LMFDB - Embedded newform 1890.2.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1890,2,Mod(1,1890)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1890, 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("1890.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1890.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:	$15.0917259820$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{97}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 24 $ x^2 - x - 24
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.42443$ of defining polynomial
Character	$\chi$	$=$	1890.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} +6.42443 q^{11} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -5.42443 q^{17} +8.42443 q^{19} +1.00000 q^{20} +6.42443 q^{22} -3.42443 q^{23} +1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{28} -3.42443 q^{29} -1.42443 q^{31} +1.00000 q^{32} -5.42443 q^{34} -1.00000 q^{35} -8.84886 q^{37} +8.42443 q^{38} +1.00000 q^{40} +10.4244 q^{41} -7.00000 q^{43} +6.42443 q^{44} -3.42443 q^{46} +10.4244 q^{47} +1.00000 q^{49} +1.00000 q^{50} +1.00000 q^{52} +5.00000 q^{53} +6.42443 q^{55} -1.00000 q^{56} -3.42443 q^{58} +7.42443 q^{59} +10.8489 q^{61} -1.42443 q^{62} +1.00000 q^{64} +1.00000 q^{65} -3.00000 q^{67} -5.42443 q^{68} -1.00000 q^{70} +5.42443 q^{71} +2.42443 q^{73} -8.84886 q^{74} +8.42443 q^{76} -6.42443 q^{77} -8.00000 q^{79} +1.00000 q^{80} +10.4244 q^{82} +7.57557 q^{83} -5.42443 q^{85} -7.00000 q^{86} +6.42443 q^{88} +1.00000 q^{89} -1.00000 q^{91} -3.42443 q^{92} +10.4244 q^{94} +8.42443 q^{95} -10.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{8} + 2 q^{10} + 3 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} - q^{17} + 7 q^{19} + 2 q^{20} + 3 q^{22} + 3 q^{23} + 2 q^{25} + 2 q^{26} - 2 q^{28} + 3 q^{29} + 7 q^{31} + 2 q^{32} - q^{34} - 2 q^{35} + 2 q^{37} + 7 q^{38} + 2 q^{40} + 11 q^{41} - 14 q^{43} + 3 q^{44} + 3 q^{46} + 11 q^{47} + 2 q^{49} + 2 q^{50} + 2 q^{52} + 10 q^{53} + 3 q^{55} - 2 q^{56} + 3 q^{58} + 5 q^{59} + 2 q^{61} + 7 q^{62} + 2 q^{64} + 2 q^{65} - 6 q^{67} - q^{68} - 2 q^{70} + q^{71} - 5 q^{73} + 2 q^{74} + 7 q^{76} - 3 q^{77} - 16 q^{79} + 2 q^{80} + 11 q^{82} + 25 q^{83} - q^{85} - 14 q^{86} + 3 q^{88} + 2 q^{89} - 2 q^{91} + 3 q^{92} + 11 q^{94} + 7 q^{95} - 20 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 2 * q^8 + 2 * q^10 + 3 * q^11 + 2 * q^13 - 2 * q^14 + 2 * q^16 - q^17 + 7 * q^19 + 2 * q^20 + 3 * q^22 + 3 * q^23 + 2 * q^25 + 2 * q^26 - 2 * q^28 + 3 * q^29 + 7 * q^31 + 2 * q^32 - q^34 - 2 * q^35 + 2 * q^37 + 7 * q^38 + 2 * q^40 + 11 * q^41 - 14 * q^43 + 3 * q^44 + 3 * q^46 + 11 * q^47 + 2 * q^49 + 2 * q^50 + 2 * q^52 + 10 * q^53 + 3 * q^55 - 2 * q^56 + 3 * q^58 + 5 * q^59 + 2 * q^61 + 7 * q^62 + 2 * q^64 + 2 * q^65 - 6 * q^67 - q^68 - 2 * q^70 + q^71 - 5 * q^73 + 2 * q^74 + 7 * q^76 - 3 * q^77 - 16 * q^79 + 2 * q^80 + 11 * q^82 + 25 * q^83 - q^85 - 14 * q^86 + 3 * q^88 + 2 * q^89 - 2 * q^91 + 3 * q^92 + 11 * q^94 + 7 * q^95 - 20 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	6.42443	1.93704	0.968519	−	0.248939i	$-0.0800820\pi$
$11$	6.42443	1.93704	0.968519	+	0.248939i	$0.0800820\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.42443	−1.31562	−0.657809	−	0.753185i	$-0.728516\pi$
$17$	−5.42443	−1.31562	−0.657809	+	0.753185i	$0.728516\pi$
$18$	0	0
$19$	8.42443	1.93270	0.966348	−	0.257237i	$-0.0828122\pi$
$19$	8.42443	1.93270	0.966348	+	0.257237i	$0.0828122\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	6.42443	1.36969
$23$	−3.42443	−0.714043	−0.357021	−	0.934096i	$-0.616208\pi$
$23$	−3.42443	−0.714043	−0.357021	+	0.934096i	$0.616208\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−3.42443	−0.635900	−0.317950	−	0.948107i	$-0.602994\pi$
$29$	−3.42443	−0.635900	−0.317950	+	0.948107i	$0.602994\pi$
$30$	0	0
$31$	−1.42443	−0.255835	−0.127917	−	0.991785i	$-0.540829\pi$
$31$	−1.42443	−0.255835	−0.127917	+	0.991785i	$0.540829\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.42443	−0.930282
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−8.84886	−1.45474	−0.727372	−	0.686244i	$-0.759258\pi$
$37$	−8.84886	−1.45474	−0.727372	+	0.686244i	$0.759258\pi$
$38$	8.42443	1.36662
$39$	0	0
$40$	1.00000	0.158114
$41$	10.4244	1.62802	0.814011	−	0.580849i	$-0.197279\pi$
$41$	10.4244	1.62802	0.814011	+	0.580849i	$0.197279\pi$
$42$	0	0
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	6.42443	0.968519
$45$	0	0
$46$	−3.42443	−0.504904
$47$	10.4244	1.52056	0.760280	−	0.649596i	$-0.225062\pi$
$47$	10.4244	1.52056	0.760280	+	0.649596i	$0.225062\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	1.00000	0.138675
$53$	5.00000	0.686803	0.343401	−	0.939189i	$-0.388421\pi$
$53$	5.00000	0.686803	0.343401	+	0.939189i	$0.388421\pi$
$54$	0	0
$55$	6.42443	0.866270
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−3.42443	−0.449650
$59$	7.42443	0.966578	0.483289	−	0.875461i	$-0.339442\pi$
$59$	7.42443	0.966578	0.483289	+	0.875461i	$0.339442\pi$
$60$	0	0
