LMFDB - Embedded newform 1935.2.a.u.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1935,2,Mod(1,1935)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1935, 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("1935.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1935 = 3^{2} \cdot 5 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1935.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$15.4510527911$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1933097.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 13x^{2} + 5x - 4 $ x^5 - 2x^4 - 7x^3 + 13x^2 + 5x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 215)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.667116$ of defining polynomial
Character	$\chi$	$=$	1935.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.667116 q^{2} -1.55496 q^{4} -1.00000 q^{5} -4.17800 q^{7} -2.37157 q^{8} +O(q^{10})$	\(q+0.667116 q^{2} -1.55496 q^{4} -1.00000 q^{5} -4.17800 q^{7} -2.37157 q^{8} -0.667116 q^{10} -2.70580 q^{11} +4.36004 q^{13} -2.78721 q^{14} +1.52780 q^{16} +2.58436 q^{17} -2.83128 q^{19} +1.55496 q^{20} -1.80508 q^{22} -5.69427 q^{23} +1.00000 q^{25} +2.90865 q^{26} +6.49661 q^{28} -5.24609 q^{29} +4.64924 q^{31} +5.76236 q^{32} +1.72407 q^{34} +4.17800 q^{35} +1.95593 q^{37} -1.88879 q^{38} +2.37157 q^{40} +10.0672 q^{41} -1.00000 q^{43} +4.20740 q^{44} -3.79874 q^{46} -8.44414 q^{47} +10.4557 q^{49} +0.667116 q^{50} -6.77967 q^{52} +5.41564 q^{53} +2.70580 q^{55} +9.90841 q^{56} -3.49975 q^{58} +3.03868 q^{59} +10.7201 q^{61} +3.10158 q^{62} +0.788558 q^{64} -4.36004 q^{65} +13.3302 q^{67} -4.01856 q^{68} +2.78721 q^{70} -9.52471 q^{71} +10.1486 q^{73} +1.30483 q^{74} +4.40252 q^{76} +11.3048 q^{77} +11.2902 q^{79} -1.52780 q^{80} +6.71598 q^{82} +9.02850 q^{83} -2.58436 q^{85} -0.667116 q^{86} +6.41699 q^{88} -2.08815 q^{89} -18.2162 q^{91} +8.85434 q^{92} -5.63322 q^{94} +2.83128 q^{95} +14.5256 q^{97} +6.97515 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 8 q^{4} - 5 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) $ 5 * q - 2 * q^2 + 8 * q^4 - 5 * q^5 + 5 * q^7 - 3 * q^8	$ 5 q - 2 q^{2} + 8 q^{4} - 5 q^{5} + 5 q^{7} - 3 q^{8} + 2 q^{10} + 6 q^{11} + 5 q^{13} - q^{14} + 14 q^{16} + 17 q^{17} - 6 q^{19} - 8 q^{20} - 8 q^{22} - q^{23} + 5 q^{25} - 22 q^{26} + 26 q^{28} - 6 q^{29} + 6 q^{31} + 7 q^{32} - 5 q^{35} + 5 q^{37} + 16 q^{38} + 3 q^{40} - 2 q^{41} - 5 q^{43} + 15 q^{44} - 14 q^{46} + 18 q^{49} - 2 q^{50} - 38 q^{52} + 23 q^{53} - 6 q^{55} + 19 q^{56} + 12 q^{58} + q^{59} + 20 q^{61} + 3 q^{62} - 25 q^{64} - 5 q^{65} + 21 q^{67} + 48 q^{68} + q^{70} - 4 q^{71} + 5 q^{73} - 24 q^{74} + 32 q^{76} + 26 q^{77} + 41 q^{79} - 14 q^{80} + 38 q^{82} + 7 q^{83} - 17 q^{85} + 2 q^{86} + 12 q^{88} - 20 q^{89} - 42 q^{91} + 52 q^{92} - 42 q^{94} + 6 q^{95} + 37 q^{97} + 26 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 8 * q^4 - 5 * q^5 + 5 * q^7 - 3 * q^8 + 2 * q^10 + 6 * q^11 + 5 * q^13 - q^14 + 14 * q^16 + 17 * q^17 - 6 * q^19 - 8 * q^20 - 8 * q^22 - q^23 + 5 * q^25 - 22 * q^26 + 26 * q^28 - 6 * q^29 + 6 * q^31 + 7 * q^32 - 5 * q^35 + 5 * q^37 + 16 * q^38 + 3 * q^40 - 2 * q^41 - 5 * q^43 + 15 * q^44 - 14 * q^46 + 18 * q^49 - 2 * q^50 - 38 * q^52 + 23 * q^53 - 6 * q^55 + 19 * q^56 + 12 * q^58 + q^59 + 20 * q^61 + 3 * q^62 - 25 * q^64 - 5 * q^65 + 21 * q^67 + 48 * q^68 + q^70 - 4 * q^71 + 5 * q^73 - 24 * q^74 + 32 * q^76 + 26 * q^77 + 41 * q^79 - 14 * q^80 + 38 * q^82 + 7 * q^83 - 17 * q^85 + 2 * q^86 + 12 * q^88 - 20 * q^89 - 42 * q^91 + 52 * q^92 - 42 * q^94 + 6 * q^95 + 37 * q^97 + 26 * 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 $

$2$	0.667116	0.471722	0.235861	−	0.971787i	$-0.424209\pi$
$2$	0.667116	0.471722	0.235861	+	0.971787i	$0.424209\pi$
$3$	0	0
$3$	0	0
$4$	−1.55496	−0.777478
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.17800	−1.57914	−0.789568	−	0.613664i	$-0.789695\pi$
$7$	−4.17800	−1.57914	−0.789568	+	0.613664i	$0.789695\pi$
$8$	−2.37157	−0.838476
$9$	0	0
$10$	−0.667116	−0.210961
$11$	−2.70580	−0.815829	−0.407915	−	0.913020i	$-0.633744\pi$
$11$	−2.70580	−0.815829	−0.407915	+	0.913020i	$0.633744\pi$
$12$	0	0
$13$	4.36004	1.20926	0.604629	−	0.796508i	$-0.293322\pi$
$13$	4.36004	1.20926	0.604629	+	0.796508i	$0.293322\pi$
$14$	−2.78721	−0.744913
$15$	0	0
$16$	1.52780	0.381950
$17$	2.58436	0.626799	0.313399	−	0.949621i	$-0.398532\pi$
$17$	2.58436	0.626799	0.313399	+	0.949621i	$0.398532\pi$
$18$	0	0
$19$	−2.83128	−0.649541	−0.324770	−	0.945793i	$-0.605287\pi$
$19$	−2.83128	−0.649541	−0.324770	+	0.945793i	$0.605287\pi$
$20$	1.55496	0.347699
$21$	0	0
$22$	−1.80508	−0.384845
$23$	−5.69427	−1.18734	−0.593669	−	0.804710i	$-0.702321\pi$
$23$	−5.69427	−1.18734	−0.593669	+	0.804710i	$0.702321\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.90865	0.570434
$27$	0	0
$28$	6.49661	1.22774
$29$	−5.24609	−0.974174	−0.487087	−	0.873354i	$-0.661940\pi$
$29$	−5.24609	−0.974174	−0.487087	+	0.873354i	$0.661940\pi$
$30$	0	0
$31$	4.64924	0.835029	0.417514	−	0.908670i	$-0.362901\pi$
$31$	4.64924	0.835029	0.417514	+	0.908670i	$0.362901\pi$
$32$	5.76236	1.01865
$33$	0	0
$34$	1.72407	0.295675
$35$	4.17800	0.706211
$36$	0	0
$37$	1.95593	0.321552	0.160776	−	0.986991i	$-0.448600\pi$
$37$	1.95593	0.321552	0.160776	+	0.986991i	$0.448600\pi$
$38$	−1.88879	−0.306403
$39$	0	0
