LMFDB - Embedded newform 8001.2.a.w.1.1

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			1.1
Root			$2.75954$ 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)

$f(q)$	$=$	$q-2.75954 q^{2} +5.61505 q^{4} -0.308409 q^{5} +1.00000 q^{7} -9.97588 q^{8} +O(q^{10})$	\(q-2.75954 q^{2} +5.61505 q^{4} -0.308409 q^{5} +1.00000 q^{7} -9.97588 q^{8} +0.851066 q^{10} -1.97479 q^{11} -1.50333 q^{13} -2.75954 q^{14} +16.2987 q^{16} +1.53428 q^{17} +4.38101 q^{19} -1.73173 q^{20} +5.44950 q^{22} +5.56049 q^{23} -4.90488 q^{25} +4.14849 q^{26} +5.61505 q^{28} -6.08293 q^{29} +4.88474 q^{31} -25.0252 q^{32} -4.23392 q^{34} -0.308409 q^{35} +4.19832 q^{37} -12.0896 q^{38} +3.07665 q^{40} -9.66235 q^{41} -5.29439 q^{43} -11.0885 q^{44} -15.3444 q^{46} +1.73288 q^{47} +1.00000 q^{49} +13.5352 q^{50} -8.44126 q^{52} +11.5579 q^{53} +0.609042 q^{55} -9.97588 q^{56} +16.7861 q^{58} -4.40906 q^{59} -14.1347 q^{61} -13.4796 q^{62} +36.4605 q^{64} +0.463639 q^{65} -11.4930 q^{67} +8.61509 q^{68} +0.851066 q^{70} +6.35836 q^{71} -12.8604 q^{73} -11.5854 q^{74} +24.5996 q^{76} -1.97479 q^{77} +0.329262 q^{79} -5.02667 q^{80} +26.6636 q^{82} +6.82127 q^{83} -0.473187 q^{85} +14.6101 q^{86} +19.7002 q^{88} +9.25437 q^{89} -1.50333 q^{91} +31.2225 q^{92} -4.78194 q^{94} -1.35114 q^{95} +4.74074 q^{97} -2.75954 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.75954	−1.95129	−0.975644	−	0.219359i	$-0.929603\pi$
$2$	−2.75954	−1.95129	−0.975644	+	0.219359i	$0.929603\pi$
$3$	0	0
$3$	0	0
$4$	5.61505	2.80753
$5$	−0.308409	−0.137925	−0.0689623	−	0.997619i	$-0.521969\pi$
$5$	−0.308409	−0.137925	−0.0689623	+	0.997619i	$0.521969\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−9.97588	−3.52701
$9$	0	0
$10$	0.851066	0.269131
$11$	−1.97479	−0.595421	−0.297710	−	0.954656i	$-0.596223\pi$
$11$	−1.97479	−0.595421	−0.297710	+	0.954656i	$0.596223\pi$
$12$	0	0
$13$	−1.50333	−0.416948	−0.208474	−	0.978028i	$-0.566850\pi$
$13$	−1.50333	−0.416948	−0.208474	+	0.978028i	$0.566850\pi$
$14$	−2.75954	−0.737518
$15$	0	0
$16$	16.2987	4.07468
$17$	1.53428	0.372119	0.186059	−	0.982539i	$-0.440428\pi$
$17$	1.53428	0.372119	0.186059	+	0.982539i	$0.440428\pi$
$18$	0	0
$19$	4.38101	1.00507	0.502536	−	0.864556i	$-0.332400\pi$
$19$	4.38101	1.00507	0.502536	+	0.864556i	$0.332400\pi$
$20$	−1.73173	−0.387227
$21$	0	0
$22$	5.44950	1.16184
$23$	5.56049	1.15944	0.579721	−	0.814815i	$-0.303161\pi$
$23$	5.56049	1.15944	0.579721	+	0.814815i	$0.303161\pi$
$24$	0	0
$25$	−4.90488	−0.980977
$26$	4.14849	0.813585
$27$	0	0
$28$	5.61505	1.06115
$29$	−6.08293	−1.12957	−0.564786	−	0.825237i	$-0.691041\pi$
$29$	−6.08293	−1.12957	−0.564786	+	0.825237i	$0.691041\pi$
$30$	0	0
$31$	4.88474	0.877324	0.438662	−	0.898652i	$-0.355452\pi$
$31$	4.88474	0.877324	0.438662	+	0.898652i	$0.355452\pi$
$32$	−25.0252	−4.42387
$33$	0	0
$34$	−4.23392	−0.726111
$35$	−0.308409	−0.0521306
$36$	0	0
$37$	4.19832	0.690200	0.345100	−	0.938566i	$-0.387845\pi$
$37$	4.19832	0.690200	0.345100	+	0.938566i	$0.387845\pi$
$38$	−12.0896	−1.96119
$39$	0	0
$40$	3.07665	0.486461
$41$	−9.66235	−1.50901	−0.754503	−	0.656297i	$-0.772122\pi$
$41$	−9.66235	−1.50901	−0.754503	+	0.656297i	$0.772122\pi$
$42$	0	0
$43$	−5.29439	−0.807387	−0.403693	−	0.914894i	$-0.632274\pi$
$43$	−5.29439	−0.807387	−0.403693	+	0.914894i	$0.632274\pi$
$44$	−11.0885	−1.67166
$45$	0	0
$46$	−15.3444	−2.26241
$47$	1.73288	0.252766	0.126383	−	0.991982i	$-0.459663\pi$
$47$	1.73288	0.252766	0.126383	+	0.991982i	$0.459663\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	13.5352	1.91417
$51$	0	0
$52$	−8.44126	−1.17059
$53$	11.5579	1.58760	0.793799	−	0.608180i	$-0.208100\pi$
$53$	11.5579	1.58760	0.793799	+	0.608180i	$0.208100\pi$
$54$	0	0
$55$	0.609042	0.0821232
$56$	−9.97588	−1.33308
$57$	0	0
$58$	16.7861	2.20412
$59$	−4.40906	−0.574011	−0.287006	−	0.957929i	$-0.592660\pi$
$59$	−4.40906	−0.574011	−0.287006	+	0.957929i	$0.592660\pi$
$60$	0	0
$61$	−14.1347	−1.80976	−0.904880	−	0.425666i	$-0.860040\pi$
$61$	−14.1347	−1.80976	−0.904880	+	0.425666i	$0.860040\pi$
$62$	−13.4796	−1.71191
$63$	0	0
$64$	36.4605	4.55757
$65$	0.463639	0.0575073
$66$	0	0
$67$	−11.4930	−1.40409	−0.702047	−	0.712131i	$-0.747730\pi$
$67$	−11.4930	−1.40409	−0.702047	+	0.712131i	$0.747730\pi$
$68$	8.61509	1.04473
$69$	0	0
$70$	0.851066	0.101722
$71$	6.35836	0.754599	0.377299	−	0.926091i	$-0.376853\pi$
$71$	6.35836	0.754599	0.377299	+	0.926091i	$0.376853\pi$
$72$	0	0
$73$	−12.8604	−1.50520	−0.752599	−	0.658479i	$-0.771200\pi$
$73$	−12.8604	−1.50520	−0.752599	+	0.658479i	$0.771200\pi$
$74$	−11.5854	−1.34678
$75$	0	0
$76$	24.5996	2.82177
$77$	−1.97479	−0.225048
$78$	0	0
$79$	0.329262	0.0370448	0.0185224	−	0.999828i	$-0.494104\pi$
$79$	0.329262	0.0370448	0.0185224	+	0.999828i	$0.494104\pi$
$80$	−5.02667	−0.561999
$81$	0	0
