LMFDB - Embedded newform 6422.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.10273.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + x + 2 $ x^4 - 2x^3 - 5x^2 + x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.673533$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -0.673533 q^{3} +1.00000 q^{4} +0.0759953 q^{5} -0.673533 q^{6} -3.29588 q^{7} +1.00000 q^{8} -2.54635 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.673533 q^{3} +1.00000 q^{4} +0.0759953 q^{5} -0.673533 q^{6} -3.29588 q^{7} +1.00000 q^{8} -2.54635 q^{9} +0.0759953 q^{10} +2.89342 q^{11} -0.673533 q^{12} -3.29588 q^{14} -0.0511853 q^{15} +1.00000 q^{16} +4.16449 q^{17} -2.54635 q^{18} +1.00000 q^{19} +0.0759953 q^{20} +2.21989 q^{21} +2.89342 q^{22} -2.67353 q^{23} -0.673533 q^{24} -4.99422 q^{25} +3.73565 q^{27} -3.29588 q^{28} +5.32069 q^{29} -0.0511853 q^{30} -2.40246 q^{31} +1.00000 q^{32} -1.94881 q^{33} +4.16449 q^{34} -0.250471 q^{35} -2.54635 q^{36} +6.16449 q^{37} +1.00000 q^{38} +0.0759953 q^{40} -0.478457 q^{41} +2.21989 q^{42} -7.31648 q^{43} +2.89342 q^{44} -0.193511 q^{45} -2.67353 q^{46} -11.8197 q^{47} -0.673533 q^{48} +3.86283 q^{49} -4.99422 q^{50} -2.80492 q^{51} +1.21989 q^{53} +3.73565 q^{54} +0.219886 q^{55} -3.29588 q^{56} -0.673533 q^{57} +5.32069 q^{58} +2.83225 q^{59} -0.0511853 q^{60} +1.99422 q^{61} -2.40246 q^{62} +8.39248 q^{63} +1.00000 q^{64} -1.94881 q^{66} -1.73722 q^{67} +4.16449 q^{68} +1.80071 q^{69} -0.250471 q^{70} +2.08021 q^{71} -2.54635 q^{72} -9.32898 q^{73} +6.16449 q^{74} +3.36378 q^{75} +1.00000 q^{76} -9.53637 q^{77} -9.61425 q^{79} +0.0759953 q^{80} +5.12297 q^{81} -0.478457 q^{82} -2.18930 q^{83} +2.21989 q^{84} +0.316482 q^{85} -7.31648 q^{86} -3.58366 q^{87} +2.89342 q^{88} +2.01231 q^{89} -0.193511 q^{90} -2.67353 q^{92} +1.61814 q^{93} -11.8197 q^{94} +0.0759953 q^{95} -0.673533 q^{96} +17.7753 q^{97} +3.86283 q^{98} -7.36767 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} + q^{5} - 2 q^{6} - q^{7} + 4 q^{8} + 2 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 + q^5 - 2 * q^6 - q^7 + 4 * q^8 + 2 * q^9	$ 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} + q^{5} - 2 q^{6} - q^{7} + 4 q^{8} + 2 q^{9} + q^{10} - 2 q^{11} - 2 q^{12} - q^{14} - 11 q^{15} + 4 q^{16} + q^{17} + 2 q^{18} + 4 q^{19} + q^{20} - 4 q^{21} - 2 q^{22} - 10 q^{23} - 2 q^{24} + 3 q^{25} - 23 q^{27} - q^{28} - q^{29} - 11 q^{30} - 11 q^{31} + 4 q^{32} + 3 q^{33} + q^{34} - q^{35} + 2 q^{36} + 9 q^{37} + 4 q^{38} + q^{40} - 4 q^{41} - 4 q^{42} - 15 q^{43} - 2 q^{44} + 33 q^{45} - 10 q^{46} - 25 q^{47} - 2 q^{48} - 11 q^{49} + 3 q^{50} - 14 q^{51} - 8 q^{53} - 23 q^{54} - 12 q^{55} - q^{56} - 2 q^{57} - q^{58} - 28 q^{59} - 11 q^{60} - 15 q^{61} - 11 q^{62} + 20 q^{63} + 4 q^{64} + 3 q^{66} + q^{68} + 18 q^{69} - q^{70} + q^{71} + 2 q^{72} - 6 q^{73} + 9 q^{74} - 13 q^{75} + 4 q^{76} - 11 q^{77} - 12 q^{79} + q^{80} + 40 q^{81} - 4 q^{82} + 17 q^{83} - 4 q^{84} - 13 q^{85} - 15 q^{86} + 25 q^{87} - 2 q^{88} + 15 q^{89} + 33 q^{90} - 10 q^{92} + 3 q^{93} - 25 q^{94} + q^{95} - 2 q^{96} - 2 q^{97} - 11 q^{98} - 26 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 + q^5 - 2 * q^6 - q^7 + 4 * q^8 + 2 * q^9 + q^10 - 2 * q^11 - 2 * q^12 - q^14 - 11 * q^15 + 4 * q^16 + q^17 + 2 * q^18 + 4 * q^19 + q^20 - 4 * q^21 - 2 * q^22 - 10 * q^23 - 2 * q^24 + 3 * q^25 - 23 * q^27 - q^28 - q^29 - 11 * q^30 - 11 * q^31 + 4 * q^32 + 3 * q^33 + q^34 - q^35 + 2 * q^36 + 9 * q^37 + 4 * q^38 + q^40 - 4 * q^41 - 4 * q^42 - 15 * q^43 - 2 * q^44 + 33 * q^45 - 10 * q^46 - 25 * q^47 - 2 * q^48 - 11 * q^49 + 3 * q^50 - 14 * q^51 - 8 * q^53 - 23 * q^54 - 12 * q^55 - q^56 - 2 * q^57 - q^58 - 28 * q^59 - 11 * q^60 - 15 * q^61 - 11 * q^62 + 20 * q^63 + 4 * q^64 + 3 * q^66 + q^68 + 18 * q^69 - q^70 + q^71 + 2 * q^72 - 6 * q^73 + 9 * q^74 - 13 * q^75 + 4 * q^76 - 11 * q^77 - 12 * q^79 + q^80 + 40 * q^81 - 4 * q^82 + 17 * q^83 - 4 * q^84 - 13 * q^85 - 15 * q^86 + 25 * q^87 - 2 * q^88 + 15 * q^89 + 33 * q^90 - 10 * q^92 + 3 * q^93 - 25 * q^94 + q^95 - 2 * q^96 - 2 * q^97 - 11 * q^98 - 26 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−0.673533	−0.388865	−0.194432	−	0.980916i	$-0.562286\pi$
$3$	−0.673533	−0.388865	−0.194432	+	0.980916i	$0.562286\pi$
$4$	1.00000	0.500000
$5$	0.0759953	0.0339861	0.0169931	−	0.999856i	$-0.494591\pi$
$5$	0.0759953	0.0339861	0.0169931	+	0.999856i	$0.494591\pi$
$6$	−0.673533	−0.274969
$7$	−3.29588	−1.24573	−0.622863	−	0.782331i	$-0.714031\pi$
$7$	−3.29588	−1.24573	−0.622863	+	0.782331i	$0.714031\pi$
$8$	1.00000	0.353553
$9$	−2.54635	−0.848784
$10$	0.0759953	0.0240318
$11$	2.89342	0.872399	0.436199	−	0.899850i	$-0.356324\pi$
$11$	2.89342	0.872399	0.436199	+	0.899850i	$0.356324\pi$
$12$	−0.673533	−0.194432
$13$	0	0
$13$	0	0
$14$	−3.29588	−0.880861
$15$	−0.0511853	−0.0132160
$16$	1.00000	0.250000
$17$	4.16449	1.01004	0.505019	−	0.863108i	$-0.331486\pi$
$17$	4.16449	1.01004	0.505019	+	0.863108i	$0.331486\pi$
$18$	−2.54635	−0.600181
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0.0759953	0.0169931
$21$	2.21989	0.484419
$22$	2.89342	0.616879
$23$	−2.67353	−0.557470	−0.278735	−	0.960368i	$-0.589915\pi$
$23$	−2.67353	−0.557470	−0.278735	+	0.960368i	$0.589915\pi$
$24$	−0.673533	−0.137484
$25$	−4.99422	−0.998845
$26$	0	0
$27$	3.73565	0.718927
$28$	−3.29588	−0.622863
$29$	5.32069	0.988028	0.494014	−	0.869454i	$-0.335529\pi$
$29$	5.32069	0.988028	0.494014	+	0.869454i	$0.335529\pi$
$30$	−0.0511853	−0.00934512
$31$	−2.40246	−0.431495	−0.215747	−	0.976449i	$-0.569219\pi$
$31$	−2.40246	−0.431495	−0.215747	+	0.976449i	$0.569219\pi$
$32$	1.00000	0.176777
$33$	−1.94881	−0.339245
$34$	4.16449	0.714204
$35$	−0.250471	−0.0423374
$36$	−2.54635	−0.424392
$37$	6.16449	1.01344	0.506718	−	0.862112i	$-0.330859\pi$
$37$	6.16449	1.01344	0.506718	+	0.862112i	$0.330859\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0.0759953	0.0120159
$41$	−0.478457	−0.0747225	−0.0373612	−	0.999302i	$-0.511895\pi$
