LMFDB - Embedded newform 4110.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4110,2,Mod(1,4110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4110, 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("4110.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4110 = 2 \cdot 3 \cdot 5 \cdot 137 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4110.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:	$32.8185152307$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1900864.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 9x^{3} + 16x^{2} + 14x - 16 $ x^5 - 2x^4 - 9x^3 + 16x^2 + 14x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.63045$ of defining polynomial
Character	$\chi$	$=$	4110.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.91928 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.91928 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.66871 q^{11} -1.00000 q^{12} -1.41814 q^{13} -1.91928 q^{14} +1.00000 q^{15} +1.00000 q^{16} -4.35121 q^{17} +1.00000 q^{18} -3.54974 q^{19} -1.00000 q^{20} +1.91928 q^{21} -2.66871 q^{22} +6.45944 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.41814 q^{26} -1.00000 q^{27} -1.91928 q^{28} -6.35121 q^{29} +1.00000 q^{30} +7.07196 q^{31} +1.00000 q^{32} +2.66871 q^{33} -4.35121 q^{34} +1.91928 q^{35} +1.00000 q^{36} +10.5188 q^{37} -3.54974 q^{38} +1.41814 q^{39} -1.00000 q^{40} +7.92962 q^{41} +1.91928 q^{42} -0.320546 q^{43} -2.66871 q^{44} -1.00000 q^{45} +6.45944 q^{46} -6.45944 q^{47} -1.00000 q^{48} -3.31635 q^{49} +1.00000 q^{50} +4.35121 q^{51} -1.41814 q^{52} +3.87343 q^{53} -1.00000 q^{54} +2.66871 q^{55} -1.91928 q^{56} +3.54974 q^{57} -6.35121 q^{58} +13.8255 q^{59} +1.00000 q^{60} -0.720755 q^{61} +7.07196 q^{62} -1.91928 q^{63} +1.00000 q^{64} +1.41814 q^{65} +2.66871 q^{66} +12.4479 q^{67} -4.35121 q^{68} -6.45944 q^{69} +1.91928 q^{70} -10.1534 q^{71} +1.00000 q^{72} +2.00000 q^{73} +10.5188 q^{74} -1.00000 q^{75} -3.54974 q^{76} +5.12202 q^{77} +1.41814 q^{78} -2.17375 q^{79} -1.00000 q^{80} +1.00000 q^{81} +7.92962 q^{82} +2.47357 q^{83} +1.91928 q^{84} +4.35121 q^{85} -0.320546 q^{86} +6.35121 q^{87} -2.66871 q^{88} +12.7311 q^{89} -1.00000 q^{90} +2.72181 q^{91} +6.45944 q^{92} -7.07196 q^{93} -6.45944 q^{94} +3.54974 q^{95} -1.00000 q^{96} -3.93382 q^{97} -3.31635 q^{98} -2.66871 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} - 5 q^{3} + 5 q^{4} - 5 q^{5} - 5 q^{6} + 3 q^{7} + 5 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^2 - 5 * q^3 + 5 * q^4 - 5 * q^5 - 5 * q^6 + 3 * q^7 + 5 * q^8 + 5 * q^9	$ 5 q + 5 q^{2} - 5 q^{3} + 5 q^{4} - 5 q^{5} - 5 q^{6} + 3 q^{7} + 5 q^{8} + 5 q^{9} - 5 q^{10} - 5 q^{12} + 7 q^{13} + 3 q^{14} + 5 q^{15} + 5 q^{16} + 5 q^{18} + 10 q^{19} - 5 q^{20} - 3 q^{21} - 4 q^{23} - 5 q^{24} + 5 q^{25} + 7 q^{26} - 5 q^{27} + 3 q^{28} - 10 q^{29} + 5 q^{30} + 7 q^{31} + 5 q^{32} - 3 q^{35} + 5 q^{36} + 16 q^{37} + 10 q^{38} - 7 q^{39} - 5 q^{40} - 4 q^{41} - 3 q^{42} + 14 q^{43} - 5 q^{45} - 4 q^{46} + 4 q^{47} - 5 q^{48} + 8 q^{49} + 5 q^{50} + 7 q^{52} - 3 q^{53} - 5 q^{54} + 3 q^{56} - 10 q^{57} - 10 q^{58} + 12 q^{59} + 5 q^{60} + 3 q^{61} + 7 q^{62} + 3 q^{63} + 5 q^{64} - 7 q^{65} + 24 q^{67} + 4 q^{69} - 3 q^{70} - 8 q^{71} + 5 q^{72} + 10 q^{73} + 16 q^{74} - 5 q^{75} + 10 q^{76} + 16 q^{77} - 7 q^{78} + 7 q^{79} - 5 q^{80} + 5 q^{81} - 4 q^{82} + 4 q^{83} - 3 q^{84} + 14 q^{86} + 10 q^{87} + 26 q^{89} - 5 q^{90} - 5 q^{91} - 4 q^{92} - 7 q^{93} + 4 q^{94} - 10 q^{95} - 5 q^{96} + 15 q^{97} + 8 q^{98}+O(q^{100}) $ 5 * q + 5 * q^2 - 5 * q^3 + 5 * q^4 - 5 * q^5 - 5 * q^6 + 3 * q^7 + 5 * q^8 + 5 * q^9 - 5 * q^10 - 5 * q^12 + 7 * q^13 + 3 * q^14 + 5 * q^15 + 5 * q^16 + 5 * q^18 + 10 * q^19 - 5 * q^20 - 3 * q^21 - 4 * q^23 - 5 * q^24 + 5 * q^25 + 7 * q^26 - 5 * q^27 + 3 * q^28 - 10 * q^29 + 5 * q^30 + 7 * q^31 + 5 * q^32 - 3 * q^35 + 5 * q^36 + 16 * q^37 + 10 * q^38 - 7 * q^39 - 5 * q^40 - 4 * q^41 - 3 * q^42 + 14 * q^43 - 5 * q^45 - 4 * q^46 + 4 * q^47 - 5 * q^48 + 8 * q^49 + 5 * q^50 + 7 * q^52 - 3 * q^53 - 5 * q^54 + 3 * q^56 - 10 * q^57 - 10 * q^58 + 12 * q^59 + 5 * q^60 + 3 * q^61 + 7 * q^62 + 3 * q^63 + 5 * q^64 - 7 * q^65 + 24 * q^67 + 4 * q^69 - 3 * q^70 - 8 * q^71 + 5 * q^72 + 10 * q^73 + 16 * q^74 - 5 * q^75 + 10 * q^76 + 16 * q^77 - 7 * q^78 + 7 * q^79 - 5 * q^80 + 5 * q^81 - 4 * q^82 + 4 * q^83 - 3 * q^84 + 14 * q^86 + 10 * q^87 + 26 * q^89 - 5 * q^90 - 5 * q^91 - 4 * q^92 - 7 * q^93 + 4 * q^94 - 10 * q^95 - 5 * q^96 + 15 * q^97 + 8 * q^98

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	−1.91928	−0.725422	−0.362711	−	0.931902i	$-0.618149\pi$
$7$	−1.91928	−0.725422	−0.362711	+	0.931902i	$0.618149\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−2.66871	−0.804647	−0.402323	−	0.915498i	$-0.631797\pi$
$11$	−2.66871	−0.804647	−0.402323	+	0.915498i	$0.631797\pi$
$12$	−1.00000	−0.288675
$13$	−1.41814	−0.393320	−0.196660	−	0.980472i	$-0.563010\pi$
$13$	−1.41814	−0.393320	−0.196660	+	0.980472i	$0.563010\pi$
$14$	−1.91928	−0.512950
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−4.35121	−1.05532	−0.527661	−	0.849455i	$-0.676931\pi$
$17$	−4.35121	−1.05532	−0.527661	+	0.849455i	$0.676931\pi$
$18$	1.00000	0.235702
$19$	−3.54974	−0.814366	−0.407183	−	0.913347i	$-0.633489\pi$
$19$	−3.54974	−0.814366	−0.407183	+	0.913347i	$0.633489\pi$
$20$	−1.00000	−0.223607
$21$	1.91928	0.418822
$22$	−2.66871	−0.568971
$23$	6.45944	1.34689	0.673443	−	0.739239i	$-0.264815\pi$
$23$	6.45944	1.34689	0.673443	+	0.739239i	$0.264815\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	−1.41814	−0.278119
$27$	−1.00000	−0.192450
$28$	−1.91928	−0.362711
$29$	−6.35121	−1.17939	−0.589695	−	0.807626i	$-0.700752\pi$
$29$	−6.35121	−1.17939	−0.589695	+	0.807626i	$0.700752\pi$
$30$	1.00000	0.182574
$31$	7.07196	1.27016	0.635081	−	0.772446i	$-0.280967\pi$
$31$	7.07196	1.27016	0.635081	+	0.772446i	$0.280967\pi$
$32$	1.00000	0.176777
$33$	2.66871	0.464563
$34$	−4.35121	−0.746226
$35$	1.91928	0.324418
$36$	1.00000	0.166667
$37$	10.5188	1.72928	0.864638	−	0.502396i	$-0.167548\pi$
