LMFDB - Embedded newform 667.4.a.a.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [667,4,Mod(1,667)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(667, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("667.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 667 = 23 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	667.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:	$39.3542739738$
Analytic rank:	$1$
Dimension:	$35$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	667.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.282966 q^{2} +8.03408 q^{3} -7.91993 q^{4} -4.50118 q^{5} +2.27337 q^{6} -16.5936 q^{7} -4.50481 q^{8} +37.5464 q^{9} +O(q^{10})$	\(q+0.282966 q^{2} +8.03408 q^{3} -7.91993 q^{4} -4.50118 q^{5} +2.27337 q^{6} -16.5936 q^{7} -4.50481 q^{8} +37.5464 q^{9} -1.27368 q^{10} +57.1558 q^{11} -63.6293 q^{12} -4.84285 q^{13} -4.69543 q^{14} -36.1628 q^{15} +62.0847 q^{16} -116.365 q^{17} +10.6244 q^{18} +27.0229 q^{19} +35.6491 q^{20} -133.314 q^{21} +16.1732 q^{22} +23.0000 q^{23} -36.1920 q^{24} -104.739 q^{25} -1.37036 q^{26} +84.7303 q^{27} +131.420 q^{28} -29.0000 q^{29} -10.2329 q^{30} -93.1191 q^{31} +53.6063 q^{32} +459.194 q^{33} -32.9274 q^{34} +74.6908 q^{35} -297.365 q^{36} -238.484 q^{37} +7.64658 q^{38} -38.9078 q^{39} +20.2770 q^{40} -263.109 q^{41} -37.7234 q^{42} +65.6750 q^{43} -452.670 q^{44} -169.003 q^{45} +6.50823 q^{46} -382.822 q^{47} +498.793 q^{48} -67.6526 q^{49} -29.6377 q^{50} -934.885 q^{51} +38.3550 q^{52} -19.3549 q^{53} +23.9758 q^{54} -257.269 q^{55} +74.7509 q^{56} +217.104 q^{57} -8.20603 q^{58} -220.081 q^{59} +286.407 q^{60} -438.419 q^{61} -26.3496 q^{62} -623.029 q^{63} -481.509 q^{64} +21.7986 q^{65} +129.936 q^{66} -1058.87 q^{67} +921.603 q^{68} +184.784 q^{69} +21.1350 q^{70} +340.106 q^{71} -169.139 q^{72} +1160.97 q^{73} -67.4830 q^{74} -841.484 q^{75} -214.020 q^{76} -948.420 q^{77} -11.0096 q^{78} -1161.14 q^{79} -279.455 q^{80} -333.022 q^{81} -74.4509 q^{82} +531.322 q^{83} +1055.84 q^{84} +523.780 q^{85} +18.5838 q^{86} -232.988 q^{87} -257.476 q^{88} +112.596 q^{89} -47.8222 q^{90} +80.3603 q^{91} -182.158 q^{92} -748.126 q^{93} -108.326 q^{94} -121.635 q^{95} +430.677 q^{96} +1881.37 q^{97} -19.1434 q^{98} +2145.99 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 35 q - 6 q^{2} - 22 q^{3} + 116 q^{4} - 80 q^{5} - 52 q^{6} - 38 q^{7} + 12 q^{8} + 231 q^{9}+O(q^{10}) $ 35 * q - 6 * q^2 - 22 * q^3 + 116 * q^4 - 80 * q^5 - 52 * q^6 - 38 * q^7 + 12 * q^8 + 231 * q^9	\( 35 q - 6 q^{2} - 22 q^{3} + 116 q^{4} - 80 q^{5} - 52 q^{6} - 38 q^{7} + 12 q^{8} + 231 q^{9} - 52 q^{10} - 126 q^{11} - 173 q^{12} - 252 q^{13} + 112 q^{14} - 32 q^{15} + 312 q^{16} - 332 q^{17} - 225 q^{18} - 2 q^{19} - 747 q^{20} - 202 q^{21} - 127 q^{22} + 805 q^{23} - 494 q^{24} + 315 q^{25} - 677 q^{26} - 694 q^{27} - 529 q^{28} - 1015 q^{29} + 389 q^{30} - 652 q^{31} + 320 q^{32} - 290 q^{33} - 455 q^{34} - 940 q^{35} + 34 q^{36} - 528 q^{37} - 1218 q^{38} - 268 q^{39} - 806 q^{40} - 68 q^{41} - 1484 q^{42} - 162 q^{43} - 1817 q^{44} - 356 q^{45} - 138 q^{46} - 1200 q^{47} - 2153 q^{48} + 93 q^{49} - 1369 q^{50} - 270 q^{51} - 3134 q^{52} - 1892 q^{53} - 1221 q^{54} - 794 q^{55} + 191 q^{56} - 1764 q^{57} + 174 q^{58} - 1354 q^{59} + 159 q^{60} - 1274 q^{61} - 5413 q^{62} - 2904 q^{63} - 926 q^{64} - 548 q^{65} - 2477 q^{66} - 3212 q^{67} - 3901 q^{68} - 506 q^{69} - 2768 q^{70} - 2342 q^{71} - 2381 q^{72} + 916 q^{73} + 661 q^{74} - 4708 q^{75} - 2810 q^{76} - 5536 q^{77} - 2434 q^{78} + 2622 q^{79} - 5444 q^{80} + 607 q^{81} - 3687 q^{82} - 2702 q^{83} + 346 q^{84} - 3304 q^{85} - 5789 q^{86} + 638 q^{87} - 2252 q^{88} - 1620 q^{89} - 3933 q^{90} - 4016 q^{91} + 2668 q^{92} - 4942 q^{93} - 1413 q^{94} - 4528 q^{95} - 7920 q^{96} + 682 q^{97} + 152 q^{98} - 582 q^{99}+O(q^{100}) \) 35 * q - 6 * q^2 - 22 * q^3 + 116 * q^4 - 80 * q^5 - 52 * q^6 - 38 * q^7 + 12 * q^8 + 231 * q^9 - 52 * q^10 - 126 * q^11 - 173 * q^12 - 252 * q^13 + 112 * q^14 - 32 * q^15 + 312 * q^16 - 332 * q^17 - 225 * q^18 - 2 * q^19 - 747 * q^20 - 202 * q^21 - 127 * q^22 + 805 * q^23 - 494 * q^24 + 315 * q^25 - 677 * q^26 - 694 * q^27 - 529 * q^28 - 1015 * q^29 + 389 * q^30 - 652 * q^31 + 320 * q^32 - 290 * q^33 - 455 * q^34 - 940 * q^35 + 34 * q^36 - 528 * q^37 - 1218 * q^38 - 268 * q^39 - 806 * q^40 - 68 * q^41 - 1484 * q^42 - 162 * q^43 - 1817 * q^44 - 356 * q^45 - 138 * q^46 - 1200 * q^47 - 2153 * q^48 + 93 * q^49 - 1369 * q^50 - 270 * q^51 - 3134 * q^52 - 1892 * q^53 - 1221 * q^54 - 794 * q^55 + 191 * q^56 - 1764 * q^57 + 174 * q^58 - 1354 * q^59 + 159 * q^60 - 1274 * q^61 - 5413 * q^62 - 2904 * q^63 - 926 * q^64 - 548 * q^65 - 2477 * q^66 - 3212 * q^67 - 3901 * q^68 - 506 * q^69 - 2768 * q^70 - 2342 * q^71 - 2381 * q^72 + 916 * q^73 + 661 * q^74 - 4708 * q^75 - 2810 * q^76 - 5536 * q^77 - 2434 * q^78 + 2622 * q^79 - 5444 * q^80 + 607 * q^81 - 3687 * q^82 - 2702 * q^83 + 346 * q^84 - 3304 * q^85 - 5789 * q^86 + 638 * q^87 - 2252 * q^88 - 1620 * q^89 - 3933 * q^90 - 4016 * q^91 + 2668 * q^92 - 4942 * q^93 - 1413 * q^94 - 4528 * q^95 - 7920 * q^96 + 682 * q^97 + 152 * q^98 - 582 * 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 $

$2$	0.282966	0.100044	0.0500219	−	0.998748i	$-0.484071\pi$
$2$	0.282966	0.100044	0.0500219	+	0.998748i	$0.484071\pi$
$3$	8.03408	1.54616	0.773079	−	0.634309i	$-0.218715\pi$
$3$	8.03408	1.54616	0.773079	+	0.634309i	$0.218715\pi$
$4$	−7.91993	−0.989991
$5$	−4.50118	−0.402598	−0.201299	−	0.979530i	$-0.564516\pi$
$5$	−4.50118	−0.402598	−0.201299	+	0.979530i	$0.564516\pi$
$6$	2.27337	0.154684
$7$	−16.5936	−0.895970	−0.447985	−	0.894041i	$-0.647858\pi$
$7$	−16.5936	−0.895970	−0.447985	+	0.894041i	$0.647858\pi$
$8$	−4.50481	−0.199086
$9$	37.5464	1.39061
$10$	−1.27368	−0.0402774
$11$	57.1558	1.56665	0.783323	−	0.621614i	$-0.213523\pi$
$11$	57.1558	1.56665	0.783323	+	0.621614i	$0.213523\pi$
$12$	−63.6293	−1.53068
$13$	−4.84285	−0.103320	−0.0516602	−	0.998665i	$-0.516451\pi$
$13$	−4.84285	−0.103320	−0.0516602	+	0.998665i	$0.516451\pi$
$14$	−4.69543	−0.0896362
$15$	−36.1628	−0.622481
$16$	62.0847	0.970074
$17$	−116.365	−1.66016	−0.830078	−	0.557647i	$-0.811704\pi$
$17$	−116.365	−1.66016	−0.830078	+	0.557647i	$0.811704\pi$
$18$	10.6244	0.139121
$19$	27.0229	0.326289	0.163144	−	0.986602i	$-0.447836\pi$
