LMFDB - Embedded newform 2013.4.a.d.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2013,4,Mod(1,2013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2013, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2013.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2013 = 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2013.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:	$118.770844842$
Analytic rank:	$1$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	2013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.80549 q^{2} -3.00000 q^{3} -4.74021 q^{4} -6.98611 q^{5} +5.41646 q^{6} +21.1508 q^{7} +23.0023 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-1.80549 q^{2} -3.00000 q^{3} -4.74021 q^{4} -6.98611 q^{5} +5.41646 q^{6} +21.1508 q^{7} +23.0023 q^{8} +9.00000 q^{9} +12.6133 q^{10} +11.0000 q^{11} +14.2206 q^{12} -45.4212 q^{13} -38.1875 q^{14} +20.9583 q^{15} -3.60868 q^{16} +84.0029 q^{17} -16.2494 q^{18} +39.1620 q^{19} +33.1157 q^{20} -63.4524 q^{21} -19.8604 q^{22} -108.284 q^{23} -69.0069 q^{24} -76.1942 q^{25} +82.0074 q^{26} -27.0000 q^{27} -100.259 q^{28} -71.1321 q^{29} -37.8400 q^{30} -109.336 q^{31} -177.503 q^{32} -33.0000 q^{33} -151.666 q^{34} -147.762 q^{35} -42.6619 q^{36} +159.103 q^{37} -70.7065 q^{38} +136.264 q^{39} -160.697 q^{40} -376.575 q^{41} +114.563 q^{42} +306.455 q^{43} -52.1423 q^{44} -62.8750 q^{45} +195.506 q^{46} +281.450 q^{47} +10.8260 q^{48} +104.356 q^{49} +137.568 q^{50} -252.009 q^{51} +215.306 q^{52} +465.621 q^{53} +48.7482 q^{54} -76.8473 q^{55} +486.517 q^{56} -117.486 q^{57} +128.428 q^{58} -659.971 q^{59} -99.3470 q^{60} -61.0000 q^{61} +197.406 q^{62} +190.357 q^{63} +349.349 q^{64} +317.318 q^{65} +59.5811 q^{66} -72.0545 q^{67} -398.192 q^{68} +324.853 q^{69} +266.782 q^{70} -2.28428 q^{71} +207.021 q^{72} +266.292 q^{73} -287.259 q^{74} +228.583 q^{75} -185.636 q^{76} +232.659 q^{77} -246.022 q^{78} +246.519 q^{79} +25.2107 q^{80} +81.0000 q^{81} +679.901 q^{82} +1007.59 q^{83} +300.778 q^{84} -586.854 q^{85} -553.300 q^{86} +213.396 q^{87} +253.025 q^{88} -717.077 q^{89} +113.520 q^{90} -960.694 q^{91} +513.291 q^{92} +328.009 q^{93} -508.155 q^{94} -273.590 q^{95} +532.509 q^{96} +126.650 q^{97} -188.414 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q - 4 q^{2} - 111 q^{3} + 158 q^{4} - 15 q^{5} + 12 q^{6} - 77 q^{7} - 69 q^{8} + 333 q^{9}+O(q^{10}) $ 37 * q - 4 * q^2 - 111 * q^3 + 158 * q^4 - 15 * q^5 + 12 * q^6 - 77 * q^7 - 69 * q^8 + 333 * q^9	\( 37 q - 4 q^{2} - 111 q^{3} + 158 q^{4} - 15 q^{5} + 12 q^{6} - 77 q^{7} - 69 q^{8} + 333 q^{9} - 45 q^{10} + 407 q^{11} - 474 q^{12} - 169 q^{13} + 102 q^{14} + 45 q^{15} + 598 q^{16} - 338 q^{17} - 36 q^{18} - 235 q^{19} - 550 q^{20} + 231 q^{21} - 44 q^{22} - 53 q^{23} + 207 q^{24} + 750 q^{25} - 75 q^{26} - 999 q^{27} - 1378 q^{28} - 30 q^{29} + 135 q^{30} - 506 q^{31} - 841 q^{32} - 1221 q^{33} - 316 q^{34} - 822 q^{35} + 1422 q^{36} - 830 q^{37} - 371 q^{38} + 507 q^{39} - 613 q^{40} + 16 q^{41} - 306 q^{42} - 1137 q^{43} + 1738 q^{44} - 135 q^{45} - 659 q^{46} - 489 q^{47} - 1794 q^{48} + 2214 q^{49} + 1066 q^{50} + 1014 q^{51} - 2342 q^{52} + 731 q^{53} + 108 q^{54} - 165 q^{55} + 3051 q^{56} + 705 q^{57} - 611 q^{58} - 425 q^{59} + 1650 q^{60} - 2257 q^{61} + 453 q^{62} - 693 q^{63} + 4919 q^{64} + 1346 q^{65} + 132 q^{66} - 1907 q^{67} - 3236 q^{68} + 159 q^{69} - 1050 q^{70} - 561 q^{71} - 621 q^{72} - 2397 q^{73} - 1840 q^{74} - 2250 q^{75} - 3868 q^{76} - 847 q^{77} + 225 q^{78} + 393 q^{79} - 4031 q^{80} + 2997 q^{81} - 1946 q^{82} - 4191 q^{83} + 4134 q^{84} - 2667 q^{85} + 2405 q^{86} + 90 q^{87} - 759 q^{88} + 1437 q^{89} - 405 q^{90} - 5192 q^{91} - 737 q^{92} + 1518 q^{93} - 1960 q^{94} + 1356 q^{95} + 2523 q^{96} - 2368 q^{97} - 3014 q^{98} + 3663 q^{99}+O(q^{100}) \) 37 * q - 4 * q^2 - 111 * q^3 + 158 * q^4 - 15 * q^5 + 12 * q^6 - 77 * q^7 - 69 * q^8 + 333 * q^9 - 45 * q^10 + 407 * q^11 - 474 * q^12 - 169 * q^13 + 102 * q^14 + 45 * q^15 + 598 * q^16 - 338 * q^17 - 36 * q^18 - 235 * q^19 - 550 * q^20 + 231 * q^21 - 44 * q^22 - 53 * q^23 + 207 * q^24 + 750 * q^25 - 75 * q^26 - 999 * q^27 - 1378 * q^28 - 30 * q^29 + 135 * q^30 - 506 * q^31 - 841 * q^32 - 1221 * q^33 - 316 * q^34 - 822 * q^35 + 1422 * q^36 - 830 * q^37 - 371 * q^38 + 507 * q^39 - 613 * q^40 + 16 * q^41 - 306 * q^42 - 1137 * q^43 + 1738 * q^44 - 135 * q^45 - 659 * q^46 - 489 * q^47 - 1794 * q^48 + 2214 * q^49 + 1066 * q^50 + 1014 * q^51 - 2342 * q^52 + 731 * q^53 + 108 * q^54 - 165 * q^55 + 3051 * q^56 + 705 * q^57 - 611 * q^58 - 425 * q^59 + 1650 * q^60 - 2257 * q^61 + 453 * q^62 - 693 * q^63 + 4919 * q^64 + 1346 * q^65 + 132 * q^66 - 1907 * q^67 - 3236 * q^68 + 159 * q^69 - 1050 * q^70 - 561 * q^71 - 621 * q^72 - 2397 * q^73 - 1840 * q^74 - 2250 * q^75 - 3868 * q^76 - 847 * q^77 + 225 * q^78 + 393 * q^79 - 4031 * q^80 + 2997 * q^81 - 1946 * q^82 - 4191 * q^83 + 4134 * q^84 - 2667 * q^85 + 2405 * q^86 + 90 * q^87 - 759 * q^88 + 1437 * q^89 - 405 * q^90 - 5192 * q^91 - 737 * q^92 + 1518 * q^93 - 1960 * q^94 + 1356 * q^95 + 2523 * q^96 - 2368 * q^97 - 3014 * q^98 + 3663 * 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 $

$2$	−1.80549	−0.638336	−0.319168	−	0.947698i	$-0.603403\pi$
$2$	−1.80549	−0.638336	−0.319168	+	0.947698i	$0.603403\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−4.74021	−0.592527
$5$	−6.98611	−0.624857	−0.312429	−	0.949941i	$-0.601142\pi$
$5$	−6.98611	−0.624857	−0.312429	+	0.949941i	$0.601142\pi$
$6$	5.41646	0.368544
$7$	21.1508	1.14204	0.571018	−	0.820938i	$-0.306549\pi$
$7$	21.1508	1.14204	0.571018	+	0.820938i	$0.306549\pi$
$8$	23.0023	1.01657
$9$	9.00000	0.333333
$10$	12.6133	0.398869
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	14.2206	0.342095
$13$	−45.4212	−0.969044	−0.484522	−	0.874779i	$-0.661006\pi$
$13$	−45.4212	−0.969044	−0.484522	+	0.874779i	$0.661006\pi$
$14$	−38.1875	−0.729003
$15$	20.9583	0.360761
$16$	−3.60868	−0.0563856
$17$	84.0029	1.19845	0.599227	−	0.800579i	$-0.295475\pi$
$17$	84.0029	1.19845	0.599227	+	0.800579i	$0.295475\pi$
$18$	−16.2494	−0.212779
$19$	39.1620	0.472862	0.236431	−	0.971648i	$-0.424022\pi$
$19$	39.1620	0.472862	0.236431	+	0.971648i	$0.424022\pi$
$20$	33.1157	0.370244
$21$	−63.4524	−0.659355
$22$	−19.8604	−0.192466
