LMFDB - Embedded newform 4019.2.a.b.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4019,2,Mod(1,4019)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4019, 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("4019.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4019.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:	$32.0918765724$
Analytic rank:	$0$
Dimension:	$186$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	4019.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56492 q^{2} -2.35383 q^{3} +4.57881 q^{4} +1.72936 q^{5} +6.03739 q^{6} -3.13382 q^{7} -6.61443 q^{8} +2.54053 q^{9} +O(q^{10})$	\(q-2.56492 q^{2} -2.35383 q^{3} +4.57881 q^{4} +1.72936 q^{5} +6.03739 q^{6} -3.13382 q^{7} -6.61443 q^{8} +2.54053 q^{9} -4.43566 q^{10} -4.23381 q^{11} -10.7777 q^{12} +2.84943 q^{13} +8.03799 q^{14} -4.07062 q^{15} +7.80786 q^{16} -1.50486 q^{17} -6.51625 q^{18} -1.85001 q^{19} +7.91840 q^{20} +7.37648 q^{21} +10.8594 q^{22} -7.40761 q^{23} +15.5693 q^{24} -2.00932 q^{25} -7.30855 q^{26} +1.08151 q^{27} -14.3491 q^{28} -1.18024 q^{29} +10.4408 q^{30} -7.01727 q^{31} -6.79766 q^{32} +9.96567 q^{33} +3.85985 q^{34} -5.41950 q^{35} +11.6326 q^{36} -0.469227 q^{37} +4.74514 q^{38} -6.70707 q^{39} -11.4387 q^{40} +3.60425 q^{41} -18.9201 q^{42} -10.0365 q^{43} -19.3858 q^{44} +4.39349 q^{45} +18.9999 q^{46} -11.2607 q^{47} -18.3784 q^{48} +2.82082 q^{49} +5.15374 q^{50} +3.54219 q^{51} +13.0470 q^{52} -6.26150 q^{53} -2.77400 q^{54} -7.32177 q^{55} +20.7284 q^{56} +4.35462 q^{57} +3.02722 q^{58} -8.36887 q^{59} -18.6386 q^{60} +14.3617 q^{61} +17.9987 q^{62} -7.96156 q^{63} +1.81973 q^{64} +4.92768 q^{65} -25.5611 q^{66} +14.3594 q^{67} -6.89047 q^{68} +17.4363 q^{69} +13.9006 q^{70} +12.7596 q^{71} -16.8042 q^{72} -12.7007 q^{73} +1.20353 q^{74} +4.72960 q^{75} -8.47086 q^{76} +13.2680 q^{77} +17.2031 q^{78} -0.827441 q^{79} +13.5026 q^{80} -10.1673 q^{81} -9.24461 q^{82} -12.6263 q^{83} +33.7755 q^{84} -2.60245 q^{85} +25.7429 q^{86} +2.77809 q^{87} +28.0042 q^{88} +16.1548 q^{89} -11.2689 q^{90} -8.92958 q^{91} -33.9180 q^{92} +16.5175 q^{93} +28.8828 q^{94} -3.19934 q^{95} +16.0006 q^{96} +5.98839 q^{97} -7.23516 q^{98} -10.7561 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9}+O(q^{10}) $ 186 * q + 6 * q^2 + 10 * q^3 + 212 * q^4 + 38 * q^5 + 47 * q^6 + 32 * q^7 + 15 * q^8 + 216 * q^9	\( 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9} + 50 q^{10} + 25 q^{11} + 17 q^{12} + 113 q^{13} + 12 q^{14} + 12 q^{15} + 252 q^{16} + 35 q^{17} + 13 q^{18} + 97 q^{19} + 55 q^{20} + 115 q^{21} + 14 q^{22} + 27 q^{23} + 122 q^{24} + 244 q^{25} + 39 q^{26} + 34 q^{27} + 66 q^{28} + 91 q^{29} + 4 q^{30} + 135 q^{31} + 21 q^{32} + 32 q^{33} + 58 q^{34} + 17 q^{35} + 273 q^{36} + 133 q^{37} - 3 q^{38} + 55 q^{39} + 142 q^{40} + 97 q^{41} - 8 q^{42} + 67 q^{43} + 44 q^{44} + 154 q^{45} + 101 q^{46} + 20 q^{47} - 7 q^{48} + 312 q^{49} + 21 q^{50} + 23 q^{51} + 193 q^{52} + 22 q^{53} + 141 q^{54} + 88 q^{55} + 28 q^{56} + 65 q^{57} + 62 q^{58} + 41 q^{59} + q^{60} + 377 q^{61} + 29 q^{62} + 39 q^{63} + 311 q^{64} + 21 q^{65} + 35 q^{66} + 42 q^{67} + 24 q^{68} + 137 q^{69} + 35 q^{70} + 17 q^{71} - 8 q^{72} + 213 q^{73} - 9 q^{74} + 2 q^{75} + 242 q^{76} + 60 q^{77} + 103 q^{79} + 80 q^{80} + 270 q^{81} + 84 q^{82} + 42 q^{83} + 137 q^{84} + 294 q^{85} - 9 q^{86} + 22 q^{87} - 13 q^{88} + 78 q^{89} + 69 q^{90} + 118 q^{91} + 49 q^{92} + 51 q^{93} + 93 q^{94} + 10 q^{95} + 260 q^{96} + 142 q^{97} - 31 q^{98} + 78 q^{99}+O(q^{100}) \) 186 * q + 6 * q^2 + 10 * q^3 + 212 * q^4 + 38 * q^5 + 47 * q^6 + 32 * q^7 + 15 * q^8 + 216 * q^9 + 50 * q^10 + 25 * q^11 + 17 * q^12 + 113 * q^13 + 12 * q^14 + 12 * q^15 + 252 * q^16 + 35 * q^17 + 13 * q^18 + 97 * q^19 + 55 * q^20 + 115 * q^21 + 14 * q^22 + 27 * q^23 + 122 * q^24 + 244 * q^25 + 39 * q^26 + 34 * q^27 + 66 * q^28 + 91 * q^29 + 4 * q^30 + 135 * q^31 + 21 * q^32 + 32 * q^33 + 58 * q^34 + 17 * q^35 + 273 * q^36 + 133 * q^37 - 3 * q^38 + 55 * q^39 + 142 * q^40 + 97 * q^41 - 8 * q^42 + 67 * q^43 + 44 * q^44 + 154 * q^45 + 101 * q^46 + 20 * q^47 - 7 * q^48 + 312 * q^49 + 21 * q^50 + 23 * q^51 + 193 * q^52 + 22 * q^53 + 141 * q^54 + 88 * q^55 + 28 * q^56 + 65 * q^57 + 62 * q^58 + 41 * q^59 + q^60 + 377 * q^61 + 29 * q^62 + 39 * q^63 + 311 * q^64 + 21 * q^65 + 35 * q^66 + 42 * q^67 + 24 * q^68 + 137 * q^69 + 35 * q^70 + 17 * q^71 - 8 * q^72 + 213 * q^73 - 9 * q^74 + 2 * q^75 + 242 * q^76 + 60 * q^77 + 103 * q^79 + 80 * q^80 + 270 * q^81 + 84 * q^82 + 42 * q^83 + 137 * q^84 + 294 * q^85 - 9 * q^86 + 22 * q^87 - 13 * q^88 + 78 * q^89 + 69 * q^90 + 118 * q^91 + 49 * q^92 + 51 * q^93 + 93 * q^94 + 10 * q^95 + 260 * q^96 + 142 * q^97 - 31 * q^98 + 78 * 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 $

$2$	−2.56492	−1.81367	−0.906836	−	0.421484i	$-0.861509\pi$
$2$	−2.56492	−1.81367	−0.906836	+	0.421484i	$0.861509\pi$
$3$	−2.35383	−1.35899	−0.679493	−	0.733682i	$-0.737800\pi$
$3$	−2.35383	−1.35899	−0.679493	+	0.733682i	$0.737800\pi$
$4$	4.57881	2.28940
$5$	1.72936	0.773393	0.386696	−	0.922207i	$-0.373616\pi$
$5$	1.72936	0.773393	0.386696	+	0.922207i	$0.373616\pi$
$6$	6.03739	2.46475
$7$	−3.13382	−1.18447	−0.592236	−	0.805765i	$-0.701755\pi$
$7$	−3.13382	−1.18447	−0.592236	+	0.805765i	$0.701755\pi$
$8$	−6.61443	−2.33855
$9$	2.54053	0.846844
$10$	−4.43566	−1.40268
$11$	−4.23381	−1.27654	−0.638270	−	0.769812i	$-0.720350\pi$
$11$	−4.23381	−1.27654	−0.638270	+	0.769812i	$0.720350\pi$
$12$	−10.7777	−3.11127
$13$	2.84943	0.790289	0.395144	−	0.918619i	$-0.370695\pi$
$13$	2.84943	0.790289	0.395144	+	0.918619i	$0.370695\pi$
$14$	8.03799	2.14824
$15$	−4.07062	−1.05103
$16$	7.80786	1.95196
$17$	−1.50486	−0.364983	−0.182491	−	0.983207i	$-0.558416\pi$
$17$	−1.50486	−0.364983	−0.182491	+	0.983207i	$0.558416\pi$
$18$	−6.51625	−1.53590
$19$	−1.85001	−0.424422	−0.212211	−	0.977224i	$-0.568066\pi$
$19$	−1.85001	−0.424422	−0.212211	+	0.977224i	$0.568066\pi$
$20$	7.91840	1.77061
$21$	7.37648	1.60968
$22$	10.8594	2.31522
$23$	−7.40761	−1.54459	−0.772297	−	0.635262i	$-0.780892\pi$
$23$	−7.40761	−1.54459	−0.772297	+	0.635262i	$0.780892\pi$
$24$	15.5693	3.17806
