LMFDB - Embedded newform 4033.2.a.d.1.77

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4033,2,Mod(1,4033)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4033, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4033.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4033 = 37 \cdot 109 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4033.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:	$32.2036671352$
Analytic rank:	$1$
Dimension:	$79$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.77
Character	$\chi$	$=$	4033.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.54331 q^{2} -0.553818 q^{3} +4.46844 q^{4} -3.89336 q^{5} -1.40853 q^{6} +3.15111 q^{7} +6.27803 q^{8} -2.69329 q^{9} +O(q^{10})$	\(q+2.54331 q^{2} -0.553818 q^{3} +4.46844 q^{4} -3.89336 q^{5} -1.40853 q^{6} +3.15111 q^{7} +6.27803 q^{8} -2.69329 q^{9} -9.90203 q^{10} +0.439543 q^{11} -2.47471 q^{12} -4.45537 q^{13} +8.01427 q^{14} +2.15621 q^{15} +7.03010 q^{16} +0.854911 q^{17} -6.84987 q^{18} +2.32532 q^{19} -17.3972 q^{20} -1.74514 q^{21} +1.11790 q^{22} -7.12324 q^{23} -3.47688 q^{24} +10.1582 q^{25} -11.3314 q^{26} +3.15304 q^{27} +14.0806 q^{28} -6.21366 q^{29} +5.48392 q^{30} -2.50357 q^{31} +5.32370 q^{32} -0.243427 q^{33} +2.17431 q^{34} -12.2684 q^{35} -12.0348 q^{36} -1.00000 q^{37} +5.91402 q^{38} +2.46747 q^{39} -24.4426 q^{40} +4.91852 q^{41} -4.43845 q^{42} -4.07895 q^{43} +1.96407 q^{44} +10.4859 q^{45} -18.1166 q^{46} -12.1110 q^{47} -3.89340 q^{48} +2.92951 q^{49} +25.8356 q^{50} -0.473465 q^{51} -19.9086 q^{52} +0.257652 q^{53} +8.01918 q^{54} -1.71130 q^{55} +19.7828 q^{56} -1.28781 q^{57} -15.8033 q^{58} -8.95821 q^{59} +9.63491 q^{60} -8.51888 q^{61} -6.36735 q^{62} -8.48685 q^{63} -0.520365 q^{64} +17.3464 q^{65} -0.619111 q^{66} +14.9954 q^{67} +3.82012 q^{68} +3.94498 q^{69} -31.2024 q^{70} -1.64622 q^{71} -16.9085 q^{72} -6.56007 q^{73} -2.54331 q^{74} -5.62581 q^{75} +10.3906 q^{76} +1.38505 q^{77} +6.27554 q^{78} +6.87708 q^{79} -27.3707 q^{80} +6.33364 q^{81} +12.5093 q^{82} +12.2348 q^{83} -7.79807 q^{84} -3.32847 q^{85} -10.3741 q^{86} +3.44124 q^{87} +2.75946 q^{88} -11.6454 q^{89} +26.6690 q^{90} -14.0394 q^{91} -31.8298 q^{92} +1.38652 q^{93} -30.8020 q^{94} -9.05331 q^{95} -2.94836 q^{96} -2.52625 q^{97} +7.45066 q^{98} -1.18382 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 79 q - 11 q^{2} - 11 q^{3} + 79 q^{4} - 16 q^{5} - 14 q^{6} - 15 q^{7} - 42 q^{8} + 76 q^{9}+O(q^{10}) $ 79 * q - 11 * q^2 - 11 * q^3 + 79 * q^4 - 16 * q^5 - 14 * q^6 - 15 * q^7 - 42 * q^8 + 76 * q^9	\( 79 q - 11 q^{2} - 11 q^{3} + 79 q^{4} - 16 q^{5} - 14 q^{6} - 15 q^{7} - 42 q^{8} + 76 q^{9} - 13 q^{10} - 5 q^{11} - 40 q^{12} - 18 q^{13} - 42 q^{14} - 49 q^{15} + 83 q^{16} - 62 q^{17} - 33 q^{18} - 25 q^{19} - 39 q^{20} - 15 q^{21} - 31 q^{22} - 94 q^{23} - 39 q^{24} + 71 q^{25} - 35 q^{26} - 47 q^{27} - 13 q^{28} - 17 q^{29} + 15 q^{30} - 37 q^{31} - 105 q^{32} - 60 q^{33} + 9 q^{34} - 60 q^{35} + 43 q^{36} - 79 q^{37} - 80 q^{38} - 41 q^{39} - 64 q^{40} - 37 q^{41} - 30 q^{42} - 20 q^{43} - 14 q^{44} - 8 q^{45} + 61 q^{46} - 148 q^{47} - 39 q^{48} + 82 q^{49} - 90 q^{50} - 45 q^{51} - 27 q^{52} - 70 q^{53} - 41 q^{54} - 105 q^{55} - 68 q^{56} - 31 q^{57} - 14 q^{58} - 96 q^{59} - 74 q^{60} - 21 q^{61} + 4 q^{62} - 60 q^{63} + 132 q^{64} - 15 q^{65} + 71 q^{66} - 44 q^{67} - 166 q^{68} - 72 q^{69} - 9 q^{70} - 55 q^{71} - 126 q^{72} - 27 q^{73} + 11 q^{74} - 39 q^{75} - 4 q^{76} - 104 q^{77} - 47 q^{78} - 49 q^{79} - 82 q^{80} + 55 q^{81} + 20 q^{82} - 52 q^{83} - 29 q^{84} + 3 q^{85} - 32 q^{86} - 113 q^{87} + 14 q^{88} - 68 q^{89} - 39 q^{90} - 30 q^{91} - 179 q^{92} - 53 q^{93} - 33 q^{94} - 86 q^{95} + 33 q^{96} - 57 q^{97} - 116 q^{98} + q^{99}+O(q^{100}) \) 79 * q - 11 * q^2 - 11 * q^3 + 79 * q^4 - 16 * q^5 - 14 * q^6 - 15 * q^7 - 42 * q^8 + 76 * q^9 - 13 * q^10 - 5 * q^11 - 40 * q^12 - 18 * q^13 - 42 * q^14 - 49 * q^15 + 83 * q^16 - 62 * q^17 - 33 * q^18 - 25 * q^19 - 39 * q^20 - 15 * q^21 - 31 * q^22 - 94 * q^23 - 39 * q^24 + 71 * q^25 - 35 * q^26 - 47 * q^27 - 13 * q^28 - 17 * q^29 + 15 * q^30 - 37 * q^31 - 105 * q^32 - 60 * q^33 + 9 * q^34 - 60 * q^35 + 43 * q^36 - 79 * q^37 - 80 * q^38 - 41 * q^39 - 64 * q^40 - 37 * q^41 - 30 * q^42 - 20 * q^43 - 14 * q^44 - 8 * q^45 + 61 * q^46 - 148 * q^47 - 39 * q^48 + 82 * q^49 - 90 * q^50 - 45 * q^51 - 27 * q^52 - 70 * q^53 - 41 * q^54 - 105 * q^55 - 68 * q^56 - 31 * q^57 - 14 * q^58 - 96 * q^59 - 74 * q^60 - 21 * q^61 + 4 * q^62 - 60 * q^63 + 132 * q^64 - 15 * q^65 + 71 * q^66 - 44 * q^67 - 166 * q^68 - 72 * q^69 - 9 * q^70 - 55 * q^71 - 126 * q^72 - 27 * q^73 + 11 * q^74 - 39 * q^75 - 4 * q^76 - 104 * q^77 - 47 * q^78 - 49 * q^79 - 82 * q^80 + 55 * q^81 + 20 * q^82 - 52 * q^83 - 29 * q^84 + 3 * q^85 - 32 * q^86 - 113 * q^87 + 14 * q^88 - 68 * q^89 - 39 * q^90 - 30 * q^91 - 179 * q^92 - 53 * q^93 - 33 * q^94 - 86 * q^95 + 33 * q^96 - 57 * q^97 - 116 * q^98 + 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 $

$2$	2.54331	1.79839	0.899197	−	0.437544i	$-0.144151\pi$
$2$	2.54331	1.79839	0.899197	+	0.437544i	$0.144151\pi$
$3$	−0.553818	−0.319747	−0.159874	−	0.987138i	$-0.551109\pi$
$3$	−0.553818	−0.319747	−0.159874	+	0.987138i	$0.551109\pi$
$4$	4.46844	2.23422
$5$	−3.89336	−1.74116	−0.870581	−	0.492025i	$-0.836257\pi$
$5$	−3.89336	−1.74116	−0.870581	+	0.492025i	$0.836257\pi$
$6$	−1.40853	−0.575031
$7$	3.15111	1.19101	0.595504	−	0.803352i	$-0.296952\pi$
$7$	3.15111	1.19101	0.595504	+	0.803352i	$0.296952\pi$
$8$	6.27803	2.21962
$9$	−2.69329	−0.897762
$10$	−9.90203	−3.13130
$11$	0.439543	0.132527	0.0662637	−	0.997802i	$-0.478892\pi$
$11$	0.439543	0.132527	0.0662637	+	0.997802i	$0.478892\pi$
$12$	−2.47471	−0.714386
$13$	−4.45537	−1.23570	−0.617849	−	0.786297i	$-0.711996\pi$
$13$	−4.45537	−1.23570	−0.617849	+	0.786297i	$0.711996\pi$
$14$	8.01427	2.14190
$15$	2.15621	0.556731
$16$	7.03010	1.75753
$17$	0.854911	0.207346	0.103673	−	0.994611i	$-0.466940\pi$
$17$	0.854911	0.207346	0.103673	+	0.994611i	$0.466940\pi$
$18$	−6.84987	−1.61453
$19$	2.32532	0.533465	0.266733	−	0.963771i	$-0.414056\pi$
$19$	2.32532	0.533465	0.266733	+	0.963771i	$0.414056\pi$
$20$	−17.3972	−3.89014
$21$	−1.74514	−0.380821
