LMFDB - Embedded newform 3654.2.a.x.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3654,2,Mod(1,3654)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3654.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3654, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3654 = 2 \cdot 3^{2} \cdot 7 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3654.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,2,-4,0,2,-2,0,4,1,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$29.1773368986$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1218)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	3654.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +2.00000 q^{10} -1.56155 q^{11} +3.56155 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -4.68466 q^{19} -2.00000 q^{20} +1.56155 q^{22} +4.68466 q^{23} -1.00000 q^{25} -3.56155 q^{26} +1.00000 q^{28} -1.00000 q^{29} +6.24621 q^{31} -1.00000 q^{32} +2.00000 q^{34} -2.00000 q^{35} +3.56155 q^{37} +4.68466 q^{38} +2.00000 q^{40} +1.12311 q^{41} -3.12311 q^{43} -1.56155 q^{44} -4.68466 q^{46} -8.68466 q^{47} +1.00000 q^{49} +1.00000 q^{50} +3.56155 q^{52} -6.68466 q^{53} +3.12311 q^{55} -1.00000 q^{56} +1.00000 q^{58} -4.68466 q^{59} +6.87689 q^{61} -6.24621 q^{62} +1.00000 q^{64} -7.12311 q^{65} -9.56155 q^{67} -2.00000 q^{68} +2.00000 q^{70} +10.2462 q^{71} -2.68466 q^{73} -3.56155 q^{74} -4.68466 q^{76} -1.56155 q^{77} +2.24621 q^{79} -2.00000 q^{80} -1.12311 q^{82} +15.8078 q^{83} +4.00000 q^{85} +3.12311 q^{86} +1.56155 q^{88} -6.87689 q^{89} +3.56155 q^{91} +4.68466 q^{92} +8.68466 q^{94} +9.36932 q^{95} -5.80776 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + 2 q^{7} - 2 q^{8} + 4 q^{10} + q^{11} + 3 q^{13} - 2 q^{14} + 2 q^{16} - 4 q^{17} + 3 q^{19} - 4 q^{20} - q^{22} - 3 q^{23} - 2 q^{25} - 3 q^{26} + 2 q^{28} - 2 q^{29}+ \cdots - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 + 2 * q^7 - 2 * q^8 + 4 * q^10 + q^11 + 3 * q^13 - 2 * q^14 + 2 * q^16 - 4 * q^17 + 3 * q^19 - 4 * q^20 - q^22 - 3 * q^23 - 2 * q^25 - 3 * q^26 + 2 * q^28 - 2 * q^29 - 4 * q^31 - 2 * q^32 + 4 * q^34 - 4 * q^35 + 3 * q^37 - 3 * q^38 + 4 * q^40 - 6 * q^41 + 2 * q^43 + q^44 + 3 * q^46 - 5 * q^47 + 2 * q^49 + 2 * q^50 + 3 * q^52 - q^53 - 2 * q^55 - 2 * q^56 + 2 * q^58 + 3 * q^59 + 22 * q^61 + 4 * q^62 + 2 * q^64 - 6 * q^65 - 15 * q^67 - 4 * q^68 + 4 * q^70 + 4 * q^71 + 7 * q^73 - 3 * q^74 + 3 * q^76 + q^77 - 12 * q^79 - 4 * q^80 + 6 * q^82 + 11 * q^83 + 8 * q^85 - 2 * q^86 - q^88 - 22 * q^89 + 3 * q^91 - 3 * q^92 + 5 * q^94 - 6 * q^95 + 9 * q^97 - 2 * q^98

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
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.00000	0.632456
$11$	−1.56155	−0.470826	−0.235413	−	0.971895i	$-0.575644\pi$
$11$	−1.56155	−0.470826	−0.235413	+	0.971895i	$0.575644\pi$
$12$	0	0
$13$	3.56155	0.987797	0.493899	−	0.869520i	$-0.335571\pi$
$13$	3.56155	0.987797	0.493899	+	0.869520i	$0.335571\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−4.68466	−1.07473	−0.537367	−	0.843348i	$-0.680581\pi$
$19$	−4.68466	−1.07473	−0.537367	+	0.843348i	$0.680581\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	1.56155	0.332924
$23$	4.68466	0.976819	0.488409	−	0.872615i	$-0.337577\pi$
$23$	4.68466	0.976819	0.488409	+	0.872615i	$0.337577\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	−3.56155	−0.698478
$27$	0	0
$28$	1.00000	0.188982
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	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
$34$	2.00000	0.342997
$35$	−2.00000	−0.338062
$36$	0	0
$37$	3.56155	0.585516	0.292758	−	0.956187i	$-0.405427\pi$
$37$	3.56155	0.585516	0.292758	+	0.956187i	$0.405427\pi$
$38$	4.68466	0.759952
$39$	0	0
$40$	2.00000	0.316228
$41$	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
$43$	−3.12311	−0.476269	−0.238135	−	0.971232i	$-0.576536\pi$
$43$	−3.12311	−0.476269	−0.238135	+	0.971232i	$0.576536\pi$
$44$	−1.56155	−0.235413
$45$	0	0
$46$	−4.68466	−0.690715
$47$	−8.68466	−1.26679	−0.633394	−	0.773830i	$-0.718339\pi$
$47$	−8.68466	−1.26679	−0.633394	+	0.773830i	$0.718339\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	3.56155	0.493899
$53$	−6.68466	−0.918208	−0.459104	−	0.888382i	$-0.651830\pi$
$53$	−6.68466	−0.918208	−0.459104	+	0.888382i	$0.651830\pi$
$54$	0	0
$55$	3.12311	0.421119
$56$	−1.00000	−0.133631
$57$	0	0
$58$	1.00000	0.131306
$59$	−4.68466	−0.609891	−0.304945	−	0.952370i	$-0.598638\pi$
$59$	−4.68466	−0.609891	−0.304945	+	0.952370i	$0.598638\pi$
$60$	0	0
$61$	6.87689	0.880496	0.440248	−	0.897876i	$-0.354891\pi$
$61$	6.87689	0.880496	0.440248	+	0.897876i	$0.354891\pi$
$62$	−6.24621	−0.793270
$63$	0	0
$64$	1.00000	0.125000
$65$	−7.12311	−0.883513
$66$	0	0
$67$	−9.56155	−1.16813	−0.584065	−	0.811707i	$-0.698539\pi$
$67$	−9.56155	−1.16813	−0.584065	+	0.811707i	$0.698539\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	2.00000	0.239046
