LMFDB - Embedded newform 625.8.a.a.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [625,8,Mod(1,625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(625, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("625.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 625 = 5^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	625.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:	$195.240640928$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.79880 q^{2} +0.219481 q^{3} -31.9834 q^{4} +2.15065 q^{6} +700.879 q^{7} -1567.65 q^{8} -2186.95 q^{9} +O(q^{10})$	\(q+9.79880 q^{2} +0.219481 q^{3} -31.9834 q^{4} +2.15065 q^{6} +700.879 q^{7} -1567.65 q^{8} -2186.95 q^{9} -3670.59 q^{11} -7.01975 q^{12} -6564.10 q^{13} +6867.78 q^{14} -11267.2 q^{16} +39231.7 q^{17} -21429.5 q^{18} +56579.9 q^{19} +153.830 q^{21} -35967.4 q^{22} +49912.5 q^{23} -344.068 q^{24} -64320.3 q^{26} -959.998 q^{27} -22416.5 q^{28} -15515.9 q^{29} -170888. q^{31} +90253.8 q^{32} -805.624 q^{33} +384424. q^{34} +69946.2 q^{36} +321714. q^{37} +554415. q^{38} -1440.69 q^{39} +91170.6 q^{41} +1507.35 q^{42} +746642. q^{43} +117398. q^{44} +489083. q^{46} -999896. q^{47} -2472.93 q^{48} -332311. q^{49} +8610.61 q^{51} +209942. q^{52} -451725. q^{53} -9406.84 q^{54} -1.09873e6 q^{56} +12418.2 q^{57} -152037. q^{58} -1.51441e6 q^{59} -2.59978e6 q^{61} -1.67450e6 q^{62} -1.53279e6 q^{63} +2.32658e6 q^{64} -7894.15 q^{66} +89193.3 q^{67} -1.25477e6 q^{68} +10954.8 q^{69} -3.63140e6 q^{71} +3.42837e6 q^{72} -1.19469e6 q^{73} +3.15241e6 q^{74} -1.80962e6 q^{76} -2.57264e6 q^{77} -14117.1 q^{78} -2.69150e6 q^{79} +4.78265e6 q^{81} +893363. q^{82} +3.12842e6 q^{83} -4920.00 q^{84} +7.31620e6 q^{86} -3405.44 q^{87} +5.75419e6 q^{88} -1.05500e7 q^{89} -4.60064e6 q^{91} -1.59637e6 q^{92} -37506.6 q^{93} -9.79779e6 q^{94} +19809.0 q^{96} +2.81105e6 q^{97} -3.25625e6 q^{98} +8.02740e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 23 q^{2} - 121 q^{3} + 1061 q^{4} - 431 q^{6} - 843 q^{7} - 4980 q^{8} + 14799 q^{9}+O(q^{10}) $ 22 * q - 23 * q^2 - 121 * q^3 + 1061 * q^4 - 431 * q^6 - 843 * q^7 - 4980 * q^8 + 14799 * q^9	\( 22 q - 23 q^{2} - 121 q^{3} + 1061 q^{4} - 431 q^{6} - 843 q^{7} - 4980 q^{8} + 14799 q^{9} - 781 q^{11} - 12608 q^{12} - 14686 q^{13} + 20762 q^{14} + 16117 q^{16} - 45648 q^{17} - 47171 q^{18} + 6185 q^{19} + 14149 q^{21} + 71124 q^{22} - 126921 q^{23} + 271570 q^{24} + 304129 q^{26} - 546520 q^{27} - 2019 q^{28} + 59330 q^{29} + 394804 q^{31} + 74397 q^{32} - 49067 q^{33} - 286938 q^{34} - 287278 q^{36} + 792122 q^{37} - 1338860 q^{38} + 635223 q^{39} - 160466 q^{41} - 3420191 q^{42} - 1527256 q^{43} - 1154853 q^{44} + 2653604 q^{46} - 1300863 q^{47} - 1885241 q^{48} + 1652981 q^{49} + 3408539 q^{51} - 1423303 q^{52} - 755656 q^{53} + 3117755 q^{54} - 2132625 q^{56} - 3026890 q^{57} + 5941470 q^{58} - 1548370 q^{59} - 6029951 q^{61} - 79936 q^{62} + 6962459 q^{63} - 5858224 q^{64} - 5380407 q^{66} - 7608838 q^{67} - 10737124 q^{68} + 13519553 q^{69} + 9483549 q^{71} - 6806340 q^{72} - 13548801 q^{73} - 15016023 q^{74} + 19635315 q^{76} + 2145019 q^{77} - 17222402 q^{78} + 10769160 q^{79} + 2757382 q^{81} - 12087571 q^{82} + 9632744 q^{83} + 19168542 q^{84} - 8511651 q^{86} - 298330 q^{87} + 21641425 q^{88} - 10850545 q^{89} - 6648131 q^{91} - 52978503 q^{92} - 51294822 q^{93} + 5863777 q^{94} + 2611654 q^{96} - 11579993 q^{97} - 7468074 q^{98} - 13552997 q^{99}+O(q^{100}) \) 22 * q - 23 * q^2 - 121 * q^3 + 1061 * q^4 - 431 * q^6 - 843 * q^7 - 4980 * q^8 + 14799 * q^9 - 781 * q^11 - 12608 * q^12 - 14686 * q^13 + 20762 * q^14 + 16117 * q^16 - 45648 * q^17 - 47171 * q^18 + 6185 * q^19 + 14149 * q^21 + 71124 * q^22 - 126921 * q^23 + 271570 * q^24 + 304129 * q^26 - 546520 * q^27 - 2019 * q^28 + 59330 * q^29 + 394804 * q^31 + 74397 * q^32 - 49067 * q^33 - 286938 * q^34 - 287278 * q^36 + 792122 * q^37 - 1338860 * q^38 + 635223 * q^39 - 160466 * q^41 - 3420191 * q^42 - 1527256 * q^43 - 1154853 * q^44 + 2653604 * q^46 - 1300863 * q^47 - 1885241 * q^48 + 1652981 * q^49 + 3408539 * q^51 - 1423303 * q^52 - 755656 * q^53 + 3117755 * q^54 - 2132625 * q^56 - 3026890 * q^57 + 5941470 * q^58 - 1548370 * q^59 - 6029951 * q^61 - 79936 * q^62 + 6962459 * q^63 - 5858224 * q^64 - 5380407 * q^66 - 7608838 * q^67 - 10737124 * q^68 + 13519553 * q^69 + 9483549 * q^71 - 6806340 * q^72 - 13548801 * q^73 - 15016023 * q^74 + 19635315 * q^76 + 2145019 * q^77 - 17222402 * q^78 + 10769160 * q^79 + 2757382 * q^81 - 12087571 * q^82 + 9632744 * q^83 + 19168542 * q^84 - 8511651 * q^86 - 298330 * q^87 + 21641425 * q^88 - 10850545 * q^89 - 6648131 * q^91 - 52978503 * q^92 - 51294822 * q^93 + 5863777 * q^94 + 2611654 * q^96 - 11579993 * q^97 - 7468074 * q^98 - 13552997 * 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 $

$2$	9.79880	0.866100	0.433050	−	0.901370i	$-0.357437\pi$
$2$	9.79880	0.866100	0.433050	+	0.901370i	$0.357437\pi$
$3$	0.219481	0.00469323	0.00234662	−	0.999997i	$-0.499253\pi$
$3$	0.219481	0.00469323	0.00234662	+	0.999997i	$0.499253\pi$
$4$	−31.9834	−0.249871
$5$	0	0
$5$	0	0
$6$	2.15065	0.00406481
$7$	700.879	0.772325	0.386163	−	0.922431i	$-0.373800\pi$
$7$	700.879	0.772325	0.386163	+	0.922431i	$0.373800\pi$