$22$	1.11790	0.238336
$23$	−7.12324	−1.48530	−0.742649	−	0.669680i	$-0.766431\pi$
$23$	−7.12324	−1.48530	−0.742649	+	0.669680i	$0.766431\pi$
$24$	−3.47688	−0.709716
$25$	10.1582	2.03165
$26$	−11.3314	−2.22227
$27$	3.15304	0.606804
$28$	14.0806	2.66098
$29$	−6.21366	−1.15385	−0.576923	−	0.816798i	$-0.695747\pi$
$29$	−6.21366	−1.15385	−0.576923	+	0.816798i	$0.695747\pi$
$30$	5.48392	1.00122
$31$	−2.50357	−0.449654	−0.224827	−	0.974399i	$-0.572182\pi$
$31$	−2.50357	−0.449654	−0.224827	+	0.974399i	$0.572182\pi$
$32$	5.32370	0.941106
$33$	−0.243427	−0.0423752
$34$	2.17431	0.372891
$35$	−12.2684	−2.07374
$36$	−12.0348	−2.00580
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	5.91402	0.959381
$39$	2.46747	0.395111
$40$	−24.4426	−3.86471
$41$	4.91852	0.768144	0.384072	−	0.923303i	$-0.374522\pi$
$41$	4.91852	0.768144	0.384072	+	0.923303i	$0.374522\pi$
$42$	−4.43845	−0.684867
$43$	−4.07895	−0.622034	−0.311017	−	0.950404i	$-0.600670\pi$
$43$	−4.07895	−0.622034	−0.311017	+	0.950404i	$0.600670\pi$
$44$	1.96407	0.296095
$45$	10.4859	1.56315
$46$	−18.1166	−2.67115
$47$	−12.1110	−1.76657	−0.883283	−	0.468840i	$-0.844672\pi$
$47$	−12.1110	−1.76657	−0.883283	+	0.468840i	$0.844672\pi$
$48$	−3.89340	−0.561964
$49$	2.92951	0.418501
$50$	25.8356	3.65370
$51$	−0.473465	−0.0662984
$52$	−19.9086	−2.76082
$53$	0.257652	0.0353912	0.0176956	−	0.999843i	$-0.494367\pi$
$53$	0.257652	0.0353912	0.0176956	+	0.999843i	$0.494367\pi$
$54$	8.01918	1.09127
$55$	−1.71130	−0.230752
$56$	19.7828	2.64358
$57$	−1.28781	−0.170574
$58$	−15.8033	−2.07507
$59$	−8.95821	−1.16626	−0.583130	−	0.812379i	$-0.698172\pi$
$59$	−8.95821	−1.16626	−0.583130	+	0.812379i	$0.698172\pi$
$60$	9.63491	1.24386
$61$	−8.51888	−1.09073	−0.545365	−	0.838198i	$-0.683609\pi$
$61$	−8.51888	−1.09073	−0.545365	+	0.838198i	$0.683609\pi$
$62$	−6.36735	−0.808654
$63$	−8.48685	−1.06924
$64$	−0.520365	−0.0650456
$65$	17.3464	2.15155
$66$	−0.619111	−0.0762073
$67$	14.9954	1.83198	0.915992	−	0.401196i	$-0.131405\pi$
$67$	14.9954	1.83198	0.915992	+	0.401196i	$0.131405\pi$
$68$	3.82012	0.463258
$69$	3.94498	0.474920
$70$	−31.2024	−3.72940
$71$	−1.64622	−0.195370	−0.0976850	−	0.995217i	$-0.531144\pi$
$71$	−1.64622	−0.195370	−0.0976850	+	0.995217i	$0.531144\pi$
$72$	−16.9085	−1.99269
$73$	−6.56007	−0.767798	−0.383899	−	0.923375i	$-0.625419\pi$
$73$	−6.56007	−0.767798	−0.383899	+	0.923375i	$0.625419\pi$
$74$	−2.54331	−0.295654
$75$	−5.62581	−0.649613
$76$	10.3906	1.19188
$77$	1.38505	0.157841
$78$	6.27554	0.710565
$79$	6.87708	0.773732	0.386866	−	0.922136i	$-0.373558\pi$
$79$	6.87708	0.773732	0.386866	+	0.922136i	$0.373558\pi$
$80$	−27.3707	−3.06014
$81$	6.33364	0.703738
$82$	12.5093	1.38143
$83$	12.2348	1.34294	0.671472	−	0.741030i	$-0.265662\pi$
$83$	12.2348	1.34294	0.671472	+	0.741030i	$0.265662\pi$
$84$	−7.79807	−0.850840
$85$	−3.32847	−0.361024
$86$	−10.3741	−1.11866
$87$	3.44124	0.368939
$88$	2.75946	0.294160
$89$	−11.6454	−1.23441	−0.617203	−	0.786804i	$-0.711734\pi$
$89$	−11.6454	−1.23441	−0.617203	+	0.786804i	$0.711734\pi$
$90$	26.6690	2.81116
$91$	−14.0394	−1.47173
$92$	−31.8298	−3.31849
$93$	1.38652	0.143775
$94$	−30.8020	−3.17698
$95$	−9.05331	−0.928850
$96$	−2.94836	−0.300916
$97$	−2.52625	−0.256502	−0.128251	−	0.991742i	$-0.540936\pi$
$97$	−2.52625	−0.256502	−0.128251	+	0.991742i	$0.540936\pi$
$98$	7.45066	0.752630
$99$	−1.18382	−0.118978
$100$	45.3915	4.53915
$101$	−16.3939	−1.63125	−0.815626	−	0.578580i	$-0.803607\pi$
$101$	−16.3939	−1.63125	−0.815626	+	0.578580i	$0.803607\pi$
$102$	−1.20417	−0.119231
$103$	−14.2207	−1.40121	−0.700604	−	0.713550i	$-0.747086\pi$
$103$	−14.2207	−1.40121	−0.700604	+	0.713550i	$0.747086\pi$
$104$	−27.9710	−2.74278
$105$	6.79446	0.663072
$106$	0.655290	0.0636474
$107$	19.8355	1.91757	0.958784	−	0.284135i	$-0.0917064\pi$
$107$	19.8355	1.91757	0.958784	+	0.284135i	$0.0917064\pi$
$108$	14.0892	1.35573
$109$	−1.00000	−0.0957826
$109$	−1.00000	−0.0957826
$110$	−4.35237	−0.414982
$111$	0.553818	0.0525661
$112$	22.1526	2.09323
$113$	−14.8394	−1.39597	−0.697986	−	0.716112i	$-0.745920\pi$
$113$	−14.8394	−1.39597	−0.697986	+	0.716112i	$0.745920\pi$
$114$	−3.27529	−0.306759
$115$	27.7333	2.58615
$116$	−27.7654	−2.57795
$117$	11.9996	1.10936
$118$	−22.7835	−2.09739
$119$	2.69392	0.246951
$120$	13.5368	1.23573
$121$	−10.8068	−0.982437
$122$	−21.6662	−1.96156
$123$	−2.72397	−0.245612
$124$	−11.1870	−1.00463
$125$	−20.0828	−1.79626
$126$	−21.5847	−1.92292
$127$	13.0597	1.15886	0.579429	−	0.815023i	$-0.303276\pi$
$127$	13.0597	1.15886	0.579429	+	0.815023i	$0.303276\pi$
$128$	−11.9709	−1.05808
$129$	2.25900	0.198894
$130$	44.1172	3.86934
$131$	12.2197	1.06764	0.533821	−	0.845597i	$-0.320755\pi$
$131$	12.2197	1.06764	0.533821	+	0.845597i	$0.320755\pi$
$132$	−1.08774	−0.0946756
$133$	7.32735	0.635362
$134$	38.1381	3.29463
$135$	−12.2759	−1.05654
$136$	5.36715	0.460230
$137$	14.0149	1.19737	0.598685	−	0.800984i	$-0.295690\pi$
$137$	14.0149	1.19737	0.598685	+	0.800984i	$0.295690\pi$
$138$	10.0333	0.854093
$139$	0.688571	0.0584039	0.0292019	−	0.999574i	$-0.490703\pi$
$139$	0.688571	0.0584039	0.0292019	+	0.999574i	$0.490703\pi$
$140$	−54.8207	−4.63319
$141$	6.70727	0.564854
$142$	−4.18685	−0.351352
$143$	−1.95833	−0.163764
$144$	−18.9341	−1.57784
$145$	24.1920	2.00903
$146$	−16.6843	−1.38080
$147$	−1.62242	−0.133815
$148$	−4.46844	−0.367304
$149$	−7.84065	−0.642331	−0.321166	−	0.947023i	$-0.604075\pi$
$149$	−7.84065	−0.642331	−0.321166	+	0.947023i	$0.604075\pi$
$150$	−14.3082	−1.16826
$151$	−3.37512	−0.274664	−0.137332	−	0.990525i	$-0.543853\pi$
$151$	−3.37512	−0.274664	−0.137332	+	0.990525i	$0.543853\pi$
$152$	14.5984	1.18409
$153$	−2.30252	−0.186148
$154$	3.52262	0.283861
$155$	9.74727	0.782920
$156$	11.0257	0.882766
$157$	16.8723	1.34655	0.673276	−	0.739391i	$-0.264887\pi$
$157$	16.8723	1.34655	0.673276	+	0.739391i	$0.264887\pi$
$158$	17.4906	1.39147
$159$	−0.142692	−0.0113162
$160$	−20.7271	−1.63862
$161$	−22.4461	−1.76900
$162$	16.1084	1.26560
$163$	−2.98776	−0.234020	−0.117010	−	0.993131i	$-0.537331\pi$
$163$	−2.98776	−0.234020	−0.117010	+	0.993131i	$0.537331\pi$
$164$	21.9781	1.71620
$165$	0.947748	0.0737821
$166$	31.1169	2.41514
$167$	−22.9095	−1.77279	−0.886394	−	0.462931i	$-0.846798\pi$