$23$	−108.284	−0.981689	−0.490845	−	0.871247i	$-0.663312\pi$
$23$	−108.284	−0.981689	−0.490845	+	0.871247i	$0.663312\pi$
$24$	−69.0069	−0.586916
$25$	−76.1942	−0.609554
$26$	82.0074	0.618576
$27$	−27.0000	−0.192450
$28$	−100.259	−0.676686
$29$	−71.1321	−0.455479	−0.227740	−	0.973722i	$-0.573134\pi$
$29$	−71.1321	−0.455479	−0.227740	+	0.973722i	$0.573134\pi$
$30$	−37.8400	−0.230287
$31$	−109.336	−0.633465	−0.316732	−	0.948515i	$-0.602586\pi$
$31$	−109.336	−0.633465	−0.316732	+	0.948515i	$0.602586\pi$
$32$	−177.503	−0.980575
$33$	−33.0000	−0.174078
$34$	−151.666	−0.765016
$35$	−147.762	−0.713609
$36$	−42.6619	−0.197509
$37$	159.103	0.706930	0.353465	−	0.935448i	$-0.385003\pi$
$37$	159.103	0.706930	0.353465	+	0.935448i	$0.385003\pi$
$38$	−70.7065	−0.301845
$39$	136.264	0.559478
$40$	−160.697	−0.635210
$41$	−376.575	−1.43442	−0.717208	−	0.696859i	$-0.754580\pi$
$41$	−376.575	−1.43442	−0.717208	+	0.696859i	$0.754580\pi$
$42$	114.563	0.420890
$43$	306.455	1.08683	0.543417	−	0.839463i	$-0.317130\pi$
$43$	306.455	1.08683	0.543417	+	0.839463i	$0.317130\pi$
$44$	−52.1423	−0.178653
$45$	−62.8750	−0.208286
$46$	195.506	0.626648
$47$	281.450	0.873483	0.436742	−	0.899587i	$-0.356132\pi$
$47$	281.450	0.873483	0.436742	+	0.899587i	$0.356132\pi$
$48$	10.8260	0.0325543
$49$	104.356	0.304245
$50$	137.568	0.389100
$51$	−252.009	−0.691927
$52$	215.306	0.574184
$53$	465.621	1.20675	0.603377	−	0.797456i	$-0.293821\pi$
$53$	465.621	1.20675	0.603377	+	0.797456i	$0.293821\pi$
$54$	48.7482	0.122848
$55$	−76.8473	−0.188402
$56$	486.517	1.16096
$57$	−117.486	−0.273007
$58$	128.428	0.290749
$59$	−659.971	−1.45629	−0.728144	−	0.685424i	$-0.759617\pi$
$59$	−659.971	−1.45629	−0.728144	+	0.685424i	$0.759617\pi$
$60$	−99.3470	−0.213761
$61$	−61.0000	−0.128037
$61$	−61.0000	−0.128037
$62$	197.406	0.404364
$63$	190.357	0.380679
$64$	349.349	0.682322
$65$	317.318	0.605514
$66$	59.5811	0.111120
$67$	−72.0545	−0.131386	−0.0656930	−	0.997840i	$-0.520926\pi$
$67$	−72.0545	−0.131386	−0.0656930	+	0.997840i	$0.520926\pi$
$68$	−398.192	−0.710115
$69$	324.853	0.566779
$70$	266.782	0.455523
$71$	−2.28428	−0.00381823	−0.00190911	−	0.999998i	$-0.500608\pi$
$71$	−2.28428	−0.00381823	−0.00190911	+	0.999998i	$0.500608\pi$
$72$	207.021	0.338856
$73$	266.292	0.426947	0.213473	−	0.976949i	$-0.431522\pi$
$73$	266.292	0.426947	0.213473	+	0.976949i	$0.431522\pi$
$74$	−287.259	−0.451259
$75$	228.583	0.351926
$76$	−185.636	−0.280183
$77$	232.659	0.344337
$78$	−246.022	−0.357135
$79$	246.519	0.351083	0.175542	−	0.984472i	$-0.443832\pi$
$79$	246.519	0.351083	0.175542	+	0.984472i	$0.443832\pi$
$80$	25.2107	0.0352330
$81$	81.0000	0.111111
$82$	679.901	0.915640
$83$	1007.59	1.33250	0.666251	−	0.745727i	$-0.267898\pi$
$83$	1007.59	1.33250	0.666251	+	0.745727i	$0.267898\pi$
$84$	300.778	0.390685
$85$	−586.854	−0.748862
$86$	−553.300	−0.693766
$87$	213.396	0.262971
$88$	253.025	0.306507
$89$	−717.077	−0.854045	−0.427023	−	0.904241i	$-0.640438\pi$
$89$	−717.077	−0.854045	−0.427023	+	0.904241i	$0.640438\pi$
$90$	113.520	0.132956
$91$	−960.694	−1.10668
$92$	513.291	0.581677
$93$	328.009	0.365731
$94$	−508.155	−0.557576
$95$	−273.590	−0.295471
$96$	532.509	0.566135
$97$	126.650	0.132571	0.0662853	−	0.997801i	$-0.478885\pi$
$97$	126.650	0.132571	0.0662853	+	0.997801i	$0.478885\pi$
$98$	−188.414	−0.194211
$99$	99.0000	0.100504
$100$	361.177	0.361177
$101$	595.056	0.586241	0.293120	−	0.956076i	$-0.405306\pi$
$101$	595.056	0.586241	0.293120	+	0.956076i	$0.405306\pi$
$102$	454.999	0.441682
$103$	−259.423	−0.248172	−0.124086	−	0.992271i	$-0.539600\pi$
$103$	−259.423	−0.248172	−0.124086	+	0.992271i	$0.539600\pi$
$104$	−1044.79	−0.985099
$105$	443.286	0.412002
$106$	−840.673	−0.770315
$107$	−700.278	−0.632696	−0.316348	−	0.948643i	$-0.602457\pi$
$107$	−700.278	−0.632696	−0.316348	+	0.948643i	$0.602457\pi$
$108$	127.986	0.114032
$109$	−65.5422	−0.0575945	−0.0287973	−	0.999585i	$-0.509168\pi$
$109$	−65.5422	−0.0575945	−0.0287973	+	0.999585i	$0.509168\pi$
$110$	138.747	0.120264
$111$	−477.309	−0.408146
$112$	−76.3265	−0.0643944
$113$	1959.46	1.63125	0.815624	−	0.578583i	$-0.196394\pi$
$113$	1959.46	1.63125	0.815624	+	0.578583i	$0.196394\pi$
$114$	212.120	0.174270
$115$	756.487	0.613416
$116$	337.181	0.269884
$117$	−408.791	−0.323015
$118$	1191.57	0.929601
$119$	1776.73	1.36868
$120$	482.090	0.366738
$121$	121.000	0.0909091
$122$	110.135	0.0817306
$123$	1129.72	0.828161
$124$	518.278	0.375345
$125$	1405.57	1.00574
$126$	−343.688	−0.243001
$127$	197.763	0.138178	0.0690890	−	0.997610i	$-0.477991\pi$
$127$	197.763	0.138178	0.0690890	+	0.997610i	$0.477991\pi$
$128$	789.278	0.545024
$129$	−919.364	−0.627484
$130$	−572.913	−0.386522
$131$	1653.04	1.10250	0.551249	−	0.834341i	$-0.314152\pi$
$131$	1653.04	1.10250	0.551249	+	0.834341i	$0.314152\pi$
$132$	156.427	0.103146
$133$	828.307	0.540025
$134$	130.094	0.0838684
$135$	188.625	0.120254
$136$	1932.26	1.21831
$137$	−1481.05	−0.923609	−0.461804	−	0.886982i	$-0.652798\pi$
$137$	−1481.05	−0.923609	−0.461804	+	0.886982i	$0.652798\pi$
$138$	−586.519	−0.361795
$139$	−2423.87	−1.47906	−0.739531	−	0.673122i	$-0.764953\pi$
$139$	−2423.87	−1.47906	−0.739531	+	0.673122i	$0.764953\pi$
$140$	700.423	0.422832
$141$	−844.350	−0.504306
$142$	4.12424	0.00243731
$143$	−499.633	−0.292178
$144$	−32.4781	−0.0187952
$145$	496.937	0.284610
$146$	−480.787	−0.272536
$147$	−313.069	−0.175656
$148$	−754.183	−0.418875
$149$	1381.56	0.759612	0.379806	−	0.925066i	$-0.375991\pi$
$149$	1381.56	0.759612	0.379806	+	0.925066i	$0.375991\pi$
$150$	−412.703	−0.224647
$151$	1719.22	0.926543	0.463272	−	0.886216i	$-0.346675\pi$
$151$	1719.22	0.926543	0.463272	+	0.886216i	$0.346675\pi$
$152$	900.816	0.480696
$153$	756.026	0.399484
$154$	−420.063	−0.219803
$155$	763.837	0.395825
$156$	−645.918	−0.331505
$157$	−2849.06	−1.44828	−0.724139	−	0.689654i	$-0.757763\pi$
$157$	−2849.06	−1.44828	−0.724139	+	0.689654i	$0.757763\pi$
$158$	−445.087	−0.224109
$159$	−1396.86	−0.696720
$160$	1240.06	0.612719
$161$	−2290.30	−1.12112
$162$	−146.245	−0.0709263
$163$	1028.31	0.494129	0.247065	−	0.968999i	$-0.420534\pi$
$163$	1028.31	0.494129	0.247065	+	0.968999i	$0.420534\pi$
$164$	1785.04	0.849930
$165$	230.542	0.108774
$166$	−1819.20	−0.850585
$167$	2985.41	1.38334	0.691671	−	0.722213i	$-0.256875\pi$