$37$	10.5188	1.72928	0.864638	+	0.502396i	$0.167548\pi$
$38$	−3.54974	−0.575844
$39$	1.41814	0.227084
$40$	−1.00000	−0.158114
$41$	7.92962	1.23840	0.619199	−	0.785234i	$-0.287457\pi$
$41$	7.92962	1.23840	0.619199	+	0.785234i	$0.287457\pi$
$42$	1.91928	0.296152
$43$	−0.320546	−0.0488828	−0.0244414	−	0.999701i	$-0.507781\pi$
$43$	−0.320546	−0.0488828	−0.0244414	+	0.999701i	$0.507781\pi$
$44$	−2.66871	−0.402323
$45$	−1.00000	−0.149071
$46$	6.45944	0.952392
$47$	−6.45944	−0.942206	−0.471103	−	0.882078i	$-0.656144\pi$
$47$	−6.45944	−0.942206	−0.471103	+	0.882078i	$0.656144\pi$
$48$	−1.00000	−0.144338
$49$	−3.31635	−0.473764
$50$	1.00000	0.141421
$51$	4.35121	0.609291
$52$	−1.41814	−0.196660
$53$	3.87343	0.532057	0.266028	−	0.963965i	$-0.414288\pi$
$53$	3.87343	0.532057	0.266028	+	0.963965i	$0.414288\pi$
$54$	−1.00000	−0.136083
$55$	2.66871	0.359849
$56$	−1.91928	−0.256475
$57$	3.54974	0.470174
$58$	−6.35121	−0.833954
$59$	13.8255	1.79993	0.899965	−	0.435962i	$-0.143592\pi$
$59$	13.8255	1.79993	0.899965	+	0.435962i	$0.143592\pi$
$60$	1.00000	0.129099
$61$	−0.720755	−0.0922832	−0.0461416	−	0.998935i	$-0.514693\pi$
$61$	−0.720755	−0.0922832	−0.0461416	+	0.998935i	$0.514693\pi$
$62$	7.07196	0.898140
$63$	−1.91928	−0.241807
$64$	1.00000	0.125000
$65$	1.41814	0.175898
$66$	2.66871	0.328496
$67$	12.4479	1.52076	0.760379	−	0.649479i	$-0.225013\pi$
$67$	12.4479	1.52076	0.760379	+	0.649479i	$0.225013\pi$
$68$	−4.35121	−0.527661
$69$	−6.45944	−0.777625
$70$	1.91928	0.229398
$71$	−10.1534	−1.20499	−0.602495	−	0.798123i	$-0.705827\pi$
$71$	−10.1534	−1.20499	−0.602495	+	0.798123i	$0.705827\pi$
$72$	1.00000	0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	10.5188	1.22278
$75$	−1.00000	−0.115470
$76$	−3.54974	−0.407183
$77$	5.12202	0.583708
$78$	1.41814	0.160572
$79$	−2.17375	−0.244566	−0.122283	−	0.992495i	$-0.539022\pi$
$79$	−2.17375	−0.244566	−0.122283	+	0.992495i	$0.539022\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	7.92962	0.875680
$83$	2.47357	0.271510	0.135755	−	0.990742i	$-0.456654\pi$
$83$	2.47357	0.271510	0.135755	+	0.990742i	$0.456654\pi$
$84$	1.91928	0.209411
$85$	4.35121	0.471955
$86$	−0.320546	−0.0345654
$87$	6.35121	0.680921
$88$	−2.66871	−0.284485
$89$	12.7311	1.34949	0.674746	−	0.738050i	$-0.264253\pi$
$89$	12.7311	1.34949	0.674746	+	0.738050i	$0.264253\pi$
$90$	−1.00000	−0.105409
$91$	2.72181	0.285323
$92$	6.45944	0.673443
$93$	−7.07196	−0.733328
$94$	−6.45944	−0.666240
$95$	3.54974	0.364195
$96$	−1.00000	−0.102062
$97$	−3.93382	−0.399419	−0.199709	−	0.979855i	$-0.564000\pi$
$97$	−3.93382	−0.399419	−0.199709	+	0.979855i	$0.564000\pi$
$98$	−3.31635	−0.335001
$99$	−2.66871	−0.268216
$100$	1.00000	0.100000
$101$	5.43532	0.540834	0.270417	−	0.962743i	$-0.412838\pi$
$101$	5.43532	0.540834	0.270417	+	0.962743i	$0.412838\pi$
$102$	4.35121	0.430834
$103$	18.8799	1.86029	0.930144	−	0.367194i	$-0.119681\pi$
$103$	18.8799	1.86029	0.930144	+	0.367194i	$0.119681\pi$
$104$	−1.41814	−0.139060
$105$	−1.91928	−0.187303
$106$	3.87343	0.376221
$107$	−9.87757	−0.954901	−0.477451	−	0.878659i	$-0.658439\pi$
$107$	−9.87757	−0.954901	−0.477451	+	0.878659i	$0.658439\pi$
$108$	−1.00000	−0.0962250
$109$	−8.91053	−0.853474	−0.426737	−	0.904376i	$-0.640337\pi$
$109$	−8.91053	−0.853474	−0.426737	+	0.904376i	$0.640337\pi$
$110$	2.66871	0.254452
$111$	−10.5188	−0.998397
$112$	−1.91928	−0.181355
$113$	−10.2746	−0.966556	−0.483278	−	0.875467i	$-0.660554\pi$
$113$	−10.2746	−0.966556	−0.483278	+	0.875467i	$0.660554\pi$
$114$	3.54974	0.332463
$115$	−6.45944	−0.602346
$116$	−6.35121	−0.589695
$117$	−1.41814	−0.131107
$118$	13.8255	1.27274
$119$	8.35121	0.765554
$120$	1.00000	0.0912871
$121$	−3.87798	−0.352544
$122$	−0.720755	−0.0652541
$123$	−7.92962	−0.714990
$124$	7.07196	0.635081
$125$	−1.00000	−0.0894427
$126$	−1.91928	−0.170983
$127$	1.99696	0.177201	0.0886006	−	0.996067i	$-0.471761\pi$
$127$	1.99696	0.177201	0.0886006	+	0.996067i	$0.471761\pi$
$128$	1.00000	0.0883883
$129$	0.320546	0.0282225
$130$	1.41814	0.124379
$131$	10.3054	0.900383	0.450192	−	0.892932i	$-0.351356\pi$
$131$	10.3054	0.900383	0.450192	+	0.892932i	$0.351356\pi$
$132$	2.66871	0.232281
$133$	6.81296	0.590758
$134$	12.4479	1.07534
$135$	1.00000	0.0860663
$136$	−4.35121	−0.373113
$137$	−1.00000	−0.0854358
$137$	−1.00000	−0.0854358
$138$	−6.45944	−0.549864
$139$	14.4901	1.22903	0.614517	−	0.788904i	$-0.289351\pi$
$139$	14.4901	1.22903	0.614517	+	0.788904i	$0.289351\pi$
$140$	1.91928	0.162209
$141$	6.45944	0.543983
$142$	−10.1534	−0.852056
$143$	3.78459	0.316484
$144$	1.00000	0.0833333
$145$	6.35121	0.527439
$146$	2.00000	0.165521
$147$	3.31635	0.273528
$148$	10.5188	0.864638
$149$	6.89451	0.564820	0.282410	−	0.959294i	$-0.408866\pi$
$149$	6.89451	0.564820	0.282410	+	0.959294i	$0.408866\pi$
$150$	−1.00000	−0.0816497
$151$	−22.1224	−1.80030	−0.900149	−	0.435582i	$-0.856543\pi$
$151$	−22.1224	−1.80030	−0.900149	+	0.435582i	$0.856543\pi$
$152$	−3.54974	−0.287922
$153$	−4.35121	−0.351774
$154$	5.12202	0.412744
$155$	−7.07196	−0.568034
$156$	1.41814	0.113542
$157$	22.2361	1.77463	0.887316	−	0.461162i	$-0.152567\pi$
$157$	22.2361	1.77463	0.887316	+	0.461162i	$0.152567\pi$
$158$	−2.17375	−0.172935
$159$	−3.87343	−0.307183
$160$	−1.00000	−0.0790569
$161$	−12.3975	−0.977060
$162$	1.00000	0.0785674
$163$	11.8386	0.927268	0.463634	−	0.886027i	$-0.346545\pi$
$163$	11.8386	0.927268	0.463634	+	0.886027i	$0.346545\pi$
$164$	7.92962	0.619199
$165$	−2.66871	−0.207759
$166$	2.47357	0.191987
$167$	5.66112	0.438070	0.219035	−	0.975717i	$-0.429709\pi$
$167$	5.66112	0.438070	0.219035	+	0.975717i	$0.429709\pi$
$168$	1.91928	0.148076
$169$	−10.9889	−0.845299
$170$	4.35121	0.333722
$171$	−3.54974	−0.271455
$172$	−0.320546	−0.0244414
$173$	12.8363	0.975924	0.487962	−	0.872865i	$-0.337741\pi$
$173$	12.8363	0.975924	0.487962	+	0.872865i	$0.337741\pi$
$174$	6.35121	0.481484
$175$	−1.91928	−0.145084
$176$	−2.66871	−0.201162
$177$	−13.8255	−1.03919
$178$	12.7311	0.954235
$179$	1.97557	0.147661	0.0738307	−	0.997271i	$-0.476478\pi$
$179$	1.97557	0.147661	0.0738307	+	0.997271i	$0.476478\pi$
$180$	−1.00000	−0.0745356
