LMFDB - Embedded newform 4029.2.a.i.1.19

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			1.19
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)

$f(q)$	$=$	$q+1.57938 q^{2} -1.00000 q^{3} +0.494430 q^{4} -1.92798 q^{5} -1.57938 q^{6} -2.95110 q^{7} -2.37786 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.57938 q^{2} -1.00000 q^{3} +0.494430 q^{4} -1.92798 q^{5} -1.57938 q^{6} -2.95110 q^{7} -2.37786 q^{8} +1.00000 q^{9} -3.04500 q^{10} -0.760159 q^{11} -0.494430 q^{12} -6.68474 q^{13} -4.66089 q^{14} +1.92798 q^{15} -4.74440 q^{16} +1.00000 q^{17} +1.57938 q^{18} -0.0491237 q^{19} -0.953250 q^{20} +2.95110 q^{21} -1.20058 q^{22} -2.97310 q^{23} +2.37786 q^{24} -1.28290 q^{25} -10.5577 q^{26} -1.00000 q^{27} -1.45911 q^{28} +5.14357 q^{29} +3.04500 q^{30} +3.50819 q^{31} -2.73747 q^{32} +0.760159 q^{33} +1.57938 q^{34} +5.68965 q^{35} +0.494430 q^{36} -11.4773 q^{37} -0.0775849 q^{38} +6.68474 q^{39} +4.58447 q^{40} +7.19768 q^{41} +4.66089 q^{42} +4.22426 q^{43} -0.375845 q^{44} -1.92798 q^{45} -4.69565 q^{46} -1.66899 q^{47} +4.74440 q^{48} +1.70897 q^{49} -2.02618 q^{50} -1.00000 q^{51} -3.30514 q^{52} +3.32198 q^{53} -1.57938 q^{54} +1.46557 q^{55} +7.01730 q^{56} +0.0491237 q^{57} +8.12363 q^{58} +7.21345 q^{59} +0.953250 q^{60} -4.68865 q^{61} +5.54076 q^{62} -2.95110 q^{63} +5.16531 q^{64} +12.8880 q^{65} +1.20058 q^{66} +4.16514 q^{67} +0.494430 q^{68} +2.97310 q^{69} +8.98610 q^{70} -8.52953 q^{71} -2.37786 q^{72} +6.25191 q^{73} -18.1270 q^{74} +1.28290 q^{75} -0.0242883 q^{76} +2.24330 q^{77} +10.5577 q^{78} -1.00000 q^{79} +9.14710 q^{80} +1.00000 q^{81} +11.3678 q^{82} -13.6398 q^{83} +1.45911 q^{84} -1.92798 q^{85} +6.67170 q^{86} -5.14357 q^{87} +1.80755 q^{88} +3.69015 q^{89} -3.04500 q^{90} +19.7273 q^{91} -1.46999 q^{92} -3.50819 q^{93} -2.63597 q^{94} +0.0947095 q^{95} +2.73747 q^{96} +9.86149 q^{97} +2.69911 q^{98} -0.760159 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.57938	1.11679	0.558394	−	0.829576i	$-0.311418\pi$
$2$	1.57938	1.11679	0.558394	+	0.829576i	$0.311418\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.494430	0.247215
$5$	−1.92798	−0.862218	−0.431109	−	0.902300i	$-0.641878\pi$
$5$	−1.92798	−0.862218	−0.431109	+	0.902300i	$0.641878\pi$
$6$	−1.57938	−0.644778
$7$	−2.95110	−1.11541	−0.557705	−	0.830039i	$-0.688318\pi$
$7$	−2.95110	−1.11541	−0.557705	+	0.830039i	$0.688318\pi$
$8$	−2.37786	−0.840701
$9$	1.00000	0.333333
$10$	−3.04500	−0.962915
$11$	−0.760159	−0.229197	−0.114598	−	0.993412i	$-0.536558\pi$
$11$	−0.760159	−0.229197	−0.114598	+	0.993412i	$0.536558\pi$
$12$	−0.494430	−0.142730
$13$	−6.68474	−1.85401	−0.927007	−	0.375043i	$-0.877628\pi$
$13$	−6.68474	−1.85401	−0.927007	+	0.375043i	$0.877628\pi$
$14$	−4.66089	−1.24568
$15$	1.92798	0.497802
$16$	−4.74440	−1.18610
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	1.57938	0.372263
$19$	−0.0491237	−0.0112698	−0.00563488	−	0.999984i	$-0.501794\pi$
$19$	−0.0491237	−0.0112698	−0.00563488	+	0.999984i	$0.501794\pi$
$20$	−0.953250	−0.213153
$21$	2.95110	0.643982
$22$	−1.20058	−0.255964
$23$	−2.97310	−0.619935	−0.309967	−	0.950747i	$-0.600318\pi$
$23$	−2.97310	−0.619935	−0.309967	+	0.950747i	$0.600318\pi$
$24$	2.37786	0.485379
$25$	−1.28290	−0.256580
$26$	−10.5577	−2.07054
$27$	−1.00000	−0.192450
$28$	−1.45911	−0.275746
$29$	5.14357	0.955137	0.477568	−	0.878595i	$-0.341518\pi$
$29$	5.14357	0.955137	0.477568	+	0.878595i	$0.341518\pi$
$30$	3.04500	0.555939
$31$	3.50819	0.630090	0.315045	−	0.949077i	$-0.397980\pi$
$31$	3.50819	0.630090	0.315045	+	0.949077i	$0.397980\pi$
$32$	−2.73747	−0.483921
$33$	0.760159	0.132327
$34$	1.57938	0.270861
$35$	5.68965	0.961727
$36$	0.494430	0.0824050
$37$	−11.4773	−1.88685	−0.943427	−	0.331579i	$-0.892419\pi$
$37$	−11.4773	−1.88685	−0.943427	+	0.331579i	$0.892419\pi$
$38$	−0.0775849	−0.0125859
$39$	6.68474	1.07042
$40$	4.58447	0.724868
$41$	7.19768	1.12409	0.562044	−	0.827107i	$-0.310015\pi$
$41$	7.19768	1.12409	0.562044	+	0.827107i	$0.310015\pi$
$42$	4.66089	0.719191
$43$	4.22426	0.644193	0.322097	−	0.946707i	$-0.395612\pi$
$43$	4.22426	0.644193	0.322097	+	0.946707i	$0.395612\pi$
$44$	−0.375845	−0.0566608
$45$	−1.92798	−0.287406
$46$	−4.69565	−0.692336
$47$	−1.66899	−0.243447	−0.121724	−	0.992564i	$-0.538842\pi$
$47$	−1.66899	−0.243447	−0.121724	+	0.992564i	$0.538842\pi$
$48$	4.74440	0.684795
$49$	1.70897	0.244139
$50$	−2.02618	−0.286545
$51$	−1.00000	−0.140028
$52$	−3.30514	−0.458340
$53$	3.32198	0.456309	0.228154	−	0.973625i	$-0.426731\pi$
$53$	3.32198	0.456309	0.228154	+	0.973625i	$0.426731\pi$
$54$	−1.57938	−0.214926
$55$	1.46557	0.197618
$56$	7.01730	0.937726
$57$	0.0491237	0.00650660
$58$	8.12363	1.06669
$59$	7.21345	0.939111	0.469555	−	0.882903i	$-0.344414\pi$
$59$	7.21345	0.939111	0.469555	+	0.882903i	$0.344414\pi$
$60$	0.953250	0.123064
$61$	−4.68865	−0.600320	−0.300160	−	0.953889i	$-0.597040\pi$
$61$	−4.68865	−0.600320	−0.300160	+	0.953889i	$0.597040\pi$
$62$	5.54076	0.703677
$63$	−2.95110	−0.371803
$64$	5.16531	0.645663
$65$	12.8880	1.59857
$66$	1.20058	0.147781
$67$	4.16514	0.508853	0.254427	−	0.967092i	$-0.418113\pi$
$67$	4.16514	0.508853	0.254427	+	0.967092i	$0.418113\pi$
$68$	0.494430	0.0599584
$69$	2.97310	0.357920
$70$	8.98610	1.07404
$71$	−8.52953	−1.01227	−0.506134	−	0.862455i	$-0.668926\pi$
$71$	−8.52953	−1.01227	−0.506134	+	0.862455i	$0.668926\pi$
$72$	−2.37786	−0.280234
$73$	6.25191	0.731730	0.365865	−	0.930668i	$-0.380773\pi$
$73$	6.25191	0.731730	0.365865	+	0.930668i	$0.380773\pi$
$74$	−18.1270	−2.10722
$75$	1.28290	0.148136
