LMFDB - Embedded newform 4005.2.a.s.1.10

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 16x^{8} + 15x^{7} + 85x^{6} - 75x^{5} - 163x^{4} + 138x^{3} + 78x^{2} - 67x + 4 $ x^10 - x^9 - 16x^8 + 15x^7 + 85x^6 - 75x^5 - 163x^4 + 138x^3 + 78x^2 - 67x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.10
Root			$-2.51771$ 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)

$f(q)$	$=$	$q+2.51771 q^{2} +4.33887 q^{4} -1.00000 q^{5} -2.77781 q^{7} +5.88859 q^{8} +O(q^{10})$	\(q+2.51771 q^{2} +4.33887 q^{4} -1.00000 q^{5} -2.77781 q^{7} +5.88859 q^{8} -2.51771 q^{10} +2.85609 q^{11} -6.80048 q^{13} -6.99372 q^{14} +6.14803 q^{16} -5.31202 q^{17} -5.88409 q^{19} -4.33887 q^{20} +7.19082 q^{22} +5.46553 q^{23} +1.00000 q^{25} -17.1216 q^{26} -12.0525 q^{28} -9.76331 q^{29} +0.0306833 q^{31} +3.70178 q^{32} -13.3741 q^{34} +2.77781 q^{35} -1.74430 q^{37} -14.8144 q^{38} -5.88859 q^{40} -4.43445 q^{41} +10.6108 q^{43} +12.3922 q^{44} +13.7606 q^{46} +9.83555 q^{47} +0.716229 q^{49} +2.51771 q^{50} -29.5064 q^{52} -3.57570 q^{53} -2.85609 q^{55} -16.3574 q^{56} -24.5812 q^{58} -5.04048 q^{59} -0.612487 q^{61} +0.0772518 q^{62} -2.97604 q^{64} +6.80048 q^{65} -12.3743 q^{67} -23.0481 q^{68} +6.99372 q^{70} +12.2609 q^{71} -3.42742 q^{73} -4.39164 q^{74} -25.5303 q^{76} -7.93368 q^{77} +12.9489 q^{79} -6.14803 q^{80} -11.1647 q^{82} -8.84466 q^{83} +5.31202 q^{85} +26.7148 q^{86} +16.8184 q^{88} +1.00000 q^{89} +18.8904 q^{91} +23.7142 q^{92} +24.7631 q^{94} +5.88409 q^{95} +13.9782 q^{97} +1.80326 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7}+O(q^{10}) $ 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7	$ 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7} + q^{10} - 10 q^{11} + 5 q^{13} - 13 q^{14} + 19 q^{16} - 9 q^{17} - 6 q^{19} - 13 q^{20} + 3 q^{23} + 10 q^{25} - 14 q^{26} - 18 q^{28} - 38 q^{29} + 2 q^{31} - 16 q^{32} - 8 q^{34} + q^{35} + 9 q^{37} - 20 q^{38} - 36 q^{41} - 7 q^{43} - 16 q^{44} + 2 q^{46} + 23 q^{47} + 25 q^{49} - q^{50} + 13 q^{52} - 27 q^{53} + 10 q^{55} - 41 q^{56} - 32 q^{58} - 20 q^{59} + 30 q^{61} + 2 q^{62} - 2 q^{64} - 5 q^{65} - 5 q^{67} + 10 q^{68} + 13 q^{70} - 24 q^{71} - 19 q^{73} - 42 q^{74} - 30 q^{76} - 18 q^{77} + 12 q^{79} - 19 q^{80} + 29 q^{82} - 3 q^{83} + 9 q^{85} - 38 q^{86} - 16 q^{88} + 10 q^{89} - 6 q^{91} - 32 q^{92} + 17 q^{94} + 6 q^{95} + 3 q^{97} + 37 q^{98}+O(q^{100}) $ 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7 + q^10 - 10 * q^11 + 5 * q^13 - 13 * q^14 + 19 * q^16 - 9 * q^17 - 6 * q^19 - 13 * q^20 + 3 * q^23 + 10 * q^25 - 14 * q^26 - 18 * q^28 - 38 * q^29 + 2 * q^31 - 16 * q^32 - 8 * q^34 + q^35 + 9 * q^37 - 20 * q^38 - 36 * q^41 - 7 * q^43 - 16 * q^44 + 2 * q^46 + 23 * q^47 + 25 * q^49 - q^50 + 13 * q^52 - 27 * q^53 + 10 * q^55 - 41 * q^56 - 32 * q^58 - 20 * q^59 + 30 * q^61 + 2 * q^62 - 2 * q^64 - 5 * q^65 - 5 * q^67 + 10 * q^68 + 13 * q^70 - 24 * q^71 - 19 * q^73 - 42 * q^74 - 30 * q^76 - 18 * q^77 + 12 * q^79 - 19 * q^80 + 29 * q^82 - 3 * q^83 + 9 * q^85 - 38 * q^86 - 16 * q^88 + 10 * q^89 - 6 * q^91 - 32 * q^92 + 17 * q^94 + 6 * q^95 + 3 * q^97 + 37 * 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 $

$2$	2.51771	1.78029	0.890145	−	0.455677i	$-0.150603\pi$
$2$	2.51771	1.78029	0.890145	+	0.455677i	$0.150603\pi$
$3$	0	0
$3$	0	0
$4$	4.33887	2.16943
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.77781	−1.04991	−0.524957	−	0.851129i	$-0.675919\pi$
$7$	−2.77781	−1.04991	−0.524957	+	0.851129i	$0.675919\pi$
$8$	5.88859	2.08193
$9$	0	0
$10$	−2.51771	−0.796170
$11$	2.85609	0.861144	0.430572	−	0.902556i	$-0.358312\pi$
$11$	2.85609	0.861144	0.430572	+	0.902556i	$0.358312\pi$
$12$	0	0
$13$	−6.80048	−1.88611	−0.943057	−	0.332632i	$-0.892063\pi$
$13$	−6.80048	−1.88611	−0.943057	+	0.332632i	$0.892063\pi$
$14$	−6.99372	−1.86915
$15$	0	0
$16$	6.14803	1.53701
$17$	−5.31202	−1.28835	−0.644177	−	0.764877i	$-0.722800\pi$
$17$	−5.31202	−1.28835	−0.644177	+	0.764877i	$0.722800\pi$
$18$	0	0
$19$	−5.88409	−1.34990	−0.674951	−	0.737862i	$-0.735835\pi$
$19$	−5.88409	−1.34990	−0.674951	+	0.737862i	$0.735835\pi$
$20$	−4.33887	−0.970200
$21$	0	0
$22$	7.19082	1.53309
$23$	5.46553	1.13964	0.569821	−	0.821769i	$-0.307013\pi$
$23$	5.46553	1.13964	0.569821	+	0.821769i	$0.307013\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−17.1216	−3.35783
$27$	0	0
$28$	−12.0525	−2.27772
$29$	−9.76331	−1.81300	−0.906500	−	0.422205i	$-0.861256\pi$
$29$	−9.76331	−1.81300	−0.906500	+	0.422205i	$0.861256\pi$
$30$	0	0
$31$	0.0306833	0.00551089	0.00275545	−	0.999996i	$-0.499123\pi$
$31$	0.0306833	0.00551089	0.00275545	+	0.999996i	$0.499123\pi$
$32$	3.70178	0.654389
$33$	0	0
$34$	−13.3741	−2.29364
$35$	2.77781	0.469536
$36$	0	0
$37$	−1.74430	−0.286761	−0.143380	−	0.989668i	$-0.545797\pi$
$37$	−1.74430	−0.286761	−0.143380	+	0.989668i	$0.545797\pi$
$38$	−14.8144	−2.40322
$39$	0	0
$40$	−5.88859	−0.931068
$41$	−4.43445	−0.692545	−0.346273	−	0.938134i	$-0.612553\pi$
$41$	−4.43445	−0.692545	−0.346273	+	0.938134i	$0.612553\pi$
$42$	0	0
$43$	10.6108	1.61813	0.809063	−	0.587721i	$-0.199975\pi$
$43$	10.6108	1.61813	0.809063	+	0.587721i	$0.199975\pi$
$44$	12.3922	1.86820
$45$	0	0
$46$	13.7606	2.02889
