LMFDB - Embedded newform 8039.2.a.b.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8039,2,Mod(1,8039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8039 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8039.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:	$64.1917381849$
Analytic rank:	$0$
Dimension:	$391$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61507 q^{2} +3.14612 q^{3} +4.83861 q^{4} +2.05578 q^{5} -8.22734 q^{6} +4.83721 q^{7} -7.42319 q^{8} +6.89808 q^{9} +O(q^{10})$	\(q-2.61507 q^{2} +3.14612 q^{3} +4.83861 q^{4} +2.05578 q^{5} -8.22734 q^{6} +4.83721 q^{7} -7.42319 q^{8} +6.89808 q^{9} -5.37602 q^{10} -0.0800961 q^{11} +15.2229 q^{12} +4.01691 q^{13} -12.6497 q^{14} +6.46773 q^{15} +9.73496 q^{16} -1.27961 q^{17} -18.0390 q^{18} +6.18088 q^{19} +9.94713 q^{20} +15.2184 q^{21} +0.209457 q^{22} +1.67230 q^{23} -23.3542 q^{24} -0.773766 q^{25} -10.5045 q^{26} +12.2638 q^{27} +23.4054 q^{28} -9.73987 q^{29} -16.9136 q^{30} -8.86776 q^{31} -10.6113 q^{32} -0.251992 q^{33} +3.34629 q^{34} +9.94423 q^{35} +33.3771 q^{36} -1.37542 q^{37} -16.1635 q^{38} +12.6377 q^{39} -15.2604 q^{40} -6.30073 q^{41} -39.7973 q^{42} -6.45319 q^{43} -0.387554 q^{44} +14.1809 q^{45} -4.37320 q^{46} +11.2331 q^{47} +30.6274 q^{48} +16.3986 q^{49} +2.02346 q^{50} -4.02582 q^{51} +19.4363 q^{52} +5.89701 q^{53} -32.0708 q^{54} -0.164660 q^{55} -35.9075 q^{56} +19.4458 q^{57} +25.4705 q^{58} -14.3207 q^{59} +31.2949 q^{60} +4.69434 q^{61} +23.1898 q^{62} +33.3674 q^{63} +8.27934 q^{64} +8.25789 q^{65} +0.658978 q^{66} -0.410139 q^{67} -6.19156 q^{68} +5.26127 q^{69} -26.0049 q^{70} -12.8106 q^{71} -51.2057 q^{72} -6.02222 q^{73} +3.59682 q^{74} -2.43436 q^{75} +29.9069 q^{76} -0.387441 q^{77} -33.0485 q^{78} +1.45674 q^{79} +20.0129 q^{80} +17.8892 q^{81} +16.4769 q^{82} +17.2639 q^{83} +73.6361 q^{84} -2.63061 q^{85} +16.8756 q^{86} -30.6428 q^{87} +0.594568 q^{88} -7.02767 q^{89} -37.0842 q^{90} +19.4306 q^{91} +8.09163 q^{92} -27.8990 q^{93} -29.3754 q^{94} +12.7065 q^{95} -33.3843 q^{96} +5.16987 q^{97} -42.8835 q^{98} -0.552509 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9}+O(q^{10}) $ 391 * q + 14 * q^2 + 12 * q^3 + 446 * q^4 + 22 * q^5 + 40 * q^6 + 63 * q^7 + 36 * q^8 + 501 * q^9	\( 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9} + 40 q^{10} + 57 q^{11} + 20 q^{12} + 83 q^{13} + 21 q^{14} + 60 q^{15} + 548 q^{16} + 59 q^{17} + 54 q^{18} + 131 q^{19} + 35 q^{20} + 121 q^{21} + 89 q^{22} + 34 q^{23} + 110 q^{24} + 609 q^{25} + 54 q^{26} + 27 q^{27} + 182 q^{28} + 102 q^{29} + 92 q^{30} + 88 q^{31} + 76 q^{32} + 131 q^{33} + 128 q^{34} + 31 q^{35} + 654 q^{36} + 135 q^{37} + 23 q^{38} + 96 q^{39} + 113 q^{40} + 128 q^{41} + 45 q^{42} + 140 q^{43} + 151 q^{44} + 77 q^{45} + 245 q^{46} + 22 q^{47} + 25 q^{48} + 712 q^{49} + 53 q^{50} + 102 q^{51} + 174 q^{52} + 54 q^{53} + 131 q^{54} + 101 q^{55} + 43 q^{56} + 226 q^{57} + 109 q^{58} + 40 q^{59} + 123 q^{60} + 249 q^{61} + 28 q^{62} + 139 q^{63} + 730 q^{64} + 227 q^{65} + 55 q^{66} + 169 q^{67} + 48 q^{68} + 89 q^{69} + 98 q^{70} + 66 q^{71} + 120 q^{72} + 324 q^{73} + 60 q^{74} + 19 q^{75} + 356 q^{76} + 83 q^{77} - 11 q^{78} + 195 q^{79} + 26 q^{80} + 807 q^{81} + 49 q^{82} + 74 q^{83} + 252 q^{84} + 373 q^{85} + 100 q^{86} + 43 q^{87} + 211 q^{88} + 207 q^{89} + 10 q^{90} + 189 q^{91} + 30 q^{92} + 172 q^{93} + 130 q^{94} + 43 q^{95} + 203 q^{96} + 254 q^{97} + 26 q^{98} + 273 q^{99}+O(q^{100}) \) 391 * q + 14 * q^2 + 12 * q^3 + 446 * q^4 + 22 * q^5 + 40 * q^6 + 63 * q^7 + 36 * q^8 + 501 * q^9 + 40 * q^10 + 57 * q^11 + 20 * q^12 + 83 * q^13 + 21 * q^14 + 60 * q^15 + 548 * q^16 + 59 * q^17 + 54 * q^18 + 131 * q^19 + 35 * q^20 + 121 * q^21 + 89 * q^22 + 34 * q^23 + 110 * q^24 + 609 * q^25 + 54 * q^26 + 27 * q^27 + 182 * q^28 + 102 * q^29 + 92 * q^30 + 88 * q^31 + 76 * q^32 + 131 * q^33 + 128 * q^34 + 31 * q^35 + 654 * q^36 + 135 * q^37 + 23 * q^38 + 96 * q^39 + 113 * q^40 + 128 * q^41 + 45 * q^42 + 140 * q^43 + 151 * q^44 + 77 * q^45 + 245 * q^46 + 22 * q^47 + 25 * q^48 + 712 * q^49 + 53 * q^50 + 102 * q^51 + 174 * q^52 + 54 * q^53 + 131 * q^54 + 101 * q^55 + 43 * q^56 + 226 * q^57 + 109 * q^58 + 40 * q^59 + 123 * q^60 + 249 * q^61 + 28 * q^62 + 139 * q^63 + 730 * q^64 + 227 * q^65 + 55 * q^66 + 169 * q^67 + 48 * q^68 + 89 * q^69 + 98 * q^70 + 66 * q^71 + 120 * q^72 + 324 * q^73 + 60 * q^74 + 19 * q^75 + 356 * q^76 + 83 * q^77 - 11 * q^78 + 195 * q^79 + 26 * q^80 + 807 * q^81 + 49 * q^82 + 74 * q^83 + 252 * q^84 + 373 * q^85 + 100 * q^86 + 43 * q^87 + 211 * q^88 + 207 * q^89 + 10 * q^90 + 189 * q^91 + 30 * q^92 + 172 * q^93 + 130 * q^94 + 43 * q^95 + 203 * q^96 + 254 * q^97 + 26 * q^98 + 273 * q^99

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 $

$2$	−2.61507	−1.84914	−0.924568	−	0.381016i	$-0.875574\pi$
$2$	−2.61507	−1.84914	−0.924568	+	0.381016i	$0.875574\pi$
$3$	3.14612	1.81641	0.908207	−	0.418521i	$-0.137452\pi$
$3$	3.14612	1.81641	0.908207	+	0.418521i	$0.137452\pi$
$4$	4.83861	2.41931
$5$	2.05578	0.919373	0.459686	−	0.888081i	$-0.347962\pi$
$5$	2.05578	0.919373	0.459686	+	0.888081i	$0.347962\pi$
$6$	−8.22734	−3.35880
$7$	4.83721	1.82829	0.914146	−	0.405385i	$-0.132863\pi$
$7$	4.83721	1.82829	0.914146	+	0.405385i	$0.132863\pi$
$8$	−7.42319	−2.62449
$9$	6.89808	2.29936
$10$	−5.37602	−1.70005
$11$	−0.0800961	−0.0241499	−0.0120749	−	0.999927i	$-0.503844\pi$
$11$	−0.0800961	−0.0241499	−0.0120749	+	0.999927i	$0.503844\pi$
$12$	15.2229	4.39446
$13$	4.01691	1.11409	0.557046	−	0.830482i	$-0.311935\pi$
$13$	4.01691	1.11409	0.557046	+	0.830482i	$0.311935\pi$
$14$	−12.6497	−3.38076
$15$	6.46773	1.66996
$16$	9.73496	2.43374
$17$	−1.27961	−0.310352	−0.155176	−	0.987887i	$-0.549594\pi$
$17$	−1.27961	−0.310352	−0.155176	+	0.987887i	$0.549594\pi$
$18$	−18.0390	−4.25183
$19$	6.18088	1.41799	0.708995	−	0.705213i	$-0.249149\pi$
$19$	6.18088	1.41799	0.708995	+	0.705213i	$0.249149\pi$
$20$	9.94713	2.22425
$21$	15.2184	3.32093
$22$	0.209457	0.0446564
$23$	1.67230	0.348700	0.174350	−	0.984684i	$-0.444218\pi$
$23$	1.67230	0.348700	0.174350	+	0.984684i	$0.444218\pi$
$24$	−23.3542	−4.76717
$25$	−0.773766	−0.154753
$26$	−10.5045	−2.06011
