LMFDB - Embedded newform 4235.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4235,2,Mod(1,4235)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4235, 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("4235.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4235 = 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4235.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:	$33.8166452560$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.32694400625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 16x^{6} + 44x^{5} + 83x^{4} - 200x^{3} - 132x^{2} + 288x - 16 $ x^8 - 3x^7 - 16x^6 + 44x^5 + 83x^4 - 200x^3 - 132x^2 + 288*x - 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 385)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.33308$ of defining polynomial
Character	$\chi$	$=$	4235.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.09529 q^{2} -0.0571808 q^{3} +2.39026 q^{4} -1.00000 q^{5} +0.119811 q^{6} -1.00000 q^{7} -0.817703 q^{8} -2.99673 q^{9} +O(q^{10})$	\(q-2.09529 q^{2} -0.0571808 q^{3} +2.39026 q^{4} -1.00000 q^{5} +0.119811 q^{6} -1.00000 q^{7} -0.817703 q^{8} -2.99673 q^{9} +2.09529 q^{10} -0.136677 q^{12} -4.09807 q^{13} +2.09529 q^{14} +0.0571808 q^{15} -3.06719 q^{16} +1.19575 q^{17} +6.27903 q^{18} +6.15525 q^{19} -2.39026 q^{20} +0.0571808 q^{21} -2.55413 q^{23} +0.0467569 q^{24} +1.00000 q^{25} +8.58666 q^{26} +0.342898 q^{27} -2.39026 q^{28} +1.25571 q^{29} -0.119811 q^{30} +0.611767 q^{31} +8.06206 q^{32} -2.50545 q^{34} +1.00000 q^{35} -7.16296 q^{36} -3.01605 q^{37} -12.8971 q^{38} +0.234331 q^{39} +0.817703 q^{40} +3.59120 q^{41} -0.119811 q^{42} +10.2533 q^{43} +2.99673 q^{45} +5.35165 q^{46} -1.09281 q^{47} +0.175384 q^{48} +1.00000 q^{49} -2.09529 q^{50} -0.0683741 q^{51} -9.79543 q^{52} +3.95416 q^{53} -0.718472 q^{54} +0.817703 q^{56} -0.351962 q^{57} -2.63109 q^{58} +4.13445 q^{59} +0.136677 q^{60} +8.83144 q^{61} -1.28183 q^{62} +2.99673 q^{63} -10.7580 q^{64} +4.09807 q^{65} +1.98712 q^{67} +2.85816 q^{68} +0.146047 q^{69} -2.09529 q^{70} -2.42917 q^{71} +2.45044 q^{72} -3.64516 q^{73} +6.31951 q^{74} -0.0571808 q^{75} +14.7126 q^{76} -0.490992 q^{78} -11.9768 q^{79} +3.06719 q^{80} +8.97058 q^{81} -7.52463 q^{82} +1.33053 q^{83} +0.136677 q^{84} -1.19575 q^{85} -21.4838 q^{86} -0.0718027 q^{87} -12.7564 q^{89} -6.27903 q^{90} +4.09807 q^{91} -6.10502 q^{92} -0.0349813 q^{93} +2.28976 q^{94} -6.15525 q^{95} -0.460995 q^{96} -13.3795 q^{97} -2.09529 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} - 3 q^{3} - 2 q^{4} - 8 q^{5} + 6 q^{6} - 8 q^{7} - 6 q^{8} + 17 q^{9}+O(q^{10}) $ 8 * q - 2 * q^2 - 3 * q^3 - 2 * q^4 - 8 * q^5 + 6 * q^6 - 8 * q^7 - 6 * q^8 + 17 * q^9	$ 8 q - 2 q^{2} - 3 q^{3} - 2 q^{4} - 8 q^{5} + 6 q^{6} - 8 q^{7} - 6 q^{8} + 17 q^{9} + 2 q^{10} - 10 q^{12} + 2 q^{14} + 3 q^{15} - 6 q^{16} - 7 q^{17} + 3 q^{18} + 19 q^{19} + 2 q^{20} + 3 q^{21} - 16 q^{23} + 5 q^{24} + 8 q^{25} + 15 q^{26} - 21 q^{27} + 2 q^{28} + 8 q^{29} - 6 q^{30} - 6 q^{31} + 16 q^{32} - 3 q^{34} + 8 q^{35} + 4 q^{36} - 26 q^{37} - 25 q^{38} - 11 q^{39} + 6 q^{40} + 9 q^{41} - 6 q^{42} - 4 q^{43} - 17 q^{45} - 20 q^{46} + 9 q^{47} + 16 q^{48} + 8 q^{49} - 2 q^{50} + 44 q^{51} - 5 q^{52} - 7 q^{53} + 21 q^{54} + 6 q^{56} - 36 q^{57} + 15 q^{58} + 20 q^{59} + 10 q^{60} + 13 q^{61} + 9 q^{62} - 17 q^{63} - 6 q^{64} - 21 q^{67} + q^{68} - 12 q^{69} - 2 q^{70} + 3 q^{71} - 38 q^{72} + 16 q^{73} - 2 q^{74} - 3 q^{75} + 11 q^{76} + 7 q^{78} - 14 q^{79} + 6 q^{80} + 12 q^{81} - 16 q^{82} + 4 q^{83} + 10 q^{84} + 7 q^{85} + 7 q^{86} + 8 q^{87} - 23 q^{89} - 3 q^{90} + 26 q^{92} - 67 q^{93} - 12 q^{94} - 19 q^{95} - 27 q^{96} - 38 q^{97} - 2 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 - 3 * q^3 - 2 * q^4 - 8 * q^5 + 6 * q^6 - 8 * q^7 - 6 * q^8 + 17 * q^9 + 2 * q^10 - 10 * q^12 + 2 * q^14 + 3 * q^15 - 6 * q^16 - 7 * q^17 + 3 * q^18 + 19 * q^19 + 2 * q^20 + 3 * q^21 - 16 * q^23 + 5 * q^24 + 8 * q^25 + 15 * q^26 - 21 * q^27 + 2 * q^28 + 8 * q^29 - 6 * q^30 - 6 * q^31 + 16 * q^32 - 3 * q^34 + 8 * q^35 + 4 * q^36 - 26 * q^37 - 25 * q^38 - 11 * q^39 + 6 * q^40 + 9 * q^41 - 6 * q^42 - 4 * q^43 - 17 * q^45 - 20 * q^46 + 9 * q^47 + 16 * q^48 + 8 * q^49 - 2 * q^50 + 44 * q^51 - 5 * q^52 - 7 * q^53 + 21 * q^54 + 6 * q^56 - 36 * q^57 + 15 * q^58 + 20 * q^59 + 10 * q^60 + 13 * q^61 + 9 * q^62 - 17 * q^63 - 6 * q^64 - 21 * q^67 + q^68 - 12 * q^69 - 2 * q^70 + 3 * q^71 - 38 * q^72 + 16 * q^73 - 2 * q^74 - 3 * q^75 + 11 * q^76 + 7 * q^78 - 14 * q^79 + 6 * q^80 + 12 * q^81 - 16 * q^82 + 4 * q^83 + 10 * q^84 + 7 * q^85 + 7 * q^86 + 8 * q^87 - 23 * q^89 - 3 * q^90 + 26 * q^92 - 67 * q^93 - 12 * q^94 - 19 * q^95 - 27 * q^96 - 38 * 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 $

$2$	−2.09529	−1.48160	−0.740798	−	0.671728i	$-0.765553\pi$
$2$	−2.09529	−1.48160	−0.740798	+	0.671728i	$0.765553\pi$
$3$	−0.0571808	−0.0330134	−0.0165067	−	0.999864i	$-0.505254\pi$
$3$	−0.0571808	−0.0330134	−0.0165067	+	0.999864i	$0.505254\pi$
$4$	2.39026	1.19513
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.119811	0.0489125
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−0.817703	−0.289102
$9$	−2.99673	−0.998910
$10$	2.09529	0.662590
$11$	0	0
$11$	0	0
$12$	−0.136677	−0.0394552
$13$	−4.09807	−1.13660	−0.568300	−	0.822822i	$-0.692399\pi$
$13$	−4.09807	−1.13660	−0.568300	+	0.822822i	$0.692399\pi$
$14$	2.09529	0.559991
$15$	0.0571808	0.0147640
$16$	−3.06719	−0.766796
$17$	1.19575	0.290013	0.145006	−	0.989431i	$-0.453680\pi$
$17$	1.19575	0.290013	0.145006	+	0.989431i	$0.453680\pi$
$18$	6.27903	1.47998
$19$	6.15525	1.41211	0.706055	−	0.708157i	$-0.250473\pi$
$19$	6.15525	1.41211	0.706055	+	0.708157i	$0.250473\pi$
$20$	−2.39026	−0.534478
$21$	0.0571808	0.0124779
$22$	0	0
$23$	−2.55413	−0.532572	−0.266286	−	0.963894i	$-0.585797\pi$
$23$	−2.55413	−0.532572	−0.266286	+	0.963894i	$0.585797\pi$
$24$	0.0467569	0.00954421
$25$	1.00000	0.200000
$26$	8.58666	1.68398
$27$	0.342898	0.0659907
$28$	−2.39026	−0.451716
$29$	1.25571	0.233180	0.116590	−	0.993180i	$-0.462804\pi$