$8$	−1567.65	−1.08251
$9$	−2186.95	−0.999978
$10$	0	0
$11$	−3670.59	−0.831498	−0.415749	−	0.909479i	$-0.636481\pi$
$11$	−3670.59	−0.831498	−0.415749	+	0.909479i	$0.636481\pi$
$12$	−7.01975	−0.00117270
$13$	−6564.10	−0.828654	−0.414327	−	0.910128i	$-0.635983\pi$
$13$	−6564.10	−0.828654	−0.414327	+	0.910128i	$0.635983\pi$
$14$	6867.78	0.668911
$15$	0	0
$16$	−11267.2	−0.687694
$17$	39231.7	1.93672	0.968358	−	0.249563i	$-0.0802871\pi$
$17$	39231.7	1.93672	0.968358	+	0.249563i	$0.0802871\pi$
$18$	−21429.5	−0.866081
$19$	56579.9	1.89245	0.946225	−	0.323511i	$-0.104863\pi$
$19$	56579.9	1.89245	0.946225	+	0.323511i	$0.104863\pi$
$20$	0	0
$21$	153.830	0.00362470
$22$	−35967.4	−0.720161
$23$	49912.5	0.855385	0.427693	−	0.903924i	$-0.359327\pi$
$23$	49912.5	0.855385	0.427693	+	0.903924i	$0.359327\pi$
$24$	−344.068	−0.00508049
$25$	0	0
$26$	−64320.3	−0.717697
$27$	−959.998	−0.00938636
$28$	−22416.5	−0.192981
$29$	−15515.9	−0.118136	−0.0590682	−	0.998254i	$-0.518813\pi$
$29$	−15515.9	−0.118136	−0.0590682	+	0.998254i	$0.518813\pi$
$30$	0	0
$31$	−170888.	−1.03026	−0.515128	−	0.857113i	$-0.672256\pi$
$31$	−170888.	−1.03026	−0.515128	+	0.857113i	$0.672256\pi$
$32$	90253.8	0.486901
$33$	−805.624	−0.00390242
$34$	384424.	1.67739
$35$	0	0
$36$	69946.2	0.249865
$37$	321714.	1.04415	0.522076	−	0.852899i	$-0.325158\pi$
$37$	321714.	1.04415	0.522076	+	0.852899i	$0.325158\pi$
$38$	554415.	1.63905
$39$	−1440.69	−0.00388907
$40$	0	0
$41$	91170.6	0.206591	0.103295	−	0.994651i	$-0.467061\pi$
$41$	91170.6	0.206591	0.103295	+	0.994651i	$0.467061\pi$
$42$	1507.35	0.00313936
$43$	746642.	1.43210	0.716050	−	0.698050i	$-0.245948\pi$
$43$	746642.	1.43210	0.716050	+	0.698050i	$0.245948\pi$
$44$	117398.	0.207767
$45$	0	0
$46$	489083.	0.740849
$47$	−999896.	−1.40479	−0.702397	−	0.711786i	$-0.747887\pi$
$47$	−999896.	−1.40479	−0.702397	+	0.711786i	$0.747887\pi$
$48$	−2472.93	−0.00322751
$49$	−332311.	−0.403514
$50$	0	0
$51$	8610.61	0.00908946
$52$	209942.	0.207056
$53$	−451725.	−0.416782	−0.208391	−	0.978046i	$-0.566823\pi$
$53$	−451725.	−0.416782	−0.208391	+	0.978046i	$0.566823\pi$
$54$	−9406.84	−0.00812953
$55$	0	0
$56$	−1.09873e6	−0.836052
$57$	12418.2	0.00888171
$58$	−152037.	−0.102318
$59$	−1.51441e6	−0.959981	−0.479990	−	0.877274i	$-0.659360\pi$
$59$	−1.51441e6	−0.959981	−0.479990	+	0.877274i	$0.659360\pi$
$60$	0	0
$61$	−2.59978e6	−1.46650	−0.733249	−	0.679960i	$-0.761997\pi$
$61$	−2.59978e6	−1.46650	−0.733249	+	0.679960i	$0.761997\pi$
$62$	−1.67450e6	−0.892305
$63$	−1.53279e6	−0.772308
$64$	2.32658e6	1.10940
$65$	0	0
$66$	−7894.15	−0.00337988
$67$	89193.3	0.0362302	0.0181151	−	0.999836i	$-0.494233\pi$
$67$	89193.3	0.0362302	0.0181151	+	0.999836i	$0.494233\pi$
$68$	−1.25477e6	−0.483929
$69$	10954.8	0.00401452
$70$	0	0
$71$	−3.63140e6	−1.20412	−0.602060	−	0.798450i	$-0.705653\pi$
$71$	−3.63140e6	−1.20412	−0.602060	+	0.798450i	$0.705653\pi$
$72$	3.42837e6	1.08249
$73$	−1.19469e6	−0.359440	−0.179720	−	0.983718i	$-0.557519\pi$
$73$	−1.19469e6	−0.359440	−0.179720	+	0.983718i	$0.557519\pi$
$74$	3.15241e6	0.904339
$75$	0	0
$76$	−1.80962e6	−0.472867
$77$	−2.57264e6	−0.642187
$78$	−14117.1	−0.00336832
$79$	−2.69150e6	−0.614186	−0.307093	−	0.951679i	$-0.599356\pi$
$79$	−2.69150e6	−0.614186	−0.307093	+	0.951679i	$0.599356\pi$
$80$	0	0
$81$	4.78265e6	0.999934
$82$	893363.	0.178928
$83$	3.12842e6	0.600554	0.300277	−	0.953852i	$-0.402921\pi$
$83$	3.12842e6	0.600554	0.300277	+	0.953852i	$0.402921\pi$
$84$	−4920.00	−0.000905707 0
$85$	0	0
$86$	7.31620e6	1.24034
$87$	−3405.44	−0.000554442 0
$88$	5.75419e6	0.900108
$89$	−1.05500e7	−1.58631	−0.793155	−	0.609020i	$-0.791563\pi$
$89$	−1.05500e7	−1.58631	−0.793155	+	0.609020i	$0.791563\pi$
$90$	0	0
$91$	−4.60064e6	−0.639990
$92$	−1.59637e6	−0.213736
$93$	−37506.6	−0.00483523
$94$	−9.79779e6	−1.21669
$95$	0	0
$96$	19809.0	0.00228514
$97$	2.81105e6	0.312729	0.156364	−	0.987699i	$-0.450023\pi$
$97$	2.81105e6	0.312729	0.156364	+	0.987699i	$0.450023\pi$
$98$	−3.25625e6	−0.349483
$99$	8.02740e6	0.831480
$100$	0	0
$101$	7.97139e6	0.769855	0.384928	−	0.922947i	$-0.374226\pi$
$101$	7.97139e6	0.769855	0.384928	+	0.922947i	$0.374226\pi$
$102$	84373.7	0.00787239
$103$	−6.30920e6	−0.568911	−0.284455	−	0.958689i	$-0.591813\pi$
$103$	−6.30920e6	−0.568911	−0.284455	+	0.958689i	$0.591813\pi$
$104$	1.02902e7	0.897029
$105$	0	0
$106$	−4.42637e6	−0.360975
$107$	−4.50974e6	−0.355884	−0.177942	−	0.984041i	$-0.556944\pi$
$107$	−4.50974e6	−0.355884	−0.177942	+	0.984041i	$0.556944\pi$
$108$	30704.1	0.00234538
$109$	−1.81237e7	−1.34046	−0.670231	−	0.742152i	$-0.733805\pi$
$109$	−1.81237e7	−1.34046	−0.670231	+	0.742152i	$0.733805\pi$
$110$	0	0
$111$	70610.0	0.00490044
$112$	−7.89693e6	−0.531123
$113$	1.52540e7	0.994508	0.497254	−	0.867605i	$-0.334342\pi$
$113$	1.52540e7	0.994508	0.497254	+	0.867605i	$0.334342\pi$
$114$	121683.	0.00769245
$115$	0	0
$116$	496251.	0.0295188
$117$	1.43554e7	0.828636
$118$	−1.48394e7	−0.831439
$119$	2.74967e7	1.49578
$120$	0	0
$121$	−6.01394e6	−0.308610
$122$	−2.54747e7	−1.27013
$123$	20010.2	0.000969579 0
$124$	5.46558e6	0.257431
$125$	0	0
$126$	−1.50195e7	−0.668896
$127$	2.88312e7	1.24896	0.624482	−	0.781039i	$-0.285310\pi$