$181$	−4.63275	−0.344350	−0.172175	−	0.985066i	$-0.555079\pi$
$181$	−4.63275	−0.344350	−0.172175	+	0.985066i	$0.555079\pi$
$182$	2.72181	0.201754
$183$	0.720755	0.0532797
$184$	6.45944	0.476196
$185$	−10.5188	−0.773355
$186$	−7.07196	−0.518541
$187$	11.6121	0.849162
$188$	−6.45944	−0.471103
$189$	1.91928	0.139607
$190$	3.54974	0.257525
$191$	−15.6266	−1.13070	−0.565352	−	0.824850i	$-0.691260\pi$
$191$	−15.6266	−1.13070	−0.565352	+	0.824850i	$0.691260\pi$
$192$	−1.00000	−0.0721688
$193$	−0.273641	−0.0196971	−0.00984856	−	0.999952i	$-0.503135\pi$
$193$	−0.273641	−0.0196971	−0.00984856	+	0.999952i	$0.503135\pi$
$194$	−3.93382	−0.282432
$195$	−1.41814	−0.101555
$196$	−3.31635	−0.236882
$197$	−5.58040	−0.397587	−0.198794	−	0.980041i	$-0.563702\pi$
$197$	−5.58040	−0.397587	−0.198794	+	0.980041i	$0.563702\pi$
$198$	−2.66871	−0.189657
$199$	27.0046	1.91431	0.957154	−	0.289581i	$-0.0935160\pi$
$199$	27.0046	1.91431	0.957154	+	0.289581i	$0.0935160\pi$
$200$	1.00000	0.0707107
$201$	−12.4479	−0.878011
$202$	5.43532	0.382428
$203$	12.1898	0.855555
$204$	4.35121	0.304646
$205$	−7.92962	−0.553828
$206$	18.8799	1.31542
$207$	6.45944	0.448962
$208$	−1.41814	−0.0983300
$209$	9.47322	0.655277
$210$	−1.91928	−0.132443
$211$	11.0188	0.758567	0.379283	−	0.925281i	$-0.376171\pi$
$211$	11.0188	0.758567	0.379283	+	0.925281i	$0.376171\pi$
$212$	3.87343	0.266028
$213$	10.1534	0.695701
$214$	−9.87757	−0.675217
$215$	0.320546	0.0218611
$216$	−1.00000	−0.0680414
$217$	−13.5731	−0.921403
$218$	−8.91053	−0.603497
$219$	−2.00000	−0.135147
$220$	2.66871	0.179924
$221$	6.17061	0.415080
$222$	−10.5188	−0.705974
$223$	24.5505	1.64402	0.822011	−	0.569472i	$-0.192852\pi$
$223$	24.5505	1.64402	0.822011	+	0.569472i	$0.192852\pi$
$224$	−1.91928	−0.128238
$225$	1.00000	0.0666667
$226$	−10.2746	−0.683458
$227$	0.497448	0.0330168	0.0165084	−	0.999864i	$-0.494745\pi$
$227$	0.497448	0.0330168	0.0165084	+	0.999864i	$0.494745\pi$
$228$	3.54974	0.235087
$229$	−1.05699	−0.0698479	−0.0349240	−	0.999390i	$-0.511119\pi$
$229$	−1.05699	−0.0698479	−0.0349240	+	0.999390i	$0.511119\pi$
$230$	−6.45944	−0.425923
$231$	−5.12202	−0.337004
$232$	−6.35121	−0.416977
$233$	15.6905	1.02792	0.513960	−	0.857814i	$-0.328178\pi$
$233$	15.6905	1.02792	0.513960	+	0.857814i	$0.328178\pi$
$234$	−1.41814	−0.0927065
$235$	6.45944	0.421367
$236$	13.8255	0.899965
$237$	2.17375	0.141200
$238$	8.35121	0.541328
$239$	−26.1543	−1.69178	−0.845889	−	0.533359i	$-0.820930\pi$
$239$	−26.1543	−1.69178	−0.845889	+	0.533359i	$0.820930\pi$
$240$	1.00000	0.0645497
$241$	13.3539	0.860203	0.430102	−	0.902780i	$-0.358478\pi$
$241$	13.3539	0.860203	0.430102	+	0.902780i	$0.358478\pi$
$242$	−3.87798	−0.249286
$243$	−1.00000	−0.0641500
$244$	−0.720755	−0.0461416
$245$	3.31635	0.211874
$246$	−7.92962	−0.505574
$247$	5.03401	0.320307
$248$	7.07196	0.449070
$249$	−2.47357	−0.156756
$250$	−1.00000	−0.0632456
$251$	20.5352	1.29617	0.648085	−	0.761568i	$-0.275570\pi$
$251$	20.5352	1.29617	0.648085	+	0.761568i	$0.275570\pi$
$252$	−1.91928	−0.120904
$253$	−17.2384	−1.08377
$254$	1.99696	0.125300
$255$	−4.35121	−0.272483
$256$	1.00000	0.0625000
$257$	6.98390	0.435644	0.217822	−	0.975989i	$-0.430105\pi$
$257$	6.98390	0.435644	0.217822	+	0.975989i	$0.430105\pi$
$258$	0.320546	0.0199563
$259$	−20.1885	−1.25445
$260$	1.41814	0.0879491
$261$	−6.35121	−0.393130
$262$	10.3054	0.636667
$263$	−0.651029	−0.0401442	−0.0200721	−	0.999799i	$-0.506390\pi$
$263$	−0.651029	−0.0401442	−0.0200721	+	0.999799i	$0.506390\pi$
$264$	2.66871	0.164248
$265$	−3.87343	−0.237943
$266$	6.81296	0.417729
$267$	−12.7311	−0.779130
$268$	12.4479	0.760379
$269$	−22.8761	−1.39478	−0.697391	−	0.716691i	$-0.745656\pi$
$269$	−22.8761	−1.39478	−0.697391	+	0.716691i	$0.745656\pi$
$270$	1.00000	0.0608581
$271$	−19.8286	−1.20450	−0.602250	−	0.798308i	$-0.705729\pi$
$271$	−19.8286	−1.20450	−0.602250	+	0.798308i	$0.705729\pi$
$272$	−4.35121	−0.263831
$273$	−2.72181	−0.164731
$274$	−1.00000	−0.0604122
$275$	−2.66871	−0.160929
$276$	−6.45944	−0.388812
$277$	16.9299	1.01722	0.508608	−	0.860998i	$-0.330160\pi$
$277$	16.9299	1.01722	0.508608	+	0.860998i	$0.330160\pi$
$278$	14.4901	0.869058
$279$	7.07196	0.423387
$280$	1.91928	0.114699
$281$	−8.56773	−0.511108	−0.255554	−	0.966795i	$-0.582258\pi$
$281$	−8.56773	−0.511108	−0.255554	+	0.966795i	$0.582258\pi$
$282$	6.45944	0.384654
$283$	2.35607	0.140054	0.0700268	−	0.997545i	$-0.477692\pi$
$283$	2.35607	0.140054	0.0700268	+	0.997545i	$0.477692\pi$
$284$	−10.1534	−0.602495
$285$	−3.54974	−0.210268
$286$	3.78459	0.223788
$287$	−15.2192	−0.898361
$288$	1.00000	0.0589256
$289$	1.93301	0.113707
$290$	6.35121	0.372956
$291$	3.93382	0.230604
$292$	2.00000	0.117041
$293$	25.4127	1.48462	0.742312	−	0.670054i	$-0.233729\pi$
$293$	25.4127	1.48462	0.742312	+	0.670054i	$0.233729\pi$
$294$	3.31635	0.193413
$295$	−13.8255	−0.804953
$296$	10.5188	0.611391
$297$	2.66871	0.154854
$298$	6.89451	0.399388
$299$	−9.16036	−0.529757
$300$	−1.00000	−0.0577350
$301$	0.615219	0.0354607
$302$	−22.1224	−1.27300
$303$	−5.43532	−0.312251
$304$	−3.54974	−0.203591
$305$	0.720755	0.0412703
$306$	−4.35121	−0.248742
$307$	24.9078	1.42156	0.710782	−	0.703412i	$-0.248341\pi$
$307$	24.9078	1.42156	0.710782	+	0.703412i	$0.248341\pi$
$308$	5.12202	0.291854
$309$	−18.8799	−1.07404
$310$	−7.07196	−0.401660
$311$	18.5441	1.05154	0.525769	−	0.850627i	$-0.323778\pi$
$311$	18.5441	1.05154	0.525769	+	0.850627i	$0.323778\pi$
$312$	1.41814	0.0802861
$313$	−13.7062	−0.774721	−0.387360	−	0.921928i	$-0.626613\pi$
$313$	−13.7062	−0.774721	−0.387360	+	0.921928i	$0.626613\pi$
$314$	22.2361	1.25485
$315$	1.91928	0.108139
$316$	−2.17375	−0.122283
$317$	−13.1870	−0.740658	−0.370329	−	0.928901i	$-0.620755\pi$
$317$	−13.1870	−0.740658	−0.370329	+	0.928901i	$0.620755\pi$
$318$	−3.87343	−0.217211
$319$	16.9495	0.948992
$320$	−1.00000	−0.0559017
$321$	9.87757	0.551312
$322$	−12.3975	−0.690886
$323$	15.4456	0.859419
$324$	1.00000	0.0555556
$325$	−1.41814	−0.0786640
$326$	11.8386	0.655678
$327$	8.91053	0.492754
$328$	7.92962	0.437840
$329$	12.3975	0.683496
$330$	−2.66871	−0.146908
