LMFDB - Embedded newform 1755.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1755,2,Mod(1,1755)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1755, 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("1755.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1755 = 3^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1755.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:	$14.0137455547$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1755.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} +1.82843 q^{7} +4.41421 q^{8} +O(q^{10})$	\(q+2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} +1.82843 q^{7} +4.41421 q^{8} -2.41421 q^{10} +3.82843 q^{11} -1.00000 q^{13} +4.41421 q^{14} +3.00000 q^{16} +4.65685 q^{17} -1.17157 q^{19} -3.82843 q^{20} +9.24264 q^{22} -5.65685 q^{23} +1.00000 q^{25} -2.41421 q^{26} +7.00000 q^{28} +5.65685 q^{29} +6.00000 q^{31} -1.58579 q^{32} +11.2426 q^{34} -1.82843 q^{35} +6.00000 q^{37} -2.82843 q^{38} -4.41421 q^{40} +0.171573 q^{41} +7.65685 q^{43} +14.6569 q^{44} -13.6569 q^{46} -7.17157 q^{47} -3.65685 q^{49} +2.41421 q^{50} -3.82843 q^{52} -9.00000 q^{53} -3.82843 q^{55} +8.07107 q^{56} +13.6569 q^{58} -11.4853 q^{59} +1.00000 q^{61} +14.4853 q^{62} -9.82843 q^{64} +1.00000 q^{65} -1.48528 q^{67} +17.8284 q^{68} -4.41421 q^{70} +9.82843 q^{71} -5.48528 q^{73} +14.4853 q^{74} -4.48528 q^{76} +7.00000 q^{77} +9.00000 q^{79} -3.00000 q^{80} +0.414214 q^{82} -8.82843 q^{83} -4.65685 q^{85} +18.4853 q^{86} +16.8995 q^{88} -6.00000 q^{89} -1.82843 q^{91} -21.6569 q^{92} -17.3137 q^{94} +1.17157 q^{95} +15.8284 q^{97} -8.82843 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 6 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 6 q^{8} - 2 q^{10} + 2 q^{11} - 2 q^{13} + 6 q^{14} + 6 q^{16} - 2 q^{17} - 8 q^{19} - 2 q^{20} + 10 q^{22} + 2 q^{25} - 2 q^{26} + 14 q^{28} + 12 q^{31} - 6 q^{32} + 14 q^{34} + 2 q^{35} + 12 q^{37} - 6 q^{40} + 6 q^{41} + 4 q^{43} + 18 q^{44} - 16 q^{46} - 20 q^{47} + 4 q^{49} + 2 q^{50} - 2 q^{52} - 18 q^{53} - 2 q^{55} + 2 q^{56} + 16 q^{58} - 6 q^{59} + 2 q^{61} + 12 q^{62} - 14 q^{64} + 2 q^{65} + 14 q^{67} + 30 q^{68} - 6 q^{70} + 14 q^{71} + 6 q^{73} + 12 q^{74} + 8 q^{76} + 14 q^{77} + 18 q^{79} - 6 q^{80} - 2 q^{82} - 12 q^{83} + 2 q^{85} + 20 q^{86} + 14 q^{88} - 12 q^{89} + 2 q^{91} - 32 q^{92} - 12 q^{94} + 8 q^{95} + 26 q^{97} - 12 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 6 * q^8 - 2 * q^10 + 2 * q^11 - 2 * q^13 + 6 * q^14 + 6 * q^16 - 2 * q^17 - 8 * q^19 - 2 * q^20 + 10 * q^22 + 2 * q^25 - 2 * q^26 + 14 * q^28 + 12 * q^31 - 6 * q^32 + 14 * q^34 + 2 * q^35 + 12 * q^37 - 6 * q^40 + 6 * q^41 + 4 * q^43 + 18 * q^44 - 16 * q^46 - 20 * q^47 + 4 * q^49 + 2 * q^50 - 2 * q^52 - 18 * q^53 - 2 * q^55 + 2 * q^56 + 16 * q^58 - 6 * q^59 + 2 * q^61 + 12 * q^62 - 14 * q^64 + 2 * q^65 + 14 * q^67 + 30 * q^68 - 6 * q^70 + 14 * q^71 + 6 * q^73 + 12 * q^74 + 8 * q^76 + 14 * q^77 + 18 * q^79 - 6 * q^80 - 2 * q^82 - 12 * q^83 + 2 * q^85 + 20 * q^86 + 14 * q^88 - 12 * q^89 + 2 * q^91 - 32 * q^92 - 12 * q^94 + 8 * q^95 + 26 * q^97 - 12 * 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 $

$2$	2.41421	1.70711	0.853553	−	0.521005i	$-0.174443\pi$
$2$	2.41421	1.70711	0.853553	+	0.521005i	$0.174443\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.82843	0.691080	0.345540	−	0.938404i	$-0.387696\pi$
$7$	1.82843	0.691080	0.345540	+	0.938404i	$0.387696\pi$
$8$	4.41421	1.56066
$9$	0	0
$10$	−2.41421	−0.763441
$11$	3.82843	1.15431	0.577157	−	0.816633i	$-0.304162\pi$
$11$	3.82843	1.15431	0.577157	+	0.816633i	$0.304162\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	4.41421	1.17975
$15$	0	0
$16$	3.00000	0.750000
$17$	4.65685	1.12945	0.564727	−	0.825278i	$-0.308982\pi$
$17$	4.65685	1.12945	0.564727	+	0.825278i	$0.308982\pi$
$18$	0	0
$19$	−1.17157	−0.268777	−0.134389	−	0.990929i	$-0.542907\pi$
$19$	−1.17157	−0.268777	−0.134389	+	0.990929i	$0.542907\pi$
$20$	−3.82843	−0.856062
$21$	0	0
$22$	9.24264	1.97054
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.41421	−0.473466
$27$	0	0
$28$	7.00000	1.32288
$29$	5.65685	1.05045	0.525226	−	0.850963i	$-0.323981\pi$
$29$	5.65685	1.05045	0.525226	+	0.850963i	$0.323981\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−1.58579	−0.280330
$33$	0	0
$34$	11.2426	1.92810
$35$	−1.82843	−0.309061
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−2.82843	−0.458831
$39$	0	0
$40$	−4.41421	−0.697948
$41$	0.171573	0.0267952	0.0133976	−	0.999910i	$-0.495735\pi$
$41$	0.171573	0.0267952	0.0133976	+	0.999910i	$0.495735\pi$
$42$	0	0
$43$	7.65685	1.16766	0.583830	−	0.811876i	$-0.301554\pi$
$43$	7.65685	1.16766	0.583830	+	0.811876i	$0.301554\pi$
$44$	14.6569	2.20960
$45$	0	0
$46$	−13.6569	−2.01359
$47$	−7.17157	−1.04608	−0.523041	−	0.852308i	$-0.675202\pi$
$47$	−7.17157	−1.04608	−0.523041	+	0.852308i	$0.675202\pi$
$48$	0	0
$49$	−3.65685	−0.522408
$50$	2.41421	0.341421
$51$	0	0
$52$	−3.82843	−0.530907
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	0	0
$55$	−3.82843	−0.516225
$56$	8.07107	1.07854
$57$	0	0
$58$	13.6569	1.79323