$76$	−0.0242883	−0.00278605
$77$	2.24330	0.255648
$78$	10.5577	1.19543
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	9.14710	1.02268
$81$	1.00000	0.111111
$82$	11.3678	1.25537
$83$	−13.6398	−1.49716	−0.748581	−	0.663043i	$-0.769265\pi$
$83$	−13.6398	−1.49716	−0.748581	+	0.663043i	$0.769265\pi$
$84$	1.45911	0.159202
$85$	−1.92798	−0.209119
$86$	6.67170	0.719427
$87$	−5.14357	−0.551449
$88$	1.80755	0.192686
$89$	3.69015	0.391155	0.195577	−	0.980688i	$-0.437342\pi$
$89$	3.69015	0.391155	0.195577	+	0.980688i	$0.437342\pi$
$90$	−3.04500	−0.320972
$91$	19.7273	2.06799
$92$	−1.46999	−0.153257
$93$	−3.50819	−0.363783
$94$	−2.63597	−0.271879
$95$	0.0947095	0.00971699
$96$	2.73747	0.279392
$97$	9.86149	1.00128	0.500641	−	0.865655i	$-0.333098\pi$
$97$	9.86149	1.00128	0.500641	+	0.865655i	$0.333098\pi$
$98$	2.69911	0.272652
$99$	−0.760159	−0.0763989
$100$	−0.634303	−0.0634303
$101$	0.450953	0.0448715	0.0224358	−	0.999748i	$-0.492858\pi$
$101$	0.450953	0.0448715	0.0224358	+	0.999748i	$0.492858\pi$
$102$	−1.57938	−0.156382
$103$	−18.9590	−1.86808	−0.934042	−	0.357163i	$-0.883744\pi$
$103$	−18.9590	−1.86808	−0.934042	+	0.357163i	$0.883744\pi$
$104$	15.8954	1.55867
$105$	−5.68965	−0.555253
$106$	5.24665	0.509600
$107$	12.1861	1.17807	0.589035	−	0.808107i	$-0.299508\pi$
$107$	12.1861	1.17807	0.589035	+	0.808107i	$0.299508\pi$
$108$	−0.494430	−0.0475765
$109$	−2.15334	−0.206253	−0.103126	−	0.994668i	$-0.532885\pi$
$109$	−2.15334	−0.206253	−0.103126	+	0.994668i	$0.532885\pi$
$110$	2.31469	0.220697
$111$	11.4773	1.08938
$112$	14.0012	1.32299
$113$	18.6541	1.75483	0.877415	−	0.479732i	$-0.159266\pi$
$113$	18.6541	1.75483	0.877415	+	0.479732i	$0.159266\pi$
$114$	0.0775849	0.00726649
$115$	5.73208	0.534519
$116$	2.54313	0.236124
$117$	−6.68474	−0.618005
$118$	11.3927	1.04879
$119$	−2.95110	−0.270527
$120$	−4.58447	−0.418503
$121$	−10.4222	−0.947469
$122$	−7.40514	−0.670430
$123$	−7.19768	−0.648993
$124$	1.73456	0.155768
$125$	12.1133	1.08345
$126$	−4.66089	−0.415225
$127$	−9.35640	−0.830246	−0.415123	−	0.909765i	$-0.636262\pi$
$127$	−9.35640	−0.830246	−0.415123	+	0.909765i	$0.636262\pi$
$128$	13.6329	1.20499
$129$	−4.22426	−0.371925
$130$	20.3551	1.78526
$131$	3.46449	0.302694	0.151347	−	0.988481i	$-0.451639\pi$
$131$	3.46449	0.302694	0.151347	+	0.988481i	$0.451639\pi$
$132$	0.375845	0.0327131
$133$	0.144969	0.0125704
$134$	6.57833	0.568281
$135$	1.92798	0.165934
$136$	−2.37786	−0.203900
$137$	−18.3449	−1.56731	−0.783657	−	0.621194i	$-0.786648\pi$
$137$	−18.3449	−1.56731	−0.783657	+	0.621194i	$0.786648\pi$
$138$	4.69565	0.399720
$139$	−2.84113	−0.240982	−0.120491	−	0.992714i	$-0.538447\pi$
$139$	−2.84113	−0.240982	−0.120491	+	0.992714i	$0.538447\pi$
$140$	2.81313	0.237753
$141$	1.66899	0.140554
$142$	−13.4713	−1.13049
$143$	5.08147	0.424934
$144$	−4.74440	−0.395367
$145$	−9.91669	−0.823537
$146$	9.87412	0.817188
$147$	−1.70897	−0.140954
$148$	−5.67472	−0.466459
$149$	−7.46886	−0.611873	−0.305937	−	0.952052i	$-0.598970\pi$
$149$	−7.46886	−0.611873	−0.305937	+	0.952052i	$0.598970\pi$
$150$	2.02618	0.165437
$151$	15.7883	1.28483	0.642417	−	0.766355i	$-0.277932\pi$
$151$	15.7883	1.28483	0.642417	+	0.766355i	$0.277932\pi$
$152$	0.116809	0.00947450
$153$	1.00000	0.0808452
$154$	3.54302	0.285505
$155$	−6.76372	−0.543275
$156$	3.30514	0.264623
$157$	15.9473	1.27273	0.636366	−	0.771387i	$-0.280437\pi$
$157$	15.9473	1.27273	0.636366	+	0.771387i	$0.280437\pi$
$158$	−1.57938	−0.125648
$159$	−3.32198	−0.263450
$160$	5.27778	0.417245
$161$	8.77392	0.691481
$162$	1.57938	0.124088
$163$	11.7930	0.923696	0.461848	−	0.886959i	$-0.347187\pi$
$163$	11.7930	0.923696	0.461848	+	0.886959i	$0.347187\pi$
$164$	3.55875	0.277891
$165$	−1.46557	−0.114095
$166$	−21.5424	−1.67201
$167$	−17.8618	−1.38219	−0.691094	−	0.722765i	$-0.742871\pi$
$167$	−17.8618	−1.38219	−0.691094	+	0.722765i	$0.742871\pi$
$168$	−7.01730	−0.541397
$169$	31.6858	2.43737
$170$	−3.04500	−0.233541
$171$	−0.0491237	−0.00375659
$172$	2.08860	0.159254
$173$	7.44000	0.565653	0.282827	−	0.959171i	$-0.408728\pi$
$173$	7.44000	0.565653	0.282827	+	0.959171i	$0.408728\pi$
$174$	−8.12363	−0.615851
$175$	3.78596	0.286191
$176$	3.60650	0.271850
$177$	−7.21345	−0.542196
$178$	5.82813	0.436837
$179$	24.5730	1.83667	0.918335	−	0.395804i	$-0.129534\pi$
$179$	24.5730	1.83667	0.918335	+	0.395804i	$0.129534\pi$
$180$	−0.953250	−0.0710511
$181$	−4.81382	−0.357808	−0.178904	−	0.983866i	$-0.557255\pi$
$181$	−4.81382	−0.357808	−0.178904	+	0.983866i	$0.557255\pi$
$182$	31.1569	2.30950
$183$	4.68865	0.346595
$184$	7.06963	0.521180
$185$	22.1280	1.62688
$186$	−5.54076	−0.406268
$187$	−0.760159	−0.0555883
$188$	−0.825199	−0.0601838
$189$	2.95110	0.214661
$190$	0.149582	0.0108518
$191$	−18.8764	−1.36585	−0.682923	−	0.730490i	$-0.739292\pi$
$191$	−18.8764	−1.36585	−0.682923	+	0.730490i	$0.739292\pi$
$192$	−5.16531	−0.372774
$193$	−11.5894	−0.834221	−0.417110	−	0.908856i	$-0.636957\pi$
$193$	−11.5894	−0.834221	−0.417110	+	0.908856i	$0.636957\pi$
$194$	15.5750	1.11822
$195$	−12.8880	−0.922932
$196$	0.844968	0.0603549
$197$	−17.0108	−1.21197	−0.605983	−	0.795478i	$-0.707220\pi$
$197$	−17.0108	−1.21197	−0.605983	+	0.795478i	$0.707220\pi$
$198$	−1.20058	−0.0853213
$199$	0.310180	0.0219881	0.0109941	−	0.999940i	$-0.496500\pi$
$199$	0.310180	0.0219881	0.0109941	+	0.999940i	$0.496500\pi$
$200$	3.05055	0.215707
$201$	−4.16514	−0.293786
$202$	0.712225	0.0501120
$203$	−15.1792	−1.06537
$204$	−0.494430	−0.0346170
$205$	−13.8770	−0.969209
$206$	−29.9434	−2.08625
$207$	−2.97310	−0.206645
$208$	31.7151	2.19905
