.254954	0.00984240
$672$	0	0
$673$	−18.1608	−0.700048	−0.350024	−	0.936741i	$-0.613827\pi$
$673$	−18.1608	−0.700048	−0.350024	+	0.936741i	$0.613827\pi$
$674$	−71.3890	−2.74980
$675$	0	0
$676$	−47.3678	−1.82184
$677$	−13.4238	−0.515920	−0.257960	−	0.966156i	$-0.583050\pi$
$677$	−13.4238	−0.515920	−0.257960	+	0.966156i	$0.583050\pi$
$678$	0	0
$679$	−0.0612387	−0.00235013
$680$	12.3029	0.471793
$681$	0	0
$682$	−38.7022	−1.48199
$683$	−37.8944	−1.44999	−0.724995	−	0.688754i	$-0.758158\pi$
$683$	−37.8944	−1.44999	−0.724995	+	0.688754i	$0.758158\pi$
$684$	0	0
$685$	−16.0733	−0.614128
$686$	2.51564	0.0960478
$687$	0	0
$688$	27.7641	1.05850
$689$	14.0900	0.536786
$690$	0	0
$691$	−16.2233	−0.617162	−0.308581	−	0.951198i	$-0.599854\pi$
$691$	−16.2233	−0.617162	−0.308581	+	0.951198i	$0.599854\pi$
$692$	−94.2348	−3.58227
$693$	0	0
$694$	−3.00972	−0.114247
$695$	−16.6328	−0.630920
$696$	0	0
$697$	14.8827	0.563721
$698$	17.3964	0.658462
$699$	0	0
$700$	15.5024	0.585936
$701$	−38.4189	−1.45106	−0.725530	−	0.688190i	$-0.758406\pi$
$701$	−38.4189	−1.45106	−0.725530	+	0.688190i	$0.758406\pi$
$702$	0	0
$703$	−6.41736	−0.242035
$704$	−13.2368	−0.498881
$705$	0	0
$706$	53.2708	2.00487
$707$	−9.53969	−0.358777
$708$	0	0
$709$	29.6978	1.11532	0.557662	−	0.830068i	$-0.311698\pi$
$709$	29.6978	1.11532	0.557662	+	0.830068i	$0.311698\pi$
$710$	19.4366	0.729444
$711$	0	0
$712$	71.0788	2.66379
$713$	2.42905	0.0909686
$714$	0	0
$715$	7.15519	0.267589
$716$	−14.1707	−0.529583
$717$	0	0
$718$	−56.5673	−2.11107
$719$	45.3102	1.68979	0.844893	−	0.534935i	$-0.179664\pi$
$719$	45.3102	1.68979	0.844893	+	0.534935i	$0.179664\pi$
$720$	0	0
$721$	2.73565	0.101881
$722$	−13.3203	−0.495728
$723$	0	0
$724$	−71.5620	−2.65958
$725$	−1.86618	−0.0693083
$726$	0	0
$727$	−39.3597	−1.45977	−0.729885	−	0.683570i	$-0.760426\pi$
$727$	−39.3597	−1.45977	−0.729885	+	0.683570i	$0.760426\pi$
$728$	8.40046	0.311342
$729$	0	0
$730$	28.8104	1.06632
$731$	−8.05461	−0.297911
$732$	0	0
$733$	−27.2312	−1.00581	−0.502903	−	0.864343i	$-0.667735\pi$
$733$	−27.2312	−1.00581	−0.502903	+	0.864343i	$0.667735\pi$
$734$	−43.2536	−1.59652
$735$	0	0
$736$	−2.36565	−0.0871989
$737$	−33.3456	−1.22830
$738$	0	0
$739$	−22.2361	−0.817967	−0.408984	−	0.912542i	$-0.634117\pi$
$739$	−22.2361	−0.817967	−0.408984	+	0.912542i	$0.634117\pi$
$740$	6.71192	0.246735
$741$	0	0
$742$	24.7159	0.907349
$743$	−28.0249	−1.02814	−0.514068	−	0.857750i	$-0.671862\pi$
$743$	−28.0249	−1.02814	−0.514068	+	0.857750i	$0.671862\pi$
$744$	0	0
