LMFDB - Embedded newform 4009.2.a.c.1.21

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:	$1$
Dimension:	$71$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.21
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)

$f(q)$	$=$	$q-1.67829 q^{2} +0.141844 q^{3} +0.816665 q^{4} +4.00216 q^{5} -0.238056 q^{6} +0.121465 q^{7} +1.98598 q^{8} -2.97988 q^{9} +O(q^{10})$	\(q-1.67829 q^{2} +0.141844 q^{3} +0.816665 q^{4} +4.00216 q^{5} -0.238056 q^{6} +0.121465 q^{7} +1.98598 q^{8} -2.97988 q^{9} -6.71679 q^{10} +3.49313 q^{11} +0.115839 q^{12} -2.27940 q^{13} -0.203854 q^{14} +0.567682 q^{15} -4.96639 q^{16} -5.47989 q^{17} +5.00111 q^{18} +1.00000 q^{19} +3.26842 q^{20} +0.0172291 q^{21} -5.86249 q^{22} +1.11389 q^{23} +0.281700 q^{24} +11.0173 q^{25} +3.82550 q^{26} -0.848210 q^{27} +0.0991964 q^{28} -8.09903 q^{29} -0.952737 q^{30} +2.71923 q^{31} +4.36309 q^{32} +0.495479 q^{33} +9.19686 q^{34} +0.486123 q^{35} -2.43356 q^{36} -11.0002 q^{37} -1.67829 q^{38} -0.323319 q^{39} +7.94822 q^{40} -3.96062 q^{41} -0.0289155 q^{42} -10.2918 q^{43} +2.85271 q^{44} -11.9260 q^{45} -1.86944 q^{46} +8.20975 q^{47} -0.704452 q^{48} -6.98525 q^{49} -18.4902 q^{50} -0.777290 q^{51} -1.86151 q^{52} +13.3336 q^{53} +1.42354 q^{54} +13.9801 q^{55} +0.241228 q^{56} +0.141844 q^{57} +13.5925 q^{58} -0.894481 q^{59} +0.463606 q^{60} -6.26576 q^{61} -4.56367 q^{62} -0.361952 q^{63} +2.61024 q^{64} -9.12253 q^{65} -0.831559 q^{66} -13.9677 q^{67} -4.47524 q^{68} +0.157999 q^{69} -0.815857 q^{70} -3.35906 q^{71} -5.91799 q^{72} -13.9154 q^{73} +18.4615 q^{74} +1.56273 q^{75} +0.816665 q^{76} +0.424294 q^{77} +0.542625 q^{78} -12.3223 q^{79} -19.8763 q^{80} +8.81933 q^{81} +6.64708 q^{82} -10.9657 q^{83} +0.0140704 q^{84} -21.9314 q^{85} +17.2727 q^{86} -1.14880 q^{87} +6.93729 q^{88} -12.5741 q^{89} +20.0152 q^{90} -0.276868 q^{91} +0.909676 q^{92} +0.385707 q^{93} -13.7784 q^{94} +4.00216 q^{95} +0.618878 q^{96} +6.80039 q^{97} +11.7233 q^{98} -10.4091 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) $ 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * 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 $

$2$	−1.67829	−1.18673	−0.593366	−	0.804933i	$-0.702201\pi$
$2$	−1.67829	−1.18673	−0.593366	+	0.804933i	$0.702201\pi$
$3$	0.141844	0.0818937	0.0409468	−	0.999161i	$-0.486963\pi$
$3$	0.141844	0.0818937	0.0409468	+	0.999161i	$0.486963\pi$
$4$	0.816665	0.408332
$5$	4.00216	1.78982	0.894910	−	0.446247i	$-0.147240\pi$
$5$	4.00216	1.78982	0.894910	+	0.446247i	$0.147240\pi$
$6$	−0.238056	−0.0971858
$7$	0.121465	0.0459096	0.0229548	−	0.999737i	$-0.492693\pi$
$7$	0.121465	0.0459096	0.0229548	+	0.999737i	$0.492693\pi$
$8$	1.98598	0.702151
$9$	−2.97988	−0.993293
$10$	−6.71679	−2.12404
$11$	3.49313	1.05322	0.526609	−	0.850108i	$-0.323463\pi$
$11$	3.49313	1.05322	0.526609	+	0.850108i	$0.323463\pi$
$12$	0.115839	0.0334398
$13$	−2.27940	−0.632192	−0.316096	−	0.948727i	$-0.602372\pi$
$13$	−2.27940	−0.632192	−0.316096	+	0.948727i	$0.602372\pi$
$14$	−0.203854	−0.0544823
$15$	0.567682	0.146575
$16$	−4.96639	−1.24160
$17$	−5.47989	−1.32907	−0.664535	−	0.747257i	$-0.731370\pi$
$17$	−5.47989	−1.32907	−0.664535	+	0.747257i	$0.731370\pi$
$18$	5.00111	1.17877
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	3.26842	0.730842
$21$	0.0172291	0.00375970
$22$	−5.86249	−1.24989
$23$	1.11389	0.232262	0.116131	−	0.993234i	$-0.462951\pi$
$23$	1.11389	0.232262	0.116131	+	0.993234i	$0.462951\pi$
$24$	0.281700	0.0575017
$25$	11.0173	2.20346
$26$	3.82550	0.750243
$27$	−0.848210	−0.163238
$28$	0.0991964	0.0187464
$29$	−8.09903	−1.50395	−0.751976	−	0.659190i	$-0.770899\pi$
$29$	−8.09903	−1.50395	−0.751976	+	0.659190i	$0.770899\pi$
$30$	−0.952737	−0.173945
$31$	2.71923	0.488388	0.244194	−	0.969726i	$-0.421477\pi$
$31$	2.71923	0.488388	0.244194	+	0.969726i	$0.421477\pi$
$32$	4.36309	0.771292
$33$	0.495479	0.0862518
$34$	9.19686	1.57725
$35$	0.486123	0.0821698
$36$	−2.43356	−0.405594
$37$	−11.0002	−1.80842	−0.904211	−	0.427087i	$-0.859540\pi$
$37$	−11.0002	−1.80842	−0.904211	+	0.427087i	$0.859540\pi$
$38$	−1.67829	−0.272255
$39$	−0.323319	−0.0517726
$40$	7.94822	1.25672
$41$	−3.96062	−0.618545	−0.309273	−	0.950973i	$-0.600086\pi$
$41$	−3.96062	−0.618545	−0.309273	+	0.950973i	$0.600086\pi$
$42$	−0.0289155	−0.00446176
$43$	−10.2918	−1.56949	−0.784746	−	0.619818i	$-0.787206\pi$
$43$	−10.2918	−1.56949	−0.784746	+	0.619818i	$0.787206\pi$
$44$	2.85271	0.430063
$45$	−11.9260	−1.77782
$46$	−1.86944	−0.275633
$47$	8.20975	1.19752	0.598758	−	0.800930i	$-0.295661\pi$
$47$	8.20975	1.19752	0.598758	+	0.800930i	$0.295661\pi$
$48$	−0.704452	−0.101679
$49$	−6.98525	−0.997892
$50$	−18.4902	−2.61491
$51$	−0.777290	−0.108842
$52$	−1.86151	−0.258145
$53$	13.3336	1.83151	0.915754	−	0.401738i	$-0.131594\pi$
$53$	13.3336	1.83151	0.915754	+	0.401738i	$0.131594\pi$
$54$	1.42354	0.193720
$55$	13.9801	1.88507
$56$	0.241228	0.0322354
$57$	0.141844	0.0187877
$58$	13.5925	1.78479
$59$	−0.894481	−0.116452	−0.0582258	−	0.998303i	$-0.518544\pi$
$59$	−0.894481	−0.116452	−0.0582258	+	0.998303i	$0.518544\pi$
$60$	0.463606	0.0598513
$61$	−6.26576	−0.802248	−0.401124	−	0.916024i	$-0.631380\pi$
$61$	−6.26576	−0.802248	−0.401124	+	0.916024i	$0.631380\pi$
$62$	−4.56367	−0.579586
$63$	−0.361952	−0.0456017
$64$	2.61024	0.326280
$65$	−9.12253	−1.13151
$66$	−0.831559	−0.102358
$67$	−13.9677	−1.70643	−0.853213	−	0.521563i	$-0.825349\pi$
$67$	−13.9677	−1.70643	−0.853213	+	0.521563i	$0.825349\pi$
$68$	−4.47524	−0.542702
$69$	0.157999	0.0190208
$70$	−0.815857	−0.0975136
$71$	−3.35906	−0.398647	−0.199323	−	0.979934i	$-0.563874\pi$