$209$	0.0373419	0.00258299
$210$	−8.98610	−0.620100
$211$	10.0737	0.693505	0.346752	−	0.937957i	$-0.387284\pi$
$211$	10.0737	0.693505	0.346752	+	0.937957i	$0.387284\pi$
$212$	1.64249	0.112806
$213$	8.52953	0.584434
$214$	19.2464	1.31565
$215$	−8.14428	−0.555435
$216$	2.37786	0.161793
$217$	−10.3530	−0.702809
$218$	−3.40094	−0.230341
$219$	−6.25191	−0.422465
$220$	0.724622	0.0488540
$221$	−6.68474	−0.449665
$222$	18.1270	1.21660
$223$	3.57541	0.239427	0.119714	−	0.992808i	$-0.461802\pi$
$223$	3.57541	0.239427	0.119714	+	0.992808i	$0.461802\pi$
$224$	8.07853	0.539770
$225$	−1.28290	−0.0855265
$226$	29.4618	1.95977
$227$	−9.56215	−0.634662	−0.317331	−	0.948315i	$-0.602787\pi$
$227$	−9.56215	−0.634662	−0.317331	+	0.948315i	$0.602787\pi$
$228$	0.0242883	0.00160853
$229$	−24.0821	−1.59139	−0.795694	−	0.605699i	$-0.792893\pi$
$229$	−24.0821	−1.59139	−0.795694	+	0.605699i	$0.792893\pi$
$230$	9.05311	0.596945
$231$	−2.24330	−0.147599
$232$	−12.2307	−0.802985
$233$	−7.98320	−0.522997	−0.261498	−	0.965204i	$-0.584217\pi$
$233$	−7.98320	−0.522997	−0.261498	+	0.965204i	$0.584217\pi$
$234$	−10.5577	−0.690180
$235$	3.21778	0.209905
$236$	3.56654	0.232162
$237$	1.00000	0.0649570
$238$	−4.66089	−0.302121
$239$	−24.4595	−1.58215	−0.791076	−	0.611718i	$-0.790479\pi$
$239$	−24.4595	−1.58215	−0.791076	+	0.611718i	$0.790479\pi$
$240$	−9.14710	−0.590443
$241$	20.5608	1.32444	0.662219	−	0.749310i	$-0.269615\pi$
$241$	20.5608	1.32444	0.662219	+	0.749310i	$0.269615\pi$
$242$	−16.4605	−1.05812
$243$	−1.00000	−0.0641500
$244$	−2.31821	−0.148408
$245$	−3.29487	−0.210501
$246$	−11.3678	−0.724787
$247$	0.328380	0.0208943
$248$	−8.34200	−0.529717
$249$	13.6398	0.864387
$250$	19.1315	1.20998
$251$	30.2363	1.90850	0.954248	−	0.299017i	$-0.0966588\pi$
$251$	30.2363	1.90850	0.954248	+	0.299017i	$0.0966588\pi$
$252$	−1.45911	−0.0919153
$253$	2.26003	0.142087
$254$	−14.7773	−0.927209
$255$	1.92798	0.120735
$256$	11.2009	0.700054
$257$	−27.1030	−1.69064	−0.845320	−	0.534260i	$-0.820590\pi$
$257$	−27.1030	−1.69064	−0.845320	+	0.534260i	$0.820590\pi$
$258$	−6.67170	−0.415362
$259$	33.8706	2.10462
$260$	6.37224	0.395189
$261$	5.14357	0.318379
$262$	5.47174	0.338045
$263$	14.0669	0.867402	0.433701	−	0.901057i	$-0.357207\pi$
$263$	14.0669	0.867402	0.433701	+	0.901057i	$0.357207\pi$
$264$	−1.80755	−0.111247
$265$	−6.40470	−0.393438
$266$	0.228961	0.0140385
$267$	−3.69015	−0.225833
$268$	2.05937	0.125796
$269$	1.93011	0.117681	0.0588406	−	0.998267i	$-0.481260\pi$
$269$	1.93011	0.117681	0.0588406	+	0.998267i	$0.481260\pi$
$270$	3.04500	0.185313
$271$	−16.1680	−0.982138	−0.491069	−	0.871121i	$-0.663394\pi$
$271$	−16.1680	−0.982138	−0.491069	+	0.871121i	$0.663394\pi$
$272$	−4.74440	−0.287671
$273$	−19.7273	−1.19395
$274$	−28.9736	−1.75036
$275$	0.975207	0.0588072
$276$	1.46999	0.0884831
$277$	24.5593	1.47562	0.737812	−	0.675006i	$-0.235859\pi$
$277$	24.5593	1.47562	0.737812	+	0.675006i	$0.235859\pi$
$278$	−4.48722	−0.269126
$279$	3.50819	0.210030
$280$	−13.5292	−0.808525
$281$	−3.02931	−0.180714	−0.0903568	−	0.995909i	$-0.528801\pi$
$281$	−3.02931	−0.180714	−0.0903568	+	0.995909i	$0.528801\pi$
$282$	2.63597	0.156969
$283$	20.8760	1.24095	0.620475	−	0.784226i	$-0.286940\pi$
$283$	20.8760	1.24095	0.620475	+	0.784226i	$0.286940\pi$
$284$	−4.21725	−0.250248
$285$	−0.0947095	−0.00561011
$286$	8.02555	0.474561
$287$	−21.2410	−1.25382
$288$	−2.73747	−0.161307
$289$	1.00000	0.0588235
$290$	−15.6622	−0.919716
$291$	−9.86149	−0.578091
$292$	3.09113	0.180895
$293$	18.9097	1.10471	0.552357	−	0.833607i	$-0.313728\pi$
$293$	18.9097	1.10471	0.552357	+	0.833607i	$0.313728\pi$
$294$	−2.69911	−0.157415
$295$	−13.9074	−0.809718
$296$	27.2914	1.58628
$297$	0.760159	0.0441089
$298$	−11.7961	−0.683332
$299$	19.8744	1.14937
$300$	0.634303	0.0366215
$301$	−12.4662	−0.718540
$302$	24.9357	1.43489
$303$	−0.450953	−0.0259066
$304$	0.233063	0.0133671
$305$	9.03961	0.517607
$306$	1.57938	0.0902869
$307$	5.19563	0.296530	0.148265	−	0.988948i	$-0.452631\pi$
$307$	5.19563	0.296530	0.148265	+	0.988948i	$0.452631\pi$
$308$	1.10916	0.0632001
$309$	18.9590	1.07854
$310$	−10.6825	−0.606723
$311$	18.1563	1.02955	0.514774	−	0.857326i	$-0.327876\pi$
$311$	18.1563	1.02955	0.514774	+	0.857326i	$0.327876\pi$
$312$	−15.8954	−0.899900
$313$	10.5030	0.593665	0.296833	−	0.954929i	$-0.404070\pi$
$313$	10.5030	0.593665	0.296833	+	0.954929i	$0.404070\pi$
$314$	25.1868	1.42137
$315$	5.68965	0.320576
$316$	−0.494430	−0.0278139
$317$	4.33435	0.243441	0.121721	−	0.992564i	$-0.461159\pi$
$317$	4.33435	0.243441	0.121721	+	0.992564i	$0.461159\pi$
$318$	−5.24665	−0.294218
$319$	−3.90993	−0.218914
$320$	−9.95860	−0.556703
$321$	−12.1861	−0.680159
$322$	13.8573	0.772238
$323$	−0.0491237	−0.00273332
$324$	0.494430	0.0274683
$325$	8.57584	0.475702
$326$	18.6255	1.03157
$327$	2.15334	0.119080
$328$	−17.1151	−0.945022
$329$	4.92535	0.271544
$330$	−2.31469	−0.127419
$331$	−3.52561	−0.193785	−0.0968925	−	0.995295i	$-0.530890\pi$
$331$	−3.52561	−0.193785	−0.0968925	+	0.995295i	$0.530890\pi$
$332$	−6.74393	−0.370121
$333$	−11.4773	−0.628952
$334$	−28.2105	−1.54361
$335$	−8.03031	−0.438742
$336$	−14.0012	−0.763827
$337$	−8.23560	−0.448622	−0.224311	−	0.974518i	$-0.572013\pi$
$337$	−8.23560	−0.448622	−0.224311	+	0.974518i	$0.572013\pi$
$338$	50.0438	2.72202
$339$	−18.6541	−1.01315
$340$	−0.953250	−0.0516973
$341$	−2.66679	−0.144415
$342$	−0.0775849	−0.00419531
$343$	15.6143	0.843095
$344$	−10.0447	−0.541574
$345$	−5.73208	−0.308605
$346$	11.7506	0.631714
$347$	−24.8370	−1.33332	−0.666661	−	0.745361i	$-0.732277\pi$
