LMFDB - Embedded newform 6023.2.a.b.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6023,2,Mod(1,6023)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6023, 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("6023.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6023 = 19 \cdot 317 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6023.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:	$48.0938971374$
Analytic rank:	$1$
Dimension:	$99$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	6023.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.96796 q^{2} -0.967508 q^{3} +1.87288 q^{4} +0.188751 q^{5} +1.90402 q^{6} -4.91446 q^{7} +0.250172 q^{8} -2.06393 q^{9} +O(q^{10})$	\(q-1.96796 q^{2} -0.967508 q^{3} +1.87288 q^{4} +0.188751 q^{5} +1.90402 q^{6} -4.91446 q^{7} +0.250172 q^{8} -2.06393 q^{9} -0.371455 q^{10} -1.60871 q^{11} -1.81202 q^{12} +3.63672 q^{13} +9.67146 q^{14} -0.182618 q^{15} -4.23808 q^{16} +3.34454 q^{17} +4.06173 q^{18} -1.00000 q^{19} +0.353508 q^{20} +4.75477 q^{21} +3.16589 q^{22} +2.14601 q^{23} -0.242044 q^{24} -4.96437 q^{25} -7.15693 q^{26} +4.89939 q^{27} -9.20417 q^{28} -3.45241 q^{29} +0.359386 q^{30} +3.44285 q^{31} +7.84005 q^{32} +1.55644 q^{33} -6.58193 q^{34} -0.927609 q^{35} -3.86548 q^{36} -5.13774 q^{37} +1.96796 q^{38} -3.51856 q^{39} +0.0472203 q^{40} -11.7806 q^{41} -9.35722 q^{42} -3.74167 q^{43} -3.01292 q^{44} -0.389569 q^{45} -4.22327 q^{46} +5.82681 q^{47} +4.10038 q^{48} +17.1519 q^{49} +9.76970 q^{50} -3.23587 q^{51} +6.81113 q^{52} -10.8588 q^{53} -9.64182 q^{54} -0.303646 q^{55} -1.22946 q^{56} +0.967508 q^{57} +6.79422 q^{58} +5.17048 q^{59} -0.342022 q^{60} +9.16826 q^{61} -6.77540 q^{62} +10.1431 q^{63} -6.95275 q^{64} +0.686435 q^{65} -3.06302 q^{66} +13.8922 q^{67} +6.26392 q^{68} -2.07628 q^{69} +1.82550 q^{70} -7.81496 q^{71} -0.516338 q^{72} +7.15869 q^{73} +10.1109 q^{74} +4.80307 q^{75} -1.87288 q^{76} +7.90594 q^{77} +6.92439 q^{78} +14.7823 q^{79} -0.799943 q^{80} +1.45159 q^{81} +23.1839 q^{82} +8.05904 q^{83} +8.90511 q^{84} +0.631286 q^{85} +7.36348 q^{86} +3.34024 q^{87} -0.402456 q^{88} +0.0958099 q^{89} +0.766657 q^{90} -17.8725 q^{91} +4.01921 q^{92} -3.33098 q^{93} -11.4670 q^{94} -0.188751 q^{95} -7.58531 q^{96} -3.34440 q^{97} -33.7542 q^{98} +3.32027 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 99 q - 4 q^{2} - 3 q^{3} + 80 q^{4} - 15 q^{5} - 12 q^{6} - 19 q^{7} - 12 q^{8} + 58 q^{9}+O(q^{10}) $ 99 * q - 4 * q^2 - 3 * q^3 + 80 * q^4 - 15 * q^5 - 12 * q^6 - 19 * q^7 - 12 * q^8 + 58 * q^9	\( 99 q - 4 q^{2} - 3 q^{3} + 80 q^{4} - 15 q^{5} - 12 q^{6} - 19 q^{7} - 12 q^{8} + 58 q^{9} - 6 q^{10} - 9 q^{11} - 27 q^{12} - 28 q^{13} - 13 q^{14} - 10 q^{15} + 38 q^{16} - 36 q^{17} - 14 q^{18} - 99 q^{19} - 34 q^{20} - 20 q^{21} - 53 q^{22} - 38 q^{23} - 25 q^{24} - 8 q^{25} - 3 q^{26} - 3 q^{27} - 63 q^{28} - 34 q^{29} - 30 q^{30} - 16 q^{31} - 43 q^{32} - 41 q^{33} - 14 q^{34} - 25 q^{35} - 16 q^{36} - 80 q^{37} + 4 q^{38} - 48 q^{39} - 10 q^{40} - 32 q^{41} - 37 q^{42} - 76 q^{43} - 21 q^{44} - 53 q^{45} - 23 q^{46} - 31 q^{47} - 74 q^{48} - 32 q^{49} - 29 q^{50} - 30 q^{51} - 71 q^{52} - 35 q^{53} - 80 q^{54} - 45 q^{55} - 33 q^{56} + 3 q^{57} - 91 q^{58} + 12 q^{59} - 56 q^{60} - 61 q^{61} - 46 q^{62} - 43 q^{63} - 30 q^{64} - 46 q^{65} - 75 q^{66} - 26 q^{67} - 55 q^{68} - 45 q^{69} - 76 q^{70} - 41 q^{71} - 77 q^{72} - 143 q^{73} - 64 q^{74} - 8 q^{75} - 80 q^{76} - 58 q^{77} - 34 q^{78} - 22 q^{79} - 36 q^{80} - 81 q^{81} - 109 q^{82} - 7 q^{83} - 6 q^{84} - 80 q^{85} + 32 q^{86} - 57 q^{87} - 120 q^{88} - 28 q^{89} - 12 q^{90} - 30 q^{91} - 107 q^{92} - 121 q^{93} + 8 q^{94} + 15 q^{95} + 4 q^{96} - 128 q^{97} + 54 q^{98} - 34 q^{99}+O(q^{100}) \) 99 * q - 4 * q^2 - 3 * q^3 + 80 * q^4 - 15 * q^5 - 12 * q^6 - 19 * q^7 - 12 * q^8 + 58 * q^9 - 6 * q^10 - 9 * q^11 - 27 * q^12 - 28 * q^13 - 13 * q^14 - 10 * q^15 + 38 * q^16 - 36 * q^17 - 14 * q^18 - 99 * q^19 - 34 * q^20 - 20 * q^21 - 53 * q^22 - 38 * q^23 - 25 * q^24 - 8 * q^25 - 3 * q^26 - 3 * q^27 - 63 * q^28 - 34 * q^29 - 30 * q^30 - 16 * q^31 - 43 * q^32 - 41 * q^33 - 14 * q^34 - 25 * q^35 - 16 * q^36 - 80 * q^37 + 4 * q^38 - 48 * q^39 - 10 * q^40 - 32 * q^41 - 37 * q^42 - 76 * q^43 - 21 * q^44 - 53 * q^45 - 23 * q^46 - 31 * q^47 - 74 * q^48 - 32 * q^49 - 29 * q^50 - 30 * q^51 - 71 * q^52 - 35 * q^53 - 80 * q^54 - 45 * q^55 - 33 * q^56 + 3 * q^57 - 91 * q^58 + 12 * q^59 - 56 * q^60 - 61 * q^61 - 46 * q^62 - 43 * q^63 - 30 * q^64 - 46 * q^65 - 75 * q^66 - 26 * q^67 - 55 * q^68 - 45 * q^69 - 76 * q^70 - 41 * q^71 - 77 * q^72 - 143 * q^73 - 64 * q^74 - 8 * q^75 - 80 * q^76 - 58 * q^77 - 34 * q^78 - 22 * q^79 - 36 * q^80 - 81 * q^81 - 109 * q^82 - 7 * q^83 - 6 * q^84 - 80 * q^85 + 32 * q^86 - 57 * q^87 - 120 * q^88 - 28 * q^89 - 12 * q^90 - 30 * q^91 - 107 * q^92 - 121 * q^93 + 8 * q^94 + 15 * q^95 + 4 * q^96 - 128 * q^97 + 54 * q^98 - 34 * 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 $

$2$	−1.96796	−1.39156	−0.695780	−	0.718255i	$-0.744941\pi$
$2$	−1.96796	−1.39156	−0.695780	+	0.718255i	$0.744941\pi$
$3$	−0.967508	−0.558591	−0.279295	−	0.960205i	$-0.590101\pi$
$3$	−0.967508	−0.558591	−0.279295	+	0.960205i	$0.590101\pi$
$4$	1.87288	0.936439
$5$	0.188751	0.0844121	0.0422060	−	0.999109i	$-0.486561\pi$
$5$	0.188751	0.0844121	0.0422060	+	0.999109i	$0.486561\pi$
$6$	1.90402	0.777313
$7$	−4.91446	−1.85749	−0.928745	−	0.370720i	$-0.879111\pi$
$7$	−4.91446	−1.85749	−0.928745	+	0.370720i	$0.879111\pi$
$8$	0.250172	0.0884493
$9$	−2.06393	−0.687976
$10$	−0.371455	−0.117464
$11$	−1.60871	−0.485045	−0.242522	−	0.970146i	$-0.577975\pi$
$11$	−1.60871	−0.485045	−0.242522	+	0.970146i	$0.577975\pi$
$12$	−1.81202	−0.523086
$13$	3.63672	1.00864	0.504322	−	0.863515i	$-0.331742\pi$
$13$	3.63672	1.00864	0.504322	+	0.863515i	$0.331742\pi$
$14$	9.67146	2.58481
$15$	−0.182618	−0.0471518
$16$	−4.23808	−1.05952
$17$	3.34454	0.811171	0.405585	−	0.914057i	$-0.367068\pi$
$17$	3.34454	0.811171	0.405585	+	0.914057i	$0.367068\pi$
$18$	4.06173	0.957360
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0.353508	0.0790467
$21$	4.75477	1.03758
$22$	3.16589	0.674969
$23$	2.14601	0.447474	0.223737	−	0.974650i	$-0.428174\pi$
$23$	2.14601	0.447474	0.223737	+	0.974650i	$0.428174\pi$
