LMFDB - Embedded newform 2005.2.a.f.1.2

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:	$0$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-2.59156 q^{2} +0.254340 q^{3} +4.71619 q^{4} -1.00000 q^{5} -0.659137 q^{6} -3.82594 q^{7} -7.03917 q^{8} -2.93531 q^{9} +O(q^{10})$	\(q-2.59156 q^{2} +0.254340 q^{3} +4.71619 q^{4} -1.00000 q^{5} -0.659137 q^{6} -3.82594 q^{7} -7.03917 q^{8} -2.93531 q^{9} +2.59156 q^{10} +5.13218 q^{11} +1.19951 q^{12} -0.728325 q^{13} +9.91516 q^{14} -0.254340 q^{15} +8.81005 q^{16} +5.28912 q^{17} +7.60704 q^{18} +6.91001 q^{19} -4.71619 q^{20} -0.973089 q^{21} -13.3004 q^{22} -7.64911 q^{23} -1.79034 q^{24} +1.00000 q^{25} +1.88750 q^{26} -1.50959 q^{27} -18.0439 q^{28} -8.96377 q^{29} +0.659137 q^{30} -3.82004 q^{31} -8.75345 q^{32} +1.30532 q^{33} -13.7071 q^{34} +3.82594 q^{35} -13.8435 q^{36} -1.51880 q^{37} -17.9077 q^{38} -0.185242 q^{39} +7.03917 q^{40} -0.177353 q^{41} +2.52182 q^{42} -9.44333 q^{43} +24.2043 q^{44} +2.93531 q^{45} +19.8231 q^{46} +1.50649 q^{47} +2.24075 q^{48} +7.63783 q^{49} -2.59156 q^{50} +1.34523 q^{51} -3.43492 q^{52} +10.1263 q^{53} +3.91218 q^{54} -5.13218 q^{55} +26.9314 q^{56} +1.75749 q^{57} +23.2302 q^{58} +7.97946 q^{59} -1.19951 q^{60} -6.88675 q^{61} +9.89986 q^{62} +11.2303 q^{63} +5.06500 q^{64} +0.728325 q^{65} -3.38281 q^{66} -3.32946 q^{67} +24.9445 q^{68} -1.94547 q^{69} -9.91516 q^{70} -0.651915 q^{71} +20.6621 q^{72} -11.3458 q^{73} +3.93605 q^{74} +0.254340 q^{75} +32.5889 q^{76} -19.6354 q^{77} +0.480066 q^{78} -9.55749 q^{79} -8.81005 q^{80} +8.42199 q^{81} +0.459621 q^{82} +6.74788 q^{83} -4.58927 q^{84} -5.28912 q^{85} +24.4730 q^{86} -2.27984 q^{87} -36.1263 q^{88} +5.33471 q^{89} -7.60704 q^{90} +2.78653 q^{91} -36.0746 q^{92} -0.971587 q^{93} -3.90417 q^{94} -6.91001 q^{95} -2.22635 q^{96} +15.3978 q^{97} -19.7939 q^{98} -15.0645 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 7 q^{2} + 3 q^{3} + 43 q^{4} - 37 q^{5} + 8 q^{6} - 16 q^{7} + 21 q^{8} + 54 q^{9}+O(q^{10}) $ 37 * q + 7 * q^2 + 3 * q^3 + 43 * q^4 - 37 * q^5 + 8 * q^6 - 16 * q^7 + 21 * q^8 + 54 * q^9	\( 37 q + 7 q^{2} + 3 q^{3} + 43 q^{4} - 37 q^{5} + 8 q^{6} - 16 q^{7} + 21 q^{8} + 54 q^{9} - 7 q^{10} + 42 q^{11} - 13 q^{13} + 14 q^{14} - 3 q^{15} + 63 q^{16} + 18 q^{17} + 22 q^{18} + 22 q^{19} - 43 q^{20} + 16 q^{21} - 10 q^{22} + 23 q^{23} + 23 q^{24} + 37 q^{25} + 21 q^{26} + 3 q^{27} - 18 q^{28} + 33 q^{29} - 8 q^{30} + 11 q^{31} + 54 q^{32} + 2 q^{33} + 8 q^{34} + 16 q^{35} + 91 q^{36} - 11 q^{37} + 29 q^{38} + 25 q^{39} - 21 q^{40} + 24 q^{41} + 4 q^{42} + 25 q^{43} + 84 q^{44} - 54 q^{45} + 31 q^{46} + 7 q^{47} + 4 q^{48} + 45 q^{49} + 7 q^{50} + 94 q^{51} - 43 q^{52} + 49 q^{53} + 38 q^{54} - 42 q^{55} + 46 q^{56} + 6 q^{57} + 15 q^{58} + 69 q^{59} + 9 q^{61} + 17 q^{62} - 38 q^{63} + 107 q^{64} + 13 q^{65} + 74 q^{66} + 13 q^{67} + 86 q^{68} - 14 q^{70} + 51 q^{71} + 81 q^{72} - 47 q^{73} + 79 q^{74} + 3 q^{75} + 59 q^{76} + 2 q^{77} + 20 q^{78} + 67 q^{79} - 63 q^{80} + 125 q^{81} - 24 q^{82} + 80 q^{83} + 50 q^{84} - 18 q^{85} + 69 q^{86} - 32 q^{87} - 12 q^{88} + 34 q^{89} - 22 q^{90} + 39 q^{91} + 85 q^{92} + q^{93} + 12 q^{94} - 22 q^{95} + 77 q^{96} - 14 q^{97} + 40 q^{98} + 133 q^{99}+O(q^{100}) \) 37 * q + 7 * q^2 + 3 * q^3 + 43 * q^4 - 37 * q^5 + 8 * q^6 - 16 * q^7 + 21 * q^8 + 54 * q^9 - 7 * q^10 + 42 * q^11 - 13 * q^13 + 14 * q^14 - 3 * q^15 + 63 * q^16 + 18 * q^17 + 22 * q^18 + 22 * q^19 - 43 * q^20 + 16 * q^21 - 10 * q^22 + 23 * q^23 + 23 * q^24 + 37 * q^25 + 21 * q^26 + 3 * q^27 - 18 * q^28 + 33 * q^29 - 8 * q^30 + 11 * q^31 + 54 * q^32 + 2 * q^33 + 8 * q^34 + 16 * q^35 + 91 * q^36 - 11 * q^37 + 29 * q^38 + 25 * q^39 - 21 * q^40 + 24 * q^41 + 4 * q^42 + 25 * q^43 + 84 * q^44 - 54 * q^45 + 31 * q^46 + 7 * q^47 + 4 * q^48 + 45 * q^49 + 7 * q^50 + 94 * q^51 - 43 * q^52 + 49 * q^53 + 38 * q^54 - 42 * q^55 + 46 * q^56 + 6 * q^57 + 15 * q^58 + 69 * q^59 + 9 * q^61 + 17 * q^62 - 38 * q^63 + 107 * q^64 + 13 * q^65 + 74 * q^66 + 13 * q^67 + 86 * q^68 - 14 * q^70 + 51 * q^71 + 81 * q^72 - 47 * q^73 + 79 * q^74 + 3 * q^75 + 59 * q^76 + 2 * q^77 + 20 * q^78 + 67 * q^79 - 63 * q^80 + 125 * q^81 - 24 * q^82 + 80 * q^83 + 50 * q^84 - 18 * q^85 + 69 * q^86 - 32 * q^87 - 12 * q^88 + 34 * q^89 - 22 * q^90 + 39 * q^91 + 85 * q^92 + q^93 + 12 * q^94 - 22 * q^95 + 77 * q^96 - 14 * q^97 + 40 * q^98 + 133 * 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 $

$2$	−2.59156	−1.83251	−0.916255	−	0.400595i	$-0.868803\pi$
$2$	−2.59156	−1.83251	−0.916255	+	0.400595i	$0.868803\pi$
$3$	0.254340	0.146843	0.0734215	−	0.997301i	$-0.476608\pi$
$3$	0.254340	0.146843	0.0734215	+	0.997301i	$0.476608\pi$
$4$	4.71619	2.35809
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.659137	−0.269091
$7$	−3.82594	−1.44607	−0.723035	−	0.690811i	$-0.757253\pi$
$7$	−3.82594	−1.44607	−0.723035	+	0.690811i	$0.757253\pi$
$8$	−7.03917	−2.48872
$9$	−2.93531	−0.978437
$10$	2.59156	0.819524
$11$	5.13218	1.54741	0.773705	−	0.633546i	$-0.218401\pi$
$11$	5.13218	1.54741	0.773705	+	0.633546i	$0.218401\pi$
$12$	1.19951	0.346270
$13$	−0.728325	−0.202001	−0.101001	−	0.994886i	$-0.532204\pi$
$13$	−0.728325	−0.202001	−0.101001	+	0.994886i	$0.532204\pi$
$14$	9.91516	2.64994
$15$	−0.254340	−0.0656702
$16$	8.81005	2.20251
$17$	5.28912	1.28280	0.641400	−	0.767207i	$-0.278354\pi$
$17$	5.28912	1.28280	0.641400	+	0.767207i	$0.278354\pi$
$18$	7.60704	1.79300
$19$	6.91001	1.58527	0.792633	−	0.609699i	$-0.208710\pi$
$19$	6.91001	1.58527	0.792633	+	0.609699i	$0.208710\pi$
$20$	−4.71619	−1.05457
$21$	−0.973089	−0.212345
$22$	−13.3004	−2.83565
$23$	−7.64911	−1.59495	−0.797474	−	0.603353i	$-0.793831\pi$
$23$	−7.64911	−1.59495	−0.797474	+	0.603353i	$0.793831\pi$
$24$	−1.79034	−0.365451
$25$	1.00000	0.200000
$26$	1.88750	0.370169
$27$	−1.50959	−0.290520
$28$	−18.0439	−3.40997
$29$	−8.96377	−1.66453	−0.832265	−	0.554377i	$-0.812956\pi$
$29$	−8.96377	−1.66453	−0.832265	+	0.554377i	$0.812956\pi$
$30$	0.659137	0.120341
$31$	−3.82004	−0.686099	−0.343049	−	0.939317i	$-0.611460\pi$
$31$	−3.82004	−0.686099	−0.343049	+	0.939317i	$0.611460\pi$
