LMFDB - Embedded newform 8001.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 22 x^{17} + 101 x^{16} + 178 x^{15} - 1035 x^{14} - 583 x^{13} + 5572 x^{12} + \cdots + 210 $ x^19 - 4x^18 - 22x^17 + 101x^16 + 178x^15 - 1035x^14 - 583x^13 + 5572x^12 + 21x^11 - 17032x^10 + 4985x^9 + 29792x^8 - 13249x^7 - 28600x^6 + 14000x^5 + 13725x^4 - 5723x^3 - 2913x^2 + 608x + 210
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.49323$ 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)

$f(q)$	$=$	$q-2.49323 q^{2} +4.21621 q^{4} +0.262973 q^{5} +1.00000 q^{7} -5.52554 q^{8} +O(q^{10})$	\(q-2.49323 q^{2} +4.21621 q^{4} +0.262973 q^{5} +1.00000 q^{7} -5.52554 q^{8} -0.655653 q^{10} -3.55277 q^{11} -2.65719 q^{13} -2.49323 q^{14} +5.34403 q^{16} -7.62237 q^{17} -1.10257 q^{19} +1.10875 q^{20} +8.85789 q^{22} +3.50334 q^{23} -4.93085 q^{25} +6.62499 q^{26} +4.21621 q^{28} +4.14426 q^{29} +7.01796 q^{31} -2.27284 q^{32} +19.0043 q^{34} +0.262973 q^{35} -0.847404 q^{37} +2.74897 q^{38} -1.45307 q^{40} +8.88998 q^{41} -11.3921 q^{43} -14.9792 q^{44} -8.73465 q^{46} -8.26272 q^{47} +1.00000 q^{49} +12.2937 q^{50} -11.2033 q^{52} -4.21727 q^{53} -0.934282 q^{55} -5.52554 q^{56} -10.3326 q^{58} -13.5072 q^{59} +1.49849 q^{61} -17.4974 q^{62} -5.02134 q^{64} -0.698769 q^{65} -2.38846 q^{67} -32.1375 q^{68} -0.655653 q^{70} +9.51155 q^{71} -1.22482 q^{73} +2.11278 q^{74} -4.64869 q^{76} -3.55277 q^{77} +4.70270 q^{79} +1.40533 q^{80} -22.1648 q^{82} -3.89809 q^{83} -2.00448 q^{85} +28.4032 q^{86} +19.6310 q^{88} -6.98825 q^{89} -2.65719 q^{91} +14.7708 q^{92} +20.6009 q^{94} -0.289947 q^{95} -10.7226 q^{97} -2.49323 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * 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 $

$2$	−2.49323	−1.76298	−0.881491	−	0.472200i	$-0.843460\pi$
$2$	−2.49323	−1.76298	−0.881491	+	0.472200i	$0.843460\pi$
$3$	0	0
$3$	0	0
$4$	4.21621	2.10811
$5$	0.262973	0.117605	0.0588025	−	0.998270i	$-0.481272\pi$
$5$	0.262973	0.117605	0.0588025	+	0.998270i	$0.481272\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−5.52554	−1.95357
$9$	0	0
$10$	−0.655653	−0.207336
$11$	−3.55277	−1.07120	−0.535601	−	0.844471i	$-0.679915\pi$
$11$	−3.55277	−1.07120	−0.535601	+	0.844471i	$0.679915\pi$
$12$	0	0
$13$	−2.65719	−0.736972	−0.368486	−	0.929633i	$-0.620124\pi$
$13$	−2.65719	−0.736972	−0.368486	+	0.929633i	$0.620124\pi$
$14$	−2.49323	−0.666345
$15$	0	0
$16$	5.34403	1.33601
$17$	−7.62237	−1.84870	−0.924348	−	0.381551i	$-0.875390\pi$
$17$	−7.62237	−1.84870	−0.924348	+	0.381551i	$0.875390\pi$
$18$	0	0
$19$	−1.10257	−0.252948	−0.126474	−	0.991970i	$-0.540366\pi$
$19$	−1.10257	−0.252948	−0.126474	+	0.991970i	$0.540366\pi$
$20$	1.10875	0.247924
$21$	0	0
$22$	8.85789	1.88851
$23$	3.50334	0.730497	0.365249	−	0.930910i	$-0.380984\pi$
$23$	3.50334	0.730497	0.365249	+	0.930910i	$0.380984\pi$
$24$	0	0
$25$	−4.93085	−0.986169
$26$	6.62499	1.29927
$27$	0	0
$28$	4.21621	0.796790
$29$	4.14426	0.769570	0.384785	−	0.923006i	$-0.374276\pi$
$29$	4.14426	0.769570	0.384785	+	0.923006i	$0.374276\pi$
$30$	0	0
$31$	7.01796	1.26046	0.630231	−	0.776408i	$-0.282960\pi$
$31$	7.01796	1.26046	0.630231	+	0.776408i	$0.282960\pi$
$32$	−2.27284	−0.401785
$33$	0	0
$34$	19.0043	3.25922
$35$	0.262973	0.0444505
$36$	0	0
$37$	−0.847404	−0.139312	−0.0696562	−	0.997571i	$-0.522190\pi$
$37$	−0.847404	−0.139312	−0.0696562	+	0.997571i	$0.522190\pi$
$38$	2.74897	0.445942
$39$	0	0
$40$	−1.45307	−0.229750
$41$	8.88998	1.38838	0.694191	−	0.719791i	$-0.255762\pi$
$41$	8.88998	1.38838	0.694191	+	0.719791i	$0.255762\pi$
$42$	0	0
$43$	−11.3921	−1.73728	−0.868642	−	0.495441i	$-0.835006\pi$
$43$	−11.3921	−1.73728	−0.868642	+	0.495441i	$0.835006\pi$
$44$	−14.9792	−2.25821
$45$	0	0
$46$	−8.73465	−1.28785
$47$	−8.26272	−1.20524	−0.602621	−	0.798028i	$-0.705877\pi$
$47$	−8.26272	−1.20524	−0.602621	+	0.798028i	$0.705877\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	12.2937	1.73860
$51$	0	0
$52$	−11.2033	−1.55362
$53$	−4.21727	−0.579287	−0.289643	−	0.957135i	$-0.593537\pi$
$53$	−4.21727	−0.579287	−0.289643	+	0.957135i	$0.593537\pi$
$54$	0	0
$55$	−0.934282	−0.125979
$56$	−5.52554	−0.738381
$57$	0	0
$58$	−10.3326	−1.35674
$59$	−13.5072	−1.75849	−0.879244	−	0.476372i	$-0.841952\pi$
$59$	−13.5072	−1.75849	−0.879244	+	0.476372i	$0.841952\pi$
$60$	0	0
$61$	1.49849	0.191862	0.0959310	−	0.995388i	$-0.469417\pi$
$61$	1.49849	0.191862	0.0959310	+	0.995388i	$0.469417\pi$
$62$	−17.4974	−2.22217
$63$	0	0
$64$	−5.02134	−0.627667
$65$	−0.698769	−0.0866716
$66$	0	0
$67$	−2.38846	−0.291797	−0.145898	−	0.989300i	$-0.546607\pi$
$67$	−2.38846	−0.291797	−0.145898	+	0.989300i	$0.546607\pi$
$68$	−32.1375	−3.89725
$69$	0	0
$70$	−0.655653	−0.0783655
$71$	9.51155	1.12881	0.564407	−	0.825497i	$-0.309105\pi$
$71$	9.51155	1.12881	0.564407	+	0.825497i	$0.309105\pi$