$40$	2.37157	0.374978
$41$	10.0672	1.57223	0.786115	−	0.618080i	$-0.212089\pi$
$41$	10.0672	1.57223	0.786115	+	0.618080i	$0.212089\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	4.20740	0.634290
$45$	0	0
$46$	−3.79874	−0.560094
$47$	−8.44414	−1.23171	−0.615853	−	0.787861i	$-0.711188\pi$
$47$	−8.44414	−1.23171	−0.615853	+	0.787861i	$0.711188\pi$
$48$	0	0
$49$	10.4557	1.49367
$50$	0.667116	0.0943444
$51$	0	0
$52$	−6.77967	−0.940171
$53$	5.41564	0.743896	0.371948	−	0.928254i	$-0.378690\pi$
$53$	5.41564	0.743896	0.371948	+	0.928254i	$0.378690\pi$
$54$	0	0
$55$	2.70580	0.364850
$56$	9.90841	1.32407
$57$	0	0
$58$	−3.49975	−0.459539
$59$	3.03868	0.395603	0.197801	−	0.980242i	$-0.436620\pi$
$59$	3.03868	0.395603	0.197801	+	0.980242i	$0.436620\pi$
$60$	0	0
$61$	10.7201	1.37257	0.686283	−	0.727335i	$-0.259241\pi$
$61$	10.7201	1.37257	0.686283	+	0.727335i	$0.259241\pi$
$62$	3.10158	0.393902
$63$	0	0
$64$	0.788558	0.0985697
$65$	−4.36004	−0.540796
$66$	0	0
$67$	13.3302	1.62854	0.814271	−	0.580485i	$-0.197137\pi$
$67$	13.3302	1.62854	0.814271	+	0.580485i	$0.197137\pi$
$68$	−4.01856	−0.487322
$69$	0	0
$70$	2.78721	0.333135
$71$	−9.52471	−1.13038	−0.565188	−	0.824962i	$-0.691196\pi$
$71$	−9.52471	−1.13038	−0.565188	+	0.824962i	$0.691196\pi$
$72$	0	0
$73$	10.1486	1.18780	0.593902	−	0.804538i	$-0.297587\pi$
$73$	10.1486	1.18780	0.593902	+	0.804538i	$0.297587\pi$
$74$	1.30483	0.151683
$75$	0	0
$76$	4.40252	0.505004
$77$	11.3048	1.28830
$78$	0	0
$79$	11.2902	1.27024	0.635121	−	0.772413i	$-0.280950\pi$
$79$	11.2902	1.27024	0.635121	+	0.772413i	$0.280950\pi$
$80$	−1.52780	−0.170813
$81$	0	0
$82$	6.71598	0.741656
$83$	9.02850	0.991007	0.495503	−	0.868606i	$-0.334984\pi$
$83$	9.02850	0.991007	0.495503	+	0.868606i	$0.334984\pi$
$84$	0	0
$85$	−2.58436	−0.280313
$86$	−0.667116	−0.0719370
$87$	0	0
$88$	6.41699	0.684053
$89$	−2.08815	−0.221343	−0.110672	−	0.993857i	$-0.535300\pi$
$89$	−2.08815	−0.221343	−0.110672	+	0.993857i	$0.535300\pi$
$90$	0	0
$91$	−18.2162	−1.90958
$92$	8.85434	0.923129
$93$	0	0
$94$	−5.63322	−0.581023
$95$	2.83128	0.290484
$96$	0	0
$97$	14.5256	1.47485	0.737423	−	0.675431i	$-0.236042\pi$
$97$	14.5256	1.47485	0.737423	+	0.675431i	$0.236042\pi$
$98$	6.97515	0.704596
$99$	0	0
$100$	−1.55496	−0.155496
$101$	2.76236	0.274865	0.137432	−	0.990511i	$-0.456115\pi$
$101$	2.76236	0.274865	0.137432	+	0.990511i	$0.456115\pi$
$102$	0	0
$103$	2.24692	0.221396	0.110698	−	0.993854i	$-0.464691\pi$
$103$	2.24692	0.221396	0.110698	+	0.993854i	$0.464691\pi$
$104$	−10.3401	−1.01393
$105$	0	0
$106$	3.61286	0.350912
$107$	−8.24288	−0.796870	−0.398435	−	0.917197i	$-0.630446\pi$
$107$	−8.24288	−0.796870	−0.398435	+	0.917197i	$0.630446\pi$
$108$	0	0
$109$	−20.2540	−1.93998	−0.969992	−	0.243138i	$-0.921823\pi$
$109$	−20.2540	−1.93998	−0.969992	+	0.243138i	$0.921823\pi$
$110$	1.80508	0.172108
$111$	0	0
$112$	−6.38315	−0.603151
$113$	3.95368	0.371931	0.185965	−	0.982556i	$-0.440459\pi$
$113$	3.95368	0.371931	0.185965	+	0.982556i	$0.440459\pi$
$114$	0	0
$115$	5.69427	0.530993
$116$	8.15743	0.757399
$117$	0	0
$118$	2.02715	0.186615
$119$	−10.7974	−0.989800
$120$	0	0
$121$	−3.67865	−0.334422
$122$	7.15154	0.647470
$123$	0	0
$124$	−7.22937	−0.649217
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−4.27459	−0.379308	−0.189654	−	0.981851i	$-0.560737\pi$
$127$	−4.27459	−0.379308	−0.189654	+	0.981851i	$0.560737\pi$
$128$	−10.9987	−0.972153
$129$	0	0
$130$	−2.90865	−0.255106
$131$	−0.301688	−0.0263586	−0.0131793	−	0.999913i	$-0.504195\pi$
$131$	−0.301688	−0.0263586	−0.0131793	+	0.999913i	$0.504195\pi$
$132$	0	0
$133$	11.8291	1.02571
$134$	8.89278	0.768219
$135$	0	0
$136$	−6.12898	−0.525556
$137$	−17.9858	−1.53663	−0.768314	−	0.640073i	$-0.778904\pi$
$137$	−17.9858	−1.53663	−0.768314	+	0.640073i	$0.778904\pi$
$138$	0	0
$139$	−11.8659	−1.00645	−0.503227	−	0.864154i	$-0.667854\pi$
$139$	−11.8659	−1.00645	−0.503227	+	0.864154i	$0.667854\pi$
$140$	−6.49661	−0.549063
$141$	0	0
$142$	−6.35409	−0.533223
$143$	−11.7974	−0.986548
$144$	0	0
$145$	5.24609	0.435664
$146$	6.77029	0.560313
$147$	0	0
$148$	−3.04138	−0.250000
$149$	15.3528	1.25775	0.628875	−	0.777506i	$-0.283516\pi$
$149$	15.3528	1.25775	0.628875	+	0.777506i	$0.283516\pi$
$150$	0	0
$151$	−6.64271	−0.540576	−0.270288	−	0.962780i	$-0.587119\pi$
$151$	−6.64271	−0.540576	−0.270288	+	0.962780i	$0.587119\pi$
$152$	6.71458	0.544624
$153$	0	0
$154$	7.54163	0.607722
$155$	−4.64924	−0.373436
$156$	0	0
$157$	−24.0631	−1.92045	−0.960224	−	0.279231i	$-0.909921\pi$
$157$	−24.0631	−1.92045	−0.960224	+	0.279231i	$0.909921\pi$
$158$	7.53185	0.599201
$159$	0	0
$160$	−5.76236	−0.455554
$161$	23.7907	1.87497
$162$	0	0
$163$	4.95273	0.387927	0.193964	−	0.981009i	$-0.437866\pi$
$163$	4.95273	0.387927	0.193964	+	0.981009i	$0.437866\pi$
$164$	−15.6540	−1.22237
$165$	0	0
$166$	6.02306	0.467480
$167$	14.8630	1.15013	0.575066	−	0.818107i	$-0.304976\pi$
$167$	14.8630	1.15013	0.575066	+	0.818107i	$0.304976\pi$
$168$	0	0
$169$	6.00994	0.462303
$170$	−1.72407	−0.132230
$171$	0	0
$172$	1.55496	0.118564
$173$	4.19048	0.318596	0.159298	−	0.987231i	$-0.449077\pi$
$173$	4.19048	0.318596	0.159298	+	0.987231i	$0.449077\pi$
$174$	0	0
$175$	−4.17800	−0.315827
$176$	−4.13393	−0.311606
$177$	0	0
$178$	−1.39304	−0.104412
$179$	−6.88559	−0.514653	−0.257327	−	0.966324i	$-0.582842\pi$
