LMFDB - Embedded newform 2013.4.a.c.1.13

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			1.13
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)

$f(q)$	$=$	$q-2.43775 q^{2} +3.00000 q^{3} -2.05739 q^{4} -9.35530 q^{5} -7.31324 q^{6} +19.3013 q^{7} +24.5174 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.43775 q^{2} +3.00000 q^{3} -2.05739 q^{4} -9.35530 q^{5} -7.31324 q^{6} +19.3013 q^{7} +24.5174 q^{8} +9.00000 q^{9} +22.8059 q^{10} -11.0000 q^{11} -6.17216 q^{12} +25.2121 q^{13} -47.0517 q^{14} -28.0659 q^{15} -43.3081 q^{16} +24.5275 q^{17} -21.9397 q^{18} -79.9793 q^{19} +19.2475 q^{20} +57.9039 q^{21} +26.8152 q^{22} -10.4680 q^{23} +73.5521 q^{24} -37.4784 q^{25} -61.4607 q^{26} +27.0000 q^{27} -39.7102 q^{28} +30.4198 q^{29} +68.4176 q^{30} +63.5596 q^{31} -90.5648 q^{32} -33.0000 q^{33} -59.7918 q^{34} -180.569 q^{35} -18.5165 q^{36} -25.8919 q^{37} +194.969 q^{38} +75.6362 q^{39} -229.367 q^{40} +236.491 q^{41} -141.155 q^{42} -69.2058 q^{43} +22.6312 q^{44} -84.1977 q^{45} +25.5182 q^{46} -323.660 q^{47} -129.924 q^{48} +29.5400 q^{49} +91.3629 q^{50} +73.5824 q^{51} -51.8710 q^{52} +161.474 q^{53} -65.8192 q^{54} +102.908 q^{55} +473.217 q^{56} -239.938 q^{57} -74.1558 q^{58} -794.740 q^{59} +57.7424 q^{60} -61.0000 q^{61} -154.942 q^{62} +173.712 q^{63} +567.239 q^{64} -235.866 q^{65} +80.4457 q^{66} +486.715 q^{67} -50.4625 q^{68} -31.4039 q^{69} +440.183 q^{70} -850.549 q^{71} +220.656 q^{72} -169.457 q^{73} +63.1180 q^{74} -112.435 q^{75} +164.548 q^{76} -212.314 q^{77} -184.382 q^{78} +481.917 q^{79} +405.160 q^{80} +81.0000 q^{81} -576.505 q^{82} -1337.68 q^{83} -119.131 q^{84} -229.462 q^{85} +168.706 q^{86} +91.2595 q^{87} -269.691 q^{88} +1325.06 q^{89} +205.253 q^{90} +486.626 q^{91} +21.5366 q^{92} +190.679 q^{93} +789.002 q^{94} +748.230 q^{95} -271.694 q^{96} +1651.94 q^{97} -72.0112 q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q - 8 q^{2} + 111 q^{3} + 130 q^{4} - 35 q^{5} - 24 q^{6} - 35 q^{7} - 117 q^{8} + 333 q^{9}+O(q^{10}) $ 37 * q - 8 * q^2 + 111 * q^3 + 130 * q^4 - 35 * q^5 - 24 * q^6 - 35 * q^7 - 117 * q^8 + 333 * q^9	\( 37 q - 8 q^{2} + 111 q^{3} + 130 q^{4} - 35 q^{5} - 24 q^{6} - 35 q^{7} - 117 q^{8} + 333 q^{9} - 41 q^{10} - 407 q^{11} + 390 q^{12} + 51 q^{13} - 228 q^{14} - 105 q^{15} + 462 q^{16} - 190 q^{17} - 72 q^{18} - 51 q^{19} - 720 q^{20} - 105 q^{21} + 88 q^{22} - 583 q^{23} - 351 q^{24} + 598 q^{25} - 1019 q^{26} + 999 q^{27} - 498 q^{28} - 566 q^{29} - 123 q^{30} - 696 q^{31} - 859 q^{32} - 1221 q^{33} - 348 q^{34} - 1102 q^{35} + 1170 q^{36} - 1022 q^{37} - 455 q^{38} + 153 q^{39} - 503 q^{40} - 790 q^{41} - 684 q^{42} - 87 q^{43} - 1430 q^{44} - 315 q^{45} - 303 q^{46} - 1603 q^{47} + 1386 q^{48} + 110 q^{49} - 1926 q^{50} - 570 q^{51} + 736 q^{52} - 2619 q^{53} - 216 q^{54} + 385 q^{55} - 4937 q^{56} - 153 q^{57} - 1099 q^{58} - 2471 q^{59} - 2160 q^{60} - 2257 q^{61} - 2909 q^{62} - 315 q^{63} - 265 q^{64} - 1970 q^{65} + 264 q^{66} - 3033 q^{67} - 1956 q^{68} - 1749 q^{69} + 2410 q^{70} - 3891 q^{71} - 1053 q^{72} + 391 q^{73} - 532 q^{74} + 1794 q^{75} + 1554 q^{76} + 385 q^{77} - 3057 q^{78} + 67 q^{79} - 5111 q^{80} + 2997 q^{81} - 4818 q^{82} - 5315 q^{83} - 1494 q^{84} - 2747 q^{85} - 5195 q^{86} - 1698 q^{87} + 1287 q^{88} - 8945 q^{89} - 369 q^{90} - 4432 q^{91} - 4701 q^{92} - 2088 q^{93} - 372 q^{94} - 3388 q^{95} - 2577 q^{96} - 3784 q^{97} - 4502 q^{98} - 3663 q^{99}+O(q^{100}) \) 37 * q - 8 * q^2 + 111 * q^3 + 130 * q^4 - 35 * q^5 - 24 * q^6 - 35 * q^7 - 117 * q^8 + 333 * q^9 - 41 * q^10 - 407 * q^11 + 390 * q^12 + 51 * q^13 - 228 * q^14 - 105 * q^15 + 462 * q^16 - 190 * q^17 - 72 * q^18 - 51 * q^19 - 720 * q^20 - 105 * q^21 + 88 * q^22 - 583 * q^23 - 351 * q^24 + 598 * q^25 - 1019 * q^26 + 999 * q^27 - 498 * q^28 - 566 * q^29 - 123 * q^30 - 696 * q^31 - 859 * q^32 - 1221 * q^33 - 348 * q^34 - 1102 * q^35 + 1170 * q^36 - 1022 * q^37 - 455 * q^38 + 153 * q^39 - 503 * q^40 - 790 * q^41 - 684 * q^42 - 87 * q^43 - 1430 * q^44 - 315 * q^45 - 303 * q^46 - 1603 * q^47 + 1386 * q^48 + 110 * q^49 - 1926 * q^50 - 570 * q^51 + 736 * q^52 - 2619 * q^53 - 216 * q^54 + 385 * q^55 - 4937 * q^56 - 153 * q^57 - 1099 * q^58 - 2471 * q^59 - 2160 * q^60 - 2257 * q^61 - 2909 * q^62 - 315 * q^63 - 265 * q^64 - 1970 * q^65 + 264 * q^66 - 3033 * q^67 - 1956 * q^68 - 1749 * q^69 + 2410 * q^70 - 3891 * q^71 - 1053 * q^72 + 391 * q^73 - 532 * q^74 + 1794 * q^75 + 1554 * q^76 + 385 * q^77 - 3057 * q^78 + 67 * q^79 - 5111 * q^80 + 2997 * q^81 - 4818 * q^82 - 5315 * q^83 - 1494 * q^84 - 2747 * q^85 - 5195 * q^86 - 1698 * q^87 + 1287 * q^88 - 8945 * q^89 - 369 * q^90 - 4432 * q^91 - 4701 * q^92 - 2088 * q^93 - 372 * q^94 - 3388 * q^95 - 2577 * q^96 - 3784 * q^97 - 4502 * 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 $

$2$	−2.43775	−0.861874	−0.430937	−	0.902382i	$-0.641817\pi$
$2$	−2.43775	−0.861874	−0.430937	+	0.902382i	$0.641817\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−2.05739	−0.257173
$5$	−9.35530	−0.836763	−0.418382	−	0.908271i	$-0.637403\pi$
$5$	−9.35530	−0.836763	−0.418382	+	0.908271i	$0.637403\pi$
$6$	−7.31324	−0.497603
$7$	19.3013	1.04217	0.521086	−	0.853504i	$-0.325527\pi$
$7$	19.3013	1.04217	0.521086	+	0.853504i	$0.325527\pi$
$8$	24.5174	1.08352
$9$	9.00000	0.333333
$10$	22.8059	0.721184
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−6.17216	−0.148479
$13$	25.2121	0.537890	0.268945	−	0.963156i	$-0.413325\pi$
$13$	25.2121	0.537890	0.268945	+	0.963156i	$0.413325\pi$
$14$	−47.0517	−0.898221
$15$	−28.0659	−0.483105
$16$	−43.3081	−0.676689
$17$	24.5275	0.349929	0.174964	−	0.984575i	$-0.444019\pi$
$17$	24.5275	0.349929	0.174964	+	0.984575i	$0.444019\pi$
$18$	−21.9397	−0.287291
$19$	−79.9793	−0.965711	−0.482855	−	0.875700i	$-0.660400\pi$
$19$	−79.9793	−0.965711	−0.482855	+	0.875700i	$0.660400\pi$
$20$	19.2475	0.215193
$21$	57.9039	0.601698
$22$	26.8152	0.259865
$23$	−10.4680	−0.0949008	−0.0474504	−	0.998874i	$-0.515110\pi$
$23$	−10.4680	−0.0949008	−0.0474504	+	0.998874i	$0.515110\pi$
$24$	73.5521	0.625573
$25$	−37.4784	−0.299827
$26$	−61.4607	−0.463593
$27$	27.0000	0.192450
$28$	−39.7102	−0.268019
$29$	30.4198	0.194787	0.0973934	−	0.995246i	$-0.468949\pi$
