LMFDB - Embedded newform 8001.2.a.n.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 3 x^{10} + 41 x^{9} - 11 x^{8} - 123 x^{7} + 44 x^{6} + 159 x^{5} - 39 x^{4} + \cdots - 1 $ x^12 - 5x^11 - 3x^10 + 41x^9 - 11x^8 - 123x^7 + 44x^6 + 159x^5 - 39x^4 - 71x^3 + 16x^2 + 7*x - 1
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			1.8
Root			$-0.329641$ 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)

$f(q)$	$=$	$q+1.32964 q^{2} -0.232054 q^{4} +0.891044 q^{5} +1.00000 q^{7} -2.96783 q^{8} +O(q^{10})$	\(q+1.32964 q^{2} -0.232054 q^{4} +0.891044 q^{5} +1.00000 q^{7} -2.96783 q^{8} +1.18477 q^{10} +1.74116 q^{11} -2.35399 q^{13} +1.32964 q^{14} -3.48204 q^{16} -1.98218 q^{17} +6.32335 q^{19} -0.206771 q^{20} +2.31512 q^{22} -4.16692 q^{23} -4.20604 q^{25} -3.12997 q^{26} -0.232054 q^{28} -5.78313 q^{29} +2.29387 q^{31} +1.30580 q^{32} -2.63559 q^{34} +0.891044 q^{35} +6.24730 q^{37} +8.40779 q^{38} -2.64447 q^{40} +8.30155 q^{41} -6.53325 q^{43} -0.404045 q^{44} -5.54051 q^{46} +9.52409 q^{47} +1.00000 q^{49} -5.59253 q^{50} +0.546255 q^{52} +12.0582 q^{53} +1.55145 q^{55} -2.96783 q^{56} -7.68949 q^{58} +9.30060 q^{59} +7.96184 q^{61} +3.05003 q^{62} +8.70032 q^{64} -2.09751 q^{65} -12.7048 q^{67} +0.459974 q^{68} +1.18477 q^{70} +12.8839 q^{71} -2.24371 q^{73} +8.30667 q^{74} -1.46736 q^{76} +1.74116 q^{77} -0.349374 q^{79} -3.10265 q^{80} +11.0381 q^{82} -10.0345 q^{83} -1.76621 q^{85} -8.68688 q^{86} -5.16748 q^{88} +13.1714 q^{89} -2.35399 q^{91} +0.966952 q^{92} +12.6636 q^{94} +5.63438 q^{95} +17.9239 q^{97} +1.32964 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) $ 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8	$ 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8} - 2 q^{10} + 22 q^{11} + 7 q^{14} + 7 q^{16} + 6 q^{17} - 7 q^{19} + 8 q^{20} + 13 q^{22} + 29 q^{23} + 3 q^{25} + 9 q^{28} + 22 q^{29} - 16 q^{31} + 27 q^{32} - 5 q^{34} + 7 q^{35} - 4 q^{37} - 2 q^{38} + 16 q^{40} + 21 q^{41} + 11 q^{43} + 11 q^{44} + 31 q^{47} + 12 q^{49} + 21 q^{50} + 3 q^{52} + 38 q^{53} - 11 q^{55} + 15 q^{56} + 20 q^{58} + 15 q^{59} - 3 q^{61} + 4 q^{62} + 29 q^{64} + 32 q^{65} - q^{67} - 17 q^{68} - 2 q^{70} + 57 q^{71} - 7 q^{73} + 42 q^{74} - 44 q^{76} + 22 q^{77} - 18 q^{79} - q^{80} + 56 q^{82} + 21 q^{83} - 5 q^{85} + 32 q^{86} - 10 q^{88} - 6 q^{89} + 15 q^{92} + 35 q^{94} + 57 q^{95} + 4 q^{97} + 7 q^{98}+O(q^{100}) $ 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8 - 2 * q^10 + 22 * q^11 + 7 * q^14 + 7 * q^16 + 6 * q^17 - 7 * q^19 + 8 * q^20 + 13 * q^22 + 29 * q^23 + 3 * q^25 + 9 * q^28 + 22 * q^29 - 16 * q^31 + 27 * q^32 - 5 * q^34 + 7 * q^35 - 4 * q^37 - 2 * q^38 + 16 * q^40 + 21 * q^41 + 11 * q^43 + 11 * q^44 + 31 * q^47 + 12 * q^49 + 21 * q^50 + 3 * q^52 + 38 * q^53 - 11 * q^55 + 15 * q^56 + 20 * q^58 + 15 * q^59 - 3 * q^61 + 4 * q^62 + 29 * q^64 + 32 * q^65 - q^67 - 17 * q^68 - 2 * q^70 + 57 * q^71 - 7 * q^73 + 42 * q^74 - 44 * q^76 + 22 * q^77 - 18 * q^79 - q^80 + 56 * q^82 + 21 * q^83 - 5 * q^85 + 32 * q^86 - 10 * q^88 - 6 * q^89 + 15 * q^92 + 35 * q^94 + 57 * q^95 + 4 * q^97 + 7 * 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 $

$2$	1.32964	0.940198	0.470099	−	0.882614i	$-0.344218\pi$
$2$	1.32964	0.940198	0.470099	+	0.882614i	$0.344218\pi$
$3$	0	0
$3$	0	0
$4$	−0.232054	−0.116027
$5$	0.891044	0.398487	0.199243	−	0.979950i	$-0.436152\pi$
$5$	0.891044	0.398487	0.199243	+	0.979950i	$0.436152\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.96783	−1.04929
$9$	0	0
$10$	1.18477	0.374657
$11$	1.74116	0.524980	0.262490	−	0.964935i	$-0.415456\pi$
$11$	1.74116	0.524980	0.262490	+	0.964935i	$0.415456\pi$
$12$	0	0
$13$	−2.35399	−0.652880	−0.326440	−	0.945218i	$-0.605849\pi$
$13$	−2.35399	−0.652880	−0.326440	+	0.945218i	$0.605849\pi$
$14$	1.32964	0.355362
$15$	0	0
$16$	−3.48204	−0.870510
$17$	−1.98218	−0.480750	−0.240375	−	0.970680i	$-0.577270\pi$
$17$	−1.98218	−0.480750	−0.240375	+	0.970680i	$0.577270\pi$
$18$	0	0
$19$	6.32335	1.45068	0.725338	−	0.688393i	$-0.241683\pi$
$19$	6.32335	1.45068	0.725338	+	0.688393i	$0.241683\pi$
$20$	−0.206771	−0.0462353
$21$	0	0
$22$	2.31512	0.493585
$23$	−4.16692	−0.868863	−0.434431	−	0.900705i	$-0.643051\pi$
$23$	−4.16692	−0.868863	−0.434431	+	0.900705i	$0.643051\pi$
$24$	0	0
$25$	−4.20604	−0.841208
$26$	−3.12997	−0.613837
$27$	0	0
$28$	−0.232054	−0.0438542
$29$	−5.78313	−1.07390	−0.536950	−	0.843614i	$-0.680424\pi$
$29$	−5.78313	−1.07390	−0.536950	+	0.843614i	$0.680424\pi$
$30$	0	0
$31$	2.29387	0.411992	0.205996	−	0.978553i	$-0.433957\pi$
$31$	2.29387	0.411992	0.205996	+	0.978553i	$0.433957\pi$
$32$	1.30580	0.230834
$33$	0	0
$34$	−2.63559	−0.452000
$35$	0.891044	0.150614
$36$	0	0
$37$	6.24730	1.02705	0.513525	−	0.858075i	$-0.328339\pi$
$37$	6.24730	1.02705	0.513525	+	0.858075i	$0.328339\pi$
$38$	8.40779	1.36392
$39$	0	0
$40$	−2.64447	−0.418127
$41$	8.30155	1.29648	0.648242	−	0.761434i	$-0.275504\pi$
$41$	8.30155	1.29648	0.648242	+	0.761434i	$0.275504\pi$
$42$	0	0
$43$	−6.53325	−0.996311	−0.498156	−	0.867088i	$-0.665989\pi$
$43$	−6.53325	−0.996311	−0.498156	+	0.867088i	$0.665989\pi$
$44$	−0.404045	−0.0609120
$45$	0	0
$46$	−5.54051	−0.816903
$47$	9.52409	1.38923	0.694615	−	0.719381i	$-0.255575\pi$