$179$	−6.88559	−0.514653	−0.257327	+	0.966324i	$0.582842\pi$
$180$	0	0
$181$	14.7262	1.09459	0.547296	−	0.836939i	$-0.315657\pi$
$181$	14.7262	1.09459	0.547296	+	0.836939i	$0.315657\pi$
$182$	−12.1523	−0.900792
$183$	0	0
$184$	13.5044	0.995554
$185$	−1.95593	−0.143803
$186$	0	0
$187$	−6.99276	−0.511361
$188$	13.1303	0.957624
$189$	0	0
$190$	1.88879	0.137028
$191$	−3.96616	−0.286981	−0.143491	−	0.989652i	$-0.545833\pi$
$191$	−3.96616	−0.286981	−0.143491	+	0.989652i	$0.545833\pi$
$192$	0	0
$193$	10.9742	0.789940	0.394970	−	0.918694i	$-0.370755\pi$
$193$	10.9742	0.789940	0.394970	+	0.918694i	$0.370755\pi$
$194$	9.69023	0.695718
$195$	0	0
$196$	−16.2581	−1.16129
$197$	−23.8477	−1.69908	−0.849538	−	0.527528i	$-0.823119\pi$
$197$	−23.8477	−1.69908	−0.849538	+	0.527528i	$0.823119\pi$
$198$	0	0
$199$	−3.09135	−0.219140	−0.109570	−	0.993979i	$-0.534947\pi$
$199$	−3.09135	−0.219140	−0.109570	+	0.993979i	$0.534947\pi$
$200$	−2.37157	−0.167695
$201$	0	0
$202$	1.84281	0.129660
$203$	21.9181	1.53835
$204$	0	0
$205$	−10.0672	−0.703123
$206$	1.49896	0.104437
$207$	0	0
$208$	6.66127	0.461876
$209$	7.66089	0.529915
$210$	0	0
$211$	−2.58391	−0.177883	−0.0889417	−	0.996037i	$-0.528348\pi$
$211$	−2.58391	−0.177883	−0.0889417	+	0.996037i	$0.528348\pi$
$212$	−8.42109	−0.578363
$213$	0	0
$214$	−5.49896	−0.375901
$215$	1.00000	0.0681994
$216$	0	0
$217$	−19.4245	−1.31862
$218$	−13.5118	−0.915133
$219$	0	0
$220$	−4.20740	−0.283663
$221$	11.2679	0.757961
$222$	0	0
$223$	7.43466	0.497862	0.248931	−	0.968521i	$-0.419921\pi$
$223$	7.43466	0.497862	0.248931	+	0.968521i	$0.419921\pi$
$224$	−24.0751	−1.60859
$225$	0	0
$226$	2.63756	0.175448
$227$	17.3002	1.14825	0.574127	−	0.818766i	$-0.305342\pi$
$227$	17.3002	1.14825	0.574127	+	0.818766i	$0.305342\pi$
$228$	0	0
$229$	8.44574	0.558110	0.279055	−	0.960275i	$-0.409979\pi$
$229$	8.44574	0.558110	0.279055	+	0.960275i	$0.409979\pi$
$230$	3.79874	0.250481
$231$	0	0
$232$	12.4414	0.816821
$233$	3.36343	0.220346	0.110173	−	0.993912i	$-0.464860\pi$
$233$	3.36343	0.220346	0.110173	+	0.993912i	$0.464860\pi$
$234$	0	0
$235$	8.44414	0.550835
$236$	−4.72502	−0.307573
$237$	0	0
$238$	−7.20315	−0.466911
$239$	−20.7483	−1.34210	−0.671049	−	0.741413i	$-0.734156\pi$
$239$	−20.7483	−1.34210	−0.671049	+	0.741413i	$0.734156\pi$
$240$	0	0
$241$	19.9689	1.28631	0.643154	−	0.765737i	$-0.277626\pi$
$241$	19.9689	1.28631	0.643154	+	0.765737i	$0.277626\pi$
$242$	−2.45408	−0.157754
$243$	0	0
$244$	−16.6693	−1.06714
$245$	−10.4557	−0.667988
$246$	0	0
$247$	−12.3445	−0.785462
$248$	−11.0260	−0.700152
$249$	0	0
$250$	−0.667116	−0.0421921
$251$	7.32749	0.462507	0.231254	−	0.972893i	$-0.425717\pi$
$251$	7.32749	0.462507	0.231254	+	0.972893i	$0.425717\pi$
$252$	0	0
$253$	15.4076	0.968665
$254$	−2.85165	−0.178928
$255$	0	0
$256$	−8.91449	−0.557156
$257$	−4.14720	−0.258695	−0.129348	−	0.991599i	$-0.541288\pi$
$257$	−4.14720	−0.258695	−0.129348	+	0.991599i	$0.541288\pi$
$258$	0	0
$259$	−8.17186	−0.507775
$260$	6.77967	0.420457
$261$	0	0
$262$	−0.201261	−0.0124339
$263$	8.18653	0.504803	0.252402	−	0.967623i	$-0.418780\pi$
$263$	8.18653	0.504803	0.252402	+	0.967623i	$0.418780\pi$
$264$	0	0
$265$	−5.41564	−0.332680
$266$	7.89138	0.483852
$267$	0	0
$268$	−20.7279	−1.26616
$269$	12.3576	0.753455	0.376728	−	0.926324i	$-0.377049\pi$
$269$	12.3576	0.753455	0.376728	+	0.926324i	$0.377049\pi$
$270$	0	0
$271$	14.6972	0.892792	0.446396	−	0.894836i	$-0.352707\pi$
$271$	14.6972	0.892792	0.446396	+	0.894836i	$0.352707\pi$
$272$	3.94839	0.239406
$273$	0	0
$274$	−11.9986	−0.724862
$275$	−2.70580	−0.163166
$276$	0	0
$277$	25.8094	1.55074	0.775369	−	0.631508i	$-0.217564\pi$
$277$	25.8094	1.55074	0.775369	+	0.631508i	$0.217564\pi$
$278$	−7.91595	−0.474767
$279$	0	0
$280$	−9.90841	−0.592141
$281$	−18.0836	−1.07877	−0.539387	−	0.842058i	$-0.681344\pi$
$281$	−18.0836	−1.07877	−0.539387	+	0.842058i	$0.681344\pi$
$282$	0	0
$283$	16.7641	0.996520	0.498260	−	0.867028i	$-0.333972\pi$
$283$	16.7641	0.996520	0.498260	+	0.867028i	$0.333972\pi$
$284$	14.8105	0.878842
$285$	0	0
$286$	−7.87023	−0.465376
$287$	−42.0607	−2.48276
$288$	0	0
$289$	−10.3211	−0.607123
$290$	3.49975	0.205512
$291$	0	0
$292$	−15.7806	−0.923491
$293$	16.4334	0.960047	0.480024	−	0.877255i	$-0.340628\pi$
$293$	16.4334	0.960047	0.480024	+	0.877255i	$0.340628\pi$
$294$	0	0
$295$	−3.03868	−0.176919
$296$	−4.63861	−0.269614
$297$	0	0
$298$	10.2421	0.593309
$299$	−24.8272	−1.43580
$300$	0	0
$301$	4.17800	0.240816
$302$	−4.43146	−0.255002
$303$	0	0
$304$	−4.32564	−0.248092
$305$	−10.7201	−0.613830
$306$	0	0
$307$	−10.6830	−0.609710	−0.304855	−	0.952399i	$-0.598608\pi$
$307$	−10.6830	−0.609710	−0.304855	+	0.952399i	$0.598608\pi$
$308$	−17.5785	−1.00163
$309$	0	0
$310$	−3.10158	−0.176158
$311$	15.8659	0.899674	0.449837	−	0.893111i	$-0.351482\pi$
$311$	15.8659	0.899674	0.449837	+	0.893111i	$0.351482\pi$
$312$	0	0
$313$	4.80484	0.271586	0.135793	−	0.990737i	$-0.456642\pi$
$313$	4.80484	0.271586	0.135793	+	0.990737i	$0.456642\pi$
$314$	−16.0529	−0.905918
$315$	0	0
$316$	−17.5557	−0.987585
$317$	10.8272	0.608118	0.304059	−	0.952653i	$-0.401658\pi$
$317$	10.8272	0.608118	0.304059	+	0.952653i	$0.401658\pi$
$318$	0	0
$319$	14.1949	0.794759
$320$	−0.788558	−0.0440817
$321$	0	0
$322$	15.8711	0.884463
$323$	−7.31705	−0.407132
$324$	0	0
$325$	4.36004	0.241851
$326$	3.30404	0.182994
$327$	0	0
