LMFDB - Embedded newform 4005.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, 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("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.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:	$31.9800860095$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 5 x^{16} - 15 x^{15} + 105 x^{14} + 45 x^{13} - 849 x^{12} + 320 x^{11} + 3371 x^{10} + \cdots + 24 $ x^17 - 5x^16 - 15x^15 + 105x^14 + 45x^13 - 849x^12 + 320x^11 + 3371x^10 - 2456x^9 - 7002x^8 + 6279x^7 + 7299x^6 - 7119x^5 - 3066x^4 + 3184x^3 + 99x^2 - 231x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.71108$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.71108 q^{2} +5.34994 q^{4} -1.00000 q^{5} -0.946340 q^{7} -9.08194 q^{8} +O(q^{10})$	\(q-2.71108 q^{2} +5.34994 q^{4} -1.00000 q^{5} -0.946340 q^{7} -9.08194 q^{8} +2.71108 q^{10} -2.67492 q^{11} +3.58607 q^{13} +2.56560 q^{14} +13.9220 q^{16} +0.102863 q^{17} -3.52745 q^{19} -5.34994 q^{20} +7.25191 q^{22} +0.253455 q^{23} +1.00000 q^{25} -9.72212 q^{26} -5.06286 q^{28} +7.95379 q^{29} +6.95521 q^{31} -19.5796 q^{32} -0.278870 q^{34} +0.946340 q^{35} -4.65911 q^{37} +9.56319 q^{38} +9.08194 q^{40} -2.67772 q^{41} -4.32595 q^{43} -14.3106 q^{44} -0.687136 q^{46} -0.628642 q^{47} -6.10444 q^{49} -2.71108 q^{50} +19.1853 q^{52} -9.62601 q^{53} +2.67492 q^{55} +8.59460 q^{56} -21.5633 q^{58} +8.30551 q^{59} +5.45346 q^{61} -18.8561 q^{62} +25.2380 q^{64} -3.58607 q^{65} -5.05000 q^{67} +0.550311 q^{68} -2.56560 q^{70} -1.17674 q^{71} -7.35153 q^{73} +12.6312 q^{74} -18.8717 q^{76} +2.53138 q^{77} +5.39325 q^{79} -13.9220 q^{80} +7.25951 q^{82} -9.85257 q^{83} -0.102863 q^{85} +11.7280 q^{86} +24.2934 q^{88} +1.00000 q^{89} -3.39364 q^{91} +1.35597 q^{92} +1.70430 q^{94} +3.52745 q^{95} -11.0214 q^{97} +16.5496 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8}+O(q^{10}) $ 17 * q - 5 * q^2 + 21 * q^4 - 17 * q^5 + 12 * q^7 - 15 * q^8	$ 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8} + 5 q^{10} - 2 q^{11} + 8 q^{13} - 4 q^{14} + 33 q^{16} - 10 q^{17} + 32 q^{19} - 21 q^{20} + 8 q^{22} - 15 q^{23} + 17 q^{25} + 15 q^{26} + 24 q^{28} - q^{29} + 18 q^{31} - 25 q^{32} + 14 q^{34} - 12 q^{35} + 12 q^{37} - 22 q^{38} + 15 q^{40} + 7 q^{41} + 28 q^{43} + 14 q^{44} + 4 q^{46} - 26 q^{47} + 41 q^{49} - 5 q^{50} + 10 q^{52} - 12 q^{53} + 2 q^{55} - 13 q^{56} + 16 q^{58} + 23 q^{59} + 26 q^{61} - 10 q^{62} + 59 q^{64} - 8 q^{65} + 31 q^{67} + q^{68} + 4 q^{70} + 2 q^{71} + 33 q^{73} + 10 q^{74} + 66 q^{76} - 12 q^{77} + 33 q^{79} - 33 q^{80} + 30 q^{82} - 13 q^{83} + 10 q^{85} + 20 q^{86} + 12 q^{88} + 17 q^{89} + 40 q^{91} - 16 q^{92} + 38 q^{94} - 32 q^{95} + 45 q^{97} - 2 q^{98}+O(q^{100}) $ 17 * q - 5 * q^2 + 21 * q^4 - 17 * q^5 + 12 * q^7 - 15 * q^8 + 5 * q^10 - 2 * q^11 + 8 * q^13 - 4 * q^14 + 33 * q^16 - 10 * q^17 + 32 * q^19 - 21 * q^20 + 8 * q^22 - 15 * q^23 + 17 * q^25 + 15 * q^26 + 24 * q^28 - q^29 + 18 * q^31 - 25 * q^32 + 14 * q^34 - 12 * q^35 + 12 * q^37 - 22 * q^38 + 15 * q^40 + 7 * q^41 + 28 * q^43 + 14 * q^44 + 4 * q^46 - 26 * q^47 + 41 * q^49 - 5 * q^50 + 10 * q^52 - 12 * q^53 + 2 * q^55 - 13 * q^56 + 16 * q^58 + 23 * q^59 + 26 * q^61 - 10 * q^62 + 59 * q^64 - 8 * q^65 + 31 * q^67 + q^68 + 4 * q^70 + 2 * q^71 + 33 * q^73 + 10 * q^74 + 66 * q^76 - 12 * q^77 + 33 * q^79 - 33 * q^80 + 30 * q^82 - 13 * q^83 + 10 * q^85 + 20 * q^86 + 12 * q^88 + 17 * q^89 + 40 * q^91 - 16 * q^92 + 38 * q^94 - 32 * q^95 + 45 * q^97 - 2 * 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 $

$2$	−2.71108	−1.91702	−0.958510	−	0.285057i	$-0.907987\pi$
$2$	−2.71108	−1.91702	−0.958510	+	0.285057i	$0.907987\pi$
$3$	0	0
$3$	0	0
$4$	5.34994	2.67497
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.946340	−0.357683	−0.178841	−	0.983878i	$-0.557235\pi$
$7$	−0.946340	−0.357683	−0.178841	+	0.983878i	$0.557235\pi$
$8$	−9.08194	−3.21095
$9$	0	0
$10$	2.71108	0.857318
$11$	−2.67492	−0.806518	−0.403259	−	0.915086i	$-0.632123\pi$
$11$	−2.67492	−0.806518	−0.403259	+	0.915086i	$0.632123\pi$
$12$	0	0
$13$	3.58607	0.994597	0.497299	−	0.867579i	$-0.334325\pi$
$13$	3.58607	0.994597	0.497299	+	0.867579i	$0.334325\pi$
$14$	2.56560	0.685685
$15$	0	0
$16$	13.9220	3.48049
$17$	0.102863	0.0249479	0.0124740	−	0.999922i	$-0.496029\pi$
$17$	0.102863	0.0249479	0.0124740	+	0.999922i	$0.496029\pi$
$18$	0	0
$19$	−3.52745	−0.809253	−0.404627	−	0.914482i	$-0.632598\pi$
$19$	−3.52745	−0.809253	−0.404627	+	0.914482i	$0.632598\pi$
$20$	−5.34994	−1.19628
$21$	0	0
$22$	7.25191	1.54611
$23$	0.253455	0.0528490	0.0264245	−	0.999651i	$-0.491588\pi$
$23$	0.253455	0.0528490	0.0264245	+	0.999651i	$0.491588\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−9.72212	−1.90666
$27$	0	0
$28$	−5.06286	−0.956790
$29$	7.95379	1.47698	0.738491	−	0.674263i	$-0.235539\pi$
$29$	7.95379	1.47698	0.738491	+	0.674263i	$0.235539\pi$
$30$	0	0
$31$	6.95521	1.24919	0.624597	−	0.780948i	$-0.285263\pi$
$31$	6.95521	1.24919	0.624597	+	0.780948i	$0.285263\pi$
$32$	−19.5796	−3.46122
$33$	0	0
$34$	−0.278870	−0.0478257
$35$	0.946340	0.159961
$36$	0	0
$37$	−4.65911	−0.765953	−0.382977	−	0.923758i	$-0.625101\pi$
$37$	−4.65911	−0.765953	−0.382977	+	0.923758i	$0.625101\pi$
$38$	9.56319	1.55135
$39$	0	0
$40$	9.08194	1.43598