$29$	1.25571	0.233180	0.116590	+	0.993180i	$0.462804\pi$
$30$	−0.119811	−0.0218743
$31$	0.611767	0.109877	0.0549383	−	0.998490i	$-0.482504\pi$
$31$	0.611767	0.109877	0.0549383	+	0.998490i	$0.482504\pi$
$32$	8.06206	1.42518
$33$	0	0
$34$	−2.50545	−0.429682
$35$	1.00000	0.169031
$36$	−7.16296	−1.19383
$37$	−3.01605	−0.495835	−0.247918	−	0.968781i	$-0.579746\pi$
$37$	−3.01605	−0.495835	−0.247918	+	0.968781i	$0.579746\pi$
$38$	−12.8971	−2.09218
$39$	0.234331	0.0375230
$40$	0.817703	0.129290
$41$	3.59120	0.560852	0.280426	−	0.959876i	$-0.409524\pi$
$41$	3.59120	0.560852	0.280426	+	0.959876i	$0.409524\pi$
$42$	−0.119811	−0.0184872
$43$	10.2533	1.56362	0.781810	−	0.623517i	$-0.214297\pi$
$43$	10.2533	1.56362	0.781810	+	0.623517i	$0.214297\pi$
$44$	0	0
$45$	2.99673	0.446726
$46$	5.35165	0.789057
$47$	−1.09281	−0.159403	−0.0797013	−	0.996819i	$-0.525397\pi$
$47$	−1.09281	−0.159403	−0.0797013	+	0.996819i	$0.525397\pi$
$48$	0.175384	0.0253145
$49$	1.00000	0.142857
$50$	−2.09529	−0.296319
$51$	−0.0683741	−0.00957429
$52$	−9.79543	−1.35838
$53$	3.95416	0.543146	0.271573	−	0.962418i	$-0.412456\pi$
$53$	3.95416	0.543146	0.271573	+	0.962418i	$0.412456\pi$
$54$	−0.718472	−0.0977716
$55$	0	0
$56$	0.817703	0.109270
$57$	−0.351962	−0.0466185
$58$	−2.63109	−0.345479
$59$	4.13445	0.538259	0.269130	−	0.963104i	$-0.413264\pi$
$59$	4.13445	0.538259	0.269130	+	0.963104i	$0.413264\pi$
$60$	0.136677	0.0176449
$61$	8.83144	1.13075	0.565375	−	0.824834i	$-0.308731\pi$
$61$	8.83144	1.13075	0.565375	+	0.824834i	$0.308731\pi$
$62$	−1.28183	−0.162793
$63$	2.99673	0.377553
$64$	−10.7580	−1.34475
$65$	4.09807	0.508303
$66$	0	0
$67$	1.98712	0.242766	0.121383	−	0.992606i	$-0.461267\pi$
$67$	1.98712	0.242766	0.121383	+	0.992606i	$0.461267\pi$
$68$	2.85816	0.346603
$69$	0.146047	0.0175820
$70$	−2.09529	−0.250436
$71$	−2.42917	−0.288289	−0.144144	−	0.989557i	$-0.546043\pi$
$71$	−2.42917	−0.288289	−0.144144	+	0.989557i	$0.546043\pi$
$72$	2.45044	0.288787
$73$	−3.64516	−0.426634	−0.213317	−	0.976983i	$-0.568427\pi$
$73$	−3.64516	−0.426634	−0.213317	+	0.976983i	$0.568427\pi$
$74$	6.31951	0.734628
$75$	−0.0571808	−0.00660267
$76$	14.7126	1.68765
$77$	0	0
$78$	−0.490992	−0.0555939
$79$	−11.9768	−1.34749	−0.673746	−	0.738963i	$-0.735316\pi$
$79$	−11.9768	−1.34749	−0.673746	+	0.738963i	$0.735316\pi$
$80$	3.06719	0.342922
$81$	8.97058	0.996732
$82$	−7.52463	−0.830956
$83$	1.33053	0.146045	0.0730224	−	0.997330i	$-0.476736\pi$
$83$	1.33053	0.146045	0.0730224	+	0.997330i	$0.476736\pi$
$84$	0.136677	0.0149127
$85$	−1.19575	−0.129698
$86$	−21.4838	−2.31665
$87$	−0.0718027	−0.00769806
$88$	0	0
$89$	−12.7564	−1.35218	−0.676088	−	0.736821i	$-0.736326\pi$
$89$	−12.7564	−1.35218	−0.676088	+	0.736821i	$0.736326\pi$
$90$	−6.27903	−0.661868
$91$	4.09807	0.429594
$92$	−6.10502	−0.636492
$93$	−0.0349813	−0.00362739
$94$	2.28976	0.236170
$95$	−6.15525	−0.631515
$96$	−0.460995	−0.0470501
$97$	−13.3795	−1.35849	−0.679244	−	0.733913i	$-0.737692\pi$
$97$	−13.3795	−1.35849	−0.679244	+	0.733913i	$0.737692\pi$
$98$	−2.09529	−0.211657
$99$	0	0
$100$	2.39026	0.239026
$101$	5.99553	0.596578	0.298289	−	0.954476i	$-0.403584\pi$
$101$	5.99553	0.596578	0.298289	+	0.954476i	$0.403584\pi$
$102$	0.143264	0.0141852
$103$	−17.9796	−1.77158	−0.885791	−	0.464085i	$-0.846383\pi$
$103$	−17.9796	−1.77158	−0.885791	+	0.464085i	$0.846383\pi$
$104$	3.35100	0.328593
$105$	−0.0571808	−0.00558028
$106$	−8.28513	−0.804723
$107$	−6.13240	−0.592842	−0.296421	−	0.955057i	$-0.595793\pi$
$107$	−6.13240	−0.592842	−0.296421	+	0.955057i	$0.595793\pi$
$108$	0.819614	0.0788674
$109$	15.9252	1.52536	0.762680	−	0.646776i	$-0.223883\pi$
$109$	15.9252	1.52536	0.762680	+	0.646776i	$0.223883\pi$
$110$	0	0
$111$	0.172460	0.0163692
$112$	3.06719	0.289822
$113$	6.56475	0.617560	0.308780	−	0.951133i	$-0.400079\pi$
$113$	6.56475	0.617560	0.308780	+	0.951133i	$0.400079\pi$
$114$	0.737464	0.0690698
$115$	2.55413	0.238174
$116$	3.00148	0.278680
$117$	12.2808	1.13536
$118$	−8.66288	−0.797483
$119$	−1.19575	−0.109615
$120$	−0.0467569	−0.00426830
$121$	0	0
$122$	−18.5045	−1.67532
$123$	−0.205348	−0.0185156
$124$	1.46228	0.131317
$125$	−1.00000	−0.0894427
$126$	−6.27903	−0.559381
$127$	−15.3579	−1.36279	−0.681397	−	0.731914i	$-0.738627\pi$
$127$	−15.3579	−1.36279	−0.681397	+	0.731914i	$0.738627\pi$
$128$	6.41709	0.567196
$129$	−0.586294	−0.0516203
$130$	−8.58666	−0.753100
$131$	14.9630	1.30733	0.653663	−	0.756785i	$-0.273231\pi$
$131$	14.9630	1.30733	0.653663	+	0.756785i	$0.273231\pi$
$132$	0	0
$133$	−6.15525	−0.533728
$134$	−4.16361	−0.359681
$135$	−0.342898	−0.0295119
$136$	−0.977771	−0.0838432
$137$	12.8092	1.09436	0.547182	−	0.837014i	$-0.315700\pi$
$137$	12.8092	1.09436	0.547182	+	0.837014i	$0.315700\pi$
$138$	−0.306012	−0.0260494
$139$	12.5186	1.06182	0.530909	−	0.847429i	$-0.321851\pi$
$139$	12.5186	1.06182	0.530909	+	0.847429i	$0.321851\pi$
$140$	2.39026	0.202014
$141$	0.0624877	0.00526241
$142$	5.08981	0.427128
$143$	0	0
$144$	9.19153	0.765961
$145$	−1.25571	−0.104281
$146$	7.63768	0.632099
$147$	−0.0571808	−0.00471619
$148$	−7.20913	−0.592587
$149$	20.8086	1.70471	0.852355	−	0.522964i	$-0.175174\pi$
$149$	20.8086	1.70471	0.852355	+	0.522964i	$0.175174\pi$
$150$	0.119811	0.00978249
$151$	−22.3397	−1.81798	−0.908991	−	0.416815i	$-0.863146\pi$
$151$	−22.3397	−1.81798	−0.908991	+	0.416815i	$0.863146\pi$
$152$	−5.03316	−0.408244
$153$	−3.58335	−0.289697
$154$	0	0
$155$	−0.611767	−0.0491383
$156$	0.560111	0.0448448
$157$	10.0373	0.801063	0.400531	−	0.916283i	$-0.368826\pi$
$157$	10.0373	0.801063	0.400531	+	0.916283i	$0.368826\pi$
$158$	25.0949	1.99644
$159$	−0.226102	−0.0179311
$160$	−8.06206	−0.637362
$161$	2.55413	0.201293
$162$	−18.7960	−1.47675
$163$	8.09554	0.634092	0.317046	−	0.948410i	$-0.397309\pi$
$163$	8.09554	0.634092	0.317046	+	0.948410i	$0.397309\pi$
$164$	8.58390	0.670290
$165$	0	0
$166$	−2.78785	−0.216379
$167$	−21.4208	−1.65759	−0.828797	−	0.559549i	$-0.810974\pi$
$167$	−21.4208	−1.65759	−0.828797	+	0.559549i	$0.810974\pi$
$168$	−0.0467569	−0.00360737
$169$	3.79416	0.291858
$170$	2.50545	0.192160