$72$	0	0
$73$	−1.22482	−0.143354	−0.0716771	−	0.997428i	$-0.522835\pi$
$73$	−1.22482	−0.143354	−0.0716771	+	0.997428i	$0.522835\pi$
$74$	2.11278	0.245605
$75$	0	0
$76$	−4.64869	−0.533241
$77$	−3.55277	−0.404876
$78$	0	0
$79$	4.70270	0.529095	0.264548	−	0.964373i	$-0.414777\pi$
$79$	4.70270	0.529095	0.264548	+	0.964373i	$0.414777\pi$
$80$	1.40533	0.157121
$81$	0	0
$82$	−22.1648	−2.44769
$83$	−3.89809	−0.427871	−0.213935	−	0.976848i	$-0.568628\pi$
$83$	−3.89809	−0.427871	−0.213935	+	0.976848i	$0.568628\pi$
$84$	0	0
$85$	−2.00448	−0.217416
$86$	28.4032	3.06280
$87$	0	0
$88$	19.6310	2.09267
$89$	−6.98825	−0.740753	−0.370377	−	0.928882i	$-0.620771\pi$
$89$	−6.98825	−0.740753	−0.370377	+	0.928882i	$0.620771\pi$
$90$	0	0
$91$	−2.65719	−0.278549
$92$	14.7708	1.53997
$93$	0	0
$94$	20.6009	2.12482
$95$	−0.289947	−0.0297479
$96$	0	0
$97$	−10.7226	−1.08872	−0.544360	−	0.838852i	$-0.683227\pi$
$97$	−10.7226	−1.08872	−0.544360	+	0.838852i	$0.683227\pi$
$98$	−2.49323	−0.251855
$99$	0	0
$100$	−20.7895	−2.07895
$101$	7.08508	0.704992	0.352496	−	0.935813i	$-0.385333\pi$
$101$	7.08508	0.704992	0.352496	+	0.935813i	$0.385333\pi$
$102$	0	0
$103$	12.8326	1.26443	0.632215	−	0.774793i	$-0.282146\pi$
$103$	12.8326	1.26443	0.632215	+	0.774793i	$0.282146\pi$
$104$	14.6824	1.43973
$105$	0	0
$106$	10.5146	1.02127
$107$	−3.49701	−0.338068	−0.169034	−	0.985610i	$-0.554065\pi$
$107$	−3.49701	−0.338068	−0.169034	+	0.985610i	$0.554065\pi$
$108$	0	0
$109$	6.77216	0.648655	0.324328	−	0.945945i	$-0.394862\pi$
$109$	6.77216	0.648655	0.324328	+	0.945945i	$0.394862\pi$
$110$	2.32938	0.222098
$111$	0	0
$112$	5.34403	0.504964
$113$	3.39124	0.319021	0.159511	−	0.987196i	$-0.449008\pi$
$113$	3.39124	0.319021	0.159511	+	0.987196i	$0.449008\pi$
$114$	0	0
$115$	0.921284	0.0859101
$116$	17.4731	1.62234
$117$	0	0
$118$	33.6766	3.10018
$119$	−7.62237	−0.698741
$120$	0	0
$121$	1.62219	0.147472
$122$	−3.73609	−0.338249
$123$	0	0
$124$	29.5892	2.65719
$125$	−2.61154	−0.233583
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	17.0651	1.50835
$129$	0	0
$130$	1.74219	0.152800
$131$	12.8441	1.12220	0.561099	−	0.827749i	$-0.310379\pi$
$131$	12.8441	1.12220	0.561099	+	0.827749i	$0.310379\pi$
$132$	0	0
$133$	−1.10257	−0.0956052
$134$	5.95499	0.514432
$135$	0	0
$136$	42.1177	3.61156
$137$	9.68042	0.827054	0.413527	−	0.910492i	$-0.364297\pi$
$137$	9.68042	0.827054	0.413527	+	0.910492i	$0.364297\pi$
$138$	0	0
$139$	−8.17409	−0.693317	−0.346659	−	0.937991i	$-0.612684\pi$
$139$	−8.17409	−0.693317	−0.346659	+	0.937991i	$0.612684\pi$
$140$	1.10875	0.0937064
$141$	0	0
$142$	−23.7145	−1.99008
$143$	9.44039	0.789445
$144$	0	0
$145$	1.08983	0.0905053
$146$	3.05376	0.252731
$147$	0	0
$148$	−3.57284	−0.293685
$149$	−4.34652	−0.356081	−0.178040	−	0.984023i	$-0.556976\pi$
$149$	−4.34652	−0.356081	−0.178040	+	0.984023i	$0.556976\pi$
$150$	0	0
$151$	−16.7253	−1.36108	−0.680542	−	0.732709i	$-0.738256\pi$
$151$	−16.7253	−1.36108	−0.680542	+	0.732709i	$0.738256\pi$
$152$	6.09231	0.494152
$153$	0	0
$154$	8.85789	0.713789
$155$	1.84553	0.148237
$156$	0	0
$157$	0.888372	0.0708998	0.0354499	−	0.999371i	$-0.488714\pi$
$157$	0.888372	0.0708998	0.0354499	+	0.999371i	$0.488714\pi$
$158$	−11.7249	−0.932786
$159$	0	0
$160$	−0.597695	−0.0472520
$161$	3.50334	0.276102
$162$	0	0
$163$	−2.55167	−0.199862	−0.0999310	−	0.994994i	$-0.531862\pi$
$163$	−2.55167	−0.199862	−0.0999310	+	0.994994i	$0.531862\pi$
$164$	37.4821	2.92686
$165$	0	0
$166$	9.71884	0.754328
$167$	18.1567	1.40500	0.702502	−	0.711681i	$-0.252066\pi$
$167$	18.1567	1.40500	0.702502	+	0.711681i	$0.252066\pi$
$168$	0	0
$169$	−5.93934	−0.456873
$170$	4.99763	0.383300
$171$	0	0
$172$	−48.0316	−3.66238
$173$	8.07410	0.613863	0.306931	−	0.951732i	$-0.400698\pi$
$173$	8.07410	0.613863	0.306931	+	0.951732i	$0.400698\pi$
$174$	0	0
$175$	−4.93085	−0.372737
$176$	−18.9861	−1.43113
$177$	0	0
$178$	17.4233	1.30593
$179$	12.3253	0.921239	0.460619	−	0.887598i	$-0.347627\pi$
$179$	12.3253	0.921239	0.460619	+	0.887598i	$0.347627\pi$
$180$	0	0
$181$	−7.94838	−0.590798	−0.295399	−	0.955374i	$-0.595453\pi$
$181$	−7.94838	−0.590798	−0.295399	+	0.955374i	$0.595453\pi$
$182$	6.62499	0.491077
$183$	0	0
$184$	−19.3579	−1.42708
$185$	−0.222844	−0.0163838
$186$	0	0
$187$	27.0805	1.98033
$188$	−34.8374	−2.54078
$189$	0	0
$190$	0.722905	0.0524451
$191$	0.672554	0.0486643	0.0243321	−	0.999704i	$-0.492254\pi$
$191$	0.672554	0.0486643	0.0243321	+	0.999704i	$0.492254\pi$
$192$	0	0
$193$	9.97229	0.717821	0.358911	−	0.933372i	$-0.383148\pi$
$193$	9.97229	0.717821	0.358911	+	0.933372i	$0.383148\pi$
$194$	26.7341	1.91939
$195$	0	0
$196$	4.21621	0.301158
$197$	−13.3551	−0.951509	−0.475755	−	0.879578i	$-0.657825\pi$
$197$	−13.3551	−0.951509	−0.475755	+	0.879578i	$0.657825\pi$
$198$	0	0
$199$	−4.93412	−0.349770	−0.174885	−	0.984589i	$-0.555955\pi$
