LMFDB - Embedded newform 8045.2.a.d.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8045,2,Mod(1,8045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8045, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8045 = 5 \cdot 1609 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8045.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:	$64.2396484261$
Analytic rank:	$0$
Dimension:	$141$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.29844 q^{2} +3.20612 q^{3} +3.28283 q^{4} -1.00000 q^{5} -7.36907 q^{6} +2.27898 q^{7} -2.94851 q^{8} +7.27918 q^{9} +O(q^{10})$	\(q-2.29844 q^{2} +3.20612 q^{3} +3.28283 q^{4} -1.00000 q^{5} -7.36907 q^{6} +2.27898 q^{7} -2.94851 q^{8} +7.27918 q^{9} +2.29844 q^{10} -3.52350 q^{11} +10.5251 q^{12} -0.233347 q^{13} -5.23810 q^{14} -3.20612 q^{15} +0.211306 q^{16} +6.57694 q^{17} -16.7308 q^{18} +5.82596 q^{19} -3.28283 q^{20} +7.30668 q^{21} +8.09855 q^{22} -3.29009 q^{23} -9.45325 q^{24} +1.00000 q^{25} +0.536334 q^{26} +13.7196 q^{27} +7.48150 q^{28} +1.13488 q^{29} +7.36907 q^{30} +9.99218 q^{31} +5.41134 q^{32} -11.2967 q^{33} -15.1167 q^{34} -2.27898 q^{35} +23.8963 q^{36} +6.44896 q^{37} -13.3906 q^{38} -0.748137 q^{39} +2.94851 q^{40} -1.29926 q^{41} -16.7940 q^{42} +9.24290 q^{43} -11.5670 q^{44} -7.27918 q^{45} +7.56208 q^{46} +3.72350 q^{47} +0.677473 q^{48} -1.80625 q^{49} -2.29844 q^{50} +21.0864 q^{51} -0.766037 q^{52} -1.74132 q^{53} -31.5336 q^{54} +3.52350 q^{55} -6.71959 q^{56} +18.6787 q^{57} -2.60845 q^{58} +3.89217 q^{59} -10.5251 q^{60} -2.71146 q^{61} -22.9664 q^{62} +16.5891 q^{63} -12.8602 q^{64} +0.233347 q^{65} +25.9649 q^{66} -6.07860 q^{67} +21.5910 q^{68} -10.5484 q^{69} +5.23810 q^{70} -10.7386 q^{71} -21.4627 q^{72} -11.5789 q^{73} -14.8226 q^{74} +3.20612 q^{75} +19.1256 q^{76} -8.02999 q^{77} +1.71955 q^{78} -11.2622 q^{79} -0.211306 q^{80} +22.1490 q^{81} +2.98627 q^{82} -7.60552 q^{83} +23.9866 q^{84} -6.57694 q^{85} -21.2443 q^{86} +3.63855 q^{87} +10.3891 q^{88} +3.14116 q^{89} +16.7308 q^{90} -0.531793 q^{91} -10.8008 q^{92} +32.0361 q^{93} -8.55825 q^{94} -5.82596 q^{95} +17.3494 q^{96} +9.11466 q^{97} +4.15155 q^{98} -25.6482 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 141 q - 8 q^{2} + 11 q^{3} + 158 q^{4} - 141 q^{5} + 23 q^{6} + 29 q^{7} - 21 q^{8} + 160 q^{9}+O(q^{10}) $ 141 * q - 8 * q^2 + 11 * q^3 + 158 * q^4 - 141 * q^5 + 23 * q^6 + 29 * q^7 - 21 * q^8 + 160 * q^9	\( 141 q - 8 q^{2} + 11 q^{3} + 158 q^{4} - 141 q^{5} + 23 q^{6} + 29 q^{7} - 21 q^{8} + 160 q^{9} + 8 q^{10} + 32 q^{11} + 17 q^{12} + 35 q^{13} + 18 q^{14} - 11 q^{15} + 188 q^{16} - 13 q^{17} - 16 q^{18} + 152 q^{19} - 158 q^{20} + 40 q^{21} + 14 q^{22} - 77 q^{23} + 69 q^{24} + 141 q^{25} + 27 q^{26} + 38 q^{27} + 67 q^{28} + 22 q^{29} - 23 q^{30} + 86 q^{31} - 65 q^{32} + 51 q^{33} + 79 q^{34} - 29 q^{35} + 191 q^{36} + 45 q^{37} - 9 q^{38} + 55 q^{39} + 21 q^{40} + 36 q^{41} + 6 q^{42} + 132 q^{43} + 74 q^{44} - 160 q^{45} + 72 q^{46} - 16 q^{47} + 22 q^{48} + 212 q^{49} - 8 q^{50} + 82 q^{51} + 106 q^{52} - 28 q^{53} + 86 q^{54} - 32 q^{55} + 27 q^{56} + 10 q^{57} + 23 q^{58} + 71 q^{59} - 17 q^{60} + 116 q^{61} - 36 q^{62} + 45 q^{63} + 237 q^{64} - 35 q^{65} + 69 q^{66} + 99 q^{67} - 7 q^{68} + 45 q^{69} - 18 q^{70} + 34 q^{71} - 53 q^{72} + 125 q^{73} + 50 q^{74} + 11 q^{75} + 271 q^{76} - 31 q^{77} + 2 q^{78} + 101 q^{79} - 188 q^{80} + 221 q^{81} + 67 q^{82} + 67 q^{83} + 141 q^{84} + 13 q^{85} + 48 q^{86} - 21 q^{87} + 71 q^{88} + 79 q^{89} + 16 q^{90} + 228 q^{91} - 198 q^{92} - 12 q^{93} + 114 q^{94} - 152 q^{95} + 129 q^{96} + 98 q^{97} - 31 q^{98} + 195 q^{99}+O(q^{100}) \) 141 * q - 8 * q^2 + 11 * q^3 + 158 * q^4 - 141 * q^5 + 23 * q^6 + 29 * q^7 - 21 * q^8 + 160 * q^9 + 8 * q^10 + 32 * q^11 + 17 * q^12 + 35 * q^13 + 18 * q^14 - 11 * q^15 + 188 * q^16 - 13 * q^17 - 16 * q^18 + 152 * q^19 - 158 * q^20 + 40 * q^21 + 14 * q^22 - 77 * q^23 + 69 * q^24 + 141 * q^25 + 27 * q^26 + 38 * q^27 + 67 * q^28 + 22 * q^29 - 23 * q^30 + 86 * q^31 - 65 * q^32 + 51 * q^33 + 79 * q^34 - 29 * q^35 + 191 * q^36 + 45 * q^37 - 9 * q^38 + 55 * q^39 + 21 * q^40 + 36 * q^41 + 6 * q^42 + 132 * q^43 + 74 * q^44 - 160 * q^45 + 72 * q^46 - 16 * q^47 + 22 * q^48 + 212 * q^49 - 8 * q^50 + 82 * q^51 + 106 * q^52 - 28 * q^53 + 86 * q^54 - 32 * q^55 + 27 * q^56 + 10 * q^57 + 23 * q^58 + 71 * q^59 - 17 * q^60 + 116 * q^61 - 36 * q^62 + 45 * q^63 + 237 * q^64 - 35 * q^65 + 69 * q^66 + 99 * q^67 - 7 * q^68 + 45 * q^69 - 18 * q^70 + 34 * q^71 - 53 * q^72 + 125 * q^73 + 50 * q^74 + 11 * q^75 + 271 * q^76 - 31 * q^77 + 2 * q^78 + 101 * q^79 - 188 * q^80 + 221 * q^81 + 67 * q^82 + 67 * q^83 + 141 * q^84 + 13 * q^85 + 48 * q^86 - 21 * q^87 + 71 * q^88 + 79 * q^89 + 16 * q^90 + 228 * q^91 - 198 * q^92 - 12 * q^93 + 114 * q^94 - 152 * q^95 + 129 * q^96 + 98 * q^97 - 31 * q^98 + 195 * 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 $

$2$	−2.29844	−1.62524	−0.812621	−	0.582792i	$-0.801960\pi$
$2$	−2.29844	−1.62524	−0.812621	+	0.582792i	$0.801960\pi$
$3$	3.20612	1.85105	0.925526	−	0.378684i	$-0.123623\pi$
$3$	3.20612	1.85105	0.925526	+	0.378684i	$0.123623\pi$
$4$	3.28283	1.64141
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−7.36907	−3.00841
$7$	2.27898	0.861374	0.430687	−	0.902501i	$-0.358271\pi$
$7$	2.27898	0.861374	0.430687	+	0.902501i	$0.358271\pi$
$8$	−2.94851	−1.04245
$9$	7.27918	2.42639
$10$	2.29844	0.726831
$11$	−3.52350	−1.06237	−0.531187	−	0.847254i	$-0.678254\pi$
$11$	−3.52350	−1.06237	−0.531187	+	0.847254i	$0.678254\pi$
$12$	10.5251	3.03834
$13$	−0.233347	−0.0647187	−0.0323594	−	0.999476i	$-0.510302\pi$
$13$	−0.233347	−0.0647187	−0.0323594	+	0.999476i	$0.510302\pi$
$14$	−5.23810	−1.39994
$15$	−3.20612	−0.827816
$16$	0.211306	0.0528266
$17$	6.57694	1.59514	0.797571	−	0.603225i	$-0.206118\pi$
$17$	6.57694	1.59514	0.797571	+	0.603225i	$0.206118\pi$
$18$	−16.7308	−3.94348
$19$	5.82596	1.33657	0.668284	−	0.743906i	$-0.267029\pi$
$19$	5.82596	1.33657	0.668284	+	0.743906i	$0.267029\pi$
$20$	−3.28283	−0.734063
$21$	7.30668	1.59445
$22$	8.09855	1.72662
$23$	−3.29009	−0.686032	−0.343016	−	0.939330i	$-0.611448\pi$
$23$	−3.29009	−0.686032	−0.343016	+	0.939330i	$0.611448\pi$