$61$	10.8489	1.38905	0.694527	−	0.719467i	$-0.255614\pi$
$61$	10.8489	1.38905	0.694527	+	0.719467i	$0.255614\pi$
$62$	−1.42443	−0.180903
$63$	0	0
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	0	0
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	−5.42443	−0.657809
$69$	0	0
$70$	−1.00000	−0.119523
$71$	5.42443	0.643761	0.321881	−	0.946780i	$-0.395685\pi$
$71$	5.42443	0.643761	0.321881	+	0.946780i	$0.395685\pi$
$72$	0	0
$73$	2.42443	0.283758	0.141879	−	0.989884i	$-0.454686\pi$
$73$	2.42443	0.283758	0.141879	+	0.989884i	$0.454686\pi$
$74$	−8.84886	−1.02866
$75$	0	0
$76$	8.42443	0.966348
$77$	−6.42443	−0.732132
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	10.4244	1.15119
$83$	7.57557	0.831527	0.415763	−	0.909473i	$-0.363514\pi$
$83$	7.57557	0.831527	0.415763	+	0.909473i	$0.363514\pi$
$84$	0	0
$85$	−5.42443	−0.588362
$86$	−7.00000	−0.754829
$87$	0	0
$88$	6.42443	0.684846
$89$	1.00000	0.106000	0.0529999	−	0.998595i	$-0.483122\pi$
$89$	1.00000	0.106000	0.0529999	+	0.998595i	$0.483122\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	−3.42443	−0.357021
$93$	0	0
$94$	10.4244	1.07520
$95$	8.42443	0.864328
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−13.2733	−1.32074	−0.660371	−	0.750940i	$-0.729601\pi$
$101$	−13.2733	−1.32074	−0.660371	+	0.750940i	$0.729601\pi$
$102$	0	0
$103$	−9.42443	−0.928617	−0.464308	−	0.885674i	$-0.653697\pi$
$103$	−9.42443	−0.928617	−0.464308	+	0.885674i	$0.653697\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	5.00000	0.485643
$107$	2.84886	0.275409	0.137705	−	0.990473i	$-0.456027\pi$
$107$	2.84886	0.275409	0.137705	+	0.990473i	$0.456027\pi$
$108$	0	0
$109$	−2.42443	−0.232218	−0.116109	−	0.993236i	$-0.537042\pi$
$109$	−2.42443	−0.232218	−0.116109	+	0.993236i	$0.537042\pi$
$110$	6.42443	0.612545
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	19.2733	1.81308	0.906539	−	0.422122i	$-0.138715\pi$
$113$	19.2733	1.81308	0.906539	+	0.422122i	$0.138715\pi$
$114$	0	0
$115$	−3.42443	−0.319330
$116$	−3.42443	−0.317950
$117$	0	0
$118$	7.42443	0.683474
$119$	5.42443	0.497257
$120$	0	0
$121$	30.2733	2.75212
$122$	10.8489	0.982209
$123$	0	0
$124$	−1.42443	−0.127917
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.5756	−1.02717	−0.513583	−	0.858040i	$-0.671682\pi$
$127$	−11.5756	−1.02717	−0.513583	+	0.858040i	$0.671682\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	1.00000	0.0877058
$131$	−20.2733	−1.77129	−0.885643	−	0.464367i	$-0.846282\pi$
$131$	−20.2733	−1.77129	−0.885643	+	0.464367i	$0.846282\pi$
$132$	0	0
$133$	−8.42443	−0.730491
$134$	−3.00000	−0.259161
$135$	0	0
$136$	−5.42443	−0.465141
$137$	2.42443	0.207133	0.103566	−	0.994623i	$-0.466975\pi$
$137$	2.42443	0.207133	0.103566	+	0.994623i	$0.466975\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	5.42443	0.455208
$143$	6.42443	0.537238
$144$	0	0
$145$	−3.42443	−0.284383
$146$	2.42443	0.200647
$147$	0	0
$148$	−8.84886	−0.727372
$149$	0.575571	0.0471526	0.0235763	−	0.999722i	$-0.492495\pi$
$149$	0.575571	0.0471526	0.0235763	+	0.999722i	$0.492495\pi$
$150$	0	0
$151$	−16.8489	−1.37114	−0.685570	−	0.728006i	$-0.740447\pi$
$151$	−16.8489	−1.37114	−0.685570	+	0.728006i	$0.740447\pi$
$152$	8.42443	0.683311
$153$	0	0
$154$	−6.42443	−0.517695
$155$	−1.42443	−0.114413
$156$	0	0
$157$	7.42443	0.592534	0.296267	−	0.955105i	$-0.404258\pi$
$157$	7.42443	0.592534	0.296267	+	0.955105i	$0.404258\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	1.00000	0.0790569
$161$	3.42443	0.269883
$162$	0	0
$163$	−2.57557	−0.201734	−0.100867	−	0.994900i	$-0.532162\pi$
$163$	−2.57557	−0.201734	−0.100867	+	0.994900i	$0.532162\pi$
$164$	10.4244	0.814011
$165$	0	0
$166$	7.57557	0.587978
$167$	1.15114	0.0890781	0.0445390	−	0.999008i	$-0.485818\pi$
$167$	1.15114	0.0890781	0.0445390	+	0.999008i	$0.485818\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	−5.42443	−0.416035
$171$	0	0
$172$	−7.00000	−0.533745
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	6.42443	0.484260
$177$	0	0
$178$	1.00000	0.0749532
$179$	−21.2733	−1.59004	−0.795020	−	0.606583i	$-0.792540\pi$
$179$	−21.2733	−1.59004	−0.795020	+	0.606583i	$0.792540\pi$
$180$	0	0
$181$	−22.2733	−1.65556	−0.827780	−	0.561053i	$-0.810397\pi$
$181$	−22.2733	−1.65556	−0.827780	+	0.561053i	$0.810397\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	−3.42443	−0.252452
$185$	−8.84886	−0.650581
$186$	0	0
$187$	−34.8489	−2.54840
$188$	10.4244	0.760280
$189$	0	0
$190$	8.42443	0.611172
$191$	−20.4244	−1.47786	−0.738930	−	0.673782i	$-0.764669\pi$
$191$	−20.4244	−1.47786	−0.738930	+	0.673782i	$0.764669\pi$
$192$	0	0
$193$	15.4244	1.11027	0.555137	−	0.831759i	$-0.312666\pi$
$193$	15.4244	1.11027	0.555137	+	0.831759i	$0.312666\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	−22.4244	−1.59767	−0.798837	−	0.601547i	$-0.794551\pi$
$197$	−22.4244	−1.59767	−0.798837	+	0.601547i	$0.794551\pi$
$198$	0	0
$199$	−3.00000	−0.212664	−0.106332	−	0.994331i	$-0.533911\pi$
