LMFDB - Embedded newform 4005.2.a.q.1.3

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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 6x^{7} + 31x^{6} + 13x^{5} - 75x^{4} - 17x^{3} + 52x^{2} + 11x - 1 $ x^9 - 4x^8 - 6x^7 + 31x^6 + 13x^5 - 75x^4 - 17x^3 + 52x^2 + 11x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1335)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.02867$ 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)

$f(q)$	$=$	$q-2.02867 q^{2} +2.11549 q^{4} +1.00000 q^{5} +2.64395 q^{7} -0.234298 q^{8} +O(q^{10})$	\(q-2.02867 q^{2} +2.11549 q^{4} +1.00000 q^{5} +2.64395 q^{7} -0.234298 q^{8} -2.02867 q^{10} -3.03195 q^{11} -2.43364 q^{13} -5.36370 q^{14} -3.75567 q^{16} -2.27321 q^{17} +8.01214 q^{19} +2.11549 q^{20} +6.15082 q^{22} -4.63872 q^{23} +1.00000 q^{25} +4.93704 q^{26} +5.59326 q^{28} +1.32647 q^{29} -4.79688 q^{31} +8.08761 q^{32} +4.61159 q^{34} +2.64395 q^{35} +1.97359 q^{37} -16.2540 q^{38} -0.234298 q^{40} -4.67852 q^{41} +10.5022 q^{43} -6.41407 q^{44} +9.41043 q^{46} -10.0075 q^{47} -0.00952776 q^{49} -2.02867 q^{50} -5.14834 q^{52} -4.56148 q^{53} -3.03195 q^{55} -0.619474 q^{56} -2.69096 q^{58} -4.45963 q^{59} -10.6617 q^{61} +9.73128 q^{62} -8.89573 q^{64} -2.43364 q^{65} +2.03361 q^{67} -4.80896 q^{68} -5.36370 q^{70} +1.11126 q^{71} -11.9538 q^{73} -4.00375 q^{74} +16.9496 q^{76} -8.01633 q^{77} -10.2654 q^{79} -3.75567 q^{80} +9.49117 q^{82} +2.69588 q^{83} -2.27321 q^{85} -21.3055 q^{86} +0.710382 q^{88} -1.00000 q^{89} -6.43442 q^{91} -9.81319 q^{92} +20.3020 q^{94} +8.01214 q^{95} -5.21230 q^{97} +0.0193287 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} + 11 q^{4} + 9 q^{5} - 3 q^{7} - 18 q^{8}+O(q^{10}) $ 9 * q - 5 * q^2 + 11 * q^4 + 9 * q^5 - 3 * q^7 - 18 * q^8	$ 9 q - 5 q^{2} + 11 q^{4} + 9 q^{5} - 3 q^{7} - 18 q^{8} - 5 q^{10} - 4 q^{11} + 5 q^{13} + q^{14} + 15 q^{16} - 21 q^{17} - 18 q^{19} + 11 q^{20} + 6 q^{22} - 16 q^{23} + 9 q^{25} - 8 q^{26} + 6 q^{28} - 3 q^{29} - 6 q^{31} - 46 q^{32} + 12 q^{34} - 3 q^{35} + 11 q^{37} - 20 q^{38} - 18 q^{40} + q^{41} - 3 q^{43} - 38 q^{44} + 16 q^{46} - 27 q^{47} + 24 q^{49} - 5 q^{50} + 17 q^{52} - 43 q^{53} - 4 q^{55} - 5 q^{56} + 34 q^{58} - 3 q^{59} - 30 q^{61} - 36 q^{62} + 50 q^{64} + 5 q^{65} - 12 q^{67} - 64 q^{68} + q^{70} + 4 q^{71} + 26 q^{73} - 2 q^{74} - 12 q^{76} - 34 q^{77} + q^{79} + 15 q^{80} - 51 q^{82} - 24 q^{83} - 21 q^{85} - 18 q^{86} + 64 q^{88} - 9 q^{89} - 50 q^{91} - 10 q^{92} - 11 q^{94} - 18 q^{95} - 4 q^{97} - 75 q^{98}+O(q^{100}) $ 9 * q - 5 * q^2 + 11 * q^4 + 9 * q^5 - 3 * q^7 - 18 * q^8 - 5 * q^10 - 4 * q^11 + 5 * q^13 + q^14 + 15 * q^16 - 21 * q^17 - 18 * q^19 + 11 * q^20 + 6 * q^22 - 16 * q^23 + 9 * q^25 - 8 * q^26 + 6 * q^28 - 3 * q^29 - 6 * q^31 - 46 * q^32 + 12 * q^34 - 3 * q^35 + 11 * q^37 - 20 * q^38 - 18 * q^40 + q^41 - 3 * q^43 - 38 * q^44 + 16 * q^46 - 27 * q^47 + 24 * q^49 - 5 * q^50 + 17 * q^52 - 43 * q^53 - 4 * q^55 - 5 * q^56 + 34 * q^58 - 3 * q^59 - 30 * q^61 - 36 * q^62 + 50 * q^64 + 5 * q^65 - 12 * q^67 - 64 * q^68 + q^70 + 4 * q^71 + 26 * q^73 - 2 * q^74 - 12 * q^76 - 34 * q^77 + q^79 + 15 * q^80 - 51 * q^82 - 24 * q^83 - 21 * q^85 - 18 * q^86 + 64 * q^88 - 9 * q^89 - 50 * q^91 - 10 * q^92 - 11 * q^94 - 18 * q^95 - 4 * q^97 - 75 * 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 $

$2$	−2.02867	−1.43448	−0.717242	−	0.696824i	$-0.754596\pi$
$2$	−2.02867	−1.43448	−0.717242	+	0.696824i	$0.754596\pi$
$3$	0	0
$3$	0	0
$4$	2.11549	1.05775
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.64395	0.999319	0.499660	−	0.866222i	$-0.333458\pi$
$7$	2.64395	0.999319	0.499660	+	0.866222i	$0.333458\pi$
$8$	−0.234298	−0.0828370
$9$	0	0
$10$	−2.02867	−0.641521
$11$	−3.03195	−0.914168	−0.457084	−	0.889424i	$-0.651106\pi$
$11$	−3.03195	−0.914168	−0.457084	+	0.889424i	$0.651106\pi$
$12$	0	0
$13$	−2.43364	−0.674969	−0.337485	−	0.941331i	$-0.609576\pi$
$13$	−2.43364	−0.674969	−0.337485	+	0.941331i	$0.609576\pi$
$14$	−5.36370	−1.43351
$15$	0	0
$16$	−3.75567	−0.938918
$17$	−2.27321	−0.551334	−0.275667	−	0.961253i	$-0.588899\pi$
$17$	−2.27321	−0.551334	−0.275667	+	0.961253i	$0.588899\pi$
$18$	0	0
$19$	8.01214	1.83811	0.919056	−	0.394128i	$-0.128953\pi$
$19$	8.01214	1.83811	0.919056	+	0.394128i	$0.128953\pi$
$20$	2.11549	0.473039
$21$	0	0
$22$	6.15082	1.31136
$23$	−4.63872	−0.967241	−0.483620	−	0.875278i	$-0.660678\pi$
$23$	−4.63872	−0.967241	−0.483620	+	0.875278i	$0.660678\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.93704	0.968234
$27$	0	0
$28$	5.59326	1.05703
$29$	1.32647	0.246319	0.123159	−	0.992387i	$-0.460697\pi$
$29$	1.32647	0.246319	0.123159	+	0.992387i	$0.460697\pi$
$30$	0	0
$31$	−4.79688	−0.861546	−0.430773	−	0.902460i	$-0.641759\pi$
$31$	−4.79688	−0.861546	−0.430773	+	0.902460i	$0.641759\pi$
$32$	8.08761	1.42970
$33$	0	0
$34$	4.61159	0.790881
$35$	2.64395	0.446909
$36$	0	0
$37$	1.97359	0.324456	0.162228	−	0.986753i	$-0.448132\pi$
$37$	1.97359	0.324456	0.162228	+	0.986753i	$0.448132\pi$
$38$	−16.2540	−2.63674
$39$	0	0
$40$	−0.234298	−0.0370458
$41$	−4.67852	−0.730663	−0.365331	−	0.930878i	$-0.619044\pi$
$41$	−4.67852	−0.730663	−0.365331	+	0.930878i	$0.619044\pi$
$42$	0	0
$43$	10.5022	1.60157	0.800786	−	0.598951i	$-0.204416\pi$
