LMFDB - Embedded newform 8001.2.a.w.1.3

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} + 152 x^{17} - 274 x^{16} - 1061 x^{15} + 3125 x^{14} + 2977 x^{13} - 15474 x^{12} + \cdots + 31 $ x^20 - 8x^19 + 152x^17 - 274x^16 - 1061x^15 + 3125x^14 + 2977x^13 - 15474x^12 - 56x^11 + 39579x^10 - 17664x^9 - 52271x^8 + 35701x^7 + 32493x^6 - 25504x^5 - 8607x^4 + 6812x^3 + 609x^2 - 425x + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.58654$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.58654 q^{2} +4.69021 q^{4} -1.18795 q^{5} +1.00000 q^{7} -6.95835 q^{8} +O(q^{10})$	\(q-2.58654 q^{2} +4.69021 q^{4} -1.18795 q^{5} +1.00000 q^{7} -6.95835 q^{8} +3.07267 q^{10} +1.95437 q^{11} -4.53210 q^{13} -2.58654 q^{14} +8.61765 q^{16} -3.82267 q^{17} -1.81798 q^{19} -5.57171 q^{20} -5.05507 q^{22} -1.66510 q^{23} -3.58879 q^{25} +11.7225 q^{26} +4.69021 q^{28} -1.48215 q^{29} -2.48876 q^{31} -8.37324 q^{32} +9.88751 q^{34} -1.18795 q^{35} +4.79658 q^{37} +4.70229 q^{38} +8.26614 q^{40} +8.09104 q^{41} +11.2964 q^{43} +9.16642 q^{44} +4.30686 q^{46} +11.2775 q^{47} +1.00000 q^{49} +9.28255 q^{50} -21.2565 q^{52} -7.14761 q^{53} -2.32169 q^{55} -6.95835 q^{56} +3.83366 q^{58} +5.62494 q^{59} +0.689991 q^{61} +6.43728 q^{62} +4.42246 q^{64} +5.38389 q^{65} +14.0307 q^{67} -17.9291 q^{68} +3.07267 q^{70} -7.44927 q^{71} +5.29270 q^{73} -12.4066 q^{74} -8.52672 q^{76} +1.95437 q^{77} -4.25002 q^{79} -10.2373 q^{80} -20.9278 q^{82} -2.64391 q^{83} +4.54113 q^{85} -29.2186 q^{86} -13.5992 q^{88} -3.43526 q^{89} -4.53210 q^{91} -7.80968 q^{92} -29.1698 q^{94} +2.15966 q^{95} +14.3656 q^{97} -2.58654 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) $ 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8	$ 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28} - 16 q^{29} + 6 q^{31} - 41 q^{32} - 10 q^{34} - 3 q^{35} + 2 q^{37} - 3 q^{38} - 38 q^{40} - 25 q^{41} + 13 q^{43} - 66 q^{44} + 20 q^{46} - 19 q^{47} + 20 q^{49} + 4 q^{50} + 20 q^{52} - 24 q^{53} - 3 q^{55} - 24 q^{56} + 12 q^{58} - 23 q^{59} - 27 q^{61} - 7 q^{62} + 2 q^{64} - 26 q^{65} + 9 q^{67} + 25 q^{68} - 8 q^{70} - 63 q^{71} - 21 q^{73} - 21 q^{74} - 10 q^{76} - 26 q^{77} + 18 q^{79} + 23 q^{80} - 42 q^{82} + q^{83} - 41 q^{85} + 12 q^{86} + 57 q^{88} + 16 q^{89} - 4 q^{91} - 17 q^{92} + 7 q^{94} - 75 q^{95} - 32 q^{97} - 8 q^{98}+O(q^{100}) $ 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8 - 8 * q^10 - 26 * q^11 - 4 * q^13 - 8 * q^14 + 24 * q^16 - 4 * q^17 + q^19 + 2 * q^20 + q^22 - 31 * q^23 + 27 * q^25 - 4 * q^26 + 24 * q^28 - 16 * q^29 + 6 * q^31 - 41 * q^32 - 10 * q^34 - 3 * q^35 + 2 * q^37 - 3 * q^38 - 38 * q^40 - 25 * q^41 + 13 * q^43 - 66 * q^44 + 20 * q^46 - 19 * q^47 + 20 * q^49 + 4 * q^50 + 20 * q^52 - 24 * q^53 - 3 * q^55 - 24 * q^56 + 12 * q^58 - 23 * q^59 - 27 * q^61 - 7 * q^62 + 2 * q^64 - 26 * q^65 + 9 * q^67 + 25 * q^68 - 8 * q^70 - 63 * q^71 - 21 * q^73 - 21 * q^74 - 10 * q^76 - 26 * q^77 + 18 * q^79 + 23 * q^80 - 42 * q^82 + q^83 - 41 * q^85 + 12 * q^86 + 57 * q^88 + 16 * q^89 - 4 * q^91 - 17 * q^92 + 7 * q^94 - 75 * q^95 - 32 * q^97 - 8 * 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 $

$2$	−2.58654	−1.82896	−0.914481	−	0.404628i	$-0.867401\pi$
$2$	−2.58654	−1.82896	−0.914481	+	0.404628i	$0.867401\pi$
$3$	0	0
$3$	0	0
$4$	4.69021	2.34511
$5$	−1.18795	−0.531265	−0.265633	−	0.964074i	$-0.585581\pi$
$5$	−1.18795	−0.531265	−0.265633	+	0.964074i	$0.585581\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−6.95835	−2.46015
$9$	0	0
$10$	3.07267	0.971665
$11$	1.95437	0.589266	0.294633	−	0.955611i	$-0.404803\pi$
$11$	1.95437	0.589266	0.294633	+	0.955611i	$0.404803\pi$
$12$	0	0
$13$	−4.53210	−1.25698	−0.628490	−	0.777818i	$-0.716327\pi$
$13$	−4.53210	−1.25698	−0.628490	+	0.777818i	$0.716327\pi$
$14$	−2.58654	−0.691283
$15$	0	0
$16$	8.61765	2.15441
$17$	−3.82267	−0.927134	−0.463567	−	0.886062i	$-0.653431\pi$
$17$	−3.82267	−0.927134	−0.463567	+	0.886062i	$0.653431\pi$
$18$	0	0
$19$	−1.81798	−0.417074	−0.208537	−	0.978015i	$-0.566870\pi$
$19$	−1.81798	−0.417074	−0.208537	+	0.978015i	$0.566870\pi$
$20$	−5.57171	−1.24587
$21$	0	0
$22$	−5.05507	−1.07775
$23$	−1.66510	−0.347198	−0.173599	−	0.984816i	$-0.555540\pi$
$23$	−1.66510	−0.347198	−0.173599	+	0.984816i	$0.555540\pi$
$24$	0	0
$25$	−3.58879	−0.717757
$26$	11.7225	2.29897
$27$	0	0
$28$	4.69021	0.886366
$29$	−1.48215	−0.275229	−0.137615	−	0.990486i	$-0.543944\pi$
$29$	−1.48215	−0.275229	−0.137615	+	0.990486i	$0.543944\pi$
$30$	0	0
$31$	−2.48876	−0.446994	−0.223497	−	0.974705i	$-0.571747\pi$
$31$	−2.48876	−0.446994	−0.223497	+	0.974705i	$0.571747\pi$
$32$	−8.37324	−1.48019
$33$	0	0
$34$	9.88751	1.69569
$35$	−1.18795	−0.200799
$36$	0	0
$37$	4.79658	0.788553	0.394276	−	0.918992i	$-0.370995\pi$
$37$	4.79658	0.788553	0.394276	+	0.918992i	$0.370995\pi$
$38$	4.70229	0.762812
$39$	0	0
$40$	8.26614	1.30699
$41$	8.09104	1.26361	0.631804	−	0.775128i	$-0.282315\pi$
$41$	8.09104	1.26361	0.631804	+	0.775128i	$0.282315\pi$
$42$	0	0
$43$	11.2964	1.72269	0.861343	−	0.508024i	$-0.169624\pi$
$43$	11.2964	1.72269	0.861343	+	0.508024i	$0.169624\pi$
$44$	9.16642	1.38189
$45$	0	0
$46$	4.30686	0.635012
