LMFDB - Embedded newform 8001.2.a.m.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:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 15 x^{9} + 25 x^{8} + 88 x^{7} - 112 x^{6} - 247 x^{5} + 215 x^{4} + 313 x^{3} + \cdots + 57 $ x^11 - 2x^10 - 15x^9 + 25x^8 + 88x^7 - 112x^6 - 247x^5 + 215x^4 + 313x^3 - 170x^2 - 121x + 57
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.81808$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.81808 q^{2} +1.30541 q^{4} +1.39765 q^{5} -1.00000 q^{7} +1.26282 q^{8} +O(q^{10})$	\(q-1.81808 q^{2} +1.30541 q^{4} +1.39765 q^{5} -1.00000 q^{7} +1.26282 q^{8} -2.54104 q^{10} -0.0844564 q^{11} -3.52450 q^{13} +1.81808 q^{14} -4.90673 q^{16} -4.93727 q^{17} -3.17772 q^{19} +1.82451 q^{20} +0.153548 q^{22} +3.91903 q^{23} -3.04657 q^{25} +6.40782 q^{26} -1.30541 q^{28} +7.80555 q^{29} -6.27152 q^{31} +6.39517 q^{32} +8.97635 q^{34} -1.39765 q^{35} +3.42797 q^{37} +5.77734 q^{38} +1.76499 q^{40} +9.78445 q^{41} +10.6350 q^{43} -0.110250 q^{44} -7.12509 q^{46} -4.50967 q^{47} +1.00000 q^{49} +5.53890 q^{50} -4.60091 q^{52} +0.234825 q^{53} -0.118041 q^{55} -1.26282 q^{56} -14.1911 q^{58} +2.23465 q^{59} +8.26515 q^{61} +11.4021 q^{62} -1.81346 q^{64} -4.92602 q^{65} -3.74734 q^{67} -6.44515 q^{68} +2.54104 q^{70} +5.09148 q^{71} +15.8865 q^{73} -6.23232 q^{74} -4.14822 q^{76} +0.0844564 q^{77} -10.3600 q^{79} -6.85789 q^{80} -17.7889 q^{82} +5.52767 q^{83} -6.90059 q^{85} -19.3353 q^{86} -0.106653 q^{88} +15.0241 q^{89} +3.52450 q^{91} +5.11593 q^{92} +8.19894 q^{94} -4.44134 q^{95} -9.13128 q^{97} -1.81808 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8}+O(q^{10}) $ 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8	$ 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8} - 12 q^{10} + 7 q^{11} - 24 q^{13} - 2 q^{14} - 6 q^{16} + 15 q^{17} - 19 q^{19} - 3 q^{20} - 3 q^{22} + 11 q^{23} + 10 q^{25} - 10 q^{26} - 12 q^{28} + 10 q^{29} - 20 q^{31} + 27 q^{32} - 9 q^{34} + q^{35} - 22 q^{37} - 8 q^{38} - 29 q^{40} - 9 q^{41} - 17 q^{43} + 9 q^{44} - 18 q^{46} + 7 q^{47} + 11 q^{49} + 47 q^{50} - 66 q^{52} + 28 q^{53} - 24 q^{55} - 15 q^{56} - 39 q^{58} - 35 q^{59} - 6 q^{61} - 18 q^{62} + 11 q^{64} + 43 q^{65} - 22 q^{67} + 12 q^{68} + 12 q^{70} + 22 q^{71} - 29 q^{73} - 14 q^{74} + 10 q^{76} - 7 q^{77} - 20 q^{79} - 66 q^{80} - 24 q^{82} - 17 q^{83} - 50 q^{85} + 12 q^{86} + 2 q^{88} + q^{89} + 24 q^{91} + 22 q^{92} + q^{94} - 10 q^{95} - 45 q^{97} + 2 q^{98}+O(q^{100}) $ 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8 - 12 * q^10 + 7 * q^11 - 24 * q^13 - 2 * q^14 - 6 * q^16 + 15 * q^17 - 19 * q^19 - 3 * q^20 - 3 * q^22 + 11 * q^23 + 10 * q^25 - 10 * q^26 - 12 * q^28 + 10 * q^29 - 20 * q^31 + 27 * q^32 - 9 * q^34 + q^35 - 22 * q^37 - 8 * q^38 - 29 * q^40 - 9 * q^41 - 17 * q^43 + 9 * q^44 - 18 * q^46 + 7 * q^47 + 11 * q^49 + 47 * q^50 - 66 * q^52 + 28 * q^53 - 24 * q^55 - 15 * q^56 - 39 * q^58 - 35 * q^59 - 6 * q^61 - 18 * q^62 + 11 * q^64 + 43 * q^65 - 22 * q^67 + 12 * q^68 + 12 * q^70 + 22 * q^71 - 29 * q^73 - 14 * q^74 + 10 * q^76 - 7 * q^77 - 20 * q^79 - 66 * q^80 - 24 * q^82 - 17 * q^83 - 50 * q^85 + 12 * q^86 + 2 * q^88 + q^89 + 24 * q^91 + 22 * q^92 + q^94 - 10 * q^95 - 45 * q^97 + 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 $

$2$	−1.81808	−1.28558	−0.642788	−	0.766044i	$-0.722222\pi$
$2$	−1.81808	−1.28558	−0.642788	+	0.766044i	$0.722222\pi$
$3$	0	0
$3$	0	0
$4$	1.30541	0.652704
$5$	1.39765	0.625049	0.312524	−	0.949910i	$-0.398825\pi$
$5$	1.39765	0.625049	0.312524	+	0.949910i	$0.398825\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.26282	0.446475
$9$	0	0
$10$	−2.54104	−0.803547
$11$	−0.0844564	−0.0254646	−0.0127323	−	0.999919i	$-0.504053\pi$
$11$	−0.0844564	−0.0254646	−0.0127323	+	0.999919i	$0.504053\pi$
$12$	0	0
$13$	−3.52450	−0.977520	−0.488760	−	0.872418i	$-0.662551\pi$
$13$	−3.52450	−0.977520	−0.488760	+	0.872418i	$0.662551\pi$
$14$	1.81808	0.485902
$15$	0	0
$16$	−4.90673	−1.22668
$17$	−4.93727	−1.19746	−0.598732	−	0.800949i	$-0.704329\pi$
$17$	−4.93727	−1.19746	−0.598732	+	0.800949i	$0.704329\pi$
$18$	0	0
$19$	−3.17772	−0.729018	−0.364509	−	0.931200i	$-0.618763\pi$
$19$	−3.17772	−0.729018	−0.364509	+	0.931200i	$0.618763\pi$
$20$	1.82451	0.407972
$21$	0	0
$22$	0.153548	0.0327366
$23$	3.91903	0.817173	0.408587	−	0.912720i	$-0.366022\pi$
$23$	3.91903	0.817173	0.408587	+	0.912720i	$0.366022\pi$
$24$	0	0
$25$	−3.04657	−0.609314
$26$	6.40782	1.25668
$27$	0	0
$28$	−1.30541	−0.246699
$29$	7.80555	1.44945	0.724727	−	0.689036i	$-0.241966\pi$
$29$	7.80555	1.44945	0.724727	+	0.689036i	$0.241966\pi$
$30$	0	0
$31$	−6.27152	−1.12640	−0.563199	−	0.826321i	$-0.690430\pi$
$31$	−6.27152	−1.12640	−0.563199	+	0.826321i	$0.690430\pi$
$32$	6.39517	1.13052
$33$	0	0
$34$	8.97635	1.53943
$35$	−1.39765	−0.236246
$36$	0	0
$37$	3.42797	0.563555	0.281778	−	0.959480i	$-0.409076\pi$
$37$	3.42797	0.563555	0.281778	+	0.959480i	$0.409076\pi$
$38$	5.77734	0.937208
$39$	0	0
$40$	1.76499	0.279069
$41$	9.78445	1.52807	0.764037	−	0.645172i	$-0.223214\pi$
$41$	9.78445	1.52807	0.764037	+	0.645172i	$0.223214\pi$
$42$	0	0
$43$	10.6350	1.62183	0.810913	−	0.585167i	$-0.198971\pi$
$43$	10.6350	1.62183	0.810913	+	0.585167i	$0.198971\pi$
