LMFDB - Embedded newform 4009.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$0$
Dimension:	$82$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64355 q^{2} -0.244924 q^{3} +4.98837 q^{4} -2.38993 q^{5} +0.647469 q^{6} -3.32754 q^{7} -7.89992 q^{8} -2.94001 q^{9} +O(q^{10})$	\(q-2.64355 q^{2} -0.244924 q^{3} +4.98837 q^{4} -2.38993 q^{5} +0.647469 q^{6} -3.32754 q^{7} -7.89992 q^{8} -2.94001 q^{9} +6.31792 q^{10} +5.93512 q^{11} -1.22177 q^{12} -1.76306 q^{13} +8.79652 q^{14} +0.585352 q^{15} +10.9071 q^{16} -6.37545 q^{17} +7.77208 q^{18} +1.00000 q^{19} -11.9219 q^{20} +0.814992 q^{21} -15.6898 q^{22} +4.84729 q^{23} +1.93488 q^{24} +0.711789 q^{25} +4.66075 q^{26} +1.45485 q^{27} -16.5990 q^{28} -0.117484 q^{29} -1.54741 q^{30} -1.24707 q^{31} -13.0337 q^{32} -1.45365 q^{33} +16.8538 q^{34} +7.95260 q^{35} -14.6659 q^{36} -2.53220 q^{37} -2.64355 q^{38} +0.431816 q^{39} +18.8803 q^{40} +4.09172 q^{41} -2.15448 q^{42} -5.25564 q^{43} +29.6066 q^{44} +7.02644 q^{45} -12.8141 q^{46} -6.93293 q^{47} -2.67141 q^{48} +4.07250 q^{49} -1.88165 q^{50} +1.56150 q^{51} -8.79481 q^{52} -3.76584 q^{53} -3.84597 q^{54} -14.1845 q^{55} +26.2873 q^{56} -0.244924 q^{57} +0.310576 q^{58} -15.1531 q^{59} +2.91995 q^{60} -6.51744 q^{61} +3.29670 q^{62} +9.78300 q^{63} +12.6410 q^{64} +4.21361 q^{65} +3.84280 q^{66} -12.2939 q^{67} -31.8031 q^{68} -1.18722 q^{69} -21.0231 q^{70} -11.0480 q^{71} +23.2259 q^{72} -9.39250 q^{73} +6.69401 q^{74} -0.174334 q^{75} +4.98837 q^{76} -19.7493 q^{77} -1.14153 q^{78} +0.0139057 q^{79} -26.0673 q^{80} +8.46371 q^{81} -10.8167 q^{82} +13.1561 q^{83} +4.06549 q^{84} +15.2369 q^{85} +13.8936 q^{86} +0.0287747 q^{87} -46.8870 q^{88} +8.66445 q^{89} -18.5748 q^{90} +5.86666 q^{91} +24.1801 q^{92} +0.305437 q^{93} +18.3276 q^{94} -2.38993 q^{95} +3.19226 q^{96} -9.32513 q^{97} -10.7659 q^{98} -17.4493 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) $ 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9 + 4 * q^10 + 41 * q^11 + 26 * q^12 + 13 * q^13 + 22 * q^14 + 41 * q^15 + 87 * q^16 + 12 * q^17 + 24 * q^18 + 82 * q^19 + 26 * q^20 + 29 * q^21 + 2 * q^22 + 59 * q^23 + 16 * q^24 + 67 * q^25 + 24 * q^26 + 42 * q^27 - 2 * q^28 + 101 * q^29 - 22 * q^30 + 48 * q^31 + 69 * q^32 + 3 * q^33 + q^34 + 38 * q^35 + 82 * q^36 + 16 * q^37 + 15 * q^38 + 82 * q^39 + 20 * q^40 + 86 * q^41 - q^42 + 9 * q^43 + 82 * q^44 - 8 * q^45 + 43 * q^46 + 24 * q^47 + 34 * q^48 + 76 * q^49 + 82 * q^50 + 57 * q^51 - 22 * q^52 + 39 * q^53 + 17 * q^54 - 21 * q^55 + 50 * q^56 + 12 * q^57 + 33 * q^58 + 79 * q^59 + 87 * q^60 + 4 * q^61 + 40 * q^62 + 44 * q^63 + 90 * q^64 + 66 * q^65 - 39 * q^66 + 33 * q^67 - 9 * q^68 + 60 * q^69 + 30 * q^70 + 168 * q^71 + 15 * q^72 - 28 * q^73 + 35 * q^74 + 55 * q^75 + 89 * q^76 + 19 * q^77 - 41 * q^78 + 121 * q^79 + 64 * q^80 + 110 * q^81 + 41 * q^82 + 28 * q^84 + 17 * q^85 + 80 * q^86 + 29 * q^87 + 49 * q^88 + 83 * q^89 - 42 * q^90 + 38 * q^91 + 71 * q^92 - q^93 + 89 * q^94 + 9 * q^95 + 35 * q^96 - 23 * q^97 + 135 * q^98 + 93 * 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 $

$2$	−2.64355	−1.86927	−0.934637	−	0.355603i	$-0.884276\pi$
$2$	−2.64355	−1.86927	−0.934637	+	0.355603i	$0.884276\pi$
$3$	−0.244924	−0.141407	−0.0707034	−	0.997497i	$-0.522524\pi$
$3$	−0.244924	−0.141407	−0.0707034	+	0.997497i	$0.522524\pi$
$4$	4.98837	2.49419
$5$	−2.38993	−1.06881	−0.534406	−	0.845228i	$-0.679465\pi$
$5$	−2.38993	−1.06881	−0.534406	+	0.845228i	$0.679465\pi$
$6$	0.647469	0.264328
$7$	−3.32754	−1.25769	−0.628845	−	0.777530i	$-0.716472\pi$
$7$	−3.32754	−1.25769	−0.628845	+	0.777530i	$0.716472\pi$
$8$	−7.89992	−2.79304
$9$	−2.94001	−0.980004
$10$	6.31792	1.99790
$11$	5.93512	1.78951	0.894753	−	0.446562i	$-0.147352\pi$
$11$	5.93512	1.78951	0.894753	+	0.446562i	$0.147352\pi$
$12$	−1.22177	−0.352695
$13$	−1.76306	−0.488986	−0.244493	−	0.969651i	$-0.578621\pi$
$13$	−1.76306	−0.488986	−0.244493	+	0.969651i	$0.578621\pi$
$14$	8.79652	2.35097
$15$	0.585352	0.151137
$16$	10.9071	2.72678
$17$	−6.37545	−1.54627	−0.773137	−	0.634240i	$-0.781313\pi$
$17$	−6.37545	−1.54627	−0.773137	+	0.634240i	$0.781313\pi$
$18$	7.77208	1.83190
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−11.9219	−2.66581
$21$	0.814992	0.177846
$22$	−15.6898	−3.34508
$23$	4.84729	1.01073	0.505365	−	0.862906i	$-0.331358\pi$
$23$	4.84729	1.01073	0.505365	+	0.862906i	$0.331358\pi$
$24$	1.93488	0.394955
$25$	0.711789	0.142358
$26$	4.66075	0.914048
$27$	1.45485	0.279986
$28$	−16.5990	−3.13691
$29$	−0.117484	−0.0218163	−0.0109081	−	0.999941i	$-0.503472\pi$
$29$	−0.117484	−0.0218163	−0.0109081	+	0.999941i	$0.503472\pi$
$30$	−1.54741	−0.282517
$31$	−1.24707	−0.223980	−0.111990	−	0.993709i	$-0.535723\pi$
$31$	−1.24707	−0.223980	−0.111990	+	0.993709i	$0.535723\pi$
$32$	−13.0337	−2.30405
$33$	−1.45365	−0.253048
$34$	16.8538	2.89041
$35$	7.95260	1.34423
$36$	−14.6659	−2.44431
$37$	−2.53220	−0.416291	−0.208146	−	0.978098i	$-0.566743\pi$
$37$	−2.53220	−0.416291	−0.208146	+	0.978098i	$0.566743\pi$
$38$	−2.64355	−0.428841
$39$	0.431816	0.0691459
$40$	18.8803	2.98524
$41$	4.09172	0.639019	0.319510	−	0.947583i	$-0.396482\pi$
$41$	4.09172	0.639019	0.319510	+	0.947583i	$0.396482\pi$
$42$	−2.15448	−0.332443
$43$	−5.25564	−0.801478	−0.400739	−	0.916192i	$-0.631247\pi$
$43$	−5.25564	−0.801478	−0.400739	+	0.916192i	$0.631247\pi$
$44$	29.6066	4.46336
$45$	7.02644	1.04744
$46$	−12.8141	−1.88933
$47$	−6.93293	−1.01127	−0.505636	−	0.862747i	$-0.668742\pi$
$47$	−6.93293	−1.01127	−0.505636	+	0.862747i	$0.668742\pi$
$48$	−2.67141	−0.385585
$49$	4.07250	0.581786
$50$	−1.88165	−0.266106
$51$	1.56150	0.218653
$52$	−8.79481	−1.21962
$53$	−3.76584	−0.517278	−0.258639	−	0.965974i	$-0.583274\pi$
$53$	−3.76584	−0.517278	−0.258639	+	0.965974i	$0.583274\pi$
$54$	−3.84597	−0.523371