$82$	26.6636	2.94451
$83$	6.82127	0.748732	0.374366	−	0.927281i	$-0.377860\pi$
$83$	6.82127	0.748732	0.374366	+	0.927281i	$0.377860\pi$
$84$	0	0
$85$	−0.473187	−0.0513243
$86$	14.6101	1.57544
$87$	0	0
$88$	19.7002	2.10005
$89$	9.25437	0.980962	0.490481	−	0.871452i	$-0.336821\pi$
$89$	9.25437	0.980962	0.490481	+	0.871452i	$0.336821\pi$
$90$	0	0
$91$	−1.50333	−0.157591
$92$	31.2225	3.25517
$93$	0	0
$94$	−4.78194	−0.493219
$95$	−1.35114	−0.138624
$96$	0	0
$97$	4.74074	0.481349	0.240674	−	0.970606i	$-0.422631\pi$
$97$	4.74074	0.481349	0.240674	+	0.970606i	$0.422631\pi$
$98$	−2.75954	−0.278756
$99$	0	0
$100$	−27.5412	−2.75412
$101$	15.1594	1.50842	0.754210	−	0.656633i	$-0.228020\pi$
$101$	15.1594	1.50842	0.754210	+	0.656633i	$0.228020\pi$
$102$	0	0
$103$	8.27078	0.814944	0.407472	−	0.913218i	$-0.366410\pi$
$103$	8.27078	0.814944	0.407472	+	0.913218i	$0.366410\pi$
$104$	14.9970	1.47058
$105$	0	0
$106$	−31.8944	−3.09786
$107$	−2.93782	−0.284010	−0.142005	−	0.989866i	$-0.545355\pi$
$107$	−2.93782	−0.284010	−0.142005	+	0.989866i	$0.545355\pi$
$108$	0	0
$109$	3.12177	0.299012	0.149506	−	0.988761i	$-0.452232\pi$
$109$	3.12177	0.299012	0.149506	+	0.988761i	$0.452232\pi$
$110$	−1.68067	−0.160246
$111$	0	0
$112$	16.2987	1.54008
$113$	−3.67619	−0.345827	−0.172913	−	0.984937i	$-0.555318\pi$
$113$	−3.67619	−0.345827	−0.172913	+	0.984937i	$0.555318\pi$
$114$	0	0
$115$	−1.71490	−0.159916
$116$	−34.1560	−3.17130
$117$	0	0
$118$	12.1670	1.12006
$119$	1.53428	0.140648
$120$	0	0
$121$	−7.10022	−0.645474
$122$	39.0052	3.53136
$123$	0	0
$124$	27.4281	2.46311
$125$	3.05475	0.273225
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−50.5639	−4.46926
$129$	0	0
$130$	−1.27943	−0.112213
$131$	−13.9223	−1.21640	−0.608199	−	0.793784i	$-0.708108\pi$
$131$	−13.9223	−1.21640	−0.608199	+	0.793784i	$0.708108\pi$
$132$	0	0
$133$	4.38101	0.379882
$134$	31.7154	2.73979
$135$	0	0
$136$	−15.3058	−1.31247
$137$	−4.12414	−0.352349	−0.176175	−	0.984359i	$-0.556372\pi$
$137$	−4.12414	−0.352349	−0.176175	+	0.984359i	$0.556372\pi$
$138$	0	0
$139$	0.0250793	0.00212720	0.00106360	−	0.999999i	$-0.499661\pi$
$139$	0.0250793	0.00212720	0.00106360	+	0.999999i	$0.499661\pi$
$140$	−1.73173	−0.146358
$141$	0	0
$142$	−17.5462	−1.47244
$143$	2.96875	0.248259
$144$	0	0
$145$	1.87603	0.155796
$146$	35.4888	2.93707
$147$	0	0
$148$	23.5738	1.93775
$149$	−17.9241	−1.46840	−0.734200	−	0.678933i	$-0.762443\pi$
$149$	−17.9241	−1.46840	−0.734200	+	0.678933i	$0.762443\pi$
$150$	0	0
$151$	19.0202	1.54784	0.773919	−	0.633284i	$-0.218294\pi$
$151$	19.0202	1.54784	0.773919	+	0.633284i	$0.218294\pi$
$152$	−43.7044	−3.54490
$153$	0	0
$154$	5.44950	0.439133
$155$	−1.50650	−0.121005
$156$	0	0
$157$	−17.7828	−1.41922	−0.709611	−	0.704594i	$-0.751129\pi$
$157$	−17.7828	−1.41922	−0.709611	+	0.704594i	$0.751129\pi$
$158$	−0.908610	−0.0722852
$159$	0	0
$160$	7.71799	0.610161
$161$	5.56049	0.438228
$162$	0	0
$163$	20.2728	1.58789	0.793945	−	0.607989i	$-0.208024\pi$
$163$	20.2728	1.58789	0.793945	+	0.607989i	$0.208024\pi$
$164$	−54.2546	−4.23657
$165$	0	0
$166$	−18.8236	−1.46099
$167$	−20.8220	−1.61126	−0.805629	−	0.592420i	$-0.798172\pi$
$167$	−20.8220	−1.61126	−0.805629	+	0.592420i	$0.798172\pi$
$168$	0	0
$169$	−10.7400	−0.826155
$170$	1.30578	0.100149
$171$	0	0
$172$	−29.7283	−2.26676
$173$	9.45710	0.719010	0.359505	−	0.933143i	$-0.382946\pi$
$173$	9.45710	0.719010	0.359505	+	0.933143i	$0.382946\pi$
$174$	0	0
$175$	−4.90488	−0.370774
$176$	−32.1865	−2.42615
$177$	0	0
$178$	−25.5378	−1.91414
$179$	8.00826	0.598565	0.299283	−	0.954165i	$-0.403253\pi$
$179$	8.00826	0.598565	0.299283	+	0.954165i	$0.403253\pi$
$180$	0	0
$181$	−4.31397	−0.320655	−0.160327	−	0.987064i	$-0.551255\pi$
$181$	−4.31397	−0.320655	−0.160327	+	0.987064i	$0.551255\pi$
$182$	4.14849	0.307506
$183$	0	0
$184$	−55.4708	−4.08936
$185$	−1.29480	−0.0951955
$186$	0	0
$187$	−3.02989	−0.221567
$188$	9.73019	0.709647
$189$	0	0
$190$	3.72853	0.270496
$191$	−1.09157	−0.0789831	−0.0394915	−	0.999220i	$-0.512574\pi$
$191$	−1.09157	−0.0789831	−0.0394915	+	0.999220i	$0.512574\pi$
$192$	0	0
$193$	−12.8329	−0.923735	−0.461868	−	0.886949i	$-0.652821\pi$
$193$	−12.8329	−0.923735	−0.461868	+	0.886949i	$0.652821\pi$
$194$	−13.0822	−0.939251
$195$	0	0
$196$	5.61505	0.401075
$197$	23.7467	1.69188	0.845942	−	0.533274i	$-0.179039\pi$
$197$	23.7467	1.69188	0.845942	+	0.533274i	$0.179039\pi$
$198$	0	0
$199$	16.3672	1.16024	0.580122	−	0.814530i	$-0.303005\pi$
$199$	16.3672	1.16024	0.580122	+	0.814530i	$0.303005\pi$
$200$	48.9305	3.45991
$201$	0	0
$202$	−41.8331	−2.94336
$203$	−6.08293	−0.426938
$204$	0	0
$205$	2.97995	0.208129
$206$	−22.8235	−1.59019
$207$	0	0
$208$	−24.5023	−1.69893
$209$	−8.65156	−0.598441
$210$	0	0
$211$	−21.2266	−1.46130	−0.730649	−	0.682754i	$-0.760782\pi$