$44$	−0.110250	−0.0166208
$45$	0	0
$46$	−7.12509	−1.05054
$47$	−4.50967	−0.657803	−0.328902	−	0.944364i	$-0.606678\pi$
$47$	−4.50967	−0.657803	−0.328902	+	0.944364i	$0.606678\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	5.53890	0.783319
$51$	0	0
$52$	−4.60091	−0.638031
$53$	0.234825	0.0322557	0.0161279	−	0.999870i	$-0.494866\pi$
$53$	0.234825	0.0322557	0.0161279	+	0.999870i	$0.494866\pi$
$54$	0	0
$55$	−0.118041	−0.0159166
$56$	−1.26282	−0.168752
$57$	0	0
$58$	−14.1911	−1.86338
$59$	2.23465	0.290927	0.145463	−	0.989364i	$-0.453533\pi$
$59$	2.23465	0.290927	0.145463	+	0.989364i	$0.453533\pi$
$60$	0	0
$61$	8.26515	1.05824	0.529122	−	0.848546i	$-0.322521\pi$
$61$	8.26515	1.05824	0.529122	+	0.848546i	$0.322521\pi$
$62$	11.4021	1.44807
$63$	0	0
$64$	−1.81346	−0.226682
$65$	−4.92602	−0.610998
$66$	0	0
$67$	−3.74734	−0.457810	−0.228905	−	0.973449i	$-0.573515\pi$
$67$	−3.74734	−0.457810	−0.228905	+	0.973449i	$0.573515\pi$
$68$	−6.44515	−0.781590
$69$	0	0
$70$	2.54104	0.303712
$71$	5.09148	0.604248	0.302124	−	0.953269i	$-0.402304\pi$
$71$	5.09148	0.604248	0.302124	+	0.953269i	$0.402304\pi$
$72$	0	0
$73$	15.8865	1.85937	0.929686	−	0.368353i	$-0.120078\pi$
$73$	15.8865	1.85937	0.929686	+	0.368353i	$0.120078\pi$
$74$	−6.23232	−0.724493
$75$	0	0
$76$	−4.14822	−0.475833
$77$	0.0844564	0.00962470
$78$	0	0
$79$	−10.3600	−1.16560	−0.582798	−	0.812617i	$-0.698042\pi$
$79$	−10.3600	−1.16560	−0.582798	+	0.812617i	$0.698042\pi$
$80$	−6.85789	−0.766736
$81$	0	0
$82$	−17.7889	−1.96446
$83$	5.52767	0.606741	0.303370	−	0.952873i	$-0.401888\pi$
$83$	5.52767	0.606741	0.303370	+	0.952873i	$0.401888\pi$
$84$	0	0
$85$	−6.90059	−0.748474
$86$	−19.3353	−2.08498
$87$	0	0
$88$	−0.106653	−0.0113693
$89$	15.0241	1.59255	0.796276	−	0.604934i	$-0.206801\pi$
$89$	15.0241	1.59255	0.796276	+	0.604934i	$0.206801\pi$
$90$	0	0
$91$	3.52450	0.369468
$92$	5.11593	0.533372
$93$	0	0
$94$	8.19894	0.845656
$95$	−4.44134	−0.455672
$96$	0	0
$97$	−9.13128	−0.927141	−0.463571	−	0.886060i	$-0.653432\pi$
$97$	−9.13128	−0.927141	−0.463571	+	0.886060i	$0.653432\pi$
$98$	−1.81808	−0.183654
$99$	0	0
$100$	−3.97702	−0.397702
$101$	−12.1017	−1.20417	−0.602084	−	0.798433i	$-0.705663\pi$
$101$	−12.1017	−1.20417	−0.602084	+	0.798433i	$0.705663\pi$
$102$	0	0
$103$	−5.80333	−0.571819	−0.285909	−	0.958257i	$-0.592296\pi$
$103$	−5.80333	−0.571819	−0.285909	+	0.958257i	$0.592296\pi$
$104$	−4.45082	−0.436439
$105$	0	0
$106$	−0.426931	−0.0414672
$107$	0.365904	0.0353732	0.0176866	−	0.999844i	$-0.494370\pi$
$107$	0.365904	0.0353732	0.0176866	+	0.999844i	$0.494370\pi$
$108$	0	0
$109$	−14.5385	−1.39253	−0.696266	−	0.717784i	$-0.745156\pi$
$109$	−14.5385	−1.39253	−0.696266	+	0.717784i	$0.745156\pi$
$110$	0.214607	0.0204620
$111$	0	0
$112$	4.90673	0.463642
$113$	7.14306	0.671963	0.335981	−	0.941869i	$-0.390932\pi$
$113$	7.14306	0.671963	0.335981	+	0.941869i	$0.390932\pi$
$114$	0	0
$115$	5.47743	0.510773
$116$	10.1894	0.946064
$117$	0	0
$118$	−4.06277	−0.374008
$119$	4.93727	0.452599
$120$	0	0
$121$	−10.9929	−0.999352
$122$	−15.0267	−1.36045
$123$	0	0
$124$	−8.18689	−0.735205
$125$	−11.2463	−1.00590
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−9.49332	−0.839099
$129$	0	0
$130$	8.95590	0.785484
$131$	−21.8579	−1.90973	−0.954866	−	0.297037i	$-0.904002\pi$
$131$	−21.8579	−1.90973	−0.954866	+	0.297037i	$0.904002\pi$
$132$	0	0
$133$	3.17772	0.275543
$134$	6.81295	0.588550
$135$	0	0
$136$	−6.23490	−0.534638
$137$	−10.8301	−0.925281	−0.462641	−	0.886546i	$-0.653098\pi$
$137$	−10.8301	−0.925281	−0.462641	+	0.886546i	$0.653098\pi$
$138$	0	0
$139$	−14.1968	−1.20416	−0.602078	−	0.798437i	$-0.705660\pi$
$139$	−14.1968	−1.20416	−0.602078	+	0.798437i	$0.705660\pi$
$140$	−1.82451	−0.154199
$141$	0	0
$142$	−9.25671	−0.776806
$143$	0.297667	0.0248921
$144$	0	0
$145$	10.9094	0.905979
$146$	−28.8829	−2.39036
$147$	0	0
$148$	4.47490	0.367835
$149$	19.7609	1.61887	0.809437	−	0.587207i	$-0.199773\pi$
$149$	19.7609	1.61887	0.809437	+	0.587207i	$0.199773\pi$
$150$	0	0
$151$	7.89570	0.642543	0.321272	−	0.946987i	$-0.395890\pi$
$151$	7.89570	0.642543	0.321272	+	0.946987i	$0.395890\pi$
$152$	−4.01289	−0.325489
$153$	0	0
$154$	−0.153548	−0.0123733
$155$	−8.76540	−0.704054
$156$	0	0
$157$	18.3445	1.46405	0.732024	−	0.681279i	$-0.238576\pi$
$157$	18.3445	1.46405	0.732024	+	0.681279i	$0.238576\pi$
$158$	18.8354	1.49846
$159$	0	0
$160$	8.93821	0.706628
$161$	−3.91903	−0.308862
$162$	0	0
$163$	20.4515	1.60189	0.800944	−	0.598739i	$-0.204331\pi$
$163$	20.4515	1.60189	0.800944	+	0.598739i	$0.204331\pi$
$164$	12.7727	0.997380
$165$	0	0
$166$	−10.0497	−0.780011
$167$	−22.4305	−1.73572	−0.867861	−	0.496807i	$-0.834506\pi$
$167$	−22.4305	−1.73572	−0.867861	+	0.496807i	$0.834506\pi$
$168$	0	0
$169$	−0.577899	−0.0444538
$170$	12.5458	0.962219
$171$	0	0
$172$	13.8830	1.05857
$173$	−23.4089	−1.77975	−0.889873	−	0.456208i	$-0.849207\pi$
$173$	−23.4089	−1.77975	−0.889873	+	0.456208i	$0.849207\pi$
$174$	0	0
$175$	3.04657	0.230299
$176$	0.414405	0.0312369
$177$	0	0
$178$	−27.3150	−2.04734
$179$	12.0263	0.898887	0.449443	−	0.893309i	$-0.351622\pi$
$179$	12.0263	0.898887	0.449443	+	0.893309i	$0.351622\pi$
$180$	0	0
$181$	−14.7462	−1.09607	−0.548037	−	0.836454i	$-0.684625\pi$