$199$	−3.00000	−0.212664	−0.106332	+	0.994331i	$0.533911\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−13.2733	−0.933905
$203$	3.42443	0.240348
$204$	0	0
$205$	10.4244	0.728074
$206$	−9.42443	−0.656631
$207$	0	0
$208$	1.00000	0.0693375
$209$	54.1221	3.74371
$210$	0	0
$211$	7.42443	0.511119	0.255559	−	0.966793i	$-0.417740\pi$
$211$	7.42443	0.511119	0.255559	+	0.966793i	$0.417740\pi$
$212$	5.00000	0.343401
$213$	0	0
$214$	2.84886	0.194744
$215$	−7.00000	−0.477396
$216$	0	0
$217$	1.42443	0.0966965
$218$	−2.42443	−0.164203
$219$	0	0
$220$	6.42443	0.433135
$221$	−5.42443	−0.364887
$222$	0	0
$223$	−7.15114	−0.478876	−0.239438	−	0.970912i	$-0.576963\pi$
$223$	−7.15114	−0.478876	−0.239438	+	0.970912i	$0.576963\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	19.2733	1.28204
$227$	−11.0000	−0.730096	−0.365048	−	0.930989i	$-0.618947\pi$
$227$	−11.0000	−0.730096	−0.365048	+	0.930989i	$0.618947\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	−3.42443	−0.225800
$231$	0	0
$232$	−3.42443	−0.224825
$233$	−14.4244	−0.944976	−0.472488	−	0.881337i	$-0.656644\pi$
$233$	−14.4244	−0.944976	−0.472488	+	0.881337i	$0.656644\pi$
$234$	0	0
$235$	10.4244	0.680015
$236$	7.42443	0.483289
$237$	0	0
$238$	5.42443	0.351614
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	16.0000	1.03065	0.515325	−	0.856995i	$-0.327671\pi$
$241$	16.0000	1.03065	0.515325	+	0.856995i	$0.327671\pi$
$242$	30.2733	1.94604
$243$	0	0
$244$	10.8489	0.694527
$245$	1.00000	0.0638877
$246$	0	0
$247$	8.42443	0.536034
$248$	−1.42443	−0.0904513
$249$	0	0
$250$	1.00000	0.0632456
$251$	−8.84886	−0.558535	−0.279267	−	0.960213i	$-0.590092\pi$
$251$	−8.84886	−0.558535	−0.279267	+	0.960213i	$0.590092\pi$
$252$	0	0
$253$	−22.0000	−1.38313
$254$	−11.5756	−0.726316
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.84886	0.427220	0.213610	−	0.976919i	$-0.431478\pi$
$257$	6.84886	0.427220	0.213610	+	0.976919i	$0.431478\pi$
$258$	0	0
$259$	8.84886	0.549841
$260$	1.00000	0.0620174
$261$	0	0
$262$	−20.2733	−1.25249
$263$	18.5756	1.14542	0.572709	−	0.819758i	$-0.305892\pi$
$263$	18.5756	1.14542	0.572709	+	0.819758i	$0.305892\pi$
$264$	0	0
$265$	5.00000	0.307148
$266$	−8.42443	−0.516535
$267$	0	0
$268$	−3.00000	−0.183254
$269$	8.84886	0.539524	0.269762	−	0.962927i	$-0.413055\pi$
$269$	8.84886	0.539524	0.269762	+	0.962927i	$0.413055\pi$
$270$	0	0
$271$	−22.6977	−1.37879	−0.689394	−	0.724387i	$-0.742123\pi$
$271$	−22.6977	−1.37879	−0.689394	+	0.724387i	$0.742123\pi$
$272$	−5.42443	−0.328904
$273$	0	0
$274$	2.42443	0.146465
$275$	6.42443	0.387408
$276$	0	0
$277$	−11.1511	−0.670007	−0.335004	−	0.942217i	$-0.608738\pi$
$277$	−11.1511	−0.670007	−0.335004	+	0.942217i	$0.608738\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	−3.15114	−0.187981	−0.0939907	−	0.995573i	$-0.529962\pi$
$281$	−3.15114	−0.187981	−0.0939907	+	0.995573i	$0.529962\pi$
$282$	0	0
$283$	22.0000	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$284$	5.42443	0.321881
$285$	0	0
$286$	6.42443	0.379884
$287$	−10.4244	−0.615335
$288$	0	0
$289$	12.4244	0.730849
$290$	−3.42443	−0.201089
$291$	0	0
$292$	2.42443	0.141879
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	0	0
$295$	7.42443	0.432267
$296$	−8.84886	−0.514329
$297$	0	0
$298$	0.575571	0.0333419
$299$	−3.42443	−0.198040
$300$	0	0
$301$	7.00000	0.403473
$302$	−16.8489	−0.969543
$303$	0	0
$304$	8.42443	0.483174
$305$	10.8489	0.621204
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	−6.42443	−0.366066
$309$	0	0
$310$	−1.42443	−0.0809021
$311$	27.6977	1.57059	0.785297	−	0.619120i	$-0.212510\pi$
$311$	27.6977	1.57059	0.785297	+	0.619120i	$0.212510\pi$
$312$	0	0
$313$	−13.2733	−0.750251	−0.375125	−	0.926974i	$-0.622400\pi$
$313$	−13.2733	−0.750251	−0.375125	+	0.926974i	$0.622400\pi$
$314$	7.42443	0.418985
$315$	0	0
$316$	−8.00000	−0.450035
$317$	−21.5756	−1.21180	−0.605902	−	0.795539i	$-0.707188\pi$
$317$	−21.5756	−1.21180	−0.605902	+	0.795539i	$0.707188\pi$
$318$	0	0
$319$	−22.0000	−1.23176
$320$	1.00000	0.0559017
$321$	0	0
$322$	3.42443	0.190836
$323$	−45.6977	−2.54269
$324$	0	0
$325$	1.00000	0.0554700
$326$	−2.57557	−0.142648
$327$	0	0
$328$	10.4244	0.575593
$329$	−10.4244	−0.574717
$330$	0	0
$331$	−25.1221	−1.38084	−0.690419	−	0.723410i	$-0.742574\pi$
$331$	−25.1221	−1.38084	−0.690419	+	0.723410i	$0.742574\pi$
$332$	7.57557	0.415763
$333$	0	0
$334$	1.15114	0.0629877
$335$	−3.00000	−0.163908
$336$	0	0
$337$	−4.27329	−0.232781	−0.116390	−	0.993204i	$-0.537132\pi$
$337$	−4.27329	−0.232781	−0.116390	+	0.993204i	$0.537132\pi$
$338$	−12.0000	−0.652714
$339$	0	0
$340$	−5.42443	−0.294181
$341$	−9.15114	−0.495562
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−7.00000	−0.377415
$345$	0	0
$346$	18.0000	0.967686
$347$	19.6977	1.05743	0.528714	−	0.848800i	$-0.322674\pi$
$347$	19.6977	1.05743	0.528714	+	0.848800i	$0.322674\pi$
$348$	0	0
$349$	−23.4244	−1.25388	−0.626940	−	0.779067i	$-0.715693\pi$
$349$	−23.4244	−1.25388	−0.626940	+	0.779067i	$0.715693\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	6.42443	0.342423