$19$	27.0229	0.326289	0.163144	+	0.986602i	$0.447836\pi$
$20$	35.6491	0.398569
$21$	−133.314	−1.38531
$22$	16.1732	0.156733
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−36.1920	−0.307819
$25$	−104.739	−0.837915
$26$	−1.37036	−0.0103366
$27$	84.7303	0.603939
$28$	131.420	0.887002
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	−10.2329	−0.0622753
$31$	−93.1191	−0.539506	−0.269753	−	0.962930i	$-0.586942\pi$
$31$	−93.1191	−0.539506	−0.269753	+	0.962930i	$0.586942\pi$
$32$	53.6063	0.296136
$33$	459.194	2.42228
$34$	−32.9274	−0.166088
$35$	74.6908	0.360716
$36$	−297.365	−1.37669
$37$	−238.484	−1.05964	−0.529818	−	0.848111i	$-0.677740\pi$
$37$	−238.484	−1.05964	−0.529818	+	0.848111i	$0.677740\pi$
$38$	7.64658	0.0326431
$39$	−38.9078	−0.159750
$40$	20.2770	0.0801517
$41$	−263.109	−1.00221	−0.501106	−	0.865386i	$-0.667073\pi$
$41$	−263.109	−1.00221	−0.501106	+	0.865386i	$0.667073\pi$
$42$	−37.7234	−0.138592
$43$	65.6750	0.232915	0.116458	−	0.993196i	$-0.462846\pi$
$43$	65.6750	0.232915	0.116458	+	0.993196i	$0.462846\pi$
$44$	−452.670	−1.55097
$45$	−169.003	−0.559855
$46$	6.50823	0.0208606
$47$	−382.822	−1.18809	−0.594046	−	0.804431i	$-0.702470\pi$
$47$	−382.822	−1.18809	−0.594046	+	0.804431i	$0.702470\pi$
$48$	498.793	1.49989
$49$	−67.6526	−0.197238
$50$	−29.6377	−0.0838281
$51$	−934.885	−2.56687
$52$	38.3550	0.102286
$53$	−19.3549	−0.0501622	−0.0250811	−	0.999685i	$-0.507984\pi$
$53$	−19.3549	−0.0501622	−0.0250811	+	0.999685i	$0.507984\pi$
$54$	23.9758	0.0604203
$55$	−257.269	−0.630729
$56$	74.7509	0.178375
$57$	217.104	0.504494
$58$	−8.20603	−0.0185777
$59$	−220.081	−0.485629	−0.242815	−	0.970073i	$-0.578071\pi$
$59$	−220.081	−0.485629	−0.242815	+	0.970073i	$0.578071\pi$
$60$	286.407	0.616250
$61$	−438.419	−0.920226	−0.460113	−	0.887860i	$-0.652191\pi$
$61$	−438.419	−0.920226	−0.460113	+	0.887860i	$0.652191\pi$
$62$	−26.3496	−0.0539742
$63$	−623.029	−1.24594
$64$	−481.509	−0.940447
$65$	21.7986	0.0415966
$66$	129.936	0.242334
$67$	−1058.87	−1.93077	−0.965387	−	0.260822i	$-0.916007\pi$
$67$	−1058.87	−1.93077	−0.965387	+	0.260822i	$0.916007\pi$
$68$	921.603	1.64354
$69$	184.784	0.322396
$70$	21.1350	0.0360874
$71$	340.106	0.568496	0.284248	−	0.958751i	$-0.408256\pi$
$71$	340.106	0.568496	0.284248	+	0.958751i	$0.408256\pi$
$72$	−169.139	−0.276851
$73$	1160.97	1.86138	0.930692	−	0.365803i	$-0.119206\pi$
$73$	1160.97	1.86138	0.930692	+	0.365803i	$0.119206\pi$
$74$	−67.4830	−0.106010
$75$	−841.484	−1.29555
$76$	−214.020	−0.323023
$77$	−948.420	−1.40367
$78$	−11.0096	−0.0159820
$79$	−1161.14	−1.65365	−0.826824	−	0.562461i	$-0.809855\pi$
$79$	−1161.14	−1.65365	−0.826824	+	0.562461i	$0.809855\pi$
$80$	−279.455	−0.390550
$81$	−333.022	−0.456821
$82$	−74.4509	−0.100265
$83$	531.322	0.702653	0.351326	−	0.936253i	$-0.385731\pi$
$83$	531.322	0.702653	0.351326	+	0.936253i	$0.385731\pi$
$84$	1055.84	1.37145
$85$	523.780	0.668376
$86$	18.5838	0.0233017
$87$	−232.988	−0.287114
$88$	−257.476	−0.311898
$89$	112.596	0.134103	0.0670516	−	0.997750i	$-0.478641\pi$
$89$	112.596	0.134103	0.0670516	+	0.997750i	$0.478641\pi$
$90$	−47.8222	−0.0560100
$91$	80.3603	0.0925719
$92$	−182.158	−0.206427
$93$	−748.126	−0.834161
$94$	−108.326	−0.118861
$95$	−121.635	−0.131363
$96$	430.677	0.457873
$97$	1881.37	1.96932	0.984662	−	0.174470i	$-0.0558212\pi$
$97$	1881.37	1.96932	0.984662	+	0.174470i	$0.0558212\pi$
$98$	−19.1434	−0.0197324
$99$	2145.99	2.17859
$100$	829.528	0.829528
$101$	154.516	0.152227	0.0761135	−	0.997099i	$-0.475749\pi$
$101$	154.516	0.152227	0.0761135	+	0.997099i	$0.475749\pi$
$102$	−264.541	−0.256799
$103$	164.095	0.156979	0.0784893	−	0.996915i	$-0.474990\pi$
$103$	164.095	0.156979	0.0784893	+	0.996915i	$0.474990\pi$
$104$	21.8161	0.0205697
$105$	600.072	0.557724
$106$	−5.47678	−0.00501841
$107$	1054.53	0.952762	0.476381	−	0.879239i	$-0.341948\pi$
$107$	1054.53	0.952762	0.476381	+	0.879239i	$0.341948\pi$
$108$	−671.058	−0.597895
$109$	−1557.25	−1.36841	−0.684207	−	0.729288i	$-0.739852\pi$
$109$	−1557.25	−1.36841	−0.684207	+	0.729288i	$0.739852\pi$
$110$	−72.7984	−0.0631005
$111$	−1916.00	−1.63837
$112$	−1030.21	−0.869157
$113$	1513.64	1.26010	0.630049	−	0.776555i	$-0.283035\pi$
$113$	1513.64	1.26010	0.630049	+	0.776555i	$0.283035\pi$
$114$	61.4332	0.0504715
$115$	−103.527	−0.0839475
$116$	229.678	0.183837
$117$	−181.831	−0.143678
$118$	−62.2756	−0.0485842
$119$	1930.91	1.48745
$120$	162.907	0.123927
$121$	1935.78	1.45438
$122$	−124.058	−0.0920629
$123$	−2113.83	−1.54958
$124$	737.496	0.534106
$125$	1034.10	0.739941
$126$	−176.296	−0.124649
$127$	−1951.20	−1.36332	−0.681658	−	0.731671i	$-0.738741\pi$
$127$	−1951.20	−1.36332	−0.681658	+	0.731671i	$0.738741\pi$
$128$	−565.102	−0.390222
$129$	527.638	0.360124
$130$	6.16826	0.00416148
$131$	1303.46	0.869344	0.434672	−	0.900589i	$-0.356864\pi$
$131$	1303.46	0.869344	0.434672	+	0.900589i	$0.356864\pi$
$132$	−3636.78	−2.39804
$133$	−448.407	−0.292345
$134$	−299.625	−0.193162
$135$	−381.387	−0.243145
$136$	524.202	0.330514
$137$	−595.339	−0.371265	−0.185632	−	0.982619i	$-0.559433\pi$
$137$	−595.339	−0.371265	−0.185632	+	0.982619i	$0.559433\pi$
$138$	52.2876	0.0322537
$139$	210.878	0.128679	0.0643396	−	0.997928i	$-0.479506\pi$
$139$	210.878	0.128679	0.0643396	+	0.997928i	$0.479506\pi$
$140$	−591.546	−0.357105
$141$	−3075.62	−1.83698
$142$	96.2387	0.0568744
$143$	−276.797	−0.161867
$144$	2331.06	1.34899
$145$	130.534	0.0747606
$146$	328.515	0.186220
$147$	−543.526	−0.304961
$148$	1888.78	1.04903
$149$	−1784.84	−0.981338	−0.490669	−	0.871346i	$-0.663248\pi$
$149$	−1784.84	−0.981338	−0.490669	+	0.871346i	$0.663248\pi$
$150$	−238.112	−0.129612
$151$	1859.49	1.00214	0.501070	−	0.865407i	$-0.332940\pi$
$151$	1859.49	1.00214	0.501070	+	0.865407i	$0.332940\pi$
$152$	−121.733	−0.0649596
$153$	−4369.08	−2.30862
$154$	−268.371	−0.140428
$155$	419.146	0.217204
$156$	308.147	0.158151
$157$	2260.89	1.14929	0.574645	−	0.818403i	$-0.305140\pi$
$157$	2260.89	1.14929	0.574645	+	0.818403i	$0.305140\pi$
$158$	−328.563	−0.165437
$159$	−155.498	−0.0775587
$160$	−241.292	−0.119224
$161$	−381.653	−0.186823
$162$	−94.2341	−0.0457020
$163$	−2219.99	−1.06677	−0.533383	−	0.845874i	$-0.679080\pi$
$163$	−2219.99	−1.06677	−0.533383	+	0.845874i	$0.679080\pi$
$164$	2083.80	0.992181
$165$	−2066.92	−0.975207