$59$	−11.4853	−1.49526	−0.747628	−	0.664118i	$-0.768807\pi$
$59$	−11.4853	−1.49526	−0.747628	+	0.664118i	$0.768807\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	14.4853	1.83963
$63$	0	0
$64$	−9.82843	−1.22855
$65$	1.00000	0.124035
$66$	0	0
$67$	−1.48528	−0.181456	−0.0907280	−	0.995876i	$-0.528919\pi$
$67$	−1.48528	−0.181456	−0.0907280	+	0.995876i	$0.528919\pi$
$68$	17.8284	2.16201
$69$	0	0
$70$	−4.41421	−0.527599
$71$	9.82843	1.16642	0.583210	−	0.812322i	$-0.301797\pi$
$71$	9.82843	1.16642	0.583210	+	0.812322i	$0.301797\pi$
$72$	0	0
$73$	−5.48528	−0.642004	−0.321002	−	0.947079i	$-0.604020\pi$
$73$	−5.48528	−0.642004	−0.321002	+	0.947079i	$0.604020\pi$
$74$	14.4853	1.68388
$75$	0	0
$76$	−4.48528	−0.514497
$77$	7.00000	0.797724
$78$	0	0
$79$	9.00000	1.01258	0.506290	−	0.862364i	$-0.331017\pi$
$79$	9.00000	1.01258	0.506290	+	0.862364i	$0.331017\pi$
$80$	−3.00000	−0.335410
$81$	0	0
$82$	0.414214	0.0457422
$83$	−8.82843	−0.969046	−0.484523	−	0.874779i	$-0.661007\pi$
$83$	−8.82843	−0.969046	−0.484523	+	0.874779i	$0.661007\pi$
$84$	0	0
$85$	−4.65685	−0.505107
$86$	18.4853	1.99332
$87$	0	0
$88$	16.8995	1.80149
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−1.82843	−0.191671
$92$	−21.6569	−2.25788
$93$	0	0
$94$	−17.3137	−1.78577
$95$	1.17157	0.120201
$96$	0	0
$97$	15.8284	1.60713	0.803567	−	0.595215i	$-0.202933\pi$
$97$	15.8284	1.60713	0.803567	+	0.595215i	$0.202933\pi$
$98$	−8.82843	−0.891806
$99$	0	0
$100$	3.82843	0.382843
$101$	−4.82843	−0.480446	−0.240223	−	0.970718i	$-0.577221\pi$
$101$	−4.82843	−0.480446	−0.240223	+	0.970718i	$0.577221\pi$
$102$	0	0
$103$	−5.17157	−0.509570	−0.254785	−	0.966998i	$-0.582005\pi$
$103$	−5.17157	−0.509570	−0.254785	+	0.966998i	$0.582005\pi$
$104$	−4.41421	−0.432849
$105$	0	0
$106$	−21.7279	−2.11040
$107$	16.6569	1.61028	0.805139	−	0.593086i	$-0.202090\pi$
$107$	16.6569	1.61028	0.805139	+	0.593086i	$0.202090\pi$
$108$	0	0
$109$	1.51472	0.145084	0.0725419	−	0.997365i	$-0.476889\pi$
$109$	1.51472	0.145084	0.0725419	+	0.997365i	$0.476889\pi$
$110$	−9.24264	−0.881251
$111$	0	0
$112$	5.48528	0.518310
$113$	−14.6569	−1.37880	−0.689400	−	0.724380i	$-0.742126\pi$
$113$	−14.6569	−1.37880	−0.689400	+	0.724380i	$0.742126\pi$
$114$	0	0
$115$	5.65685	0.527504
$116$	21.6569	2.01079
$117$	0	0
$118$	−27.7279	−2.55256
$119$	8.51472	0.780543
$120$	0	0
$121$	3.65685	0.332441
$122$	2.41421	0.218573
$123$	0	0
$124$	22.9706	2.06282
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−16.8284	−1.49328	−0.746641	−	0.665228i	$-0.768335\pi$
$127$	−16.8284	−1.49328	−0.746641	+	0.665228i	$0.768335\pi$
$128$	−20.5563	−1.81694
$129$	0	0
$130$	2.41421	0.211741
$131$	−8.48528	−0.741362	−0.370681	−	0.928760i	$-0.620876\pi$
$131$	−8.48528	−0.741362	−0.370681	+	0.928760i	$0.620876\pi$
$132$	0	0
$133$	−2.14214	−0.185747
$134$	−3.58579	−0.309765
$135$	0	0
$136$	20.5563	1.76269
$137$	−11.6569	−0.995912	−0.497956	−	0.867202i	$-0.665916\pi$
$137$	−11.6569	−0.995912	−0.497956	+	0.867202i	$0.665916\pi$
$138$	0	0
$139$	−7.00000	−0.593732	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000	−0.593732	−0.296866	+	0.954919i	$0.595942\pi$
$140$	−7.00000	−0.591608
$141$	0	0
$142$	23.7279	1.99120
$143$	−3.82843	−0.320149
$144$	0	0
$145$	−5.65685	−0.469776
$146$	−13.2426	−1.09597
$147$	0	0
$148$	22.9706	1.88817
$149$	−5.31371	−0.435316	−0.217658	−	0.976025i	$-0.569842\pi$
$149$	−5.31371	−0.435316	−0.217658	+	0.976025i	$0.569842\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	−5.17157	−0.419470
$153$	0	0
$154$	16.8995	1.36180
$155$	−6.00000	−0.481932
$156$	0	0
$157$	8.48528	0.677199	0.338600	−	0.940931i	$-0.390047\pi$
$157$	8.48528	0.677199	0.338600	+	0.940931i	$0.390047\pi$
$158$	21.7279	1.72858
$159$	0	0
$160$	1.58579	0.125367
$161$	−10.3431	−0.815154
$162$	0	0
$163$	8.51472	0.666924	0.333462	−	0.942763i	$-0.391783\pi$
$163$	8.51472	0.666924	0.333462	+	0.942763i	$0.391783\pi$
$164$	0.656854	0.0512917
$165$	0	0
$166$	−21.3137	−1.65426
$167$	−6.34315	−0.490847	−0.245424	−	0.969416i	$-0.578927\pi$
$167$	−6.34315	−0.490847	−0.245424	+	0.969416i	$0.578927\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−11.2426	−0.862271
$171$	0	0
$172$	29.3137	2.23515
$173$	−1.31371	−0.0998794	−0.0499397	−	0.998752i	$-0.515903\pi$
$173$	−1.31371	−0.0998794	−0.0499397	+	0.998752i	$0.515903\pi$
$174$	0	0
$175$	1.82843	0.138216
$176$	11.4853	0.865736
$177$	0	0
$178$	−14.4853	−1.08572
$179$	−9.17157	−0.685516	−0.342758	−	0.939424i	$-0.611361\pi$
$179$	−9.17157	−0.685516	−0.342758	+	0.939424i	$0.611361\pi$
$180$	0	0
$181$	−19.9706	−1.48440	−0.742200	−	0.670178i	$-0.766218\pi$
$181$	−19.9706	−1.48440	−0.742200	+	0.670178i	$0.766218\pi$
$182$	−4.41421	−0.327203
$183$	0	0
$184$	−24.9706	−1.84085
$185$	−6.00000	−0.441129
$186$	0	0
$187$	17.8284	1.30374
$188$	−27.4558	−2.00242
$189$	0	0
$190$	2.82843	0.205196
$191$	18.1421	1.31272	0.656359	−	0.754448i	$-0.272096\pi$
$191$	18.1421	1.31272	0.656359	+	0.754448i	$0.272096\pi$
$192$	0	0
$193$	9.31371	0.670415	0.335208	−	0.942144i	$-0.391194\pi$
$193$	9.31371	0.670415	0.335208	+	0.942144i	$0.391194\pi$
$194$	38.2132	2.74355