$353$	17.4244	0.927409	0.463704	−	0.885990i	$-0.346520\pi$
$353$	17.4244	0.927409	0.463704	+	0.885990i	$0.346520\pi$
$354$	0	0
$355$	5.42443	0.287899
$356$	1.00000	0.0529999
$357$	0	0
$358$	−21.2733	−1.12433
$359$	−23.8489	−1.25869	−0.629347	−	0.777124i	$-0.716678\pi$
$359$	−23.8489	−1.25869	−0.629347	+	0.777124i	$0.716678\pi$
$360$	0	0
$361$	51.9710	2.73532
$362$	−22.2733	−1.17066
$363$	0	0
$364$	−1.00000	−0.0524142
$365$	2.42443	0.126900
$366$	0	0
$367$	2.27329	0.118665	0.0593323	−	0.998238i	$-0.481103\pi$
$367$	2.27329	0.118665	0.0593323	+	0.998238i	$0.481103\pi$
$368$	−3.42443	−0.178511
$369$	0	0
$370$	−8.84886	−0.460030
$371$	−5.00000	−0.259587
$372$	0	0
$373$	−9.69772	−0.502129	−0.251064	−	0.967970i	$-0.580781\pi$
$373$	−9.69772	−0.502129	−0.251064	+	0.967970i	$0.580781\pi$
$374$	−34.8489	−1.80199
$375$	0	0
$376$	10.4244	0.537599
$377$	−3.42443	−0.176367
$378$	0	0
$379$	−37.6977	−1.93640	−0.968201	−	0.250174i	$-0.919512\pi$
$379$	−37.6977	−1.93640	−0.968201	+	0.250174i	$0.919512\pi$
$380$	8.42443	0.432164
$381$	0	0
$382$	−20.4244	−1.04500
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	−6.42443	−0.327419
$386$	15.4244	0.785083
$387$	0	0
$388$	−10.0000	−0.507673
$389$	−31.6977	−1.60714	−0.803569	−	0.595212i	$-0.797068\pi$
$389$	−31.6977	−1.60714	−0.803569	+	0.595212i	$0.797068\pi$
$390$	0	0
$391$	18.5756	0.939407
$392$	1.00000	0.0505076
$393$	0	0
$394$	−22.4244	−1.12973
$395$	−8.00000	−0.402524
$396$	0	0
$397$	11.6977	0.587092	0.293546	−	0.955945i	$-0.405165\pi$
$397$	11.6977	0.587092	0.293546	+	0.955945i	$0.405165\pi$
$398$	−3.00000	−0.150376
$399$	0	0
$400$	1.00000	0.0500000
$401$	−5.15114	−0.257236	−0.128618	−	0.991694i	$-0.541054\pi$
$401$	−5.15114	−0.257236	−0.128618	+	0.991694i	$0.541054\pi$
$402$	0	0
$403$	−1.42443	−0.0709559
$404$	−13.2733	−0.660371
$405$	0	0
$406$	3.42443	0.169952
$407$	−56.8489	−2.81789
$408$	0	0
$409$	−7.69772	−0.380628	−0.190314	−	0.981723i	$-0.560951\pi$
$409$	−7.69772	−0.380628	−0.190314	+	0.981723i	$0.560951\pi$
$410$	10.4244	0.514826
$411$	0	0
$412$	−9.42443	−0.464308
$413$	−7.42443	−0.365332
$414$	0	0
$415$	7.57557	0.371870
$416$	1.00000	0.0490290
$417$	0	0
$418$	54.1221	2.64720
$419$	−2.27329	−0.111057	−0.0555287	−	0.998457i	$-0.517684\pi$
$419$	−2.27329	−0.111057	−0.0555287	+	0.998457i	$0.517684\pi$
$420$	0	0
$421$	35.2733	1.71911	0.859557	−	0.511039i	$-0.170739\pi$
$421$	35.2733	1.71911	0.859557	+	0.511039i	$0.170739\pi$
$422$	7.42443	0.361416
$423$	0	0
$424$	5.00000	0.242821
$425$	−5.42443	−0.263123
$426$	0	0
$427$	−10.8489	−0.525013
$428$	2.84886	0.137705
$429$	0	0
$430$	−7.00000	−0.337570
$431$	37.2733	1.79539	0.897696	−	0.440616i	$-0.145240\pi$
$431$	37.2733	1.79539	0.897696	+	0.440616i	$0.145240\pi$
$432$	0	0
$433$	−33.2733	−1.59901	−0.799506	−	0.600658i	$-0.794905\pi$
$433$	−33.2733	−1.59901	−0.799506	+	0.600658i	$0.794905\pi$
$434$	1.42443	0.0683748
$435$	0	0
$436$	−2.42443	−0.116109
$437$	−28.8489	−1.38003
$438$	0	0
$439$	−14.1511	−0.675397	−0.337699	−	0.941254i	$-0.609648\pi$
$439$	−14.1511	−0.675397	−0.337699	+	0.941254i	$0.609648\pi$
$440$	6.42443	0.306273
$441$	0	0
$442$	−5.42443	−0.258014
$443$	5.69772	0.270707	0.135353	−	0.990797i	$-0.456783\pi$
$443$	5.69772	0.270707	0.135353	+	0.990797i	$0.456783\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−7.15114	−0.338616
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	13.6977	0.646435	0.323218	−	0.946325i	$-0.395235\pi$
$449$	13.6977	0.646435	0.323218	+	0.946325i	$0.395235\pi$
$450$	0	0
$451$	66.9710	3.15354
$452$	19.2733	0.906539
$453$	0	0
$454$	−11.0000	−0.516256
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	17.4244	0.815080	0.407540	−	0.913187i	$-0.366387\pi$
$457$	17.4244	0.815080	0.407540	+	0.913187i	$0.366387\pi$
$458$	2.00000	0.0934539
$459$	0	0
$460$	−3.42443	−0.159665
$461$	−26.4244	−1.23071	−0.615354	−	0.788251i	$-0.710987\pi$
$461$	−26.4244	−1.23071	−0.615354	+	0.788251i	$0.710987\pi$
$462$	0	0
$463$	42.9710	1.99703	0.998516	−	0.0544606i	$-0.0173439\pi$
$463$	42.9710	1.99703	0.998516	+	0.0544606i	$0.0173439\pi$
$464$	−3.42443	−0.158975
$465$	0	0
$466$	−14.4244	−0.668199
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	3.00000	0.138527
$470$	10.4244	0.480843
$471$	0	0
$472$	7.42443	0.341737
$473$	−44.9710	−2.06777
$474$	0	0
$475$	8.42443	0.386539
$476$	5.42443	0.248628
$477$	0	0
$478$	0	0
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	−8.84886	−0.403473
$482$	16.0000	0.728780
$483$	0	0
$484$	30.2733	1.37606
$485$	−10.0000	−0.454077
$486$	0	0
$487$	−6.42443	−0.291119	−0.145559	−	0.989350i	$-0.546498\pi$
$487$	−6.42443	−0.291119	−0.145559	+	0.989350i	$0.546498\pi$
$488$	10.8489	0.491105
$489$	0	0
$490$	1.00000	0.0451754
$491$	−17.6977	−0.798687	−0.399343	−	0.916801i	$-0.630762\pi$
$491$	−17.6977	−0.798687	−0.399343	+	0.916801i	$0.630762\pi$
$492$	0	0
$493$	18.5756	0.836602
$494$	8.42443	0.379033
$495$	0	0
$496$	−1.42443	−0.0639587
$497$	−5.42443	−0.243319
$498$	0	0
$499$	−3.15114	−0.141064	−0.0705322	−	0.997509i	$-0.522470\pi$
