LMFDB - Embedded newform 1638.2.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, 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("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.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:	$13.0794958511$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	1638.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.56155 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -3.56155 q^{5} -1.00000 q^{7} -1.00000 q^{8} +3.56155 q^{10} +1.56155 q^{11} +1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.68466 q^{17} -4.68466 q^{19} -3.56155 q^{20} -1.56155 q^{22} +5.56155 q^{23} +7.68466 q^{25} -1.00000 q^{26} -1.00000 q^{28} -6.68466 q^{29} +6.24621 q^{31} -1.00000 q^{32} -6.68466 q^{34} +3.56155 q^{35} -7.56155 q^{37} +4.68466 q^{38} +3.56155 q^{40} +1.12311 q^{41} -6.43845 q^{43} +1.56155 q^{44} -5.56155 q^{46} +1.00000 q^{49} -7.68466 q^{50} +1.00000 q^{52} -12.2462 q^{53} -5.56155 q^{55} +1.00000 q^{56} +6.68466 q^{58} -2.24621 q^{59} +6.68466 q^{61} -6.24621 q^{62} +1.00000 q^{64} -3.56155 q^{65} -7.12311 q^{67} +6.68466 q^{68} -3.56155 q^{70} -8.00000 q^{71} -3.56155 q^{73} +7.56155 q^{74} -4.68466 q^{76} -1.56155 q^{77} -11.1231 q^{79} -3.56155 q^{80} -1.12311 q^{82} -8.87689 q^{83} -23.8078 q^{85} +6.43845 q^{86} -1.56155 q^{88} -10.0000 q^{89} -1.00000 q^{91} +5.56155 q^{92} +16.6847 q^{95} +14.4924 q^{97} -1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 3 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 3 * q^5 - 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 3 q^{5} - 2 q^{7} - 2 q^{8} + 3 q^{10} - q^{11} + 2 q^{13} + 2 q^{14} + 2 q^{16} + q^{17} + 3 q^{19} - 3 q^{20} + q^{22} + 7 q^{23} + 3 q^{25} - 2 q^{26} - 2 q^{28} - q^{29} - 4 q^{31} - 2 q^{32} - q^{34} + 3 q^{35} - 11 q^{37} - 3 q^{38} + 3 q^{40} - 6 q^{41} - 17 q^{43} - q^{44} - 7 q^{46} + 2 q^{49} - 3 q^{50} + 2 q^{52} - 8 q^{53} - 7 q^{55} + 2 q^{56} + q^{58} + 12 q^{59} + q^{61} + 4 q^{62} + 2 q^{64} - 3 q^{65} - 6 q^{67} + q^{68} - 3 q^{70} - 16 q^{71} - 3 q^{73} + 11 q^{74} + 3 q^{76} + q^{77} - 14 q^{79} - 3 q^{80} + 6 q^{82} - 26 q^{83} - 27 q^{85} + 17 q^{86} + q^{88} - 20 q^{89} - 2 q^{91} + 7 q^{92} + 21 q^{95} - 4 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 3 * q^5 - 2 * q^7 - 2 * q^8 + 3 * q^10 - q^11 + 2 * q^13 + 2 * q^14 + 2 * q^16 + q^17 + 3 * q^19 - 3 * q^20 + q^22 + 7 * q^23 + 3 * q^25 - 2 * q^26 - 2 * q^28 - q^29 - 4 * q^31 - 2 * q^32 - q^34 + 3 * q^35 - 11 * q^37 - 3 * q^38 + 3 * q^40 - 6 * q^41 - 17 * q^43 - q^44 - 7 * q^46 + 2 * q^49 - 3 * q^50 + 2 * q^52 - 8 * q^53 - 7 * q^55 + 2 * q^56 + q^58 + 12 * q^59 + q^61 + 4 * q^62 + 2 * q^64 - 3 * q^65 - 6 * q^67 + q^68 - 3 * q^70 - 16 * q^71 - 3 * q^73 + 11 * q^74 + 3 * q^76 + q^77 - 14 * q^79 - 3 * q^80 + 6 * q^82 - 26 * q^83 - 27 * q^85 + 17 * q^86 + q^88 - 20 * q^89 - 2 * q^91 + 7 * q^92 + 21 * q^95 - 4 * 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
$5$	−3.56155	−1.59277	−0.796387	−	0.604787i	$-0.793258\pi$
$5$	−3.56155	−1.59277	−0.796387	+	0.604787i	$0.793258\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.56155	1.12626
$11$	1.56155	0.470826	0.235413	−	0.971895i	$-0.424356\pi$
$11$	1.56155	0.470826	0.235413	+	0.971895i	$0.424356\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	6.68466	1.62127	0.810634	−	0.585553i	$-0.199123\pi$
$17$	6.68466	1.62127	0.810634	+	0.585553i	$0.199123\pi$
$18$	0	0
$19$	−4.68466	−1.07473	−0.537367	−	0.843348i	$-0.680581\pi$
$19$	−4.68466	−1.07473	−0.537367	+	0.843348i	$0.680581\pi$
$20$	−3.56155	−0.796387
$21$	0	0
$22$	−1.56155	−0.332924
$23$	5.56155	1.15966	0.579832	−	0.814736i	$-0.303118\pi$
$23$	5.56155	1.15966	0.579832	+	0.814736i	$0.303118\pi$
$24$	0	0
$25$	7.68466	1.53693
$26$	−1.00000	−0.196116
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−6.68466	−1.24131	−0.620655	−	0.784084i	$-0.713133\pi$
$29$	−6.68466	−1.24131	−0.620655	+	0.784084i	$0.713133\pi$
$30$	0	0
$31$	6.24621	1.12185	0.560926	−	0.827866i	$-0.310445\pi$
$31$	6.24621	1.12185	0.560926	+	0.827866i	$0.310445\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.68466	−1.14641
$35$	3.56155	0.602012
$36$	0	0
$37$	−7.56155	−1.24311	−0.621556	−	0.783370i	$-0.713499\pi$
$37$	−7.56155	−1.24311	−0.621556	+	0.783370i	$0.713499\pi$
$38$	4.68466	0.759952
$39$	0	0
$40$	3.56155	0.563131
$41$	1.12311	0.175400	0.0876998	−	0.996147i	$-0.472048\pi$
$41$	1.12311	0.175400	0.0876998	+	0.996147i	$0.472048\pi$
$42$	0	0
$43$	−6.43845	−0.981854	−0.490927	−	0.871201i	$-0.663342\pi$
$43$	−6.43845	−0.981854	−0.490927	+	0.871201i	$0.663342\pi$
$44$	1.56155	0.235413
$45$	0	0
$46$	−5.56155	−0.820006
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−7.68466	−1.08677
$51$	0	0
$52$	1.00000	0.138675
$53$	−12.2462	−1.68215	−0.841073	−	0.540921i	$-0.818076\pi$
$53$	−12.2462	−1.68215	−0.841073	+	0.540921i	$0.818076\pi$
$54$	0	0
$55$	−5.56155	−0.749920
$56$	1.00000	0.133631
$57$	0	0
$58$	6.68466	0.877739
$59$	−2.24621	−0.292432	−0.146216	−	0.989253i	$-0.546709\pi$
