LMFDB - Embedded newform 8002.2.a.d.1.45

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.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:	$63.8962916974$
Analytic rank:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.45
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.613105 q^{3} +1.00000 q^{4} +1.67602 q^{5} +0.613105 q^{6} +2.56938 q^{7} +1.00000 q^{8} -2.62410 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.613105 q^{3} +1.00000 q^{4} +1.67602 q^{5} +0.613105 q^{6} +2.56938 q^{7} +1.00000 q^{8} -2.62410 q^{9} +1.67602 q^{10} -5.50207 q^{11} +0.613105 q^{12} -3.62861 q^{13} +2.56938 q^{14} +1.02757 q^{15} +1.00000 q^{16} +3.34578 q^{17} -2.62410 q^{18} -7.43149 q^{19} +1.67602 q^{20} +1.57530 q^{21} -5.50207 q^{22} +3.76407 q^{23} +0.613105 q^{24} -2.19097 q^{25} -3.62861 q^{26} -3.44816 q^{27} +2.56938 q^{28} +0.0692995 q^{29} +1.02757 q^{30} -5.47929 q^{31} +1.00000 q^{32} -3.37335 q^{33} +3.34578 q^{34} +4.30633 q^{35} -2.62410 q^{36} -5.26078 q^{37} -7.43149 q^{38} -2.22472 q^{39} +1.67602 q^{40} +3.91818 q^{41} +1.57530 q^{42} -4.71763 q^{43} -5.50207 q^{44} -4.39804 q^{45} +3.76407 q^{46} +8.48557 q^{47} +0.613105 q^{48} -0.398265 q^{49} -2.19097 q^{50} +2.05131 q^{51} -3.62861 q^{52} -3.08448 q^{53} -3.44816 q^{54} -9.22156 q^{55} +2.56938 q^{56} -4.55628 q^{57} +0.0692995 q^{58} +9.84849 q^{59} +1.02757 q^{60} +5.40321 q^{61} -5.47929 q^{62} -6.74233 q^{63} +1.00000 q^{64} -6.08161 q^{65} -3.37335 q^{66} -8.71325 q^{67} +3.34578 q^{68} +2.30777 q^{69} +4.30633 q^{70} +0.215441 q^{71} -2.62410 q^{72} -3.39453 q^{73} -5.26078 q^{74} -1.34329 q^{75} -7.43149 q^{76} -14.1369 q^{77} -2.22472 q^{78} +8.59173 q^{79} +1.67602 q^{80} +5.75822 q^{81} +3.91818 q^{82} -13.0358 q^{83} +1.57530 q^{84} +5.60758 q^{85} -4.71763 q^{86} +0.0424879 q^{87} -5.50207 q^{88} -8.95446 q^{89} -4.39804 q^{90} -9.32330 q^{91} +3.76407 q^{92} -3.35938 q^{93} +8.48557 q^{94} -12.4553 q^{95} +0.613105 q^{96} -11.0984 q^{97} -0.398265 q^{98} +14.4380 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.613105	0.353976	0.176988	−	0.984213i	$-0.443365\pi$
$3$	0.613105	0.353976	0.176988	+	0.984213i	$0.443365\pi$
$4$	1.00000	0.500000
$5$	1.67602	0.749537	0.374769	−	0.927118i	$-0.377722\pi$
$5$	1.67602	0.749537	0.374769	+	0.927118i	$0.377722\pi$
$6$	0.613105	0.250299
$7$	2.56938	0.971136	0.485568	−	0.874199i	$-0.338613\pi$
$7$	2.56938	0.971136	0.485568	+	0.874199i	$0.338613\pi$
$8$	1.00000	0.353553
$9$	−2.62410	−0.874701
$10$	1.67602	0.530003
$11$	−5.50207	−1.65894	−0.829468	−	0.558554i	$-0.811356\pi$
$11$	−5.50207	−1.65894	−0.829468	+	0.558554i	$0.811356\pi$
$12$	0.613105	0.176988
$13$	−3.62861	−1.00640	−0.503198	−	0.864171i	$-0.667843\pi$
$13$	−3.62861	−1.00640	−0.503198	+	0.864171i	$0.667843\pi$
$14$	2.56938	0.686697
$15$	1.02757	0.265318
$16$	1.00000	0.250000
$17$	3.34578	0.811471	0.405735	−	0.913991i	$-0.367015\pi$
$17$	3.34578	0.811471	0.405735	+	0.913991i	$0.367015\pi$
$18$	−2.62410	−0.618507
$19$	−7.43149	−1.70490	−0.852450	−	0.522809i	$-0.824884\pi$
$19$	−7.43149	−1.70490	−0.852450	+	0.522809i	$0.824884\pi$
$20$	1.67602	0.374769
$21$	1.57530	0.343759
$22$	−5.50207	−1.17305
$23$	3.76407	0.784864	0.392432	−	0.919781i	$-0.371634\pi$
$23$	3.76407	0.784864	0.392432	+	0.919781i	$0.371634\pi$
$24$	0.613105	0.125150
$25$	−2.19097	−0.438194
$26$	−3.62861	−0.711630
$27$	−3.44816	−0.663600
$28$	2.56938	0.485568
$29$	0.0692995	0.0128686	0.00643430	−	0.999979i	$-0.497952\pi$
$29$	0.0692995	0.0128686	0.00643430	+	0.999979i	$0.497952\pi$
$30$	1.02757	0.187608
$31$	−5.47929	−0.984109	−0.492055	−	0.870564i	$-0.663754\pi$
$31$	−5.47929	−0.984109	−0.492055	+	0.870564i	$0.663754\pi$
$32$	1.00000	0.176777
$33$	−3.37335	−0.587224
$34$	3.34578	0.573797
$35$	4.30633	0.727902
$36$	−2.62410	−0.437350
$37$	−5.26078	−0.864867	−0.432433	−	0.901666i	$-0.642345\pi$
$37$	−5.26078	−0.864867	−0.432433	+	0.901666i	$0.642345\pi$
$38$	−7.43149	−1.20555
$39$	−2.22472	−0.356240
$40$	1.67602	0.265001
$41$	3.91818	0.611917	0.305959	−	0.952045i	$-0.401023\pi$
$41$	3.91818	0.611917	0.305959	+	0.952045i	$0.401023\pi$
$42$	1.57530	0.243074
$43$	−4.71763	−0.719433	−0.359716	−	0.933062i	$-0.617127\pi$
$43$	−4.71763	−0.719433	−0.359716	+	0.933062i	$0.617127\pi$
$44$	−5.50207	−0.829468
$45$	−4.39804	−0.655621
$46$	3.76407	0.554982
$47$	8.48557	1.23775	0.618874	−	0.785491i	$-0.287589\pi$
$47$	8.48557	1.23775	0.618874	+	0.785491i	$0.287589\pi$
$48$	0.613105	0.0884941
$49$	−0.398265	−0.0568950
$50$	−2.19097	−0.309850
$51$	2.05131	0.287241
$52$	−3.62861	−0.503198
$53$	−3.08448	−0.423686	−0.211843	−	0.977304i	$-0.567947\pi$
$53$	−3.08448	−0.423686	−0.211843	+	0.977304i	$0.567947\pi$
$54$	−3.44816	−0.469236
$55$	−9.22156	−1.24343
$56$	2.56938	0.343348
$57$	−4.55628	−0.603494
$58$	0.0692995	0.00909947
$59$	9.84849	1.28216	0.641082	−	0.767472i	$-0.278486\pi$
$59$	9.84849	1.28216	0.641082	+	0.767472i	$0.278486\pi$
$60$	1.02757	0.132659
$61$	5.40321	0.691810	0.345905	−	0.938270i	$-0.387572\pi$
$61$	5.40321	0.691810	0.345905	+	0.938270i	$0.387572\pi$
$62$	−5.47929	−0.695870
$63$	−6.74233	−0.849453
$64$	1.00000	0.125000
$65$	−6.08161	−0.754331
$66$	−3.37335	−0.415230
$67$	−8.71325	−1.06449	−0.532247	−	0.846589i	$-0.678652\pi$
$67$	−8.71325	−1.06449	−0.532247	+	0.846589i	$0.678652\pi$
$68$	3.34578	0.405735
$69$	2.30777	0.277823
$70$	4.30633	0.514705
$71$	0.215441	0.0255681	0.0127841	−	0.999918i	$-0.495931\pi$
$71$	0.215441	0.0255681	0.0127841	+	0.999918i	$0.495931\pi$
$72$	−2.62410	−0.309253
$73$	−3.39453	−0.397299	−0.198650	−	0.980071i	$-0.563656\pi$