$24$	−0.242044	−0.0494070
$25$	−4.96437	−0.992875
$26$	−7.15693	−1.40359
$27$	4.89939	0.942888
$28$	−9.20417	−1.73943
$29$	−3.45241	−0.641097	−0.320549	−	0.947232i	$-0.603867\pi$
$29$	−3.45241	−0.641097	−0.320549	+	0.947232i	$0.603867\pi$
$30$	0.359386	0.0656146
$31$	3.44285	0.618354	0.309177	−	0.951005i	$-0.399946\pi$
$31$	3.44285	0.618354	0.309177	+	0.951005i	$0.399946\pi$
$32$	7.84005	1.38594
$33$	1.55644	0.270942
$34$	−6.58193	−1.12879
$35$	−0.927609	−0.156795
$36$	−3.86548	−0.644247
$37$	−5.13774	−0.844639	−0.422319	−	0.906447i	$-0.638784\pi$
$37$	−5.13774	−0.844639	−0.422319	+	0.906447i	$0.638784\pi$
$38$	1.96796	0.319246
$39$	−3.51856	−0.563420
$40$	0.0472203	0.00746619
$41$	−11.7806	−1.83983	−0.919914	−	0.392121i	$-0.871741\pi$
$41$	−11.7806	−1.83983	−0.919914	+	0.392121i	$0.871741\pi$
$42$	−9.35722	−1.44385
$43$	−3.74167	−0.570600	−0.285300	−	0.958438i	$-0.592093\pi$
$43$	−3.74167	−0.570600	−0.285300	+	0.958438i	$0.592093\pi$
$44$	−3.01292	−0.454215
$45$	−0.389569	−0.0580735
$46$	−4.22327	−0.622687
$47$	5.82681	0.849928	0.424964	−	0.905210i	$-0.360287\pi$
$47$	5.82681	0.849928	0.424964	+	0.905210i	$0.360287\pi$
$48$	4.10038	0.591839
$49$	17.1519	2.45027
$50$	9.76970	1.38164
$51$	−3.23587	−0.453113
$52$	6.81113	0.944534
$53$	−10.8588	−1.49157	−0.745785	−	0.666186i	$-0.767926\pi$
$53$	−10.8588	−1.49157	−0.745785	+	0.666186i	$0.767926\pi$
$54$	−9.64182	−1.31209
$55$	−0.303646	−0.0409436
$56$	−1.22946	−0.164294
$57$	0.967508	0.128150
$58$	6.79422	0.892125
$59$	5.17048	0.673139	0.336569	−	0.941659i	$-0.390733\pi$
$59$	5.17048	0.673139	0.336569	+	0.941659i	$0.390733\pi$
$60$	−0.342022	−0.0441548
$61$	9.16826	1.17387	0.586937	−	0.809632i	$-0.300333\pi$
$61$	9.16826	1.17387	0.586937	+	0.809632i	$0.300333\pi$
$62$	−6.77540	−0.860477
$63$	10.1431	1.27791
$64$	−6.95275	−0.869094
$65$	0.686435	0.0851418
$66$	−3.06302	−0.377032
$67$	13.8922	1.69720	0.848601	−	0.529034i	$-0.177445\pi$
$67$	13.8922	1.69720	0.848601	+	0.529034i	$0.177445\pi$
$68$	6.26392	0.759612
$69$	−2.07628	−0.249955
$70$	1.82550	0.218189
$71$	−7.81496	−0.927465	−0.463733	−	0.885975i	$-0.653490\pi$
$71$	−7.81496	−0.927465	−0.463733	+	0.885975i	$0.653490\pi$
$72$	−0.516338	−0.0608510
$73$	7.15869	0.837861	0.418931	−	0.908018i	$-0.362405\pi$
$73$	7.15869	0.837861	0.418931	+	0.908018i	$0.362405\pi$
$74$	10.1109	1.17537
$75$	4.80307	0.554611
$76$	−1.87288	−0.214834
$77$	7.90594	0.900966
$78$	6.92439	0.784032
$79$	14.7823	1.66314	0.831571	−	0.555419i	$-0.187442\pi$
$79$	14.7823	1.66314	0.831571	+	0.555419i	$0.187442\pi$
$80$	−0.799943	−0.0894364
$81$	1.45159	0.161287
$82$	23.1839	2.56023
$83$	8.05904	0.884595	0.442297	−	0.896868i	$-0.354164\pi$
$83$	8.05904	0.884595	0.442297	+	0.896868i	$0.354164\pi$
$84$	8.90511	0.971627
$85$	0.631286	0.0684726
$86$	7.36348	0.794024
$87$	3.34024	0.358111
$88$	−0.402456	−0.0429019
$89$	0.0958099	0.0101558	0.00507792	−	0.999987i	$-0.498384\pi$
$89$	0.0958099	0.0101558	0.00507792	+	0.999987i	$0.498384\pi$
$90$	0.766657	0.0808127
$91$	−17.8725	−1.87355
$92$	4.01921	0.419032
$93$	−3.33098	−0.345407
$94$	−11.4670	−1.18273
$95$	−0.188751	−0.0193655
$96$	−7.58531	−0.774172
$97$	−3.34440	−0.339572	−0.169786	−	0.985481i	$-0.554308\pi$
$97$	−3.34440	−0.339572	−0.169786	+	0.985481i	$0.554308\pi$
$98$	−33.7542	−3.40969
$99$	3.32027	0.333699
$100$	−9.29766	−0.929766
$101$	−5.26784	−0.524170	−0.262085	−	0.965045i	$-0.584410\pi$
$101$	−5.26784	−0.524170	−0.262085	+	0.965045i	$0.584410\pi$
$102$	6.36807	0.630533
$103$	1.78581	0.175961	0.0879805	−	0.996122i	$-0.471959\pi$
$103$	1.78581	0.175961	0.0879805	+	0.996122i	$0.471959\pi$
$104$	0.909807	0.0892139
$105$	0.897469	0.0875840
$106$	21.3697	2.07561
$107$	−11.9476	−1.15502	−0.577508	−	0.816385i	$-0.695975\pi$
$107$	−11.9476	−1.15502	−0.577508	+	0.816385i	$0.695975\pi$
$108$	9.17596	0.882957
$109$	10.7995	1.03441	0.517204	−	0.855862i	$-0.326973\pi$
$109$	10.7995	1.03441	0.517204	+	0.855862i	$0.326973\pi$
$110$	0.597564	0.0569755
$111$	4.97080	0.471808
$112$	20.8279	1.96805
$113$	18.0149	1.69470	0.847350	−	0.531034i	$-0.178197\pi$
$113$	18.0149	1.69470	0.847350	+	0.531034i	$0.178197\pi$
$114$	−1.90402	−0.178328
$115$	0.405062	0.0377722
$116$	−6.46595	−0.600348
$117$	−7.50593	−0.693923
$118$	−10.1753	−0.936713
$119$	−16.4366	−1.50674
$120$	−0.0456861	−0.00417055
$121$	−8.41205	−0.764731
$122$	−18.0428	−1.63352
$123$	11.3979	1.02771
$124$	6.44803	0.579051
$125$	−1.88079	−0.168223
$126$	−19.9612	−1.77829
$127$	−14.8043	−1.31367	−0.656836	−	0.754033i	$-0.728106\pi$
$127$	−14.8043	−1.31367	−0.656836	+	0.754033i	$0.728106\pi$
$128$	−1.99734	−0.176541
$129$	3.62010	0.318732
$130$	−1.35088	−0.118480
$131$	10.0720	0.879996	0.439998	−	0.897999i	$-0.354979\pi$
$131$	10.0720	0.879996	0.439998	+	0.897999i	$0.354979\pi$
$132$	2.91502	0.253720
$133$	4.91446	0.426137
$134$	−27.3393	−2.36176
$135$	0.924765	0.0795911
$136$	0.836712	0.0717475
$137$	10.9503	0.935543	0.467772	−	0.883849i	$-0.345057\pi$
$137$	10.9503	0.935543	0.467772	+	0.883849i	$0.345057\pi$
$138$	4.08604	0.347827
$139$	14.7580	1.25175	0.625877	−	0.779922i	$-0.284741\pi$
$139$	14.7580	1.25175	0.625877	+	0.779922i	$0.284741\pi$
$140$	−1.73730	−0.146828
$141$	−5.63749	−0.474762
$142$	15.3796	1.29062
$143$	−5.85044	−0.489238
$144$	8.74710	0.728925
$145$	−0.651647	−0.0541163
$146$	−14.0880	−1.16593
$147$	−16.5946	−1.36870
$148$	−9.62235	−0.790952
$149$	14.6769	1.20238	0.601188	−	0.799108i	$-0.294694\pi$
$149$	14.6769	1.20238	0.601188	+	0.799108i	$0.294694\pi$
$150$	−9.45226	−0.771774
$151$	−9.88759	−0.804641	−0.402320	−	0.915499i	$-0.631796\pi$
$151$	−9.88759	−0.804641	−0.402320	+	0.915499i	$0.631796\pi$
$152$	−0.250172	−0.0202917
$153$	−6.90290	−0.558066
$154$	−15.5586	−1.25375
$155$	0.649842	0.0521965
$156$	−6.58982	−0.527608
$157$	−6.72897	−0.537030	−0.268515	−	0.963275i	$-0.586533\pi$
$157$	−6.72897	−0.537030	−0.268515	+	0.963275i	$0.586533\pi$
$158$	−29.0911	−2.31436
$159$	10.5060	0.833178
$160$	1.47982	0.116990
$161$	−10.5465	−0.831178
$162$	−2.85667	−0.224441
$163$	−5.75158	−0.450498	−0.225249	−	0.974301i	$-0.572320\pi$
$163$	−5.75158	−0.450498	−0.225249	+	0.974301i	$0.572320\pi$
$164$	−22.0637	−1.72289
$165$	0.293780	0.0228708
$166$	−15.8599	−1.23097
$167$	0.849574	0.0657420	0.0328710	−	0.999460i	$-0.489535\pi$