$27$	12.2638	2.36017
$28$	23.4054	4.42320
$29$	−9.73987	−1.80865	−0.904324	−	0.426847i	$-0.859624\pi$
$29$	−9.73987	−1.80865	−0.904324	+	0.426847i	$0.859624\pi$
$30$	−16.9136	−3.08799
$31$	−8.86776	−1.59270	−0.796348	−	0.604838i	$-0.793238\pi$
$31$	−8.86776	−1.59270	−0.796348	+	0.604838i	$0.793238\pi$
$32$	−10.6113	−1.87582
$33$	−0.251992	−0.0438662
$34$	3.34629	0.573883
$35$	9.94423	1.68088
$36$	33.3771	5.56286
$37$	−1.37542	−0.226117	−0.113059	−	0.993588i	$-0.536065\pi$
$37$	−1.37542	−0.226117	−0.113059	+	0.993588i	$0.536065\pi$
$38$	−16.1635	−2.62206
$39$	12.6377	2.02365
$40$	−15.2604	−2.41289
$41$	−6.30073	−0.984009	−0.492004	−	0.870593i	$-0.663736\pi$
$41$	−6.30073	−0.984009	−0.492004	+	0.870593i	$0.663736\pi$
$42$	−39.7973	−6.14086
$43$	−6.45319	−0.984102	−0.492051	−	0.870566i	$-0.663753\pi$
$43$	−6.45319	−0.984102	−0.492051	+	0.870566i	$0.663753\pi$
$44$	−0.387554	−0.0584260
$45$	14.1809	2.11397
$46$	−4.37320	−0.644793
$47$	11.2331	1.63852	0.819258	−	0.573425i	$-0.194386\pi$
$47$	11.2331	1.63852	0.819258	+	0.573425i	$0.194386\pi$
$48$	30.6274	4.42068
$49$	16.3986	2.34265
$50$	2.02346	0.286160
$51$	−4.02582	−0.563728
$52$	19.4363	2.69533
$53$	5.89701	0.810016	0.405008	−	0.914313i	$-0.367269\pi$
$53$	5.89701	0.810016	0.405008	+	0.914313i	$0.367269\pi$
$54$	−32.0708	−4.36429
$55$	−0.164660	−0.0222028
$56$	−35.9075	−4.79834
$57$	19.4458	2.57566
$58$	25.4705	3.34444
$59$	−14.3207	−1.86439	−0.932197	−	0.361952i	$-0.882110\pi$
$59$	−14.3207	−1.86439	−0.932197	+	0.361952i	$0.882110\pi$
$60$	31.2949	4.04015
$61$	4.69434	0.601049	0.300524	−	0.953774i	$-0.402838\pi$
$61$	4.69434	0.601049	0.300524	+	0.953774i	$0.402838\pi$
$62$	23.1898	2.94511
$63$	33.3674	4.20390
$64$	8.27934	1.03492
$65$	8.25789	1.02427
$66$	0.658978	0.0811146
$67$	−0.410139	−0.0501065	−0.0250532	−	0.999686i	$-0.507976\pi$
$67$	−0.410139	−0.0501065	−0.0250532	+	0.999686i	$0.507976\pi$
$68$	−6.19156	−0.750837
$69$	5.26127	0.633383
$70$	−26.0049	−3.10818
$71$	−12.8106	−1.52034	−0.760170	−	0.649725i	$-0.774884\pi$
$71$	−12.8106	−1.52034	−0.760170	+	0.649725i	$0.774884\pi$
$72$	−51.2057	−6.03465
$73$	−6.02222	−0.704848	−0.352424	−	0.935840i	$-0.614643\pi$
$73$	−6.02222	−0.704848	−0.352424	+	0.935840i	$0.614643\pi$
$74$	3.59682	0.418122
$75$	−2.43436	−0.281096
$76$	29.9069	3.43055
$77$	−0.387441	−0.0441530
$78$	−33.0485	−3.74201
$79$	1.45674	0.163896	0.0819481	−	0.996637i	$-0.473886\pi$
$79$	1.45674	0.163896	0.0819481	+	0.996637i	$0.473886\pi$
$80$	20.0129	2.23751
$81$	17.8892	1.98769
$82$	16.4769	1.81957
$83$	17.2639	1.89496	0.947479	−	0.319817i	$-0.103621\pi$
$83$	17.2639	1.89496	0.947479	+	0.319817i	$0.103621\pi$
$84$	73.6361	8.03436
$85$	−2.63061	−0.285329
$86$	16.8756	1.81974
$87$	−30.6428	−3.28525
$88$	0.594568	0.0633812
$89$	−7.02767	−0.744931	−0.372466	−	0.928046i	$-0.621488\pi$
$89$	−7.02767	−0.744931	−0.372466	+	0.928046i	$0.621488\pi$
$90$	−37.0842	−3.90902
$91$	19.4306	2.03688
$92$	8.09163	0.843611
$93$	−27.8990	−2.89300
$94$	−29.3754	−3.02984
$95$	12.7065	1.30366
$96$	−33.3843	−3.40727
$97$	5.16987	0.524921	0.262460	−	0.964943i	$-0.415466\pi$
$97$	5.16987	0.524921	0.262460	+	0.964943i	$0.415466\pi$
$98$	−42.8835	−4.33188
$99$	−0.552509	−0.0555293
$100$	−3.74396	−0.374396
$101$	0.644496	0.0641297	0.0320649	−	0.999486i	$-0.489792\pi$
$101$	0.644496	0.0641297	0.0320649	+	0.999486i	$0.489792\pi$
$102$	10.5278	1.04241
$103$	5.99869	0.591068	0.295534	−	0.955332i	$-0.404502\pi$
$103$	5.99869	0.591068	0.295534	+	0.955332i	$0.404502\pi$
$104$	−29.8183	−2.92392
$105$	31.2858	3.05318
$106$	−15.4211	−1.49783
$107$	−0.866273	−0.0837458	−0.0418729	−	0.999123i	$-0.513332\pi$
$107$	−0.866273	−0.0837458	−0.0418729	+	0.999123i	$0.513332\pi$
$108$	59.3399	5.70999
$109$	15.5000	1.48463	0.742315	−	0.670051i	$-0.233728\pi$
$109$	15.5000	1.48463	0.742315	+	0.670051i	$0.233728\pi$
$110$	0.430598	0.0410559
$111$	−4.32723	−0.410723
$112$	47.0900	4.44959
$113$	−0.479359	−0.0450943	−0.0225472	−	0.999746i	$-0.507178\pi$
$113$	−0.479359	−0.0450943	−0.0225472	+	0.999746i	$0.507178\pi$
$114$	−50.8522	−4.76274
$115$	3.43789	0.320585
$116$	−47.1275	−4.37568
$117$	27.7090	2.56170
$118$	37.4496	3.44752
$119$	−6.18976	−0.567414
$120$	−48.0112	−4.38280
$121$	−10.9936	−0.999417
$122$	−12.2760	−1.11142
$123$	−19.8229	−1.78737
$124$	−42.9077	−3.85322
$125$	−11.8696	−1.06165
$126$	−87.2583	−7.77359
$127$	7.91851	0.702654	0.351327	−	0.936253i	$-0.385731\pi$
$127$	7.91851	0.702654	0.351327	+	0.936253i	$0.385731\pi$
$128$	−0.428546	−0.0378785
$129$	−20.3025	−1.78754
$130$	−21.5950	−1.89401
$131$	−17.8995	−1.56389	−0.781944	−	0.623349i	$-0.785772\pi$
$131$	−17.8995	−1.56389	−0.781944	+	0.623349i	$0.785772\pi$
$132$	−1.21929	−0.106126
$133$	29.8982	2.59250
$134$	1.07254	0.0926537
$135$	25.2117	2.16988
$136$	9.49882	0.814517
$137$	−3.63150	−0.310260	−0.155130	−	0.987894i	$-0.549580\pi$
$137$	−3.63150	−0.310260	−0.155130	+	0.987894i	$0.549580\pi$
$138$	−13.7586	−1.17121
$139$	12.1112	1.02726	0.513630	−	0.858012i	$-0.328301\pi$
$139$	12.1112	1.02726	0.513630	+	0.858012i	$0.328301\pi$
$140$	48.1163	4.06657
$141$	35.3407	2.97622
$142$	33.5007	2.81132
$143$	−0.321739	−0.0269052
$144$	67.1525	5.59604
$145$	−20.0230	−1.66282
$146$	15.7486	1.30336
$147$	51.5919	4.25522
$148$	−6.65512	−0.547047
$149$	15.4796	1.26814	0.634068	−	0.773277i	$-0.281384\pi$
$149$	15.4796	1.26814	0.634068	+	0.773277i	$0.281384\pi$
$150$	6.36604	0.519785
$151$	−19.4704	−1.58448	−0.792239	−	0.610211i	$-0.791085\pi$
$151$	−19.4704	−1.58448	−0.792239	+	0.610211i	$0.791085\pi$
$152$	−45.8818	−3.72151
$153$	−8.82688	−0.713611
$154$	1.01319	0.0816450
$155$	−18.2302	−1.46428
$156$	61.1489	4.89583
$157$	11.7325	0.936358	0.468179	−	0.883634i	$-0.344910\pi$
$157$	11.7325	0.936358	0.468179	+	0.883634i	$0.344910\pi$
$158$	−3.80948	−0.303066
$159$	18.5527	1.47132
$160$	−21.8144	−1.72458
$161$	8.08928	0.637525
$162$	−46.7817	−3.67552
$163$	7.29898	0.571700	0.285850	−	0.958274i	$-0.407724\pi$
$163$	7.29898	0.571700	0.285850	+	0.958274i	$0.407724\pi$
$164$	−30.4868	−2.38062
$165$	−0.518040	−0.0403294
$166$	−45.1464	−3.50404
$167$	0.873427	0.0675878	0.0337939	−	0.999429i	$-0.489241\pi$
$167$	0.873427	0.0675878	0.0337939	+	0.999429i	$0.489241\pi$
