LMFDB - Embedded newform 4029.2.a.h.1.15

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			1.15
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)

$f(q)$	$=$	$q+0.0993996 q^{2} +1.00000 q^{3} -1.99012 q^{4} +1.12985 q^{5} +0.0993996 q^{6} +0.411227 q^{7} -0.396616 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.0993996 q^{2} +1.00000 q^{3} -1.99012 q^{4} +1.12985 q^{5} +0.0993996 q^{6} +0.411227 q^{7} -0.396616 q^{8} +1.00000 q^{9} +0.112306 q^{10} +2.25332 q^{11} -1.99012 q^{12} -2.41275 q^{13} +0.0408758 q^{14} +1.12985 q^{15} +3.94082 q^{16} -1.00000 q^{17} +0.0993996 q^{18} -6.49295 q^{19} -2.24853 q^{20} +0.411227 q^{21} +0.223979 q^{22} -6.81345 q^{23} -0.396616 q^{24} -3.72344 q^{25} -0.239827 q^{26} +1.00000 q^{27} -0.818390 q^{28} +10.0375 q^{29} +0.112306 q^{30} -4.31951 q^{31} +1.18495 q^{32} +2.25332 q^{33} -0.0993996 q^{34} +0.464624 q^{35} -1.99012 q^{36} -1.60752 q^{37} -0.645397 q^{38} -2.41275 q^{39} -0.448116 q^{40} +4.87459 q^{41} +0.0408758 q^{42} +2.25493 q^{43} -4.48438 q^{44} +1.12985 q^{45} -0.677255 q^{46} -11.7078 q^{47} +3.94082 q^{48} -6.83089 q^{49} -0.370109 q^{50} -1.00000 q^{51} +4.80166 q^{52} +7.65957 q^{53} +0.0993996 q^{54} +2.54591 q^{55} -0.163099 q^{56} -6.49295 q^{57} +0.997720 q^{58} -7.20091 q^{59} -2.24853 q^{60} +1.35017 q^{61} -0.429358 q^{62} +0.411227 q^{63} -7.76385 q^{64} -2.72604 q^{65} +0.223979 q^{66} -12.9776 q^{67} +1.99012 q^{68} -6.81345 q^{69} +0.0461834 q^{70} -14.3582 q^{71} -0.396616 q^{72} +1.10037 q^{73} -0.159787 q^{74} -3.72344 q^{75} +12.9217 q^{76} +0.926626 q^{77} -0.239827 q^{78} +1.00000 q^{79} +4.45252 q^{80} +1.00000 q^{81} +0.484532 q^{82} +2.81499 q^{83} -0.818390 q^{84} -1.12985 q^{85} +0.224139 q^{86} +10.0375 q^{87} -0.893705 q^{88} +17.0324 q^{89} +0.112306 q^{90} -0.992188 q^{91} +13.5596 q^{92} -4.31951 q^{93} -1.16375 q^{94} -7.33604 q^{95} +1.18495 q^{96} -9.24683 q^{97} -0.678988 q^{98} +2.25332 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.0993996	0.0702862	0.0351431	−	0.999382i	$-0.488811\pi$
$2$	0.0993996	0.0702862	0.0351431	+	0.999382i	$0.488811\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.99012	−0.995060
$5$	1.12985	0.505283	0.252642	−	0.967560i	$-0.418701\pi$
$5$	1.12985	0.505283	0.252642	+	0.967560i	$0.418701\pi$
$6$	0.0993996	0.0405797
$7$	0.411227	0.155429	0.0777145	−	0.996976i	$-0.475238\pi$
$7$	0.411227	0.155429	0.0777145	+	0.996976i	$0.475238\pi$
$8$	−0.396616	−0.140225
$9$	1.00000	0.333333
$10$	0.112306	0.0355144
$11$	2.25332	0.679402	0.339701	−	0.940533i	$-0.389674\pi$
$11$	2.25332	0.679402	0.339701	+	0.940533i	$0.389674\pi$
$12$	−1.99012	−0.574498
$13$	−2.41275	−0.669177	−0.334588	−	0.942364i	$-0.608597\pi$
$13$	−2.41275	−0.669177	−0.334588	+	0.942364i	$0.608597\pi$
$14$	0.0408758	0.0109245
$15$	1.12985	0.291725
$16$	3.94082	0.985204
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0.0993996	0.0234287
$19$	−6.49295	−1.48958	−0.744792	−	0.667297i	$-0.767451\pi$
$19$	−6.49295	−1.48958	−0.744792	+	0.667297i	$0.767451\pi$
$20$	−2.24853	−0.502787
$21$	0.411227	0.0897370
$22$	0.223979	0.0477526
$23$	−6.81345	−1.42070	−0.710352	−	0.703847i	$-0.751464\pi$
$23$	−6.81345	−1.42070	−0.710352	+	0.703847i	$0.751464\pi$
$24$	−0.396616	−0.0809590
$25$	−3.72344	−0.744689
$26$	−0.239827	−0.0470339
$27$	1.00000	0.192450
$28$	−0.818390	−0.154661
$29$	10.0375	1.86391	0.931955	−	0.362574i	$-0.118102\pi$
$29$	10.0375	1.86391	0.931955	+	0.362574i	$0.118102\pi$
$30$	0.112306	0.0205043
$31$	−4.31951	−0.775807	−0.387904	−	0.921700i	$-0.626801\pi$
$31$	−4.31951	−0.775807	−0.387904	+	0.921700i	$0.626801\pi$
$32$	1.18495	0.209471
$33$	2.25332	0.392253
$34$	−0.0993996	−0.0170469
$35$	0.464624	0.0785357
$36$	−1.99012	−0.331687
$37$	−1.60752	−0.264274	−0.132137	−	0.991231i	$-0.542184\pi$
$37$	−1.60752	−0.264274	−0.132137	+	0.991231i	$0.542184\pi$
$38$	−0.645397	−0.104697
$39$	−2.41275	−0.386349
$40$	−0.448116	−0.0708534
$41$	4.87459	0.761283	0.380641	−	0.924723i	$-0.375703\pi$
$41$	4.87459	0.761283	0.380641	+	0.924723i	$0.375703\pi$
$42$	0.0408758	0.00630727
$43$	2.25493	0.343873	0.171937	−	0.985108i	$-0.444998\pi$
$43$	2.25493	0.343873	0.171937	+	0.985108i	$0.444998\pi$
$44$	−4.48438	−0.676046
$45$	1.12985	0.168428
$46$	−0.677255	−0.0998558
$47$	−11.7078	−1.70776	−0.853879	−	0.520472i	$-0.825756\pi$
$47$	−11.7078	−1.70776	−0.853879	+	0.520472i	$0.825756\pi$
$48$	3.94082	0.568808
$49$	−6.83089	−0.975842
$50$	−0.370109	−0.0523413
$51$	−1.00000	−0.140028
$52$	4.80166	0.665871
$53$	7.65957	1.05212	0.526061	−	0.850447i	$-0.323668\pi$
$53$	7.65957	1.05212	0.526061	+	0.850447i	$0.323668\pi$
$54$	0.0993996	0.0135266
$55$	2.54591	0.343291
$56$	−0.163099	−0.0217951
$57$	−6.49295	−0.860012
$58$	0.997720	0.131007
$59$	−7.20091	−0.937479	−0.468739	−	0.883336i	$-0.655292\pi$
$59$	−7.20091	−0.937479	−0.468739	+	0.883336i	$0.655292\pi$
$60$	−2.24853	−0.290284
$61$	1.35017	0.172871	0.0864357	−	0.996257i	$-0.472452\pi$
$61$	1.35017	0.172871	0.0864357	+	0.996257i	$0.472452\pi$
$62$	−0.429358	−0.0545285
$63$	0.411227	0.0518097
$64$	−7.76385	−0.970481
$65$	−2.72604	−0.338124
$66$	0.223979	0.0275700
$67$	−12.9776	−1.58546	−0.792732	−	0.609570i	$-0.791342\pi$
$67$	−12.9776	−1.58546	−0.792732	+	0.609570i	$0.791342\pi$
$68$	1.99012	0.241337
$69$	−6.81345	−0.820243
$70$	0.0461834	0.00551997
$71$	−14.3582	−1.70400	−0.852001	−	0.523539i	$-0.824611\pi$
$71$	−14.3582	−1.70400	−0.852001	+	0.523539i	$0.824611\pi$
$72$	−0.396616	−0.0467417
$73$	1.10037	0.128789	0.0643945	−	0.997925i	$-0.479488\pi$
$73$	1.10037	0.128789	0.0643945	+	0.997925i	$0.479488\pi$
$74$	−0.159787	−0.0185748
