LMFDB - Embedded newform 4029.2.a.i.1.4

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:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-2.24485 q^{2} -1.00000 q^{3} +3.03936 q^{4} -1.59596 q^{5} +2.24485 q^{6} +3.55304 q^{7} -2.33321 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.24485 q^{2} -1.00000 q^{3} +3.03936 q^{4} -1.59596 q^{5} +2.24485 q^{6} +3.55304 q^{7} -2.33321 q^{8} +1.00000 q^{9} +3.58269 q^{10} -3.09021 q^{11} -3.03936 q^{12} +3.26659 q^{13} -7.97604 q^{14} +1.59596 q^{15} -0.841010 q^{16} +1.00000 q^{17} -2.24485 q^{18} -5.67115 q^{19} -4.85069 q^{20} -3.55304 q^{21} +6.93706 q^{22} -4.49157 q^{23} +2.33321 q^{24} -2.45292 q^{25} -7.33302 q^{26} -1.00000 q^{27} +10.7990 q^{28} +1.22338 q^{29} -3.58269 q^{30} -2.86983 q^{31} +6.55436 q^{32} +3.09021 q^{33} -2.24485 q^{34} -5.67050 q^{35} +3.03936 q^{36} +9.10279 q^{37} +12.7309 q^{38} -3.26659 q^{39} +3.72370 q^{40} -0.946335 q^{41} +7.97604 q^{42} +11.4540 q^{43} -9.39226 q^{44} -1.59596 q^{45} +10.0829 q^{46} +6.83413 q^{47} +0.841010 q^{48} +5.62408 q^{49} +5.50644 q^{50} -1.00000 q^{51} +9.92835 q^{52} -8.45723 q^{53} +2.24485 q^{54} +4.93184 q^{55} -8.28998 q^{56} +5.67115 q^{57} -2.74631 q^{58} -5.17305 q^{59} +4.85069 q^{60} +2.20823 q^{61} +6.44234 q^{62} +3.55304 q^{63} -13.0316 q^{64} -5.21334 q^{65} -6.93706 q^{66} +9.02616 q^{67} +3.03936 q^{68} +4.49157 q^{69} +12.7294 q^{70} +1.93460 q^{71} -2.33321 q^{72} -10.6017 q^{73} -20.4344 q^{74} +2.45292 q^{75} -17.2367 q^{76} -10.9796 q^{77} +7.33302 q^{78} -1.00000 q^{79} +1.34222 q^{80} +1.00000 q^{81} +2.12438 q^{82} +2.57393 q^{83} -10.7990 q^{84} -1.59596 q^{85} -25.7126 q^{86} -1.22338 q^{87} +7.21011 q^{88} -1.73736 q^{89} +3.58269 q^{90} +11.6063 q^{91} -13.6515 q^{92} +2.86983 q^{93} -15.3416 q^{94} +9.05091 q^{95} -6.55436 q^{96} -0.683468 q^{97} -12.6252 q^{98} -3.09021 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 2 q^{2} - 25 q^{3} + 26 q^{4} - 2 q^{5} + 2 q^{6} + 12 q^{7} + 25 q^{9}+O(q^{10}) $ 25 * q - 2 * q^2 - 25 * q^3 + 26 * q^4 - 2 * q^5 + 2 * q^6 + 12 * q^7 + 25 * q^9	\( 25 q - 2 q^{2} - 25 q^{3} + 26 q^{4} - 2 q^{5} + 2 q^{6} + 12 q^{7} + 25 q^{9} + 19 q^{10} + 19 q^{11} - 26 q^{12} + 4 q^{13} + 15 q^{14} + 2 q^{15} + 32 q^{16} + 25 q^{17} - 2 q^{18} + 29 q^{19} - 8 q^{20} - 12 q^{21} + 23 q^{22} + 6 q^{23} + 15 q^{25} - 8 q^{26} - 25 q^{27} + 23 q^{28} + 11 q^{29} - 19 q^{30} + 38 q^{31} - 27 q^{32} - 19 q^{33} - 2 q^{34} + 20 q^{35} + 26 q^{36} + 8 q^{37} - 25 q^{38} - 4 q^{39} + 48 q^{40} + 24 q^{41} - 15 q^{42} + 11 q^{43} + 6 q^{44} - 2 q^{45} + 25 q^{46} + 23 q^{47} - 32 q^{48} + 21 q^{49} - 21 q^{50} - 25 q^{51} + 31 q^{52} - 16 q^{53} + 2 q^{54} - 11 q^{55} + 18 q^{56} - 29 q^{57} - 5 q^{58} + 27 q^{59} + 8 q^{60} + 40 q^{61} - 34 q^{62} + 12 q^{63} + 46 q^{64} - 19 q^{65} - 23 q^{66} + 24 q^{67} + 26 q^{68} - 6 q^{69} + 17 q^{70} + 19 q^{71} + 13 q^{73} - 56 q^{74} - 15 q^{75} + 21 q^{76} - 30 q^{77} + 8 q^{78} - 25 q^{79} - 40 q^{80} + 25 q^{81} + 61 q^{82} + q^{83} - 23 q^{84} - 2 q^{85} + 62 q^{86} - 11 q^{87} - q^{88} - 10 q^{89} + 19 q^{90} + 50 q^{91} + 18 q^{92} - 38 q^{93} + 15 q^{94} + 14 q^{95} + 27 q^{96} + 19 q^{97} - 23 q^{98} + 19 q^{99}+O(q^{100}) \) 25 * q - 2 * q^2 - 25 * q^3 + 26 * q^4 - 2 * q^5 + 2 * q^6 + 12 * q^7 + 25 * q^9 + 19 * q^10 + 19 * q^11 - 26 * q^12 + 4 * q^13 + 15 * q^14 + 2 * q^15 + 32 * q^16 + 25 * q^17 - 2 * q^18 + 29 * q^19 - 8 * q^20 - 12 * q^21 + 23 * q^22 + 6 * q^23 + 15 * q^25 - 8 * q^26 - 25 * q^27 + 23 * q^28 + 11 * q^29 - 19 * q^30 + 38 * q^31 - 27 * q^32 - 19 * q^33 - 2 * q^34 + 20 * q^35 + 26 * q^36 + 8 * q^37 - 25 * q^38 - 4 * q^39 + 48 * q^40 + 24 * q^41 - 15 * q^42 + 11 * q^43 + 6 * q^44 - 2 * q^45 + 25 * q^46 + 23 * q^47 - 32 * q^48 + 21 * q^49 - 21 * q^50 - 25 * q^51 + 31 * q^52 - 16 * q^53 + 2 * q^54 - 11 * q^55 + 18 * q^56 - 29 * q^57 - 5 * q^58 + 27 * q^59 + 8 * q^60 + 40 * q^61 - 34 * q^62 + 12 * q^63 + 46 * q^64 - 19 * q^65 - 23 * q^66 + 24 * q^67 + 26 * q^68 - 6 * q^69 + 17 * q^70 + 19 * q^71 + 13 * q^73 - 56 * q^74 - 15 * q^75 + 21 * q^76 - 30 * q^77 + 8 * q^78 - 25 * q^79 - 40 * q^80 + 25 * q^81 + 61 * q^82 + q^83 - 23 * q^84 - 2 * q^85 + 62 * q^86 - 11 * q^87 - q^88 - 10 * q^89 + 19 * q^90 + 50 * q^91 + 18 * q^92 - 38 * q^93 + 15 * q^94 + 14 * q^95 + 27 * q^96 + 19 * q^97 - 23 * 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 $

$2$	−2.24485	−1.58735	−0.793675	−	0.608342i	$-0.791835\pi$
$2$	−2.24485	−1.58735	−0.793675	+	0.608342i	$0.791835\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.03936	1.51968
$5$	−1.59596	−0.713734	−0.356867	−	0.934155i	$-0.616155\pi$
$5$	−1.59596	−0.713734	−0.356867	+	0.934155i	$0.616155\pi$
$6$	2.24485	0.916457
$7$	3.55304	1.34292	0.671461	−	0.741040i	$-0.265667\pi$
$7$	3.55304	1.34292	0.671461	+	0.741040i	$0.265667\pi$
$8$	−2.33321	−0.824914
$9$	1.00000	0.333333
$10$	3.58269	1.13295
$11$	−3.09021	−0.931733	−0.465866	−	0.884855i	$-0.654257\pi$
$11$	−3.09021	−0.931733	−0.465866	+	0.884855i	$0.654257\pi$
$12$	−3.03936	−0.877388
$13$	3.26659	0.905990	0.452995	−	0.891513i	$-0.350356\pi$
$13$	3.26659	0.905990	0.452995	+	0.891513i	$0.350356\pi$
$14$	−7.97604	−2.13169
$15$	1.59596	0.412074
$16$	−0.841010	−0.210252
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−2.24485	−0.529117
$19$	−5.67115	−1.30105	−0.650525	−	0.759485i	$-0.725451\pi$
$19$	−5.67115	−1.30105	−0.650525	+	0.759485i	$0.725451\pi$
$20$	−4.85069	−1.08465
$21$	−3.55304	−0.775336
$22$	6.93706	1.47899
$23$	−4.49157	−0.936557	−0.468278	−	0.883581i	$-0.655126\pi$
$23$	−4.49157	−0.936557	−0.468278	+	0.883581i	$0.655126\pi$
$24$	2.33321	0.476264
$25$	−2.45292	−0.490584
$26$	−7.33302	−1.43812
$27$	−1.00000	−0.192450
$28$	10.7990	2.04081
$29$	1.22338	0.227176	0.113588	−	0.993528i	$-0.463766\pi$
$29$	1.22338	0.227176	0.113588	+	0.993528i	$0.463766\pi$
$30$	−3.58269	−0.654106
$31$	−2.86983	−0.515437	−0.257718	−	0.966220i	$-0.582971\pi$
$31$	−2.86983	−0.515437	−0.257718	+	0.966220i	$0.582971\pi$
$32$	6.55436	1.15866
$33$	3.09021	0.537936
$34$	−2.24485	−0.384989
$35$	−5.67050	−0.958489
$36$	3.03936	0.506560
