LMFDB - Embedded newform 4015.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4015,2,Mod(1,4015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4015, 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("4015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4015 = 5 \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4015.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:	$32.0599364115$
Analytic rank:	$0$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.34661 q^{2} +2.98114 q^{3} +3.50659 q^{4} -1.00000 q^{5} -6.99558 q^{6} +0.286679 q^{7} -3.53537 q^{8} +5.88720 q^{9} +O(q^{10})$	\(q-2.34661 q^{2} +2.98114 q^{3} +3.50659 q^{4} -1.00000 q^{5} -6.99558 q^{6} +0.286679 q^{7} -3.53537 q^{8} +5.88720 q^{9} +2.34661 q^{10} +1.00000 q^{11} +10.4536 q^{12} -0.143931 q^{13} -0.672724 q^{14} -2.98114 q^{15} +1.28298 q^{16} +5.28532 q^{17} -13.8150 q^{18} +5.34840 q^{19} -3.50659 q^{20} +0.854630 q^{21} -2.34661 q^{22} -0.898127 q^{23} -10.5394 q^{24} +1.00000 q^{25} +0.337750 q^{26} +8.60714 q^{27} +1.00526 q^{28} +1.38453 q^{29} +6.99558 q^{30} -4.84089 q^{31} +4.06010 q^{32} +2.98114 q^{33} -12.4026 q^{34} -0.286679 q^{35} +20.6440 q^{36} +0.737843 q^{37} -12.5506 q^{38} -0.429078 q^{39} +3.53537 q^{40} +8.18243 q^{41} -2.00548 q^{42} +2.25717 q^{43} +3.50659 q^{44} -5.88720 q^{45} +2.10756 q^{46} +0.731108 q^{47} +3.82474 q^{48} -6.91782 q^{49} -2.34661 q^{50} +15.7563 q^{51} -0.504706 q^{52} -2.72342 q^{53} -20.1976 q^{54} -1.00000 q^{55} -1.01352 q^{56} +15.9443 q^{57} -3.24895 q^{58} +9.25857 q^{59} -10.4536 q^{60} -2.78000 q^{61} +11.3597 q^{62} +1.68774 q^{63} -12.0934 q^{64} +0.143931 q^{65} -6.99558 q^{66} +7.49820 q^{67} +18.5334 q^{68} -2.67744 q^{69} +0.672724 q^{70} -6.55767 q^{71} -20.8135 q^{72} -1.00000 q^{73} -1.73143 q^{74} +2.98114 q^{75} +18.7546 q^{76} +0.286679 q^{77} +1.00688 q^{78} -6.30608 q^{79} -1.28298 q^{80} +7.99751 q^{81} -19.2010 q^{82} +2.13987 q^{83} +2.99683 q^{84} -5.28532 q^{85} -5.29670 q^{86} +4.12748 q^{87} -3.53537 q^{88} -9.21151 q^{89} +13.8150 q^{90} -0.0412619 q^{91} -3.14936 q^{92} -14.4314 q^{93} -1.71563 q^{94} -5.34840 q^{95} +12.1037 q^{96} -11.1456 q^{97} +16.2334 q^{98} +5.88720 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q + 6 q^{2} + 5 q^{3} + 22 q^{4} - 27 q^{5} + 7 q^{6} + 10 q^{7} + 15 q^{8} + 24 q^{9}+O(q^{10}) $ 27 * q + 6 * q^2 + 5 * q^3 + 22 * q^4 - 27 * q^5 + 7 * q^6 + 10 * q^7 + 15 * q^8 + 24 * q^9	\( 27 q + 6 q^{2} + 5 q^{3} + 22 q^{4} - 27 q^{5} + 7 q^{6} + 10 q^{7} + 15 q^{8} + 24 q^{9} - 6 q^{10} + 27 q^{11} + 28 q^{12} + 21 q^{13} + 11 q^{14} - 5 q^{15} + 12 q^{16} + 30 q^{17} + 10 q^{18} + 20 q^{19} - 22 q^{20} - 14 q^{21} + 6 q^{22} + 16 q^{23} + 23 q^{24} + 27 q^{25} + 13 q^{26} + 17 q^{27} + 6 q^{28} - 2 q^{29} - 7 q^{30} - 6 q^{31} + 41 q^{32} + 5 q^{33} + 8 q^{34} - 10 q^{35} + 13 q^{36} + 20 q^{37} + 11 q^{38} - 10 q^{39} - 15 q^{40} + 38 q^{41} - 33 q^{42} + 29 q^{43} + 22 q^{44} - 24 q^{45} + 23 q^{46} + 17 q^{47} + 37 q^{48} + 21 q^{49} + 6 q^{50} + 13 q^{51} + 29 q^{52} + 4 q^{53} + q^{54} - 27 q^{55} + 28 q^{56} + 51 q^{57} - 6 q^{58} + 35 q^{59} - 28 q^{60} - 13 q^{61} + 28 q^{62} + 41 q^{63} - 5 q^{64} - 21 q^{65} + 7 q^{66} + 10 q^{67} + 65 q^{68} - 12 q^{69} - 11 q^{70} + 22 q^{71} - 12 q^{72} - 27 q^{73} - 5 q^{74} + 5 q^{75} + 13 q^{76} + 10 q^{77} - 9 q^{78} - 18 q^{79} - 12 q^{80} + 39 q^{81} + 20 q^{82} + 54 q^{83} - 50 q^{84} - 30 q^{85} + 3 q^{86} + 15 q^{87} + 15 q^{88} + 89 q^{89} - 10 q^{90} - 18 q^{91} + 57 q^{92} + 20 q^{93} - 29 q^{94} - 20 q^{95} + 105 q^{96} + 35 q^{97} + 10 q^{98} + 24 q^{99}+O(q^{100}) \) 27 * q + 6 * q^2 + 5 * q^3 + 22 * q^4 - 27 * q^5 + 7 * q^6 + 10 * q^7 + 15 * q^8 + 24 * q^9 - 6 * q^10 + 27 * q^11 + 28 * q^12 + 21 * q^13 + 11 * q^14 - 5 * q^15 + 12 * q^16 + 30 * q^17 + 10 * q^18 + 20 * q^19 - 22 * q^20 - 14 * q^21 + 6 * q^22 + 16 * q^23 + 23 * q^24 + 27 * q^25 + 13 * q^26 + 17 * q^27 + 6 * q^28 - 2 * q^29 - 7 * q^30 - 6 * q^31 + 41 * q^32 + 5 * q^33 + 8 * q^34 - 10 * q^35 + 13 * q^36 + 20 * q^37 + 11 * q^38 - 10 * q^39 - 15 * q^40 + 38 * q^41 - 33 * q^42 + 29 * q^43 + 22 * q^44 - 24 * q^45 + 23 * q^46 + 17 * q^47 + 37 * q^48 + 21 * q^49 + 6 * q^50 + 13 * q^51 + 29 * q^52 + 4 * q^53 + q^54 - 27 * q^55 + 28 * q^56 + 51 * q^57 - 6 * q^58 + 35 * q^59 - 28 * q^60 - 13 * q^61 + 28 * q^62 + 41 * q^63 - 5 * q^64 - 21 * q^65 + 7 * q^66 + 10 * q^67 + 65 * q^68 - 12 * q^69 - 11 * q^70 + 22 * q^71 - 12 * q^72 - 27 * q^73 - 5 * q^74 + 5 * q^75 + 13 * q^76 + 10 * q^77 - 9 * q^78 - 18 * q^79 - 12 * q^80 + 39 * q^81 + 20 * q^82 + 54 * q^83 - 50 * q^84 - 30 * q^85 + 3 * q^86 + 15 * q^87 + 15 * q^88 + 89 * q^89 - 10 * q^90 - 18 * q^91 + 57 * q^92 + 20 * q^93 - 29 * q^94 - 20 * q^95 + 105 * q^96 + 35 * q^97 + 10 * q^98 + 24 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.34661	−1.65931	−0.829653	−	0.558280i	$-0.811461\pi$
$2$	−2.34661	−1.65931	−0.829653	+	0.558280i	$0.811461\pi$
$3$	2.98114	1.72116	0.860581	−	0.509313i	$-0.170101\pi$
$3$	2.98114	1.72116	0.860581	+	0.509313i	$0.170101\pi$
$4$	3.50659	1.75329
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−6.99558	−2.85593
$7$	0.286679	0.108354	0.0541772	−	0.998531i	$-0.482746\pi$
$7$	0.286679	0.108354	0.0541772	+	0.998531i	$0.482746\pi$
$8$	−3.53537	−1.24994
$9$	5.88720	1.96240
$10$	2.34661	0.742064
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	10.4536	3.01770
$13$	−0.143931	−0.0399192	−0.0199596	−	0.999801i	$-0.506354\pi$
$13$	−0.143931	−0.0399192	−0.0199596	+	0.999801i	$0.506354\pi$
$14$	−0.672724	−0.179793
$15$	−2.98114	−0.769727
$16$	1.28298	0.320744
$17$	5.28532	1.28188	0.640939	−	0.767592i	$-0.278545\pi$
$17$	5.28532	1.28188	0.640939	+	0.767592i	$0.278545\pi$
$18$	−13.8150	−3.25622
$19$	5.34840	1.22701	0.613504	−	0.789692i	$-0.289760\pi$
$19$	5.34840	1.22701	0.613504	+	0.789692i	$0.289760\pi$
$20$	−3.50659	−0.784097
$21$	0.854630	0.186496
$22$	−2.34661	−0.500299
$23$	−0.898127	−0.187272	−0.0936362	−	0.995606i	$-0.529849\pi$
$23$	−0.898127	−0.187272	−0.0936362	+	0.995606i	$0.529849\pi$
$24$	−10.5394	−2.15136
$25$	1.00000	0.200000
$26$	0.337750	0.0662381
$27$	8.60714	1.65645
$28$	1.00526	0.189977
$29$	1.38453	0.257101	0.128550	−	0.991703i	$-0.458968\pi$