$73$	−3.39453	−0.397299	−0.198650	+	0.980071i	$0.563656\pi$
$74$	−5.26078	−0.611553
$75$	−1.34329	−0.155110
$76$	−7.43149	−0.852450
$77$	−14.1369	−1.61105
$78$	−2.22472	−0.251900
$79$	8.59173	0.966645	0.483323	−	0.875442i	$-0.339430\pi$
$79$	8.59173	0.966645	0.483323	+	0.875442i	$0.339430\pi$
$80$	1.67602	0.187384
$81$	5.75822	0.639802
$82$	3.91818	0.432691
$83$	−13.0358	−1.43087	−0.715435	−	0.698680i	$-0.753771\pi$
$83$	−13.0358	−1.43087	−0.715435	+	0.698680i	$0.753771\pi$
$84$	1.57530	0.171880
$85$	5.60758	0.608228
$86$	−4.71763	−0.508716
$87$	0.0424879	0.00455518
$88$	−5.50207	−0.586523
$89$	−8.95446	−0.949171	−0.474585	−	0.880209i	$-0.657402\pi$
$89$	−8.95446	−0.949171	−0.474585	+	0.880209i	$0.657402\pi$
$90$	−4.39804	−0.463594
$91$	−9.32330	−0.977347
$92$	3.76407	0.392432
$93$	−3.35938	−0.348351
$94$	8.48557	0.875220
$95$	−12.4553	−1.27789
$96$	0.613105	0.0625748
$97$	−11.0984	−1.12687	−0.563434	−	0.826161i	$-0.690520\pi$
$97$	−11.0984	−1.12687	−0.563434	+	0.826161i	$0.690520\pi$
$98$	−0.398265	−0.0402309
$99$	14.4380	1.45107
$100$	−2.19097	−0.219097
$101$	−11.2102	−1.11546	−0.557728	−	0.830024i	$-0.688327\pi$
$101$	−11.2102	−1.11546	−0.557728	+	0.830024i	$0.688327\pi$
$102$	2.05131	0.203110
$103$	−1.46164	−0.144020	−0.0720100	−	0.997404i	$-0.522941\pi$
$103$	−1.46164	−0.144020	−0.0720100	+	0.997404i	$0.522941\pi$
$104$	−3.62861	−0.355815
$105$	2.64023	0.257660
$106$	−3.08448	−0.299591
$107$	−13.5109	−1.30615	−0.653075	−	0.757293i	$-0.726521\pi$
$107$	−13.5109	−1.30615	−0.653075	+	0.757293i	$0.726521\pi$
$108$	−3.44816	−0.331800
$109$	−13.6244	−1.30498	−0.652491	−	0.757797i	$-0.726276\pi$
$109$	−13.6244	−1.30498	−0.652491	+	0.757797i	$0.726276\pi$
$110$	−9.22156	−0.879241
$111$	−3.22541	−0.306142
$112$	2.56938	0.242784
$113$	7.71230	0.725512	0.362756	−	0.931884i	$-0.381836\pi$
$113$	7.71230	0.725512	0.362756	+	0.931884i	$0.381836\pi$
$114$	−4.55628	−0.426735
$115$	6.30865	0.588284
$116$	0.0692995	0.00643430
$117$	9.52185	0.880296
$118$	9.84849	0.906627
$119$	8.59660	0.788049
$120$	1.02757	0.0938042
$121$	19.2728	1.75207
$122$	5.40321	0.489183
$123$	2.40226	0.216604
$124$	−5.47929	−0.492055
$125$	−12.0522	−1.07798
$126$	−6.74233	−0.600654
$127$	−13.3046	−1.18059	−0.590294	−	0.807188i	$-0.700988\pi$
$127$	−13.3046	−1.18059	−0.590294	+	0.807188i	$0.700988\pi$
$128$	1.00000	0.0883883
$129$	−2.89240	−0.254662
$130$	−6.08161	−0.533393
$131$	21.0266	1.83710	0.918550	−	0.395306i	$-0.129361\pi$
$131$	21.0266	1.83710	0.918550	+	0.395306i	$0.129361\pi$
$132$	−3.37335	−0.293612
$133$	−19.0943	−1.65569
$134$	−8.71325	−0.752711
$135$	−5.77918	−0.497393
$136$	3.34578	0.286898
$137$	3.90949	0.334010	0.167005	−	0.985956i	$-0.446590\pi$
$137$	3.90949	0.334010	0.167005	+	0.985956i	$0.446590\pi$
$138$	2.30777	0.196451
$139$	−0.812347	−0.0689024	−0.0344512	−	0.999406i	$-0.510968\pi$
$139$	−0.812347	−0.0689024	−0.0344512	+	0.999406i	$0.510968\pi$
$140$	4.30633	0.363951
$141$	5.20254	0.438133
$142$	0.215441	0.0180794
$143$	19.9649	1.66955
$144$	−2.62410	−0.218675
$145$	0.116147	0.00964549
$146$	−3.39453	−0.280933
$147$	−0.244178	−0.0201395
$148$	−5.26078	−0.432433
$149$	0.439332	0.0359915	0.0179957	−	0.999838i	$-0.494271\pi$
$149$	0.439332	0.0359915	0.0179957	+	0.999838i	$0.494271\pi$
$150$	−1.34329	−0.109680
$151$	−12.1838	−0.991505	−0.495752	−	0.868464i	$-0.665108\pi$
$151$	−12.1838	−0.991505	−0.495752	+	0.868464i	$0.665108\pi$
$152$	−7.43149	−0.602773
$153$	−8.77967	−0.709794
$154$	−14.1369	−1.13919
$155$	−9.18337	−0.737626
$156$	−2.22472	−0.178120
$157$	5.62221	0.448701	0.224350	−	0.974509i	$-0.427974\pi$
$157$	5.62221	0.448701	0.224350	+	0.974509i	$0.427974\pi$
$158$	8.59173	0.683521
$159$	−1.89111	−0.149975
$160$	1.67602	0.132501
$161$	9.67135	0.762209
$162$	5.75822	0.452409
$163$	−7.42956	−0.581928	−0.290964	−	0.956734i	$-0.593976\pi$
$163$	−7.42956	−0.581928	−0.290964	+	0.956734i	$0.593976\pi$
$164$	3.91818	0.305959
$165$	−5.65378	−0.440146
$166$	−13.0358	−1.01178
$167$	5.38215	0.416483	0.208242	−	0.978077i	$-0.433226\pi$
$167$	5.38215	0.416483	0.208242	+	0.978077i	$0.433226\pi$
$168$	1.57530	0.121537
$169$	0.166832	0.0128333
$170$	5.60758	0.430082
$171$	19.5010	1.49128
$172$	−4.71763	−0.359716
$173$	10.3512	0.786990	0.393495	−	0.919327i	$-0.371266\pi$
$173$	10.3512	0.786990	0.393495	+	0.919327i	$0.371266\pi$
$174$	0.0424879	0.00322100
$175$	−5.62944	−0.425546
$176$	−5.50207	−0.414734
$177$	6.03816	0.453856
$178$	−8.95446	−0.671165
$179$	−10.5580	−0.789140	−0.394570	−	0.918866i	$-0.629106\pi$
$179$	−10.5580	−0.789140	−0.394570	+	0.918866i	$0.629106\pi$
$180$	−4.39804	−0.327810
$181$	−5.28416	−0.392769	−0.196384	−	0.980527i	$-0.562920\pi$
$181$	−5.28416	−0.392769	−0.196384	+	0.980527i	$0.562920\pi$
$182$	−9.32330	−0.691089
$183$	3.31273	0.244884
$184$	3.76407	0.277491
$185$	−8.81715	−0.648250
$186$	−3.35938	−0.246322
$187$	−18.4087	−1.34618
$188$	8.48557	0.618874
$189$	−8.85966	−0.644445
$190$	−12.4553	−0.903602
$191$	20.5384	1.48611	0.743055	−	0.669230i	$-0.233376\pi$
$191$	20.5384	1.48611	0.743055	+	0.669230i	$0.233376\pi$
$192$	0.613105	0.0442470
$193$	−16.7861	−1.20829	−0.604145	−	0.796874i	$-0.706485\pi$
$193$	−16.7861	−1.20829	−0.604145	+	0.796874i	$0.706485\pi$
$194$	−11.0984	−0.796815
$195$	−3.72867	−0.267015
$196$	−0.398265	−0.0284475
$197$	18.7025	1.33250	0.666248	−	0.745730i	$-0.267899\pi$
$197$	18.7025	1.33250	0.666248	+	0.745730i	$0.267899\pi$
$198$	14.4380	1.02606
$199$	20.6556	1.46424	0.732118	−	0.681178i	$-0.238532\pi$
$199$	20.6556	1.46424	0.732118	+	0.681178i	$0.238532\pi$
$200$	−2.19097	−0.154925