$25$	−2.00932	−0.401864
$26$	−7.30855	−1.43332
$27$	1.08151	0.208138
$28$	−14.3491	−2.71173
$29$	−1.18024	−0.219165	−0.109583	−	0.993978i	$-0.534951\pi$
$29$	−1.18024	−0.219165	−0.109583	+	0.993978i	$0.534951\pi$
$30$	10.4408	1.90622
$31$	−7.01727	−1.26034	−0.630169	−	0.776458i	$-0.717015\pi$
$31$	−7.01727	−1.26034	−0.630169	+	0.776458i	$0.717015\pi$
$32$	−6.79766	−1.20167
$33$	9.96567	1.73480
$34$	3.85985	0.661958
$35$	−5.41950	−0.916062
$36$	11.6326	1.93877
$37$	−0.469227	−0.0771404	−0.0385702	−	0.999256i	$-0.512280\pi$
$37$	−0.469227	−0.0771404	−0.0385702	+	0.999256i	$0.512280\pi$
$38$	4.74514	0.769763
$39$	−6.70707	−1.07399
$40$	−11.4387	−1.80862
$41$	3.60425	0.562890	0.281445	−	0.959577i	$-0.409186\pi$
$41$	3.60425	0.562890	0.281445	+	0.959577i	$0.409186\pi$
$42$	−18.9201	−2.91943
$43$	−10.0365	−1.53055	−0.765277	−	0.643701i	$-0.777398\pi$
$43$	−10.0365	−1.53055	−0.765277	+	0.643701i	$0.777398\pi$
$44$	−19.3858	−2.92252
$45$	4.39349	0.654943
$46$	18.9999	2.80138
$47$	−11.2607	−1.64254	−0.821270	−	0.570539i	$-0.806734\pi$
$47$	−11.2607	−1.64254	−0.821270	+	0.570539i	$0.806734\pi$
$48$	−18.3784	−2.65269
$49$	2.82082	0.402974
$50$	5.15374	0.728849
$51$	3.54219	0.496006
$52$	13.0470	1.80929
$53$	−6.26150	−0.860083	−0.430042	−	0.902809i	$-0.641501\pi$
$53$	−6.26150	−0.860083	−0.430042	+	0.902809i	$0.641501\pi$
$54$	−2.77400	−0.377493
$55$	−7.32177	−0.987267
$56$	20.7284	2.76995
$57$	4.35462	0.576784
$58$	3.02722	0.397493
$59$	−8.36887	−1.08953	−0.544767	−	0.838587i	$-0.683382\pi$
$59$	−8.36887	−1.08953	−0.544767	+	0.838587i	$0.683382\pi$
$60$	−18.6386	−2.40623
$61$	14.3617	1.83883	0.919416	−	0.393286i	$-0.128662\pi$
$61$	14.3617	1.83883	0.919416	+	0.393286i	$0.128662\pi$
$62$	17.9987	2.28584
$63$	−7.96156	−1.00306
$64$	1.81973	0.227466
$65$	4.92768	0.611203
$66$	−25.5611	−3.14636
$67$	14.3594	1.75429	0.877143	−	0.480230i	$-0.159447\pi$
$67$	14.3594	1.75429	0.877143	+	0.480230i	$0.159447\pi$
$68$	−6.89047	−0.835592
$69$	17.4363	2.09908
$70$	13.9006	1.66144
$71$	12.7596	1.51429	0.757144	−	0.653248i	$-0.226594\pi$
$71$	12.7596	1.51429	0.757144	+	0.653248i	$0.226594\pi$
$72$	−16.8042	−1.98039
$73$	−12.7007	−1.48651	−0.743253	−	0.669010i	$-0.766718\pi$
$73$	−12.7007	−1.48651	−0.743253	+	0.669010i	$0.766718\pi$
$74$	1.20353	0.139907
$75$	4.72960	0.546127
$76$	−8.47086	−0.971674
$77$	13.2680	1.51203
$78$	17.2031	1.94787
$79$	−0.827441	−0.0930944	−0.0465472	−	0.998916i	$-0.514822\pi$
$79$	−0.827441	−0.0930944	−0.0465472	+	0.998916i	$0.514822\pi$
$80$	13.5026	1.50963
$81$	−10.1673	−1.12970
$82$	−9.24461	−1.02090
$83$	−12.6263	−1.38592	−0.692959	−	0.720977i	$-0.743694\pi$
$83$	−12.6263	−1.38592	−0.692959	+	0.720977i	$0.743694\pi$
$84$	33.7755	3.68521
$85$	−2.60245	−0.282275
$86$	25.7429	2.77592
$87$	2.77809	0.297842
$88$	28.0042	2.98526
$89$	16.1548	1.71241	0.856205	−	0.516636i	$-0.172816\pi$
$89$	16.1548	1.71241	0.856205	+	0.516636i	$0.172816\pi$
$90$	−11.2689	−1.18785
$91$	−8.92958	−0.936075
$92$	−33.9180	−3.53620
$93$	16.5175	1.71278
$94$	28.8828	2.97903
$95$	−3.19934	−0.328245
$96$	16.0006	1.63305
$97$	5.98839	0.608029	0.304015	−	0.952667i	$-0.401673\pi$
$97$	5.98839	0.608029	0.304015	+	0.952667i	$0.401673\pi$
$98$	−7.23516	−0.730862
$99$	−10.7561	−1.08103
$100$	−9.20028	−0.920028
$101$	−9.67112	−0.962313	−0.481156	−	0.876635i	$-0.659783\pi$
$101$	−9.67112	−0.962313	−0.481156	+	0.876635i	$0.659783\pi$
$102$	−9.08544	−0.899592
$103$	14.9180	1.46991	0.734955	−	0.678116i	$-0.237203\pi$
$103$	14.9180	1.46991	0.734955	+	0.678116i	$0.237203\pi$
$104$	−18.8473	−1.84813
$105$	12.7566	1.24492
$106$	16.0602	1.55991
$107$	−14.4129	−1.39335	−0.696673	−	0.717388i	$-0.745337\pi$
$107$	−14.4129	−1.39335	−0.696673	+	0.717388i	$0.745337\pi$
$108$	4.95205	0.476511
$109$	−3.42377	−0.327938	−0.163969	−	0.986465i	$-0.552430\pi$
$109$	−3.42377	−0.327938	−0.163969	+	0.986465i	$0.552430\pi$
$110$	18.7797	1.79058
$111$	1.10448	0.104833
$112$	−24.4684	−2.31205
$113$	−20.0631	−1.88738	−0.943690	−	0.330831i	$-0.892671\pi$
$113$	−20.0631	−1.88738	−0.943690	+	0.330831i	$0.892671\pi$
$114$	−11.1693	−1.04610
$115$	−12.8104	−1.19458
$116$	−5.40409	−0.501757
$117$	7.23905	0.669251
$118$	21.4655	1.97606
$119$	4.71596	0.432312
$120$	26.9248	2.45789
$121$	6.92511	0.629555
$122$	−36.8367	−3.33504
$123$	−8.48381	−0.764959
$124$	−32.1307	−2.88542
$125$	−12.1216	−1.08419
$126$	20.4208	1.81923
$127$	−0.833610	−0.0739709	−0.0369854	−	0.999316i	$-0.511776\pi$
$127$	−0.833610	−0.0739709	−0.0369854	+	0.999316i	$0.511776\pi$
$128$	8.92787	0.789120
$129$	23.6243	2.08000
$130$	−12.6391	−1.10852
$131$	13.5317	1.18227	0.591136	−	0.806572i	$-0.298680\pi$
$131$	13.5317	1.18227	0.591136	+	0.806572i	$0.298680\pi$
$132$	45.6309	3.97166
$133$	5.79761	0.502716
$134$	−36.8308	−3.18170
$135$	1.87033	0.160972
$136$	9.95380	0.853531
$137$	6.36466	0.543770	0.271885	−	0.962330i	$-0.412353\pi$
$137$	6.36466	0.543770	0.271885	+	0.962330i	$0.412353\pi$
$138$	−44.7226	−3.80704
$139$	0.239341	0.0203006	0.0101503	−	0.999948i	$-0.496769\pi$
$139$	0.239341	0.0203006	0.0101503	+	0.999948i	$0.496769\pi$
$140$	−24.8148	−2.09724
$141$	26.5058	2.23219
$142$	−32.7274	−2.74642
$143$	−12.0639	−1.00884
$144$	19.8361	1.65301
$145$	−2.04106	−0.169501
$146$	32.5763	2.69603
$147$	−6.63973	−0.547636
$148$	−2.14850	−0.176606
$149$	−14.3535	−1.17588	−0.587942	−	0.808903i	$-0.700062\pi$
$149$	−14.3535	−1.17588	−0.587942	+	0.808903i	$0.700062\pi$
$150$	−12.1310	−0.990495
$151$	−8.54922	−0.695725	−0.347863	−	0.937546i	$-0.613092\pi$
$151$	−8.54922	−0.695725	−0.347863	+	0.937546i	$0.613092\pi$
$152$	12.2368	0.992534
$153$	−3.82315	−0.309083
$154$	−34.0313	−2.74232
$155$	−12.1354	−0.974737
$156$	−30.7104	−2.45880
$157$	−11.7819	−0.940299	−0.470150	−	0.882587i	$-0.655800\pi$
$157$	−11.7819	−0.940299	−0.470150	+	0.882587i	$0.655800\pi$
$158$	2.12232	0.168843
$159$	14.7385	1.16884
$160$	−11.7556	−0.929361
$161$	23.2141	1.82953
$162$	26.0783	2.04890
$163$	−17.7086	−1.38705	−0.693523	−	0.720435i	$-0.743942\pi$
$163$	−17.7086	−1.38705	−0.693523	+	0.720435i	$0.743942\pi$
$164$	16.5032	1.28868
$165$	17.2342	1.34168
$166$	32.3855	2.51360
$167$	−14.8486	−1.14902	−0.574508	−	0.818499i	$-0.694807\pi$
