LMFDB - Embedded newform 5355.2.a.bf.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5355,2,Mod(1,5355)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5355, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5355.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5355 = 3^{2} \cdot 5 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5355.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:	$42.7598902824$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.169.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x - 1 $ x^3 - x^2 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 595)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.37720$ of defining polynomial
Character	$\chi$	$=$	5355.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.37720 q^{2} +3.65109 q^{4} +1.00000 q^{5} +1.00000 q^{7} +3.92498 q^{8} +O(q^{10})$	\(q+2.37720 q^{2} +3.65109 q^{4} +1.00000 q^{5} +1.00000 q^{7} +3.92498 q^{8} +2.37720 q^{10} +3.00000 q^{11} -3.92498 q^{13} +2.37720 q^{14} +2.02830 q^{16} -1.00000 q^{17} -4.30219 q^{19} +3.65109 q^{20} +7.13161 q^{22} +4.09556 q^{23} +1.00000 q^{25} -9.33048 q^{26} +3.65109 q^{28} +9.37720 q^{29} +5.57608 q^{31} -3.02830 q^{32} -2.37720 q^{34} +1.00000 q^{35} +10.2555 q^{37} -10.2272 q^{38} +3.92498 q^{40} +7.58383 q^{41} +7.37720 q^{43} +10.9533 q^{44} +9.73598 q^{46} -5.95328 q^{47} +1.00000 q^{49} +2.37720 q^{50} -14.3305 q^{52} +7.93273 q^{53} +3.00000 q^{55} +3.92498 q^{56} +22.2915 q^{58} +8.50106 q^{59} -4.44447 q^{61} +13.2555 q^{62} -11.2555 q^{64} -3.92498 q^{65} -7.61212 q^{67} -3.65109 q^{68} +2.37720 q^{70} +2.44447 q^{71} +7.17833 q^{73} +24.3793 q^{74} -15.7077 q^{76} +3.00000 q^{77} -14.6121 q^{79} +2.02830 q^{80} +18.0283 q^{82} +10.7827 q^{83} -1.00000 q^{85} +17.5371 q^{86} +11.7750 q^{88} -7.75441 q^{89} -3.92498 q^{91} +14.9533 q^{92} -14.1522 q^{94} -4.30219 q^{95} +0.726109 q^{97} +2.37720 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 4 q^{4} + 3 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 2 * q^2 + 4 * q^4 + 3 * q^5 + 3 * q^7 + 3 * q^8	$ 3 q + 2 q^{2} + 4 q^{4} + 3 q^{5} + 3 q^{7} + 3 q^{8} + 2 q^{10} + 9 q^{11} - 3 q^{13} + 2 q^{14} - 6 q^{16} - 3 q^{17} + q^{19} + 4 q^{20} + 6 q^{22} + 5 q^{23} + 3 q^{25} - 2 q^{26} + 4 q^{28} + 23 q^{29} + q^{31} + 3 q^{32} - 2 q^{34} + 3 q^{35} - 4 q^{37} - 8 q^{38} + 3 q^{40} + 11 q^{41} + 17 q^{43} + 12 q^{44} - 14 q^{46} + 3 q^{47} + 3 q^{49} + 2 q^{50} - 17 q^{52} + 19 q^{53} + 9 q^{55} + 3 q^{56} + 24 q^{58} + q^{59} - 13 q^{61} + 5 q^{62} + q^{64} - 3 q^{65} + q^{67} - 4 q^{68} + 2 q^{70} + 7 q^{71} + 27 q^{73} + 19 q^{74} - 16 q^{76} + 9 q^{77} - 20 q^{79} - 6 q^{80} + 42 q^{82} + 10 q^{83} - 3 q^{85} + 20 q^{86} + 9 q^{88} - 13 q^{89} - 3 q^{91} + 24 q^{92} - 11 q^{94} + q^{95} + 4 q^{97} + 2 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 4 * q^4 + 3 * q^5 + 3 * q^7 + 3 * q^8 + 2 * q^10 + 9 * q^11 - 3 * q^13 + 2 * q^14 - 6 * q^16 - 3 * q^17 + q^19 + 4 * q^20 + 6 * q^22 + 5 * q^23 + 3 * q^25 - 2 * q^26 + 4 * q^28 + 23 * q^29 + q^31 + 3 * q^32 - 2 * q^34 + 3 * q^35 - 4 * q^37 - 8 * q^38 + 3 * q^40 + 11 * q^41 + 17 * q^43 + 12 * q^44 - 14 * q^46 + 3 * q^47 + 3 * q^49 + 2 * q^50 - 17 * q^52 + 19 * q^53 + 9 * q^55 + 3 * q^56 + 24 * q^58 + q^59 - 13 * q^61 + 5 * q^62 + q^64 - 3 * q^65 + q^67 - 4 * q^68 + 2 * q^70 + 7 * q^71 + 27 * q^73 + 19 * q^74 - 16 * q^76 + 9 * q^77 - 20 * q^79 - 6 * q^80 + 42 * q^82 + 10 * q^83 - 3 * q^85 + 20 * q^86 + 9 * q^88 - 13 * q^89 - 3 * q^91 + 24 * q^92 - 11 * q^94 + 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 $

$2$	2.37720	1.68094	0.840468	−	0.541861i	$-0.182280\pi$
$2$	2.37720	1.68094	0.840468	+	0.541861i	$0.182280\pi$
$3$	0	0
$3$	0	0
$4$	3.65109	1.82555
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	3.92498	1.38769
$9$	0	0
$10$	2.37720	0.751738
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	−3.92498	−1.08859	−0.544297	−	0.838892i	$-0.683204\pi$
$13$	−3.92498	−1.08859	−0.544297	+	0.838892i	$0.683204\pi$
$14$	2.37720	0.635334
$15$	0	0
$16$	2.02830	0.507074
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−4.30219	−0.986989	−0.493495	−	0.869749i	$-0.664281\pi$
$19$	−4.30219	−0.986989	−0.493495	+	0.869749i	$0.664281\pi$
$20$	3.65109	0.816409
$21$	0	0
$22$	7.13161	1.52046
$23$	4.09556	0.853984	0.426992	−	0.904255i	$-0.359573\pi$
$23$	4.09556	0.853984	0.426992	+	0.904255i	$0.359573\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−9.33048	−1.82986
$27$	0	0
$28$	3.65109	0.689992
$29$	9.37720	1.74130	0.870651	−	0.491900i	$-0.163698\pi$
$29$	9.37720	1.74130	0.870651	+	0.491900i	$0.163698\pi$
$30$	0	0
$31$	5.57608	1.00149	0.500747	−	0.865594i	$-0.333059\pi$
$31$	5.57608	1.00149	0.500747	+	0.865594i	$0.333059\pi$
$32$	−3.02830	−0.535332
$33$	0	0
$34$	−2.37720	−0.407687
$35$	1.00000	0.169031
$36$	0	0
$37$	10.2555	1.68599	0.842994	−	0.537923i	$-0.180791\pi$
$37$	10.2555	1.68599	0.842994	+	0.537923i	$0.180791\pi$
$38$	−10.2272	−1.65907
$39$	0	0
$40$	3.92498	0.620594
$41$	7.58383	1.18439	0.592197	−	0.805793i	$-0.298261\pi$
$41$	7.58383	1.18439	0.592197	+	0.805793i	$0.298261\pi$
$42$	0	0
$43$	7.37720	1.12501	0.562506	−	0.826793i	$-0.309837\pi$
$43$	7.37720	1.12501	0.562506	+	0.826793i	$0.309837\pi$
$44$	10.9533	1.65127
$45$	0	0