$32$	−8.75345	−1.54741
$33$	1.30532	0.227227
$34$	−13.7071	−2.35074
$35$	3.82594	0.646702
$36$	−13.8435	−2.30725
$37$	−1.51880	−0.249688	−0.124844	−	0.992176i	$-0.539843\pi$
$37$	−1.51880	−0.249688	−0.124844	+	0.992176i	$0.539843\pi$
$38$	−17.9077	−2.90502
$39$	−0.185242	−0.0296625
$40$	7.03917	1.11299
$41$	−0.177353	−0.0276979	−0.0138489	−	0.999904i	$-0.504408\pi$
$41$	−0.177353	−0.0276979	−0.0138489	+	0.999904i	$0.504408\pi$
$42$	2.52182	0.389125
$43$	−9.44333	−1.44009	−0.720047	−	0.693925i	$-0.755880\pi$
$43$	−9.44333	−1.44009	−0.720047	+	0.693925i	$0.755880\pi$
$44$	24.2043	3.64894
$45$	2.93531	0.437570
$46$	19.8231	2.92276
$47$	1.50649	0.219745	0.109872	−	0.993946i	$-0.464956\pi$
$47$	1.50649	0.219745	0.109872	+	0.993946i	$0.464956\pi$
$48$	2.24075	0.323424
$49$	7.63783	1.09112
$50$	−2.59156	−0.366502
$51$	1.34523	0.188370
$52$	−3.43492	−0.476337
$53$	10.1263	1.39095	0.695477	−	0.718548i	$-0.255193\pi$
$53$	10.1263	1.39095	0.695477	+	0.718548i	$0.255193\pi$
$54$	3.91218	0.532380
$55$	−5.13218	−0.692023
$56$	26.9314	3.59886
$57$	1.75749	0.232785
$58$	23.2302	3.05027
$59$	7.97946	1.03884	0.519419	−	0.854520i	$-0.326148\pi$
$59$	7.97946	1.03884	0.519419	+	0.854520i	$0.326148\pi$
$60$	−1.19951	−0.154857
$61$	−6.88675	−0.881758	−0.440879	−	0.897566i	$-0.645333\pi$
$61$	−6.88675	−0.881758	−0.440879	+	0.897566i	$0.645333\pi$
$62$	9.89986	1.25728
$63$	11.2303	1.41489
$64$	5.06500	0.633125
$65$	0.728325	0.0903376
$66$	−3.38281	−0.416395
$67$	−3.32946	−0.406759	−0.203379	−	0.979100i	$-0.565192\pi$
$67$	−3.32946	−0.406759	−0.203379	+	0.979100i	$0.565192\pi$
$68$	24.9445	3.02496
$69$	−1.94547	−0.234207
$70$	−9.91516	−1.18509
$71$	−0.651915	−0.0773680	−0.0386840	−	0.999251i	$-0.512317\pi$
$71$	−0.651915	−0.0773680	−0.0386840	+	0.999251i	$0.512317\pi$
$72$	20.6621	2.43506
$73$	−11.3458	−1.32792	−0.663961	−	0.747767i	$-0.731126\pi$
$73$	−11.3458	−1.32792	−0.663961	+	0.747767i	$0.731126\pi$
$74$	3.93605	0.457556
$75$	0.254340	0.0293686
$76$	32.5889	3.73821
$77$	−19.6354	−2.23766
$78$	0.480066	0.0543568
$79$	−9.55749	−1.07530	−0.537651	−	0.843168i	$-0.680688\pi$
$79$	−9.55749	−1.07530	−0.537651	+	0.843168i	$0.680688\pi$
$80$	−8.81005	−0.984994
$81$	8.42199	0.935776
$82$	0.459621	0.0507566
$83$	6.74788	0.740677	0.370338	−	0.928897i	$-0.379242\pi$
$83$	6.74788	0.740677	0.370338	+	0.928897i	$0.379242\pi$
$84$	−4.58927	−0.500730
$85$	−5.28912	−0.573686
$86$	24.4730	2.63899
$87$	−2.27984	−0.244425
$88$	−36.1263	−3.85107
$89$	5.33471	0.565478	0.282739	−	0.959197i	$-0.408757\pi$
$89$	5.33471	0.565478	0.282739	+	0.959197i	$0.408757\pi$
$90$	−7.60704	−0.801852
$91$	2.78653	0.292108
$92$	−36.0746	−3.76104
$93$	−0.971587	−0.100749
$94$	−3.90417	−0.402684
$95$	−6.91001	−0.708953
$96$	−2.22635	−0.227226
$97$	15.3978	1.56341	0.781705	−	0.623649i	$-0.214350\pi$
$97$	15.3978	1.56341	0.781705	+	0.623649i	$0.214350\pi$
$98$	−19.7939	−1.99949
$99$	−15.0645	−1.51404
$100$	4.71619	0.471619
$101$	19.5865	1.94893	0.974463	−	0.224547i	$-0.0720903\pi$
$101$	19.5865	1.94893	0.974463	+	0.224547i	$0.0720903\pi$
$102$	−3.48625	−0.345191
$103$	10.5543	1.03995	0.519973	−	0.854183i	$-0.325942\pi$
$103$	10.5543	1.03995	0.519973	+	0.854183i	$0.325942\pi$
$104$	5.12680	0.502724
$105$	0.973089	0.0949637
$106$	−26.2429	−2.54894
$107$	17.7731	1.71819	0.859093	−	0.511819i	$-0.171028\pi$
$107$	17.7731	1.71819	0.859093	+	0.511819i	$0.171028\pi$
$108$	−7.11949	−0.685073
$109$	−1.18283	−0.113294	−0.0566471	−	0.998394i	$-0.518041\pi$
$109$	−1.18283	−0.113294	−0.0566471	+	0.998394i	$0.518041\pi$
$110$	13.3004	1.26814
$111$	−0.386290	−0.0366650
$112$	−33.7067	−3.18499
$113$	−4.60229	−0.432947	−0.216474	−	0.976288i	$-0.569456\pi$
$113$	−4.60229	−0.432947	−0.216474	+	0.976288i	$0.569456\pi$
$114$	−4.55464	−0.426581
$115$	7.64911	0.713283
$116$	−42.2748	−3.92512
$117$	2.13786	0.197645
$118$	−20.6793	−1.90368
$119$	−20.2359	−1.85502
$120$	1.79034	0.163435
$121$	15.3393	1.39448
$122$	17.8474	1.61583
$123$	−0.0451079	−0.00406724
$124$	−18.0160	−1.61789
$125$	−1.00000	−0.0894427
$126$	−29.1041	−2.59280
$127$	14.9557	1.32711	0.663553	−	0.748129i	$-0.269048\pi$
$127$	14.9557	1.32711	0.663553	+	0.748129i	$0.269048\pi$
$128$	4.38064	0.387198
$129$	−2.40181	−0.211468
$130$	−1.88750	−0.165545
$131$	17.3573	1.51651	0.758256	−	0.651956i	$-0.226051\pi$
$131$	17.3573	1.51651	0.758256	+	0.651956i	$0.226051\pi$
$132$	6.15612	0.535821
$133$	−26.4373	−2.29241
$134$	8.62851	0.745390
$135$	1.50959	0.129924
$136$	−37.2310	−3.19253
$137$	5.99717	0.512373	0.256187	−	0.966627i	$-0.417534\pi$
$137$	5.99717	0.512373	0.256187	+	0.966627i	$0.417534\pi$
$138$	5.04181	0.429187
$139$	8.91700	0.756330	0.378165	−	0.925738i	$-0.376555\pi$
$139$	8.91700	0.756330	0.378165	+	0.925738i	$0.376555\pi$
$140$	18.0439	1.52498
$141$	0.383161	0.0322680
$142$	1.68948	0.141778
$143$	−3.73790	−0.312579
$144$	−25.8602	−2.15502
$145$	8.96377	0.744401
$146$	29.4032	2.43343
$147$	1.94260	0.160223
$148$	−7.16292	−0.588789
$149$	−6.67049	−0.546468	−0.273234	−	0.961948i	$-0.588093\pi$
$149$	−6.67049	−0.546468	−0.273234	+	0.961948i	$0.588093\pi$
$150$	−0.659137	−0.0538183
$151$	0.363731	0.0296000	0.0148000	−	0.999890i	$-0.495289\pi$
$151$	0.363731	0.0296000	0.0148000	+	0.999890i	$0.495289\pi$
$152$	−48.6407	−3.94528
$153$	−15.5252	−1.25514
$154$	50.8864	4.10054
$155$	3.82004	0.306833
$156$	−0.873636	−0.0699469
$157$	12.3762	0.987730	0.493865	−	0.869539i	$-0.335584\pi$
$157$	12.3762	0.987730	0.493865	+	0.869539i	$0.335584\pi$
$158$	24.7688	1.97050
$159$	2.57552	0.204252
$160$	8.75345	0.692021
$161$	29.2650	2.30641
$162$	−21.8261	−1.71482
$163$	−24.7824	−1.94110	−0.970552	−	0.240891i	$-0.922560\pi$
$163$	−24.7824	−1.94110	−0.970552	+	0.240891i	$0.922560\pi$
$164$	−0.836430	−0.0653142
$165$	−1.30532	−0.101619
$166$	−17.4876	−1.35730
$167$	−20.3967	−1.57835	−0.789174	−	0.614170i	$-0.789491\pi$
$167$	−20.3967	−1.57835	−0.789174	+	0.614170i	$0.789491\pi$
$168$	6.84973	0.528468
$169$	−12.4695	−0.959196
$170$	13.7071	1.05128
$171$	−20.2830	−1.55108
$172$	−44.5365	−3.39588
$173$	1.98995	0.151293	0.0756465	−	0.997135i	$-0.475898\pi$
$173$	1.98995	0.151293	0.0756465	+	0.997135i	$0.475898\pi$