$195$	0	0
$196$	−14.0000	−1.00000
$197$	−26.4853	−1.88700	−0.943499	−	0.331375i	$-0.892487\pi$
$197$	−26.4853	−1.88700	−0.943499	+	0.331375i	$0.892487\pi$
$198$	0	0
$199$	17.9706	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$199$	17.9706	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$200$	4.41421	0.312132
$201$	0	0
$202$	−11.6569	−0.820173
$203$	10.3431	0.725947
$204$	0	0
$205$	−0.171573	−0.0119832
$206$	−12.4853	−0.869891
$207$	0	0
$208$	−3.00000	−0.208013
$209$	−4.48528	−0.310253
$210$	0	0
$211$	1.34315	0.0924660	0.0462330	−	0.998931i	$-0.485278\pi$
$211$	1.34315	0.0924660	0.0462330	+	0.998931i	$0.485278\pi$
$212$	−34.4558	−2.36644
$213$	0	0
$214$	40.2132	2.74892
$215$	−7.65685	−0.522193
$216$	0	0
$217$	10.9706	0.744730
$218$	3.65685	0.247673
$219$	0	0
$220$	−14.6569	−0.988165
$221$	−4.65685	−0.313254
$222$	0	0
$223$	−17.4853	−1.17090	−0.585451	−	0.810708i	$-0.699082\pi$
$223$	−17.4853	−1.17090	−0.585451	+	0.810708i	$0.699082\pi$
$224$	−2.89949	−0.193731
$225$	0	0
$226$	−35.3848	−2.35376
$227$	−25.6569	−1.70291	−0.851453	−	0.524432i	$-0.824278\pi$
$227$	−25.6569	−1.70291	−0.851453	+	0.524432i	$0.824278\pi$
$228$	0	0
$229$	21.6569	1.43113	0.715563	−	0.698549i	$-0.246170\pi$
$229$	21.6569	1.43113	0.715563	+	0.698549i	$0.246170\pi$
$230$	13.6569	0.900506
$231$	0	0
$232$	24.9706	1.63940
$233$	−22.3137	−1.46182	−0.730910	−	0.682474i	$-0.760904\pi$
$233$	−22.3137	−1.46182	−0.730910	+	0.682474i	$0.760904\pi$
$234$	0	0
$235$	7.17157	0.467822
$236$	−43.9706	−2.86224
$237$	0	0
$238$	20.5563	1.33247
$239$	−3.31371	−0.214346	−0.107173	−	0.994240i	$-0.534180\pi$
$239$	−3.31371	−0.214346	−0.107173	+	0.994240i	$0.534180\pi$
$240$	0	0
$241$	−8.48528	−0.546585	−0.273293	−	0.961931i	$-0.588113\pi$
$241$	−8.48528	−0.546585	−0.273293	+	0.961931i	$0.588113\pi$
$242$	8.82843	0.567513
$243$	0	0
$244$	3.82843	0.245090
$245$	3.65685	0.233628
$246$	0	0
$247$	1.17157	0.0745454
$248$	26.4853	1.68182
$249$	0	0
$250$	−2.41421	−0.152688
$251$	−12.3431	−0.779092	−0.389546	−	0.921007i	$-0.627368\pi$
$251$	−12.3431	−0.779092	−0.389546	+	0.921007i	$0.627368\pi$
$252$	0	0
$253$	−21.6569	−1.36155
$254$	−40.6274	−2.54919
$255$	0	0
$256$	−29.9706	−1.87316
$257$	5.34315	0.333296	0.166648	−	0.986016i	$-0.446706\pi$
$257$	5.34315	0.333296	0.166648	+	0.986016i	$0.446706\pi$
$258$	0	0
$259$	10.9706	0.681678
$260$	3.82843	0.237429
$261$	0	0
$262$	−20.4853	−1.26558
$263$	31.2843	1.92907	0.964535	−	0.263954i	$-0.0850266\pi$
$263$	31.2843	1.92907	0.964535	+	0.263954i	$0.0850266\pi$
$264$	0	0
$265$	9.00000	0.552866
$266$	−5.17157	−0.317089
$267$	0	0
$268$	−5.68629	−0.347346
$269$	−0.142136	−0.00866616	−0.00433308	−	0.999991i	$-0.501379\pi$
$269$	−0.142136	−0.00866616	−0.00433308	+	0.999991i	$0.501379\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	13.9706	0.847090
$273$	0	0
$274$	−28.1421	−1.70013
$275$	3.82843	0.230863
$276$	0	0
$277$	28.1421	1.69090	0.845449	−	0.534057i	$-0.179333\pi$
$277$	28.1421	1.69090	0.845449	+	0.534057i	$0.179333\pi$
$278$	−16.8995	−1.01356
$279$	0	0
$280$	−8.07107	−0.482339
$281$	4.17157	0.248855	0.124428	−	0.992229i	$-0.460291\pi$
$281$	4.17157	0.248855	0.124428	+	0.992229i	$0.460291\pi$
$282$	0	0
$283$	−1.51472	−0.0900407	−0.0450203	−	0.998986i	$-0.514335\pi$
$283$	−1.51472	−0.0900407	−0.0450203	+	0.998986i	$0.514335\pi$
$284$	37.6274	2.23278
$285$	0	0
$286$	−9.24264	−0.546529
$287$	0.313708	0.0185176
$288$	0	0
$289$	4.68629	0.275664
$290$	−13.6569	−0.801958
$291$	0	0
$292$	−21.0000	−1.22893
$293$	28.2843	1.65238	0.826192	−	0.563388i	$-0.190502\pi$
$293$	28.2843	1.65238	0.826192	+	0.563388i	$0.190502\pi$
$294$	0	0
$295$	11.4853	0.668699
$296$	26.4853	1.53943
$297$	0	0
$298$	−12.8284	−0.743131
$299$	5.65685	0.327144
$300$	0	0
$301$	14.0000	0.806947
$302$	−24.1421	−1.38922
$303$	0	0
$304$	−3.51472	−0.201583
$305$	−1.00000	−0.0572598
$306$	0	0
$307$	−8.68629	−0.495753	−0.247876	−	0.968792i	$-0.579733\pi$
$307$	−8.68629	−0.495753	−0.247876	+	0.968792i	$0.579733\pi$
$308$	26.7990	1.52701
$309$	0	0
$310$	−14.4853	−0.822709
$311$	14.4853	0.821385	0.410692	−	0.911774i	$-0.365287\pi$
$311$	14.4853	0.821385	0.410692	+	0.911774i	$0.365287\pi$
$312$	0	0
$313$	−33.7990	−1.91043	−0.955216	−	0.295910i	$-0.904377\pi$
$313$	−33.7990	−1.91043	−0.955216	+	0.295910i	$0.904377\pi$
$314$	20.4853	1.15605
$315$	0	0
$316$	34.4558	1.93829
$317$	−2.82843	−0.158860	−0.0794301	−	0.996840i	$-0.525310\pi$
$317$	−2.82843	−0.158860	−0.0794301	+	0.996840i	$0.525310\pi$
$318$	0	0
$319$	21.6569	1.21255
$320$	9.82843	0.549426
$321$	0	0
$322$	−24.9706	−1.39156
$323$	−5.45584	−0.303571
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	20.5563	1.13851
$327$	0	0
$328$	0.757359	0.0418182
$329$	−13.1127	−0.722926
$330$	0	0
$331$	−4.82843	−0.265394	−0.132697	−	0.991157i	$-0.542364\pi$
$331$	−4.82843	−0.265394	−0.132697	+	0.991157i	$0.542364\pi$
$332$	−33.7990	−1.85496
$333$	0	0
$334$	−15.3137	−0.837929
$335$	1.48528	0.0811496
$336$	0	0
$337$	35.7990	1.95010	0.975048	−	0.221996i	$-0.0712571\pi$
$337$	35.7990	1.95010	0.975048	+	0.221996i	$0.0712571\pi$
$338$	2.41421	0.131316
