LMFDB - Embedded newform 8001.2.a.o.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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 10 x^{11} + 53 x^{10} + 19 x^{9} - 242 x^{8} + 61 x^{7} + 467 x^{6} - 211 x^{5} + \cdots - 8 $ x^13 - 4x^12 - 10x^11 + 53x^10 + 19x^9 - 242x^8 + 61x^7 + 467x^6 - 211x^5 - 372x^4 + 146x^3 + 116x^2 - 12x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
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			1.3
Root			$2.28328$ 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)

$f(q)$	$=$	$q-2.28328 q^{2} +3.21337 q^{4} -2.58645 q^{5} +1.00000 q^{7} -2.77047 q^{8} +O(q^{10})$	\(q-2.28328 q^{2} +3.21337 q^{4} -2.58645 q^{5} +1.00000 q^{7} -2.77047 q^{8} +5.90560 q^{10} -3.21396 q^{11} -1.17753 q^{13} -2.28328 q^{14} -0.100977 q^{16} -1.07054 q^{17} +5.70698 q^{19} -8.31124 q^{20} +7.33837 q^{22} +0.390812 q^{23} +1.68974 q^{25} +2.68862 q^{26} +3.21337 q^{28} +1.39899 q^{29} -0.821549 q^{31} +5.77151 q^{32} +2.44435 q^{34} -2.58645 q^{35} +2.05441 q^{37} -13.0306 q^{38} +7.16570 q^{40} +0.263558 q^{41} -8.34093 q^{43} -10.3276 q^{44} -0.892333 q^{46} -8.74495 q^{47} +1.00000 q^{49} -3.85814 q^{50} -3.78383 q^{52} -0.256854 q^{53} +8.31275 q^{55} -2.77047 q^{56} -3.19429 q^{58} +5.69629 q^{59} -2.53528 q^{61} +1.87583 q^{62} -12.9760 q^{64} +3.04561 q^{65} +14.2247 q^{67} -3.44005 q^{68} +5.90560 q^{70} +7.31522 q^{71} +9.20546 q^{73} -4.69081 q^{74} +18.3386 q^{76} -3.21396 q^{77} +0.721272 q^{79} +0.261173 q^{80} -0.601776 q^{82} +5.52329 q^{83} +2.76890 q^{85} +19.0447 q^{86} +8.90418 q^{88} -5.73214 q^{89} -1.17753 q^{91} +1.25582 q^{92} +19.9672 q^{94} -14.7608 q^{95} +12.1101 q^{97} -2.28328 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8	$ 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8} + 6 q^{10} - 3 q^{11} + 21 q^{13} - 4 q^{14} + 8 q^{16} - 17 q^{17} + 5 q^{19} - 29 q^{20} + q^{22} - 4 q^{23} + q^{25} - 22 q^{26} + 10 q^{28} - 21 q^{29} - 7 q^{31} - 12 q^{32} + 2 q^{34} - 12 q^{35} + 7 q^{37} + 9 q^{38} + 29 q^{40} - 21 q^{41} - 9 q^{43} + 2 q^{44} - 28 q^{46} - 23 q^{47} + 13 q^{49} - 15 q^{50} + 15 q^{52} - 31 q^{53} - 8 q^{55} - 9 q^{56} - 25 q^{58} - 28 q^{59} + 29 q^{61} + 3 q^{62} + 9 q^{64} - 30 q^{65} - 18 q^{67} - 34 q^{68} + 6 q^{70} - 10 q^{71} + 24 q^{73} + 19 q^{74} - 3 q^{77} - 28 q^{79} - 26 q^{80} + 18 q^{82} - 26 q^{83} + 20 q^{85} + 2 q^{86} - 17 q^{88} - 44 q^{89} + 21 q^{91} - 6 q^{92} - 9 q^{94} + 2 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8 + 6 * q^10 - 3 * q^11 + 21 * q^13 - 4 * q^14 + 8 * q^16 - 17 * q^17 + 5 * q^19 - 29 * q^20 + q^22 - 4 * q^23 + q^25 - 22 * q^26 + 10 * q^28 - 21 * q^29 - 7 * q^31 - 12 * q^32 + 2 * q^34 - 12 * q^35 + 7 * q^37 + 9 * q^38 + 29 * q^40 - 21 * q^41 - 9 * q^43 + 2 * q^44 - 28 * q^46 - 23 * q^47 + 13 * q^49 - 15 * q^50 + 15 * q^52 - 31 * q^53 - 8 * q^55 - 9 * q^56 - 25 * q^58 - 28 * q^59 + 29 * q^61 + 3 * q^62 + 9 * q^64 - 30 * q^65 - 18 * q^67 - 34 * q^68 + 6 * q^70 - 10 * q^71 + 24 * q^73 + 19 * q^74 - 3 * q^77 - 28 * q^79 - 26 * q^80 + 18 * q^82 - 26 * q^83 + 20 * q^85 + 2 * q^86 - 17 * q^88 - 44 * q^89 + 21 * q^91 - 6 * q^92 - 9 * q^94 + 2 * q^95 + 17 * 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 $

$2$	−2.28328	−1.61452	−0.807262	−	0.590193i	$-0.799051\pi$
$2$	−2.28328	−1.61452	−0.807262	+	0.590193i	$0.799051\pi$
$3$	0	0
$3$	0	0
$4$	3.21337	1.60669
$5$	−2.58645	−1.15670	−0.578348	−	0.815790i	$-0.696303\pi$
$5$	−2.58645	−1.15670	−0.578348	+	0.815790i	$0.696303\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.77047	−0.979510
$9$	0	0
$10$	5.90560	1.86751
$11$	−3.21396	−0.969045	−0.484522	−	0.874779i	$-0.661007\pi$
$11$	−3.21396	−0.969045	−0.484522	+	0.874779i	$0.661007\pi$
$12$	0	0
$13$	−1.17753	−0.326587	−0.163293	−	0.986578i	$-0.552212\pi$
$13$	−1.17753	−0.326587	−0.163293	+	0.986578i	$0.552212\pi$
$14$	−2.28328	−0.610233
$15$	0	0
$16$	−0.100977	−0.0252444
$17$	−1.07054	−0.259644	−0.129822	−	0.991537i	$-0.541441\pi$
$17$	−1.07054	−0.259644	−0.129822	+	0.991537i	$0.541441\pi$
$18$	0	0
$19$	5.70698	1.30927	0.654635	−	0.755945i	$-0.272822\pi$
$19$	5.70698	1.30927	0.654635	+	0.755945i	$0.272822\pi$
$20$	−8.31124	−1.85845
$21$	0	0
$22$	7.33837	1.56455
$23$	0.390812	0.0814899	0.0407449	−	0.999170i	$-0.487027\pi$
$23$	0.390812	0.0814899	0.0407449	+	0.999170i	$0.487027\pi$
$24$	0	0
$25$	1.68974	0.337947
$26$	2.68862	0.527282
$27$	0	0
$28$	3.21337	0.607271
$29$	1.39899	0.259786	0.129893	−	0.991528i	$-0.458537\pi$
$29$	1.39899	0.259786	0.129893	+	0.991528i	$0.458537\pi$
$30$	0	0
$31$	−0.821549	−0.147555	−0.0737773	−	0.997275i	$-0.523505\pi$
$31$	−0.821549	−0.147555	−0.0737773	+	0.997275i	$0.523505\pi$
$32$	5.77151	1.02027
$33$	0	0
$34$	2.44435	0.419202
$35$	−2.58645	−0.437190
$36$	0	0
$37$	2.05441	0.337744	0.168872	−	0.985638i	$-0.445988\pi$
$37$	2.05441	0.337744	0.168872	+	0.985638i	$0.445988\pi$
$38$	−13.0306	−2.11385
$39$	0	0
$40$	7.16570	1.13300
$41$	0.263558	0.0411608	0.0205804	−	0.999788i	$-0.493449\pi$
$41$	0.263558	0.0411608	0.0205804	+	0.999788i	$0.493449\pi$
$42$	0	0
$43$	−8.34093	−1.27198	−0.635990	−	0.771697i	$-0.719408\pi$
$43$	−8.34093	−1.27198	−0.635990	+	0.771697i	$0.719408\pi$
$44$	−10.3276	−1.55695
$45$	0	0