$59$	−2.24621	−0.292432	−0.146216	+	0.989253i	$0.546709\pi$
$60$	0	0
$61$	6.68466	0.855883	0.427941	−	0.903806i	$-0.359239\pi$
$61$	6.68466	0.855883	0.427941	+	0.903806i	$0.359239\pi$
$62$	−6.24621	−0.793270
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.56155	−0.441756
$66$	0	0
$67$	−7.12311	−0.870226	−0.435113	−	0.900376i	$-0.643292\pi$
$67$	−7.12311	−0.870226	−0.435113	+	0.900376i	$0.643292\pi$
$68$	6.68466	0.810634
$69$	0	0
$70$	−3.56155	−0.425687
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−3.56155	−0.416848	−0.208424	−	0.978039i	$-0.566833\pi$
$73$	−3.56155	−0.416848	−0.208424	+	0.978039i	$0.566833\pi$
$74$	7.56155	0.879013
$75$	0	0
$76$	−4.68466	−0.537367
$77$	−1.56155	−0.177955
$78$	0	0
$79$	−11.1231	−1.25145	−0.625724	−	0.780045i	$-0.715196\pi$
$79$	−11.1231	−1.25145	−0.625724	+	0.780045i	$0.715196\pi$
$80$	−3.56155	−0.398194
$81$	0	0
$82$	−1.12311	−0.124026
$83$	−8.87689	−0.974366	−0.487183	−	0.873300i	$-0.661975\pi$
$83$	−8.87689	−0.974366	−0.487183	+	0.873300i	$0.661975\pi$
$84$	0	0
$85$	−23.8078	−2.58231
$86$	6.43845	0.694276
$87$	0	0
$88$	−1.56155	−0.166462
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	5.56155	0.579832
$93$	0	0
$94$	0	0
$95$	16.6847	1.71181
$96$	0	0
$97$	14.4924	1.47148	0.735741	−	0.677263i	$-0.236834\pi$
$97$	14.4924	1.47148	0.735741	+	0.677263i	$0.236834\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	7.68466	0.768466
$101$	5.12311	0.509768	0.254884	−	0.966972i	$-0.417963\pi$
$101$	5.12311	0.509768	0.254884	+	0.966972i	$0.417963\pi$
$102$	0	0
$103$	−0.684658	−0.0674614	−0.0337307	−	0.999431i	$-0.510739\pi$
$103$	−0.684658	−0.0674614	−0.0337307	+	0.999431i	$0.510739\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	12.2462	1.18946
$107$	8.87689	0.858162	0.429081	−	0.903266i	$-0.358838\pi$
$107$	8.87689	0.858162	0.429081	+	0.903266i	$0.358838\pi$
$108$	0	0
$109$	−16.9309	−1.62168	−0.810842	−	0.585266i	$-0.800990\pi$
$109$	−16.9309	−1.62168	−0.810842	+	0.585266i	$0.800990\pi$
$110$	5.56155	0.530273
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	4.24621	0.399450	0.199725	−	0.979852i	$-0.435995\pi$
$113$	4.24621	0.399450	0.199725	+	0.979852i	$0.435995\pi$
$114$	0	0
$115$	−19.8078	−1.84708
$116$	−6.68466	−0.620655
$117$	0	0
$118$	2.24621	0.206781
$119$	−6.68466	−0.612782
$120$	0	0
$121$	−8.56155	−0.778323
$122$	−6.68466	−0.605201
$123$	0	0
$124$	6.24621	0.560926
$125$	−9.56155	−0.855211
$126$	0	0
$127$	−4.87689	−0.432754	−0.216377	−	0.976310i	$-0.569424\pi$
$127$	−4.87689	−0.432754	−0.216377	+	0.976310i	$0.569424\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	3.56155	0.312369
$131$	9.56155	0.835397	0.417698	−	0.908586i	$-0.362837\pi$
$131$	9.56155	0.835397	0.417698	+	0.908586i	$0.362837\pi$
$132$	0	0
$133$	4.68466	0.406211
$134$	7.12311	0.615343
$135$	0	0
$136$	−6.68466	−0.573205
$137$	3.56155	0.304284	0.152142	−	0.988359i	$-0.451383\pi$
$137$	3.56155	0.304284	0.152142	+	0.988359i	$0.451383\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	3.56155	0.301006
$141$	0	0
$142$	8.00000	0.671345
$143$	1.56155	0.130584
$144$	0	0
$145$	23.8078	1.97713
$146$	3.56155	0.294756
$147$	0	0
$148$	−7.56155	−0.621556
$149$	17.6155	1.44312	0.721560	−	0.692352i	$-0.243425\pi$
$149$	17.6155	1.44312	0.721560	+	0.692352i	$0.243425\pi$
$150$	0	0
$151$	11.8078	0.960902	0.480451	−	0.877022i	$-0.340473\pi$
$151$	11.8078	0.960902	0.480451	+	0.877022i	$0.340473\pi$
$152$	4.68466	0.379976
$153$	0	0
$154$	1.56155	0.125834
$155$	−22.2462	−1.78686
$156$	0	0
$157$	−15.5616	−1.24195	−0.620974	−	0.783832i	$-0.713263\pi$
$157$	−15.5616	−1.24195	−0.620974	+	0.783832i	$0.713263\pi$
$158$	11.1231	0.884907
$159$	0	0
$160$	3.56155	0.281565
$161$	−5.56155	−0.438312
$162$	0	0
$163$	−16.8769	−1.32190	−0.660950	−	0.750430i	$-0.729847\pi$
$163$	−16.8769	−1.32190	−0.660950	+	0.750430i	$0.729847\pi$
$164$	1.12311	0.0876998
$165$	0	0
$166$	8.87689	0.688981
$167$	−22.9309	−1.77444	−0.887222	−	0.461343i	$-0.847368\pi$
$167$	−22.9309	−1.77444	−0.887222	+	0.461343i	$0.847368\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	23.8078	1.82597
$171$	0	0
$172$	−6.43845	−0.490927
$173$	−20.2462	−1.53929	−0.769645	−	0.638471i	$-0.779567\pi$
$173$	−20.2462	−1.53929	−0.769645	+	0.638471i	$0.779567\pi$
$174$	0	0
$175$	−7.68466	−0.580906
$176$	1.56155	0.117706
$177$	0	0
$178$	10.0000	0.749532
$179$	−16.4924	−1.23270	−0.616351	−	0.787472i	$-0.711390\pi$
$179$	−16.4924	−1.23270	−0.616351	+	0.787472i	$0.711390\pi$
$180$	0	0
$181$	−0.246211	−0.0183007	−0.00915037	−	0.999958i	$-0.502913\pi$
$181$	−0.246211	−0.0183007	−0.00915037	+	0.999958i	$0.502913\pi$
$182$	1.00000	0.0741249
$183$	0	0
$184$	−5.56155	−0.410003
$185$	26.9309	1.98000
$186$	0	0
$187$	10.4384	0.763335
$188$	0	0
$189$	0	0
$190$	−16.6847	−1.21043
$191$	−14.9309	−1.08036	−0.540180	−	0.841550i	$-0.681644\pi$
$191$	−14.9309	−1.08036	−0.540180	+	0.841550i	$0.681644\pi$
$192$	0	0
$193$	19.3693	1.39423	0.697117	−	0.716957i	$-0.254466\pi$
$193$	19.3693	1.39423	0.697117	+	0.716957i	$0.254466\pi$