$41$	−2.67772	−0.418190	−0.209095	−	0.977895i	$-0.567052\pi$
$41$	−2.67772	−0.418190	−0.209095	+	0.977895i	$0.567052\pi$
$42$	0	0
$43$	−4.32595	−0.659701	−0.329850	−	0.944033i	$-0.606998\pi$
$43$	−4.32595	−0.659701	−0.329850	+	0.944033i	$0.606998\pi$
$44$	−14.3106	−2.15741
$45$	0	0
$46$	−0.687136	−0.101313
$47$	−0.628642	−0.0916968	−0.0458484	−	0.998948i	$-0.514599\pi$
$47$	−0.628642	−0.0916968	−0.0458484	+	0.998948i	$0.514599\pi$
$48$	0	0
$49$	−6.10444	−0.872063
$50$	−2.71108	−0.383404
$51$	0	0
$52$	19.1853	2.66052
$53$	−9.62601	−1.32223	−0.661117	−	0.750283i	$-0.729917\pi$
$53$	−9.62601	−1.32223	−0.661117	+	0.750283i	$0.729917\pi$
$54$	0	0
$55$	2.67492	0.360686
$56$	8.59460	1.14850
$57$	0	0
$58$	−21.5633	−2.83141
$59$	8.30551	1.08128	0.540642	−	0.841253i	$-0.318181\pi$
$59$	8.30551	1.08128	0.540642	+	0.841253i	$0.318181\pi$
$60$	0	0
$61$	5.45346	0.698244	0.349122	−	0.937077i	$-0.386480\pi$
$61$	5.45346	0.698244	0.349122	+	0.937077i	$0.386480\pi$
$62$	−18.8561	−2.39473
$63$	0	0
$64$	25.2380	3.15475
$65$	−3.58607	−0.444797
$66$	0	0
$67$	−5.05000	−0.616956	−0.308478	−	0.951232i	$-0.599820\pi$
$67$	−5.05000	−0.616956	−0.308478	+	0.951232i	$0.599820\pi$
$68$	0.550311	0.0667350
$69$	0	0
$70$	−2.56560	−0.306648
$71$	−1.17674	−0.139653	−0.0698264	−	0.997559i	$-0.522245\pi$
$71$	−1.17674	−0.139653	−0.0698264	+	0.997559i	$0.522245\pi$
$72$	0	0
$73$	−7.35153	−0.860431	−0.430216	−	0.902726i	$-0.641562\pi$
$73$	−7.35153	−0.860431	−0.430216	+	0.902726i	$0.641562\pi$
$74$	12.6312	1.46835
$75$	0	0
$76$	−18.8717	−2.16473
$77$	2.53138	0.288478
$78$	0	0
$79$	5.39325	0.606788	0.303394	−	0.952865i	$-0.401880\pi$
$79$	5.39325	0.606788	0.303394	+	0.952865i	$0.401880\pi$
$80$	−13.9220	−1.55652
$81$	0	0
$82$	7.25951	0.801679
$83$	−9.85257	−1.08146	−0.540730	−	0.841196i	$-0.681852\pi$
$83$	−9.85257	−1.08146	−0.540730	+	0.841196i	$0.681852\pi$
$84$	0	0
$85$	−0.102863	−0.0111571
$86$	11.7280	1.26466
$87$	0	0
$88$	24.2934	2.58969
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	−3.39364	−0.355750
$92$	1.35597	0.141369
$93$	0	0
$94$	1.70430	0.175785
$95$	3.52745	0.361909
$96$	0	0
$97$	−11.0214	−1.11906	−0.559528	−	0.828811i	$-0.689018\pi$
$97$	−11.0214	−1.11906	−0.559528	+	0.828811i	$0.689018\pi$
$98$	16.5496	1.67176
$99$	0	0
$100$	5.34994	0.534994
$101$	3.55443	0.353679	0.176840	−	0.984240i	$-0.443413\pi$
$101$	3.55443	0.353679	0.176840	+	0.984240i	$0.443413\pi$
$102$	0	0
$103$	9.33221	0.919530	0.459765	−	0.888041i	$-0.347934\pi$
$103$	9.33221	0.919530	0.459765	+	0.888041i	$0.347934\pi$
$104$	−32.5685	−3.19360
$105$	0	0
$106$	26.0968	2.53475
$107$	6.01056	0.581063	0.290531	−	0.956865i	$-0.406168\pi$
$107$	6.01056	0.581063	0.290531	+	0.956865i	$0.406168\pi$
$108$	0	0
$109$	16.2040	1.55206	0.776032	−	0.630693i	$-0.217229\pi$
$109$	16.2040	1.55206	0.776032	+	0.630693i	$0.217229\pi$
$110$	−7.25191	−0.691442
$111$	0	0
$112$	−13.1749	−1.24491
$113$	0.713014	0.0670747	0.0335373	−	0.999437i	$-0.489323\pi$
$113$	0.713014	0.0670747	0.0335373	+	0.999437i	$0.489323\pi$
$114$	0	0
$115$	−0.253455	−0.0236348
$116$	42.5523	3.95088
$117$	0	0
$118$	−22.5169	−2.07285
$119$	−0.0973433	−0.00892345
$120$	0	0
$121$	−3.84482	−0.349529
$122$	−14.7847	−1.33855
$123$	0	0
$124$	37.2100	3.34155
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	11.9833	1.06335	0.531674	−	0.846949i	$-0.321563\pi$
$127$	11.9833	1.06335	0.531674	+	0.846949i	$0.321563\pi$
$128$	−29.2628	−2.58649
$129$	0	0
$130$	9.72212	0.852686
$131$	9.26971	0.809898	0.404949	−	0.914339i	$-0.367289\pi$
$131$	9.26971	0.809898	0.404949	+	0.914339i	$0.367289\pi$
$132$	0	0
$133$	3.33817	0.289456
$134$	13.6909	1.18272
$135$	0	0
$136$	−0.934196	−0.0801066
$137$	14.9676	1.27877	0.639386	−	0.768886i	$-0.279189\pi$
$137$	14.9676	1.27877	0.639386	+	0.768886i	$0.279189\pi$
$138$	0	0
$139$	0.126982	0.0107705	0.00538526	−	0.999985i	$-0.498286\pi$
$139$	0.126982	0.0107705	0.00538526	+	0.999985i	$0.498286\pi$
$140$	5.06286	0.427890
$141$	0	0
$142$	3.19022	0.267717
$143$	−9.59244	−0.802160
$144$	0	0
$145$	−7.95379	−0.660527
$146$	19.9306	1.64946
$147$	0	0
$148$	−24.9260	−2.04890
$149$	−5.42391	−0.444344	−0.222172	−	0.975008i	$-0.571315\pi$
$149$	−5.42391	−0.444344	−0.222172	+	0.975008i	$0.571315\pi$
$150$	0	0
$151$	−2.07850	−0.169146	−0.0845729	−	0.996417i	$-0.526953\pi$
$151$	−2.07850	−0.169146	−0.0845729	+	0.996417i	$0.526953\pi$
$152$	32.0361	2.59847
$153$	0	0
$154$	−6.86277	−0.553018
$155$	−6.95521	−0.558656
$156$	0	0
$157$	7.76806	0.619959	0.309979	−	0.950743i	$-0.399678\pi$
$157$	7.76806	0.619959	0.309979	+	0.950743i	$0.399678\pi$
$158$	−14.6215	−1.16323
$159$	0	0
$160$	19.5796	1.54791
$161$	−0.239854	−0.0189032
$162$	0	0
$163$	−8.38997	−0.657153	−0.328577	−	0.944477i	$-0.606569\pi$
$163$	−8.38997	−0.657153	−0.328577	+	0.944477i	$0.606569\pi$
$164$	−14.3256	−1.11865
$165$	0	0
$166$	26.7111	2.07318
$167$	3.70218	0.286483	0.143241	−	0.989688i	$-0.454247\pi$
$167$	3.70218	0.286483	0.143241	+	0.989688i	$0.454247\pi$
$168$	0	0
$169$	−0.140094	−0.0107764
$170$	0.278870	0.0213883
$171$	0	0
$172$	−23.1436	−1.76468
$173$	−14.1646	−1.07691	−0.538456	−	0.842653i	$-0.680992\pi$
$173$	−14.1646	−1.07691	−0.538456	+	0.842653i	$0.680992\pi$
$174$	0	0
$175$	−0.946340	−0.0715366
$176$	−37.2401	−2.80708
$177$	0	0
$178$	−2.71108	−0.203204