$37$	9.10279	1.49649	0.748244	−	0.663423i	$-0.230897\pi$
$37$	9.10279	1.49649	0.748244	+	0.663423i	$0.230897\pi$
$38$	12.7309	2.06522
$39$	−3.26659	−0.523073
$40$	3.72370	0.588769
$41$	−0.946335	−0.147793	−0.0738963	−	0.997266i	$-0.523543\pi$
$41$	−0.946335	−0.147793	−0.0738963	+	0.997266i	$0.523543\pi$
$42$	7.97604	1.23073
$43$	11.4540	1.74672	0.873362	−	0.487071i	$-0.161935\pi$
$43$	11.4540	1.74672	0.873362	+	0.487071i	$0.161935\pi$
$44$	−9.39226	−1.41594
$45$	−1.59596	−0.237911
$46$	10.0829	1.48664
$47$	6.83413	0.996860	0.498430	−	0.866930i	$-0.333910\pi$
$47$	6.83413	0.996860	0.498430	+	0.866930i	$0.333910\pi$
$48$	0.841010	0.121389
$49$	5.62408	0.803440
$50$	5.50644	0.778729
$51$	−1.00000	−0.140028
$52$	9.92835	1.37681
$53$	−8.45723	−1.16169	−0.580845	−	0.814014i	$-0.697278\pi$
$53$	−8.45723	−1.16169	−0.580845	+	0.814014i	$0.697278\pi$
$54$	2.24485	0.305486
$55$	4.93184	0.665009
$56$	−8.28998	−1.10780
$57$	5.67115	0.751162
$58$	−2.74631	−0.360608
$59$	−5.17305	−0.673474	−0.336737	−	0.941599i	$-0.609323\pi$
$59$	−5.17305	−0.673474	−0.336737	+	0.941599i	$0.609323\pi$
$60$	4.85069	0.626221
$61$	2.20823	0.282734	0.141367	−	0.989957i	$-0.454850\pi$
$61$	2.20823	0.282734	0.141367	+	0.989957i	$0.454850\pi$
$62$	6.44234	0.818179
$63$	3.55304	0.447641
$64$	−13.0316	−1.62894
$65$	−5.21334	−0.646635
$66$	−6.93706	−0.853893
$67$	9.02616	1.10272	0.551360	−	0.834267i	$-0.314109\pi$
$67$	9.02616	1.10272	0.551360	+	0.834267i	$0.314109\pi$
$68$	3.03936	0.368577
$69$	4.49157	0.540721
$70$	12.7294	1.52146
$71$	1.93460	0.229595	0.114797	−	0.993389i	$-0.463378\pi$
$71$	1.93460	0.229595	0.114797	+	0.993389i	$0.463378\pi$
$72$	−2.33321	−0.274971
$73$	−10.6017	−1.24084	−0.620418	−	0.784271i	$-0.713037\pi$
$73$	−10.6017	−1.24084	−0.620418	+	0.784271i	$0.713037\pi$
$74$	−20.4344	−2.37545
$75$	2.45292	0.283239
$76$	−17.2367	−1.97718
$77$	−10.9796	−1.25124
$78$	7.33302	0.830301
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	1.34222	0.150064
$81$	1.00000	0.111111
$82$	2.12438	0.234599
$83$	2.57393	0.282526	0.141263	−	0.989972i	$-0.454884\pi$
$83$	2.57393	0.282526	0.141263	+	0.989972i	$0.454884\pi$
$84$	−10.7990	−1.17826
$85$	−1.59596	−0.173106
$86$	−25.7126	−2.77266
$87$	−1.22338	−0.131160
$88$	7.21011	0.768600
$89$	−1.73736	−0.184159	−0.0920796	−	0.995752i	$-0.529351\pi$
$89$	−1.73736	−0.184159	−0.0920796	+	0.995752i	$0.529351\pi$
$90$	3.58269	0.377648
$91$	11.6063	1.21667
$92$	−13.6515	−1.42327
$93$	2.86983	0.297588
$94$	−15.3416	−1.58237
$95$	9.05091	0.928604
$96$	−6.55436	−0.668952
$97$	−0.683468	−0.0693956	−0.0346978	−	0.999398i	$-0.511047\pi$
$97$	−0.683468	−0.0693956	−0.0346978	+	0.999398i	$0.511047\pi$
$98$	−12.6252	−1.27534
$99$	−3.09021	−0.310578
$100$	−7.45531	−0.745531
$101$	−3.93243	−0.391291	−0.195646	−	0.980675i	$-0.562680\pi$
$101$	−3.93243	−0.391291	−0.195646	+	0.980675i	$0.562680\pi$
$102$	2.24485	0.222273
$103$	9.99968	0.985297	0.492649	−	0.870228i	$-0.336029\pi$
$103$	9.99968	0.985297	0.492649	+	0.870228i	$0.336029\pi$
$104$	−7.62165	−0.747364
$105$	5.67050	0.553384
$106$	18.9852	1.84401
$107$	−0.187612	−0.0181371	−0.00906855	−	0.999959i	$-0.502887\pi$
$107$	−0.187612	−0.0181371	−0.00906855	+	0.999959i	$0.502887\pi$
$108$	−3.03936	−0.292463
$109$	8.26945	0.792070	0.396035	−	0.918235i	$-0.370386\pi$
$109$	8.26945	0.792070	0.396035	+	0.918235i	$0.370386\pi$
$110$	−11.0713	−1.05560
$111$	−9.10279	−0.863998
$112$	−2.98814	−0.282353
$113$	−4.39850	−0.413776	−0.206888	−	0.978365i	$-0.566334\pi$
$113$	−4.39850	−0.413776	−0.206888	+	0.978365i	$0.566334\pi$
$114$	−12.7309	−1.19236
$115$	7.16835	0.668452
$116$	3.71829	0.345235
$117$	3.26659	0.301997
$118$	11.6127	1.06904
$119$	3.55304	0.325706
$120$	−3.72370	−0.339926
$121$	−1.45061	−0.131874
$122$	−4.95714	−0.448798
$123$	0.946335	0.0853281
$124$	−8.72245	−0.783299
$125$	11.8945	1.06388
$126$	−7.97604	−0.710563
$127$	−8.83571	−0.784043	−0.392021	−	0.919956i	$-0.628224\pi$
$127$	−8.83571	−0.784043	−0.392021	+	0.919956i	$0.628224\pi$
$128$	16.1452	1.42705
$129$	−11.4540	−1.00847
$130$	11.7032	1.02644
$131$	21.9991	1.92207	0.961037	−	0.276421i	$-0.0891483\pi$
$131$	21.9991	1.92207	0.961037	+	0.276421i	$0.0891483\pi$
$132$	9.39226	0.817491
$133$	−20.1498	−1.74721
$134$	−20.2624	−1.75040
$135$	1.59596	0.137358
$136$	−2.33321	−0.200071
$137$	−13.8000	−1.17902	−0.589508	−	0.807762i	$-0.700678\pi$
$137$	−13.8000	−1.17902	−0.589508	+	0.807762i	$0.700678\pi$
$138$	−10.0829	−0.858314
$139$	−7.21939	−0.612341	−0.306170	−	0.951977i	$-0.599048\pi$
$139$	−7.21939	−0.612341	−0.306170	+	0.951977i	$0.599048\pi$
$140$	−17.2347	−1.45660
$141$	−6.83413	−0.575537
$142$	−4.34289	−0.364447
$143$	−10.0945	−0.844140
$144$	−0.841010	−0.0700841
$145$	−1.95246	−0.162143
$146$	23.7993	1.96964
$147$	−5.62408	−0.463866
$148$	27.6666	2.27418
$149$	−1.02846	−0.0842548	−0.0421274	−	0.999112i	$-0.513414\pi$
$149$	−1.02846	−0.0842548	−0.0421274	+	0.999112i	$0.513414\pi$
$150$	−5.50644	−0.449599
$151$	1.14360	0.0930652	0.0465326	−	0.998917i	$-0.485183\pi$
$151$	1.14360	0.0930652	0.0465326	+	0.998917i	$0.485183\pi$
$152$	13.2320	1.07326
$153$	1.00000	0.0808452
$154$	24.6476	1.98616
$155$	4.58013	0.367885
$156$	−9.92835	−0.794904
$157$	0.823120	0.0656922	0.0328461	−	0.999460i	$-0.489543\pi$
$157$	0.823120	0.0656922	0.0328461	+	0.999460i	$0.489543\pi$
$158$	2.24485	0.178591
$159$	8.45723	0.670702
$160$	−10.4605	−0.826974
$161$	−15.9587	−1.25772
$162$	−2.24485	−0.176372
$163$	−4.60012	−0.360309	−0.180155	−	0.983638i	$-0.557660\pi$
$163$	−4.60012	−0.360309	−0.180155	+	0.983638i	$0.557660\pi$
$164$	−2.87625	−0.224598
$165$	−4.93184	−0.383943
$166$	−5.77810	−0.448467
$167$	9.13572	0.706943	0.353471	−	0.935445i	$-0.385001\pi$
$167$	9.13572	0.706943	0.353471	+	0.935445i	$0.385001\pi$
$168$	8.28998	0.639586
$169$	−2.32938	−0.179183
$170$	3.58269	0.274780
$171$	−5.67115	−0.433684
$172$	34.8129	2.65446
$173$	−6.97071	−0.529973	−0.264987	−	0.964252i	$-0.585368\pi$
$173$	−6.97071	−0.529973	−0.264987	+	0.964252i	$0.585368\pi$
$174$	2.74631	0.208197
$175$	−8.71532	−0.658816
$176$	2.59890	0.195899
$177$	5.17305	0.388831
$178$	3.90011	0.292325
