LMFDB - Embedded newform 4029.2.a.h.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4029,2,Mod(1,4029)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4029, 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("4029.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4029 = 3 \cdot 17 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4029.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:	$32.1717269744$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	4029.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.319374 q^{2} +1.00000 q^{3} -1.89800 q^{4} -2.81066 q^{5} -0.319374 q^{6} -2.62902 q^{7} +1.24492 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.319374 q^{2} +1.00000 q^{3} -1.89800 q^{4} -2.81066 q^{5} -0.319374 q^{6} -2.62902 q^{7} +1.24492 q^{8} +1.00000 q^{9} +0.897652 q^{10} -1.49606 q^{11} -1.89800 q^{12} +1.38418 q^{13} +0.839638 q^{14} -2.81066 q^{15} +3.39841 q^{16} -1.00000 q^{17} -0.319374 q^{18} +5.81194 q^{19} +5.33464 q^{20} -2.62902 q^{21} +0.477803 q^{22} +2.82044 q^{23} +1.24492 q^{24} +2.89984 q^{25} -0.442071 q^{26} +1.00000 q^{27} +4.98987 q^{28} +0.415697 q^{29} +0.897652 q^{30} -0.794801 q^{31} -3.57520 q^{32} -1.49606 q^{33} +0.319374 q^{34} +7.38928 q^{35} -1.89800 q^{36} +3.13051 q^{37} -1.85618 q^{38} +1.38418 q^{39} -3.49905 q^{40} +7.83998 q^{41} +0.839638 q^{42} -4.83939 q^{43} +2.83953 q^{44} -2.81066 q^{45} -0.900774 q^{46} +6.29636 q^{47} +3.39841 q^{48} -0.0882782 q^{49} -0.926131 q^{50} -1.00000 q^{51} -2.62718 q^{52} -13.5734 q^{53} -0.319374 q^{54} +4.20493 q^{55} -3.27291 q^{56} +5.81194 q^{57} -0.132763 q^{58} -1.18458 q^{59} +5.33464 q^{60} +12.9021 q^{61} +0.253839 q^{62} -2.62902 q^{63} -5.65499 q^{64} -3.89047 q^{65} +0.477803 q^{66} +2.97741 q^{67} +1.89800 q^{68} +2.82044 q^{69} -2.35994 q^{70} -9.04738 q^{71} +1.24492 q^{72} +1.42726 q^{73} -0.999804 q^{74} +2.89984 q^{75} -11.0311 q^{76} +3.93317 q^{77} -0.442071 q^{78} +1.00000 q^{79} -9.55178 q^{80} +1.00000 q^{81} -2.50388 q^{82} -1.32704 q^{83} +4.98987 q^{84} +2.81066 q^{85} +1.54557 q^{86} +0.415697 q^{87} -1.86248 q^{88} -18.3491 q^{89} +0.897652 q^{90} -3.63904 q^{91} -5.35320 q^{92} -0.794801 q^{93} -2.01089 q^{94} -16.3354 q^{95} -3.57520 q^{96} -5.95421 q^{97} +0.0281937 q^{98} -1.49606 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 7 q^{2} + 25 q^{3} + 21 q^{4} - 12 q^{5} - 7 q^{6} - 4 q^{7} - 21 q^{8} + 25 q^{9}+O(q^{10}) $ 25 * q - 7 * q^2 + 25 * q^3 + 21 * q^4 - 12 * q^5 - 7 * q^6 - 4 * q^7 - 21 * q^8 + 25 * q^9	\( 25 q - 7 q^{2} + 25 q^{3} + 21 q^{4} - 12 q^{5} - 7 q^{6} - 4 q^{7} - 21 q^{8} + 25 q^{9} - 9 q^{10} - 19 q^{11} + 21 q^{12} - 12 q^{13} - 15 q^{14} - 12 q^{15} + q^{16} - 25 q^{17} - 7 q^{18} - 35 q^{19} - 11 q^{20} - 4 q^{21} - 2 q^{22} - 16 q^{23} - 21 q^{24} + 19 q^{25} - 5 q^{26} + 25 q^{27} + 3 q^{28} - 37 q^{29} - 9 q^{30} - 28 q^{31} - 19 q^{32} - 19 q^{33} + 7 q^{34} - 42 q^{35} + 21 q^{36} + 8 q^{37} - 35 q^{38} - 12 q^{39} - 9 q^{40} - 34 q^{41} - 15 q^{42} - 19 q^{43} - 56 q^{44} - 12 q^{45} + q^{46} - 25 q^{47} + q^{48} + 25 q^{49} - 7 q^{50} - 25 q^{51} - 37 q^{52} - 44 q^{53} - 7 q^{54} - 11 q^{55} - 18 q^{56} - 35 q^{57} - 3 q^{58} - 47 q^{59} - 11 q^{60} - 28 q^{61} + 11 q^{62} - 4 q^{63} - 9 q^{64} - 63 q^{65} - 2 q^{66} - 28 q^{67} - 21 q^{68} - 16 q^{69} + 5 q^{70} - 27 q^{71} - 21 q^{72} - 21 q^{73} - 18 q^{74} + 19 q^{75} - 50 q^{76} - 58 q^{77} - 5 q^{78} + 25 q^{79} - 56 q^{80} + 25 q^{81} - 5 q^{82} - 61 q^{83} + 3 q^{84} + 12 q^{85} - 28 q^{86} - 37 q^{87} + 15 q^{88} - 34 q^{89} - 9 q^{90} - 30 q^{91} - 31 q^{92} - 28 q^{93} + q^{94} - 32 q^{95} - 19 q^{96} - 11 q^{97} - 66 q^{98} - 19 q^{99}+O(q^{100}) \) 25 * q - 7 * q^2 + 25 * q^3 + 21 * q^4 - 12 * q^5 - 7 * q^6 - 4 * q^7 - 21 * q^8 + 25 * q^9 - 9 * q^10 - 19 * q^11 + 21 * q^12 - 12 * q^13 - 15 * q^14 - 12 * q^15 + q^16 - 25 * q^17 - 7 * q^18 - 35 * q^19 - 11 * q^20 - 4 * q^21 - 2 * q^22 - 16 * q^23 - 21 * q^24 + 19 * q^25 - 5 * q^26 + 25 * q^27 + 3 * q^28 - 37 * q^29 - 9 * q^30 - 28 * q^31 - 19 * q^32 - 19 * q^33 + 7 * q^34 - 42 * q^35 + 21 * q^36 + 8 * q^37 - 35 * q^38 - 12 * q^39 - 9 * q^40 - 34 * q^41 - 15 * q^42 - 19 * q^43 - 56 * q^44 - 12 * q^45 + q^46 - 25 * q^47 + q^48 + 25 * q^49 - 7 * q^50 - 25 * q^51 - 37 * q^52 - 44 * q^53 - 7 * q^54 - 11 * q^55 - 18 * q^56 - 35 * q^57 - 3 * q^58 - 47 * q^59 - 11 * q^60 - 28 * q^61 + 11 * q^62 - 4 * q^63 - 9 * q^64 - 63 * q^65 - 2 * q^66 - 28 * q^67 - 21 * q^68 - 16 * q^69 + 5 * q^70 - 27 * q^71 - 21 * q^72 - 21 * q^73 - 18 * q^74 + 19 * q^75 - 50 * q^76 - 58 * q^77 - 5 * q^78 + 25 * q^79 - 56 * q^80 + 25 * q^81 - 5 * q^82 - 61 * q^83 + 3 * q^84 + 12 * q^85 - 28 * q^86 - 37 * q^87 + 15 * q^88 - 34 * q^89 - 9 * q^90 - 30 * q^91 - 31 * q^92 - 28 * q^93 + q^94 - 32 * q^95 - 19 * q^96 - 11 * q^97 - 66 * q^98 - 19 * 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 $

$2$	−0.319374	−0.225831	−0.112916	−	0.993605i	$-0.536019\pi$
$2$	−0.319374	−0.225831	−0.112916	+	0.993605i	$0.536019\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.89800	−0.949000
$5$	−2.81066	−1.25697	−0.628484	−	0.777823i	$-0.716324\pi$
$5$	−2.81066	−1.25697	−0.628484	+	0.777823i	$0.716324\pi$
$6$	−0.319374	−0.130384
$7$	−2.62902	−0.993674	−0.496837	−	0.867844i	$-0.665505\pi$
$7$	−2.62902	−0.993674	−0.496837	+	0.867844i	$0.665505\pi$
$8$	1.24492	0.440145
$9$	1.00000	0.333333
$10$	0.897652	0.283863
$11$	−1.49606	−0.451080	−0.225540	−	0.974234i	$-0.572415\pi$
$11$	−1.49606	−0.451080	−0.225540	+	0.974234i	$0.572415\pi$
$12$	−1.89800	−0.547906
$13$	1.38418	0.383903	0.191951	−	0.981404i	$-0.438518\pi$
$13$	1.38418	0.383903	0.191951	+	0.981404i	$0.438518\pi$
$14$	0.839638	0.224403
$15$	−2.81066	−0.725711
$16$	3.39841	0.849602
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−0.319374	−0.0752771
$19$	5.81194	1.33335	0.666675	−	0.745349i	$-0.267717\pi$
$19$	5.81194	1.33335	0.666675	+	0.745349i	$0.267717\pi$
$20$	5.33464	1.19286
$21$	−2.62902	−0.573698
$22$	0.477803	0.101868
$23$	2.82044	0.588102	0.294051	−	0.955790i	$-0.404996\pi$
$23$	2.82044	0.588102	0.294051	+	0.955790i	$0.404996\pi$
$24$	1.24492	0.254118
$25$	2.89984	0.579967
$26$	−0.442071	−0.0866973
$27$	1.00000	0.192450
$28$	4.98987	0.942997