$24$	−9.45325	−1.92964
$25$	1.00000	0.200000
$26$	0.536334	0.105184
$27$	13.7196	2.64033
$28$	7.48150	1.41387
$29$	1.13488	0.210742	0.105371	−	0.994433i	$-0.466397\pi$
$29$	1.13488	0.210742	0.105371	+	0.994433i	$0.466397\pi$
$30$	7.36907	1.34540
$31$	9.99218	1.79465	0.897324	−	0.441372i	$-0.145508\pi$
$31$	9.99218	1.79465	0.897324	+	0.441372i	$0.145508\pi$
$32$	5.41134	0.956598
$33$	−11.2967	−1.96651
$34$	−15.1167	−2.59249
$35$	−2.27898	−0.385218
$36$	23.8963	3.98272
$37$	6.44896	1.06020	0.530101	−	0.847934i	$-0.322154\pi$
$37$	6.44896	1.06020	0.530101	+	0.847934i	$0.322154\pi$
$38$	−13.3906	−2.17225
$39$	−0.748137	−0.119798
$40$	2.94851	0.466200
$41$	−1.29926	−0.202910	−0.101455	−	0.994840i	$-0.532350\pi$
$41$	−1.29926	−0.202910	−0.101455	+	0.994840i	$0.532350\pi$
$42$	−16.7940	−2.59136
$43$	9.24290	1.40953	0.704765	−	0.709441i	$-0.251053\pi$
$43$	9.24290	1.40953	0.704765	+	0.709441i	$0.251053\pi$
$44$	−11.5670	−1.74380
$45$	−7.27918	−1.08512
$46$	7.56208	1.11497
$47$	3.72350	0.543129	0.271564	−	0.962420i	$-0.412459\pi$
$47$	3.72350	0.543129	0.271564	+	0.962420i	$0.412459\pi$
$48$	0.677473	0.0977847
$49$	−1.80625	−0.258035
$50$	−2.29844	−0.325049
$51$	21.0864	2.95269
$52$	−0.766037	−0.106230
$53$	−1.74132	−0.239189	−0.119594	−	0.992823i	$-0.538159\pi$
$53$	−1.74132	−0.239189	−0.119594	+	0.992823i	$0.538159\pi$
$54$	−31.5336	−4.29118
$55$	3.52350	0.475109
$56$	−6.71959	−0.897943
$57$	18.6787	2.47406
$58$	−2.60845	−0.342506
$59$	3.89217	0.506718	0.253359	−	0.967372i	$-0.418465\pi$
$59$	3.89217	0.506718	0.253359	+	0.967372i	$0.418465\pi$
$60$	−10.5251	−1.35879
$61$	−2.71146	−0.347167	−0.173584	−	0.984819i	$-0.555535\pi$
$61$	−2.71146	−0.347167	−0.173584	+	0.984819i	$0.555535\pi$
$62$	−22.9664	−2.91674
$63$	16.5891	2.09003
$64$	−12.8602	−1.60753
$65$	0.233347	0.0289431
$66$	25.9649	3.19606
$67$	−6.07860	−0.742619	−0.371309	−	0.928509i	$-0.621091\pi$
$67$	−6.07860	−0.742619	−0.371309	+	0.928509i	$0.621091\pi$
$68$	21.5910	2.61829
$69$	−10.5484	−1.26988
$70$	5.23810	0.626073
$71$	−10.7386	−1.27443	−0.637216	−	0.770685i	$-0.719914\pi$
$71$	−10.7386	−1.27443	−0.637216	+	0.770685i	$0.719914\pi$
$72$	−21.4627	−2.52940
$73$	−11.5789	−1.35521	−0.677605	−	0.735426i	$-0.736982\pi$
$73$	−11.5789	−1.35521	−0.677605	+	0.735426i	$0.736982\pi$
$74$	−14.8226	−1.72309
$75$	3.20612	0.370210
$76$	19.1256	2.19386
$77$	−8.02999	−0.915102
$78$	1.71955	0.194700
$79$	−11.2622	−1.26709	−0.633547	−	0.773704i	$-0.718402\pi$
$79$	−11.2622	−1.26709	−0.633547	+	0.773704i	$0.718402\pi$
$80$	−0.211306	−0.0236248
$81$	22.1490	2.46099
$82$	2.98627	0.329779
$83$	−7.60552	−0.834814	−0.417407	−	0.908720i	$-0.637061\pi$
$83$	−7.60552	−0.834814	−0.417407	+	0.908720i	$0.637061\pi$
$84$	23.9866	2.61715
$85$	−6.57694	−0.713369
$86$	−21.2443	−2.29083
$87$	3.63855	0.390094
$88$	10.3891	1.10748
$89$	3.14116	0.332963	0.166481	−	0.986045i	$-0.446759\pi$
$89$	3.14116	0.332963	0.166481	+	0.986045i	$0.446759\pi$
$90$	16.7308	1.76358
$91$	−0.531793	−0.0557470
$92$	−10.8008	−1.12606
$93$	32.0361	3.32199
$94$	−8.55825	−0.882716
$95$	−5.82596	−0.597731
$96$	17.3494	1.77071
$97$	9.11466	0.925454	0.462727	−	0.886501i	$-0.346871\pi$
$97$	9.11466	0.925454	0.462727	+	0.886501i	$0.346871\pi$
$98$	4.15155	0.419370
$99$	−25.6482	−2.57774
$100$	3.28283	0.328283
$101$	1.20890	0.120290	0.0601448	−	0.998190i	$-0.480844\pi$
$101$	1.20890	0.120290	0.0601448	+	0.998190i	$0.480844\pi$
$102$	−48.4659	−4.79884
$103$	17.3214	1.70672	0.853362	−	0.521318i	$-0.174560\pi$
$103$	17.3214	1.70672	0.853362	+	0.521318i	$0.174560\pi$
$104$	0.688024	0.0674663
$105$	−7.30668	−0.713059
$106$	4.00232	0.388740
$107$	−13.6715	−1.32168	−0.660839	−	0.750528i	$-0.729799\pi$
$107$	−13.6715	−1.32168	−0.660839	+	0.750528i	$0.729799\pi$
$108$	45.0390	4.33388
$109$	−3.38393	−0.324122	−0.162061	−	0.986781i	$-0.551814\pi$
$109$	−3.38393	−0.324122	−0.162061	+	0.986781i	$0.551814\pi$
$110$	−8.09855	−0.772167
$111$	20.6761	1.96249
$112$	0.481563	0.0455034
$113$	−12.1286	−1.14097	−0.570484	−	0.821309i	$-0.693244\pi$
$113$	−12.1286	−1.14097	−0.570484	+	0.821309i	$0.693244\pi$
$114$	−42.9319	−4.02094
$115$	3.29009	0.306803
$116$	3.72561	0.345914
$117$	−1.69857	−0.157033
$118$	−8.94592	−0.823539
$119$	14.9887	1.37401
$120$	9.45325	0.862960
$121$	1.41505	0.128640
$122$	6.23213	0.564231
$123$	−4.16558	−0.375598
$124$	32.8026	2.94576
$125$	−1.00000	−0.0894427
$126$	−38.1291	−3.39681
$127$	0.829286	0.0735872	0.0367936	−	0.999323i	$-0.488286\pi$
$127$	0.829286	0.0735872	0.0367936	+	0.999323i	$0.488286\pi$
$128$	18.7358	1.65603
$129$	29.6338	2.60911
$130$	−0.536334	−0.0470396
$131$	−15.2763	−1.33470	−0.667350	−	0.744744i	$-0.732572\pi$
$131$	−15.2763	−1.33470	−0.667350	+	0.744744i	$0.732572\pi$
$132$	−37.0853	−3.22786
$133$	13.2773	1.15128
$134$	13.9713	1.20694
$135$	−13.7196	−1.18079
$136$	−19.3921	−1.66286
$137$	3.66954	0.313510	0.156755	−	0.987637i	$-0.449897\pi$
$137$	3.66954	0.313510	0.156755	+	0.987637i	$0.449897\pi$
$138$	24.2449	2.06386
$139$	2.34824	0.199175	0.0995876	−	0.995029i	$-0.468248\pi$
$139$	2.34824	0.199175	0.0995876	+	0.995029i	$0.468248\pi$
$140$	−7.48150	−0.632302
$141$	11.9380	1.00536
$142$	24.6819	2.07126
$143$	0.822197	0.0687556
$144$	1.53814	0.128178
$145$	−1.13488	−0.0942465
$146$	26.6134	2.20254
$147$	−5.79104	−0.477637
$148$	21.1708	1.74023
$149$	7.61598	0.623925	0.311963	−	0.950094i	$-0.399014\pi$
$149$	7.61598	0.623925	0.311963	+	0.950094i	$0.399014\pi$
$150$	−7.36907	−0.601682
$151$	−16.6340	−1.35366	−0.676830	−	0.736140i	$-0.736647\pi$
$151$	−16.6340	−1.35366	−0.676830	+	0.736140i	$0.736647\pi$
$152$	−17.1779	−1.39331
$153$	47.8747	3.87044
$154$	18.4564	1.48726
$155$	−9.99218	−0.802591
$156$	−2.45601	−0.196638
$157$	16.6072	1.32540	0.662699	−	0.748886i	$-0.269411\pi$
$157$	16.6072	1.32540	0.662699	+	0.748886i	$0.269411\pi$
$158$	25.8854	2.05934
$159$	−5.58288	−0.442751
$160$	−5.41134	−0.427804
$161$	−7.49806	−0.590930
$162$	−50.9081	−3.99971
$163$	−0.752193	−0.0589163	−0.0294582	−	0.999566i	$-0.509378\pi$
$163$	−0.752193	−0.0589163	−0.0294582	+	0.999566i	$0.509378\pi$
$164$	−4.26525	−0.333060
$165$	11.2967	0.879451
$166$	17.4808	1.35678
$167$	4.79997	0.371433	0.185716	−	0.982603i	$-0.440539\pi$