$199$	−4.93412	−0.349770	−0.174885	+	0.984589i	$0.555955\pi$
$200$	27.2456	1.92655
$201$	0	0
$202$	−17.6648	−1.24289
$203$	4.14426	0.290870
$204$	0	0
$205$	2.33782	0.163281
$206$	−31.9946	−2.22917
$207$	0	0
$208$	−14.2001	−0.984600
$209$	3.91719	0.270958
$210$	0	0
$211$	19.9155	1.37104	0.685518	−	0.728055i	$-0.259576\pi$
$211$	19.9155	1.37104	0.685518	+	0.728055i	$0.259576\pi$
$212$	−17.7809	−1.22120
$213$	0	0
$214$	8.71885	0.596009
$215$	−2.99582	−0.204313
$216$	0	0
$217$	7.01796	0.476410
$218$	−16.8846	−1.14357
$219$	0	0
$220$	−3.93913	−0.265576
$221$	20.2541	1.36244
$222$	0	0
$223$	−8.03732	−0.538219	−0.269109	−	0.963110i	$-0.586729\pi$
$223$	−8.03732	−0.538219	−0.269109	+	0.963110i	$0.586729\pi$
$224$	−2.27284	−0.151861
$225$	0	0
$226$	−8.45515	−0.562428
$227$	8.51416	0.565104	0.282552	−	0.959252i	$-0.408819\pi$
$227$	8.51416	0.565104	0.282552	+	0.959252i	$0.408819\pi$
$228$	0	0
$229$	−5.72906	−0.378587	−0.189294	−	0.981921i	$-0.560620\pi$
$229$	−5.72906	−0.378587	−0.189294	+	0.981921i	$0.560620\pi$
$230$	−2.29698	−0.151458
$231$	0	0
$232$	−22.8993	−1.50341
$233$	10.0593	0.659004	0.329502	−	0.944155i	$-0.393119\pi$
$233$	10.0593	0.659004	0.329502	+	0.944155i	$0.393119\pi$
$234$	0	0
$235$	−2.17287	−0.141742
$236$	−56.9492	−3.70708
$237$	0	0
$238$	19.0043	1.23187
$239$	−13.4107	−0.867463	−0.433732	−	0.901042i	$-0.642803\pi$
$239$	−13.4107	−0.867463	−0.433732	+	0.901042i	$0.642803\pi$
$240$	0	0
$241$	8.69479	0.560080	0.280040	−	0.959988i	$-0.409652\pi$
$241$	8.69479	0.560080	0.280040	+	0.959988i	$0.409652\pi$
$242$	−4.04450	−0.259991
$243$	0	0
$244$	6.31795	0.404466
$245$	0.262973	0.0168007
$246$	0	0
$247$	2.92975	0.186415
$248$	−38.7780	−2.46240
$249$	0	0
$250$	6.51118	0.411803
$251$	20.7428	1.30927	0.654636	−	0.755944i	$-0.272822\pi$
$251$	20.7428	1.30927	0.654636	+	0.755944i	$0.272822\pi$
$252$	0	0
$253$	−12.4466	−0.782510
$254$	−2.49323	−0.156439
$255$	0	0
$256$	−32.5045	−2.03153
$257$	28.4551	1.77498	0.887491	−	0.460825i	$-0.152446\pi$
$257$	28.4551	1.77498	0.887491	+	0.460825i	$0.152446\pi$
$258$	0	0
$259$	−0.847404	−0.0526551
$260$	−2.94616	−0.182713
$261$	0	0
$262$	−32.0235	−1.97842
$263$	−24.8938	−1.53502	−0.767508	−	0.641039i	$-0.778503\pi$
$263$	−24.8938	−1.53502	−0.767508	+	0.641039i	$0.778503\pi$
$264$	0	0
$265$	−1.10903	−0.0681270
$266$	2.74897	0.168550
$267$	0	0
$268$	−10.0703	−0.615139
$269$	−10.3169	−0.629034	−0.314517	−	0.949252i	$-0.601843\pi$
$269$	−10.3169	−0.629034	−0.314517	+	0.949252i	$0.601843\pi$
$270$	0	0
$271$	9.92171	0.602701	0.301350	−	0.953513i	$-0.402563\pi$
$271$	9.92171	0.602701	0.301350	+	0.953513i	$0.402563\pi$
$272$	−40.7342	−2.46987
$273$	0	0
$274$	−24.1356	−1.45808
$275$	17.5182	1.05639
$276$	0	0
$277$	2.98940	0.179615	0.0898077	−	0.995959i	$-0.471375\pi$
$277$	2.98940	0.179615	0.0898077	+	0.995959i	$0.471375\pi$
$278$	20.3799	1.22231
$279$	0	0
$280$	−1.45307	−0.0868373
$281$	30.9297	1.84511	0.922556	−	0.385864i	$-0.126097\pi$
$281$	30.9297	1.84511	0.922556	+	0.385864i	$0.126097\pi$
$282$	0	0
$283$	26.0239	1.54696	0.773479	−	0.633822i	$-0.218515\pi$
$283$	26.0239	1.54696	0.773479	+	0.633822i	$0.218515\pi$
$284$	40.1027	2.37966
$285$	0	0
$286$	−23.5371	−1.39178
$287$	8.88998	0.524759
$288$	0	0
$289$	41.1005	2.41768
$290$	−2.71720	−0.159559
$291$	0	0
$292$	−5.16409	−0.302206
$293$	−28.1794	−1.64626	−0.823128	−	0.567855i	$-0.807773\pi$
$293$	−28.1794	−1.64626	−0.823128	+	0.567855i	$0.807773\pi$
$294$	0	0
$295$	−3.55203	−0.206807
$296$	4.68237	0.272157
$297$	0	0
$298$	10.8369	0.627764
$299$	−9.30905	−0.538356
$300$	0	0
$301$	−11.3921	−0.656631
$302$	41.7001	2.39957
$303$	0	0
$304$	−5.89219	−0.337940
$305$	0.394062	0.0225639
$306$	0	0
$307$	−14.2085	−0.810924	−0.405462	−	0.914112i	$-0.632889\pi$
$307$	−14.2085	−0.810924	−0.405462	+	0.914112i	$0.632889\pi$
$308$	−14.9792	−0.853522
$309$	0	0
$310$	−4.60134	−0.261339
$311$	−19.2348	−1.09071	−0.545353	−	0.838206i	$-0.683605\pi$
$311$	−19.2348	−1.09071	−0.545353	+	0.838206i	$0.683605\pi$
$312$	0	0
$313$	−21.8899	−1.23729	−0.618646	−	0.785670i	$-0.712318\pi$
$313$	−21.8899	−1.23729	−0.618646	+	0.785670i	$0.712318\pi$
$314$	−2.21492	−0.124995
$315$	0	0
$316$	19.8276	1.11539
$317$	26.0453	1.46285	0.731426	−	0.681921i	$-0.238855\pi$
$317$	26.0453	1.46285	0.731426	+	0.681921i	$0.238855\pi$
$318$	0	0
$319$	−14.7236	−0.824365
$320$	−1.32048	−0.0738168
$321$	0	0
$322$	−8.73465	−0.486763
$323$	8.40422	0.467623
$324$	0	0
$325$	13.1022	0.726779
$326$	6.36190	0.352353
$327$	0	0
$328$	−49.1219	−2.71230
$329$	−8.26272	−0.455538
$330$	0	0
$331$	3.29836	0.181294	0.0906470	−	0.995883i	$-0.471106\pi$
$331$	3.29836	0.181294	0.0906470	+	0.995883i	$0.471106\pi$
$332$	−16.4352	−0.901997
$333$	0	0
$334$	−45.2688	−2.47700
$335$	−0.628100	−0.0343168
$336$	0	0
$337$	17.1260	0.932912	0.466456	−	0.884545i	$-0.345531\pi$