$43$	10.5022	1.60157	0.800786	+	0.598951i	$0.204416\pi$
$44$	−6.41407	−0.966958
$45$	0	0
$46$	9.41043	1.38749
$47$	−10.0075	−1.45975	−0.729875	−	0.683581i	$-0.760422\pi$
$47$	−10.0075	−1.45975	−0.729875	+	0.683581i	$0.760422\pi$
$48$	0	0
$49$	−0.00952776	−0.00136111
$50$	−2.02867	−0.286897
$51$	0	0
$52$	−5.14834	−0.713947
$53$	−4.56148	−0.626567	−0.313283	−	0.949660i	$-0.601429\pi$
$53$	−4.56148	−0.626567	−0.313283	+	0.949660i	$0.601429\pi$
$54$	0	0
$55$	−3.03195	−0.408828
$56$	−0.619474	−0.0827806
$57$	0	0
$58$	−2.69096	−0.353340
$59$	−4.45963	−0.580595	−0.290297	−	0.956936i	$-0.593754\pi$
$59$	−4.45963	−0.580595	−0.290297	+	0.956936i	$0.593754\pi$
$60$	0	0
$61$	−10.6617	−1.36510	−0.682548	−	0.730841i	$-0.739128\pi$
$61$	−10.6617	−1.36510	−0.682548	+	0.730841i	$0.739128\pi$
$62$	9.73128	1.23587
$63$	0	0
$64$	−8.89573	−1.11197
$65$	−2.43364	−0.301856
$66$	0	0
$67$	2.03361	0.248445	0.124222	−	0.992254i	$-0.460356\pi$
$67$	2.03361	0.248445	0.124222	+	0.992254i	$0.460356\pi$
$68$	−4.80896	−0.583172
$69$	0	0
$70$	−5.36370	−0.641084
$71$	1.11126	0.131882	0.0659411	−	0.997824i	$-0.478995\pi$
$71$	1.11126	0.131882	0.0659411	+	0.997824i	$0.478995\pi$
$72$	0	0
$73$	−11.9538	−1.39909	−0.699546	−	0.714588i	$-0.746614\pi$
$73$	−11.9538	−1.39909	−0.699546	+	0.714588i	$0.746614\pi$
$74$	−4.00375	−0.465427
$75$	0	0
$76$	16.9496	1.94426
$77$	−8.01633	−0.913545
$78$	0	0
$79$	−10.2654	−1.15494	−0.577472	−	0.816411i	$-0.695961\pi$
$79$	−10.2654	−1.15494	−0.577472	+	0.816411i	$0.695961\pi$
$80$	−3.75567	−0.419897
$81$	0	0
$82$	9.49117	1.04812
$83$	2.69588	0.295911	0.147956	−	0.988994i	$-0.452731\pi$
$83$	2.69588	0.295911	0.147956	+	0.988994i	$0.452731\pi$
$84$	0	0
$85$	−2.27321	−0.246564
$86$	−21.3055	−2.29743
$87$	0	0
$88$	0.710382	0.0757269
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−6.43442	−0.674510
$92$	−9.81319	−1.02310
$93$	0	0
$94$	20.3020	2.09399
$95$	8.01214	0.822028
$96$	0	0
$97$	−5.21230	−0.529229	−0.264614	−	0.964354i	$-0.585245\pi$
$97$	−5.21230	−0.529229	−0.264614	+	0.964354i	$0.585245\pi$
$98$	0.0193287	0.00195249
$99$	0	0
$100$	2.11549	0.211549
$101$	18.1763	1.80861	0.904303	−	0.426890i	$-0.140391\pi$
$101$	18.1763	1.80861	0.904303	+	0.426890i	$0.140391\pi$
$102$	0	0
$103$	−7.70073	−0.758775	−0.379388	−	0.925238i	$-0.623865\pi$
$103$	−7.70073	−0.758775	−0.379388	+	0.925238i	$0.623865\pi$
$104$	0.570198	0.0559125
$105$	0	0
$106$	9.25372	0.898801
$107$	4.33629	0.419205	0.209602	−	0.977787i	$-0.432783\pi$
$107$	4.33629	0.419205	0.209602	+	0.977787i	$0.432783\pi$
$108$	0	0
$109$	12.6688	1.21346	0.606728	−	0.794910i	$-0.292482\pi$
$109$	12.6688	1.21346	0.606728	+	0.794910i	$0.292482\pi$
$110$	6.15082	0.586458
$111$	0	0
$112$	−9.92981	−0.938279
$113$	−17.7668	−1.67136	−0.835679	−	0.549218i	$-0.814926\pi$
$113$	−17.7668	−1.67136	−0.835679	+	0.549218i	$0.814926\pi$
$114$	0	0
$115$	−4.63872	−0.432563
$116$	2.80613	0.260543
$117$	0	0
$118$	9.04712	0.832855
$119$	−6.01025	−0.550959
$120$	0	0
$121$	−1.80727	−0.164297
$122$	21.6291	1.95821
$123$	0	0
$124$	−10.1478	−0.911297
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.2796	−1.00090	−0.500451	−	0.865765i	$-0.666832\pi$
$127$	−11.2796	−1.00090	−0.500451	+	0.865765i	$0.666832\pi$
$128$	1.87126	0.165398
$129$	0	0
$130$	4.93704	0.433007
$131$	3.55553	0.310648	0.155324	−	0.987864i	$-0.450358\pi$
$131$	3.55553	0.310648	0.155324	+	0.987864i	$0.450358\pi$
$132$	0	0
$133$	21.1837	1.83686
$134$	−4.12552	−0.356391
$135$	0	0
$136$	0.532610	0.0456709
$137$	4.36059	0.372550	0.186275	−	0.982498i	$-0.440358\pi$
$137$	4.36059	0.372550	0.186275	+	0.982498i	$0.440358\pi$
$138$	0	0
$139$	13.5096	1.14587	0.572934	−	0.819602i	$-0.305805\pi$
$139$	13.5096	1.14587	0.572934	+	0.819602i	$0.305805\pi$
$140$	5.59326	0.472717
$141$	0	0
$142$	−2.25437	−0.189183
$143$	7.37867	0.617035
$144$	0	0
$145$	1.32647	0.110157
$146$	24.2504	2.00698
$147$	0	0
$148$	4.17511	0.343192
$149$	4.84043	0.396544	0.198272	−	0.980147i	$-0.436467\pi$
$149$	4.84043	0.396544	0.198272	+	0.980147i	$0.436467\pi$
$150$	0	0
$151$	15.4785	1.25962	0.629812	−	0.776747i	$-0.283132\pi$
$151$	15.4785	1.25962	0.629812	+	0.776747i	$0.283132\pi$
$152$	−1.87723	−0.152264
$153$	0	0
$154$	16.2625	1.31047
$155$	−4.79688	−0.385295
$156$	0	0
$157$	−15.5762	−1.24312	−0.621558	−	0.783368i	$-0.713500\pi$
$157$	−15.5762	−1.24312	−0.621558	+	0.783368i	$0.713500\pi$
$158$	20.8250	1.65675
$159$	0	0
$160$	8.08761	0.639382
$161$	−12.2646	−0.966582
$162$	0	0
$163$	−17.2159	−1.34845	−0.674227	−	0.738524i	$-0.735523\pi$
$163$	−17.2159	−1.34845	−0.674227	+	0.738524i	$0.735523\pi$
$164$	−9.89739	−0.772856
$165$	0	0
$166$	−5.46905	−0.424480
$167$	−3.98859	−0.308646	−0.154323	−	0.988020i	$-0.549320\pi$
$167$	−3.98859	−0.308646	−0.154323	+	0.988020i	$0.549320\pi$
$168$	0	0
$169$	−7.07741	−0.544416
$170$	4.61159	0.353693
$171$	0	0
$172$	22.2174	1.69406
$173$	19.2592	1.46425	0.732126	−	0.681169i	$-0.238528\pi$
$173$	19.2592	1.46425	0.732126	+	0.681169i	$0.238528\pi$
$174$	0	0
$175$	2.64395	0.199864
$176$	11.3870	0.858329
$177$	0	0
$178$	2.02867	0.152055
$179$	13.5370	1.01180	0.505900	−	0.862592i	$-0.331161\pi$
$179$	13.5370	1.01180	0.505900	+	0.862592i	$0.331161\pi$
$180$	0	0
$181$	−15.5357	−1.15476	−0.577378	−	0.816477i	$-0.695924\pi$