$167$	4.79997	0.371433	0.185716	+	0.982603i	$0.440539\pi$
$168$	−21.5438	−1.66214
$169$	−12.9455	−0.995811
$170$	15.1167	1.15940
$171$	42.4083	3.24304
$172$	30.3429	2.31362
$173$	−4.60512	−0.350121	−0.175060	−	0.984558i	$-0.556012\pi$
$173$	−4.60512	−0.350121	−0.175060	+	0.984558i	$0.556012\pi$
$174$	−8.36300	−0.633997
$175$	2.27898	0.172275
$176$	−0.744537	−0.0561216
$177$	12.4788	0.937961
$178$	−7.21978	−0.541145
$179$	17.9702	1.34315	0.671577	−	0.740935i	$-0.265617\pi$
$179$	17.9702	1.34315	0.671577	+	0.740935i	$0.265617\pi$
$180$	−23.8963	−1.78113
$181$	−10.1165	−0.751950	−0.375975	−	0.926630i	$-0.622692\pi$
$181$	−10.1165	−0.751950	−0.375975	+	0.926630i	$0.622692\pi$
$182$	1.22229	0.0906025
$183$	−8.69326	−0.642625
$184$	9.70085	0.715156
$185$	−6.44896	−0.474137
$186$	−73.6330	−5.39904
$187$	−23.1738	−1.69464
$188$	12.2236	0.891499
$189$	31.2666	2.27431
$190$	13.3906	0.971458
$191$	−9.83850	−0.711889	−0.355944	−	0.934507i	$-0.615841\pi$
$191$	−9.83850	−0.711889	−0.355944	+	0.934507i	$0.615841\pi$
$192$	−41.2314	−2.97562
$193$	7.71481	0.555324	0.277662	−	0.960679i	$-0.410440\pi$
$193$	7.71481	0.555324	0.277662	+	0.960679i	$0.410440\pi$
$194$	−20.9495	−1.50409
$195$	0.748137	0.0535752
$196$	−5.92960	−0.423543
$197$	−10.9385	−0.779339	−0.389669	−	0.920955i	$-0.627411\pi$
$197$	−10.9385	−0.779339	−0.389669	+	0.920955i	$0.627411\pi$
$198$	58.9508	4.18945
$199$	−7.64156	−0.541696	−0.270848	−	0.962622i	$-0.587304\pi$
$199$	−7.64156	−0.541696	−0.270848	+	0.962622i	$0.587304\pi$
$200$	−2.94851	−0.208491
$201$	−19.4887	−1.37463
$202$	−2.77857	−0.195500
$203$	2.58637	0.181527
$204$	69.2232	4.84659
$205$	1.29926	0.0907443
$206$	−39.8121	−2.77384
$207$	−23.9492	−1.66458
$208$	−0.0493076	−0.00341887
$209$	−20.5278	−1.41994
$210$	16.7940	1.15889
$211$	18.1875	1.25208	0.626041	−	0.779790i	$-0.284674\pi$
$211$	18.1875	1.25208	0.626041	+	0.779790i	$0.284674\pi$
$212$	−5.71646	−0.392608
$213$	−34.4290	−2.35904
$214$	31.4232	2.14805
$215$	−9.24290	−0.630361
$216$	−40.4522	−2.75242
$217$	22.7720	1.54586
$218$	7.77776	0.526777
$219$	−37.1233	−2.50856
$220$	11.5670	0.779850
$221$	−1.53471	−0.103236
$222$	−47.5228	−3.18952
$223$	−0.573538	−0.0384070	−0.0192035	−	0.999816i	$-0.506113\pi$
$223$	−0.573538	−0.0384070	−0.0192035	+	0.999816i	$0.506113\pi$
$224$	12.3323	0.823988
$225$	7.27918	0.485279
$226$	27.8770	1.85435
$227$	26.8032	1.77899	0.889497	−	0.456941i	$-0.151055\pi$
$227$	26.8032	1.77899	0.889497	+	0.456941i	$0.151055\pi$
$228$	61.3190	4.06095
$229$	−1.12005	−0.0740148	−0.0370074	−	0.999315i	$-0.511783\pi$
$229$	−1.12005	−0.0740148	−0.0370074	+	0.999315i	$0.511783\pi$
$230$	−7.56208	−0.498629
$231$	−25.7451	−1.69390
$232$	−3.34619	−0.219688
$233$	−11.9715	−0.784281	−0.392140	−	0.919905i	$-0.628265\pi$
$233$	−11.9715	−0.784281	−0.392140	+	0.919905i	$0.628265\pi$
$234$	3.90407	0.255217
$235$	−3.72350	−0.242894
$236$	12.7773	0.831733
$237$	−36.1078	−2.34546
$238$	−34.4507	−2.23311
$239$	−7.05098	−0.456090	−0.228045	−	0.973651i	$-0.573233\pi$
$239$	−7.05098	−0.456090	−0.228045	+	0.973651i	$0.573233\pi$
$240$	−0.677473	−0.0437307
$241$	19.1469	1.23336	0.616681	−	0.787213i	$-0.288477\pi$
$241$	19.1469	1.23336	0.616681	+	0.787213i	$0.288477\pi$
$242$	−3.25240	−0.209072
$243$	29.8535	1.91510
$244$	−8.90127	−0.569845
$245$	1.80625	0.115397
$246$	9.57434	0.610438
$247$	−1.35947	−0.0865010
$248$	−29.4620	−1.87084
$249$	−24.3842	−1.54528
$250$	2.29844	0.145366
$251$	−20.1741	−1.27338	−0.636688	−	0.771122i	$-0.719696\pi$
$251$	−20.1741	−1.27338	−0.636688	+	0.771122i	$0.719696\pi$
$252$	54.4592	3.43061
$253$	11.5926	0.728823
$254$	−1.90606	−0.119597
$255$	−21.0864	−1.32048
$256$	−17.3427	−1.08392
$257$	29.9013	1.86519	0.932597	−	0.360920i	$-0.117537\pi$
$257$	29.9013	1.86519	0.932597	+	0.360920i	$0.117537\pi$
$258$	−68.1116	−4.24044
$259$	14.6971	0.913231
$260$	0.766037	0.0475076
$261$	8.26099	0.511342
$262$	35.1118	2.16921
$263$	8.04979	0.496371	0.248186	−	0.968712i	$-0.420166\pi$
$263$	8.04979	0.496371	0.248186	+	0.968712i	$0.420166\pi$
$264$	33.3085	2.05000
$265$	1.74132	0.106969
$266$	−30.5170	−1.87112
$267$	10.0709	0.616331
$268$	−19.9550	−1.21895
$269$	11.6580	0.710801	0.355400	−	0.934714i	$-0.384345\pi$
$269$	11.6580	0.710801	0.355400	+	0.934714i	$0.384345\pi$
$270$	31.5336	1.91907
$271$	13.5780	0.824805	0.412403	−	0.911002i	$-0.364690\pi$
$271$	13.5780	0.824805	0.412403	+	0.911002i	$0.364690\pi$
$272$	1.38975	0.0842659
$273$	−1.70499	−0.103191
$274$	−8.43423	−0.509530
$275$	−3.52350	−0.212475
$276$	−34.6286	−2.08440
$277$	−20.5848	−1.23682	−0.618411	−	0.785855i	$-0.712223\pi$
$277$	−20.5848	−1.23682	−0.618411	+	0.785855i	$0.712223\pi$
$278$	−5.39729	−0.323708
$279$	72.7349	4.35452
$280$	6.71959	0.401572
$281$	24.1507	1.44071	0.720356	−	0.693604i	$-0.243978\pi$
$281$	24.1507	1.44071	0.720356	+	0.693604i	$0.243978\pi$
$282$	−27.4387	−1.63395
$283$	10.0547	0.597692	0.298846	−	0.954301i	$-0.403398\pi$
$283$	10.0547	0.597692	0.298846	+	0.954301i	$0.403398\pi$
$284$	−35.2528	−2.09187
$285$	−18.6787	−1.10643
$286$	−1.88977	−0.111744
$287$	−2.96099	−0.174782
$288$	39.3901	2.32108
$289$	26.2561	1.54448
$290$	2.60845	0.153173
$291$	29.2227	1.71306
$292$	−38.0116	−2.22446
$293$	15.1289	0.883837	0.441918	−	0.897055i	$-0.354298\pi$
$293$	15.1289	0.883837	0.441918	+	0.897055i	$0.354298\pi$
$294$	13.3104	0.776276
$295$	−3.89217	−0.226611
$296$	−19.0148	−1.10521
$297$	−48.3409	−2.80502
$298$	−17.5049	−1.01403
$299$	0.767732	0.0443991
$300$	10.5251	0.607669
$301$	21.0644	1.21413
$302$	38.2324	2.20003
$303$	3.87586	0.222662
$304$	1.23106	0.0706063
$305$	2.71146	0.155258
$306$	−110.037	−6.29041
$307$	16.0646	0.916857	0.458428	−	0.888731i	$-0.348413\pi$
$307$	16.0646	0.916857	0.458428	+	0.888731i	$0.348413\pi$
$308$	−26.3611	−1.50206
$309$	55.5343	3.15924
$310$	22.9664	1.30441
$311$	2.73238	0.154939	0.0774696	−	0.996995i	$-0.475316\pi$
$311$	2.73238	0.154939	0.0774696	+	0.996995i	$0.475316\pi$
$312$	2.20589	0.124884
$313$	−12.6883	−0.717186	−0.358593	−	0.933494i	$-0.616743\pi$
$313$	−12.6883	−0.717186	−0.358593	+	0.933494i	$0.616743\pi$
$314$	−38.1706	−2.15409
$315$	−16.5891	−0.934691
$316$	−36.9718	−2.07983
$317$	21.1236	1.18642	0.593210	−	0.805048i	$-0.297860\pi$