$211$	−21.2266	−1.46130	−0.730649	+	0.682754i	$0.760782\pi$
$212$	64.8982	4.45722
$213$	0	0
$214$	8.10702	0.554185
$215$	1.63284	0.111359
$216$	0	0
$217$	4.88474	0.331597
$218$	−8.61466	−0.583458
$219$	0	0
$220$	3.41980	0.230563
$221$	−2.30653	−0.155154
$222$	0	0
$223$	12.7914	0.856576	0.428288	−	0.903642i	$-0.359117\pi$
$223$	12.7914	0.856576	0.428288	+	0.903642i	$0.359117\pi$
$224$	−25.0252	−1.67207
$225$	0	0
$226$	10.1446	0.674808
$227$	−5.96574	−0.395960	−0.197980	−	0.980206i	$-0.563438\pi$
$227$	−5.96574	−0.395960	−0.197980	+	0.980206i	$0.563438\pi$
$228$	0	0
$229$	−12.1405	−0.802270	−0.401135	−	0.916019i	$-0.631384\pi$
$229$	−12.1405	−0.802270	−0.401135	+	0.916019i	$0.631384\pi$
$230$	4.73234	0.312042
$231$	0	0
$232$	60.6826	3.98401
$233$	−27.2091	−1.78253	−0.891263	−	0.453488i	$-0.850180\pi$
$233$	−27.2091	−1.78253	−0.891263	+	0.453488i	$0.850180\pi$
$234$	0	0
$235$	−0.534434	−0.0348626
$236$	−24.7571	−1.61155
$237$	0	0
$238$	−4.23392	−0.274444
$239$	−6.50151	−0.420548	−0.210274	−	0.977643i	$-0.567436\pi$
$239$	−6.50151	−0.420548	−0.210274	+	0.977643i	$0.567436\pi$
$240$	0	0
$241$	−21.0546	−1.35624	−0.678122	−	0.734950i	$-0.737206\pi$
$241$	−21.0546	−1.35624	−0.678122	+	0.734950i	$0.737206\pi$
$242$	19.5933	1.25951
$243$	0	0
$244$	−79.3670	−5.08095
$245$	−0.308409	−0.0197035
$246$	0	0
$247$	−6.58609	−0.419063
$248$	−48.7295	−3.09433
$249$	0	0
$250$	−8.42971	−0.533142
$251$	20.3697	1.28572	0.642861	−	0.765983i	$-0.277747\pi$
$251$	20.3697	1.28572	0.642861	+	0.765983i	$0.277747\pi$
$252$	0	0
$253$	−10.9808	−0.690356
$254$	2.75954	0.173149
$255$	0	0
$256$	66.6120	4.16325
$257$	5.38398	0.335843	0.167922	−	0.985800i	$-0.446294\pi$
$257$	5.38398	0.335843	0.167922	+	0.985800i	$0.446294\pi$
$258$	0	0
$259$	4.19832	0.260871
$260$	2.60336	0.161453
$261$	0	0
$262$	38.4192	2.37354
$263$	3.53018	0.217680	0.108840	−	0.994059i	$-0.465286\pi$
$263$	3.53018	0.217680	0.108840	+	0.994059i	$0.465286\pi$
$264$	0	0
$265$	−3.56455	−0.218969
$266$	−12.0896	−0.741259
$267$	0	0
$268$	−64.5338	−3.94203
$269$	−17.8009	−1.08534	−0.542671	−	0.839946i	$-0.682587\pi$
$269$	−17.8009	−1.08534	−0.542671	+	0.839946i	$0.682587\pi$
$270$	0	0
$271$	3.28732	0.199690	0.0998451	−	0.995003i	$-0.468165\pi$
$271$	3.28732	0.199690	0.0998451	+	0.995003i	$0.468165\pi$
$272$	25.0069	1.51626
$273$	0	0
$274$	11.3807	0.687535
$275$	9.68610	0.584094
$276$	0	0
$277$	27.1056	1.62862	0.814308	−	0.580434i	$-0.197117\pi$
$277$	27.1056	1.62862	0.814308	+	0.580434i	$0.197117\pi$
$278$	−0.0692074	−0.00415079
$279$	0	0
$280$	3.07665	0.183865
$281$	−20.4693	−1.22109	−0.610547	−	0.791980i	$-0.709050\pi$
$281$	−20.4693	−1.22109	−0.610547	+	0.791980i	$0.709050\pi$
$282$	0	0
$283$	2.16403	0.128639	0.0643193	−	0.997929i	$-0.479512\pi$
$283$	2.16403	0.128639	0.0643193	+	0.997929i	$0.479512\pi$
$284$	35.7026	2.11856
$285$	0	0
$286$	−8.19238	−0.484426
$287$	−9.66235	−0.570351
$288$	0	0
$289$	−14.6460	−0.861528
$290$	−5.17698	−0.304003
$291$	0	0
$292$	−72.2119	−4.22588
$293$	26.9859	1.57653	0.788265	−	0.615336i	$-0.210980\pi$
$293$	26.9859	1.57653	0.788265	+	0.615336i	$0.210980\pi$
$294$	0	0
$295$	1.35979	0.0791703
$296$	−41.8820	−2.43434
$297$	0	0
$298$	49.4623	2.86527
$299$	−8.35923	−0.483427
$300$	0	0
$301$	−5.29439	−0.305164
$302$	−52.4869	−3.02028
$303$	0	0
$304$	71.4049	4.09535
$305$	4.35926	0.249610
$306$	0	0
$307$	−1.23124	−0.0702707	−0.0351353	−	0.999383i	$-0.511186\pi$
$307$	−1.23124	−0.0702707	−0.0351353	+	0.999383i	$0.511186\pi$
$308$	−11.0885	−0.631828
$309$	0	0
$310$	4.15723	0.236115
$311$	−13.0193	−0.738256	−0.369128	−	0.929379i	$-0.620344\pi$
$311$	−13.0193	−0.738256	−0.369128	+	0.929379i	$0.620344\pi$
$312$	0	0
$313$	−20.1422	−1.13851	−0.569253	−	0.822163i	$-0.692767\pi$
$313$	−20.1422	−1.13851	−0.569253	+	0.822163i	$0.692767\pi$
$314$	49.0723	2.76931
$315$	0	0
$316$	1.84882	0.104004
$317$	−5.78161	−0.324728	−0.162364	−	0.986731i	$-0.551912\pi$
$317$	−5.78161	−0.324728	−0.162364	+	0.986731i	$0.551912\pi$
$318$	0	0
$319$	12.0125	0.672571
$320$	−11.2448	−0.628601
$321$	0	0
$322$	−15.3444	−0.855109
$323$	6.72172	0.374006
$324$	0	0
$325$	7.37364	0.409016
$326$	−55.9437	−3.09843
$327$	0	0
$328$	96.3905	5.32227
$329$	1.73288	0.0955365
$330$	0	0
$331$	28.6048	1.57226	0.786130	−	0.618061i	$-0.212082\pi$
$331$	28.6048	1.57226	0.786130	+	0.618061i	$0.212082\pi$
$332$	38.3018	2.10208
$333$	0	0
$334$	57.4592	3.14403
$335$	3.54454	0.193659
$336$	0	0
$337$	4.78711	0.260771	0.130385	−	0.991463i	$-0.458379\pi$
$337$	4.78711	0.260771	0.130385	+	0.991463i	$0.458379\pi$
$338$	29.6375	1.61207
$339$	0	0
$340$	−2.65697	−0.144094
$341$	−9.64631	−0.522377
$342$	0	0
$343$	1.00000	0.0539949
$344$	52.8162	2.84766
$345$	0	0
$346$	−26.0972	−1.40300
$347$	0.522899	0.0280707	0.0140353	−	0.999901i	$-0.495532\pi$
$347$	0.522899	0.0280707	0.0140353	+	0.999901i	$0.495532\pi$