$47$	11.2775	1.64500	0.822498	−	0.568768i	$-0.192580\pi$
$47$	11.2775	1.64500	0.822498	+	0.568768i	$0.192580\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	9.28255	1.31275
$51$	0	0
$52$	−21.2565	−2.94775
$53$	−7.14761	−0.981800	−0.490900	−	0.871216i	$-0.663332\pi$
$53$	−7.14761	−0.981800	−0.490900	+	0.871216i	$0.663332\pi$
$54$	0	0
$55$	−2.32169	−0.313057
$56$	−6.95835	−0.929848
$57$	0	0
$58$	3.83366	0.503384
$59$	5.62494	0.732304	0.366152	−	0.930555i	$-0.380675\pi$
$59$	5.62494	0.732304	0.366152	+	0.930555i	$0.380675\pi$
$60$	0	0
$61$	0.689991	0.0883443	0.0441721	−	0.999024i	$-0.485935\pi$
$61$	0.689991	0.0883443	0.0441721	+	0.999024i	$0.485935\pi$
$62$	6.43728	0.817535
$63$	0	0
$64$	4.42246	0.552807
$65$	5.38389	0.667789
$66$	0	0
$67$	14.0307	1.71412	0.857062	−	0.515214i	$-0.172287\pi$
$67$	14.0307	1.71412	0.857062	+	0.515214i	$0.172287\pi$
$68$	−17.9291	−2.17423
$69$	0	0
$70$	3.07267	0.367255
$71$	−7.44927	−0.884065	−0.442033	−	0.896999i	$-0.645742\pi$
$71$	−7.44927	−0.884065	−0.442033	+	0.896999i	$0.645742\pi$
$72$	0	0
$73$	5.29270	0.619464	0.309732	−	0.950824i	$-0.399761\pi$
$73$	5.29270	0.619464	0.309732	+	0.950824i	$0.399761\pi$
$74$	−12.4066	−1.44223
$75$	0	0
$76$	−8.52672	−0.978081
$77$	1.95437	0.222722
$78$	0	0
$79$	−4.25002	−0.478165	−0.239082	−	0.970999i	$-0.576847\pi$
$79$	−4.25002	−0.478165	−0.239082	+	0.970999i	$0.576847\pi$
$80$	−10.2373	−1.14457
$81$	0	0
$82$	−20.9278	−2.31109
$83$	−2.64391	−0.290207	−0.145103	−	0.989417i	$-0.546351\pi$
$83$	−2.64391	−0.290207	−0.145103	+	0.989417i	$0.546351\pi$
$84$	0	0
$85$	4.54113	0.492554
$86$	−29.2186	−3.15073
$87$	0	0
$88$	−13.5992	−1.44968
$89$	−3.43526	−0.364137	−0.182068	−	0.983286i	$-0.558279\pi$
$89$	−3.43526	−0.364137	−0.182068	+	0.983286i	$0.558279\pi$
$90$	0	0
$91$	−4.53210	−0.475093
$92$	−7.80968	−0.814215
$93$	0	0
$94$	−29.1698	−3.00864
$95$	2.15966	0.221577
$96$	0	0
$97$	14.3656	1.45861	0.729304	−	0.684189i	$-0.239844\pi$
$97$	14.3656	1.45861	0.729304	+	0.684189i	$0.239844\pi$
$98$	−2.58654	−0.261280
$99$	0	0
$100$	−16.8322	−1.68322
$101$	−5.40980	−0.538295	−0.269148	−	0.963099i	$-0.586742\pi$
$101$	−5.40980	−0.538295	−0.269148	+	0.963099i	$0.586742\pi$
$102$	0	0
$103$	6.97135	0.686907	0.343454	−	0.939170i	$-0.388403\pi$
$103$	6.97135	0.686907	0.343454	+	0.939170i	$0.388403\pi$
$104$	31.5359	3.09235
$105$	0	0
$106$	18.4876	1.79568
$107$	5.36742	0.518888	0.259444	−	0.965758i	$-0.416461\pi$
$107$	5.36742	0.518888	0.259444	+	0.965758i	$0.416461\pi$
$108$	0	0
$109$	−12.8606	−1.23182	−0.615912	−	0.787815i	$-0.711212\pi$
$109$	−12.8606	−1.23182	−0.615912	+	0.787815i	$0.711212\pi$
$110$	6.00515	0.572569
$111$	0	0
$112$	8.61765	0.814292
$113$	7.74153	0.728262	0.364131	−	0.931348i	$-0.381366\pi$
$113$	7.74153	0.728262	0.364131	+	0.931348i	$0.381366\pi$
$114$	0	0
$115$	1.97805	0.184454
$116$	−6.95162	−0.645441
$117$	0	0
$118$	−14.5491	−1.33936
$119$	−3.82267	−0.350424
$120$	0	0
$121$	−7.18042	−0.652766
$122$	−1.78469	−0.161578
$123$	0	0
$124$	−11.6728	−1.04825
$125$	10.2030	0.912585
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	5.30760	0.469130
$129$	0	0
$130$	−13.9257	−1.22136
$131$	7.06508	0.617279	0.308639	−	0.951179i	$-0.400126\pi$
$131$	7.06508	0.617279	0.308639	+	0.951179i	$0.400126\pi$
$132$	0	0
$133$	−1.81798	−0.157639
$134$	−36.2910	−3.13507
$135$	0	0
$136$	26.5995	2.28089
$137$	5.26269	0.449622	0.224811	−	0.974402i	$-0.427824\pi$
$137$	5.26269	0.449622	0.224811	+	0.974402i	$0.427824\pi$
$138$	0	0
$139$	3.34254	0.283510	0.141755	−	0.989902i	$-0.454725\pi$
$139$	3.34254	0.283510	0.141755	+	0.989902i	$0.454725\pi$
$140$	−5.57171	−0.470896
$141$	0	0
$142$	19.2679	1.61692
$143$	−8.85742	−0.740695
$144$	0	0
$145$	1.76072	0.146220
$146$	−13.6898	−1.13298
$147$	0	0
$148$	22.4970	1.84924
$149$	−22.6184	−1.85297	−0.926487	−	0.376327i	$-0.877187\pi$
$149$	−22.6184	−1.85297	−0.926487	+	0.376327i	$0.877187\pi$
$150$	0	0
$151$	−10.4673	−0.851813	−0.425906	−	0.904767i	$-0.640045\pi$
$151$	−10.4673	−0.851813	−0.425906	+	0.904767i	$0.640045\pi$
$152$	12.6501	1.02606
$153$	0	0
$154$	−5.05507	−0.407349
$155$	2.95651	0.237472
$156$	0	0
$157$	−18.1467	−1.44826	−0.724131	−	0.689663i	$-0.757759\pi$
$157$	−18.1467	−1.44826	−0.724131	+	0.689663i	$0.757759\pi$
$158$	10.9929	0.874546
$159$	0	0
$160$	9.94696	0.786376
$161$	−1.66510	−0.131228
$162$	0	0
$163$	5.66211	0.443491	0.221745	−	0.975105i	$-0.428825\pi$
$163$	5.66211	0.443491	0.221745	+	0.975105i	$0.428825\pi$
$164$	37.9487	2.96330
$165$	0	0
$166$	6.83859	0.530777
$167$	16.6450	1.28803	0.644014	−	0.765014i	$-0.277268\pi$
$167$	16.6450	1.28803	0.644014	+	0.765014i	$0.277268\pi$
$168$	0	0
$169$	7.53995	0.579996
$170$	−11.7458	−0.900863
$171$	0	0
$172$	52.9825	4.03988
$173$	−18.4240	−1.40075	−0.700376	−	0.713774i	$-0.746984\pi$
$173$	−18.4240	−1.40075	−0.700376	+	0.713774i	$0.746984\pi$
$174$	0	0
$175$	−3.58879	−0.271287
$176$	16.8421	1.26952
$177$	0	0
$178$	8.88545	0.665992
$179$	6.65351	0.497307	0.248653	−	0.968593i	$-0.420012\pi$
$179$	6.65351	0.497307	0.248653	+	0.968593i	$0.420012\pi$
$180$	0	0
$181$	−22.5225	−1.67409	−0.837043	−	0.547137i	$-0.815718\pi$
$181$	−22.5225	−1.67409	−0.837043	+	0.547137i	$0.815718\pi$
$182$	11.7225	0.868928
