LMFDB - Embedded newform 2005.2.a.d.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2005,2,Mod(1,2005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2005, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2005 = 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2005.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:	$16.0100056053$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	2005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.42545 q^{2} -3.28864 q^{3} +0.0319048 q^{4} +1.00000 q^{5} +4.68779 q^{6} -2.81826 q^{7} +2.80542 q^{8} +7.81515 q^{9} +O(q^{10})$	\(q-1.42545 q^{2} -3.28864 q^{3} +0.0319048 q^{4} +1.00000 q^{5} +4.68779 q^{6} -2.81826 q^{7} +2.80542 q^{8} +7.81515 q^{9} -1.42545 q^{10} -1.87141 q^{11} -0.104923 q^{12} -5.83313 q^{13} +4.01728 q^{14} -3.28864 q^{15} -4.06279 q^{16} +4.26439 q^{17} -11.1401 q^{18} -0.928184 q^{19} +0.0319048 q^{20} +9.26824 q^{21} +2.66759 q^{22} -4.32680 q^{23} -9.22601 q^{24} +1.00000 q^{25} +8.31482 q^{26} -15.8353 q^{27} -0.0899160 q^{28} +8.39369 q^{29} +4.68779 q^{30} -2.12253 q^{31} +0.180464 q^{32} +6.15438 q^{33} -6.07868 q^{34} -2.81826 q^{35} +0.249341 q^{36} -1.68954 q^{37} +1.32308 q^{38} +19.1830 q^{39} +2.80542 q^{40} +1.25590 q^{41} -13.2114 q^{42} +6.56150 q^{43} -0.0597068 q^{44} +7.81515 q^{45} +6.16764 q^{46} +8.81781 q^{47} +13.3611 q^{48} +0.942584 q^{49} -1.42545 q^{50} -14.0241 q^{51} -0.186105 q^{52} +7.79498 q^{53} +22.5724 q^{54} -1.87141 q^{55} -7.90640 q^{56} +3.05246 q^{57} -11.9648 q^{58} +3.90646 q^{59} -0.104923 q^{60} +10.0457 q^{61} +3.02555 q^{62} -22.0251 q^{63} +7.86834 q^{64} -5.83313 q^{65} -8.77275 q^{66} -10.0523 q^{67} +0.136055 q^{68} +14.2293 q^{69} +4.01728 q^{70} +14.6245 q^{71} +21.9248 q^{72} -13.7213 q^{73} +2.40835 q^{74} -3.28864 q^{75} -0.0296135 q^{76} +5.27410 q^{77} -27.3445 q^{78} -8.07291 q^{79} -4.06279 q^{80} +28.6311 q^{81} -1.79021 q^{82} +10.7583 q^{83} +0.295701 q^{84} +4.26439 q^{85} -9.35308 q^{86} -27.6038 q^{87} -5.25008 q^{88} +8.76904 q^{89} -11.1401 q^{90} +16.4393 q^{91} -0.138046 q^{92} +6.98022 q^{93} -12.5693 q^{94} -0.928184 q^{95} -0.593480 q^{96} +8.43982 q^{97} -1.34361 q^{98} -14.6253 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 5 q^{2} - 10 q^{3} + 25 q^{4} + 25 q^{5} + 2 q^{6} - 31 q^{7} - 30 q^{8} + 17 q^{9}+O(q^{10}) $ 25 * q - 5 * q^2 - 10 * q^3 + 25 * q^4 + 25 * q^5 + 2 * q^6 - 31 * q^7 - 30 * q^8 + 17 * q^9	\( 25 q - 5 q^{2} - 10 q^{3} + 25 q^{4} + 25 q^{5} + 2 q^{6} - 31 q^{7} - 30 q^{8} + 17 q^{9} - 5 q^{10} - 30 q^{11} - 29 q^{12} - 18 q^{13} + 6 q^{14} - 10 q^{15} + 21 q^{16} - 18 q^{17} - 30 q^{18} - 17 q^{19} + 25 q^{20} + 6 q^{21} - 2 q^{22} - 44 q^{23} + 11 q^{24} + 25 q^{25} - 14 q^{26} - 25 q^{27} - 50 q^{28} - 9 q^{29} + 2 q^{30} - 13 q^{31} - 45 q^{32} - 21 q^{33} - 21 q^{34} - 31 q^{35} + 5 q^{36} - 28 q^{37} - 32 q^{38} + 9 q^{39} - 30 q^{40} + 28 q^{41} - 67 q^{42} - 61 q^{43} - 49 q^{44} + 17 q^{45} + 18 q^{46} - 53 q^{47} - 44 q^{48} + 28 q^{49} - 5 q^{50} - 30 q^{51} - 3 q^{52} - 36 q^{53} + 17 q^{54} - 30 q^{55} - 3 q^{56} - 13 q^{57} + 2 q^{58} - 39 q^{59} - 29 q^{60} + 10 q^{61} - 30 q^{62} - 44 q^{63} - 4 q^{64} - 18 q^{65} + 33 q^{66} - 10 q^{67} - 18 q^{68} - 6 q^{69} + 6 q^{70} - 7 q^{71} - q^{72} - 26 q^{73} - 3 q^{74} - 10 q^{75} + 12 q^{76} + 29 q^{77} - 5 q^{78} - 6 q^{79} + 21 q^{80} + 13 q^{81} - 30 q^{82} - 35 q^{83} + 117 q^{84} - 18 q^{85} + 14 q^{86} - 104 q^{87} + 53 q^{88} + 7 q^{89} - 30 q^{90} - 25 q^{91} - 31 q^{92} + 2 q^{93} + 68 q^{94} - 17 q^{95} + 92 q^{96} + 6 q^{97} + 15 q^{98} - 36 q^{99}+O(q^{100}) \) 25 * q - 5 * q^2 - 10 * q^3 + 25 * q^4 + 25 * q^5 + 2 * q^6 - 31 * q^7 - 30 * q^8 + 17 * q^9 - 5 * q^10 - 30 * q^11 - 29 * q^12 - 18 * q^13 + 6 * q^14 - 10 * q^15 + 21 * q^16 - 18 * q^17 - 30 * q^18 - 17 * q^19 + 25 * q^20 + 6 * q^21 - 2 * q^22 - 44 * q^23 + 11 * q^24 + 25 * q^25 - 14 * q^26 - 25 * q^27 - 50 * q^28 - 9 * q^29 + 2 * q^30 - 13 * q^31 - 45 * q^32 - 21 * q^33 - 21 * q^34 - 31 * q^35 + 5 * q^36 - 28 * q^37 - 32 * q^38 + 9 * q^39 - 30 * q^40 + 28 * q^41 - 67 * q^42 - 61 * q^43 - 49 * q^44 + 17 * q^45 + 18 * q^46 - 53 * q^47 - 44 * q^48 + 28 * q^49 - 5 * q^50 - 30 * q^51 - 3 * q^52 - 36 * q^53 + 17 * q^54 - 30 * q^55 - 3 * q^56 - 13 * q^57 + 2 * q^58 - 39 * q^59 - 29 * q^60 + 10 * q^61 - 30 * q^62 - 44 * q^63 - 4 * q^64 - 18 * q^65 + 33 * q^66 - 10 * q^67 - 18 * q^68 - 6 * q^69 + 6 * q^70 - 7 * q^71 - q^72 - 26 * q^73 - 3 * q^74 - 10 * q^75 + 12 * q^76 + 29 * q^77 - 5 * q^78 - 6 * q^79 + 21 * q^80 + 13 * q^81 - 30 * q^82 - 35 * q^83 + 117 * q^84 - 18 * q^85 + 14 * q^86 - 104 * q^87 + 53 * q^88 + 7 * q^89 - 30 * q^90 - 25 * q^91 - 31 * q^92 + 2 * q^93 + 68 * q^94 - 17 * q^95 + 92 * q^96 + 6 * q^97 + 15 * q^98 - 36 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.42545	−1.00794	−0.503972	−	0.863720i	$-0.668129\pi$
$2$	−1.42545	−1.00794	−0.503972	+	0.863720i	$0.668129\pi$
$3$	−3.28864	−1.89870	−0.949348	−	0.314226i	$-0.898255\pi$
$3$	−3.28864	−1.89870	−0.949348	+	0.314226i	$0.898255\pi$
$4$	0.0319048	0.0159524
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	4.68779	1.91378
$7$	−2.81826	−1.06520	−0.532601	−	0.846367i	$-0.678785\pi$
$7$	−2.81826	−1.06520	−0.532601	+	0.846367i	$0.678785\pi$
$8$	2.80542	0.991866
$9$	7.81515	2.60505
$10$	−1.42545	−0.450767
$11$	−1.87141	−0.564250	−0.282125	−	0.959378i	$-0.591039\pi$
$11$	−1.87141	−0.564250	−0.282125	+	0.959378i	$0.591039\pi$
$12$	−0.104923	−0.0302888
$13$	−5.83313	−1.61782	−0.808909	−	0.587934i	$-0.799942\pi$
$13$	−5.83313	−1.61782	−0.808909	+	0.587934i	$0.799942\pi$
$14$	4.01728	1.07366
$15$	−3.28864	−0.849123
$16$	−4.06279	−1.01570
$17$	4.26439	1.03427	0.517134	−	0.855905i	$-0.326999\pi$
$17$	4.26439	1.03427	0.517134	+	0.855905i	$0.326999\pi$
$18$	−11.1401	−2.62575
$19$	−0.928184	−0.212940	−0.106470	−	0.994316i	$-0.533955\pi$
$19$	−0.928184	−0.212940	−0.106470	+	0.994316i	$0.533955\pi$
$20$	0.0319048	0.00713413
$21$	9.26824	2.02250
$22$	2.66759	0.568733
$23$	−4.32680	−0.902201	−0.451100	−	0.892473i	$-0.648968\pi$
$23$	−4.32680	−0.902201	−0.451100	+	0.892473i	$0.648968\pi$
$24$	−9.22601	−1.88325
$25$	1.00000	0.200000
$26$	8.31482	1.63067
$27$	−15.8353	−3.04750