$328$	−23.8750	−1.31828
$329$	35.2796	1.94503
$330$	0	0
$331$	1.47399	0.0810180	0.0405090	−	0.999179i	$-0.487102\pi$
$331$	1.47399	0.0810180	0.0405090	+	0.999179i	$0.487102\pi$
$332$	−14.0389	−0.770486
$333$	0	0
$334$	9.91534	0.542543
$335$	−13.3302	−0.728306
$336$	0	0
$337$	32.6953	1.78102	0.890512	−	0.454959i	$-0.150346\pi$
$337$	32.6953	1.78102	0.890512	+	0.454959i	$0.150346\pi$
$338$	4.00933	0.218079
$339$	0	0
$340$	4.01856	0.217937
$341$	−12.5799	−0.681241
$342$	0	0
$343$	−14.4378	−0.779568
$344$	2.37157	0.127866
$345$	0	0
$346$	2.79554	0.150289
$347$	16.3413	0.877245	0.438623	−	0.898671i	$-0.355467\pi$
$347$	16.3413	0.877245	0.438623	+	0.898671i	$0.355467\pi$
$348$	0	0
$349$	−1.01986	−0.0545917	−0.0272959	−	0.999627i	$-0.508690\pi$
$349$	−1.01986	−0.0545917	−0.0272959	+	0.999627i	$0.508690\pi$
$350$	−2.78721	−0.148983
$351$	0	0
$352$	−15.5918	−0.831045
$353$	19.4726	1.03642	0.518212	−	0.855252i	$-0.326598\pi$
$353$	19.4726	1.03642	0.518212	+	0.855252i	$0.326598\pi$
$354$	0	0
$355$	9.52471	0.505519
$356$	3.24698	0.172089
$357$	0	0
$358$	−4.59349	−0.242773
$359$	−3.90631	−0.206167	−0.103084	−	0.994673i	$-0.532871\pi$
$359$	−3.90631	−0.206167	−0.103084	+	0.994673i	$0.532871\pi$
$360$	0	0
$361$	−10.9838	−0.578097
$362$	9.82410	0.516343
$363$	0	0
$364$	28.3255	1.48466
$365$	−10.1486	−0.531202
$366$	0	0
$367$	8.27672	0.432041	0.216021	−	0.976389i	$-0.430692\pi$
$367$	8.27672	0.432041	0.216021	+	0.976389i	$0.430692\pi$
$368$	−8.69971	−0.453504
$369$	0	0
$370$	−1.30483	−0.0678349
$371$	−22.6265	−1.17471
$372$	0	0
$373$	−24.3808	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$373$	−24.3808	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$374$	−4.66498	−0.241220
$375$	0	0
$376$	20.0259	1.03276
$377$	−22.8731	−1.17803
$378$	0	0
$379$	29.5290	1.51680	0.758401	−	0.651788i	$-0.225981\pi$
$379$	29.5290	1.51680	0.758401	+	0.651788i	$0.225981\pi$
$380$	−4.40252	−0.225845
$381$	0	0
$382$	−2.64589	−0.135376
$383$	32.1155	1.64103	0.820514	−	0.571627i	$-0.193687\pi$
$383$	32.1155	1.64103	0.820514	+	0.571627i	$0.193687\pi$
$384$	0	0
$385$	−11.3048	−0.576147
$386$	7.32106	0.372632
$387$	0	0
$388$	−22.5866	−1.14666
$389$	−23.1846	−1.17551	−0.587753	−	0.809041i	$-0.699987\pi$
$389$	−23.1846	−1.17551	−0.587753	+	0.809041i	$0.699987\pi$
$390$	0	0
$391$	−14.7160	−0.744222
$392$	−24.7963	−1.25240
$393$	0	0
$394$	−15.9092	−0.801492
$395$	−11.2902	−0.568070
$396$	0	0
$397$	−23.4162	−1.17523	−0.587613	−	0.809142i	$-0.699932\pi$
$397$	−23.4162	−1.17523	−0.587613	+	0.809142i	$0.699932\pi$
$398$	−2.06229	−0.103373
$399$	0	0
$400$	1.52780	0.0763901
$401$	−10.9387	−0.546250	−0.273125	−	0.961979i	$-0.588057\pi$
$401$	−10.9387	−0.546250	−0.273125	+	0.961979i	$0.588057\pi$
$402$	0	0
$403$	20.2709	1.00976
$404$	−4.29534	−0.213701
$405$	0	0
$406$	14.6219	0.725675
$407$	−5.29235	−0.262332
$408$	0	0
$409$	10.4441	0.516430	0.258215	−	0.966088i	$-0.416866\pi$
$409$	10.4441	0.516430	0.258215	+	0.966088i	$0.416866\pi$
$410$	−6.71598	−0.331679
$411$	0	0
$412$	−3.49387	−0.172131
$413$	−12.6956	−0.624711
$414$	0	0
$415$	−9.02850	−0.443192
$416$	25.1241	1.23181
$417$	0	0
$418$	5.11070	0.249973
$419$	−20.0258	−0.978322	−0.489161	−	0.872193i	$-0.662697\pi$
$419$	−20.0258	−0.978322	−0.489161	+	0.872193i	$0.662697\pi$
$420$	0	0
$421$	−2.85793	−0.139287	−0.0696435	−	0.997572i	$-0.522186\pi$
$421$	−2.85793	−0.139287	−0.0696435	+	0.997572i	$0.522186\pi$
$422$	−1.72376	−0.0839115
$423$	0	0
$424$	−12.8436	−0.623739
$425$	2.58436	0.125360
$426$	0	0
$427$	−44.7885	−2.16747
$428$	12.8173	0.619549
$429$	0	0
$430$	0.667116	0.0321712
$431$	24.4570	1.17805	0.589027	−	0.808114i	$-0.299511\pi$
$431$	24.4570	1.17805	0.589027	+	0.808114i	$0.299511\pi$
$432$	0	0
$433$	15.2128	0.731080	0.365540	−	0.930796i	$-0.380884\pi$
$433$	15.2128	0.731080	0.365540	+	0.930796i	$0.380884\pi$
$434$	−12.9584	−0.622024
$435$	0	0
$436$	31.4941	1.50829
$437$	16.1221	0.771224
$438$	0	0
$439$	5.90297	0.281733	0.140867	−	0.990029i	$-0.455011\pi$
$439$	5.90297	0.281733	0.140867	+	0.990029i	$0.455011\pi$
$440$	−6.41699	−0.305918
$441$	0	0
$442$	7.51700	0.357547
$443$	24.3026	1.15465	0.577325	−	0.816514i	$-0.304096\pi$
$443$	24.3026	1.15465	0.577325	+	0.816514i	$0.304096\pi$
$444$	0	0
$445$	2.08815	0.0989877
$446$	4.95978	0.234852
$447$	0	0
$448$	−3.29459	−0.155655
$449$	3.64670	0.172098	0.0860492	−	0.996291i	$-0.472576\pi$
$449$	3.64670	0.172098	0.0860492	+	0.996291i	$0.472576\pi$
$450$	0	0
$451$	−27.2398	−1.28267
$452$	−6.14780	−0.289168
$453$	0	0
$454$	11.5412	0.541657
$455$	18.2162	0.853990
$456$	0	0
$457$	−23.2754	−1.08878	−0.544388	−	0.838833i	$-0.683238\pi$
$457$	−23.2754	−1.08878	−0.544388	+	0.838833i	$0.683238\pi$
$458$	5.63429	0.263273
$459$	0	0
$460$	−8.85434	−0.412836
$461$	−8.87536	−0.413367	−0.206683	−	0.978408i	$-0.566267\pi$
$461$	−8.87536	−0.413367	−0.206683	+	0.978408i	$0.566267\pi$
$462$	0	0
$463$	6.86358	0.318978	0.159489	−	0.987200i	$-0.449015\pi$
$463$	6.86358	0.318978	0.159489	+	0.987200i	$0.449015\pi$
$464$	−8.01498	−0.372086
$465$	0	0
$466$	2.24380	0.103942
$467$	21.3288	0.986981	0.493491	−	0.869751i	$-0.335721\pi$
$467$	21.3288	0.986981	0.493491	+	0.869751i	$0.335721\pi$
$468$	0	0
$469$	−55.6935	−2.57169
$470$	5.63322	0.259841
$471$	0	0
$472$	−7.20645	−0.331704
$473$	2.70580	0.124413
$474$	0	0
$475$	−2.83128	−0.129908
$476$	16.7896	0.769548
