LMFDB - Embedded newform 8041.2.a.g.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8041,2,Mod(1,8041)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8041, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8041.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8041 = 11 \cdot 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8041.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:	$64.2077082653$
Analytic rank:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.42698 q^{2} +0.265461 q^{3} +3.89025 q^{4} +3.56663 q^{5} -0.644270 q^{6} -2.27175 q^{7} -4.58762 q^{8} -2.92953 q^{9} +O(q^{10})$	\(q-2.42698 q^{2} +0.265461 q^{3} +3.89025 q^{4} +3.56663 q^{5} -0.644270 q^{6} -2.27175 q^{7} -4.58762 q^{8} -2.92953 q^{9} -8.65616 q^{10} -1.00000 q^{11} +1.03271 q^{12} +0.560903 q^{13} +5.51350 q^{14} +0.946802 q^{15} +3.35357 q^{16} +1.00000 q^{17} +7.10992 q^{18} -5.53686 q^{19} +13.8751 q^{20} -0.603061 q^{21} +2.42698 q^{22} +0.0941365 q^{23} -1.21783 q^{24} +7.72087 q^{25} -1.36130 q^{26} -1.57406 q^{27} -8.83768 q^{28} +3.09247 q^{29} -2.29787 q^{30} +0.672520 q^{31} +1.03618 q^{32} -0.265461 q^{33} -2.42698 q^{34} -8.10249 q^{35} -11.3966 q^{36} +1.72926 q^{37} +13.4379 q^{38} +0.148898 q^{39} -16.3623 q^{40} -0.709853 q^{41} +1.46362 q^{42} +1.00000 q^{43} -3.89025 q^{44} -10.4486 q^{45} -0.228468 q^{46} -5.78116 q^{47} +0.890241 q^{48} -1.83916 q^{49} -18.7384 q^{50} +0.265461 q^{51} +2.18206 q^{52} +4.54147 q^{53} +3.82022 q^{54} -3.56663 q^{55} +10.4219 q^{56} -1.46982 q^{57} -7.50538 q^{58} +14.9209 q^{59} +3.68330 q^{60} -3.26994 q^{61} -1.63220 q^{62} +6.65516 q^{63} -9.22192 q^{64} +2.00054 q^{65} +0.644270 q^{66} -1.08719 q^{67} +3.89025 q^{68} +0.0249896 q^{69} +19.6646 q^{70} +12.9951 q^{71} +13.4396 q^{72} -2.05221 q^{73} -4.19690 q^{74} +2.04959 q^{75} -21.5398 q^{76} +2.27175 q^{77} -0.361373 q^{78} -5.52944 q^{79} +11.9609 q^{80} +8.37074 q^{81} +1.72280 q^{82} -5.42574 q^{83} -2.34606 q^{84} +3.56663 q^{85} -2.42698 q^{86} +0.820931 q^{87} +4.58762 q^{88} +13.3425 q^{89} +25.3585 q^{90} -1.27423 q^{91} +0.366215 q^{92} +0.178528 q^{93} +14.0308 q^{94} -19.7479 q^{95} +0.275065 q^{96} +6.01597 q^{97} +4.46361 q^{98} +2.92953 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q - 11 q^{2} - 3 q^{3} + 65 q^{4} - 6 q^{5} - 10 q^{6} - 11 q^{7} - 33 q^{8} + 56 q^{9}+O(q^{10}) $ 69 * q - 11 * q^2 - 3 * q^3 + 65 * q^4 - 6 * q^5 - 10 * q^6 - 11 * q^7 - 33 * q^8 + 56 * q^9	\( 69 q - 11 q^{2} - 3 q^{3} + 65 q^{4} - 6 q^{5} - 10 q^{6} - 11 q^{7} - 33 q^{8} + 56 q^{9} - q^{10} - 69 q^{11} - 3 q^{12} - 28 q^{13} - 15 q^{14} - 45 q^{15} + 53 q^{16} + 69 q^{17} - 17 q^{18} - 32 q^{19} - 21 q^{20} - 38 q^{21} + 11 q^{22} - 41 q^{23} - 11 q^{24} + 67 q^{25} - 6 q^{26} - 3 q^{27} - 21 q^{28} - 22 q^{29} - 22 q^{30} - 27 q^{31} - 87 q^{32} + 3 q^{33} - 11 q^{34} - 44 q^{35} + 59 q^{36} + 24 q^{37} - 22 q^{38} - 59 q^{39} + q^{40} - 43 q^{41} - 15 q^{42} + 69 q^{43} - 65 q^{44} - 12 q^{45} - 21 q^{46} - 99 q^{47} + 2 q^{48} + 64 q^{49} - 78 q^{50} - 3 q^{51} - 57 q^{52} - 50 q^{53} + 20 q^{54} + 6 q^{55} - 59 q^{56} - 15 q^{57} + 22 q^{58} - 82 q^{59} - 86 q^{60} - 24 q^{61} + 15 q^{62} - 63 q^{63} + 63 q^{64} - 23 q^{65} + 10 q^{66} - 54 q^{67} + 65 q^{68} + 36 q^{69} + 9 q^{70} - 128 q^{71} - 69 q^{72} + 2 q^{73} - 58 q^{74} - 31 q^{75} - 76 q^{76} + 11 q^{77} - 19 q^{78} - 43 q^{79} - 19 q^{80} + 49 q^{81} - 2 q^{82} - 62 q^{83} - 82 q^{84} - 6 q^{85} - 11 q^{86} - 62 q^{87} + 33 q^{88} - 49 q^{89} - 37 q^{90} - 2 q^{91} - 96 q^{92} - 29 q^{93} - 75 q^{94} - 133 q^{95} - 86 q^{96} + 5 q^{97} - 72 q^{98} - 56 q^{99}+O(q^{100}) \) 69 * q - 11 * q^2 - 3 * q^3 + 65 * q^4 - 6 * q^5 - 10 * q^6 - 11 * q^7 - 33 * q^8 + 56 * q^9 - q^10 - 69 * q^11 - 3 * q^12 - 28 * q^13 - 15 * q^14 - 45 * q^15 + 53 * q^16 + 69 * q^17 - 17 * q^18 - 32 * q^19 - 21 * q^20 - 38 * q^21 + 11 * q^22 - 41 * q^23 - 11 * q^24 + 67 * q^25 - 6 * q^26 - 3 * q^27 - 21 * q^28 - 22 * q^29 - 22 * q^30 - 27 * q^31 - 87 * q^32 + 3 * q^33 - 11 * q^34 - 44 * q^35 + 59 * q^36 + 24 * q^37 - 22 * q^38 - 59 * q^39 + q^40 - 43 * q^41 - 15 * q^42 + 69 * q^43 - 65 * q^44 - 12 * q^45 - 21 * q^46 - 99 * q^47 + 2 * q^48 + 64 * q^49 - 78 * q^50 - 3 * q^51 - 57 * q^52 - 50 * q^53 + 20 * q^54 + 6 * q^55 - 59 * q^56 - 15 * q^57 + 22 * q^58 - 82 * q^59 - 86 * q^60 - 24 * q^61 + 15 * q^62 - 63 * q^63 + 63 * q^64 - 23 * q^65 + 10 * q^66 - 54 * q^67 + 65 * q^68 + 36 * q^69 + 9 * q^70 - 128 * q^71 - 69 * q^72 + 2 * q^73 - 58 * q^74 - 31 * q^75 - 76 * q^76 + 11 * q^77 - 19 * q^78 - 43 * q^79 - 19 * q^80 + 49 * q^81 - 2 * q^82 - 62 * q^83 - 82 * q^84 - 6 * q^85 - 11 * q^86 - 62 * q^87 + 33 * q^88 - 49 * q^89 - 37 * q^90 - 2 * q^91 - 96 * q^92 - 29 * q^93 - 75 * q^94 - 133 * q^95 - 86 * q^96 + 5 * q^97 - 72 * q^98 - 56 * 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 $

$2$	−2.42698	−1.71614	−0.858069	−	0.513535i	$-0.828336\pi$
$2$	−2.42698	−1.71614	−0.858069	+	0.513535i	$0.828336\pi$
$3$	0.265461	0.153264	0.0766320	−	0.997059i	$-0.475583\pi$
$3$	0.265461	0.153264	0.0766320	+	0.997059i	$0.475583\pi$
$4$	3.89025	1.94513
$5$	3.56663	1.59505	0.797523	−	0.603288i	$-0.206143\pi$
$5$	3.56663	1.59505	0.797523	+	0.603288i	$0.206143\pi$
$6$	−0.644270	−0.263022
$7$	−2.27175	−0.858640	−0.429320	−	0.903152i	$-0.641247\pi$
$7$	−2.27175	−0.858640	−0.429320	+	0.903152i	$0.641247\pi$
$8$	−4.58762	−1.62197
$9$	−2.92953	−0.976510
$10$	−8.65616	−2.73732
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.03271	0.298118
$13$	0.560903	0.155567	0.0777833	−	0.996970i	$-0.475216\pi$
$13$	0.560903	0.155567	0.0777833	+	0.996970i	$0.475216\pi$
$14$	5.51350	1.47354
$15$	0.946802	0.244463
$16$	3.35357	0.838392
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	7.10992	1.67583
$19$	−5.53686	−1.27024	−0.635121	−	0.772413i	$-0.719050\pi$
$19$	−5.53686	−1.27024	−0.635121	+	0.772413i	$0.719050\pi$
$20$	13.8751	3.10257
$21$	−0.603061	−0.131599
$22$	2.42698	0.517435
$23$	0.0941365	0.0196288	0.00981441	−	0.999952i	$-0.496876\pi$
$23$	0.0941365	0.0196288	0.00981441	+	0.999952i	$0.496876\pi$
$24$	−1.21783	−0.248589
$25$	7.72087	1.54417
$26$	−1.36130	−0.266974
$27$	−1.57406	−0.302928
$28$	−8.83768	−1.67016
$29$	3.09247	0.574258	0.287129	−	0.957892i	$-0.407299\pi$