$201$	−5.34214	−0.376805
$202$	−11.2102	−0.788746
$203$	0.178057	0.0124972
$204$	2.05131	0.143621
$205$	6.56694	0.458655
$206$	−1.46164	−0.101838
$207$	−9.87731	−0.686521
$208$	−3.62861	−0.251599
$209$	40.8886	2.82832
$210$	2.64023	0.182193
$211$	−10.7621	−0.740895	−0.370447	−	0.928853i	$-0.620796\pi$
$211$	−10.7621	−0.740895	−0.370447	+	0.928853i	$0.620796\pi$
$212$	−3.08448	−0.211843
$213$	0.132088	0.00905051
$214$	−13.5109	−0.923588
$215$	−7.90683	−0.539241
$216$	−3.44816	−0.234618
$217$	−14.0784	−0.955704
$218$	−13.6244	−0.922761
$219$	−2.08120	−0.140635
$220$	−9.22156	−0.621717
$221$	−12.1405	−0.816661
$222$	−3.22541	−0.216475
$223$	9.39463	0.629111	0.314555	−	0.949239i	$-0.398145\pi$
$223$	9.39463	0.629111	0.314555	+	0.949239i	$0.398145\pi$
$224$	2.56938	0.171674
$225$	5.74933	0.383289
$226$	7.71230	0.513015
$227$	−14.3504	−0.952469	−0.476235	−	0.879318i	$-0.657999\pi$
$227$	−14.3504	−0.952469	−0.476235	+	0.879318i	$0.657999\pi$
$228$	−4.55628	−0.301747
$229$	−21.2665	−1.40533	−0.702664	−	0.711522i	$-0.748006\pi$
$229$	−21.2665	−1.40533	−0.702664	+	0.711522i	$0.748006\pi$
$230$	6.30865	0.415980
$231$	−8.66742	−0.570274
$232$	0.0692995	0.00454974
$233$	17.9337	1.17487	0.587437	−	0.809270i	$-0.300137\pi$
$233$	17.9337	1.17487	0.587437	+	0.809270i	$0.300137\pi$
$234$	9.52185	0.622463
$235$	14.2219	0.927738
$236$	9.84849	0.641082
$237$	5.26763	0.342170
$238$	8.59660	0.557234
$239$	−1.58857	−0.102756	−0.0513781	−	0.998679i	$-0.516361\pi$
$239$	−1.58857	−0.102756	−0.0513781	+	0.998679i	$0.516361\pi$
$240$	1.02757	0.0663296
$241$	−24.4529	−1.57515	−0.787576	−	0.616218i	$-0.788664\pi$
$241$	−24.4529	−1.57515	−0.787576	+	0.616218i	$0.788664\pi$
$242$	19.2728	1.23890
$243$	13.8749	0.890074
$244$	5.40321	0.345905
$245$	−0.667499	−0.0426450
$246$	2.40226	0.153162
$247$	26.9660	1.71580
$248$	−5.47929	−0.347935
$249$	−7.99234	−0.506494
$250$	−12.0522	−0.762247
$251$	25.2552	1.59410	0.797048	−	0.603916i	$-0.206394\pi$
$251$	25.2552	1.59410	0.797048	+	0.603916i	$0.206394\pi$
$252$	−6.74233	−0.424727
$253$	−20.7102	−1.30204
$254$	−13.3046	−0.834802
$255$	3.43804	0.215298
$256$	1.00000	0.0625000
$257$	−16.7529	−1.04502	−0.522509	−	0.852634i	$-0.675004\pi$
$257$	−16.7529	−1.04502	−0.522509	+	0.852634i	$0.675004\pi$
$258$	−2.89240	−0.180073
$259$	−13.5170	−0.839903
$260$	−6.08161	−0.377166
$261$	−0.181849	−0.0112562
$262$	21.0266	1.29903
$263$	−2.46101	−0.151752	−0.0758761	−	0.997117i	$-0.524175\pi$
$263$	−2.46101	−0.151752	−0.0758761	+	0.997117i	$0.524175\pi$
$264$	−3.37335	−0.207615
$265$	−5.16964	−0.317568
$266$	−19.0943	−1.17075
$267$	−5.49002	−0.335984
$268$	−8.71325	−0.532247
$269$	26.9079	1.64060	0.820301	−	0.571932i	$-0.193806\pi$
$269$	26.9079	1.64060	0.820301	+	0.571932i	$0.193806\pi$
$270$	−5.77918	−0.351710
$271$	11.6407	0.707123	0.353561	−	0.935411i	$-0.384971\pi$
$271$	11.6407	0.707123	0.353561	+	0.935411i	$0.384971\pi$
$272$	3.34578	0.202868
$273$	−5.71616	−0.345958
$274$	3.90949	0.236181
$275$	12.0549	0.726936
$276$	2.30777	0.138912
$277$	14.4447	0.867896	0.433948	−	0.900938i	$-0.357120\pi$
$277$	14.4447	0.867896	0.433948	+	0.900938i	$0.357120\pi$
$278$	−0.812347	−0.0487214
$279$	14.3782	0.860801
$280$	4.30633	0.257352
$281$	0.236277	0.0140951	0.00704754	−	0.999975i	$-0.497757\pi$
$281$	0.236277	0.0140951	0.00704754	+	0.999975i	$0.497757\pi$
$282$	5.20254	0.309807
$283$	−1.55117	−0.0922077	−0.0461038	−	0.998937i	$-0.514681\pi$
$283$	−1.55117	−0.0922077	−0.0461038	+	0.998937i	$0.514681\pi$
$284$	0.215441	0.0127841
$285$	−7.63640	−0.452341
$286$	19.9649	1.18055
$287$	10.0673	0.594255
$288$	−2.62410	−0.154627
$289$	−5.80575	−0.341515
$290$	0.116147	0.00682039
$291$	−6.80445	−0.398884
$292$	−3.39453	−0.198650
$293$	−15.9306	−0.930676	−0.465338	−	0.885133i	$-0.654067\pi$
$293$	−15.9306	−0.930676	−0.465338	+	0.885133i	$0.654067\pi$
$294$	−0.244178	−0.0142408
$295$	16.5062	0.961030
$296$	−5.26078	−0.305777
$297$	18.9720	1.10087
$298$	0.439332	0.0254498
$299$	−13.6584	−0.789884
$300$	−1.34329	−0.0775551
$301$	−12.1214	−0.698667
$302$	−12.1838	−0.701100
$303$	−6.87302	−0.394845
$304$	−7.43149	−0.426225
$305$	9.05586	0.518537
$306$	−8.77967	−0.501900
$307$	29.7564	1.69829	0.849145	−	0.528160i	$-0.177118\pi$
$307$	29.7564	1.69829	0.849145	+	0.528160i	$0.177118\pi$
$308$	−14.1369	−0.805526
$309$	−0.896141	−0.0509797
$310$	−9.18337	−0.521581
$311$	−30.6154	−1.73604	−0.868020	−	0.496529i	$-0.834608\pi$
$311$	−30.6154	−1.73604	−0.868020	+	0.496529i	$0.834608\pi$
$312$	−2.22472	−0.125950
$313$	5.52521	0.312303	0.156152	−	0.987733i	$-0.450091\pi$
$313$	5.52521	0.312303	0.156152	+	0.987733i	$0.450091\pi$
$314$	5.62221	0.317279
$315$	−11.3002	−0.636697
$316$	8.59173	0.483323
$317$	10.8654	0.610262	0.305131	−	0.952310i	$-0.401300\pi$
$317$	10.8654	0.610262	0.305131	+	0.952310i	$0.401300\pi$
$318$	−1.89111	−0.106048
$319$	−0.381291	−0.0213482
$320$	1.67602	0.0936921
$321$	−8.28362	−0.462346
$322$	9.67135	0.538963
$323$	−24.8641	−1.38348
$324$	5.75822	0.319901
$325$	7.95018	0.440997
$326$	−7.42956	−0.411485
$327$	−8.35319	−0.461932
$328$	3.91818	0.216345
$329$	21.8027	1.20202
$330$	−5.65378	−0.311230
$331$	19.2842	1.05996	0.529978	−	0.848012i	$-0.322200\pi$
$331$	19.2842	1.05996	0.529978	+	0.848012i	$0.322200\pi$
$332$	−13.0358	−0.715435
$333$	13.8048	0.756500
$334$	5.38215	0.294498
$335$	−14.6036	−0.797877
$336$	1.57530	0.0859398
$337$	8.23976	0.448848	0.224424	−	0.974492i	$-0.427950\pi$
$337$	8.23976	0.448848	0.224424	+	0.974492i	$0.427950\pi$
$338$	0.166832	0.00907448
$339$	4.72845	0.256814
$340$	5.60758	0.304114