$29$	1.38453	0.257101	0.128550	+	0.991703i	$0.458968\pi$
$30$	6.99558	1.27721
$31$	−4.84089	−0.869450	−0.434725	−	0.900563i	$-0.643154\pi$
$31$	−4.84089	−0.869450	−0.434725	+	0.900563i	$0.643154\pi$
$32$	4.06010	0.717731
$33$	2.98114	0.518950
$34$	−12.4026	−2.12703
$35$	−0.286679	−0.0484576
$36$	20.6440	3.44066
$37$	0.737843	0.121301	0.0606503	−	0.998159i	$-0.480683\pi$
$37$	0.737843	0.121301	0.0606503	+	0.998159i	$0.480683\pi$
$38$	−12.5506	−2.03598
$39$	−0.429078	−0.0687074
$40$	3.53537	0.558992
$41$	8.18243	1.27788	0.638940	−	0.769256i	$-0.279373\pi$
$41$	8.18243	1.27788	0.638940	+	0.769256i	$0.279373\pi$
$42$	−2.00548	−0.309453
$43$	2.25717	0.344215	0.172107	−	0.985078i	$-0.444942\pi$
$43$	2.25717	0.344215	0.172107	+	0.985078i	$0.444942\pi$
$44$	3.50659	0.528638
$45$	−5.88720	−0.877612
$46$	2.10756	0.310742
$47$	0.731108	0.106643	0.0533215	−	0.998577i	$-0.483019\pi$
$47$	0.731108	0.106643	0.0533215	+	0.998577i	$0.483019\pi$
$48$	3.82474	0.552053
$49$	−6.91782	−0.988259
$50$	−2.34661	−0.331861
$51$	15.7563	2.20632
$52$	−0.504706	−0.0699901
$53$	−2.72342	−0.374090	−0.187045	−	0.982351i	$-0.559891\pi$
$53$	−2.72342	−0.374090	−0.187045	+	0.982351i	$0.559891\pi$
$54$	−20.1976	−2.74855
$55$	−1.00000	−0.134840
$56$	−1.01352	−0.135437
$57$	15.9443	2.11188
$58$	−3.24895	−0.426608
$59$	9.25857	1.20536	0.602682	−	0.797982i	$-0.294099\pi$
$59$	9.25857	1.20536	0.602682	+	0.797982i	$0.294099\pi$
$60$	−10.4536	−1.34956
$61$	−2.78000	−0.355942	−0.177971	−	0.984036i	$-0.556953\pi$
$61$	−2.78000	−0.355942	−0.177971	+	0.984036i	$0.556953\pi$
$62$	11.3597	1.44268
$63$	1.68774	0.212635
$64$	−12.0934	−1.51168
$65$	0.143931	0.0178524
$66$	−6.99558	−0.861096
$67$	7.49820	0.916051	0.458025	−	0.888939i	$-0.348557\pi$
$67$	7.49820	0.916051	0.458025	+	0.888939i	$0.348557\pi$
$68$	18.5334	2.24751
$69$	−2.67744	−0.322326
$70$	0.672724	0.0804059
$71$	−6.55767	−0.778252	−0.389126	−	0.921185i	$-0.627223\pi$
$71$	−6.55767	−0.778252	−0.389126	+	0.921185i	$0.627223\pi$
$72$	−20.8135	−2.45289
$73$	−1.00000	−0.117041
$73$	−1.00000	−0.117041
$74$	−1.73143	−0.201275
$75$	2.98114	0.344232
$76$	18.7546	2.15130
$77$	0.286679	0.0326701
$78$	1.00688	0.114007
$79$	−6.30608	−0.709490	−0.354745	−	0.934963i	$-0.615432\pi$
$79$	−6.30608	−0.709490	−0.354745	+	0.934963i	$0.615432\pi$
$80$	−1.28298	−0.143441
$81$	7.99751	0.888612
$82$	−19.2010	−2.12039
$83$	2.13987	0.234881	0.117441	−	0.993080i	$-0.462531\pi$
$83$	2.13987	0.234881	0.117441	+	0.993080i	$0.462531\pi$
$84$	2.99683	0.326981
$85$	−5.28532	−0.573273
$86$	−5.29670	−0.571157
$87$	4.12748	0.442512
$88$	−3.53537	−0.376872
$89$	−9.21151	−0.976418	−0.488209	−	0.872727i	$-0.662350\pi$
$89$	−9.21151	−0.976418	−0.488209	+	0.872727i	$0.662350\pi$
$90$	13.8150	1.45623
$91$	−0.0412619	−0.00432542
$92$	−3.14936	−0.328344
$93$	−14.4314	−1.49646
$94$	−1.71563	−0.176953
$95$	−5.34840	−0.548734
$96$	12.1037	1.23533
$97$	−11.1456	−1.13166	−0.565831	−	0.824521i	$-0.691444\pi$
$97$	−11.1456	−1.13166	−0.565831	+	0.824521i	$0.691444\pi$
$98$	16.2334	1.63982
$99$	5.88720	0.591686
$100$	3.50659	0.350659
$101$	6.75736	0.672382	0.336191	−	0.941794i	$-0.390861\pi$
$101$	6.75736	0.672382	0.336191	+	0.941794i	$0.390861\pi$
$102$	−36.9739	−3.66096
$103$	−8.13200	−0.801270	−0.400635	−	0.916238i	$-0.631210\pi$
$103$	−8.13200	−0.801270	−0.400635	+	0.916238i	$0.631210\pi$
$104$	0.508849	0.0498968
$105$	−0.854630	−0.0834033
$106$	6.39081	0.620730
$107$	−4.31764	−0.417402	−0.208701	−	0.977979i	$-0.566924\pi$
$107$	−4.31764	−0.417402	−0.208701	+	0.977979i	$0.566924\pi$
$108$	30.1817	2.90424
$109$	7.14209	0.684088	0.342044	−	0.939684i	$-0.388881\pi$
$109$	7.14209	0.684088	0.342044	+	0.939684i	$0.388881\pi$
$110$	2.34661	0.223741
$111$	2.19961	0.208778
$112$	0.367802	0.0347541
$113$	11.7157	1.10212	0.551062	−	0.834464i	$-0.314223\pi$
$113$	11.7157	1.10212	0.551062	+	0.834464i	$0.314223\pi$
$114$	−37.4152	−3.50425
$115$	0.898127	0.0837508
$116$	4.85497	0.450773
$117$	−0.847349	−0.0783374
$118$	−21.7263	−2.00007
$119$	1.51519	0.138897
$120$	10.5394	0.962116
$121$	1.00000	0.0909091
$122$	6.52357	0.590616
$123$	24.3930	2.19944
$124$	−16.9750	−1.52440
$125$	−1.00000	−0.0894427
$126$	−3.96046	−0.352826
$127$	−8.33093	−0.739250	−0.369625	−	0.929181i	$-0.620514\pi$
$127$	−8.33093	−0.739250	−0.369625	+	0.929181i	$0.620514\pi$
$128$	20.2584	1.79061
$129$	6.72893	0.592450
$130$	−0.337750	−0.0296226
$131$	11.1856	0.977292	0.488646	−	0.872482i	$-0.337491\pi$
$131$	11.1856	0.977292	0.488646	+	0.872482i	$0.337491\pi$
$132$	10.4536	0.909872
$133$	1.53327	0.132952
$134$	−17.5954	−1.52001
$135$	−8.60714	−0.740785
$136$	−18.6856	−1.60228
$137$	22.2073	1.89730	0.948648	−	0.316333i	$-0.102452\pi$
$137$	22.2073	1.89730	0.948648	+	0.316333i	$0.102452\pi$
$138$	6.28292	0.534838
$139$	−8.85549	−0.751113	−0.375556	−	0.926800i	$-0.622548\pi$
$139$	−8.85549	−0.751113	−0.375556	+	0.926800i	$0.622548\pi$
$140$	−1.00526	−0.0849603
$141$	2.17953	0.183550
$142$	15.3883	1.29136
$143$	−0.143931	−0.0120361
$144$	7.55314	0.629429
$145$	−1.38453	−0.114979
$146$	2.34661	0.194207
$147$	−20.6230	−1.70095
$148$	2.58731	0.212676
$149$	7.29784	0.597863	0.298931	−	0.954275i	$-0.403370\pi$
$149$	7.29784	0.597863	0.298931	+	0.954275i	$0.403370\pi$
$150$	−6.99558	−0.571187
$151$	14.4230	1.17373	0.586865	−	0.809685i	$-0.300362\pi$
$151$	14.4230	1.17373	0.586865	+	0.809685i	$0.300362\pi$
$152$	−18.9086	−1.53369
$153$	31.1157	2.51556
$154$	−0.672724	−0.0542096
$155$	4.84089	0.388830
$156$	−1.50460	−0.120464
$157$	12.1340	0.968400	0.484200	−	0.874957i	$-0.339111\pi$
$157$	12.1340	0.968400	0.484200	+	0.874957i	$0.339111\pi$
$158$	14.7979	1.17726
$159$	−8.11889	−0.643870
$160$	−4.06010	−0.320979
$161$	−0.257474	−0.0202918
$162$	−18.7671	−1.47448
$163$	15.4797	1.21247	0.606233	−	0.795287i	$-0.292680\pi$
$163$	15.4797	1.21247	0.606233	+	0.795287i	$0.292680\pi$
$164$	28.6924	2.24050
$165$	−2.98114	−0.232081
$166$	−5.02144	−0.389739
$167$	0.781677	0.0604880	0.0302440	−	0.999543i	$-0.490372\pi$
$167$	0.781677	0.0604880	0.0302440	+	0.999543i	$0.490372\pi$
$168$	−3.02144	−0.233109
$169$	−12.9793	−0.998406
$170$	12.4026	0.951236
$171$	31.4871	2.40788
$172$	7.91495	0.603510