$348$	0	0
$349$	17.5273	0.938216	0.469108	−	0.883141i	$-0.344575\pi$
$349$	17.5273	0.938216	0.469108	+	0.883141i	$0.344575\pi$
$350$	13.5352	0.723488
$351$	0	0
$352$	49.4194	2.63406
$353$	−13.8534	−0.737341	−0.368671	−	0.929560i	$-0.620187\pi$
$353$	−13.8534	−0.737341	−0.368671	+	0.929560i	$0.620187\pi$
$354$	0	0
$355$	−1.96098	−0.104078
$356$	51.9638	2.75408
$357$	0	0
$358$	−22.0991	−1.16797
$359$	−4.94008	−0.260727	−0.130364	−	0.991466i	$-0.541614\pi$
$359$	−4.94008	−0.260727	−0.130364	+	0.991466i	$0.541614\pi$
$360$	0	0
$361$	0.193250	0.0101711
$362$	11.9046	0.625690
$363$	0	0
$364$	−8.44126	−0.442442
$365$	3.96626	0.207604
$366$	0	0
$367$	16.4313	0.857708	0.428854	−	0.903374i	$-0.358918\pi$
$367$	16.4313	0.857708	0.428854	+	0.903374i	$0.358918\pi$
$368$	90.6289	4.72436
$369$	0	0
$370$	3.57305	0.185754
$371$	11.5579	0.600056
$372$	0	0
$373$	18.5850	0.962297	0.481148	−	0.876639i	$-0.340220\pi$
$373$	18.5850	0.962297	0.481148	+	0.876639i	$0.340220\pi$
$374$	8.36109	0.432342
$375$	0	0
$376$	−17.2870	−0.891507
$377$	9.14463	0.470973
$378$	0	0
$379$	−10.2500	−0.526505	−0.263253	−	0.964727i	$-0.584795\pi$
$379$	−10.2500	−0.526505	−0.263253	+	0.964727i	$0.584795\pi$
$380$	−7.58674	−0.389191
$381$	0	0
$382$	3.01222	0.154119
$383$	−25.5087	−1.30344	−0.651718	−	0.758462i	$-0.725951\pi$
$383$	−25.5087	−1.30344	−0.651718	+	0.758462i	$0.725951\pi$
$384$	0	0
$385$	0.609042	0.0310396
$386$	35.4130	1.80247
$387$	0	0
$388$	26.6195	1.35140
$389$	1.62422	0.0823511	0.0411756	−	0.999152i	$-0.486890\pi$
$389$	1.62422	0.0823511	0.0411756	+	0.999152i	$0.486890\pi$
$390$	0	0
$391$	8.53138	0.431450
$392$	−9.97588	−0.503858
$393$	0	0
$394$	−65.5300	−3.30136
$395$	−0.101547	−0.00510939
$396$	0	0
$397$	20.7976	1.04380	0.521902	−	0.853006i	$-0.325223\pi$
$397$	20.7976	1.04380	0.521902	+	0.853006i	$0.325223\pi$
$398$	−45.1660	−2.26397
$399$	0	0
$400$	−79.9433	−3.99717
$401$	−34.6555	−1.73061	−0.865306	−	0.501245i	$-0.832876\pi$
$401$	−34.6555	−1.73061	−0.865306	+	0.501245i	$0.832876\pi$
$402$	0	0
$403$	−7.34335	−0.365798
$404$	85.1211	4.23493
$405$	0	0
$406$	16.7861	0.833080
$407$	−8.29079	−0.410959
$408$	0	0
$409$	−28.7690	−1.42253	−0.711267	−	0.702922i	$-0.751878\pi$
$409$	−28.7690	−1.42253	−0.711267	+	0.702922i	$0.751878\pi$
$410$	−8.22330	−0.406120
$411$	0	0
$412$	46.4409	2.28798
$413$	−4.40906	−0.216956
$414$	0	0
$415$	−2.10374	−0.103269
$416$	37.6210	1.84452
$417$	0	0
$418$	23.8743	1.16773
$419$	5.64930	0.275986	0.137993	−	0.990433i	$-0.455935\pi$
$419$	5.64930	0.275986	0.137993	+	0.990433i	$0.455935\pi$
$420$	0	0
$421$	−19.1531	−0.933464	−0.466732	−	0.884399i	$-0.654569\pi$
$421$	−19.1531	−0.933464	−0.466732	+	0.884399i	$0.654569\pi$
$422$	58.5755	2.85141
$423$	0	0
$424$	−115.300	−5.59947
$425$	−7.52549	−0.365040
$426$	0	0
$427$	−14.1347	−0.684025
$428$	−16.4960	−0.797364
$429$	0	0
$430$	−4.50587	−0.217293
$431$	39.3607	1.89594	0.947970	−	0.318359i	$-0.103132\pi$
$431$	39.3607	1.89594	0.947970	+	0.318359i	$0.103132\pi$
$432$	0	0
$433$	28.1173	1.35123	0.675615	−	0.737254i	$-0.263878\pi$
$433$	28.1173	1.35123	0.675615	+	0.737254i	$0.263878\pi$
$434$	−13.4796	−0.647042
$435$	0	0
$436$	17.5289	0.839483
$437$	24.3606	1.16532
$438$	0	0
$439$	−28.7294	−1.37118	−0.685590	−	0.727988i	$-0.740456\pi$
$439$	−28.7294	−1.37118	−0.685590	+	0.727988i	$0.740456\pi$
$440$	−6.07573	−0.289649
$441$	0	0
$442$	6.36496	0.302750
$443$	−7.89956	−0.375319	−0.187660	−	0.982234i	$-0.560090\pi$
$443$	−7.89956	−0.375319	−0.187660	+	0.982234i	$0.560090\pi$
$444$	0	0
$445$	−2.85413	−0.135299
$446$	−35.2984	−1.67143
$447$	0	0
$448$	36.4605	1.72260
$449$	−29.4416	−1.38943	−0.694717	−	0.719283i	$-0.744471\pi$
$449$	−29.4416	−1.38943	−0.694717	+	0.719283i	$0.744471\pi$
$450$	0	0
$451$	19.0811	0.898493
$452$	−20.6420	−0.970918
$453$	0	0
$454$	16.4627	0.772632
$455$	0.463639	0.0217357
$456$	0	0
$457$	24.4590	1.14414	0.572072	−	0.820203i	$-0.306140\pi$
$457$	24.4590	1.14414	0.572072	+	0.820203i	$0.306140\pi$
$458$	33.5023	1.56546
$459$	0	0
$460$	−9.62928	−0.448968
$461$	−11.5852	−0.539578	−0.269789	−	0.962920i	$-0.586954\pi$
$461$	−11.5852	−0.539578	−0.269789	+	0.962920i	$0.586954\pi$
$462$	0	0
$463$	5.17913	0.240694	0.120347	−	0.992732i	$-0.461599\pi$
$463$	5.17913	0.240694	0.120347	+	0.992732i	$0.461599\pi$
$464$	−99.1440	−4.60265
$465$	0	0
$466$	75.0845	3.47822
$467$	11.3176	0.523716	0.261858	−	0.965106i	$-0.415665\pi$
$467$	11.3176	0.523716	0.261858	+	0.965106i	$0.415665\pi$
$468$	0	0
$469$	−11.4930	−0.530697
$470$	1.47479	0.0680271
$471$	0	0
$472$	43.9843	2.02454
$473$	10.4553	0.480735
$474$	0	0
$475$	−21.4883	−0.985953
$476$	8.61509	0.394872
$477$	0	0
$478$	17.9412	0.820610
$479$	−38.1976	−1.74529	−0.872646	−	0.488354i	$-0.837598\pi$
$479$	−38.1976	−1.74529	−0.872646	+	0.488354i	$0.837598\pi$