$183$	0	0
$184$	11.5864	0.854158
$185$	−5.69807	−0.418931
$186$	0	0
$187$	−7.47093	−0.546329
$188$	52.8940	3.85769
$189$	0	0
$190$	−5.58606	−0.405256
$191$	−24.2129	−1.75198	−0.875992	−	0.482326i	$-0.839792\pi$
$191$	−24.2129	−1.75198	−0.875992	+	0.482326i	$0.839792\pi$
$192$	0	0
$193$	2.20716	0.158875	0.0794373	−	0.996840i	$-0.474688\pi$
$193$	2.20716	0.158875	0.0794373	+	0.996840i	$0.474688\pi$
$194$	−37.1573	−2.66774
$195$	0	0
$196$	4.69021	0.335015
$197$	3.09584	0.220570	0.110285	−	0.993900i	$-0.464824\pi$
$197$	3.09584	0.220570	0.110285	+	0.993900i	$0.464824\pi$
$198$	0	0
$199$	6.88412	0.488002	0.244001	−	0.969775i	$-0.421540\pi$
$199$	6.88412	0.488002	0.244001	+	0.969775i	$0.421540\pi$
$200$	24.9720	1.76579
$201$	0	0
$202$	13.9927	0.984522
$203$	−1.48215	−0.104027
$204$	0	0
$205$	−9.61172	−0.671312
$206$	−18.0317	−1.25633
$207$	0	0
$208$	−39.0561	−2.70805
$209$	−3.55302	−0.245767
$210$	0	0
$211$	−7.84220	−0.539879	−0.269940	−	0.962877i	$-0.587004\pi$
$211$	−7.84220	−0.539879	−0.269940	+	0.962877i	$0.587004\pi$
$212$	−33.5238	−2.30242
$213$	0	0
$214$	−13.8831	−0.949028
$215$	−13.4195	−0.915203
$216$	0	0
$217$	−2.48876	−0.168948
$218$	33.2646	2.25296
$219$	0	0
$220$	−10.8892	−0.734151
$221$	17.3247	1.16539
$222$	0	0
$223$	8.38216	0.561311	0.280655	−	0.959809i	$-0.409448\pi$
$223$	8.38216	0.561311	0.280655	+	0.959809i	$0.409448\pi$
$224$	−8.37324	−0.559461
$225$	0	0
$226$	−20.0238	−1.33196
$227$	−11.5434	−0.766164	−0.383082	−	0.923714i	$-0.625137\pi$
$227$	−11.5434	−0.766164	−0.383082	+	0.923714i	$0.625137\pi$
$228$	0	0
$229$	−14.5266	−0.959948	−0.479974	−	0.877283i	$-0.659354\pi$
$229$	−14.5266	−0.959948	−0.479974	+	0.877283i	$0.659354\pi$
$230$	−5.11632	−0.337360
$231$	0	0
$232$	10.3133	0.677104
$233$	−14.8006	−0.969620	−0.484810	−	0.874619i	$-0.661111\pi$
$233$	−14.8006	−0.969620	−0.484810	+	0.874619i	$0.661111\pi$
$234$	0	0
$235$	−13.3971	−0.873929
$236$	26.3821	1.71733
$237$	0	0
$238$	9.88751	0.640912
$239$	3.91897	0.253497	0.126748	−	0.991935i	$-0.459546\pi$
$239$	3.91897	0.253497	0.126748	+	0.991935i	$0.459546\pi$
$240$	0	0
$241$	−8.70319	−0.560622	−0.280311	−	0.959909i	$-0.590438\pi$
$241$	−8.70319	−0.560622	−0.280311	+	0.959909i	$0.590438\pi$
$242$	18.5725	1.19388
$243$	0	0
$244$	3.23620	0.207177
$245$	−1.18795	−0.0758951
$246$	0	0
$247$	8.23928	0.524253
$248$	17.3176	1.09967
$249$	0	0
$250$	−26.3905	−1.66908
$251$	5.54946	0.350279	0.175139	−	0.984544i	$-0.443962\pi$
$251$	5.54946	0.350279	0.175139	+	0.984544i	$0.443962\pi$
$252$	0	0
$253$	−3.25423	−0.204592
$254$	2.58654	0.162294
$255$	0	0
$256$	−22.5733	−1.41083
$257$	−0.287396	−0.0179273	−0.00896364	−	0.999960i	$-0.502853\pi$
$257$	−0.287396	−0.0179273	−0.00896364	+	0.999960i	$0.502853\pi$
$258$	0	0
$259$	4.79658	0.298045
$260$	25.2516	1.56604
$261$	0	0
$262$	−18.2741	−1.12898
$263$	−21.9781	−1.35523	−0.677615	−	0.735417i	$-0.736986\pi$
$263$	−21.9781	−1.35523	−0.677615	+	0.735417i	$0.736986\pi$
$264$	0	0
$265$	8.49097	0.521596
$266$	4.70229	0.288316
$267$	0	0
$268$	65.8070	4.01980
$269$	28.0186	1.70832	0.854161	−	0.520009i	$-0.174071\pi$
$269$	28.0186	1.70832	0.854161	+	0.520009i	$0.174071\pi$
$270$	0	0
$271$	3.83266	0.232817	0.116409	−	0.993201i	$-0.462862\pi$
$271$	3.83266	0.232817	0.116409	+	0.993201i	$0.462862\pi$
$272$	−32.9425	−1.99743
$273$	0	0
$274$	−13.6122	−0.822342
$275$	−7.01383	−0.422950
$276$	0	0
$277$	17.9035	1.07572	0.537859	−	0.843035i	$-0.319233\pi$
$277$	17.9035	1.07572	0.537859	+	0.843035i	$0.319233\pi$
$278$	−8.64562	−0.518530
$279$	0	0
$280$	8.26614	0.493996
$281$	−26.0485	−1.55392	−0.776961	−	0.629549i	$-0.783240\pi$
$281$	−26.0485	−1.55392	−0.776961	+	0.629549i	$0.783240\pi$
$282$	0	0
$283$	8.74541	0.519861	0.259930	−	0.965627i	$-0.416300\pi$
$283$	8.74541	0.519861	0.259930	+	0.965627i	$0.416300\pi$
$284$	−34.9386	−2.07323
$285$	0	0
$286$	22.9101	1.35470
$287$	8.09104	0.477599
$288$	0	0
$289$	−2.38718	−0.140422
$290$	−4.55418	−0.267430
$291$	0	0
$292$	24.8239	1.45271
$293$	4.90307	0.286440	0.143220	−	0.989691i	$-0.454254\pi$
$293$	4.90307	0.286440	0.143220	+	0.989691i	$0.454254\pi$
$294$	0	0
$295$	−6.68212	−0.389048
$296$	−33.3763	−1.93996
$297$	0	0
$298$	58.5036	3.38902
$299$	7.54641	0.436420
$300$	0	0
$301$	11.2964	0.651114
$302$	27.0740	1.55793
$303$	0	0
$304$	−15.6667	−0.898549
$305$	−0.819671	−0.0469342
$306$	0	0
$307$	17.4163	0.994003	0.497001	−	0.867750i	$-0.334434\pi$
$307$	17.4163	0.994003	0.497001	+	0.867750i	$0.334434\pi$
$308$	9.16642	0.522305
$309$	0	0
$310$	−7.64714	−0.434328
$311$	−31.0678	−1.76170	−0.880848	−	0.473400i	$-0.843027\pi$
$311$	−31.0678	−1.76170	−0.880848	+	0.473400i	$0.843027\pi$
$312$	0	0
$313$	5.49443	0.310564	0.155282	−	0.987870i	$-0.450371\pi$
$313$	5.49443	0.310564	0.155282	+	0.987870i	$0.450371\pi$
$314$	46.9371	2.64882
$315$	0	0
$316$	−19.9335	−1.12135
$317$	−16.0147	−0.899474	−0.449737	−	0.893161i	$-0.648482\pi$
$317$	−16.0147	−0.899474	−0.449737	+	0.893161i	$0.648482\pi$
$318$	0	0
$319$	−2.89668	−0.162183
$320$	−5.25364	−0.293687
$321$	0	0
$322$	4.30686	0.240012
$323$	6.94955	0.386683
$324$	0	0
$325$	16.2647	0.902206
$326$	−14.6453	−0.811128
$327$	0	0
$328$	−56.3003	−3.10866
$329$	11.2775	0.621750