$75$	−3.72344	−0.429946
$76$	12.9217	1.48223
$77$	0.926626	0.105599
$78$	−0.239827	−0.0271550
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	4.45252	0.497807
$81$	1.00000	0.111111
$82$	0.484532	0.0535076
$83$	2.81499	0.308985	0.154492	−	0.987994i	$-0.450626\pi$
$83$	2.81499	0.308985	0.154492	+	0.987994i	$0.450626\pi$
$84$	−0.818390	−0.0892937
$85$	−1.12985	−0.122549
$86$	0.224139	0.0241695
$87$	10.0375	1.07613
$88$	−0.893705	−0.0952692
$89$	17.0324	1.80544	0.902718	−	0.430233i	$-0.141569\pi$
$89$	17.0324	1.80544	0.902718	+	0.430233i	$0.141569\pi$
$90$	0.112306	0.0118381
$91$	−0.992188	−0.104010
$92$	13.5596	1.41368
$93$	−4.31951	−0.447913
$94$	−1.16375	−0.120032
$95$	−7.33604	−0.752662
$96$	1.18495	0.120938
$97$	−9.24683	−0.938874	−0.469437	−	0.882966i	$-0.655543\pi$
$97$	−9.24683	−0.938874	−0.469437	+	0.882966i	$0.655543\pi$
$98$	−0.678988	−0.0685882
$99$	2.25332	0.226467
$100$	7.41010	0.741010
$101$	13.2836	1.32176	0.660881	−	0.750490i	$-0.270183\pi$
$101$	13.2836	1.32176	0.660881	+	0.750490i	$0.270183\pi$
$102$	−0.0993996	−0.00984203
$103$	8.85908	0.872911	0.436456	−	0.899726i	$-0.356234\pi$
$103$	8.85908	0.872911	0.436456	+	0.899726i	$0.356234\pi$
$104$	0.956937	0.0938354
$105$	0.464624	0.0453426
$106$	0.761358	0.0739497
$107$	−3.62874	−0.350803	−0.175402	−	0.984497i	$-0.556122\pi$
$107$	−3.62874	−0.350803	−0.175402	+	0.984497i	$0.556122\pi$
$108$	−1.99012	−0.191499
$109$	3.11898	0.298744	0.149372	−	0.988781i	$-0.452275\pi$
$109$	3.11898	0.298744	0.149372	+	0.988781i	$0.452275\pi$
$110$	0.253063	0.0241286
$111$	−1.60752	−0.152579
$112$	1.62057	0.153129
$113$	−3.69883	−0.347956	−0.173978	−	0.984750i	$-0.555662\pi$
$113$	−3.69883	−0.347956	−0.173978	+	0.984750i	$0.555662\pi$
$114$	−0.645397	−0.0604469
$115$	−7.69816	−0.717858
$116$	−19.9757	−1.85470
$117$	−2.41275	−0.223059
$118$	−0.715768	−0.0658918
$119$	−0.411227	−0.0376971
$120$	−0.448116	−0.0409072
$121$	−5.92254	−0.538413
$122$	0.134206	0.0121505
$123$	4.87459	0.439527
$124$	8.59635	0.771975
$125$	−9.85616	−0.881562
$126$	0.0408758	0.00364150
$127$	−0.489967	−0.0434775	−0.0217388	−	0.999764i	$-0.506920\pi$
$127$	−0.489967	−0.0434775	−0.0217388	+	0.999764i	$0.506920\pi$
$128$	−3.14162	−0.277683
$129$	2.25493	0.198535
$130$	−0.270968	−0.0237654
$131$	−10.1073	−0.883077	−0.441538	−	0.897242i	$-0.645567\pi$
$131$	−10.1073	−0.883077	−0.441538	+	0.897242i	$0.645567\pi$
$132$	−4.48438	−0.390315
$133$	−2.67007	−0.231525
$134$	−1.28997	−0.111436
$135$	1.12985	0.0972418
$136$	0.396616	0.0340096
$137$	4.37903	0.374126	0.187063	−	0.982348i	$-0.440103\pi$
$137$	4.37903	0.374126	0.187063	+	0.982348i	$0.440103\pi$
$138$	−0.677255	−0.0576518
$139$	5.77337	0.489691	0.244845	−	0.969562i	$-0.421263\pi$
$139$	5.77337	0.489691	0.244845	+	0.969562i	$0.421263\pi$
$140$	−0.924657	−0.0781477
$141$	−11.7078	−0.985974
$142$	−1.42720	−0.119768
$143$	−5.43671	−0.454640
$144$	3.94082	0.328401
$145$	11.3408	0.941803
$146$	0.109377	0.00905208
$147$	−6.83089	−0.563403
$148$	3.19915	0.262969
$149$	−4.52516	−0.370715	−0.185358	−	0.982671i	$-0.559344\pi$
$149$	−4.52516	−0.370715	−0.185358	+	0.982671i	$0.559344\pi$
$150$	−0.370109	−0.0302193
$151$	−17.9859	−1.46367	−0.731836	−	0.681481i	$-0.761336\pi$
$151$	−17.9859	−1.46367	−0.731836	+	0.681481i	$0.761336\pi$
$152$	2.57521	0.208877
$153$	−1.00000	−0.0808452
$154$	0.0921063	0.00742214
$155$	−4.88039	−0.392003
$156$	4.80166	0.384441
$157$	−19.5734	−1.56213	−0.781064	−	0.624451i	$-0.785323\pi$
$157$	−19.5734	−1.56213	−0.781064	+	0.624451i	$0.785323\pi$
$158$	0.0993996	0.00790781
$159$	7.65957	0.607443
$160$	1.33881	0.105842
$161$	−2.80187	−0.220819
$162$	0.0993996	0.00780957
$163$	19.7395	1.54612	0.773060	−	0.634333i	$-0.218725\pi$
$163$	19.7395	1.54612	0.773060	+	0.634333i	$0.218725\pi$
$164$	−9.70101	−0.757522
$165$	2.54591	0.198199
$166$	0.279809	0.0217174
$167$	−18.4760	−1.42972	−0.714858	−	0.699270i	$-0.753508\pi$
$167$	−18.4760	−1.42972	−0.714858	+	0.699270i	$0.753508\pi$
$168$	−0.163099	−0.0125834
$169$	−7.17863	−0.552202
$170$	−0.112306	−0.00861351
$171$	−6.49295	−0.496528
$172$	−4.48757	−0.342174
$173$	25.2896	1.92274	0.961368	−	0.275266i	$-0.0887659\pi$
$173$	25.2896	1.92274	0.961368	+	0.275266i	$0.0887659\pi$
$174$	0.997720	0.0756370
$175$	−1.53118	−0.115746
$176$	8.87993	0.669350
$177$	−7.20091	−0.541254
$178$	1.69302	0.126897
$179$	16.9559	1.26735	0.633673	−	0.773601i	$-0.281547\pi$
$179$	16.9559	1.26735	0.633673	+	0.773601i	$0.281547\pi$
$180$	−2.24853	−0.167596
$181$	−21.5750	−1.60365	−0.801827	−	0.597556i	$-0.796139\pi$
$181$	−21.5750	−1.60365	−0.801827	+	0.597556i	$0.796139\pi$
$182$	−0.0986231	−0.00731043
$183$	1.35017	0.0998074
$184$	2.70233	0.199218
$185$	−1.81625	−0.133533
$186$	−0.429358	−0.0314821
$187$	−2.25332	−0.164779
$188$	23.2999	1.69932
$189$	0.411227	0.0299123
$190$	−0.729200	−0.0529017
$191$	−13.6940	−0.990862	−0.495431	−	0.868647i	$-0.664990\pi$
$191$	−13.6940	−0.990862	−0.495431	+	0.868647i	$0.664990\pi$
$192$	−7.76385	−0.560307
$193$	−19.6320	−1.41314	−0.706572	−	0.707641i	$-0.749759\pi$
$193$	−19.6320	−1.41314	−0.706572	+	0.707641i	$0.749759\pi$
$194$	−0.919132	−0.0659898
$195$	−2.72604	−0.195216
$196$	13.5943	0.971021
$197$	12.4370	0.886103	0.443051	−	0.896496i	$-0.353896\pi$
$197$	12.4370	0.886103	0.443051	+	0.896496i	$0.353896\pi$
$198$	0.223979	0.0159175
$199$	−11.3603	−0.805314	−0.402657	−	0.915351i	$-0.631913\pi$
$199$	−11.3603	−0.805314	−0.402657	+	0.915351i	$0.631913\pi$
$200$	1.47678	0.104424
$201$	−12.9776	−0.915368
$202$	1.32038	0.0929016
$203$	4.12767	0.289706
$204$	1.99012	0.139336
$205$	5.50754	0.384663
$206$	0.880589	0.0613536