$171$	−18.4456	−1.41057
$172$	24.5081	1.86873
$173$	−10.4487	−0.794400	−0.397200	−	0.917732i	$-0.630018\pi$
$173$	−10.4487	−0.794400	−0.397200	+	0.917732i	$0.630018\pi$
$174$	0.150448	0.0114054
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−0.236411	−0.0177697
$178$	26.7284	2.00338
$179$	−17.2407	−1.28863	−0.644316	−	0.764760i	$-0.722858\pi$
$179$	−17.2407	−1.28863	−0.644316	+	0.764760i	$0.722858\pi$
$180$	7.16296	0.533895
$181$	−25.3929	−1.88744	−0.943718	−	0.330751i	$-0.892698\pi$
$181$	−25.3929	−1.88744	−0.943718	+	0.330751i	$0.892698\pi$
$182$	−8.58666	−0.636485
$183$	−0.504989	−0.0373298
$184$	2.08852	0.153968
$185$	3.01605	0.221744
$186$	0.0732962	0.00537434
$187$	0	0
$188$	−2.61209	−0.190507
$189$	−0.342898	−0.0249421
$190$	12.8971	0.935651
$191$	13.4631	0.974153	0.487076	−	0.873359i	$-0.338063\pi$
$191$	13.4631	0.974153	0.487076	+	0.873359i	$0.338063\pi$
$192$	0.615152	0.0443948
$193$	9.25577	0.666245	0.333123	−	0.942884i	$-0.391898\pi$
$193$	9.25577	0.666245	0.333123	+	0.942884i	$0.391898\pi$
$194$	28.0341	2.01273
$195$	−0.234331	−0.0167808
$196$	2.39026	0.170733
$197$	−20.5858	−1.46668	−0.733340	−	0.679862i	$-0.762040\pi$
$197$	−20.5858	−1.46668	−0.733340	+	0.679862i	$0.762040\pi$
$198$	0	0
$199$	−16.5271	−1.17158	−0.585788	−	0.810464i	$-0.699215\pi$
$199$	−16.5271	−1.17158	−0.585788	+	0.810464i	$0.699215\pi$
$200$	−0.817703	−0.0578203
$201$	−0.113625	−0.00801452
$202$	−12.5624	−0.883887
$203$	−1.25571	−0.0881338
$204$	−0.163432	−0.0114425
$205$	−3.59120	−0.250820
$206$	37.6725	2.62477
$207$	7.65403	0.531992
$208$	12.5695	0.871540
$209$	0	0
$210$	0.119811	0.00826772
$211$	4.49150	0.309207	0.154604	−	0.987977i	$-0.450590\pi$
$211$	4.49150	0.309207	0.154604	+	0.987977i	$0.450590\pi$
$212$	9.45146	0.649129
$213$	0.138902	0.00951738
$214$	12.8492	0.878352
$215$	−10.2533	−0.699272
$216$	−0.280389	−0.0190780
$217$	−0.611767	−0.0415294
$218$	−33.3680	−2.25997
$219$	0.208433	0.0140846
$220$	0	0
$221$	−4.90028	−0.329628
$222$	−0.361355	−0.0242525
$223$	−2.29800	−0.153886	−0.0769428	−	0.997036i	$-0.524516\pi$
$223$	−2.29800	−0.153886	−0.0769428	+	0.997036i	$0.524516\pi$
$224$	−8.06206	−0.538669
$225$	−2.99673	−0.199782
$226$	−13.7551	−0.914975
$227$	13.0833	0.868370	0.434185	−	0.900824i	$-0.357036\pi$
$227$	13.0833	0.868370	0.434185	+	0.900824i	$0.357036\pi$
$228$	−0.841280	−0.0557151
$229$	13.2359	0.874650	0.437325	−	0.899304i	$-0.355926\pi$
$229$	13.2359	0.874650	0.437325	+	0.899304i	$0.355926\pi$
$230$	−5.35165	−0.352877
$231$	0	0
$232$	−1.02680	−0.0674127
$233$	−0.464562	−0.0304345	−0.0152172	−	0.999884i	$-0.504844\pi$
$233$	−0.464562	−0.0304345	−0.0152172	+	0.999884i	$0.504844\pi$
$234$	−25.7319	−1.68215
$235$	1.09281	0.0712870
$236$	9.88239	0.643289
$237$	0.684841	0.0444852
$238$	2.50545	0.162405
$239$	20.6602	1.33640	0.668199	−	0.743982i	$-0.267065\pi$
$239$	20.6602	1.33640	0.668199	+	0.743982i	$0.267065\pi$
$240$	−0.175384	−0.0113210
$241$	−11.6253	−0.748851	−0.374425	−	0.927257i	$-0.622160\pi$
$241$	−11.6253	−0.748851	−0.374425	+	0.927257i	$0.622160\pi$
$242$	0	0
$243$	−1.54164	−0.0988962
$244$	21.1094	1.35139
$245$	−1.00000	−0.0638877
$246$	0.430264	0.0274326
$247$	−25.2246	−1.60500
$248$	−0.500244	−0.0317655
$249$	−0.0760809	−0.00482143
$250$	2.09529	0.132518
$251$	13.0328	0.822621	0.411310	−	0.911495i	$-0.365071\pi$
$251$	13.0328	0.822621	0.411310	+	0.911495i	$0.365071\pi$
$252$	7.16296	0.451224
$253$	0	0
$254$	32.1793	2.01911
$255$	0.0683741	0.00428175
$256$	8.07035	0.504397
$257$	−19.7366	−1.23114	−0.615569	−	0.788083i	$-0.711074\pi$
$257$	−19.7366	−1.23114	−0.615569	+	0.788083i	$0.711074\pi$
$258$	1.22846	0.0764805
$259$	3.01605	0.187408
$260$	9.79543	0.607487
$261$	−3.76303	−0.232926
$262$	−31.3520	−1.93693
$263$	12.7113	0.783814	0.391907	−	0.920005i	$-0.371816\pi$
$263$	12.7113	0.783814	0.391907	+	0.920005i	$0.371816\pi$
$264$	0	0
$265$	−3.95416	−0.242902
$266$	12.8971	0.790769
$267$	0.729421	0.0446398
$268$	4.74974	0.290136
$269$	26.8953	1.63984	0.819919	−	0.572479i	$-0.194018\pi$
$269$	26.8953	1.63984	0.819919	+	0.572479i	$0.194018\pi$
$270$	0.718472	0.0437248
$271$	−11.5469	−0.701424	−0.350712	−	0.936483i	$-0.614060\pi$
$271$	−11.5469	−0.701424	−0.350712	+	0.936483i	$0.614060\pi$
$272$	−3.66760	−0.222381
$273$	−0.234331	−0.0141823
$274$	−26.8391	−1.62141
$275$	0	0
$276$	0.349090	0.0210127
$277$	−27.3877	−1.64557	−0.822785	−	0.568353i	$-0.807581\pi$
$277$	−27.3877	−1.64557	−0.822785	+	0.568353i	$0.807581\pi$
$278$	−26.2302	−1.57318
$279$	−1.83330	−0.109757
$280$	−0.817703	−0.0488671
$281$	−2.12720	−0.126898	−0.0634492	−	0.997985i	$-0.520210\pi$
$281$	−2.12720	−0.126898	−0.0634492	+	0.997985i	$0.520210\pi$
$282$	−0.130930	−0.00779677
$283$	3.85764	0.229313	0.114656	−	0.993405i	$-0.463423\pi$
$283$	3.85764	0.229313	0.114656	+	0.993405i	$0.463423\pi$
$284$	−5.80633	−0.344542
$285$	0.351962	0.0208484
$286$	0	0
$287$	−3.59120	−0.211982
$288$	−24.1598	−1.42363
$289$	−15.5702	−0.915893
$290$	2.63109	0.154503
$291$	0.765053	0.0448482
$292$	−8.71287	−0.509882
$293$	−14.0288	−0.819571	−0.409785	−	0.912182i	$-0.634396\pi$
$293$	−14.0288	−0.819571	−0.409785	+	0.912182i	$0.634396\pi$
$294$	0.119811	0.00698750
$295$	−4.13445	−0.240717
$296$	2.46623	0.143347
$297$	0	0
$298$	−43.6002	−2.52569
$299$	10.4670	0.605321
$300$	−0.136677	−0.00789104
$301$	−10.2533	−0.590993
$302$	46.8083	2.69352
$303$	−0.342829	−0.0196950
$304$	−18.8793	−1.08280
$305$	−8.83144	−0.505687
$306$	7.50817	0.429214
$307$	−21.7805	−1.24308	−0.621540	−	0.783382i	$-0.713493\pi$
$307$	−21.7805	−1.24308	−0.621540	+	0.783382i	$0.713493\pi$
$308$	0	0
$309$	1.02809	0.0584858
$310$	1.28183	0.0728031
$311$	26.9041	1.52559	0.762795	−	0.646640i	$-0.223826\pi$
$311$	26.9041	1.52559	0.762795	+	0.646640i	$0.223826\pi$
$312$	−0.191613	−0.0108479
$313$	31.0141	1.75302	0.876509	−	0.481385i	$-0.159866\pi$
$313$	31.0141	1.75302	0.876509	+	0.481385i	$0.159866\pi$
$314$	−21.0311	−1.18685
$315$	−2.99673	−0.168847
$316$	−28.6276	−1.61043
$317$	8.47948	0.476255	0.238128	−	0.971234i	$-0.423466\pi$