$331$	35.9387	1.97537	0.987684	−	0.156459i	$-0.0500079\pi$
$331$	35.9387	1.97537	0.987684	+	0.156459i	$0.0500079\pi$
$332$	2.47357	0.135755
$333$	10.5188	0.576425
$334$	5.66112	0.309762
$335$	−12.4479	−0.680104
$336$	1.91928	0.104706
$337$	−34.3389	−1.87056	−0.935279	−	0.353910i	$-0.884852\pi$
$337$	−34.3389	−1.87056	−0.935279	+	0.353910i	$0.884852\pi$
$338$	−10.9889	−0.597717
$339$	10.2746	0.558041
$340$	4.35121	0.235977
$341$	−18.8730	−1.02203
$342$	−3.54974	−0.191948
$343$	19.8000	1.06910
$344$	−0.320546	−0.0172827
$345$	6.45944	0.347764
$346$	12.8363	0.690082
$347$	−7.88498	−0.423288	−0.211644	−	0.977347i	$-0.567882\pi$
$347$	−7.88498	−0.423288	−0.211644	+	0.977347i	$0.567882\pi$
$348$	6.35121	0.340460
$349$	−18.8609	−1.00960	−0.504801	−	0.863236i	$-0.668434\pi$
$349$	−18.8609	−1.00960	−0.504801	+	0.863236i	$0.668434\pi$
$350$	−1.91928	−0.102590
$351$	1.41814	0.0756945
$352$	−2.66871	−0.142243
$353$	−13.5382	−0.720565	−0.360283	−	0.932843i	$-0.617320\pi$
$353$	−13.5382	−0.720565	−0.360283	+	0.932843i	$0.617320\pi$
$354$	−13.8255	−0.734819
$355$	10.1534	0.538888
$356$	12.7311	0.674746
$357$	−8.35121	−0.441993
$358$	1.97557	0.104412
$359$	−3.41465	−0.180218	−0.0901092	−	0.995932i	$-0.528722\pi$
$359$	−3.41465	−0.180218	−0.0901092	+	0.995932i	$0.528722\pi$
$360$	−1.00000	−0.0527046
$361$	−6.39936	−0.336808
$362$	−4.63275	−0.243492
$363$	3.87798	0.203541
$364$	2.72181	0.142661
$365$	−2.00000	−0.104685
$366$	0.720755	0.0376745
$367$	20.3339	1.06142	0.530709	−	0.847554i	$-0.321926\pi$
$367$	20.3339	1.06142	0.530709	+	0.847554i	$0.321926\pi$
$368$	6.45944	0.336721
$369$	7.92962	0.412799
$370$	−10.5188	−0.546845
$371$	−7.43422	−0.385966
$372$	−7.07196	−0.366664
$373$	0.132759	0.00687401	0.00343701	−	0.999994i	$-0.498906\pi$
$373$	0.132759	0.00687401	0.00343701	+	0.999994i	$0.498906\pi$
$374$	11.6121	0.600448
$375$	1.00000	0.0516398
$376$	−6.45944	−0.333120
$377$	9.00688	0.463878
$378$	1.91928	0.0987174
$379$	8.86153	0.455186	0.227593	−	0.973756i	$-0.426914\pi$
$379$	8.86153	0.455186	0.227593	+	0.973756i	$0.426914\pi$
$380$	3.54974	0.182098
$381$	−1.99696	−0.102307
$382$	−15.6266	−0.799529
$383$	−21.0517	−1.07569	−0.537847	−	0.843042i	$-0.680762\pi$
$383$	−21.0517	−1.07569	−0.537847	+	0.843042i	$0.680762\pi$
$384$	−1.00000	−0.0510310
$385$	−5.12202	−0.261042
$386$	−0.273641	−0.0139280
$387$	−0.320546	−0.0162943
$388$	−3.93382	−0.199709
$389$	−3.25471	−0.165021	−0.0825103	−	0.996590i	$-0.526294\pi$
$389$	−3.25471	−0.165021	−0.0825103	+	0.996590i	$0.526294\pi$
$390$	−1.41814	−0.0718101
$391$	−28.1064	−1.42140
$392$	−3.31635	−0.167501
$393$	−10.3054	−0.519836
$394$	−5.58040	−0.281137
$395$	2.17375	0.109373
$396$	−2.66871	−0.134108
$397$	−34.5247	−1.73275	−0.866373	−	0.499397i	$-0.833555\pi$
$397$	−34.5247	−1.73275	−0.866373	+	0.499397i	$0.833555\pi$
$398$	27.0046	1.35362
$399$	−6.81296	−0.341075
$400$	1.00000	0.0500000
$401$	−11.5623	−0.577395	−0.288697	−	0.957420i	$-0.593222\pi$
$401$	−11.5623	−0.577395	−0.288697	+	0.957420i	$0.593222\pi$
$402$	−12.4479	−0.620847
$403$	−10.0290	−0.499580
$404$	5.43532	0.270417
$405$	−1.00000	−0.0496904
$406$	12.1898	0.604969
$407$	−28.0716	−1.39146
$408$	4.35121	0.215417
$409$	−2.16332	−0.106969	−0.0534846	−	0.998569i	$-0.517033\pi$
$409$	−2.16332	−0.106969	−0.0534846	+	0.998569i	$0.517033\pi$
$410$	−7.92962	−0.391616
$411$	1.00000	0.0493264
$412$	18.8799	0.930144
$413$	−26.5351	−1.30571
$414$	6.45944	0.317464
$415$	−2.47357	−0.121423
$416$	−1.41814	−0.0695298
$417$	−14.4901	−0.709583
$418$	9.47322	0.463351
$419$	4.54545	0.222060	0.111030	−	0.993817i	$-0.464585\pi$
$419$	4.54545	0.222060	0.111030	+	0.993817i	$0.464585\pi$
$420$	−1.91928	−0.0936515
$421$	−0.798382	−0.0389108	−0.0194554	−	0.999811i	$-0.506193\pi$
$421$	−0.798382	−0.0389108	−0.0194554	+	0.999811i	$0.506193\pi$
$422$	11.0188	0.536388
$423$	−6.45944	−0.314069
$424$	3.87343	0.188111
$425$	−4.35121	−0.211065
$426$	10.1534	0.491935
$427$	1.38333	0.0669442
$428$	−9.87757	−0.477451
$429$	−3.78459	−0.182722
$430$	0.320546	0.0154581
$431$	9.88841	0.476308	0.238154	−	0.971227i	$-0.423458\pi$
$431$	9.88841	0.476308	0.238154	+	0.971227i	$0.423458\pi$
$432$	−1.00000	−0.0481125
$433$	23.5177	1.13019	0.565093	−	0.825027i	$-0.308840\pi$
$433$	23.5177	1.13019	0.565093	+	0.825027i	$0.308840\pi$
$434$	−13.5731	−0.651530
$435$	−6.35121	−0.304517
$436$	−8.91053	−0.426737
$437$	−22.9293	−1.09686
$438$	−2.00000	−0.0955637
$439$	14.4686	0.690549	0.345275	−	0.938502i	$-0.387786\pi$
$439$	14.4686	0.690549	0.345275	+	0.938502i	$0.387786\pi$
$440$	2.66871	0.127226
$441$	−3.31635	−0.157921
$442$	6.17061	0.293506
$443$	15.8121	0.751256	0.375628	−	0.926771i	$-0.377427\pi$
$443$	15.8121	0.751256	0.375628	+	0.926771i	$0.377427\pi$
$444$	−10.5188	−0.499199
$445$	−12.7311	−0.603511
$446$	24.5505	1.16250
$447$	−6.89451	−0.326099
$448$	−1.91928	−0.0906777
$449$	−17.2406	−0.813635	−0.406818	−	0.913509i	$-0.633362\pi$
$449$	−17.2406	−0.813635	−0.406818	+	0.913509i	$0.633362\pi$
$450$	1.00000	0.0471405
$451$	−21.1619	−0.996473
$452$	−10.2746	−0.483278
$453$	22.1224	1.03940
$454$	0.497448	0.0233464
$455$	−2.72181	−0.127600
$456$	3.54974	0.166232
$457$	34.7182	1.62405	0.812024	−	0.583625i	$-0.198366\pi$
$457$	34.7182	1.62405	0.812024	+	0.583625i	$0.198366\pi$
$458$	−1.05699	−0.0493899
$459$	4.35121	0.203097
$460$	−6.45944	−0.301173
$461$	−8.08420	−0.376519	−0.188259	−	0.982119i	$-0.560285\pi$
$461$	−8.08420	−0.376519	−0.188259	+	0.982119i	$0.560285\pi$
$462$	−5.12202	−0.238298
$463$	−20.9530	−0.973769	−0.486885	−	0.873466i	$-0.661867\pi$
$463$	−20.9530	−0.973769	−0.486885	+	0.873466i	$0.661867\pi$
$464$	−6.35121	−0.294847
$465$	7.07196	0.327954
$466$	15.6905	0.726849
$467$	−30.3295	−1.40348	−0.701741	−	0.712432i	$-0.747594\pi$
$467$	−30.3295	−1.40348	−0.701741	+	0.712432i	$0.747594\pi$
$468$	−1.41814	−0.0655534
$469$	−23.8912	−1.10319
$470$	6.45944	0.297952
$471$	−22.2361	−1.02458
$472$	13.8255	0.636372
$473$	0.855445	0.0393334
$474$	2.17375	0.0998438
$475$	−3.54974	−0.162873
$476$	8.35121	0.382777
$477$	3.87343	0.177352
$478$	−26.1543	−1.19627