$46$	−0.892333	−0.131567
$47$	−8.74495	−1.27558	−0.637791	−	0.770209i	$-0.720152\pi$
$47$	−8.74495	−1.27558	−0.637791	+	0.770209i	$0.720152\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.85814	−0.545624
$51$	0	0
$52$	−3.78383	−0.524723
$53$	−0.256854	−0.0352815	−0.0176408	−	0.999844i	$-0.505616\pi$
$53$	−0.256854	−0.0352815	−0.0176408	+	0.999844i	$0.505616\pi$
$54$	0	0
$55$	8.31275	1.12089
$56$	−2.77047	−0.370220
$57$	0	0
$58$	−3.19429	−0.419430
$59$	5.69629	0.741593	0.370797	−	0.928714i	$-0.379085\pi$
$59$	5.69629	0.741593	0.370797	+	0.928714i	$0.379085\pi$
$60$	0	0
$61$	−2.53528	−0.324609	−0.162304	−	0.986741i	$-0.551893\pi$
$61$	−2.53528	−0.324609	−0.162304	+	0.986741i	$0.551893\pi$
$62$	1.87583	0.238230
$63$	0	0
$64$	−12.9760	−1.62200
$65$	3.04561	0.377762
$66$	0	0
$67$	14.2247	1.73782	0.868912	−	0.494967i	$-0.164820\pi$
$67$	14.2247	1.73782	0.868912	+	0.494967i	$0.164820\pi$
$68$	−3.44005	−0.417167
$69$	0	0
$70$	5.90560	0.705854
$71$	7.31522	0.868157	0.434079	−	0.900875i	$-0.357074\pi$
$71$	7.31522	0.868157	0.434079	+	0.900875i	$0.357074\pi$
$72$	0	0
$73$	9.20546	1.07742	0.538709	−	0.842492i	$-0.318912\pi$
$73$	9.20546	1.07742	0.538709	+	0.842492i	$0.318912\pi$
$74$	−4.69081	−0.545295
$75$	0	0
$76$	18.3386	2.10359
$77$	−3.21396	−0.366265
$78$	0	0
$79$	0.721272	0.0811495	0.0405747	−	0.999177i	$-0.487081\pi$
$79$	0.721272	0.0811495	0.0405747	+	0.999177i	$0.487081\pi$
$80$	0.261173	0.0292001
$81$	0	0
$82$	−0.601776	−0.0664551
$83$	5.52329	0.606260	0.303130	−	0.952949i	$-0.401968\pi$
$83$	5.52329	0.606260	0.303130	+	0.952949i	$0.401968\pi$
$84$	0	0
$85$	2.76890	0.300330
$86$	19.0447	2.05364
$87$	0	0
$88$	8.90418	0.949189
$89$	−5.73214	−0.607606	−0.303803	−	0.952735i	$-0.598256\pi$
$89$	−5.73214	−0.607606	−0.303803	+	0.952735i	$0.598256\pi$
$90$	0	0
$91$	−1.17753	−0.123438
$92$	1.25582	0.130929
$93$	0	0
$94$	19.9672	2.05946
$95$	−14.7608	−1.51443
$96$	0	0
$97$	12.1101	1.22959	0.614797	−	0.788686i	$-0.289238\pi$
$97$	12.1101	1.22959	0.614797	+	0.788686i	$0.289238\pi$
$98$	−2.28328	−0.230646
$99$	0	0
$100$	5.42975	0.542975
$101$	−15.0055	−1.49311	−0.746553	−	0.665326i	$-0.768293\pi$
$101$	−15.0055	−1.49311	−0.746553	+	0.665326i	$0.768293\pi$
$102$	0	0
$103$	−15.8235	−1.55913	−0.779566	−	0.626320i	$-0.784560\pi$
$103$	−15.8235	−1.55913	−0.779566	+	0.626320i	$0.784560\pi$
$104$	3.26230	0.319895
$105$	0	0
$106$	0.586469	0.0569629
$107$	3.24337	0.313549	0.156774	−	0.987634i	$-0.449890\pi$
$107$	3.24337	0.313549	0.156774	+	0.987634i	$0.449890\pi$
$108$	0	0
$109$	−12.2937	−1.17752	−0.588760	−	0.808308i	$-0.700384\pi$
$109$	−12.2937	−1.17752	−0.588760	+	0.808308i	$0.700384\pi$
$110$	−18.9803	−1.80971
$111$	0	0
$112$	−0.100977	−0.00954147
$113$	18.7120	1.76028	0.880138	−	0.474717i	$-0.157450\pi$
$113$	18.7120	1.76028	0.880138	+	0.474717i	$0.157450\pi$
$114$	0	0
$115$	−1.01082	−0.0942591
$116$	4.49547	0.417394
$117$	0	0
$118$	−13.0062	−1.19732
$119$	−1.07054	−0.0981364
$120$	0	0
$121$	−0.670473	−0.0609521
$122$	5.78875	0.524089
$123$	0	0
$124$	−2.63994	−0.237074
$125$	8.56184	0.765794
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	18.0849	1.59849
$129$	0	0
$130$	−6.95400	−0.609906
$131$	5.84698	0.510853	0.255426	−	0.966829i	$-0.417784\pi$
$131$	5.84698	0.510853	0.255426	+	0.966829i	$0.417784\pi$
$132$	0	0
$133$	5.70698	0.494858
$134$	−32.4790	−2.80576
$135$	0	0
$136$	2.96591	0.254324
$137$	−1.43625	−0.122707	−0.0613537	−	0.998116i	$-0.519542\pi$
$137$	−1.43625	−0.122707	−0.0613537	+	0.998116i	$0.519542\pi$
$138$	0	0
$139$	9.24219	0.783912	0.391956	−	0.919984i	$-0.371799\pi$
$139$	9.24219	0.783912	0.391956	+	0.919984i	$0.371799\pi$
$140$	−8.31124	−0.702428
$141$	0	0
$142$	−16.7027	−1.40166
$143$	3.78452	0.316477
$144$	0	0
$145$	−3.61842	−0.300493
$146$	−21.0187	−1.73952
$147$	0	0
$148$	6.60160	0.542648
$149$	4.33821	0.355400	0.177700	−	0.984085i	$-0.443134\pi$
$149$	4.33821	0.355400	0.177700	+	0.984085i	$0.443134\pi$
$150$	0	0
$151$	−1.12546	−0.0915883	−0.0457942	−	0.998951i	$-0.514582\pi$
$151$	−1.12546	−0.0915883	−0.0457942	+	0.998951i	$0.514582\pi$
$152$	−15.8110	−1.28244
$153$	0	0
$154$	7.33837	0.591343
$155$	2.12490	0.170676
$156$	0	0
$157$	4.14724	0.330985	0.165493	−	0.986211i	$-0.447079\pi$
$157$	4.14724	0.330985	0.165493	+	0.986211i	$0.447079\pi$
$158$	−1.64687	−0.131018
$159$	0	0
$160$	−14.9277	−1.18014
$161$	0.390812	0.0308003
$162$	0	0
$163$	−9.28635	−0.727363	−0.363682	−	0.931523i	$-0.618480\pi$
$163$	−9.28635	−0.727363	−0.363682	+	0.931523i	$0.618480\pi$
$164$	0.846909	0.0661325
$165$	0	0
$166$	−12.6112	−0.978821
$167$	−7.82020	−0.605145	−0.302573	−	0.953126i	$-0.597845\pi$
$167$	−7.82020	−0.605145	−0.302573	+	0.953126i	$0.597845\pi$
$168$	0	0
$169$	−11.6134	−0.893341
$170$	−6.32219	−0.484890
$171$	0	0
$172$	−26.8025	−2.04367
$173$	1.32777	0.100948	0.0504741	−	0.998725i	$-0.483927\pi$
$173$	1.32777	0.100948	0.0504741	+	0.998725i	$0.483927\pi$
$174$	0	0
$175$	1.68974	0.127732
$176$	0.324537	0.0244629
$177$	0	0
$178$	13.0881	0.980994
$179$	−12.6185	−0.943148	−0.471574	−	0.881826i	$-0.656314\pi$
$179$	−12.6185	−0.943148	−0.471574	+	0.881826i	$0.656314\pi$
$180$	0	0
$181$	16.1667	1.20166	0.600832	−	0.799375i	$-0.294836\pi$
$181$	16.1667	1.20166	0.600832	+	0.799375i	$0.294836\pi$