$339$	0	0
$340$	−17.8284	−0.966882
$341$	22.9706	1.24393
$342$	0	0
$343$	−19.4853	−1.05211
$344$	33.7990	1.82232
$345$	0	0
$346$	−3.17157	−0.170505
$347$	28.9706	1.55522	0.777611	−	0.628746i	$-0.216432\pi$
$347$	28.9706	1.55522	0.777611	+	0.628746i	$0.216432\pi$
$348$	0	0
$349$	20.9706	1.12253	0.561264	−	0.827637i	$-0.310315\pi$
$349$	20.9706	1.12253	0.561264	+	0.827637i	$0.310315\pi$
$350$	4.41421	0.235950
$351$	0	0
$352$	−6.07107	−0.323589
$353$	−6.34315	−0.337612	−0.168806	−	0.985649i	$-0.553991\pi$
$353$	−6.34315	−0.337612	−0.168806	+	0.985649i	$0.553991\pi$
$354$	0	0
$355$	−9.82843	−0.521639
$356$	−22.9706	−1.21744
$357$	0	0
$358$	−22.1421	−1.17025
$359$	29.8284	1.57428	0.787142	−	0.616772i	$-0.211560\pi$
$359$	29.8284	1.57428	0.787142	+	0.616772i	$0.211560\pi$
$360$	0	0
$361$	−17.6274	−0.927759
$362$	−48.2132	−2.53403
$363$	0	0
$364$	−7.00000	−0.366900
$365$	5.48528	0.287113
$366$	0	0
$367$	−32.6274	−1.70314	−0.851569	−	0.524243i	$-0.824348\pi$
$367$	−32.6274	−1.70314	−0.851569	+	0.524243i	$0.824348\pi$
$368$	−16.9706	−0.884652
$369$	0	0
$370$	−14.4853	−0.753054
$371$	−16.4558	−0.854345
$372$	0	0
$373$	27.3137	1.41425	0.707125	−	0.707088i	$-0.249992\pi$
$373$	27.3137	1.41425	0.707125	+	0.707088i	$0.249992\pi$
$374$	43.0416	2.22563
$375$	0	0
$376$	−31.6569	−1.63258
$377$	−5.65685	−0.291343
$378$	0	0
$379$	11.7990	0.606073	0.303037	−	0.952979i	$-0.402000\pi$
$379$	11.7990	0.606073	0.303037	+	0.952979i	$0.402000\pi$
$380$	4.48528	0.230090
$381$	0	0
$382$	43.7990	2.24095
$383$	−4.82843	−0.246721	−0.123361	−	0.992362i	$-0.539367\pi$
$383$	−4.82843	−0.246721	−0.123361	+	0.992362i	$0.539367\pi$
$384$	0	0
$385$	−7.00000	−0.356753
$386$	22.4853	1.14447
$387$	0	0
$388$	60.5980	3.07640
$389$	−8.14214	−0.412823	−0.206411	−	0.978465i	$-0.566178\pi$
$389$	−8.14214	−0.412823	−0.206411	+	0.978465i	$0.566178\pi$
$390$	0	0
$391$	−26.3431	−1.33223
$392$	−16.1421	−0.815301
$393$	0	0
$394$	−63.9411	−3.22131
$395$	−9.00000	−0.452839
$396$	0	0
$397$	−14.1716	−0.711251	−0.355625	−	0.934629i	$-0.615732\pi$
$397$	−14.1716	−0.711251	−0.355625	+	0.934629i	$0.615732\pi$
$398$	43.3848	2.17468
$399$	0	0
$400$	3.00000	0.150000
$401$	5.82843	0.291058	0.145529	−	0.989354i	$-0.453512\pi$
$401$	5.82843	0.291058	0.145529	+	0.989354i	$0.453512\pi$
$402$	0	0
$403$	−6.00000	−0.298881
$404$	−18.4853	−0.919677
$405$	0	0
$406$	24.9706	1.23927
$407$	22.9706	1.13861
$408$	0	0
$409$	15.6569	0.774182	0.387091	−	0.922042i	$-0.373480\pi$
$409$	15.6569	0.774182	0.387091	+	0.922042i	$0.373480\pi$
$410$	−0.414214	−0.0204565
$411$	0	0
$412$	−19.7990	−0.975426
$413$	−21.0000	−1.03334
$414$	0	0
$415$	8.82843	0.433370
$416$	1.58579	0.0777496
$417$	0	0
$418$	−10.8284	−0.529636
$419$	−10.8284	−0.529003	−0.264502	−	0.964385i	$-0.585207\pi$
$419$	−10.8284	−0.529003	−0.264502	+	0.964385i	$0.585207\pi$
$420$	0	0
$421$	3.51472	0.171297	0.0856485	−	0.996325i	$-0.472704\pi$
$421$	3.51472	0.171297	0.0856485	+	0.996325i	$0.472704\pi$
$422$	3.24264	0.157849
$423$	0	0
$424$	−39.7279	−1.92936
$425$	4.65685	0.225891
$426$	0	0
$427$	1.82843	0.0884838
$428$	63.7696	3.08242
$429$	0	0
$430$	−18.4853	−0.891439
$431$	2.34315	0.112865	0.0564327	−	0.998406i	$-0.482027\pi$
$431$	2.34315	0.112865	0.0564327	+	0.998406i	$0.482027\pi$
$432$	0	0
$433$	−4.34315	−0.208718	−0.104359	−	0.994540i	$-0.533279\pi$
$433$	−4.34315	−0.208718	−0.104359	+	0.994540i	$0.533279\pi$
$434$	26.4853	1.27133
$435$	0	0
$436$	5.79899	0.277721
$437$	6.62742	0.317032
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	−16.8995	−0.805652
$441$	0	0
$442$	−11.2426	−0.534758
$443$	21.6274	1.02755	0.513775	−	0.857925i	$-0.328247\pi$
$443$	21.6274	1.02755	0.513775	+	0.857925i	$0.328247\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	−42.2132	−1.99885
$447$	0	0
$448$	−17.9706	−0.849029
$449$	−9.31371	−0.439541	−0.219771	−	0.975552i	$-0.570531\pi$
$449$	−9.31371	−0.439541	−0.219771	+	0.975552i	$0.570531\pi$
$450$	0	0
$451$	0.656854	0.0309301
$452$	−56.1127	−2.63932
$453$	0	0
$454$	−61.9411	−2.90704
$455$	1.82843	0.0857180
$456$	0	0
$457$	−23.4853	−1.09860	−0.549298	−	0.835627i	$-0.685105\pi$
$457$	−23.4853	−1.09860	−0.549298	+	0.835627i	$0.685105\pi$
$458$	52.2843	2.44308
$459$	0	0
$460$	21.6569	1.00976
$461$	23.4853	1.09382	0.546909	−	0.837192i	$-0.315804\pi$
$461$	23.4853	1.09382	0.546909	+	0.837192i	$0.315804\pi$
$462$	0	0
$463$	30.6274	1.42338	0.711688	−	0.702495i	$-0.247931\pi$
$463$	30.6274	1.42338	0.711688	+	0.702495i	$0.247931\pi$
$464$	16.9706	0.787839
$465$	0	0
$466$	−53.8701	−2.49548
$467$	−11.3137	−0.523536	−0.261768	−	0.965131i	$-0.584306\pi$
$467$	−11.3137	−0.523536	−0.261768	+	0.965131i	$0.584306\pi$
$468$	0	0
$469$	−2.71573	−0.125401
$470$	17.3137	0.798622
$471$	0	0
$472$	−50.6985	−2.33359
$473$	29.3137	1.34785
$474$	0	0
$475$	−1.17157	−0.0537555
$476$	32.5980	1.49413
$477$	0	0
$478$	−8.00000	−0.365911
$479$	−18.5147	−0.845959	−0.422980	−	0.906139i	$-0.639016\pi$
$479$	−18.5147	−0.845959	−0.422980	+	0.906139i	$0.639016\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	−20.4853	−0.933079
$483$	0	0
$484$	14.0000	0.636364