$167$	2985.41	1.38334	0.691671	+	0.722213i	$0.256875\pi$
$168$	−1459.55	−0.670279
$169$	−133.916	−0.0609539
$170$	1059.56	0.478026
$171$	352.458	0.157621
$172$	−1452.66	−0.643978
$173$	−3406.85	−1.49721	−0.748607	−	0.663014i	$-0.769277\pi$
$173$	−3406.85	−1.49721	−0.748607	+	0.663014i	$0.769277\pi$
$174$	−385.285	−0.167864
$175$	−1611.57	−0.696132
$176$	−39.6955	−0.0170009
$177$	1979.91	0.840788
$178$	1294.67	0.545168
$179$	3727.15	1.55631	0.778156	−	0.628071i	$-0.216155\pi$
$179$	3727.15	1.55631	0.778156	+	0.628071i	$0.216155\pi$
$180$	298.041	0.123415
$181$	−2043.17	−0.839049	−0.419525	−	0.907744i	$-0.637803\pi$
$181$	−2043.17	−0.839049	−0.419525	+	0.907744i	$0.637803\pi$
$182$	1734.52	0.706436
$183$	183.000	0.0739221
$184$	−2490.79	−0.997954
$185$	−1111.51	−0.441730
$186$	−592.217	−0.233460
$187$	924.032	0.361347
$188$	−1334.13	−0.517562
$189$	−571.072	−0.219785
$190$	493.964	0.188610
$191$	−3687.80	−1.39707	−0.698535	−	0.715576i	$-0.746164\pi$
$191$	−3687.80	−1.39707	−0.698535	+	0.715576i	$0.746164\pi$
$192$	−1048.05	−0.393939
$193$	−2097.77	−0.782387	−0.391194	−	0.920308i	$-0.627938\pi$
$193$	−2097.77	−0.782387	−0.391194	+	0.920308i	$0.627938\pi$
$194$	−228.665	−0.0846246
$195$	−951.953	−0.349594
$196$	−494.671	−0.180274
$197$	−878.716	−0.317796	−0.158898	−	0.987295i	$-0.550794\pi$
$197$	−878.716	−0.317796	−0.158898	+	0.987295i	$0.550794\pi$
$198$	−178.743	−0.0641552
$199$	−1722.32	−0.613526	−0.306763	−	0.951786i	$-0.599246\pi$
$199$	−1722.32	−0.613526	−0.306763	+	0.951786i	$0.599246\pi$
$200$	−1752.64	−0.619653
$201$	216.163	0.0758557
$202$	−1074.37	−0.374219
$203$	−1504.50	−0.520174
$204$	1194.58	0.409985
$205$	2630.79	0.896305
$206$	468.386	0.158417
$207$	−974.560	−0.327230
$208$	163.911	0.0546402
$209$	430.782	0.142573
$210$	−800.347	−0.262996
$211$	1209.26	0.394543	0.197272	−	0.980349i	$-0.436792\pi$
$211$	1209.26	0.394543	0.197272	+	0.980349i	$0.436792\pi$
$212$	−2207.14	−0.715034
$213$	6.85284	0.00220445
$214$	1264.34	0.403873
$215$	−2140.93	−0.679116
$216$	−621.062	−0.195639
$217$	−2312.55	−0.723439
$218$	118.336	0.0367647
$219$	−798.876	−0.246498
$220$	364.272	0.111633
$221$	−3815.51	−1.16135
$222$	861.776	0.260534
$223$	−188.769	−0.0566858	−0.0283429	−	0.999598i	$-0.509023\pi$
$223$	−188.769	−0.0566858	−0.0283429	+	0.999598i	$0.509023\pi$
$224$	−3754.33	−1.11985
$225$	−685.748	−0.203185
$226$	−3537.79	−1.04128
$227$	−1046.61	−0.306017	−0.153008	−	0.988225i	$-0.548896\pi$
$227$	−1046.61	−0.306017	−0.153008	+	0.988225i	$0.548896\pi$
$228$	556.908	0.161764
$229$	−2354.92	−0.679553	−0.339777	−	0.940506i	$-0.610351\pi$
$229$	−2354.92	−0.679553	−0.339777	+	0.940506i	$0.610351\pi$
$230$	−1365.83	−0.391566
$231$	−697.976	−0.198803
$232$	−1636.20	−0.463026
$233$	5160.33	1.45092	0.725460	−	0.688264i	$-0.241627\pi$
$233$	5160.33	1.45092	0.725460	+	0.688264i	$0.241627\pi$
$234$	738.067	0.206192
$235$	−1966.24	−0.545802
$236$	3128.41	0.862889
$237$	−739.557	−0.202698
$238$	−3207.86	−0.873676
$239$	−1758.77	−0.476006	−0.238003	−	0.971264i	$-0.576493\pi$
$239$	−1758.77	−0.476006	−0.238003	+	0.971264i	$0.576493\pi$
$240$	−75.6320	−0.0203418
$241$	2642.55	0.706315	0.353157	−	0.935564i	$-0.385108\pi$
$241$	2642.55	0.706315	0.353157	+	0.935564i	$0.385108\pi$
$242$	−218.464	−0.0580306
$243$	−243.000	−0.0641500
$244$	289.153	0.0758653
$245$	−729.044	−0.190110
$246$	−2039.70	−0.528645
$247$	−1778.78	−0.458224
$248$	−2514.99	−0.643960
$249$	−3022.78	−0.769321
$250$	−2537.73	−0.642001
$251$	−4647.05	−1.16860	−0.584301	−	0.811537i	$-0.698631\pi$
$251$	−4647.05	−1.16860	−0.584301	+	0.811537i	$0.698631\pi$
$252$	−902.333	−0.225562
$253$	−1191.13	−0.295991
$254$	−357.058	−0.0882041
$255$	1760.56	0.432356
$256$	−4219.82	−1.03023
$257$	−7147.84	−1.73490	−0.867451	−	0.497523i	$-0.834243\pi$
$257$	−7147.84	−1.73490	−0.867451	+	0.497523i	$0.834243\pi$
$258$	1659.90	0.400546
$259$	3365.16	0.807339
$260$	−1504.15	−0.358783
$261$	−640.189	−0.151826
$262$	−2984.55	−0.703764
$263$	8093.50	1.89759	0.948796	−	0.315889i	$-0.102303\pi$
$263$	8093.50	1.89759	0.948796	+	0.315889i	$0.102303\pi$
$264$	−759.076	−0.176962
$265$	−3252.88	−0.754049
$266$	−1495.50	−0.344718
$267$	2151.23	0.493083
$268$	341.554	0.0778497
$269$	−2209.23	−0.500740	−0.250370	−	0.968150i	$-0.580552\pi$
$269$	−2209.23	−0.500740	−0.250370	+	0.968150i	$0.580552\pi$
$270$	−340.560	−0.0767624
$271$	−2504.99	−0.561504	−0.280752	−	0.959780i	$-0.590584\pi$
$271$	−2504.99	−0.561504	−0.280752	+	0.959780i	$0.590584\pi$
$272$	−303.140	−0.0675756
$273$	2882.08	0.638944
$274$	2674.01	0.589573
$275$	−838.136	−0.183787
$276$	−1539.87	−0.335831
$277$	−6318.06	−1.37045	−0.685227	−	0.728330i	$-0.740297\pi$
$277$	−6318.06	−1.37045	−0.685227	+	0.728330i	$0.740297\pi$
$278$	4376.26	0.944139
$279$	−984.028	−0.211155
$280$	−3398.86	−0.725432
$281$	−635.979	−0.135015	−0.0675077	−	0.997719i	$-0.521505\pi$
$281$	−635.979	−0.135015	−0.0675077	+	0.997719i	$0.521505\pi$
$282$	1524.46	0.321917
$283$	−319.449	−0.0671000	−0.0335500	−	0.999437i	$-0.510681\pi$
$283$	−319.449	−0.0671000	−0.0335500	+	0.999437i	$0.510681\pi$
$284$	10.8280	0.00226240
$285$	820.770	0.170590
$286$	902.082	0.186508
$287$	−7964.85	−1.63815
$288$	−1597.53	−0.326858
$289$	2143.49	0.436290
$290$	−897.214	−0.181677
$291$	−379.950	−0.0765397
$292$	−1262.28	−0.252977
$293$	5572.29	1.11105	0.555523	−	0.831501i	$-0.312518\pi$
$293$	5572.29	1.11105	0.555523	+	0.831501i	$0.312518\pi$
$294$	565.242	0.112128
$295$	4610.64	0.909972
$296$	3659.74	0.718642
$297$	−297.000	−0.0580259
$298$	−2494.40	−0.484888
$299$	4918.41	0.951300
$300$	−1083.53	−0.208525
$301$	6481.76	1.24120
$302$	−3104.03	−0.591446
$303$	−1785.17	−0.338466
$304$	−141.323	−0.0266626
$305$	426.153	0.0800048
$306$	−1365.00	−0.255005
$307$	−4393.69	−0.816811	−0.408405	−	0.912801i	$-0.633915\pi$
$307$	−4393.69	−0.816811	−0.408405	+	0.912801i	$0.633915\pi$
$308$	−1102.85	−0.204029
$309$	778.270	0.143282
$310$	−1379.10	−0.252670
$311$	6081.85	1.10891	0.554453	−	0.832215i	$-0.312927\pi$
$311$	6081.85	1.10891	0.554453	+	0.832215i	$0.312927\pi$
$312$	3134.38	0.568747
$313$	−1581.97	−0.285682	−0.142841	−	0.989746i	$-0.545624\pi$
$313$	−1581.97	−0.285682	−0.142841	+	0.989746i	$0.545624\pi$
$314$	5143.94	0.924488
$315$	−1329.86	−0.237870
$316$	−1168.55	−0.208026
$317$	−6494.28	−1.15065	−0.575324	−	0.817926i	$-0.695124\pi$