$55$	−14.1845	−1.91264
$56$	26.2873	3.51279
$57$	−0.244924	−0.0324409
$58$	0.310576	0.0407806
$59$	−15.1531	−1.97276	−0.986380	−	0.164480i	$-0.947406\pi$
$59$	−15.1531	−1.97276	−0.986380	+	0.164480i	$0.947406\pi$
$60$	2.91995	0.376964
$61$	−6.51744	−0.834472	−0.417236	−	0.908798i	$-0.637001\pi$
$61$	−6.51744	−0.834472	−0.417236	+	0.908798i	$0.637001\pi$
$62$	3.29670	0.418681
$63$	9.78300	1.23254
$64$	12.6410	1.58013
$65$	4.21361	0.522633
$66$	3.84280	0.473016
$67$	−12.2939	−1.50194	−0.750968	−	0.660338i	$-0.770413\pi$
$67$	−12.2939	−1.50194	−0.750968	+	0.660338i	$0.770413\pi$
$68$	−31.8031	−3.85669
$69$	−1.18722	−0.142924
$70$	−21.0231	−2.51274
$71$	−11.0480	−1.31116	−0.655579	−	0.755126i	$-0.727575\pi$
$71$	−11.0480	−1.31116	−0.655579	+	0.755126i	$0.727575\pi$
$72$	23.2259	2.73719
$73$	−9.39250	−1.09931	−0.549655	−	0.835392i	$-0.685241\pi$
$73$	−9.39250	−1.09931	−0.549655	+	0.835392i	$0.685241\pi$
$74$	6.69401	0.778163
$75$	−0.174334	−0.0201304
$76$	4.98837	0.572206
$77$	−19.7493	−2.25064
$78$	−1.14153	−0.129253
$79$	0.0139057	0.00156451	0.000782254	−	1.00000i	$-0.499751\pi$
$79$	0.0139057	0.00156451	0.000782254		1.00000i	$0.499751\pi$
$80$	−26.0673	−2.91441
$81$	8.46371	0.940412
$82$	−10.8167	−1.19450
$83$	13.1561	1.44407	0.722033	−	0.691859i	$-0.243208\pi$
$83$	13.1561	1.44407	0.722033	+	0.691859i	$0.243208\pi$
$84$	4.06549	0.443581
$85$	15.2369	1.65267
$86$	13.8936	1.49818
$87$	0.0287747	0.00308497
$88$	−46.8870	−4.99817
$89$	8.66445	0.918430	0.459215	−	0.888325i	$-0.348131\pi$
$89$	8.66445	0.918430	0.459215	+	0.888325i	$0.348131\pi$
$90$	−18.5748	−1.95795
$91$	5.86666	0.614993
$92$	24.1801	2.52095
$93$	0.305437	0.0316723
$94$	18.3276	1.89034
$95$	−2.38993	−0.245202
$96$	3.19226	0.325809
$97$	−9.32513	−0.946824	−0.473412	−	0.880841i	$-0.656978\pi$
$97$	−9.32513	−0.946824	−0.473412	+	0.880841i	$0.656978\pi$
$98$	−10.7659	−1.08752
$99$	−17.4493	−1.75372
$100$	3.55067	0.355067
$101$	−10.0899	−1.00399	−0.501993	−	0.864872i	$-0.667400\pi$
$101$	−10.0899	−1.00399	−0.501993	+	0.864872i	$0.667400\pi$
$102$	−4.12790	−0.408723
$103$	3.15306	0.310680	0.155340	−	0.987861i	$-0.450353\pi$
$103$	3.15306	0.310680	0.155340	+	0.987861i	$0.450353\pi$
$104$	13.9281	1.36576
$105$	−1.94778	−0.190084
$106$	9.95520	0.966935
$107$	−2.79937	−0.270625	−0.135313	−	0.990803i	$-0.543204\pi$
$107$	−2.79937	−0.270625	−0.135313	+	0.990803i	$0.543204\pi$
$108$	7.25733	0.698337
$109$	11.3730	1.08934	0.544668	−	0.838652i	$-0.316656\pi$
$109$	11.3730	1.08934	0.544668	+	0.838652i	$0.316656\pi$
$110$	37.4976	3.57526
$111$	0.620196	0.0588664
$112$	−36.2938	−3.42944
$113$	9.53553	0.897027	0.448513	−	0.893776i	$-0.351954\pi$
$113$	9.53553	0.897027	0.448513	+	0.893776i	$0.351954\pi$
$114$	0.647469	0.0606410
$115$	−11.5847	−1.08028
$116$	−0.586055	−0.0544138
$117$	5.18343	0.479208
$118$	40.0579	3.68763
$119$	21.2145	1.94473
$120$	−4.62423	−0.422133
$121$	24.2256	2.20233
$122$	17.2292	1.55986
$123$	−1.00216	−0.0903616
$124$	−6.22085	−0.558649
$125$	10.2485	0.916658
$126$	−25.8619	−2.30396
$127$	−13.3492	−1.18455	−0.592274	−	0.805736i	$-0.701770\pi$
$127$	−13.3492	−1.18455	−0.592274	+	0.805736i	$0.701770\pi$
$128$	−7.34986	−0.649642
$129$	1.28723	0.113334
$130$	−11.1389	−0.976945
$131$	−10.4910	−0.916602	−0.458301	−	0.888797i	$-0.651542\pi$
$131$	−10.4910	−0.916602	−0.458301	+	0.888797i	$0.651542\pi$
$132$	−7.25135	−0.631149
$133$	−3.32754	−0.288534
$134$	32.4995	2.80753
$135$	−3.47700	−0.299252
$136$	50.3655	4.31881
$137$	−8.41299	−0.718770	−0.359385	−	0.933189i	$-0.617014\pi$
$137$	−8.41299	−0.718770	−0.359385	+	0.933189i	$0.617014\pi$
$138$	3.13847	0.267164
$139$	−13.7403	−1.16544	−0.582720	−	0.812673i	$-0.698011\pi$
$139$	−13.7403	−1.16544	−0.582720	+	0.812673i	$0.698011\pi$
$140$	39.6705	3.35277
$141$	1.69804	0.143001
$142$	29.2060	2.45091
$143$	−10.4640	−0.875042
$144$	−32.0671	−2.67225
$145$	0.280780	0.0233175
$146$	24.8296	2.05491
$147$	−0.997451	−0.0822684
$148$	−12.6316	−1.03831
$149$	18.5466	1.51940	0.759698	−	0.650276i	$-0.225347\pi$
$149$	18.5466	1.51940	0.759698	+	0.650276i	$0.225347\pi$
$150$	0.460861	0.0376292
$151$	−17.0274	−1.38567	−0.692834	−	0.721097i	$-0.743638\pi$
$151$	−17.0274	−1.38567	−0.692834	+	0.721097i	$0.743638\pi$
$152$	−7.89992	−0.640768
$153$	18.7439	1.51535
$154$	52.2084	4.20707
$155$	2.98042	0.239393
$156$	2.15406	0.172463
$157$	9.26465	0.739400	0.369700	−	0.929151i	$-0.379460\pi$
$157$	9.26465	0.739400	0.369700	+	0.929151i	$0.379460\pi$
$158$	−0.0367603	−0.00292450
$159$	0.922344	0.0731466
$160$	31.1497	2.46260
$161$	−16.1295	−1.27119
$162$	−22.3743	−1.75789
$163$	−4.42782	−0.346814	−0.173407	−	0.984850i	$-0.555478\pi$
$163$	−4.42782	−0.346814	−0.173407	+	0.984850i	$0.555478\pi$
$164$	20.4110	1.59383
$165$	3.47413	0.270461
$166$	−34.7788	−2.69936
$167$	17.0241	1.31737	0.658683	−	0.752420i	$-0.271114\pi$
$167$	17.0241	1.31737	0.658683	+	0.752420i	$0.271114\pi$
$168$	−6.43838	−0.496732
$169$	−9.89161	−0.760893
$170$	−40.2796	−3.08930
$171$	−2.94001	−0.224828
$172$	−26.2171	−1.99904
$173$	15.4360	1.17358	0.586788	−	0.809740i	$-0.300392\pi$
$173$	15.4360	1.17358	0.586788	+	0.809740i	$0.300392\pi$
$174$	−0.0760673	−0.00576665
$175$	−2.36850	−0.179042
$176$	64.7350	4.87959
$177$	3.71134	0.278962
$178$	−22.9049	−1.71680
$179$	19.7353	1.47508	0.737542	−	0.675302i	$-0.235987\pi$
$179$	19.7353	1.47508	0.737542	+	0.675302i	$0.235987\pi$
$180$	35.0505	2.61251
$181$	8.96878	0.666644	0.333322	−	0.942813i	$-0.391830\pi$
$181$	8.96878	0.666644	0.333322	+	0.942813i	$0.391830\pi$
$182$	−15.5088	−1.14959
$183$	1.59627	0.118000
$184$	−38.2932	−2.82301
$185$	6.05180	0.444937
$186$	−0.807439	−0.0592043
$187$	−37.8390	−2.76706
$188$	−34.5840	−2.52230
$189$	−4.84106	−0.352136
$190$	6.31792	0.458350
$191$	−2.56885	−0.185875	−0.0929376	−	0.995672i	$-0.529626\pi$
