LMFDB - Embedded newform 8041.2.a.i.1.19

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:	$0$
Dimension:	$78$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q-1.53655 q^{2} +0.0485562 q^{3} +0.360996 q^{4} -0.584069 q^{5} -0.0746091 q^{6} -1.23435 q^{7} +2.51842 q^{8} -2.99764 q^{9} +O(q^{10})$	\(q-1.53655 q^{2} +0.0485562 q^{3} +0.360996 q^{4} -0.584069 q^{5} -0.0746091 q^{6} -1.23435 q^{7} +2.51842 q^{8} -2.99764 q^{9} +0.897453 q^{10} +1.00000 q^{11} +0.0175286 q^{12} -0.643118 q^{13} +1.89665 q^{14} -0.0283601 q^{15} -4.59167 q^{16} -1.00000 q^{17} +4.60604 q^{18} -5.42219 q^{19} -0.210846 q^{20} -0.0599355 q^{21} -1.53655 q^{22} -9.24603 q^{23} +0.122285 q^{24} -4.65886 q^{25} +0.988185 q^{26} -0.291222 q^{27} -0.445597 q^{28} -8.11165 q^{29} +0.0435769 q^{30} -6.20028 q^{31} +2.01852 q^{32} +0.0485562 q^{33} +1.53655 q^{34} +0.720948 q^{35} -1.08214 q^{36} +5.04229 q^{37} +8.33148 q^{38} -0.0312273 q^{39} -1.47093 q^{40} -4.44113 q^{41} +0.0920941 q^{42} -1.00000 q^{43} +0.360996 q^{44} +1.75083 q^{45} +14.2070 q^{46} -7.15793 q^{47} -0.222954 q^{48} -5.47637 q^{49} +7.15859 q^{50} -0.0485562 q^{51} -0.232163 q^{52} +9.14969 q^{53} +0.447479 q^{54} -0.584069 q^{55} -3.10862 q^{56} -0.263281 q^{57} +12.4640 q^{58} +0.525978 q^{59} -0.0102379 q^{60} -11.6655 q^{61} +9.52706 q^{62} +3.70015 q^{63} +6.08179 q^{64} +0.375625 q^{65} -0.0746091 q^{66} -0.835380 q^{67} -0.360996 q^{68} -0.448952 q^{69} -1.10777 q^{70} +0.263645 q^{71} -7.54931 q^{72} +13.9133 q^{73} -7.74775 q^{74} -0.226216 q^{75} -1.95739 q^{76} -1.23435 q^{77} +0.0479825 q^{78} +0.175797 q^{79} +2.68185 q^{80} +8.97879 q^{81} +6.82404 q^{82} +7.40067 q^{83} -0.0216365 q^{84} +0.584069 q^{85} +1.53655 q^{86} -0.393870 q^{87} +2.51842 q^{88} -10.3951 q^{89} -2.69024 q^{90} +0.793836 q^{91} -3.33778 q^{92} -0.301062 q^{93} +10.9985 q^{94} +3.16693 q^{95} +0.0980114 q^{96} -1.48711 q^{97} +8.41473 q^{98} -2.99764 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 78 q + 7 q^{2} + 10 q^{3} + 91 q^{4} + 17 q^{5} + 12 q^{6} + 11 q^{7} + 33 q^{8} + 102 q^{9}+O(q^{10}) $ 78 * q + 7 * q^2 + 10 * q^3 + 91 * q^4 + 17 * q^5 + 12 * q^6 + 11 * q^7 + 33 * q^8 + 102 * q^9	\( 78 q + 7 q^{2} + 10 q^{3} + 91 q^{4} + 17 q^{5} + 12 q^{6} + 11 q^{7} + 33 q^{8} + 102 q^{9} + 3 q^{10} + 78 q^{11} + 31 q^{12} - 16 q^{13} + 31 q^{14} + 38 q^{15} + 121 q^{16} - 78 q^{17} + 11 q^{18} + 51 q^{20} + 6 q^{21} + 7 q^{22} + 48 q^{23} + 11 q^{24} + 101 q^{25} + 18 q^{26} + 46 q^{27} + 27 q^{28} + 22 q^{29} + 14 q^{30} + 56 q^{31} + 83 q^{32} + 10 q^{33} - 7 q^{34} + 24 q^{35} + 139 q^{36} + 53 q^{37} + 10 q^{38} + 79 q^{39} - q^{40} + 23 q^{41} + 17 q^{42} - 78 q^{43} + 91 q^{44} + 76 q^{45} + 21 q^{46} + 57 q^{47} + 78 q^{48} + 115 q^{49} + 58 q^{50} - 10 q^{51} - 63 q^{52} + 22 q^{53} - 18 q^{54} + 17 q^{55} + 111 q^{56} - 11 q^{57} + 36 q^{58} + 71 q^{59} + 36 q^{60} + 4 q^{61} - 5 q^{62} + 71 q^{63} + 183 q^{64} + 47 q^{65} + 12 q^{66} + 11 q^{67} - 91 q^{68} + 31 q^{69} + 33 q^{70} + 159 q^{71} + 59 q^{72} + 2 q^{73} - 4 q^{74} + 83 q^{75} - 44 q^{76} + 11 q^{77} + 101 q^{78} + 35 q^{79} + 85 q^{80} + 170 q^{81} + 98 q^{82} - 32 q^{83} + 44 q^{84} - 17 q^{85} - 7 q^{86} - 6 q^{87} + 33 q^{88} + 50 q^{89} - 5 q^{90} + 86 q^{91} + 106 q^{92} + 68 q^{93} - q^{94} + 109 q^{95} - 50 q^{96} + 40 q^{97} + 106 q^{98} + 102 q^{99}+O(q^{100}) \) 78 * q + 7 * q^2 + 10 * q^3 + 91 * q^4 + 17 * q^5 + 12 * q^6 + 11 * q^7 + 33 * q^8 + 102 * q^9 + 3 * q^10 + 78 * q^11 + 31 * q^12 - 16 * q^13 + 31 * q^14 + 38 * q^15 + 121 * q^16 - 78 * q^17 + 11 * q^18 + 51 * q^20 + 6 * q^21 + 7 * q^22 + 48 * q^23 + 11 * q^24 + 101 * q^25 + 18 * q^26 + 46 * q^27 + 27 * q^28 + 22 * q^29 + 14 * q^30 + 56 * q^31 + 83 * q^32 + 10 * q^33 - 7 * q^34 + 24 * q^35 + 139 * q^36 + 53 * q^37 + 10 * q^38 + 79 * q^39 - q^40 + 23 * q^41 + 17 * q^42 - 78 * q^43 + 91 * q^44 + 76 * q^45 + 21 * q^46 + 57 * q^47 + 78 * q^48 + 115 * q^49 + 58 * q^50 - 10 * q^51 - 63 * q^52 + 22 * q^53 - 18 * q^54 + 17 * q^55 + 111 * q^56 - 11 * q^57 + 36 * q^58 + 71 * q^59 + 36 * q^60 + 4 * q^61 - 5 * q^62 + 71 * q^63 + 183 * q^64 + 47 * q^65 + 12 * q^66 + 11 * q^67 - 91 * q^68 + 31 * q^69 + 33 * q^70 + 159 * q^71 + 59 * q^72 + 2 * q^73 - 4 * q^74 + 83 * q^75 - 44 * q^76 + 11 * q^77 + 101 * q^78 + 35 * q^79 + 85 * q^80 + 170 * q^81 + 98 * q^82 - 32 * q^83 + 44 * q^84 - 17 * q^85 - 7 * q^86 - 6 * q^87 + 33 * q^88 + 50 * q^89 - 5 * q^90 + 86 * q^91 + 106 * q^92 + 68 * q^93 - q^94 + 109 * q^95 - 50 * q^96 + 40 * q^97 + 106 * q^98 + 102 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.53655	−1.08651	−0.543254	−	0.839569i	$-0.682808\pi$
$2$	−1.53655	−1.08651	−0.543254	+	0.839569i	$0.682808\pi$
$3$	0.0485562	0.0280339	0.0140170	−	0.999902i	$-0.495538\pi$
$3$	0.0485562	0.0280339	0.0140170	+	0.999902i	$0.495538\pi$
$4$	0.360996	0.180498
$5$	−0.584069	−0.261203	−0.130602	−	0.991435i	$-0.541691\pi$
$5$	−0.584069	−0.261203	−0.130602	+	0.991435i	$0.541691\pi$
$6$	−0.0746091	−0.0304590
$7$	−1.23435	−0.466542	−0.233271	−	0.972412i	$-0.574943\pi$
$7$	−1.23435	−0.466542	−0.233271	+	0.972412i	$0.574943\pi$
$8$	2.51842	0.890395
$9$	−2.99764	−0.999214
$10$	0.897453	0.283799
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0.0175286	0.00506006
$13$	−0.643118	−0.178369	−0.0891845	−	0.996015i	$-0.528426\pi$
$13$	−0.643118	−0.178369	−0.0891845	+	0.996015i	$0.528426\pi$
$14$	1.89665	0.506901
$15$	−0.0283601	−0.00732255
$16$	−4.59167	−1.14792
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	4.60604	1.08565
$19$	−5.42219	−1.24393	−0.621967	−	0.783043i	$-0.713666\pi$
$19$	−5.42219	−1.24393	−0.621967	+	0.783043i	$0.713666\pi$
$20$	−0.210846	−0.0471467
$21$	−0.0599355	−0.0130790
$22$	−1.53655	−0.327594
$23$	−9.24603	−1.92793	−0.963965	−	0.266028i	$-0.914289\pi$
$23$	−9.24603	−1.92793	−0.963965	+	0.266028i	$0.914289\pi$
$24$	0.122285	0.0249613
$25$	−4.65886	−0.931773
$26$	0.988185	0.193799
$27$	−0.291222	−0.0560458
$28$	−0.445597	−0.0842099
$29$	−8.11165	−1.50629	−0.753147	−	0.657852i	$-0.771465\pi$
$29$	−8.11165	−1.50629	−0.753147	+	0.657852i	$0.771465\pi$
$30$	0.0435769	0.00795601
$31$	−6.20028	−1.11360	−0.556802	−	0.830645i	$-0.687972\pi$
$31$	−6.20028	−1.11360	−0.556802	+	0.830645i	$0.687972\pi$
$32$	2.01852	0.356827
$33$	0.0485562	0.00845254