$166$	150.346	0.0702960
$167$	−1344.00	−0.622764	−0.311382	−	0.950285i	$-0.600792\pi$
$167$	−1344.00	−0.622764	−0.311382	+	0.950285i	$0.600792\pi$
$168$	600.555	0.275796
$169$	−2173.55	−0.989325
$170$	148.212	0.0668668
$171$	1014.61	0.453739
$172$	−520.142	−0.230584
$173$	2375.12	1.04380	0.521899	−	0.853007i	$-0.325224\pi$
$173$	2375.12	1.04380	0.521899	+	0.853007i	$0.325224\pi$
$174$	−65.9278	−0.0287240
$175$	1738.00	0.750746
$176$	3548.50	1.51976
$177$	−1768.15	−0.750860
$178$	31.8610	0.0134162
$179$	2279.18	0.951699	0.475849	−	0.879527i	$-0.342141\pi$
$179$	2279.18	0.951699	0.475849	+	0.879527i	$0.342141\pi$
$180$	1338.49	0.554252
$181$	−1574.68	−0.646657	−0.323328	−	0.946287i	$-0.604802\pi$
$181$	−1574.68	−0.646657	−0.323328	+	0.946287i	$0.604802\pi$
$182$	22.7393	0.00926124
$183$	−3522.29	−1.42282
$184$	−103.611	−0.0415123
$185$	1073.46	0.426608
$186$	−211.694	−0.0834526
$187$	−6650.93	−2.60088
$188$	3031.93	1.17620
$189$	−1405.98	−0.541111
$190$	−34.4187	−0.0131421
$191$	2955.85	1.11978	0.559889	−	0.828568i	$-0.310844\pi$
$191$	2955.85	1.11978	0.559889	+	0.828568i	$0.310844\pi$
$192$	−3868.48	−1.45408
$193$	−3762.09	−1.40311	−0.701557	−	0.712613i	$-0.747511\pi$
$193$	−3762.09	−1.40311	−0.701557	+	0.712613i	$0.747511\pi$
$194$	532.366	0.197019
$195$	175.131	0.0643149
$196$	535.804	0.195264
$197$	2023.95	0.731981	0.365991	−	0.930619i	$-0.380730\pi$
$197$	2023.95	0.731981	0.365991	+	0.930619i	$0.380730\pi$
$198$	607.244	0.217954
$199$	2037.89	0.725941	0.362970	−	0.931801i	$-0.381763\pi$
$199$	2037.89	0.725941	0.362970	+	0.931801i	$0.381763\pi$
$200$	471.830	0.166817
$201$	−8507.06	−2.98528
$202$	43.7229	0.0152294
$203$	481.214	0.166377
$204$	7404.23	2.54117
$205$	1184.30	0.403488
$206$	46.4335	0.0157047
$207$	863.566	0.289961
$208$	−300.667	−0.100228
$209$	1544.52	0.511179
$210$	169.800	0.0557968
$211$	−1254.09	−0.409171	−0.204586	−	0.978849i	$-0.565585\pi$
$211$	−1254.09	−0.409171	−0.204586	+	0.978849i	$0.565585\pi$
$212$	153.289	0.0496601
$213$	2732.44	0.878984
$214$	298.397	0.0953179
$215$	−295.615	−0.0937712
$216$	−381.694	−0.120236
$217$	1545.18	0.483381
$218$	−440.649	−0.136901
$219$	9327.31	2.87800
$220$	2037.55	0.624416
$221$	563.538	0.171528
$222$	−542.164	−0.163908
$223$	−4996.49	−1.50040	−0.750201	−	0.661210i	$-0.770043\pi$
$223$	−4996.49	−1.50040	−0.750201	+	0.661210i	$0.770043\pi$
$224$	−889.522	−0.265329
$225$	−3932.58	−1.16521
$226$	428.309	0.126065
$227$	−2927.81	−0.856060	−0.428030	−	0.903764i	$-0.640792\pi$
$227$	−2927.81	−0.856060	−0.428030	+	0.903764i	$0.640792\pi$
$228$	−1719.45	−0.499445
$229$	5169.41	1.49172	0.745861	−	0.666102i	$-0.232038\pi$
$229$	5169.41	1.49172	0.745861	+	0.666102i	$0.232038\pi$
$230$	−29.2947	−0.00839842
$231$	−7619.68	−2.17029
$232$	130.639	0.0369694
$233$	−6143.10	−1.72724	−0.863622	−	0.504139i	$-0.831810\pi$
$233$	−6143.10	−1.72724	−0.863622	+	0.504139i	$0.831810\pi$
$234$	−51.4522	−0.0143741
$235$	1723.15	0.478324
$236$	1743.03	0.480769
$237$	−9328.67	−2.55680
$238$	546.384	0.148810
$239$	−6070.25	−1.64289	−0.821447	−	0.570285i	$-0.806833\pi$
$239$	−6070.25	−1.64289	−0.821447	+	0.570285i	$0.806833\pi$
$240$	−2245.16	−0.603852
$241$	5266.04	1.40753	0.703767	−	0.710431i	$-0.251500\pi$
$241$	5266.04	1.40753	0.703767	+	0.710431i	$0.251500\pi$
$242$	547.762	0.145502
$243$	−4963.24	−1.31026
$244$	3472.25	0.911016
$245$	304.517	0.0794076
$246$	−598.144	−0.155026
$247$	−130.868	−0.0337123
$248$	419.483	0.107408
$249$	4268.68	1.08641
$250$	292.615	0.0740265
$251$	−590.652	−0.148532	−0.0742662	−	0.997238i	$-0.523661\pi$
$251$	−590.652	−0.148532	−0.0742662	+	0.997238i	$0.523661\pi$
$252$	4934.35	1.23347
$253$	1314.58	0.326668
$254$	−552.125	−0.136391
$255$	4208.09	1.03342
$256$	3692.17	0.901408
$257$	−431.995	−0.104852	−0.0524262	−	0.998625i	$-0.516695\pi$
$257$	−431.995	−0.104852	−0.0524262	+	0.998625i	$0.516695\pi$
$258$	149.304	0.0360281
$259$	3957.31	0.949402
$260$	−172.643	−0.0411803
$261$	−1088.84	−0.258229
$262$	368.836	0.0869724
$263$	4742.46	1.11191	0.555955	−	0.831212i	$-0.312352\pi$
$263$	4742.46	1.11191	0.555955	+	0.831212i	$0.312352\pi$
$264$	−2068.58	−0.482243
$265$	87.1198	0.0201952
$266$	−126.884	−0.0292473
$267$	904.607	0.207345
$268$	8386.20	1.91145
$269$	−3334.46	−0.755784	−0.377892	−	0.925850i	$-0.623351\pi$
$269$	−3334.46	−0.755784	−0.377892	+	0.925850i	$0.623351\pi$
$270$	−107.920	−0.0243251
$271$	−4423.93	−0.991642	−0.495821	−	0.868425i	$-0.665133\pi$
$271$	−4423.93	−0.991642	−0.495821	+	0.868425i	$0.665133\pi$
$272$	−7224.49	−1.61047
$273$	645.621	0.143131
$274$	−168.461	−0.0371427
$275$	−5986.46	−1.31272
$276$	−1463.47	−0.319170
$277$	−1249.20	−0.270965	−0.135482	−	0.990780i	$-0.543258\pi$
$277$	−1249.20	−0.270965	−0.135482	+	0.990780i	$0.543258\pi$
$278$	59.6713	0.0128736
$279$	−3496.28	−0.750240
$280$	−336.468	−0.0718135
$281$	327.056	0.0694325	0.0347162	−	0.999397i	$-0.488947\pi$
$281$	327.056	0.0694325	0.0347162	+	0.999397i	$0.488947\pi$
$282$	−870.298	−0.183778
$283$	−423.975	−0.0890555	−0.0445277	−	0.999008i	$-0.514178\pi$
$283$	−423.975	−0.0890555	−0.0445277	+	0.999008i	$0.514178\pi$
$284$	−2693.62	−0.562806
$285$	−977.226	−0.203108
$286$	−78.3242	−0.0161937
$287$	4365.92	0.897951
$288$	2012.72	0.411809
$289$	8627.82	1.75612
$290$	36.9368	0.00747933
$291$	15115.1	3.04489
$292$	−9194.79	−1.84275
$293$	−4369.22	−0.871170	−0.435585	−	0.900148i	$-0.643458\pi$
$293$	−4369.22	−0.871170	−0.435585	+	0.900148i	$0.643458\pi$
$294$	−153.800	−0.0305095
$295$	990.625	0.195513
$296$	1074.32	0.210959
$297$	4842.83	0.946160
$298$	−505.048	−0.0981768
$299$	−111.386	−0.0215438
$300$	6664.49	1.28258
$301$	−1089.78	−0.208685
$302$	526.173	0.100258
$303$	1241.39	0.235367
$304$	1677.71	0.316524
$305$	1973.40	0.370481
$306$	−1236.30	−0.230963
$307$	6815.36	1.26701	0.633507	−	0.773737i	$-0.281615\pi$
$307$	6815.36	1.26701	0.633507	+	0.773737i	$0.281615\pi$
$308$	7511.42	1.38962
$309$	1318.35	0.242714
$310$	118.604	0.0217299
$311$	2750.89	0.501571	0.250785	−	0.968043i	$-0.419311\pi$
$311$	2750.89	0.501571	0.250785	+	0.968043i	$0.419311\pi$
$312$	175.272	0.0318040
$313$	−3926.70	−0.709106	−0.354553	−	0.935036i	$-0.615367\pi$
$313$	−3926.70	−0.709106	−0.354553	+	0.935036i	$0.615367\pi$
$314$	639.756	0.114979
$315$	2804.37	0.501614
$316$	9196.13	1.63710
$317$	2266.16	0.401515	0.200757	−	0.979641i	$-0.435660\pi$