$41$	−0.478457	−0.0747225	−0.0373612	+	0.999302i	$0.511895\pi$
$42$	2.21989	0.342536
$43$	−7.31648	−1.11575	−0.557876	−	0.829924i	$-0.688384\pi$
$43$	−7.31648	−1.11575	−0.557876	+	0.829924i	$0.688384\pi$
$44$	2.89342	0.436199
$45$	−0.193511	−0.0288469
$46$	−2.67353	−0.394191
$47$	−11.8197	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$47$	−11.8197	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$48$	−0.673533	−0.0972162
$49$	3.86283	0.551833
$50$	−4.99422	−0.706290
$51$	−2.80492	−0.392768
$52$	0	0
$53$	1.21989	0.167564	0.0837821	−	0.996484i	$-0.473300\pi$
$53$	1.21989	0.167564	0.0837821	+	0.996484i	$0.473300\pi$
$54$	3.73565	0.508358
$55$	0.219886	0.0296494
$56$	−3.29588	−0.440431
$57$	−0.673533	−0.0892117
$58$	5.32069	0.698641
$59$	2.83225	0.368727	0.184364	−	0.982858i	$-0.440978\pi$
$59$	2.83225	0.368727	0.184364	+	0.982858i	$0.440978\pi$
$60$	−0.0511853	−0.00660800
$61$	1.99422	0.255334	0.127667	−	0.991817i	$-0.459251\pi$
$61$	1.99422	0.255334	0.127667	+	0.991817i	$0.459251\pi$
$62$	−2.40246	−0.305113
$63$	8.39248	1.05735
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.94881	−0.239883
$67$	−1.73722	−0.212235	−0.106118	−	0.994354i	$-0.533842\pi$
$67$	−1.73722	−0.212235	−0.106118	+	0.994354i	$0.533842\pi$
$68$	4.16449	0.505019
$69$	1.80071	0.216780
$70$	−0.250471	−0.0299371
$71$	2.08021	0.246875	0.123437	−	0.992352i	$-0.460608\pi$
$71$	2.08021	0.246875	0.123437	+	0.992352i	$0.460608\pi$
$72$	−2.54635	−0.300091
$73$	−9.32898	−1.09187	−0.545937	−	0.837826i	$-0.683826\pi$
$73$	−9.32898	−1.09187	−0.545937	+	0.837826i	$0.683826\pi$
$74$	6.16449	0.716608
$75$	3.36378	0.388416
$76$	1.00000	0.114708
$77$	−9.53637	−1.08677
$78$	0	0
$79$	−9.61425	−1.08169	−0.540844	−	0.841123i	$-0.681895\pi$
$79$	−9.61425	−1.08169	−0.540844	+	0.841123i	$0.681895\pi$
$80$	0.0759953	0.00849653
$81$	5.12297	0.569219
$82$	−0.478457	−0.0528368
$83$	−2.18930	−0.240307	−0.120154	−	0.992755i	$-0.538339\pi$
$83$	−2.18930	−0.240307	−0.120154	+	0.992755i	$0.538339\pi$
$84$	2.21989	0.242209
$85$	0.316482	0.0343272
$86$	−7.31648	−0.788956
$87$	−3.58366	−0.384209
$88$	2.89342	0.308440
$89$	2.01231	0.213304	0.106652	−	0.994296i	$-0.465987\pi$
$89$	2.01231	0.213304	0.106652	+	0.994296i	$0.465987\pi$
$90$	−0.193511	−0.0203978
$91$	0	0
$92$	−2.67353	−0.278735
$93$	1.61814	0.167793
$94$	−11.8197	−1.21911
$95$	0.0759953	0.00779695
$96$	−0.673533	−0.0687422
$97$	17.7753	1.80481	0.902404	−	0.430892i	$-0.141801\pi$
$97$	17.7753	1.80481	0.902404	+	0.430892i	$0.141801\pi$
$98$	3.86283	0.390205
$99$	−7.36767	−0.740478
$100$	−4.99422	−0.499422
$101$	−17.7688	−1.76806	−0.884029	−	0.467433i	$-0.845179\pi$
$101$	−17.7688	−1.76806	−0.884029	+	0.467433i	$0.845179\pi$
$102$	−2.80492	−0.277729
$103$	13.9150	1.37108	0.685541	−	0.728034i	$-0.259566\pi$
$103$	13.9150	1.37108	0.685541	+	0.728034i	$0.259566\pi$
$104$	0	0
$105$	0.168701	0.0164635
$106$	1.21989	0.118486
$107$	−6.38859	−0.617608	−0.308804	−	0.951126i	$-0.599929\pi$
$107$	−6.38859	−0.617608	−0.308804	+	0.951126i	$0.599929\pi$
$108$	3.73565	0.359463
$109$	−1.57926	−0.151266	−0.0756329	−	0.997136i	$-0.524098\pi$
$109$	−1.57926	−0.151266	−0.0756329	+	0.997136i	$0.524098\pi$
$110$	0.219886	0.0209653
$111$	−4.15199	−0.394089
$112$	−3.29588	−0.311432
$113$	−5.11079	−0.480783	−0.240391	−	0.970676i	$-0.577276\pi$
$113$	−5.11079	−0.480783	−0.240391	+	0.970676i	$0.577276\pi$
$114$	−0.673533	−0.0630822
$115$	−0.203176	−0.0189462
$116$	5.32069	0.494014
$117$	0	0
$118$	2.83225	0.260730
$119$	−13.7257	−1.25823
$120$	−0.0511853	−0.00467256
$121$	−2.62812	−0.238920
$122$	1.99422	0.180549
$123$	0.322257	0.0290569
$124$	−2.40246	−0.215747
$125$	−0.759514	−0.0679330
$126$	8.39248	0.747661
$127$	0.0373105	0.00331077	0.00165538	−	0.999999i	$-0.499473\pi$
$127$	0.0373105	0.00331077	0.00165538	+	0.999999i	$0.499473\pi$
$128$	1.00000	0.0883883
$129$	4.92789	0.433877
$130$	0	0
$131$	−14.6993	−1.28428	−0.642142	−	0.766586i	$-0.721954\pi$
$131$	−14.6993	−1.28428	−0.642142	+	0.766586i	$0.721954\pi$
$132$	−1.94881	−0.169623
$133$	−3.29588	−0.285789
$134$	−1.73722	−0.150073
$135$	0.283892	0.0244335
$136$	4.16449	0.357102
$137$	−16.2843	−1.39126	−0.695632	−	0.718398i	$-0.744876\pi$
$137$	−16.2843	−1.39126	−0.695632	+	0.718398i	$0.744876\pi$
$138$	1.80071	0.153287
$139$	−19.9761	−1.69435	−0.847177	−	0.531311i	$-0.821700\pi$
$139$	−19.9761	−1.69435	−0.847177	+	0.531311i	$0.821700\pi$
$140$	−0.250471	−0.0211687
$141$	7.96100	0.670437
$142$	2.08021	0.174567
$143$	0	0
$144$	−2.54635	−0.212196
$145$	0.404347	0.0335792
$146$	−9.32898	−0.772072
$147$	−2.60175	−0.214589
$148$	6.16449	0.506718
$149$	−4.67448	−0.382949	−0.191474	−	0.981498i	$-0.561327\pi$
$149$	−4.67448	−0.382949	−0.191474	+	0.981498i	$0.561327\pi$
$150$	3.36378	0.274651
$151$	−4.47192	−0.363920	−0.181960	−	0.983306i	$-0.558244\pi$
$151$	−4.47192	−0.363920	−0.181960	+	0.983306i	$0.558244\pi$
$152$	1.00000	0.0811107
$153$	−10.6043	−0.857304
$154$	−9.53637	−0.768462
$155$	−0.182576	−0.0146648
$156$	0	0
$157$	−16.4134	−1.30993	−0.654966	−	0.755658i	$-0.727317\pi$
$157$	−16.4134	−1.30993	−0.654966	+	0.755658i	$0.727317\pi$
$158$	−9.61425	−0.764869
$159$	−0.821634	−0.0651598
$160$	0.0759953	0.00600795
$161$	8.81165	0.694455
$162$	5.12297	0.402499
$163$	7.70896	0.603812	0.301906	−	0.953338i	$-0.402377\pi$
$163$	7.70896	0.603812	0.301906	+	0.953338i	$0.402377\pi$
$164$	−0.478457	−0.0373612
$165$	−0.148101	−0.0115296
$166$	−2.18930	−0.169923
$167$	−7.70896	−0.596537	−0.298269	−	0.954482i	$-0.596409\pi$
$167$	−7.70896	−0.596537	−0.298269	+	0.954482i	$0.596409\pi$
$168$	2.21989	0.171268
$169$	0	0
$170$	0.316482	0.0242730
$171$	−2.54635	−0.194724
$172$	−7.31648	−0.557876
$173$	−1.11331	−0.0846431	−0.0423215	−	0.999104i	$-0.513475\pi$
$173$	−1.11331	−0.0846431	−0.0423215	+	0.999104i	$0.513475\pi$
$174$	−3.58366	−0.271677
$175$	16.4604	1.24429
$176$	2.89342	0.218100
$177$	−1.90761	−0.143385
$178$	2.01231	0.150829
$179$	−16.4571	−1.23006	−0.615031	−	0.788503i	$-0.710857\pi$
$179$	−16.4571	−1.23006	−0.615031	+	0.788503i	$0.710857\pi$