$745$	17.1644	0.628855
$746$	−21.5818	−0.790165
$747$	0	0
$748$	−31.9764	−1.16917
$749$	−6.98917	−0.255379
$750$	0	0
$751$	9.19486	0.335525	0.167763	−	0.985827i	$-0.446346\pi$
$751$	9.19486	0.335525	0.167763	+	0.985827i	$0.446346\pi$
$752$	−4.84834	−0.176801
$753$	0	0
$754$	−1.87985	−0.0684599
$755$	−10.8828	−0.396064
$756$	0	0
$757$	8.04507	0.292403	0.146202	−	0.989255i	$-0.453295\pi$
$757$	8.04507	0.292403	0.146202	+	0.989255i	$0.453295\pi$
$758$	11.5889	0.420928
$759$	0	0
$760$	34.3868	1.24734
$761$	−7.85377	−0.284699	−0.142350	−	0.989816i	$-0.545466\pi$
$761$	−7.85377	−0.284699	−0.142350	+	0.989816i	$0.545466\pi$
$762$	0	0
$763$	1.71721	0.0621672
$764$	−103.209	−3.73396
$765$	0	0
$766$	−53.7589	−1.94239
$767$	−18.2962	−0.660638
$768$	0	0
$769$	−20.0308	−0.722331	−0.361165	−	0.932502i	$-0.617621\pi$
$769$	−20.0308	−0.722331	−0.361165	+	0.932502i	$0.617621\pi$
$770$	12.5512	0.452316
$771$	0	0
$772$	9.66441	0.347830
$773$	28.9847	1.04251	0.521253	−	0.853402i	$-0.325465\pi$
$773$	28.9847	1.04251	0.521253	+	0.853402i	$0.325465\pi$
$774$	0	0
$775$	−13.1531	−0.472474
$776$	−0.358712	−0.0128770
$777$	0	0
$778$	−62.0887	−2.22599
$779$	41.5974	1.49038
$780$	0	0
$781$	−27.1757	−0.972422
$782$	2.93423	0.104928
$783$	0	0
$784$	6.07870	0.217097
$785$	11.7795	0.420430
$786$	0	0
$787$	−24.5826	−0.876276	−0.438138	−	0.898908i	$-0.644362\pi$
$787$	−24.5826	−0.876276	−0.438138	+	0.898908i	$0.644362\pi$
$788$	−88.3671	−3.14795
$789$	0	0
$790$	−45.9973	−1.63651
$791$	−19.9992	−0.711089
$792$	0	0
$793$	0.0872816	0.00309946
$794$	−52.9381	−1.87870
$795$	0	0
$796$	41.2002	1.46030
$797$	36.3224	1.28661	0.643303	−	0.765611i	$-0.277563\pi$
$797$	36.3224	1.28661	0.643303	+	0.765611i	$0.277563\pi$
$798$	0	0
$799$	1.40655	0.0497600
$800$	12.8098	0.452895
$801$	0	0
$802$	32.0564	1.13195
$803$	−40.2818	−1.42151
$804$	0	0
$805$	−0.787747	−0.0277644
$806$	−13.2494	−0.466691
$807$	0	0
$808$	−55.8797	−1.96584
$809$	−36.0132	−1.26616	−0.633079	−	0.774087i	$-0.718209\pi$
$809$	−36.0132	−1.26616	−0.633079	+	0.774087i	$0.718209\pi$
$810$	0	0
$811$	−26.6085	−0.934350	−0.467175	−	0.884165i	$-0.654728\pi$
$811$	−26.6085	−0.934350	−0.467175	+	0.884165i	$0.654728\pi$
$812$	−2.25540	−0.0791490
$813$	0	0
$814$	−13.7205	−0.480904
$815$	15.1838	0.531864
$816$	0	0
$817$	−22.5129	−0.787625
$818$	26.2876	0.919126
$819$	0	0
$820$	−43.5068	−1.51932
$821$	26.1926	0.914130	0.457065	−	0.889433i	$-0.348901\pi$
$821$	26.1926	0.914130	0.457065	+	0.889433i	$0.348901\pi$
$822$	0	0