$173$	5.74690	0.436929	0.218464	−	0.975845i	$-0.429895\pi$
$173$	5.74690	0.436929	0.218464	+	0.975845i	$0.429895\pi$
$174$	−9.68558	−0.734262
$175$	0.286679	0.0216709
$176$	1.28298	0.0967081
$177$	27.6011	2.07463
$178$	21.6158	1.62018
$179$	9.15286	0.684117	0.342058	−	0.939679i	$-0.388876\pi$
$179$	9.15286	0.684117	0.342058	+	0.939679i	$0.388876\pi$
$180$	−20.6440	−1.53871
$181$	16.5274	1.22847	0.614237	−	0.789121i	$-0.289464\pi$
$181$	16.5274	1.22847	0.614237	+	0.789121i	$0.289464\pi$
$182$	0.0968257	0.00717720
$183$	−8.28756	−0.612634
$184$	3.17522	0.234080
$185$	−0.737843	−0.0542473
$186$	33.8649	2.48309
$187$	5.28532	0.386501
$188$	2.56369	0.186976
$189$	2.46749	0.179483
$190$	12.5506	0.910518
$191$	−14.9010	−1.07820	−0.539101	−	0.842241i	$-0.681236\pi$
$191$	−14.9010	−1.07820	−0.539101	+	0.842241i	$0.681236\pi$
$192$	−36.0522	−2.60184
$193$	13.5464	0.975090	0.487545	−	0.873098i	$-0.337893\pi$
$193$	13.5464	0.975090	0.487545	+	0.873098i	$0.337893\pi$
$194$	26.1543	1.87777
$195$	0.429078	0.0307269
$196$	−24.2579	−1.73271
$197$	−12.1909	−0.868562	−0.434281	−	0.900777i	$-0.642998\pi$
$197$	−12.1909	−0.868562	−0.434281	+	0.900777i	$0.642998\pi$
$198$	−13.8150	−0.981787
$199$	−23.9640	−1.69876	−0.849381	−	0.527781i	$-0.823024\pi$
$199$	−23.9640	−1.69876	−0.849381	+	0.527781i	$0.823024\pi$
$200$	−3.53537	−0.249989
$201$	22.3532	1.57667
$202$	−15.8569	−1.11569
$203$	0.396915	0.0278580
$204$	55.2508	3.86833
$205$	−8.18243	−0.571486
$206$	19.0827	1.32955
$207$	−5.28745	−0.367503
$208$	−0.184660	−0.0128039
$209$	5.34840	0.369957
$210$	2.00548	0.138392
$211$	3.09774	0.213257	0.106628	−	0.994299i	$-0.465994\pi$
$211$	3.09774	0.213257	0.106628	+	0.994299i	$0.465994\pi$
$212$	−9.54990	−0.655890
$213$	−19.5493	−1.33950
$214$	10.1318	0.692598
$215$	−2.25717	−0.153938
$216$	−30.4295	−2.07046
$217$	−1.38778	−0.0942088
$218$	−16.7597	−1.13511
$219$	−2.98114	−0.201447
$220$	−3.50659	−0.236414
$221$	−0.760720	−0.0511716
$222$	−5.16164	−0.346426
$223$	0.609582	0.0408206	0.0204103	−	0.999792i	$-0.493503\pi$
$223$	0.609582	0.0408206	0.0204103	+	0.999792i	$0.493503\pi$
$224$	1.16394	0.0777693
$225$	5.88720	0.392480
$226$	−27.4923	−1.82876
$227$	17.9631	1.19225	0.596125	−	0.802892i	$-0.296706\pi$
$227$	17.9631	1.19225	0.596125	+	0.802892i	$0.296706\pi$
$228$	55.9102	3.70274
$229$	9.01832	0.595948	0.297974	−	0.954574i	$-0.403689\pi$
$229$	9.01832	0.595948	0.297974	+	0.954574i	$0.403689\pi$
$230$	−2.10756	−0.138968
$231$	0.854630	0.0562305
$232$	−4.89483	−0.321361
$233$	−10.2409	−0.670905	−0.335452	−	0.942057i	$-0.608889\pi$
$233$	−10.2409	−0.670905	−0.335452	+	0.942057i	$0.608889\pi$
$234$	1.98840	0.129986
$235$	−0.731108	−0.0476922
$236$	32.4660	2.11336
$237$	−18.7993	−1.22115
$238$	−3.55556	−0.230473
$239$	2.96702	0.191920	0.0959602	−	0.995385i	$-0.469408\pi$
$239$	2.96702	0.191920	0.0959602	+	0.995385i	$0.469408\pi$
$240$	−3.82474	−0.246886
$241$	2.70644	0.174337	0.0871686	−	0.996194i	$-0.472218\pi$
$241$	2.70644	0.174337	0.0871686	+	0.996194i	$0.472218\pi$
$242$	−2.34661	−0.150846
$243$	−1.97973	−0.127000
$244$	−9.74829	−0.624071
$245$	6.91782	0.441963
$246$	−57.2408	−3.64954
$247$	−0.769800	−0.0489812
$248$	17.1144	1.08676
$249$	6.37925	0.404268
$250$	2.34661	0.148413
$251$	−3.50733	−0.221381	−0.110690	−	0.993855i	$-0.535306\pi$
$251$	−3.50733	−0.221381	−0.110690	+	0.993855i	$0.535306\pi$
$252$	5.91819	0.372811
$253$	−0.898127	−0.0564648
$254$	19.5494	1.22664
$255$	−15.7563	−0.986697
$256$	−23.3517	−1.45948
$257$	−12.6949	−0.791887	−0.395943	−	0.918275i	$-0.629582\pi$
$257$	−12.6949	−0.791887	−0.395943	+	0.918275i	$0.629582\pi$
$258$	−15.7902	−0.983055
$259$	0.211524	0.0131435
$260$	0.504706	0.0313005
$261$	8.15100	0.504534
$262$	−26.2483	−1.62163
$263$	10.7989	0.665891	0.332945	−	0.942946i	$-0.391957\pi$
$263$	10.7989	0.665891	0.332945	+	0.942946i	$0.391957\pi$
$264$	−10.5394	−0.648658
$265$	2.72342	0.167298
$266$	−3.59800	−0.220607
$267$	−27.4608	−1.68057
$268$	26.2931	1.60611
$269$	−1.66376	−0.101441	−0.0507206	−	0.998713i	$-0.516152\pi$
$269$	−1.66376	−0.101441	−0.0507206	+	0.998713i	$0.516152\pi$
$270$	20.1976	1.22919
$271$	−20.6514	−1.25448	−0.627242	−	0.778824i	$-0.715816\pi$
$271$	−20.6514	−1.25448	−0.627242	+	0.778824i	$0.715816\pi$
$272$	6.78095	0.411155
$273$	−0.123008	−0.00744475
$274$	−52.1119	−3.14819
$275$	1.00000	0.0603023
$276$	−9.38869	−0.565133
$277$	−0.409007	−0.0245748	−0.0122874	−	0.999925i	$-0.503911\pi$
$277$	−0.409007	−0.0245748	−0.0122874	+	0.999925i	$0.503911\pi$
$278$	20.7804	1.24632
$279$	−28.4993	−1.70621
$280$	1.01352	0.0605692
$281$	−15.2778	−0.911396	−0.455698	−	0.890134i	$-0.650610\pi$
$281$	−15.2778	−0.911396	−0.455698	+	0.890134i	$0.650610\pi$
$282$	−5.11452	−0.304565
$283$	1.64944	0.0980490	0.0490245	−	0.998798i	$-0.484389\pi$
$283$	1.64944	0.0980490	0.0490245	+	0.998798i	$0.484389\pi$
$284$	−22.9950	−1.36450
$285$	−15.9443	−0.944461
$286$	0.337750	0.0199716
$287$	2.34573	0.138464
$288$	23.9026	1.40847
$289$	10.9346	0.643212
$290$	3.24895	0.190785
$291$	−33.2265	−1.94777
$292$	−3.50659	−0.205207
$293$	6.05938	0.353992	0.176996	−	0.984212i	$-0.443362\pi$
$293$	6.05938	0.353992	0.176996	+	0.984212i	$0.443362\pi$
$294$	48.3941	2.82240
$295$	−9.25857	−0.539055
$296$	−2.60855	−0.151619
$297$	8.60714	0.499437
$298$	−17.1252	−0.992036
$299$	0.129268	0.00747577
$300$	10.4536	0.603540
$301$	0.647082	0.0372972
$302$	−33.8453	−1.94758
$303$	20.1446	1.15728
$304$	6.86188	0.393556
$305$	2.78000	0.159182
$306$	−73.0165	−4.17408
$307$	−10.4420	−0.595954	−0.297977	−	0.954573i	$-0.596312\pi$
$307$	−10.4420	−0.595954	−0.297977	+	0.954573i	$0.596312\pi$
$308$	1.00526	0.0572802
$309$	−24.2426	−1.37912
$310$	−11.3597	−0.645188
$311$	27.2608	1.54582	0.772908	−	0.634518i	$-0.218801\pi$
$311$	27.2608	1.54582	0.772908	+	0.634518i	$0.218801\pi$
$312$	1.51695	0.0858804
$313$	−18.7928	−1.06223	−0.531115	−	0.847300i	$-0.678227\pi$
$313$	−18.7928	−1.06223	−0.531115	+	0.847300i	$0.678227\pi$
$314$	−28.4738	−1.60687
$315$	−1.68774	−0.0950931
$316$	−22.1128	−1.24394
$317$	5.03513	0.282801	0.141401	−	0.989952i	$-0.454839\pi$
$317$	5.03513	0.282801	0.141401	+	0.989952i	$0.454839\pi$
$318$	19.0519	1.06838
$319$	1.38453	0.0775188
$320$	12.0934	0.676043