$180$	−0.193511	−0.0144234
$181$	−1.69181	−0.125751	−0.0628756	−	0.998021i	$-0.520027\pi$
$181$	−1.69181	−0.125751	−0.0628756	+	0.998021i	$0.520027\pi$
$182$	0	0
$183$	−1.34318	−0.0992905
$184$	−2.67353	−0.197095
$185$	0.468472	0.0344428
$186$	1.61814	0.118648
$187$	12.0496	0.881155
$188$	−11.8197	−0.862044
$189$	−12.3123	−0.895586
$190$	0.0759953	0.00551328
$191$	25.4002	1.83789	0.918945	−	0.394385i	$-0.129042\pi$
$191$	25.4002	1.83789	0.918945	+	0.394385i	$0.129042\pi$
$192$	−0.673533	−0.0486081
$193$	−12.1216	−0.872532	−0.436266	−	0.899818i	$-0.643699\pi$
$193$	−12.1216	−0.872532	−0.436266	+	0.899818i	$0.643699\pi$
$194$	17.7753	1.27619
$195$	0	0
$196$	3.86283	0.275917
$197$	−12.1916	−0.868617	−0.434309	−	0.900764i	$-0.643007\pi$
$197$	−12.1916	−0.868617	−0.434309	+	0.900764i	$0.643007\pi$
$198$	−7.36767	−0.523597
$199$	−21.5454	−1.52731	−0.763657	−	0.645622i	$-0.776598\pi$
$199$	−21.5454	−1.52731	−0.763657	+	0.645622i	$0.776598\pi$
$200$	−4.99422	−0.353145
$201$	1.17008	0.0825307
$202$	−17.7688	−1.25021
$203$	−17.5364	−1.23081
$204$	−2.80492	−0.196384
$205$	−0.0363605	−0.00253953
$206$	13.9150	0.969502
$207$	6.80776	0.473172
$208$	0	0
$209$	2.89342	0.200142
$210$	0.168701	0.0116415
$211$	8.69087	0.598304	0.299152	−	0.954205i	$-0.403296\pi$
$211$	8.69087	0.598304	0.299152	+	0.954205i	$0.403296\pi$
$212$	1.21989	0.0837821
$213$	−1.40109	−0.0960009
$214$	−6.38859	−0.436715
$215$	−0.556018	−0.0379201
$216$	3.73565	0.254179
$217$	7.91823	0.537524
$218$	−1.57926	−0.106961
$219$	6.28338	0.424592
$220$	0.219886	0.0148247
$221$	0	0
$222$	−4.15199	−0.278663
$223$	20.1710	1.35075	0.675376	−	0.737474i	$-0.263981\pi$
$223$	20.1710	1.35075	0.675376	+	0.737474i	$0.263981\pi$
$224$	−3.29588	−0.220215
$225$	12.7171	0.847804
$226$	−5.11079	−0.339965
$227$	−16.8925	−1.12119	−0.560597	−	0.828089i	$-0.689428\pi$
$227$	−16.8925	−1.12119	−0.560597	+	0.828089i	$0.689428\pi$
$228$	−0.673533	−0.0446058
$229$	8.47099	0.559779	0.279889	−	0.960032i	$-0.409702\pi$
$229$	8.47099	0.559779	0.279889	+	0.960032i	$0.409702\pi$
$230$	−0.203176	−0.0133970
$231$	6.42306	0.422606
$232$	5.32069	0.349320
$233$	−23.6522	−1.54951	−0.774753	−	0.632264i	$-0.782126\pi$
$233$	−23.6522	−1.54951	−0.774753	+	0.632264i	$0.782126\pi$
$234$	0	0
$235$	−0.898245	−0.0585950
$236$	2.83225	0.184364
$237$	6.47552	0.420630
$238$	−13.7257	−0.889703
$239$	9.77685	0.632412	0.316206	−	0.948691i	$-0.397591\pi$
$239$	9.77685	0.632412	0.316206	+	0.948691i	$0.397591\pi$
$240$	−0.0511853	−0.00330400
$241$	−2.19275	−0.141248	−0.0706238	−	0.997503i	$-0.522499\pi$
$241$	−2.19275	−0.141248	−0.0706238	+	0.997503i	$0.522499\pi$
$242$	−2.62812	−0.168942
$243$	−14.6575	−0.940276
$244$	1.99422	0.127667
$245$	0.293557	0.0187547
$246$	0.322257	0.0205463
$247$	0	0
$248$	−2.40246	−0.152556
$249$	1.47457	0.0934469
$250$	−0.759514	−0.0480359
$251$	4.09239	0.258309	0.129155	−	0.991624i	$-0.458774\pi$
$251$	4.09239	0.258309	0.129155	+	0.991624i	$0.458774\pi$
$252$	8.39248	0.528676
$253$	−7.73565	−0.486336
$254$	0.0373105	0.00234107
$255$	−0.213161	−0.0133487
$256$	1.00000	0.0625000
$257$	26.5910	1.65870	0.829351	−	0.558728i	$-0.188710\pi$
$257$	26.5910	1.65870	0.829351	+	0.558728i	$0.188710\pi$
$258$	4.92789	0.306797
$259$	−20.3174	−1.26246
$260$	0	0
$261$	−13.5484	−0.838622
$262$	−14.6993	−0.908125
$263$	−16.4877	−1.01668	−0.508338	−	0.861158i	$-0.669740\pi$
$263$	−16.4877	−1.01668	−0.508338	+	0.861158i	$0.669740\pi$
$264$	−1.94881	−0.119941
$265$	0.0927056	0.00569486
$266$	−3.29588	−0.202083
$267$	−1.35536	−0.0829465
$268$	−1.73722	−0.106118
$269$	−9.83802	−0.599835	−0.299917	−	0.953965i	$-0.596959\pi$
$269$	−9.83802	−0.599835	−0.299917	+	0.953965i	$0.596959\pi$
$270$	0.283892	0.0172771
$271$	−16.1678	−0.982122	−0.491061	−	0.871125i	$-0.663391\pi$
$271$	−16.1678	−0.982122	−0.491061	+	0.871125i	$0.663391\pi$
$272$	4.16449	0.252509
$273$	0	0
$274$	−16.2843	−0.983772
$275$	−14.4504	−0.871391
$276$	1.80071	0.108390
$277$	25.5762	1.53672	0.768362	−	0.640015i	$-0.221072\pi$
$277$	25.5762	1.53672	0.768362	+	0.640015i	$0.221072\pi$
$278$	−19.9761	−1.19809
$279$	6.11752	0.366246
$280$	−0.250471	−0.0149685
$281$	15.5885	0.929932	0.464966	−	0.885329i	$-0.346067\pi$
$281$	15.5885	0.929932	0.464966	+	0.885329i	$0.346067\pi$
$282$	7.96100	0.474070
$283$	−4.81397	−0.286161	−0.143081	−	0.989711i	$-0.545701\pi$
$283$	−4.81397	−0.286161	−0.143081	+	0.989711i	$0.545701\pi$
$284$	2.08021	0.123437
$285$	−0.0511853	−0.00303196
$286$	0	0
$287$	1.57694	0.0930837
$288$	−2.54635	−0.150045
$289$	0.342986	0.0201756
$290$	0.404347	0.0237441
$291$	−11.9723	−0.701826
$292$	−9.32898	−0.545937
$293$	−26.7470	−1.56258	−0.781289	−	0.624169i	$-0.785437\pi$
$293$	−26.7470	−1.56258	−0.781289	+	0.624169i	$0.785437\pi$
$294$	−2.60175	−0.151737
$295$	0.215238	0.0125316
$296$	6.16449	0.358304
$297$	10.8088	0.627191
$298$	−4.67448	−0.270786
$299$	0	0
$300$	3.36378	0.194208
$301$	24.1143	1.38992
$302$	−4.47192	−0.257330
$303$	11.9678	0.687535
$304$	1.00000	0.0573539
$305$	0.151552	0.00867782
$306$	−10.6043	−0.606205
$307$	−15.5546	−0.887750	−0.443875	−	0.896089i	$-0.646397\pi$
$307$	−15.5546	−0.887750	−0.443875	+	0.896089i	$0.646397\pi$
$308$	−9.53637	−0.543385
$309$	−9.37220	−0.533166
$310$	−0.182576	−0.0103696
$311$	−1.28527	−0.0728808	−0.0364404	−	0.999336i	$-0.511602\pi$
$311$	−1.28527	−0.0728808	−0.0364404	+	0.999336i	$0.511602\pi$
$312$	0	0
$313$	−22.3334	−1.26236	−0.631179	−	0.775637i	$-0.717429\pi$
$313$	−22.3334	−1.26236	−0.631179	+	0.775637i	$0.717429\pi$
$314$	−16.4134	−0.926262
$315$	0.637788	0.0359353
$316$	−9.61425	−0.540844
$317$	−22.3093	−1.25302	−0.626508	−	0.779415i	$-0.715517\pi$
$317$	−22.3093	−1.25302	−0.626508	+	0.779415i	$0.715517\pi$
$318$	−0.821634	−0.0460749
$319$	15.3950	0.861954
$320$	0.0759953	0.00424826
$321$	4.30293	0.240166
$322$	8.81165	0.491054
$323$	4.16449	0.231718
$324$	5.12297	0.284609
$325$	0	0
$326$	7.70896	0.426960
$327$	1.06369	0.0588220
$328$	−0.478457	−0.0264184
$329$	38.9565	2.14774