$337$	17.1260	0.932912	0.466456	+	0.884545i	$0.345531\pi$
$338$	14.8082	0.805458
$339$	0	0
$340$	−8.45130	−0.458336
$341$	−24.9332	−1.35021
$342$	0	0
$343$	1.00000	0.0539949
$344$	62.9476	3.39391
$345$	0	0
$346$	−20.1306	−1.08223
$347$	−14.8310	−0.796171	−0.398085	−	0.917348i	$-0.630325\pi$
$347$	−14.8310	−0.796171	−0.398085	+	0.917348i	$0.630325\pi$
$348$	0	0
$349$	−13.6783	−0.732180	−0.366090	−	0.930579i	$-0.619304\pi$
$349$	−13.6783	−0.732180	−0.366090	+	0.930579i	$0.619304\pi$
$350$	12.2937	0.657129
$351$	0	0
$352$	8.07489	0.430393
$353$	30.9696	1.64835	0.824174	−	0.566337i	$-0.191640\pi$
$353$	30.9696	1.64835	0.824174	+	0.566337i	$0.191640\pi$
$354$	0	0
$355$	2.50128	0.132754
$356$	−29.4640	−1.56159
$357$	0	0
$358$	−30.7299	−1.62413
$359$	20.9788	1.10722	0.553610	−	0.832776i	$-0.313250\pi$
$359$	20.9788	1.10722	0.553610	+	0.832776i	$0.313250\pi$
$360$	0	0
$361$	−17.7843	−0.936017
$362$	19.8172	1.04157
$363$	0	0
$364$	−11.2033	−0.587211
$365$	−0.322094	−0.0168592
$366$	0	0
$367$	8.13181	0.424477	0.212239	−	0.977218i	$-0.431925\pi$
$367$	8.13181	0.424477	0.212239	+	0.977218i	$0.431925\pi$
$368$	18.7220	0.975950
$369$	0	0
$370$	0.555603	0.0288844
$371$	−4.21727	−0.218950
$372$	0	0
$373$	32.8632	1.70159	0.850795	−	0.525497i	$-0.176121\pi$
$373$	32.8632	1.70159	0.850795	+	0.525497i	$0.176121\pi$
$374$	−67.5181	−3.49128
$375$	0	0
$376$	45.6560	2.35453
$377$	−11.0121	−0.567152
$378$	0	0
$379$	16.8929	0.867731	0.433866	−	0.900978i	$-0.357149\pi$
$379$	16.8929	0.867731	0.433866	+	0.900978i	$0.357149\pi$
$380$	−1.22248	−0.0627118
$381$	0	0
$382$	−1.67683	−0.0857943
$383$	−8.23533	−0.420806	−0.210403	−	0.977615i	$-0.567478\pi$
$383$	−8.23533	−0.420806	−0.210403	+	0.977615i	$0.567478\pi$
$384$	0	0
$385$	−0.934282	−0.0476154
$386$	−24.8632	−1.26551
$387$	0	0
$388$	−45.2090	−2.29514
$389$	−30.8529	−1.56430	−0.782152	−	0.623087i	$-0.785878\pi$
$389$	−30.8529	−1.56430	−0.782152	+	0.623087i	$0.785878\pi$
$390$	0	0
$391$	−26.7038	−1.35047
$392$	−5.52554	−0.279082
$393$	0	0
$394$	33.2973	1.67749
$395$	1.23668	0.0622242
$396$	0	0
$397$	12.4669	0.625698	0.312849	−	0.949803i	$-0.398717\pi$
$397$	12.4669	0.625698	0.312849	+	0.949803i	$0.398717\pi$
$398$	12.3019	0.616639
$399$	0	0
$400$	−26.3506	−1.31753
$401$	11.9396	0.596237	0.298118	−	0.954529i	$-0.403641\pi$
$401$	11.9396	0.596237	0.298118	+	0.954529i	$0.403641\pi$
$402$	0	0
$403$	−18.6480	−0.928925
$404$	29.8722	1.48620
$405$	0	0
$406$	−10.3326	−0.512799
$407$	3.01064	0.149232
$408$	0	0
$409$	40.0631	1.98099	0.990497	−	0.137538i	$-0.0439189\pi$
$409$	40.0631	1.98099	0.990497	+	0.137538i	$0.0439189\pi$
$410$	−5.82874	−0.287861
$411$	0	0
$412$	54.1048	2.66555
$413$	−13.5072	−0.664646
$414$	0	0
$415$	−1.02509	−0.0503197
$416$	6.03937	0.296104
$417$	0	0
$418$	−9.76648	−0.477694
$419$	−38.5864	−1.88507	−0.942535	−	0.334107i	$-0.891565\pi$
$419$	−38.5864	−1.88507	−0.942535	+	0.334107i	$0.891565\pi$
$420$	0	0
$421$	27.8460	1.35713	0.678566	−	0.734539i	$-0.262602\pi$
$421$	27.8460	1.35713	0.678566	+	0.734539i	$0.262602\pi$
$422$	−49.6539	−2.41711
$423$	0	0
$424$	23.3027	1.13168
$425$	37.5847	1.82313
$426$	0	0
$427$	1.49849	0.0725170
$428$	−14.7441	−0.712684
$429$	0	0
$430$	7.46928	0.360201
$431$	−4.80037	−0.231226	−0.115613	−	0.993294i	$-0.536883\pi$
$431$	−4.80037	−0.231226	−0.115613	+	0.993294i	$0.536883\pi$
$432$	0	0
$433$	−20.7252	−0.995992	−0.497996	−	0.867179i	$-0.665931\pi$
$433$	−20.7252	−0.995992	−0.497996	+	0.867179i	$0.665931\pi$
$434$	−17.4974	−0.839902
$435$	0	0
$436$	28.5529	1.36743
$437$	−3.86269	−0.184778
$438$	0	0
$439$	16.3776	0.781662	0.390831	−	0.920462i	$-0.372188\pi$
$439$	16.3776	0.781662	0.390831	+	0.920462i	$0.372188\pi$
$440$	5.16241	0.246108
$441$	0	0
$442$	−50.4982	−2.40195
$443$	−20.1883	−0.959175	−0.479588	−	0.877494i	$-0.659214\pi$
$443$	−20.1883	−0.959175	−0.479588	+	0.877494i	$0.659214\pi$
$444$	0	0
$445$	−1.83772	−0.0871163
$446$	20.0389	0.948870
$447$	0	0
$448$	−5.02134	−0.237236
$449$	10.0200	0.472873	0.236437	−	0.971647i	$-0.424020\pi$
$449$	10.0200	0.472873	0.236437	+	0.971647i	$0.424020\pi$
$450$	0	0
$451$	−31.5841	−1.48724
$452$	14.2982	0.672530
$453$	0	0
$454$	−21.2278	−0.996269
$455$	−0.698769	−0.0327588
$456$	0	0
$457$	−12.2395	−0.572538	−0.286269	−	0.958149i	$-0.592415\pi$
$457$	−12.2395	−0.572538	−0.286269	+	0.958149i	$0.592415\pi$
$458$	14.2839	0.667442
$459$	0	0
$460$	3.88433	0.181108
$461$	−33.2139	−1.54693	−0.773464	−	0.633840i	$-0.781478\pi$
$461$	−33.2139	−1.54693	−0.773464	+	0.633840i	$0.781478\pi$
$462$	0	0
$463$	1.80160	0.0837272	0.0418636	−	0.999123i	$-0.486671\pi$
$463$	1.80160	0.0837272	0.0418636	+	0.999123i	$0.486671\pi$
$464$	22.1471	1.02815
$465$	0	0
$466$	−25.0801	−1.16181
$467$	26.2032	1.21254	0.606270	−	0.795259i	$-0.292665\pi$
$467$	26.2032	1.21254	0.606270	+	0.795259i	$0.292665\pi$
$468$	0	0
$469$	−2.38846	−0.110289