$29$	3.09247	0.574258	0.287129	+	0.957892i	$0.407299\pi$
$30$	−2.29787	−0.419532
$31$	0.672520	0.120788	0.0603941	−	0.998175i	$-0.480764\pi$
$31$	0.672520	0.120788	0.0603941	+	0.998175i	$0.480764\pi$
$32$	1.03618	0.183172
$33$	−0.265461	−0.0462108
$34$	−2.42698	−0.416224
$35$	−8.10249	−1.36957
$36$	−11.3966	−1.89944
$37$	1.72926	0.284289	0.142145	−	0.989846i	$-0.454600\pi$
$37$	1.72926	0.284289	0.142145	+	0.989846i	$0.454600\pi$
$38$	13.4379	2.17991
$39$	0.148898	0.0238428
$40$	−16.3623	−2.58711
$41$	−0.709853	−0.110860	−0.0554302	−	0.998463i	$-0.517653\pi$
$41$	−0.709853	−0.110860	−0.0554302	+	0.998463i	$0.517653\pi$
$42$	1.46362	0.225841
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	−3.89025	−0.586478
$45$	−10.4486	−1.55758
$46$	−0.228468	−0.0336857
$47$	−5.78116	−0.843269	−0.421634	−	0.906766i	$-0.638543\pi$
$47$	−5.78116	−0.843269	−0.421634	+	0.906766i	$0.638543\pi$
$48$	0.890241	0.128495
$49$	−1.83916	−0.262737
$50$	−18.7384	−2.65001
$51$	0.265461	0.0371720
$52$	2.18206	0.302597
$53$	4.54147	0.623818	0.311909	−	0.950112i	$-0.399032\pi$
$53$	4.54147	0.623818	0.311909	+	0.950112i	$0.399032\pi$
$54$	3.82022	0.519866
$55$	−3.56663	−0.480925
$56$	10.4219	1.39269
$57$	−1.46982	−0.194682
$58$	−7.50538	−0.985505
$59$	14.9209	1.94253	0.971266	−	0.237998i	$-0.0764912\pi$
$59$	14.9209	1.94253	0.971266	+	0.237998i	$0.0764912\pi$
$60$	3.68330	0.475512
$61$	−3.26994	−0.418673	−0.209337	−	0.977844i	$-0.567130\pi$
$61$	−3.26994	−0.418673	−0.209337	+	0.977844i	$0.567130\pi$
$62$	−1.63220	−0.207289
$63$	6.65516	0.838471
$64$	−9.22192	−1.15274
$65$	2.00054	0.248136
$66$	0.644270	0.0793041
$67$	−1.08719	−0.132822	−0.0664108	−	0.997792i	$-0.521155\pi$
$67$	−1.08719	−0.132822	−0.0664108	+	0.997792i	$0.521155\pi$
$68$	3.89025	0.471763
$69$	0.0249896	0.00300839
$70$	19.6646	2.35037
$71$	12.9951	1.54223	0.771116	−	0.636695i	$-0.219699\pi$
$71$	12.9951	1.54223	0.771116	+	0.636695i	$0.219699\pi$
$72$	13.4396	1.58387
$73$	−2.05221	−0.240192	−0.120096	−	0.992762i	$-0.538320\pi$
$73$	−2.05221	−0.240192	−0.120096	+	0.992762i	$0.538320\pi$
$74$	−4.19690	−0.487880
$75$	2.04959	0.236666
$76$	−21.5398	−2.47078
$77$	2.27175	0.258890
$78$	−0.361373	−0.0409175
$79$	−5.52944	−0.622110	−0.311055	−	0.950392i	$-0.600682\pi$
$79$	−5.52944	−0.622110	−0.311055	+	0.950392i	$0.600682\pi$
$80$	11.9609	1.33727
$81$	8.37074	0.930082
$82$	1.72280	0.190252
$83$	−5.42574	−0.595553	−0.297776	−	0.954636i	$-0.596245\pi$
$83$	−5.42574	−0.595553	−0.297776	+	0.954636i	$0.596245\pi$
$84$	−2.34606	−0.255976
$85$	3.56663	0.386856
$86$	−2.42698	−0.261708
$87$	0.820931	0.0880131
$88$	4.58762	0.489042
$89$	13.3425	1.41431	0.707154	−	0.707060i	$-0.249979\pi$
$89$	13.3425	1.41431	0.707154	+	0.707060i	$0.249979\pi$
$90$	25.3585	2.67302
$91$	−1.27423	−0.133576
$92$	0.366215	0.0381805
$93$	0.178528	0.0185125
$94$	14.0308	1.44716
$95$	−19.7479	−2.02610
$96$	0.275065	0.0280737
$97$	6.01597	0.610830	0.305415	−	0.952219i	$-0.401205\pi$
$97$	6.01597	0.610830	0.305415	+	0.952219i	$0.401205\pi$
$98$	4.46361	0.450893
$99$	2.92953	0.294429
$100$	30.0361	3.00361
$101$	−4.48745	−0.446518	−0.223259	−	0.974759i	$-0.571670\pi$
$101$	−4.48745	−0.446518	−0.223259	+	0.974759i	$0.571670\pi$
$102$	−0.644270	−0.0637922
$103$	9.56546	0.942512	0.471256	−	0.881996i	$-0.343801\pi$
$103$	9.56546	0.942512	0.471256	+	0.881996i	$0.343801\pi$
$104$	−2.57321	−0.252324
$105$	−2.15090	−0.209906
$106$	−11.0221	−1.07056
$107$	1.95259	0.188764	0.0943821	−	0.995536i	$-0.469912\pi$
$107$	1.95259	0.188764	0.0943821	+	0.995536i	$0.469912\pi$
$108$	−6.12349	−0.589233
$109$	16.9918	1.62752	0.813759	−	0.581203i	$-0.197418\pi$
$109$	16.9918	1.62752	0.813759	+	0.581203i	$0.197418\pi$
$110$	8.65616	0.825333
$111$	0.459052	0.0435713
$112$	−7.61846	−0.719877
$113$	−4.77576	−0.449266	−0.224633	−	0.974443i	$-0.572118\pi$
$113$	−4.77576	−0.449266	−0.224633	+	0.974443i	$0.572118\pi$
$114$	3.56723	0.334102
$115$	0.335750	0.0313089
$116$	12.0305	1.11700
$117$	−1.64318	−0.151912
$118$	−36.2127	−3.33365
$119$	−2.27175	−0.208251
$120$	−4.34356	−0.396511
$121$	1.00000	0.0909091
$122$	7.93610	0.718501
$123$	−0.188438	−0.0169909
$124$	2.61627	0.234948
$125$	9.70434	0.867982
$126$	−16.1520	−1.43893
$127$	−14.6468	−1.29970	−0.649849	−	0.760063i	$-0.725168\pi$
$127$	−14.6468	−1.29970	−0.649849	+	0.760063i	$0.725168\pi$
$128$	20.3091	1.79509
$129$	0.265461	0.0233725
$130$	−4.85527	−0.425835
$131$	−10.2817	−0.898313	−0.449156	−	0.893453i	$-0.648275\pi$
$131$	−10.2817	−0.898313	−0.449156	+	0.893453i	$0.648275\pi$
$132$	−1.03271	−0.0898859
$133$	12.5783	1.09068
$134$	2.63860	0.227940
$135$	−5.61409	−0.483184
$136$	−4.58762	−0.393385
$137$	−11.5639	−0.987973	−0.493987	−	0.869470i	$-0.664461\pi$
$137$	−11.5639	−0.987973	−0.493987	+	0.869470i	$0.664461\pi$
$138$	−0.0606493	−0.00516281
$139$	−8.22923	−0.697994	−0.348997	−	0.937124i	$-0.613478\pi$
$139$	−8.22923	−0.697994	−0.348997	+	0.937124i	$0.613478\pi$
$140$	−31.5207	−2.66399
$141$	−1.53467	−0.129243
$142$	−31.5388	−2.64668
$143$	−0.560903	−0.0469051
$144$	−9.82437	−0.818698
$145$	11.0297	0.915968
$146$	4.98067	0.412203
$147$	−0.488225	−0.0402682
$148$	6.72728	0.552979
$149$	−12.3816	−1.01434	−0.507169	−	0.861847i	$-0.669308\pi$
$149$	−12.3816	−1.01434	−0.507169	+	0.861847i	$0.669308\pi$
$150$	−4.97432	−0.406152
$151$	7.09973	0.577768	0.288884	−	0.957364i	$-0.406716\pi$
$151$	7.09973	0.577768	0.288884	+	0.957364i	$0.406716\pi$
$152$	25.4010	2.06029
$153$	−2.92953	−0.236838
$154$	−5.51350	−0.444290
$155$	2.39863	0.192663
$156$	0.579251	0.0463772
$157$	−21.0258	−1.67804	−0.839021	−	0.544100i	$-0.816871\pi$
$157$	−21.0258	−1.67804	−0.839021	+	0.544100i	$0.816871\pi$
$158$	13.4199	1.06763
$159$	1.20558	0.0956089
$160$	3.69567	0.292169
$161$	−0.213854	−0.0168541
$162$	−20.3157	−1.59615
$163$	−4.43888	−0.347680	−0.173840	−	0.984774i	$-0.555618\pi$
$163$	−4.43888	−0.347680	−0.173840	+	0.984774i	$0.555618\pi$
$164$	−2.76151	−0.215638
$165$	−0.946802	−0.0737084
$166$	13.1682	1.02205
$167$	−22.7629	−1.76144	−0.880722	−	0.473633i	$-0.842942\pi$
$167$	−22.7629	−1.76144	−0.880722	+	0.473633i	$0.842942\pi$
$168$	2.76661	0.213449
$169$	−12.6854	−0.975799
$170$	−8.65616	−0.663897
$171$	16.2204	1.24040
$172$	3.89025	0.296629