$29$	30.4198	0.194787	0.0973934	+	0.995246i	$0.468949\pi$
$30$	68.4176	0.416376
$31$	63.5596	0.368246	0.184123	−	0.982903i	$-0.441055\pi$
$31$	63.5596	0.368246	0.184123	+	0.982903i	$0.441055\pi$
$32$	−90.5648	−0.500305
$33$	−33.0000	−0.174078
$34$	−59.7918	−0.301594
$35$	−180.569	−0.872051
$36$	−18.5165	−0.0857244
$37$	−25.8919	−0.115044	−0.0575218	−	0.998344i	$-0.518320\pi$
$37$	−25.8919	−0.115044	−0.0575218	+	0.998344i	$0.518320\pi$
$38$	194.969	0.832321
$39$	75.6362	0.310551
$40$	−229.367	−0.906654
$41$	236.491	0.900821	0.450410	−	0.892822i	$-0.351278\pi$
$41$	236.491	0.900821	0.450410	+	0.892822i	$0.351278\pi$
$42$	−141.155	−0.518588
$43$	−69.2058	−0.245437	−0.122718	−	0.992442i	$-0.539161\pi$
$43$	−69.2058	−0.245437	−0.122718	+	0.992442i	$0.539161\pi$
$44$	22.6312	0.0775407
$45$	−84.1977	−0.278921
$46$	25.5182	0.0817926
$47$	−323.660	−1.00448	−0.502242	−	0.864727i	$-0.667491\pi$
$47$	−323.660	−1.00448	−0.502242	+	0.864727i	$0.667491\pi$
$48$	−129.924	−0.390686
$49$	29.5400	0.0861226
$50$	91.3629	0.258413
$51$	73.5824	0.202031
$52$	−51.8710	−0.138331
$53$	161.474	0.418495	0.209247	−	0.977863i	$-0.432899\pi$
$53$	161.474	0.418495	0.209247	+	0.977863i	$0.432899\pi$
$54$	−65.8192	−0.165868
$55$	102.908	0.252294
$56$	473.217	1.12922
$57$	−239.938	−0.557553
$58$	−74.1558	−0.167882
$59$	−794.740	−1.75367	−0.876833	−	0.480795i	$-0.840348\pi$
$59$	−794.740	−1.75367	−0.876833	+	0.480795i	$0.840348\pi$
$60$	57.7424	0.124242
$61$	−61.0000	−0.128037
$61$	−61.0000	−0.128037
$62$	−154.942	−0.317382
$63$	173.712	0.347391
$64$	567.239	1.10789
$65$	−235.866	−0.450087
$66$	80.4457	0.150033
$67$	486.715	0.887487	0.443744	−	0.896154i	$-0.353650\pi$
$67$	486.715	0.887487	0.443744	+	0.896154i	$0.353650\pi$
$68$	−50.4625	−0.0899923
$69$	−31.4039	−0.0547910
$70$	440.183	0.751598
$71$	−850.549	−1.42171	−0.710856	−	0.703337i	$-0.751692\pi$
$71$	−850.549	−1.42171	−0.710856	+	0.703337i	$0.751692\pi$
$72$	220.656	0.361175
$73$	−169.457	−0.271691	−0.135846	−	0.990730i	$-0.543375\pi$
$73$	−169.457	−0.271691	−0.135846	+	0.990730i	$0.543375\pi$
$74$	63.1180	0.0991530
$75$	−112.435	−0.173105
$76$	164.548	0.248355
$77$	−212.314	−0.314227
$78$	−184.382	−0.267656
$79$	481.917	0.686328	0.343164	−	0.939275i	$-0.388501\pi$
$79$	481.917	0.686328	0.343164	+	0.939275i	$0.388501\pi$
$80$	405.160	0.566228
$81$	81.0000	0.111111
$82$	−576.505	−0.776394
$83$	−1337.68	−1.76903	−0.884516	−	0.466509i	$-0.845511\pi$
$83$	−1337.68	−1.76903	−0.884516	+	0.466509i	$0.845511\pi$
$84$	−119.131	−0.154741
$85$	−229.462	−0.292807
$86$	168.706	0.211536
$87$	91.2595	0.112460
$88$	−269.691	−0.326695
$89$	1325.06	1.57816	0.789082	−	0.614288i	$-0.210557\pi$
$89$	1325.06	1.57816	0.789082	+	0.614288i	$0.210557\pi$
$90$	205.253	0.240395
$91$	486.626	0.560574
$92$	21.5366	0.0244060
$93$	190.679	0.212607
$94$	789.002	0.865738
$95$	748.230	0.808071
$96$	−271.694	−0.288851
$97$	1651.94	1.72917	0.864584	−	0.502489i	$-0.167582\pi$
$97$	1651.94	1.72917	0.864584	+	0.502489i	$0.167582\pi$
$98$	−72.0112	−0.0742268
$99$	−99.0000	−0.100504
$100$	77.1076	0.0771076
$101$	968.113	0.953771	0.476885	−	0.878965i	$-0.341766\pi$
$101$	968.113	0.953771	0.476885	+	0.878965i	$0.341766\pi$
$102$	−179.375	−0.174126
$103$	1481.75	1.41749	0.708746	−	0.705464i	$-0.249261\pi$
$103$	1481.75	1.41749	0.708746	+	0.705464i	$0.249261\pi$
$104$	618.134	0.582817
$105$	−541.708	−0.503479
$106$	−393.634	−0.360690
$107$	−595.400	−0.537939	−0.268970	−	0.963149i	$-0.586683\pi$
$107$	−595.400	−0.537939	−0.268970	+	0.963149i	$0.586683\pi$
$108$	−55.5494	−0.0494930
$109$	337.124	0.296245	0.148122	−	0.988969i	$-0.452677\pi$
$109$	337.124	0.296245	0.148122	+	0.988969i	$0.452677\pi$
$110$	−250.864	−0.217445
$111$	−77.6758	−0.0664204
$112$	−835.902	−0.705226
$113$	−759.036	−0.631894	−0.315947	−	0.948777i	$-0.602322\pi$
$113$	−759.036	−0.631894	−0.315947	+	0.948777i	$0.602322\pi$
$114$	584.908	0.480541
$115$	97.9308	0.0794095
$116$	−62.5853	−0.0500940
$117$	226.909	0.179297
$118$	1937.38	1.51144
$119$	473.412	0.364686
$120$	−688.102	−0.523457
$121$	121.000	0.0909091
$122$	148.703	0.110352
$123$	709.472	0.520089
$124$	−130.767	−0.0947031
$125$	1520.03	1.08765
$126$	−423.465	−0.299407
$127$	−1329.38	−0.928844	−0.464422	−	0.885614i	$-0.653738\pi$
$127$	−1329.38	−0.928844	−0.464422	+	0.885614i	$0.653738\pi$
$128$	−658.266	−0.454555
$129$	−207.617	−0.141703
$130$	574.983	0.387918
$131$	75.8007	0.0505552	0.0252776	−	0.999680i	$-0.491953\pi$
$131$	75.8007	0.0505552	0.0252776	+	0.999680i	$0.491953\pi$
$132$	67.8937	0.0447681
$133$	−1543.70	−1.00644
$134$	−1186.49	−0.764902
$135$	−252.593	−0.161035
$136$	601.349	0.379156
$137$	2894.39	1.80500	0.902498	−	0.430695i	$-0.141731\pi$
$137$	2894.39	1.80500	0.902498	+	0.430695i	$0.141731\pi$
$138$	76.5547	0.0472230
$139$	−1094.81	−0.668059	−0.334029	−	0.942563i	$-0.608408\pi$
$139$	−1094.81	−0.668059	−0.334029	+	0.942563i	$0.608408\pi$
$140$	371.501	0.224268
$141$	−970.981	−0.579939
$142$	2073.42	1.22534
$143$	−277.333	−0.162180
$144$	−389.773	−0.225563
$145$	−284.586	−0.162990
$146$	413.094	0.234163
$147$	88.6201	0.0497229
$148$	53.2697	0.0295861
$149$	−1662.36	−0.913999	−0.457000	−	0.889467i	$-0.651076\pi$
$149$	−1662.36	−0.913999	−0.457000	+	0.889467i	$0.651076\pi$
$150$	274.089	0.149195
$151$	897.406	0.483641	0.241821	−	0.970321i	$-0.422255\pi$
$151$	897.406	0.483641	0.241821	+	0.970321i	$0.422255\pi$
$152$	−1960.88	−1.04637
$153$	220.747	0.116643
$154$	517.569	0.270824
$155$	−594.619	−0.308135
$156$	−155.613	−0.0798654
$157$	418.461	0.212719	0.106359	−	0.994328i	$-0.466081\pi$
$157$	418.461	0.212719	0.106359	+	0.994328i	$0.466081\pi$
$158$	−1174.79	−0.591528
$159$	484.423	0.241618
$160$	847.261	0.418637
$161$	−202.045	−0.0989030
$162$	−197.458	−0.0957638
$163$	−1661.60	−0.798443	−0.399222	−	0.916854i	$-0.630720\pi$
$163$	−1661.60	−0.798443	−0.399222	+	0.916854i	$0.630720\pi$
$164$	−486.553	−0.231667
$165$	308.725	0.145662
$166$	3260.93	1.52468
$167$	−1563.55	−0.724498	−0.362249	−	0.932081i	$-0.617991\pi$
$167$	−1563.55	−0.724498	−0.362249	+	0.932081i	$0.617991\pi$
$168$	1419.65	0.651955
$169$	−1561.35	−0.710674
$170$	559.370	0.252363
$171$	−719.814	−0.321904
$172$	142.383	0.0631198
$173$	−3591.45	−1.57834	−0.789171	−	0.614173i	$-0.789490\pi$