$330$	−0.148101	−0.00815267
$331$	30.9268	1.69989	0.849946	−	0.526869i	$-0.176634\pi$
$331$	30.9268	1.69989	0.849946	+	0.526869i	$0.176634\pi$
$332$	−2.18930	−0.120154
$333$	−15.6970	−0.860189
$334$	−7.70896	−0.421815
$335$	−0.132020	−0.00721305
$336$	2.21989	0.121105
$337$	−23.6429	−1.28791	−0.643957	−	0.765062i	$-0.722708\pi$
$337$	−23.6429	−1.28791	−0.643957	+	0.765062i	$0.722708\pi$
$338$	0	0
$339$	3.44229	0.186959
$340$	0.316482	0.0171636
$341$	−6.95133	−0.376436
$342$	−2.54635	−0.137691
$343$	10.3397	0.558293
$344$	−7.31648	−0.394478
$345$	0.136846	0.00736753
$346$	−1.11331	−0.0598517
$347$	18.2143	0.977795	0.488897	−	0.872341i	$-0.337399\pi$
$347$	18.2143	0.977795	0.488897	+	0.872341i	$0.337399\pi$
$348$	−3.58366	−0.192105
$349$	27.9134	1.49417	0.747085	−	0.664728i	$-0.231453\pi$
$349$	27.9134	1.49417	0.747085	+	0.664728i	$0.231453\pi$
$350$	16.4604	0.879844
$351$	0	0
$352$	2.89342	0.154220
$353$	8.95943	0.476862	0.238431	−	0.971159i	$-0.423367\pi$
$353$	8.95943	0.476862	0.238431	+	0.971159i	$0.423367\pi$
$354$	−1.90761	−0.101389
$355$	0.158086	0.00839032
$356$	2.01231	0.106652
$357$	9.24470	0.489281
$358$	−16.4571	−0.869785
$359$	−29.5820	−1.56128	−0.780638	−	0.624983i	$-0.785106\pi$
$359$	−29.5820	−1.56128	−0.780638	+	0.624983i	$0.785106\pi$
$360$	−0.193511	−0.0101989
$361$	1.00000	0.0526316
$362$	−1.69181	−0.0889195
$363$	1.77013	0.0929077
$364$	0	0
$365$	−0.708958	−0.0371086
$366$	−1.34318	−0.0702090
$367$	6.66951	0.348146	0.174073	−	0.984733i	$-0.444307\pi$
$367$	6.66951	0.348146	0.174073	+	0.984733i	$0.444307\pi$
$368$	−2.67353	−0.139368
$369$	1.21832	0.0634232
$370$	0.468472	0.0243547
$371$	−4.02060	−0.208739
$372$	1.61814	0.0838966
$373$	−21.0273	−1.08875	−0.544377	−	0.838841i	$-0.683234\pi$
$373$	−21.0273	−1.08875	−0.544377	+	0.838841i	$0.683234\pi$
$374$	12.0496	0.623071
$375$	0.511558	0.0264167
$376$	−11.8197	−0.609557
$377$	0	0
$378$	−12.3123	−0.633275
$379$	13.8447	0.711157	0.355579	−	0.934646i	$-0.384284\pi$
$379$	13.8447	0.711157	0.355579	+	0.934646i	$0.384284\pi$
$380$	0.0759953	0.00389847
$381$	−0.0251298	−0.00128744
$382$	25.4002	1.29958
$383$	30.6977	1.56858	0.784290	−	0.620394i	$-0.213027\pi$
$383$	30.6977	1.56858	0.784290	+	0.620394i	$0.213027\pi$
$384$	−0.673533	−0.0343711
$385$	−0.724719	−0.0369351
$386$	−12.1216	−0.616973
$387$	18.6303	0.947034
$388$	17.7753	0.902404
$389$	14.5966	0.740077	0.370039	−	0.929016i	$-0.379344\pi$
$389$	14.5966	0.740077	0.370039	+	0.929016i	$0.379344\pi$
$390$	0	0
$391$	−11.1339	−0.563066
$392$	3.86283	0.195103
$393$	9.90046	0.499412
$394$	−12.1916	−0.614205
$395$	−0.730637	−0.0367624
$396$	−7.36767	−0.370239
$397$	−6.27572	−0.314969	−0.157485	−	0.987521i	$-0.550338\pi$
$397$	−6.27572	−0.314969	−0.157485	+	0.987521i	$0.550338\pi$
$398$	−21.5454	−1.07997
$399$	2.21989	0.111133
$400$	−4.99422	−0.249711
$401$	3.66135	0.182839	0.0914196	−	0.995812i	$-0.470860\pi$
$401$	3.66135	0.182839	0.0914196	+	0.995812i	$0.470860\pi$
$402$	1.17008	0.0583580
$403$	0	0
$404$	−17.7688	−0.884029
$405$	0.389321	0.0193455
$406$	−17.5364	−0.870315
$407$	17.8365	0.884120
$408$	−2.80492	−0.138864
$409$	36.0206	1.78110	0.890552	−	0.454881i	$-0.150318\pi$
$409$	36.0206	1.78110	0.890552	+	0.454881i	$0.150318\pi$
$410$	−0.0363605	−0.00179572
$411$	10.9680	0.541014
$412$	13.9150	0.685541
$413$	−9.33476	−0.459333
$414$	6.80776	0.334583
$415$	−0.166376	−0.00816710
$416$	0	0
$417$	13.4546	0.658874
$418$	2.89342	0.141522
$419$	−16.6885	−0.815289	−0.407644	−	0.913141i	$-0.633650\pi$
$419$	−16.6885	−0.815289	−0.407644	+	0.913141i	$0.633650\pi$
$420$	0.168701	0.00823176
$421$	11.1526	0.543545	0.271773	−	0.962361i	$-0.412390\pi$
$421$	11.1526	0.543545	0.271773	+	0.962361i	$0.412390\pi$
$422$	8.69087	0.423065
$423$	30.0972	1.46338
$424$	1.21989	0.0592429
$425$	−20.7984	−1.00887
$426$	−1.40109	−0.0678829
$427$	−6.57273	−0.318077
$428$	−6.38859	−0.308804
$429$	0	0
$430$	−0.556018	−0.0268136
$431$	−0.0878657	−0.00423234	−0.00211617	−	0.999998i	$-0.500674\pi$
$431$	−0.0878657	−0.00423234	−0.00211617	+	0.999998i	$0.500674\pi$
$432$	3.73565	0.179732
$433$	−30.6001	−1.47055	−0.735273	−	0.677771i	$-0.762946\pi$
$433$	−30.6001	−1.47055	−0.735273	+	0.677771i	$0.762946\pi$
$434$	7.91823	0.380087
$435$	−0.272341	−0.0130578
$436$	−1.57926	−0.0756329
$437$	−2.67353	−0.127892
$438$	6.28338	0.300232
$439$	33.5208	1.59986	0.799930	−	0.600093i	$-0.204870\pi$
$439$	33.5208	1.59986	0.799930	+	0.600093i	$0.204870\pi$
$440$	0.219886	0.0104827
$441$	−9.83614	−0.468388
$442$	0	0
$443$	−8.94147	−0.424822	−0.212411	−	0.977180i	$-0.568132\pi$
$443$	−8.94147	−0.424822	−0.212411	+	0.977180i	$0.568132\pi$
$444$	−4.15199	−0.197045
$445$	0.152926	0.00724939
$446$	20.1710	0.955126
$447$	3.14842	0.148915
$448$	−3.29588	−0.155716
$449$	10.4594	0.493611	0.246806	−	0.969065i	$-0.420619\pi$
$449$	10.4594	0.493611	0.246806	+	0.969065i	$0.420619\pi$
$450$	12.7171	0.599488
$451$	−1.38438	−0.0651878
$452$	−5.11079	−0.240391
$453$	3.01199	0.141516
$454$	−16.8925	−0.792804
$455$	0	0
$456$	−0.673533	−0.0315411
$457$	25.6435	1.19955	0.599775	−	0.800169i	$-0.295257\pi$
$457$	25.6435	1.19955	0.599775	+	0.800169i	$0.295257\pi$
$458$	8.47099	0.395823
$459$	15.5571	0.726143
$460$	−0.203176	−0.00947312
$461$	31.3107	1.45828	0.729142	−	0.684362i	$-0.239919\pi$
$461$	31.3107	1.45828	0.729142	+	0.684362i	$0.239919\pi$
$462$	6.42306	0.298828
$463$	−22.2909	−1.03594	−0.517972	−	0.855397i	$-0.673313\pi$
$463$	−22.2909	−1.03594	−0.517972	+	0.855397i	$0.673313\pi$
$464$	5.32069	0.247007
$465$	0.122971	0.00570264
$466$	−23.6522	−1.09567
$467$	−16.8378	−0.779162	−0.389581	−	0.920992i	$-0.627380\pi$
$467$	−16.8378	−0.779162	−0.389581	+	0.920992i	$0.627380\pi$
$468$	0	0
$469$	5.72567	0.264387
$470$	−0.898245	−0.0414329
$471$	11.0550	0.509386
$472$	2.83225	0.130365
$473$	−21.1697	−0.973382
$474$	6.47552	0.297430
$475$	−4.99422	−0.229151
$476$	−13.7257	−0.629115
$477$	−3.10626	−0.142226
$478$	9.77685	0.447183
$479$	17.9504	0.820174	0.410087	−	0.912046i	$-0.365498\pi$