$317$	8.47948	0.476255	0.238128	+	0.971234i	$0.423466\pi$
$318$	0.473750	0.0265666
$319$	0	0
$320$	10.7580	0.601391
$321$	0.350656	0.0195717
$322$	−5.35165	−0.298236
$323$	7.36016	0.409530
$324$	21.4420	1.19122
$325$	−4.09807	−0.227320
$326$	−16.9625	−0.939468
$327$	−0.910618	−0.0503573
$328$	−2.93654	−0.162143
$329$	1.09281	0.0602485
$330$	0	0
$331$	1.38456	0.0761022	0.0380511	−	0.999276i	$-0.487885\pi$
$331$	1.38456	0.0761022	0.0380511	+	0.999276i	$0.487885\pi$
$332$	3.18031	0.174542
$333$	9.03828	0.495295
$334$	44.8830	2.45589
$335$	−1.98712	−0.108568
$336$	−0.175384	−0.00956799
$337$	−0.215014	−0.0117126	−0.00585628	−	0.999983i	$-0.501864\pi$
$337$	−0.215014	−0.0117126	−0.00585628	+	0.999983i	$0.501864\pi$
$338$	−7.94987	−0.432416
$339$	−0.375378	−0.0203877
$340$	−2.85816	−0.155005
$341$	0	0
$342$	38.6490	2.08990
$343$	−1.00000	−0.0539949
$344$	−8.38419	−0.452045
$345$	−0.146047	−0.00786291
$346$	21.8931	1.17698
$347$	−30.4731	−1.63588	−0.817942	−	0.575301i	$-0.804885\pi$
$347$	−30.4731	−1.63588	−0.817942	+	0.575301i	$0.804885\pi$
$348$	−0.171627	−0.00920017
$349$	17.3784	0.930242	0.465121	−	0.885247i	$-0.346011\pi$
$349$	17.3784	0.930242	0.465121	+	0.885247i	$0.346011\pi$
$350$	2.09529	0.111998
$351$	−1.40522	−0.0750050
$352$	0	0
$353$	−22.8429	−1.21581	−0.607903	−	0.794011i	$-0.707989\pi$
$353$	−22.8429	−1.21581	−0.607903	+	0.794011i	$0.707989\pi$
$354$	0.495351	0.0263276
$355$	2.42917	0.128927
$356$	−30.4911	−1.61602
$357$	0.0683741	0.00361874
$358$	36.1244	1.90923
$359$	−6.35088	−0.335187	−0.167593	−	0.985856i	$-0.553600\pi$
$359$	−6.35088	−0.335187	−0.167593	+	0.985856i	$0.553600\pi$
$360$	−2.45044	−0.129149
$361$	18.8871	0.994057
$362$	53.2055	2.79642
$363$	0	0
$364$	9.79543	0.513420
$365$	3.64516	0.190796
$366$	1.05810	0.0553078
$367$	−25.3139	−1.32138	−0.660688	−	0.750660i	$-0.729736\pi$
$367$	−25.3139	−1.32138	−0.660688	+	0.750660i	$0.729736\pi$
$368$	7.83398	0.408375
$369$	−10.7619	−0.560240
$370$	−6.31951	−0.328536
$371$	−3.95416	−0.205290
$372$	−0.0836144	−0.00433520
$373$	−34.5066	−1.78668	−0.893341	−	0.449380i	$-0.851645\pi$
$373$	−34.5066	−1.78668	−0.893341	+	0.449380i	$0.851645\pi$
$374$	0	0
$375$	0.0571808	0.00295280
$376$	0.893593	0.0460835
$377$	−5.14600	−0.265032
$378$	0.718472	0.0369542
$379$	−36.5817	−1.87908	−0.939539	−	0.342443i	$-0.888746\pi$
$379$	−36.5817	−1.87908	−0.939539	+	0.342443i	$0.888746\pi$
$380$	−14.7126	−0.754742
$381$	0.878177	0.0449904
$382$	−28.2091	−1.44330
$383$	14.8520	0.758899	0.379450	−	0.925212i	$-0.376113\pi$
$383$	14.8520	0.758899	0.379450	+	0.925212i	$0.376113\pi$
$384$	−0.366934	−0.0187250
$385$	0	0
$386$	−19.3936	−0.987106
$387$	−30.7265	−1.56192
$388$	−31.9806	−1.62357
$389$	11.5137	0.583769	0.291884	−	0.956454i	$-0.405718\pi$
$389$	11.5137	0.583769	0.291884	+	0.956454i	$0.405718\pi$
$390$	0.490992	0.0248623
$391$	−3.05411	−0.154453
$392$	−0.817703	−0.0413002
$393$	−0.855599	−0.0431592
$394$	43.1334	2.17303
$395$	11.9768	0.602617
$396$	0	0
$397$	−27.7213	−1.39129	−0.695647	−	0.718384i	$-0.744882\pi$
$397$	−27.7213	−1.39129	−0.695647	+	0.718384i	$0.744882\pi$
$398$	34.6292	1.73580
$399$	0.351962	0.0176201
$400$	−3.06719	−0.153359
$401$	10.3818	0.518440	0.259220	−	0.965818i	$-0.416534\pi$
$401$	10.3818	0.518440	0.259220	+	0.965818i	$0.416534\pi$
$402$	0.238079	0.0118743
$403$	−2.50706	−0.124886
$404$	14.3309	0.712987
$405$	−8.97058	−0.445752
$406$	2.63109	0.130579
$407$	0	0
$408$	0.0559097	0.00276794
$409$	5.61194	0.277493	0.138746	−	0.990328i	$-0.455693\pi$
$409$	5.61194	0.277493	0.138746	+	0.990328i	$0.455693\pi$
$410$	7.52463	0.371615
$411$	−0.732441	−0.0361286
$412$	−42.9758	−2.11727
$413$	−4.13445	−0.203443
$414$	−16.0374	−0.788197
$415$	−1.33053	−0.0653132
$416$	−33.0389	−1.61986
$417$	−0.715826	−0.0350541
$418$	0	0
$419$	−14.6484	−0.715623	−0.357811	−	0.933794i	$-0.616477\pi$
$419$	−14.6484	−0.715623	−0.357811	+	0.933794i	$0.616477\pi$
$420$	−0.136677	−0.00666915
$421$	−17.0589	−0.831399	−0.415700	−	0.909502i	$-0.636463\pi$
$421$	−17.0589	−0.831399	−0.415700	+	0.909502i	$0.636463\pi$
$422$	−9.41100	−0.458121
$423$	3.27485	0.159229
$424$	−3.23333	−0.157024
$425$	1.19575	0.0580026
$426$	−0.291040	−0.0141009
$427$	−8.83144	−0.427383
$428$	−14.6580	−0.708522
$429$	0	0
$430$	21.4838	1.03604
$431$	1.07742	0.0518973	0.0259487	−	0.999663i	$-0.491739\pi$
$431$	1.07742	0.0518973	0.0259487	+	0.999663i	$0.491739\pi$
$432$	−1.05173	−0.0506015
$433$	−21.6590	−1.04086	−0.520432	−	0.853903i	$-0.674229\pi$
$433$	−21.6590	−1.04086	−0.520432	+	0.853903i	$0.674229\pi$
$434$	1.28183	0.0615299
$435$	0.0718027	0.00344268
$436$	38.0654	1.82300
$437$	−15.7213	−0.752051
$438$	−0.436729	−0.0208677
$439$	−20.2299	−0.965518	−0.482759	−	0.875753i	$-0.660365\pi$
$439$	−20.2299	−0.965518	−0.482759	+	0.875753i	$0.660365\pi$
$440$	0	0
$441$	−2.99673	−0.142701
$442$	10.2675	0.488376
$443$	−36.4763	−1.73304	−0.866520	−	0.499142i	$-0.833649\pi$
$443$	−36.4763	−1.73304	−0.866520	+	0.499142i	$0.833649\pi$
$444$	0.412224	0.0195633
$445$	12.7564	0.604711
$446$	4.81499	0.227996
$447$	−1.18985	−0.0562782
$448$	10.7580	0.508269
$449$	0.533526	0.0251786	0.0125893	−	0.999921i	$-0.495993\pi$
$449$	0.533526	0.0251786	0.0125893	+	0.999921i	$0.495993\pi$
$450$	6.27903	0.295996
$451$	0	0
$452$	15.6914	0.738063
$453$	1.27740	0.0600177
$454$	−27.4134	−1.28657
$455$	−4.09807	−0.192120
$456$	0.287800	0.0134775
$457$	3.34237	0.156349	0.0781747	−	0.996940i	$-0.475091\pi$
$457$	3.34237	0.156349	0.0781747	+	0.996940i	$0.475091\pi$
$458$	−27.7330	−1.29588
$459$	0.410021	0.0191382
$460$	6.10502	0.284648
$461$	−15.0967	−0.703123	−0.351562	−	0.936165i	$-0.614349\pi$
$461$	−15.0967	−0.703123	−0.351562	+	0.936165i	$0.614349\pi$
$462$	0	0
$463$	−14.4740	−0.672664	−0.336332	−	0.941743i	$-0.609187\pi$
$463$	−14.4740	−0.672664	−0.336332	+	0.941743i	$0.609187\pi$
$464$	−3.85151	−0.178802
$465$	0.0349813	0.00162222
$466$	0.973394	0.0450916
$467$	17.1799	0.794989	0.397494	−	0.917605i	$-0.369880\pi$
$467$	17.1799	0.794989	0.397494	+	0.917605i	$0.369880\pi$
$468$	29.3543	1.35690
$469$	−1.98712	−0.0917569
$470$	−2.28976	−0.105619