$330$	0	0
$331$	0.944806	0.0519312	0.0259656	−	0.999663i	$-0.491734\pi$
$331$	0.944806	0.0519312	0.0259656	+	0.999663i	$0.491734\pi$
$332$	−12.4005	−0.680565
$333$	0	0
$334$	−43.0530	−2.35576
$335$	−16.6677	−0.910654
$336$	0	0
$337$	−27.2884	−1.48649	−0.743246	−	0.669018i	$-0.766715\pi$
$337$	−27.2884	−1.48649	−0.743246	+	0.669018i	$0.766715\pi$
$338$	−19.5024	−1.06079
$339$	0	0
$340$	21.2988	1.15509
$341$	−4.86396	−0.263398
$342$	0	0
$343$	1.00000	0.0539949
$344$	−78.6043	−4.23806
$345$	0	0
$346$	47.6545	2.56192
$347$	25.2582	1.35593	0.677966	−	0.735093i	$-0.262862\pi$
$347$	25.2582	1.35593	0.677966	+	0.735093i	$0.262862\pi$
$348$	0	0
$349$	−15.4822	−0.828745	−0.414372	−	0.910107i	$-0.635999\pi$
$349$	−15.4822	−0.828745	−0.414372	+	0.910107i	$0.635999\pi$
$350$	9.28255	0.496173
$351$	0	0
$352$	−16.3644	−0.872228
$353$	−20.3899	−1.08525	−0.542623	−	0.839976i	$-0.682569\pi$
$353$	−20.3899	−1.08525	−0.542623	+	0.839976i	$0.682569\pi$
$354$	0	0
$355$	8.84932	0.469673
$356$	−16.1121	−0.853939
$357$	0	0
$358$	−17.2096	−0.909555
$359$	−12.3986	−0.654375	−0.327188	−	0.944959i	$-0.606101\pi$
$359$	−12.3986	−0.654375	−0.327188	+	0.944959i	$0.606101\pi$
$360$	0	0
$361$	−15.6949	−0.826050
$362$	58.2555	3.06184
$363$	0	0
$364$	−21.2565	−1.11414
$365$	−6.28744	−0.329100
$366$	0	0
$367$	18.4016	0.960555	0.480277	−	0.877117i	$-0.340536\pi$
$367$	18.4016	0.960555	0.480277	+	0.877117i	$0.340536\pi$
$368$	−14.3493	−0.748008
$369$	0	0
$370$	14.7383	0.766209
$371$	−7.14761	−0.371085
$372$	0	0
$373$	−36.0243	−1.86527	−0.932635	−	0.360822i	$-0.882496\pi$
$373$	−36.0243	−1.86527	−0.932635	+	0.360822i	$0.882496\pi$
$374$	19.3239	0.999215
$375$	0	0
$376$	−78.4729	−4.04693
$377$	6.71728	0.345957
$378$	0	0
$379$	13.4489	0.690825	0.345412	−	0.938451i	$-0.387739\pi$
$379$	13.4489	0.690825	0.345412	+	0.938451i	$0.387739\pi$
$380$	10.1293	0.519621
$381$	0	0
$382$	62.6277	3.20431
$383$	−19.0263	−0.972200	−0.486100	−	0.873903i	$-0.661581\pi$
$383$	−19.0263	−0.972200	−0.486100	+	0.873903i	$0.661581\pi$
$384$	0	0
$385$	−2.32169	−0.118324
$386$	−5.70891	−0.290576
$387$	0	0
$388$	67.3778	3.42059
$389$	−9.87341	−0.500602	−0.250301	−	0.968168i	$-0.580530\pi$
$389$	−9.87341	−0.500602	−0.250301	+	0.968168i	$0.580530\pi$
$390$	0	0
$391$	6.36514	0.321899
$392$	−6.95835	−0.351450
$393$	0	0
$394$	−8.00753	−0.403413
$395$	5.04879	0.254032
$396$	0	0
$397$	5.02735	0.252315	0.126158	−	0.992010i	$-0.459735\pi$
$397$	5.02735	0.252315	0.126158	+	0.992010i	$0.459735\pi$
$398$	−17.8061	−0.892538
$399$	0	0
$400$	−30.9269	−1.54635
$401$	−5.24824	−0.262085	−0.131042	−	0.991377i	$-0.541832\pi$
$401$	−5.24824	−0.262085	−0.131042	+	0.991377i	$0.541832\pi$
$402$	0	0
$403$	11.2793	0.561862
$404$	−25.3731	−1.26236
$405$	0	0
$406$	3.83366	0.190261
$407$	9.37431	0.464667
$408$	0	0
$409$	20.0832	0.993049	0.496525	−	0.868023i	$-0.334609\pi$
$409$	20.0832	0.993049	0.496525	+	0.868023i	$0.334609\pi$
$410$	24.8611	1.22780
$411$	0	0
$412$	32.6971	1.61087
$413$	5.62494	0.276785
$414$	0	0
$415$	3.14082	0.154177
$416$	37.9484	1.86057
$417$	0	0
$418$	9.19003	0.449499
$419$	27.8199	1.35909	0.679544	−	0.733634i	$-0.262178\pi$
$419$	27.8199	1.35909	0.679544	+	0.733634i	$0.262178\pi$
$420$	0	0
$421$	−20.0436	−0.976866	−0.488433	−	0.872602i	$-0.662431\pi$
$421$	−20.0436	−0.976866	−0.488433	+	0.872602i	$0.662431\pi$
$422$	20.2842	0.987419
$423$	0	0
$424$	49.7356	2.41537
$425$	13.7188	0.665457
$426$	0	0
$427$	0.689991	0.0333910
$428$	25.1743	1.21685
$429$	0	0
$430$	34.7102	1.67387
$431$	26.2131	1.26264	0.631319	−	0.775523i	$-0.282514\pi$
$431$	26.2131	1.26264	0.631319	+	0.775523i	$0.282514\pi$
$432$	0	0
$433$	−7.15126	−0.343668	−0.171834	−	0.985126i	$-0.554969\pi$
$433$	−7.15126	−0.343668	−0.171834	+	0.985126i	$0.554969\pi$
$434$	6.43728	0.308999
$435$	0	0
$436$	−60.3190	−2.88876
$437$	3.02712	0.144807
$438$	0	0
$439$	−6.53862	−0.312071	−0.156036	−	0.987751i	$-0.549871\pi$
$439$	−6.53862	−0.312071	−0.156036	+	0.987751i	$0.549871\pi$
$440$	16.1551	0.770165
$441$	0	0
$442$	−44.8112	−2.13145
$443$	−28.0826	−1.33424	−0.667122	−	0.744948i	$-0.732474\pi$
$443$	−28.0826	−1.33424	−0.667122	+	0.744948i	$0.732474\pi$
$444$	0	0
$445$	4.08090	0.193453
$446$	−21.6808	−1.02662
$447$	0	0
$448$	4.42246	0.208942
$449$	11.2476	0.530809	0.265405	−	0.964137i	$-0.414494\pi$
$449$	11.2476	0.530809	0.265405	+	0.964137i	$0.414494\pi$
$450$	0	0
$451$	15.8129	0.744601
$452$	36.3094	1.70785
$453$	0	0
$454$	29.8576	1.40129
$455$	5.38389	0.252401
$456$	0	0
$457$	−23.5615	−1.10216	−0.551081	−	0.834452i	$-0.685784\pi$
$457$	−23.5615	−1.10216	−0.551081	+	0.834452i	$0.685784\pi$
$458$	37.5738	1.75571
$459$	0	0
$460$	9.27747	0.432564
$461$	30.8006	1.43453	0.717263	−	0.696803i	$-0.245395\pi$
$461$	30.8006	1.43453	0.717263	+	0.696803i	$0.245395\pi$
$462$	0	0
$463$	−30.4122	−1.41337	−0.706687	−	0.707526i	$-0.749811\pi$
$463$	−30.4122	−1.41337	−0.706687	+	0.707526i	$0.749811\pi$
$464$	−12.7727	−0.592957
$465$	0	0
$466$	38.2824	1.77340
$467$	20.5176	0.949442	0.474721	−	0.880136i	$-0.342549\pi$
$467$	20.5176	0.949442	0.474721	+	0.880136i	$0.342549\pi$
$468$	0	0
$469$	14.0307	0.647878
$470$	34.6521	1.59838
$471$	0	0
$472$	−39.1403	−1.80158