$167$	−14.8486	−1.14902	−0.574508	+	0.818499i	$0.694807\pi$
$168$	−48.7912	−3.76433
$169$	−4.88077	−0.375444
$170$	6.67506	0.511954
$171$	−4.70002	−0.359419
$172$	−45.9553	−3.50406
$173$	−7.43849	−0.565538	−0.282769	−	0.959188i	$-0.591253\pi$
$173$	−7.43849	−0.565538	−0.282769	+	0.959188i	$0.591253\pi$
$174$	−7.12557	−0.540188
$175$	6.29684	0.475996
$176$	−33.0569	−2.49176
$177$	19.6989	1.48066
$178$	−41.4359	−3.10575
$179$	12.3096	0.920063	0.460031	−	0.887903i	$-0.347838\pi$
$179$	12.3096	0.920063	0.460031	+	0.887903i	$0.347838\pi$
$180$	20.1169	1.49943
$181$	−24.0222	−1.78556	−0.892780	−	0.450493i	$-0.851248\pi$
$181$	−24.0222	−1.78556	−0.892780	+	0.450493i	$0.851248\pi$
$182$	22.9037	1.69773
$183$	−33.8051	−2.49895
$184$	48.9971	3.61212
$185$	−0.811462	−0.0596598
$186$	−42.3660	−3.10642
$187$	6.37129	0.465915
$188$	−51.5605	−3.76044
$189$	−3.38927	−0.246533
$190$	8.20604	0.595329
$191$	21.9545	1.58857	0.794286	−	0.607544i	$-0.207845\pi$
$191$	21.9545	1.58857	0.794286	+	0.607544i	$0.207845\pi$
$192$	−4.28333	−0.309123
$193$	14.9290	1.07461	0.537306	−	0.843387i	$-0.319442\pi$
$193$	14.9290	1.07461	0.537306	+	0.843387i	$0.319442\pi$
$194$	−15.3597	−1.10276
$195$	−11.5989	−0.830617
$196$	12.9160	0.922569
$197$	−11.1127	−0.791748	−0.395874	−	0.918305i	$-0.629558\pi$
$197$	−11.1127	−0.791748	−0.395874	+	0.918305i	$0.629558\pi$
$198$	27.5886	1.96063
$199$	−9.01753	−0.639236	−0.319618	−	0.947547i	$-0.603555\pi$
$199$	−9.01753	−0.639236	−0.319618	+	0.947547i	$0.603555\pi$
$200$	13.2905	0.939780
$201$	−33.7997	−2.38405
$202$	24.8056	1.74532
$203$	3.69866	0.259595
$204$	16.2190	1.13556
$205$	6.23304	0.435335
$206$	−38.2633	−2.66593
$207$	−18.8193	−1.30803
$208$	22.2479	1.54262
$209$	7.83260	0.541792
$210$	−32.7196	−2.25787
$211$	25.4175	1.74981	0.874906	−	0.484292i	$-0.160923\pi$
$211$	25.4175	1.74981	0.874906	+	0.484292i	$0.160923\pi$
$212$	−28.6702	−1.96908
$213$	−30.0340	−2.05790
$214$	36.9679	2.52707
$215$	−17.3567	−1.18372
$216$	−7.15360	−0.486741
$217$	21.9908	1.49284
$218$	8.78170	0.594772
$219$	29.8954	2.02014
$220$	−33.5250	−2.26025
$221$	−4.28799	−0.288442
$222$	−2.83291	−0.190132
$223$	−8.35112	−0.559232	−0.279616	−	0.960112i	$-0.590207\pi$
$223$	−8.35112	−0.559232	−0.279616	+	0.960112i	$0.590207\pi$
$224$	21.3026	1.42334
$225$	−5.10473	−0.340316
$226$	51.4603	3.42309
$227$	−22.2708	−1.47816	−0.739082	−	0.673616i	$-0.764740\pi$
$227$	−22.2708	−1.47816	−0.739082	+	0.673616i	$0.764740\pi$
$228$	19.9390	1.32049
$229$	5.03103	0.332460	0.166230	−	0.986087i	$-0.446841\pi$
$229$	5.03103	0.332460	0.166230	+	0.986087i	$0.446841\pi$
$230$	32.8577	2.16657
$231$	−31.2306	−2.05482
$232$	7.80661	0.512529
$233$	0.216936	0.0142119	0.00710597	−	0.999975i	$-0.497738\pi$
$233$	0.216936	0.0142119	0.00710597	+	0.999975i	$0.497738\pi$
$234$	−18.5676	−1.21380
$235$	−19.4738	−1.27033
$236$	−38.3194	−2.49438
$237$	1.94766	0.126514
$238$	−12.0961	−0.784071
$239$	4.06053	0.262654	0.131327	−	0.991339i	$-0.458076\pi$
$239$	4.06053	0.262654	0.131327	+	0.991339i	$0.458076\pi$
$240$	−31.7828	−2.05157
$241$	3.31497	0.213536	0.106768	−	0.994284i	$-0.465950\pi$
$241$	3.31497	0.213536	0.106768	+	0.994284i	$0.465950\pi$
$242$	−17.7623	−1.14181
$243$	20.6876	1.32711
$244$	65.7596	4.20983
$245$	4.87820	0.311657
$246$	21.7603	1.38738
$247$	−5.27148	−0.335416
$248$	46.4152	2.94737
$249$	29.7203	1.88344
$250$	31.0910	1.96637
$251$	−24.0627	−1.51883	−0.759413	−	0.650609i	$-0.774514\pi$
$251$	−24.0627	−1.51883	−0.759413	+	0.650609i	$0.774514\pi$
$252$	−36.4544	−2.29641
$253$	31.3624	1.97174
$254$	2.13814	0.134159
$255$	6.12572	0.383608
$256$	−26.5387	−1.65867
$257$	−13.2377	−0.825747	−0.412874	−	0.910788i	$-0.635475\pi$
$257$	−13.2377	−0.825747	−0.412874	+	0.910788i	$0.635475\pi$
$258$	−60.5944	−3.77244
$259$	1.47047	0.0913707
$260$	22.5629	1.39929
$261$	−2.99844	−0.185599
$262$	−34.7078	−2.14425
$263$	22.3921	1.38075	0.690377	−	0.723450i	$-0.257445\pi$
$263$	22.3921	1.38075	0.690377	+	0.723450i	$0.257445\pi$
$264$	−65.9172	−4.05692
$265$	−10.8284	−0.665182
$266$	−14.8704	−0.911762
$267$	−38.0258	−2.32714
$268$	65.7491	4.01627
$269$	−11.6268	−0.708897	−0.354449	−	0.935075i	$-0.615331\pi$
$269$	−11.6268	−0.708897	−0.354449	+	0.935075i	$0.615331\pi$
$270$	−4.79723	−0.291950
$271$	20.3251	1.23466	0.617331	−	0.786704i	$-0.288214\pi$
$271$	20.3251	1.23466	0.617331	+	0.786704i	$0.288214\pi$
$272$	−11.7497	−0.712433
$273$	21.0187	1.27211
$274$	−16.3248	−0.986219
$275$	8.50706	0.512995
$276$	79.8374	4.80564
$277$	−22.8110	−1.37058	−0.685291	−	0.728269i	$-0.740325\pi$
$277$	−22.8110	−1.37058	−0.685291	+	0.728269i	$0.740325\pi$
$278$	−0.613889	−0.0368186
$279$	−17.8276	−1.06731
$280$	35.8469	2.14226
$281$	−27.8052	−1.65872	−0.829361	−	0.558713i	$-0.811295\pi$
$281$	−27.8052	−1.65872	−0.829361	+	0.558713i	$0.811295\pi$
$282$	−67.9852	−4.04846
$283$	2.11607	0.125787	0.0628937	−	0.998020i	$-0.479967\pi$
$283$	2.11607	0.125787	0.0628937	+	0.998020i	$0.479967\pi$
$284$	58.4238	3.46682
$285$	7.53071	0.446081
$286$	30.9430	1.82970
$287$	−11.2951	−0.666727
$288$	−17.2697	−1.01762
$289$	−14.7354	−0.866788
$290$	5.23515	0.307419
$291$	−14.0957	−0.826303
$292$	−58.1541	−3.40321
$293$	2.87678	0.168063	0.0840317	−	0.996463i	$-0.473220\pi$
$293$	2.87678	0.168063	0.0840317	+	0.996463i	$0.473220\pi$
$294$	17.0304	0.993231
$295$	−14.4728	−0.842638
$296$	3.10367	0.180397
$297$	−4.57892	−0.265696
$298$	36.8155	2.13267
$299$	−21.1074	−1.22067
$300$	21.6559	1.25031
$301$	31.4526	1.81290
$302$	21.9280	1.26182
$303$	22.7642	1.30777
$304$	−14.4446	−0.828457
$305$	24.8366	1.42214
$306$	9.80606	0.560575
$307$	25.2953	1.44368	0.721840	−	0.692060i	$-0.243297\pi$
$307$	25.2953	1.44368	0.721840	+	0.692060i	$0.243297\pi$
$308$	60.7515	3.46164
$309$	−35.1144	−1.99759
$310$	31.1262	1.76785
$311$	0.219709	0.0124585	0.00622927	−	0.999981i	$-0.498017\pi$
$311$	0.219709	0.0124585	0.00622927	+	0.999981i	$0.498017\pi$
$312$	44.3635	2.51159
$313$	30.9954	1.75197	0.875983	−	0.482342i	$-0.160214\pi$
$313$	30.9954	1.75197	0.875983	+	0.482342i	$0.160214\pi$
$314$	30.2197	1.70539
$315$	−13.7684	−0.775761
$316$	−3.78869	−0.213131
$317$	19.5578	1.09848	0.549238	−	0.835666i	$-0.314918\pi$