$29$	0.415697	0.0771930	0.0385965	−	0.999255i	$-0.487711\pi$
$29$	0.415697	0.0771930	0.0385965	+	0.999255i	$0.487711\pi$
$30$	0.897652	0.163888
$31$	−0.794801	−0.142751	−0.0713753	−	0.997450i	$-0.522739\pi$
$31$	−0.794801	−0.142751	−0.0713753	+	0.997450i	$0.522739\pi$
$32$	−3.57520	−0.632012
$33$	−1.49606	−0.260431
$34$	0.319374	0.0547721
$35$	7.38928	1.24902
$36$	−1.89800	−0.316333
$37$	3.13051	0.514653	0.257327	−	0.966324i	$-0.417158\pi$
$37$	3.13051	0.514653	0.257327	+	0.966324i	$0.417158\pi$
$38$	−1.85618	−0.301112
$39$	1.38418	0.221646
$40$	−3.49905	−0.553248
$41$	7.83998	1.22440	0.612200	−	0.790703i	$-0.290285\pi$
$41$	7.83998	1.22440	0.612200	+	0.790703i	$0.290285\pi$
$42$	0.839638	0.129559
$43$	−4.83939	−0.737999	−0.369000	−	0.929430i	$-0.620300\pi$
$43$	−4.83939	−0.737999	−0.369000	+	0.929430i	$0.620300\pi$
$44$	2.83953	0.428075
$45$	−2.81066	−0.418989
$46$	−0.900774	−0.132812
$47$	6.29636	0.918419	0.459209	−	0.888328i	$-0.348133\pi$
$47$	6.29636	0.918419	0.459209	+	0.888328i	$0.348133\pi$
$48$	3.39841	0.490518
$49$	−0.0882782	−0.0126112
$50$	−0.926131	−0.130975
$51$	−1.00000	−0.140028
$52$	−2.62718	−0.364324
$53$	−13.5734	−1.86445	−0.932224	−	0.361883i	$-0.882134\pi$
$53$	−13.5734	−1.86445	−0.932224	+	0.361883i	$0.882134\pi$
$54$	−0.319374	−0.0434612
$55$	4.20493	0.566993
$56$	−3.27291	−0.437361
$57$	5.81194	0.769810
$58$	−0.132763	−0.0174326
$59$	−1.18458	−0.154219	−0.0771094	−	0.997023i	$-0.524569\pi$
$59$	−1.18458	−0.154219	−0.0771094	+	0.997023i	$0.524569\pi$
$60$	5.33464	0.688699
$61$	12.9021	1.65195	0.825975	−	0.563707i	$-0.190625\pi$
$61$	12.9021	1.65195	0.825975	+	0.563707i	$0.190625\pi$
$62$	0.253839	0.0322375
$63$	−2.62902	−0.331225
$64$	−5.65499	−0.706874
$65$	−3.89047	−0.482554
$66$	0.477803	0.0588135
$67$	2.97741	0.363749	0.181874	−	0.983322i	$-0.441784\pi$
$67$	2.97741	0.363749	0.181874	+	0.983322i	$0.441784\pi$
$68$	1.89800	0.230166
$69$	2.82044	0.339541
$70$	−2.35994	−0.282067
$71$	−9.04738	−1.07373	−0.536863	−	0.843669i	$-0.680391\pi$
$71$	−9.04738	−1.07373	−0.536863	+	0.843669i	$0.680391\pi$
$72$	1.24492	0.146715
$73$	1.42726	0.167048	0.0835239	−	0.996506i	$-0.473383\pi$
$73$	1.42726	0.167048	0.0835239	+	0.996506i	$0.473383\pi$
$74$	−0.999804	−0.116225
$75$	2.89984	0.334844
$76$	−11.0311	−1.26535
$77$	3.93317	0.448227
$78$	−0.442071	−0.0500547
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	−9.55178	−1.06792
$81$	1.00000	0.111111
$82$	−2.50388	−0.276508
$83$	−1.32704	−0.145662	−0.0728308	−	0.997344i	$-0.523203\pi$
$83$	−1.32704	−0.145662	−0.0728308	+	0.997344i	$0.523203\pi$
$84$	4.98987	0.544440
$85$	2.81066	0.304859
$86$	1.54557	0.166663
$87$	0.415697	0.0445674
$88$	−1.86248	−0.198541
$89$	−18.3491	−1.94500	−0.972499	−	0.232906i	$-0.925177\pi$
$89$	−18.3491	−1.94500	−0.972499	+	0.232906i	$0.925177\pi$
$90$	0.897652	0.0946209
$91$	−3.63904	−0.381475
$92$	−5.35320	−0.558109
$93$	−0.794801	−0.0824171
$94$	−2.01089	−0.207408
$95$	−16.3354	−1.67598
$96$	−3.57520	−0.364892
$97$	−5.95421	−0.604558	−0.302279	−	0.953219i	$-0.597747\pi$
$97$	−5.95421	−0.604558	−0.302279	+	0.953219i	$0.597747\pi$
$98$	0.0281937	0.00284800
$99$	−1.49606	−0.150360
$100$	−5.50389	−0.550389
$101$	8.04818	0.800823	0.400412	−	0.916335i	$-0.368867\pi$
$101$	8.04818	0.800823	0.400412	+	0.916335i	$0.368867\pi$
$102$	0.319374	0.0316227
$103$	7.19735	0.709176	0.354588	−	0.935023i	$-0.384621\pi$
$103$	7.19735	0.709176	0.354588	+	0.935023i	$0.384621\pi$
$104$	1.72319	0.168973
$105$	7.38928	0.721120
$106$	4.33498	0.421050
$107$	−6.78409	−0.655843	−0.327921	−	0.944705i	$-0.606348\pi$
$107$	−6.78409	−0.655843	−0.327921	+	0.944705i	$0.606348\pi$
$108$	−1.89800	−0.182635
$109$	−17.5383	−1.67987	−0.839934	−	0.542688i	$-0.817406\pi$
$109$	−17.5383	−1.67987	−0.839934	+	0.542688i	$0.817406\pi$
$110$	−1.34294	−0.128045
$111$	3.13051	0.297135
$112$	−8.93446	−0.844227
$113$	−2.40236	−0.225995	−0.112998	−	0.993595i	$-0.536045\pi$
$113$	−2.40236	−0.225995	−0.112998	+	0.993595i	$0.536045\pi$
$114$	−1.85618	−0.173847
$115$	−7.92731	−0.739225
$116$	−0.788993	−0.0732562
$117$	1.38418	0.127968
$118$	0.378323	0.0348274
$119$	2.62902	0.241001
$120$	−3.49905	−0.319418
$121$	−8.76180	−0.796527
$122$	−4.12060	−0.373062
$123$	7.83998	0.706908
$124$	1.50853	0.135470
$125$	5.90286	0.527967
$126$	0.839638	0.0748009
$127$	16.1487	1.43296	0.716481	−	0.697606i	$-0.245752\pi$
$127$	16.1487	1.43296	0.716481	+	0.697606i	$0.245752\pi$
$128$	8.95645	0.791646
$129$	−4.83939	−0.426084
$130$	1.24251	0.108976
$131$	−0.0174094	−0.00152107	−0.000760534	−	1.00000i	$-0.500242\pi$
$131$	−0.0174094	−0.00152107	−0.000760534		1.00000i	$0.500242\pi$
$132$	2.83953	0.247149
$133$	−15.2797	−1.32492
$134$	−0.950907	−0.0821459
$135$	−2.81066	−0.241904
$136$	−1.24492	−0.106751
$137$	−6.77125	−0.578507	−0.289253	−	0.957253i	$-0.593407\pi$
$137$	−6.77125	−0.578507	−0.289253	+	0.957253i	$0.593407\pi$
$138$	−0.900774	−0.0766790
$139$	−0.700267	−0.0593959	−0.0296979	−	0.999559i	$-0.509455\pi$
$139$	−0.700267	−0.0593959	−0.0296979	+	0.999559i	$0.509455\pi$
$140$	−14.0249	−1.18532
$141$	6.29636	0.530249
$142$	2.88949	0.242481
$143$	−2.07082	−0.173171
$144$	3.39841	0.283201
$145$	−1.16839	−0.0970291
$146$	−0.455828	−0.0377246
$147$	−0.0882782	−0.00728106
$148$	−5.94172	−0.488406
$149$	15.4836	1.26846	0.634232	−	0.773143i	$-0.281316\pi$
$149$	15.4836	1.26846	0.634232	+	0.773143i	$0.281316\pi$
$150$	−0.926131	−0.0756183
$151$	−12.7221	−1.03531	−0.517656	−	0.855589i	$-0.673195\pi$
$151$	−12.7221	−1.03531	−0.517656	+	0.855589i	$0.673195\pi$
$152$	7.23539	0.586867
$153$	−1.00000	−0.0808452
$154$	−1.25615	−0.101224
$155$	2.23392	0.179433
$156$	−2.62718	−0.210343
$157$	−20.1142	−1.60529	−0.802643	−	0.596459i	$-0.796574\pi$
$157$	−20.1142	−1.60529	−0.802643	+	0.596459i	$0.796574\pi$
$158$	−0.319374	−0.0254080
$159$	−13.5734	−1.07644
$160$	10.0487	0.794418
$161$	−7.41498	−0.584382
$162$	−0.319374	−0.0250924
$163$	−22.3517	−1.75072	−0.875359	−	0.483474i	$-0.839375\pi$
$163$	−22.3517	−1.75072	−0.875359	+	0.483474i	$0.839375\pi$
$164$	−14.8803	−1.16196
$165$	4.20493	0.327354
$166$	0.423822	0.0328949
$167$	15.9225	1.23212	0.616060	−	0.787699i	$-0.288728\pi$
$167$	15.9225	1.23212	0.616060	+	0.787699i	$0.288728\pi$
$168$	−3.27291	−0.252511
$169$	−11.0840	−0.852619