$347$	−24.8370	−1.33332	−0.666661	+	0.745361i	$0.732277\pi$
$348$	−2.54313	−0.136326
$349$	18.3358	0.981494	0.490747	−	0.871302i	$-0.336724\pi$
$349$	18.3358	0.981494	0.490747	+	0.871302i	$0.336724\pi$
$350$	5.97945	0.319615
$351$	6.68474	0.356805
$352$	2.08091	0.110913
$353$	−11.0099	−0.585998	−0.292999	−	0.956113i	$-0.594653\pi$
$353$	−11.0099	−0.585998	−0.292999	+	0.956113i	$0.594653\pi$
$354$	−11.3927	−0.605518
$355$	16.4447	0.872797
$356$	1.82452	0.0966993
$357$	2.95110	0.156189
$358$	38.8100	2.05117
$359$	5.06119	0.267119	0.133560	−	0.991041i	$-0.457359\pi$
$359$	5.06119	0.267119	0.133560	+	0.991041i	$0.457359\pi$
$360$	4.58447	0.241623
$361$	−18.9976	−0.999873
$362$	−7.60283	−0.399596
$363$	10.4222	0.547021
$364$	9.75378	0.511237
$365$	−12.0535	−0.630911
$366$	7.40514	0.387073
$367$	−28.4647	−1.48585	−0.742923	−	0.669377i	$-0.766561\pi$
$367$	−28.4647	−1.48585	−0.742923	+	0.669377i	$0.766561\pi$
$368$	14.1056	0.735305
$369$	7.19768	0.374696
$370$	34.9484	1.81688
$371$	−9.80348	−0.508971
$372$	−1.73456	−0.0899325
$373$	−36.7021	−1.90037	−0.950183	−	0.311694i	$-0.899104\pi$
$373$	−36.7021	−1.90037	−0.950183	+	0.311694i	$0.899104\pi$
$374$	−1.20058	−0.0620804
$375$	−12.1133	−0.625528
$376$	3.96863	0.204666
$377$	−34.3834	−1.77084
$378$	4.66089	0.239730
$379$	6.02654	0.309563	0.154781	−	0.987949i	$-0.450533\pi$
$379$	6.02654	0.309563	0.154781	+	0.987949i	$0.450533\pi$
$380$	0.0468272	0.00240219
$381$	9.35640	0.479343
$382$	−29.8129	−1.52536
$383$	15.6543	0.799898	0.399949	−	0.916537i	$-0.369028\pi$
$383$	15.6543	0.799898	0.399949	+	0.916537i	$0.369028\pi$
$384$	−13.6329	−0.695701
$385$	−4.32504	−0.220425
$386$	−18.3040	−0.931648
$387$	4.22426	0.214731
$388$	4.87582	0.247532
$389$	0.108533	0.00550283	0.00275141	−	0.999996i	$-0.499124\pi$
$389$	0.108533	0.00550283	0.00275141	+	0.999996i	$0.499124\pi$
$390$	−20.3551	−1.03072
$391$	−2.97310	−0.150356
$392$	−4.06370	−0.205248
$393$	−3.46449	−0.174761
$394$	−26.8664	−1.35351
$395$	1.92798	0.0970071
$396$	−0.375845	−0.0188869
$397$	−13.5692	−0.681018	−0.340509	−	0.940241i	$-0.610599\pi$
$397$	−13.5692	−0.681018	−0.340509	+	0.940241i	$0.610599\pi$
$398$	0.489892	0.0245561
$399$	−0.144969	−0.00725752
$400$	6.08658	0.304329
$401$	1.94602	0.0971796	0.0485898	−	0.998819i	$-0.484527\pi$
$401$	1.94602	0.0971796	0.0485898	+	0.998819i	$0.484527\pi$
$402$	−6.57833	−0.328097
$403$	−23.4514	−1.16820
$404$	0.222965	0.0110929
$405$	−1.92798	−0.0958020
$406$	−23.9736	−1.18979
$407$	8.72457	0.432461
$408$	2.37786	0.117722
$409$	−14.9669	−0.740067	−0.370033	−	0.929019i	$-0.620654\pi$
$409$	−14.9669	−0.740067	−0.370033	+	0.929019i	$0.620654\pi$
$410$	−21.9170	−1.08240
$411$	18.3449	0.904889
$412$	−9.37389	−0.461818
$413$	−21.2876	−1.04749
$414$	−4.69565	−0.230779
$415$	26.2972	1.29088
$416$	18.2993	0.897196
$417$	2.84113	0.139131
$418$	0.0589769	0.00288465
$419$	18.6305	0.910159	0.455080	−	0.890451i	$-0.349611\pi$
$419$	18.6305	0.910159	0.455080	+	0.890451i	$0.349611\pi$
$420$	−2.81313	−0.137267
$421$	−3.16400	−0.154204	−0.0771020	−	0.997023i	$-0.524567\pi$
$421$	−3.16400	−0.154204	−0.0771020	+	0.997023i	$0.524567\pi$
$422$	15.9102	0.774497
$423$	−1.66899	−0.0811491
$424$	−7.89921	−0.383619
$425$	−1.28290	−0.0622297
$426$	13.4713	0.652688
$427$	13.8367	0.669602
$428$	6.02515	0.291237
$429$	−5.08147	−0.245336
$430$	−12.8629	−0.620303
$431$	−23.4018	−1.12723	−0.563613	−	0.826039i	$-0.690589\pi$
$431$	−23.4018	−1.12723	−0.563613	+	0.826039i	$0.690589\pi$
$432$	4.74440	0.228265
$433$	−29.9008	−1.43694	−0.718470	−	0.695558i	$-0.755157\pi$
$433$	−29.9008	−1.43694	−0.718470	+	0.695558i	$0.755157\pi$
$434$	−16.3513	−0.784888
$435$	9.91669	0.475469
$436$	−1.06468	−0.0509888
$437$	0.146050	0.00698652
$438$	−9.87412	−0.471804
$439$	33.3980	1.59400	0.797000	−	0.603979i	$-0.206419\pi$
$439$	33.3980	1.59400	0.797000	+	0.603979i	$0.206419\pi$
$440$	−3.48492	−0.166137
$441$	1.70897	0.0813797
$442$	−10.5577	−0.502180
$443$	−16.8326	−0.799742	−0.399871	−	0.916571i	$-0.630945\pi$
$443$	−16.8326	−0.799742	−0.399871	+	0.916571i	$0.630945\pi$
$444$	5.67472	0.269310
$445$	−7.11453	−0.337261
$446$	5.64692	0.267389
$447$	7.46886	0.353265
$448$	−15.2433	−0.720179
$449$	−21.3527	−1.00770	−0.503849	−	0.863792i	$-0.668083\pi$
$449$	−21.3527	−1.00770	−0.503849	+	0.863792i	$0.668083\pi$
$450$	−2.02618	−0.0955150
$451$	−5.47138	−0.257637
$452$	9.22314	0.433820
$453$	−15.7883	−0.741799
$454$	−15.1022	−0.708783
$455$	−38.0339	−1.78306
$456$	−0.116809	−0.00547011
$457$	24.8374	1.16184	0.580922	−	0.813959i	$-0.302692\pi$
$457$	24.8374	1.16184	0.580922	+	0.813959i	$0.302692\pi$
$458$	−38.0346	−1.77724
$459$	−1.00000	−0.0466760
$460$	2.83411	0.132141
$461$	18.8277	0.876895	0.438448	−	0.898757i	$-0.355529\pi$
$461$	18.8277	0.876895	0.438448	+	0.898757i	$0.355529\pi$
$462$	−3.54302	−0.164836
$463$	−2.90577	−0.135043	−0.0675213	−	0.997718i	$-0.521509\pi$
$463$	−2.90577	−0.135043	−0.0675213	+	0.997718i	$0.521509\pi$
$464$	−24.4031	−1.13289
$465$	6.76372	0.313660
$466$	−12.6085	−0.584076
$467$	0.0668766	0.00309468	0.00154734	−	0.999999i	$-0.499507\pi$
$467$	0.0668766	0.00309468	0.00154734	+	0.999999i	$0.499507\pi$
$468$	−3.30514	−0.152780
$469$	−12.2917	−0.567580
$470$	5.08208	0.234419
$471$	−15.9473	−0.734813
$472$	−17.1526	−0.789511
$473$	−3.21111	−0.147647
$474$	1.57938	0.0725432
$475$	0.0630208	0.00289159
$476$	−1.45911	−0.0668782
$477$	3.32198	0.152103
$478$	−38.6307	−1.76693
$479$	11.4922	0.525093	0.262546	−	0.964919i	$-0.415438\pi$
$479$	11.4922	0.525093	0.262546	+	0.964919i	$0.415438\pi$