$317$	19.5578	1.09848	0.549238	+	0.835666i	$0.314918\pi$
$318$	−37.8031	−2.11989
$319$	4.99691	0.279773
$320$	3.14696	0.175920
$321$	33.9255	1.89354
$322$	−59.5423	−3.31816
$323$	2.78402	0.154907
$324$	−46.5541	−2.58634
$325$	−5.72540	−0.317588
$326$	45.4211	2.51564
$327$	8.05899	0.445663
$328$	−23.8401	−1.31635
$329$	35.2890	1.94554
$330$	−44.2044	−2.43337
$331$	0.257369	0.0141463	0.00707314	−	0.999975i	$-0.497749\pi$
$331$	0.257369	0.0141463	0.00707314	+	0.999975i	$0.497749\pi$
$332$	−57.8135	−3.17293
$333$	−1.19209	−0.0653259
$334$	38.0854	2.08394
$335$	24.8326	1.35675
$336$	57.5945	3.14204
$337$	13.2799	0.723400	0.361700	−	0.932295i	$-0.382196\pi$
$337$	13.2799	0.723400	0.361700	+	0.932295i	$0.382196\pi$
$338$	12.5188	0.680932
$339$	47.2252	2.56492
$340$	−11.9161	−0.646241
$341$	29.7098	1.60887
$342$	12.0552	0.651868
$343$	13.0968	0.707161
$344$	66.3858	3.57928
$345$	30.1536	1.62341
$346$	19.0791	1.02570
$347$	9.05813	0.486266	0.243133	−	0.969993i	$-0.421825\pi$
$347$	9.05813	0.486266	0.243133	+	0.969993i	$0.421825\pi$
$348$	12.7203	0.681881
$349$	−1.78507	−0.0955525	−0.0477763	−	0.998858i	$-0.515213\pi$
$349$	−1.78507	−0.0955525	−0.0477763	+	0.998858i	$0.515213\pi$
$350$	−16.1509	−0.863301
$351$	3.08170	0.164489
$352$	28.7800	1.53398
$353$	−10.4495	−0.556172	−0.278086	−	0.960556i	$-0.589700\pi$
$353$	−10.4495	−0.556172	−0.278086	+	0.960556i	$0.589700\pi$
$354$	−50.5261	−2.68543
$355$	22.0660	1.17114
$356$	73.9699	3.92040
$357$	−11.1006	−0.587505
$358$	−31.5731	−1.66869
$359$	25.1178	1.32567	0.662833	−	0.748768i	$-0.269354\pi$
$359$	25.1178	1.32567	0.662833	+	0.748768i	$0.269354\pi$
$360$	−29.0604	−1.53162
$361$	−15.5774	−0.819866
$362$	61.6151	3.23842
$363$	−16.3006	−0.855557
$364$	−40.8868	−2.14305
$365$	−21.9641	−1.14965
$366$	86.7074	4.53227
$367$	15.7527	0.822284	0.411142	−	0.911571i	$-0.365130\pi$
$367$	15.7527	0.822284	0.411142	+	0.911571i	$0.365130\pi$
$368$	−57.8376	−3.01499
$369$	9.15671	0.476679
$370$	2.08133	0.108203
$371$	19.6224	1.01874
$372$	75.6303	3.92125
$373$	8.45360	0.437711	0.218855	−	0.975757i	$-0.429768\pi$
$373$	8.45360	0.437711	0.218855	+	0.975757i	$0.429768\pi$
$374$	−16.3418	−0.845017
$375$	28.5323	1.47340
$376$	74.4830	3.84117
$377$	−3.36301	−0.173204
$378$	8.69320	0.447130
$379$	22.5437	1.15799	0.578996	−	0.815330i	$-0.303445\pi$
$379$	22.5437	1.15799	0.578996	+	0.815330i	$0.303445\pi$
$380$	−14.6491	−0.751486
$381$	1.96218	0.100525
$382$	−56.3115	−2.88115
$383$	−29.1214	−1.48803	−0.744017	−	0.668161i	$-0.767082\pi$
$383$	−29.1214	−1.48803	−0.744017	+	0.668161i	$0.767082\pi$
$384$	−21.0147	−1.07240
$385$	22.9451	1.16939
$386$	−38.2916	−1.94899
$387$	−25.4981	−1.29614
$388$	27.4197	1.39202
$389$	27.9047	1.41483	0.707413	−	0.706800i	$-0.249862\pi$
$389$	27.9047	1.41483	0.707413	+	0.706800i	$0.249862\pi$
$390$	29.7503	1.50647
$391$	11.1474	0.563750
$392$	−18.6581	−0.942375
$393$	−31.8514	−1.60669
$394$	28.5032	1.43597
$395$	−1.43094	−0.0719985
$396$	−49.2502	−2.47491
$397$	0.653695	0.0328080	0.0164040	−	0.999865i	$-0.494778\pi$
$397$	0.653695	0.0328080	0.0164040	+	0.999865i	$0.494778\pi$
$398$	23.1292	1.15936
$399$	−13.6466	−0.683185
$400$	−15.6885	−0.784423
$401$	−11.4916	−0.573861	−0.286930	−	0.957951i	$-0.592635\pi$
$401$	−11.4916	−0.573861	−0.286930	+	0.957951i	$0.592635\pi$
$402$	86.6936	4.32388
$403$	−19.9952	−0.996031
$404$	−44.2822	−2.20312
$405$	−17.5829	−0.873701
$406$	−9.48676	−0.470820
$407$	1.98662	0.0984729
$408$	−23.4296	−1.15994
$409$	−17.6216	−0.871334	−0.435667	−	0.900108i	$-0.643488\pi$
$409$	−17.6216	−0.871334	−0.435667	+	0.900108i	$0.643488\pi$
$410$	−15.9873	−0.789554
$411$	−14.9813	−0.738975
$412$	68.3064	3.36522
$413$	26.2265	1.29052
$414$	48.2699	2.37233
$415$	−21.8354	−1.07186
$416$	−19.3694	−0.949664
$417$	−0.563368	−0.0275882
$418$	−20.0900	−0.982633
$419$	29.9448	1.46290	0.731450	−	0.681896i	$-0.238844\pi$
$419$	29.9448	1.46290	0.731450	+	0.681896i	$0.238844\pi$
$420$	58.4100	2.85011
$421$	12.5826	0.613238	0.306619	−	0.951832i	$-0.400802\pi$
$421$	12.5826	0.613238	0.306619	+	0.951832i	$0.400802\pi$
$422$	−65.1938	−3.17359
$423$	−28.6081	−1.39098
$424$	41.4162	2.01135
$425$	3.02375	0.146673
$426$	77.0348	3.73235
$427$	−45.0071	−2.17804
$428$	−65.9939	−3.18993
$429$	28.3964	1.37099
$430$	44.5186	2.14688
$431$	4.02343	0.193802	0.0969010	−	0.995294i	$-0.469107\pi$
$431$	4.02343	0.193802	0.0969010	+	0.995294i	$0.469107\pi$
$432$	8.44431	0.406277
$433$	3.50484	0.168432	0.0842159	−	0.996448i	$-0.473161\pi$
$433$	3.50484	0.168432	0.0842159	+	0.996448i	$0.473161\pi$
$434$	−56.4047	−2.70751
$435$	4.80431	0.230349
$436$	−15.6768	−0.750782
$437$	13.7042	0.655560
$438$	−76.6792	−3.66387
$439$	26.2372	1.25223	0.626116	−	0.779730i	$-0.284644\pi$
$439$	26.2372	1.25223	0.626116	+	0.779730i	$0.284644\pi$
$440$	48.4293	2.30878
$441$	7.16637	0.341256
$442$	10.9983	0.523138
$443$	0.516363	0.0245331	0.0122666	−	0.999925i	$-0.496095\pi$
$443$	0.516363	0.0245331	0.0122666	+	0.999925i	$0.496095\pi$
$444$	5.05721	0.240004
$445$	27.9375	1.32437
$446$	21.4199	1.01426
$447$	33.7857	1.59801
$448$	−5.70269	−0.269427
$449$	−18.1358	−0.855882	−0.427941	−	0.903807i	$-0.640761\pi$
$449$	−18.1358	−0.855882	−0.427941	+	0.903807i	$0.640761\pi$
$450$	13.0932	0.617221
$451$	−15.2597	−0.718551
$452$	−91.8652	−4.32097
$453$	20.1234	0.945481
$454$	57.1227	2.68090
$455$	−15.4425	−0.723953
$456$	−28.8034	−1.34884
$457$	13.9130	0.650825	0.325412	−	0.945572i	$-0.394497\pi$
$457$	13.9130	0.650825	0.325412	+	0.945572i	$0.394497\pi$
$458$	−12.9042	−0.602973
$459$	−1.62753	−0.0759666
$460$	−58.6564	−2.73487
$461$	−41.6907	−1.94173	−0.970865	−	0.239626i	$-0.922975\pi$
$461$	−41.6907	−1.94173	−0.970865	+	0.239626i	$0.922975\pi$
$462$	80.1040	3.72677
$463$	8.96811	0.416783	0.208392	−	0.978045i	$-0.433177\pi$
$463$	8.96811	0.416783	0.208392	+	0.978045i	$0.433177\pi$
$464$	−9.21515	−0.427802
$465$	28.5646	1.32465
$466$	−0.556423	−0.0257758
$467$	−40.1977	−1.86013	−0.930064	−	0.367397i	$-0.880249\pi$
$467$	−40.1977	−1.86013	−0.930064	+	0.367397i	$0.880249\pi$
$468$	33.1462	1.53218
$469$	−44.9999	−2.07790
$470$	49.9487	2.30396
$471$	27.7327	1.27785
$472$	55.3553	2.54793