$28$	−0.0899160	−0.0169925
$29$	8.39369	1.55867	0.779334	−	0.626609i	$-0.215557\pi$
$29$	8.39369	1.55867	0.779334	+	0.626609i	$0.215557\pi$
$30$	4.68779	0.855869
$31$	−2.12253	−0.381217	−0.190608	−	0.981666i	$-0.561046\pi$
$31$	−2.12253	−0.381217	−0.190608	+	0.981666i	$0.561046\pi$
$32$	0.180464	0.0319018
$33$	6.15438	1.07134
$34$	−6.07868	−1.04248
$35$	−2.81826	−0.476373
$36$	0.249341	0.0415568
$37$	−1.68954	−0.277758	−0.138879	−	0.990309i	$-0.544350\pi$
$37$	−1.68954	−0.277758	−0.138879	+	0.990309i	$0.544350\pi$
$38$	1.32308	0.214632
$39$	19.1830	3.07175
$40$	2.80542	0.443576
$41$	1.25590	0.196138	0.0980689	−	0.995180i	$-0.468733\pi$
$41$	1.25590	0.196138	0.0980689	+	0.995180i	$0.468733\pi$
$42$	−13.2114	−2.03856
$43$	6.56150	1.00062	0.500309	−	0.865847i	$-0.333220\pi$
$43$	6.56150	1.00062	0.500309	+	0.865847i	$0.333220\pi$
$44$	−0.0597068	−0.00900114
$45$	7.81515	1.16501
$46$	6.16764	0.909369
$47$	8.81781	1.28621	0.643105	−	0.765778i	$-0.277646\pi$
$47$	8.81781	1.28621	0.643105	+	0.765778i	$0.277646\pi$
$48$	13.3611	1.92850
$49$	0.942584	0.134655
$50$	−1.42545	−0.201589
$51$	−14.0241	−1.96376
$52$	−0.186105	−0.0258081
$53$	7.79498	1.07072	0.535361	−	0.844623i	$-0.320175\pi$
$53$	7.79498	1.07072	0.535361	+	0.844623i	$0.320175\pi$
$54$	22.5724	3.07171
$55$	−1.87141	−0.252340
$56$	−7.90640	−1.05654
$57$	3.05246	0.404308
$58$	−11.9648	−1.57105
$59$	3.90646	0.508578	0.254289	−	0.967128i	$-0.418159\pi$
$59$	3.90646	0.508578	0.254289	+	0.967128i	$0.418159\pi$
$60$	−0.104923	−0.0135455
$61$	10.0457	1.28623	0.643113	−	0.765772i	$-0.277643\pi$
$61$	10.0457	1.28623	0.643113	+	0.765772i	$0.277643\pi$
$62$	3.02555	0.384245
$63$	−22.0251	−2.77490
$64$	7.86834	0.983543
$65$	−5.83313	−0.723510
$66$	−8.77275	−1.07985
$67$	−10.0523	−1.22808	−0.614041	−	0.789274i	$-0.710457\pi$
$67$	−10.0523	−1.22808	−0.614041	+	0.789274i	$0.710457\pi$
$68$	0.136055	0.0164990
$69$	14.2293	1.71301
$70$	4.01728	0.480157
$71$	14.6245	1.73561	0.867804	−	0.496907i	$-0.165531\pi$
$71$	14.6245	1.73561	0.867804	+	0.496907i	$0.165531\pi$
$72$	21.9248	2.58386
$73$	−13.7213	−1.60595	−0.802977	−	0.596010i	$-0.796752\pi$
$73$	−13.7213	−1.60595	−0.802977	+	0.596010i	$0.796752\pi$
$74$	2.40835	0.279965
$75$	−3.28864	−0.379739
$76$	−0.0296135	−0.00339690
$77$	5.27410	0.601040
$78$	−27.3445	−3.09615
$79$	−8.07291	−0.908274	−0.454137	−	0.890932i	$-0.650052\pi$
$79$	−8.07291	−0.908274	−0.454137	+	0.890932i	$0.650052\pi$
$80$	−4.06279	−0.454234
$81$	28.6311	3.18123
$82$	−1.79021	−0.197696
$83$	10.7583	1.18088	0.590438	−	0.807083i	$-0.298955\pi$
$83$	10.7583	1.18088	0.590438	+	0.807083i	$0.298955\pi$
$84$	0.295701	0.0322636
$85$	4.26439	0.462538
$86$	−9.35308	−1.00857
$87$	−27.6038	−2.95944
$88$	−5.25008	−0.559660
$89$	8.76904	0.929517	0.464758	−	0.885438i	$-0.346141\pi$
$89$	8.76904	0.929517	0.464758	+	0.885438i	$0.346141\pi$
$90$	−11.1401	−1.17427
$91$	16.4393	1.72330
$92$	−0.138046	−0.0143923
$93$	6.98022	0.723815
$94$	−12.5693	−1.29643
$95$	−0.928184	−0.0952296
$96$	−0.593480	−0.0605718
$97$	8.43982	0.856934	0.428467	−	0.903558i	$-0.359054\pi$
$97$	8.43982	0.856934	0.428467	+	0.903558i	$0.359054\pi$
$98$	−1.34361	−0.135725
$99$	−14.6253	−1.46990
$100$	0.0319048	0.00319048
$101$	−6.36863	−0.633702	−0.316851	−	0.948475i	$-0.602626\pi$
$101$	−6.36863	−0.633702	−0.316851	+	0.948475i	$0.602626\pi$
$102$	19.9906	1.97936
$103$	6.92264	0.682108	0.341054	−	0.940044i	$-0.389216\pi$
$103$	6.92264	0.682108	0.341054	+	0.940044i	$0.389216\pi$
$104$	−16.3644	−1.60466
$105$	9.26824	0.904487
$106$	−11.1113	−1.07923
$107$	−12.1296	−1.17261	−0.586304	−	0.810091i	$-0.699418\pi$
$107$	−12.1296	−1.17261	−0.586304	+	0.810091i	$0.699418\pi$
$108$	−0.505221	−0.0486150
$109$	−8.51009	−0.815119	−0.407560	−	0.913179i	$-0.633620\pi$
$109$	−8.51009	−0.815119	−0.407560	+	0.913179i	$0.633620\pi$
$110$	2.66759	0.254345
$111$	5.55628	0.527378
$112$	11.4500	1.08192
$113$	8.55427	0.804718	0.402359	−	0.915482i	$-0.368190\pi$
$113$	8.55427	0.804718	0.402359	+	0.915482i	$0.368190\pi$
$114$	−4.35113	−0.407520
$115$	−4.32680	−0.403477
$116$	0.267799	0.0248645
$117$	−45.5867	−4.21450
$118$	−5.56846	−0.512618
$119$	−12.0182	−1.10170
$120$	−9.22601	−0.842216
$121$	−7.49784	−0.681622
$122$	−14.3197	−1.29644
$123$	−4.13019	−0.372406
$124$	−0.0677187	−0.00608132
$125$	1.00000	0.0894427
$126$	31.3957	2.79695
$127$	−4.91773	−0.436378	−0.218189	−	0.975907i	$-0.570015\pi$
$127$	−4.91773	−0.436378	−0.218189	+	0.975907i	$0.570015\pi$
$128$	−11.5768	−1.02326
$129$	−21.5784	−1.89987
$130$	8.31482	0.729258
$131$	−9.90470	−0.865378	−0.432689	−	0.901543i	$-0.642435\pi$
$131$	−9.90470	−0.865378	−0.432689	+	0.901543i	$0.642435\pi$
$132$	0.196354	0.0170904
$133$	2.61586	0.226824
$134$	14.3290	1.23784
$135$	−15.8353	−1.36288
$136$	11.9634	1.02585
$137$	−19.3954	−1.65706	−0.828529	−	0.559947i	$-0.810822\pi$
$137$	−19.3954	−1.65706	−0.828529	+	0.559947i	$0.810822\pi$
$138$	−20.2831	−1.72662
$139$	8.47604	0.718929	0.359464	−	0.933159i	$-0.382959\pi$
$139$	8.47604	0.718929	0.359464	+	0.933159i	$0.382959\pi$
$140$	−0.0899160	−0.00759929
$141$	−28.9986	−2.44212
$142$	−20.8465	−1.74940
$143$	10.9161	0.912854
$144$	−31.7513	−2.64594
$145$	8.39369	0.697058
$146$	19.5590	1.61871
$147$	−3.09982	−0.255669
$148$	−0.0539043	−0.00443091
$149$	−13.7541	−1.12678	−0.563388	−	0.826192i	$-0.690502\pi$
$149$	−13.7541	−1.12678	−0.563388	+	0.826192i	$0.690502\pi$
$150$	4.68779	0.382756
$151$	−21.5242	−1.75161	−0.875806	−	0.482663i	$-0.839670\pi$
$151$	−21.5242	−1.75161	−0.875806	+	0.482663i	$0.839670\pi$
$152$	−2.60394	−0.211208
$153$	33.3269	2.69432
$154$	−7.51797	−0.605815
$155$	−2.12253	−0.170485
$156$	0.612031	0.0490017
$157$	−3.22311	−0.257232	−0.128616	−	0.991694i	$-0.541053\pi$
$157$	−3.22311	−0.257232	−0.128616	+	0.991694i	$0.541053\pi$
$158$	11.5075	0.915490
$159$	−25.6349	−2.03298
$160$	0.180464	0.0142669
$161$	12.1941	0.961026
$162$	−40.8122	−3.20651
$163$	−4.01546	−0.314515	−0.157257	−	0.987558i	$-0.550265\pi$
$163$	−4.01546	−0.314515	−0.157257	+	0.987558i	$0.550265\pi$
$164$	0.0400691	0.00312887
$165$	6.15438	0.479118
$166$	−15.3354	−1.19026
$167$	12.4411	0.962718	0.481359	−	0.876524i	$-0.340143\pi$
$167$	12.4411	0.962718	0.481359	+	0.876524i	$0.340143\pi$