$207$	−6.81345	−0.473568
$208$	−9.50821	−0.659276
$209$	−14.6307	−1.01203
$210$	0.0461834	0.00318696
$211$	−15.7500	−1.08428	−0.542138	−	0.840290i	$-0.682385\pi$
$211$	−15.7500	−1.08428	−0.542138	+	0.840290i	$0.682385\pi$
$212$	−15.2435	−1.04693
$213$	−14.3582	−0.983807
$214$	−0.360695	−0.0246566
$215$	2.54772	0.173753
$216$	−0.396616	−0.0269863
$217$	−1.77630	−0.120583
$218$	0.310025	0.0209975
$219$	1.10037	0.0743563
$220$	−5.06667	−0.341595
$221$	2.41275	0.162299
$222$	−0.159787	−0.0107242
$223$	−2.12609	−0.142373	−0.0711867	−	0.997463i	$-0.522679\pi$
$223$	−2.12609	−0.142373	−0.0711867	+	0.997463i	$0.522679\pi$
$224$	0.487283	0.0325579
$225$	−3.72344	−0.248230
$226$	−0.367662	−0.0244565
$227$	14.1013	0.935938	0.467969	−	0.883745i	$-0.344986\pi$
$227$	14.1013	0.935938	0.467969	+	0.883745i	$0.344986\pi$
$228$	12.9217	0.855763
$229$	−0.272709	−0.0180211	−0.00901057	−	0.999959i	$-0.502868\pi$
$229$	−0.272709	−0.0180211	−0.00901057	+	0.999959i	$0.502868\pi$
$230$	−0.765195	−0.0504555
$231$	0.926626	0.0609675
$232$	−3.98102	−0.261367
$233$	−25.4716	−1.66870	−0.834350	−	0.551235i	$-0.814157\pi$
$233$	−25.4716	−1.66870	−0.834350	+	0.551235i	$0.814157\pi$
$234$	−0.239827	−0.0156780
$235$	−13.2280	−0.862901
$236$	14.3307	0.932848
$237$	1.00000	0.0649570
$238$	−0.0408758	−0.00264958
$239$	−25.1388	−1.62609	−0.813046	−	0.582200i	$-0.802192\pi$
$239$	−25.1388	−1.62609	−0.813046	+	0.582200i	$0.802192\pi$
$240$	4.45252	0.287409
$241$	19.6407	1.26517	0.632583	−	0.774492i	$-0.281995\pi$
$241$	19.6407	1.26517	0.632583	+	0.774492i	$0.281995\pi$
$242$	−0.588698	−0.0378430
$243$	1.00000	0.0641500
$244$	−2.68700	−0.172017
$245$	−7.71787	−0.493077
$246$	0.484532	0.0308926
$247$	15.6659	0.996795
$248$	1.71319	0.108788
$249$	2.81499	0.178393
$250$	−0.979699	−0.0619616
$251$	−30.7228	−1.93921	−0.969603	−	0.244685i	$-0.921315\pi$
$251$	−30.7228	−1.93921	−0.969603	+	0.244685i	$0.921315\pi$
$252$	−0.818390	−0.0515538
$253$	−15.3529	−0.965229
$254$	−0.0487025	−0.00305587
$255$	−1.12985	−0.0707538
$256$	15.2154	0.950964
$257$	−5.65845	−0.352964	−0.176482	−	0.984304i	$-0.556472\pi$
$257$	−5.65845	−0.352964	−0.176482	+	0.984304i	$0.556472\pi$
$258$	0.224139	0.0139543
$259$	−0.661054	−0.0410759
$260$	5.42515	0.336454
$261$	10.0375	0.621303
$262$	−1.00466	−0.0620681
$263$	23.1776	1.42919	0.714597	−	0.699537i	$-0.246610\pi$
$263$	23.1776	1.42919	0.714597	+	0.699537i	$0.246610\pi$
$264$	−0.893705	−0.0550037
$265$	8.65415	0.531620
$266$	−0.265404	−0.0162730
$267$	17.0324	1.04237
$268$	25.8270	1.57763
$269$	−2.95911	−0.180420	−0.0902100	−	0.995923i	$-0.528754\pi$
$269$	−2.95911	−0.180420	−0.0902100	+	0.995923i	$0.528754\pi$
$270$	0.112306	0.00683475
$271$	19.1491	1.16323	0.581614	−	0.813465i	$-0.302421\pi$
$271$	19.1491	1.16323	0.581614	+	0.813465i	$0.302421\pi$
$272$	−3.94082	−0.238947
$273$	−0.992188	−0.0600499
$274$	0.435274	0.0262959
$275$	−8.39012	−0.505943
$276$	13.5596	0.816191
$277$	2.50785	0.150682	0.0753409	−	0.997158i	$-0.475995\pi$
$277$	2.50785	0.150682	0.0753409	+	0.997158i	$0.475995\pi$
$278$	0.573871	0.0344185
$279$	−4.31951	−0.258602
$280$	−0.184277	−0.0110127
$281$	−13.5725	−0.809666	−0.404833	−	0.914391i	$-0.632670\pi$
$281$	−13.5725	−0.809666	−0.404833	+	0.914391i	$0.632670\pi$
$282$	−1.16375	−0.0693004
$283$	3.50100	0.208113	0.104056	−	0.994571i	$-0.466818\pi$
$283$	3.50100	0.208113	0.104056	+	0.994571i	$0.466818\pi$
$284$	28.5745	1.69558
$285$	−7.33604	−0.434550
$286$	−0.540407	−0.0319549
$287$	2.00456	0.118325
$288$	1.18495	0.0698238
$289$	1.00000	0.0588235
$290$	1.12727	0.0661957
$291$	−9.24683	−0.542059
$292$	−2.18987	−0.128153
$293$	−17.1524	−1.00205	−0.501026	−	0.865433i	$-0.667044\pi$
$293$	−17.1524	−1.00205	−0.501026	+	0.865433i	$0.667044\pi$
$294$	−0.678988	−0.0395994
$295$	−8.13593	−0.473692
$296$	0.637568	0.0370579
$297$	2.25332	0.130751
$298$	−0.449799	−0.0260562
$299$	16.4392	0.950702
$300$	7.41010	0.427822
$301$	0.927286	0.0534479
$302$	−1.78779	−0.102876
$303$	13.2836	0.763120
$304$	−25.5875	−1.46754
$305$	1.52549	0.0873491
$306$	−0.0993996	−0.00568230
$307$	27.8589	1.58999	0.794997	−	0.606614i	$-0.207473\pi$
$307$	27.8589	1.58999	0.794997	+	0.606614i	$0.207473\pi$
$308$	−1.84410	−0.105077
$309$	8.85908	0.503976
$310$	−0.485109	−0.0275524
$311$	−29.1640	−1.65374	−0.826869	−	0.562395i	$-0.809880\pi$
$311$	−29.1640	−1.65374	−0.826869	+	0.562395i	$0.809880\pi$
$312$	0.956937	0.0541759
$313$	5.60793	0.316979	0.158489	−	0.987361i	$-0.449338\pi$
$313$	5.60793	0.316979	0.158489	+	0.987361i	$0.449338\pi$
$314$	−1.94559	−0.109796
$315$	0.464624	0.0261786
$316$	−1.99012	−0.111953
$317$	7.19342	0.404023	0.202011	−	0.979383i	$-0.435252\pi$
$317$	7.19342	0.404023	0.202011	+	0.979383i	$0.435252\pi$
$318$	0.761358	0.0426949
$319$	22.6176	1.26634
$320$	−8.77197	−0.490368
$321$	−3.62874	−0.202536
$322$	−0.278505	−0.0155205
$323$	6.49295	0.361277
$324$	−1.99012	−0.110562
$325$	8.98374	0.498328
$326$	1.96210	0.108671
$327$	3.11898	0.172480
$328$	−1.93334	−0.106751
$329$	−4.81456	−0.265435
$330$	0.253063	0.0139306
$331$	−13.1886	−0.724913	−0.362457	−	0.932001i	$-0.618062\pi$
$331$	−13.1886	−0.724913	−0.362457	+	0.932001i	$0.618062\pi$
$332$	−5.60216	−0.307459
$333$	−1.60752	−0.0880914
$334$	−1.83651	−0.100489
$335$	−14.6627	−0.801109
$336$	1.62057	0.0884093
$337$	0.0506168	0.00275728	0.00137864	−	0.999999i	$-0.499561\pi$
$337$	0.0506168	0.00275728	0.00137864	+	0.999999i	$0.499561\pi$
$338$	−0.713553	−0.0388122
$339$	−3.69883	−0.200893
$340$	2.24853	0.121944
$341$	−9.73325	−0.527085
$342$	−0.645397	−0.0348991
$343$	−5.68763	−0.307103
$344$	−0.894341	−0.0482196
$345$	−7.69816	−0.414455