$174$	5.90835	0.447911
$175$	−3.82594	−0.289214
$176$	45.2148	3.40819
$177$	2.02949	0.152546
$178$	−13.8252	−1.03624
$179$	−13.0517	−0.975532	−0.487766	−	0.872974i	$-0.662188\pi$
$179$	−13.0517	−0.975532	−0.487766	+	0.872974i	$0.662188\pi$
$180$	13.8435	1.03183
$181$	14.3331	1.06537	0.532685	−	0.846313i	$-0.321183\pi$
$181$	14.3331	1.06537	0.532685	+	0.846313i	$0.321183\pi$
$182$	−7.22146	−0.535290
$183$	−1.75157	−0.129480
$184$	53.8433	3.96938
$185$	1.51880	0.111664
$186$	2.51793	0.184623
$187$	27.1447	1.98502
$188$	7.10491	0.518179
$189$	5.77558	0.420112
$190$	17.9077	1.29916
$191$	22.5010	1.62812	0.814058	−	0.580784i	$-0.197254\pi$
$191$	22.5010	1.62812	0.814058	+	0.580784i	$0.197254\pi$
$192$	1.28823	0.0929701
$193$	23.1538	1.66665	0.833325	−	0.552784i	$-0.186434\pi$
$193$	23.1538	1.66665	0.833325	+	0.552784i	$0.186434\pi$
$194$	−39.9043	−2.86496
$195$	0.185242	0.0132655
$196$	36.0214	2.57296
$197$	17.3588	1.23676	0.618381	−	0.785879i	$-0.287789\pi$
$197$	17.3588	1.23676	0.618381	+	0.785879i	$0.287789\pi$
$198$	39.0407	2.77450
$199$	−1.03880	−0.0736387	−0.0368193	−	0.999322i	$-0.511723\pi$
$199$	−1.03880	−0.0736387	−0.0368193	+	0.999322i	$0.511723\pi$
$200$	−7.03917	−0.497744
$201$	−0.846815	−0.0597297
$202$	−50.7595	−3.57143
$203$	34.2949	2.40703
$204$	6.34437	0.444195
$205$	0.177353	0.0123869
$206$	−27.3521	−1.90571
$207$	22.4525	1.56056
$208$	−6.41658	−0.444910
$209$	35.4634	2.45306
$210$	−2.52182	−0.174022
$211$	8.85656	0.609710	0.304855	−	0.952399i	$-0.401392\pi$
$211$	8.85656	0.609710	0.304855	+	0.952399i	$0.401392\pi$
$212$	47.7575	3.28000
$213$	−0.165808	−0.0113610
$214$	−46.0600	−3.14859
$215$	9.44333	0.644030
$216$	10.6262	0.723023
$217$	14.6152	0.992147
$218$	3.06536	0.207613
$219$	−2.88568	−0.194996
$220$	−24.2043	−1.63186
$221$	−3.85220	−0.259127
$222$	1.00109	0.0671890
$223$	−3.37125	−0.225755	−0.112878	−	0.993609i	$-0.536007\pi$
$223$	−3.37125	−0.225755	−0.112878	+	0.993609i	$0.536007\pi$
$224$	33.4902	2.23766
$225$	−2.93531	−0.195687
$226$	11.9271	0.793380
$227$	0.563265	0.0373852	0.0186926	−	0.999825i	$-0.494050\pi$
$227$	0.563265	0.0373852	0.0186926	+	0.999825i	$0.494050\pi$
$228$	8.28866	0.548930
$229$	−7.89711	−0.521856	−0.260928	−	0.965358i	$-0.584028\pi$
$229$	−7.89711	−0.521856	−0.260928	+	0.965358i	$0.584028\pi$
$230$	−19.8231	−1.30710
$231$	−4.99407	−0.328585
$232$	63.0975	4.14255
$233$	14.0098	0.917812	0.458906	−	0.888485i	$-0.348241\pi$
$233$	14.0098	0.917812	0.458906	+	0.888485i	$0.348241\pi$
$234$	−5.54040	−0.362187
$235$	−1.50649	−0.0982728
$236$	37.6327	2.44968
$237$	−2.43085	−0.157901
$238$	52.4425	3.39934
$239$	−2.87310	−0.185845	−0.0929227	−	0.995673i	$-0.529621\pi$
$239$	−2.87310	−0.185845	−0.0929227	+	0.995673i	$0.529621\pi$
$240$	−2.24075	−0.144639
$241$	2.41408	0.155504	0.0777522	−	0.996973i	$-0.475226\pi$
$241$	2.41408	0.155504	0.0777522	+	0.996973i	$0.475226\pi$
$242$	−39.7527	−2.55540
$243$	6.67080	0.427932
$244$	−32.4792	−2.07927
$245$	−7.63783	−0.487963
$246$	0.116900	0.00745326
$247$	−5.03274	−0.320225
$248$	26.8899	1.70751
$249$	1.71625	0.108763
$250$	2.59156	0.163905
$251$	−6.63799	−0.418986	−0.209493	−	0.977810i	$-0.567181\pi$
$251$	−6.63799	−0.418986	−0.209493	+	0.977810i	$0.567181\pi$
$252$	52.9643	3.33644
$253$	−39.2566	−2.46804
$254$	−38.7587	−2.43193
$255$	−1.34523	−0.0842418
$256$	−21.4827	−1.34267
$257$	−0.0247653	−0.00154482	−0.000772409	−	1.00000i	$-0.500246\pi$
$257$	−0.0247653	−0.00154482	−0.000772409		1.00000i	$0.500246\pi$
$258$	6.22444	0.387517
$259$	5.81082	0.361067
$260$	3.43492	0.213025
$261$	26.3115	1.62864
$262$	−44.9825	−2.77903
$263$	22.3831	1.38020	0.690102	−	0.723712i	$-0.257566\pi$
$263$	22.3831	1.38020	0.690102	+	0.723712i	$0.257566\pi$
$264$	−9.18834	−0.565503
$265$	−10.1263	−0.622054
$266$	68.5139	4.20086
$267$	1.35683	0.0830366
$268$	−15.7024	−0.959176
$269$	11.7205	0.714609	0.357304	−	0.933988i	$-0.383696\pi$
$269$	11.7205	0.714609	0.357304	+	0.933988i	$0.383696\pi$
$270$	−3.91218	−0.238088
$271$	20.2818	1.23203	0.616017	−	0.787733i	$-0.288745\pi$
$271$	20.2818	1.23203	0.616017	+	0.787733i	$0.288745\pi$
$272$	46.5974	2.82538
$273$	0.708725	0.0428940
$274$	−15.5420	−0.938929
$275$	5.13218	0.309482
$276$	−9.17521	−0.552283
$277$	−23.2950	−1.39966	−0.699830	−	0.714310i	$-0.746741\pi$
$277$	−23.2950	−1.39966	−0.699830	+	0.714310i	$0.746741\pi$
$278$	−23.1090	−1.38598
$279$	11.2130	0.671304
$280$	−26.9314	−1.60946
$281$	−13.8432	−0.825816	−0.412908	−	0.910773i	$-0.635487\pi$
$281$	−13.8432	−0.825816	−0.412908	+	0.910773i	$0.635487\pi$
$282$	−0.992985	−0.0591314
$283$	1.04547	0.0621467	0.0310734	−	0.999517i	$-0.490107\pi$
$283$	1.04547	0.0621467	0.0310734	+	0.999517i	$0.490107\pi$
$284$	−3.07455	−0.182441
$285$	−1.75749	−0.104105
$286$	9.68698	0.572803
$287$	0.678542	0.0400531
$288$	25.6941	1.51404
$289$	10.9748	0.645576
$290$	−23.2302	−1.36412
$291$	3.91627	0.229576
$292$	−53.5088	−3.13136
$293$	4.83060	0.282207	0.141103	−	0.989995i	$-0.454935\pi$
$293$	4.83060	0.282207	0.141103	+	0.989995i	$0.454935\pi$
$294$	−5.03437	−0.293611
$295$	−7.97946	−0.464582
$296$	10.6911	0.621405
$297$	−7.74746	−0.449553
$298$	17.2870	1.00141
$299$	5.57104	0.322181
$300$	1.19951	0.0692539
$301$	36.1296	2.08248
$302$	−0.942631	−0.0542423
$303$	4.98162	0.286186
$304$	60.8776	3.49157
$305$	6.88675	0.394334
$306$	40.2345	2.30006
$307$	−8.09216	−0.461844	−0.230922	−	0.972972i	$-0.574174\pi$
$307$	−8.09216	−0.461844	−0.230922	+	0.972972i	$0.574174\pi$
$308$	−92.6043	−5.27662
$309$	2.68438	0.152709
$310$	−9.89986	−0.562274
$311$	10.9179	0.619098	0.309549	−	0.950884i	$-0.399822\pi$
$311$	10.9179	0.619098	0.309549	+	0.950884i	$0.399822\pi$
$312$	1.30395	0.0738216
$313$	−10.7304	−0.606515	−0.303258	−	0.952909i	$-0.598074\pi$
$313$	−10.7304	−0.606515	−0.303258	+	0.952909i	$0.598074\pi$
$314$	−32.0737	−1.81002
$315$	−11.2303	−0.632757
$316$	−45.0749	−2.53566
$317$	−19.9117	−1.11835	−0.559175	−	0.829049i	$-0.688882\pi$
$317$	−19.9117	−1.11835	−0.559175	+	0.829049i	$0.688882\pi$
$318$	−6.67462	−0.374294
$319$	−46.0037	−2.57571
$320$	−5.06500	−0.283142
$321$	4.52039	0.252304
$322$	−75.8421	−4.22652
$323$	36.5479	2.03358