$127$	2.88312e7	1.24896	0.624482	+	0.781039i	$0.285310\pi$
$128$	1.12452e7	0.473950
$129$	163874.	0.00672118
$130$	0	0
$131$	−4.19911e7	−1.63195	−0.815977	−	0.578084i	$-0.803800\pi$
$131$	−4.19911e7	−1.63195	−0.815977	+	0.578084i	$0.803800\pi$
$132$	25766.6	0.000975099 0
$133$	3.96557e7	1.46159
$134$	873988.	0.0313790
$135$	0	0
$136$	−6.15015e7	−2.09652
$137$	5.64551e6	0.187578	0.0937888	−	0.995592i	$-0.470102\pi$
$137$	5.64551e6	0.187578	0.0937888	+	0.995592i	$0.470102\pi$
$138$	107344.	0.00347698
$139$	−2.74293e7	−0.866290	−0.433145	−	0.901324i	$-0.642596\pi$
$139$	−2.74293e7	−0.866290	−0.433145	+	0.901324i	$0.642596\pi$
$140$	0	0
$141$	−219458.	−0.00659302
$142$	−3.55834e7	−1.04289
$143$	2.40941e7	0.689024
$144$	2.46408e7	0.687679
$145$	0	0
$146$	−1.17066e7	−0.311311
$147$	−72935.9	−0.00189378
$148$	−1.02895e7	−0.260903
$149$	4.35855e7	1.07942	0.539709	−	0.841851i	$-0.318534\pi$
$149$	4.35855e7	1.07942	0.539709	+	0.841851i	$0.318534\pi$
$150$	0	0
$151$	−1.14365e7	−0.270317	−0.135159	−	0.990824i	$-0.543154\pi$
$151$	−1.14365e7	−0.270317	−0.135159	+	0.990824i	$0.543154\pi$
$152$	−8.86972e7	−2.04860
$153$	−8.57979e7	−1.93667
$154$	−2.52088e7	−0.556198
$155$	0	0
$156$	46078.3	0.000971763 0
$157$	3.78923e6	0.0781452	0.0390726	−	0.999236i	$-0.487560\pi$
$157$	3.78923e6	0.0781452	0.0390726	+	0.999236i	$0.487560\pi$
$158$	−2.63735e7	−0.531947
$159$	−99145.0	−0.00195605
$160$	0	0
$161$	3.49826e7	0.660636
$162$	4.68643e7	0.866043
$163$	9.92134e7	1.79438	0.897188	−	0.441648i	$-0.145606\pi$
$163$	9.92134e7	1.79438	0.897188	+	0.441648i	$0.145606\pi$
$164$	−2.91595e6	−0.0516210
$165$	0	0
$166$	3.06548e7	0.520140
$167$	−1.59430e7	−0.264889	−0.132444	−	0.991190i	$-0.542283\pi$
$167$	−1.59430e7	−0.264889	−0.132444	+	0.991190i	$0.542283\pi$
$168$	−241150.	−0.00392379
$169$	−1.96612e7	−0.313333
$170$	0	0
$171$	−1.23737e8	−1.89241
$172$	−2.38802e7	−0.357839
$173$	−8.17729e7	−1.20074	−0.600369	−	0.799723i	$-0.704980\pi$
$173$	−8.17729e7	−1.20074	−0.600369	+	0.799723i	$0.704980\pi$
$174$	−33369.2	−0.000480202 0
$175$	0	0
$176$	4.13572e7	0.571817
$177$	−332385.	−0.00450541
$178$	−1.03377e8	−1.37390
$179$	8.17148e7	1.06492	0.532458	−	0.846457i	$-0.321268\pi$
$179$	8.17148e7	1.06492	0.532458	+	0.846457i	$0.321268\pi$
$180$	0	0
$181$	−6.82951e7	−0.856080	−0.428040	−	0.903760i	$-0.640796\pi$
$181$	−6.82951e7	−0.856080	−0.428040	+	0.903760i	$0.640796\pi$
$182$	−4.50808e7	−0.554296
$183$	−570601.	−0.00688262
$184$	−7.82451e7	−0.925966
$185$	0	0
$186$	−367520.	−0.00418780
$187$	−1.44004e8	−1.61038
$188$	3.19801e7	0.351017
$189$	−672843.	−0.00724932
$190$	0	0
$191$	−1.24235e8	−1.29011	−0.645056	−	0.764135i	$-0.723166\pi$
$191$	−1.24235e8	−1.29011	−0.645056	+	0.764135i	$0.723166\pi$
$192$	510639.	0.00520667
$193$	−1.36552e8	−1.36725	−0.683624	−	0.729834i	$-0.739597\pi$
$193$	−1.36552e8	−1.36725	−0.683624	+	0.729834i	$0.739597\pi$
$194$	2.75450e7	0.270855
$195$	0	0
$196$	1.06284e7	0.100826
$197$	8.64681e7	0.805795	0.402897	−	0.915245i	$-0.368003\pi$
$197$	8.64681e7	0.805795	0.402897	+	0.915245i	$0.368003\pi$
$198$	7.86589e7	0.720145
$199$	5.43348e7	0.488756	0.244378	−	0.969680i	$-0.421416\pi$
$199$	5.43348e7	0.488756	0.244378	+	0.969680i	$0.421416\pi$
$200$	0	0
$201$	19576.2	0.000170037 0
$202$	7.81101e7	0.666772
$203$	−1.08748e7	−0.0912397
$204$	−275397.	−0.00227119
$205$	0	0
$206$	−6.18226e7	−0.492734
$207$	−1.09156e8	−0.855367
$208$	7.39588e7	0.569860
$209$	−2.07681e8	−1.57357
$210$	0	0
$211$	−7.01204e7	−0.513873	−0.256937	−	0.966428i	$-0.582713\pi$
$211$	−7.01204e7	−0.513873	−0.256937	+	0.966428i	$0.582713\pi$
$212$	1.44477e7	0.104142
$213$	−797023.	−0.00565122
$214$	−4.41901e7	−0.308231
$215$	0	0
$216$	1.50494e6	0.0101609
$217$	−1.19772e8	−0.795693
$218$	−1.77591e8	−1.16097
$219$	−262213.	−0.00168694
$220$	0	0
$221$	−2.57521e8	−1.60487
$222$	691893.	0.00424428
$223$	1.46591e8	0.885195	0.442598	−	0.896720i	$-0.354057\pi$
$223$	1.46591e8	0.885195	0.442598	+	0.896720i	$0.354057\pi$
$224$	6.32571e7	0.376046
$225$	0	0
$226$	1.49471e8	0.861343
$227$	−1.25022e8	−0.709405	−0.354703	−	0.934979i	$-0.615418\pi$
$227$	−1.25022e8	−0.709405	−0.354703	+	0.934979i	$0.615418\pi$
$228$	−397176.	−0.00221928
$229$	−2.28983e8	−1.26003	−0.630013	−	0.776584i	$-0.716951\pi$
$229$	−2.28983e8	−1.26003	−0.630013	+	0.776584i	$0.716951\pi$
$230$	0	0
$231$	−564645.	−0.00301393
$232$	2.43234e7	0.127884
$233$	1.94722e8	1.00848	0.504241	−	0.863563i	$-0.331772\pi$
$233$	1.94722e8	1.00848	0.504241	+	0.863563i	$0.331772\pi$
$234$	1.40665e8	0.717681
$235$	0	0
$236$	4.84362e7	0.239871
$237$	−590733.	−0.00288252
$238$	2.69435e8	1.29549
$239$	−3.60549e8	−1.70833	−0.854165	−	0.520002i	$-0.825931\pi$
$239$	−3.60549e8	−1.70833	−0.854165	+	0.520002i	$0.825931\pi$
$240$	0	0
$241$	2.95185e8	1.35842	0.679211	−	0.733943i	$-0.262322\pi$
$241$	2.95185e8	1.35842	0.679211	+	0.733943i	$0.262322\pi$
$242$	−5.89294e7	−0.267287
$243$	3.14922e6	0.0140793
$244$	8.31498e7	0.366435
$245$	0	0
$246$	196076.	0.000839753 0
$247$	−3.71396e8	−1.56819
$248$	2.67892e8	1.11527
$249$	686629.	0.00281854
$250$	0	0
$251$	3.78099e8	1.50920	0.754600	−	0.656185i	$-0.227831\pi$
$251$	3.78099e8	1.50920	0.754600	+	0.656185i	$0.227831\pi$
$252$	4.90239e7	0.192977
$253$	−1.83208e8	−0.711252