$191$	−2.56885	−0.185875	−0.0929376	+	0.995672i	$0.529626\pi$
$192$	−3.09609	−0.223441
$193$	−9.67022	−0.696078	−0.348039	−	0.937480i	$-0.613152\pi$
$193$	−9.67022	−0.696078	−0.348039	+	0.937480i	$0.613152\pi$
$194$	24.6515	1.76987
$195$	−1.03201	−0.0739039
$196$	20.3151	1.45108
$197$	−10.1040	−0.719882	−0.359941	−	0.932975i	$-0.617203\pi$
$197$	−10.1040	−0.719882	−0.359941	+	0.932975i	$0.617203\pi$
$198$	46.1282	3.27819
$199$	−18.1986	−1.29007	−0.645033	−	0.764155i	$-0.723156\pi$
$199$	−18.1986	−1.29007	−0.645033	+	0.764155i	$0.723156\pi$
$200$	−5.62308	−0.397612
$201$	3.01106	0.212384
$202$	26.6733	1.87673
$203$	0.390933	0.0274381
$204$	7.78933	0.545362
$205$	−9.77894	−0.682991
$206$	−8.33528	−0.580746
$207$	−14.2511	−0.990519
$208$	−19.2299	−1.33336
$209$	5.93512	0.410541
$210$	5.14906	0.355319
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	−18.7854	−1.29019
$213$	2.70592	0.185407
$214$	7.40028	0.505872
$215$	12.5606	0.856629
$216$	−11.4932	−0.782013
$217$	4.14967	0.281698
$218$	−30.0652	−2.03627
$219$	2.30045	0.155450
$220$	−70.7578	−4.77049
$221$	11.2403	0.756105
$222$	−1.63952	−0.110037
$223$	0.829449	0.0555440	0.0277720	−	0.999614i	$-0.491159\pi$
$223$	0.829449	0.0555440	0.0277720	+	0.999614i	$0.491159\pi$
$224$	43.3701	2.89779
$225$	−2.09267	−0.139511
$226$	−25.2077	−1.67679
$227$	26.8534	1.78232	0.891160	−	0.453689i	$-0.149892\pi$
$227$	26.8534	1.78232	0.891160	+	0.453689i	$0.149892\pi$
$228$	−1.22177	−0.0809137
$229$	−6.95790	−0.459791	−0.229895	−	0.973215i	$-0.573838\pi$
$229$	−6.95790	−0.459791	−0.229895	+	0.973215i	$0.573838\pi$
$230$	30.6248	2.01934
$231$	4.83708	0.318256
$232$	0.928116	0.0609338
$233$	−22.7731	−1.49191	−0.745957	−	0.665994i	$-0.768007\pi$
$233$	−22.7731	−1.49191	−0.745957	+	0.665994i	$0.768007\pi$
$234$	−13.7027	−0.895771
$235$	16.5693	1.08086
$236$	−75.5891	−4.92043
$237$	−0.00340582	−0.000221232 0
$238$	−56.0817	−3.63524
$239$	21.2176	1.37245	0.686226	−	0.727388i	$-0.259266\pi$
$239$	21.2176	1.37245	0.686226	+	0.727388i	$0.259266\pi$
$240$	6.38450	0.412118
$241$	−12.1440	−0.782261	−0.391131	−	0.920335i	$-0.627916\pi$
$241$	−12.1440	−0.782261	−0.391131	+	0.920335i	$0.627916\pi$
$242$	−64.0417	−4.11676
$243$	−6.43751	−0.412967
$244$	−32.5114	−2.08133
$245$	−9.73301	−0.621819
$246$	2.64926	0.168911
$247$	−1.76306	−0.112181
$248$	9.85176	0.625587
$249$	−3.22223	−0.204201
$250$	−27.0926	−1.71348
$251$	−18.3303	−1.15700	−0.578498	−	0.815684i	$-0.696361\pi$
$251$	−18.3303	−1.15700	−0.578498	+	0.815684i	$0.696361\pi$
$252$	48.8012	3.07419
$253$	28.7692	1.80871
$254$	35.2893	2.21425
$255$	−3.73188	−0.233699
$256$	−5.85233	−0.365771
$257$	−4.08921	−0.255078	−0.127539	−	0.991834i	$-0.540708\pi$
$257$	−4.08921	−0.255078	−0.127539	+	0.991834i	$0.540708\pi$
$258$	−3.40286	−0.211853
$259$	8.42599	0.523566
$260$	21.0190	1.30355
$261$	0.345405	0.0213800
$262$	27.7335	1.71338
$263$	14.4796	0.892849	0.446424	−	0.894821i	$-0.352697\pi$
$263$	14.4796	0.892849	0.446424	+	0.894821i	$0.352697\pi$
$264$	11.4837	0.706775
$265$	9.00012	0.552873
$266$	8.79652	0.539349
$267$	−2.12213	−0.129872
$268$	−61.3265	−3.74611
$269$	−2.60018	−0.158536	−0.0792680	−	0.996853i	$-0.525258\pi$
$269$	−2.60018	−0.158536	−0.0792680	+	0.996853i	$0.525258\pi$
$270$	9.19162	0.559384
$271$	3.53395	0.214672	0.107336	−	0.994223i	$-0.465768\pi$
$271$	3.53395	0.214672	0.107336	+	0.994223i	$0.465768\pi$
$272$	−69.5377	−4.21635
$273$	−1.43688	−0.0869641
$274$	22.2402	1.34358
$275$	4.22455	0.254750
$276$	−5.92228	−0.356479
$277$	23.3049	1.40025	0.700127	−	0.714018i	$-0.253127\pi$
$277$	23.3049	1.40025	0.700127	+	0.714018i	$0.253127\pi$
$278$	36.3233	2.17853
$279$	3.66640	0.219502
$280$	−62.8249	−3.75450
$281$	3.07950	0.183708	0.0918538	−	0.995773i	$-0.470721\pi$
$281$	3.07950	0.183708	0.0918538	+	0.995773i	$0.470721\pi$
$282$	−4.48886	−0.267308
$283$	3.27640	0.194762	0.0973808	−	0.995247i	$-0.468954\pi$
$283$	3.27640	0.194762	0.0973808	+	0.995247i	$0.468954\pi$
$284$	−55.1116	−3.27027
$285$	0.585352	0.0346732
$286$	27.6621	1.63569
$287$	−13.6153	−0.803688
$288$	38.3192	2.25798
$289$	23.6463	1.39096
$290$	−0.742256	−0.0435868
$291$	2.28395	0.133887
$292$	−46.8533	−2.74188
$293$	15.4335	0.901636	0.450818	−	0.892616i	$-0.351132\pi$
$293$	15.4335	0.901636	0.450818	+	0.892616i	$0.351132\pi$
$294$	2.63682	0.153782
$295$	36.2148	2.10851
$296$	20.0042	1.16272
$297$	8.63470	0.501036
$298$	−49.0289	−2.84017
$299$	−8.54608	−0.494232
$300$	−0.869643	−0.0502089
$301$	17.4883	1.00801
$302$	45.0127	2.59019
$303$	2.47126	0.141970
$304$	10.9071	0.625566
$305$	15.5763	0.891894
$306$	−49.5505	−2.83261
$307$	−19.4096	−1.10777	−0.553883	−	0.832595i	$-0.686854\pi$
$307$	−19.4096	−1.10777	−0.553883	+	0.832595i	$0.686854\pi$
$308$	−98.5170	−5.61353
$309$	−0.772259	−0.0439323
$310$	−7.87889	−0.447491
$311$	19.3409	1.09672	0.548360	−	0.836242i	$-0.315252\pi$
$311$	19.3409	1.09672	0.548360	+	0.836242i	$0.315252\pi$
$312$	−3.41131	−0.193127
$313$	−29.3705	−1.66012	−0.830060	−	0.557674i	$-0.811694\pi$
$313$	−29.3705	−1.66012	−0.830060	+	0.557674i	$0.811694\pi$
$314$	−24.4916	−1.38214
$315$	−23.3807	−1.31735
$316$	0.0693666	0.00390218
$317$	25.1014	1.40984	0.704919	−	0.709288i	$-0.250983\pi$
$317$	25.1014	1.40984	0.704919	+	0.709288i	$0.250983\pi$
$318$	−2.43826	−0.136731
$319$	−0.697282	−0.0390403
$320$	−30.2113	−1.68886
$321$	0.685631	0.0382682
$322$	42.6393	2.37619
$323$	−6.37545	−0.354739
$324$	42.2201	2.34556
$325$	−1.25493	−0.0696109
$326$	11.7052	0.648290
$327$	−2.78552	−0.154040
$328$	−32.3243	−1.78481
$329$	23.0696	1.27187
$330$	−9.18405	−0.505565
$331$	9.02935	0.496298	0.248149	−	0.968722i	$-0.420178\pi$
$331$	9.02935	0.496298	0.248149	+	0.968722i	$0.420178\pi$
$332$	65.6274	3.60177
$333$	7.44470	0.407967
$334$	−45.0042	−2.46252