$480$	−5.27778	−0.240897
$481$	76.7227	3.49826
$482$	32.4733	1.47912
$483$	−8.77392	−0.399227
$484$	−5.15303	−0.234229
$485$	−19.0127	−0.863324
$486$	−1.57938	−0.0716420
$487$	29.9523	1.35727	0.678634	−	0.734477i	$-0.262572\pi$
$487$	29.9523	1.35727	0.678634	+	0.734477i	$0.262572\pi$
$488$	11.1490	0.504689
$489$	−11.7930	−0.533296
$490$	−5.20383	−0.235085
$491$	−9.23777	−0.416895	−0.208447	−	0.978034i	$-0.566841\pi$
$491$	−9.23777	−0.416895	−0.208447	+	0.978034i	$0.566841\pi$
$492$	−3.55875	−0.160441
$493$	5.14357	0.231655
$494$	0.518635	0.0233345
$495$	1.46557	0.0658725
$496$	−16.6443	−0.747350
$497$	25.1715	1.12909
$498$	21.5424	0.965337
$499$	−0.352380	−0.0157747	−0.00788734	−	0.999969i	$-0.502511\pi$
$499$	−0.352380	−0.0157747	−0.00788734	+	0.999969i	$0.502511\pi$
$500$	5.98918	0.267844
$501$	17.8618	0.798007
$502$	47.7544	2.13138
$503$	35.7136	1.59239	0.796196	−	0.605038i	$-0.206842\pi$
$503$	35.7136	1.59239	0.796196	+	0.605038i	$0.206842\pi$
$504$	7.01730	0.312575
$505$	−0.869428	−0.0386890
$506$	3.56944	0.158681
$507$	−31.6858	−1.40722
$508$	−4.62609	−0.205249
$509$	−15.2828	−0.677399	−0.338699	−	0.940895i	$-0.609987\pi$
$509$	−15.2828	−0.677399	−0.338699	+	0.940895i	$0.609987\pi$
$510$	3.04500	0.134835
$511$	−18.4500	−0.816179
$512$	−9.57541	−0.423177
$513$	0.0491237	0.00216887
$514$	−42.8059	−1.88809
$515$	36.5525	1.61070
$516$	−2.08860	−0.0919455
$517$	1.26870	0.0557973
$518$	53.4944	2.35041
$519$	−7.44000	−0.326580
$520$	−30.6460	−1.34392
$521$	35.1431	1.53965	0.769823	−	0.638257i	$-0.220344\pi$
$521$	35.1431	1.53965	0.769823	+	0.638257i	$0.220344\pi$
$522$	8.12363	0.355562
$523$	14.8544	0.649540	0.324770	−	0.945793i	$-0.394713\pi$
$523$	14.8544	0.649540	0.324770	+	0.945793i	$0.394713\pi$
$524$	1.71295	0.0748305
$525$	−3.78596	−0.165233
$526$	22.2169	0.968704
$527$	3.50819	0.152819
$528$	−3.60650	−0.156953
$529$	−14.1607	−0.615681
$530$	−10.1154	−0.439387
$531$	7.21345	0.313037
$532$	0.0716770	0.00310759
$533$	−48.1146	−2.08408
$534$	−5.82813	−0.252208
$535$	−23.4944	−1.01575
$536$	−9.90413	−0.427793
$537$	−24.5730	−1.06040
$538$	3.04838	0.131425
$539$	−1.29909	−0.0559559
$540$	0.953250	0.0410214
$541$	14.1579	0.608694	0.304347	−	0.952561i	$-0.401562\pi$
$541$	14.1579	0.608694	0.304347	+	0.952561i	$0.401562\pi$
$542$	−25.5354	−1.09684
$543$	4.81382	0.206581
$544$	−2.73747	−0.117368
$545$	4.15160	0.177835
$546$	−31.1569	−1.33339
$547$	1.64241	0.0702244	0.0351122	−	0.999383i	$-0.488821\pi$
$547$	1.64241	0.0702244	0.0351122	+	0.999383i	$0.488821\pi$
$548$	−9.07029	−0.387463
$549$	−4.68865	−0.200107
$550$	1.54022	0.0656751
$551$	−0.252671	−0.0107642
$552$	−7.06963	−0.300903
$553$	2.95110	0.125493
$554$	38.7884	1.64796
$555$	−22.1280	−0.939280
$556$	−1.40474	−0.0595743
$557$	−35.1226	−1.48819	−0.744096	−	0.668072i	$-0.767120\pi$
$557$	−35.1226	−1.48819	−0.744096	+	0.668072i	$0.767120\pi$
$558$	5.54076	0.234559
$559$	−28.2381	−1.19434
$560$	−26.9940	−1.14070
$561$	0.760159	0.0320939
$562$	−4.78442	−0.201819
$563$	−11.7923	−0.496988	−0.248494	−	0.968633i	$-0.579936\pi$
$563$	−11.7923	−0.496988	−0.248494	+	0.968633i	$0.579936\pi$
$564$	0.825199	0.0347472
$565$	−35.9647	−1.51305
$566$	32.9711	1.38588
$567$	−2.95110	−0.123934
$568$	20.2820	0.851015
$569$	0.426828	0.0178936	0.00894679	−	0.999960i	$-0.497152\pi$
$569$	0.426828	0.0178936	0.00894679	+	0.999960i	$0.497152\pi$
$570$	−0.149582	−0.00626530
$571$	1.39027	0.0581811	0.0290905	−	0.999577i	$-0.490739\pi$
$571$	1.39027	0.0581811	0.0290905	+	0.999577i	$0.490739\pi$
$572$	2.51243	0.105050
$573$	18.8764	0.788572
$574$	−33.5476	−1.40025
$575$	3.81419	0.159063
$576$	5.16531	0.215221
$577$	−18.6345	−0.775765	−0.387882	−	0.921709i	$-0.626793\pi$
$577$	−18.6345	−0.775765	−0.387882	+	0.921709i	$0.626793\pi$
$578$	1.57938	0.0656934
$579$	11.5894	0.481638
$580$	−4.90311	−0.203591
$581$	40.2524	1.66995
$582$	−15.5750	−0.645605
$583$	−2.52523	−0.104584
$584$	−14.8662	−0.615167
$585$	12.8880	0.532855
$586$	29.8655	1.23373
$587$	34.5608	1.42648	0.713239	−	0.700921i	$-0.247227\pi$
$587$	34.5608	1.42648	0.713239	+	0.700921i	$0.247227\pi$
$588$	−0.844968	−0.0348459
$589$	−0.172336	−0.00710096
$590$	−21.9650	−0.904284
$591$	17.0108	0.699729
$592$	54.4528	2.23800
$593$	−47.4023	−1.94658	−0.973290	−	0.229579i	$-0.926265\pi$
$593$	−47.4023	−1.94658	−0.973290	+	0.229579i	$0.926265\pi$
$594$	1.20058	0.0492603
$595$	5.68965	0.233253
$596$	−3.69283	−0.151264
$597$	−0.310180	−0.0126948
$598$	31.3892	1.28360
$599$	1.11888	0.0457161	0.0228581	−	0.999739i	$-0.492723\pi$
$599$	1.11888	0.0457161	0.0228581	+	0.999739i	$0.492723\pi$
$600$	−3.05055	−0.124538
$601$	−22.3557	−0.911906	−0.455953	−	0.890004i	$-0.650702\pi$
$601$	−22.3557	−0.911906	−0.455953	+	0.890004i	$0.650702\pi$
$602$	−19.6888	−0.802456
$603$	4.16514	0.169618
$604$	7.80621	0.317630
$605$	20.0937	0.816925
$606$	−0.712225	−0.0289321
$607$	48.1868	1.95584	0.977920	−	0.208978i	$-0.0670138\pi$
$607$	48.1868	1.95584	0.977920	+	0.208978i	$0.0670138\pi$
$608$	0.134475	0.00545367
$609$	15.1792	0.615091
$610$	14.2769	0.578057
$611$	11.1568	0.451355
$612$	0.494430	0.0199861
$613$	−20.1729	−0.814776	−0.407388	−	0.913255i	$-0.633560\pi$
$613$	−20.1729	−0.814776	−0.407388	+	0.913255i	$0.633560\pi$
$614$	8.20586	0.331161
$615$	13.8770	0.559573
$616$	−5.33427	−0.214924
$617$	−40.0613	−1.61281	−0.806404	−	0.591365i	$-0.798589\pi$
$617$	−40.0613	−1.61281	−0.806404	+	0.591365i	$0.798589\pi$
$618$	29.9434	1.20450
$619$	−17.9781	−0.722602	−0.361301	−	0.932449i	$-0.617667\pi$
$619$	−17.9781	−0.722602	−0.361301	+	0.932449i	$0.617667\pi$