$480$	0	0
$481$	−6.31145	−0.287777
$482$	58.1009	2.64642
$483$	0	0
$484$	−39.8681	−1.81219
$485$	−1.46208	−0.0663899
$486$	0	0
$487$	−15.7287	−0.712735	−0.356368	−	0.934346i	$-0.615985\pi$
$487$	−15.7287	−0.712735	−0.356368	+	0.934346i	$0.615985\pi$
$488$	141.006	6.38304
$489$	0	0
$490$	0.851066	0.0384472
$491$	26.9989	1.21844	0.609222	−	0.792999i	$-0.291482\pi$
$491$	26.9989	1.21844	0.609222	+	0.792999i	$0.291482\pi$
$492$	0	0
$493$	−9.33295	−0.420335
$494$	18.1746	0.817712
$495$	0	0
$496$	79.6150	3.57482
$497$	6.35836	0.285212
$498$	0	0
$499$	−42.2261	−1.89030	−0.945149	−	0.326639i	$-0.894084\pi$
$499$	−42.2261	−1.89030	−0.945149	+	0.326639i	$0.894084\pi$
$500$	17.1526	0.767088
$501$	0	0
$502$	−56.2109	−2.50881
$503$	−14.1095	−0.629113	−0.314557	−	0.949239i	$-0.601856\pi$
$503$	−14.1095	−0.629113	−0.314557	+	0.949239i	$0.601856\pi$
$504$	0	0
$505$	−4.67530	−0.208048
$506$	30.3019	1.34708
$507$	0	0
$508$	−5.61505	−0.249128
$509$	−18.0732	−0.801079	−0.400539	−	0.916280i	$-0.631177\pi$
$509$	−18.0732	−0.801079	−0.400539	+	0.916280i	$0.631177\pi$
$510$	0	0
$511$	−12.8604	−0.568911
$512$	−82.6905	−3.65444
$513$	0	0
$514$	−14.8573	−0.655327
$515$	−2.55078	−0.112401
$516$	0	0
$517$	−3.42206	−0.150502
$518$	−11.5854	−0.509035
$519$	0	0
$520$	−4.62521	−0.202829
$521$	−7.36327	−0.322591	−0.161295	−	0.986906i	$-0.551567\pi$
$521$	−7.36327	−0.322591	−0.161295	+	0.986906i	$0.551567\pi$
$522$	0	0
$523$	−0.668564	−0.0292343	−0.0146171	−	0.999893i	$-0.504653\pi$
$523$	−0.668564	−0.0292343	−0.0146171	+	0.999893i	$0.504653\pi$
$524$	−78.1746	−3.41507
$525$	0	0
$526$	−9.74168	−0.424757
$527$	7.49458	0.326469
$528$	0	0
$529$	7.91906	0.344307
$530$	9.83653	0.427271
$531$	0	0
$532$	24.5996	1.06653
$533$	14.5257	0.629177
$534$	0	0
$535$	0.906049	0.0391719
$536$	114.653	4.95225
$537$	0	0
$538$	49.1223	2.11781
$539$	−1.97479	−0.0850601
$540$	0	0
$541$	−36.7082	−1.57821	−0.789104	−	0.614260i	$-0.789455\pi$
$541$	−36.7082	−1.57821	−0.789104	+	0.614260i	$0.789455\pi$
$542$	−9.07147	−0.389653
$543$	0	0
$544$	−38.3958	−1.64621
$545$	−0.962782	−0.0412411
$546$	0	0
$547$	26.3575	1.12697	0.563483	−	0.826128i	$-0.309461\pi$
$547$	26.3575	1.12697	0.563483	+	0.826128i	$0.309461\pi$
$548$	−23.1573	−0.989230
$549$	0	0
$550$	−26.7292	−1.13974
$551$	−26.6494	−1.13530
$552$	0	0
$553$	0.329262	0.0140016
$554$	−74.7988	−3.17790
$555$	0	0
$556$	0.140822	0.00597218
$557$	22.1623	0.939047	0.469524	−	0.882920i	$-0.344426\pi$
$557$	22.1623	0.939047	0.469524	+	0.882920i	$0.344426\pi$
$558$	0	0
$559$	7.95920	0.336638
$560$	−5.02667	−0.212416
$561$	0	0
$562$	56.4858	2.38271
$563$	16.1850	0.682118	0.341059	−	0.940042i	$-0.389214\pi$
$563$	16.1850	0.682118	0.341059	+	0.940042i	$0.389214\pi$
$564$	0	0
$565$	1.13377	0.0476980
$566$	−5.97174	−0.251011
$567$	0	0
$568$	−63.4303	−2.66148
$569$	14.6693	0.614969	0.307485	−	0.951553i	$-0.400513\pi$
$569$	14.6693	0.614969	0.307485	+	0.951553i	$0.400513\pi$
$570$	0	0
$571$	43.0790	1.80280	0.901399	−	0.432989i	$-0.142541\pi$
$571$	43.0790	1.80280	0.901399	+	0.432989i	$0.142541\pi$
$572$	16.6697	0.696995
$573$	0	0
$574$	26.6636	1.11292
$575$	−27.2736	−1.13739
$576$	0	0
$577$	8.86799	0.369179	0.184590	−	0.982816i	$-0.440904\pi$
$577$	8.86799	0.369179	0.184590	+	0.982816i	$0.440904\pi$
$578$	40.4161	1.68109
$579$	0	0
$580$	10.5340	0.437401
$581$	6.82127	0.282994
$582$	0	0
$583$	−22.8244	−0.945289
$584$	128.294	5.30884
$585$	0	0
$586$	−74.4685	−3.07627
$587$	−16.7226	−0.690214	−0.345107	−	0.938563i	$-0.612157\pi$
$587$	−16.7226	−0.690214	−0.345107	+	0.938563i	$0.612157\pi$
$588$	0	0
$589$	21.4001	0.881775
$590$	−3.75240	−0.154484
$591$	0	0
$592$	68.4273	2.81234
$593$	−36.7541	−1.50931	−0.754655	−	0.656122i	$-0.772196\pi$
$593$	−36.7541	−1.50931	−0.754655	+	0.656122i	$0.772196\pi$
$594$	0	0
$595$	−0.473187	−0.0193988
$596$	−100.645	−4.12258
$597$	0	0
$598$	23.0676	0.943305
$599$	39.3871	1.60931	0.804656	−	0.593742i	$-0.202350\pi$
$599$	39.3871	1.60931	0.804656	+	0.593742i	$0.202350\pi$
$600$	0	0
$601$	−6.93379	−0.282835	−0.141418	−	0.989950i	$-0.545166\pi$
$601$	−6.93379	−0.282835	−0.141418	+	0.989950i	$0.545166\pi$
$602$	14.6101	0.595462
$603$	0	0
$604$	106.799	4.34560
$605$	2.18977	0.0890268
$606$	0	0
$607$	−15.1875	−0.616441	−0.308221	−	0.951315i	$-0.599734\pi$
$607$	−15.1875	−0.616441	−0.308221	+	0.951315i	$0.599734\pi$
$608$	−109.636	−4.44631
$609$	0	0
$610$	−12.0295	−0.487062
$611$	−2.60508	−0.105390
$612$	0	0
$613$	−35.2659	−1.42438	−0.712188	−	0.701989i	$-0.752296\pi$
$613$	−35.2659	−1.42438	−0.712188	+	0.701989i	$0.752296\pi$
$614$	3.39766	0.137118
$615$	0	0
$616$	19.7002	0.793745
$617$	−3.48007	−0.140102	−0.0700512	−	0.997543i	$-0.522316\pi$
$617$	−3.48007	−0.140102	−0.0700512	+	0.997543i	$0.522316\pi$
$618$	0	0
$619$	−47.5147	−1.90978	−0.954888	−	0.296965i	$-0.904026\pi$