$346$	2.51378	0.135142
$347$	11.6176	0.623665	0.311833	−	0.950137i	$-0.399057\pi$
$347$	11.6176	0.623665	0.311833	+	0.950137i	$0.399057\pi$
$348$	−19.9757	−1.07081
$349$	12.4627	0.667114	0.333557	−	0.942730i	$-0.391751\pi$
$349$	12.4627	0.667114	0.333557	+	0.942730i	$0.391751\pi$
$350$	−0.152199	−0.00813536
$351$	−2.41275	−0.128783
$352$	2.67007	0.142315
$353$	6.14494	0.327062	0.163531	−	0.986538i	$-0.447712\pi$
$353$	6.14494	0.327062	0.163531	+	0.986538i	$0.447712\pi$
$354$	−0.715768	−0.0380426
$355$	−16.2226	−0.861004
$356$	−33.8966	−1.79652
$357$	−0.411227	−0.0217644
$358$	1.68541	0.0890768
$359$	19.3788	1.02277	0.511387	−	0.859350i	$-0.329132\pi$
$359$	19.3788	1.02277	0.511387	+	0.859350i	$0.329132\pi$
$360$	−0.448116	−0.0236178
$361$	23.1584	1.21886
$362$	−2.14454	−0.112715
$363$	−5.92254	−0.310853
$364$	1.97457	0.103496
$365$	1.24325	0.0650749
$366$	0.134206	0.00701508
$367$	−8.62984	−0.450474	−0.225237	−	0.974304i	$-0.572316\pi$
$367$	−8.62984	−0.450474	−0.225237	+	0.974304i	$0.572316\pi$
$368$	−26.8506	−1.39968
$369$	4.87459	0.253761
$370$	−0.180535	−0.00938555
$371$	3.14982	0.163530
$372$	8.59635	0.445700
$373$	5.72233	0.296291	0.148146	−	0.988966i	$-0.452670\pi$
$373$	5.72233	0.296291	0.148146	+	0.988966i	$0.452670\pi$
$374$	−0.223979	−0.0115817
$375$	−9.85616	−0.508970
$376$	4.64351	0.239470
$377$	−24.2179	−1.24729
$378$	0.0408758	0.00210242
$379$	−3.97150	−0.204002	−0.102001	−	0.994784i	$-0.532525\pi$
$379$	−3.97150	−0.204002	−0.102001	+	0.994784i	$0.532525\pi$
$380$	14.5996	0.748944
$381$	−0.489967	−0.0251017
$382$	−1.36118	−0.0696439
$383$	16.1078	0.823070	0.411535	−	0.911394i	$-0.364993\pi$
$383$	16.1078	0.823070	0.411535	+	0.911394i	$0.364993\pi$
$384$	−3.14162	−0.160320
$385$	1.04695	0.0533573
$386$	−1.95142	−0.0993244
$387$	2.25493	0.114624
$388$	18.4023	0.934235
$389$	−27.6072	−1.39974	−0.699870	−	0.714271i	$-0.746759\pi$
$389$	−27.6072	−1.39974	−0.699870	+	0.714271i	$0.746759\pi$
$390$	−0.270968	−0.0137210
$391$	6.81345	0.344571
$392$	2.70924	0.136838
$393$	−10.1073	−0.509844
$394$	1.23624	0.0622808
$395$	1.12985	0.0568488
$396$	−4.48438	−0.225349
$397$	−12.5045	−0.627582	−0.313791	−	0.949492i	$-0.601599\pi$
$397$	−12.5045	−0.627582	−0.313791	+	0.949492i	$0.601599\pi$
$398$	−1.12921	−0.0566024
$399$	−2.67007	−0.133671
$400$	−14.6734	−0.733670
$401$	−1.89762	−0.0947624	−0.0473812	−	0.998877i	$-0.515088\pi$
$401$	−1.89762	−0.0947624	−0.0473812	+	0.998877i	$0.515088\pi$
$402$	−1.28997	−0.0643377
$403$	10.4219	0.519152
$404$	−26.4359	−1.31523
$405$	1.12985	0.0561426
$406$	0.410289	0.0203623
$407$	−3.62225	−0.179548
$408$	0.396616	0.0196354
$409$	−33.6991	−1.66632	−0.833158	−	0.553035i	$-0.813469\pi$
$409$	−33.6991	−1.66632	−0.833158	+	0.553035i	$0.813469\pi$
$410$	0.547448	0.0270365
$411$	4.37903	0.216002
$412$	−17.6306	−0.868599
$413$	−2.96121	−0.145711
$414$	−0.677255	−0.0332853
$415$	3.18051	0.156125
$416$	−2.85899	−0.140173
$417$	5.77337	0.282723
$418$	−1.45429	−0.0711315
$419$	2.31356	0.113025	0.0565125	−	0.998402i	$-0.482002\pi$
$419$	2.31356	0.113025	0.0565125	+	0.998402i	$0.482002\pi$
$420$	−0.924657	−0.0451186
$421$	35.7431	1.74201	0.871006	−	0.491273i	$-0.163468\pi$
$421$	35.7431	1.74201	0.871006	+	0.491273i	$0.163468\pi$
$422$	−1.56554	−0.0762095
$423$	−11.7078	−0.569253
$424$	−3.03791	−0.147534
$425$	3.72344	0.180614
$426$	−1.42720	−0.0691480
$427$	0.555226	0.0268693
$428$	7.22163	0.349070
$429$	−5.43671	−0.262487
$430$	0.253243	0.0122125
$431$	0.482987	0.0232647	0.0116323	−	0.999932i	$-0.496297\pi$
$431$	0.482987	0.0232647	0.0116323	+	0.999932i	$0.496297\pi$
$432$	3.94082	0.189603
$433$	18.5469	0.891307	0.445653	−	0.895206i	$-0.352971\pi$
$433$	18.5469	0.891307	0.445653	+	0.895206i	$0.352971\pi$
$434$	−0.176564	−0.00847532
$435$	11.3408	0.543750
$436$	−6.20713	−0.297268
$437$	44.2394	2.11626
$438$	0.109377	0.00522622
$439$	25.7681	1.22985	0.614923	−	0.788587i	$-0.289187\pi$
$439$	25.7681	1.22985	0.614923	+	0.788587i	$0.289187\pi$
$440$	−1.00975	−0.0481379
$441$	−6.83089	−0.325281
$442$	0.239827	0.0114074
$443$	10.2362	0.486338	0.243169	−	0.969984i	$-0.421813\pi$
$443$	10.2362	0.486338	0.243169	+	0.969984i	$0.421813\pi$
$444$	3.19915	0.151825
$445$	19.2441	0.912257
$446$	−0.211332	−0.0100069
$447$	−4.52516	−0.214033
$448$	−3.19270	−0.150841
$449$	−23.4188	−1.10520	−0.552600	−	0.833447i	$-0.686364\pi$
$449$	−23.4188	−1.10520	−0.552600	+	0.833447i	$0.686364\pi$
$450$	−0.370109	−0.0174471
$451$	10.9840	0.517217
$452$	7.36111	0.346237
$453$	−17.9859	−0.845051
$454$	1.40167	0.0657835
$455$	−1.12102	−0.0525543
$456$	2.57521	0.120595
$457$	19.7042	0.921724	0.460862	−	0.887472i	$-0.347540\pi$
$457$	19.7042	0.921724	0.460862	+	0.887472i	$0.347540\pi$
$458$	−0.0271072	−0.00126664
$459$	−1.00000	−0.0466760
$460$	15.3203	0.714311
$461$	−9.55913	−0.445213	−0.222607	−	0.974908i	$-0.571457\pi$
$461$	−9.55913	−0.445213	−0.222607	+	0.974908i	$0.571457\pi$
$462$	0.0921063	0.00428517
$463$	13.1101	0.609278	0.304639	−	0.952468i	$-0.401464\pi$
$463$	13.1101	0.609278	0.304639	+	0.952468i	$0.401464\pi$
$464$	39.5558	1.83633
$465$	−4.88039	−0.226323
$466$	−2.53187	−0.117286
$467$	16.1137	0.745655	0.372828	−	0.927901i	$-0.378388\pi$
$467$	16.1137	0.745655	0.372828	+	0.927901i	$0.378388\pi$
$468$	4.80166	0.221957
$469$	−5.33673	−0.246427
$470$	−1.31486	−0.0606500
$471$	−19.5734	−0.901895
$472$	2.85600	0.131458
$473$	5.08108	0.233628
$474$	0.0993996	0.00456558
$475$	24.1761	1.10928
$476$	0.818390	0.0375109
$477$	7.65957	0.350708
$478$	−2.49878	−0.114292
$479$	−24.1934	−1.10542	−0.552711	−	0.833373i	$-0.686407\pi$
$479$	−24.1934	−1.10542	−0.552711	+	0.833373i	$0.686407\pi$