$479$	17.9504	0.820174	0.410087	+	0.912046i	$0.365498\pi$
$480$	−0.0511853	−0.00233628
$481$	0	0
$482$	−2.19275	−0.0998771
$483$	−5.93494	−0.270049
$484$	−2.62812	−0.119460
$485$	1.35084	0.0613384
$486$	−14.6575	−0.664876
$487$	−5.67850	−0.257317	−0.128659	−	0.991689i	$-0.541067\pi$
$487$	−5.67850	−0.257317	−0.128659	+	0.991689i	$0.541067\pi$
$488$	1.99422	0.0902743
$489$	−5.19224	−0.234801
$490$	0.293557	0.0132616
$491$	−34.0090	−1.53481	−0.767403	−	0.641165i	$-0.778452\pi$
$491$	−34.0090	−1.53481	−0.767403	+	0.641165i	$0.778452\pi$
$492$	0.322257	0.0145285
$493$	22.1580	0.997945
$494$	0	0
$495$	−0.559908	−0.0251660
$496$	−2.40246	−0.107874
$497$	−6.85611	−0.307539
$498$	1.47457	0.0660769
$499$	−30.4388	−1.36263	−0.681315	−	0.731991i	$-0.738591\pi$
$499$	−30.4388	−1.36263	−0.681315	+	0.731991i	$0.738591\pi$
$500$	−0.759514	−0.0339665
$501$	5.19224	0.231972
$502$	4.09239	0.182652
$503$	−43.6733	−1.94730	−0.973648	−	0.228056i	$-0.926763\pi$
$503$	−43.6733	−1.94730	−0.973648	+	0.228056i	$0.926763\pi$
$504$	8.39248	0.373831
$505$	−1.35034	−0.0600894
$506$	−7.73565	−0.343892
$507$	0	0
$508$	0.0373105	0.00165538
$509$	−6.64470	−0.294521	−0.147261	−	0.989098i	$-0.547046\pi$
$509$	−6.64470	−0.294521	−0.147261	+	0.989098i	$0.547046\pi$
$510$	−0.213161	−0.00943892
$511$	30.7472	1.36018
$512$	1.00000	0.0441942
$513$	3.73565	0.164933
$514$	26.5910	1.17288
$515$	1.05747	0.0465978
$516$	4.92789	0.216938
$517$	−34.1995	−1.50409
$518$	−20.3174	−0.892697
$519$	0.749849	0.0329147
$520$	0	0
$521$	26.3316	1.15361	0.576805	−	0.816882i	$-0.304299\pi$
$521$	26.3316	1.15361	0.576805	+	0.816882i	$0.304299\pi$
$522$	−13.5484	−0.592995
$523$	−4.07580	−0.178222	−0.0891112	−	0.996022i	$-0.528403\pi$
$523$	−4.07580	−0.178222	−0.0891112	+	0.996022i	$0.528403\pi$
$524$	−14.6993	−0.642142
$525$	−11.0866	−0.483859
$526$	−16.4877	−0.718898
$527$	−10.0050	−0.435826
$528$	−1.94881	−0.0848113
$529$	−15.8522	−0.689227
$530$	0.0927056	0.00402687
$531$	−7.21191	−0.312970
$532$	−3.29588	−0.142895
$533$	0	0
$534$	−1.35536	−0.0586521
$535$	−0.485502	−0.0209901
$536$	−1.73722	−0.0750364
$537$	11.0844	0.478328
$538$	−9.83802	−0.424147
$539$	11.1768	0.481419
$540$	0.283892	0.0122168
$541$	24.5637	1.05608	0.528038	−	0.849221i	$-0.322928\pi$
$541$	24.5637	1.05608	0.528038	+	0.849221i	$0.322928\pi$
$542$	−16.1678	−0.694465
$543$	1.13949	0.0489002
$544$	4.16449	0.178551
$545$	−0.120016	−0.00514094
$546$	0	0
$547$	−7.86019	−0.336077	−0.168039	−	0.985780i	$-0.553743\pi$
$547$	−7.86019	−0.336077	−0.168039	+	0.985780i	$0.553743\pi$
$548$	−16.2843	−0.695632
$549$	−5.07800	−0.216724
$550$	−14.4504	−0.616167
$551$	5.32069	0.226669
$552$	1.80071	0.0766435
$553$	31.6874	1.34749
$554$	25.5762	1.08663
$555$	−0.315532	−0.0133936
$556$	−19.9761	−0.847177
$557$	14.2513	0.603847	0.301923	−	0.953332i	$-0.402371\pi$
$557$	14.2513	0.603847	0.301923	+	0.953332i	$0.402371\pi$
$558$	6.11752	0.258975
$559$	0	0
$560$	−0.250471	−0.0105843
$561$	−8.11582	−0.342650
$562$	15.5885	0.657561
$563$	−21.6918	−0.914201	−0.457101	−	0.889415i	$-0.651112\pi$
$563$	−21.6918	−0.914201	−0.457101	+	0.889415i	$0.651112\pi$
$564$	7.96100	0.335218
$565$	−0.388396	−0.0163399
$566$	−4.81397	−0.202346
$567$	−16.8847	−0.709091
$568$	2.08021	0.0872835
$569$	41.6272	1.74510	0.872551	−	0.488523i	$-0.162464\pi$
$569$	41.6272	1.74510	0.872551	+	0.488523i	$0.162464\pi$
$570$	−0.0511853	−0.00214392
$571$	−36.2480	−1.51693	−0.758465	−	0.651714i	$-0.774050\pi$
$571$	−36.2480	−1.51693	−0.758465	+	0.651714i	$0.774050\pi$
$572$	0	0
$573$	−17.1079	−0.714691
$574$	1.57694	0.0658201
$575$	13.3522	0.556826
$576$	−2.54635	−0.106098
$577$	31.1366	1.29623	0.648116	−	0.761542i	$-0.275557\pi$
$577$	31.1366	1.29623	0.648116	+	0.761542i	$0.275557\pi$
$578$	0.342986	0.0142663
$579$	8.16430	0.339297
$580$	0.404347	0.0167896
$581$	7.21568	0.299357
$582$	−11.9723	−0.496266
$583$	3.52964	0.146183
$584$	−9.32898	−0.386036
$585$	0	0
$586$	−26.7470	−1.10491
$587$	29.9775	1.23731	0.618653	−	0.785665i	$-0.287679\pi$
$587$	29.9775	1.23731	0.618653	+	0.785665i	$0.287679\pi$
$588$	−2.60175	−0.107294
$589$	−2.40246	−0.0989917
$590$	0.215238	0.00886119
$591$	8.21147	0.337775
$592$	6.16449	0.253359
$593$	23.5038	0.965184	0.482592	−	0.875845i	$-0.339695\pi$
$593$	23.5038	0.965184	0.482592	+	0.875845i	$0.339695\pi$
$594$	10.8088	0.443491
$595$	−1.04309	−0.0427623
$596$	−4.67448	−0.191474
$597$	14.5116	0.593919
$598$	0	0
$599$	0.941651	0.0384748	0.0192374	−	0.999815i	$-0.493876\pi$
$599$	0.941651	0.0384748	0.0192374	+	0.999815i	$0.493876\pi$
$600$	3.36378	0.137326
$601$	−15.1225	−0.616862	−0.308431	−	0.951247i	$-0.599804\pi$
$601$	−15.1225	−0.616862	−0.308431	+	0.951247i	$0.599804\pi$
$602$	24.1143	0.982824
$603$	4.42357	0.180142
$604$	−4.47192	−0.181960
$605$	−0.199725	−0.00811997
$606$	11.9678	0.486161
$607$	−14.8316	−0.601997	−0.300998	−	0.953625i	$-0.597320\pi$
$607$	−14.8316	−0.601997	−0.300998	+	0.953625i	$0.597320\pi$
$608$	1.00000	0.0405554
$609$	11.8113	0.478619
$610$	0.151552	0.00613615
$611$	0	0
$612$	−10.6043	−0.428652
$613$	23.2213	0.937898	0.468949	−	0.883225i	$-0.344633\pi$
$613$	23.2213	0.937898	0.468949	+	0.883225i	$0.344633\pi$
$614$	−15.5546	−0.627734
$615$	0.0244900	0.000987532 0
$616$	−9.53637	−0.384231
$617$	29.0799	1.17071	0.585356	−	0.810776i	$-0.300955\pi$
$617$	29.0799	1.17071	0.585356	+	0.810776i	$0.300955\pi$
$618$	−9.37220	−0.377005
$619$	22.3725	0.899227	0.449613	−	0.893223i	$-0.351562\pi$
$619$	22.3725	0.899227	0.449613	+	0.893223i	$0.351562\pi$
$620$	−0.182576	−0.00733242
$621$	−9.98740	−0.400780
$622$	−1.28527	−0.0515345
$623$	−6.63233	−0.265719
$624$	0	0
$625$	24.9134	0.996536
$626$	−22.3334	−0.892622
$627$	−1.94881	−0.0778282
$628$	−16.4134	−0.654966
$629$	25.6720	1.02361
$630$	0.637788	0.0254101
$631$	−32.2622	−1.28434	−0.642170	−	0.766563i	$-0.721965\pi$
$631$	−32.2622	−1.28434	−0.642170	+	0.766563i	$0.721965\pi$
$632$	−9.61425	−0.382434
$633$	−5.85359	−0.232659
$634$	−22.3093	−0.886017
$635$	0.00283542	0.000112520 0
$636$	−0.821634	−0.0325799
