LMFDB - Embedded newform 8001.2.a.r.1.9

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 5 x^{15} - 13 x^{14} + 98 x^{13} + 9 x^{12} - 712 x^{11} + 565 x^{10} + 2282 x^{9} - 3082 x^{8} + \cdots + 4 $ x^16 - 5x^15 - 13x^14 + 98x^13 + 9x^12 - 712x^11 + 565x^10 + 2282x^9 - 3082x^8 - 2747x^7 + 5821x^6 - 158x^5 - 3341x^4 + 1002x^3 + 416x^2 - 148*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.298113$ 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)

$f(q)$	$=$	$q-0.298113 q^{2} -1.91113 q^{4} -2.54722 q^{5} -1.00000 q^{7} +1.16596 q^{8} +O(q^{10})$	\(q-0.298113 q^{2} -1.91113 q^{4} -2.54722 q^{5} -1.00000 q^{7} +1.16596 q^{8} +0.759360 q^{10} -6.15296 q^{11} +4.68194 q^{13} +0.298113 q^{14} +3.47467 q^{16} +4.41002 q^{17} -5.03205 q^{19} +4.86807 q^{20} +1.83427 q^{22} -9.11699 q^{23} +1.48835 q^{25} -1.39575 q^{26} +1.91113 q^{28} +4.28790 q^{29} +0.0487464 q^{31} -3.36776 q^{32} -1.31468 q^{34} +2.54722 q^{35} +3.02081 q^{37} +1.50012 q^{38} -2.96995 q^{40} +3.31879 q^{41} +1.89550 q^{43} +11.7591 q^{44} +2.71789 q^{46} +10.1769 q^{47} +1.00000 q^{49} -0.443697 q^{50} -8.94780 q^{52} -6.07411 q^{53} +15.6730 q^{55} -1.16596 q^{56} -1.27828 q^{58} +9.37602 q^{59} +14.3700 q^{61} -0.0145319 q^{62} -5.94537 q^{64} -11.9260 q^{65} +5.59222 q^{67} -8.42813 q^{68} -0.759360 q^{70} -7.54064 q^{71} +5.40772 q^{73} -0.900543 q^{74} +9.61689 q^{76} +6.15296 q^{77} -8.18983 q^{79} -8.85077 q^{80} -0.989374 q^{82} +12.5627 q^{83} -11.2333 q^{85} -0.565072 q^{86} -7.17408 q^{88} -17.6526 q^{89} -4.68194 q^{91} +17.4237 q^{92} -3.03386 q^{94} +12.8178 q^{95} -0.133391 q^{97} -0.298113 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) $ 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8	$ 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8} - 12 q^{10} - 11 q^{11} + 18 q^{13} + 5 q^{14} + 25 q^{16} + 5 q^{17} - 11 q^{19} + q^{20} + q^{22} - 13 q^{23} + 33 q^{25} - 8 q^{26} - 19 q^{28} - 24 q^{29} - 42 q^{31} - 42 q^{32} + 9 q^{34} - q^{35} + 40 q^{37} - 38 q^{38} - 61 q^{40} - 9 q^{41} + 7 q^{43} - 3 q^{44} + 24 q^{46} - 31 q^{47} + 16 q^{49} - 6 q^{50} + 52 q^{52} - 66 q^{53} - 36 q^{55} + 6 q^{56} + 19 q^{58} + 7 q^{59} + 6 q^{61} - 52 q^{62} + 10 q^{64} - 51 q^{65} + 16 q^{67} - 14 q^{68} + 12 q^{70} - 46 q^{71} + 39 q^{73} - 72 q^{74} + 24 q^{76} + 11 q^{77} + 4 q^{79} + 2 q^{80} - 18 q^{82} - 15 q^{83} - 4 q^{85} - 14 q^{86} + 58 q^{88} + q^{89} - 18 q^{91} - 26 q^{92} + 5 q^{94} - 44 q^{95} + 41 q^{97} - 5 q^{98}+O(q^{100}) $ 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8 - 12 * q^10 - 11 * q^11 + 18 * q^13 + 5 * q^14 + 25 * q^16 + 5 * q^17 - 11 * q^19 + q^20 + q^22 - 13 * q^23 + 33 * q^25 - 8 * q^26 - 19 * q^28 - 24 * q^29 - 42 * q^31 - 42 * q^32 + 9 * q^34 - q^35 + 40 * q^37 - 38 * q^38 - 61 * q^40 - 9 * q^41 + 7 * q^43 - 3 * q^44 + 24 * q^46 - 31 * q^47 + 16 * q^49 - 6 * q^50 + 52 * q^52 - 66 * q^53 - 36 * q^55 + 6 * q^56 + 19 * q^58 + 7 * q^59 + 6 * q^61 - 52 * q^62 + 10 * q^64 - 51 * q^65 + 16 * q^67 - 14 * q^68 + 12 * q^70 - 46 * q^71 + 39 * q^73 - 72 * q^74 + 24 * q^76 + 11 * q^77 + 4 * q^79 + 2 * q^80 - 18 * q^82 - 15 * q^83 - 4 * q^85 - 14 * q^86 + 58 * q^88 + q^89 - 18 * q^91 - 26 * q^92 + 5 * q^94 - 44 * q^95 + 41 * q^97 - 5 * 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 $

$2$	−0.298113	−0.210797	−0.105399	−	0.994430i	$-0.533612\pi$
$2$	−0.298113	−0.210797	−0.105399	+	0.994430i	$0.533612\pi$
$3$	0	0
$3$	0	0
$4$	−1.91113	−0.955564
$5$	−2.54722	−1.13915	−0.569577	−	0.821938i	$-0.692893\pi$
$5$	−2.54722	−1.13915	−0.569577	+	0.821938i	$0.692893\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.16596	0.412228
$9$	0	0
$10$	0.759360	0.240131
$11$	−6.15296	−1.85519	−0.927593	−	0.373592i	$-0.878126\pi$
$11$	−6.15296	−1.85519	−0.927593	+	0.373592i	$0.878126\pi$
$12$	0	0
$13$	4.68194	1.29854	0.649269	−	0.760559i	$-0.275075\pi$
$13$	4.68194	1.29854	0.649269	+	0.760559i	$0.275075\pi$
$14$	0.298113	0.0796740
$15$	0	0
$16$	3.47467	0.868668
$17$	4.41002	1.06959	0.534794	−	0.844982i	$-0.320389\pi$
$17$	4.41002	1.06959	0.534794	+	0.844982i	$0.320389\pi$
$18$	0	0
$19$	−5.03205	−1.15443	−0.577215	−	0.816592i	$-0.695861\pi$
$19$	−5.03205	−1.15443	−0.577215	+	0.816592i	$0.695861\pi$
$20$	4.86807	1.08853
$21$	0	0
$22$	1.83427	0.391069
$23$	−9.11699	−1.90102	−0.950512	−	0.310688i	$-0.899441\pi$
$23$	−9.11699	−1.90102	−0.950512	+	0.310688i	$0.899441\pi$
$24$	0	0
$25$	1.48835	0.297670
$26$	−1.39575	−0.273728
$27$	0	0
$28$	1.91113	0.361169
$29$	4.28790	0.796243	0.398121	−	0.917333i	$-0.369662\pi$
$29$	4.28790	0.796243	0.398121	+	0.917333i	$0.369662\pi$
$30$	0	0
$31$	0.0487464	0.00875512	0.00437756	−	0.999990i	$-0.498607\pi$
$31$	0.0487464	0.00875512	0.00437756	+	0.999990i	$0.498607\pi$
$32$	−3.36776	−0.595341
$33$	0	0
$34$	−1.31468	−0.225466
$35$	2.54722	0.430560
$36$	0	0
$37$	3.02081	0.496619	0.248309	−	0.968681i	$-0.420125\pi$
$37$	3.02081	0.496619	0.248309	+	0.968681i	$0.420125\pi$
$38$	1.50012	0.243351
$39$	0	0
$40$	−2.96995	−0.469591
$41$	3.31879	0.518309	0.259154	−	0.965836i	$-0.416556\pi$
$41$	3.31879	0.518309	0.259154	+	0.965836i	$0.416556\pi$
$42$	0	0
$43$	1.89550	0.289061	0.144531	−	0.989500i	$-0.453833\pi$
$43$	1.89550	0.289061	0.144531	+	0.989500i	$0.453833\pi$