$317$	2266.16	0.401515	0.200757	+	0.979641i	$0.435660\pi$
$318$	−44.0009	−0.00775926
$319$	−1657.52	−0.290919
$320$	2167.36	0.378622
$321$	8472.20	1.47312
$322$	−107.995	−0.0186904
$323$	−3144.52	−0.541690
$324$	2637.51	0.452248
$325$	507.237	0.0865737
$326$	−628.182	−0.106723
$327$	−12511.0	−2.11579
$328$	1185.25	0.199526
$329$	6352.40	1.06450
$330$	−584.868	−0.0975634
$331$	9490.15	1.57591	0.787954	−	0.615734i	$-0.211140\pi$
$331$	9490.15	1.57591	0.787954	+	0.615734i	$0.211140\pi$
$332$	−4208.03	−0.695620
$333$	−8954.21	−1.47354
$334$	−380.306	−0.0623037
$335$	4766.18	0.777326
$336$	−8276.78	−1.34385
$337$	9014.68	1.45715	0.728577	−	0.684964i	$-0.240182\pi$
$337$	9014.68	1.45715	0.728577	+	0.684964i	$0.240182\pi$
$338$	−615.041	−0.0989758
$339$	12160.7	1.94831
$340$	−4148.30	−0.661686
$341$	−5322.29	−0.845215
$342$	287.101	0.0453938
$343$	6814.20	1.07269
$344$	−295.853	−0.0463702
$345$	−831.746	−0.129796
$346$	672.080	0.104426
$347$	7345.84	1.13644	0.568221	−	0.822876i	$-0.307632\pi$
$347$	7345.84	1.13644	0.568221	+	0.822876i	$0.307632\pi$
$348$	1845.25	0.284241
$349$	6674.75	1.02376	0.511878	−	0.859058i	$-0.328950\pi$
$349$	6674.75	1.02376	0.511878	+	0.859058i	$0.328950\pi$
$350$	491.796	0.0751075
$351$	−410.336	−0.0623992
$352$	3063.91	0.463941
$353$	11767.0	1.77421	0.887104	−	0.461570i	$-0.152714\pi$
$353$	11767.0	1.77421	0.887104	+	0.461570i	$0.152714\pi$
$354$	−500.327	−0.0751188
$355$	−1530.88	−0.228875
$356$	−891.755	−0.132761
$357$	15513.1	2.29983
$358$	644.932	0.0952115
$359$	3753.43	0.551806	0.275903	−	0.961186i	$-0.411023\pi$
$359$	3753.43	0.551806	0.275903	+	0.961186i	$0.411023\pi$
$360$	761.326	0.111459
$361$	−6128.76	−0.893536
$362$	−445.581	−0.0646940
$363$	15552.2	2.24871
$364$	−636.448	−0.0916454
$365$	−5225.73	−0.749390
$366$	−996.690	−0.142344
$367$	−4964.33	−0.706092	−0.353046	−	0.935606i	$-0.614854\pi$
$367$	−4964.33	−0.706092	−0.353046	+	0.935606i	$0.614854\pi$
$368$	1427.95	0.202274
$369$	−9878.78	−1.39368
$370$	303.753	0.0426794
$371$	321.167	0.0449438
$372$	5925.10	0.825812
$373$	13738.8	1.90716	0.953579	−	0.301142i	$-0.0973679\pi$
$373$	13738.8	1.90716	0.953579	+	0.301142i	$0.0973679\pi$
$374$	−1881.99	−0.260202
$375$	8308.03	1.14407
$376$	1724.54	0.236533
$377$	140.443	0.0191861
$378$	−397.845	−0.0541348
$379$	−4641.28	−0.629041	−0.314520	−	0.949251i	$-0.601844\pi$
$379$	−4641.28	−0.629041	−0.314520	+	0.949251i	$0.601844\pi$
$380$	963.342	0.130048
$381$	−15676.1	−2.10790
$382$	836.405	0.112027
$383$	−10325.6	−1.37758	−0.688791	−	0.724960i	$-0.741858\pi$
$383$	−10325.6	−1.37758	−0.688791	+	0.724960i	$0.741858\pi$
$384$	−4540.07	−0.603345
$385$	4269.01	0.565114
$386$	−1064.54	−0.140373
$387$	2465.86	0.323893
$388$	−14900.3	−1.94961
$389$	5833.43	0.760326	0.380163	−	0.924920i	$-0.375868\pi$
$389$	5833.43	0.760326	0.380163	+	0.924920i	$0.375868\pi$
$390$	49.5563	0.00643431
$391$	−2676.40	−0.346167
$392$	304.762	0.0392674
$393$	10472.1	1.34414
$394$	572.709	0.0732302
$395$	5226.49	0.665756
$396$	−16996.1	−2.15678
$397$	−2681.72	−0.339021	−0.169511	−	0.985528i	$-0.554219\pi$
$397$	−2681.72	−0.339021	−0.169511	+	0.985528i	$0.554219\pi$
$398$	576.655	0.0726258
$399$	−3602.54	−0.452011
$400$	−6502.71	−0.812839
$401$	1412.79	0.175939	0.0879693	−	0.996123i	$-0.471962\pi$
$401$	1412.79	0.175939	0.0879693	+	0.996123i	$0.471962\pi$
$402$	−2407.21	−0.298659
$403$	450.962	0.0557419
$404$	−1223.76	−0.150703
$405$	1498.99	0.183915
$406$	136.167	0.0166450
$407$	−13630.7	−1.66008
$408$	4211.48	0.511027
$409$	−14921.4	−1.80395	−0.901974	−	0.431791i	$-0.857882\pi$
$409$	−14921.4	−1.80395	−0.901974	+	0.431791i	$0.857882\pi$
$410$	335.117	0.0403665
$411$	−4783.00	−0.574034
$412$	−1299.62	−0.155407
$413$	3651.94	0.435109
$414$	244.360	0.0290088
$415$	−2391.58	−0.282887
$416$	−259.607	−0.0305969
$417$	1694.21	0.198959
$418$	437.046	0.0511403
$419$	−3839.28	−0.447640	−0.223820	−	0.974630i	$-0.571853\pi$
$419$	−3839.28	−0.447640	−0.223820	+	0.974630i	$0.571853\pi$
$420$	−4752.53	−0.552142
$421$	2351.11	0.272176	0.136088	−	0.990697i	$-0.456547\pi$
$421$	2351.11	0.272176	0.136088	+	0.990697i	$0.456547\pi$
$422$	−354.865	−0.0409350
$423$	−14373.6	−1.65217
$424$	87.1899	0.00998660
$425$	12188.0	1.39107
$426$	773.189	0.0879369
$427$	7274.94	0.824495
$428$	−8351.83	−0.943226
$429$	−2223.81	−0.250271
$430$	−83.6492	−0.00938122
$431$	−11922.2	−1.33241	−0.666207	−	0.745767i	$-0.732083\pi$
$431$	−11922.2	−1.33241	−0.666207	+	0.745767i	$0.732083\pi$
$432$	5260.46	0.585866
$433$	−8346.24	−0.926315	−0.463158	−	0.886276i	$-0.653284\pi$
$433$	−8346.24	−0.926315	−0.463158	+	0.886276i	$0.653284\pi$
$434$	437.234	0.0483592
$435$	1048.72	0.115592
$436$	12333.3	1.35472
$437$	621.527	0.0680359
$438$	2639.32	0.287926
$439$	−2856.64	−0.310570	−0.155285	−	0.987870i	$-0.549630\pi$
$439$	−2856.64	−0.310570	−0.155285	+	0.987870i	$0.549630\pi$
$440$	1158.95	0.125569
$441$	−2540.11	−0.274280
$442$	159.462	0.0171603
$443$	−1370.17	−0.146950	−0.0734749	−	0.997297i	$-0.523409\pi$
$443$	−1370.17	−0.146950	−0.0734749	+	0.997297i	$0.523409\pi$
$444$	15174.6	1.62197
$445$	−506.817	−0.0539897
$446$	−1413.84	−0.150106
$447$	−14339.5	−1.51730
$448$	7989.97	0.842613
$449$	442.101	0.0464678	0.0232339	−	0.999730i	$-0.492604\pi$
$449$	442.101	0.0464678	0.0232339	+	0.999730i	$0.492604\pi$
$450$	−1112.79	−0.116572
$451$	−15038.2	−1.57011
$452$	−11987.9	−1.24749
$453$	14939.3	1.54947
$454$	−828.472	−0.0856435
$455$	−361.716	−0.0372693
$456$	−978.012	−0.100438
$457$	−7403.01	−0.757764	−0.378882	−	0.925445i	$-0.623691\pi$
$457$	−7403.01	−0.757764	−0.378882	+	0.925445i	$0.623691\pi$
$458$	1462.77	0.149237
$459$	−9859.65	−1.00263
$460$	819.928	0.0831073
$461$	8547.71	0.863571	0.431786	−	0.901976i	$-0.357884\pi$
$461$	8547.71	0.863571	0.431786	+	0.901976i	$0.357884\pi$
$462$	−2156.11	−0.217124
$463$	−12869.2	−1.29176	−0.645878	−	0.763441i	$-0.723508\pi$
$463$	−12869.2	−1.29176	−0.645878	+	0.763441i	$0.723508\pi$
$464$	−1800.46	−0.180138
$465$	3367.45	0.335832
$466$	−1738.29	−0.172800
$467$	−3636.97	−0.360383	−0.180192	−	0.983632i	$-0.557672\pi$
$467$	−3636.97	−0.360383	−0.180192	+	0.983632i	$0.557672\pi$
$468$	1440.09	0.142240
$469$	17570.5	1.72992
$470$	487.595	0.0478533
$471$	18164.1	1.77698
$472$	991.423	0.0966821
$473$	3753.71	0.364896
$474$	−2639.70	−0.255792