$335$	29.3816	1.60529
$336$	8.88922	0.484947
$337$	11.4925	0.626035	0.313018	−	0.949747i	$-0.398660\pi$
$337$	11.4925	0.626035	0.313018	+	0.949747i	$0.398660\pi$
$338$	26.1490	1.42232
$339$	−2.33548	−0.126846
$340$	76.0074	4.12208
$341$	−7.40151	−0.400814
$342$	7.77208	0.420266
$343$	9.74137	0.525984
$344$	41.5192	2.23856
$345$	2.83737	0.152759
$346$	−40.8059	−2.19374
$347$	−7.52968	−0.404215	−0.202107	−	0.979363i	$-0.564779\pi$
$347$	−7.52968	−0.404215	−0.202107	+	0.979363i	$0.564779\pi$
$348$	0.143539	0.00769448
$349$	4.28641	0.229446	0.114723	−	0.993398i	$-0.463402\pi$
$349$	4.28641	0.229446	0.114723	+	0.993398i	$0.463402\pi$
$350$	6.26127	0.334679
$351$	−2.56499	−0.136909
$352$	−77.3565	−4.12312
$353$	−36.1077	−1.92182	−0.960911	−	0.276858i	$-0.910707\pi$
$353$	−36.1077	−1.92182	−0.960911	+	0.276858i	$0.910707\pi$
$354$	−9.81113	−0.521456
$355$	26.4040	1.40138
$356$	43.2215	2.29073
$357$	−5.19594	−0.274998
$358$	−52.1712	−2.75734
$359$	−8.79140	−0.463992	−0.231996	−	0.972717i	$-0.574526\pi$
$359$	−8.79140	−0.463992	−0.231996	+	0.972717i	$0.574526\pi$
$360$	−55.5083	−2.92555
$361$	1.00000	0.0526316
$362$	−23.7094	−1.24614
$363$	−5.93343	−0.311424
$364$	29.2651	1.53391
$365$	22.4475	1.17495
$366$	−4.21984	−0.220574
$367$	−9.25549	−0.483132	−0.241566	−	0.970384i	$-0.577661\pi$
$367$	−9.25549	−0.483132	−0.241566	+	0.970384i	$0.577661\pi$
$368$	52.8700	2.75604
$369$	−12.0297	−0.626241
$370$	−15.9982	−0.831709
$371$	12.5310	0.650576
$372$	1.52363	0.0789967
$373$	25.2350	1.30662	0.653309	−	0.757091i	$-0.273380\pi$
$373$	25.2350	1.30662	0.653309	+	0.757091i	$0.273380\pi$
$374$	100.029	5.17240
$375$	−2.51011	−0.129622
$376$	54.7696	2.82453
$377$	0.207132	0.0106678
$378$	12.7976	0.658238
$379$	9.79177	0.502970	0.251485	−	0.967861i	$-0.419081\pi$
$379$	9.79177	0.502970	0.251485	+	0.967861i	$0.419081\pi$
$380$	−11.9219	−0.611580
$381$	3.26953	0.167503
$382$	6.79088	0.347452
$383$	−22.7993	−1.16499	−0.582495	−	0.812835i	$-0.697923\pi$
$383$	−22.7993	−1.16499	−0.582495	+	0.812835i	$0.697923\pi$
$384$	1.80016	0.0918638
$385$	47.1996	2.40551
$386$	25.5637	1.30116
$387$	15.4517	0.785452
$388$	−46.5172	−2.36155
$389$	−19.3781	−0.982510	−0.491255	−	0.871016i	$-0.663462\pi$
$389$	−19.3781	−0.982510	−0.491255	+	0.871016i	$0.663462\pi$
$390$	2.72818	0.138147
$391$	−30.9036	−1.56286
$392$	−32.1724	−1.62495
$393$	2.56949	0.129614
$394$	26.7105	1.34566
$395$	−0.0332336	−0.00167216
$396$	−87.0437	−4.37411
$397$	18.1285	0.909844	0.454922	−	0.890531i	$-0.349667\pi$
$397$	18.1285	0.909844	0.454922	+	0.890531i	$0.349667\pi$
$398$	48.1090	2.41149
$399$	0.814992	0.0408007
$400$	7.76357	0.388178
$401$	−32.3748	−1.61672	−0.808360	−	0.588688i	$-0.799645\pi$
$401$	−32.3748	−1.61672	−0.808360	+	0.588688i	$0.799645\pi$
$402$	−7.95990	−0.397004
$403$	2.19866	0.109523
$404$	−50.3324	−2.50413
$405$	−20.2277	−1.00512
$406$	−1.03345	−0.0512894
$407$	−15.0289	−0.744955
$408$	−12.3357	−0.610709
$409$	21.0058	1.03867	0.519335	−	0.854571i	$-0.326180\pi$
$409$	21.0058	1.03867	0.519335	+	0.854571i	$0.326180\pi$
$410$	25.8511	1.27670
$411$	2.06054	0.101639
$412$	15.7286	0.774894
$413$	50.4224	2.48112
$414$	37.6735	1.85155
$415$	−31.4421	−1.54343
$416$	22.9792	1.12665
$417$	3.36533	0.164801
$418$	−15.6898	−0.767413
$419$	12.1081	0.591521	0.295760	−	0.955262i	$-0.404427\pi$
$419$	12.1081	0.591521	0.295760	+	0.955262i	$0.404427\pi$
$420$	−9.71625	−0.474104
$421$	0.0752569	0.00366780	0.00183390	−	0.999998i	$-0.499416\pi$
$421$	0.0752569	0.00366780	0.00183390	+	0.999998i	$0.499416\pi$
$422$	2.64355	0.128686
$423$	20.3829	0.991051
$424$	29.7499	1.44478
$425$	−4.53797	−0.220124
$426$	−7.15325	−0.346576
$427$	21.6870	1.04951
$428$	−13.9643	−0.674989
$429$	2.56288	0.123737
$430$	−33.2047	−1.60127
$431$	37.8102	1.82125	0.910627	−	0.413229i	$-0.135599\pi$
$431$	37.8102	1.82125	0.910627	+	0.413229i	$0.135599\pi$
$432$	15.8682	0.763460
$433$	27.2787	1.31093	0.655466	−	0.755224i	$-0.272472\pi$
$433$	27.2787	1.31093	0.655466	+	0.755224i	$0.272472\pi$
$434$	−10.9699	−0.526571
$435$	−0.0687696	−0.00329725
$436$	56.7328	2.71701
$437$	4.84729	0.231877
$438$	−6.08135	−0.290578
$439$	14.8867	0.710501	0.355251	−	0.934771i	$-0.384395\pi$
$439$	14.8867	0.710501	0.355251	+	0.934771i	$0.384395\pi$
$440$	112.057	5.34210
$441$	−11.9732	−0.570152
$442$	−29.7144	−1.41337
$443$	−17.8114	−0.846244	−0.423122	−	0.906073i	$-0.639066\pi$
$443$	−17.8114	−0.846244	−0.423122	+	0.906073i	$0.639066\pi$
$444$	3.09377	0.146824
$445$	−20.7075	−0.981628
$446$	−2.19269	−0.103827
$447$	−4.54250	−0.214853
$448$	−42.0635	−1.98731
$449$	1.01058	0.0476924	0.0238462	−	0.999716i	$-0.492409\pi$
$449$	1.01058	0.0476924	0.0238462	+	0.999716i	$0.492409\pi$
$450$	5.53208	0.260785
$451$	24.2848	1.14353
$452$	47.5668	2.23735
$453$	4.17040	0.195943
$454$	−70.9883	−3.33164
$455$	−14.0209	−0.657311
$456$	1.93488	0.0906090
$457$	21.7491	1.01738	0.508690	−	0.860950i	$-0.330130\pi$
$457$	21.7491	1.01738	0.508690	+	0.860950i	$0.330130\pi$
$458$	18.3936	0.859475
$459$	−9.27532	−0.432935
$460$	−57.7888	−2.69442
$461$	25.2922	1.17798	0.588988	−	0.808142i	$-0.299527\pi$
$461$	25.2922	1.17798	0.588988	+	0.808142i	$0.299527\pi$
$462$	−12.7871	−0.594908
$463$	20.1364	0.935816	0.467908	−	0.883777i	$-0.345008\pi$
$463$	20.1364	0.935816	0.467908	+	0.883777i	$0.345008\pi$
$464$	−1.28141	−0.0594881
$465$	−0.729975	−0.0338518
$466$	60.2018	2.78880
$467$	−14.2980	−0.661633	−0.330817	−	0.943695i	$-0.607324\pi$
$467$	−14.2980	−0.661633	−0.330817	+	0.943695i	$0.607324\pi$
$468$	25.8569	1.19523
$469$	40.9083	1.88897
$470$	−43.8017	−2.02042
$471$	−2.26913	−0.104556
$472$	119.708	5.51001
$473$	−31.1929	−1.43425
$474$	0.00900348	0.000413543 0
$475$	0.711789	0.0326591
$476$	105.826	4.85053
$477$	11.0716	0.506935
$478$	−56.0899	−2.56549