$44$	11.7591	1.77275
$45$	0	0
$46$	2.71789	0.400731
$47$	10.1769	1.48445	0.742227	−	0.670149i	$-0.233770\pi$
$47$	10.1769	1.48445	0.742227	+	0.670149i	$0.233770\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.443697	−0.0627482
$51$	0	0
$52$	−8.94780	−1.24084
$53$	−6.07411	−0.834343	−0.417171	−	0.908828i	$-0.636979\pi$
$53$	−6.07411	−0.834343	−0.417171	+	0.908828i	$0.636979\pi$
$54$	0	0
$55$	15.6730	2.11334
$56$	−1.16596	−0.155808
$57$	0	0
$58$	−1.27828	−0.167846
$59$	9.37602	1.22065	0.610327	−	0.792150i	$-0.291038\pi$
$59$	9.37602	1.22065	0.610327	+	0.792150i	$0.291038\pi$
$60$	0	0
$61$	14.3700	1.83989	0.919943	−	0.392052i	$-0.128235\pi$
$61$	14.3700	1.83989	0.919943	+	0.392052i	$0.128235\pi$
$62$	−0.0145319	−0.00184556
$63$	0	0
$64$	−5.94537	−0.743171
$65$	−11.9260	−1.47923
$66$	0	0
$67$	5.59222	0.683199	0.341599	−	0.939846i	$-0.389031\pi$
$67$	5.59222	0.683199	0.341599	+	0.939846i	$0.389031\pi$
$68$	−8.42813	−1.02206
$69$	0	0
$70$	−0.759360	−0.0907609
$71$	−7.54064	−0.894909	−0.447455	−	0.894307i	$-0.647669\pi$
$71$	−7.54064	−0.894909	−0.447455	+	0.894307i	$0.647669\pi$
$72$	0	0
$73$	5.40772	0.632926	0.316463	−	0.948605i	$-0.397505\pi$
$73$	5.40772	0.632926	0.316463	+	0.948605i	$0.397505\pi$
$74$	−0.900543	−0.104686
$75$	0	0
$76$	9.61689	1.10313
$77$	6.15296	0.701195
$78$	0	0
$79$	−8.18983	−0.921427	−0.460714	−	0.887549i	$-0.652407\pi$
$79$	−8.18983	−0.921427	−0.460714	+	0.887549i	$0.652407\pi$
$80$	−8.85077	−0.989546
$81$	0	0
$82$	−0.989374	−0.109258
$83$	12.5627	1.37894	0.689470	−	0.724314i	$-0.257843\pi$
$83$	12.5627	1.37894	0.689470	+	0.724314i	$0.257843\pi$
$84$	0	0
$85$	−11.2333	−1.21842
$86$	−0.565072	−0.0609333
$87$	0	0
$88$	−7.17408	−0.764760
$89$	−17.6526	−1.87118	−0.935588	−	0.353093i	$-0.885130\pi$
$89$	−17.6526	−1.87118	−0.935588	+	0.353093i	$0.885130\pi$
$90$	0	0
$91$	−4.68194	−0.490801
$92$	17.4237	1.81655
$93$	0	0
$94$	−3.03386	−0.312919
$95$	12.8178	1.31507
$96$	0	0
$97$	−0.133391	−0.0135438	−0.00677188	−	0.999977i	$-0.502156\pi$
$97$	−0.133391	−0.0135438	−0.00677188	+	0.999977i	$0.502156\pi$
$98$	−0.298113	−0.0301139
$99$	0	0
$100$	−2.84443	−0.284443
$101$	−17.4204	−1.73340	−0.866699	−	0.498831i	$-0.833763\pi$
$101$	−17.4204	−1.73340	−0.866699	+	0.498831i	$0.833763\pi$
$102$	0	0
$103$	5.99870	0.591069	0.295534	−	0.955332i	$-0.404502\pi$
$103$	5.99870	0.591069	0.295534	+	0.955332i	$0.404502\pi$
$104$	5.45894	0.535293
$105$	0	0
$106$	1.81077	0.175877
$107$	−16.9503	−1.63865	−0.819323	−	0.573332i	$-0.805650\pi$
$107$	−16.9503	−1.63865	−0.819323	+	0.573332i	$0.805650\pi$
$108$	0	0
$109$	−1.13699	−0.108904	−0.0544522	−	0.998516i	$-0.517341\pi$
$109$	−1.13699	−0.108904	−0.0544522	+	0.998516i	$0.517341\pi$
$110$	−4.67231	−0.445487
$111$	0	0
$112$	−3.47467	−0.328326
$113$	7.37766	0.694032	0.347016	−	0.937859i	$-0.387195\pi$
$113$	7.37766	0.694032	0.347016	+	0.937859i	$0.387195\pi$
$114$	0	0
$115$	23.2230	2.16556
$116$	−8.19472	−0.760861
$117$	0	0
$118$	−2.79511	−0.257311
$119$	−4.41002	−0.404266
$120$	0	0
$121$	26.8589	2.44172
$122$	−4.28387	−0.387843
$123$	0	0
$124$	−0.0931607	−0.00836608
$125$	8.94496	0.800061
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	8.50790	0.752000
$129$	0	0
$130$	3.55528	0.311819
$131$	0.589991	0.0515478	0.0257739	−	0.999668i	$-0.491795\pi$
$131$	0.589991	0.0515478	0.0257739	+	0.999668i	$0.491795\pi$
$132$	0	0
$133$	5.03205	0.436334
$134$	−1.66711	−0.144017
$135$	0	0
$136$	5.14190	0.440914
$137$	6.88031	0.587824	0.293912	−	0.955832i	$-0.405043\pi$
$137$	6.88031	0.587824	0.293912	+	0.955832i	$0.405043\pi$
$138$	0	0
$139$	23.2136	1.96895	0.984476	−	0.175521i	$-0.0561610\pi$
$139$	23.2136	1.96895	0.984476	+	0.175521i	$0.0561610\pi$
$140$	−4.86807	−0.411427
$141$	0	0
$142$	2.24796	0.188645
$143$	−28.8078	−2.40903
$144$	0	0
$145$	−10.9222	−0.907042
$146$	−1.61211	−0.133419
$147$	0	0
$148$	−5.77316	−0.474551
$149$	5.01259	0.410647	0.205324	−	0.978694i	$-0.434175\pi$
$149$	5.01259	0.410647	0.205324	+	0.978694i	$0.434175\pi$
$150$	0	0
$151$	10.9460	0.890769	0.445384	−	0.895339i	$-0.353067\pi$
$151$	10.9460	0.890769	0.445384	+	0.895339i	$0.353067\pi$
$152$	−5.86715	−0.475889
$153$	0	0
$154$	−1.83427	−0.147810
$155$	−0.124168	−0.00997342
$156$	0	0
$157$	4.71794	0.376532	0.188266	−	0.982118i	$-0.439713\pi$
$157$	4.71794	0.376532	0.188266	+	0.982118i	$0.439713\pi$
$158$	2.44149	0.194235
$159$	0	0
$160$	8.57843	0.678185
$161$	9.11699	0.718520
$162$	0	0
$163$	−2.39560	−0.187638	−0.0938188	−	0.995589i	$-0.529907\pi$
$163$	−2.39560	−0.187638	−0.0938188	+	0.995589i	$0.529907\pi$
$164$	−6.34264	−0.495277
$165$	0	0
$166$	−3.74511	−0.290677
$167$	6.18252	0.478418	0.239209	−	0.970968i	$-0.423112\pi$
$167$	6.18252	0.478418	0.239209	+	0.970968i	$0.423112\pi$
$168$	0	0
$169$	8.92059	0.686199
$170$	3.34880	0.256841
$171$	0	0
$172$	−3.62254	−0.276216
$173$	16.3023	1.23944	0.619719	−	0.784824i	$-0.287247\pi$
$173$	16.3023	1.23944	0.619719	+	0.784824i	$0.287247\pi$
$174$	0	0
$175$	−1.48835	−0.112509
$176$	−21.3795	−1.61154
$177$	0	0
$178$	5.26248	0.394439
$179$	−22.0264	−1.64633	−0.823166	−	0.567801i	$-0.807794\pi$
$179$	−22.0264	−1.64633	−0.823166	+	0.567801i	$0.807794\pi$
$180$	0	0
$181$	7.90350	0.587463	0.293731	−	0.955888i	$-0.405103\pi$