$168$	−112.969	−8.71577
$169$	3.13559	0.241199
$170$	6.87923	0.527613
$171$	42.6362	3.26047
$172$	−31.2245	−2.38084
$173$	−1.44118	−0.109571	−0.0547855	−	0.998498i	$-0.517447\pi$
$173$	−1.44118	−0.109571	−0.0547855	+	0.998498i	$0.517447\pi$
$174$	80.1332	6.07488
$175$	−3.74287	−0.282934
$176$	−0.779732	−0.0587745
$177$	−45.0546	−3.38651
$178$	18.3779	1.37748
$179$	6.33051	0.473164	0.236582	−	0.971611i	$-0.423973\pi$
$179$	6.33051	0.473164	0.236582	+	0.971611i	$0.423973\pi$
$180$	68.6161	5.11434
$181$	8.75210	0.650539	0.325269	−	0.945621i	$-0.394545\pi$
$181$	8.75210	0.650539	0.325269	+	0.945621i	$0.394545\pi$
$182$	−50.8126	−3.76648
$183$	14.7690	1.09175
$184$	−12.4138	−0.915159
$185$	−2.82756	−0.207886
$186$	72.9581	5.34955
$187$	0.102492	0.00749497
$188$	54.3526	3.96407
$189$	59.3227	4.31509
$190$	−33.2285	−2.41065
$191$	−0.514149	−0.0372025	−0.0186012	−	0.999827i	$-0.505921\pi$
$191$	−0.514149	−0.0372025	−0.0186012	+	0.999827i	$0.505921\pi$
$192$	26.0478	1.87984
$193$	−22.3425	−1.60825	−0.804123	−	0.594462i	$-0.797365\pi$
$193$	−22.3425	−1.60825	−0.804123	+	0.594462i	$0.797365\pi$
$194$	−13.5196	−0.970650
$195$	25.9803	1.86049
$196$	79.3463	5.66759
$197$	10.7817	0.768163	0.384082	−	0.923299i	$-0.374518\pi$
$197$	10.7817	0.768163	0.384082	+	0.923299i	$0.374518\pi$
$198$	1.44485	0.102681
$199$	−26.0476	−1.84647	−0.923233	−	0.384241i	$-0.874463\pi$
$199$	−26.0476	−1.84647	−0.923233	+	0.384241i	$0.874463\pi$
$200$	5.74381	0.406149
$201$	−1.29035	−0.0910141
$202$	−1.68540	−0.118585
$203$	−47.1137	−3.30674
$204$	−19.4794	−1.36383
$205$	−12.9529	−0.904671
$206$	−15.6870	−1.09297
$207$	11.5357	0.801786
$208$	39.1045	2.71141
$209$	−0.495064	−0.0342443
$210$	−81.8146	−5.64574
$211$	28.6612	1.97312	0.986559	−	0.163407i	$-0.0522485\pi$
$211$	28.6612	1.97312	0.986559	+	0.163407i	$0.0522485\pi$
$212$	28.5333	1.95968
$213$	−40.3037	−2.76157
$214$	2.26537	0.154857
$215$	−13.2663	−0.904757
$216$	−91.0367	−6.19426
$217$	−42.8952	−2.91191
$218$	−40.5336	−2.74529
$219$	−18.9466	−1.28030
$220$	−0.796726	−0.0537153
$221$	−5.14010	−0.345760
$222$	11.3160	0.759483
$223$	5.50494	0.368638	0.184319	−	0.982866i	$-0.440992\pi$
$223$	5.50494	0.368638	0.184319	+	0.982866i	$0.440992\pi$
$224$	−51.3289	−3.42956
$225$	−5.33750	−0.355833
$226$	1.25356	0.0833856
$227$	16.1446	1.07155	0.535777	−	0.844360i	$-0.320019\pi$
$227$	16.1446	1.07155	0.535777	+	0.844360i	$0.320019\pi$
$228$	94.0907	6.23131
$229$	17.1296	1.13195	0.565976	−	0.824422i	$-0.308499\pi$
$229$	17.1296	1.13195	0.565976	+	0.824422i	$0.308499\pi$
$230$	−8.99034	−0.592805
$231$	−1.21894	−0.0802002
$232$	72.3009	4.74678
$233$	−7.93042	−0.519539	−0.259770	−	0.965671i	$-0.583647\pi$
$233$	−7.93042	−0.519539	−0.259770	+	0.965671i	$0.583647\pi$
$234$	−72.4610	−4.73693
$235$	23.0928	1.50641
$236$	−69.2922	−4.51054
$237$	4.58308	0.297703
$238$	16.1867	1.04923
$239$	−9.08092	−0.587396	−0.293698	−	0.955898i	$-0.594886\pi$
$239$	−9.08092	−0.587396	−0.293698	+	0.955898i	$0.594886\pi$
$240$	62.9631	4.06425
$241$	−24.3875	−1.57093	−0.785467	−	0.618904i	$-0.787577\pi$
$241$	−24.3875	−1.57093	−0.785467	+	0.618904i	$0.787577\pi$
$242$	28.7490	1.84806
$243$	19.4903	1.25030
$244$	22.7141	1.45412
$245$	33.7118	2.15377
$246$	51.8383	3.30509
$247$	24.8280	1.57977
$248$	65.8270	4.18002
$249$	54.3143	3.44203
$250$	31.0399	1.96313
$251$	16.0085	1.01045	0.505224	−	0.862988i	$-0.331410\pi$
$251$	16.0085	1.01045	0.505224	+	0.862988i	$0.331410\pi$
$252$	161.452	10.1705
$253$	−0.133945	−0.00842105
$254$	−20.7075	−1.29930
$255$	−8.27621	−0.518276
$256$	−15.4380	−0.964875
$257$	22.1184	1.37971	0.689853	−	0.723950i	$-0.257675\pi$
$257$	22.1184	1.37971	0.689853	+	0.723950i	$0.257675\pi$
$258$	53.0926	3.30540
$259$	−6.65318	−0.413409
$260$	39.9567	2.47801
$261$	−67.1864	−4.15873
$262$	46.8085	2.89184
$263$	−4.97647	−0.306862	−0.153431	−	0.988159i	$-0.549032\pi$
$263$	−4.97647	−0.306862	−0.153431	+	0.988159i	$0.549032\pi$
$264$	1.87058	0.115127
$265$	12.1230	0.744707
$266$	−78.1860	−4.79389
$267$	−22.1099	−1.35310
$268$	−1.98451	−0.121223
$269$	−25.3331	−1.54459	−0.772293	−	0.635267i	$-0.780890\pi$
$269$	−25.3331	−1.54459	−0.772293	+	0.635267i	$0.780890\pi$
$270$	−65.9306	−4.01241
$271$	−13.2296	−0.803641	−0.401820	−	0.915718i	$-0.631622\pi$
$271$	−13.2296	−0.803641	−0.401820	+	0.915718i	$0.631622\pi$
$272$	−12.4570	−0.755316
$273$	61.1311	3.69982
$274$	9.49663	0.573712
$275$	0.0619757	0.00373727
$276$	25.4573	1.53235
$277$	17.2971	1.03928	0.519641	−	0.854384i	$-0.326066\pi$
$277$	17.2971	1.03928	0.519641	+	0.854384i	$0.326066\pi$
$278$	−31.6717	−1.89954
$279$	−61.1705	−3.66218
$280$	−73.8179	−4.41146
$281$	31.7835	1.89605	0.948023	−	0.318201i	$-0.103079\pi$
$281$	31.7835	1.89605	0.948023	+	0.318201i	$0.103079\pi$
$282$	−92.4185	−5.50344
$283$	19.0253	1.13094	0.565469	−	0.824769i	$-0.308695\pi$
$283$	19.0253	1.13094	0.565469	+	0.824769i	$0.308695\pi$
$284$	−61.9856	−3.67817
$285$	39.9763	2.36799
$286$	0.841372	0.0497514
$287$	−30.4779	−1.79906
$288$	−73.1974	−4.31320
$289$	−15.3626	−0.903682
$290$	52.3617	3.07479
$291$	16.2650	0.953473
$292$	−29.1392	−1.70524
$293$	3.92626	0.229374	0.114687	−	0.993402i	$-0.463413\pi$
$293$	3.92626	0.229374	0.114687	+	0.993402i	$0.463413\pi$
$294$	−134.917	−7.86849
$295$	−29.4402	−1.71407
$296$	10.2100	0.593444
$297$	−0.982285	−0.0569979
$298$	−40.4802	−2.34496
$299$	6.71750	0.388483
$300$	−11.7789	−0.680058
$301$	−31.2154	−1.79923
$302$	50.9165	2.92992
$303$	2.02766	0.116486
$304$	60.1706	3.45102
$305$	9.65053	0.552588
$306$	23.0829	1.31956
$307$	17.7750	1.01447	0.507237	−	0.861807i	$-0.330667\pi$
$307$	17.7750	1.01447	0.507237	+	0.861807i	$0.330667\pi$
$308$	−1.87468	−0.106820
$309$	18.8726	1.07362
$310$	47.6732	2.70766
$311$	−0.880024	−0.0499016	−0.0249508	−	0.999689i	$-0.507943\pi$
$311$	−0.880024	−0.0499016	−0.0249508	+	0.999689i	$0.507943\pi$
$312$	−93.8120	−5.31106
$313$	−6.10504	−0.345077	−0.172539	−	0.985003i	$-0.555197\pi$
$313$	−6.10504	−0.345077	−0.172539	+	0.985003i	$0.555197\pi$
$314$	−30.6815	−1.73145
$315$	68.5961	3.86495
$316$	7.04861	0.396515
$317$	−23.0881	−1.29676	−0.648379	−	0.761318i	$-0.724553\pi$
$317$	−23.0881	−1.29676	−0.648379	+	0.761318i	$0.724553\pi$