$179$	2.14305	0.160179	0.0800897	−	0.996788i	$-0.474479\pi$
$179$	2.14305	0.160179	0.0800897	+	0.996788i	$0.474479\pi$
$180$	0	0
$181$	4.53493	0.337079	0.168539	−	0.985695i	$-0.446095\pi$
$181$	4.53493	0.337079	0.168539	+	0.985695i	$0.446095\pi$
$182$	9.20042	0.681981
$183$	0	0
$184$	−2.30186	−0.169696
$185$	4.65911	0.342545
$186$	0	0
$187$	−0.275150	−0.0201210
$188$	−3.36319	−0.245286
$189$	0	0
$190$	−9.56319	−0.693787
$191$	21.6395	1.56578	0.782890	−	0.622160i	$-0.213745\pi$
$191$	21.6395	1.56578	0.782890	+	0.622160i	$0.213745\pi$
$192$	0	0
$193$	−23.7049	−1.70632	−0.853160	−	0.521650i	$-0.825317\pi$
$193$	−23.7049	−1.70632	−0.853160	+	0.521650i	$0.825317\pi$
$194$	29.8799	2.14525
$195$	0	0
$196$	−32.6584	−2.33274
$197$	5.46681	0.389494	0.194747	−	0.980854i	$-0.437611\pi$
$197$	5.46681	0.389494	0.194747	+	0.980854i	$0.437611\pi$
$198$	0	0
$199$	23.5099	1.66657	0.833286	−	0.552843i	$-0.186457\pi$
$199$	23.5099	1.66657	0.833286	+	0.552843i	$0.186457\pi$
$200$	−9.08194	−0.642190
$201$	0	0
$202$	−9.63633	−0.678010
$203$	−7.52699	−0.528291
$204$	0	0
$205$	2.67772	0.187020
$206$	−25.3003	−1.76276
$207$	0	0
$208$	49.9252	3.46169
$209$	9.43564	0.652677
$210$	0	0
$211$	6.95422	0.478748	0.239374	−	0.970927i	$-0.423058\pi$
$211$	6.95422	0.478748	0.239374	+	0.970927i	$0.423058\pi$
$212$	−51.4986	−3.53693
$213$	0	0
$214$	−16.2951	−1.11391
$215$	4.32595	0.295027
$216$	0	0
$217$	−6.58199	−0.446815
$218$	−43.9304	−2.97534
$219$	0	0
$220$	14.3106	0.964823
$221$	0.368874	0.0248132
$222$	0	0
$223$	1.14106	0.0764108	0.0382054	−	0.999270i	$-0.487836\pi$
$223$	1.14106	0.0764108	0.0382054	+	0.999270i	$0.487836\pi$
$224$	18.5290	1.23802
$225$	0	0
$226$	−1.93304	−0.128584
$227$	21.6611	1.43770	0.718849	−	0.695167i	$-0.244669\pi$
$227$	21.6611	1.43770	0.718849	+	0.695167i	$0.244669\pi$
$228$	0	0
$229$	−25.2495	−1.66854	−0.834268	−	0.551360i	$-0.814109\pi$
$229$	−25.2495	−1.66854	−0.834268	+	0.551360i	$0.814109\pi$
$230$	0.687136	0.0453084
$231$	0	0
$232$	−72.2359	−4.74252
$233$	−20.3535	−1.33340	−0.666702	−	0.745325i	$-0.732295\pi$
$233$	−20.3535	−1.33340	−0.666702	+	0.745325i	$0.732295\pi$
$234$	0	0
$235$	0.628642	0.0410081
$236$	44.4339	2.89240
$237$	0	0
$238$	0.263905	0.0171064
$239$	−4.42693	−0.286354	−0.143177	−	0.989697i	$-0.545732\pi$
$239$	−4.42693	−0.286354	−0.143177	+	0.989697i	$0.545732\pi$
$240$	0	0
$241$	−22.5066	−1.44977	−0.724887	−	0.688868i	$-0.758108\pi$
$241$	−22.5066	−1.44977	−0.724887	+	0.688868i	$0.758108\pi$
$242$	10.4236	0.670054
$243$	0	0
$244$	29.1757	1.86778
$245$	6.10444	0.389998
$246$	0	0
$247$	−12.6497	−0.804881
$248$	−63.1668	−4.01110
$249$	0	0
$250$	2.71108	0.171464
$251$	4.72305	0.298116	0.149058	−	0.988828i	$-0.452376\pi$
$251$	4.72305	0.298116	0.149058	+	0.988828i	$0.452376\pi$
$252$	0	0
$253$	−0.677971	−0.0426237
$254$	−32.4877	−2.03846
$255$	0	0
$256$	28.8578	1.80361
$257$	−15.1709	−0.946336	−0.473168	−	0.880972i	$-0.656890\pi$
$257$	−15.1709	−0.946336	−0.473168	+	0.880972i	$0.656890\pi$
$258$	0	0
$259$	4.40910	0.273968
$260$	−19.1853	−1.18982
$261$	0	0
$262$	−25.1309	−1.55259
$263$	23.5006	1.44911	0.724554	−	0.689218i	$-0.242046\pi$
$263$	23.5006	1.44911	0.724554	+	0.689218i	$0.242046\pi$
$264$	0	0
$265$	9.62601	0.591321
$266$	−9.05003	−0.554893
$267$	0	0
$268$	−27.0172	−1.65034
$269$	−16.0368	−0.977781	−0.488890	−	0.872345i	$-0.662598\pi$
$269$	−16.0368	−0.977781	−0.488890	+	0.872345i	$0.662598\pi$
$270$	0	0
$271$	8.15090	0.495132	0.247566	−	0.968871i	$-0.420369\pi$
$271$	8.15090	0.495132	0.247566	+	0.968871i	$0.420369\pi$
$272$	1.43206	0.0868311
$273$	0	0
$274$	−40.5784	−2.45143
$275$	−2.67492	−0.161304
$276$	0	0
$277$	3.82153	0.229614	0.114807	−	0.993388i	$-0.463375\pi$
$277$	3.82153	0.229614	0.114807	+	0.993388i	$0.463375\pi$
$278$	−0.344259	−0.0206473
$279$	0	0
$280$	−8.59460	−0.513626
$281$	6.04345	0.360522	0.180261	−	0.983619i	$-0.442306\pi$
$281$	6.04345	0.360522	0.180261	+	0.983619i	$0.442306\pi$
$282$	0	0
$283$	6.85831	0.407684	0.203842	−	0.979004i	$-0.434657\pi$
$283$	6.85831	0.407684	0.203842	+	0.979004i	$0.434657\pi$
$284$	−6.29546	−0.373567
$285$	0	0
$286$	26.0059	1.53776
$287$	2.53403	0.149579
$288$	0	0
$289$	−16.9894	−0.999378
$290$	21.5633	1.26624
$291$	0	0
$292$	−39.3302	−2.30163
$293$	31.8333	1.85972	0.929860	−	0.367913i	$-0.119927\pi$
$293$	31.8333	1.85972	0.929860	+	0.367913i	$0.119927\pi$
$294$	0	0
$295$	−8.30551	−0.483565
$296$	42.3138	2.45944
$297$	0	0
$298$	14.7046	0.851816
$299$	0.908907	0.0525635
$300$	0	0
$301$	4.09382	0.235964
$302$	5.63497	0.324256
$303$	0	0
$304$	−49.1091	−2.81660
$305$	−5.45346	−0.312264
$306$	0	0
$307$	23.6759	1.35125	0.675627	−	0.737243i	$-0.263873\pi$
$307$	23.6759	1.35125	0.675627	+	0.737243i	$0.263873\pi$
$308$	13.5427	0.771669
$309$	0	0
$310$	18.8561	1.07096
$311$	−10.2360	−0.580431	−0.290215	−	0.956961i	$-0.593727\pi$
$311$	−10.2360	−0.580431	−0.290215	+	0.956961i	$0.593727\pi$
$312$	0	0
$313$	26.7560	1.51234	0.756168	−	0.654378i	$-0.227069\pi$
$313$	26.7560	1.51234	0.756168	+	0.654378i	$0.227069\pi$
$314$	−21.0598	−1.18847
$315$	0	0
$316$	28.8536	1.62314
$317$	22.9123	1.28688	0.643440	−	0.765496i	$-0.277507\pi$
$317$	22.9123	1.28688	0.643440	+	0.765496i	$0.277507\pi$
$318$	0	0
$319$	−21.2757	−1.19121
$320$	−25.2380	−1.41085
$321$	0	0