$168$	26.0013	2.00604
$169$	21.0254	1.61734
$170$	−6.07868	−0.466213
$171$	−7.25389	−0.554719
$172$	0.209343	0.0159623
$173$	7.65232	0.581795	0.290898	−	0.956754i	$-0.406046\pi$
$173$	7.65232	0.581795	0.290898	+	0.956754i	$0.406046\pi$
$174$	39.3478	2.98295
$175$	−2.81826	−0.213040
$176$	7.60313	0.573107
$177$	−12.8469	−0.965634
$178$	−12.4998	−0.936901
$179$	4.93298	0.368708	0.184354	−	0.982860i	$-0.440981\pi$
$179$	4.93298	0.368708	0.184354	+	0.982860i	$0.440981\pi$
$180$	0.249341	0.0185848
$181$	−0.866499	−0.0644064	−0.0322032	−	0.999481i	$-0.510252\pi$
$181$	−0.866499	−0.0644064	−0.0322032	+	0.999481i	$0.510252\pi$
$182$	−23.4333	−1.73699
$183$	−33.0368	−2.44215
$184$	−12.1385	−0.894862
$185$	−1.68954	−0.124217
$186$	−9.94995	−0.729566
$187$	−7.98041	−0.583585
$188$	0.281330	0.0205181
$189$	44.6279	3.24620
$190$	1.32308	0.0959862
$191$	−11.1577	−0.807344	−0.403672	−	0.914904i	$-0.632266\pi$
$191$	−11.1577	−0.807344	−0.403672	+	0.914904i	$0.632266\pi$
$192$	−25.8761	−1.86745
$193$	17.7252	1.27589	0.637945	−	0.770082i	$-0.279785\pi$
$193$	17.7252	1.27589	0.637945	+	0.770082i	$0.279785\pi$
$194$	−12.0305	−0.863742
$195$	19.1830	1.37373
$196$	0.0300729	0.00214807
$197$	−21.2065	−1.51090	−0.755449	−	0.655208i	$-0.772581\pi$
$197$	−21.2065	−1.51090	−0.755449	+	0.655208i	$0.772581\pi$
$198$	20.8476	1.48158
$199$	19.0113	1.34767	0.673837	−	0.738880i	$-0.264645\pi$
$199$	19.0113	1.34767	0.673837	+	0.738880i	$0.264645\pi$
$200$	2.80542	0.198373
$201$	33.0583	2.33175
$202$	9.07816	0.638737
$203$	−23.6556	−1.66030
$204$	−0.447434	−0.0313267
$205$	1.25590	0.0877155
$206$	−9.86786	−0.687527
$207$	−33.8146	−2.35028
$208$	23.6988	1.64321
$209$	1.73701	0.120151
$210$	−13.2114	−0.911673
$211$	−5.43465	−0.374137	−0.187068	−	0.982347i	$-0.559899\pi$
$211$	−5.43465	−0.374137	−0.187068	+	0.982347i	$0.559899\pi$
$212$	0.248697	0.0170806
$213$	−48.0947	−3.29539
$214$	17.2901	1.18192
$215$	6.56150	0.447490
$216$	−44.4246	−3.02271
$217$	5.98183	0.406073
$218$	12.1307	0.821595
$219$	45.1243	3.04922
$220$	−0.0597068	−0.00402543
$221$	−24.8747	−1.67326
$222$	−7.92019	−0.531568
$223$	13.8859	0.929869	0.464934	−	0.885345i	$-0.346078\pi$
$223$	13.8859	0.929869	0.464934	+	0.885345i	$0.346078\pi$
$224$	−0.508593	−0.0339818
$225$	7.81515	0.521010
$226$	−12.1937	−0.811111
$227$	−4.23766	−0.281263	−0.140632	−	0.990062i	$-0.544913\pi$
$227$	−4.23766	−0.281263	−0.140632	+	0.990062i	$0.544913\pi$
$228$	0.0973881	0.00644969
$229$	−8.11320	−0.536135	−0.268068	−	0.963400i	$-0.586385\pi$
$229$	−8.11320	−0.536135	−0.268068	+	0.963400i	$0.586385\pi$
$230$	6.16764	0.406682
$231$	−17.3446	−1.14119
$232$	23.5478	1.54599
$233$	−6.60408	−0.432647	−0.216324	−	0.976322i	$-0.569407\pi$
$233$	−6.60408	−0.432647	−0.216324	+	0.976322i	$0.569407\pi$
$234$	64.9816	4.24798
$235$	8.81781	0.575210
$236$	0.124635	0.00811303
$237$	26.5489	1.72454
$238$	17.1313	1.11046
$239$	10.0797	0.652004	0.326002	−	0.945369i	$-0.394298\pi$
$239$	10.0797	0.652004	0.326002	+	0.945369i	$0.394298\pi$
$240$	13.3611	0.862452
$241$	21.4146	1.37944	0.689719	−	0.724077i	$-0.257734\pi$
$241$	21.4146	1.37944	0.689719	+	0.724077i	$0.257734\pi$
$242$	10.6878	0.687037
$243$	−46.6515	−2.99269
$244$	0.320507	0.0205184
$245$	0.942584	0.0602195
$246$	5.88737	0.375365
$247$	5.41421	0.344498
$248$	−5.95457	−0.378116
$249$	−35.3801	−2.24212
$250$	−1.42545	−0.0901533
$251$	−27.0458	−1.70712	−0.853559	−	0.520996i	$-0.825561\pi$
$251$	−27.0458	−1.70712	−0.853559	+	0.520996i	$0.825561\pi$
$252$	−0.702707	−0.0442664
$253$	8.09720	0.509067
$254$	7.00997	0.439845
$255$	−14.0241	−0.878220
$256$	0.765521	0.0478451
$257$	−3.59531	−0.224269	−0.112135	−	0.993693i	$-0.535769\pi$
$257$	−3.59531	−0.224269	−0.112135	+	0.993693i	$0.535769\pi$
$258$	30.7589	1.91497
$259$	4.76155	0.295868
$260$	−0.186105	−0.0115417
$261$	65.5979	4.06041
$262$	14.1186	0.872253
$263$	−5.54272	−0.341779	−0.170890	−	0.985290i	$-0.554664\pi$
$263$	−5.54272	−0.341779	−0.170890	+	0.985290i	$0.554664\pi$
$264$	17.2656	1.06262
$265$	7.79498	0.478842
$266$	−3.72878	−0.228626
$267$	−28.8382	−1.76487
$268$	−0.320716	−0.0195908
$269$	−17.4009	−1.06095	−0.530476	−	0.847700i	$-0.677987\pi$
$269$	−17.4009	−1.06095	−0.530476	+	0.847700i	$0.677987\pi$
$270$	22.5724	1.37371
$271$	−1.01752	−0.0618097	−0.0309049	−	0.999522i	$-0.509839\pi$
$271$	−1.01752	−0.0618097	−0.0309049	+	0.999522i	$0.509839\pi$
$272$	−17.3253	−1.05050
$273$	−54.0628	−3.27203
$274$	27.6471	1.67022
$275$	−1.87141	−0.112850
$276$	0.453983	0.0273265
$277$	−25.2646	−1.51801	−0.759003	−	0.651088i	$-0.774313\pi$
$277$	−25.2646	−1.51801	−0.759003	+	0.651088i	$0.774313\pi$
$278$	−12.0822	−0.724640
$279$	−16.5878	−0.993089
$280$	−7.90640	−0.472498
$281$	−17.2668	−1.03005	−0.515026	−	0.857175i	$-0.672218\pi$
$281$	−17.2668	−1.03005	−0.515026	+	0.857175i	$0.672218\pi$
$282$	41.3360	2.46152
$283$	17.9156	1.06497	0.532487	−	0.846438i	$-0.321257\pi$
$283$	17.9156	1.06497	0.532487	+	0.846438i	$0.321257\pi$
$284$	0.466591	0.0276871
$285$	3.05246	0.180812
$286$	−15.5604	−0.920106
$287$	−3.53944	−0.208926
$288$	1.41035	0.0831057
$289$	1.18505	0.0697089
$290$	−11.9648	−0.702596
$291$	−27.7555	−1.62706
$292$	−0.437774	−0.0256188
$293$	13.7055	0.800684	0.400342	−	0.916366i	$-0.368891\pi$
$293$	13.7055	0.800684	0.400342	+	0.916366i	$0.368891\pi$
$294$	4.41863	0.257700
$295$	3.90646	0.227443
$296$	−4.73986	−0.275499
$297$	29.6342	1.71955
$298$	19.6057	1.13573
$299$	25.2388	1.45960
$300$	−0.104923	−0.00605775
$301$	−18.4920	−1.06586
$302$	30.6816	1.76553
$303$	20.9441	1.20321
$304$	3.77102	0.216283
$305$	10.0457	0.575217
$306$	−47.5057	−2.71572
$307$	−31.6149	−1.80436	−0.902178	−	0.431363i	$-0.858033\pi$
$307$	−31.6149	−1.80436	−0.902178	+	0.431363i	$0.858033\pi$
$308$	0.168269	0.00958803
$309$	−22.7660	−1.29512
$310$	3.02555	0.171840
$311$	−11.2736	−0.639266	−0.319633	−	0.947541i	$-0.603560\pi$
$311$	−11.2736	−0.639266	−0.319633	+	0.947541i	$0.603560\pi$
$312$	53.8165	3.04676
$313$	19.3327	1.09275	0.546374	−	0.837541i	$-0.316008\pi$
$313$	19.3327	1.09275	0.546374	+	0.837541i	$0.316008\pi$
$314$	4.59438	0.259276
$315$	−22.0251	−1.24097
$316$	−0.257565	−0.0144891
$317$	21.1190	1.18616	0.593081	−	0.805143i	$-0.297911\pi$