$179$	−20.9002	−1.56216	−0.781078	−	0.624434i	$-0.785330\pi$
$179$	−20.9002	−1.56216	−0.781078	+	0.624434i	$0.785330\pi$
$180$	−4.85069	−0.361549
$181$	4.14763	0.308291	0.154146	−	0.988048i	$-0.450738\pi$
$181$	4.14763	0.308291	0.154146	+	0.988048i	$0.450738\pi$
$182$	−26.0545	−1.93129
$183$	−2.20823	−0.163237
$184$	10.4798	0.772579
$185$	−14.5277	−1.06809
$186$	−6.44234	−0.472376
$187$	−3.09021	−0.225978
$188$	20.7714	1.51491
$189$	−3.55304	−0.258445
$190$	−20.3180	−1.47402
$191$	12.4563	0.901305	0.450653	−	0.892699i	$-0.351191\pi$
$191$	12.4563	0.901305	0.450653	+	0.892699i	$0.351191\pi$
$192$	13.0316	0.940471
$193$	9.40701	0.677131	0.338566	−	0.940943i	$-0.390058\pi$
$193$	9.40701	0.677131	0.338566	+	0.940943i	$0.390058\pi$
$194$	1.53428	0.110155
$195$	5.21334	0.373335
$196$	17.0936	1.22097
$197$	24.9944	1.78078	0.890390	−	0.455198i	$-0.150432\pi$
$197$	24.9944	1.78078	0.890390	+	0.455198i	$0.150432\pi$
$198$	6.93706	0.492995
$199$	20.1373	1.42750	0.713749	−	0.700401i	$-0.246996\pi$
$199$	20.1373	1.42750	0.713749	+	0.700401i	$0.246996\pi$
$200$	5.72318	0.404690
$201$	−9.02616	−0.636656
$202$	8.82772	0.621116
$203$	4.34672	0.305080
$204$	−3.03936	−0.212798
$205$	1.51031	0.105485
$206$	−22.4478	−1.56401
$207$	−4.49157	−0.312186
$208$	−2.74724	−0.190487
$209$	17.5250	1.21223
$210$	−12.7294	−0.878414
$211$	3.19125	0.219695	0.109847	−	0.993948i	$-0.464964\pi$
$211$	3.19125	0.219695	0.109847	+	0.993948i	$0.464964\pi$
$212$	−25.7046	−1.76540
$213$	−1.93460	−0.132557
$214$	0.421160	0.0287899
$215$	−18.2802	−1.24670
$216$	2.33321	0.158755
$217$	−10.1966	−0.692191
$218$	−18.5637	−1.25729
$219$	10.6017	0.716397
$220$	14.9896	1.01060
$221$	3.26659	0.219735
$222$	20.4344	1.37147
$223$	−0.563868	−0.0377594	−0.0188797	−	0.999822i	$-0.506010\pi$
$223$	−0.563868	−0.0377594	−0.0188797	+	0.999822i	$0.506010\pi$
$224$	23.2879	1.55599
$225$	−2.45292	−0.163528
$226$	9.87399	0.656808
$227$	0.409740	0.0271954	0.0135977	−	0.999908i	$-0.495672\pi$
$227$	0.409740	0.0271954	0.0135977	+	0.999908i	$0.495672\pi$
$228$	17.2367	1.14153
$229$	−23.1849	−1.53210	−0.766051	−	0.642780i	$-0.777781\pi$
$229$	−23.1849	−1.53210	−0.766051	+	0.642780i	$0.777781\pi$
$230$	−16.0919	−1.06107
$231$	10.9796	0.722407
$232$	−2.85440	−0.187401
$233$	18.6324	1.22065	0.610325	−	0.792151i	$-0.291039\pi$
$233$	18.6324	1.22065	0.610325	+	0.792151i	$0.291039\pi$
$234$	−7.33302	−0.479374
$235$	−10.9070	−0.711493
$236$	−15.7228	−1.02347
$237$	1.00000	0.0649570
$238$	−7.97604	−0.517010
$239$	19.3632	1.25250	0.626251	−	0.779621i	$-0.284588\pi$
$239$	19.3632	1.25250	0.626251	+	0.779621i	$0.284588\pi$
$240$	−1.34222	−0.0866396
$241$	−14.0838	−0.907215	−0.453607	−	0.891202i	$-0.649863\pi$
$241$	−14.0838	−0.907215	−0.453607	+	0.891202i	$0.649863\pi$
$242$	3.25641	0.209330
$243$	−1.00000	−0.0641500
$244$	6.71160	0.429666
$245$	−8.97579	−0.573442
$246$	−2.12438	−0.135446
$247$	−18.5253	−1.17874
$248$	6.69592	0.425191
$249$	−2.57393	−0.163116
$250$	−26.7015	−1.68875
$251$	25.1845	1.58963	0.794817	−	0.606849i	$-0.207567\pi$
$251$	25.1845	1.58963	0.794817	+	0.606849i	$0.207567\pi$
$252$	10.7990	0.680271
$253$	13.8799	0.872621
$254$	19.8349	1.24455
$255$	1.59596	0.0999427
$256$	−10.1804	−0.636278
$257$	18.5165	1.15503	0.577514	−	0.816380i	$-0.304023\pi$
$257$	18.5165	1.15503	0.577514	+	0.816380i	$0.304023\pi$
$258$	25.7126	1.60080
$259$	32.3425	2.00967
$260$	−15.8452	−0.982679
$261$	1.22338	0.0757254
$262$	−49.3848	−3.05100
$263$	−11.4543	−0.706301	−0.353151	−	0.935566i	$-0.614890\pi$
$263$	−11.4543	−0.706301	−0.353151	+	0.935566i	$0.614890\pi$
$264$	−7.21011	−0.443751
$265$	13.4974	0.829137
$266$	45.2333	2.77343
$267$	1.73736	0.106324
$268$	27.4337	1.67578
$269$	1.75022	0.106713	0.0533563	−	0.998576i	$-0.483008\pi$
$269$	1.75022	0.106713	0.0533563	+	0.998576i	$0.483008\pi$
$270$	−3.58269	−0.218035
$271$	9.07247	0.551114	0.275557	−	0.961285i	$-0.411138\pi$
$271$	9.07247	0.551114	0.275557	+	0.961285i	$0.411138\pi$
$272$	−0.841010	−0.0509937
$273$	−11.6063	−0.702447
$274$	30.9790	1.87151
$275$	7.58004	0.457093
$276$	13.6515	0.821724
$277$	−12.9754	−0.779617	−0.389809	−	0.920896i	$-0.627459\pi$
$277$	−12.9754	−0.779617	−0.389809	+	0.920896i	$0.627459\pi$
$278$	16.2065	0.971999
$279$	−2.86983	−0.171812
$280$	13.2305	0.790671
$281$	−22.0524	−1.31554	−0.657768	−	0.753221i	$-0.728499\pi$
$281$	−22.0524	−1.31554	−0.657768	+	0.753221i	$0.728499\pi$
$282$	15.3416	0.913579
$283$	−15.1425	−0.900130	−0.450065	−	0.892996i	$-0.648599\pi$
$283$	−15.1425	−0.900130	−0.450065	+	0.892996i	$0.648599\pi$
$284$	5.87995	0.348911
$285$	−9.05091	−0.536130
$286$	22.6605	1.33995
$287$	−3.36236	−0.198474
$288$	6.55436	0.386219
$289$	1.00000	0.0588235
$290$	4.38299	0.257378
$291$	0.683468	0.0400656
$292$	−32.2224	−1.88567
$293$	−1.97175	−0.115191	−0.0575954	−	0.998340i	$-0.518343\pi$
$293$	−1.97175	−0.115191	−0.0575954	+	0.998340i	$0.518343\pi$
$294$	12.6252	0.736318
$295$	8.25597	0.480681
$296$	−21.2387	−1.23448
$297$	3.09021	0.179312
$298$	2.30874	0.133742
$299$	−14.6721	−0.848511
$300$	7.45531	0.430432
$301$	40.6966	2.34571
$302$	−2.56722	−0.147727
$303$	3.93243	0.225912
$304$	4.76949	0.273549
$305$	−3.52423	−0.201797
$306$	−2.24485	−0.128330
$307$	20.1176	1.14817	0.574087	−	0.818794i	$-0.305357\pi$
$307$	20.1176	1.14817	0.574087	+	0.818794i	$0.305357\pi$
$308$	−33.3710	−1.90149
$309$	−9.99968	−0.568862
$310$	−10.2817	−0.583962
$311$	13.3394	0.756409	0.378205	−	0.925722i	$-0.376542\pi$
$311$	13.3394	0.756409	0.378205	+	0.925722i	$0.376542\pi$
$312$	7.62165	0.431491
$313$	−1.23078	−0.0695677	−0.0347838	−	0.999395i	$-0.511074\pi$
$313$	−1.23078	−0.0695677	−0.0347838	+	0.999395i	$0.511074\pi$
$314$	−1.84778	−0.104276
$315$	−5.67050	−0.319496
$316$	−3.03936	−0.170977
$317$	−5.58121	−0.313472	−0.156736	−	0.987641i	$-0.550097\pi$
$317$	−5.58121	−0.313472	−0.156736	+	0.987641i	$0.550097\pi$
$318$	−18.9852	−1.06464
$319$	−3.78050	−0.211667
$320$	20.7978	1.16263
$321$	0.187612	0.0104715
$322$	35.8250	1.99645
$323$	−5.67115	−0.315551
$324$	3.03936	0.168853
$325$	−8.01269	−0.444464
$326$	10.3266	0.571937
$327$	−8.26945	−0.457302