$173$	−3591.45	−1.57834	−0.789171	+	0.614173i	$0.789490\pi$
$174$	−222.468	−0.0969266
$175$	−723.382	−0.312472
$176$	476.389	0.204029
$177$	−2384.22	−1.01248
$178$	−3230.17	−1.36018
$179$	−1983.39	−0.828186	−0.414093	−	0.910235i	$-0.635901\pi$
$179$	−1983.39	−0.828186	−0.414093	+	0.910235i	$0.635901\pi$
$180$	173.227	0.0717311
$181$	−1620.94	−0.665655	−0.332828	−	0.942988i	$-0.608003\pi$
$181$	−1620.94	−0.665655	−0.332828	+	0.942988i	$0.608003\pi$
$182$	−1186.27	−0.483144
$183$	−183.000	−0.0739221
$184$	−256.647	−0.102827
$185$	242.227	0.0962642
$186$	−464.827	−0.183241
$187$	−269.802	−0.105507
$188$	665.894	0.258326
$189$	521.135	0.200566
$190$	−1824.00	−0.696456
$191$	−2977.25	−1.12789	−0.563943	−	0.825813i	$-0.690716\pi$
$191$	−2977.25	−1.12789	−0.563943	+	0.825813i	$0.690716\pi$
$192$	1701.72	0.639640
$193$	1551.43	0.578623	0.289312	−	0.957235i	$-0.406574\pi$
$193$	1551.43	0.578623	0.289312	+	0.957235i	$0.406574\pi$
$194$	−4027.02	−1.49032
$195$	−707.599	−0.259858
$196$	−60.7753	−0.0221484
$197$	120.009	0.0434024	0.0217012	−	0.999765i	$-0.493092\pi$
$197$	120.009	0.0434024	0.0217012	+	0.999765i	$0.493092\pi$
$198$	241.337	0.0866216
$199$	−598.326	−0.213137	−0.106568	−	0.994305i	$-0.533986\pi$
$199$	−598.326	−0.213137	−0.106568	+	0.994305i	$0.533986\pi$
$200$	−918.872	−0.324870
$201$	1460.14	0.512391
$202$	−2360.02	−0.822030
$203$	587.142	0.203001
$204$	−151.387	−0.0519571
$205$	−2212.44	−0.753774
$206$	−3612.14	−1.22170
$207$	−94.2116	−0.0316336
$208$	−1091.89	−0.363984
$209$	879.772	0.291173
$210$	1320.55	0.433935
$211$	2681.43	0.874869	0.437435	−	0.899250i	$-0.355887\pi$
$211$	2681.43	0.874869	0.437435	+	0.899250i	$0.355887\pi$
$212$	−332.215	−0.107626
$213$	−2551.65	−0.820826
$214$	1451.43	0.463636
$215$	647.441	0.205373
$216$	661.969	0.208524
$217$	1226.78	0.383776
$218$	−821.824	−0.255325
$219$	−508.371	−0.156861
$220$	−211.722	−0.0648832
$221$	618.388	0.188223
$222$	189.354	0.0572460
$223$	1029.01	0.309002	0.154501	−	0.987993i	$-0.450623\pi$
$223$	1029.01	0.309002	0.154501	+	0.987993i	$0.450623\pi$
$224$	−1748.02	−0.521403
$225$	−337.306	−0.0999424
$226$	1850.34	0.544613
$227$	−4608.04	−1.34734	−0.673671	−	0.739031i	$-0.735283\pi$
$227$	−4608.04	−1.34734	−0.673671	+	0.739031i	$0.735283\pi$
$228$	493.645	0.143388
$229$	−860.816	−0.248403	−0.124202	−	0.992257i	$-0.539637\pi$
$229$	−860.816	−0.248403	−0.124202	+	0.992257i	$0.539637\pi$
$230$	−238.731	−0.0684410
$231$	−636.943	−0.181419
$232$	745.814	0.211056
$233$	2232.48	0.627702	0.313851	−	0.949472i	$-0.398381\pi$
$233$	2232.48	0.627702	0.313851	+	0.949472i	$0.398381\pi$
$234$	−553.146	−0.154531
$235$	3027.94	0.840514
$236$	1635.09	0.450996
$237$	1445.75	0.396252
$238$	−1154.06	−0.314313
$239$	−4806.24	−1.30079	−0.650397	−	0.759594i	$-0.725398\pi$
$239$	−4806.24	−1.30079	−0.650397	+	0.759594i	$0.725398\pi$
$240$	1215.48	0.326912
$241$	193.450	0.0517063	0.0258532	−	0.999666i	$-0.491770\pi$
$241$	193.450	0.0517063	0.0258532	+	0.999666i	$0.491770\pi$
$242$	−294.967	−0.0783522
$243$	243.000	0.0641500
$244$	125.501	0.0329277
$245$	−276.356	−0.0720642
$246$	−1729.51	−0.448251
$247$	−2016.44	−0.519446
$248$	1558.31	0.399004
$249$	−4013.05	−1.02135
$250$	−3705.46	−0.937415
$251$	−4480.94	−1.12683	−0.563416	−	0.826174i	$-0.690513\pi$
$251$	−4480.94	−1.12683	−0.563416	+	0.826174i	$0.690513\pi$
$252$	−357.392	−0.0893396
$253$	115.147	0.0286137
$254$	3240.69	0.800547
$255$	−688.385	−0.169052
$256$	−2933.22	−0.716119
$257$	2003.22	0.486217	0.243108	−	0.969999i	$-0.421833\pi$
$257$	2003.22	0.486217	0.243108	+	0.969999i	$0.421833\pi$
$258$	506.119	0.122130
$259$	−499.748	−0.119895
$260$	485.268	0.115750
$261$	273.778	0.0649290
$262$	−184.783	−0.0435722
$263$	−1873.12	−0.439169	−0.219584	−	0.975594i	$-0.570470\pi$
$263$	−1873.12	−0.439169	−0.219584	+	0.975594i	$0.570470\pi$
$264$	−809.073	−0.188617
$265$	−1510.64	−0.350181
$266$	3763.16	0.867422
$267$	3975.19	0.911153
$268$	−1001.36	−0.228238
$269$	3881.65	0.879808	0.439904	−	0.898045i	$-0.355013\pi$
$269$	3881.65	0.879808	0.439904	+	0.898045i	$0.355013\pi$
$270$	615.758	0.138792
$271$	−1731.82	−0.388193	−0.194097	−	0.980982i	$-0.562178\pi$
$271$	−1731.82	−0.388193	−0.194097	+	0.980982i	$0.562178\pi$
$272$	−1062.24	−0.236793
$273$	1459.88	0.323648
$274$	−7055.79	−1.55568
$275$	412.262	0.0904013
$276$	64.6099	0.0140908
$277$	−2773.02	−0.601496	−0.300748	−	0.953704i	$-0.597236\pi$
$277$	−2773.02	−0.601496	−0.300748	+	0.953704i	$0.597236\pi$
$278$	2668.86	0.575783
$279$	572.036	0.122749
$280$	−4427.09	−0.944889
$281$	6224.51	1.32144	0.660718	−	0.750634i	$-0.270252\pi$
$281$	6224.51	1.32144	0.660718	+	0.750634i	$0.270252\pi$
$282$	2367.01	0.499834
$283$	5703.11	1.19793	0.598966	−	0.800774i	$-0.295578\pi$
$283$	5703.11	1.19793	0.598966	+	0.800774i	$0.295578\pi$
$284$	1749.91	0.365627
$285$	2244.69	0.466540
$286$	676.067	0.139779
$287$	4564.58	0.938810
$288$	−815.083	−0.166768
$289$	−4311.40	−0.877550
$290$	693.750	0.140477
$291$	4955.82	0.998335
$292$	348.639	0.0698717
$293$	−8134.89	−1.62200	−0.810999	−	0.585047i	$-0.801076\pi$
$293$	−8134.89	−1.62200	−0.810999	+	0.585047i	$0.801076\pi$
$294$	−216.034	−0.0428549
$295$	7435.03	1.46740
$296$	−634.802	−0.124653
$297$	−297.000	−0.0580259
$298$	4052.42	0.787752
$299$	−263.919	−0.0510462
$300$	231.323	0.0445181
$301$	−1335.76	−0.255787
$302$	−2187.65	−0.416838
$303$	2904.34	0.550660
$304$	3463.75	0.653486
$305$	570.673	0.107137
$306$	−538.126	−0.100531
$307$	−9212.33	−1.71262	−0.856312	−	0.516460i	$-0.827250\pi$
$307$	−9212.33	−1.71262	−0.856312	+	0.516460i	$0.827250\pi$
$308$	436.812	0.0808107
$309$	4445.26	0.818389
$310$	1449.53	0.265574
$311$	−6807.24	−1.24117	−0.620584	−	0.784140i	$-0.713104\pi$
$311$	−6807.24	−1.24117	−0.620584	+	0.784140i	$0.713104\pi$
$312$	1854.40	0.336490
$313$	6681.23	1.20653	0.603267	−	0.797539i	$-0.293865\pi$
$313$	6681.23	1.20653	0.603267	+	0.797539i	$0.293865\pi$
$314$	−1020.10	−0.183337
$315$	−1625.12	−0.290684
$316$	−991.490	−0.176505
$317$	557.507	0.0987782	0.0493891	−	0.998780i	$-0.484273\pi$
$317$	557.507	0.0987782	0.0493891	+	0.998780i	$0.484273\pi$
$318$	−1180.90	−0.208244
$319$	−334.618	−0.0587304
$320$	−5306.69	−0.927040
$321$	−1786.20	−0.310579
$322$	492.535	0.0852419
$323$	−1961.69	−0.337930
$324$	−166.648	−0.0285748