$317$	21.1236	1.18642	0.593210	+	0.805048i	$0.297860\pi$
$318$	12.8319	0.719578
$319$	−3.99874	−0.223887
$320$	12.8602	0.718910
$321$	−43.8326	−2.44650
$322$	17.2338	0.960404
$323$	38.3170	2.13202
$324$	72.7112	4.03951
$325$	−0.233347	−0.0129437
$326$	1.72887	0.0957533
$327$	−10.8493	−0.599966
$328$	3.83088	0.211525
$329$	8.48579	0.467837
$330$	−25.9649	−1.42932
$331$	30.6746	1.68603	0.843013	−	0.537893i	$-0.180780\pi$
$331$	30.6746	1.68603	0.843013	+	0.537893i	$0.180780\pi$
$332$	−24.9676	−1.37028
$333$	46.9432	2.57247
$334$	−11.0324	−0.603669
$335$	6.07860	0.332109
$336$	1.54395	0.0842292
$337$	13.4266	0.731393	0.365696	−	0.930734i	$-0.380831\pi$
$337$	13.4266	0.731393	0.365696	+	0.930734i	$0.380831\pi$
$338$	29.7546	1.61844
$339$	−38.8859	−2.11199
$340$	−21.5910	−1.17093
$341$	−35.2074	−1.90659
$342$	−97.4728	−5.27073
$343$	−20.0693	−1.08364
$344$	−27.2527	−1.46937
$345$	10.5484	0.567908
$346$	10.5846	0.569031
$347$	−29.4082	−1.57871	−0.789357	−	0.613934i	$-0.789586\pi$
$347$	−29.4082	−1.57871	−0.789357	+	0.613934i	$0.789586\pi$
$348$	11.9447	0.640305
$349$	29.4335	1.57554	0.787771	−	0.615968i	$-0.211235\pi$
$349$	29.4335	1.57554	0.787771	+	0.615968i	$0.211235\pi$
$350$	−5.23810	−0.279988
$351$	−3.20141	−0.170879
$352$	−19.0668	−1.01627
$353$	13.7006	0.729212	0.364606	−	0.931162i	$-0.381204\pi$
$353$	13.7006	0.729212	0.364606	+	0.931162i	$0.381204\pi$
$354$	−28.6817	−1.52441
$355$	10.7386	0.569943
$356$	10.3119	0.546530
$357$	48.0556	2.54337
$358$	−41.3034	−2.18295
$359$	11.9836	0.632472	0.316236	−	0.948680i	$-0.397581\pi$
$359$	11.9836	0.632472	0.316236	+	0.948680i	$0.397581\pi$
$360$	21.4627	1.13118
$361$	14.9418	0.786413
$362$	23.2521	1.22210
$363$	4.53680	0.238120
$364$	−1.74578	−0.0915040
$365$	11.5789	0.606068
$366$	19.9810	1.04442
$367$	−35.4466	−1.85030	−0.925149	−	0.379605i	$-0.876060\pi$
$367$	−35.4466	−1.85030	−0.925149	+	0.379605i	$0.876060\pi$
$368$	−0.695217	−0.0362407
$369$	−9.45756	−0.492341
$370$	14.8226	0.770588
$371$	−3.96844	−0.206031
$372$	105.169	5.45276
$373$	17.9872	0.931342	0.465671	−	0.884958i	$-0.345813\pi$
$373$	17.9872	0.931342	0.465671	+	0.884958i	$0.345813\pi$
$374$	53.2637	2.75420
$375$	−3.20612	−0.165563
$376$	−10.9788	−0.566187
$377$	−0.264820	−0.0136389
$378$	−71.8644	−3.69631
$379$	11.6853	0.600231	0.300116	−	0.953903i	$-0.402975\pi$
$379$	11.6853	0.600231	0.300116	+	0.953903i	$0.402975\pi$
$380$	−19.1256	−0.981125
$381$	2.65879	0.136214
$382$	22.6132	1.15699
$383$	−0.696606	−0.0355949	−0.0177975	−	0.999842i	$-0.505665\pi$
$383$	−0.696606	−0.0355949	−0.0177975	+	0.999842i	$0.505665\pi$
$384$	60.0693	3.06540
$385$	8.02999	0.409246
$386$	−17.7320	−0.902537
$387$	67.2808	3.42007
$388$	29.9219	1.51905
$389$	−18.3940	−0.932611	−0.466306	−	0.884624i	$-0.654415\pi$
$389$	−18.3940	−0.932611	−0.466306	+	0.884624i	$0.654415\pi$
$390$	−1.71955	−0.0870727
$391$	−21.6387	−1.09432
$392$	5.32573	0.268990
$393$	−48.9777	−2.47060
$394$	25.1416	1.26661
$395$	11.2622	0.566662
$396$	−84.1986	−4.23114
$397$	−23.2392	−1.16634	−0.583170	−	0.812350i	$-0.698188\pi$
$397$	−23.2392	−1.16634	−0.583170	+	0.812350i	$0.698188\pi$
$398$	17.5637	0.880388
$399$	42.5684	2.13109
$400$	0.211306	0.0105653
$401$	4.54389	0.226911	0.113455	−	0.993543i	$-0.463808\pi$
$401$	4.54389	0.226911	0.113455	+	0.993543i	$0.463808\pi$
$402$	44.7936	2.23410
$403$	−2.33164	−0.116147
$404$	3.96860	0.197445
$405$	−22.1490	−1.10059
$406$	−5.94461	−0.295026
$407$	−22.7229	−1.12633
$408$	−62.1735	−3.07804
$409$	13.0520	0.645382	0.322691	−	0.946504i	$-0.395413\pi$
$409$	13.0520	0.645382	0.322691	+	0.946504i	$0.395413\pi$
$410$	−2.98627	−0.147482
$411$	11.7650	0.580324
$412$	56.8631	2.80144
$413$	8.87018	0.436473
$414$	55.0458	2.70535
$415$	7.60552	0.373340
$416$	−1.26272	−0.0619098
$417$	7.52874	0.368684
$418$	47.1819	2.30774
$419$	1.21585	0.0593982	0.0296991	−	0.999559i	$-0.490545\pi$
$419$	1.21585	0.0593982	0.0296991	+	0.999559i	$0.490545\pi$
$420$	−23.9866	−1.17042
$421$	−1.41127	−0.0687809	−0.0343904	−	0.999408i	$-0.510949\pi$
$421$	−1.41127	−0.0687809	−0.0343904	+	0.999408i	$0.510949\pi$
$422$	−41.8030	−2.03494
$423$	27.1041	1.31784
$424$	5.13430	0.249343
$425$	6.57694	0.319028
$426$	79.1331	3.83401
$427$	−6.17937	−0.299041
$428$	−44.8813	−2.16942
$429$	2.63606	0.127270
$430$	21.2443	1.02449
$431$	7.11179	0.342563	0.171282	−	0.985222i	$-0.445209\pi$
$431$	7.11179	0.342563	0.171282	+	0.985222i	$0.445209\pi$
$432$	2.89903	0.139480
$433$	−6.90193	−0.331686	−0.165843	−	0.986152i	$-0.553034\pi$
$433$	−6.90193	−0.331686	−0.165843	+	0.986152i	$0.553034\pi$
$434$	−52.3401	−2.51240
$435$	−3.63855	−0.174455
$436$	−11.1089	−0.532018
$437$	−19.1680	−0.916928
$438$	85.3258	4.07702
$439$	6.97348	0.332826	0.166413	−	0.986056i	$-0.446782\pi$
$439$	6.97348	0.332826	0.166413	+	0.986056i	$0.446782\pi$
$440$	−10.3891	−0.495279
$441$	−13.1480	−0.626095
$442$	3.52743	0.167783
$443$	22.6981	1.07842	0.539209	−	0.842172i	$-0.318723\pi$
$443$	22.6981	1.07842	0.539209	+	0.842172i	$0.318723\pi$
$444$	67.8762	3.22126
$445$	−3.14116	−0.148905
$446$	1.31824	0.0624206
$447$	24.4177	1.15492
$448$	−29.3083	−1.38468
$449$	7.02914	0.331726	0.165863	−	0.986149i	$-0.446959\pi$
$449$	7.02914	0.331726	0.165863	+	0.986149i	$0.446959\pi$
$450$	−16.7308	−0.788696
$451$	4.57794	0.215567
$452$	−39.8163	−1.87280
$453$	−53.3307	−2.50569
$454$	−61.6057	−2.89130
$455$	0.531793	0.0249308
$456$	−55.0743	−2.57909
$457$	23.8528	1.11578	0.557892	−	0.829913i	$-0.311610\pi$
$457$	23.8528	1.11578	0.557892	+	0.829913i	$0.311610\pi$
$458$	2.57436	0.120292
$459$	90.2327	4.21170
$460$	10.8008	0.503590
$461$	−14.4972	−0.675202	−0.337601	−	0.941289i	$-0.609615\pi$
$461$	−14.4972	−0.675202	−0.337601	+	0.941289i	$0.609615\pi$
$462$	59.1735	2.75300
$463$	15.3917	0.715312	0.357656	−	0.933853i	$-0.383576\pi$
$463$	15.3917	0.715312	0.357656	+	0.933853i	$0.383576\pi$
$464$	0.239807	0.0111328
$465$	−32.0361	−1.48564
$466$	27.5158	1.27465
$467$	−23.7643	−1.09968	−0.549839	−	0.835270i	$-0.685311\pi$
$467$	−23.7643	−1.09968	−0.549839	+	0.835270i	$0.685311\pi$
$468$	−5.57613	−0.257757
$469$	−13.8530	−0.639672
$470$	8.55825	0.394763
$471$	53.2446	2.45338
$472$	−11.4761	−0.528230
$473$	−32.5674	−1.49745