$167$	0.849574	0.0657420	0.0328710	+	0.999460i	$0.489535\pi$
$168$	1.18951	0.0917730
$169$	0.225734	0.0173641
$170$	−1.24235	−0.0952837
$171$	2.06393	0.157833
$172$	−7.00770	−0.534332
$173$	14.2419	1.08279	0.541395	−	0.840768i	$-0.317896\pi$
$173$	14.2419	1.08279	0.541395	+	0.840768i	$0.317896\pi$
$174$	−6.57347	−0.498333
$175$	24.3972	1.84425
$176$	6.81786	0.513915
$177$	−5.00248	−0.376009
$178$	−0.188550	−0.0141324
$179$	8.03224	0.600358	0.300179	−	0.953883i	$-0.402954\pi$
$179$	8.03224	0.600358	0.300179	+	0.953883i	$0.402954\pi$
$180$	−0.729615	−0.0543823
$181$	2.09618	0.155808	0.0779040	−	0.996961i	$-0.475177\pi$
$181$	2.09618	0.155808	0.0779040	+	0.996961i	$0.475177\pi$
$182$	35.1724	2.60715
$183$	−8.87036	−0.655716
$184$	0.536872	0.0395788
$185$	−0.969754	−0.0712977
$186$	6.55525	0.480654
$187$	−5.38041	−0.393454
$188$	10.9129	0.795906
$189$	−24.0778	−1.75140
$190$	0.371455	0.0269482
$191$	−1.76351	−0.127603	−0.0638014	−	0.997963i	$-0.520322\pi$
$191$	−1.76351	−0.127603	−0.0638014	+	0.997963i	$0.520322\pi$
$192$	6.72684	0.485468
$193$	−0.887799	−0.0639052	−0.0319526	−	0.999489i	$-0.510173\pi$
$193$	−0.887799	−0.0639052	−0.0319526	+	0.999489i	$0.510173\pi$
$194$	6.58166	0.472535
$195$	−0.664131	−0.0475594
$196$	32.1233	2.29452
$197$	14.4349	1.02844	0.514221	−	0.857658i	$-0.328081\pi$
$197$	14.4349	1.02844	0.514221	+	0.857658i	$0.328081\pi$
$198$	−6.53416	−0.464363
$199$	0.796173	0.0564392	0.0282196	−	0.999602i	$-0.491016\pi$
$199$	0.796173	0.0564392	0.0282196	+	0.999602i	$0.491016\pi$
$200$	−1.24195	−0.0878191
$201$	−13.4408	−0.948042
$202$	10.3669	0.729414
$203$	16.9667	1.19083
$204$	−6.06039	−0.424312
$205$	−2.22361	−0.155304
$206$	−3.51441	−0.244860
$207$	−4.42921	−0.307851
$208$	−15.4127	−1.06868
$209$	1.60871	0.111277
$210$	−1.76619	−0.121878
$211$	4.91514	0.338372	0.169186	−	0.985584i	$-0.445886\pi$
$211$	4.91514	0.338372	0.169186	+	0.985584i	$0.445886\pi$
$212$	−20.3372	−1.39676
$213$	7.56104	0.518074
$214$	23.5124	1.60727
$215$	−0.706245	−0.0481655
$216$	1.22569	0.0833978
$217$	−16.9197	−1.14859
$218$	−21.2531	−1.43944
$219$	−6.92609	−0.468022
$220$	−0.568692	−0.0383412
$221$	12.1632	0.818183
$222$	−9.78235	−0.656548
$223$	11.1315	0.745424	0.372712	−	0.927947i	$-0.378428\pi$
$223$	11.1315	0.745424	0.372712	+	0.927947i	$0.378428\pi$
$224$	−38.5296	−2.57436
$225$	10.2461	0.683074
$226$	−35.4527	−2.35828
$227$	−22.8786	−1.51851	−0.759253	−	0.650796i	$-0.774435\pi$
$227$	−22.8786	−1.51851	−0.759253	+	0.650796i	$0.774435\pi$
$228$	1.81202	0.120004
$229$	−23.5220	−1.55438	−0.777189	−	0.629267i	$-0.783355\pi$
$229$	−23.5220	−1.55438	−0.777189	+	0.629267i	$0.783355\pi$
$230$	−0.797146	−0.0525623
$231$	−7.64906	−0.503271
$232$	−0.863699	−0.0567046
$233$	−10.1296	−0.663613	−0.331806	−	0.943347i	$-0.607658\pi$
$233$	−10.1296	−0.663613	−0.331806	+	0.943347i	$0.607658\pi$
$234$	14.7714	0.965636
$235$	1.09982	0.0717442
$236$	9.68367	0.630353
$237$	−14.3020	−0.929016
$238$	32.3466	2.09672
$239$	−4.49190	−0.290557	−0.145278	−	0.989391i	$-0.546408\pi$
$239$	−4.49190	−0.290557	−0.145278	+	0.989391i	$0.546408\pi$
$240$	0.773951	0.0499584
$241$	−6.25584	−0.402974	−0.201487	−	0.979491i	$-0.564577\pi$
$241$	−6.25584	−0.402974	−0.201487	+	0.979491i	$0.564577\pi$
$242$	16.5546	1.06417
$243$	−16.1026	−1.03298
$244$	17.1710	1.09926
$245$	3.23743	0.206832
$246$	−22.4306	−1.43012
$247$	−3.63672	−0.231399
$248$	0.861306	0.0546930
$249$	−7.79719	−0.494127
$250$	3.70132	0.234092
$251$	−7.32121	−0.462111	−0.231055	−	0.972941i	$-0.574218\pi$
$251$	−7.32121	−0.462111	−0.231055	+	0.972941i	$0.574218\pi$
$252$	18.9968	1.19668
$253$	−3.45231	−0.217045
$254$	29.1344	1.82805
$255$	−0.610774	−0.0382482
$256$	17.8362	1.11476
$257$	−17.8582	−1.11396	−0.556981	−	0.830525i	$-0.688040\pi$
$257$	−17.8582	−1.11396	−0.556981	+	0.830525i	$0.688040\pi$
$258$	−7.12422	−0.443535
$259$	25.2492	1.56891
$260$	1.28561	0.0797301
$261$	7.12554	0.441060
$262$	−19.8213	−1.22457
$263$	11.6234	0.716728	0.358364	−	0.933582i	$-0.383335\pi$
$263$	11.6234	0.716728	0.358364	+	0.933582i	$0.383335\pi$
$264$	0.389379	0.0239646
$265$	−2.04961	−0.125907
$266$	−9.67146	−0.592996
$267$	−0.0926969	−0.00567296
$268$	26.0184	1.58933
$269$	21.0073	1.28084	0.640419	−	0.768025i	$-0.278761\pi$
$269$	21.0073	1.28084	0.640419	+	0.768025i	$0.278761\pi$
$270$	−1.81990	−0.110756
$271$	−3.97821	−0.241659	−0.120830	−	0.992673i	$-0.538555\pi$
$271$	−3.97821	−0.241659	−0.120830	+	0.992673i	$0.538555\pi$
$272$	−14.1745	−0.859453
$273$	17.2918	1.04655
$274$	−21.5497	−1.30186
$275$	7.98625	0.481589
$276$	−3.88862	−0.234067
$277$	−5.35621	−0.321824	−0.160912	−	0.986969i	$-0.551443\pi$
$277$	−5.35621	−0.321824	−0.160912	+	0.986969i	$0.551443\pi$
$278$	−29.0431	−1.74189
$279$	−7.10579	−0.425413
$280$	−0.232062	−0.0138684
$281$	17.6593	1.05346	0.526732	−	0.850032i	$-0.323417\pi$
$281$	17.6593	1.05346	0.526732	+	0.850032i	$0.323417\pi$
$282$	11.0944	0.660660
$283$	−3.63125	−0.215856	−0.107928	−	0.994159i	$-0.534422\pi$
$283$	−3.63125	−0.215856	−0.107928	+	0.994159i	$0.534422\pi$
$284$	−14.6365	−0.868514
$285$	0.182618	0.0108174
$286$	11.5134	0.680804
$287$	57.8954	3.41746
$288$	−16.1813	−0.953492
$289$	−5.81404	−0.342002
$290$	1.28242	0.0753061
$291$	3.23573	0.189682
$292$	13.4073	0.784606
$293$	−7.40251	−0.432459	−0.216230	−	0.976343i	$-0.569376\pi$
$293$	−7.40251	−0.432459	−0.216230	+	0.976343i	$0.569376\pi$
$294$	32.6575	1.90462
$295$	0.975933	0.0568210
$296$	−1.28532	−0.0747077
$297$	−7.88171	−0.457343
$298$	−28.8835	−1.67318
$299$	7.80443	0.451342
$300$	8.99556	0.519359
$301$	18.3883	1.05988
$302$	19.4584	1.11971
$303$	5.09668	0.292797
$304$	4.23808	0.243071
$305$	1.73052	0.0990892
$306$	13.5846	0.776582
$307$	−13.0501	−0.744807	−0.372404	−	0.928071i	$-0.621466\pi$
$307$	−13.0501	−0.744807	−0.372404	+	0.928071i	$0.621466\pi$
$308$	14.8069	0.843699
$309$	−1.72778	−0.0982902
$310$	−1.27886	−0.0726346
$311$	33.5659	1.90335	0.951673	−	0.307113i	$-0.0993629\pi$
$311$	33.5659	1.90335	0.951673	+	0.307113i	$0.0993629\pi$
$312$	−0.880246	−0.0498341
$313$	−19.6588	−1.11118	−0.555592	−	0.831455i	$-0.687508\pi$
$313$	−19.6588	−1.11118	−0.555592	+	0.831455i	$0.687508\pi$
$314$	13.2424	0.747310
$315$	1.91452	0.107871
$316$	27.6855	1.55743
$317$	−1.00000	−0.0561656