$324$	39.7197	2.20665
$325$	−0.728325	−0.0404002
$326$	64.2250	3.55709
$327$	−0.300839	−0.0166365
$328$	1.24842	0.0689323
$329$	−5.76376	−0.317766
$330$	3.38281	0.186217
$331$	−18.6105	−1.02293	−0.511463	−	0.859305i	$-0.670896\pi$
$331$	−18.6105	−1.02293	−0.511463	+	0.859305i	$0.670896\pi$
$332$	31.8243	1.74658
$333$	4.45814	0.244304
$334$	52.8594	2.89234
$335$	3.32946	0.181908
$336$	−8.57296	−0.467693
$337$	15.2657	0.831577	0.415788	−	0.909461i	$-0.363506\pi$
$337$	15.2657	0.831577	0.415788	+	0.909461i	$0.363506\pi$
$338$	32.3156	1.75774
$339$	−1.17055	−0.0635753
$340$	−24.9445	−1.35280
$341$	−19.6051	−1.06168
$342$	52.5647	2.84238
$343$	−2.44029	−0.131763
$344$	66.4731	3.58399
$345$	1.94547	0.104741
$346$	−5.15707	−0.277246
$347$	24.9831	1.34116	0.670581	−	0.741837i	$-0.266045\pi$
$347$	24.9831	1.34116	0.670581	+	0.741837i	$0.266045\pi$
$348$	−10.7522	−0.576377
$349$	24.3956	1.30587	0.652933	−	0.757416i	$-0.273538\pi$
$349$	24.3956	1.30587	0.652933	+	0.757416i	$0.273538\pi$
$350$	9.91516	0.529988
$351$	1.09947	0.0586853
$352$	−44.9243	−2.39447
$353$	−17.8555	−0.950354	−0.475177	−	0.879890i	$-0.657616\pi$
$353$	−17.8555	−0.950354	−0.475177	+	0.879890i	$0.657616\pi$
$354$	−5.25956	−0.279542
$355$	0.651915	0.0346000
$356$	25.1595	1.33345
$357$	−5.14678	−0.272397
$358$	33.8244	1.78767
$359$	14.6875	0.775176	0.387588	−	0.921833i	$-0.373308\pi$
$359$	14.6875	0.775176	0.387588	+	0.921833i	$0.373308\pi$
$360$	−20.6621	−1.08899
$361$	28.7483	1.51307
$362$	−37.1451	−1.95230
$363$	3.90138	0.204770
$364$	13.1418	0.688817
$365$	11.3458	0.593865
$366$	4.53931	0.237274
$367$	16.9997	0.887379	0.443690	−	0.896181i	$-0.353669\pi$
$367$	16.9997	0.887379	0.443690	+	0.896181i	$0.353669\pi$
$368$	−67.3890	−3.51290
$369$	0.520586	0.0271006
$370$	−3.93605	−0.204625
$371$	−38.7426	−2.01142
$372$	−4.58218	−0.237575
$373$	−32.4418	−1.67977	−0.839887	−	0.542761i	$-0.817379\pi$
$373$	−32.4418	−1.67977	−0.839887	+	0.542761i	$0.817379\pi$
$374$	−70.3472	−3.63757
$375$	−0.254340	−0.0131340
$376$	−10.6045	−0.546883
$377$	6.52854	0.336237
$378$	−14.9678	−0.769859
$379$	4.37643	0.224802	0.112401	−	0.993663i	$-0.464146\pi$
$379$	4.37643	0.224802	0.112401	+	0.993663i	$0.464146\pi$
$380$	−32.5889	−1.67178
$381$	3.80383	0.194876
$382$	−58.3127	−2.98354
$383$	22.9275	1.17154	0.585771	−	0.810477i	$-0.300792\pi$
$383$	22.9275	1.17154	0.585771	+	0.810477i	$0.300792\pi$
$384$	1.11417	0.0568573
$385$	19.6354	1.00071
$386$	−60.0046	−3.05415
$387$	27.7191	1.40904
$388$	72.6189	3.68667
$389$	−13.2435	−0.671470	−0.335735	−	0.941956i	$-0.608985\pi$
$389$	−13.2435	−0.671470	−0.335735	+	0.941956i	$0.608985\pi$
$390$	−0.480066	−0.0243091
$391$	−40.4570	−2.04600
$392$	−53.7639	−2.71549
$393$	4.41465	0.222689
$394$	−44.9863	−2.26638
$395$	9.55749	0.480889
$396$	−71.0472	−3.57026
$397$	−25.2690	−1.26822	−0.634108	−	0.773245i	$-0.718632\pi$
$397$	−25.2690	−1.26822	−0.634108	+	0.773245i	$0.718632\pi$
$398$	2.69212	0.134944
$399$	−6.72406	−0.336624
$400$	8.81005	0.440503
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	2.19457	0.109455
$403$	2.78223	0.138593
$404$	92.3735	4.59575
$405$	−8.42199	−0.418492
$406$	−88.8772	−4.41090
$407$	−7.79473	−0.386370
$408$	−9.46932	−0.468801
$409$	28.2588	1.39731	0.698655	−	0.715459i	$-0.253782\pi$
$409$	28.2588	1.39731	0.698655	+	0.715459i	$0.253782\pi$
$410$	−0.459621	−0.0226991
$411$	1.52532	0.0752384
$412$	49.7760	2.45229
$413$	−30.5290	−1.50223
$414$	−58.1870	−2.85974
$415$	−6.74788	−0.331241
$416$	6.37536	0.312578
$417$	2.26795	0.111062
$418$	−91.9057	−4.49525
$419$	−10.3554	−0.505893	−0.252946	−	0.967480i	$-0.581400\pi$
$419$	−10.3554	−0.505893	−0.252946	+	0.967480i	$0.581400\pi$
$420$	4.58927	0.223933
$421$	−2.82241	−0.137556	−0.0687779	−	0.997632i	$-0.521910\pi$
$421$	−2.82241	−0.137556	−0.0687779	+	0.997632i	$0.521910\pi$
$422$	−22.9523	−1.11730
$423$	−4.42203	−0.215006
$424$	−71.2807	−3.46170
$425$	5.28912	0.256560
$426$	0.429701	0.0208191
$427$	26.3483	1.27508
$428$	83.8211	4.05164
$429$	−0.950695	−0.0459000
$430$	−24.4730	−1.18019
$431$	−11.3181	−0.545173	−0.272586	−	0.962131i	$-0.587879\pi$
$431$	−11.3181	−0.545173	−0.272586	+	0.962131i	$0.587879\pi$
$432$	−13.2995	−0.639874
$433$	35.2501	1.69401	0.847006	−	0.531584i	$-0.178403\pi$
$433$	35.2501	1.69401	0.847006	+	0.531584i	$0.178403\pi$
$434$	−37.8763	−1.81812
$435$	2.27984	0.109310
$436$	−5.57843	−0.267158
$437$	−52.8554	−2.52842
$438$	7.47841	0.357332
$439$	15.6331	0.746126	0.373063	−	0.927806i	$-0.378308\pi$
$439$	15.6331	0.746126	0.373063	+	0.927806i	$0.378308\pi$
$440$	36.1263	1.72225
$441$	−22.4194	−1.06759
$442$	9.98321	0.474853
$443$	−4.59972	−0.218539	−0.109270	−	0.994012i	$-0.534851\pi$
$443$	−4.59972	−0.218539	−0.109270	+	0.994012i	$0.534851\pi$
$444$	−1.82182	−0.0864595
$445$	−5.33471	−0.252890
$446$	8.73679	0.413699
$447$	−1.69657	−0.0802450
$448$	−19.3784	−0.915544
$449$	−37.2618	−1.75849	−0.879246	−	0.476368i	$-0.841953\pi$
$449$	−37.2618	−1.75849	−0.879246	+	0.476368i	$0.841953\pi$
$450$	7.60704	0.358599
$451$	−0.910207	−0.0428600
$452$	−21.7053	−1.02093
$453$	0.0925113	0.00434656
$454$	−1.45973	−0.0685088
$455$	−2.78653	−0.130635
$456$	−12.3713	−0.579338
$457$	−1.52950	−0.0715470	−0.0357735	−	0.999360i	$-0.511389\pi$
$457$	−1.52950	−0.0715470	−0.0357735	+	0.999360i	$0.511389\pi$
$458$	20.4658	0.956306
$459$	−7.98438	−0.372679
$460$	36.0746	1.68199
$461$	32.0399	1.49225	0.746123	−	0.665808i	$-0.231913\pi$
$461$	32.0399	1.49225	0.746123	+	0.665808i	$0.231913\pi$
$462$	12.9424	0.602136
$463$	−22.7990	−1.05956	−0.529779	−	0.848136i	$-0.677725\pi$
$463$	−22.7990	−1.05956	−0.529779	+	0.848136i	$0.677725\pi$
$464$	−78.9713	−3.66615
$465$	0.971587	0.0450562
$466$	−36.3072	−1.68190
$467$	17.3590	0.803278	0.401639	−	0.915798i	$-0.368441\pi$
$467$	17.3590	0.803278	0.401639	+	0.915798i	$0.368441\pi$
$468$	10.0826	0.466066
$469$	12.7383	0.588202
$470$	3.90417	0.180086
$471$	3.14776	0.145041
$472$	−56.1688	−2.58538
$473$	−48.4649	−2.22842
$474$	6.29969	0.289354
$475$	6.91001	0.317053
$476$	−95.4361	−4.37431
$477$	−29.7238	−1.36096
$478$	7.44581	0.340563
$479$	6.67049	0.304782	0.152391	−	0.988320i	$-0.451303\pi$