$321$	−12.8715	−0.718417
$322$	0.604192	0.0336703
$323$	28.2680	1.57287
$324$	28.0440	1.55800
$325$	−0.143931	−0.00798384
$326$	−36.3249	−2.01185
$327$	21.2916	1.17743
$328$	−28.9280	−1.59728
$329$	0.209593	0.0115552
$330$	6.99558	0.385094
$331$	−24.2819	−1.33466	−0.667328	−	0.744764i	$-0.732562\pi$
$331$	−24.2819	−1.33466	−0.667328	+	0.744764i	$0.732562\pi$
$332$	7.50363	0.411815
$333$	4.34383	0.238040
$334$	−1.83429	−0.100368
$335$	−7.49820	−0.409670
$336$	1.09647	0.0598174
$337$	−6.20569	−0.338045	−0.169023	−	0.985612i	$-0.554061\pi$
$337$	−6.20569	−0.338045	−0.169023	+	0.985612i	$0.554061\pi$
$338$	30.4573	1.65666
$339$	34.9263	1.89694
$340$	−18.5334	−1.00512
$341$	−4.84089	−0.262149
$342$	−73.8880	−3.99541
$343$	−3.98994	−0.215437
$344$	−7.97993	−0.430249
$345$	2.67744	0.144149
$346$	−13.4857	−0.724998
$347$	16.0694	0.862652	0.431326	−	0.902196i	$-0.358046\pi$
$347$	16.0694	0.862652	0.431326	+	0.902196i	$0.358046\pi$
$348$	14.4734	0.775853
$349$	6.95560	0.372325	0.186162	−	0.982519i	$-0.440395\pi$
$349$	6.95560	0.372325	0.186162	+	0.982519i	$0.440395\pi$
$350$	−0.672724	−0.0359586
$351$	−1.23883	−0.0661240
$352$	4.06010	0.216404
$353$	25.1524	1.33873	0.669364	−	0.742934i	$-0.266567\pi$
$353$	25.1524	1.33873	0.669364	+	0.742934i	$0.266567\pi$
$354$	−64.7691	−3.44244
$355$	6.55767	0.348045
$356$	−32.3010	−1.71195
$357$	4.51699	0.239065
$358$	−21.4782	−1.13516
$359$	−28.8066	−1.52035	−0.760176	−	0.649717i	$-0.774887\pi$
$359$	−28.8066	−1.52035	−0.760176	+	0.649717i	$0.774887\pi$
$360$	20.8135	1.09697
$361$	9.60540	0.505547
$362$	−38.7835	−2.03841
$363$	2.98114	0.156469
$364$	−0.144688	−0.00758373
$365$	1.00000	0.0523424
$366$	19.4477	1.01655
$367$	−7.37067	−0.384746	−0.192373	−	0.981322i	$-0.561618\pi$
$367$	−7.37067	−0.384746	−0.192373	+	0.981322i	$0.561618\pi$
$368$	−1.15228	−0.0600666
$369$	48.1716	2.50771
$370$	1.73143	0.0900128
$371$	−0.780746	−0.0405343
$372$	−50.6049	−2.62374
$373$	−0.895851	−0.0463854	−0.0231927	−	0.999731i	$-0.507383\pi$
$373$	−0.895851	−0.0463854	−0.0231927	+	0.999731i	$0.507383\pi$
$374$	−12.4026	−0.641323
$375$	−2.98114	−0.153945
$376$	−2.58474	−0.133298
$377$	−0.199276	−0.0102633
$378$	−5.79023	−0.297817
$379$	5.82110	0.299010	0.149505	−	0.988761i	$-0.452232\pi$
$379$	5.82110	0.299010	0.149505	+	0.988761i	$0.452232\pi$
$380$	−18.7546	−0.962092
$381$	−24.8357	−1.27237
$382$	34.9669	1.78906
$383$	35.0149	1.78918	0.894588	−	0.446892i	$-0.147469\pi$
$383$	35.0149	1.78918	0.894588	+	0.446892i	$0.147469\pi$
$384$	60.3931	3.08192
$385$	−0.286679	−0.0146105
$386$	−31.7881	−1.61797
$387$	13.2884	0.675487
$388$	−39.0829	−1.98413
$389$	−2.40076	−0.121723	−0.0608617	−	0.998146i	$-0.519385\pi$
$389$	−2.40076	−0.121723	−0.0608617	+	0.998146i	$0.519385\pi$
$390$	−1.00688	−0.0509853
$391$	−4.74689	−0.240060
$392$	24.4571	1.23527
$393$	33.3459	1.68208
$394$	28.6072	1.44121
$395$	6.30608	0.317294
$396$	20.6440	1.03740
$397$	8.80087	0.441703	0.220851	−	0.975307i	$-0.429116\pi$
$397$	8.80087	0.441703	0.220851	+	0.975307i	$0.429116\pi$
$398$	56.2341	2.81876
$399$	4.57090	0.228831
$400$	1.28298	0.0641489
$401$	−25.4711	−1.27197	−0.635983	−	0.771703i	$-0.719405\pi$
$401$	−25.4711	−1.27197	−0.635983	+	0.771703i	$0.719405\pi$
$402$	−52.4542	−2.61618
$403$	0.696754	0.0347078
$404$	23.6953	1.17888
$405$	−7.99751	−0.397400
$406$	−0.931406	−0.0462249
$407$	0.737843	0.0365735
$408$	−55.7044	−2.75778
$409$	−12.8812	−0.636934	−0.318467	−	0.947934i	$-0.603168\pi$
$409$	−12.8812	−0.636934	−0.318467	+	0.947934i	$0.603168\pi$
$410$	19.2010	0.948269
$411$	66.2030	3.26555
$412$	−28.5156	−1.40486
$413$	2.65424	0.130606
$414$	12.4076	0.609800
$415$	−2.13987	−0.105042
$416$	−0.584373	−0.0286512
$417$	−26.3995	−1.29279
$418$	−12.5506	−0.613871
$419$	−23.7100	−1.15831	−0.579155	−	0.815217i	$-0.696617\pi$
$419$	−23.7100	−1.15831	−0.579155	+	0.815217i	$0.696617\pi$
$420$	−2.99683	−0.146231
$421$	23.4645	1.14359	0.571794	−	0.820397i	$-0.306248\pi$
$421$	23.4645	1.14359	0.571794	+	0.820397i	$0.306248\pi$
$422$	−7.26918	−0.353858
$423$	4.30418	0.209276
$424$	9.62830	0.467592
$425$	5.28532	0.256376
$426$	45.8747	2.22263
$427$	−0.796966	−0.0385679
$428$	−15.1402	−0.731829
$429$	−0.429078	−0.0207161
$430$	5.29670	0.255429
$431$	−13.7206	−0.660900	−0.330450	−	0.943824i	$-0.607200\pi$
$431$	−13.7206	−0.660900	−0.330450	+	0.943824i	$0.607200\pi$
$432$	11.0428	0.531296
$433$	15.2911	0.734843	0.367422	−	0.930055i	$-0.380241\pi$
$433$	15.2911	0.734843	0.367422	+	0.930055i	$0.380241\pi$
$434$	3.25658	0.156321
$435$	−4.12748	−0.197897
$436$	25.0444	1.19941
$437$	−4.80354	−0.229785
$438$	6.99558	0.334262
$439$	24.1606	1.15312	0.576560	−	0.817055i	$-0.304395\pi$
$439$	24.1606	1.15312	0.576560	+	0.817055i	$0.304395\pi$
$440$	3.53537	0.168542
$441$	−40.7266	−1.93936
$442$	1.78511	0.0849092
$443$	35.2267	1.67367	0.836835	−	0.547455i	$-0.184403\pi$
$443$	35.2267	1.67367	0.836835	+	0.547455i	$0.184403\pi$
$444$	7.71313	0.366049
$445$	9.21151	0.436667
$446$	−1.43045	−0.0677338
$447$	21.7559	1.02902
$448$	−3.46693	−0.163797
$449$	−23.0745	−1.08895	−0.544477	−	0.838776i	$-0.683272\pi$
$449$	−23.0745	−1.08895	−0.544477	+	0.838776i	$0.683272\pi$
$450$	−13.8150	−0.651244
$451$	8.18243	0.385296
$452$	41.0823	1.93235
$453$	42.9971	2.02018
$454$	−42.1523	−1.97831
$455$	0.0412619	0.00193439
$456$	−56.3692	−2.63973
$457$	21.6613	1.01327	0.506637	−	0.862159i	$-0.330888\pi$
$457$	21.6613	1.01327	0.506637	+	0.862159i	$0.330888\pi$
$458$	−21.1625	−0.988859
$459$	45.4915	2.12336
$460$	3.14936	0.146840
$461$	38.4387	1.79027	0.895134	−	0.445797i	$-0.147080\pi$
$461$	38.4387	1.79027	0.895134	+	0.445797i	$0.147080\pi$
$462$	−2.00548	−0.0933036
$463$	−38.7759	−1.80207	−0.901035	−	0.433747i	$-0.857191\pi$
$463$	−38.7759	−1.80207	−0.901035	+	0.433747i	$0.857191\pi$
$464$	1.77632	0.0824636
$465$	14.4314	0.669239
$466$	24.0315	1.11324
$467$	5.20401	0.240813	0.120406	−	0.992725i	$-0.461580\pi$
$467$	5.20401	0.240813	0.120406	+	0.992725i	$0.461580\pi$
$468$	−2.97130	−0.137349
$469$	2.14957	0.0992581
$470$	1.71563	0.0791359
$471$	36.1732	1.66677
$472$	−32.7325	−1.50664
$473$	2.25717	0.103785
$474$	44.1147	2.02626
$475$	5.34840	0.245401
$476$	5.31314	0.243527
$477$	−16.0333	−0.734115