$71$	10.2462	1.21600	0.608001	−	0.793936i	$-0.291972\pi$
$71$	10.2462	1.21600	0.608001	+	0.793936i	$0.291972\pi$
$72$	0	0
$73$	−2.68466	−0.314216	−0.157108	−	0.987581i	$-0.550217\pi$
$73$	−2.68466	−0.314216	−0.157108	+	0.987581i	$0.550217\pi$
$74$	−3.56155	−0.414022
$75$	0	0
$76$	−4.68466	−0.537367
$77$	−1.56155	−0.177955
$78$	0	0
$79$	2.24621	0.252719	0.126359	−	0.991985i	$-0.459671\pi$
$79$	2.24621	0.252719	0.126359	+	0.991985i	$0.459671\pi$
$80$	−2.00000	−0.223607
$81$	0	0
$82$	−1.12311	−0.124026
$83$	15.8078	1.73513	0.867564	−	0.497326i	$-0.165685\pi$
$83$	15.8078	1.73513	0.867564	+	0.497326i	$0.165685\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	3.12311	0.336773
$87$	0	0
$88$	1.56155	0.166462
$89$	−6.87689	−0.728949	−0.364475	−	0.931213i	$-0.618751\pi$
$89$	−6.87689	−0.728949	−0.364475	+	0.931213i	$0.618751\pi$
$90$	0	0
$91$	3.56155	0.373352
$92$	4.68466	0.488409
$93$	0	0
$94$	8.68466	0.895754
$95$	9.36932	0.961272
$96$	0	0
$97$	−5.80776	−0.589689	−0.294845	−	0.955545i	$-0.595268\pi$
$97$	−5.80776	−0.589689	−0.294845	+	0.955545i	$0.595268\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−1.00000	−0.100000
$101$	0.246211	0.0244989	0.0122495	−	0.999925i	$-0.496101\pi$
$101$	0.246211	0.0244989	0.0122495	+	0.999925i	$0.496101\pi$
$102$	0	0
$103$	14.9309	1.47118	0.735591	−	0.677426i	$-0.236904\pi$
$103$	14.9309	1.47118	0.735591	+	0.677426i	$0.236904\pi$
$104$	−3.56155	−0.349239
$105$	0	0
$106$	6.68466	0.649271
$107$	−9.36932	−0.905766	−0.452883	−	0.891570i	$-0.649604\pi$
$107$	−9.36932	−0.905766	−0.452883	+	0.891570i	$0.649604\pi$
$108$	0	0
$109$	−13.1231	−1.25697	−0.628483	−	0.777824i	$-0.716324\pi$
$109$	−13.1231	−1.25697	−0.628483	+	0.777824i	$0.716324\pi$
$110$	−3.12311	−0.297776
$111$	0	0
$112$	1.00000	0.0944911
$113$	−5.80776	−0.546348	−0.273174	−	0.961965i	$-0.588074\pi$
$113$	−5.80776	−0.546348	−0.273174	+	0.961965i	$0.588074\pi$
$114$	0	0
$115$	−9.36932	−0.873693
$116$	−1.00000	−0.0928477
$117$	0	0
$118$	4.68466	0.431258
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−8.56155	−0.778323
$122$	−6.87689	−0.622605
$123$	0	0
$124$	6.24621	0.560926
$125$	12.0000	1.07331
$126$	0	0
$127$	−12.6847	−1.12558	−0.562791	−	0.826599i	$-0.690272\pi$
$127$	−12.6847	−1.12558	−0.562791	+	0.826599i	$0.690272\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	7.12311	0.624738
$131$	−15.1231	−1.32131	−0.660656	−	0.750689i	$-0.729722\pi$
$131$	−15.1231	−1.32131	−0.660656	+	0.750689i	$0.729722\pi$
$132$	0	0
$133$	−4.68466	−0.406211
$134$	9.56155	0.825992
$135$	0	0
$136$	2.00000	0.171499
$137$	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	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	−10.2462	−0.859843
$143$	−5.56155	−0.465080
$144$	0	0
$145$	2.00000	0.166091
$146$	2.68466	0.222184
$147$	0	0
$148$	3.56155	0.292758
$149$	−0.438447	−0.0359190	−0.0179595	−	0.999839i	$-0.505717\pi$
$149$	−0.438447	−0.0359190	−0.0179595	+	0.999839i	$0.505717\pi$
$150$	0	0
$151$	1.36932	0.111433	0.0557167	−	0.998447i	$-0.482256\pi$
$151$	1.36932	0.111433	0.0557167	+	0.998447i	$0.482256\pi$
$152$	4.68466	0.379976
$153$	0	0
$154$	1.56155	0.125834
$155$	−12.4924	−1.00342
$156$	0	0
$157$	8.24621	0.658119	0.329060	−	0.944309i	$-0.393268\pi$
$157$	8.24621	0.658119	0.329060	+	0.944309i	$0.393268\pi$
$158$	−2.24621	−0.178699
$159$	0	0
$160$	2.00000	0.158114
$161$	4.68466	0.369203
$162$	0	0
$163$	−19.1231	−1.49784	−0.748919	−	0.662662i	$-0.769427\pi$
$163$	−19.1231	−1.49784	−0.748919	+	0.662662i	$0.769427\pi$
$164$	1.12311	0.0876998
$165$	0	0
$166$	−15.8078	−1.22692
$167$	1.75379	0.135712	0.0678561	−	0.997695i	$-0.478384\pi$
$167$	1.75379	0.135712	0.0678561	+	0.997695i	$0.478384\pi$
$168$	0	0
$169$	−0.315342	−0.0242570
$170$	−4.00000	−0.306786
$171$	0	0
$172$	−3.12311	−0.238135
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−1.56155	−0.117706
$177$	0	0
$178$	6.87689	0.515445
$179$	3.12311	0.233432	0.116716	−	0.993165i	$-0.462763\pi$
$179$	3.12311	0.233432	0.116716	+	0.993165i	$0.462763\pi$
$180$	0	0
$181$	10.4924	0.779896	0.389948	−	0.920837i	$-0.372493\pi$
$181$	10.4924	0.779896	0.389948	+	0.920837i	$0.372493\pi$
$182$	−3.56155	−0.264000
$183$	0	0
$184$	−4.68466	−0.345358
$185$	−7.12311	−0.523701
$186$	0	0
$187$	3.12311	0.228384
$188$	−8.68466	−0.633394
$189$	0	0
$190$	−9.36932	−0.679722
$191$	−3.12311	−0.225980	−0.112990	−	0.993596i	$-0.536043\pi$
$191$	−3.12311	−0.225980	−0.112990	+	0.993596i	$0.536043\pi$
$192$	0	0
$193$	−7.75379	−0.558130	−0.279065	−	0.960272i	$-0.590024\pi$
$193$	−7.75379	−0.558130	−0.279065	+	0.960272i	$0.590024\pi$
$194$	5.80776	0.416973
$195$	0	0
$196$	1.00000	0.0714286
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	−21.5616	−1.52846	−0.764229	−	0.644945i	$-0.776880\pi$
$199$	−21.5616	−1.52846	−0.764229	+	0.644945i	$0.776880\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−0.246211	−0.0173234