$470$	5.41747	0.249889
$471$	0	0
$472$	74.6346	3.43533
$473$	40.4736	1.86098
$474$	0	0
$475$	5.43662	0.249449
$476$	−32.1375	−1.47302
$477$	0	0
$478$	33.4359	1.52932
$479$	21.1860	0.968013	0.484007	−	0.875064i	$-0.339181\pi$
$479$	21.1860	0.968013	0.484007	+	0.875064i	$0.339181\pi$
$480$	0	0
$481$	2.25171	0.102669
$482$	−21.6781	−0.987412
$483$	0	0
$484$	6.83951	0.310887
$485$	−2.81976	−0.128039
$486$	0	0
$487$	6.70021	0.303615	0.151808	−	0.988410i	$-0.451491\pi$
$487$	6.70021	0.303615	0.151808	+	0.988410i	$0.451491\pi$
$488$	−8.27996	−0.374816
$489$	0	0
$490$	−0.655653	−0.0296194
$491$	28.8514	1.30204	0.651022	−	0.759059i	$-0.274340\pi$
$491$	28.8514	1.30204	0.651022	+	0.759059i	$0.274340\pi$
$492$	0	0
$493$	−31.5891	−1.42270
$494$	−7.30454	−0.328647
$495$	0	0
$496$	37.5042	1.68399
$497$	9.51155	0.426651
$498$	0	0
$499$	−19.4113	−0.868971	−0.434485	−	0.900679i	$-0.643070\pi$
$499$	−19.4113	−0.868971	−0.434485	+	0.900679i	$0.643070\pi$
$500$	−11.0108	−0.492419
$501$	0	0
$502$	−51.7166	−2.30822
$503$	−31.6324	−1.41042	−0.705209	−	0.708999i	$-0.749147\pi$
$503$	−31.6324	−1.41042	−0.705209	+	0.708999i	$0.749147\pi$
$504$	0	0
$505$	1.86318	0.0829105
$506$	31.0322	1.37955
$507$	0	0
$508$	4.21621	0.187064
$509$	−2.59835	−0.115170	−0.0575848	−	0.998341i	$-0.518340\pi$
$509$	−2.59835	−0.115170	−0.0575848	+	0.998341i	$0.518340\pi$
$510$	0	0
$511$	−1.22482	−0.0541828
$512$	46.9112	2.07320
$513$	0	0
$514$	−70.9453	−3.12926
$515$	3.37461	0.148703
$516$	0	0
$517$	29.3556	1.29106
$518$	2.11278	0.0928301
$519$	0	0
$520$	3.86107	0.169319
$521$	−34.5776	−1.51487	−0.757436	−	0.652909i	$-0.773548\pi$
$521$	−34.5776	−1.51487	−0.757436	+	0.652909i	$0.773548\pi$
$522$	0	0
$523$	10.4040	0.454935	0.227467	−	0.973786i	$-0.426955\pi$
$523$	10.4040	0.454935	0.227467	+	0.973786i	$0.426955\pi$
$524$	54.1537	2.36571
$525$	0	0
$526$	62.0660	2.70621
$527$	−53.4934	−2.33021
$528$	0	0
$529$	−10.7266	−0.466374
$530$	2.76506	0.120107
$531$	0	0
$532$	−4.64869	−0.201546
$533$	−23.6224	−1.02320
$534$	0	0
$535$	−0.919618	−0.0397585
$536$	13.1975	0.570046
$537$	0	0
$538$	25.7225	1.10898
$539$	−3.55277	−0.153029
$540$	0	0
$541$	−22.7755	−0.979196	−0.489598	−	0.871948i	$-0.662856\pi$
$541$	−22.7755	−0.979196	−0.489598	+	0.871948i	$0.662856\pi$
$542$	−24.7371	−1.06255
$543$	0	0
$544$	17.3244	0.742779
$545$	1.78089	0.0762851
$546$	0	0
$547$	25.2963	1.08159	0.540796	−	0.841154i	$-0.318123\pi$
$547$	25.2963	1.08159	0.540796	+	0.841154i	$0.318123\pi$
$548$	40.8147	1.74352
$549$	0	0
$550$	−43.6769	−1.86239
$551$	−4.56935	−0.194661
$552$	0	0
$553$	4.70270	0.199979
$554$	−7.45326	−0.316659
$555$	0	0
$556$	−34.4637	−1.46159
$557$	−36.4909	−1.54617	−0.773085	−	0.634302i	$-0.781288\pi$
$557$	−36.4909	−1.54617	−0.773085	+	0.634302i	$0.781288\pi$
$558$	0	0
$559$	30.2710	1.28033
$560$	1.40533	0.0593862
$561$	0	0
$562$	−77.1150	−3.25290
$563$	−31.8227	−1.34116	−0.670582	−	0.741835i	$-0.733956\pi$
$563$	−31.8227	−1.34116	−0.670582	+	0.741835i	$0.733956\pi$
$564$	0	0
$565$	0.891804	0.0375185
$566$	−64.8836	−2.72726
$567$	0	0
$568$	−52.5565	−2.20522
$569$	25.3998	1.06482	0.532408	−	0.846488i	$-0.321287\pi$
$569$	25.3998	1.06482	0.532408	+	0.846488i	$0.321287\pi$
$570$	0	0
$571$	34.2769	1.43444	0.717222	−	0.696845i	$-0.245413\pi$
$571$	34.2769	1.43444	0.717222	+	0.696845i	$0.245413\pi$
$572$	39.8027	1.66423
$573$	0	0
$574$	−22.1648	−0.925141
$575$	−17.2744	−0.720394
$576$	0	0
$577$	30.4223	1.26650	0.633248	−	0.773949i	$-0.281721\pi$
$577$	30.4223	1.26650	0.633248	+	0.773949i	$0.281721\pi$
$578$	−102.473	−4.26232
$579$	0	0
$580$	4.59495	0.190795
$581$	−3.89809	−0.161720
$582$	0	0
$583$	14.9830	0.620533
$584$	6.76778	0.280053
$585$	0	0
$586$	70.2578	2.90232
$587$	−9.57137	−0.395053	−0.197526	−	0.980298i	$-0.563291\pi$
$587$	−9.57137	−0.395053	−0.197526	+	0.980298i	$0.563291\pi$
$588$	0	0
$589$	−7.73781	−0.318831
$590$	8.85603	0.364597
$591$	0	0
$592$	−4.52856	−0.186123
$593$	30.8249	1.26583	0.632914	−	0.774222i	$-0.281859\pi$
$593$	30.8249	1.26583	0.632914	+	0.774222i	$0.281859\pi$
$594$	0	0
$595$	−2.00448	−0.0821755
$596$	−18.3259	−0.750657
$597$	0	0
$598$	23.2096	0.949112
$599$	35.6568	1.45690	0.728449	−	0.685100i	$-0.240241\pi$
$599$	35.6568	1.45690	0.728449	+	0.685100i	$0.240241\pi$
$600$	0	0
$601$	32.9469	1.34393	0.671967	−	0.740582i	$-0.265450\pi$
$601$	32.9469	1.34393	0.671967	+	0.740582i	$0.265450\pi$
$602$	28.4032	1.15763
$603$	0	0
$604$	−70.5174	−2.86931
$605$	0.426592	0.0173434
$606$	0	0
$607$	8.81706	0.357873	0.178937	−	0.983861i	$-0.442734\pi$
$607$	8.81706	0.357873	0.178937	+	0.983861i	$0.442734\pi$
$608$	2.50597	0.101631
$609$	0	0
$610$	−0.982489	−0.0397798
$611$	21.9556	0.888229
$612$	0	0
$613$	−42.7883	−1.72820	−0.864102	−	0.503316i	$-0.832113\pi$
$613$	−42.7883	−1.72820	−0.864102	+	0.503316i	$0.832113\pi$
$614$	35.4252	1.42964