$317$	21.1190	1.18616	0.593081	+	0.805143i	$0.297911\pi$
$318$	36.5412	2.04913
$319$	−15.7080	−0.879478
$320$	7.86834	0.439854
$321$	39.8897	2.22643
$322$	−17.3820	−0.968661
$323$	−3.95814	−0.220237
$324$	0.913469	0.0507483
$325$	−5.83313	−0.323564
$326$	5.72383	0.317013
$327$	27.9866	1.54766
$328$	3.52331	0.194542
$329$	−24.8509	−1.37007
$330$	−8.77275	−0.482924
$331$	13.3259	0.732460	0.366230	−	0.930524i	$-0.380648\pi$
$331$	13.3259	0.732460	0.366230	+	0.930524i	$0.380648\pi$
$332$	0.343241	0.0188378
$333$	−13.2040	−0.723573
$334$	−17.7341	−0.970367
$335$	−10.0523	−0.549215
$336$	−37.6549	−2.05424
$337$	13.3894	0.729366	0.364683	−	0.931132i	$-0.381177\pi$
$337$	13.3894	0.729366	0.364683	+	0.931132i	$0.381177\pi$
$338$	−29.9706	−1.63018
$339$	−28.1319	−1.52792
$340$	0.136055	0.00737860
$341$	3.97210	0.215102
$342$	10.3401	0.559126
$343$	17.0714	0.921767
$344$	18.4077	0.992479
$345$	14.2293	0.766080
$346$	−10.9080	−0.586417
$347$	−19.4505	−1.04416	−0.522079	−	0.852897i	$-0.674843\pi$
$347$	−19.4505	−1.04416	−0.522079	+	0.852897i	$0.674843\pi$
$348$	−0.880694	−0.0472101
$349$	−15.5717	−0.833533	−0.416766	−	0.909014i	$-0.636837\pi$
$349$	−15.5717	−0.833533	−0.416766	+	0.909014i	$0.636837\pi$
$350$	4.01728	0.214733
$351$	92.3692	4.93030
$352$	−0.337721	−0.0180006
$353$	−15.9125	−0.846937	−0.423469	−	0.905911i	$-0.639188\pi$
$353$	−15.9125	−0.846937	−0.423469	+	0.905911i	$0.639188\pi$
$354$	18.3126	0.973306
$355$	14.6245	0.776187
$356$	0.279775	0.0148280
$357$	39.5234	2.09180
$358$	−7.03171	−0.371638
$359$	29.2587	1.54422	0.772108	−	0.635492i	$-0.219203\pi$
$359$	29.2587	1.54422	0.772108	+	0.635492i	$0.219203\pi$
$360$	21.9248	1.15554
$361$	−18.1385	−0.954657
$362$	1.23515	0.0649181
$363$	24.6577	1.29419
$364$	0.524491	0.0274908
$365$	−13.7213	−0.718204
$366$	47.0923	2.46155
$367$	−8.30286	−0.433406	−0.216703	−	0.976238i	$-0.569530\pi$
$367$	−8.30286	−0.433406	−0.216703	+	0.976238i	$0.569530\pi$
$368$	17.5789	0.916364
$369$	9.81501	0.510949
$370$	2.40835	0.125204
$371$	−21.9683	−1.14054
$372$	0.222702	0.0115466
$373$	−21.6699	−1.12202	−0.561012	−	0.827808i	$-0.689588\pi$
$373$	−21.6699	−1.12202	−0.561012	+	0.827808i	$0.689588\pi$
$374$	11.3757	0.588222
$375$	−3.28864	−0.169825
$376$	24.7376	1.27575
$377$	−48.9614	−2.52164
$378$	−63.6148	−3.27199
$379$	−2.34082	−0.120240	−0.0601200	−	0.998191i	$-0.519148\pi$
$379$	−2.34082	−0.120240	−0.0601200	+	0.998191i	$0.519148\pi$
$380$	−0.0296135	−0.00151914
$381$	16.1726	0.828549
$382$	15.9047	0.813758
$383$	11.7122	0.598464	0.299232	−	0.954180i	$-0.403269\pi$
$383$	11.7122	0.598464	0.299232	+	0.954180i	$0.403269\pi$
$384$	38.0721	1.94286
$385$	5.27410	0.268793
$386$	−25.2664	−1.28603
$387$	51.2791	2.60666
$388$	0.269271	0.0136701
$389$	6.30162	0.319505	0.159752	−	0.987157i	$-0.448930\pi$
$389$	6.30162	0.319505	0.159752	+	0.987157i	$0.448930\pi$
$390$	−27.3445	−1.38464
$391$	−18.4512	−0.933117
$392$	2.64434	0.133560
$393$	32.5730	1.64309
$394$	30.2287	1.52290
$395$	−8.07291	−0.406192
$396$	−0.466617	−0.0234484
$397$	−14.3918	−0.722304	−0.361152	−	0.932507i	$-0.617616\pi$
$397$	−14.3918	−0.722304	−0.361152	+	0.932507i	$0.617616\pi$
$398$	−27.0996	−1.35838
$399$	−8.60263	−0.430670
$400$	−4.06279	−0.203140
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	−47.1230	−2.35028
$403$	12.3810	0.616739
$404$	−0.203190	−0.0101091
$405$	28.6311	1.42269
$406$	33.7198	1.67349
$407$	3.16181	0.156725
$408$	−39.3433	−1.94779
$409$	6.24404	0.308748	0.154374	−	0.988013i	$-0.450664\pi$
$409$	6.24404	0.308748	0.154374	+	0.988013i	$0.450664\pi$
$410$	−1.79021	−0.0884124
$411$	63.7843	3.14625
$412$	0.220865	0.0108812
$413$	−11.0094	−0.541738
$414$	48.2010	2.36895
$415$	10.7583	0.528104
$416$	−1.05267	−0.0516112
$417$	−27.8747	−1.36503
$418$	−2.47602	−0.121106
$419$	8.99650	0.439508	0.219754	−	0.975555i	$-0.429475\pi$
$419$	8.99650	0.439508	0.219754	+	0.975555i	$0.429475\pi$
$420$	0.295701	0.0144287
$421$	−32.9477	−1.60577	−0.802887	−	0.596131i	$-0.796704\pi$
$421$	−32.9477	−1.60577	−0.802887	+	0.596131i	$0.796704\pi$
$422$	7.74682	0.377109
$423$	68.9125	3.35064
$424$	21.8682	1.06201
$425$	4.26439	0.206853
$426$	68.5565	3.32157
$427$	−28.3115	−1.37009
$428$	−0.386991	−0.0187059
$429$	−35.8993	−1.73323
$430$	−9.35308	−0.451045
$431$	−12.0395	−0.579924	−0.289962	−	0.957038i	$-0.593643\pi$
$431$	−12.0395	−0.579924	−0.289962	+	0.957038i	$0.593643\pi$
$432$	64.3355	3.09534
$433$	−5.65833	−0.271922	−0.135961	−	0.990714i	$-0.543412\pi$
$433$	−5.65833	−0.271922	−0.135961	+	0.990714i	$0.543412\pi$
$434$	−8.52679	−0.409299
$435$	−27.6038	−1.32350
$436$	−0.271513	−0.0130031
$437$	4.01607	0.192115
$438$	−64.3224	−3.07344
$439$	0.608505	0.0290423	0.0145212	−	0.999895i	$-0.495378\pi$
$439$	0.608505	0.0290423	0.0145212	+	0.999895i	$0.495378\pi$
$440$	−5.25008	−0.250288
$441$	7.36643	0.350783
$442$	35.4577	1.68655
$443$	3.24006	0.153940	0.0769700	−	0.997033i	$-0.475475\pi$
$443$	3.24006	0.153940	0.0769700	+	0.997033i	$0.475475\pi$
$444$	0.177272	0.00841295
$445$	8.76904	0.415693
$446$	−19.7936	−0.937256
$447$	45.2321	2.13941
$448$	−22.1750	−1.04767
$449$	−29.2507	−1.38043	−0.690213	−	0.723606i	$-0.742483\pi$
$449$	−29.2507	−1.38043	−0.690213	+	0.723606i	$0.742483\pi$
$450$	−11.1401	−0.525149
$451$	−2.35029	−0.110671
$452$	0.272922	0.0128372
$453$	70.7853	3.32578
$454$	6.04057	0.283498
$455$	16.4393	0.770684
$456$	8.56343	0.401020
$457$	27.2646	1.27538	0.637692	−	0.770292i	$-0.279889\pi$
$457$	27.2646	1.27538	0.637692	+	0.770292i	$0.279889\pi$
$458$	11.5650	0.540395
$459$	−67.5279	−3.15193
$460$	−0.138046	−0.00643642
$461$	−24.4318	−1.13790	−0.568952	−	0.822371i	$-0.692651\pi$
$461$	−24.4318	−1.13790	−0.568952	+	0.822371i	$0.692651\pi$
$462$	24.7239	1.15026
$463$	−36.2268	−1.68360	−0.841802	−	0.539787i	$-0.818505\pi$
$463$	−36.2268	−1.68360	−0.841802	+	0.539787i	$0.818505\pi$
$464$	−34.1018	−1.58314
$465$	6.98022	0.323700
$466$	9.41377	0.436085
$467$	1.34196	0.0620983	0.0310492	−	0.999518i	$-0.490115\pi$
$467$	1.34196	0.0620983	0.0310492	+	0.999518i	$0.490115\pi$
$468$	−1.45444	−0.0672313
$469$	28.3299	1.30815
$470$	−12.5693	−0.579780
$471$	10.5996	0.488406
$472$	10.9593	0.504441
$473$	−12.2792	−0.564599
$474$	−37.8441	−1.73824
$475$	−0.928184	−0.0425880