$474$	82.9917	3.81194
$475$	5.82596	0.267314
$476$	49.2054	2.25533
$477$	−12.6754	−0.580367
$478$	16.2063	0.741257
$479$	10.6622	0.487167	0.243584	−	0.969880i	$-0.421677\pi$
$479$	10.6622	0.487167	0.243584	+	0.969880i	$0.421677\pi$
$480$	−17.3494	−0.791887
$481$	−1.50484	−0.0686150
$482$	−44.0081	−2.00451
$483$	−24.0396	−1.09384
$484$	4.64535	0.211152
$485$	−9.11466	−0.413876
$486$	−68.6164	−3.11250
$487$	−27.9276	−1.26552	−0.632761	−	0.774347i	$-0.718078\pi$
$487$	−27.9276	−1.26552	−0.632761	+	0.774347i	$0.718078\pi$
$488$	7.99476	0.361906
$489$	−2.41162	−0.109057
$490$	−4.15155	−0.187548
$491$	5.98384	0.270047	0.135023	−	0.990842i	$-0.456889\pi$
$491$	5.98384	0.270047	0.135023	+	0.990842i	$0.456889\pi$
$492$	−13.6749	−0.616512
$493$	7.46403	0.336163
$494$	3.12466	0.140585
$495$	25.6482	1.15280
$496$	2.11141	0.0948051
$497$	−24.4729	−1.09776
$498$	56.0456	2.51146
$499$	16.7606	0.750310	0.375155	−	0.926962i	$-0.377590\pi$
$499$	16.7606	0.750310	0.375155	+	0.926962i	$0.377590\pi$
$500$	−3.28283	−0.146813
$501$	15.3893	0.687542
$502$	46.3689	2.06955
$503$	−8.56607	−0.381942	−0.190971	−	0.981596i	$-0.561164\pi$
$503$	−8.56607	−0.381942	−0.190971	+	0.981596i	$0.561164\pi$
$504$	−48.9131	−2.17876
$505$	−1.20890	−0.0537951
$506$	−26.6450	−1.18451
$507$	−41.5049	−1.84330
$508$	2.72240	0.120787
$509$	−33.1141	−1.46776	−0.733879	−	0.679280i	$-0.762292\pi$
$509$	−33.1141	−1.46776	−0.733879	+	0.679280i	$0.762292\pi$
$510$	48.4659	2.14611
$511$	−26.3881	−1.16734
$512$	2.38952	0.105603
$513$	79.9296	3.52898
$514$	−68.7264	−3.03139
$515$	−17.3214	−0.763270
$516$	97.2828	4.28263
$517$	−13.1198	−0.577006
$518$	−33.7803	−1.48422
$519$	−14.7646	−0.648092
$520$	−0.688024	−0.0301719
$521$	−37.1669	−1.62831	−0.814155	−	0.580648i	$-0.802799\pi$
$521$	−37.1669	−1.62831	−0.814155	+	0.580648i	$0.802799\pi$
$522$	−18.9874	−0.831055
$523$	21.5124	0.940674	0.470337	−	0.882487i	$-0.344132\pi$
$523$	21.5124	0.940674	0.470337	+	0.882487i	$0.344132\pi$
$524$	−50.1496	−2.19080
$525$	7.30668	0.318890
$526$	−18.5020	−0.806724
$527$	65.7180	2.86272
$528$	−2.38707	−0.103884
$529$	−12.1753	−0.529361
$530$	−4.00232	−0.173850
$531$	28.3318	1.22950
$532$	43.5870	1.88973
$533$	0.303178	0.0131321
$534$	−23.1475	−1.00169
$535$	13.6715	0.591072
$536$	17.9228	0.774146
$537$	57.6145	2.48625
$538$	−26.7952	−1.15522
$539$	6.36431	0.274130
$540$	−45.0390	−1.93817
$541$	−42.3511	−1.82081	−0.910407	−	0.413714i	$-0.864231\pi$
$541$	−42.3511	−1.82081	−0.910407	+	0.413714i	$0.864231\pi$
$542$	−31.2082	−1.34051
$543$	−32.4345	−1.39190
$544$	35.5900	1.52591
$545$	3.38393	0.144952
$546$	3.91882	0.167710
$547$	−5.10392	−0.218228	−0.109114	−	0.994029i	$-0.534801\pi$
$547$	−5.10392	−0.218228	−0.109114	+	0.994029i	$0.534801\pi$
$548$	12.0465	0.514600
$549$	−19.7372	−0.842364
$550$	8.09855	0.345323
$551$	6.61176	0.281670
$552$	31.1021	1.32379
$553$	−25.6663	−1.09144
$554$	47.3130	2.01014
$555$	−20.6761	−0.877652
$556$	7.70888	0.326929
$557$	5.32109	0.225462	0.112731	−	0.993626i	$-0.464040\pi$
$557$	5.32109	0.225462	0.112731	+	0.993626i	$0.464040\pi$
$558$	−167.177	−7.07716
$559$	−2.15680	−0.0912230
$560$	−0.481563	−0.0203497
$561$	−74.2980	−3.13687
$562$	−55.5090	−2.34151
$563$	−29.0974	−1.22631	−0.613154	−	0.789963i	$-0.710100\pi$
$563$	−29.0974	−1.22631	−0.613154	+	0.789963i	$0.710100\pi$
$564$	39.1903	1.65021
$565$	12.1286	0.510256
$566$	−23.1102	−0.971395
$567$	50.4770	2.11984
$568$	31.6627	1.32854
$569$	−17.4523	−0.731640	−0.365820	−	0.930686i	$-0.619211\pi$
$569$	−17.4523	−0.731640	−0.365820	+	0.930686i	$0.619211\pi$
$570$	42.9319	1.79822
$571$	5.87040	0.245669	0.122834	−	0.992427i	$-0.460802\pi$
$571$	5.87040	0.245669	0.122834	+	0.992427i	$0.460802\pi$
$572$	2.69913	0.112856
$573$	−31.5434	−1.31774
$574$	6.80566	0.284063
$575$	−3.29009	−0.137206
$576$	−93.6121	−3.90050
$577$	−9.68190	−0.403063	−0.201531	−	0.979482i	$-0.564592\pi$
$577$	−9.68190	−0.403063	−0.201531	+	0.979482i	$0.564592\pi$
$578$	−60.3482	−2.51015
$579$	24.7346	1.02793
$580$	−3.72561	−0.154698
$581$	−17.3328	−0.719087
$582$	−67.1666	−2.78414
$583$	6.13554	0.254108
$584$	34.1405	1.41274
$585$	1.69857	0.0702274
$586$	−34.7728	−1.43645
$587$	33.2641	1.37296	0.686479	−	0.727150i	$-0.259155\pi$
$587$	33.2641	1.37296	0.686479	+	0.727150i	$0.259155\pi$
$588$	−19.0110	−0.784000
$589$	58.2141	2.39867
$590$	8.94592	0.368298
$591$	−35.0702	−1.44260
$592$	1.36271	0.0560069
$593$	−16.7452	−0.687645	−0.343822	−	0.939035i	$-0.611722\pi$
$593$	−16.7452	−0.687645	−0.343822	+	0.939035i	$0.611722\pi$
$594$	111.109	4.55884
$595$	−14.9887	−0.614478
$596$	25.0020	1.02412
$597$	−24.4997	−1.00271
$598$	−1.76459	−0.0721593
$599$	3.54065	0.144667	0.0723334	−	0.997381i	$-0.476955\pi$
$599$	3.54065	0.144667	0.0723334	+	0.997381i	$0.476955\pi$
$600$	−9.45325	−0.385927
$601$	24.0869	0.982525	0.491263	−	0.871012i	$-0.336536\pi$
$601$	24.0869	0.982525	0.491263	+	0.871012i	$0.336536\pi$
$602$	−48.4153	−1.97326
$603$	−44.2472	−1.80189
$604$	−54.6067	−2.22192
$605$	−1.41505	−0.0575298
$606$	−8.90843	−0.361880
$607$	34.8699	1.41533	0.707663	−	0.706550i	$-0.249749\pi$
$607$	34.8699	1.41533	0.707663	+	0.706550i	$0.249749\pi$
$608$	31.5262	1.27856
$609$	8.29219	0.336017
$610$	−6.23213	−0.252332
$611$	−0.868867	−0.0351506
$612$	157.165	6.35300
$613$	−41.2642	−1.66664	−0.833322	−	0.552787i	$-0.813564\pi$
$613$	−41.2642	−1.66664	−0.833322	+	0.552787i	$0.813564\pi$
$614$	−36.9236	−1.49011
$615$	4.16558	0.167972
$616$	23.6765	0.953952
$617$	−34.7572	−1.39927	−0.699636	−	0.714499i	$-0.746654\pi$
$617$	−34.7572	−1.39927	−0.699636	+	0.714499i	$0.746654\pi$
$618$	−127.642	−5.13453
$619$	34.8989	1.40271	0.701353	−	0.712814i	$-0.252580\pi$
$619$	34.8989	1.40271	0.701353	+	0.712814i	$0.252580\pi$
$620$	−32.8026	−1.31738
$621$	−45.1386	−1.81135
$622$	−6.28022	−0.251814
$623$	7.15865	0.286805
$624$	−0.158086	−0.00632851
$625$	1.00000	0.0400000
$626$	29.1633	1.16560
$627$	−65.8144	−2.62838
$628$	54.5185	2.17553
$629$	42.4144	1.69117
$630$	38.1291	1.51910
$631$	−12.5291	−0.498774	−0.249387	−	0.968404i	$-0.580229\pi$
$631$	−12.5291	−0.498774	−0.249387	+	0.968404i	$0.580229\pi$
$632$	33.2066	1.32089
$633$	58.3114	2.31767
$634$	−48.5513	−1.92822
$635$	−0.829286	−0.0329092