$194$	−14.4924	−1.04050
$195$	0	0
$196$	1.00000	0.0714286
$197$	−10.8769	−0.774947	−0.387473	−	0.921881i	$-0.626652\pi$
$197$	−10.8769	−0.774947	−0.387473	+	0.921881i	$0.626652\pi$
$198$	0	0
$199$	6.93087	0.491316	0.245658	−	0.969357i	$-0.420996\pi$
$199$	6.93087	0.491316	0.245658	+	0.969357i	$0.420996\pi$
$200$	−7.68466	−0.543387
$201$	0	0
$202$	−5.12311	−0.360460
$203$	6.68466	0.469171
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0.684658	0.0477024
$207$	0	0
$208$	1.00000	0.0693375
$209$	−7.31534	−0.506013
$210$	0	0
$211$	−15.8078	−1.08825	−0.544126	−	0.839004i	$-0.683139\pi$
$211$	−15.8078	−1.08825	−0.544126	+	0.839004i	$0.683139\pi$
$212$	−12.2462	−0.841073
$213$	0	0
$214$	−8.87689	−0.606812
$215$	22.9309	1.56387
$216$	0	0
$217$	−6.24621	−0.424020
$218$	16.9309	1.14670
$219$	0	0
$220$	−5.56155	−0.374960
$221$	6.68466	0.449659
$222$	0	0
$223$	−23.6155	−1.58141	−0.790706	−	0.612196i	$-0.790287\pi$
$223$	−23.6155	−1.58141	−0.790706	+	0.612196i	$0.790287\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−4.24621	−0.282454
$227$	7.12311	0.472777	0.236389	−	0.971659i	$-0.424036\pi$
$227$	7.12311	0.472777	0.236389	+	0.971659i	$0.424036\pi$
$228$	0	0
$229$	28.2462	1.86656	0.933281	−	0.359147i	$-0.116932\pi$
$229$	28.2462	1.86656	0.933281	+	0.359147i	$0.116932\pi$
$230$	19.8078	1.30609
$231$	0	0
$232$	6.68466	0.438869
$233$	−27.3693	−1.79302	−0.896512	−	0.443020i	$-0.853907\pi$
$233$	−27.3693	−1.79302	−0.896512	+	0.443020i	$0.853907\pi$
$234$	0	0
$235$	0	0
$236$	−2.24621	−0.146216
$237$	0	0
$238$	6.68466	0.433302
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	8.56155	0.550357
$243$	0	0
$244$	6.68466	0.427941
$245$	−3.56155	−0.227539
$246$	0	0
$247$	−4.68466	−0.298078
$248$	−6.24621	−0.396635
$249$	0	0
$250$	9.56155	0.604726
$251$	7.80776	0.492822	0.246411	−	0.969165i	$-0.420749\pi$
$251$	7.80776	0.492822	0.246411	+	0.969165i	$0.420749\pi$
$252$	0	0
$253$	8.68466	0.546000
$254$	4.87689	0.306004
$255$	0	0
$256$	1.00000	0.0625000
$257$	4.24621	0.264871	0.132436	−	0.991192i	$-0.457720\pi$
$257$	4.24621	0.264871	0.132436	+	0.991192i	$0.457720\pi$
$258$	0	0
$259$	7.56155	0.469852
$260$	−3.56155	−0.220878
$261$	0	0
$262$	−9.56155	−0.590715
$263$	30.2462	1.86506	0.932531	−	0.361091i	$-0.117596\pi$
$263$	30.2462	1.86506	0.932531	+	0.361091i	$0.117596\pi$
$264$	0	0
$265$	43.6155	2.67928
$266$	−4.68466	−0.287235
$267$	0	0
$268$	−7.12311	−0.435113
$269$	−20.2462	−1.23443	−0.617217	−	0.786793i	$-0.711740\pi$
$269$	−20.2462	−1.23443	−0.617217	+	0.786793i	$0.711740\pi$
$270$	0	0
$271$	4.87689	0.296250	0.148125	−	0.988969i	$-0.452676\pi$
$271$	4.87689	0.296250	0.148125	+	0.988969i	$0.452676\pi$
$272$	6.68466	0.405317
$273$	0	0
$274$	−3.56155	−0.215161
$275$	12.0000	0.723627
$276$	0	0
$277$	−22.4924	−1.35144	−0.675719	−	0.737159i	$-0.736167\pi$
$277$	−22.4924	−1.35144	−0.675719	+	0.737159i	$0.736167\pi$
$278$	−12.0000	−0.719712
$279$	0	0
$280$	−3.56155	−0.212843
$281$	−16.2462	−0.969168	−0.484584	−	0.874745i	$-0.661029\pi$
$281$	−16.2462	−0.969168	−0.484584	+	0.874745i	$0.661029\pi$
$282$	0	0
$283$	18.2462	1.08462	0.542312	−	0.840177i	$-0.317549\pi$
$283$	18.2462	1.08462	0.542312	+	0.840177i	$0.317549\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	−1.56155	−0.0923366
$287$	−1.12311	−0.0662948
$288$	0	0
$289$	27.6847	1.62851
$290$	−23.8078	−1.39804
$291$	0	0
$292$	−3.56155	−0.208424
$293$	−24.7386	−1.44525	−0.722623	−	0.691242i	$-0.757064\pi$
$293$	−24.7386	−1.44525	−0.722623	+	0.691242i	$0.757064\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	7.56155	0.439506
$297$	0	0
$298$	−17.6155	−1.02044
$299$	5.56155	0.321633
$300$	0	0
$301$	6.43845	0.371106
$302$	−11.8078	−0.679460
$303$	0	0
$304$	−4.68466	−0.268684
$305$	−23.8078	−1.36323
$306$	0	0
$307$	26.2462	1.49795	0.748975	−	0.662598i	$-0.230546\pi$
$307$	26.2462	1.49795	0.748975	+	0.662598i	$0.230546\pi$
$308$	−1.56155	−0.0889777
$309$	0	0
$310$	22.2462	1.26350
$311$	12.8769	0.730182	0.365091	−	0.930972i	$-0.381038\pi$
$311$	12.8769	0.730182	0.365091	+	0.930972i	$0.381038\pi$
$312$	0	0
$313$	−13.6155	−0.769595	−0.384798	−	0.923001i	$-0.625729\pi$
$313$	−13.6155	−0.769595	−0.384798	+	0.923001i	$0.625729\pi$
$314$	15.5616	0.878189
$315$	0	0
$316$	−11.1231	−0.625724
$317$	−2.87689	−0.161582	−0.0807912	−	0.996731i	$-0.525745\pi$
$317$	−2.87689	−0.161582	−0.0807912	+	0.996731i	$0.525745\pi$
$318$	0	0
$319$	−10.4384	−0.584441
$320$	−3.56155	−0.199097
$321$	0	0
$322$	5.56155	0.309933
$323$	−31.3153	−1.74243
$324$	0	0
$325$	7.68466	0.426268
$326$	16.8769	0.934725
$327$	0	0
$328$	−1.12311	−0.0620131
$329$	0	0
$330$	0	0
$331$	13.3693	0.734844	0.367422	−	0.930054i	$-0.380240\pi$
$331$	13.3693	0.734844	0.367422	+	0.930054i	$0.380240\pi$
$332$	−8.87689	−0.487183
$333$	0	0
$334$	22.9309	1.25472
$335$	25.3693	1.38607
$336$	0	0
$337$	4.43845	0.241778	0.120889	−	0.992666i	$-0.461426\pi$
$337$	4.43845	0.241778	0.120889	+	0.992666i	$0.461426\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	−23.8078	−1.29116