$47$	9.83555	1.43466	0.717332	−	0.696732i	$-0.245363\pi$
$47$	9.83555	1.43466	0.717332	+	0.696732i	$0.245363\pi$
$48$	0	0
$49$	0.716229	0.102318
$50$	2.51771	0.356058
$51$	0	0
$52$	−29.5064	−4.09180
$53$	−3.57570	−0.491160	−0.245580	−	0.969376i	$-0.578978\pi$
$53$	−3.57570	−0.491160	−0.245580	+	0.969376i	$0.578978\pi$
$54$	0	0
$55$	−2.85609	−0.385115
$56$	−16.3574	−2.18585
$57$	0	0
$58$	−24.5812	−3.22767
$59$	−5.04048	−0.656214	−0.328107	−	0.944640i	$-0.606411\pi$
$59$	−5.04048	−0.656214	−0.328107	+	0.944640i	$0.606411\pi$
$60$	0	0
$61$	−0.612487	−0.0784209	−0.0392104	−	0.999231i	$-0.512484\pi$
$61$	−0.612487	−0.0784209	−0.0392104	+	0.999231i	$0.512484\pi$
$62$	0.0772518	0.00981098
$63$	0	0
$64$	−2.97604	−0.372005
$65$	6.80048	0.843496
$66$	0	0
$67$	−12.3743	−1.51176	−0.755881	−	0.654709i	$-0.772791\pi$
$67$	−12.3743	−1.51176	−0.755881	+	0.654709i	$0.772791\pi$
$68$	−23.0481	−2.79500
$69$	0	0
$70$	6.99372	0.835910
$71$	12.2609	1.45511	0.727553	−	0.686051i	$-0.240657\pi$
$71$	12.2609	1.45511	0.727553	+	0.686051i	$0.240657\pi$
$72$	0	0
$73$	−3.42742	−0.401149	−0.200574	−	0.979678i	$-0.564281\pi$
$73$	−3.42742	−0.401149	−0.200574	+	0.979678i	$0.564281\pi$
$74$	−4.39164	−0.510518
$75$	0	0
$76$	−25.5303	−2.92852
$77$	−7.93368	−0.904127
$78$	0	0
$79$	12.9489	1.45686	0.728432	−	0.685118i	$-0.240249\pi$
$79$	12.9489	1.45686	0.728432	+	0.685118i	$0.240249\pi$
$80$	−6.14803	−0.687371
$81$	0	0
$82$	−11.1647	−1.23293
$83$	−8.84466	−0.970828	−0.485414	−	0.874284i	$-0.661331\pi$
$83$	−8.84466	−0.970828	−0.485414	+	0.874284i	$0.661331\pi$
$84$	0	0
$85$	5.31202	0.576169
$86$	26.7148	2.88074
$87$	0	0
$88$	16.8184	1.79284
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	18.8904	1.98026
$92$	23.7142	2.47238
$93$	0	0
$94$	24.7631	2.55412
$95$	5.88409	0.603695
$96$	0	0
$97$	13.9782	1.41927	0.709634	−	0.704571i	$-0.248860\pi$
$97$	13.9782	1.41927	0.709634	+	0.704571i	$0.248860\pi$
$98$	1.80326	0.182157
$99$	0	0
$100$	4.33887	0.433887
$101$	−12.6730	−1.26101	−0.630506	−	0.776184i	$-0.717153\pi$
$101$	−12.6730	−1.26101	−0.630506	+	0.776184i	$0.717153\pi$
$102$	0	0
$103$	18.1078	1.78422	0.892109	−	0.451821i	$-0.149225\pi$
$103$	18.1078	1.78422	0.892109	+	0.451821i	$0.149225\pi$
$104$	−40.0452	−3.92676
$105$	0	0
$106$	−9.00258	−0.874408
$107$	−1.28620	−0.124342	−0.0621710	−	0.998066i	$-0.519802\pi$
$107$	−1.28620	−0.124342	−0.0621710	+	0.998066i	$0.519802\pi$
$108$	0	0
$109$	−14.5541	−1.39403	−0.697014	−	0.717057i	$-0.745489\pi$
$109$	−14.5541	−1.39403	−0.697014	+	0.717057i	$0.745489\pi$
$110$	−7.19082	−0.685617
$111$	0	0
$112$	−17.0781	−1.61373
$113$	−5.25141	−0.494011	−0.247005	−	0.969014i	$-0.579447\pi$
$113$	−5.25141	−0.494011	−0.247005	+	0.969014i	$0.579447\pi$
$114$	0	0
$115$	−5.46553	−0.509663
$116$	−42.3617	−3.93318
$117$	0	0
$118$	−12.6905	−1.16825
$119$	14.7558	1.35266
$120$	0	0
$121$	−2.84273	−0.258430
$122$	−1.54206	−0.139612
$123$	0	0
$124$	0.133131	0.0119555
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	7.27231	0.645313	0.322657	−	0.946516i	$-0.395424\pi$
$127$	7.27231	0.645313	0.322657	+	0.946516i	$0.395424\pi$
$128$	−14.8964	−1.31667
$129$	0	0
$130$	17.1216	1.50167
$131$	−6.59942	−0.576594	−0.288297	−	0.957541i	$-0.593089\pi$
$131$	−6.59942	−0.576594	−0.288297	+	0.957541i	$0.593089\pi$
$132$	0	0
$133$	16.3449	1.41728
$134$	−31.1549	−2.69138
$135$	0	0
$136$	−31.2803	−2.68226
$137$	0.490578	0.0419129	0.0209564	−	0.999780i	$-0.493329\pi$
$137$	0.490578	0.0419129	0.0209564	+	0.999780i	$0.493329\pi$
$138$	0	0
$139$	−0.787100	−0.0667609	−0.0333805	−	0.999443i	$-0.510627\pi$
$139$	−0.787100	−0.0667609	−0.0333805	+	0.999443i	$0.510627\pi$
$140$	12.0525	1.01863
$141$	0	0
$142$	30.8695	2.59051
$143$	−19.4228	−1.62422
$144$	0	0
$145$	9.76331	0.810799
$146$	−8.62924	−0.714161
$147$	0	0
$148$	−7.56828	−0.622109
$149$	−3.22962	−0.264581	−0.132291	−	0.991211i	$-0.542233\pi$
$149$	−3.22962	−0.264581	−0.132291	+	0.991211i	$0.542233\pi$
$150$	0	0
$151$	2.17924	0.177344	0.0886719	−	0.996061i	$-0.471738\pi$
$151$	2.17924	0.177344	0.0886719	+	0.996061i	$0.471738\pi$
$152$	−34.6490	−2.81040
$153$	0	0
$154$	−19.9747	−1.60961
$155$	−0.0306833	−0.00246454
$156$	0	0
$157$	−11.4153	−0.911038	−0.455519	−	0.890226i	$-0.650546\pi$
$157$	−11.4153	−0.911038	−0.455519	+	0.890226i	$0.650546\pi$
$158$	32.6016	2.59364
$159$	0	0
$160$	−3.70178	−0.292652
$161$	−15.1822	−1.19652
$162$	0	0
$163$	23.8418	1.86743	0.933716	−	0.358015i	$-0.116546\pi$
$163$	23.8418	1.86743	0.933716	+	0.358015i	$0.116546\pi$
$164$	−19.2405	−1.50243
$165$	0	0
$166$	−22.2683	−1.72836
$167$	3.47961	0.269261	0.134630	−	0.990896i	$-0.457015\pi$
$167$	3.47961	0.269261	0.134630	+	0.990896i	$0.457015\pi$
$168$	0	0
$169$	33.2465	2.55742
$170$	13.3741	1.02575
$171$	0	0
$172$	46.0387	3.51042
$173$	13.4571	1.02313	0.511563	−	0.859246i	$-0.329067\pi$
$173$	13.4571	1.02313	0.511563	+	0.859246i	$0.329067\pi$
$174$	0	0
$175$	−2.77781	−0.209983
$176$	17.5593	1.32359
$177$	0	0
$178$	2.51771	0.188710
$179$	−9.29534	−0.694766	−0.347383	−	0.937723i	$-0.612930\pi$
$179$	−9.29534	−0.694766	−0.347383	+	0.937723i	$0.612930\pi$
$180$	0	0
$181$	−4.64846	−0.345518	−0.172759	−	0.984964i	$-0.555268\pi$
$181$	−4.64846	−0.345518	−0.172759	+	0.984964i	$0.555268\pi$