$71$	−3.35906	−0.398647	−0.199323	+	0.979934i	$0.563874\pi$
$72$	−5.91799	−0.697442
$73$	−13.9154	−1.62868	−0.814340	−	0.580388i	$-0.802901\pi$
$73$	−13.9154	−1.62868	−0.814340	+	0.580388i	$0.802901\pi$
$74$	18.4615	2.14611
$75$	1.56273	0.180449
$76$	0.816665	0.0936779
$77$	0.424294	0.0483528
$78$	0.542625	0.0614401
$79$	−12.3223	−1.38637	−0.693183	−	0.720762i	$-0.743792\pi$
$79$	−12.3223	−1.38637	−0.693183	+	0.720762i	$0.743792\pi$
$80$	−19.8763	−2.22224
$81$	8.81933	0.979925
$82$	6.64708	0.734047
$83$	−10.9657	−1.20365	−0.601823	−	0.798630i	$-0.705559\pi$
$83$	−10.9657	−1.20365	−0.601823	+	0.798630i	$0.705559\pi$
$84$	0.0140704	0.00153521
$85$	−21.9314	−2.37879
$86$	17.2727	1.86257
$87$	−1.14880	−0.123164
$88$	6.93729	0.739517
$89$	−12.5741	−1.33285	−0.666425	−	0.745572i	$-0.732176\pi$
$89$	−12.5741	−1.33285	−0.666425	+	0.745572i	$0.732176\pi$
$90$	20.0152	2.10979
$91$	−0.276868	−0.0290237
$92$	0.909676	0.0948403
$93$	0.385707	0.0399959
$94$	−13.7784	−1.42113
$95$	4.00216	0.410613
$96$	0.618878	0.0631639
$97$	6.80039	0.690475	0.345237	−	0.938515i	$-0.387798\pi$
$97$	6.80039	0.690475	0.345237	+	0.938515i	$0.387798\pi$
$98$	11.7233	1.18423
$99$	−10.4091	−1.04615
$100$	8.99742	0.899742
$101$	12.6407	1.25780	0.628899	−	0.777487i	$-0.283506\pi$
$101$	12.6407	1.25780	0.628899	+	0.777487i	$0.283506\pi$
$102$	1.30452	0.129167
$103$	12.3105	1.21299	0.606493	−	0.795089i	$-0.292576\pi$
$103$	12.3105	1.21299	0.606493	+	0.795089i	$0.292576\pi$
$104$	−4.52685	−0.443894
$105$	0.0689537	0.00672919
$106$	−22.3777	−2.17351
$107$	12.4303	1.20169	0.600843	−	0.799367i	$-0.294832\pi$
$107$	12.4303	1.20169	0.600843	+	0.799367i	$0.294832\pi$
$108$	−0.692703	−0.0666554
$109$	−11.1418	−1.06719	−0.533597	−	0.845739i	$-0.679160\pi$
$109$	−11.1418	−1.06719	−0.533597	+	0.845739i	$0.679160\pi$
$110$	−23.4626	−2.23707
$111$	−1.56031	−0.148098
$112$	−0.603244	−0.0570012
$113$	−16.5158	−1.55368	−0.776839	−	0.629700i	$-0.783178\pi$
$113$	−16.5158	−1.55368	−0.776839	+	0.629700i	$0.783178\pi$
$114$	−0.238056	−0.0222960
$115$	4.45797	0.415708
$116$	−6.61420	−0.614113
$117$	6.79235	0.627953
$118$	1.50120	0.138197
$119$	−0.665617	−0.0610170
$120$	1.12741	0.102918
$121$	1.20194	0.109267
$122$	10.5158	0.952054
$123$	−0.561790	−0.0506549
$124$	2.22070	0.199425
$125$	24.0821	2.15397
$126$	0.607461	0.0541169
$127$	7.07559	0.627857	0.313929	−	0.949447i	$-0.398355\pi$
$127$	7.07559	0.627857	0.313929	+	0.949447i	$0.398355\pi$
$128$	−13.1069	−1.15850
$129$	−1.45984	−0.128531
$130$	15.3103	1.34280
$131$	15.5068	1.35484	0.677419	−	0.735597i	$-0.263098\pi$
$131$	15.5068	1.35484	0.677419	+	0.735597i	$0.263098\pi$
$132$	0.404640	0.0352194
$133$	0.121465	0.0105324
$134$	23.4419	2.02507
$135$	−3.39467	−0.292167
$136$	−10.8830	−0.933207
$137$	12.4347	1.06237	0.531183	−	0.847257i	$-0.321748\pi$
$137$	12.4347	1.06237	0.531183	+	0.847257i	$0.321748\pi$
$138$	−0.265168	−0.0225726
$139$	−9.66190	−0.819511	−0.409756	−	0.912195i	$-0.634386\pi$
$139$	−9.66190	−0.819511	−0.409756	+	0.912195i	$0.634386\pi$
$140$	0.397000	0.0335526
$141$	1.16450	0.0980689
$142$	5.63748	0.473087
$143$	−7.96224	−0.665836
$144$	14.7992	1.23327
$145$	−32.4136	−2.69180
$146$	23.3542	1.93281
$147$	−0.990815	−0.0817211
$148$	−8.98348	−0.738437
$149$	−9.40544	−0.770524	−0.385262	−	0.922807i	$-0.625889\pi$
$149$	−9.40544	−0.770524	−0.385262	+	0.922807i	$0.625889\pi$
$150$	−2.62273	−0.214145
$151$	−1.24503	−0.101319	−0.0506594	−	0.998716i	$-0.516132\pi$
$151$	−1.24503	−0.101319	−0.0506594	+	0.998716i	$0.516132\pi$
$152$	1.98598	0.161084
$153$	16.3294	1.32016
$154$	−0.712089	−0.0573818
$155$	10.8828	0.874127
$156$	−0.264044	−0.0211404
$157$	18.0184	1.43802	0.719011	−	0.694999i	$-0.244595\pi$
$157$	18.0184	1.43802	0.719011	+	0.694999i	$0.244595\pi$
$158$	20.6804	1.64524
$159$	1.89129	0.149989
$160$	17.4618	1.38047
$161$	0.135299	0.0106631
$162$	−14.8014	−1.16291
$163$	8.10776	0.635049	0.317525	−	0.948250i	$-0.397148\pi$
$163$	8.10776	0.635049	0.317525	+	0.948250i	$0.397148\pi$
$164$	−3.23450	−0.252572
$165$	1.98299	0.154375
$166$	18.4037	1.42840
$167$	−8.86958	−0.686348	−0.343174	−	0.939272i	$-0.611502\pi$
$167$	−8.86958	−0.686348	−0.343174	+	0.939272i	$0.611502\pi$
$168$	0.0342167	0.00263988
$169$	−7.80433	−0.600333
$170$	36.8073	2.82299
$171$	−2.97988	−0.227877
$172$	−8.40499	−0.640874
$173$	13.4198	1.02029	0.510143	−	0.860090i	$-0.329592\pi$
$173$	13.4198	1.02029	0.510143	+	0.860090i	$0.329592\pi$
$174$	1.92802	0.146163
$175$	1.33822	0.101160
$176$	−17.3482	−1.30767
$177$	−0.126877	−0.00953664
$178$	21.1030	1.58173
$179$	6.10284	0.456148	0.228074	−	0.973644i	$-0.426757\pi$
$179$	6.10284	0.456148	0.228074	+	0.973644i	$0.426757\pi$
$180$	−9.73951	−0.725940
$181$	10.9440	0.813459	0.406729	−	0.913549i	$-0.366669\pi$
$181$	10.9440	0.813459	0.406729	+	0.913549i	$0.366669\pi$
$182$	0.464666	0.0344433
$183$	−0.888760	−0.0656990
$184$	2.21217	0.163083
$185$	−44.0245	−3.23675
$186$	−0.647329	−0.0474644
$187$	−19.1420	−1.39980
$188$	6.70462	0.488984
$189$	−0.103028	−0.00749419
$190$	−6.71679	−0.487287
$191$	21.7100	1.57088	0.785441	−	0.618936i	$-0.212436\pi$
$191$	21.7100	1.57088	0.785441	+	0.618936i	$0.212436\pi$
$192$	0.370247	0.0267203
$193$	−6.05974	−0.436190	−0.218095	−	0.975928i	$-0.569984\pi$
$193$	−6.05974	−0.436190	−0.218095	+	0.975928i	$0.569984\pi$
$194$	−11.4130	−0.819408
$195$	−1.29398	−0.0926635
$196$	−5.70461	−0.407472
$197$	10.9453	0.779821	0.389911	−	0.920853i	$-0.372506\pi$
$197$	10.9453	0.779821	0.389911	+	0.920853i	$0.372506\pi$
$198$	17.4695	1.24150