$167$	−22.9095	−1.77279	−0.886394	+	0.462931i	$0.846798\pi$
$168$	−10.9561	−0.845278
$169$	6.85037	0.526951
$170$	−8.46535	−0.649263
$171$	−6.26276	−0.478925
$172$	−18.2266	−1.38976
$173$	−25.9997	−1.97672	−0.988360	−	0.152135i	$-0.951385\pi$
$173$	−25.9997	−1.97672	−0.988360	+	0.152135i	$0.951385\pi$
$174$	8.75214	0.663498
$175$	32.0097	2.41971
$176$	3.09003	0.232920
$177$	4.96122	0.372908
$178$	−29.6178	−2.21995
$179$	14.3289	1.07099	0.535494	−	0.844539i	$-0.320125\pi$
$179$	14.3289	1.07099	0.535494	+	0.844539i	$0.320125\pi$
$180$	46.8557	3.49242
$181$	17.2492	1.28212	0.641061	−	0.767490i	$-0.278495\pi$
$181$	17.2492	1.28212	0.641061	+	0.767490i	$0.278495\pi$
$182$	−35.7066	−2.64675
$183$	4.71791	0.348758
$184$	−44.7199	−3.29679
$185$	3.89336	0.286245
$186$	3.52635	0.258565
$187$	0.375770	0.0274791
$188$	−54.1172	−3.94690
$189$	9.93560	0.722708
$190$	−23.0254	−1.67044
$191$	−10.5988	−0.766902	−0.383451	−	0.923561i	$-0.625265\pi$
$191$	−10.5988	−0.766902	−0.383451	+	0.923561i	$0.625265\pi$
$192$	0.288188	0.0207982
$193$	16.1418	1.16191	0.580957	−	0.813935i	$-0.302679\pi$
$193$	16.1418	1.16191	0.580957	+	0.813935i	$0.302679\pi$
$194$	−6.42505	−0.461292
$195$	−9.60673	−0.687952
$196$	13.0903	0.935025
$197$	3.45990	0.246508	0.123254	−	0.992375i	$-0.460667\pi$
$197$	3.45990	0.246508	0.123254	+	0.992375i	$0.460667\pi$
$198$	−3.01081	−0.213969
$199$	−10.1114	−0.716775	−0.358388	−	0.933573i	$-0.616673\pi$
$199$	−10.1114	−0.716775	−0.358388	+	0.933573i	$0.616673\pi$
$200$	63.7736	4.50947
$201$	−8.30475	−0.585772
$202$	−41.6948	−2.93363
$203$	−19.5799	−1.37424
$204$	−2.11565	−0.148125
$205$	−19.1495	−1.33746
$206$	−36.1677	−2.51993
$207$	19.1849	1.33344
$208$	−31.3217	−2.17177
$209$	1.02208	0.0706987
$210$	17.2805	1.19246
$211$	11.3639	0.782322	0.391161	−	0.920322i	$-0.372074\pi$
$211$	11.3639	0.782322	0.391161	+	0.920322i	$0.372074\pi$
$212$	1.15130	0.0790719
$213$	0.911705	0.0624690
$214$	50.4479	3.44854
$215$	15.8808	1.08306
$216$	19.7949	1.34687
$217$	−7.88901	−0.535541
$218$	−2.54331	−0.172255
$219$	3.63309	0.245501
$220$	−7.64684	−0.515550
$221$	−3.80895	−0.256218
$222$	1.40853	0.0945346
$223$	0.0455267	0.00304870	0.00152435	−	0.999999i	$-0.499515\pi$
$223$	0.0455267	0.00304870	0.00152435	+	0.999999i	$0.499515\pi$
$224$	16.7756	1.12087
$225$	−27.3590	−1.82393
$226$	−37.7412	−2.51051
$227$	19.1634	1.27192	0.635959	−	0.771723i	$-0.280605\pi$
$227$	19.1634	1.27192	0.635959	+	0.771723i	$0.280605\pi$
$228$	−5.75449	−0.381100
$229$	21.9972	1.45361	0.726807	−	0.686842i	$-0.241004\pi$
$229$	21.9972	1.45361	0.726807	+	0.686842i	$0.241004\pi$
$230$	70.5345	4.65091
$231$	−0.767066	−0.0504692
$232$	−39.0095	−2.56110
$233$	17.2796	1.13203	0.566014	−	0.824396i	$-0.308485\pi$
$233$	17.2796	1.13203	0.566014	+	0.824396i	$0.308485\pi$
$234$	30.5187	1.99507
$235$	47.1523	3.07588
$236$	−40.0293	−2.60568
$237$	−3.80865	−0.247398
$238$	6.85149	0.444116
$239$	−6.91073	−0.447018	−0.223509	−	0.974702i	$-0.571751\pi$
$239$	−6.91073	−0.447018	−0.223509	+	0.974702i	$0.571751\pi$
$240$	15.1584	0.978470
$241$	11.2975	0.727736	0.363868	−	0.931450i	$-0.381456\pi$
$241$	11.2975	0.727736	0.363868	+	0.931450i	$0.381456\pi$
$242$	−27.4851	−1.76681
$243$	−12.9668	−0.831822
$244$	−38.0661	−2.43693
$245$	−11.4056	−0.728679
$246$	−6.92790	−0.441707
$247$	−10.3602	−0.659203
$248$	−15.7174	−0.998059
$249$	−6.77586	−0.429403
$250$	−51.0769	−3.23039
$251$	18.3496	1.15822	0.579109	−	0.815250i	$-0.303401\pi$
$251$	18.3496	1.15822	0.579109	+	0.815250i	$0.303401\pi$
$252$	−37.9230	−2.38892
$253$	−3.13097	−0.196843
$254$	33.2148	2.08408
$255$	1.84337	0.115436
$256$	−29.4049	−1.83781
$257$	−24.1340	−1.50543	−0.752717	−	0.658344i	$-0.771257\pi$
$257$	−24.1340	−1.50543	−0.752717	+	0.658344i	$0.771257\pi$
$258$	5.74534	0.357689
$259$	−3.15111	−0.195801
$260$	77.5112	4.80704
$261$	16.7351	1.03588
$262$	31.0786	1.92004
$263$	16.4684	1.01548	0.507742	−	0.861509i	$-0.330480\pi$
$263$	16.4684	1.01548	0.507742	+	0.861509i	$0.330480\pi$
$264$	−1.52824	−0.0940568
$265$	−1.00313	−0.0616219
$266$	18.6358	1.14263
$267$	6.44942	0.394698
$268$	67.0063	4.09306
$269$	−0.627153	−0.0382382	−0.0191191	−	0.999817i	$-0.506086\pi$
$269$	−0.627153	−0.0382382	−0.0191191	+	0.999817i	$0.506086\pi$
$270$	−31.2215	−1.90008
$271$	3.91953	0.238094	0.119047	−	0.992889i	$-0.462016\pi$
$271$	3.91953	0.238094	0.119047	+	0.992889i	$0.462016\pi$
$272$	6.01011	0.364417
$273$	7.77527	0.470581
$274$	35.6442	2.15334
$275$	4.46498	0.269248
$276$	17.6279	1.06108
$277$	−16.1732	−0.971754	−0.485877	−	0.874027i	$-0.661500\pi$
$277$	−16.1732	−0.971754	−0.485877	+	0.874027i	$0.661500\pi$
$278$	1.75125	0.105033
$279$	6.74282	0.403682
$280$	−77.0214	−4.60291
$281$	15.2379	0.909015	0.454507	−	0.890743i	$-0.349815\pi$
$281$	15.2379	0.909015	0.454507	+	0.890743i	$0.349815\pi$
$282$	17.0587	1.01583
$283$	25.4283	1.51156	0.755779	−	0.654827i	$-0.227259\pi$
$283$	25.4283	1.51156	0.755779	+	0.654827i	$0.227259\pi$
$284$	−7.35603	−0.436500
$285$	5.01389	0.296997
$286$	−4.98065	−0.294512
$287$	15.4988	0.914866
$288$	−14.3382	−0.844889
$289$	−16.2691	−0.957007
$290$	61.5278	3.61304
$291$	1.39908	0.0820157
$292$	−29.3133	−1.71543
$293$	−25.5269	−1.49129	−0.745647	−	0.666341i	$-0.767860\pi$
$293$	−25.5269	−1.49129	−0.745647	+	0.666341i	$0.767860\pi$
$294$	−4.12631	−0.240651
$295$	34.8775	2.03065
$296$	−6.27803	−0.364903
$297$	1.38590	0.0804181
$298$	−19.9412	−1.15516
$299$	31.7367	1.83538
$300$	−25.1386	−1.45138
$301$	−12.8532	−0.740848
$302$	−8.58399	−0.493953
$303$	9.07923	0.521588
$304$	16.3473	0.937579
$305$	33.1670	1.89914
$306$	−5.85603	−0.334767
$307$	1.29575	0.0739521	0.0369761	−	0.999316i	$-0.488227\pi$
$307$	1.29575	0.0739521	0.0369761	+	0.999316i	$0.488227\pi$
$308$	6.18902	0.352652
$309$	7.87569	0.448032
$310$	24.7904	1.40800
$311$	27.9768	1.58642	0.793210	−	0.608948i	$-0.208408\pi$
$311$	27.9768	1.58642	0.793210	+	0.608948i	$0.208408\pi$
$312$	15.4908	0.876995
$313$	−17.2264	−0.973691	−0.486846	−	0.873488i	$-0.661853\pi$
$313$	−17.2264	−0.973691	−0.486846	+	0.873488i	$0.661853\pi$
$314$	42.9114	2.42163
$315$	33.0423	1.86172
$316$	30.7298	1.72869
$317$	7.98631	0.448556	0.224278	−	0.974525i	$-0.427998\pi$