$473$	22.0774	1.01512
$474$	0	0
$475$	6.52435	0.299357
$476$	−17.9291	−0.821781
$477$	0	0
$478$	−10.1366	−0.463636
$479$	−28.7665	−1.31437	−0.657187	−	0.753727i	$-0.728254\pi$
$479$	−28.7665	−1.31437	−0.657187	+	0.753727i	$0.728254\pi$
$480$	0	0
$481$	−21.7386	−0.991194
$482$	22.5112	1.02536
$483$	0	0
$484$	−33.6777	−1.53080
$485$	−17.0656	−0.774908
$486$	0	0
$487$	15.5923	0.706555	0.353278	−	0.935519i	$-0.385067\pi$
$487$	15.5923	0.706555	0.353278	+	0.935519i	$0.385067\pi$
$488$	−4.80120	−0.217340
$489$	0	0
$490$	3.07267	0.138809
$491$	−8.92896	−0.402958	−0.201479	−	0.979493i	$-0.564575\pi$
$491$	−8.92896	−0.402958	−0.201479	+	0.979493i	$0.564575\pi$
$492$	0	0
$493$	5.66579	0.255174
$494$	−21.3113	−0.958839
$495$	0	0
$496$	−21.4472	−0.963010
$497$	−7.44927	−0.334145
$498$	0	0
$499$	23.8250	1.06655	0.533277	−	0.845940i	$-0.320960\pi$
$499$	23.8250	1.06655	0.533277	+	0.845940i	$0.320960\pi$
$500$	47.8543	2.14011
$501$	0	0
$502$	−14.3539	−0.640647
$503$	39.5472	1.76332	0.881661	−	0.471883i	$-0.156426\pi$
$503$	39.5472	1.76332	0.881661	+	0.471883i	$0.156426\pi$
$504$	0	0
$505$	6.42655	0.285978
$506$	8.41722	0.374191
$507$	0	0
$508$	−4.69021	−0.208094
$509$	−29.4561	−1.30562	−0.652808	−	0.757523i	$-0.726409\pi$
$509$	−29.4561	−1.30562	−0.652808	+	0.757523i	$0.726409\pi$
$510$	0	0
$511$	5.29270	0.234135
$512$	47.7715	2.11122
$513$	0	0
$514$	0.743363	0.0327883
$515$	−8.28158	−0.364930
$516$	0	0
$517$	22.0405	0.969340
$518$	−12.4066	−0.545113
$519$	0	0
$520$	−37.4630	−1.64286
$521$	5.91511	0.259146	0.129573	−	0.991570i	$-0.458639\pi$
$521$	5.91511	0.259146	0.129573	+	0.991570i	$0.458639\pi$
$522$	0	0
$523$	29.3043	1.28139	0.640694	−	0.767796i	$-0.278647\pi$
$523$	29.3043	1.28139	0.640694	+	0.767796i	$0.278647\pi$
$524$	33.1367	1.44758
$525$	0	0
$526$	56.8474	2.47867
$527$	9.51370	0.414423
$528$	0	0
$529$	−20.2274	−0.879454
$530$	−21.9623	−0.953980
$531$	0	0
$532$	−8.52672	−0.369680
$533$	−36.6694	−1.58833
$534$	0	0
$535$	−6.37621	−0.275667
$536$	−97.6305	−4.21700
$537$	0	0
$538$	−72.4712	−3.12446
$539$	1.95437	0.0841808
$540$	0	0
$541$	5.91807	0.254438	0.127219	−	0.991875i	$-0.459395\pi$
$541$	5.91807	0.254438	0.127219	+	0.991875i	$0.459395\pi$
$542$	−9.91333	−0.425814
$543$	0	0
$544$	32.0082	1.37234
$545$	15.2777	0.654426
$546$	0	0
$547$	−38.1336	−1.63047	−0.815237	−	0.579127i	$-0.803393\pi$
$547$	−38.1336	−1.63047	−0.815237	+	0.579127i	$0.803393\pi$
$548$	24.6831	1.05441
$549$	0	0
$550$	18.1416	0.773559
$551$	2.69453	0.114791
$552$	0	0
$553$	−4.25002	−0.180729
$554$	−46.3082	−1.96745
$555$	0	0
$556$	15.6772	0.664861
$557$	−40.7583	−1.72698	−0.863492	−	0.504363i	$-0.831727\pi$
$557$	−40.7583	−1.72698	−0.863492	+	0.504363i	$0.831727\pi$
$558$	0	0
$559$	−51.1965	−2.16538
$560$	−10.2373	−0.432605
$561$	0	0
$562$	67.3755	2.84207
$563$	−11.5983	−0.488808	−0.244404	−	0.969673i	$-0.578592\pi$
$563$	−11.5983	−0.488808	−0.244404	+	0.969673i	$0.578592\pi$
$564$	0	0
$565$	−9.19652	−0.386901
$566$	−22.6204	−0.950806
$567$	0	0
$568$	51.8346	2.17493
$569$	29.8881	1.25297	0.626487	−	0.779432i	$-0.284492\pi$
$569$	29.8881	1.25297	0.626487	+	0.779432i	$0.284492\pi$
$570$	0	0
$571$	4.44774	0.186132	0.0930660	−	0.995660i	$-0.470333\pi$
$571$	4.44774	0.186132	0.0930660	+	0.995660i	$0.470333\pi$
$572$	−41.5432	−1.73701
$573$	0	0
$574$	−20.9278	−0.873511
$575$	5.97569	0.249204
$576$	0	0
$577$	−12.0274	−0.500708	−0.250354	−	0.968154i	$-0.580547\pi$
$577$	−12.0274	−0.500708	−0.250354	+	0.968154i	$0.580547\pi$
$578$	6.17454	0.256827
$579$	0	0
$580$	8.25814	0.342901
$581$	−2.64391	−0.109688
$582$	0	0
$583$	−13.9691	−0.578541
$584$	−36.8285	−1.52397
$585$	0	0
$586$	−12.6820	−0.523888
$587$	−9.45273	−0.390156	−0.195078	−	0.980788i	$-0.562496\pi$
$587$	−9.45273	−0.390156	−0.195078	+	0.980788i	$0.562496\pi$
$588$	0	0
$589$	4.52451	0.186429
$590$	17.2836	0.711554
$591$	0	0
$592$	41.3352	1.69887
$593$	2.27664	0.0934905	0.0467453	−	0.998907i	$-0.485115\pi$
$593$	2.27664	0.0934905	0.0467453	+	0.998907i	$0.485115\pi$
$594$	0	0
$595$	4.54113	0.186168
$596$	−106.085	−4.34542
$597$	0	0
$598$	−19.5191	−0.798197
$599$	−10.7367	−0.438690	−0.219345	−	0.975647i	$-0.570392\pi$
$599$	−10.7367	−0.438690	−0.219345	+	0.975647i	$0.570392\pi$
$600$	0	0
$601$	1.56750	0.0639397	0.0319699	−	0.999489i	$-0.489822\pi$
$601$	1.56750	0.0639397	0.0319699	+	0.999489i	$0.489822\pi$
$602$	−29.2186	−1.19086
$603$	0	0
$604$	−49.0936	−1.99759
$605$	8.52995	0.346792
$606$	0	0
$607$	37.6814	1.52944	0.764719	−	0.644363i	$-0.222878\pi$
$607$	37.6814	1.52944	0.764719	+	0.644363i	$0.222878\pi$
$608$	15.2224	0.617350
$609$	0	0
$610$	2.12012	0.0858410
$611$	−51.1109	−2.06773
$612$	0	0
$613$	5.63184	0.227468	0.113734	−	0.993511i	$-0.463719\pi$
$613$	5.63184	0.227468	0.113734	+	0.993511i	$0.463719\pi$
$614$	−45.0481	−1.81799
$615$	0	0
$616$	−13.5992	−0.547928
$617$	5.94711	0.239422	0.119711	−	0.992809i	$-0.461803\pi$
$617$	5.94711	0.239422	0.119711	+	0.992809i	$0.461803\pi$
$618$	0	0
$619$	−22.8228	−0.917325	−0.458663	−	0.888611i	$-0.651671\pi$
$619$	−22.8228	−0.917325	−0.458663	+	0.888611i	$0.651671\pi$
$620$	13.8666	0.556898
$621$	0	0
$622$	80.3583	3.22208
$623$	−3.43526	−0.137631
$624$	0	0