$619$	−47.5147	−1.90978	−0.954888	+	0.296965i	$0.904026\pi$
$620$	−8.45905	−0.339724
$621$	0	0
$622$	35.9272	1.44055
$623$	9.25437	0.370769
$624$	0	0
$625$	23.5823	0.943292
$626$	55.5832	2.22155
$627$	0	0
$628$	−99.8514	−3.98450
$629$	6.44142	0.256836
$630$	0	0
$631$	4.36804	0.173889	0.0869446	−	0.996213i	$-0.472290\pi$
$631$	4.36804	0.173889	0.0869446	+	0.996213i	$0.472290\pi$
$632$	−3.28468	−0.130657
$633$	0	0
$634$	15.9546	0.633637
$635$	0.308409	0.0122388
$636$	0	0
$637$	−1.50333	−0.0595640
$638$	−33.1489	−1.31238
$639$	0	0
$640$	15.5944	0.616421
$641$	16.2774	0.642918	0.321459	−	0.946924i	$-0.395827\pi$
$641$	16.2774	0.642918	0.321459	+	0.946924i	$0.395827\pi$
$642$	0	0
$643$	−37.7904	−1.49031	−0.745155	−	0.666891i	$-0.767624\pi$
$643$	−37.7904	−1.49031	−0.745155	+	0.666891i	$0.767624\pi$
$644$	31.2225	1.23034
$645$	0	0
$646$	−18.5488	−0.729794
$647$	21.8216	0.857894	0.428947	−	0.903330i	$-0.358885\pi$
$647$	21.8216	0.857894	0.428947	+	0.903330i	$0.358885\pi$
$648$	0	0
$649$	8.70696	0.341778
$650$	−20.3478	−0.798108
$651$	0	0
$652$	113.833	4.45805
$653$	−30.9287	−1.21033	−0.605167	−	0.796099i	$-0.706894\pi$
$653$	−30.9287	−1.21033	−0.605167	+	0.796099i	$0.706894\pi$
$654$	0	0
$655$	4.29376	0.167771
$656$	−157.484	−6.14872
$657$	0	0
$658$	−4.78194	−0.186419
$659$	−15.3388	−0.597513	−0.298757	−	0.954329i	$-0.596572\pi$
$659$	−15.3388	−0.597513	−0.298757	+	0.954329i	$0.596572\pi$
$660$	0	0
$661$	48.9904	1.90550	0.952752	−	0.303749i	$-0.0982385\pi$
$661$	48.9904	1.90550	0.952752	+	0.303749i	$0.0982385\pi$
$662$	−78.9359	−3.06793
$663$	0	0
$664$	−68.0482	−2.64078
$665$	−1.35114	−0.0523950
$666$	0	0
$667$	−33.8241	−1.30967
$668$	−116.917	−4.52365
$669$	0	0
$670$	−9.78130	−0.377885
$671$	27.9130	1.07757
$672$	0	0
$673$	29.8519	1.15070	0.575352	−	0.817906i	$-0.304865\pi$
$673$	29.8519	1.15070	0.575352	+	0.817906i	$0.304865\pi$
$674$	−13.2102	−0.508839
$675$	0	0
$676$	−60.3057	−2.31945
$677$	47.9132	1.84146	0.920728	−	0.390206i	$-0.127596\pi$
$677$	47.9132	1.84146	0.920728	+	0.390206i	$0.127596\pi$
$678$	0	0
$679$	4.74074	0.181933
$680$	4.72046	0.181021
$681$	0	0
$682$	26.6194	1.01931
$683$	41.6076	1.59207	0.796034	−	0.605251i	$-0.206927\pi$
$683$	41.6076	1.59207	0.796034	+	0.605251i	$0.206927\pi$
$684$	0	0
$685$	1.27192	0.0485976
$686$	−2.75954	−0.105360
$687$	0	0
$688$	−86.2918	−3.28984
$689$	−17.3753	−0.661945
$690$	0	0
$691$	−8.91082	−0.338984	−0.169492	−	0.985532i	$-0.554213\pi$
$691$	−8.91082	−0.338984	−0.169492	+	0.985532i	$0.554213\pi$
$692$	53.1021	2.01864
$693$	0	0
$694$	−1.44296	−0.0547740
$695$	−0.00773469	−0.000293394 0
$696$	0	0
$697$	−14.8248	−0.561529
$698$	−48.3673	−1.83073
$699$	0	0
$700$	−27.5412	−1.04096
$701$	−13.9575	−0.527167	−0.263583	−	0.964637i	$-0.584904\pi$
$701$	−13.9575	−0.527167	−0.263583	+	0.964637i	$0.584904\pi$
$702$	0	0
$703$	18.3929	0.693701
$704$	−72.0018	−2.71367
$705$	0	0
$706$	38.2289	1.43877
$707$	15.1594	0.570129
$708$	0	0
$709$	15.2898	0.574221	0.287111	−	0.957897i	$-0.407305\pi$
$709$	15.2898	0.574221	0.287111	+	0.957897i	$0.407305\pi$
$710$	5.41139	0.203086
$711$	0	0
$712$	−92.3205	−3.45986
$713$	27.1615	1.01721
$714$	0	0
$715$	−0.915588	−0.0342411
$716$	44.9668	1.68049
$717$	0	0
$718$	13.6323	0.508754
$719$	−23.0445	−0.859416	−0.429708	−	0.902968i	$-0.641383\pi$
$719$	−23.0445	−0.859416	−0.429708	+	0.902968i	$0.641383\pi$
$720$	0	0
$721$	8.27078	0.308020
$722$	−0.533282	−0.0198467
$723$	0	0
$724$	−24.2232	−0.900247
$725$	29.8361	1.10808
$726$	0	0
$727$	31.2173	1.15778	0.578892	−	0.815404i	$-0.303485\pi$
$727$	31.2173	1.15778	0.578892	+	0.815404i	$0.303485\pi$
$728$	14.9970	0.555826
$729$	0	0
$730$	−10.9451	−0.405095
$731$	−8.12310	−0.300444
$732$	0	0
$733$	−27.4495	−1.01387	−0.506935	−	0.861984i	$-0.669222\pi$
$733$	−27.4495	−1.01387	−0.506935	+	0.861984i	$0.669222\pi$
$734$	−45.3429	−1.67364
$735$	0	0
$736$	−139.152	−5.12922
$737$	22.6962	0.836026
$738$	0	0
$739$	29.2357	1.07545	0.537727	−	0.843119i	$-0.319283\pi$
$739$	29.2357	1.07545	0.537727	+	0.843119i	$0.319283\pi$
$740$	−7.27037	−0.267264
$741$	0	0
$742$	−31.8944	−1.17088
$743$	12.2500	0.449410	0.224705	−	0.974427i	$-0.427858\pi$
$743$	12.2500	0.449410	0.224705	+	0.974427i	$0.427858\pi$
$744$	0	0
$745$	5.52796	0.202529
$746$	−51.2861	−1.87772
$747$	0	0
$748$	−17.0130	−0.622056
$749$	−2.93782	−0.107346
$750$	0	0
$751$	−8.29748	−0.302779	−0.151390	−	0.988474i	$-0.548375\pi$
$751$	−8.29748	−0.302779	−0.151390	+	0.988474i	$0.548375\pi$
$752$	28.2437	1.02994
$753$	0	0
$754$	−25.2350	−0.919003
$755$	−5.86598	−0.213485
$756$	0	0
$757$	−14.0388	−0.510250	−0.255125	−	0.966908i	$-0.582117\pi$
$757$	−14.0388	−0.510250	−0.255125	+	0.966908i	$0.582117\pi$
$758$	28.2852	1.02736
$759$	0	0
$760$	13.4788	0.488929
$761$	−39.3218	−1.42542	−0.712708	−	0.701461i	$-0.752531\pi$