$181$	−14.7462	−1.09607	−0.548037	+	0.836454i	$0.684625\pi$
$182$	−6.40782	−0.474979
$183$	0	0
$184$	4.94904	0.364848
$185$	4.79111	0.352250
$186$	0	0
$187$	0.416984	0.0304929
$188$	−5.88696	−0.429351
$189$	0	0
$190$	8.07471	0.585801
$191$	15.1077	1.09315	0.546577	−	0.837409i	$-0.315931\pi$
$191$	15.1077	1.09315	0.546577	+	0.837409i	$0.315931\pi$
$192$	0	0
$193$	−4.11572	−0.296256	−0.148128	−	0.988968i	$-0.547325\pi$
$193$	−4.11572	−0.296256	−0.148128	+	0.988968i	$0.547325\pi$
$194$	16.6014	1.19191
$195$	0	0
$196$	1.30541	0.0932434
$197$	8.76996	0.624834	0.312417	−	0.949945i	$-0.398861\pi$
$197$	8.76996	0.624834	0.312417	+	0.949945i	$0.398861\pi$
$198$	0	0
$199$	−19.9853	−1.41672	−0.708360	−	0.705851i	$-0.750565\pi$
$199$	−19.9853	−1.41672	−0.708360	+	0.705851i	$0.750565\pi$
$200$	−3.84728	−0.272044
$201$	0	0
$202$	22.0019	1.54805
$203$	−7.80555	−0.547842
$204$	0	0
$205$	13.6753	0.955122
$206$	10.5509	0.735116
$207$	0	0
$208$	17.2938	1.19911
$209$	0.268379	0.0185641
$210$	0	0
$211$	6.14263	0.422876	0.211438	−	0.977391i	$-0.432185\pi$
$211$	6.14263	0.422876	0.211438	+	0.977391i	$0.432185\pi$
$212$	0.306543	0.0210534
$213$	0	0
$214$	−0.665241	−0.0454750
$215$	14.8641	1.01372
$216$	0	0
$217$	6.27152	0.425739
$218$	26.4320	1.79020
$219$	0	0
$220$	−0.154091	−0.0103888
$221$	17.4014	1.17055
$222$	0	0
$223$	−6.16516	−0.412850	−0.206425	−	0.978462i	$-0.566183\pi$
$223$	−6.16516	−0.412850	−0.206425	+	0.978462i	$0.566183\pi$
$224$	−6.39517	−0.427295
$225$	0	0
$226$	−12.9866	−0.863858
$227$	−24.5032	−1.62633	−0.813167	−	0.582030i	$-0.802258\pi$
$227$	−24.5032	−1.62633	−0.813167	+	0.582030i	$0.802258\pi$
$228$	0	0
$229$	−1.24111	−0.0820148	−0.0410074	−	0.999159i	$-0.513057\pi$
$229$	−1.24111	−0.0820148	−0.0410074	+	0.999159i	$0.513057\pi$
$230$	−9.95840	−0.656638
$231$	0	0
$232$	9.85702	0.647145
$233$	−24.8114	−1.62545	−0.812723	−	0.582650i	$-0.802016\pi$
$233$	−24.8114	−1.62545	−0.812723	+	0.582650i	$0.802016\pi$
$234$	0	0
$235$	−6.30295	−0.411159
$236$	2.91713	0.189889
$237$	0	0
$238$	−8.97635	−0.581850
$239$	−19.9316	−1.28927	−0.644633	−	0.764492i	$-0.722990\pi$
$239$	−19.9316	−1.28927	−0.644633	+	0.764492i	$0.722990\pi$
$240$	0	0
$241$	1.78751	0.115144	0.0575718	−	0.998341i	$-0.481664\pi$
$241$	1.78751	0.115144	0.0575718	+	0.998341i	$0.481664\pi$
$242$	19.9859	1.28474
$243$	0	0
$244$	10.7894	0.690720
$245$	1.39765	0.0892927
$246$	0	0
$247$	11.1999	0.712630
$248$	−7.91982	−0.502909
$249$	0	0
$250$	20.4467	1.29316
$251$	−19.0011	−1.19934	−0.599671	−	0.800247i	$-0.704702\pi$
$251$	−19.0011	−1.19934	−0.599671	+	0.800247i	$0.704702\pi$
$252$	0	0
$253$	−0.330987	−0.0208090
$254$	−1.81808	−0.114076
$255$	0	0
$256$	20.8865	1.30541
$257$	5.27896	0.329293	0.164646	−	0.986353i	$-0.447352\pi$
$257$	5.27896	0.329293	0.164646	+	0.986353i	$0.447352\pi$
$258$	0	0
$259$	−3.42797	−0.213004
$260$	−6.43047	−0.398801
$261$	0	0
$262$	39.7393	2.45510
$263$	22.2560	1.37236	0.686181	−	0.727431i	$-0.259286\pi$
$263$	22.2560	1.37236	0.686181	+	0.727431i	$0.259286\pi$
$264$	0	0
$265$	0.328204	0.0201614
$266$	−5.77734	−0.354231
$267$	0	0
$268$	−4.89181	−0.298815
$269$	−8.70678	−0.530862	−0.265431	−	0.964130i	$-0.585514\pi$
$269$	−8.70678	−0.530862	−0.265431	+	0.964130i	$0.585514\pi$
$270$	0	0
$271$	10.9473	0.665002	0.332501	−	0.943103i	$-0.392107\pi$
$271$	10.9473	0.665002	0.332501	+	0.943103i	$0.392107\pi$
$272$	24.2258	1.46891
$273$	0	0
$274$	19.6900	1.18952
$275$	0.257302	0.0155159
$276$	0	0
$277$	−21.4950	−1.29151	−0.645756	−	0.763544i	$-0.723458\pi$
$277$	−21.4950	−1.29151	−0.645756	+	0.763544i	$0.723458\pi$
$278$	25.8109	1.54803
$279$	0	0
$280$	−1.76499	−0.105478
$281$	−16.9237	−1.00958	−0.504790	−	0.863242i	$-0.668430\pi$
$281$	−16.9237	−1.00958	−0.504790	+	0.863242i	$0.668430\pi$
$282$	0	0
$283$	−11.0667	−0.657850	−0.328925	−	0.944356i	$-0.606686\pi$
$283$	−11.0667	−0.657850	−0.328925	+	0.944356i	$0.606686\pi$
$284$	6.64646	0.394395
$285$	0	0
$286$	−0.541181	−0.0320007
$287$	−9.78445	−0.577558
$288$	0	0
$289$	7.37666	0.433921
$290$	−19.8342	−1.16470
$291$	0	0
$292$	20.7383	1.21362
$293$	24.7793	1.44762	0.723811	−	0.689999i	$-0.242389\pi$
$293$	24.7793	1.44762	0.723811	+	0.689999i	$0.242389\pi$
$294$	0	0
$295$	3.12326	0.181843
$296$	4.32892	0.251614
$297$	0	0
$298$	−35.9268	−2.08118
$299$	−13.8126	−0.798804
$300$	0	0
$301$	−10.6350	−0.612992
$302$	−14.3550	−0.826038
$303$	0	0
$304$	15.5922	0.894273
$305$	11.5518	0.661454
$306$	0	0
$307$	22.2886	1.27208	0.636038	−	0.771658i	$-0.280572\pi$
$307$	22.2886	1.27208	0.636038	+	0.771658i	$0.280572\pi$
$308$	0.110250	0.00628208
$309$	0	0
$310$	15.9362	0.905115
$311$	−9.48282	−0.537721	−0.268861	−	0.963179i	$-0.586647\pi$
$311$	−9.48282	−0.537721	−0.268861	+	0.963179i	$0.586647\pi$
$312$	0	0
$313$	5.82224	0.329092	0.164546	−	0.986369i	$-0.447384\pi$
$313$	5.82224	0.329092	0.164546	+	0.986369i	$0.447384\pi$
$314$	−33.3517	−1.88214
$315$	0	0
$316$	−13.5241	−0.760789
$317$	−28.5643	−1.60433	−0.802166	−	0.597100i	$-0.796319\pi$
$317$	−28.5643	−1.60433	−0.802166	+	0.597100i	$0.796319\pi$
$318$	0	0
$319$	−0.659228	−0.0369097
$320$	−2.53458	−0.141687
$321$	0	0
$322$	7.12509	0.397066
$323$	15.6893	0.872973
$324$	0	0
$325$	10.7376	0.595617
$326$	−37.1825	−2.05935
$327$	0	0
$328$	12.3560	0.682248