$480$	1.33881	0.0611081
$481$	3.87854	0.176846
$482$	1.95228	0.0889237
$483$	−2.80187	−0.127490
$484$	11.7866	0.535753
$485$	−10.4475	−0.474397
$486$	0.0993996	0.00450886
$487$	5.44720	0.246836	0.123418	−	0.992355i	$-0.460614\pi$
$487$	5.44720	0.246836	0.123418	+	0.992355i	$0.460614\pi$
$488$	−0.535499	−0.0242409
$489$	19.7395	0.892653
$490$	−0.767153	−0.0346565
$491$	−16.2133	−0.731695	−0.365847	−	0.930675i	$-0.619221\pi$
$491$	−16.2133	−0.731695	−0.365847	+	0.930675i	$0.619221\pi$
$492$	−9.70101	−0.437355
$493$	−10.0375	−0.452065
$494$	1.55718	0.0700609
$495$	2.54591	0.114430
$496$	−17.0224	−0.764329
$497$	−5.90447	−0.264852
$498$	0.279809	0.0125385
$499$	−0.720536	−0.0322556	−0.0161278	−	0.999870i	$-0.505134\pi$
$499$	−0.720536	−0.0322556	−0.0161278	+	0.999870i	$0.505134\pi$
$500$	19.6149	0.877207
$501$	−18.4760	−0.825446
$502$	−3.05383	−0.136299
$503$	−5.12765	−0.228631	−0.114315	−	0.993445i	$-0.536467\pi$
$503$	−5.12765	−0.228631	−0.114315	+	0.993445i	$0.536467\pi$
$504$	−0.163099	−0.00726502
$505$	15.0084	0.667865
$506$	−1.52607	−0.0678422
$507$	−7.17863	−0.318814
$508$	0.975092	0.0432627
$509$	0.410459	0.0181933	0.00909663	−	0.999959i	$-0.497104\pi$
$509$	0.410459	0.0181933	0.00909663	+	0.999959i	$0.497104\pi$
$510$	−0.112306	−0.00497301
$511$	0.452503	0.0200175
$512$	7.79565	0.344522
$513$	−6.49295	−0.286671
$514$	−0.562448	−0.0248085
$515$	10.0094	0.441067
$516$	−4.48757	−0.197554
$517$	−26.3814	−1.16025
$518$	−0.0657086	−0.00288707
$519$	25.2896	1.11009
$520$	1.08119	0.0474135
$521$	−1.99457	−0.0873837	−0.0436918	−	0.999045i	$-0.513912\pi$
$521$	−1.99457	−0.0873837	−0.0436918	+	0.999045i	$0.513912\pi$
$522$	0.997720	0.0436690
$523$	26.9193	1.17710	0.588549	−	0.808462i	$-0.299700\pi$
$523$	26.9193	1.17710	0.588549	+	0.808462i	$0.299700\pi$
$524$	20.1147	0.878714
$525$	−1.53118	−0.0668262
$526$	2.30385	0.100453
$527$	4.31951	0.188161
$528$	8.87993	0.386449
$529$	23.4231	1.01840
$530$	0.860219	0.0373655
$531$	−7.20091	−0.312493
$532$	5.31377	0.230381
$533$	−11.7612	−0.509433
$534$	1.69302	0.0732641
$535$	−4.09992	−0.177255
$536$	5.14713	0.222322
$537$	16.9559	0.731702
$538$	−0.294134	−0.0126810
$539$	−15.3922	−0.662989
$540$	−2.24853	−0.0967614
$541$	36.0804	1.55122	0.775608	−	0.631215i	$-0.217443\pi$
$541$	36.0804	1.55122	0.775608	+	0.631215i	$0.217443\pi$
$542$	1.90342	0.0817588
$543$	−21.5750	−0.925870
$544$	−1.18495	−0.0508043
$545$	3.52397	0.150950
$546$	−0.0986231	−0.00422068
$547$	31.4291	1.34381	0.671906	−	0.740636i	$-0.265476\pi$
$547$	31.4291	1.34381	0.671906	+	0.740636i	$0.265476\pi$
$548$	−8.71479	−0.372278
$549$	1.35017	0.0576238
$550$	−0.833975	−0.0355608
$551$	−65.1727	−2.77645
$552$	2.70233	0.115019
$553$	0.411227	0.0174871
$554$	0.249279	0.0105908
$555$	−1.81625	−0.0770955
$556$	−11.4897	−0.487271
$557$	23.7209	1.00509	0.502544	−	0.864551i	$-0.332397\pi$
$557$	23.7209	1.00509	0.502544	+	0.864551i	$0.332397\pi$
$558$	−0.429358	−0.0181762
$559$	−5.44058	−0.230112
$560$	1.83100	0.0773737
$561$	−2.25332	−0.0951353
$562$	−1.34910	−0.0569083
$563$	5.55796	0.234240	0.117120	−	0.993118i	$-0.462634\pi$
$563$	5.55796	0.234240	0.117120	+	0.993118i	$0.462634\pi$
$564$	23.2999	0.981104
$565$	−4.17911	−0.175817
$566$	0.347998	0.0146275
$567$	0.411227	0.0172699
$568$	5.69469	0.238944
$569$	38.9970	1.63484	0.817419	−	0.576044i	$-0.195404\pi$
$569$	38.9970	1.63484	0.817419	+	0.576044i	$0.195404\pi$
$570$	−0.729200	−0.0305428
$571$	−11.8644	−0.496511	−0.248256	−	0.968695i	$-0.579857\pi$
$571$	−11.8644	−0.496511	−0.248256	+	0.968695i	$0.579857\pi$
$572$	10.8197	0.452394
$573$	−13.6940	−0.572074
$574$	0.199253	0.00831664
$575$	25.3695	1.05798
$576$	−7.76385	−0.323494
$577$	−43.1418	−1.79602	−0.898008	−	0.439978i	$-0.854986\pi$
$577$	−43.1418	−1.79602	−0.898008	+	0.439978i	$0.854986\pi$
$578$	0.0993996	0.00413448
$579$	−19.6320	−0.815879
$580$	−22.5696	−0.937150
$581$	1.15760	0.0480253
$582$	−0.919132	−0.0380992
$583$	17.2595	0.714814
$584$	−0.436426	−0.0180594
$585$	−2.72604	−0.112708
$586$	−1.70494	−0.0704303
$587$	41.7837	1.72460	0.862298	−	0.506401i	$-0.169024\pi$
$587$	41.7837	1.72460	0.862298	+	0.506401i	$0.169024\pi$
$588$	13.5943	0.560619
$589$	28.0464	1.15563
$590$	−0.808709	−0.0332940
$591$	12.4370	0.511592
$592$	−6.33493	−0.260364
$593$	12.4870	0.512779	0.256390	−	0.966574i	$-0.417467\pi$
$593$	12.4870	0.512779	0.256390	+	0.966574i	$0.417467\pi$
$594$	0.223979	0.00918999
$595$	−0.464624	−0.0190477
$596$	9.00561	0.368884
$597$	−11.3603	−0.464948
$598$	1.63405	0.0668212
$599$	−11.3567	−0.464021	−0.232011	−	0.972713i	$-0.574530\pi$
$599$	−11.3567	−0.464021	−0.232011	+	0.972713i	$0.574530\pi$
$600$	1.47678	0.0602893
$601$	−34.2289	−1.39623	−0.698113	−	0.715987i	$-0.745977\pi$
$601$	−34.2289	−1.39623	−0.698113	+	0.715987i	$0.745977\pi$
$602$	0.0921719	0.00375665
$603$	−12.9776	−0.528488
$604$	35.7941	1.45644
$605$	−6.69157	−0.272051
$606$	1.32038	0.0536368
$607$	−8.52231	−0.345910	−0.172955	−	0.984930i	$-0.555332\pi$
$607$	−8.52231	−0.345910	−0.172955	+	0.984930i	$0.555332\pi$
$608$	−7.69381	−0.312025
$609$	4.12767	0.167262
$610$	0.151633	0.00613943
$611$	28.2480	1.14279
$612$	1.99012	0.0804458
$613$	37.4952	1.51442	0.757209	−	0.653172i	$-0.226562\pi$
$613$	37.4952	1.51442	0.757209	+	0.653172i	$0.226562\pi$
$614$	2.76917	0.111755
$615$	5.50754	0.222085
$616$	−0.367515	−0.0148076
$617$	13.8772	0.558677	0.279338	−	0.960193i	$-0.409885\pi$
$617$	13.8772	0.558677	0.279338	+	0.960193i	$0.409885\pi$
$618$	0.880589	0.0354225
$619$	31.9740	1.28514	0.642572	−	0.766225i	$-0.277867\pi$
$619$	31.9740	1.28514	0.642572	+	0.766225i	$0.277867\pi$