$199$	−19.5650	−1.38693	−0.693463	−	0.720492i	$-0.743916\pi$
$199$	−19.5650	−1.38693	−0.693463	+	0.720492i	$0.743916\pi$
$200$	21.8801	1.54716
$201$	−1.98123	−0.139745
$202$	−21.2148	−1.49267
$203$	−0.983751	−0.0690458
$204$	−0.634785	−0.0444439
$205$	−15.8510	−1.10708
$206$	−20.6606	−1.43949
$207$	−3.31926	−0.230705
$208$	11.3204	0.784928
$209$	3.49313	0.241625
$210$	−0.115724	−0.00798574
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	10.8891	0.747865
$213$	−0.476462	−0.0326466
$214$	−20.8617	−1.42608
$215$	−41.1896	−2.80911
$216$	−1.68453	−0.114618
$217$	0.330292	0.0224217
$218$	18.6992	1.26647
$219$	−1.97382	−0.133379
$220$	11.4170	0.769735
$221$	12.4909	0.840227
$222$	2.61866	0.175753
$223$	−10.6169	−0.710958	−0.355479	−	0.934684i	$-0.615682\pi$
$223$	−10.6169	−0.710958	−0.355479	+	0.934684i	$0.615682\pi$
$224$	0.529964	0.0354097
$225$	−32.8302	−2.18868
$226$	27.7184	1.84380
$227$	−4.84289	−0.321434	−0.160717	−	0.987001i	$-0.551381\pi$
$227$	−4.84289	−0.321434	−0.160717	+	0.987001i	$0.551381\pi$
$228$	0.115839	0.00767163
$229$	2.85479	0.188650	0.0943251	−	0.995541i	$-0.469931\pi$
$229$	2.85479	0.188650	0.0943251	+	0.995541i	$0.469931\pi$
$230$	−7.48178	−0.493334
$231$	0.0601835	0.00395978
$232$	−16.0845	−1.05600
$233$	0.766414	0.0502095	0.0251047	−	0.999685i	$-0.492008\pi$
$233$	0.766414	0.0502095	0.0251047	+	0.999685i	$0.492008\pi$
$234$	−11.3995	−0.745211
$235$	32.8567	2.14334
$236$	−0.730492	−0.0475510
$237$	−1.74784	−0.113535
$238$	1.11710	0.0724108
$239$	24.9901	1.61647	0.808236	−	0.588858i	$-0.200422\pi$
$239$	24.9901	1.61647	0.808236	+	0.588858i	$0.200422\pi$
$240$	−2.81933	−0.181987
$241$	6.52580	0.420364	0.210182	−	0.977662i	$-0.432594\pi$
$241$	6.52580	0.420364	0.210182	+	0.977662i	$0.432594\pi$
$242$	−2.01721	−0.129671
$243$	3.79560	0.243488
$244$	−5.11703	−0.327584
$245$	−27.9561	−1.78605
$246$	0.942849	0.0601138
$247$	−2.27940	−0.145035
$248$	5.40035	0.342922
$249$	−1.55542	−0.0985709
$250$	−40.4168	−2.55618
$251$	−18.4518	−1.16467	−0.582334	−	0.812950i	$-0.697860\pi$
$251$	−18.4518	−1.16467	−0.582334	+	0.812950i	$0.697860\pi$
$252$	−0.295594	−0.0186206
$253$	3.89096	0.244623
$254$	−11.8749	−0.745098
$255$	−3.11084	−0.194808
$256$	16.7768	1.04855
$257$	−5.87349	−0.366378	−0.183189	−	0.983078i	$-0.558642\pi$
$257$	−5.87349	−0.366378	−0.183189	+	0.983078i	$0.558642\pi$
$258$	2.45003	0.152532
$259$	−1.33614	−0.0830238
$260$	−7.45005	−0.462033
$261$	24.1341	1.49387
$262$	−26.0250	−1.60783
$263$	−8.26974	−0.509934	−0.254967	−	0.966950i	$-0.582065\pi$
$263$	−8.26974	−0.509934	−0.254967	+	0.966950i	$0.582065\pi$
$264$	0.984013	0.0605618
$265$	53.3631	3.27807
$266$	−0.203854	−0.0124991
$267$	−1.78356	−0.109152
$268$	−11.4069	−0.696789
$269$	−1.01149	−0.0616718	−0.0308359	−	0.999524i	$-0.509817\pi$
$269$	−1.01149	−0.0616718	−0.0308359	+	0.999524i	$0.509817\pi$
$270$	5.69725	0.346724
$271$	14.8532	0.902265	0.451132	−	0.892457i	$-0.351020\pi$
$271$	14.8532	0.902265	0.451132	+	0.892457i	$0.351020\pi$
$272$	27.2153	1.65017
$273$	−0.0392721	−0.00237686
$274$	−20.8690	−1.26074
$275$	38.4848	2.32072
$276$	0.129032	0.00776682
$277$	23.5621	1.41571	0.707855	−	0.706358i	$-0.249663\pi$
$277$	23.5621	1.41571	0.707855	+	0.706358i	$0.249663\pi$
$278$	16.2155	0.972540
$279$	−8.10299	−0.485113
$280$	0.965432	0.0576956
$281$	2.56742	0.153160	0.0765799	−	0.997063i	$-0.475600\pi$
$281$	2.56742	0.153160	0.0765799	+	0.997063i	$0.475600\pi$
$282$	−1.95438	−0.116381
$283$	−7.55478	−0.449085	−0.224543	−	0.974464i	$-0.572089\pi$
$283$	−7.55478	−0.449085	−0.224543	+	0.974464i	$0.572089\pi$
$284$	−2.74322	−0.162780
$285$	0.567682	0.0336266
$286$	13.3630	0.790169
$287$	−0.481078	−0.0283971
$288$	−13.0015	−0.766119
$289$	13.0292	0.766425
$290$	54.3995	3.19445
$291$	0.964594	0.0565455
$292$	−11.3643	−0.665043
$293$	−13.7107	−0.800989	−0.400494	−	0.916299i	$-0.631162\pi$
$293$	−13.7107	−0.800989	−0.400494	+	0.916299i	$0.631162\pi$
$294$	1.66288	0.0969810
$295$	−3.57986	−0.208427
$296$	−21.8462	−1.26978
$297$	−2.96291	−0.171925
$298$	15.7851	0.914405
$299$	−2.53901	−0.146835
$300$	1.27623	0.0736832
$301$	−1.25010	−0.0720547
$302$	2.08952	0.120238
$303$	1.79301	0.103006
$304$	−4.96639	−0.284842
$305$	−25.0766	−1.43588
$306$	−27.4055	−1.56667
$307$	−25.6247	−1.46248	−0.731238	−	0.682122i	$-0.761057\pi$
$307$	−25.6247	−1.46248	−0.731238	+	0.682122i	$0.761057\pi$
$308$	0.346506	0.0197440
$309$	1.74616	0.0993358
$310$	−18.2645	−1.03735
$311$	−4.97766	−0.282257	−0.141128	−	0.989991i	$-0.545073\pi$
$311$	−4.97766	−0.282257	−0.141128	+	0.989991i	$0.545073\pi$
$312$	−0.642107	−0.0363521
$313$	−14.2267	−0.804138	−0.402069	−	0.915609i	$-0.631709\pi$
$313$	−14.2267	−0.804138	−0.402069	+	0.915609i	$0.631709\pi$
$314$	−30.2401	−1.70655
$315$	−1.44859	−0.0816188
$316$	−10.0632	−0.566098
$317$	11.3070	0.635063	0.317532	−	0.948248i	$-0.397146\pi$
$317$	11.3070	0.635063	0.317532	+	0.948248i	$0.397146\pi$
$318$	−3.17414	−0.177997
$319$	−28.2909	−1.58399
$320$	10.4466	0.583983
$321$	1.76317	0.0984104
$322$	−0.227071	−0.0126542
$323$	−5.47989	−0.304909
$324$	7.20244	0.400135
$325$	−25.1128	−1.39301
$326$	−13.6072	−0.753633
$327$	−1.58040	−0.0873964
$328$	−7.86573	−0.434312
$329$	0.997200	0.0549774
$330$	−3.32803	−0.183202
$331$	9.04960	0.497411	0.248705	−	0.968579i	$-0.419995\pi$
$331$	9.04960	0.497411	0.248705	+	0.968579i	$0.419995\pi$
$332$	−8.95533	−0.491487
$333$	32.7793	1.79629
$334$	14.8857	0.814511
$335$	−55.9009	−3.05419
$336$	−0.0855665	−0.00466804
$337$	−25.0866	−1.36655	−0.683277	−	0.730160i	$-0.739446\pi$
$337$	−25.0866	−1.36655	−0.683277	+	0.730160i	$0.739446\pi$