$329$	4.50967	0.248626
$330$	0	0
$331$	−27.6246	−1.51839	−0.759193	−	0.650866i	$-0.774406\pi$
$331$	−27.6246	−1.51839	−0.759193	+	0.650866i	$0.774406\pi$
$332$	7.21586	0.396022
$333$	0	0
$334$	40.7803	2.23140
$335$	−5.23748	−0.286154
$336$	0	0
$337$	12.0879	0.658467	0.329234	−	0.944248i	$-0.393210\pi$
$337$	12.0879	0.658467	0.329234	+	0.944248i	$0.393210\pi$
$338$	1.05067	0.0571487
$339$	0	0
$340$	−9.00808	−0.488532
$341$	0.529670	0.0286833
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	13.4301	0.724105
$345$	0	0
$346$	42.5592	2.28800
$347$	11.5304	0.618985	0.309492	−	0.950902i	$-0.399841\pi$
$347$	11.5304	0.618985	0.309492	+	0.950902i	$0.399841\pi$
$348$	0	0
$349$	−32.9671	−1.76469	−0.882345	−	0.470603i	$-0.844036\pi$
$349$	−32.9671	−1.76469	−0.882345	+	0.470603i	$0.844036\pi$
$350$	−5.53890	−0.296067
$351$	0	0
$352$	−0.540113	−0.0287881
$353$	−12.8554	−0.684222	−0.342111	−	0.939660i	$-0.611142\pi$
$353$	−12.8554	−0.684222	−0.342111	+	0.939660i	$0.611142\pi$
$354$	0	0
$355$	7.11612	0.377684
$356$	19.6126	1.03946
$357$	0	0
$358$	−21.8647	−1.15559
$359$	−21.9370	−1.15779	−0.578896	−	0.815402i	$-0.696516\pi$
$359$	−21.9370	−1.15779	−0.578896	+	0.815402i	$0.696516\pi$
$360$	0	0
$361$	−8.90212	−0.468532
$362$	26.8097	1.40909
$363$	0	0
$364$	4.60091	0.241153
$365$	22.2038	1.16220
$366$	0	0
$367$	−16.7528	−0.874487	−0.437244	−	0.899343i	$-0.644045\pi$
$367$	−16.7528	−0.874487	−0.437244	+	0.899343i	$0.644045\pi$
$368$	−19.2296	−1.00241
$369$	0	0
$370$	−8.71062	−0.452843
$371$	−0.234825	−0.0121915
$372$	0	0
$373$	−14.6000	−0.755961	−0.377980	−	0.925814i	$-0.623381\pi$
$373$	−14.6000	−0.755961	−0.377980	+	0.925814i	$0.623381\pi$
$374$	−0.758110	−0.0392009
$375$	0	0
$376$	−5.69492	−0.293693
$377$	−27.5106	−1.41687
$378$	0	0
$379$	1.91231	0.0982286	0.0491143	−	0.998793i	$-0.484360\pi$
$379$	1.91231	0.0982286	0.0491143	+	0.998793i	$0.484360\pi$
$380$	−5.79776	−0.297419
$381$	0	0
$382$	−27.4669	−1.40533
$383$	14.6000	0.746025	0.373012	−	0.927826i	$-0.378325\pi$
$383$	14.6000	0.746025	0.373012	+	0.927826i	$0.378325\pi$
$384$	0	0
$385$	0.118041	0.00601591
$386$	7.48270	0.380859
$387$	0	0
$388$	−11.9200	−0.605149
$389$	1.62130	0.0822033	0.0411017	−	0.999155i	$-0.486913\pi$
$389$	1.62130	0.0822033	0.0411017	+	0.999155i	$0.486913\pi$
$390$	0	0
$391$	−19.3493	−0.978536
$392$	1.26282	0.0637822
$393$	0	0
$394$	−15.9445	−0.803271
$395$	−14.4797	−0.728554
$396$	0	0
$397$	−38.7222	−1.94341	−0.971705	−	0.236197i	$-0.924099\pi$
$397$	−38.7222	−1.94341	−0.971705	+	0.236197i	$0.924099\pi$
$398$	36.3348	1.82130
$399$	0	0
$400$	14.9487	0.747434
$401$	12.5099	0.624716	0.312358	−	0.949964i	$-0.398881\pi$
$401$	12.5099	0.624716	0.312358	+	0.949964i	$0.398881\pi$
$402$	0	0
$403$	22.1040	1.10108
$404$	−15.7977	−0.785965
$405$	0	0
$406$	14.1911	0.704292
$407$	−0.289514	−0.0143507
$408$	0	0
$409$	−1.09186	−0.0539888	−0.0269944	−	0.999636i	$-0.508594\pi$
$409$	−1.09186	−0.0539888	−0.0269944	+	0.999636i	$0.508594\pi$
$410$	−24.8627	−1.22788
$411$	0	0
$412$	−7.57571	−0.373228
$413$	−2.23465	−0.109960
$414$	0	0
$415$	7.72576	0.379243
$416$	−22.5398	−1.10510
$417$	0	0
$418$	−0.487933	−0.0238656
$419$	−25.5953	−1.25041	−0.625207	−	0.780459i	$-0.714985\pi$
$419$	−25.5953	−1.25041	−0.625207	+	0.780459i	$0.714985\pi$
$420$	0	0
$421$	26.7270	1.30260	0.651298	−	0.758822i	$-0.274225\pi$
$421$	26.7270	1.30260	0.651298	+	0.758822i	$0.274225\pi$
$422$	−11.1678	−0.543639
$423$	0	0
$424$	0.296543	0.0144014
$425$	15.0417	0.729632
$426$	0	0
$427$	−8.26515	−0.399979
$428$	0.477653	0.0230883
$429$	0	0
$430$	−27.0240	−1.30321
$431$	20.1905	0.972544	0.486272	−	0.873808i	$-0.338356\pi$
$431$	20.1905	0.972544	0.486272	+	0.873808i	$0.338356\pi$
$432$	0	0
$433$	21.9651	1.05557	0.527787	−	0.849377i	$-0.323022\pi$
$433$	21.9651	1.05557	0.527787	+	0.849377i	$0.323022\pi$
$434$	−11.4021	−0.547319
$435$	0	0
$436$	−18.9786	−0.908911
$437$	−12.4536	−0.595734
$438$	0	0
$439$	12.8006	0.610939	0.305470	−	0.952202i	$-0.401187\pi$
$439$	12.8006	0.610939	0.305470	+	0.952202i	$0.401187\pi$
$440$	−0.149064	−0.00710637
$441$	0	0
$442$	−31.6371	−1.50482
$443$	35.3169	1.67795	0.838977	−	0.544166i	$-0.183154\pi$
$443$	35.3169	1.67795	0.838977	+	0.544166i	$0.183154\pi$
$444$	0	0
$445$	20.9985	0.995422
$446$	11.2087	0.530749
$447$	0	0
$448$	1.81346	0.0856778
$449$	9.48179	0.447473	0.223737	−	0.974650i	$-0.428174\pi$
$449$	9.48179	0.447473	0.223737	+	0.974650i	$0.428174\pi$
$450$	0	0
$451$	−0.826360	−0.0389118
$452$	9.32461	0.438593
$453$	0	0
$454$	44.5487	2.09077
$455$	4.92602	0.230936
$456$	0	0
$457$	14.4625	0.676528	0.338264	−	0.941051i	$-0.390160\pi$
$457$	14.4625	0.676528	0.338264	+	0.941051i	$0.390160\pi$
$458$	2.25643	0.105436
$459$	0	0
$460$	7.15028	0.333384
$461$	16.5198	0.769404	0.384702	−	0.923041i	$-0.374304\pi$
$461$	16.5198	0.769404	0.384702	+	0.923041i	$0.374304\pi$
$462$	0	0
$463$	−21.0251	−0.977120	−0.488560	−	0.872530i	$-0.662478\pi$
$463$	−21.0251	−0.977120	−0.488560	+	0.872530i	$0.662478\pi$
$464$	−38.2997	−1.77802
$465$	0	0
$466$	45.1090	2.08963
$467$	3.24697	0.150252	0.0751260	−	0.997174i	$-0.476064\pi$
$467$	3.24697	0.150252	0.0751260	+	0.997174i	$0.476064\pi$
$468$	0	0
$469$	3.74734	0.173036
$470$	11.4593	0.528576
$471$	0	0