$317$	−1.00000	−0.0561656
$318$	−20.6754	−1.15942
$319$	5.55394	0.310961
$320$	−1.31234	−0.0733620
$321$	11.5594	0.645182
$322$	20.7550	1.15663
$323$	−3.34454	−0.186095
$324$	2.71864	0.151036
$325$	−18.0540	−1.00146
$326$	11.3189	0.626895
$327$	−10.4486	−0.577811
$328$	−2.94719	−0.162731
$329$	−28.6356	−1.57873
$330$	−0.578148	−0.0318260
$331$	27.0196	1.48513	0.742565	−	0.669774i	$-0.233609\pi$
$331$	27.0196	1.48513	0.742565	+	0.669774i	$0.233609\pi$
$332$	15.0936	0.828369
$333$	10.6039	0.581091
$334$	−1.67193	−0.0914840
$335$	2.62217	0.143264
$336$	−20.1511	−1.09933
$337$	−6.55603	−0.357130	−0.178565	−	0.983928i	$-0.557145\pi$
$337$	−6.55603	−0.357130	−0.178565	+	0.983928i	$0.557145\pi$
$338$	−0.444235	−0.0241632
$339$	−17.4296	−0.946644
$340$	1.18232	0.0641204
$341$	−5.53855	−0.299929
$342$	−4.06173	−0.219633
$343$	−49.8909	−2.69386
$344$	−0.936064	−0.0504692
$345$	−0.391900	−0.0210992
$346$	−28.0275	−1.50677
$347$	−16.5458	−0.888227	−0.444113	−	0.895971i	$-0.646481\pi$
$347$	−16.5458	−0.888227	−0.444113	+	0.895971i	$0.646481\pi$
$348$	6.25586	0.335349
$349$	−35.4680	−1.89856	−0.949279	−	0.314436i	$-0.898184\pi$
$349$	−35.4680	−1.89856	−0.949279	+	0.314436i	$0.898184\pi$
$350$	−48.0128	−2.56639
$351$	17.8177	0.951039
$352$	−12.6124	−0.672242
$353$	−5.82385	−0.309973	−0.154986	−	0.987917i	$-0.549533\pi$
$353$	−5.82385	−0.309973	−0.154986	+	0.987917i	$0.549533\pi$
$354$	9.84469	0.523239
$355$	−1.47508	−0.0782893
$356$	0.179440	0.00951031
$357$	15.9025	0.841652
$358$	−15.8071	−0.835434
$359$	23.3455	1.23213	0.616063	−	0.787697i	$-0.288727\pi$
$359$	23.3455	1.23213	0.616063	+	0.787697i	$0.288727\pi$
$360$	−0.0974594	−0.00513656
$361$	1.00000	0.0526316
$362$	−4.12521	−0.216816
$363$	8.13872	0.427172
$364$	−33.4730	−1.75446
$365$	1.35121	0.0707256
$366$	17.4565	0.912468
$367$	−0.883059	−0.0460953	−0.0230477	−	0.999734i	$-0.507337\pi$
$367$	−0.883059	−0.0460953	−0.0230477	+	0.999734i	$0.507337\pi$
$368$	−9.09497	−0.474108
$369$	24.3144	1.26576
$370$	1.90844	0.0992150
$371$	53.3651	2.77058
$372$	−6.23852	−0.323452
$373$	5.06087	0.262042	0.131021	−	0.991380i	$-0.458174\pi$
$373$	5.06087	0.262042	0.131021	+	0.991380i	$0.458174\pi$
$374$	10.5884	0.547515
$375$	1.81968	0.0939677
$376$	1.45771	0.0751756
$377$	−12.5555	−0.646639
$378$	47.3843	2.43718
$379$	2.77259	0.142418	0.0712092	−	0.997461i	$-0.477314\pi$
$379$	2.77259	0.142418	0.0712092	+	0.997461i	$0.477314\pi$
$380$	−0.353508	−0.0181346
$381$	14.3233	0.733806
$382$	3.47051	0.177567
$383$	10.3211	0.527382	0.263691	−	0.964607i	$-0.415060\pi$
$383$	10.3211	0.527382	0.263691	+	0.964607i	$0.415060\pi$
$384$	1.93244	0.0986144
$385$	1.49226	0.0760524
$386$	1.74716	0.0889279
$387$	7.72255	0.392559
$388$	−6.26365	−0.317989
$389$	−10.0931	−0.511740	−0.255870	−	0.966711i	$-0.582362\pi$
$389$	−10.0931	−0.511740	−0.255870	+	0.966711i	$0.582362\pi$
$390$	1.30699	0.0661818
$391$	7.17742	0.362978
$392$	4.29093	0.216724
$393$	−9.74475	−0.491558
$394$	−28.4073	−1.43114
$395$	2.79018	0.140389
$396$	6.21845	0.312489
$397$	4.21868	0.211729	0.105865	−	0.994381i	$-0.466239\pi$
$397$	4.21868	0.211729	0.105865	+	0.994381i	$0.466239\pi$
$398$	−1.56684	−0.0785385
$399$	−4.75477	−0.238036
$400$	21.0394	1.05197
$401$	−22.5470	−1.12594	−0.562971	−	0.826477i	$-0.690342\pi$
$401$	−22.5470	−1.12594	−0.562971	+	0.826477i	$0.690342\pi$
$402$	26.4510	1.31926
$403$	12.5207	0.623699
$404$	−9.86603	−0.490853
$405$	0.273988	0.0136146
$406$	−33.3899	−1.65711
$407$	8.26514	0.409688
$408$	−0.809526	−0.0400775
$409$	28.8575	1.42691	0.713456	−	0.700700i	$-0.247129\pi$
$409$	28.8575	1.42691	0.713456	+	0.700700i	$0.247129\pi$
$410$	4.37598	0.216114
$411$	−10.5945	−0.522586
$412$	3.34460	0.164777
$413$	−25.4101	−1.25035
$414$	8.71652	0.428393
$415$	1.52115	0.0746705
$416$	28.5121	1.39792
$417$	−14.2784	−0.699218
$418$	−3.16589	−0.154849
$419$	−1.85598	−0.0906707	−0.0453354	−	0.998972i	$-0.514436\pi$
$419$	−1.85598	−0.0906707	−0.0453354	+	0.998972i	$0.514436\pi$
$420$	1.68085	0.0820171
$421$	−27.4874	−1.33965	−0.669827	−	0.742517i	$-0.733632\pi$
$421$	−27.4874	−1.33965	−0.669827	+	0.742517i	$0.733632\pi$
$422$	−9.67282	−0.470865
$423$	−12.0261	−0.584730
$424$	−2.71657	−0.131928
$425$	−16.6036	−0.805391
$426$	−14.8798	−0.720931
$427$	−45.0570	−2.18046
$428$	−22.3764	−1.08160
$429$	5.66034	0.273284
$430$	1.38986	0.0670252
$431$	−12.9904	−0.625727	−0.312863	−	0.949798i	$-0.601288\pi$
$431$	−12.9904	−0.625727	−0.312863	+	0.949798i	$0.601288\pi$
$432$	−20.7640	−0.999010
$433$	−11.7942	−0.566793	−0.283397	−	0.959003i	$-0.591461\pi$
$433$	−11.7942	−0.566793	−0.283397	+	0.959003i	$0.591461\pi$
$434$	33.2974	1.59833
$435$	0.630474	0.0302289
$436$	20.2262	0.968659
$437$	−2.14601	−0.102658
$438$	13.6303	0.651280
$439$	−6.60344	−0.315165	−0.157582	−	0.987506i	$-0.550370\pi$
$439$	−6.60344	−0.315165	−0.157582	+	0.987506i	$0.550370\pi$
$440$	−0.0759639	−0.00362144
$441$	−35.4002	−1.68573
$442$	−23.9367	−1.13855
$443$	−3.92798	−0.186624	−0.0933120	−	0.995637i	$-0.529745\pi$
$443$	−3.92798	−0.186624	−0.0933120	+	0.995637i	$0.529745\pi$
$444$	9.30970	0.441819
$445$	0.0180842	0.000857275 0
$446$	−21.9065	−1.03730
$447$	−14.2000	−0.671637
$448$	34.1690	1.61433
$449$	−13.5326	−0.638644	−0.319322	−	0.947646i	$-0.603455\pi$
$449$	−13.5326	−0.638644	−0.319322	+	0.947646i	$0.603455\pi$
$450$	−20.1640	−0.950538
$451$	18.9517	0.892399
$452$	33.7397	1.58698
$453$	9.56632	0.449465
$454$	45.0242	2.11309
$455$	−3.37345	−0.158150
$456$	0.242044	0.0113347
$457$	7.58845	0.354973	0.177486	−	0.984123i	$-0.443203\pi$
$457$	7.58845	0.354973	0.177486	+	0.984123i	$0.443203\pi$
$458$	46.2905	2.16301
$459$	16.3862	0.764843
$460$	0.758631	0.0353713
$461$	−5.04183	−0.234822	−0.117411	−	0.993083i	$-0.537459\pi$
$461$	−5.04183	−0.234822	−0.117411	+	0.993083i	$0.537459\pi$
$462$	15.0531	0.700332
$463$	10.1341	0.470970	0.235485	−	0.971878i	$-0.424332\pi$
$463$	10.1341	0.470970	0.235485	+	0.971878i	$0.424332\pi$
$464$	14.6316	0.679256
$465$	−0.628727	−0.0291565
$466$	19.9347	0.923457
$467$	22.4356	1.03820	0.519098	−	0.854715i	$-0.326268\pi$
$467$	22.4356	1.03820	0.519098	+	0.854715i	$0.326268\pi$
$468$	−14.0577	−0.649817
$469$	−68.2726	−3.15253
$470$	−2.16440	−0.0998363
$471$	6.51033	0.299980
$472$	1.29351	0.0595387
$473$	6.01928	0.276767