$46$	9.73598	1.43549
$47$	−5.95328	−0.868375	−0.434188	−	0.900822i	$-0.642964\pi$
$47$	−5.95328	−0.868375	−0.434188	+	0.900822i	$0.642964\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.37720	0.336187
$51$	0	0
$52$	−14.3305	−1.98728
$53$	7.93273	1.08964	0.544822	−	0.838551i	$-0.316597\pi$
$53$	7.93273	1.08964	0.544822	+	0.838551i	$0.316597\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	3.92498	0.524498
$57$	0	0
$58$	22.2915	2.92702
$59$	8.50106	1.10674	0.553372	−	0.832934i	$-0.313341\pi$
$59$	8.50106	1.10674	0.553372	+	0.832934i	$0.313341\pi$
$60$	0	0
$61$	−4.44447	−0.569056	−0.284528	−	0.958668i	$-0.591837\pi$
$61$	−4.44447	−0.569056	−0.284528	+	0.958668i	$0.591837\pi$
$62$	13.2555	1.68345
$63$	0	0
$64$	−11.2555	−1.40693
$65$	−3.92498	−0.486834
$66$	0	0
$67$	−7.61212	−0.929969	−0.464984	−	0.885319i	$-0.653940\pi$
$67$	−7.61212	−0.929969	−0.464984	+	0.885319i	$0.653940\pi$
$68$	−3.65109	−0.442760
$69$	0	0
$70$	2.37720	0.284130
$71$	2.44447	0.290105	0.145053	−	0.989424i	$-0.453665\pi$
$71$	2.44447	0.290105	0.145053	+	0.989424i	$0.453665\pi$
$72$	0	0
$73$	7.17833	0.840160	0.420080	−	0.907487i	$-0.362002\pi$
$73$	7.17833	0.840160	0.420080	+	0.907487i	$0.362002\pi$
$74$	24.3793	2.83404
$75$	0	0
$76$	−15.7077	−1.80180
$77$	3.00000	0.341882
$78$	0	0
$79$	−14.6121	−1.64399	−0.821996	−	0.569493i	$-0.807140\pi$
$79$	−14.6121	−1.64399	−0.821996	+	0.569493i	$0.807140\pi$
$80$	2.02830	0.226770
$81$	0	0
$82$	18.0283	1.99089
$83$	10.7827	1.18356	0.591778	−	0.806101i	$-0.298426\pi$
$83$	10.7827	1.18356	0.591778	+	0.806101i	$0.298426\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	17.5371	1.89107
$87$	0	0
$88$	11.7750	1.25521
$89$	−7.75441	−0.821965	−0.410983	−	0.911643i	$-0.634814\pi$
$89$	−7.75441	−0.821965	−0.410983	+	0.911643i	$0.634814\pi$
$90$	0	0
$91$	−3.92498	−0.411450
$92$	14.9533	1.55899
$93$	0	0
$94$	−14.1522	−1.45968
$95$	−4.30219	−0.441395
$96$	0	0
$97$	0.726109	0.0737252	0.0368626	−	0.999320i	$-0.488264\pi$
$97$	0.726109	0.0737252	0.0368626	+	0.999320i	$0.488264\pi$
$98$	2.37720	0.240134
$99$	0	0
$100$	3.65109	0.365109
$101$	−10.9816	−1.09271	−0.546354	−	0.837554i	$-0.683985\pi$
$101$	−10.9816	−1.09271	−0.546354	+	0.837554i	$0.683985\pi$
$102$	0	0
$103$	−7.78270	−0.766852	−0.383426	−	0.923572i	$-0.625256\pi$
$103$	−7.78270	−0.766852	−0.383426	+	0.923572i	$0.625256\pi$
$104$	−15.4055	−1.51063
$105$	0	0
$106$	18.8577	1.83162
$107$	2.51948	0.243568	0.121784	−	0.992557i	$-0.461139\pi$
$107$	2.51948	0.243568	0.121784	+	0.992557i	$0.461139\pi$
$108$	0	0
$109$	−2.42392	−0.232170	−0.116085	−	0.993239i	$-0.537034\pi$
$109$	−2.42392	−0.232170	−0.116085	+	0.993239i	$0.537034\pi$
$110$	7.13161	0.679972
$111$	0	0
$112$	2.02830	0.191656
$113$	−15.9816	−1.50342	−0.751710	−	0.659494i	$-0.770771\pi$
$113$	−15.9816	−1.50342	−0.751710	+	0.659494i	$0.770771\pi$
$114$	0	0
$115$	4.09556	0.381913
$116$	34.2370	3.17883
$117$	0	0
$118$	20.2087	1.86037
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−10.5654	−0.956547
$123$	0	0
$124$	20.3588	1.82827
$125$	1.00000	0.0894427
$126$	0	0
$127$	−0.508811	−0.0451497	−0.0225749	−	0.999745i	$-0.507186\pi$
$127$	−0.508811	−0.0451497	−0.0225749	+	0.999745i	$0.507186\pi$
$128$	−20.6999	−1.82963
$129$	0	0
$130$	−9.33048	−0.818338
$131$	−13.5577	−1.18454	−0.592269	−	0.805740i	$-0.701768\pi$
$131$	−13.5577	−1.18454	−0.592269	+	0.805740i	$0.701768\pi$
$132$	0	0
$133$	−4.30219	−0.373047
$134$	−18.0956	−1.56322
$135$	0	0
$136$	−3.92498	−0.336565
$137$	−12.6249	−1.07862	−0.539310	−	0.842107i	$-0.681315\pi$
$137$	−12.6249	−1.07862	−0.539310	+	0.842107i	$0.681315\pi$
$138$	0	0
$139$	3.39775	0.288193	0.144097	−	0.989564i	$-0.453972\pi$
$139$	3.39775	0.288193	0.144097	+	0.989564i	$0.453972\pi$
$140$	3.65109	0.308574
$141$	0	0
$142$	5.81100	0.487648
$143$	−11.7750	−0.984671
$144$	0	0
$145$	9.37720	0.778734
$146$	17.0643	1.41226
$147$	0	0
$148$	37.4437	3.07785
$149$	−1.13161	−0.0927050	−0.0463525	−	0.998925i	$-0.514760\pi$
$149$	−1.13161	−0.0927050	−0.0463525	+	0.998925i	$0.514760\pi$
$150$	0	0
$151$	8.86547	0.721462	0.360731	−	0.932670i	$-0.382527\pi$
$151$	8.86547	0.721462	0.360731	+	0.932670i	$0.382527\pi$
$152$	−16.8860	−1.36964
$153$	0	0
$154$	7.13161	0.574681
$155$	5.57608	0.447881
$156$	0	0
$157$	−16.2944	−1.30044	−0.650219	−	0.759747i	$-0.725323\pi$
$157$	−16.2944	−1.30044	−0.650219	+	0.759747i	$0.725323\pi$
$158$	−34.7360	−2.76345
$159$	0	0
$160$	−3.02830	−0.239408
$161$	4.09556	0.322776
$162$	0	0
$163$	−2.66177	−0.208486	−0.104243	−	0.994552i	$-0.533242\pi$
$163$	−2.66177	−0.208486	−0.104243	+	0.994552i	$0.533242\pi$
$164$	27.6893	2.16217
$165$	0	0
$166$	25.6327	1.98948
$167$	−2.31286	−0.178974	−0.0894872	−	0.995988i	$-0.528523\pi$
$167$	−2.31286	−0.178974	−0.0894872	+	0.995988i	$0.528523\pi$
$168$	0	0
$169$	2.40550	0.185038
$170$	−2.37720	−0.182323
$171$	0	0
$172$	26.9349	2.05376
$173$	25.2165	1.91717	0.958587	−	0.284799i	$-0.0919267\pi$
$173$	25.2165	1.91717	0.958587	+	0.284799i	$0.0919267\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	6.08489	0.458666
$177$	0	0
$178$	−18.4338	−1.38167
$179$	−19.8315	−1.48228	−0.741140	−	0.671351i	$-0.765714\pi$
$179$	−19.8315	−1.48228	−0.741140	+	0.671351i	$0.765714\pi$
$180$	0	0
$181$	−16.7750	−1.24687	−0.623436	−	0.781874i	$-0.714264\pi$