$341$	30.1474	1.63257
$342$	19.5010	1.05449
$343$	−19.0090	−1.02639
$344$	−4.71763	−0.254358
$345$	3.86786	0.208239
$346$	10.3512	0.556486
$347$	−9.29645	−0.499059	−0.249530	−	0.968367i	$-0.580276\pi$
$347$	−9.29645	−0.499059	−0.249530	+	0.968367i	$0.580276\pi$
$348$	0.0424879	0.00227759
$349$	−10.2918	−0.550909	−0.275454	−	0.961314i	$-0.588828\pi$
$349$	−10.2918	−0.550909	−0.275454	+	0.961314i	$0.588828\pi$
$350$	−5.62944	−0.300906
$351$	12.5121	0.667844
$352$	−5.50207	−0.293261
$353$	−19.9409	−1.06134	−0.530672	−	0.847577i	$-0.678061\pi$
$353$	−19.9409	−1.06134	−0.530672	+	0.847577i	$0.678061\pi$
$354$	6.03816	0.320925
$355$	0.361082	0.0191643
$356$	−8.95446	−0.474585
$357$	5.27061	0.278950
$358$	−10.5580	−0.558006
$359$	−7.29664	−0.385102	−0.192551	−	0.981287i	$-0.561676\pi$
$359$	−7.29664	−0.385102	−0.192551	+	0.981287i	$0.561676\pi$
$360$	−4.39804	−0.231797
$361$	36.2270	1.90668
$362$	−5.28416	−0.277729
$363$	11.8162	0.620191
$364$	−9.32330	−0.488674
$365$	−5.68928	−0.297791
$366$	3.31273	0.173159
$367$	25.3547	1.32350	0.661752	−	0.749723i	$-0.269813\pi$
$367$	25.3547	1.32350	0.661752	+	0.749723i	$0.269813\pi$
$368$	3.76407	0.196216
$369$	−10.2817	−0.535245
$370$	−8.81715	−0.458382
$371$	−7.92521	−0.411456
$372$	−3.35938	−0.174176
$373$	−2.59731	−0.134484	−0.0672419	−	0.997737i	$-0.521420\pi$
$373$	−2.59731	−0.134484	−0.0672419	+	0.997737i	$0.521420\pi$
$374$	−18.4087	−0.951892
$375$	−7.38925	−0.381579
$376$	8.48557	0.437610
$377$	−0.251461	−0.0129509
$378$	−8.85966	−0.455692
$379$	30.4255	1.56285	0.781426	−	0.623997i	$-0.214492\pi$
$379$	30.4255	1.56285	0.781426	+	0.623997i	$0.214492\pi$
$380$	−12.4553	−0.638943
$381$	−8.15709	−0.417900
$382$	20.5384	1.05084
$383$	−7.05080	−0.360279	−0.180139	−	0.983641i	$-0.557655\pi$
$383$	−7.05080	−0.360279	−0.180139	+	0.983641i	$0.557655\pi$
$384$	0.613105	0.0312874
$385$	−23.6937	−1.20754
$386$	−16.7861	−0.854390
$387$	12.3796	0.629288
$388$	−11.0984	−0.563434
$389$	−25.9783	−1.31715	−0.658577	−	0.752514i	$-0.728841\pi$
$389$	−25.9783	−1.31715	−0.658577	+	0.752514i	$0.728841\pi$
$390$	−3.72867	−0.188808
$391$	12.5938	0.636894
$392$	−0.398265	−0.0201154
$393$	12.8915	0.650290
$394$	18.7025	0.942217
$395$	14.3999	0.724537
$396$	14.4380	0.725536
$397$	21.6965	1.08892	0.544458	−	0.838788i	$-0.316735\pi$
$397$	21.6965	1.08892	0.544458	+	0.838788i	$0.316735\pi$
$398$	20.6556	1.03537
$399$	−11.7068	−0.586075
$400$	−2.19097	−0.109549
$401$	29.7249	1.48439	0.742195	−	0.670184i	$-0.233785\pi$
$401$	29.7249	1.48439	0.742195	+	0.670184i	$0.233785\pi$
$402$	−5.34214	−0.266442
$403$	19.8822	0.990404
$404$	−11.2102	−0.557728
$405$	9.65087	0.479556
$406$	0.178057	0.00883682
$407$	28.9452	1.43476
$408$	2.05131	0.101555
$409$	−3.88165	−0.191935	−0.0959676	−	0.995384i	$-0.530595\pi$
$409$	−3.88165	−0.191935	−0.0959676	+	0.995384i	$0.530595\pi$
$410$	6.56694	0.324318
$411$	2.39693	0.118232
$412$	−1.46164	−0.0720100
$413$	25.3046	1.24516
$414$	−9.87731	−0.485443
$415$	−21.8483	−1.07249
$416$	−3.62861	−0.177907
$417$	−0.498054	−0.0243898
$418$	40.8886	1.99992
$419$	−5.06014	−0.247204	−0.123602	−	0.992332i	$-0.539445\pi$
$419$	−5.06014	−0.247204	−0.123602	+	0.992332i	$0.539445\pi$
$420$	2.64023	0.128830
$421$	2.61006	0.127206	0.0636032	−	0.997975i	$-0.479741\pi$
$421$	2.61006	0.127206	0.0636032	+	0.997975i	$0.479741\pi$
$422$	−10.7621	−0.523892
$423$	−22.2670	−1.08266
$424$	−3.08448	−0.149796
$425$	−7.33051	−0.355582
$426$	0.132088	0.00639967
$427$	13.8829	0.671841
$428$	−13.5109	−0.653075
$429$	12.2406	0.590980
$430$	−7.90683	−0.381301
$431$	7.88933	0.380016	0.190008	−	0.981783i	$-0.439149\pi$
$431$	7.88933	0.380016	0.190008	+	0.981783i	$0.439149\pi$
$432$	−3.44816	−0.165900
$433$	21.6935	1.04252	0.521261	−	0.853398i	$-0.325462\pi$
$433$	21.6935	1.04252	0.521261	+	0.853398i	$0.325462\pi$
$434$	−14.0784	−0.675785
$435$	0.0712104	0.00341428
$436$	−13.6244	−0.652491
$437$	−27.9727	−1.33811
$438$	−2.08120	−0.0994437
$439$	−4.72568	−0.225545	−0.112772	−	0.993621i	$-0.535973\pi$
$439$	−4.72568	−0.225545	−0.112772	+	0.993621i	$0.535973\pi$
$440$	−9.22156	−0.439620
$441$	1.04509	0.0497661
$442$	−12.1405	−0.577467
$443$	18.4481	0.876493	0.438247	−	0.898855i	$-0.355600\pi$
$443$	18.4481	0.876493	0.438247	+	0.898855i	$0.355600\pi$
$444$	−3.22541	−0.153071
$445$	−15.0078	−0.711439
$446$	9.39463	0.444848
$447$	0.269356	0.0127401
$448$	2.56938	0.121392
$449$	−14.9783	−0.706869	−0.353434	−	0.935459i	$-0.614986\pi$
$449$	−14.9783	−0.706869	−0.353434	+	0.935459i	$0.614986\pi$
$450$	5.74933	0.271026
$451$	−21.5581	−1.01513
$452$	7.71230	0.362756
$453$	−7.46996	−0.350969
$454$	−14.3504	−0.673498
$455$	−15.6260	−0.732558
$456$	−4.55628	−0.213367
$457$	28.0178	1.31062	0.655308	−	0.755362i	$-0.272539\pi$
$457$	28.0178	1.31062	0.655308	+	0.755362i	$0.272539\pi$
$458$	−21.2665	−0.993717
$459$	−11.5368	−0.538492
$460$	6.30865	0.294142
$461$	−8.28194	−0.385728	−0.192864	−	0.981225i	$-0.561778\pi$
$461$	−8.28194	−0.385728	−0.192864	+	0.981225i	$0.561778\pi$
$462$	−8.66742	−0.403245
$463$	−15.1456	−0.703874	−0.351937	−	0.936024i	$-0.614477\pi$
$463$	−15.1456	−0.703874	−0.351937	+	0.936024i	$0.614477\pi$
$464$	0.0692995	0.00321715
$465$	−5.63037	−0.261102
$466$	17.9337	0.830762
$467$	16.7460	0.774912	0.387456	−	0.921888i	$-0.373354\pi$
$467$	16.7460	0.774912	0.387456	+	0.921888i	$0.373354\pi$
$468$	9.52185	0.440148
$469$	−22.3877	−1.03377
$470$	14.2219	0.656010
$471$	3.44700	0.158829
$472$	9.84849	0.453314
$473$	25.9568	1.19349
$474$	5.26763	0.241950
$475$	16.2822	0.747077
$476$	8.59660	0.394024
$477$	8.09399	0.370598