$181$	−15.5357	−1.15476	−0.577378	+	0.816477i	$0.695924\pi$
$182$	13.0533	0.967574
$183$	0	0
$184$	1.08685	0.0801233
$185$	1.97359	0.145101
$186$	0	0
$187$	6.89226	0.504012
$188$	−21.1709	−1.54405
$189$	0	0
$190$	−16.2540	−1.17919
$191$	9.20856	0.666308	0.333154	−	0.942872i	$-0.391887\pi$
$191$	9.20856	0.666308	0.333154	+	0.942872i	$0.391887\pi$
$192$	0	0
$193$	19.7727	1.42327	0.711635	−	0.702549i	$-0.247955\pi$
$193$	19.7727	1.42327	0.711635	+	0.702549i	$0.247955\pi$
$194$	10.5740	0.759171
$195$	0	0
$196$	−0.0201559	−0.00143971
$197$	1.71571	0.122239	0.0611197	−	0.998130i	$-0.480533\pi$
$197$	1.71571	0.122239	0.0611197	+	0.998130i	$0.480533\pi$
$198$	0	0
$199$	4.00724	0.284066	0.142033	−	0.989862i	$-0.454636\pi$
$199$	4.00724	0.284066	0.142033	+	0.989862i	$0.454636\pi$
$200$	−0.234298	−0.0165674
$201$	0	0
$202$	−36.8736	−2.59442
$203$	3.50711	0.246151
$204$	0	0
$205$	−4.67852	−0.326762
$206$	15.6222	1.08845
$207$	0	0
$208$	9.13995	0.633741
$209$	−24.2924	−1.68034
$210$	0	0
$211$	−17.8069	−1.22588	−0.612940	−	0.790129i	$-0.710013\pi$
$211$	−17.8069	−1.22588	−0.612940	+	0.790129i	$0.710013\pi$
$212$	−9.64977	−0.662749
$213$	0	0
$214$	−8.79689	−0.601343
$215$	10.5022	0.716244
$216$	0	0
$217$	−12.6827	−0.860959
$218$	−25.7009	−1.74068
$219$	0	0
$220$	−6.41407	−0.432437
$221$	5.53217	0.372134
$222$	0	0
$223$	3.03763	0.203415	0.101707	−	0.994814i	$-0.467569\pi$
$223$	3.03763	0.203415	0.101707	+	0.994814i	$0.467569\pi$
$224$	21.3832	1.42873
$225$	0	0
$226$	36.0429	2.39754
$227$	−25.7386	−1.70833	−0.854167	−	0.519999i	$-0.825932\pi$
$227$	−25.7386	−1.70833	−0.854167	+	0.519999i	$0.825932\pi$
$228$	0	0
$229$	−21.7913	−1.44001	−0.720003	−	0.693971i	$-0.755860\pi$
$229$	−21.7913	−1.44001	−0.720003	+	0.693971i	$0.755860\pi$
$230$	9.41043	0.620505
$231$	0	0
$232$	−0.310789	−0.0204043
$233$	−14.9312	−0.978175	−0.489087	−	0.872235i	$-0.662670\pi$
$233$	−14.9312	−0.978175	−0.489087	+	0.872235i	$0.662670\pi$
$234$	0	0
$235$	−10.0075	−0.652820
$236$	−9.43433	−0.614122
$237$	0	0
$238$	12.1928	0.790342
$239$	−2.96011	−0.191474	−0.0957368	−	0.995407i	$-0.530521\pi$
$239$	−2.96011	−0.191474	−0.0957368	+	0.995407i	$0.530521\pi$
$240$	0	0
$241$	19.5590	1.25990	0.629952	−	0.776634i	$-0.283074\pi$
$241$	19.5590	1.25990	0.629952	+	0.776634i	$0.283074\pi$
$242$	3.66635	0.235682
$243$	0	0
$244$	−22.5549	−1.44393
$245$	−0.00952776	−0.000608706 0
$246$	0	0
$247$	−19.4986	−1.24067
$248$	1.12390	0.0713679
$249$	0	0
$250$	−2.02867	−0.128304
$251$	15.2363	0.961709	0.480854	−	0.876800i	$-0.340327\pi$
$251$	15.2363	0.961709	0.480854	+	0.876800i	$0.340327\pi$
$252$	0	0
$253$	14.0644	0.884220
$254$	22.8826	1.43578
$255$	0	0
$256$	13.9953	0.874706
$257$	−24.4137	−1.52288	−0.761442	−	0.648233i	$-0.775508\pi$
$257$	−24.4137	−1.52288	−0.761442	+	0.648233i	$0.775508\pi$
$258$	0	0
$259$	5.21806	0.324235
$260$	−5.14834	−0.319287
$261$	0	0
$262$	−7.21299	−0.445620
$263$	−14.3972	−0.887768	−0.443884	−	0.896084i	$-0.646400\pi$
$263$	−14.3972	−0.887768	−0.443884	+	0.896084i	$0.646400\pi$
$264$	0	0
$265$	−4.56148	−0.280209
$266$	−42.9747	−2.63495
$267$	0	0
$268$	4.30209	0.262792
$269$	13.2901	0.810312	0.405156	−	0.914248i	$-0.367217\pi$
$269$	13.2901	0.810312	0.405156	+	0.914248i	$0.367217\pi$
$270$	0	0
$271$	−1.32540	−0.0805123	−0.0402561	−	0.999189i	$-0.512817\pi$
$271$	−1.32540	−0.0805123	−0.0402561	+	0.999189i	$0.512817\pi$
$272$	8.53744	0.517658
$273$	0	0
$274$	−8.84618	−0.534417
$275$	−3.03195	−0.182834
$276$	0	0
$277$	18.0179	1.08259	0.541294	−	0.840834i	$-0.317935\pi$
$277$	18.0179	1.08259	0.541294	+	0.840834i	$0.317935\pi$
$278$	−27.4065	−1.64373
$279$	0	0
$280$	−0.619474	−0.0370206
$281$	−6.84516	−0.408348	−0.204174	−	0.978935i	$-0.565451\pi$
$281$	−6.84516	−0.408348	−0.204174	+	0.978935i	$0.565451\pi$
$282$	0	0
$283$	−30.4752	−1.81156	−0.905781	−	0.423745i	$-0.860715\pi$
$283$	−30.4752	−1.81156	−0.905781	+	0.423745i	$0.860715\pi$
$284$	2.35086	0.139498
$285$	0	0
$286$	−14.9689	−0.885128
$287$	−12.3698	−0.730165
$288$	0	0
$289$	−11.8325	−0.696030
$290$	−2.69096	−0.158019
$291$	0	0
$292$	−25.2883	−1.47988
$293$	4.89203	0.285796	0.142898	−	0.989737i	$-0.454358\pi$
$293$	4.89203	0.285796	0.142898	+	0.989737i	$0.454358\pi$
$294$	0	0
$295$	−4.45963	−0.259650
$296$	−0.462408	−0.0268769
$297$	0	0
$298$	−9.81963	−0.568836
$299$	11.2890	0.652858
$300$	0	0
$301$	27.7673	1.60048
$302$	−31.4008	−1.80691
$303$	0	0
$304$	−30.0910	−1.72584
$305$	−10.6617	−0.610490
$306$	0	0
$307$	−25.7913	−1.47199	−0.735995	−	0.676987i	$-0.763285\pi$
$307$	−25.7913	−1.47199	−0.735995	+	0.676987i	$0.763285\pi$
$308$	−16.9585	−0.966300
$309$	0	0
$310$	9.73128	0.552700
$311$	14.4372	0.818659	0.409329	−	0.912387i	$-0.365763\pi$
$311$	14.4372	0.818659	0.409329	+	0.912387i	$0.365763\pi$
$312$	0	0
$313$	−3.80586	−0.215120	−0.107560	−	0.994199i	$-0.534304\pi$
$313$	−3.80586	−0.215120	−0.107560	+	0.994199i	$0.534304\pi$
$314$	31.5990	1.78323
$315$	0	0
$316$	−21.7163	−1.22164
$317$	−11.3284	−0.636268	−0.318134	−	0.948046i	$-0.603056\pi$
$317$	−11.3284	−0.636268	−0.318134	+	0.948046i	$0.603056\pi$
$318$	0	0
$319$	−4.02178	−0.225177
$320$	−8.89573	−0.497287
$321$	0	0
$322$	24.8807	1.38655
$323$	−18.2133	−1.01341
$324$	0	0
$325$	−2.43364	−0.134994
$326$	34.9254	1.93434
$327$	0	0
$328$	1.09617	0.0605259