$485$	−15.8284	−0.718732
$486$	0	0
$487$	−10.4558	−0.473800	−0.236900	−	0.971534i	$-0.576131\pi$
$487$	−10.4558	−0.473800	−0.236900	+	0.971534i	$0.576131\pi$
$488$	4.41421	0.199822
$489$	0	0
$490$	8.82843	0.398828
$491$	−41.5980	−1.87729	−0.938645	−	0.344884i	$-0.887918\pi$
$491$	−41.5980	−1.87729	−0.938645	+	0.344884i	$0.887918\pi$
$492$	0	0
$493$	26.3431	1.18644
$494$	2.82843	0.127257
$495$	0	0
$496$	18.0000	0.808224
$497$	17.9706	0.806090
$498$	0	0
$499$	2.34315	0.104894	0.0524468	−	0.998624i	$-0.483298\pi$
$499$	2.34315	0.104894	0.0524468	+	0.998624i	$0.483298\pi$
$500$	−3.82843	−0.171212
$501$	0	0
$502$	−29.7990	−1.32999
$503$	12.2843	0.547729	0.273864	−	0.961768i	$-0.411698\pi$
$503$	12.2843	0.547729	0.273864	+	0.961768i	$0.411698\pi$
$504$	0	0
$505$	4.82843	0.214862
$506$	−52.2843	−2.32432
$507$	0	0
$508$	−64.4264	−2.85846
$509$	−4.17157	−0.184902	−0.0924509	−	0.995717i	$-0.529470\pi$
$509$	−4.17157	−0.184902	−0.0924509	+	0.995717i	$0.529470\pi$
$510$	0	0
$511$	−10.0294	−0.443676
$512$	−31.2426	−1.38074
$513$	0	0
$514$	12.8995	0.568972
$515$	5.17157	0.227887
$516$	0	0
$517$	−27.4558	−1.20751
$518$	26.4853	1.16370
$519$	0	0
$520$	4.41421	0.193576
$521$	33.1716	1.45327	0.726636	−	0.687022i	$-0.241082\pi$
$521$	33.1716	1.45327	0.726636	+	0.687022i	$0.241082\pi$
$522$	0	0
$523$	−19.4558	−0.850745	−0.425372	−	0.905018i	$-0.639857\pi$
$523$	−19.4558	−0.850745	−0.425372	+	0.905018i	$0.639857\pi$
$524$	−32.4853	−1.41913
$525$	0	0
$526$	75.5269	3.29313
$527$	27.9411	1.21713
$528$	0	0
$529$	9.00000	0.391304
$530$	21.7279	0.943801
$531$	0	0
$532$	−8.20101	−0.355559
$533$	−0.171573	−0.00743165
$534$	0	0
$535$	−16.6569	−0.720139
$536$	−6.55635	−0.283191
$537$	0	0
$538$	−0.343146	−0.0147941
$539$	−14.0000	−0.603023
$540$	0	0
$541$	28.8284	1.23943	0.619715	−	0.784827i	$-0.287248\pi$
$541$	28.8284	1.23943	0.619715	+	0.784827i	$0.287248\pi$
$542$	19.3137	0.829595
$543$	0	0
$544$	−7.38478	−0.316620
$545$	−1.51472	−0.0648834
$546$	0	0
$547$	35.4558	1.51598	0.757991	−	0.652265i	$-0.226181\pi$
$547$	35.4558	1.51598	0.757991	+	0.652265i	$0.226181\pi$
$548$	−44.6274	−1.90639
$549$	0	0
$550$	9.24264	0.394108
$551$	−6.62742	−0.282337
$552$	0	0
$553$	16.4558	0.699774
$554$	67.9411	2.88654
$555$	0	0
$556$	−26.7990	−1.13653
$557$	41.7990	1.77108	0.885540	−	0.464563i	$-0.153789\pi$
$557$	41.7990	1.77108	0.885540	+	0.464563i	$0.153789\pi$
$558$	0	0
$559$	−7.65685	−0.323850
$560$	−5.48528	−0.231795
$561$	0	0
$562$	10.0711	0.424822
$563$	−28.9706	−1.22096	−0.610482	−	0.792030i	$-0.709024\pi$
$563$	−28.9706	−1.22096	−0.610482	+	0.792030i	$0.709024\pi$
$564$	0	0
$565$	14.6569	0.616618
$566$	−3.65685	−0.153709
$567$	0	0
$568$	43.3848	1.82038
$569$	41.7990	1.75230	0.876152	−	0.482034i	$-0.160102\pi$
$569$	41.7990	1.75230	0.876152	+	0.482034i	$0.160102\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	−14.6569	−0.612834
$573$	0	0
$574$	0.757359	0.0316116
$575$	−5.65685	−0.235907
$576$	0	0
$577$	22.1716	0.923015	0.461507	−	0.887136i	$-0.347309\pi$
$577$	22.1716	0.923015	0.461507	+	0.887136i	$0.347309\pi$
$578$	11.3137	0.470588
$579$	0	0
$580$	−21.6569	−0.899252
$581$	−16.1421	−0.669689
$582$	0	0
$583$	−34.4558	−1.42702
$584$	−24.2132	−1.00195
$585$	0	0
$586$	68.2843	2.82080
$587$	13.1716	0.543649	0.271824	−	0.962347i	$-0.412373\pi$
$587$	13.1716	0.543649	0.271824	+	0.962347i	$0.412373\pi$
$588$	0	0
$589$	−7.02944	−0.289643
$590$	27.7279	1.14154
$591$	0	0
$592$	18.0000	0.739795
$593$	24.6274	1.01133	0.505663	−	0.862731i	$-0.331248\pi$
$593$	24.6274	1.01133	0.505663	+	0.862731i	$0.331248\pi$
$594$	0	0
$595$	−8.51472	−0.349069
$596$	−20.3431	−0.833288
$597$	0	0
$598$	13.6569	0.558470
$599$	−33.9411	−1.38680	−0.693398	−	0.720554i	$-0.743887\pi$
$599$	−33.9411	−1.38680	−0.693398	+	0.720554i	$0.743887\pi$
$600$	0	0
$601$	−13.6863	−0.558275	−0.279138	−	0.960251i	$-0.590049\pi$
$601$	−13.6863	−0.558275	−0.279138	+	0.960251i	$0.590049\pi$
$602$	33.7990	1.37754
$603$	0	0
$604$	−38.2843	−1.55776
$605$	−3.65685	−0.148672
$606$	0	0
$607$	−25.4558	−1.03322	−0.516610	−	0.856221i	$-0.672806\pi$
$607$	−25.4558	−1.03322	−0.516610	+	0.856221i	$0.672806\pi$
$608$	1.85786	0.0753463
$609$	0	0
$610$	−2.41421	−0.0977486
$611$	7.17157	0.290131
$612$	0	0
$613$	44.7990	1.80941	0.904707	−	0.426034i	$-0.140090\pi$
$613$	44.7990	1.80941	0.904707	+	0.426034i	$0.140090\pi$
$614$	−20.9706	−0.846303
$615$	0	0
$616$	30.8995	1.24498
$617$	−34.9706	−1.40786	−0.703931	−	0.710268i	$-0.748574\pi$
$617$	−34.9706	−1.40786	−0.703931	+	0.710268i	$0.748574\pi$
$618$	0	0
$619$	10.9706	0.440944	0.220472	−	0.975393i	$-0.429240\pi$
$619$	10.9706	0.440944	0.220472	+	0.975393i	$0.429240\pi$
$620$	−22.9706	−0.922520
$621$	0	0
$622$	34.9706	1.40219
$623$	−10.9706	−0.439526
$624$	0	0
$625$	1.00000	0.0400000
$626$	−81.5980	−3.26131
$627$	0	0
$628$	32.4853	1.29630
$629$	27.9411	1.11409
$630$	0	0
$631$	−6.97056	−0.277494	−0.138747	−	0.990328i	$-0.544307\pi$
$631$	−6.97056	−0.277494	−0.138747	+	0.990328i	$0.544307\pi$
$632$	39.7279	1.58029
$633$	0	0
$634$	−6.82843	−0.271191
$635$	16.8284	0.667816
$636$	0	0
$637$	3.65685	0.144890