$478$	−1.58857	−0.0726596
$479$	5.54103	0.253176	0.126588	−	0.991955i	$-0.459597\pi$
$479$	5.54103	0.253176	0.126588	+	0.991955i	$0.459597\pi$
$480$	1.02757	0.0469021
$481$	19.0893	0.870399
$482$	−24.4529	−1.11380
$483$	5.92955	0.269804
$484$	19.2728	0.876035
$485$	−18.6010	−0.844629
$486$	13.8749	0.629378
$487$	−10.2260	−0.463385	−0.231692	−	0.972789i	$-0.574426\pi$
$487$	−10.2260	−0.463385	−0.231692	+	0.972789i	$0.574426\pi$
$488$	5.40321	0.244592
$489$	−4.55510	−0.205989
$490$	−0.667499	−0.0301545
$491$	−32.5720	−1.46996	−0.734978	−	0.678091i	$-0.762807\pi$
$491$	−32.5720	−1.46996	−0.734978	+	0.678091i	$0.762807\pi$
$492$	2.40226	0.108302
$493$	0.231861	0.0104425
$494$	26.9660	1.21326
$495$	24.1983	1.08763
$496$	−5.47929	−0.246027
$497$	0.553550	0.0248301
$498$	−7.99234	−0.358145
$499$	−15.8146	−0.707958	−0.353979	−	0.935253i	$-0.615172\pi$
$499$	−15.8146	−0.707958	−0.353979	+	0.935253i	$0.615172\pi$
$500$	−12.0522	−0.538990
$501$	3.29982	0.147425
$502$	25.2552	1.12720
$503$	14.1378	0.630373	0.315186	−	0.949030i	$-0.397933\pi$
$503$	14.1378	0.630373	0.315186	+	0.949030i	$0.397933\pi$
$504$	−6.74233	−0.300327
$505$	−18.7885	−0.836075
$506$	−20.7102	−0.920680
$507$	0.102286	0.00454267
$508$	−13.3046	−0.590294
$509$	−4.78721	−0.212189	−0.106095	−	0.994356i	$-0.533835\pi$
$509$	−4.78721	−0.212189	−0.106095	+	0.994356i	$0.533835\pi$
$510$	3.43804	0.152239
$511$	−8.72185	−0.385832
$512$	1.00000	0.0441942
$513$	25.6250	1.13137
$514$	−16.7529	−0.738939
$515$	−2.44974	−0.107948
$516$	−2.89240	−0.127331
$517$	−46.6882	−2.05334
$518$	−13.5170	−0.593901
$519$	6.34639	0.278576
$520$	−6.08161	−0.266696
$521$	43.6878	1.91400	0.957000	−	0.290090i	$-0.0936851\pi$
$521$	43.6878	1.91400	0.957000	+	0.290090i	$0.0936851\pi$
$522$	−0.181849	−0.00795932
$523$	37.2498	1.62882	0.814410	−	0.580290i	$-0.197061\pi$
$523$	37.2498	1.62882	0.814410	+	0.580290i	$0.197061\pi$
$524$	21.0266	0.918550
$525$	−3.45144	−0.150633
$526$	−2.46101	−0.107305
$527$	−18.3325	−0.798576
$528$	−3.37335	−0.146806
$529$	−8.83175	−0.383989
$530$	−5.16964	−0.224555
$531$	−25.8435	−1.12151
$532$	−19.0943	−0.827845
$533$	−14.2176	−0.615831
$534$	−5.49002	−0.237577
$535$	−22.6445	−0.979008
$536$	−8.71325	−0.376355
$537$	−6.47314	−0.279337
$538$	26.9079	1.16008
$539$	2.19128	0.0943853
$540$	−5.77918	−0.248696
$541$	−35.8023	−1.53926	−0.769631	−	0.638489i	$-0.779560\pi$
$541$	−35.8023	−1.53926	−0.769631	+	0.638489i	$0.779560\pi$
$542$	11.6407	0.500011
$543$	−3.23975	−0.139031
$544$	3.34578	0.143449
$545$	−22.8347	−0.978132
$546$	−5.71616	−0.244629
$547$	−1.91728	−0.0819771	−0.0409885	−	0.999160i	$-0.513051\pi$
$547$	−1.91728	−0.0819771	−0.0409885	+	0.999160i	$0.513051\pi$
$548$	3.90949	0.167005
$549$	−14.1786	−0.605126
$550$	12.0549	0.514021
$551$	−0.514998	−0.0219397
$552$	2.30777	0.0982253
$553$	22.0755	0.938744
$554$	14.4447	0.613695
$555$	−5.40584	−0.229465
$556$	−0.812347	−0.0344512
$557$	−12.9092	−0.546982	−0.273491	−	0.961875i	$-0.588178\pi$
$557$	−12.9092	−0.546982	−0.273491	+	0.961875i	$0.588178\pi$
$558$	14.3782	0.608678
$559$	17.1185	0.724034
$560$	4.30633	0.181976
$561$	−11.2865	−0.476515
$562$	0.236277	0.00996673
$563$	14.7506	0.621663	0.310831	−	0.950465i	$-0.399393\pi$
$563$	14.7506	0.621663	0.310831	+	0.950465i	$0.399393\pi$
$564$	5.20254	0.219067
$565$	12.9259	0.543799
$566$	−1.55117	−0.0652007
$567$	14.7951	0.621335
$568$	0.215441	0.00903969
$569$	−4.57833	−0.191934	−0.0959668	−	0.995385i	$-0.530594\pi$
$569$	−4.57833	−0.191934	−0.0959668	+	0.995385i	$0.530594\pi$
$570$	−7.63640	−0.319854
$571$	18.7010	0.782611	0.391306	−	0.920261i	$-0.372024\pi$
$571$	18.7010	0.782611	0.391306	+	0.920261i	$0.372024\pi$
$572$	19.9649	0.834774
$573$	12.5922	0.526048
$574$	10.0673	0.420202
$575$	−8.24697	−0.343923
$576$	−2.62410	−0.109338
$577$	46.8559	1.95064	0.975318	−	0.220803i	$-0.0708678\pi$
$577$	46.8559	1.95064	0.975318	+	0.220803i	$0.0708678\pi$
$578$	−5.80575	−0.241487
$579$	−10.2916	−0.427706
$580$	0.116147	0.00482275
$581$	−33.4941	−1.38957
$582$	−6.80445	−0.282054
$583$	16.9710	0.702868
$584$	−3.39453	−0.140467
$585$	15.9588	0.659814
$586$	−15.9306	−0.658088
$587$	−34.3278	−1.41686	−0.708430	−	0.705782i	$-0.750596\pi$
$587$	−34.3278	−1.41686	−0.708430	+	0.705782i	$0.750596\pi$
$588$	−0.244178	−0.0100697
$589$	40.7193	1.67781
$590$	16.5062	0.679551
$591$	11.4666	0.471672
$592$	−5.26078	−0.216217
$593$	−13.3233	−0.547122	−0.273561	−	0.961855i	$-0.588202\pi$
$593$	−13.3233	−0.547122	−0.273561	+	0.961855i	$0.588202\pi$
$594$	18.9720	0.778432
$595$	14.4080	0.590672
$596$	0.439332	0.0179957
$597$	12.6640	0.518305
$598$	−13.6584	−0.558532
$599$	−24.6907	−1.00883	−0.504417	−	0.863460i	$-0.668293\pi$
$599$	−24.6907	−1.00883	−0.504417	+	0.863460i	$0.668293\pi$
$600$	−1.34329	−0.0548398
$601$	26.2979	1.07271	0.536356	−	0.843992i	$-0.319800\pi$
$601$	26.2979	1.07271	0.536356	+	0.843992i	$0.319800\pi$
$602$	−12.1214	−0.494032
$603$	22.8645	0.931113
$604$	−12.1838	−0.495752
$605$	32.3015	1.31324
$606$	−6.87302	−0.279197
$607$	35.7025	1.44912	0.724560	−	0.689212i	$-0.242043\pi$
$607$	35.7025	1.44912	0.724560	+	0.689212i	$0.242043\pi$
$608$	−7.43149	−0.301387
$609$	0.109168	0.00442370
$610$	9.05586	0.366661
$611$	−30.7908	−1.24566
$612$	−8.77967	−0.354897
$613$	−47.3272	−1.91153	−0.955765	−	0.294132i	$-0.904969\pi$
$613$	−47.3272	−1.91153	−0.955765	+	0.294132i	$0.904969\pi$
$614$	29.7564	1.20087
$615$	4.02622	0.162353
$616$	−14.1369	−0.569593
$617$	−0.837568	−0.0337192	−0.0168596	−	0.999858i	$-0.505367\pi$
$617$	−0.837568	−0.0337192	−0.0168596	+	0.999858i	$0.505367\pi$