$475$	−2830.36	−0.273402
$476$	−15292.7	−1.47256
$477$	−726.705	−0.0697558
$478$	−1717.68	−0.164361
$479$	1515.29	0.144541	0.0722706	−	0.997385i	$-0.476975\pi$
$479$	1515.29	0.144541	0.0722706	+	0.997385i	$0.476975\pi$
$480$	−1938.56	−0.184339
$481$	1154.94	0.109482
$482$	1490.11	0.140815
$483$	−3066.23	−0.288857
$484$	−15331.3	−1.43983
$485$	−8468.41	−0.792847
$486$	−1404.43	−0.131083
$487$	−8460.93	−0.787271	−0.393636	−	0.919267i	$-0.628783\pi$
$487$	−8460.93	−0.787271	−0.393636	+	0.919267i	$0.628783\pi$
$488$	1974.99	0.183204
$489$	−17835.6	−1.64939
$490$	86.1681	0.00794424
$491$	−13560.4	−1.24638	−0.623189	−	0.782071i	$-0.714163\pi$
$491$	−13560.4	−1.24638	−0.623189	+	0.782071i	$0.714163\pi$
$492$	16741.4	1.53407
$493$	3374.59	0.308283
$494$	−37.0312	−0.00337270
$495$	−9659.50	−0.877096
$496$	−5781.27	−0.523360
$497$	−5643.59	−0.509355
$498$	1207.89	0.108689
$499$	3547.11	0.318217	0.159109	−	0.987261i	$-0.449138\pi$
$499$	3547.11	0.318217	0.159109	+	0.987261i	$0.449138\pi$
$500$	−8189.99	−0.732535
$501$	−10797.8	−0.962892
$502$	−167.135	−0.0148597
$503$	−11317.2	−1.00320	−0.501600	−	0.865100i	$-0.667255\pi$
$503$	−11317.2	−1.00320	−0.501600	+	0.865100i	$0.667255\pi$
$504$	2806.63	0.248050
$505$	−695.505	−0.0612863
$506$	371.983	0.0326811
$507$	−17462.4	−1.52965
$508$	15453.4	1.34967
$509$	−9327.87	−0.812280	−0.406140	−	0.913811i	$-0.633125\pi$
$509$	−9327.87	−0.812280	−0.406140	+	0.913811i	$0.633125\pi$
$510$	1190.75	0.103387
$511$	−19264.6	−1.66774
$512$	5565.57	0.480402
$513$	2289.66	0.197058
$514$	−122.240	−0.0104898
$515$	−738.624	−0.0631993
$516$	−4178.86	−0.356519
$517$	−21880.5	−1.86132
$518$	1119.79	0.0949818
$519$	19081.9	1.61388
$520$	−98.1983	−0.00828131
$521$	−810.620	−0.0681649	−0.0340824	−	0.999419i	$-0.510851\pi$
$521$	−810.620	−0.0681649	−0.0340824	+	0.999419i	$0.510851\pi$
$522$	−308.107	−0.0258342
$523$	14286.9	1.19450	0.597249	−	0.802056i	$-0.296260\pi$
$523$	14286.9	1.19450	0.597249	+	0.802056i	$0.296260\pi$
$524$	−10323.3	−0.860643
$525$	13963.2	1.16077
$526$	1341.96	0.111240
$527$	10835.8	0.895664
$528$	28508.9	2.34980
$529$	529.000	0.0434783
$530$	24.6520	0.00202040
$531$	−8263.25	−0.675319
$532$	3551.36	0.289419
$533$	1274.20	0.103549
$534$	255.974	0.0207436
$535$	−4746.65	−0.383580
$536$	4770.02	0.384390
$537$	18311.1	1.47148
$538$	−943.541	−0.0756114
$539$	−3866.74	−0.309002
$540$	3020.56	0.240711
$541$	−2169.41	−0.172403	−0.0862017	−	0.996278i	$-0.527473\pi$
$541$	−2169.41	−0.172403	−0.0862017	+	0.996278i	$0.527473\pi$
$542$	−1251.83	−0.0992075
$543$	−12651.1	−0.999834
$544$	−6237.90	−0.491632
$545$	7009.45	0.550921
$546$	182.689	0.0143194
$547$	11721.6	0.916232	0.458116	−	0.888892i	$-0.348524\pi$
$547$	11721.6	0.916232	0.458116	+	0.888892i	$0.348524\pi$
$548$	4715.04	0.367549
$549$	−16461.0	−1.27967
$550$	−1693.97	−0.131329
$551$	−783.665	−0.0605903
$552$	−832.415	−0.0641847
$553$	19267.4	1.48162
$554$	−353.482	−0.0271083
$555$	8624.27	0.659603
$556$	−1670.14	−0.127391
$557$	−1021.82	−0.0777306	−0.0388653	−	0.999244i	$-0.512374\pi$
$557$	−1021.82	−0.0777306	−0.0388653	+	0.999244i	$0.512374\pi$
$558$	−989.331	−0.0750568
$559$	−318.054	−0.0240649
$560$	4637.16	0.349921
$561$	−53434.1	−4.02137
$562$	92.5459	0.00694629
$563$	−20274.5	−1.51771	−0.758853	−	0.651262i	$-0.774240\pi$
$563$	−20274.5	−1.51771	−0.758853	+	0.651262i	$0.774240\pi$
$564$	24358.7	1.81859
$565$	−6813.16	−0.507313
$566$	−119.971	−0.00890944
$567$	5526.03	0.409297
$568$	−1532.11	−0.113180
$569$	23722.8	1.74782	0.873911	−	0.486087i	$-0.161576\pi$
$569$	23722.8	1.74782	0.873911	+	0.486087i	$0.161576\pi$
$570$	−276.522	−0.0203197
$571$	10534.6	0.772082	0.386041	−	0.922482i	$-0.373842\pi$
$571$	10534.6	0.772082	0.386041	+	0.922482i	$0.373842\pi$
$572$	2192.21	0.160246
$573$	23747.5	1.73135
$574$	1235.41	0.0898344
$575$	−2409.00	−0.174717
$576$	−18078.9	−1.30779
$577$	−11469.1	−0.827497	−0.413749	−	0.910391i	$-0.635781\pi$
$577$	−11469.1	−0.827497	−0.413749	+	0.910391i	$0.635781\pi$
$578$	2441.38	0.175689
$579$	−30224.9	−2.16944
$580$	−1033.82	−0.0740123
$581$	−8816.55	−0.629556
$582$	4277.07	0.304622
$583$	−1106.24	−0.0785864
$584$	−5229.94	−0.370576
$585$	818.457	0.0578445
$586$	−1236.34	−0.0871551
$587$	15639.4	1.09967	0.549837	−	0.835272i	$-0.314690\pi$
$587$	15639.4	1.09967	0.549837	+	0.835272i	$0.314690\pi$
$588$	4304.69	0.301909
$589$	−2516.35	−0.176035
$590$	280.314	0.0195599
$591$	16260.6	1.13176
$592$	−14806.2	−1.02793
$593$	−17826.2	−1.23446	−0.617231	−	0.786782i	$-0.711745\pi$
$593$	−17826.2	−1.23446	−0.617231	+	0.786782i	$0.711745\pi$
$594$	1370.36	0.0946574
$595$	−8691.40	−0.598845
$596$	14135.8	0.971516
$597$	16372.6	1.12242
$598$	−31.5184	−0.00215532
$599$	−27777.0	−1.89472	−0.947360	−	0.320172i	$-0.896259\pi$
$599$	−27777.0	−1.89472	−0.947360	+	0.320172i	$0.896259\pi$
$600$	3790.72	0.257926
$601$	11429.8	0.775760	0.387880	−	0.921710i	$-0.373207\pi$
$601$	11429.8	0.775760	0.387880	+	0.921710i	$0.373207\pi$
$602$	−308.373	−0.0208776
$603$	−39756.8	−2.68495
$604$	−14727.0	−0.992110
$605$	−8713.32	−0.585532
$606$	351.273	0.0235470
$607$	2871.14	0.191987	0.0959935	−	0.995382i	$-0.469397\pi$
$607$	2871.14	0.191987	0.0959935	+	0.995382i	$0.469397\pi$
$608$	1448.60	0.0966258
$609$	3866.11	0.257246
$610$	558.407	0.0370643
$611$	1853.95	0.122754
$612$	34602.8	2.28552
$613$	22152.4	1.45959	0.729795	−	0.683666i	$-0.239616\pi$
$613$	22152.4	1.45959	0.729795	+	0.683666i	$0.239616\pi$
$614$	1928.52	0.126757
$615$	9514.76	0.623857
$616$	4272.45	0.279451
$617$	25430.5	1.65931	0.829654	−	0.558278i	$-0.188538\pi$
$617$	25430.5	1.65931	0.829654	+	0.558278i	$0.188538\pi$
$618$	373.050	0.0242820
$619$	−20661.1	−1.34158	−0.670792	−	0.741645i	$-0.734046\pi$
$619$	−20661.1	−1.34158	−0.670792	+	0.741645i	$0.734046\pi$
$620$	−3319.61	−0.215030
$621$	1948.80	0.125930
$622$	778.409	0.0501790
$623$	−1868.38	−0.120152
$624$	−2415.58	−0.154969
$625$	8437.75	0.540016
$626$	−1111.12	−0.0709416
$627$	12408.8	0.790364
$628$	−17906.1	−1.13779
$629$	27751.2	1.75916
$630$	793.542	0.0501833
$631$	5832.89	0.367993	0.183997	−	0.982927i	$-0.441096\pi$
$631$	5832.89	0.367993	0.183997	+	0.982927i	$0.441096\pi$
$632$	5230.70	0.329218
$633$	−10075.5	−0.632643
$634$	641.247	0.0401691
$635$	8782.72	0.548869
$636$	1231.54	0.0767824
$637$	327.631	0.0203787