$479$	−35.8189	−1.63661	−0.818303	−	0.574788i	$-0.805085\pi$
$479$	−35.8189	−1.63661	−0.818303	+	0.574788i	$0.805085\pi$
$480$	1.00000	0.0456435
$481$	−14.9170	−0.680159
$482$	13.3539	0.608256
$483$	12.3975	0.564106
$484$	−3.87798	−0.176272
$485$	3.93382	0.178625
$486$	−1.00000	−0.0453609
$487$	24.9385	1.13007	0.565035	−	0.825067i	$-0.308863\pi$
$487$	24.9385	1.13007	0.565035	+	0.825067i	$0.308863\pi$
$488$	−0.720755	−0.0326270
$489$	−11.8386	−0.535359
$490$	3.31635	0.149817
$491$	−1.46714	−0.0662110	−0.0331055	−	0.999452i	$-0.510540\pi$
$491$	−1.46714	−0.0662110	−0.0331055	+	0.999452i	$0.510540\pi$
$492$	−7.92962	−0.357495
$493$	27.6354	1.24464
$494$	5.03401	0.226491
$495$	2.66871	0.119950
$496$	7.07196	0.317541
$497$	19.4873	0.874125
$498$	−2.47357	−0.110844
$499$	13.1967	0.590763	0.295382	−	0.955379i	$-0.404553\pi$
$499$	13.1967	0.590763	0.295382	+	0.955379i	$0.404553\pi$
$500$	−1.00000	−0.0447214
$501$	−5.66112	−0.252920
$502$	20.5352	0.916530
$503$	6.94759	0.309778	0.154889	−	0.987932i	$-0.450498\pi$
$503$	6.94759	0.309778	0.154889	+	0.987932i	$0.450498\pi$
$504$	−1.91928	−0.0854917
$505$	−5.43532	−0.241868
$506$	−17.2384	−0.766339
$507$	10.9889	0.488034
$508$	1.99696	0.0886006
$509$	−2.65407	−0.117640	−0.0588199	−	0.998269i	$-0.518734\pi$
$509$	−2.65407	−0.117640	−0.0588199	+	0.998269i	$0.518734\pi$
$510$	−4.35121	−0.192675
$511$	−3.83857	−0.169808
$512$	1.00000	0.0441942
$513$	3.54974	0.156725
$514$	6.98390	0.308047
$515$	−18.8799	−0.831947
$516$	0.320546	0.0141113
$517$	17.2384	0.758143
$518$	−20.1885	−0.887032
$519$	−12.8363	−0.563450
$520$	1.41814	0.0621894
$521$	−31.6174	−1.38518	−0.692591	−	0.721331i	$-0.743531\pi$
$521$	−31.6174	−1.38518	−0.692591	+	0.721331i	$0.743531\pi$
$522$	−6.35121	−0.277985
$523$	16.4542	0.719492	0.359746	−	0.933050i	$-0.382863\pi$
$523$	16.4542	0.719492	0.359746	+	0.933050i	$0.382863\pi$
$524$	10.3054	0.450192
$525$	1.91928	0.0837645
$526$	−0.651029	−0.0283862
$527$	−30.7716	−1.34043
$528$	2.66871	0.116141
$529$	18.7243	0.814101
$530$	−3.87343	−0.168251
$531$	13.8255	0.599977
$532$	6.81296	0.295379
$533$	−11.2453	−0.487087
$534$	−12.7311	−0.550928
$535$	9.87757	0.427045
$536$	12.4479	0.537669
$537$	−1.97557	−0.0852523
$538$	−22.8761	−0.986259
$539$	8.85037	0.381212
$540$	1.00000	0.0430331
$541$	26.9565	1.15895	0.579474	−	0.814991i	$-0.303258\pi$
$541$	26.9565	1.15895	0.579474	+	0.814991i	$0.303258\pi$
$542$	−19.8286	−0.851710
$543$	4.63275	0.198810
$544$	−4.35121	−0.186557
$545$	8.91053	0.381685
$546$	−2.72181	−0.116483
$547$	30.6278	1.30955	0.654776	−	0.755823i	$-0.272763\pi$
$547$	30.6278	1.30955	0.654776	+	0.755823i	$0.272763\pi$
$548$	−1.00000	−0.0427179
$549$	−0.720755	−0.0307611
$550$	−2.66871	−0.113794
$551$	22.5451	0.960455
$552$	−6.45944	−0.274932
$553$	4.17205	0.177414
$554$	16.9299	0.719281
$555$	10.5188	0.446497
$556$	14.4901	0.614517
$557$	20.3081	0.860481	0.430240	−	0.902714i	$-0.358429\pi$
$557$	20.3081	0.860481	0.430240	+	0.902714i	$0.358429\pi$
$558$	7.07196	0.299380
$559$	0.454578	0.0192266
$560$	1.91928	0.0811046
$561$	−11.6121	−0.490264
$562$	−8.56773	−0.361408
$563$	29.4815	1.24250	0.621249	−	0.783613i	$-0.286626\pi$
$563$	29.4815	1.24250	0.621249	+	0.783613i	$0.286626\pi$
$564$	6.45944	0.271991
$565$	10.2746	0.432257
$566$	2.35607	0.0990328
$567$	−1.91928	−0.0806024
$568$	−10.1534	−0.426028
$569$	−13.0799	−0.548338	−0.274169	−	0.961681i	$-0.588403\pi$
$569$	−13.0799	−0.548338	−0.274169	+	0.961681i	$0.588403\pi$
$570$	−3.54974	−0.148682
$571$	−15.7021	−0.657114	−0.328557	−	0.944484i	$-0.606562\pi$
$571$	−15.7021	−0.657114	−0.328557	+	0.944484i	$0.606562\pi$
$572$	3.78459	0.158242
$573$	15.6266	0.652813
$574$	−15.2192	−0.635237
$575$	6.45944	0.269377
$576$	1.00000	0.0416667
$577$	−38.2285	−1.59147	−0.795736	−	0.605644i	$-0.792916\pi$
$577$	−38.2285	−1.59147	−0.795736	+	0.605644i	$0.792916\pi$
$578$	1.93301	0.0804027
$579$	0.273641	0.0113721
$580$	6.35121	0.263720
$581$	−4.74749	−0.196959
$582$	3.93382	0.163062
$583$	−10.3371	−0.428118
$584$	2.00000	0.0827606
$585$	1.41814	0.0586327
$586$	25.4127	1.04979
$587$	−14.6020	−0.602688	−0.301344	−	0.953516i	$-0.597435\pi$
$587$	−14.6020	−0.602688	−0.301344	+	0.953516i	$0.597435\pi$
$588$	3.31635	0.136764
$589$	−25.1036	−1.03438
$590$	−13.8255	−0.569188
$591$	5.58040	0.229547
$592$	10.5188	0.432319
$593$	−24.9243	−1.02352	−0.511760	−	0.859128i	$-0.671006\pi$
$593$	−24.9243	−1.02352	−0.511760	+	0.859128i	$0.671006\pi$
$594$	2.66871	0.109499
$595$	−8.35121	−0.342366
$596$	6.89451	0.282410
$597$	−27.0046	−1.10523
$598$	−9.16036	−0.374595
$599$	−23.7215	−0.969234	−0.484617	−	0.874726i	$-0.661041\pi$
$599$	−23.7215	−0.969234	−0.484617	+	0.874726i	$0.661041\pi$
$600$	−1.00000	−0.0408248
$601$	−0.0148562	−0.000605997 0	−0.000302999	−	1.00000i	$-0.500096\pi$
$601$	−0.0148562	−0.000605997 0	−0.000302999		1.00000i	$0.500096\pi$
$602$	0.615219	0.0250745
$603$	12.4479	0.506920
$604$	−22.1224	−0.900149
$605$	3.87798	0.157662
$606$	−5.43532	−0.220795
$607$	−44.1493	−1.79196	−0.895982	−	0.444090i	$-0.853527\pi$
$607$	−44.1493	−1.79196	−0.895982	+	0.444090i	$0.853527\pi$
$608$	−3.54974	−0.143961
$609$	−12.1898	−0.493955
$610$	0.720755	0.0291825
$611$	9.16036	0.370589
$612$	−4.35121	−0.175887
$613$	4.90709	0.198196	0.0990978	−	0.995078i	$-0.468404\pi$
$613$	4.90709	0.198196	0.0990978	+	0.995078i	$0.468404\pi$
$614$	24.9078	1.00520
$615$	7.92962	0.319753
$616$	5.12202	0.206372
$617$	−1.37719	−0.0554434	−0.0277217	−	0.999616i	$-0.508825\pi$
$617$	−1.37719	−0.0554434	−0.0277217	+	0.999616i	$0.508825\pi$
$618$	−18.8799	−0.759460
$619$	−17.1984	−0.691262	−0.345631	−	0.938370i	$-0.612335\pi$
$619$	−17.1984	−0.691262	−0.345631	+	0.938370i	$0.612335\pi$
$620$	−7.07196	−0.284017
$621$	−6.45944	−0.259208
$622$	18.5441	0.743550
$623$	−24.4346	−0.978951
$624$	1.41814	0.0567709
$625$	1.00000	0.0400000
$626$	−13.7062	−0.547810
$627$	−9.47322	−0.378324
$628$	22.2361	0.887316
$629$	−45.7694	−1.82494
$630$	1.91928	0.0764661
$631$	32.6901	1.30137	0.650686	−	0.759347i	$-0.274481\pi$
$631$	32.6901	1.30137	0.650686	+	0.759347i	$0.274481\pi$