$638$	52.2843	2.06995
$639$	0	0
$640$	20.5563	0.812561
$641$	−24.6863	−0.975050	−0.487525	−	0.873109i	$-0.662100\pi$
$641$	−24.6863	−0.975050	−0.487525	+	0.873109i	$0.662100\pi$
$642$	0	0
$643$	−4.45584	−0.175721	−0.0878607	−	0.996133i	$-0.528003\pi$
$643$	−4.45584	−0.175721	−0.0878607	+	0.996133i	$0.528003\pi$
$644$	−39.5980	−1.56038
$645$	0	0
$646$	−13.1716	−0.518229
$647$	−22.6569	−0.890733	−0.445366	−	0.895348i	$-0.646927\pi$
$647$	−22.6569	−0.890733	−0.445366	+	0.895348i	$0.646927\pi$
$648$	0	0
$649$	−43.9706	−1.72600
$650$	−2.41421	−0.0946932
$651$	0	0
$652$	32.5980	1.27664
$653$	42.9411	1.68042	0.840208	−	0.542264i	$-0.182433\pi$
$653$	42.9411	1.68042	0.840208	+	0.542264i	$0.182433\pi$
$654$	0	0
$655$	8.48528	0.331547
$656$	0.514719	0.0200964
$657$	0	0
$658$	−31.6569	−1.23411
$659$	27.6569	1.07736	0.538679	−	0.842511i	$-0.318924\pi$
$659$	27.6569	1.07736	0.538679	+	0.842511i	$0.318924\pi$
$660$	0	0
$661$	34.1421	1.32798	0.663988	−	0.747744i	$-0.268863\pi$
$661$	34.1421	1.32798	0.663988	+	0.747744i	$0.268863\pi$
$662$	−11.6569	−0.453057
$663$	0	0
$664$	−38.9706	−1.51235
$665$	2.14214	0.0830685
$666$	0	0
$667$	−32.0000	−1.23904
$668$	−24.2843	−0.939587
$669$	0	0
$670$	3.58579	0.138531
$671$	3.82843	0.147795
$672$	0	0
$673$	30.4853	1.17512	0.587561	−	0.809180i	$-0.300088\pi$
$673$	30.4853	1.17512	0.587561	+	0.809180i	$0.300088\pi$
$674$	86.4264	3.32902
$675$	0	0
$676$	3.82843	0.147247
$677$	15.6863	0.602873	0.301437	−	0.953486i	$-0.402534\pi$
$677$	15.6863	0.602873	0.301437	+	0.953486i	$0.402534\pi$
$678$	0	0
$679$	28.9411	1.11066
$680$	−20.5563	−0.788300
$681$	0	0
$682$	55.4558	2.12351
$683$	−0.142136	−0.00543867	−0.00271933	−	0.999996i	$-0.500866\pi$
$683$	−0.142136	−0.00543867	−0.00271933	+	0.999996i	$0.500866\pi$
$684$	0	0
$685$	11.6569	0.445386
$686$	−47.0416	−1.79606
$687$	0	0
$688$	22.9706	0.875744
$689$	9.00000	0.342873
$690$	0	0
$691$	4.14214	0.157574	0.0787871	−	0.996891i	$-0.474895\pi$
$691$	4.14214	0.157574	0.0787871	+	0.996891i	$0.474895\pi$
$692$	−5.02944	−0.191191
$693$	0	0
$694$	69.9411	2.65493
$695$	7.00000	0.265525
$696$	0	0
$697$	0.798990	0.0302639
$698$	50.6274	1.91628
$699$	0	0
$700$	7.00000	0.264575
$701$	−6.20101	−0.234209	−0.117104	−	0.993120i	$-0.537361\pi$
$701$	−6.20101	−0.234209	−0.117104	+	0.993120i	$0.537361\pi$
$702$	0	0
$703$	−7.02944	−0.265120
$704$	−37.6274	−1.41814
$705$	0	0
$706$	−15.3137	−0.576339
$707$	−8.82843	−0.332027
$708$	0	0
$709$	29.1716	1.09556	0.547781	−	0.836622i	$-0.315473\pi$
$709$	29.1716	1.09556	0.547781	+	0.836622i	$0.315473\pi$
$710$	−23.7279	−0.890493
$711$	0	0
$712$	−26.4853	−0.992578
$713$	−33.9411	−1.27111
$714$	0	0
$715$	3.82843	0.143175
$716$	−35.1127	−1.31222
$717$	0	0
$718$	72.0122	2.68747
$719$	37.4558	1.39687	0.698434	−	0.715675i	$-0.253881\pi$
$719$	37.4558	1.39687	0.698434	+	0.715675i	$0.253881\pi$
$720$	0	0
$721$	−9.45584	−0.352154
$722$	−42.5563	−1.58378
$723$	0	0
$724$	−76.4558	−2.84146
$725$	5.65685	0.210090
$726$	0	0
$727$	−37.5980	−1.39443	−0.697216	−	0.716861i	$-0.745578\pi$
$727$	−37.5980	−1.39443	−0.697216	+	0.716861i	$0.745578\pi$
$728$	−8.07107	−0.299134
$729$	0	0
$730$	13.2426	0.490132
$731$	35.6569	1.31882
$732$	0	0
$733$	−12.1127	−0.447393	−0.223696	−	0.974659i	$-0.571812\pi$
$733$	−12.1127	−0.447393	−0.223696	+	0.974659i	$0.571812\pi$
$734$	−78.7696	−2.90744
$735$	0	0
$736$	8.97056	0.330659
$737$	−5.68629	−0.209457
$738$	0	0
$739$	16.9706	0.624272	0.312136	−	0.950037i	$-0.398955\pi$
$739$	16.9706	0.624272	0.312136	+	0.950037i	$0.398955\pi$
$740$	−22.9706	−0.844415
$741$	0	0
$742$	−39.7279	−1.45846
$743$	−30.0000	−1.10059	−0.550297	−	0.834969i	$-0.685485\pi$
$743$	−30.0000	−1.10059	−0.550297	+	0.834969i	$0.685485\pi$
$744$	0	0
$745$	5.31371	0.194679
$746$	65.9411	2.41428
$747$	0	0
$748$	68.2548	2.49564
$749$	30.4558	1.11283
$750$	0	0
$751$	9.00000	0.328415	0.164207	−	0.986426i	$-0.447493\pi$
$751$	9.00000	0.328415	0.164207	+	0.986426i	$0.447493\pi$
$752$	−21.5147	−0.784561
$753$	0	0
$754$	−13.6569	−0.497353
$755$	10.0000	0.363937
$756$	0	0
$757$	−3.65685	−0.132911	−0.0664553	−	0.997789i	$-0.521169\pi$
$757$	−3.65685	−0.132911	−0.0664553	+	0.997789i	$0.521169\pi$
$758$	28.4853	1.03463
$759$	0	0
$760$	5.17157	0.187593
$761$	41.7696	1.51415	0.757073	−	0.653331i	$-0.226629\pi$
$761$	41.7696	1.51415	0.757073	+	0.653331i	$0.226629\pi$
$762$	0	0
$763$	2.76955	0.100265
$764$	69.4558	2.51282
$765$	0	0
$766$	−11.6569	−0.421179
$767$	11.4853	0.414709
$768$	0	0
$769$	38.4853	1.38781	0.693907	−	0.720064i	$-0.255888\pi$
$769$	38.4853	1.38781	0.693907	+	0.720064i	$0.255888\pi$
$770$	−16.8995	−0.609016
$771$	0	0
$772$	35.6569	1.28332
$773$	−38.2843	−1.37699	−0.688495	−	0.725241i	$-0.741728\pi$
$773$	−38.2843	−1.37699	−0.688495	+	0.725241i	$0.741728\pi$
$774$	0	0
$775$	6.00000	0.215526
$776$	69.8701	2.50819
$777$	0	0
$778$	−19.6569	−0.704732
$779$	−0.201010	−0.00720194
$780$	0	0
$781$	37.6274	1.34641
$782$	−63.5980	−2.27426
$783$	0	0
$784$	−10.9706	−0.391806
$785$	−8.48528	−0.302853
$786$	0	0
$787$	7.48528	0.266821	0.133411	−	0.991061i	$-0.457407\pi$
$787$	7.48528	0.266821	0.133411	+	0.991061i	$0.457407\pi$