$182$	2.68862	0.199294
$183$	0	0
$184$	−1.08273	−0.0798202
$185$	−5.31365	−0.390667
$186$	0	0
$187$	3.44068	0.251607
$188$	−28.1008	−2.04946
$189$	0	0
$190$	33.7031	2.44508
$191$	3.33542	0.241343	0.120671	−	0.992693i	$-0.461495\pi$
$191$	3.33542	0.241343	0.120671	+	0.992693i	$0.461495\pi$
$192$	0	0
$193$	3.32925	0.239645	0.119822	−	0.992795i	$-0.461767\pi$
$193$	3.32925	0.239645	0.119822	+	0.992795i	$0.461767\pi$
$194$	−27.6507	−1.98521
$195$	0	0
$196$	3.21337	0.229527
$197$	−15.9172	−1.13405	−0.567026	−	0.823700i	$-0.691906\pi$
$197$	−15.9172	−1.13405	−0.567026	+	0.823700i	$0.691906\pi$
$198$	0	0
$199$	−20.6832	−1.46620	−0.733098	−	0.680123i	$-0.761926\pi$
$199$	−20.6832	−1.46620	−0.733098	+	0.680123i	$0.761926\pi$
$200$	−4.68137	−0.331023
$201$	0	0
$202$	34.2619	2.41066
$203$	1.39899	0.0981898
$204$	0	0
$205$	−0.681679	−0.0476106
$206$	36.1294	2.51726
$207$	0	0
$208$	0.118904	0.00824448
$209$	−18.3420	−1.26874
$210$	0	0
$211$	2.72646	0.187697	0.0938487	−	0.995586i	$-0.470083\pi$
$211$	2.72646	0.187697	0.0938487	+	0.995586i	$0.470083\pi$
$212$	−0.825366	−0.0566864
$213$	0	0
$214$	−7.40554	−0.506232
$215$	21.5734	1.47130
$216$	0	0
$217$	−0.821549	−0.0557704
$218$	28.0699	1.90113
$219$	0	0
$220$	26.7120	1.80092
$221$	1.26059	0.0847965
$222$	0	0
$223$	10.6503	0.713197	0.356598	−	0.934258i	$-0.383936\pi$
$223$	10.6503	0.713197	0.356598	+	0.934258i	$0.383936\pi$
$224$	5.77151	0.385625
$225$	0	0
$226$	−42.7248	−2.84201
$227$	3.10452	0.206054	0.103027	−	0.994679i	$-0.467147\pi$
$227$	3.10452	0.206054	0.103027	+	0.994679i	$0.467147\pi$
$228$	0	0
$229$	−0.123627	−0.00816948	−0.00408474	−	0.999992i	$-0.501300\pi$
$229$	−0.123627	−0.00816948	−0.00408474	+	0.999992i	$0.501300\pi$
$230$	2.30798	0.152184
$231$	0	0
$232$	−3.87586	−0.254463
$233$	7.62240	0.499360	0.249680	−	0.968328i	$-0.419675\pi$
$233$	7.62240	0.499360	0.249680	+	0.968328i	$0.419675\pi$
$234$	0	0
$235$	22.6184	1.47546
$236$	18.3043	1.19151
$237$	0	0
$238$	2.44435	0.158443
$239$	22.7517	1.47168	0.735841	−	0.677154i	$-0.236787\pi$
$239$	22.7517	1.47168	0.735841	+	0.677154i	$0.236787\pi$
$240$	0	0
$241$	12.7612	0.822020	0.411010	−	0.911631i	$-0.365176\pi$
$241$	12.7612	0.822020	0.411010	+	0.911631i	$0.365176\pi$
$242$	1.53088	0.0984086
$243$	0	0
$244$	−8.14679	−0.521545
$245$	−2.58645	−0.165242
$246$	0	0
$247$	−6.72011	−0.427590
$248$	2.27608	0.144531
$249$	0	0
$250$	−19.5491	−1.23639
$251$	21.1413	1.33443	0.667213	−	0.744867i	$-0.267487\pi$
$251$	21.1413	1.33443	0.667213	+	0.744867i	$0.267487\pi$
$252$	0	0
$253$	−1.25605	−0.0789674
$254$	2.28328	0.143266
$255$	0	0
$256$	−15.3408	−0.958803
$257$	−2.75313	−0.171736	−0.0858678	−	0.996307i	$-0.527366\pi$
$257$	−2.75313	−0.171736	−0.0858678	+	0.996307i	$0.527366\pi$
$258$	0	0
$259$	2.05441	0.127655
$260$	9.78670	0.606945
$261$	0	0
$262$	−13.3503	−0.824784
$263$	10.0766	0.621352	0.310676	−	0.950516i	$-0.399445\pi$
$263$	10.0766	0.621352	0.310676	+	0.950516i	$0.399445\pi$
$264$	0	0
$265$	0.664340	0.0408101
$266$	−13.0306	−0.798959
$267$	0	0
$268$	45.7093	2.79214
$269$	14.2329	0.867798	0.433899	−	0.900962i	$-0.357137\pi$
$269$	14.2329	0.867798	0.433899	+	0.900962i	$0.357137\pi$
$270$	0	0
$271$	0.448422	0.0272397	0.0136199	−	0.999907i	$-0.495665\pi$
$271$	0.448422	0.0272397	0.0136199	+	0.999907i	$0.495665\pi$
$272$	0.108101	0.00655456
$273$	0	0
$274$	3.27937	0.198114
$275$	−5.43074	−0.327486
$276$	0	0
$277$	−0.932903	−0.0560527	−0.0280264	−	0.999607i	$-0.508922\pi$
$277$	−0.932903	−0.0560527	−0.0280264	+	0.999607i	$0.508922\pi$
$278$	−21.1025	−1.26564
$279$	0	0
$280$	7.16570	0.428232
$281$	−3.75909	−0.224249	−0.112124	−	0.993694i	$-0.535766\pi$
$281$	−3.75909	−0.224249	−0.112124	+	0.993694i	$0.535766\pi$
$282$	0	0
$283$	−14.5739	−0.866330	−0.433165	−	0.901315i	$-0.642603\pi$
$283$	−14.5739	−0.866330	−0.433165	+	0.901315i	$0.642603\pi$
$284$	23.5065	1.39486
$285$	0	0
$286$	−8.64112	−0.510960
$287$	0.263558	0.0155573
$288$	0	0
$289$	−15.8539	−0.932585
$290$	8.26187	0.485154
$291$	0	0
$292$	29.5806	1.73107
$293$	0.169691	0.00991343	0.00495672	−	0.999988i	$-0.498422\pi$
$293$	0.169691	0.00991343	0.00495672	+	0.999988i	$0.498422\pi$
$294$	0	0
$295$	−14.7332	−0.857799
$296$	−5.69170	−0.330823
$297$	0	0
$298$	−9.90536	−0.573802
$299$	−0.460191	−0.0266135
$300$	0	0
$301$	−8.34093	−0.480763
$302$	2.56973	0.147871
$303$	0	0
$304$	−0.576276	−0.0330517
$305$	6.55737	0.375474
$306$	0	0
$307$	−18.2671	−1.04256	−0.521279	−	0.853386i	$-0.674545\pi$
$307$	−18.2671	−1.04256	−0.521279	+	0.853386i	$0.674545\pi$
$308$	−10.3276	−0.588472
$309$	0	0
$310$	−4.85174	−0.275560
$311$	14.0534	0.796894	0.398447	−	0.917191i	$-0.369549\pi$
$311$	14.0534	0.796894	0.398447	+	0.917191i	$0.369549\pi$
$312$	0	0
$313$	2.27563	0.128626	0.0643130	−	0.997930i	$-0.479514\pi$
$313$	2.27563	0.128626	0.0643130	+	0.997930i	$0.479514\pi$
$314$	−9.46931	−0.534384
$315$	0	0
$316$	2.31772	0.130382
$317$	−33.0480	−1.85616	−0.928080	−	0.372382i	$-0.878541\pi$
$317$	−33.0480	−1.85616	−0.928080	+	0.372382i	$0.878541\pi$
$318$	0	0
$319$	−4.49629	−0.251744
$320$	33.5619	1.87616
$321$	0	0
$322$	−0.892333	−0.0497278
$323$	−6.10955	−0.339945
$324$	0	0
$325$	−1.98971	−0.110369
$326$	21.2034	1.17435