$329$	−26.4594	−1.45876
$330$	0	0
$331$	16.6934	0.917550	0.458775	−	0.888553i	$-0.348288\pi$
$331$	16.6934	0.917550	0.458775	+	0.888553i	$0.348288\pi$
$332$	5.70312	0.312999
$333$	0	0
$334$	8.09152	0.442748
$335$	2.03361	0.111108
$336$	0	0
$337$	19.7102	1.07368	0.536842	−	0.843683i	$-0.319617\pi$
$337$	19.7102	1.07368	0.536842	+	0.843683i	$0.319617\pi$
$338$	14.3577	0.780957
$339$	0	0
$340$	−4.80896	−0.260803
$341$	14.5439	0.787597
$342$	0	0
$343$	−18.5328	−1.00068
$344$	−2.46065	−0.132669
$345$	0	0
$346$	−39.0706	−2.10045
$347$	−8.14542	−0.437269	−0.218634	−	0.975807i	$-0.570160\pi$
$347$	−8.14542	−0.437269	−0.218634	+	0.975807i	$0.570160\pi$
$348$	0	0
$349$	6.96805	0.372991	0.186496	−	0.982456i	$-0.440287\pi$
$349$	6.96805	0.372991	0.186496	+	0.982456i	$0.440287\pi$
$350$	−5.36370	−0.286702
$351$	0	0
$352$	−24.5212	−1.30699
$353$	0.389588	0.0207357	0.0103678	−	0.999946i	$-0.496700\pi$
$353$	0.389588	0.0207357	0.0103678	+	0.999946i	$0.496700\pi$
$354$	0	0
$355$	1.11126	0.0589795
$356$	−2.11549	−0.112121
$357$	0	0
$358$	−27.4620	−1.45141
$359$	13.5494	0.715110	0.357555	−	0.933892i	$-0.383611\pi$
$359$	13.5494	0.715110	0.357555	+	0.933892i	$0.383611\pi$
$360$	0	0
$361$	45.1944	2.37865
$362$	31.5167	1.65648
$363$	0	0
$364$	−13.6120	−0.713461
$365$	−11.9538	−0.625693
$366$	0	0
$367$	7.25694	0.378809	0.189405	−	0.981899i	$-0.439344\pi$
$367$	7.25694	0.378809	0.189405	+	0.981899i	$0.439344\pi$
$368$	17.4215	0.908160
$369$	0	0
$370$	−4.00375	−0.208145
$371$	−12.0603	−0.626140
$372$	0	0
$373$	12.2420	0.633868	0.316934	−	0.948448i	$-0.397347\pi$
$373$	12.2420	0.633868	0.316934	+	0.948448i	$0.397347\pi$
$374$	−13.9821	−0.722998
$375$	0	0
$376$	2.34475	0.120921
$377$	−3.22814	−0.166258
$378$	0	0
$379$	−2.21482	−0.113768	−0.0568838	−	0.998381i	$-0.518116\pi$
$379$	−2.21482	−0.113768	−0.0568838	+	0.998381i	$0.518116\pi$
$380$	16.9496	0.869498
$381$	0	0
$382$	−18.6811	−0.955808
$383$	−37.9381	−1.93855	−0.969274	−	0.245985i	$-0.920889\pi$
$383$	−37.9381	−1.93855	−0.969274	+	0.245985i	$0.920889\pi$
$384$	0	0
$385$	−8.01633	−0.408550
$386$	−40.1123	−2.04166
$387$	0	0
$388$	−11.0266	−0.559790
$389$	−10.0836	−0.511261	−0.255630	−	0.966775i	$-0.582283\pi$
$389$	−10.0836	−0.511261	−0.255630	+	0.966775i	$0.582283\pi$
$390$	0	0
$391$	10.5448	0.533273
$392$	0.00223234	0.000112750 0
$393$	0	0
$394$	−3.48061	−0.175351
$395$	−10.2654	−0.516507
$396$	0	0
$397$	−27.2880	−1.36955	−0.684773	−	0.728756i	$-0.740099\pi$
$397$	−27.2880	−1.36955	−0.684773	+	0.728756i	$0.740099\pi$
$398$	−8.12936	−0.407488
$399$	0	0
$400$	−3.75567	−0.187784
$401$	−17.5451	−0.876162	−0.438081	−	0.898935i	$-0.644342\pi$
$401$	−17.5451	−0.876162	−0.438081	+	0.898935i	$0.644342\pi$
$402$	0	0
$403$	11.6739	0.581517
$404$	38.4518	1.91305
$405$	0	0
$406$	−7.11476	−0.353100
$407$	−5.98382	−0.296607
$408$	0	0
$409$	−14.5068	−0.717317	−0.358659	−	0.933469i	$-0.616766\pi$
$409$	−14.5068	−0.717317	−0.358659	+	0.933469i	$0.616766\pi$
$410$	9.49117	0.468735
$411$	0	0
$412$	−16.2908	−0.802592
$413$	−11.7910	−0.580200
$414$	0	0
$415$	2.69588	0.132336
$416$	−19.6823	−0.965005
$417$	0	0
$418$	49.2813	2.41043
$419$	−7.33565	−0.358370	−0.179185	−	0.983815i	$-0.557346\pi$
$419$	−7.33565	−0.358370	−0.179185	+	0.983815i	$0.557346\pi$
$420$	0	0
$421$	5.03251	0.245269	0.122635	−	0.992452i	$-0.460866\pi$
$421$	5.03251	0.245269	0.122635	+	0.992452i	$0.460866\pi$
$422$	36.1244	1.75851
$423$	0	0
$424$	1.06875	0.0519029
$425$	−2.27321	−0.110267
$426$	0	0
$427$	−28.1891	−1.36417
$428$	9.17339	0.443413
$429$	0	0
$430$	−21.3055	−1.02744
$431$	36.1710	1.74230	0.871148	−	0.491020i	$-0.163376\pi$
$431$	36.1710	1.74230	0.871148	+	0.491020i	$0.163376\pi$
$432$	0	0
$433$	−27.5801	−1.32542	−0.662708	−	0.748878i	$-0.730593\pi$
$433$	−27.5801	−1.32542	−0.662708	+	0.748878i	$0.730593\pi$
$434$	25.7290	1.23503
$435$	0	0
$436$	26.8009	1.28353
$437$	−37.1661	−1.77790
$438$	0	0
$439$	−17.4183	−0.831331	−0.415666	−	0.909517i	$-0.636451\pi$
$439$	−17.4183	−0.831331	−0.415666	+	0.909517i	$0.636451\pi$
$440$	0.710382	0.0338661
$441$	0	0
$442$	−11.2229	−0.533820
$443$	28.5122	1.35466	0.677328	−	0.735681i	$-0.263138\pi$
$443$	28.5122	1.35466	0.677328	+	0.735681i	$0.263138\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−6.16234	−0.291795
$447$	0	0
$448$	−23.5199	−1.11121
$449$	−25.4082	−1.19909	−0.599543	−	0.800342i	$-0.704651\pi$
$449$	−25.4082	−1.19909	−0.599543	+	0.800342i	$0.704651\pi$
$450$	0	0
$451$	14.1851	0.667948
$452$	−37.5855	−1.76787
$453$	0	0
$454$	52.2152	2.45058
$455$	−6.43442	−0.301650
$456$	0	0
$457$	15.9336	0.745343	0.372672	−	0.927963i	$-0.378442\pi$
$457$	15.9336	0.745343	0.372672	+	0.927963i	$0.378442\pi$
$458$	44.2072	2.06567
$459$	0	0
$460$	−9.81319	−0.457542
$461$	12.7626	0.594415	0.297207	−	0.954813i	$-0.403945\pi$
$461$	12.7626	0.594415	0.297207	+	0.954813i	$0.403945\pi$
$462$	0	0
$463$	42.2109	1.96171	0.980854	−	0.194746i	$-0.0623883\pi$
$463$	42.2109	1.96171	0.980854	+	0.194746i	$0.0623883\pi$
$464$	−4.98178	−0.231273
$465$	0	0
$466$	30.2904	1.40318
$467$	3.17680	0.147005	0.0735023	−	0.997295i	$-0.476582\pi$
$467$	3.17680	0.147005	0.0735023	+	0.997295i	$0.476582\pi$
$468$	0	0
$469$	5.37676	0.248276
$470$	20.3020	0.936460
$471$	0	0
$472$	1.04489	0.0480948