$499$	−3.15114	−0.141064	−0.0705322	+	0.997509i	$0.522470\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−8.84886	−0.394944
$503$	28.9710	1.29175	0.645877	−	0.763442i	$-0.276492\pi$
$503$	28.9710	1.29175	0.645877	+	0.763442i	$0.276492\pi$
$504$	0	0
$505$	−13.2733	−0.590654
$506$	−22.0000	−0.978019
$507$	0	0
$508$	−11.5756	−0.513583
$509$	−3.15114	−0.139672	−0.0698360	−	0.997558i	$-0.522248\pi$
$509$	−3.15114	−0.139672	−0.0698360	+	0.997558i	$0.522248\pi$
$510$	0	0
$511$	−2.42443	−0.107250
$512$	1.00000	0.0441942
$513$	0	0
$514$	6.84886	0.302090
$515$	−9.42443	−0.415290
$516$	0	0
$517$	66.9710	2.94538
$518$	8.84886	0.388796
$519$	0	0
$520$	1.00000	0.0438529
$521$	21.0000	0.920027	0.460013	−	0.887912i	$-0.347845\pi$
$521$	21.0000	0.920027	0.460013	+	0.887912i	$0.347845\pi$
$522$	0	0
$523$	12.8489	0.561841	0.280921	−	0.959731i	$-0.409360\pi$
$523$	12.8489	0.561841	0.280921	+	0.959731i	$0.409360\pi$
$524$	−20.2733	−0.885643
$525$	0	0
$526$	18.5756	0.809933
$527$	7.72671	0.336581
$528$	0	0
$529$	−11.2733	−0.490143
$530$	5.00000	0.217186
$531$	0	0
$532$	−8.42443	−0.365245
$533$	10.4244	0.451532
$534$	0	0
$535$	2.84886	0.123167
$536$	−3.00000	−0.129580
$537$	0	0
$538$	8.84886	0.381501
$539$	6.42443	0.276720
$540$	0	0
$541$	12.7267	0.547164	0.273582	−	0.961849i	$-0.411791\pi$
$541$	12.7267	0.547164	0.273582	+	0.961849i	$0.411791\pi$
$542$	−22.6977	−0.974950
$543$	0	0
$544$	−5.42443	−0.232570
$545$	−2.42443	−0.103851
$546$	0	0
$547$	−29.6977	−1.26978	−0.634891	−	0.772601i	$-0.718955\pi$
$547$	−29.6977	−1.26978	−0.634891	+	0.772601i	$0.718955\pi$
$548$	2.42443	0.103566
$549$	0	0
$550$	6.42443	0.273939
$551$	−28.8489	−1.22900
$552$	0	0
$553$	8.00000	0.340195
$554$	−11.1511	−0.473767
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−10.2733	−0.435293	−0.217647	−	0.976028i	$-0.569838\pi$
$557$	−10.2733	−0.435293	−0.217647	+	0.976028i	$0.569838\pi$
$558$	0	0
$559$	−7.00000	−0.296068
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−3.15114	−0.132923
$563$	5.00000	0.210725	0.105362	−	0.994434i	$-0.466400\pi$
$563$	5.00000	0.210725	0.105362	+	0.994434i	$0.466400\pi$
$564$	0	0
$565$	19.2733	0.810833
$566$	22.0000	0.924729
$567$	0	0
$568$	5.42443	0.227604
$569$	−27.6977	−1.16115	−0.580574	−	0.814207i	$-0.697172\pi$
$569$	−27.6977	−1.16115	−0.580574	+	0.814207i	$0.697172\pi$
$570$	0	0
$571$	−24.2733	−1.01581	−0.507903	−	0.861414i	$-0.669579\pi$
$571$	−24.2733	−1.01581	−0.507903	+	0.861414i	$0.669579\pi$
$572$	6.42443	0.268619
$573$	0	0
$574$	−10.4244	−0.435107
$575$	−3.42443	−0.142809
$576$	0	0
$577$	−33.2733	−1.38519	−0.692593	−	0.721329i	$-0.743532\pi$
$577$	−33.2733	−1.38519	−0.692593	+	0.721329i	$0.743532\pi$
$578$	12.4244	0.516788
$579$	0	0
$580$	−3.42443	−0.142192
$581$	−7.57557	−0.314288
$582$	0	0
$583$	32.1221	1.33036
$584$	2.42443	0.100324
$585$	0	0
$586$	18.0000	0.743573
$587$	−4.27329	−0.176377	−0.0881887	−	0.996104i	$-0.528108\pi$
$587$	−4.27329	−0.176377	−0.0881887	+	0.996104i	$0.528108\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	7.42443	0.305659
$591$	0	0
$592$	−8.84886	−0.363686
$593$	−38.8489	−1.59533	−0.797666	−	0.603100i	$-0.793932\pi$
$593$	−38.8489	−1.59533	−0.797666	+	0.603100i	$0.793932\pi$
$594$	0	0
$595$	5.42443	0.222380
$596$	0.575571	0.0235763
$597$	0	0
$598$	−3.42443	−0.140035
$599$	29.8489	1.21959	0.609796	−	0.792559i	$-0.291252\pi$
$599$	29.8489	1.21959	0.609796	+	0.792559i	$0.291252\pi$
$600$	0	0
$601$	28.0000	1.14214	0.571072	−	0.820900i	$-0.306528\pi$
$601$	28.0000	1.14214	0.571072	+	0.820900i	$0.306528\pi$
$602$	7.00000	0.285299
$603$	0	0
$604$	−16.8489	−0.685570
$605$	30.2733	1.23078
$606$	0	0
$607$	−39.4244	−1.60019	−0.800094	−	0.599875i	$-0.795217\pi$
$607$	−39.4244	−1.60019	−0.800094	+	0.599875i	$0.795217\pi$
$608$	8.42443	0.341656
$609$	0	0
$610$	10.8489	0.439257
$611$	10.4244	0.421727
$612$	0	0
$613$	6.30228	0.254547	0.127273	−	0.991868i	$-0.459377\pi$
$613$	6.30228	0.254547	0.127273	+	0.991868i	$0.459377\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	−6.42443	−0.258848
$617$	−10.0000	−0.402585	−0.201292	−	0.979531i	$-0.564514\pi$
$617$	−10.0000	−0.402585	−0.201292	+	0.979531i	$0.564514\pi$
$618$	0	0
$619$	12.7267	0.511530	0.255765	−	0.966739i	$-0.417673\pi$
$619$	12.7267	0.511530	0.255765	+	0.966739i	$0.417673\pi$
$620$	−1.42443	−0.0572064
$621$	0	0
$622$	27.6977	1.11058
$623$	−1.00000	−0.0400642
$624$	0	0
$625$	1.00000	0.0400000
$626$	−13.2733	−0.530507
$627$	0	0
$628$	7.42443	0.296267
$629$	48.0000	1.91389
$630$	0	0
$631$	−24.8489	−0.989217	−0.494609	−	0.869116i	$-0.664689\pi$
$631$	−24.8489	−0.989217	−0.494609	+	0.869116i	$0.664689\pi$
$632$	−8.00000	−0.318223
$633$	0	0
$634$	−21.5756	−0.856875
$635$	−11.5756	−0.459363
$636$	0	0
$637$	1.00000	0.0396214
$638$	−22.0000	−0.870988
$639$	0	0
$640$	1.00000	0.0395285
$641$	−10.8489	−0.428504	−0.214252	−	0.976778i	$-0.568731\pi$
$641$	−10.8489	−0.428504	−0.214252	+	0.976778i	$0.568731\pi$
$642$	0	0
$643$	−36.5466	−1.44126	−0.720628	−	0.693322i	$-0.756146\pi$