$638$	−469.022	−0.0291046
$639$	12769.8	0.790554
$640$	2543.63	0.157103
$641$	−4762.90	−0.293484	−0.146742	−	0.989175i	$-0.546879\pi$
$641$	−4762.90	−0.293484	−0.146742	+	0.989175i	$0.546879\pi$
$642$	2397.35	0.147377
$643$	23736.6	1.45580	0.727901	−	0.685682i	$-0.240496\pi$
$643$	23736.6	1.45580	0.727901	+	0.685682i	$0.240496\pi$
$644$	3022.66	0.184953
$645$	−2375.00	−0.144985
$646$	−889.795	−0.0541927
$647$	−22915.4	−1.39242	−0.696212	−	0.717836i	$-0.745133\pi$
$647$	−22915.4	−1.39242	−0.696212	+	0.717836i	$0.745133\pi$
$648$	1500.20	0.0909467
$649$	−12578.9	−0.760809
$650$	143.531	0.00866115
$651$	12414.1	0.747383
$652$	17582.2	1.05609
$653$	−15975.2	−0.957364	−0.478682	−	0.877988i	$-0.658885\pi$
$653$	−15975.2	−0.957364	−0.478682	+	0.877988i	$0.658885\pi$
$654$	−3540.20	−0.211671
$655$	−5867.12	−0.349996
$656$	−16335.0	−0.972219
$657$	43590.1	2.58845
$658$	1797.52	0.106496
$659$	19729.2	1.16622	0.583110	−	0.812393i	$-0.301835\pi$
$659$	19729.2	1.16622	0.583110	+	0.812393i	$0.301835\pi$
$660$	16369.8	0.965447
$661$	32725.5	1.92568	0.962839	−	0.270076i	$-0.0870490\pi$
$661$	32725.5	1.92568	0.962839	+	0.270076i	$0.0870490\pi$
$662$	2685.39	0.157660
$663$	4527.51	0.265209
$664$	−2393.50	−0.139888
$665$	2018.36	0.117697
$666$	−2533.74	−0.147418
$667$	−667.000	−0.0387202
$668$	10644.4	0.616531
$669$	−40142.2	−2.31986
$670$	1348.67	0.0777666
$671$	−25058.2	−1.44167
$672$	−7146.49	−0.410241
$673$	−8692.79	−0.497893	−0.248947	−	0.968517i	$-0.580084\pi$
$673$	−8692.79	−0.497893	−0.248947	+	0.968517i	$0.580084\pi$
$674$	2550.85	0.145779
$675$	−8874.60	−0.506050
$676$	17214.3	0.979423
$677$	−2999.81	−0.170298	−0.0851491	−	0.996368i	$-0.527137\pi$
$677$	−2999.81	−0.170298	−0.0851491	+	0.996368i	$0.527137\pi$
$678$	3441.07	0.194916
$679$	−31218.7	−1.76446
$680$	−2359.53	−0.133064
$681$	−23522.3	−1.32360
$682$	−1506.03	−0.0845585
$683$	6698.30	0.375261	0.187631	−	0.982240i	$-0.439919\pi$
$683$	6698.30	0.375261	0.187631	+	0.982240i	$0.439919\pi$
$684$	−8035.66	−0.449198
$685$	2679.73	0.149470
$686$	1928.19	0.107316
$687$	41531.4	2.30644
$688$	4077.42	0.225945
$689$	93.7327	0.00518278
$690$	−235.356	−0.0129853
$691$	1882.16	0.103619	0.0518095	−	0.998657i	$-0.483501\pi$
$691$	1882.16	0.103619	0.0518095	+	0.998657i	$0.483501\pi$
$692$	−18810.8	−1.03335
$693$	−35609.7	−1.95195
$694$	2078.63	0.113694
$695$	−949.200	−0.0518060
$696$	1049.57	0.0571605
$697$	30616.6	1.66383
$698$	1888.73	0.102420
$699$	−49354.1	−2.67059
$700$	−13764.9	−0.743232
$701$	−1266.41	−0.0682333	−0.0341166	−	0.999418i	$-0.510862\pi$
$701$	−1266.41	−0.0682333	−0.0341166	+	0.999418i	$0.510862\pi$
$702$	−116.111	−0.00624265
$703$	−6444.54	−0.345747
$704$	−27521.0	−1.47335
$705$	13843.9	0.739565
$706$	3329.67	0.177498
$707$	−2563.98	−0.136391
$708$	14003.6	0.743344
$709$	16158.2	0.855899	0.427950	−	0.903803i	$-0.359236\pi$
$709$	16158.2	0.855899	0.427950	+	0.903803i	$0.359236\pi$
$710$	−433.188	−0.0228975
$711$	−43596.5	−2.29957
$712$	−507.225	−0.0266981
$713$	−2141.74	−0.112495
$714$	4389.69	0.230084
$715$	1245.91	0.0651672
$716$	−18051.0	−0.942174
$717$	−48768.8	−2.54018
$718$	1062.09	0.0552047
$719$	22012.8	1.14178	0.570890	−	0.821027i	$-0.306598\pi$
$719$	22012.8	1.14178	0.570890	+	0.821027i	$0.306598\pi$
$720$	−10492.5	−0.543101
$721$	−2722.93	−0.140648
$722$	−1734.23	−0.0893927
$723$	42307.8	2.17627
$724$	12471.3	0.640185
$725$	3037.44	0.155597
$726$	4400.76	0.224969
$727$	−6523.57	−0.332800	−0.166400	−	0.986058i	$-0.553214\pi$
$727$	−6523.57	−0.332800	−0.166400	+	0.986058i	$0.553214\pi$
$728$	−362.008	−0.0184298
$729$	−30883.5	−1.56904
$730$	−1478.71	−0.0749718
$731$	−7642.28	−0.386676
$732$	27896.3	1.40857
$733$	−11823.4	−0.595782	−0.297891	−	0.954600i	$-0.596283\pi$
$733$	−11823.4	−0.595782	−0.297891	+	0.954600i	$0.596283\pi$
$734$	−1404.74	−0.0706401
$735$	2446.51	0.122777
$736$	1232.95	0.0617486
$737$	−60520.7	−3.02484
$738$	−2795.36	−0.139429
$739$	18605.1	0.926116	0.463058	−	0.886328i	$-0.346752\pi$
$739$	18605.1	0.926116	0.463058	+	0.886328i	$0.346752\pi$
$740$	−8501.73	−0.422338
$741$	−1051.40	−0.0521245
$742$	90.8794	0.00449635
$743$	8858.51	0.437399	0.218699	−	0.975792i	$-0.429819\pi$
$743$	8858.51	0.437399	0.218699	+	0.975792i	$0.429819\pi$
$744$	3370.16	0.166070
$745$	8033.87	0.395085
$746$	3887.63	0.190799
$747$	19949.2	0.977113
$748$	52674.9	2.57485
$749$	−17498.5	−0.853646
$750$	2350.89	0.114457
$751$	−18857.8	−0.916287	−0.458144	−	0.888878i	$-0.651485\pi$
$751$	−18857.8	−0.916287	−0.458144	+	0.888878i	$0.651485\pi$
$752$	−23767.4	−1.15254
$753$	−4745.35	−0.229655
$754$	39.7406	0.00191945
$755$	−8369.90	−0.403460
$756$	11135.3	0.535695
$757$	−28242.0	−1.35598	−0.677988	−	0.735073i	$-0.737148\pi$
$757$	−28242.0	−1.35598	−0.677988	+	0.735073i	$0.737148\pi$
$758$	−1313.33	−0.0629316
$759$	10561.5	0.505081
$760$	547.943	0.0261526
$761$	15151.0	0.721714	0.360857	−	0.932621i	$-0.382484\pi$
$761$	15151.0	0.721714	0.360857	+	0.932621i	$0.382484\pi$
$762$	−4435.81	−0.210883
$763$	25840.3	1.22606
$764$	−23410.1	−1.10857
$765$	19666.0	0.929448
$766$	−2921.80	−0.137818
$767$	1065.82	0.0501754
$768$	29663.2	1.39372
$769$	10742.8	0.503764	0.251882	−	0.967758i	$-0.418951\pi$
$769$	10742.8	0.503764	0.251882	+	0.967758i	$0.418951\pi$
$770$	1207.99	0.0565362
$771$	−3470.68	−0.162118
$772$	29795.5	1.38907
$773$	−19614.1	−0.912638	−0.456319	−	0.889816i	$-0.650832\pi$
$773$	−19614.1	−0.912638	−0.456319	+	0.889816i	$0.650832\pi$
$774$	697.755	0.0324035
$775$	9753.23	0.452060
$776$	−8475.22	−0.392065
$777$	31793.3	1.46793
$778$	1650.67	0.0760658
$779$	−7109.97	−0.327010
$780$	−1387.03	−0.0636712
$781$	19439.0	0.890632
$782$	−757.330	−0.0346318
$783$	−2457.18	−0.112149
$784$	−4200.19	−0.191335
$785$	−10176.7	−0.462702
$786$	2963.26	0.134473
$787$	2762.33	0.125116	0.0625580	−	0.998041i	$-0.480074\pi$
$787$	2762.33	0.125116	0.0625580	+	0.998041i	$0.480074\pi$
$788$	−16029.5	−0.724655
$789$	38101.3	1.71919
$790$	1478.92	0.0666047
$791$	−25116.7	−1.12901
$792$	−9667.28	−0.433727
$793$	2123.20	0.0950781
$794$	−758.836	−0.0339170
$795$	699.927	0.0312250
$796$	−16139.9	−0.718675
$797$	−16296.9	−0.724296	−0.362148	−	0.932121i	$-0.617957\pi$
$797$	−16296.9	−0.724296	−0.362148	+	0.932121i	$0.617957\pi$
$798$	−1019.40	−0.0452209
$799$	44547.1	1.97242
$800$	−5614.69	−0.248137
$801$	4227.58	0.186485