$328$	2.20800	0.121916
$329$	24.2819	1.33871
$330$	11.0713	0.609452
$331$	4.48749	0.246655	0.123327	−	0.992366i	$-0.460643\pi$
$331$	4.48749	0.246655	0.123327	+	0.992366i	$0.460643\pi$
$332$	7.82311	0.429349
$333$	9.10279	0.498830
$334$	−20.5083	−1.12217
$335$	−14.4054	−0.787049
$336$	2.98814	0.163016
$337$	−2.37934	−0.129611	−0.0648056	−	0.997898i	$-0.520643\pi$
$337$	−2.37934	−0.129611	−0.0648056	+	0.997898i	$0.520643\pi$
$338$	5.22910	0.284426
$339$	4.39850	0.238894
$340$	−4.85069	−0.263066
$341$	8.86838	0.480249
$342$	12.7309	0.688408
$343$	−4.88870	−0.263965
$344$	−26.7247	−1.44090
$345$	−7.16835	−0.385931
$346$	15.6482	0.841253
$347$	−7.43900	−0.399347	−0.199673	−	0.979863i	$-0.563988\pi$
$347$	−7.43900	−0.399347	−0.199673	+	0.979863i	$0.563988\pi$
$348$	−3.71829	−0.199322
$349$	10.3932	0.556337	0.278168	−	0.960532i	$-0.410273\pi$
$349$	10.3932	0.556337	0.278168	+	0.960532i	$0.410273\pi$
$350$	19.5646	1.04577
$351$	−3.26659	−0.174358
$352$	−20.2543	−1.07956
$353$	6.55787	0.349040	0.174520	−	0.984654i	$-0.444163\pi$
$353$	6.55787	0.349040	0.174520	+	0.984654i	$0.444163\pi$
$354$	−11.6127	−0.617210
$355$	−3.08754	−0.163870
$356$	−5.28045	−0.279863
$357$	−3.55304	−0.188047
$358$	46.9179	2.47969
$359$	−9.73058	−0.513560	−0.256780	−	0.966470i	$-0.582662\pi$
$359$	−9.73058	−0.513560	−0.256780	+	0.966470i	$0.582662\pi$
$360$	3.72370	0.196256
$361$	13.1619	0.692733
$362$	−9.31082	−0.489366
$363$	1.45061	0.0761373
$364$	35.2758	1.84895
$365$	16.9199	0.885627
$366$	4.95714	0.259114
$367$	0.723510	0.0377669	0.0188835	−	0.999822i	$-0.493989\pi$
$367$	0.723510	0.0377669	0.0188835	+	0.999822i	$0.493989\pi$
$368$	3.77745	0.196913
$369$	−0.946335	−0.0492642
$370$	32.6124	1.69544
$371$	−30.0488	−1.56006
$372$	8.72245	0.452238
$373$	18.6035	0.963255	0.481628	−	0.876376i	$-0.340046\pi$
$373$	18.6035	0.963255	0.481628	+	0.876376i	$0.340046\pi$
$374$	6.93706	0.358707
$375$	−11.8945	−0.614232
$376$	−15.9455	−0.822324
$377$	3.99629	0.205819
$378$	7.97604	0.410243
$379$	−3.74427	−0.192330	−0.0961652	−	0.995365i	$-0.530658\pi$
$379$	−3.74427	−0.192330	−0.0961652	+	0.995365i	$0.530658\pi$
$380$	27.5090	1.41118
$381$	8.83571	0.452667
$382$	−27.9625	−1.43069
$383$	24.3746	1.24548	0.622742	−	0.782427i	$-0.286019\pi$
$383$	24.3746	1.24548	0.622742	+	0.782427i	$0.286019\pi$
$384$	−16.1452	−0.823905
$385$	17.5230	0.893056
$386$	−21.1173	−1.07484
$387$	11.4540	0.582241
$388$	−2.07731	−0.105459
$389$	25.3252	1.28404	0.642018	−	0.766689i	$-0.278097\pi$
$389$	25.3252	1.28404	0.642018	+	0.766689i	$0.278097\pi$
$390$	−11.7032	−0.592614
$391$	−4.49157	−0.227148
$392$	−13.1222	−0.662769
$393$	−21.9991	−1.10971
$394$	−56.1088	−2.82672
$395$	1.59596	0.0803013
$396$	−9.39226	−0.471979
$397$	29.2987	1.47046	0.735230	−	0.677818i	$-0.237074\pi$
$397$	29.2987	1.47046	0.735230	+	0.677818i	$0.237074\pi$
$398$	−45.2054	−2.26594
$399$	20.1498	1.00875
$400$	2.06293	0.103146
$401$	−5.86852	−0.293060	−0.146530	−	0.989206i	$-0.546810\pi$
$401$	−5.86852	−0.293060	−0.146530	+	0.989206i	$0.546810\pi$
$402$	20.2624	1.01060
$403$	−9.37457	−0.466980
$404$	−11.9521	−0.594637
$405$	−1.59596	−0.0793038
$406$	−9.75774	−0.484268
$407$	−28.1295	−1.39433
$408$	2.33321	0.115511
$409$	−5.34049	−0.264070	−0.132035	−	0.991245i	$-0.542151\pi$
$409$	−5.34049	−0.264070	−0.132035	+	0.991245i	$0.542151\pi$
$410$	−3.39042	−0.167441
$411$	13.8000	0.680706
$412$	30.3926	1.49734
$413$	−18.3801	−0.904424
$414$	10.0829	0.495548
$415$	−4.10789	−0.201648
$416$	21.4104	1.04973
$417$	7.21939	0.353535
$418$	−39.3411	−1.92424
$419$	11.1227	0.543378	0.271689	−	0.962385i	$-0.412418\pi$
$419$	11.1227	0.543378	0.271689	+	0.962385i	$0.412418\pi$
$420$	17.2347	0.840966
$421$	30.9522	1.50852	0.754260	−	0.656576i	$-0.227996\pi$
$421$	30.9522	1.50852	0.754260	+	0.656576i	$0.227996\pi$
$422$	−7.16389	−0.348733
$423$	6.83413	0.332287
$424$	19.7325	0.958294
$425$	−2.45292	−0.118984
$426$	4.34289	0.210414
$427$	7.84591	0.379690
$428$	−0.570219	−0.0275626
$429$	10.0945	0.487365
$430$	41.0362	1.97894
$431$	7.25831	0.349621	0.174810	−	0.984602i	$-0.444069\pi$
$431$	7.25831	0.349621	0.174810	+	0.984602i	$0.444069\pi$
$432$	0.841010	0.0404631
$433$	−36.6106	−1.75939	−0.879697	−	0.475534i	$-0.842255\pi$
$433$	−36.6106	−1.75939	−0.879697	+	0.475534i	$0.842255\pi$
$434$	22.8899	1.09875
$435$	1.95246	0.0936135
$436$	25.1339	1.20369
$437$	25.4724	1.21851
$438$	−23.7993	−1.13717
$439$	13.8507	0.661059	0.330529	−	0.943796i	$-0.392773\pi$
$439$	13.8507	0.661059	0.330529	+	0.943796i	$0.392773\pi$
$440$	−11.5070	−0.548576
$441$	5.62408	0.267813
$442$	−7.33302	−0.348796
$443$	9.70824	0.461252	0.230626	−	0.973042i	$-0.425923\pi$
$443$	9.70824	0.461252	0.230626	+	0.973042i	$0.425923\pi$
$444$	−27.6666	−1.31300
$445$	2.77274	0.131441
$446$	1.26580	0.0599374
$447$	1.02846	0.0486445
$448$	−46.3016	−2.18755
$449$	4.74856	0.224098	0.112049	−	0.993703i	$-0.464259\pi$
$449$	4.74856	0.224098	0.112049	+	0.993703i	$0.464259\pi$
$450$	5.50644	0.259576
$451$	2.92437	0.137703
$452$	−13.3686	−0.628808
$453$	−1.14360	−0.0537312
$454$	−0.919805	−0.0431686
$455$	−18.5232	−0.868381
$456$	−13.2320	−0.619644
$457$	7.81690	0.365659	0.182830	−	0.983145i	$-0.441474\pi$
$457$	7.81690	0.365659	0.182830	+	0.983145i	$0.441474\pi$
$458$	52.0467	2.43198
$459$	−1.00000	−0.0466760
$460$	21.7872	1.01583
$461$	18.4009	0.857015	0.428508	−	0.903538i	$-0.359039\pi$
$461$	18.4009	0.857015	0.428508	+	0.903538i	$0.359039\pi$
$462$	−24.6476	−1.14671
$463$	24.2133	1.12529	0.562645	−	0.826699i	$-0.309784\pi$
$463$	24.2133	1.12529	0.562645	+	0.826699i	$0.309784\pi$
$464$	−1.02887	−0.0477643
$465$	−4.58013	−0.212398
$466$	−41.8270	−1.93760
$467$	1.85394	0.0857903	0.0428951	−	0.999080i	$-0.486342\pi$
$467$	1.85394	0.0857903	0.0428951	+	0.999080i	$0.486342\pi$
$468$	9.92835	0.458938
$469$	32.0703	1.48087
$470$	24.4845	1.12939
$471$	−0.823120	−0.0379274
$472$	12.0698	0.555559
$473$	−35.3954	−1.62748
$474$	−2.24485	−0.103109
$475$	13.9109	0.638275
$476$	10.7990	0.494970
$477$	−8.45723	−0.387230
$478$	−43.4676	−1.98816
$479$	−14.2655	−0.651805	−0.325903	−	0.945403i	$-0.605668\pi$