$620$	−3.34419	−0.134306
$621$	2.97310	0.119307
$622$	28.6756	1.14979
$623$	−10.8900	−0.436298
$624$	−31.7151	−1.26962
$625$	−16.9397	−0.677587
$626$	16.5882	0.662998
$627$	−0.0373419	−0.00149129
$628$	7.88482	0.314639
$629$	−11.4773	−0.457629
$630$	8.98610	0.358015
$631$	5.17581	0.206046	0.103023	−	0.994679i	$-0.467148\pi$
$631$	5.17581	0.206046	0.103023	+	0.994679i	$0.467148\pi$
$632$	2.37786	0.0945863
$633$	−10.0737	−0.400395
$634$	6.84556	0.271872
$635$	18.0389	0.715854
$636$	−1.64249	−0.0651288
$637$	−11.4241	−0.452638
$638$	−6.17525	−0.244481
$639$	−8.52953	−0.337423
$640$	−26.2839	−1.03896
$641$	24.3860	0.963187	0.481594	−	0.876395i	$-0.340058\pi$
$641$	24.3860	0.963187	0.481594	+	0.876395i	$0.340058\pi$
$642$	−19.2464	−0.759593
$643$	14.8551	0.585828	0.292914	−	0.956139i	$-0.405375\pi$
$643$	14.8551	0.585828	0.292914	+	0.956139i	$0.405375\pi$
$644$	4.33809	0.170945
$645$	8.14428	0.320681
$646$	−0.0775849	−0.00305254
$647$	1.04461	0.0410677	0.0205338	−	0.999789i	$-0.493463\pi$
$647$	1.04461	0.0410677	0.0205338	+	0.999789i	$0.493463\pi$
$648$	−2.37786	−0.0934112
$649$	−5.48337	−0.215241
$650$	13.5445	0.531258
$651$	10.3530	0.405767
$652$	5.83079	0.228351
$653$	49.0306	1.91872	0.959358	−	0.282191i	$-0.0910611\pi$
$653$	49.0306	1.91872	0.959358	+	0.282191i	$0.0910611\pi$
$654$	3.40094	0.132987
$655$	−6.67947	−0.260988
$656$	−34.1486	−1.33328
$657$	6.25191	0.243910
$658$	7.77899	0.303257
$659$	−2.45561	−0.0956570	−0.0478285	−	0.998856i	$-0.515230\pi$
$659$	−2.45561	−0.0956570	−0.0478285	+	0.998856i	$0.515230\pi$
$660$	−0.724622	−0.0282059
$661$	13.6176	0.529662	0.264831	−	0.964295i	$-0.414684\pi$
$661$	13.6176	0.529662	0.264831	+	0.964295i	$0.414684\pi$
$662$	−5.56827	−0.216417
$663$	6.68474	0.259614
$664$	32.4336	1.25867
$665$	−0.279497	−0.0108384
$666$	−18.1270	−0.702405
$667$	−15.2924	−0.592123
$668$	−8.83141	−0.341698
$669$	−3.57541	−0.138233
$670$	−12.6829	−0.489982
$671$	3.56412	0.137591
$672$	−8.07853	−0.311636
$673$	−12.0609	−0.464914	−0.232457	−	0.972607i	$-0.574677\pi$
$673$	−12.0609	−0.464914	−0.232457	+	0.972607i	$0.574677\pi$
$674$	−13.0071	−0.501015
$675$	1.28290	0.0493788
$676$	15.6664	0.602554
$677$	23.5258	0.904169	0.452084	−	0.891975i	$-0.350681\pi$
$677$	23.5258	0.904169	0.452084	+	0.891975i	$0.350681\pi$
$678$	−29.4618	−1.13148
$679$	−29.1022	−1.11684
$680$	4.58447	0.175806
$681$	9.56215	0.366422
$682$	−4.21186	−0.161280
$683$	19.3554	0.740615	0.370307	−	0.928909i	$-0.379252\pi$
$683$	19.3554	0.740615	0.370307	+	0.928909i	$0.379252\pi$
$684$	−0.0242883	−0.000928685 0
$685$	35.3686	1.35137
$686$	24.6609	0.941558
$687$	24.0821	0.918788
$688$	−20.0416	−0.764078
$689$	−22.2066	−0.846003
$690$	−9.05311	−0.344646
$691$	43.9472	1.67183	0.835915	−	0.548859i	$-0.184937\pi$
$691$	43.9472	1.67183	0.835915	+	0.548859i	$0.184937\pi$
$692$	3.67856	0.139838
$693$	2.24330	0.0852161
$694$	−39.2270	−1.48904
$695$	5.47764	0.207779
$696$	12.2307	0.463603
$697$	7.19768	0.272631
$698$	28.9592	1.09612
$699$	7.98320	0.301952
$700$	1.87189	0.0707508
$701$	2.88118	0.108821	0.0544104	−	0.998519i	$-0.482672\pi$
$701$	2.88118	0.108821	0.0544104	+	0.998519i	$0.482672\pi$
$702$	10.5577	0.398476
$703$	0.563807	0.0212644
$704$	−3.92645	−0.147984
$705$	−3.21778	−0.121189
$706$	−17.3888	−0.654435
$707$	−1.33081	−0.0500501
$708$	−3.56654	−0.134039
$709$	32.8683	1.23439	0.617197	−	0.786809i	$-0.288268\pi$
$709$	32.8683	1.23439	0.617197	+	0.786809i	$0.288268\pi$
$710$	25.9724	0.974729
$711$	−1.00000	−0.0375029
$712$	−8.77466	−0.328844
$713$	−10.4302	−0.390615
$714$	4.66089	0.174430
$715$	−9.79697	−0.366386
$716$	12.1496	0.454052
$717$	24.4595	0.913456
$718$	7.99353	0.298316
$719$	−32.7634	−1.22187	−0.610935	−	0.791681i	$-0.709206\pi$
$719$	−32.7634	−1.22187	−0.610935	+	0.791681i	$0.709206\pi$
$720$	9.14710	0.340892
$721$	55.9498	2.08368
$722$	−30.0043	−1.11665
$723$	−20.5608	−0.764665
$724$	−2.38010	−0.0884556
$725$	−6.59867	−0.245069
$726$	16.4605	0.610907
$727$	35.2404	1.30699	0.653497	−	0.756930i	$-0.273301\pi$
$727$	35.2404	1.30699	0.653497	+	0.756930i	$0.273301\pi$
$728$	−46.9089	−1.73856
$729$	1.00000	0.0370370
$730$	−19.0371	−0.704594
$731$	4.22426	0.156240
$732$	2.31821	0.0856834
$733$	38.4003	1.41835	0.709174	−	0.705033i	$-0.249068\pi$
$733$	38.4003	1.41835	0.709174	+	0.705033i	$0.249068\pi$
$734$	−44.9565	−1.65938
$735$	3.29487	0.121533
$736$	8.13878	0.299999
$737$	−3.16617	−0.116627
$738$	11.3678	0.418456
$739$	−11.2185	−0.412679	−0.206339	−	0.978480i	$-0.566155\pi$
$739$	−11.2185	−0.412679	−0.206339	+	0.978480i	$0.566155\pi$
$740$	10.9407	0.402189
$741$	−0.328380	−0.0120633
$742$	−15.4834	−0.568413
$743$	30.2492	1.10974	0.554868	−	0.831939i	$-0.312769\pi$
$743$	30.2492	1.10974	0.554868	+	0.831939i	$0.312769\pi$
$744$	8.34200	0.305833
$745$	14.3998	0.527568
$746$	−57.9665	−2.12230
$747$	−13.6398	−0.499054
$748$	−0.375845	−0.0137423
$749$	−35.9622	−1.31403
$750$	−19.1315	−0.698582
$751$	−37.4400	−1.36620	−0.683102	−	0.730323i	$-0.739370\pi$
$751$	−37.4400	−1.36620	−0.683102	+	0.730323i	$0.739370\pi$
$752$	7.91836	0.288753
$753$	−30.2363	−1.10187
$754$	−54.3044	−1.97765
$755$	−30.4395	−1.10781
$756$	1.45911	0.0530673
$757$	5.03086	0.182850	0.0914249	−	0.995812i	$-0.470858\pi$
$757$	5.03086	0.182850	0.0914249	+	0.995812i	$0.470858\pi$
$758$	9.51818	0.345716
$759$	−2.26003	−0.0820340
$760$	−0.225206	−0.00816909
$761$	−23.8815	−0.865702	−0.432851	−	0.901465i	$-0.642492\pi$
$761$	−23.8815	−0.865702	−0.432851	+	0.901465i	$0.642492\pi$
$762$	14.7773	0.535324
$763$	6.35473	0.230057
$764$	−9.33304	−0.337658