$181$	−16.7750	−1.24687	−0.623436	+	0.781874i	$0.714264\pi$
$182$	−9.33048	−0.691621
$183$	0	0
$184$	16.0750	1.18507
$185$	10.2555	0.753997
$186$	0	0
$187$	−3.00000	−0.219382
$188$	−21.7360	−1.58526
$189$	0	0
$190$	−10.2272	−0.741957
$191$	−8.80325	−0.636981	−0.318490	−	0.947926i	$-0.603176\pi$
$191$	−8.80325	−0.636981	−0.318490	+	0.947926i	$0.603176\pi$
$192$	0	0
$193$	19.1882	1.38120	0.690598	−	0.723238i	$-0.257347\pi$
$193$	19.1882	1.38120	0.690598	+	0.723238i	$0.257347\pi$
$194$	1.72611	0.123927
$195$	0	0
$196$	3.65109	0.260792
$197$	−16.4437	−1.17156	−0.585781	−	0.810469i	$-0.699212\pi$
$197$	−16.4437	−1.17156	−0.585781	+	0.810469i	$0.699212\pi$
$198$	0	0
$199$	11.6482	0.825717	0.412858	−	0.910795i	$-0.364530\pi$
$199$	11.6482	0.825717	0.412858	+	0.910795i	$0.364530\pi$
$200$	3.92498	0.277538
$201$	0	0
$202$	−26.1054	−1.83677
$203$	9.37720	0.658151
$204$	0	0
$205$	7.58383	0.529677
$206$	−18.5011	−1.28903
$207$	0	0
$208$	−7.96103	−0.551998
$209$	−12.9066	−0.892765
$210$	0	0
$211$	−18.7437	−1.29037	−0.645186	−	0.764026i	$-0.723220\pi$
$211$	−18.7437	−1.29037	−0.645186	+	0.764026i	$0.723220\pi$
$212$	28.9632	1.98920
$213$	0	0
$214$	5.98933	0.409422
$215$	7.37720	0.503121
$216$	0	0
$217$	5.57608	0.378529
$218$	−5.76216	−0.390262
$219$	0	0
$220$	10.9533	0.738470
$221$	3.92498	0.264023
$222$	0	0
$223$	4.34116	0.290705	0.145353	−	0.989380i	$-0.453568\pi$
$223$	4.34116	0.290705	0.145353	+	0.989380i	$0.453568\pi$
$224$	−3.02830	−0.202337
$225$	0	0
$226$	−37.9914	−2.52715
$227$	−18.9349	−1.25675	−0.628375	−	0.777910i	$-0.716280\pi$
$227$	−18.9349	−1.25675	−0.628375	+	0.777910i	$0.716280\pi$
$228$	0	0
$229$	−15.7077	−1.03799	−0.518997	−	0.854776i	$-0.673694\pi$
$229$	−15.7077	−1.03799	−0.518997	+	0.854776i	$0.673694\pi$
$230$	9.73598	0.641972
$231$	0	0
$232$	36.8054	2.41639
$233$	18.4933	1.21154	0.605769	−	0.795641i	$-0.292866\pi$
$233$	18.4933	1.21154	0.605769	+	0.795641i	$0.292866\pi$
$234$	0	0
$235$	−5.95328	−0.388349
$236$	31.0382	2.02041
$237$	0	0
$238$	−2.37720	−0.154091
$239$	18.8988	1.22246	0.611231	−	0.791452i	$-0.290675\pi$
$239$	18.8988	1.22246	0.611231	+	0.791452i	$0.290675\pi$
$240$	0	0
$241$	7.29151	0.469688	0.234844	−	0.972033i	$-0.424542\pi$
$241$	7.29151	0.469688	0.234844	+	0.972033i	$0.424542\pi$
$242$	−4.75441	−0.305625
$243$	0	0
$244$	−16.2272	−1.03884
$245$	1.00000	0.0638877
$246$	0	0
$247$	16.8860	1.07443
$248$	21.8860	1.38976
$249$	0	0
$250$	2.37720	0.150348
$251$	−17.6249	−1.11248	−0.556238	−	0.831023i	$-0.687756\pi$
$251$	−17.6249	−1.11248	−0.556238	+	0.831023i	$0.687756\pi$
$252$	0	0
$253$	12.2867	0.772457
$254$	−1.20955	−0.0758938
$255$	0	0
$256$	−26.6970	−1.66856
$257$	−9.75441	−0.608463	−0.304232	−	0.952598i	$-0.598400\pi$
$257$	−9.75441	−0.608463	−0.304232	+	0.952598i	$0.598400\pi$
$258$	0	0
$259$	10.2555	0.637244
$260$	−14.3305	−0.888739
$261$	0	0
$262$	−32.2293	−1.99113
$263$	27.9504	1.72349	0.861746	−	0.507339i	$-0.169371\pi$
$263$	27.9504	1.72349	0.861746	+	0.507339i	$0.169371\pi$
$264$	0	0
$265$	7.93273	0.487304
$266$	−10.2272	−0.627068
$267$	0	0
$268$	−27.7926	−1.69770
$269$	2.09344	0.127639	0.0638196	−	0.997961i	$-0.479672\pi$
$269$	2.09344	0.127639	0.0638196	+	0.997961i	$0.479672\pi$
$270$	0	0
$271$	2.11399	0.128415	0.0642077	−	0.997937i	$-0.479548\pi$
$271$	2.11399	0.128415	0.0642077	+	0.997937i	$0.479548\pi$
$272$	−2.02830	−0.122984
$273$	0	0
$274$	−30.0120	−1.81309
$275$	3.00000	0.180907
$276$	0	0
$277$	−10.9709	−0.659178	−0.329589	−	0.944125i	$-0.606910\pi$
$277$	−10.9709	−0.659178	−0.329589	+	0.944125i	$0.606910\pi$
$278$	8.07714	0.484435
$279$	0	0
$280$	3.92498	0.234563
$281$	−2.59450	−0.154775	−0.0773875	−	0.997001i	$-0.524658\pi$
$281$	−2.59450	−0.154775	−0.0773875	+	0.997001i	$0.524658\pi$
$282$	0	0
$283$	−20.1620	−1.19851	−0.599254	−	0.800559i	$-0.704536\pi$
$283$	−20.1620	−1.19851	−0.599254	+	0.800559i	$0.704536\pi$
$284$	8.92498	0.529600
$285$	0	0
$286$	−27.9914	−1.65517
$287$	7.58383	0.447659
$288$	0	0
$289$	1.00000	0.0588235
$290$	22.2915	1.30900
$291$	0	0
$292$	26.2087	1.53375
$293$	−14.8548	−0.867826	−0.433913	−	0.900955i	$-0.642868\pi$
$293$	−14.8548	−0.867826	−0.433913	+	0.900955i	$0.642868\pi$
$294$	0	0
$295$	8.50106	0.494951
$296$	40.2525	2.33963
$297$	0	0
$298$	−2.69006	−0.155831
$299$	−16.0750	−0.929642
$300$	0	0
$301$	7.37720	0.425215
$302$	21.0750	1.21273
$303$	0	0
$304$	−8.72611	−0.500477
$305$	−4.44447	−0.254490
$306$	0	0
$307$	24.0099	1.37032	0.685158	−	0.728395i	$-0.259733\pi$
$307$	24.0099	1.37032	0.685158	+	0.728395i	$0.259733\pi$
$308$	10.9533	0.624121
$309$	0	0
$310$	13.2555	0.752860
$311$	−24.1239	−1.36794	−0.683969	−	0.729511i	$-0.739748\pi$
$311$	−24.1239	−1.36794	−0.683969	+	0.729511i	$0.739748\pi$
$312$	0	0
$313$	29.5315	1.66922	0.834609	−	0.550843i	$-0.185694\pi$
$313$	29.5315	1.66922	0.834609	+	0.550843i	$0.185694\pi$
$314$	−38.7352	−2.18595
$315$	0	0
$316$	−53.3502	−3.00118
$317$	3.53499	0.198545	0.0992723	−	0.995060i	$-0.468349\pi$
$317$	3.53499	0.198545	0.0992723	+	0.995060i	$0.468349\pi$
$318$	0	0
$319$	28.1316	1.57507
$320$	−11.2555	−0.629200
$321$	0	0
$322$	9.73598	0.542565
$323$	4.30219	0.239380
$324$	0	0
$325$	−3.92498	−0.217719
$326$	−6.32756	−0.350451
$327$	0	0
$328$	29.7664	1.64357
$329$	−5.95328	−0.328215
$330$	0	0