$317$	−6494.28	−1.15065	−0.575324	+	0.817926i	$0.695124\pi$
$318$	2522.02	0.444742
$319$	−782.453	−0.137332
$320$	−2440.59	−0.426354
$321$	2100.83	0.365287
$322$	4135.11	0.715655
$323$	3289.72	0.566703
$324$	−383.957	−0.0658363
$325$	3460.83	0.590684
$326$	−1856.59	−0.315421
$327$	196.627	0.0332522
$328$	−8662.08	−1.45818
$329$	5952.89	0.997549
$330$	−416.240	−0.0694342
$331$	−1631.41	−0.270907	−0.135454	−	0.990784i	$-0.543249\pi$
$331$	−1631.41	−0.270907	−0.135454	+	0.990784i	$0.543249\pi$
$332$	−4776.21	−0.789543
$333$	1431.93	0.235643
$334$	−5390.12	−0.883037
$335$	503.381	0.0820974
$336$	228.979	0.0371781
$337$	8554.38	1.38275	0.691375	−	0.722496i	$-0.257005\pi$
$337$	8554.38	1.38275	0.691375	+	0.722496i	$0.257005\pi$
$338$	241.783	0.0389091
$339$	−5878.39	−0.941801
$340$	2781.81	0.443721
$341$	−1202.70	−0.190997
$342$	−636.359	−0.100615
$343$	−5047.51	−0.794577
$344$	7049.16	1.10484
$345$	−2269.46	−0.354156
$346$	6151.02	0.955726
$347$	3017.41	0.466811	0.233405	−	0.972380i	$-0.425013\pi$
$347$	3017.41	0.466811	0.233405	+	0.972380i	$0.425013\pi$
$348$	−1011.54	−0.155817
$349$	−3644.69	−0.559014	−0.279507	−	0.960144i	$-0.590171\pi$
$349$	−3644.69	−0.559014	−0.279507	+	0.960144i	$0.590171\pi$
$350$	2909.67	0.444366
$351$	1226.37	0.186493
$352$	−1952.53	−0.295654
$353$	5509.59	0.830725	0.415362	−	0.909656i	$-0.363655\pi$
$353$	5509.59	0.830725	0.415362	+	0.909656i	$0.363655\pi$
$354$	−3574.71	−0.536706
$355$	15.9582	0.00238584
$356$	3399.10	0.506045
$357$	−5330.19	−0.790206
$358$	−6729.32	−0.993451
$359$	6501.97	0.955880	0.477940	−	0.878393i	$-0.341384\pi$
$359$	6501.97	0.955880	0.477940	+	0.878393i	$0.341384\pi$
$360$	−1446.27	−0.211737
$361$	−5325.34	−0.776402
$362$	3688.92	0.535596
$363$	−363.000	−0.0524864
$364$	4553.90	0.655739
$365$	−1860.35	−0.266781
$366$	−330.404	−0.0471872
$367$	4386.03	0.623839	0.311919	−	0.950109i	$-0.399028\pi$
$367$	4386.03	0.623839	0.311919	+	0.950109i	$0.399028\pi$
$368$	390.764	0.0553532
$369$	−3389.17	−0.478139
$370$	2006.82	0.281972
$371$	9848.26	1.37816
$372$	−1554.83	−0.216705
$373$	5516.10	0.765718	0.382859	−	0.923807i	$-0.374940\pi$
$373$	5516.10	0.765718	0.382859	+	0.923807i	$0.374940\pi$
$374$	−1668.33	−0.230661
$375$	−4216.70	−0.580665
$376$	6474.00	0.887955
$377$	3230.91	0.441380
$378$	1031.06	0.140297
$379$	−1483.89	−0.201114	−0.100557	−	0.994931i	$-0.532062\pi$
$379$	−1483.89	−0.201114	−0.100557	+	0.994931i	$0.532062\pi$
$380$	1296.88	0.175074
$381$	−593.289	−0.0797771
$382$	6658.29	0.891800
$383$	−5759.86	−0.768447	−0.384224	−	0.923240i	$-0.625531\pi$
$383$	−5759.86	−0.768447	−0.384224	+	0.923240i	$0.625531\pi$
$384$	−2367.84	−0.314670
$385$	−1625.38	−0.215161
$386$	3787.50	0.499426
$387$	2758.09	0.362278
$388$	−600.347	−0.0785516
$389$	−13294.4	−1.73278	−0.866392	−	0.499365i	$-0.833567\pi$
$389$	−13294.4	−1.73278	−0.866392	+	0.499365i	$0.833567\pi$
$390$	1718.74	0.223158
$391$	−9096.21	−1.17651
$392$	2400.43	0.309286
$393$	−4959.13	−0.636527
$394$	1586.51	0.202861
$395$	−1722.21	−0.219377
$396$	−469.281	−0.0595512
$397$	5824.00	0.736268	0.368134	−	0.929773i	$-0.379997\pi$
$397$	5824.00	0.736268	0.368134	+	0.929773i	$0.379997\pi$
$398$	3109.62	0.391636
$399$	−2484.92	−0.311784
$400$	274.961	0.0343701
$401$	−4672.22	−0.581844	−0.290922	−	0.956747i	$-0.593962\pi$
$401$	−4672.22	−0.581844	−0.290922	+	0.956747i	$0.593962\pi$
$402$	−390.281	−0.0484215
$403$	4966.19	0.613855
$404$	−2820.69	−0.347363
$405$	−565.875	−0.0694286
$406$	2716.36	0.332046
$407$	1750.13	0.213147
$408$	−5796.78	−0.703391
$409$	−13757.8	−1.66328	−0.831640	−	0.555316i	$-0.812597\pi$
$409$	−13757.8	−1.66328	−0.831640	+	0.555316i	$0.812597\pi$
$410$	−4749.86	−0.572144
$411$	4443.14	0.533246
$412$	1229.72	0.147049
$413$	−13958.9	−1.66313
$414$	1759.56	0.208883
$415$	−7039.16	−0.832624
$416$	8062.40	0.950220
$417$	7271.60	0.853937
$418$	−777.772	−0.0910097
$419$	−10323.2	−1.20363	−0.601817	−	0.798634i	$-0.705556\pi$
$419$	−10323.2	−1.20363	−0.601817	+	0.798634i	$0.705556\pi$
$420$	−2101.27	−0.244122
$421$	−7740.44	−0.896071	−0.448036	−	0.894016i	$-0.647876\pi$
$421$	−7740.44	−0.896071	−0.448036	+	0.894016i	$0.647876\pi$
$422$	−2183.30	−0.251851
$423$	2533.05	0.291161
$424$	10710.4	1.22675
$425$	−6400.54	−0.730521
$426$	−12.3727	−0.00140718
$427$	−1290.20	−0.146223
$428$	3319.47	0.374889
$429$	1498.90	0.168689
$430$	3865.42	0.433505
$431$	−9637.85	−1.07712	−0.538560	−	0.842587i	$-0.681032\pi$
$431$	−9637.85	−1.07712	−0.538560	+	0.842587i	$0.681032\pi$
$432$	97.4344	0.0108514
$433$	−723.323	−0.0802787	−0.0401393	−	0.999194i	$-0.512780\pi$
$433$	−723.323	−0.0802787	−0.0401393	+	0.999194i	$0.512780\pi$
$434$	4175.29	0.461798
$435$	−1490.81	−0.164319
$436$	310.684	0.0341263
$437$	−4240.63	−0.464204
$438$	1442.36	0.157349
$439$	2675.21	0.290845	0.145422	−	0.989370i	$-0.453546\pi$
$439$	2675.21	0.290845	0.145422	+	0.989370i	$0.453546\pi$
$440$	−1767.66	−0.191523
$441$	939.206	0.101415
$442$	6888.86	0.741334
$443$	7658.27	0.821344	0.410672	−	0.911783i	$-0.365294\pi$
$443$	7658.27	0.821344	0.410672	+	0.911783i	$0.365294\pi$
$444$	2262.55	0.241837
$445$	5009.58	0.533656
$446$	340.821	0.0361846
$447$	−4144.69	−0.438562
$448$	7389.01	0.779236
$449$	5831.58	0.612938	0.306469	−	0.951881i	$-0.400852\pi$
$449$	5831.58	0.612938	0.306469	+	0.951881i	$0.400852\pi$
$450$	1238.11	0.129700
$451$	−4142.32	−0.432493
$452$	−9288.28	−0.966557
$453$	−5157.65	−0.534940
$454$	1889.64	0.195342
$455$	6711.52	0.691519
$456$	−2702.45	−0.277530
$457$	−5599.31	−0.573140	−0.286570	−	0.958059i	$-0.592515\pi$
$457$	−5599.31	−0.573140	−0.286570	+	0.958059i	$0.592515\pi$
$458$	4251.79	0.433784
$459$	−2268.08	−0.230642
$460$	−3585.91	−0.363465
$461$	−6887.08	−0.695799	−0.347899	−	0.937532i	$-0.613105\pi$
$461$	−6887.08	−0.695799	−0.347899	+	0.937532i	$0.613105\pi$
$462$	1260.19	0.126903
$463$	−9050.25	−0.908425	−0.454212	−	0.890893i	$-0.650079\pi$
$463$	−9050.25	−0.908425	−0.454212	+	0.890893i	$0.650079\pi$
$464$	256.693	0.0256825
$465$	−2291.51	−0.228530
$466$	−9316.92	−0.926176
$467$	−17851.3	−1.76886	−0.884432	−	0.466669i	$-0.845454\pi$
$467$	−17851.3	−1.76886	−0.884432	+	0.466669i	$0.845454\pi$
$468$	1937.75	0.191395
$469$	−1524.01	−0.150047
$470$	3550.03	0.348406
$471$	8547.17	0.836164
$472$	−15180.9	−1.48041
$473$	3371.00	0.327693