$170$	−0.897652	−0.0688468
$171$	5.81194	0.444450
$172$	9.18516	0.700362
$173$	18.2569	1.38805	0.694024	−	0.719952i	$-0.255836\pi$
$173$	18.2569	1.38805	0.694024	+	0.719952i	$0.255836\pi$
$174$	−0.132763	−0.0100647
$175$	−7.62371	−0.576299
$176$	−5.08423	−0.383238
$177$	−1.18458	−0.0890383
$178$	5.86021	0.439241
$179$	−13.0132	−0.972650	−0.486325	−	0.873778i	$-0.661663\pi$
$179$	−13.0132	−0.972650	−0.486325	+	0.873778i	$0.661663\pi$
$180$	5.33464	0.397621
$181$	8.34896	0.620574	0.310287	−	0.950643i	$-0.399575\pi$
$181$	8.34896	0.620574	0.310287	+	0.950643i	$0.399575\pi$
$182$	1.16221	0.0861489
$183$	12.9021	0.953754
$184$	3.51122	0.258850
$185$	−8.79883	−0.646903
$186$	0.253839	0.0186123
$187$	1.49606	0.109403
$188$	−11.9505	−0.871579
$189$	−2.62902	−0.191233
$190$	5.21710	0.378488
$191$	−26.6533	−1.92857	−0.964283	−	0.264873i	$-0.914670\pi$
$191$	−26.6533	−1.92857	−0.964283	+	0.264873i	$0.914670\pi$
$192$	−5.65499	−0.408114
$193$	19.1634	1.37941	0.689707	−	0.724089i	$-0.257739\pi$
$193$	19.1634	1.37941	0.689707	+	0.724089i	$0.257739\pi$
$194$	1.90162	0.136528
$195$	−3.89047	−0.278602
$196$	0.167552	0.0119680
$197$	−17.8889	−1.27453	−0.637264	−	0.770645i	$-0.719934\pi$
$197$	−17.8889	−1.27453	−0.637264	+	0.770645i	$0.719934\pi$
$198$	0.477803	0.0339560
$199$	−17.0872	−1.21128	−0.605640	−	0.795739i	$-0.707083\pi$
$199$	−17.0872	−1.21128	−0.605640	+	0.795739i	$0.707083\pi$
$200$	3.61006	0.255270
$201$	2.97741	0.210011
$202$	−2.57038	−0.180851
$203$	−1.09287	−0.0767047
$204$	1.89800	0.132887
$205$	−22.0356	−1.53903
$206$	−2.29864	−0.160154
$207$	2.82044	0.196034
$208$	4.70401	0.326165
$209$	−8.69502	−0.601447
$210$	−2.35994	−0.162851
$211$	17.3211	1.19243	0.596215	−	0.802825i	$-0.296670\pi$
$211$	17.3211	1.19243	0.596215	+	0.802825i	$0.296670\pi$
$212$	25.7623	1.76936
$213$	−9.04738	−0.619916
$214$	2.16666	0.148110
$215$	13.6019	0.927641
$216$	1.24492	0.0847060
$217$	2.08954	0.141848
$218$	5.60128	0.379367
$219$	1.42726	0.0964451
$220$	−7.98096	−0.538076
$221$	−1.38418	−0.0931101
$222$	−0.999804	−0.0671024
$223$	11.5467	0.773222	0.386611	−	0.922243i	$-0.373646\pi$
$223$	11.5467	0.773222	0.386611	+	0.922243i	$0.373646\pi$
$224$	9.39925	0.628014
$225$	2.89984	0.193322
$226$	0.767251	0.0510368
$227$	−7.70659	−0.511504	−0.255752	−	0.966742i	$-0.582323\pi$
$227$	−7.70659	−0.511504	−0.255752	+	0.966742i	$0.582323\pi$
$228$	−11.0311	−0.730550
$229$	−1.30667	−0.0863474	−0.0431737	−	0.999068i	$-0.513747\pi$
$229$	−1.30667	−0.0863474	−0.0431737	+	0.999068i	$0.513747\pi$
$230$	2.53177	0.166940
$231$	3.93317	0.258784
$232$	0.517509	0.0339761
$233$	−27.0181	−1.77001	−0.885007	−	0.465577i	$-0.845847\pi$
$233$	−27.0181	−1.77001	−0.885007	+	0.465577i	$0.845847\pi$
$234$	−0.442071	−0.0288991
$235$	−17.6970	−1.15442
$236$	2.24833	0.146354
$237$	1.00000	0.0649570
$238$	−0.839638	−0.0544257
$239$	−21.4308	−1.38624	−0.693122	−	0.720821i	$-0.743765\pi$
$239$	−21.4308	−1.38624	−0.693122	+	0.720821i	$0.743765\pi$
$240$	−9.55178	−0.616565
$241$	21.1773	1.36415	0.682075	−	0.731282i	$-0.261078\pi$
$241$	21.1773	1.36415	0.682075	+	0.731282i	$0.261078\pi$
$242$	2.79829	0.179881
$243$	1.00000	0.0641500
$244$	−24.4883	−1.56770
$245$	0.248120	0.0158518
$246$	−2.50388	−0.159642
$247$	8.04478	0.511877
$248$	−0.989463	−0.0628310
$249$	−1.32704	−0.0840978
$250$	−1.88522	−0.119232
$251$	1.97507	0.124665	0.0623326	−	0.998055i	$-0.480146\pi$
$251$	1.97507	0.124665	0.0623326	+	0.998055i	$0.480146\pi$
$252$	4.98987	0.314332
$253$	−4.21956	−0.265281
$254$	−5.15746	−0.323608
$255$	2.81066	0.176011
$256$	8.44952	0.528095
$257$	−28.9488	−1.80578	−0.902888	−	0.429876i	$-0.858557\pi$
$257$	−28.9488	−1.80578	−0.902888	+	0.429876i	$0.858557\pi$
$258$	1.54557	0.0962231
$259$	−8.23017	−0.511398
$260$	7.38412	0.457943
$261$	0.415697	0.0257310
$262$	0.00556011	0.000343505 0
$263$	24.4206	1.50584	0.752920	−	0.658112i	$-0.228644\pi$
$263$	24.4206	1.50584	0.752920	+	0.658112i	$0.228644\pi$
$264$	−1.86248	−0.114628
$265$	38.1502	2.34355
$266$	4.87992	0.299207
$267$	−18.3491	−1.12295
$268$	−5.65113	−0.345198
$269$	−3.14643	−0.191841	−0.0959206	−	0.995389i	$-0.530580\pi$
$269$	−3.14643	−0.191841	−0.0959206	+	0.995389i	$0.530580\pi$
$270$	0.897652	0.0546294
$271$	−7.10047	−0.431323	−0.215661	−	0.976468i	$-0.569191\pi$
$271$	−7.10047	−0.431323	−0.215661	+	0.976468i	$0.569191\pi$
$272$	−3.39841	−0.206059
$273$	−3.63904	−0.220244
$274$	2.16256	0.130645
$275$	−4.33834	−0.261612
$276$	−5.35320	−0.322225
$277$	27.5955	1.65805	0.829026	−	0.559209i	$-0.188895\pi$
$277$	27.5955	1.65805	0.829026	+	0.559209i	$0.188895\pi$
$278$	0.223647	0.0134134
$279$	−0.794801	−0.0475835
$280$	9.19905	0.549749
$281$	−12.9632	−0.773320	−0.386660	−	0.922222i	$-0.626371\pi$
$281$	−12.9632	−0.773320	−0.386660	+	0.922222i	$0.626371\pi$
$282$	−2.01089	−0.119747
$283$	−16.3266	−0.970516	−0.485258	−	0.874371i	$-0.661274\pi$
$283$	−16.3266	−0.970516	−0.485258	+	0.874371i	$0.661274\pi$
$284$	17.1719	1.01897
$285$	−16.3354	−0.967626
$286$	0.661366	0.0391074
$287$	−20.6114	−1.21665
$288$	−3.57520	−0.210671
$289$	1.00000	0.0588235
$290$	0.373151	0.0219122
$291$	−5.95421	−0.349042
$292$	−2.70893	−0.158528
$293$	−20.2950	−1.18565	−0.592825	−	0.805332i	$-0.701987\pi$
$293$	−20.2950	−1.18565	−0.592825	+	0.805332i	$0.701987\pi$
$294$	0.0281937	0.00164429
$295$	3.32945	0.193848
$296$	3.89724	0.226522
$297$	−1.49606	−0.0868104
$298$	−4.94505	−0.286459
$299$	3.90400	0.225774
$300$	−5.50389	−0.317767
$301$	12.7228	0.733331
$302$	4.06311	0.233806
$303$	8.04818	0.462356
$304$	19.7513	1.13282
$305$	−36.2636	−2.07645
$306$	0.319374	0.0182574
$307$	2.35371	0.134334	0.0671668	−	0.997742i	$-0.478604\pi$
$307$	2.35371	0.134334	0.0671668	+	0.997742i	$0.478604\pi$
$308$	−7.46516	−0.425367
$309$	7.19735	0.409443
$310$	−0.713455	−0.0405215
$311$	12.5034	0.709004	0.354502	−	0.935055i	$-0.384650\pi$
$311$	12.5034	0.709004	0.354502	+	0.935055i	$0.384650\pi$
$312$	1.72319	0.0975566
$313$	23.2861	1.31621	0.658105	−	0.752926i	$-0.271358\pi$
$313$	23.2861	1.31621	0.658105	+	0.752926i	$0.271358\pi$
$314$	6.42394	0.362524
$315$	7.38928	0.416339
$316$	−1.89800	−0.106771
$317$	−17.5501	−0.985710	−0.492855	−	0.870111i	$-0.664047\pi$
$317$	−17.5501	−0.985710	−0.492855	+	0.870111i	$0.664047\pi$
$318$	4.33498	0.243094