$331$	22.8598	1.25649	0.628245	−	0.778015i	$-0.283773\pi$
$331$	22.8598	1.25649	0.628245	+	0.778015i	$0.283773\pi$
$332$	39.3687	2.16064
$333$	0	0
$334$	−5.49814	−0.300845
$335$	−7.61212	−0.415895
$336$	0	0
$337$	−26.6580	−1.45216	−0.726078	−	0.687612i	$-0.758659\pi$
$337$	−26.6580	−1.45216	−0.726078	+	0.687612i	$0.758659\pi$
$338$	5.71836	0.311038
$339$	0	0
$340$	−3.65109	−0.198008
$341$	16.7282	0.905885
$342$	0	0
$343$	1.00000	0.0539949
$344$	28.9554	1.56117
$345$	0	0
$346$	59.9447	3.22265
$347$	21.7771	1.16905	0.584527	−	0.811374i	$-0.301280\pi$
$347$	21.7771	1.16905	0.584527	+	0.811374i	$0.301280\pi$
$348$	0	0
$349$	−10.3206	−0.552450	−0.276225	−	0.961093i	$-0.589083\pi$
$349$	−10.3206	−0.552450	−0.276225	+	0.961093i	$0.589083\pi$
$350$	2.37720	0.127067
$351$	0	0
$352$	−9.08489	−0.484226
$353$	29.7176	1.58171	0.790853	−	0.612006i	$-0.209637\pi$
$353$	29.7176	1.58171	0.790853	+	0.612006i	$0.209637\pi$
$354$	0	0
$355$	2.44447	0.129739
$356$	−28.3121	−1.50054
$357$	0	0
$358$	−47.1436	−2.49162
$359$	20.2010	1.06617	0.533084	−	0.846062i	$-0.321033\pi$
$359$	20.2010	1.06617	0.533084	+	0.846062i	$0.321033\pi$
$360$	0	0
$361$	−0.491189	−0.0258520
$362$	−39.8775	−2.09591
$363$	0	0
$364$	−14.3305	−0.751121
$365$	7.17833	0.375731
$366$	0	0
$367$	11.0595	0.577302	0.288651	−	0.957434i	$-0.406793\pi$
$367$	11.0595	0.577302	0.288651	+	0.957434i	$0.406793\pi$
$368$	8.30701	0.433033
$369$	0	0
$370$	24.3793	1.26742
$371$	7.93273	0.411847
$372$	0	0
$373$	−27.3043	−1.41376	−0.706882	−	0.707332i	$-0.749899\pi$
$373$	−27.3043	−1.41376	−0.706882	+	0.707332i	$0.749899\pi$
$374$	−7.13161	−0.368767
$375$	0	0
$376$	−23.3665	−1.20504
$377$	−36.8054	−1.89557
$378$	0	0
$379$	8.92498	0.458446	0.229223	−	0.973374i	$-0.426382\pi$
$379$	8.92498	0.458446	0.229223	+	0.973374i	$0.426382\pi$
$380$	−15.7077	−0.805787
$381$	0	0
$382$	−20.9271	−1.07072
$383$	−0.0955622	−0.00488300	−0.00244150	−	0.999997i	$-0.500777\pi$
$383$	−0.0955622	−0.00488300	−0.00244150	+	0.999997i	$0.500777\pi$
$384$	0	0
$385$	3.00000	0.152894
$386$	45.6142	2.32170
$387$	0	0
$388$	2.65109	0.134589
$389$	−16.2632	−0.824578	−0.412289	−	0.911053i	$-0.635271\pi$
$389$	−16.2632	−0.824578	−0.412289	+	0.911053i	$0.635271\pi$
$390$	0	0
$391$	−4.09556	−0.207121
$392$	3.92498	0.198242
$393$	0	0
$394$	−39.0899	−1.96932
$395$	−14.6121	−0.735216
$396$	0	0
$397$	3.16283	0.158738	0.0793689	−	0.996845i	$-0.474709\pi$
$397$	3.16283	0.158738	0.0793689	+	0.996845i	$0.474709\pi$
$398$	27.6901	1.38798
$399$	0	0
$400$	2.02830	0.101415
$401$	15.4728	0.772673	0.386337	−	0.922358i	$-0.373740\pi$
$401$	15.4728	0.772673	0.386337	+	0.922358i	$0.373740\pi$
$402$	0	0
$403$	−21.8860	−1.09022
$404$	−40.0948	−1.99479
$405$	0	0
$406$	22.2915	1.10631
$407$	30.7664	1.52503
$408$	0	0
$409$	−29.2739	−1.44750	−0.723750	−	0.690062i	$-0.757583\pi$
$409$	−29.2739	−1.44750	−0.723750	+	0.690062i	$0.757583\pi$
$410$	18.0283	0.890354
$411$	0	0
$412$	−28.4154	−1.39992
$413$	8.50106	0.418310
$414$	0	0
$415$	10.7827	0.529302
$416$	11.8860	0.582760
$417$	0	0
$418$	−30.6815	−1.50068
$419$	−3.80595	−0.185933	−0.0929665	−	0.995669i	$-0.529635\pi$
$419$	−3.80595	−0.185933	−0.0929665	+	0.995669i	$0.529635\pi$
$420$	0	0
$421$	−0.395626	−0.0192816	−0.00964082	−	0.999954i	$-0.503069\pi$
$421$	−0.395626	−0.0192816	−0.00964082	+	0.999954i	$0.503069\pi$
$422$	−44.5577	−2.16903
$423$	0	0
$424$	31.1359	1.51209
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	−4.44447	−0.215083
$428$	9.19887	0.444644
$429$	0	0
$430$	17.5371	0.845714
$431$	−15.1337	−0.728966	−0.364483	−	0.931210i	$-0.618754\pi$
$431$	−15.1337	−0.728966	−0.364483	+	0.931210i	$0.618754\pi$
$432$	0	0
$433$	34.5422	1.65999	0.829995	−	0.557771i	$-0.188343\pi$
$433$	34.5422	1.65999	0.829995	+	0.557771i	$0.188343\pi$
$434$	13.2555	0.636283
$435$	0	0
$436$	−8.84997	−0.423837
$437$	−17.6199	−0.842873
$438$	0	0
$439$	−24.9709	−1.19180	−0.595898	−	0.803060i	$-0.703204\pi$
$439$	−24.9709	−1.19180	−0.595898	+	0.803060i	$0.703204\pi$
$440$	11.7750	0.561349
$441$	0	0
$442$	9.33048	0.443806
$443$	−8.75653	−0.416035	−0.208018	−	0.978125i	$-0.566701\pi$
$443$	−8.75653	−0.416035	−0.208018	+	0.978125i	$0.566701\pi$
$444$	0	0
$445$	−7.75441	−0.367594
$446$	10.3198	0.488657
$447$	0	0
$448$	−11.2555	−0.531771
$449$	29.2944	1.38249	0.691245	−	0.722621i	$-0.257063\pi$
$449$	29.2944	1.38249	0.691245	+	0.722621i	$0.257063\pi$
$450$	0	0
$451$	22.7515	1.07133
$452$	−58.3502	−2.74456
$453$	0	0
$454$	−45.0120	−2.11252
$455$	−3.92498	−0.184006
$456$	0	0
$457$	−24.8131	−1.16071	−0.580354	−	0.814364i	$-0.697086\pi$
$457$	−24.8131	−1.16071	−0.580354	+	0.814364i	$0.697086\pi$
$458$	−37.3404	−1.74480
$459$	0	0
$460$	14.9533	0.697200
$461$	4.15990	0.193746	0.0968730	−	0.995297i	$-0.469116\pi$
$461$	4.15990	0.193746	0.0968730	+	0.995297i	$0.469116\pi$
$462$	0	0
$463$	4.87534	0.226576	0.113288	−	0.993562i	$-0.463862\pi$
$463$	4.87534	0.226576	0.113288	+	0.993562i	$0.463862\pi$
$464$	19.0197	0.882970
$465$	0	0
$466$	43.9624	2.03652
$467$	10.1834	0.471230	0.235615	−	0.971846i	$-0.424290\pi$
$467$	10.1834	0.471230	0.235615	+	0.971846i	$0.424290\pi$
$468$	0	0
$469$	−7.61212	−0.351495
$470$	−14.1522	−0.652790
$471$	0	0
$472$	33.3665	1.53582
$473$	22.1316	1.01761
$474$	0	0
$475$	−4.30219	−0.197398
$476$	−3.65109	−0.167348