$761$	−39.3218	−1.42542	−0.712708	+	0.701461i	$0.752531\pi$
$762$	0	0
$763$	3.12177	0.113016
$764$	−6.12921	−0.221747
$765$	0	0
$766$	70.3923	2.54338
$767$	6.62826	0.239333
$768$	0	0
$769$	−13.0656	−0.471156	−0.235578	−	0.971855i	$-0.575698\pi$
$769$	−13.0656	−0.471156	−0.235578	+	0.971855i	$0.575698\pi$
$770$	−1.68067	−0.0605673
$771$	0	0
$772$	−72.0577	−2.59341
$773$	10.3852	0.373529	0.186764	−	0.982405i	$-0.440200\pi$
$773$	10.3852	0.373529	0.186764	+	0.982405i	$0.440200\pi$
$774$	0	0
$775$	−23.9591	−0.860635
$776$	−47.2930	−1.69772
$777$	0	0
$778$	−4.48209	−0.160691
$779$	−42.3309	−1.51666
$780$	0	0
$781$	−12.5564	−0.449304
$782$	−23.5427	−0.841884
$783$	0	0
$784$	16.2987	0.582097
$785$	5.48437	0.195746
$786$	0	0
$787$	−51.2006	−1.82510	−0.912552	−	0.408960i	$-0.865892\pi$
$787$	−51.2006	−1.82510	−0.912552	+	0.408960i	$0.865892\pi$
$788$	133.339	4.75001
$789$	0	0
$790$	0.280223	0.00996990
$791$	−3.67619	−0.130710
$792$	0	0
$793$	21.2490	0.754576
$794$	−57.3919	−2.03676
$795$	0	0
$796$	91.9029	3.25741
$797$	9.61879	0.340715	0.170358	−	0.985382i	$-0.445508\pi$
$797$	9.61879	0.340715	0.170358	+	0.985382i	$0.445508\pi$
$798$	0	0
$799$	2.65872	0.0940589
$800$	122.746	4.33972
$801$	0	0
$802$	95.6331	3.37692
$803$	25.3966	0.896226
$804$	0	0
$805$	−1.71490	−0.0604424
$806$	20.2643	0.713778
$807$	0	0
$808$	−151.229	−5.32021
$809$	−35.1085	−1.23435	−0.617174	−	0.786827i	$-0.711723\pi$
$809$	−35.1085	−1.23435	−0.617174	+	0.786827i	$0.711723\pi$
$810$	0	0
$811$	−37.9552	−1.33279	−0.666394	−	0.745599i	$-0.732163\pi$
$811$	−37.9552	−1.33279	−0.666394	+	0.745599i	$0.732163\pi$
$812$	−34.1560	−1.19864
$813$	0	0
$814$	22.8788	0.801900
$815$	−6.25232	−0.219009
$816$	0	0
$817$	−23.1948	−0.811482
$818$	79.3891	2.77578
$819$	0	0
$820$	16.7326	0.584328
$821$	−27.6250	−0.964119	−0.482060	−	0.876138i	$-0.660111\pi$
$821$	−27.6250	−0.964119	−0.482060	+	0.876138i	$0.660111\pi$
$822$	0	0
$823$	−27.4533	−0.956963	−0.478481	−	0.878098i	$-0.658813\pi$
$823$	−27.4533	−0.956963	−0.478481	+	0.878098i	$0.658813\pi$
$824$	−82.5083	−2.87431
$825$	0	0
$826$	12.1670	0.423343
$827$	−41.0840	−1.42863	−0.714315	−	0.699824i	$-0.753262\pi$
$827$	−41.0840	−1.42863	−0.714315	+	0.699824i	$0.753262\pi$
$828$	0	0
$829$	5.58985	0.194144	0.0970718	−	0.995277i	$-0.469052\pi$
$829$	5.58985	0.194144	0.0970718	+	0.995277i	$0.469052\pi$
$830$	5.80535	0.201507
$831$	0	0
$832$	−54.8121	−1.90027
$833$	1.53428	0.0531598
$834$	0	0
$835$	6.42170	0.222232
$836$	−48.5790	−1.68014
$837$	0	0
$838$	−15.5895	−0.538529
$839$	−22.6479	−0.781893	−0.390947	−	0.920413i	$-0.627852\pi$
$839$	−22.6479	−0.781893	−0.390947	+	0.920413i	$0.627852\pi$
$840$	0	0
$841$	8.00207	0.275933
$842$	52.8537	1.82146
$843$	0	0
$844$	−119.188	−4.10263
$845$	3.31231	0.113947
$846$	0	0
$847$	−7.10022	−0.243966
$848$	188.379	6.46896
$849$	0	0
$850$	20.7669	0.712298
$851$	23.3447	0.800247
$852$	0	0
$853$	20.4312	0.699551	0.349776	−	0.936833i	$-0.386258\pi$
$853$	20.4312	0.699551	0.349776	+	0.936833i	$0.386258\pi$
$854$	39.0052	1.33473
$855$	0	0
$856$	29.3073	1.00170
$857$	−24.8797	−0.849876	−0.424938	−	0.905223i	$-0.639704\pi$
$857$	−24.8797	−0.849876	−0.424938	+	0.905223i	$0.639704\pi$
$858$	0	0
$859$	−1.79392	−0.0612078	−0.0306039	−	0.999532i	$-0.509743\pi$
$859$	−1.79392	−0.0612078	−0.0306039	+	0.999532i	$0.509743\pi$
$860$	9.16846	0.312642
$861$	0	0
$862$	−108.617	−3.69953
$863$	−50.4753	−1.71820	−0.859100	−	0.511807i	$-0.828976\pi$
$863$	−50.4753	−1.71820	−0.859100	+	0.511807i	$0.828976\pi$
$864$	0	0
$865$	−2.91665	−0.0991692
$866$	−77.5907	−2.63664
$867$	0	0
$868$	27.4281	0.930969
$869$	−0.650222	−0.0220573
$870$	0	0
$871$	17.2777	0.585434
$872$	−31.1424	−1.05462
$873$	0	0
$874$	−67.2239	−2.27388
$875$	3.05475	0.103270
$876$	0	0
$877$	−48.1817	−1.62698	−0.813491	−	0.581578i	$-0.802435\pi$
$877$	−48.1817	−1.62698	−0.813491	+	0.581578i	$0.802435\pi$
$878$	79.2799	2.67557
$879$	0	0
$880$	9.92660	0.334626
$881$	−49.2632	−1.65972	−0.829859	−	0.557972i	$-0.811579\pi$
$881$	−49.2632	−1.65972	−0.829859	+	0.557972i	$0.811579\pi$
$882$	0	0
$883$	−5.95775	−0.200494	−0.100247	−	0.994963i	$-0.531963\pi$
$883$	−5.95775	−0.200494	−0.100247	+	0.994963i	$0.531963\pi$
$884$	−12.9513	−0.435599
$885$	0	0
$886$	21.7991	0.732356
$887$	−0.858528	−0.0288265	−0.0144133	−	0.999896i	$-0.504588\pi$
$887$	−0.858528	−0.0288265	−0.0144133	+	0.999896i	$0.504588\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	7.87608	0.264007
$891$	0	0
$892$	71.8245	2.40486
$893$	7.59174	0.254048
$894$	0	0
$895$	−2.46982	−0.0825569
$896$	−50.5639	−1.68922
$897$	0	0
$898$	81.2452	2.71119
$899$	−29.7135	−0.991001
$900$	0	0
$901$	17.7331	0.590775
$902$	−52.6550	−1.75322
$903$	0	0
$904$	36.6732	1.21973
$905$	1.33047	0.0442262
$906$	0	0
$907$	−1.52255	−0.0505555	−0.0252777	−	0.999680i	$-0.508047\pi$