$182$	47.5607	3.52543
$183$	0	0
$184$	32.1842	2.37265
$185$	1.74430	0.128243
$186$	0	0
$187$	−15.1716	−1.10946
$188$	42.6752	3.11241
$189$	0	0
$190$	14.8144	1.07475
$191$	−20.6023	−1.49073	−0.745366	−	0.666655i	$-0.767725\pi$
$191$	−20.6023	−1.49073	−0.745366	+	0.666655i	$0.767725\pi$
$192$	0	0
$193$	12.9010	0.928635	0.464318	−	0.885669i	$-0.346300\pi$
$193$	12.9010	0.928635	0.464318	+	0.885669i	$0.346300\pi$
$194$	35.1930	2.52671
$195$	0	0
$196$	3.10762	0.221973
$197$	−11.6778	−0.832008	−0.416004	−	0.909363i	$-0.636570\pi$
$197$	−11.6778	−0.832008	−0.416004	+	0.909363i	$0.636570\pi$
$198$	0	0
$199$	−8.35160	−0.592029	−0.296014	−	0.955183i	$-0.595658\pi$
$199$	−8.35160	−0.592029	−0.296014	+	0.955183i	$0.595658\pi$
$200$	5.88859	0.416386
$201$	0	0
$202$	−31.9070	−2.24497
$203$	27.1206	1.90349
$204$	0	0
$205$	4.43445	0.309716
$206$	45.5903	3.17642
$207$	0	0
$208$	−41.8096	−2.89897
$209$	−16.8055	−1.16246
$210$	0	0
$211$	−24.4762	−1.68501	−0.842507	−	0.538685i	$-0.818921\pi$
$211$	−24.4762	−1.68501	−0.842507	+	0.538685i	$0.818921\pi$
$212$	−15.5145	−1.06554
$213$	0	0
$214$	−3.23829	−0.221365
$215$	−10.6108	−0.723648
$216$	0	0
$217$	−0.0852325	−0.00578596
$218$	−36.6430	−2.48178
$219$	0	0
$220$	−12.3922	−0.835482
$221$	36.1243	2.42998
$222$	0	0
$223$	23.4518	1.57045	0.785226	−	0.619210i	$-0.212547\pi$
$223$	23.4518	1.57045	0.785226	+	0.619210i	$0.212547\pi$
$224$	−10.2829	−0.687052
$225$	0	0
$226$	−13.2215	−0.879482
$227$	−14.6444	−0.971981	−0.485991	−	0.873964i	$-0.661541\pi$
$227$	−14.6444	−0.971981	−0.485991	+	0.873964i	$0.661541\pi$
$228$	0	0
$229$	−25.9785	−1.71671	−0.858354	−	0.513057i	$-0.828513\pi$
$229$	−25.9785	−1.71671	−0.858354	+	0.513057i	$0.828513\pi$
$230$	−13.7606	−0.907348
$231$	0	0
$232$	−57.4921	−3.77454
$233$	7.03490	0.460872	0.230436	−	0.973088i	$-0.425985\pi$
$233$	7.03490	0.460872	0.230436	+	0.973088i	$0.425985\pi$
$234$	0	0
$235$	−9.83555	−0.641601
$236$	−21.8700	−1.42361
$237$	0	0
$238$	37.1508	2.40813
$239$	−3.56159	−0.230380	−0.115190	−	0.993343i	$-0.536748\pi$
$239$	−3.56159	−0.230380	−0.115190	+	0.993343i	$0.536748\pi$
$240$	0	0
$241$	5.92903	0.381922	0.190961	−	0.981598i	$-0.438840\pi$
$241$	5.92903	0.381922	0.190961	+	0.981598i	$0.438840\pi$
$242$	−7.15718	−0.460081
$243$	0	0
$244$	−2.65750	−0.170129
$245$	−0.716229	−0.0457582
$246$	0	0
$247$	40.0146	2.54607
$248$	0.180682	0.0114733
$249$	0	0
$250$	−2.51771	−0.159234
$251$	18.6776	1.17892	0.589459	−	0.807798i	$-0.299341\pi$
$251$	18.6776	1.17892	0.589459	+	0.807798i	$0.299341\pi$
$252$	0	0
$253$	15.6101	0.981395
$254$	18.3096	1.14884
$255$	0	0
$256$	−31.5527	−1.97204
$257$	4.93490	0.307830	0.153915	−	0.988084i	$-0.450812\pi$
$257$	4.93490	0.307830	0.153915	+	0.988084i	$0.450812\pi$
$258$	0	0
$259$	4.84533	0.301074
$260$	29.5064	1.82991
$261$	0	0
$262$	−16.6154	−1.02650
$263$	−10.9474	−0.675043	−0.337521	−	0.941318i	$-0.609589\pi$
$263$	−10.9474	−0.675043	−0.337521	+	0.941318i	$0.609589\pi$
$264$	0	0
$265$	3.57570	0.219653
$266$	41.1517	2.52317
$267$	0	0
$268$	−53.6905	−3.27967
$269$	−10.9543	−0.667896	−0.333948	−	0.942591i	$-0.608381\pi$
$269$	−10.9543	−0.667896	−0.333948	+	0.942591i	$0.608381\pi$
$270$	0	0
$271$	2.03573	0.123662	0.0618310	−	0.998087i	$-0.480306\pi$
$271$	2.03573	0.123662	0.0618310	+	0.998087i	$0.480306\pi$
$272$	−32.6584	−1.98021
$273$	0	0
$274$	1.23513	0.0746171
$275$	2.85609	0.172229
$276$	0	0
$277$	5.76278	0.346252	0.173126	−	0.984900i	$-0.444613\pi$
$277$	5.76278	0.346252	0.173126	+	0.984900i	$0.444613\pi$
$278$	−1.98169	−0.118854
$279$	0	0
$280$	16.3574	0.977541
$281$	−27.4657	−1.63846	−0.819232	−	0.573462i	$-0.805600\pi$
$281$	−27.4657	−1.63846	−0.819232	+	0.573462i	$0.805600\pi$
$282$	0	0
$283$	16.9700	1.00876	0.504380	−	0.863482i	$-0.331721\pi$
$283$	16.9700	1.00876	0.504380	+	0.863482i	$0.331721\pi$
$284$	53.1986	3.15676
$285$	0	0
$286$	−48.9010	−2.89158
$287$	12.3181	0.727113
$288$	0	0
$289$	11.2175	0.659854
$290$	24.5812	1.44346
$291$	0	0
$292$	−14.8711	−0.870266
$293$	3.41240	0.199354	0.0996771	−	0.995020i	$-0.468219\pi$
$293$	3.41240	0.199354	0.0996771	+	0.995020i	$0.468219\pi$
$294$	0	0
$295$	5.04048	0.293468
$296$	−10.2715	−0.597016
$297$	0	0
$298$	−8.13126	−0.471031
$299$	−37.1682	−2.14949
$300$	0	0
$301$	−29.4747	−1.69889
$302$	5.48669	0.315724
$303$	0	0
$304$	−36.1756	−2.07481
$305$	0.612487	0.0350709
$306$	0	0
$307$	−26.4408	−1.50906	−0.754529	−	0.656266i	$-0.772135\pi$
$307$	−26.4408	−1.50906	−0.754529	+	0.656266i	$0.772135\pi$
$308$	−34.4232	−1.96144
$309$	0	0
$310$	−0.0772518	−0.00438761
$311$	18.2211	1.03323	0.516613	−	0.856219i	$-0.327193\pi$
$311$	18.2211	1.03323	0.516613	+	0.856219i	$0.327193\pi$
$312$	0	0
$313$	−28.4164	−1.60619	−0.803096	−	0.595849i	$-0.796815\pi$
$313$	−28.4164	−1.60619	−0.803096	+	0.595849i	$0.796815\pi$
$314$	−28.7404	−1.62191
$315$	0	0
$316$	56.1835	3.16057
$317$	4.55222	0.255678	0.127839	−	0.991795i	$-0.459196\pi$
$317$	4.55222	0.255678	0.127839	+	0.991795i	$0.459196\pi$
$318$	0	0
$319$	−27.8849	−1.56126
$320$	2.97604	0.166366
$321$	0	0
$322$	−38.2244	−2.13016
$323$	31.2564	1.73915
$324$	0	0
$325$	−6.80048	−0.377223
$326$	60.0267	3.32457
$327$	0	0
$328$	−26.1127	−1.44183
$329$	−27.3213	−1.50627
$330$	0	0