$47$	9.52409	1.38923	0.694615	+	0.719381i	$0.255575\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−5.59253	−0.790903
$51$	0	0
$52$	0.546255	0.0757519
$53$	12.0582	1.65633	0.828163	−	0.560487i	$-0.189386\pi$
$53$	12.0582	1.65633	0.828163	+	0.560487i	$0.189386\pi$
$54$	0	0
$55$	1.55145	0.209198
$56$	−2.96783	−0.396593
$57$	0	0
$58$	−7.68949	−1.00968
$59$	9.30060	1.21084	0.605418	−	0.795908i	$-0.293006\pi$
$59$	9.30060	1.21084	0.605418	+	0.795908i	$0.293006\pi$
$60$	0	0
$61$	7.96184	1.01941	0.509705	−	0.860349i	$-0.329755\pi$
$61$	7.96184	1.01941	0.509705	+	0.860349i	$0.329755\pi$
$62$	3.05003	0.387354
$63$	0	0
$64$	8.70032	1.08754
$65$	−2.09751	−0.260164
$66$	0	0
$67$	−12.7048	−1.55214	−0.776068	−	0.630650i	$-0.782789\pi$
$67$	−12.7048	−1.55214	−0.776068	+	0.630650i	$0.782789\pi$
$68$	0.459974	0.0557800
$69$	0	0
$70$	1.18477	0.141607
$71$	12.8839	1.52904	0.764520	−	0.644600i	$-0.222976\pi$
$71$	12.8839	1.52904	0.764520	+	0.644600i	$0.222976\pi$
$72$	0	0
$73$	−2.24371	−0.262606	−0.131303	−	0.991342i	$-0.541916\pi$
$73$	−2.24371	−0.262606	−0.131303	+	0.991342i	$0.541916\pi$
$74$	8.30667	0.965631
$75$	0	0
$76$	−1.46736	−0.168318
$77$	1.74116	0.198424
$78$	0	0
$79$	−0.349374	−0.0393077	−0.0196538	−	0.999807i	$-0.506256\pi$
$79$	−0.349374	−0.0393077	−0.0196538	+	0.999807i	$0.506256\pi$
$80$	−3.10265	−0.346887
$81$	0	0
$82$	11.0381	1.21895
$83$	−10.0345	−1.10143	−0.550715	−	0.834693i	$-0.685645\pi$
$83$	−10.0345	−1.10143	−0.550715	+	0.834693i	$0.685645\pi$
$84$	0	0
$85$	−1.76621	−0.191572
$86$	−8.68688	−0.936730
$87$	0	0
$88$	−5.16748	−0.550855
$89$	13.1714	1.39616	0.698081	−	0.716019i	$-0.254038\pi$
$89$	13.1714	1.39616	0.698081	+	0.716019i	$0.254038\pi$
$90$	0	0
$91$	−2.35399	−0.246765
$92$	0.966952	0.100812
$93$	0	0
$94$	12.6636	1.30615
$95$	5.63438	0.578075
$96$	0	0
$97$	17.9239	1.81989	0.909947	−	0.414726i	$-0.136122\pi$
$97$	17.9239	1.81989	0.909947	+	0.414726i	$0.136122\pi$
$98$	1.32964	0.134314
$99$	0	0
$100$	0.976031	0.0976031
$101$	−15.0736	−1.49988	−0.749938	−	0.661508i	$-0.769917\pi$
$101$	−15.0736	−1.49988	−0.749938	+	0.661508i	$0.769917\pi$
$102$	0	0
$103$	3.39796	0.334810	0.167405	−	0.985888i	$-0.446461\pi$
$103$	3.39796	0.334810	0.167405	+	0.985888i	$0.446461\pi$
$104$	6.98625	0.685059
$105$	0	0
$106$	16.0331	1.55727
$107$	4.34570	0.420115	0.210058	−	0.977689i	$-0.432635\pi$
$107$	4.34570	0.420115	0.210058	+	0.977689i	$0.432635\pi$
$108$	0	0
$109$	0.198733	0.0190352	0.00951759	−	0.999955i	$-0.496970\pi$
$109$	0.198733	0.0190352	0.00951759	+	0.999955i	$0.496970\pi$
$110$	2.06287	0.196687
$111$	0	0
$112$	−3.48204	−0.329022
$113$	−15.9179	−1.49743	−0.748715	−	0.662892i	$-0.769329\pi$
$113$	−15.9179	−1.49743	−0.748715	+	0.662892i	$0.769329\pi$
$114$	0	0
$115$	−3.71291	−0.346230
$116$	1.34200	0.124602
$117$	0	0
$118$	12.3665	1.13842
$119$	−1.98218	−0.181706
$120$	0	0
$121$	−7.96835	−0.724396
$122$	10.5864	0.958447
$123$	0	0
$124$	−0.532304	−0.0478023
$125$	−8.20298	−0.733697
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	8.95672	0.791669
$129$	0	0
$130$	−2.78894	−0.244606
$131$	16.9403	1.48008	0.740039	−	0.672564i	$-0.234807\pi$
$131$	16.9403	1.48008	0.740039	+	0.672564i	$0.234807\pi$
$132$	0	0
$133$	6.32335	0.548304
$134$	−16.8928	−1.45931
$135$	0	0
$136$	5.88278	0.504444
$137$	20.1803	1.72412	0.862060	−	0.506806i	$-0.169174\pi$
$137$	20.1803	1.72412	0.862060	+	0.506806i	$0.169174\pi$
$138$	0	0
$139$	−21.4560	−1.81987	−0.909936	−	0.414748i	$-0.863870\pi$
$139$	−21.4560	−1.81987	−0.909936	+	0.414748i	$0.863870\pi$
$140$	−0.206771	−0.0174753
$141$	0	0
$142$	17.1310	1.43760
$143$	−4.09868	−0.342749
$144$	0	0
$145$	−5.15302	−0.427935
$146$	−2.98333	−0.246902
$147$	0	0
$148$	−1.44971	−0.119166
$149$	24.0178	1.96762	0.983808	−	0.179228i	$-0.0573599\pi$
$149$	24.0178	1.96762	0.983808	+	0.179228i	$0.0573599\pi$
$150$	0	0
$151$	−12.1235	−0.986595	−0.493298	−	0.869861i	$-0.664209\pi$
$151$	−12.1235	−0.986595	−0.493298	+	0.869861i	$0.664209\pi$
$152$	−18.7666	−1.52218
$153$	0	0
$154$	2.31512	0.186558
$155$	2.04394	0.164173
$156$	0	0
$157$	0.844951	0.0674345	0.0337172	−	0.999431i	$-0.489265\pi$
$157$	0.844951	0.0674345	0.0337172	+	0.999431i	$0.489265\pi$
$158$	−0.464543	−0.0369570
$159$	0	0
$160$	1.16352	0.0919845
$161$	−4.16692	−0.328399
$162$	0	0
$163$	10.5548	0.826716	0.413358	−	0.910569i	$-0.364356\pi$
$163$	10.5548	0.826716	0.413358	+	0.910569i	$0.364356\pi$
$164$	−1.92641	−0.150428
$165$	0	0
$166$	−13.3423	−1.03556
$167$	11.0088	0.851886	0.425943	−	0.904750i	$-0.359942\pi$
$167$	11.0088	0.851886	0.425943	+	0.904750i	$0.359942\pi$
$168$	0	0
$169$	−7.45872	−0.573748
$170$	−2.34843	−0.180116
$171$	0	0
$172$	1.51607	0.115599
$173$	−24.9743	−1.89876	−0.949382	−	0.314124i	$-0.898289\pi$
$173$	−24.9743	−1.89876	−0.949382	+	0.314124i	$0.898289\pi$
$174$	0	0
$175$	−4.20604	−0.317947
$176$	−6.06280	−0.457001
$177$	0	0
$178$	17.5132	1.31267
$179$	−0.281804	−0.0210630	−0.0105315	−	0.999945i	$-0.503352\pi$
$179$	−0.281804	−0.0210630	−0.0105315	+	0.999945i	$0.503352\pi$
$180$	0	0
$181$	−3.70500	−0.275391	−0.137695	−	0.990475i	$-0.543969\pi$
$181$	−3.70500	−0.275391	−0.137695	+	0.990475i	$0.543969\pi$
$182$	−3.12997	−0.232008
$183$	0	0
$184$	12.3667	0.911686