$479$	−0.989604	−0.0452162	−0.0226081	−	0.999744i	$-0.507197\pi$
$479$	−0.989604	−0.0452162	−0.0226081	+	0.999744i	$0.507197\pi$
$480$	−7.62930	−0.348228
$481$	4.46443	0.203560
$482$	32.1032	1.46226
$483$	3.95050	0.179754
$484$	120.846	5.49302
$485$	22.2865	1.01198
$486$	17.0179	0.771948
$487$	−8.40812	−0.381008	−0.190504	−	0.981686i	$-0.561012\pi$
$487$	−8.40812	−0.381008	−0.190504	+	0.981686i	$0.561012\pi$
$488$	51.4872	2.33072
$489$	1.08448	0.0490418
$490$	25.7297	1.16235
$491$	24.6054	1.11043	0.555214	−	0.831708i	$-0.312636\pi$
$491$	24.6054	1.11043	0.555214	+	0.831708i	$0.312636\pi$
$492$	−4.99914	−0.225379
$493$	0.749014	0.0337339
$494$	4.66075	0.209697
$495$	41.7027	1.87440
$496$	−13.6019	−0.610745
$497$	36.7627	1.64903
$498$	8.51814	0.381707
$499$	−2.50301	−0.112050	−0.0560251	−	0.998429i	$-0.517843\pi$
$499$	−2.50301	−0.112050	−0.0560251	+	0.998429i	$0.517843\pi$
$500$	51.1236	2.28632
$501$	−4.16961	−0.186285
$502$	48.4570	2.16274
$503$	−5.87648	−0.262019	−0.131010	−	0.991381i	$-0.541822\pi$
$503$	−5.87648	−0.262019	−0.131010	+	0.991381i	$0.541822\pi$
$504$	−77.2849	−3.44254
$505$	24.1143	1.07307
$506$	−76.0530	−3.38097
$507$	2.42269	0.107595
$508$	−66.5907	−2.95449
$509$	20.4929	0.908330	0.454165	−	0.890917i	$-0.349938\pi$
$509$	20.4929	0.908330	0.454165	+	0.890917i	$0.349938\pi$
$510$	9.86542	0.436848
$511$	31.2539	1.38259
$512$	30.1707	1.33337
$513$	1.45485	0.0642332
$514$	10.8100	0.476810
$515$	−7.53561	−0.332058
$516$	6.42119	0.282677
$517$	−41.1478	−1.80968
$518$	−22.2746	−0.978688
$519$	−3.78064	−0.165952
$520$	−33.2872	−1.45974
$521$	0.953697	0.0417822	0.0208911	−	0.999782i	$-0.493350\pi$
$521$	0.953697	0.0417822	0.0208911	+	0.999782i	$0.493350\pi$
$522$	−0.913096	−0.0399651
$523$	41.7703	1.82649	0.913244	−	0.407412i	$-0.133569\pi$
$523$	41.7703	1.82649	0.913244	+	0.407412i	$0.133569\pi$
$524$	−52.3330	−2.28618
$525$	0.580103	0.0253178
$526$	−38.2775	−1.66898
$527$	7.95063	0.346335
$528$	−15.8551	−0.690006
$529$	0.496223	0.0215749
$530$	−23.7923	−1.03347
$531$	44.5502	1.93331
$532$	−16.5990	−0.719658
$533$	−7.21396	−0.312471
$534$	5.60996	0.242767
$535$	6.69031	0.289247
$536$	97.1207	4.19498
$537$	−4.83363	−0.208587
$538$	6.87372	0.296347
$539$	24.1708	1.04111
$540$	−17.3446	−0.746391
$541$	−30.3728	−1.30583	−0.652914	−	0.757432i	$-0.726454\pi$
$541$	−30.3728	−1.30583	−0.652914	+	0.757432i	$0.726454\pi$
$542$	−9.34218	−0.401281
$543$	−2.19667	−0.0942680
$544$	83.0957	3.56270
$545$	−27.1808	−1.16430
$546$	3.79848	0.162560
$547$	22.3344	0.954952	0.477476	−	0.878645i	$-0.341552\pi$
$547$	22.3344	0.954952	0.477476	+	0.878645i	$0.341552\pi$
$548$	−41.9671	−1.79275
$549$	19.1613	0.817786
$550$	−11.1678	−0.476198
$551$	−0.117484	−0.00500499
$552$	9.37891	0.399193
$553$	−0.0462716	−0.00196767
$554$	−61.6077	−2.61746
$555$	−1.48223	−0.0629171
$556$	−68.5419	−2.90682
$557$	−14.5258	−0.615477	−0.307739	−	0.951471i	$-0.599572\pi$
$557$	−14.5258	−0.615477	−0.307739	+	0.951471i	$0.599572\pi$
$558$	−9.69233	−0.410309
$559$	9.26603	0.391911
$560$	86.7399	3.66543
$561$	9.26767	0.391281
$562$	−8.14082	−0.343400
$563$	−15.5676	−0.656094	−0.328047	−	0.944661i	$-0.606390\pi$
$563$	−15.5676	−0.656094	−0.328047	+	0.944661i	$0.606390\pi$
$564$	8.47045	0.356670
$565$	−22.7893	−0.958753
$566$	−8.66133	−0.364063
$567$	−28.1633	−1.18275
$568$	87.2785	3.66212
$569$	−39.8300	−1.66976	−0.834880	−	0.550433i	$-0.814463\pi$
$569$	−39.8300	−1.66976	−0.834880	+	0.550433i	$0.814463\pi$
$570$	−1.54741	−0.0648138
$571$	−11.4672	−0.479886	−0.239943	−	0.970787i	$-0.577129\pi$
$571$	−11.4672	−0.479886	−0.239943	+	0.970787i	$0.577129\pi$
$572$	−52.1983	−2.18252
$573$	0.629171	0.0262840
$574$	35.9929	1.50231
$575$	3.45025	0.143885
$576$	−37.1648	−1.54853
$577$	−26.0146	−1.08300	−0.541500	−	0.840701i	$-0.682143\pi$
$577$	−26.0146	−1.08300	−0.541500	+	0.840701i	$0.682143\pi$
$578$	−62.5103	−2.60009
$579$	2.36847	0.0984301
$580$	1.40063	0.0581581
$581$	−43.7773	−1.81619
$582$	−6.03773	−0.250272
$583$	−22.3507	−0.925672
$584$	74.2000	3.07042
$585$	−12.3881	−0.512183
$586$	−40.7993	−1.68540
$587$	34.2639	1.41422	0.707111	−	0.707103i	$-0.249998\pi$
$587$	34.2639	1.41422	0.707111	+	0.707103i	$0.249998\pi$
$588$	−4.97566	−0.205193
$589$	−1.24707	−0.0513846
$590$	−95.7358	−3.94138
$591$	2.47471	0.101796
$592$	−27.6190	−1.13513
$593$	39.9902	1.64220	0.821101	−	0.570783i	$-0.193360\pi$
$593$	39.9902	1.64220	0.821101	+	0.570783i	$0.193360\pi$
$594$	−22.8263	−0.936574
$595$	−50.7014	−2.07855
$596$	92.5173	3.78966
$597$	4.45727	0.182424
$598$	22.5920	0.923856
$599$	0.770920	0.0314989	0.0157495	−	0.999876i	$-0.494987\pi$
$599$	0.770920	0.0314989	0.0157495	+	0.999876i	$0.494987\pi$
$600$	1.37723	0.0562250
$601$	−28.9902	−1.18253	−0.591267	−	0.806476i	$-0.701372\pi$
$601$	−28.9902	−1.18253	−0.591267	+	0.806476i	$0.701372\pi$
$602$	−46.2314	−1.88425
$603$	36.1442	1.47190
$604$	−84.9388	−3.45611
$605$	−57.8977	−2.35387
$606$	−6.53292	−0.265382
$607$	−33.6807	−1.36706	−0.683529	−	0.729924i	$-0.739556\pi$
$607$	−33.6807	−1.36706	−0.683529	+	0.729924i	$0.739556\pi$
$608$	−13.0337	−0.528586
$609$	−0.0957487	−0.00387993
$610$	−41.1766	−1.66719
$611$	12.2232	0.494498
$612$	93.5015	3.77958
$613$	26.8573	1.08476	0.542378	−	0.840135i	$-0.317524\pi$
$613$	26.8573	1.08476	0.542378	+	0.840135i	$0.317524\pi$
$614$	51.3103	2.07072
$615$	2.39509	0.0965795
$616$	156.018	6.28615
$617$	−7.41195	−0.298394	−0.149197	−	0.988808i	$-0.547669\pi$
$617$	−7.41195	−0.298394	−0.149197	+	0.988808i	$0.547669\pi$
$618$	2.04151	0.0821214
$619$	−21.7204	−0.873017	−0.436508	−	0.899700i	$-0.643785\pi$
$619$	−21.7204	−0.873017	−0.436508	+	0.899700i	$0.643785\pi$
$620$	14.8674	0.597090
$621$	7.05208	0.282990
$622$	−51.1286	−2.05007
$623$	−28.8313	−1.15510
$624$	4.70987	0.188546
$625$	−28.0523	−1.12209