$331$	−1.28059	−0.0703875	−0.0351938	−	0.999381i	$-0.511205\pi$
$331$	−1.28059	−0.0703875	−0.0351938	+	0.999381i	$0.511205\pi$
$332$	−38.3758	−2.10615
$333$	0	0
$334$	8.76066	0.479362
$335$	12.3743	0.676081
$336$	0	0
$337$	14.8269	0.807673	0.403836	−	0.914831i	$-0.367676\pi$
$337$	14.8269	0.807673	0.403836	+	0.914831i	$0.367676\pi$
$338$	83.7051	4.55296
$339$	0	0
$340$	23.0481	1.24996
$341$	0.0876345	0.00474567
$342$	0	0
$343$	17.4551	0.942488
$344$	62.4825	3.36883
$345$	0	0
$346$	33.8811	1.82146
$347$	−14.5876	−0.783104	−0.391552	−	0.920156i	$-0.628062\pi$
$347$	−14.5876	−0.783104	−0.391552	+	0.920156i	$0.628062\pi$
$348$	0	0
$349$	6.31455	0.338010	0.169005	−	0.985615i	$-0.445945\pi$
$349$	6.31455	0.338010	0.169005	+	0.985615i	$0.445945\pi$
$350$	−6.99372	−0.373830
$351$	0	0
$352$	10.5726	0.563523
$353$	−25.5911	−1.36208	−0.681038	−	0.732248i	$-0.738471\pi$
$353$	−25.5911	−1.36208	−0.681038	+	0.732248i	$0.738471\pi$
$354$	0	0
$355$	−12.2609	−0.650744
$356$	4.33887	0.229959
$357$	0	0
$358$	−23.4030	−1.23689
$359$	27.1692	1.43394	0.716968	−	0.697106i	$-0.245529\pi$
$359$	27.1692	1.43394	0.716968	+	0.697106i	$0.245529\pi$
$360$	0	0
$361$	15.6225	0.822236
$362$	−11.7035	−0.615121
$363$	0	0
$364$	81.9631	4.29603
$365$	3.42742	0.179399
$366$	0	0
$367$	−12.3854	−0.646511	−0.323256	−	0.946312i	$-0.604777\pi$
$367$	−12.3854	−0.646511	−0.323256	+	0.946312i	$0.604777\pi$
$368$	33.6022	1.75164
$369$	0	0
$370$	4.39164	0.228310
$371$	9.93262	0.515676
$372$	0	0
$373$	15.8802	0.822248	0.411124	−	0.911579i	$-0.365136\pi$
$373$	15.8802	0.822248	0.411124	+	0.911579i	$0.365136\pi$
$374$	−38.1977	−1.97516
$375$	0	0
$376$	57.9175	2.98687
$377$	66.3952	3.41953
$378$	0	0
$379$	28.0666	1.44169	0.720843	−	0.693098i	$-0.243755\pi$
$379$	28.0666	1.44169	0.720843	+	0.693098i	$0.243755\pi$
$380$	25.5303	1.30968
$381$	0	0
$382$	−51.8707	−2.65394
$383$	23.1604	1.18344	0.591721	−	0.806143i	$-0.298449\pi$
$383$	23.1604	1.18344	0.591721	+	0.806143i	$0.298449\pi$
$384$	0	0
$385$	7.93368	0.404338
$386$	32.4810	1.65324
$387$	0	0
$388$	60.6494	3.07901
$389$	7.38515	0.374442	0.187221	−	0.982318i	$-0.440052\pi$
$389$	7.38515	0.374442	0.187221	+	0.982318i	$0.440052\pi$
$390$	0	0
$391$	−29.0330	−1.46826
$392$	4.21758	0.213020
$393$	0	0
$394$	−29.4013	−1.48122
$395$	−12.9489	−0.651530
$396$	0	0
$397$	−20.5366	−1.03070	−0.515350	−	0.856980i	$-0.672338\pi$
$397$	−20.5366	−1.03070	−0.515350	+	0.856980i	$0.672338\pi$
$398$	−21.0269	−1.05398
$399$	0	0
$400$	6.14803	0.307402
$401$	−14.8529	−0.741720	−0.370860	−	0.928689i	$-0.620937\pi$
$401$	−14.8529	−0.741720	−0.370860	+	0.928689i	$0.620937\pi$
$402$	0	0
$403$	−0.208661	−0.0103942
$404$	−54.9865	−2.73568
$405$	0	0
$406$	68.2819	3.38877
$407$	−4.98188	−0.246943
$408$	0	0
$409$	33.6299	1.66289	0.831446	−	0.555605i	$-0.187513\pi$
$409$	33.6299	1.66289	0.831446	+	0.555605i	$0.187513\pi$
$410$	11.1647	0.551384
$411$	0	0
$412$	78.5674	3.87074
$413$	14.0015	0.688968
$414$	0	0
$415$	8.84466	0.434167
$416$	−25.1739	−1.23425
$417$	0	0
$418$	−42.3114	−2.06952
$419$	−25.7058	−1.25581	−0.627906	−	0.778289i	$-0.716088\pi$
$419$	−25.7058	−1.25581	−0.627906	+	0.778289i	$0.716088\pi$
$420$	0	0
$421$	15.3530	0.748259	0.374129	−	0.927377i	$-0.377942\pi$
$421$	15.3530	0.748259	0.374129	+	0.927377i	$0.377942\pi$
$422$	−61.6241	−2.99981
$423$	0	0
$424$	−21.0558	−1.02256
$425$	−5.31202	−0.257671
$426$	0	0
$427$	1.70137	0.0823351
$428$	−5.58067	−0.269752
$429$	0	0
$430$	−26.7148	−1.28830
$431$	−20.6692	−0.995601	−0.497801	−	0.867292i	$-0.665859\pi$
$431$	−20.6692	−0.995601	−0.497801	+	0.867292i	$0.665859\pi$
$432$	0	0
$433$	12.1627	0.584500	0.292250	−	0.956342i	$-0.405596\pi$
$433$	12.1627	0.584500	0.292250	+	0.956342i	$0.405596\pi$
$434$	−0.214591	−0.0103007
$435$	0	0
$436$	−63.1483	−3.02425
$437$	−32.1596	−1.53840
$438$	0	0
$439$	0.320546	0.0152988	0.00764942	−	0.999971i	$-0.497565\pi$
$439$	0.320546	0.0152988	0.00764942	+	0.999971i	$0.497565\pi$
$440$	−16.8184	−0.801784
$441$	0	0
$442$	90.9504	4.32607
$443$	19.6750	0.934789	0.467394	−	0.884049i	$-0.345193\pi$
$443$	19.6750	0.934789	0.467394	+	0.884049i	$0.345193\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	59.0449	2.79586
$447$	0	0
$448$	8.26688	0.390573
$449$	−17.0901	−0.806531	−0.403266	−	0.915083i	$-0.632125\pi$
$449$	−17.0901	−0.806531	−0.403266	+	0.915083i	$0.632125\pi$
$450$	0	0
$451$	−12.6652	−0.596382
$452$	−22.7851	−1.07172
$453$	0	0
$454$	−36.8703	−1.73041
$455$	−18.8904	−0.885598
$456$	0	0
$457$	4.26437	0.199479	0.0997394	−	0.995014i	$-0.468199\pi$
$457$	4.26437	0.199479	0.0997394	+	0.995014i	$0.468199\pi$
$458$	−65.4064	−3.05624
$459$	0	0
$460$	−23.7142	−1.10568
$461$	−18.4256	−0.858168	−0.429084	−	0.903265i	$-0.641164\pi$
$461$	−18.4256	−0.858168	−0.429084	+	0.903265i	$0.641164\pi$
$462$	0	0
$463$	17.4273	0.809914	0.404957	−	0.914336i	$-0.367286\pi$
$463$	17.4273	0.809914	0.404957	+	0.914336i	$0.367286\pi$
$464$	−60.0251	−2.78660
$465$	0	0
$466$	17.7118	0.820486
$467$	4.15014	0.192045	0.0960227	−	0.995379i	$-0.469388\pi$
$467$	4.15014	0.192045	0.0960227	+	0.995379i	$0.469388\pi$
$468$	0	0
$469$	34.3735	1.58722
$470$	−24.7631	−1.14224
$471$	0	0
$472$	−29.6813	−1.36619
$473$	30.3053	1.39344
$474$	0	0
$475$	−5.88409	−0.269980