$637$	0	0
$638$	15.3950	0.609494
$639$	−5.29694	−0.209544
$640$	0.0759953	0.00300398
$641$	−32.9011	−1.29952	−0.649758	−	0.760142i	$-0.725130\pi$
$641$	−32.9011	−1.29952	−0.649758	+	0.760142i	$0.725130\pi$
$642$	4.30293	0.169823
$643$	16.0258	0.631994	0.315997	−	0.948760i	$-0.397661\pi$
$643$	16.0258	0.631994	0.315997	+	0.948760i	$0.397661\pi$
$644$	8.81165	0.347228
$645$	0.374497	0.0147458
$646$	4.16449	0.163850
$647$	−28.6833	−1.12766	−0.563829	−	0.825891i	$-0.690672\pi$
$647$	−28.6833	−1.12766	−0.563829	+	0.825891i	$0.690672\pi$
$648$	5.12297	0.201249
$649$	8.19489	0.321677
$650$	0	0
$651$	−5.33319	−0.209024
$652$	7.70896	0.301906
$653$	38.5753	1.50957	0.754783	−	0.655974i	$-0.227742\pi$
$653$	38.5753	1.50957	0.754783	+	0.655974i	$0.227742\pi$
$654$	1.06369	0.0415934
$655$	−1.11708	−0.0436478
$656$	−0.478457	−0.0186806
$657$	23.7549	0.926766
$658$	38.9565	1.51868
$659$	13.7014	0.533732	0.266866	−	0.963734i	$-0.414012\pi$
$659$	13.7014	0.533732	0.266866	+	0.963734i	$0.414012\pi$
$660$	−0.148101	−0.00576481
$661$	8.84412	0.343996	0.171998	−	0.985097i	$-0.444978\pi$
$661$	8.84412	0.343996	0.171998	+	0.985097i	$0.444978\pi$
$662$	30.9268	1.20201
$663$	0	0
$664$	−2.18930	−0.0849614
$665$	−0.250471	−0.00971286
$666$	−15.6970	−0.608245
$667$	−14.2250	−0.550796
$668$	−7.70896	−0.298269
$669$	−13.5859	−0.525260
$670$	−0.132020	−0.00510039
$671$	5.77013	0.222753
$672$	2.21989	0.0856340
$673$	40.1243	1.54668	0.773338	−	0.633994i	$-0.218586\pi$
$673$	40.1243	1.54668	0.773338	+	0.633994i	$0.218586\pi$
$674$	−23.6429	−0.910692
$675$	−18.6567	−0.718096
$676$	0	0
$677$	34.4388	1.32359	0.661796	−	0.749684i	$-0.269794\pi$
$677$	34.4388	1.32359	0.661796	+	0.749684i	$0.269794\pi$
$678$	3.44229	0.132200
$679$	−58.5852	−2.24830
$680$	0.316482	0.0121365
$681$	11.3777	0.435993
$682$	−6.95133	−0.266180
$683$	−15.4248	−0.590212	−0.295106	−	0.955464i	$-0.595355\pi$
$683$	−15.4248	−0.590212	−0.295106	+	0.955464i	$0.595355\pi$
$684$	−2.54635	−0.0973622
$685$	−1.23753	−0.0472837
$686$	10.3397	0.394773
$687$	−5.70549	−0.217678
$688$	−7.31648	−0.278938
$689$	0	0
$690$	0.136846	0.00520963
$691$	32.5075	1.23665	0.618323	−	0.785924i	$-0.287813\pi$
$691$	32.5075	1.23665	0.618323	+	0.785924i	$0.287813\pi$
$692$	−1.11331	−0.0423215
$693$	24.2830	0.922433
$694$	18.2143	0.691405
$695$	−1.51809	−0.0575845
$696$	−3.58366	−0.135838
$697$	−1.99253	−0.0754725
$698$	27.9134	1.05654
$699$	15.9305	0.602548
$700$	16.4604	0.622144
$701$	−45.9415	−1.73519	−0.867593	−	0.497275i	$-0.834334\pi$
$701$	−45.9415	−1.73519	−0.867593	+	0.497275i	$0.834334\pi$
$702$	0	0
$703$	6.16449	0.232498
$704$	2.89342	0.109050
$705$	0.604998	0.0227855
$706$	8.95943	0.337193
$707$	58.5637	2.20251
$708$	−1.90761	−0.0716925
$709$	−1.53292	−0.0575699	−0.0287849	−	0.999586i	$-0.509164\pi$
$709$	−1.53292	−0.0575699	−0.0287849	+	0.999586i	$0.509164\pi$
$710$	0.158086	0.00593285
$711$	24.4813	0.918119
$712$	2.01231	0.0754145
$713$	6.42306	0.240546
$714$	9.24470	0.345974
$715$	0	0
$716$	−16.4571	−0.615031
$717$	−6.58504	−0.245923
$718$	−29.5820	−1.10399
$719$	−2.53178	−0.0944193	−0.0472096	−	0.998885i	$-0.515033\pi$
$719$	−2.53178	−0.0944193	−0.0472096	+	0.998885i	$0.515033\pi$
$720$	−0.193511	−0.00721172
$721$	−45.8621	−1.70799
$722$	1.00000	0.0372161
$723$	1.47689	0.0549262
$724$	−1.69181	−0.0628756
$725$	−26.5727	−0.986886
$726$	1.77013	0.0656956
$727$	8.64282	0.320544	0.160272	−	0.987073i	$-0.448763\pi$
$727$	8.64282	0.320544	0.160272	+	0.987073i	$0.448763\pi$
$728$	0	0
$729$	−5.49663	−0.203579
$730$	−0.708958	−0.0262397
$731$	−30.4694	−1.12695
$732$	−1.34318	−0.0496452
$733$	−3.13076	−0.115637	−0.0578186	−	0.998327i	$-0.518415\pi$
$733$	−3.13076	−0.115637	−0.0578186	+	0.998327i	$0.518415\pi$
$734$	6.66951	0.246176
$735$	−0.197721	−0.00729303
$736$	−2.67353	−0.0985477
$737$	−5.02650	−0.185154
$738$	1.21832	0.0448470
$739$	−22.1222	−0.813777	−0.406888	−	0.913478i	$-0.633386\pi$
$739$	−22.1222	−0.813777	−0.406888	+	0.913478i	$0.633386\pi$
$740$	0.468472	0.0172214
$741$	0	0
$742$	−4.02060	−0.147601
$743$	−2.40322	−0.0881656	−0.0440828	−	0.999028i	$-0.514037\pi$
$743$	−2.40322	−0.0881656	−0.0440828	+	0.999028i	$0.514037\pi$
$744$	1.61814	0.0593238
$745$	−0.355239	−0.0130149
$746$	−21.0273	−0.769865
$747$	5.57473	0.203969
$748$	12.0496	0.440578
$749$	21.0560	0.769370
$750$	0.511558	0.0186795
$751$	48.9841	1.78745	0.893727	−	0.448611i	$-0.148081\pi$
$751$	48.9841	1.78745	0.893727	+	0.448611i	$0.148081\pi$
$752$	−11.8197	−0.431022
$753$	−2.75636	−0.100447
$754$	0	0
$755$	−0.339845	−0.0123682
$756$	−12.3123	−0.447793
$757$	−26.1494	−0.950415	−0.475207	−	0.879874i	$-0.657627\pi$
$757$	−26.1494	−0.950415	−0.475207	+	0.879874i	$0.657627\pi$
$758$	13.8447	0.502864
$759$	5.21022	0.189119
$760$	0.0759953	0.00275664
$761$	2.22779	0.0807575	0.0403787	−	0.999184i	$-0.487144\pi$
$761$	2.22779	0.0807575	0.0403787	+	0.999184i	$0.487144\pi$
$762$	−0.0251298	−0.000910358 0
$763$	5.20506	0.188436
$764$	25.4002	0.918945
$765$	−0.805874	−0.0291364
$766$	30.6977	1.10915
$767$	0	0
$768$	−0.673533	−0.0243040
$769$	−19.7911	−0.713684	−0.356842	−	0.934165i	$-0.616147\pi$
$769$	−19.7911	−0.713684	−0.356842	+	0.934165i	$0.616147\pi$
$770$	−0.724719	−0.0261171
$771$	−17.9099	−0.645011
$772$	−12.1216	−0.436266
$773$	−32.6108	−1.17293	−0.586464	−	0.809975i	$-0.699481\pi$
$773$	−32.6108	−1.17293	−0.586464	+	0.809975i	$0.699481\pi$
$774$	18.6303	0.669654
$775$	11.9984	0.430996
$776$	17.7753	0.638096
$777$	13.6845	0.490928
$778$	14.5966	0.523314
$779$	−0.478457	−0.0171425
$780$	0	0
$781$	6.01891	0.215373
$782$	−11.1339	−0.398148
$783$	19.8763	0.710320
$784$	3.86283	0.137958
$785$	−1.24734	−0.0445195
$786$	9.90046	0.353138
$787$	−31.1306	−1.10969	−0.554844	−	0.831955i	$-0.687222\pi$
$787$	−31.1306	−1.10969	−0.554844	+	0.831955i	$0.687222\pi$
$788$	−12.1916	−0.434309
$789$	11.1050	0.395349
$790$	−0.730637	−0.0259949
$791$	16.8446	0.598924
$792$	−7.36767	−0.261799
$793$	0	0
$794$	−6.27572	−0.222717
$795$	−0.0624403	−0.00221453
$796$	−21.5454	−0.763657