$625$	5.82331	0.232932
$626$	−14.2116	−0.568009
$627$	0	0
$628$	−85.1117	−3.39633
$629$	−18.3357	−0.731094
$630$	0	0
$631$	28.1155	1.11926	0.559629	−	0.828743i	$-0.310944\pi$
$631$	28.1155	1.11926	0.559629	+	0.828743i	$0.310944\pi$
$632$	29.5731	1.17636
$633$	0	0
$634$	41.4227	1.64510
$635$	1.18795	0.0471422
$636$	0	0
$637$	−4.53210	−0.179568
$638$	7.49240	0.296627
$639$	0	0
$640$	−6.30514	−0.249233
$641$	1.02141	0.0403432	0.0201716	−	0.999797i	$-0.493579\pi$
$641$	1.02141	0.0403432	0.0201716	+	0.999797i	$0.493579\pi$
$642$	0	0
$643$	11.0890	0.437306	0.218653	−	0.975803i	$-0.429834\pi$
$643$	11.0890	0.437306	0.218653	+	0.975803i	$0.429834\pi$
$644$	−7.80968	−0.307745
$645$	0	0
$646$	−17.9753	−0.707229
$647$	9.09832	0.357692	0.178846	−	0.983877i	$-0.442764\pi$
$647$	9.09832	0.357692	0.178846	+	0.983877i	$0.442764\pi$
$648$	0	0
$649$	10.9932	0.431522
$650$	−42.0695	−1.65010
$651$	0	0
$652$	26.5565	1.04003
$653$	15.0943	0.590686	0.295343	−	0.955391i	$-0.404566\pi$
$653$	15.0943	0.590686	0.295343	+	0.955391i	$0.404566\pi$
$654$	0	0
$655$	−8.39293	−0.327939
$656$	69.7258	2.72234
$657$	0	0
$658$	−29.1698	−1.13716
$659$	30.5564	1.19031	0.595154	−	0.803611i	$-0.297091\pi$
$659$	30.5564	1.19031	0.595154	+	0.803611i	$0.297091\pi$
$660$	0	0
$661$	19.8386	0.771632	0.385816	−	0.922576i	$-0.373920\pi$
$661$	19.8386	0.771632	0.385816	+	0.922576i	$0.373920\pi$
$662$	−2.44378	−0.0949803
$663$	0	0
$664$	18.3972	0.713951
$665$	2.15966	0.0837481
$666$	0	0
$667$	2.46794	0.0955590
$668$	78.0685	3.02056
$669$	0	0
$670$	43.1118	1.66555
$671$	1.34850	0.0520583
$672$	0	0
$673$	−8.66123	−0.333866	−0.166933	−	0.985968i	$-0.553386\pi$
$673$	−8.66123	−0.333866	−0.166933	+	0.985968i	$0.553386\pi$
$674$	70.5825	2.71874
$675$	0	0
$676$	35.3640	1.36015
$677$	−45.2598	−1.73948	−0.869738	−	0.493513i	$-0.835713\pi$
$677$	−45.2598	−1.73948	−0.869738	+	0.493513i	$0.835713\pi$
$678$	0	0
$679$	14.3656	0.551302
$680$	−31.5987	−1.21176
$681$	0	0
$682$	12.5808	0.481746
$683$	−26.8087	−1.02581	−0.512903	−	0.858446i	$-0.671430\pi$
$683$	−26.8087	−1.02581	−0.512903	+	0.858446i	$0.671430\pi$
$684$	0	0
$685$	−6.25179	−0.238869
$686$	−2.58654	−0.0987547
$687$	0	0
$688$	97.3485	3.71138
$689$	32.3937	1.23410
$690$	0	0
$691$	−14.0806	−0.535652	−0.267826	−	0.963467i	$-0.586305\pi$
$691$	−14.0806	−0.535652	−0.267826	+	0.963467i	$0.586305\pi$
$692$	−86.4125	−3.28491
$693$	0	0
$694$	−65.3315	−2.47995
$695$	−3.97075	−0.150619
$696$	0	0
$697$	−30.9294	−1.17153
$698$	40.0455	1.51574
$699$	0	0
$700$	−16.8322	−0.636196
$701$	−0.193911	−0.00732391	−0.00366196	−	0.999993i	$-0.501166\pi$
$701$	−0.193911	−0.00732391	−0.00366196	+	0.999993i	$0.501166\pi$
$702$	0	0
$703$	−8.72009	−0.328884
$704$	8.64314	0.325750
$705$	0	0
$706$	52.7394	1.98487
$707$	−5.40980	−0.203456
$708$	0	0
$709$	−7.47338	−0.280669	−0.140334	−	0.990104i	$-0.544818\pi$
$709$	−7.47338	−0.280669	−0.140334	+	0.990104i	$0.544818\pi$
$710$	−22.8892	−0.859015
$711$	0	0
$712$	23.9037	0.895830
$713$	4.14403	0.155195
$714$	0	0
$715$	10.5221	0.393506
$716$	31.2064	1.16624
$717$	0	0
$718$	32.0696	1.19683
$719$	9.02040	0.336404	0.168202	−	0.985753i	$-0.446204\pi$
$719$	9.02040	0.336404	0.168202	+	0.985753i	$0.446204\pi$
$720$	0	0
$721$	6.97135	0.259627
$722$	40.5957	1.51081
$723$	0	0
$724$	−105.635	−3.92591
$725$	5.31914	0.197548
$726$	0	0
$727$	9.87897	0.366391	0.183195	−	0.983077i	$-0.441356\pi$
$727$	9.87897	0.366391	0.183195	+	0.983077i	$0.441356\pi$
$728$	31.5359	1.16880
$729$	0	0
$730$	16.2628	0.601911
$731$	−43.1824	−1.59716
$732$	0	0
$733$	−26.2401	−0.969201	−0.484601	−	0.874736i	$-0.661035\pi$
$733$	−26.2401	−0.969201	−0.484601	+	0.874736i	$0.661035\pi$
$734$	−47.5965	−1.75682
$735$	0	0
$736$	13.9423	0.513920
$737$	27.4212	1.01007
$738$	0	0
$739$	−48.1388	−1.77081	−0.885406	−	0.464818i	$-0.846120\pi$
$739$	−48.1388	−1.77081	−0.885406	+	0.464818i	$0.846120\pi$
$740$	−26.7252	−0.982437
$741$	0	0
$742$	18.4876	0.678701
$743$	−19.6535	−0.721017	−0.360509	−	0.932756i	$-0.617397\pi$
$743$	−19.6535	−0.721017	−0.360509	+	0.932756i	$0.617397\pi$
$744$	0	0
$745$	26.8695	0.984421
$746$	93.1785	3.41151
$747$	0	0
$748$	−35.0402	−1.28120
$749$	5.36742	0.196121
$750$	0	0
$751$	39.0126	1.42359	0.711794	−	0.702388i	$-0.247883\pi$
$751$	39.0126	1.42359	0.711794	+	0.702388i	$0.247883\pi$
$752$	97.1858	3.54400
$753$	0	0
$754$	−17.3745	−0.632743
$755$	12.4345	0.452539
$756$	0	0
$757$	−0.484793	−0.0176201	−0.00881004	−	0.999961i	$-0.502804\pi$
$757$	−0.484793	−0.0176201	−0.00881004	+	0.999961i	$0.502804\pi$
$758$	−34.7862	−1.26349
$759$	0	0
$760$	−15.0277	−0.545111
$761$	−50.4633	−1.82929	−0.914647	−	0.404253i	$-0.867532\pi$
$761$	−50.4633	−1.82929	−0.914647	+	0.404253i	$0.867532\pi$
$762$	0	0
$763$	−12.8606	−0.465586
$764$	−113.564	−4.10859
$765$	0	0
$766$	49.2125	1.77812
$767$	−25.4928	−0.920491
$768$	0	0
$769$	−19.7231	−0.711232	−0.355616	−	0.934632i	$-0.615729\pi$
$769$	−19.7231	−0.711232	−0.355616	+	0.934632i	$0.615729\pi$
$770$	6.00515	0.216411
$771$	0	0
$772$	10.3520	0.372578
$773$	33.5331	1.20610	0.603051	−	0.797703i	$-0.293952\pi$
$773$	33.5331	1.20610	0.603051	+	0.797703i	$0.293952\pi$
$774$	0	0
$775$	8.93161	0.320833
$776$	−99.9611	−3.58839