$173$	11.0979	0.843759	0.421880	−	0.906652i	$-0.361370\pi$
$173$	11.0979	0.843759	0.421880	+	0.906652i	$0.361370\pi$
$174$	−1.99239	−0.151042
$175$	−17.5399	−1.32589
$176$	−3.35357	−0.252785
$177$	3.96091	0.297720
$178$	−32.3822	−2.42715
$179$	−9.24728	−0.691174	−0.345587	−	0.938387i	$-0.612320\pi$
$179$	−9.24728	−0.691174	−0.345587	+	0.938387i	$0.612320\pi$
$180$	−40.6475	−3.02969
$181$	−21.0477	−1.56446	−0.782232	−	0.622987i	$-0.785919\pi$
$181$	−21.0477	−1.56446	−0.782232	+	0.622987i	$0.785919\pi$
$182$	3.09254	0.229234
$183$	−0.868042	−0.0641675
$184$	−0.431862	−0.0318373
$185$	6.16765	0.453455
$186$	−0.433284	−0.0317699
$187$	−1.00000	−0.0731272
$188$	−22.4902	−1.64026
$189$	3.57587	0.260106
$190$	47.9279	3.47706
$191$	12.4313	0.899500	0.449750	−	0.893154i	$-0.351513\pi$
$191$	12.4313	0.899500	0.449750	+	0.893154i	$0.351513\pi$
$192$	−2.44806	−0.176674
$193$	−0.198958	−0.0143213	−0.00716064	−	0.999974i	$-0.502279\pi$
$193$	−0.198958	−0.0143213	−0.00716064	+	0.999974i	$0.502279\pi$
$194$	−14.6007	−1.04827
$195$	0.531065	0.0380303
$196$	−7.15480	−0.511057
$197$	0.475159	0.0338537	0.0169268	−	0.999857i	$-0.494612\pi$
$197$	0.475159	0.0338537	0.0169268	+	0.999857i	$0.494612\pi$
$198$	−7.10992	−0.505280
$199$	−4.61260	−0.326979	−0.163489	−	0.986545i	$-0.552275\pi$
$199$	−4.61260	−0.326979	−0.163489	+	0.986545i	$0.552275\pi$
$200$	−35.4204	−2.50460
$201$	−0.288607	−0.0203568
$202$	10.8910	0.766287
$203$	−7.02532	−0.493081
$204$	1.03271	0.0723042
$205$	−2.53179	−0.176828
$206$	−23.2152	−1.61748
$207$	−0.275776	−0.0191677
$208$	1.88103	0.130426
$209$	5.53686	0.382992
$210$	5.22019	0.360227
$211$	−4.98729	−0.343339	−0.171669	−	0.985155i	$-0.554916\pi$
$211$	−4.98729	−0.343339	−0.171669	+	0.985155i	$0.554916\pi$
$212$	17.6675	1.21341
$213$	3.44969	0.236369
$214$	−4.73891	−0.323945
$215$	3.56663	0.243242
$216$	7.22118	0.491339
$217$	−1.52780	−0.103714
$218$	−41.2388	−2.79304
$219$	−0.544781	−0.0368129
$220$	−13.8751	−0.935459
$221$	0.560903	0.0377304
$222$	−1.11411	−0.0747744
$223$	−4.70781	−0.315259	−0.157629	−	0.987498i	$-0.550385\pi$
$223$	−4.70781	−0.315259	−0.157629	+	0.987498i	$0.550385\pi$
$224$	−2.35394	−0.157279
$225$	−22.6185	−1.50790
$226$	11.5907	0.771001
$227$	−1.63824	−0.108734	−0.0543669	−	0.998521i	$-0.517314\pi$
$227$	−1.63824	−0.108734	−0.0543669	+	0.998521i	$0.517314\pi$
$228$	−5.71797	−0.378682
$229$	−11.0325	−0.729045	−0.364523	−	0.931195i	$-0.618768\pi$
$229$	−11.0325	−0.729045	−0.364523	+	0.931195i	$0.618768\pi$
$230$	−0.814861	−0.0537303
$231$	0.603061	0.0396785
$232$	−14.1871	−0.931427
$233$	−15.0223	−0.984144	−0.492072	−	0.870554i	$-0.663760\pi$
$233$	−15.0223	−0.984144	−0.492072	+	0.870554i	$0.663760\pi$
$234$	3.98798	0.260702
$235$	−20.6193	−1.34505
$236$	58.0460	3.77847
$237$	−1.46785	−0.0953471
$238$	5.51350	0.357387
$239$	−24.6565	−1.59489	−0.797447	−	0.603389i	$-0.793817\pi$
$239$	−24.6565	−1.59489	−0.797447	+	0.603389i	$0.793817\pi$
$240$	3.17516	0.204956
$241$	−9.99166	−0.643620	−0.321810	−	0.946804i	$-0.604291\pi$
$241$	−9.99166	−0.643620	−0.321810	+	0.946804i	$0.604291\pi$
$242$	−2.42698	−0.156012
$243$	6.94428	0.445476
$244$	−12.7209	−0.814372
$245$	−6.55961	−0.419078
$246$	0.457337	0.0291587
$247$	−3.10564	−0.197607
$248$	−3.08526	−0.195914
$249$	−1.44032	−0.0912768
$250$	−23.5523	−1.48958
$251$	−0.301048	−0.0190020	−0.00950099	−	0.999955i	$-0.503024\pi$
$251$	−0.301048	−0.0190020	−0.00950099	+	0.999955i	$0.503024\pi$
$252$	25.8902	1.63093
$253$	−0.0941365	−0.00591831
$254$	35.5477	2.23046
$255$	0.946802	0.0592911
$256$	−30.8460	−1.92788
$257$	−15.0251	−0.937242	−0.468621	−	0.883399i	$-0.655249\pi$
$257$	−15.0251	−0.937242	−0.468621	+	0.883399i	$0.655249\pi$
$258$	−0.644270	−0.0401105
$259$	−3.92845	−0.244102
$260$	7.78259	0.482656
$261$	−9.05949	−0.560769
$262$	24.9534	1.54163
$263$	−14.6691	−0.904536	−0.452268	−	0.891882i	$-0.649385\pi$
$263$	−14.6691	−0.904536	−0.452268	+	0.891882i	$0.649385\pi$
$264$	1.21783	0.0749525
$265$	16.1977	0.995019
$266$	−30.5274	−1.87176
$267$	3.54193	0.216762
$268$	−4.22945	−0.258355
$269$	−1.54034	−0.0939161	−0.0469581	−	0.998897i	$-0.514953\pi$
$269$	−1.54034	−0.0939161	−0.0469581	+	0.998897i	$0.514953\pi$
$270$	13.6253	0.829210
$271$	−20.7782	−1.26218	−0.631091	−	0.775708i	$-0.717393\pi$
$271$	−20.7782	−1.26218	−0.631091	+	0.775708i	$0.717393\pi$
$272$	3.35357	0.203340
$273$	−0.338259	−0.0204724
$274$	28.0655	1.69550
$275$	−7.72087	−0.465586
$276$	0.0972158	0.00585170
$277$	12.1567	0.730424	0.365212	−	0.930924i	$-0.380996\pi$
$277$	12.1567	0.730424	0.365212	+	0.930924i	$0.380996\pi$
$278$	19.9722	1.19785
$279$	−1.97017	−0.117951
$280$	37.1711	2.22140
$281$	2.10752	0.125724	0.0628620	−	0.998022i	$-0.479977\pi$
$281$	2.10752	0.125724	0.0628620	+	0.998022i	$0.479977\pi$
$282$	3.72463	0.221798
$283$	−21.6008	−1.28404	−0.642018	−	0.766689i	$-0.721903\pi$
$283$	−21.6008	−1.28404	−0.642018	+	0.766689i	$0.721903\pi$
$284$	50.5541	2.99984
$285$	−5.24231	−0.310528
$286$	1.36130	0.0804956
$287$	1.61261	0.0951892
$288$	−3.03552	−0.178870
$289$	1.00000	0.0588235
$290$	−26.7689	−1.57193
$291$	1.59701	0.0936182
$292$	−7.98360	−0.467205
$293$	4.77366	0.278880	0.139440	−	0.990230i	$-0.455470\pi$
$293$	4.77366	0.278880	0.139440	+	0.990230i	$0.455470\pi$
$294$	1.18492	0.0691057
$295$	53.2172	3.09843
$296$	−7.93320	−0.461108
$297$	1.57406	0.0913362
$298$	30.0499	1.74074
$299$	0.0528015	0.00305359
$300$	7.97343	0.460346
$301$	−2.27175	−0.130941
$302$	−17.2309	−0.991529
$303$	−1.19124	−0.0684352
$304$	−18.5682	−1.06496
$305$	−11.6627	−0.667803
$306$	7.10992	0.406447
$307$	−21.3459	−1.21828	−0.609138	−	0.793064i	$-0.708484\pi$
$307$	−21.3459	−1.21828	−0.609138	+	0.793064i	$0.708484\pi$
$308$	8.83768	0.503573
$309$	2.53926	0.144453
$310$	−5.82144	−0.330636
$311$	−25.7859	−1.46219	−0.731093	−	0.682278i	$-0.760990\pi$
$311$	−25.7859	−1.46219	−0.731093	+	0.682278i	$0.760990\pi$
$312$	−0.683087	−0.0386722
$313$	11.1647	0.631068	0.315534	−	0.948914i	$-0.397816\pi$
$313$	11.1647	0.631068	0.315534	+	0.948914i	$0.397816\pi$
$314$	51.0293	2.87975
$315$	23.7365	1.33740
$316$	−21.5109	−1.21008
$317$	6.58877	0.370062	0.185031	−	0.982733i	$-0.440761\pi$
$317$	6.58877	0.370062	0.185031	+	0.982733i	$0.440761\pi$
$318$	−2.92593	−0.164078