$615$	0	0
$616$	19.6310	0.790955
$617$	−1.39640	−0.0562170	−0.0281085	−	0.999605i	$-0.508948\pi$
$617$	−1.39640	−0.0562170	−0.0281085	+	0.999605i	$0.508948\pi$
$618$	0	0
$619$	9.82893	0.395058	0.197529	−	0.980297i	$-0.436708\pi$
$619$	9.82893	0.395058	0.197529	+	0.980297i	$0.436708\pi$
$620$	7.78115	0.312499
$621$	0	0
$622$	47.9569	1.92290
$623$	−6.98825	−0.279978
$624$	0	0
$625$	23.9675	0.958698
$626$	54.5767	2.18132
$627$	0	0
$628$	3.74557	0.149464
$629$	6.45923	0.257546
$630$	0	0
$631$	−4.16758	−0.165909	−0.0829544	−	0.996553i	$-0.526436\pi$
$631$	−4.16758	−0.165909	−0.0829544	+	0.996553i	$0.526436\pi$
$632$	−25.9850	−1.03363
$633$	0	0
$634$	−64.9371	−2.57898
$635$	0.262973	0.0104358
$636$	0	0
$637$	−2.65719	−0.105282
$638$	36.7094	1.45334
$639$	0	0
$640$	4.48764	0.177390
$641$	0.957332	0.0378123	0.0189062	−	0.999821i	$-0.493982\pi$
$641$	0.957332	0.0378123	0.0189062	+	0.999821i	$0.493982\pi$
$642$	0	0
$643$	−12.6509	−0.498905	−0.249452	−	0.968387i	$-0.580251\pi$
$643$	−12.6509	−0.498905	−0.249452	+	0.968387i	$0.580251\pi$
$644$	14.7708	0.582053
$645$	0	0
$646$	−20.9537	−0.824412
$647$	−26.6327	−1.04704	−0.523519	−	0.852014i	$-0.675381\pi$
$647$	−26.6327	−1.04704	−0.523519	+	0.852014i	$0.675381\pi$
$648$	0	0
$649$	47.9880	1.88369
$650$	−32.6668	−1.28130
$651$	0	0
$652$	−10.7584	−0.421331
$653$	−8.00679	−0.313330	−0.156665	−	0.987652i	$-0.550074\pi$
$653$	−8.00679	−0.313330	−0.156665	+	0.987652i	$0.550074\pi$
$654$	0	0
$655$	3.37766	0.131976
$656$	47.5083	1.85489
$657$	0	0
$658$	20.6009	0.803106
$659$	−9.88982	−0.385253	−0.192626	−	0.981272i	$-0.561701\pi$
$659$	−9.88982	−0.385253	−0.192626	+	0.981272i	$0.561701\pi$
$660$	0	0
$661$	28.6257	1.11341	0.556706	−	0.830709i	$-0.312065\pi$
$661$	28.6257	1.11341	0.556706	+	0.830709i	$0.312065\pi$
$662$	−8.22357	−0.319618
$663$	0	0
$664$	21.5390	0.835876
$665$	−0.289947	−0.0112437
$666$	0	0
$667$	14.5188	0.562169
$668$	76.5524	2.96190
$669$	0	0
$670$	1.56600	0.0604998
$671$	−5.32379	−0.205523
$672$	0	0
$673$	40.8176	1.57340	0.786701	−	0.617335i	$-0.211788\pi$
$673$	40.8176	1.57340	0.786701	+	0.617335i	$0.211788\pi$
$674$	−42.6991	−1.64471
$675$	0	0
$676$	−25.0415	−0.963136
$677$	37.1987	1.42966	0.714831	−	0.699297i	$-0.246503\pi$
$677$	37.1987	1.42966	0.714831	+	0.699297i	$0.246503\pi$
$678$	0	0
$679$	−10.7226	−0.411497
$680$	11.0758	0.424738
$681$	0	0
$682$	62.1643	2.38039
$683$	10.1783	0.389461	0.194731	−	0.980857i	$-0.437617\pi$
$683$	10.1783	0.389461	0.194731	+	0.980857i	$0.437617\pi$
$684$	0	0
$685$	2.54569	0.0972657
$686$	−2.49323	−0.0951921
$687$	0	0
$688$	−60.8799	−2.32102
$689$	11.2061	0.426918
$690$	0	0
$691$	−18.8159	−0.715789	−0.357895	−	0.933762i	$-0.616505\pi$
$691$	−18.8159	−0.715789	−0.357895	+	0.933762i	$0.616505\pi$
$692$	34.0421	1.29409
$693$	0	0
$694$	36.9772	1.40363
$695$	−2.14956	−0.0815376
$696$	0	0
$697$	−67.7627	−2.56670
$698$	34.1031	1.29082
$699$	0	0
$700$	−20.7895	−0.785769
$701$	40.7417	1.53879	0.769396	−	0.638772i	$-0.220557\pi$
$701$	40.7417	1.53879	0.769396	+	0.638772i	$0.220557\pi$
$702$	0	0
$703$	0.934326	0.0352388
$704$	17.8397	0.672358
$705$	0	0
$706$	−77.2145	−2.90601
$707$	7.08508	0.266462
$708$	0	0
$709$	24.0156	0.901925	0.450962	−	0.892543i	$-0.351081\pi$
$709$	24.0156	0.901925	0.450962	+	0.892543i	$0.351081\pi$
$710$	−6.23627	−0.234043
$711$	0	0
$712$	38.6139	1.44712
$713$	24.5863	0.920764
$714$	0	0
$715$	2.48257	0.0928427
$716$	51.9662	1.94207
$717$	0	0
$718$	−52.3051	−1.95201
$719$	2.05844	0.0767667	0.0383834	−	0.999263i	$-0.487779\pi$
$719$	2.05844	0.0767667	0.0383834	+	0.999263i	$0.487779\pi$
$720$	0	0
$721$	12.8326	0.477909
$722$	44.3405	1.65018
$723$	0	0
$724$	−33.5121	−1.24547
$725$	−20.4347	−0.758926
$726$	0	0
$727$	23.1073	0.857004	0.428502	−	0.903541i	$-0.359042\pi$
$727$	23.1073	0.857004	0.428502	+	0.903541i	$0.359042\pi$
$728$	14.6824	0.544166
$729$	0	0
$730$	0.803055	0.0297224
$731$	86.8350	3.21171
$732$	0	0
$733$	8.13013	0.300293	0.150147	−	0.988664i	$-0.452025\pi$
$733$	8.13013	0.300293	0.150147	+	0.988664i	$0.452025\pi$
$734$	−20.2745	−0.748346
$735$	0	0
$736$	−7.96254	−0.293503
$737$	8.48565	0.312573
$738$	0	0
$739$	18.6947	0.687695	0.343848	−	0.939025i	$-0.388270\pi$
$739$	18.6947	0.687695	0.343848	+	0.939025i	$0.388270\pi$
$740$	−0.939559	−0.0345389
$741$	0	0
$742$	10.5146	0.386005
$743$	6.23303	0.228668	0.114334	−	0.993442i	$-0.463527\pi$
$743$	6.23303	0.228668	0.114334	+	0.993442i	$0.463527\pi$
$744$	0	0
$745$	−1.14302	−0.0418769
$746$	−81.9356	−2.99987
$747$	0	0
$748$	114.177	4.17474
$749$	−3.49701	−0.127778
$750$	0	0
$751$	−11.2328	−0.409889	−0.204945	−	0.978774i	$-0.565701\pi$
$751$	−11.2328	−0.409889	−0.204945	+	0.978774i	$0.565701\pi$
$752$	−44.1562	−1.61021
$753$	0	0
$754$	27.4557	0.999878
$755$	−4.39830	−0.160070
$756$	0	0
$757$	35.0765	1.27488	0.637439	−	0.770501i	$-0.279994\pi$