$254$	2.82511e8	1.08173
$255$	0	0
$256$	−1.87613e8	−0.698911
$257$	−1.51682e8	−0.557400	−0.278700	−	0.960378i	$-0.589904\pi$
$257$	−1.51682e8	−0.557400	−0.278700	+	0.960378i	$0.589904\pi$
$258$	1.60577e6	0.00582121
$259$	2.25482e8	0.806424
$260$	0	0
$261$	3.39325e7	0.118134
$262$	−4.11463e8	−1.41344
$263$	−4.50891e7	−0.152836	−0.0764181	−	0.997076i	$-0.524348\pi$
$263$	−4.50891e7	−0.152836	−0.0764181	+	0.997076i	$0.524348\pi$
$264$	1.26293e6	0.00422442
$265$	0	0
$266$	3.88578e8	1.26588
$267$	−2.31552e6	−0.00744492
$268$	−2.85271e6	−0.00905286
$269$	−1.08935e8	−0.341219	−0.170609	−	0.985339i	$-0.554574\pi$
$269$	−1.08935e8	−0.341219	−0.170609	+	0.985339i	$0.554574\pi$
$270$	0	0
$271$	−1.44539e8	−0.441157	−0.220579	−	0.975369i	$-0.570795\pi$
$271$	−1.44539e8	−0.441157	−0.220579	+	0.975369i	$0.570795\pi$
$272$	−4.42031e8	−1.33187
$273$	−1.00975e6	−0.00300362
$274$	5.53192e7	0.162461
$275$	0	0
$276$	−350373.	−0.00100311
$277$	−6.20385e8	−1.75381	−0.876905	−	0.480664i	$-0.840396\pi$
$277$	−6.20385e8	−1.75381	−0.876905	+	0.480664i	$0.840396\pi$
$278$	−2.68775e8	−0.750294
$279$	3.73723e8	1.03023
$280$	0	0
$281$	−8.42401e7	−0.226489	−0.113244	−	0.993567i	$-0.536124\pi$
$281$	−8.42401e7	−0.226489	−0.113244	+	0.993567i	$0.536124\pi$
$282$	−2.15043e6	−0.00571022
$283$	−1.72782e8	−0.453153	−0.226577	−	0.973993i	$-0.572753\pi$
$283$	−1.72782e8	−0.453153	−0.226577	+	0.973993i	$0.572753\pi$
$284$	1.16145e8	0.300874
$285$	0	0
$286$	2.36093e8	0.596764
$287$	6.38996e7	0.159555
$288$	−1.97381e8	−0.486890
$289$	1.12879e9	2.75087
$290$	0	0
$291$	616972.	0.00146771
$292$	3.82104e7	0.0898136
$293$	−7.87224e8	−1.82836	−0.914179	−	0.405310i	$-0.867164\pi$
$293$	−7.87224e8	−1.82836	−0.914179	+	0.405310i	$0.867164\pi$
$294$	−714684.	−0.00164021
$295$	0	0
$296$	−5.04333e8	−1.13031
$297$	3.52376e6	0.00780475
$298$	4.27086e8	0.934885
$299$	−3.27630e8	−0.708818
$300$	0	0
$301$	5.23306e8	1.10605
$302$	−1.12064e8	−0.234122
$303$	1.74957e6	0.00361311
$304$	−6.37495e8	−1.30143
$305$	0	0
$306$	−8.40717e8	−1.67735
$307$	−3.26890e8	−0.644789	−0.322394	−	0.946605i	$-0.604488\pi$
$307$	−3.26890e8	−0.644789	−0.322394	+	0.946605i	$0.604488\pi$
$308$	8.22819e7	0.160464
$309$	−1.38475e6	−0.00267003
$310$	0	0
$311$	−4.26599e8	−0.804190	−0.402095	−	0.915598i	$-0.631718\pi$
$311$	−4.26599e8	−0.804190	−0.402095	+	0.915598i	$0.631718\pi$
$312$	2.25850e6	0.00420996
$313$	3.26460e8	0.601762	0.300881	−	0.953662i	$-0.402719\pi$
$313$	3.26460e8	0.601762	0.300881	+	0.953662i	$0.402719\pi$
$314$	3.71299e7	0.0676816
$315$	0	0
$316$	8.60835e7	0.153467
$317$	−4.75370e8	−0.838155	−0.419078	−	0.907950i	$-0.637646\pi$
$317$	−4.75370e8	−0.838155	−0.419078	+	0.907950i	$0.637646\pi$
$318$	−971502.	−0.00169414
$319$	5.69525e7	0.0982302
$320$	0	0
$321$	−989802.	−0.00167025
$322$	3.42788e8	0.572177
$323$	2.21973e9	3.66514
$324$	−1.52966e8	−0.249854
$325$	0	0
$326$	9.72172e8	1.55411
$327$	−3.97781e6	−0.00629110
$328$	−1.42923e8	−0.223637
$329$	−7.00807e8	−1.08496
$330$	0	0
$331$	2.48921e8	0.377280	0.188640	−	0.982046i	$-0.439592\pi$
$331$	2.48921e8	0.377280	0.188640	+	0.982046i	$0.439592\pi$
$332$	−1.00058e8	−0.150061
$333$	−7.03572e8	−1.04413
$334$	−1.56223e8	−0.229420
$335$	0	0
$336$	−1.73323e6	−0.00249269
$337$	−3.19702e8	−0.455031	−0.227515	−	0.973774i	$-0.573060\pi$
$337$	−3.19702e8	−0.455031	−0.227515	+	0.973774i	$0.573060\pi$
$338$	−1.92656e8	−0.271378
$339$	3.34795e6	0.00466746
$340$	0	0
$341$	6.27259e8	0.856656
$342$	−1.21248e9	−1.63901
$343$	−8.10114e8	−1.08397
$344$	−1.17047e9	−1.55027
$345$	0	0
$346$	−8.01277e8	−1.03996
$347$	−1.49562e8	−0.192162	−0.0960810	−	0.995374i	$-0.530631\pi$
$347$	−1.49562e8	−0.192162	−0.0960810	+	0.995374i	$0.530631\pi$
$348$	108918.	0.000138539 0
$349$	−5.36098e8	−0.675080	−0.337540	−	0.941311i	$-0.609595\pi$
$349$	−5.36098e8	−0.675080	−0.337540	+	0.941311i	$0.609595\pi$
$350$	0	0
$351$	6.30152e6	0.00777805
$352$	−3.31285e8	−0.404858
$353$	−1.13288e9	−1.37080	−0.685400	−	0.728167i	$-0.740373\pi$
$353$	−1.13288e9	−1.37080	−0.685400	+	0.728167i	$0.740373\pi$
$354$	−3.25697e6	−0.00390214
$355$	0	0
$356$	3.37426e8	0.396372
$357$	6.03500e6	0.00702002
$358$	8.00707e8	0.922324
$359$	1.05998e9	1.20911	0.604556	−	0.796563i	$-0.293351\pi$
$359$	1.05998e9	1.20911	0.604556	+	0.796563i	$0.293351\pi$
$360$	0	0
$361$	2.30741e9	2.58136
$362$	−6.69210e8	−0.741451
$363$	−1.31994e6	−0.00144838
$364$	1.47144e8	0.159915
$365$	0	0
$366$	−5.59121e6	−0.00596104
$367$	4.82267e8	0.509280	0.254640	−	0.967036i	$-0.418043\pi$
$367$	4.82267e8	0.509280	0.254640	+	0.967036i	$0.418043\pi$
$368$	−5.62373e8	−0.588244
$369$	−1.99386e8	−0.206586
$370$	0	0
$371$	−3.16605e8	−0.321891
$372$	1.19959e6	0.00120818
$373$	3.04508e8	0.303821	0.151910	−	0.988394i	$-0.451457\pi$
$373$	3.04508e8	0.303821	0.151910	+	0.988394i	$0.451457\pi$
$374$	−1.41106e9	−1.39475
$375$	0	0
$376$	1.56748e9	1.52071
$377$	1.01848e8	0.0978942
$378$	−6.59306e6	−0.00627864
$379$	1.36048e9	1.28367	0.641837	−	0.766841i	$-0.278172\pi$
$379$	1.36048e9	1.28367	0.641837	+	0.766841i	$0.278172\pi$
$380$	0	0
$381$	6.32790e6	0.00586168
$382$	−1.21736e9	−1.11737
$383$	−9.43189e8	−0.857833	−0.428917	−	0.903344i	$-0.641105\pi$
$383$	−9.43189e8	−0.857833	−0.428917	+	0.903344i	$0.641105\pi$