$474$	1335.26	0.129389
$475$	−2983.92	−0.288235
$476$	−8422.07	−0.810977
$477$	4190.59	0.402251
$478$	3175.44	0.303852
$479$	−1159.94	−0.110645	−0.0553225	−	0.998469i	$-0.517619\pi$
$479$	−1159.94	−0.110645	−0.0553225	+	0.998469i	$0.517619\pi$
$480$	−3720.17	−0.353754
$481$	−7226.65	−0.685046
$482$	−4771.10	−0.450867
$483$	6870.90	0.647281
$484$	−573.566	−0.0538661
$485$	−884.790	−0.0828377
$486$	438.734	0.0409493
$487$	−7201.01	−0.670039	−0.335020	−	0.942211i	$-0.608743\pi$
$487$	−7201.01	−0.670039	−0.335020	+	0.942211i	$0.608743\pi$
$488$	−1403.14	−0.130158
$489$	−3084.92	−0.285286
$490$	1316.28	0.121354
$491$	7343.71	0.674984	0.337492	−	0.941328i	$-0.390421\pi$
$491$	7343.71	0.674984	0.337492	+	0.941328i	$0.390421\pi$
$492$	−5355.13	−0.490707
$493$	−5975.31	−0.545871
$494$	3211.57	0.292501
$495$	−691.625	−0.0628005
$496$	394.560	0.0357183
$497$	−48.3143	−0.00436055
$498$	5457.59	0.491086
$499$	−19022.2	−1.70652	−0.853258	−	0.521488i	$-0.825377\pi$
$499$	−19022.2	−1.70652	−0.853258	+	0.521488i	$0.825377\pi$
$500$	−6662.68	−0.595928
$501$	−8956.23	−0.798673
$502$	8390.19	0.745961
$503$	20189.8	1.78970	0.894849	−	0.446369i	$-0.147283\pi$
$503$	20189.8	1.78970	0.894849	+	0.446369i	$0.147283\pi$
$504$	4378.65	0.386986
$505$	−4157.13	−0.366317
$506$	2150.57	0.188942
$507$	401.747	0.0351918
$508$	−937.438	−0.0818742
$509$	1015.34	0.0884166	0.0442083	−	0.999022i	$-0.485923\pi$
$509$	1015.34	0.0884166	0.0442083	+	0.999022i	$0.485923\pi$
$510$	−3178.67	−0.275988
$511$	5632.29	0.487588
$512$	1304.62	0.112610
$513$	−1057.37	−0.0910023
$514$	12905.3	1.10745
$515$	1812.36	0.155072
$516$	4357.98	0.371801
$517$	3095.95	0.263365
$518$	−6075.75	−0.515354
$519$	10220.5	0.864416
$520$	7299.04	0.615546
$521$	8055.56	0.677391	0.338695	−	0.940896i	$-0.390014\pi$
$521$	8055.56	0.677391	0.338695	+	0.940896i	$0.390014\pi$
$522$	1155.85	0.0969164
$523$	13465.6	1.12583	0.562914	−	0.826515i	$-0.309680\pi$
$523$	13465.6	1.12583	0.562914	+	0.826515i	$0.309680\pi$
$524$	−7835.78	−0.653259
$525$	4834.70	0.401912
$526$	−14612.7	−1.21130
$527$	−9184.59	−0.759178
$528$	119.086	0.00981548
$529$	−441.489	−0.0362857
$530$	5873.04	0.481337
$531$	−5939.74	−0.485429
$532$	−3926.35	−0.319979
$533$	17104.5	1.39001
$534$	−3884.02	−0.314753
$535$	4892.22	0.395344
$536$	−1657.42	−0.133563
$537$	−11181.4	−0.898537
$538$	3988.74	0.319641
$539$	1147.92	0.0917335
$540$	−894.123	−0.0712536
$541$	−5680.56	−0.451435	−0.225717	−	0.974193i	$-0.572473\pi$
$541$	−5680.56	−0.451435	−0.225717	+	0.974193i	$0.572473\pi$
$542$	4522.73	0.358428
$543$	6129.52	0.484425
$544$	−14910.8	−1.17517
$545$	457.885	0.0359883
$546$	−5203.57	−0.407861
$547$	−2693.83	−0.210566	−0.105283	−	0.994442i	$-0.533575\pi$
$547$	−2693.83	−0.210566	−0.105283	+	0.994442i	$0.533575\pi$
$548$	7020.48	0.547263
$549$	−549.000	−0.0426790
$550$	1513.24	0.117318
$551$	−2785.68	−0.215379
$552$	7472.37	0.576169
$553$	5214.08	0.400949
$554$	11407.2	0.874811
$555$	3334.54	0.255033
$556$	11489.6	0.876384
$557$	760.631	0.0578617	0.0289309	−	0.999581i	$-0.490790\pi$
$557$	760.631	0.0578617	0.0289309	+	0.999581i	$0.490790\pi$
$558$	1776.65	0.134788
$559$	−13919.5	−1.05319
$560$	533.226	0.0402373
$561$	−2772.10	−0.208624
$562$	1148.25	0.0861853
$563$	19554.3	1.46379	0.731897	−	0.681415i	$-0.238635\pi$
$563$	19554.3	1.46379	0.731897	+	0.681415i	$0.238635\pi$
$564$	4002.40	0.298815
$565$	−13689.0	−1.01930
$566$	576.762	0.0428324
$567$	1713.21	0.126893
$568$	−52.5437	−0.00388148
$569$	16302.7	1.20113	0.600565	−	0.799576i	$-0.294942\pi$
$569$	16302.7	1.20113	0.600565	+	0.799576i	$0.294942\pi$
$570$	−1481.89	−0.108894
$571$	−747.371	−0.0547749	−0.0273875	−	0.999625i	$-0.508719\pi$
$571$	−747.371	−0.0547749	−0.0273875	+	0.999625i	$0.508719\pi$
$572$	2368.37	0.173123
$573$	11063.4	0.806598
$574$	14380.4	1.04569
$575$	8250.64	0.598392
$576$	3144.14	0.227441
$577$	−8779.32	−0.633428	−0.316714	−	0.948521i	$-0.602579\pi$
$577$	−8779.32	−0.633428	−0.316714	+	0.948521i	$0.602579\pi$
$578$	−3870.05	−0.278500
$579$	6293.31	0.451712
$580$	−2355.59	−0.168639
$581$	21311.4	1.52177
$582$	685.994	0.0488580
$583$	5121.83	0.363850
$584$	6125.33	0.434020
$585$	2855.86	0.201838
$586$	−10060.7	−0.709222
$587$	−13037.9	−0.916750	−0.458375	−	0.888759i	$-0.651568\pi$
$587$	−13037.9	−0.916750	−0.458375	+	0.888759i	$0.651568\pi$
$588$	1484.01	0.104081
$589$	−4281.83	−0.299541
$590$	−8324.45	−0.580868
$591$	2636.15	0.183480
$592$	−574.152	−0.0398607
$593$	7760.11	0.537386	0.268693	−	0.963226i	$-0.413408\pi$
$593$	7760.11	0.537386	0.268693	+	0.963226i	$0.413408\pi$
$594$	536.230	0.0370400
$595$	−12412.4	−0.855227
$596$	−6548.91	−0.450090
$597$	5166.95	0.354220
$598$	−8880.12	−0.607250
$599$	−17237.6	−1.17581	−0.587904	−	0.808930i	$-0.700047\pi$
$599$	−17237.6	−1.17581	−0.587904	+	0.808930i	$0.700047\pi$
$600$	5257.93	0.357757
$601$	−4025.85	−0.273241	−0.136620	−	0.990623i	$-0.543624\pi$
$601$	−4025.85	−0.273241	−0.136620	+	0.990623i	$0.543624\pi$
$602$	−11702.7	−0.792306
$603$	−648.490	−0.0437953
$604$	−8149.46	−0.549001
$605$	−845.320	−0.0568052
$606$	3223.10	0.216055
$607$	3234.85	0.216308	0.108154	−	0.994134i	$-0.465506\pi$
$607$	3234.85	0.216308	0.108154	+	0.994134i	$0.465506\pi$
$608$	−6951.37	−0.463676
$609$	4513.50	0.300322
$610$	−769.414	−0.0510699
$611$	−12783.8	−0.846444
$612$	−3583.73	−0.236705
$613$	−5418.08	−0.356989	−0.178494	−	0.983941i	$-0.557123\pi$
$613$	−5418.08	−0.356989	−0.178494	+	0.983941i	$0.557123\pi$
$614$	7932.75	0.521400
$615$	−7892.38	−0.517482
$616$	5351.69	0.350042
$617$	−14607.2	−0.953100	−0.476550	−	0.879147i	$-0.658113\pi$
$617$	−14607.2	−0.953100	−0.476550	+	0.879147i	$0.658113\pi$
$618$	−1405.16	−0.0914623
$619$	−12649.5	−0.821367	−0.410683	−	0.911778i	$-0.634710\pi$
$619$	−12649.5	−0.821367	−0.410683	+	0.911778i	$0.634710\pi$
$620$	−3620.75	−0.234537
$621$	2923.68	0.188926
$622$	−10980.7	−0.707855
$623$	−15166.8	−0.975350
$624$	−491.732	−0.0315465
$625$	−295.168	−0.0188908
$626$	2856.23	0.182361
$627$	−1292.35	−0.0823147
$628$	13505.1	0.858143
$629$	13365.1	0.847222
$630$	2401.04	0.151841
$631$	−11872.6	−0.749031	−0.374516	−	0.927221i	$-0.622191\pi$
$631$	−11872.6	−0.749031	−0.374516	+	0.927221i	$0.622191\pi$
$632$	5670.51	0.356900
$633$	−3627.77	−0.227790
$634$	11725.3	0.734500
$635$	−1381.59	−0.0863415
$636$	6621.43	0.412825