$757$	35.0765	1.27488	0.637439	+	0.770501i	$0.279994\pi$
$758$	−42.1180	−1.52980
$759$	0	0
$760$	1.60211	0.0581147
$761$	−12.1870	−0.441779	−0.220890	−	0.975299i	$-0.570896\pi$
$761$	−12.1870	−0.441779	−0.220890	+	0.975299i	$0.570896\pi$
$762$	0	0
$763$	6.77216	0.245169
$764$	2.83563	0.102589
$765$	0	0
$766$	20.5326	0.741873
$767$	35.8912	1.29596
$768$	0	0
$769$	3.45765	0.124686	0.0623430	−	0.998055i	$-0.480143\pi$
$769$	3.45765	0.124686	0.0623430	+	0.998055i	$0.480143\pi$
$770$	2.32938	0.0839452
$771$	0	0
$772$	42.0453	1.51324
$773$	13.7406	0.494215	0.247108	−	0.968988i	$-0.420520\pi$
$773$	13.7406	0.494215	0.247108	+	0.968988i	$0.420520\pi$
$774$	0	0
$775$	−34.6045	−1.24303
$776$	59.2484	2.12689
$777$	0	0
$778$	76.9235	2.75784
$779$	−9.80186	−0.351188
$780$	0	0
$781$	−33.7924	−1.20919
$782$	66.5787	2.38085
$783$	0	0
$784$	5.34403	0.190858
$785$	0.233618	0.00833817
$786$	0	0
$787$	−33.2567	−1.18547	−0.592736	−	0.805397i	$-0.701952\pi$
$787$	−33.2567	−1.18547	−0.592736	+	0.805397i	$0.701952\pi$
$788$	−56.3078	−2.00588
$789$	0	0
$790$	−3.08334	−0.109700
$791$	3.39124	0.120579
$792$	0	0
$793$	−3.98177	−0.141397
$794$	−31.0830	−1.10309
$795$	0	0
$796$	−20.8033	−0.737353
$797$	−40.0553	−1.41883	−0.709416	−	0.704790i	$-0.751041\pi$
$797$	−40.0553	−1.41883	−0.709416	+	0.704790i	$0.751041\pi$
$798$	0	0
$799$	62.9815	2.22812
$800$	11.2070	0.396228
$801$	0	0
$802$	−29.7683	−1.05115
$803$	4.35150	0.153561
$804$	0	0
$805$	0.921284	0.0324710
$806$	46.4939	1.63768
$807$	0	0
$808$	−39.1489	−1.37725
$809$	34.2875	1.20548	0.602742	−	0.797936i	$-0.294075\pi$
$809$	34.2875	1.20548	0.602742	+	0.797936i	$0.294075\pi$
$810$	0	0
$811$	13.9247	0.488963	0.244481	−	0.969654i	$-0.421382\pi$
$811$	13.9247	0.488963	0.244481	+	0.969654i	$0.421382\pi$
$812$	17.4731	0.613186
$813$	0	0
$814$	−7.50622	−0.263093
$815$	−0.671019	−0.0235048
$816$	0	0
$817$	12.5607	0.439442
$818$	−99.8867	−3.49246
$819$	0	0
$820$	9.85676	0.344213
$821$	−11.5694	−0.403775	−0.201888	−	0.979409i	$-0.564708\pi$
$821$	−11.5694	−0.403775	−0.201888	+	0.979409i	$0.564708\pi$
$822$	0	0
$823$	−29.7145	−1.03578	−0.517891	−	0.855447i	$-0.673283\pi$
$823$	−29.7145	−1.03578	−0.517891	+	0.855447i	$0.673283\pi$
$824$	−70.9068	−2.47015
$825$	0	0
$826$	33.6766	1.17176
$827$	−9.76698	−0.339631	−0.169816	−	0.985476i	$-0.554317\pi$
$827$	−9.76698	−0.339631	−0.169816	+	0.985476i	$0.554317\pi$
$828$	0	0
$829$	−32.2336	−1.11952	−0.559759	−	0.828655i	$-0.689106\pi$
$829$	−32.2336	−1.11952	−0.559759	+	0.828655i	$0.689106\pi$
$830$	2.55579	0.0887128
$831$	0	0
$832$	13.3426	0.462573
$833$	−7.62237	−0.264099
$834$	0	0
$835$	4.77471	0.165236
$836$	16.5157	0.571208
$837$	0	0
$838$	96.2050	3.32335
$839$	39.8825	1.37690	0.688448	−	0.725286i	$-0.258292\pi$
$839$	39.8825	1.37690	0.688448	+	0.725286i	$0.258292\pi$
$840$	0	0
$841$	−11.8251	−0.407762
$842$	−69.4267	−2.39260
$843$	0	0
$844$	83.9678	2.89029
$845$	−1.56189	−0.0537305
$846$	0	0
$847$	1.62219	0.0557392
$848$	−22.5372	−0.773932
$849$	0	0
$850$	−93.7075	−3.21414
$851$	−2.96875	−0.101767
$852$	0	0
$853$	26.2567	0.899011	0.449505	−	0.893278i	$-0.351600\pi$
$853$	26.2567	0.899011	0.449505	+	0.893278i	$0.351600\pi$
$854$	−3.73609	−0.127846
$855$	0	0
$856$	19.3228	0.660441
$857$	−14.4750	−0.494458	−0.247229	−	0.968957i	$-0.579520\pi$
$857$	−14.4750	−0.494458	−0.247229	+	0.968957i	$0.579520\pi$
$858$	0	0
$859$	−1.76811	−0.0603271	−0.0301636	−	0.999545i	$-0.509603\pi$
$859$	−1.76811	−0.0603271	−0.0301636	+	0.999545i	$0.509603\pi$
$860$	−12.6310	−0.430714
$861$	0	0
$862$	11.9685	0.407647
$863$	51.3029	1.74637	0.873186	−	0.487388i	$-0.162050\pi$
$863$	51.3029	1.74637	0.873186	+	0.487388i	$0.162050\pi$
$864$	0	0
$865$	2.12327	0.0721933
$866$	51.6729	1.75592
$867$	0	0
$868$	29.5892	1.00432
$869$	−16.7076	−0.566767
$870$	0	0
$871$	6.34659	0.215046
$872$	−37.4198	−1.26720
$873$	0	0
$874$	9.63060	0.325760
$875$	−2.61154	−0.0882862
$876$	0	0
$877$	58.7886	1.98515	0.992575	−	0.121633i	$-0.0388130\pi$
$877$	58.7886	1.98515	0.992575	+	0.121633i	$0.0388130\pi$
$878$	−40.8333	−1.37806
$879$	0	0
$880$	−4.99283	−0.168308
$881$	6.41629	0.216170	0.108085	−	0.994142i	$-0.465528\pi$
$881$	6.41629	0.216170	0.108085	+	0.994142i	$0.465528\pi$
$882$	0	0
$883$	−15.8316	−0.532776	−0.266388	−	0.963866i	$-0.585830\pi$
$883$	−15.8316	−0.532776	−0.266388	+	0.963866i	$0.585830\pi$
$884$	85.3955	2.87216
$885$	0	0
$886$	50.3341	1.69101
$887$	−33.1828	−1.11417	−0.557084	−	0.830456i	$-0.688080\pi$
$887$	−33.1828	−1.11417	−0.557084	+	0.830456i	$0.688080\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	4.58186	0.153584
$891$	0	0
$892$	−33.8871	−1.13462
$893$	9.11025	0.304863
$894$	0	0
$895$	3.24123	0.108342
$896$	17.0651	0.570103
$897$	0	0
$898$	−24.9822	−0.833667
$899$	29.0843	0.970014
$900$	0	0
$901$	32.1456	1.07092
$902$	78.7465	2.62197
$903$	0	0