$478$	−6.96244	−0.318455
$479$	18.1617	0.829827	0.414914	−	0.909861i	$-0.363812\pi$
$479$	18.1617	0.829827	0.414914	+	0.909861i	$0.363812\pi$
$480$	−12.1037	−0.552457
$481$	−0.106198	−0.00484222
$482$	−6.35096	−0.289278
$483$	−0.767566	−0.0349255
$484$	3.50659	0.159390
$485$	11.1456	0.506094
$486$	4.64565	0.210731
$487$	20.5546	0.931418	0.465709	−	0.884938i	$-0.345799\pi$
$487$	20.5546	0.931418	0.465709	+	0.884938i	$0.345799\pi$
$488$	9.82832	0.444907
$489$	46.1472	2.08685
$490$	−16.2334	−0.733351
$491$	−1.93246	−0.0872109	−0.0436055	−	0.999049i	$-0.513884\pi$
$491$	−1.93246	−0.0872109	−0.0436055	+	0.999049i	$0.513884\pi$
$492$	85.5361	3.85626
$493$	7.31768	0.329572
$494$	1.80642	0.0812747
$495$	−5.88720	−0.264610
$496$	−6.21076	−0.278871
$497$	−1.87994	−0.0843270
$498$	−14.9696	−0.670805
$499$	−17.9950	−0.805568	−0.402784	−	0.915295i	$-0.631957\pi$
$499$	−17.9950	−0.805568	−0.402784	+	0.915295i	$0.631957\pi$
$500$	−3.50659	−0.156819
$501$	2.33029	0.104110
$502$	8.23035	0.367338
$503$	4.06922	0.181437	0.0907187	−	0.995877i	$-0.471084\pi$
$503$	4.06922	0.181437	0.0907187	+	0.995877i	$0.471084\pi$
$504$	−5.96678	−0.265781
$505$	−6.75736	−0.300699
$506$	2.10756	0.0936923
$507$	−38.6931	−1.71842
$508$	−29.2131	−1.29612
$509$	−15.5762	−0.690405	−0.345202	−	0.938528i	$-0.612190\pi$
$509$	−15.5762	−0.690405	−0.345202	+	0.938528i	$0.612190\pi$
$510$	36.9739	1.63723
$511$	−0.286679	−0.0126819
$512$	14.2806	0.631120
$513$	46.0345	2.03247
$514$	29.7900	1.31398
$515$	8.13200	0.358339
$516$	23.5956	1.03874
$517$	0.731108	0.0321541
$518$	−0.496364	−0.0218090
$519$	17.1323	0.752025
$520$	−0.508849	−0.0223145
$521$	−38.5440	−1.68864	−0.844322	−	0.535836i	$-0.819996\pi$
$521$	−38.5440	−1.68864	−0.844322	+	0.535836i	$0.819996\pi$
$522$	−19.1272	−0.837176
$523$	−13.2987	−0.581514	−0.290757	−	0.956797i	$-0.593907\pi$
$523$	−13.2987	−0.581514	−0.290757	+	0.956797i	$0.593907\pi$
$524$	39.2233	1.71348
$525$	0.854630	0.0372991
$526$	−25.3409	−1.10492
$527$	−25.5857	−1.11453
$528$	3.82474	0.166450
$529$	−22.1934	−0.964929
$530$	−6.39081	−0.277599
$531$	54.5070	2.36540
$532$	5.37656	0.233103
$533$	−1.17770	−0.0510120
$534$	64.4398	2.78858
$535$	4.31764	0.186668
$536$	−26.5089	−1.14501
$537$	27.2859	1.17748
$538$	3.90420	0.168322
$539$	−6.91782	−0.297971
$540$	−30.1817	−1.29881
$541$	−30.2660	−1.30124	−0.650619	−	0.759404i	$-0.725490\pi$
$541$	−30.2660	−1.30124	−0.650619	+	0.759404i	$0.725490\pi$
$542$	48.4609	2.08157
$543$	49.2706	2.11440
$544$	21.4589	0.920044
$545$	−7.14209	−0.305934
$546$	0.288651	0.0123531
$547$	−8.53376	−0.364877	−0.182439	−	0.983217i	$-0.558399\pi$
$547$	−8.53376	−0.364877	−0.182439	+	0.983217i	$0.558399\pi$
$548$	77.8718	3.32652
$549$	−16.3664	−0.698500
$550$	−2.34661	−0.100060
$551$	7.40502	0.315464
$552$	9.46576	0.402890
$553$	−1.80782	−0.0768763
$554$	0.959781	0.0407772
$555$	−2.19961	−0.0933684
$556$	−31.0525	−1.31692
$557$	14.4966	0.614240	0.307120	−	0.951671i	$-0.400635\pi$
$557$	14.4966	0.614240	0.307120	+	0.951671i	$0.400635\pi$
$558$	66.8768	2.83112
$559$	−0.324876	−0.0137408
$560$	−0.367802	−0.0155425
$561$	15.7563	0.665231
$562$	35.8510	1.51228
$563$	−39.4079	−1.66085	−0.830423	−	0.557134i	$-0.811901\pi$
$563$	−39.4079	−1.66085	−0.830423	+	0.557134i	$0.811901\pi$
$564$	7.64273	0.321817
$565$	−11.7157	−0.492885
$566$	−3.87059	−0.162693
$567$	2.29272	0.0962851
$568$	23.1838	0.972771
$569$	−3.40415	−0.142709	−0.0713547	−	0.997451i	$-0.522732\pi$
$569$	−3.40415	−0.142709	−0.0713547	+	0.997451i	$0.522732\pi$
$570$	37.4152	1.56715
$571$	−7.42716	−0.310817	−0.155408	−	0.987850i	$-0.549669\pi$
$571$	−7.42716	−0.310817	−0.155408	+	0.987850i	$0.549669\pi$
$572$	−0.504706	−0.0211028
$573$	−44.4221	−1.85576
$574$	−5.50452	−0.229754
$575$	−0.898127	−0.0374545
$576$	−71.1964	−2.96652
$577$	44.3433	1.84604	0.923019	−	0.384755i	$-0.125714\pi$
$577$	44.3433	1.84604	0.923019	+	0.384755i	$0.125714\pi$
$578$	−25.6593	−1.06729
$579$	40.3837	1.67829
$580$	−4.85497	−0.201592
$581$	0.613455	0.0254504
$582$	77.9697	3.23195
$583$	−2.72342	−0.112792
$584$	3.53537	0.146295
$585$	0.847349	0.0350336
$586$	−14.2190	−0.587382
$587$	6.92857	0.285973	0.142986	−	0.989725i	$-0.454329\pi$
$587$	6.92857	0.285973	0.142986	+	0.989725i	$0.454329\pi$
$588$	−72.3163	−2.98227
$589$	−25.8910	−1.06682
$590$	21.7263	0.894456
$591$	−36.3426	−1.49494
$592$	0.946636	0.0389065
$593$	6.65342	0.273223	0.136612	−	0.990625i	$-0.456379\pi$
$593$	6.65342	0.273223	0.136612	+	0.990625i	$0.456379\pi$
$594$	−20.1976	−0.828719
$595$	−1.51519	−0.0621167
$596$	25.5905	1.04823
$597$	−71.4400	−2.92384
$598$	−0.303342	−0.0124046
$599$	21.1232	0.863072	0.431536	−	0.902096i	$-0.357972\pi$
$599$	21.1232	0.863072	0.431536	+	0.902096i	$0.357972\pi$
$600$	−10.5394	−0.430271
$601$	30.0318	1.22502	0.612511	−	0.790462i	$-0.290159\pi$
$601$	30.0318	1.22502	0.612511	+	0.790462i	$0.290159\pi$
$602$	−1.51845	−0.0618874
$603$	44.1434	1.79766
$604$	50.5756	2.05789
$605$	−1.00000	−0.0406558
$606$	−47.2716	−1.92028
$607$	−32.1574	−1.30523	−0.652614	−	0.757691i	$-0.726327\pi$
$607$	−32.1574	−1.30523	−0.652614	+	0.757691i	$0.726327\pi$
$608$	21.7150	0.880661
$609$	1.18326	0.0479481
$610$	−6.52357	−0.264132
$611$	−0.105229	−0.00425710
$612$	109.110	4.41051
$613$	1.29262	0.0522085	0.0261043	−	0.999659i	$-0.491690\pi$
$613$	1.29262	0.0522085	0.0261043	+	0.999659i	$0.491690\pi$
$614$	24.5032	0.988870
$615$	−24.3930	−0.983620
$616$	−1.01352	−0.0408358
$617$	9.90879	0.398913	0.199456	−	0.979907i	$-0.436082\pi$
$617$	9.90879	0.398913	0.199456	+	0.979907i	$0.436082\pi$
$618$	56.8881	2.28837
$619$	34.4072	1.38294	0.691470	−	0.722405i	$-0.256963\pi$
$619$	34.4072	1.38294	0.691470	+	0.722405i	$0.256963\pi$
$620$	16.9750	0.681733
$621$	−7.73031	−0.310207
$622$	−63.9704	−2.56498
$623$	−2.64074	−0.105799
$624$	−0.550497	−0.0220375
$625$	1.00000	0.0400000
$626$	44.0993	1.76256
$627$	15.9443	0.636755
$628$	42.5490	1.69789
$629$	3.89974	0.155493
$630$	3.96046	0.157788
$631$	30.5683	1.21691	0.608453	−	0.793590i	$-0.291790\pi$
$631$	30.5683	1.21691	0.608453	+	0.793590i	$0.291790\pi$
$632$	22.2944	0.886822
$633$	9.23479	0.367050
$634$	−11.8155	−0.469253
$635$	8.33093	0.330603
$636$	−28.4696	−1.12889
$637$	0.995686	0.0394505