$471$	−0.573940	−0.0264458
$472$	−3.38075	−0.155612
$473$	0	0
$474$	−1.43494	−0.0659092
$475$	6.15525	0.282422
$476$	−2.85816	−0.131003
$477$	−11.8495	−0.542554
$478$	−43.2892	−1.98000
$479$	−20.9735	−0.958303	−0.479152	−	0.877732i	$-0.659056\pi$
$479$	−20.9735	−0.958303	−0.479152	+	0.877732i	$0.659056\pi$
$480$	0.460995	0.0210415
$481$	12.3600	0.563566
$482$	24.3584	1.10950
$483$	−0.146047	−0.00664537
$484$	0	0
$485$	13.3795	0.607534
$486$	3.23019	0.146524
$487$	−22.3835	−1.01429	−0.507147	−	0.861859i	$-0.669300\pi$
$487$	−22.3835	−1.01429	−0.507147	+	0.861859i	$0.669300\pi$
$488$	−7.22149	−0.326902
$489$	−0.462910	−0.0209335
$490$	2.09529	0.0946557
$491$	−42.1327	−1.90142	−0.950711	−	0.310078i	$-0.899645\pi$
$491$	−42.1327	−1.90142	−0.950711	+	0.310078i	$0.899645\pi$
$492$	−0.490834	−0.0221285
$493$	1.50152	0.0676252
$494$	52.8530	2.37797
$495$	0	0
$496$	−1.87640	−0.0842530
$497$	2.42917	0.108963
$498$	0.159412	0.00714341
$499$	25.7695	1.15360	0.576800	−	0.816886i	$-0.304301\pi$
$499$	25.7695	1.15360	0.576800	+	0.816886i	$0.304301\pi$
$500$	−2.39026	−0.106896
$501$	1.22486	0.0547227
$502$	−27.3075	−1.21879
$503$	36.2190	1.61493	0.807464	−	0.589917i	$-0.200840\pi$
$503$	36.2190	1.61493	0.807464	+	0.589917i	$0.200840\pi$
$504$	−2.45044	−0.109151
$505$	−5.99553	−0.266798
$506$	0	0
$507$	−0.216953	−0.00963522
$508$	−36.7093	−1.62871
$509$	1.63941	0.0726655	0.0363328	−	0.999340i	$-0.488432\pi$
$509$	1.63941	0.0726655	0.0363328	+	0.999340i	$0.488432\pi$
$510$	−0.143264	−0.00634383
$511$	3.64516	0.161252
$512$	−29.7439	−1.31451
$513$	2.11062	0.0931862
$514$	41.3541	1.82405
$515$	17.9796	0.792275
$516$	−1.40139	−0.0616929
$517$	0	0
$518$	−6.31951	−0.277663
$519$	0.597465	0.0262258
$520$	−3.35100	−0.146951
$521$	8.80650	0.385820	0.192910	−	0.981216i	$-0.438208\pi$
$521$	8.80650	0.385820	0.192910	+	0.981216i	$0.438208\pi$
$522$	7.88466	0.345102
$523$	19.6121	0.857576	0.428788	−	0.903405i	$-0.358941\pi$
$523$	19.6121	0.857576	0.428788	+	0.903405i	$0.358941\pi$
$524$	35.7655	1.56242
$525$	0.0571808	0.00249557
$526$	−26.6340	−1.16130
$527$	0.731522	0.0318656
$528$	0	0
$529$	−16.4764	−0.716367
$530$	8.28513	0.359883
$531$	−12.3898	−0.537673
$532$	−14.7126	−0.637873
$533$	−14.7170	−0.637464
$534$	−1.52835	−0.0661382
$535$	6.13240	0.265127
$536$	−1.62488	−0.0701840
$537$	0.985838	0.0425420
$538$	−56.3537	−2.42958
$539$	0	0
$540$	−0.819614	−0.0352706
$541$	−38.9849	−1.67609	−0.838046	−	0.545599i	$-0.816302\pi$
$541$	−38.9849	−1.67609	−0.838046	+	0.545599i	$0.816302\pi$
$542$	24.1942	1.03923
$543$	1.45198	0.0623106
$544$	9.64024	0.413322
$545$	−15.9252	−0.682162
$546$	0.490992	0.0210125
$547$	−21.1669	−0.905032	−0.452516	−	0.891756i	$-0.649473\pi$
$547$	−21.1669	−0.905032	−0.452516	+	0.891756i	$0.649473\pi$
$548$	30.6173	1.30791
$549$	−26.4654	−1.12952
$550$	0	0
$551$	7.72923	0.329276
$552$	−0.119423	−0.00508298
$553$	11.9768	0.509304
$554$	57.3854	2.43807
$555$	−0.172460	−0.00732052
$556$	29.9228	1.26901
$557$	−42.0751	−1.78278	−0.891389	−	0.453240i	$-0.850268\pi$
$557$	−42.0751	−1.78278	−0.891389	+	0.453240i	$0.850268\pi$
$558$	3.84130	0.162615
$559$	−42.0189	−1.77721
$560$	−3.06719	−0.129612
$561$	0	0
$562$	4.45712	0.188012
$563$	29.6421	1.24927	0.624633	−	0.780918i	$-0.285249\pi$
$563$	29.6421	1.24927	0.624633	+	0.780918i	$0.285249\pi$
$564$	0.149362	0.00628926
$565$	−6.56475	−0.276181
$566$	−8.08288	−0.339749
$567$	−8.97058	−0.376729
$568$	1.98634	0.0833448
$569$	9.49107	0.397886	0.198943	−	0.980011i	$-0.436249\pi$
$569$	9.49107	0.397886	0.198943	+	0.980011i	$0.436249\pi$
$570$	−0.737464	−0.0308890
$571$	18.8966	0.790799	0.395400	−	0.918509i	$-0.370606\pi$
$571$	18.8966	0.790799	0.395400	+	0.918509i	$0.370606\pi$
$572$	0	0
$573$	−0.769828	−0.0321600
$574$	7.52463	0.314072
$575$	−2.55413	−0.106514
$576$	32.2389	1.34329
$577$	−42.4498	−1.76721	−0.883604	−	0.468235i	$-0.844890\pi$
$577$	−42.4498	−1.76721	−0.883604	+	0.468235i	$0.844890\pi$
$578$	32.6241	1.35698
$579$	−0.529252	−0.0219950
$580$	−3.00148	−0.124630
$581$	−1.33053	−0.0551997
$582$	−1.60301	−0.0664470
$583$	0	0
$584$	2.98066	0.123341
$585$	−12.2808	−0.507749
$586$	29.3944	1.21427
$587$	−31.1458	−1.28552	−0.642762	−	0.766066i	$-0.722212\pi$
$587$	−31.1458	−1.28552	−0.642762	+	0.766066i	$0.722212\pi$
$588$	−0.136677	−0.00563646
$589$	3.76558	0.155158
$590$	8.66288	0.356645
$591$	1.17711	0.0484200
$592$	9.25078	0.380205
$593$	27.2374	1.11851	0.559253	−	0.828997i	$-0.311088\pi$
$593$	27.2374	1.11851	0.559253	+	0.828997i	$0.311088\pi$
$594$	0	0
$595$	1.19575	0.0490211
$596$	49.7380	2.03735
$597$	0.945033	0.0386776
$598$	−21.9314	−0.896842
$599$	43.7670	1.78827	0.894136	−	0.447796i	$-0.147791\pi$
$599$	43.7670	1.78827	0.894136	+	0.447796i	$0.147791\pi$
$600$	0.0467569	0.00190884
$601$	3.25149	0.132631	0.0663154	−	0.997799i	$-0.478876\pi$
$601$	3.25149	0.132631	0.0663154	+	0.997799i	$0.478876\pi$
$602$	21.4838	0.875613
$603$	−5.95488	−0.242501
$604$	−53.3977	−2.17272
$605$	0	0
$606$	0.718328	0.0291801
$607$	−36.4653	−1.48008	−0.740041	−	0.672562i	$-0.765194\pi$
$607$	−36.4653	−1.48008	−0.740041	+	0.672562i	$0.765194\pi$
$608$	49.6240	2.01252
$609$	0.0718027	0.00290959
$610$	18.5045	0.749224
$611$	4.47841	0.181177
$612$	−8.56513	−0.346225
$613$	21.7753	0.879497	0.439749	−	0.898121i	$-0.355068\pi$
$613$	21.7753	0.879497	0.439749	+	0.898121i	$0.355068\pi$
$614$	45.6366	1.84174
$615$	0.205348	0.00828042
$616$	0	0
$617$	1.58269	0.0637167	0.0318583	−	0.999492i	$-0.489857\pi$
$617$	1.58269	0.0637167	0.0318583	+	0.999492i	$0.489857\pi$
$618$	−2.15414	−0.0866524
$619$	−36.1941	−1.45476	−0.727382	−	0.686233i	$-0.759263\pi$
$619$	−36.1941	−1.45476	−0.727382	+	0.686233i	$0.759263\pi$
$620$	−1.46228	−0.0587266
$621$	−0.875805	−0.0351448
$622$	−56.3720	−2.26031
$623$	12.7564	0.511074
$624$	−0.718736	−0.0287725
$625$	1.00000	0.0400000
$626$	−64.9836	−2.59727
$627$	0	0
$628$	23.9917	0.957373
$629$	−3.60645	−0.143799
$630$	6.27903	0.250163
$631$	0.442379	0.0176108	0.00880542	−	0.999961i	$-0.497197\pi$
$631$	0.442379	0.0176108	0.00880542	+	0.999961i	$0.497197\pi$