$318$	−48.5167	−2.72068
$319$	0.780126	0.0436786
$320$	17.0205	0.951475
$321$	−2.72540	−0.152117
$322$	−21.1541	−1.17887
$323$	−7.90914	−0.440076
$324$	86.5592	4.80884
$325$	−3.10815	−0.172409
$326$	−19.0874	−1.05715
$327$	48.7649	2.69670
$328$	46.7715	2.58252
$329$	54.3368	2.99568
$330$	1.35471	0.0745746
$331$	8.43140	0.463432	0.231716	−	0.972784i	$-0.425566\pi$
$331$	8.43140	0.463432	0.231716	+	0.972784i	$0.425566\pi$
$332$	83.5333	4.58449
$333$	−9.48774	−0.519925
$334$	−2.28408	−0.124979
$335$	−0.843156	−0.0460665
$336$	148.151	8.08229
$337$	7.29147	0.397192	0.198596	−	0.980081i	$-0.436362\pi$
$337$	7.29147	0.397192	0.198596	+	0.980081i	$0.436362\pi$
$338$	−8.19980	−0.446010
$339$	−1.50812	−0.0819100
$340$	−12.7285	−0.690299
$341$	0.710273	0.0384634
$342$	−111.497	−6.02906
$343$	45.4628	2.45476
$344$	47.9032	2.58277
$345$	10.8160	0.582315
$346$	3.76879	0.202612
$347$	9.88347	0.530573	0.265286	−	0.964170i	$-0.414534\pi$
$347$	9.88347	0.530573	0.265286	+	0.964170i	$0.414534\pi$
$348$	−148.269	−7.94804
$349$	25.4977	1.36486	0.682429	−	0.730952i	$-0.260924\pi$
$349$	25.4977	1.36486	0.682429	+	0.730952i	$0.260924\pi$
$350$	9.78788	0.523184
$351$	49.2627	2.62945
$352$	0.849921	0.0453010
$353$	−15.2302	−0.810620	−0.405310	−	0.914179i	$-0.632836\pi$
$353$	−15.2302	−0.810620	−0.405310	+	0.914179i	$0.632836\pi$
$354$	117.821	6.26212
$355$	−26.3358	−1.39776
$356$	−34.0042	−1.80222
$357$	−19.4737	−1.03066
$358$	−16.5547	−0.874946
$359$	−17.0807	−0.901484	−0.450742	−	0.892654i	$-0.648841\pi$
$359$	−17.0807	−0.901484	−0.450742	+	0.892654i	$0.648841\pi$
$360$	−105.268	−5.54810
$361$	19.2033	1.01070
$362$	−22.8874	−1.20294
$363$	−34.5871	−1.81535
$364$	94.0173	4.92785
$365$	−12.3804	−0.648018
$366$	−38.6219	−2.01880
$367$	13.4439	0.701767	0.350883	−	0.936419i	$-0.385881\pi$
$367$	13.4439	0.701767	0.350883	+	0.936419i	$0.385881\pi$
$368$	16.2798	0.848644
$369$	−43.4629	−2.26259
$370$	7.39428	0.384410
$371$	28.5250	1.48095
$372$	−134.993	−6.99905
$373$	−13.4600	−0.696931	−0.348466	−	0.937322i	$-0.613297\pi$
$373$	−13.4600	−0.696931	−0.348466	+	0.937322i	$0.613297\pi$
$374$	−0.268025	−0.0138592
$375$	−37.3432	−1.92839
$376$	−83.3854	−4.30027
$377$	−39.1242	−2.01500
$378$	−155.133	−7.97919
$379$	−13.3277	−0.684596	−0.342298	−	0.939592i	$-0.611205\pi$
$379$	−13.3277	−0.684596	−0.342298	+	0.939592i	$0.611205\pi$
$380$	61.4820	3.15396
$381$	24.9126	1.27631
$382$	1.34454	0.0687925
$383$	−11.7041	−0.598049	−0.299025	−	0.954245i	$-0.596661\pi$
$383$	−11.7041	−0.598049	−0.299025	+	0.954245i	$0.596661\pi$
$384$	−1.34826	−0.0688031
$385$	−0.796494	−0.0405931
$386$	58.4272	2.97387
$387$	−44.5146	−2.26280
$388$	25.0150	1.26994
$389$	−30.7901	−1.56112	−0.780561	−	0.625080i	$-0.785066\pi$
$389$	−30.7901	−1.56112	−0.780561	+	0.625080i	$0.785066\pi$
$390$	−67.9405	−3.44030
$391$	−2.13990	−0.108220
$392$	−121.730	−6.14827
$393$	−56.3140	−2.84067
$394$	−28.1949	−1.42044
$395$	2.99474	0.150682
$396$	−2.67338	−0.134342
$397$	14.4575	0.725599	0.362800	−	0.931867i	$-0.381821\pi$
$397$	14.4575	0.725599	0.362800	+	0.931867i	$0.381821\pi$
$398$	68.1164	3.41437
$399$	94.0633	4.70905
$400$	−7.53259	−0.376629
$401$	10.0784	0.503289	0.251645	−	0.967820i	$-0.419029\pi$
$401$	10.0784	0.503289	0.251645	+	0.967820i	$0.419029\pi$
$402$	3.37435	0.168298
$403$	−35.6210	−1.77441
$404$	3.11847	0.155149
$405$	36.7764	1.82743
$406$	123.206	6.11461
$407$	0.110166	0.00546071
$408$	29.8844	1.47950
$409$	−17.2002	−0.850495	−0.425248	−	0.905077i	$-0.639813\pi$
$409$	−17.2002	−0.850495	−0.425248	+	0.905077i	$0.639813\pi$
$410$	33.8728	1.67286
$411$	−11.4251	−0.563560
$412$	29.0253	1.42998
$413$	−69.2721	−3.40866
$414$	−30.1667	−1.48261
$415$	35.4908	1.74217
$416$	−42.6245	−2.08984
$417$	38.1033	1.86593
$418$	1.29463	0.0633224
$419$	21.3347	1.04227	0.521134	−	0.853475i	$-0.325509\pi$
$419$	21.3347	1.04227	0.521134	+	0.853475i	$0.325509\pi$
$420$	151.380	7.38658
$421$	3.99462	0.194686	0.0973429	−	0.995251i	$-0.468966\pi$
$421$	3.99462	0.194686	0.0973429	+	0.995251i	$0.468966\pi$
$422$	−74.9511	−3.64856
$423$	77.4868	3.76754
$424$	−43.7746	−2.12588
$425$	0.990123	0.0480280
$426$	105.397	5.10651
$427$	22.7075	1.09889
$428$	−4.19156	−0.202607
$429$	−1.01223	−0.0488709
$430$	34.6925	1.67302
$431$	−3.41846	−0.164661	−0.0823306	−	0.996605i	$-0.526236\pi$
$431$	−3.41846	−0.164661	−0.0823306	+	0.996605i	$0.526236\pi$
$432$	119.388	5.74405
$433$	−41.2368	−1.98171	−0.990857	−	0.134920i	$-0.956922\pi$
$433$	−41.2368	−1.98171	−0.990857	+	0.134920i	$0.956922\pi$
$434$	112.174	5.38453
$435$	−62.9949	−3.02037
$436$	74.9985	3.59178
$437$	10.3363	0.494453
$438$	49.5469	2.36744
$439$	−24.2316	−1.15651	−0.578255	−	0.815856i	$-0.696266\pi$
$439$	−24.2316	−1.15651	−0.578255	+	0.815856i	$0.696266\pi$
$440$	1.22230	0.0582710
$441$	113.119	5.38660
$442$	13.4417	0.639358
$443$	−0.945911	−0.0449416	−0.0224708	−	0.999748i	$-0.507153\pi$
$443$	−0.945911	−0.0449416	−0.0224708	+	0.999748i	$0.507153\pi$
$444$	−20.9378	−0.993664
$445$	−14.4473	−0.684870
$446$	−14.3958	−0.681662
$447$	48.7006	2.30346
$448$	40.0489	1.89213
$449$	−31.2262	−1.47366	−0.736828	−	0.676080i	$-0.763677\pi$
$449$	−31.2262	−1.47366	−0.736828	+	0.676080i	$0.763677\pi$
$450$	13.9580	0.657985
$451$	0.504664	0.0237637
$452$	−2.31943	−0.109097
$453$	−61.2562	−2.87807
$454$	−42.2193	−1.98145
$455$	39.9451	1.87266
$456$	−144.350	−6.75980
$457$	−3.53777	−0.165490	−0.0827449	−	0.996571i	$-0.526369\pi$
$457$	−3.53777	−0.165490	−0.0827449	+	0.996571i	$0.526369\pi$
$458$	−44.7951	−2.09314
$459$	−15.6930	−0.732485
$460$	16.6346	0.775593
$461$	−12.9822	−0.604642	−0.302321	−	0.953206i	$-0.597761\pi$
$461$	−12.9822	−0.604642	−0.302321	+	0.953206i	$0.597761\pi$
$462$	3.18761	0.148301
$463$	19.0927	0.887315	0.443658	−	0.896196i	$-0.353681\pi$
$463$	19.0927	0.887315	0.443658	+	0.896196i	$0.353681\pi$
$464$	−94.8172	−4.40178
$465$	−57.3543	−2.65974
$466$	20.7386	0.960699
$467$	−20.9623	−0.970018	−0.485009	−	0.874509i	$-0.661184\pi$
$467$	−20.9623	−0.970018	−0.485009	+	0.874509i	$0.661184\pi$
$468$	134.073	6.19753
$469$	−1.98393	−0.0916093
$470$	−60.3893	−2.78555
$471$	36.9120	1.70081
$472$	106.305	4.89309
$473$	0.516875	0.0237659