$472$	2.82197	0.129892
$473$	−0.898196	−0.0412991
$474$	0	0
$475$	9.68113	0.444201
$476$	6.44515	0.295413
$477$	0	0
$478$	36.2372	1.65745
$479$	37.5282	1.71471	0.857353	−	0.514729i	$-0.172108\pi$
$479$	37.5282	1.71471	0.857353	+	0.514729i	$0.172108\pi$
$480$	0	0
$481$	−12.0819	−0.550887
$482$	−3.24983	−0.148026
$483$	0	0
$484$	−14.3502	−0.652281
$485$	−12.7624	−0.579509
$486$	0	0
$487$	−40.2962	−1.82600	−0.912998	−	0.407964i	$-0.866239\pi$
$487$	−40.2962	−1.82600	−0.912998	+	0.407964i	$0.866239\pi$
$488$	10.4374	0.472480
$489$	0	0
$490$	−2.54104	−0.114792
$491$	23.9030	1.07873	0.539364	−	0.842072i	$-0.318665\pi$
$491$	23.9030	1.07873	0.539364	+	0.842072i	$0.318665\pi$
$492$	0	0
$493$	−38.5381	−1.73567
$494$	−20.3622	−0.916140
$495$	0	0
$496$	30.7726	1.38173
$497$	−5.09148	−0.228384
$498$	0	0
$499$	4.38382	0.196247	0.0981233	−	0.995174i	$-0.468716\pi$
$499$	4.38382	0.196247	0.0981233	+	0.995174i	$0.468716\pi$
$500$	−14.6810	−0.656555
$501$	0	0
$502$	34.5456	1.54184
$503$	−26.1847	−1.16752	−0.583758	−	0.811927i	$-0.698418\pi$
$503$	−26.1847	−1.16752	−0.583758	+	0.811927i	$0.698418\pi$
$504$	0	0
$505$	−16.9140	−0.752663
$506$	0.601760	0.0267515
$507$	0	0
$508$	1.30541	0.0579181
$509$	−3.82171	−0.169394	−0.0846972	−	0.996407i	$-0.526992\pi$
$509$	−3.82171	−0.169394	−0.0846972	+	0.996407i	$0.526992\pi$
$510$	0	0
$511$	−15.8865	−0.702777
$512$	−18.9867	−0.839100
$513$	0	0
$514$	−9.59757	−0.423331
$515$	−8.11103	−0.357415
$516$	0	0
$517$	0.380871	0.0167507
$518$	6.23232	0.273833
$519$	0	0
$520$	−6.22070	−0.272796
$521$	−15.3809	−0.673850	−0.336925	−	0.941531i	$-0.609387\pi$
$521$	−15.3809	−0.673850	−0.336925	+	0.941531i	$0.609387\pi$
$522$	0	0
$523$	10.9314	0.477998	0.238999	−	0.971020i	$-0.423181\pi$
$523$	10.9314	0.477998	0.238999	+	0.971020i	$0.423181\pi$
$524$	−28.5335	−1.24649
$525$	0	0
$526$	−40.4631	−1.76427
$527$	30.9642	1.34882
$528$	0	0
$529$	−7.64124	−0.332228
$530$	−0.596701	−0.0259190
$531$	0	0
$532$	4.14822	0.179848
$533$	−34.4853	−1.49372
$534$	0	0
$535$	0.511406	0.0221100
$536$	−4.73223	−0.204401
$537$	0	0
$538$	15.8296	0.682463
$539$	−0.0844564	−0.00363780
$540$	0	0
$541$	−40.3414	−1.73441	−0.867206	−	0.497949i	$-0.834087\pi$
$541$	−40.3414	−1.73441	−0.867206	+	0.497949i	$0.834087\pi$
$542$	−19.9031	−0.854910
$543$	0	0
$544$	−31.5747	−1.35375
$545$	−20.3197	−0.870400
$546$	0	0
$547$	−38.2345	−1.63479	−0.817396	−	0.576077i	$-0.804583\pi$
$547$	−38.2345	−1.63479	−0.817396	+	0.576077i	$0.804583\pi$
$548$	−14.1377	−0.603935
$549$	0	0
$550$	−0.467796	−0.0199469
$551$	−24.8038	−1.05668
$552$	0	0
$553$	10.3600	0.440554
$554$	39.0797	1.66034
$555$	0	0
$556$	−18.5326	−0.785957
$557$	−24.3805	−1.03304	−0.516518	−	0.856276i	$-0.672772\pi$
$557$	−24.3805	−1.03304	−0.516518	+	0.856276i	$0.672772\pi$
$558$	0	0
$559$	−37.4831	−1.58537
$560$	6.85789	0.289799
$561$	0	0
$562$	30.7685	1.29789
$563$	11.0319	0.464938	0.232469	−	0.972604i	$-0.425320\pi$
$563$	11.0319	0.464938	0.232469	+	0.972604i	$0.425320\pi$
$564$	0	0
$565$	9.98351	0.420009
$566$	20.1202	0.845716
$567$	0	0
$568$	6.42964	0.269782
$569$	−41.4077	−1.73590	−0.867950	−	0.496651i	$-0.834563\pi$
$569$	−41.4077	−1.73590	−0.867950	+	0.496651i	$0.834563\pi$
$570$	0	0
$571$	−20.0004	−0.836991	−0.418495	−	0.908219i	$-0.637442\pi$
$571$	−20.0004	−0.836991	−0.418495	+	0.908219i	$0.637442\pi$
$572$	0.388576	0.0162472
$573$	0	0
$574$	17.7889	0.742494
$575$	−11.9396	−0.497915
$576$	0	0
$577$	29.2882	1.21928	0.609642	−	0.792677i	$-0.291313\pi$
$577$	29.2882	1.21928	0.609642	+	0.792677i	$0.291313\pi$
$578$	−13.4113	−0.557838
$579$	0	0
$580$	14.2413	0.591336
$581$	−5.52767	−0.229326
$582$	0	0
$583$	−0.0198325	−0.000821378 0
$584$	20.0618	0.830164
$585$	0	0
$586$	−45.0507	−1.86103
$587$	−21.2041	−0.875187	−0.437593	−	0.899173i	$-0.644169\pi$
$587$	−21.2041	−0.875187	−0.437593	+	0.899173i	$0.644169\pi$
$588$	0	0
$589$	19.9291	0.821165
$590$	−5.67834	−0.233773
$591$	0	0
$592$	−16.8201	−0.691303
$593$	37.6147	1.54465	0.772325	−	0.635227i	$-0.219094\pi$
$593$	37.6147	1.54465	0.772325	+	0.635227i	$0.219094\pi$
$594$	0	0
$595$	6.90059	0.282897
$596$	25.7960	1.05664
$597$	0	0
$598$	25.1124	1.02692
$599$	25.3004	1.03375	0.516873	−	0.856062i	$-0.327096\pi$
$599$	25.3004	1.03375	0.516873	+	0.856062i	$0.327096\pi$
$600$	0	0
$601$	22.2166	0.906235	0.453118	−	0.891451i	$-0.350312\pi$
$601$	22.2166	0.906235	0.453118	+	0.891451i	$0.350312\pi$
$602$	19.3353	0.788048
$603$	0	0
$604$	10.3071	0.419390
$605$	−15.3642	−0.624644
$606$	0	0
$607$	26.6139	1.08023	0.540113	−	0.841592i	$-0.318381\pi$
$607$	26.6139	1.08023	0.540113	+	0.841592i	$0.318381\pi$
$608$	−20.3220	−0.824167
$609$	0	0
$610$	−21.0021	−0.850349
$611$	15.8943	0.643016
$612$	0	0
$613$	−32.9527	−1.33095	−0.665474	−	0.746421i	$-0.731771\pi$
$613$	−32.9527	−1.33095	−0.665474	+	0.746421i	$0.731771\pi$
$614$	−40.5224	−1.63535
$615$	0	0
$616$	0.106653	0.00429719
$617$	42.6655	1.71765	0.858824	−	0.512271i	$-0.171196\pi$
$617$	42.6655	1.71765	0.858824	+	0.512271i	$0.171196\pi$
$618$	0	0
$619$	−27.3570	−1.09957	−0.549786	−	0.835306i	$-0.685291\pi$
$619$	−27.3570	−1.09957	−0.549786	+	0.835306i	$0.685291\pi$
$620$	−11.4424	−0.459539
$621$	0	0
$622$	17.2405	0.691281
$623$	−15.0241	−0.601928
$624$	0	0
$625$	−0.485568	−0.0194227