$474$	28.1458	1.29278
$475$	4.96437	0.227781
$476$	−30.7837	−1.41097
$477$	22.4118	1.02617
$478$	8.83990	0.404327
$479$	−15.6650	−0.715752	−0.357876	−	0.933769i	$-0.616499\pi$
$479$	−15.6650	−0.715752	−0.357876	+	0.933769i	$0.616499\pi$
$480$	−1.43174	−0.0653495
$481$	−18.6845	−0.851940
$482$	12.3113	0.560762
$483$	10.2038	0.464288
$484$	−15.7547	−0.716124
$485$	−0.631259	−0.0286640
$486$	31.6893	1.43746
$487$	29.6945	1.34559	0.672793	−	0.739831i	$-0.265094\pi$
$487$	29.6945	1.34559	0.672793	+	0.739831i	$0.265094\pi$
$488$	2.29365	0.103828
$489$	5.56470	0.251644
$490$	−6.37115	−0.287819
$491$	−6.71702	−0.303135	−0.151567	−	0.988447i	$-0.548432\pi$
$491$	−6.71702	−0.303135	−0.151567	+	0.988447i	$0.548432\pi$
$492$	21.3468	0.962388
$493$	−11.5467	−0.520039
$494$	7.15693	0.322005
$495$	0.626704	0.0281682
$496$	−14.5911	−0.655159
$497$	38.4063	1.72276
$498$	15.3446	0.687607
$499$	27.2615	1.22039	0.610197	−	0.792250i	$-0.291090\pi$
$499$	27.2615	1.22039	0.610197	+	0.792250i	$0.291090\pi$
$500$	−3.52248	−0.157530
$501$	−0.821970	−0.0367229
$502$	14.4079	0.643055
$503$	−14.3469	−0.639698	−0.319849	−	0.947469i	$-0.603632\pi$
$503$	−14.3469	−0.639698	−0.319849	+	0.947469i	$0.603632\pi$
$504$	2.53752	0.113030
$505$	−0.994311	−0.0442463
$506$	6.79402	0.302031
$507$	−0.218399	−0.00969944
$508$	−27.7267	−1.23017
$509$	−6.37551	−0.282590	−0.141295	−	0.989968i	$-0.545127\pi$
$509$	−6.37551	−0.282590	−0.141295	+	0.989968i	$0.545127\pi$
$510$	1.20198	0.0532246
$511$	−35.1811	−1.55632
$512$	−31.1063	−1.37472
$513$	−4.89939	−0.216313
$514$	35.1442	1.55015
$515$	0.337073	0.0148532
$516$	6.78000	0.298473
$517$	−9.37366	−0.412253
$518$	−49.6894	−2.18323
$519$	−13.7791	−0.604837
$520$	0.171727	0.00753073
$521$	−15.5851	−0.682796	−0.341398	−	0.939919i	$-0.610900\pi$
$521$	−15.5851	−0.682796	−0.341398	+	0.939919i	$0.610900\pi$
$522$	−14.0228	−0.613761
$523$	−26.4795	−1.15787	−0.578933	−	0.815375i	$-0.696531\pi$
$523$	−26.4795	−1.15787	−0.578933	+	0.815375i	$0.696531\pi$
$524$	18.8636	0.824062
$525$	−23.6045	−1.03018
$526$	−22.8744	−0.997370
$527$	11.5148	0.501591
$528$	−6.59633	−0.287069
$529$	−18.3946	−0.799767
$530$	4.03356	0.175207
$531$	−10.6715	−0.463103
$532$	9.20417	0.399051
$533$	−42.8429	−1.85573
$534$	0.182424	0.00789426
$535$	−2.25512	−0.0974973
$536$	3.47545	0.150116
$537$	−7.77125	−0.335354
$538$	−41.3416	−1.78236
$539$	−27.5924	−1.18849
$540$	1.73197	0.0745322
$541$	−9.54249	−0.410264	−0.205132	−	0.978734i	$-0.565762\pi$
$541$	−9.54249	−0.410264	−0.205132	+	0.978734i	$0.565762\pi$
$542$	7.82898	0.336283
$543$	−2.02807	−0.0870329
$544$	26.2214	1.12423
$545$	2.03842	0.0873165
$546$	−34.0296	−1.45633
$547$	−18.2343	−0.779641	−0.389820	−	0.920891i	$-0.627463\pi$
$547$	−18.2343	−0.779641	−0.389820	+	0.920891i	$0.627463\pi$
$548$	20.5085	0.876079
$549$	−18.9226	−0.807598
$550$	−15.7166	−0.670160
$551$	3.45241	0.147078
$552$	−0.519428	−0.0221083
$553$	−72.6471	−3.08927
$554$	10.5408	0.447837
$555$	0.938244	0.0398263
$556$	27.6398	1.17219
$557$	6.84967	0.290230	0.145115	−	0.989415i	$-0.453645\pi$
$557$	6.84967	0.290230	0.145115	+	0.989415i	$0.453645\pi$
$558$	13.9839	0.591987
$559$	−13.6074	−0.575533
$560$	3.93129	0.166127
$561$	5.20559	0.219780
$562$	−34.7528	−1.46596
$563$	−38.4048	−1.61857	−0.809284	−	0.587418i	$-0.800145\pi$
$563$	−38.4048	−1.61857	−0.809284	+	0.587418i	$0.800145\pi$
$564$	−10.5583	−0.444586
$565$	3.40033	0.143053
$566$	7.14617	0.300376
$567$	−7.13375	−0.299589
$568$	−1.95509	−0.0820337
$569$	−4.08204	−0.171128	−0.0855639	−	0.996333i	$-0.527269\pi$
$569$	−4.08204	−0.171128	−0.0855639	+	0.996333i	$0.527269\pi$
$570$	−0.359386	−0.0150530
$571$	−1.39252	−0.0582752	−0.0291376	−	0.999575i	$-0.509276\pi$
$571$	−1.39252	−0.0582752	−0.0291376	+	0.999575i	$0.509276\pi$
$572$	−10.9571	−0.458141
$573$	1.70621	0.0712778
$574$	−113.936	−4.75560
$575$	−10.6536	−0.444285
$576$	14.3500	0.597916
$577$	−36.8132	−1.53255	−0.766277	−	0.642511i	$-0.777893\pi$
$577$	−36.8132	−1.53255	−0.766277	+	0.642511i	$0.777893\pi$
$578$	11.4418	0.475916
$579$	0.858952	0.0356968
$580$	−1.22046	−0.0506766
$581$	−39.6058	−1.64313
$582$	−6.36780	−0.263954
$583$	17.4687	0.723479
$584$	1.79091	0.0741083
$585$	−1.41675	−0.0585755
$586$	14.5679	0.601793
$587$	2.29919	0.0948978	0.0474489	−	0.998874i	$-0.484891\pi$
$587$	2.29919	0.0948978	0.0474489	+	0.998874i	$0.484891\pi$
$588$	−31.0796	−1.28170
$589$	−3.44285	−0.141860
$590$	−1.92060	−0.0790699
$591$	−13.9658	−0.574478
$592$	21.7742	0.894913
$593$	29.5571	1.21377	0.606883	−	0.794791i	$-0.292420\pi$
$593$	29.5571	1.21377	0.606883	+	0.794791i	$0.292420\pi$
$594$	15.5109	0.636420
$595$	−3.10243	−0.127187
$596$	27.4880	1.12595
$597$	−0.770304	−0.0315264
$598$	−15.3588	−0.628069
$599$	−28.0205	−1.14488	−0.572442	−	0.819945i	$-0.694004\pi$
$599$	−28.0205	−1.14488	−0.572442	+	0.819945i	$0.694004\pi$
$600$	1.20160	0.0490550
$601$	−30.6197	−1.24900	−0.624501	−	0.781024i	$-0.714698\pi$
$601$	−30.6197	−1.24900	−0.624501	+	0.781024i	$0.714698\pi$
$602$	−36.1875	−1.47489
$603$	−28.6725	−1.16763
$604$	−18.5182	−0.753497
$605$	−1.58778	−0.0645526
$606$	−10.0301	−0.407444
$607$	−0.931814	−0.0378212	−0.0189106	−	0.999821i	$-0.506020\pi$
$607$	−0.931814	−0.0378212	−0.0189106	+	0.999821i	$0.506020\pi$
$608$	−7.84005	−0.317956
$609$	−16.4155	−0.665188
$610$	−3.40560	−0.137889
$611$	21.1905	0.857275
$612$	−12.9283	−0.522595
$613$	−46.0062	−1.85817	−0.929086	−	0.369864i	$-0.879404\pi$
$613$	−46.0062	−1.85817	−0.929086	+	0.369864i	$0.879404\pi$
$614$	25.6821	1.03644
$615$	2.15136	0.0867512
$616$	1.97785	0.0796898
$617$	−33.0705	−1.33137	−0.665685	−	0.746233i	$-0.731861\pi$
$617$	−33.0705	−1.33137	−0.665685	+	0.746233i	$0.731861\pi$
$618$	3.40022	0.136777
$619$	9.02535	0.362760	0.181380	−	0.983413i	$-0.441944\pi$
$619$	9.02535	0.362760	0.181380	+	0.983413i	$0.441944\pi$
$620$	1.21707	0.0488789
$621$	10.5141	0.421918
$622$	−66.0564	−2.64862
$623$	−0.470854	−0.0188644
$624$	14.9119	0.596955
$625$	24.4669	0.978675
$626$	38.6879	1.54628
$627$	−1.55644	−0.0621583
$628$	−12.6025	−0.502896
$629$	−17.1834	−0.685146
$630$	−3.76770	−0.150109
$631$	−12.5967	−0.501465	−0.250733	−	0.968056i	$-0.580672\pi$
$631$	−12.5967	−0.501465	−0.250733	+	0.968056i	$0.580672\pi$
$632$	3.69813	0.147104
$633$	−4.75544	−0.189012