$384$	2.46810e6	0.00222436
$385$	0	0
$386$	−1.33805e9	−1.18417
$387$	−1.63287e9	−1.43207
$388$	−8.99072e7	−0.0781418
$389$	−4.89822e8	−0.421905	−0.210952	−	0.977496i	$-0.567656\pi$
$389$	−4.89822e8	−0.421905	−0.210952	+	0.977496i	$0.567656\pi$
$390$	0	0
$391$	1.95815e9	1.65664
$392$	5.20946e8	0.436809
$393$	−9.21625e6	−0.00765914
$394$	8.47284e8	0.697899
$395$	0	0
$396$	−2.56744e8	−0.207762
$397$	−1.83502e9	−1.47188	−0.735941	−	0.677046i	$-0.763260\pi$
$397$	−1.83502e9	−1.47188	−0.735941	+	0.677046i	$0.763260\pi$
$398$	5.32416e8	0.423311
$399$	8.70366e6	0.00685956
$400$	0	0
$401$	1.57176e9	1.21726	0.608628	−	0.793456i	$-0.291720\pi$
$401$	1.57176e9	1.21726	0.608628	+	0.793456i	$0.291720\pi$
$402$	191824.	0.000147269 0
$403$	1.12172e9	0.853726
$404$	−2.54952e8	−0.192364
$405$	0	0
$406$	−1.06560e8	−0.0790227
$407$	−1.18088e9	−0.868210
$408$	−1.34984e7	−0.00983946
$409$	1.09853e9	0.793923	0.396962	−	0.917835i	$-0.370065\pi$
$409$	1.09853e9	0.793923	0.396962	+	0.917835i	$0.370065\pi$
$410$	0	0
$411$	1.23908e6	0.000880346 0
$412$	2.01790e8	0.142154
$413$	−1.06142e9	−0.741417
$414$	−1.06960e9	−0.740833
$415$	0	0
$416$	−5.92435e8	−0.403473
$417$	−6.02021e6	−0.00406570
$418$	−2.03503e9	−1.36287
$419$	1.27820e9	0.848885	0.424443	−	0.905455i	$-0.360470\pi$
$419$	1.27820e9	0.848885	0.424443	+	0.905455i	$0.360470\pi$
$420$	0	0
$421$	−1.69366e9	−1.10622	−0.553108	−	0.833109i	$-0.686558\pi$
$421$	−1.69366e9	−1.10622	−0.553108	+	0.833109i	$0.686558\pi$
$422$	−6.87097e8	−0.445066
$423$	2.18673e9	1.40476
$424$	7.08145e8	0.451172
$425$	0	0
$426$	−7.80987e6	−0.00489452
$427$	−1.82213e9	−1.13261
$428$	1.44237e8	0.0889250
$429$	5.28819e6	0.00323375
$430$	0	0
$431$	−1.28297e9	−0.771872	−0.385936	−	0.922526i	$-0.626121\pi$
$431$	−1.28297e9	−0.771872	−0.385936	+	0.922526i	$0.626121\pi$
$432$	1.08165e7	0.00645495
$433$	3.04331e9	1.80152	0.900760	−	0.434317i	$-0.143010\pi$
$433$	3.04331e9	1.80152	0.900760	+	0.434317i	$0.143010\pi$
$434$	−1.17362e9	−0.689150
$435$	0	0
$436$	5.79659e8	0.334942
$437$	2.82404e9	1.61877
$438$	−2.56937e6	−0.00146106
$439$	1.74522e9	0.984520	0.492260	−	0.870448i	$-0.336171\pi$
$439$	1.74522e9	0.984520	0.492260	+	0.870448i	$0.336171\pi$
$440$	0	0
$441$	7.26748e8	0.403505
$442$	−2.52340e9	−1.38998
$443$	3.27301e8	0.178869	0.0894343	−	0.995993i	$-0.471494\pi$
$443$	3.27301e8	0.178869	0.0894343	+	0.995993i	$0.471494\pi$
$444$	−2.25835e6	−0.00122448
$445$	0	0
$446$	1.43641e9	0.766668
$447$	9.56618e6	0.00506597
$448$	1.63065e9	0.856817
$449$	1.76801e9	0.921770	0.460885	−	0.887460i	$-0.347532\pi$
$449$	1.76801e9	0.921770	0.460885	+	0.887460i	$0.347532\pi$
$450$	0	0
$451$	−3.34650e8	−0.171780
$452$	−4.87874e8	−0.248498
$453$	−2.51009e6	−0.00126866
$454$	−1.22506e9	−0.614416
$455$	0	0
$456$	−1.94673e7	−0.00961456
$457$	3.66718e8	0.179732	0.0898659	−	0.995954i	$-0.471356\pi$
$457$	3.66718e8	0.179732	0.0898659	+	0.995954i	$0.471356\pi$
$458$	−2.24376e9	−1.09131
$459$	−3.76624e7	−0.0181787
$460$	0	0
$461$	−8.85414e8	−0.420914	−0.210457	−	0.977603i	$-0.567495\pi$
$461$	−8.85414e8	−0.420914	−0.210457	+	0.977603i	$0.567495\pi$
$462$	−5.53285e6	−0.00261037
$463$	6.85195e8	0.320834	0.160417	−	0.987049i	$-0.448716\pi$
$463$	6.85195e8	0.320834	0.160417	+	0.987049i	$0.448716\pi$
$464$	1.74820e8	0.0812417
$465$	0	0
$466$	1.90804e9	0.873447
$467$	−4.60978e8	−0.209445	−0.104723	−	0.994501i	$-0.533396\pi$
$467$	−4.60978e8	−0.209445	−0.104723	+	0.994501i	$0.533396\pi$
$468$	−4.59134e8	−0.207052
$469$	6.25138e7	0.0279815
$470$	0	0
$471$	831663.	0.000366754 0
$472$	2.37407e9	1.03919
$473$	−2.74062e9	−1.19079
$474$	−5.78848e6	−0.00249655
$475$	0	0
$476$	−8.79439e8	−0.373750
$477$	9.87901e8	0.416773
$478$	−3.53295e9	−1.47958
$479$	1.15019e9	0.478185	0.239093	−	0.970997i	$-0.423150\pi$
$479$	1.15019e9	0.478185	0.239093	+	0.970997i	$0.423150\pi$
$480$	0	0
$481$	−2.11176e9	−0.865240
$482$	2.89246e9	1.17653
$483$	7.67802e6	0.00310052
$484$	1.92347e8	0.0771127
$485$	0	0
$486$	3.08586e7	0.0121941
$487$	−3.09171e8	−0.121296	−0.0606480	−	0.998159i	$-0.519317\pi$
$487$	−3.09171e8	−0.121296	−0.0606480	+	0.998159i	$0.519317\pi$
$488$	4.07553e9	1.58750
$489$	2.17754e7	0.00842143
$490$	0	0
$491$	−2.78751e9	−1.06275	−0.531376	−	0.847136i	$-0.678325\pi$
$491$	−2.78751e9	−1.06275	−0.531376	+	0.847136i	$0.678325\pi$
$492$	−639995.	−0.000242269 0
$493$	−6.08715e8	−0.228797
$494$	−3.63923e9	−1.35821
$495$	0	0
$496$	1.92542e9	0.708501
$497$	−2.54518e9	−0.929973
$498$	6.72814e6	0.00244114
$499$	−2.41672e9	−0.870711	−0.435356	−	0.900259i	$-0.643377\pi$
$499$	−2.41672e9	−0.870711	−0.435356	+	0.900259i	$0.643377\pi$
$500$	0	0
$501$	−3.49919e6	−0.00124318
$502$	3.70491e9	1.30712
$503$	−1.94911e9	−0.682886	−0.341443	−	0.939903i	$-0.610916\pi$
$503$	−1.94911e9	−0.682886	−0.341443	+	0.939903i	$0.610916\pi$
$504$	2.40287e9	0.836034
$505$	0	0
$506$	−1.79522e9	−0.616015
$507$	−4.31525e6	−0.00147054
$508$	−9.22121e8	−0.312079
$509$	−2.13295e9	−0.716917	−0.358458	−	0.933546i	$-0.616698\pi$
$509$	−2.13295e9	−0.716917	−0.358458	+	0.933546i	$0.616698\pi$
$510$	0	0
$511$	−8.37337e8	−0.277605
$512$	−3.27776e9	−1.07928
$513$	−5.43166e7	−0.0177632
$514$	−1.48630e9	−0.482765
$515$	0	0
$516$	−5.24124e6	−0.00167942