$322$	0.650264	0.0362378
$323$	−0.362844	−0.0201892
$324$	0	0
$325$	3.58607	0.198919
$326$	22.7459	1.25978
$327$	0	0
$328$	24.3189	1.34279
$329$	0.594909	0.0327984
$330$	0	0
$331$	17.5122	0.962558	0.481279	−	0.876568i	$-0.340172\pi$
$331$	17.5122	0.962558	0.481279	+	0.876568i	$0.340172\pi$
$332$	−52.7107	−2.89287
$333$	0	0
$334$	−10.0369	−0.549194
$335$	5.05000	0.275911
$336$	0	0
$337$	0.618352	0.0336838	0.0168419	−	0.999858i	$-0.494639\pi$
$337$	0.618352	0.0336838	0.0168419	+	0.999858i	$0.494639\pi$
$338$	0.379804	0.0206586
$339$	0	0
$340$	−0.550311	−0.0298448
$341$	−18.6046	−1.00750
$342$	0	0
$343$	12.4013	0.669605
$344$	39.2880	2.11827
$345$	0	0
$346$	38.4013	2.06446
$347$	20.6657	1.10939	0.554696	−	0.832053i	$-0.312835\pi$
$347$	20.6657	1.10939	0.554696	+	0.832053i	$0.312835\pi$
$348$	0	0
$349$	−31.3997	−1.68079	−0.840395	−	0.541975i	$-0.817677\pi$
$349$	−31.3997	−1.68079	−0.840395	+	0.541975i	$0.817677\pi$
$350$	2.56560	0.137137
$351$	0	0
$352$	52.3739	2.79154
$353$	15.1931	0.808650	0.404325	−	0.914615i	$-0.367507\pi$
$353$	15.1931	0.808650	0.404325	+	0.914615i	$0.367507\pi$
$354$	0	0
$355$	1.17674	0.0624546
$356$	5.34994	0.283546
$357$	0	0
$358$	−5.80998	−0.307067
$359$	27.0990	1.43023	0.715116	−	0.699006i	$-0.246374\pi$
$359$	27.0990	1.43023	0.715116	+	0.699006i	$0.246374\pi$
$360$	0	0
$361$	−6.55708	−0.345110
$362$	−12.2945	−0.646187
$363$	0	0
$364$	−18.1558	−0.951621
$365$	7.35153	0.384797
$366$	0	0
$367$	23.5127	1.22735	0.613676	−	0.789558i	$-0.289690\pi$
$367$	23.5127	1.22735	0.613676	+	0.789558i	$0.289690\pi$
$368$	3.52859	0.183941
$369$	0	0
$370$	−12.6312	−0.656665
$371$	9.10947	0.472940
$372$	0	0
$373$	26.1076	1.35180	0.675900	−	0.736993i	$-0.263755\pi$
$373$	26.1076	1.35180	0.675900	+	0.736993i	$0.263755\pi$
$374$	0.745953	0.0385723
$375$	0	0
$376$	5.70929	0.294434
$377$	28.5229	1.46900
$378$	0	0
$379$	16.5729	0.851293	0.425646	−	0.904890i	$-0.360047\pi$
$379$	16.5729	0.851293	0.425646	+	0.904890i	$0.360047\pi$
$380$	18.8717	0.968095
$381$	0	0
$382$	−58.6664	−3.00163
$383$	8.24534	0.421317	0.210659	−	0.977560i	$-0.432439\pi$
$383$	8.24534	0.421317	0.210659	+	0.977560i	$0.432439\pi$
$384$	0	0
$385$	−2.53138	−0.129011
$386$	64.2659	3.27105
$387$	0	0
$388$	−58.9639	−2.99344
$389$	25.7935	1.30778	0.653892	−	0.756588i	$-0.273135\pi$
$389$	25.7935	1.30778	0.653892	+	0.756588i	$0.273135\pi$
$390$	0	0
$391$	0.0260711	0.00131847
$392$	55.4402	2.80015
$393$	0	0
$394$	−14.8209	−0.746668
$395$	−5.39325	−0.271364
$396$	0	0
$397$	−20.2155	−1.01458	−0.507292	−	0.861774i	$-0.669354\pi$
$397$	−20.2155	−1.01458	−0.507292	+	0.861774i	$0.669354\pi$
$398$	−63.7371	−3.19485
$399$	0	0
$400$	13.9220	0.696098
$401$	0.271321	0.0135491	0.00677456	−	0.999977i	$-0.497844\pi$
$401$	0.271321	0.0135491	0.00677456	+	0.999977i	$0.497844\pi$
$402$	0	0
$403$	24.9419	1.24244
$404$	19.0160	0.946081
$405$	0	0
$406$	20.4063	1.01275
$407$	12.4627	0.617755
$408$	0	0
$409$	12.3766	0.611985	0.305992	−	0.952034i	$-0.401012\pi$
$409$	12.3766	0.611985	0.305992	+	0.952034i	$0.401012\pi$
$410$	−7.25951	−0.358522
$411$	0	0
$412$	49.9268	2.45971
$413$	−7.85983	−0.386757
$414$	0	0
$415$	9.85257	0.483644
$416$	−70.2140	−3.44252
$417$	0	0
$418$	−25.5808	−1.25120
$419$	−22.8239	−1.11502	−0.557511	−	0.830170i	$-0.688243\pi$
$419$	−22.8239	−1.11502	−0.557511	+	0.830170i	$0.688243\pi$
$420$	0	0
$421$	26.1381	1.27389	0.636947	−	0.770908i	$-0.280197\pi$
$421$	26.1381	1.27389	0.636947	+	0.770908i	$0.280197\pi$
$422$	−18.8534	−0.917770
$423$	0	0
$424$	87.4228	4.24563
$425$	0.102863	0.00498959
$426$	0	0
$427$	−5.16082	−0.249750
$428$	32.1561	1.55433
$429$	0	0
$430$	−11.7280	−0.565573
$431$	41.0872	1.97910	0.989551	−	0.144184i	$-0.0460556\pi$
$431$	41.0872	1.97910	0.989551	+	0.144184i	$0.0460556\pi$
$432$	0	0
$433$	−3.87251	−0.186101	−0.0930504	−	0.995661i	$-0.529662\pi$
$433$	−3.87251	−0.186101	−0.0930504	+	0.995661i	$0.529662\pi$
$434$	17.8443	0.856553
$435$	0	0
$436$	86.6905	4.15172
$437$	−0.894050	−0.0427682
$438$	0	0
$439$	−1.43929	−0.0686934	−0.0343467	−	0.999410i	$-0.510935\pi$
$439$	−1.43929	−0.0686934	−0.0343467	+	0.999410i	$0.510935\pi$
$440$	−24.2934	−1.15814
$441$	0	0
$442$	−1.00005	−0.0475673
$443$	−7.06341	−0.335593	−0.167796	−	0.985822i	$-0.553665\pi$
$443$	−7.06341	−0.335593	−0.167796	+	0.985822i	$0.553665\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−3.09349	−0.146481
$447$	0	0
$448$	−23.8837	−1.12840
$449$	21.9505	1.03591	0.517953	−	0.855409i	$-0.326694\pi$
$449$	21.9505	1.03591	0.517953	+	0.855409i	$0.326694\pi$
$450$	0	0
$451$	7.16269	0.337278
$452$	3.81458	0.179423
$453$	0	0
$454$	−58.7249	−2.75610
$455$	3.39364	0.159096
$456$	0	0
$457$	33.1808	1.55213	0.776066	−	0.630652i	$-0.217212\pi$
$457$	33.1808	1.55213	0.776066	+	0.630652i	$0.217212\pi$
$458$	68.4534	3.19862
$459$	0	0
$460$	−1.35597	−0.0632223
$461$	29.9410	1.39449	0.697246	−	0.716832i	$-0.254409\pi$
$461$	29.9410	1.39449	0.697246	+	0.716832i	$0.254409\pi$
$462$	0	0
$463$	−12.5648	−0.583934	−0.291967	−	0.956428i	$-0.594310\pi$
$463$	−12.5648	−0.583934	−0.291967	+	0.956428i	$0.594310\pi$
$464$	110.732	5.14062
$465$	0	0
$466$	55.1800	2.55616
$467$	0.853080	0.0394758	0.0197379	−	0.999805i	$-0.493717\pi$
$467$	0.853080	0.0394758	0.0197379	+	0.999805i	$0.493717\pi$
$468$	0	0
$469$	4.77901	0.220674