$618$	−0.896141	−0.0360481
$619$	6.20812	0.249526	0.124763	−	0.992187i	$-0.460183\pi$
$619$	6.20812	0.249526	0.124763	+	0.992187i	$0.460183\pi$
$620$	−9.18337	−0.368813
$621$	−12.9791	−0.520835
$622$	−30.6154	−1.22757
$623$	−23.0074	−0.921774
$624$	−2.22472	−0.0890601
$625$	−9.24480	−0.369792
$626$	5.52521	0.220832
$627$	25.0690	1.00116
$628$	5.62221	0.224350
$629$	−17.6014	−0.701814
$630$	−11.3002	−0.450213
$631$	−17.4622	−0.695159	−0.347580	−	0.937650i	$-0.612996\pi$
$631$	−17.4622	−0.695159	−0.347580	+	0.937650i	$0.612996\pi$
$632$	8.59173	0.341761
$633$	−6.59831	−0.262259
$634$	10.8654	0.431520
$635$	−22.2986	−0.884895
$636$	−1.89111	−0.0749873
$637$	1.44515	0.0572590
$638$	−0.381291	−0.0150954
$639$	−0.565339	−0.0223645
$640$	1.67602	0.0662504
$641$	24.1369	0.953350	0.476675	−	0.879080i	$-0.341842\pi$
$641$	24.1369	0.953350	0.476675	+	0.879080i	$0.341842\pi$
$642$	−8.28362	−0.326928
$643$	4.67158	0.184229	0.0921146	−	0.995748i	$-0.470637\pi$
$643$	4.67158	0.184229	0.0921146	+	0.995748i	$0.470637\pi$
$644$	9.67135	0.381105
$645$	−4.84772	−0.190879
$646$	−24.8641	−0.978266
$647$	12.0872	0.475198	0.237599	−	0.971363i	$-0.423640\pi$
$647$	12.0872	0.475198	0.237599	+	0.971363i	$0.423640\pi$
$648$	5.75822	0.226204
$649$	−54.1871	−2.12703
$650$	7.95018	0.311832
$651$	−8.63153	−0.338296
$652$	−7.42956	−0.290964
$653$	−18.2428	−0.713894	−0.356947	−	0.934125i	$-0.616182\pi$
$653$	−18.2428	−0.713894	−0.356947	+	0.934125i	$0.616182\pi$
$654$	−8.35319	−0.326636
$655$	35.2409	1.37697
$656$	3.91818	0.152979
$657$	8.90759	0.347518
$658$	21.8027	0.849957
$659$	−3.98918	−0.155396	−0.0776981	−	0.996977i	$-0.524757\pi$
$659$	−3.98918	−0.155396	−0.0776981	+	0.996977i	$0.524757\pi$
$660$	−5.65378	−0.220073
$661$	−48.7088	−1.89455	−0.947276	−	0.320420i	$-0.896176\pi$
$661$	−48.7088	−1.89455	−0.947276	+	0.320420i	$0.896176\pi$
$662$	19.2842	0.749502
$663$	−7.44343	−0.289079
$664$	−13.0358	−0.505889
$665$	−32.0024	−1.24100
$666$	13.8048	0.534926
$667$	0.260848	0.0101001
$668$	5.38215	0.208242
$669$	5.75989	0.222690
$670$	−14.6036	−0.564185
$671$	−29.7288	−1.14767
$672$	1.57530	0.0607686
$673$	13.2262	0.509834	0.254917	−	0.966963i	$-0.417952\pi$
$673$	13.2262	0.509834	0.254917	+	0.966963i	$0.417952\pi$
$674$	8.23976	0.317384
$675$	7.55483	0.290785
$676$	0.166832	0.00641663
$677$	−40.6489	−1.56226	−0.781132	−	0.624366i	$-0.785357\pi$
$677$	−40.6489	−1.56226	−0.781132	+	0.624366i	$0.785357\pi$
$678$	4.72845	0.181595
$679$	−28.5159	−1.09434
$680$	5.60758	0.215041
$681$	−8.79830	−0.337152
$682$	30.1474	1.15440
$683$	21.0810	0.806643	0.403321	−	0.915058i	$-0.367856\pi$
$683$	21.0810	0.806643	0.403321	+	0.915058i	$0.367856\pi$
$684$	19.5010	0.745639
$685$	6.55237	0.250353
$686$	−19.0090	−0.725766
$687$	−13.0386	−0.497453
$688$	−4.71763	−0.179858
$689$	11.1924	0.426396
$690$	3.86786	0.147247
$691$	−38.1866	−1.45269	−0.726343	−	0.687332i	$-0.758782\pi$
$691$	−38.1866	−1.45269	−0.726343	+	0.687332i	$0.758782\pi$
$692$	10.3512	0.393495
$693$	37.0967	1.40919
$694$	−9.29645	−0.352888
$695$	−1.36151	−0.0516449
$696$	0.0424879	0.00161050
$697$	13.1094	0.496553
$698$	−10.2918	−0.389551
$699$	10.9952	0.415878
$700$	−5.62944	−0.212773
$701$	50.8857	1.92193	0.960964	−	0.276675i	$-0.0892324\pi$
$701$	50.8857	1.92193	0.960964	+	0.276675i	$0.0892324\pi$
$702$	12.5121	0.472237
$703$	39.0954	1.47451
$704$	−5.50207	−0.207367
$705$	8.71955	0.328397
$706$	−19.9409	−0.750484
$707$	−28.8033	−1.08326
$708$	6.03816	0.226928
$709$	10.8403	0.407117	0.203558	−	0.979063i	$-0.434749\pi$
$709$	10.8403	0.407117	0.203558	+	0.979063i	$0.434749\pi$
$710$	0.361082	0.0135512
$711$	−22.5456	−0.845525
$712$	−8.95446	−0.335583
$713$	−20.6244	−0.772391
$714$	5.27061	0.197248
$715$	33.4615	1.25139
$716$	−10.5580	−0.394570
$717$	−0.973961	−0.0363733
$718$	−7.29664	−0.272308
$719$	−46.0898	−1.71886	−0.859429	−	0.511255i	$-0.829181\pi$
$719$	−46.0898	−1.71886	−0.859429	+	0.511255i	$0.829181\pi$
$720$	−4.39804	−0.163905
$721$	−3.75553	−0.139863
$722$	36.2270	1.34823
$723$	−14.9922	−0.557566
$724$	−5.28416	−0.196384
$725$	−0.151833	−0.00563894
$726$	11.8162	0.438541
$727$	−40.7800	−1.51245	−0.756223	−	0.654314i	$-0.772957\pi$
$727$	−40.7800	−1.51245	−0.756223	+	0.654314i	$0.772957\pi$
$728$	−9.32330	−0.345545
$729$	−8.76790	−0.324737
$730$	−5.68928	−0.210570
$731$	−15.7842	−0.583799
$732$	3.31273	0.122442
$733$	2.20589	0.0814763	0.0407382	−	0.999170i	$-0.487029\pi$
$733$	2.20589	0.0814763	0.0407382	+	0.999170i	$0.487029\pi$
$734$	25.3547	0.935858
$735$	−0.409247	−0.0150953
$736$	3.76407	0.138746
$737$	47.9409	1.76593
$738$	−10.2817	−0.378475
$739$	−40.0736	−1.47413	−0.737065	−	0.675821i	$-0.763789\pi$
$739$	−40.0736	−1.47413	−0.737065	+	0.675821i	$0.763789\pi$
$740$	−8.81715	−0.324125
$741$	16.5330	0.607354
$742$	−7.92521	−0.290944
$743$	−13.7482	−0.504372	−0.252186	−	0.967679i	$-0.581150\pi$
$743$	−13.7482	−0.504372	−0.252186	+	0.967679i	$0.581150\pi$
$744$	−3.35938	−0.123161
$745$	0.736327	0.0269769
$746$	−2.59731	−0.0950944
$747$	34.2074	1.25158
$748$	−18.4087	−0.673089
$749$	−34.7148	−1.26845
$750$	−7.38925	−0.269817
$751$	−30.7584	−1.12239	−0.561195	−	0.827683i	$-0.689658\pi$
$751$	−30.7584	−1.12239	−0.561195	+	0.827683i	$0.689658\pi$
$752$	8.48557	0.309437
$753$	15.4841	0.564272
$754$	−0.251461	−0.00915767
$755$	−20.4203	−0.743170
$756$	−8.85966	−0.322223
$757$	−2.73492	−0.0994024	−0.0497012	−	0.998764i	$-0.515827\pi$
$757$	−2.73492	−0.0994024	−0.0497012	+	0.998764i	$0.515827\pi$
$758$	30.4255	1.10510
$759$	−12.6975	−0.460891
$760$	−12.4553	−0.451801