$338$	13.0979	0.712434
$339$	−2.34267	−0.127236
$340$	−17.9106	−0.971339
$341$	9.49862	0.514379
$342$	5.00111	0.270429
$343$	−1.69872	−0.0917224
$344$	−20.4394	−1.10202
$345$	0.632336	0.0340438
$346$	−22.5223	−1.21081
$347$	4.12317	0.221344	0.110672	−	0.993857i	$-0.464700\pi$
$347$	4.12317	0.221344	0.110672	+	0.993857i	$0.464700\pi$
$348$	−0.938184	−0.0502919
$349$	−14.0857	−0.753990	−0.376995	−	0.926215i	$-0.623043\pi$
$349$	−14.0857	−0.753990	−0.376995	+	0.926215i	$0.623043\pi$
$350$	−2.24592	−0.120049
$351$	1.93341	0.103198
$352$	15.2408	0.812338
$353$	4.59021	0.244312	0.122156	−	0.992511i	$-0.461019\pi$
$353$	4.59021	0.244312	0.122156	+	0.992511i	$0.461019\pi$
$354$	0.212936	0.0113174
$355$	−13.4435	−0.713506
$356$	−10.2688	−0.544246
$357$	−0.0944137	−0.00499690
$358$	−10.2424	−0.541325
$359$	−3.06356	−0.161689	−0.0808443	−	0.996727i	$-0.525762\pi$
$359$	−3.06356	−0.161689	−0.0808443	+	0.996727i	$0.525762\pi$
$360$	−23.6847	−1.24829
$361$	1.00000	0.0526316
$362$	−18.3672	−0.965358
$363$	0.170488	0.00894830
$364$	−0.226109	−0.0118513
$365$	−55.6918	−2.91504
$366$	1.49160	0.0779671
$367$	37.6778	1.96677	0.983383	−	0.181546i	$-0.0581100\pi$
$367$	37.6778	1.96677	0.983383	+	0.181546i	$0.0581100\pi$
$368$	−5.53202	−0.288376
$369$	11.8022	0.614397
$370$	73.8860	3.84115
$371$	1.61957	0.0840838
$372$	0.314993	0.0163316
$373$	−15.0159	−0.777492	−0.388746	−	0.921345i	$-0.627092\pi$
$373$	−15.0159	−0.777492	−0.388746	+	0.921345i	$0.627092\pi$
$374$	32.1258	1.66119
$375$	3.41590	0.176396
$376$	16.3044	0.840836
$377$	18.4609	0.950787
$378$	0.172911	0.00889359
$379$	9.84919	0.505919	0.252960	−	0.967477i	$-0.418596\pi$
$379$	9.84919	0.505919	0.252960	+	0.967477i	$0.418596\pi$
$380$	3.26842	0.167667
$381$	1.00363	0.0514175
$382$	−36.4358	−1.86422
$383$	−14.6652	−0.749356	−0.374678	−	0.927155i	$-0.622247\pi$
$383$	−14.6652	−0.749356	−0.374678	+	0.927155i	$0.622247\pi$
$384$	−1.85914	−0.0948737
$385$	1.69809	0.0865427
$386$	10.1700	0.517640
$387$	30.6685	1.55897
$388$	5.55364	0.281943
$389$	−11.9118	−0.603954	−0.301977	−	0.953315i	$-0.597647\pi$
$389$	−11.9118	−0.603954	−0.301977	+	0.953315i	$0.597647\pi$
$390$	2.17167	0.109967
$391$	−6.10401	−0.308693
$392$	−13.8726	−0.700671
$393$	2.19955	0.110953
$394$	−18.3694	−0.925439
$395$	−49.3158	−2.48135
$396$	−8.50075	−0.427179
$397$	10.5903	0.531513	0.265756	−	0.964040i	$-0.414378\pi$
$397$	10.5903	0.531513	0.265756	+	0.964040i	$0.414378\pi$
$398$	32.8358	1.64591
$399$	0.0172291	0.000862535 0
$400$	−54.7161	−2.73580
$401$	−20.2143	−1.00946	−0.504728	−	0.863279i	$-0.668407\pi$
$401$	−20.2143	−1.00946	−0.504728	+	0.863279i	$0.668407\pi$
$402$	3.32509	0.165840
$403$	−6.19822	−0.308755
$404$	10.3232	0.513600
$405$	35.2964	1.75389
$406$	1.65102	0.0819388
$407$	−38.4251	−1.90466
$408$	−1.54368	−0.0764237
$409$	8.72284	0.431317	0.215658	−	0.976469i	$-0.430810\pi$
$409$	8.72284	0.431317	0.215658	+	0.976469i	$0.430810\pi$
$410$	26.6027	1.31381
$411$	1.76378	0.0870010
$412$	10.0535	0.495301
$413$	−0.108648	−0.00534624
$414$	5.57069	0.273785
$415$	−43.8866	−2.15431
$416$	−9.94523	−0.487605
$417$	−1.37048	−0.0671128
$418$	−5.86249	−0.286744
$419$	23.5492	1.15046	0.575228	−	0.817993i	$-0.304913\pi$
$419$	23.5492	1.15046	0.575228	+	0.817993i	$0.304913\pi$
$420$	0.0563121	0.00274775
$421$	−19.0853	−0.930159	−0.465079	−	0.885269i	$-0.653974\pi$
$421$	−19.0853	−0.930159	−0.465079	+	0.885269i	$0.653974\pi$
$422$	−1.67829	−0.0816980
$423$	−24.4641	−1.18948
$424$	26.4803	1.28600
$425$	−60.3735	−2.92854
$426$	0.799642	0.0387428
$427$	−0.761072	−0.0368309
$428$	10.1514	0.490687
$429$	−1.12940	−0.0545278
$430$	69.1282	3.33366
$431$	−31.1597	−1.50091	−0.750455	−	0.660921i	$-0.770166\pi$
$431$	−31.1597	−1.50091	−0.750455	+	0.660921i	$0.770166\pi$
$432$	4.21254	0.202676
$433$	−1.97700	−0.0950087	−0.0475044	−	0.998871i	$-0.515127\pi$
$433$	−1.97700	−0.0950087	−0.0475044	+	0.998871i	$0.515127\pi$
$434$	−0.554327	−0.0266085
$435$	−4.59768	−0.220442
$436$	−9.09914	−0.435770
$437$	1.11389	0.0532846
$438$	3.31265	0.158285
$439$	−6.24558	−0.298085	−0.149043	−	0.988831i	$-0.547619\pi$
$439$	−6.24558	−0.298085	−0.149043	+	0.988831i	$0.547619\pi$
$440$	27.7641	1.32360
$441$	20.8152	0.991200
$442$	−20.9633	−0.997125
$443$	−12.0958	−0.574689	−0.287345	−	0.957827i	$-0.592773\pi$
$443$	−12.0958	−0.574689	−0.287345	+	0.957827i	$0.592773\pi$
$444$	−1.27425	−0.0604733
$445$	−50.3235	−2.38556
$446$	17.8182	0.843716
$447$	−1.33410	−0.0631010
$448$	0.317054	0.0149794
$449$	11.5162	0.543482	0.271741	−	0.962370i	$-0.412401\pi$
$449$	11.5162	0.543482	0.271741	+	0.962370i	$0.412401\pi$
$450$	55.0986	2.59737
$451$	−13.8350	−0.651463
$452$	−13.4879	−0.634417
$453$	−0.176600	−0.00829737
$454$	8.12779	0.381456
$455$	−1.10807	−0.0519472
$456$	0.281700	0.0131918
$457$	−12.2047	−0.570913	−0.285457	−	0.958392i	$-0.592145\pi$
$457$	−12.2047	−0.570913	−0.285457	+	0.958392i	$0.592145\pi$
$458$	−4.79118	−0.223877
$459$	4.64810	0.216955
$460$	3.64067	0.169747
$461$	−15.4120	−0.717809	−0.358905	−	0.933374i	$-0.616850\pi$
$461$	−15.4120	−0.717809	−0.358905	+	0.933374i	$0.616850\pi$
$462$	−0.101006	−0.00469920
$463$	13.8610	0.644177	0.322089	−	0.946710i	$-0.395615\pi$
$463$	13.8610	0.644177	0.322089	+	0.946710i	$0.395615\pi$
$464$	40.2229	1.86730
$465$	1.54366	0.0715855
$466$	−1.28627	−0.0595852
$467$	−24.1597	−1.11798	−0.558989	−	0.829175i	$-0.688811\pi$
$467$	−24.1597	−1.11798	−0.558989	+	0.829175i	$0.688811\pi$
$468$	5.54707	0.256413
$469$	−1.69659	−0.0783412
$470$	−55.1432	−2.54357
$471$	2.55580	0.117765
$472$	−1.77642	−0.0817665
$473$	−35.9507	−1.65302