$341$	9.75379	0.528197
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	6.43845	0.347138
$345$	0	0
$346$	20.2462	1.08844
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	7.75379	0.415051	0.207525	−	0.978230i	$-0.433459\pi$
$349$	7.75379	0.415051	0.207525	+	0.978230i	$0.433459\pi$
$350$	7.68466	0.410762
$351$	0	0
$352$	−1.56155	−0.0832310
$353$	−30.4924	−1.62295	−0.811474	−	0.584389i	$-0.801334\pi$
$353$	−30.4924	−1.62295	−0.811474	+	0.584389i	$0.801334\pi$
$354$	0	0
$355$	28.4924	1.51222
$356$	−10.0000	−0.529999
$357$	0	0
$358$	16.4924	0.871652
$359$	26.7386	1.41121	0.705606	−	0.708605i	$-0.250675\pi$
$359$	26.7386	1.41121	0.705606	+	0.708605i	$0.250675\pi$
$360$	0	0
$361$	2.94602	0.155054
$362$	0.246211	0.0129406
$363$	0	0
$364$	−1.00000	−0.0524142
$365$	12.6847	0.663945
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	5.56155	0.289916
$369$	0	0
$370$	−26.9309	−1.40007
$371$	12.2462	0.635792
$372$	0	0
$373$	−17.6155	−0.912097	−0.456049	−	0.889955i	$-0.650736\pi$
$373$	−17.6155	−0.912097	−0.456049	+	0.889955i	$0.650736\pi$
$374$	−10.4384	−0.539759
$375$	0	0
$376$	0	0
$377$	−6.68466	−0.344277
$378$	0	0
$379$	34.2462	1.75911	0.879555	−	0.475797i	$-0.157840\pi$
$379$	34.2462	1.75911	0.879555	+	0.475797i	$0.157840\pi$
$380$	16.6847	0.855905
$381$	0	0
$382$	14.9309	0.763930
$383$	−27.4233	−1.40126	−0.700632	−	0.713522i	$-0.747099\pi$
$383$	−27.4233	−1.40126	−0.700632	+	0.713522i	$0.747099\pi$
$384$	0	0
$385$	5.56155	0.283443
$386$	−19.3693	−0.985872
$387$	0	0
$388$	14.4924	0.735741
$389$	28.7386	1.45711	0.728553	−	0.684989i	$-0.240193\pi$
$389$	28.7386	1.45711	0.728553	+	0.684989i	$0.240193\pi$
$390$	0	0
$391$	37.1771	1.88013
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	10.8769	0.547970
$395$	39.6155	1.99327
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	−6.93087	−0.347413
$399$	0	0
$400$	7.68466	0.384233
$401$	−8.24621	−0.411796	−0.205898	−	0.978573i	$-0.566012\pi$
$401$	−8.24621	−0.411796	−0.205898	+	0.978573i	$0.566012\pi$
$402$	0	0
$403$	6.24621	0.311146
$404$	5.12311	0.254884
$405$	0	0
$406$	−6.68466	−0.331754
$407$	−11.8078	−0.585289
$408$	0	0
$409$	−4.93087	−0.243816	−0.121908	−	0.992541i	$-0.538901\pi$
$409$	−4.93087	−0.243816	−0.121908	+	0.992541i	$0.538901\pi$
$410$	4.00000	0.197546
$411$	0	0
$412$	−0.684658	−0.0337307
$413$	2.24621	0.110529
$414$	0	0
$415$	31.6155	1.55195
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	7.31534	0.357805
$419$	36.6847	1.79216	0.896081	−	0.443890i	$-0.146402\pi$
$419$	36.6847	1.79216	0.896081	+	0.443890i	$0.146402\pi$
$420$	0	0
$421$	−3.75379	−0.182948	−0.0914742	−	0.995807i	$-0.529158\pi$
$421$	−3.75379	−0.182948	−0.0914742	+	0.995807i	$0.529158\pi$
$422$	15.8078	0.769510
$423$	0	0
$424$	12.2462	0.594729
$425$	51.3693	2.49178
$426$	0	0
$427$	−6.68466	−0.323493
$428$	8.87689	0.429081
$429$	0	0
$430$	−22.9309	−1.10582
$431$	33.3693	1.60734	0.803672	−	0.595073i	$-0.202877\pi$
$431$	33.3693	1.60734	0.803672	+	0.595073i	$0.202877\pi$
$432$	0	0
$433$	−31.3693	−1.50751	−0.753757	−	0.657154i	$-0.771760\pi$
$433$	−31.3693	−1.50751	−0.753757	+	0.657154i	$0.771760\pi$
$434$	6.24621	0.299828
$435$	0	0
$436$	−16.9309	−0.810842
$437$	−26.0540	−1.24633
$438$	0	0
$439$	−6.93087	−0.330792	−0.165396	−	0.986227i	$-0.552890\pi$
$439$	−6.93087	−0.330792	−0.165396	+	0.986227i	$0.552890\pi$
$440$	5.56155	0.265137
$441$	0	0
$442$	−6.68466	−0.317957
$443$	5.36932	0.255104	0.127552	−	0.991832i	$-0.459288\pi$
$443$	5.36932	0.255104	0.127552	+	0.991832i	$0.459288\pi$
$444$	0	0
$445$	35.6155	1.68834
$446$	23.6155	1.11823
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−10.6847	−0.504240	−0.252120	−	0.967696i	$-0.581128\pi$
$449$	−10.6847	−0.504240	−0.252120	+	0.967696i	$0.581128\pi$
$450$	0	0
$451$	1.75379	0.0825827
$452$	4.24621	0.199725
$453$	0	0
$454$	−7.12311	−0.334304
$455$	3.56155	0.166968
$456$	0	0
$457$	27.3693	1.28028	0.640141	−	0.768257i	$-0.278876\pi$
$457$	27.3693	1.28028	0.640141	+	0.768257i	$0.278876\pi$
$458$	−28.2462	−1.31986
$459$	0	0
$460$	−19.8078	−0.923542
$461$	28.0540	1.30660	0.653302	−	0.757097i	$-0.273383\pi$
$461$	28.0540	1.30660	0.653302	+	0.757097i	$0.273383\pi$
$462$	0	0
$463$	−8.68466	−0.403610	−0.201805	−	0.979426i	$-0.564681\pi$
$463$	−8.68466	−0.403610	−0.201805	+	0.979426i	$0.564681\pi$
$464$	−6.68466	−0.310327
$465$	0	0
$466$	27.3693	1.26786
$467$	3.31534	0.153416	0.0767079	−	0.997054i	$-0.475559\pi$
$467$	3.31534	0.153416	0.0767079	+	0.997054i	$0.475559\pi$
$468$	0	0
$469$	7.12311	0.328914
$470$	0	0
$471$	0	0
$472$	2.24621	0.103390
$473$	−10.0540	−0.462282
$474$	0	0
$475$	−36.0000	−1.65179
$476$	−6.68466	−0.306391
$477$	0	0
$478$	16.0000	0.731823
$479$	−32.3002	−1.47583	−0.737917	−	0.674892i	$-0.764190\pi$
$479$	−32.3002	−1.47583	−0.737917	+	0.674892i	$0.764190\pi$
$480$	0	0
$481$	−7.56155	−0.344777
$482$	14.0000	0.637683
$483$	0	0
$484$	−8.56155	−0.389161
$485$	−51.6155	−2.34374
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−6.68466	−0.302600
$489$	0	0