$626$	−10.5853	−0.423073
$627$	0	0
$628$	23.9470	0.955590
$629$	−16.9248	−0.674838
$630$	0	0
$631$	17.0826	0.680049	0.340024	−	0.940417i	$-0.389565\pi$
$631$	17.0826	0.680049	0.340024	+	0.940417i	$0.389565\pi$
$632$	−13.0829	−0.520410
$633$	0	0
$634$	51.9322	2.06249
$635$	1.39765	0.0554641
$636$	0	0
$637$	−3.52450	−0.139646
$638$	1.19853	0.0474502
$639$	0	0
$640$	−13.2684	−0.524478
$641$	−32.1820	−1.27111	−0.635557	−	0.772054i	$-0.719230\pi$
$641$	−32.1820	−1.27111	−0.635557	+	0.772054i	$0.719230\pi$
$642$	0	0
$643$	−5.32118	−0.209847	−0.104923	−	0.994480i	$-0.533460\pi$
$643$	−5.32118	−0.209847	−0.104923	+	0.994480i	$0.533460\pi$
$644$	−5.11593	−0.201596
$645$	0	0
$646$	−28.5243	−1.12227
$647$	−2.44907	−0.0962827	−0.0481414	−	0.998841i	$-0.515330\pi$
$647$	−2.44907	−0.0962827	−0.0481414	+	0.998841i	$0.515330\pi$
$648$	0	0
$649$	−0.188731	−0.00740832
$650$	−19.5219	−0.765710
$651$	0	0
$652$	26.6976	1.04556
$653$	30.1572	1.18014	0.590070	−	0.807352i	$-0.299100\pi$
$653$	30.1572	1.18014	0.590070	+	0.807352i	$0.299100\pi$
$654$	0	0
$655$	−30.5497	−1.19368
$656$	−48.0096	−1.87446
$657$	0	0
$658$	−8.19894	−0.319628
$659$	33.1094	1.28976	0.644880	−	0.764284i	$-0.276907\pi$
$659$	33.1094	1.28976	0.644880	+	0.764284i	$0.276907\pi$
$660$	0	0
$661$	−38.7669	−1.50786	−0.753928	−	0.656957i	$-0.771843\pi$
$661$	−38.7669	−1.50786	−0.753928	+	0.656957i	$0.771843\pi$
$662$	50.2237	1.95200
$663$	0	0
$664$	6.98047	0.270895
$665$	4.44134	0.172228
$666$	0	0
$667$	30.5901	1.18445
$668$	−29.2809	−1.13291
$669$	0	0
$670$	9.52214	0.367872
$671$	−0.698045	−0.0269477
$672$	0	0
$673$	−21.3247	−0.822006	−0.411003	−	0.911634i	$-0.634821\pi$
$673$	−21.3247	−0.822006	−0.411003	+	0.911634i	$0.634821\pi$
$674$	−21.9767	−0.846509
$675$	0	0
$676$	−0.754394	−0.0290152
$677$	−10.7266	−0.412257	−0.206129	−	0.978525i	$-0.566087\pi$
$677$	−10.7266	−0.412257	−0.206129	+	0.978525i	$0.566087\pi$
$678$	0	0
$679$	9.13128	0.350426
$680$	−8.71422	−0.334175
$681$	0	0
$682$	−0.962982	−0.0368745
$683$	14.7316	0.563688	0.281844	−	0.959460i	$-0.409054\pi$
$683$	14.7316	0.563688	0.281844	+	0.959460i	$0.409054\pi$
$684$	0	0
$685$	−15.1368	−0.578346
$686$	1.81808	0.0694145
$687$	0	0
$688$	−52.1831	−1.98946
$689$	−0.827642	−0.0315306
$690$	0	0
$691$	−26.0749	−0.991934	−0.495967	−	0.868341i	$-0.665186\pi$
$691$	−26.0749	−0.991934	−0.495967	+	0.868341i	$0.665186\pi$
$692$	−30.5582	−1.16165
$693$	0	0
$694$	−20.9632	−0.795751
$695$	−19.8422	−0.752656
$696$	0	0
$697$	−48.3085	−1.82982
$698$	59.9368	2.26864
$699$	0	0
$700$	3.97702	0.150317
$701$	3.42272	0.129275	0.0646373	−	0.997909i	$-0.479411\pi$
$701$	3.42272	0.129275	0.0646373	+	0.997909i	$0.479411\pi$
$702$	0	0
$703$	−10.8931	−0.410842
$704$	0.153158	0.00577236
$705$	0	0
$706$	23.3721	0.879619
$707$	12.1017	0.455132
$708$	0	0
$709$	15.4309	0.579520	0.289760	−	0.957099i	$-0.406425\pi$
$709$	15.4309	0.579520	0.289760	+	0.957099i	$0.406425\pi$
$710$	−12.9377	−0.485542
$711$	0	0
$712$	18.9728	0.711035
$713$	−24.5783	−0.920463
$714$	0	0
$715$	0.416034	0.0155588
$716$	15.6992	0.586707
$717$	0	0
$718$	39.8832	1.48843
$719$	−0.647747	−0.0241569	−0.0120784	−	0.999927i	$-0.503845\pi$
$719$	−0.647747	−0.0241569	−0.0120784	+	0.999927i	$0.503845\pi$
$720$	0	0
$721$	5.80333	0.216127
$722$	16.1847	0.602334
$723$	0	0
$724$	−19.2498	−0.715412
$725$	−23.7801	−0.883172
$726$	0	0
$727$	22.4181	0.831442	0.415721	−	0.909492i	$-0.363529\pi$
$727$	22.4181	0.831442	0.415721	+	0.909492i	$0.363529\pi$
$728$	4.45082	0.164958
$729$	0	0
$730$	−40.3682	−1.49409
$731$	−52.5080	−1.94208
$732$	0	0
$733$	−22.0750	−0.815358	−0.407679	−	0.913125i	$-0.633662\pi$
$733$	−22.0750	−0.815358	−0.407679	+	0.913125i	$0.633662\pi$
$734$	30.4578	1.12422
$735$	0	0
$736$	25.0628	0.923828
$737$	0.316487	0.0116579
$738$	0	0
$739$	−15.2742	−0.561871	−0.280936	−	0.959727i	$-0.590645\pi$
$739$	−15.2742	−0.561871	−0.280936	+	0.959727i	$0.590645\pi$
$740$	6.25436	0.229915
$741$	0	0
$742$	0.426931	0.0156731
$743$	16.5688	0.607851	0.303925	−	0.952696i	$-0.401703\pi$
$743$	16.5688	0.607851	0.303925	+	0.952696i	$0.401703\pi$
$744$	0	0
$745$	27.6188	1.01187
$746$	26.5440	0.971845
$747$	0	0
$748$	0.544335	0.0199028
$749$	−0.365904	−0.0133698
$750$	0	0
$751$	−25.6803	−0.937089	−0.468544	−	0.883440i	$-0.655221\pi$
$751$	−25.6803	−0.937089	−0.468544	+	0.883440i	$0.655221\pi$
$752$	22.1277	0.806915
$753$	0	0
$754$	50.0165	1.82149
$755$	11.0354	0.401621
$756$	0	0
$757$	2.68605	0.0976262	0.0488131	−	0.998808i	$-0.484456\pi$
$757$	2.68605	0.0976262	0.0488131	+	0.998808i	$0.484456\pi$
$758$	−3.47672	−0.126280
$759$	0	0
$760$	−5.60863	−0.203446
$761$	−32.2716	−1.16984	−0.584922	−	0.811090i	$-0.698875\pi$
$761$	−32.2716	−1.16984	−0.584922	+	0.811090i	$0.698875\pi$
$762$	0	0
$763$	14.5385	0.526327
$764$	19.7217	0.713505
$765$	0	0
$766$	−26.5439	−0.959071
$767$	−7.87602	−0.284387
$768$	0	0
$769$	33.0914	1.19331	0.596654	−	0.802499i	$-0.296497\pi$
$769$	33.0914	1.19331	0.596654	+	0.802499i	$0.296497\pi$
$770$	−0.214607	−0.00773390
$771$	0	0
$772$	−5.37269	−0.193367
$773$	41.8362	1.50474	0.752372	−	0.658738i	$-0.228909\pi$
$773$	41.8362	1.50474	0.752372	+	0.658738i	$0.228909\pi$
$774$	0	0
$775$	19.1066	0.686330
$776$	−11.5312	−0.413946
$777$	0	0