$181$	7.90350	0.587463	0.293731	+	0.955888i	$0.405103\pi$
$182$	1.39575	0.103460
$183$	0	0
$184$	−10.6300	−0.783655
$185$	−7.69469	−0.565725
$186$	0	0
$187$	−27.1347	−1.98429
$188$	−19.4494	−1.41849
$189$	0	0
$190$	−3.82113	−0.277214
$191$	18.4005	1.33141	0.665707	−	0.746213i	$-0.268130\pi$
$191$	18.4005	1.33141	0.665707	+	0.746213i	$0.268130\pi$
$192$	0	0
$193$	6.99017	0.503163	0.251582	−	0.967836i	$-0.419049\pi$
$193$	6.99017	0.503163	0.251582	+	0.967836i	$0.419049\pi$
$194$	0.0397654	0.00285499
$195$	0	0
$196$	−1.91113	−0.136509
$197$	−19.1489	−1.36430	−0.682152	−	0.731210i	$-0.738956\pi$
$197$	−19.1489	−1.36430	−0.682152	+	0.731210i	$0.738956\pi$
$198$	0	0
$199$	11.7831	0.835280	0.417640	−	0.908613i	$-0.362857\pi$
$199$	11.7831	0.835280	0.417640	+	0.908613i	$0.362857\pi$
$200$	1.73535	0.122708
$201$	0	0
$202$	5.19325	0.365396
$203$	−4.28790	−0.300951
$204$	0	0
$205$	−8.45371	−0.590433
$206$	−1.78829	−0.124596
$207$	0	0
$208$	16.2682	1.12800
$209$	30.9620	2.14168
$210$	0	0
$211$	−25.5442	−1.75854	−0.879268	−	0.476327i	$-0.841968\pi$
$211$	−25.5442	−1.75854	−0.879268	+	0.476327i	$0.841968\pi$
$212$	11.6084	0.797268
$213$	0	0
$214$	5.05309	0.345422
$215$	−4.82826	−0.329285
$216$	0	0
$217$	−0.0487464	−0.00330912
$218$	0.338953	0.0229568
$219$	0	0
$220$	−29.9531	−2.01943
$221$	20.6475	1.38890
$222$	0	0
$223$	−17.0666	−1.14286	−0.571432	−	0.820650i	$-0.693612\pi$
$223$	−17.0666	−1.14286	−0.571432	+	0.820650i	$0.693612\pi$
$224$	3.36776	0.225018
$225$	0	0
$226$	−2.19937	−0.146300
$227$	−19.1172	−1.26885	−0.634427	−	0.772982i	$-0.718764\pi$
$227$	−19.1172	−1.26885	−0.634427	+	0.772982i	$0.718764\pi$
$228$	0	0
$229$	−17.4537	−1.15337	−0.576686	−	0.816966i	$-0.695654\pi$
$229$	−17.4537	−1.15337	−0.576686	+	0.816966i	$0.695654\pi$
$230$	−6.92308	−0.456494
$231$	0	0
$232$	4.99950	0.328233
$233$	−18.3373	−1.20132	−0.600659	−	0.799506i	$-0.705095\pi$
$233$	−18.3373	−1.20132	−0.600659	+	0.799506i	$0.705095\pi$
$234$	0	0
$235$	−25.9229	−1.69102
$236$	−17.9188	−1.16641
$237$	0	0
$238$	1.31468	0.0852183
$239$	−12.5721	−0.813222	−0.406611	−	0.913601i	$-0.633290\pi$
$239$	−12.5721	−0.813222	−0.406611	+	0.913601i	$0.633290\pi$
$240$	0	0
$241$	1.76945	0.113980	0.0569900	−	0.998375i	$-0.481850\pi$
$241$	1.76945	0.113980	0.0569900	+	0.998375i	$0.481850\pi$
$242$	−8.00697	−0.514708
$243$	0	0
$244$	−27.4629	−1.75813
$245$	−2.54722	−0.162736
$246$	0	0
$247$	−23.5598	−1.49907
$248$	0.0568362	0.00360910
$249$	0	0
$250$	−2.66660	−0.168651
$251$	30.8131	1.94491	0.972453	−	0.233099i	$-0.0748865\pi$
$251$	30.8131	1.94491	0.972453	+	0.233099i	$0.0748865\pi$
$252$	0	0
$253$	56.0965	3.52675
$254$	−0.298113	−0.0187052
$255$	0	0
$256$	9.35443	0.584652
$257$	−13.9647	−0.871092	−0.435546	−	0.900166i	$-0.643445\pi$
$257$	−13.9647	−0.871092	−0.435546	+	0.900166i	$0.643445\pi$
$258$	0	0
$259$	−3.02081	−0.187704
$260$	22.7920	1.41350
$261$	0	0
$262$	−0.175884	−0.0108661
$263$	−2.37995	−0.146754	−0.0733770	−	0.997304i	$-0.523378\pi$
$263$	−2.37995	−0.146754	−0.0733770	+	0.997304i	$0.523378\pi$
$264$	0	0
$265$	15.4721	0.950445
$266$	−1.50012	−0.0919781
$267$	0	0
$268$	−10.6875	−0.652840
$269$	−6.90652	−0.421098	−0.210549	−	0.977583i	$-0.567525\pi$
$269$	−6.90652	−0.421098	−0.210549	+	0.977583i	$0.567525\pi$
$270$	0	0
$271$	−15.2627	−0.927141	−0.463570	−	0.886060i	$-0.653432\pi$
$271$	−15.2627	−0.927141	−0.463570	+	0.886060i	$0.653432\pi$
$272$	15.3234	0.929117
$273$	0	0
$274$	−2.05111	−0.123912
$275$	−9.15777	−0.552234
$276$	0	0
$277$	−20.0356	−1.20382	−0.601910	−	0.798564i	$-0.705593\pi$
$277$	−20.0356	−1.20382	−0.601910	+	0.798564i	$0.705593\pi$
$278$	−6.92027	−0.415050
$279$	0	0
$280$	2.96995	0.177489
$281$	−19.7110	−1.17586	−0.587929	−	0.808912i	$-0.700057\pi$
$281$	−19.7110	−1.17586	−0.587929	+	0.808912i	$0.700057\pi$
$282$	0	0
$283$	−16.3259	−0.970473	−0.485236	−	0.874383i	$-0.661266\pi$
$283$	−16.3259	−0.970473	−0.485236	+	0.874383i	$0.661266\pi$
$284$	14.4111	0.855143
$285$	0	0
$286$	8.58797	0.507817
$287$	−3.31879	−0.195902
$288$	0	0
$289$	2.44832	0.144019
$290$	3.25606	0.191202
$291$	0	0
$292$	−10.3349	−0.604802
$293$	7.46323	0.436006	0.218003	−	0.975948i	$-0.430046\pi$
$293$	7.46323	0.436006	0.218003	+	0.975948i	$0.430046\pi$
$294$	0	0
$295$	−23.8828	−1.39051
$296$	3.52214	0.204720
$297$	0	0
$298$	−1.49432	−0.0865634
$299$	−42.6852	−2.46855
$300$	0	0
$301$	−1.89550	−0.109255
$302$	−3.26313	−0.187772
$303$	0	0
$304$	−17.4847	−1.00282
$305$	−36.6035	−2.09591
$306$	0	0
$307$	10.3378	0.590010	0.295005	−	0.955496i	$-0.404679\pi$
$307$	10.3378	0.590010	0.295005	+	0.955496i	$0.404679\pi$
$308$	−11.7591	−0.670037
$309$	0	0
$310$	0.0370161	0.00210237
$311$	−18.9888	−1.07676	−0.538379	−	0.842703i	$-0.680963\pi$
$311$	−18.9888	−1.07676	−0.538379	+	0.842703i	$0.680963\pi$
$312$	0	0
$313$	21.6377	1.22304	0.611519	−	0.791230i	$-0.290559\pi$
$313$	21.6377	1.22304	0.611519	+	0.791230i	$0.290559\pi$
$314$	−1.40648	−0.0793720
$315$	0	0
$316$	15.6518	0.880483
$317$	−5.64090	−0.316824	−0.158412	−	0.987373i	$-0.550637\pi$
$317$	−5.64090	−0.316824	−0.158412	+	0.987373i	$0.550637\pi$
$318$	0	0
$319$	−26.3832	−1.47718
$320$	15.1442	0.846586
$321$	0	0
$322$	−2.71789	−0.151462
$323$	−22.1915	−1.23477
$324$	0	0
$325$	6.96838	0.386536
$326$	0.714158	0.0395535
$327$	0	0