$761$	−4.54284	−0.164678	−0.0823390	−	0.996604i	$-0.526239\pi$
$761$	−4.54284	−0.164678	−0.0823390	+	0.996604i	$0.526239\pi$
$762$	−8.15709	−0.295500
$763$	−35.0063	−1.26731
$764$	20.5384	0.743055
$765$	−14.7149	−0.532017
$766$	−7.05080	−0.254756
$767$	−35.7364	−1.29037
$768$	0.613105	0.0221235
$769$	47.6136	1.71699	0.858495	−	0.512822i	$-0.171400\pi$
$769$	47.6136	1.71699	0.858495	+	0.512822i	$0.171400\pi$
$770$	−23.6937	−0.853862
$771$	−10.2713	−0.369912
$772$	−16.7861	−0.604145
$773$	−25.5629	−0.919433	−0.459716	−	0.888066i	$-0.652049\pi$
$773$	−25.5629	−0.919433	−0.459716	+	0.888066i	$0.652049\pi$
$774$	12.3796	0.444974
$775$	12.0050	0.431231
$776$	−11.0984	−0.398408
$777$	−8.28731	−0.297306
$778$	−25.9783	−0.931368
$779$	−29.1179	−1.04326
$780$	−3.72867	−0.133508
$781$	−1.18537	−0.0424159
$782$	12.5938	0.450352
$783$	−0.238956	−0.00853960
$784$	−0.398265	−0.0142238
$785$	9.42291	0.336318
$786$	12.8915	0.459824
$787$	−4.28673	−0.152806	−0.0764028	−	0.997077i	$-0.524343\pi$
$787$	−4.28673	−0.152806	−0.0764028	+	0.997077i	$0.524343\pi$
$788$	18.7025	0.666248
$789$	−1.50885	−0.0537167
$790$	14.3999	0.512325
$791$	19.8159	0.704571
$792$	14.4380	0.513032
$793$	−19.6061	−0.696235
$794$	21.6965	0.769980
$795$	−3.16953	−0.112412
$796$	20.6556	0.732118
$797$	−14.3600	−0.508656	−0.254328	−	0.967118i	$-0.581854\pi$
$797$	−14.3600	−0.508656	−0.254328	+	0.967118i	$0.581854\pi$
$798$	−11.7068	−0.414418
$799$	28.3909	1.00440
$800$	−2.19097	−0.0774625
$801$	23.4974	0.830241
$802$	29.7249	1.04962
$803$	18.6769	0.659094
$804$	−5.34214	−0.188403
$805$	16.2093	0.571304
$806$	19.8822	0.700321
$807$	16.4973	0.580734
$808$	−11.2102	−0.394373
$809$	1.80757	0.0635507	0.0317754	−	0.999495i	$-0.489884\pi$
$809$	1.80757	0.0635507	0.0317754	+	0.999495i	$0.489884\pi$
$810$	9.65087	0.339097
$811$	−38.4379	−1.34974	−0.674869	−	0.737938i	$-0.735800\pi$
$811$	−38.4379	−1.34974	−0.674869	+	0.737938i	$0.735800\pi$
$812$	0.178057	0.00624858
$813$	7.13697	0.250305
$814$	28.9452	1.01453
$815$	−12.4521	−0.436177
$816$	2.05131	0.0718104
$817$	35.0590	1.22656
$818$	−3.88165	−0.135719
$819$	24.4653	0.854887
$820$	6.56694	0.229327
$821$	−26.2454	−0.915970	−0.457985	−	0.888960i	$-0.651429\pi$
$821$	−26.2454	−0.915970	−0.457985	+	0.888960i	$0.651429\pi$
$822$	2.39693	0.0836025
$823$	−30.4320	−1.06079	−0.530396	−	0.847750i	$-0.677957\pi$
$823$	−30.4320	−1.06079	−0.530396	+	0.847750i	$0.677957\pi$
$824$	−1.46164	−0.0509188
$825$	7.39090	0.257318
$826$	25.3046	0.880458
$827$	−18.0113	−0.626316	−0.313158	−	0.949701i	$-0.601387\pi$
$827$	−18.0113	−0.626316	−0.313158	+	0.949701i	$0.601387\pi$
$828$	−9.87731	−0.343260
$829$	−4.52339	−0.157104	−0.0785520	−	0.996910i	$-0.525030\pi$
$829$	−4.52339	−0.157104	−0.0785520	+	0.996910i	$0.525030\pi$
$830$	−21.8483	−0.758365
$831$	8.85610	0.307215
$832$	−3.62861	−0.125800
$833$	−1.33251	−0.0461687
$834$	−0.498054	−0.0172462
$835$	9.02057	0.312170
$836$	40.8886	1.41416
$837$	18.8935	0.653054
$838$	−5.06014	−0.174800
$839$	8.08501	0.279125	0.139563	−	0.990213i	$-0.455430\pi$
$839$	8.08501	0.279125	0.139563	+	0.990213i	$0.455430\pi$
$840$	2.64023	0.0910966
$841$	−28.9952	−0.999834
$842$	2.61006	0.0899486
$843$	0.144862	0.00498932
$844$	−10.7621	−0.370447
$845$	0.279614	0.00961900
$846$	−22.2670	−0.765555
$847$	49.5191	1.70150
$848$	−3.08448	−0.105921
$849$	−0.951032	−0.0326393
$850$	−7.33051	−0.251434
$851$	−19.8020	−0.678802
$852$	0.132088	0.00452525
$853$	−36.2849	−1.24237	−0.621186	−	0.783663i	$-0.713349\pi$
$853$	−36.2849	−1.24237	−0.621186	+	0.783663i	$0.713349\pi$
$854$	13.8829	0.475063
$855$	32.6840	1.11777
$856$	−13.5109	−0.461794
$857$	30.1594	1.03023	0.515113	−	0.857122i	$-0.327750\pi$
$857$	30.1594	1.03023	0.515113	+	0.857122i	$0.327750\pi$
$858$	12.2406	0.417886
$859$	−32.3393	−1.10340	−0.551701	−	0.834042i	$-0.686021\pi$
$859$	−32.3393	−1.10340	−0.551701	+	0.834042i	$0.686021\pi$
$860$	−7.90683	−0.269621
$861$	6.17232	0.210352
$862$	7.88933	0.268712
$863$	42.1563	1.43502	0.717509	−	0.696549i	$-0.245282\pi$
$863$	42.1563	1.43502	0.717509	+	0.696549i	$0.245282\pi$
$864$	−3.44816	−0.117309
$865$	17.3488	0.589878
$866$	21.6935	0.737174
$867$	−3.55954	−0.120888
$868$	−14.0784	−0.477852
$869$	−47.2723	−1.60360
$870$	0.0712104	0.00241426
$871$	31.6170	1.07130
$872$	−13.6244	−0.461381
$873$	29.1232	0.985672
$874$	−27.9727	−0.946189
$875$	−30.9667	−1.04686
$876$	−2.08120	−0.0703173
$877$	−4.99673	−0.168727	−0.0843637	−	0.996435i	$-0.526886\pi$
$877$	−4.99673	−0.168727	−0.0843637	+	0.996435i	$0.526886\pi$
$878$	−4.72568	−0.159484
$879$	−9.76714	−0.329437
$880$	−9.22156	−0.310859
$881$	−37.8020	−1.27358	−0.636792	−	0.771036i	$-0.719739\pi$
$881$	−37.8020	−1.27358	−0.636792	+	0.771036i	$0.719739\pi$
$882$	1.04509	0.0351900
$883$	20.0029	0.673151	0.336576	−	0.941656i	$-0.390731\pi$
$883$	20.0029	0.673151	0.336576	+	0.941656i	$0.390731\pi$
$884$	−12.1405	−0.408331
$885$	10.1201	0.340182
$886$	18.4481	0.619774
$887$	45.8702	1.54017	0.770085	−	0.637942i	$-0.220214\pi$
$887$	45.8702	1.54017	0.770085	+	0.637942i	$0.220214\pi$
$888$	−3.22541	−0.108238
$889$	−34.1845	−1.14651
$890$	−15.0078	−0.503063
$891$	−31.6821	−1.06139
$892$	9.39463	0.314555
$893$	−63.0604	−2.11024
$894$	0.269356	0.00900863
$895$	−17.6953	−0.591490
$896$	2.56938	0.0858371
$897$	−8.37401	−0.279600
$898$	−14.9783	−0.499832
$899$	−0.379712	−0.0126641
$900$	5.74933	0.191644
$901$	−10.3200	−0.343809
$902$	−21.5581	−0.717807
$903$	−7.43170	−0.247311
$904$	7.71230	0.256507
$905$	−8.85634	−0.294395
$906$	−7.46996	−0.248173
$907$	−10.8930	−0.361695	−0.180847	−	0.983511i	$-0.557884\pi$