$823$	11.7091	0.408155	0.204077	−	0.978955i	$-0.434581\pi$
$823$	11.7091	0.408155	0.204077	+	0.978955i	$0.434581\pi$
$824$	16.0244	0.558235
$825$	0	0
$826$	−32.0942	−1.11670
$827$	−18.5235	−0.644126	−0.322063	−	0.946718i	$-0.604376\pi$
$827$	−18.5235	−0.644126	−0.322063	+	0.946718i	$0.604376\pi$
$828$	0	0
$829$	−42.3461	−1.47074	−0.735370	−	0.677666i	$-0.762992\pi$
$829$	−42.3461	−1.47074	−0.735370	+	0.677666i	$0.762992\pi$
$830$	17.4462	0.605567
$831$	0	0
$832$	−4.53152	−0.157102
$833$	−1.76349	−0.0611012
$834$	0	0
$835$	−24.3529	−0.842766
$836$	−89.3749	−3.09110
$837$	0	0
$838$	90.2428	3.11738
$839$	44.7000	1.54322	0.771608	−	0.636098i	$-0.219453\pi$
$839$	44.7000	1.54322	0.771608	+	0.636098i	$0.219453\pi$
$840$	0	0
$841$	−28.7285	−0.990638
$842$	−5.84875	−0.201561
$843$	0	0
$844$	−52.4860	−1.80664
$845$	−13.0336	−0.448369
$846$	0	0
$847$	−6.54875	−0.225018
$848$	59.7225	2.05088
$849$	0	0
$850$	−15.8886	−0.544976
$851$	0.861133	0.0295193
$852$	0	0
$853$	7.80530	0.267248	0.133624	−	0.991032i	$-0.457338\pi$
$853$	7.80530	0.267248	0.133624	+	0.991032i	$0.457338\pi$
$854$	0.153105	0.00523914
$855$	0	0
$856$	−40.9398	−1.39929
$857$	1.90271	0.0649954	0.0324977	−	0.999472i	$-0.489654\pi$
$857$	1.90271	0.0649954	0.0324977	+	0.999472i	$0.489654\pi$
$858$	0	0
$859$	−28.4688	−0.971343	−0.485671	−	0.874142i	$-0.661425\pi$
$859$	−28.4688	−0.971343	−0.485671	+	0.874142i	$0.661425\pi$
$860$	23.5462	0.802919
$861$	0	0
$862$	87.6499	2.98537
$863$	13.4857	0.459058	0.229529	−	0.973302i	$-0.426281\pi$
$863$	13.4857	0.459058	0.229529	+	0.973302i	$0.426281\pi$
$864$	0	0
$865$	−25.9293	−0.881624
$866$	6.19067	0.210368
$867$	0	0
$868$	−15.8964	−0.539558
$869$	64.3120	2.18163
$870$	0	0
$871$	−11.4156	−0.386803
$872$	10.0587	0.340632
$873$	0	0
$874$	8.20125	0.277411
$875$	10.2206	0.345520
$876$	0	0
$877$	−44.3003	−1.49592	−0.747958	−	0.663746i	$-0.768965\pi$
$877$	−44.3003	−1.49592	−0.747958	+	0.663746i	$0.768965\pi$
$878$	−12.2435	−0.413198
$879$	0	0
$880$	30.3283	1.02237
$881$	6.63739	0.223619	0.111810	−	0.993730i	$-0.464335\pi$
$881$	6.63739	0.223619	0.111810	+	0.993730i	$0.464335\pi$
$882$	0	0
$883$	−2.05879	−0.0692840	−0.0346420	−	0.999400i	$-0.511029\pi$
$883$	−2.05879	−0.0692840	−0.0346420	+	0.999400i	$0.511029\pi$
$884$	−10.9469	−0.368183
$885$	0	0
$886$	14.9619	0.502655
$887$	30.8935	1.03730	0.518650	−	0.854986i	$-0.326435\pi$
$887$	30.8935	1.03730	0.518650	+	0.854986i	$0.326435\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	36.3567	1.21868
$891$	0	0
$892$	26.0292	0.871522