$643$	−36.5466	−1.44126	−0.720628	+	0.693322i	$0.756146\pi$
$644$	3.42443	0.134941
$645$	0	0
$646$	−45.6977	−1.79795
$647$	10.4244	0.409827	0.204913	−	0.978780i	$-0.434309\pi$
$647$	10.4244	0.409827	0.204913	+	0.978780i	$0.434309\pi$
$648$	0	0
$649$	47.6977	1.87230
$650$	1.00000	0.0392232
$651$	0	0
$652$	−2.57557	−0.100867
$653$	18.2733	0.715089	0.357544	−	0.933896i	$-0.383614\pi$
$653$	18.2733	0.715089	0.357544	+	0.933896i	$0.383614\pi$
$654$	0	0
$655$	−20.2733	−0.792143
$656$	10.4244	0.407006
$657$	0	0
$658$	−10.4244	−0.406387
$659$	8.54657	0.332927	0.166464	−	0.986048i	$-0.446765\pi$
$659$	8.54657	0.332927	0.166464	+	0.986048i	$0.446765\pi$
$660$	0	0
$661$	8.30228	0.322921	0.161461	−	0.986879i	$-0.448379\pi$
$661$	8.30228	0.322921	0.161461	+	0.986879i	$0.448379\pi$
$662$	−25.1221	−0.976400
$663$	0	0
$664$	7.57557	0.293989
$665$	−8.42443	−0.326685
$666$	0	0
$667$	11.7267	0.454060
$668$	1.15114	0.0445390
$669$	0	0
$670$	−3.00000	−0.115900
$671$	69.6977	2.69065
$672$	0	0
$673$	43.1221	1.66224	0.831118	−	0.556096i	$-0.187701\pi$
$673$	43.1221	1.66224	0.831118	+	0.556096i	$0.187701\pi$
$674$	−4.27329	−0.164601
$675$	0	0
$676$	−12.0000	−0.461538
$677$	30.5466	1.17400	0.587000	−	0.809587i	$-0.300309\pi$
$677$	30.5466	1.17400	0.587000	+	0.809587i	$0.300309\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	−5.42443	−0.208017
$681$	0	0
$682$	−9.15114	−0.350415
$683$	23.6977	0.906768	0.453384	−	0.891315i	$-0.350217\pi$
$683$	23.6977	0.906768	0.453384	+	0.891315i	$0.350217\pi$
$684$	0	0
$685$	2.42443	0.0926327
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−7.00000	−0.266872
$689$	5.00000	0.190485
$690$	0	0
$691$	51.8199	1.97132	0.985660	−	0.168742i	$-0.0539706\pi$
$691$	51.8199	1.97132	0.985660	+	0.168742i	$0.0539706\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	19.6977	0.747715
$695$	−4.00000	−0.151729
$696$	0	0
$697$	−56.5466	−2.14185
$698$	−23.4244	−0.886628
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	−8.54657	−0.322800	−0.161400	−	0.986889i	$-0.551601\pi$
$701$	−8.54657	−0.322800	−0.161400	+	0.986889i	$0.551601\pi$
$702$	0	0
$703$	−74.5466	−2.81158
$704$	6.42443	0.242130
$705$	0	0
$706$	17.4244	0.655777
$707$	13.2733	0.499193
$708$	0	0
$709$	8.12214	0.305034	0.152517	−	0.988301i	$-0.451262\pi$
$709$	8.12214	0.305034	0.152517	+	0.988301i	$0.451262\pi$
$710$	5.42443	0.203575
$711$	0	0
$712$	1.00000	0.0374766
$713$	4.87786	0.182677
$714$	0	0
$715$	6.42443	0.240260
$716$	−21.2733	−0.795020
$717$	0	0
$718$	−23.8489	−0.890031
$719$	−6.84886	−0.255419	−0.127710	−	0.991812i	$-0.540763\pi$
$719$	−6.84886	−0.255419	−0.127710	+	0.991812i	$0.540763\pi$
$720$	0	0
$721$	9.42443	0.350984
$722$	51.9710	1.93416
$723$	0	0
$724$	−22.2733	−0.827780
$725$	−3.42443	−0.127180
$726$	0	0
$727$	3.72671	0.138216	0.0691081	−	0.997609i	$-0.477985\pi$
$727$	3.72671	0.138216	0.0691081	+	0.997609i	$0.477985\pi$
$728$	−1.00000	−0.0370625
$729$	0	0
$730$	2.42443	0.0897321
$731$	37.9710	1.40441
$732$	0	0
$733$	3.84886	0.142161	0.0710804	−	0.997471i	$-0.477355\pi$
$733$	3.84886	0.142161	0.0710804	+	0.997471i	$0.477355\pi$
$734$	2.27329	0.0839085
$735$	0	0
$736$	−3.42443	−0.126226
$737$	−19.2733	−0.709941
$738$	0	0
$739$	32.0000	1.17714	0.588570	−	0.808447i	$-0.299691\pi$
$739$	32.0000	1.17714	0.588570	+	0.808447i	$0.299691\pi$
$740$	−8.84886	−0.325290
$741$	0	0
$742$	−5.00000	−0.183556
$743$	−25.1221	−0.921642	−0.460821	−	0.887493i	$-0.652445\pi$
$743$	−25.1221	−0.921642	−0.460821	+	0.887493i	$0.652445\pi$
$744$	0	0
$745$	0.575571	0.0210873
$746$	−9.69772	−0.355059
$747$	0	0
$748$	−34.8489	−1.27420
$749$	−2.84886	−0.104095
$750$	0	0
$751$	−11.1511	−0.406911	−0.203455	−	0.979084i	$-0.565217\pi$
$751$	−11.1511	−0.406911	−0.203455	+	0.979084i	$0.565217\pi$
$752$	10.4244	0.380140
$753$	0	0
$754$	−3.42443	−0.124710
$755$	−16.8489	−0.613193
$756$	0	0
$757$	39.3954	1.43185	0.715926	−	0.698177i	$-0.246005\pi$
$757$	39.3954	1.43185	0.715926	+	0.698177i	$0.246005\pi$
$758$	−37.6977	−1.36924
$759$	0	0
$760$	8.42443	0.305586
$761$	−12.5756	−0.455864	−0.227932	−	0.973677i	$-0.573196\pi$
$761$	−12.5756	−0.455864	−0.227932	+	0.973677i	$0.573196\pi$
$762$	0	0
$763$	2.42443	0.0877702
$764$	−20.4244	−0.738930
$765$	0	0
$766$	−8.00000	−0.289052
$767$	7.42443	0.268081
$768$	0	0
$769$	24.5466	0.885172	0.442586	−	0.896726i	$-0.354061\pi$
$769$	24.5466	0.885172	0.442586	+	0.896726i	$0.354061\pi$
$770$	−6.42443	−0.231520
$771$	0	0
$772$	15.4244	0.555137
$773$	−16.8489	−0.606011	−0.303006	−	0.952989i	$-0.597990\pi$
$773$	−16.8489	−0.606011	−0.303006	+	0.952989i	$0.597990\pi$
$774$	0	0
$775$	−1.42443	−0.0511670
$776$	−10.0000	−0.358979
$777$	0	0
$778$	−31.6977	−1.13642
$779$	87.8199	3.14647
$780$	0	0
$781$	34.8489	1.24699
$782$	18.5756	0.664261
$783$	0	0
$784$	1.00000	0.0357143
$785$	7.42443	0.264989
$786$	0	0
$787$	−18.8489	−0.671889	−0.335945	−	0.941882i	$-0.609055\pi$
$787$	−18.8489	−0.671889	−0.335945	+	0.941882i	$0.609055\pi$
$788$	−22.4244	−0.798837
$789$	0	0
$790$	−8.00000	−0.284627