$319$	−0.621909	−0.0348202
$320$	15.8943	0.888517
$321$	−6.78409	−0.378651
$322$	2.36815	0.131972
$323$	−5.81194	−0.323385
$324$	−1.89800	−0.105444
$325$	4.01390	0.222651
$326$	7.13853	0.395367
$327$	−17.5383	−0.969872
$328$	9.76014	0.538914
$329$	−16.5532	−0.912609
$330$	−1.34294	−0.0739267
$331$	−20.1755	−1.10895	−0.554474	−	0.832201i	$-0.687081\pi$
$331$	−20.1755	−1.10895	−0.554474	+	0.832201i	$0.687081\pi$
$332$	2.51872	0.138233
$333$	3.13051	0.171551
$334$	−5.08523	−0.278251
$335$	−8.36851	−0.457221
$336$	−8.93446	−0.487415
$337$	−14.7521	−0.803601	−0.401800	−	0.915727i	$-0.631615\pi$
$337$	−14.7521	−0.803601	−0.401800	+	0.915727i	$0.631615\pi$
$338$	3.53995	0.192548
$339$	−2.40236	−0.130478
$340$	−5.33464	−0.289312
$341$	1.18907	0.0643919
$342$	−1.85618	−0.100371
$343$	18.6352	1.00621
$344$	−6.02464	−0.324827
$345$	−7.92731	−0.426792
$346$	−5.83078	−0.313465
$347$	4.11289	0.220792	0.110396	−	0.993888i	$-0.464788\pi$
$347$	4.11289	0.220792	0.110396	+	0.993888i	$0.464788\pi$
$348$	−0.788993	−0.0422945
$349$	32.3678	1.73261	0.866304	−	0.499517i	$-0.166489\pi$
$349$	32.3678	1.73261	0.866304	+	0.499517i	$0.166489\pi$
$350$	2.43481	0.130146
$351$	1.38418	0.0738822
$352$	5.34872	0.285088
$353$	−7.69164	−0.409385	−0.204692	−	0.978826i	$-0.565619\pi$
$353$	−7.69164	−0.409385	−0.204692	+	0.978826i	$0.565619\pi$
$354$	0.378323	0.0201076
$355$	25.4291	1.34964
$356$	34.8266	1.84580
$357$	2.62902	0.139142
$358$	4.15606	0.219655
$359$	−15.4920	−0.817636	−0.408818	−	0.912616i	$-0.634059\pi$
$359$	−15.4920	−0.817636	−0.408818	+	0.912616i	$0.634059\pi$
$360$	−3.49905	−0.184416
$361$	14.7786	0.777821
$362$	−2.66644	−0.140145
$363$	−8.76180	−0.459875
$364$	6.90689	0.362019
$365$	−4.01154	−0.209974
$366$	−4.12060	−0.215387
$367$	−9.13280	−0.476728	−0.238364	−	0.971176i	$-0.576611\pi$
$367$	−9.13280	−0.476728	−0.238364	+	0.971176i	$0.576611\pi$
$368$	9.58500	0.499653
$369$	7.83998	0.408133
$370$	2.81011	0.146091
$371$	35.6846	1.85265
$372$	1.50853	0.0782138
$373$	−2.86236	−0.148208	−0.0741038	−	0.997251i	$-0.523610\pi$
$373$	−2.86236	−0.148208	−0.0741038	+	0.997251i	$0.523610\pi$
$374$	−0.477803	−0.0247066
$375$	5.90286	0.304822
$376$	7.83846	0.404238
$377$	0.575400	0.0296346
$378$	0.839638	0.0431863
$379$	−10.8131	−0.555434	−0.277717	−	0.960663i	$-0.589578\pi$
$379$	−10.8131	−0.555434	−0.277717	+	0.960663i	$0.589578\pi$
$380$	31.0046	1.59050
$381$	16.1487	0.827321
$382$	8.51237	0.435531
$383$	18.5682	0.948793	0.474396	−	0.880311i	$-0.342666\pi$
$383$	18.5682	0.948793	0.474396	+	0.880311i	$0.342666\pi$
$384$	8.95645	0.457057
$385$	−11.0548	−0.563406
$386$	−6.12029	−0.311515
$387$	−4.83939	−0.246000
$388$	11.3011	0.573726
$389$	−28.5592	−1.44801	−0.724006	−	0.689794i	$-0.757701\pi$
$389$	−28.5592	−1.44801	−0.724006	+	0.689794i	$0.757701\pi$
$390$	1.24251	0.0629171
$391$	−2.82044	−0.142636
$392$	−0.109899	−0.00555075
$393$	−0.0174094	−0.000878189 0
$394$	5.71323	0.287828
$395$	−2.81066	−0.141420
$396$	2.83953	0.142692
$397$	26.8710	1.34862	0.674309	−	0.738449i	$-0.264442\pi$
$397$	26.8710	1.34862	0.674309	+	0.738449i	$0.264442\pi$
$398$	5.45721	0.273545
$399$	−15.2797	−0.764940
$400$	9.85482	0.492741
$401$	30.0961	1.50293	0.751464	−	0.659774i	$-0.229348\pi$
$401$	30.0961	1.50293	0.751464	+	0.659774i	$0.229348\pi$
$402$	−0.950907	−0.0474269
$403$	−1.10015	−0.0548024
$404$	−15.2754	−0.759982
$405$	−2.81066	−0.139663
$406$	0.349035	0.0173223
$407$	−4.68345	−0.232150
$408$	−1.24492	−0.0616327
$409$	−14.2570	−0.704962	−0.352481	−	0.935819i	$-0.614662\pi$
$409$	−14.2570	−0.704962	−0.352481	+	0.935819i	$0.614662\pi$
$410$	7.03758	0.347561
$411$	−6.77125	−0.334001
$412$	−13.6606	−0.673008
$413$	3.11427	0.153243
$414$	−0.900774	−0.0442706
$415$	3.72987	0.183092
$416$	−4.94873	−0.242631
$417$	−0.700267	−0.0342922
$418$	2.77696	0.135826
$419$	−4.35242	−0.212630	−0.106315	−	0.994333i	$-0.533905\pi$
$419$	−4.35242	−0.212630	−0.106315	+	0.994333i	$0.533905\pi$
$420$	−14.0249	−0.684343
$421$	29.4627	1.43592	0.717961	−	0.696083i	$-0.245076\pi$
$421$	29.4627	1.43592	0.717961	+	0.696083i	$0.245076\pi$
$422$	−5.53189	−0.269288
$423$	6.29636	0.306140
$424$	−16.8978	−0.820627
$425$	−2.89984	−0.140663
$426$	2.88949	0.139996
$427$	−33.9199	−1.64150
$428$	12.8762	0.622395
$429$	−2.07082	−0.0999803
$430$	−4.34409	−0.209490
$431$	10.8813	0.524133	0.262067	−	0.965050i	$-0.415596\pi$
$431$	10.8813	0.524133	0.262067	+	0.965050i	$0.415596\pi$
$432$	3.39841	0.163506
$433$	−15.1008	−0.725696	−0.362848	−	0.931848i	$-0.618196\pi$
$433$	−15.1008	−0.725696	−0.362848	+	0.931848i	$0.618196\pi$
$434$	−0.667346	−0.0320336
$435$	−1.16839	−0.0560198
$436$	33.2878	1.59420
$437$	16.3922	0.784146
$438$	−0.455828	−0.0217803
$439$	−18.1708	−0.867246	−0.433623	−	0.901094i	$-0.642765\pi$
$439$	−18.1708	−0.867246	−0.433623	+	0.901094i	$0.642765\pi$
$440$	5.23480	0.249559
$441$	−0.0882782	−0.00420372
$442$	0.442071	0.0210272
$443$	19.7169	0.936777	0.468388	−	0.883523i	$-0.344835\pi$
$443$	19.7169	0.936777	0.468388	+	0.883523i	$0.344835\pi$
$444$	−5.94172	−0.281981
$445$	51.5731	2.44480
$446$	−3.68770	−0.174618
$447$	15.4836	0.732348
$448$	14.8671	0.702402
$449$	−13.7536	−0.649073	−0.324536	−	0.945873i	$-0.605208\pi$
$449$	−13.7536	−0.649073	−0.324536	+	0.945873i	$0.605208\pi$
$450$	−0.926131	−0.0436583
$451$	−11.7291	−0.552302
$452$	4.55968	0.214469
$453$	−12.7221	−0.597738
$454$	2.46128	0.115514
$455$	10.2281	0.479501
$456$	7.23539	0.338828
$457$	32.3618	1.51382	0.756910	−	0.653519i	$-0.226708\pi$
$457$	32.3618	1.51382	0.756910	+	0.653519i	$0.226708\pi$
$458$	0.417317	0.0194999
$459$	−1.00000	−0.0466760
$460$	15.0460	0.701525
$461$	−12.0928	−0.563218	−0.281609	−	0.959529i	$-0.590868\pi$
$461$	−12.0928	−0.563218	−0.281609	+	0.959529i	$0.590868\pi$
$462$	−1.25615	−0.0584415
$463$	29.4048	1.36656	0.683278	−	0.730159i	$-0.260554\pi$
$463$	29.4048	1.36656	0.683278	+	0.730159i	$0.260554\pi$
$464$	1.41271	0.0655833
$465$	2.23392	0.103596
$466$	8.62887	0.399725
$467$	−4.06177	−0.187956	−0.0939781	−	0.995574i	$-0.529958\pi$
$467$	−4.06177	−0.187956	−0.0939781	+	0.995574i	$0.529958\pi$
$468$	−2.62718	−0.121441
$469$	−7.82767	−0.361448
$470$	5.65194	0.260705
$471$	−20.1142	−0.926813
$472$	−1.47470	−0.0678787
$473$	7.24003	0.332897
$474$	−0.319374	−0.0146693