$325$	−944.908	−0.161274
$326$	4050.55	0.688157
$327$	1011.37	0.171037
$328$	5798.13	0.976061
$329$	−6247.06	−1.04684
$330$	−752.593	−0.125542
$331$	−8868.23	−1.47263	−0.736317	−	0.676637i	$-0.763437\pi$
$331$	−8868.23	−1.47263	−0.736317	+	0.676637i	$0.763437\pi$
$332$	2752.13	0.454948
$333$	−233.028	−0.0383478
$334$	3811.54	0.624426
$335$	−4553.36	−0.742617
$336$	−2507.71	−0.407162
$337$	6526.52	1.05496	0.527481	−	0.849567i	$-0.323137\pi$
$337$	6526.52	1.05496	0.527481	+	0.849567i	$0.323137\pi$
$338$	3806.18	0.612512
$339$	−2277.11	−0.364824
$340$	472.092	0.0753022
$341$	−699.155	−0.111030
$342$	1754.72	0.277440
$343$	−6050.18	−0.952417
$344$	−1696.74	−0.265937
$345$	293.792	0.0458471
$346$	8755.06	1.36033
$347$	−7078.26	−1.09505	−0.547523	−	0.836791i	$-0.684429\pi$
$347$	−7078.26	−1.09505	−0.547523	+	0.836791i	$0.684429\pi$
$348$	−187.756	−0.0289218
$349$	−10807.3	−1.65759	−0.828797	−	0.559549i	$-0.810974\pi$
$349$	−10807.3	−1.65759	−0.828797	+	0.559549i	$0.810974\pi$
$350$	1763.42	0.269311
$351$	680.726	0.103517
$352$	996.213	0.150848
$353$	−7112.24	−1.07237	−0.536184	−	0.844101i	$-0.680135\pi$
$353$	−7112.24	−1.07237	−0.536184	+	0.844101i	$0.680135\pi$
$354$	5812.13	0.872630
$355$	7957.14	1.18964
$356$	−2726.17	−0.405862
$357$	1420.24	0.210551
$358$	4835.00	0.713792
$359$	−8187.37	−1.20366	−0.601829	−	0.798625i	$-0.705561\pi$
$359$	−8187.37	−1.20366	−0.601829	+	0.798625i	$0.705561\pi$
$360$	−2064.31	−0.302218
$361$	−462.312	−0.0674023
$362$	3951.44	0.573711
$363$	363.000	0.0524864
$364$	−1001.18	−0.144165
$365$	1585.32	0.227341
$366$	446.108	0.0637116
$367$	874.861	0.124434	0.0622171	−	0.998063i	$-0.480183\pi$
$367$	874.861	0.124434	0.0622171	+	0.998063i	$0.480183\pi$
$368$	453.347	0.0642183
$369$	2128.42	0.300274
$370$	−590.488	−0.0829676
$371$	3116.67	0.436143
$372$	−392.300	−0.0546769
$373$	4262.79	0.591739	0.295870	−	0.955228i	$-0.404391\pi$
$373$	4262.79	0.591739	0.295870	+	0.955228i	$0.404391\pi$
$374$	657.710	0.0909341
$375$	4560.10	0.627954
$376$	−7935.30	−1.08838
$377$	766.947	0.104774
$378$	−1270.40	−0.172863
$379$	4168.19	0.564923	0.282461	−	0.959279i	$-0.408849\pi$
$379$	4168.19	0.564923	0.282461	+	0.959279i	$0.408849\pi$
$380$	−1539.40	−0.207814
$381$	−3988.13	−0.536268
$382$	7257.79	0.972096
$383$	3517.68	0.469308	0.234654	−	0.972079i	$-0.424604\pi$
$383$	3517.68	0.469308	0.234654	+	0.972079i	$0.424604\pi$
$384$	−1974.80	−0.262438
$385$	1986.26	0.262933
$386$	−3781.99	−0.498701
$387$	−622.852	−0.0818123
$388$	−3398.68	−0.444696
$389$	6143.45	0.800733	0.400366	−	0.916355i	$-0.368883\pi$
$389$	6143.45	0.800733	0.400366	+	0.916355i	$0.368883\pi$
$390$	1724.95	0.223965
$391$	−256.752	−0.0332085
$392$	724.244	0.0933160
$393$	227.402	0.0291881
$394$	−292.552	−0.0374074
$395$	−4508.48	−0.574294
$396$	203.681	0.0258469
$397$	10121.0	1.27949	0.639747	−	0.768586i	$-0.279039\pi$
$397$	10121.0	1.27949	0.639747	+	0.768586i	$0.279039\pi$
$398$	1458.57	0.183697
$399$	−4631.11	−0.581067
$400$	1623.12	0.202890
$401$	−246.201	−0.0306601	−0.0153301	−	0.999882i	$-0.504880\pi$
$401$	−246.201	−0.0306601	−0.0153301	+	0.999882i	$0.504880\pi$
$402$	−3559.46	−0.441616
$403$	1602.47	0.198076
$404$	−1991.78	−0.245284
$405$	−757.779	−0.0929737
$406$	−1431.30	−0.174962
$407$	284.811	0.0346869
$408$	1804.05	0.218906
$409$	−7688.08	−0.929465	−0.464733	−	0.885451i	$-0.653850\pi$
$409$	−7688.08	−0.929465	−0.464733	+	0.885451i	$0.653850\pi$
$410$	5393.37	0.649658
$411$	8683.17	1.04211
$412$	−3048.54	−0.364541
$413$	−15339.5	−1.82762
$414$	229.664	0.0272642
$415$	12514.4	1.48026
$416$	−2283.33	−0.269109
$417$	−3284.42	−0.385704
$418$	−2144.66	−0.250954
$419$	633.150	0.0738219	0.0369110	−	0.999319i	$-0.488248\pi$
$419$	633.150	0.0738219	0.0369110	+	0.999319i	$0.488248\pi$
$420$	1114.50	0.129481
$421$	−13391.6	−1.55027	−0.775136	−	0.631794i	$-0.782319\pi$
$421$	−13391.6	−1.55027	−0.775136	+	0.631794i	$0.782319\pi$
$422$	−6536.65	−0.754027
$423$	−2912.94	−0.334828
$424$	3958.93	0.453449
$425$	−919.251	−0.104918
$426$	6220.27	0.707449
$427$	−1177.38	−0.133436
$428$	1224.97	0.138344
$429$	−831.998	−0.0936346
$430$	−1578.30	−0.177005
$431$	−12098.5	−1.35212	−0.676059	−	0.736848i	$-0.736313\pi$
$431$	−12098.5	−1.35212	−0.676059	+	0.736848i	$0.736313\pi$
$432$	−1169.32	−0.130229
$433$	−4583.48	−0.508702	−0.254351	−	0.967112i	$-0.581862\pi$
$433$	−4583.48	−0.508702	−0.254351	+	0.967112i	$0.581862\pi$
$434$	−2990.59	−0.330767
$435$	−853.759	−0.0941026
$436$	−693.595	−0.0761862
$437$	837.220	0.0916468
$438$	1239.28	0.135194
$439$	−4821.31	−0.524166	−0.262083	−	0.965045i	$-0.584409\pi$
$439$	−4821.31	−0.524166	−0.262083	+	0.965045i	$0.584409\pi$
$440$	2523.04	0.273366
$441$	265.860	0.0287075
$442$	−1507.47	−0.162225
$443$	−12197.2	−1.30814	−0.654069	−	0.756435i	$-0.726940\pi$
$443$	−12197.2	−1.30814	−0.654069	+	0.756435i	$0.726940\pi$
$444$	159.809	0.0170816
$445$	−12396.4	−1.32055
$446$	−2508.46	−0.266321
$447$	−4987.08	−0.527698
$448$	10948.4	1.15461
$449$	438.161	0.0460537	0.0230268	−	0.999735i	$-0.492670\pi$
$449$	438.161	0.0460537	0.0230268	+	0.999735i	$0.492670\pi$
$450$	822.266	0.0861378
$451$	−2601.40	−0.271608
$452$	1561.63	0.162506
$453$	2692.22	0.279231
$454$	11233.2	1.16124
$455$	−4552.53	−0.469068
$456$	−5882.65	−0.604123
$457$	15785.9	1.61583	0.807916	−	0.589298i	$-0.200596\pi$
$457$	15785.9	1.61583	0.807916	+	0.589298i	$0.200596\pi$
$458$	2098.45	0.214092
$459$	662.242	0.0673438
$460$	−201.482	−0.0204220
$461$	−11481.4	−1.15996	−0.579979	−	0.814632i	$-0.696939\pi$
$461$	−11481.4	−1.15996	−0.579979	+	0.814632i	$0.696939\pi$
$462$	1552.71	0.156360
$463$	−2456.54	−0.246577	−0.123288	−	0.992371i	$-0.539344\pi$
$463$	−2456.54	−0.246577	−0.123288	+	0.992371i	$0.539344\pi$
$464$	−1317.42	−0.131810
$465$	−1783.86	−0.177902
$466$	−5442.22	−0.541000
$467$	16669.5	1.65176	0.825881	−	0.563844i	$-0.190678\pi$
$467$	16669.5	1.65176	0.825881	+	0.563844i	$0.190678\pi$
$468$	−466.839	−0.0461103
$469$	9394.22	0.924914
$470$	−7381.35	−0.724418
$471$	1255.38	0.122813
$472$	−19484.9	−1.90014
$473$	761.264	0.0740020
$474$	−3524.38	−0.341519
$475$	2997.50	0.289546
$476$	−973.991	−0.0937874
$477$	1453.27	0.139498
$478$	11716.4	1.12112
$479$	5041.44	0.480896	0.240448	−	0.970662i	$-0.422706\pi$