$473$	42.4927	1.95381
$474$	−4.99559	−0.229455
$475$	3.71727	0.170560
$476$	21.5935	0.989736
$477$	−15.9075	−0.728356
$478$	−10.4149	−0.476367
$479$	3.64681	0.166627	0.0833134	−	0.996523i	$-0.473450\pi$
$479$	3.64681	0.166627	0.0833134	+	0.996523i	$0.473450\pi$
$480$	27.6707	1.26299
$481$	−1.33703	−0.0609632
$482$	−8.50262	−0.387284
$483$	−54.6421	−2.48630
$484$	31.7087	1.44131
$485$	10.3561	0.470245
$486$	−53.0619	−2.40694
$487$	14.1341	0.640479	0.320239	−	0.947337i	$-0.396237\pi$
$487$	14.1341	0.640479	0.320239	+	0.947337i	$0.396237\pi$
$488$	−94.9947	−4.30021
$489$	41.6831	1.88498
$490$	−12.5122	−0.565243
$491$	−7.46697	−0.336980	−0.168490	−	0.985703i	$-0.553889\pi$
$491$	−7.46697	−0.336980	−0.168490	+	0.985703i	$0.553889\pi$
$492$	−38.8457	−1.75130
$493$	1.77610	0.0799914
$494$	13.5209	0.608335
$495$	−18.6012	−0.836061
$496$	−54.7898	−2.46014
$497$	−39.9863	−1.79363
$498$	−76.2300	−3.41595
$499$	40.3277	1.80531	0.902657	−	0.430360i	$-0.141613\pi$
$499$	40.3277	1.80531	0.902657	+	0.430360i	$0.141613\pi$
$500$	−55.5026	−2.48215
$501$	34.9510	1.56150
$502$	61.7189	2.75465
$503$	−17.4210	−0.776765	−0.388383	−	0.921498i	$-0.626966\pi$
$503$	−17.4210	−0.776765	−0.388383	+	0.921498i	$0.626966\pi$
$504$	52.6612	2.34572
$505$	−16.7248	−0.744246
$506$	−80.4420	−3.57608
$507$	11.4885	0.510223
$508$	−3.81694	−0.169349
$509$	7.67650	0.340255	0.170127	−	0.985422i	$-0.445582\pi$
$509$	7.67650	0.340255	0.170127	+	0.985422i	$0.445582\pi$
$510$	−15.7120	−0.695738
$511$	39.8017	1.76073
$512$	50.2139	2.21916
$513$	−2.00082	−0.0883382
$514$	33.9537	1.49763
$515$	25.7985	1.13682
$516$	108.171	4.76197
$517$	47.6756	2.09677
$518$	−3.77164	−0.165716
$519$	17.5090	0.768558
$520$	−32.5938	−1.42933
$521$	−25.6705	−1.12465	−0.562323	−	0.826918i	$-0.690092\pi$
$521$	−25.6705	−1.12465	−0.562323	+	0.826918i	$0.690092\pi$
$522$	7.69074	0.336615
$523$	10.3769	0.453749	0.226874	−	0.973924i	$-0.427149\pi$
$523$	10.3769	0.453749	0.226874	+	0.973924i	$0.427149\pi$
$524$	61.9591	2.70670
$525$	−14.8217	−0.646872
$526$	−57.4338	−2.50423
$527$	10.5600	0.460002
$528$	77.8105	3.38627
$529$	31.8727	1.38577
$530$	27.7739	1.20642
$531$	−21.2614	−0.922665
$532$	26.5461	1.15092
$533$	10.2700	0.444845
$534$	97.5331	4.22067
$535$	−24.9251	−1.07760
$536$	−94.9795	−4.10249
$537$	−28.9747	−1.25035
$538$	29.8218	1.28571
$539$	−11.9428	−0.514412
$540$	8.56386	0.368530
$541$	−33.8666	−1.45604	−0.728018	−	0.685558i	$-0.759558\pi$
$541$	−33.8666	−1.45604	−0.728018	+	0.685558i	$0.759558\pi$
$542$	−52.1322	−2.23927
$543$	56.5444	2.42655
$544$	10.2295	0.438588
$545$	−5.92093	−0.253625
$546$	−53.9114	−2.30719
$547$	−28.0498	−1.19932	−0.599662	−	0.800254i	$-0.704698\pi$
$547$	−28.0498	−1.19932	−0.599662	+	0.800254i	$0.704698\pi$
$548$	29.1425	1.24491
$549$	36.4864	1.55720
$550$	−21.8199	−0.930405
$551$	2.18346	0.0930186
$552$	−115.331	−4.90881
$553$	2.59305	0.110268
$554$	58.5085	2.48579
$555$	1.91005	0.0810769
$556$	1.09589	0.0464763
$557$	9.44587	0.400234	0.200117	−	0.979772i	$-0.435868\pi$
$557$	9.44587	0.400234	0.200117	+	0.979772i	$0.435868\pi$
$558$	45.7263	1.93575
$559$	−28.5983	−1.20958
$560$	−42.3146	−1.78812
$561$	−14.9970	−0.633172
$562$	71.3182	3.00838
$563$	20.5234	0.864960	0.432480	−	0.901643i	$-0.357639\pi$
$563$	20.5234	0.864960	0.432480	+	0.901643i	$0.357639\pi$
$564$	121.365	5.11038
$565$	−34.6963	−1.45969
$566$	−5.42755	−0.228137
$567$	31.8625	1.33810
$568$	−84.3976	−3.54124
$569$	−36.4038	−1.52613	−0.763064	−	0.646323i	$-0.776306\pi$
$569$	−36.4038	−1.52613	−0.763064	+	0.646323i	$0.776306\pi$
$570$	−19.3157	−0.809044
$571$	−1.13713	−0.0475872	−0.0237936	−	0.999717i	$-0.507574\pi$
$571$	−1.13713	−0.0475872	−0.0237936	+	0.999717i	$0.507574\pi$
$572$	−55.2383	−2.30963
$573$	−51.6773	−2.15885
$574$	28.9709	1.20922
$575$	14.8842	0.620716
$576$	4.62307	0.192628
$577$	5.13363	0.213716	0.106858	−	0.994274i	$-0.465921\pi$
$577$	5.13363	0.213716	0.106858	+	0.994274i	$0.465921\pi$
$578$	37.7951	1.57207
$579$	−35.1404	−1.46038
$580$	−9.34561	−0.388055
$581$	39.5686	1.64158
$582$	36.1543	1.49864
$583$	26.5100	1.09793
$584$	84.0080	3.47628
$585$	12.5189	0.517594
$586$	−7.37871	−0.304812
$587$	19.5210	0.805716	0.402858	−	0.915263i	$-0.368017\pi$
$587$	19.5210	0.805716	0.402858	+	0.915263i	$0.368017\pi$
$588$	−30.4020	−1.25376
$589$	12.9820	0.534916
$590$	37.1215	1.52827
$591$	26.1575	1.07597
$592$	−3.66366	−0.150575
$593$	36.7443	1.50891	0.754453	−	0.656354i	$-0.227902\pi$
$593$	36.7443	1.50891	0.754453	+	0.656354i	$0.227902\pi$
$594$	11.7446	0.481885
$595$	8.15559	0.334347
$596$	−65.7219	−2.69207
$597$	21.2258	0.868712
$598$	54.1389	2.21390
$599$	−15.2825	−0.624425	−0.312212	−	0.950012i	$-0.601070\pi$
$599$	−15.2825	−0.624425	−0.312212	+	0.950012i	$0.601070\pi$
$600$	−31.2836	−1.27715
$601$	30.3715	1.23888	0.619440	−	0.785044i	$-0.287360\pi$
$601$	30.3715	1.23888	0.619440	+	0.785044i	$0.287360\pi$
$602$	−80.6734	−3.28800
$603$	36.4806	1.48561
$604$	−39.1452	−1.59280
$605$	11.9760	0.486894
$606$	−58.3883	−2.37186
$607$	33.1723	1.34642	0.673212	−	0.739450i	$-0.264914\pi$
$607$	33.1723	1.34642	0.673212	+	0.739450i	$0.264914\pi$
$608$	12.5758	0.510015
$609$	−8.70602	−0.352786
$610$	−63.7038	−2.57929
$611$	−32.0865	−1.29808
$612$	−17.5055	−0.707616
$613$	2.52504	0.101985	0.0509927	−	0.998699i	$-0.483761\pi$
$613$	2.52504	0.101985	0.0509927	+	0.998699i	$0.483761\pi$
$614$	−64.8804	−2.61836
$615$	−14.6715	−0.591614
$616$	−87.7601	−3.53595
$617$	12.3309	0.496424	0.248212	−	0.968706i	$-0.420157\pi$
$617$	12.3309	0.496424	0.248212	+	0.968706i	$0.420157\pi$
$618$	90.0655	3.62297
$619$	−3.69101	−0.148354	−0.0741771	−	0.997245i	$-0.523633\pi$
$619$	−3.69101	−0.148354	−0.0741771	+	0.997245i	$0.523633\pi$
$620$	−55.5655	−2.23157
$621$	−8.01144	−0.321488
$622$	−0.563535	−0.0225957
$623$	−50.6263	−2.02830
$624$	−52.3679	−2.09639
$625$	−10.9160	−0.436642
$626$	−79.5008	−3.17749
$627$	−18.4366	−0.736288
$628$	−53.9471	−2.15272
$629$	0.706121	0.0281549
$630$	35.3148	1.40698
$631$	28.4517	1.13264	0.566321	−	0.824185i	$-0.308366\pi$
$631$	28.4517	1.13264	0.566321	+	0.824185i	$0.308366\pi$
$632$	5.47305	0.217706
$633$	−59.8286	−2.37797
$634$	−50.1642	−1.99227