$185$	5.56662	0.409266
$186$	0	0
$187$	−3.45130	−0.252384
$188$	−2.21011	−0.161189
$189$	0	0
$190$	7.49171	0.543505
$191$	19.2668	1.39410	0.697049	−	0.717023i	$-0.254496\pi$
$191$	19.2668	1.39410	0.697049	+	0.717023i	$0.254496\pi$
$192$	0	0
$193$	23.8155	1.71428	0.857139	−	0.515084i	$-0.172239\pi$
$193$	23.8155	1.71428	0.857139	+	0.515084i	$0.172239\pi$
$194$	23.8323	1.71106
$195$	0	0
$196$	−0.232054	−0.0165753
$197$	−11.3255	−0.806909	−0.403454	−	0.915000i	$-0.632191\pi$
$197$	−11.3255	−0.806909	−0.403454	+	0.915000i	$0.632191\pi$
$198$	0	0
$199$	−5.25596	−0.372585	−0.186293	−	0.982494i	$-0.559647\pi$
$199$	−5.25596	−0.372585	−0.186293	+	0.982494i	$0.559647\pi$
$200$	12.4828	0.882669
$201$	0	0
$202$	−20.0424	−1.41018
$203$	−5.78313	−0.405896
$204$	0	0
$205$	7.39705	0.516632
$206$	4.51806	0.314788
$207$	0	0
$208$	8.19670	0.568339
$209$	11.0100	0.761576
$210$	0	0
$211$	6.41685	0.441754	0.220877	−	0.975302i	$-0.429108\pi$
$211$	6.41685	0.441754	0.220877	+	0.975302i	$0.429108\pi$
$212$	−2.79817	−0.192179
$213$	0	0
$214$	5.77823	0.394991
$215$	−5.82141	−0.397017
$216$	0	0
$217$	2.29387	0.155718
$218$	0.264244	0.0178969
$219$	0	0
$220$	−0.360021	−0.0242726
$221$	4.66604	0.313872
$222$	0	0
$223$	−10.0200	−0.670986	−0.335493	−	0.942043i	$-0.608903\pi$
$223$	−10.0200	−0.670986	−0.335493	+	0.942043i	$0.608903\pi$
$224$	1.30580	0.0872472
$225$	0	0
$226$	−21.1651	−1.40788
$227$	0.197889	0.0131344	0.00656719	−	0.999978i	$-0.497910\pi$
$227$	0.197889	0.0131344	0.00656719	+	0.999978i	$0.497910\pi$
$228$	0	0
$229$	13.2729	0.877095	0.438548	−	0.898708i	$-0.355493\pi$
$229$	13.2729	0.877095	0.438548	+	0.898708i	$0.355493\pi$
$230$	−4.93683	−0.325525
$231$	0	0
$232$	17.1634	1.12683
$233$	5.18128	0.339437	0.169719	−	0.985493i	$-0.445714\pi$
$233$	5.18128	0.339437	0.169719	+	0.985493i	$0.445714\pi$
$234$	0	0
$235$	8.48638	0.553590
$236$	−2.15825	−0.140490
$237$	0	0
$238$	−2.63559	−0.170840
$239$	27.7675	1.79613	0.898065	−	0.439864i	$-0.144973\pi$
$239$	27.7675	1.79613	0.898065	+	0.439864i	$0.144973\pi$
$240$	0	0
$241$	1.15833	0.0746144	0.0373072	−	0.999304i	$-0.488122\pi$
$241$	1.15833	0.0746144	0.0373072	+	0.999304i	$0.488122\pi$
$242$	−10.5951	−0.681076
$243$	0	0
$244$	−1.84758	−0.118279
$245$	0.891044	0.0569267
$246$	0	0
$247$	−14.8851	−0.947118
$248$	−6.80783	−0.432298
$249$	0	0
$250$	−10.9070	−0.689821
$251$	−1.43178	−0.0903729	−0.0451865	−	0.998979i	$-0.514388\pi$
$251$	−1.43178	−0.0903729	−0.0451865	+	0.998979i	$0.514388\pi$
$252$	0	0
$253$	−7.25528	−0.456136
$254$	1.32964	0.0834291
$255$	0	0
$256$	−5.49143	−0.343214
$257$	13.9520	0.870304	0.435152	−	0.900357i	$-0.356695\pi$
$257$	13.9520	0.870304	0.435152	+	0.900357i	$0.356695\pi$
$258$	0	0
$259$	6.24730	0.388188
$260$	0.486737	0.0301861
$261$	0	0
$262$	22.5245	1.39157
$263$	−18.3120	−1.12917	−0.564584	−	0.825375i	$-0.690964\pi$
$263$	−18.3120	−1.12917	−0.564584	+	0.825375i	$0.690964\pi$
$264$	0	0
$265$	10.7444	0.660024
$266$	8.40779	0.515515
$267$	0	0
$268$	2.94820	0.180090
$269$	8.29130	0.505529	0.252765	−	0.967528i	$-0.418660\pi$
$269$	8.29130	0.505529	0.252765	+	0.967528i	$0.418660\pi$
$270$	0	0
$271$	17.9422	1.08991	0.544956	−	0.838465i	$-0.316546\pi$
$271$	17.9422	1.08991	0.544956	+	0.838465i	$0.316546\pi$
$272$	6.90204	0.418497
$273$	0	0
$274$	26.8326	1.62102
$275$	−7.32340	−0.441618
$276$	0	0
$277$	−1.02094	−0.0613421	−0.0306710	−	0.999530i	$-0.509764\pi$
$277$	−1.02094	−0.0613421	−0.0306710	+	0.999530i	$0.509764\pi$
$278$	−28.5288	−1.71104
$279$	0	0
$280$	−2.64447	−0.158037
$281$	10.4187	0.621526	0.310763	−	0.950487i	$-0.399415\pi$
$281$	10.4187	0.621526	0.310763	+	0.950487i	$0.399415\pi$
$282$	0	0
$283$	21.3682	1.27021	0.635103	−	0.772428i	$-0.280958\pi$
$283$	21.3682	1.27021	0.635103	+	0.772428i	$0.280958\pi$
$284$	−2.98977	−0.177410
$285$	0	0
$286$	−5.44978	−0.322252
$287$	8.30155	0.490025
$288$	0	0
$289$	−13.0710	−0.768880
$290$	−6.85167	−0.402344
$291$	0	0
$292$	0.520663	0.0304695
$293$	17.5070	1.02277	0.511384	−	0.859353i	$-0.329133\pi$
$293$	17.5070	1.02277	0.511384	+	0.859353i	$0.329133\pi$
$294$	0	0
$295$	8.28724	0.482502
$296$	−18.5409	−1.07767
$297$	0	0
$298$	31.9351	1.84995
$299$	9.80890	0.567263
$300$	0	0
$301$	−6.53325	−0.376570
$302$	−16.1199	−0.927595
$303$	0	0
$304$	−22.0182	−1.26283
$305$	7.09435	0.406221
$306$	0	0
$307$	8.50981	0.485680	0.242840	−	0.970066i	$-0.421921\pi$
$307$	8.50981	0.485680	0.242840	+	0.970066i	$0.421921\pi$
$308$	−0.404045	−0.0230226
$309$	0	0
$310$	2.71771	0.154356
$311$	21.3926	1.21306	0.606531	−	0.795060i	$-0.292561\pi$
$311$	21.3926	1.21306	0.606531	+	0.795060i	$0.292561\pi$
$312$	0	0
$313$	27.7990	1.57129	0.785647	−	0.618674i	$-0.212330\pi$
$313$	27.7990	1.57129	0.785647	+	0.618674i	$0.212330\pi$
$314$	1.12348	0.0634018
$315$	0	0
$316$	0.0810739	0.00456076
$317$	12.9622	0.728032	0.364016	−	0.931393i	$-0.381405\pi$
$317$	12.9622	0.728032	0.364016	+	0.931393i	$0.381405\pi$
$318$	0	0
$319$	−10.0694	−0.563776
$320$	7.75237	0.433371
$321$	0	0
$322$	−5.54051	−0.308760
$323$	−12.5340	−0.697412
$324$	0	0
$325$	9.90099	0.549208
$326$	14.0341	0.777277
$327$	0	0
$328$	−24.6376	−1.36038
$329$	9.52409	0.525080
$330$	0	0
$331$	−13.3429	−0.733394	−0.366697	−	0.930340i	$-0.619511\pi$