$620$	9.71257	0.390066
$621$	−6.81345	−0.273414
$622$	−2.89889	−0.116235
$623$	7.00420	0.280617
$624$	−9.50821	−0.380633
$625$	7.48125	0.299250
$626$	0.557426	0.0222792
$627$	−14.6307	−0.584294
$628$	38.9534	1.55441
$629$	1.60752	0.0640959
$630$	0.0461834	0.00183999
$631$	43.6127	1.73619	0.868097	−	0.496395i	$-0.165343\pi$
$631$	43.6127	1.73619	0.868097	+	0.496395i	$0.165343\pi$
$632$	−0.396616	−0.0157766
$633$	−15.7500	−0.626006
$634$	0.715024	0.0283972
$635$	−0.553588	−0.0219685
$636$	−15.2435	−0.604442
$637$	16.4812	0.653011
$638$	2.24818	0.0890065
$639$	−14.3582	−0.568001
$640$	−3.54955	−0.140308
$641$	−33.0633	−1.30592	−0.652961	−	0.757392i	$-0.726473\pi$
$641$	−33.0633	−1.30592	−0.652961	+	0.757392i	$0.726473\pi$
$642$	−0.360695	−0.0142355
$643$	13.4270	0.529510	0.264755	−	0.964316i	$-0.414709\pi$
$643$	13.4270	0.529510	0.264755	+	0.964316i	$0.414709\pi$
$644$	5.57606	0.219728
$645$	2.54772	0.100317
$646$	0.645397	0.0253928
$647$	−21.5056	−0.845471	−0.422736	−	0.906253i	$-0.638930\pi$
$647$	−21.5056	−0.845471	−0.422736	+	0.906253i	$0.638930\pi$
$648$	−0.396616	−0.0155806
$649$	−16.2260	−0.636925
$650$	0.892981	0.0350256
$651$	−1.77630	−0.0696187
$652$	−39.2840	−1.53848
$653$	13.8762	0.543017	0.271508	−	0.962436i	$-0.412478\pi$
$653$	13.8762	0.543017	0.271508	+	0.962436i	$0.412478\pi$
$654$	0.310025	0.0121229
$655$	−11.4197	−0.446204
$656$	19.2098	0.750019
$657$	1.10037	0.0429296
$658$	−0.478565	−0.0186564
$659$	23.8299	0.928280	0.464140	−	0.885762i	$-0.346363\pi$
$659$	23.8299	0.928280	0.464140	+	0.885762i	$0.346363\pi$
$660$	−5.06667	−0.197220
$661$	17.9615	0.698621	0.349310	−	0.937007i	$-0.386416\pi$
$661$	17.9615	0.698621	0.349310	+	0.937007i	$0.386416\pi$
$662$	−1.31095	−0.0509514
$663$	2.41275	0.0937035
$664$	−1.11647	−0.0433274
$665$	−3.01678	−0.116986
$666$	−0.159787	−0.00619161
$667$	−68.3898	−2.64806
$668$	36.7694	1.42265
$669$	−2.12609	−0.0821993
$670$	−1.45747	−0.0563069
$671$	3.04237	0.117449
$672$	0.487283	0.0187973
$673$	−28.6211	−1.10326	−0.551632	−	0.834088i	$-0.685995\pi$
$673$	−28.6211	−1.10326	−0.551632	+	0.834088i	$0.685995\pi$
$674$	0.00503130	0.000193798 0
$675$	−3.72344	−0.143315
$676$	14.2863	0.549474
$677$	−24.4412	−0.939353	−0.469677	−	0.882838i	$-0.655630\pi$
$677$	−24.4412	−0.939353	−0.469677	+	0.882838i	$0.655630\pi$
$678$	−0.367662	−0.0141200
$679$	−3.80254	−0.145928
$680$	0.448116	0.0171845
$681$	14.1013	0.540364
$682$	−0.967482	−0.0370468
$683$	−10.7506	−0.411362	−0.205681	−	0.978619i	$-0.565941\pi$
$683$	−10.7506	−0.411362	−0.205681	+	0.978619i	$0.565941\pi$
$684$	12.9217	0.494075
$685$	4.94764	0.189040
$686$	−0.565349	−0.0215851
$687$	−0.272709	−0.0104045
$688$	8.88625	0.338785
$689$	−18.4806	−0.704056
$690$	−0.765195	−0.0291305
$691$	−31.9126	−1.21401	−0.607006	−	0.794697i	$-0.707630\pi$
$691$	−31.9126	−1.21401	−0.607006	+	0.794697i	$0.707630\pi$
$692$	−50.3294	−1.91324
$693$	0.926626	0.0351996
$694$	1.15478	0.0438350
$695$	6.52303	0.247432
$696$	−3.98102	−0.150900
$697$	−4.87459	−0.184638
$698$	1.23879	0.0468889
$699$	−25.4716	−0.963424
$700$	3.04723	0.115174
$701$	7.67287	0.289800	0.144900	−	0.989446i	$-0.453714\pi$
$701$	7.67287	0.289800	0.144900	+	0.989446i	$0.453714\pi$
$702$	−0.239827	−0.00905167
$703$	10.4375	0.393659
$704$	−17.4944	−0.659347
$705$	−13.2280	−0.498196
$706$	0.610805	0.0229880
$707$	5.46255	0.205440
$708$	14.3307	0.538580
$709$	5.88038	0.220842	0.110421	−	0.993885i	$-0.464780\pi$
$709$	5.88038	0.220842	0.110421	+	0.993885i	$0.464780\pi$
$710$	−1.61252	−0.0605167
$711$	1.00000	0.0375029
$712$	−6.75535	−0.253167
$713$	29.4308	1.10219
$714$	−0.0408758	−0.00152974
$715$	−6.14265	−0.229722
$716$	−33.7443	−1.26108
$717$	−25.1388	−0.938824
$718$	1.92625	0.0718869
$719$	−11.9248	−0.444719	−0.222360	−	0.974965i	$-0.571376\pi$
$719$	−11.9248	−0.444719	−0.222360	+	0.974965i	$0.571376\pi$
$720$	4.45252	0.165936
$721$	3.64309	0.135676
$722$	2.30193	0.0856691
$723$	19.6407	0.730444
$724$	42.9368	1.59573
$725$	−37.3739	−1.38803
$726$	−0.588698	−0.0218487
$727$	−15.2458	−0.565435	−0.282718	−	0.959203i	$-0.591236\pi$
$727$	−15.2458	−0.565435	−0.282718	+	0.959203i	$0.591236\pi$
$728$	0.393518	0.0145847
$729$	1.00000	0.0370370
$730$	0.123579	0.00457387
$731$	−2.25493	−0.0834015
$732$	−2.68700	−0.0993143
$733$	49.7318	1.83689	0.918443	−	0.395554i	$-0.129447\pi$
$733$	49.7318	1.83689	0.918443	+	0.395554i	$0.129447\pi$
$734$	−0.857803	−0.0316621
$735$	−7.71787	−0.284678
$736$	−8.07359	−0.297597
$737$	−29.2427	−1.07717
$738$	0.484532	0.0178359
$739$	5.75987	0.211880	0.105940	−	0.994373i	$-0.466215\pi$
$739$	5.75987	0.211880	0.105940	+	0.994373i	$0.466215\pi$
$740$	3.61456	0.132874
$741$	15.6659	0.575500
$742$	0.313091	0.0114939
$743$	11.7849	0.432344	0.216172	−	0.976355i	$-0.430643\pi$
$743$	11.7849	0.432344	0.216172	+	0.976355i	$0.430643\pi$
$744$	1.71319	0.0628086
$745$	−5.11274	−0.187316
$746$	0.568798	0.0208252
$747$	2.81499	0.102995
$748$	4.48438	0.163965
$749$	−1.49223	−0.0545251
$750$	−0.979699	−0.0357736
$751$	3.00203	0.109546	0.0547728	−	0.998499i	$-0.482557\pi$
$751$	3.00203	0.109546	0.0547728	+	0.998499i	$0.482557\pi$
$752$	−46.1383	−1.68249
$753$	−30.7228	−1.11960
$754$	−2.40725	−0.0876669
$755$	−20.3213	−0.739569
$756$	−0.818390	−0.0297646
$757$	45.8814	1.66759	0.833794	−	0.552075i	$-0.186164\pi$
$757$	45.8814	1.66759	0.833794	+	0.552075i	$0.186164\pi$
$758$	−0.394766	−0.0143385
$759$	−15.3529	−0.557275
$760$	2.90959	0.105542
$761$	13.5554	0.491382	0.245691	−	0.969348i	$-0.420985\pi$
$761$	13.5554	0.491382	0.245691	+	0.969348i	$0.420985\pi$
$762$	−0.0487025	−0.00176431
$763$	1.28261	0.0464335