$476$	64.0233	2.93450
$477$	0	0
$478$	−8.96705	−0.410143
$479$	−9.91083	−0.452837	−0.226419	−	0.974030i	$-0.572702\pi$
$479$	−9.91083	−0.452837	−0.226419	+	0.974030i	$0.572702\pi$
$480$	0	0
$481$	11.8621	0.540864
$482$	14.9276	0.679933
$483$	0	0
$484$	−12.3342	−0.560647
$485$	−13.9782	−0.634716
$486$	0	0
$487$	−19.6261	−0.889345	−0.444673	−	0.895693i	$-0.646680\pi$
$487$	−19.6261	−0.889345	−0.444673	+	0.895693i	$0.646680\pi$
$488$	−3.60668	−0.163267
$489$	0	0
$490$	−1.80326	−0.0814629
$491$	35.9771	1.62362	0.811812	−	0.583919i	$-0.198481\pi$
$491$	35.9771	1.62362	0.811812	+	0.583919i	$0.198481\pi$
$492$	0	0
$493$	51.8629	2.33579
$494$	100.745	4.53274
$495$	0	0
$496$	0.188642	0.00847028
$497$	−34.0586	−1.52774
$498$	0	0
$499$	−28.4569	−1.27390	−0.636952	−	0.770903i	$-0.719805\pi$
$499$	−28.4569	−1.27390	−0.636952	+	0.770903i	$0.719805\pi$
$500$	−4.33887	−0.194040
$501$	0	0
$502$	47.0247	2.09882
$503$	−13.6065	−0.606684	−0.303342	−	0.952882i	$-0.598102\pi$
$503$	−13.6065	−0.606684	−0.303342	+	0.952882i	$0.598102\pi$
$504$	0	0
$505$	12.6730	0.563942
$506$	39.3016	1.74717
$507$	0	0
$508$	31.5536	1.39996
$509$	33.4364	1.48204	0.741021	−	0.671481i	$-0.234342\pi$
$509$	33.4364	1.48204	0.741021	+	0.671481i	$0.234342\pi$
$510$	0	0
$511$	9.52071	0.421172
$512$	−49.6478	−2.19414
$513$	0	0
$514$	12.4246	0.548028
$515$	−18.1078	−0.797926
$516$	0	0
$517$	28.0913	1.23545
$518$	12.1991	0.535999
$519$	0	0
$520$	40.0452	1.75610
$521$	28.4974	1.24850	0.624248	−	0.781226i	$-0.285406\pi$
$521$	28.4974	1.24850	0.624248	+	0.781226i	$0.285406\pi$
$522$	0	0
$523$	−2.14488	−0.0937889	−0.0468944	−	0.998900i	$-0.514932\pi$
$523$	−2.14488	−0.0937889	−0.0468944	+	0.998900i	$0.514932\pi$
$524$	−28.6340	−1.25088
$525$	0	0
$526$	−27.5623	−1.20177
$527$	−0.162990	−0.00709997
$528$	0	0
$529$	6.87198	0.298782
$530$	9.00258	0.391047
$531$	0	0
$532$	70.9183	3.07470
$533$	30.1564	1.30622
$534$	0	0
$535$	1.28620	0.0556075
$536$	−72.8672	−3.14738
$537$	0	0
$538$	−27.5798	−1.18905
$539$	2.04562	0.0881110
$540$	0	0
$541$	−5.73301	−0.246481	−0.123241	−	0.992377i	$-0.539329\pi$
$541$	−5.73301	−0.246481	−0.123241	+	0.992377i	$0.539329\pi$
$542$	5.12539	0.220154
$543$	0	0
$544$	−19.6639	−0.843084
$545$	14.5541	0.623429
$546$	0	0
$547$	−0.841704	−0.0359887	−0.0179943	−	0.999838i	$-0.505728\pi$
$547$	−0.841704	−0.0359887	−0.0179943	+	0.999838i	$0.505728\pi$
$548$	2.12855	0.0909272
$549$	0	0
$550$	7.19082	0.306617
$551$	57.4482	2.44737
$552$	0	0
$553$	−35.9696	−1.52958
$554$	14.5090	0.616429
$555$	0	0
$556$	−3.41512	−0.144833
$557$	0.824108	0.0349186	0.0174593	−	0.999848i	$-0.494442\pi$
$557$	0.824108	0.0349186	0.0174593	+	0.999848i	$0.494442\pi$
$558$	0	0
$559$	−72.1583	−3.05197
$560$	17.0781	0.721680
$561$	0	0
$562$	−69.1506	−2.91694
$563$	−2.12487	−0.0895527	−0.0447763	−	0.998997i	$-0.514258\pi$
$563$	−2.12487	−0.0895527	−0.0447763	+	0.998997i	$0.514258\pi$
$564$	0	0
$565$	5.25141	0.220928
$566$	42.7254	1.79588
$567$	0	0
$568$	72.1997	3.02943
$569$	−4.90747	−0.205732	−0.102866	−	0.994695i	$-0.532801\pi$
$569$	−4.90747	−0.205732	−0.102866	+	0.994695i	$0.532801\pi$
$570$	0	0
$571$	−13.2225	−0.553346	−0.276673	−	0.960964i	$-0.589232\pi$
$571$	−13.2225	−0.553346	−0.276673	+	0.960964i	$0.589232\pi$
$572$	−84.2729	−3.52363
$573$	0	0
$574$	31.0133	1.29447
$575$	5.46553	0.227928
$576$	0	0
$577$	19.2457	0.801210	0.400605	−	0.916251i	$-0.368800\pi$
$577$	19.2457	0.801210	0.400605	+	0.916251i	$0.368800\pi$
$578$	28.2425	1.17473
$579$	0	0
$580$	42.3617	1.75897
$581$	24.5688	1.01929
$582$	0	0
$583$	−10.2125	−0.422960
$584$	−20.1827	−0.835164
$585$	0	0
$586$	8.59143	0.354909
$587$	38.7761	1.60046	0.800231	−	0.599692i	$-0.204710\pi$
$587$	38.7761	1.60046	0.800231	+	0.599692i	$0.204710\pi$
$588$	0	0
$589$	−0.180543	−0.00743916
$590$	12.6905	0.522458
$591$	0	0
$592$	−10.7240	−0.440754
$593$	−38.7512	−1.59132	−0.795660	−	0.605744i	$-0.792876\pi$
$593$	−38.7512	−1.59132	−0.795660	+	0.605744i	$0.792876\pi$
$594$	0	0
$595$	−14.7558	−0.604928
$596$	−14.0129	−0.573991
$597$	0	0
$598$	−93.5788	−3.82672
$599$	−31.1313	−1.27199	−0.635994	−	0.771694i	$-0.719410\pi$
$599$	−31.1313	−1.27199	−0.635994	+	0.771694i	$0.719410\pi$
$600$	0	0
$601$	−36.7318	−1.49832	−0.749160	−	0.662389i	$-0.769543\pi$
$601$	−36.7318	−1.49832	−0.749160	+	0.662389i	$0.769543\pi$
$602$	−74.2088	−3.02452
$603$	0	0
$604$	9.45542	0.384736
$605$	2.84273	0.115574
$606$	0	0
$607$	28.4169	1.15341	0.576703	−	0.816954i	$-0.304339\pi$
$607$	28.4169	1.15341	0.576703	+	0.816954i	$0.304339\pi$
$608$	−21.7816	−0.883361
$609$	0	0
$610$	1.54206	0.0624364
$611$	−66.8865	−2.70594
$612$	0	0
$613$	−39.5155	−1.59602	−0.798009	−	0.602646i	$-0.794113\pi$
$613$	−39.5155	−1.59602	−0.798009	+	0.602646i	$0.794113\pi$
$614$	−66.5704	−2.68656
$615$	0	0
$616$	−46.7182	−1.88233
$617$	1.22182	0.0491887	0.0245943	−	0.999698i	$-0.492171\pi$
$617$	1.22182	0.0491887	0.0245943	+	0.999698i	$0.492171\pi$
$618$	0	0
$619$	11.3839	0.457556	0.228778	−	0.973479i	$-0.426527\pi$
$619$	11.3839	0.457556	0.228778	+	0.973479i	$0.426527\pi$
$620$	−0.133131	−0.00534667
$621$	0	0
$622$	45.8755	1.83944
$623$	−2.77781	−0.111291
$624$	0	0
$625$	1.00000	0.0400000
$626$	−71.5444	−2.85949
$627$	0	0