$788$	−101.397	−3.61212
$789$	0	0
$790$	−21.7279	−0.773045
$791$	−26.7990	−0.952862
$792$	0	0
$793$	−1.00000	−0.0355110
$794$	−34.2132	−1.21418
$795$	0	0
$796$	68.7990	2.43852
$797$	−43.9706	−1.55752	−0.778759	−	0.627324i	$-0.784150\pi$
$797$	−43.9706	−1.55752	−0.778759	+	0.627324i	$0.784150\pi$
$798$	0	0
$799$	−33.3970	−1.18150
$800$	−1.58579	−0.0560660
$801$	0	0
$802$	14.0711	0.496867
$803$	−21.0000	−0.741074
$804$	0	0
$805$	10.3431	0.364548
$806$	−14.4853	−0.510222
$807$	0	0
$808$	−21.3137	−0.749814
$809$	−24.7696	−0.870851	−0.435426	−	0.900225i	$-0.643402\pi$
$809$	−24.7696	−0.870851	−0.435426	+	0.900225i	$0.643402\pi$
$810$	0	0
$811$	−9.65685	−0.339098	−0.169549	−	0.985522i	$-0.554231\pi$
$811$	−9.65685	−0.339098	−0.169549	+	0.985522i	$0.554231\pi$
$812$	39.5980	1.38962
$813$	0	0
$814$	55.4558	1.94373
$815$	−8.51472	−0.298258
$816$	0	0
$817$	−8.97056	−0.313840
$818$	37.7990	1.32161
$819$	0	0
$820$	−0.656854	−0.0229383
$821$	36.3431	1.26838	0.634192	−	0.773175i	$-0.281333\pi$
$821$	36.3431	1.26838	0.634192	+	0.773175i	$0.281333\pi$
$822$	0	0
$823$	−33.1127	−1.15424	−0.577118	−	0.816661i	$-0.695823\pi$
$823$	−33.1127	−1.15424	−0.577118	+	0.816661i	$0.695823\pi$
$824$	−22.8284	−0.795266
$825$	0	0
$826$	−50.6985	−1.76403
$827$	7.79899	0.271197	0.135599	−	0.990764i	$-0.456704\pi$
$827$	7.79899	0.271197	0.135599	+	0.990764i	$0.456704\pi$
$828$	0	0
$829$	55.9706	1.94394	0.971969	−	0.235109i	$-0.0755447\pi$
$829$	55.9706	1.94394	0.971969	+	0.235109i	$0.0755447\pi$
$830$	21.3137	0.739810
$831$	0	0
$832$	9.82843	0.340739
$833$	−17.0294	−0.590035
$834$	0	0
$835$	6.34315	0.219514
$836$	−17.1716	−0.593891
$837$	0	0
$838$	−26.1421	−0.903065
$839$	−33.4264	−1.15401	−0.577004	−	0.816741i	$-0.695778\pi$
$839$	−33.4264	−1.15401	−0.577004	+	0.816741i	$0.695778\pi$
$840$	0	0
$841$	3.00000	0.103448
$842$	8.48528	0.292422
$843$	0	0
$844$	5.14214	0.177000
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	6.68629	0.229744
$848$	−27.0000	−0.927184
$849$	0	0
$850$	11.2426	0.385619
$851$	−33.9411	−1.16349
$852$	0	0
$853$	−51.9411	−1.77843	−0.889215	−	0.457489i	$-0.848749\pi$
$853$	−51.9411	−1.77843	−0.889215	+	0.457489i	$0.848749\pi$
$854$	4.41421	0.151051
$855$	0	0
$856$	73.5269	2.51310
$857$	−40.6274	−1.38781	−0.693903	−	0.720068i	$-0.744110\pi$
$857$	−40.6274	−1.38781	−0.693903	+	0.720068i	$0.744110\pi$
$858$	0	0
$859$	−7.31371	−0.249541	−0.124770	−	0.992186i	$-0.539819\pi$
$859$	−7.31371	−0.249541	−0.124770	+	0.992186i	$0.539819\pi$
$860$	−29.3137	−0.999589
$861$	0	0
$862$	5.65685	0.192673
$863$	−26.4853	−0.901569	−0.450785	−	0.892633i	$-0.648856\pi$
$863$	−26.4853	−0.901569	−0.450785	+	0.892633i	$0.648856\pi$
$864$	0	0
$865$	1.31371	0.0446674
$866$	−10.4853	−0.356304
$867$	0	0
$868$	42.0000	1.42557
$869$	34.4558	1.16883
$870$	0	0
$871$	1.48528	0.0503268
$872$	6.68629	0.226426
$873$	0	0
$874$	16.0000	0.541208
$875$	−1.82843	−0.0618121
$876$	0	0
$877$	13.8284	0.466953	0.233476	−	0.972362i	$-0.424990\pi$
$877$	13.8284	0.466953	0.233476	+	0.972362i	$0.424990\pi$
$878$	48.2843	1.62952
$879$	0	0
$880$	−11.4853	−0.387169
$881$	−1.45584	−0.0490486	−0.0245243	−	0.999699i	$-0.507807\pi$
$881$	−1.45584	−0.0490486	−0.0245243	+	0.999699i	$0.507807\pi$
$882$	0	0
$883$	−44.9706	−1.51338	−0.756690	−	0.653774i	$-0.773185\pi$
$883$	−44.9706	−1.51338	−0.756690	+	0.653774i	$0.773185\pi$
$884$	−17.8284	−0.599635
$885$	0	0
$886$	52.2132	1.75414
$887$	19.6863	0.661001	0.330500	−	0.943806i	$-0.392782\pi$
$887$	19.6863	0.661001	0.330500	+	0.943806i	$0.392782\pi$
$888$	0	0
$889$	−30.7696	−1.03198
$890$	14.4853	0.485548
$891$	0	0
$892$	−66.9411	−2.24135
$893$	8.40202	0.281163
$894$	0	0
$895$	9.17157	0.306572
$896$	−37.5858	−1.25565
$897$	0	0
$898$	−22.4853	−0.750344
$899$	33.9411	1.13200
$900$	0	0
$901$	−41.9117	−1.39628
$902$	1.58579	0.0528009
$903$	0	0
$904$	−64.6985	−2.15184
$905$	19.9706	0.663844
$906$	0	0
$907$	−52.9706	−1.75886	−0.879429	−	0.476029i	$-0.842076\pi$
$907$	−52.9706	−1.75886	−0.879429	+	0.476029i	$0.842076\pi$
$908$	−98.2254	−3.25972
$909$	0	0
$910$	4.41421	0.146330
$911$	16.9706	0.562260	0.281130	−	0.959670i	$-0.409291\pi$
$911$	16.9706	0.562260	0.281130	+	0.959670i	$0.409291\pi$
$912$	0	0
$913$	−33.7990	−1.11858
$914$	−56.6985	−1.87542
$915$	0	0
$916$	82.9117	2.73948
$917$	−15.5147	−0.512341
$918$	0	0
$919$	−56.9411	−1.87831	−0.939157	−	0.343488i	$-0.888392\pi$
$919$	−56.9411	−1.87831	−0.939157	+	0.343488i	$0.888392\pi$
$920$	24.9706	0.823255
$921$	0	0
$922$	56.6985	1.86727
$923$	−9.82843	−0.323507
$924$	0	0
$925$	6.00000	0.197279
$926$	73.9411	2.42986
$927$	0	0
$928$	−8.97056	−0.294473
$929$	39.1421	1.28421	0.642106	−	0.766616i	$-0.278061\pi$
$929$	39.1421	1.28421	0.642106	+	0.766616i	$0.278061\pi$
$930$	0	0
$931$	4.28427	0.140411
$932$	−85.4264	−2.79823
$933$	0	0
$934$	−27.3137	−0.893732
$935$	−17.8284	−0.583052
$936$	0	0
$937$	25.4558	0.831606	0.415803	−	0.909455i	$-0.363501\pi$
$937$	25.4558	0.831606	0.415803	+	0.909455i	$0.363501\pi$
$938$	−6.55635	−0.214072
$939$	0	0
$940$	27.4558	0.895511
$941$	35.6569	1.16238	0.581190	−	0.813768i	$-0.302587\pi$