$479$	5041.44	0.480896	0.240448	+	0.970662i	$0.422706\pi$
$480$	2541.78	0.241700
$481$	−652.790	−0.0618808
$482$	−471.583	−0.0445643
$483$	−606.135	−0.0571017
$484$	−248.944	−0.0233794
$485$	−15454.4	−1.44690
$486$	−592.373	−0.0552892
$487$	−2728.41	−0.253873	−0.126936	−	0.991911i	$-0.540514\pi$
$487$	−2728.41	−0.253873	−0.126936	+	0.991911i	$0.540514\pi$
$488$	−1495.56	−0.138731
$489$	−4984.79	−0.460981
$490$	673.686	0.0621103
$491$	−20512.0	−1.88533	−0.942663	−	0.333745i	$-0.891688\pi$
$491$	−20512.0	−1.88533	−0.942663	+	0.333745i	$0.891688\pi$
$492$	−1459.66	−0.133753
$493$	746.121	0.0681615
$494$	4915.58	0.447697
$495$	926.174	0.0840979
$496$	−2752.64	−0.249188
$497$	−16416.7	−1.48167
$498$	9782.79	0.880276
$499$	15546.5	1.39471	0.697353	−	0.716728i	$-0.254361\pi$
$499$	15546.5	1.39471	0.697353	+	0.716728i	$0.254361\pi$
$500$	−3127.30	−0.279714
$501$	−4690.65	−0.418289
$502$	10923.4	0.971187
$503$	−5507.82	−0.488233	−0.244117	−	0.969746i	$-0.578498\pi$
$503$	−5507.82	−0.488233	−0.244117	+	0.969746i	$0.578498\pi$
$504$	4258.95	0.376406
$505$	−9056.99	−0.798081
$506$	−280.701	−0.0246614
$507$	−4684.05	−0.410308
$508$	2735.04	0.238874
$509$	−13561.5	−1.18095	−0.590476	−	0.807055i	$-0.701060\pi$
$509$	−13561.5	−1.18095	−0.590476	+	0.807055i	$0.701060\pi$
$510$	1678.11	0.145702
$511$	−3270.74	−0.283149
$512$	12416.6	1.07176
$513$	−2159.44	−0.185851
$514$	−4883.36	−0.419058
$515$	−13862.3	−1.18610
$516$	427.149	0.0364422
$517$	3560.26	0.302863
$518$	1218.26	0.103334
$519$	−10774.4	−0.911256
$520$	−5782.82	−0.487680
$521$	−2845.67	−0.239292	−0.119646	−	0.992817i	$-0.538176\pi$
$521$	−2845.67	−0.239292	−0.119646	+	0.992817i	$0.538176\pi$
$522$	−667.403	−0.0559606
$523$	3875.42	0.324016	0.162008	−	0.986789i	$-0.448203\pi$
$523$	3875.42	0.324016	0.162008	+	0.986789i	$0.448203\pi$
$524$	−155.951	−0.0130015
$525$	−2170.15	−0.180406
$526$	4566.19	0.378508
$527$	1558.96	0.128860
$528$	1429.17	0.117796
$529$	−12057.4	−0.990994
$530$	3682.56	0.301812
$531$	−7152.66	−0.584556
$532$	3176.00	0.258829
$533$	5962.42	0.484542
$534$	−9690.52	−0.785299
$535$	5570.14	0.450128
$536$	11933.0	0.961615
$537$	−5950.16	−0.478153
$538$	−9462.48	−0.758283
$539$	−324.941	−0.0259669
$540$	519.681	0.0414139
$541$	23262.3	1.84866	0.924328	−	0.381600i	$-0.124627\pi$
$541$	23262.3	1.84866	0.924328	+	0.381600i	$0.124627\pi$
$542$	4221.73	0.334574
$543$	−4862.82	−0.384316
$544$	−2221.33	−0.175071
$545$	−3153.90	−0.247887
$546$	−3558.81	−0.278943
$547$	−10108.0	−0.790103	−0.395051	−	0.918659i	$-0.629273\pi$
$547$	−10108.0	−0.790103	−0.395051	+	0.918659i	$0.629273\pi$
$548$	−5954.88	−0.464197
$549$	−549.000	−0.0426790
$550$	−1004.99	−0.0779145
$551$	−2432.96	−0.188108
$552$	−769.940	−0.0593674
$553$	9301.63	0.715272
$554$	6759.91	0.518414
$555$	726.681	0.0555781
$556$	2252.44	0.171807
$557$	8886.37	0.675992	0.337996	−	0.941148i	$-0.390251\pi$
$557$	8886.37	0.675992	0.337996	+	0.941148i	$0.390251\pi$
$558$	−1394.48	−0.105794
$559$	−1744.82	−0.132018
$560$	7820.11	0.590107
$561$	−809.407	−0.0609147
$562$	−15173.8	−1.13891
$563$	12274.3	0.918826	0.459413	−	0.888223i	$-0.348060\pi$
$563$	12274.3	0.918826	0.459413	+	0.888223i	$0.348060\pi$
$564$	1997.68	0.149145
$565$	7101.00	0.528746
$566$	−13902.7	−1.03247
$567$	1563.41	0.115797
$568$	−20853.2	−1.54046
$569$	9143.05	0.673632	0.336816	−	0.941571i	$-0.390650\pi$
$569$	9143.05	0.673632	0.336816	+	0.941571i	$0.390650\pi$
$570$	−5471.99	−0.402099
$571$	−3136.17	−0.229851	−0.114925	−	0.993374i	$-0.536663\pi$
$571$	−3136.17	−0.229851	−0.114925	+	0.993374i	$0.536663\pi$
$572$	570.581	0.0417084
$573$	−8931.75	−0.651186
$574$	−11127.3	−0.809136
$575$	392.322	0.0284539
$576$	5105.15	0.369296
$577$	15119.1	1.09084	0.545422	−	0.838162i	$-0.316369\pi$
$577$	15119.1	1.09084	0.545422	+	0.838162i	$0.316369\pi$
$578$	10510.1	0.756337
$579$	4654.29	0.334068
$580$	585.504	0.0419168
$581$	−25819.0	−1.84364
$582$	−12081.0	−0.860439
$583$	−1776.22	−0.126181
$584$	−4154.64	−0.294384
$585$	−2122.80	−0.150029
$586$	19830.8	1.39796
$587$	−17618.7	−1.23884	−0.619420	−	0.785059i	$-0.712632\pi$
$587$	−17618.7	−1.23884	−0.619420	+	0.785059i	$0.712632\pi$
$588$	−182.326	−0.0127874
$589$	−5083.45	−0.355620
$590$	−18124.7	−1.26472
$591$	360.027	0.0250584
$592$	1121.33	0.0778486
$593$	−25639.4	−1.77552	−0.887759	−	0.460308i	$-0.847739\pi$
$593$	−25639.4	−1.77552	−0.887759	+	0.460308i	$0.847739\pi$
$594$	724.011	0.0500110
$595$	−4428.91	−0.305156
$596$	3420.12	0.235056
$597$	−1794.98	−0.123055
$598$	643.367	0.0439954
$599$	−5438.57	−0.370975	−0.185487	−	0.982647i	$-0.559386\pi$
$599$	−5438.57	−0.370975	−0.185487	+	0.982647i	$0.559386\pi$
$600$	−2756.62	−0.187564
$601$	−12866.4	−0.873263	−0.436632	−	0.899640i	$-0.643829\pi$
$601$	−12866.4	−0.873263	−0.436632	+	0.899640i	$0.643829\pi$
$602$	3256.25	0.220457
$603$	4380.43	0.295829
$604$	−1846.31	−0.124380
$605$	−1131.99	−0.0760694
$606$	−7080.05	−0.474599
$607$	−23030.6	−1.54000	−0.770001	−	0.638042i	$-0.779745\pi$
$607$	−23030.6	−1.54000	−0.770001	+	0.638042i	$0.779745\pi$
$608$	7243.31	0.483150
$609$	1761.43	0.117203
$610$	−1391.16	−0.0923382
$611$	−8160.15	−0.540301
$612$	−454.162	−0.0299974
$613$	1056.92	0.0696388	0.0348194	−	0.999394i	$-0.488914\pi$
$613$	1056.92	0.0696388	0.0348194	+	0.999394i	$0.488914\pi$
$614$	22457.3	1.47607
$615$	−6637.32	−0.435191
$616$	−5205.39	−0.340472
$617$	−27847.2	−1.81699	−0.908496	−	0.417893i	$-0.862769\pi$
$617$	−27847.2	−1.81699	−0.908496	+	0.417893i	$0.862769\pi$
$618$	−10836.4	−0.705348
$619$	−10128.1	−0.657648	−0.328824	−	0.944391i	$-0.606652\pi$
$619$	−10128.1	−0.657648	−0.328824	+	0.944391i	$0.606652\pi$
$620$	1223.36	0.0792441
$621$	−282.635	−0.0182637
$622$	16594.3	1.06973
$623$	25575.5	1.64472
$624$	−3275.66	−0.210146
$625$	−9535.57	−0.610276
$626$	−16287.2	−1.03988
$627$	2639.32	0.168109
$628$	−860.936	−0.0547056
$629$	−635.064	−0.0402570
$630$	3961.64	0.250533
$631$	−562.834	−0.0355088	−0.0177544	−	0.999842i	$-0.505652\pi$
$631$	−562.834	−0.0355088	−0.0177544	+	0.999842i	$0.505652\pi$
$632$	11815.3	0.743654
$633$	8044.30	0.505106
$634$	−1359.06	−0.0851344
$635$	12436.7	0.777223
$636$	−996.646	−0.0621377
$637$	744.766	0.0463245
$638$	815.714	0.0506182
$639$	−7654.94	−0.473904
$640$	6158.28	0.380355