$474$	2.93339	0.134735
$475$	11.0173	0.505507
$476$	−0.543586	−0.0249152
$477$	−39.7325	−1.81923
$478$	−41.9406	−1.91832
$479$	−38.9079	−1.77775	−0.888873	−	0.458154i	$-0.848511\pi$
$479$	−38.9079	−1.77775	−0.888873	+	0.458154i	$0.848511\pi$
$480$	2.47685	0.113052
$481$	25.0739	1.14327
$482$	−10.9522	−0.498859
$483$	0.0191914	0.000873237 0
$484$	0.981582	0.0446174
$485$	27.2162	1.23583
$486$	−6.37012	−0.288955
$487$	13.2847	0.601989	0.300994	−	0.953626i	$-0.402681\pi$
$487$	13.2847	0.601989	0.300994	+	0.953626i	$0.402681\pi$
$488$	−12.4437	−0.563299
$489$	1.15004	0.0520065
$490$	46.9185	2.11956
$491$	19.0604	0.860184	0.430092	−	0.902785i	$-0.358481\pi$
$491$	19.0604	0.860184	0.430092	+	0.902785i	$0.358481\pi$
$492$	−0.458795	−0.0206841
$493$	44.3818	1.99886
$494$	3.82550	0.172118
$495$	−41.6589	−1.87243
$496$	−13.5048	−0.606382
$497$	−0.408009	−0.0183017
$498$	2.61045	0.116977
$499$	10.7631	0.481824	0.240912	−	0.970547i	$-0.422554\pi$
$499$	10.7631	0.481824	0.240912	+	0.970547i	$0.422554\pi$
$500$	19.6670	0.879535
$501$	−1.25810	−0.0562076
$502$	30.9675	1.38215
$503$	0.684425	0.0305170	0.0152585	−	0.999884i	$-0.495143\pi$
$503$	0.684425	0.0305170	0.0152585	+	0.999884i	$0.495143\pi$
$504$	−0.718830	−0.0320192
$505$	50.5902	2.25123
$506$	−6.53018	−0.290302
$507$	−1.10700	−0.0491634
$508$	5.77839	0.256374
$509$	7.06531	0.313164	0.156582	−	0.987665i	$-0.449952\pi$
$509$	7.06531	0.313164	0.156582	+	0.987665i	$0.449952\pi$
$510$	5.22089	0.231185
$511$	−1.69024	−0.0747720
$512$	−1.94247	−0.0858458
$513$	−0.848210	−0.0374494
$514$	9.85743	0.434793
$515$	49.2684	2.17103
$516$	−1.19220	−0.0524835
$517$	28.6777	1.26124
$518$	2.24244	0.0985270
$519$	1.90351	0.0835549
$520$	−18.1172	−0.794491
$521$	−19.2069	−0.841472	−0.420736	−	0.907183i	$-0.638228\pi$
$521$	−19.2069	−0.841472	−0.420736	+	0.907183i	$0.638228\pi$
$522$	−40.5041	−1.77282
$523$	−25.2189	−1.10275	−0.551373	−	0.834259i	$-0.685896\pi$
$523$	−25.2189	−1.10275	−0.551373	+	0.834259i	$0.685896\pi$
$524$	12.6639	0.553225
$525$	0.189818	0.00828434
$526$	13.8790	0.605155
$527$	−14.9011	−0.649102
$528$	−2.46074	−0.107090
$529$	−21.7592	−0.946054
$530$	−89.5589	−3.89019
$531$	2.66545	0.115671
$532$	0.0991964	0.00430071
$533$	9.02785	0.391040
$534$	2.99333	0.129534
$535$	49.7482	2.15080
$536$	−27.7396	−1.19817
$537$	0.865652	0.0373556
$538$	1.69758	0.0731879
$539$	−24.4004	−1.05100
$540$	−2.77231	−0.119301
$541$	−2.71375	−0.116673	−0.0583367	−	0.998297i	$-0.518580\pi$
$541$	−2.71375	−0.116673	−0.0583367	+	0.998297i	$0.518580\pi$
$542$	−24.9279	−1.07075
$543$	1.55234	0.0666171
$544$	−23.9092	−1.02510
$545$	−44.5914	−1.91008
$546$	0.0659100	0.00282069
$547$	−7.31606	−0.312812	−0.156406	−	0.987693i	$-0.549991\pi$
$547$	−7.31606	−0.312812	−0.156406	+	0.987693i	$0.549991\pi$
$548$	10.1550	0.433799
$549$	18.6712	0.796868
$550$	−64.5887	−2.75407
$551$	−8.09903	−0.345030
$552$	0.313783	0.0133555
$553$	−1.49673	−0.0636475
$554$	−39.5441	−1.68007
$555$	−6.24462	−0.265069
$556$	−7.89053	−0.334633
$557$	−0.0578598	−0.00245160	−0.00122580	−	0.999999i	$-0.500390\pi$
$557$	−0.0578598	−0.00245160	−0.00122580	+	0.999999i	$0.500390\pi$
$558$	13.5992	0.575699
$559$	23.4592	0.992220
$560$	−2.41428	−0.102022
$561$	−2.71517	−0.114635
$562$	−4.30889	−0.181760
$563$	20.4236	0.860754	0.430377	−	0.902649i	$-0.358381\pi$
$563$	20.4236	0.860754	0.430377	+	0.902649i	$0.358381\pi$
$564$	0.951009	0.0400447
$565$	−66.0989	−2.78080
$566$	12.6791	0.532944
$567$	1.07124	0.0449879
$568$	−6.67102	−0.279910
$569$	36.1043	1.51357	0.756785	−	0.653664i	$-0.226769\pi$
$569$	36.1043	1.51357	0.756785	+	0.653664i	$0.226769\pi$
$570$	−0.952737	−0.0399057
$571$	−36.5973	−1.53155	−0.765774	−	0.643110i	$-0.777644\pi$
$571$	−36.5973	−1.53155	−0.765774	+	0.643110i	$0.777644\pi$
$572$	−6.50248	−0.271883
$573$	3.07944	0.128645
$574$	0.807390	0.0336998
$575$	12.2721	0.511780
$576$	−7.77821	−0.324092
$577$	15.4395	0.642756	0.321378	−	0.946951i	$-0.395854\pi$
$577$	15.4395	0.642756	0.321378	+	0.946951i	$0.395854\pi$
$578$	−21.8668	−0.909541
$579$	−0.859538	−0.0357212
$580$	−26.4711	−1.09915
$581$	−1.33196	−0.0552588
$582$	−1.61887	−0.0671043
$583$	46.5759	1.92898
$584$	−27.6358	−1.14358
$585$	27.1840	1.12392
$586$	23.0106	0.950559
$587$	−35.0082	−1.44494	−0.722472	−	0.691400i	$-0.756994\pi$
$587$	−35.0082	−1.44494	−0.722472	+	0.691400i	$0.756994\pi$
$588$	−0.809164	−0.0333694
$589$	2.71923	0.112044
$590$	6.00805	0.247347
$591$	1.55253	0.0638624
$592$	54.6313	2.24533
$593$	27.5626	1.13186	0.565930	−	0.824454i	$-0.308517\pi$
$593$	27.5626	1.13186	0.565930	+	0.824454i	$0.308517\pi$
$594$	4.97262	0.204029
$595$	−2.66390	−0.109209
$596$	−7.68109	−0.314630
$597$	−2.77518	−0.113580
$598$	4.26119	0.174253
$599$	3.60134	0.147147	0.0735735	−	0.997290i	$-0.476560\pi$
$599$	3.60134	0.147147	0.0735735	+	0.997290i	$0.476560\pi$
$600$	3.10356	0.126702
$601$	12.4464	0.507698	0.253849	−	0.967244i	$-0.418303\pi$
$601$	12.4464	0.507698	0.253849	+	0.967244i	$0.418303\pi$
$602$	2.09804	0.0855096
$603$	41.6221	1.69498
$604$	−1.01677	−0.0413718
$605$	4.81036	0.195569
$606$	−3.00919	−0.122240
$607$	11.4182	0.463449	0.231724	−	0.972782i	$-0.425563\pi$
$607$	11.4182	0.463449	0.231724	+	0.972782i	$0.425563\pi$
$608$	4.36309	0.176947
$609$	−0.139539	−0.00565441
$610$	42.0858	1.70400
$611$	−18.7133	−0.757060
$612$	13.3357	0.539062
$613$	14.5363	0.587115	0.293558	−	0.955941i	$-0.405161\pi$
$613$	14.5363	0.587115	0.293558	+	0.955941i	$0.405161\pi$
$614$	43.0057	1.73557
$615$	−2.24837	−0.0906632
$616$	0.842640	0.0339509
$617$	28.3931	1.14306	0.571532	−	0.820580i	$-0.306349\pi$