$632$	−2.17375	−0.0864673
$633$	−11.0188	−0.437959
$634$	−13.1870	−0.523724
$635$	−1.99696	−0.0792468
$636$	−3.87343	−0.153592
$637$	4.70303	0.186341
$638$	16.9495	0.671039
$639$	−10.1534	−0.401663
$640$	−1.00000	−0.0395285
$641$	10.0849	0.398331	0.199165	−	0.979966i	$-0.436177\pi$
$641$	10.0849	0.398331	0.199165	+	0.979966i	$0.436177\pi$
$642$	9.87757	0.389837
$643$	31.3751	1.23731	0.618656	−	0.785662i	$-0.287678\pi$
$643$	31.3751	1.23731	0.618656	+	0.785662i	$0.287678\pi$
$644$	−12.3975	−0.488530
$645$	−0.320546	−0.0126215
$646$	15.4456	0.607701
$647$	34.7897	1.36772	0.683861	−	0.729612i	$-0.260299\pi$
$647$	34.7897	1.36772	0.683861	+	0.729612i	$0.260299\pi$
$648$	1.00000	0.0392837
$649$	−36.8963	−1.44831
$650$	−1.41814	−0.0556239
$651$	13.5731	0.531972
$652$	11.8386	0.463634
$653$	−8.32939	−0.325954	−0.162977	−	0.986630i	$-0.552110\pi$
$653$	−8.32939	−0.325954	−0.162977	+	0.986630i	$0.552110\pi$
$654$	8.91053	0.348429
$655$	−10.3054	−0.402664
$656$	7.92962	0.309600
$657$	2.00000	0.0780274
$658$	12.3975	0.483305
$659$	11.1494	0.434318	0.217159	−	0.976136i	$-0.430321\pi$
$659$	11.1494	0.434318	0.217159	+	0.976136i	$0.430321\pi$
$660$	−2.66871	−0.103879
$661$	−29.4553	−1.14568	−0.572839	−	0.819668i	$-0.694158\pi$
$661$	−29.4553	−1.14568	−0.572839	+	0.819668i	$0.694158\pi$
$662$	35.9387	1.39680
$663$	−6.17061	−0.239646
$664$	2.47357	0.0959933
$665$	−6.81296	−0.264195
$666$	10.5188	0.407594
$667$	−41.0252	−1.58850
$668$	5.66112	0.219035
$669$	−24.5505	−0.949177
$670$	−12.4479	−0.480906
$671$	1.92349	0.0742553
$672$	1.91928	0.0740380
$673$	48.3882	1.86523	0.932615	−	0.360874i	$-0.117522\pi$
$673$	48.3882	1.86523	0.932615	+	0.360874i	$0.117522\pi$
$674$	−34.3389	−1.32268
$675$	−1.00000	−0.0384900
$676$	−10.9889	−0.422650
$677$	8.04938	0.309363	0.154681	−	0.987964i	$-0.450565\pi$
$677$	8.04938	0.309363	0.154681	+	0.987964i	$0.450565\pi$
$678$	10.2746	0.394595
$679$	7.55012	0.289747
$680$	4.35121	0.166861
$681$	−0.497448	−0.0190622
$682$	−18.8730	−0.722685
$683$	32.0704	1.22714	0.613571	−	0.789640i	$-0.289733\pi$
$683$	32.0704	1.22714	0.613571	+	0.789640i	$0.289733\pi$
$684$	−3.54974	−0.135728
$685$	1.00000	0.0382080
$686$	19.8000	0.755968
$687$	1.05699	0.0403267
$688$	−0.320546	−0.0122207
$689$	−5.49305	−0.209269
$690$	6.45944	0.245907
$691$	−31.4343	−1.19582	−0.597908	−	0.801564i	$-0.704001\pi$
$691$	−31.4343	−1.19582	−0.597908	+	0.801564i	$0.704001\pi$
$692$	12.8363	0.487962
$693$	5.12202	0.194569
$694$	−7.88498	−0.299310
$695$	−14.4901	−0.549641
$696$	6.35121	0.240742
$697$	−34.5034	−1.30691
$698$	−18.8609	−0.713896
$699$	−15.6905	−0.593470
$700$	−1.91928	−0.0725422
$701$	23.1454	0.874189	0.437094	−	0.899416i	$-0.356008\pi$
$701$	23.1454	0.874189	0.437094	+	0.899416i	$0.356008\pi$
$702$	1.41814	0.0535241
$703$	−37.3389	−1.40826
$704$	−2.66871	−0.100581
$705$	−6.45944	−0.243276
$706$	−13.5382	−0.509516
$707$	−10.4319	−0.392333
$708$	−13.8255	−0.519595
$709$	38.7671	1.45593	0.727964	−	0.685616i	$-0.240467\pi$
$709$	38.7671	1.45593	0.727964	+	0.685616i	$0.240467\pi$
$710$	10.1534	0.381051
$711$	−2.17375	−0.0815221
$712$	12.7311	0.477118
$713$	45.6809	1.71076
$714$	−8.35121	−0.312536
$715$	−3.78459	−0.141536
$716$	1.97557	0.0738307
$717$	26.1543	0.976749
$718$	−3.41465	−0.127434
$719$	−3.18979	−0.118959	−0.0594796	−	0.998230i	$-0.518944\pi$
$719$	−3.18979	−0.118959	−0.0594796	+	0.998230i	$0.518944\pi$
$720$	−1.00000	−0.0372678
$721$	−36.2359	−1.34949
$722$	−6.39936	−0.238159
$723$	−13.3539	−0.496639
$724$	−4.63275	−0.172175
$725$	−6.35121	−0.235878
$726$	3.87798	0.143925
$727$	−7.23990	−0.268513	−0.134257	−	0.990947i	$-0.542865\pi$
$727$	−7.23990	−0.268513	−0.134257	+	0.990947i	$0.542865\pi$
$728$	2.72181	0.100877
$729$	1.00000	0.0370370
$730$	−2.00000	−0.0740233
$731$	1.39476	0.0515872
$732$	0.720755	0.0266399
$733$	5.65757	0.208967	0.104484	−	0.994527i	$-0.466681\pi$
$733$	5.65757	0.208967	0.104484	+	0.994527i	$0.466681\pi$
$734$	20.3339	0.750537
$735$	−3.31635	−0.122325
$736$	6.45944	0.238098
$737$	−33.2200	−1.22367
$738$	7.92962	0.291893
$739$	3.20072	0.117741	0.0588703	−	0.998266i	$-0.481250\pi$
$739$	3.20072	0.117741	0.0588703	+	0.998266i	$0.481250\pi$
$740$	−10.5188	−0.386678
$741$	−5.03401	−0.184929
$742$	−7.43422	−0.272919
$743$	10.6883	0.392115	0.196057	−	0.980592i	$-0.437186\pi$
$743$	10.6883	0.392115	0.196057	+	0.980592i	$0.437186\pi$
$744$	−7.07196	−0.259271
$745$	−6.89451	−0.252595
$746$	0.132759	0.00486066
$747$	2.47357	0.0905034
$748$	11.6121	0.424581
$749$	18.9579	0.692706
$750$	1.00000	0.0365148
$751$	−19.8822	−0.725511	−0.362755	−	0.931884i	$-0.618164\pi$
$751$	−19.8822	−0.725511	−0.362755	+	0.931884i	$0.618164\pi$
$752$	−6.45944	−0.235551
$753$	−20.5352	−0.748344
$754$	9.00688	0.328011
$755$	22.1224	0.805118
$756$	1.91928	0.0698037
$757$	19.4550	0.707104	0.353552	−	0.935415i	$-0.384974\pi$
$757$	19.4550	0.707104	0.353552	+	0.935415i	$0.384974\pi$
$758$	8.86153	0.321865
$759$	17.2384	0.625713
$760$	3.54974	0.128763
$761$	−11.1005	−0.402392	−0.201196	−	0.979551i	$-0.564483\pi$
$761$	−11.1005	−0.402392	−0.201196	+	0.979551i	$0.564483\pi$
$762$	−1.99696	−0.0723421
$763$	17.1019	0.619129
$764$	−15.6266	−0.565352
$765$	4.35121	0.157318
$766$	−21.0517	−0.760631
$767$	−19.6065	−0.707949
$768$	−1.00000	−0.0360844
$769$	43.4509	1.56688	0.783440	−	0.621467i	$-0.213463\pi$
$769$	43.4509	1.56688	0.783440	+	0.621467i	$0.213463\pi$
$770$	−5.12202	−0.184585
$771$	−6.98390	−0.251519
$772$	−0.273641	−0.00984856
$773$	43.8833	1.57837	0.789187	−	0.614153i	$-0.210502\pi$
$773$	43.8833	1.57837	0.789187	+	0.614153i	$0.210502\pi$
$774$	−0.320546	−0.0115218
$775$	7.07196	0.254032
$776$	−3.93382	−0.141216
$777$	20.1885	0.724259
$778$	−3.25471	−0.116687
$779$	−28.1481	−1.00851
$780$	−1.41814	−0.0507774
$781$	27.0965	0.969591
$782$	−28.1064	−1.00508
$783$	6.35121	0.226974
$784$	−3.31635	−0.118441
$785$	−22.2361	−0.793639
$786$	−10.3054	−0.367580
$787$	−47.4173	−1.69024	−0.845121	−	0.534575i	$-0.820472\pi$
$787$	−47.4173	−1.69024	−0.845121	+	0.534575i	$0.820472\pi$
$788$	−5.58040	−0.198794
$789$	0.651029	0.0231773
$790$	2.17375	0.0773387