$203$	−1.00000	−0.0701862
$204$	0	0
$205$	−2.24621	−0.156882
$206$	−14.9309	−1.04028
$207$	0	0
$208$	3.56155	0.246949
$209$	7.31534	0.506013
$210$	0	0
$211$	−11.1231	−0.765746	−0.382873	−	0.923801i	$-0.625065\pi$
$211$	−11.1231	−0.765746	−0.382873	+	0.923801i	$0.625065\pi$
$212$	−6.68466	−0.459104
$213$	0	0
$214$	9.36932	0.640473
$215$	6.24621	0.425988
$216$	0	0
$217$	6.24621	0.424020
$218$	13.1231	0.888809
$219$	0	0
$220$	3.12311	0.210560
$221$	−7.12311	−0.479152
$222$	0	0
$223$	−18.0540	−1.20898	−0.604492	−	0.796611i	$-0.706624\pi$
$223$	−18.0540	−1.20898	−0.604492	+	0.796611i	$0.706624\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	5.80776	0.386327
$227$	8.49242	0.563662	0.281831	−	0.959464i	$-0.409058\pi$
$227$	8.49242	0.563662	0.281831	+	0.959464i	$0.409058\pi$
$228$	0	0
$229$	−26.4924	−1.75067	−0.875334	−	0.483518i	$-0.839359\pi$
$229$	−26.4924	−1.75067	−0.875334	+	0.483518i	$0.839359\pi$
$230$	9.36932	0.617794
$231$	0	0
$232$	1.00000	0.0656532
$233$	15.3693	1.00688	0.503439	−	0.864031i	$-0.332068\pi$
$233$	15.3693	1.00688	0.503439	+	0.864031i	$0.332068\pi$
$234$	0	0
$235$	17.3693	1.13305
$236$	−4.68466	−0.304945
$237$	0	0
$238$	2.00000	0.129641
$239$	22.0540	1.42655	0.713277	−	0.700883i	$-0.247210\pi$
$239$	22.0540	1.42655	0.713277	+	0.700883i	$0.247210\pi$
$240$	0	0
$241$	−7.75379	−0.499465	−0.249733	−	0.968315i	$-0.580343\pi$
$241$	−7.75379	−0.499465	−0.249733	+	0.968315i	$0.580343\pi$
$242$	8.56155	0.550357
$243$	0	0
$244$	6.87689	0.440248
$245$	−2.00000	−0.127775
$246$	0	0
$247$	−16.6847	−1.06162
$248$	−6.24621	−0.396635
$249$	0	0
$250$	−12.0000	−0.758947
$251$	16.8769	1.06526	0.532630	−	0.846348i	$-0.321204\pi$
$251$	16.8769	1.06526	0.532630	+	0.846348i	$0.321204\pi$
$252$	0	0
$253$	−7.31534	−0.459912
$254$	12.6847	0.795906
$255$	0	0
$256$	1.00000	0.0625000
$257$	−10.4924	−0.654499	−0.327250	−	0.944938i	$-0.606122\pi$
$257$	−10.4924	−0.654499	−0.327250	+	0.944938i	$0.606122\pi$
$258$	0	0
$259$	3.56155	0.221304
$260$	−7.12311	−0.441756
$261$	0	0
$262$	15.1231	0.934309
$263$	−1.75379	−0.108143	−0.0540716	−	0.998537i	$-0.517220\pi$
$263$	−1.75379	−0.108143	−0.0540716	+	0.998537i	$0.517220\pi$
$264$	0	0
$265$	13.3693	0.821271
$266$	4.68466	0.287235
$267$	0	0
$268$	−9.56155	−0.584065
$269$	−11.5616	−0.704920	−0.352460	−	0.935827i	$-0.614655\pi$
$269$	−11.5616	−0.704920	−0.352460	+	0.935827i	$0.614655\pi$
$270$	0	0
$271$	14.2462	0.865396	0.432698	−	0.901539i	$-0.357562\pi$
$271$	14.2462	0.865396	0.432698	+	0.901539i	$0.357562\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−3.56155	−0.215161
$275$	1.56155	0.0941652
$276$	0	0
$277$	−17.6155	−1.05841	−0.529207	−	0.848493i	$-0.677511\pi$
$277$	−17.6155	−1.05841	−0.529207	+	0.848493i	$0.677511\pi$
$278$	12.0000	0.719712
$279$	0	0
$280$	2.00000	0.119523
$281$	9.12311	0.544239	0.272119	−	0.962263i	$-0.412275\pi$
$281$	9.12311	0.544239	0.272119	+	0.962263i	$0.412275\pi$
$282$	0	0
$283$	−30.7386	−1.82722	−0.913611	−	0.406589i	$-0.866718\pi$
$283$	−30.7386	−1.82722	−0.913611	+	0.406589i	$0.866718\pi$
$284$	10.2462	0.608001
$285$	0	0
$286$	5.56155	0.328862
$287$	1.12311	0.0662948
$288$	0	0
$289$	−13.0000	−0.764706
$290$	−2.00000	−0.117444
$291$	0	0
$292$	−2.68466	−0.157108
$293$	−24.0540	−1.40525	−0.702624	−	0.711561i	$-0.747988\pi$
$293$	−24.0540	−1.40525	−0.702624	+	0.711561i	$0.747988\pi$
$294$	0	0
$295$	9.36932	0.545503
$296$	−3.56155	−0.207011
$297$	0	0
$298$	0.438447	0.0253986
$299$	16.6847	0.964899
$300$	0	0
$301$	−3.12311	−0.180013
$302$	−1.36932	−0.0787953
$303$	0	0
$304$	−4.68466	−0.268684
$305$	−13.7538	−0.787540
$306$	0	0
$307$	−1.56155	−0.0891225	−0.0445613	−	0.999007i	$-0.514189\pi$
$307$	−1.56155	−0.0891225	−0.0445613	+	0.999007i	$0.514189\pi$
$308$	−1.56155	−0.0889777
$309$	0	0
$310$	12.4924	0.709522
$311$	−12.4924	−0.708380	−0.354190	−	0.935173i	$-0.615243\pi$
$311$	−12.4924	−0.708380	−0.354190	+	0.935173i	$0.615243\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	−8.24621	−0.465361
$315$	0	0
$316$	2.24621	0.126359
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	0	0
$319$	1.56155	0.0874302
$320$	−2.00000	−0.111803
$321$	0	0
$322$	−4.68466	−0.261066
$323$	9.36932	0.521323
$324$	0	0
$325$	−3.56155	−0.197559
$326$	19.1231	1.05913
$327$	0	0
$328$	−1.12311	−0.0620131
$329$	−8.68466	−0.478801
$330$	0	0
$331$	1.36932	0.0752645	0.0376322	−	0.999292i	$-0.488018\pi$
$331$	1.36932	0.0752645	0.0376322	+	0.999292i	$0.488018\pi$
$332$	15.8078	0.867564
$333$	0	0
$334$	−1.75379	−0.0959631
$335$	19.1231	1.04481
$336$	0	0
$337$	−7.36932	−0.401432	−0.200716	−	0.979649i	$-0.564327\pi$
$337$	−7.36932	−0.401432	−0.200716	+	0.979649i	$0.564327\pi$
$338$	0.315342	0.0171523
$339$	0	0
$340$	4.00000	0.216930
$341$	−9.75379	−0.528197
$342$	0	0
$343$	1.00000	0.0539949
$344$	3.12311	0.168387
$345$	0	0
$346$	10.0000	0.537603