$636$	−18.3276	−0.726738
$637$	0.421482	0.0166997
$638$	9.19087	0.363870
$639$	−78.1679	−3.09227
$640$	−18.7358	−0.740599
$641$	−35.8871	−1.41746	−0.708728	−	0.705482i	$-0.750730\pi$
$641$	−35.8871	−1.41746	−0.708728	+	0.705482i	$0.750730\pi$
$642$	100.747	3.97615
$643$	−49.8663	−1.96654	−0.983268	−	0.182167i	$-0.941689\pi$
$643$	−49.8663	−1.96654	−0.983268	+	0.182167i	$0.941689\pi$
$644$	−24.6148	−0.969960
$645$	−29.6338	−1.16683
$646$	−88.0694	−3.46504
$647$	−7.21332	−0.283585	−0.141792	−	0.989896i	$-0.545287\pi$
$647$	−7.21332	−0.283585	−0.141792	+	0.989896i	$0.545287\pi$
$648$	−65.3063	−2.56547
$649$	−13.7141	−0.538324
$650$	0.536334	0.0210367
$651$	73.0096	2.86147
$652$	−2.46932	−0.0967061
$653$	−4.94065	−0.193343	−0.0966713	−	0.995316i	$-0.530820\pi$
$653$	−4.94065	−0.193343	−0.0966713	+	0.995316i	$0.530820\pi$
$654$	24.9364	0.975091
$655$	15.2763	0.596896
$656$	−0.274542	−0.0107191
$657$	−84.2850	−3.28827
$658$	−19.5041	−0.760348
$659$	17.4906	0.681337	0.340669	−	0.940183i	$-0.389347\pi$
$659$	17.4906	0.681337	0.340669	+	0.940183i	$0.389347\pi$
$660$	37.0853	1.44354
$661$	−38.2300	−1.48697	−0.743487	−	0.668751i	$-0.766829\pi$
$661$	−38.2300	−1.48697	−0.743487	+	0.668751i	$0.766829\pi$
$662$	−70.5036	−2.74020
$663$	−4.92045	−0.191094
$664$	22.4249	0.870255
$665$	−13.2773	−0.514870
$666$	−107.896	−4.18089
$667$	−3.73385	−0.144575
$668$	15.7575	0.609675
$669$	−1.83883	−0.0710933
$670$	−13.9713	−0.539758
$671$	9.55384	0.368822
$672$	39.5389	1.52525
$673$	−6.60013	−0.254416	−0.127208	−	0.991876i	$-0.540602\pi$
$673$	−6.60013	−0.254416	−0.127208	+	0.991876i	$0.540602\pi$
$674$	−30.8602	−1.18869
$675$	13.7196	0.528066
$676$	−42.4980	−1.63454
$677$	−29.5439	−1.13546	−0.567732	−	0.823213i	$-0.692179\pi$
$677$	−29.5439	−1.13546	−0.567732	+	0.823213i	$0.692179\pi$
$678$	89.3768	3.43250
$679$	20.7721	0.797162
$680$	19.3921	0.743655
$681$	85.9343	3.29301
$682$	80.9222	3.09867
$683$	13.1220	0.502098	0.251049	−	0.967974i	$-0.419224\pi$
$683$	13.1220	0.502098	0.251049	+	0.967974i	$0.419224\pi$
$684$	139.219	5.32317
$685$	−3.66954	−0.140206
$686$	46.1280	1.76118
$687$	−3.59100	−0.137005
$688$	1.95308	0.0744606
$689$	0.406332	0.0154800
$690$	−24.2449	−0.922988
$691$	36.6285	1.39341	0.696707	−	0.717356i	$-0.254648\pi$
$691$	36.6285	1.39341	0.696707	+	0.717356i	$0.254648\pi$
$692$	−15.1178	−0.574693
$693$	−58.4517	−2.22040
$694$	67.5930	2.56579
$695$	−2.34824	−0.0890739
$696$	−10.7283	−0.406655
$697$	−8.54516	−0.323671
$698$	−67.6513	−2.56064
$699$	−38.3821	−1.45174
$700$	7.48150	0.282774
$701$	38.6122	1.45836	0.729182	−	0.684320i	$-0.239901\pi$
$701$	38.6122	1.45836	0.729182	+	0.684320i	$0.239901\pi$
$702$	7.35826	0.277720
$703$	37.5714	1.41703
$704$	45.3131	1.70780
$705$	−11.9380	−0.449610
$706$	−31.4901	−1.18515
$707$	2.75505	0.103614
$708$	40.9656	1.53958
$709$	3.96442	0.148887	0.0744434	−	0.997225i	$-0.476282\pi$
$709$	3.96442	0.148887	0.0744434	+	0.997225i	$0.476282\pi$
$710$	−24.6819	−0.926296
$711$	−81.9794	−3.07447
$712$	−9.26174	−0.347098
$713$	−32.8752	−1.23119
$714$	−110.453	−4.13360
$715$	−0.822197	−0.0307484
$716$	58.9930	2.20467
$717$	−22.6063	−0.844246
$718$	−27.5437	−1.02792
$719$	−1.74142	−0.0649440	−0.0324720	−	0.999473i	$-0.510338\pi$
$719$	−1.74142	−0.0649440	−0.0324720	+	0.999473i	$0.510338\pi$
$720$	−1.53814	−0.0573230
$721$	39.4751	1.47013
$722$	−34.3429	−1.27811
$723$	61.3873	2.28302
$724$	−33.2106	−1.23426
$725$	1.13488	0.0421483
$726$	−10.4276	−0.387003
$727$	−34.7348	−1.28824	−0.644121	−	0.764924i	$-0.722777\pi$
$727$	−34.7348	−1.28824	−0.644121	+	0.764924i	$0.722777\pi$
$728$	1.56799	0.0581137
$729$	29.2668	1.08395
$730$	−26.6134	−0.985008
$731$	60.7900	2.24840
$732$	−28.5385	−1.05481
$733$	14.5017	0.535632	0.267816	−	0.963470i	$-0.413698\pi$
$733$	14.5017	0.535632	0.267816	+	0.963470i	$0.413698\pi$
$734$	81.4719	3.00718
$735$	5.79104	0.213606
$736$	−17.8038	−0.656256
$737$	21.4179	0.788940
$738$	21.7376	0.800173
$739$	−40.3193	−1.48317	−0.741584	−	0.670860i	$-0.765925\pi$
$739$	−40.3193	−1.48317	−0.741584	+	0.670860i	$0.765925\pi$
$740$	−21.1708	−0.778255
$741$	−4.35862	−0.160118
$742$	9.12122	0.334850
$743$	−40.0199	−1.46819	−0.734093	−	0.679049i	$-0.762392\pi$
$743$	−40.0199	−1.46819	−0.734093	+	0.679049i	$0.762392\pi$
$744$	−94.4586	−3.46302
$745$	−7.61598	−0.279028
$746$	−41.3425	−1.51366
$747$	−55.3620	−2.02559
$748$	−76.0757	−2.78160
$749$	−31.1572	−1.13846
$750$	7.36907	0.269080
$751$	50.9488	1.85915	0.929573	−	0.368637i	$-0.120175\pi$
$751$	50.9488	1.85915	0.929573	+	0.368637i	$0.120175\pi$
$752$	0.786799	0.0286916
$753$	−64.6804	−2.35709
$754$	0.608673	0.0221666
$755$	16.6340	0.605375
$756$	102.643	3.73309
$757$	−29.1921	−1.06101	−0.530503	−	0.847683i	$-0.677997\pi$
$757$	−29.1921	−1.06101	−0.530503	+	0.847683i	$0.677997\pi$
$758$	−26.8579	−0.975522
$759$	37.1673	1.34909
$760$	17.1779	0.623107
$761$	42.5616	1.54286	0.771428	−	0.636317i	$-0.219543\pi$
$761$	42.5616	1.54286	0.771428	+	0.636317i	$0.219543\pi$
$762$	−6.11106	−0.221381
$763$	−7.71191	−0.279190
$764$	−32.2981	−1.16850
$765$	−47.8747	−1.73092
$766$	1.60111	0.0578504
$767$	−0.908225	−0.0327941
$768$	−55.6028	−2.00639
$769$	25.2967	0.912221	0.456110	−	0.889923i	$-0.349242\pi$
$769$	25.2967	0.912221	0.456110	+	0.889923i	$0.349242\pi$
$770$	−18.4564	−0.665124
$771$	95.8671	3.45257
$772$	25.3264	0.911517
$773$	−11.9672	−0.430431	−0.215215	−	0.976567i	$-0.569045\pi$
$773$	−11.9672	−0.430431	−0.215215	+	0.976567i	$0.569045\pi$
$774$	−154.641	−5.55845
$775$	9.99218	0.358930
$776$	−26.8746	−0.964743
$777$	47.1205	1.69044
$778$	42.2774	1.51572
$779$	−7.56945	−0.271204
$780$	2.45601	0.0879391
$781$	37.8373	1.35392
$782$	49.7353	1.77853
$783$	15.5700	0.556427
$784$	−0.381671	−0.0136311
$785$	−16.6072	−0.592736
$786$	112.572	4.01532
$787$	−39.9528	−1.42416	−0.712082	−	0.702096i	$-0.752248\pi$
$787$	−39.9528	−1.42416	−0.712082	+	0.702096i	$0.752248\pi$
$788$	−35.9093	−1.27922
$789$	25.8086	0.918809
$790$	−25.8854	−0.920963
$791$	−27.6410	−0.982799
$792$	75.6238	2.68718
$793$	0.632711	0.0224682
$794$	53.4139	1.89559
$795$	5.58288	0.198004
$796$	−25.0859	−0.889148
$797$	−12.1926	−0.431884	−0.215942	−	0.976406i	$-0.569282\pi$