$34$	1.53655	0.263517
$35$	0.720948	0.121862
$36$	−1.08214	−0.180356
$37$	5.04229	0.828948	0.414474	−	0.910061i	$-0.363966\pi$
$37$	5.04229	0.828948	0.414474	+	0.910061i	$0.363966\pi$
$38$	8.33148	1.35154
$39$	−0.0312273	−0.00500038
$40$	−1.47093	−0.232574
$41$	−4.44113	−0.693588	−0.346794	−	0.937941i	$-0.612730\pi$
$41$	−4.44113	−0.693588	−0.346794	+	0.937941i	$0.612730\pi$
$42$	0.0920941	0.0142104
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0.360996	0.0544221
$45$	1.75083	0.260998
$46$	14.2070	2.09471
$47$	−7.15793	−1.04409	−0.522046	−	0.852917i	$-0.674831\pi$
$47$	−7.15793	−1.04409	−0.522046	+	0.852917i	$0.674831\pi$
$48$	−0.222954	−0.0321806
$49$	−5.47637	−0.782338
$50$	7.15859	1.01238
$51$	−0.0485562	−0.00679922
$52$	−0.232163	−0.0321952
$53$	9.14969	1.25681	0.628403	−	0.777888i	$-0.283709\pi$
$53$	9.14969	1.25681	0.628403	+	0.777888i	$0.283709\pi$
$54$	0.447479	0.0608941
$55$	−0.584069	−0.0787558
$56$	−3.10862	−0.415407
$57$	−0.263281	−0.0348724
$58$	12.4640	1.63660
$59$	0.525978	0.0684766	0.0342383	−	0.999414i	$-0.489099\pi$
$59$	0.525978	0.0684766	0.0342383	+	0.999414i	$0.489099\pi$
$60$	−0.0102379	−0.00132171
$61$	−11.6655	−1.49361	−0.746806	−	0.665042i	$-0.768414\pi$
$61$	−11.6655	−1.49361	−0.746806	+	0.665042i	$0.768414\pi$
$62$	9.52706	1.20994
$63$	3.70015	0.466176
$64$	6.08179	0.760224
$65$	0.375625	0.0465906
$66$	−0.0746091	−0.00918375
$67$	−0.835380	−0.102058	−0.0510290	−	0.998697i	$-0.516250\pi$
$67$	−0.835380	−0.102058	−0.0510290	+	0.998697i	$0.516250\pi$
$68$	−0.360996	−0.0437772
$69$	−0.448952	−0.0540474
$70$	−1.10777	−0.132404
$71$	0.263645	0.0312889	0.0156445	−	0.999878i	$-0.495020\pi$
$71$	0.263645	0.0312889	0.0156445	+	0.999878i	$0.495020\pi$
$72$	−7.54931	−0.889695
$73$	13.9133	1.62843	0.814215	−	0.580563i	$-0.197168\pi$
$73$	13.9133	1.62843	0.814215	+	0.580563i	$0.197168\pi$
$74$	−7.74775	−0.900658
$75$	−0.226216	−0.0261212
$76$	−1.95739	−0.224528
$77$	−1.23435	−0.140668
$78$	0.0479825	0.00543295
$79$	0.175797	0.0197787	0.00988935	−	0.999951i	$-0.496852\pi$
$79$	0.175797	0.0197787	0.00988935	+	0.999951i	$0.496852\pi$
$80$	2.68185	0.299840
$81$	8.97879	0.997643
$82$	6.82404	0.753589
$83$	7.40067	0.812329	0.406165	−	0.913800i	$-0.366866\pi$
$83$	7.40067	0.812329	0.406165	+	0.913800i	$0.366866\pi$
$84$	−0.0216365	−0.00236073
$85$	0.584069	0.0633512
$86$	1.53655	0.165691
$87$	−0.393870	−0.0422273
$88$	2.51842	0.268464
$89$	−10.3951	−1.10188	−0.550940	−	0.834545i	$-0.685731\pi$
$89$	−10.3951	−1.10188	−0.550940	+	0.834545i	$0.685731\pi$
$90$	−2.69024	−0.283576
$91$	0.793836	0.0832166
$92$	−3.33778	−0.347987
$93$	−0.301062	−0.0312187
$94$	10.9985	1.13441
$95$	3.16693	0.324920
$96$	0.0980114	0.0100032
$97$	−1.48711	−0.150993	−0.0754967	−	0.997146i	$-0.524054\pi$
$97$	−1.48711	−0.150993	−0.0754967	+	0.997146i	$0.524054\pi$
$98$	8.41473	0.850016
$99$	−2.99764	−0.301274
$100$	−1.68183	−0.168183
$101$	−6.32102	−0.628965	−0.314482	−	0.949263i	$-0.601831\pi$
$101$	−6.32102	−0.628965	−0.314482	+	0.949263i	$0.601831\pi$
$102$	0.0746091	0.00738740
$103$	−4.78599	−0.471578	−0.235789	−	0.971804i	$-0.575767\pi$
$103$	−4.78599	−0.471578	−0.235789	+	0.971804i	$0.575767\pi$
$104$	−1.61964	−0.158819
$105$	0.0350065	0.00341628
$106$	−14.0590	−1.36553
$107$	9.13716	0.883323	0.441661	−	0.897182i	$-0.354389\pi$
$107$	9.13716	0.883323	0.441661	+	0.897182i	$0.354389\pi$
$108$	−0.105130	−0.0101161
$109$	3.94113	0.377492	0.188746	−	0.982026i	$-0.439558\pi$
$109$	3.94113	0.377492	0.188746	+	0.982026i	$0.439558\pi$
$110$	0.897453	0.0855688
$111$	0.244834	0.0232386
$112$	5.66775	0.535552
$113$	−8.68162	−0.816698	−0.408349	−	0.912826i	$-0.633895\pi$
$113$	−8.68162	−0.816698	−0.408349	+	0.912826i	$0.633895\pi$
$114$	0.404545	0.0378891
$115$	5.40032	0.503582
$116$	−2.92827	−0.271883
$117$	1.92784	0.178229
$118$	−0.808194	−0.0744003
$119$	1.23435	0.113153
$120$	−0.0714226	−0.00651997
$121$	1.00000	0.0909091
$122$	17.9246	1.62282
$123$	−0.215644	−0.0194440
$124$	−2.23827	−0.201003
$125$	5.64144	0.504586
$126$	−5.68548	−0.506503
$127$	13.6417	1.21050	0.605251	−	0.796035i	$-0.293073\pi$
$127$	13.6417	1.21050	0.605251	+	0.796035i	$0.293073\pi$
$128$	−13.3820	−1.18282
$129$	−0.0485562	−0.00427513
$130$	−0.577168	−0.0506210
$131$	−4.29946	−0.375646	−0.187823	−	0.982203i	$-0.560143\pi$
$131$	−4.29946	−0.375646	−0.187823	+	0.982203i	$0.560143\pi$
$132$	0.0175286	0.00152567
$133$	6.69290	0.580348
$134$	1.28361	0.110887
$135$	0.170094	0.0146394
$136$	−2.51842	−0.215953
$137$	−18.4732	−1.57827	−0.789136	−	0.614219i	$-0.789471\pi$
$137$	−18.4732	−1.57827	−0.789136	+	0.614219i	$0.789471\pi$
$138$	0.689838	0.0587229
$139$	−21.7475	−1.84460	−0.922299	−	0.386477i	$-0.873692\pi$
$139$	−21.7475	−1.84460	−0.922299	+	0.386477i	$0.873692\pi$
$140$	0.260259	0.0219959
$141$	−0.347562	−0.0292700
$142$	−0.405105	−0.0339957
$143$	−0.643118	−0.0537803
$144$	13.7642	1.14702
$145$	4.73776	0.393449
$146$	−21.3785	−1.76930
$147$	−0.265911	−0.0219320
$148$	1.82025	0.149623
$149$	−18.4626	−1.51252	−0.756259	−	0.654272i	$-0.772975\pi$
$149$	−18.4626	−1.51252	−0.756259	+	0.654272i	$0.772975\pi$
$150$	0.347594	0.0283809
$151$	1.00422	0.0817220	0.0408610	−	0.999165i	$-0.486990\pi$
$151$	1.00422	0.0817220	0.0408610	+	0.999165i	$0.486990\pi$
$152$	−13.6553	−1.10759
$153$	2.99764	0.242345
$154$	1.89665	0.152837
$155$	3.62139	0.290877
$156$	−0.0112729	−0.000902557 0
$157$	−0.666392	−0.0531839	−0.0265920	−	0.999646i	$-0.508465\pi$
$157$	−0.666392	−0.0531839	−0.0265920	+	0.999646i	$0.508465\pi$
$158$	−0.270121	−0.0214897
$159$	0.444274	0.0352332
$160$	−1.17895	−0.0932043
$161$	11.4129	0.899461
$162$	−13.7964	−1.08395
$163$	−15.0794	−1.18111	−0.590555	−	0.806997i	$-0.701091\pi$
$163$	−15.0794	−1.18111	−0.590555	+	0.806997i	$0.701091\pi$
$164$	−1.60323	−0.125191
$165$	−0.0283601	−0.00220783
$166$	−11.3715	−0.882601
$167$	−0.955468	−0.0739363	−0.0369682	−	0.999316i	$-0.511770\pi$
$167$	−0.955468	−0.0739363	−0.0369682	+	0.999316i	$0.511770\pi$
$168$	−0.150943	−0.0116455
$169$	−12.5864	−0.968185
$170$	−0.897453	−0.0688315
$171$	16.2538	1.24296
$172$	−0.360996	−0.0275257
$173$	−14.4563	−1.09909	−0.549544	−	0.835465i	$-0.685199\pi$
$173$	−14.4563	−1.09909	−0.549544	+	0.835465i	$0.685199\pi$
$174$	0.605203	0.0458803
$175$	5.75069	0.434711