$317$	7.98631	0.448556	0.224278	+	0.974525i	$0.427998\pi$
$318$	−0.362911	−0.0203511
$319$	−2.73117	−0.152916
$320$	2.02597	0.113255
$321$	−10.9852	−0.613137
$322$	−57.0876	−3.18137
$323$	1.98794	0.110612
$324$	28.3015	1.57231
$325$	−45.2587	−2.51050
$326$	−7.59882	−0.420860
$327$	0.553818	0.0306262
$328$	30.8786	1.70498
$329$	−38.1630	−2.10399
$330$	2.41042	0.132689
$331$	34.2728	1.88381	0.941903	−	0.335885i	$-0.109035\pi$
$331$	34.2728	1.88381	0.941903	+	0.335885i	$0.109035\pi$
$332$	54.6705	3.00044
$333$	2.69329	0.147591
$334$	−58.2660	−3.18817
$335$	−58.3826	−3.18978
$336$	−12.2685	−0.669303
$337$	−22.9814	−1.25188	−0.625938	−	0.779873i	$-0.715284\pi$
$337$	−22.9814	−1.25188	−0.625938	+	0.779873i	$0.715284\pi$
$338$	17.4226	0.947666
$339$	8.21832	0.446358
$340$	−14.8731	−0.806607
$341$	−1.10043	−0.0595914
$342$	−15.9282	−0.861296
$343$	−12.8266	−0.692570
$344$	−25.6078	−1.38068
$345$	−15.3592	−0.826912
$346$	−66.1253	−3.55492
$347$	9.39864	0.504546	0.252273	−	0.967656i	$-0.418822\pi$
$347$	9.39864	0.504546	0.252273	+	0.967656i	$0.418822\pi$
$348$	15.3770	0.824292
$349$	−16.6847	−0.893109	−0.446555	−	0.894756i	$-0.647349\pi$
$349$	−16.6847	−0.893109	−0.446555	+	0.894756i	$0.647349\pi$
$350$	81.4107	4.35159
$351$	−14.0480	−0.749827
$352$	2.34000	0.124722
$353$	−30.0716	−1.60055	−0.800276	−	0.599632i	$-0.795313\pi$
$353$	−30.0716	−1.60055	−0.800276	+	0.599632i	$0.795313\pi$
$354$	12.6179	0.670636
$355$	6.40931	0.340171
$356$	−52.0367	−2.75794
$357$	−1.49194	−0.0789620
$358$	36.4428	1.92606
$359$	16.5752	0.874806	0.437403	−	0.899266i	$-0.355898\pi$
$359$	16.5752	0.874806	0.437403	+	0.899266i	$0.355898\pi$
$360$	65.8309	3.46959
$361$	−13.5929	−0.715415
$362$	43.8701	2.30576
$363$	5.98500	0.314131
$364$	−62.7342	−3.28817
$365$	25.5407	1.33686
$366$	11.9991	0.627204
$367$	−9.79581	−0.511337	−0.255668	−	0.966765i	$-0.582296\pi$
$367$	−9.79581	−0.511337	−0.255668	+	0.966765i	$0.582296\pi$
$368$	−50.0771	−2.61045
$369$	−13.2470	−0.689610
$370$	9.90203	0.514782
$371$	0.811891	0.0421513
$372$	6.19559	0.321226
$373$	10.4833	0.542802	0.271401	−	0.962466i	$-0.412513\pi$
$373$	10.4833	0.542802	0.271401	+	0.962466i	$0.412513\pi$
$374$	0.955702	0.0494182
$375$	11.1222	0.574349
$376$	−76.0330	−3.92110
$377$	27.6842	1.42581
$378$	25.2693	1.29971
$379$	7.79982	0.400650	0.200325	−	0.979730i	$-0.435800\pi$
$379$	7.79982	0.400650	0.200325	+	0.979730i	$0.435800\pi$
$380$	−40.4542	−2.07526
$381$	−7.23268	−0.370541
$382$	−26.9561	−1.37919
$383$	−0.579067	−0.0295890	−0.0147945	−	0.999891i	$-0.504709\pi$
$383$	−0.579067	−0.0295890	−0.0147945	+	0.999891i	$0.504709\pi$
$384$	6.62967	0.338319
$385$	−5.39250	−0.274827
$386$	41.0537	2.08958
$387$	10.9858	0.558439
$388$	−11.2884	−0.573082
$389$	5.02559	0.254807	0.127404	−	0.991851i	$-0.459336\pi$
$389$	5.02559	0.254807	0.127404	+	0.991851i	$0.459336\pi$
$390$	−24.4329	−1.23721
$391$	−6.08974	−0.307971
$392$	18.3915	0.928913
$393$	−6.76751	−0.341376
$394$	8.79960	0.443318
$395$	−26.7749	−1.34719
$396$	−5.28981	−0.265823
$397$	−9.17221	−0.460340	−0.230170	−	0.973150i	$-0.573928\pi$
$397$	−9.17221	−0.460340	−0.230170	+	0.973150i	$0.573928\pi$
$398$	−25.7163	−1.28904
$399$	−4.05802	−0.203155
$400$	71.4134	3.57067
$401$	−22.8407	−1.14061	−0.570306	−	0.821433i	$-0.693175\pi$
$401$	−22.8407	−1.14061	−0.570306	+	0.821433i	$0.693175\pi$
$402$	−21.1216	−1.05345
$403$	11.1543	0.555636
$404$	−73.2551	−3.64458
$405$	−24.6591	−1.22532
$406$	−49.7979	−2.47143
$407$	−0.439543	−0.0217874
$408$	−2.97243	−0.147157
$409$	−8.40076	−0.415391	−0.207696	−	0.978194i	$-0.566596\pi$
$409$	−8.40076	−0.415391	−0.207696	+	0.978194i	$0.566596\pi$
$410$	−48.7033	−2.40528
$411$	−7.76168	−0.382856
$412$	−63.5445	−3.13061
$413$	−28.2283	−1.38903
$414$	48.7933	2.39806
$415$	−47.6345	−2.33828
$416$	−23.7191	−1.16292
$417$	−0.381343	−0.0186745
$418$	2.59947	0.127144
$419$	−6.46180	−0.315680	−0.157840	−	0.987465i	$-0.550453\pi$
$419$	−6.46180	−0.315680	−0.157840	+	0.987465i	$0.550453\pi$
$420$	30.3607	1.48145
$421$	−10.1814	−0.496213	−0.248106	−	0.968733i	$-0.579808\pi$
$421$	−10.1814	−0.496213	−0.248106	+	0.968733i	$0.579808\pi$
$422$	28.9019	1.40692
$423$	32.6183	1.58596
$424$	1.61755	0.0785550
$425$	8.68438	0.421254
$426$	2.31875	0.112344
$427$	−26.8439	−1.29907
$428$	88.6337	4.28427
$429$	1.08456	0.0523630
$430$	40.3899	1.94777
$431$	−16.8604	−0.812139	−0.406070	−	0.913842i	$-0.633101\pi$
$431$	−16.8604	−0.812139	−0.406070	+	0.913842i	$0.633101\pi$
$432$	22.1662	1.06647
$433$	14.7797	0.710269	0.355134	−	0.934815i	$-0.384435\pi$
$433$	14.7797	0.710269	0.355134	+	0.934815i	$0.384435\pi$
$434$	−20.0642	−0.963114
$435$	−13.3980	−0.642383
$436$	−4.46844	−0.214000
$437$	−16.5638	−0.792356
$438$	9.24008	0.441508
$439$	−22.3736	−1.06783	−0.533917	−	0.845537i	$-0.679280\pi$
$439$	−22.3736	−1.06783	−0.533917	+	0.845537i	$0.679280\pi$
$440$	−10.7436	−0.512180
$441$	−7.89000	−0.375714
$442$	−9.68735	−0.460780
$443$	−6.13552	−0.291507	−0.145754	−	0.989321i	$-0.546561\pi$
$443$	−6.13552	−0.291507	−0.145754	+	0.989321i	$0.546561\pi$
$444$	2.47471	0.117444
$445$	45.3396	2.14930
$446$	0.115789	0.00548276
$447$	4.34230	0.205384
$448$	−1.63973	−0.0774699
$449$	7.69686	0.363237	0.181619	−	0.983369i	$-0.441866\pi$
$449$	7.69686	0.363237	0.181619	+	0.983369i	$0.441866\pi$
$450$	−69.5825	−3.28015
$451$	2.16190	0.101800
$452$	−66.3089	−3.11891
$453$	1.86920	0.0878229
$454$	48.7385	2.28741
$455$	54.6603	2.56252
$456$	−8.08488	−0.378609
$457$	−14.9086	−0.697393	−0.348697	−	0.937236i	$-0.613376\pi$
$457$	−14.9086	−0.697393	−0.348697	+	0.937236i	$0.613376\pi$
$458$	55.9457	2.61417
$459$	2.69557	0.125819
$460$	123.925	5.77802
$461$	−12.0202	−0.559836	−0.279918	−	0.960024i	$-0.590307\pi$
$461$	−12.0202	−0.559836	−0.279918	+	0.960024i	$0.590307\pi$
$462$	−1.95089	−0.0907636
$463$	4.10208	0.190640	0.0953199	−	0.995447i	$-0.469613\pi$
$463$	4.10208	0.190640	0.0953199	+	0.995447i	$0.469613\pi$
$464$	−43.6826	−2.02792
$465$	−5.39822	−0.250336
$466$	43.9476	2.03583
$467$	19.0011	0.879267	0.439633	−	0.898177i	$-0.355108\pi$
$467$	19.0011	0.879267	0.439633	+	0.898177i	$0.355108\pi$
$468$	53.6195	2.47856
$469$	47.2523	2.18191
$470$	119.923	5.53164
$471$	−9.34416	−0.430556
$472$	−56.2399	−2.58865
$473$	−1.79288	−0.0824366