$327$	0	0
$328$	−0.730179	−0.0403174
$329$	−8.74495	−0.482125
$330$	0	0
$331$	−24.6358	−1.35411	−0.677054	−	0.735933i	$-0.736744\pi$
$331$	−24.6358	−1.35411	−0.677054	+	0.735933i	$0.736744\pi$
$332$	17.7484	0.974070
$333$	0	0
$334$	17.8557	0.977021
$335$	−36.7915	−2.01013
$336$	0	0
$337$	−7.74642	−0.421974	−0.210987	−	0.977489i	$-0.567668\pi$
$337$	−7.74642	−0.421974	−0.210987	+	0.977489i	$0.567668\pi$
$338$	26.5167	1.44232
$339$	0	0
$340$	8.89752	0.482536
$341$	2.64043	0.142987
$342$	0	0
$343$	1.00000	0.0539949
$344$	23.1083	1.24592
$345$	0	0
$346$	−3.03166	−0.162983
$347$	33.8629	1.81786	0.908928	−	0.416953i	$-0.136902\pi$
$347$	33.8629	1.81786	0.908928	+	0.416953i	$0.136902\pi$
$348$	0	0
$349$	−20.8238	−1.11467	−0.557337	−	0.830286i	$-0.688177\pi$
$349$	−20.8238	−1.11467	−0.557337	+	0.830286i	$0.688177\pi$
$350$	−3.85814	−0.206226
$351$	0	0
$352$	−18.5494	−0.988685
$353$	8.51240	0.453069	0.226535	−	0.974003i	$-0.427260\pi$
$353$	8.51240	0.453069	0.226535	+	0.974003i	$0.427260\pi$
$354$	0	0
$355$	−18.9205	−1.00419
$356$	−18.4195	−0.976232
$357$	0	0
$358$	28.8115	1.52274
$359$	−3.33523	−0.176027	−0.0880135	−	0.996119i	$-0.528052\pi$
$359$	−3.33523	−0.176027	−0.0880135	+	0.996119i	$0.528052\pi$
$360$	0	0
$361$	13.5696	0.714188
$362$	−36.9132	−1.94012
$363$	0	0
$364$	−3.78383	−0.198327
$365$	−23.8095	−1.24625
$366$	0	0
$367$	24.0171	1.25368	0.626842	−	0.779147i	$-0.284347\pi$
$367$	24.0171	1.25368	0.626842	+	0.779147i	$0.284347\pi$
$368$	−0.0394632	−0.00205716
$369$	0	0
$370$	12.1326	0.630741
$371$	−0.256854	−0.0133352
$372$	0	0
$373$	−3.43047	−0.177623	−0.0888115	−	0.996048i	$-0.528307\pi$
$373$	−3.43047	−0.177623	−0.0888115	+	0.996048i	$0.528307\pi$
$374$	−7.85603	−0.406226
$375$	0	0
$376$	24.2277	1.24945
$377$	−1.64735	−0.0848426
$378$	0	0
$379$	−7.05174	−0.362223	−0.181112	−	0.983463i	$-0.557970\pi$
$379$	−7.05174	−0.362223	−0.181112	+	0.983463i	$0.557970\pi$
$380$	−47.4320	−2.43321
$381$	0	0
$382$	−7.61571	−0.389654
$383$	−37.6211	−1.92235	−0.961174	−	0.275943i	$-0.911010\pi$
$383$	−37.6211	−1.92235	−0.961174	+	0.275943i	$0.911010\pi$
$384$	0	0
$385$	8.31275	0.423657
$386$	−7.60162	−0.386912
$387$	0	0
$388$	38.9142	1.97557
$389$	14.5892	0.739704	0.369852	−	0.929091i	$-0.379408\pi$
$389$	14.5892	0.739704	0.369852	+	0.929091i	$0.379408\pi$
$390$	0	0
$391$	−0.418380	−0.0211584
$392$	−2.77047	−0.139930
$393$	0	0
$394$	36.3434	1.83095
$395$	−1.86554	−0.0938653
$396$	0	0
$397$	−1.75296	−0.0879787	−0.0439894	−	0.999032i	$-0.514007\pi$
$397$	−1.75296	−0.0879787	−0.0439894	+	0.999032i	$0.514007\pi$
$398$	47.2257	2.36721
$399$	0	0
$400$	−0.170625	−0.00853126
$401$	7.49775	0.374420	0.187210	−	0.982320i	$-0.440056\pi$
$401$	7.49775	0.374420	0.187210	+	0.982320i	$0.440056\pi$
$402$	0	0
$403$	0.967396	0.0481894
$404$	−48.2184	−2.39895
$405$	0	0
$406$	−3.19429	−0.158530
$407$	−6.60280	−0.327289
$408$	0	0
$409$	−17.8323	−0.881750	−0.440875	−	0.897568i	$-0.645332\pi$
$409$	−17.8323	−0.881750	−0.440875	+	0.897568i	$0.645332\pi$
$410$	1.55647	0.0768684
$411$	0	0
$412$	−50.8467	−2.50504
$413$	5.69629	0.280296
$414$	0	0
$415$	−14.2857	−0.701259
$416$	−6.79610	−0.333206
$417$	0	0
$418$	41.8799	2.04841
$419$	7.83938	0.382979	0.191490	−	0.981495i	$-0.438668\pi$
$419$	7.83938	0.382979	0.191490	+	0.981495i	$0.438668\pi$
$420$	0	0
$421$	−14.0995	−0.687167	−0.343584	−	0.939122i	$-0.611641\pi$
$421$	−14.0995	−0.687167	−0.343584	+	0.939122i	$0.611641\pi$
$422$	−6.22528	−0.303042
$423$	0	0
$424$	0.711606	0.0345586
$425$	−1.80893	−0.0877461
$426$	0	0
$427$	−2.53528	−0.122691
$428$	10.4222	0.503775
$429$	0	0
$430$	−49.2582	−2.37544
$431$	14.5822	0.702400	0.351200	−	0.936301i	$-0.385774\pi$
$431$	14.5822	0.702400	0.351200	+	0.936301i	$0.385774\pi$
$432$	0	0
$433$	−4.78704	−0.230051	−0.115025	−	0.993363i	$-0.536695\pi$
$433$	−4.78704	−0.230051	−0.115025	+	0.993363i	$0.536695\pi$
$434$	1.87583	0.0900426
$435$	0	0
$436$	−39.5041	−1.89190
$437$	2.23035	0.106692
$438$	0	0
$439$	−5.34817	−0.255254	−0.127627	−	0.991822i	$-0.540736\pi$
$439$	−5.34817	−0.255254	−0.127627	+	0.991822i	$0.540736\pi$
$440$	−23.0303	−1.09792
$441$	0	0
$442$	−2.87828	−0.136906
$443$	26.3283	1.25089	0.625447	−	0.780267i	$-0.284917\pi$
$443$	26.3283	1.25089	0.625447	+	0.780267i	$0.284917\pi$
$444$	0	0
$445$	14.8259	0.702816
$446$	−24.3176	−1.15147
$447$	0	0
$448$	−12.9760	−0.613059
$449$	−20.1615	−0.951482	−0.475741	−	0.879585i	$-0.657820\pi$
$449$	−20.1615	−0.951482	−0.475741	+	0.879585i	$0.657820\pi$
$450$	0	0
$451$	−0.847063	−0.0398867
$452$	60.1286	2.82821
$453$	0	0
$454$	−7.08850	−0.332680
$455$	3.04561	0.142781
$456$	0	0
$457$	20.4546	0.956826	0.478413	−	0.878135i	$-0.341212\pi$
$457$	20.4546	0.956826	0.478413	+	0.878135i	$0.341212\pi$
$458$	0.282275	0.0131898
$459$	0	0
$460$	−3.24813	−0.151445
$461$	−2.10294	−0.0979436	−0.0489718	−	0.998800i	$-0.515594\pi$
$461$	−2.10294	−0.0979436	−0.0489718	+	0.998800i	$0.515594\pi$
$462$	0	0
$463$	7.38019	0.342986	0.171493	−	0.985185i	$-0.445141\pi$
$463$	7.38019	0.342986	0.171493	+	0.985185i	$0.445141\pi$
$464$	−0.141266	−0.00655813
$465$	0	0
$466$	−17.4041	−0.806229
$467$	5.28658	0.244634	0.122317	−	0.992491i	$-0.460968\pi$
$467$	5.28658	0.244634	0.122317	+	0.992491i	$0.460968\pi$