$628$	−49.5294	−1.97644
$629$	9.26574	0.369449
$630$	0	0
$631$	−18.2862	−0.727962	−0.363981	−	0.931406i	$-0.618583\pi$
$631$	−18.2862	−0.727962	−0.363981	+	0.931406i	$0.618583\pi$
$632$	76.2507	3.03309
$633$	0	0
$634$	11.4612	0.455181
$635$	−7.27231	−0.288593
$636$	0	0
$637$	−4.87070	−0.192984
$638$	−70.2061	−2.77949
$639$	0	0
$640$	14.8964	0.588831
$641$	−1.10478	−0.0436364	−0.0218182	−	0.999762i	$-0.506945\pi$
$641$	−1.10478	−0.0436364	−0.0218182	+	0.999762i	$0.506945\pi$
$642$	0	0
$643$	−1.22107	−0.0481544	−0.0240772	−	0.999710i	$-0.507665\pi$
$643$	−1.22107	−0.0481544	−0.0240772	+	0.999710i	$0.507665\pi$
$644$	−65.8735	−2.59578
$645$	0	0
$646$	78.6945	3.09619
$647$	17.0143	0.668902	0.334451	−	0.942413i	$-0.391449\pi$
$647$	17.0143	0.668902	0.334451	+	0.942413i	$0.391449\pi$
$648$	0	0
$649$	−14.3961	−0.565095
$650$	−17.1216	−0.671566
$651$	0	0
$652$	103.446	4.05127
$653$	−32.9774	−1.29051	−0.645253	−	0.763969i	$-0.723248\pi$
$653$	−32.9774	−1.29051	−0.645253	+	0.763969i	$0.723248\pi$
$654$	0	0
$655$	6.59942	0.257861
$656$	−27.2632	−1.06445
$657$	0	0
$658$	−68.7871	−2.68160
$659$	−22.0253	−0.857984	−0.428992	−	0.903308i	$-0.641131\pi$
$659$	−22.0253	−0.857984	−0.428992	+	0.903308i	$0.641131\pi$
$660$	0	0
$661$	40.9970	1.59460	0.797300	−	0.603584i	$-0.206261\pi$
$661$	40.9970	1.59460	0.797300	+	0.603584i	$0.206261\pi$
$662$	−3.22415	−0.125310
$663$	0	0
$664$	−52.0826	−2.02120
$665$	−16.3449	−0.633827
$666$	0	0
$667$	−53.3616	−2.06617
$668$	15.0976	0.584143
$669$	0	0
$670$	31.1549	1.20362
$671$	−1.74932	−0.0675317
$672$	0	0
$673$	−24.6460	−0.950033	−0.475017	−	0.879977i	$-0.657558\pi$
$673$	−24.6460	−0.950033	−0.475017	+	0.879977i	$0.657558\pi$
$674$	37.3299	1.43789
$675$	0	0
$676$	144.252	5.54816
$677$	31.4935	1.21039	0.605196	−	0.796077i	$-0.293095\pi$
$677$	31.4935	1.21039	0.605196	+	0.796077i	$0.293095\pi$
$678$	0	0
$679$	−38.8287	−1.49011
$680$	31.2803	1.19954
$681$	0	0
$682$	0.220638	0.00844867
$683$	−44.7780	−1.71338	−0.856691	−	0.515829i	$-0.827484\pi$
$683$	−44.7780	−1.71338	−0.856691	+	0.515829i	$0.827484\pi$
$684$	0	0
$685$	−0.490578	−0.0187440
$686$	43.9469	1.67790
$687$	0	0
$688$	65.2353	2.48707
$689$	24.3165	0.926384
$690$	0	0
$691$	49.2648	1.87412	0.937061	−	0.349166i	$-0.113535\pi$
$691$	49.2648	1.87412	0.937061	+	0.349166i	$0.113535\pi$
$692$	58.3886	2.21960
$693$	0	0
$694$	−36.7274	−1.39415
$695$	0.787100	0.0298564
$696$	0	0
$697$	23.5559	0.892243
$698$	15.8982	0.601757
$699$	0	0
$700$	−12.0525	−0.455543
$701$	19.3837	0.732111	0.366055	−	0.930593i	$-0.380708\pi$
$701$	19.3837	0.732111	0.366055	+	0.930593i	$0.380708\pi$
$702$	0	0
$703$	10.2636	0.387099
$704$	−8.49985	−0.320350
$705$	0	0
$706$	−64.4309	−2.42489
$707$	35.2032	1.32395
$708$	0	0
$709$	−20.4538	−0.768157	−0.384078	−	0.923300i	$-0.625481\pi$
$709$	−20.4538	−0.768157	−0.384078	+	0.923300i	$0.625481\pi$
$710$	−30.8695	−1.15851
$711$	0	0
$712$	5.88859	0.220684
$713$	0.167701	0.00628044
$714$	0	0
$715$	19.4228	0.726372
$716$	−40.3312	−1.50725
$717$	0	0
$718$	68.4042	2.55282
$719$	−30.7515	−1.14684	−0.573418	−	0.819263i	$-0.694383\pi$
$719$	−30.7515	−1.14684	−0.573418	+	0.819263i	$0.694383\pi$
$720$	0	0
$721$	−50.3001	−1.87327
$722$	39.3329	1.46382
$723$	0	0
$724$	−20.1691	−0.749577
$725$	−9.76331	−0.362600
$726$	0	0
$727$	−17.9302	−0.664995	−0.332497	−	0.943104i	$-0.607891\pi$
$727$	−17.9302	−0.664995	−0.332497	+	0.943104i	$0.607891\pi$
$728$	111.238	4.12276
$729$	0	0
$730$	8.62924	0.319383
$731$	−56.3646	−2.08472
$732$	0	0
$733$	−25.6271	−0.946558	−0.473279	−	0.880913i	$-0.656930\pi$
$733$	−25.6271	−0.946558	−0.473279	+	0.880913i	$0.656930\pi$
$734$	−31.1828	−1.15098
$735$	0	0
$736$	20.2322	0.745769
$737$	−35.3422	−1.30185
$738$	0	0
$739$	−31.6967	−1.16598	−0.582991	−	0.812478i	$-0.698118\pi$
$739$	−31.6967	−1.16598	−0.582991	+	0.812478i	$0.698118\pi$
$740$	7.56828	0.278215
$741$	0	0
$742$	25.0075	0.918052
$743$	6.13738	0.225159	0.112579	−	0.993643i	$-0.464089\pi$
$743$	6.13738	0.225159	0.112579	+	0.993643i	$0.464089\pi$
$744$	0	0
$745$	3.22962	0.118324
$746$	39.9819	1.46384
$747$	0	0
$748$	−65.8276	−2.40690
$749$	3.57283	0.130548
$750$	0	0
$751$	−41.2598	−1.50559	−0.752796	−	0.658254i	$-0.771295\pi$
$751$	−41.2598	−1.50559	−0.752796	+	0.658254i	$0.771295\pi$
$752$	60.4693	2.20509
$753$	0	0
$754$	167.164	6.08775
$755$	−2.17924	−0.0793106
$756$	0	0
$757$	−20.9578	−0.761725	−0.380863	−	0.924632i	$-0.624373\pi$
$757$	−20.9578	−0.761725	−0.380863	+	0.924632i	$0.624373\pi$
$758$	70.6637	2.56662
$759$	0	0
$760$	34.6490	1.25685
$761$	−18.0674	−0.654941	−0.327471	−	0.944861i	$-0.606196\pi$
$761$	−18.0674	−0.654941	−0.327471	+	0.944861i	$0.606196\pi$
$762$	0	0
$763$	40.4285	1.46361
$764$	−89.3908	−3.23404
$765$	0	0
$766$	58.3112	2.10687
$767$	34.2777	1.23770
$768$	0	0
$769$	−32.1645	−1.15988	−0.579942	−	0.814658i	$-0.696925\pi$
$769$	−32.1645	−1.15988	−0.579942	+	0.814658i	$0.696925\pi$
$770$	19.9747	0.719839
$771$	0	0
$772$	55.9758	2.01461
$773$	0.0694202	0.00249687	0.00124844	−	0.999999i	$-0.499603\pi$
$773$	0.0694202	0.00249687	0.00124844	+	0.999999i	$0.499603\pi$
$774$	0	0
$775$	0.0306833	0.00110218
$776$	82.3117	2.95482
$777$	0	0
$778$	18.5937	0.666615
$779$	26.0927	0.934869
$780$	0	0