$328$	3.86957	0.213661
$329$	−10.1769	−0.561071
$330$	0	0
$331$	−11.1486	−0.612781	−0.306391	−	0.951906i	$-0.599121\pi$
$331$	−11.1486	−0.612781	−0.306391	+	0.951906i	$0.599121\pi$
$332$	−24.0090	−1.31767
$333$	0	0
$334$	−1.84309	−0.100849
$335$	−14.2446	−0.778268
$336$	0	0
$337$	26.5229	1.44480	0.722398	−	0.691477i	$-0.243040\pi$
$337$	26.5229	1.44480	0.722398	+	0.691477i	$0.243040\pi$
$338$	−2.65934	−0.144649
$339$	0	0
$340$	21.4683	1.16428
$341$	−0.299935	−0.0162424
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	2.21007	0.119159
$345$	0	0
$346$	−4.85991	−0.261270
$347$	−8.02759	−0.430944	−0.215472	−	0.976510i	$-0.569129\pi$
$347$	−8.02759	−0.430944	−0.215472	+	0.976510i	$0.569129\pi$
$348$	0	0
$349$	4.22025	0.225905	0.112952	−	0.993600i	$-0.463969\pi$
$349$	4.22025	0.225905	0.112952	+	0.993600i	$0.463969\pi$
$350$	0.443697	0.0237166
$351$	0	0
$352$	20.7217	1.10447
$353$	−27.0656	−1.44056	−0.720278	−	0.693686i	$-0.755986\pi$
$353$	−27.0656	−1.44056	−0.720278	+	0.693686i	$0.755986\pi$
$354$	0	0
$355$	19.2077	1.01944
$356$	33.7365	1.78803
$357$	0	0
$358$	6.56636	0.347043
$359$	20.6950	1.09224	0.546121	−	0.837706i	$-0.316104\pi$
$359$	20.6950	1.09224	0.546121	+	0.837706i	$0.316104\pi$
$360$	0	0
$361$	6.32150	0.332711
$362$	−2.35613	−0.123836
$363$	0	0
$364$	8.94780	0.468992
$365$	−13.7747	−0.721000
$366$	0	0
$367$	−0.789317	−0.0412020	−0.0206010	−	0.999788i	$-0.506558\pi$
$367$	−0.789317	−0.0412020	−0.0206010	+	0.999788i	$0.506558\pi$
$368$	−31.6785	−1.65136
$369$	0	0
$370$	2.29388	0.119253
$371$	6.07411	0.315352
$372$	0	0
$373$	−6.27308	−0.324808	−0.162404	−	0.986724i	$-0.551925\pi$
$373$	−6.27308	−0.324808	−0.162404	+	0.986724i	$0.551925\pi$
$374$	8.08919	0.418282
$375$	0	0
$376$	11.8658	0.611934
$377$	20.0757	1.03395
$378$	0	0
$379$	−25.4550	−1.30754	−0.653768	−	0.756695i	$-0.726813\pi$
$379$	−25.4550	−1.30754	−0.653768	+	0.756695i	$0.726813\pi$
$380$	−24.4964	−1.25664
$381$	0	0
$382$	−5.48542	−0.280659
$383$	1.91780	0.0979952	0.0489976	−	0.998799i	$-0.484397\pi$
$383$	1.91780	0.0979952	0.0489976	+	0.998799i	$0.484397\pi$
$384$	0	0
$385$	−15.6730	−0.798768
$386$	−2.08386	−0.106066
$387$	0	0
$388$	0.254926	0.0129419
$389$	30.4853	1.54567	0.772833	−	0.634610i	$-0.218839\pi$
$389$	30.4853	1.54567	0.772833	+	0.634610i	$0.218839\pi$
$390$	0	0
$391$	−40.2062	−2.03331
$392$	1.16596	0.0588897
$393$	0	0
$394$	5.70853	0.287592
$395$	20.8613	1.04965
$396$	0	0
$397$	−1.00903	−0.0506415	−0.0253208	−	0.999679i	$-0.508061\pi$
$397$	−1.00903	−0.0506415	−0.0253208	+	0.999679i	$0.508061\pi$
$398$	−3.51268	−0.176075
$399$	0	0
$400$	5.17153	0.258577
$401$	−18.9677	−0.947203	−0.473602	−	0.880739i	$-0.657046\pi$
$401$	−18.9677	−0.947203	−0.473602	+	0.880739i	$0.657046\pi$
$402$	0	0
$403$	0.228228	0.0113688
$404$	33.2927	1.65637
$405$	0	0
$406$	1.27828	0.0634398
$407$	−18.5869	−0.921320
$408$	0	0
$409$	21.3336	1.05488	0.527439	−	0.849593i	$-0.323152\pi$
$409$	21.3336	1.05488	0.527439	+	0.849593i	$0.323152\pi$
$410$	2.52016	0.124462
$411$	0	0
$412$	−11.4643	−0.564805
$413$	−9.37602	−0.461364
$414$	0	0
$415$	−32.0001	−1.57083
$416$	−15.7676	−0.773072
$417$	0	0
$418$	−9.23015	−0.451462
$419$	−1.91650	−0.0936270	−0.0468135	−	0.998904i	$-0.514907\pi$
$419$	−1.91650	−0.0936270	−0.0468135	+	0.998904i	$0.514907\pi$
$420$	0	0
$421$	−11.1631	−0.544055	−0.272028	−	0.962289i	$-0.587694\pi$
$421$	−11.1631	−0.544055	−0.272028	+	0.962289i	$0.587694\pi$
$422$	7.61505	0.370695
$423$	0	0
$424$	−7.08215	−0.343940
$425$	6.56367	0.318385
$426$	0	0
$427$	−14.3700	−0.695412
$428$	32.3942	1.56583
$429$	0	0
$430$	1.43937	0.0694124
$431$	−25.2766	−1.21753	−0.608766	−	0.793350i	$-0.708335\pi$
$431$	−25.2766	−1.21753	−0.608766	+	0.793350i	$0.708335\pi$
$432$	0	0
$433$	−24.4814	−1.17650	−0.588251	−	0.808678i	$-0.700183\pi$
$433$	−24.4814	−1.17650	−0.588251	+	0.808678i	$0.700183\pi$
$434$	0.0145319	0.000697555 0
$435$	0	0
$436$	2.17294	0.104065
$437$	45.8771	2.19460
$438$	0	0
$439$	−33.1662	−1.58294	−0.791469	−	0.611209i	$-0.790683\pi$
$439$	−33.1662	−1.58294	−0.791469	+	0.611209i	$0.790683\pi$
$440$	18.2740	0.871179
$441$	0	0
$442$	−6.15528	−0.292777
$443$	−22.9199	−1.08896	−0.544478	−	0.838775i	$-0.683272\pi$
$443$	−22.9199	−1.08896	−0.544478	+	0.838775i	$0.683272\pi$
$444$	0	0
$445$	44.9652	2.13156
$446$	5.08777	0.240913
$447$	0	0
$448$	5.94537	0.280892
$449$	10.8422	0.511676	0.255838	−	0.966720i	$-0.417649\pi$
$449$	10.8422	0.511676	0.255838	+	0.966720i	$0.417649\pi$
$450$	0	0
$451$	−20.4204	−0.961559
$452$	−14.0997	−0.663192
$453$	0	0
$454$	5.69909	0.267471
$455$	11.9260	0.559098
$456$	0	0
$457$	20.5538	0.961466	0.480733	−	0.876867i	$-0.340371\pi$
$457$	20.5538	0.961466	0.480733	+	0.876867i	$0.340371\pi$
$458$	5.20317	0.243128
$459$	0	0
$460$	−44.3822	−2.06933
$461$	30.0440	1.39929	0.699645	−	0.714491i	$-0.253342\pi$
$461$	30.0440	1.39929	0.699645	+	0.714491i	$0.253342\pi$
$462$	0	0
$463$	−25.9689	−1.20688	−0.603439	−	0.797409i	$-0.706203\pi$
$463$	−25.9689	−1.20688	−0.603439	+	0.797409i	$0.706203\pi$
$464$	14.8990	0.691670
$465$	0	0
$466$	5.46659	0.253235
$467$	28.5728	1.32219	0.661096	−	0.750301i	$-0.270091\pi$
$467$	28.5728	1.32219	0.661096	+	0.750301i	$0.270091\pi$
$468$	0	0
$469$	−5.59222	−0.258225
$470$	7.72793	0.356463