$176$	−4.59167	−0.346110
$177$	0.0255395	0.00191967
$178$	15.9727	1.19720
$179$	6.92508	0.517605	0.258802	−	0.965930i	$-0.416672\pi$
$179$	6.92508	0.517605	0.258802	+	0.965930i	$0.416672\pi$
$180$	0.632042	0.0471096
$181$	11.4946	0.854390	0.427195	−	0.904159i	$-0.359502\pi$
$181$	11.4946	0.854390	0.427195	+	0.904159i	$0.359502\pi$
$182$	−1.21977	−0.0904155
$183$	−0.566431	−0.0418718
$184$	−23.2854	−1.71662
$185$	−2.94505	−0.216524
$186$	0.462597	0.0339193
$187$	−1.00000	−0.0731272
$188$	−2.58398	−0.188456
$189$	0.359472	0.0261477
$190$	−4.86616	−0.353028
$191$	−0.906511	−0.0655928	−0.0327964	−	0.999462i	$-0.510441\pi$
$191$	−0.906511	−0.0655928	−0.0327964	+	0.999462i	$0.510441\pi$
$192$	0.295308	0.0213120
$193$	−8.49348	−0.611374	−0.305687	−	0.952132i	$-0.598886\pi$
$193$	−8.49348	−0.611374	−0.305687	+	0.952132i	$0.598886\pi$
$194$	2.28503	0.164055
$195$	0.0182389	0.00130612
$196$	−1.97695	−0.141210
$197$	17.9019	1.27545	0.637727	−	0.770262i	$-0.279875\pi$
$197$	17.9019	1.27545	0.637727	+	0.770262i	$0.279875\pi$
$198$	4.60604	0.327337
$199$	−26.7661	−1.89740	−0.948698	−	0.316184i	$-0.897598\pi$
$199$	−26.7661	−1.89740	−0.948698	+	0.316184i	$0.897598\pi$
$200$	−11.7330	−0.829646
$201$	−0.0405628	−0.00286108
$202$	9.71258	0.683375
$203$	10.0126	0.702750
$204$	−0.0175286	−0.00122724
$205$	2.59393	0.181168
$206$	7.35393	0.512372
$207$	27.7163	1.92642
$208$	2.95299	0.204753
$209$	−5.42219	−0.375060
$210$	−0.0537893	−0.00371181
$211$	23.1076	1.59079	0.795395	−	0.606091i	$-0.207263\pi$
$211$	23.1076	1.59079	0.795395	+	0.606091i	$0.207263\pi$
$212$	3.30300	0.226851
$213$	0.0128016	0.000877152 0
$214$	−14.0397	−0.959736
$215$	0.584069	0.0398332
$216$	−0.733420	−0.0499029
$217$	7.65335	0.519543
$218$	−6.05576	−0.410148
$219$	0.675577	0.0456513
$220$	−0.210846	−0.0142153
$221$	0.643118	0.0432608
$222$	−0.376201	−0.0252490
$223$	24.8419	1.66353	0.831767	−	0.555125i	$-0.187329\pi$
$223$	24.8419	1.66353	0.831767	+	0.555125i	$0.187329\pi$
$224$	−2.49156	−0.166475
$225$	13.9656	0.931040
$226$	13.3398	0.887349
$227$	−10.5292	−0.698850	−0.349425	−	0.936964i	$-0.613623\pi$
$227$	−10.5292	−0.698850	−0.349425	+	0.936964i	$0.613623\pi$
$228$	−0.0950431	−0.00629438
$229$	−18.4123	−1.21672	−0.608359	−	0.793662i	$-0.708172\pi$
$229$	−18.4123	−1.21672	−0.608359	+	0.793662i	$0.708172\pi$
$230$	−8.29787	−0.547146
$231$	−0.0599355	−0.00394347
$232$	−20.4285	−1.34120
$233$	0.107721	0.00705702	0.00352851	−	0.999994i	$-0.498877\pi$
$233$	0.107721	0.00705702	0.00352851	+	0.999994i	$0.498877\pi$
$234$	−2.96223	−0.193647
$235$	4.18072	0.272720
$236$	0.189876	0.0123599
$237$	0.00853602	0.000554474 0
$238$	−1.89665	−0.122942
$239$	7.84106	0.507196	0.253598	−	0.967310i	$-0.418386\pi$
$239$	7.84106	0.507196	0.253598	+	0.967310i	$0.418386\pi$
$240$	0.130220	0.00840570
$241$	−7.45893	−0.480472	−0.240236	−	0.970715i	$-0.577225\pi$
$241$	−7.45893	−0.480472	−0.240236	+	0.970715i	$0.577225\pi$
$242$	−1.53655	−0.0987734
$243$	1.30964	0.0840136
$244$	−4.21119	−0.269594
$245$	3.19858	0.204350
$246$	0.331349	0.0211260
$247$	3.48711	0.221879
$248$	−15.6149	−0.991547
$249$	0.359348	0.0227728
$250$	−8.66837	−0.548236
$251$	24.0496	1.51800	0.758998	−	0.651093i	$-0.225689\pi$
$251$	24.0496	1.51800	0.758998	+	0.651093i	$0.225689\pi$
$252$	1.33574	0.0841437
$253$	−9.24603	−0.581293
$254$	−20.9611	−1.31522
$255$	0.0283601	0.00177598
$256$	8.39861	0.524913
$257$	−21.5641	−1.34513	−0.672565	−	0.740038i	$-0.734807\pi$
$257$	−21.5641	−1.34513	−0.672565	+	0.740038i	$0.734807\pi$
$258$	0.0746091	0.00464496
$259$	−6.22398	−0.386739
$260$	0.135599	0.00840950
$261$	24.3158	1.50511
$262$	6.60635	0.408142
$263$	−17.4113	−1.07363	−0.536813	−	0.843702i	$-0.680372\pi$
$263$	−17.4113	−1.07363	−0.536813	+	0.843702i	$0.680372\pi$
$264$	0.122285	0.00752610
$265$	−5.34405	−0.328282
$266$	−10.2840	−0.630552
$267$	−0.504747	−0.0308900
$268$	−0.301569	−0.0184212
$269$	3.90551	0.238123	0.119061	−	0.992887i	$-0.462011\pi$
$269$	3.90551	0.238123	0.119061	+	0.992887i	$0.462011\pi$
$270$	−0.261358	−0.0159058
$271$	−8.12883	−0.493791	−0.246895	−	0.969042i	$-0.579410\pi$
$271$	−8.12883	−0.493791	−0.246895	+	0.969042i	$0.579410\pi$
$272$	4.59167	0.278411
$273$	0.0385456	0.00233289
$274$	28.3851	1.71480
$275$	−4.65886	−0.280940
$276$	−0.162070	−0.00975544
$277$	21.4082	1.28629	0.643146	−	0.765744i	$-0.277629\pi$
$277$	21.4082	1.28629	0.643146	+	0.765744i	$0.277629\pi$
$278$	33.4162	2.00417
$279$	18.5862	1.11273
$280$	1.81565	0.108506
$281$	14.5081	0.865479	0.432739	−	0.901519i	$-0.357547\pi$
$281$	14.5081	0.865479	0.432739	+	0.901519i	$0.357547\pi$
$282$	0.534047	0.0318020
$283$	23.1042	1.37340	0.686702	−	0.726939i	$-0.259058\pi$
$283$	23.1042	1.37340	0.686702	+	0.726939i	$0.259058\pi$
$284$	0.0951748	0.00564759
$285$	0.153774	0.00910878
$286$	0.988185	0.0584326
$287$	5.48193	0.323588
$288$	−6.05079	−0.356546
$289$	1.00000	0.0588235
$290$	−7.27982	−0.427486
$291$	−0.0722085	−0.00423294
$292$	5.02265	0.293928
$293$	−11.1265	−0.650019	−0.325010	−	0.945711i	$-0.605368\pi$
$293$	−11.1265	−0.650019	−0.325010	+	0.945711i	$0.605368\pi$
$294$	0.408587	0.0238293
$295$	−0.307208	−0.0178863
$296$	12.6986	0.738091
$297$	−0.291222	−0.0168984
$298$	28.3688	1.64336
$299$	5.94629	0.343883
$300$	−0.0816632	−0.00471483
$301$	1.23435	0.0711470
$302$	−1.54303	−0.0887916
$303$	−0.306924	−0.0176323
$304$	24.8969	1.42794
$305$	6.81345	0.390137
$306$	−4.60604	−0.263310
$307$	−20.2711	−1.15693	−0.578466	−	0.815706i	$-0.696348\pi$
$307$	−20.2711	−1.15693	−0.578466	+	0.815706i	$0.696348\pi$
$308$	−0.445597	−0.0253902
$309$	−0.232389	−0.0132202
$310$	−5.56446	−0.316040
$311$	−9.01884	−0.511411	−0.255706	−	0.966755i	$-0.582308\pi$
$311$	−9.01884	−0.511411	−0.255706	+	0.966755i	$0.582308\pi$
$312$	−0.0786435	−0.00445231
$313$	10.7103	0.605381	0.302690	−	0.953089i	$-0.402115\pi$
$313$	10.7103	0.605381	0.302690	+	0.953089i	$0.402115\pi$
$314$	1.02395	0.0577847
$315$	−2.16114	−0.121767
$316$	0.0634619	0.00357001
$317$	24.1675	1.35738	0.678692	−	0.734423i	$-0.262547\pi$
$317$	24.1675	1.35738	0.678692	+	0.734423i	$0.262547\pi$
$318$	−0.682650	−0.0382811
$319$	−8.11165	−0.454165
$320$	−3.55218	−0.198573
$321$	0.443665	0.0247630
$322$	−17.5365	−0.977271
$323$	5.42219	0.301699
$324$	3.24130	0.180072
$325$	2.99620	0.166199
$326$	23.1703	1.28329