$477$	0	0
$478$	44.9263	2.05488
$479$	−14.1890	−0.648312	−0.324156	−	0.946004i	$-0.605080\pi$
$479$	−14.1890	−0.648312	−0.324156	+	0.946004i	$0.605080\pi$
$480$	0	0
$481$	−40.2525	−1.83536
$482$	17.3334	0.789515
$483$	0	0
$484$	−7.30219	−0.331918
$485$	0.726109	0.0329709
$486$	0	0
$487$	0.235722	0.0106816	0.00534078	−	0.999986i	$-0.498300\pi$
$487$	0.235722	0.0106816	0.00534078	+	0.999986i	$0.498300\pi$
$488$	−17.4445	−0.789674
$489$	0	0
$490$	2.37720	0.107391
$491$	16.6220	0.750140	0.375070	−	0.926996i	$-0.377619\pi$
$491$	16.6220	0.750140	0.375070	+	0.926996i	$0.377619\pi$
$492$	0	0
$493$	−9.37720	−0.422328
$494$	40.1415	1.80605
$495$	0	0
$496$	11.3099	0.507831
$497$	2.44447	0.109649
$498$	0	0
$499$	30.4671	1.36390	0.681948	−	0.731400i	$-0.261133\pi$
$499$	30.4671	1.36390	0.681948	+	0.731400i	$0.261133\pi$
$500$	3.65109	0.163282
$501$	0	0
$502$	−41.8980	−1.87000
$503$	−15.1209	−0.674209	−0.337105	−	0.941467i	$-0.609448\pi$
$503$	−15.1209	−0.674209	−0.337105	+	0.941467i	$0.609448\pi$
$504$	0	0
$505$	−10.9816	−0.488674
$506$	29.2079	1.29845
$507$	0	0
$508$	−1.85772	−0.0824229
$509$	20.8959	0.926194	0.463097	−	0.886308i	$-0.346738\pi$
$509$	20.8959	0.926194	0.463097	+	0.886308i	$0.346738\pi$
$510$	0	0
$511$	7.17833	0.317551
$512$	−22.0643	−0.975115
$513$	0	0
$514$	−23.1882	−1.02279
$515$	−7.78270	−0.342947
$516$	0	0
$517$	−17.8598	−0.785475
$518$	24.3793	1.07117
$519$	0	0
$520$	−15.4055	−0.675576
$521$	−34.2165	−1.49905	−0.749526	−	0.661975i	$-0.769719\pi$
$521$	−34.2165	−1.49905	−0.749526	+	0.661975i	$0.769719\pi$
$522$	0	0
$523$	7.15508	0.312870	0.156435	−	0.987688i	$-0.450000\pi$
$523$	7.15508	0.312870	0.156435	+	0.987688i	$0.450000\pi$
$524$	−49.5003	−2.16243
$525$	0	0
$526$	66.4437	2.89708
$527$	−5.57608	−0.242898
$528$	0	0
$529$	−6.22637	−0.270712
$530$	18.8577	0.819127
$531$	0	0
$532$	−15.7077	−0.681015
$533$	−29.7664	−1.28933
$534$	0	0
$535$	2.51948	0.108927
$536$	−29.8775	−1.29051
$537$	0	0
$538$	4.97653	0.214553
$539$	3.00000	0.129219
$540$	0	0
$541$	−21.7154	−0.933620	−0.466810	−	0.884358i	$-0.654597\pi$
$541$	−21.7154	−0.933620	−0.466810	+	0.884358i	$0.654597\pi$
$542$	5.02537	0.215858
$543$	0	0
$544$	3.02830	0.129837
$545$	−2.42392	−0.103829
$546$	0	0
$547$	−43.1124	−1.84335	−0.921676	−	0.387960i	$-0.873180\pi$
$547$	−43.1124	−1.84335	−0.921676	+	0.387960i	$0.873180\pi$
$548$	−46.0948	−1.96907
$549$	0	0
$550$	7.13161	0.304093
$551$	−40.3425	−1.71865
$552$	0	0
$553$	−14.6121	−0.621371
$554$	−26.0801	−1.10804
$555$	0	0
$556$	12.4055	0.526110
$557$	−1.40045	−0.0593391	−0.0296696	−	0.999560i	$-0.509445\pi$
$557$	−1.40045	−0.0593391	−0.0296696	+	0.999560i	$0.509445\pi$
$558$	0	0
$559$	−28.9554	−1.22468
$560$	2.02830	0.0857112
$561$	0	0
$562$	−6.16765	−0.260167
$563$	−21.0438	−0.886890	−0.443445	−	0.896302i	$-0.646244\pi$
$563$	−21.0438	−0.886890	−0.443445	+	0.896302i	$0.646244\pi$
$564$	0	0
$565$	−15.9816	−0.672350
$566$	−47.9292	−2.01462
$567$	0	0
$568$	9.59450	0.402576
$569$	−11.7467	−0.492445	−0.246223	−	0.969213i	$-0.579189\pi$
$569$	−11.7467	−0.492445	−0.246223	+	0.969213i	$0.579189\pi$
$570$	0	0
$571$	15.8470	0.663178	0.331589	−	0.943424i	$-0.392415\pi$
$571$	15.8470	0.663178	0.331589	+	0.943424i	$0.392415\pi$
$572$	−42.9914	−1.79756
$573$	0	0
$574$	18.0283	0.752487
$575$	4.09556	0.170797
$576$	0	0
$577$	−2.47277	−0.102943	−0.0514713	−	0.998674i	$-0.516391\pi$
$577$	−2.47277	−0.102943	−0.0514713	+	0.998674i	$0.516391\pi$
$578$	2.37720	0.0988786
$579$	0	0
$580$	34.2370	1.42162
$581$	10.7827	0.447342
$582$	0	0
$583$	23.7982	0.985621
$584$	28.1748	1.16588
$585$	0	0
$586$	−35.3129	−1.45876
$587$	−20.4162	−0.842666	−0.421333	−	0.906906i	$-0.638438\pi$
$587$	−20.4162	−0.842666	−0.421333	+	0.906906i	$0.638438\pi$
$588$	0	0
$589$	−23.9893	−0.988463
$590$	20.2087	0.831981
$591$	0	0
$592$	20.8011	0.854921
$593$	24.2760	0.996896	0.498448	−	0.866919i	$-0.333903\pi$
$593$	24.2760	0.996896	0.498448	+	0.866919i	$0.333903\pi$
$594$	0	0
$595$	−1.00000	−0.0409960
$596$	−4.13161	−0.169237
$597$	0	0
$598$	−38.2136	−1.56267
$599$	−1.50398	−0.0614511	−0.0307256	−	0.999528i	$-0.509782\pi$
$599$	−1.50398	−0.0614511	−0.0307256	+	0.999528i	$0.509782\pi$
$600$	0	0
$601$	−13.6241	−0.555739	−0.277870	−	0.960619i	$-0.589628\pi$
$601$	−13.6241	−0.555739	−0.277870	+	0.960619i	$0.589628\pi$
$602$	17.5371	0.714759
$603$	0	0
$604$	32.3687	1.31706
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	−19.2349	−0.780721	−0.390361	−	0.920662i	$-0.627650\pi$
$607$	−19.2349	−0.780721	−0.390361	+	0.920662i	$0.627650\pi$
$608$	13.0283	0.528367
$609$	0	0
$610$	−10.5654	−0.427781
$611$	23.3665	0.945309
$612$	0	0
$613$	42.2341	1.70582	0.852910	−	0.522058i	$-0.174836\pi$
$613$	42.2341	1.70582	0.852910	+	0.522058i	$0.174836\pi$
$614$	57.0763	2.30341
$615$	0	0
$616$	11.7750	0.474426
$617$	35.5675	1.43189	0.715947	−	0.698154i	$-0.245995\pi$
$617$	35.5675	1.43189	0.715947	+	0.698154i	$0.245995\pi$
$618$	0	0
$619$	−8.60145	−0.345721	−0.172861	−	0.984946i	$-0.555301\pi$
$619$	−8.60145	−0.345721	−0.172861	+	0.984946i	$0.555301\pi$
$620$	20.3588	0.817628
$621$	0	0
$622$	−57.3473	−2.29942
$623$	−7.75441	−0.310674
$624$	0	0
$625$	1.00000	0.0400000
$626$	70.2023	2.80585
$627$	0	0
$628$	−59.4925	−2.37401
$629$	−10.2555	−0.408912
$630$	0	0