$470$	−1.70430	−0.0786133
$471$	0	0
$472$	−75.4301	−3.47195
$473$	11.5716	0.532061
$474$	0	0
$475$	−3.52745	−0.161851
$476$	−0.520781	−0.0238700
$477$	0	0
$478$	12.0017	0.548947
$479$	−24.5425	−1.12137	−0.560687	−	0.828028i	$-0.689463\pi$
$479$	−24.5425	−1.12137	−0.560687	+	0.828028i	$0.689463\pi$
$480$	0	0
$481$	−16.7079	−0.761815
$482$	61.0170	2.77925
$483$	0	0
$484$	−20.5695	−0.934979
$485$	11.0214	0.500457
$486$	0	0
$487$	−1.53324	−0.0694775	−0.0347388	−	0.999396i	$-0.511060\pi$
$487$	−1.53324	−0.0694775	−0.0347388	+	0.999396i	$0.511060\pi$
$488$	−49.5280	−2.24203
$489$	0	0
$490$	−16.5496	−0.747635
$491$	−15.7953	−0.712834	−0.356417	−	0.934327i	$-0.616002\pi$
$491$	−15.7953	−0.712834	−0.356417	+	0.934327i	$0.616002\pi$
$492$	0	0
$493$	0.818151	0.0368477
$494$	34.2943	1.54297
$495$	0	0
$496$	96.8302	4.34781
$497$	1.11359	0.0499514
$498$	0	0
$499$	−31.0562	−1.39027	−0.695133	−	0.718882i	$-0.744654\pi$
$499$	−31.0562	−1.39027	−0.695133	+	0.718882i	$0.744654\pi$
$500$	−5.34994	−0.239257
$501$	0	0
$502$	−12.8046	−0.571495
$503$	−15.4036	−0.686812	−0.343406	−	0.939187i	$-0.611581\pi$
$503$	−15.4036	−0.686812	−0.343406	+	0.939187i	$0.611581\pi$
$504$	0	0
$505$	−3.55443	−0.158170
$506$	1.83803	0.0817105
$507$	0	0
$508$	64.1100	2.84442
$509$	20.8319	0.923359	0.461680	−	0.887047i	$-0.347247\pi$
$509$	20.8319	0.923359	0.461680	+	0.887047i	$0.347247\pi$
$510$	0	0
$511$	6.95704	0.307761
$512$	−19.7101	−0.871071
$513$	0	0
$514$	41.1295	1.81415
$515$	−9.33221	−0.411226
$516$	0	0
$517$	1.68156	0.0739551
$518$	−11.9534	−0.525203
$519$	0	0
$520$	32.5685	1.42822
$521$	−13.8518	−0.606860	−0.303430	−	0.952854i	$-0.598132\pi$
$521$	−13.8518	−0.606860	−0.303430	+	0.952854i	$0.598132\pi$
$522$	0	0
$523$	18.4645	0.807398	0.403699	−	0.914892i	$-0.367724\pi$
$523$	18.4645	0.807398	0.403699	+	0.914892i	$0.367724\pi$
$524$	49.5924	2.16645
$525$	0	0
$526$	−63.7119	−2.77797
$527$	0.715434	0.0311648
$528$	0	0
$529$	−22.9358	−0.997207
$530$	−26.0968	−1.13357
$531$	0	0
$532$	17.8590	0.774286
$533$	−9.60250	−0.415931
$534$	0	0
$535$	−6.01056	−0.259859
$536$	45.8638	1.98101
$537$	0	0
$538$	43.4770	1.87443
$539$	16.3289	0.703334
$540$	0	0
$541$	−25.9948	−1.11761	−0.558803	−	0.829301i	$-0.688739\pi$
$541$	−25.9948	−1.11761	−0.558803	+	0.829301i	$0.688739\pi$
$542$	−22.0977	−0.949178
$543$	0	0
$544$	−2.01402	−0.0863504
$545$	−16.2040	−0.694104
$546$	0	0
$547$	−1.59132	−0.0680397	−0.0340199	−	0.999421i	$-0.510831\pi$
$547$	−1.59132	−0.0680397	−0.0340199	+	0.999421i	$0.510831\pi$
$548$	80.0760	3.42068
$549$	0	0
$550$	7.25191	0.309222
$551$	−28.0566	−1.19525
$552$	0	0
$553$	−5.10385	−0.217038
$554$	−10.3605	−0.440174
$555$	0	0
$556$	0.679348	0.0288108
$557$	−28.4649	−1.20610	−0.603049	−	0.797704i	$-0.706048\pi$
$557$	−28.4649	−1.20610	−0.603049	+	0.797704i	$0.706048\pi$
$558$	0	0
$559$	−15.5132	−0.656137
$560$	13.1749	0.556741
$561$	0	0
$562$	−16.3843	−0.691129
$563$	23.0401	0.971024	0.485512	−	0.874230i	$-0.338633\pi$
$563$	23.0401	0.971024	0.485512	+	0.874230i	$0.338633\pi$
$564$	0	0
$565$	−0.713014	−0.0299967
$566$	−18.5934	−0.781539
$567$	0	0
$568$	10.6870	0.448418
$569$	39.4567	1.65411	0.827055	−	0.562121i	$-0.190015\pi$
$569$	39.4567	1.65411	0.827055	+	0.562121i	$0.190015\pi$
$570$	0	0
$571$	34.8536	1.45858	0.729289	−	0.684206i	$-0.239851\pi$
$571$	34.8536	1.45858	0.729289	+	0.684206i	$0.239851\pi$
$572$	−51.3190	−2.14575
$573$	0	0
$574$	−6.86996	−0.286747
$575$	0.253455	0.0105698
$576$	0	0
$577$	29.8337	1.24199	0.620997	−	0.783813i	$-0.286728\pi$
$577$	29.8337	1.24199	0.620997	+	0.783813i	$0.286728\pi$
$578$	46.0596	1.91583
$579$	0	0
$580$	−42.5523	−1.76689
$581$	9.32388	0.386820
$582$	0	0
$583$	25.7488	1.06641
$584$	66.7662	2.76280
$585$	0	0
$586$	−86.3025	−3.56512
$587$	−44.0779	−1.81929	−0.909645	−	0.415386i	$-0.863646\pi$
$587$	−44.0779	−1.81929	−0.909645	+	0.415386i	$0.863646\pi$
$588$	0	0
$589$	−24.5342	−1.01091
$590$	22.5169	0.927005
$591$	0	0
$592$	−64.8640	−2.66589
$593$	−25.3294	−1.04016	−0.520078	−	0.854119i	$-0.674097\pi$
$593$	−25.3294	−1.04016	−0.520078	+	0.854119i	$0.674097\pi$
$594$	0	0
$595$	0.0973433	0.00399069
$596$	−29.0176	−1.18861
$597$	0	0
$598$	−2.46412	−0.100765
$599$	−2.82158	−0.115287	−0.0576433	−	0.998337i	$-0.518359\pi$
$599$	−2.82158	−0.115287	−0.0576433	+	0.998337i	$0.518359\pi$
$600$	0	0
$601$	18.7131	0.763323	0.381661	−	0.924302i	$-0.375352\pi$
$601$	18.7131	0.763323	0.381661	+	0.924302i	$0.375352\pi$
$602$	−11.0987	−0.452347
$603$	0	0
$604$	−11.1198	−0.452460
$605$	3.84482	0.156314
$606$	0	0
$607$	33.0581	1.34179	0.670893	−	0.741554i	$-0.265911\pi$
$607$	33.0581	1.34179	0.670893	+	0.741554i	$0.265911\pi$
$608$	69.0662	2.80101
$609$	0	0
$610$	14.7847	0.598617
$611$	−2.25435	−0.0912014
$612$	0	0
$613$	42.5594	1.71896	0.859479	−	0.511172i	$-0.170788\pi$
$613$	42.5594	1.71896	0.859479	+	0.511172i	$0.170788\pi$
$614$	−64.1872	−2.59038
$615$	0	0
$616$	−22.9898	−0.926287
$617$	−21.0221	−0.846318	−0.423159	−	0.906056i	$-0.639079\pi$
$617$	−21.0221	−0.846318	−0.423159	+	0.906056i	$0.639079\pi$
$618$	0	0
$619$	24.7010	0.992816	0.496408	−	0.868089i	$-0.334652\pi$
$619$	24.7010	0.992816	0.496408	+	0.868089i	$0.334652\pi$
$620$	−37.2100	−1.49439
$621$	0	0
$622$	27.7506	1.11270
$623$	−0.946340	−0.0379143
$624$	0	0