$893$	3.93134	0.131557
$894$	0	0
$895$	−3.89916	−0.130335
$896$	−15.1023	−0.504531
$897$	0	0
$898$	−92.3340	−3.08122
$899$	1.91361	0.0638224
$900$	0	0
$901$	−17.3260	−0.577214
$902$	88.9367	2.96127
$903$	0	0
$904$	−117.147	−3.89625
$905$	−19.6908	−0.654544
$906$	0	0
$907$	−41.0485	−1.36299	−0.681496	−	0.731822i	$-0.738670\pi$
$907$	−41.0485	−1.36299	−0.681496	+	0.731822i	$0.738670\pi$
$908$	87.3652	2.89931
$909$	0	0
$910$	4.29682	0.142438
$911$	−26.5615	−0.880022	−0.440011	−	0.897992i	$-0.645025\pi$
$911$	−26.5615	−0.880022	−0.440011	+	0.897992i	$0.645025\pi$
$912$	0	0
$913$	−24.3928	−0.807282
$914$	76.7348	2.53816
$915$	0	0
$916$	46.7710	1.54536
$917$	10.3792	0.342750
$918$	0	0
$919$	17.7417	0.585244	0.292622	−	0.956228i	$-0.405472\pi$
$919$	17.7417	0.585244	0.292622	+	0.956228i	$0.405472\pi$
$920$	−4.61431	−0.152129
$921$	0	0
$922$	65.8533	2.16876
$923$	−9.30338	−0.306225
$924$	0	0
$925$	−4.66297	−0.153318
$926$	−75.0470	−2.46620
$927$	0	0
$928$	−1.86366	−0.0611776
$929$	−7.81515	−0.256407	−0.128203	−	0.991748i	$-0.540921\pi$
$929$	−7.81515	−0.256407	−0.128203	+	0.991748i	$0.540921\pi$
$930$	0	0
$931$	−4.92899	−0.161541
$932$	76.7167	2.51294
$933$	0	0
$934$	−66.4500	−2.17431
$935$	−8.79852	−0.287742
$936$	0	0
$937$	8.84523	0.288961	0.144481	−	0.989508i	$-0.453849\pi$
$937$	8.84523	0.288961	0.144481	+	0.989508i	$0.453849\pi$
$938$	−20.0246	−0.653828
$939$	0	0
$940$	−4.11179	−0.134112
$941$	−49.2706	−1.60618	−0.803089	−	0.595860i	$-0.796811\pi$
$941$	−49.2706	−1.60618	−0.803089	+	0.595860i	$0.796811\pi$
$942$	0	0
$943$	−5.58188	−0.181771
$944$	−77.5512	−2.52408
$945$	0	0
$946$	−48.1332	−1.56495
$947$	−13.1496	−0.427304	−0.213652	−	0.976910i	$-0.568536\pi$
$947$	−13.1496	−0.427304	−0.213652	+	0.976910i	$0.568536\pi$
$948$	0	0
$949$	−13.7901	−0.447647
$950$	−44.4092	−1.44082
$951$	0	0
$952$	−10.3298	−0.334791
$953$	28.3848	0.919474	0.459737	−	0.888055i	$-0.347944\pi$
$953$	28.3848	0.919474	0.459737	+	0.888055i	$0.347944\pi$
$954$	0	0
$955$	−28.3986	−0.918956
$956$	−70.5552	−2.28192
$957$	0	0
$958$	75.0456	2.42461
$959$	13.4955	0.435793
$960$	0	0
$961$	−17.5126	−0.564924
$962$	−4.69711	−0.151441
$963$	0	0
$964$	−71.0959	−2.28985
$965$	2.65923	0.0856035
$966$	0	0
$967$	0.655032	0.0210644	0.0105322	−	0.999945i	$-0.496647\pi$
$967$	0.655032	0.0210644	0.0105322	+	0.999945i	$0.496647\pi$
$968$	−38.3600	−1.23294
$969$	0	0
$970$	−0.183481	−0.00589120
$971$	−13.2088	−0.423892	−0.211946	−	0.977281i	$-0.567980\pi$
$971$	−13.2088	−0.423892	−0.211946	+	0.977281i	$0.567980\pi$