$331$	−13.3429	−0.733394	−0.366697	+	0.930340i	$0.619511\pi$
$332$	2.32855	0.127796
$333$	0	0
$334$	14.6377	0.800941
$335$	−11.3205	−0.618505
$336$	0	0
$337$	−11.9541	−0.651184	−0.325592	−	0.945510i	$-0.605564\pi$
$337$	−11.9541	−0.651184	−0.325592	+	0.945510i	$0.605564\pi$
$338$	−9.91742	−0.539436
$339$	0	0
$340$	0.409857	0.0222276
$341$	3.99401	0.216288
$342$	0	0
$343$	1.00000	0.0539949
$344$	19.3896	1.04542
$345$	0	0
$346$	−33.2069	−1.78521
$347$	12.1646	0.653028	0.326514	−	0.945192i	$-0.394126\pi$
$347$	12.1646	0.653028	0.326514	+	0.945192i	$0.394126\pi$
$348$	0	0
$349$	28.0980	1.50405	0.752027	−	0.659133i	$-0.229077\pi$
$349$	28.0980	1.50405	0.752027	+	0.659133i	$0.229077\pi$
$350$	−5.59253	−0.298933
$351$	0	0
$352$	2.27360	0.121184
$353$	−9.54377	−0.507963	−0.253982	−	0.967209i	$-0.581740\pi$
$353$	−9.54377	−0.507963	−0.253982	+	0.967209i	$0.581740\pi$
$354$	0	0
$355$	11.4801	0.609302
$356$	−3.05647	−0.161993
$357$	0	0
$358$	−0.374698	−0.0198034
$359$	−35.1275	−1.85396	−0.926981	−	0.375109i	$-0.877605\pi$
$359$	−35.1275	−1.85396	−0.926981	+	0.375109i	$0.877605\pi$
$360$	0	0
$361$	20.9848	1.10446
$362$	−4.92633	−0.258922
$363$	0	0
$364$	0.546255	0.0286315
$365$	−1.99924	−0.104645
$366$	0	0
$367$	−29.2177	−1.52515	−0.762576	−	0.646899i	$-0.776066\pi$
$367$	−29.2177	−1.52515	−0.762576	+	0.646899i	$0.776066\pi$
$368$	14.5094	0.756354
$369$	0	0
$370$	7.40161	0.384791
$371$	12.0582	0.626032
$372$	0	0
$373$	−9.60378	−0.497265	−0.248632	−	0.968598i	$-0.579981\pi$
$373$	−9.60378	−0.497265	−0.248632	+	0.968598i	$0.579981\pi$
$374$	−4.58899	−0.237291
$375$	0	0
$376$	−28.2659	−1.45770
$377$	13.6134	0.701128
$378$	0	0
$379$	7.93256	0.407468	0.203734	−	0.979026i	$-0.434692\pi$
$379$	7.93256	0.407468	0.203734	+	0.979026i	$0.434692\pi$
$380$	−1.30748	−0.0670725
$381$	0	0
$382$	25.6180	1.31073
$383$	8.92126	0.455855	0.227928	−	0.973678i	$-0.426805\pi$
$383$	8.92126	0.455855	0.227928	+	0.973678i	$0.426805\pi$
$384$	0	0
$385$	1.55145	0.0790693
$386$	31.6661	1.61176
$387$	0	0
$388$	−4.15931	−0.211157
$389$	−12.8068	−0.649330	−0.324665	−	0.945829i	$-0.605252\pi$
$389$	−12.8068	−0.649330	−0.324665	+	0.945829i	$0.605252\pi$
$390$	0	0
$391$	8.25959	0.417705
$392$	−2.96783	−0.149898
$393$	0	0
$394$	−15.0589	−0.758654
$395$	−0.311308	−0.0156636
$396$	0	0
$397$	−27.7701	−1.39374	−0.696871	−	0.717197i	$-0.745425\pi$
$397$	−27.7701	−1.39374	−0.696871	+	0.717197i	$0.745425\pi$
$398$	−6.98854	−0.350304
$399$	0	0
$400$	14.6456	0.732281
$401$	32.3888	1.61742	0.808710	−	0.588208i	$-0.200166\pi$
$401$	32.3888	1.61742	0.808710	+	0.588208i	$0.200166\pi$
$402$	0	0
$403$	−5.39976	−0.268981
$404$	3.49789	0.174026
$405$	0	0
$406$	−7.68949	−0.381623
$407$	10.8776	0.539181
$408$	0	0
$409$	30.5014	1.50819	0.754097	−	0.656763i	$-0.228075\pi$
$409$	30.5014	1.50819	0.754097	+	0.656763i	$0.228075\pi$
$410$	9.83542	0.485737
$411$	0	0
$412$	−0.788511	−0.0388471
$413$	9.30060	0.457653
$414$	0	0
$415$	−8.94118	−0.438905
$416$	−3.07384	−0.150707
$417$	0	0
$418$	14.6393	0.716033
$419$	−12.7782	−0.624254	−0.312127	−	0.950040i	$-0.601041\pi$
$419$	−12.7782	−0.624254	−0.312127	+	0.950040i	$0.601041\pi$
$420$	0	0
$421$	−8.23066	−0.401138	−0.200569	−	0.979680i	$-0.564279\pi$
$421$	−8.23066	−0.401138	−0.200569	+	0.979680i	$0.564279\pi$
$422$	8.53211	0.415337
$423$	0	0
$424$	−35.7868	−1.73796
$425$	8.33714	0.404410
$426$	0	0
$427$	7.96184	0.385301
$428$	−1.00844	−0.0487448
$429$	0	0
$430$	−7.74039	−0.373275
$431$	17.6664	0.850960	0.425480	−	0.904968i	$-0.360105\pi$
$431$	17.6664	0.850960	0.425480	+	0.904968i	$0.360105\pi$
$432$	0	0
$433$	−26.8943	−1.29246	−0.646230	−	0.763143i	$-0.723655\pi$
$433$	−26.8943	−1.29246	−0.646230	+	0.763143i	$0.723655\pi$
$434$	3.05003	0.146406
$435$	0	0
$436$	−0.0461169	−0.00220860
$437$	−26.3489	−1.26044
$438$	0	0
$439$	−8.89438	−0.424506	−0.212253	−	0.977215i	$-0.568080\pi$
$439$	−8.89438	−0.424506	−0.212253	+	0.977215i	$0.568080\pi$
$440$	−4.60445	−0.219508
$441$	0	0
$442$	6.20416	0.295102
$443$	−0.610058	−0.0289847	−0.0144924	−	0.999895i	$-0.504613\pi$
$443$	−0.610058	−0.0289847	−0.0144924	+	0.999895i	$0.504613\pi$
$444$	0	0
$445$	11.7363	0.556352
$446$	−13.3229	−0.630860
$447$	0	0
$448$	8.70032	0.411052
$449$	14.6147	0.689711	0.344855	−	0.938656i	$-0.387928\pi$
$449$	14.6147	0.689711	0.344855	+	0.938656i	$0.387928\pi$
$450$	0	0
$451$	14.4544	0.680629
$452$	3.69382	0.173743
$453$	0	0
$454$	0.263122	0.0123489
$455$	−2.09751	−0.0983328
$456$	0	0
$457$	−16.4481	−0.769411	−0.384706	−	0.923039i	$-0.625697\pi$
$457$	−16.4481	−0.769411	−0.384706	+	0.923039i	$0.625697\pi$
$458$	17.6481	0.824643
$459$	0	0
$460$	0.861597	0.0401722
$461$	−3.53864	−0.164811	−0.0824055	−	0.996599i	$-0.526260\pi$
$461$	−3.53864	−0.164811	−0.0824055	+	0.996599i	$0.526260\pi$
$462$	0	0
$463$	−27.7212	−1.28831	−0.644156	−	0.764894i	$-0.722791\pi$
$463$	−27.7212	−1.28831	−0.644156	+	0.764894i	$0.722791\pi$
$464$	20.1371	0.934841
$465$	0	0
$466$	6.88925	0.319138
$467$	26.2118	1.21294	0.606469	−	0.795107i	$-0.292585\pi$
$467$	26.2118	1.21294	0.606469	+	0.795107i	$0.292585\pi$
$468$	0	0
$469$	−12.7048	−0.586652
$470$	11.2838	0.520485
$471$	0	0
$472$	−27.6026	−1.27051
$473$	−11.3755	−0.523044
$474$	0	0
$475$	−26.5963	−1.22032