$635$	−1.44161	−0.0572086
$636$	67.4849	2.67595
$637$	8.03771	0.318465
$638$	−12.8167	−0.507416
$639$	32.4162	1.28237
$640$	15.4395	0.610299
$641$	25.8084	1.01937	0.509684	−	0.860361i	$-0.329762\pi$
$641$	25.8084	1.01937	0.509684	+	0.860361i	$0.329762\pi$
$642$	−87.0163	−3.43426
$643$	21.3973	0.843828	0.421914	−	0.906636i	$-0.361358\pi$
$643$	21.3973	0.843828	0.421914	+	0.906636i	$0.361358\pi$
$644$	106.293	4.18853
$645$	40.8549	1.60866
$646$	−7.14077	−0.280950
$647$	−15.4759	−0.608419	−0.304210	−	0.952605i	$-0.598392\pi$
$647$	−15.4759	−0.608419	−0.304210	+	0.952605i	$0.598392\pi$
$648$	67.2509	2.64186
$649$	35.4322	1.39083
$650$	14.6852	0.576001
$651$	−51.7628	−2.02874
$652$	−81.0843	−3.17551
$653$	19.5037	0.763238	0.381619	−	0.924320i	$-0.375367\pi$
$653$	19.5037	0.763238	0.381619	+	0.924320i	$0.375367\pi$
$654$	−20.6706	−0.808286
$655$	23.4012	0.914361
$656$	28.1415	1.09874
$657$	−32.2666	−1.25884
$658$	−90.5133	−3.52858
$659$	0.261503	0.0101867	0.00509336	−	0.999987i	$-0.498379\pi$
$659$	0.261503	0.0101867	0.00509336	+	0.999987i	$0.498379\pi$
$660$	78.9122	3.07165
$661$	−3.09137	−0.120240	−0.0601202	−	0.998191i	$-0.519148\pi$
$661$	−3.09137	−0.120240	−0.0601202	+	0.998191i	$0.519148\pi$
$662$	−0.660131	−0.0256567
$663$	10.0932	0.391988
$664$	83.5159	3.24105
$665$	10.0261	0.388797
$666$	3.05760	0.118480
$667$	8.74276	0.338521
$668$	−67.9887	−2.63056
$669$	19.6571	0.759989
$670$	−63.6937	−2.46070
$671$	−60.8048	−2.34734
$672$	−50.1428	−1.93430
$673$	33.6025	1.29528	0.647640	−	0.761946i	$-0.275756\pi$
$673$	33.6025	1.29528	0.647640	+	0.761946i	$0.275756\pi$
$674$	−34.0617	−1.31201
$675$	−2.17311	−0.0836429
$676$	−22.3481	−0.859543
$677$	0.107883	0.00414628	0.00207314	−	0.999998i	$-0.499340\pi$
$677$	0.107883	0.00414628	0.00207314	+	0.999998i	$0.499340\pi$
$678$	−121.129	−4.65193
$679$	−18.7665	−0.720193
$680$	17.2137	0.660115
$681$	52.4217	2.00880
$682$	−76.2031	−2.91797
$683$	−41.5019	−1.58802	−0.794012	−	0.607902i	$-0.792011\pi$
$683$	−41.5019	−1.58802	−0.794012	+	0.607902i	$0.792011\pi$
$684$	−21.5205	−0.822856
$685$	11.0068	0.420548
$686$	−33.5922	−1.28256
$687$	−11.8422	−0.451808
$688$	−78.3637	−2.98759
$689$	−17.8417	−0.679714
$690$	−77.3415	−2.94434
$691$	−29.4036	−1.11856	−0.559282	−	0.828977i	$-0.688923\pi$
$691$	−29.4036	−1.11856	−0.559282	+	0.828977i	$0.688923\pi$
$692$	−34.0594	−1.29474
$693$	33.7077	1.28045
$694$	−23.2334	−0.881926
$695$	0.413906	0.0157003
$696$	−18.3755	−0.696520
$697$	−5.42390	−0.205445
$698$	4.57856	0.173301
$699$	−0.510631	−0.0193138
$700$	28.8320	1.08975
$701$	32.0049	1.20881	0.604404	−	0.796678i	$-0.293411\pi$
$701$	32.0049	1.20881	0.604404	+	0.796678i	$0.293411\pi$
$702$	−7.90430	−0.298328
$703$	0.868076	0.0327401
$704$	−7.70437	−0.290369
$705$	45.8380	1.72636
$706$	26.8022	1.00871
$707$	30.3075	1.13983
$708$	90.1976	3.38983
$709$	16.2222	0.609238	0.304619	−	0.952474i	$-0.401471\pi$
$709$	16.2222	0.609238	0.304619	+	0.952474i	$0.401471\pi$
$710$	−56.5974	−2.12406
$711$	−2.10214	−0.0788364
$712$	−106.855	−4.00456
$713$	51.9812	1.94671
$714$	28.4721	1.06554
$715$	−20.8628	−0.780226
$716$	56.3633	2.10639
$717$	−9.55780	−0.356943
$718$	−64.4250	−2.40432
$719$	−27.5372	−1.02696	−0.513482	−	0.858101i	$-0.671645\pi$
$719$	−27.5372	−1.02696	−0.513482	+	0.858101i	$0.671645\pi$
$720$	34.3037	1.27842
$721$	−46.7502	−1.74107
$722$	39.9549	1.48697
$723$	−7.80288	−0.290192
$724$	−109.993	−4.08787
$725$	2.37148	0.0880745
$726$	41.8096	1.55170
$727$	2.58108	0.0957268	0.0478634	−	0.998854i	$-0.484759\pi$
$727$	2.58108	0.0957268	0.0478634	+	0.998854i	$0.484759\pi$
$728$	59.0641	2.18906
$729$	−18.1932	−0.673823
$730$	56.3361	2.08509
$731$	15.1036	0.558626
$732$	−154.787	−5.72110
$733$	−20.2967	−0.749675	−0.374837	−	0.927091i	$-0.622301\pi$
$733$	−20.2967	−0.749675	−0.374837	+	0.927091i	$0.622301\pi$
$734$	−40.4044	−1.49135
$735$	−11.4825	−0.423537
$736$	50.3544	1.85609
$737$	−60.7951	−2.23942
$738$	−23.4862	−0.864540
$739$	−32.4696	−1.19441	−0.597207	−	0.802087i	$-0.703723\pi$
$739$	−32.4696	−1.19441	−0.597207	+	0.802087i	$0.703723\pi$
$740$	−3.71553	−0.136585
$741$	12.4082	0.455826
$742$	−50.3299	−1.84767
$743$	−4.23816	−0.155483	−0.0777416	−	0.996974i	$-0.524771\pi$
$743$	−4.23816	−0.155483	−0.0777416	+	0.996974i	$0.524771\pi$
$744$	−109.254	−4.00543
$745$	−24.8223	−0.909420
$746$	−21.6828	−0.793863
$747$	−32.0776	−1.17366
$748$	29.1729	1.06667
$749$	45.1674	1.65038
$750$	−73.1830	−2.67226
$751$	−43.3553	−1.58206	−0.791028	−	0.611780i	$-0.790454\pi$
$751$	−43.3553	−1.58206	−0.791028	+	0.611780i	$0.790454\pi$
$752$	−87.9219	−3.20618
$753$	56.6397	2.06406
$754$	8.62584	0.314135
$755$	−14.7847	−0.538069
$756$	−15.5188	−0.564414
$757$	21.8697	0.794867	0.397434	−	0.917631i	$-0.369901\pi$
$757$	21.8697	0.794867	0.397434	+	0.917631i	$0.369901\pi$
$758$	−57.8227	−2.10022
$759$	−73.8218	−2.67956
$760$	21.1618	0.767619
$761$	1.57884	0.0572328	0.0286164	−	0.999590i	$-0.490890\pi$
$761$	1.57884	0.0572328	0.0286164	+	0.999590i	$0.490890\pi$
$762$	−5.03283	−0.182320
$763$	10.7295	0.388433
$764$	100.525	3.63688
$765$	−6.61159	−0.239043
$766$	74.6940	2.69880
$767$	−23.8465	−0.861046
$768$	62.4677	2.25411
$769$	24.0855	0.868545	0.434272	−	0.900782i	$-0.357006\pi$
$769$	24.0855	0.868545	0.434272	+	0.900782i	$0.357006\pi$
$770$	−58.8523	−2.12089
$771$	31.1594	1.12218
$772$	68.3570	2.46022
$773$	−28.4294	−1.02253	−0.511266	−	0.859422i	$-0.670823\pi$
$773$	−28.4294	−1.02253	−0.511266	+	0.859422i	$0.670823\pi$
$774$	65.4005	2.35077
$775$	14.0999	0.506484
$776$	−39.6098	−1.42191
$777$	−3.46124	−0.124171
$778$	−71.5734	−2.56603
$779$	−6.66792	−0.238903
$780$	−53.1093	−1.90162
$781$	−54.0217	−1.93305
$782$	−28.5922	−1.02246
$783$	−1.27645	−0.0456165
$784$	22.0245	0.786590
$785$	−20.3752	−0.727221
$786$	81.6963	2.91401
$787$	10.3541	0.369083	0.184541	−	0.982825i	$-0.440920\pi$
$787$	10.3541	0.369083	0.184541	+	0.982825i	$0.440920\pi$
$788$	−50.8830	−1.81263
$789$	−52.7072	−1.87643
$790$	3.67025	0.130582
$791$	62.8742	2.23555
$792$	71.1455	2.52805
$793$	40.9227	1.45321
$794$	−1.67667	−0.0595029
$795$	25.4882	0.903973
$796$	−41.2895	−1.46347
$797$	17.6804	0.626273	0.313137	−	0.949708i	$-0.398620\pi$