$791$	19.7199	0.701161
$792$	−2.66871	−0.0948285
$793$	1.02213	0.0362968
$794$	−34.5247	−1.22524
$795$	3.87343	0.137376
$796$	27.0046	0.957154
$797$	−12.3491	−0.437426	−0.218713	−	0.975789i	$-0.570186\pi$
$797$	−12.3491	−0.437426	−0.218713	+	0.975789i	$0.570186\pi$
$798$	−6.81296	−0.241176
$799$	28.1064	0.994331
$800$	1.00000	0.0353553
$801$	12.7311	0.449831
$802$	−11.5623	−0.408280
$803$	−5.33742	−0.188353
$804$	−12.4479	−0.439005
$805$	12.3975	0.436954
$806$	−10.0290	−0.353257
$807$	22.8761	0.805277
$808$	5.43532	0.191214
$809$	−3.12543	−0.109884	−0.0549421	−	0.998490i	$-0.517497\pi$
$809$	−3.12543	−0.109884	−0.0549421	+	0.998490i	$0.517497\pi$
$810$	−1.00000	−0.0351364
$811$	−33.7884	−1.18647	−0.593235	−	0.805029i	$-0.702150\pi$
$811$	−33.7884	−1.18647	−0.593235	+	0.805029i	$0.702150\pi$
$812$	12.1898	0.427777
$813$	19.8286	0.695418
$814$	−28.0716	−0.983907
$815$	−11.8386	−0.414687
$816$	4.35121	0.152323
$817$	1.13785	0.0398085
$818$	−2.16332	−0.0756386
$819$	2.72181	0.0951076
$820$	−7.92962	−0.276914
$821$	−33.6019	−1.17272	−0.586358	−	0.810052i	$-0.699439\pi$
$821$	−33.6019	−1.17272	−0.586358	+	0.810052i	$0.699439\pi$
$822$	1.00000	0.0348790
$823$	−38.0560	−1.32655	−0.663274	−	0.748376i	$-0.730834\pi$
$823$	−38.0560	−1.32655	−0.663274	+	0.748376i	$0.730834\pi$
$824$	18.8799	0.657711
$825$	2.66871	0.0929126
$826$	−26.5351	−0.923275
$827$	−42.7362	−1.48608	−0.743042	−	0.669244i	$-0.766618\pi$
$827$	−42.7362	−1.48608	−0.743042	+	0.669244i	$0.766618\pi$
$828$	6.45944	0.224481
$829$	−7.85006	−0.272644	−0.136322	−	0.990665i	$-0.543528\pi$
$829$	−7.85006	−0.272644	−0.136322	+	0.990665i	$0.543528\pi$
$830$	−2.47357	−0.0858590
$831$	−16.9299	−0.587290
$832$	−1.41814	−0.0491650
$833$	14.4301	0.499974
$834$	−14.4901	−0.501751
$835$	−5.66112	−0.195911
$836$	9.47322	0.327638
$837$	−7.07196	−0.244443
$838$	4.54545	0.157020
$839$	46.9197	1.61985	0.809924	−	0.586534i	$-0.199508\pi$
$839$	46.9197	1.61985	0.809924	+	0.586534i	$0.199508\pi$
$840$	−1.91928	−0.0662216
$841$	11.3378	0.390960
$842$	−0.798382	−0.0275141
$843$	8.56773	0.295088
$844$	11.0188	0.379283
$845$	10.9889	0.378029
$846$	−6.45944	−0.222080
$847$	7.44296	0.255743
$848$	3.87343	0.133014
$849$	−2.35607	−0.0808600
$850$	−4.35121	−0.149245
$851$	67.9453	2.32914
$852$	10.1534	0.347851
$853$	45.7694	1.56711	0.783557	−	0.621319i	$-0.213403\pi$
$853$	45.7694	1.56711	0.783557	+	0.621319i	$0.213403\pi$
$854$	1.38333	0.0473367
$855$	3.54974	0.121398
$856$	−9.87757	−0.337608
$857$	−28.7974	−0.983700	−0.491850	−	0.870680i	$-0.663679\pi$
$857$	−28.7974	−0.983700	−0.491850	+	0.870680i	$0.663679\pi$
$858$	−3.78459	−0.129204
$859$	38.0167	1.29711	0.648557	−	0.761166i	$-0.275373\pi$
$859$	38.0167	1.29711	0.648557	+	0.761166i	$0.275373\pi$
$860$	0.320546	0.0109305
$861$	15.2192	0.518669
$862$	9.88841	0.336800
$863$	−44.4688	−1.51373	−0.756867	−	0.653569i	$-0.773271\pi$
$863$	−44.4688	−1.51373	−0.756867	+	0.653569i	$0.773271\pi$
$864$	−1.00000	−0.0340207
$865$	−12.8363	−0.436446
$866$	23.5177	0.799162
$867$	−1.93301	−0.0656485
$868$	−13.5731	−0.460701
$869$	5.80112	0.196789
$870$	−6.35121	−0.215326
$871$	−17.6529	−0.598145
$872$	−8.91053	−0.301749
$873$	−3.93382	−0.133140
$874$	−22.9293	−0.775596
$875$	1.91928	0.0648837
$876$	−2.00000	−0.0675737
$877$	14.6639	0.495163	0.247582	−	0.968867i	$-0.420364\pi$
$877$	14.6639	0.495163	0.247582	+	0.968867i	$0.420364\pi$
$878$	14.4686	0.488292
$879$	−25.4127	−0.857148
$880$	2.66871	0.0899622
$881$	7.21876	0.243206	0.121603	−	0.992579i	$-0.461197\pi$
$881$	7.21876	0.243206	0.121603	+	0.992579i	$0.461197\pi$
$882$	−3.31635	−0.111667
$883$	39.3408	1.32392	0.661962	−	0.749537i	$-0.269724\pi$
$883$	39.3408	1.32392	0.661962	+	0.749537i	$0.269724\pi$
$884$	6.17061	0.207540
$885$	13.8255	0.464740
$886$	15.8121	0.531218
$887$	−19.5155	−0.655266	−0.327633	−	0.944805i	$-0.606251\pi$
$887$	−19.5155	−0.655266	−0.327633	+	0.944805i	$0.606251\pi$
$888$	−10.5188	−0.352987
$889$	−3.83273	−0.128546
$890$	−12.7311	−0.426747
$891$	−2.66871	−0.0894052
$892$	24.5505	0.822011
$893$	22.9293	0.767300
$894$	−6.89451	−0.230587
$895$	−1.97557	−0.0660362
$896$	−1.91928	−0.0641188
$897$	9.16036	0.305856
$898$	−17.2406	−0.575327
$899$	−44.9155	−1.49802
$900$	1.00000	0.0333333
$901$	−16.8541	−0.561492
$902$	−21.1619	−0.704613
$903$	−0.615219	−0.0204732
$904$	−10.2746	−0.341729
$905$	4.63275	0.153998
$906$	22.1224	0.734969
$907$	−43.0328	−1.42888	−0.714440	−	0.699697i	$-0.753318\pi$
$907$	−43.0328	−1.42888	−0.714440	+	0.699697i	$0.753318\pi$
$908$	0.497448	0.0165084
$909$	5.43532	0.180278
$910$	−2.72181	−0.0902270
$911$	32.5783	1.07937	0.539683	−	0.841868i	$-0.318544\pi$
$911$	32.5783	1.07937	0.539683	+	0.841868i	$0.318544\pi$
$912$	3.54974	0.117544
$913$	−6.60126	−0.218470
$914$	34.7182	1.14837
$915$	−0.720755	−0.0238274
$916$	−1.05699	−0.0349240
$917$	−19.7789	−0.653157
$918$	4.35121	0.143611
$919$	56.4271	1.86136	0.930678	−	0.365838i	$-0.119218\pi$
$919$	56.4271	1.86136	0.930678	+	0.365838i	$0.119218\pi$
$920$	−6.45944	−0.212961
$921$	−24.9078	−0.820741
$922$	−8.08420	−0.266239
$923$	14.3989	0.473947
$924$	−5.12202	−0.168502
$925$	10.5188	0.345855
$926$	−20.9530	−0.688559
$927$	18.8799	0.620096
$928$	−6.35121	−0.208489
$929$	−16.4919	−0.541082	−0.270541	−	0.962708i	$-0.587203\pi$
$929$	−16.4919	−0.541082	−0.270541	+	0.962708i	$0.587203\pi$
$930$	7.07196	0.231899
$931$	11.7722	0.385817
$932$	15.6905	0.513960
$933$	−18.5441	−0.607106
$934$	−30.3295	−0.992412
$935$	−11.6121	−0.379757
$936$	−1.41814	−0.0463532
$937$	−15.9777	−0.521970	−0.260985	−	0.965343i	$-0.584047\pi$
$937$	−15.9777	−0.521970	−0.260985	+	0.965343i	$0.584047\pi$
$938$	−23.8912	−0.780074
$939$	13.7062	0.447285
$940$	6.45944	0.210684
$941$	11.7251	0.382229	0.191114	−	0.981568i	$-0.438790\pi$
$941$	11.7251	0.382229	0.191114	+	0.981568i	$0.438790\pi$
$942$	−22.2361	−0.724490
$943$	51.2209	1.66798
$944$	13.8255	0.449983
$945$	−1.91928	−0.0624343
$946$	0.855445	0.0278129
$947$	10.3374	0.335921	0.167960	−	0.985794i	$-0.446282\pi$
$947$	10.3374	0.335921	0.167960	+	0.985794i	$0.446282\pi$
$948$	2.17375	0.0706002