$517$	3.67021e9	1.16808
$518$	2.20946e9	0.698444
$519$	−1.79476e7	−0.00563534
$520$	0	0
$521$	−4.96329e9	−1.53758	−0.768790	−	0.639501i	$-0.779141\pi$
$521$	−4.96329e9	−1.53758	−0.768790	+	0.639501i	$0.779141\pi$
$522$	3.32498e8	0.102316
$523$	−5.92414e9	−1.81080	−0.905398	−	0.424564i	$-0.860427\pi$
$523$	−5.92414e9	−1.81080	−0.905398	+	0.424564i	$0.860427\pi$
$524$	1.34302e9	0.407777
$525$	0	0
$526$	−4.41819e8	−0.132371
$527$	−6.70422e9	−1.99531
$528$	9.07711e6	0.00268367
$529$	−9.13568e8	−0.268316
$530$	0	0
$531$	3.31195e9	0.959960
$532$	−1.26832e9	−0.365207
$533$	−5.98452e8	−0.171192
$534$	−2.26894e7	−0.00644805
$535$	0	0
$536$	−1.39824e8	−0.0392196
$537$	1.79348e7	0.00499790
$538$	−1.06743e9	−0.295530
$539$	1.21978e9	0.335521
$540$	0	0
$541$	−6.01356e9	−1.63283	−0.816416	−	0.577465i	$-0.804042\pi$
$541$	−6.01356e9	−1.63283	−0.816416	+	0.577465i	$0.804042\pi$
$542$	−1.41631e9	−0.382086
$543$	−1.49895e7	−0.00401778
$544$	3.54081e9	0.942990
$545$	0	0
$546$	−9.89436e6	−0.00260144
$547$	5.25827e9	1.37369	0.686843	−	0.726806i	$-0.258996\pi$
$547$	5.25827e9	1.37369	0.686843	+	0.726806i	$0.258996\pi$
$548$	−1.80563e8	−0.0468701
$549$	5.68559e9	1.46647
$550$	0	0
$551$	−8.77887e8	−0.223567
$552$	−1.71733e7	−0.00434577
$553$	−1.88642e9	−0.474352
$554$	−6.07904e9	−1.51897
$555$	0	0
$556$	8.77284e8	0.216460
$557$	−2.46930e9	−0.605453	−0.302727	−	0.953077i	$-0.597897\pi$
$557$	−2.46930e9	−0.605453	−0.302727	+	0.953077i	$0.597897\pi$
$558$	3.66204e9	0.892285
$559$	−4.90103e9	−1.18671
$560$	0	0
$561$	−3.16060e7	−0.00755788
$562$	−8.25452e8	−0.196162
$563$	2.05583e9	0.485521	0.242760	−	0.970086i	$-0.421947\pi$
$563$	2.05583e9	0.485521	0.242760	+	0.970086i	$0.421947\pi$
$564$	7.01902e6	0.00164740
$565$	0	0
$566$	−1.69305e9	−0.392476
$567$	3.35206e9	0.772274
$568$	5.69275e9	1.30348
$569$	4.29899e9	0.978304	0.489152	−	0.872199i	$-0.337306\pi$
$569$	4.29899e9	0.978304	0.489152	+	0.872199i	$0.337306\pi$
$570$	0	0
$571$	−9.60657e7	−0.0215944	−0.0107972	−	0.999942i	$-0.503437\pi$
$571$	−9.60657e7	−0.0215944	−0.0107972	+	0.999942i	$0.503437\pi$
$572$	−7.70612e8	−0.172167
$573$	−2.72672e7	−0.00605480
$574$	6.26140e8	0.138191
$575$	0	0
$576$	−5.08812e9	−1.10937
$577$	6.92828e9	1.50145	0.750723	−	0.660617i	$-0.229705\pi$
$577$	6.92828e9	1.50145	0.750723	+	0.660617i	$0.229705\pi$
$578$	1.10608e10	2.38253
$579$	−2.99706e7	−0.00641682
$580$	0	0
$581$	2.19265e9	0.463823
$582$	6.04559e6	0.00127118
$583$	1.65810e9	0.346553
$584$	1.87286e9	0.389099
$585$	0	0
$586$	−7.71385e9	−1.58354
$587$	3.30841e9	0.675128	0.337564	−	0.941303i	$-0.390397\pi$
$587$	3.30841e9	0.675128	0.337564	+	0.941303i	$0.390397\pi$
$588$	2.33274e6	0.000473201 0
$589$	−9.66881e9	−1.94971
$590$	0	0
$591$	1.89781e7	0.00378178
$592$	−3.62480e9	−0.718057
$593$	4.28371e9	0.843584	0.421792	−	0.906693i	$-0.361401\pi$
$593$	4.28371e9	0.843584	0.421792	+	0.906693i	$0.361401\pi$
$594$	3.45286e7	0.00675969
$595$	0	0
$596$	−1.39401e9	−0.269715
$597$	1.19254e7	0.00229385
$598$	−3.21039e9	−0.613908
$599$	7.33889e8	0.139520	0.0697601	−	0.997564i	$-0.477777\pi$
$599$	7.33889e8	0.139520	0.0697601	+	0.997564i	$0.477777\pi$
$600$	0	0
$601$	−1.02592e10	−1.92776	−0.963882	−	0.266331i	$-0.914189\pi$
$601$	−1.02592e10	−1.92776	−0.963882	+	0.266331i	$0.914189\pi$
$602$	5.12778e9	0.957947
$603$	−1.95062e8	−0.0362294
$604$	3.65779e8	0.0675443
$605$	0	0
$606$	1.71437e7	0.00312932
$607$	3.91444e9	0.710411	0.355205	−	0.934788i	$-0.384411\pi$
$607$	3.91444e9	0.710411	0.355205	+	0.934788i	$0.384411\pi$
$608$	5.10655e9	0.921436
$609$	−2.38680e6	−0.000428209 0
$610$	0	0
$611$	6.56342e9	1.16409
$612$	2.74411e9	0.483918
$613$	9.83595e8	0.172466	0.0862332	−	0.996275i	$-0.472517\pi$
$613$	9.83595e8	0.172466	0.0862332	+	0.996275i	$0.472517\pi$
$614$	−3.20313e9	−0.558452
$615$	0	0
$616$	4.03299e9	0.695176
$617$	8.78082e9	1.50500	0.752501	−	0.658591i	$-0.228847\pi$
$617$	8.78082e9	1.50500	0.752501	+	0.658591i	$0.228847\pi$
$618$	−1.35689e7	−0.00231251
$619$	−3.71365e9	−0.629337	−0.314668	−	0.949202i	$-0.601893\pi$
$619$	−3.71365e9	−0.629337	−0.314668	+	0.949202i	$0.601893\pi$
$620$	0	0
$621$	−4.79159e7	−0.00802896
$622$	−4.18016e9	−0.696509
$623$	−7.39429e9	−1.22515
$624$	1.62325e7	0.00267449
$625$	0	0
$626$	3.19892e9	0.521186
$627$	−4.55821e7	−0.00738512
$628$	−1.21193e8	−0.0195262
$629$	1.26214e10	2.02223
$630$	0	0
$631$	3.50323e9	0.555093	0.277546	−	0.960712i	$-0.410479\pi$
$631$	3.50323e9	0.555093	0.277546	+	0.960712i	$0.410479\pi$
$632$	4.21933e9	0.664865
$633$	−1.53901e7	−0.00241173
$634$	−4.65806e9	−0.725926
$635$	0	0
$636$	3.17100e6	0.000488760 0
$637$	2.18132e9	0.334373
$638$	5.58066e8	0.0850772
$639$	7.94170e9	1.20409
$640$	0	0
$641$	−3.48025e9	−0.521925	−0.260962	−	0.965349i	$-0.584040\pi$
$641$	−3.48025e9	−0.521925	−0.260962	+	0.965349i	$0.584040\pi$
$642$	−9.69888e6	−0.00144660
$643$	−2.24393e9	−0.332867	−0.166434	−	0.986053i	$-0.553225\pi$
$643$	−2.24393e9	−0.332867	−0.166434	+	0.986053i	$0.553225\pi$
$644$	−1.11887e9	−0.165073
$645$	0	0
$646$	2.17507e10	3.17438
$647$	4.26552e9	0.619166	0.309583	−	0.950872i	$-0.399811\pi$
$647$	4.26552e9	0.619166	0.309583	+	0.950872i	$0.399811\pi$
$648$	−7.49751e9	−1.08244
$649$	5.55879e9	0.798223
$650$	0	0