$474$	−11.9851	−0.550494
$475$	−4.78256	−0.219439
$476$	−29.9499	−1.37275
$477$	40.6780	1.86252
$478$	23.7473	1.08618
$479$	20.9891	0.959014	0.479507	−	0.877538i	$-0.340815\pi$
$479$	20.9891	0.959014	0.479507	+	0.877538i	$0.340815\pi$
$480$	−68.6309	−3.13256
$481$	−5.52494	−0.251915
$482$	63.7750	2.90487
$483$	25.4499	1.15801
$484$	−53.1937	−2.41790
$485$	10.6281	0.482598
$486$	−50.9685	−2.31198
$487$	−16.7017	−0.756824	−0.378412	−	0.925637i	$-0.623530\pi$
$487$	−16.7017	−0.756824	−0.378412	+	0.925637i	$0.623530\pi$
$488$	−34.8470	−1.57745
$489$	22.9635	1.03844
$490$	−88.1590	−3.98262
$491$	18.8166	0.849183	0.424591	−	0.905385i	$-0.360418\pi$
$491$	18.8166	0.849183	0.424591	+	0.905385i	$0.360418\pi$
$492$	−95.9152	−4.32419
$493$	12.4633	0.561318
$494$	−64.9272	−2.92121
$495$	−1.13584	−0.0510521
$496$	−86.3273	−3.87621
$497$	−61.9675	−2.77962
$498$	−142.036	−6.36478
$499$	−6.94926	−0.311092	−0.155546	−	0.987829i	$-0.549714\pi$
$499$	−6.94926	−0.311092	−0.155546	+	0.987829i	$0.549714\pi$
$500$	−57.4324	−2.56845
$501$	2.74791	0.122767
$502$	−41.8635	−1.86846
$503$	−4.47219	−0.199405	−0.0997025	−	0.995017i	$-0.531789\pi$
$503$	−4.47219	−0.199405	−0.0997025	+	0.995017i	$0.531789\pi$
$504$	−247.693	−11.0331
$505$	1.32494	0.0589591
$506$	0.350276	0.0155717
$507$	9.86494	0.438117
$508$	38.3146	1.69994
$509$	−19.0228	−0.843170	−0.421585	−	0.906789i	$-0.638526\pi$
$509$	−19.0228	−0.843170	−0.421585	+	0.906789i	$0.638526\pi$
$510$	21.6429	0.958363
$511$	−29.1307	−1.28867
$512$	41.2286	1.82206
$513$	75.8012	3.34671
$514$	−57.8412	−2.55126
$515$	12.3320	0.543412
$516$	−98.2360	−4.32460
$517$	−0.899727	−0.0395700
$518$	17.3986	0.764449
$519$	−4.53413	−0.199026
$520$	−61.2999	−2.68818
$521$	−43.6555	−1.91258	−0.956291	−	0.292416i	$-0.905541\pi$
$521$	−43.6555	−1.91258	−0.956291	+	0.292416i	$0.905541\pi$
$522$	175.697	7.69006
$523$	−28.5326	−1.24765	−0.623823	−	0.781566i	$-0.714421\pi$
$523$	−28.5326	−1.24765	−0.623823	+	0.781566i	$0.714421\pi$
$524$	−86.6088	−3.78352
$525$	−11.7755	−0.513926
$526$	13.0138	0.567430
$527$	11.3473	0.494297
$528$	−2.45313	−0.106759
$529$	−20.2034	−0.878409
$530$	−31.7024	−1.37707
$531$	−98.7852	−4.28691
$532$	144.666	6.27206
$533$	−25.3095	−1.09628
$534$	57.8190	2.50207
$535$	−1.78087	−0.0769936
$536$	3.04454	0.131504
$537$	19.9165	0.859462
$538$	66.2479	2.85615
$539$	−1.31346	−0.0565748
$540$	121.990	5.24961
$541$	−29.2820	−1.25893	−0.629466	−	0.777028i	$-0.716726\pi$
$541$	−29.2820	−1.25893	−0.629466	+	0.777028i	$0.716726\pi$
$542$	34.5964	1.48604
$543$	27.5352	1.18165
$544$	13.5783	0.582166
$545$	31.8646	1.36493
$546$	−159.862	−6.84148
$547$	−15.9999	−0.684107	−0.342054	−	0.939680i	$-0.611122\pi$
$547$	−15.9999	−0.684107	−0.342054	+	0.939680i	$0.611122\pi$
$548$	−17.5714	−0.750613
$549$	32.3819	1.38203
$550$	−0.162071	−0.00691073
$551$	−60.2009	−2.56465
$552$	−39.0554	−1.66231
$553$	7.04655	0.299650
$554$	−45.2332	−1.92178
$555$	−8.89584	−0.377607
$556$	58.6014	2.48526
$557$	−8.77000	−0.371597	−0.185798	−	0.982588i	$-0.559487\pi$
$557$	−8.77000	−0.371597	−0.185798	+	0.982588i	$0.559487\pi$
$558$	159.965	6.77188
$559$	−25.9219	−1.09638
$560$	96.8067	4.09083
$561$	0.322453	0.0136140
$562$	−83.1163	−3.50605
$563$	−21.6444	−0.912203	−0.456102	−	0.889928i	$-0.650755\pi$
$563$	−21.6444	−0.912203	−0.456102	+	0.889928i	$0.650755\pi$
$564$	171.000	7.20040
$565$	−0.985457	−0.0414585
$566$	−49.7527	−2.09126
$567$	86.5340	3.63409
$568$	95.0955	3.99012
$569$	18.6288	0.780959	0.390480	−	0.920612i	$-0.372309\pi$
$569$	18.6288	0.780959	0.390480	+	0.920612i	$0.372309\pi$
$570$	−104.541	−4.37874
$571$	−22.7799	−0.953308	−0.476654	−	0.879091i	$-0.658150\pi$
$571$	−22.7799	−0.953308	−0.476654	+	0.879091i	$0.658150\pi$
$572$	−1.55677	−0.0650919
$573$	−1.61757	−0.0675751
$574$	79.7021	3.32670
$575$	−1.29397	−0.0539624
$576$	57.1115	2.37965
$577$	−47.5855	−1.98101	−0.990505	−	0.137479i	$-0.956100\pi$
$577$	−47.5855	−1.98101	−0.990505	+	0.137479i	$0.956100\pi$
$578$	40.1743	1.67103
$579$	−70.2921	−2.92124
$580$	−96.8837	−4.02288
$581$	83.5090	3.46454
$582$	−42.5343	−1.76310
$583$	−0.472327	−0.0195618
$584$	44.7041	1.84987
$585$	56.9636	2.35515
$586$	−10.2675	−0.424145
$587$	−6.68320	−0.275845	−0.137922	−	0.990443i	$-0.544042\pi$
$587$	−6.68320	−0.275845	−0.137922	+	0.990443i	$0.544042\pi$
$588$	249.633	10.2947
$589$	−54.8105	−2.25843
$590$	76.9882	3.16956
$591$	33.9205	1.39530
$592$	−13.3896	−0.550311
$593$	34.8157	1.42971	0.714854	−	0.699274i	$-0.246493\pi$
$593$	34.8157	1.42971	0.714854	+	0.699274i	$0.246493\pi$
$594$	2.56875	0.105397
$595$	−12.7248	−0.521665
$596$	74.8997	3.06801
$597$	−81.9489	−3.35395
$598$	−17.5668	−0.718358
$599$	1.38559	0.0566138	0.0283069	−	0.999599i	$-0.490988\pi$
$599$	1.38559	0.0566138	0.0283069	+	0.999599i	$0.490988\pi$
$600$	18.0707	0.737735
$601$	−0.131888	−0.00537984	−0.00268992	−	0.999996i	$-0.500856\pi$
$601$	−0.131888	−0.00537984	−0.00268992	+	0.999996i	$0.500856\pi$
$602$	81.6306	3.32701
$603$	−2.82917	−0.115213
$604$	−94.2097	−3.83334
$605$	−22.6004	−0.918837
$606$	−5.30248	−0.215399
$607$	−22.6478	−0.919245	−0.459622	−	0.888114i	$-0.652015\pi$
$607$	−22.6478	−0.919245	−0.459622	+	0.888114i	$0.652015\pi$
$608$	−65.5870	−2.65990
$609$	−148.226	−6.00640
$610$	−25.2369	−1.02181
$611$	45.1224	1.82546
$612$	−42.7099	−1.72644
$613$	1.95857	0.0791057	0.0395529	−	0.999217i	$-0.487407\pi$
$613$	1.95857	0.0791057	0.0395529	+	0.999217i	$0.487407\pi$
$614$	−46.4830	−1.87590
$615$	−40.7514	−1.64326
$616$	2.87605	0.115879
$617$	−6.32218	−0.254521	−0.127261	−	0.991869i	$-0.540618\pi$
$617$	−6.32218	−0.254521	−0.127261	+	0.991869i	$0.540618\pi$
$618$	−49.3532	−1.98528
$619$	6.02715	0.242252	0.121126	−	0.992637i	$-0.461350\pi$
$619$	6.02715	0.242252	0.121126	+	0.992637i	$0.461350\pi$
$620$	−88.2087	−3.54255
$621$	20.5088	0.822992
$622$	2.30133	0.0922749
$623$	−33.9943	−1.36195
$624$	123.027	4.92504
$625$	−20.5325	−0.821298
$626$	15.9651	0.638095
$627$	−1.55753	−0.0622018
$628$	56.7692	2.26534
$629$	1.76001	0.0701760
$630$	−179.384	−7.14683
$631$	38.4694	1.53144	0.765722	−	0.643172i	$-0.222382\pi$
$631$	38.4694	1.53144	0.765722	+	0.643172i	$0.222382\pi$
$632$	−10.8137	−0.430144
$633$	90.1716	3.58400