$479$	6.67049	0.304782	0.152391	+	0.988320i	$0.451303\pi$
$480$	2.22635	0.101619
$481$	1.10618	0.0504373
$482$	−6.25623	−0.284963
$483$	7.44326	0.338680
$484$	72.3429	3.28831
$485$	−15.3978	−0.699178
$486$	−17.2878	−0.784190
$487$	10.6763	0.483791	0.241896	−	0.970302i	$-0.422231\pi$
$487$	10.6763	0.483791	0.241896	+	0.970302i	$0.422231\pi$
$488$	48.4770	2.19445
$489$	−6.30314	−0.285038
$490$	19.7939	0.894197
$491$	−37.0792	−1.67336	−0.836681	−	0.547690i	$-0.815507\pi$
$491$	−37.0792	−1.67336	−0.836681	+	0.547690i	$0.815507\pi$
$492$	−0.212737	−0.00959093
$493$	−47.4105	−2.13526
$494$	13.0426	0.586816
$495$	15.0645	0.677101
$496$	−33.6547	−1.51114
$497$	2.49419	0.111880
$498$	−4.44778	−0.199310
$499$	−0.400892	−0.0179464	−0.00897320	−	0.999960i	$-0.502856\pi$
$499$	−0.400892	−0.0179464	−0.00897320	+	0.999960i	$0.502856\pi$
$500$	−4.71619	−0.210914
$501$	−5.18770	−0.231769
$502$	17.2028	0.767797
$503$	43.1490	1.92392	0.961960	−	0.273190i	$-0.0880788\pi$
$503$	43.1490	1.92392	0.961960	+	0.273190i	$0.0880788\pi$
$504$	−79.0522	−3.52126
$505$	−19.5865	−0.871586
$506$	101.736	4.52271
$507$	−3.17150	−0.140851
$508$	70.5340	3.12944
$509$	−27.1738	−1.20446	−0.602228	−	0.798324i	$-0.705720\pi$
$509$	−27.1738	−1.20446	−0.602228	+	0.798324i	$0.705720\pi$
$510$	3.48625	0.154374
$511$	43.4082	1.92027
$512$	46.9125	2.07326
$513$	−10.4313	−0.460551
$514$	0.0641808	0.00283089
$515$	−10.5543	−0.465078
$516$	−11.3274	−0.498661
$517$	7.73160	0.340035
$518$	−15.0591	−0.661659
$519$	0.506123	0.0222163
$520$	−5.12680	−0.224825
$521$	−9.59969	−0.420570	−0.210285	−	0.977640i	$-0.567439\pi$
$521$	−9.59969	−0.420570	−0.210285	+	0.977640i	$0.567439\pi$
$522$	−68.1878	−2.98450
$523$	16.8699	0.737671	0.368836	−	0.929495i	$-0.379757\pi$
$523$	16.8699	0.737671	0.368836	+	0.929495i	$0.379757\pi$
$524$	81.8602	3.57608
$525$	−0.973089	−0.0424691
$526$	−58.0073	−2.52924
$527$	−20.2046	−0.880128
$528$	11.4999	0.500469
$529$	35.5088	1.54386
$530$	26.2429	1.13992
$531$	−23.4222	−1.01644
$532$	−124.683	−5.40571
$533$	0.129171	0.00559500
$534$	−3.51630	−0.152165
$535$	−17.7731	−0.768396
$536$	23.4367	1.01231
$537$	−3.31957	−0.143250
$538$	−30.3743	−1.30953
$539$	39.1987	1.68841
$540$	7.11949	0.306374
$541$	−26.9664	−1.15938	−0.579688	−	0.814838i	$-0.696826\pi$
$541$	−26.9664	−1.15938	−0.579688	+	0.814838i	$0.696826\pi$
$542$	−52.5616	−2.25772
$543$	3.64547	0.156442
$544$	−46.2981	−1.98501
$545$	1.18283	0.0506667
$546$	−1.83670	−0.0786037
$547$	−18.8906	−0.807704	−0.403852	−	0.914824i	$-0.632329\pi$
$547$	−18.8906	−0.807704	−0.403852	+	0.914824i	$0.632329\pi$
$548$	28.2838	1.20822
$549$	20.2148	0.862745
$550$	−13.3004	−0.567129
$551$	−61.9398	−2.63872
$552$	13.6945	0.582876
$553$	36.5664	1.55496
$554$	60.3703	2.56489
$555$	0.386290	0.0163971
$556$	42.0543	1.78350
$557$	31.8915	1.35128	0.675642	−	0.737229i	$-0.263866\pi$
$557$	31.8915	1.35128	0.675642	+	0.737229i	$0.263866\pi$
$558$	−29.0592	−1.23017
$559$	6.87781	0.290900
$560$	33.7067	1.42437
$561$	6.90398	0.291486
$562$	35.8755	1.51332
$563$	27.0761	1.14112	0.570560	−	0.821256i	$-0.306726\pi$
$563$	27.0761	1.14112	0.570560	+	0.821256i	$0.306726\pi$
$564$	1.80706	0.0760909
$565$	4.60229	0.193620
$566$	−2.70940	−0.113885
$567$	−32.2220	−1.35320
$568$	4.58894	0.192547
$569$	−32.8415	−1.37679	−0.688395	−	0.725336i	$-0.741684\pi$
$569$	−32.8415	−1.37679	−0.688395	+	0.725336i	$0.741684\pi$
$570$	4.55464	0.190773
$571$	−2.15452	−0.0901637	−0.0450818	−	0.998983i	$-0.514355\pi$
$571$	−2.15452	−0.0901637	−0.0450818	+	0.998983i	$0.514355\pi$
$572$	−17.6286	−0.737090
$573$	5.72290	0.239077
$574$	−1.75848	−0.0733976
$575$	−7.64911	−0.318990
$576$	−14.8674	−0.619473
$577$	43.0317	1.79143	0.895716	−	0.444627i	$-0.146664\pi$
$577$	43.0317	1.79143	0.895716	+	0.444627i	$0.146664\pi$
$578$	−28.4419	−1.18303
$579$	5.88894	0.244736
$580$	42.2748	1.75537
$581$	−25.8170	−1.07107
$582$	−10.1493	−0.420700
$583$	51.9700	2.15238
$584$	79.8647	3.30483
$585$	−2.13786	−0.0883897
$586$	−12.5188	−0.517147
$587$	6.88506	0.284177	0.142088	−	0.989854i	$-0.454618\pi$
$587$	6.88506	0.284177	0.142088	+	0.989854i	$0.454618\pi$
$588$	9.16168	0.377821
$589$	−26.3965	−1.08765
$590$	20.6793	0.851352
$591$	4.41503	0.181610
$592$	−13.3807	−0.549942
$593$	14.3120	0.587722	0.293861	−	0.955848i	$-0.405060\pi$
$593$	14.3120	0.587722	0.293861	+	0.955848i	$0.405060\pi$
$594$	20.0780	0.823811
$595$	20.2359	0.829590
$596$	−31.4593	−1.28862
$597$	−0.264208	−0.0108133
$598$	−14.4377	−0.590401
$599$	24.6240	1.00611	0.503054	−	0.864255i	$-0.332209\pi$
$599$	24.6240	1.00611	0.503054	+	0.864255i	$0.332209\pi$
$600$	−1.79034	−0.0730903
$601$	15.8413	0.646178	0.323089	−	0.946369i	$-0.395279\pi$
$601$	15.8413	0.646178	0.323089	+	0.946369i	$0.395279\pi$
$602$	−93.6321	−3.81616
$603$	9.77302	0.397988
$604$	1.71542	0.0697996
$605$	−15.3393	−0.623630
$606$	−12.9102	−0.524439
$607$	13.3997	0.543878	0.271939	−	0.962314i	$-0.412335\pi$
$607$	13.3997	0.543878	0.271939	+	0.962314i	$0.412335\pi$
$608$	−60.4865	−2.45305
$609$	8.72254	0.353455
$610$	−17.8474	−0.722622
$611$	−1.09722	−0.0443886
$612$	−73.2198	−2.95974
$613$	−34.7497	−1.40353	−0.701764	−	0.712410i	$-0.747604\pi$
$613$	−34.7497	−1.40353	−0.701764	+	0.712410i	$0.747604\pi$
$614$	20.9713	0.846334
$615$	0.0451079	0.00181893
$616$	138.217	5.56892
$617$	−10.4991	−0.422678	−0.211339	−	0.977413i	$-0.567782\pi$
$617$	−10.4991	−0.422678	−0.211339	+	0.977413i	$0.567782\pi$
$618$	−6.95672	−0.279840
$619$	34.3749	1.38164	0.690822	−	0.723025i	$-0.257249\pi$
$619$	34.3749	1.38164	0.690822	+	0.723025i	$0.257249\pi$
$620$	18.0160	0.723540
$621$	11.5470	0.463364
$622$	−28.2944	−1.13450
$623$	−20.4103	−0.817721
$624$	−1.63199	−0.0653319
$625$	1.00000	0.0400000
$626$	27.8084	1.11145
$627$	9.01976	0.360214
$628$	58.3686	2.32916
$629$	−8.03309	−0.320300
$630$	29.1041	1.15953
$631$	45.7834	1.82261	0.911304	−	0.411735i	$-0.135077\pi$
$631$	45.7834	1.82261	0.911304	+	0.411735i	$0.135077\pi$
$632$	67.2768	2.67613
$633$	2.25257	0.0895318
$634$	51.6023	2.04939
$635$	−14.9557	−0.593500
$636$	12.1466	0.481645
$637$	−5.56282	−0.220407
$638$	119.221	4.72002