$637$	−4739.98	−0.294827
$638$	1412.71	0.0876642
$639$	−20.5585	−0.00127274
$640$	−5513.99	−0.340562
$641$	646.626	0.0398443	0.0199221	−	0.999802i	$-0.493658\pi$
$641$	646.626	0.0398443	0.0199221	+	0.999802i	$0.493658\pi$
$642$	−3793.03	−0.233176
$643$	−21716.6	−1.33191	−0.665956	−	0.745991i	$-0.731976\pi$
$643$	−21716.6	−1.33191	−0.665956	+	0.745991i	$0.731976\pi$
$644$	10856.5	0.664296
$645$	6422.78	0.392088
$646$	−5939.55	−0.361747
$647$	25853.8	1.57097	0.785485	−	0.618881i	$-0.212414\pi$
$647$	25853.8	1.57097	0.785485	+	0.618881i	$0.212414\pi$
$648$	1863.19	0.112952
$649$	−7259.69	−0.439087
$650$	−6248.49	−0.377055
$651$	6937.66	0.417678
$652$	−4874.39	−0.292785
$653$	−17993.1	−1.07829	−0.539147	−	0.842211i	$-0.681253\pi$
$653$	−17993.1	−1.07829	−0.539147	+	0.842211i	$0.681253\pi$
$654$	−355.007	−0.0212261
$655$	−11548.4	−0.688903
$656$	1358.94	0.0808805
$657$	2396.63	0.142316
$658$	−10747.9	−0.636772
$659$	32711.7	1.93363	0.966817	−	0.255469i	$-0.0822299\pi$
$659$	32711.7	1.93363	0.966817	+	0.255469i	$0.0822299\pi$
$660$	−1092.82	−0.0644513
$661$	4959.85	0.291854	0.145927	−	0.989295i	$-0.453383\pi$
$661$	4959.85	0.291854	0.145927	+	0.989295i	$0.453383\pi$
$662$	2945.49	0.172930
$663$	11446.5	0.670508
$664$	23177.0	1.35458
$665$	−5786.65	−0.337439
$666$	−2585.33	−0.150420
$667$	7702.50	0.447139
$668$	−14151.5	−0.819667
$669$	566.308	0.0327276
$670$	−908.848	−0.0524058
$671$	−671.000	−0.0386046
$672$	11263.0	0.646546
$673$	2173.95	0.124516	0.0622581	−	0.998060i	$-0.480170\pi$
$673$	2173.95	0.124516	0.0622581	+	0.998060i	$0.480170\pi$
$674$	−15444.8	−0.882660
$675$	2057.24	0.117309
$676$	634.789	0.0361168
$677$	−25464.0	−1.44558	−0.722792	−	0.691066i	$-0.757141\pi$
$677$	−25464.0	−1.44558	−0.722792	+	0.691066i	$0.757141\pi$
$678$	10613.4	0.601186
$679$	2678.75	0.151400
$680$	−13499.0	−0.761269
$681$	3139.82	0.176679
$682$	2171.46	0.121920
$683$	8671.36	0.485798	0.242899	−	0.970052i	$-0.421902\pi$
$683$	8671.36	0.485798	0.242899	+	0.970052i	$0.421902\pi$
$684$	−1670.73	−0.0933944
$685$	10346.8	0.577124
$686$	9113.21	0.507207
$687$	7064.77	0.392340
$688$	−1105.90	−0.0612819
$689$	−21149.1	−1.16940
$690$	4097.49	0.226070
$691$	−5827.32	−0.320813	−0.160406	−	0.987051i	$-0.551280\pi$
$691$	−5827.32	−0.320813	−0.160406	+	0.987051i	$0.551280\pi$
$692$	16149.2	0.887139
$693$	2093.93	0.114779
$694$	−5447.91	−0.297982
$695$	16933.4	0.924203
$696$	4908.61	0.267328
$697$	−31633.4	−1.71908
$698$	6580.45	0.356839
$699$	−15481.0	−0.837690
$700$	7639.18	0.412477
$701$	−16701.2	−0.899853	−0.449927	−	0.893065i	$-0.648550\pi$
$701$	−16701.2	−0.899853	−0.449927	+	0.893065i	$0.648550\pi$
$702$	−2214.20	−0.119045
$703$	6230.79	0.334280
$704$	3842.84	0.205728
$705$	5898.73	0.315119
$706$	−9947.50	−0.530282
$707$	12585.9	0.669508
$708$	−9385.22	−0.498189
$709$	−16128.8	−0.854345	−0.427173	−	0.904170i	$-0.640490\pi$
$709$	−16128.8	−0.854345	−0.427173	+	0.904170i	$0.640490\pi$
$710$	−28.8124	−0.00152297
$711$	2218.67	0.117028
$712$	−16494.4	−0.868195
$713$	11839.4	0.621866
$714$	9623.59	0.504417
$715$	3490.49	0.182569
$716$	−17667.5	−0.922156
$717$	5276.32	0.274822
$718$	−11739.2	−0.610173
$719$	21517.8	1.11610	0.558051	−	0.829807i	$-0.311549\pi$
$719$	21517.8	1.11610	0.558051	+	0.829807i	$0.311549\pi$
$720$	226.896	0.0117443
$721$	−5487.01	−0.283421
$722$	9614.84	0.495605
$723$	−7927.66	−0.407791
$724$	9685.08	0.497159
$725$	5419.86	0.277639
$726$	655.392	0.0335040
$727$	−10515.0	−0.536425	−0.268213	−	0.963360i	$-0.586433\pi$
$727$	−10515.0	−0.536425	−0.268213	+	0.963360i	$0.586433\pi$
$728$	−22098.2	−1.12502
$729$	729.000	0.0370370
$730$	3358.83	0.170296
$731$	25743.1	1.30252
$732$	−867.459	−0.0438008
$733$	36375.1	1.83294	0.916469	−	0.400106i	$-0.131027\pi$
$733$	36375.1	1.83294	0.916469	+	0.400106i	$0.131027\pi$
$734$	−7918.92	−0.398219
$735$	2187.13	0.109760
$736$	19220.8	0.962620
$737$	−792.599	−0.0396144
$738$	6119.11	0.305213
$739$	−16922.2	−0.842347	−0.421173	−	0.906980i	$-0.638382\pi$
$739$	−16922.2	−0.842347	−0.421173	+	0.906980i	$0.638382\pi$
$740$	5268.81	0.261737
$741$	5336.35	0.264556
$742$	−17780.9	−0.879728
$743$	−5912.56	−0.291939	−0.145970	−	0.989289i	$-0.546630\pi$
$743$	−5912.56	−0.291939	−0.145970	+	0.989289i	$0.546630\pi$
$744$	7544.97	0.371790
$745$	−9651.77	−0.474649
$746$	−9959.25	−0.488786
$747$	9068.34	0.444168
$748$	−4380.11	−0.214108
$749$	−14811.4	−0.722561
$750$	7613.20	0.370660
$751$	12125.9	0.589190	0.294595	−	0.955622i	$-0.404815\pi$
$751$	12125.9	0.589190	0.294595	+	0.955622i	$0.404815\pi$
$752$	−1015.66	−0.0492519
$753$	13941.1	0.674693
$754$	−5833.36	−0.281749
$755$	−12010.7	−0.578957
$756$	2707.00	0.130228
$757$	−15113.4	−0.725635	−0.362818	−	0.931860i	$-0.618185\pi$
$757$	−15113.4	−0.725635	−0.362818	+	0.931860i	$0.618185\pi$
$758$	2679.14	0.128378
$759$	3573.39	0.170890
$760$	−6293.20	−0.300366
$761$	−18329.3	−0.873108	−0.436554	−	0.899678i	$-0.643801\pi$
$761$	−18329.3	−0.873108	−0.436554	+	0.899678i	$0.643801\pi$
$762$	1071.18	0.0509247
$763$	−1386.27	−0.0657750
$764$	17481.0	0.827801
$765$	−5281.69	−0.249621
$766$	10399.4	0.490528
$767$	29976.7	1.41121
$768$	12659.5	0.594804
$769$	−36726.5	−1.72223	−0.861114	−	0.508412i	$-0.830233\pi$
$769$	−36726.5	−1.72223	−0.861114	+	0.508412i	$0.830233\pi$
$770$	2934.61	0.137345
$771$	21443.5	1.00165
$772$	9943.87	0.463585
$773$	−9316.97	−0.433516	−0.216758	−	0.976225i	$-0.569548\pi$
$773$	−9316.97	−0.433516	−0.216758	+	0.976225i	$0.569548\pi$
$774$	−4979.70	−0.231255
$775$	8330.81	0.386131
$776$	2913.24	0.134767
$777$	−10095.5	−0.466117
$778$	24002.9	1.10610
$779$	−14747.4	−0.678281
$780$	4512.46	0.207144
$781$	−25.1271	−0.00115124
$782$	16423.1	0.751009
$783$	1920.57	0.0876571
$784$	−376.588	−0.0171551
$785$	19903.8	0.904967
$786$	8953.66	0.406319
$787$	29675.5	1.34411	0.672056	−	0.740501i	$-0.265412\pi$
$787$	29675.5	1.34411	0.672056	+	0.740501i	$0.265412\pi$
$788$	4165.30	0.188303
$789$	−24280.5	−1.09558
$790$	3109.43	0.140036
$791$	41444.2	1.86294
$792$	2277.23	0.102169
$793$	2770.69	0.124073
$794$	−10515.2	−0.469987
$795$	9758.65	0.435350
$796$	8164.14	0.363531
$797$	31273.6	1.38992	0.694961	−	0.719047i	$-0.255422\pi$
$797$	31273.6	1.38992	0.694961	+	0.719047i	$0.255422\pi$
$798$	4486.50	0.199023
$799$	23642.6	1.04683
$800$	13524.7	0.597713
$801$	−6453.70	−0.284682