$631$	−39.4407	−1.57011	−0.785056	−	0.619425i	$-0.787366\pi$
$631$	−39.4407	−1.57011	−0.785056	+	0.619425i	$0.787366\pi$
$632$	−57.3524	−2.28135
$633$	0	0
$634$	8.40338	0.333741
$635$	−0.508811	−0.0201916
$636$	0	0
$637$	−3.92498	−0.155514
$638$	66.8745	2.64759
$639$	0	0
$640$	−20.6999	−0.818237
$641$	42.1690	1.66557	0.832787	−	0.553593i	$-0.186744\pi$
$641$	42.1690	1.66557	0.832787	+	0.553593i	$0.186744\pi$
$642$	0	0
$643$	22.1981	0.875407	0.437703	−	0.899119i	$-0.355792\pi$
$643$	22.1981	0.875407	0.437703	+	0.899119i	$0.355792\pi$
$644$	14.9533	0.589242
$645$	0	0
$646$	10.2272	0.402383
$647$	−41.7557	−1.64159	−0.820794	−	0.571225i	$-0.806468\pi$
$647$	−41.7557	−1.64159	−0.820794	+	0.571225i	$0.806468\pi$
$648$	0	0
$649$	25.5032	1.00109
$650$	−9.33048	−0.365972
$651$	0	0
$652$	−9.71836	−0.380600
$653$	19.9250	0.779725	0.389862	−	0.920873i	$-0.372523\pi$
$653$	19.9250	0.779725	0.389862	+	0.920873i	$0.372523\pi$
$654$	0	0
$655$	−13.5577	−0.529741
$656$	15.3822	0.600576
$657$	0	0
$658$	−14.1522	−0.551708
$659$	8.45726	0.329448	0.164724	−	0.986340i	$-0.447327\pi$
$659$	8.45726	0.329448	0.164724	+	0.986340i	$0.447327\pi$
$660$	0	0
$661$	−21.6172	−0.840810	−0.420405	−	0.907336i	$-0.638112\pi$
$661$	−21.6172	−0.840810	−0.420405	+	0.907336i	$0.638112\pi$
$662$	54.3425	2.11208
$663$	0	0
$664$	42.3219	1.64241
$665$	−4.30219	−0.166832
$666$	0	0
$667$	38.4049	1.48704
$668$	−8.44447	−0.326726
$669$	0	0
$670$	−18.0956	−0.699093
$671$	−13.3334	−0.514730
$672$	0	0
$673$	−37.4097	−1.44204	−0.721020	−	0.692914i	$-0.756326\pi$
$673$	−37.4097	−1.44204	−0.721020	+	0.692914i	$0.756326\pi$
$674$	−63.3716	−2.44098
$675$	0	0
$676$	8.78270	0.337796
$677$	14.9765	0.575595	0.287797	−	0.957691i	$-0.407077\pi$
$677$	14.9765	0.575595	0.287797	+	0.957691i	$0.407077\pi$
$678$	0	0
$679$	0.726109	0.0278655
$680$	−3.92498	−0.150516
$681$	0	0
$682$	39.7664	1.52273
$683$	−24.5011	−0.937507	−0.468754	−	0.883329i	$-0.655297\pi$
$683$	−24.5011	−0.937507	−0.468754	+	0.883329i	$0.655297\pi$
$684$	0	0
$685$	−12.6249	−0.482373
$686$	2.37720	0.0907620
$687$	0	0
$688$	14.9632	0.570465
$689$	−31.1359	−1.18618
$690$	0	0
$691$	37.5958	1.43021	0.715106	−	0.699016i	$-0.246378\pi$
$691$	37.5958	1.43021	0.715106	+	0.699016i	$0.246378\pi$
$692$	92.0678	3.49989
$693$	0	0
$694$	51.7685	1.96511
$695$	3.39775	0.128884
$696$	0	0
$697$	−7.58383	−0.287258
$698$	−24.5342	−0.928633
$699$	0	0
$700$	3.65109	0.137998
$701$	41.1386	1.55378	0.776891	−	0.629635i	$-0.216796\pi$
$701$	41.1386	1.55378	0.776891	+	0.629635i	$0.216796\pi$
$702$	0	0
$703$	−44.1209	−1.66405
$704$	−33.7664	−1.27262
$705$	0	0
$706$	70.6447	2.65875
$707$	−10.9816	−0.413005
$708$	0	0
$709$	−36.1677	−1.35830	−0.679152	−	0.733997i	$-0.737652\pi$
$709$	−36.1677	−1.35830	−0.679152	+	0.733997i	$0.737652\pi$
$710$	5.81100	0.218083
$711$	0	0
$712$	−30.4359	−1.14063
$713$	22.8372	0.855259
$714$	0	0
$715$	−11.7750	−0.440358
$716$	−72.4068	−2.70597
$717$	0	0
$718$	48.0219	1.79216
$719$	3.31074	0.123470	0.0617348	−	0.998093i	$-0.480337\pi$
$719$	3.31074	0.123470	0.0617348	+	0.998093i	$0.480337\pi$
$720$	0	0
$721$	−7.78270	−0.289843
$722$	−1.16765	−0.0434556
$723$	0	0
$724$	−61.2469	−2.27622
$725$	9.37720	0.348261
$726$	0	0
$727$	15.3404	0.568942	0.284471	−	0.958685i	$-0.408182\pi$
$727$	15.3404	0.568942	0.284471	+	0.958685i	$0.408182\pi$
$728$	−15.4055	−0.570966
$729$	0	0
$730$	17.0643	0.631580
$731$	−7.37720	−0.272856
$732$	0	0
$733$	−14.3665	−0.530640	−0.265320	−	0.964160i	$-0.585478\pi$
$733$	−14.3665	−0.530640	−0.265320	+	0.964160i	$0.585478\pi$
$734$	26.2907	0.970408
$735$	0	0
$736$	−12.4026	−0.457165
$737$	−22.8364	−0.841189
$738$	0	0
$739$	−0.198875	−0.00731572	−0.00365786	−	0.999993i	$-0.501164\pi$
$739$	−0.198875	−0.00731572	−0.00365786	+	0.999993i	$0.501164\pi$
$740$	37.4437	1.37646
$741$	0	0
$742$	18.8577	0.692289
$743$	−2.04804	−0.0751354	−0.0375677	−	0.999294i	$-0.511961\pi$
$743$	−2.04804	−0.0751354	−0.0375677	+	0.999294i	$0.511961\pi$
$744$	0	0
$745$	−1.13161	−0.0414589
$746$	−64.9079	−2.37645
$747$	0	0
$748$	−10.9533	−0.400492
$749$	2.51948	0.0920600
$750$	0	0
$751$	−47.1415	−1.72022	−0.860109	−	0.510111i	$-0.829604\pi$
$751$	−47.1415	−1.72022	−0.860109	+	0.510111i	$0.829604\pi$
$752$	−12.0750	−0.440331
$753$	0	0
$754$	−87.4938	−3.18634
$755$	8.86547	0.322647
$756$	0	0
$757$	5.46209	0.198523	0.0992615	−	0.995061i	$-0.468352\pi$
$757$	5.46209	0.198523	0.0992615	+	0.995061i	$0.468352\pi$
$758$	21.2165	0.770618
$759$	0	0
$760$	−16.8860	−0.612520
$761$	−23.5080	−0.852165	−0.426082	−	0.904684i	$-0.640107\pi$
$761$	−23.5080	−0.852165	−0.426082	+	0.904684i	$0.640107\pi$
$762$	0	0
$763$	−2.42392	−0.0877519
$764$	−32.1415	−1.16284
$765$	0	0
$766$	−0.227171	−0.00820801
$767$	−33.3665	−1.20480
$768$	0	0
$769$	−1.98933	−0.0717370	−0.0358685	−	0.999357i	$-0.511420\pi$
$769$	−1.98933	−0.0717370	−0.0358685	+	0.999357i	$0.511420\pi$
$770$	7.13161	0.257005
$771$	0	0
$772$	70.0579	2.52144
$773$	14.0977	0.507058	0.253529	−	0.967328i	$-0.418409\pi$
$773$	14.0977	0.507058	0.253529	+	0.967328i	$0.418409\pi$
$774$	0	0
$775$	5.57608	0.200299
$776$	2.84997	0.102308
$777$	0	0
$778$	−38.6610	−1.38606
$779$	−32.6270	−1.16899
$780$	0	0
$781$	7.33341	0.262410
$782$	−9.73598	−0.348158
$783$	0	0
$784$	2.02830	0.0724392
$785$	−16.2944	−0.581573