$476$	0.459974	0.0210829
$477$	0	0
$478$	36.9208	1.68872
$479$	−15.3275	−0.700333	−0.350166	−	0.936688i	$-0.613875\pi$
$479$	−15.3275	−0.700333	−0.350166	+	0.936688i	$0.613875\pi$
$480$	0	0
$481$	−14.7061	−0.670541
$482$	1.54016	0.0701524
$483$	0	0
$484$	1.84909	0.0840496
$485$	15.9709	0.725203
$486$	0	0
$487$	−24.0534	−1.08996	−0.544981	−	0.838448i	$-0.683463\pi$
$487$	−24.0534	−1.08996	−0.544981	+	0.838448i	$0.683463\pi$
$488$	−23.6294	−1.06965
$489$	0	0
$490$	1.18477	0.0535224
$491$	−30.2922	−1.36707	−0.683534	−	0.729919i	$-0.739558\pi$
$491$	−30.2922	−1.36707	−0.683534	+	0.729919i	$0.739558\pi$
$492$	0	0
$493$	11.4632	0.516277
$494$	−19.7919	−0.890478
$495$	0	0
$496$	−7.98737	−0.358643
$497$	12.8839	0.577923
$498$	0	0
$499$	−2.30139	−0.103024	−0.0515122	−	0.998672i	$-0.516404\pi$
$499$	−2.30139	−0.103024	−0.0515122	+	0.998672i	$0.516404\pi$
$500$	1.90354	0.0851289
$501$	0	0
$502$	−1.90375	−0.0849685
$503$	−15.7789	−0.703548	−0.351774	−	0.936085i	$-0.614421\pi$
$503$	−15.7789	−0.703548	−0.351774	+	0.936085i	$0.614421\pi$
$504$	0	0
$505$	−13.4312	−0.597681
$506$	−9.64692	−0.428858
$507$	0	0
$508$	−0.232054	−0.0102958
$509$	−22.5782	−1.00076	−0.500380	−	0.865806i	$-0.666807\pi$
$509$	−22.5782	−1.00076	−0.500380	+	0.865806i	$0.666807\pi$
$510$	0	0
$511$	−2.24371	−0.0992559
$512$	−25.2151	−1.11436
$513$	0	0
$514$	18.5512	0.818258
$515$	3.02773	0.133418
$516$	0	0
$517$	16.5830	0.729319
$518$	8.30667	0.364974
$519$	0	0
$520$	6.22506	0.272987
$521$	2.14532	0.0939884	0.0469942	−	0.998895i	$-0.485036\pi$
$521$	2.14532	0.0939884	0.0469942	+	0.998895i	$0.485036\pi$
$522$	0	0
$523$	−28.3034	−1.23762	−0.618811	−	0.785540i	$-0.712385\pi$
$523$	−28.3034	−1.23762	−0.618811	+	0.785540i	$0.712385\pi$
$524$	−3.93107	−0.171729
$525$	0	0
$526$	−24.3484	−1.06164
$527$	−4.54687	−0.198065
$528$	0	0
$529$	−5.63678	−0.245077
$530$	14.2862	0.620553
$531$	0	0
$532$	−1.46736	−0.0636182
$533$	−19.5418	−0.846449
$534$	0	0
$535$	3.87221	0.167410
$536$	37.7056	1.62864
$537$	0	0
$538$	11.0245	0.475298
$539$	1.74116	0.0749972
$540$	0	0
$541$	36.5757	1.57251	0.786257	−	0.617900i	$-0.212016\pi$
$541$	36.5757	1.57251	0.786257	+	0.617900i	$0.212016\pi$
$542$	23.8567	1.02473
$543$	0	0
$544$	−2.58833	−0.110974
$545$	0.177080	0.00758527
$546$	0	0
$547$	24.7766	1.05937	0.529686	−	0.848194i	$-0.322310\pi$
$547$	24.7766	1.05937	0.529686	+	0.848194i	$0.322310\pi$
$548$	−4.68293	−0.200045
$549$	0	0
$550$	−9.73750	−0.415208
$551$	−36.5688	−1.55788
$552$	0	0
$553$	−0.349374	−0.0148569
$554$	−1.35748	−0.0576737
$555$	0	0
$556$	4.97896	0.211155
$557$	1.09255	0.0462929	0.0231464	−	0.999732i	$-0.492632\pi$
$557$	1.09255	0.0462929	0.0231464	+	0.999732i	$0.492632\pi$
$558$	0	0
$559$	15.3792	0.650472
$560$	−3.10265	−0.131111
$561$	0	0
$562$	13.8531	0.584358
$563$	−5.52201	−0.232725	−0.116362	−	0.993207i	$-0.537123\pi$
$563$	−5.52201	−0.232725	−0.116362	+	0.993207i	$0.537123\pi$
$564$	0	0
$565$	−14.1835	−0.596706
$566$	28.4120	1.19425
$567$	0	0
$568$	−38.2373	−1.60440
$569$	41.9895	1.76029	0.880146	−	0.474703i	$-0.157445\pi$
$569$	41.9895	1.76029	0.880146	+	0.474703i	$0.157445\pi$
$570$	0	0
$571$	14.1207	0.590933	0.295467	−	0.955353i	$-0.404525\pi$
$571$	14.1207	0.590933	0.295467	+	0.955353i	$0.404525\pi$
$572$	0.951118	0.0397682
$573$	0	0
$574$	11.0381	0.460721
$575$	17.5262	0.730895
$576$	0	0
$577$	6.04799	0.251781	0.125890	−	0.992044i	$-0.459821\pi$
$577$	6.04799	0.251781	0.125890	+	0.992044i	$0.459821\pi$
$578$	−17.3797	−0.722900
$579$	0	0
$580$	1.19578	0.0496521
$581$	−10.0345	−0.416301
$582$	0	0
$583$	20.9953	0.869538
$584$	6.65895	0.275549
$585$	0	0
$586$	23.2780	0.961604
$587$	20.0650	0.828172	0.414086	−	0.910238i	$-0.364101\pi$
$587$	20.0650	0.828172	0.414086	+	0.910238i	$0.364101\pi$
$588$	0	0
$589$	14.5050	0.597667
$590$	11.0191	0.453647
$591$	0	0
$592$	−21.7534	−0.894058
$593$	−6.58626	−0.270465	−0.135233	−	0.990814i	$-0.543178\pi$
$593$	−6.58626	−0.270465	−0.135233	+	0.990814i	$0.543178\pi$
$594$	0	0
$595$	−1.76621	−0.0724075
$596$	−5.57344	−0.228297
$597$	0	0
$598$	13.0423	0.533340
$599$	27.7887	1.13542	0.567708	−	0.823230i	$-0.307830\pi$
$599$	27.7887	1.13542	0.567708	+	0.823230i	$0.307830\pi$
$600$	0	0
$601$	−27.5514	−1.12384	−0.561922	−	0.827190i	$-0.689938\pi$
$601$	−27.5514	−1.12384	−0.561922	+	0.827190i	$0.689938\pi$
$602$	−8.68688	−0.354051
$603$	0	0
$604$	2.81331	0.114472
$605$	−7.10015	−0.288662
$606$	0	0
$607$	17.9515	0.728629	0.364314	−	0.931276i	$-0.381303\pi$
$607$	17.9515	0.728629	0.364314	+	0.931276i	$0.381303\pi$
$608$	8.25701	0.334866
$609$	0	0
$610$	9.43294	0.381928
$611$	−22.4196	−0.907001
$612$	0	0
$613$	44.5323	1.79864	0.899321	−	0.437289i	$-0.144061\pi$
$613$	44.5323	1.79864	0.899321	+	0.437289i	$0.144061\pi$
$614$	11.3150	0.456636
$615$	0	0
$616$	−5.16748	−0.208204
$617$	−14.8812	−0.599095	−0.299548	−	0.954081i	$-0.596836\pi$
$617$	−14.8812	−0.599095	−0.299548	+	0.954081i	$0.596836\pi$
$618$	0	0
$619$	−47.9534	−1.92741	−0.963706	−	0.266967i	$-0.913979\pi$
$619$	−47.9534	−1.92741	−0.963706	+	0.266967i	$0.913979\pi$
$620$	−0.474306	−0.0190486
$621$	0	0
$622$	28.4444	1.14052
$623$	13.1714	0.527700
$624$	0	0
$625$	13.7210	0.548840
$626$	36.9628	1.47733
$627$	0	0
$628$	−0.196075	−0.00782423