$490$	3.56155	0.160895
$491$	8.87689	0.400609	0.200304	−	0.979734i	$-0.435807\pi$
$491$	8.87689	0.400609	0.200304	+	0.979734i	$0.435807\pi$
$492$	0	0
$493$	−44.6847	−2.01250
$494$	4.68466	0.210773
$495$	0	0
$496$	6.24621	0.280463
$497$	8.00000	0.358849
$498$	0	0
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	−9.56155	−0.427606
$501$	0	0
$502$	−7.80776	−0.348478
$503$	3.12311	0.139252	0.0696262	−	0.997573i	$-0.477819\pi$
$503$	3.12311	0.139252	0.0696262	+	0.997573i	$0.477819\pi$
$504$	0	0
$505$	−18.2462	−0.811946
$506$	−8.68466	−0.386080
$507$	0	0
$508$	−4.87689	−0.216377
$509$	12.0540	0.534283	0.267142	−	0.963657i	$-0.413921\pi$
$509$	12.0540	0.534283	0.267142	+	0.963657i	$0.413921\pi$
$510$	0	0
$511$	3.56155	0.157554
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−4.24621	−0.187292
$515$	2.43845	0.107451
$516$	0	0
$517$	0	0
$518$	−7.56155	−0.332236
$519$	0	0
$520$	3.56155	0.156184
$521$	−2.68466	−0.117617	−0.0588085	−	0.998269i	$-0.518730\pi$
$521$	−2.68466	−0.117617	−0.0588085	+	0.998269i	$0.518730\pi$
$522$	0	0
$523$	21.7538	0.951227	0.475613	−	0.879654i	$-0.342226\pi$
$523$	21.7538	0.951227	0.475613	+	0.879654i	$0.342226\pi$
$524$	9.56155	0.417698
$525$	0	0
$526$	−30.2462	−1.31880
$527$	41.7538	1.81882
$528$	0	0
$529$	7.93087	0.344820
$530$	−43.6155	−1.89454
$531$	0	0
$532$	4.68466	0.203106
$533$	1.12311	0.0486471
$534$	0	0
$535$	−31.6155	−1.36686
$536$	7.12311	0.307671
$537$	0	0
$538$	20.2462	0.872876
$539$	1.56155	0.0672608
$540$	0	0
$541$	−32.9309	−1.41581	−0.707904	−	0.706308i	$-0.750359\pi$
$541$	−32.9309	−1.41581	−0.707904	+	0.706308i	$0.750359\pi$
$542$	−4.87689	−0.209481
$543$	0	0
$544$	−6.68466	−0.286602
$545$	60.3002	2.58298
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	3.56155	0.152142
$549$	0	0
$550$	−12.0000	−0.511682
$551$	31.3153	1.33408
$552$	0	0
$553$	11.1231	0.473003
$554$	22.4924	0.955611
$555$	0	0
$556$	12.0000	0.508913
$557$	3.36932	0.142763	0.0713813	−	0.997449i	$-0.477259\pi$
$557$	3.36932	0.142763	0.0713813	+	0.997449i	$0.477259\pi$
$558$	0	0
$559$	−6.43845	−0.272317
$560$	3.56155	0.150503
$561$	0	0
$562$	16.2462	0.685305
$563$	6.05398	0.255145	0.127572	−	0.991829i	$-0.459282\pi$
$563$	6.05398	0.255145	0.127572	+	0.991829i	$0.459282\pi$
$564$	0	0
$565$	−15.1231	−0.636234
$566$	−18.2462	−0.766945
$567$	0	0
$568$	8.00000	0.335673
$569$	−41.2311	−1.72850	−0.864248	−	0.503066i	$-0.832205\pi$
$569$	−41.2311	−1.72850	−0.864248	+	0.503066i	$0.832205\pi$
$570$	0	0
$571$	18.2462	0.763580	0.381790	−	0.924249i	$-0.375308\pi$
$571$	18.2462	0.763580	0.381790	+	0.924249i	$0.375308\pi$
$572$	1.56155	0.0652918
$573$	0	0
$574$	1.12311	0.0468775
$575$	42.7386	1.78232
$576$	0	0
$577$	−30.0000	−1.24892	−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000	−1.24892	−0.624458	+	0.781058i	$0.714680\pi$
$578$	−27.6847	−1.15153
$579$	0	0
$580$	23.8078	0.988564
$581$	8.87689	0.368276
$582$	0	0
$583$	−19.1231	−0.791998
$584$	3.56155	0.147378
$585$	0	0
$586$	24.7386	1.02194
$587$	−13.3693	−0.551811	−0.275905	−	0.961185i	$-0.588978\pi$
$587$	−13.3693	−0.551811	−0.275905	+	0.961185i	$0.588978\pi$
$588$	0	0
$589$	−29.2614	−1.20569
$590$	−8.00000	−0.329355
$591$	0	0
$592$	−7.56155	−0.310778
$593$	20.2462	0.831412	0.415706	−	0.909499i	$-0.363534\pi$
$593$	20.2462	0.831412	0.415706	+	0.909499i	$0.363534\pi$
$594$	0	0
$595$	23.8078	0.976023
$596$	17.6155	0.721560
$597$	0	0
$598$	−5.56155	−0.227429
$599$	−21.5616	−0.880981	−0.440491	−	0.897757i	$-0.645195\pi$
$599$	−21.5616	−0.880981	−0.440491	+	0.897757i	$0.645195\pi$
$600$	0	0
$601$	−10.8769	−0.443678	−0.221839	−	0.975083i	$-0.571206\pi$
$601$	−10.8769	−0.443678	−0.221839	+	0.975083i	$0.571206\pi$
$602$	−6.43845	−0.262412
$603$	0	0
$604$	11.8078	0.480451
$605$	30.4924	1.23969
$606$	0	0
$607$	−18.4384	−0.748393	−0.374197	−	0.927349i	$-0.622082\pi$
$607$	−18.4384	−0.748393	−0.374197	+	0.927349i	$0.622082\pi$
$608$	4.68466	0.189988
$609$	0	0
$610$	23.8078	0.963948
$611$	0	0
$612$	0	0
$613$	44.5464	1.79921	0.899606	−	0.436702i	$-0.143854\pi$
$613$	44.5464	1.79921	0.899606	+	0.436702i	$0.143854\pi$
$614$	−26.2462	−1.05921
$615$	0	0
$616$	1.56155	0.0629168
$617$	33.4233	1.34557	0.672786	−	0.739838i	$-0.265098\pi$
$617$	33.4233	1.34557	0.672786	+	0.739838i	$0.265098\pi$
$618$	0	0
$619$	19.3153	0.776349	0.388175	−	0.921586i	$-0.373106\pi$
$619$	19.3153	0.776349	0.388175	+	0.921586i	$0.373106\pi$
$620$	−22.2462	−0.893429
$621$	0	0
$622$	−12.8769	−0.516316
$623$	10.0000	0.400642
$624$	0	0
$625$	−4.36932	−0.174773
$626$	13.6155	0.544186
$627$	0	0
$628$	−15.5616	−0.620974
$629$	−50.5464	−2.01542
$630$	0	0
$631$	−22.9309	−0.912864	−0.456432	−	0.889758i	$-0.650873\pi$
$631$	−22.9309	−0.912864	−0.456432	+	0.889758i	$0.650873\pi$
$632$	11.1231	0.442453
$633$	0	0
$634$	2.87689	0.114256
$635$	17.3693	0.689280
$636$	0	0
$637$	1.00000	0.0396214
$638$	10.4384	0.413262
$639$	0	0
$640$	3.56155	0.140783
$641$	20.2462	0.799677	0.399839	−	0.916586i	$-0.369066\pi$
$641$	20.2462	0.799677	0.399839	+	0.916586i	$0.369066\pi$
$642$	0	0