$634$	1.96796	0.0781578
$635$	−2.79433	−0.110890
$636$	19.6764	0.780220
$637$	62.3765	2.47145
$638$	−10.9299	−0.432721
$639$	16.1295	0.638074
$640$	−0.377000	−0.0149022
$641$	47.1585	1.86265	0.931324	−	0.364191i	$-0.118655\pi$
$641$	47.1585	1.86265	0.931324	+	0.364191i	$0.118655\pi$
$642$	−22.7484	−0.897809
$643$	−43.1064	−1.69995	−0.849976	−	0.526821i	$-0.823384\pi$
$643$	−43.1064	−1.69995	−0.849976	+	0.526821i	$0.823384\pi$
$644$	−19.7522	−0.778347
$645$	0.683298	0.0269048
$646$	6.58193	0.258963
$647$	46.8352	1.84128	0.920640	−	0.390412i	$-0.127667\pi$
$647$	46.8352	1.84128	0.920640	+	0.390412i	$0.127667\pi$
$648$	0.363147	0.0142657
$649$	−8.31781	−0.326503
$650$	35.5297	1.39359
$651$	16.3700	0.641590
$652$	−10.7720	−0.421864
$653$	−24.8394	−0.972039	−0.486020	−	0.873948i	$-0.661552\pi$
$653$	−24.8394	−0.972039	−0.486020	+	0.873948i	$0.661552\pi$
$654$	20.5625	0.804058
$655$	1.90110	0.0742823
$656$	49.9274	1.94934
$657$	−14.7750	−0.576428
$658$	56.3538	2.19690
$659$	40.5713	1.58043	0.790217	−	0.612827i	$-0.209968\pi$
$659$	40.5713	1.58043	0.790217	+	0.612827i	$0.209968\pi$
$660$	0.550214	0.0214171
$661$	−9.50686	−0.369774	−0.184887	−	0.982760i	$-0.559192\pi$
$661$	−9.50686	−0.369774	−0.184887	+	0.982760i	$0.559192\pi$
$662$	−53.1735	−2.06665
$663$	−11.7680	−0.457030
$664$	2.01615	0.0782418
$665$	0.927609	0.0359711
$666$	−20.8681	−0.808623
$667$	−7.40891	−0.286874
$668$	1.59115	0.0615634
$669$	−10.7699	−0.416387
$670$	−5.16033	−0.199361
$671$	−14.7491	−0.569382
$672$	37.2777	1.43802
$673$	−30.0316	−1.15763	−0.578816	−	0.815458i	$-0.696485\pi$
$673$	−30.0316	−1.15763	−0.578816	+	0.815458i	$0.696485\pi$
$674$	12.9020	0.496967
$675$	−24.3224	−0.936170
$676$	0.422771	0.0162604
$677$	−13.3937	−0.514761	−0.257380	−	0.966310i	$-0.582859\pi$
$677$	−13.3937	−0.514761	−0.257380	+	0.966310i	$0.582859\pi$
$678$	34.3007	1.31731
$679$	16.4359	0.630752
$680$	0.157930	0.00605635
$681$	22.1352	0.848224
$682$	10.8997	0.417370
$683$	−2.22871	−0.0852793	−0.0426396	−	0.999091i	$-0.513577\pi$
$683$	−2.22871	−0.0852793	−0.0426396	+	0.999091i	$0.513577\pi$
$684$	3.86548	0.147801
$685$	2.06687	0.0789711
$686$	98.1834	3.74866
$687$	22.7577	0.868262
$688$	15.8575	0.604563
$689$	−39.4904	−1.50447
$690$	0.771245	0.0293608
$691$	25.7189	0.978391	0.489196	−	0.872174i	$-0.337290\pi$
$691$	25.7189	0.978391	0.489196	+	0.872174i	$0.337290\pi$
$692$	26.6733	1.01397
$693$	−16.3173	−0.619843
$694$	32.5616	1.23602
$695$	2.78558	0.105663
$696$	0.835636	0.0316747
$697$	−39.4009	−1.49241
$698$	69.7997	2.64196
$699$	9.80048	0.370688
$700$	45.6929	1.72703
$701$	10.8324	0.409135	0.204567	−	0.978853i	$-0.434421\pi$
$701$	10.8324	0.409135	0.204567	+	0.978853i	$0.434421\pi$
$702$	−35.0646	−1.32343
$703$	5.13774	0.193773
$704$	11.1850	0.421550
$705$	−1.06408	−0.0400757
$706$	11.4611	0.431345
$707$	25.8886	0.973640
$708$	−9.36903	−0.352110
$709$	−7.60880	−0.285755	−0.142877	−	0.989740i	$-0.545635\pi$
$709$	−7.60880	−0.285755	−0.142877	+	0.989740i	$0.545635\pi$
$710$	2.90291	0.108944
$711$	−30.5097	−1.14420
$712$	0.0239690	0.000898277 0
$713$	7.38838	0.276697
$714$	−31.2956	−1.17121
$715$	−1.10428	−0.0412976
$716$	15.0434	0.562198
$717$	4.34595	0.162303
$718$	−45.9430	−1.71458
$719$	−49.0630	−1.82974	−0.914870	−	0.403749i	$-0.867707\pi$
$719$	−49.0630	−1.82974	−0.914870	+	0.403749i	$0.867707\pi$
$720$	1.65103	0.0615301
$721$	−8.77628	−0.326846
$722$	−1.96796	−0.0732400
$723$	6.05257	0.225098
$724$	3.92589	0.145905
$725$	17.1391	0.636529
$726$	−16.0167	−0.594435
$727$	−38.6064	−1.43183	−0.715915	−	0.698187i	$-0.753990\pi$
$727$	−38.6064	−1.43183	−0.715915	+	0.698187i	$0.753990\pi$
$728$	−4.47121	−0.165714
$729$	11.2246	0.415727
$730$	−2.65913	−0.0984189
$731$	−12.5142	−0.462854
$732$	−16.6131	−0.614038
$733$	−21.7522	−0.803436	−0.401718	−	0.915763i	$-0.631587\pi$
$733$	−21.7522	−0.803436	−0.401718	+	0.915763i	$0.631587\pi$
$734$	1.73783	0.0641444
$735$	−3.13224	−0.115535
$736$	16.8248	0.620171
$737$	−22.3485	−0.823219
$738$	−47.8498	−1.76138
$739$	10.7551	0.395632	0.197816	−	0.980239i	$-0.436615\pi$
$739$	10.7551	0.395632	0.197816	+	0.980239i	$0.436615\pi$
$740$	−1.81623	−0.0667659
$741$	3.51856	0.129257
$742$	−105.020	−3.85542
$743$	1.46444	0.0537253	0.0268626	−	0.999639i	$-0.491448\pi$
$743$	1.46444	0.0537253	0.0268626	+	0.999639i	$0.491448\pi$
$744$	−0.833321	−0.0305510
$745$	2.77028	0.101495
$746$	−9.95961	−0.364647
$747$	−16.6333	−0.608580
$748$	−10.0768	−0.368446
$749$	58.7159	2.14543
$750$	−3.58105	−0.130762
$751$	40.9213	1.49324	0.746619	−	0.665252i	$-0.231676\pi$
$751$	40.9213	1.49324	0.746619	+	0.665252i	$0.231676\pi$
$752$	−24.6945	−0.900517
$753$	7.08333	0.258131
$754$	24.7087	0.899837
$755$	−1.86629	−0.0679214
$756$	−45.0948	−1.64008
$757$	36.9995	1.34477	0.672384	−	0.740202i	$-0.265270\pi$
$757$	36.9995	1.34477	0.672384	+	0.740202i	$0.265270\pi$
$758$	−5.45635	−0.198184
$759$	3.34014	0.121239
$760$	−0.0472203	−0.00171286
$761$	51.8764	1.88052	0.940259	−	0.340460i	$-0.110583\pi$
$761$	51.8764	1.88052	0.940259	+	0.340460i	$0.110583\pi$
$762$	−28.1877	−1.02113
$763$	−53.0738	−1.92140
$764$	−3.30283	−0.119492
$765$	−1.30293	−0.0471075
$766$	−20.3115	−0.733884
$767$	18.8036	0.678958
$768$	−17.2567	−0.622696
$769$	−33.2617	−1.19945	−0.599724	−	0.800207i	$-0.704723\pi$
$769$	−33.2617	−1.19945	−0.599724	+	0.800207i	$0.704723\pi$
$770$	−2.93670	−0.105831
$771$	17.2779	0.622249
$772$	−1.66274	−0.0598433
$773$	−19.9933	−0.719109	−0.359554	−	0.933124i	$-0.617071\pi$
$773$	−19.9933	−0.719109	−0.359554	+	0.933124i	$0.617071\pi$
$774$	−15.1977	−0.546270
$775$	−17.0916	−0.613948
$776$	−0.836677	−0.0300350
$777$	−24.4288	−0.876378
$778$	19.8628	0.712117
$779$	11.7806	0.422085
$780$	−1.24384	−0.0445365
$781$	12.5720	0.449862
$782$	−14.1249	−0.505105
$783$	−16.9147	−0.604483
$784$	−72.6911	−2.59611
$785$	−1.27010	−0.0453318
$786$	19.1773	0.684032
$787$	54.5076	1.94299	0.971493	−	0.237066i	$-0.0761859\pi$
$787$	54.5076	1.94299	0.971493	+	0.237066i	$0.0761859\pi$
$788$	27.0347	0.963072
$789$	−11.2457	−0.400358
$790$	−5.49097	−0.195360
$791$	−88.5335	−3.14789
$792$	0.830639	0.0295155
$793$	33.3424	1.18402
$794$	−8.30220	−0.294634
$795$	1.98301	0.0703303
$796$	1.49113	0.0528519
$797$	−41.5152	−1.47055	−0.735273	−	0.677772i	$-0.762946\pi$