$617$	28.3931	1.14306	0.571532	+	0.820580i	$0.306349\pi$
$618$	−2.93057	−0.117885
$619$	−39.0767	−1.57063	−0.785313	−	0.619099i	$-0.787498\pi$
$619$	−39.0767	−1.57063	−0.785313	+	0.619099i	$0.787498\pi$
$620$	8.88760	0.356935
$621$	−0.944814	−0.0379141
$622$	8.35396	0.334963
$623$	−1.52731	−0.0611905
$624$	1.60573	0.0642806
$625$	41.2940	1.65176
$626$	23.8765	0.954297
$627$	0.495479	0.0197875
$628$	14.7150	0.587191
$629$	60.2799	2.40352
$630$	2.43116	0.0968596
$631$	34.0909	1.35714	0.678568	−	0.734538i	$-0.262601\pi$
$631$	34.0909	1.35714	0.678568	+	0.734538i	$0.262601\pi$
$632$	−24.4719	−0.973438
$633$	0.141844	0.00563779
$634$	−18.9764	−0.753650
$635$	28.3176	1.12375
$636$	1.54455	0.0612454
$637$	15.9222	0.630860
$638$	47.4805	1.87977
$639$	10.0096	0.395973
$640$	−52.4560	−2.07350
$641$	−20.8502	−0.823535	−0.411768	−	0.911289i	$-0.635088\pi$
$641$	−20.8502	−0.823535	−0.411768	+	0.911289i	$0.635088\pi$
$642$	−2.95911	−0.116787
$643$	−11.7027	−0.461510	−0.230755	−	0.973012i	$-0.574120\pi$
$643$	−11.7027	−0.461510	−0.230755	+	0.973012i	$0.574120\pi$
$644$	0.110494	0.00435408
$645$	−5.84250	−0.230048
$646$	9.19686	0.361846
$647$	−14.8474	−0.583712	−0.291856	−	0.956462i	$-0.594273\pi$
$647$	−14.8474	−0.583712	−0.291856	+	0.956462i	$0.594273\pi$
$648$	17.5150	0.688055
$649$	−3.12454	−0.122649
$650$	42.1466	1.65313
$651$	0.0468500	0.00183620
$652$	6.62133	0.259311
$653$	−23.3761	−0.914776	−0.457388	−	0.889267i	$-0.651215\pi$
$653$	−23.3761	−0.914776	−0.457388	+	0.889267i	$0.651215\pi$
$654$	2.65238	0.103716
$655$	62.0608	2.42492
$656$	19.6700	0.767984
$657$	41.4664	1.61776
$658$	−1.67359	−0.0652434
$659$	6.35818	0.247680	0.123840	−	0.992302i	$-0.460479\pi$
$659$	6.35818	0.247680	0.123840	+	0.992302i	$0.460479\pi$
$660$	1.61944	0.0630364
$661$	34.6787	1.34885	0.674423	−	0.738346i	$-0.264393\pi$
$661$	34.6787	1.34885	0.674423	+	0.738346i	$0.264393\pi$
$662$	−15.1879	−0.590293
$663$	1.77176	0.0688093
$664$	−21.7777	−0.845140
$665$	0.486123	0.0188511
$666$	−55.0132	−2.13172
$667$	−9.02144	−0.349312
$668$	−7.24347	−0.280258
$669$	−1.50594	−0.0582229
$670$	93.8181	3.62451
$671$	−21.8871	−0.844942
$672$	0.0751721	0.00289983
$673$	20.4526	0.788391	0.394195	−	0.919027i	$-0.371023\pi$
$673$	20.4526	0.788391	0.394195	+	0.919027i	$0.371023\pi$
$674$	42.1026	1.62173
$675$	−9.34497	−0.359688
$676$	−6.37352	−0.245135
$677$	−40.5289	−1.55765	−0.778826	−	0.627240i	$-0.784185\pi$
$677$	−40.5289	−1.55765	−0.778826	+	0.627240i	$0.784185\pi$
$678$	3.93168	0.150995
$679$	0.826011	0.0316994
$680$	−43.5554	−1.67027
$681$	−0.686935	−0.0263234
$682$	−15.9415	−0.610430
$683$	−22.3564	−0.855445	−0.427723	−	0.903910i	$-0.640684\pi$
$683$	−22.3564	−0.855445	−0.427723	+	0.903910i	$0.640684\pi$
$684$	−2.43356	−0.0930496
$685$	49.7656	1.90144
$686$	2.85095	0.108850
$687$	0.404935	0.0154493
$688$	51.1133	1.94868
$689$	−30.3926	−1.15787
$690$	−1.06125	−0.0404009
$691$	2.78868	0.106087	0.0530433	−	0.998592i	$-0.483108\pi$
$691$	2.78868	0.106087	0.0530433	+	0.998592i	$0.483108\pi$
$692$	10.9595	0.416616
$693$	−1.26434	−0.0480285
$694$	−6.91989	−0.262675
$695$	−38.6684	−1.46678
$696$	−2.28149	−0.0864798
$697$	21.7038	0.822090
$698$	23.6399	0.894784
$699$	0.108711	0.00411184
$700$	1.09287	0.0413068
$701$	20.1494	0.761031	0.380515	−	0.924775i	$-0.375747\pi$
$701$	20.1494	0.761031	0.380515	+	0.924775i	$0.375747\pi$
$702$	−3.24483	−0.122468
$703$	−11.0002	−0.414880
$704$	9.11791	0.343644
$705$	4.66053	0.175526
$706$	−7.70371	−0.289933
$707$	1.53541	0.0577450
$708$	−0.103616	−0.00389412
$709$	−35.3617	−1.32803	−0.664017	−	0.747717i	$-0.731150\pi$
$709$	−35.3617	−1.32803	−0.664017	+	0.747717i	$0.731150\pi$
$710$	22.5621	0.846740
$711$	36.7190	1.37707
$712$	−24.9719	−0.935861
$713$	3.02893	0.113434
$714$	0.158454	0.00592999
$715$	−31.8662	−1.19173
$716$	4.98398	0.186260
$717$	3.54469	0.132379
$718$	5.14155	0.191881
$719$	−28.7691	−1.07291	−0.536453	−	0.843930i	$-0.680236\pi$
$719$	−28.7691	−1.07291	−0.536453	+	0.843930i	$0.680236\pi$
$720$	59.2289	2.20733
$721$	1.49529	0.0556876
$722$	−1.67829	−0.0624596
$723$	0.925646	0.0344251
$724$	8.93756	0.332162
$725$	−89.2293	−3.31389
$726$	−0.286129	−0.0106192
$727$	7.36862	0.273287	0.136644	−	0.990620i	$-0.456369\pi$
$727$	7.36862	0.273287	0.136644	+	0.990620i	$0.456369\pi$
$728$	−0.549855	−0.0203790
$729$	−25.9196	−0.959985
$730$	93.4672	3.45938
$731$	56.3982	2.08596
$732$	−0.725819	−0.0268271
$733$	41.4515	1.53105	0.765523	−	0.643409i	$-0.222480\pi$
$733$	41.4515	1.53105	0.765523	+	0.643409i	$0.222480\pi$
$734$	−63.2344	−2.33402
$735$	−3.96540	−0.146266
$736$	4.86000	0.179142
$737$	−48.7909	−1.79724
$738$	−19.8075	−0.729125
$739$	39.3253	1.44660	0.723302	−	0.690532i	$-0.242623\pi$
$739$	39.3253	1.44660	0.723302	+	0.690532i	$0.242623\pi$
$740$	−35.9533	−1.32167
$741$	−0.323319	−0.0118774
$742$	−2.71811	−0.0997849
$743$	28.2532	1.03651	0.518255	−	0.855226i	$-0.326582\pi$
$743$	28.2532	1.03651	0.518255	+	0.855226i	$0.326582\pi$
$744$	0.766006	0.0280832
$745$	−37.6421	−1.37910
$746$	25.2010	0.922675
$747$	32.6766	1.19557
$748$	−15.6326	−0.571583
$749$	1.50985	0.0551688
$750$	−5.73288	−0.209335
$751$	−35.8783	−1.30922	−0.654610	−	0.755967i	$-0.727167\pi$
$751$	−35.8783	−1.30922	−0.654610	+	0.755967i	$0.727167\pi$
$752$	−40.7728	−1.48683
$753$	−2.61728	−0.0953788
$754$	−30.9829	−1.12833
$755$	−4.98280	−0.181343
$756$	−0.0841394	−0.00306012
$757$	42.0920	1.52986	0.764929	−	0.644114i	$-0.222774\pi$
$757$	42.0920	1.52986	0.764929	+	0.644114i	$0.222774\pi$
$758$	−16.5298	−0.600390
$759$	0.551910	0.0200331