$641$	−11425.9	−0.704052	−0.352026	−	0.935990i	$-0.614507\pi$
$641$	−11425.9	−0.704052	−0.352026	+	0.935990i	$0.614507\pi$
$642$	4354.30	0.267680
$643$	11939.9	0.732291	0.366146	−	0.930558i	$-0.380677\pi$
$643$	11939.9	0.732291	0.366146	+	0.930558i	$0.380677\pi$
$644$	415.685	0.0254352
$645$	1942.32	0.118572
$646$	4782.11	0.291253
$647$	29306.6	1.78077	0.890386	−	0.455206i	$-0.150434\pi$
$647$	29306.6	1.78077	0.890386	+	0.455206i	$0.150434\pi$
$648$	1985.91	0.120392
$649$	8742.14	0.528750
$650$	2303.45	0.138998
$651$	3680.35	0.221573
$652$	3418.55	0.205338
$653$	−5269.65	−0.315800	−0.157900	−	0.987455i	$-0.550472\pi$
$653$	−5269.65	−0.315800	−0.157900	+	0.987455i	$0.550472\pi$
$654$	−2465.47	−0.147412
$655$	−709.138	−0.0423028
$656$	−10242.0	−0.609575
$657$	−1525.11	−0.0905637
$658$	15228.8	0.902248
$659$	−1543.80	−0.0912562	−0.0456281	−	0.998958i	$-0.514529\pi$
$659$	−1543.80	−0.0912562	−0.0456281	+	0.998958i	$0.514529\pi$
$660$	−635.166	−0.0374603
$661$	22016.0	1.29550	0.647750	−	0.761853i	$-0.275710\pi$
$661$	22016.0	1.29550	0.647750	+	0.761853i	$0.275710\pi$
$662$	21618.5	1.26922
$663$	1855.17	0.108671
$664$	−32796.4	−1.91679
$665$	14441.8	0.842149
$666$	568.062	0.0330510
$667$	−318.433	−0.0184854
$668$	3216.83	0.186322
$669$	3087.02	0.178402
$670$	11099.9	0.640042
$671$	671.000	0.0386046
$672$	−5244.05	−0.301032
$673$	−2912.33	−0.166809	−0.0834043	−	0.996516i	$-0.526579\pi$
$673$	−2912.33	−0.166809	−0.0834043	+	0.996516i	$0.526579\pi$
$674$	−15910.0	−0.909244
$675$	−1011.92	−0.0577018
$676$	3212.30	0.182766
$677$	16171.1	0.918029	0.459015	−	0.888429i	$-0.348203\pi$
$677$	16171.1	0.918029	0.459015	+	0.888429i	$0.348203\pi$
$678$	5551.01	0.314433
$679$	31884.6	1.80209
$680$	−5625.80	−0.317264
$681$	−13824.1	−0.777888
$682$	1704.36	0.0956943
$683$	−728.551	−0.0408159	−0.0204079	−	0.999792i	$-0.506497\pi$
$683$	−728.551	−0.0408159	−0.0204079	+	0.999792i	$0.506497\pi$
$684$	1480.93	0.0827850
$685$	−27077.9	−1.51035
$686$	14748.8	0.820864
$687$	−2582.45	−0.143416
$688$	2997.17	0.166084
$689$	4071.10	0.225104
$690$	−716.192	−0.0395144
$691$	−24145.0	−1.32926	−0.664629	−	0.747173i	$-0.731411\pi$
$691$	−24145.0	−1.32926	−0.664629	+	0.747173i	$0.731411\pi$
$692$	7389.01	0.405907
$693$	−1910.83	−0.104742
$694$	17255.0	0.943791
$695$	10242.2	0.559007
$696$	2237.44	0.121853
$697$	5800.52	0.315223
$698$	26345.4	1.42864
$699$	6697.44	0.362404
$700$	1488.28	0.0803593
$701$	2326.40	0.125345	0.0626725	−	0.998034i	$-0.480038\pi$
$701$	2326.40	0.125345	0.0626725	+	0.998034i	$0.480038\pi$
$702$	−1659.44	−0.0892186
$703$	2070.82	0.111099
$704$	−6239.63	−0.334041
$705$	9083.81	0.485271
$706$	17337.8	0.924247
$707$	18685.8	0.993993
$708$	4905.26	0.260383
$709$	−35473.0	−1.87901	−0.939505	−	0.342536i	$-0.888714\pi$
$709$	−35473.0	−1.87901	−0.939505	+	0.342536i	$0.888714\pi$
$710$	−19397.5	−1.02532
$711$	4337.26	0.228776
$712$	32487.1	1.70998
$713$	−665.339	−0.0349469
$714$	−3462.18	−0.181469
$715$	2594.53	0.135706
$716$	4080.59	0.212987
$717$	−14418.7	−0.751014
$718$	19958.8	1.03740
$719$	8893.60	0.461301	0.230651	−	0.973037i	$-0.425915\pi$
$719$	8893.60	0.461301	0.230651	+	0.973037i	$0.425915\pi$
$720$	3646.44	0.188743
$721$	28599.8	1.47727
$722$	1127.00	0.0580923
$723$	580.351	0.0298527
$724$	3334.90	0.171189
$725$	−1140.09	−0.0584024
$726$	−884.902	−0.0452367
$727$	315.955	0.0161184	0.00805922	−	0.999968i	$-0.497435\pi$
$727$	315.955	0.0161184	0.00805922	+	0.999968i	$0.497435\pi$
$728$	11930.8	0.607396
$729$	729.000	0.0370370
$730$	−3864.61	−0.195939
$731$	−1697.44	−0.0858854
$732$	376.502	0.0190108
$733$	34968.7	1.76207	0.881037	−	0.473048i	$-0.156846\pi$
$733$	34968.7	1.76207	0.881037	+	0.473048i	$0.156846\pi$
$734$	−2132.69	−0.107247
$735$	−829.068	−0.0416063
$736$	948.028	0.0474793
$737$	−5353.86	−0.267587
$738$	−5188.54	−0.258798
$739$	26899.3	1.33898	0.669490	−	0.742821i	$-0.266513\pi$
$739$	26899.3	1.33898	0.669490	+	0.742821i	$0.266513\pi$
$740$	−498.354	−0.0247566
$741$	−6049.33	−0.299902
$742$	−7597.64	−0.375901
$743$	10709.4	0.528790	0.264395	−	0.964415i	$-0.414828\pi$
$743$	10709.4	0.528790	0.264395	+	0.964415i	$0.414828\pi$
$744$	4674.94	0.230365
$745$	15551.9	0.764801
$746$	−10391.6	−0.510005
$747$	−12039.1	−0.589677
$748$	555.087	0.0271337
$749$	−11492.0	−0.560625
$750$	−11116.4	−0.541217
$751$	8840.96	0.429576	0.214788	−	0.976661i	$-0.431094\pi$
$751$	8840.96	0.429576	0.214788	+	0.976661i	$0.431094\pi$
$752$	14017.1	0.679722
$753$	−13442.8	−0.650576
$754$	−1869.62	−0.0903019
$755$	−8395.50	−0.404693
$756$	−1072.18	−0.0515802
$757$	−5354.13	−0.257066	−0.128533	−	0.991705i	$-0.541027\pi$
$757$	−5354.13	−0.257066	−0.128533	+	0.991705i	$0.541027\pi$
$758$	−10161.0	−0.486892
$759$	345.442	0.0165201
$760$	18344.6	0.875566
$761$	21656.3	1.03159	0.515796	−	0.856711i	$-0.327496\pi$
$761$	21656.3	1.03159	0.515796	+	0.856711i	$0.327496\pi$
$762$	9722.06	0.462196
$763$	6506.94	0.308738
$764$	6125.36	0.290062
$765$	−2065.16	−0.0976025
$766$	−8575.21	−0.404484
$767$	−20037.0	−0.943280
$768$	−8799.67	−0.413451
$769$	−10389.0	−0.487176	−0.243588	−	0.969879i	$-0.578325\pi$
$769$	−10389.0	−0.487176	−0.243588	+	0.969879i	$0.578325\pi$
$770$	−4842.01	−0.226615
$771$	6009.67	0.280717
$772$	−3191.89	−0.148807
$773$	−19016.8	−0.884849	−0.442425	−	0.896806i	$-0.645882\pi$
$773$	−19016.8	−0.884849	−0.442425	+	0.896806i	$0.645882\pi$
$774$	1518.36	0.0705119
$775$	−2382.11	−0.110410
$776$	40501.3	1.87360
$777$	−1499.24	−0.0692215
$778$	−14976.2	−0.690131
$779$	−18914.4	−0.869932
$780$	1455.81	0.0668284
$781$	9356.04	0.428663
$782$	625.898	0.0286216
$783$	821.335	0.0374867
$784$	−1279.32	−0.0582782
$785$	−3914.83	−0.177995
$786$	−554.349	−0.0251564
$787$	24225.8	1.09728	0.548638	−	0.836060i	$-0.315147\pi$
$787$	24225.8	1.09728	0.548638	+	0.836060i	$0.315147\pi$
$788$	−246.905	−0.0111620
$789$	−5619.35	−0.253554
$790$	10990.5	0.494969
$791$	−14650.4	−0.658543
$792$	−2427.22	−0.108898
$793$	−1537.94	−0.0688698
$794$	−24672.5	−1.10276
$795$	−4531.92	−0.202177
$796$	1230.99	0.0548131
$797$	19688.4	0.875030	0.437515	−	0.899211i	$-0.355859\pi$
$797$	19688.4	0.875030	0.437515	+	0.899211i	$0.355859\pi$
$798$	11289.5	0.500806
$799$	−7938.57	−0.351497
$800$	3394.22	0.150005
$801$	11925.6	0.526055
$802$	600.177	0.0264252