$632$	9.79344	0.389562
$633$	−0.256827	−0.0102080
$634$	−17.7670	−0.705618
$635$	15.3579	0.609460
$636$	−0.540442	−0.0214299
$637$	−4.09807	−0.162371
$638$	0	0
$639$	7.27955	0.287975
$640$	−6.41709	−0.253658
$641$	−21.1225	−0.834287	−0.417144	−	0.908841i	$-0.636969\pi$
$641$	−21.1225	−0.834287	−0.417144	+	0.908841i	$0.636969\pi$
$642$	−0.734727	−0.0289974
$643$	2.52865	0.0997201	0.0498600	−	0.998756i	$-0.484122\pi$
$643$	2.52865	0.0997201	0.0498600	+	0.998756i	$0.484122\pi$
$644$	6.10502	0.240572
$645$	0.586294	0.0230853
$646$	−15.4217	−0.606759
$647$	−21.9134	−0.861506	−0.430753	−	0.902470i	$-0.641752\pi$
$647$	−21.9134	−0.861506	−0.430753	+	0.902470i	$0.641752\pi$
$648$	−7.33527	−0.288157
$649$	0	0
$650$	8.58666	0.336796
$651$	0.0349813	0.00137103
$652$	19.3504	0.757821
$653$	−1.40725	−0.0550699	−0.0275349	−	0.999621i	$-0.508766\pi$
$653$	−1.40725	−0.0550699	−0.0275349	+	0.999621i	$0.508766\pi$
$654$	1.90801	0.0746092
$655$	−14.9630	−0.584654
$656$	−11.0149	−0.430059
$657$	10.9236	0.426169
$658$	−2.28976	−0.0892640
$659$	−1.87549	−0.0730589	−0.0365295	−	0.999333i	$-0.511630\pi$
$659$	−1.87549	−0.0730589	−0.0365295	+	0.999333i	$0.511630\pi$
$660$	0	0
$661$	21.1558	0.822867	0.411433	−	0.911440i	$-0.365028\pi$
$661$	21.1558	0.822867	0.411433	+	0.911440i	$0.365028\pi$
$662$	−2.90106	−0.112753
$663$	0.280202	0.0108821
$664$	−1.08798	−0.0422218
$665$	6.15525	0.238690
$666$	−18.9379	−0.733827
$667$	−3.20725	−0.124185
$668$	−51.2013	−1.98104
$669$	0.131402	0.00508028
$670$	4.16361	0.160854
$671$	0	0
$672$	0.460995	0.0177833
$673$	35.6387	1.37377	0.686885	−	0.726766i	$-0.258978\pi$
$673$	35.6387	1.37377	0.686885	+	0.726766i	$0.258978\pi$
$674$	0.450517	0.0173533
$675$	0.342898	0.0131981
$676$	9.06901	0.348808
$677$	−13.9366	−0.535626	−0.267813	−	0.963471i	$-0.586301\pi$
$677$	−13.9366	−0.535626	−0.267813	+	0.963471i	$0.586301\pi$
$678$	0.786527	0.0302064
$679$	13.3795	0.513460
$680$	0.977771	0.0374958
$681$	−0.748115	−0.0286678
$682$	0	0
$683$	7.18083	0.274767	0.137383	−	0.990518i	$-0.456131\pi$
$683$	7.18083	0.274767	0.137383	+	0.990518i	$0.456131\pi$
$684$	−44.0898	−1.68581
$685$	−12.8092	−0.489415
$686$	2.09529	0.0799987
$687$	−0.756837	−0.0288751
$688$	−31.4489	−1.19898
$689$	−16.2044	−0.617339
$690$	0.306012	0.0116497
$691$	11.9246	0.453633	0.226816	−	0.973938i	$-0.427168\pi$
$691$	11.9246	0.453633	0.226816	+	0.973938i	$0.427168\pi$
$692$	−24.9751	−0.949410
$693$	0	0
$694$	63.8502	2.42372
$695$	−12.5186	−0.474859
$696$	0.0587133	0.00222552
$697$	4.29419	0.162654
$698$	−36.4128	−1.37824
$699$	0.0265640	0.00100474
$700$	−2.39026	−0.0903432
$701$	−45.0819	−1.70272	−0.851359	−	0.524583i	$-0.824221\pi$
$701$	−45.0819	−1.70272	−0.851359	+	0.524583i	$0.824221\pi$
$702$	2.94435	0.111127
$703$	−18.5645	−0.700174
$704$	0	0
$705$	−0.0624877	−0.00235342
$706$	47.8626	1.80133
$707$	−5.99553	−0.225485
$708$	−0.565083	−0.0212371
$709$	−37.1782	−1.39626	−0.698128	−	0.715973i	$-0.745983\pi$
$709$	−37.1782	−1.39626	−0.698128	+	0.715973i	$0.745983\pi$
$710$	−5.08981	−0.191017
$711$	35.8911	1.34602
$712$	10.4309	0.390916
$713$	−1.56253	−0.0585172
$714$	−0.143264	−0.00536152
$715$	0	0
$716$	−41.2097	−1.54008
$717$	−1.18137	−0.0441190
$718$	13.3070	0.496611
$719$	−15.7912	−0.588913	−0.294456	−	0.955665i	$-0.595139\pi$
$719$	−15.7912	−0.588913	−0.294456	+	0.955665i	$0.595139\pi$
$720$	−9.19153	−0.342548
$721$	17.9796	0.669595
$722$	−39.5740	−1.47279
$723$	0.664744	0.0247221
$724$	−60.6954	−2.25573
$725$	1.25571	0.0466360
$726$	0	0
$727$	36.1942	1.34237	0.671184	−	0.741291i	$-0.265786\pi$
$727$	36.1942	1.34237	0.671184	+	0.741291i	$0.265786\pi$
$728$	−3.35100	−0.124196
$729$	−26.8236	−0.993467
$730$	−7.63768	−0.282683
$731$	12.2605	0.453470
$732$	−1.20705	−0.0446140
$733$	3.86848	0.142886	0.0714429	−	0.997445i	$-0.477240\pi$
$733$	3.86848	0.142886	0.0714429	+	0.997445i	$0.477240\pi$
$734$	53.0401	1.95775
$735$	0.0571808	0.00210915
$736$	−20.5915	−0.759014
$737$	0	0
$738$	22.5493	0.830050
$739$	−29.0560	−1.06884	−0.534421	−	0.845218i	$-0.679470\pi$
$739$	−29.0560	−1.06884	−0.534421	+	0.845218i	$0.679470\pi$
$740$	7.20913	0.265013
$741$	1.44236	0.0529866
$742$	8.28513	0.304157
$743$	−7.25968	−0.266332	−0.133166	−	0.991094i	$-0.542514\pi$
$743$	−7.25968	−0.266332	−0.133166	+	0.991094i	$0.542514\pi$
$744$	0.0286043	0.00104869
$745$	−20.8086	−0.762369
$746$	72.3014	2.64714
$747$	−3.98724	−0.145886
$748$	0	0
$749$	6.13240	0.224073
$750$	−0.119811	−0.00437486
$751$	−30.7067	−1.12050	−0.560252	−	0.828322i	$-0.689296\pi$
$751$	−30.7067	−1.12050	−0.560252	+	0.828322i	$0.689296\pi$
$752$	3.35185	0.122229
$753$	−0.745224	−0.0271575
$754$	10.7824	0.392671
$755$	22.3397	0.813026
$756$	−0.819614	−0.0298091
$757$	5.54009	0.201358	0.100679	−	0.994919i	$-0.467898\pi$
$757$	5.54009	0.201358	0.100679	+	0.994919i	$0.467898\pi$
$758$	76.6495	2.78403
$759$	0	0
$760$	5.03316	0.182572
$761$	−51.1014	−1.85243	−0.926213	−	0.377001i	$-0.876955\pi$
$761$	−51.1014	−1.85243	−0.926213	+	0.377001i	$0.876955\pi$
$762$	−1.84004	−0.0666576
$763$	−15.9252	−0.576532
$764$	32.1802	1.16424
$765$	3.58335	0.129556
$766$	−31.1192	−1.12438
$767$	−16.9432	−0.611785
$768$	−0.461469	−0.0166518
$769$	42.3505	1.52720	0.763599	−	0.645690i	$-0.223430\pi$
$769$	42.3505	1.52720	0.763599	+	0.645690i	$0.223430\pi$
$770$	0	0
$771$	1.12856	0.0406440
$772$	22.1237	0.796248
$773$	0.269977	0.00971039	0.00485520	−	0.999988i	$-0.498455\pi$
$773$	0.269977	0.00971039	0.00485520	+	0.999988i	$0.498455\pi$
$774$	64.3811	2.31413
$775$	0.611767	0.0219753
$776$	10.9405	0.392741
$777$	−0.172460	−0.00618697
$778$	−24.1246	−0.864910
$779$	22.1047	0.791985
$780$	−0.560111	−0.0200552
$781$	0	0
$782$	6.39925	0.228837
$783$	0.430581	0.0153877
$784$	−3.06719	−0.109542
$785$	−10.0373	−0.358246
$786$	1.79273	0.0639446
$787$	51.3718	1.83121	0.915603	−	0.402082i	$-0.131713\pi$
$787$	51.3718	1.83121	0.915603	+	0.402082i	$0.131713\pi$
$788$	−49.2054	−1.75287
$789$	−0.726844	−0.0258763
$790$	−25.0949	−0.892835
$791$	−6.56475	−0.233416
$792$	0	0
$793$	−36.1918	−1.28521
$794$	58.0843	2.06134