$760$	7.94822	0.288312
$761$	12.3086	0.446186	0.223093	−	0.974797i	$-0.428385\pi$
$761$	12.3086	0.446186	0.223093	+	0.974797i	$0.428385\pi$
$762$	−1.68438	−0.0610188
$763$	−1.35335	−0.0489944
$764$	17.7298	0.641442
$765$	65.3530	2.36284
$766$	24.6125	0.889284
$767$	2.03888	0.0736198
$768$	2.37968	0.0858694
$769$	−21.2484	−0.766236	−0.383118	−	0.923699i	$-0.625150\pi$
$769$	−21.2484	−0.766236	−0.383118	+	0.923699i	$0.625150\pi$
$770$	−2.84989	−0.102703
$771$	−0.833119	−0.0300040
$772$	−4.94878	−0.178110
$773$	−44.2228	−1.59058	−0.795292	−	0.606226i	$-0.792683\pi$
$773$	−44.2228	−1.59058	−0.795292	+	0.606226i	$0.792683\pi$
$774$	−51.4706	−1.85007
$775$	29.9585	1.07614
$776$	13.5054	0.484817
$777$	−0.189524	−0.00679913
$778$	19.9915	0.716731
$779$	−3.96062	−0.141904
$780$	−1.05674	−0.0378375
$781$	−11.7336	−0.419862
$782$	10.2443	0.366336
$783$	6.86968	0.245502
$784$	34.6914	1.23898
$785$	72.1123	2.57380
$786$	−3.69149	−0.131671
$787$	−31.6454	−1.12804	−0.564018	−	0.825762i	$-0.690745\pi$
$787$	−31.6454	−1.12804	−0.564018	+	0.825762i	$0.690745\pi$
$788$	8.93865	0.318426
$789$	−1.17301	−0.0417604
$790$	82.7663	2.94469
$791$	−2.00610	−0.0713286
$792$	−20.6723	−0.734558
$793$	14.2822	0.507175
$794$	−17.7736	−0.630763
$795$	7.56924	0.268453
$796$	−15.9780	−0.566327
$797$	54.0861	1.91583	0.957914	−	0.287056i	$-0.0926765\pi$
$797$	54.0861	1.91583	0.957914	+	0.287056i	$0.0926765\pi$
$798$	−0.0289155	−0.00102360
$799$	−44.9885	−1.59158
$800$	48.0693	1.69951
$801$	37.4692	1.32391
$802$	33.9256	1.19795
$803$	−48.6084	−1.71535
$804$	−1.61800	−0.0570626
$805$	0.541489	0.0190850
$806$	10.4024	0.366410
$807$	−0.143474	−0.00505053
$808$	25.1042	0.883164
$809$	−13.0101	−0.457411	−0.228705	−	0.973496i	$-0.573449\pi$
$809$	−13.0101	−0.457411	−0.228705	+	0.973496i	$0.573449\pi$
$810$	−59.2376	−2.08140
$811$	−32.1148	−1.12770	−0.563851	−	0.825876i	$-0.690681\pi$
$811$	−32.1148	−1.12770	−0.563851	+	0.825876i	$0.690681\pi$
$812$	−0.803395	−0.0281936
$813$	2.10683	0.0738898
$814$	64.4885	2.26032
$815$	32.4486	1.13662
$816$	3.86032	0.135138
$817$	−10.2918	−0.360066
$818$	−14.6395	−0.511857
$819$	0.825034	0.0288290
$820$	−12.9450	−0.452059
$821$	−51.1909	−1.78657	−0.893287	−	0.449487i	$-0.851607\pi$
$821$	−51.1909	−1.78657	−0.893287	+	0.449487i	$0.851607\pi$
$822$	−2.96015	−0.103247
$823$	14.6150	0.509446	0.254723	−	0.967014i	$-0.418016\pi$
$823$	14.6150	0.509446	0.254723	+	0.967014i	$0.418016\pi$
$824$	24.4484	0.851699
$825$	5.45883	0.190052
$826$	0.182344	0.00634455
$827$	−19.1212	−0.664909	−0.332454	−	0.943119i	$-0.607877\pi$
$827$	−19.1212	−0.664909	−0.332454	+	0.943119i	$0.607877\pi$
$828$	−2.71073	−0.0942042
$829$	−9.54673	−0.331572	−0.165786	−	0.986162i	$-0.553016\pi$
$829$	−9.54673	−0.331572	−0.165786	+	0.986162i	$0.553016\pi$
$830$	73.6545	2.55659
$831$	3.34214	0.115938
$832$	−5.94979	−0.206272
$833$	38.2784	1.32627
$834$	2.30007	0.0796449
$835$	−35.4975	−1.22844
$836$	2.85271	0.0986632
$837$	−2.30648	−0.0797236
$838$	−39.5225	−1.36528
$839$	−25.6323	−0.884926	−0.442463	−	0.896787i	$-0.645895\pi$
$839$	−25.6323	−0.884926	−0.442463	+	0.896787i	$0.645895\pi$
$840$	0.136941	0.00472491
$841$	36.5943	1.26187
$842$	32.0306	1.10385
$843$	0.364174	0.0125428
$844$	0.816665	0.0281108
$845$	−31.2342	−1.07449
$846$	41.0579	1.41160
$847$	0.145994	0.00501641
$848$	−66.2198	−2.27400
$849$	−1.07160	−0.0367772
$850$	101.324	3.47540
$851$	−12.2530	−0.420028
$852$	−0.389110	−0.0133307
$853$	32.8884	1.12608	0.563038	−	0.826431i	$-0.309632\pi$
$853$	32.8884	1.12608	0.563038	+	0.826431i	$0.309632\pi$
$854$	1.27730	0.0437084
$855$	−11.9260	−0.407859
$856$	24.6864	0.843764
$857$	15.4235	0.526858	0.263429	−	0.964679i	$-0.415146\pi$
$857$	15.4235	0.526858	0.263429	+	0.964679i	$0.415146\pi$
$858$	1.89546	0.0647098
$859$	8.88213	0.303054	0.151527	−	0.988453i	$-0.451581\pi$
$859$	8.88213	0.303054	0.151527	+	0.988453i	$0.451581\pi$
$860$	−33.6381	−1.14705
$861$	−0.0682380	−0.00232555
$862$	52.2951	1.78118
$863$	50.8421	1.73068	0.865342	−	0.501182i	$-0.167101\pi$
$863$	50.8421	1.73068	0.865342	+	0.501182i	$0.167101\pi$
$864$	−3.70081	−0.125904
$865$	53.7080	1.82613
$866$	3.31799	0.112750
$867$	1.84812	0.0627653
$868$	0.269738	0.00915551
$869$	−43.0433	−1.46015
$870$	7.71624	0.261605
$871$	31.8380	1.07879
$872$	−22.1275	−0.749331
$873$	−20.2643	−0.685844
$874$	−1.86944	−0.0632346
$875$	2.92514	0.0988878
$876$	−1.61195	−0.0544628
$877$	−13.7915	−0.465706	−0.232853	−	0.972512i	$-0.574806\pi$
$877$	−13.7915	−0.465706	−0.232853	+	0.972512i	$0.574806\pi$
$878$	10.4819	0.353747
$879$	−1.94478	−0.0655959
$880$	−69.4304	−2.34050
$881$	41.1822	1.38746	0.693732	−	0.720233i	$-0.255965\pi$
$881$	41.1822	1.38746	0.693732	+	0.720233i	$0.255965\pi$
$882$	−34.9340	−1.17629
$883$	9.22607	0.310482	0.155241	−	0.987877i	$-0.450385\pi$
$883$	9.22607	0.310482	0.155241	+	0.987877i	$0.450385\pi$
$884$	10.2009	0.343092
$885$	−0.507781	−0.0170689
$886$	20.3003	0.682002
$887$	−35.6635	−1.19746	−0.598732	−	0.800949i	$-0.704329\pi$
$887$	−35.6635	−1.19746	−0.598732	+	0.800949i	$0.704329\pi$
$888$	−3.09875	−0.103987
$889$	0.859439	0.0288246
$890$	84.4575	2.83102
$891$	30.8070	1.03207
$892$	−8.67042	−0.290307
$893$	8.20975	0.274729
$894$	2.23902	0.0748840
$895$	24.4246	0.816423
$896$	−1.59204	−0.0531862
$897$	−0.360143	−0.0120248
$898$	−19.3275	−0.644967
$899$	−22.0231	−0.734513
$900$	−26.8112	−0.893708
$901$	−73.0666	−2.43420
$902$	23.2191	0.773112
$903$	−0.177319	−0.00590082
$904$	−32.8001	−1.09092
$905$	43.7995	1.45595
$906$	0.296386	0.00984676
$907$	9.97860	0.331334	0.165667	−	0.986182i	$-0.447022\pi$