$475$	16.8537	0.773299
$476$	−4.98987	−0.228710
$477$	−13.5734	−0.621482
$478$	6.84443	0.313057
$479$	4.57055	0.208834	0.104417	−	0.994534i	$-0.466702\pi$
$479$	4.57055	0.208834	0.104417	+	0.994534i	$0.466702\pi$
$480$	10.0487	0.458658
$481$	4.33320	0.197577
$482$	−6.76347	−0.308068
$483$	−7.41498	−0.337393
$484$	16.6299	0.755904
$485$	16.7353	0.759910
$486$	−0.319374	−0.0144871
$487$	−11.4593	−0.519272	−0.259636	−	0.965707i	$-0.583603\pi$
$487$	−11.4593	−0.519272	−0.259636	+	0.965707i	$0.583603\pi$
$488$	16.0621	0.727098
$489$	−22.3517	−1.01078
$490$	−0.0792431	−0.00357984
$491$	−41.0936	−1.85453	−0.927265	−	0.374405i	$-0.877847\pi$
$491$	−41.0936	−1.85453	−0.927265	+	0.374405i	$0.877847\pi$
$492$	−14.8803	−0.670855
$493$	−0.415697	−0.0187221
$494$	−2.56929	−0.115598
$495$	4.20493	0.188998
$496$	−2.70106	−0.121281
$497$	23.7857	1.06693
$498$	0.423822	0.0189919
$499$	−17.0537	−0.763429	−0.381715	−	0.924280i	$-0.624666\pi$
$499$	−17.0537	−0.763429	−0.381715	+	0.924280i	$0.624666\pi$
$500$	−11.2036	−0.501041
$501$	15.9225	0.711365
$502$	−0.630784	−0.0281533
$503$	−9.89607	−0.441244	−0.220622	−	0.975359i	$-0.570809\pi$
$503$	−9.89607	−0.441244	−0.220622	+	0.975359i	$0.570809\pi$
$504$	−3.27291	−0.145787
$505$	−22.6207	−1.00661
$506$	1.34761	0.0599088
$507$	−11.0840	−0.492260
$508$	−30.6502	−1.35988
$509$	8.77202	0.388813	0.194406	−	0.980921i	$-0.437722\pi$
$509$	8.77202	0.388813	0.194406	+	0.980921i	$0.437722\pi$
$510$	−0.897652	−0.0397487
$511$	−3.75228	−0.165991
$512$	−20.6115	−0.910906
$513$	5.81194	0.256603
$514$	9.24548	0.407801
$515$	−20.2293	−0.891411
$516$	9.18516	0.404354
$517$	−9.41975	−0.414280
$518$	2.62850	0.115490
$519$	18.2569	0.801390
$520$	−4.84332	−0.212394
$521$	−7.99744	−0.350374	−0.175187	−	0.984535i	$-0.556053\pi$
$521$	−7.99744	−0.350374	−0.175187	+	0.984535i	$0.556053\pi$
$522$	−0.132763	−0.00581087
$523$	8.87583	0.388113	0.194056	−	0.980990i	$-0.437835\pi$
$523$	8.87583	0.388113	0.194056	+	0.980990i	$0.437835\pi$
$524$	0.0330431	0.00144349
$525$	−7.62371	−0.332726
$526$	−7.79931	−0.340066
$527$	0.794801	0.0346221
$528$	−5.08423	−0.221263
$529$	−15.0451	−0.654136
$530$	−12.1842	−0.529247
$531$	−1.18458	−0.0514063
$532$	29.0008	1.25735
$533$	10.8520	0.470051
$534$	5.86021	0.253596
$535$	19.0678	0.824373
$536$	3.70664	0.160102
$537$	−13.0132	−0.561560
$538$	1.00489	0.0433238
$539$	0.132070	0.00568865
$540$	5.33464	0.229566
$541$	−12.5360	−0.538965	−0.269483	−	0.963005i	$-0.586853\pi$
$541$	−12.5360	−0.538965	−0.269483	+	0.963005i	$0.586853\pi$
$542$	2.26770	0.0974062
$543$	8.34896	0.358288
$544$	3.57520	0.153285
$545$	49.2944	2.11154
$546$	1.16221	0.0497381
$547$	24.1394	1.03213	0.516063	−	0.856551i	$-0.327397\pi$
$547$	24.1394	1.03213	0.516063	+	0.856551i	$0.327397\pi$
$548$	12.8518	0.549003
$549$	12.9021	0.550650
$550$	1.38555	0.0590801
$551$	2.41601	0.102925
$552$	3.51122	0.149447
$553$	−2.62902	−0.111797
$554$	−8.81328	−0.374440
$555$	−8.79883	−0.373489
$556$	1.32911	0.0563667
$557$	0.367284	0.0155623	0.00778117	−	0.999970i	$-0.497523\pi$
$557$	0.367284	0.0155623	0.00778117	+	0.999970i	$0.497523\pi$
$558$	0.253839	0.0107458
$559$	−6.69859	−0.283320
$560$	25.1118	1.06117
$561$	1.49606	0.0631638
$562$	4.14011	0.174640
$563$	−15.7736	−0.664778	−0.332389	−	0.943142i	$-0.607855\pi$
$563$	−15.7736	−0.664778	−0.332389	+	0.943142i	$0.607855\pi$
$564$	−11.9505	−0.503207
$565$	6.75223	0.284068
$566$	5.21429	0.219173
$567$	−2.62902	−0.110408
$568$	−11.2632	−0.472595
$569$	−12.0171	−0.503783	−0.251892	−	0.967755i	$-0.581053\pi$
$569$	−12.0171	−0.503783	−0.251892	+	0.967755i	$0.581053\pi$
$570$	5.21710	0.218520
$571$	32.4211	1.35678	0.678390	−	0.734702i	$-0.262678\pi$
$571$	32.4211	1.35678	0.678390	+	0.734702i	$0.262678\pi$
$572$	3.93042	0.164339
$573$	−26.6533	−1.11346
$574$	6.58275	0.274759
$575$	8.17881	0.341080
$576$	−5.65499	−0.235625
$577$	−17.9349	−0.746642	−0.373321	−	0.927702i	$-0.621781\pi$
$577$	−17.9349	−0.746642	−0.373321	+	0.927702i	$0.621781\pi$
$578$	−0.319374	−0.0132842
$579$	19.1634	0.796405
$580$	2.21760	0.0920807
$581$	3.48881	0.144740
$582$	1.90162	0.0788246
$583$	20.3066	0.841015
$584$	1.77682	0.0735253
$585$	−3.89047	−0.160851
$586$	6.48170	0.267757
$587$	−42.0050	−1.73373	−0.866867	−	0.498540i	$-0.833870\pi$
$587$	−42.0050	−1.73373	−0.866867	+	0.498540i	$0.833870\pi$
$588$	0.167552	0.00690973
$589$	−4.61933	−0.190336
$590$	−1.06334	−0.0437769
$591$	−17.8889	−0.735849
$592$	10.6388	0.437250
$593$	13.3900	0.549863	0.274931	−	0.961464i	$-0.411345\pi$
$593$	13.3900	0.549863	0.274931	+	0.961464i	$0.411345\pi$
$594$	0.477803	0.0196045
$595$	−7.38928	−0.302931
$596$	−29.3878	−1.20377
$597$	−17.0872	−0.699333
$598$	−1.24684	−0.0509869
$599$	−23.7703	−0.971228	−0.485614	−	0.874173i	$-0.661404\pi$
$599$	−23.7703	−0.971228	−0.485614	+	0.874173i	$0.661404\pi$
$600$	3.61006	0.147380
$601$	46.6146	1.90145	0.950725	−	0.310037i	$-0.100341\pi$
$601$	46.6146	1.90145	0.950725	+	0.310037i	$0.100341\pi$
$602$	−4.06333	−0.165609
$603$	2.97741	0.121250
$604$	24.1466	0.982511
$605$	24.6265	1.00121
$606$	−2.57038	−0.104414
$607$	−34.8757	−1.41556	−0.707780	−	0.706433i	$-0.750303\pi$
$607$	−34.8757	−1.41556	−0.707780	+	0.706433i	$0.750303\pi$
$608$	−20.7788	−0.842693
$609$	−1.09287	−0.0442855
$610$	11.5816	0.468927
$611$	8.71531	0.352584
$612$	1.89800	0.0767221
$613$	−12.8952	−0.520833	−0.260417	−	0.965496i	$-0.583860\pi$
$613$	−12.8952	−0.520833	−0.260417	+	0.965496i	$0.583860\pi$
$614$	−0.751714	−0.0303367
$615$	−22.0356	−0.888560
$616$	4.89648	0.197285
$617$	−5.12587	−0.206360	−0.103180	−	0.994663i	$-0.532902\pi$
$617$	−5.12587	−0.206360	−0.103180	+	0.994663i	$0.532902\pi$
$618$	−2.29864	−0.0924650
$619$	−28.6276	−1.15064	−0.575321	−	0.817928i	$-0.695123\pi$
$619$	−28.6276	−1.15064	−0.575321	+	0.817928i	$0.695123\pi$
$620$	−4.23998	−0.170282
$621$	2.82044	0.113180
$622$	−3.99326	−0.160115
$623$	48.2400	1.93270
$624$	4.70401	0.188311
$625$	−31.0901	−1.24361
$626$	−7.43698	−0.297241
$627$	−8.69502	−0.347246
$628$	38.1767	1.52342
$629$	−3.13051	−0.124822
$630$	−2.35994	−0.0940223
$631$	44.7794	1.78264	0.891320	−	0.453375i	$-0.149780\pi$
$631$	44.7794	1.78264	0.891320	+	0.453375i	$0.149780\pi$
$632$	1.24492	0.0495202
$633$	17.3211	0.688450
$634$	5.60503	0.222604