$477$	0	0
$478$	−13.8415	−0.633097
$479$	19.3177	0.882649	0.441324	−	0.897348i	$-0.354509\pi$
$479$	19.3177	0.882649	0.441324	+	0.897348i	$0.354509\pi$
$480$	0	0
$481$	8.52792	0.388839
$482$	13.3215	0.606780
$483$	0	0
$484$	5.72013	0.260006
$485$	−14.5256	−0.659571
$486$	0	0
$487$	1.05482	0.0477982	0.0238991	−	0.999714i	$-0.492392\pi$
$487$	1.05482	0.0477982	0.0238991	+	0.999714i	$0.492392\pi$
$488$	−25.4234	−1.15086
$489$	0	0
$490$	−6.97515	−0.315105
$491$	−13.3031	−0.600360	−0.300180	−	0.953883i	$-0.597047\pi$
$491$	−13.3031	−0.600360	−0.300180	+	0.953883i	$0.597047\pi$
$492$	0	0
$493$	−13.5578	−0.610611
$494$	−8.23522	−0.370520
$495$	0	0
$496$	7.10312	0.318940
$497$	39.7942	1.78502
$498$	0	0
$499$	16.6442	0.745097	0.372549	−	0.928013i	$-0.378484\pi$
$499$	16.6442	0.745097	0.372549	+	0.928013i	$0.378484\pi$
$500$	1.55496	0.0695398
$501$	0	0
$502$	4.88829	0.218175
$503$	−0.626638	−0.0279404	−0.0139702	−	0.999902i	$-0.504447\pi$
$503$	−0.626638	−0.0279404	−0.0139702	+	0.999902i	$0.504447\pi$
$504$	0	0
$505$	−2.76236	−0.122923
$506$	10.2786	0.456941
$507$	0	0
$508$	6.64680	0.294904
$509$	6.26851	0.277847	0.138923	−	0.990303i	$-0.455636\pi$
$509$	6.26851	0.277847	0.138923	+	0.990303i	$0.455636\pi$
$510$	0	0
$511$	−42.4008	−1.87570
$512$	16.0503	0.709330
$513$	0	0
$514$	−2.76666	−0.122032
$515$	−2.24692	−0.0990113
$516$	0	0
$517$	22.8482	1.00486
$518$	−5.45158	−0.239529
$519$	0	0
$520$	10.3401	0.453445
$521$	−13.0619	−0.572252	−0.286126	−	0.958192i	$-0.592368\pi$
$521$	−13.0619	−0.572252	−0.286126	+	0.958192i	$0.592368\pi$
$522$	0	0
$523$	−2.34385	−0.102490	−0.0512448	−	0.998686i	$-0.516319\pi$
$523$	−2.34385	−0.102490	−0.0512448	+	0.998686i	$0.516319\pi$
$524$	0.469111	0.0204932
$525$	0	0
$526$	5.46137	0.238127
$527$	12.0153	0.523395
$528$	0	0
$529$	9.42472	0.409770
$530$	−3.61286	−0.156933
$531$	0	0
$532$	−18.3937	−0.797469
$533$	43.8933	1.90123
$534$	0	0
$535$	8.24288	0.356371
$536$	−31.6135	−1.36549
$537$	0	0
$538$	8.24394	0.355422
$539$	−28.2910	−1.21858
$540$	0	0
$541$	−27.6634	−1.18934	−0.594671	−	0.803969i	$-0.702718\pi$
$541$	−27.6634	−1.18934	−0.594671	+	0.803969i	$0.702718\pi$
$542$	9.80474	0.421150
$543$	0	0
$544$	14.8920	0.638489
$545$	20.2540	0.867587
$546$	0	0
$547$	24.1791	1.03383	0.516913	−	0.856038i	$-0.327081\pi$
$547$	24.1791	1.03383	0.516913	+	0.856038i	$0.327081\pi$
$548$	27.9671	1.19470
$549$	0	0
$550$	−1.80508	−0.0769690
$551$	14.8532	0.632766
$552$	0	0
$553$	−47.1703	−2.00588
$554$	17.2179	0.731518
$555$	0	0
$556$	18.4510	0.782497
$557$	21.9737	0.931055	0.465528	−	0.885033i	$-0.345865\pi$
$557$	21.9737	0.931055	0.465528	+	0.885033i	$0.345865\pi$
$558$	0	0
$559$	−4.36004	−0.184410
$560$	6.38315	0.269737
$561$	0	0
$562$	−12.0638	−0.508882
$563$	1.06155	0.0447391	0.0223695	−	0.999750i	$-0.492879\pi$
$563$	1.06155	0.0447391	0.0223695	+	0.999750i	$0.492879\pi$
$564$	0	0
$565$	−3.95368	−0.166333
$566$	11.1836	0.470081
$567$	0	0
$568$	22.5885	0.947793
$569$	−28.6235	−1.19996	−0.599980	−	0.800015i	$-0.704825\pi$
$569$	−28.6235	−1.19996	−0.599980	+	0.800015i	$0.704825\pi$
$570$	0	0
$571$	26.6889	1.11690	0.558449	−	0.829539i	$-0.311397\pi$
$571$	26.6889	1.11690	0.558449	+	0.829539i	$0.311397\pi$
$572$	18.3444	0.767019
$573$	0	0
$574$	−28.0594	−1.17118
$575$	−5.69427	−0.237468
$576$	0	0
$577$	−3.26004	−0.135717	−0.0678587	−	0.997695i	$-0.521617\pi$
$577$	−3.26004	−0.135717	−0.0678587	+	0.997695i	$0.521617\pi$
$578$	−6.88537	−0.286393
$579$	0	0
$580$	−8.15743	−0.338719
$581$	−37.7211	−1.56493
$582$	0	0
$583$	−14.6536	−0.606892
$584$	−24.0681	−0.995945
$585$	0	0
$586$	10.9630	0.452876
$587$	32.9611	1.36045	0.680225	−	0.733004i	$-0.261882\pi$
$587$	32.9611	1.36045	0.680225	+	0.733004i	$0.261882\pi$
$588$	0	0
$589$	−13.1633	−0.542385
$590$	−2.02715	−0.0834566
$591$	0	0
$592$	2.98827	0.122817
$593$	−16.3019	−0.669440	−0.334720	−	0.942318i	$-0.608642\pi$
$593$	−16.3019	−0.669440	−0.334720	+	0.942318i	$0.608642\pi$
$594$	0	0
$595$	10.7974	0.442652
$596$	−23.8729	−0.977873
$597$	0	0
$598$	−16.5627	−0.677297
$599$	15.5570	0.635642	0.317821	−	0.948151i	$-0.397049\pi$
$599$	15.5570	0.635642	0.317821	+	0.948151i	$0.397049\pi$
$600$	0	0
$601$	0.0149762	0.000610890 0	0.000305445	−	1.00000i	$-0.499903\pi$
$601$	0.0149762	0.000610890 0	0.000305445		1.00000i	$0.499903\pi$
$602$	2.78721	0.113598
$603$	0	0
$604$	10.3291	0.420286
$605$	3.67865	0.149558
$606$	0	0
$607$	27.4196	1.11293	0.556465	−	0.830871i	$-0.312157\pi$
$607$	27.4196	1.11293	0.556465	+	0.830871i	$0.312157\pi$
$608$	−16.3149	−0.661655
$609$	0	0
$610$	−7.15154	−0.289557
$611$	−36.8168	−1.48945
$612$	0	0
$613$	31.4158	1.26887	0.634436	−	0.772975i	$-0.281232\pi$
$613$	31.4158	1.26887	0.634436	+	0.772975i	$0.281232\pi$
$614$	−7.12679	−0.287614
$615$	0	0
$616$	−26.8102	−1.08021
$617$	−27.4600	−1.10550	−0.552748	−	0.833348i	$-0.686421\pi$
$617$	−27.4600	−1.10550	−0.552748	+	0.833348i	$0.686421\pi$
$618$	0	0
$619$	−8.94419	−0.359498	−0.179749	−	0.983713i	$-0.557528\pi$
$619$	−8.94419	−0.359498	−0.179749	+	0.983713i	$0.557528\pi$
$620$	7.22937	0.290338
$621$	0	0
$622$	10.5844	0.424396
$623$	8.72428	0.349531
$624$	0	0
$625$	1.00000	0.0400000
$626$	3.20539	0.128113
$627$	0	0
$628$	37.4171	1.49311
$629$	5.05482	0.201549
$630$	0	0
$631$	−0.678048	−0.0269927	−0.0134963	−	0.999909i	$-0.504296\pi$
$631$	−0.678048	−0.0269927	−0.0134963	+	0.999909i	$0.504296\pi$