$474$	−9.68659	−0.444920
$475$	23.6211	1.08381
$476$	12.0376	0.551744
$477$	−0.693930	−0.0317729
$478$	−17.5761	−0.803914
$479$	35.3364	1.61456	0.807280	−	0.590168i	$-0.200939\pi$
$479$	35.3364	1.61456	0.807280	+	0.590168i	$0.200939\pi$
$480$	11.4790	0.523943
$481$	4.45537	0.203148
$482$	28.7331	1.30876
$483$	12.4311	0.565634
$484$	−48.2896	−2.19498
$485$	9.83559	0.446611
$486$	−32.9787	−1.49594
$487$	23.7289	1.07526	0.537629	−	0.843181i	$-0.319320\pi$
$487$	23.7289	1.07526	0.537629	+	0.843181i	$0.319320\pi$
$488$	−53.4817	−2.42100
$489$	1.65468	0.0748271
$490$	−29.0081	−1.31045
$491$	−10.9944	−0.496171	−0.248086	−	0.968738i	$-0.579801\pi$
$491$	−10.9944	−0.496171	−0.248086	+	0.968738i	$0.579801\pi$
$492$	−12.1719	−0.548751
$493$	−5.31212	−0.239246
$494$	−26.3492	−1.18551
$495$	4.60902	0.207160
$496$	−17.6003	−0.790278
$497$	−5.18741	−0.232687
$498$	−17.2331	−0.772235
$499$	−23.1569	−1.03664	−0.518322	−	0.855186i	$-0.673443\pi$
$499$	−23.1569	−1.03664	−0.518322	+	0.855186i	$0.673443\pi$
$500$	−89.7389	−4.01325
$501$	12.6877	0.566844
$502$	46.6689	2.08293
$503$	33.4615	1.49198	0.745988	−	0.665960i	$-0.231978\pi$
$503$	33.4615	1.49198	0.745988	+	0.665960i	$0.231978\pi$
$504$	−53.2806	−2.37331
$505$	63.8272	2.84027
$506$	−7.96305	−0.354001
$507$	−3.79386	−0.168491
$508$	58.3564	2.58914
$509$	5.80218	0.257177	0.128589	−	0.991698i	$-0.458955\pi$
$509$	5.80218	0.257177	0.128589	+	0.991698i	$0.458955\pi$
$510$	4.68827	0.207600
$511$	−20.6715	−0.914454
$512$	−50.8442	−2.24702
$513$	7.33184	0.323709
$514$	−61.3802	−2.70737
$515$	55.3663	2.43973
$516$	10.0942	0.444373
$517$	−5.32329	−0.234118
$518$	−8.01427	−0.352127
$519$	14.3991	0.632050
$520$	108.901	4.77562
$521$	−18.1223	−0.793954	−0.396977	−	0.917829i	$-0.629941\pi$
$521$	−18.1223	−0.793954	−0.396977	+	0.917829i	$0.629941\pi$
$522$	42.5627	1.86292
$523$	10.4543	0.457136	0.228568	−	0.973528i	$-0.426596\pi$
$523$	10.4543	0.457136	0.228568	+	0.973528i	$0.426596\pi$
$524$	54.6032	2.38535
$525$	−17.7276	−0.773694
$526$	41.8843	1.82624
$527$	−2.14033	−0.0932340
$528$	−1.71132	−0.0744755
$529$	27.7406	1.20611
$530$	−2.55128	−0.110820
$531$	24.1270	1.04702
$532$	32.7419	1.41954
$533$	−21.9138	−0.949194
$534$	16.4029	0.709822
$535$	−77.2266	−3.33880
$536$	94.1418	4.06631
$537$	−7.93558	−0.342446
$538$	−1.59505	−0.0687673
$539$	1.28765	0.0554628
$540$	−54.8543	−2.36055
$541$	14.0411	0.603674	0.301837	−	0.953359i	$-0.402400\pi$
$541$	14.0411	0.603674	0.301837	+	0.953359i	$0.402400\pi$
$542$	9.96859	0.428187
$543$	−9.55291	−0.409955
$544$	4.55129	0.195135
$545$	3.89336	0.166773
$546$	19.7749	0.846289
$547$	11.5326	0.493099	0.246550	−	0.969130i	$-0.420703\pi$
$547$	11.5326	0.493099	0.246550	+	0.969130i	$0.420703\pi$
$548$	62.6246	2.67519
$549$	22.9438	0.979216
$550$	11.3558	0.484215
$551$	−14.4488	−0.615537
$552$	24.7667	1.05414
$553$	21.6704	0.921521
$554$	−41.1336	−1.74760
$555$	−2.15621	−0.0915261
$556$	3.07684	0.130487
$557$	−0.440767	−0.0186759	−0.00933796	−	0.999956i	$-0.502972\pi$
$557$	−0.440767	−0.0186759	−0.00933796	+	0.999956i	$0.502972\pi$
$558$	17.1491	0.725979
$559$	18.1733	0.768647
$560$	−86.2481	−3.64465
$561$	−0.208109	−0.00878635
$562$	38.7547	1.63477
$563$	−27.1083	−1.14248	−0.571239	−	0.820784i	$-0.693537\pi$
$563$	−27.1083	−1.14248	−0.571239	+	0.820784i	$0.693537\pi$
$564$	29.9711	1.26201
$565$	57.7750	2.43061
$566$	64.6722	2.71838
$567$	19.9580	0.838158
$568$	−10.3350	−0.433647
$569$	−8.55110	−0.358481	−0.179240	−	0.983805i	$-0.557364\pi$
$569$	−8.55110	−0.358481	−0.179240	+	0.983805i	$0.557364\pi$
$570$	12.7519	0.534118
$571$	13.8035	0.577660	0.288830	−	0.957380i	$-0.406734\pi$
$571$	13.8035	0.577660	0.288830	+	0.957380i	$0.406734\pi$
$572$	−8.75069	−0.365885
$573$	5.86981	0.245215
$574$	39.4183	1.64529
$575$	−72.3595	−3.01760
$576$	1.40149	0.0583955
$577$	11.6321	0.484251	0.242125	−	0.970245i	$-0.422155\pi$
$577$	11.6321	0.484251	0.242125	+	0.970245i	$0.422155\pi$
$578$	−41.3775	−1.72108
$579$	−8.93963	−0.371518
$580$	108.100	4.48863
$581$	38.5532	1.59946
$582$	3.55831	0.147497
$583$	0.113249	0.00469030
$584$	−41.1843	−1.70422
$585$	−46.7187	−1.93158
$586$	−64.9228	−2.68194
$587$	−2.44126	−0.100761	−0.0503807	−	0.998730i	$-0.516043\pi$
$587$	−2.44126	−0.100761	−0.0503807	+	0.998730i	$0.516043\pi$
$588$	−7.24967	−0.298971
$589$	−5.82159	−0.239875
$590$	88.7044	3.65190
$591$	−1.91615	−0.0788200
$592$	−7.03010	−0.288935
$593$	−18.1455	−0.745147	−0.372573	−	0.928003i	$-0.621525\pi$
$593$	−18.1455	−0.745147	−0.372573	+	0.928003i	$0.621525\pi$
$594$	3.52478	0.144623
$595$	−10.4884	−0.429982
$596$	−35.0355	−1.43511
$597$	5.59985	0.229187
$598$	80.7164	3.30074
$599$	5.64311	0.230571	0.115286	−	0.993332i	$-0.463222\pi$
$599$	5.64311	0.230571	0.115286	+	0.993332i	$0.463222\pi$
$600$	−35.3190	−1.44189
$601$	35.2645	1.43847	0.719234	−	0.694768i	$-0.244493\pi$
$601$	35.2645	1.43847	0.719234	+	0.694768i	$0.244493\pi$
$602$	−32.6898	−1.33234
$603$	−40.3870	−1.64469
$604$	−15.0815	−0.613659
$605$	42.0747	1.71058
$606$	23.0913	0.938021
$607$	5.94852	0.241443	0.120722	−	0.992686i	$-0.461479\pi$
$607$	5.94852	0.241443	0.120722	+	0.992686i	$0.461479\pi$
$608$	12.3793	0.502048
$609$	10.8437	0.439410
$610$	84.3542	3.41540
$611$	53.9589	2.18294
$612$	−10.2887	−0.415895
$613$	−13.6764	−0.552384	−0.276192	−	0.961102i	$-0.589073\pi$
$613$	−13.6764	−0.552384	−0.276192	+	0.961102i	$0.589073\pi$
$614$	3.29549	0.132995
$615$	10.6054	0.427650
$616$	8.69538	0.350347
$617$	−10.1194	−0.407393	−0.203696	−	0.979034i	$-0.565296\pi$
$617$	−10.1194	−0.407393	−0.203696	+	0.979034i	$0.565296\pi$
$618$	20.0304	0.805739
$619$	34.6305	1.39192	0.695959	−	0.718082i	$-0.254980\pi$
$619$	34.6305	1.39192	0.695959	+	0.718082i	$0.254980\pi$
$620$	43.5551	1.74922
$621$	−22.4599	−0.901285
$622$	71.1538	2.85301
$623$	−36.6959	−1.47019
$624$	17.3465	0.694418
$625$	27.3984	1.09594
$626$	−43.8120	−1.75108
$627$	−0.566046	−0.0226057
$628$	75.3927	3.00850
$629$	−0.854911	−0.0340875
$630$	84.0370	3.34811
$631$	20.8616	0.830489	0.415244	−	0.909710i	$-0.363696\pi$
$631$	20.8616	0.830489	0.415244	+	0.909710i	$0.363696\pi$
$632$	43.1745	1.71739
$633$	−6.29352	−0.250145
$634$	20.3117	0.806680
$635$	−50.8459	−2.01776
$636$	−0.637613	−0.0252830