$781$	35.0184	1.25306
$782$	−73.0966	−2.61393
$783$	0	0
$784$	4.40340	0.157264
$785$	11.4153	0.407429
$786$	0	0
$787$	7.06720	0.251918	0.125959	−	0.992035i	$-0.459799\pi$
$787$	7.06720	0.251918	0.125959	+	0.992035i	$0.459799\pi$
$788$	−50.6684	−1.80499
$789$	0	0
$790$	−32.6016	−1.15991
$791$	14.5874	0.518668
$792$	0	0
$793$	4.16520	0.147911
$794$	−51.7051	−1.83495
$795$	0	0
$796$	−36.2365	−1.28437
$797$	7.16964	0.253962	0.126981	−	0.991905i	$-0.459471\pi$
$797$	7.16964	0.253962	0.126981	+	0.991905i	$0.459471\pi$
$798$	0	0
$799$	−52.2466	−1.84835
$800$	3.70178	0.130878
$801$	0	0
$802$	−37.3954	−1.32048
$803$	−9.78902	−0.345447
$804$	0	0
$805$	15.1822	0.535102
$806$	−0.525349	−0.0185046
$807$	0	0
$808$	−74.6262	−2.62534
$809$	−0.877876	−0.0308645	−0.0154322	−	0.999881i	$-0.504912\pi$
$809$	−0.877876	−0.0308645	−0.0154322	+	0.999881i	$0.504912\pi$
$810$	0	0
$811$	0.401924	0.0141135	0.00705674	−	0.999975i	$-0.497754\pi$
$811$	0.401924	0.0141135	0.00705674	+	0.999975i	$0.497754\pi$
$812$	117.673	4.12950
$813$	0	0
$814$	−12.5429	−0.439629
$815$	−23.8418	−0.835141
$816$	0	0
$817$	−62.4347	−2.18431
$818$	84.6704	2.96043
$819$	0	0
$820$	19.2405	0.671908
$821$	31.4406	1.09729	0.548643	−	0.836057i	$-0.315145\pi$
$821$	31.4406	1.09729	0.548643	+	0.836057i	$0.315145\pi$
$822$	0	0
$823$	30.7400	1.07153	0.535764	−	0.844368i	$-0.320024\pi$
$823$	30.7400	1.07153	0.535764	+	0.844368i	$0.320024\pi$
$824$	106.630	3.71462
$825$	0	0
$826$	35.2517	1.22656
$827$	0.855585	0.0297516	0.0148758	−	0.999889i	$-0.495265\pi$
$827$	0.855585	0.0297516	0.0148758	+	0.999889i	$0.495265\pi$
$828$	0	0
$829$	−48.0406	−1.66852	−0.834260	−	0.551372i	$-0.814105\pi$
$829$	−48.0406	−1.66852	−0.834260	+	0.551372i	$0.814105\pi$
$830$	22.2683	0.772944
$831$	0	0
$832$	20.2385	0.701644
$833$	−3.80462	−0.131822
$834$	0	0
$835$	−3.47961	−0.120417
$836$	−72.9168	−2.52188
$837$	0	0
$838$	−64.7199	−2.23571
$839$	8.54835	0.295122	0.147561	−	0.989053i	$-0.452858\pi$
$839$	8.54835	0.295122	0.147561	+	0.989053i	$0.452858\pi$
$840$	0	0
$841$	66.3222	2.28697
$842$	38.6544	1.33212
$843$	0	0
$844$	−106.199	−3.65553
$845$	−33.2465	−1.14372
$846$	0	0
$847$	7.89658	0.271330
$848$	−21.9835	−0.754917
$849$	0	0
$850$	−13.3741	−0.458729
$851$	−9.53351	−0.326805
$852$	0	0
$853$	37.0856	1.26979	0.634893	−	0.772600i	$-0.281044\pi$
$853$	37.0856	1.26979	0.634893	+	0.772600i	$0.281044\pi$
$854$	4.28356	0.146580
$855$	0	0
$856$	−7.57393	−0.258872
$857$	−24.2278	−0.827606	−0.413803	−	0.910366i	$-0.635800\pi$
$857$	−24.2278	−0.827606	−0.413803	+	0.910366i	$0.635800\pi$
$858$	0	0
$859$	−39.2891	−1.34053	−0.670263	−	0.742124i	$-0.733819\pi$
$859$	−39.2891	−1.34053	−0.670263	+	0.742124i	$0.733819\pi$
$860$	−46.0387	−1.56991
$861$	0	0
$862$	−52.0391	−1.77246
$863$	−33.4535	−1.13877	−0.569386	−	0.822070i	$-0.692819\pi$
$863$	−33.4535	−1.13877	−0.569386	+	0.822070i	$0.692819\pi$
$864$	0	0
$865$	−13.4571	−0.457555
$866$	30.6220	1.04058
$867$	0	0
$868$	−0.369812	−0.0125523
$869$	36.9832	1.25457
$870$	0	0
$871$	84.1512	2.85136
$872$	−85.7031	−2.90227
$873$	0	0
$874$	−80.9687	−2.73881
$875$	2.77781	0.0939071
$876$	0	0
$877$	5.93291	0.200340	0.100170	−	0.994970i	$-0.468061\pi$
$877$	5.93291	0.200340	0.100170	+	0.994970i	$0.468061\pi$
$878$	0.807043	0.0272364
$879$	0	0
$880$	−17.5593	−0.591925
$881$	34.0065	1.14571	0.572855	−	0.819657i	$-0.305836\pi$
$881$	34.0065	1.14571	0.572855	+	0.819657i	$0.305836\pi$
$882$	0	0
$883$	−6.32330	−0.212796	−0.106398	−	0.994324i	$-0.533932\pi$
$883$	−6.32330	−0.212796	−0.106398	+	0.994324i	$0.533932\pi$
$884$	156.738	5.27168
$885$	0	0
$886$	49.5360	1.66420
$887$	17.3570	0.582791	0.291395	−	0.956603i	$-0.405881\pi$
$887$	17.3570	0.582791	0.291395	+	0.956603i	$0.405881\pi$
$888$	0	0
$889$	−20.2011	−0.677523
$890$	−2.51771	−0.0843939
$891$	0	0
$892$	101.754	3.40699
$893$	−57.8733	−1.93666
$894$	0	0
$895$	9.29534	0.310709
$896$	41.3793	1.38239
$897$	0	0
$898$	−43.0279	−1.43586
$899$	−0.299571	−0.00999125
$900$	0	0
$901$	18.9942	0.632788
$902$	−31.8873	−1.06173
$903$	0	0
$904$	−30.9234	−1.02850
$905$	4.64846	0.154520
$906$	0	0
$907$	9.42582	0.312979	0.156490	−	0.987680i	$-0.449982\pi$
$907$	9.42582	0.312979	0.156490	+	0.987680i	$0.449982\pi$
$908$	−63.5400	−2.10865
$909$	0	0
$910$	−47.5607	−1.57662
$911$	−25.3755	−0.840728	−0.420364	−	0.907355i	$-0.638098\pi$
$911$	−25.3755	−0.840728	−0.420364	+	0.907355i	$0.638098\pi$
$912$	0	0
$913$	−25.2612	−0.836023
$914$	10.7365	0.355130
$915$	0	0
$916$	−112.717	−3.72428
$917$	18.3319	0.605374
$918$	0	0
$919$	20.6698	0.681832	0.340916	−	0.940094i	$-0.389263\pi$
$919$	20.6698	0.681832	0.340916	+	0.940094i	$0.389263\pi$
$920$	−32.1842	−1.06108
$921$	0	0
$922$	−46.3904	−1.52779
$923$	−83.3803	−2.74450
$924$	0	0
$925$	−1.74430	−0.0573522
$926$	43.8768	1.44188
$927$	0	0
$928$	−36.1417	−1.18641
$929$	−29.5950	−0.970982	−0.485491	−	0.874242i	$-0.661359\pi$
$929$	−29.5950	−0.970982	−0.485491	+	0.874242i	$0.661359\pi$
$930$	0	0
$931$	−4.21436	−0.138120
$932$	30.5235	0.999831
$933$	0	0
$934$	10.4488	0.341897
$935$	15.1716	0.496165
$936$	0	0
$937$	−12.8373	−0.419377	−0.209688	−	0.977768i	$-0.567245\pi$
$937$	−12.8373	−0.419377	−0.209688	+	0.977768i	$0.567245\pi$
$938$	86.5425	2.82571
$939$	0	0