$629$	−12.3833	−0.493754
$630$	0	0
$631$	−41.0672	−1.63486	−0.817430	−	0.576028i	$-0.804602\pi$
$631$	−41.0672	−1.63486	−0.817430	+	0.576028i	$0.804602\pi$
$632$	1.03688	0.0412451
$633$	0	0
$634$	17.2351	0.684494
$635$	0.891044	0.0353600
$636$	0	0
$637$	−2.35399	−0.0932686
$638$	−13.3886	−0.530061
$639$	0	0
$640$	7.98082	0.315470
$641$	−13.0288	−0.514606	−0.257303	−	0.966331i	$-0.582834\pi$
$641$	−13.0288	−0.514606	−0.257303	+	0.966331i	$0.582834\pi$
$642$	0	0
$643$	−5.39620	−0.212805	−0.106403	−	0.994323i	$-0.533933\pi$
$643$	−5.39620	−0.212805	−0.106403	+	0.994323i	$0.533933\pi$
$644$	0.966952	0.0381033
$645$	0	0
$646$	−16.6658	−0.655705
$647$	7.75006	0.304686	0.152343	−	0.988328i	$-0.451318\pi$
$647$	7.75006	0.304686	0.152343	+	0.988328i	$0.451318\pi$
$648$	0	0
$649$	16.1939	0.635664
$650$	13.1648	0.516365
$651$	0	0
$652$	−2.44929	−0.0959216
$653$	28.9925	1.13456	0.567282	−	0.823524i	$-0.307995\pi$
$653$	28.9925	1.13456	0.567282	+	0.823524i	$0.307995\pi$
$654$	0	0
$655$	15.0945	0.589792
$656$	−28.9064	−1.12860
$657$	0	0
$658$	12.6636	0.493679
$659$	2.66102	0.103658	0.0518292	−	0.998656i	$-0.483495\pi$
$659$	2.66102	0.103658	0.0518292	+	0.998656i	$0.483495\pi$
$660$	0	0
$661$	−7.12423	−0.277100	−0.138550	−	0.990355i	$-0.544244\pi$
$661$	−7.12423	−0.277100	−0.138550	+	0.990355i	$0.544244\pi$
$662$	−17.7413	−0.689536
$663$	0	0
$664$	29.7807	1.15572
$665$	5.63438	0.218492
$666$	0	0
$667$	24.0978	0.933072
$668$	−2.55464	−0.0988419
$669$	0	0
$670$	−15.0522	−0.581518
$671$	13.8629	0.535170
$672$	0	0
$673$	−20.4373	−0.787800	−0.393900	−	0.919153i	$-0.628874\pi$
$673$	−20.4373	−0.787800	−0.393900	+	0.919153i	$0.628874\pi$
$674$	−15.8947	−0.612242
$675$	0	0
$676$	1.73083	0.0665703
$677$	−23.2712	−0.894384	−0.447192	−	0.894438i	$-0.647576\pi$
$677$	−23.2712	−0.894384	−0.447192	+	0.894438i	$0.647576\pi$
$678$	0	0
$679$	17.9239	0.687855
$680$	5.24181	0.201014
$681$	0	0
$682$	5.31060	0.203353
$683$	−9.60870	−0.367667	−0.183833	−	0.982957i	$-0.558851\pi$
$683$	−9.60870	−0.367667	−0.183833	+	0.982957i	$0.558851\pi$
$684$	0	0
$685$	17.9815	0.687039
$686$	1.32964	0.0507659
$687$	0	0
$688$	22.7491	0.867300
$689$	−28.3850	−1.08138
$690$	0	0
$691$	−36.7861	−1.39941	−0.699704	−	0.714433i	$-0.746685\pi$
$691$	−36.7861	−1.39941	−0.699704	+	0.714433i	$0.746685\pi$
$692$	5.79541	0.220308
$693$	0	0
$694$	16.1745	0.613976
$695$	−19.1182	−0.725195
$696$	0	0
$697$	−16.4552	−0.623284
$698$	37.3603	1.41411
$699$	0	0
$700$	0.976031	0.0368905
$701$	−0.0135994	−0.000513643 0	−0.000256822	−	1.00000i	$-0.500082\pi$
$701$	−0.0135994	−0.000513643 0	−0.000256822		1.00000i	$0.500082\pi$
$702$	0	0
$703$	39.5039	1.48992
$704$	15.1487	0.570937
$705$	0	0
$706$	−12.6898	−0.477586
$707$	−15.0736	−0.566900
$708$	0	0
$709$	−29.5464	−1.10964	−0.554818	−	0.831971i	$-0.687212\pi$
$709$	−29.5464	−1.10964	−0.554818	+	0.831971i	$0.687212\pi$
$710$	15.2645	0.572865
$711$	0	0
$712$	−39.0904	−1.46497
$713$	−9.55839	−0.357965
$714$	0	0
$715$	−3.65211	−0.136581
$716$	0.0653939	0.00244389
$717$	0	0
$718$	−46.7070	−1.74309
$719$	−41.8589	−1.56107	−0.780537	−	0.625109i	$-0.785054\pi$
$719$	−41.8589	−1.56107	−0.780537	+	0.625109i	$0.785054\pi$
$720$	0	0
$721$	3.39796	0.126546
$722$	27.9022	1.03841
$723$	0	0
$724$	0.859763	0.0319528
$725$	24.3241	0.903374
$726$	0	0
$727$	−20.7340	−0.768981	−0.384491	−	0.923129i	$-0.625623\pi$
$727$	−20.7340	−0.768981	−0.384491	+	0.923129i	$0.625623\pi$
$728$	6.98625	0.258928
$729$	0	0
$730$	−2.65828	−0.0983872
$731$	12.9501	0.478976
$732$	0	0
$733$	−9.42966	−0.348292	−0.174146	−	0.984720i	$-0.555717\pi$
$733$	−9.42966	−0.348292	−0.174146	+	0.984720i	$0.555717\pi$
$734$	−38.8491	−1.43395
$735$	0	0
$736$	−5.44115	−0.200563
$737$	−22.1211	−0.814840
$738$	0	0
$739$	1.90344	0.0700191	0.0350096	−	0.999387i	$-0.488854\pi$
$739$	1.90344	0.0700191	0.0350096	+	0.999387i	$0.488854\pi$
$740$	−1.29176	−0.0474860
$741$	0	0
$742$	16.0331	0.588595
$743$	26.3740	0.967569	0.483785	−	0.875187i	$-0.339262\pi$
$743$	26.3740	0.967569	0.483785	+	0.875187i	$0.339262\pi$
$744$	0	0
$745$	21.4009	0.784069
$746$	−12.7696	−0.467528
$747$	0	0
$748$	0.800889	0.0292834
$749$	4.34570	0.158789
$750$	0	0
$751$	−18.8159	−0.686603	−0.343301	−	0.939225i	$-0.611545\pi$
$751$	−18.8159	−0.686603	−0.343301	+	0.939225i	$0.611545\pi$
$752$	−33.1633	−1.20934
$753$	0	0
$754$	18.1010	0.659199
$755$	−10.8026	−0.393145
$756$	0	0
$757$	13.7623	0.500199	0.250099	−	0.968220i	$-0.419537\pi$
$757$	13.7623	0.500199	0.250099	+	0.968220i	$0.419537\pi$
$758$	10.5475	0.383101
$759$	0	0
$760$	−16.7219	−0.606567
$761$	26.1030	0.946234	0.473117	−	0.881000i	$-0.343129\pi$
$761$	26.1030	0.946234	0.473117	+	0.881000i	$0.343129\pi$
$762$	0	0
$763$	0.198733	0.00719462
$764$	−4.47095	−0.161753
$765$	0	0
$766$	11.8621	0.428594
$767$	−21.8935	−0.790530
$768$	0	0
$769$	28.9323	1.04333	0.521663	−	0.853152i	$-0.325312\pi$
$769$	28.9323	1.04333	0.521663	+	0.853152i	$0.325312\pi$
$770$	2.06287	0.0743408
$771$	0	0
$772$	−5.52650	−0.198903
$773$	−34.8964	−1.25514	−0.627568	−	0.778561i	$-0.715950\pi$
$773$	−34.8964	−1.25514	−0.627568	+	0.778561i	$0.715950\pi$
$774$	0	0
$775$	−9.64813	−0.346571
$776$	−53.1950	−1.90959
$777$	0	0
$778$	−17.0284	−0.610499
$779$	52.4936	1.88078