$637$	−13.0521	−0.517141
$638$	−6.94622	−0.275004
$639$	4.43373	0.175396
$640$	46.6068	1.84230
$641$	−6.80107	−0.268626	−0.134313	−	0.990939i	$-0.542883\pi$
$641$	−6.80107	−0.268626	−0.134313	+	0.990939i	$0.542883\pi$
$642$	−27.9389	−1.10266
$643$	−0.621658	−0.0245158	−0.0122579	−	0.999925i	$-0.503902\pi$
$643$	−0.621658	−0.0245158	−0.0122579	+	0.999925i	$0.503902\pi$
$644$	−100.299	−3.95235
$645$	−8.79509	−0.346306
$646$	5.05596	0.198924
$647$	−31.8138	−1.25073	−0.625365	−	0.780333i	$-0.715050\pi$
$647$	−31.8138	−1.25073	−0.625365	+	0.780333i	$0.715050\pi$
$648$	39.7628	1.56203
$649$	−3.93752	−0.154561
$650$	−115.107	−4.51487
$651$	4.36908	0.171238
$652$	−13.3507	−0.522852
$653$	−42.7077	−1.67128	−0.835640	−	0.549277i	$-0.814903\pi$
$653$	−42.7077	−1.67128	−0.835640	+	0.549277i	$0.814903\pi$
$654$	1.40853	0.0550780
$655$	−47.5758	−1.85894
$656$	34.5777	1.35003
$657$	17.6681	0.689300
$658$	−97.0605	−3.78381
$659$	−36.7465	−1.43144	−0.715720	−	0.698388i	$-0.753901\pi$
$659$	−36.7465	−1.43144	−0.715720	+	0.698388i	$0.753901\pi$
$660$	4.23496	0.164846
$661$	−48.5336	−1.88774	−0.943869	−	0.330319i	$-0.892844\pi$
$661$	−48.5336	−1.88774	−0.943869	+	0.330319i	$0.892844\pi$
$662$	87.1666	3.38783
$663$	2.10947	0.0819248
$664$	76.8104	2.98082
$665$	−28.5280	−1.10627
$666$	6.84987	0.265427
$667$	44.2614	1.71381
$668$	−102.370	−3.96080
$669$	−0.0252135	−0.000974812 0
$670$	−148.485	−5.73649
$671$	−3.74442	−0.144552
$672$	−9.29062	−0.358393
$673$	−41.5889	−1.60313	−0.801566	−	0.597906i	$-0.796000\pi$
$673$	−41.5889	−1.60313	−0.801566	+	0.597906i	$0.796000\pi$
$674$	−58.4489	−2.25137
$675$	32.0293	1.23281
$676$	30.6105	1.17733
$677$	50.9401	1.95779	0.978894	−	0.204370i	$-0.0655145\pi$
$677$	50.9401	1.95779	0.978894	+	0.204370i	$0.0655145\pi$
$678$	20.9018	0.802727
$679$	−7.96050	−0.305496
$680$	−20.8962	−0.801334
$681$	−10.6130	−0.406692
$682$	−2.79873	−0.107169
$683$	−7.18870	−0.275068	−0.137534	−	0.990497i	$-0.543918\pi$
$683$	−7.18870	−0.275068	−0.137534	+	0.990497i	$0.543918\pi$
$684$	−27.9848	−1.07002
$685$	−54.5649	−2.08482
$686$	−32.6220	−1.24551
$687$	−12.1824	−0.464789
$688$	−28.6755	−1.09324
$689$	−1.14794	−0.0437329
$690$	−39.0633	−1.48711
$691$	46.5394	1.77044	0.885222	−	0.465169i	$-0.154007\pi$
$691$	46.5394	1.77044	0.885222	+	0.465169i	$0.154007\pi$
$692$	−116.178	−4.41643
$693$	−3.73034	−0.141704
$694$	23.9037	0.907372
$695$	−2.68085	−0.101691
$696$	21.6042	0.818904
$697$	4.20490	0.159272
$698$	−42.4343	−1.60616
$699$	−9.56978	−0.361962
$700$	143.034	5.40616
$701$	23.7363	0.896508	0.448254	−	0.893906i	$-0.352046\pi$
$701$	23.7363	0.896508	0.448254	+	0.893906i	$0.352046\pi$
$702$	−35.7285	−1.34848
$703$	−2.32532	−0.0877012
$704$	−0.228723	−0.00862032
$705$	−26.1138	−0.983503
$706$	−76.4816	−2.87842
$707$	−51.6589	−1.94283
$708$	22.1689	0.833159
$709$	33.1647	1.24553	0.622763	−	0.782411i	$-0.286010\pi$
$709$	33.1647	1.24553	0.622763	+	0.782411i	$0.286010\pi$
$710$	16.3009	0.611761
$711$	−18.5219	−0.694627
$712$	−73.1099	−2.73991
$713$	17.8335	0.667870
$714$	−3.79448	−0.142005
$715$	7.62448	0.285139
$716$	64.0277	2.39283
$717$	3.82729	0.142933
$718$	42.1559	1.57325
$719$	−3.35542	−0.125136	−0.0625680	−	0.998041i	$-0.519929\pi$
$719$	−3.35542	−0.125136	−0.0625680	+	0.998041i	$0.519929\pi$
$720$	73.7171	2.74727
$721$	−44.8111	−1.66885
$722$	−34.5709	−1.28660
$723$	−6.25676	−0.232692
$724$	77.0770	2.86454
$725$	−63.1197	−2.34421
$726$	15.2217	0.564932
$727$	28.7181	1.06510	0.532549	−	0.846399i	$-0.321234\pi$
$727$	28.7181	1.06510	0.532549	+	0.846399i	$0.321234\pi$
$728$	−88.1396	−3.26667
$729$	−11.8197	−0.437766
$730$	64.9580	2.40420
$731$	−3.48714	−0.128977
$732$	21.0817	0.779203
$733$	9.53196	0.352071	0.176035	−	0.984384i	$-0.443673\pi$
$733$	9.53196	0.352071	0.176035	+	0.984384i	$0.443673\pi$
$734$	−24.9138	−0.919585
$735$	6.31664	0.232993
$736$	−37.9220	−1.39782
$737$	6.59115	0.242788
$738$	−33.6912	−1.24019
$739$	−18.0672	−0.664611	−0.332306	−	0.943172i	$-0.607827\pi$
$739$	−18.0672	−0.664611	−0.332306	+	0.943172i	$0.607827\pi$
$740$	17.3972	0.639535
$741$	5.73766	0.210778
$742$	2.06489	0.0758046
$743$	−5.89592	−0.216300	−0.108150	−	0.994135i	$-0.534493\pi$
$743$	−5.89592	−0.216300	−0.108150	+	0.994135i	$0.534493\pi$
$744$	8.70461	0.319126
$745$	30.5265	1.11840
$746$	26.6622	0.976172
$747$	−32.9518	−1.20564
$748$	1.67911	0.0613943
$749$	62.5038	2.28384
$750$	28.2873	1.03291
$751$	−38.6632	−1.41084	−0.705420	−	0.708790i	$-0.749242\pi$
$751$	−38.6632	−1.41084	−0.705420	+	0.708790i	$0.749242\pi$
$752$	−85.1413	−3.10478
$753$	−10.1624	−0.370337
$754$	70.4095	2.56416
$755$	13.1406	0.478234
$756$	44.3967	1.61469
$757$	5.82216	0.211610	0.105805	−	0.994387i	$-0.466258\pi$
$757$	5.82216	0.211610	0.105805	+	0.994387i	$0.466258\pi$
$758$	19.8374	0.720526
$759$	1.73399	0.0629399
$760$	−56.8369	−2.06169
$761$	−32.0721	−1.16261	−0.581306	−	0.813685i	$-0.697458\pi$
$761$	−32.0721	−1.16261	−0.581306	+	0.813685i	$0.697458\pi$
$762$	−18.3950	−0.666379
$763$	−3.15111	−0.114078
$764$	−47.3601	−1.71343
$765$	8.96453	0.324113
$766$	−1.47275	−0.0532126
$767$	39.9122	1.44115
$768$	16.2850	0.587633
$769$	1.50081	0.0541205	0.0270602	−	0.999634i	$-0.491385\pi$
$769$	1.50081	0.0541205	0.0270602	+	0.999634i	$0.491385\pi$
$770$	−13.7148	−0.494247
$771$	13.3658	0.481358
$772$	72.1288	2.59597
$773$	3.80678	0.136920	0.0684601	−	0.997654i	$-0.478191\pi$
$773$	3.80678	0.136920	0.0684601	+	0.997654i	$0.478191\pi$
$774$	27.9403	1.00429
$775$	−25.4318	−0.913536
$776$	−15.8599	−0.569336
$777$	1.74514	0.0626067
$778$	12.7816	0.458244
$779$	11.4371	0.409778
$780$	−42.9271	−1.53704
$781$	−0.723584	−0.0258919
$782$	−15.4881	−0.553854
$783$	−19.5919	−0.700159
$784$	20.5947	0.735527
$785$	−65.6897	−2.34457
$786$	−17.2119	−0.613928
$787$	−37.9434	−1.35254	−0.676268	−	0.736655i	$-0.736404\pi$
$787$	−37.9434	−1.35254	−0.676268	+	0.736655i	$0.736404\pi$
$788$	15.4604	0.550752
$789$	−9.12049	−0.324698
$790$	−68.0970	−2.42278
$791$	−46.7605	−1.66261
$792$	−7.43203	−0.264086
$793$	37.9548	1.34781
$794$	−23.3278	−0.827873
$795$	0.555552	0.0197034
$796$	−45.1820	−1.60143
$797$	20.2890	0.718673	0.359336	−	0.933208i	$-0.383003\pi$
$797$	20.2890	0.718673	0.359336	+	0.933208i	$0.383003\pi$