$473$	−31.8422	−1.46410
$474$	0	0
$475$	8.01214	0.367622
$476$	−12.7147	−0.582775
$477$	0	0
$478$	6.00508	0.274666
$479$	−23.8482	−1.08965	−0.544825	−	0.838549i	$-0.683404\pi$
$479$	−23.8482	−1.08965	−0.544825	+	0.838549i	$0.683404\pi$
$480$	0	0
$481$	−4.80299	−0.218998
$482$	−39.6787	−1.80731
$483$	0	0
$484$	−3.82327	−0.173785
$485$	−5.21230	−0.236678
$486$	0	0
$487$	−3.31844	−0.150373	−0.0751864	−	0.997169i	$-0.523955\pi$
$487$	−3.31844	−0.150373	−0.0751864	+	0.997169i	$0.523955\pi$
$488$	2.49803	0.113081
$489$	0	0
$490$	0.0193287	0.000873180 0
$491$	3.06667	0.138397	0.0691985	−	0.997603i	$-0.477956\pi$
$491$	3.06667	0.138397	0.0691985	+	0.997603i	$0.477956\pi$
$492$	0	0
$493$	−3.01534	−0.135804
$494$	39.5563	1.77972
$495$	0	0
$496$	18.0155	0.808921
$497$	2.93811	0.131792
$498$	0	0
$499$	8.21917	0.367940	0.183970	−	0.982932i	$-0.441105\pi$
$499$	8.21917	0.367940	0.183970	+	0.982932i	$0.441105\pi$
$500$	2.11549	0.0946078
$501$	0	0
$502$	−30.9095	−1.37956
$503$	12.2586	0.546585	0.273292	−	0.961931i	$-0.411887\pi$
$503$	12.2586	0.546585	0.273292	+	0.961931i	$0.411887\pi$
$504$	0	0
$505$	18.1763	0.808833
$506$	−28.5320	−1.26840
$507$	0	0
$508$	−23.8619	−1.05870
$509$	−25.5598	−1.13292	−0.566460	−	0.824089i	$-0.691687\pi$
$509$	−25.5598	−1.13292	−0.566460	+	0.824089i	$0.691687\pi$
$510$	0	0
$511$	−31.6054	−1.39814
$512$	−32.1343	−1.42015
$513$	0	0
$514$	49.5272	2.18455
$515$	−7.70073	−0.339335
$516$	0	0
$517$	30.3424	1.33446
$518$	−10.5857	−0.465110
$519$	0	0
$520$	0.570198	0.0250048
$521$	12.1498	0.532294	0.266147	−	0.963932i	$-0.414249\pi$
$521$	12.1498	0.532294	0.266147	+	0.963932i	$0.414249\pi$
$522$	0	0
$523$	−6.74027	−0.294731	−0.147366	−	0.989082i	$-0.547079\pi$
$523$	−6.74027	−0.294731	−0.147366	+	0.989082i	$0.547079\pi$
$524$	7.52170	0.328587
$525$	0	0
$526$	29.2071	1.27349
$527$	10.9043	0.475000
$528$	0	0
$529$	−1.48225	−0.0644457
$530$	9.25372	0.401956
$531$	0	0
$532$	44.8140	1.94293
$533$	11.3858	0.493175
$534$	0	0
$535$	4.33629	0.187474
$536$	−0.476472	−0.0205804
$537$	0	0
$538$	−26.9612	−1.16238
$539$	0.0288877	0.00124428
$540$	0	0
$541$	−13.8066	−0.593591	−0.296796	−	0.954941i	$-0.595918\pi$
$541$	−13.8066	−0.593591	−0.296796	+	0.954941i	$0.595918\pi$
$542$	2.68879	0.115494
$543$	0	0
$544$	−18.3848	−0.788244
$545$	12.6688	0.542674
$546$	0	0
$547$	−13.1120	−0.560628	−0.280314	−	0.959908i	$-0.590439\pi$
$547$	−13.1120	−0.560628	−0.280314	+	0.959908i	$0.590439\pi$
$548$	9.22479	0.394064
$549$	0	0
$550$	6.15082	0.262272
$551$	10.6278	0.452761
$552$	0	0
$553$	−27.1411	−1.15416
$554$	−36.5522	−1.55296
$555$	0	0
$556$	28.5794	1.21204
$557$	4.30529	0.182421	0.0912104	−	0.995832i	$-0.470926\pi$
$557$	4.30529	0.182421	0.0912104	+	0.995832i	$0.470926\pi$
$558$	0	0
$559$	−25.5586	−1.08101
$560$	−9.92981	−0.419611
$561$	0	0
$562$	13.8866	0.585769
$563$	−24.7354	−1.04247	−0.521236	−	0.853413i	$-0.674529\pi$
$563$	−24.7354	−1.04247	−0.521236	+	0.853413i	$0.674529\pi$
$564$	0	0
$565$	−17.7668	−0.747454
$566$	61.8241	2.59866
$567$	0	0
$568$	−0.260366	−0.0109247
$569$	−8.34045	−0.349650	−0.174825	−	0.984600i	$-0.555936\pi$
$569$	−8.34045	−0.349650	−0.174825	+	0.984600i	$0.555936\pi$
$570$	0	0
$571$	−33.8630	−1.41712	−0.708562	−	0.705648i	$-0.750656\pi$
$571$	−33.8630	−1.41712	−0.708562	+	0.705648i	$0.750656\pi$
$572$	15.6095	0.652667
$573$	0	0
$574$	25.0942	1.04741
$575$	−4.63872	−0.193448
$576$	0	0
$577$	−33.4783	−1.39372	−0.696861	−	0.717206i	$-0.745420\pi$
$577$	−33.4783	−1.39372	−0.696861	+	0.717206i	$0.745420\pi$
$578$	24.0042	0.998445
$579$	0	0
$580$	2.80613	0.116518
$581$	7.12777	0.295710
$582$	0	0
$583$	13.8302	0.572787
$584$	2.80077	0.115897
$585$	0	0
$586$	−9.92431	−0.409970
$587$	−44.9264	−1.85431	−0.927155	−	0.374677i	$-0.877754\pi$
$587$	−44.9264	−1.85431	−0.927155	+	0.374677i	$0.877754\pi$
$588$	0	0
$589$	−38.4333	−1.58362
$590$	9.04712	0.372464
$591$	0	0
$592$	−7.41215	−0.304637
$593$	−38.5751	−1.58409	−0.792044	−	0.610463i	$-0.790983\pi$
$593$	−38.5751	−1.58409	−0.792044	+	0.610463i	$0.790983\pi$
$594$	0	0
$595$	−6.01025	−0.246396
$596$	10.2399	0.419443
$597$	0	0
$598$	−22.9016	−0.936515
$599$	−5.92336	−0.242022	−0.121011	−	0.992651i	$-0.538614\pi$
$599$	−5.92336	−0.242022	−0.121011	+	0.992651i	$0.538614\pi$
$600$	0	0
$601$	36.8725	1.50406	0.752031	−	0.659128i	$-0.229074\pi$
$601$	36.8725	1.50406	0.752031	+	0.659128i	$0.229074\pi$
$602$	−56.3306	−2.29587
$603$	0	0
$604$	32.7447	1.33236
$605$	−1.80727	−0.0734760
$606$	0	0
$607$	−1.90908	−0.0774871	−0.0387436	−	0.999249i	$-0.512336\pi$
$607$	−1.90908	−0.0774871	−0.0387436	+	0.999249i	$0.512336\pi$
$608$	64.7991	2.62795
$609$	0	0
$610$	21.6291	0.875738
$611$	24.3547	0.985286
$612$	0	0
$613$	−4.23402	−0.171011	−0.0855053	−	0.996338i	$-0.527250\pi$
$613$	−4.23402	−0.171011	−0.0855053	+	0.996338i	$0.527250\pi$
$614$	52.3221	2.11155
$615$	0	0
$616$	1.87821	0.0756754
$617$	−10.5992	−0.426709	−0.213354	−	0.976975i	$-0.568439\pi$
$617$	−10.5992	−0.426709	−0.213354	+	0.976975i	$0.568439\pi$
$618$	0	0
$619$	−37.4228	−1.50415	−0.752075	−	0.659077i	$-0.770947\pi$
$619$	−37.4228	−1.50415	−0.752075	+	0.659077i	$0.770947\pi$
$620$	−10.1478	−0.407544
$621$	0	0
$622$	−29.2883	−1.17435
$623$	−2.64395	−0.105928
$624$	0	0
$625$	1.00000	0.0400000
$626$	7.72082	0.308586