$786$	0	0
$787$	33.1444	1.18147	0.590735	−	0.806865i	$-0.298838\pi$
$787$	33.1444	1.18147	0.590735	+	0.806865i	$0.298838\pi$
$788$	−60.0374	−2.13874
$789$	0	0
$790$	−34.7360	−1.23585
$791$	−15.9816	−0.568239
$792$	0	0
$793$	17.4445	0.619471
$794$	7.51868	0.266828
$795$	0	0
$796$	42.5286	1.50738
$797$	8.07019	0.285861	0.142930	−	0.989733i	$-0.454347\pi$
$797$	8.07019	0.285861	0.142930	+	0.989733i	$0.454347\pi$
$798$	0	0
$799$	5.95328	0.210612
$800$	−3.02830	−0.107066
$801$	0	0
$802$	36.7819	1.29881
$803$	21.5350	0.759953
$804$	0	0
$805$	4.09556	0.144350
$806$	−52.0275	−1.83259
$807$	0	0
$808$	−43.1025	−1.51634
$809$	−20.8783	−0.734041	−0.367020	−	0.930213i	$-0.619622\pi$
$809$	−20.8783	−0.734041	−0.367020	+	0.930213i	$0.619622\pi$
$810$	0	0
$811$	42.3318	1.48647	0.743235	−	0.669030i	$-0.233290\pi$
$811$	42.3318	1.48647	0.743235	+	0.669030i	$0.233290\pi$
$812$	34.2370	1.20148
$813$	0	0
$814$	73.1380	2.56348
$815$	−2.66177	−0.0932376
$816$	0	0
$817$	−31.7381	−1.11038
$818$	−69.5900	−2.43316
$819$	0	0
$820$	27.6893	0.966951
$821$	31.7918	1.10954	0.554770	−	0.832004i	$-0.312806\pi$
$821$	31.7918	1.10954	0.554770	+	0.832004i	$0.312806\pi$
$822$	0	0
$823$	29.1463	1.01598	0.507988	−	0.861364i	$-0.330389\pi$
$823$	29.1463	1.01598	0.507988	+	0.861364i	$0.330389\pi$
$824$	−30.5470	−1.06415
$825$	0	0
$826$	20.2087	0.703152
$827$	9.99225	0.347465	0.173732	−	0.984793i	$-0.444417\pi$
$827$	9.99225	0.347465	0.173732	+	0.984793i	$0.444417\pi$
$828$	0	0
$829$	45.5675	1.58263	0.791313	−	0.611412i	$-0.209398\pi$
$829$	45.5675	1.58263	0.791313	+	0.611412i	$0.209398\pi$
$830$	25.6327	0.889723
$831$	0	0
$832$	44.1775	1.53158
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−2.31286	−0.0800398
$836$	−47.1231	−1.62979
$837$	0	0
$838$	−9.04752	−0.312541
$839$	49.8131	1.71974	0.859870	−	0.510513i	$-0.170545\pi$
$839$	49.8131	1.71974	0.859870	+	0.510513i	$0.170545\pi$
$840$	0	0
$841$	58.9319	2.03214
$842$	−0.940484	−0.0324112
$843$	0	0
$844$	−68.4351	−2.35563
$845$	2.40550	0.0827517
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	16.0899	0.552531
$849$	0	0
$850$	−2.37720	−0.0815374
$851$	42.0019	1.43981
$852$	0	0
$853$	−21.1442	−0.723963	−0.361982	−	0.932185i	$-0.617900\pi$
$853$	−21.1442	−0.723963	−0.361982	+	0.932185i	$0.617900\pi$
$854$	−10.5654	−0.361541
$855$	0	0
$856$	9.88894	0.337997
$857$	45.1669	1.54287	0.771435	−	0.636308i	$-0.219539\pi$
$857$	45.1669	1.54287	0.771435	+	0.636308i	$0.219539\pi$
$858$	0	0
$859$	−29.4309	−1.00417	−0.502084	−	0.864819i	$-0.667433\pi$
$859$	−29.4309	−1.00417	−0.502084	+	0.864819i	$0.667433\pi$
$860$	26.9349	0.918471
$861$	0	0
$862$	−35.9759	−1.22535
$863$	16.2322	0.552551	0.276276	−	0.961078i	$-0.410900\pi$
$863$	16.2322	0.552551	0.276276	+	0.961078i	$0.410900\pi$
$864$	0	0
$865$	25.2165	0.857387
$866$	82.1137	2.79034
$867$	0	0
$868$	20.3588	0.691022
$869$	−43.8364	−1.48705
$870$	0	0
$871$	29.8775	1.01236
$872$	−9.51386	−0.322180
$873$	0	0
$874$	−41.8860	−1.41682
$875$	1.00000	0.0338062
$876$	0	0
$877$	26.6842	0.901062	0.450531	−	0.892761i	$-0.351235\pi$
$877$	26.6842	0.901062	0.450531	+	0.892761i	$0.351235\pi$
$878$	−59.3609	−2.00333
$879$	0	0
$880$	6.08489	0.205122
$881$	−10.0184	−0.337529	−0.168765	−	0.985656i	$-0.553978\pi$
$881$	−10.0184	−0.337529	−0.168765	+	0.985656i	$0.553978\pi$
$882$	0	0
$883$	−35.7976	−1.20469	−0.602343	−	0.798237i	$-0.705766\pi$
$883$	−35.7976	−1.20469	−0.602343	+	0.798237i	$0.705766\pi$
$884$	14.3305	0.481986
$885$	0	0
$886$	−20.8160	−0.699329
$887$	−57.4202	−1.92798	−0.963991	−	0.265936i	$-0.914319\pi$
$887$	−57.4202	−1.92798	−0.963991	+	0.265936i	$0.914319\pi$
$888$	0	0
$889$	−0.508811	−0.0170650
$890$	−18.4338	−0.617902
$891$	0	0
$892$	15.8500	0.530696
$893$	25.6121	0.857077
$894$	0	0
$895$	−19.8315	−0.662895
$896$	−20.6999	−0.691536
$897$	0	0
$898$	69.6388	2.32388
$899$	52.2880	1.74390
$900$	0	0
$901$	−7.93273	−0.264278
$902$	54.0849	1.80083
$903$	0	0
$904$	−62.7274	−2.08628
$905$	−16.7750	−0.557618
$906$	0	0
$907$	−48.5088	−1.61071	−0.805354	−	0.592794i	$-0.798025\pi$
$907$	−48.5088	−1.61071	−0.805354	+	0.592794i	$0.798025\pi$
$908$	−69.1329	−2.29426
$909$	0	0
$910$	−9.33048	−0.309303
$911$	37.8881	1.25529	0.627645	−	0.778500i	$-0.284019\pi$
$911$	37.8881	1.25529	0.627645	+	0.778500i	$0.284019\pi$
$912$	0	0
$913$	32.3481	1.07057
$914$	−58.9858	−1.95108
$915$	0	0
$916$	−57.3502	−1.89490
$917$	−13.5577	−0.447713
$918$	0	0
$919$	−43.8521	−1.44655	−0.723273	−	0.690562i	$-0.757363\pi$
$919$	−43.8521	−1.44655	−0.723273	+	0.690562i	$0.757363\pi$
$920$	16.0750	0.529978
$921$	0	0
$922$	9.88894	0.325675
$923$	−9.59450	−0.315807
$924$	0	0
$925$	10.2555	0.337198
$926$	11.5897	0.380860
$927$	0	0
$928$	−28.3969	−0.932175
$929$	−15.7614	−0.517113	−0.258557	−	0.965996i	$-0.583247\pi$
$929$	−15.7614	−0.517113	−0.258557	+	0.965996i	$0.583247\pi$
$930$	0	0
$931$	−4.30219	−0.140998
$932$	67.5208	2.21172
$933$	0	0
$934$	24.2079	0.792108
$935$	−3.00000	−0.0981105
$936$	0	0
$937$	58.0502	1.89642	0.948208	−	0.317650i	$-0.102894\pi$
$937$	58.0502	1.89642	0.948208	+	0.317650i	$0.102894\pi$
$938$	−18.0956	−0.590841
$939$	0	0
$940$	−21.7360	−0.708950
$941$	52.8435	1.72265	0.861325	−	0.508054i	$-0.169635\pi$
$941$	52.8435	1.72265	0.861325	+	0.508054i	$0.169635\pi$
$942$	0	0