$797$	−41.5152	−1.47055	−0.735273	+	0.677772i	$0.762946\pi$
$798$	9.35722	0.331242
$799$	19.4880	0.689437
$800$	−38.9209	−1.37606
$801$	−0.197745	−0.00698697
$802$	44.3716	1.56682
$803$	−11.5163	−0.406400
$804$	−25.1730	−0.887783
$805$	−1.99066	−0.0701614
$806$	−24.6402	−0.867915
$807$	−20.3247	−0.715465
$808$	−1.31787	−0.0463625
$809$	36.3019	1.27631	0.638154	−	0.769909i	$-0.279698\pi$
$809$	36.3019	1.27631	0.638154	+	0.769909i	$0.279698\pi$
$810$	−0.539199	−0.0189455
$811$	−7.54260	−0.264856	−0.132428	−	0.991193i	$-0.542277\pi$
$811$	−7.54260	−0.264856	−0.132428	+	0.991193i	$0.542277\pi$
$812$	31.7766	1.11514
$813$	3.84895	0.134989
$814$	−16.2655	−0.570105
$815$	−1.08562	−0.0380275
$816$	13.7139	0.480082
$817$	3.74167	0.130905
$818$	−56.7905	−1.98563
$819$	36.8876	1.28896
$820$	−4.16455	−0.145432
$821$	−46.0521	−1.60723	−0.803614	−	0.595151i	$-0.797092\pi$
$821$	−46.0521	−1.60723	−0.803614	+	0.595151i	$0.797092\pi$
$822$	20.8495	0.727210
$823$	−23.4774	−0.818370	−0.409185	−	0.912451i	$-0.634187\pi$
$823$	−23.4774	−0.818370	−0.409185	+	0.912451i	$0.634187\pi$
$824$	0.446760	0.0155636
$825$	−7.72676	−0.269011
$826$	50.0061	1.73993
$827$	−5.68901	−0.197826	−0.0989132	−	0.995096i	$-0.531537\pi$
$827$	−5.68901	−0.197826	−0.0989132	+	0.995096i	$0.531537\pi$
$828$	−8.29536	−0.288284
$829$	−0.0641062	−0.00222650	−0.00111325	−	0.999999i	$-0.500354\pi$
$829$	−0.0641062	−0.00222650	−0.00111325	+	0.999999i	$0.500354\pi$
$830$	−2.99357	−0.103908
$831$	5.18218	0.179768
$832$	−25.2852	−0.876607
$833$	57.3652	1.98758
$834$	28.0994	0.973004
$835$	0.160358	0.00554942
$836$	3.01292	0.104204
$837$	16.8679	0.583039
$838$	3.65251	0.126174
$839$	−27.8242	−0.960599	−0.480300	−	0.877104i	$-0.659472\pi$
$839$	−27.8242	−0.960599	−0.480300	+	0.877104i	$0.659472\pi$
$840$	0.224522	0.00774675
$841$	−17.0808	−0.588994
$842$	54.0942	1.86421
$843$	−17.0855	−0.588455
$844$	9.20546	0.316865
$845$	0.0426075	0.00146574
$846$	23.6670	0.813687
$847$	41.3406	1.42048
$848$	46.0205	1.58035
$849$	3.51327	0.120575
$850$	32.6752	1.12075
$851$	−11.0256	−0.377954
$852$	14.1609	0.485144
$853$	−40.4409	−1.38467	−0.692335	−	0.721577i	$-0.743418\pi$
$853$	−40.4409	−1.38467	−0.692335	+	0.721577i	$0.743418\pi$
$854$	88.6705	3.03424
$855$	0.389569	0.0133230
$856$	−2.98896	−0.102160
$857$	−48.8738	−1.66950	−0.834749	−	0.550631i	$-0.814387\pi$
$857$	−48.8738	−1.66950	−0.834749	+	0.550631i	$0.814387\pi$
$858$	−11.1393	−0.380291
$859$	−9.65359	−0.329376	−0.164688	−	0.986346i	$-0.552662\pi$
$859$	−9.65359	−0.329376	−0.164688	+	0.986346i	$0.552662\pi$
$860$	−1.32271	−0.0451041
$861$	−56.0143	−1.90896
$862$	25.5647	0.870736
$863$	50.9009	1.73269	0.866344	−	0.499448i	$-0.166464\pi$
$863$	50.9009	1.73269	0.866344	+	0.499448i	$0.166464\pi$
$864$	38.4115	1.30678
$865$	2.68817	0.0914006
$866$	23.2105	0.788726
$867$	5.62513	0.191039
$868$	−31.6886	−1.07558
$869$	−23.7805	−0.806698
$870$	−1.24075	−0.0420653
$871$	50.5220	1.71187
$872$	2.70175	0.0914927
$873$	6.90260	0.233618
$874$	4.22327	0.142854
$875$	9.24304	0.312472
$876$	−12.9717	−0.438274
$877$	−33.6231	−1.13537	−0.567686	−	0.823245i	$-0.692161\pi$
$877$	−33.6231	−1.13537	−0.567686	+	0.823245i	$0.692161\pi$
$878$	12.9953	0.438571
$879$	7.16198	0.241568
$880$	1.28688	0.0433807
$881$	3.15075	0.106151	0.0530757	−	0.998590i	$-0.483098\pi$
$881$	3.15075	0.106151	0.0530757	+	0.998590i	$0.483098\pi$
$882$	69.6663	2.34579
$883$	−48.8822	−1.64502	−0.822508	−	0.568753i	$-0.807426\pi$
$883$	−48.8822	−1.64502	−0.822508	+	0.568753i	$0.807426\pi$
$884$	22.7801	0.766178
$885$	−0.944223	−0.0317397
$886$	7.73012	0.259698
$887$	16.2806	0.546648	0.273324	−	0.961922i	$-0.411877\pi$
$887$	16.2806	0.546648	0.273324	+	0.961922i	$0.411877\pi$
$888$	1.24356	0.0417311
$889$	72.7552	2.44013
$890$	−0.0355891	−0.00119295
$891$	−2.33518	−0.0782316
$892$	20.8480	0.698044
$893$	−5.82681	−0.194987
$894$	27.9451	0.934622
$895$	1.51609	0.0506774
$896$	9.81582	0.327924
$897$	−7.55085	−0.252116
$898$	26.6317	0.888711
$899$	−11.8861	−0.396425
$900$	19.1897	0.639657
$901$	−36.3177	−1.20992
$902$	−37.2962	−1.24183
$903$	−17.7908	−0.592041
$904$	4.50684	0.149895
$905$	0.395657	0.0131521
$906$	−18.8262	−0.625457
$907$	31.3658	1.04148	0.520742	−	0.853714i	$-0.325655\pi$
$907$	31.3658	1.04148	0.520742	+	0.853714i	$0.325655\pi$
$908$	−42.8488	−1.42199
$909$	10.8725	0.360616
$910$	6.63883	0.220075
$911$	−3.23866	−0.107302	−0.0536509	−	0.998560i	$-0.517086\pi$
$911$	−3.23866	−0.107302	−0.0536509	+	0.998560i	$0.517086\pi$
$912$	−4.10038	−0.135777
$913$	−12.9647	−0.429068
$914$	−14.9338	−0.493965
$915$	−1.67429	−0.0553503
$916$	−44.0539	−1.45558
$917$	−49.4985	−1.63458
$918$	−32.2475	−1.06433
$919$	44.9041	1.48125	0.740625	−	0.671918i	$-0.234529\pi$
$919$	44.9041	1.48125	0.740625	+	0.671918i	$0.234529\pi$
$920$	0.101335	0.00334092
$921$	12.6261	0.416043
$922$	9.92214	0.326768
$923$	−28.4208	−0.935483
$924$	−14.3258	−0.471283
$925$	25.5056	0.838620
$926$	−19.9435	−0.655383
$927$	−3.68578	−0.121057
$928$	−27.0671	−0.888521
$929$	43.0558	1.41261	0.706307	−	0.707906i	$-0.250360\pi$
$929$	43.0558	1.41261	0.706307	+	0.707906i	$0.250360\pi$
$930$	1.23731	0.0405730
$931$	−17.1519	−0.562130
$932$	−18.9715	−0.621433
$933$	−32.4753	−1.06319
$934$	−44.1524	−1.44471
$935$	−1.01556	−0.0332123
$936$	−1.87778	−0.0613771
$937$	21.2253	0.693399	0.346699	−	0.937976i	$-0.387302\pi$
$937$	21.2253	0.693399	0.346699	+	0.937976i	$0.387302\pi$
$938$	134.358	4.38694
$939$	19.0201	0.620697
$940$	2.05982	0.0671840
$941$	9.73246	0.317269	0.158635	−	0.987337i	$-0.449291\pi$
$941$	9.73246	0.317269	0.158635	+	0.987337i	$0.449291\pi$
$942$	−12.8121	−0.417440
$943$	−25.2814	−0.823274
$944$	−21.9129	−0.713205
$945$	−4.54472	−0.147840
$946$	−11.8457	−0.385137
$947$	−36.0309	−1.17085	−0.585423	−	0.810728i	$-0.699071\pi$
$947$	−36.0309	−1.17085	−0.585423	+	0.810728i	$0.699071\pi$
$948$	−26.7859	−0.869967
$949$	26.0341	0.845104
$950$	−9.76970	−0.316971
$951$	0.967508	0.0313736
$952$	−4.11199	−0.133270
$953$	17.6960	0.573228	0.286614	−	0.958046i	$-0.407470\pi$
$953$	17.6960	0.573228	0.286614	+	0.958046i	$0.407470\pi$
$954$	−44.1056	−1.42797
$955$	−0.332864	−0.0107712
$956$	−8.41278	−0.272089
$957$	−5.37348	−0.173700
$958$	30.8281	0.996012
$959$	−53.8145	−1.73776
$960$	1.26970	0.0409794