$319$	−3.09247	−0.173145
$320$	−32.8912	−1.83867
$321$	0.518337	0.0289308
$322$	0.519021	0.0289239
$323$	−5.53686	−0.308079
$324$	32.5643	1.80913
$325$	4.33066	0.240222
$326$	10.7731	0.596666
$327$	4.51066	0.249440
$328$	3.25653	0.179812
$329$	13.1333	0.724064
$330$	2.29787	0.126494
$331$	−5.40053	−0.296840	−0.148420	−	0.988924i	$-0.547419\pi$
$331$	−5.40053	−0.296840	−0.148420	+	0.988924i	$0.547419\pi$
$332$	−21.1075	−1.15843
$333$	−5.06593	−0.277611
$334$	55.2452	3.02288
$335$	−3.87762	−0.211857
$336$	−2.02240	−0.110331
$337$	22.5485	1.22829	0.614147	−	0.789191i	$-0.289500\pi$
$337$	22.5485	1.22829	0.614147	+	0.789191i	$0.289500\pi$
$338$	30.7872	1.67460
$339$	−1.26778	−0.0688563
$340$	13.8751	0.752483
$341$	−0.672520	−0.0364190
$342$	−39.3666	−2.12870
$343$	20.0803	1.08424
$344$	−4.58762	−0.247348
$345$	0.0891286	0.00479852
$346$	−26.9345	−1.44801
$347$	−27.2409	−1.46237	−0.731185	−	0.682179i	$-0.761033\pi$
$347$	−27.2409	−1.46237	−0.731185	+	0.682179i	$0.761033\pi$
$348$	3.19363	0.171197
$349$	−16.4422	−0.880130	−0.440065	−	0.897966i	$-0.645045\pi$
$349$	−16.4422	−0.880130	−0.440065	+	0.897966i	$0.645045\pi$
$350$	42.5690	2.27541
$351$	−0.882895	−0.0471255
$352$	−1.03618	−0.0552286
$353$	5.60309	0.298222	0.149111	−	0.988820i	$-0.452359\pi$
$353$	5.60309	0.298222	0.149111	+	0.988820i	$0.452359\pi$
$354$	−9.61306	−0.510929
$355$	46.3487	2.45993
$356$	51.9059	2.75101
$357$	−0.603061	−0.0319174
$358$	22.4430	1.18615
$359$	−5.37033	−0.283435	−0.141718	−	0.989907i	$-0.545262\pi$
$359$	−5.37033	−0.283435	−0.141718	+	0.989907i	$0.545262\pi$
$360$	47.9340	2.52634
$361$	11.6568	0.613515
$362$	51.0825	2.68483
$363$	0.265461	0.0139331
$364$	−4.95708	−0.259822
$365$	−7.31946	−0.383118
$366$	2.10673	0.110120
$367$	−28.0390	−1.46362	−0.731811	−	0.681508i	$-0.761325\pi$
$367$	−28.0390	−1.46362	−0.731811	+	0.681508i	$0.761325\pi$
$368$	0.315693	0.0164566
$369$	2.07954	0.108256
$370$	−14.9688	−0.778191
$371$	−10.3171	−0.535635
$372$	0.694519	0.0360091
$373$	2.74950	0.142364	0.0711819	−	0.997463i	$-0.477323\pi$
$373$	2.74950	0.142364	0.0711819	+	0.997463i	$0.477323\pi$
$374$	2.42698	0.125496
$375$	2.57612	0.133030
$376$	26.5217	1.36775
$377$	1.73458	0.0893353
$378$	−8.67857	−0.446378
$379$	−18.1234	−0.930936	−0.465468	−	0.885065i	$-0.654114\pi$
$379$	−18.1234	−0.930936	−0.465468	+	0.885065i	$0.654114\pi$
$380$	−76.8245	−3.94101
$381$	−3.88817	−0.199197
$382$	−30.1707	−1.54367
$383$	4.58617	0.234342	0.117171	−	0.993112i	$-0.462617\pi$
$383$	4.58617	0.234342	0.117171	+	0.993112i	$0.462617\pi$
$384$	5.39128	0.275122
$385$	8.10249	0.412941
$386$	0.482867	0.0245773
$387$	−2.92953	−0.148916
$388$	23.4037	1.18814
$389$	17.0800	0.865990	0.432995	−	0.901396i	$-0.357457\pi$
$389$	17.0800	0.865990	0.432995	+	0.901396i	$0.357457\pi$
$390$	−1.28889	−0.0652653
$391$	0.0941365	0.00476069
$392$	8.43736	0.426151
$393$	−2.72938	−0.137679
$394$	−1.15320	−0.0580976
$395$	−19.7215	−0.992295
$396$	11.3966	0.572702
$397$	36.6708	1.84045	0.920226	−	0.391387i	$-0.128005\pi$
$397$	36.6708	1.84045	0.920226	+	0.391387i	$0.128005\pi$
$398$	11.1947	0.561140
$399$	3.33906	0.167162
$400$	25.8924	1.29462
$401$	1.32855	0.0663445	0.0331722	−	0.999450i	$-0.489439\pi$
$401$	1.32855	0.0663445	0.0331722	+	0.999450i	$0.489439\pi$
$402$	0.700445	0.0349350
$403$	0.377219	0.0187906
$404$	−17.4573	−0.868535
$405$	29.8554	1.48352
$406$	17.0503	0.846194
$407$	−1.72926	−0.0857165
$408$	−1.21783	−0.0602917
$409$	−22.7390	−1.12437	−0.562186	−	0.827011i	$-0.690040\pi$
$409$	−22.7390	−1.12437	−0.562186	+	0.827011i	$0.690040\pi$
$410$	6.14460	0.303460
$411$	−3.06977	−0.151421
$412$	37.2121	1.83331
$413$	−33.8964	−1.66794
$414$	0.669303	0.0328945
$415$	−19.3516	−0.949934
$416$	0.581197	0.0284955
$417$	−2.18454	−0.106977
$418$	−13.4379	−0.657268
$419$	−24.0820	−1.17648	−0.588242	−	0.808685i	$-0.700180\pi$
$419$	−24.0820	−1.17648	−0.588242	+	0.808685i	$0.700180\pi$
$420$	−8.36753	−0.408294
$421$	−35.2258	−1.71680	−0.858401	−	0.512980i	$-0.828541\pi$
$421$	−35.2258	−1.71680	−0.858401	+	0.512980i	$0.828541\pi$
$422$	12.1041	0.589217
$423$	16.9361	0.823461
$424$	−20.8345	−1.01181
$425$	7.72087	0.374517
$426$	−8.37234	−0.405641
$427$	7.42848	0.359490
$428$	7.59608	0.367170
$429$	−0.148898	−0.00718886
$430$	−8.65616	−0.417437
$431$	12.4389	0.599158	0.299579	−	0.954071i	$-0.403154\pi$
$431$	12.4389	0.599158	0.299579	+	0.954071i	$0.403154\pi$
$432$	−5.27871	−0.253972
$433$	21.5156	1.03398	0.516988	−	0.855993i	$-0.327053\pi$
$433$	21.5156	1.03398	0.516988	+	0.855993i	$0.327053\pi$
$434$	3.70794	0.177987
$435$	2.92796	0.140385
$436$	66.1023	3.16573
$437$	−0.521220	−0.0249333
$438$	1.32217	0.0631759
$439$	14.2630	0.680736	0.340368	−	0.940292i	$-0.389448\pi$
$439$	14.2630	0.680736	0.340368	+	0.940292i	$0.389448\pi$
$440$	16.3623	0.780044
$441$	5.38788	0.256566
$442$	−1.36130	−0.0647506
$443$	30.6206	1.45483	0.727415	−	0.686198i	$-0.240722\pi$
$443$	30.6206	1.45483	0.727415	+	0.686198i	$0.240722\pi$
$444$	1.78583	0.0847518
$445$	47.5880	2.25589
$446$	11.4258	0.541027
$447$	−3.28682	−0.155461
$448$	20.9499	0.989789
$449$	−25.3899	−1.19822	−0.599112	−	0.800665i	$-0.704479\pi$
$449$	−25.3899	−1.19822	−0.599112	+	0.800665i	$0.704479\pi$
$450$	54.8948	2.58777
$451$	0.709853	0.0334257
$452$	−18.5789	−0.873879
$453$	1.88470	0.0885510
$454$	3.97599	0.186602
$455$	−4.54472	−0.213060
$456$	6.74297	0.315769
$457$	33.6698	1.57501	0.787503	−	0.616311i	$-0.211374\pi$
$457$	33.6698	1.57501	0.787503	+	0.616311i	$0.211374\pi$
$458$	26.7756	1.25114
$459$	−1.57406	−0.0734708
$460$	1.30615	0.0608997
$461$	2.67596	0.124632	0.0623159	−	0.998056i	$-0.480151\pi$
$461$	2.67596	0.124632	0.0623159	+	0.998056i	$0.480151\pi$
$462$	−1.46362	−0.0680937
$463$	20.9166	0.972076	0.486038	−	0.873938i	$-0.338442\pi$
$463$	20.9166	0.972076	0.486038	+	0.873938i	$0.338442\pi$
$464$	10.3708	0.481453
$465$	0.636743	0.0295283
$466$	36.4589	1.68893
$467$	35.6656	1.65041	0.825204	−	0.564835i	$-0.191060\pi$
$467$	35.6656	1.65041	0.825204	+	0.564835i	$0.191060\pi$
$468$	−6.39240	−0.295489
$469$	2.46983	0.114046
$470$	50.0427	2.30830
$471$	−5.58153	−0.257183
$472$	−68.4512	−3.15072
$473$	−1.00000	−0.0459800
$474$	3.56245	0.163629
$475$	−42.7493	−1.96147
$476$	−8.83768	−0.405074