$639$	1.91357	0.0756998
$640$	−4.38064	−0.173160
$641$	−5.93168	−0.234287	−0.117144	−	0.993115i	$-0.537374\pi$
$641$	−5.93168	−0.234287	−0.117144	+	0.993115i	$0.537374\pi$
$642$	−11.7149	−0.462349
$643$	−40.0396	−1.57901	−0.789505	−	0.613745i	$-0.789662\pi$
$643$	−40.0396	−1.57901	−0.789505	+	0.613745i	$0.789662\pi$
$644$	138.019	5.43873
$645$	2.40181	0.0945713
$646$	−94.7161	−3.72656
$647$	14.4067	0.566386	0.283193	−	0.959063i	$-0.408606\pi$
$647$	14.4067	0.566386	0.283193	+	0.959063i	$0.408606\pi$
$648$	−59.2838	−2.32889
$649$	40.9520	1.60751
$650$	1.88750	0.0740338
$651$	3.71723	0.145690
$652$	−116.878	−4.57731
$653$	0.248386	0.00972010	0.00486005	−	0.999988i	$-0.498453\pi$
$653$	0.248386	0.00972010	0.00486005	+	0.999988i	$0.498453\pi$
$654$	0.779644	0.0304865
$655$	−17.3573	−0.678205
$656$	−1.56249	−0.0610049
$657$	33.3034	1.29929
$658$	14.9371	0.582310
$659$	−21.7134	−0.845836	−0.422918	−	0.906168i	$-0.638994\pi$
$659$	−21.7134	−0.845836	−0.422918	+	0.906168i	$0.638994\pi$
$660$	−6.15612	−0.239627
$661$	−12.7405	−0.495549	−0.247775	−	0.968818i	$-0.579699\pi$
$661$	−12.7405	−0.495549	−0.247775	+	0.968818i	$0.579699\pi$
$662$	48.2303	1.87452
$663$	−0.979767	−0.0380510
$664$	−47.4995	−1.84334
$665$	26.4373	1.02519
$666$	−11.5535	−0.447690
$667$	68.5648	2.65484
$668$	−96.1949	−3.72189
$669$	−0.857441	−0.0331506
$670$	−8.62851	−0.333348
$671$	−35.3441	−1.36444
$672$	8.51789	0.328585
$673$	13.9667	0.538378	0.269189	−	0.963087i	$-0.413244\pi$
$673$	13.9667	0.538378	0.269189	+	0.963087i	$0.413244\pi$
$674$	−39.5620	−1.52387
$675$	−1.50959	−0.0581040
$676$	−58.8087	−2.26187
$677$	1.91964	0.0737776	0.0368888	−	0.999319i	$-0.488255\pi$
$677$	1.91964	0.0737776	0.0368888	+	0.999319i	$0.488255\pi$
$678$	3.03354	0.116502
$679$	−58.9111	−2.26080
$680$	37.2310	1.42774
$681$	0.143261	0.00548976
$682$	50.8078	1.94553
$683$	0.657629	0.0251635	0.0125817	−	0.999921i	$-0.495995\pi$
$683$	0.657629	0.0251635	0.0125817	+	0.999921i	$0.495995\pi$
$684$	−95.6586	−3.65760
$685$	−5.99717	−0.229140
$686$	6.32416	0.241458
$687$	−2.00855	−0.0766309
$688$	−83.1962	−3.17183
$689$	−7.37524	−0.280974
$690$	−5.04181	−0.191938
$691$	−44.2104	−1.68184	−0.840922	−	0.541157i	$-0.817987\pi$
$691$	−44.2104	−1.68184	−0.840922	+	0.541157i	$0.817987\pi$
$692$	9.38497	0.356763
$693$	57.6361	2.18941
$694$	−64.7452	−2.45769
$695$	−8.91700	−0.338241
$696$	16.0482	0.608305
$697$	−0.938041	−0.0355308
$698$	−63.2226	−2.39301
$699$	3.56325	0.134774
$700$	−18.0439	−0.681994
$701$	34.5577	1.30523	0.652613	−	0.757692i	$-0.273673\pi$
$701$	34.5577	1.30523	0.652613	+	0.757692i	$0.273673\pi$
$702$	−2.84934	−0.107541
$703$	−10.4949	−0.395822
$704$	25.9945	0.979705
$705$	−0.383161	−0.0144307
$706$	46.2737	1.74153
$707$	−74.9367	−2.81828
$708$	9.57148	0.359718
$709$	5.80197	0.217898	0.108949	−	0.994047i	$-0.465252\pi$
$709$	5.80197	0.217898	0.108949	+	0.994047i	$0.465252\pi$
$710$	−1.68948	−0.0634049
$711$	28.0542	1.05211
$712$	−37.5519	−1.40732
$713$	29.2199	1.09429
$714$	13.3382	0.499170
$715$	3.73790	0.139789
$716$	−61.5544	−2.30040
$717$	−0.730743	−0.0272901
$718$	−38.0635	−1.42052
$719$	−3.10161	−0.115671	−0.0578353	−	0.998326i	$-0.518420\pi$
$719$	−3.10161	−0.115671	−0.0578353	+	0.998326i	$0.518420\pi$
$720$	25.8602	0.963754
$721$	−40.3801	−1.50383
$722$	−74.5030	−2.77271
$723$	0.613996	0.0228347
$724$	67.5976	2.51224
$725$	−8.96377	−0.332906
$726$	−10.1107	−0.375242
$727$	11.1507	0.413557	0.206779	−	0.978388i	$-0.433702\pi$
$727$	11.1507	0.413557	0.206779	+	0.978388i	$0.433702\pi$
$728$	−19.6148	−0.726974
$729$	−23.5693	−0.872937
$730$	−29.4032	−1.08826
$731$	−49.9469	−1.84735
$732$	−8.26075	−0.305326
$733$	9.87194	0.364628	0.182314	−	0.983240i	$-0.441641\pi$
$733$	9.87194	0.364628	0.182314	+	0.983240i	$0.441641\pi$
$734$	−44.0559	−1.62613
$735$	−1.94260	−0.0716540
$736$	66.9561	2.46803
$737$	−17.0874	−0.629423
$738$	−1.34913	−0.0496622
$739$	37.2437	1.37003	0.685016	−	0.728528i	$-0.259795\pi$
$739$	37.2437	1.37003	0.685016	+	0.728528i	$0.259795\pi$
$740$	7.16292	0.263314
$741$	−1.28002	−0.0470229
$742$	100.404	3.68594
$743$	−21.6686	−0.794946	−0.397473	−	0.917614i	$-0.630113\pi$
$743$	−21.6686	−0.794946	−0.397473	+	0.917614i	$0.630113\pi$
$744$	6.83916	0.250736
$745$	6.67049	0.244388
$746$	84.0750	3.07820
$747$	−19.8071	−0.724705
$748$	128.020	4.68086
$749$	−67.9987	−2.48462
$750$	0.659137	0.0240683
$751$	−1.13693	−0.0414873	−0.0207437	−	0.999785i	$-0.506603\pi$
$751$	−1.13693	−0.0414873	−0.0207437	+	0.999785i	$0.506603\pi$
$752$	13.2723	0.483990
$753$	−1.68830	−0.0615252
$754$	−16.9191	−0.616158
$755$	−0.363731	−0.0132375
$756$	27.2387	0.990663
$757$	−24.2820	−0.882543	−0.441272	−	0.897374i	$-0.645473\pi$
$757$	−24.2820	−0.882543	−0.441272	+	0.897374i	$0.645473\pi$
$758$	−11.3418	−0.411952
$759$	−9.98451	−0.362415
$760$	48.6407	1.76438
$761$	15.0434	0.545322	0.272661	−	0.962110i	$-0.412096\pi$
$761$	15.0434	0.545322	0.272661	+	0.962110i	$0.412096\pi$
$762$	−9.85787	−0.357113
$763$	4.52542	0.163831
$764$	106.119	3.83925
$765$	15.5252	0.561315
$766$	−59.4181	−2.14686
$767$	−5.81164	−0.209846
$768$	−5.46390	−0.197162
$769$	−4.83382	−0.174312	−0.0871561	−	0.996195i	$-0.527778\pi$
$769$	−4.83382	−0.174312	−0.0871561	+	0.996195i	$0.527778\pi$
$770$	−50.8864	−1.83382
$771$	−0.00629880	−0.000226846 0
$772$	109.198	3.93012
$773$	−25.6721	−0.923362	−0.461681	−	0.887046i	$-0.652754\pi$
$773$	−25.6721	−0.923362	−0.461681	+	0.887046i	$0.652754\pi$
$774$	−71.8357	−2.58208
$775$	−3.82004	−0.137220
$776$	−108.388	−3.89089
$777$	1.47792	0.0530202
$778$	34.3212	1.23048
$779$	−1.22551	−0.0439085
$780$	0.873636	0.0312812
$781$	−3.34574	−0.119720
$782$	104.847	3.74932
$783$	13.5316	0.483579
$784$	67.2897	2.40320
$785$	−12.3762	−0.441726
$786$	−11.4408	−0.408081
$787$	13.9958	0.498897	0.249449	−	0.968388i	$-0.419751\pi$
$787$	13.9958	0.498897	0.249449	+	0.968388i	$0.419751\pi$
$788$	81.8673	2.91640
$789$	5.69292	0.202673
$790$	−24.7688	−0.881235
$791$	17.6081	0.626072
$792$	106.042	3.76803
$793$	5.01579	0.178116
$794$	65.4862	2.32402
$795$	−2.57552	−0.0913443
$796$	−4.89918	−0.173647
$797$	11.6787	0.413683	0.206841	−	0.978375i	$-0.433682\pi$