$797$	47.1342	1.66958	0.834790	−	0.550568i	$-0.185589\pi$
$797$	47.1342	1.66958	0.834790	+	0.550568i	$0.185589\pi$
$798$	2.21989	0.0785831
$799$	−49.2232	−1.74139
$800$	−4.99422	−0.176573
$801$	−5.12405	−0.181049
$802$	3.66135	0.129287
$803$	−26.9927	−0.952550
$804$	1.17008	0.0412654
$805$	0.669644	0.0236018
$806$	0	0
$807$	6.62624	0.233255
$808$	−17.7688	−0.625103
$809$	0.936505	0.0329258	0.0164629	−	0.999864i	$-0.494759\pi$
$809$	0.936505	0.0329258	0.0164629	+	0.999864i	$0.494759\pi$
$810$	0.389321	0.0136794
$811$	−42.6987	−1.49935	−0.749676	−	0.661805i	$-0.769791\pi$
$811$	−42.6987	−1.49935	−0.749676	+	0.661805i	$0.769791\pi$
$812$	−17.5364	−0.615406
$813$	10.8895	0.381912
$814$	17.8365	0.625168
$815$	0.585844	0.0205212
$816$	−2.80492	−0.0981920
$817$	−7.31648	−0.255971
$818$	36.0206	1.25943
$819$	0	0
$820$	−0.0363605	−0.00126976
$821$	−9.42960	−0.329095	−0.164548	−	0.986369i	$-0.552616\pi$
$821$	−9.42960	−0.329095	−0.164548	+	0.986369i	$0.552616\pi$
$822$	10.9680	0.382554
$823$	−1.22189	−0.0425924	−0.0212962	−	0.999773i	$-0.506779\pi$
$823$	−1.22189	−0.0425924	−0.0212962	+	0.999773i	$0.506779\pi$
$824$	13.9150	0.484751
$825$	9.73282	0.338853
$826$	−9.33476	−0.324798
$827$	−5.74720	−0.199850	−0.0999249	−	0.994995i	$-0.531860\pi$
$827$	−5.74720	−0.199850	−0.0999249	+	0.994995i	$0.531860\pi$
$828$	6.80776	0.236586
$829$	15.1831	0.527330	0.263665	−	0.964614i	$-0.415069\pi$
$829$	15.1831	0.527330	0.263665	+	0.964614i	$0.415069\pi$
$830$	−0.166376	−0.00577501
$831$	−17.2264	−0.597578
$832$	0	0
$833$	16.0867	0.557372
$834$	13.4546	0.465895
$835$	−0.585844	−0.0202740
$836$	2.89342	0.100071
$837$	−8.97477	−0.310213
$838$	−16.6885	−0.576496
$839$	−29.1124	−1.00507	−0.502535	−	0.864557i	$-0.667599\pi$
$839$	−29.1124	−1.00507	−0.502535	+	0.864557i	$0.667599\pi$
$840$	0.168701	0.00582073
$841$	−0.690244	−0.0238015
$842$	11.1526	0.384345
$843$	−10.4994	−0.361618
$844$	8.69087	0.299152
$845$	0	0
$846$	30.0972	1.03476
$847$	8.66198	0.297629
$848$	1.21989	0.0418911
$849$	3.24237	0.111278
$850$	−20.7984	−0.713379
$851$	−16.4810	−0.564960
$852$	−1.40109	−0.0480005
$853$	−9.51346	−0.325735	−0.162867	−	0.986648i	$-0.552074\pi$
$853$	−9.51346	−0.325735	−0.162867	+	0.986648i	$0.552074\pi$
$854$	−6.57273	−0.224914
$855$	−0.193511	−0.00661793
$856$	−6.38859	−0.218357
$857$	16.6863	0.569993	0.284996	−	0.958529i	$-0.408008\pi$
$857$	16.6863	0.569993	0.284996	+	0.958529i	$0.408008\pi$
$858$	0	0
$859$	2.73314	0.0932535	0.0466267	−	0.998912i	$-0.485153\pi$
$859$	2.73314	0.0932535	0.0466267	+	0.998912i	$0.485153\pi$
$860$	−0.556018	−0.0189601
$861$	−1.06212	−0.0361970
$862$	−0.0878657	−0.00299272
$863$	3.62875	0.123524	0.0617621	−	0.998091i	$-0.480328\pi$
$863$	3.62875	0.123524	0.0617621	+	0.998091i	$0.480328\pi$
$864$	3.73565	0.127090
$865$	−0.0846060	−0.00287669
$866$	−30.6001	−1.03983
$867$	−0.231012	−0.00784559
$868$	7.91823	0.268762
$869$	−27.8181	−0.943663
$870$	−0.272341	−0.00923324
$871$	0	0
$872$	−1.57926	−0.0534806
$873$	−45.2622	−1.53189
$874$	−2.67353	−0.0904336
$875$	2.50327	0.0846259
$876$	6.28338	0.212296
$877$	−5.47771	−0.184969	−0.0924846	−	0.995714i	$-0.529481\pi$
$877$	−5.47771	−0.184969	−0.0924846	+	0.995714i	$0.529481\pi$
$878$	33.5208	1.13127
$879$	18.0150	0.607631
$880$	0.219886	0.00741236
$881$	−14.5260	−0.489393	−0.244697	−	0.969600i	$-0.578688\pi$
$881$	−14.5260	−0.489393	−0.244697	+	0.969600i	$0.578688\pi$
$882$	−9.83614	−0.331200
$883$	−21.5058	−0.723727	−0.361864	−	0.932231i	$-0.617859\pi$
$883$	−21.5058	−0.723727	−0.361864	+	0.932231i	$0.617859\pi$
$884$	0	0
$885$	−0.144970	−0.00487310
$886$	−8.94147	−0.300395
$887$	26.7314	0.897554	0.448777	−	0.893644i	$-0.351860\pi$
$887$	26.7314	0.897554	0.448777	+	0.893644i	$0.351860\pi$
$888$	−4.15199	−0.139332
$889$	−0.122971	−0.00412431
$890$	0.152926	0.00512609
$891$	14.8229	0.496586
$892$	20.1710	0.675376
$893$	−11.8197	−0.395533
$894$	3.14842	0.105299
$895$	−1.25066	−0.0418050
$896$	−3.29588	−0.110108
$897$	0	0
$898$	10.4594	0.349036
$899$	−12.7828	−0.426329
$900$	12.7171	0.423902
$901$	5.08021	0.169246
$902$	−1.38438	−0.0460947
$903$	−16.2418	−0.540492
$904$	−5.11079	−0.169982
$905$	−0.128569	−0.00427379
$906$	3.01199	0.100067
$907$	−47.1265	−1.56481	−0.782404	−	0.622771i	$-0.786007\pi$
$907$	−47.1265	−1.56481	−0.782404	+	0.622771i	$0.786007\pi$
$908$	−16.8925	−0.560597
$909$	45.2455	1.50070
$910$	0	0
$911$	8.39511	0.278142	0.139071	−	0.990282i	$-0.455588\pi$
$911$	8.39511	0.278142	0.139071	+	0.990282i	$0.455588\pi$
$912$	−0.673533	−0.0223029
$913$	−6.33457	−0.209644
$914$	25.6435	0.848210
$915$	−0.102075	−0.00337450
$916$	8.47099	0.279889
$917$	48.4471	1.59987
$918$	15.5571	0.513461
$919$	38.9579	1.28510	0.642551	−	0.766243i	$-0.277876\pi$
$919$	38.9579	1.28510	0.642551	+	0.766243i	$0.277876\pi$
$920$	−0.203176	−0.00669851
$921$	10.4766	0.345215
$922$	31.3107	1.03116
$923$	0	0
$924$	6.42306	0.211303
$925$	−30.7869	−1.01227
$926$	−22.2909	−0.732523
$927$	−35.4324	−1.16375
$928$	5.32069	0.174660
$929$	5.93525	0.194729	0.0973646	−	0.995249i	$-0.468959\pi$
$929$	5.93525	0.194729	0.0973646	+	0.995249i	$0.468959\pi$
$930$	0.122971	0.00403237
$931$	3.86283	0.126599
$932$	−23.6522	−0.774753
$933$	0.865670	0.0283408
$934$	−16.8378	−0.550951
$935$	0.915714	0.0299470
$936$	0	0
$937$	10.6879	0.349160	0.174580	−	0.984643i	$-0.444143\pi$
$937$	10.6879	0.349160	0.174580	+	0.984643i	$0.444143\pi$
$938$	5.72567	0.186950
$939$	15.0423	0.490886
$940$	−0.898245	−0.0292975
$941$	19.5108	0.636034	0.318017	−	0.948085i	$-0.396983\pi$
$941$	19.5108	0.636034	0.318017	+	0.948085i	$0.396983\pi$
$942$	11.0550	0.360190
$943$	1.27917	0.0416555
$944$	2.83225	0.0921819
$945$	−0.935674	−0.0304375
$946$	−21.1697	−0.688285
$947$	−40.2868	−1.30915	−0.654573	−	0.755999i	$-0.727152\pi$
$947$	−40.2868	−1.30915	−0.654573	+	0.755999i	$0.727152\pi$
$948$	6.47552	0.210315
$949$	0	0
$950$	−4.99422	−0.162034
$951$	15.0261	0.487254
$952$	−13.7257	−0.444851
$953$	−53.9354	−1.74714	−0.873569	−	0.486700i	$-0.838201\pi$