$972$	0	0
$973$	13.9654	0.447709
$974$	−76.6965	−2.45751
$975$	0	0
$976$	0.369956	0.0118420
$977$	41.1030	1.31500	0.657501	−	0.753454i	$-0.271614\pi$
$977$	41.1030	1.31500	0.657501	+	0.753454i	$0.271614\pi$
$978$	0	0
$979$	−50.8327	−1.62462
$980$	5.15524	0.164678
$981$	0	0
$982$	31.8012	1.01482
$983$	−31.5908	−1.00759	−0.503795	−	0.863823i	$-0.668063\pi$
$983$	−31.5908	−1.00759	−0.503795	+	0.863823i	$0.668063\pi$
$984$	0	0
$985$	−24.3148	−0.774735
$986$	2.31159	0.0736160
$987$	0	0
$988$	−30.5968	−0.973414
$989$	3.02096	0.0960609
$990$	0	0
$991$	−18.8619	−0.599168	−0.299584	−	0.954070i	$-0.596848\pi$
$991$	−18.8619	−0.599168	−0.299584	+	0.954070i	$0.596848\pi$
$992$	−13.1353	−0.417047
$993$	0	0
$994$	−16.3195	−0.517623
$995$	11.3365	0.359392
$996$	0	0
$997$	10.7025	0.338952	0.169476	−	0.985534i	$-0.445793\pi$
$997$	10.7025	0.338952	0.169476	+	0.985534i	$0.445793\pi$
$998$	82.8967	2.62405
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.q.1.2		15
3.2	odd	2		889.2.a.b.1.14	✓	15
21.20	even	2		6223.2.a.j.1.14		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.b.1.14	✓	15	3.2	odd	2
6223.2.a.j.1.14		15	21.20	even	2
8001.2.a.q.1.2		15	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)

\(f(q)\)	\(=\)	\(q-2.51564 q^{2} +4.32847 q^{4} +1.19101 q^{5} -1.00000 q^{7} -5.85760 q^{8} +O(q^{10})\)	\(q-2.51564 q^{2} +4.32847 q^{4} +1.19101 q^{5} -1.00000 q^{7} -5.85760 q^{8} -2.99615 q^{10} +4.18912 q^{11} +1.43411 q^{13} +2.51564 q^{14} +6.07870 q^{16} -1.76349 q^{17} -4.92899 q^{19} +5.15524 q^{20} -10.5383 q^{22} +0.661413 q^{23} -3.58150 q^{25} -3.60772 q^{26} -4.32847 q^{28} +0.521062 q^{29} +3.67252 q^{31} -3.57666 q^{32} +4.43630 q^{34} -1.19101 q^{35} +1.30196 q^{37} +12.3996 q^{38} -6.97644 q^{40} -8.43934 q^{41} +4.56744 q^{43} +18.1325 q^{44} -1.66388 q^{46} -0.797594 q^{47} +1.00000 q^{49} +9.00978 q^{50} +6.20752 q^{52} +9.82487 q^{53} +4.98928 q^{55} +5.85760 q^{56} -1.31081 q^{58} -12.7579 q^{59} +0.0608610 q^{61} -9.23874 q^{62} -3.15981 q^{64} +1.70804 q^{65} -7.96005 q^{67} -7.63319 q^{68} +2.99615 q^{70} -6.48720 q^{71} -9.61580 q^{73} -3.27527 q^{74} -21.3350 q^{76} -4.18912 q^{77} +15.3521 q^{79} +7.23978 q^{80} +21.2304 q^{82} -5.82288 q^{83} -2.10033 q^{85} -11.4900 q^{86} -24.5382 q^{88} -12.1345 q^{89} -1.43411 q^{91} +2.86290 q^{92} +2.00646 q^{94} -5.87047 q^{95} +0.0612387 q^{97} -2.51564 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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