$797$	−12.1926	−0.431884	−0.215942	+	0.976406i	$0.569282\pi$
$798$	−97.8410	−3.46353
$799$	24.4892	0.866367
$800$	5.41134	0.191320
$801$	22.8651	0.807899
$802$	−10.4438	−0.368785
$803$	40.7983	1.43974
$804$	−63.9780	−2.25633
$805$	7.49806	0.264272
$806$	5.35914	0.188768
$807$	37.3769	1.31573
$808$	−3.56443	−0.125396
$809$	−18.1242	−0.637213	−0.318607	−	0.947887i	$-0.603215\pi$
$809$	−18.1242	−0.637213	−0.318607	+	0.947887i	$0.603215\pi$
$810$	50.9081	1.78873
$811$	−46.6248	−1.63722	−0.818609	−	0.574351i	$-0.805255\pi$
$811$	−46.6248	−1.63722	−0.818609	+	0.574351i	$0.805255\pi$
$812$	8.49060	0.297962
$813$	43.5327	1.52676
$814$	52.2272	1.83056
$815$	0.752193	0.0263482
$816$	4.45570	0.155981
$817$	53.8488	1.88393
$818$	−29.9993	−1.04890
$819$	−3.87102	−0.135264
$820$	4.26525	0.148949
$821$	24.1971	0.844485	0.422243	−	0.906483i	$-0.361243\pi$
$821$	24.1971	0.844485	0.422243	+	0.906483i	$0.361243\pi$
$822$	−27.0411	−0.943167
$823$	22.1060	0.770566	0.385283	−	0.922799i	$-0.374104\pi$
$823$	22.1060	0.770566	0.385283	+	0.922799i	$0.374104\pi$
$824$	−51.0721	−1.77918
$825$	−11.2967	−0.393302
$826$	−20.3876	−0.709375
$827$	−1.48255	−0.0515532	−0.0257766	−	0.999668i	$-0.508206\pi$
$827$	−1.48255	−0.0515532	−0.0257766	+	0.999668i	$0.508206\pi$
$828$	−78.6211	−2.73227
$829$	−8.43414	−0.292930	−0.146465	−	0.989216i	$-0.546790\pi$
$829$	−8.43414	−0.292930	−0.146465	+	0.989216i	$0.546790\pi$
$830$	−17.4808	−0.606769
$831$	−65.9974	−2.28942
$832$	3.00090	0.104037
$833$	−11.8796	−0.411603
$834$	−17.3044	−0.599201
$835$	−4.79997	−0.166110
$836$	−67.3892	−2.33070
$837$	137.088	4.73846
$838$	−2.79456	−0.0965366
$839$	3.53942	0.122194	0.0610971	−	0.998132i	$-0.480540\pi$
$839$	3.53942	0.122194	0.0610971	+	0.998132i	$0.480540\pi$
$840$	21.5438	0.743331
$841$	−27.7121	−0.955588
$842$	3.24371	0.111786
$843$	77.4301	2.66683
$844$	59.7066	2.05518
$845$	12.9455	0.445340
$846$	−62.2971	−2.14182
$847$	3.22486	0.110808
$848$	−0.367952	−0.0126355
$849$	32.2367	1.10636
$850$	−15.1167	−0.518499
$851$	−21.2177	−0.727332
$852$	−113.025	−3.87216
$853$	17.3507	0.594077	0.297038	−	0.954866i	$-0.404001\pi$
$853$	17.3507	0.594077	0.297038	+	0.954866i	$0.404001\pi$
$854$	14.2029	0.486014
$855$	−42.4083	−1.45033
$856$	40.3106	1.37779
$857$	44.0097	1.50334	0.751671	−	0.659538i	$-0.229248\pi$
$857$	44.0097	1.50334	0.751671	+	0.659538i	$0.229248\pi$
$858$	−6.05883	−0.206845
$859$	7.58004	0.258628	0.129314	−	0.991604i	$-0.458723\pi$
$859$	7.58004	0.258628	0.129314	+	0.991604i	$0.458723\pi$
$860$	−30.3429	−1.03468
$861$	−9.49328	−0.323530
$862$	−16.3460	−0.556748
$863$	−49.6760	−1.69099	−0.845496	−	0.533982i	$-0.820695\pi$
$863$	−49.6760	−1.69099	−0.845496	+	0.533982i	$0.820695\pi$
$864$	74.2411	2.52573
$865$	4.60512	0.156579
$866$	15.8637	0.539070
$867$	84.1802	2.85891
$868$	74.7565	2.53740
$869$	39.6823	1.34613
$870$	8.36300	0.283532
$871$	1.41842	0.0480614
$872$	9.97754	0.337882
$873$	66.3473	2.24552
$874$	44.0564	1.49023
$875$	−2.27898	−0.0770436
$876$	−121.870	−4.11759
$877$	42.2288	1.42596	0.712982	−	0.701182i	$-0.247344\pi$
$877$	42.2288	1.42596	0.712982	+	0.701182i	$0.247344\pi$
$878$	−16.0281	−0.540923
$879$	48.5049	1.63603
$880$	0.744537	0.0250984
$881$	−22.5936	−0.761199	−0.380600	−	0.924740i	$-0.624282\pi$
$881$	−22.5936	−0.761199	−0.380600	+	0.924740i	$0.624282\pi$
$882$	30.2199	1.01756
$883$	6.84107	0.230220	0.115110	−	0.993353i	$-0.463278\pi$
$883$	6.84107	0.230220	0.115110	+	0.993353i	$0.463278\pi$
$884$	−5.03818	−0.169452
$885$	−12.4788	−0.419469
$886$	−52.1702	−1.75269
$887$	−43.1410	−1.44853	−0.724267	−	0.689520i	$-0.757822\pi$
$887$	−43.1410	−1.44853	−0.724267	+	0.689520i	$0.757822\pi$
$888$	−60.9636	−2.04581
$889$	1.88993	0.0633861
$890$	7.21978	0.242008
$891$	−78.0418	−2.61450
$892$	−1.88283	−0.0630417
$893$	21.6930	0.725928
$894$	−56.1227	−1.87702
$895$	−17.9702	−0.600677
$896$	42.6986	1.42646
$897$	2.46144	0.0821851
$898$	−16.1561	−0.539135
$899$	11.3399	0.378207
$900$	23.8963	0.796544
$901$	−11.4526	−0.381540
$902$	−10.5221	−0.350349
$903$	67.5349	2.24742
$904$	35.7614	1.18941
$905$	10.1165	0.336282
$906$	122.577	4.07236
$907$	−32.7804	−1.08846	−0.544228	−	0.838937i	$-0.683177\pi$
$907$	−32.7804	−1.08846	−0.544228	+	0.838937i	$0.683177\pi$
$908$	87.9904	2.92007
$909$	8.79977	0.291870
$910$	−1.22229	−0.0405187
$911$	52.3942	1.73590	0.867948	−	0.496654i	$-0.165438\pi$
$911$	52.3942	1.73590	0.867948	+	0.496654i	$0.165438\pi$
$912$	3.94693	0.130696
$913$	26.7980	0.886886
$914$	−54.8241	−1.81342
$915$	8.69326	0.287390
$916$	−3.67692	−0.121489
$917$	−34.8145	−1.14968
$918$	−207.395	−6.84504
$919$	−5.67921	−0.187340	−0.0936699	−	0.995603i	$-0.529860\pi$
$919$	−5.67921	−0.187340	−0.0936699	+	0.995603i	$0.529860\pi$
$920$	−9.70085	−0.319828
$921$	51.5051	1.69715
$922$	33.3209	1.09737
$923$	2.50581	0.0824796
$924$	−84.5167	−2.78039
$925$	6.44896	0.212041
$926$	−35.3769	−1.16256
$927$	126.085	4.14119
$928$	6.14121	0.201595
$929$	38.7445	1.27116	0.635582	−	0.772033i	$-0.280760\pi$
$929$	38.7445	1.27116	0.635582	+	0.772033i	$0.280760\pi$
$930$	73.6330	2.41452
$931$	−10.5231	−0.344882
$932$	−39.3005	−1.28733
$933$	8.76033	0.286800
$934$	54.6207	1.78725
$935$	23.1738	0.757866
$936$	5.00825	0.163700
$937$	30.8289	1.00714	0.503568	−	0.863955i	$-0.332020\pi$
$937$	30.8289	1.00714	0.503568	+	0.863955i	$0.332020\pi$
$938$	31.8403	1.03962
$939$	−40.6802	−1.32755
$940$	−12.2236	−0.398690
$941$	−20.6959	−0.674665	−0.337333	−	0.941385i	$-0.609525\pi$
$941$	−20.6959	−0.674665	−0.337333	+	0.941385i	$0.609525\pi$
$942$	−122.379	−3.98734
$943$	4.27469	0.139203
$944$	0.822440	0.0267681
$945$	−31.2666	−1.01710
$946$	74.8541	2.43372
$947$	−9.02159	−0.293162	−0.146581	−	0.989199i	$-0.546827\pi$
$947$	−9.02159	−0.293162	−0.146581	+	0.989199i	$0.546827\pi$
$948$	−118.536	−3.84987
$949$	2.70190	0.0877075
$950$	−13.3906	−0.434449
$951$	67.7247	2.19612
$952$	−44.1943	−1.43235
$953$	9.03585	0.292700	0.146350	−	0.989233i	$-0.453247\pi$
$953$	9.03585	0.292700	0.146350	+	0.989233i	$0.453247\pi$
$954$	29.1336	0.943237
$955$	9.83850	0.318366
$956$	−23.1472	−0.748633
$957$	−12.8204	−0.414426
$958$	−24.5064	−0.791765
$959$	8.36282	0.270050
$960$	41.2314	1.33074
$961$	68.8436	2.22076
$962$	3.45879	0.111516