$625$	1.00000	0.0400000
$626$	−72.5374	−2.89918
$627$	0	0
$628$	41.5586	1.65837
$629$	−0.479250	−0.0191090
$630$	0	0
$631$	−25.9051	−1.03127	−0.515633	−	0.856809i	$-0.672443\pi$
$631$	−25.9051	−1.03127	−0.515633	+	0.856809i	$0.672443\pi$
$632$	−48.9812	−1.94837
$633$	0	0
$634$	−62.1169	−2.46698
$635$	−11.9833	−0.475544
$636$	0	0
$637$	−21.8910	−0.867351
$638$	57.6802	2.28358
$639$	0	0
$640$	29.2628	1.15671
$641$	5.40721	0.213572	0.106786	−	0.994282i	$-0.465944\pi$
$641$	5.40721	0.213572	0.106786	+	0.994282i	$0.465944\pi$
$642$	0	0
$643$	35.0406	1.38186	0.690932	−	0.722919i	$-0.257200\pi$
$643$	35.0406	1.38186	0.690932	+	0.722919i	$0.257200\pi$
$644$	−1.28321	−0.0505654
$645$	0	0
$646$	0.983699	0.0387031
$647$	16.4750	0.647699	0.323849	−	0.946109i	$-0.395023\pi$
$647$	16.4750	0.647699	0.323849	+	0.946109i	$0.395023\pi$
$648$	0	0
$649$	−22.2165	−0.872076
$650$	−9.72212	−0.381333
$651$	0	0
$652$	−44.8858	−1.75787
$653$	−16.9029	−0.661460	−0.330730	−	0.943725i	$-0.607295\pi$
$653$	−16.9029	−0.661460	−0.330730	+	0.943725i	$0.607295\pi$
$654$	0	0
$655$	−9.26971	−0.362197
$656$	−37.2792	−1.45551
$657$	0	0
$658$	−1.61284	−0.0628752
$659$	−49.8374	−1.94139	−0.970694	−	0.240318i	$-0.922748\pi$
$659$	−49.8374	−1.94139	−0.970694	+	0.240318i	$0.922748\pi$
$660$	0	0
$661$	16.3895	0.637476	0.318738	−	0.947843i	$-0.396741\pi$
$661$	16.3895	0.637476	0.318738	+	0.947843i	$0.396741\pi$
$662$	−47.4769	−1.84524
$663$	0	0
$664$	89.4805	3.47252
$665$	−3.33817	−0.129449
$666$	0	0
$667$	2.01593	0.0780571
$668$	19.8064	0.766333
$669$	0	0
$670$	−13.6909	−0.528927
$671$	−14.5875	−0.563146
$672$	0	0
$673$	32.6039	1.25679	0.628394	−	0.777895i	$-0.283713\pi$
$673$	32.6039	1.25679	0.628394	+	0.777895i	$0.283713\pi$
$674$	−1.67640	−0.0645725
$675$	0	0
$676$	−0.749492	−0.0288266
$677$	23.3164	0.896123	0.448061	−	0.894003i	$-0.352115\pi$
$677$	23.3164	0.896123	0.448061	+	0.894003i	$0.352115\pi$
$678$	0	0
$679$	10.4300	0.400267
$680$	0.934196	0.0358248
$681$	0	0
$682$	50.4386	1.93139
$683$	34.8820	1.33472	0.667362	−	0.744733i	$-0.267423\pi$
$683$	34.8820	1.33472	0.667362	+	0.744733i	$0.267423\pi$
$684$	0	0
$685$	−14.9676	−0.571884
$686$	−33.6208	−1.28365
$687$	0	0
$688$	−60.2257	−2.29608
$689$	−34.5195	−1.31509
$690$	0	0
$691$	0.972062	0.0369790	0.0184895	−	0.999829i	$-0.494114\pi$
$691$	0.972062	0.0369790	0.0184895	+	0.999829i	$0.494114\pi$
$692$	−75.7796	−2.88071
$693$	0	0
$694$	−56.0263	−2.12673
$695$	−0.126982	−0.00481672
$696$	0	0
$697$	−0.275439	−0.0104330
$698$	85.1271	3.22211
$699$	0	0
$700$	−5.06286	−0.191358
$701$	19.5858	0.739745	0.369873	−	0.929082i	$-0.379401\pi$
$701$	19.5858	0.739745	0.369873	+	0.929082i	$0.379401\pi$
$702$	0	0
$703$	16.4348	0.619850
$704$	−67.5095	−2.54436
$705$	0	0
$706$	−41.1898	−1.55020
$707$	−3.36370	−0.126505
$708$	0	0
$709$	−9.14963	−0.343622	−0.171811	−	0.985130i	$-0.554962\pi$
$709$	−9.14963	−0.343622	−0.171811	+	0.985130i	$0.554962\pi$
$710$	−3.19022	−0.119727
$711$	0	0
$712$	−9.08194	−0.340360
$713$	1.76283	0.0660186
$714$	0	0
$715$	9.59244	0.358737
$716$	11.4652	0.428475
$717$	0	0
$718$	−73.4676	−2.74178
$719$	−13.2073	−0.492549	−0.246275	−	0.969200i	$-0.579207\pi$
$719$	−13.2073	−0.492549	−0.246275	+	0.969200i	$0.579207\pi$
$720$	0	0
$721$	−8.83144	−0.328900
$722$	17.7768	0.661582
$723$	0	0
$724$	24.2616	0.901675
$725$	7.95379	0.295396
$726$	0	0
$727$	−2.40432	−0.0891714	−0.0445857	−	0.999006i	$-0.514197\pi$
$727$	−2.40432	−0.0891714	−0.0445857	+	0.999006i	$0.514197\pi$
$728$	30.8209	1.14230
$729$	0	0
$730$	−19.9306	−0.737663
$731$	−0.444980	−0.0164582
$732$	0	0
$733$	22.4775	0.830225	0.415113	−	0.909770i	$-0.363742\pi$
$733$	22.4775	0.830225	0.415113	+	0.909770i	$0.363742\pi$
$734$	−63.7447	−2.35286
$735$	0	0
$736$	−4.96256	−0.182922
$737$	13.5083	0.497586
$738$	0	0
$739$	21.1568	0.778266	0.389133	−	0.921182i	$-0.372775\pi$
$739$	21.1568	0.778266	0.389133	+	0.921182i	$0.372775\pi$
$740$	24.9260	0.916297
$741$	0	0
$742$	−24.6965	−0.906636
$743$	−17.1962	−0.630866	−0.315433	−	0.948948i	$-0.602150\pi$
$743$	−17.1962	−0.630866	−0.315433	+	0.948948i	$0.602150\pi$
$744$	0	0
$745$	5.42391	0.198717
$746$	−70.7797	−2.59143
$747$	0	0
$748$	−1.47204	−0.0538230
$749$	−5.68803	−0.207836
$750$	0	0
$751$	42.0249	1.53351	0.766755	−	0.641940i	$-0.221870\pi$
$751$	42.0249	1.53351	0.766755	+	0.641940i	$0.221870\pi$
$752$	−8.75193	−0.319150
$753$	0	0
$754$	−77.3277	−2.81611
$755$	2.07850	0.0756443
$756$	0	0
$757$	36.7284	1.33492	0.667458	−	0.744647i	$-0.267382\pi$
$757$	36.7284	1.33492	0.667458	+	0.744647i	$0.267382\pi$
$758$	−44.9304	−1.63195
$759$	0	0
$760$	−32.0361	−1.16207
$761$	27.5494	0.998664	0.499332	−	0.866411i	$-0.333579\pi$
$761$	27.5494	0.998664	0.499332	+	0.866411i	$0.333579\pi$
$762$	0	0
$763$	−15.3345	−0.555147
$764$	115.770	4.18841
$765$	0	0
$766$	−22.3537	−0.807674
$767$	29.7841	1.07544
$768$	0	0
$769$	−31.7867	−1.14626	−0.573129	−	0.819465i	$-0.694271\pi$
$769$	−31.7867	−1.14626	−0.573129	+	0.819465i	$0.694271\pi$
$770$	6.86277	0.247317
$771$	0	0
$772$	−126.820	−4.56435
$773$	36.6598	1.31856	0.659280	−	0.751897i	$-0.270861\pi$
$773$	36.6598	1.31856	0.659280	+	0.751897i	$0.270861\pi$
$774$	0	0
$775$	6.95521	0.249839
$776$	100.096	3.59323
$777$	0	0
$778$	−69.9282	−2.50705
$779$	9.44554	0.338421
$780$	0	0
$781$	3.14767	0.112632