$471$	0	0
$472$	10.9320	0.503188
$473$	−11.6629	−0.536262
$474$	0	0
$475$	−7.48946	−0.343640
$476$	8.42813	0.386302
$477$	0	0
$478$	3.74791	0.171425
$479$	29.2436	1.33618	0.668088	−	0.744082i	$-0.267113\pi$
$479$	29.2436	1.33618	0.668088	+	0.744082i	$0.267113\pi$
$480$	0	0
$481$	14.1433	0.644878
$482$	−0.527494	−0.0240267
$483$	0	0
$484$	−51.3308	−2.33322
$485$	0.339776	0.0154284
$486$	0	0
$487$	−18.8801	−0.855537	−0.427769	−	0.903888i	$-0.640700\pi$
$487$	−18.8801	−0.855537	−0.427769	+	0.903888i	$0.640700\pi$
$488$	16.7548	0.758453
$489$	0	0
$490$	0.759360	0.0343044
$491$	−24.1789	−1.09118	−0.545589	−	0.838053i	$-0.683694\pi$
$491$	−24.1789	−1.09118	−0.545589	+	0.838053i	$0.683694\pi$
$492$	0	0
$493$	18.9097	0.851652
$494$	7.02346	0.316000
$495$	0	0
$496$	0.169378	0.00760529
$497$	7.54064	0.338244
$498$	0	0
$499$	−6.67434	−0.298784	−0.149392	−	0.988778i	$-0.547732\pi$
$499$	−6.67434	−0.298784	−0.149392	+	0.988778i	$0.547732\pi$
$500$	−17.0950	−0.764510
$501$	0	0
$502$	−9.18578	−0.409981
$503$	−8.56876	−0.382062	−0.191031	−	0.981584i	$-0.561183\pi$
$503$	−8.56876	−0.382062	−0.191031	+	0.981584i	$0.561183\pi$
$504$	0	0
$505$	44.3738	1.97461
$506$	−16.7231	−0.743431
$507$	0	0
$508$	−1.91113	−0.0847926
$509$	−22.5337	−0.998791	−0.499395	−	0.866374i	$-0.666444\pi$
$509$	−22.5337	−0.998791	−0.499395	+	0.866374i	$0.666444\pi$
$510$	0	0
$511$	−5.40772	−0.239224
$512$	−19.8045	−0.875243
$513$	0	0
$514$	4.16305	0.183624
$515$	−15.2800	−0.673318
$516$	0	0
$517$	−62.6181	−2.75394
$518$	0.900543	0.0395676
$519$	0	0
$520$	−13.9052	−0.609781
$521$	32.0705	1.40503	0.702516	−	0.711668i	$-0.252060\pi$
$521$	32.0705	1.40503	0.702516	+	0.711668i	$0.252060\pi$
$522$	0	0
$523$	−36.3672	−1.59023	−0.795113	−	0.606461i	$-0.792589\pi$
$523$	−36.3672	−1.59023	−0.795113	+	0.606461i	$0.792589\pi$
$524$	−1.12755	−0.0492572
$525$	0	0
$526$	0.709493	0.0309354
$527$	0.214973	0.00936437
$528$	0	0
$529$	60.1195	2.61389
$530$	−4.61243	−0.200351
$531$	0	0
$532$	−9.61689	−0.416945
$533$	15.5384	0.673043
$534$	0	0
$535$	43.1762	1.86667
$536$	6.52029	0.281634
$537$	0	0
$538$	2.05892	0.0887664
$539$	−6.15296	−0.265027
$540$	0	0
$541$	−35.7283	−1.53608	−0.768039	−	0.640403i	$-0.778767\pi$
$541$	−35.7283	−1.53608	−0.768039	+	0.640403i	$0.778767\pi$
$542$	4.54999	0.195439
$543$	0	0
$544$	−14.8519	−0.636770
$545$	2.89618	0.124059
$546$	0	0
$547$	6.53314	0.279337	0.139668	−	0.990198i	$-0.455396\pi$
$547$	6.53314	0.279337	0.139668	+	0.990198i	$0.455396\pi$
$548$	−13.1492	−0.561704
$549$	0	0
$550$	2.73005	0.116410
$551$	−21.5769	−0.919207
$552$	0	0
$553$	8.18983	0.348267
$554$	5.97285	0.253762
$555$	0	0
$556$	−44.3642	−1.88146
$557$	−22.4418	−0.950891	−0.475446	−	0.879745i	$-0.657713\pi$
$557$	−22.4418	−0.950891	−0.475446	+	0.879745i	$0.657713\pi$
$558$	0	0
$559$	8.87462	0.375356
$560$	8.85077	0.374013
$561$	0	0
$562$	5.87609	0.247868
$563$	−1.91786	−0.0808282	−0.0404141	−	0.999183i	$-0.512868\pi$
$563$	−1.91786	−0.0808282	−0.0404141	+	0.999183i	$0.512868\pi$
$564$	0	0
$565$	−18.7926	−0.790609
$566$	4.86695	0.204573
$567$	0	0
$568$	−8.79206	−0.368907
$569$	5.35626	0.224546	0.112273	−	0.993677i	$-0.464187\pi$
$569$	5.35626	0.224546	0.112273	+	0.993677i	$0.464187\pi$
$570$	0	0
$571$	9.68343	0.405239	0.202619	−	0.979258i	$-0.435055\pi$
$571$	9.68343	0.405239	0.202619	+	0.979258i	$0.435055\pi$
$572$	55.0554	2.30198
$573$	0	0
$574$	0.989374	0.0412957
$575$	−13.5693	−0.565879
$576$	0	0
$577$	−25.6603	−1.06825	−0.534127	−	0.845404i	$-0.679360\pi$
$577$	−25.6603	−1.06825	−0.534127	+	0.845404i	$0.679360\pi$
$578$	−0.729874	−0.0303588
$579$	0	0
$580$	20.8738	0.866737
$581$	−12.5627	−0.521191
$582$	0	0
$583$	37.3737	1.54786
$584$	6.30517	0.260910
$585$	0	0
$586$	−2.22488	−0.0919091
$587$	15.0132	0.619663	0.309832	−	0.950791i	$-0.399727\pi$
$587$	15.0132	0.619663	0.309832	+	0.950791i	$0.399727\pi$
$588$	0	0
$589$	−0.245294	−0.0101072
$590$	7.11977	0.293116
$591$	0	0
$592$	10.4963	0.431397
$593$	−25.4740	−1.04609	−0.523046	−	0.852305i	$-0.675204\pi$
$593$	−25.4740	−1.04609	−0.523046	+	0.852305i	$0.675204\pi$
$594$	0	0
$595$	11.2333	0.460521
$596$	−9.57970	−0.392400
$597$	0	0
$598$	12.7250	0.520364
$599$	43.7657	1.78822	0.894109	−	0.447850i	$-0.147810\pi$
$599$	43.7657	1.78822	0.894109	+	0.447850i	$0.147810\pi$
$600$	0	0
$601$	31.5090	1.28528	0.642640	−	0.766168i	$-0.277839\pi$
$601$	31.5090	1.28528	0.642640	+	0.766168i	$0.277839\pi$
$602$	0.565072	0.0230306
$603$	0	0
$604$	−20.9191	−0.851187
$605$	−68.4156	−2.78149
$606$	0	0
$607$	25.6191	1.03985	0.519923	−	0.854213i	$-0.325961\pi$
$607$	25.6191	1.03985	0.519923	+	0.854213i	$0.325961\pi$
$608$	16.9467	0.687280
$609$	0	0
$610$	10.9120	0.441813
$611$	47.6477	1.92762
$612$	0	0
$613$	14.1667	0.572188	0.286094	−	0.958201i	$-0.407643\pi$
$613$	14.1667	0.572188	0.286094	+	0.958201i	$0.407643\pi$
$614$	−3.08183	−0.124373
$615$	0	0
$616$	7.17408	0.289052
$617$	−31.0958	−1.25187	−0.625934	−	0.779876i	$-0.715282\pi$
$617$	−31.0958	−1.25187	−0.625934	+	0.779876i	$0.715282\pi$
$618$	0	0
$619$	−1.80860	−0.0726939	−0.0363469	−	0.999339i	$-0.511572\pi$
$619$	−1.80860	−0.0726939	−0.0363469	+	0.999339i	$0.511572\pi$
$620$	0.237301	0.00953024
$621$	0	0
$622$	5.66081	0.226978
$623$	17.6526	0.707238
$624$	0	0
$625$	−30.2266	−1.20906