$626$	77.6425	3.10322
$627$	−1.45365	−0.0580532
$628$	46.2155	1.84420
$629$	16.1439	0.643700
$630$	61.8082	2.46250
$631$	9.55817	0.380505	0.190252	−	0.981735i	$-0.439069\pi$
$631$	9.55817	0.380505	0.190252	+	0.981735i	$0.439069\pi$
$632$	−0.109854	−0.00436974
$633$	0.244924	0.00973484
$634$	−66.3570	−2.63537
$635$	31.9037	1.26606
$636$	4.60099	0.182441
$637$	−7.18007	−0.284485
$638$	1.84330	0.0729771
$639$	32.4813	1.28494
$640$	17.5657	0.694345
$641$	20.1510	0.795916	0.397958	−	0.917404i	$-0.369719\pi$
$641$	20.1510	0.795916	0.397958	+	0.917404i	$0.369719\pi$
$642$	−1.81250	−0.0715338
$643$	24.5141	0.966743	0.483371	−	0.875415i	$-0.339412\pi$
$643$	24.5141	0.966743	0.483371	+	0.875415i	$0.339412\pi$
$644$	−80.4601	−3.17057
$645$	−3.07640	−0.121133
$646$	16.8538	0.663105
$647$	7.82423	0.307602	0.153801	−	0.988102i	$-0.450849\pi$
$647$	7.82423	0.307602	0.153801	+	0.988102i	$0.450849\pi$
$648$	−66.8626	−2.62661
$649$	−89.9352	−3.53027
$650$	3.31747	0.130122
$651$	−1.01635	−0.0398340
$652$	−22.0876	−0.865018
$653$	−1.33630	−0.0522935	−0.0261467	−	0.999658i	$-0.508324\pi$
$653$	−1.33630	−0.0522935	−0.0261467	+	0.999658i	$0.508324\pi$
$654$	7.36367	0.287942
$655$	25.0728	0.979675
$656$	44.6289	1.74246
$657$	27.6141	1.07733
$658$	−60.9857	−2.37747
$659$	−3.36906	−0.131240	−0.0656200	−	0.997845i	$-0.520903\pi$
$659$	−3.36906	−0.131240	−0.0656200	+	0.997845i	$0.520903\pi$
$660$	17.3303	0.674579
$661$	21.5717	0.839044	0.419522	−	0.907745i	$-0.362198\pi$
$661$	21.5717	0.839044	0.419522	+	0.907745i	$0.362198\pi$
$662$	−23.8696	−0.927717
$663$	−2.75302	−0.106918
$664$	−103.932	−4.03334
$665$	7.95260	0.308388
$666$	−19.6805	−0.762603
$667$	−0.569480	−0.0220504
$668$	84.9227	3.28576
$669$	−0.203152	−0.00785430
$670$	−77.6718	−3.00072
$671$	−38.6818	−1.49329
$672$	−10.6224	−0.409767
$673$	24.3672	0.939285	0.469642	−	0.882857i	$-0.344383\pi$
$673$	24.3672	0.939285	0.469642	+	0.882857i	$0.344383\pi$
$674$	−30.3810	−1.17023
$675$	1.03555	0.0398582
$676$	−49.3430	−1.89781
$677$	−3.68553	−0.141646	−0.0708232	−	0.997489i	$-0.522563\pi$
$677$	−3.68553	−0.141646	−0.0708232	+	0.997489i	$0.522563\pi$
$678$	6.17396	0.237109
$679$	31.0297	1.19081
$680$	−120.370	−4.61599
$681$	−6.57702	−0.252032
$682$	19.5663	0.749232
$683$	41.5098	1.58833	0.794165	−	0.607703i	$-0.207909\pi$
$683$	41.5098	1.58833	0.794165	+	0.607703i	$0.207909\pi$
$684$	−14.6659	−0.560764
$685$	20.1065	0.768230
$686$	−25.7518	−0.983209
$687$	1.70415	0.0650175
$688$	−57.3239	−2.18545
$689$	6.63941	0.252942
$690$	−7.50074	−0.285548
$691$	−33.6014	−1.27826	−0.639129	−	0.769100i	$-0.720705\pi$
$691$	−33.6014	−1.27826	−0.639129	+	0.769100i	$0.720705\pi$
$692$	77.0005	2.92712
$693$	58.0632	2.20564
$694$	19.9051	0.755588
$695$	32.8385	1.24563
$696$	−0.227318	−0.00861645
$697$	−26.0865	−0.988098
$698$	−11.3314	−0.428898
$699$	5.57766	0.210967
$700$	−11.8150	−0.446564
$701$	−34.0920	−1.28764	−0.643818	−	0.765179i	$-0.722651\pi$
$701$	−34.0920	−1.28764	−0.643818	+	0.765179i	$0.722651\pi$
$702$	6.78069	0.255921
$703$	−2.53220	−0.0955038
$704$	75.0261	2.82765
$705$	−4.05820	−0.152841
$706$	95.4527	3.59241
$707$	33.5746	1.26270
$708$	18.5136	0.695782
$709$	49.3325	1.85272	0.926360	−	0.376640i	$-0.122921\pi$
$709$	49.3325	1.85272	0.926360	+	0.376640i	$0.122921\pi$
$710$	−69.8005	−2.61957
$711$	−0.0408828	−0.00153323
$712$	−68.4485	−2.56521
$713$	−6.04491	−0.226384
$714$	13.7357	0.514047
$715$	25.0082	0.935255
$716$	98.4469	3.67913
$717$	−5.19669	−0.194074
$718$	23.2405	0.867329
$719$	−12.2934	−0.458465	−0.229232	−	0.973372i	$-0.573622\pi$
$719$	−12.2934	−0.458465	−0.229232	+	0.973372i	$0.573622\pi$
$720$	76.6382	2.85614
$721$	−10.4919	−0.390739
$722$	−2.64355	−0.0983829
$723$	2.97434	0.110617
$724$	44.7396	1.66273
$725$	−0.0836240	−0.00310572
$726$	15.6853	0.582137
$727$	51.6728	1.91644	0.958219	−	0.286037i	$-0.0923379\pi$
$727$	51.6728	1.91644	0.958219	+	0.286037i	$0.0923379\pi$
$728$	−46.3461	−1.71770
$729$	−23.8144	−0.882016
$730$	−59.3411	−2.19631
$731$	33.5071	1.23930
$732$	7.96281	0.294314
$733$	−46.2304	−1.70756	−0.853779	−	0.520635i	$-0.825695\pi$
$733$	−46.2304	−1.70756	−0.853779	+	0.520635i	$0.825695\pi$
$734$	24.4674	0.903107
$735$	2.38384	0.0879294
$736$	−63.1781	−2.32878
$737$	−72.9656	−2.68772
$738$	31.8012	1.17062
$739$	25.8363	0.950403	0.475201	−	0.879877i	$-0.342375\pi$
$739$	25.8363	0.950403	0.475201	+	0.879877i	$0.342375\pi$
$740$	30.1886	1.10976
$741$	0.431816	0.0158632
$742$	−33.1263	−1.21610
$743$	26.4008	0.968551	0.484275	−	0.874916i	$-0.339083\pi$
$743$	26.4008	0.968551	0.484275	+	0.874916i	$0.339083\pi$
$744$	−2.41293	−0.0884623
$745$	−44.3251	−1.62395
$746$	−66.7100	−2.44243
$747$	−38.6790	−1.41519
$748$	−188.755	−6.90157
$749$	9.31500	0.340363
$750$	6.63561	0.242298
$751$	42.5718	1.55347	0.776734	−	0.629829i	$-0.216875\pi$
$751$	42.5718	1.55347	0.776734	+	0.629829i	$0.216875\pi$
$752$	−75.6183	−2.75752
$753$	4.48952	0.163607
$754$	−0.547564	−0.0199411
$755$	40.6943	1.48102
$756$	−24.1490	−0.878292
$757$	12.0894	0.439396	0.219698	−	0.975568i	$-0.429493\pi$
$757$	12.0894	0.439396	0.219698	+	0.975568i	$0.429493\pi$
$758$	−25.8851	−0.940188
$759$	−7.04627	−0.255763
$760$	18.8803	0.684860
$761$	10.4457	0.378656	0.189328	−	0.981914i	$-0.439369\pi$
$761$	10.4457	0.378656	0.189328	+	0.981914i	$0.439369\pi$
$762$	−8.64318	−0.313109
$763$	−37.8441	−1.37005
$764$	−12.8144	−0.463607
$765$	−44.7967	−1.61963
$766$	60.2711	2.17768
$767$	26.7158	0.964652
$768$	1.43337	0.0517224
$769$	−6.91354	−0.249309	−0.124654	−	0.992200i	$-0.539782\pi$
$769$	−6.91354	−0.249309	−0.124654	+	0.992200i	$0.539782\pi$
$770$	−124.775	−4.49657
$771$	1.00154	0.0360697
$772$	−48.2387	−1.73615
$773$	20.6527	0.742826	0.371413	−	0.928468i	$-0.378873\pi$
$773$	20.6527	0.742826	0.371413	+	0.928468i	$0.378873\pi$