$635$	−45.3885	−1.80119
$636$	25.7623	1.02154
$637$	−0.122193	−0.00484147
$638$	0.198621	0.00786350
$639$	−9.04738	−0.357909
$640$	−25.1736	−0.995073
$641$	−34.1973	−1.35071	−0.675356	−	0.737492i	$-0.736010\pi$
$641$	−34.1973	−1.35071	−0.675356	+	0.737492i	$0.736010\pi$
$642$	2.16666	0.0855113
$643$	7.37823	0.290969	0.145485	−	0.989361i	$-0.453526\pi$
$643$	7.37823	0.290969	0.145485	+	0.989361i	$0.453526\pi$
$644$	14.0736	0.554579
$645$	13.6019	0.535574
$646$	1.85618	0.0730304
$647$	6.53589	0.256952	0.128476	−	0.991713i	$-0.458991\pi$
$647$	6.53589	0.256952	0.128476	+	0.991713i	$0.458991\pi$
$648$	1.24492	0.0489050
$649$	1.77220	0.0695650
$650$	−1.28193	−0.0502816
$651$	2.08954	0.0818957
$652$	42.4235	1.66143
$653$	−33.5123	−1.31144	−0.655719	−	0.755005i	$-0.727635\pi$
$653$	−33.5123	−1.31144	−0.655719	+	0.755005i	$0.727635\pi$
$654$	5.60128	0.219028
$655$	0.0489320	0.00191193
$656$	26.6435	1.04025
$657$	1.42726	0.0556826
$658$	5.28667	0.206096
$659$	−49.3297	−1.92161	−0.960807	−	0.277219i	$-0.910587\pi$
$659$	−49.3297	−1.92161	−0.960807	+	0.277219i	$0.910587\pi$
$660$	−7.98096	−0.310659
$661$	−19.2359	−0.748191	−0.374096	−	0.927390i	$-0.622047\pi$
$661$	−19.2359	−0.748191	−0.374096	+	0.927390i	$0.622047\pi$
$662$	6.44354	0.250435
$663$	−1.38418	−0.0537572
$664$	−1.65206	−0.0641122
$665$	42.9460	1.66538
$666$	−0.999804	−0.0387416
$667$	1.17245	0.0453974
$668$	−30.2209	−1.16928
$669$	11.5467	0.446420
$670$	2.67268	0.103255
$671$	−19.3024	−0.745162
$672$	9.39925	0.362584
$673$	−8.52865	−0.328755	−0.164378	−	0.986397i	$-0.552562\pi$
$673$	−8.52865	−0.328755	−0.164378	+	0.986397i	$0.552562\pi$
$674$	4.71145	0.181478
$675$	2.89984	0.111615
$676$	21.0375	0.809135
$677$	−33.5220	−1.28835	−0.644177	−	0.764876i	$-0.722800\pi$
$677$	−33.5220	−1.28835	−0.644177	+	0.764876i	$0.722800\pi$
$678$	0.767251	0.0294661
$679$	15.6537	0.600734
$680$	3.49905	0.134182
$681$	−7.70659	−0.295317
$682$	−0.379759	−0.0145417
$683$	−16.3693	−0.626354	−0.313177	−	0.949695i	$-0.601393\pi$
$683$	−16.3693	−0.626354	−0.313177	+	0.949695i	$0.601393\pi$
$684$	−11.0311	−0.421783
$685$	19.0317	0.727164
$686$	−5.95159	−0.227233
$687$	−1.30667	−0.0498527
$688$	−16.4462	−0.627005
$689$	−18.7880	−0.715767
$690$	2.53177	0.0963830
$691$	−36.0210	−1.37030	−0.685151	−	0.728401i	$-0.740264\pi$
$691$	−36.0210	−1.37030	−0.685151	+	0.728401i	$0.740264\pi$
$692$	−34.6517	−1.31726
$693$	3.93317	0.149409
$694$	−1.31355	−0.0498617
$695$	1.96822	0.0746587
$696$	0.517509	0.0196161
$697$	−7.83998	−0.296961
$698$	−10.3374	−0.391277
$699$	−27.0181	−1.02192
$700$	14.4698	0.546908
$701$	−47.1813	−1.78201	−0.891007	−	0.453990i	$-0.850000\pi$
$701$	−47.1813	−1.78201	−0.891007	+	0.453990i	$0.850000\pi$
$702$	−0.442071	−0.0166849
$703$	18.1944	0.686213
$704$	8.46022	0.318857
$705$	−17.6970	−0.666506
$706$	2.45651	0.0924519
$707$	−21.1588	−0.795758
$708$	2.24833	0.0844973
$709$	29.9494	1.12477	0.562386	−	0.826875i	$-0.309884\pi$
$709$	29.9494	1.12477	0.562386	+	0.826875i	$0.309884\pi$
$710$	−8.12140	−0.304791
$711$	1.00000	0.0375029
$712$	−22.8431	−0.856082
$713$	−2.24169	−0.0839519
$714$	−0.839638	−0.0314227
$715$	5.82039	0.217670
$716$	24.6990	0.923045
$717$	−21.4308	−0.800348
$718$	4.94773	0.184648
$719$	34.8296	1.29893	0.649463	−	0.760393i	$-0.274994\pi$
$719$	34.8296	1.29893	0.649463	+	0.760393i	$0.274994\pi$
$720$	−9.55178	−0.355974
$721$	−18.9219	−0.704690
$722$	−4.71990	−0.175656
$723$	21.1773	0.787592
$724$	−15.8463	−0.588925
$725$	1.20545	0.0447694
$726$	2.79829	0.103854
$727$	−23.3397	−0.865623	−0.432812	−	0.901484i	$-0.642478\pi$
$727$	−23.3397	−0.865623	−0.432812	+	0.901484i	$0.642478\pi$
$728$	−4.53030	−0.167904
$729$	1.00000	0.0370370
$730$	1.28118	0.0474186
$731$	4.83939	0.178991
$732$	−24.4883	−0.905113
$733$	28.3989	1.04894	0.524469	−	0.851430i	$-0.324264\pi$
$733$	28.3989	1.04894	0.524469	+	0.851430i	$0.324264\pi$
$734$	2.91678	0.107660
$735$	0.248120	0.00915206
$736$	−10.0836	−0.371688
$737$	−4.45440	−0.164080
$738$	−2.50388	−0.0921693
$739$	−16.0809	−0.591544	−0.295772	−	0.955259i	$-0.595577\pi$
$739$	−16.0809	−0.591544	−0.295772	+	0.955259i	$0.595577\pi$
$740$	16.7002	0.613911
$741$	8.04478	0.295532
$742$	−11.3967	−0.418387
$743$	11.6094	0.425908	0.212954	−	0.977062i	$-0.431692\pi$
$743$	11.6094	0.425908	0.212954	+	0.977062i	$0.431692\pi$
$744$	−0.989463	−0.0362755
$745$	−43.5192	−1.59442
$746$	0.914163	0.0334699
$747$	−1.32704	−0.0485539
$748$	−2.83953	−0.103823
$749$	17.8355	0.651694
$750$	−1.88522	−0.0688384
$751$	−19.9536	−0.728117	−0.364058	−	0.931376i	$-0.618609\pi$
$751$	−19.9536	−0.728117	−0.364058	+	0.931376i	$0.618609\pi$
$752$	21.3976	0.780290
$753$	1.97507	0.0719755
$754$	−0.183768	−0.00669243
$755$	35.7576	1.30135
$756$	4.98987	0.181480
$757$	7.51950	0.273301	0.136650	−	0.990619i	$-0.456366\pi$
$757$	7.51950	0.273301	0.136650	+	0.990619i	$0.456366\pi$
$758$	3.45343	0.125434
$759$	−4.21956	−0.153160
$760$	−20.3362	−0.737673
$761$	−3.42303	−0.124085	−0.0620424	−	0.998074i	$-0.519761\pi$
$761$	−3.42303	−0.124085	−0.0620424	+	0.998074i	$0.519761\pi$
$762$	−5.15746	−0.186835
$763$	46.1086	1.66924
$764$	50.5880	1.83021
$765$	2.81066	0.101620
$766$	−5.93021	−0.214267
$767$	−1.63967	−0.0592051
$768$	8.44952	0.304896
$769$	−23.3941	−0.843611	−0.421806	−	0.906686i	$-0.638604\pi$
$769$	−23.3941	−0.843611	−0.421806	+	0.906686i	$0.638604\pi$
$770$	3.53062	0.127235
$771$	−28.9488	−1.04257
$772$	−36.3722	−1.30906
$773$	−52.1969	−1.87739	−0.938696	−	0.344747i	$-0.887965\pi$
$773$	−52.1969	−1.87739	−0.938696	+	0.344747i	$0.887965\pi$
$774$	1.54557	0.0555544
$775$	−2.30479	−0.0827906
$776$	−7.41250	−0.266093
$777$	−8.23017	−0.295256
$778$	9.12107	0.327006
$779$	45.5655	1.63255
$780$	7.38412	0.264394
$781$	13.5354	0.484336
$782$	0.900774	0.0322116
$783$	0.415697	0.0148558
$784$	−0.300005	−0.0107145
$785$	56.5342	2.01779
$786$	0.00556011	0.000198322 0
$787$	−25.9345	−0.924466	−0.462233	−	0.886759i	$-0.652952\pi$
$787$	−25.9345	−0.924466	−0.462233	+	0.886759i	$0.652952\pi$
$788$	33.9531	1.20953
$789$	24.4206	0.869398
$790$	0.897652	0.0319370
$791$	6.31584	0.224566
$792$	−1.86248	−0.0661802
$793$	17.8589	0.634189
$794$	−8.58190	−0.304560
$795$	38.1502	1.35305
$796$	32.4315	1.14951
$797$	−3.69743	−0.130970	−0.0654849	−	0.997854i	$-0.520859\pi$