$638$	−3.24895	−0.128627
$639$	−38.6063	−1.52724
$640$	−20.2584	−0.800783
$641$	2.45255	0.0968697	0.0484349	−	0.998826i	$-0.484577\pi$
$641$	2.45255	0.0968697	0.0484349	+	0.998826i	$0.484577\pi$
$642$	30.2044	1.19207
$643$	14.6390	0.577307	0.288654	−	0.957434i	$-0.406792\pi$
$643$	14.6390	0.577307	0.288654	+	0.957434i	$0.406792\pi$
$644$	−0.902855	−0.0355775
$645$	−6.72893	−0.264952
$646$	−66.3341	−2.60988
$647$	−10.1842	−0.400382	−0.200191	−	0.979757i	$-0.564156\pi$
$647$	−10.1842	−0.400382	−0.200191	+	0.979757i	$0.564156\pi$
$648$	−28.2742	−1.11072
$649$	9.25857	0.363431
$650$	0.337750	0.0132476
$651$	−4.13717	−0.162149
$652$	54.2810	2.12581
$653$	31.4844	1.23208	0.616041	−	0.787714i	$-0.288736\pi$
$653$	31.4844	1.23208	0.616041	+	0.787714i	$0.288736\pi$
$654$	−49.9631	−1.95371
$655$	−11.1856	−0.437058
$656$	10.4979	0.409873
$657$	−5.88720	−0.229681
$658$	−0.491834	−0.0191737
$659$	−11.1891	−0.435864	−0.217932	−	0.975964i	$-0.569931\pi$
$659$	−11.1891	−0.435864	−0.217932	+	0.975964i	$0.569931\pi$
$660$	−10.4536	−0.406907
$661$	−18.1546	−0.706132	−0.353066	−	0.935598i	$-0.614861\pi$
$661$	−18.1546	−0.706132	−0.353066	+	0.935598i	$0.614861\pi$
$662$	56.9802	2.21460
$663$	−2.26781	−0.0880746
$664$	−7.56524	−0.293588
$665$	−1.53327	−0.0594578
$666$	−10.1933	−0.394981
$667$	−1.24348	−0.0481479
$668$	2.74102	0.106053
$669$	1.81725	0.0702589
$670$	17.5954	0.679768
$671$	−2.78000	−0.107321
$672$	3.46988	0.133854
$673$	26.9005	1.03694	0.518468	−	0.855097i	$-0.326502\pi$
$673$	26.9005	1.03694	0.518468	+	0.855097i	$0.326502\pi$
$674$	14.5623	0.560921
$675$	8.60714	0.331289
$676$	−45.5130	−1.75050
$677$	20.3536	0.782253	0.391127	−	0.920337i	$-0.372085\pi$
$677$	20.3536	0.782253	0.391127	+	0.920337i	$0.372085\pi$
$678$	−81.9584	−3.14759
$679$	−3.19520	−0.122620
$680$	18.6856	0.716559
$681$	53.5504	2.05206
$682$	11.3597	0.434985
$683$	23.8401	0.912217	0.456109	−	0.889924i	$-0.349243\pi$
$683$	23.8401	0.912217	0.456109	+	0.889924i	$0.349243\pi$
$684$	110.412	4.22172
$685$	−22.2073	−0.848497
$686$	9.36285	0.357475
$687$	26.8849	1.02572
$688$	2.89590	0.110405
$689$	0.391984	0.0149334
$690$	−6.28292	−0.239187
$691$	−16.3709	−0.622780	−0.311390	−	0.950282i	$-0.600795\pi$
$691$	−16.3709	−0.622780	−0.311390	+	0.950282i	$0.600795\pi$
$692$	20.1520	0.766064
$693$	1.68774	0.0641118
$694$	−37.7087	−1.43140
$695$	8.85549	0.335908
$696$	−14.5922	−0.553115
$697$	43.2468	1.63809
$698$	−16.3221	−0.617801
$699$	−30.5296	−1.15474
$700$	1.00526	0.0379954
$701$	−26.0469	−0.983776	−0.491888	−	0.870658i	$-0.663693\pi$
$701$	−26.0469	−0.983776	−0.491888	+	0.870658i	$0.663693\pi$
$702$	2.90706	0.109720
$703$	3.94628	0.148837
$704$	−12.0934	−0.455788
$705$	−2.17953	−0.0820860
$706$	−59.0230	−2.22136
$707$	1.93719	0.0728556
$708$	96.7857	3.63743
$709$	−49.2512	−1.84967	−0.924834	−	0.380372i	$-0.875796\pi$
$709$	−49.2512	−1.84967	−0.924834	+	0.380372i	$0.875796\pi$
$710$	−15.3883	−0.577512
$711$	−37.1252	−1.39230
$712$	32.5661	1.22047
$713$	4.34774	0.162824
$714$	−10.5996	−0.396681
$715$	0.143931	0.00538271
$716$	32.0953	1.19946
$717$	8.84510	0.330326
$718$	67.5978	2.52273
$719$	−24.7446	−0.922817	−0.461409	−	0.887188i	$-0.652656\pi$
$719$	−24.7446	−0.922817	−0.461409	+	0.887188i	$0.652656\pi$
$720$	−7.55314	−0.281489
$721$	−2.33127	−0.0868211
$722$	−22.5401	−0.838857
$723$	8.06828	0.300062
$724$	57.9549	2.15388
$725$	1.38453	0.0514201
$726$	−6.99558	−0.259630
$727$	−45.8463	−1.70034	−0.850172	−	0.526504i	$-0.823502\pi$
$727$	−45.8463	−1.70034	−0.850172	+	0.526504i	$0.823502\pi$
$728$	0.145876	0.00540653
$729$	−29.8944	−1.10720
$730$	−2.34661	−0.0868520
$731$	11.9299	0.441242
$732$	−29.0610	−1.07413
$733$	18.9851	0.701231	0.350616	−	0.936519i	$-0.385972\pi$
$733$	18.9851	0.701231	0.350616	+	0.936519i	$0.385972\pi$
$734$	17.2961	0.638410
$735$	20.6230	0.760690
$736$	−3.64648	−0.134411
$737$	7.49820	0.276200
$738$	−113.040	−4.16106
$739$	−16.8238	−0.618873	−0.309436	−	0.950920i	$-0.600140\pi$
$739$	−16.8238	−0.618873	−0.309436	+	0.950920i	$0.600140\pi$
$740$	−2.58731	−0.0951114
$741$	−2.29488	−0.0843045
$742$	1.83211	0.0672588
$743$	−34.7356	−1.27433	−0.637163	−	0.770729i	$-0.719892\pi$
$743$	−34.7356	−1.27433	−0.637163	+	0.770729i	$0.719892\pi$
$744$	51.0203	1.87050
$745$	−7.29784	−0.267372
$746$	2.10221	0.0769675
$747$	12.5978	0.460930
$748$	18.5334	0.677649
$749$	−1.23778	−0.0452274
$750$	6.99558	0.255442
$751$	38.1495	1.39210	0.696048	−	0.717995i	$-0.254940\pi$
$751$	38.1495	1.39210	0.696048	+	0.717995i	$0.254940\pi$
$752$	0.937995	0.0342051
$753$	−10.4558	−0.381032
$754$	0.467624	0.0170299
$755$	−14.4230	−0.524908
$756$	8.65245	0.314687
$757$	8.65195	0.314461	0.157230	−	0.987562i	$-0.449743\pi$
$757$	8.65195	0.314461	0.157230	+	0.987562i	$0.449743\pi$
$758$	−13.6599	−0.496148
$759$	−2.67744	−0.0971850
$760$	18.9086	0.685887
$761$	−52.0656	−1.88738	−0.943688	−	0.330837i	$-0.892669\pi$
$761$	−52.0656	−1.88738	−0.943688	+	0.330837i	$0.892669\pi$
$762$	58.2797	2.11125
$763$	2.04749	0.0741240
$764$	−52.2518	−1.89040
$765$	−31.1157	−1.12499
$766$	−82.1663	−2.96879
$767$	−1.33259	−0.0481171
$768$	−69.6147	−2.51201
$769$	15.7648	0.568492	0.284246	−	0.958751i	$-0.408257\pi$
$769$	15.7648	0.568492	0.284246	+	0.958751i	$0.408257\pi$
$770$	0.672724	0.0242433
$771$	−37.8453	−1.36297
$772$	47.5016	1.70962
$773$	−11.9635	−0.430299	−0.215149	−	0.976581i	$-0.569024\pi$
$773$	−11.9635	−0.430299	−0.215149	+	0.976581i	$0.569024\pi$
$774$	−31.1827	−1.12084
$775$	−4.84089	−0.173890
$776$	39.4038	1.41451
$777$	0.630582	0.0226220
$778$	5.63365	0.201976
$779$	43.7629	1.56797
$780$	1.50460	0.0538733
$781$	−6.55767	−0.234652
$782$	11.1391	0.398334
$783$	11.9168	0.425873
$784$	−8.87540	−0.316979
$785$	−12.1340	−0.433082
$786$	−78.2499	−2.79108
$787$	−5.67447	−0.202273	−0.101136	−	0.994873i	$-0.532248\pi$
$787$	−5.67447	−0.202273	−0.101136	+	0.994873i	$0.532248\pi$
$788$	−42.7483	−1.52284
$789$	32.1931	1.14611
$790$	−14.7979	−0.526487
$791$	3.35866	0.119420
$792$	−20.8135	−0.739574
$793$	0.400127	0.0142089
$794$	−20.6522	−0.732920
$795$	8.11889	0.287947
$796$	−84.0318	−2.97843
$797$	5.30916	0.188060	0.0940301	−	0.995569i	$-0.470025\pi$
$797$	5.30916	0.188060	0.0940301	+	0.995569i	$0.470025\pi$