$476$	−0.383437	−0.0175748
$477$	60.9189	2.78929
$478$	−14.3682	−0.657184
$479$	28.8420	1.31782	0.658911	−	0.752221i	$-0.271017\pi$
$479$	28.8420	1.31782	0.658911	+	0.752221i	$0.271017\pi$
$480$	−0.593480	−0.0270885
$481$	9.85528	0.449362
$482$	−30.5255	−1.39040
$483$	−40.1018	−1.82470
$484$	−0.239217	−0.0108735
$485$	8.43982	0.383232
$486$	66.4993	3.01647
$487$	39.1234	1.77285	0.886425	−	0.462873i	$-0.153181\pi$
$487$	39.1234	1.77285	0.886425	+	0.462873i	$0.153181\pi$
$488$	28.1825	1.27576
$489$	13.2054	0.597168
$490$	−1.34361	−0.0606979
$491$	−37.6242	−1.69796	−0.848978	−	0.528428i	$-0.822782\pi$
$491$	−37.6242	−1.69796	−0.848978	+	0.528428i	$0.822782\pi$
$492$	−0.131773	−0.00594077
$493$	35.7940	1.61208
$494$	−7.71768	−0.347235
$495$	−14.6253	−0.657359
$496$	8.62338	0.387201
$497$	−41.2156	−1.84877
$498$	50.4326	2.25994
$499$	−20.9349	−0.937177	−0.468588	−	0.883417i	$-0.655237\pi$
$499$	−20.9349	−0.937177	−0.468588	+	0.883417i	$0.655237\pi$
$500$	0.0319048	0.00142683
$501$	−40.9142	−1.82791
$502$	38.5525	1.72068
$503$	12.5845	0.561114	0.280557	−	0.959837i	$-0.409481\pi$
$503$	12.5845	0.561114	0.280557	+	0.959837i	$0.409481\pi$
$504$	−61.7897	−2.75233
$505$	−6.36863	−0.283400
$506$	−11.5421	−0.513111
$507$	−69.1448	−3.07083
$508$	−0.156899	−0.00696127
$509$	−37.9076	−1.68022	−0.840112	−	0.542413i	$-0.817511\pi$
$509$	−37.9076	−1.68022	−0.840112	+	0.542413i	$0.817511\pi$
$510$	19.9906	0.885197
$511$	38.6701	1.71066
$512$	22.0625	0.975033
$513$	14.6981	0.648935
$514$	5.12493	0.226051
$515$	6.92264	0.305048
$516$	−0.688454	−0.0303075
$517$	−16.5017	−0.725743
$518$	−6.78735	−0.298219
$519$	−25.1657	−1.10465
$520$	−16.3644	−0.717625
$521$	41.1604	1.80327	0.901635	−	0.432498i	$-0.142368\pi$
$521$	41.1604	1.80327	0.901635	+	0.432498i	$0.142368\pi$
$522$	−93.5065	−4.09267
$523$	−28.4574	−1.24435	−0.622177	−	0.782877i	$-0.713752\pi$
$523$	−28.4574	−1.24435	−0.622177	+	0.782877i	$0.713752\pi$
$524$	−0.316007	−0.0138048
$525$	9.26824	0.404499
$526$	7.90087	0.344494
$527$	−9.05128	−0.394280
$528$	−25.0039	−1.08816
$529$	−4.27877	−0.186033
$530$	−11.1113	−0.482646
$531$	30.5295	1.32487
$532$	0.0834585	0.00361839
$533$	−7.32580	−0.317315
$534$	41.1074	1.77889
$535$	−12.1296	−0.524407
$536$	−28.2009	−1.21809
$537$	−16.2228	−0.700065
$538$	24.8041	1.06938
$539$	−1.76396	−0.0759790
$540$	−0.505221	−0.0217413
$541$	6.36652	0.273718	0.136859	−	0.990591i	$-0.456299\pi$
$541$	6.36652	0.273718	0.136859	+	0.990591i	$0.456299\pi$
$542$	1.45042	0.0623008
$543$	2.84960	0.122288
$544$	0.769568	0.0329949
$545$	−8.51009	−0.364532
$546$	77.0638	3.29802
$547$	−38.4973	−1.64602	−0.823012	−	0.568024i	$-0.807708\pi$
$547$	−38.4973	−1.64602	−0.823012	+	0.568024i	$0.807708\pi$
$548$	−0.618805	−0.0264340
$549$	78.5089	3.35068
$550$	2.66759	0.113747
$551$	−7.79088	−0.331903
$552$	39.9191	1.69907
$553$	22.7516	0.967495
$554$	36.0135	1.53007
$555$	5.55628	0.235851
$556$	0.270426	0.0114686
$557$	40.9288	1.73421	0.867105	−	0.498125i	$-0.165978\pi$
$557$	40.9288	1.73421	0.867105	+	0.498125i	$0.165978\pi$
$558$	23.6451	1.00098
$559$	−38.2740	−1.61882
$560$	11.4500	0.483851
$561$	26.2447	1.10805
$562$	24.6130	1.03824
$563$	32.2733	1.36016	0.680078	−	0.733140i	$-0.261946\pi$
$563$	32.2733	1.36016	0.680078	+	0.733140i	$0.261946\pi$
$564$	−0.925194	−0.0389577
$565$	8.55427	0.359881
$566$	−25.5378	−1.07344
$567$	−80.6898	−3.38865
$568$	41.0278	1.72149
$569$	5.91792	0.248092	0.124046	−	0.992276i	$-0.460413\pi$
$569$	5.91792	0.248092	0.124046	+	0.992276i	$0.460413\pi$
$570$	−4.35113	−0.182249
$571$	21.3689	0.894261	0.447131	−	0.894469i	$-0.352446\pi$
$571$	21.3689	0.894261	0.447131	+	0.894469i	$0.352446\pi$
$572$	0.348277	0.0145622
$573$	36.6937	1.53290
$574$	5.04529	0.210586
$575$	−4.32680	−0.180440
$576$	61.4923	2.56218
$577$	−36.7617	−1.53041	−0.765205	−	0.643787i	$-0.777362\pi$
$577$	−36.7617	−1.53041	−0.765205	+	0.643787i	$0.777362\pi$
$578$	−1.68923	−0.0702627
$579$	−58.2919	−2.42253
$580$	0.267799	0.0111197
$581$	−30.3196	−1.25787
$582$	39.5641	1.63998
$583$	−14.5876	−0.604155
$584$	−38.4939	−1.59289
$585$	−45.5867	−1.88478
$586$	−19.5365	−0.807045
$587$	24.1271	0.995832	0.497916	−	0.867225i	$-0.334099\pi$
$587$	24.1271	0.995832	0.497916	+	0.867225i	$0.334099\pi$
$588$	−0.0988991	−0.00407853
$589$	1.97009	0.0811763
$590$	−5.56846	−0.229250
$591$	69.7404	2.86874
$592$	6.86423	0.282118
$593$	12.0854	0.496287	0.248143	−	0.968723i	$-0.420180\pi$
$593$	12.0854	0.496287	0.248143	+	0.968723i	$0.420180\pi$
$594$	−42.2421	−1.73321
$595$	−12.0182	−0.492697
$596$	−0.438820	−0.0179748
$597$	−62.5213	−2.55883
$598$	−35.9766	−1.47119
$599$	−34.6091	−1.41409	−0.707045	−	0.707169i	$-0.749972\pi$
$599$	−34.6091	−1.41409	−0.707045	+	0.707169i	$0.749972\pi$
$600$	−9.22601	−0.376650
$601$	14.2905	0.582921	0.291460	−	0.956583i	$-0.405859\pi$
$601$	14.2905	0.582921	0.291460	+	0.956583i	$0.405859\pi$
$602$	26.3594	1.07433
$603$	−78.5601	−3.19921
$604$	−0.686724	−0.0279424
$605$	−7.49784	−0.304831
$606$	−29.8548	−1.21277
$607$	−25.1738	−1.02177	−0.510887	−	0.859648i	$-0.670683\pi$
$607$	−25.1738	−1.02177	−0.510887	+	0.859648i	$0.670683\pi$
$608$	−0.167503	−0.00679316
$609$	77.7947	3.15240
$610$	−14.3197	−0.579787
$611$	−51.4354	−2.08085
$612$	1.06329	0.0429808
$613$	10.1959	0.411807	0.205904	−	0.978572i	$-0.433987\pi$
$613$	10.1959	0.411807	0.205904	+	0.978572i	$0.433987\pi$
$614$	45.0654	1.81869
$615$	−4.13019	−0.166545
$616$	14.7961	0.596151
$617$	29.5722	1.19053	0.595267	−	0.803528i	$-0.297046\pi$
$617$	29.5722	1.19053	0.595267	+	0.803528i	$0.297046\pi$
$618$	32.4518	1.30540
$619$	−25.3452	−1.01871	−0.509355	−	0.860556i	$-0.670116\pi$
$619$	−25.3452	−1.01871	−0.509355	+	0.860556i	$0.670116\pi$
$620$	−0.0677187	−0.00271965
$621$	68.5162	2.74946
$622$	16.0699	0.644345
$623$	−24.7134	−0.990123
$624$	−77.9367	−3.11997
$625$	1.00000	0.0400000
$626$	−27.5577	−1.10143
$627$	−5.71239	−0.228131
$628$	−0.102833	−0.00410347
$629$	−7.20485	−0.287276
$630$	31.3957	1.25083
$631$	24.4425	0.973039	0.486519	−	0.873670i	$-0.338266\pi$
$631$	24.4425	0.973039	0.486519	+	0.873670i	$0.338266\pi$
$632$	−22.6479	−0.900885
$633$	17.8726	0.710372
$634$	−30.1041	−1.19559
$635$	−4.91773	−0.195154
$636$	−0.817875	−0.0324309