$907$	−1.52255	−0.0505555	−0.0252777	+	0.999680i	$0.508047\pi$
$908$	−33.4979	−1.11167
$909$	0	0
$910$	−1.27943	−0.0424127
$911$	−35.4227	−1.17361	−0.586803	−	0.809729i	$-0.699614\pi$
$911$	−35.4227	−1.17361	−0.586803	+	0.809729i	$0.699614\pi$
$912$	0	0
$913$	−13.4706	−0.445810
$914$	−67.4956	−2.23256
$915$	0	0
$916$	−68.1698	−2.25239
$917$	−13.9223	−0.459755
$918$	0	0
$919$	5.78057	0.190683	0.0953416	−	0.995445i	$-0.469606\pi$
$919$	5.78057	0.190683	0.0953416	+	0.995445i	$0.469606\pi$
$920$	17.1077	0.564024
$921$	0	0
$922$	31.9699	1.05287
$923$	−9.55870	−0.314628
$924$	0	0
$925$	−20.5923	−0.677070
$926$	−14.2920	−0.469664
$927$	0	0
$928$	152.227	4.99708
$929$	−4.86127	−0.159493	−0.0797465	−	0.996815i	$-0.525411\pi$
$929$	−4.86127	−0.159493	−0.0797465	+	0.996815i	$0.525411\pi$
$930$	0	0
$931$	4.38101	0.143582
$932$	−152.780	−5.00449
$933$	0	0
$934$	−31.2314	−1.02192
$935$	0.934443	0.0305596
$936$	0	0
$937$	46.8532	1.53063	0.765313	−	0.643658i	$-0.222584\pi$
$937$	46.8532	1.53063	0.765313	+	0.643658i	$0.222584\pi$
$938$	31.7154	1.03554
$939$	0	0
$940$	−3.00088	−0.0978778
$941$	15.2517	0.497190	0.248595	−	0.968608i	$-0.420031\pi$
$941$	15.2517	0.497190	0.248595	+	0.968608i	$0.420031\pi$
$942$	0	0
$943$	−53.7274	−1.74961
$944$	−71.8621	−2.33891
$945$	0	0
$946$	−28.8518	−0.938052
$947$	−42.7983	−1.39076	−0.695379	−	0.718643i	$-0.744764\pi$
$947$	−42.7983	−1.39076	−0.695379	+	0.718643i	$0.744764\pi$
$948$	0	0
$949$	19.3334	0.627589
$950$	59.2979	1.92388
$951$	0	0
$952$	−15.3058	−0.496065
$953$	−0.841810	−0.0272689	−0.0136344	−	0.999907i	$-0.504340\pi$
$953$	−0.841810	−0.0272689	−0.0136344	+	0.999907i	$0.504340\pi$
$954$	0	0
$955$	0.336649	0.0108937
$956$	−36.5063	−1.18070
$957$	0	0
$958$	105.408	3.40557
$959$	−4.12414	−0.133175
$960$	0	0
$961$	−7.13936	−0.230302
$962$	17.4167	0.561536
$963$	0	0
$964$	−118.222	−3.80769
$965$	3.95779	0.127406
$966$	0	0
$967$	−49.3967	−1.58849	−0.794245	−	0.607598i	$-0.792133\pi$
$967$	−49.3967	−1.58849	−0.794245	+	0.607598i	$0.792133\pi$
$968$	70.8309	2.27659
$969$	0	0
$970$	4.03468	0.129546
$971$	−33.6708	−1.08055	−0.540273	−	0.841490i	$-0.681679\pi$
$971$	−33.6708	−1.08055	−0.540273	+	0.841490i	$0.681679\pi$
$972$	0	0
$973$	0.0250793	0.000804007 0
$974$	43.4040	1.39075
$975$	0	0
$976$	−230.377	−7.37420
$977$	−41.2147	−1.31857	−0.659287	−	0.751891i	$-0.729142\pi$
$977$	−41.2147	−1.31857	−0.659287	+	0.751891i	$0.729142\pi$
$978$	0	0
$979$	−18.2754	−0.584085
$980$	−1.73173	−0.0553181
$981$	0	0
$982$	−74.5046	−2.37754
$983$	−45.2976	−1.44477	−0.722385	−	0.691491i	$-0.756954\pi$
$983$	−45.2976	−1.44477	−0.722385	+	0.691491i	$0.756954\pi$
$984$	0	0
$985$	−7.32370	−0.233353
$986$	25.7546	0.820195
$987$	0	0
$988$	−36.9812	−1.17653
$989$	−29.4394	−0.936119
$990$	0	0
$991$	−13.6569	−0.433826	−0.216913	−	0.976191i	$-0.569599\pi$
$991$	−13.6569	−0.433826	−0.216913	+	0.976191i	$0.569599\pi$
$992$	−122.241	−3.88117
$993$	0	0
$994$	−17.5462	−0.556530
$995$	−5.04780	−0.160026
$996$	0	0
$997$	−41.0717	−1.30075	−0.650376	−	0.759612i	$-0.725389\pi$
$997$	−41.0717	−1.30075	−0.650376	+	0.759612i	$0.725389\pi$
$998$	116.524	3.68852
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.w.1.1		20
3.2	odd	2		889.2.a.d.1.20	✓	20
21.20	even	2		6223.2.a.l.1.20		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.d.1.20	✓	20	3.2	odd	2
6223.2.a.l.1.20		20	21.20	even	2
8001.2.a.w.1.1		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)

\(f(q)\)	\(=\)	\(q-2.75954 q^{2} +5.61505 q^{4} -0.308409 q^{5} +1.00000 q^{7} -9.97588 q^{8} +O(q^{10})\)	\(q-2.75954 q^{2} +5.61505 q^{4} -0.308409 q^{5} +1.00000 q^{7} -9.97588 q^{8} +0.851066 q^{10} -1.97479 q^{11} -1.50333 q^{13} -2.75954 q^{14} +16.2987 q^{16} +1.53428 q^{17} +4.38101 q^{19} -1.73173 q^{20} +5.44950 q^{22} +5.56049 q^{23} -4.90488 q^{25} +4.14849 q^{26} +5.61505 q^{28} -6.08293 q^{29} +4.88474 q^{31} -25.0252 q^{32} -4.23392 q^{34} -0.308409 q^{35} +4.19832 q^{37} -12.0896 q^{38} +3.07665 q^{40} -9.66235 q^{41} -5.29439 q^{43} -11.0885 q^{44} -15.3444 q^{46} +1.73288 q^{47} +1.00000 q^{49} +13.5352 q^{50} -8.44126 q^{52} +11.5579 q^{53} +0.609042 q^{55} -9.97588 q^{56} +16.7861 q^{58} -4.40906 q^{59} -14.1347 q^{61} -13.4796 q^{62} +36.4605 q^{64} +0.463639 q^{65} -11.4930 q^{67} +8.61509 q^{68} +0.851066 q^{70} +6.35836 q^{71} -12.8604 q^{73} -11.5854 q^{74} +24.5996 q^{76} -1.97479 q^{77} +0.329262 q^{79} -5.02667 q^{80} +26.6636 q^{82} +6.82127 q^{83} -0.473187 q^{85} +14.6101 q^{86} +19.7002 q^{88} +9.25437 q^{89} -1.50333 q^{91} +31.2225 q^{92} -4.78194 q^{94} -1.35114 q^{95} +4.74074 q^{97} -2.75954 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) \) 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

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