$797$	−3.69743	−0.130970	−0.0654849	+	0.997854i	$0.520859\pi$
$798$	4.87992	0.172747
$799$	−6.29636	−0.222749
$800$	−10.3675	−0.366546
$801$	−18.3491	−0.648333
$802$	−9.61191	−0.339408
$803$	−2.13527	−0.0753519
$804$	−5.65113	−0.199300
$805$	20.8410	0.734549
$806$	0.351359	0.0123761
$807$	−3.14643	−0.110760
$808$	10.0193	0.352479
$809$	47.1518	1.65777	0.828884	−	0.559420i	$-0.188976\pi$
$809$	47.1518	1.65777	0.828884	+	0.559420i	$0.188976\pi$
$810$	0.897652	0.0315403
$811$	−14.9562	−0.525183	−0.262591	−	0.964907i	$-0.584577\pi$
$811$	−14.9562	−0.525183	−0.262591	+	0.964907i	$0.584577\pi$
$812$	2.07428	0.0727928
$813$	−7.10047	−0.249024
$814$	1.49577	0.0524267
$815$	62.8230	2.20060
$816$	−3.39841	−0.118968
$817$	−28.1262	−0.984011
$818$	4.55330	0.159202
$819$	−3.63904	−0.127158
$820$	41.8235	1.46054
$821$	26.5940	0.928137	0.464068	−	0.885799i	$-0.346389\pi$
$821$	26.5940	0.928137	0.464068	+	0.885799i	$0.346389\pi$
$822$	2.16256	0.0754279
$823$	−5.29370	−0.184527	−0.0922634	−	0.995735i	$-0.529410\pi$
$823$	−5.29370	−0.184527	−0.0922634	+	0.995735i	$0.529410\pi$
$824$	8.96011	0.312140
$825$	−4.33834	−0.151042
$826$	−0.994616	−0.0346071
$827$	0.490548	0.0170580	0.00852901	−	0.999964i	$-0.497285\pi$
$827$	0.490548	0.0170580	0.00852901	+	0.999964i	$0.497285\pi$
$828$	−5.35320	−0.186036
$829$	7.57026	0.262926	0.131463	−	0.991321i	$-0.458033\pi$
$829$	7.57026	0.262926	0.131463	+	0.991321i	$0.458033\pi$
$830$	−1.19122	−0.0413479
$831$	27.5955	0.957277
$832$	−7.82753	−0.271371
$833$	0.0882782	0.00305866
$834$	0.223647	0.00774426
$835$	−44.7528	−1.54873
$836$	16.5032	0.570774
$837$	−0.794801	−0.0274724
$838$	1.39005	0.0480184
$839$	17.9395	0.619339	0.309670	−	0.950844i	$-0.399782\pi$
$839$	17.9395	0.619339	0.309670	+	0.950844i	$0.399782\pi$
$840$	9.19905	0.317397
$841$	−28.8272	−0.994041
$842$	−9.40960	−0.324276
$843$	−12.9632	−0.446477
$844$	−32.8754	−1.13162
$845$	31.1535	1.07171
$846$	−2.01089	−0.0691359
$847$	23.0349	0.791488
$848$	−46.1279	−1.58404
$849$	−16.3266	−0.560328
$850$	0.926131	0.0317660
$851$	8.82943	0.302669
$852$	17.1719	0.588300
$853$	−8.13272	−0.278459	−0.139230	−	0.990260i	$-0.544463\pi$
$853$	−8.13272	−0.278459	−0.139230	+	0.990260i	$0.544463\pi$
$854$	10.8331	0.370702
$855$	−16.3354	−0.558659
$856$	−8.44564	−0.288666
$857$	1.51596	0.0517842	0.0258921	−	0.999665i	$-0.491757\pi$
$857$	1.51596	0.0517842	0.0258921	+	0.999665i	$0.491757\pi$
$858$	0.661366	0.0225787
$859$	35.4421	1.20927	0.604634	−	0.796503i	$-0.293319\pi$
$859$	35.4421	1.20927	0.604634	+	0.796503i	$0.293319\pi$
$860$	−25.8164	−0.880332
$861$	−20.6114	−0.702436
$862$	−3.47520	−0.118366
$863$	−53.6720	−1.82702	−0.913509	−	0.406819i	$-0.866638\pi$
$863$	−53.6720	−1.82702	−0.913509	+	0.406819i	$0.866638\pi$
$864$	−3.57520	−0.121631
$865$	−51.3141	−1.74473
$866$	4.82279	0.163885
$867$	1.00000	0.0339618
$868$	−3.96596	−0.134613
$869$	−1.49606	−0.0507505
$870$	0.373151	0.0126510
$871$	4.12128	0.139644
$872$	−21.8338	−0.739386
$873$	−5.95421	−0.201519
$874$	−5.23524	−0.177085
$875$	−15.5187	−0.524628
$876$	−2.70893	−0.0915264
$877$	25.3478	0.855934	0.427967	−	0.903794i	$-0.359230\pi$
$877$	25.3478	0.855934	0.427967	+	0.903794i	$0.359230\pi$
$878$	5.80328	0.195851
$879$	−20.2950	−0.684535
$880$	14.2901	0.481718
$881$	−26.1003	−0.879341	−0.439670	−	0.898159i	$-0.644905\pi$
$881$	−26.1003	−0.879341	−0.439670	+	0.898159i	$0.644905\pi$
$882$	0.0281937	0.000949332 0
$883$	9.84963	0.331466	0.165733	−	0.986171i	$-0.447001\pi$
$883$	9.84963	0.331466	0.165733	+	0.986171i	$0.447001\pi$
$884$	2.62718	0.0883616
$885$	3.32945	0.111918
$886$	−6.29705	−0.211553
$887$	21.4586	0.720509	0.360255	−	0.932854i	$-0.382690\pi$
$887$	21.4586	0.720509	0.360255	+	0.932854i	$0.382690\pi$
$888$	3.89724	0.130783
$889$	−42.4551	−1.42390
$890$	−16.4711	−0.552112
$891$	−1.49606	−0.0501200
$892$	−21.9156	−0.733788
$893$	36.5940	1.22457
$894$	−4.94505	−0.165387
$895$	36.5757	1.22259
$896$	−23.5467	−0.786638
$897$	3.90400	0.130351
$898$	4.39254	0.146581
$899$	−0.330397	−0.0110193
$900$	−5.50389	−0.183463
$901$	13.5734	0.452195
$902$	3.74597	0.124727
$903$	12.7228	0.423389
$904$	−2.99074	−0.0994706
$905$	−23.4661	−0.780041
$906$	4.06311	0.134988
$907$	43.2380	1.43570	0.717848	−	0.696200i	$-0.245127\pi$
$907$	43.2380	1.43570	0.717848	+	0.696200i	$0.245127\pi$
$908$	14.6271	0.485418
$909$	8.04818	0.266941
$910$	−3.26659	−0.108286
$911$	−1.56275	−0.0517762	−0.0258881	−	0.999665i	$-0.508241\pi$
$911$	−1.56275	−0.0517762	−0.0258881	+	0.999665i	$0.508241\pi$
$912$	19.7513	0.654032
$913$	1.98534	0.0657050
$914$	−10.3355	−0.341868
$915$	−36.2636	−1.19884
$916$	2.48007	0.0819437
$917$	0.0457696	0.00151145
$918$	0.319374	0.0105409
$919$	4.04786	0.133527	0.0667633	−	0.997769i	$-0.478733\pi$
$919$	4.04786	0.133527	0.0667633	+	0.997769i	$0.478733\pi$
$920$	−9.86886	−0.325367
$921$	2.35371	0.0775575
$922$	3.86213	0.127192
$923$	−12.5232	−0.412207
$924$	−7.46516	−0.245586
$925$	9.07798	0.298482
$926$	−9.39111	−0.308611
$927$	7.19735	0.236392
$928$	−1.48620	−0.0487869
$929$	15.8223	0.519111	0.259556	−	0.965728i	$-0.416424\pi$
$929$	15.8223	0.519111	0.259556	+	0.965728i	$0.416424\pi$
$930$	−0.713455	−0.0233951
$931$	−0.513067	−0.0168151
$932$	51.2804	1.67974
$933$	12.5034	0.409344
$934$	1.29722	0.0424464
$935$	−4.20493	−0.137516
$936$	1.72319	0.0563243
$937$	29.8021	0.973591	0.486795	−	0.873516i	$-0.338166\pi$
$937$	29.8021	0.973591	0.486795	+	0.873516i	$0.338166\pi$
$938$	2.49995	0.0816263
$939$	23.2861	0.759914
$940$	33.5888	1.09555
$941$	−25.0336	−0.816072	−0.408036	−	0.912966i	$-0.633786\pi$
$941$	−25.0336	−0.816072	−0.408036	+	0.912966i	$0.633786\pi$
$942$	6.42394	0.209303
$943$	22.1122	0.720072
$944$	−4.02568	−0.131025
$945$	7.38928	0.240373
$946$	−2.31227	−0.0751785
$947$	54.4618	1.76977	0.884886	−	0.465808i	$-0.154236\pi$
$947$	54.4618	1.76977	0.884886	+	0.465808i	$0.154236\pi$
$948$	−1.89800	−0.0616442
$949$	1.97558	0.0641301
$950$	−5.38262	−0.174635
$951$	−17.5501	−0.569100
$952$	3.27291	0.106076
$953$	−36.3280	−1.17678	−0.588389	−	0.808578i	$-0.700238\pi$
$953$	−36.3280	−1.17678	−0.588389	+	0.808578i	$0.700238\pi$
$954$	4.33498	0.140350
$955$	74.9135	2.42415
$956$	40.6757	1.31555
$957$	−0.621909	−0.0201035
$958$	−1.45971	−0.0471612