$627$	0	0
$628$	−32.9514	−1.31490
$629$	−4.48638	−0.178884
$630$	0	0
$631$	−28.3622	−1.12908	−0.564540	−	0.825405i	$-0.690947\pi$
$631$	−28.3622	−1.12908	−0.564540	+	0.825405i	$0.690947\pi$
$632$	2.40516	0.0956721
$633$	0	0
$634$	22.9816	0.912716
$635$	−11.2796	−0.447617
$636$	0	0
$637$	0.0231871	0.000918707 0
$638$	8.15886	0.323012
$639$	0	0
$640$	1.87126	0.0739682
$641$	5.18435	0.204769	0.102385	−	0.994745i	$-0.467353\pi$
$641$	5.18435	0.204769	0.102385	+	0.994745i	$0.467353\pi$
$642$	0	0
$643$	9.74254	0.384208	0.192104	−	0.981375i	$-0.438469\pi$
$643$	9.74254	0.384208	0.192104	+	0.981375i	$0.438469\pi$
$644$	−25.9456	−1.02240
$645$	0	0
$646$	36.9487	1.45373
$647$	−40.4894	−1.59180	−0.795901	−	0.605427i	$-0.793002\pi$
$647$	−40.4894	−1.59180	−0.795901	+	0.605427i	$0.793002\pi$
$648$	0	0
$649$	13.5214	0.530761
$650$	4.93704	0.193647
$651$	0	0
$652$	−36.4202	−1.42632
$653$	−3.43301	−0.134344	−0.0671720	−	0.997741i	$-0.521398\pi$
$653$	−3.43301	−0.134344	−0.0671720	+	0.997741i	$0.521398\pi$
$654$	0	0
$655$	3.55553	0.138926
$656$	17.5710	0.686033
$657$	0	0
$658$	53.6774	2.09256
$659$	−24.3967	−0.950362	−0.475181	−	0.879888i	$-0.657617\pi$
$659$	−24.3967	−0.950362	−0.475181	+	0.879888i	$0.657617\pi$
$660$	0	0
$661$	−7.04139	−0.273878	−0.136939	−	0.990579i	$-0.543726\pi$
$661$	−7.04139	−0.273878	−0.136939	+	0.990579i	$0.543726\pi$
$662$	−33.8653	−1.31621
$663$	0	0
$664$	−0.631641	−0.0245124
$665$	21.1837	0.821469
$666$	0	0
$667$	−6.15311	−0.238249
$668$	−8.43783	−0.326470
$669$	0	0
$670$	−4.12552	−0.159383
$671$	32.3259	1.24793
$672$	0	0
$673$	14.2100	0.547754	0.273877	−	0.961765i	$-0.411694\pi$
$673$	14.2100	0.547754	0.273877	+	0.961765i	$0.411694\pi$
$674$	−39.9855	−1.54018
$675$	0	0
$676$	−14.9722	−0.575855
$677$	−2.62262	−0.100796	−0.0503978	−	0.998729i	$-0.516049\pi$
$677$	−2.62262	−0.100796	−0.0503978	+	0.998729i	$0.516049\pi$
$678$	0	0
$679$	−13.7811	−0.528869
$680$	0.532610	0.0204246
$681$	0	0
$682$	−29.5048	−1.12980
$683$	31.2569	1.19601	0.598005	−	0.801492i	$-0.295960\pi$
$683$	31.2569	1.19601	0.598005	+	0.801492i	$0.295960\pi$
$684$	0	0
$685$	4.36059	0.166609
$686$	37.5970	1.43546
$687$	0	0
$688$	−39.4429	−1.50374
$689$	11.1010	0.422914
$690$	0	0
$691$	15.4563	0.587986	0.293993	−	0.955808i	$-0.405016\pi$
$691$	15.4563	0.587986	0.293993	+	0.955808i	$0.405016\pi$
$692$	40.7428	1.54881
$693$	0	0
$694$	16.5243	0.627256
$695$	13.5096	0.512448
$696$	0	0
$697$	10.6353	0.402839
$698$	−14.1359	−0.535050
$699$	0	0
$700$	5.59326	0.211405
$701$	34.0584	1.28637	0.643185	−	0.765711i	$-0.277613\pi$
$701$	34.0584	1.28637	0.643185	+	0.765711i	$0.277613\pi$
$702$	0	0
$703$	15.8127	0.596386
$704$	26.9714	1.01652
$705$	0	0
$706$	−0.790345	−0.0297450
$707$	48.0572	1.80738
$708$	0	0
$709$	−41.7478	−1.56787	−0.783935	−	0.620843i	$-0.786790\pi$
$709$	−41.7478	−1.56787	−0.783935	+	0.620843i	$0.786790\pi$
$710$	−2.25437	−0.0846052
$711$	0	0
$712$	0.234298	0.00878071
$713$	22.2514	0.833322
$714$	0	0
$715$	7.37867	0.275947
$716$	28.6373	1.07023
$717$	0	0
$718$	−27.4872	−1.02581
$719$	38.5951	1.43936	0.719678	−	0.694308i	$-0.244290\pi$
$719$	38.5951	1.43936	0.719678	+	0.694308i	$0.244290\pi$
$720$	0	0
$721$	−20.3603	−0.758259
$722$	−91.6845	−3.41214
$723$	0	0
$724$	−32.8656	−1.22144
$725$	1.32647	0.0492637
$726$	0	0
$727$	−30.7526	−1.14055	−0.570275	−	0.821454i	$-0.693163\pi$
$727$	−30.7526	−1.14055	−0.570275	+	0.821454i	$0.693163\pi$
$728$	1.50757	0.0558744
$729$	0	0
$730$	24.2504	0.897547
$731$	−23.8737	−0.883001
$732$	0	0
$733$	−9.51936	−0.351605	−0.175803	−	0.984425i	$-0.556252\pi$
$733$	−9.51936	−0.351605	−0.175803	+	0.984425i	$0.556252\pi$
$734$	−14.7219	−0.543396
$735$	0	0
$736$	−37.5162	−1.38287
$737$	−6.16580	−0.227120
$738$	0	0
$739$	−4.35120	−0.160061	−0.0800307	−	0.996792i	$-0.525502\pi$
$739$	−4.35120	−0.160061	−0.0800307	+	0.996792i	$0.525502\pi$
$740$	4.17511	0.153480
$741$	0	0
$742$	24.4664	0.898189
$743$	12.0177	0.440887	0.220444	−	0.975400i	$-0.429250\pi$
$743$	12.0177	0.440887	0.220444	+	0.975400i	$0.429250\pi$
$744$	0	0
$745$	4.84043	0.177340
$746$	−24.8350	−0.909274
$747$	0	0
$748$	14.5805	0.533117
$749$	11.4649	0.418920
$750$	0	0
$751$	−6.75035	−0.246324	−0.123162	−	0.992387i	$-0.539303\pi$
$751$	−6.75035	−0.246324	−0.123162	+	0.992387i	$0.539303\pi$
$752$	37.5850	1.37059
$753$	0	0
$754$	6.54882	0.238494
$755$	15.4785	0.563321
$756$	0	0
$757$	45.1106	1.63957	0.819786	−	0.572670i	$-0.194092\pi$
$757$	45.1106	1.63957	0.819786	+	0.572670i	$0.194092\pi$
$758$	4.49313	0.163198
$759$	0	0
$760$	−1.87723	−0.0680944
$761$	38.5108	1.39602	0.698008	−	0.716090i	$-0.254070\pi$
$761$	38.5108	1.39602	0.698008	+	0.716090i	$0.254070\pi$
$762$	0	0
$763$	33.4958	1.21263
$764$	19.4806	0.704785
$765$	0	0
$766$	76.9638	2.78082
$767$	10.8531	0.391884
$768$	0	0
$769$	30.6316	1.10460	0.552301	−	0.833645i	$-0.313750\pi$
$769$	30.6316	1.10460	0.552301	+	0.833645i	$0.313750\pi$
$770$	16.2625	0.586059
$771$	0	0
$772$	41.8290	1.50546
$773$	38.5610	1.38694	0.693471	−	0.720485i	$-0.256081\pi$
$773$	38.5610	1.38694	0.693471	+	0.720485i	$0.256081\pi$
$774$	0	0
$775$	−4.79688	−0.172309
$776$	1.22123	0.0438397
$777$	0	0
$778$	20.4563	0.733396
$779$	−37.4850	−1.34304
$780$	0	0
$781$	−3.36928	−0.120562
$782$	−21.3919	−0.764972
$783$	0	0