$347$	−35.1231	−1.88551	−0.942754	−	0.333490i	$-0.891774\pi$
$347$	−35.1231	−1.88551	−0.942754	+	0.333490i	$0.891774\pi$
$348$	0	0
$349$	34.4924	1.84634	0.923169	−	0.384395i	$-0.125590\pi$
$349$	34.4924	1.84634	0.923169	+	0.384395i	$0.125590\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	1.56155	0.0832310
$353$	−10.4924	−0.558455	−0.279228	−	0.960225i	$-0.590078\pi$
$353$	−10.4924	−0.558455	−0.279228	+	0.960225i	$0.590078\pi$
$354$	0	0
$355$	−20.4924	−1.08762
$356$	−6.87689	−0.364475
$357$	0	0
$358$	−3.12311	−0.165061
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	2.94602	0.155054
$362$	−10.4924	−0.551469
$363$	0	0
$364$	3.56155	0.186676
$365$	5.36932	0.281043
$366$	0	0
$367$	14.2462	0.743646	0.371823	−	0.928304i	$-0.378733\pi$
$367$	14.2462	0.743646	0.371823	+	0.928304i	$0.378733\pi$
$368$	4.68466	0.244205
$369$	0	0
$370$	7.12311	0.370313
$371$	−6.68466	−0.347050
$372$	0	0
$373$	2.49242	0.129053	0.0645264	−	0.997916i	$-0.479446\pi$
$373$	2.49242	0.129053	0.0645264	+	0.997916i	$0.479446\pi$
$374$	−3.12311	−0.161492
$375$	0	0
$376$	8.68466	0.447877
$377$	−3.56155	−0.183429
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	9.36932	0.480636
$381$	0	0
$382$	3.12311	0.159792
$383$	4.49242	0.229552	0.114776	−	0.993391i	$-0.463385\pi$
$383$	4.49242	0.229552	0.114776	+	0.993391i	$0.463385\pi$
$384$	0	0
$385$	3.12311	0.159168
$386$	7.75379	0.394657
$387$	0	0
$388$	−5.80776	−0.294845
$389$	−23.3693	−1.18487	−0.592436	−	0.805618i	$-0.701834\pi$
$389$	−23.3693	−1.18487	−0.592436	+	0.805618i	$0.701834\pi$
$390$	0	0
$391$	−9.36932	−0.473827
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−2.00000	−0.100759
$395$	−4.49242	−0.226038
$396$	0	0
$397$	28.9309	1.45200	0.725999	−	0.687695i	$-0.241378\pi$
$397$	28.9309	1.45200	0.725999	+	0.687695i	$0.241378\pi$
$398$	21.5616	1.08078
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	2.49242	0.124466	0.0622328	−	0.998062i	$-0.480178\pi$
$401$	2.49242	0.124466	0.0622328	+	0.998062i	$0.480178\pi$
$402$	0	0
$403$	22.2462	1.10816
$404$	0.246211	0.0122495
$405$	0	0
$406$	1.00000	0.0496292
$407$	−5.56155	−0.275676
$408$	0	0
$409$	−22.4924	−1.11218	−0.556089	−	0.831123i	$-0.687699\pi$
$409$	−22.4924	−1.11218	−0.556089	+	0.831123i	$0.687699\pi$
$410$	2.24621	0.110932
$411$	0	0
$412$	14.9309	0.735591
$413$	−4.68466	−0.230517
$414$	0	0
$415$	−31.6155	−1.55195
$416$	−3.56155	−0.174619
$417$	0	0
$418$	−7.31534	−0.357805
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	11.1231	0.541464
$423$	0	0
$424$	6.68466	0.324636
$425$	2.00000	0.0970143
$426$	0	0
$427$	6.87689	0.332796
$428$	−9.36932	−0.452883
$429$	0	0
$430$	−6.24621	−0.301219
$431$	−15.8078	−0.761433	−0.380717	−	0.924692i	$-0.624323\pi$
$431$	−15.8078	−0.761433	−0.380717	+	0.924692i	$0.624323\pi$
$432$	0	0
$433$	−8.24621	−0.396288	−0.198144	−	0.980173i	$-0.563491\pi$
$433$	−8.24621	−0.396288	−0.198144	+	0.980173i	$0.563491\pi$
$434$	−6.24621	−0.299828
$435$	0	0
$436$	−13.1231	−0.628483
$437$	−21.9460	−1.04982
$438$	0	0
$439$	7.31534	0.349142	0.174571	−	0.984645i	$-0.444146\pi$
$439$	7.31534	0.349142	0.174571	+	0.984645i	$0.444146\pi$
$440$	−3.12311	−0.148888
$441$	0	0
$442$	7.12311	0.338812
$443$	−10.9309	−0.519341	−0.259671	−	0.965697i	$-0.583614\pi$
$443$	−10.9309	−0.519341	−0.259671	+	0.965697i	$0.583614\pi$
$444$	0	0
$445$	13.7538	0.651992
$446$	18.0540	0.854881
$447$	0	0
$448$	1.00000	0.0472456
$449$	−23.1771	−1.09379	−0.546897	−	0.837200i	$-0.684191\pi$
$449$	−23.1771	−1.09379	−0.546897	+	0.837200i	$0.684191\pi$
$450$	0	0
$451$	−1.75379	−0.0825827
$452$	−5.80776	−0.273174
$453$	0	0
$454$	−8.49242	−0.398569
$455$	−7.12311	−0.333936
$456$	0	0
$457$	15.1771	0.709954	0.354977	−	0.934875i	$-0.384489\pi$
$457$	15.1771	0.709954	0.354977	+	0.934875i	$0.384489\pi$
$458$	26.4924	1.23791
$459$	0	0
$460$	−9.36932	−0.436847
$461$	32.5464	1.51584	0.757918	−	0.652349i	$-0.226216\pi$
$461$	32.5464	1.51584	0.757918	+	0.652349i	$0.226216\pi$
$462$	0	0
$463$	32.9848	1.53294	0.766468	−	0.642283i	$-0.222012\pi$
$463$	32.9848	1.53294	0.766468	+	0.642283i	$0.222012\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	−15.3693	−0.711970
$467$	−42.2462	−1.95492	−0.977461	−	0.211117i	$-0.932290\pi$
$467$	−42.2462	−1.95492	−0.977461	+	0.211117i	$0.932290\pi$
$468$	0	0
$469$	−9.56155	−0.441511
$470$	−17.3693	−0.801187
$471$	0	0
$472$	4.68466	0.215629
$473$	4.87689	0.224240
$474$	0	0
$475$	4.68466	0.214947
$476$	−2.00000	−0.0916698
$477$	0	0
$478$	−22.0540	−1.00873
$479$	−19.8078	−0.905040	−0.452520	−	0.891754i	$-0.649475\pi$
$479$	−19.8078	−0.905040	−0.452520	+	0.891754i	$0.649475\pi$
$480$	0	0
$481$	12.6847	0.578371
$482$	7.75379	0.353175
$483$	0	0
$484$	−8.56155	−0.389161
$485$	11.6155	0.527434
$486$	0	0
$487$	−6.24621	−0.283043	−0.141521	−	0.989935i	$-0.545199\pi$
$487$	−6.24621	−0.283043	−0.141521	+	0.989935i	$0.545199\pi$
$488$	−6.87689	−0.311302
$489$	0	0
$490$	2.00000	0.0903508