$651$	−2.62876e7	−0.00373437
$652$	−3.17318e9	−0.448362
$653$	−3.92073e9	−0.551025	−0.275512	−	0.961298i	$-0.588848\pi$
$653$	−3.92073e9	−0.551025	−0.275512	+	0.961298i	$0.588848\pi$
$654$	−3.89778e7	−0.00544873
$655$	0	0
$656$	−1.02724e9	−0.142071
$657$	2.61274e9	0.359432
$658$	−6.86707e9	−0.939682
$659$	−1.58837e9	−0.216199	−0.108099	−	0.994140i	$-0.534476\pi$
$659$	−1.58837e9	−0.216199	−0.108099	+	0.994140i	$0.534476\pi$
$660$	0	0
$661$	−5.11970e7	−0.00689508	−0.00344754	−	0.999994i	$-0.501097\pi$
$661$	−5.11970e7	−0.00689508	−0.00344754	+	0.999994i	$0.501097\pi$
$662$	2.43913e9	0.326762
$663$	−5.65209e7	−0.00753202
$664$	−4.90426e9	−0.650108
$665$	0	0
$666$	−6.89417e9	−0.904319
$667$	−7.74437e8	−0.101052
$668$	5.09913e8	0.0661879
$669$	3.21738e7	0.00415443
$670$	0	0
$671$	9.54271e9	1.21939
$672$	1.38837e7	0.00176487
$673$	−4.15444e9	−0.525363	−0.262681	−	0.964883i	$-0.584607\pi$
$673$	−4.15444e9	−0.525363	−0.262681	+	0.964883i	$0.584607\pi$
$674$	−3.13270e9	−0.394102
$675$	0	0
$676$	6.28832e8	0.0782926
$677$	5.00823e9	0.620332	0.310166	−	0.950682i	$-0.399615\pi$
$677$	5.00823e9	0.620332	0.310166	+	0.950682i	$0.399615\pi$
$678$	3.28059e7	0.00404249
$679$	1.97021e9	0.241528
$680$	0	0
$681$	−2.74398e7	−0.00332940
$682$	6.14639e9	0.741950
$683$	−5.60971e9	−0.673702	−0.336851	−	0.941558i	$-0.609362\pi$
$683$	−5.60971e9	−0.673702	−0.336851	+	0.941558i	$0.609362\pi$
$684$	3.95755e9	0.472857
$685$	0	0
$686$	−7.93815e9	−0.938826
$687$	−5.02574e7	−0.00591360
$688$	−8.41255e9	−0.984846
$689$	2.96517e9	0.345368
$690$	0	0
$691$	−3.70584e8	−0.0427281	−0.0213641	−	0.999772i	$-0.506801\pi$
$691$	−3.70584e8	−0.0427281	−0.0213641	+	0.999772i	$0.506801\pi$
$692$	2.61538e9	0.300029
$693$	5.62624e9	0.642173
$694$	−1.46553e9	−0.166432
$695$	0	0
$696$	5.33853e6	0.000600190 0
$697$	3.57678e9	0.400108
$698$	−5.25312e9	−0.584686
$699$	4.27376e7	0.00473304
$700$	0	0
$701$	−1.63842e10	−1.79644	−0.898221	−	0.439545i	$-0.855140\pi$
$701$	−1.63842e10	−1.79644	−0.898221	+	0.439545i	$0.855140\pi$
$702$	6.17474e7	0.00673657
$703$	1.82025e10	1.97600
$704$	−8.53992e9	−0.922464
$705$	0	0
$706$	−1.11009e10	−1.18725
$707$	5.58698e9	0.594579
$708$	1.06308e7	0.00112577
$709$	3.39683e9	0.357942	0.178971	−	0.983854i	$-0.442723\pi$
$709$	3.39683e9	0.357942	0.178971	+	0.983854i	$0.442723\pi$
$710$	0	0
$711$	5.88619e9	0.614173
$712$	1.65387e10	1.71720
$713$	−8.52944e9	−0.881266
$714$	5.91358e7	0.00608004
$715$	0	0
$716$	−2.61352e9	−0.266091
$717$	−7.91336e7	−0.00801759
$718$	1.03865e10	1.04721
$719$	1.33634e10	1.34081	0.670403	−	0.741997i	$-0.266121\pi$
$719$	1.33634e10	1.34081	0.670403	+	0.741997i	$0.266121\pi$
$720$	0	0
$721$	−4.42199e9	−0.439384
$722$	2.26098e10	2.23572
$723$	6.47875e7	0.00637539
$724$	2.18431e9	0.213909
$725$	0	0
$726$	−1.29339e7	−0.00125444
$727$	−8.06750e9	−0.778698	−0.389349	−	0.921090i	$-0.627300\pi$
$727$	−8.06750e9	−0.778698	−0.389349	+	0.921090i	$0.627300\pi$
$728$	7.21218e9	0.692798
$729$	−1.04590e10	−0.999868
$730$	0	0
$731$	2.92921e10	2.77357
$732$	1.82498e7	0.00171976
$733$	−1.32403e10	−1.24175	−0.620876	−	0.783909i	$-0.713223\pi$
$733$	−1.32403e10	−1.24175	−0.620876	+	0.783909i	$0.713223\pi$
$734$	4.72564e9	0.441087
$735$	0	0
$736$	4.50479e9	0.416488
$737$	−3.27392e8	−0.0301253
$738$	−1.95374e9	−0.178924
$739$	1.48443e10	1.35302	0.676509	−	0.736434i	$-0.263492\pi$
$739$	1.48443e10	1.35302	0.676509	+	0.736434i	$0.263492\pi$
$740$	0	0
$741$	−8.15142e7	−0.00735986
$742$	−3.10235e9	−0.278790
$743$	3.94575e9	0.352914	0.176457	−	0.984308i	$-0.443536\pi$
$743$	3.94575e9	0.352914	0.176457	+	0.984308i	$0.443536\pi$
$744$	5.87971e7	0.00523420
$745$	0	0
$746$	2.98381e9	0.263139
$747$	−6.84171e9	−0.600541
$748$	4.60573e9	0.402386
$749$	−3.16079e9	−0.274858
$750$	0	0
$751$	−1.61442e10	−1.39084	−0.695421	−	0.718602i	$-0.744782\pi$
$751$	−1.61442e10	−1.39084	−0.695421	+	0.718602i	$0.744782\pi$
$752$	1.12660e10	0.966068
$753$	8.29854e7	0.00708303
$754$	9.97986e8	0.0847862
$755$	0	0
$756$	2.15198e7	0.00181139
$757$	−2.02147e10	−1.69368	−0.846840	−	0.531847i	$-0.821498\pi$
$757$	−2.02147e10	−1.69368	−0.846840	+	0.531847i	$0.821498\pi$
$758$	1.33311e10	1.11179
$759$	−4.02107e7	−0.00333807
$760$	0	0
$761$	−3.58878e9	−0.295189	−0.147595	−	0.989048i	$-0.547153\pi$
$761$	−3.58878e9	−0.295189	−0.147595	+	0.989048i	$0.547153\pi$
$762$	6.20058e7	0.00507680
$763$	−1.27025e10	−1.03527
$764$	3.97347e9	0.322361
$765$	0	0
$766$	−9.24212e9	−0.742969
$767$	9.94076e9	0.795492
$768$	−4.11774e7	−0.00328015
$769$	−1.43011e10	−1.13403	−0.567017	−	0.823706i	$-0.691903\pi$
$769$	−1.43011e10	−1.13403	−0.567017	+	0.823706i	$0.691903\pi$
$770$	0	0
$771$	−3.32912e7	−0.00261601
$772$	4.36740e9	0.341635
$773$	−1.14315e10	−0.890172	−0.445086	−	0.895488i	$-0.646827\pi$
$773$	−1.14315e10	−0.890172	−0.445086	+	0.895488i	$0.646827\pi$
$774$	−1.60002e10	−1.24031
$775$	0	0
$776$	−4.40674e9	−0.338533
$777$	4.94891e7	0.00378474
$778$	−4.79967e9	−0.365412
$779$	5.15842e9	0.390963
$780$	0	0
$781$	1.33294e10	1.00122
$782$	1.91876e10	1.43482
$783$	1.48952e7	0.00110887
$784$	3.74421e9	0.277494
$785$	0	0
$786$	−9.03082e7	−0.00663358
$787$	−8.94991e9	−0.654496	−0.327248	−	0.944938i	$-0.606121\pi$
$787$	−8.94991e9	−0.654496	−0.327248	+	0.944938i	$0.606121\pi$