$632$	−26.7754	−1.06507
$633$	0	0
$634$	7.22303	0.286863
$635$	4.27459	0.169632
$636$	0	0
$637$	45.5871	1.80623
$638$	9.46962	0.374906
$639$	0	0
$640$	10.9987	0.434760
$641$	−24.7590	−0.977922	−0.488961	−	0.872306i	$-0.662624\pi$
$641$	−24.7590	−0.977922	−0.488961	+	0.872306i	$0.662624\pi$
$642$	0	0
$643$	−13.3021	−0.524583	−0.262291	−	0.964989i	$-0.584478\pi$
$643$	−13.3021	−0.524583	−0.262291	+	0.964989i	$0.584478\pi$
$644$	−36.9934	−1.45775
$645$	0	0
$646$	−4.88132	−0.192053
$647$	38.4734	1.51255	0.756273	−	0.654257i	$-0.227018\pi$
$647$	38.4734	1.51255	0.756273	+	0.654257i	$0.227018\pi$
$648$	0	0
$649$	−8.22207	−0.322745
$650$	2.90865	0.114087
$651$	0	0
$652$	−7.70127	−0.301605
$653$	47.7958	1.87040	0.935198	−	0.354125i	$-0.115221\pi$
$653$	47.7958	1.87040	0.935198	+	0.354125i	$0.115221\pi$
$654$	0	0
$655$	0.301688	0.0117879
$656$	15.3807	0.600514
$657$	0	0
$658$	23.5356	0.917513
$659$	−0.962771	−0.0375042	−0.0187521	−	0.999824i	$-0.505969\pi$
$659$	−0.962771	−0.0375042	−0.0187521	+	0.999824i	$0.505969\pi$
$660$	0	0
$661$	21.5317	0.837486	0.418743	−	0.908105i	$-0.362471\pi$
$661$	21.5317	0.837486	0.418743	+	0.908105i	$0.362471\pi$
$662$	0.983324	0.0382180
$663$	0	0
$664$	−21.4117	−0.830936
$665$	−11.8291	−0.458713
$666$	0	0
$667$	29.8726	1.15667
$668$	−23.1113	−0.894203
$669$	0	0
$670$	−8.89278	−0.343558
$671$	−29.0064	−1.11978
$672$	0	0
$673$	−17.8883	−0.689541	−0.344771	−	0.938687i	$-0.612043\pi$
$673$	−17.8883	−0.689541	−0.344771	+	0.938687i	$0.612043\pi$
$674$	21.8115	0.840149
$675$	0	0
$676$	−9.34519	−0.359430
$677$	−8.85919	−0.340486	−0.170243	−	0.985402i	$-0.554455\pi$
$677$	−8.85919	−0.340486	−0.170243	+	0.985402i	$0.554455\pi$
$678$	0	0
$679$	−60.6877	−2.32898
$680$	6.12898	0.235036
$681$	0	0
$682$	−8.39227	−0.321357
$683$	28.8869	1.10533	0.552664	−	0.833404i	$-0.313611\pi$
$683$	28.8869	1.10533	0.552664	+	0.833404i	$0.313611\pi$
$684$	0	0
$685$	17.9858	0.687201
$686$	−9.63169	−0.367740
$687$	0	0
$688$	−1.52780	−0.0582469
$689$	23.6124	0.899561
$690$	0	0
$691$	−43.8504	−1.66815	−0.834073	−	0.551654i	$-0.813997\pi$
$691$	−43.8504	−1.66815	−0.834073	+	0.551654i	$0.813997\pi$
$692$	−6.51602	−0.247702
$693$	0	0
$694$	10.9015	0.413816
$695$	11.8659	0.450100
$696$	0	0
$697$	26.0172	0.985472
$698$	−0.680363	−0.0257521
$699$	0	0
$700$	6.49661	0.245549
$701$	−21.8597	−0.825629	−0.412815	−	0.910815i	$-0.635454\pi$
$701$	−21.8597	−0.825629	−0.412815	+	0.910815i	$0.635454\pi$
$702$	0	0
$703$	−5.53778	−0.208861
$704$	−2.13368	−0.0804161
$705$	0	0
$706$	12.9905	0.488904
$707$	−11.5411	−0.434049
$708$	0	0
$709$	−35.0273	−1.31548	−0.657740	−	0.753245i	$-0.728487\pi$
$709$	−35.0273	−1.31548	−0.657740	+	0.753245i	$0.728487\pi$
$710$	6.35409	0.238465
$711$	0	0
$712$	4.95218	0.185591
$713$	−26.4741	−0.991461
$714$	0	0
$715$	11.7974	0.441198
$716$	10.7068	0.400132
$717$	0	0
$718$	−2.60596	−0.0972536
$719$	−0.792596	−0.0295589	−0.0147794	−	0.999891i	$-0.504705\pi$
$719$	−0.792596	−0.0295589	−0.0147794	+	0.999891i	$0.504705\pi$
$720$	0	0
$721$	−9.38765	−0.349614
$722$	−7.32749	−0.272701
$723$	0	0
$724$	−22.8986	−0.851021
$725$	−5.24609	−0.194835
$726$	0	0
$727$	11.3417	0.420639	0.210320	−	0.977633i	$-0.432550\pi$
$727$	11.3417	0.420639	0.210320	+	0.977633i	$0.432550\pi$
$728$	43.2010	1.60114
$729$	0	0
$730$	−6.77029	−0.250580
$731$	−2.58436	−0.0955859
$732$	0	0
$733$	−30.5953	−1.13006	−0.565031	−	0.825069i	$-0.691136\pi$
$733$	−30.5953	−1.13006	−0.565031	+	0.825069i	$0.691136\pi$
$734$	5.52153	0.203803
$735$	0	0
$736$	−32.8124	−1.20948
$737$	−36.0688	−1.32861
$738$	0	0
$739$	−14.0643	−0.517364	−0.258682	−	0.965963i	$-0.583288\pi$
$739$	−14.0643	−0.517364	−0.258682	+	0.965963i	$0.583288\pi$
$740$	3.04138	0.111803
$741$	0	0
$742$	−15.0945	−0.554138
$743$	2.26441	0.0830730	0.0415365	−	0.999137i	$-0.486775\pi$
$743$	2.26441	0.0830730	0.0415365	+	0.999137i	$0.486775\pi$
$744$	0	0
$745$	−15.3528	−0.562483
$746$	−16.2648	−0.595496
$747$	0	0
$748$	10.8734	0.397572
$749$	34.4388	1.25836
$750$	0	0
$751$	29.0652	1.06060	0.530302	−	0.847809i	$-0.322079\pi$
$751$	29.0652	1.06060	0.530302	+	0.847809i	$0.322079\pi$
$752$	−12.9010	−0.470450
$753$	0	0
$754$	−15.2590	−0.555701
$755$	6.64271	0.241753
$756$	0	0
$757$	−35.3092	−1.28333	−0.641667	−	0.766983i	$-0.721757\pi$
$757$	−35.3092	−1.28333	−0.641667	+	0.766983i	$0.721757\pi$
$758$	19.6993	0.715510
$759$	0	0
$760$	−6.71458	−0.243563
$761$	33.6653	1.22036	0.610182	−	0.792261i	$-0.291096\pi$
$761$	33.6653	1.22036	0.610182	+	0.792261i	$0.291096\pi$
$762$	0	0
$763$	84.6213	3.06350
$764$	6.16721	0.223122
$765$	0	0
$766$	21.4248	0.774109
$767$	13.2488	0.478386
$768$	0	0
$769$	2.70131	0.0974117	0.0487059	−	0.998813i	$-0.484490\pi$
$769$	2.70131	0.0974117	0.0487059	+	0.998813i	$0.484490\pi$
$770$	−7.54163	−0.271782
$771$	0	0
$772$	−17.0644	−0.614161
$773$	3.35121	0.120535	0.0602673	−	0.998182i	$-0.480805\pi$
$773$	3.35121	0.120535	0.0602673	+	0.998182i	$0.480805\pi$
$774$	0	0
$775$	4.64924	0.167006
$776$	−34.4483	−1.23662
$777$	0	0
$778$	−15.4668	−0.554512
$779$	−28.5031	−1.02123
$780$	0	0
$781$	25.7720	0.922194
$782$	−9.81730	−0.351066
$783$	0	0
$784$	15.9742	0.570507
$785$	24.0631	0.858851
$786$	0	0
$787$	38.2795	1.36452	0.682259	−	0.731111i	$-0.260998\pi$
$787$	38.2795	1.36452	0.682259	+	0.731111i	$0.260998\pi$
$788$	37.0821	1.32099
$789$	0	0