$468$	0	0
$469$	14.2247	0.656835
$470$	−51.6442	−2.38217
$471$	0	0
$472$	−15.7814	−0.726398
$473$	26.8074	1.23261
$474$	0	0
$475$	9.64328	0.442464
$476$	−3.44005	−0.157674
$477$	0	0
$478$	−51.9484	−2.37607
$479$	9.01045	0.411698	0.205849	−	0.978584i	$-0.434004\pi$
$479$	9.01045	0.411698	0.205849	+	0.978584i	$0.434004\pi$
$480$	0	0
$481$	−2.41913	−0.110303
$482$	−29.1374	−1.32717
$483$	0	0
$484$	−2.15448	−0.0979309
$485$	−31.3222	−1.42227
$486$	0	0
$487$	−2.84576	−0.128954	−0.0644768	−	0.997919i	$-0.520538\pi$
$487$	−2.84576	−0.128954	−0.0644768	+	0.997919i	$0.520538\pi$
$488$	7.02391	0.317958
$489$	0	0
$490$	5.90560	0.266788
$491$	13.1851	0.595034	0.297517	−	0.954716i	$-0.403841\pi$
$491$	13.1851	0.595034	0.297517	+	0.954716i	$0.403841\pi$
$492$	0	0
$493$	−1.49768	−0.0674519
$494$	15.3439	0.690355
$495$	0	0
$496$	0.0829580	0.00372492
$497$	7.31522	0.328133
$498$	0	0
$499$	2.96311	0.132647	0.0663234	−	0.997798i	$-0.478873\pi$
$499$	2.96311	0.132647	0.0663234	+	0.997798i	$0.478873\pi$
$500$	27.5124	1.23039
$501$	0	0
$502$	−48.2715	−2.15446
$503$	−38.2091	−1.70366	−0.851830	−	0.523819i	$-0.824507\pi$
$503$	−38.2091	−1.70366	−0.851830	+	0.523819i	$0.824507\pi$
$504$	0	0
$505$	38.8111	1.72707
$506$	2.86792	0.127495
$507$	0	0
$508$	−3.21337	−0.142570
$509$	−23.2050	−1.02854	−0.514272	−	0.857627i	$-0.671938\pi$
$509$	−23.2050	−1.02854	−0.514272	+	0.857627i	$0.671938\pi$
$510$	0	0
$511$	9.20546	0.407226
$512$	−1.14230	−0.0504831
$513$	0	0
$514$	6.28617	0.277271
$515$	40.9266	1.80344
$516$	0	0
$517$	28.1059	1.23610
$518$	−4.69081	−0.206102
$519$	0	0
$520$	−8.43779	−0.370022
$521$	17.4609	0.764977	0.382488	−	0.923960i	$-0.375067\pi$
$521$	17.4609	0.764977	0.382488	+	0.923960i	$0.375067\pi$
$522$	0	0
$523$	−22.5165	−0.984576	−0.492288	−	0.870432i	$-0.663839\pi$
$523$	−22.5165	−0.984576	−0.492288	+	0.870432i	$0.663839\pi$
$524$	18.7885	0.820780
$525$	0	0
$526$	−23.0078	−1.00319
$527$	0.879503	0.0383117
$528$	0	0
$529$	−22.8473	−0.993359
$530$	−1.51687	−0.0658888
$531$	0	0
$532$	18.3386	0.795081
$533$	−0.310346	−0.0134426
$534$	0	0
$535$	−8.38883	−0.362681
$536$	−39.4091	−1.70222
$537$	0	0
$538$	−32.4978	−1.40108
$539$	−3.21396	−0.138435
$540$	0	0
$541$	5.43712	0.233760	0.116880	−	0.993146i	$-0.462711\pi$
$541$	5.43712	0.233760	0.116880	+	0.993146i	$0.462711\pi$
$542$	−1.02387	−0.0439791
$543$	0	0
$544$	−6.17864	−0.264907
$545$	31.7970	1.36203
$546$	0	0
$547$	14.1214	0.603788	0.301894	−	0.953342i	$-0.402381\pi$
$547$	14.1214	0.603788	0.301894	+	0.953342i	$0.402381\pi$
$548$	−4.61521	−0.197152
$549$	0	0
$550$	12.3999	0.528734
$551$	7.98400	0.340130
$552$	0	0
$553$	0.721272	0.0306716
$554$	2.13008	0.0904984
$555$	0	0
$556$	29.6986	1.25950
$557$	33.5497	1.42155	0.710773	−	0.703421i	$-0.248345\pi$
$557$	33.5497	1.42155	0.710773	+	0.703421i	$0.248345\pi$
$558$	0	0
$559$	9.82166	0.415412
$560$	0.261173	0.0110366
$561$	0	0
$562$	8.58307	0.362055
$563$	−11.7812	−0.496518	−0.248259	−	0.968694i	$-0.579858\pi$
$563$	−11.7812	−0.496518	−0.248259	+	0.968694i	$0.579858\pi$
$564$	0	0
$565$	−48.3977	−2.03611
$566$	33.2764	1.39871
$567$	0	0
$568$	−20.2666	−0.850369
$569$	−35.0339	−1.46870	−0.734349	−	0.678772i	$-0.762513\pi$
$569$	−35.0339	−1.46870	−0.734349	+	0.678772i	$0.762513\pi$
$570$	0	0
$571$	2.79170	0.116829	0.0584145	−	0.998292i	$-0.481395\pi$
$571$	2.79170	0.116829	0.0584145	+	0.998292i	$0.481395\pi$
$572$	12.1611	0.508480
$573$	0	0
$574$	−0.601776	−0.0251177
$575$	0.660369	0.0275393
$576$	0	0
$577$	25.1595	1.04741	0.523703	−	0.851901i	$-0.324550\pi$
$577$	25.1595	1.04741	0.523703	+	0.851901i	$0.324550\pi$
$578$	36.1990	1.50568
$579$	0	0
$580$	−11.6273	−0.482799
$581$	5.52329	0.229145
$582$	0	0
$583$	0.825517	0.0341894
$584$	−25.5035	−1.05534
$585$	0	0
$586$	−0.387451	−0.0160055
$587$	13.4468	0.555010	0.277505	−	0.960724i	$-0.410493\pi$
$587$	13.4468	0.555010	0.277505	+	0.960724i	$0.410493\pi$
$588$	0	0
$589$	−4.68856	−0.193189
$590$	33.6400	1.38494
$591$	0	0
$592$	−0.207450	−0.00852613
$593$	−20.6997	−0.850034	−0.425017	−	0.905185i	$-0.639732\pi$
$593$	−20.6997	−0.850034	−0.425017	+	0.905185i	$0.639732\pi$
$594$	0	0
$595$	2.76890	0.113514
$596$	13.9403	0.571017
$597$	0	0
$598$	1.05075	0.0429682
$599$	−25.5749	−1.04496	−0.522481	−	0.852651i	$-0.674993\pi$
$599$	−25.5749	−1.04496	−0.522481	+	0.852651i	$0.674993\pi$
$600$	0	0
$601$	48.4029	1.97440	0.987198	−	0.159497i	$-0.0509873\pi$
$601$	48.4029	1.97440	0.987198	+	0.159497i	$0.0509873\pi$
$602$	19.0447	0.776204
$603$	0	0
$604$	−3.61651	−0.147154
$605$	1.73415	0.0705031
$606$	0	0
$607$	−40.8197	−1.65682	−0.828410	−	0.560122i	$-0.810754\pi$
$607$	−40.8197	−1.65682	−0.828410	+	0.560122i	$0.810754\pi$
$608$	32.9378	1.33581
$609$	0	0
$610$	−14.9723	−0.606211
$611$	10.2974	0.416589
$612$	0	0
$613$	24.4892	0.989110	0.494555	−	0.869146i	$-0.335331\pi$
$613$	24.4892	0.989110	0.494555	+	0.869146i	$0.335331\pi$
$614$	41.7089	1.68324
$615$	0	0
$616$	8.90418	0.358760
$617$	−42.3749	−1.70595	−0.852974	−	0.521953i	$-0.825204\pi$
$617$	−42.3749	−1.70595	−0.852974	+	0.521953i	$0.825204\pi$
$618$	0	0
$619$	−33.9298	−1.36375	−0.681877	−	0.731467i	$-0.738836\pi$
$619$	−33.9298	−1.36375	−0.681877	+	0.731467i	$0.738836\pi$
$620$	6.82809	0.274223
$621$	0	0