$795$	0.226102	0.00801901
$796$	−39.5040	−1.40018
$797$	−10.0279	−0.355205	−0.177603	−	0.984102i	$-0.556834\pi$
$797$	−10.0279	−0.355205	−0.177603	+	0.984102i	$0.556834\pi$
$798$	−0.737464	−0.0261059
$799$	−1.30673	−0.0462288
$800$	8.06206	0.285037
$801$	38.2275	1.35070
$802$	−21.7528	−0.768120
$803$	0	0
$804$	−0.271594	−0.00957838
$805$	−2.55413	−0.0900212
$806$	5.25303	0.185030
$807$	−1.53790	−0.0541366
$808$	−4.90256	−0.172472
$809$	25.4809	0.895860	0.447930	−	0.894069i	$-0.352161\pi$
$809$	25.4809	0.895860	0.447930	+	0.894069i	$0.352161\pi$
$810$	18.7960	0.660424
$811$	6.76423	0.237524	0.118762	−	0.992923i	$-0.462107\pi$
$811$	6.76423	0.237524	0.118762	+	0.992923i	$0.462107\pi$
$812$	−3.00148	−0.105331
$813$	0.660261	0.0231564
$814$	0	0
$815$	−8.09554	−0.283575
$816$	0.209716	0.00734154
$817$	63.1119	2.20800
$818$	−11.7587	−0.411132
$819$	−12.2808	−0.429126
$820$	−8.58390	−0.299763
$821$	7.66524	0.267519	0.133759	−	0.991014i	$-0.457295\pi$
$821$	7.66524	0.267519	0.133759	+	0.991014i	$0.457295\pi$
$822$	1.53468	0.0535281
$823$	42.5565	1.48343	0.741713	−	0.670718i	$-0.234014\pi$
$823$	42.5565	1.48343	0.741713	+	0.670718i	$0.234014\pi$
$824$	14.7020	0.512167
$825$	0	0
$826$	8.66288	0.301420
$827$	16.4858	0.573268	0.286634	−	0.958040i	$-0.407464\pi$
$827$	16.4858	0.573268	0.286634	+	0.958040i	$0.407464\pi$
$828$	18.2951	0.635799
$829$	−52.9934	−1.84054	−0.920269	−	0.391287i	$-0.872030\pi$
$829$	−52.9934	−1.84054	−0.920269	+	0.391287i	$0.872030\pi$
$830$	2.78785	0.0967678
$831$	1.56605	0.0543258
$832$	44.0871	1.52844
$833$	1.19575	0.0414304
$834$	1.49987	0.0519361
$835$	21.4208	0.741299
$836$	0	0
$837$	0.209774	0.00725084
$838$	30.6928	1.06026
$839$	35.7084	1.23279	0.616396	−	0.787436i	$-0.288592\pi$
$839$	35.7084	1.23279	0.616396	+	0.787436i	$0.288592\pi$
$840$	0.0467569	0.00161327
$841$	−27.4232	−0.945627
$842$	35.7434	1.23180
$843$	0.121635	0.00418934
$844$	10.7358	0.369542
$845$	−3.79416	−0.130523
$846$	−6.86178	−0.235913
$847$	0	0
$848$	−12.1281	−0.416482
$849$	−0.220583	−0.00757038
$850$	−2.50545	−0.0859364
$851$	7.70337	0.264068
$852$	0.332011	0.0113745
$853$	26.0070	0.890462	0.445231	−	0.895416i	$-0.353121\pi$
$853$	26.0070	0.890462	0.445231	+	0.895416i	$0.353121\pi$
$854$	18.5045	0.633210
$855$	18.4456	0.630827
$856$	5.01448	0.171392
$857$	38.2872	1.30786	0.653932	−	0.756553i	$-0.273118\pi$
$857$	38.2872	1.30786	0.653932	+	0.756553i	$0.273118\pi$
$858$	0	0
$859$	45.8563	1.56460	0.782298	−	0.622904i	$-0.214047\pi$
$859$	45.8563	1.56460	0.782298	+	0.622904i	$0.214047\pi$
$860$	−24.5081	−0.835720
$861$	0.205348	0.00699824
$862$	−2.25750	−0.0768909
$863$	8.90417	0.303102	0.151551	−	0.988449i	$-0.451573\pi$
$863$	8.90417	0.303102	0.151551	+	0.988449i	$0.451573\pi$
$864$	2.76446	0.0940490
$865$	10.4487	0.355266
$866$	45.3819	1.54214
$867$	0.890315	0.0302367
$868$	−1.46228	−0.0496330
$869$	0	0
$870$	−0.150448	−0.00510066
$871$	−8.14337	−0.275928
$872$	−13.0221	−0.440984
$873$	40.0949	1.35701
$874$	32.9407	1.11424
$875$	1.00000	0.0338062
$876$	0.498209	0.0168329
$877$	28.7347	0.970303	0.485152	−	0.874430i	$-0.338764\pi$
$877$	28.7347	0.970303	0.485152	+	0.874430i	$0.338764\pi$
$878$	42.3875	1.43051
$879$	0.802177	0.0270568
$880$	0	0
$881$	1.30963	0.0441226	0.0220613	−	0.999757i	$-0.492977\pi$
$881$	1.30963	0.0441226	0.0220613	+	0.999757i	$0.492977\pi$
$882$	6.27903	0.211426
$883$	−18.0455	−0.607281	−0.303640	−	0.952787i	$-0.598202\pi$
$883$	−18.0455	−0.607281	−0.303640	+	0.952787i	$0.598202\pi$
$884$	−11.7129	−0.393948
$885$	0.236411	0.00794687
$886$	76.4285	2.56767
$887$	28.0919	0.943233	0.471616	−	0.881804i	$-0.343671\pi$
$887$	28.0919	0.943233	0.471616	+	0.881804i	$0.343671\pi$
$888$	−0.141021	−0.00473236
$889$	15.3579	0.515087
$890$	−26.7284	−0.895938
$891$	0	0
$892$	−5.49281	−0.183913
$893$	−6.72651	−0.225094
$894$	2.49309	0.0833815
$895$	17.2407	0.576294
$896$	−6.41709	−0.214380
$897$	−0.598511	−0.0199837
$898$	−1.11789	−0.0373046
$899$	0.768204	0.0256210
$900$	−7.16296	−0.238765
$901$	4.72820	0.157519
$902$	0	0
$903$	0.586294	0.0195107
$904$	−5.36802	−0.178538
$905$	25.3929	0.844087
$906$	−2.67654	−0.0889220
$907$	3.83246	0.127255	0.0636275	−	0.997974i	$-0.479733\pi$
$907$	3.83246	0.127255	0.0636275	+	0.997974i	$0.479733\pi$
$908$	31.2725	1.03781
$909$	−17.9670	−0.595927
$910$	8.58666	0.284645
$911$	−17.7616	−0.588467	−0.294233	−	0.955734i	$-0.595064\pi$
$911$	−17.7616	−0.588467	−0.294233	+	0.955734i	$0.595064\pi$
$912$	1.07953	0.0357469
$913$	0	0
$914$	−7.00325	−0.231647
$915$	0.504989	0.0166944
$916$	31.6371	1.04532
$917$	−14.9630	−0.494123
$918$	−0.859115	−0.0283550
$919$	−8.06010	−0.265878	−0.132939	−	0.991124i	$-0.542441\pi$
$919$	−8.06010	−0.265878	−0.132939	+	0.991124i	$0.542441\pi$
$920$	−2.08852	−0.0688564
$921$	1.24543	0.0410383
$922$	31.6320	1.04174
$923$	9.95488	0.327669
$924$	0	0
$925$	−3.01605	−0.0991671
$926$	30.3273	0.996617
$927$	53.8800	1.76965
$928$	10.1236	0.332325
$929$	48.4546	1.58974	0.794872	−	0.606777i	$-0.207538\pi$
$929$	48.4546	1.58974	0.794872	+	0.606777i	$0.207538\pi$
$930$	−0.0732962	−0.00240348
$931$	6.15525	0.201730
$932$	−1.11042	−0.0363731
$933$	−1.53840	−0.0503649
$934$	−35.9968	−1.17785
$935$	0	0
$936$	−10.0420	−0.328235
$937$	36.4421	1.19051	0.595256	−	0.803536i	$-0.297051\pi$
$937$	36.4421	1.19051	0.595256	+	0.803536i	$0.297051\pi$
$938$	4.16361	0.135947
$939$	−1.77341	−0.0578730
$940$	2.61209	0.0851971
$941$	50.3800	1.64234	0.821171	−	0.570682i	$-0.193321\pi$
$941$	50.3800	1.64234	0.821171	+	0.570682i	$0.193321\pi$
$942$	1.20257	0.0391820
$943$	−9.17239	−0.298694
$944$	−12.6811	−0.412735
$945$	0.342898	0.0111545
$946$	0	0
$947$	1.09209	0.0354882	0.0177441	−	0.999843i	$-0.494352\pi$
$947$	1.09209	0.0354882	0.0177441	+	0.999843i	$0.494352\pi$
$948$	1.63695	0.0531656
$949$	14.9381	0.484912
$950$	−12.8971	−0.418436
$951$	−0.484863	−0.0157228
$952$	0.977771	0.0316897
$953$	−7.72204	−0.250141	−0.125071	−	0.992148i	$-0.539916\pi$
$953$	−7.72204	−0.250141	−0.125071	+	0.992148i	$0.539916\pi$
$954$	24.8283	0.803846
$955$	−13.4631	−0.435654