$477$	−13.3044	−0.609165
$478$	59.8409	2.73706
$479$	−22.3899	−1.02302	−0.511511	−	0.859277i	$-0.670914\pi$
$479$	−22.3899	−1.02302	−0.511511	+	0.859277i	$0.670914\pi$
$480$	0.981057	0.0447789
$481$	0.969950	0.0442259
$482$	24.2496	1.10454
$483$	−0.0567700	−0.00258312
$484$	3.89025	0.176830
$485$	21.4568	0.974302
$486$	−16.8537	−0.764498
$487$	8.46910	0.383771	0.191886	−	0.981417i	$-0.438540\pi$
$487$	8.46910	0.383771	0.191886	+	0.981417i	$0.438540\pi$
$488$	15.0012	0.679074
$489$	−1.17835	−0.0532868
$490$	15.9201	0.719195
$491$	−5.97261	−0.269540	−0.134770	−	0.990877i	$-0.543030\pi$
$491$	−5.97261	−0.269540	−0.134770	+	0.990877i	$0.543030\pi$
$492$	−0.733073	−0.0330495
$493$	3.09247	0.139278
$494$	7.53735	0.339121
$495$	10.4486	0.469628
$496$	2.25534	0.101268
$497$	−29.5215	−1.32422
$498$	3.49564	0.156644
$499$	−27.7930	−1.24419	−0.622093	−	0.782943i	$-0.713718\pi$
$499$	−27.7930	−1.24419	−0.622093	+	0.782943i	$0.713718\pi$
$500$	37.7523	1.68834
$501$	−6.04266	−0.269966
$502$	0.730639	0.0326100
$503$	−30.0007	−1.33767	−0.668833	−	0.743412i	$-0.733206\pi$
$503$	−30.0007	−1.33767	−0.668833	+	0.743412i	$0.733206\pi$
$504$	−30.5313	−1.35997
$505$	−16.0051	−0.712217
$506$	0.228468	0.0101566
$507$	−3.36748	−0.149555
$508$	−56.9800	−2.52808
$509$	7.48214	0.331640	0.165820	−	0.986156i	$-0.446973\pi$
$509$	7.48214	0.331640	0.165820	+	0.986156i	$0.446973\pi$
$510$	−2.29787	−0.101752
$511$	4.66209	0.206239
$512$	34.2446	1.51341
$513$	8.71534	0.384792
$514$	36.4658	1.60844
$515$	34.1165	1.50335
$516$	1.03271	0.0454626
$517$	5.78116	0.254255
$518$	9.53430	0.418913
$519$	2.94607	0.129318
$520$	−9.17769	−0.402468
$521$	7.12546	0.312172	0.156086	−	0.987743i	$-0.450112\pi$
$521$	7.12546	0.312172	0.156086	+	0.987743i	$0.450112\pi$
$522$	21.9872	0.962356
$523$	30.3756	1.32823	0.664117	−	0.747628i	$-0.268808\pi$
$523$	30.3756	1.32823	0.664117	+	0.747628i	$0.268808\pi$
$524$	−39.9983	−1.74733
$525$	−4.65615	−0.203211
$526$	35.6017	1.55231
$527$	0.672520	0.0292954
$528$	−0.890241	−0.0387428
$529$	−22.9911	−0.999615
$530$	−39.3117	−1.70759
$531$	−43.7111	−1.89690
$532$	48.9330	2.12151
$533$	−0.398159	−0.0172462
$534$	−8.59620	−0.371994
$535$	6.96418	0.301088
$536$	4.98762	0.215432
$537$	−2.45479	−0.105932
$538$	3.73838	0.161173
$539$	1.83916	0.0792182
$540$	−21.8402	−0.939854
$541$	40.2347	1.72983	0.864913	−	0.501922i	$-0.167373\pi$
$541$	40.2347	1.72983	0.864913	+	0.501922i	$0.167373\pi$
$542$	50.4283	2.16608
$543$	−5.58735	−0.239776
$544$	1.03618	0.0444258
$545$	60.6034	2.59597
$546$	0.820949	0.0351334
$547$	2.24499	0.0959890	0.0479945	−	0.998848i	$-0.484717\pi$
$547$	2.24499	0.0959890	0.0479945	+	0.998848i	$0.484717\pi$
$548$	−44.9866	−1.92173
$549$	9.57939	0.408839
$550$	18.7384	0.799009
$551$	−17.1226	−0.729446
$552$	−0.114643	−0.00487951
$553$	12.5615	0.534169
$554$	−29.5041	−1.25351
$555$	1.63727	0.0694983
$556$	−32.0138	−1.35769
$557$	−3.78858	−0.160527	−0.0802636	−	0.996774i	$-0.525576\pi$
$557$	−3.78858	−0.160527	−0.0802636	+	0.996774i	$0.525576\pi$
$558$	4.78157	0.202420
$559$	0.560903	0.0237237
$560$	−27.1722	−1.14824
$561$	−0.265461	−0.0112078
$562$	−5.11491	−0.215760
$563$	−16.4438	−0.693022	−0.346511	−	0.938046i	$-0.612634\pi$
$563$	−16.4438	−0.693022	−0.346511	+	0.938046i	$0.612634\pi$
$564$	−5.97027	−0.251394
$565$	−17.0334	−0.716600
$566$	52.4249	2.20358
$567$	−19.0162	−0.798606
$568$	−59.6164	−2.50145
$569$	−29.4783	−1.23579	−0.617896	−	0.786259i	$-0.712015\pi$
$569$	−29.4783	−1.23579	−0.617896	+	0.786259i	$0.712015\pi$
$570$	12.7230	0.532908
$571$	4.56327	0.190967	0.0954834	−	0.995431i	$-0.469560\pi$
$571$	4.56327	0.190967	0.0954834	+	0.995431i	$0.469560\pi$
$572$	−2.18206	−0.0912364
$573$	3.30004	0.137861
$574$	−3.91377	−0.163358
$575$	0.726815	0.0303103
$576$	27.0159	1.12566
$577$	17.0384	0.709318	0.354659	−	0.934996i	$-0.384597\pi$
$577$	17.0384	0.709318	0.354659	+	0.934996i	$0.384597\pi$
$578$	−2.42698	−0.100949
$579$	−0.0528155	−0.00219494
$580$	42.9084	1.78167
$581$	12.3259	0.511365
$582$	−3.87591	−0.160662
$583$	−4.54147	−0.188088
$584$	9.41473	0.389584
$585$	−5.86063	−0.242307
$586$	−11.5856	−0.478597
$587$	−10.4283	−0.430424	−0.215212	−	0.976567i	$-0.569044\pi$
$587$	−10.4283	−0.430424	−0.215212	+	0.976567i	$0.569044\pi$
$588$	−1.89932	−0.0783267
$589$	−3.72365	−0.153430
$590$	−129.157	−5.31733
$591$	0.126136	0.00518855
$592$	5.79920	0.238346
$593$	9.00871	0.369943	0.184972	−	0.982744i	$-0.440781\pi$
$593$	9.00871	0.369943	0.184972	+	0.982744i	$0.440781\pi$
$594$	−3.82022	−0.156745
$595$	−8.10249	−0.332170
$596$	−48.1674	−1.97301
$597$	−1.22447	−0.0501141
$598$	−0.128148	−0.00524038
$599$	−0.515451	−0.0210608	−0.0105304	−	0.999945i	$-0.503352\pi$
$599$	−0.515451	−0.0210608	−0.0105304	+	0.999945i	$0.503352\pi$
$600$	−9.40273	−0.383865
$601$	5.18341	0.211436	0.105718	−	0.994396i	$-0.466286\pi$
$601$	5.18341	0.211436	0.105718	+	0.994396i	$0.466286\pi$
$602$	5.51350	0.224713
$603$	3.18496	0.129702
$604$	27.6197	1.12383
$605$	3.56663	0.145004
$606$	2.89113	0.117444
$607$	3.25608	0.132160	0.0660801	−	0.997814i	$-0.478951\pi$
$607$	3.25608	0.132160	0.0660801	+	0.997814i	$0.478951\pi$
$608$	−5.73718	−0.232673
$609$	−1.86495	−0.0755715
$610$	28.3051	1.14604
$611$	−3.24267	−0.131184
$612$	−11.3966	−0.460681
$613$	9.21578	0.372222	0.186111	−	0.982529i	$-0.440412\pi$
$613$	9.21578	0.372222	0.186111	+	0.982529i	$0.440412\pi$
$614$	51.8062	2.09073
$615$	−0.672090	−0.0271013
$616$	−10.4219	−0.419911
$617$	−15.5452	−0.625826	−0.312913	−	0.949782i	$-0.601305\pi$
$617$	−15.5452	−0.625826	−0.312913	+	0.949782i	$0.601305\pi$
$618$	−6.16274	−0.247902
$619$	16.6783	0.670356	0.335178	−	0.942155i	$-0.391204\pi$
$619$	16.6783	0.670356	0.335178	+	0.942155i	$0.391204\pi$
$620$	9.33128	0.374753
$621$	−0.148176	−0.00594611
$622$	62.5821	2.50931
$623$	−30.3109	−1.21438
$624$	0.499339	0.0199896
$625$	−3.99253	−0.159701
$626$	−27.0966	−1.08300
$627$	1.46982	0.0586990
$628$	−81.7957	−3.26400
$629$	1.72926	0.0689503
$630$	−57.6081	−2.29516
$631$	−22.5981	−0.899616	−0.449808	−	0.893125i	$-0.648508\pi$
$631$	−22.5981	−0.899616	−0.449808	+	0.893125i	$0.648508\pi$
$632$	25.3669	1.00904
$633$	−1.32393	−0.0526215
$634$	−15.9908	−0.635078
$635$	−52.2399	−2.07308
$636$	4.69002	0.185971