$907$	−10.8930	−0.361695	−0.180847	+	0.983511i	$0.557884\pi$
$908$	−14.3504	−0.476235
$909$	29.4167	0.975690
$910$	−15.6260	−0.517997
$911$	38.8667	1.28771	0.643855	−	0.765148i	$-0.277334\pi$
$911$	38.8667	1.28771	0.643855	+	0.765148i	$0.277334\pi$
$912$	−4.55628	−0.150874
$913$	71.7241	2.37372
$914$	28.0178	0.926745
$915$	5.55219	0.183550
$916$	−21.2665	−0.702664
$917$	54.0253	1.78407
$918$	−11.5368	−0.380771
$919$	−43.3796	−1.43096	−0.715480	−	0.698633i	$-0.753792\pi$
$919$	−43.3796	−1.43096	−0.715480	+	0.698633i	$0.753792\pi$
$920$	6.30865	0.207990
$921$	18.2438	0.601154
$922$	−8.28194	−0.272751
$923$	−0.781751	−0.0257317
$924$	−8.66742	−0.285137
$925$	11.5262	0.378979
$926$	−15.1456	−0.497714
$927$	3.83550	0.125974
$928$	0.0692995	0.00227487
$929$	−53.6868	−1.76141	−0.880704	−	0.473668i	$-0.842930\pi$
$929$	−53.6868	−1.76141	−0.880704	+	0.473668i	$0.842930\pi$
$930$	−5.63037	−0.184627
$931$	2.95970	0.0970004
$932$	17.9337	0.587437
$933$	−18.7705	−0.614517
$934$	16.7460	0.547945
$935$	−30.8533	−1.00901
$936$	9.52185	0.311231
$937$	−36.0656	−1.17821	−0.589106	−	0.808056i	$-0.700520\pi$
$937$	−36.0656	−1.17821	−0.589106	+	0.808056i	$0.700520\pi$
$938$	−22.3877	−0.730984
$939$	3.38753	0.110548
$940$	14.2219	0.463869
$941$	−29.5255	−0.962503	−0.481252	−	0.876583i	$-0.659818\pi$
$941$	−29.5255	−0.962503	−0.481252	+	0.876583i	$0.659818\pi$
$942$	3.44700	0.112309
$943$	14.7483	0.480272
$944$	9.84849	0.320541
$945$	−14.8489	−0.483036
$946$	25.9568	0.843927
$947$	−36.6812	−1.19198	−0.595989	−	0.802992i	$-0.703240\pi$
$947$	−36.6812	−1.19198	−0.595989	+	0.802992i	$0.703240\pi$
$948$	5.26763	0.171085
$949$	12.3174	0.399841
$950$	16.2822	0.528263
$951$	6.66163	0.216018
$952$	8.59660	0.278617
$953$	−26.6795	−0.864233	−0.432116	−	0.901818i	$-0.642233\pi$
$953$	−26.6795	−0.864233	−0.432116	+	0.901818i	$0.642233\pi$
$954$	8.09399	0.262053
$955$	34.4228	1.11389
$956$	−1.58857	−0.0513781
$957$	−0.233771	−0.00755675
$958$	5.54103	0.179023
$959$	10.0450	0.324369
$960$	1.02757	0.0331648
$961$	−0.977405	−0.0315292
$962$	19.0893	0.615465
$963$	35.4541	1.14249
$964$	−24.4529	−0.787576
$965$	−28.1338	−0.905658
$966$	5.92955	0.190780
$967$	42.0644	1.35270	0.676349	−	0.736581i	$-0.263561\pi$
$967$	42.0644	1.35270	0.676349	+	0.736581i	$0.263561\pi$
$968$	19.2728	0.619450
$969$	−15.2443	−0.489718
$970$	−18.6010	−0.597243
$971$	18.5438	0.595098	0.297549	−	0.954706i	$-0.403831\pi$
$971$	18.5438	0.595098	0.297549	+	0.954706i	$0.403831\pi$
$972$	13.8749	0.445037
$973$	−2.08723	−0.0669136
$974$	−10.2260	−0.327663
$975$	4.87430	0.156102
$976$	5.40321	0.172952
$977$	29.8440	0.954795	0.477398	−	0.878687i	$-0.341580\pi$
$977$	29.8440	0.954795	0.477398	+	0.878687i	$0.341580\pi$
$978$	−4.55510	−0.145656
$979$	49.2681	1.57461
$980$	−0.667499	−0.0213225
$981$	35.7518	1.14147
$982$	−32.5720	−1.03942
$983$	−1.37567	−0.0438770	−0.0219385	−	0.999759i	$-0.506984\pi$
$983$	−1.37567	−0.0438770	−0.0219385	+	0.999759i	$0.506984\pi$
$984$	2.40226	0.0765812
$985$	31.3456	0.998755
$986$	0.231861	0.00738396
$987$	13.3673	0.425487
$988$	26.9660	0.857902
$989$	−17.7575	−0.564656
$990$	24.1983	0.769073
$991$	−53.6864	−1.70540	−0.852702	−	0.522398i	$-0.825037\pi$
$991$	−53.6864	−1.70540	−0.852702	+	0.522398i	$0.825037\pi$
$992$	−5.47929	−0.173968
$993$	11.8232	0.375199
$994$	0.553550	0.0175575
$995$	34.6191	1.09750
$996$	−7.99234	−0.253247
$997$	19.2056	0.608247	0.304123	−	0.952633i	$-0.401636\pi$
$997$	19.2056	0.608247	0.304123	+	0.952633i	$0.401636\pi$
$998$	−15.8146	−0.500602
$999$	18.1400	0.573925

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.45	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.45	✓	69	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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.613105 q^{3} +1.00000 q^{4} +1.67602 q^{5} +0.613105 q^{6} +2.56938 q^{7} +1.00000 q^{8} -2.62410 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.613105 q^{3} +1.00000 q^{4} +1.67602 q^{5} +0.613105 q^{6} +2.56938 q^{7} +1.00000 q^{8} -2.62410 q^{9} +1.67602 q^{10} -5.50207 q^{11} +0.613105 q^{12} -3.62861 q^{13} +2.56938 q^{14} +1.02757 q^{15} +1.00000 q^{16} +3.34578 q^{17} -2.62410 q^{18} -7.43149 q^{19} +1.67602 q^{20} +1.57530 q^{21} -5.50207 q^{22} +3.76407 q^{23} +0.613105 q^{24} -2.19097 q^{25} -3.62861 q^{26} -3.44816 q^{27} +2.56938 q^{28} +0.0692995 q^{29} +1.02757 q^{30} -5.47929 q^{31} +1.00000 q^{32} -3.37335 q^{33} +3.34578 q^{34} +4.30633 q^{35} -2.62410 q^{36} -5.26078 q^{37} -7.43149 q^{38} -2.22472 q^{39} +1.67602 q^{40} +3.91818 q^{41} +1.57530 q^{42} -4.71763 q^{43} -5.50207 q^{44} -4.39804 q^{45} +3.76407 q^{46} +8.48557 q^{47} +0.613105 q^{48} -0.398265 q^{49} -2.19097 q^{50} +2.05131 q^{51} -3.62861 q^{52} -3.08448 q^{53} -3.44816 q^{54} -9.22156 q^{55} +2.56938 q^{56} -4.55628 q^{57} +0.0692995 q^{58} +9.84849 q^{59} +1.02757 q^{60} +5.40321 q^{61} -5.47929 q^{62} -6.74233 q^{63} +1.00000 q^{64} -6.08161 q^{65} -3.37335 q^{66} -8.71325 q^{67} +3.34578 q^{68} +2.30777 q^{69} +4.30633 q^{70} +0.215441 q^{71} -2.62410 q^{72} -3.39453 q^{73} -5.26078 q^{74} -1.34329 q^{75} -7.43149 q^{76} -14.1369 q^{77} -2.22472 q^{78} +8.59173 q^{79} +1.67602 q^{80} +5.75822 q^{81} +3.91818 q^{82} -13.0358 q^{83} +1.57530 q^{84} +5.60758 q^{85} -4.71763 q^{86} +0.0424879 q^{87} -5.50207 q^{88} -8.95446 q^{89} -4.39804 q^{90} -9.32330 q^{91} +3.76407 q^{92} -3.35938 q^{93} +8.48557 q^{94} -12.4553 q^{95} +0.613105 q^{96} -11.0984 q^{97} -0.398265 q^{98} +14.4380 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more