$797$	17.6804	0.626273	0.313137	+	0.949708i	$0.398620\pi$
$798$	35.0024	1.23907
$799$	16.9458	0.599499
$800$	13.6587	0.482907
$801$	41.0419	1.45014
$802$	29.4749	1.04079
$803$	53.7724	1.89759
$804$	−154.762	−5.45805
$805$	40.1455	1.41494
$806$	51.2860	1.80647
$807$	27.3675	0.963382
$808$	63.9690	2.25042
$809$	−2.30859	−0.0811657	−0.0405829	−	0.999176i	$-0.512921\pi$
$809$	−2.30859	−0.0811657	−0.0405829	+	0.999176i	$0.512921\pi$
$810$	45.0987	1.58461
$811$	−32.7612	−1.15040	−0.575201	−	0.818012i	$-0.695076\pi$
$811$	−32.7612	−1.15040	−0.575201	+	0.818012i	$0.695076\pi$
$812$	16.9354	0.594317
$813$	−47.8419	−1.67789
$814$	−5.09551	−0.178597
$815$	−30.6245	−1.07273
$816$	27.6569	0.968186
$817$	18.5677	0.649602
$818$	45.1981	1.58031
$819$	−22.6859	−0.792709
$820$	28.5399	0.996657
$821$	38.2226	1.33398	0.666989	−	0.745068i	$-0.267583\pi$
$821$	38.2226	1.33398	0.666989	+	0.745068i	$0.267583\pi$
$822$	38.4259	1.34026
$823$	21.1413	0.736940	0.368470	−	0.929640i	$-0.379882\pi$
$823$	21.1413	0.736940	0.368470	+	0.929640i	$0.379882\pi$
$824$	−98.6738	−3.43746
$825$	−20.0242	−0.697153
$826$	−67.2689	−2.34058
$827$	29.9798	1.04250	0.521250	−	0.853404i	$-0.325466\pi$
$827$	29.9798	1.04250	0.521250	+	0.853404i	$0.325466\pi$
$828$	−86.1698	−2.99461
$829$	7.93885	0.275728	0.137864	−	0.990451i	$-0.455976\pi$
$829$	7.93885	0.275728	0.137864	+	0.990451i	$0.455976\pi$
$830$	56.0061	1.94400
$831$	53.6934	1.86260
$832$	5.18517	0.179764
$833$	−4.24494	−0.147078
$834$	1.44499	0.0500360
$835$	−25.6785	−0.888641
$836$	35.8640	1.24038
$837$	−7.58928	−0.262324
$838$	−76.8060	−2.65322
$839$	42.6530	1.47254	0.736272	−	0.676686i	$-0.236584\pi$
$839$	42.6530	1.47254	0.736272	+	0.676686i	$0.236584\pi$
$840$	−84.3775	−2.91130
$841$	−27.6070	−0.951967
$842$	−32.2733	−1.11221
$843$	65.4489	2.25418
$844$	116.382	4.00603
$845$	−8.44060	−0.290366
$846$	73.3775	2.52277
$847$	−21.7020	−0.745691
$848$	−48.8889	−1.67885
$849$	−4.98088	−0.170943
$850$	−7.75566	−0.266017
$851$	3.47585	0.119151
$852$	−137.520	−4.71136
$853$	46.8850	1.60531	0.802657	−	0.596442i	$-0.203419\pi$
$853$	46.8850	1.60531	0.802657	+	0.596442i	$0.203419\pi$
$854$	115.439	3.95026
$855$	−8.12802	−0.277972
$856$	95.3331	3.25842
$857$	0.963805	0.0329230	0.0164615	−	0.999865i	$-0.494760\pi$
$857$	0.963805	0.0329230	0.0164615	+	0.999865i	$0.494760\pi$
$858$	−72.8346	−2.48653
$859$	4.46872	0.152471	0.0762353	−	0.997090i	$-0.475710\pi$
$859$	4.46872	0.152471	0.0762353	+	0.997090i	$0.475710\pi$
$860$	−79.4732	−2.71001
$861$	26.5867	0.906073
$862$	−10.3198	−0.351493
$863$	13.6613	0.465038	0.232519	−	0.972592i	$-0.425303\pi$
$863$	13.6613	0.465038	0.232519	+	0.972592i	$0.425303\pi$
$864$	−7.35177	−0.250112
$865$	−12.8638	−0.437383
$866$	−8.98963	−0.305480
$867$	34.6847	1.17795
$868$	100.692	3.41770
$869$	3.50323	0.118839
$870$	−12.3227	−0.417778
$871$	40.9162	1.38639
$872$	22.6463	0.766900
$873$	15.2137	0.514905
$874$	−35.1501	−1.18897
$875$	37.9870	1.28419
$876$	136.885	4.62492
$877$	−2.14149	−0.0723129	−0.0361565	−	0.999346i	$-0.511511\pi$
$877$	−2.14149	−0.0723129	−0.0361565	+	0.999346i	$0.511511\pi$
$878$	−67.2962	−2.27114
$879$	−6.77146	−0.228396
$880$	−57.1673	−1.92711
$881$	7.40246	0.249395	0.124698	−	0.992195i	$-0.460204\pi$
$881$	7.40246	0.249395	0.124698	+	0.992195i	$0.460204\pi$
$882$	−18.3811	−0.618925
$883$	−35.5025	−1.19476	−0.597378	−	0.801960i	$-0.703791\pi$
$883$	−35.5025	−1.19476	−0.597378	+	0.801960i	$0.703791\pi$
$884$	−19.6339	−0.660359
$885$	34.0665	1.14513
$886$	−1.32443	−0.0444950
$887$	−37.9021	−1.27263	−0.636314	−	0.771430i	$-0.719542\pi$
$887$	−37.9021	−1.27263	−0.636314	+	0.771430i	$0.719542\pi$
$888$	−7.30552	−0.245157
$889$	2.61238	0.0876164
$890$	−71.6575	−2.40196
$891$	43.0464	1.44211
$892$	−38.2381	−1.28031
$893$	20.8324	0.697131
$894$	−86.6576	−2.89826
$895$	21.2877	0.711570
$896$	−27.9783	−0.934690
$897$	49.6834	1.65888
$898$	46.5169	1.55229
$899$	8.28206	0.276222
$900$	−23.3736	−0.779120
$901$	9.42269	0.313915
$902$	39.1399	1.30322
$903$	−74.0342	−2.46370
$904$	132.706	4.41374
$905$	−41.5431	−1.38094
$906$	−51.6149	−1.71479
$907$	33.0502	1.09741	0.548707	−	0.836015i	$-0.315120\pi$
$907$	33.0502	1.09741	0.548707	+	0.836015i	$0.315120\pi$
$908$	−101.974	−3.38411
$909$	−24.5698	−0.814928
$910$	39.6086	1.31301
$911$	−16.3806	−0.542715	−0.271357	−	0.962479i	$-0.587473\pi$
$911$	−16.3806	−0.542715	−0.271357	+	0.962479i	$0.587473\pi$
$912$	34.0003	1.12586
$913$	53.4574	1.76918
$914$	−35.6858	−1.18038
$915$	−58.4612	−1.93267
$916$	23.0361	0.761135
$917$	−42.4060	−1.40037
$918$	4.17448	0.137778
$919$	11.4723	0.378436	0.189218	−	0.981935i	$-0.439405\pi$
$919$	11.4723	0.378436	0.189218	+	0.981935i	$0.439405\pi$
$920$	84.7336	2.79358
$921$	−59.5409	−1.96194
$922$	106.933	3.52166
$923$	36.3576	1.19672
$924$	−142.999	−4.70432
$925$	0.942826	0.0309999
$926$	−23.0025	−0.755908
$927$	37.8995	1.24478
$928$	8.02287	0.263364
$929$	−34.0419	−1.11688	−0.558439	−	0.829545i	$-0.688600\pi$
$929$	−34.0419	−1.11688	−0.558439	+	0.829545i	$0.688600\pi$
$930$	−73.2660	−2.40249
$931$	−5.21855	−0.171031
$932$	0.993307	0.0325369
$933$	−0.517158	−0.0169310
$934$	103.104	3.37366
$935$	11.0182	0.360335
$936$	−47.8822	−1.56508
$937$	−32.5403	−1.06305	−0.531523	−	0.847044i	$-0.678380\pi$
$937$	−32.5403	−1.06305	−0.531523	+	0.847044i	$0.678380\pi$
$938$	115.421	3.76863
$939$	−72.9581	−2.38090
$940$	−89.1667	−2.90830
$941$	12.8669	0.419449	0.209724	−	0.977761i	$-0.432743\pi$
$941$	12.8669	0.419449	0.209724	+	0.977761i	$0.432743\pi$
$942$	−71.1320	−2.31761
$943$	−26.6989	−0.869436
$944$	−65.3429	−2.12673
$945$	−5.86126	−0.190667
$946$	−108.990	−3.54358
$947$	−8.06349	−0.262028	−0.131014	−	0.991381i	$-0.541823\pi$
$947$	−8.06349	−0.262028	−0.131014	+	0.991381i	$0.541823\pi$
$948$	8.91795	0.289642
$949$	−36.1898	−1.17477
$950$	−9.53449	−0.309340
$951$	−46.0358	−1.49281
$952$	−31.1934	−1.01098
$953$	−16.7651	−0.543075	−0.271537	−	0.962428i	$-0.587532\pi$
$953$	−16.7651	−0.543075	−0.271537	+	0.962428i	$0.587532\pi$
$954$	40.8015	1.32100
$955$	37.9672	1.22859
$956$	18.5924	0.601320
$957$	−11.7619	−0.380208
$958$	−9.35376	−0.302206
$959$	−19.9457	−0.644080