$637$	−1.03159	−0.0408731
$638$	7.50538	0.297141
$639$	−38.0695	−1.50601
$640$	72.4351	2.86325
$641$	−5.97068	−0.235828	−0.117914	−	0.993024i	$-0.537621\pi$
$641$	−5.97068	−0.235828	−0.117914	+	0.993024i	$0.537621\pi$
$642$	−1.25800	−0.0496492
$643$	17.8129	0.702472	0.351236	−	0.936287i	$-0.385761\pi$
$643$	17.8129	0.702472	0.351236	+	0.936287i	$0.385761\pi$
$644$	−0.831948	−0.0327833
$645$	0.946802	0.0372803
$646$	13.4379	0.528706
$647$	−31.2569	−1.22884	−0.614419	−	0.788980i	$-0.710609\pi$
$647$	−31.2569	−1.22884	−0.614419	+	0.788980i	$0.710609\pi$
$648$	−38.4017	−1.50856
$649$	−14.9209	−0.585695
$650$	−10.5104	−0.412254
$651$	−0.405570	−0.0158956
$652$	−17.2684	−0.676281
$653$	−0.160921	−0.00629733	−0.00314867	−	0.999995i	$-0.501002\pi$
$653$	−0.160921	−0.00629733	−0.00314867	+	0.999995i	$0.501002\pi$
$654$	−10.9473	−0.428073
$655$	−36.6709	−1.43285
$656$	−2.38054	−0.0929444
$657$	6.01200	0.234550
$658$	−31.8744	−1.24259
$659$	−14.2421	−0.554793	−0.277397	−	0.960756i	$-0.589472\pi$
$659$	−14.2421	−0.554793	−0.277397	+	0.960756i	$0.589472\pi$
$660$	−3.68330	−0.143372
$661$	10.6546	0.414418	0.207209	−	0.978297i	$-0.433562\pi$
$661$	10.6546	0.414418	0.207209	+	0.978297i	$0.433562\pi$
$662$	13.1070	0.509418
$663$	0.148898	0.00578272
$664$	24.8912	0.965967
$665$	44.8623	1.73969
$666$	12.2949	0.476419
$667$	0.291114	0.0112720
$668$	−88.5534	−3.42623
$669$	−1.24974	−0.0483178
$670$	9.41091	0.363575
$671$	3.26994	0.126235
$672$	−0.624879	−0.0241052
$673$	19.6795	0.758590	0.379295	−	0.925276i	$-0.376167\pi$
$673$	19.6795	0.758590	0.379295	+	0.925276i	$0.376167\pi$
$674$	−54.7249	−2.10792
$675$	−12.1531	−0.467773
$676$	−49.3494	−1.89805
$677$	41.9652	1.61285	0.806426	−	0.591335i	$-0.201399\pi$
$677$	41.9652	1.61285	0.806426	+	0.591335i	$0.201399\pi$
$678$	3.07688	0.118167
$679$	−13.6668	−0.524483
$680$	−16.3623	−0.627467
$681$	−0.434889	−0.0166650
$682$	1.63220	0.0625000
$683$	1.41580	0.0541742	0.0270871	−	0.999633i	$-0.491377\pi$
$683$	1.41580	0.0541742	0.0270871	+	0.999633i	$0.491377\pi$
$684$	63.1014	2.41274
$685$	−41.2443	−1.57586
$686$	−48.7347	−1.86070
$687$	−2.92869	−0.111736
$688$	3.35357	0.127854
$689$	2.54732	0.0970453
$690$	−0.216314	−0.00823492
$691$	−46.1662	−1.75624	−0.878122	−	0.478437i	$-0.841203\pi$
$691$	−46.1662	−1.75624	−0.878122	+	0.478437i	$0.841203\pi$
$692$	43.1737	1.64122
$693$	−6.65516	−0.252808
$694$	66.1134	2.50963
$695$	−29.3506	−1.11333
$696$	−3.76612	−0.142754
$697$	−0.709853	−0.0268876
$698$	39.9049	1.51042
$699$	−3.98784	−0.150834
$700$	−68.2345	−2.57902
$701$	31.3644	1.18462	0.592308	−	0.805712i	$-0.298217\pi$
$701$	31.3644	1.18462	0.592308	+	0.805712i	$0.298217\pi$
$702$	2.14277	0.0808738
$703$	−9.57469	−0.361116
$704$	9.22192	0.347564
$705$	−5.47361	−0.206148
$706$	−13.5986	−0.511791
$707$	10.1944	0.383398
$708$	15.4089	0.579103
$709$	−16.1781	−0.607583	−0.303791	−	0.952739i	$-0.598253\pi$
$709$	−16.1781	−0.607583	−0.303791	+	0.952739i	$0.598253\pi$
$710$	−112.487	−4.22158
$711$	16.1987	0.607497
$712$	−61.2105	−2.29396
$713$	0.0633086	0.00237093
$714$	1.46362	0.0547746
$715$	−2.00054	−0.0748158
$716$	−35.9743	−1.34442
$717$	−6.54533	−0.244440
$718$	13.0337	0.486414
$719$	14.7977	0.551861	0.275931	−	0.961178i	$-0.411014\pi$
$719$	14.7977	0.551861	0.275931	+	0.961178i	$0.411014\pi$
$720$	−35.0399	−1.30586
$721$	−21.7303	−0.809279
$722$	−28.2908	−1.05288
$723$	−2.65240	−0.0986437
$724$	−81.8809	−3.04308
$725$	23.8766	0.886754
$726$	−0.644270	−0.0239111
$727$	4.00979	0.148715	0.0743574	−	0.997232i	$-0.476309\pi$
$727$	4.00979	0.148715	0.0743574	+	0.997232i	$0.476309\pi$
$728$	5.84568	0.216655
$729$	−23.2688	−0.861807
$730$	17.7642	0.657483
$731$	1.00000	0.0369863
$732$	−3.37690	−0.124814
$733$	−31.8557	−1.17662	−0.588308	−	0.808637i	$-0.700206\pi$
$733$	−31.8557	−1.17662	−0.588308	+	0.808637i	$0.700206\pi$
$734$	68.0501	2.51178
$735$	−1.74132	−0.0642296
$736$	0.0975423	0.00359546
$737$	1.08719	0.0400472
$738$	−5.04700	−0.185783
$739$	16.2278	0.596951	0.298475	−	0.954417i	$-0.403522\pi$
$739$	16.2278	0.596951	0.298475	+	0.954417i	$0.403522\pi$
$740$	23.9937	0.882027
$741$	−0.824427	−0.0302861
$742$	25.0394	0.919224
$743$	−20.1018	−0.737465	−0.368733	−	0.929536i	$-0.620208\pi$
$743$	−20.1018	−0.737465	−0.368733	+	0.929536i	$0.620208\pi$
$744$	−0.819017	−0.0300266
$745$	−44.1605	−1.61792
$746$	−6.67300	−0.244316
$747$	15.8949	0.581563
$748$	−3.89025	−0.142242
$749$	−4.43580	−0.162081
$750$	−6.25221	−0.228299
$751$	−30.3326	−1.10685	−0.553426	−	0.832898i	$-0.686680\pi$
$751$	−30.3326	−1.10685	−0.553426	+	0.832898i	$0.686680\pi$
$752$	−19.3875	−0.706989
$753$	−0.0799166	−0.00291232
$754$	−4.20980	−0.153312
$755$	25.3221	0.921566
$756$	13.9110	0.505939
$757$	11.0651	0.402167	0.201083	−	0.979574i	$-0.435554\pi$
$757$	11.0651	0.402167	0.201083	+	0.979574i	$0.435554\pi$
$758$	43.9852	1.59761
$759$	−0.0249896	−0.000907064 0
$760$	90.5959	3.28626
$761$	−11.8063	−0.427979	−0.213989	−	0.976836i	$-0.568646\pi$
$761$	−11.8063	−0.427979	−0.213989	+	0.976836i	$0.568646\pi$
$762$	9.43652	0.341849
$763$	−38.6010	−1.39745
$764$	48.3611	1.74964
$765$	−10.4486	−0.377768
$766$	−11.1306	−0.402163
$767$	8.36916	0.302193
$768$	−8.18842	−0.295474
$769$	−14.8518	−0.535570	−0.267785	−	0.963479i	$-0.586292\pi$
$769$	−14.8518	−0.535570	−0.267785	+	0.963479i	$0.586292\pi$
$770$	−19.6646	−0.708664
$771$	−3.98859	−0.143645
$772$	−0.773995	−0.0278567
$773$	−47.2794	−1.70052	−0.850260	−	0.526363i	$-0.823555\pi$
$773$	−47.2794	−1.70052	−0.850260	+	0.526363i	$0.823555\pi$
$774$	7.10992	0.255561
$775$	5.19244	0.186518
$776$	−27.5990	−0.990746
$777$	−1.04285	−0.0374121
$778$	−41.4529	−1.48616
$779$	3.93036	0.140820
$780$	2.06598	0.0739738
$781$	−12.9951	−0.465000
$782$	−0.228468	−0.00816999
$783$	−4.86774	−0.173959
$784$	−6.16775	−0.220277
$785$	−74.9913	−2.67655
$786$	6.62416	0.236276
$787$	7.36580	0.262563	0.131281	−	0.991345i	$-0.458091\pi$
$787$	7.36580	0.262563	0.131281	+	0.991345i	$0.458091\pi$
$788$	1.84849	0.0658497
$789$	−3.89408	−0.138633
$790$	47.8637	1.70291
$791$	10.8493	0.385757
$792$	−13.4396	−0.477554
$793$	−1.83412	−0.0651316
$794$	−88.9993	−3.15847
$795$	4.29987	0.152501
$796$	−17.9442	−0.636015
$797$	20.3167	0.719654	0.359827	−	0.933019i	$-0.382836\pi$