$643$	16.1922	0.638559	0.319280	−	0.947661i	$-0.396559\pi$
$643$	16.1922	0.638559	0.319280	+	0.947661i	$0.396559\pi$
$644$	−5.56155	−0.219156
$645$	0	0
$646$	31.3153	1.23209
$647$	14.2462	0.560076	0.280038	−	0.959989i	$-0.409653\pi$
$647$	14.2462	0.560076	0.280038	+	0.959989i	$0.409653\pi$
$648$	0	0
$649$	−3.50758	−0.137684
$650$	−7.68466	−0.301417
$651$	0	0
$652$	−16.8769	−0.660950
$653$	16.9309	0.662556	0.331278	−	0.943533i	$-0.392520\pi$
$653$	16.9309	0.662556	0.331278	+	0.943533i	$0.392520\pi$
$654$	0	0
$655$	−34.0540	−1.33060
$656$	1.12311	0.0438499
$657$	0	0
$658$	0	0
$659$	−21.3693	−0.832430	−0.416215	−	0.909266i	$-0.636644\pi$
$659$	−21.3693	−0.832430	−0.416215	+	0.909266i	$0.636644\pi$
$660$	0	0
$661$	−0.246211	−0.00957651	−0.00478825	−	0.999989i	$-0.501524\pi$
$661$	−0.246211	−0.00957651	−0.00478825	+	0.999989i	$0.501524\pi$
$662$	−13.3693	−0.519613
$663$	0	0
$664$	8.87689	0.344490
$665$	−16.6847	−0.647003
$666$	0	0
$667$	−37.1771	−1.43950
$668$	−22.9309	−0.887222
$669$	0	0
$670$	−25.3693	−0.980102
$671$	10.4384	0.402972
$672$	0	0
$673$	−33.8078	−1.30319	−0.651597	−	0.758566i	$-0.725901\pi$
$673$	−33.8078	−1.30319	−0.651597	+	0.758566i	$0.725901\pi$
$674$	−4.43845	−0.170963
$675$	0	0
$676$	1.00000	0.0384615
$677$	−2.49242	−0.0957916	−0.0478958	−	0.998852i	$-0.515252\pi$
$677$	−2.49242	−0.0957916	−0.0478958	+	0.998852i	$0.515252\pi$
$678$	0	0
$679$	−14.4924	−0.556168
$680$	23.8078	0.912986
$681$	0	0
$682$	−9.75379	−0.373492
$683$	−35.3153	−1.35130	−0.675652	−	0.737221i	$-0.736138\pi$
$683$	−35.3153	−1.35130	−0.675652	+	0.737221i	$0.736138\pi$
$684$	0	0
$685$	−12.6847	−0.484656
$686$	1.00000	0.0381802
$687$	0	0
$688$	−6.43845	−0.245463
$689$	−12.2462	−0.466543
$690$	0	0
$691$	16.4924	0.627401	0.313701	−	0.949522i	$-0.398431\pi$
$691$	16.4924	0.627401	0.313701	+	0.949522i	$0.398431\pi$
$692$	−20.2462	−0.769645
$693$	0	0
$694$	−20.0000	−0.759190
$695$	−42.7386	−1.62117
$696$	0	0
$697$	7.50758	0.284370
$698$	−7.75379	−0.293485
$699$	0	0
$700$	−7.68466	−0.290453
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	35.4233	1.33601
$704$	1.56155	0.0588532
$705$	0	0
$706$	30.4924	1.14760
$707$	−5.12311	−0.192674
$708$	0	0
$709$	−16.2462	−0.610139	−0.305070	−	0.952330i	$-0.598680\pi$
$709$	−16.2462	−0.610139	−0.305070	+	0.952330i	$0.598680\pi$
$710$	−28.4924	−1.06930
$711$	0	0
$712$	10.0000	0.374766
$713$	34.7386	1.30097
$714$	0	0
$715$	−5.56155	−0.207990
$716$	−16.4924	−0.616351
$717$	0	0
$718$	−26.7386	−0.997877
$719$	−43.1231	−1.60822	−0.804110	−	0.594480i	$-0.797358\pi$
$719$	−43.1231	−1.60822	−0.804110	+	0.594480i	$0.797358\pi$
$720$	0	0
$721$	0.684658	0.0254980
$722$	−2.94602	−0.109640
$723$	0	0
$724$	−0.246211	−0.00915037
$725$	−51.3693	−1.90781
$726$	0	0
$727$	−6.93087	−0.257052	−0.128526	−	0.991706i	$-0.541025\pi$
$727$	−6.93087	−0.257052	−0.128526	+	0.991706i	$0.541025\pi$
$728$	1.00000	0.0370625
$729$	0	0
$730$	−12.6847	−0.469480
$731$	−43.0388	−1.59185
$732$	0	0
$733$	−41.6155	−1.53710	−0.768552	−	0.639787i	$-0.779023\pi$
$733$	−41.6155	−1.53710	−0.768552	+	0.639787i	$0.779023\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	−5.56155	−0.205002
$737$	−11.1231	−0.409725
$738$	0	0
$739$	−32.1080	−1.18111	−0.590555	−	0.806997i	$-0.701091\pi$
$739$	−32.1080	−1.18111	−0.590555	+	0.806997i	$0.701091\pi$
$740$	26.9309	0.989998
$741$	0	0
$742$	−12.2462	−0.449573
$743$	−28.1080	−1.03118	−0.515590	−	0.856835i	$-0.672427\pi$
$743$	−28.1080	−1.03118	−0.515590	+	0.856835i	$0.672427\pi$
$744$	0	0
$745$	−62.7386	−2.29857
$746$	17.6155	0.644950
$747$	0	0
$748$	10.4384	0.381667
$749$	−8.87689	−0.324355
$750$	0	0
$751$	−6.24621	−0.227927	−0.113964	−	0.993485i	$-0.536355\pi$
$751$	−6.24621	−0.227927	−0.113964	+	0.993485i	$0.536355\pi$
$752$	0	0
$753$	0	0
$754$	6.68466	0.243441
$755$	−42.0540	−1.53050
$756$	0	0
$757$	34.4924	1.25365	0.626824	−	0.779161i	$-0.284354\pi$
$757$	34.4924	1.25365	0.626824	+	0.779161i	$0.284354\pi$
$758$	−34.2462	−1.24388
$759$	0	0
$760$	−16.6847	−0.605216
$761$	−5.12311	−0.185712	−0.0928562	−	0.995680i	$-0.529600\pi$
$761$	−5.12311	−0.185712	−0.0928562	+	0.995680i	$0.529600\pi$
$762$	0	0
$763$	16.9309	0.612939
$764$	−14.9309	−0.540180
$765$	0	0
$766$	27.4233	0.990844
$767$	−2.24621	−0.0811060
$768$	0	0
$769$	−16.4384	−0.592786	−0.296393	−	0.955066i	$-0.595784\pi$
$769$	−16.4384	−0.592786	−0.296393	+	0.955066i	$0.595784\pi$
$770$	−5.56155	−0.200424
$771$	0	0
$772$	19.3693	0.697117
$773$	−52.9309	−1.90379	−0.951896	−	0.306423i	$-0.900868\pi$
$773$	−52.9309	−1.90379	−0.951896	+	0.306423i	$0.900868\pi$
$774$	0	0
$775$	48.0000	1.72421
$776$	−14.4924	−0.520248
$777$	0	0
$778$	−28.7386	−1.03033
$779$	−5.26137	−0.188508
$780$	0	0
$781$	−12.4924	−0.447014
$782$	−37.1771	−1.32945
$783$	0	0
$784$	1.00000	0.0357143
$785$	55.4233	1.97814
$786$	0	0
$787$	20.3002	0.723624	0.361812	−	0.932251i	$-0.382158\pi$
$787$	20.3002	0.723624	0.361812	+	0.932251i	$0.382158\pi$
$788$	−10.8769	−0.387473
$789$	0	0
$790$	−39.6155	−1.40946
$791$	−4.24621	−0.150978