$907$	9.97860	0.331334	0.165667	+	0.986182i	$0.447022\pi$
$908$	−3.95502	−0.131252
$909$	−37.6678	−1.24936
$910$	1.85967	0.0616473
$911$	−16.5912	−0.549690	−0.274845	−	0.961489i	$-0.588627\pi$
$911$	−16.5912	−0.549690	−0.274845	+	0.961489i	$0.588627\pi$
$912$	−0.704452	−0.0233267
$913$	−38.3047	−1.26770
$914$	20.4831	0.677521
$915$	−3.55696	−0.117589
$916$	2.33141	0.0770320
$917$	1.88354	0.0622000
$918$	−7.80087	−0.257467
$919$	59.4125	1.95984	0.979919	−	0.199394i	$-0.0638975\pi$
$919$	59.4125	1.95984	0.979919	+	0.199394i	$0.0638975\pi$
$920$	8.85345	0.291890
$921$	−3.63470	−0.119768
$922$	25.8659	0.851847
$923$	7.65664	0.252021
$924$	0.0491498	0.00161691
$925$	−121.192	−3.98478
$926$	−23.2629	−0.764466
$927$	−36.6837	−1.20485
$928$	−35.3368	−1.15999
$929$	−3.64695	−0.119653	−0.0598263	−	0.998209i	$-0.519055\pi$
$929$	−3.64695	−0.119653	−0.0598263	+	0.998209i	$0.519055\pi$
$930$	−2.59071	−0.0849528
$931$	−6.98525	−0.228932
$932$	0.625904	0.0205022
$933$	−0.706050	−0.0231151
$934$	40.5471	1.32674
$935$	−76.6092	−2.50539
$936$	13.4895	0.440917
$937$	30.1543	0.985097	0.492549	−	0.870285i	$-0.336065\pi$
$937$	30.1543	0.985097	0.492549	+	0.870285i	$0.336065\pi$
$938$	2.84737	0.0929700
$939$	−2.01797	−0.0658538
$940$	26.8329	0.875194
$941$	−28.5393	−0.930354	−0.465177	−	0.885218i	$-0.654009\pi$
$941$	−28.5393	−0.930354	−0.465177	+	0.885218i	$0.654009\pi$
$942$	−4.28937	−0.139755
$943$	−4.41170	−0.143665
$944$	4.44234	0.144586
$945$	−0.412335	−0.0134132
$946$	60.3358	1.96169
$947$	10.2751	0.333894	0.166947	−	0.985966i	$-0.446609\pi$
$947$	10.2751	0.333894	0.166947	+	0.985966i	$0.446609\pi$
$948$	−1.42740	−0.0463599
$949$	31.7189	1.02964
$950$	−18.4902	−0.599902
$951$	1.60383	0.0520077
$952$	−1.32190	−0.0428431
$953$	42.0649	1.36262	0.681308	−	0.731997i	$-0.261412\pi$
$953$	42.0649	1.36262	0.681308	+	0.731997i	$0.261412\pi$
$954$	66.6827	2.15893
$955$	86.8870	2.81160
$956$	20.4085	0.660058
$957$	−4.01290	−0.129719
$958$	65.2988	2.10971
$959$	1.51038	0.0487728
$960$	1.48179	0.0478245
$961$	−23.6058	−0.761477
$962$	−42.0813	−1.35676
$963$	−37.0409	−1.19363
$964$	5.32939	0.171648
$965$	−24.2520	−0.780701
$966$	−0.0322087	−0.00103630
$967$	37.6600	1.21106	0.605531	−	0.795821i	$-0.292961\pi$
$967$	37.6600	1.21106	0.605531	+	0.795821i	$0.292961\pi$
$968$	2.38703	0.0767221
$969$	−0.777290	−0.0249701
$970$	−45.6768	−1.46659
$971$	−25.5646	−0.820407	−0.410203	−	0.911994i	$-0.634542\pi$
$971$	−25.5646	−0.820407	−0.410203	+	0.911994i	$0.634542\pi$
$972$	3.09973	0.0994240
$973$	−1.17358	−0.0376234
$974$	−22.2957	−0.714399
$975$	−3.56210	−0.114079
$976$	31.1182	0.996069
$977$	−31.5286	−1.00869	−0.504344	−	0.863503i	$-0.668266\pi$
$977$	−31.5286	−1.00869	−0.504344	+	0.863503i	$0.668266\pi$
$978$	−1.93010	−0.0617178
$979$	−43.9229	−1.40378
$980$	−22.8307	−0.729301
$981$	33.2013	1.06004
$982$	−31.9889	−1.02081
$983$	19.3168	0.616111	0.308056	−	0.951368i	$-0.400322\pi$
$983$	19.3168	0.616111	0.308056	+	0.951368i	$0.400322\pi$
$984$	−1.11571	−0.0355674
$985$	43.8049	1.39574
$986$	−74.4857	−2.37211
$987$	0.141447	0.00450230
$988$	−1.86151	−0.0592225
$989$	−11.4640	−0.364534
$990$	69.9158	2.22207
$991$	6.92511	0.219983	0.109992	−	0.993933i	$-0.464918\pi$
$991$	6.92511	0.219983	0.109992	+	0.993933i	$0.464918\pi$
$992$	11.8642	0.376690
$993$	1.28363	0.0407348
$994$	0.684758	0.0217192
$995$	−78.3022	−2.48235
$996$	−1.27026	−0.0402497
$997$	45.1725	1.43063	0.715314	−	0.698803i	$-0.246284\pi$
$997$	45.1725	1.43063	0.715314	+	0.698803i	$0.246284\pi$
$998$	−18.0637	−0.571796
$999$	9.33048	0.295203

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.21	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.21	✓	71	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)

\(f(q)\)	\(=\)	\(q-1.67829 q^{2} +0.141844 q^{3} +0.816665 q^{4} +4.00216 q^{5} -0.238056 q^{6} +0.121465 q^{7} +1.98598 q^{8} -2.97988 q^{9} +O(q^{10})\)	\(q-1.67829 q^{2} +0.141844 q^{3} +0.816665 q^{4} +4.00216 q^{5} -0.238056 q^{6} +0.121465 q^{7} +1.98598 q^{8} -2.97988 q^{9} -6.71679 q^{10} +3.49313 q^{11} +0.115839 q^{12} -2.27940 q^{13} -0.203854 q^{14} +0.567682 q^{15} -4.96639 q^{16} -5.47989 q^{17} +5.00111 q^{18} +1.00000 q^{19} +3.26842 q^{20} +0.0172291 q^{21} -5.86249 q^{22} +1.11389 q^{23} +0.281700 q^{24} +11.0173 q^{25} +3.82550 q^{26} -0.848210 q^{27} +0.0991964 q^{28} -8.09903 q^{29} -0.952737 q^{30} +2.71923 q^{31} +4.36309 q^{32} +0.495479 q^{33} +9.19686 q^{34} +0.486123 q^{35} -2.43356 q^{36} -11.0002 q^{37} -1.67829 q^{38} -0.323319 q^{39} +7.94822 q^{40} -3.96062 q^{41} -0.0289155 q^{42} -10.2918 q^{43} +2.85271 q^{44} -11.9260 q^{45} -1.86944 q^{46} +8.20975 q^{47} -0.704452 q^{48} -6.98525 q^{49} -18.4902 q^{50} -0.777290 q^{51} -1.86151 q^{52} +13.3336 q^{53} +1.42354 q^{54} +13.9801 q^{55} +0.241228 q^{56} +0.141844 q^{57} +13.5925 q^{58} -0.894481 q^{59} +0.463606 q^{60} -6.26576 q^{61} -4.56367 q^{62} -0.361952 q^{63} +2.61024 q^{64} -9.12253 q^{65} -0.831559 q^{66} -13.9677 q^{67} -4.47524 q^{68} +0.157999 q^{69} -0.815857 q^{70} -3.35906 q^{71} -5.91799 q^{72} -13.9154 q^{73} +18.4615 q^{74} +1.56273 q^{75} +0.816665 q^{76} +0.424294 q^{77} +0.542625 q^{78} -12.3223 q^{79} -19.8763 q^{80} +8.81933 q^{81} +6.64708 q^{82} -10.9657 q^{83} +0.0140704 q^{84} -21.9314 q^{85} +17.2727 q^{86} -1.14880 q^{87} +6.93729 q^{88} -12.5741 q^{89} +20.0152 q^{90} -0.276868 q^{91} +0.909676 q^{92} +0.385707 q^{93} -13.7784 q^{94} +4.00216 q^{95} +0.618878 q^{96} +6.80039 q^{97} +11.7233 q^{98} -10.4091 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more