$634$	60.3772	2.39788
$635$	16.2787	0.646001
$636$	89.7694	3.55959
$637$	65.8716	2.60993
$638$	−2.04009	−0.0807678
$639$	−88.3686	−3.49581
$640$	−0.880997	−0.0348245
$641$	−30.0685	−1.18763	−0.593816	−	0.804601i	$-0.702379\pi$
$641$	−30.0685	−1.18763	−0.593816	+	0.804601i	$0.702379\pi$
$642$	7.12712	0.281285
$643$	−8.51884	−0.335950	−0.167975	−	0.985791i	$-0.553723\pi$
$643$	−8.51884	−0.335950	−0.167975	+	0.985791i	$0.553723\pi$
$644$	39.1409	1.54237
$645$	−41.7375	−1.64341
$646$	20.6830	0.813761
$647$	36.4791	1.43414	0.717070	−	0.697001i	$-0.245483\pi$
$647$	36.4791	1.43414	0.717070	+	0.697001i	$0.245483\pi$
$648$	−132.795	−5.21669
$649$	1.14703	0.0450249
$650$	8.12805	0.318808
$651$	−134.953	−5.28924
$652$	35.3170	1.38312
$653$	−10.4586	−0.409278	−0.204639	−	0.978838i	$-0.565602\pi$
$653$	−10.4586	−0.409278	−0.204639	+	0.978838i	$0.565602\pi$
$654$	−127.524	−4.98657
$655$	−36.7975	−1.43780
$656$	−61.3374	−2.39482
$657$	−41.5418	−1.62070
$658$	−142.095	−5.53943
$659$	−26.1778	−1.01974	−0.509872	−	0.860250i	$-0.670307\pi$
$659$	−26.1778	−1.01974	−0.509872	+	0.860250i	$0.670307\pi$
$660$	−2.50660	−0.0975692
$661$	3.81184	0.148263	0.0741317	−	0.997248i	$-0.476381\pi$
$661$	3.81184	0.148263	0.0741317	+	0.997248i	$0.476381\pi$
$662$	−22.0487	−0.856949
$663$	−16.1714	−0.628044
$664$	−128.153	−4.97331
$665$	61.4641	2.38348
$666$	24.8112	0.961413
$667$	−16.2880	−0.630675
$668$	4.22618	0.163516
$669$	17.3192	0.669599
$670$	2.20492	0.0851833
$671$	−0.375998	−0.0145153
$672$	−161.487	−6.22949
$673$	16.4296	0.633314	0.316657	−	0.948540i	$-0.397440\pi$
$673$	16.4296	0.633314	0.316657	+	0.948540i	$0.397440\pi$
$674$	−19.0677	−0.734462
$675$	−9.48934	−0.365245
$676$	15.1719	0.583535
$677$	29.7211	1.14227	0.571137	−	0.820855i	$-0.306502\pi$
$677$	29.7211	1.14227	0.571137	+	0.820855i	$0.306502\pi$
$678$	3.94385	0.151463
$679$	25.0077	0.959708
$680$	19.5275	0.748845
$681$	50.7928	1.94638
$682$	−1.85742	−0.0711242
$683$	47.4106	1.81412	0.907059	−	0.421004i	$-0.138322\pi$
$683$	47.4106	1.81412	0.907059	+	0.421004i	$0.138322\pi$
$684$	206.300	7.88808
$685$	−7.46556	−0.285244
$686$	−118.889	−4.53919
$687$	53.8916	2.05609
$688$	−62.8215	−2.39505
$689$	23.6878	0.902432
$690$	−28.2847	−1.07678
$691$	−37.8128	−1.43847	−0.719234	−	0.694768i	$-0.755507\pi$
$691$	−37.8128	−1.43847	−0.719234	+	0.694768i	$0.755507\pi$
$692$	−6.97332	−0.265086
$693$	−2.67260	−0.101524
$694$	−25.8460	−0.981102
$695$	24.8980	0.944434
$696$	227.467	8.62212
$697$	8.06250	0.305389
$698$	−66.6783	−2.52381
$699$	−24.9501	−0.943698
$700$	−18.1103	−0.684505
$701$	6.69726	0.252952	0.126476	−	0.991970i	$-0.459633\pi$
$701$	6.69726	0.252952	0.126476	+	0.991970i	$0.459633\pi$
$702$	−128.826	−4.86221
$703$	−8.50129	−0.320632
$704$	−0.663143	−0.0249931
$705$	72.6527	2.73626
$706$	39.8280	1.49895
$707$	3.11756	0.117248
$708$	−218.002	−8.19301
$709$	−3.66097	−0.137491	−0.0687453	−	0.997634i	$-0.521900\pi$
$709$	−3.66097	−0.137491	−0.0687453	+	0.997634i	$0.521900\pi$
$710$	68.8701	2.58465
$711$	10.0487	0.376856
$712$	52.1677	1.95507
$713$	−14.8296	−0.555373
$714$	50.9252	1.90583
$715$	−0.661425	−0.0247359
$716$	30.6309	1.14473
$717$	−28.5697	−1.06695
$718$	44.6673	1.66697
$719$	40.2141	1.49973	0.749866	−	0.661589i	$-0.230118\pi$
$719$	40.2141	1.49973	0.749866	+	0.661589i	$0.230118\pi$
$720$	138.051	5.14485
$721$	29.0169	1.08065
$722$	−50.2179	−1.86892
$723$	−76.7259	−2.85347
$724$	42.3481	1.57385
$725$	7.53638	0.279894
$726$	90.4480	3.35684
$727$	−40.6111	−1.50618	−0.753092	−	0.657916i	$-0.771438\pi$
$727$	−40.6111	−1.50618	−0.753092	+	0.657916i	$0.771438\pi$
$728$	−144.237	−5.34579
$729$	7.65098	0.283370
$730$	32.3756	1.19827
$731$	8.25759	0.305418
$732$	71.4613	2.64129
$733$	14.2337	0.525735	0.262868	−	0.964832i	$-0.415332\pi$
$733$	14.2337	0.525735	0.262868	+	0.964832i	$0.415332\pi$
$734$	−35.1569	−1.29766
$735$	106.062	3.91214
$736$	−17.7453	−0.654099
$737$	0.0328506	0.00121007
$738$	113.659	4.18384
$739$	−25.4727	−0.937030	−0.468515	−	0.883456i	$-0.655211\pi$
$739$	−25.4727	−0.937030	−0.468515	+	0.883456i	$0.655211\pi$
$740$	−13.6815	−0.502941
$741$	78.1120	2.86952
$742$	−74.5951	−2.73847
$743$	20.1069	0.737652	0.368826	−	0.929499i	$-0.379760\pi$
$743$	20.1069	0.737652	0.368826	+	0.929499i	$0.379760\pi$
$744$	207.100	7.59265
$745$	31.8226	1.16589
$746$	35.1989	1.28872
$747$	119.088	4.35719
$748$	0.495920	0.0181326
$749$	−4.19034	−0.153112
$750$	97.6552	3.56586
$751$	−40.8001	−1.48882	−0.744409	−	0.667724i	$-0.767269\pi$
$751$	−40.8001	−1.48882	−0.744409	+	0.667724i	$0.767269\pi$
$752$	109.354	3.98772
$753$	50.3648	1.83539
$754$	102.313	3.72601
$755$	−40.0269	−1.45673
$756$	287.039	10.4395
$757$	17.8518	0.648836	0.324418	−	0.945914i	$-0.394831\pi$
$757$	17.8518	0.648836	0.324418	+	0.945914i	$0.394831\pi$
$758$	34.8528	1.26591
$759$	−0.421407	−0.0152961
$760$	−94.3229	−3.42145
$761$	−7.49828	−0.271812	−0.135906	−	0.990722i	$-0.543395\pi$
$761$	−7.49828	−0.271812	−0.135906	+	0.990722i	$0.543395\pi$
$762$	−65.1483	−2.36007
$763$	74.9767	2.71434
$764$	−2.48777	−0.0900043
$765$	−18.1461	−0.656075
$766$	30.6070	1.10587
$767$	−57.5249	−2.07710
$768$	−48.5698	−1.75261
$769$	−12.9668	−0.467596	−0.233798	−	0.972285i	$-0.575115\pi$
$769$	−12.9668	−0.467596	−0.233798	+	0.972285i	$0.575115\pi$
$770$	2.08289	0.0750622
$771$	69.5871	2.50612
$772$	−108.107	−3.89084
$773$	−51.3783	−1.84795	−0.923974	−	0.382454i	$-0.875079\pi$
$773$	−51.3783	−1.84795	−0.923974	+	0.382454i	$0.875079\pi$
$774$	116.409	4.18423
$775$	6.86157	0.246475
$776$	−38.3769	−1.37765
$777$	−20.9317	−0.750921
$778$	80.5185	2.88673
$779$	−38.9440	−1.39532
$780$	125.709	4.50110
$781$	1.02608	0.0367160
$782$	5.59601	0.200113
$783$	−119.448	−4.26872
$784$	159.639	5.70140
$785$	24.1195	0.860862
$786$	147.265	5.25278
$787$	−1.07853	−0.0384456	−0.0192228	−	0.999815i	$-0.506119\pi$
$787$	−1.07853	−0.0384456	−0.0192228	+	0.999815i	$0.506119\pi$
$788$	52.1684	1.85842
$789$	−15.6566	−0.557389
$790$	−7.83146	−0.278631
$791$	−2.31876	−0.0824456
$792$	4.10138	0.145736
$793$	18.8568	0.669623
$794$	−37.8073	−1.34173
$795$	38.1403	1.35270
$796$	−126.034	−4.46717
$797$	31.4963	1.11566	0.557828	−	0.829957i	$-0.311635\pi$