$802$	8435.64	0.371412
$803$	2929.21	0.128729
$804$	−1024.66	−0.0449465
$805$	16000.3	0.700543
$806$	−8966.40	−0.391846
$807$	6627.69	0.289103
$808$	13687.7	0.595954
$809$	9908.62	0.430616	0.215308	−	0.976546i	$-0.430924\pi$
$809$	9908.62	0.430616	0.215308	+	0.976546i	$0.430924\pi$
$810$	1021.68	0.0443188
$811$	10623.5	0.459979	0.229989	−	0.973193i	$-0.426131\pi$
$811$	10623.5	0.459979	0.229989	+	0.973193i	$0.426131\pi$
$812$	7131.66	0.308217
$813$	7514.98	0.324184
$814$	−3159.85	−0.136060
$815$	−7183.86	−0.308760
$816$	909.419	0.0390148
$817$	12001.4	0.513923
$818$	24839.6	1.06173
$819$	−8646.25	−0.368894
$820$	−12470.5	−0.531085
$821$	−1972.08	−0.0838320	−0.0419160	−	0.999121i	$-0.513346\pi$
$821$	−1972.08	−0.0838320	−0.0419160	+	0.999121i	$0.513346\pi$
$822$	−8022.04	−0.340390
$823$	−37003.2	−1.56725	−0.783627	−	0.621232i	$-0.786632\pi$
$823$	−37003.2	−1.56725	−0.783627	+	0.621232i	$0.786632\pi$
$824$	−5967.33	−0.252284
$825$	2514.41	0.106110
$826$	25202.7	1.06164
$827$	576.571	0.0242434	0.0121217	−	0.999927i	$-0.496141\pi$
$827$	576.571	0.0242434	0.0121217	+	0.999927i	$0.496141\pi$
$828$	4619.62	0.193892
$829$	2620.12	0.109771	0.0548857	−	0.998493i	$-0.482521\pi$
$829$	2620.12	0.109771	0.0548857	+	0.998493i	$0.482521\pi$
$830$	12709.1	0.531494
$831$	18954.2	0.791232
$832$	−15867.8	−0.661200
$833$	8766.23	0.364624
$834$	−13128.8	−0.545099
$835$	−20856.4	−0.864391
$836$	−2042.00	−0.0844784
$837$	2952.08	0.121910
$838$	18638.4	0.768323
$839$	−1481.10	−0.0609456	−0.0304728	−	0.999536i	$-0.509701\pi$
$839$	−1481.10	−0.0609456	−0.0304728	+	0.999536i	$0.509701\pi$
$840$	10196.6	0.418828
$841$	−19329.2	−0.792538
$842$	13975.3	0.571995
$843$	1907.94	0.0779512
$844$	−5732.13	−0.233777
$845$	935.551	0.0380875
$846$	−4573.39	−0.185859
$847$	2559.25	0.103821
$848$	−1680.28	−0.0680436
$849$	958.348	0.0387402
$850$	11556.1	0.466318
$851$	−17228.4	−0.693985
$852$	−32.4839	−0.00130620
$853$	6213.50	0.249410	0.124705	−	0.992194i	$-0.460202\pi$
$853$	6213.50	0.249410	0.124705	+	0.992194i	$0.460202\pi$
$854$	2329.44	0.0933393
$855$	−2462.31	−0.0984904
$856$	−16108.0	−0.643178
$857$	−4020.79	−0.160265	−0.0801327	−	0.996784i	$-0.525534\pi$
$857$	−4020.79	−0.160265	−0.0801327	+	0.996784i	$0.525534\pi$
$858$	−2706.24	−0.107680
$859$	−44536.3	−1.76898	−0.884492	−	0.466555i	$-0.845495\pi$
$859$	−44536.3	−1.76898	−0.884492	+	0.466555i	$0.845495\pi$
$860$	10148.5	0.402395
$861$	23894.6	0.945789
$862$	17401.0	0.687565
$863$	9559.91	0.377084	0.188542	−	0.982065i	$-0.439624\pi$
$863$	9559.91	0.377084	0.188542	+	0.982065i	$0.439624\pi$
$864$	4792.58	0.188712
$865$	23800.6	0.935544
$866$	1305.95	0.0512448
$867$	−6430.48	−0.251892
$868$	10962.0	0.428657
$869$	2711.71	0.105856
$870$	2691.64	0.104891
$871$	3272.80	0.127319
$872$	−1507.62	−0.0585487
$873$	1139.85	0.0441902
$874$	7656.41	0.296318
$875$	29728.8	1.14859
$876$	3786.84	0.146056
$877$	−33222.4	−1.27918	−0.639590	−	0.768716i	$-0.720896\pi$
$877$	−33222.4	−1.27918	−0.639590	+	0.768716i	$0.720896\pi$
$878$	−4830.06	−0.185657
$879$	−16716.9	−0.641463
$880$	277.317	0.0106231
$881$	−2683.97	−0.102639	−0.0513197	−	0.998682i	$-0.516343\pi$
$881$	−2683.97	−0.102639	−0.0513197	+	0.998682i	$0.516343\pi$
$882$	−1695.72	−0.0647370
$883$	−22446.2	−0.855463	−0.427732	−	0.903906i	$-0.640687\pi$
$883$	−22446.2	−0.855463	−0.427732	+	0.903906i	$0.640687\pi$
$884$	18086.3	0.688133
$885$	−13831.9	−0.525372
$886$	−13826.9	−0.524294
$887$	34172.1	1.29356	0.646780	−	0.762677i	$-0.276115\pi$
$887$	34172.1	1.29356	0.646780	+	0.762677i	$0.276115\pi$
$888$	−10979.2	−0.414908
$889$	4182.84	0.157804
$890$	−9044.74	−0.340652
$891$	891.000	0.0335013
$892$	894.807	0.0335878
$893$	11022.1	0.413037
$894$	7483.20	0.279950
$895$	−26038.3	−0.972473
$896$	16693.9	0.622436
$897$	−14755.2	−0.549233
$898$	−10528.8	−0.391261
$899$	7777.34	0.288530
$900$	3250.59	0.120392
$901$	39113.5	1.44624
$902$	7478.91	0.276076
$903$	−19445.3	−0.716609
$904$	45072.2	1.65827
$905$	14273.8	0.524286
$906$	9312.08	0.341472
$907$	−29935.2	−1.09590	−0.547950	−	0.836511i	$-0.684591\pi$
$907$	−29935.2	−1.09590	−0.547950	+	0.836511i	$0.684591\pi$
$908$	4961.14	0.181323
$909$	5355.51	0.195414
$910$	−12117.6	−0.441421
$911$	−25976.1	−0.944706	−0.472353	−	0.881410i	$-0.656595\pi$
$911$	−25976.1	−0.944706	−0.472353	+	0.881410i	$0.656595\pi$
$912$	423.969	0.0153937
$913$	11083.5	0.401765
$914$	10109.5	0.365856
$915$	−1278.46	−0.0461908
$916$	11162.8	0.402653
$917$	34963.2	1.25909
$918$	4094.99	0.147227
$919$	−20218.2	−0.725719	−0.362860	−	0.931844i	$-0.618200\pi$
$919$	−20218.2	−0.725719	−0.362860	+	0.931844i	$0.618200\pi$
$920$	17400.9	0.623579
$921$	13181.1	0.471586
$922$	12434.5	0.444154
$923$	103.755	0.00370003
$924$	3308.56	0.117796
$925$	−12122.7	−0.430911
$926$	16340.1	0.579881
$927$	−2334.81	−0.0827241
$928$	12626.2	0.446632
$929$	−16195.0	−0.571948	−0.285974	−	0.958237i	$-0.592317\pi$
$929$	−16195.0	−0.571948	−0.285974	+	0.958237i	$0.592317\pi$
$930$	4137.30	0.145879
$931$	4086.80	0.143866
$932$	−24461.1	−0.859709
$933$	−18245.5	−0.640227
$934$	32230.3	1.12913
$935$	−6455.40	−0.225790
$936$	−9403.13	−0.328366
$937$	−371.320	−0.0129461	−0.00647305	−	0.999979i	$-0.502060\pi$
$937$	−371.320	−0.0129461	−0.00647305	+	0.999979i	$0.502060\pi$
$938$	2751.58	0.0957807
$939$	4745.92	0.164938
$940$	9320.41	0.323402
$941$	−14889.6	−0.515820	−0.257910	−	0.966169i	$-0.583034\pi$
$941$	−14889.6	−0.515820	−0.257910	+	0.966169i	$0.583034\pi$
$942$	−15431.8	−0.533754
$943$	40777.1	1.40815
$944$	2381.63	0.0821137
$945$	3989.57	0.137334
$946$	−6086.30	−0.209178
$947$	24864.8	0.853217	0.426608	−	0.904436i	$-0.359708\pi$
$947$	24864.8	0.853217	0.426608	+	0.904436i	$0.359708\pi$
$948$	3505.66	0.120104
$949$	−12095.3	−0.413730
$950$	5387.43	0.183991
$951$	19482.8	0.664326
$952$	40868.9	1.39135
$953$	−15487.2	−0.526422	−0.263211	−	0.964738i	$-0.584782\pi$
$953$	−15487.2	−0.526422	−0.263211	+	0.964738i	$0.584782\pi$
$954$	−7566.06	−0.256772
$955$	25763.4	0.872969
$956$	8336.96	0.282046
$957$	2347.36	0.0792888
$958$	2094.26	0.0706287
$959$	−31325.3	−1.05479
$960$	7321.78	0.246156
$961$	−17836.5	−0.598722
$962$	13047.6	0.437290
$963$	−6302.50	−0.210899
$964$	−12526.3	−0.418510
$965$	14655.3	0.488880
$966$	−12405.3	−0.413183
$967$	−3935.44	−0.130874	−0.0654371	−	0.997857i	$-0.520844\pi$