$479$	−14.2655	−0.651805	−0.325903	+	0.945403i	$0.605668\pi$
$480$	10.4605	0.477453
$481$	29.7351	1.35580
$482$	31.6160	1.44007
$483$	15.9587	0.726147
$484$	−4.40893	−0.200406
$485$	1.09079	0.0495300
$486$	2.24485	0.101829
$487$	14.1820	0.642650	0.321325	−	0.946969i	$-0.395872\pi$
$487$	14.1820	0.642650	0.321325	+	0.946969i	$0.395872\pi$
$488$	−5.15226	−0.233232
$489$	4.60012	0.208025
$490$	20.1493	0.910254
$491$	3.98355	0.179775	0.0898875	−	0.995952i	$-0.471349\pi$
$491$	3.98355	0.179775	0.0898875	+	0.995952i	$0.471349\pi$
$492$	2.87625	0.129671
$493$	1.22338	0.0550983
$494$	41.5866	1.87107
$495$	4.93184	0.221670
$496$	2.41356	0.108372
$497$	6.87371	0.308328
$498$	5.77810	0.258923
$499$	16.6359	0.744725	0.372362	−	0.928087i	$-0.378548\pi$
$499$	16.6359	0.744725	0.372362	+	0.928087i	$0.378548\pi$
$500$	36.1518	1.61676
$501$	−9.13572	−0.408154
$502$	−56.5356	−2.52331
$503$	26.8690	1.19803	0.599014	−	0.800739i	$-0.295560\pi$
$503$	26.8690	1.19803	0.599014	+	0.800739i	$0.295560\pi$
$504$	−8.28998	−0.369265
$505$	6.27599	0.279278
$506$	−31.1583	−1.38515
$507$	2.32938	0.103451
$508$	−26.8549	−1.19149
$509$	−11.4169	−0.506046	−0.253023	−	0.967460i	$-0.581425\pi$
$509$	−11.4169	−0.506046	−0.253023	+	0.967460i	$0.581425\pi$
$510$	−3.58269	−0.158644
$511$	−37.6683	−1.66635
$512$	−9.43679	−0.417051
$513$	5.67115	0.250387
$514$	−41.5669	−1.83344
$515$	−15.9591	−0.703240
$516$	−34.8129	−1.53255
$517$	−21.1189	−0.928807
$518$	−72.6042	−3.19005
$519$	6.97071	0.305980
$520$	12.1638	0.533419
$521$	39.0608	1.71129	0.855643	−	0.517566i	$-0.173162\pi$
$521$	39.0608	1.71129	0.855643	+	0.517566i	$0.173162\pi$
$522$	−2.74631	−0.120203
$523$	−26.0621	−1.13961	−0.569807	−	0.821779i	$-0.692982\pi$
$523$	−26.0621	−1.13961	−0.569807	+	0.821779i	$0.692982\pi$
$524$	66.8633	2.92094
$525$	8.71532	0.380368
$526$	25.7132	1.12115
$527$	−2.86983	−0.125012
$528$	−2.59890	−0.113102
$529$	−2.82580	−0.122861
$530$	−30.2996	−1.31613
$531$	−5.17305	−0.224491
$532$	−61.2425	−2.65520
$533$	−3.09129	−0.133899
$534$	−3.90011	−0.168774
$535$	0.299420	0.0129451
$536$	−21.0599	−0.909650
$537$	20.9002	0.901911
$538$	−3.92898	−0.169390
$539$	−17.3796	−0.748591
$540$	4.85069	0.208740
$541$	26.4468	1.13704	0.568519	−	0.822670i	$-0.307517\pi$
$541$	26.4468	1.13704	0.568519	+	0.822670i	$0.307517\pi$
$542$	−20.3664	−0.874810
$543$	−4.14763	−0.177992
$544$	6.55436	0.281016
$545$	−13.1977	−0.565327
$546$	26.0545	1.11503
$547$	21.1355	0.903691	0.451845	−	0.892096i	$-0.350766\pi$
$547$	21.1355	0.903691	0.451845	+	0.892096i	$0.350766\pi$
$548$	−41.9433	−1.79173
$549$	2.20823	0.0942448
$550$	−17.0161	−0.725567
$551$	−6.93797	−0.295568
$552$	−10.4798	−0.446049
$553$	−3.55304	−0.151091
$554$	29.1279	1.23753
$555$	14.5277	0.616665
$556$	−21.9423	−0.930562
$557$	−3.11002	−0.131776	−0.0658879	−	0.997827i	$-0.520988\pi$
$557$	−3.11002	−0.131776	−0.0658879	+	0.997827i	$0.520988\pi$
$558$	6.44234	0.272726
$559$	37.4157	1.58251
$560$	4.76894	0.201525
$561$	3.09021	0.130469
$562$	49.5044	2.08822
$563$	−26.9509	−1.13584	−0.567922	−	0.823083i	$-0.692252\pi$
$563$	−26.9509	−1.13584	−0.567922	+	0.823083i	$0.692252\pi$
$564$	−20.7714	−0.874633
$565$	7.01982	0.295326
$566$	33.9927	1.42882
$567$	3.55304	0.149214
$568$	−4.51383	−0.189396
$569$	20.9032	0.876309	0.438154	−	0.898900i	$-0.355632\pi$
$569$	20.9032	0.876309	0.438154	+	0.898900i	$0.355632\pi$
$570$	20.3180	0.851025
$571$	35.9899	1.50613	0.753066	−	0.657945i	$-0.228574\pi$
$571$	35.9899	1.50613	0.753066	+	0.657945i	$0.228574\pi$
$572$	−30.6807	−1.28282
$573$	−12.4563	−0.520369
$574$	7.54801	0.315048
$575$	11.0175	0.459460
$576$	−13.0316	−0.542981
$577$	14.9377	0.621866	0.310933	−	0.950432i	$-0.399359\pi$
$577$	14.9377	0.621866	0.310933	+	0.950432i	$0.399359\pi$
$578$	−2.24485	−0.0933735
$579$	−9.40701	−0.390942
$580$	−5.93424	−0.246406
$581$	9.14528	0.379410
$582$	−1.53428	−0.0635981
$583$	26.1346	1.08238
$584$	24.7360	1.02358
$585$	−5.21334	−0.215545
$586$	4.42628	0.182848
$587$	17.1716	0.708749	0.354374	−	0.935104i	$-0.384694\pi$
$587$	17.1716	0.708749	0.354374	+	0.935104i	$0.384694\pi$
$588$	−17.0936	−0.704928
$589$	16.2752	0.670609
$590$	−18.5334	−0.763010
$591$	−24.9944	−1.02813
$592$	−7.65553	−0.314640
$593$	5.77762	0.237258	0.118629	−	0.992939i	$-0.462150\pi$
$593$	5.77762	0.237258	0.118629	+	0.992939i	$0.462150\pi$
$594$	−6.93706	−0.284631
$595$	−5.67050	−0.232468
$596$	−3.12586	−0.128040
$597$	−20.1373	−0.824167
$598$	32.9367	1.34688
$599$	43.4218	1.77417	0.887083	−	0.461610i	$-0.152728\pi$
$599$	43.4218	1.77417	0.887083	+	0.461610i	$0.152728\pi$
$600$	−5.72318	−0.233648
$601$	−17.0285	−0.694607	−0.347303	−	0.937753i	$-0.612903\pi$
$601$	−17.0285	−0.694607	−0.347303	+	0.937753i	$0.612903\pi$
$602$	−91.3579	−3.72347
$603$	9.02616	0.367573
$604$	3.47582	0.141429
$605$	2.31511	0.0941227
$606$	−8.82772	−0.358602
$607$	−35.8971	−1.45702	−0.728509	−	0.685036i	$-0.759787\pi$
$607$	−35.8971	−1.45702	−0.728509	+	0.685036i	$0.759787\pi$
$608$	−37.1708	−1.50747
$609$	−4.34672	−0.176138
$610$	7.91139	0.320323
$611$	22.3243	0.903145
$612$	3.03936	0.122859
$613$	−26.4636	−1.06886	−0.534428	−	0.845214i	$-0.679473\pi$
$613$	−26.4636	−1.06886	−0.534428	+	0.845214i	$0.679473\pi$
$614$	−45.1611	−1.82255
$615$	−1.51031	−0.0609016
$616$	25.6178	1.03217
$617$	11.0040	0.443004	0.221502	−	0.975160i	$-0.428904\pi$
$617$	11.0040	0.443004	0.221502	+	0.975160i	$0.428904\pi$
$618$	22.4478	0.902983
$619$	28.2760	1.13651	0.568255	−	0.822853i	$-0.307619\pi$
$619$	28.2760	1.13651	0.568255	+	0.822853i	$0.307619\pi$
$620$	13.9207	0.559067
$621$	4.49157	0.180240
$622$	−29.9450	−1.20069
$623$	−6.17289	−0.247312
$624$	2.74724	0.109977
$625$	−6.71858	−0.268743
$626$	2.76291	0.110428
$627$	−17.5250	−0.699882
$628$	2.50176	0.0998311
$629$	9.10279	0.362952
$630$	12.7294	0.507152
$631$	−43.3766	−1.72679	−0.863397	−	0.504526i	$-0.831667\pi$
$631$	−43.3766	−1.72679	−0.863397	+	0.504526i	$0.831667\pi$
$632$	2.33321	0.0928101
$633$	−3.19125	−0.126841
$634$	12.5290	0.497590
$635$	14.1014	0.559598
$636$	25.7046	1.01925
$637$	18.3716	0.727908
$638$	8.48667	0.335990
$639$	1.93460	0.0765316
$640$	−25.7670	−1.01853