$626$	−6.45049	−0.257813
$627$	0	0
$628$	−9.01658	−0.359801
$629$	13.3219	0.531177
$630$	0	0
$631$	4.27550	0.170205	0.0851025	−	0.996372i	$-0.472878\pi$
$631$	4.27550	0.170205	0.0851025	+	0.996372i	$0.472878\pi$
$632$	−9.54898	−0.379838
$633$	0	0
$634$	1.68162	0.0667858
$635$	−2.54722	−0.101084
$636$	0	0
$637$	4.68194	0.185505
$638$	7.86518	0.311385
$639$	0	0
$640$	−21.6715	−0.856643
$641$	21.0567	0.831688	0.415844	−	0.909436i	$-0.363486\pi$
$641$	21.0567	0.831688	0.415844	+	0.909436i	$0.363486\pi$
$642$	0	0
$643$	−23.4286	−0.923935	−0.461968	−	0.886897i	$-0.652856\pi$
$643$	−23.4286	−0.923935	−0.461968	+	0.886897i	$0.652856\pi$
$644$	−17.4237	−0.686592
$645$	0	0
$646$	6.61555	0.260285
$647$	30.4852	1.19850	0.599248	−	0.800564i	$-0.295467\pi$
$647$	30.4852	1.19850	0.599248	+	0.800564i	$0.295467\pi$
$648$	0	0
$649$	−57.6903	−2.26454
$650$	−2.07736	−0.0814808
$651$	0	0
$652$	4.57829	0.179300
$653$	11.7448	0.459609	0.229805	−	0.973237i	$-0.426191\pi$
$653$	11.7448	0.459609	0.229805	+	0.973237i	$0.426191\pi$
$654$	0	0
$655$	−1.50284	−0.0587208
$656$	11.5317	0.450238
$657$	0	0
$658$	3.03386	0.118272
$659$	23.1760	0.902809	0.451405	−	0.892319i	$-0.350923\pi$
$659$	23.1760	0.902809	0.451405	+	0.892319i	$0.350923\pi$
$660$	0	0
$661$	−11.9144	−0.463416	−0.231708	−	0.972785i	$-0.574431\pi$
$661$	−11.9144	−0.463416	−0.231708	+	0.972785i	$0.574431\pi$
$662$	3.32353	0.129173
$663$	0	0
$664$	14.6476	0.568438
$665$	−12.8178	−0.497051
$666$	0	0
$667$	−39.0927	−1.51368
$668$	−11.8156	−0.457159
$669$	0	0
$670$	4.24651	0.164057
$671$	−88.4178	−3.41333
$672$	0	0
$673$	−12.2102	−0.470668	−0.235334	−	0.971915i	$-0.575618\pi$
$673$	−12.2102	−0.470668	−0.235334	+	0.971915i	$0.575618\pi$
$674$	−7.90682	−0.304559
$675$	0	0
$676$	−17.0484	−0.655707
$677$	−30.4234	−1.16927	−0.584633	−	0.811298i	$-0.698761\pi$
$677$	−30.4234	−1.16927	−0.584633	+	0.811298i	$0.698761\pi$
$678$	0	0
$679$	0.133391	0.00511906
$680$	−13.0976	−0.502269
$681$	0	0
$682$	0.0894143	0.00342385
$683$	10.7503	0.411349	0.205674	−	0.978620i	$-0.434061\pi$
$683$	10.7503	0.411349	0.205674	+	0.978620i	$0.434061\pi$
$684$	0	0
$685$	−17.5257	−0.669622
$686$	0.298113	0.0113820
$687$	0	0
$688$	6.58624	0.251098
$689$	−28.4386	−1.08343
$690$	0	0
$691$	−24.4202	−0.928987	−0.464493	−	0.885577i	$-0.653764\pi$
$691$	−24.4202	−0.928987	−0.464493	+	0.885577i	$0.653764\pi$
$692$	−31.1557	−1.18436
$693$	0	0
$694$	2.39313	0.0908418
$695$	−59.1302	−2.24294
$696$	0	0
$697$	14.6360	0.554377
$698$	−1.25811	−0.0476201
$699$	0	0
$700$	2.84443	0.107509
$701$	10.6603	0.402636	0.201318	−	0.979526i	$-0.435478\pi$
$701$	10.6603	0.402636	0.201318	+	0.979526i	$0.435478\pi$
$702$	0	0
$703$	−15.2009	−0.573312
$704$	36.5816	1.37872
$705$	0	0
$706$	8.06859	0.303665
$707$	17.4204	0.655163
$708$	0	0
$709$	8.83062	0.331641	0.165820	−	0.986156i	$-0.446973\pi$
$709$	8.83062	0.331641	0.165820	+	0.986156i	$0.446973\pi$
$710$	−5.72606	−0.214895
$711$	0	0
$712$	−20.5822	−0.771351
$713$	−0.444421	−0.0166437
$714$	0	0
$715$	73.3799	2.74425
$716$	42.0953	1.57318
$717$	0	0
$718$	−6.16945	−0.230242
$719$	11.2112	0.418109	0.209054	−	0.977904i	$-0.432961\pi$
$719$	11.2112	0.418109	0.209054	+	0.977904i	$0.432961\pi$
$720$	0	0
$721$	−5.99870	−0.223403
$722$	−1.88452	−0.0701346
$723$	0	0
$724$	−15.1046	−0.561359
$725$	6.38190	0.237018
$726$	0	0
$727$	−30.6785	−1.13780	−0.568901	−	0.822406i	$-0.692631\pi$
$727$	−30.6785	−1.13780	−0.568901	+	0.822406i	$0.692631\pi$
$728$	−5.45894	−0.202322
$729$	0	0
$730$	4.10641	0.151985
$731$	8.35920	0.309176
$732$	0	0
$733$	42.1689	1.55754	0.778772	−	0.627307i	$-0.215843\pi$
$733$	42.1689	1.55754	0.778772	+	0.627307i	$0.215843\pi$
$734$	0.235305	0.00868528
$735$	0	0
$736$	30.7038	1.13176
$737$	−34.4087	−1.26746
$738$	0	0
$739$	−21.9476	−0.807354	−0.403677	−	0.914902i	$-0.632268\pi$
$739$	−21.9476	−0.807354	−0.403677	+	0.914902i	$0.632268\pi$
$740$	14.7055	0.540587
$741$	0	0
$742$	−1.81077	−0.0664754
$743$	−34.0869	−1.25053	−0.625264	−	0.780414i	$-0.715009\pi$
$743$	−34.0869	−1.25053	−0.625264	+	0.780414i	$0.715009\pi$
$744$	0	0
$745$	−12.7682	−0.467790
$746$	1.87008	0.0684686
$747$	0	0
$748$	51.8579	1.89611
$749$	16.9503	0.619350
$750$	0	0
$751$	3.36757	0.122884	0.0614422	−	0.998111i	$-0.480430\pi$
$751$	3.36757	0.122884	0.0614422	+	0.998111i	$0.480430\pi$
$752$	35.3614	1.28950
$753$	0	0
$754$	−5.98482	−0.217954
$755$	−27.8818	−1.01472
$756$	0	0
$757$	4.97069	0.180663	0.0903315	−	0.995912i	$-0.471207\pi$
$757$	4.97069	0.180663	0.0903315	+	0.995912i	$0.471207\pi$
$758$	7.58846	0.275625
$759$	0	0
$760$	14.9449	0.542110
$761$	34.7856	1.26098	0.630489	−	0.776198i	$-0.282854\pi$
$761$	34.7856	1.26098	0.630489	+	0.776198i	$0.282854\pi$
$762$	0	0
$763$	1.13699	0.0411620
$764$	−35.1657	−1.27225
$765$	0	0
$766$	−0.571721	−0.0206571
$767$	43.8980	1.58506
$768$	0	0
$769$	9.55583	0.344592	0.172296	−	0.985045i	$-0.444881\pi$
$769$	9.55583	0.344592	0.172296	+	0.985045i	$0.444881\pi$
$770$	4.67231	0.168378
$771$	0	0
$772$	−13.3591	−0.480805
$773$	−28.8187	−1.03654	−0.518268	−	0.855218i	$-0.673423\pi$
$773$	−28.8187	−1.03654	−0.518268	+	0.855218i	$0.673423\pi$
$774$	0	0
$775$	0.0725518	0.00260614
$776$	−0.155528	−0.00558312
$777$	0	0
$778$	−9.08805	−0.325822
$779$	−16.7003	−0.598351