$959$	17.8017	0.574847
$960$	15.8943	0.512986
$961$	−30.3683	−0.979622
$962$	−1.38391	−0.0446191
$963$	−6.78409	−0.218614
$964$	−40.1945	−1.29458
$965$	−53.8620	−1.73388
$966$	2.36815	0.0761939
$967$	−40.2478	−1.29428	−0.647141	−	0.762370i	$-0.724036\pi$
$967$	−40.2478	−1.29428	−0.647141	+	0.762370i	$0.724036\pi$
$968$	−10.9077	−0.350587
$969$	−5.81194	−0.186706
$970$	−5.34481	−0.171611
$971$	46.8066	1.50209	0.751047	−	0.660249i	$-0.229549\pi$
$971$	46.8066	1.50209	0.751047	+	0.660249i	$0.229549\pi$
$972$	−1.89800	−0.0608784
$973$	1.84101	0.0590202
$974$	3.65981	0.117268
$975$	4.01390	0.128548
$976$	43.8467	1.40350
$977$	50.2028	1.60613	0.803064	−	0.595892i	$-0.203201\pi$
$977$	50.2028	1.60613	0.803064	+	0.595892i	$0.203201\pi$
$978$	7.13853	0.228265
$979$	27.4514	0.877350
$980$	−0.470933	−0.0150434
$981$	−17.5383	−0.559956
$982$	13.1242	0.418811
$983$	−1.12476	−0.0358744	−0.0179372	−	0.999839i	$-0.505710\pi$
$983$	−1.12476	−0.0358744	−0.0179372	+	0.999839i	$0.505710\pi$
$984$	9.76014	0.311142
$985$	50.2796	1.60204
$986$	0.132763	0.00422803
$987$	−16.5532	−0.526895
$988$	−15.2690	−0.485771
$989$	−13.6492	−0.434019
$990$	−1.34294	−0.0426816
$991$	15.8151	0.502382	0.251191	−	0.967937i	$-0.419178\pi$
$991$	15.8151	0.502382	0.251191	+	0.967937i	$0.419178\pi$
$992$	2.84157	0.0902200
$993$	−20.1755	−0.640252
$994$	−7.59652	−0.240947
$995$	48.0264	1.52254
$996$	2.51872	0.0798088
$997$	38.9337	1.23304	0.616521	−	0.787339i	$-0.288542\pi$
$997$	38.9337	1.23304	0.616521	+	0.787339i	$0.288542\pi$
$998$	5.44651	0.172406
$999$	3.13051	0.0990451

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4029.2.a.h.1.13	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4029.2.a.h.1.13	✓	25	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:	\( N \)	\(=\)	\( 4029 = 3 \cdot 17 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4029.a (trivial)

\(f(q)\)	\(=\)	\(q-0.319374 q^{2} +1.00000 q^{3} -1.89800 q^{4} -2.81066 q^{5} -0.319374 q^{6} -2.62902 q^{7} +1.24492 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.319374 q^{2} +1.00000 q^{3} -1.89800 q^{4} -2.81066 q^{5} -0.319374 q^{6} -2.62902 q^{7} +1.24492 q^{8} +1.00000 q^{9} +0.897652 q^{10} -1.49606 q^{11} -1.89800 q^{12} +1.38418 q^{13} +0.839638 q^{14} -2.81066 q^{15} +3.39841 q^{16} -1.00000 q^{17} -0.319374 q^{18} +5.81194 q^{19} +5.33464 q^{20} -2.62902 q^{21} +0.477803 q^{22} +2.82044 q^{23} +1.24492 q^{24} +2.89984 q^{25} -0.442071 q^{26} +1.00000 q^{27} +4.98987 q^{28} +0.415697 q^{29} +0.897652 q^{30} -0.794801 q^{31} -3.57520 q^{32} -1.49606 q^{33} +0.319374 q^{34} +7.38928 q^{35} -1.89800 q^{36} +3.13051 q^{37} -1.85618 q^{38} +1.38418 q^{39} -3.49905 q^{40} +7.83998 q^{41} +0.839638 q^{42} -4.83939 q^{43} +2.83953 q^{44} -2.81066 q^{45} -0.900774 q^{46} +6.29636 q^{47} +3.39841 q^{48} -0.0882782 q^{49} -0.926131 q^{50} -1.00000 q^{51} -2.62718 q^{52} -13.5734 q^{53} -0.319374 q^{54} +4.20493 q^{55} -3.27291 q^{56} +5.81194 q^{57} -0.132763 q^{58} -1.18458 q^{59} +5.33464 q^{60} +12.9021 q^{61} +0.253839 q^{62} -2.62902 q^{63} -5.65499 q^{64} -3.89047 q^{65} +0.477803 q^{66} +2.97741 q^{67} +1.89800 q^{68} +2.82044 q^{69} -2.35994 q^{70} -9.04738 q^{71} +1.24492 q^{72} +1.42726 q^{73} -0.999804 q^{74} +2.89984 q^{75} -11.0311 q^{76} +3.93317 q^{77} -0.442071 q^{78} +1.00000 q^{79} -9.55178 q^{80} +1.00000 q^{81} -2.50388 q^{82} -1.32704 q^{83} +4.98987 q^{84} +2.81066 q^{85} +1.54557 q^{86} +0.415697 q^{87} -1.86248 q^{88} -18.3491 q^{89} +0.897652 q^{90} -3.63904 q^{91} -5.35320 q^{92} -0.794801 q^{93} -2.01089 q^{94} -16.3354 q^{95} -3.57520 q^{96} -5.95421 q^{97} +0.0281937 q^{98} -1.49606 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 7 q^{2} + 25 q^{3} + 21 q^{4} - 12 q^{5} - 7 q^{6} - 4 q^{7} - 21 q^{8} + 25 q^{9}+O(q^{10}) \) 25 * q - 7 * q^2 + 25 * q^3 + 21 * q^4 - 12 * q^5 - 7 * q^6 - 4 * q^7 - 21 * q^8 + 25 * q^9	\( 25 q - 7 q^{2} + 25 q^{3} + 21 q^{4} - 12 q^{5} - 7 q^{6} - 4 q^{7} - 21 q^{8} + 25 q^{9} - 9 q^{10} - 19 q^{11} + 21 q^{12} - 12 q^{13} - 15 q^{14} - 12 q^{15} + q^{16} - 25 q^{17} - 7 q^{18} - 35 q^{19} - 11 q^{20} - 4 q^{21} - 2 q^{22} - 16 q^{23} - 21 q^{24} + 19 q^{25} - 5 q^{26} + 25 q^{27} + 3 q^{28} - 37 q^{29} - 9 q^{30} - 28 q^{31} - 19 q^{32} - 19 q^{33} + 7 q^{34} - 42 q^{35} + 21 q^{36} + 8 q^{37} - 35 q^{38} - 12 q^{39} - 9 q^{40} - 34 q^{41} - 15 q^{42} - 19 q^{43} - 56 q^{44} - 12 q^{45} + q^{46} - 25 q^{47} + q^{48} + 25 q^{49} - 7 q^{50} - 25 q^{51} - 37 q^{52} - 44 q^{53} - 7 q^{54} - 11 q^{55} - 18 q^{56} - 35 q^{57} - 3 q^{58} - 47 q^{59} - 11 q^{60} - 28 q^{61} + 11 q^{62} - 4 q^{63} - 9 q^{64} - 63 q^{65} - 2 q^{66} - 28 q^{67} - 21 q^{68} - 16 q^{69} + 5 q^{70} - 27 q^{71} - 21 q^{72} - 21 q^{73} - 18 q^{74} + 19 q^{75} - 50 q^{76} - 58 q^{77} - 5 q^{78} + 25 q^{79} - 56 q^{80} + 25 q^{81} - 5 q^{82} - 61 q^{83} + 3 q^{84} + 12 q^{85} - 28 q^{86} - 37 q^{87} + 15 q^{88} - 34 q^{89} - 9 q^{90} - 30 q^{91} - 31 q^{92} - 28 q^{93} + q^{94} - 32 q^{95} - 19 q^{96} - 11 q^{97} - 66 q^{98} - 19 q^{99}+O(q^{100}) \) 25 * q - 7 * q^2 + 25 * q^3 + 21 * q^4 - 12 * q^5 - 7 * q^6 - 4 * q^7 - 21 * q^8 + 25 * q^9 - 9 * q^10 - 19 * q^11 + 21 * q^12 - 12 * q^13 - 15 * q^14 - 12 * q^15 + q^16 - 25 * q^17 - 7 * q^18 - 35 * q^19 - 11 * q^20 - 4 * q^21 - 2 * q^22 - 16 * q^23 - 21 * q^24 + 19 * q^25 - 5 * q^26 + 25 * q^27 + 3 * q^28 - 37 * q^29 - 9 * q^30 - 28 * q^31 - 19 * q^32 - 19 * q^33 + 7 * q^34 - 42 * q^35 + 21 * q^36 + 8 * q^37 - 35 * q^38 - 12 * q^39 - 9 * q^40 - 34 * q^41 - 15 * q^42 - 19 * q^43 - 56 * q^44 - 12 * q^45 + q^46 - 25 * q^47 + q^48 + 25 * q^49 - 7 * q^50 - 25 * q^51 - 37 * q^52 - 44 * q^53 - 7 * q^54 - 11 * q^55 - 18 * q^56 - 35 * q^57 - 3 * q^58 - 47 * q^59 - 11 * q^60 - 28 * q^61 + 11 * q^62 - 4 * q^63 - 9 * q^64 - 63 * q^65 - 2 * q^66 - 28 * q^67 - 21 * q^68 - 16 * q^69 + 5 * q^70 - 27 * q^71 - 21 * q^72 - 21 * q^73 - 18 * q^74 + 19 * q^75 - 50 * q^76 - 58 * q^77 - 5 * q^78 + 25 * q^79 - 56 * q^80 + 25 * q^81 - 5 * q^82 - 61 * q^83 + 3 * q^84 + 12 * q^85 - 28 * q^86 - 37 * q^87 + 15 * q^88 - 34 * q^89 - 9 * q^90 - 30 * q^91 - 31 * q^92 - 28 * q^93 + q^94 - 32 * q^95 - 19 * q^96 - 11 * q^97 - 66 * q^98 - 19 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more