$637$	−5.49821	−0.217847
$638$	22.3909	0.886465
$639$	114.293	4.52134
$640$	−11.5768	−0.457615
$641$	−27.7600	−1.09645	−0.548227	−	0.836329i	$-0.684697\pi$
$641$	−27.7600	−1.09645	−0.548227	+	0.836329i	$0.684697\pi$
$642$	−56.8608	−2.24412
$643$	−49.3645	−1.94674	−0.973372	−	0.229229i	$-0.926379\pi$
$643$	−49.3645	−1.94674	−0.973372	+	0.229229i	$0.926379\pi$
$644$	0.389049	0.0153307
$645$	−21.5784	−0.849648
$646$	5.64213	0.221987
$647$	−38.8380	−1.52688	−0.763441	−	0.645878i	$-0.776491\pi$
$647$	−38.8380	−1.52688	−0.763441	+	0.645878i	$0.776491\pi$
$648$	80.3222	3.15535
$649$	−7.31057	−0.286965
$650$	8.31482	0.326134
$651$	−19.6721	−0.771009
$652$	−0.128112	−0.00501726
$653$	−45.5380	−1.78204	−0.891019	−	0.453965i	$-0.850009\pi$
$653$	−45.5380	−1.78204	−0.891019	+	0.453965i	$0.850009\pi$
$654$	−39.8935	−1.55996
$655$	−9.90470	−0.387009
$656$	−5.10244	−0.199217
$657$	−107.234	−4.18359
$658$	35.4236	1.38096
$659$	−24.3979	−0.950406	−0.475203	−	0.879876i	$-0.657625\pi$
$659$	−24.3979	−0.950406	−0.475203	+	0.879876i	$0.657625\pi$
$660$	0.196354	0.00764307
$661$	−2.67170	−0.103917	−0.0519586	−	0.998649i	$-0.516546\pi$
$661$	−2.67170	−0.103917	−0.0519586	+	0.998649i	$0.516546\pi$
$662$	−18.9954	−0.738279
$663$	81.8041	3.17701
$664$	30.1815	1.17127
$665$	2.61586	0.101439
$666$	18.8216	0.729322
$667$	−36.3178	−1.40623
$668$	0.396929	0.0153577
$669$	−45.6657	−1.76554
$670$	14.3290	0.553578
$671$	−18.7996	−0.725752
$672$	1.67258	0.0645212
$673$	38.1733	1.47147	0.735736	−	0.677268i	$-0.236836\pi$
$673$	38.1733	1.47147	0.735736	+	0.677268i	$0.236836\pi$
$674$	−19.0859	−0.735160
$675$	−15.8353	−0.609500
$676$	0.670810	0.0258004
$677$	8.52163	0.327513	0.163756	−	0.986501i	$-0.447639\pi$
$677$	8.52163	0.327513	0.163756	+	0.986501i	$0.447639\pi$
$678$	40.1006	1.54005
$679$	−23.7856	−0.912807
$680$	11.9634	0.458776
$681$	13.9361	0.534034
$682$	−5.66203	−0.216810
$683$	0.105560	0.00403914	0.00201957	−	0.999998i	$-0.499357\pi$
$683$	0.105560	0.00403914	0.00201957	+	0.999998i	$0.499357\pi$
$684$	−0.231434	−0.00884910
$685$	−19.3954	−0.741059
$686$	−24.3344	−0.929090
$687$	26.6814	1.01796
$688$	−26.6580	−1.01633
$689$	−45.4691	−1.73223
$690$	−20.2831	−0.772166
$691$	−5.02824	−0.191283	−0.0956416	−	0.995416i	$-0.530490\pi$
$691$	−5.02824	−0.191283	−0.0956416	+	0.995416i	$0.530490\pi$
$692$	0.244146	0.00928103
$693$	41.2179	1.56574
$694$	27.7257	1.05245
$695$	8.47604	0.321515
$696$	−77.4402	−2.93536
$697$	5.35563	0.202859
$698$	22.1966	0.840155
$699$	21.7184	0.821466
$700$	−0.0899160	−0.00339850
$701$	−20.7414	−0.783390	−0.391695	−	0.920095i	$-0.628111\pi$
$701$	−20.7414	−0.783390	−0.391695	+	0.920095i	$0.628111\pi$
$702$	−131.668	−4.96947
$703$	1.56820	0.0591458
$704$	−14.7249	−0.554964
$705$	−28.9986	−1.09215
$706$	22.6825	0.853666
$707$	17.9485	0.675021
$708$	−0.409879	−0.0154042
$709$	28.0278	1.05261	0.526303	−	0.850297i	$-0.323578\pi$
$709$	28.0278	1.05261	0.526303	+	0.850297i	$0.323578\pi$
$710$	−20.8465	−0.782354
$711$	−63.0910	−2.36610
$712$	24.6008	0.921956
$713$	9.18375	0.343934
$714$	−56.3386	−2.10842
$715$	10.9161	0.408241
$716$	0.157386	0.00588178
$717$	−33.1486	−1.23796
$718$	−41.7068	−1.55648
$719$	1.35893	0.0506796	0.0253398	−	0.999679i	$-0.491933\pi$
$719$	1.35893	0.0506796	0.0253398	+	0.999679i	$0.491933\pi$
$720$	−31.7513	−1.18330
$721$	−19.5098	−0.726582
$722$	25.8555	0.962241
$723$	−70.4250	−2.61914
$724$	−0.0276455	−0.00102744
$725$	8.39369	0.311734
$726$	−35.1483	−1.30448
$727$	−3.36587	−0.124833	−0.0624167	−	0.998050i	$-0.519881\pi$
$727$	−3.36587	−0.124833	−0.0624167	+	0.998050i	$0.519881\pi$
$728$	46.1190	1.70928
$729$	67.5266	2.50099
$730$	19.5590	0.723910
$731$	27.9808	1.03491
$732$	−1.05403	−0.0389582
$733$	−39.6579	−1.46480	−0.732398	−	0.680876i	$-0.761599\pi$
$733$	−39.6579	−1.46480	−0.732398	+	0.680876i	$0.761599\pi$
$734$	11.8353	0.436849
$735$	−3.09982	−0.114339
$736$	−0.780831	−0.0287818
$737$	18.8119	0.692945
$738$	−13.9908	−0.515008
$739$	2.15131	0.0791372	0.0395686	−	0.999217i	$-0.487402\pi$
$739$	2.15131	0.0791372	0.0395686	+	0.999217i	$0.487402\pi$
$740$	−0.0539043	−0.00198156
$741$	−17.8054	−0.654097
$742$	31.3147	1.14960
$743$	−6.70921	−0.246137	−0.123069	−	0.992398i	$-0.539274\pi$
$743$	−6.70921	−0.246137	−0.123069	+	0.992398i	$0.539274\pi$
$744$	19.5824	0.717927
$745$	−13.7541	−0.503910
$746$	30.8893	1.13094
$747$	84.0776	3.07624
$748$	−0.254613	−0.00930958
$749$	34.1842	1.24906
$750$	4.68779	0.171174
$751$	−15.8683	−0.579042	−0.289521	−	0.957172i	$-0.593496\pi$
$751$	−15.8683	−0.579042	−0.289521	+	0.957172i	$0.593496\pi$
$752$	−35.8249	−1.30640
$753$	88.9440	3.24130
$754$	69.7920	2.54168
$755$	−21.5242	−0.783345
$756$	1.42384	0.0517847
$757$	28.0231	1.01852	0.509259	−	0.860614i	$-0.329920\pi$
$757$	28.0231	1.01852	0.509259	+	0.860614i	$0.329920\pi$
$758$	3.33672	0.121195
$759$	−26.6288	−0.966563
$760$	−2.60394	−0.0944550
$761$	−5.10436	−0.185033	−0.0925165	−	0.995711i	$-0.529491\pi$
$761$	−5.10436	−0.185033	−0.0925165	+	0.995711i	$0.529491\pi$
$762$	−23.0533	−0.835132
$763$	23.9836	0.868266
$764$	−0.355984	−0.0128791
$765$	33.3269	1.20494
$766$	−16.6951	−0.603219
$767$	−22.7869	−0.822786
$768$	−2.51752	−0.0908433
$769$	−35.1513	−1.26759	−0.633794	−	0.773502i	$-0.718503\pi$
$769$	−35.1513	−1.26759	−0.633794	+	0.773502i	$0.718503\pi$
$770$	−7.51797	−0.270929
$771$	11.8237	0.425819
$772$	0.565520	0.0203535
$773$	34.4936	1.24065	0.620323	−	0.784346i	$-0.287001\pi$
$773$	34.4936	1.24065	0.620323	+	0.784346i	$0.287001\pi$
$774$	−73.0957	−2.62737
$775$	−2.12253	−0.0762434
$776$	23.6772	0.849963
$777$	−15.6590	−0.561764
$778$	−8.98263	−0.322043
$779$	−1.16570	−0.0417656
$780$	0.612031	0.0219142
$781$	−27.3683	−0.979316
$782$	26.3012	0.940530
$783$	−132.916	−4.75004
$784$	−3.82952	−0.136769
$785$	−3.22311	−0.115038
$786$	−46.4311	−1.65614
$787$	−8.54304	−0.304527	−0.152263	−	0.988340i	$-0.548656\pi$
$787$	−8.54304	−0.304527	−0.152263	+	0.988340i	$0.548656\pi$
$788$	−0.676587	−0.0241024
$789$	18.2280	0.648935
$790$	11.5075	0.409419
$791$	−24.1082	−0.857187
$792$	−41.0301	−1.45794
$793$	−58.5981	−2.08088
$794$	20.5148	0.728043
$795$	−25.6349	−0.909175
$796$	0.606551	0.0214986
$797$	33.0035	1.16904	0.584521	−	0.811378i	$-0.301282\pi$