$774$	−40.8473	−1.46822
$775$	−0.887651	−0.0318854
$776$	73.6678	2.64452
$777$	−2.06372	−0.0740357
$778$	51.2271	1.83658
$779$	4.09172	0.146601
$780$	−5.14806	−0.184330
$781$	−65.5713	−2.34633
$782$	81.6954	2.92142
$783$	−0.170922	−0.00610825
$784$	44.4192	1.58640
$785$	−22.1419	−0.790279
$786$	−6.79259	−0.242284
$787$	−16.1694	−0.576375	−0.288188	−	0.957574i	$-0.593053\pi$
$787$	−16.1694	−0.576375	−0.288188	+	0.957574i	$0.593053\pi$
$788$	−50.4026	−1.79552
$789$	−3.54639	−0.126255
$790$	0.0878548	0.00312573
$791$	−31.7298	−1.12818
$792$	137.848	4.89822
$793$	11.4907	0.408045
$794$	−47.9237	−1.70075
$795$	−2.20434	−0.0781799
$796$	−90.7815	−3.21766
$797$	−22.1799	−0.785652	−0.392826	−	0.919613i	$-0.628503\pi$
$797$	−22.1799	−0.785652	−0.392826	+	0.919613i	$0.628503\pi$
$798$	−2.15448	−0.0762676
$799$	44.2005	1.56370
$800$	−9.27725	−0.328000
$801$	−25.4736	−0.900065
$802$	85.5845	3.02209
$803$	−55.7456	−1.96722
$804$	15.0203	0.529725
$805$	38.5485	1.35866
$806$	−5.81228	−0.204729
$807$	0.636846	0.0224180
$808$	79.7097	2.80418
$809$	−17.2807	−0.607556	−0.303778	−	0.952743i	$-0.598248\pi$
$809$	−17.2807	−0.607556	−0.303778	+	0.952743i	$0.598248\pi$
$810$	53.4730	1.87885
$811$	40.0336	1.40577	0.702886	−	0.711303i	$-0.251894\pi$
$811$	40.0336	1.40577	0.702886	+	0.711303i	$0.251894\pi$
$812$	1.95012	0.0684358
$813$	−0.865547	−0.0303561
$814$	39.7297	1.39253
$815$	10.5822	0.370679
$816$	17.0314	0.596220
$817$	−5.25564	−0.183872
$818$	−55.5300	−1.94156
$819$	−17.2480	−0.602695
$820$	−48.7810	−1.70351
$821$	37.0460	1.29291	0.646457	−	0.762950i	$-0.276250\pi$
$821$	37.0460	1.29291	0.646457	+	0.762950i	$0.276250\pi$
$822$	−5.44715	−0.189991
$823$	−10.1388	−0.353418	−0.176709	−	0.984263i	$-0.556545\pi$
$823$	−10.1388	−0.353418	−0.176709	+	0.984263i	$0.556545\pi$
$824$	−24.9089	−0.867743
$825$	−1.03469	−0.0360234
$826$	−133.294	−4.63790
$827$	22.2942	0.775247	0.387623	−	0.921818i	$-0.373296\pi$
$827$	22.2942	0.775247	0.387623	+	0.921818i	$0.373296\pi$
$828$	−71.0898	−2.47054
$829$	11.4489	0.397635	0.198818	−	0.980036i	$-0.436290\pi$
$829$	11.4489	0.397635	0.198818	+	0.980036i	$0.436290\pi$
$830$	83.1190	2.88510
$831$	−5.70792	−0.198005
$832$	−22.2869	−0.772661
$833$	−25.9640	−0.899599
$834$	−8.89643	−0.308058
$835$	−40.6866	−1.40802
$836$	29.6066	1.02396
$837$	−1.81430	−0.0627114
$838$	−32.0085	−1.10571
$839$	−12.9556	−0.447277	−0.223638	−	0.974672i	$-0.571793\pi$
$839$	−12.9556	−0.447277	−0.223638	+	0.974672i	$0.571793\pi$
$840$	15.3873	0.530912
$841$	−28.9862	−0.999524
$842$	−0.198946	−0.00685612
$843$	−0.754242	−0.0259775
$844$	−4.98837	−0.171707
$845$	23.6403	0.813251
$846$	−53.8833	−1.85255
$847$	−80.6116	−2.76985
$848$	−41.0745	−1.41050
$849$	−0.802467	−0.0275406
$850$	11.9964	0.411472
$851$	−12.2743	−0.420758
$852$	13.4981	0.462439
$853$	−25.1889	−0.862450	−0.431225	−	0.902244i	$-0.641919\pi$
$853$	−25.1889	−0.862450	−0.431225	+	0.902244i	$0.641919\pi$
$854$	−57.3308	−1.96182
$855$	7.02644	0.240299
$856$	22.1148	0.755868
$857$	−33.6582	−1.14974	−0.574871	−	0.818244i	$-0.694948\pi$
$857$	−33.6582	−1.14974	−0.574871	+	0.818244i	$0.694948\pi$
$858$	−6.77510	−0.231298
$859$	−49.0700	−1.67425	−0.837124	−	0.547013i	$-0.815765\pi$
$859$	−49.0700	−1.67425	−0.837124	+	0.547013i	$0.815765\pi$
$860$	62.6572	2.13659
$861$	3.33472	0.113647
$862$	−99.9533	−3.40442
$863$	−53.0180	−1.80475	−0.902377	−	0.430947i	$-0.858180\pi$
$863$	−53.0180	−1.80475	−0.902377	+	0.430947i	$0.858180\pi$
$864$	−18.9621	−0.645103
$865$	−36.8910	−1.25433
$866$	−72.1128	−2.45049
$867$	−5.79154	−0.196691
$868$	20.7001	0.702607
$869$	0.0825317	0.00279970
$870$	0.181796	0.00616346
$871$	21.6749	0.734426
$872$	−89.8459	−3.04257
$873$	27.4160	0.927891
$874$	−12.8141	−0.433442
$875$	−34.1024	−1.15287
$876$	11.4755	0.387721
$877$	−0.105788	−0.00357220	−0.00178610	−	0.999998i	$-0.500569\pi$
$877$	−0.105788	−0.00357220	−0.00178610	+	0.999998i	$0.500569\pi$
$878$	−39.3537	−1.32812
$879$	−3.78003	−0.127497
$880$	−154.712	−5.21536
$881$	24.0269	0.809488	0.404744	−	0.914430i	$-0.367361\pi$
$881$	24.0269	0.809488	0.404744	+	0.914430i	$0.367361\pi$
$882$	31.6518	1.06577
$883$	−1.66266	−0.0559529	−0.0279764	−	0.999609i	$-0.508906\pi$
$883$	−1.66266	−0.0559529	−0.0279764	+	0.999609i	$0.508906\pi$
$884$	56.0709	1.88587
$885$	−8.86987	−0.298157
$886$	47.0853	1.58186
$887$	−57.3984	−1.92725	−0.963624	−	0.267261i	$-0.913881\pi$
$887$	−57.3984	−1.92725	−0.963624	+	0.267261i	$0.913881\pi$
$888$	−4.89950	−0.164416
$889$	44.4199	1.48980
$890$	54.7413	1.83493
$891$	50.2331	1.68287
$892$	4.13760	0.138537
$893$	−6.93293	−0.232002
$894$	12.0083	0.401619
$895$	−47.1660	−1.57659
$896$	24.4569	0.817049
$897$	2.09314	0.0698878
$898$	−2.67153	−0.0891501
$899$	0.146511	0.00488642
$900$	−10.4390	−0.347967
$901$	24.0089	0.799853
$902$	−64.1982	−2.13757
$903$	−4.28331	−0.142540
$904$	−75.3299	−2.50544
$905$	−21.4348	−0.712517
$906$	−11.0247	−0.366271
$907$	20.7975	0.690570	0.345285	−	0.938498i	$-0.387782\pi$
$907$	20.7975	0.690570	0.345285	+	0.938498i	$0.387782\pi$
$908$	133.955	4.44544
$909$	29.6645	0.983911
$910$	37.0651	1.22869
$911$	40.4854	1.34134	0.670671	−	0.741755i	$-0.266006\pi$
$911$	40.4854	1.34134	0.670671	+	0.741755i	$0.266006\pi$
$912$	−2.67141	−0.0884593
$913$	78.0828	2.58416
$914$	−57.4948	−1.90176
$915$	−3.81499	−0.126120
$916$	−34.7086	−1.14680
$917$	34.9092	1.15280
$918$	24.5198	0.809274
$919$	−1.92941	−0.0636454	−0.0318227	−	0.999494i	$-0.510131\pi$
$919$	−1.92941	−0.0636454	−0.0318227	+	0.999494i	$0.510131\pi$
$920$	91.5183	3.01727
$921$	4.75387	0.156645
$922$	−66.8613	−2.20196
$923$	19.4784	0.641138
$924$	24.1291	0.793790
$925$	−1.80239	−0.0592623
$926$	−53.2315	−1.74930
$927$	−9.27003	−0.304468
$928$	1.53125	0.0502659
$929$	−25.1367	−0.824709	−0.412354	−	0.911024i	$-0.635293\pi$