$782$	−0.0706809	−0.00252754
$783$	0	0
$784$	−84.9858	−3.03521
$785$	−7.76806	−0.277254
$786$	0	0
$787$	−21.2386	−0.757075	−0.378537	−	0.925586i	$-0.623573\pi$
$787$	−21.2386	−0.757075	−0.378537	+	0.925586i	$0.623573\pi$
$788$	29.2471	1.04188
$789$	0	0
$790$	14.6215	0.520210
$791$	−0.674753	−0.0239915
$792$	0	0
$793$	19.5565	0.694471
$794$	54.8057	1.94498
$795$	0	0
$796$	125.776	4.45803
$797$	−2.19829	−0.0778675	−0.0389338	−	0.999242i	$-0.512396\pi$
$797$	−2.19829	−0.0778675	−0.0389338	+	0.999242i	$0.512396\pi$
$798$	0	0
$799$	−0.0646640	−0.00228765
$800$	−19.5796	−0.692245
$801$	0	0
$802$	−0.735572	−0.0259739
$803$	19.6647	0.693953
$804$	0	0
$805$	0.239854	0.00845376
$806$	−67.6194	−2.38179
$807$	0	0
$808$	−32.2811	−1.13565
$809$	52.3147	1.83929	0.919644	−	0.392754i	$-0.128477\pi$
$809$	52.3147	1.83929	0.919644	+	0.392754i	$0.128477\pi$
$810$	0	0
$811$	23.3463	0.819800	0.409900	−	0.912130i	$-0.365564\pi$
$811$	23.3463	0.819800	0.409900	+	0.912130i	$0.365564\pi$
$812$	−40.2689	−1.41316
$813$	0	0
$814$	−33.7874	−1.18425
$815$	8.38997	0.293888
$816$	0	0
$817$	15.2596	0.533865
$818$	−33.5540	−1.17319
$819$	0	0
$820$	14.3256	0.500273
$821$	17.7826	0.620617	0.310309	−	0.950636i	$-0.399568\pi$
$821$	17.7826	0.620617	0.310309	+	0.950636i	$0.399568\pi$
$822$	0	0
$823$	2.84723	0.0992483	0.0496241	−	0.998768i	$-0.484198\pi$
$823$	2.84723	0.0992483	0.0496241	+	0.998768i	$0.484198\pi$
$824$	−84.7546	−2.95257
$825$	0	0
$826$	21.3086	0.741421
$827$	−0.390574	−0.0135816	−0.00679080	−	0.999977i	$-0.502162\pi$
$827$	−0.390574	−0.0135816	−0.00679080	+	0.999977i	$0.502162\pi$
$828$	0	0
$829$	49.3799	1.71503	0.857517	−	0.514455i	$-0.172006\pi$
$829$	49.3799	1.71503	0.857517	+	0.514455i	$0.172006\pi$
$830$	−26.7111	−0.927155
$831$	0	0
$832$	90.5052	3.13770
$833$	−0.627921	−0.0217562
$834$	0	0
$835$	−3.70218	−0.128119
$836$	50.4801	1.74589
$837$	0	0
$838$	61.8774	2.13752
$839$	−53.0414	−1.83119	−0.915596	−	0.402098i	$-0.868281\pi$
$839$	−53.0414	−1.83119	−0.915596	+	0.402098i	$0.868281\pi$
$840$	0	0
$841$	34.2628	1.18148
$842$	−70.8625	−2.44208
$843$	0	0
$844$	37.2046	1.28064
$845$	0.140094	0.00481937
$846$	0	0
$847$	3.63850	0.125020
$848$	−134.013	−4.60202
$849$	0	0
$850$	−0.278870	−0.00956515
$851$	−1.18087	−0.0404799
$852$	0	0
$853$	28.9351	0.990720	0.495360	−	0.868688i	$-0.335036\pi$
$853$	28.9351	0.990720	0.495360	+	0.868688i	$0.335036\pi$
$854$	13.9914	0.478775
$855$	0	0
$856$	−54.5876	−1.86576
$857$	−14.3048	−0.488642	−0.244321	−	0.969694i	$-0.578565\pi$
$857$	−14.3048	−0.488642	−0.244321	+	0.969694i	$0.578565\pi$
$858$	0	0
$859$	−19.3098	−0.658842	−0.329421	−	0.944183i	$-0.606854\pi$
$859$	−19.3098	−0.658842	−0.329421	+	0.944183i	$0.606854\pi$
$860$	23.1436	0.789189
$861$	0	0
$862$	−111.391	−3.79398
$863$	−46.5130	−1.58332	−0.791660	−	0.610962i	$-0.790783\pi$
$863$	−46.5130	−1.58332	−0.791660	+	0.610962i	$0.790783\pi$
$864$	0	0
$865$	14.1646	0.481610
$866$	10.4987	0.356759
$867$	0	0
$868$	−35.2133	−1.19522
$869$	−14.4265	−0.489386
$870$	0	0
$871$	−18.1097	−0.613622
$872$	−147.164	−4.98360
$873$	0	0
$874$	2.42384	0.0819876
$875$	0.946340	0.0319921
$876$	0	0
$877$	7.44528	0.251409	0.125705	−	0.992068i	$-0.459881\pi$
$877$	7.44528	0.251409	0.125705	+	0.992068i	$0.459881\pi$
$878$	3.90201	0.131687
$879$	0	0
$880$	37.2401	1.25536
$881$	−22.8284	−0.769107	−0.384553	−	0.923103i	$-0.625645\pi$
$881$	−22.8284	−0.769107	−0.384553	+	0.923103i	$0.625645\pi$
$882$	0	0
$883$	−45.7912	−1.54100	−0.770499	−	0.637442i	$-0.779993\pi$
$883$	−45.7912	−1.54100	−0.770499	+	0.637442i	$0.779993\pi$
$884$	1.97345	0.0663744
$885$	0	0
$886$	19.1495	0.643339
$887$	−28.5165	−0.957492	−0.478746	−	0.877953i	$-0.658909\pi$
$887$	−28.5165	−0.957492	−0.478746	+	0.877953i	$0.658909\pi$
$888$	0	0
$889$	−11.3403	−0.380341
$890$	2.71108	0.0908755
$891$	0	0
$892$	6.10458	0.204396
$893$	2.21750	0.0742059
$894$	0	0
$895$	−2.14305	−0.0716344
$896$	27.6926	0.925144
$897$	0	0
$898$	−59.5094	−1.98585
$899$	55.3203	1.84504
$900$	0	0
$901$	−0.990160	−0.0329870
$902$	−19.4186	−0.646568
$903$	0	0
$904$	−6.47555	−0.215374
$905$	−4.53493	−0.150746
$906$	0	0
$907$	−38.2007	−1.26843	−0.634216	−	0.773156i	$-0.718677\pi$
$907$	−38.2007	−1.26843	−0.634216	+	0.773156i	$0.718677\pi$
$908$	115.886	3.84580
$909$	0	0
$910$	−9.20042	−0.304991
$911$	19.4721	0.645140	0.322570	−	0.946546i	$-0.395453\pi$
$911$	19.4721	0.645140	0.322570	+	0.946546i	$0.395453\pi$
$912$	0	0
$913$	26.3548	0.872217
$914$	−89.9557	−2.97547
$915$	0	0
$916$	−135.083	−4.46328
$917$	−8.77229	−0.289687
$918$	0	0
$919$	−39.4086	−1.29997	−0.649986	−	0.759947i	$-0.725225\pi$
$919$	−39.4086	−1.29997	−0.649986	+	0.759947i	$0.725225\pi$
$920$	2.30186	0.0758902
$921$	0	0
$922$	−81.1724	−2.67327
$923$	−4.21986	−0.138898
$924$	0	0
$925$	−4.65911	−0.153191
$926$	34.0640	1.11941
$927$	0	0
$928$	−155.732	−5.11217
$929$	−27.7768	−0.911328	−0.455664	−	0.890152i	$-0.650598\pi$
$929$	−27.7768	−0.911328	−0.455664	+	0.890152i	$0.650598\pi$
$930$	0	0
$931$	21.5331	0.705720
$932$	−108.890	−3.56681
$933$	0	0
$934$	−2.31276	−0.0756760
$935$	0.275150	0.00899837
$936$	0	0
$937$	−49.8044	−1.62704	−0.813520	−	0.581537i	$-0.802451\pi$
$937$	−49.8044	−1.62704	−0.813520	+	0.581537i	$0.802451\pi$
$938$	−12.9563	−0.423037
$939$	0	0
$940$	3.36319	0.109695