$641$	−19.0341	−0.751802	−0.375901	−	0.926660i	$-0.622667\pi$
$641$	−19.0341	−0.751802	−0.375901	+	0.926660i	$0.622667\pi$
$642$	−0.421160	−0.0166219
$643$	26.7910	1.05654	0.528268	−	0.849078i	$-0.322842\pi$
$643$	26.7910	1.05654	0.528268	+	0.849078i	$0.322842\pi$
$644$	−48.5043	−1.91134
$645$	18.2802	0.719780
$646$	12.7309	0.500890
$647$	−6.25742	−0.246005	−0.123002	−	0.992406i	$-0.539252\pi$
$647$	−6.25742	−0.246005	−0.123002	+	0.992406i	$0.539252\pi$
$648$	−2.33321	−0.0916571
$649$	15.9858	0.627498
$650$	17.9873	0.705520
$651$	10.1966	0.399637
$652$	−13.9814	−0.547555
$653$	−1.90344	−0.0744872	−0.0372436	−	0.999306i	$-0.511858\pi$
$653$	−1.90344	−0.0744872	−0.0372436	+	0.999306i	$0.511858\pi$
$654$	18.5637	0.725898
$655$	−35.1097	−1.37185
$656$	0.795876	0.0310738
$657$	−10.6017	−0.413612
$658$	−54.5093	−2.12499
$659$	−41.5944	−1.62029	−0.810144	−	0.586232i	$-0.800611\pi$
$659$	−41.5944	−1.62029	−0.810144	+	0.586232i	$0.800611\pi$
$660$	−14.9896	−0.583471
$661$	39.1146	1.52138	0.760691	−	0.649114i	$-0.224860\pi$
$661$	39.1146	1.52138	0.760691	+	0.649114i	$0.224860\pi$
$662$	−10.0738	−0.391528
$663$	−3.26659	−0.126864
$664$	−6.00553	−0.233060
$665$	32.1582	1.24704
$666$	−20.4344	−0.791817
$667$	−5.49490	−0.212763
$668$	27.7667	1.07433
$669$	0.563868	0.0218004
$670$	32.3379	1.24932
$671$	−6.82388	−0.263433
$672$	−23.2879	−0.898350
$673$	4.42302	0.170495	0.0852474	−	0.996360i	$-0.472832\pi$
$673$	4.42302	0.170495	0.0852474	+	0.996360i	$0.472832\pi$
$674$	5.34128	0.205738
$675$	2.45292	0.0944129
$676$	−7.07981	−0.272300
$677$	35.8066	1.37616	0.688081	−	0.725634i	$-0.258454\pi$
$677$	35.8066	1.37616	0.688081	+	0.725634i	$0.258454\pi$
$678$	−9.87399	−0.379208
$679$	−2.42839	−0.0931929
$680$	3.72370	0.142798
$681$	−0.409740	−0.0157013
$682$	−19.9082	−0.762324
$683$	19.9171	0.762105	0.381053	−	0.924553i	$-0.375562\pi$
$683$	19.9171	0.762105	0.381053	+	0.924553i	$0.375562\pi$
$684$	−17.2367	−0.659060
$685$	22.0243	0.841504
$686$	10.9744	0.419005
$687$	23.1849	0.884559
$688$	−9.63295	−0.367253
$689$	−27.6263	−1.05248
$690$	16.0919	0.612608
$691$	18.9879	0.722333	0.361167	−	0.932501i	$-0.382379\pi$
$691$	18.9879	0.722333	0.361167	+	0.932501i	$0.382379\pi$
$692$	−21.1865	−0.805390
$693$	−10.9796	−0.417082
$694$	16.6995	0.633903
$695$	11.5218	0.437048
$696$	2.85440	0.108196
$697$	−0.946335	−0.0358450
$698$	−23.3312	−0.883101
$699$	−18.6324	−0.704742
$700$	−26.4890	−1.00119
$701$	17.4527	0.659178	0.329589	−	0.944124i	$-0.393090\pi$
$701$	17.4527	0.659178	0.329589	+	0.944124i	$0.393090\pi$
$702$	7.33302	0.276767
$703$	−51.6233	−1.94701
$704$	40.2702	1.51774
$705$	10.9070	0.410780
$706$	−14.7215	−0.554049
$707$	−13.9721	−0.525474
$708$	15.7228	0.590898
$709$	−21.4402	−0.805202	−0.402601	−	0.915376i	$-0.631894\pi$
$709$	−21.4402	−0.805202	−0.402601	+	0.915376i	$0.631894\pi$
$710$	6.93107	0.260118
$711$	−1.00000	−0.0375029
$712$	4.05361	0.151916
$713$	12.8900	0.482736
$714$	7.97604	0.298496
$715$	16.1103	0.602492
$716$	−63.5233	−2.37398
$717$	−19.3632	−0.723133
$718$	21.8437	0.815200
$719$	−18.1628	−0.677357	−0.338679	−	0.940902i	$-0.609980\pi$
$719$	−18.1628	−0.677357	−0.338679	+	0.940902i	$0.609980\pi$
$720$	1.34222	0.0500214
$721$	35.5292	1.32318
$722$	−29.5466	−1.09961
$723$	14.0838	0.523781
$724$	12.6061	0.468504
$725$	−3.00086	−0.111449
$726$	−3.25641	−0.120857
$727$	15.9497	0.591542	0.295771	−	0.955259i	$-0.404423\pi$
$727$	15.9497	0.591542	0.295771	+	0.955259i	$0.404423\pi$
$728$	−27.0800	−1.00365
$729$	1.00000	0.0370370
$730$	−37.9826	−1.40580
$731$	11.4540	0.423643
$732$	−6.71160	−0.248068
$733$	49.0261	1.81082	0.905410	−	0.424538i	$-0.139564\pi$
$733$	49.0261	1.81082	0.905410	+	0.424538i	$0.139564\pi$
$734$	−1.62417	−0.0599493
$735$	8.97579	0.331077
$736$	−29.4394	−1.08515
$737$	−27.8927	−1.02744
$738$	2.12438	0.0781995
$739$	−4.53539	−0.166837	−0.0834185	−	0.996515i	$-0.526584\pi$
$739$	−4.53539	−0.166837	−0.0834185	+	0.996515i	$0.526584\pi$
$740$	−44.1548	−1.62316
$741$	18.5253	0.680545
$742$	67.4552	2.47636
$743$	18.8960	0.693227	0.346613	−	0.938008i	$-0.387332\pi$
$743$	18.8960	0.693227	0.346613	+	0.938008i	$0.387332\pi$
$744$	−6.69592	−0.245484
$745$	1.64138	0.0601355
$746$	−41.7622	−1.52902
$747$	2.57393	0.0941753
$748$	−9.39226	−0.343415
$749$	−0.666591	−0.0243567
$750$	26.7015	0.975000
$751$	11.6294	0.424364	0.212182	−	0.977230i	$-0.431943\pi$
$751$	11.6294	0.424364	0.212182	+	0.977230i	$0.431943\pi$
$752$	−5.74757	−0.209592
$753$	−25.1845	−0.917776
$754$	−8.97107	−0.326707
$755$	−1.82514	−0.0664237
$756$	−10.7990	−0.392754
$757$	27.4589	0.998010	0.499005	−	0.866599i	$-0.333699\pi$
$757$	27.4589	0.998010	0.499005	+	0.866599i	$0.333699\pi$
$758$	8.40533	0.305296
$759$	−13.8799	−0.503808
$760$	−21.1177	−0.766019
$761$	−38.7512	−1.40473	−0.702365	−	0.711817i	$-0.747873\pi$
$761$	−38.7512	−1.40473	−0.702365	+	0.711817i	$0.747873\pi$
$762$	−19.8349	−0.718541
$763$	29.3817	1.06369
$764$	37.8591	1.36970
$765$	−1.59596	−0.0577020
$766$	−54.7173	−1.97702
$767$	−16.8983	−0.610161
$768$	10.1804	0.367355
$769$	−28.4135	−1.02462	−0.512308	−	0.858802i	$-0.671210\pi$
$769$	−28.4135	−1.02462	−0.512308	+	0.858802i	$0.671210\pi$
$770$	−39.3366	−1.41759
$771$	−18.5165	−0.666856
$772$	28.5913	1.02902
$773$	−28.3072	−1.01814	−0.509070	−	0.860725i	$-0.670011\pi$
$773$	−28.3072	−1.01814	−0.509070	+	0.860725i	$0.670011\pi$
$774$	−25.7126	−0.924221
$775$	7.03947	0.252865
$776$	1.59467	0.0572455
$777$	−32.3425	−1.16028
$778$	−56.8512	−2.03821
$779$	5.36680	0.192286
$780$	15.8452	0.567350
$781$	−5.97832	−0.213921
$782$	10.0829	0.360564
$783$	−1.22338	−0.0437201
$784$	−4.72990	−0.168925
$785$	−1.31367	−0.0468867
$786$	49.3848	1.76150
$787$	−22.3941	−0.798263	−0.399132	−	0.916894i	$-0.630688\pi$
$787$	−22.3941	−0.798263	−0.399132	+	0.916894i	$0.630688\pi$
$788$	75.9671	2.70622
$789$	11.4543	0.407783
$790$	−3.58269	−0.127466
$791$	−15.6281	−0.555670
$792$	7.21011	0.256200
$793$	7.21337	0.256154
$794$	−65.7713	−2.33413
$795$	−13.4974	−0.478702
$796$	61.2046	2.16934
$797$	51.0996	1.81004	0.905020	−	0.425369i	$-0.139856\pi$
$797$	51.0996	1.81004	0.905020	+	0.425369i	$0.139856\pi$