$949$	−2.83627	−0.0920693
$950$	−3.54974	−0.115169
$951$	13.1870	0.427619
$952$	8.35121	0.270664
$953$	20.8459	0.675265	0.337632	−	0.941278i	$-0.390374\pi$
$953$	20.8459	0.675265	0.337632	+	0.941278i	$0.390374\pi$
$954$	3.87343	0.125407
$955$	15.6266	0.505666
$956$	−26.1543	−0.845889
$957$	−16.9495	−0.547901
$958$	−35.8189	−1.15725
$959$	1.91928	0.0619769
$960$	1.00000	0.0322749
$961$	19.0127	0.613312
$962$	−14.9170	−0.480945
$963$	−9.87757	−0.318300
$964$	13.3539	0.430102
$965$	0.273641	0.00880882
$966$	12.3975	0.398883
$967$	7.11843	0.228913	0.114457	−	0.993428i	$-0.463487\pi$
$967$	7.11843	0.228913	0.114457	+	0.993428i	$0.463487\pi$
$968$	−3.87798	−0.124643
$969$	−15.4456	−0.496186
$970$	3.93382	0.126307
$971$	−40.3284	−1.29420	−0.647100	−	0.762405i	$-0.724018\pi$
$971$	−40.3284	−1.29420	−0.647100	+	0.762405i	$0.724018\pi$
$972$	−1.00000	−0.0320750
$973$	−27.8106	−0.891568
$974$	24.9385	0.799081
$975$	1.41814	0.0454167
$976$	−0.720755	−0.0230708
$977$	−14.2173	−0.454851	−0.227426	−	0.973795i	$-0.573031\pi$
$977$	−14.2173	−0.454851	−0.227426	+	0.973795i	$0.573031\pi$
$978$	−11.8386	−0.378556
$979$	−33.9756	−1.08586
$980$	3.31635	0.105937
$981$	−8.91053	−0.284491
$982$	−1.46714	−0.0468182
$983$	−23.6460	−0.754191	−0.377096	−	0.926174i	$-0.623077\pi$
$983$	−23.6460	−0.754191	−0.377096	+	0.926174i	$0.623077\pi$
$984$	−7.92962	−0.252787
$985$	5.58040	0.177806
$986$	27.6354	0.880091
$987$	−12.3975	−0.394617
$988$	5.03401	0.160153
$989$	−2.07055	−0.0658396
$990$	2.66871	0.0848172
$991$	−33.4910	−1.06388	−0.531938	−	0.846783i	$-0.678536\pi$
$991$	−33.4910	−1.06388	−0.531938	+	0.846783i	$0.678536\pi$
$992$	7.07196	0.224535
$993$	−35.9387	−1.14048
$994$	19.4873	0.618100
$995$	−27.0046	−0.856104
$996$	−2.47357	−0.0783782
$997$	−32.5589	−1.03115	−0.515576	−	0.856844i	$-0.672422\pi$
$997$	−32.5589	−1.03115	−0.515576	+	0.856844i	$0.672422\pi$
$998$	13.1967	0.417733
$999$	−10.5188	−0.332799

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4110.2.a.bb.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4110.2.a.bb.1.2	✓	5	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:	\( N \)	\(=\)	\( 4110 = 2 \cdot 3 \cdot 5 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4110.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.91928 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.91928 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.66871 q^{11} -1.00000 q^{12} -1.41814 q^{13} -1.91928 q^{14} +1.00000 q^{15} +1.00000 q^{16} -4.35121 q^{17} +1.00000 q^{18} -3.54974 q^{19} -1.00000 q^{20} +1.91928 q^{21} -2.66871 q^{22} +6.45944 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.41814 q^{26} -1.00000 q^{27} -1.91928 q^{28} -6.35121 q^{29} +1.00000 q^{30} +7.07196 q^{31} +1.00000 q^{32} +2.66871 q^{33} -4.35121 q^{34} +1.91928 q^{35} +1.00000 q^{36} +10.5188 q^{37} -3.54974 q^{38} +1.41814 q^{39} -1.00000 q^{40} +7.92962 q^{41} +1.91928 q^{42} -0.320546 q^{43} -2.66871 q^{44} -1.00000 q^{45} +6.45944 q^{46} -6.45944 q^{47} -1.00000 q^{48} -3.31635 q^{49} +1.00000 q^{50} +4.35121 q^{51} -1.41814 q^{52} +3.87343 q^{53} -1.00000 q^{54} +2.66871 q^{55} -1.91928 q^{56} +3.54974 q^{57} -6.35121 q^{58} +13.8255 q^{59} +1.00000 q^{60} -0.720755 q^{61} +7.07196 q^{62} -1.91928 q^{63} +1.00000 q^{64} +1.41814 q^{65} +2.66871 q^{66} +12.4479 q^{67} -4.35121 q^{68} -6.45944 q^{69} +1.91928 q^{70} -10.1534 q^{71} +1.00000 q^{72} +2.00000 q^{73} +10.5188 q^{74} -1.00000 q^{75} -3.54974 q^{76} +5.12202 q^{77} +1.41814 q^{78} -2.17375 q^{79} -1.00000 q^{80} +1.00000 q^{81} +7.92962 q^{82} +2.47357 q^{83} +1.91928 q^{84} +4.35121 q^{85} -0.320546 q^{86} +6.35121 q^{87} -2.66871 q^{88} +12.7311 q^{89} -1.00000 q^{90} +2.72181 q^{91} +6.45944 q^{92} -7.07196 q^{93} -6.45944 q^{94} +3.54974 q^{95} -1.00000 q^{96} -3.93382 q^{97} -3.31635 q^{98} -2.66871 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} - 5 q^{3} + 5 q^{4} - 5 q^{5} - 5 q^{6} + 3 q^{7} + 5 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^2 - 5 * q^3 + 5 * q^4 - 5 * q^5 - 5 * q^6 + 3 * q^7 + 5 * q^8 + 5 * q^9	\( 5 q + 5 q^{2} - 5 q^{3} + 5 q^{4} - 5 q^{5} - 5 q^{6} + 3 q^{7} + 5 q^{8} + 5 q^{9} - 5 q^{10} - 5 q^{12} + 7 q^{13} + 3 q^{14} + 5 q^{15} + 5 q^{16} + 5 q^{18} + 10 q^{19} - 5 q^{20} - 3 q^{21} - 4 q^{23} - 5 q^{24} + 5 q^{25} + 7 q^{26} - 5 q^{27} + 3 q^{28} - 10 q^{29} + 5 q^{30} + 7 q^{31} + 5 q^{32} - 3 q^{35} + 5 q^{36} + 16 q^{37} + 10 q^{38} - 7 q^{39} - 5 q^{40} - 4 q^{41} - 3 q^{42} + 14 q^{43} - 5 q^{45} - 4 q^{46} + 4 q^{47} - 5 q^{48} + 8 q^{49} + 5 q^{50} + 7 q^{52} - 3 q^{53} - 5 q^{54} + 3 q^{56} - 10 q^{57} - 10 q^{58} + 12 q^{59} + 5 q^{60} + 3 q^{61} + 7 q^{62} + 3 q^{63} + 5 q^{64} - 7 q^{65} + 24 q^{67} + 4 q^{69} - 3 q^{70} - 8 q^{71} + 5 q^{72} + 10 q^{73} + 16 q^{74} - 5 q^{75} + 10 q^{76} + 16 q^{77} - 7 q^{78} + 7 q^{79} - 5 q^{80} + 5 q^{81} - 4 q^{82} + 4 q^{83} - 3 q^{84} + 14 q^{86} + 10 q^{87} + 26 q^{89} - 5 q^{90} - 5 q^{91} - 4 q^{92} - 7 q^{93} + 4 q^{94} - 10 q^{95} - 5 q^{96} + 15 q^{97} + 8 q^{98}+O(q^{100}) \) 5 * q + 5 * q^2 - 5 * q^3 + 5 * q^4 - 5 * q^5 - 5 * q^6 + 3 * q^7 + 5 * q^8 + 5 * q^9 - 5 * q^10 - 5 * q^12 + 7 * q^13 + 3 * q^14 + 5 * q^15 + 5 * q^16 + 5 * q^18 + 10 * q^19 - 5 * q^20 - 3 * q^21 - 4 * q^23 - 5 * q^24 + 5 * q^25 + 7 * q^26 - 5 * q^27 + 3 * q^28 - 10 * q^29 + 5 * q^30 + 7 * q^31 + 5 * q^32 - 3 * q^35 + 5 * q^36 + 16 * q^37 + 10 * q^38 - 7 * q^39 - 5 * q^40 - 4 * q^41 - 3 * q^42 + 14 * q^43 - 5 * q^45 - 4 * q^46 + 4 * q^47 - 5 * q^48 + 8 * q^49 + 5 * q^50 + 7 * q^52 - 3 * q^53 - 5 * q^54 + 3 * q^56 - 10 * q^57 - 10 * q^58 + 12 * q^59 + 5 * q^60 + 3 * q^61 + 7 * q^62 + 3 * q^63 + 5 * q^64 - 7 * q^65 + 24 * q^67 + 4 * q^69 - 3 * q^70 - 8 * q^71 + 5 * q^72 + 10 * q^73 + 16 * q^74 - 5 * q^75 + 10 * q^76 + 16 * q^77 - 7 * q^78 + 7 * q^79 - 5 * q^80 + 5 * q^81 - 4 * q^82 + 4 * q^83 - 3 * q^84 + 14 * q^86 + 10 * q^87 + 26 * q^89 - 5 * q^90 - 5 * q^91 - 4 * q^92 - 7 * q^93 + 4 * q^94 - 10 * q^95 - 5 * q^96 + 15 * q^97 + 8 * q^98

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