$797$	33.0035	1.16904	0.584521	+	0.811378i	$0.301282\pi$
$798$	12.2626	0.434092
$799$	37.6026	1.33028
$800$	0.180464	0.00638035
$801$	68.5314	2.42144
$802$	−1.42545	−0.0503344
$803$	25.6781	0.906159
$804$	1.05472	0.0371971
$805$	12.1941	0.429784
$806$	−17.6484	−0.621639
$807$	57.2253	2.01443
$808$	−17.8667	−0.628548
$809$	−37.0989	−1.30433	−0.652165	−	0.758077i	$-0.726139\pi$
$809$	−37.0989	−1.30433	−0.652165	+	0.758077i	$0.726139\pi$
$810$	−40.8122	−1.43399
$811$	26.9667	0.946929	0.473465	−	0.880813i	$-0.343003\pi$
$811$	26.9667	0.946929	0.473465	+	0.880813i	$0.343003\pi$
$812$	−0.754726	−0.0264857
$813$	3.34625	0.117358
$814$	−4.50699	−0.157970
$815$	−4.01546	−0.140655
$816$	56.9768	1.99459
$817$	−6.09027	−0.213072
$818$	−8.90055	−0.311201
$819$	128.475	4.48929
$820$	0.0400691	0.00139927
$821$	54.5767	1.90474	0.952370	−	0.304944i	$-0.0986378\pi$
$821$	54.5767	1.90474	0.952370	+	0.304944i	$0.0986378\pi$
$822$	−90.9213	−3.17125
$823$	18.0280	0.628417	0.314208	−	0.949354i	$-0.398261\pi$
$823$	18.0280	0.628417	0.314208	+	0.949354i	$0.398261\pi$
$824$	19.4209	0.676559
$825$	6.15438	0.214268
$826$	15.6934	0.546042
$827$	55.8587	1.94240	0.971198	−	0.238274i	$-0.0765816\pi$
$827$	55.8587	1.94240	0.971198	+	0.238274i	$0.0765816\pi$
$828$	−1.07885	−0.0374926
$829$	9.89477	0.343659	0.171830	−	0.985127i	$-0.445032\pi$
$829$	9.89477	0.343659	0.171830	+	0.985127i	$0.445032\pi$
$830$	−15.3354	−0.532299
$831$	83.0863	2.88223
$832$	−45.8970	−1.59119
$833$	4.01955	0.139269
$834$	39.7339	1.37587
$835$	12.4411	0.430541
$836$	0.0554189	0.00191670
$837$	33.6108	1.16176
$838$	−12.8240	−0.442999
$839$	39.2536	1.35518	0.677592	−	0.735438i	$-0.263024\pi$
$839$	39.2536	1.35518	0.677592	+	0.735438i	$0.263024\pi$
$840$	26.0013	0.897130
$841$	41.4540	1.42945
$842$	46.9653	1.61853
$843$	56.7843	1.95576
$844$	−0.173391	−0.00596838
$845$	21.0254	0.723294
$846$	−98.2312	−3.37726
$847$	21.1309	0.726065
$848$	−31.6694	−1.08753
$849$	−58.9181	−2.02206
$850$	−6.07868	−0.208497
$851$	7.31029	0.250594
$852$	−1.53445	−0.0525694
$853$	41.9330	1.43576	0.717880	−	0.696167i	$-0.245113\pi$
$853$	41.9330	1.43576	0.717880	+	0.696167i	$0.245113\pi$
$854$	40.3566	1.38097
$855$	−7.25389	−0.248078
$856$	−34.0285	−1.16307
$857$	33.0772	1.12990	0.564948	−	0.825126i	$-0.308896\pi$
$857$	33.0772	1.12990	0.564948	+	0.825126i	$0.308896\pi$
$858$	51.1726	1.74700
$859$	17.8899	0.610395	0.305198	−	0.952289i	$-0.401277\pi$
$859$	17.8899	0.610395	0.305198	+	0.952289i	$0.401277\pi$
$860$	0.209343	0.00713854
$861$	11.6399	0.396688
$862$	17.1617	0.584531
$863$	41.6682	1.41840	0.709201	−	0.705006i	$-0.249056\pi$
$863$	41.6682	1.41840	0.709201	+	0.705006i	$0.249056\pi$
$864$	−2.85769	−0.0972207
$865$	7.65232	0.260187
$866$	8.06566	0.274082
$867$	−3.89721	−0.132356
$868$	0.190849	0.00647783
$869$	15.1077	0.512493
$870$	39.3478	1.33402
$871$	58.6362	1.98681
$872$	−23.8744	−0.808488
$873$	65.9584	2.23235
$874$	−5.72470	−0.193641
$875$	−2.81826	−0.0952745
$876$	1.43968	0.0486423
$877$	28.7008	0.969158	0.484579	−	0.874748i	$-0.338973\pi$
$877$	28.7008	0.969158	0.484579	+	0.874748i	$0.338973\pi$
$878$	−0.867392	−0.0292731
$879$	−45.0724	−1.52026
$880$	7.60313	0.256301
$881$	−31.6562	−1.06652	−0.533262	−	0.845950i	$-0.679034\pi$
$881$	−31.6562	−1.06652	−0.533262	+	0.845950i	$0.679034\pi$
$882$	−10.5005	−0.353569
$883$	−41.3549	−1.39170	−0.695852	−	0.718185i	$-0.744973\pi$
$883$	−41.3549	−1.39170	−0.695852	+	0.718185i	$0.744973\pi$
$884$	−0.793624	−0.0266924
$885$	−12.8469	−0.431845
$886$	−4.61855	−0.155163
$887$	−16.4067	−0.550881	−0.275441	−	0.961318i	$-0.588824\pi$
$887$	−16.4067	−0.550881	−0.275441	+	0.961318i	$0.588824\pi$
$888$	15.5877	0.523088
$889$	13.8594	0.464831
$890$	−12.4998	−0.418995
$891$	−53.5804	−1.79501
$892$	0.443027	0.0148336
$893$	−8.18455	−0.273885
$894$	−64.4761	−2.15640
$895$	4.93298	0.164891
$896$	32.6266	1.08998
$897$	−83.0013	−2.77133
$898$	41.6954	1.39139
$899$	−17.8158	−0.594191
$900$	0.249341	0.00831136
$901$	33.2409	1.10741
$902$	3.35022	0.111550
$903$	60.8135	2.02375
$904$	23.9983	0.798172
$905$	−0.866499	−0.0288034
$906$	−100.901	−3.35220
$907$	−33.2747	−1.10487	−0.552435	−	0.833556i	$-0.686301\pi$
$907$	−33.2747	−1.10487	−0.552435	+	0.833556i	$0.686301\pi$
$908$	−0.135202	−0.00448683
$909$	−49.7718	−1.65083
$910$	−23.4333	−0.776807
$911$	−35.8874	−1.18900	−0.594502	−	0.804094i	$-0.702651\pi$
$911$	−35.8874	−1.18900	−0.594502	+	0.804094i	$0.702651\pi$
$912$	−12.4015	−0.410655
$913$	−20.1331	−0.666309
$914$	−38.8643	−1.28552
$915$	−33.0368	−1.09216
$916$	−0.258850	−0.00855264
$917$	27.9140	0.921802
$918$	96.2575	3.17697
$919$	−4.90679	−0.161860	−0.0809300	−	0.996720i	$-0.525789\pi$
$919$	−4.90679	−0.161860	−0.0809300	+	0.996720i	$0.525789\pi$
$920$	−12.1385	−0.400194
$921$	103.970	3.42593
$922$	34.8263	1.14694
$923$	−85.3065	−2.80790
$924$	−0.553377	−0.0182048
$925$	−1.68954	−0.0555516
$926$	51.6395	1.69698
$927$	54.1014	1.77692
$928$	1.51475	0.0497243
$929$	1.28331	0.0421041	0.0210521	−	0.999778i	$-0.493298\pi$
$929$	1.28331	0.0421041	0.0210521	+	0.999778i	$0.493298\pi$
$930$	−9.94995	−0.326272
$931$	−0.874891	−0.0286734
$932$	−0.210702	−0.00690176
$933$	37.0747	1.21377
$934$	−1.91289	−0.0625917
$935$	−7.98041	−0.260987
$936$	−127.890	−4.18021
$937$	−45.9859	−1.50229	−0.751147	−	0.660135i	$-0.770499\pi$
$937$	−45.9859	−1.50229	−0.751147	+	0.660135i	$0.770499\pi$
$938$	−40.3829	−1.31855
$939$	−63.5782	−2.07480
$940$	0.281330	0.00917598
$941$	−48.3137	−1.57498	−0.787491	−	0.616326i	$-0.788620\pi$
$941$	−48.3137	−1.57498	−0.787491	+	0.616326i	$0.788620\pi$
$942$	−15.1092	−0.492286
$943$	−5.43401	−0.176956
$944$	−15.8711	−0.516561
$945$	44.6279	1.45175
$946$	17.5034	0.569085
$947$	34.1639	1.11018	0.555089	−	0.831791i	$-0.312684\pi$
$947$	34.1639	1.11018	0.555089	+	0.831791i	$0.312684\pi$
$948$	0.847037	0.0275105
$949$	80.0379	2.59814
$950$	1.32308	0.0429263
$951$	−69.4528	−2.25216
$952$	−33.7160	−1.09274
$953$	−42.9025	−1.38975	−0.694873	−	0.719132i	$-0.744540\pi$
$953$	−42.9025	−1.38975	−0.694873	+	0.719132i	$0.744540\pi$
$954$	−86.8368	−2.81145
$955$	−11.1577	−0.361055
$956$	0.321592	0.0104010
$957$	51.6579	1.66986
$958$	−41.1127	−1.32829
$959$	54.6611	1.76510