$792$	0	0
$793$	6.68466	0.237379
$794$	18.0000	0.638796
$795$	0	0
$796$	6.93087	0.245658
$797$	−31.3693	−1.11116	−0.555579	−	0.831464i	$-0.687503\pi$
$797$	−31.3693	−1.11116	−0.555579	+	0.831464i	$0.687503\pi$
$798$	0	0
$799$	0	0
$800$	−7.68466	−0.271694
$801$	0	0
$802$	8.24621	0.291184
$803$	−5.56155	−0.196263
$804$	0	0
$805$	19.8078	0.698132
$806$	−6.24621	−0.220013
$807$	0	0
$808$	−5.12311	−0.180230
$809$	2.49242	0.0876289	0.0438145	−	0.999040i	$-0.486049\pi$
$809$	2.49242	0.0876289	0.0438145	+	0.999040i	$0.486049\pi$
$810$	0	0
$811$	41.5616	1.45942	0.729712	−	0.683755i	$-0.239654\pi$
$811$	41.5616	1.45942	0.729712	+	0.683755i	$0.239654\pi$
$812$	6.68466	0.234586
$813$	0	0
$814$	11.8078	0.413862
$815$	60.1080	2.10549
$816$	0	0
$817$	30.1619	1.05523
$818$	4.93087	0.172404
$819$	0	0
$820$	−4.00000	−0.139686
$821$	−14.3845	−0.502022	−0.251011	−	0.967984i	$-0.580763\pi$
$821$	−14.3845	−0.502022	−0.251011	+	0.967984i	$0.580763\pi$
$822$	0	0
$823$	4.49242	0.156596	0.0782980	−	0.996930i	$-0.475051\pi$
$823$	4.49242	0.156596	0.0782980	+	0.996930i	$0.475051\pi$
$824$	0.684658	0.0238512
$825$	0	0
$826$	−2.24621	−0.0781557
$827$	−26.9309	−0.936478	−0.468239	−	0.883602i	$-0.655111\pi$
$827$	−26.9309	−0.936478	−0.468239	+	0.883602i	$0.655111\pi$
$828$	0	0
$829$	38.6847	1.34357	0.671787	−	0.740744i	$-0.265527\pi$
$829$	38.6847	1.34357	0.671787	+	0.740744i	$0.265527\pi$
$830$	−31.6155	−1.09739
$831$	0	0
$832$	1.00000	0.0346688
$833$	6.68466	0.231610
$834$	0	0
$835$	81.6695	2.82629
$836$	−7.31534	−0.253006
$837$	0	0
$838$	−36.6847	−1.26725
$839$	10.7386	0.370739	0.185369	−	0.982669i	$-0.440652\pi$
$839$	10.7386	0.370739	0.185369	+	0.982669i	$0.440652\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	3.75379	0.129364
$843$	0	0
$844$	−15.8078	−0.544126
$845$	−3.56155	−0.122521
$846$	0	0
$847$	8.56155	0.294178
$848$	−12.2462	−0.420537
$849$	0	0
$850$	−51.3693	−1.76195
$851$	−42.0540	−1.44159
$852$	0	0
$853$	−27.3693	−0.937108	−0.468554	−	0.883435i	$-0.655225\pi$
$853$	−27.3693	−0.937108	−0.468554	+	0.883435i	$0.655225\pi$
$854$	6.68466	0.228744
$855$	0	0
$856$	−8.87689	−0.303406
$857$	34.4924	1.17824	0.589119	−	0.808046i	$-0.299475\pi$
$857$	34.4924	1.17824	0.589119	+	0.808046i	$0.299475\pi$
$858$	0	0
$859$	5.75379	0.196317	0.0981584	−	0.995171i	$-0.468705\pi$
$859$	5.75379	0.196317	0.0981584	+	0.995171i	$0.468705\pi$
$860$	22.9309	0.781936
$861$	0	0
$862$	−33.3693	−1.13656
$863$	17.3693	0.591258	0.295629	−	0.955303i	$-0.404471\pi$
$863$	17.3693	0.591258	0.295629	+	0.955303i	$0.404471\pi$
$864$	0	0
$865$	72.1080	2.45174
$866$	31.3693	1.06597
$867$	0	0
$868$	−6.24621	−0.212010
$869$	−17.3693	−0.589214
$870$	0	0
$871$	−7.12311	−0.241357
$872$	16.9309	0.573352
$873$	0	0
$874$	26.0540	0.881289
$875$	9.56155	0.323239
$876$	0	0
$877$	−40.2462	−1.35902	−0.679509	−	0.733667i	$-0.737807\pi$
$877$	−40.2462	−1.35902	−0.679509	+	0.733667i	$0.737807\pi$
$878$	6.93087	0.233906
$879$	0	0
$880$	−5.56155	−0.187480
$881$	19.1771	0.646092	0.323046	−	0.946383i	$-0.395293\pi$
$881$	19.1771	0.646092	0.323046	+	0.946383i	$0.395293\pi$
$882$	0	0
$883$	1.56155	0.0525504	0.0262752	−	0.999655i	$-0.491635\pi$
$883$	1.56155	0.0525504	0.0262752	+	0.999655i	$0.491635\pi$
$884$	6.68466	0.224829
$885$	0	0
$886$	−5.36932	−0.180386
$887$	52.4924	1.76252	0.881262	−	0.472629i	$-0.156695\pi$
$887$	52.4924	1.76252	0.881262	+	0.472629i	$0.156695\pi$
$888$	0	0
$889$	4.87689	0.163566
$890$	−35.6155	−1.19384
$891$	0	0
$892$	−23.6155	−0.790706
$893$	0	0
$894$	0	0
$895$	58.7386	1.96342
$896$	1.00000	0.0334077
$897$	0	0
$898$	10.6847	0.356552
$899$	−41.7538	−1.39257
$900$	0	0
$901$	−81.8617	−2.72721
$902$	−1.75379	−0.0583948
$903$	0	0
$904$	−4.24621	−0.141227
$905$	0.876894	0.0291490
$906$	0	0
$907$	−7.50758	−0.249285	−0.124643	−	0.992202i	$-0.539778\pi$
$907$	−7.50758	−0.249285	−0.124643	+	0.992202i	$0.539778\pi$
$908$	7.12311	0.236389
$909$	0	0
$910$	−3.56155	−0.118064
$911$	11.4233	0.378471	0.189235	−	0.981932i	$-0.439399\pi$
$911$	11.4233	0.378471	0.189235	+	0.981932i	$0.439399\pi$
$912$	0	0
$913$	−13.8617	−0.458757
$914$	−27.3693	−0.905297
$915$	0	0
$916$	28.2462	0.933281
$917$	−9.56155	−0.315750
$918$	0	0
$919$	19.1231	0.630813	0.315407	−	0.948957i	$-0.397859\pi$
$919$	19.1231	0.630813	0.315407	+	0.948957i	$0.397859\pi$
$920$	19.8078	0.653043
$921$	0	0
$922$	−28.0540	−0.923908
$923$	−8.00000	−0.263323
$924$	0	0
$925$	−58.1080	−1.91058
$926$	8.68466	0.285396
$927$	0	0
$928$	6.68466	0.219435
$929$	−25.6155	−0.840418	−0.420209	−	0.907427i	$-0.638043\pi$
$929$	−25.6155	−0.840418	−0.420209	+	0.907427i	$0.638043\pi$
$930$	0	0
$931$	−4.68466	−0.153533
$932$	−27.3693	−0.896512
$933$	0	0
$934$	−3.31534	−0.108481
$935$	−37.1771	−1.21582
$936$	0	0
$937$	−46.9848	−1.53493	−0.767464	−	0.641092i	$-0.778482\pi$
$937$	−46.9848	−1.53493	−0.767464	+	0.641092i	$0.778482\pi$
$938$	−7.12311	−0.232578
$939$	0	0
$940$	0	0
$941$	34.0000	1.10837	0.554184	−	0.832394i	$-0.313030\pi$
$941$	34.0000	1.10837	0.554184	+	0.832394i	$0.313030\pi$