$327$	0.191366	0.0105826
$328$	−11.1846	−0.617568
$329$	8.83543	0.487113
$330$	0.0435769	0.00239883
$331$	−24.6210	−1.35329	−0.676646	−	0.736309i	$-0.736567\pi$
$331$	−24.6210	−1.35329	−0.676646	+	0.736309i	$0.736567\pi$
$332$	2.67161	0.146624
$333$	−15.1150	−0.828296
$334$	1.46813	0.0803324
$335$	0.487919	0.0266579
$336$	0.275204	0.0150136
$337$	−4.47958	−0.244018	−0.122009	−	0.992529i	$-0.538934\pi$
$337$	−4.47958	−0.244018	−0.122009	+	0.992529i	$0.538934\pi$
$338$	19.3397	1.05194
$339$	−0.421546	−0.0228952
$340$	0.210846	0.0114347
$341$	−6.20028	−0.335764
$342$	−24.9748	−1.35048
$343$	15.4003	0.831536
$344$	−2.51842	−0.135784
$345$	0.262219	0.0141174
$346$	22.2128	1.19417
$347$	5.22982	0.280752	0.140376	−	0.990098i	$-0.455169\pi$
$347$	5.22982	0.280752	0.140376	+	0.990098i	$0.455169\pi$
$348$	−0.142185	−0.00762194
$349$	18.2439	0.976573	0.488287	−	0.872683i	$-0.337622\pi$
$349$	18.2439	0.976573	0.488287	+	0.872683i	$0.337622\pi$
$350$	−8.83624	−0.472317
$351$	0.187290	0.00999683
$352$	2.01852	0.107587
$353$	10.7747	0.573477	0.286738	−	0.958009i	$-0.407429\pi$
$353$	10.7747	0.573477	0.286738	+	0.958009i	$0.407429\pi$
$354$	−0.0392428	−0.00208573
$355$	−0.153987	−0.00817278
$356$	−3.75260	−0.198887
$357$	0.0599355	0.00317212
$358$	−10.6407	−0.562381
$359$	−4.90610	−0.258934	−0.129467	−	0.991584i	$-0.541327\pi$
$359$	−4.90610	−0.258934	−0.129467	+	0.991584i	$0.541327\pi$
$360$	4.40932	0.232391
$361$	10.4001	0.547374
$362$	−17.6621	−0.928301
$363$	0.0485562	0.00254854
$364$	0.286571	0.0150204
$365$	−8.12633	−0.425352
$366$	0.870351	0.0454940
$367$	−28.4388	−1.48450	−0.742248	−	0.670126i	$-0.766240\pi$
$367$	−28.4388	−1.48450	−0.742248	+	0.670126i	$0.766240\pi$
$368$	42.4547	2.21311
$369$	13.3129	0.693043
$370$	4.52522	0.235255
$371$	−11.2940	−0.586353
$372$	−0.108682	−0.00563490
$373$	−8.53764	−0.442062	−0.221031	−	0.975267i	$-0.570942\pi$
$373$	−8.53764	−0.442062	−0.221031	+	0.975267i	$0.570942\pi$
$374$	1.53655	0.0794533
$375$	0.273927	0.0141455
$376$	−18.0267	−0.929654
$377$	5.21675	0.268676
$378$	−0.552347	−0.0284097
$379$	20.9665	1.07698	0.538488	−	0.842633i	$-0.318996\pi$
$379$	20.9665	1.07698	0.538488	+	0.842633i	$0.318996\pi$
$380$	1.14325	0.0586474
$381$	0.662387	0.0339351
$382$	1.39290	0.0712671
$383$	9.42317	0.481502	0.240751	−	0.970587i	$-0.422606\pi$
$383$	9.42317	0.481502	0.240751	+	0.970587i	$0.422606\pi$
$384$	−0.649780	−0.0331589
$385$	0.720948	0.0367429
$386$	13.0507	0.664262
$387$	2.99764	0.152379
$388$	−0.536841	−0.0272540
$389$	−1.74050	−0.0882470	−0.0441235	−	0.999026i	$-0.514050\pi$
$389$	−1.74050	−0.0882470	−0.0441235	+	0.999026i	$0.514050\pi$
$390$	−0.0280251	−0.00141910
$391$	9.24603	0.467592
$392$	−13.7918	−0.696590
$393$	−0.208765	−0.0105308
$394$	−27.5071	−1.38579
$395$	−0.102677	−0.00516626
$396$	−1.08214	−0.0543794
$397$	23.1547	1.16210	0.581051	−	0.813867i	$-0.302642\pi$
$397$	23.1547	1.16210	0.581051	+	0.813867i	$0.302642\pi$
$398$	41.1275	2.06153
$399$	0.324982	0.0162694
$400$	21.3920	1.06960
$401$	4.36539	0.217997	0.108999	−	0.994042i	$-0.465236\pi$
$401$	4.36539	0.217997	0.108999	+	0.994042i	$0.465236\pi$
$402$	0.0623270	0.00310859
$403$	3.98751	0.198632
$404$	−2.28186	−0.113527
$405$	−5.24423	−0.260588
$406$	−15.3850	−0.763543
$407$	5.04229	0.249937
$408$	−0.122285	−0.00605399
$409$	−2.94948	−0.145842	−0.0729211	−	0.997338i	$-0.523232\pi$
$409$	−2.94948	−0.145842	−0.0729211	+	0.997338i	$0.523232\pi$
$410$	−3.98571	−0.196840
$411$	−0.896987	−0.0442451
$412$	−1.72772	−0.0851187
$413$	−0.649244	−0.0319472
$414$	−42.5876	−2.09306
$415$	−4.32250	−0.212183
$416$	−1.29814	−0.0636468
$417$	−1.05597	−0.0517113
$418$	8.33148	0.407506
$419$	−23.0926	−1.12815	−0.564073	−	0.825725i	$-0.690766\pi$
$419$	−23.0926	−1.12815	−0.564073	+	0.825725i	$0.690766\pi$
$420$	0.0126372	0.000616631 0
$421$	13.1345	0.640137	0.320068	−	0.947394i	$-0.396294\pi$
$421$	13.1345	0.640137	0.320068	+	0.947394i	$0.396294\pi$
$422$	−35.5060	−1.72841
$423$	21.4569	1.04327
$424$	23.0427	1.11905
$425$	4.65886	0.225988
$426$	−0.0196703	−0.000953031 0
$427$	14.3993	0.696833
$428$	3.29848	0.159438
$429$	−0.0312273	−0.00150767
$430$	−0.897453	−0.0432790
$431$	−34.3596	−1.65504	−0.827521	−	0.561435i	$-0.810250\pi$
$431$	−34.3596	−1.65504	−0.827521	+	0.561435i	$0.810250\pi$
$432$	1.33720	0.0643360
$433$	−36.5213	−1.75510	−0.877552	−	0.479482i	$-0.840825\pi$
$433$	−36.5213	−1.75510	−0.877552	+	0.479482i	$0.840825\pi$
$434$	−11.7598	−0.564487
$435$	0.230047	0.0110299
$436$	1.42273	0.0681365
$437$	50.1337	2.39822
$438$	−1.03806	−0.0496004
$439$	−9.54196	−0.455413	−0.227706	−	0.973730i	$-0.573123\pi$
$439$	−9.54196	−0.455413	−0.227706	+	0.973730i	$0.573123\pi$
$440$	−1.47093	−0.0701238
$441$	16.4162	0.781724
$442$	−0.988185	−0.0470032
$443$	13.2121	0.627727	0.313863	−	0.949468i	$-0.398377\pi$
$443$	13.2121	0.627727	0.313863	+	0.949468i	$0.398377\pi$
$444$	0.0883841	0.00419452
$445$	6.07147	0.287815
$446$	−38.1708	−1.80744
$447$	−0.896475	−0.0424018
$448$	−7.50709	−0.354676
$449$	0.00520906	0.000245831 0	0.000122915	−	1.00000i	$-0.499961\pi$
$449$	0.00520906	0.000245831 0	0.000122915		1.00000i	$0.499961\pi$
$450$	−21.4589	−1.01158
$451$	−4.44113	−0.209125
$452$	−3.13403	−0.147412
$453$	0.0487609	0.00229099
$454$	16.1787	0.759306
$455$	−0.463655	−0.0217365
$456$	−0.663050	−0.0310502
$457$	−31.5018	−1.47359	−0.736797	−	0.676114i	$-0.763663\pi$
$457$	−31.5018	−1.47359	−0.736797	+	0.676114i	$0.763663\pi$
$458$	28.2914	1.32197
$459$	0.291222	0.0135931
$460$	1.94949	0.0908955
$461$	0.607061	0.0282737	0.0141368	−	0.999900i	$-0.495500\pi$
$461$	0.607061	0.0282737	0.0141368	+	0.999900i	$0.495500\pi$
$462$	0.0920941	0.00428461
$463$	−37.8214	−1.75771	−0.878854	−	0.477091i	$-0.841691\pi$
$463$	−37.8214	−1.75771	−0.878854	+	0.477091i	$0.841691\pi$
$464$	37.2460	1.72910
$465$	0.175841	0.00815442
$466$	−0.165519	−0.00766750
$467$	26.6899	1.23506	0.617530	−	0.786547i	$-0.288133\pi$
$467$	26.6899	1.23506	0.617530	+	0.786547i	$0.288133\pi$
$468$	0.695941	0.0321699
$469$	1.03116	0.0476143
$470$	−6.42391	−0.296313
$471$	−0.0323575	−0.00149095
$472$	1.32463	0.0609712
$473$	−1.00000	−0.0459800
$474$	−0.0131160	−0.000602440 0
$475$	25.2612	1.15906
$476$	0.445597	0.0204239
$477$	−27.4275	−1.25582
$478$	−12.0482	−0.551072