$784$	0.0357832	0.00127797
$785$	−15.5762	−0.555939
$786$	0	0
$787$	−47.2535	−1.68440	−0.842202	−	0.539162i	$-0.818741\pi$
$787$	−47.2535	−1.68440	−0.842202	+	0.539162i	$0.818741\pi$
$788$	3.62958	0.129298
$789$	0	0
$790$	20.8250	0.740921
$791$	−46.9745	−1.67022
$792$	0	0
$793$	25.9468	0.921398
$794$	55.3583	1.96459
$795$	0	0
$796$	8.47729	0.300470
$797$	−4.88138	−0.172907	−0.0864537	−	0.996256i	$-0.527553\pi$
$797$	−4.88138	−0.172907	−0.0864537	+	0.996256i	$0.527553\pi$
$798$	0	0
$799$	22.7492	0.804810
$800$	8.08761	0.285940
$801$	0	0
$802$	35.5933	1.25684
$803$	36.2435	1.27900
$804$	0	0
$805$	−12.2646	−0.432269
$806$	−23.6824	−0.834177
$807$	0	0
$808$	−4.25867	−0.149820
$809$	45.1918	1.58886	0.794430	−	0.607356i	$-0.207770\pi$
$809$	45.1918	1.58886	0.794430	+	0.607356i	$0.207770\pi$
$810$	0	0
$811$	−27.0269	−0.949043	−0.474522	−	0.880244i	$-0.657379\pi$
$811$	−27.0269	−0.949043	−0.474522	+	0.880244i	$0.657379\pi$
$812$	7.41927	0.260365
$813$	0	0
$814$	12.1392	0.425478
$815$	−17.2159	−0.603047
$816$	0	0
$817$	84.1452	2.94387
$818$	29.4296	1.02898
$819$	0	0
$820$	−9.89739	−0.345632
$821$	−40.3268	−1.40741	−0.703707	−	0.710490i	$-0.748473\pi$
$821$	−40.3268	−1.40741	−0.703707	+	0.710490i	$0.748473\pi$
$822$	0	0
$823$	45.1779	1.57480	0.787402	−	0.616440i	$-0.211426\pi$
$823$	45.1779	1.57480	0.787402	+	0.616440i	$0.211426\pi$
$824$	1.80427	0.0628547
$825$	0	0
$826$	23.9201	0.832288
$827$	49.5807	1.72409	0.862045	−	0.506831i	$-0.169183\pi$
$827$	49.5807	1.72409	0.862045	+	0.506831i	$0.169183\pi$
$828$	0	0
$829$	41.3349	1.43562	0.717810	−	0.696239i	$-0.245144\pi$
$829$	41.3349	1.43562	0.717810	+	0.696239i	$0.245144\pi$
$830$	−5.46905	−0.189833
$831$	0	0
$832$	21.6490	0.750543
$833$	0.0216586	0.000750426 0
$834$	0	0
$835$	−3.98859	−0.138031
$836$	−51.3905	−1.77738
$837$	0	0
$838$	14.8816	0.514076
$839$	−42.2843	−1.45982	−0.729908	−	0.683546i	$-0.760437\pi$
$839$	−42.2843	−1.45982	−0.729908	+	0.683546i	$0.760437\pi$
$840$	0	0
$841$	−27.2405	−0.939327
$842$	−10.2093	−0.351835
$843$	0	0
$844$	−37.6705	−1.29667
$845$	−7.07741	−0.243470
$846$	0	0
$847$	−4.77834	−0.164186
$848$	17.1314	0.588295
$849$	0	0
$850$	4.61159	0.158176
$851$	−9.15492	−0.313827
$852$	0	0
$853$	24.0869	0.824721	0.412361	−	0.911021i	$-0.364704\pi$
$853$	24.0869	0.824721	0.412361	+	0.911021i	$0.364704\pi$
$854$	57.1864	1.95688
$855$	0	0
$856$	−1.01599	−0.0347257
$857$	−34.8656	−1.19099	−0.595494	−	0.803360i	$-0.703043\pi$
$857$	−34.8656	−1.19099	−0.595494	+	0.803360i	$0.703043\pi$
$858$	0	0
$859$	−12.0809	−0.412194	−0.206097	−	0.978532i	$-0.566076\pi$
$859$	−12.0809	−0.412194	−0.206097	+	0.978532i	$0.566076\pi$
$860$	22.2174	0.757605
$861$	0	0
$862$	−73.3790	−2.49930
$863$	−17.9213	−0.610049	−0.305024	−	0.952345i	$-0.598665\pi$
$863$	−17.9213	−0.610049	−0.305024	+	0.952345i	$0.598665\pi$
$864$	0	0
$865$	19.2592	0.654833
$866$	55.9509	1.90129
$867$	0	0
$868$	−26.8302	−0.910677
$869$	31.1241	1.05581
$870$	0	0
$871$	−4.94907	−0.167693
$872$	−2.96829	−0.100519
$873$	0	0
$874$	75.3977	2.55036
$875$	2.64395	0.0893818
$876$	0	0
$877$	39.0430	1.31839	0.659194	−	0.751973i	$-0.270897\pi$
$877$	39.0430	1.31839	0.659194	+	0.751973i	$0.270897\pi$
$878$	35.3360	1.19253
$879$	0	0
$880$	11.3870	0.383856
$881$	41.3307	1.39247	0.696233	−	0.717816i	$-0.254858\pi$
$881$	41.3307	1.39247	0.696233	+	0.717816i	$0.254858\pi$
$882$	0	0
$883$	−9.68535	−0.325938	−0.162969	−	0.986631i	$-0.552107\pi$
$883$	−9.68535	−0.325938	−0.162969	+	0.986631i	$0.552107\pi$
$884$	11.7033	0.393623
$885$	0	0
$886$	−57.8418	−1.94323
$887$	−5.19674	−0.174490	−0.0872448	−	0.996187i	$-0.527806\pi$
$887$	−5.19674	−0.174490	−0.0872448	+	0.996187i	$0.527806\pi$
$888$	0	0
$889$	−29.8227	−1.00022
$890$	2.02867	0.0680011
$891$	0	0
$892$	6.42609	0.215161
$893$	−80.1818	−2.68318
$894$	0	0
$895$	13.5370	0.452491
$896$	4.94753	0.165285
$897$	0	0
$898$	51.5448	1.72007
$899$	−6.36290	−0.212215
$900$	0	0
$901$	10.3692	0.345448
$902$	−28.7768	−0.958161
$903$	0	0
$904$	4.16273	0.138450
$905$	−15.5357	−0.516423
$906$	0	0
$907$	−4.34639	−0.144319	−0.0721597	−	0.997393i	$-0.522989\pi$
$907$	−4.34639	−0.144319	−0.0721597	+	0.997393i	$0.522989\pi$
$908$	−54.4499	−1.80698
$909$	0	0
$910$	13.0533	0.432712
$911$	17.6597	0.585090	0.292545	−	0.956252i	$-0.405498\pi$
$911$	17.6597	0.585090	0.292545	+	0.956252i	$0.405498\pi$
$912$	0	0
$913$	−8.17378	−0.270513
$914$	−32.3240	−1.06918
$915$	0	0
$916$	−46.0993	−1.52316
$917$	9.40064	0.310436
$918$	0	0
$919$	−18.6608	−0.615561	−0.307781	−	0.951457i	$-0.599586\pi$
$919$	−18.6608	−0.615561	−0.307781	+	0.951457i	$0.599586\pi$
$920$	1.08685	0.0358322
$921$	0	0
$922$	−25.8911	−0.852679
$923$	−2.70440	−0.0890164
$924$	0	0
$925$	1.97359	0.0648911
$926$	−85.6319	−2.81404
$927$	0	0
$928$	10.7279	0.352162
$929$	28.5953	0.938183	0.469091	−	0.883150i	$-0.344582\pi$
$929$	28.5953	0.938183	0.469091	+	0.883150i	$0.344582\pi$
$930$	0	0
$931$	−0.0763378	−0.00250187
$932$	−31.5868	−1.03466
$933$	0	0
$934$	−6.44466	−0.210876
$935$	6.89226	0.225401
$936$	0	0
$937$	54.2502	1.77228	0.886139	−	0.463420i	$-0.153378\pi$
$937$	54.2502	1.77228	0.886139	+	0.463420i	$0.153378\pi$
$938$	−10.9077	−0.356148
$939$	0	0
$940$	−21.1709	−0.690518
$941$	14.9583	0.487627	0.243814	−	0.969822i	$-0.421601\pi$