$780$	0	0
$781$	22.4330	0.802716
$782$	10.9823	0.392726
$783$	0	0
$784$	−3.48204	−0.124359
$785$	0.752889	0.0268717
$786$	0	0
$787$	19.2015	0.684460	0.342230	−	0.939616i	$-0.388818\pi$
$787$	19.2015	0.684460	0.342230	+	0.939616i	$0.388818\pi$
$788$	2.62813	0.0936234
$789$	0	0
$790$	−0.413928	−0.0147269
$791$	−15.9179	−0.565975
$792$	0	0
$793$	−18.7421	−0.665552
$794$	−36.9243	−1.31039
$795$	0	0
$796$	1.21967	0.0432300
$797$	−17.6228	−0.624233	−0.312116	−	0.950044i	$-0.601038\pi$
$797$	−17.6228	−0.624233	−0.312116	+	0.950044i	$0.601038\pi$
$798$	0	0
$799$	−18.8785	−0.667872
$800$	−5.49224	−0.194180
$801$	0	0
$802$	43.0655	1.52070
$803$	−3.90666	−0.137863
$804$	0	0
$805$	−3.71291	−0.130863
$806$	−7.17975	−0.252896
$807$	0	0
$808$	44.7358	1.57380
$809$	−44.6656	−1.57036	−0.785179	−	0.619269i	$-0.787429\pi$
$809$	−44.6656	−1.57036	−0.785179	+	0.619269i	$0.787429\pi$
$810$	0	0
$811$	19.2674	0.676571	0.338286	−	0.941043i	$-0.390153\pi$
$811$	19.2674	0.676571	0.338286	+	0.941043i	$0.390153\pi$
$812$	1.34200	0.0470950
$813$	0	0
$814$	14.4633	0.506937
$815$	9.40479	0.329436
$816$	0	0
$817$	−41.3120	−1.44533
$818$	40.5559	1.41800
$819$	0	0
$820$	−1.71652	−0.0599434
$821$	1.67173	0.0583437	0.0291719	−	0.999574i	$-0.490713\pi$
$821$	1.67173	0.0583437	0.0291719	+	0.999574i	$0.490713\pi$
$822$	0	0
$823$	−1.20807	−0.0421107	−0.0210553	−	0.999778i	$-0.506703\pi$
$823$	−1.20807	−0.0421107	−0.0210553	+	0.999778i	$0.506703\pi$
$824$	−10.0846	−0.351312
$825$	0	0
$826$	12.3665	0.430284
$827$	−22.6076	−0.786145	−0.393072	−	0.919507i	$-0.628588\pi$
$827$	−22.6076	−0.786145	−0.393072	+	0.919507i	$0.628588\pi$
$828$	0	0
$829$	−7.94224	−0.275845	−0.137923	−	0.990443i	$-0.544043\pi$
$829$	−7.94224	−0.275845	−0.137923	+	0.990443i	$0.544043\pi$
$830$	−11.8886	−0.412658
$831$	0	0
$832$	−20.4805	−0.710034
$833$	−1.98218	−0.0686785
$834$	0	0
$835$	9.80931	0.339465
$836$	−2.55492	−0.0883636
$837$	0	0
$838$	−16.9904	−0.586923
$839$	8.39820	0.289938	0.144969	−	0.989436i	$-0.453692\pi$
$839$	8.39820	0.289938	0.144969	+	0.989436i	$0.453692\pi$
$840$	0	0
$841$	4.44457	0.153261
$842$	−10.9438	−0.377149
$843$	0	0
$844$	−1.48906	−0.0512555
$845$	−6.64604	−0.228631
$846$	0	0
$847$	−7.96835	−0.273796
$848$	−41.9873	−1.44185
$849$	0	0
$850$	11.0854	0.380226
$851$	−26.0320	−0.892366
$852$	0	0
$853$	24.9125	0.852989	0.426494	−	0.904490i	$-0.359748\pi$
$853$	24.9125	0.852989	0.426494	+	0.904490i	$0.359748\pi$
$854$	10.5864	0.362259
$855$	0	0
$856$	−12.8973	−0.440821
$857$	−38.1565	−1.30340	−0.651701	−	0.758476i	$-0.725944\pi$
$857$	−38.1565	−1.30340	−0.651701	+	0.758476i	$0.725944\pi$
$858$	0	0
$859$	7.67246	0.261781	0.130891	−	0.991397i	$-0.458216\pi$
$859$	7.67246	0.261781	0.130891	+	0.991397i	$0.458216\pi$
$860$	1.35088	0.0460648
$861$	0	0
$862$	23.4899	0.800071
$863$	−15.6995	−0.534418	−0.267209	−	0.963639i	$-0.586101\pi$
$863$	−15.6995	−0.534418	−0.267209	+	0.963639i	$0.586101\pi$
$864$	0	0
$865$	−22.2532	−0.756632
$866$	−35.7598	−1.21517
$867$	0	0
$868$	−0.532304	−0.0180676
$869$	−0.608318	−0.0206358
$870$	0	0
$871$	29.9069	1.01336
$872$	−0.589807	−0.0199734
$873$	0	0
$874$	−35.0346	−1.18506
$875$	−8.20298	−0.277311
$876$	0	0
$877$	5.71826	0.193092	0.0965459	−	0.995329i	$-0.469221\pi$
$877$	5.71826	0.193092	0.0965459	+	0.995329i	$0.469221\pi$
$878$	−11.8263	−0.399119
$879$	0	0
$880$	−5.40222	−0.182109
$881$	−29.2128	−0.984203	−0.492102	−	0.870538i	$-0.663771\pi$
$881$	−29.2128	−0.984203	−0.492102	+	0.870538i	$0.663771\pi$
$882$	0	0
$883$	−34.0516	−1.14593	−0.572964	−	0.819581i	$-0.694206\pi$
$883$	−34.0516	−1.14593	−0.572964	+	0.819581i	$0.694206\pi$
$884$	−1.08278	−0.0364177
$885$	0	0
$886$	−0.811158	−0.0272514
$887$	14.9316	0.501355	0.250678	−	0.968071i	$-0.419347\pi$
$887$	14.9316	0.501355	0.250678	+	0.968071i	$0.419347\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	15.6050	0.523081
$891$	0	0
$892$	2.32517	0.0778526
$893$	60.2242	2.01532
$894$	0	0
$895$	−0.251100	−0.00839334
$896$	8.95672	0.299223
$897$	0	0
$898$	19.4323	0.648465
$899$	−13.2658	−0.442438
$900$	0	0
$901$	−23.9016	−0.796278
$902$	19.2191	0.639926
$903$	0	0
$904$	47.2417	1.57123
$905$	−3.30132	−0.109740
$906$	0	0
$907$	26.5236	0.880703	0.440352	−	0.897825i	$-0.354854\pi$
$907$	26.5236	0.880703	0.440352	+	0.897825i	$0.354854\pi$
$908$	−0.0459211	−0.00152395
$909$	0	0
$910$	−2.78894	−0.0924523
$911$	−43.5118	−1.44161	−0.720805	−	0.693138i	$-0.756228\pi$
$911$	−43.5118	−1.44161	−0.720805	+	0.693138i	$0.756228\pi$
$912$	0	0
$913$	−17.4717	−0.578229
$914$	−21.8701	−0.723399
$915$	0	0
$916$	−3.08003	−0.101767
$917$	16.9403	0.559417
$918$	0	0
$919$	−4.55465	−0.150244	−0.0751221	−	0.997174i	$-0.523935\pi$
$919$	−4.55465	−0.150244	−0.0751221	+	0.997174i	$0.523935\pi$
$920$	11.0193	0.363295
$921$	0	0
$922$	−4.70513	−0.154955
$923$	−30.3287	−0.998280
$924$	0	0
$925$	−26.2764	−0.863963
$926$	−36.8592	−1.21127
$927$	0	0
$928$	−7.55159	−0.247893
$929$	5.82471	0.191103	0.0955513	−	0.995425i	$-0.469539\pi$
$929$	5.82471	0.191103	0.0955513	+	0.995425i	$0.469539\pi$
$930$	0	0
$931$	6.32335	0.207239
$932$	−1.20234	−0.0393840
$933$	0	0
$934$	34.8523	1.14040
$935$	−3.07526	−0.100572
$936$	0	0
$937$	26.8638	0.877603	0.438802	−	0.898584i	$-0.355403\pi$