$802$	399.772	0.0176016
$803$	66356.1	2.91613
$804$	67375.3	2.95540
$805$	1717.89	0.0752144
$806$	127.607	0.00557663
$807$	−26789.3	−1.16856
$808$	−696.065	−0.0303063
$809$	15587.5	0.677412	0.338706	−	0.940892i	$-0.390011\pi$
$809$	15587.5	0.677412	0.338706	+	0.940892i	$0.390011\pi$
$810$	424.165	0.0183996
$811$	−12226.4	−0.529380	−0.264690	−	0.964333i	$-0.585270\pi$
$811$	−12226.4	−0.529380	−0.264690	+	0.964333i	$0.585270\pi$
$812$	−3811.18	−0.164712
$813$	−35542.2	−1.53324
$814$	−3857.04	−0.166080
$815$	9992.58	0.429478
$816$	−58042.1	−2.49005
$817$	1774.73	0.0759975
$818$	−4222.25	−0.180474
$819$	3017.24	0.128731
$820$	−9379.58	−0.399450
$821$	5925.66	0.251897	0.125948	−	0.992037i	$-0.459803\pi$
$821$	5925.66	0.251897	0.125948	+	0.992037i	$0.459803\pi$
$822$	−1353.43	−0.0574285
$823$	−24244.8	−1.02688	−0.513438	−	0.858126i	$-0.671629\pi$
$823$	−24244.8	−1.02688	−0.513438	+	0.858126i	$0.671629\pi$
$824$	−739.218	−0.0312523
$825$	−48095.7	−2.02967
$826$	1033.38	0.0435299
$827$	−6794.60	−0.285697	−0.142848	−	0.989745i	$-0.545626\pi$
$827$	−6794.60	−0.285697	−0.142848	+	0.989745i	$0.545626\pi$
$828$	−6839.39	−0.287059
$829$	39145.1	1.64001	0.820004	−	0.572358i	$-0.193971\pi$
$829$	39145.1	1.64001	0.820004	+	0.572358i	$0.193971\pi$
$830$	−676.737	−0.0283010
$831$	−10036.2	−0.418954
$832$	2331.88	0.0971674
$833$	7872.40	0.327446
$834$	479.404	0.0199046
$835$	6049.58	0.250724
$836$	−12232.5	−0.506063
$837$	−7890.01	−0.325829
$838$	−1086.39	−0.0447836
$839$	34464.0	1.41815	0.709076	−	0.705132i	$-0.249112\pi$
$839$	34464.0	1.41815	0.709076	+	0.705132i	$0.249112\pi$
$840$	−2703.21	−0.111035
$841$	841.000	0.0344828
$842$	665.285	0.0272295
$843$	2627.59	0.107354
$844$	9932.30	0.405076
$845$	9783.53	0.398300
$846$	−4067.24	−0.165289
$847$	−32121.6	−1.30308
$848$	−1201.64	−0.0486610
$849$	−3406.25	−0.137694
$850$	3448.79	0.139168
$851$	−5485.14	−0.220949
$852$	−21640.7	−0.870187
$853$	12609.4	0.506139	0.253069	−	0.967448i	$-0.418560\pi$
$853$	12609.4	0.506139	0.253069	+	0.967448i	$0.418560\pi$
$854$	2058.57	0.0824855
$855$	−4566.96	−0.182674
$856$	−4750.47	−0.189682
$857$	−31538.3	−1.25709	−0.628547	−	0.777772i	$-0.716350\pi$
$857$	−31538.3	−1.25709	−0.628547	+	0.777772i	$0.716350\pi$
$858$	−629.263	−0.0250381
$859$	30445.5	1.20930	0.604649	−	0.796492i	$-0.293313\pi$
$859$	30445.5	1.20930	0.604649	+	0.796492i	$0.293313\pi$
$860$	2341.25	0.0928326
$861$	35076.1	1.38838
$862$	−3373.57	−0.133300
$863$	−16428.7	−0.648019	−0.324010	−	0.946054i	$-0.605031\pi$
$863$	−16428.7	−0.648019	−0.324010	+	0.946054i	$0.605031\pi$
$864$	4542.08	0.178848
$865$	−10690.9	−0.420231
$866$	−2361.71	−0.0926721
$867$	69316.5	2.71524
$868$	−12237.7	−0.478543
$869$	−66365.7	−2.59068
$870$	296.753	0.0115642
$871$	5127.96	0.199488
$872$	7015.10	0.272432
$873$	70638.7	2.73856
$874$	175.871	0.00680657
$875$	−17159.4	−0.662965
$876$	−73871.6	−2.84919
$877$	−33168.1	−1.27709	−0.638545	−	0.769584i	$-0.720464\pi$
$877$	−33168.1	−1.27709	−0.638545	+	0.769584i	$0.720464\pi$
$878$	−808.335	−0.0310706
$879$	−35102.7	−1.34697
$880$	−15972.5	−0.611854
$881$	−18188.6	−0.695562	−0.347781	−	0.937576i	$-0.613065\pi$
$881$	−18188.6	−0.695562	−0.347781	+	0.937576i	$0.613065\pi$
$882$	−718.766	−0.0274400
$883$	−37482.2	−1.42851	−0.714256	−	0.699884i	$-0.753235\pi$
$883$	−37482.2	−1.42851	−0.714256	+	0.699884i	$0.753235\pi$
$884$	−4463.18	−0.169811
$885$	7958.76	0.302295
$886$	−387.712	−0.0147014
$887$	−2726.65	−0.103215	−0.0516077	−	0.998667i	$-0.516435\pi$
$887$	−2726.65	−0.103215	−0.0516077	+	0.998667i	$0.516435\pi$
$888$	8631.21	0.326176
$889$	32377.5	1.22149
$890$	−143.412	−0.00540133
$891$	−19034.1	−0.715676
$892$	39571.8	1.48538
$893$	−10345.0	−0.387661
$894$	−4057.60	−0.151797
$895$	−10259.0	−0.383152
$896$	9377.07	0.349627
$897$	−894.880	−0.0333101
$898$	125.100	0.00464882
$899$	2700.45	0.100184
$900$	31145.8	1.15355
$901$	2252.23	0.0832771
$902$	−4255.30	−0.157080
$903$	−8755.41	−0.322660
$904$	−6818.65	−0.250868
$905$	7087.92	0.260343
$906$	4227.31	0.155014
$907$	−1072.82	−0.0392750	−0.0196375	−	0.999807i	$-0.506251\pi$
$907$	−1072.82	−0.0392750	−0.0196375	+	0.999807i	$0.506251\pi$
$908$	23188.1	0.847492
$909$	5801.52	0.211688
$910$	−102.354	−0.00372856
$911$	−40961.3	−1.48969	−0.744845	−	0.667237i	$-0.767477\pi$
$911$	−40961.3	−1.48969	−0.744845	+	0.667237i	$0.767477\pi$
$912$	13478.9	0.489396
$913$	30368.1	1.10081
$914$	−2094.80	−0.0758095
$915$	15854.5	0.572823
$916$	−40941.4	−1.47679
$917$	−21629.1	−0.778906
$918$	−2789.95	−0.100307
$919$	40912.1	1.46851	0.734257	−	0.678871i	$-0.237531\pi$
$919$	40912.1	1.46851	0.734257	+	0.678871i	$0.237531\pi$
$920$	466.370	0.0167128
$921$	54755.1	1.95900
$922$	2418.71	0.0863949
$923$	−1647.08	−0.0587372
$924$	60347.3	2.14857
$925$	24978.7	0.887885
$926$	−3641.55	−0.129232
$927$	6161.19	0.218295
$928$	−1554.58	−0.0549911
$929$	−10065.8	−0.355489	−0.177744	−	0.984077i	$-0.556880\pi$
$929$	−10065.8	−0.355489	−0.177744	+	0.984077i	$0.556880\pi$
$930$	952.876	0.0335979
$931$	−1828.17	−0.0643565
$932$	48652.9	1.70996
$933$	22100.8	0.775508
$934$	−1029.14	−0.0360541
$935$	29937.1	1.04711
$936$	819.115	0.0286043
$937$	8116.08	0.282968	0.141484	−	0.989941i	$-0.454813\pi$
$937$	8116.08	0.282968	0.141484	+	0.989941i	$0.454813\pi$
$938$	4971.86	0.173067
$939$	−31547.4	−1.09639
$940$	−13647.3	−0.473537
$941$	−44832.1	−1.55312	−0.776559	−	0.630045i	$-0.783037\pi$
$941$	−44832.1	−1.55312	−0.776559	+	0.630045i	$0.783037\pi$
$942$	5139.85	0.177776
$943$	−6051.50	−0.208976
$944$	−13663.7	−0.471096
$945$	6328.58	0.217850
$946$	1062.17	0.0365055
$947$	−20188.8	−0.692763	−0.346382	−	0.938094i	$-0.612590\pi$
$947$	−20188.8	−0.692763	−0.346382	+	0.938094i	$0.612590\pi$
$948$	73882.4	2.53121
$949$	−5622.40	−0.192319
$950$	−800.898	−0.0273522
$951$	18206.5	0.620806
$952$	−8698.39	−0.296131
$953$	−11646.4	−0.395871	−0.197935	−	0.980215i	$-0.563424\pi$
$953$	−11646.4	−0.395871	−0.197935	+	0.980215i	$0.563424\pi$
$954$	−205.633	−0.00697864
$955$	−13304.8	−0.450820
$956$	48075.9	1.62645
$957$	−13316.6	−0.449807
$958$	428.775	0.0144604
$959$	9878.81	0.332642
$960$	17412.7	0.585410
$961$	−21119.8	−0.708934
$962$	326.810	0.0109530
$963$	39593.9	1.32492
$964$	−41706.7	−1.39345
$965$	16933.9	0.564891
$966$	−867.639	−0.0288984
$967$	3024.50	0.100581	0.0502903	−	0.998735i	$-0.483985\pi$