$778$	−2.94766	−0.105679
$779$	−31.0922	−1.11399
$780$	0	0
$781$	−0.430008	−0.0153869
$782$	35.1785	1.25798
$783$	0	0
$784$	−4.90673	−0.175240
$785$	25.6392	0.915101
$786$	0	0
$787$	−49.6735	−1.77067	−0.885335	−	0.464954i	$-0.846071\pi$
$787$	−49.6735	−1.77067	−0.885335	+	0.464954i	$0.846071\pi$
$788$	11.4484	0.407831
$789$	0	0
$790$	26.3253	0.936611
$791$	−7.14306	−0.253978
$792$	0	0
$793$	−29.1305	−1.03445
$794$	70.3999	2.49840
$795$	0	0
$796$	−26.0890	−0.924699
$797$	47.6456	1.68769	0.843847	−	0.536584i	$-0.180286\pi$
$797$	47.6456	1.68769	0.843847	+	0.536584i	$0.180286\pi$
$798$	0	0
$799$	22.2655	0.787696
$800$	−19.4833	−0.688839
$801$	0	0
$802$	−22.7440	−0.803120
$803$	−1.34172	−0.0473481
$804$	0	0
$805$	−5.47743	−0.193054
$806$	−40.1868	−1.41552
$807$	0	0
$808$	−15.2823	−0.537631
$809$	25.8849	0.910064	0.455032	−	0.890475i	$-0.349628\pi$
$809$	25.8849	0.910064	0.455032	+	0.890475i	$0.349628\pi$
$810$	0	0
$811$	−10.3199	−0.362382	−0.181191	−	0.983448i	$-0.557995\pi$
$811$	−10.3199	−0.362382	−0.181191	+	0.983448i	$0.557995\pi$
$812$	−10.1894	−0.357579
$813$	0	0
$814$	0.526360	0.0184489
$815$	28.5841	1.00126
$816$	0	0
$817$	−33.7951	−1.18234
$818$	1.98508	0.0694066
$819$	0	0
$820$	17.8518	0.623412
$821$	14.2032	0.495696	0.247848	−	0.968799i	$-0.420277\pi$
$821$	14.2032	0.495696	0.247848	+	0.968799i	$0.420277\pi$
$822$	0	0
$823$	56.7644	1.97868	0.989342	−	0.145612i	$-0.0465152\pi$
$823$	56.7644	1.97868	0.989342	+	0.145612i	$0.0465152\pi$
$824$	−7.32857	−0.255303
$825$	0	0
$826$	4.06277	0.141362
$827$	−19.9274	−0.692944	−0.346472	−	0.938060i	$-0.612620\pi$
$827$	−19.9274	−0.692944	−0.346472	+	0.938060i	$0.612620\pi$
$828$	0	0
$829$	−9.92254	−0.344624	−0.172312	−	0.985042i	$-0.555124\pi$
$829$	−9.92254	−0.344624	−0.172312	+	0.985042i	$0.555124\pi$
$830$	−14.0460	−0.487545
$831$	0	0
$832$	6.39153	0.221586
$833$	−4.93727	−0.171066
$834$	0	0
$835$	−31.3500	−1.08491
$836$	0.350343	0.0121169
$837$	0	0
$838$	46.5343	1.60750
$839$	−51.1367	−1.76543	−0.882717	−	0.469904i	$-0.844288\pi$
$839$	−51.1367	−1.76543	−0.882717	+	0.469904i	$0.844288\pi$
$840$	0	0
$841$	31.9265	1.10092
$842$	−48.5918	−1.67459
$843$	0	0
$844$	8.01864	0.276013
$845$	−0.807702	−0.0277858
$846$	0	0
$847$	10.9929	0.377719
$848$	−1.15222	−0.0395675
$849$	0	0
$850$	−27.3471	−0.937997
$851$	13.4343	0.460522
$852$	0	0
$853$	−37.5651	−1.28620	−0.643102	−	0.765781i	$-0.722353\pi$
$853$	−37.5651	−1.28620	−0.643102	+	0.765781i	$0.722353\pi$
$854$	15.0267	0.514203
$855$	0	0
$856$	0.462071	0.0157933
$857$	21.4442	0.732521	0.366261	−	0.930512i	$-0.380638\pi$
$857$	21.4442	0.732521	0.366261	+	0.930512i	$0.380638\pi$
$858$	0	0
$859$	−3.57663	−0.122033	−0.0610166	−	0.998137i	$-0.519434\pi$
$859$	−3.57663	−0.122033	−0.0610166	+	0.998137i	$0.519434\pi$
$860$	19.4037	0.661659
$861$	0	0
$862$	−36.7080	−1.25028
$863$	−2.03377	−0.0692303	−0.0346151	−	0.999401i	$-0.511021\pi$
$863$	−2.03377	−0.0692303	−0.0346151	+	0.999401i	$0.511021\pi$
$864$	0	0
$865$	−32.7175	−1.11243
$866$	−39.9342	−1.35702
$867$	0	0
$868$	8.18689	0.277881
$869$	0.874972	0.0296814
$870$	0	0
$871$	13.2075	0.447519
$872$	−18.3595	−0.621731
$873$	0	0
$874$	22.6415	0.765861
$875$	11.2463	0.380194
$876$	0	0
$877$	34.8009	1.17514	0.587572	−	0.809172i	$-0.300084\pi$
$877$	34.8009	1.17514	0.587572	+	0.809172i	$0.300084\pi$
$878$	−23.2725	−0.785409
$879$	0	0
$880$	0.579193	0.0195246
$881$	−46.1355	−1.55435	−0.777173	−	0.629287i	$-0.783347\pi$
$881$	−46.1355	−1.55435	−0.777173	+	0.629287i	$0.783347\pi$
$882$	0	0
$883$	2.96916	0.0999203	0.0499602	−	0.998751i	$-0.484091\pi$
$883$	2.96916	0.0999203	0.0499602	+	0.998751i	$0.484091\pi$
$884$	22.7159	0.764020
$885$	0	0
$886$	−64.2088	−2.15714
$887$	−41.6884	−1.39976	−0.699880	−	0.714260i	$-0.746763\pi$
$887$	−41.6884	−1.39976	−0.699880	+	0.714260i	$0.746763\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−38.1768	−1.27969
$891$	0	0
$892$	−8.04805	−0.269469
$893$	14.3305	0.479551
$894$	0	0
$895$	16.8086	0.561848
$896$	9.49332	0.317150
$897$	0	0
$898$	−17.2386	−0.575261
$899$	−48.9526	−1.63266
$900$	0	0
$901$	−1.15940	−0.0386251
$902$	1.50239	0.0500240
$903$	0	0
$904$	9.02042	0.300015
$905$	−20.6100	−0.685100
$906$	0	0
$907$	−33.2816	−1.10510	−0.552549	−	0.833480i	$-0.686345\pi$
$907$	−33.2816	−1.10510	−0.552549	+	0.833480i	$0.686345\pi$
$908$	−31.9867	−1.06151
$909$	0	0
$910$	−8.95590	−0.296885
$911$	−41.8018	−1.38496	−0.692478	−	0.721439i	$-0.743481\pi$
$911$	−41.8018	−1.38496	−0.692478	+	0.721439i	$0.743481\pi$
$912$	0	0
$913$	−0.466847	−0.0154504
$914$	−26.2940	−0.869728
$915$	0	0
$916$	−1.62015	−0.0535314
$917$	21.8579	0.721811
$918$	0	0
$919$	8.93340	0.294686	0.147343	−	0.989085i	$-0.452928\pi$
$919$	8.93340	0.294686	0.147343	+	0.989085i	$0.452928\pi$
$920$	6.91703	0.228048
$921$	0	0
$922$	−30.0343	−0.989127
$923$	−17.9449	−0.590665
$924$	0	0
$925$	−10.4436	−0.343382
$926$	38.2253	1.25616
$927$	0	0
$928$	49.9178	1.63863
$929$	−43.2613	−1.41936	−0.709679	−	0.704525i	$-0.751160\pi$
$929$	−43.2613	−1.41936	−0.709679	+	0.704525i	$0.751160\pi$
$930$	0	0
$931$	−3.17772	−0.104145
$932$	−32.3890	−1.06094
$933$	0	0
$934$	−5.90325	−0.193160
$935$	0.582799	0.0190596
$936$	0	0
$937$	7.34205	0.239854	0.119927	−	0.992783i	$-0.461734\pi$