$797$	31.4963	1.11566	0.557828	+	0.829957i	$0.311635\pi$
$798$	−245.983	−8.70769
$799$	−14.3740	−0.508517
$800$	8.21064	0.290290
$801$	−48.4774	−1.71287
$802$	−26.3556	−0.930650
$803$	0.482357	0.0170220
$804$	−6.24349	−0.220191
$805$	16.6298	0.586123
$806$	93.1516	3.28113
$807$	−79.7009	−2.80561
$808$	−4.78421	−0.168308
$809$	−44.1209	−1.55121	−0.775605	−	0.631219i	$-0.782555\pi$
$809$	−44.1209	−1.55121	−0.775605	+	0.631219i	$0.782555\pi$
$810$	−96.1729	−3.37917
$811$	−28.1830	−0.989639	−0.494819	−	0.868996i	$-0.664766\pi$
$811$	−28.1830	−0.989639	−0.494819	+	0.868996i	$0.664766\pi$
$812$	−227.965	−8.00001
$813$	−41.6219	−1.45974
$814$	−0.288091	−0.0100976
$815$	15.0051	0.525606
$816$	−39.1912	−1.37197
$817$	−39.8864	−1.39545
$818$	44.9798	1.57268
$819$	134.034	4.68353
$820$	−62.6742	−2.18868
$821$	55.2763	1.92916	0.964578	−	0.263799i	$-0.0849755\pi$
$821$	55.2763	1.92916	0.964578	+	0.263799i	$0.0849755\pi$
$822$	29.8776	1.04210
$823$	7.82491	0.272759	0.136380	−	0.990657i	$-0.456453\pi$
$823$	7.82491	0.272759	0.136380	+	0.990657i	$0.456453\pi$
$824$	−44.5294	−1.55125
$825$	0.194983	0.00678844
$826$	181.152	6.30307
$827$	−21.3020	−0.740743	−0.370371	−	0.928884i	$-0.620770\pi$
$827$	−21.3020	−0.740743	−0.370371	+	0.928884i	$0.620770\pi$
$828$	55.8167	1.93977
$829$	40.5881	1.40968	0.704842	−	0.709365i	$-0.251018\pi$
$829$	40.5881	1.40968	0.704842	+	0.709365i	$0.251018\pi$
$830$	−92.8110	−3.22152
$831$	54.4188	1.88777
$832$	33.2574	1.15299
$833$	−20.9838	−0.727047
$834$	−99.6430	−3.45036
$835$	1.79557	0.0621384
$836$	−2.39543	−0.0828475
$837$	−108.753	−3.75904
$838$	−55.7918	−1.92730
$839$	2.19275	0.0757023	0.0378511	−	0.999283i	$-0.487949\pi$
$839$	2.19275	0.0757023	0.0378511	+	0.999283i	$0.487949\pi$
$840$	−232.240	−8.01304
$841$	65.8650	2.27121
$842$	−10.4462	−0.360001
$843$	99.9948	3.44401
$844$	138.680	4.77358
$845$	6.44608	0.221752
$846$	−202.634	−6.96669
$847$	−53.1782	−1.82723
$848$	57.4071	1.97137
$849$	59.8560	2.05425
$850$	−2.58924	−0.0888104
$851$	−2.30012	−0.0788470
$852$	−195.014	−6.68108
$853$	21.4500	0.734435	0.367218	−	0.930135i	$-0.380310\pi$
$853$	21.4500	0.734435	0.367218	+	0.930135i	$0.380310\pi$
$854$	−59.3818	−2.03200
$855$	87.6506	2.99759
$856$	6.43051	0.219790
$857$	24.6074	0.840574	0.420287	−	0.907391i	$-0.361929\pi$
$857$	24.6074	0.840574	0.420287	+	0.907391i	$0.361929\pi$
$858$	2.64706	0.0903690
$859$	−4.07711	−0.139109	−0.0695546	−	0.997578i	$-0.522158\pi$
$859$	−4.07711	−0.139109	−0.0695546	+	0.997578i	$0.522158\pi$
$860$	−64.1907	−2.18888
$861$	−95.8873	−3.26783
$862$	8.93952	0.304481
$863$	3.97417	0.135282	0.0676412	−	0.997710i	$-0.478453\pi$
$863$	3.97417	0.135282	0.0676412	+	0.997710i	$0.478453\pi$
$864$	−130.135	−4.42727
$865$	−2.96275	−0.100737
$866$	107.837	3.66446
$867$	−48.3326	−1.64146
$868$	−207.553	−7.04481
$869$	−0.116679	−0.00395807
$870$	164.736	5.58508
$871$	−1.64749	−0.0558232
$872$	−115.059	−3.89640
$873$	35.6622	1.20698
$874$	−27.0302	−0.914311
$875$	−57.4157	−1.94100
$876$	−91.6755	−3.09743
$877$	−24.8118	−0.837836	−0.418918	−	0.908024i	$-0.637591\pi$
$877$	−24.8118	−0.837836	−0.418918	+	0.908024i	$0.637591\pi$
$878$	63.3673	2.13854
$879$	12.3525	0.416639
$880$	−1.60296	−0.0540357
$881$	−15.3087	−0.515765	−0.257882	−	0.966176i	$-0.583025\pi$
$881$	−15.3087	−0.515765	−0.257882	+	0.966176i	$0.583025\pi$
$882$	−295.813	−9.96056
$883$	−9.57583	−0.322252	−0.161126	−	0.986934i	$-0.551513\pi$
$883$	−9.57583	−0.322252	−0.161126	+	0.986934i	$0.551513\pi$
$884$	−24.8710	−0.836501
$885$	−92.6223	−3.11347
$886$	2.47363	0.0831031
$887$	22.7077	0.762451	0.381226	−	0.924482i	$-0.375502\pi$
$887$	22.7077	0.762451	0.381226	+	0.924482i	$0.375502\pi$
$888$	32.1219	1.07794
$889$	38.3035	1.28466
$890$	37.7809	1.26642
$891$	−1.43286	−0.0480026
$892$	26.6363	0.891848
$893$	69.4304	2.32340
$894$	−127.356	−4.25941
$895$	13.0141	0.435014
$896$	−2.07297	−0.0692530
$897$	21.1341	0.705646
$898$	81.6589	2.72499
$899$	86.3708	2.88063
$900$	−25.8261	−0.860870
$901$	−7.54590	−0.251390
$902$	−1.31973	−0.0439423
$903$	−98.2074	−3.26814
$904$	3.55837	0.118350
$905$	17.9924	0.598088
$906$	160.190	5.32194
$907$	44.2110	1.46800	0.734001	−	0.679149i	$-0.237651\pi$
$907$	44.2110	1.46800	0.734001	+	0.679149i	$0.237651\pi$
$908$	78.1174	2.59242
$909$	4.44578	0.147457
$910$	−104.459	−3.46280
$911$	−49.0082	−1.62371	−0.811857	−	0.583856i	$-0.801543\pi$
$911$	−49.0082	−1.62371	−0.811857	+	0.583856i	$0.801543\pi$
$912$	189.304	6.26848
$913$	−1.38277	−0.0457630
$914$	9.25153	0.306013
$915$	30.3617	1.00373
$916$	82.8833	2.73854
$917$	−86.5836	−2.85924
$918$	41.0383	1.35446
$919$	17.7367	0.585079	0.292540	−	0.956253i	$-0.405500\pi$
$919$	17.7367	0.585079	0.292540	+	0.956253i	$0.405500\pi$
$920$	−25.5201	−0.841373
$921$	55.9224	1.84270
$922$	33.9494	1.11807
$923$	−51.4591	−1.69380
$924$	−5.89797	−0.194029
$925$	1.06425	0.0349924
$926$	−49.9289	−1.64077
$927$	41.3794	1.35908
$928$	103.352	3.39271
$929$	−28.3175	−0.929068	−0.464534	−	0.885555i	$-0.653778\pi$
$929$	−28.3175	−0.929068	−0.464534	+	0.885555i	$0.653778\pi$
$930$	149.986	4.91823
$931$	101.358	3.32186
$932$	−38.3723	−1.25693
$933$	−2.76866	−0.0906420
$934$	54.8179	1.79370
$935$	0.210701	0.00689067
$936$	−205.689	−6.72315
$937$	−29.5052	−0.963892	−0.481946	−	0.876201i	$-0.660070\pi$
$937$	−29.5052	−0.963892	−0.481946	+	0.876201i	$0.660070\pi$
$938$	5.18812	0.169398
$939$	−19.2072	−0.626804
$940$	111.737	3.64446
$941$	−13.1958	−0.430171	−0.215085	−	0.976595i	$-0.569003\pi$
$941$	−13.1958	−0.430171	−0.215085	+	0.976595i	$0.569003\pi$
$942$	−96.5276	−3.14504
$943$	−10.5367	−0.343123
$944$	−139.411	−4.53745
$945$	121.954	3.96718
$946$	−1.35167	−0.0439465
$947$	20.1737	0.655556	0.327778	−	0.944755i	$-0.393700\pi$
$947$	20.1737	0.655556	0.327778	+	0.944755i	$0.393700\pi$
$948$	22.1758	0.720235
$949$	−24.1908	−0.785265
$950$	12.5067	0.405772
$951$	−72.6380	−2.35545
$952$	45.9477	1.48917
$953$	35.1581	1.13888	0.569441	−	0.822033i	$-0.307160\pi$
$953$	35.1581	1.13888	0.569441	+	0.822033i	$0.307160\pi$
$954$	−106.376	−3.44405
$955$	−1.05698	−0.0342030
$956$	−43.9391	−1.42109
$957$	2.45437	0.0793385
$958$	−54.8879	−1.77335
$959$	−17.5663	−0.567245
$960$	53.5486	1.72827
$961$	47.6371	1.53668