$780$	0	0
$781$	46.3972	1.66022
$782$	11.9860	0.428617
$783$	0	0
$784$	3.47467	0.124095
$785$	−12.0176	−0.428928
$786$	0	0
$787$	−16.4957	−0.588009	−0.294004	−	0.955804i	$-0.594988\pi$
$787$	−16.4957	−0.588009	−0.294004	+	0.955804i	$0.594988\pi$
$788$	36.5960	1.30368
$789$	0	0
$790$	−6.21902	−0.221263
$791$	−7.37766	−0.262319
$792$	0	0
$793$	67.2794	2.38916
$794$	0.300803	0.0106751
$795$	0	0
$796$	−22.5190	−0.798164
$797$	38.3239	1.35750	0.678752	−	0.734368i	$-0.262521\pi$
$797$	38.3239	1.35750	0.678752	+	0.734368i	$0.262521\pi$
$798$	0	0
$799$	44.8804	1.58775
$800$	−5.01241	−0.177215
$801$	0	0
$802$	5.65452	0.199668
$803$	−33.2735	−1.17420
$804$	0	0
$805$	−23.2230	−0.818504
$806$	−0.0680376	−0.00239652
$807$	0	0
$808$	−20.3115	−0.714555
$809$	−1.14289	−0.0401819	−0.0200910	−	0.999798i	$-0.506396\pi$
$809$	−1.14289	−0.0401819	−0.0200910	+	0.999798i	$0.506396\pi$
$810$	0	0
$811$	3.16944	0.111294	0.0556471	−	0.998450i	$-0.482278\pi$
$811$	3.16944	0.111294	0.0556471	+	0.998450i	$0.482278\pi$
$812$	8.19472	0.287578
$813$	0	0
$814$	5.54100	0.194212
$815$	6.10212	0.213748
$816$	0	0
$817$	−9.53825	−0.333701
$818$	−6.35981	−0.222365
$819$	0	0
$820$	16.1561	0.564197
$821$	14.3760	0.501725	0.250863	−	0.968023i	$-0.419286\pi$
$821$	14.3760	0.501725	0.250863	+	0.968023i	$0.419286\pi$
$822$	0	0
$823$	12.2041	0.425409	0.212705	−	0.977117i	$-0.431773\pi$
$823$	12.2041	0.425409	0.212705	+	0.977117i	$0.431773\pi$
$824$	6.99422	0.243655
$825$	0	0
$826$	2.79511	0.0972543
$827$	−17.7731	−0.618030	−0.309015	−	0.951057i	$-0.599999\pi$
$827$	−17.7731	−0.618030	−0.309015	+	0.951057i	$0.599999\pi$
$828$	0	0
$829$	12.9454	0.449613	0.224807	−	0.974403i	$-0.427825\pi$
$829$	12.9454	0.449613	0.224807	+	0.974403i	$0.427825\pi$
$830$	9.53965	0.331126
$831$	0	0
$832$	−27.8359	−0.965036
$833$	4.41002	0.152798
$834$	0	0
$835$	−15.7483	−0.544991
$836$	−59.1723	−2.04652
$837$	0	0
$838$	0.571332	0.0197363
$839$	−35.7687	−1.23487	−0.617437	−	0.786621i	$-0.711829\pi$
$839$	−35.7687	−1.23487	−0.617437	+	0.786621i	$0.711829\pi$
$840$	0	0
$841$	−10.6139	−0.365998
$842$	3.32786	0.114685
$843$	0	0
$844$	48.8183	1.68039
$845$	−22.7227	−0.781686
$846$	0	0
$847$	−26.8589	−0.922882
$848$	−21.1055	−0.724767
$849$	0	0
$850$	−1.95671	−0.0671147
$851$	−27.5407	−0.944084
$852$	0	0
$853$	−42.0346	−1.43924	−0.719619	−	0.694370i	$-0.755683\pi$
$853$	−42.0346	−1.43924	−0.719619	+	0.694370i	$0.755683\pi$
$854$	4.28387	0.146591
$855$	0	0
$856$	−19.7633	−0.675496
$857$	−39.5497	−1.35099	−0.675496	−	0.737364i	$-0.736070\pi$
$857$	−39.5497	−1.35099	−0.675496	+	0.737364i	$0.736070\pi$
$858$	0	0
$859$	5.41594	0.184789	0.0923947	−	0.995722i	$-0.470548\pi$
$859$	5.41594	0.184789	0.0923947	+	0.995722i	$0.470548\pi$
$860$	9.22743	0.314653
$861$	0	0
$862$	7.53527	0.256652
$863$	−30.5858	−1.04115	−0.520576	−	0.853815i	$-0.674283\pi$
$863$	−30.5858	−1.04115	−0.520576	+	0.853815i	$0.674283\pi$
$864$	0	0
$865$	−41.5255	−1.41191
$866$	7.29822	0.248004
$867$	0	0
$868$	0.0931607	0.00316208
$869$	50.3916	1.70942
$870$	0	0
$871$	26.1825	0.887159
$872$	−1.32569	−0.0448934
$873$	0	0
$874$	−13.6766	−0.462616
$875$	−8.94496	−0.302395
$876$	0	0
$877$	40.7400	1.37569	0.687846	−	0.725857i	$-0.258556\pi$
$877$	40.7400	1.37569	0.687846	+	0.725857i	$0.258556\pi$
$878$	9.88727	0.333679
$879$	0	0
$880$	54.4584	1.83579
$881$	13.0783	0.440620	0.220310	−	0.975430i	$-0.429293\pi$
$881$	13.0783	0.440620	0.220310	+	0.975430i	$0.429293\pi$
$882$	0	0
$883$	−46.1932	−1.55452	−0.777262	−	0.629177i	$-0.783392\pi$
$883$	−46.1932	−1.55452	−0.777262	+	0.629177i	$0.783392\pi$
$884$	−39.4600	−1.32718
$885$	0	0
$886$	6.83270	0.229549
$887$	−23.4042	−0.785838	−0.392919	−	0.919573i	$-0.628535\pi$
$887$	−23.4042	−0.785838	−0.392919	+	0.919573i	$0.628535\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−13.4047	−0.449327
$891$	0	0
$892$	32.6165	1.09208
$893$	−51.2107	−1.71370
$894$	0	0
$895$	56.1062	1.87542
$896$	−8.50790	−0.284229
$897$	0	0
$898$	−3.23220	−0.107860
$899$	0.209020	0.00697119
$900$	0	0
$901$	−26.7870	−0.892403
$902$	6.08758	0.202694
$903$	0	0
$904$	8.60203	0.286099
$905$	−20.1320	−0.669210
$906$	0	0
$907$	−0.601645	−0.0199773	−0.00998864	−	0.999950i	$-0.503180\pi$
$907$	−0.601645	−0.0199773	−0.00998864	+	0.999950i	$0.503180\pi$
$908$	36.5355	1.21247
$909$	0	0
$910$	−3.55528	−0.117856
$911$	55.2607	1.83087	0.915435	−	0.402467i	$-0.131847\pi$
$911$	55.2607	1.83087	0.915435	+	0.402467i	$0.131847\pi$
$912$	0	0
$913$	−77.2980	−2.55819
$914$	−6.12734	−0.202675
$915$	0	0
$916$	33.3563	1.10212
$917$	−0.589991	−0.0194832
$918$	0	0
$919$	40.3608	1.33138	0.665689	−	0.746229i	$-0.268138\pi$
$919$	40.3608	1.33138	0.665689	+	0.746229i	$0.268138\pi$
$920$	27.0770	0.892704
$921$	0	0
$922$	−8.95650	−0.294967
$923$	−35.3048	−1.16207
$924$	0	0
$925$	4.49603	0.147829
$926$	7.74166	0.254407
$927$	0	0
$928$	−14.4406	−0.474036
$929$	17.1233	0.561798	0.280899	−	0.959737i	$-0.409367\pi$
$929$	17.1233	0.561798	0.280899	+	0.959737i	$0.409367\pi$
$930$	0	0
$931$	−5.03205	−0.164919
$932$	35.0450	1.14794
$933$	0	0
$934$	−8.51792	−0.278715
$935$	69.1181	2.26041
$936$	0	0
$937$	−27.9645	−0.913560	−0.456780	−	0.889580i	$-0.650997\pi$
$937$	−27.9645	−0.913560	−0.456780	+	0.889580i	$0.650997\pi$
$938$	1.66711	0.0544331