$491$	−23.8078	−1.07443	−0.537215	−	0.843446i	$-0.680524\pi$
$491$	−23.8078	−1.07443	−0.537215	+	0.843446i	$0.680524\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	16.6847	0.750678
$495$	0	0
$496$	6.24621	0.280463
$497$	10.2462	0.459605
$498$	0	0
$499$	−42.9309	−1.92185	−0.960925	−	0.276809i	$-0.910723\pi$
$499$	−42.9309	−1.92185	−0.960925	+	0.276809i	$0.910723\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	−16.8769	−0.753253
$503$	30.9309	1.37914	0.689570	−	0.724219i	$-0.257800\pi$
$503$	30.9309	1.37914	0.689570	+	0.724219i	$0.257800\pi$
$504$	0	0
$505$	−0.492423	−0.0219125
$506$	7.31534	0.325207
$507$	0	0
$508$	−12.6847	−0.562791
$509$	15.7538	0.698274	0.349137	−	0.937072i	$-0.386475\pi$
$509$	15.7538	0.698274	0.349137	+	0.937072i	$0.386475\pi$
$510$	0	0
$511$	−2.68466	−0.118762
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	10.4924	0.462801
$515$	−29.8617	−1.31587
$516$	0	0
$517$	13.5616	0.596436
$518$	−3.56155	−0.156486
$519$	0	0
$520$	7.12311	0.312369
$521$	2.68466	0.117617	0.0588085	−	0.998269i	$-0.481270\pi$
$521$	2.68466	0.117617	0.0588085	+	0.998269i	$0.481270\pi$
$522$	0	0
$523$	0.876894	0.0383439	0.0191720	−	0.999816i	$-0.493897\pi$
$523$	0.876894	0.0383439	0.0191720	+	0.999816i	$0.493897\pi$
$524$	−15.1231	−0.660656
$525$	0	0
$526$	1.75379	0.0764688
$527$	−12.4924	−0.544178
$528$	0	0
$529$	−1.05398	−0.0458250
$530$	−13.3693	−0.580726
$531$	0	0
$532$	−4.68466	−0.203106
$533$	4.00000	0.173259
$534$	0	0
$535$	18.7386	0.810142
$536$	9.56155	0.412996
$537$	0	0
$538$	11.5616	0.498454
$539$	−1.56155	−0.0672608
$540$	0	0
$541$	−33.3153	−1.43234	−0.716169	−	0.697927i	$-0.754106\pi$
$541$	−33.3153	−1.43234	−0.716169	+	0.697927i	$0.754106\pi$
$542$	−14.2462	−0.611927
$543$	0	0
$544$	2.00000	0.0857493
$545$	26.2462	1.12426
$546$	0	0
$547$	11.3153	0.483809	0.241905	−	0.970300i	$-0.422228\pi$
$547$	11.3153	0.483809	0.241905	+	0.970300i	$0.422228\pi$
$548$	3.56155	0.152142
$549$	0	0
$550$	−1.56155	−0.0665848
$551$	4.68466	0.199573
$552$	0	0
$553$	2.24621	0.0955186
$554$	17.6155	0.748412
$555$	0	0
$556$	−12.0000	−0.508913
$557$	−19.5616	−0.828850	−0.414425	−	0.910084i	$-0.636017\pi$
$557$	−19.5616	−0.828850	−0.414425	+	0.910084i	$0.636017\pi$
$558$	0	0
$559$	−11.1231	−0.470457
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	−9.12311	−0.384835
$563$	29.3693	1.23777	0.618885	−	0.785482i	$-0.287585\pi$
$563$	29.3693	1.23777	0.618885	+	0.785482i	$0.287585\pi$
$564$	0	0
$565$	11.6155	0.488669
$566$	30.7386	1.29204
$567$	0	0
$568$	−10.2462	−0.429921
$569$	25.8078	1.08192	0.540959	−	0.841049i	$-0.318061\pi$
$569$	25.8078	1.08192	0.540959	+	0.841049i	$0.318061\pi$
$570$	0	0
$571$	17.5616	0.734928	0.367464	−	0.930038i	$-0.380226\pi$
$571$	17.5616	0.734928	0.367464	+	0.930038i	$0.380226\pi$
$572$	−5.56155	−0.232540
$573$	0	0
$574$	−1.12311	−0.0468775
$575$	−4.68466	−0.195364
$576$	0	0
$577$	11.1771	0.465308	0.232654	−	0.972560i	$-0.425259\pi$
$577$	11.1771	0.465308	0.232654	+	0.972560i	$0.425259\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	2.00000	0.0830455
$581$	15.8078	0.655817
$582$	0	0
$583$	10.4384	0.432316
$584$	2.68466	0.111092
$585$	0	0
$586$	24.0540	0.993661
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	0	0
$589$	−29.2614	−1.20569
$590$	−9.36932	−0.385729
$591$	0	0
$592$	3.56155	0.146379
$593$	−0.438447	−0.0180049	−0.00900243	−	0.999959i	$-0.502866\pi$
$593$	−0.438447	−0.0180049	−0.00900243	+	0.999959i	$0.502866\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	−0.438447	−0.0179595
$597$	0	0
$598$	−16.6847	−0.682286
$599$	34.7386	1.41938	0.709691	−	0.704513i	$-0.248835\pi$
$599$	34.7386	1.41938	0.709691	+	0.704513i	$0.248835\pi$
$600$	0	0
$601$	−9.31534	−0.379981	−0.189990	−	0.981786i	$-0.560846\pi$
$601$	−9.31534	−0.379981	−0.189990	+	0.981786i	$0.560846\pi$
$602$	3.12311	0.127288
$603$	0	0
$604$	1.36932	0.0557167
$605$	17.1231	0.696153
$606$	0	0
$607$	23.6155	0.958525	0.479262	−	0.877672i	$-0.340904\pi$
$607$	23.6155	0.958525	0.479262	+	0.877672i	$0.340904\pi$
$608$	4.68466	0.189988
$609$	0	0
$610$	13.7538	0.556875
$611$	−30.9309	−1.25133
$612$	0	0
$613$	45.6155	1.84239	0.921197	−	0.389097i	$-0.127213\pi$
$613$	45.6155	1.84239	0.921197	+	0.389097i	$0.127213\pi$
$614$	1.56155	0.0630191
$615$	0	0
$616$	1.56155	0.0629168
$617$	−19.7538	−0.795258	−0.397629	−	0.917546i	$-0.630167\pi$
$617$	−19.7538	−0.795258	−0.397629	+	0.917546i	$0.630167\pi$
$618$	0	0
$619$	19.3153	0.776349	0.388175	−	0.921586i	$-0.373106\pi$
$619$	19.3153	0.776349	0.388175	+	0.921586i	$0.373106\pi$
$620$	−12.4924	−0.501708
$621$	0	0
$622$	12.4924	0.500901
$623$	−6.87689	−0.275517
$624$	0	0
$625$	−19.0000	−0.760000
$626$	−2.00000	−0.0799361
$627$	0	0
$628$	8.24621	0.329060
$629$	−7.12311	−0.284017
$630$	0	0
$631$	−32.9848	−1.31311	−0.656553	−	0.754280i	$-0.727986\pi$
$631$	−32.9848	−1.31311	−0.656553	+	0.754280i	$0.727986\pi$