$797$	11.6787	0.413683	0.206841	+	0.978375i	$0.433682\pi$
$798$	17.4258	0.616867
$799$	7.96803	0.281888
$800$	−8.75345	−0.309481
$801$	−15.6590	−0.553285
$802$	−2.59156	−0.0915112
$803$	−58.2285	−2.05484
$804$	−3.99374	−0.140848
$805$	−29.2650	−1.03146
$806$	−7.21031	−0.253972
$807$	2.98098	0.104935
$808$	−137.872	−4.85033
$809$	−48.2926	−1.69788	−0.848939	−	0.528490i	$-0.822758\pi$
$809$	−48.2926	−1.69788	−0.848939	+	0.528490i	$0.822758\pi$
$810$	21.8261	0.766891
$811$	−12.8436	−0.451001	−0.225500	−	0.974243i	$-0.572402\pi$
$811$	−12.8436	−0.451001	−0.225500	+	0.974243i	$0.572402\pi$
$812$	161.741	5.67600
$813$	5.15848	0.180916
$814$	20.2005	0.708028
$815$	24.7824	0.868088
$816$	11.8516	0.414888
$817$	−65.2535	−2.28293
$818$	−73.2345	−2.56059
$819$	−8.17933	−0.285809
$820$	0.836430	0.0292094
$821$	24.1930	0.844341	0.422171	−	0.906516i	$-0.361268\pi$
$821$	24.1930	0.844341	0.422171	+	0.906516i	$0.361268\pi$
$822$	−3.95296	−0.137875
$823$	24.1701	0.842515	0.421258	−	0.906941i	$-0.361589\pi$
$823$	24.1701	0.842515	0.421258	+	0.906941i	$0.361589\pi$
$824$	−74.2934	−2.58813
$825$	1.30532	0.0454453
$826$	79.1177	2.75286
$827$	25.1170	0.873402	0.436701	−	0.899607i	$-0.356147\pi$
$827$	25.1170	0.873402	0.436701	+	0.899607i	$0.356147\pi$
$828$	105.890	3.67994
$829$	−9.04450	−0.314128	−0.157064	−	0.987588i	$-0.550203\pi$
$829$	−9.04450	−0.314128	−0.157064	+	0.987588i	$0.550203\pi$
$830$	17.4876	0.607002
$831$	−5.92484	−0.205530
$832$	−3.68897	−0.127892
$833$	40.3974	1.39969
$834$	−5.87752	−0.203522
$835$	20.3967	0.705858
$836$	167.252	5.78454
$837$	5.76667	0.199325
$838$	26.8366	0.927053
$839$	7.58607	0.261900	0.130950	−	0.991389i	$-0.458197\pi$
$839$	7.58607	0.261900	0.130950	+	0.991389i	$0.458197\pi$
$840$	−6.84973	−0.236338
$841$	51.3492	1.77066
$842$	7.31444	0.252072
$843$	−3.52088	−0.121265
$844$	41.7692	1.43775
$845$	12.4695	0.428965
$846$	11.4600	0.394001
$847$	−58.6872	−2.01651
$848$	89.2132	3.06359
$849$	0.265904	0.00912582
$850$	−13.7071	−0.470149
$851$	11.6174	0.398240
$852$	−0.781981	−0.0267902
$853$	31.9368	1.09350	0.546748	−	0.837297i	$-0.315866\pi$
$853$	31.9368	1.09350	0.546748	+	0.837297i	$0.315866\pi$
$854$	−68.2833	−2.33660
$855$	20.2830	0.693665
$856$	−125.107	−4.27609
$857$	−47.1526	−1.61070	−0.805350	−	0.592799i	$-0.798023\pi$
$857$	−47.1526	−1.61070	−0.805350	+	0.592799i	$0.798023\pi$
$858$	2.46378	0.0841122
$859$	14.9623	0.510507	0.255254	−	0.966874i	$-0.417841\pi$
$859$	14.9623	0.510507	0.255254	+	0.966874i	$0.417841\pi$
$860$	44.5365	1.51868
$861$	0.172580	0.00588151
$862$	29.3315	0.999035
$863$	−19.6085	−0.667483	−0.333741	−	0.942665i	$-0.608311\pi$
$863$	−19.6085	−0.667483	−0.333741	+	0.942665i	$0.608311\pi$
$864$	13.2141	0.449552
$865$	−1.98995	−0.0676603
$866$	−91.3528	−3.10429
$867$	2.79133	0.0947984
$868$	68.9282	2.33958
$869$	−49.0508	−1.66393
$870$	−5.90835	−0.200312
$871$	2.42493	0.0821657
$872$	8.32611	0.281958
$873$	−45.1973	−1.52970
$874$	136.978	4.63335
$875$	3.82594	0.129340
$876$	−13.6094	−0.459819
$877$	−24.1671	−0.816065	−0.408032	−	0.912967i	$-0.633785\pi$
$877$	−24.1671	−0.816065	−0.408032	+	0.912967i	$0.633785\pi$
$878$	−40.5140	−1.36728
$879$	1.22861	0.0414401
$880$	−45.2148	−1.52419
$881$	11.1029	0.374066	0.187033	−	0.982354i	$-0.440113\pi$
$881$	11.1029	0.374066	0.187033	+	0.982354i	$0.440113\pi$
$882$	58.1012	1.95637
$883$	23.2877	0.783695	0.391847	−	0.920030i	$-0.371836\pi$
$883$	23.2877	0.783695	0.391847	+	0.920030i	$0.371836\pi$
$884$	−18.1677	−0.611046
$885$	−2.02949	−0.0682207
$886$	11.9205	0.400476
$887$	26.7213	0.897214	0.448607	−	0.893729i	$-0.351920\pi$
$887$	26.7213	0.897214	0.448607	+	0.893729i	$0.351920\pi$
$888$	2.71916	0.0912490
$889$	−57.2197	−1.91909
$890$	13.8252	0.463423
$891$	43.2232	1.44803
$892$	−15.8994	−0.532352
$893$	10.4099	0.348354
$894$	4.39677	0.147050
$895$	13.0517	0.436271
$896$	−16.7601	−0.559915
$897$	1.41694	0.0473101
$898$	96.5662	3.22246
$899$	34.2419	1.14203
$900$	−13.8435	−0.461449
$901$	53.5592	1.78432
$902$	2.35886	0.0785413
$903$	9.18919	0.305797
$904$	32.3963	1.07749
$905$	−14.3331	−0.476448
$906$	−0.239749	−0.00796511
$907$	−19.8344	−0.658591	−0.329295	−	0.944227i	$-0.606811\pi$
$907$	−19.8344	−0.658591	−0.329295	+	0.944227i	$0.606811\pi$
$908$	2.65646	0.0881578
$909$	−57.4924	−1.90690
$910$	7.22146	0.239389
$911$	−21.4088	−0.709306	−0.354653	−	0.934998i	$-0.615401\pi$
$911$	−21.4088	−0.709306	−0.354653	+	0.934998i	$0.615401\pi$
$912$	15.4836	0.512713
$913$	34.6314	1.14613
$914$	3.96380	0.131111
$915$	1.75157	0.0579053
$916$	−37.2442	−1.23058
$917$	−66.4080	−2.19298
$918$	20.6920	0.682938
$919$	−10.2061	−0.336667	−0.168334	−	0.985730i	$-0.553839\pi$
$919$	−10.2061	−0.336667	−0.168334	+	0.985730i	$0.553839\pi$
$920$	−53.8433	−1.77516
$921$	−2.05816	−0.0678186
$922$	−83.0333	−2.73456
$923$	0.474806	0.0156284
$924$	−23.5530	−0.774835
$925$	−1.51880	−0.0499377
$926$	59.0849	1.94165
$927$	−30.9801	−1.01752
$928$	78.4640	2.57571
$929$	−11.6334	−0.381680	−0.190840	−	0.981621i	$-0.561121\pi$
$929$	−11.6334	−0.381680	−0.190840	+	0.981621i	$0.561121\pi$
$930$	−2.51793	−0.0825660
$931$	52.7775	1.72971
$932$	66.0728	2.16429
$933$	2.77686	0.0909102
$934$	−44.9868	−1.47201
$935$	−27.1447	−0.887727
$936$	−15.0488	−0.491884
$937$	9.97379	0.325830	0.162915	−	0.986640i	$-0.447910\pi$
$937$	9.97379	0.325830	0.162915	+	0.986640i	$0.447910\pi$
$938$	−33.0122	−1.07789
$939$	−2.72915	−0.0890626
$940$	−7.10491	−0.231736
$941$	−37.2635	−1.21476	−0.607378	−	0.794413i	$-0.707779\pi$
$941$	−37.2635	−1.21476	−0.607378	+	0.794413i	$0.707779\pi$
$942$	−8.15762	−0.265790
$943$	1.35659	0.0441767
$944$	70.2995	2.28805
$945$	−5.77558	−0.187880
$946$	125.600	4.08360
$947$	−21.1714	−0.687979	−0.343990	−	0.938973i	$-0.611779\pi$
$947$	−21.1714	−0.687979	−0.343990	+	0.938973i	$0.611779\pi$
$948$	−11.4643	−0.372344
$949$	8.26341	0.268242
$950$	−17.9077	−0.581003
$951$	−5.06433	−0.164222
$952$	142.444	4.61662
$953$	6.65141	0.215460	0.107730	−	0.994180i	$-0.465642\pi$
$953$	6.65141	0.215460	0.107730	+	0.994180i	$0.465642\pi$
$954$	77.0311	2.49398
$955$	−22.5010	−0.728115
$956$	−13.5501	−0.438241