$956$	49.3832	1.59717
$957$	0	0
$958$	43.9456	1.41982
$959$	−12.8092	−0.413631
$960$	−0.615152	−0.0198539
$961$	−30.6257	−0.987927
$962$	−25.8978	−0.834978
$963$	18.3772	0.592196
$964$	−27.7874	−0.894973
$965$	−9.25577	−0.297954
$966$	0.306012	0.00984576
$967$	−24.3719	−0.783749	−0.391874	−	0.920019i	$-0.628173\pi$
$967$	−24.3719	−0.783749	−0.391874	+	0.920019i	$0.628173\pi$
$968$	0	0
$969$	−0.420860	−0.0135200
$970$	−28.0341	−0.900120
$971$	−27.3538	−0.877825	−0.438913	−	0.898530i	$-0.644636\pi$
$971$	−27.3538	−0.877825	−0.438913	+	0.898530i	$0.644636\pi$
$972$	−3.68491	−0.118194
$973$	−12.5186	−0.401329
$974$	46.9001	1.50278
$975$	0.234331	0.00750459
$976$	−27.0877	−0.867055
$977$	5.28607	0.169116	0.0845581	−	0.996419i	$-0.473052\pi$
$977$	5.28607	0.169116	0.0845581	+	0.996419i	$0.473052\pi$
$978$	0.969932	0.0310150
$979$	0	0
$980$	−2.39026	−0.0763540
$981$	−47.7236	−1.52370
$982$	88.2804	2.81714
$983$	−24.8110	−0.791350	−0.395675	−	0.918391i	$-0.629489\pi$
$983$	−24.8110	−0.791350	−0.395675	+	0.918391i	$0.629489\pi$
$984$	0.167914	0.00535289
$985$	20.5858	0.655919
$986$	−3.14613	−0.100193
$987$	−0.0624877	−0.00198901
$988$	−60.2933	−1.91819
$989$	−26.1883	−0.832741
$990$	0	0
$991$	−4.91568	−0.156152	−0.0780758	−	0.996947i	$-0.524878\pi$
$991$	−4.91568	−0.156152	−0.0780758	+	0.996947i	$0.524878\pi$
$992$	4.93210	0.156594
$993$	−0.0791701	−0.00251239
$994$	−5.08981	−0.161439
$995$	16.5271	0.523945
$996$	−0.181853	−0.00576223
$997$	47.3119	1.49838	0.749191	−	0.662354i	$-0.230443\pi$
$997$	47.3119	1.49838	0.749191	+	0.662354i	$0.230443\pi$
$998$	−53.9946	−1.70917
$999$	−1.03420	−0.0327205

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4235.2.a.bg.1.2		8
11.3	even	5		385.2.n.d.141.1	yes	16
11.4	even	5		385.2.n.d.71.1	✓	16
11.10	odd	2		4235.2.a.bh.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.2.n.d.71.1	✓	16	11.4	even	5
385.2.n.d.141.1	yes	16	11.3	even	5
4235.2.a.bg.1.2		8	1.1	even	1	trivial
4235.2.a.bh.1.8		8	11.10	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:	\( N \)	\(=\)	\( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4235.a (trivial)

\(f(q)\)	\(=\)	\(q-2.09529 q^{2} -0.0571808 q^{3} +2.39026 q^{4} -1.00000 q^{5} +0.119811 q^{6} -1.00000 q^{7} -0.817703 q^{8} -2.99673 q^{9} +O(q^{10})\)	\(q-2.09529 q^{2} -0.0571808 q^{3} +2.39026 q^{4} -1.00000 q^{5} +0.119811 q^{6} -1.00000 q^{7} -0.817703 q^{8} -2.99673 q^{9} +2.09529 q^{10} -0.136677 q^{12} -4.09807 q^{13} +2.09529 q^{14} +0.0571808 q^{15} -3.06719 q^{16} +1.19575 q^{17} +6.27903 q^{18} +6.15525 q^{19} -2.39026 q^{20} +0.0571808 q^{21} -2.55413 q^{23} +0.0467569 q^{24} +1.00000 q^{25} +8.58666 q^{26} +0.342898 q^{27} -2.39026 q^{28} +1.25571 q^{29} -0.119811 q^{30} +0.611767 q^{31} +8.06206 q^{32} -2.50545 q^{34} +1.00000 q^{35} -7.16296 q^{36} -3.01605 q^{37} -12.8971 q^{38} +0.234331 q^{39} +0.817703 q^{40} +3.59120 q^{41} -0.119811 q^{42} +10.2533 q^{43} +2.99673 q^{45} +5.35165 q^{46} -1.09281 q^{47} +0.175384 q^{48} +1.00000 q^{49} -2.09529 q^{50} -0.0683741 q^{51} -9.79543 q^{52} +3.95416 q^{53} -0.718472 q^{54} +0.817703 q^{56} -0.351962 q^{57} -2.63109 q^{58} +4.13445 q^{59} +0.136677 q^{60} +8.83144 q^{61} -1.28183 q^{62} +2.99673 q^{63} -10.7580 q^{64} +4.09807 q^{65} +1.98712 q^{67} +2.85816 q^{68} +0.146047 q^{69} -2.09529 q^{70} -2.42917 q^{71} +2.45044 q^{72} -3.64516 q^{73} +6.31951 q^{74} -0.0571808 q^{75} +14.7126 q^{76} -0.490992 q^{78} -11.9768 q^{79} +3.06719 q^{80} +8.97058 q^{81} -7.52463 q^{82} +1.33053 q^{83} +0.136677 q^{84} -1.19575 q^{85} -21.4838 q^{86} -0.0718027 q^{87} -12.7564 q^{89} -6.27903 q^{90} +4.09807 q^{91} -6.10502 q^{92} -0.0349813 q^{93} +2.28976 q^{94} -6.15525 q^{95} -0.460995 q^{96} -13.3795 q^{97} -2.09529 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} - 3 q^{3} - 2 q^{4} - 8 q^{5} + 6 q^{6} - 8 q^{7} - 6 q^{8} + 17 q^{9}+O(q^{10}) \) 8 * q - 2 * q^2 - 3 * q^3 - 2 * q^4 - 8 * q^5 + 6 * q^6 - 8 * q^7 - 6 * q^8 + 17 * q^9	\( 8 q - 2 q^{2} - 3 q^{3} - 2 q^{4} - 8 q^{5} + 6 q^{6} - 8 q^{7} - 6 q^{8} + 17 q^{9} + 2 q^{10} - 10 q^{12} + 2 q^{14} + 3 q^{15} - 6 q^{16} - 7 q^{17} + 3 q^{18} + 19 q^{19} + 2 q^{20} + 3 q^{21} - 16 q^{23} + 5 q^{24} + 8 q^{25} + 15 q^{26} - 21 q^{27} + 2 q^{28} + 8 q^{29} - 6 q^{30} - 6 q^{31} + 16 q^{32} - 3 q^{34} + 8 q^{35} + 4 q^{36} - 26 q^{37} - 25 q^{38} - 11 q^{39} + 6 q^{40} + 9 q^{41} - 6 q^{42} - 4 q^{43} - 17 q^{45} - 20 q^{46} + 9 q^{47} + 16 q^{48} + 8 q^{49} - 2 q^{50} + 44 q^{51} - 5 q^{52} - 7 q^{53} + 21 q^{54} + 6 q^{56} - 36 q^{57} + 15 q^{58} + 20 q^{59} + 10 q^{60} + 13 q^{61} + 9 q^{62} - 17 q^{63} - 6 q^{64} - 21 q^{67} + q^{68} - 12 q^{69} - 2 q^{70} + 3 q^{71} - 38 q^{72} + 16 q^{73} - 2 q^{74} - 3 q^{75} + 11 q^{76} + 7 q^{78} - 14 q^{79} + 6 q^{80} + 12 q^{81} - 16 q^{82} + 4 q^{83} + 10 q^{84} + 7 q^{85} + 7 q^{86} + 8 q^{87} - 23 q^{89} - 3 q^{90} + 26 q^{92} - 67 q^{93} - 12 q^{94} - 19 q^{95} - 27 q^{96} - 38 q^{97} - 2 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 - 3 * q^3 - 2 * q^4 - 8 * q^5 + 6 * q^6 - 8 * q^7 - 6 * q^8 + 17 * q^9 + 2 * q^10 - 10 * q^12 + 2 * q^14 + 3 * q^15 - 6 * q^16 - 7 * q^17 + 3 * q^18 + 19 * q^19 + 2 * q^20 + 3 * q^21 - 16 * q^23 + 5 * q^24 + 8 * q^25 + 15 * q^26 - 21 * q^27 + 2 * q^28 + 8 * q^29 - 6 * q^30 - 6 * q^31 + 16 * q^32 - 3 * q^34 + 8 * q^35 + 4 * q^36 - 26 * q^37 - 25 * q^38 - 11 * q^39 + 6 * q^40 + 9 * q^41 - 6 * q^42 - 4 * q^43 - 17 * q^45 - 20 * q^46 + 9 * q^47 + 16 * q^48 + 8 * q^49 - 2 * q^50 + 44 * q^51 - 5 * q^52 - 7 * q^53 + 21 * q^54 + 6 * q^56 - 36 * q^57 + 15 * q^58 + 20 * q^59 + 10 * q^60 + 13 * q^61 + 9 * q^62 - 17 * q^63 - 6 * q^64 - 21 * q^67 + q^68 - 12 * q^69 - 2 * q^70 + 3 * q^71 - 38 * q^72 + 16 * q^73 - 2 * q^74 - 3 * q^75 + 11 * q^76 + 7 * q^78 - 14 * q^79 + 6 * q^80 + 12 * q^81 - 16 * q^82 + 4 * q^83 + 10 * q^84 + 7 * q^85 + 7 * q^86 + 8 * q^87 - 23 * q^89 - 3 * q^90 + 26 * q^92 - 67 * q^93 - 12 * q^94 - 19 * q^95 - 27 * q^96 - 38 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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