$479$	24.3703	1.11351	0.556755	−	0.830677i	$-0.312046\pi$
$479$	24.3703	1.11351	0.556755	+	0.830677i	$0.312046\pi$
$480$	−0.0572454	−0.00261288
$481$	−3.24279	−0.147859
$482$	11.4610	0.522036
$483$	0.554166	0.0252154
$484$	0.360996	0.0164089
$485$	0.868576	0.0394400
$486$	−2.01234	−0.0912814
$487$	16.2709	0.737306	0.368653	−	0.929567i	$-0.379819\pi$
$487$	16.2709	0.737306	0.368653	+	0.929567i	$0.379819\pi$
$488$	−29.3786	−1.32990
$489$	−0.732198	−0.0331111
$490$	−4.91478	−0.222027
$491$	32.9265	1.48595	0.742975	−	0.669319i	$-0.233414\pi$
$491$	32.9265	1.48595	0.742975	+	0.669319i	$0.233414\pi$
$492$	−0.0778467	−0.00350960
$493$	8.11165	0.365330
$494$	−5.35813	−0.241073
$495$	1.75083	0.0786939
$496$	28.4697	1.27833
$497$	−0.325432	−0.0145976
$498$	−0.552157	−0.0247428
$499$	−3.77275	−0.168892	−0.0844458	−	0.996428i	$-0.526912\pi$
$499$	−3.77275	−0.168892	−0.0844458	+	0.996428i	$0.526912\pi$
$500$	2.03654	0.0910766
$501$	−0.0463939	−0.00207272
$502$	−36.9534	−1.64931
$503$	−0.984764	−0.0439084	−0.0219542	−	0.999759i	$-0.506989\pi$
$503$	−0.984764	−0.0439084	−0.0219542	+	0.999759i	$0.506989\pi$
$504$	9.31853	0.415080
$505$	3.69191	0.164288
$506$	14.2070	0.631579
$507$	−0.611147	−0.0271420
$508$	4.92458	0.218493
$509$	−23.4354	−1.03876	−0.519378	−	0.854545i	$-0.673836\pi$
$509$	−23.4354	−1.03876	−0.519378	+	0.854545i	$0.673836\pi$
$510$	−0.0435769	−0.00192962
$511$	−17.1740	−0.759731
$512$	13.8591	0.612493
$513$	1.57906	0.0697173
$514$	33.1343	1.46149
$515$	2.79535	0.123178
$516$	−0.0175286	−0.000771652 0
$517$	−7.15793	−0.314805
$518$	9.56347	0.420195
$519$	−0.701940	−0.0308118
$520$	0.945981	0.0414840
$521$	−26.1850	−1.14718	−0.573592	−	0.819141i	$-0.694451\pi$
$521$	−26.1850	−1.14718	−0.573592	+	0.819141i	$0.694451\pi$
$522$	−37.3625	−1.63531
$523$	20.0920	0.878560	0.439280	−	0.898350i	$-0.355234\pi$
$523$	20.0920	0.878560	0.439280	+	0.898350i	$0.355234\pi$
$524$	−1.55209	−0.0678033
$525$	0.279231	0.0121867
$526$	26.7533	1.16650
$527$	6.20028	0.270088
$528$	−0.222954	−0.00970283
$529$	62.4891	2.71692
$530$	8.21141	0.356681
$531$	−1.57670	−0.0684227
$532$	2.41611	0.104752
$533$	2.85617	0.123715
$534$	0.775571	0.0335622
$535$	−5.33673	−0.230727
$536$	−2.10384	−0.0908719
$537$	0.336255	0.0145105
$538$	−6.00102	−0.258722
$539$	−5.47637	−0.235884
$540$	0.0614032	0.00264237
$541$	−8.89476	−0.382415	−0.191208	−	0.981550i	$-0.561240\pi$
$541$	−8.89476	−0.382415	−0.191208	+	0.981550i	$0.561240\pi$
$542$	12.4904	0.536507
$543$	0.558136	0.0239519
$544$	−2.01852	−0.0865432
$545$	−2.30189	−0.0986022
$546$	−0.0592274	−0.00253470
$547$	−22.9935	−0.983130	−0.491565	−	0.870841i	$-0.663575\pi$
$547$	−22.9935	−0.983130	−0.491565	+	0.870841i	$0.663575\pi$
$548$	−6.66874	−0.284875
$549$	34.9690	1.49244
$550$	7.15859	0.305243
$551$	43.9829	1.87373
$552$	−1.13065	−0.0481236
$553$	−0.216996	−0.00922759
$554$	−32.8948	−1.39757
$555$	−0.143000	−0.00607001
$556$	−7.85075	−0.332946
$557$	−36.7708	−1.55803	−0.779014	−	0.627006i	$-0.784280\pi$
$557$	−36.7708	−1.55803	−0.779014	+	0.627006i	$0.784280\pi$
$558$	−28.5587	−1.20899
$559$	0.643118	0.0272010
$560$	−3.31036	−0.139888
$561$	−0.0485562	−0.00205004
$562$	−22.2924	−0.940349
$563$	−27.5432	−1.16081	−0.580403	−	0.814330i	$-0.697105\pi$
$563$	−27.5432	−1.16081	−0.580403	+	0.814330i	$0.697105\pi$
$564$	−0.125468	−0.00528317
$565$	5.07066	0.213324
$566$	−35.5009	−1.49221
$567$	−11.0830	−0.465443
$568$	0.663969	0.0278595
$569$	−36.3524	−1.52397	−0.761986	−	0.647594i	$-0.775775\pi$
$569$	−36.3524	−1.52397	−0.761986	+	0.647594i	$0.775775\pi$
$570$	−0.236282	−0.00989676
$571$	−17.5535	−0.734592	−0.367296	−	0.930104i	$-0.619716\pi$
$571$	−17.5535	−0.734592	−0.367296	+	0.930104i	$0.619716\pi$
$572$	−0.232163	−0.00970722
$573$	−0.0440167	−0.00183882
$574$	−8.42328	−0.351581
$575$	43.0760	1.79639
$576$	−18.2310	−0.759626
$577$	17.6108	0.733149	0.366574	−	0.930389i	$-0.380531\pi$
$577$	17.6108	0.733149	0.366574	+	0.930389i	$0.380531\pi$
$578$	−1.53655	−0.0639122
$579$	−0.412411	−0.0171392
$580$	1.71031	0.0710168
$581$	−9.13505	−0.378986
$582$	0.110952	0.00459912
$583$	9.14969	0.378941
$584$	35.0395	1.44995
$585$	−1.12599	−0.0465540
$586$	17.0965	0.706251
$587$	−16.4426	−0.678658	−0.339329	−	0.940668i	$-0.610200\pi$
$587$	−16.4426	−0.678658	−0.339329	+	0.940668i	$0.610200\pi$
$588$	−0.0959929	−0.00395868
$589$	33.6191	1.38525
$590$	0.472041	0.0194336
$591$	0.869245	0.0357560
$592$	−23.1526	−0.951564
$593$	10.3327	0.424315	0.212157	−	0.977236i	$-0.431951\pi$
$593$	10.3327	0.424315	0.212157	+	0.977236i	$0.431951\pi$
$594$	0.447479	0.0183603
$595$	−0.720948	−0.0295560
$596$	−6.66493	−0.273006
$597$	−1.29966	−0.0531914
$598$	−9.13679	−0.373631
$599$	40.2005	1.64255	0.821274	−	0.570535i	$-0.193264\pi$
$599$	40.2005	1.64255	0.821274	+	0.570535i	$0.193264\pi$
$600$	−0.569708	−0.0232582
$601$	−40.1107	−1.63615	−0.818075	−	0.575112i	$-0.804959\pi$
$601$	−40.1107	−1.63615	−0.818075	+	0.575112i	$0.804959\pi$
$602$	−1.89665	−0.0773017
$603$	2.50417	0.101978
$604$	0.362518	0.0147507
$605$	−0.584069	−0.0237458
$606$	0.471605	0.0191577
$607$	1.08336	0.0439720	0.0219860	−	0.999758i	$-0.493001\pi$
$607$	1.08336	0.0439720	0.0219860	+	0.999758i	$0.493001\pi$
$608$	−10.9448	−0.443869
$609$	0.486176	0.0197008
$610$	−10.4692	−0.423886
$611$	4.60340	0.186233
$612$	1.08214	0.0437428
$613$	10.3660	0.418680	0.209340	−	0.977843i	$-0.432868\pi$
$613$	10.3660	0.418680	0.209340	+	0.977843i	$0.432868\pi$
$614$	31.1476	1.25701
$615$	0.125951	0.00507884
$616$	−3.10862	−0.125250
$617$	18.1931	0.732428	0.366214	−	0.930531i	$-0.380654\pi$
$617$	18.1931	0.732428	0.366214	+	0.930531i	$0.380654\pi$
$618$	0.357078	0.0143638
$619$	−25.4061	−1.02116	−0.510578	−	0.859832i	$-0.670568\pi$
$619$	−25.4061	−1.02116	−0.510578	+	0.859832i	$0.670568\pi$
$620$	1.30731	0.0525027
$621$	2.69265	0.108052
$622$	13.8579	0.555652
$623$	12.8313	0.514074
$624$	0.143386	0.00574003
$625$	19.9993	0.799973
$626$	−16.4569	−0.657751
$627$	−0.263281	−0.0105144
$628$	−0.240565	−0.00959958
$629$	−5.04229	−0.201049
$630$	3.32071	0.132300
$631$	28.3582	1.12892	0.564461	−	0.825460i	$-0.309084\pi$
$631$	28.3582	1.12892	0.564461	+	0.825460i	$0.309084\pi$
$632$	0.442730	0.0176108
$633$	1.12201	0.0445961
$634$	−37.1347	−1.47481
$635$	−7.96767	−0.316187
$636$	0.160381	0.00635952