$967$	−3935.44	−0.130874	−0.0654371	+	0.997857i	$0.520844\pi$
$968$	2783.28	0.0924153
$969$	−9869.17	−0.327186
$970$	1597.48	0.0528783
$971$	22495.1	0.743461	0.371731	−	0.928341i	$-0.378765\pi$
$971$	22495.1	0.743461	0.371731	+	0.928341i	$0.378765\pi$
$972$	1151.87	0.0380106
$973$	−51266.7	−1.68914
$974$	13001.3	0.427710
$975$	−10382.5	−0.341032
$976$	220.130	0.00721944
$977$	13829.8	0.452870	0.226435	−	0.974026i	$-0.427293\pi$
$977$	13829.8	0.452870	0.226435	+	0.974026i	$0.427293\pi$
$978$	5569.78	0.182108
$979$	−7887.85	−0.257504
$980$	3455.82	0.112645
$981$	−589.880	−0.0191982
$982$	−13259.0	−0.430867
$983$	−16983.2	−0.551049	−0.275524	−	0.961294i	$-0.588851\pi$
$983$	−16983.2	−0.551049	−0.275524	+	0.961294i	$0.588851\pi$
$984$	25986.2	0.841881
$985$	6138.81	0.198577
$986$	10788.3	0.348449
$987$	−17858.7	−0.575935
$988$	8431.81	0.271510
$989$	−33184.3	−1.06693
$990$	1248.72	0.0400878
$991$	−48879.3	−1.56680	−0.783401	−	0.621517i	$-0.786517\pi$
$991$	−48879.3	−1.56680	−0.783401	+	0.621517i	$0.786517\pi$
$992$	19407.6	0.621160
$993$	4894.23	0.156408
$994$	87.2309	0.00278350
$995$	12032.3	0.383366
$996$	14328.6	0.455843
$997$	−11929.6	−0.378950	−0.189475	−	0.981886i	$-0.560679\pi$
$997$	−11929.6	−0.378950	−0.189475	+	0.981886i	$0.560679\pi$
$998$	34344.4	1.08933
$999$	−4295.78	−0.136049

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2013.4.a.d.1.15	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.4.a.d.1.15	✓	37	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:	\( N \)	\(=\)	\( 2013 = 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2013.a (trivial)

\(f(q)\)	\(=\)	\(q-1.80549 q^{2} -3.00000 q^{3} -4.74021 q^{4} -6.98611 q^{5} +5.41646 q^{6} +21.1508 q^{7} +23.0023 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-1.80549 q^{2} -3.00000 q^{3} -4.74021 q^{4} -6.98611 q^{5} +5.41646 q^{6} +21.1508 q^{7} +23.0023 q^{8} +9.00000 q^{9} +12.6133 q^{10} +11.0000 q^{11} +14.2206 q^{12} -45.4212 q^{13} -38.1875 q^{14} +20.9583 q^{15} -3.60868 q^{16} +84.0029 q^{17} -16.2494 q^{18} +39.1620 q^{19} +33.1157 q^{20} -63.4524 q^{21} -19.8604 q^{22} -108.284 q^{23} -69.0069 q^{24} -76.1942 q^{25} +82.0074 q^{26} -27.0000 q^{27} -100.259 q^{28} -71.1321 q^{29} -37.8400 q^{30} -109.336 q^{31} -177.503 q^{32} -33.0000 q^{33} -151.666 q^{34} -147.762 q^{35} -42.6619 q^{36} +159.103 q^{37} -70.7065 q^{38} +136.264 q^{39} -160.697 q^{40} -376.575 q^{41} +114.563 q^{42} +306.455 q^{43} -52.1423 q^{44} -62.8750 q^{45} +195.506 q^{46} +281.450 q^{47} +10.8260 q^{48} +104.356 q^{49} +137.568 q^{50} -252.009 q^{51} +215.306 q^{52} +465.621 q^{53} +48.7482 q^{54} -76.8473 q^{55} +486.517 q^{56} -117.486 q^{57} +128.428 q^{58} -659.971 q^{59} -99.3470 q^{60} -61.0000 q^{61} +197.406 q^{62} +190.357 q^{63} +349.349 q^{64} +317.318 q^{65} +59.5811 q^{66} -72.0545 q^{67} -398.192 q^{68} +324.853 q^{69} +266.782 q^{70} -2.28428 q^{71} +207.021 q^{72} +266.292 q^{73} -287.259 q^{74} +228.583 q^{75} -185.636 q^{76} +232.659 q^{77} -246.022 q^{78} +246.519 q^{79} +25.2107 q^{80} +81.0000 q^{81} +679.901 q^{82} +1007.59 q^{83} +300.778 q^{84} -586.854 q^{85} -553.300 q^{86} +213.396 q^{87} +253.025 q^{88} -717.077 q^{89} +113.520 q^{90} -960.694 q^{91} +513.291 q^{92} +328.009 q^{93} -508.155 q^{94} -273.590 q^{95} +532.509 q^{96} +126.650 q^{97} -188.414 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q - 4 q^{2} - 111 q^{3} + 158 q^{4} - 15 q^{5} + 12 q^{6} - 77 q^{7} - 69 q^{8} + 333 q^{9}+O(q^{10}) \) 37 * q - 4 * q^2 - 111 * q^3 + 158 * q^4 - 15 * q^5 + 12 * q^6 - 77 * q^7 - 69 * q^8 + 333 * q^9	\( 37 q - 4 q^{2} - 111 q^{3} + 158 q^{4} - 15 q^{5} + 12 q^{6} - 77 q^{7} - 69 q^{8} + 333 q^{9} - 45 q^{10} + 407 q^{11} - 474 q^{12} - 169 q^{13} + 102 q^{14} + 45 q^{15} + 598 q^{16} - 338 q^{17} - 36 q^{18} - 235 q^{19} - 550 q^{20} + 231 q^{21} - 44 q^{22} - 53 q^{23} + 207 q^{24} + 750 q^{25} - 75 q^{26} - 999 q^{27} - 1378 q^{28} - 30 q^{29} + 135 q^{30} - 506 q^{31} - 841 q^{32} - 1221 q^{33} - 316 q^{34} - 822 q^{35} + 1422 q^{36} - 830 q^{37} - 371 q^{38} + 507 q^{39} - 613 q^{40} + 16 q^{41} - 306 q^{42} - 1137 q^{43} + 1738 q^{44} - 135 q^{45} - 659 q^{46} - 489 q^{47} - 1794 q^{48} + 2214 q^{49} + 1066 q^{50} + 1014 q^{51} - 2342 q^{52} + 731 q^{53} + 108 q^{54} - 165 q^{55} + 3051 q^{56} + 705 q^{57} - 611 q^{58} - 425 q^{59} + 1650 q^{60} - 2257 q^{61} + 453 q^{62} - 693 q^{63} + 4919 q^{64} + 1346 q^{65} + 132 q^{66} - 1907 q^{67} - 3236 q^{68} + 159 q^{69} - 1050 q^{70} - 561 q^{71} - 621 q^{72} - 2397 q^{73} - 1840 q^{74} - 2250 q^{75} - 3868 q^{76} - 847 q^{77} + 225 q^{78} + 393 q^{79} - 4031 q^{80} + 2997 q^{81} - 1946 q^{82} - 4191 q^{83} + 4134 q^{84} - 2667 q^{85} + 2405 q^{86} + 90 q^{87} - 759 q^{88} + 1437 q^{89} - 405 q^{90} - 5192 q^{91} - 737 q^{92} + 1518 q^{93} - 1960 q^{94} + 1356 q^{95} + 2523 q^{96} - 2368 q^{97} - 3014 q^{98} + 3663 q^{99}+O(q^{100}) \) 37 * q - 4 * q^2 - 111 * q^3 + 158 * q^4 - 15 * q^5 + 12 * q^6 - 77 * q^7 - 69 * q^8 + 333 * q^9 - 45 * q^10 + 407 * q^11 - 474 * q^12 - 169 * q^13 + 102 * q^14 + 45 * q^15 + 598 * q^16 - 338 * q^17 - 36 * q^18 - 235 * q^19 - 550 * q^20 + 231 * q^21 - 44 * q^22 - 53 * q^23 + 207 * q^24 + 750 * q^25 - 75 * q^26 - 999 * q^27 - 1378 * q^28 - 30 * q^29 + 135 * q^30 - 506 * q^31 - 841 * q^32 - 1221 * q^33 - 316 * q^34 - 822 * q^35 + 1422 * q^36 - 830 * q^37 - 371 * q^38 + 507 * q^39 - 613 * q^40 + 16 * q^41 - 306 * q^42 - 1137 * q^43 + 1738 * q^44 - 135 * q^45 - 659 * q^46 - 489 * q^47 - 1794 * q^48 + 2214 * q^49 + 1066 * q^50 + 1014 * q^51 - 2342 * q^52 + 731 * q^53 + 108 * q^54 - 165 * q^55 + 3051 * q^56 + 705 * q^57 - 611 * q^58 - 425 * q^59 + 1650 * q^60 - 2257 * q^61 + 453 * q^62 - 693 * q^63 + 4919 * q^64 + 1346 * q^65 + 132 * q^66 - 1907 * q^67 - 3236 * q^68 + 159 * q^69 - 1050 * q^70 - 561 * q^71 - 621 * q^72 - 2397 * q^73 - 1840 * q^74 - 2250 * q^75 - 3868 * q^76 - 847 * q^77 + 225 * q^78 + 393 * q^79 - 4031 * q^80 + 2997 * q^81 - 1946 * q^82 - 4191 * q^83 + 4134 * q^84 - 2667 * q^85 + 2405 * q^86 + 90 * q^87 - 759 * q^88 + 1437 * q^89 - 405 * q^90 - 5192 * q^91 - 737 * q^92 + 1518 * q^93 - 1960 * q^94 + 1356 * q^95 + 2523 * q^96 - 2368 * q^97 - 3014 * q^98 + 3663 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more