$777$	0	0
$778$	25.5380	0.915582
$779$	−14.7094	−0.527018
$780$	0	0
$781$	−14.5587	−0.520949
$782$	−16.4637	−0.588741
$783$	0	0
$784$	8.61765	0.307773
$785$	21.5572	0.769411
$786$	0	0
$787$	−47.7392	−1.70172	−0.850859	−	0.525395i	$-0.823918\pi$
$787$	−47.7392	−1.70172	−0.850859	+	0.525395i	$0.823918\pi$
$788$	14.5201	0.517259
$789$	0	0
$790$	−13.0589	−0.464616
$791$	7.74153	0.275257
$792$	0	0
$793$	−3.12711	−0.111047
$794$	−13.0035	−0.461475
$795$	0	0
$796$	32.2880	1.14442
$797$	37.0701	1.31309	0.656545	−	0.754287i	$-0.272017\pi$
$797$	37.0701	1.31309	0.656545	+	0.754287i	$0.272017\pi$
$798$	0	0
$799$	−43.1103	−1.52513
$800$	30.0498	1.06242
$801$	0	0
$802$	13.5748	0.479343
$803$	10.3439	0.365029
$804$	0	0
$805$	1.97805	0.0697171
$806$	−29.1744	−1.02762
$807$	0	0
$808$	37.6433	1.32429
$809$	44.1243	1.55133	0.775664	−	0.631146i	$-0.217415\pi$
$809$	44.1243	1.55133	0.775664	+	0.631146i	$0.217415\pi$
$810$	0	0
$811$	38.4803	1.35122	0.675612	−	0.737257i	$-0.263879\pi$
$811$	38.4803	1.35122	0.675612	+	0.737257i	$0.263879\pi$
$812$	−6.95162	−0.243954
$813$	0	0
$814$	−24.2471	−0.849859
$815$	−6.72628	−0.235611
$816$	0	0
$817$	−20.5367	−0.718486
$818$	−51.9460	−1.81625
$819$	0	0
$820$	−45.0810	−1.57430
$821$	24.8431	0.867032	0.433516	−	0.901146i	$-0.357273\pi$
$821$	24.8431	0.867032	0.433516	+	0.901146i	$0.357273\pi$
$822$	0	0
$823$	1.55570	0.0542282	0.0271141	−	0.999632i	$-0.491368\pi$
$823$	1.55570	0.0542282	0.0271141	+	0.999632i	$0.491368\pi$
$824$	−48.5091	−1.68989
$825$	0	0
$826$	−14.5491	−0.506230
$827$	−41.4950	−1.44292	−0.721460	−	0.692456i	$-0.756529\pi$
$827$	−41.4950	−1.44292	−0.721460	+	0.692456i	$0.756529\pi$
$828$	0	0
$829$	−48.1042	−1.67073	−0.835365	−	0.549696i	$-0.814744\pi$
$829$	−48.1042	−1.67073	−0.835365	+	0.549696i	$0.814744\pi$
$830$	−8.12387	−0.281984
$831$	0	0
$832$	−20.0430	−0.694867
$833$	−3.82267	−0.132448
$834$	0	0
$835$	−19.7733	−0.684285
$836$	−16.6644	−0.576350
$837$	0	0
$838$	−71.9573	−2.48572
$839$	6.43495	0.222159	0.111079	−	0.993812i	$-0.464569\pi$
$839$	6.43495	0.222159	0.111079	+	0.993812i	$0.464569\pi$
$840$	0	0
$841$	−26.8032	−0.924249
$842$	51.8437	1.78665
$843$	0	0
$844$	−36.7816	−1.26607
$845$	−8.95705	−0.308132
$846$	0	0
$847$	−7.18042	−0.246722
$848$	−61.5956	−2.11520
$849$	0	0
$850$	−35.4842	−1.21710
$851$	−7.98679	−0.273784
$852$	0	0
$853$	−18.8479	−0.645339	−0.322670	−	0.946512i	$-0.604580\pi$
$853$	−18.8479	−0.645339	−0.322670	+	0.946512i	$0.604580\pi$
$854$	−1.78469	−0.0610709
$855$	0	0
$856$	−37.3484	−1.27654
$857$	−3.90483	−0.133386	−0.0666932	−	0.997774i	$-0.521245\pi$
$857$	−3.90483	−0.133386	−0.0666932	+	0.997774i	$0.521245\pi$
$858$	0	0
$859$	2.46614	0.0841435	0.0420717	−	0.999115i	$-0.486604\pi$
$859$	2.46614	0.0841435	0.0420717	+	0.999115i	$0.486604\pi$
$860$	−62.9403	−2.14625
$861$	0	0
$862$	−67.8012	−2.30932
$863$	7.58030	0.258037	0.129018	−	0.991642i	$-0.458817\pi$
$863$	7.58030	0.258037	0.129018	+	0.991642i	$0.458817\pi$
$864$	0	0
$865$	21.8867	0.744171
$866$	18.4971	0.628555
$867$	0	0
$868$	−11.6728	−0.396200
$869$	−8.30613	−0.281766
$870$	0	0
$871$	−63.5886	−2.15462
$872$	89.4887	3.03047
$873$	0	0
$874$	−7.82979	−0.264847
$875$	10.2030	0.344925
$876$	0	0
$877$	−40.5357	−1.36879	−0.684396	−	0.729111i	$-0.739934\pi$
$877$	−40.5357	−1.36879	−0.684396	+	0.729111i	$0.739934\pi$
$878$	16.9124	0.570766
$879$	0	0
$880$	−20.0075	−0.674453
$881$	30.0161	1.01127	0.505635	−	0.862748i	$-0.331258\pi$
$881$	30.0161	1.01127	0.505635	+	0.862748i	$0.331258\pi$
$882$	0	0
$883$	−56.7984	−1.91142	−0.955709	−	0.294314i	$-0.904909\pi$
$883$	−56.7984	−1.91142	−0.955709	+	0.294314i	$0.904909\pi$
$884$	81.2567	2.73296
$885$	0	0
$886$	72.6369	2.44028
$887$	−0.943805	−0.0316899	−0.0158449	−	0.999874i	$-0.505044\pi$
$887$	−0.943805	−0.0316899	−0.0158449	+	0.999874i	$0.505044\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−10.5554	−0.353819
$891$	0	0
$892$	39.3141	1.31633
$893$	−20.5023	−0.686084
$894$	0	0
$895$	−7.90401	−0.264202
$896$	5.30760	0.177315
$897$	0	0
$898$	−29.0925	−0.970830
$899$	3.68872	0.123026
$900$	0	0
$901$	27.3230	0.910260
$902$	−40.9008	−1.36185
$903$	0	0
$904$	−53.8683	−1.79163
$905$	26.7555	0.889384
$906$	0	0
$907$	−36.8933	−1.22502	−0.612510	−	0.790463i	$-0.709840\pi$
$907$	−36.8933	−1.22502	−0.612510	+	0.790463i	$0.709840\pi$
$908$	−54.1411	−1.79674
$909$	0	0
$910$	−13.9257	−0.461632
$911$	39.3195	1.30271	0.651356	−	0.758772i	$-0.274200\pi$
$911$	39.3195	1.30271	0.651356	+	0.758772i	$0.274200\pi$
$912$	0	0
$913$	−5.16719	−0.171009
$914$	60.9429	2.01581
$915$	0	0
$916$	−68.1330	−2.25118
$917$	7.06508	0.233309
$918$	0	0
$919$	20.4646	0.675066	0.337533	−	0.941314i	$-0.390408\pi$
$919$	20.4646	0.675066	0.337533	+	0.941314i	$0.390408\pi$
$920$	−13.7640	−0.453784
$921$	0	0
$922$	−79.6670	−2.62369
$923$	33.7608	1.11125
$924$	0	0
$925$	−17.2139	−0.565989
$926$	78.6624	2.58501
$927$	0	0
$928$	12.4104	0.407393
$929$	−10.5962	−0.347649	−0.173824	−	0.984777i	$-0.555613\pi$
$929$	−10.5962	−0.347649	−0.173824	+	0.984777i	$0.555613\pi$
$930$	0	0
$931$	−1.81798	−0.0595819
$932$	−69.4180	−2.27386
$933$	0	0
$934$	−53.0697	−1.73649
$935$	8.87506	0.290245
$936$	0	0