$632$	−2.24621	−0.0893495
$633$	0	0
$634$	14.0000	0.556011
$635$	25.3693	1.00675
$636$	0	0
$637$	3.56155	0.141114
$638$	−1.56155	−0.0618225
$639$	0	0
$640$	2.00000	0.0790569
$641$	32.7386	1.29310	0.646549	−	0.762872i	$-0.276212\pi$
$641$	32.7386	1.29310	0.646549	+	0.762872i	$0.276212\pi$
$642$	0	0
$643$	−0.492423	−0.0194192	−0.00970962	−	0.999953i	$-0.503091\pi$
$643$	−0.492423	−0.0194192	−0.00970962	+	0.999953i	$0.503091\pi$
$644$	4.68466	0.184601
$645$	0	0
$646$	−9.36932	−0.368631
$647$	17.3693	0.682858	0.341429	−	0.939908i	$-0.389089\pi$
$647$	17.3693	0.682858	0.341429	+	0.939908i	$0.389089\pi$
$648$	0	0
$649$	7.31534	0.287152
$650$	3.56155	0.139696
$651$	0	0
$652$	−19.1231	−0.748919
$653$	−43.8617	−1.71644	−0.858221	−	0.513280i	$-0.828430\pi$
$653$	−43.8617	−1.71644	−0.858221	+	0.513280i	$0.828430\pi$
$654$	0	0
$655$	30.2462	1.18182
$656$	1.12311	0.0438499
$657$	0	0
$658$	8.68466	0.338563
$659$	−5.75379	−0.224136	−0.112068	−	0.993701i	$-0.535747\pi$
$659$	−5.75379	−0.224136	−0.112068	+	0.993701i	$0.535747\pi$
$660$	0	0
$661$	−25.3153	−0.984653	−0.492326	−	0.870411i	$-0.663853\pi$
$661$	−25.3153	−0.984653	−0.492326	+	0.870411i	$0.663853\pi$
$662$	−1.36932	−0.0532200
$663$	0	0
$664$	−15.8078	−0.613460
$665$	9.36932	0.363327
$666$	0	0
$667$	−4.68466	−0.181391
$668$	1.75379	0.0678561
$669$	0	0
$670$	−19.1231	−0.738790
$671$	−10.7386	−0.414560
$672$	0	0
$673$	−12.9309	−0.498448	−0.249224	−	0.968446i	$-0.580176\pi$
$673$	−12.9309	−0.498448	−0.249224	+	0.968446i	$0.580176\pi$
$674$	7.36932	0.283855
$675$	0	0
$676$	−0.315342	−0.0121285
$677$	8.93087	0.343241	0.171621	−	0.985163i	$-0.445100\pi$
$677$	8.93087	0.343241	0.171621	+	0.985163i	$0.445100\pi$
$678$	0	0
$679$	−5.80776	−0.222882
$680$	−4.00000	−0.153393
$681$	0	0
$682$	9.75379	0.373492
$683$	28.8769	1.10494	0.552472	−	0.833532i	$-0.313685\pi$
$683$	28.8769	1.10494	0.552472	+	0.833532i	$0.313685\pi$
$684$	0	0
$685$	−7.12311	−0.272160
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−3.12311	−0.119067
$689$	−23.8078	−0.907004
$690$	0	0
$691$	13.7538	0.523219	0.261609	−	0.965174i	$-0.415747\pi$
$691$	13.7538	0.523219	0.261609	+	0.965174i	$0.415747\pi$
$692$	−10.0000	−0.380143
$693$	0	0
$694$	35.1231	1.33325
$695$	24.0000	0.910372
$696$	0	0
$697$	−2.24621	−0.0850813
$698$	−34.4924	−1.30556
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	−14.0000	−0.528773	−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000	−0.528773	−0.264386	+	0.964417i	$0.585169\pi$
$702$	0	0
$703$	−16.6847	−0.629274
$704$	−1.56155	−0.0588532
$705$	0	0
$706$	10.4924	0.394888
$707$	0.246211	0.00925973
$708$	0	0
$709$	3.26137	0.122483	0.0612416	−	0.998123i	$-0.480494\pi$
$709$	3.26137	0.122483	0.0612416	+	0.998123i	$0.480494\pi$
$710$	20.4924	0.769067
$711$	0	0
$712$	6.87689	0.257723
$713$	29.2614	1.09585
$714$	0	0
$715$	11.1231	0.415981
$716$	3.12311	0.116716
$717$	0	0
$718$	0	0
$719$	−32.9848	−1.23013	−0.615064	−	0.788478i	$-0.710870\pi$
$719$	−32.9848	−1.23013	−0.615064	+	0.788478i	$0.710870\pi$
$720$	0	0
$721$	14.9309	0.556055
$722$	−2.94602	−0.109640
$723$	0	0
$724$	10.4924	0.389948
$725$	1.00000	0.0371391
$726$	0	0
$727$	27.1231	1.00594	0.502970	−	0.864304i	$-0.332241\pi$
$727$	27.1231	1.00594	0.502970	+	0.864304i	$0.332241\pi$
$728$	−3.56155	−0.132000
$729$	0	0
$730$	−5.36932	−0.198727
$731$	6.24621	0.231024
$732$	0	0
$733$	−19.8617	−0.733610	−0.366805	−	0.930298i	$-0.619548\pi$
$733$	−19.8617	−0.733610	−0.366805	+	0.930298i	$0.619548\pi$
$734$	−14.2462	−0.525837
$735$	0	0
$736$	−4.68466	−0.172679
$737$	14.9309	0.549986
$738$	0	0
$739$	14.2462	0.524055	0.262028	−	0.965060i	$-0.415609\pi$
$739$	14.2462	0.524055	0.262028	+	0.965060i	$0.415609\pi$
$740$	−7.12311	−0.261851
$741$	0	0
$742$	6.68466	0.245402
$743$	28.4924	1.04529	0.522643	−	0.852552i	$-0.324946\pi$
$743$	28.4924	1.04529	0.522643	+	0.852552i	$0.324946\pi$
$744$	0	0
$745$	0.876894	0.0321269
$746$	−2.49242	−0.0912541
$747$	0	0
$748$	3.12311	0.114192
$749$	−9.36932	−0.342347
$750$	0	0
$751$	−33.1771	−1.21065	−0.605324	−	0.795979i	$-0.706957\pi$
$751$	−33.1771	−1.21065	−0.605324	+	0.795979i	$0.706957\pi$
$752$	−8.68466	−0.316697
$753$	0	0
$754$	3.56155	0.129704
$755$	−2.73863	−0.0996691
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	8.00000	0.290573
$759$	0	0
$760$	−9.36932	−0.339861
$761$	−0.0539753	−0.00195660	−0.000978302	−	1.00000i	$-0.500311\pi$
$761$	−0.0539753	−0.00195660	−0.000978302		1.00000i	$0.500311\pi$
$762$	0	0
$763$	−13.1231	−0.475088
$764$	−3.12311	−0.112990
$765$	0	0
$766$	−4.49242	−0.162318
$767$	−16.6847	−0.602448
$768$	0	0
$769$	12.2462	0.441610	0.220805	−	0.975318i	$-0.429132\pi$
$769$	12.2462	0.441610	0.220805	+	0.975318i	$0.429132\pi$
$770$	−3.12311	−0.112549
$771$	0	0
$772$	−7.75379	−0.279065
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	0	0
$775$	−6.24621	−0.224371
$776$	5.80776	0.208487