$960$	−7.40742	−0.239073
$961$	18.2421	0.588453
$962$	3.42937	0.110567
$963$	−36.6164	−1.17995
$964$	15.1786	0.488870
$965$	25.8176	0.831097
$966$	140.153	4.50934
$967$	−21.4038	−0.688301	−0.344150	−	0.938915i	$-0.611833\pi$
$967$	−21.4038	−0.688301	−0.344150	+	0.938915i	$0.611833\pi$
$968$	−45.8056	−1.47225
$969$	−6.55311	−0.210516
$970$	−26.5625	−0.852870
$971$	28.5292	0.915546	0.457773	−	0.889069i	$-0.348647\pi$
$971$	28.5292	0.915546	0.457773	+	0.889069i	$0.348647\pi$
$972$	94.7244	3.03829
$973$	−0.750050	−0.0240455
$974$	−36.2529	−1.16162
$975$	13.4766	0.431598
$976$	112.134	3.58933
$977$	−27.9594	−0.894501	−0.447250	−	0.894409i	$-0.647597\pi$
$977$	−27.9594	−0.894501	−0.447250	+	0.894409i	$0.647597\pi$
$978$	−106.914	−3.41873
$979$	−68.3965	−2.18596
$980$	22.3363	0.713508
$981$	−8.69820	−0.277712
$982$	19.1522	0.611171
$983$	−27.2372	−0.868730	−0.434365	−	0.900737i	$-0.643027\pi$
$983$	−27.2372	−0.868730	−0.434365	+	0.900737i	$0.643027\pi$
$984$	56.1155	1.78890
$985$	−19.2179	−0.612332
$986$	−4.55555	−0.145078
$987$	−83.0643	−2.64397
$988$	−24.1371	−0.767903
$989$	74.3466	2.36408
$990$	47.7105	1.51634
$991$	−36.4980	−1.15940	−0.579698	−	0.814831i	$-0.696830\pi$
$991$	−36.4980	−1.15940	−0.579698	+	0.814831i	$0.696830\pi$
$992$	47.7010	1.51451
$993$	−0.605804	−0.0192246
$994$	102.562	3.25306
$995$	−15.5945	−0.494380
$996$	136.083	4.31196
$997$	43.9585	1.39218	0.696091	−	0.717954i	$-0.254921\pi$
$997$	43.9585	1.39218	0.696091	+	0.717954i	$0.254921\pi$
$998$	−103.437	−3.27425
$999$	−0.507476	−0.0160558

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4019.2.a.b.1.13	✓	186

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4019.2.a.b.1.13	✓	186	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:	\( N \)	\(=\)	\( 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4019.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56492 q^{2} -2.35383 q^{3} +4.57881 q^{4} +1.72936 q^{5} +6.03739 q^{6} -3.13382 q^{7} -6.61443 q^{8} +2.54053 q^{9} +O(q^{10})\)	\(q-2.56492 q^{2} -2.35383 q^{3} +4.57881 q^{4} +1.72936 q^{5} +6.03739 q^{6} -3.13382 q^{7} -6.61443 q^{8} +2.54053 q^{9} -4.43566 q^{10} -4.23381 q^{11} -10.7777 q^{12} +2.84943 q^{13} +8.03799 q^{14} -4.07062 q^{15} +7.80786 q^{16} -1.50486 q^{17} -6.51625 q^{18} -1.85001 q^{19} +7.91840 q^{20} +7.37648 q^{21} +10.8594 q^{22} -7.40761 q^{23} +15.5693 q^{24} -2.00932 q^{25} -7.30855 q^{26} +1.08151 q^{27} -14.3491 q^{28} -1.18024 q^{29} +10.4408 q^{30} -7.01727 q^{31} -6.79766 q^{32} +9.96567 q^{33} +3.85985 q^{34} -5.41950 q^{35} +11.6326 q^{36} -0.469227 q^{37} +4.74514 q^{38} -6.70707 q^{39} -11.4387 q^{40} +3.60425 q^{41} -18.9201 q^{42} -10.0365 q^{43} -19.3858 q^{44} +4.39349 q^{45} +18.9999 q^{46} -11.2607 q^{47} -18.3784 q^{48} +2.82082 q^{49} +5.15374 q^{50} +3.54219 q^{51} +13.0470 q^{52} -6.26150 q^{53} -2.77400 q^{54} -7.32177 q^{55} +20.7284 q^{56} +4.35462 q^{57} +3.02722 q^{58} -8.36887 q^{59} -18.6386 q^{60} +14.3617 q^{61} +17.9987 q^{62} -7.96156 q^{63} +1.81973 q^{64} +4.92768 q^{65} -25.5611 q^{66} +14.3594 q^{67} -6.89047 q^{68} +17.4363 q^{69} +13.9006 q^{70} +12.7596 q^{71} -16.8042 q^{72} -12.7007 q^{73} +1.20353 q^{74} +4.72960 q^{75} -8.47086 q^{76} +13.2680 q^{77} +17.2031 q^{78} -0.827441 q^{79} +13.5026 q^{80} -10.1673 q^{81} -9.24461 q^{82} -12.6263 q^{83} +33.7755 q^{84} -2.60245 q^{85} +25.7429 q^{86} +2.77809 q^{87} +28.0042 q^{88} +16.1548 q^{89} -11.2689 q^{90} -8.92958 q^{91} -33.9180 q^{92} +16.5175 q^{93} +28.8828 q^{94} -3.19934 q^{95} +16.0006 q^{96} +5.98839 q^{97} -7.23516 q^{98} -10.7561 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9}+O(q^{10}) \) 186 * q + 6 * q^2 + 10 * q^3 + 212 * q^4 + 38 * q^5 + 47 * q^6 + 32 * q^7 + 15 * q^8 + 216 * q^9	\( 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9} + 50 q^{10} + 25 q^{11} + 17 q^{12} + 113 q^{13} + 12 q^{14} + 12 q^{15} + 252 q^{16} + 35 q^{17} + 13 q^{18} + 97 q^{19} + 55 q^{20} + 115 q^{21} + 14 q^{22} + 27 q^{23} + 122 q^{24} + 244 q^{25} + 39 q^{26} + 34 q^{27} + 66 q^{28} + 91 q^{29} + 4 q^{30} + 135 q^{31} + 21 q^{32} + 32 q^{33} + 58 q^{34} + 17 q^{35} + 273 q^{36} + 133 q^{37} - 3 q^{38} + 55 q^{39} + 142 q^{40} + 97 q^{41} - 8 q^{42} + 67 q^{43} + 44 q^{44} + 154 q^{45} + 101 q^{46} + 20 q^{47} - 7 q^{48} + 312 q^{49} + 21 q^{50} + 23 q^{51} + 193 q^{52} + 22 q^{53} + 141 q^{54} + 88 q^{55} + 28 q^{56} + 65 q^{57} + 62 q^{58} + 41 q^{59} + q^{60} + 377 q^{61} + 29 q^{62} + 39 q^{63} + 311 q^{64} + 21 q^{65} + 35 q^{66} + 42 q^{67} + 24 q^{68} + 137 q^{69} + 35 q^{70} + 17 q^{71} - 8 q^{72} + 213 q^{73} - 9 q^{74} + 2 q^{75} + 242 q^{76} + 60 q^{77} + 103 q^{79} + 80 q^{80} + 270 q^{81} + 84 q^{82} + 42 q^{83} + 137 q^{84} + 294 q^{85} - 9 q^{86} + 22 q^{87} - 13 q^{88} + 78 q^{89} + 69 q^{90} + 118 q^{91} + 49 q^{92} + 51 q^{93} + 93 q^{94} + 10 q^{95} + 260 q^{96} + 142 q^{97} - 31 q^{98} + 78 q^{99}+O(q^{100}) \) 186 * q + 6 * q^2 + 10 * q^3 + 212 * q^4 + 38 * q^5 + 47 * q^6 + 32 * q^7 + 15 * q^8 + 216 * q^9 + 50 * q^10 + 25 * q^11 + 17 * q^12 + 113 * q^13 + 12 * q^14 + 12 * q^15 + 252 * q^16 + 35 * q^17 + 13 * q^18 + 97 * q^19 + 55 * q^20 + 115 * q^21 + 14 * q^22 + 27 * q^23 + 122 * q^24 + 244 * q^25 + 39 * q^26 + 34 * q^27 + 66 * q^28 + 91 * q^29 + 4 * q^30 + 135 * q^31 + 21 * q^32 + 32 * q^33 + 58 * q^34 + 17 * q^35 + 273 * q^36 + 133 * q^37 - 3 * q^38 + 55 * q^39 + 142 * q^40 + 97 * q^41 - 8 * q^42 + 67 * q^43 + 44 * q^44 + 154 * q^45 + 101 * q^46 + 20 * q^47 - 7 * q^48 + 312 * q^49 + 21 * q^50 + 23 * q^51 + 193 * q^52 + 22 * q^53 + 141 * q^54 + 88 * q^55 + 28 * q^56 + 65 * q^57 + 62 * q^58 + 41 * q^59 + q^60 + 377 * q^61 + 29 * q^62 + 39 * q^63 + 311 * q^64 + 21 * q^65 + 35 * q^66 + 42 * q^67 + 24 * q^68 + 137 * q^69 + 35 * q^70 + 17 * q^71 - 8 * q^72 + 213 * q^73 - 9 * q^74 + 2 * q^75 + 242 * q^76 + 60 * q^77 + 103 * q^79 + 80 * q^80 + 270 * q^81 + 84 * q^82 + 42 * q^83 + 137 * q^84 + 294 * q^85 - 9 * q^86 + 22 * q^87 - 13 * q^88 + 78 * q^89 + 69 * q^90 + 118 * q^91 + 49 * q^92 + 51 * q^93 + 93 * q^94 + 10 * q^95 + 260 * q^96 + 142 * q^97 - 31 * q^98 + 78 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more