$961$	−19.1468	−0.617638
$962$	36.7704	1.18553
$963$	24.6590	0.794624
$964$	−11.7164	−0.377360
$965$	−0.167573	−0.00539437
$966$	−20.0807	−0.646085
$967$	1.36674	0.0439515	0.0219757	−	0.999759i	$-0.493004\pi$
$967$	1.36674	0.0439515	0.0219757	+	0.999759i	$0.493004\pi$
$968$	−2.10446	−0.0676400
$969$	3.23587	0.103951
$970$	1.24229	0.0398877
$971$	43.9793	1.41136	0.705682	−	0.708529i	$-0.250641\pi$
$971$	43.9793	1.41136	0.705682	+	0.708529i	$0.250641\pi$
$972$	−30.1582	−0.967324
$973$	−72.5273	−2.32512
$974$	−58.4377	−1.87246
$975$	17.4674	0.559405
$976$	−38.8558	−1.24375
$977$	14.0611	0.449854	0.224927	−	0.974376i	$-0.427786\pi$
$977$	14.0611	0.449854	0.224927	+	0.974376i	$0.427786\pi$
$978$	−10.9511	−0.350178
$979$	−0.154131	−0.00492603
$980$	6.06332	0.193686
$981$	−22.2895	−0.711648
$982$	13.2188	0.421830
$983$	−56.3939	−1.79869	−0.899344	−	0.437242i	$-0.855955\pi$
$983$	−56.3939	−1.79869	−0.899344	+	0.437242i	$0.855955\pi$
$984$	2.85143	0.0909003
$985$	2.72460	0.0868129
$986$	22.7236	0.723666
$987$	27.7052	0.881866
$988$	−6.81113	−0.216691
$989$	−8.02967	−0.255329
$990$	−1.23333	−0.0391978
$991$	24.6063	0.781644	0.390822	−	0.920466i	$-0.372191\pi$
$991$	24.6063	0.781644	0.390822	+	0.920466i	$0.372191\pi$
$992$	26.9921	0.857000
$993$	−26.1416	−0.829580
$994$	−75.5821	−2.39732
$995$	0.150279	0.00476415
$996$	−14.6032	−0.462719
$997$	39.5072	1.25120	0.625602	−	0.780142i	$-0.284853\pi$
$997$	39.5072	1.25120	0.625602	+	0.780142i	$0.284853\pi$
$998$	−53.6497	−1.69825
$999$	−25.1718	−0.796400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6023.2.a.b.1.17	✓	99

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6023.2.a.b.1.17	✓	99	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:	\( N \)	\(=\)	\( 6023 = 19 \cdot 317 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6023.a (trivial)

\(f(q)\)	\(=\)	\(q-1.96796 q^{2} -0.967508 q^{3} +1.87288 q^{4} +0.188751 q^{5} +1.90402 q^{6} -4.91446 q^{7} +0.250172 q^{8} -2.06393 q^{9} +O(q^{10})\)	\(q-1.96796 q^{2} -0.967508 q^{3} +1.87288 q^{4} +0.188751 q^{5} +1.90402 q^{6} -4.91446 q^{7} +0.250172 q^{8} -2.06393 q^{9} -0.371455 q^{10} -1.60871 q^{11} -1.81202 q^{12} +3.63672 q^{13} +9.67146 q^{14} -0.182618 q^{15} -4.23808 q^{16} +3.34454 q^{17} +4.06173 q^{18} -1.00000 q^{19} +0.353508 q^{20} +4.75477 q^{21} +3.16589 q^{22} +2.14601 q^{23} -0.242044 q^{24} -4.96437 q^{25} -7.15693 q^{26} +4.89939 q^{27} -9.20417 q^{28} -3.45241 q^{29} +0.359386 q^{30} +3.44285 q^{31} +7.84005 q^{32} +1.55644 q^{33} -6.58193 q^{34} -0.927609 q^{35} -3.86548 q^{36} -5.13774 q^{37} +1.96796 q^{38} -3.51856 q^{39} +0.0472203 q^{40} -11.7806 q^{41} -9.35722 q^{42} -3.74167 q^{43} -3.01292 q^{44} -0.389569 q^{45} -4.22327 q^{46} +5.82681 q^{47} +4.10038 q^{48} +17.1519 q^{49} +9.76970 q^{50} -3.23587 q^{51} +6.81113 q^{52} -10.8588 q^{53} -9.64182 q^{54} -0.303646 q^{55} -1.22946 q^{56} +0.967508 q^{57} +6.79422 q^{58} +5.17048 q^{59} -0.342022 q^{60} +9.16826 q^{61} -6.77540 q^{62} +10.1431 q^{63} -6.95275 q^{64} +0.686435 q^{65} -3.06302 q^{66} +13.8922 q^{67} +6.26392 q^{68} -2.07628 q^{69} +1.82550 q^{70} -7.81496 q^{71} -0.516338 q^{72} +7.15869 q^{73} +10.1109 q^{74} +4.80307 q^{75} -1.87288 q^{76} +7.90594 q^{77} +6.92439 q^{78} +14.7823 q^{79} -0.799943 q^{80} +1.45159 q^{81} +23.1839 q^{82} +8.05904 q^{83} +8.90511 q^{84} +0.631286 q^{85} +7.36348 q^{86} +3.34024 q^{87} -0.402456 q^{88} +0.0958099 q^{89} +0.766657 q^{90} -17.8725 q^{91} +4.01921 q^{92} -3.33098 q^{93} -11.4670 q^{94} -0.188751 q^{95} -7.58531 q^{96} -3.34440 q^{97} -33.7542 q^{98} +3.32027 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 99 q - 4 q^{2} - 3 q^{3} + 80 q^{4} - 15 q^{5} - 12 q^{6} - 19 q^{7} - 12 q^{8} + 58 q^{9}+O(q^{10}) \) 99 * q - 4 * q^2 - 3 * q^3 + 80 * q^4 - 15 * q^5 - 12 * q^6 - 19 * q^7 - 12 * q^8 + 58 * q^9	\( 99 q - 4 q^{2} - 3 q^{3} + 80 q^{4} - 15 q^{5} - 12 q^{6} - 19 q^{7} - 12 q^{8} + 58 q^{9} - 6 q^{10} - 9 q^{11} - 27 q^{12} - 28 q^{13} - 13 q^{14} - 10 q^{15} + 38 q^{16} - 36 q^{17} - 14 q^{18} - 99 q^{19} - 34 q^{20} - 20 q^{21} - 53 q^{22} - 38 q^{23} - 25 q^{24} - 8 q^{25} - 3 q^{26} - 3 q^{27} - 63 q^{28} - 34 q^{29} - 30 q^{30} - 16 q^{31} - 43 q^{32} - 41 q^{33} - 14 q^{34} - 25 q^{35} - 16 q^{36} - 80 q^{37} + 4 q^{38} - 48 q^{39} - 10 q^{40} - 32 q^{41} - 37 q^{42} - 76 q^{43} - 21 q^{44} - 53 q^{45} - 23 q^{46} - 31 q^{47} - 74 q^{48} - 32 q^{49} - 29 q^{50} - 30 q^{51} - 71 q^{52} - 35 q^{53} - 80 q^{54} - 45 q^{55} - 33 q^{56} + 3 q^{57} - 91 q^{58} + 12 q^{59} - 56 q^{60} - 61 q^{61} - 46 q^{62} - 43 q^{63} - 30 q^{64} - 46 q^{65} - 75 q^{66} - 26 q^{67} - 55 q^{68} - 45 q^{69} - 76 q^{70} - 41 q^{71} - 77 q^{72} - 143 q^{73} - 64 q^{74} - 8 q^{75} - 80 q^{76} - 58 q^{77} - 34 q^{78} - 22 q^{79} - 36 q^{80} - 81 q^{81} - 109 q^{82} - 7 q^{83} - 6 q^{84} - 80 q^{85} + 32 q^{86} - 57 q^{87} - 120 q^{88} - 28 q^{89} - 12 q^{90} - 30 q^{91} - 107 q^{92} - 121 q^{93} + 8 q^{94} + 15 q^{95} + 4 q^{96} - 128 q^{97} + 54 q^{98} - 34 q^{99}+O(q^{100}) \) 99 * q - 4 * q^2 - 3 * q^3 + 80 * q^4 - 15 * q^5 - 12 * q^6 - 19 * q^7 - 12 * q^8 + 58 * q^9 - 6 * q^10 - 9 * q^11 - 27 * q^12 - 28 * q^13 - 13 * q^14 - 10 * q^15 + 38 * q^16 - 36 * q^17 - 14 * q^18 - 99 * q^19 - 34 * q^20 - 20 * q^21 - 53 * q^22 - 38 * q^23 - 25 * q^24 - 8 * q^25 - 3 * q^26 - 3 * q^27 - 63 * q^28 - 34 * q^29 - 30 * q^30 - 16 * q^31 - 43 * q^32 - 41 * q^33 - 14 * q^34 - 25 * q^35 - 16 * q^36 - 80 * q^37 + 4 * q^38 - 48 * q^39 - 10 * q^40 - 32 * q^41 - 37 * q^42 - 76 * q^43 - 21 * q^44 - 53 * q^45 - 23 * q^46 - 31 * q^47 - 74 * q^48 - 32 * q^49 - 29 * q^50 - 30 * q^51 - 71 * q^52 - 35 * q^53 - 80 * q^54 - 45 * q^55 - 33 * q^56 + 3 * q^57 - 91 * q^58 + 12 * q^59 - 56 * q^60 - 61 * q^61 - 46 * q^62 - 43 * q^63 - 30 * q^64 - 46 * q^65 - 75 * q^66 - 26 * q^67 - 55 * q^68 - 45 * q^69 - 76 * q^70 - 41 * q^71 - 77 * q^72 - 143 * q^73 - 64 * q^74 - 8 * q^75 - 80 * q^76 - 58 * q^77 - 34 * q^78 - 22 * q^79 - 36 * q^80 - 81 * q^81 - 109 * q^82 - 7 * q^83 - 6 * q^84 - 80 * q^85 + 32 * q^86 - 57 * q^87 - 120 * q^88 - 28 * q^89 - 12 * q^90 - 30 * q^91 - 107 * q^92 - 121 * q^93 + 8 * q^94 + 15 * q^95 + 4 * q^96 - 128 * q^97 + 54 * q^98 - 34 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more