$798$	−10.7261	−0.379701
$799$	3.86414	0.136703
$800$	4.06010	0.143546
$801$	−54.2300	−1.91612
$802$	59.7708	2.11058
$803$	−1.00000	−0.0352892
$804$	78.3834	2.76437
$805$	0.257474	0.00907477
$806$	−1.63501	−0.0575908
$807$	−4.95990	−0.174597
$808$	−23.8898	−0.840440
$809$	−45.5210	−1.60043	−0.800217	−	0.599711i	$-0.795282\pi$
$809$	−45.5210	−1.60043	−0.800217	+	0.599711i	$0.795282\pi$
$810$	18.7671	0.659407
$811$	11.0170	0.386858	0.193429	−	0.981114i	$-0.438039\pi$
$811$	11.0170	0.386858	0.193429	+	0.981114i	$0.438039\pi$
$812$	1.39182	0.0488432
$813$	−61.5648	−2.15917
$814$	−1.73143	−0.0606866
$815$	−15.4797	−0.542231
$816$	20.2150	0.707665
$817$	12.0722	0.422354
$818$	30.2271	1.05687
$819$	−0.242917	−0.00848821
$820$	−28.6924	−1.00198
$821$	33.4200	1.16636	0.583182	−	0.812341i	$-0.301807\pi$
$821$	33.4200	1.16636	0.583182	+	0.812341i	$0.301807\pi$
$822$	−155.353	−5.41855
$823$	−8.69068	−0.302938	−0.151469	−	0.988462i	$-0.548400\pi$
$823$	−8.69068	−0.302938	−0.151469	+	0.988462i	$0.548400\pi$
$824$	28.7497	1.00154
$825$	2.98114	0.103790
$826$	−6.22846	−0.216716
$827$	0.236461	0.00822257	0.00411128	−	0.999992i	$-0.498691\pi$
$827$	0.236461	0.00822257	0.00411128	+	0.999992i	$0.498691\pi$
$828$	−18.5409	−0.644341
$829$	−55.6214	−1.93181	−0.965906	−	0.258894i	$-0.916642\pi$
$829$	−55.6214	−1.93181	−0.965906	+	0.258894i	$0.916642\pi$
$830$	5.02144	0.174297
$831$	−1.21931	−0.0422973
$832$	1.74062	0.0603450
$833$	−36.5629	−1.26683
$834$	61.9493	2.14513
$835$	−0.781677	−0.0270511
$836$	18.7546	0.648643
$837$	−41.6663	−1.44020
$838$	55.6382	1.92199
$839$	0.0316496	0.00109267	0.000546333	−	1.00000i	$-0.499826\pi$
$839$	0.0316496	0.00109267	0.000546333		1.00000i	$0.499826\pi$
$840$	3.02144	0.104249
$841$	−27.0831	−0.933899
$842$	−55.0620	−1.89756
$843$	−45.5452	−1.56866
$844$	10.8625	0.373902
$845$	12.9793	0.446501
$846$	−10.1002	−0.347253
$847$	0.286679	0.00985040
$848$	−3.49408	−0.119987
$849$	4.91721	0.168758
$850$	−12.4026	−0.425405
$851$	−0.662677	−0.0227163
$852$	−68.5514	−2.34853
$853$	−10.4598	−0.358138	−0.179069	−	0.983837i	$-0.557308\pi$
$853$	−10.4598	−0.358138	−0.179069	+	0.983837i	$0.557308\pi$
$854$	1.87017	0.0639959
$855$	−31.4871	−1.07684
$856$	15.2645	0.521729
$857$	3.62944	0.123979	0.0619896	−	0.998077i	$-0.480255\pi$
$857$	3.62944	0.123979	0.0619896	+	0.998077i	$0.480255\pi$
$858$	1.00688	0.0343743
$859$	−34.9073	−1.19102	−0.595511	−	0.803347i	$-0.703050\pi$
$859$	−34.9073	−1.19102	−0.595511	+	0.803347i	$0.703050\pi$
$860$	−7.91495	−0.269898
$861$	6.99295	0.238319
$862$	32.1970	1.09663
$863$	−27.0362	−0.920324	−0.460162	−	0.887835i	$-0.652209\pi$
$863$	−27.0362	−0.920324	−0.460162	+	0.887835i	$0.652209\pi$
$864$	34.9459	1.18888
$865$	−5.74690	−0.195400
$866$	−35.8823	−1.21933
$867$	32.5976	1.10707
$868$	−4.86638	−0.165176
$869$	−6.30608	−0.213919
$870$	9.68558	0.328372
$871$	−1.07922	−0.0365680
$872$	−25.2500	−0.855072
$873$	−65.6162	−2.22077
$874$	11.2721	0.381283
$875$	−0.286679	−0.00969151
$876$	−10.4536	−0.353195
$877$	13.2650	0.447927	0.223964	−	0.974598i	$-0.428100\pi$
$877$	13.2650	0.447927	0.223964	+	0.974598i	$0.428100\pi$
$878$	−56.6955	−1.91338
$879$	18.0638	0.609278
$880$	−1.28298	−0.0432492
$881$	33.7591	1.13737	0.568686	−	0.822555i	$-0.307452\pi$
$881$	33.7591	1.13737	0.568686	+	0.822555i	$0.307452\pi$
$882$	95.5694	3.21799
$883$	−2.03231	−0.0683928	−0.0341964	−	0.999415i	$-0.510887\pi$
$883$	−2.03231	−0.0683928	−0.0341964	+	0.999415i	$0.510887\pi$
$884$	−2.66753	−0.0897188
$885$	−27.6011	−0.927801
$886$	−82.6633	−2.77713
$887$	48.5024	1.62855	0.814275	−	0.580479i	$-0.197135\pi$
$887$	48.5024	1.62855	0.814275	+	0.580479i	$0.197135\pi$
$888$	−7.77646	−0.260961
$889$	−2.38830	−0.0801010
$890$	−21.6158	−0.724564
$891$	7.99751	0.267927
$892$	2.13755	0.0715705
$893$	3.91026	0.130852
$894$	−51.0526	−1.70746
$895$	−9.15286	−0.305946
$896$	5.80765	0.194020
$897$	0.385366	0.0128670
$898$	54.1470	1.80691
$899$	−6.70236	−0.223536
$900$	20.6440	0.688132
$901$	−14.3941	−0.479538
$902$	−19.2010	−0.639323
$903$	1.92904	0.0641945
$904$	−41.4196	−1.37759
$905$	−16.5274	−0.549391
$906$	−100.897	−3.35209
$907$	20.1248	0.668233	0.334117	−	0.942532i	$-0.391562\pi$
$907$	20.1248	0.668233	0.334117	+	0.942532i	$0.391562\pi$
$908$	62.9890	2.09036
$909$	39.7819	1.31948
$910$	−0.0968257	−0.00320974
$911$	−9.13254	−0.302575	−0.151287	−	0.988490i	$-0.548342\pi$
$911$	−9.13254	−0.302575	−0.151287	+	0.988490i	$0.548342\pi$
$912$	20.4562	0.677373
$913$	2.13987	0.0708193
$914$	−50.8307	−1.68133
$915$	8.28756	0.273978
$916$	31.6235	1.04487
$917$	3.20668	0.105894
$918$	−106.751	−3.52331
$919$	35.6169	1.17489	0.587447	−	0.809263i	$-0.300133\pi$
$919$	35.6169	1.17489	0.587447	+	0.809263i	$0.300133\pi$
$920$	−3.17522	−0.104684
$921$	−31.1290	−1.02573
$922$	−90.2007	−2.97060
$923$	0.943850	0.0310672
$924$	2.99683	0.0985886
$925$	0.737843	0.0242601
$926$	90.9920	2.99018
$927$	−47.8747	−1.57241
$928$	5.62133	0.184529
$929$	−44.0214	−1.44429	−0.722147	−	0.691739i	$-0.756845\pi$
$929$	−44.0214	−1.44429	−0.722147	+	0.691739i	$0.756845\pi$
$930$	−33.8649	−1.11047
$931$	−36.9993	−1.21260
$932$	−35.9107	−1.17629
$933$	81.2681	2.66060
$934$	−12.2118	−0.399582
$935$	−5.28532	−0.172848
$936$	2.99570	0.0979174
$937$	−8.02245	−0.262082	−0.131041	−	0.991377i	$-0.541832\pi$
$937$	−8.02245	−0.262082	−0.131041	+	0.991377i	$0.541832\pi$
$938$	−5.04422	−0.164700
$939$	−56.0239	−1.82827
$940$	−2.56369	−0.0836184
$941$	33.6827	1.09802	0.549012	−	0.835815i	$-0.315004\pi$
$941$	33.6827	1.09802	0.549012	+	0.835815i	$0.315004\pi$
$942$	−84.8845	−2.76569
$943$	−7.34886	−0.239312
$944$	11.8785	0.386613
$945$	−2.46749	−0.0802673
$946$	−5.29670	−0.172210
$947$	17.5168	0.569220	0.284610	−	0.958643i	$-0.408136\pi$
$947$	17.5168	0.569220	0.284610	+	0.958643i	$0.408136\pi$
$948$	−65.9215	−2.14103
$949$	0.143931	0.00467219
$950$	−12.5506	−0.407196
$951$	15.0104	0.486747
$952$	−5.35676	−0.173614
$953$	−7.11956	−0.230625	−0.115313	−	0.993329i	$-0.536787\pi$
$953$	−7.11956	−0.230625	−0.115313	+	0.993329i	$0.536787\pi$
$954$	37.6239	1.21812
$955$	14.9010	0.482186
$956$	10.4041	0.336493
$957$	4.12748	0.133422
$958$	−42.6184	−1.37694
$959$	6.36636	0.205580
$960$	36.0522	1.16358