$637$	3.52195	0.139545
$638$	12.4640	0.493453
$639$	−0.790315	−0.0312644
$640$	7.81602	0.308955
$641$	6.40696	0.253060	0.126530	−	0.991963i	$-0.459616\pi$
$641$	6.40696	0.253060	0.126530	+	0.991963i	$0.459616\pi$
$642$	−0.681715	−0.0269052
$643$	−7.84093	−0.309216	−0.154608	−	0.987976i	$-0.549411\pi$
$643$	−7.84093	−0.309216	−0.154608	+	0.987976i	$0.549411\pi$
$644$	4.12000	0.162351
$645$	0.0283601	0.00111668
$646$	−8.33148	−0.327798
$647$	−20.1715	−0.793023	−0.396512	−	0.918030i	$-0.629779\pi$
$647$	−20.1715	−0.793023	−0.396512	+	0.918030i	$0.629779\pi$
$648$	22.6123	0.888296
$649$	0.525978	0.0206465
$650$	−4.60382	−0.180577
$651$	0.371617	0.0145648
$652$	−5.44360	−0.213188
$653$	34.1642	1.33695	0.668474	−	0.743735i	$-0.266948\pi$
$653$	34.1642	1.33695	0.668474	+	0.743735i	$0.266948\pi$
$654$	−0.294044	−0.0114980
$655$	2.51118	0.0981200
$656$	20.3922	0.796183
$657$	−41.7071	−1.62715
$658$	−13.5761	−0.529252
$659$	29.7774	1.15996	0.579982	−	0.814629i	$-0.303060\pi$
$659$	29.7774	1.15996	0.579982	+	0.814629i	$0.303060\pi$
$660$	−0.0102379	−0.000398509 0
$661$	17.9228	0.697117	0.348558	−	0.937287i	$-0.386671\pi$
$661$	17.9228	0.697117	0.348558	+	0.937287i	$0.386671\pi$
$662$	37.8314	1.47036
$663$	0.0312273	0.00121277
$664$	18.6380	0.723294
$665$	−3.90911	−0.151589
$666$	23.2250	0.899950
$667$	75.0005	2.90403
$668$	−0.344920	−0.0133453
$669$	1.20623	0.0466354
$670$	−0.749714	−0.0289640
$671$	−11.6655	−0.450341
$672$	−0.120981	−0.00466694
$673$	0.495551	0.0191021	0.00955105	−	0.999954i	$-0.496960\pi$
$673$	0.495551	0.0191021	0.00955105	+	0.999954i	$0.496960\pi$
$674$	6.88311	0.265128
$675$	1.35677	0.0522219
$676$	−4.54364	−0.174755
$677$	27.5306	1.05809	0.529043	−	0.848595i	$-0.322551\pi$
$677$	27.5306	1.05809	0.529043	+	0.848595i	$0.322551\pi$
$678$	0.647728	0.0248758
$679$	1.83562	0.0704448
$680$	1.47093	0.0564075
$681$	−0.511259	−0.0195915
$682$	9.52706	0.364810
$683$	6.60143	0.252597	0.126298	−	0.991992i	$-0.459690\pi$
$683$	6.60143	0.252597	0.126298	+	0.991992i	$0.459690\pi$
$684$	5.86754	0.224351
$685$	10.7896	0.412250
$686$	−23.6633	−0.903470
$687$	−0.894029	−0.0341093
$688$	4.59167	0.175056
$689$	−5.88433	−0.224175
$690$	−0.402913	−0.0153386
$691$	10.7782	0.410023	0.205012	−	0.978760i	$-0.434277\pi$
$691$	10.7782	0.410023	0.205012	+	0.978760i	$0.434277\pi$
$692$	−5.21865	−0.198383
$693$	3.70015	0.140557
$694$	−8.03590	−0.305039
$695$	12.7020	0.481815
$696$	−0.991930	−0.0375990
$697$	4.44113	0.168220
$698$	−28.0327	−1.06105
$699$	0.00523050	0.000197836 0
$700$	2.07597	0.0784645
$701$	27.9348	1.05508	0.527542	−	0.849529i	$-0.323114\pi$
$701$	27.9348	1.05508	0.527542	+	0.849529i	$0.323114\pi$
$702$	−0.287782	−0.0108616
$703$	−27.3402	−1.03116
$704$	6.08179	0.229216
$705$	0.203000	0.00764542
$706$	−16.5558	−0.623087
$707$	7.80238	0.293439
$708$	0.00921965	0.000346495 0
$709$	11.1462	0.418604	0.209302	−	0.977851i	$-0.432881\pi$
$709$	11.1462	0.418604	0.209302	+	0.977851i	$0.432881\pi$
$710$	0.236609	0.00887979
$711$	−0.526976	−0.0197631
$712$	−26.1793	−0.981109
$713$	57.3280	2.14695
$714$	−0.0920941	−0.00344654
$715$	0.375625	0.0140476
$716$	2.49992	0.0934265
$717$	0.380732	0.0142187
$718$	7.53848	0.281334
$719$	−23.0778	−0.860657	−0.430328	−	0.902672i	$-0.641602\pi$
$719$	−23.0778	−0.860657	−0.430328	+	0.902672i	$0.641602\pi$
$720$	−8.03924	−0.299605
$721$	5.90761	0.220011
$722$	−15.9803	−0.594726
$723$	−0.362177	−0.0134695
$724$	4.14952	0.154216
$725$	37.7910	1.40352
$726$	−0.0746091	−0.00276900
$727$	−25.1854	−0.934075	−0.467037	−	0.884238i	$-0.654679\pi$
$727$	−25.1854	−0.934075	−0.467037	+	0.884238i	$0.654679\pi$
$728$	1.99921	0.0740957
$729$	−26.8728	−0.995288
$730$	12.4865	0.462148
$731$	1.00000	0.0369863
$732$	−0.204479	−0.00755777
$733$	0.246073	0.00908893	0.00454446	−	0.999990i	$-0.498553\pi$
$733$	0.246073	0.00908893	0.00454446	+	0.999990i	$0.498553\pi$
$734$	43.6978	1.61291
$735$	0.155311	0.00572872
$736$	−18.6633	−0.687937
$737$	−0.835380	−0.0307716
$738$	−20.4560	−0.752996
$739$	−21.0038	−0.772636	−0.386318	−	0.922366i	$-0.626253\pi$
$739$	−21.0038	−0.772636	−0.386318	+	0.922366i	$0.626253\pi$
$740$	−1.06315	−0.0390821
$741$	0.169321	0.00622014
$742$	17.3538	0.637077
$743$	21.5072	0.789022	0.394511	−	0.918891i	$-0.370914\pi$
$743$	21.5072	0.789022	0.394511	+	0.918891i	$0.370914\pi$
$744$	−0.758199	−0.0277969
$745$	10.7835	0.395075
$746$	13.1185	0.480304
$747$	−22.1846	−0.811691
$748$	−0.360996	−0.0131993
$749$	−11.2785	−0.412107
$750$	−0.420903	−0.0153692
$751$	−36.5914	−1.33524	−0.667620	−	0.744502i	$-0.732687\pi$
$751$	−36.5914	−1.33524	−0.667620	+	0.744502i	$0.732687\pi$
$752$	32.8669	1.19853
$753$	1.16775	0.0425553
$754$	−8.01581	−0.291919
$755$	−0.586532	−0.0213461
$756$	0.129768	0.00471961
$757$	40.2922	1.46444	0.732222	−	0.681066i	$-0.238483\pi$
$757$	40.2922	1.46444	0.732222	+	0.681066i	$0.238483\pi$
$758$	−32.2161	−1.17014
$759$	−0.448952	−0.0162959
$760$	7.97565	0.289307
$761$	−13.2345	−0.479750	−0.239875	−	0.970804i	$-0.577107\pi$
$761$	−13.2345	−0.479750	−0.239875	+	0.970804i	$0.577107\pi$
$762$	−1.01779	−0.0368707
$763$	−4.86475	−0.176116
$764$	−0.327247	−0.0118394
$765$	−1.75083	−0.0633014
$766$	−14.4792	−0.523155
$767$	−0.338266	−0.0122141
$768$	0.407804	0.0147154
$769$	−48.7786	−1.75900	−0.879501	−	0.475897i	$-0.842123\pi$
$769$	−48.7786	−1.75900	−0.879501	+	0.475897i	$0.842123\pi$
$770$	−1.10777	−0.0399214
$771$	−1.04707	−0.0377092
$772$	−3.06611	−0.110352
$773$	20.8613	0.750330	0.375165	−	0.926958i	$-0.377586\pi$
$773$	20.8613	0.750330	0.375165	+	0.926958i	$0.377586\pi$
$774$	−4.60604	−0.165561
$775$	28.8863	1.03763
$776$	−3.74517	−0.134444
$777$	−0.302212	−0.0108418
$778$	2.67437	0.0958810
$779$	24.0806	0.862779
$780$	0.00658417	0.000235751 0
$781$	0.263645	0.00943397
$782$	−14.2070	−0.508042
$783$	2.36229	0.0844215
$784$	25.1457	0.898061
$785$	0.389219	0.0138918
$786$	0.320779	0.0114418
$787$	−12.4917	−0.445283	−0.222641	−	0.974900i	$-0.571468\pi$
$787$	−12.4917	−0.445283	−0.222641	+	0.974900i	$0.571468\pi$
$788$	6.46249	0.230217
$789$	−0.845424	−0.0300979
$790$	0.157769	0.00561318
$791$	10.7162	0.381024
$792$	−7.54931	−0.268253
$793$	7.50229	0.266414
$794$	−35.5785	−1.26263
$795$	−0.259486	−0.00920303
$796$	−9.66243	−0.342476
$797$	48.1209	1.70453	0.852265	−	0.523111i	$-0.175229\pi$