$798$	−45.2333	−1.60124
$799$	6.83413	0.241774
$800$	−16.0773	−0.568419
$801$	−1.73736	−0.0613864
$802$	13.1740	0.465189
$803$	32.7615	1.15613
$804$	−27.4337	−0.967513
$805$	25.4694	0.897679
$806$	21.0445	0.741261
$807$	−1.75022	−0.0616106
$808$	9.17518	0.322782
$809$	22.1199	0.777696	0.388848	−	0.921302i	$-0.372873\pi$
$809$	22.1199	0.777696	0.388848	+	0.921302i	$0.372873\pi$
$810$	3.58269	0.125883
$811$	−3.78196	−0.132803	−0.0664014	−	0.997793i	$-0.521152\pi$
$811$	−3.78196	−0.132803	−0.0664014	+	0.997793i	$0.521152\pi$
$812$	13.2112	0.463624
$813$	−9.07247	−0.318186
$814$	63.1466	2.21329
$815$	7.34159	0.257165
$816$	0.841010	0.0294412
$817$	−64.9575	−2.27258
$818$	11.9886	0.419172
$819$	11.6063	0.405558
$820$	4.59038	0.160303
$821$	−32.1079	−1.12057	−0.560287	−	0.828298i	$-0.689309\pi$
$821$	−32.1079	−1.12057	−0.560287	+	0.828298i	$0.689309\pi$
$822$	−30.9790	−1.08052
$823$	−4.08193	−0.142287	−0.0711435	−	0.997466i	$-0.522665\pi$
$823$	−4.08193	−0.142287	−0.0711435	+	0.997466i	$0.522665\pi$
$824$	−23.3313	−0.812786
$825$	−7.58004	−0.263903
$826$	41.2605	1.43564
$827$	−10.7240	−0.372909	−0.186454	−	0.982464i	$-0.559700\pi$
$827$	−10.7240	−0.372909	−0.186454	+	0.982464i	$0.559700\pi$
$828$	−13.6515	−0.474422
$829$	5.57250	0.193541	0.0967704	−	0.995307i	$-0.469149\pi$
$829$	5.57250	0.193541	0.0967704	+	0.995307i	$0.469149\pi$
$830$	9.22160	0.320086
$831$	12.9754	0.450112
$832$	−42.5688	−1.47581
$833$	5.62408	0.194863
$834$	−16.2065	−0.561184
$835$	−14.5802	−0.504569
$836$	53.2649	1.84220
$837$	2.86983	0.0991959
$838$	−24.9688	−0.862531
$839$	24.8708	0.858637	0.429318	−	0.903153i	$-0.358754\pi$
$839$	24.8708	0.858637	0.429318	+	0.903153i	$0.358754\pi$
$840$	−13.2305	−0.456494
$841$	−27.5033	−0.948391
$842$	−69.4832	−2.39455
$843$	22.0524	0.759525
$844$	9.69937	0.333866
$845$	3.71758	0.127889
$846$	−15.3416	−0.527455
$847$	−5.15408	−0.177096
$848$	7.11261	0.244248
$849$	15.1425	0.519690
$850$	5.50644	0.188869
$851$	−40.8858	−1.40155
$852$	−5.87995	−0.201444
$853$	35.0060	1.19858	0.599292	−	0.800531i	$-0.295449\pi$
$853$	35.0060	1.19858	0.599292	+	0.800531i	$0.295449\pi$
$854$	−17.6129	−0.602701
$855$	9.05091	0.309535
$856$	0.437737	0.0149616
$857$	−33.2883	−1.13711	−0.568553	−	0.822646i	$-0.692497\pi$
$857$	−33.2883	−1.13711	−0.568553	+	0.822646i	$0.692497\pi$
$858$	−22.6605	−0.773618
$859$	32.0201	1.09251	0.546257	−	0.837618i	$-0.316052\pi$
$859$	32.0201	1.09251	0.546257	+	0.837618i	$0.316052\pi$
$860$	−55.5600	−1.89458
$861$	3.36236	0.114589
$862$	−16.2938	−0.554970
$863$	−5.98537	−0.203744	−0.101872	−	0.994797i	$-0.532483\pi$
$863$	−5.98537	−0.203744	−0.101872	+	0.994797i	$0.532483\pi$
$864$	−6.55436	−0.222984
$865$	11.1250	0.378260
$866$	82.1855	2.79278
$867$	−1.00000	−0.0339618
$868$	−30.9912	−1.05191
$869$	3.09021	0.104828
$870$	−4.38299	−0.148597
$871$	29.4848	0.999053
$872$	−19.2944	−0.653390
$873$	−0.683468	−0.0231319
$874$	−57.1817	−1.93420
$875$	42.2618	1.42871
$876$	32.2224	1.08869
$877$	14.9886	0.506129	0.253064	−	0.967449i	$-0.418562\pi$
$877$	14.9886	0.506129	0.253064	+	0.967449i	$0.418562\pi$
$878$	−31.0928	−1.04933
$879$	1.97175	0.0665054
$880$	−4.14773	−0.139820
$881$	17.5827	0.592376	0.296188	−	0.955130i	$-0.404284\pi$
$881$	17.5827	0.592376	0.296188	+	0.955130i	$0.404284\pi$
$882$	−12.6252	−0.425113
$883$	−48.7987	−1.64221	−0.821104	−	0.570778i	$-0.806642\pi$
$883$	−48.7987	−1.64221	−0.821104	+	0.570778i	$0.806642\pi$
$884$	9.92835	0.333927
$885$	−8.25597	−0.277522
$886$	−21.7936	−0.732169
$887$	16.7636	0.562865	0.281433	−	0.959581i	$-0.409190\pi$
$887$	16.7636	0.562865	0.281433	+	0.959581i	$0.409190\pi$
$888$	21.2387	0.712725
$889$	−31.3936	−1.05291
$890$	−6.22440	−0.208642
$891$	−3.09021	−0.103526
$892$	−1.71380	−0.0573822
$893$	−38.7574	−1.29697
$894$	−2.30874	−0.0772159
$895$	33.3559	1.11496
$896$	57.3645	1.91641
$897$	14.6721	0.489888
$898$	−10.6598	−0.355723
$899$	−3.51090	−0.117095
$900$	−7.45531	−0.248510
$901$	−8.45723	−0.281751
$902$	−6.56478	−0.218583
$903$	−40.6966	−1.35430
$904$	10.2626	0.341330
$905$	−6.61944	−0.220038
$906$	2.56722	0.0852902
$907$	−22.3875	−0.743366	−0.371683	−	0.928360i	$-0.621219\pi$
$907$	−22.3875	−0.743366	−0.371683	+	0.928360i	$0.621219\pi$
$908$	1.24535	0.0413283
$909$	−3.93243	−0.130430
$910$	41.5818	1.37842
$911$	41.7794	1.38421	0.692107	−	0.721795i	$-0.256683\pi$
$911$	41.7794	1.38421	0.692107	+	0.721795i	$0.256683\pi$
$912$	−4.76949	−0.157934
$913$	−7.95399	−0.263239
$914$	−17.5478	−0.580429
$915$	3.52423	0.116508
$916$	−70.4673	−2.32830
$917$	78.1638	2.58120
$918$	2.24485	0.0740912
$919$	7.05591	0.232753	0.116377	−	0.993205i	$-0.462872\pi$
$919$	7.05591	0.232753	0.116377	+	0.993205i	$0.462872\pi$
$920$	−16.7253	−0.551416
$921$	−20.1176	−0.662899
$922$	−41.3073	−1.36038
$923$	6.31955	0.208011
$924$	33.3710	1.09783
$925$	−22.3284	−0.734154
$926$	−54.3554	−1.78623
$927$	9.99968	0.328432
$928$	8.01848	0.263220
$929$	−17.5587	−0.576084	−0.288042	−	0.957618i	$-0.593004\pi$
$929$	−17.5587	−0.576084	−0.288042	+	0.957618i	$0.593004\pi$
$930$	10.2817	0.337150
$931$	−31.8950	−1.04532
$932$	56.6306	1.85500
$933$	−13.3394	−0.436713
$934$	−4.16183	−0.136179
$935$	4.93184	0.161288
$936$	−7.62165	−0.249121
$937$	−51.5916	−1.68542	−0.842712	−	0.538365i	$-0.819042\pi$
$937$	−51.5916	−1.68542	−0.842712	+	0.538365i	$0.819042\pi$
$938$	−71.9930	−2.35066
$939$	1.23078	0.0401649
$940$	−33.1502	−1.08124
$941$	−49.5932	−1.61669	−0.808346	−	0.588707i	$-0.799637\pi$
$941$	−49.5932	−1.61669	−0.808346	+	0.588707i	$0.799637\pi$
$942$	1.84778	0.0602040
$943$	4.25053	0.138416
$944$	4.35059	0.141600
$945$	5.67050	0.184461
$946$	79.4573	2.58338
$947$	−17.9077	−0.581924	−0.290962	−	0.956735i	$-0.593975\pi$
$947$	−17.9077	−0.581924	−0.290962	+	0.956735i	$0.593975\pi$
$948$	3.03936	0.0987138
$949$	−34.6315	−1.12418
$950$	−31.2279	−1.01317
$951$	5.58121	0.180983
$952$	−8.28998	−0.268680
$953$	56.4882	1.82983	0.914917	−	0.403643i	$-0.132256\pi$
$953$	56.4882	1.82983	0.914917	+	0.403643i	$0.132256\pi$
$954$	18.9852	0.614669
$955$	−19.8797	−0.643292
$956$	58.8518	1.90340
$957$	3.78050	0.122206