$929$	−25.1367	−0.824709	−0.412354	+	0.911024i	$0.635293\pi$
$930$	1.92973	0.0632782
$931$	4.07250	0.133471
$932$	−113.601	−3.72111
$933$	−4.73704	−0.155084
$934$	37.7976	1.23677
$935$	90.4328	2.95747
$936$	−40.9487	−1.33845
$937$	49.0881	1.60364	0.801819	−	0.597567i	$-0.203866\pi$
$937$	49.0881	1.60364	0.801819	+	0.597567i	$0.203866\pi$
$938$	−108.143	−3.53101
$939$	7.19354	0.234752
$940$	82.6536	2.69586
$941$	−11.4827	−0.374325	−0.187163	−	0.982329i	$-0.559929\pi$
$941$	−11.4827	−0.374325	−0.187163	+	0.982329i	$0.559929\pi$
$942$	5.99857	0.195444
$943$	19.8337	0.645876
$944$	−165.276	−5.37928
$945$	11.5698	0.376367
$946$	82.4600	2.68100
$947$	10.4149	0.338438	0.169219	−	0.985578i	$-0.445875\pi$
$947$	10.4149	0.338438	0.169219	+	0.985578i	$0.445875\pi$
$948$	−0.0169895	−0.000551794 0
$949$	16.5596	0.537546
$950$	−1.88165	−0.0610489
$951$	−6.14794	−0.199361
$952$	−167.593	−5.43173
$953$	−3.40685	−0.110359	−0.0551793	−	0.998476i	$-0.517573\pi$
$953$	−3.40685	−0.110359	−0.0551793	+	0.998476i	$0.517573\pi$
$954$	−29.2684	−0.947600
$955$	6.13937	0.198665
$956$	105.841	3.42315
$957$	0.170781	0.00552056
$958$	2.61607	0.0845214
$959$	27.9945	0.903991
$960$	7.39945	0.238816
$961$	−29.4448	−0.949833
$962$	−11.8020	−0.380510
$963$	8.23017	0.265214
$964$	−60.5786	−1.95111
$965$	23.1112	0.743976
$966$	−10.4434	−0.336010
$967$	−19.9586	−0.641824	−0.320912	−	0.947109i	$-0.603989\pi$
$967$	−19.9586	−0.641824	−0.320912	+	0.947109i	$0.603989\pi$
$968$	−191.380	−6.15120
$969$	1.56150	0.0501625
$970$	−58.9154	−1.89166
$971$	26.7986	0.860007	0.430003	−	0.902827i	$-0.358512\pi$
$971$	26.7986	0.860007	0.430003	+	0.902827i	$0.358512\pi$
$972$	−32.1127	−1.03002
$973$	45.7214	1.46576
$974$	22.2273	0.712209
$975$	0.307362	0.00984346
$976$	−71.0865	−2.27542
$977$	11.3281	0.362419	0.181210	−	0.983445i	$-0.441999\pi$
$977$	11.3281	0.362419	0.181210	+	0.983445i	$0.441999\pi$
$978$	−2.86688	−0.0916726
$979$	51.4245	1.64353
$980$	−48.5519	−1.55093
$981$	−33.4368	−1.06755
$982$	−65.0457	−2.07569
$983$	18.4841	0.589552	0.294776	−	0.955566i	$-0.404755\pi$
$983$	18.4841	0.589552	0.294776	+	0.955566i	$0.404755\pi$
$984$	7.91697	0.252384
$985$	24.1480	0.769418
$986$	−1.98006	−0.0630579
$987$	−5.65029	−0.179851
$988$	−8.79481	−0.279800
$989$	−25.4756	−0.810078
$990$	−110.243	−3.50377
$991$	16.6922	0.530247	0.265123	−	0.964214i	$-0.414587\pi$
$991$	16.6922	0.530247	0.265123	+	0.964214i	$0.414587\pi$
$992$	16.2539	0.516063
$993$	−2.21150	−0.0701799
$994$	−97.1841	−3.08249
$995$	43.4935	1.37884
$996$	−16.0737	−0.509315
$997$	13.5165	0.428071	0.214035	−	0.976826i	$-0.431339\pi$
$997$	13.5165	0.428071	0.214035	+	0.976826i	$0.431339\pi$
$998$	6.61684	0.209452
$999$	−3.68397	−0.116556

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.e.1.2	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.e.1.2	✓	82	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.64355 q^{2} -0.244924 q^{3} +4.98837 q^{4} -2.38993 q^{5} +0.647469 q^{6} -3.32754 q^{7} -7.89992 q^{8} -2.94001 q^{9} +O(q^{10})\)	\(q-2.64355 q^{2} -0.244924 q^{3} +4.98837 q^{4} -2.38993 q^{5} +0.647469 q^{6} -3.32754 q^{7} -7.89992 q^{8} -2.94001 q^{9} +6.31792 q^{10} +5.93512 q^{11} -1.22177 q^{12} -1.76306 q^{13} +8.79652 q^{14} +0.585352 q^{15} +10.9071 q^{16} -6.37545 q^{17} +7.77208 q^{18} +1.00000 q^{19} -11.9219 q^{20} +0.814992 q^{21} -15.6898 q^{22} +4.84729 q^{23} +1.93488 q^{24} +0.711789 q^{25} +4.66075 q^{26} +1.45485 q^{27} -16.5990 q^{28} -0.117484 q^{29} -1.54741 q^{30} -1.24707 q^{31} -13.0337 q^{32} -1.45365 q^{33} +16.8538 q^{34} +7.95260 q^{35} -14.6659 q^{36} -2.53220 q^{37} -2.64355 q^{38} +0.431816 q^{39} +18.8803 q^{40} +4.09172 q^{41} -2.15448 q^{42} -5.25564 q^{43} +29.6066 q^{44} +7.02644 q^{45} -12.8141 q^{46} -6.93293 q^{47} -2.67141 q^{48} +4.07250 q^{49} -1.88165 q^{50} +1.56150 q^{51} -8.79481 q^{52} -3.76584 q^{53} -3.84597 q^{54} -14.1845 q^{55} +26.2873 q^{56} -0.244924 q^{57} +0.310576 q^{58} -15.1531 q^{59} +2.91995 q^{60} -6.51744 q^{61} +3.29670 q^{62} +9.78300 q^{63} +12.6410 q^{64} +4.21361 q^{65} +3.84280 q^{66} -12.2939 q^{67} -31.8031 q^{68} -1.18722 q^{69} -21.0231 q^{70} -11.0480 q^{71} +23.2259 q^{72} -9.39250 q^{73} +6.69401 q^{74} -0.174334 q^{75} +4.98837 q^{76} -19.7493 q^{77} -1.14153 q^{78} +0.0139057 q^{79} -26.0673 q^{80} +8.46371 q^{81} -10.8167 q^{82} +13.1561 q^{83} +4.06549 q^{84} +15.2369 q^{85} +13.8936 q^{86} +0.0287747 q^{87} -46.8870 q^{88} +8.66445 q^{89} -18.5748 q^{90} +5.86666 q^{91} +24.1801 q^{92} +0.305437 q^{93} +18.3276 q^{94} -2.38993 q^{95} +3.19226 q^{96} -9.32513 q^{97} -10.7659 q^{98} -17.4493 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9 + 4 * q^10 + 41 * q^11 + 26 * q^12 + 13 * q^13 + 22 * q^14 + 41 * q^15 + 87 * q^16 + 12 * q^17 + 24 * q^18 + 82 * q^19 + 26 * q^20 + 29 * q^21 + 2 * q^22 + 59 * q^23 + 16 * q^24 + 67 * q^25 + 24 * q^26 + 42 * q^27 - 2 * q^28 + 101 * q^29 - 22 * q^30 + 48 * q^31 + 69 * q^32 + 3 * q^33 + q^34 + 38 * q^35 + 82 * q^36 + 16 * q^37 + 15 * q^38 + 82 * q^39 + 20 * q^40 + 86 * q^41 - q^42 + 9 * q^43 + 82 * q^44 - 8 * q^45 + 43 * q^46 + 24 * q^47 + 34 * q^48 + 76 * q^49 + 82 * q^50 + 57 * q^51 - 22 * q^52 + 39 * q^53 + 17 * q^54 - 21 * q^55 + 50 * q^56 + 12 * q^57 + 33 * q^58 + 79 * q^59 + 87 * q^60 + 4 * q^61 + 40 * q^62 + 44 * q^63 + 90 * q^64 + 66 * q^65 - 39 * q^66 + 33 * q^67 - 9 * q^68 + 60 * q^69 + 30 * q^70 + 168 * q^71 + 15 * q^72 - 28 * q^73 + 35 * q^74 + 55 * q^75 + 89 * q^76 + 19 * q^77 - 41 * q^78 + 121 * q^79 + 64 * q^80 + 110 * q^81 + 41 * q^82 + 28 * q^84 + 17 * q^85 + 80 * q^86 + 29 * q^87 + 49 * q^88 + 83 * q^89 - 42 * q^90 + 38 * q^91 + 71 * q^92 - q^93 + 89 * q^94 + 9 * q^95 + 35 * q^96 - 23 * q^97 + 135 * q^98 + 93 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more