$764$	27.2527	0.985967
$765$	−1.12985	−0.0408497
$766$	1.60111	0.0578504
$767$	17.3740	0.627339
$768$	15.2154	0.549039
$769$	−1.09838	−0.0396087	−0.0198043	−	0.999804i	$-0.506304\pi$
$769$	−1.09838	−0.0396087	−0.0198043	+	0.999804i	$0.506304\pi$
$770$	0.104066	0.00375028
$771$	−5.65845	−0.203784
$772$	39.0701	1.40616
$773$	13.0799	0.470452	0.235226	−	0.971941i	$-0.424417\pi$
$773$	13.0799	0.470452	0.235226	+	0.971941i	$0.424417\pi$
$774$	0.224139	0.00805651
$775$	16.0835	0.577735
$776$	3.66745	0.131654
$777$	−0.661054	−0.0237152
$778$	−2.74414	−0.0983823
$779$	−31.6504	−1.13399
$780$	5.42515	0.194252
$781$	−32.3536	−1.15770
$782$	0.677255	0.0242186
$783$	10.0375	0.358710
$784$	−26.9193	−0.961403
$785$	−22.1150	−0.789318
$786$	−1.00466	−0.0358350
$787$	−31.9777	−1.13988	−0.569941	−	0.821686i	$-0.693034\pi$
$787$	−31.9777	−1.13988	−0.569941	+	0.821686i	$0.693034\pi$
$788$	−24.7512	−0.881725
$789$	23.1776	0.825145
$790$	0.112306	0.00399568
$791$	−1.52106	−0.0540825
$792$	−0.893705	−0.0317564
$793$	−3.25762	−0.115682
$794$	−1.24294	−0.0441103
$795$	8.65415	0.306931
$796$	22.6085	0.801335
$797$	−45.7820	−1.62168	−0.810841	−	0.585267i	$-0.800990\pi$
$797$	−45.7820	−1.62168	−0.810841	+	0.585267i	$0.800990\pi$
$798$	−0.265404	−0.00939521
$799$	11.7078	0.414192
$800$	−4.41209	−0.155991
$801$	17.0324	0.601812
$802$	−0.188622	−0.00666049
$803$	2.47949	0.0874995
$804$	25.8270	0.910846
$805$	−3.16569	−0.111576
$806$	1.03593	0.0364892
$807$	−2.95911	−0.104166
$808$	−5.26848	−0.185344
$809$	−36.2795	−1.27552	−0.637759	−	0.770236i	$-0.720139\pi$
$809$	−36.2795	−1.27552	−0.637759	+	0.770236i	$0.720139\pi$
$810$	0.112306	0.00394605
$811$	54.1445	1.90127	0.950635	−	0.310312i	$-0.100433\pi$
$811$	54.1445	1.90127	0.950635	+	0.310312i	$0.100433\pi$
$812$	−8.21456	−0.288275
$813$	19.1491	0.671590
$814$	−0.360051	−0.0126198
$815$	22.3027	0.781229
$816$	−3.94082	−0.137956
$817$	−14.6411	−0.512228
$818$	−3.34968	−0.117119
$819$	−0.992188	−0.0346699
$820$	−10.9607	−0.382763
$821$	−14.1297	−0.493129	−0.246564	−	0.969126i	$-0.579302\pi$
$821$	−14.1297	−0.493129	−0.246564	+	0.969126i	$0.579302\pi$
$822$	0.435274	0.0151819
$823$	17.2612	0.601689	0.300844	−	0.953673i	$-0.402732\pi$
$823$	17.2612	0.601689	0.300844	+	0.953673i	$0.402732\pi$
$824$	−3.51366	−0.122404
$825$	−8.39012	−0.292106
$826$	−0.294343	−0.0102415
$827$	11.4568	0.398393	0.199197	−	0.979960i	$-0.436167\pi$
$827$	11.4568	0.398393	0.199197	+	0.979960i	$0.436167\pi$
$828$	13.5596	0.471228
$829$	10.2218	0.355018	0.177509	−	0.984119i	$-0.443196\pi$
$829$	10.2218	0.355018	0.177509	+	0.984119i	$0.443196\pi$
$830$	0.316141	0.0109734
$831$	2.50785	0.0869962
$832$	18.7322	0.649423
$833$	6.83089	0.236676
$834$	0.573871	0.0198715
$835$	−20.8751	−0.722411
$836$	29.1168	1.00703
$837$	−4.31951	−0.149304
$838$	0.229967	0.00794409
$839$	−15.5091	−0.535435	−0.267718	−	0.963497i	$-0.586269\pi$
$839$	−15.5091	−0.535435	−0.267718	+	0.963497i	$0.586269\pi$
$840$	−0.184277	−0.00635817
$841$	71.7506	2.47416
$842$	3.55285	0.122439
$843$	−13.5725	−0.467461
$844$	31.3444	1.07892
$845$	−8.11076	−0.279019
$846$	−1.16375	−0.0400106
$847$	−2.43551	−0.0836850
$848$	30.1850	1.03656
$849$	3.50100	0.120154
$850$	0.370109	0.0126946
$851$	10.9527	0.375455
$852$	28.5745	0.978946
$853$	25.3662	0.868522	0.434261	−	0.900787i	$-0.357010\pi$
$853$	25.3662	0.868522	0.434261	+	0.900787i	$0.357010\pi$
$854$	0.0551892	0.00188854
$855$	−7.33604	−0.250887
$856$	1.43922	0.0491914
$857$	−38.6987	−1.32192	−0.660961	−	0.750420i	$-0.729851\pi$
$857$	−38.6987	−1.32192	−0.660961	+	0.750420i	$0.729851\pi$
$858$	−0.540407	−0.0184492
$859$	−14.1710	−0.483508	−0.241754	−	0.970338i	$-0.577723\pi$
$859$	−14.1710	−0.483508	−0.241754	+	0.970338i	$0.577723\pi$
$860$	−5.07028	−0.172895
$861$	2.00456	0.0683152
$862$	0.0480087	0.00163518
$863$	−17.8380	−0.607212	−0.303606	−	0.952798i	$-0.598191\pi$
$863$	−17.8380	−0.607212	−0.303606	+	0.952798i	$0.598191\pi$
$864$	1.18495	0.0403128
$865$	28.5735	0.971527
$866$	1.84355	0.0626465
$867$	1.00000	0.0339618
$868$	3.53505	0.119987
$869$	2.25332	0.0764387
$870$	1.12727	0.0382181
$871$	31.3117	1.06096
$872$	−1.23704	−0.0418914
$873$	−9.24683	−0.312958
$874$	4.39738	0.148744
$875$	−4.05312	−0.137020
$876$	−2.18987	−0.0739890
$877$	26.7930	0.904735	0.452368	−	0.891832i	$-0.350580\pi$
$877$	26.7930	0.904735	0.452368	+	0.891832i	$0.350580\pi$
$878$	2.56134	0.0864411
$879$	−17.1524	−0.578534
$880$	10.0330	0.338211
$881$	−26.4486	−0.891076	−0.445538	−	0.895263i	$-0.646988\pi$
$881$	−26.4486	−0.891076	−0.445538	+	0.895263i	$0.646988\pi$
$882$	−0.678988	−0.0228627
$883$	−26.8253	−0.902744	−0.451372	−	0.892336i	$-0.649065\pi$
$883$	−26.8253	−0.902744	−0.451372	+	0.892336i	$0.649065\pi$
$884$	−4.80166	−0.161497
$885$	−8.13593	−0.273486
$886$	1.01748	0.0341828
$887$	−34.0582	−1.14356	−0.571782	−	0.820406i	$-0.693748\pi$
$887$	−34.0582	−1.14356	−0.571782	+	0.820406i	$0.693748\pi$
$888$	0.637568	0.0213954
$889$	−0.201487	−0.00675767
$890$	1.91285	0.0641190
$891$	2.25332	0.0754891
$892$	4.23117	0.141670
$893$	76.0181	2.54385
$894$	−0.449799	−0.0150435
$895$	19.1576	0.640368
$896$	−1.29192	−0.0431600
$897$	16.4392	0.548888
$898$	−2.32782	−0.0776803
$899$	−43.3569	−1.44604
$900$	7.41010	0.247003
$901$	−7.65957	−0.255177
$902$	1.09181	0.0363532
$903$	0.927286	0.0308582
$904$	1.46702	0.0487922
$905$	−24.3764	−0.810300
$906$	−1.78779	−0.0593954
$907$	−52.5056	−1.74342	−0.871710	−	0.490023i	$-0.836988\pi$
$907$	−52.5056	−1.74342	−0.871710	+	0.490023i	$0.836988\pi$
$908$	−28.0633	−0.931314
$909$	13.2836	0.440588
$910$	−0.111429	−0.00369384
$911$	54.2646	1.79787	0.898934	−	0.438085i	$-0.144343\pi$