$797$	20.3167	0.719654	0.359827	+	0.933019i	$0.382836\pi$
$798$	−8.10385	−0.286873
$799$	−5.78116	−0.204523
$800$	8.00021	0.282850
$801$	−39.0874	−1.38109
$802$	−3.22436	−0.113856
$803$	2.05221	0.0724207
$804$	−1.12276	−0.0395965
$805$	−0.762740	−0.0268830
$806$	−0.915504	−0.0322472
$807$	−0.408900	−0.0143940
$808$	20.5867	0.724238
$809$	−28.3218	−0.995742	−0.497871	−	0.867251i	$-0.665885\pi$
$809$	−28.3218	−0.995742	−0.497871	+	0.867251i	$0.665885\pi$
$810$	−72.4585	−2.54593
$811$	2.29089	0.0804441	0.0402221	−	0.999191i	$-0.487193\pi$
$811$	2.29089	0.0804441	0.0402221	+	0.999191i	$0.487193\pi$
$812$	−27.3303	−0.959105
$813$	−5.51579	−0.193447
$814$	4.19690	0.147101
$815$	−15.8319	−0.554566
$816$	0.890241	0.0311647
$817$	−5.53686	−0.193710
$818$	55.1873	1.92958
$819$	3.73290	0.130438
$820$	−9.84929	−0.343952
$821$	−53.0608	−1.85184	−0.925918	−	0.377725i	$-0.876706\pi$
$821$	−53.0608	−1.85184	−0.925918	+	0.377725i	$0.876706\pi$
$822$	7.45029	0.259859
$823$	25.0667	0.873772	0.436886	−	0.899517i	$-0.356081\pi$
$823$	25.0667	0.873772	0.436886	+	0.899517i	$0.356081\pi$
$824$	−43.8826	−1.52872
$825$	−2.04959	−0.0713576
$826$	82.2662	2.86241
$827$	−23.7100	−0.824477	−0.412238	−	0.911076i	$-0.635253\pi$
$827$	−23.7100	−0.824477	−0.412238	+	0.911076i	$0.635253\pi$
$828$	−1.07284	−0.0372837
$829$	23.9376	0.831386	0.415693	−	0.909505i	$-0.363539\pi$
$829$	23.9376	0.831386	0.415693	+	0.909505i	$0.363539\pi$
$830$	46.9661	1.63022
$831$	3.22713	0.111948
$832$	−5.17261	−0.179328
$833$	−1.83916	−0.0637231
$834$	5.30184	0.183588
$835$	−81.1868	−2.80959
$836$	21.5398	0.744969
$837$	−1.05859	−0.0365901
$838$	58.4467	2.01901
$839$	44.6869	1.54276	0.771382	−	0.636373i	$-0.219566\pi$
$839$	44.6869	1.54276	0.771382	+	0.636373i	$0.219566\pi$
$840$	9.86749	0.340461
$841$	−19.4366	−0.670228
$842$	85.4925	2.94627
$843$	0.559464	0.0192690
$844$	−19.4018	−0.667838
$845$	−45.2441	−1.55644
$846$	−41.1036	−1.41317
$847$	−2.27175	−0.0780582
$848$	15.2301	0.523004
$849$	−5.73418	−0.196797
$850$	−18.7384	−0.642723
$851$	0.162787	0.00558026
$852$	13.4202	0.459767
$853$	33.8753	1.15987	0.579935	−	0.814663i	$-0.303078\pi$
$853$	33.8753	1.15987	0.579935	+	0.814663i	$0.303078\pi$
$854$	−18.0288	−0.616933
$855$	57.8522	1.97850
$856$	−8.95775	−0.306169
$857$	−44.2756	−1.51242	−0.756212	−	0.654327i	$-0.772952\pi$
$857$	−44.2756	−1.51242	−0.756212	+	0.654327i	$0.772952\pi$
$858$	0.361373	0.0123371
$859$	1.80274	0.0615086	0.0307543	−	0.999527i	$-0.490209\pi$
$859$	1.80274	0.0615086	0.0307543	+	0.999527i	$0.490209\pi$
$860$	13.8751	0.473137
$861$	0.428085	0.0145891
$862$	−30.1889	−1.02824
$863$	23.5042	0.800092	0.400046	−	0.916495i	$-0.368994\pi$
$863$	23.5042	0.800092	0.400046	+	0.916495i	$0.368994\pi$
$864$	−1.63101	−0.0554880
$865$	39.5822	1.34584
$866$	−52.2181	−1.77444
$867$	0.265461	0.00901553
$868$	−5.94351	−0.201736
$869$	5.52944	0.187573
$870$	−7.10611	−0.240920
$871$	−0.609810	−0.0206626
$872$	−77.9518	−2.63978
$873$	−17.6240	−0.596481
$874$	1.26499	0.0427890
$875$	−22.0458	−0.745284
$876$	−2.11933	−0.0716057
$877$	−6.32715	−0.213653	−0.106826	−	0.994278i	$-0.534069\pi$
$877$	−6.32715	−0.213653	−0.106826	+	0.994278i	$0.534069\pi$
$878$	−34.6161	−1.16824
$879$	1.26722	0.0427423
$880$	−11.9609	−0.403203
$881$	−2.60245	−0.0876786	−0.0438393	−	0.999039i	$-0.513959\pi$
$881$	−2.60245	−0.0876786	−0.0438393	+	0.999039i	$0.513959\pi$
$882$	−13.0763	−0.440302
$883$	30.2445	1.01781	0.508904	−	0.860824i	$-0.330051\pi$
$883$	30.2445	1.01781	0.508904	+	0.860824i	$0.330051\pi$
$884$	2.18206	0.0733905
$885$	14.1271	0.474878
$886$	−74.3158	−2.49669
$887$	−15.3076	−0.513979	−0.256989	−	0.966414i	$-0.582731\pi$
$887$	−15.3076	−0.513979	−0.256989	+	0.966414i	$0.582731\pi$
$888$	−2.10596	−0.0706713
$889$	33.2740	1.11597
$890$	−115.495	−3.87141
$891$	−8.37074	−0.280430
$892$	−18.3146	−0.613218
$893$	32.0095	1.07116
$894$	7.97707	0.266793
$895$	−32.9816	−1.10245
$896$	−46.1372	−1.54133
$897$	0.0140167	0.000468005 0
$898$	61.6209	2.05632
$899$	2.07975	0.0693635
$900$	−87.9918	−2.93306
$901$	4.54147	0.151298
$902$	−1.72280	−0.0573630
$903$	−0.603061	−0.0200686
$904$	21.9093	0.728694
$905$	−75.0694	−2.49539
$906$	−4.57414	−0.151966
$907$	−9.14766	−0.303743	−0.151872	−	0.988400i	$-0.548530\pi$
$907$	−9.14766	−0.303743	−0.151872	+	0.988400i	$0.548530\pi$
$908$	−6.37317	−0.211501
$909$	13.1461	0.436030
$910$	11.0300	0.365639
$911$	−44.7738	−1.48342	−0.741712	−	0.670719i	$-0.765986\pi$
$911$	−44.7738	−1.48342	−0.741712	+	0.670719i	$0.765986\pi$
$912$	−4.92914	−0.163220
$913$	5.42574	0.179566
$914$	−81.7160	−2.70293
$915$	−3.09599	−0.102350
$916$	−42.9191	−1.41809
$917$	23.3573	0.771327
$918$	3.82022	0.126086
$919$	6.41709	0.211680	0.105840	−	0.994383i	$-0.466247\pi$
$919$	6.41709	0.211680	0.105840	+	0.994383i	$0.466247\pi$
$920$	−1.54029	−0.0507820
$921$	−5.66651	−0.186718
$922$	−6.49451	−0.213885
$923$	7.28898	0.239920
$924$	2.34606	0.0771797
$925$	13.3514	0.438992
$926$	−50.7642	−1.66822
$927$	−28.0223	−0.920373
$928$	3.20436	0.105188
$929$	3.19854	0.104941	0.0524704	−	0.998622i	$-0.483290\pi$
$929$	3.19854	0.104941	0.0524704	+	0.998622i	$0.483290\pi$
$930$	−1.54537	−0.0506745
$931$	10.1832	0.333740
$932$	−58.4406	−1.91429
$933$	−6.84516	−0.224101
$934$	−86.5599	−2.83233
$935$	−3.56663	−0.116641
$936$	7.53830	0.246397
$937$	49.4621	1.61586	0.807928	−	0.589281i	$-0.200589\pi$
$937$	49.4621	1.61586	0.807928	+	0.589281i	$0.200589\pi$
$938$	−5.99423	−0.195719
$939$	2.96380	0.0967200
$940$	−80.2142	−2.61630
$941$	44.7320	1.45822	0.729111	−	0.684396i	$-0.239934\pi$
$941$	44.7320	1.45822	0.729111	+	0.684396i	$0.239934\pi$
$942$	13.5463	0.441362
$943$	−0.0668231	−0.00217606
$944$	50.0381	1.62860
$945$	12.7538	0.414881
$946$	2.42698	0.0789081
$947$	−41.5474	−1.35011	−0.675054	−	0.737768i	$-0.735880\pi$
$947$	−41.5474	−1.35011	−0.675054	+	0.737768i	$0.735880\pi$
$948$	−5.71031	−0.185462
$949$	−1.15109	−0.0373659
$950$	103.752	3.36616
$951$	1.74906	0.0567172
$952$	10.4219	0.337776
$953$	16.5458	0.535970	0.267985	−	0.963423i	$-0.413642\pi$
$953$	16.5458	0.535970	0.267985	+	0.963423i	$0.413642\pi$
$954$	32.2895	1.04541
$955$	44.3380	1.43474
$956$	−95.9199	−3.10227
$957$	−0.820931	−0.0265369
$958$	54.3400	1.75564
$959$	26.2703	0.848313
$960$	−8.73134	−0.281803