$622$	−32.0878	−1.28660
$623$	−5.73214	−0.229653
$624$	0	0
$625$	−30.5935	−1.22374
$626$	−5.19590	−0.207670
$627$	0	0
$628$	13.3266	0.531790
$629$	−2.19934	−0.0876933
$630$	0	0
$631$	−40.1689	−1.59910	−0.799549	−	0.600601i	$-0.794928\pi$
$631$	−40.1689	−1.59910	−0.799549	+	0.600601i	$0.794928\pi$
$632$	−1.99827	−0.0794867
$633$	0	0
$634$	75.4578	2.99681
$635$	2.58645	0.102640
$636$	0	0
$637$	−1.17753	−0.0466553
$638$	10.2663	0.406447
$639$	0	0
$640$	−46.7757	−1.84897
$641$	−37.9472	−1.49882	−0.749411	−	0.662105i	$-0.769663\pi$
$641$	−37.9472	−1.49882	−0.749411	+	0.662105i	$0.769663\pi$
$642$	0	0
$643$	26.0146	1.02591	0.512957	−	0.858414i	$-0.328550\pi$
$643$	26.0146	1.02591	0.512957	+	0.858414i	$0.328550\pi$
$644$	1.25582	0.0494864
$645$	0	0
$646$	13.9498	0.548849
$647$	−30.8493	−1.21281	−0.606405	−	0.795156i	$-0.707389\pi$
$647$	−30.8493	−1.21281	−0.606405	+	0.795156i	$0.707389\pi$
$648$	0	0
$649$	−18.3076	−0.718637
$650$	4.54306	0.178194
$651$	0	0
$652$	−29.8405	−1.16864
$653$	−30.2629	−1.18428	−0.592140	−	0.805835i	$-0.701717\pi$
$653$	−30.2629	−1.18428	−0.592140	+	0.805835i	$0.701717\pi$
$654$	0	0
$655$	−15.1229	−0.590902
$656$	−0.0266134	−0.00103908
$657$	0	0
$658$	19.9672	0.778402
$659$	−4.03804	−0.157300	−0.0786498	−	0.996902i	$-0.525061\pi$
$659$	−4.03804	−0.157300	−0.0786498	+	0.996902i	$0.525061\pi$
$660$	0	0
$661$	−7.92170	−0.308118	−0.154059	−	0.988062i	$-0.549235\pi$
$661$	−7.92170	−0.308118	−0.154059	+	0.988062i	$0.549235\pi$
$662$	56.2505	2.18624
$663$	0	0
$664$	−15.3021	−0.593838
$665$	−14.7608	−0.572400
$666$	0	0
$667$	0.546741	0.0211699
$668$	−25.1292	−0.972279
$669$	0	0
$670$	84.0053	3.24541
$671$	8.14827	0.314560
$672$	0	0
$673$	−17.1617	−0.661535	−0.330768	−	0.943712i	$-0.607308\pi$
$673$	−17.1617	−0.661535	−0.330768	+	0.943712i	$0.607308\pi$
$674$	17.6873	0.681287
$675$	0	0
$676$	−37.3183	−1.43532
$677$	−34.0415	−1.30832	−0.654160	−	0.756356i	$-0.726978\pi$
$677$	−34.0415	−1.30832	−0.654160	+	0.756356i	$0.726978\pi$
$678$	0	0
$679$	12.1101	0.464743
$680$	−7.67118	−0.294176
$681$	0	0
$682$	−6.02883	−0.230856
$683$	−2.51831	−0.0963604	−0.0481802	−	0.998839i	$-0.515342\pi$
$683$	−2.51831	−0.0963604	−0.0481802	+	0.998839i	$0.515342\pi$
$684$	0	0
$685$	3.71480	0.141935
$686$	−2.28328	−0.0871761
$687$	0	0
$688$	0.842246	0.0321103
$689$	0.302452	0.0115225
$690$	0	0
$691$	−45.6601	−1.73699	−0.868497	−	0.495695i	$-0.834914\pi$
$691$	−45.6601	−1.73699	−0.868497	+	0.495695i	$0.834914\pi$
$692$	4.26661	0.162192
$693$	0	0
$694$	−77.3185	−2.93497
$695$	−23.9045	−0.906749
$696$	0	0
$697$	−0.282149	−0.0106872
$698$	47.5467	1.79967
$699$	0	0
$700$	5.42975	0.205225
$701$	−16.7859	−0.633996	−0.316998	−	0.948426i	$-0.602675\pi$
$701$	−16.7859	−0.633996	−0.316998	+	0.948426i	$0.602675\pi$
$702$	0	0
$703$	11.7245	0.442198
$704$	41.7044	1.57179
$705$	0	0
$706$	−19.4362	−0.731491
$707$	−15.0055	−0.564341
$708$	0	0
$709$	17.0746	0.641250	0.320625	−	0.947206i	$-0.396107\pi$
$709$	17.0746	0.641250	0.320625	+	0.947206i	$0.396107\pi$
$710$	43.2008	1.62130
$711$	0	0
$712$	15.8807	0.595156
$713$	−0.321071	−0.0120242
$714$	0	0
$715$	−9.78848	−0.366068
$716$	−40.5478	−1.51534
$717$	0	0
$718$	7.61528	0.284200
$719$	28.6856	1.06979	0.534897	−	0.844918i	$-0.320351\pi$
$719$	28.6856	1.06979	0.534897	+	0.844918i	$0.320351\pi$
$720$	0	0
$721$	−15.8235	−0.589297
$722$	−30.9831	−1.15307
$723$	0	0
$724$	51.9498	1.93070
$725$	2.36392	0.0877939
$726$	0	0
$727$	−26.2955	−0.975245	−0.487623	−	0.873055i	$-0.662136\pi$
$727$	−26.2955	−0.975245	−0.487623	+	0.873055i	$0.662136\pi$
$728$	3.26230	0.120909
$729$	0	0
$730$	54.3638	2.01209
$731$	8.92931	0.330263
$732$	0	0
$733$	−16.2742	−0.601101	−0.300550	−	0.953766i	$-0.597170\pi$
$733$	−16.2742	−0.601101	−0.300550	+	0.953766i	$0.597170\pi$
$734$	−54.8379	−2.02410
$735$	0	0
$736$	2.25557	0.0831415
$737$	−45.7176	−1.68403
$738$	0	0
$739$	17.4545	0.642074	0.321037	−	0.947067i	$-0.395969\pi$
$739$	17.4545	0.642074	0.321037	+	0.947067i	$0.395969\pi$
$740$	−17.0747	−0.627680
$741$	0	0
$742$	0.586469	0.0215300
$743$	6.62388	0.243007	0.121503	−	0.992591i	$-0.461228\pi$
$743$	6.62388	0.243007	0.121503	+	0.992591i	$0.461228\pi$
$744$	0	0
$745$	−11.2206	−0.411090
$746$	7.83273	0.286777
$747$	0	0
$748$	11.0562	0.404254
$749$	3.24337	0.118510
$750$	0	0
$751$	−11.6724	−0.425930	−0.212965	−	0.977060i	$-0.568312\pi$
$751$	−11.6724	−0.425930	−0.212965	+	0.977060i	$0.568312\pi$
$752$	0.883043	0.0322013
$753$	0	0
$754$	3.76135	0.136980
$755$	2.91094	0.105940
$756$	0	0
$757$	3.63056	0.131955	0.0659775	−	0.997821i	$-0.478983\pi$
$757$	3.63056	0.131955	0.0659775	+	0.997821i	$0.478983\pi$
$758$	16.1011	0.584818
$759$	0	0
$760$	40.8945	1.48340
$761$	1.51369	0.0548711	0.0274355	−	0.999624i	$-0.491266\pi$
$761$	1.51369	0.0548711	0.0274355	+	0.999624i	$0.491266\pi$
$762$	0	0
$763$	−12.2937	−0.445060
$764$	10.7180	0.387762
$765$	0	0
$766$	85.8995	3.10368
$767$	−6.70753	−0.242195
$768$	0	0
$769$	4.01256	0.144697	0.0723484	−	0.997379i	$-0.476951\pi$
$769$	4.01256	0.144697	0.0723484	+	0.997379i	$0.476951\pi$
$770$	−18.9803	−0.684004
$771$	0	0
$772$	10.6981	0.385034
$773$	−13.1068	−0.471418	−0.235709	−	0.971824i	$-0.575741\pi$
$773$	−13.1068	−0.471418	−0.235709	+	0.971824i	$0.575741\pi$