$791$	−19.2733	−0.685279
$792$	0	0
$793$	10.8489	0.385254
$794$	11.6977	0.415136
$795$	0	0
$796$	−3.00000	−0.106332
$797$	44.5466	1.57792	0.788960	−	0.614444i	$-0.210620\pi$
$797$	44.5466	1.57792	0.788960	+	0.614444i	$0.210620\pi$
$798$	0	0
$799$	−56.5466	−2.00047
$800$	1.00000	0.0353553
$801$	0	0
$802$	−5.15114	−0.181893
$803$	15.5756	0.549650
$804$	0	0
$805$	3.42443	0.120695
$806$	−1.42443	−0.0501734
$807$	0	0
$808$	−13.2733	−0.466953
$809$	28.8489	1.01427	0.507136	−	0.861866i	$-0.330704\pi$
$809$	28.8489	1.01427	0.507136	+	0.861866i	$0.330704\pi$
$810$	0	0
$811$	40.9710	1.43869	0.719343	−	0.694655i	$-0.244443\pi$
$811$	40.9710	1.43869	0.719343	+	0.694655i	$0.244443\pi$
$812$	3.42443	0.120174
$813$	0	0
$814$	−56.8489	−1.99255
$815$	−2.57557	−0.0902183
$816$	0	0
$817$	−58.9710	−2.06313
$818$	−7.69772	−0.269144
$819$	0	0
$820$	10.4244	0.364037
$821$	−12.5756	−0.438890	−0.219445	−	0.975625i	$-0.570425\pi$
$821$	−12.5756	−0.438890	−0.219445	+	0.975625i	$0.570425\pi$
$822$	0	0
$823$	25.2733	0.880971	0.440486	−	0.897760i	$-0.354806\pi$
$823$	25.2733	0.880971	0.440486	+	0.897760i	$0.354806\pi$
$824$	−9.42443	−0.328316
$825$	0	0
$826$	−7.42443	−0.258329
$827$	36.5466	1.27085	0.635424	−	0.772163i	$-0.280825\pi$
$827$	36.5466	1.27085	0.635424	+	0.772163i	$0.280825\pi$
$828$	0	0
$829$	38.8489	1.34928	0.674638	−	0.738148i	$-0.264300\pi$
$829$	38.8489	1.34928	0.674638	+	0.738148i	$0.264300\pi$
$830$	7.57557	0.262952
$831$	0	0
$832$	1.00000	0.0346688
$833$	−5.42443	−0.187945
$834$	0	0
$835$	1.15114	0.0398369
$836$	54.1221	1.87185
$837$	0	0
$838$	−2.27329	−0.0785294
$839$	−0.848858	−0.0293058	−0.0146529	−	0.999893i	$-0.504664\pi$
$839$	−0.848858	−0.0293058	−0.0146529	+	0.999893i	$0.504664\pi$
$840$	0	0
$841$	−17.2733	−0.595631
$842$	35.2733	1.21560
$843$	0	0
$844$	7.42443	0.255559
$845$	−12.0000	−0.412813
$846$	0	0
$847$	−30.2733	−1.04020
$848$	5.00000	0.171701
$849$	0	0
$850$	−5.42443	−0.186056
$851$	30.3023	1.03875
$852$	0	0
$853$	−18.2733	−0.625665	−0.312833	−	0.949808i	$-0.601278\pi$
$853$	−18.2733	−0.625665	−0.312833	+	0.949808i	$0.601278\pi$
$854$	−10.8489	−0.371240
$855$	0	0
$856$	2.84886	0.0973720
$857$	18.2733	0.624204	0.312102	−	0.950049i	$-0.398967\pi$
$857$	18.2733	0.624204	0.312102	+	0.950049i	$0.398967\pi$
$858$	0	0
$859$	−15.5756	−0.531432	−0.265716	−	0.964051i	$-0.585608\pi$
$859$	−15.5756	−0.531432	−0.265716	+	0.964051i	$0.585608\pi$
$860$	−7.00000	−0.238698
$861$	0	0
$862$	37.2733	1.26953
$863$	−50.2733	−1.71132	−0.855661	−	0.517536i	$-0.826849\pi$
$863$	−50.2733	−1.71132	−0.855661	+	0.517536i	$0.826849\pi$
$864$	0	0
$865$	18.0000	0.612018
$866$	−33.2733	−1.13067
$867$	0	0
$868$	1.42443	0.0483483
$869$	−51.3954	−1.74347
$870$	0	0
$871$	−3.00000	−0.101651
$872$	−2.42443	−0.0821015
$873$	0	0
$874$	−28.8489	−0.975827
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	32.0000	1.08056	0.540282	−	0.841484i	$-0.318318\pi$
$877$	32.0000	1.08056	0.540282	+	0.841484i	$0.318318\pi$
$878$	−14.1511	−0.477578
$879$	0	0
$880$	6.42443	0.216567
$881$	−26.2733	−0.885170	−0.442585	−	0.896727i	$-0.645938\pi$
$881$	−26.2733	−0.885170	−0.442585	+	0.896727i	$0.645938\pi$
$882$	0	0
$883$	−17.4244	−0.586379	−0.293189	−	0.956054i	$-0.594717\pi$
$883$	−17.4244	−0.586379	−0.293189	+	0.956054i	$0.594717\pi$
$884$	−5.42443	−0.182443
$885$	0	0
$886$	5.69772	0.191418
$887$	−2.12214	−0.0712546	−0.0356273	−	0.999365i	$-0.511343\pi$
$887$	−2.12214	−0.0712546	−0.0356273	+	0.999365i	$0.511343\pi$
$888$	0	0
$889$	11.5756	0.388232
$890$	1.00000	0.0335201
$891$	0	0
$892$	−7.15114	−0.239438
$893$	87.8199	2.93878
$894$	0	0
$895$	−21.2733	−0.711088
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	13.6977	0.457099
$899$	4.87786	0.162686
$900$	0	0
$901$	−27.1221	−0.903570
$902$	66.9710	2.22989
$903$	0	0
$904$	19.2733	0.641020
$905$	−22.2733	−0.740389
$906$	0	0
$907$	52.6687	1.74884	0.874418	−	0.485173i	$-0.161243\pi$
$907$	52.6687	1.74884	0.874418	+	0.485173i	$0.161243\pi$
$908$	−11.0000	−0.365048
$909$	0	0
$910$	−1.00000	−0.0331497
$911$	−5.27329	−0.174712	−0.0873559	−	0.996177i	$-0.527842\pi$
$911$	−5.27329	−0.174712	−0.0873559	+	0.996177i	$0.527842\pi$
$912$	0	0
$913$	48.6687	1.61070
$914$	17.4244	0.576349
$915$	0	0
$916$	2.00000	0.0660819
$917$	20.2733	0.669483
$918$	0	0
$919$	−34.8489	−1.14956	−0.574779	−	0.818309i	$-0.694912\pi$
$919$	−34.8489	−1.14956	−0.574779	+	0.818309i	$0.694912\pi$
$920$	−3.42443	−0.112900
$921$	0	0
$922$	−26.4244	−0.870242
$923$	5.42443	0.178547
$924$	0	0
$925$	−8.84886	−0.290949
$926$	42.9710	1.41211
$927$	0	0
$928$	−3.42443	−0.112412
$929$	−24.1221	−0.791422	−0.395711	−	0.918375i	$-0.629502\pi$
$929$	−24.1221	−0.791422	−0.395711	+	0.918375i	$0.629502\pi$
$930$	0	0
$931$	8.42443	0.276100
$932$	−14.4244	−0.472488
$933$	0	0
$934$	−12.0000	−0.392652
$935$	−34.8489	−1.13968
$936$	0	0
$937$	2.42443	0.0792026	0.0396013	−	0.999216i	$-0.487391\pi$
$937$	2.42443	0.0792026	0.0396013	+	0.999216i	$0.487391\pi$
$938$	3.00000	0.0979535
$939$	0	0
$940$	10.4244	0.340007