$777$	0	0
$778$	23.3693	0.837831
$779$	−5.26137	−0.188508
$780$	0	0
$781$	−16.0000	−0.572525
$782$	9.36932	0.335046
$783$	0	0
$784$	1.00000	0.0357143
$785$	−16.4924	−0.588640
$786$	0	0
$787$	53.3693	1.90241	0.951205	−	0.308559i	$-0.0998466\pi$
$787$	53.3693	1.90241	0.951205	+	0.308559i	$0.0998466\pi$
$788$	2.00000	0.0712470
$789$	0	0
$790$	4.49242	0.159833
$791$	−5.80776	−0.206500
$792$	0	0
$793$	24.4924	0.869751
$794$	−28.9309	−1.02672
$795$	0	0
$796$	−21.5616	−0.764229
$797$	14.4924	0.513348	0.256674	−	0.966498i	$-0.417373\pi$
$797$	14.4924	0.513348	0.256674	+	0.966498i	$0.417373\pi$
$798$	0	0
$799$	17.3693	0.614482
$800$	1.00000	0.0353553
$801$	0	0
$802$	−2.49242	−0.0880105
$803$	4.19224	0.147941
$804$	0	0
$805$	−9.36932	−0.330225
$806$	−22.2462	−0.783589
$807$	0	0
$808$	−0.246211	−0.00866168
$809$	−41.2311	−1.44961	−0.724803	−	0.688956i	$-0.758069\pi$
$809$	−41.2311	−1.44961	−0.724803	+	0.688956i	$0.758069\pi$
$810$	0	0
$811$	29.3693	1.03130	0.515648	−	0.856800i	$-0.327551\pi$
$811$	29.3693	1.03130	0.515648	+	0.856800i	$0.327551\pi$
$812$	−1.00000	−0.0350931
$813$	0	0
$814$	5.56155	0.194932
$815$	38.2462	1.33971
$816$	0	0
$817$	14.6307	0.511863
$818$	22.4924	0.786429
$819$	0	0
$820$	−2.24621	−0.0784411
$821$	−36.5464	−1.27548	−0.637739	−	0.770253i	$-0.720130\pi$
$821$	−36.5464	−1.27548	−0.637739	+	0.770253i	$0.720130\pi$
$822$	0	0
$823$	8.19224	0.285563	0.142782	−	0.989754i	$-0.454395\pi$
$823$	8.19224	0.285563	0.142782	+	0.989754i	$0.454395\pi$
$824$	−14.9309	−0.520141
$825$	0	0
$826$	4.68466	0.163000
$827$	−5.06913	−0.176271	−0.0881355	−	0.996108i	$-0.528091\pi$
$827$	−5.06913	−0.176271	−0.0881355	+	0.996108i	$0.528091\pi$
$828$	0	0
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	31.6155	1.09739
$831$	0	0
$832$	3.56155	0.123475
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	−3.50758	−0.121385
$836$	7.31534	0.253006
$837$	0	0
$838$	−4.00000	−0.138178
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	2.00000	0.0689246
$843$	0	0
$844$	−11.1231	−0.382873
$845$	0.630683	0.0216962
$846$	0	0
$847$	−8.56155	−0.294178
$848$	−6.68466	−0.229552
$849$	0	0
$850$	−2.00000	−0.0685994
$851$	16.6847	0.571943
$852$	0	0
$853$	−11.8617	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$853$	−11.8617	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$854$	−6.87689	−0.235322
$855$	0	0
$856$	9.36932	0.320237
$857$	43.6695	1.49172	0.745861	−	0.666102i	$-0.232038\pi$
$857$	43.6695	1.49172	0.745861	+	0.666102i	$0.232038\pi$
$858$	0	0
$859$	15.4233	0.526236	0.263118	−	0.964764i	$-0.415249\pi$
$859$	15.4233	0.526236	0.263118	+	0.964764i	$0.415249\pi$
$860$	6.24621	0.212994
$861$	0	0
$862$	15.8078	0.538415
$863$	−6.82292	−0.232255	−0.116127	−	0.993234i	$-0.537048\pi$
$863$	−6.82292	−0.232255	−0.116127	+	0.993234i	$0.537048\pi$
$864$	0	0
$865$	20.0000	0.680020
$866$	8.24621	0.280218
$867$	0	0
$868$	6.24621	0.212010
$869$	−3.50758	−0.118986
$870$	0	0
$871$	−34.0540	−1.15387
$872$	13.1231	0.444404
$873$	0	0
$874$	21.9460	0.742335
$875$	12.0000	0.405674
$876$	0	0
$877$	31.3693	1.05927	0.529633	−	0.848227i	$-0.322330\pi$
$877$	31.3693	1.05927	0.529633	+	0.848227i	$0.322330\pi$
$878$	−7.31534	−0.246881
$879$	0	0
$880$	3.12311	0.105280
$881$	−21.1231	−0.711656	−0.355828	−	0.934552i	$-0.615801\pi$
$881$	−21.1231	−0.711656	−0.355828	+	0.934552i	$0.615801\pi$
$882$	0	0
$883$	42.9309	1.44474	0.722369	−	0.691507i	$-0.243053\pi$
$883$	42.9309	1.44474	0.722369	+	0.691507i	$0.243053\pi$
$884$	−7.12311	−0.239576
$885$	0	0
$886$	10.9309	0.367230
$887$	27.4233	0.920784	0.460392	−	0.887716i	$-0.347709\pi$
$887$	27.4233	0.920784	0.460392	+	0.887716i	$0.347709\pi$
$888$	0	0
$889$	−12.6847	−0.425430
$890$	−13.7538	−0.461028
$891$	0	0
$892$	−18.0540	−0.604492
$893$	40.6847	1.36146
$894$	0	0
$895$	−6.24621	−0.208788
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	23.1771	0.773429
$899$	−6.24621	−0.208323
$900$	0	0
$901$	13.3693	0.445397
$902$	1.75379	0.0583948
$903$	0	0
$904$	5.80776	0.193163
$905$	−20.9848	−0.697560
$906$	0	0
$907$	52.1080	1.73022	0.865108	−	0.501586i	$-0.167250\pi$
$907$	52.1080	1.73022	0.865108	+	0.501586i	$0.167250\pi$
$908$	8.49242	0.281831
$909$	0	0
$910$	7.12311	0.236129
$911$	−47.2311	−1.56483	−0.782417	−	0.622754i	$-0.786014\pi$
$911$	−47.2311	−1.56483	−0.782417	+	0.622754i	$0.786014\pi$
$912$	0	0
$913$	−24.6847	−0.816943
$914$	−15.1771	−0.502013
$915$	0	0
$916$	−26.4924	−0.875334
$917$	−15.1231	−0.499409
$918$	0	0
$919$	−33.3693	−1.10075	−0.550376	−	0.834917i	$-0.685516\pi$
$919$	−33.3693	−1.10075	−0.550376	+	0.834917i	$0.685516\pi$
$920$	9.36932	0.308897
$921$	0	0
$922$	−32.5464	−1.07186
$923$	36.4924	1.20116
$924$	0	0
$925$	−3.56155	−0.117103
$926$	−32.9848	−1.08395
$927$	0	0
$928$	1.00000	0.0328266
$929$	40.2462	1.32044	0.660218	−	0.751074i	$-0.270464\pi$
$929$	40.2462	1.32044	0.660218	+	0.751074i	$0.270464\pi$
$930$	0	0