$961$	−7.56575	−0.244056
$962$	0.249206	0.00803473
$963$	−25.4188	−0.819110
$964$	9.49037	0.305664
$965$	−13.5464	−0.436073
$966$	1.80118	0.0579520
$967$	41.1269	1.32255	0.661275	−	0.750143i	$-0.270015\pi$
$967$	41.1269	1.32255	0.661275	+	0.750143i	$0.270015\pi$
$968$	−3.53537	−0.113631
$969$	84.2709	2.70717
$970$	−26.1543	−0.839765
$971$	−37.9643	−1.21833	−0.609166	−	0.793043i	$-0.708496\pi$
$971$	−37.9643	−1.21833	−0.609166	+	0.793043i	$0.708496\pi$
$972$	−6.94209	−0.222668
$973$	−2.53868	−0.0813864
$974$	−48.2337	−1.54551
$975$	−0.429078	−0.0137415
$976$	−3.56667	−0.114166
$977$	−25.2769	−0.808679	−0.404339	−	0.914609i	$-0.632499\pi$
$977$	−25.2769	−0.808679	−0.404339	+	0.914609i	$0.632499\pi$
$978$	−108.290	−3.46272
$979$	−9.21151	−0.294401
$980$	24.2579	0.774891
$981$	42.0469	1.34245
$982$	4.53474	0.144710
$983$	−11.5468	−0.368285	−0.184142	−	0.982900i	$-0.558951\pi$
$983$	−11.5468	−0.368285	−0.184142	+	0.982900i	$0.558951\pi$
$984$	−86.2383	−2.74918
$985$	12.1909	0.388433
$986$	−17.1718	−0.546860
$987$	0.624826	0.0198884
$988$	−2.69937	−0.0858784
$989$	−2.02722	−0.0644620
$990$	13.8150	0.439069
$991$	−1.85593	−0.0589556	−0.0294778	−	0.999565i	$-0.509384\pi$
$991$	−1.85593	−0.0589556	−0.0294778	+	0.999565i	$0.509384\pi$
$992$	−19.6545	−0.624031
$993$	−72.3878	−2.29716
$994$	4.41150	0.139924
$995$	23.9640	0.759709
$996$	22.3694	0.708801
$997$	−31.2914	−0.991010	−0.495505	−	0.868605i	$-0.665017\pi$
$997$	−31.2914	−0.991010	−0.495505	+	0.868605i	$0.665017\pi$
$998$	42.2273	1.33668
$999$	6.35072	0.200928

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.e.1.1	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.e.1.1	✓	27	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:	\( N \)	\(=\)	\( 4015 = 5 \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4015.a (trivial)

\(f(q)\)	\(=\)	\(q-2.34661 q^{2} +2.98114 q^{3} +3.50659 q^{4} -1.00000 q^{5} -6.99558 q^{6} +0.286679 q^{7} -3.53537 q^{8} +5.88720 q^{9} +O(q^{10})\)	\(q-2.34661 q^{2} +2.98114 q^{3} +3.50659 q^{4} -1.00000 q^{5} -6.99558 q^{6} +0.286679 q^{7} -3.53537 q^{8} +5.88720 q^{9} +2.34661 q^{10} +1.00000 q^{11} +10.4536 q^{12} -0.143931 q^{13} -0.672724 q^{14} -2.98114 q^{15} +1.28298 q^{16} +5.28532 q^{17} -13.8150 q^{18} +5.34840 q^{19} -3.50659 q^{20} +0.854630 q^{21} -2.34661 q^{22} -0.898127 q^{23} -10.5394 q^{24} +1.00000 q^{25} +0.337750 q^{26} +8.60714 q^{27} +1.00526 q^{28} +1.38453 q^{29} +6.99558 q^{30} -4.84089 q^{31} +4.06010 q^{32} +2.98114 q^{33} -12.4026 q^{34} -0.286679 q^{35} +20.6440 q^{36} +0.737843 q^{37} -12.5506 q^{38} -0.429078 q^{39} +3.53537 q^{40} +8.18243 q^{41} -2.00548 q^{42} +2.25717 q^{43} +3.50659 q^{44} -5.88720 q^{45} +2.10756 q^{46} +0.731108 q^{47} +3.82474 q^{48} -6.91782 q^{49} -2.34661 q^{50} +15.7563 q^{51} -0.504706 q^{52} -2.72342 q^{53} -20.1976 q^{54} -1.00000 q^{55} -1.01352 q^{56} +15.9443 q^{57} -3.24895 q^{58} +9.25857 q^{59} -10.4536 q^{60} -2.78000 q^{61} +11.3597 q^{62} +1.68774 q^{63} -12.0934 q^{64} +0.143931 q^{65} -6.99558 q^{66} +7.49820 q^{67} +18.5334 q^{68} -2.67744 q^{69} +0.672724 q^{70} -6.55767 q^{71} -20.8135 q^{72} -1.00000 q^{73} -1.73143 q^{74} +2.98114 q^{75} +18.7546 q^{76} +0.286679 q^{77} +1.00688 q^{78} -6.30608 q^{79} -1.28298 q^{80} +7.99751 q^{81} -19.2010 q^{82} +2.13987 q^{83} +2.99683 q^{84} -5.28532 q^{85} -5.29670 q^{86} +4.12748 q^{87} -3.53537 q^{88} -9.21151 q^{89} +13.8150 q^{90} -0.0412619 q^{91} -3.14936 q^{92} -14.4314 q^{93} -1.71563 q^{94} -5.34840 q^{95} +12.1037 q^{96} -11.1456 q^{97} +16.2334 q^{98} +5.88720 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q + 6 q^{2} + 5 q^{3} + 22 q^{4} - 27 q^{5} + 7 q^{6} + 10 q^{7} + 15 q^{8} + 24 q^{9}+O(q^{10}) \) 27 * q + 6 * q^2 + 5 * q^3 + 22 * q^4 - 27 * q^5 + 7 * q^6 + 10 * q^7 + 15 * q^8 + 24 * q^9	\( 27 q + 6 q^{2} + 5 q^{3} + 22 q^{4} - 27 q^{5} + 7 q^{6} + 10 q^{7} + 15 q^{8} + 24 q^{9} - 6 q^{10} + 27 q^{11} + 28 q^{12} + 21 q^{13} + 11 q^{14} - 5 q^{15} + 12 q^{16} + 30 q^{17} + 10 q^{18} + 20 q^{19} - 22 q^{20} - 14 q^{21} + 6 q^{22} + 16 q^{23} + 23 q^{24} + 27 q^{25} + 13 q^{26} + 17 q^{27} + 6 q^{28} - 2 q^{29} - 7 q^{30} - 6 q^{31} + 41 q^{32} + 5 q^{33} + 8 q^{34} - 10 q^{35} + 13 q^{36} + 20 q^{37} + 11 q^{38} - 10 q^{39} - 15 q^{40} + 38 q^{41} - 33 q^{42} + 29 q^{43} + 22 q^{44} - 24 q^{45} + 23 q^{46} + 17 q^{47} + 37 q^{48} + 21 q^{49} + 6 q^{50} + 13 q^{51} + 29 q^{52} + 4 q^{53} + q^{54} - 27 q^{55} + 28 q^{56} + 51 q^{57} - 6 q^{58} + 35 q^{59} - 28 q^{60} - 13 q^{61} + 28 q^{62} + 41 q^{63} - 5 q^{64} - 21 q^{65} + 7 q^{66} + 10 q^{67} + 65 q^{68} - 12 q^{69} - 11 q^{70} + 22 q^{71} - 12 q^{72} - 27 q^{73} - 5 q^{74} + 5 q^{75} + 13 q^{76} + 10 q^{77} - 9 q^{78} - 18 q^{79} - 12 q^{80} + 39 q^{81} + 20 q^{82} + 54 q^{83} - 50 q^{84} - 30 q^{85} + 3 q^{86} + 15 q^{87} + 15 q^{88} + 89 q^{89} - 10 q^{90} - 18 q^{91} + 57 q^{92} + 20 q^{93} - 29 q^{94} - 20 q^{95} + 105 q^{96} + 35 q^{97} + 10 q^{98} + 24 q^{99}+O(q^{100}) \) 27 * q + 6 * q^2 + 5 * q^3 + 22 * q^4 - 27 * q^5 + 7 * q^6 + 10 * q^7 + 15 * q^8 + 24 * q^9 - 6 * q^10 + 27 * q^11 + 28 * q^12 + 21 * q^13 + 11 * q^14 - 5 * q^15 + 12 * q^16 + 30 * q^17 + 10 * q^18 + 20 * q^19 - 22 * q^20 - 14 * q^21 + 6 * q^22 + 16 * q^23 + 23 * q^24 + 27 * q^25 + 13 * q^26 + 17 * q^27 + 6 * q^28 - 2 * q^29 - 7 * q^30 - 6 * q^31 + 41 * q^32 + 5 * q^33 + 8 * q^34 - 10 * q^35 + 13 * q^36 + 20 * q^37 + 11 * q^38 - 10 * q^39 - 15 * q^40 + 38 * q^41 - 33 * q^42 + 29 * q^43 + 22 * q^44 - 24 * q^45 + 23 * q^46 + 17 * q^47 + 37 * q^48 + 21 * q^49 + 6 * q^50 + 13 * q^51 + 29 * q^52 + 4 * q^53 + q^54 - 27 * q^55 + 28 * q^56 + 51 * q^57 - 6 * q^58 + 35 * q^59 - 28 * q^60 - 13 * q^61 + 28 * q^62 + 41 * q^63 - 5 * q^64 - 21 * q^65 + 7 * q^66 + 10 * q^67 + 65 * q^68 - 12 * q^69 - 11 * q^70 + 22 * q^71 - 12 * q^72 - 27 * q^73 - 5 * q^74 + 5 * q^75 + 13 * q^76 + 10 * q^77 - 9 * q^78 - 18 * q^79 - 12 * q^80 + 39 * q^81 + 20 * q^82 + 54 * q^83 - 50 * q^84 - 30 * q^85 + 3 * q^86 + 15 * q^87 + 15 * q^88 + 89 * q^89 - 10 * q^90 - 18 * q^91 + 57 * q^92 + 20 * q^93 - 29 * q^94 - 20 * q^95 + 105 * q^96 + 35 * q^97 + 10 * q^98 + 24 * q^99

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