$958$	32.0238	1.03464
$959$	−49.0321	−1.58333
$960$	−20.7978	−0.671246
$961$	−22.7641	−0.734325
$962$	−66.7509	−2.15213
$963$	−0.187612	−0.00604570
$964$	−42.8056	−1.37868
$965$	−15.0132	−0.483292
$966$	−35.8250	−1.15265
$967$	−12.6612	−0.407158	−0.203579	−	0.979058i	$-0.565257\pi$
$967$	−12.6612	−0.407158	−0.203579	+	0.979058i	$0.565257\pi$
$968$	3.38458	0.108785
$969$	5.67115	0.182184
$970$	−2.44865	−0.0786215
$971$	37.9931	1.21926	0.609628	−	0.792688i	$-0.291319\pi$
$971$	37.9931	1.21926	0.609628	+	0.792688i	$0.291319\pi$
$972$	−3.03936	−0.0974875
$973$	−25.6508	−0.822326
$974$	−31.8366	−1.02011
$975$	8.01269	0.256611
$976$	−1.85714	−0.0594456
$977$	−21.9977	−0.703767	−0.351884	−	0.936044i	$-0.614459\pi$
$977$	−21.9977	−0.703767	−0.351884	+	0.936044i	$0.614459\pi$
$978$	−10.3266	−0.330208
$979$	5.36879	0.171587
$980$	−27.2807	−0.871449
$981$	8.26945	0.264023
$982$	−8.94247	−0.285366
$983$	33.6771	1.07413	0.537067	−	0.843540i	$-0.319532\pi$
$983$	33.6771	1.07413	0.537067	+	0.843540i	$0.319532\pi$
$984$	−2.20800	−0.0703884
$985$	−39.8901	−1.27100
$986$	−2.74631	−0.0874603
$987$	−24.2819	−0.772902
$988$	−56.3051	−1.79131
$989$	−51.4466	−1.63591
$990$	−11.0713	−0.351868
$991$	−29.7740	−0.945804	−0.472902	−	0.881115i	$-0.656793\pi$
$991$	−29.7740	−0.945804	−0.472902	+	0.881115i	$0.656793\pi$
$992$	−18.8099	−0.597215
$993$	−4.48749	−0.142406
$994$	−15.4305	−0.489424
$995$	−32.1383	−1.01885
$996$	−7.82311	−0.247885
$997$	−14.3038	−0.453005	−0.226503	−	0.974011i	$-0.572729\pi$
$997$	−14.3038	−0.453005	−0.226503	+	0.974011i	$0.572729\pi$
$998$	−37.3451	−1.18214
$999$	−9.10279	−0.287999

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4029.2.a.i.1.4	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4029.2.a.i.1.4	✓	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)

\(f(q)\)	\(=\)	\(q-2.24485 q^{2} -1.00000 q^{3} +3.03936 q^{4} -1.59596 q^{5} +2.24485 q^{6} +3.55304 q^{7} -2.33321 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.24485 q^{2} -1.00000 q^{3} +3.03936 q^{4} -1.59596 q^{5} +2.24485 q^{6} +3.55304 q^{7} -2.33321 q^{8} +1.00000 q^{9} +3.58269 q^{10} -3.09021 q^{11} -3.03936 q^{12} +3.26659 q^{13} -7.97604 q^{14} +1.59596 q^{15} -0.841010 q^{16} +1.00000 q^{17} -2.24485 q^{18} -5.67115 q^{19} -4.85069 q^{20} -3.55304 q^{21} +6.93706 q^{22} -4.49157 q^{23} +2.33321 q^{24} -2.45292 q^{25} -7.33302 q^{26} -1.00000 q^{27} +10.7990 q^{28} +1.22338 q^{29} -3.58269 q^{30} -2.86983 q^{31} +6.55436 q^{32} +3.09021 q^{33} -2.24485 q^{34} -5.67050 q^{35} +3.03936 q^{36} +9.10279 q^{37} +12.7309 q^{38} -3.26659 q^{39} +3.72370 q^{40} -0.946335 q^{41} +7.97604 q^{42} +11.4540 q^{43} -9.39226 q^{44} -1.59596 q^{45} +10.0829 q^{46} +6.83413 q^{47} +0.841010 q^{48} +5.62408 q^{49} +5.50644 q^{50} -1.00000 q^{51} +9.92835 q^{52} -8.45723 q^{53} +2.24485 q^{54} +4.93184 q^{55} -8.28998 q^{56} +5.67115 q^{57} -2.74631 q^{58} -5.17305 q^{59} +4.85069 q^{60} +2.20823 q^{61} +6.44234 q^{62} +3.55304 q^{63} -13.0316 q^{64} -5.21334 q^{65} -6.93706 q^{66} +9.02616 q^{67} +3.03936 q^{68} +4.49157 q^{69} +12.7294 q^{70} +1.93460 q^{71} -2.33321 q^{72} -10.6017 q^{73} -20.4344 q^{74} +2.45292 q^{75} -17.2367 q^{76} -10.9796 q^{77} +7.33302 q^{78} -1.00000 q^{79} +1.34222 q^{80} +1.00000 q^{81} +2.12438 q^{82} +2.57393 q^{83} -10.7990 q^{84} -1.59596 q^{85} -25.7126 q^{86} -1.22338 q^{87} +7.21011 q^{88} -1.73736 q^{89} +3.58269 q^{90} +11.6063 q^{91} -13.6515 q^{92} +2.86983 q^{93} -15.3416 q^{94} +9.05091 q^{95} -6.55436 q^{96} -0.683468 q^{97} -12.6252 q^{98} -3.09021 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 2 q^{2} - 25 q^{3} + 26 q^{4} - 2 q^{5} + 2 q^{6} + 12 q^{7} + 25 q^{9}+O(q^{10}) \) 25 * q - 2 * q^2 - 25 * q^3 + 26 * q^4 - 2 * q^5 + 2 * q^6 + 12 * q^7 + 25 * q^9	\( 25 q - 2 q^{2} - 25 q^{3} + 26 q^{4} - 2 q^{5} + 2 q^{6} + 12 q^{7} + 25 q^{9} + 19 q^{10} + 19 q^{11} - 26 q^{12} + 4 q^{13} + 15 q^{14} + 2 q^{15} + 32 q^{16} + 25 q^{17} - 2 q^{18} + 29 q^{19} - 8 q^{20} - 12 q^{21} + 23 q^{22} + 6 q^{23} + 15 q^{25} - 8 q^{26} - 25 q^{27} + 23 q^{28} + 11 q^{29} - 19 q^{30} + 38 q^{31} - 27 q^{32} - 19 q^{33} - 2 q^{34} + 20 q^{35} + 26 q^{36} + 8 q^{37} - 25 q^{38} - 4 q^{39} + 48 q^{40} + 24 q^{41} - 15 q^{42} + 11 q^{43} + 6 q^{44} - 2 q^{45} + 25 q^{46} + 23 q^{47} - 32 q^{48} + 21 q^{49} - 21 q^{50} - 25 q^{51} + 31 q^{52} - 16 q^{53} + 2 q^{54} - 11 q^{55} + 18 q^{56} - 29 q^{57} - 5 q^{58} + 27 q^{59} + 8 q^{60} + 40 q^{61} - 34 q^{62} + 12 q^{63} + 46 q^{64} - 19 q^{65} - 23 q^{66} + 24 q^{67} + 26 q^{68} - 6 q^{69} + 17 q^{70} + 19 q^{71} + 13 q^{73} - 56 q^{74} - 15 q^{75} + 21 q^{76} - 30 q^{77} + 8 q^{78} - 25 q^{79} - 40 q^{80} + 25 q^{81} + 61 q^{82} + q^{83} - 23 q^{84} - 2 q^{85} + 62 q^{86} - 11 q^{87} - q^{88} - 10 q^{89} + 19 q^{90} + 50 q^{91} + 18 q^{92} - 38 q^{93} + 15 q^{94} + 14 q^{95} + 27 q^{96} + 19 q^{97} - 23 q^{98} + 19 q^{99}+O(q^{100}) \) 25 * q - 2 * q^2 - 25 * q^3 + 26 * q^4 - 2 * q^5 + 2 * q^6 + 12 * q^7 + 25 * q^9 + 19 * q^10 + 19 * q^11 - 26 * q^12 + 4 * q^13 + 15 * q^14 + 2 * q^15 + 32 * q^16 + 25 * q^17 - 2 * q^18 + 29 * q^19 - 8 * q^20 - 12 * q^21 + 23 * q^22 + 6 * q^23 + 15 * q^25 - 8 * q^26 - 25 * q^27 + 23 * q^28 + 11 * q^29 - 19 * q^30 + 38 * q^31 - 27 * q^32 - 19 * q^33 - 2 * q^34 + 20 * q^35 + 26 * q^36 + 8 * q^37 - 25 * q^38 - 4 * q^39 + 48 * q^40 + 24 * q^41 - 15 * q^42 + 11 * q^43 + 6 * q^44 - 2 * q^45 + 25 * q^46 + 23 * q^47 - 32 * q^48 + 21 * q^49 - 21 * q^50 - 25 * q^51 + 31 * q^52 - 16 * q^53 + 2 * q^54 - 11 * q^55 + 18 * q^56 - 29 * q^57 - 5 * q^58 + 27 * q^59 + 8 * q^60 + 40 * q^61 - 34 * q^62 + 12 * q^63 + 46 * q^64 - 19 * q^65 - 23 * q^66 + 24 * q^67 + 26 * q^68 - 6 * q^69 + 17 * q^70 + 19 * q^71 + 13 * q^73 - 56 * q^74 - 15 * q^75 + 21 * q^76 - 30 * q^77 + 8 * q^78 - 25 * q^79 - 40 * q^80 + 25 * q^81 + 61 * q^82 + q^83 - 23 * q^84 - 2 * q^85 + 62 * q^86 - 11 * q^87 - q^88 - 10 * q^89 + 19 * q^90 + 50 * q^91 + 18 * q^92 - 38 * q^93 + 15 * q^94 + 14 * q^95 + 27 * q^96 + 19 * q^97 - 23 * q^98 + 19 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more