$962$	14.4481	0.465826
$963$	−5.97562	−0.192562
$964$	−118.001	−3.80057
$965$	−45.9312	−1.47858
$966$	−66.5533	−2.14132
$967$	5.68959	0.182965	0.0914824	−	0.995807i	$-0.470839\pi$
$967$	5.68959	0.182965	0.0914824	+	0.995807i	$0.470839\pi$
$968$	81.6074	2.62296
$969$	−24.8831	−0.799361
$970$	−27.7933	−0.892390
$971$	14.3493	0.460491	0.230246	−	0.973133i	$-0.426047\pi$
$971$	14.3493	0.460491	0.230246	+	0.973133i	$0.426047\pi$
$972$	94.3059	3.02486
$973$	58.5844	1.87813
$974$	43.6761	1.39947
$975$	−9.77862	−0.313167
$976$	45.6992	1.46280
$977$	31.9540	1.02230	0.511150	−	0.859492i	$-0.329220\pi$
$977$	31.9540	1.02230	0.511150	+	0.859492i	$0.329220\pi$
$978$	−60.0512	−1.92023
$979$	0.562889	0.0179900
$980$	163.119	5.21063
$981$	106.920	3.41370
$982$	−49.2069	−1.57026
$983$	−11.4976	−0.366717	−0.183358	−	0.983046i	$-0.558697\pi$
$983$	−11.4976	−0.366717	−0.183358	+	0.983046i	$0.558697\pi$
$984$	147.149	4.69093
$985$	22.1648	0.706228
$986$	−32.5924	−1.03795
$987$	170.950	5.44140
$988$	120.133	3.82195
$989$	−10.7917	−0.343156
$990$	2.97030	0.0944023
$991$	16.3368	0.518957	0.259478	−	0.965749i	$-0.416449\pi$
$991$	16.3368	0.518957	0.259478	+	0.965749i	$0.416449\pi$
$992$	94.0982	2.98762
$993$	26.5262	0.841784
$994$	162.050	5.13991
$995$	−53.5482	−1.69759
$996$	262.806	8.32733
$997$	−35.8038	−1.13392	−0.566959	−	0.823746i	$-0.691880\pi$
$997$	−35.8038	−1.13392	−0.566959	+	0.823746i	$0.691880\pi$
$998$	18.1728	0.575251
$999$	−16.8679	−0.533676

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8039.2.a.b.1.20	✓	391

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8039.2.a.b.1.20	✓	391	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8039 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8039.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61507 q^{2} +3.14612 q^{3} +4.83861 q^{4} +2.05578 q^{5} -8.22734 q^{6} +4.83721 q^{7} -7.42319 q^{8} +6.89808 q^{9} +O(q^{10})\)	\(q-2.61507 q^{2} +3.14612 q^{3} +4.83861 q^{4} +2.05578 q^{5} -8.22734 q^{6} +4.83721 q^{7} -7.42319 q^{8} +6.89808 q^{9} -5.37602 q^{10} -0.0800961 q^{11} +15.2229 q^{12} +4.01691 q^{13} -12.6497 q^{14} +6.46773 q^{15} +9.73496 q^{16} -1.27961 q^{17} -18.0390 q^{18} +6.18088 q^{19} +9.94713 q^{20} +15.2184 q^{21} +0.209457 q^{22} +1.67230 q^{23} -23.3542 q^{24} -0.773766 q^{25} -10.5045 q^{26} +12.2638 q^{27} +23.4054 q^{28} -9.73987 q^{29} -16.9136 q^{30} -8.86776 q^{31} -10.6113 q^{32} -0.251992 q^{33} +3.34629 q^{34} +9.94423 q^{35} +33.3771 q^{36} -1.37542 q^{37} -16.1635 q^{38} +12.6377 q^{39} -15.2604 q^{40} -6.30073 q^{41} -39.7973 q^{42} -6.45319 q^{43} -0.387554 q^{44} +14.1809 q^{45} -4.37320 q^{46} +11.2331 q^{47} +30.6274 q^{48} +16.3986 q^{49} +2.02346 q^{50} -4.02582 q^{51} +19.4363 q^{52} +5.89701 q^{53} -32.0708 q^{54} -0.164660 q^{55} -35.9075 q^{56} +19.4458 q^{57} +25.4705 q^{58} -14.3207 q^{59} +31.2949 q^{60} +4.69434 q^{61} +23.1898 q^{62} +33.3674 q^{63} +8.27934 q^{64} +8.25789 q^{65} +0.658978 q^{66} -0.410139 q^{67} -6.19156 q^{68} +5.26127 q^{69} -26.0049 q^{70} -12.8106 q^{71} -51.2057 q^{72} -6.02222 q^{73} +3.59682 q^{74} -2.43436 q^{75} +29.9069 q^{76} -0.387441 q^{77} -33.0485 q^{78} +1.45674 q^{79} +20.0129 q^{80} +17.8892 q^{81} +16.4769 q^{82} +17.2639 q^{83} +73.6361 q^{84} -2.63061 q^{85} +16.8756 q^{86} -30.6428 q^{87} +0.594568 q^{88} -7.02767 q^{89} -37.0842 q^{90} +19.4306 q^{91} +8.09163 q^{92} -27.8990 q^{93} -29.3754 q^{94} +12.7065 q^{95} -33.3843 q^{96} +5.16987 q^{97} -42.8835 q^{98} -0.552509 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9}+O(q^{10}) \) 391 * q + 14 * q^2 + 12 * q^3 + 446 * q^4 + 22 * q^5 + 40 * q^6 + 63 * q^7 + 36 * q^8 + 501 * q^9	\( 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9} + 40 q^{10} + 57 q^{11} + 20 q^{12} + 83 q^{13} + 21 q^{14} + 60 q^{15} + 548 q^{16} + 59 q^{17} + 54 q^{18} + 131 q^{19} + 35 q^{20} + 121 q^{21} + 89 q^{22} + 34 q^{23} + 110 q^{24} + 609 q^{25} + 54 q^{26} + 27 q^{27} + 182 q^{28} + 102 q^{29} + 92 q^{30} + 88 q^{31} + 76 q^{32} + 131 q^{33} + 128 q^{34} + 31 q^{35} + 654 q^{36} + 135 q^{37} + 23 q^{38} + 96 q^{39} + 113 q^{40} + 128 q^{41} + 45 q^{42} + 140 q^{43} + 151 q^{44} + 77 q^{45} + 245 q^{46} + 22 q^{47} + 25 q^{48} + 712 q^{49} + 53 q^{50} + 102 q^{51} + 174 q^{52} + 54 q^{53} + 131 q^{54} + 101 q^{55} + 43 q^{56} + 226 q^{57} + 109 q^{58} + 40 q^{59} + 123 q^{60} + 249 q^{61} + 28 q^{62} + 139 q^{63} + 730 q^{64} + 227 q^{65} + 55 q^{66} + 169 q^{67} + 48 q^{68} + 89 q^{69} + 98 q^{70} + 66 q^{71} + 120 q^{72} + 324 q^{73} + 60 q^{74} + 19 q^{75} + 356 q^{76} + 83 q^{77} - 11 q^{78} + 195 q^{79} + 26 q^{80} + 807 q^{81} + 49 q^{82} + 74 q^{83} + 252 q^{84} + 373 q^{85} + 100 q^{86} + 43 q^{87} + 211 q^{88} + 207 q^{89} + 10 q^{90} + 189 q^{91} + 30 q^{92} + 172 q^{93} + 130 q^{94} + 43 q^{95} + 203 q^{96} + 254 q^{97} + 26 q^{98} + 273 q^{99}+O(q^{100}) \) 391 * q + 14 * q^2 + 12 * q^3 + 446 * q^4 + 22 * q^5 + 40 * q^6 + 63 * q^7 + 36 * q^8 + 501 * q^9 + 40 * q^10 + 57 * q^11 + 20 * q^12 + 83 * q^13 + 21 * q^14 + 60 * q^15 + 548 * q^16 + 59 * q^17 + 54 * q^18 + 131 * q^19 + 35 * q^20 + 121 * q^21 + 89 * q^22 + 34 * q^23 + 110 * q^24 + 609 * q^25 + 54 * q^26 + 27 * q^27 + 182 * q^28 + 102 * q^29 + 92 * q^30 + 88 * q^31 + 76 * q^32 + 131 * q^33 + 128 * q^34 + 31 * q^35 + 654 * q^36 + 135 * q^37 + 23 * q^38 + 96 * q^39 + 113 * q^40 + 128 * q^41 + 45 * q^42 + 140 * q^43 + 151 * q^44 + 77 * q^45 + 245 * q^46 + 22 * q^47 + 25 * q^48 + 712 * q^49 + 53 * q^50 + 102 * q^51 + 174 * q^52 + 54 * q^53 + 131 * q^54 + 101 * q^55 + 43 * q^56 + 226 * q^57 + 109 * q^58 + 40 * q^59 + 123 * q^60 + 249 * q^61 + 28 * q^62 + 139 * q^63 + 730 * q^64 + 227 * q^65 + 55 * q^66 + 169 * q^67 + 48 * q^68 + 89 * q^69 + 98 * q^70 + 66 * q^71 + 120 * q^72 + 324 * q^73 + 60 * q^74 + 19 * q^75 + 356 * q^76 + 83 * q^77 - 11 * q^78 + 195 * q^79 + 26 * q^80 + 807 * q^81 + 49 * q^82 + 74 * q^83 + 252 * q^84 + 373 * q^85 + 100 * q^86 + 43 * q^87 + 211 * q^88 + 207 * q^89 + 10 * q^90 + 189 * q^91 + 30 * q^92 + 172 * q^93 + 130 * q^94 + 43 * q^95 + 203 * q^96 + 254 * q^97 + 26 * q^98 + 273 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more