$960$	−25.8761	−0.835149
$961$	−26.4949	−0.854674
$962$	−14.0482	−0.452932
$963$	−94.7943	−3.05470
$964$	0.683230	0.0220053
$965$	17.7252	0.570595
$966$	57.1631	1.83919
$967$	−25.3753	−0.816016	−0.408008	−	0.912978i	$-0.633776\pi$
$967$	−25.3753	−0.816016	−0.408008	+	0.912978i	$0.633776\pi$
$968$	−21.0346	−0.676077
$969$	13.0169	0.418163
$970$	−12.0305	−0.386277
$971$	−19.6478	−0.630529	−0.315264	−	0.949004i	$-0.602093\pi$
$971$	−19.6478	−0.630529	−0.315264	+	0.949004i	$0.602093\pi$
$972$	−1.48841	−0.0477406
$973$	−23.8877	−0.765804
$974$	−55.7684	−1.78693
$975$	19.1830	0.614349
$976$	−40.8137	−1.30642
$977$	−12.7529	−0.408003	−0.204001	−	0.978971i	$-0.565395\pi$
$977$	−12.7529	−0.408003	−0.204001	+	0.978971i	$0.565395\pi$
$978$	−18.8236	−0.601912
$979$	−16.4104	−0.524480
$980$	0.0300729	0.000960645 0
$981$	−66.5076	−2.12343
$982$	53.6314	1.71145
$983$	5.64317	0.179989	0.0899945	−	0.995942i	$-0.471315\pi$
$983$	5.64317	0.179989	0.0899945	+	0.995942i	$0.471315\pi$
$984$	−11.5869	−0.369377
$985$	−21.2065	−0.675694
$986$	−51.0225	−1.62489
$987$	81.7255	2.60135
$988$	0.172739	0.00549557
$989$	−28.3903	−0.902759
$990$	20.8476	0.662581
$991$	16.5582	0.525988	0.262994	−	0.964797i	$-0.415290\pi$
$991$	16.5582	0.525988	0.262994	+	0.964797i	$0.415290\pi$
$992$	−0.383039	−0.0121615
$993$	−43.8242	−1.39072
$994$	58.7507	1.86346
$995$	19.0113	0.602698
$996$	−1.12880	−0.0357673
$997$	−31.0047	−0.981928	−0.490964	−	0.871180i	$-0.663355\pi$
$997$	−31.0047	−0.981928	−0.490964	+	0.871180i	$0.663355\pi$
$998$	29.8417	0.944622
$999$	26.7543	0.846468

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2005.2.a.d.1.9	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2005.2.a.d.1.9	✓	25	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2005 = 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2005.a (trivial)

\(f(q)\)	\(=\)	\(q-1.42545 q^{2} -3.28864 q^{3} +0.0319048 q^{4} +1.00000 q^{5} +4.68779 q^{6} -2.81826 q^{7} +2.80542 q^{8} +7.81515 q^{9} +O(q^{10})\)	\(q-1.42545 q^{2} -3.28864 q^{3} +0.0319048 q^{4} +1.00000 q^{5} +4.68779 q^{6} -2.81826 q^{7} +2.80542 q^{8} +7.81515 q^{9} -1.42545 q^{10} -1.87141 q^{11} -0.104923 q^{12} -5.83313 q^{13} +4.01728 q^{14} -3.28864 q^{15} -4.06279 q^{16} +4.26439 q^{17} -11.1401 q^{18} -0.928184 q^{19} +0.0319048 q^{20} +9.26824 q^{21} +2.66759 q^{22} -4.32680 q^{23} -9.22601 q^{24} +1.00000 q^{25} +8.31482 q^{26} -15.8353 q^{27} -0.0899160 q^{28} +8.39369 q^{29} +4.68779 q^{30} -2.12253 q^{31} +0.180464 q^{32} +6.15438 q^{33} -6.07868 q^{34} -2.81826 q^{35} +0.249341 q^{36} -1.68954 q^{37} +1.32308 q^{38} +19.1830 q^{39} +2.80542 q^{40} +1.25590 q^{41} -13.2114 q^{42} +6.56150 q^{43} -0.0597068 q^{44} +7.81515 q^{45} +6.16764 q^{46} +8.81781 q^{47} +13.3611 q^{48} +0.942584 q^{49} -1.42545 q^{50} -14.0241 q^{51} -0.186105 q^{52} +7.79498 q^{53} +22.5724 q^{54} -1.87141 q^{55} -7.90640 q^{56} +3.05246 q^{57} -11.9648 q^{58} +3.90646 q^{59} -0.104923 q^{60} +10.0457 q^{61} +3.02555 q^{62} -22.0251 q^{63} +7.86834 q^{64} -5.83313 q^{65} -8.77275 q^{66} -10.0523 q^{67} +0.136055 q^{68} +14.2293 q^{69} +4.01728 q^{70} +14.6245 q^{71} +21.9248 q^{72} -13.7213 q^{73} +2.40835 q^{74} -3.28864 q^{75} -0.0296135 q^{76} +5.27410 q^{77} -27.3445 q^{78} -8.07291 q^{79} -4.06279 q^{80} +28.6311 q^{81} -1.79021 q^{82} +10.7583 q^{83} +0.295701 q^{84} +4.26439 q^{85} -9.35308 q^{86} -27.6038 q^{87} -5.25008 q^{88} +8.76904 q^{89} -11.1401 q^{90} +16.4393 q^{91} -0.138046 q^{92} +6.98022 q^{93} -12.5693 q^{94} -0.928184 q^{95} -0.593480 q^{96} +8.43982 q^{97} -1.34361 q^{98} -14.6253 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} - 10 q^{3} + 25 q^{4} + 25 q^{5} + 2 q^{6} - 31 q^{7} - 30 q^{8} + 17 q^{9}+O(q^{10}) \) 25 * q - 5 * q^2 - 10 * q^3 + 25 * q^4 + 25 * q^5 + 2 * q^6 - 31 * q^7 - 30 * q^8 + 17 * q^9	\( 25 q - 5 q^{2} - 10 q^{3} + 25 q^{4} + 25 q^{5} + 2 q^{6} - 31 q^{7} - 30 q^{8} + 17 q^{9} - 5 q^{10} - 30 q^{11} - 29 q^{12} - 18 q^{13} + 6 q^{14} - 10 q^{15} + 21 q^{16} - 18 q^{17} - 30 q^{18} - 17 q^{19} + 25 q^{20} + 6 q^{21} - 2 q^{22} - 44 q^{23} + 11 q^{24} + 25 q^{25} - 14 q^{26} - 25 q^{27} - 50 q^{28} - 9 q^{29} + 2 q^{30} - 13 q^{31} - 45 q^{32} - 21 q^{33} - 21 q^{34} - 31 q^{35} + 5 q^{36} - 28 q^{37} - 32 q^{38} + 9 q^{39} - 30 q^{40} + 28 q^{41} - 67 q^{42} - 61 q^{43} - 49 q^{44} + 17 q^{45} + 18 q^{46} - 53 q^{47} - 44 q^{48} + 28 q^{49} - 5 q^{50} - 30 q^{51} - 3 q^{52} - 36 q^{53} + 17 q^{54} - 30 q^{55} - 3 q^{56} - 13 q^{57} + 2 q^{58} - 39 q^{59} - 29 q^{60} + 10 q^{61} - 30 q^{62} - 44 q^{63} - 4 q^{64} - 18 q^{65} + 33 q^{66} - 10 q^{67} - 18 q^{68} - 6 q^{69} + 6 q^{70} - 7 q^{71} - q^{72} - 26 q^{73} - 3 q^{74} - 10 q^{75} + 12 q^{76} + 29 q^{77} - 5 q^{78} - 6 q^{79} + 21 q^{80} + 13 q^{81} - 30 q^{82} - 35 q^{83} + 117 q^{84} - 18 q^{85} + 14 q^{86} - 104 q^{87} + 53 q^{88} + 7 q^{89} - 30 q^{90} - 25 q^{91} - 31 q^{92} + 2 q^{93} + 68 q^{94} - 17 q^{95} + 92 q^{96} + 6 q^{97} + 15 q^{98} - 36 q^{99}+O(q^{100}) \) 25 * q - 5 * q^2 - 10 * q^3 + 25 * q^4 + 25 * q^5 + 2 * q^6 - 31 * q^7 - 30 * q^8 + 17 * q^9 - 5 * q^10 - 30 * q^11 - 29 * q^12 - 18 * q^13 + 6 * q^14 - 10 * q^15 + 21 * q^16 - 18 * q^17 - 30 * q^18 - 17 * q^19 + 25 * q^20 + 6 * q^21 - 2 * q^22 - 44 * q^23 + 11 * q^24 + 25 * q^25 - 14 * q^26 - 25 * q^27 - 50 * q^28 - 9 * q^29 + 2 * q^30 - 13 * q^31 - 45 * q^32 - 21 * q^33 - 21 * q^34 - 31 * q^35 + 5 * q^36 - 28 * q^37 - 32 * q^38 + 9 * q^39 - 30 * q^40 + 28 * q^41 - 67 * q^42 - 61 * q^43 - 49 * q^44 + 17 * q^45 + 18 * q^46 - 53 * q^47 - 44 * q^48 + 28 * q^49 - 5 * q^50 - 30 * q^51 - 3 * q^52 - 36 * q^53 + 17 * q^54 - 30 * q^55 - 3 * q^56 - 13 * q^57 + 2 * q^58 - 39 * q^59 - 29 * q^60 + 10 * q^61 - 30 * q^62 - 44 * q^63 - 4 * q^64 - 18 * q^65 + 33 * q^66 - 10 * q^67 - 18 * q^68 - 6 * q^69 + 6 * q^70 - 7 * q^71 - q^72 - 26 * q^73 - 3 * q^74 - 10 * q^75 + 12 * q^76 + 29 * q^77 - 5 * q^78 - 6 * q^79 + 21 * q^80 + 13 * q^81 - 30 * q^82 - 35 * q^83 + 117 * q^84 - 18 * q^85 + 14 * q^86 - 104 * q^87 + 53 * q^88 + 7 * q^89 - 30 * q^90 - 25 * q^91 - 31 * q^92 + 2 * q^93 + 68 * q^94 - 17 * q^95 + 92 * q^96 + 6 * q^97 + 15 * q^98 - 36 * q^99

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