$765$	−1.92798	−0.0697062
$766$	24.7241	0.893316
$767$	−48.2200	−1.74112
$768$	−11.2009	−0.404176
$769$	50.0853	1.80612	0.903061	−	0.429513i	$-0.141315\pi$
$769$	50.0853	1.80612	0.903061	+	0.429513i	$0.141315\pi$
$770$	−6.83087	−0.246167
$771$	27.1030	0.976092
$772$	−5.73013	−0.206232
$773$	14.3679	0.516776	0.258388	−	0.966041i	$-0.416809\pi$
$773$	14.3679	0.516776	0.258388	+	0.966041i	$0.416809\pi$
$774$	6.67170	0.239809
$775$	−4.50065	−0.161668
$776$	−23.4493	−0.841779
$777$	−33.8706	−1.21510
$778$	0.171414	0.00614549
$779$	−0.353577	−0.0126682
$780$	−6.37224	−0.228163
$781$	6.48380	0.232009
$782$	−4.69565	−0.167916
$783$	−5.14357	−0.183816
$784$	−8.10805	−0.289573
$785$	−30.7460	−1.09737
$786$	−5.47174	−0.195170
$787$	−15.7286	−0.560666	−0.280333	−	0.959903i	$-0.590445\pi$
$787$	−15.7286	−0.560666	−0.280333	+	0.959903i	$0.590445\pi$
$788$	−8.41063	−0.299616
$789$	−14.0669	−0.500795
$790$	3.04500	0.108336
$791$	−55.0501	−1.95735
$792$	1.80755	0.0642286
$793$	31.3424	1.11300
$794$	−21.4309	−0.760553
$795$	6.40470	0.227151
$796$	0.153363	0.00543579
$797$	3.13786	0.111149	0.0555743	−	0.998455i	$-0.482301\pi$
$797$	3.13786	0.111149	0.0555743	+	0.998455i	$0.482301\pi$
$798$	−0.228961	−0.00810512
$799$	−1.66899	−0.0590447
$800$	3.51189	0.124164
$801$	3.69015	0.130385
$802$	3.07350	0.108529
$803$	−4.75245	−0.167710
$804$	−2.05937	−0.0726284
$805$	−16.9159	−0.596208
$806$	−37.0385	−1.30463
$807$	−1.93011	−0.0679432
$808$	−1.07230	−0.0377235
$809$	37.1011	1.30441	0.652203	−	0.758045i	$-0.273845\pi$
$809$	37.1011	1.30441	0.652203	+	0.758045i	$0.273845\pi$
$810$	−3.04500	−0.106991
$811$	−19.5796	−0.687534	−0.343767	−	0.939055i	$-0.611703\pi$
$811$	−19.5796	−0.687534	−0.343767	+	0.939055i	$0.611703\pi$
$812$	−7.50504	−0.263375
$813$	16.1680	0.567037
$814$	13.7794	0.482967
$815$	−22.7366	−0.796427
$816$	4.74440	0.166087
$817$	−0.207511	−0.00725991
$818$	−23.6384	−0.826497
$819$	19.7273	0.689329
$820$	−6.86119	−0.239603
$821$	−2.20883	−0.0770888	−0.0385444	−	0.999257i	$-0.512272\pi$
$821$	−2.20883	−0.0770888	−0.0385444	+	0.999257i	$0.512272\pi$
$822$	28.9736	1.01057
$823$	−33.8605	−1.18030	−0.590152	−	0.807292i	$-0.700932\pi$
$823$	−33.8605	−1.18030	−0.590152	+	0.807292i	$0.700932\pi$
$824$	45.0818	1.57050
$825$	−0.975207	−0.0339523
$826$	−33.6211	−1.16983
$827$	−43.9911	−1.52972	−0.764861	−	0.644195i	$-0.777192\pi$
$827$	−43.9911	−1.52972	−0.764861	+	0.644195i	$0.777192\pi$
$828$	−1.46999	−0.0510857
$829$	10.0313	0.348401	0.174201	−	0.984710i	$-0.444266\pi$
$829$	10.0313	0.348401	0.174201	+	0.984710i	$0.444266\pi$
$830$	41.5332	1.44164
$831$	−24.5593	−0.851952
$832$	−34.5287	−1.19707
$833$	1.70897	0.0592124
$834$	4.48722	0.155380
$835$	34.4372	1.19175
$836$	0.0184629	0.000638554 0
$837$	−3.50819	−0.121261
$838$	29.4246	1.01645
$839$	32.0729	1.10728	0.553640	−	0.832756i	$-0.313238\pi$
$839$	32.0729	1.10728	0.553640	+	0.832756i	$0.313238\pi$
$840$	13.5292	0.466802
$841$	−2.54369	−0.0877135
$842$	−4.99715	−0.172213
$843$	3.02931	0.104335
$844$	4.98076	0.171445
$845$	−61.0896	−2.10154
$846$	−2.63597	−0.0906263
$847$	30.7568	1.05682
$848$	−15.7608	−0.541228
$849$	−20.8760	−0.716463
$850$	−2.02618	−0.0694974
$851$	34.1232	1.16973
$852$	4.21725	0.144481
$853$	35.5533	1.21732	0.608661	−	0.793430i	$-0.291707\pi$
$853$	35.5533	1.21732	0.608661	+	0.793430i	$0.291707\pi$
$854$	21.8533	0.747804
$855$	0.0947095	0.00323900
$856$	−28.9767	−0.990405
$857$	1.65425	0.0565082	0.0282541	−	0.999601i	$-0.491005\pi$
$857$	1.65425	0.0565082	0.0282541	+	0.999601i	$0.491005\pi$
$858$	−8.02555	−0.273988
$859$	27.8559	0.950431	0.475216	−	0.879869i	$-0.342370\pi$
$859$	27.8559	0.950431	0.475216	+	0.879869i	$0.342370\pi$
$860$	−4.02678	−0.137312
$861$	21.2410	0.723893
$862$	−36.9603	−1.25887
$863$	−10.1950	−0.347044	−0.173522	−	0.984830i	$-0.555515\pi$
$863$	−10.1950	−0.347044	−0.173522	+	0.984830i	$0.555515\pi$
$864$	2.73747	0.0931306
$865$	−14.3442	−0.487716
$866$	−47.2246	−1.60476
$867$	−1.00000	−0.0339618
$868$	−5.11884	−0.173745
$869$	0.760159	0.0257866
$870$	15.6622	0.530998
$871$	−27.8429	−0.943421
$872$	5.12035	0.173397
$873$	9.86149	0.333761
$874$	0.230668	0.00780246
$875$	−35.7475	−1.20849
$876$	−3.09113	−0.104440
$877$	−21.3527	−0.721031	−0.360515	−	0.932753i	$-0.617399\pi$
$877$	−21.3527	−0.721031	−0.360515	+	0.932753i	$0.617399\pi$
$878$	52.7480	1.78016
$879$	−18.9097	−0.637807
$880$	−6.95325	−0.234394
$881$	20.3072	0.684166	0.342083	−	0.939670i	$-0.388868\pi$
$881$	20.3072	0.684166	0.342083	+	0.939670i	$0.388868\pi$
$882$	2.69911	0.0908839
$883$	5.04646	0.169827	0.0849135	−	0.996388i	$-0.472939\pi$
$883$	5.04646	0.169827	0.0849135	+	0.996388i	$0.472939\pi$
$884$	−3.30514	−0.111164
$885$	13.9074	0.467491
$886$	−26.5850	−0.893142
$887$	−28.1316	−0.944568	−0.472284	−	0.881446i	$-0.656570\pi$
$887$	−28.1316	−0.944568	−0.472284	+	0.881446i	$0.656570\pi$
$888$	−27.2914	−0.915840
$889$	27.6116	0.926065
$890$	−11.2365	−0.376649
$891$	−0.760159	−0.0254663
$892$	1.76779	0.0591900
$893$	0.0819871	0.00274359
$894$	11.7961	0.394522
$895$	−47.3762	−1.58361
$896$	−40.2320	−1.34406
$897$	−19.8744	−0.663588
$898$	−33.7240	−1.12538
$899$	18.0446	0.601822
$900$	−0.634303	−0.0211434
$901$	3.32198	0.110671
$902$	−8.64137	−0.287726
$903$	12.4662	0.414849
$904$	−44.3569	−1.47529
$905$	9.28094	0.308509
$906$	−24.9357	−0.828432
$907$	−1.13816	−0.0377919	−0.0188960	−	0.999821i	$-0.506015\pi$
$907$	−1.13816	−0.0377919	−0.0188960	+	0.999821i	$0.506015\pi$
$908$	−4.72781	−0.156898
$909$	0.450953	0.0149572
$910$	−60.0698	−1.99129