$957$	−11.7006	−0.378225
$958$	−17.2870	−0.558517
$959$	−22.9448	−0.740927
$960$	−1.28823	−0.0415775
$961$	−16.4073	−0.529269
$962$	−2.86672	−0.0924269
$963$	−52.1695	−1.68114
$964$	11.3852	0.366694
$965$	−23.1538	−0.745348
$966$	−19.2897	−0.620635
$967$	47.0298	1.51238	0.756188	−	0.654355i	$-0.227060\pi$
$967$	47.0298	1.51238	0.756188	+	0.654355i	$0.227060\pi$
$968$	−107.976	−3.47047
$969$	9.29558	0.298617
$970$	39.9043	1.28125
$971$	10.0580	0.322777	0.161388	−	0.986891i	$-0.448403\pi$
$971$	10.0580	0.322777	0.161388	+	0.986891i	$0.448403\pi$
$972$	31.4607	1.00910
$973$	−34.1159	−1.09371
$974$	−27.6684	−0.886552
$975$	−0.185242	−0.00593249
$976$	−60.6726	−1.94208
$977$	−26.5870	−0.850594	−0.425297	−	0.905054i	$-0.639830\pi$
$977$	−26.5870	−0.850594	−0.425297	+	0.905054i	$0.639830\pi$
$978$	16.3350	0.522335
$979$	27.3787	0.875027
$980$	−36.0214	−1.15066
$981$	3.47196	0.110851
$982$	96.0931	3.06645
$983$	18.8380	0.600840	0.300420	−	0.953807i	$-0.402873\pi$
$983$	18.8380	0.600840	0.300420	+	0.953807i	$0.402873\pi$
$984$	0.317522	0.0101222
$985$	−17.3588	−0.553097
$986$	122.867	3.91289
$987$	−1.46595	−0.0466618
$988$	−23.7353	−0.755122
$989$	72.2330	2.29688
$990$	−39.0407	−1.24079
$991$	5.49985	0.174709	0.0873543	−	0.996177i	$-0.472159\pi$
$991$	5.49985	0.174709	0.0873543	+	0.996177i	$0.472159\pi$
$992$	33.4385	1.06167
$993$	−4.73339	−0.150210
$994$	−6.46384	−0.205021
$995$	1.03880	0.0329322
$996$	8.09418	0.256474
$997$	30.0012	0.950147	0.475073	−	0.879946i	$-0.342421\pi$
$997$	30.0012	0.950147	0.475073	+	0.879946i	$0.342421\pi$
$998$	1.03894	0.0328870
$999$	2.29275	0.0725394

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2005.2.a.f.1.2	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2005.2.a.f.1.2	✓	37	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)

\(f(q)\)	\(=\)	\(q-2.59156 q^{2} +0.254340 q^{3} +4.71619 q^{4} -1.00000 q^{5} -0.659137 q^{6} -3.82594 q^{7} -7.03917 q^{8} -2.93531 q^{9} +O(q^{10})\)	\(q-2.59156 q^{2} +0.254340 q^{3} +4.71619 q^{4} -1.00000 q^{5} -0.659137 q^{6} -3.82594 q^{7} -7.03917 q^{8} -2.93531 q^{9} +2.59156 q^{10} +5.13218 q^{11} +1.19951 q^{12} -0.728325 q^{13} +9.91516 q^{14} -0.254340 q^{15} +8.81005 q^{16} +5.28912 q^{17} +7.60704 q^{18} +6.91001 q^{19} -4.71619 q^{20} -0.973089 q^{21} -13.3004 q^{22} -7.64911 q^{23} -1.79034 q^{24} +1.00000 q^{25} +1.88750 q^{26} -1.50959 q^{27} -18.0439 q^{28} -8.96377 q^{29} +0.659137 q^{30} -3.82004 q^{31} -8.75345 q^{32} +1.30532 q^{33} -13.7071 q^{34} +3.82594 q^{35} -13.8435 q^{36} -1.51880 q^{37} -17.9077 q^{38} -0.185242 q^{39} +7.03917 q^{40} -0.177353 q^{41} +2.52182 q^{42} -9.44333 q^{43} +24.2043 q^{44} +2.93531 q^{45} +19.8231 q^{46} +1.50649 q^{47} +2.24075 q^{48} +7.63783 q^{49} -2.59156 q^{50} +1.34523 q^{51} -3.43492 q^{52} +10.1263 q^{53} +3.91218 q^{54} -5.13218 q^{55} +26.9314 q^{56} +1.75749 q^{57} +23.2302 q^{58} +7.97946 q^{59} -1.19951 q^{60} -6.88675 q^{61} +9.89986 q^{62} +11.2303 q^{63} +5.06500 q^{64} +0.728325 q^{65} -3.38281 q^{66} -3.32946 q^{67} +24.9445 q^{68} -1.94547 q^{69} -9.91516 q^{70} -0.651915 q^{71} +20.6621 q^{72} -11.3458 q^{73} +3.93605 q^{74} +0.254340 q^{75} +32.5889 q^{76} -19.6354 q^{77} +0.480066 q^{78} -9.55749 q^{79} -8.81005 q^{80} +8.42199 q^{81} +0.459621 q^{82} +6.74788 q^{83} -4.58927 q^{84} -5.28912 q^{85} +24.4730 q^{86} -2.27984 q^{87} -36.1263 q^{88} +5.33471 q^{89} -7.60704 q^{90} +2.78653 q^{91} -36.0746 q^{92} -0.971587 q^{93} -3.90417 q^{94} -6.91001 q^{95} -2.22635 q^{96} +15.3978 q^{97} -19.7939 q^{98} -15.0645 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 7 q^{2} + 3 q^{3} + 43 q^{4} - 37 q^{5} + 8 q^{6} - 16 q^{7} + 21 q^{8} + 54 q^{9}+O(q^{10}) \) 37 * q + 7 * q^2 + 3 * q^3 + 43 * q^4 - 37 * q^5 + 8 * q^6 - 16 * q^7 + 21 * q^8 + 54 * q^9	\( 37 q + 7 q^{2} + 3 q^{3} + 43 q^{4} - 37 q^{5} + 8 q^{6} - 16 q^{7} + 21 q^{8} + 54 q^{9} - 7 q^{10} + 42 q^{11} - 13 q^{13} + 14 q^{14} - 3 q^{15} + 63 q^{16} + 18 q^{17} + 22 q^{18} + 22 q^{19} - 43 q^{20} + 16 q^{21} - 10 q^{22} + 23 q^{23} + 23 q^{24} + 37 q^{25} + 21 q^{26} + 3 q^{27} - 18 q^{28} + 33 q^{29} - 8 q^{30} + 11 q^{31} + 54 q^{32} + 2 q^{33} + 8 q^{34} + 16 q^{35} + 91 q^{36} - 11 q^{37} + 29 q^{38} + 25 q^{39} - 21 q^{40} + 24 q^{41} + 4 q^{42} + 25 q^{43} + 84 q^{44} - 54 q^{45} + 31 q^{46} + 7 q^{47} + 4 q^{48} + 45 q^{49} + 7 q^{50} + 94 q^{51} - 43 q^{52} + 49 q^{53} + 38 q^{54} - 42 q^{55} + 46 q^{56} + 6 q^{57} + 15 q^{58} + 69 q^{59} + 9 q^{61} + 17 q^{62} - 38 q^{63} + 107 q^{64} + 13 q^{65} + 74 q^{66} + 13 q^{67} + 86 q^{68} - 14 q^{70} + 51 q^{71} + 81 q^{72} - 47 q^{73} + 79 q^{74} + 3 q^{75} + 59 q^{76} + 2 q^{77} + 20 q^{78} + 67 q^{79} - 63 q^{80} + 125 q^{81} - 24 q^{82} + 80 q^{83} + 50 q^{84} - 18 q^{85} + 69 q^{86} - 32 q^{87} - 12 q^{88} + 34 q^{89} - 22 q^{90} + 39 q^{91} + 85 q^{92} + q^{93} + 12 q^{94} - 22 q^{95} + 77 q^{96} - 14 q^{97} + 40 q^{98} + 133 q^{99}+O(q^{100}) \) 37 * q + 7 * q^2 + 3 * q^3 + 43 * q^4 - 37 * q^5 + 8 * q^6 - 16 * q^7 + 21 * q^8 + 54 * q^9 - 7 * q^10 + 42 * q^11 - 13 * q^13 + 14 * q^14 - 3 * q^15 + 63 * q^16 + 18 * q^17 + 22 * q^18 + 22 * q^19 - 43 * q^20 + 16 * q^21 - 10 * q^22 + 23 * q^23 + 23 * q^24 + 37 * q^25 + 21 * q^26 + 3 * q^27 - 18 * q^28 + 33 * q^29 - 8 * q^30 + 11 * q^31 + 54 * q^32 + 2 * q^33 + 8 * q^34 + 16 * q^35 + 91 * q^36 - 11 * q^37 + 29 * q^38 + 25 * q^39 - 21 * q^40 + 24 * q^41 + 4 * q^42 + 25 * q^43 + 84 * q^44 - 54 * q^45 + 31 * q^46 + 7 * q^47 + 4 * q^48 + 45 * q^49 + 7 * q^50 + 94 * q^51 - 43 * q^52 + 49 * q^53 + 38 * q^54 - 42 * q^55 + 46 * q^56 + 6 * q^57 + 15 * q^58 + 69 * q^59 + 9 * q^61 + 17 * q^62 - 38 * q^63 + 107 * q^64 + 13 * q^65 + 74 * q^66 + 13 * q^67 + 86 * q^68 - 14 * q^70 + 51 * q^71 + 81 * q^72 - 47 * q^73 + 79 * q^74 + 3 * q^75 + 59 * q^76 + 2 * q^77 + 20 * q^78 + 67 * q^79 - 63 * q^80 + 125 * q^81 - 24 * q^82 + 80 * q^83 + 50 * q^84 - 18 * q^85 + 69 * q^86 - 32 * q^87 - 12 * q^88 + 34 * q^89 - 22 * q^90 + 39 * q^91 + 85 * q^92 + q^93 + 12 * q^94 - 22 * q^95 + 77 * q^96 - 14 * q^97 + 40 * q^98 + 133 * q^99

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