$803$	1864.03	0.0819179
$804$	−3004.08	−0.131773
$805$	1890.19	0.0827584
$806$	−3906.41	−0.170717
$807$	11644.9	0.507957
$808$	23735.6	1.03343
$809$	−78.5921	−0.00341551	−0.00170776	−	0.999999i	$-0.500544\pi$
$809$	−78.5921	−0.00341551	−0.00170776	+	0.999999i	$0.500544\pi$
$810$	1847.27	0.0801316
$811$	18895.7	0.818149	0.409075	−	0.912501i	$-0.365852\pi$
$811$	18895.7	0.818149	0.409075	+	0.912501i	$0.365852\pi$
$812$	−1207.98	−0.0522065
$813$	−5195.45	−0.224123
$814$	−694.298	−0.0298958
$815$	15544.7	0.668108
$816$	−3186.71	−0.136712
$817$	5535.03	0.237021
$818$	18741.6	0.801082
$819$	4379.63	0.186858
$820$	4551.85	0.193850
$821$	−45015.8	−1.91360	−0.956798	−	0.290753i	$-0.906094\pi$
$821$	−45015.8	−1.91360	−0.956798	+	0.290753i	$0.906094\pi$
$822$	−21167.4	−0.898171
$823$	34267.7	1.45139	0.725696	−	0.688015i	$-0.241518\pi$
$823$	34267.7	1.45139	0.725696	+	0.688015i	$0.241518\pi$
$824$	36328.7	1.53589
$825$	1236.79	0.0521932
$826$	37393.9	1.57518
$827$	40700.3	1.71135	0.855677	−	0.517511i	$-0.173141\pi$
$827$	40700.3	1.71135	0.855677	+	0.517511i	$0.173141\pi$
$828$	193.830	0.00813532
$829$	−15277.2	−0.640047	−0.320024	−	0.947410i	$-0.603691\pi$
$829$	−15277.2	−0.640047	−0.320024	+	0.947410i	$0.603691\pi$
$830$	−30507.0	−1.27580
$831$	−8319.05	−0.347274
$832$	14301.3	0.595922
$833$	724.543	0.0301368
$834$	8006.58	0.332428
$835$	14627.5	0.606233
$836$	−1810.03	−0.0748819
$837$	1716.11	0.0708691
$838$	−1543.46	−0.0636252
$839$	−43210.0	−1.77804	−0.889020	−	0.457868i	$-0.848613\pi$
$839$	−43210.0	−1.77804	−0.889020	+	0.457868i	$0.848613\pi$
$840$	−13281.3	−0.545532
$841$	−23463.6	−0.962058
$842$	32645.2	1.33614
$843$	18673.5	0.762931
$844$	−5516.74	−0.224993
$845$	14606.9	0.594666
$846$	7101.02	0.288579
$847$	2335.46	0.0947429
$848$	−6993.15	−0.283191
$849$	17109.3	0.691627
$850$	2240.90	0.0904262
$851$	271.036	0.0109177
$852$	5249.72	0.211095
$853$	3608.27	0.144836	0.0724179	−	0.997374i	$-0.476928\pi$
$853$	3608.27	0.144836	0.0724179	+	0.997374i	$0.476928\pi$
$854$	2870.15	0.115005
$855$	6734.07	0.269357
$856$	−14597.6	−0.582870
$857$	−1960.66	−0.0781503	−0.0390752	−	0.999236i	$-0.512441\pi$
$857$	−1960.66	−0.0781503	−0.0390752	+	0.999236i	$0.512441\pi$
$858$	2028.20	0.0807013
$859$	23756.0	0.943590	0.471795	−	0.881708i	$-0.343606\pi$
$859$	23756.0	0.943590	0.471795	+	0.881708i	$0.343606\pi$
$860$	−1332.04	−0.0528163
$861$	13693.7	0.542022
$862$	29493.0	1.16535
$863$	−8356.29	−0.329608	−0.164804	−	0.986326i	$-0.552699\pi$
$863$	−8356.29	−0.329608	−0.164804	+	0.986326i	$0.552699\pi$
$864$	−2445.25	−0.0962837
$865$	33599.1	1.32070
$866$	11173.4	0.438437
$867$	−12934.2	−0.506654
$868$	−2523.97	−0.0986970
$869$	−5301.09	−0.206936
$870$	2081.25	0.0811046
$871$	12271.1	0.477371
$872$	8265.40	0.320988
$873$	14867.5	0.576389
$874$	−2040.93	−0.0789880
$875$	29338.6	1.13352
$876$	1045.92	0.0403404
$877$	4289.33	0.165154	0.0825772	−	0.996585i	$-0.473685\pi$
$877$	4289.33	0.165154	0.0825772	+	0.996585i	$0.473685\pi$
$878$	11753.1	0.451765
$879$	−24404.7	−0.936461
$880$	−4456.76	−0.170724
$881$	25909.5	0.990820	0.495410	−	0.868659i	$-0.335018\pi$
$881$	25909.5	0.990820	0.495410	+	0.868659i	$0.335018\pi$
$882$	−648.101	−0.0247423
$883$	−17652.2	−0.672755	−0.336378	−	0.941727i	$-0.609202\pi$
$883$	−17652.2	−0.672755	−0.336378	+	0.941727i	$0.609202\pi$
$884$	−1272.26	−0.0484060
$885$	22305.1	0.847206
$886$	29733.6	1.12745
$887$	12843.0	0.486161	0.243081	−	0.970006i	$-0.421842\pi$
$887$	12843.0	0.486161	0.243081	+	0.970006i	$0.421842\pi$
$888$	−1904.41	−0.0719682
$889$	−25658.7	−0.968015
$890$	30219.2	1.13815
$891$	−891.000	−0.0335013
$892$	−2117.07	−0.0794671
$893$	25886.1	0.970040
$894$	12157.2	0.454809
$895$	18555.2	0.692995
$896$	−12705.4	−0.473725
$897$	−791.756	−0.0294715
$898$	−1068.13	−0.0396925
$899$	1933.47	0.0717296
$900$	693.968	0.0257025
$901$	3960.56	0.146443
$902$	6341.55	0.234092
$903$	−4007.28	−0.147679
$904$	−18609.6	−0.684673
$905$	15164.4	0.556996
$906$	−6562.95	−0.240662
$907$	−3969.04	−0.145303	−0.0726516	−	0.997357i	$-0.523146\pi$
$907$	−3969.04	−0.145303	−0.0726516	+	0.997357i	$0.523146\pi$
$908$	9480.53	0.346500
$909$	8713.02	0.317924
$910$	11097.9	0.404277
$911$	−5844.73	−0.212563	−0.106281	−	0.994336i	$-0.533894\pi$
$911$	−5844.73	−0.212563	−0.106281	+	0.994336i	$0.533894\pi$
$912$	10391.2	0.377290
$913$	14714.5	0.533383
$914$	−38482.1	−1.39264
$915$	1712.02	0.0618553
$916$	1771.03	0.0638826
$917$	1463.05	0.0526872
$918$	−1614.38	−0.0580419
$919$	−9650.26	−0.346390	−0.173195	−	0.984888i	$-0.555409\pi$
$919$	−9650.26	−0.346390	−0.173195	+	0.984888i	$0.555409\pi$
$920$	2401.01	0.0860422
$921$	−27637.0	−0.988783
$922$	27988.7	0.999737
$923$	−21444.1	−0.764725
$924$	1310.44	0.0466561
$925$	970.389	0.0344932
$926$	5988.42	0.212518
$927$	13335.8	0.472497
$928$	−2754.97	−0.0974528
$929$	−37880.8	−1.33781	−0.668907	−	0.743346i	$-0.733238\pi$
$929$	−37880.8	−1.33781	−0.668907	+	0.743346i	$0.733238\pi$
$930$	4348.59	0.153329
$931$	−2362.59	−0.0831695
$932$	−4593.07	−0.161428
$933$	−20421.7	−0.716588
$934$	−40636.1	−1.42361
$935$	2524.08	0.0882847
$936$	5563.20	0.194272
$937$	55205.7	1.92475	0.962375	−	0.271724i	$-0.0875939\pi$
$937$	55205.7	1.92475	0.962375	+	0.271724i	$0.0875939\pi$
$938$	−22900.7	−0.797160
$939$	20043.7	0.696593
$940$	−6229.64	−0.216158
$941$	−40380.0	−1.39889	−0.699443	−	0.714688i	$-0.746569\pi$
$941$	−40380.0	−1.39889	−0.699443	+	0.714688i	$0.746569\pi$
$942$	−3060.31	−0.105849
$943$	−2475.57	−0.0854886
$944$	34418.7	1.18669
$945$	−4875.37	−0.167826
$946$	−1855.77	−0.0637804
$947$	10595.3	0.363571	0.181785	−	0.983338i	$-0.441812\pi$
$947$	10595.3	0.363571	0.181785	+	0.983338i	$0.441812\pi$
$948$	−2974.47	−0.101905
$949$	−4272.36	−0.146140
$950$	−7307.14	−0.249553
$951$	1672.52	0.0570296
$952$	11606.8	0.395146
$953$	−34160.9	−1.16116	−0.580578	−	0.814205i	$-0.697173\pi$
$953$	−34160.9	−1.16116	−0.580578	+	0.814205i	$0.697173\pi$
$954$	−3542.71	−0.120230
$955$	27853.1	0.943774
$956$	9888.30	0.334530
$957$	−1003.85	−0.0339080
$958$	−12289.8	−0.414472
$959$	55865.5	1.88112
$960$	−15920.1	−0.535227
$961$	−25751.2	−0.864395
$962$	1591.34	0.0533334
$963$	−5358.60	−0.179313
$964$	−398.002	−0.0132975
$965$	−14514.1	−0.484171
$966$	1477.60	0.0492144