$790$	−7.53185	−0.267971
$791$	−16.5185	−0.587329
$792$	0	0
$793$	46.7400	1.65978
$794$	−15.6213	−0.554380
$795$	0	0
$796$	4.80691	0.170376
$797$	−0.585709	−0.0207469	−0.0103734	−	0.999946i	$-0.503302\pi$
$797$	−0.585709	−0.0207469	−0.0103734	+	0.999946i	$0.503302\pi$
$798$	0	0
$799$	−21.8227	−0.772031
$800$	5.76236	0.203730
$801$	0	0
$802$	−7.29735	−0.257678
$803$	−27.4601	−0.969045
$804$	0	0
$805$	−23.7907	−0.838510
$806$	13.5230	0.476328
$807$	0	0
$808$	−6.55112	−0.230468
$809$	45.1595	1.58772	0.793861	−	0.608099i	$-0.208068\pi$
$809$	45.1595	1.58772	0.793861	+	0.608099i	$0.208068\pi$
$810$	0	0
$811$	−17.4128	−0.611446	−0.305723	−	0.952121i	$-0.598898\pi$
$811$	−17.4128	−0.611446	−0.305723	+	0.952121i	$0.598898\pi$
$812$	−34.0817	−1.19603
$813$	0	0
$814$	−3.53061	−0.123748
$815$	−4.95273	−0.173486
$816$	0	0
$817$	2.83128	0.0990541
$818$	6.96746	0.243611
$819$	0	0
$820$	15.6540	0.546663
$821$	36.2816	1.26624	0.633119	−	0.774055i	$-0.281775\pi$
$821$	36.2816	1.26624	0.633119	+	0.774055i	$0.281775\pi$
$822$	0	0
$823$	−50.3506	−1.75511	−0.877557	−	0.479473i	$-0.840828\pi$
$823$	−50.3506	−1.75511	−0.877557	+	0.479473i	$0.840828\pi$
$824$	−5.32874	−0.185635
$825$	0	0
$826$	−8.46945	−0.294690
$827$	−0.00449454	−0.000156291 0	−7.81453e−5	−	1.00000i	$-0.500025\pi$
$827$	−0.00449454	−0.000156291 0	−7.81453e−5		1.00000i	$0.500025\pi$
$828$	0	0
$829$	7.34740	0.255186	0.127593	−	0.991827i	$-0.459275\pi$
$829$	7.34740	0.255186	0.127593	+	0.991827i	$0.459275\pi$
$830$	−6.02306	−0.209063
$831$	0	0
$832$	3.43814	0.119196
$833$	27.0212	0.936229
$834$	0	0
$835$	−14.8630	−0.514355
$836$	−11.9123	−0.411997
$837$	0	0
$838$	−13.3595	−0.461496
$839$	−37.6999	−1.30155	−0.650773	−	0.759272i	$-0.725555\pi$
$839$	−37.6999	−1.30155	−0.650773	+	0.759272i	$0.725555\pi$
$840$	0	0
$841$	−1.47859	−0.0509859
$842$	−1.90657	−0.0657048
$843$	0	0
$844$	4.01786	0.138300
$845$	−6.00994	−0.206748
$846$	0	0
$847$	15.3694	0.528098
$848$	8.27402	0.284131
$849$	0	0
$850$	1.72407	0.0591350
$851$	−11.1376	−0.381791
$852$	0	0
$853$	−30.3162	−1.03801	−0.519004	−	0.854772i	$-0.673697\pi$
$853$	−30.3162	−1.03801	−0.519004	+	0.854772i	$0.673697\pi$
$854$	−29.8791	−1.02244
$855$	0	0
$856$	19.5486	0.668156
$857$	−1.69382	−0.0578597	−0.0289299	−	0.999581i	$-0.509210\pi$
$857$	−1.69382	−0.0578597	−0.0289299	+	0.999581i	$0.509210\pi$
$858$	0	0
$859$	11.9716	0.408464	0.204232	−	0.978922i	$-0.434530\pi$
$859$	11.9716	0.408464	0.204232	+	0.978922i	$0.434530\pi$
$860$	−1.55496	−0.0530236
$861$	0	0
$862$	16.3157	0.555714
$863$	−22.9766	−0.782134	−0.391067	−	0.920362i	$-0.627894\pi$
$863$	−22.9766	−0.782134	−0.391067	+	0.920362i	$0.627894\pi$
$864$	0	0
$865$	−4.19048	−0.142481
$866$	10.1487	0.344867
$867$	0	0
$868$	30.2043	1.02520
$869$	−30.5489	−1.03630
$870$	0	0
$871$	58.1201	1.96933
$872$	48.0338	1.62663
$873$	0	0
$874$	10.7553	0.363804
$875$	4.17800	0.141242
$876$	0	0
$877$	19.8240	0.669410	0.334705	−	0.942323i	$-0.391363\pi$
$877$	19.8240	0.669410	0.334705	+	0.942323i	$0.391363\pi$
$878$	3.93796	0.132900
$879$	0	0
$880$	4.13393	0.139355
$881$	23.2616	0.783703	0.391852	−	0.920028i	$-0.371835\pi$
$881$	23.2616	0.783703	0.391852	+	0.920028i	$0.371835\pi$
$882$	0	0
$883$	0.416638	0.0140210	0.00701049	−	0.999975i	$-0.497768\pi$
$883$	0.416638	0.0140210	0.00701049	+	0.999975i	$0.497768\pi$
$884$	−17.5211	−0.589298
$885$	0	0
$886$	16.2126	0.544674
$887$	−0.146654	−0.00492417	−0.00246208	−	0.999997i	$-0.500784\pi$
$887$	−0.146654	−0.00492417	−0.00246208	+	0.999997i	$0.500784\pi$
$888$	0	0
$889$	17.8592	0.598979
$890$	1.39304	0.0466947
$891$	0	0
$892$	−11.5606	−0.387076
$893$	23.9078	0.800043
$894$	0	0
$895$	6.88559	0.230160
$896$	45.9524	1.53516
$897$	0	0
$898$	2.43277	0.0811826
$899$	−24.3903	−0.813463
$900$	0	0
$901$	13.9960	0.466273
$902$	−18.1721	−0.605065
$903$	0	0
$904$	−9.37642	−0.311855
$905$	−14.7262	−0.489516
$906$	0	0
$907$	48.5718	1.61280	0.806400	−	0.591371i	$-0.201413\pi$
$907$	48.5718	1.61280	0.806400	+	0.591371i	$0.201413\pi$
$908$	−26.9010	−0.892742
$909$	0	0
$910$	12.1523	0.402846
$911$	−46.1787	−1.52997	−0.764984	−	0.644049i	$-0.777253\pi$
$911$	−46.1787	−1.52997	−0.764984	+	0.644049i	$0.777253\pi$
$912$	0	0
$913$	−24.4293	−0.808493
$914$	−15.5274	−0.513600
$915$	0	0
$916$	−13.1327	−0.433918
$917$	1.26045	0.0416238
$918$	0	0
$919$	−25.9487	−0.855968	−0.427984	−	0.903786i	$-0.640776\pi$
$919$	−25.9487	−0.855968	−0.427984	+	0.903786i	$0.640776\pi$
$920$	−13.5044	−0.445225
$921$	0	0
$922$	−5.92089	−0.194994
$923$	−41.5281	−1.36691
$924$	0	0
$925$	1.95593	0.0643105
$926$	4.57881	0.150469
$927$	0	0
$928$	−30.2298	−0.992342
$929$	−43.5930	−1.43024	−0.715120	−	0.699002i	$-0.753628\pi$
$929$	−43.5930	−1.43024	−0.715120	+	0.699002i	$0.753628\pi$
$930$	0	0
$931$	−29.6030	−0.970198
$932$	−5.22999	−0.171314
$933$	0	0
$934$	14.2288	0.465581
$935$	6.99276	0.228688
$936$	0	0
$937$	−23.2125	−0.758318	−0.379159	−	0.925332i	$-0.623787\pi$
$937$	−23.2125	−0.758318	−0.379159	+	0.925332i	$0.623787\pi$
$938$	−37.1540	−1.21312
$939$	0	0
$940$	−13.1303	−0.428262
$941$	19.2543	0.627673	0.313837	−	0.949477i	$-0.398386\pi$
$941$	19.2543	0.627673	0.313837	+	0.949477i	$0.398386\pi$
$942$	0	0
$943$	−57.3253	−1.86677
$944$	4.64251	0.151101
$945$	0	0
$946$	1.80508	0.0586883
$947$	−2.25602	−0.0733109	−0.0366555	−	0.999328i	$-0.511670\pi$