$961$	−30.5477	−0.985410
$962$	−2.35405	−0.0758978
$963$	−5.72018	−0.184330
$964$	−38.8701	−1.25192
$965$	−0.709609	−0.0228431
$966$	0.137780	0.00443300
$967$	53.0556	1.70615	0.853076	−	0.521786i	$-0.174734\pi$
$967$	53.0556	1.70615	0.853076	+	0.521786i	$0.174734\pi$
$968$	−4.58762	−0.147452
$969$	−1.46982	−0.0472174
$970$	−52.0752	−1.67204
$971$	2.80146	0.0899031	0.0449516	−	0.998989i	$-0.485687\pi$
$971$	2.80146	0.0899031	0.0449516	+	0.998989i	$0.485687\pi$
$972$	27.0150	0.866507
$973$	18.6947	0.599326
$974$	−20.5544	−0.658605
$975$	1.14962	0.0368174
$976$	−10.9660	−0.351012
$977$	−3.18958	−0.102044	−0.0510218	−	0.998698i	$-0.516248\pi$
$977$	−3.18958	−0.102044	−0.0510218	+	0.998698i	$0.516248\pi$
$978$	2.85984	0.0914475
$979$	−13.3425	−0.426430
$980$	−25.5185	−0.815160
$981$	−49.7779	−1.58929
$982$	14.4954	0.462568
$983$	9.06196	0.289032	0.144516	−	0.989502i	$-0.453838\pi$
$983$	9.06196	0.289032	0.144516	+	0.989502i	$0.453838\pi$
$984$	0.864483	0.0275587
$985$	1.69472	0.0539982
$986$	−7.50538	−0.239020
$987$	3.48639	0.110973
$988$	−12.0817	−0.384371
$989$	0.0941365	0.00299337
$990$	−25.3585	−0.805946
$991$	21.3956	0.679652	0.339826	−	0.940488i	$-0.389632\pi$
$991$	21.3956	0.679652	0.339826	+	0.940488i	$0.389632\pi$
$992$	0.696851	0.0221251
$993$	−1.43363	−0.0454949
$994$	71.6483	2.27255
$995$	−16.4515	−0.521546
$996$	−5.60322	−0.177545
$997$	−2.71816	−0.0860850	−0.0430425	−	0.999073i	$-0.513705\pi$
$997$	−2.71816	−0.0860850	−0.0430425	+	0.999073i	$0.513705\pi$
$998$	67.4532	2.13519
$999$	−2.72197	−0.0861192

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.g.1.9	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.a.g.1.9	✓	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:	\( N \)	\(=\)	\( 8041 = 11 \cdot 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8041.a (trivial)

\(f(q)\)	\(=\)	\(q-2.42698 q^{2} +0.265461 q^{3} +3.89025 q^{4} +3.56663 q^{5} -0.644270 q^{6} -2.27175 q^{7} -4.58762 q^{8} -2.92953 q^{9} +O(q^{10})\)	\(q-2.42698 q^{2} +0.265461 q^{3} +3.89025 q^{4} +3.56663 q^{5} -0.644270 q^{6} -2.27175 q^{7} -4.58762 q^{8} -2.92953 q^{9} -8.65616 q^{10} -1.00000 q^{11} +1.03271 q^{12} +0.560903 q^{13} +5.51350 q^{14} +0.946802 q^{15} +3.35357 q^{16} +1.00000 q^{17} +7.10992 q^{18} -5.53686 q^{19} +13.8751 q^{20} -0.603061 q^{21} +2.42698 q^{22} +0.0941365 q^{23} -1.21783 q^{24} +7.72087 q^{25} -1.36130 q^{26} -1.57406 q^{27} -8.83768 q^{28} +3.09247 q^{29} -2.29787 q^{30} +0.672520 q^{31} +1.03618 q^{32} -0.265461 q^{33} -2.42698 q^{34} -8.10249 q^{35} -11.3966 q^{36} +1.72926 q^{37} +13.4379 q^{38} +0.148898 q^{39} -16.3623 q^{40} -0.709853 q^{41} +1.46362 q^{42} +1.00000 q^{43} -3.89025 q^{44} -10.4486 q^{45} -0.228468 q^{46} -5.78116 q^{47} +0.890241 q^{48} -1.83916 q^{49} -18.7384 q^{50} +0.265461 q^{51} +2.18206 q^{52} +4.54147 q^{53} +3.82022 q^{54} -3.56663 q^{55} +10.4219 q^{56} -1.46982 q^{57} -7.50538 q^{58} +14.9209 q^{59} +3.68330 q^{60} -3.26994 q^{61} -1.63220 q^{62} +6.65516 q^{63} -9.22192 q^{64} +2.00054 q^{65} +0.644270 q^{66} -1.08719 q^{67} +3.89025 q^{68} +0.0249896 q^{69} +19.6646 q^{70} +12.9951 q^{71} +13.4396 q^{72} -2.05221 q^{73} -4.19690 q^{74} +2.04959 q^{75} -21.5398 q^{76} +2.27175 q^{77} -0.361373 q^{78} -5.52944 q^{79} +11.9609 q^{80} +8.37074 q^{81} +1.72280 q^{82} -5.42574 q^{83} -2.34606 q^{84} +3.56663 q^{85} -2.42698 q^{86} +0.820931 q^{87} +4.58762 q^{88} +13.3425 q^{89} +25.3585 q^{90} -1.27423 q^{91} +0.366215 q^{92} +0.178528 q^{93} +14.0308 q^{94} -19.7479 q^{95} +0.275065 q^{96} +6.01597 q^{97} +4.46361 q^{98} +2.92953 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q - 11 q^{2} - 3 q^{3} + 65 q^{4} - 6 q^{5} - 10 q^{6} - 11 q^{7} - 33 q^{8} + 56 q^{9}+O(q^{10}) \) 69 * q - 11 * q^2 - 3 * q^3 + 65 * q^4 - 6 * q^5 - 10 * q^6 - 11 * q^7 - 33 * q^8 + 56 * q^9	\( 69 q - 11 q^{2} - 3 q^{3} + 65 q^{4} - 6 q^{5} - 10 q^{6} - 11 q^{7} - 33 q^{8} + 56 q^{9} - q^{10} - 69 q^{11} - 3 q^{12} - 28 q^{13} - 15 q^{14} - 45 q^{15} + 53 q^{16} + 69 q^{17} - 17 q^{18} - 32 q^{19} - 21 q^{20} - 38 q^{21} + 11 q^{22} - 41 q^{23} - 11 q^{24} + 67 q^{25} - 6 q^{26} - 3 q^{27} - 21 q^{28} - 22 q^{29} - 22 q^{30} - 27 q^{31} - 87 q^{32} + 3 q^{33} - 11 q^{34} - 44 q^{35} + 59 q^{36} + 24 q^{37} - 22 q^{38} - 59 q^{39} + q^{40} - 43 q^{41} - 15 q^{42} + 69 q^{43} - 65 q^{44} - 12 q^{45} - 21 q^{46} - 99 q^{47} + 2 q^{48} + 64 q^{49} - 78 q^{50} - 3 q^{51} - 57 q^{52} - 50 q^{53} + 20 q^{54} + 6 q^{55} - 59 q^{56} - 15 q^{57} + 22 q^{58} - 82 q^{59} - 86 q^{60} - 24 q^{61} + 15 q^{62} - 63 q^{63} + 63 q^{64} - 23 q^{65} + 10 q^{66} - 54 q^{67} + 65 q^{68} + 36 q^{69} + 9 q^{70} - 128 q^{71} - 69 q^{72} + 2 q^{73} - 58 q^{74} - 31 q^{75} - 76 q^{76} + 11 q^{77} - 19 q^{78} - 43 q^{79} - 19 q^{80} + 49 q^{81} - 2 q^{82} - 62 q^{83} - 82 q^{84} - 6 q^{85} - 11 q^{86} - 62 q^{87} + 33 q^{88} - 49 q^{89} - 37 q^{90} - 2 q^{91} - 96 q^{92} - 29 q^{93} - 75 q^{94} - 133 q^{95} - 86 q^{96} + 5 q^{97} - 72 q^{98} - 56 q^{99}+O(q^{100}) \) 69 * q - 11 * q^2 - 3 * q^3 + 65 * q^4 - 6 * q^5 - 10 * q^6 - 11 * q^7 - 33 * q^8 + 56 * q^9 - q^10 - 69 * q^11 - 3 * q^12 - 28 * q^13 - 15 * q^14 - 45 * q^15 + 53 * q^16 + 69 * q^17 - 17 * q^18 - 32 * q^19 - 21 * q^20 - 38 * q^21 + 11 * q^22 - 41 * q^23 - 11 * q^24 + 67 * q^25 - 6 * q^26 - 3 * q^27 - 21 * q^28 - 22 * q^29 - 22 * q^30 - 27 * q^31 - 87 * q^32 + 3 * q^33 - 11 * q^34 - 44 * q^35 + 59 * q^36 + 24 * q^37 - 22 * q^38 - 59 * q^39 + q^40 - 43 * q^41 - 15 * q^42 + 69 * q^43 - 65 * q^44 - 12 * q^45 - 21 * q^46 - 99 * q^47 + 2 * q^48 + 64 * q^49 - 78 * q^50 - 3 * q^51 - 57 * q^52 - 50 * q^53 + 20 * q^54 + 6 * q^55 - 59 * q^56 - 15 * q^57 + 22 * q^58 - 82 * q^59 - 86 * q^60 - 24 * q^61 + 15 * q^62 - 63 * q^63 + 63 * q^64 - 23 * q^65 + 10 * q^66 - 54 * q^67 + 65 * q^68 + 36 * q^69 + 9 * q^70 - 128 * q^71 - 69 * q^72 + 2 * q^73 - 58 * q^74 - 31 * q^75 - 76 * q^76 + 11 * q^77 - 19 * q^78 - 43 * q^79 - 19 * q^80 + 49 * q^81 - 2 * q^82 - 62 * q^83 - 82 * q^84 - 6 * q^85 - 11 * q^86 - 62 * q^87 + 33 * q^88 - 49 * q^89 - 37 * q^90 - 2 * q^91 - 96 * q^92 - 29 * q^93 - 75 * q^94 - 133 * q^95 - 86 * q^96 + 5 * q^97 - 72 * q^98 - 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more