$911$	7.32977	0.242846	0.121423	−	0.992601i	$-0.461254\pi$
$911$	7.32977	0.242846	0.121423	+	0.992601i	$0.461254\pi$
$912$	−0.233063	−0.00771748
$913$	10.3684	0.343145
$914$	39.2276	1.29753
$915$	−9.03961	−0.298840
$916$	−11.9069	−0.393415
$917$	−10.2241	−0.337628
$918$	−1.57938	−0.0521272
$919$	35.9920	1.18727	0.593633	−	0.804736i	$-0.297693\pi$
$919$	35.9920	1.18727	0.593633	+	0.804736i	$0.297693\pi$
$920$	−13.6301	−0.449371
$921$	−5.19563	−0.171202
$922$	29.7361	0.979306
$923$	57.0177	1.87676
$924$	−1.10916	−0.0364886
$925$	14.7242	0.484128
$926$	−4.58930	−0.150814
$927$	−18.9590	−0.622695
$928$	−14.0804	−0.462210
$929$	−27.8623	−0.914134	−0.457067	−	0.889432i	$-0.651100\pi$
$929$	−27.8623	−0.914134	−0.457067	+	0.889432i	$0.651100\pi$
$930$	10.6825	0.350292
$931$	−0.0839512	−0.00275139
$932$	−3.94713	−0.129293
$933$	−18.1563	−0.594409
$934$	0.105623	0.00345610
$935$	1.46557	0.0479293
$936$	15.8954	0.519557
$937$	25.0720	0.819067	0.409533	−	0.912295i	$-0.365692\pi$
$937$	25.0720	0.819067	0.409533	+	0.912295i	$0.365692\pi$
$938$	−19.4133	−0.633866
$939$	−10.5030	−0.342753
$940$	1.59097	0.0518916
$941$	3.87071	0.126182	0.0630908	−	0.998008i	$-0.479904\pi$
$941$	3.87071	0.126182	0.0630908	+	0.998008i	$0.479904\pi$
$942$	−25.1868	−0.820630
$943$	−21.3994	−0.696862
$944$	−34.2235	−1.11388
$945$	−5.68965	−0.185084
$946$	−5.07155	−0.164890
$947$	26.0036	0.845004	0.422502	−	0.906362i	$-0.361152\pi$
$947$	26.0036	0.845004	0.422502	+	0.906362i	$0.361152\pi$
$948$	0.494430	0.0160583
$949$	−41.7924	−1.35664
$950$	0.0995335	0.00322929
$951$	−4.33435	−0.140551
$952$	7.01730	0.227432
$953$	−30.4954	−0.987843	−0.493922	−	0.869506i	$-0.664437\pi$
$953$	−30.4954	−0.987843	−0.493922	+	0.869506i	$0.664437\pi$
$954$	5.24665	0.169867
$955$	36.3932	1.17766
$956$	−12.0935	−0.391132
$957$	3.90993	0.126390
$958$	18.1505	0.586417
$959$	54.1377	1.74820
$960$	9.95860	0.321412
$961$	−18.6926	−0.602986
$962$	121.174	3.90681
$963$	12.1861	0.392690
$964$	10.1659	0.327421
$965$	22.3441	0.719281
$966$	−13.8573	−0.445852
$967$	45.1347	1.45144	0.725718	−	0.687993i	$-0.241508\pi$
$967$	45.1347	1.45144	0.725718	+	0.687993i	$0.241508\pi$
$968$	24.7825	0.796538
$969$	0.0491237	0.00157808
$970$	−30.0283	−0.964150
$971$	23.8356	0.764921	0.382461	−	0.923972i	$-0.375077\pi$
$971$	23.8356	0.764921	0.382461	+	0.923972i	$0.375077\pi$
$972$	−0.494430	−0.0158588
$973$	8.38446	0.268793
$974$	47.3059	1.51578
$975$	−8.57584	−0.274647
$976$	22.2448	0.712039
$977$	−50.1349	−1.60396	−0.801979	−	0.597352i	$-0.796219\pi$
$977$	−50.1349	−1.60396	−0.801979	+	0.597352i	$0.796219\pi$
$978$	−18.6255	−0.595578
$979$	−2.80510	−0.0896514
$980$	−1.62908	−0.0520391
$981$	−2.15334	−0.0687510
$982$	−14.5899	−0.465583
$983$	14.1305	0.450692	0.225346	−	0.974279i	$-0.427649\pi$
$983$	14.1305	0.450692	0.225346	+	0.974279i	$0.427649\pi$
$984$	17.1151	0.545609
$985$	32.7964	1.04498
$986$	8.12363	0.258709
$987$	−4.92535	−0.156776
$988$	0.162361	0.00516538
$989$	−12.5592	−0.399358
$990$	2.31469	0.0735656
$991$	7.97359	0.253289	0.126645	−	0.991948i	$-0.459579\pi$
$991$	7.97359	0.253289	0.126645	+	0.991948i	$0.459579\pi$
$992$	−9.60357	−0.304914
$993$	3.52561	0.111882
$994$	39.7552	1.26096
$995$	−0.598021	−0.0189586
$996$	6.74393	0.213689
$997$	40.2940	1.27612	0.638062	−	0.769985i	$-0.279736\pi$
$997$	40.2940	1.27612	0.638062	+	0.769985i	$0.279736\pi$
$998$	−0.556540	−0.0176170
$999$	11.4773	0.363125

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4029.2.a.i.1.19	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4029.2.a.i.1.19	✓	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)

\(f(q)\)	\(=\)	\(q+1.57938 q^{2} -1.00000 q^{3} +0.494430 q^{4} -1.92798 q^{5} -1.57938 q^{6} -2.95110 q^{7} -2.37786 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.57938 q^{2} -1.00000 q^{3} +0.494430 q^{4} -1.92798 q^{5} -1.57938 q^{6} -2.95110 q^{7} -2.37786 q^{8} +1.00000 q^{9} -3.04500 q^{10} -0.760159 q^{11} -0.494430 q^{12} -6.68474 q^{13} -4.66089 q^{14} +1.92798 q^{15} -4.74440 q^{16} +1.00000 q^{17} +1.57938 q^{18} -0.0491237 q^{19} -0.953250 q^{20} +2.95110 q^{21} -1.20058 q^{22} -2.97310 q^{23} +2.37786 q^{24} -1.28290 q^{25} -10.5577 q^{26} -1.00000 q^{27} -1.45911 q^{28} +5.14357 q^{29} +3.04500 q^{30} +3.50819 q^{31} -2.73747 q^{32} +0.760159 q^{33} +1.57938 q^{34} +5.68965 q^{35} +0.494430 q^{36} -11.4773 q^{37} -0.0775849 q^{38} +6.68474 q^{39} +4.58447 q^{40} +7.19768 q^{41} +4.66089 q^{42} +4.22426 q^{43} -0.375845 q^{44} -1.92798 q^{45} -4.69565 q^{46} -1.66899 q^{47} +4.74440 q^{48} +1.70897 q^{49} -2.02618 q^{50} -1.00000 q^{51} -3.30514 q^{52} +3.32198 q^{53} -1.57938 q^{54} +1.46557 q^{55} +7.01730 q^{56} +0.0491237 q^{57} +8.12363 q^{58} +7.21345 q^{59} +0.953250 q^{60} -4.68865 q^{61} +5.54076 q^{62} -2.95110 q^{63} +5.16531 q^{64} +12.8880 q^{65} +1.20058 q^{66} +4.16514 q^{67} +0.494430 q^{68} +2.97310 q^{69} +8.98610 q^{70} -8.52953 q^{71} -2.37786 q^{72} +6.25191 q^{73} -18.1270 q^{74} +1.28290 q^{75} -0.0242883 q^{76} +2.24330 q^{77} +10.5577 q^{78} -1.00000 q^{79} +9.14710 q^{80} +1.00000 q^{81} +11.3678 q^{82} -13.6398 q^{83} +1.45911 q^{84} -1.92798 q^{85} +6.67170 q^{86} -5.14357 q^{87} +1.80755 q^{88} +3.69015 q^{89} -3.04500 q^{90} +19.7273 q^{91} -1.46999 q^{92} -3.50819 q^{93} -2.63597 q^{94} +0.0947095 q^{95} +2.73747 q^{96} +9.86149 q^{97} +2.69911 q^{98} -0.760159 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 2 q^{2} - 25 q^{3} + 26 q^{4} - 2 q^{5} + 2 q^{6} + 12 q^{7} + 25 q^{9}+O(q^{10}) \) 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