$940$	−42.6752	−1.39191
$941$	−20.4237	−0.665792	−0.332896	−	0.942963i	$-0.608026\pi$
$941$	−20.4237	−0.665792	−0.332896	+	0.942963i	$0.608026\pi$
$942$	0	0
$943$	−24.2366	−0.789253
$944$	−30.9890	−1.00861
$945$	0	0
$946$	76.3001	2.48073
$947$	−21.4870	−0.698233	−0.349116	−	0.937079i	$-0.613518\pi$
$947$	−21.4870	−0.698233	−0.349116	+	0.937079i	$0.613518\pi$
$948$	0	0
$949$	23.3081	0.756612
$950$	−14.8144	−0.480644
$951$	0	0
$952$	86.8907	2.81614
$953$	−16.7057	−0.541151	−0.270576	−	0.962699i	$-0.587214\pi$
$953$	−16.7057	−0.541151	−0.270576	+	0.962699i	$0.587214\pi$
$954$	0	0
$955$	20.6023	0.666676
$956$	−15.4533	−0.499794
$957$	0	0
$958$	−24.9526	−0.806182
$959$	−1.36273	−0.0440049
$960$	0	0
$961$	−30.9991	−0.999970
$962$	29.8653	0.962894
$963$	0	0
$964$	25.7253	0.828555
$965$	−12.9010	−0.415298
$966$	0	0
$967$	2.13830	0.0687631	0.0343815	−	0.999409i	$-0.489054\pi$
$967$	2.13830	0.0687631	0.0343815	+	0.999409i	$0.489054\pi$
$968$	−16.7397	−0.538034
$969$	0	0
$970$	−35.1930	−1.12998
$971$	−31.2796	−1.00381	−0.501904	−	0.864923i	$-0.667367\pi$
$971$	−31.2796	−1.00381	−0.501904	+	0.864923i	$0.667367\pi$
$972$	0	0
$973$	2.18641	0.0700932
$974$	−49.4129	−1.58329
$975$	0	0
$976$	−3.76559	−0.120534
$977$	12.3738	0.395873	0.197936	−	0.980215i	$-0.436576\pi$
$977$	12.3738	0.395873	0.197936	+	0.980215i	$0.436576\pi$
$978$	0	0
$979$	2.85609	0.0912811
$980$	−3.10762	−0.0992694
$981$	0	0
$982$	90.5799	2.89052
$983$	−24.5785	−0.783932	−0.391966	−	0.919980i	$-0.628205\pi$
$983$	−24.5785	−0.783932	−0.391966	+	0.919980i	$0.628205\pi$
$984$	0	0
$985$	11.6778	0.372085
$986$	130.576	4.15838
$987$	0	0
$988$	173.618	5.52353
$989$	57.9934	1.84408
$990$	0	0
$991$	30.5101	0.969185	0.484592	−	0.874740i	$-0.338968\pi$
$991$	30.5101	0.969185	0.484592	+	0.874740i	$0.338968\pi$
$992$	0.113583	0.00360627
$993$	0	0
$994$	−85.7497	−2.71981
$995$	8.35160	0.264763
$996$	0	0
$997$	−51.1439	−1.61974	−0.809871	−	0.586608i	$-0.800463\pi$
$997$	−51.1439	−1.61974	−0.809871	+	0.586608i	$0.800463\pi$
$998$	−71.6461	−2.26792
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.s.1.10		10
3.2	odd	2		1335.2.a.j.1.1	✓	10
15.14	odd	2		6675.2.a.z.1.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.j.1.1	✓	10	3.2	odd	2
4005.2.a.s.1.10		10	1.1	even	1	trivial
6675.2.a.z.1.10		10	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)

\(f(q)\)	\(=\)	\(q+2.51771 q^{2} +4.33887 q^{4} -1.00000 q^{5} -2.77781 q^{7} +5.88859 q^{8} +O(q^{10})\)	\(q+2.51771 q^{2} +4.33887 q^{4} -1.00000 q^{5} -2.77781 q^{7} +5.88859 q^{8} -2.51771 q^{10} +2.85609 q^{11} -6.80048 q^{13} -6.99372 q^{14} +6.14803 q^{16} -5.31202 q^{17} -5.88409 q^{19} -4.33887 q^{20} +7.19082 q^{22} +5.46553 q^{23} +1.00000 q^{25} -17.1216 q^{26} -12.0525 q^{28} -9.76331 q^{29} +0.0306833 q^{31} +3.70178 q^{32} -13.3741 q^{34} +2.77781 q^{35} -1.74430 q^{37} -14.8144 q^{38} -5.88859 q^{40} -4.43445 q^{41} +10.6108 q^{43} +12.3922 q^{44} +13.7606 q^{46} +9.83555 q^{47} +0.716229 q^{49} +2.51771 q^{50} -29.5064 q^{52} -3.57570 q^{53} -2.85609 q^{55} -16.3574 q^{56} -24.5812 q^{58} -5.04048 q^{59} -0.612487 q^{61} +0.0772518 q^{62} -2.97604 q^{64} +6.80048 q^{65} -12.3743 q^{67} -23.0481 q^{68} +6.99372 q^{70} +12.2609 q^{71} -3.42742 q^{73} -4.39164 q^{74} -25.5303 q^{76} -7.93368 q^{77} +12.9489 q^{79} -6.14803 q^{80} -11.1647 q^{82} -8.84466 q^{83} +5.31202 q^{85} +26.7148 q^{86} +16.8184 q^{88} +1.00000 q^{89} +18.8904 q^{91} +23.7142 q^{92} +24.7631 q^{94} +5.88409 q^{95} +13.9782 q^{97} +1.80326 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7}+O(q^{10}) \) 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7	\( 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7} + q^{10} - 10 q^{11} + 5 q^{13} - 13 q^{14} + 19 q^{16} - 9 q^{17} - 6 q^{19} - 13 q^{20} + 3 q^{23} + 10 q^{25} - 14 q^{26} - 18 q^{28} - 38 q^{29} + 2 q^{31} - 16 q^{32} - 8 q^{34} + q^{35} + 9 q^{37} - 20 q^{38} - 36 q^{41} - 7 q^{43} - 16 q^{44} + 2 q^{46} + 23 q^{47} + 25 q^{49} - q^{50} + 13 q^{52} - 27 q^{53} + 10 q^{55} - 41 q^{56} - 32 q^{58} - 20 q^{59} + 30 q^{61} + 2 q^{62} - 2 q^{64} - 5 q^{65} - 5 q^{67} + 10 q^{68} + 13 q^{70} - 24 q^{71} - 19 q^{73} - 42 q^{74} - 30 q^{76} - 18 q^{77} + 12 q^{79} - 19 q^{80} + 29 q^{82} - 3 q^{83} + 9 q^{85} - 38 q^{86} - 16 q^{88} + 10 q^{89} - 6 q^{91} - 32 q^{92} + 17 q^{94} + 6 q^{95} + 3 q^{97} + 37 q^{98}+O(q^{100}) \) 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7 + q^10 - 10 * q^11 + 5 * q^13 - 13 * q^14 + 19 * q^16 - 9 * q^17 - 6 * q^19 - 13 * q^20 + 3 * q^23 + 10 * q^25 - 14 * q^26 - 18 * q^28 - 38 * q^29 + 2 * q^31 - 16 * q^32 - 8 * q^34 + q^35 + 9 * q^37 - 20 * q^38 - 36 * q^41 - 7 * q^43 - 16 * q^44 + 2 * q^46 + 23 * q^47 + 25 * q^49 - q^50 + 13 * q^52 - 27 * q^53 + 10 * q^55 - 41 * q^56 - 32 * q^58 - 20 * q^59 + 30 * q^61 + 2 * q^62 - 2 * q^64 - 5 * q^65 - 5 * q^67 + 10 * q^68 + 13 * q^70 - 24 * q^71 - 19 * q^73 - 42 * q^74 - 30 * q^76 - 18 * q^77 + 12 * q^79 - 19 * q^80 + 29 * q^82 - 3 * q^83 + 9 * q^85 - 38 * q^86 - 16 * q^88 + 10 * q^89 - 6 * q^91 - 32 * q^92 + 17 * q^94 + 6 * q^95 + 3 * q^97 + 37 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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