$797$	48.1209	1.70453	0.852265	+	0.523111i	$0.175229\pi$
$798$	−0.499351	−0.0176768
$799$	7.15793	0.253229
$800$	−9.40399	−0.332481
$801$	31.1609	1.10102
$802$	−6.70766	−0.236856
$803$	13.9133	0.490990
$804$	−0.0146430	−0.000516419 0
$805$	−6.66591	−0.234942
$806$	−6.12703	−0.215815
$807$	0.189636	0.00667552
$808$	−15.9190	−0.560027
$809$	1.73362	0.0609508	0.0304754	−	0.999536i	$-0.490298\pi$
$809$	1.73362	0.0609508	0.0304754	+	0.999536i	$0.490298\pi$
$810$	8.05804	0.283131
$811$	1.05726	0.0371256	0.0185628	−	0.999828i	$-0.494091\pi$
$811$	1.05726	0.0371256	0.0185628	+	0.999828i	$0.494091\pi$
$812$	3.61452	0.126845
$813$	−0.394704	−0.0138429
$814$	−7.74775	−0.271558
$815$	8.80741	0.308510
$816$	0.222954	0.00780495
$817$	5.42219	0.189698
$818$	4.53203	0.158459
$819$	−2.37964	−0.0831512
$820$	0.936396	0.0327004
$821$	−13.2416	−0.462135	−0.231067	−	0.972938i	$-0.574222\pi$
$821$	−13.2416	−0.462135	−0.231067	+	0.972938i	$0.574222\pi$
$822$	1.37827	0.0480726
$823$	−45.7234	−1.59382	−0.796908	−	0.604100i	$-0.793533\pi$
$823$	−45.7234	−1.59382	−0.796908	+	0.604100i	$0.793533\pi$
$824$	−12.0531	−0.419890
$825$	−0.226216	−0.00787585
$826$	0.997598	0.0347109
$827$	−7.45307	−0.259169	−0.129584	−	0.991568i	$-0.541364\pi$
$827$	−7.45307	−0.259169	−0.129584	+	0.991568i	$0.541364\pi$
$828$	10.0055	0.347714
$829$	−17.4547	−0.606226	−0.303113	−	0.952955i	$-0.598026\pi$
$829$	−17.4547	−0.606226	−0.303113	+	0.952955i	$0.598026\pi$
$830$	6.64175	0.230539
$831$	1.03950	0.0360598
$832$	−3.91131	−0.135600
$833$	5.47637	0.189745
$834$	1.62256	0.0561847
$835$	0.558059	0.0193124
$836$	−1.95739	−0.0676976
$837$	1.80566	0.0624128
$838$	35.4829	1.22574
$839$	−0.0581059	−0.00200604	−0.00100302	−	0.999999i	$-0.500319\pi$
$839$	−0.0581059	−0.00200604	−0.00100302	+	0.999999i	$0.500319\pi$
$840$	0.0881609	0.00304184
$841$	36.7988	1.26892
$842$	−20.1819	−0.695513
$843$	0.704456	0.0242628
$844$	8.34173	0.287134
$845$	7.35132	0.252893
$846$	−32.9697	−1.13352
$847$	−1.23435	−0.0424129
$848$	−42.0124	−1.44271
$849$	1.12185	0.0385019
$850$	−7.15859	−0.245538
$851$	−46.6212	−1.59815
$852$	0.00462132	0.000158324 0
$853$	−54.1779	−1.85502	−0.927509	−	0.373801i	$-0.878054\pi$
$853$	−54.1779	−1.85502	−0.927509	+	0.373801i	$0.878054\pi$
$854$	−22.1254	−0.757114
$855$	−9.49332	−0.324665
$856$	23.0112	0.786506
$857$	29.3108	1.00124	0.500619	−	0.865668i	$-0.333106\pi$
$857$	29.3108	1.00124	0.500619	+	0.865668i	$0.333106\pi$
$858$	0.0479825	0.00163809
$859$	17.8831	0.610165	0.305082	−	0.952326i	$-0.401316\pi$
$859$	17.8831	0.610165	0.305082	+	0.952326i	$0.401316\pi$
$860$	0.210846	0.00718980
$861$	0.266182	0.00907144
$862$	52.7953	1.79821
$863$	−49.5510	−1.68673	−0.843367	−	0.537337i	$-0.819430\pi$
$863$	−49.5510	−1.68673	−0.843367	+	0.537337i	$0.819430\pi$
$864$	−0.587837	−0.0199986
$865$	8.44345	0.287086
$866$	56.1170	1.90693
$867$	0.0485562	0.00164905
$868$	2.76282	0.0937764
$869$	0.175797	0.00596350
$870$	−0.353480	−0.0119841
$871$	0.537248	0.0182040
$872$	9.92541	0.336117
$873$	4.45783	0.150875
$874$	−77.0331	−2.60568
$875$	−6.96354	−0.235411
$876$	0.243880	0.00823996
$877$	−32.5846	−1.10030	−0.550151	−	0.835065i	$-0.685430\pi$
$877$	−32.5846	−1.10030	−0.550151	+	0.835065i	$0.685430\pi$
$878$	14.6617	0.494809
$879$	−0.540262	−0.0182226
$880$	2.68185	0.0904052
$881$	−18.1553	−0.611667	−0.305833	−	0.952085i	$-0.598935\pi$
$881$	−18.1553	−0.611667	−0.305833	+	0.952085i	$0.598935\pi$
$882$	−25.2244	−0.849348
$883$	−18.1889	−0.612104	−0.306052	−	0.952015i	$-0.599008\pi$
$883$	−18.1889	−0.612104	−0.306052	+	0.952015i	$0.599008\pi$
$884$	0.232163	0.00780848
$885$	−0.0149168	−0.000501423 0
$886$	−20.3011	−0.682029
$887$	25.1134	0.843225	0.421613	−	0.906776i	$-0.361464\pi$
$887$	25.1134	0.843225	0.421613	+	0.906776i	$0.361464\pi$
$888$	0.616595	0.0206916
$889$	−16.8387	−0.564750
$890$	−9.32913	−0.312713
$891$	8.97879	0.300801
$892$	8.96781	0.300264
$893$	38.8116	1.29878
$894$	1.37748	0.0460699
$895$	−4.04472	−0.135200
$896$	16.5182	0.551833
$897$	0.288729	0.00964038
$898$	−0.00800400	−0.000267097 0
$899$	50.2945	1.67741
$900$	5.04152	0.168051
$901$	−9.14969	−0.304820
$902$	6.82404	0.227216
$903$	0.0599355	0.00199453
$904$	−21.8639	−0.727184
$905$	−6.71366	−0.223170
$906$	−0.0749238	−0.00248918
$907$	18.7163	0.621466	0.310733	−	0.950497i	$-0.399426\pi$
$907$	18.7163	0.621466	0.310733	+	0.950497i	$0.399426\pi$
$908$	−3.80101	−0.126141
$909$	18.9481	0.628470
$910$	0.712430	0.0236168
$911$	14.5867	0.483278	0.241639	−	0.970366i	$-0.422315\pi$
$911$	14.5867	0.483278	0.241639	+	0.970366i	$0.422315\pi$
$912$	1.20890	0.0400306
$913$	7.40067	0.244926
$914$	48.4042	1.60107
$915$	0.330835	0.0109371
$916$	−6.64675	−0.219615
$917$	5.30706	0.175255
$918$	−0.447479	−0.0147690
$919$	14.4411	0.476366	0.238183	−	0.971220i	$-0.423448\pi$
$919$	14.4411	0.476366	0.238183	+	0.971220i	$0.423448\pi$
$920$	13.6003	0.448387
$921$	−0.984286	−0.0324333
$922$	−0.932782	−0.0307195
$923$	−0.169555	−0.00558098
$924$	−0.0216365	−0.000711787 0
$925$	−23.4914	−0.772391
$926$	58.1145	1.90976
$927$	14.3467	0.471207
$928$	−16.3735	−0.537486
$929$	24.5091	0.804118	0.402059	−	0.915614i	$-0.368295\pi$
$929$	24.5091	0.804118	0.402059	+	0.915614i	$0.368295\pi$
$930$	−0.270189	−0.00885984
$931$	29.6939	0.973178
$932$	0.0388867	0.00127378
$933$	−0.437920	−0.0143369
$934$	−41.0104	−1.34190
$935$	0.584069	0.0191011
$936$	4.85510	0.158694
$937$	−56.8471	−1.85711	−0.928556	−	0.371192i	$-0.878949\pi$
$937$	−56.8471	−1.85711	−0.928556	+	0.371192i	$0.878949\pi$
$938$	−1.58443	−0.0517333
$939$	0.520050	0.0169712
$940$	1.50922	0.0492254
$941$	−39.4793	−1.28699	−0.643494	−	0.765451i	$-0.722516\pi$
$941$	−39.4793	−1.28699	−0.643494	+	0.765451i	$0.722516\pi$
$942$	0.0497189	0.00161993
$943$	41.0628	1.33719
$944$	−2.41512	−0.0786055
$945$	−0.209956	−0.00682988
$946$	1.53655	0.0499577
$947$	20.3490	0.661255	0.330628	−	0.943761i	$-0.392740\pi$
$947$	20.3490	0.661255	0.330628	+	0.943761i	$0.392740\pi$
$948$	0.00308147	0.000100081 0
$949$	−8.94791	−0.290461
$950$	−38.8152	−1.25933
$951$	1.17348	0.0380528
$952$	3.10862	0.100751
$953$	−24.9795	−0.809167	−0.404583	−	0.914501i	$-0.632583\pi$
$953$	−24.9795	−0.809167	−0.404583	+	0.914501i	$0.632583\pi$
$954$	42.1438	1.36446
$955$	0.529465	0.0171331
$956$	2.83059	0.0915478