$774$	0	0
$775$	−1.38820	−0.0498657
$776$	−33.5507	−1.20440
$777$	0	0
$778$	−33.3113	−1.19427
$779$	1.50412	0.0538906
$780$	0	0
$781$	−23.5108	−0.841283
$782$	0.955280	0.0341607
$783$	0	0
$784$	−0.100977	−0.00360634
$785$	−10.7266	−0.382850
$786$	0	0
$787$	29.6322	1.05627	0.528136	−	0.849160i	$-0.322891\pi$
$787$	29.6322	1.05627	0.528136	+	0.849160i	$0.322891\pi$
$788$	−51.1478	−1.82207
$789$	0	0
$790$	4.25954	0.151548
$791$	18.7120	0.665322
$792$	0	0
$793$	2.98535	0.106013
$794$	4.00251	0.142044
$795$	0	0
$796$	−66.4630	−2.35572
$797$	−45.0974	−1.59743	−0.798715	−	0.601709i	$-0.794487\pi$
$797$	−45.0974	−1.59743	−0.798715	+	0.601709i	$0.794487\pi$
$798$	0	0
$799$	9.36183	0.331198
$800$	9.75232	0.344797
$801$	0	0
$802$	−17.1195	−0.604510
$803$	−29.5860	−1.04407
$804$	0	0
$805$	−1.01082	−0.0356266
$806$	−2.20884	−0.0778029
$807$	0	0
$808$	41.5724	1.46251
$809$	−19.5504	−0.687356	−0.343678	−	0.939088i	$-0.611673\pi$
$809$	−19.5504	−0.687356	−0.343678	+	0.939088i	$0.611673\pi$
$810$	0	0
$811$	−1.85130	−0.0650078	−0.0325039	−	0.999472i	$-0.510348\pi$
$811$	−1.85130	−0.0650078	−0.0325039	+	0.999472i	$0.510348\pi$
$812$	4.49547	0.157760
$813$	0	0
$814$	15.0761	0.528416
$815$	24.0187	0.841339
$816$	0	0
$817$	−47.6015	−1.66537
$818$	40.7161	1.42361
$819$	0	0
$820$	−2.19049	−0.0764952
$821$	−6.10030	−0.212902	−0.106451	−	0.994318i	$-0.533949\pi$
$821$	−6.10030	−0.212902	−0.106451	+	0.994318i	$0.533949\pi$
$822$	0	0
$823$	27.7252	0.966440	0.483220	−	0.875499i	$-0.339467\pi$
$823$	27.7252	0.966440	0.483220	+	0.875499i	$0.339467\pi$
$824$	43.8385	1.52719
$825$	0	0
$826$	−13.0062	−0.452545
$827$	−21.2307	−0.738266	−0.369133	−	0.929377i	$-0.620345\pi$
$827$	−21.2307	−0.738266	−0.369133	+	0.929377i	$0.620345\pi$
$828$	0	0
$829$	17.4420	0.605787	0.302894	−	0.953024i	$-0.402047\pi$
$829$	17.4420	0.605787	0.302894	+	0.953024i	$0.402047\pi$
$830$	32.6183	1.13220
$831$	0	0
$832$	15.2796	0.529725
$833$	−1.07054	−0.0370921
$834$	0	0
$835$	20.2266	0.699970
$836$	−58.9396	−2.03847
$837$	0	0
$838$	−17.8995	−0.618329
$839$	−32.6569	−1.12744	−0.563721	−	0.825965i	$-0.690631\pi$
$839$	−32.6569	−1.12744	−0.563721	+	0.825965i	$0.690631\pi$
$840$	0	0
$841$	−27.0428	−0.932511
$842$	32.1931	1.10945
$843$	0	0
$844$	8.76115	0.301571
$845$	30.0376	1.03332
$846$	0	0
$847$	−0.670473	−0.0230377
$848$	0.0259364	0.000890660 0
$849$	0	0
$850$	4.13030	0.141668
$851$	0.802889	0.0275227
$852$	0	0
$853$	37.2064	1.27392	0.636961	−	0.770896i	$-0.280191\pi$
$853$	37.2064	1.27392	0.636961	+	0.770896i	$0.280191\pi$
$854$	5.78875	0.198087
$855$	0	0
$856$	−8.98568	−0.307124
$857$	−14.5500	−0.497020	−0.248510	−	0.968629i	$-0.579941\pi$
$857$	−14.5500	−0.497020	−0.248510	+	0.968629i	$0.579941\pi$
$858$	0	0
$859$	8.66396	0.295611	0.147805	−	0.989016i	$-0.452779\pi$
$859$	8.66396	0.295611	0.147805	+	0.989016i	$0.452779\pi$
$860$	69.3235	2.36391
$861$	0	0
$862$	−33.2953	−1.13404
$863$	−40.2943	−1.37163	−0.685816	−	0.727775i	$-0.740555\pi$
$863$	−40.2943	−1.37163	−0.685816	+	0.727775i	$0.740555\pi$
$864$	0	0
$865$	−3.43420	−0.116766
$866$	10.9302	0.371422
$867$	0	0
$868$	−2.63994	−0.0896056
$869$	−2.31814	−0.0786375
$870$	0	0
$871$	−16.7499	−0.567550
$872$	34.0593	1.15339
$873$	0	0
$874$	−5.09252	−0.172257
$875$	8.56184	0.289443
$876$	0	0
$877$	14.7099	0.496718	0.248359	−	0.968668i	$-0.420109\pi$
$877$	14.7099	0.496718	0.248359	+	0.968668i	$0.420109\pi$
$878$	12.2114	0.412114
$879$	0	0
$880$	−0.839400	−0.0282962
$881$	4.99439	0.168265	0.0841327	−	0.996455i	$-0.473188\pi$
$881$	4.99439	0.168265	0.0841327	+	0.996455i	$0.473188\pi$
$882$	0	0
$883$	−19.8711	−0.668716	−0.334358	−	0.942446i	$-0.608520\pi$
$883$	−19.8711	−0.668716	−0.334358	+	0.942446i	$0.608520\pi$
$884$	4.05075	0.136241
$885$	0	0
$886$	−60.1148	−2.01960
$887$	20.6755	0.694214	0.347107	−	0.937825i	$-0.387164\pi$
$887$	20.6755	0.694214	0.347107	+	0.937825i	$0.387164\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−33.8517	−1.13471
$891$	0	0
$892$	34.2234	1.14588
$893$	−49.9072	−1.67008
$894$	0	0
$895$	32.6371	1.09094
$896$	18.0849	0.604174
$897$	0	0
$898$	46.0345	1.53619
$899$	−1.14934	−0.0383326
$900$	0	0
$901$	0.274972	0.00916066
$902$	1.93408	0.0643979
$903$	0	0
$904$	−51.8411	−1.72421
$905$	−41.8145	−1.38996
$906$	0	0
$907$	−21.2937	−0.707044	−0.353522	−	0.935426i	$-0.615016\pi$
$907$	−21.2937	−0.707044	−0.353522	+	0.935426i	$0.615016\pi$
$908$	9.97600	0.331065
$909$	0	0
$910$	−6.95400	−0.230523
$911$	−48.1047	−1.59378	−0.796891	−	0.604124i	$-0.793523\pi$
$911$	−48.1047	−1.59378	−0.796891	+	0.604124i	$0.793523\pi$
$912$	0	0
$913$	−17.7516	−0.587493
$914$	−46.7036	−1.54482
$915$	0	0
$916$	−0.397259	−0.0131258
$917$	5.84698	0.193084
$918$	0	0
$919$	47.6807	1.57284	0.786421	−	0.617691i	$-0.211932\pi$
$919$	47.6807	1.57284	0.786421	+	0.617691i	$0.211932\pi$
$920$	2.80044	0.0923277
$921$	0	0
$922$	4.80160	0.158132
$923$	−8.61386	−0.283529
$924$	0	0
$925$	3.47142	0.114140
$926$	−16.8510	−0.553760
$927$	0	0
$928$	8.07427	0.265051
$929$	−42.9353	−1.40866	−0.704332	−	0.709871i	$-0.748753\pi$
$929$	−42.9353	−1.40866	−0.704332	+	0.709871i	$0.748753\pi$
$930$	0	0
$931$	5.70698	0.187039
$932$	24.4936	0.802316
$933$	0	0
$934$	−12.0707	−0.394967
$935$	−8.89914	−0.291033