$798$	−10.3208	−0.365353
$799$	−10.3538	−0.366291
$800$	54.0793	1.91199
$801$	31.3643	1.10820
$802$	−58.0911	−2.05127
$803$	−2.88344	−0.101754
$804$	−37.1093	−1.30874
$805$	87.3908	3.08012
$806$	28.3689	0.999253
$807$	0.347329	0.0122265
$808$	−102.921	−3.62075
$809$	30.2745	1.06439	0.532197	−	0.846620i	$-0.321367\pi$
$809$	30.2745	1.06439	0.532197	+	0.846620i	$0.321367\pi$
$810$	−62.7159	−2.20361
$811$	−35.0323	−1.23015	−0.615076	−	0.788468i	$-0.710875\pi$
$811$	−35.0323	−1.23015	−0.615076	+	0.788468i	$0.710875\pi$
$812$	−87.4918	−3.07036
$813$	−2.17071	−0.0761300
$814$	−1.11790	−0.0391823
$815$	11.6324	0.407466
$816$	−3.32851	−0.116521
$817$	−9.48488	−0.331834
$818$	−21.3658	−0.747037
$819$	37.8121	1.32126
$820$	−85.5687	−2.98819
$821$	−18.0571	−0.630196	−0.315098	−	0.949059i	$-0.602037\pi$
$821$	−18.0571	−0.630196	−0.315098	+	0.949059i	$0.602037\pi$
$822$	−19.7404	−0.688525
$823$	−4.60941	−0.160674	−0.0803370	−	0.996768i	$-0.525600\pi$
$823$	−4.60941	−0.160674	−0.0803370	+	0.996768i	$0.525600\pi$
$824$	−89.2780	−3.11015
$825$	−2.47279	−0.0860914
$826$	−71.7935	−2.49801
$827$	14.1171	0.490899	0.245450	−	0.969409i	$-0.421064\pi$
$827$	14.1171	0.490899	0.245450	+	0.969409i	$0.421064\pi$
$828$	85.7268	2.97921
$829$	8.83305	0.306785	0.153392	−	0.988165i	$-0.450980\pi$
$829$	8.83305	0.306785	0.153392	+	0.988165i	$0.450980\pi$
$830$	−121.149	−4.20516
$831$	8.95702	0.310716
$832$	2.31842	0.0803768
$833$	2.50447	0.0867747
$834$	−0.969876	−0.0335840
$835$	89.1948	3.08671
$836$	4.56711	0.157957
$837$	−7.89385	−0.272851
$838$	−16.4344	−0.567717
$839$	−24.8669	−0.858501	−0.429251	−	0.903185i	$-0.641222\pi$
$839$	−24.8669	−0.858501	−0.429251	+	0.903185i	$0.641222\pi$
$840$	42.6558	1.47177
$841$	9.60952	0.331363
$842$	−25.8946	−0.892386
$843$	−8.43900	−0.290655
$844$	50.7789	1.74788
$845$	−26.6709	−0.917507
$846$	82.9585	2.85217
$847$	−34.0534	−1.17009
$848$	1.81132	0.0622010
$849$	−14.0827	−0.483316
$850$	22.0871	0.757581
$851$	7.12324	0.244182
$852$	4.07390	0.139570
$853$	41.0840	1.40669	0.703345	−	0.710849i	$-0.251689\pi$
$853$	41.0840	1.40669	0.703345	+	0.710849i	$0.251689\pi$
$854$	−68.2726	−2.33624
$855$	24.3831	0.833886
$856$	124.528	4.25627
$857$	−48.1069	−1.64330	−0.821650	−	0.569993i	$-0.806946\pi$
$857$	−48.1069	−1.64330	−0.821650	+	0.569993i	$0.806946\pi$
$858$	2.75837	0.0941693
$859$	−29.1817	−0.995667	−0.497833	−	0.867273i	$-0.665871\pi$
$859$	−29.1817	−0.995667	−0.497833	+	0.867273i	$0.665871\pi$
$860$	70.9625	2.41980
$861$	−8.58352	−0.292526
$862$	−42.8814	−1.46055
$863$	4.14838	0.141213	0.0706063	−	0.997504i	$-0.477507\pi$
$863$	4.14838	0.141213	0.0706063	+	0.997504i	$0.477507\pi$
$864$	16.7859	0.571067
$865$	101.226	3.44179
$866$	37.5895	1.27734
$867$	9.01014	0.306000
$868$	−35.2516	−1.19652
$869$	3.02277	0.102541
$870$	−34.0752	−1.15526
$871$	−66.8103	−2.26378
$872$	−6.27803	−0.212601
$873$	6.80391	0.230278
$874$	−42.1270	−1.42497
$875$	−63.2832	−2.13936
$876$	16.2342	0.548504
$877$	−5.67636	−0.191677	−0.0958386	−	0.995397i	$-0.530553\pi$
$877$	−5.67636	−0.191677	−0.0958386	+	0.995397i	$0.530553\pi$
$878$	−56.9031	−1.92039
$879$	14.1372	0.476837
$880$	−12.0306	−0.405552
$881$	−35.8383	−1.20742	−0.603711	−	0.797203i	$-0.706312\pi$
$881$	−35.8383	−1.20742	−0.603711	+	0.797203i	$0.706312\pi$
$882$	−20.0668	−0.675683
$883$	−44.9066	−1.51123	−0.755614	−	0.655017i	$-0.772662\pi$
$883$	−44.9066	−1.51123	−0.755614	+	0.655017i	$0.772662\pi$
$884$	−17.0201	−0.572447
$885$	−19.3158	−0.649293
$886$	−15.6046	−0.524245
$887$	−20.6181	−0.692289	−0.346145	−	0.938181i	$-0.612509\pi$
$887$	−20.6181	−0.692289	−0.346145	+	0.938181i	$0.612509\pi$
$888$	3.47688	0.116677
$889$	41.1525	1.38021
$890$	115.313	3.86529
$891$	2.78391	0.0932645
$892$	0.203434	0.00681146
$893$	−28.1619	−0.942402
$894$	11.0438	0.369361
$895$	−55.7873	−1.86477
$896$	−37.7215	−1.26019
$897$	−17.5764	−0.586858
$898$	19.5755	0.653244
$899$	15.5563	0.518831
$900$	−122.252	−4.07507
$901$	0.220270	0.00733824
$902$	5.49840	0.183077
$903$	7.11836	0.236884
$904$	−93.1620	−3.09852
$905$	−67.1572	−2.23238
$906$	4.75397	0.157940
$907$	10.6138	0.352425	0.176213	−	0.984352i	$-0.443615\pi$
$907$	10.6138	0.352425	0.176213	+	0.984352i	$0.443615\pi$
$908$	85.6305	2.84175
$909$	44.1534	1.46448
$910$	139.018	4.60841
$911$	−15.8042	−0.523615	−0.261807	−	0.965120i	$-0.584319\pi$
$911$	−15.8042	−0.523615	−0.261807	+	0.965120i	$0.584319\pi$
$912$	−9.05340	−0.299788
$913$	5.37773	0.177977
$914$	−37.9171	−1.25419
$915$	−18.3685	−0.607244
$916$	98.2931	3.24769
$917$	38.5057	1.27157
$918$	6.85569	0.226271
$919$	20.8127	0.686549	0.343275	−	0.939235i	$-0.388464\pi$
$919$	20.8127	0.686549	0.343275	+	0.939235i	$0.388464\pi$
$920$	174.111	5.74025
$921$	−0.717608	−0.0236460
$922$	−30.5711	−1.00681
$923$	7.33451	0.241418
$924$	−3.42759	−0.112759
$925$	−10.1582	−0.334000
$926$	10.4329	0.342846
$927$	38.3005	1.25795
$928$	−33.0796	−1.08589
$929$	−11.7546	−0.385656	−0.192828	−	0.981233i	$-0.561766\pi$
$929$	−11.7546	−0.385656	−0.192828	+	0.981233i	$0.561766\pi$
$930$	−13.7294	−0.450203
$931$	6.81205	0.223256
$932$	77.2131	2.52920
$933$	−15.4941	−0.507253
$934$	48.3258	1.58127
$935$	−1.46301	−0.0478455
$936$	75.3338	2.46236
$937$	−54.5077	−1.78069	−0.890345	−	0.455287i	$-0.849537\pi$
$937$	−54.5077	−1.78069	−0.890345	+	0.455287i	$0.849537\pi$
$938$	120.177	3.92393
$939$	9.54027	0.311335
$940$	210.697	6.87219
$941$	−12.3343	−0.402087	−0.201044	−	0.979582i	$-0.564433\pi$
$941$	−12.3343	−0.402087	−0.201044	+	0.979582i	$0.564433\pi$
$942$	−23.7651	−0.774310
$943$	−35.0358	−1.14092
$944$	−62.9771	−2.04973
$945$	−38.6828	−1.25835
$946$	−4.55985	−0.148253
$947$	16.8131	0.546354	0.273177	−	0.961964i	$-0.411926\pi$
$947$	16.8131	0.546354	0.273177	+	0.961964i	$0.411926\pi$
$948$	−17.0187	−0.552743
$949$	29.2276	0.948767
$950$	60.0760	1.94912
$951$	−4.42296	−0.143424
$952$	16.9125	0.548137
$953$	−3.33871	−0.108151	−0.0540757	−	0.998537i	$-0.517221\pi$
$953$	−3.33871	−0.108151	−0.0540757	+	0.998537i	$0.517221\pi$
$954$	−1.76488	−0.0571402
$955$	41.2649	1.33530
$956$	−30.8802	−0.998737
$957$	1.51257	0.0488945
$958$	89.8715	2.90362
$959$	44.1624	1.42608
$960$	−1.12202	−0.0362129
$961$	−24.7322	−0.797812
$962$	11.3314	0.365339