$943$	31.0600	1.01145
$944$	17.2427	0.561201
$945$	0	0
$946$	52.6113	1.71054
$947$	43.7402	1.42137	0.710683	−	0.703512i	$-0.248386\pi$
$947$	43.7402	1.42137	0.710683	+	0.703512i	$0.248386\pi$
$948$	0	0
$949$	−28.1748	−0.914593
$950$	−10.2272	−0.331813
$951$	0	0
$952$	−3.92498	−0.127209
$953$	−44.3863	−1.43781	−0.718906	−	0.695107i	$-0.755357\pi$
$953$	−44.3863	−1.43781	−0.718906	+	0.695107i	$0.755357\pi$
$954$	0	0
$955$	−8.80325	−0.284866
$956$	69.0013	2.23166
$957$	0	0
$958$	−33.7301	−1.08977
$959$	−12.6249	−0.407680
$960$	0	0
$961$	0.0926389	0.00298835
$962$	−95.6885	−3.08512
$963$	0	0
$964$	26.6220	0.857437
$965$	19.1882	0.617690
$966$	0	0
$967$	12.9512	0.416481	0.208241	−	0.978078i	$-0.433226\pi$
$967$	12.9512	0.416481	0.208241	+	0.978078i	$0.433226\pi$
$968$	−7.84997	−0.252308
$969$	0	0
$970$	1.72611	0.0554220
$971$	−35.3969	−1.13594	−0.567971	−	0.823049i	$-0.692271\pi$
$971$	−35.3969	−1.13594	−0.567971	+	0.823049i	$0.692271\pi$
$972$	0	0
$973$	3.39775	0.108927
$974$	0.560358	0.0179550
$975$	0	0
$976$	−9.01470	−0.288553
$977$	−40.3072	−1.28954	−0.644771	−	0.764376i	$-0.723047\pi$
$977$	−40.3072	−1.28954	−0.644771	+	0.764376i	$0.723047\pi$
$978$	0	0
$979$	−23.2632	−0.743496
$980$	3.65109	0.116630
$981$	0	0
$982$	39.5139	1.26094
$983$	33.0275	1.05341	0.526707	−	0.850047i	$-0.323427\pi$
$983$	33.0275	1.05341	0.526707	+	0.850047i	$0.323427\pi$
$984$	0	0
$985$	−16.4437	−0.523939
$986$	−22.2915	−0.709906
$987$	0	0
$988$	61.6524	1.96142
$989$	30.2138	0.960743
$990$	0	0
$991$	3.30994	0.105144	0.0525718	−	0.998617i	$-0.483258\pi$
$991$	3.30994	0.105144	0.0525718	+	0.998617i	$0.483258\pi$
$992$	−16.8860	−0.536131
$993$	0	0
$994$	5.81100	0.184314
$995$	11.6482	0.369272
$996$	0	0
$997$	29.5667	0.936388	0.468194	−	0.883626i	$-0.344905\pi$
$997$	29.5667	0.936388	0.468194	+	0.883626i	$0.344905\pi$
$998$	72.4266	2.29262
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5355.2.a.bf.1.3		3
3.2	odd	2		595.2.a.f.1.1	✓	3
12.11	even	2		9520.2.a.ba.1.1		3
15.14	odd	2		2975.2.a.f.1.3		3
21.20	even	2		4165.2.a.z.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
595.2.a.f.1.1	✓	3	3.2	odd	2
2975.2.a.f.1.3		3	15.14	odd	2
4165.2.a.z.1.1		3	21.20	even	2
5355.2.a.bf.1.3		3	1.1	even	1	trivial
9520.2.a.ba.1.1		3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5355 = 3^{2} \cdot 5 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5355.a (trivial)

\(f(q)\)	\(=\)	\(q+2.37720 q^{2} +3.65109 q^{4} +1.00000 q^{5} +1.00000 q^{7} +3.92498 q^{8} +O(q^{10})\)	\(q+2.37720 q^{2} +3.65109 q^{4} +1.00000 q^{5} +1.00000 q^{7} +3.92498 q^{8} +2.37720 q^{10} +3.00000 q^{11} -3.92498 q^{13} +2.37720 q^{14} +2.02830 q^{16} -1.00000 q^{17} -4.30219 q^{19} +3.65109 q^{20} +7.13161 q^{22} +4.09556 q^{23} +1.00000 q^{25} -9.33048 q^{26} +3.65109 q^{28} +9.37720 q^{29} +5.57608 q^{31} -3.02830 q^{32} -2.37720 q^{34} +1.00000 q^{35} +10.2555 q^{37} -10.2272 q^{38} +3.92498 q^{40} +7.58383 q^{41} +7.37720 q^{43} +10.9533 q^{44} +9.73598 q^{46} -5.95328 q^{47} +1.00000 q^{49} +2.37720 q^{50} -14.3305 q^{52} +7.93273 q^{53} +3.00000 q^{55} +3.92498 q^{56} +22.2915 q^{58} +8.50106 q^{59} -4.44447 q^{61} +13.2555 q^{62} -11.2555 q^{64} -3.92498 q^{65} -7.61212 q^{67} -3.65109 q^{68} +2.37720 q^{70} +2.44447 q^{71} +7.17833 q^{73} +24.3793 q^{74} -15.7077 q^{76} +3.00000 q^{77} -14.6121 q^{79} +2.02830 q^{80} +18.0283 q^{82} +10.7827 q^{83} -1.00000 q^{85} +17.5371 q^{86} +11.7750 q^{88} -7.75441 q^{89} -3.92498 q^{91} +14.9533 q^{92} -14.1522 q^{94} -4.30219 q^{95} +0.726109 q^{97} +2.37720 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 4 q^{4} + 3 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 2 * q^2 + 4 * q^4 + 3 * q^5 + 3 * q^7 + 3 * q^8	\( 3 q + 2 q^{2} + 4 q^{4} + 3 q^{5} + 3 q^{7} + 3 q^{8} + 2 q^{10} + 9 q^{11} - 3 q^{13} + 2 q^{14} - 6 q^{16} - 3 q^{17} + q^{19} + 4 q^{20} + 6 q^{22} + 5 q^{23} + 3 q^{25} - 2 q^{26} + 4 q^{28} + 23 q^{29} + q^{31} + 3 q^{32} - 2 q^{34} + 3 q^{35} - 4 q^{37} - 8 q^{38} + 3 q^{40} + 11 q^{41} + 17 q^{43} + 12 q^{44} - 14 q^{46} + 3 q^{47} + 3 q^{49} + 2 q^{50} - 17 q^{52} + 19 q^{53} + 9 q^{55} + 3 q^{56} + 24 q^{58} + q^{59} - 13 q^{61} + 5 q^{62} + q^{64} - 3 q^{65} + q^{67} - 4 q^{68} + 2 q^{70} + 7 q^{71} + 27 q^{73} + 19 q^{74} - 16 q^{76} + 9 q^{77} - 20 q^{79} - 6 q^{80} + 42 q^{82} + 10 q^{83} - 3 q^{85} + 20 q^{86} + 9 q^{88} - 13 q^{89} - 3 q^{91} + 24 q^{92} - 11 q^{94} + q^{95} + 4 q^{97} + 2 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 4 * q^4 + 3 * q^5 + 3 * q^7 + 3 * q^8 + 2 * q^10 + 9 * q^11 - 3 * q^13 + 2 * q^14 - 6 * q^16 - 3 * q^17 + q^19 + 4 * q^20 + 6 * q^22 + 5 * q^23 + 3 * q^25 - 2 * q^26 + 4 * q^28 + 23 * q^29 + q^31 + 3 * q^32 - 2 * q^34 + 3 * q^35 - 4 * q^37 - 8 * q^38 + 3 * q^40 + 11 * q^41 + 17 * q^43 + 12 * q^44 - 14 * q^46 + 3 * q^47 + 3 * q^49 + 2 * q^50 - 17 * q^52 + 19 * q^53 + 9 * q^55 + 3 * q^56 + 24 * q^58 + q^59 - 13 * q^61 + 5 * q^62 + q^64 - 3 * q^65 + q^67 - 4 * q^68 + 2 * q^70 + 7 * q^71 + 27 * q^73 + 19 * q^74 - 16 * q^76 + 9 * q^77 - 20 * q^79 - 6 * q^80 + 42 * q^82 + 10 * q^83 - 3 * q^85 + 20 * q^86 + 9 * q^88 - 13 * q^89 - 3 * q^91 + 24 * q^92 - 11 * q^94 + q^95 + 4 * q^97 + 2 * q^98

Introduction

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