$967$	48956.6	1.62806	0.814031	−	0.580821i	$-0.197268\pi$
$967$	48956.6	1.62806	0.814031	+	0.580821i	$0.197268\pi$
$968$	2966.60	0.0985023
$969$	−5885.07	−0.195104
$970$	37673.9	1.24705
$971$	46260.3	1.52890	0.764451	−	0.644682i	$-0.223010\pi$
$971$	46260.3	1.52890	0.764451	+	0.644682i	$0.223010\pi$
$972$	−499.945	−0.0164977
$973$	−21131.2	−0.696232
$974$	6651.18	0.218806
$975$	−2834.73	−0.0931117
$976$	2641.79	0.0866411
$977$	−18039.5	−0.590721	−0.295360	−	0.955386i	$-0.595440\pi$
$977$	−18039.5	−0.590721	−0.295360	+	0.955386i	$0.595440\pi$
$978$	12151.7	0.397308
$979$	−14575.7	−0.475834
$980$	568.571	0.0185330
$981$	3034.12	0.0987482
$982$	50003.2	1.62491
$983$	1897.37	0.0615634	0.0307817	−	0.999526i	$-0.490200\pi$
$983$	1897.37	0.0615634	0.0307817	+	0.999526i	$0.490200\pi$
$984$	17394.4	0.563529
$985$	−1122.72	−0.0363176
$986$	−1818.86	−0.0587466
$987$	−18741.2	−0.604396
$988$	4148.60	0.133588
$989$	724.443	0.0232922
$990$	−2257.78	−0.0724818
$991$	29563.4	0.947642	0.473821	−	0.880621i	$-0.342874\pi$
$991$	29563.4	0.947642	0.473821	+	0.880621i	$0.342874\pi$
$992$	−5756.26	−0.184235
$993$	−26604.7	−0.850226
$994$	40019.8	1.27701
$995$	5597.52	0.178345
$996$	8256.39	0.262664
$997$	10658.7	0.338580	0.169290	−	0.985566i	$-0.445853\pi$
$997$	10658.7	0.338580	0.169290	+	0.985566i	$0.445853\pi$
$998$	−37898.5	−1.20206
$999$	−699.083	−0.0221401

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2013.4.a.c.1.13	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.4.a.c.1.13	✓	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)

\(f(q)\)	\(=\)	\(q-2.43775 q^{2} +3.00000 q^{3} -2.05739 q^{4} -9.35530 q^{5} -7.31324 q^{6} +19.3013 q^{7} +24.5174 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.43775 q^{2} +3.00000 q^{3} -2.05739 q^{4} -9.35530 q^{5} -7.31324 q^{6} +19.3013 q^{7} +24.5174 q^{8} +9.00000 q^{9} +22.8059 q^{10} -11.0000 q^{11} -6.17216 q^{12} +25.2121 q^{13} -47.0517 q^{14} -28.0659 q^{15} -43.3081 q^{16} +24.5275 q^{17} -21.9397 q^{18} -79.9793 q^{19} +19.2475 q^{20} +57.9039 q^{21} +26.8152 q^{22} -10.4680 q^{23} +73.5521 q^{24} -37.4784 q^{25} -61.4607 q^{26} +27.0000 q^{27} -39.7102 q^{28} +30.4198 q^{29} +68.4176 q^{30} +63.5596 q^{31} -90.5648 q^{32} -33.0000 q^{33} -59.7918 q^{34} -180.569 q^{35} -18.5165 q^{36} -25.8919 q^{37} +194.969 q^{38} +75.6362 q^{39} -229.367 q^{40} +236.491 q^{41} -141.155 q^{42} -69.2058 q^{43} +22.6312 q^{44} -84.1977 q^{45} +25.5182 q^{46} -323.660 q^{47} -129.924 q^{48} +29.5400 q^{49} +91.3629 q^{50} +73.5824 q^{51} -51.8710 q^{52} +161.474 q^{53} -65.8192 q^{54} +102.908 q^{55} +473.217 q^{56} -239.938 q^{57} -74.1558 q^{58} -794.740 q^{59} +57.7424 q^{60} -61.0000 q^{61} -154.942 q^{62} +173.712 q^{63} +567.239 q^{64} -235.866 q^{65} +80.4457 q^{66} +486.715 q^{67} -50.4625 q^{68} -31.4039 q^{69} +440.183 q^{70} -850.549 q^{71} +220.656 q^{72} -169.457 q^{73} +63.1180 q^{74} -112.435 q^{75} +164.548 q^{76} -212.314 q^{77} -184.382 q^{78} +481.917 q^{79} +405.160 q^{80} +81.0000 q^{81} -576.505 q^{82} -1337.68 q^{83} -119.131 q^{84} -229.462 q^{85} +168.706 q^{86} +91.2595 q^{87} -269.691 q^{88} +1325.06 q^{89} +205.253 q^{90} +486.626 q^{91} +21.5366 q^{92} +190.679 q^{93} +789.002 q^{94} +748.230 q^{95} -271.694 q^{96} +1651.94 q^{97} -72.0112 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q - 8 q^{2} + 111 q^{3} + 130 q^{4} - 35 q^{5} - 24 q^{6} - 35 q^{7} - 117 q^{8} + 333 q^{9}+O(q^{10}) \) 37 * q - 8 * q^2 + 111 * q^3 + 130 * q^4 - 35 * q^5 - 24 * q^6 - 35 * q^7 - 117 * q^8 + 333 * q^9	\( 37 q - 8 q^{2} + 111 q^{3} + 130 q^{4} - 35 q^{5} - 24 q^{6} - 35 q^{7} - 117 q^{8} + 333 q^{9} - 41 q^{10} - 407 q^{11} + 390 q^{12} + 51 q^{13} - 228 q^{14} - 105 q^{15} + 462 q^{16} - 190 q^{17} - 72 q^{18} - 51 q^{19} - 720 q^{20} - 105 q^{21} + 88 q^{22} - 583 q^{23} - 351 q^{24} + 598 q^{25} - 1019 q^{26} + 999 q^{27} - 498 q^{28} - 566 q^{29} - 123 q^{30} - 696 q^{31} - 859 q^{32} - 1221 q^{33} - 348 q^{34} - 1102 q^{35} + 1170 q^{36} - 1022 q^{37} - 455 q^{38} + 153 q^{39} - 503 q^{40} - 790 q^{41} - 684 q^{42} - 87 q^{43} - 1430 q^{44} - 315 q^{45} - 303 q^{46} - 1603 q^{47} + 1386 q^{48} + 110 q^{49} - 1926 q^{50} - 570 q^{51} + 736 q^{52} - 2619 q^{53} - 216 q^{54} + 385 q^{55} - 4937 q^{56} - 153 q^{57} - 1099 q^{58} - 2471 q^{59} - 2160 q^{60} - 2257 q^{61} - 2909 q^{62} - 315 q^{63} - 265 q^{64} - 1970 q^{65} + 264 q^{66} - 3033 q^{67} - 1956 q^{68} - 1749 q^{69} + 2410 q^{70} - 3891 q^{71} - 1053 q^{72} + 391 q^{73} - 532 q^{74} + 1794 q^{75} + 1554 q^{76} + 385 q^{77} - 3057 q^{78} + 67 q^{79} - 5111 q^{80} + 2997 q^{81} - 4818 q^{82} - 5315 q^{83} - 1494 q^{84} - 2747 q^{85} - 5195 q^{86} - 1698 q^{87} + 1287 q^{88} - 8945 q^{89} - 369 q^{90} - 4432 q^{91} - 4701 q^{92} - 2088 q^{93} - 372 q^{94} - 3388 q^{95} - 2577 q^{96} - 3784 q^{97} - 4502 q^{98} - 3663 q^{99}+O(q^{100}) \) 37 * q - 8 * q^2 + 111 * q^3 + 130 * q^4 - 35 * q^5 - 24 * q^6 - 35 * q^7 - 117 * q^8 + 333 * q^9 - 41 * q^10 - 407 * q^11 + 390 * q^12 + 51 * q^13 - 228 * q^14 - 105 * q^15 + 462 * q^16 - 190 * q^17 - 72 * q^18 - 51 * q^19 - 720 * q^20 - 105 * q^21 + 88 * q^22 - 583 * q^23 - 351 * q^24 + 598 * q^25 - 1019 * q^26 + 999 * q^27 - 498 * q^28 - 566 * q^29 - 123 * q^30 - 696 * q^31 - 859 * q^32 - 1221 * q^33 - 348 * q^34 - 1102 * q^35 + 1170 * q^36 - 1022 * q^37 - 455 * q^38 + 153 * q^39 - 503 * q^40 - 790 * q^41 - 684 * q^42 - 87 * q^43 - 1430 * q^44 - 315 * q^45 - 303 * q^46 - 1603 * q^47 + 1386 * q^48 + 110 * q^49 - 1926 * q^50 - 570 * q^51 + 736 * q^52 - 2619 * q^53 - 216 * q^54 + 385 * q^55 - 4937 * q^56 - 153 * q^57 - 1099 * q^58 - 2471 * q^59 - 2160 * q^60 - 2257 * q^61 - 2909 * q^62 - 315 * q^63 - 265 * q^64 - 1970 * q^65 + 264 * q^66 - 3033 * q^67 - 1956 * q^68 - 1749 * q^69 + 2410 * q^70 - 3891 * q^71 - 1053 * q^72 + 391 * q^73 - 532 * q^74 + 1794 * q^75 + 1554 * q^76 + 385 * q^77 - 3057 * q^78 + 67 * q^79 - 5111 * q^80 + 2997 * q^81 - 4818 * q^82 - 5315 * q^83 - 1494 * q^84 - 2747 * q^85 - 5195 * q^86 - 1698 * q^87 + 1287 * q^88 - 8945 * q^89 - 369 * q^90 - 4432 * q^91 - 4701 * q^92 - 2088 * q^93 - 372 * q^94 - 3388 * q^95 - 2577 * q^96 - 3784 * q^97 - 4502 * q^98 - 3663 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more