$963$	−53.4226	−1.72152
$964$	50.4823	1.62592
$965$	−62.8458	−2.02308
$966$	31.6161	1.01723
$967$	29.8084	0.958574	0.479287	−	0.877658i	$-0.340895\pi$
$967$	29.8084	0.958574	0.479287	+	0.877658i	$0.340895\pi$
$968$	−67.8454	−2.18063
$969$	−1.10096	−0.0353679
$970$	25.0150	0.803183
$971$	−12.5129	−0.401558	−0.200779	−	0.979637i	$-0.564347\pi$
$971$	−12.5129	−0.401558	−0.200779	+	0.979637i	$0.564347\pi$
$972$	−57.9415	−1.85847
$973$	2.16977	0.0695595
$974$	60.3500	1.93374
$975$	25.0651	0.802725
$976$	−59.8886	−1.91699
$977$	−11.2228	−0.359049	−0.179524	−	0.983754i	$-0.557456\pi$
$977$	−11.2228	−0.359049	−0.179524	+	0.983754i	$0.557456\pi$
$978$	4.20836	0.134569
$979$	−5.11864	−0.163593
$980$	−50.9654	−1.62803
$981$	2.69329	0.0859900
$982$	−27.9622	−0.892311
$983$	29.4658	0.939813	0.469907	−	0.882716i	$-0.344288\pi$
$983$	29.4658	0.939813	0.469907	+	0.882716i	$0.344288\pi$
$984$	−17.1011	−0.545164
$985$	−13.4706	−0.429210
$986$	−13.5104	−0.430259
$987$	21.1354	0.672746
$988$	−46.2939	−1.47280
$989$	29.0554	0.923907
$990$	11.7222	0.372555
$991$	14.3829	0.456887	0.228444	−	0.973557i	$-0.426636\pi$
$991$	14.3829	0.456887	0.228444	+	0.973557i	$0.426636\pi$
$992$	−13.3282	−0.423172
$993$	−18.9809	−0.602341
$994$	−13.1932	−0.418464
$995$	39.3671	1.24802
$996$	−30.2775	−0.959381
$997$	−62.6330	−1.98361	−0.991804	−	0.127772i	$-0.959217\pi$
$997$	−62.6330	−1.98361	−0.991804	+	0.127772i	$0.959217\pi$
$998$	−58.8952	−1.86429
$999$	−3.15304	−0.0997579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4033.2.a.d.1.77	✓	79

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4033.2.a.d.1.77	✓	79	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:	\( N \)	\(=\)	\( 4033 = 37 \cdot 109 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4033.a (trivial)

\(f(q)\)	\(=\)	\(q+2.54331 q^{2} -0.553818 q^{3} +4.46844 q^{4} -3.89336 q^{5} -1.40853 q^{6} +3.15111 q^{7} +6.27803 q^{8} -2.69329 q^{9} +O(q^{10})\)	\(q+2.54331 q^{2} -0.553818 q^{3} +4.46844 q^{4} -3.89336 q^{5} -1.40853 q^{6} +3.15111 q^{7} +6.27803 q^{8} -2.69329 q^{9} -9.90203 q^{10} +0.439543 q^{11} -2.47471 q^{12} -4.45537 q^{13} +8.01427 q^{14} +2.15621 q^{15} +7.03010 q^{16} +0.854911 q^{17} -6.84987 q^{18} +2.32532 q^{19} -17.3972 q^{20} -1.74514 q^{21} +1.11790 q^{22} -7.12324 q^{23} -3.47688 q^{24} +10.1582 q^{25} -11.3314 q^{26} +3.15304 q^{27} +14.0806 q^{28} -6.21366 q^{29} +5.48392 q^{30} -2.50357 q^{31} +5.32370 q^{32} -0.243427 q^{33} +2.17431 q^{34} -12.2684 q^{35} -12.0348 q^{36} -1.00000 q^{37} +5.91402 q^{38} +2.46747 q^{39} -24.4426 q^{40} +4.91852 q^{41} -4.43845 q^{42} -4.07895 q^{43} +1.96407 q^{44} +10.4859 q^{45} -18.1166 q^{46} -12.1110 q^{47} -3.89340 q^{48} +2.92951 q^{49} +25.8356 q^{50} -0.473465 q^{51} -19.9086 q^{52} +0.257652 q^{53} +8.01918 q^{54} -1.71130 q^{55} +19.7828 q^{56} -1.28781 q^{57} -15.8033 q^{58} -8.95821 q^{59} +9.63491 q^{60} -8.51888 q^{61} -6.36735 q^{62} -8.48685 q^{63} -0.520365 q^{64} +17.3464 q^{65} -0.619111 q^{66} +14.9954 q^{67} +3.82012 q^{68} +3.94498 q^{69} -31.2024 q^{70} -1.64622 q^{71} -16.9085 q^{72} -6.56007 q^{73} -2.54331 q^{74} -5.62581 q^{75} +10.3906 q^{76} +1.38505 q^{77} +6.27554 q^{78} +6.87708 q^{79} -27.3707 q^{80} +6.33364 q^{81} +12.5093 q^{82} +12.2348 q^{83} -7.79807 q^{84} -3.32847 q^{85} -10.3741 q^{86} +3.44124 q^{87} +2.75946 q^{88} -11.6454 q^{89} +26.6690 q^{90} -14.0394 q^{91} -31.8298 q^{92} +1.38652 q^{93} -30.8020 q^{94} -9.05331 q^{95} -2.94836 q^{96} -2.52625 q^{97} +7.45066 q^{98} -1.18382 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 79 q - 11 q^{2} - 11 q^{3} + 79 q^{4} - 16 q^{5} - 14 q^{6} - 15 q^{7} - 42 q^{8} + 76 q^{9}+O(q^{10}) \) 79 * q - 11 * q^2 - 11 * q^3 + 79 * q^4 - 16 * q^5 - 14 * q^6 - 15 * q^7 - 42 * q^8 + 76 * q^9	\( 79 q - 11 q^{2} - 11 q^{3} + 79 q^{4} - 16 q^{5} - 14 q^{6} - 15 q^{7} - 42 q^{8} + 76 q^{9} - 13 q^{10} - 5 q^{11} - 40 q^{12} - 18 q^{13} - 42 q^{14} - 49 q^{15} + 83 q^{16} - 62 q^{17} - 33 q^{18} - 25 q^{19} - 39 q^{20} - 15 q^{21} - 31 q^{22} - 94 q^{23} - 39 q^{24} + 71 q^{25} - 35 q^{26} - 47 q^{27} - 13 q^{28} - 17 q^{29} + 15 q^{30} - 37 q^{31} - 105 q^{32} - 60 q^{33} + 9 q^{34} - 60 q^{35} + 43 q^{36} - 79 q^{37} - 80 q^{38} - 41 q^{39} - 64 q^{40} - 37 q^{41} - 30 q^{42} - 20 q^{43} - 14 q^{44} - 8 q^{45} + 61 q^{46} - 148 q^{47} - 39 q^{48} + 82 q^{49} - 90 q^{50} - 45 q^{51} - 27 q^{52} - 70 q^{53} - 41 q^{54} - 105 q^{55} - 68 q^{56} - 31 q^{57} - 14 q^{58} - 96 q^{59} - 74 q^{60} - 21 q^{61} + 4 q^{62} - 60 q^{63} + 132 q^{64} - 15 q^{65} + 71 q^{66} - 44 q^{67} - 166 q^{68} - 72 q^{69} - 9 q^{70} - 55 q^{71} - 126 q^{72} - 27 q^{73} + 11 q^{74} - 39 q^{75} - 4 q^{76} - 104 q^{77} - 47 q^{78} - 49 q^{79} - 82 q^{80} + 55 q^{81} + 20 q^{82} - 52 q^{83} - 29 q^{84} + 3 q^{85} - 32 q^{86} - 113 q^{87} + 14 q^{88} - 68 q^{89} - 39 q^{90} - 30 q^{91} - 179 q^{92} - 53 q^{93} - 33 q^{94} - 86 q^{95} + 33 q^{96} - 57 q^{97} - 116 q^{98} + q^{99}+O(q^{100}) \) 79 * q - 11 * q^2 - 11 * q^3 + 79 * q^4 - 16 * q^5 - 14 * q^6 - 15 * q^7 - 42 * q^8 + 76 * q^9 - 13 * q^10 - 5 * q^11 - 40 * q^12 - 18 * q^13 - 42 * q^14 - 49 * q^15 + 83 * q^16 - 62 * q^17 - 33 * q^18 - 25 * q^19 - 39 * q^20 - 15 * q^21 - 31 * q^22 - 94 * q^23 - 39 * q^24 + 71 * q^25 - 35 * q^26 - 47 * q^27 - 13 * q^28 - 17 * q^29 + 15 * q^30 - 37 * q^31 - 105 * q^32 - 60 * q^33 + 9 * q^34 - 60 * q^35 + 43 * q^36 - 79 * q^37 - 80 * q^38 - 41 * q^39 - 64 * q^40 - 37 * q^41 - 30 * q^42 - 20 * q^43 - 14 * q^44 - 8 * q^45 + 61 * q^46 - 148 * q^47 - 39 * q^48 + 82 * q^49 - 90 * q^50 - 45 * q^51 - 27 * q^52 - 70 * q^53 - 41 * q^54 - 105 * q^55 - 68 * q^56 - 31 * q^57 - 14 * q^58 - 96 * q^59 - 74 * q^60 - 21 * q^61 + 4 * q^62 - 60 * q^63 + 132 * q^64 - 15 * q^65 + 71 * q^66 - 44 * q^67 - 166 * q^68 - 72 * q^69 - 9 * q^70 - 55 * q^71 - 126 * q^72 - 27 * q^73 + 11 * q^74 - 39 * q^75 - 4 * q^76 - 104 * q^77 - 47 * q^78 - 49 * q^79 - 82 * q^80 + 55 * q^81 + 20 * q^82 - 52 * q^83 - 29 * q^84 + 3 * q^85 - 32 * q^86 - 113 * q^87 + 14 * q^88 - 68 * q^89 - 39 * q^90 - 30 * q^91 - 179 * q^92 - 53 * q^93 - 33 * q^94 - 86 * q^95 + 33 * q^96 - 57 * q^97 - 116 * q^98 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more