$942$	0	0
$943$	6.24621	0.203405
$944$	−2.24621	−0.0731079
$945$	0	0
$946$	10.0540	0.326883
$947$	40.7926	1.32558	0.662791	−	0.748805i	$-0.269372\pi$
$947$	40.7926	1.32558	0.662791	+	0.748805i	$0.269372\pi$
$948$	0	0
$949$	−3.56155	−0.115613
$950$	36.0000	1.16799
$951$	0	0
$952$	6.68466	0.216651
$953$	−11.3693	−0.368288	−0.184144	−	0.982899i	$-0.558951\pi$
$953$	−11.3693	−0.368288	−0.184144	+	0.982899i	$0.558951\pi$
$954$	0	0
$955$	53.1771	1.72077
$956$	−16.0000	−0.517477
$957$	0	0
$958$	32.3002	1.04357
$959$	−3.56155	−0.115009
$960$	0	0
$961$	8.01515	0.258553
$962$	7.56155	0.243794
$963$	0	0
$964$	−14.0000	−0.450910
$965$	−68.9848	−2.22070
$966$	0	0
$967$	−21.5616	−0.693373	−0.346686	−	0.937981i	$-0.612693\pi$
$967$	−21.5616	−0.693373	−0.346686	+	0.937981i	$0.612693\pi$
$968$	8.56155	0.275179
$969$	0	0
$970$	51.6155	1.65727
$971$	−2.24621	−0.0720843	−0.0360422	−	0.999350i	$-0.511475\pi$
$971$	−2.24621	−0.0720843	−0.0360422	+	0.999350i	$0.511475\pi$
$972$	0	0
$973$	−12.0000	−0.384702
$974$	8.00000	0.256337
$975$	0	0
$976$	6.68466	0.213971
$977$	22.6847	0.725747	0.362873	−	0.931839i	$-0.381796\pi$
$977$	22.6847	0.725747	0.362873	+	0.931839i	$0.381796\pi$
$978$	0	0
$979$	−15.6155	−0.499074
$980$	−3.56155	−0.113770
$981$	0	0
$982$	−8.87689	−0.283273
$983$	13.1771	0.420284	0.210142	−	0.977671i	$-0.432607\pi$
$983$	13.1771	0.420284	0.210142	+	0.977671i	$0.432607\pi$
$984$	0	0
$985$	38.7386	1.23432
$986$	44.6847	1.42305
$987$	0	0
$988$	−4.68466	−0.149039
$989$	−35.8078	−1.13862
$990$	0	0
$991$	−7.61553	−0.241915	−0.120958	−	0.992658i	$-0.538597\pi$
$991$	−7.61553	−0.241915	−0.120958	+	0.992658i	$0.538597\pi$
$992$	−6.24621	−0.198317
$993$	0	0
$994$	−8.00000	−0.253745
$995$	−24.6847	−0.782556
$996$	0	0
$997$	40.7386	1.29021	0.645103	−	0.764096i	$-0.276815\pi$
$997$	40.7386	1.29021	0.645103	+	0.764096i	$0.276815\pi$
$998$	−36.0000	−1.13956
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.a.u.1.1		2
3.2	odd	2		546.2.a.j.1.2	✓	2
12.11	even	2		4368.2.a.be.1.2		2
21.20	even	2		3822.2.a.bo.1.1		2
39.38	odd	2		7098.2.a.bl.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.a.j.1.2	✓	2	3.2	odd	2
1638.2.a.u.1.1		2	1.1	even	1	trivial
3822.2.a.bo.1.1		2	21.20	even	2
4368.2.a.be.1.2		2	12.11	even	2
7098.2.a.bl.1.1		2	39.38	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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.56155 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.56155 q^{5} -1.00000 q^{7} -1.00000 q^{8} +3.56155 q^{10} +1.56155 q^{11} +1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.68466 q^{17} -4.68466 q^{19} -3.56155 q^{20} -1.56155 q^{22} +5.56155 q^{23} +7.68466 q^{25} -1.00000 q^{26} -1.00000 q^{28} -6.68466 q^{29} +6.24621 q^{31} -1.00000 q^{32} -6.68466 q^{34} +3.56155 q^{35} -7.56155 q^{37} +4.68466 q^{38} +3.56155 q^{40} +1.12311 q^{41} -6.43845 q^{43} +1.56155 q^{44} -5.56155 q^{46} +1.00000 q^{49} -7.68466 q^{50} +1.00000 q^{52} -12.2462 q^{53} -5.56155 q^{55} +1.00000 q^{56} +6.68466 q^{58} -2.24621 q^{59} +6.68466 q^{61} -6.24621 q^{62} +1.00000 q^{64} -3.56155 q^{65} -7.12311 q^{67} +6.68466 q^{68} -3.56155 q^{70} -8.00000 q^{71} -3.56155 q^{73} +7.56155 q^{74} -4.68466 q^{76} -1.56155 q^{77} -11.1231 q^{79} -3.56155 q^{80} -1.12311 q^{82} -8.87689 q^{83} -23.8078 q^{85} +6.43845 q^{86} -1.56155 q^{88} -10.0000 q^{89} -1.00000 q^{91} +5.56155 q^{92} +16.6847 q^{95} +14.4924 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 3 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 3 * q^5 - 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 3 q^{5} - 2 q^{7} - 2 q^{8} + 3 q^{10} - q^{11} + 2 q^{13} + 2 q^{14} + 2 q^{16} + q^{17} + 3 q^{19} - 3 q^{20} + q^{22} + 7 q^{23} + 3 q^{25} - 2 q^{26} - 2 q^{28} - q^{29} - 4 q^{31} - 2 q^{32} - q^{34} + 3 q^{35} - 11 q^{37} - 3 q^{38} + 3 q^{40} - 6 q^{41} - 17 q^{43} - q^{44} - 7 q^{46} + 2 q^{49} - 3 q^{50} + 2 q^{52} - 8 q^{53} - 7 q^{55} + 2 q^{56} + q^{58} + 12 q^{59} + q^{61} + 4 q^{62} + 2 q^{64} - 3 q^{65} - 6 q^{67} + q^{68} - 3 q^{70} - 16 q^{71} - 3 q^{73} + 11 q^{74} + 3 q^{76} + q^{77} - 14 q^{79} - 3 q^{80} + 6 q^{82} - 26 q^{83} - 27 q^{85} + 17 q^{86} + q^{88} - 20 q^{89} - 2 q^{91} + 7 q^{92} + 21 q^{95} - 4 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 3 * q^5 - 2 * q^7 - 2 * q^8 + 3 * q^10 - q^11 + 2 * q^13 + 2 * q^14 + 2 * q^16 + q^17 + 3 * q^19 - 3 * q^20 + q^22 + 7 * q^23 + 3 * q^25 - 2 * q^26 - 2 * q^28 - q^29 - 4 * q^31 - 2 * q^32 - q^34 + 3 * q^35 - 11 * q^37 - 3 * q^38 + 3 * q^40 - 6 * q^41 - 17 * q^43 - q^44 - 7 * q^46 + 2 * q^49 - 3 * q^50 + 2 * q^52 - 8 * q^53 - 7 * q^55 + 2 * q^56 + q^58 + 12 * q^59 + q^61 + 4 * q^62 + 2 * q^64 - 3 * q^65 - 6 * q^67 + q^68 - 3 * q^70 - 16 * q^71 - 3 * q^73 + 11 * q^74 + 3 * q^76 + q^77 - 14 * q^79 - 3 * q^80 + 6 * q^82 - 26 * q^83 - 27 * q^85 + 17 * q^86 + q^88 - 20 * q^89 - 2 * q^91 + 7 * q^92 + 21 * q^95 - 4 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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