$963$	−99.5177	−3.20691
$964$	62.8561	2.02446
$965$	−7.71481	−0.248349
$966$	55.2537	1.77776
$967$	−35.3138	−1.13561	−0.567807	−	0.823161i	$-0.692208\pi$
$967$	−35.3138	−1.13561	−0.567807	+	0.823161i	$0.692208\pi$
$968$	−4.17227	−0.134102
$969$	122.849	3.94647
$970$	20.9495	0.672648
$971$	31.9469	1.02523	0.512613	−	0.858620i	$-0.328678\pi$
$971$	31.9469	1.02523	0.512613	+	0.858620i	$0.328678\pi$
$972$	98.0038	3.14347
$973$	5.35160	0.171564
$974$	64.1900	2.05678
$975$	−0.748137	−0.0239596
$976$	−0.572949	−0.0183397
$977$	17.4079	0.556929	0.278464	−	0.960447i	$-0.410175\pi$
$977$	17.4079	0.556929	0.278464	+	0.960447i	$0.410175\pi$
$978$	5.54296	0.177244
$979$	−11.0679	−0.353731
$980$	5.92960	0.189414
$981$	−24.6322	−0.786447
$982$	−13.7535	−0.438892
$983$	16.2914	0.519616	0.259808	−	0.965660i	$-0.416341\pi$
$983$	16.2914	0.519616	0.259808	+	0.965660i	$0.416341\pi$
$984$	12.2822	0.391544
$985$	10.9385	0.348531
$986$	−17.1556	−0.546346
$987$	27.2064	0.865990
$988$	−4.46291	−0.141984
$989$	−30.4100	−0.966982
$990$	−58.9508	−1.87358
$991$	18.7729	0.596340	0.298170	−	0.954513i	$-0.403624\pi$
$991$	18.7729	0.596340	0.298170	+	0.954513i	$0.403624\pi$
$992$	54.0710	1.71676
$993$	98.3462	3.12092
$994$	56.2496	1.78413
$995$	7.64156	0.242254
$996$	−80.0491	−2.53645
$997$	−0.965151	−0.0305666	−0.0152833	−	0.999883i	$-0.504865\pi$
$997$	−0.965151	−0.0305666	−0.0152833	+	0.999883i	$0.504865\pi$
$998$	−38.5234	−1.21944
$999$	88.4769	2.79928

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.d.1.20	✓	141

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.d.1.20	✓	141	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:	\( N \)	\(=\)	\( 8045 = 5 \cdot 1609 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.29844 q^{2} +3.20612 q^{3} +3.28283 q^{4} -1.00000 q^{5} -7.36907 q^{6} +2.27898 q^{7} -2.94851 q^{8} +7.27918 q^{9} +O(q^{10})\)	\(q-2.29844 q^{2} +3.20612 q^{3} +3.28283 q^{4} -1.00000 q^{5} -7.36907 q^{6} +2.27898 q^{7} -2.94851 q^{8} +7.27918 q^{9} +2.29844 q^{10} -3.52350 q^{11} +10.5251 q^{12} -0.233347 q^{13} -5.23810 q^{14} -3.20612 q^{15} +0.211306 q^{16} +6.57694 q^{17} -16.7308 q^{18} +5.82596 q^{19} -3.28283 q^{20} +7.30668 q^{21} +8.09855 q^{22} -3.29009 q^{23} -9.45325 q^{24} +1.00000 q^{25} +0.536334 q^{26} +13.7196 q^{27} +7.48150 q^{28} +1.13488 q^{29} +7.36907 q^{30} +9.99218 q^{31} +5.41134 q^{32} -11.2967 q^{33} -15.1167 q^{34} -2.27898 q^{35} +23.8963 q^{36} +6.44896 q^{37} -13.3906 q^{38} -0.748137 q^{39} +2.94851 q^{40} -1.29926 q^{41} -16.7940 q^{42} +9.24290 q^{43} -11.5670 q^{44} -7.27918 q^{45} +7.56208 q^{46} +3.72350 q^{47} +0.677473 q^{48} -1.80625 q^{49} -2.29844 q^{50} +21.0864 q^{51} -0.766037 q^{52} -1.74132 q^{53} -31.5336 q^{54} +3.52350 q^{55} -6.71959 q^{56} +18.6787 q^{57} -2.60845 q^{58} +3.89217 q^{59} -10.5251 q^{60} -2.71146 q^{61} -22.9664 q^{62} +16.5891 q^{63} -12.8602 q^{64} +0.233347 q^{65} +25.9649 q^{66} -6.07860 q^{67} +21.5910 q^{68} -10.5484 q^{69} +5.23810 q^{70} -10.7386 q^{71} -21.4627 q^{72} -11.5789 q^{73} -14.8226 q^{74} +3.20612 q^{75} +19.1256 q^{76} -8.02999 q^{77} +1.71955 q^{78} -11.2622 q^{79} -0.211306 q^{80} +22.1490 q^{81} +2.98627 q^{82} -7.60552 q^{83} +23.9866 q^{84} -6.57694 q^{85} -21.2443 q^{86} +3.63855 q^{87} +10.3891 q^{88} +3.14116 q^{89} +16.7308 q^{90} -0.531793 q^{91} -10.8008 q^{92} +32.0361 q^{93} -8.55825 q^{94} -5.82596 q^{95} +17.3494 q^{96} +9.11466 q^{97} +4.15155 q^{98} -25.6482 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 141 q - 8 q^{2} + 11 q^{3} + 158 q^{4} - 141 q^{5} + 23 q^{6} + 29 q^{7} - 21 q^{8} + 160 q^{9}+O(q^{10}) \) 141 * q - 8 * q^2 + 11 * q^3 + 158 * q^4 - 141 * q^5 + 23 * q^6 + 29 * q^7 - 21 * q^8 + 160 * q^9	\( 141 q - 8 q^{2} + 11 q^{3} + 158 q^{4} - 141 q^{5} + 23 q^{6} + 29 q^{7} - 21 q^{8} + 160 q^{9} + 8 q^{10} + 32 q^{11} + 17 q^{12} + 35 q^{13} + 18 q^{14} - 11 q^{15} + 188 q^{16} - 13 q^{17} - 16 q^{18} + 152 q^{19} - 158 q^{20} + 40 q^{21} + 14 q^{22} - 77 q^{23} + 69 q^{24} + 141 q^{25} + 27 q^{26} + 38 q^{27} + 67 q^{28} + 22 q^{29} - 23 q^{30} + 86 q^{31} - 65 q^{32} + 51 q^{33} + 79 q^{34} - 29 q^{35} + 191 q^{36} + 45 q^{37} - 9 q^{38} + 55 q^{39} + 21 q^{40} + 36 q^{41} + 6 q^{42} + 132 q^{43} + 74 q^{44} - 160 q^{45} + 72 q^{46} - 16 q^{47} + 22 q^{48} + 212 q^{49} - 8 q^{50} + 82 q^{51} + 106 q^{52} - 28 q^{53} + 86 q^{54} - 32 q^{55} + 27 q^{56} + 10 q^{57} + 23 q^{58} + 71 q^{59} - 17 q^{60} + 116 q^{61} - 36 q^{62} + 45 q^{63} + 237 q^{64} - 35 q^{65} + 69 q^{66} + 99 q^{67} - 7 q^{68} + 45 q^{69} - 18 q^{70} + 34 q^{71} - 53 q^{72} + 125 q^{73} + 50 q^{74} + 11 q^{75} + 271 q^{76} - 31 q^{77} + 2 q^{78} + 101 q^{79} - 188 q^{80} + 221 q^{81} + 67 q^{82} + 67 q^{83} + 141 q^{84} + 13 q^{85} + 48 q^{86} - 21 q^{87} + 71 q^{88} + 79 q^{89} + 16 q^{90} + 228 q^{91} - 198 q^{92} - 12 q^{93} + 114 q^{94} - 152 q^{95} + 129 q^{96} + 98 q^{97} - 31 q^{98} + 195 q^{99}+O(q^{100}) \) 141 * q - 8 * q^2 + 11 * q^3 + 158 * q^4 - 141 * q^5 + 23 * q^6 + 29 * q^7 - 21 * q^8 + 160 * q^9 + 8 * q^10 + 32 * q^11 + 17 * q^12 + 35 * q^13 + 18 * q^14 - 11 * q^15 + 188 * q^16 - 13 * q^17 - 16 * q^18 + 152 * q^19 - 158 * q^20 + 40 * q^21 + 14 * q^22 - 77 * q^23 + 69 * q^24 + 141 * q^25 + 27 * q^26 + 38 * q^27 + 67 * q^28 + 22 * q^29 - 23 * q^30 + 86 * q^31 - 65 * q^32 + 51 * q^33 + 79 * q^34 - 29 * q^35 + 191 * q^36 + 45 * q^37 - 9 * q^38 + 55 * q^39 + 21 * q^40 + 36 * q^41 + 6 * q^42 + 132 * q^43 + 74 * q^44 - 160 * q^45 + 72 * q^46 - 16 * q^47 + 22 * q^48 + 212 * q^49 - 8 * q^50 + 82 * q^51 + 106 * q^52 - 28 * q^53 + 86 * q^54 - 32 * q^55 + 27 * q^56 + 10 * q^57 + 23 * q^58 + 71 * q^59 - 17 * q^60 + 116 * q^61 - 36 * q^62 + 45 * q^63 + 237 * q^64 - 35 * q^65 + 69 * q^66 + 99 * q^67 - 7 * q^68 + 45 * q^69 - 18 * q^70 + 34 * q^71 - 53 * q^72 + 125 * q^73 + 50 * q^74 + 11 * q^75 + 271 * q^76 - 31 * q^77 + 2 * q^78 + 101 * q^79 - 188 * q^80 + 221 * q^81 + 67 * q^82 + 67 * q^83 + 141 * q^84 + 13 * q^85 + 48 * q^86 - 21 * q^87 + 71 * q^88 + 79 * q^89 + 16 * q^90 + 228 * q^91 - 198 * q^92 - 12 * q^93 + 114 * q^94 - 152 * q^95 + 129 * q^96 + 98 * q^97 - 31 * q^98 + 195 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more