$941$	14.9583	0.487627	0.243814	+	0.969822i	$0.421601\pi$
$942$	0	0
$943$	21.7024	0.706726
$944$	16.7489	0.545131
$945$	0	0
$946$	64.5972	2.10024
$947$	−22.6756	−0.736856	−0.368428	−	0.929656i	$-0.620104\pi$
$947$	−22.6756	−0.736856	−0.368428	+	0.929656i	$0.620104\pi$
$948$	0	0
$949$	29.0913	0.944344
$950$	−16.2540	−0.527349
$951$	0	0
$952$	1.40819	0.0456398
$953$	29.9830	0.971243	0.485622	−	0.874169i	$-0.338593\pi$
$953$	29.9830	0.971243	0.485622	+	0.874169i	$0.338593\pi$
$954$	0	0
$955$	9.20856	0.297982
$956$	−6.26209	−0.202531
$957$	0	0
$958$	48.3800	1.56309
$959$	11.5292	0.372296
$960$	0	0
$961$	−7.98992	−0.257739
$962$	9.74368	0.314149
$963$	0	0
$964$	41.3769	1.33266
$965$	19.7727	0.636506
$966$	0	0
$967$	3.39303	0.109113	0.0545563	−	0.998511i	$-0.482626\pi$
$967$	3.39303	0.109113	0.0545563	+	0.998511i	$0.482626\pi$
$968$	0.423441	0.0136099
$969$	0	0
$970$	10.5740	0.339511
$971$	−22.8520	−0.733355	−0.366678	−	0.930348i	$-0.619505\pi$
$971$	−22.8520	−0.733355	−0.366678	+	0.930348i	$0.619505\pi$
$972$	0	0
$973$	35.7187	1.14509
$974$	6.73201	0.215707
$975$	0	0
$976$	40.0420	1.28171
$977$	−1.64896	−0.0527548	−0.0263774	−	0.999652i	$-0.508397\pi$
$977$	−1.64896	−0.0527548	−0.0263774	+	0.999652i	$0.508397\pi$
$978$	0	0
$979$	3.03195	0.0969016
$980$	−0.0201559	−0.000643857 0
$981$	0	0
$982$	−6.22126	−0.198528
$983$	20.0858	0.640638	0.320319	−	0.947310i	$-0.396210\pi$
$983$	20.0858	0.640638	0.320319	+	0.947310i	$0.396210\pi$
$984$	0	0
$985$	1.71571	0.0546671
$986$	6.11712	0.194809
$987$	0	0
$988$	−41.2493	−1.31231
$989$	−48.7168	−1.54910
$990$	0	0
$991$	18.1400	0.576236	0.288118	−	0.957595i	$-0.406970\pi$
$991$	18.1400	0.576236	0.288118	+	0.957595i	$0.406970\pi$
$992$	−38.7953	−1.23175
$993$	0	0
$994$	−5.96045	−0.189054
$995$	4.00724	0.127038
$996$	0	0
$997$	11.4118	0.361416	0.180708	−	0.983537i	$-0.442161\pi$
$997$	11.4118	0.361416	0.180708	+	0.983537i	$0.442161\pi$
$998$	−16.6740	−0.527805
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.q.1.3		9
3.2	odd	2		1335.2.a.h.1.7	✓	9
15.14	odd	2		6675.2.a.x.1.3		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.h.1.7	✓	9	3.2	odd	2
4005.2.a.q.1.3		9	1.1	even	1	trivial
6675.2.a.x.1.3		9	15.14	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)

\(f(q)\)	\(=\)	\(q-2.02867 q^{2} +2.11549 q^{4} +1.00000 q^{5} +2.64395 q^{7} -0.234298 q^{8} +O(q^{10})\)	\(q-2.02867 q^{2} +2.11549 q^{4} +1.00000 q^{5} +2.64395 q^{7} -0.234298 q^{8} -2.02867 q^{10} -3.03195 q^{11} -2.43364 q^{13} -5.36370 q^{14} -3.75567 q^{16} -2.27321 q^{17} +8.01214 q^{19} +2.11549 q^{20} +6.15082 q^{22} -4.63872 q^{23} +1.00000 q^{25} +4.93704 q^{26} +5.59326 q^{28} +1.32647 q^{29} -4.79688 q^{31} +8.08761 q^{32} +4.61159 q^{34} +2.64395 q^{35} +1.97359 q^{37} -16.2540 q^{38} -0.234298 q^{40} -4.67852 q^{41} +10.5022 q^{43} -6.41407 q^{44} +9.41043 q^{46} -10.0075 q^{47} -0.00952776 q^{49} -2.02867 q^{50} -5.14834 q^{52} -4.56148 q^{53} -3.03195 q^{55} -0.619474 q^{56} -2.69096 q^{58} -4.45963 q^{59} -10.6617 q^{61} +9.73128 q^{62} -8.89573 q^{64} -2.43364 q^{65} +2.03361 q^{67} -4.80896 q^{68} -5.36370 q^{70} +1.11126 q^{71} -11.9538 q^{73} -4.00375 q^{74} +16.9496 q^{76} -8.01633 q^{77} -10.2654 q^{79} -3.75567 q^{80} +9.49117 q^{82} +2.69588 q^{83} -2.27321 q^{85} -21.3055 q^{86} +0.710382 q^{88} -1.00000 q^{89} -6.43442 q^{91} -9.81319 q^{92} +20.3020 q^{94} +8.01214 q^{95} -5.21230 q^{97} +0.0193287 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} + 11 q^{4} + 9 q^{5} - 3 q^{7} - 18 q^{8}+O(q^{10}) \) 9 * q - 5 * q^2 + 11 * q^4 + 9 * q^5 - 3 * q^7 - 18 * q^8	\( 9 q - 5 q^{2} + 11 q^{4} + 9 q^{5} - 3 q^{7} - 18 q^{8} - 5 q^{10} - 4 q^{11} + 5 q^{13} + q^{14} + 15 q^{16} - 21 q^{17} - 18 q^{19} + 11 q^{20} + 6 q^{22} - 16 q^{23} + 9 q^{25} - 8 q^{26} + 6 q^{28} - 3 q^{29} - 6 q^{31} - 46 q^{32} + 12 q^{34} - 3 q^{35} + 11 q^{37} - 20 q^{38} - 18 q^{40} + q^{41} - 3 q^{43} - 38 q^{44} + 16 q^{46} - 27 q^{47} + 24 q^{49} - 5 q^{50} + 17 q^{52} - 43 q^{53} - 4 q^{55} - 5 q^{56} + 34 q^{58} - 3 q^{59} - 30 q^{61} - 36 q^{62} + 50 q^{64} + 5 q^{65} - 12 q^{67} - 64 q^{68} + q^{70} + 4 q^{71} + 26 q^{73} - 2 q^{74} - 12 q^{76} - 34 q^{77} + q^{79} + 15 q^{80} - 51 q^{82} - 24 q^{83} - 21 q^{85} - 18 q^{86} + 64 q^{88} - 9 q^{89} - 50 q^{91} - 10 q^{92} - 11 q^{94} - 18 q^{95} - 4 q^{97} - 75 q^{98}+O(q^{100}) \) 9 * q - 5 * q^2 + 11 * q^4 + 9 * q^5 - 3 * q^7 - 18 * q^8 - 5 * q^10 - 4 * q^11 + 5 * q^13 + q^14 + 15 * q^16 - 21 * q^17 - 18 * q^19 + 11 * q^20 + 6 * q^22 - 16 * q^23 + 9 * q^25 - 8 * q^26 + 6 * q^28 - 3 * q^29 - 6 * q^31 - 46 * q^32 + 12 * q^34 - 3 * q^35 + 11 * q^37 - 20 * q^38 - 18 * q^40 + q^41 - 3 * q^43 - 38 * q^44 + 16 * q^46 - 27 * q^47 + 24 * q^49 - 5 * q^50 + 17 * q^52 - 43 * q^53 - 4 * q^55 - 5 * q^56 + 34 * q^58 - 3 * q^59 - 30 * q^61 - 36 * q^62 + 50 * q^64 + 5 * q^65 - 12 * q^67 - 64 * q^68 + q^70 + 4 * q^71 + 26 * q^73 - 2 * q^74 - 12 * q^76 - 34 * q^77 + q^79 + 15 * q^80 - 51 * q^82 - 24 * q^83 - 21 * q^85 - 18 * q^86 + 64 * q^88 - 9 * q^89 - 50 * q^91 - 10 * q^92 - 11 * q^94 - 18 * q^95 - 4 * q^97 - 75 * q^98

Introduction

L-functions

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Database

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