$937$	−59.9050	−1.95701	−0.978506	−	0.206218i	$-0.933884\pi$
$937$	−59.9050	−1.95701	−0.978506	+	0.206218i	$0.933884\pi$
$938$	−36.2910	−1.18494
$939$	0	0
$940$	−62.8351	−2.04946
$941$	−9.70923	−0.316512	−0.158256	−	0.987398i	$-0.550587\pi$
$941$	−9.70923	−0.316512	−0.158256	+	0.987398i	$0.550587\pi$
$942$	0	0
$943$	−13.4724	−0.438722
$944$	48.4738	1.57769
$945$	0	0
$946$	−57.1042	−1.85662
$947$	−14.2612	−0.463425	−0.231713	−	0.972784i	$-0.574433\pi$
$947$	−14.2612	−0.463425	−0.231713	+	0.972784i	$0.574433\pi$
$948$	0	0
$949$	−23.9871	−0.778654
$950$	−16.8755	−0.547514
$951$	0	0
$952$	26.5995	0.862094
$953$	−41.2302	−1.33558	−0.667789	−	0.744351i	$-0.732759\pi$
$953$	−41.2302	−1.33558	−0.667789	+	0.744351i	$0.732759\pi$
$954$	0	0
$955$	28.7636	0.930768
$956$	18.3808	0.594477
$957$	0	0
$958$	74.4058	2.40394
$959$	5.26269	0.169941
$960$	0	0
$961$	−24.8061	−0.800196
$962$	56.2278	1.81286
$963$	0	0
$964$	−40.8198	−1.31472
$965$	−2.62198	−0.0844045
$966$	0	0
$967$	12.2790	0.394867	0.197434	−	0.980316i	$-0.436739\pi$
$967$	12.2790	0.394867	0.197434	+	0.980316i	$0.436739\pi$
$968$	49.9639	1.60590
$969$	0	0
$970$	44.1409	1.41728
$971$	0.150973	0.00484497	0.00242248	−	0.999997i	$-0.499229\pi$
$971$	0.150973	0.00484497	0.00242248	+	0.999997i	$0.499229\pi$
$972$	0	0
$973$	3.34254	0.107157
$974$	−40.3302	−1.29226
$975$	0	0
$976$	5.94610	0.190330
$977$	−53.1450	−1.70026	−0.850129	−	0.526574i	$-0.823476\pi$
$977$	−53.1450	−1.70026	−0.850129	+	0.526574i	$0.823476\pi$
$978$	0	0
$979$	−6.71378	−0.214573
$980$	−5.57171	−0.177982
$981$	0	0
$982$	23.0952	0.736996
$983$	−26.6214	−0.849090	−0.424545	−	0.905407i	$-0.639566\pi$
$983$	−26.6214	−0.849090	−0.424545	+	0.905407i	$0.639566\pi$
$984$	0	0
$985$	−3.67769	−0.117181
$986$	−14.6548	−0.466705
$987$	0	0
$988$	38.6439	1.22943
$989$	−18.8097	−0.598113
$990$	0	0
$991$	44.0224	1.39842	0.699208	−	0.714918i	$-0.253536\pi$
$991$	44.0224	1.39842	0.699208	+	0.714918i	$0.253536\pi$
$992$	20.8390	0.661638
$993$	0	0
$994$	19.2679	0.611139
$995$	−8.17795	−0.259259
$996$	0	0
$997$	44.5236	1.41008	0.705038	−	0.709169i	$-0.250930\pi$
$997$	44.5236	1.41008	0.705038	+	0.709169i	$0.250930\pi$
$998$	−61.6245	−1.95069
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.w.1.3		20
3.2	odd	2		889.2.a.d.1.18	✓	20
21.20	even	2		6223.2.a.l.1.18		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.d.1.18	✓	20	3.2	odd	2
6223.2.a.l.1.18		20	21.20	even	2
8001.2.a.w.1.3		20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.58654 q^{2} +4.69021 q^{4} -1.18795 q^{5} +1.00000 q^{7} -6.95835 q^{8} +O(q^{10})\)	\(q-2.58654 q^{2} +4.69021 q^{4} -1.18795 q^{5} +1.00000 q^{7} -6.95835 q^{8} +3.07267 q^{10} +1.95437 q^{11} -4.53210 q^{13} -2.58654 q^{14} +8.61765 q^{16} -3.82267 q^{17} -1.81798 q^{19} -5.57171 q^{20} -5.05507 q^{22} -1.66510 q^{23} -3.58879 q^{25} +11.7225 q^{26} +4.69021 q^{28} -1.48215 q^{29} -2.48876 q^{31} -8.37324 q^{32} +9.88751 q^{34} -1.18795 q^{35} +4.79658 q^{37} +4.70229 q^{38} +8.26614 q^{40} +8.09104 q^{41} +11.2964 q^{43} +9.16642 q^{44} +4.30686 q^{46} +11.2775 q^{47} +1.00000 q^{49} +9.28255 q^{50} -21.2565 q^{52} -7.14761 q^{53} -2.32169 q^{55} -6.95835 q^{56} +3.83366 q^{58} +5.62494 q^{59} +0.689991 q^{61} +6.43728 q^{62} +4.42246 q^{64} +5.38389 q^{65} +14.0307 q^{67} -17.9291 q^{68} +3.07267 q^{70} -7.44927 q^{71} +5.29270 q^{73} -12.4066 q^{74} -8.52672 q^{76} +1.95437 q^{77} -4.25002 q^{79} -10.2373 q^{80} -20.9278 q^{82} -2.64391 q^{83} +4.54113 q^{85} -29.2186 q^{86} -13.5992 q^{88} -3.43526 q^{89} -4.53210 q^{91} -7.80968 q^{92} -29.1698 q^{94} +2.15966 q^{95} +14.3656 q^{97} -2.58654 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) \) 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28} - 16 q^{29} + 6 q^{31} - 41 q^{32} - 10 q^{34} - 3 q^{35} + 2 q^{37} - 3 q^{38} - 38 q^{40} - 25 q^{41} + 13 q^{43} - 66 q^{44} + 20 q^{46} - 19 q^{47} + 20 q^{49} + 4 q^{50} + 20 q^{52} - 24 q^{53} - 3 q^{55} - 24 q^{56} + 12 q^{58} - 23 q^{59} - 27 q^{61} - 7 q^{62} + 2 q^{64} - 26 q^{65} + 9 q^{67} + 25 q^{68} - 8 q^{70} - 63 q^{71} - 21 q^{73} - 21 q^{74} - 10 q^{76} - 26 q^{77} + 18 q^{79} + 23 q^{80} - 42 q^{82} + q^{83} - 41 q^{85} + 12 q^{86} + 57 q^{88} + 16 q^{89} - 4 q^{91} - 17 q^{92} + 7 q^{94} - 75 q^{95} - 32 q^{97} - 8 q^{98}+O(q^{100}) \) 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8 - 8 * q^10 - 26 * q^11 - 4 * q^13 - 8 * q^14 + 24 * q^16 - 4 * q^17 + q^19 + 2 * q^20 + q^22 - 31 * q^23 + 27 * q^25 - 4 * q^26 + 24 * q^28 - 16 * q^29 + 6 * q^31 - 41 * q^32 - 10 * q^34 - 3 * q^35 + 2 * q^37 - 3 * q^38 - 38 * q^40 - 25 * q^41 + 13 * q^43 - 66 * q^44 + 20 * q^46 - 19 * q^47 + 20 * q^49 + 4 * q^50 + 20 * q^52 - 24 * q^53 - 3 * q^55 - 24 * q^56 + 12 * q^58 - 23 * q^59 - 27 * q^61 - 7 * q^62 + 2 * q^64 - 26 * q^65 + 9 * q^67 + 25 * q^68 - 8 * q^70 - 63 * q^71 - 21 * q^73 - 21 * q^74 - 10 * q^76 - 26 * q^77 + 18 * q^79 + 23 * q^80 - 42 * q^82 + q^83 - 41 * q^85 + 12 * q^86 + 57 * q^88 + 16 * q^89 - 4 * q^91 - 17 * q^92 + 7 * q^94 - 75 * q^95 - 32 * q^97 - 8 * q^98

Introduction

L-functions

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Database

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