$939$	0	0
$940$	49.5419	1.61588
$941$	37.0801	1.20878	0.604389	−	0.796689i	$-0.293417\pi$
$941$	37.0801	1.20878	0.604389	+	0.796689i	$0.293417\pi$
$942$	0	0
$943$	−30.2574	−0.985317
$944$	32.5786	1.06034
$945$	0	0
$946$	3.47687	0.113043
$947$	47.2707	1.53609	0.768045	−	0.640396i	$-0.221230\pi$
$947$	47.2707	1.53609	0.768045	+	0.640396i	$0.221230\pi$
$948$	0	0
$949$	25.3187	0.821878
$950$	2.23270	0.0724384
$951$	0	0
$952$	−5.14190	−0.166650
$953$	−1.78295	−0.0577553	−0.0288776	−	0.999583i	$-0.509193\pi$
$953$	−1.78295	−0.0577553	−0.0288776	+	0.999583i	$0.509193\pi$
$954$	0	0
$955$	−46.8702	−1.51668
$956$	24.0269	0.777086
$957$	0	0
$958$	−8.71790	−0.281663
$959$	−6.88031	−0.222177
$960$	0	0
$961$	−30.9976	−0.999923
$962$	−4.21629	−0.135939
$963$	0	0
$964$	−3.38164	−0.108915
$965$	−17.8055	−0.573180
$966$	0	0
$967$	19.3686	0.622851	0.311426	−	0.950271i	$-0.399194\pi$
$967$	19.3686	0.622851	0.311426	+	0.950271i	$0.399194\pi$
$968$	31.3163	1.00654
$969$	0	0
$970$	−0.101291	−0.00325227
$971$	56.6777	1.81888	0.909438	−	0.415840i	$-0.136513\pi$
$971$	56.6777	1.81888	0.909438	+	0.415840i	$0.136513\pi$
$972$	0	0
$973$	−23.2136	−0.744194
$974$	5.62838	0.180345
$975$	0	0
$976$	49.9309	1.59825
$977$	34.5076	1.10400	0.551998	−	0.833845i	$-0.313866\pi$
$977$	34.5076	1.10400	0.551998	+	0.833845i	$0.313866\pi$
$978$	0	0
$979$	108.616	3.47138
$980$	4.86807	0.155505
$981$	0	0
$982$	7.20803	0.230018
$983$	−30.3657	−0.968514	−0.484257	−	0.874926i	$-0.660910\pi$
$983$	−30.3657	−0.968514	−0.484257	+	0.874926i	$0.660910\pi$
$984$	0	0
$985$	48.7766	1.55415
$986$	−5.63723	−0.179526
$987$	0	0
$988$	45.0257	1.43246
$989$	−17.2813	−0.549512
$990$	0	0
$991$	−25.7213	−0.817063	−0.408532	−	0.912744i	$-0.633959\pi$
$991$	−25.7213	−0.817063	−0.408532	+	0.912744i	$0.633959\pi$
$992$	−0.164166	−0.00521228
$993$	0	0
$994$	−2.24796	−0.0713009
$995$	−30.0141	−0.951512
$996$	0	0
$997$	13.4555	0.426139	0.213069	−	0.977037i	$-0.431654\pi$
$997$	13.4555	0.426139	0.213069	+	0.977037i	$0.431654\pi$
$998$	1.98970	0.0629830
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.r.1.9		16
3.2	odd	2		2667.2.a.o.1.8	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.o.1.8	✓	16	3.2	odd	2
8001.2.a.r.1.9		16	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)

\(f(q)\)	\(=\)	\(q-0.298113 q^{2} -1.91113 q^{4} -2.54722 q^{5} -1.00000 q^{7} +1.16596 q^{8} +O(q^{10})\)	\(q-0.298113 q^{2} -1.91113 q^{4} -2.54722 q^{5} -1.00000 q^{7} +1.16596 q^{8} +0.759360 q^{10} -6.15296 q^{11} +4.68194 q^{13} +0.298113 q^{14} +3.47467 q^{16} +4.41002 q^{17} -5.03205 q^{19} +4.86807 q^{20} +1.83427 q^{22} -9.11699 q^{23} +1.48835 q^{25} -1.39575 q^{26} +1.91113 q^{28} +4.28790 q^{29} +0.0487464 q^{31} -3.36776 q^{32} -1.31468 q^{34} +2.54722 q^{35} +3.02081 q^{37} +1.50012 q^{38} -2.96995 q^{40} +3.31879 q^{41} +1.89550 q^{43} +11.7591 q^{44} +2.71789 q^{46} +10.1769 q^{47} +1.00000 q^{49} -0.443697 q^{50} -8.94780 q^{52} -6.07411 q^{53} +15.6730 q^{55} -1.16596 q^{56} -1.27828 q^{58} +9.37602 q^{59} +14.3700 q^{61} -0.0145319 q^{62} -5.94537 q^{64} -11.9260 q^{65} +5.59222 q^{67} -8.42813 q^{68} -0.759360 q^{70} -7.54064 q^{71} +5.40772 q^{73} -0.900543 q^{74} +9.61689 q^{76} +6.15296 q^{77} -8.18983 q^{79} -8.85077 q^{80} -0.989374 q^{82} +12.5627 q^{83} -11.2333 q^{85} -0.565072 q^{86} -7.17408 q^{88} -17.6526 q^{89} -4.68194 q^{91} +17.4237 q^{92} -3.03386 q^{94} +12.8178 q^{95} -0.133391 q^{97} -0.298113 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) \) 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8	\( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8} - 12 q^{10} - 11 q^{11} + 18 q^{13} + 5 q^{14} + 25 q^{16} + 5 q^{17} - 11 q^{19} + q^{20} + q^{22} - 13 q^{23} + 33 q^{25} - 8 q^{26} - 19 q^{28} - 24 q^{29} - 42 q^{31} - 42 q^{32} + 9 q^{34} - q^{35} + 40 q^{37} - 38 q^{38} - 61 q^{40} - 9 q^{41} + 7 q^{43} - 3 q^{44} + 24 q^{46} - 31 q^{47} + 16 q^{49} - 6 q^{50} + 52 q^{52} - 66 q^{53} - 36 q^{55} + 6 q^{56} + 19 q^{58} + 7 q^{59} + 6 q^{61} - 52 q^{62} + 10 q^{64} - 51 q^{65} + 16 q^{67} - 14 q^{68} + 12 q^{70} - 46 q^{71} + 39 q^{73} - 72 q^{74} + 24 q^{76} + 11 q^{77} + 4 q^{79} + 2 q^{80} - 18 q^{82} - 15 q^{83} - 4 q^{85} - 14 q^{86} + 58 q^{88} + q^{89} - 18 q^{91} - 26 q^{92} + 5 q^{94} - 44 q^{95} + 41 q^{97} - 5 q^{98}+O(q^{100}) \) 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8 - 12 * q^10 - 11 * q^11 + 18 * q^13 + 5 * q^14 + 25 * q^16 + 5 * q^17 - 11 * q^19 + q^20 + q^22 - 13 * q^23 + 33 * q^25 - 8 * q^26 - 19 * q^28 - 24 * q^29 - 42 * q^31 - 42 * q^32 + 9 * q^34 - q^35 + 40 * q^37 - 38 * q^38 - 61 * q^40 - 9 * q^41 + 7 * q^43 - 3 * q^44 + 24 * q^46 - 31 * q^47 + 16 * q^49 - 6 * q^50 + 52 * q^52 - 66 * q^53 - 36 * q^55 + 6 * q^56 + 19 * q^58 + 7 * q^59 + 6 * q^61 - 52 * q^62 + 10 * q^64 - 51 * q^65 + 16 * q^67 - 14 * q^68 + 12 * q^70 - 46 * q^71 + 39 * q^73 - 72 * q^74 + 24 * q^76 + 11 * q^77 + 4 * q^79 + 2 * q^80 - 18 * q^82 - 15 * q^83 - 4 * q^85 - 14 * q^86 + 58 * q^88 + q^89 - 18 * q^91 - 26 * q^92 + 5 * q^94 - 44 * q^95 + 41 * q^97 - 5 * q^98

Introduction

L-functions

Modular forms

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Database

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