$967$	3024.50	0.100581	0.0502903	+	0.998735i	$0.483985\pi$
$968$	−8720.33	−0.289548
$969$	−25263.3	−0.837539
$970$	−2396.28	−0.0793193
$971$	7721.56	0.255197	0.127599	−	0.991826i	$-0.459273\pi$
$971$	7721.56	0.255197	0.127599	+	0.991826i	$0.459273\pi$
$972$	39308.5	1.29714
$973$	−3499.22	−0.115293
$974$	−2394.16	−0.0787616
$975$	4075.18	0.133857
$976$	−27219.1	−0.892687
$977$	−57257.7	−1.87496	−0.937480	−	0.348040i	$-0.886847\pi$
$977$	−57257.7	−1.87496	−0.937480	+	0.348040i	$0.886847\pi$
$978$	−5046.86	−0.165011
$979$	6435.53	0.210092
$980$	−2411.75	−0.0786129
$981$	−58469.0	−1.90293
$982$	−3837.14	−0.124692
$983$	−57030.6	−1.85045	−0.925226	−	0.379417i	$-0.876125\pi$
$983$	−57030.6	−1.85045	−0.925226	+	0.379417i	$0.876125\pi$
$984$	9522.42	0.308500
$985$	−9110.16	−0.294694
$986$	954.895	0.0308418
$987$	51035.6	1.64588
$988$	1036.47	0.0333748
$989$	1510.53	0.0485662
$990$	−2733.32	−0.0877480
$991$	8360.62	0.267996	0.133998	−	0.990982i	$-0.457218\pi$
$991$	8360.62	0.267996	0.133998	+	0.990982i	$0.457218\pi$
$992$	−4991.77	−0.159767
$993$	76244.5	2.43660
$994$	−1596.95	−0.0509578
$995$	−9172.92	−0.292262
$996$	−33807.7	−1.07554
$997$	−24629.0	−0.782357	−0.391178	−	0.920315i	$-0.627932\pi$
$997$	−24629.0	−0.782357	−0.391178	+	0.920315i	$0.627932\pi$
$998$	1003.71	0.0318357
$999$	−20206.8	−0.639956

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	667.4.a.a.1.20	✓	35

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.4.a.a.1.20	✓	35	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:	\( N \)	\(=\)	\( 667 = 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	667.a (trivial)

\(f(q)\)	\(=\)	\(q+0.282966 q^{2} +8.03408 q^{3} -7.91993 q^{4} -4.50118 q^{5} +2.27337 q^{6} -16.5936 q^{7} -4.50481 q^{8} +37.5464 q^{9} +O(q^{10})\)	\(q+0.282966 q^{2} +8.03408 q^{3} -7.91993 q^{4} -4.50118 q^{5} +2.27337 q^{6} -16.5936 q^{7} -4.50481 q^{8} +37.5464 q^{9} -1.27368 q^{10} +57.1558 q^{11} -63.6293 q^{12} -4.84285 q^{13} -4.69543 q^{14} -36.1628 q^{15} +62.0847 q^{16} -116.365 q^{17} +10.6244 q^{18} +27.0229 q^{19} +35.6491 q^{20} -133.314 q^{21} +16.1732 q^{22} +23.0000 q^{23} -36.1920 q^{24} -104.739 q^{25} -1.37036 q^{26} +84.7303 q^{27} +131.420 q^{28} -29.0000 q^{29} -10.2329 q^{30} -93.1191 q^{31} +53.6063 q^{32} +459.194 q^{33} -32.9274 q^{34} +74.6908 q^{35} -297.365 q^{36} -238.484 q^{37} +7.64658 q^{38} -38.9078 q^{39} +20.2770 q^{40} -263.109 q^{41} -37.7234 q^{42} +65.6750 q^{43} -452.670 q^{44} -169.003 q^{45} +6.50823 q^{46} -382.822 q^{47} +498.793 q^{48} -67.6526 q^{49} -29.6377 q^{50} -934.885 q^{51} +38.3550 q^{52} -19.3549 q^{53} +23.9758 q^{54} -257.269 q^{55} +74.7509 q^{56} +217.104 q^{57} -8.20603 q^{58} -220.081 q^{59} +286.407 q^{60} -438.419 q^{61} -26.3496 q^{62} -623.029 q^{63} -481.509 q^{64} +21.7986 q^{65} +129.936 q^{66} -1058.87 q^{67} +921.603 q^{68} +184.784 q^{69} +21.1350 q^{70} +340.106 q^{71} -169.139 q^{72} +1160.97 q^{73} -67.4830 q^{74} -841.484 q^{75} -214.020 q^{76} -948.420 q^{77} -11.0096 q^{78} -1161.14 q^{79} -279.455 q^{80} -333.022 q^{81} -74.4509 q^{82} +531.322 q^{83} +1055.84 q^{84} +523.780 q^{85} +18.5838 q^{86} -232.988 q^{87} -257.476 q^{88} +112.596 q^{89} -47.8222 q^{90} +80.3603 q^{91} -182.158 q^{92} -748.126 q^{93} -108.326 q^{94} -121.635 q^{95} +430.677 q^{96} +1881.37 q^{97} -19.1434 q^{98} +2145.99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 35 q - 6 q^{2} - 22 q^{3} + 116 q^{4} - 80 q^{5} - 52 q^{6} - 38 q^{7} + 12 q^{8} + 231 q^{9}+O(q^{10}) \) 35 * q - 6 * q^2 - 22 * q^3 + 116 * q^4 - 80 * q^5 - 52 * q^6 - 38 * q^7 + 12 * q^8 + 231 * q^9	\( 35 q - 6 q^{2} - 22 q^{3} + 116 q^{4} - 80 q^{5} - 52 q^{6} - 38 q^{7} + 12 q^{8} + 231 q^{9} - 52 q^{10} - 126 q^{11} - 173 q^{12} - 252 q^{13} + 112 q^{14} - 32 q^{15} + 312 q^{16} - 332 q^{17} - 225 q^{18} - 2 q^{19} - 747 q^{20} - 202 q^{21} - 127 q^{22} + 805 q^{23} - 494 q^{24} + 315 q^{25} - 677 q^{26} - 694 q^{27} - 529 q^{28} - 1015 q^{29} + 389 q^{30} - 652 q^{31} + 320 q^{32} - 290 q^{33} - 455 q^{34} - 940 q^{35} + 34 q^{36} - 528 q^{37} - 1218 q^{38} - 268 q^{39} - 806 q^{40} - 68 q^{41} - 1484 q^{42} - 162 q^{43} - 1817 q^{44} - 356 q^{45} - 138 q^{46} - 1200 q^{47} - 2153 q^{48} + 93 q^{49} - 1369 q^{50} - 270 q^{51} - 3134 q^{52} - 1892 q^{53} - 1221 q^{54} - 794 q^{55} + 191 q^{56} - 1764 q^{57} + 174 q^{58} - 1354 q^{59} + 159 q^{60} - 1274 q^{61} - 5413 q^{62} - 2904 q^{63} - 926 q^{64} - 548 q^{65} - 2477 q^{66} - 3212 q^{67} - 3901 q^{68} - 506 q^{69} - 2768 q^{70} - 2342 q^{71} - 2381 q^{72} + 916 q^{73} + 661 q^{74} - 4708 q^{75} - 2810 q^{76} - 5536 q^{77} - 2434 q^{78} + 2622 q^{79} - 5444 q^{80} + 607 q^{81} - 3687 q^{82} - 2702 q^{83} + 346 q^{84} - 3304 q^{85} - 5789 q^{86} + 638 q^{87} - 2252 q^{88} - 1620 q^{89} - 3933 q^{90} - 4016 q^{91} + 2668 q^{92} - 4942 q^{93} - 1413 q^{94} - 4528 q^{95} - 7920 q^{96} + 682 q^{97} + 152 q^{98} - 582 q^{99}+O(q^{100}) \) 35 * q - 6 * q^2 - 22 * q^3 + 116 * q^4 - 80 * q^5 - 52 * q^6 - 38 * q^7 + 12 * q^8 + 231 * q^9 - 52 * q^10 - 126 * q^11 - 173 * q^12 - 252 * q^13 + 112 * q^14 - 32 * q^15 + 312 * q^16 - 332 * q^17 - 225 * q^18 - 2 * q^19 - 747 * q^20 - 202 * q^21 - 127 * q^22 + 805 * q^23 - 494 * q^24 + 315 * q^25 - 677 * q^26 - 694 * q^27 - 529 * q^28 - 1015 * q^29 + 389 * q^30 - 652 * q^31 + 320 * q^32 - 290 * q^33 - 455 * q^34 - 940 * q^35 + 34 * q^36 - 528 * q^37 - 1218 * q^38 - 268 * q^39 - 806 * q^40 - 68 * q^41 - 1484 * q^42 - 162 * q^43 - 1817 * q^44 - 356 * q^45 - 138 * q^46 - 1200 * q^47 - 2153 * q^48 + 93 * q^49 - 1369 * q^50 - 270 * q^51 - 3134 * q^52 - 1892 * q^53 - 1221 * q^54 - 794 * q^55 + 191 * q^56 - 1764 * q^57 + 174 * q^58 - 1354 * q^59 + 159 * q^60 - 1274 * q^61 - 5413 * q^62 - 2904 * q^63 - 926 * q^64 - 548 * q^65 - 2477 * q^66 - 3212 * q^67 - 3901 * q^68 - 506 * q^69 - 2768 * q^70 - 2342 * q^71 - 2381 * q^72 + 916 * q^73 + 661 * q^74 - 4708 * q^75 - 2810 * q^76 - 5536 * q^77 - 2434 * q^78 + 2622 * q^79 - 5444 * q^80 + 607 * q^81 - 3687 * q^82 - 2702 * q^83 + 346 * q^84 - 3304 * q^85 - 5789 * q^86 + 638 * q^87 - 2252 * q^88 - 1620 * q^89 - 3933 * q^90 - 4016 * q^91 + 2668 * q^92 - 4942 * q^93 - 1413 * q^94 - 4528 * q^95 - 7920 * q^96 + 682 * q^97 + 152 * q^98 - 582 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more