$957$	−0.393870	−0.0127320
$958$	−37.4463	−1.20984
$959$	22.8025	0.736330
$960$	−0.172480	−0.00556678
$961$	7.44348	0.240112
$962$	4.98272	0.160649
$963$	−27.3899	−0.882628
$964$	−2.69264	−0.0867241
$965$	4.96077	0.159693
$966$	−0.851505	−0.0273967
$967$	−28.6863	−0.922489	−0.461244	−	0.887273i	$-0.652597\pi$
$967$	−28.6863	−0.922489	−0.461244	+	0.887273i	$0.652597\pi$
$968$	2.51842	0.0809450
$969$	0.263281	0.00845779
$970$	−1.33461	−0.0428519
$971$	27.6408	0.887035	0.443518	−	0.896266i	$-0.353730\pi$
$971$	27.6408	0.887035	0.443518	+	0.896266i	$0.353730\pi$
$972$	0.472775	0.0151643
$973$	26.8441	0.860583
$974$	−25.0011	−0.801088
$975$	0.145484	0.00465922
$976$	53.5641	1.71455
$977$	22.7559	0.728027	0.364013	−	0.931394i	$-0.381406\pi$
$977$	22.7559	0.728027	0.364013	+	0.931394i	$0.381406\pi$
$978$	1.12506	0.0359755
$979$	−10.3951	−0.332230
$980$	1.15467	0.0368846
$981$	−11.8141	−0.377195
$982$	−50.5932	−1.61450
$983$	−31.8023	−1.01434	−0.507168	−	0.861847i	$-0.669307\pi$
$983$	−31.8023	−1.01434	−0.507168	+	0.861847i	$0.669307\pi$
$984$	−0.543082	−0.0173128
$985$	−10.4559	−0.333153
$986$	−12.4640	−0.396934
$987$	0.429014	0.0136557
$988$	1.25883	0.0400487
$989$	9.24603	0.294007
$990$	−2.69024	−0.0855015
$991$	30.4310	0.966672	0.483336	−	0.875435i	$-0.339425\pi$
$991$	30.4310	0.966672	0.483336	+	0.875435i	$0.339425\pi$
$992$	−12.5154	−0.397363
$993$	−1.19550	−0.0379380
$994$	0.500043	0.0158604
$995$	15.6332	0.495606
$996$	0.129723	0.00411043
$997$	7.74013	0.245132	0.122566	−	0.992460i	$-0.460888\pi$
$997$	7.74013	0.245132	0.122566	+	0.992460i	$0.460888\pi$
$998$	5.79703	0.183502
$999$	−1.46843	−0.0464590

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.i.1.19	✓	78

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.a.i.1.19	✓	78	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)

\(f(q)\)	\(=\)	\(q-1.53655 q^{2} +0.0485562 q^{3} +0.360996 q^{4} -0.584069 q^{5} -0.0746091 q^{6} -1.23435 q^{7} +2.51842 q^{8} -2.99764 q^{9} +O(q^{10})\)	\(q-1.53655 q^{2} +0.0485562 q^{3} +0.360996 q^{4} -0.584069 q^{5} -0.0746091 q^{6} -1.23435 q^{7} +2.51842 q^{8} -2.99764 q^{9} +0.897453 q^{10} +1.00000 q^{11} +0.0175286 q^{12} -0.643118 q^{13} +1.89665 q^{14} -0.0283601 q^{15} -4.59167 q^{16} -1.00000 q^{17} +4.60604 q^{18} -5.42219 q^{19} -0.210846 q^{20} -0.0599355 q^{21} -1.53655 q^{22} -9.24603 q^{23} +0.122285 q^{24} -4.65886 q^{25} +0.988185 q^{26} -0.291222 q^{27} -0.445597 q^{28} -8.11165 q^{29} +0.0435769 q^{30} -6.20028 q^{31} +2.01852 q^{32} +0.0485562 q^{33} +1.53655 q^{34} +0.720948 q^{35} -1.08214 q^{36} +5.04229 q^{37} +8.33148 q^{38} -0.0312273 q^{39} -1.47093 q^{40} -4.44113 q^{41} +0.0920941 q^{42} -1.00000 q^{43} +0.360996 q^{44} +1.75083 q^{45} +14.2070 q^{46} -7.15793 q^{47} -0.222954 q^{48} -5.47637 q^{49} +7.15859 q^{50} -0.0485562 q^{51} -0.232163 q^{52} +9.14969 q^{53} +0.447479 q^{54} -0.584069 q^{55} -3.10862 q^{56} -0.263281 q^{57} +12.4640 q^{58} +0.525978 q^{59} -0.0102379 q^{60} -11.6655 q^{61} +9.52706 q^{62} +3.70015 q^{63} +6.08179 q^{64} +0.375625 q^{65} -0.0746091 q^{66} -0.835380 q^{67} -0.360996 q^{68} -0.448952 q^{69} -1.10777 q^{70} +0.263645 q^{71} -7.54931 q^{72} +13.9133 q^{73} -7.74775 q^{74} -0.226216 q^{75} -1.95739 q^{76} -1.23435 q^{77} +0.0479825 q^{78} +0.175797 q^{79} +2.68185 q^{80} +8.97879 q^{81} +6.82404 q^{82} +7.40067 q^{83} -0.0216365 q^{84} +0.584069 q^{85} +1.53655 q^{86} -0.393870 q^{87} +2.51842 q^{88} -10.3951 q^{89} -2.69024 q^{90} +0.793836 q^{91} -3.33778 q^{92} -0.301062 q^{93} +10.9985 q^{94} +3.16693 q^{95} +0.0980114 q^{96} -1.48711 q^{97} +8.41473 q^{98} -2.99764 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q + 7 q^{2} + 10 q^{3} + 91 q^{4} + 17 q^{5} + 12 q^{6} + 11 q^{7} + 33 q^{8} + 102 q^{9}+O(q^{10}) \) 78 * q + 7 * q^2 + 10 * q^3 + 91 * q^4 + 17 * q^5 + 12 * q^6 + 11 * q^7 + 33 * q^8 + 102 * q^9	\( 78 q + 7 q^{2} + 10 q^{3} + 91 q^{4} + 17 q^{5} + 12 q^{6} + 11 q^{7} + 33 q^{8} + 102 q^{9} + 3 q^{10} + 78 q^{11} + 31 q^{12} - 16 q^{13} + 31 q^{14} + 38 q^{15} + 121 q^{16} - 78 q^{17} + 11 q^{18} + 51 q^{20} + 6 q^{21} + 7 q^{22} + 48 q^{23} + 11 q^{24} + 101 q^{25} + 18 q^{26} + 46 q^{27} + 27 q^{28} + 22 q^{29} + 14 q^{30} + 56 q^{31} + 83 q^{32} + 10 q^{33} - 7 q^{34} + 24 q^{35} + 139 q^{36} + 53 q^{37} + 10 q^{38} + 79 q^{39} - q^{40} + 23 q^{41} + 17 q^{42} - 78 q^{43} + 91 q^{44} + 76 q^{45} + 21 q^{46} + 57 q^{47} + 78 q^{48} + 115 q^{49} + 58 q^{50} - 10 q^{51} - 63 q^{52} + 22 q^{53} - 18 q^{54} + 17 q^{55} + 111 q^{56} - 11 q^{57} + 36 q^{58} + 71 q^{59} + 36 q^{60} + 4 q^{61} - 5 q^{62} + 71 q^{63} + 183 q^{64} + 47 q^{65} + 12 q^{66} + 11 q^{67} - 91 q^{68} + 31 q^{69} + 33 q^{70} + 159 q^{71} + 59 q^{72} + 2 q^{73} - 4 q^{74} + 83 q^{75} - 44 q^{76} + 11 q^{77} + 101 q^{78} + 35 q^{79} + 85 q^{80} + 170 q^{81} + 98 q^{82} - 32 q^{83} + 44 q^{84} - 17 q^{85} - 7 q^{86} - 6 q^{87} + 33 q^{88} + 50 q^{89} - 5 q^{90} + 86 q^{91} + 106 q^{92} + 68 q^{93} - q^{94} + 109 q^{95} - 50 q^{96} + 40 q^{97} + 106 q^{98} + 102 q^{99}+O(q^{100}) \) 78 * q + 7 * q^2 + 10 * q^3 + 91 * q^4 + 17 * q^5 + 12 * q^6 + 11 * q^7 + 33 * q^8 + 102 * q^9 + 3 * q^10 + 78 * q^11 + 31 * q^12 - 16 * q^13 + 31 * q^14 + 38 * q^15 + 121 * q^16 - 78 * q^17 + 11 * q^18 + 51 * q^20 + 6 * q^21 + 7 * q^22 + 48 * q^23 + 11 * q^24 + 101 * q^25 + 18 * q^26 + 46 * q^27 + 27 * q^28 + 22 * q^29 + 14 * q^30 + 56 * q^31 + 83 * q^32 + 10 * q^33 - 7 * q^34 + 24 * q^35 + 139 * q^36 + 53 * q^37 + 10 * q^38 + 79 * q^39 - q^40 + 23 * q^41 + 17 * q^42 - 78 * q^43 + 91 * q^44 + 76 * q^45 + 21 * q^46 + 57 * q^47 + 78 * q^48 + 115 * q^49 + 58 * q^50 - 10 * q^51 - 63 * q^52 + 22 * q^53 - 18 * q^54 + 17 * q^55 + 111 * q^56 - 11 * q^57 + 36 * q^58 + 71 * q^59 + 36 * q^60 + 4 * q^61 - 5 * q^62 + 71 * q^63 + 183 * q^64 + 47 * q^65 + 12 * q^66 + 11 * q^67 - 91 * q^68 + 31 * q^69 + 33 * q^70 + 159 * q^71 + 59 * q^72 + 2 * q^73 - 4 * q^74 + 83 * q^75 - 44 * q^76 + 11 * q^77 + 101 * q^78 + 35 * q^79 + 85 * q^80 + 170 * q^81 + 98 * q^82 - 32 * q^83 + 44 * q^84 - 17 * q^85 - 7 * q^86 - 6 * q^87 + 33 * q^88 + 50 * q^89 - 5 * q^90 + 86 * q^91 + 106 * q^92 + 68 * q^93 - q^94 + 109 * q^95 - 50 * q^96 + 40 * q^97 + 106 * q^98 + 102 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more