$931$	−4.68466	−0.153533
$932$	15.3693	0.503439
$933$	0	0
$934$	42.2462	1.38234
$935$	−6.24621	−0.204273
$936$	0	0
$937$	−53.2311	−1.73898	−0.869491	−	0.493948i	$-0.835553\pi$
$937$	−53.2311	−1.73898	−0.869491	+	0.493948i	$0.835553\pi$
$938$	9.56155	0.312196
$939$	0	0
$940$	17.3693	0.566525
$941$	18.4924	0.602836	0.301418	−	0.953492i	$-0.402540\pi$
$941$	18.4924	0.602836	0.301418	+	0.953492i	$0.402540\pi$
$942$	0	0
$943$	5.26137	0.171334
$944$	−4.68466	−0.152473
$945$	0	0
$946$	−4.87689	−0.158562
$947$	2.24621	0.0729921	0.0364960	−	0.999334i	$-0.488380\pi$
$947$	2.24621	0.0729921	0.0364960	+	0.999334i	$0.488380\pi$
$948$	0	0
$949$	−9.56155	−0.310381
$950$	−4.68466	−0.151990
$951$	0	0
$952$	2.00000	0.0648204
$953$	−10.0000	−0.323932	−0.161966	−	0.986796i	$-0.551783\pi$
$953$	−10.0000	−0.323932	−0.161966	+	0.986796i	$0.551783\pi$
$954$	0	0
$955$	6.24621	0.202123
$956$	22.0540	0.713277
$957$	0	0
$958$	19.8078	0.639960
$959$	3.56155	0.115009
$960$	0	0
$961$	8.01515	0.258553
$962$	−12.6847	−0.408970
$963$	0	0
$964$	−7.75379	−0.249733
$965$	15.5076	0.499207
$966$	0	0
$967$	52.3002	1.68186	0.840930	−	0.541143i	$-0.182008\pi$
$967$	52.3002	1.68186	0.840930	+	0.541143i	$0.182008\pi$
$968$	8.56155	0.275179
$969$	0	0
$970$	−11.6155	−0.372952
$971$	−13.7538	−0.441380	−0.220690	−	0.975344i	$-0.570831\pi$
$971$	−13.7538	−0.441380	−0.220690	+	0.975344i	$0.570831\pi$
$972$	0	0
$973$	−12.0000	−0.384702
$974$	6.24621	0.200142
$975$	0	0
$976$	6.87689	0.220124
$977$	40.7386	1.30334	0.651672	−	0.758501i	$-0.274068\pi$
$977$	40.7386	1.30334	0.651672	+	0.758501i	$0.274068\pi$
$978$	0	0
$979$	10.7386	0.343208
$980$	−2.00000	−0.0638877
$981$	0	0
$982$	23.8078	0.759736
$983$	17.7538	0.566258	0.283129	−	0.959082i	$-0.408628\pi$
$983$	17.7538	0.566258	0.283129	+	0.959082i	$0.408628\pi$
$984$	0	0
$985$	−4.00000	−0.127451
$986$	−2.00000	−0.0636930
$987$	0	0
$988$	−16.6847	−0.530810
$989$	−14.6307	−0.465229
$990$	0	0
$991$	28.8769	0.917305	0.458652	−	0.888616i	$-0.348332\pi$
$991$	28.8769	0.917305	0.458652	+	0.888616i	$0.348332\pi$
$992$	−6.24621	−0.198317
$993$	0	0
$994$	−10.2462	−0.324990
$995$	43.1231	1.36709
$996$	0	0
$997$	54.4924	1.72579	0.862896	−	0.505381i	$-0.168648\pi$
$997$	54.4924	1.72579	0.862896	+	0.505381i	$0.168648\pi$
$998$	42.9309	1.35895
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3654.2.a.x.1.1		2
3.2	odd	2		1218.2.a.n.1.2	✓	2
12.11	even	2		9744.2.a.be.1.1		2
21.20	even	2		8526.2.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1218.2.a.n.1.2	✓	2	3.2	odd	2
3654.2.a.x.1.1		2	1.1	even	1	trivial
8526.2.a.bi.1.2		2	21.20	even	2
9744.2.a.be.1.1		2	12.11	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3654 = 2 \cdot 3^{2} \cdot 7 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3654.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +2.00000 q^{10} -1.56155 q^{11} +3.56155 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -4.68466 q^{19} -2.00000 q^{20} +1.56155 q^{22} +4.68466 q^{23} -1.00000 q^{25} -3.56155 q^{26} +1.00000 q^{28} -1.00000 q^{29} +6.24621 q^{31} -1.00000 q^{32} +2.00000 q^{34} -2.00000 q^{35} +3.56155 q^{37} +4.68466 q^{38} +2.00000 q^{40} +1.12311 q^{41} -3.12311 q^{43} -1.56155 q^{44} -4.68466 q^{46} -8.68466 q^{47} +1.00000 q^{49} +1.00000 q^{50} +3.56155 q^{52} -6.68466 q^{53} +3.12311 q^{55} -1.00000 q^{56} +1.00000 q^{58} -4.68466 q^{59} +6.87689 q^{61} -6.24621 q^{62} +1.00000 q^{64} -7.12311 q^{65} -9.56155 q^{67} -2.00000 q^{68} +2.00000 q^{70} +10.2462 q^{71} -2.68466 q^{73} -3.56155 q^{74} -4.68466 q^{76} -1.56155 q^{77} +2.24621 q^{79} -2.00000 q^{80} -1.12311 q^{82} +15.8078 q^{83} +4.00000 q^{85} +3.12311 q^{86} +1.56155 q^{88} -6.87689 q^{89} +3.56155 q^{91} +4.68466 q^{92} +8.68466 q^{94} +9.36932 q^{95} -5.80776 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + 2 q^{7} - 2 q^{8} + 4 q^{10} + q^{11} + 3 q^{13} - 2 q^{14} + 2 q^{16} - 4 q^{17} + 3 q^{19} - 4 q^{20} - q^{22} - 3 q^{23} - 2 q^{25} - 3 q^{26} + 2 q^{28} - 2 q^{29}+ \cdots - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 + 2 * q^7 - 2 * q^8 + 4 * q^10 + q^11 + 3 * q^13 - 2 * q^14 + 2 * q^16 - 4 * q^17 + 3 * q^19 - 4 * q^20 - q^22 - 3 * q^23 - 2 * q^25 - 3 * q^26 + 2 * q^28 - 2 * q^29 - 4 * q^31 - 2 * q^32 + 4 * q^34 - 4 * q^35 + 3 * q^37 - 3 * q^38 + 4 * q^40 - 6 * q^41 + 2 * q^43 + q^44 + 3 * q^46 - 5 * q^47 + 2 * q^49 + 2 * q^50 + 3 * q^52 - q^53 - 2 * q^55 - 2 * q^56 + 2 * q^58 + 3 * q^59 + 22 * q^61 + 4 * q^62 + 2 * q^64 - 6 * q^65 - 15 * q^67 - 4 * q^68 + 4 * q^70 + 4 * q^71 + 7 * q^73 - 3 * q^74 + 3 * q^76 + q^77 - 12 * q^79 - 4 * q^80 + 6 * q^82 + 11 * q^83 + 8 * q^85 - 2 * q^86 - q^88 - 22 * q^89 + 3 * q^91 - 3 * q^92 + 5 * q^94 - 6 * q^95 + 9 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

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Database

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