$941$	4.72671	0.154086	0.0770432	−	0.997028i	$-0.475452\pi$
$941$	4.72671	0.154086	0.0770432	+	0.997028i	$0.475452\pi$
$942$	0	0
$943$	−35.6977	−1.16248
$944$	7.42443	0.241645
$945$	0	0
$946$	−44.9710	−1.46213
$947$	−49.6977	−1.61496	−0.807479	−	0.589896i	$-0.799169\pi$
$947$	−49.6977	−1.61496	−0.807479	+	0.589896i	$0.799169\pi$
$948$	0	0
$949$	2.42443	0.0787003
$950$	8.42443	0.273325
$951$	0	0
$952$	5.42443	0.175807
$953$	22.5466	0.730355	0.365178	−	0.930938i	$-0.381008\pi$
$953$	22.5466	0.730355	0.365178	+	0.930938i	$0.381008\pi$
$954$	0	0
$955$	−20.4244	−0.660919
$956$	0	0
$957$	0	0
$958$	8.00000	0.258468
$959$	−2.42443	−0.0782889
$960$	0	0
$961$	−28.9710	−0.934548
$962$	−8.84886	−0.285299
$963$	0	0
$964$	16.0000	0.515325
$965$	15.4244	0.496530
$966$	0	0
$967$	−6.72671	−0.216317	−0.108158	−	0.994134i	$-0.534495\pi$
$967$	−6.72671	−0.216317	−0.108158	+	0.994134i	$0.534495\pi$
$968$	30.2733	0.973020
$969$	0	0
$970$	−10.0000	−0.321081
$971$	17.4244	0.559177	0.279588	−	0.960120i	$-0.409802\pi$
$971$	17.4244	0.559177	0.279588	+	0.960120i	$0.409802\pi$
$972$	0	0
$973$	4.00000	0.128234
$974$	−6.42443	−0.205852
$975$	0	0
$976$	10.8489	0.347263
$977$	−4.72671	−0.151221	−0.0756105	−	0.997137i	$-0.524091\pi$
$977$	−4.72671	−0.151221	−0.0756105	+	0.997137i	$0.524091\pi$
$978$	0	0
$979$	6.42443	0.205326
$980$	1.00000	0.0319438
$981$	0	0
$982$	−17.6977	−0.564757
$983$	−35.3954	−1.12894	−0.564469	−	0.825454i	$-0.690919\pi$
$983$	−35.3954	−1.12894	−0.564469	+	0.825454i	$0.690919\pi$
$984$	0	0
$985$	−22.4244	−0.714502
$986$	18.5756	0.591567
$987$	0	0
$988$	8.42443	0.268017
$989$	23.9710	0.762234
$990$	0	0
$991$	−4.54657	−0.144427	−0.0722133	−	0.997389i	$-0.523006\pi$
$991$	−4.54657	−0.144427	−0.0722133	+	0.997389i	$0.523006\pi$
$992$	−1.42443	−0.0452257
$993$	0	0
$994$	−5.42443	−0.172052
$995$	−3.00000	−0.0951064
$996$	0	0
$997$	−3.42443	−0.108453	−0.0542264	−	0.998529i	$-0.517269\pi$
$997$	−3.42443	−0.108453	−0.0542264	+	0.998529i	$0.517269\pi$
$998$	−3.15114	−0.0997477
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1890.2.a.ba.1.2	yes	2
3.2	odd	2		1890.2.a.y.1.1	✓	2
5.4	even	2		9450.2.a.el.1.2		2
15.14	odd	2		9450.2.a.es.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1890.2.a.y.1.1	✓	2	3.2	odd	2
1890.2.a.ba.1.2	yes	2	1.1	even	1	trivial
9450.2.a.el.1.2		2	5.4	even	2
9450.2.a.es.1.1		2	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:	\( N \)	\(=\)	\( 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1890.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} +6.42443 q^{11} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -5.42443 q^{17} +8.42443 q^{19} +1.00000 q^{20} +6.42443 q^{22} -3.42443 q^{23} +1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{28} -3.42443 q^{29} -1.42443 q^{31} +1.00000 q^{32} -5.42443 q^{34} -1.00000 q^{35} -8.84886 q^{37} +8.42443 q^{38} +1.00000 q^{40} +10.4244 q^{41} -7.00000 q^{43} +6.42443 q^{44} -3.42443 q^{46} +10.4244 q^{47} +1.00000 q^{49} +1.00000 q^{50} +1.00000 q^{52} +5.00000 q^{53} +6.42443 q^{55} -1.00000 q^{56} -3.42443 q^{58} +7.42443 q^{59} +10.8489 q^{61} -1.42443 q^{62} +1.00000 q^{64} +1.00000 q^{65} -3.00000 q^{67} -5.42443 q^{68} -1.00000 q^{70} +5.42443 q^{71} +2.42443 q^{73} -8.84886 q^{74} +8.42443 q^{76} -6.42443 q^{77} -8.00000 q^{79} +1.00000 q^{80} +10.4244 q^{82} +7.57557 q^{83} -5.42443 q^{85} -7.00000 q^{86} +6.42443 q^{88} +1.00000 q^{89} -1.00000 q^{91} -3.42443 q^{92} +10.4244 q^{94} +8.42443 q^{95} -10.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{8} + 2 q^{10} + 3 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} - q^{17} + 7 q^{19} + 2 q^{20} + 3 q^{22} + 3 q^{23} + 2 q^{25} + 2 q^{26} - 2 q^{28} + 3 q^{29} + 7 q^{31} + 2 q^{32} - q^{34} - 2 q^{35} + 2 q^{37} + 7 q^{38} + 2 q^{40} + 11 q^{41} - 14 q^{43} + 3 q^{44} + 3 q^{46} + 11 q^{47} + 2 q^{49} + 2 q^{50} + 2 q^{52} + 10 q^{53} + 3 q^{55} - 2 q^{56} + 3 q^{58} + 5 q^{59} + 2 q^{61} + 7 q^{62} + 2 q^{64} + 2 q^{65} - 6 q^{67} - q^{68} - 2 q^{70} + q^{71} - 5 q^{73} + 2 q^{74} + 7 q^{76} - 3 q^{77} - 16 q^{79} + 2 q^{80} + 11 q^{82} + 25 q^{83} - q^{85} - 14 q^{86} + 3 q^{88} + 2 q^{89} - 2 q^{91} + 3 q^{92} + 11 q^{94} + 7 q^{95} - 20 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 2 * q^8 + 2 * q^10 + 3 * q^11 + 2 * q^13 - 2 * q^14 + 2 * q^16 - q^17 + 7 * q^19 + 2 * q^20 + 3 * q^22 + 3 * q^23 + 2 * q^25 + 2 * q^26 - 2 * q^28 + 3 * q^29 + 7 * q^31 + 2 * q^32 - q^34 - 2 * q^35 + 2 * q^37 + 7 * q^38 + 2 * q^40 + 11 * q^41 - 14 * q^43 + 3 * q^44 + 3 * q^46 + 11 * q^47 + 2 * q^49 + 2 * q^50 + 2 * q^52 + 10 * q^53 + 3 * q^55 - 2 * q^56 + 3 * q^58 + 5 * q^59 + 2 * q^61 + 7 * q^62 + 2 * q^64 + 2 * q^65 - 6 * q^67 - q^68 - 2 * q^70 + q^71 - 5 * q^73 + 2 * q^74 + 7 * q^76 - 3 * q^77 - 16 * q^79 + 2 * q^80 + 11 * q^82 + 25 * q^83 - q^85 - 14 * q^86 + 3 * q^88 + 2 * q^89 - 2 * q^91 + 3 * q^92 + 11 * q^94 + 7 * q^95 - 20 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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