$904$	−18.7384	−0.623231
$905$	−2.09021	−0.0694808
$906$	0	0
$907$	−37.1547	−1.23370	−0.616851	−	0.787080i	$-0.711592\pi$
$907$	−37.1547	−1.23370	−0.616851	+	0.787080i	$0.711592\pi$
$908$	35.8975	1.19130
$909$	0	0
$910$	1.74219	0.0577531
$911$	14.3754	0.476279	0.238139	−	0.971231i	$-0.423462\pi$
$911$	14.3754	0.476279	0.238139	+	0.971231i	$0.423462\pi$
$912$	0	0
$913$	13.8490	0.458335
$914$	30.5159	1.00938
$915$	0	0
$916$	−24.1549	−0.798102
$917$	12.8441	0.424151
$918$	0	0
$919$	17.1784	0.566662	0.283331	−	0.959022i	$-0.408560\pi$
$919$	17.1784	0.566662	0.283331	+	0.959022i	$0.408560\pi$
$920$	−5.09059	−0.167832
$921$	0	0
$922$	82.8101	2.72721
$923$	−25.2740	−0.831904
$924$	0	0
$925$	4.17842	0.137386
$926$	−4.49180	−0.147610
$927$	0	0
$928$	−9.41925	−0.309202
$929$	−2.11145	−0.0692745	−0.0346373	−	0.999400i	$-0.511028\pi$
$929$	−2.11145	−0.0692745	−0.0346373	+	0.999400i	$0.511028\pi$
$930$	0	0
$931$	−1.10257	−0.0361354
$932$	42.4120	1.38925
$933$	0	0
$934$	−65.3308	−2.13769
$935$	7.12145	0.232896
$936$	0	0
$937$	9.84177	0.321516	0.160758	−	0.986994i	$-0.448606\pi$
$937$	9.84177	0.321516	0.160758	+	0.986994i	$0.448606\pi$
$938$	5.95499	0.194437
$939$	0	0
$940$	−9.16128	−0.298808
$941$	−50.6893	−1.65242	−0.826211	−	0.563360i	$-0.809508\pi$
$941$	−50.6893	−1.65242	−0.826211	+	0.563360i	$0.809508\pi$
$942$	0	0
$943$	31.1446	1.01421
$944$	−72.1829	−2.34935
$945$	0	0
$946$	−100.910	−3.28087
$947$	−25.5880	−0.831497	−0.415748	−	0.909480i	$-0.636480\pi$
$947$	−25.5880	−0.831497	−0.415748	+	0.909480i	$0.636480\pi$
$948$	0	0
$949$	3.25457	0.105648
$950$	−13.5548	−0.439775
$951$	0	0
$952$	42.1177	1.36504
$953$	14.8512	0.481077	0.240539	−	0.970640i	$-0.422676\pi$
$953$	14.8512	0.481077	0.240539	+	0.970640i	$0.422676\pi$
$954$	0	0
$955$	0.176863	0.00572316
$956$	−56.5422	−1.82871
$957$	0	0
$958$	−52.8216	−1.70659
$959$	9.68042	0.312597
$960$	0	0
$961$	18.2517	0.588765
$962$	−5.61405	−0.181004
$963$	0	0
$964$	36.6591	1.18071
$965$	2.62244	0.0844194
$966$	0	0
$967$	−35.2407	−1.13326	−0.566632	−	0.823971i	$-0.691754\pi$
$967$	−35.2407	−1.13326	−0.566632	+	0.823971i	$0.691754\pi$
$968$	−8.96349	−0.288097
$969$	0	0
$970$	7.03033	0.225730
$971$	23.7014	0.760614	0.380307	−	0.924860i	$-0.375818\pi$
$971$	23.7014	0.760614	0.380307	+	0.924860i	$0.375818\pi$
$972$	0	0
$973$	−8.17409	−0.262049
$974$	−16.7052	−0.535269
$975$	0	0
$976$	8.00798	0.256329
$977$	−40.3025	−1.28939	−0.644696	−	0.764439i	$-0.723016\pi$
$977$	−40.3025	−1.28939	−0.644696	+	0.764439i	$0.723016\pi$
$978$	0	0
$979$	24.8277	0.793496
$980$	1.10875	0.0354177
$981$	0	0
$982$	−71.9332	−2.29548
$983$	−21.8967	−0.698395	−0.349197	−	0.937049i	$-0.613546\pi$
$983$	−21.8967	−0.698395	−0.349197	+	0.937049i	$0.613546\pi$
$984$	0	0
$985$	−3.51202	−0.111902
$986$	78.7590	2.50820
$987$	0	0
$988$	12.3524	0.392983
$989$	−39.9105	−1.26908
$990$	0	0
$991$	14.7659	0.469054	0.234527	−	0.972110i	$-0.424646\pi$
$991$	14.7659	0.469054	0.234527	+	0.972110i	$0.424646\pi$
$992$	−15.9507	−0.506435
$993$	0	0
$994$	−23.7145	−0.752179
$995$	−1.29754	−0.0411347
$996$	0	0
$997$	29.3782	0.930416	0.465208	−	0.885201i	$-0.345980\pi$
$997$	29.3782	0.930416	0.465208	+	0.885201i	$0.345980\pi$
$998$	48.3970	1.53198
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.v.1.2		19
3.2	odd	2		2667.2.a.q.1.18	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.q.1.18	✓	19	3.2	odd	2
8001.2.a.v.1.2		19	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)

\(f(q)\)	\(=\)	\(q-2.49323 q^{2} +4.21621 q^{4} +0.262973 q^{5} +1.00000 q^{7} -5.52554 q^{8} +O(q^{10})\)	\(q-2.49323 q^{2} +4.21621 q^{4} +0.262973 q^{5} +1.00000 q^{7} -5.52554 q^{8} -0.655653 q^{10} -3.55277 q^{11} -2.65719 q^{13} -2.49323 q^{14} +5.34403 q^{16} -7.62237 q^{17} -1.10257 q^{19} +1.10875 q^{20} +8.85789 q^{22} +3.50334 q^{23} -4.93085 q^{25} +6.62499 q^{26} +4.21621 q^{28} +4.14426 q^{29} +7.01796 q^{31} -2.27284 q^{32} +19.0043 q^{34} +0.262973 q^{35} -0.847404 q^{37} +2.74897 q^{38} -1.45307 q^{40} +8.88998 q^{41} -11.3921 q^{43} -14.9792 q^{44} -8.73465 q^{46} -8.26272 q^{47} +1.00000 q^{49} +12.2937 q^{50} -11.2033 q^{52} -4.21727 q^{53} -0.934282 q^{55} -5.52554 q^{56} -10.3326 q^{58} -13.5072 q^{59} +1.49849 q^{61} -17.4974 q^{62} -5.02134 q^{64} -0.698769 q^{65} -2.38846 q^{67} -32.1375 q^{68} -0.655653 q^{70} +9.51155 q^{71} -1.22482 q^{73} +2.11278 q^{74} -4.64869 q^{76} -3.55277 q^{77} +4.70270 q^{79} +1.40533 q^{80} -22.1648 q^{82} -3.89809 q^{83} -2.00448 q^{85} +28.4032 q^{86} +19.6310 q^{88} -6.98825 q^{89} -2.65719 q^{91} +14.7708 q^{92} +20.6009 q^{94} -0.289947 q^{95} -10.7226 q^{97} -2.49323 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * q^98

Introduction

L-functions

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Database

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