$936$	0	0
$937$	−10.3116	−0.336865	−0.168432	−	0.985713i	$-0.553870\pi$
$937$	−10.3116	−0.336865	−0.168432	+	0.985713i	$0.553870\pi$
$938$	−32.4790	−1.06048
$939$	0	0
$940$	72.6814	2.37061
$941$	−48.1796	−1.57061	−0.785305	−	0.619109i	$-0.787494\pi$
$941$	−48.1796	−1.57061	−0.785305	+	0.619109i	$0.787494\pi$
$942$	0	0
$943$	0.103001	0.00335419
$944$	−0.575197	−0.0187211
$945$	0	0
$946$	−61.2088	−1.99007
$947$	18.3826	0.597354	0.298677	−	0.954354i	$-0.403455\pi$
$947$	18.3826	0.597354	0.298677	+	0.954354i	$0.403455\pi$
$948$	0	0
$949$	−10.8397	−0.351871
$950$	−22.0183	−0.714369
$951$	0	0
$952$	2.96591	0.0961256
$953$	−48.2826	−1.56403	−0.782014	−	0.623261i	$-0.785807\pi$
$953$	−48.2826	−1.56403	−0.782014	+	0.623261i	$0.785807\pi$
$954$	0	0
$955$	−8.62691	−0.279160
$956$	73.1096	2.36453
$957$	0	0
$958$	−20.5734	−0.664696
$959$	−1.43625	−0.0463790
$960$	0	0
$961$	−30.3251	−0.978228
$962$	5.52355	0.178086
$963$	0	0
$964$	41.0065	1.32073
$965$	−8.61095	−0.277196
$966$	0	0
$967$	4.90647	0.157781	0.0788907	−	0.996883i	$-0.474862\pi$
$967$	4.90647	0.157781	0.0788907	+	0.996883i	$0.474862\pi$
$968$	1.85753	0.0597032
$969$	0	0
$970$	71.5173	2.29628
$971$	−38.4381	−1.23354	−0.616768	−	0.787145i	$-0.711558\pi$
$971$	−38.4381	−1.23354	−0.616768	+	0.787145i	$0.711558\pi$
$972$	0	0
$973$	9.24219	0.296291
$974$	6.49767	0.208199
$975$	0	0
$976$	0.256006	0.00819454
$977$	8.48769	0.271545	0.135773	−	0.990740i	$-0.456648\pi$
$977$	8.48769	0.271545	0.135773	+	0.990740i	$0.456648\pi$
$978$	0	0
$979$	18.4229	0.588797
$980$	−8.31124	−0.265493
$981$	0	0
$982$	−30.1052	−0.960697
$983$	11.7097	0.373480	0.186740	−	0.982409i	$-0.440208\pi$
$983$	11.7097	0.373480	0.186740	+	0.982409i	$0.440208\pi$
$984$	0	0
$985$	41.1690	1.31175
$986$	3.41961	0.108903
$987$	0	0
$988$	−21.5942	−0.687004
$989$	−3.25973	−0.103654
$990$	0	0
$991$	−51.1937	−1.62622	−0.813110	−	0.582110i	$-0.802227\pi$
$991$	−51.1937	−1.62622	−0.813110	+	0.582110i	$0.802227\pi$
$992$	−4.74158	−0.150545
$993$	0	0
$994$	−16.7027	−0.529778
$995$	53.4962	1.69594
$996$	0	0
$997$	−12.6330	−0.400090	−0.200045	−	0.979787i	$-0.564109\pi$
$997$	−12.6330	−0.400090	−0.200045	+	0.979787i	$0.564109\pi$
$998$	−6.76561	−0.214162
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.o.1.3		13
3.2	odd	2		2667.2.a.l.1.11	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.l.1.11	✓	13	3.2	odd	2
8001.2.a.o.1.3		13	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)

\(f(q)\)	\(=\)	\(q-2.28328 q^{2} +3.21337 q^{4} -2.58645 q^{5} +1.00000 q^{7} -2.77047 q^{8} +O(q^{10})\)	\(q-2.28328 q^{2} +3.21337 q^{4} -2.58645 q^{5} +1.00000 q^{7} -2.77047 q^{8} +5.90560 q^{10} -3.21396 q^{11} -1.17753 q^{13} -2.28328 q^{14} -0.100977 q^{16} -1.07054 q^{17} +5.70698 q^{19} -8.31124 q^{20} +7.33837 q^{22} +0.390812 q^{23} +1.68974 q^{25} +2.68862 q^{26} +3.21337 q^{28} +1.39899 q^{29} -0.821549 q^{31} +5.77151 q^{32} +2.44435 q^{34} -2.58645 q^{35} +2.05441 q^{37} -13.0306 q^{38} +7.16570 q^{40} +0.263558 q^{41} -8.34093 q^{43} -10.3276 q^{44} -0.892333 q^{46} -8.74495 q^{47} +1.00000 q^{49} -3.85814 q^{50} -3.78383 q^{52} -0.256854 q^{53} +8.31275 q^{55} -2.77047 q^{56} -3.19429 q^{58} +5.69629 q^{59} -2.53528 q^{61} +1.87583 q^{62} -12.9760 q^{64} +3.04561 q^{65} +14.2247 q^{67} -3.44005 q^{68} +5.90560 q^{70} +7.31522 q^{71} +9.20546 q^{73} -4.69081 q^{74} +18.3386 q^{76} -3.21396 q^{77} +0.721272 q^{79} +0.261173 q^{80} -0.601776 q^{82} +5.52329 q^{83} +2.76890 q^{85} +19.0447 q^{86} +8.90418 q^{88} -5.73214 q^{89} -1.17753 q^{91} +1.25582 q^{92} +19.9672 q^{94} -14.7608 q^{95} +12.1101 q^{97} -2.28328 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8	\( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8} + 6 q^{10} - 3 q^{11} + 21 q^{13} - 4 q^{14} + 8 q^{16} - 17 q^{17} + 5 q^{19} - 29 q^{20} + q^{22} - 4 q^{23} + q^{25} - 22 q^{26} + 10 q^{28} - 21 q^{29} - 7 q^{31} - 12 q^{32} + 2 q^{34} - 12 q^{35} + 7 q^{37} + 9 q^{38} + 29 q^{40} - 21 q^{41} - 9 q^{43} + 2 q^{44} - 28 q^{46} - 23 q^{47} + 13 q^{49} - 15 q^{50} + 15 q^{52} - 31 q^{53} - 8 q^{55} - 9 q^{56} - 25 q^{58} - 28 q^{59} + 29 q^{61} + 3 q^{62} + 9 q^{64} - 30 q^{65} - 18 q^{67} - 34 q^{68} + 6 q^{70} - 10 q^{71} + 24 q^{73} + 19 q^{74} - 3 q^{77} - 28 q^{79} - 26 q^{80} + 18 q^{82} - 26 q^{83} + 20 q^{85} + 2 q^{86} - 17 q^{88} - 44 q^{89} + 21 q^{91} - 6 q^{92} - 9 q^{94} + 2 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8 + 6 * q^10 - 3 * q^11 + 21 * q^13 - 4 * q^14 + 8 * q^16 - 17 * q^17 + 5 * q^19 - 29 * q^20 + q^22 - 4 * q^23 + q^25 - 22 * q^26 + 10 * q^28 - 21 * q^29 - 7 * q^31 - 12 * q^32 + 2 * q^34 - 12 * q^35 + 7 * q^37 + 9 * q^38 + 29 * q^40 - 21 * q^41 - 9 * q^43 + 2 * q^44 - 28 * q^46 - 23 * q^47 + 13 * q^49 - 15 * q^50 + 15 * q^52 - 31 * q^53 - 8 * q^55 - 9 * q^56 - 25 * q^58 - 28 * q^59 + 29 * q^61 + 3 * q^62 + 9 * q^64 - 30 * q^65 - 18 * q^67 - 34 * q^68 + 6 * q^70 - 10 * q^71 + 24 * q^73 + 19 * q^74 - 3 * q^77 - 28 * q^79 - 26 * q^80 + 18 * q^82 - 26 * q^83 + 20 * q^85 + 2 * q^86 - 17 * q^88 - 44 * q^89 + 21 * q^91 - 6 * q^92 - 9 * q^94 + 2 * q^95 + 17 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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