$941$	−24.1645	−0.787739	−0.393870	−	0.919166i	$-0.628864\pi$
$941$	−24.1645	−0.787739	−0.393870	+	0.919166i	$0.628864\pi$
$942$	0	0
$943$	−0.678682	−0.0221009
$944$	115.629	3.76340
$945$	0	0
$946$	−31.3714	−1.01997
$947$	34.4113	1.11822	0.559108	−	0.829095i	$-0.311144\pi$
$947$	34.4113	1.11822	0.559108	+	0.829095i	$0.311144\pi$
$948$	0	0
$949$	−26.3631	−0.855783
$950$	9.56319	0.310271
$951$	0	0
$952$	0.884067	0.0286528
$953$	5.61867	0.182007	0.0910033	−	0.995851i	$-0.470993\pi$
$953$	5.61867	0.182007	0.0910033	+	0.995851i	$0.470993\pi$
$954$	0	0
$955$	−21.6395	−0.700238
$956$	−23.6838	−0.765988
$957$	0	0
$958$	66.5365	2.14970
$959$	−14.1645	−0.457395
$960$	0	0
$961$	17.3750	0.560484
$962$	45.2964	1.46042
$963$	0	0
$964$	−120.409	−3.87810
$965$	23.7049	0.763089
$966$	0	0
$967$	−21.9303	−0.705231	−0.352616	−	0.935768i	$-0.614708\pi$
$967$	−21.9303	−0.705231	−0.352616	+	0.935768i	$0.614708\pi$
$968$	34.9184	1.12232
$969$	0	0
$970$	−29.8799	−0.959387
$971$	−37.1087	−1.19087	−0.595437	−	0.803402i	$-0.703021\pi$
$971$	−37.1087	−1.19087	−0.595437	+	0.803402i	$0.703021\pi$
$972$	0	0
$973$	−0.120169	−0.00385243
$974$	4.15672	0.133190
$975$	0	0
$976$	75.9228	2.43023
$977$	0.214176	0.00685209	0.00342604	−	0.999994i	$-0.498909\pi$
$977$	0.214176	0.00685209	0.00342604	+	0.999994i	$0.498909\pi$
$978$	0	0
$979$	−2.67492	−0.0854907
$980$	32.6584	1.04323
$981$	0	0
$982$	42.8224	1.36652
$983$	12.4786	0.398006	0.199003	−	0.979999i	$-0.436230\pi$
$983$	12.4786	0.398006	0.199003	+	0.979999i	$0.436230\pi$
$984$	0	0
$985$	−5.46681	−0.174187
$986$	−2.21807	−0.0706378
$987$	0	0
$988$	−67.6751	−2.15303
$989$	−1.09643	−0.0348645
$990$	0	0
$991$	−6.25349	−0.198649	−0.0993243	−	0.995055i	$-0.531668\pi$
$991$	−6.25349	−0.198649	−0.0993243	+	0.995055i	$0.531668\pi$
$992$	−136.181	−4.32374
$993$	0	0
$994$	−3.01903	−0.0957579
$995$	−23.5099	−0.745313
$996$	0	0
$997$	−41.6014	−1.31753	−0.658764	−	0.752349i	$-0.728921\pi$
$997$	−41.6014	−1.31753	−0.658764	+	0.752349i	$0.728921\pi$
$998$	84.1957	2.66517
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.w.1.2	✓	17
3.2	odd	2		4005.2.a.x.1.16	yes	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4005.2.a.w.1.2	✓	17	1.1	even	1	trivial
4005.2.a.x.1.16	yes	17	3.2	odd	2

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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q-2.71108 q^{2} +5.34994 q^{4} -1.00000 q^{5} -0.946340 q^{7} -9.08194 q^{8} +O(q^{10})\)	\(q-2.71108 q^{2} +5.34994 q^{4} -1.00000 q^{5} -0.946340 q^{7} -9.08194 q^{8} +2.71108 q^{10} -2.67492 q^{11} +3.58607 q^{13} +2.56560 q^{14} +13.9220 q^{16} +0.102863 q^{17} -3.52745 q^{19} -5.34994 q^{20} +7.25191 q^{22} +0.253455 q^{23} +1.00000 q^{25} -9.72212 q^{26} -5.06286 q^{28} +7.95379 q^{29} +6.95521 q^{31} -19.5796 q^{32} -0.278870 q^{34} +0.946340 q^{35} -4.65911 q^{37} +9.56319 q^{38} +9.08194 q^{40} -2.67772 q^{41} -4.32595 q^{43} -14.3106 q^{44} -0.687136 q^{46} -0.628642 q^{47} -6.10444 q^{49} -2.71108 q^{50} +19.1853 q^{52} -9.62601 q^{53} +2.67492 q^{55} +8.59460 q^{56} -21.5633 q^{58} +8.30551 q^{59} +5.45346 q^{61} -18.8561 q^{62} +25.2380 q^{64} -3.58607 q^{65} -5.05000 q^{67} +0.550311 q^{68} -2.56560 q^{70} -1.17674 q^{71} -7.35153 q^{73} +12.6312 q^{74} -18.8717 q^{76} +2.53138 q^{77} +5.39325 q^{79} -13.9220 q^{80} +7.25951 q^{82} -9.85257 q^{83} -0.102863 q^{85} +11.7280 q^{86} +24.2934 q^{88} +1.00000 q^{89} -3.39364 q^{91} +1.35597 q^{92} +1.70430 q^{94} +3.52745 q^{95} -11.0214 q^{97} +16.5496 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8}+O(q^{10}) \) 17 * q - 5 * q^2 + 21 * q^4 - 17 * q^5 + 12 * q^7 - 15 * q^8	\( 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8} + 5 q^{10} - 2 q^{11} + 8 q^{13} - 4 q^{14} + 33 q^{16} - 10 q^{17} + 32 q^{19} - 21 q^{20} + 8 q^{22} - 15 q^{23} + 17 q^{25} + 15 q^{26} + 24 q^{28} - q^{29} + 18 q^{31} - 25 q^{32} + 14 q^{34} - 12 q^{35} + 12 q^{37} - 22 q^{38} + 15 q^{40} + 7 q^{41} + 28 q^{43} + 14 q^{44} + 4 q^{46} - 26 q^{47} + 41 q^{49} - 5 q^{50} + 10 q^{52} - 12 q^{53} + 2 q^{55} - 13 q^{56} + 16 q^{58} + 23 q^{59} + 26 q^{61} - 10 q^{62} + 59 q^{64} - 8 q^{65} + 31 q^{67} + q^{68} + 4 q^{70} + 2 q^{71} + 33 q^{73} + 10 q^{74} + 66 q^{76} - 12 q^{77} + 33 q^{79} - 33 q^{80} + 30 q^{82} - 13 q^{83} + 10 q^{85} + 20 q^{86} + 12 q^{88} + 17 q^{89} + 40 q^{91} - 16 q^{92} + 38 q^{94} - 32 q^{95} + 45 q^{97} - 2 q^{98}+O(q^{100}) \) 17 * q - 5 * q^2 + 21 * q^4 - 17 * q^5 + 12 * q^7 - 15 * q^8 + 5 * q^10 - 2 * q^11 + 8 * q^13 - 4 * q^14 + 33 * q^16 - 10 * q^17 + 32 * q^19 - 21 * q^20 + 8 * q^22 - 15 * q^23 + 17 * q^25 + 15 * q^26 + 24 * q^28 - q^29 + 18 * q^31 - 25 * q^32 + 14 * q^34 - 12 * q^35 + 12 * q^37 - 22 * q^38 + 15 * q^40 + 7 * q^41 + 28 * q^43 + 14 * q^44 + 4 * q^46 - 26 * q^47 + 41 * q^49 - 5 * q^50 + 10 * q^52 - 12 * q^53 + 2 * q^55 - 13 * q^56 + 16 * q^58 + 23 * q^59 + 26 * q^61 - 10 * q^62 + 59 * q^64 - 8 * q^65 + 31 * q^67 + q^68 + 4 * q^70 + 2 * q^71 + 33 * q^73 + 10 * q^74 + 66 * q^76 - 12 * q^77 + 33 * q^79 - 33 * q^80 + 30 * q^82 - 13 * q^83 + 10 * q^85 + 20 * q^86 + 12 * q^88 + 17 * q^89 + 40 * q^91 - 16 * q^92 + 38 * q^94 - 32 * q^95 + 45 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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