$937$	7.34205	0.239854	0.119927	+	0.992783i	$0.461734\pi$
$938$	−6.81295	−0.222451
$939$	0	0
$940$	−8.22792	−0.268365
$941$	50.7589	1.65469	0.827346	−	0.561692i	$-0.189849\pi$
$941$	50.7589	1.65469	0.827346	+	0.561692i	$0.189849\pi$
$942$	0	0
$943$	38.3455	1.24870
$944$	−10.9648	−0.356874
$945$	0	0
$946$	1.63299	0.0530931
$947$	−26.6858	−0.867171	−0.433585	−	0.901112i	$-0.642752\pi$
$947$	−26.6858	−0.867171	−0.433585	+	0.901112i	$0.642752\pi$
$948$	0	0
$949$	−55.9919	−1.81757
$950$	−17.6011	−0.571054
$951$	0	0
$952$	6.23490	0.202074
$953$	6.47952	0.209892	0.104946	−	0.994478i	$-0.466533\pi$
$953$	6.47952	0.209892	0.104946	+	0.994478i	$0.466533\pi$
$954$	0	0
$955$	21.1153	0.683274
$956$	−26.0188	−0.841509
$957$	0	0
$958$	−68.2291	−2.20438
$959$	10.8301	0.349723
$960$	0	0
$961$	8.33198	0.268773
$962$	21.9658	0.708207
$963$	0	0
$964$	2.33343	0.0751547
$965$	−5.75234	−0.185174
$966$	0	0
$967$	14.2378	0.457857	0.228929	−	0.973443i	$-0.426478\pi$
$967$	14.2378	0.457857	0.228929	+	0.973443i	$0.426478\pi$
$968$	−13.8820	−0.446186
$969$	0	0
$970$	23.2030	0.745002
$971$	19.2850	0.618885	0.309442	−	0.950918i	$-0.399858\pi$
$971$	19.2850	0.618885	0.309442	+	0.950918i	$0.399858\pi$
$972$	0	0
$973$	14.1968	0.455128
$974$	73.2617	2.34746
$975$	0	0
$976$	−40.5548	−1.29813
$977$	−3.08393	−0.0986638	−0.0493319	−	0.998782i	$-0.515709\pi$
$977$	−3.08393	−0.0986638	−0.0493319	+	0.998782i	$0.515709\pi$
$978$	0	0
$979$	−1.26888	−0.0405536
$980$	1.82451	0.0582817
$981$	0	0
$982$	−43.4576	−1.38679
$983$	−25.0025	−0.797454	−0.398727	−	0.917070i	$-0.630548\pi$
$983$	−25.0025	−0.797454	−0.398727	+	0.917070i	$0.630548\pi$
$984$	0	0
$985$	12.2574	0.390552
$986$	70.0653	2.23133
$987$	0	0
$988$	14.6204	0.465137
$989$	41.6789	1.32531
$990$	0	0
$991$	8.99880	0.285856	0.142928	−	0.989733i	$-0.454348\pi$
$991$	8.99880	0.285856	0.142928	+	0.989733i	$0.454348\pi$
$992$	−40.1074	−1.27341
$993$	0	0
$994$	9.25671	0.293605
$995$	−27.9325	−0.885519
$996$	0	0
$997$	−32.2384	−1.02100	−0.510501	−	0.859877i	$-0.670540\pi$
$997$	−32.2384	−1.02100	−0.510501	+	0.859877i	$0.670540\pi$
$998$	−7.97012	−0.252290
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.m.1.2		11
3.2	odd	2		2667.2.a.k.1.10	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.k.1.10	✓	11	3.2	odd	2
8001.2.a.m.1.2		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.81808 q^{2} +1.30541 q^{4} +1.39765 q^{5} -1.00000 q^{7} +1.26282 q^{8} +O(q^{10})\)	\(q-1.81808 q^{2} +1.30541 q^{4} +1.39765 q^{5} -1.00000 q^{7} +1.26282 q^{8} -2.54104 q^{10} -0.0844564 q^{11} -3.52450 q^{13} +1.81808 q^{14} -4.90673 q^{16} -4.93727 q^{17} -3.17772 q^{19} +1.82451 q^{20} +0.153548 q^{22} +3.91903 q^{23} -3.04657 q^{25} +6.40782 q^{26} -1.30541 q^{28} +7.80555 q^{29} -6.27152 q^{31} +6.39517 q^{32} +8.97635 q^{34} -1.39765 q^{35} +3.42797 q^{37} +5.77734 q^{38} +1.76499 q^{40} +9.78445 q^{41} +10.6350 q^{43} -0.110250 q^{44} -7.12509 q^{46} -4.50967 q^{47} +1.00000 q^{49} +5.53890 q^{50} -4.60091 q^{52} +0.234825 q^{53} -0.118041 q^{55} -1.26282 q^{56} -14.1911 q^{58} +2.23465 q^{59} +8.26515 q^{61} +11.4021 q^{62} -1.81346 q^{64} -4.92602 q^{65} -3.74734 q^{67} -6.44515 q^{68} +2.54104 q^{70} +5.09148 q^{71} +15.8865 q^{73} -6.23232 q^{74} -4.14822 q^{76} +0.0844564 q^{77} -10.3600 q^{79} -6.85789 q^{80} -17.7889 q^{82} +5.52767 q^{83} -6.90059 q^{85} -19.3353 q^{86} -0.106653 q^{88} +15.0241 q^{89} +3.52450 q^{91} +5.11593 q^{92} +8.19894 q^{94} -4.44134 q^{95} -9.13128 q^{97} -1.81808 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8}+O(q^{10}) \) 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8	\( 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8} - 12 q^{10} + 7 q^{11} - 24 q^{13} - 2 q^{14} - 6 q^{16} + 15 q^{17} - 19 q^{19} - 3 q^{20} - 3 q^{22} + 11 q^{23} + 10 q^{25} - 10 q^{26} - 12 q^{28} + 10 q^{29} - 20 q^{31} + 27 q^{32} - 9 q^{34} + q^{35} - 22 q^{37} - 8 q^{38} - 29 q^{40} - 9 q^{41} - 17 q^{43} + 9 q^{44} - 18 q^{46} + 7 q^{47} + 11 q^{49} + 47 q^{50} - 66 q^{52} + 28 q^{53} - 24 q^{55} - 15 q^{56} - 39 q^{58} - 35 q^{59} - 6 q^{61} - 18 q^{62} + 11 q^{64} + 43 q^{65} - 22 q^{67} + 12 q^{68} + 12 q^{70} + 22 q^{71} - 29 q^{73} - 14 q^{74} + 10 q^{76} - 7 q^{77} - 20 q^{79} - 66 q^{80} - 24 q^{82} - 17 q^{83} - 50 q^{85} + 12 q^{86} + 2 q^{88} + q^{89} + 24 q^{91} + 22 q^{92} + q^{94} - 10 q^{95} - 45 q^{97} + 2 q^{98}+O(q^{100}) \) 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8 - 12 * q^10 + 7 * q^11 - 24 * q^13 - 2 * q^14 - 6 * q^16 + 15 * q^17 - 19 * q^19 - 3 * q^20 - 3 * q^22 + 11 * q^23 + 10 * q^25 - 10 * q^26 - 12 * q^28 + 10 * q^29 - 20 * q^31 + 27 * q^32 - 9 * q^34 + q^35 - 22 * q^37 - 8 * q^38 - 29 * q^40 - 9 * q^41 - 17 * q^43 + 9 * q^44 - 18 * q^46 + 7 * q^47 + 11 * q^49 + 47 * q^50 - 66 * q^52 + 28 * q^53 - 24 * q^55 - 15 * q^56 - 39 * q^58 - 35 * q^59 - 6 * q^61 - 18 * q^62 + 11 * q^64 + 43 * q^65 - 22 * q^67 + 12 * q^68 + 12 * q^70 + 22 * q^71 - 29 * q^73 - 14 * q^74 + 10 * q^76 - 7 * q^77 - 20 * q^79 - 66 * q^80 - 24 * q^82 - 17 * q^83 - 50 * q^85 + 12 * q^86 + 2 * q^88 + q^89 + 24 * q^91 + 22 * q^92 + q^94 - 10 * q^95 - 45 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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