$941$	35.6569	1.16238	0.581190	+	0.813768i	$0.302587\pi$
$942$	0	0
$943$	−0.970563	−0.0316059
$944$	−34.4558	−1.12144
$945$	0	0
$946$	70.7696	2.30092
$947$	14.3431	0.466090	0.233045	−	0.972466i	$-0.425131\pi$
$947$	14.3431	0.466090	0.233045	+	0.972466i	$0.425131\pi$
$948$	0	0
$949$	5.48528	0.178060
$950$	−2.82843	−0.0917663
$951$	0	0
$952$	37.5858	1.21816
$953$	−47.2548	−1.53073	−0.765367	−	0.643594i	$-0.777443\pi$
$953$	−47.2548	−1.53073	−0.765367	+	0.643594i	$0.777443\pi$
$954$	0	0
$955$	−18.1421	−0.587066
$956$	−12.6863	−0.410304
$957$	0	0
$958$	−44.6985	−1.44414
$959$	−21.3137	−0.688256
$960$	0	0
$961$	5.00000	0.161290
$962$	−14.4853	−0.467024
$963$	0	0
$964$	−32.4853	−1.04628
$965$	−9.31371	−0.299819
$966$	0	0
$967$	−44.9706	−1.44616	−0.723078	−	0.690766i	$-0.757273\pi$
$967$	−44.9706	−1.44616	−0.723078	+	0.690766i	$0.757273\pi$
$968$	16.1421	0.518828
$969$	0	0
$970$	−38.2132	−1.22695
$971$	−22.3431	−0.717026	−0.358513	−	0.933525i	$-0.616716\pi$
$971$	−22.3431	−0.717026	−0.358513	+	0.933525i	$0.616716\pi$
$972$	0	0
$973$	−12.7990	−0.410317
$974$	−25.2426	−0.808826
$975$	0	0
$976$	3.00000	0.0960277
$977$	27.5147	0.880274	0.440137	−	0.897931i	$-0.354930\pi$
$977$	27.5147	0.880274	0.440137	+	0.897931i	$0.354930\pi$
$978$	0	0
$979$	−22.9706	−0.734142
$980$	14.0000	0.447214
$981$	0	0
$982$	−100.426	−3.20474
$983$	−16.9706	−0.541277	−0.270638	−	0.962681i	$-0.587235\pi$
$983$	−16.9706	−0.541277	−0.270638	+	0.962681i	$0.587235\pi$
$984$	0	0
$985$	26.4853	0.843891
$986$	63.5980	2.02537
$987$	0	0
$988$	4.48528	0.142696
$989$	−43.3137	−1.37730
$990$	0	0
$991$	15.0294	0.477426	0.238713	−	0.971090i	$-0.423275\pi$
$991$	15.0294	0.477426	0.238713	+	0.971090i	$0.423275\pi$
$992$	−9.51472	−0.302093
$993$	0	0
$994$	43.3848	1.37608
$995$	−17.9706	−0.569705
$996$	0	0
$997$	−5.11270	−0.161921	−0.0809604	−	0.996717i	$-0.525799\pi$
$997$	−5.11270	−0.161921	−0.0809604	+	0.996717i	$0.525799\pi$
$998$	5.65685	0.179065
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1755.2.a.k.1.2	yes	2
3.2	odd	2		1755.2.a.g.1.1	✓	2
5.4	even	2		8775.2.a.s.1.1		2
15.14	odd	2		8775.2.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.2.a.g.1.1	✓	2	3.2	odd	2
1755.2.a.k.1.2	yes	2	1.1	even	1	trivial
8775.2.a.s.1.1		2	5.4	even	2
8775.2.a.be.1.2		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1755.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} +1.82843 q^{7} +4.41421 q^{8} +O(q^{10})\)	\(q+2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} +1.82843 q^{7} +4.41421 q^{8} -2.41421 q^{10} +3.82843 q^{11} -1.00000 q^{13} +4.41421 q^{14} +3.00000 q^{16} +4.65685 q^{17} -1.17157 q^{19} -3.82843 q^{20} +9.24264 q^{22} -5.65685 q^{23} +1.00000 q^{25} -2.41421 q^{26} +7.00000 q^{28} +5.65685 q^{29} +6.00000 q^{31} -1.58579 q^{32} +11.2426 q^{34} -1.82843 q^{35} +6.00000 q^{37} -2.82843 q^{38} -4.41421 q^{40} +0.171573 q^{41} +7.65685 q^{43} +14.6569 q^{44} -13.6569 q^{46} -7.17157 q^{47} -3.65685 q^{49} +2.41421 q^{50} -3.82843 q^{52} -9.00000 q^{53} -3.82843 q^{55} +8.07107 q^{56} +13.6569 q^{58} -11.4853 q^{59} +1.00000 q^{61} +14.4853 q^{62} -9.82843 q^{64} +1.00000 q^{65} -1.48528 q^{67} +17.8284 q^{68} -4.41421 q^{70} +9.82843 q^{71} -5.48528 q^{73} +14.4853 q^{74} -4.48528 q^{76} +7.00000 q^{77} +9.00000 q^{79} -3.00000 q^{80} +0.414214 q^{82} -8.82843 q^{83} -4.65685 q^{85} +18.4853 q^{86} +16.8995 q^{88} -6.00000 q^{89} -1.82843 q^{91} -21.6569 q^{92} -17.3137 q^{94} +1.17157 q^{95} +15.8284 q^{97} -8.82843 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 6 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 6 q^{8} - 2 q^{10} + 2 q^{11} - 2 q^{13} + 6 q^{14} + 6 q^{16} - 2 q^{17} - 8 q^{19} - 2 q^{20} + 10 q^{22} + 2 q^{25} - 2 q^{26} + 14 q^{28} + 12 q^{31} - 6 q^{32} + 14 q^{34} + 2 q^{35} + 12 q^{37} - 6 q^{40} + 6 q^{41} + 4 q^{43} + 18 q^{44} - 16 q^{46} - 20 q^{47} + 4 q^{49} + 2 q^{50} - 2 q^{52} - 18 q^{53} - 2 q^{55} + 2 q^{56} + 16 q^{58} - 6 q^{59} + 2 q^{61} + 12 q^{62} - 14 q^{64} + 2 q^{65} + 14 q^{67} + 30 q^{68} - 6 q^{70} + 14 q^{71} + 6 q^{73} + 12 q^{74} + 8 q^{76} + 14 q^{77} + 18 q^{79} - 6 q^{80} - 2 q^{82} - 12 q^{83} + 2 q^{85} + 20 q^{86} + 14 q^{88} - 12 q^{89} + 2 q^{91} - 32 q^{92} - 12 q^{94} + 8 q^{95} + 26 q^{97} - 12 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 6 * q^8 - 2 * q^10 + 2 * q^11 - 2 * q^13 + 6 * q^14 + 6 * q^16 - 2 * q^17 - 8 * q^19 - 2 * q^20 + 10 * q^22 + 2 * q^25 - 2 * q^26 + 14 * q^28 + 12 * q^31 - 6 * q^32 + 14 * q^34 + 2 * q^35 + 12 * q^37 - 6 * q^40 + 6 * q^41 + 4 * q^43 + 18 * q^44 - 16 * q^46 - 20 * q^47 + 4 * q^49 + 2 * q^50 - 2 * q^52 - 18 * q^53 - 2 * q^55 + 2 * q^56 + 16 * q^58 - 6 * q^59 + 2 * q^61 + 12 * q^62 - 14 * q^64 + 2 * q^65 + 14 * q^67 + 30 * q^68 - 6 * q^70 + 14 * q^71 + 6 * q^73 + 12 * q^74 + 8 * q^76 + 14 * q^77 + 18 * q^79 - 6 * q^80 - 2 * q^82 - 12 * q^83 + 2 * q^85 + 20 * q^86 + 14 * q^88 - 12 * q^89 + 2 * q^91 - 32 * q^92 - 12 * q^94 + 8 * q^95 + 26 * q^97 - 12 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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