$937$	26.8638	0.877603	0.438802	+	0.898584i	$0.355403\pi$
$938$	−16.8928	−0.551569
$939$	0	0
$940$	−1.96930	−0.0642315
$941$	30.4426	0.992400	0.496200	−	0.868208i	$-0.334728\pi$
$941$	30.4426	0.992400	0.496200	+	0.868208i	$0.334728\pi$
$942$	0	0
$943$	−34.5919	−1.12647
$944$	−32.3851	−1.05404
$945$	0	0
$946$	−15.1253	−0.491765
$947$	42.2318	1.37235	0.686174	−	0.727437i	$-0.259289\pi$
$947$	42.2318	1.37235	0.686174	+	0.727437i	$0.259289\pi$
$948$	0	0
$949$	5.28168	0.171451
$950$	−35.3635	−1.14734
$951$	0	0
$952$	5.88278	0.190662
$953$	8.00459	0.259294	0.129647	−	0.991560i	$-0.458616\pi$
$953$	8.00459	0.259294	0.129647	+	0.991560i	$0.458616\pi$
$954$	0	0
$955$	17.1676	0.555530
$956$	−6.44357	−0.208400
$957$	0	0
$958$	−20.3801	−0.658452
$959$	20.1803	0.651656
$960$	0	0
$961$	−25.7381	−0.830263
$962$	−19.5538	−0.630441
$963$	0	0
$964$	−0.268795	−0.00865731
$965$	21.2207	0.683118
$966$	0	0
$967$	24.9595	0.802645	0.401322	−	0.915937i	$-0.368551\pi$
$967$	24.9595	0.802645	0.401322	+	0.915937i	$0.368551\pi$
$968$	23.6487	0.760099
$969$	0	0
$970$	21.2356	0.681835
$971$	−16.8290	−0.540070	−0.270035	−	0.962851i	$-0.587035\pi$
$971$	−16.8290	−0.540070	−0.270035	+	0.962851i	$0.587035\pi$
$972$	0	0
$973$	−21.4560	−0.687847
$974$	−31.9824	−1.02478
$975$	0	0
$976$	−27.7235	−0.887407
$977$	−38.7429	−1.23949	−0.619747	−	0.784802i	$-0.712765\pi$
$977$	−38.7429	−1.23949	−0.619747	+	0.784802i	$0.712765\pi$
$978$	0	0
$979$	22.9335	0.732957
$980$	−0.206771	−0.00660505
$981$	0	0
$982$	−40.2778	−1.28531
$983$	−29.8344	−0.951571	−0.475785	−	0.879561i	$-0.657836\pi$
$983$	−29.8344	−0.951571	−0.475785	+	0.879561i	$0.657836\pi$
$984$	0	0
$985$	−10.0915	−0.321542
$986$	15.2420	0.485403
$987$	0	0
$988$	3.45416	0.109891
$989$	27.2235	0.865658
$990$	0	0
$991$	34.7227	1.10300	0.551501	−	0.834174i	$-0.314055\pi$
$991$	34.7227	1.10300	0.551501	+	0.834174i	$0.314055\pi$
$992$	2.99533	0.0951020
$993$	0	0
$994$	17.1310	0.543362
$995$	−4.68329	−0.148470
$996$	0	0
$997$	−43.7566	−1.38579	−0.692893	−	0.721040i	$-0.743664\pi$
$997$	−43.7566	−1.38579	−0.692893	+	0.721040i	$0.743664\pi$
$998$	−3.06002	−0.0968634
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.n.1.8		12
3.2	odd	2		889.2.a.a.1.5	✓	12
21.20	even	2		6223.2.a.i.1.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.a.1.5	✓	12	3.2	odd	2
6223.2.a.i.1.5		12	21.20	even	2
8001.2.a.n.1.8		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.32964 q^{2} -0.232054 q^{4} +0.891044 q^{5} +1.00000 q^{7} -2.96783 q^{8} +O(q^{10})\)	\(q+1.32964 q^{2} -0.232054 q^{4} +0.891044 q^{5} +1.00000 q^{7} -2.96783 q^{8} +1.18477 q^{10} +1.74116 q^{11} -2.35399 q^{13} +1.32964 q^{14} -3.48204 q^{16} -1.98218 q^{17} +6.32335 q^{19} -0.206771 q^{20} +2.31512 q^{22} -4.16692 q^{23} -4.20604 q^{25} -3.12997 q^{26} -0.232054 q^{28} -5.78313 q^{29} +2.29387 q^{31} +1.30580 q^{32} -2.63559 q^{34} +0.891044 q^{35} +6.24730 q^{37} +8.40779 q^{38} -2.64447 q^{40} +8.30155 q^{41} -6.53325 q^{43} -0.404045 q^{44} -5.54051 q^{46} +9.52409 q^{47} +1.00000 q^{49} -5.59253 q^{50} +0.546255 q^{52} +12.0582 q^{53} +1.55145 q^{55} -2.96783 q^{56} -7.68949 q^{58} +9.30060 q^{59} +7.96184 q^{61} +3.05003 q^{62} +8.70032 q^{64} -2.09751 q^{65} -12.7048 q^{67} +0.459974 q^{68} +1.18477 q^{70} +12.8839 q^{71} -2.24371 q^{73} +8.30667 q^{74} -1.46736 q^{76} +1.74116 q^{77} -0.349374 q^{79} -3.10265 q^{80} +11.0381 q^{82} -10.0345 q^{83} -1.76621 q^{85} -8.68688 q^{86} -5.16748 q^{88} +13.1714 q^{89} -2.35399 q^{91} +0.966952 q^{92} +12.6636 q^{94} +5.63438 q^{95} +17.9239 q^{97} +1.32964 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) \) 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8	\( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8} - 2 q^{10} + 22 q^{11} + 7 q^{14} + 7 q^{16} + 6 q^{17} - 7 q^{19} + 8 q^{20} + 13 q^{22} + 29 q^{23} + 3 q^{25} + 9 q^{28} + 22 q^{29} - 16 q^{31} + 27 q^{32} - 5 q^{34} + 7 q^{35} - 4 q^{37} - 2 q^{38} + 16 q^{40} + 21 q^{41} + 11 q^{43} + 11 q^{44} + 31 q^{47} + 12 q^{49} + 21 q^{50} + 3 q^{52} + 38 q^{53} - 11 q^{55} + 15 q^{56} + 20 q^{58} + 15 q^{59} - 3 q^{61} + 4 q^{62} + 29 q^{64} + 32 q^{65} - q^{67} - 17 q^{68} - 2 q^{70} + 57 q^{71} - 7 q^{73} + 42 q^{74} - 44 q^{76} + 22 q^{77} - 18 q^{79} - q^{80} + 56 q^{82} + 21 q^{83} - 5 q^{85} + 32 q^{86} - 10 q^{88} - 6 q^{89} + 15 q^{92} + 35 q^{94} + 57 q^{95} + 4 q^{97} + 7 q^{98}+O(q^{100}) \) 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8 - 2 * q^10 + 22 * q^11 + 7 * q^14 + 7 * q^16 + 6 * q^17 - 7 * q^19 + 8 * q^20 + 13 * q^22 + 29 * q^23 + 3 * q^25 + 9 * q^28 + 22 * q^29 - 16 * q^31 + 27 * q^32 - 5 * q^34 + 7 * q^35 - 4 * q^37 - 2 * q^38 + 16 * q^40 + 21 * q^41 + 11 * q^43 + 11 * q^44 + 31 * q^47 + 12 * q^49 + 21 * q^50 + 3 * q^52 + 38 * q^53 - 11 * q^55 + 15 * q^56 + 20 * q^58 + 15 * q^59 - 3 * q^61 + 4 * q^62 + 29 * q^64 + 32 * q^65 - q^67 - 17 * q^68 - 2 * q^70 + 57 * q^71 - 7 * q^73 + 42 * q^74 - 44 * q^76 + 22 * q^77 - 18 * q^79 - q^80 + 56 * q^82 + 21 * q^83 - 5 * q^85 + 32 * q^86 - 10 * q^88 - 6 * q^89 + 15 * q^92 + 35 * q^94 + 57 * q^95 + 4 * q^97 + 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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