$947$	−2.25602	−0.0733109	−0.0366555	+	0.999328i	$0.511670\pi$
$948$	0	0
$949$	44.2483	1.43636
$950$	−1.88879	−0.0612806
$951$	0	0
$952$	25.6069	0.829924
$953$	32.7387	1.06051	0.530256	−	0.847838i	$-0.322096\pi$
$953$	32.7387	1.06051	0.530256	+	0.847838i	$0.322096\pi$
$954$	0	0
$955$	3.96616	0.128342
$956$	32.2627	1.04345
$957$	0	0
$958$	12.8872	0.416365
$959$	75.1446	2.42654
$960$	0	0
$961$	−9.38454	−0.302727
$962$	5.68911	0.183424
$963$	0	0
$964$	−31.0507	−1.00008
$965$	−10.9742	−0.353272
$966$	0	0
$967$	−5.89512	−0.189574	−0.0947872	−	0.995498i	$-0.530217\pi$
$967$	−5.89512	−0.189574	−0.0947872	+	0.995498i	$0.530217\pi$
$968$	8.72416	0.280405
$969$	0	0
$970$	−9.69023	−0.311135
$971$	9.73071	0.312273	0.156137	−	0.987735i	$-0.450096\pi$
$971$	9.73071	0.312273	0.156137	+	0.987735i	$0.450096\pi$
$972$	0	0
$973$	49.5758	1.58933
$974$	0.703684	0.0225475
$975$	0	0
$976$	16.3781	0.524252
$977$	57.7248	1.84678	0.923390	−	0.383864i	$-0.125407\pi$
$977$	57.7248	1.84678	0.923390	+	0.383864i	$0.125407\pi$
$978$	0	0
$979$	5.65011	0.180578
$980$	16.2581	0.519346
$981$	0	0
$982$	−8.87470	−0.283203
$983$	−44.1150	−1.40705	−0.703526	−	0.710670i	$-0.748392\pi$
$983$	−44.1150	−1.40705	−0.703526	+	0.710670i	$0.748392\pi$
$984$	0	0
$985$	23.8477	0.759850
$986$	−9.04460	−0.288039
$987$	0	0
$988$	19.1952	0.610680
$989$	5.69427	0.181067
$990$	0	0
$991$	9.65909	0.306831	0.153416	−	0.988162i	$-0.450973\pi$
$991$	9.65909	0.306831	0.153416	+	0.988162i	$0.450973\pi$
$992$	26.7906	0.850602
$993$	0	0
$994$	26.5474	0.842032
$995$	3.09135	0.0980023
$996$	0	0
$997$	−8.30193	−0.262925	−0.131462	−	0.991321i	$-0.541967\pi$
$997$	−8.30193	−0.262925	−0.131462	+	0.991321i	$0.541967\pi$
$998$	11.1036	0.351479
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1935.2.a.u.1.4		5
3.2	odd	2		215.2.a.c.1.2	✓	5
5.4	even	2		9675.2.a.ch.1.2		5
12.11	even	2		3440.2.a.w.1.4		5
15.2	even	4		1075.2.b.h.474.4		10
15.8	even	4		1075.2.b.h.474.7		10
15.14	odd	2		1075.2.a.m.1.4		5
129.128	even	2		9245.2.a.l.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
215.2.a.c.1.2	✓	5	3.2	odd	2
1075.2.a.m.1.4		5	15.14	odd	2
1075.2.b.h.474.4		10	15.2	even	4
1075.2.b.h.474.7		10	15.8	even	4
1935.2.a.u.1.4		5	1.1	even	1	trivial
3440.2.a.w.1.4		5	12.11	even	2
9245.2.a.l.1.4		5	129.128	even	2
9675.2.a.ch.1.2		5	5.4	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:	\( N \)	\(=\)	\( 1935 = 3^{2} \cdot 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1935.a (trivial)

\(f(q)\)	\(=\)	\(q+0.667116 q^{2} -1.55496 q^{4} -1.00000 q^{5} -4.17800 q^{7} -2.37157 q^{8} +O(q^{10})\)	\(q+0.667116 q^{2} -1.55496 q^{4} -1.00000 q^{5} -4.17800 q^{7} -2.37157 q^{8} -0.667116 q^{10} -2.70580 q^{11} +4.36004 q^{13} -2.78721 q^{14} +1.52780 q^{16} +2.58436 q^{17} -2.83128 q^{19} +1.55496 q^{20} -1.80508 q^{22} -5.69427 q^{23} +1.00000 q^{25} +2.90865 q^{26} +6.49661 q^{28} -5.24609 q^{29} +4.64924 q^{31} +5.76236 q^{32} +1.72407 q^{34} +4.17800 q^{35} +1.95593 q^{37} -1.88879 q^{38} +2.37157 q^{40} +10.0672 q^{41} -1.00000 q^{43} +4.20740 q^{44} -3.79874 q^{46} -8.44414 q^{47} +10.4557 q^{49} +0.667116 q^{50} -6.77967 q^{52} +5.41564 q^{53} +2.70580 q^{55} +9.90841 q^{56} -3.49975 q^{58} +3.03868 q^{59} +10.7201 q^{61} +3.10158 q^{62} +0.788558 q^{64} -4.36004 q^{65} +13.3302 q^{67} -4.01856 q^{68} +2.78721 q^{70} -9.52471 q^{71} +10.1486 q^{73} +1.30483 q^{74} +4.40252 q^{76} +11.3048 q^{77} +11.2902 q^{79} -1.52780 q^{80} +6.71598 q^{82} +9.02850 q^{83} -2.58436 q^{85} -0.667116 q^{86} +6.41699 q^{88} -2.08815 q^{89} -18.2162 q^{91} +8.85434 q^{92} -5.63322 q^{94} +2.83128 q^{95} +14.5256 q^{97} +6.97515 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 8 q^{4} - 5 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) \) 5 * q - 2 * q^2 + 8 * q^4 - 5 * q^5 + 5 * q^7 - 3 * q^8	\( 5 q - 2 q^{2} + 8 q^{4} - 5 q^{5} + 5 q^{7} - 3 q^{8} + 2 q^{10} + 6 q^{11} + 5 q^{13} - q^{14} + 14 q^{16} + 17 q^{17} - 6 q^{19} - 8 q^{20} - 8 q^{22} - q^{23} + 5 q^{25} - 22 q^{26} + 26 q^{28} - 6 q^{29} + 6 q^{31} + 7 q^{32} - 5 q^{35} + 5 q^{37} + 16 q^{38} + 3 q^{40} - 2 q^{41} - 5 q^{43} + 15 q^{44} - 14 q^{46} + 18 q^{49} - 2 q^{50} - 38 q^{52} + 23 q^{53} - 6 q^{55} + 19 q^{56} + 12 q^{58} + q^{59} + 20 q^{61} + 3 q^{62} - 25 q^{64} - 5 q^{65} + 21 q^{67} + 48 q^{68} + q^{70} - 4 q^{71} + 5 q^{73} - 24 q^{74} + 32 q^{76} + 26 q^{77} + 41 q^{79} - 14 q^{80} + 38 q^{82} + 7 q^{83} - 17 q^{85} + 2 q^{86} + 12 q^{88} - 20 q^{89} - 42 q^{91} + 52 q^{92} - 42 q^{94} + 6 q^{95} + 37 q^{97} + 26 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 8 * q^4 - 5 * q^5 + 5 * q^7 - 3 * q^8 + 2 * q^10 + 6 * q^11 + 5 * q^13 - q^14 + 14 * q^16 + 17 * q^17 - 6 * q^19 - 8 * q^20 - 8 * q^22 - q^23 + 5 * q^25 - 22 * q^26 + 26 * q^28 - 6 * q^29 + 6 * q^31 + 7 * q^32 - 5 * q^35 + 5 * q^37 + 16 * q^38 + 3 * q^40 - 2 * q^41 - 5 * q^43 + 15 * q^44 - 14 * q^46 + 18 * q^49 - 2 * q^50 - 38 * q^52 + 23 * q^53 - 6 * q^55 + 19 * q^56 + 12 * q^58 + q^59 + 20 * q^61 + 3 * q^62 - 25 * q^64 - 5 * q^65 + 21 * q^67 + 48 * q^68 + q^70 - 4 * q^71 + 5 * q^73 - 24 * q^74 + 32 * q^76 + 26 * q^77 + 41 * q^79 - 14 * q^80 + 38 * q^82 + 7 * q^83 - 17 * q^85 + 2 * q^86 + 12 * q^88 - 20 * q^89 - 42 * q^91 + 52 * q^92 - 42 * q^94 + 6 * q^95 + 37 * q^97 + 26 * q^98

Introduction

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