$953$	−53.9354	−1.74714	−0.873569	+	0.486700i	$0.838201\pi$
$954$	−3.10626	−0.100569
$955$	1.93029	0.0624628
$956$	9.77685	0.316206
$957$	−10.3690	−0.335184
$958$	17.9504	0.579950
$959$	53.6712	1.73313
$960$	−0.0511853	−0.00165200
$961$	−25.2282	−0.813812
$962$	0	0
$963$	16.2676	0.524216
$964$	−2.19275	−0.0706238
$965$	−0.921184	−0.0296540
$966$	−5.93494	−0.190954
$967$	−13.8959	−0.446863	−0.223432	−	0.974720i	$-0.571726\pi$
$967$	−13.8959	−0.446863	−0.223432	+	0.974720i	$0.571726\pi$
$968$	−2.62812	−0.0844711
$969$	−2.80492	−0.0901071
$970$	1.35084	0.0433728
$971$	−28.1935	−0.904773	−0.452387	−	0.891822i	$-0.649427\pi$
$971$	−28.1935	−0.904773	−0.452387	+	0.891822i	$0.649427\pi$
$972$	−14.6575	−0.470138
$973$	65.8390	2.11070
$974$	−5.67850	−0.181951
$975$	0	0
$976$	1.99422	0.0638336
$977$	−47.6882	−1.52568	−0.762841	−	0.646586i	$-0.776196\pi$
$977$	−47.6882	−1.52568	−0.762841	+	0.646586i	$0.776196\pi$
$978$	−5.19224	−0.166030
$979$	5.82246	0.186086
$980$	0.293557	0.00937734
$981$	4.02136	0.128392
$982$	−34.0090	−1.08527
$983$	−16.1289	−0.514433	−0.257217	−	0.966354i	$-0.582805\pi$
$983$	−16.1289	−0.514433	−0.257217	+	0.966354i	$0.582805\pi$
$984$	0.322257	0.0102732
$985$	−0.926506	−0.0295209
$986$	22.1580	0.705654
$987$	−26.2385	−0.835181
$988$	0	0
$989$	19.5609	0.621999
$990$	−0.559908	−0.0177950
$991$	54.0751	1.71775	0.858876	−	0.512184i	$-0.171163\pi$
$991$	54.0751	1.71775	0.858876	+	0.512184i	$0.171163\pi$
$992$	−2.40246	−0.0762782
$993$	−20.8303	−0.661028
$994$	−6.85611	−0.217463
$995$	−1.63735	−0.0519075
$996$	1.47457	0.0467235
$997$	−1.35547	−0.0429282	−0.0214641	−	0.999770i	$-0.506833\pi$
$997$	−1.35547	−0.0429282	−0.0214641	+	0.999770i	$0.506833\pi$
$998$	−30.4388	−0.963525
$999$	23.0284	0.728586

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bb.1.2		4
13.4	even	6		494.2.g.e.419.3	yes	8
13.10	even	6		494.2.g.e.191.3	✓	8
13.12	even	2		6422.2.a.z.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.g.e.191.3	✓	8	13.10	even	6
494.2.g.e.419.3	yes	8	13.4	even	6
6422.2.a.z.1.2		4	13.12	even	2
6422.2.a.bb.1.2		4	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.673533 q^{3} +1.00000 q^{4} +0.0759953 q^{5} -0.673533 q^{6} -3.29588 q^{7} +1.00000 q^{8} -2.54635 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.673533 q^{3} +1.00000 q^{4} +0.0759953 q^{5} -0.673533 q^{6} -3.29588 q^{7} +1.00000 q^{8} -2.54635 q^{9} +0.0759953 q^{10} +2.89342 q^{11} -0.673533 q^{12} -3.29588 q^{14} -0.0511853 q^{15} +1.00000 q^{16} +4.16449 q^{17} -2.54635 q^{18} +1.00000 q^{19} +0.0759953 q^{20} +2.21989 q^{21} +2.89342 q^{22} -2.67353 q^{23} -0.673533 q^{24} -4.99422 q^{25} +3.73565 q^{27} -3.29588 q^{28} +5.32069 q^{29} -0.0511853 q^{30} -2.40246 q^{31} +1.00000 q^{32} -1.94881 q^{33} +4.16449 q^{34} -0.250471 q^{35} -2.54635 q^{36} +6.16449 q^{37} +1.00000 q^{38} +0.0759953 q^{40} -0.478457 q^{41} +2.21989 q^{42} -7.31648 q^{43} +2.89342 q^{44} -0.193511 q^{45} -2.67353 q^{46} -11.8197 q^{47} -0.673533 q^{48} +3.86283 q^{49} -4.99422 q^{50} -2.80492 q^{51} +1.21989 q^{53} +3.73565 q^{54} +0.219886 q^{55} -3.29588 q^{56} -0.673533 q^{57} +5.32069 q^{58} +2.83225 q^{59} -0.0511853 q^{60} +1.99422 q^{61} -2.40246 q^{62} +8.39248 q^{63} +1.00000 q^{64} -1.94881 q^{66} -1.73722 q^{67} +4.16449 q^{68} +1.80071 q^{69} -0.250471 q^{70} +2.08021 q^{71} -2.54635 q^{72} -9.32898 q^{73} +6.16449 q^{74} +3.36378 q^{75} +1.00000 q^{76} -9.53637 q^{77} -9.61425 q^{79} +0.0759953 q^{80} +5.12297 q^{81} -0.478457 q^{82} -2.18930 q^{83} +2.21989 q^{84} +0.316482 q^{85} -7.31648 q^{86} -3.58366 q^{87} +2.89342 q^{88} +2.01231 q^{89} -0.193511 q^{90} -2.67353 q^{92} +1.61814 q^{93} -11.8197 q^{94} +0.0759953 q^{95} -0.673533 q^{96} +17.7753 q^{97} +3.86283 q^{98} -7.36767 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} + q^{5} - 2 q^{6} - q^{7} + 4 q^{8} + 2 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 + q^5 - 2 * q^6 - q^7 + 4 * q^8 + 2 * q^9	\( 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} + q^{5} - 2 q^{6} - q^{7} + 4 q^{8} + 2 q^{9} + q^{10} - 2 q^{11} - 2 q^{12} - q^{14} - 11 q^{15} + 4 q^{16} + q^{17} + 2 q^{18} + 4 q^{19} + q^{20} - 4 q^{21} - 2 q^{22} - 10 q^{23} - 2 q^{24} + 3 q^{25} - 23 q^{27} - q^{28} - q^{29} - 11 q^{30} - 11 q^{31} + 4 q^{32} + 3 q^{33} + q^{34} - q^{35} + 2 q^{36} + 9 q^{37} + 4 q^{38} + q^{40} - 4 q^{41} - 4 q^{42} - 15 q^{43} - 2 q^{44} + 33 q^{45} - 10 q^{46} - 25 q^{47} - 2 q^{48} - 11 q^{49} + 3 q^{50} - 14 q^{51} - 8 q^{53} - 23 q^{54} - 12 q^{55} - q^{56} - 2 q^{57} - q^{58} - 28 q^{59} - 11 q^{60} - 15 q^{61} - 11 q^{62} + 20 q^{63} + 4 q^{64} + 3 q^{66} + q^{68} + 18 q^{69} - q^{70} + q^{71} + 2 q^{72} - 6 q^{73} + 9 q^{74} - 13 q^{75} + 4 q^{76} - 11 q^{77} - 12 q^{79} + q^{80} + 40 q^{81} - 4 q^{82} + 17 q^{83} - 4 q^{84} - 13 q^{85} - 15 q^{86} + 25 q^{87} - 2 q^{88} + 15 q^{89} + 33 q^{90} - 10 q^{92} + 3 q^{93} - 25 q^{94} + q^{95} - 2 q^{96} - 2 q^{97} - 11 q^{98} - 26 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 + q^5 - 2 * q^6 - q^7 + 4 * q^8 + 2 * q^9 + q^10 - 2 * q^11 - 2 * q^12 - q^14 - 11 * q^15 + 4 * q^16 + q^17 + 2 * q^18 + 4 * q^19 + q^20 - 4 * q^21 - 2 * q^22 - 10 * q^23 - 2 * q^24 + 3 * q^25 - 23 * q^27 - q^28 - q^29 - 11 * q^30 - 11 * q^31 + 4 * q^32 + 3 * q^33 + q^34 - q^35 + 2 * q^36 + 9 * q^37 + 4 * q^38 + q^40 - 4 * q^41 - 4 * q^42 - 15 * q^43 - 2 * q^44 + 33 * q^45 - 10 * q^46 - 25 * q^47 - 2 * q^48 - 11 * q^49 + 3 * q^50 - 14 * q^51 - 8 * q^53 - 23 * q^54 - 12 * q^55 - q^56 - 2 * q^57 - q^58 - 28 * q^59 - 11 * q^60 - 15 * q^61 - 11 * q^62 + 20 * q^63 + 4 * q^64 + 3 * q^66 + q^68 + 18 * q^69 - q^70 + q^71 + 2 * q^72 - 6 * q^73 + 9 * q^74 - 13 * q^75 + 4 * q^76 - 11 * q^77 - 12 * q^79 + q^80 + 40 * q^81 - 4 * q^82 + 17 * q^83 - 4 * q^84 - 13 * q^85 - 15 * q^86 + 25 * q^87 - 2 * q^88 + 15 * q^89 + 33 * q^90 - 10 * q^92 + 3 * q^93 - 25 * q^94 + q^95 - 2 * q^96 - 2 * q^97 - 11 * q^98 - 26 * q^99

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