$911$	54.2646	1.79787	0.898934	+	0.438085i	$0.144343\pi$
$912$	−25.5875	−0.847287
$913$	6.34307	0.209925
$914$	1.95859	0.0647844
$915$	1.52549	0.0504310
$916$	0.542724	0.0179321
$917$	−4.15638	−0.137256
$918$	−0.0993996	−0.00328068
$919$	−31.9389	−1.05357	−0.526784	−	0.849999i	$-0.676602\pi$
$919$	−31.9389	−1.05357	−0.526784	+	0.849999i	$0.676602\pi$
$920$	3.05322	0.100662
$921$	27.8589	0.917983
$922$	−0.950174	−0.0312923
$923$	34.6427	1.14028
$924$	−1.84410	−0.0606663
$925$	5.98550	0.196802
$926$	1.30314	0.0428238
$927$	8.85908	0.290970
$928$	11.8939	0.390436
$929$	57.4415	1.88459	0.942297	−	0.334777i	$-0.108661\pi$
$929$	57.4415	1.88459	0.942297	+	0.334777i	$0.108661\pi$
$930$	−0.485109	−0.0159074
$931$	44.3526	1.45360
$932$	50.6915	1.66046
$933$	−29.1640	−0.954786
$934$	1.60170	0.0524092
$935$	−2.54591	−0.0832602
$936$	0.956937	0.0312785
$937$	−34.8455	−1.13835	−0.569176	−	0.822216i	$-0.692738\pi$
$937$	−34.8455	−1.13835	−0.569176	+	0.822216i	$0.692738\pi$
$938$	−0.530469	−0.0173204
$939$	5.60793	0.183008
$940$	26.3254	0.858639
$941$	34.6453	1.12940	0.564702	−	0.825295i	$-0.308991\pi$
$941$	34.6453	1.12940	0.564702	+	0.825295i	$0.308991\pi$
$942$	−1.94559	−0.0633908
$943$	−33.2128	−1.08156
$944$	−28.3775	−0.923608
$945$	0.464624	0.0151142
$946$	0.505057	0.0164208
$947$	23.1304	0.751637	0.375819	−	0.926693i	$-0.377362\pi$
$947$	23.1304	0.751637	0.375819	+	0.926693i	$0.377362\pi$
$948$	−1.99012	−0.0646361
$949$	−2.65493	−0.0861826
$950$	2.40310	0.0779668
$951$	7.19342	0.233263
$952$	0.163099	0.00528608
$953$	−6.32775	−0.204976	−0.102488	−	0.994734i	$-0.532680\pi$
$953$	−6.32775	−0.204976	−0.102488	+	0.994734i	$0.532680\pi$
$954$	0.761358	0.0246499
$955$	−15.4721	−0.500666
$956$	50.0291	1.61806
$957$	22.6176	0.731124
$958$	−2.40481	−0.0776959
$959$	1.80077	0.0581500
$960$	−8.77197	−0.283114
$961$	−12.3418	−0.398123
$962$	0.385526	0.0124298
$963$	−3.62874	−0.116934
$964$	−39.0873	−1.25892
$965$	−22.1812	−0.714038
$966$	−0.278505	−0.00896076
$967$	−12.4565	−0.400575	−0.200287	−	0.979737i	$-0.564188\pi$
$967$	−12.4565	−0.400575	−0.200287	+	0.979737i	$0.564188\pi$
$968$	2.34898	0.0754990
$969$	6.49295	0.208583
$970$	−1.03848	−0.0333436
$971$	−7.35694	−0.236095	−0.118048	−	0.993008i	$-0.537664\pi$
$971$	−7.35694	−0.236095	−0.118048	+	0.993008i	$0.537664\pi$
$972$	−1.99012	−0.0638331
$973$	2.37416	0.0761122
$974$	0.541449	0.0173492
$975$	8.98374	0.287710
$976$	5.32077	0.170314
$977$	17.0134	0.544307	0.272154	−	0.962254i	$-0.412264\pi$
$977$	17.0134	0.544307	0.272154	+	0.962254i	$0.412264\pi$
$978$	1.96210	0.0627411
$979$	38.3796	1.22662
$980$	15.3595	0.490641
$981$	3.11898	0.0995812
$982$	−1.61159	−0.0514280
$983$	−46.9836	−1.49854	−0.749272	−	0.662262i	$-0.769596\pi$
$983$	−46.9836	−1.49854	−0.749272	+	0.662262i	$0.769596\pi$
$984$	−1.93334	−0.0616327
$985$	14.0520	0.447733
$986$	−0.997720	−0.0317739
$987$	−4.81456	−0.153249
$988$	−31.1769	−0.991871
$989$	−15.3638	−0.488542
$990$	0.253063	0.00804286
$991$	12.5131	0.397493	0.198746	−	0.980051i	$-0.436313\pi$
$991$	12.5131	0.397493	0.198746	+	0.980051i	$0.436313\pi$
$992$	−5.11840	−0.162509
$993$	−13.1886	−0.418529
$994$	−0.586902	−0.0186154
$995$	−12.8355	−0.406912
$996$	−5.60216	−0.177511
$997$	25.9838	0.822915	0.411458	−	0.911429i	$-0.365020\pi$
$997$	25.9838	0.822915	0.411458	+	0.911429i	$0.365020\pi$
$998$	−0.0716210	−0.00226712
$999$	−1.60752	−0.0508596

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4029.2.a.h.1.15	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4029.2.a.h.1.15	✓	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)

\(f(q)\)	\(=\)	\(q+0.0993996 q^{2} +1.00000 q^{3} -1.99012 q^{4} +1.12985 q^{5} +0.0993996 q^{6} +0.411227 q^{7} -0.396616 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.0993996 q^{2} +1.00000 q^{3} -1.99012 q^{4} +1.12985 q^{5} +0.0993996 q^{6} +0.411227 q^{7} -0.396616 q^{8} +1.00000 q^{9} +0.112306 q^{10} +2.25332 q^{11} -1.99012 q^{12} -2.41275 q^{13} +0.0408758 q^{14} +1.12985 q^{15} +3.94082 q^{16} -1.00000 q^{17} +0.0993996 q^{18} -6.49295 q^{19} -2.24853 q^{20} +0.411227 q^{21} +0.223979 q^{22} -6.81345 q^{23} -0.396616 q^{24} -3.72344 q^{25} -0.239827 q^{26} +1.00000 q^{27} -0.818390 q^{28} +10.0375 q^{29} +0.112306 q^{30} -4.31951 q^{31} +1.18495 q^{32} +2.25332 q^{33} -0.0993996 q^{34} +0.464624 q^{35} -1.99012 q^{36} -1.60752 q^{37} -0.645397 q^{38} -2.41275 q^{39} -0.448116 q^{40} +4.87459 q^{41} +0.0408758 q^{42} +2.25493 q^{43} -4.48438 q^{44} +1.12985 q^{45} -0.677255 q^{46} -11.7078 q^{47} +3.94082 q^{48} -6.83089 q^{49} -0.370109 q^{50} -1.00000 q^{51} +4.80166 q^{52} +7.65957 q^{53} +0.0993996 q^{54} +2.54591 q^{55} -0.163099 q^{56} -6.49295 q^{57} +0.997720 q^{58} -7.20091 q^{59} -2.24853 q^{60} +1.35017 q^{61} -0.429358 q^{62} +0.411227 q^{63} -7.76385 q^{64} -2.72604 q^{65} +0.223979 q^{66} -12.9776 q^{67} +1.99012 q^{68} -6.81345 q^{69} +0.0461834 q^{70} -14.3582 q^{71} -0.396616 q^{72} +1.10037 q^{73} -0.159787 q^{74} -3.72344 q^{75} +12.9217 q^{76} +0.926626 q^{77} -0.239827 q^{78} +1.00000 q^{79} +4.45252 q^{80} +1.00000 q^{81} +0.484532 q^{82} +2.81499 q^{83} -0.818390 q^{84} -1.12985 q^{85} +0.224139 q^{86} +10.0375 q^{87} -0.893705 q^{88} +17.0324 q^{89} +0.112306 q^{90} -0.992188 q^{91} +13.5596 q^{92} -4.31951 q^{93} -1.16375 q^{94} -7.33604 q^{95} +1.18495 q^{96} -9.24683 q^{97} -0.678988 q^{98} +2.25332 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 7 q^{2} + 25 q^{3} + 21 q^{4} - 12 q^{5} - 7 q^{6} - 4 q^{7} - 21 q^{8} + 25 q^{9}+O(q^{10}) \) 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