$788$	−2.76555e9	−0.201344
$789$	−9.89618e6	−0.000717296 0
$790$	0	0
$791$	1.06912e10	0.768083
$792$	−1.25841e10	−0.900088
$793$	1.70652e10	1.21522
$794$	−1.79810e10	−1.27480
$795$	0	0
$796$	−1.73781e9	−0.122126
$797$	−1.21850e10	−0.852556	−0.426278	−	0.904592i	$-0.640175\pi$
$797$	−1.21850e10	−0.852556	−0.426278	+	0.904592i	$0.640175\pi$
$798$	8.52854e7	0.00594107
$799$	−3.92277e10	−2.72069
$800$	0	0
$801$	2.30724e10	1.58627
$802$	1.54014e10	1.05426
$803$	4.38523e9	0.298874
$804$	−626115.	−4.24872e−5 0
$805$	0	0
$806$	1.09916e10	0.739412
$807$	−2.39091e7	−0.00160142
$808$	−1.24963e10	−0.833378
$809$	4.06386e9	0.269848	0.134924	−	0.990856i	$-0.456921\pi$
$809$	4.06386e9	0.269848	0.134924	+	0.990856i	$0.456921\pi$
$810$	0	0
$811$	5.23185e9	0.344415	0.172208	−	0.985061i	$-0.444910\pi$
$811$	5.23185e9	0.344415	0.172208	+	0.985061i	$0.444910\pi$
$812$	3.47812e8	0.0227981
$813$	−3.17236e7	−0.00207045
$814$	−1.15712e10	−0.751957
$815$	0	0
$816$	−9.70173e7	−0.00625077
$817$	4.22449e10	2.71017
$818$	1.07642e10	0.687617
$819$	1.00614e10	0.639976
$820$	0	0
$821$	1.98989e10	1.25495	0.627476	−	0.778636i	$-0.284088\pi$
$821$	1.98989e10	1.25495	0.627476	+	0.778636i	$0.284088\pi$
$822$	1.21415e7	0.000762468 0
$823$	1.06783e10	0.667734	0.333867	−	0.942620i	$-0.391646\pi$
$823$	1.06783e10	0.667734	0.333867	+	0.942620i	$0.391646\pi$
$824$	9.89060e9	0.615853
$825$	0	0
$826$	−1.04007e10	−0.642142
$827$	1.97775e10	1.21591	0.607955	−	0.793971i	$-0.291990\pi$
$827$	1.97775e10	1.21591	0.607955	+	0.793971i	$0.291990\pi$
$828$	3.49119e9	0.213731
$829$	1.16935e10	0.712862	0.356431	−	0.934322i	$-0.383993\pi$
$829$	1.16935e10	0.712862	0.356431	+	0.934322i	$0.383993\pi$
$830$	0	0
$831$	−1.36163e8	−0.00823104
$832$	−1.52719e10	−0.919308
$833$	−1.30371e10	−0.781492
$834$	−5.89909e7	−0.00352131
$835$	0	0
$836$	6.64237e9	0.393189
$837$	1.64052e8	0.00967036
$838$	1.25248e10	0.735220
$839$	−4.24101e8	−0.0247915	−0.0123957	−	0.999923i	$-0.503946\pi$
$839$	−4.24101e8	−0.0247915	−0.0123957	+	0.999923i	$0.503946\pi$
$840$	0	0
$841$	−1.70091e10	−0.986044
$842$	−1.65959e10	−0.958094
$843$	−1.84891e7	−0.00106296
$844$	2.24269e9	0.128402
$845$	0	0
$846$	2.14273e10	1.21667
$847$	−4.21505e9	−0.238348
$848$	5.08967e9	0.286618
$849$	−3.79223e7	−0.00212675
$850$	0	0
$851$	1.60575e10	0.893152
$852$	2.54915e7	0.00141207
$853$	2.46819e10	1.36163	0.680813	−	0.732458i	$-0.261627\pi$
$853$	2.46819e10	1.36163	0.680813	+	0.732458i	$0.261627\pi$
$854$	−1.78547e10	−0.980957
$855$	0	0
$856$	7.06969e9	0.385249
$857$	1.02379e10	0.555623	0.277811	−	0.960636i	$-0.410391\pi$
$857$	1.02379e10	0.555623	0.277811	+	0.960636i	$0.410391\pi$
$858$	5.18180e7	0.00280075
$859$	2.20251e10	1.18561	0.592806	−	0.805346i	$-0.298020\pi$
$859$	2.20251e10	1.18561	0.592806	+	0.805346i	$0.298020\pi$
$860$	0	0
$861$	1.40247e7	0.000748831 0
$862$	−1.25716e10	−0.668518
$863$	−2.78674e10	−1.47591	−0.737953	−	0.674852i	$-0.764207\pi$
$863$	−2.78674e10	−1.47591	−0.737953	+	0.674852i	$0.764207\pi$
$864$	−8.66436e7	−0.00457023
$865$	0	0
$866$	2.98208e10	1.56030
$867$	2.47748e8	0.0129105
$868$	3.83071e9	0.198820
$869$	9.87940e9	0.510695
$870$	0	0
$871$	−5.85473e8	−0.0300223
$872$	2.84116e10	1.45107
$873$	−6.14764e9	−0.312722
$874$	2.76722e10	1.40202
$875$	0	0
$876$	8.38646e6	0.000421516 0
$877$	−3.48660e10	−1.74543	−0.872717	−	0.488227i	$-0.837644\pi$
$877$	−3.48660e10	−1.74543	−0.872717	+	0.488227i	$0.837644\pi$
$878$	1.71011e10	0.852693
$879$	−1.72780e8	−0.00858091
$880$	0	0
$881$	−2.66479e10	−1.31295	−0.656473	−	0.754349i	$-0.727953\pi$
$881$	−2.66479e10	−1.31295	−0.656473	+	0.754349i	$0.727953\pi$
$882$	7.12126e9	0.349476
$883$	1.27771e10	0.624554	0.312277	−	0.949991i	$-0.398908\pi$
$883$	1.27771e10	0.624554	0.312277	+	0.949991i	$0.398908\pi$
$884$	8.23640e9	0.401009
$885$	0	0
$886$	3.20716e9	0.154918
$887$	2.02999e10	0.976698	0.488349	−	0.872648i	$-0.337599\pi$
$887$	2.02999e10	0.976698	0.488349	+	0.872648i	$0.337599\pi$
$888$	−1.10691e8	−0.00530480
$889$	2.02072e10	0.964606
$890$	0	0
$891$	−1.75552e10	−0.831443
$892$	−4.68847e9	−0.221184
$893$	−5.65740e10	−2.65850
$894$	9.37371e7	0.00438763
$895$	0	0
$896$	7.88153e9	0.366043
$897$	−7.19086e7	−0.00332665
$898$	1.73244e10	0.798345
$899$	2.65148e9	0.121711
$900$	0	0
$901$	−1.77220e10	−0.807188
$902$	−3.27917e9	−0.148779
$903$	1.14856e8	0.00519093
$904$	−2.39128e10	−1.07657
$905$	0	0
$906$	−2.45959e7	−0.00109879
$907$	2.64331e10	1.17631	0.588155	−	0.808748i	$-0.299854\pi$
$907$	2.64331e10	1.17631	0.588155	+	0.808748i	$0.299854\pi$
$908$	3.99862e9	0.177260
$909$	−1.74330e10	−0.769838
$910$	0	0
$911$	1.94727e10	0.853320	0.426660	−	0.904412i	$-0.359690\pi$
$911$	1.94727e10	0.853320	0.426660	+	0.904412i	$0.359690\pi$
$912$	−1.39918e8	−0.00610790
$913$	−1.14832e10	−0.499360
$914$	3.59339e9	0.155666
$915$	0	0
$916$	7.32367e9	0.314844
$917$	−2.94307e10	−1.26040
$918$	−3.69046e8	−0.0157446
$919$	−3.05800e10	−1.29967	−0.649836	−	0.760074i	$-0.725163\pi$
$919$	−3.05800e10	−1.29967	−0.649836	+	0.760074i	$0.725163\pi$
$920$	0	0
$921$	−7.17461e7	−0.00302614
$922$	−8.67600e9	−0.364553
$923$	2.38369e10	0.997799
$924$	1.80593e7	0.000753094 0
$925$	0	0
$926$	6.71409e9	0.277874
$927$	1.37979e10	0.568898
$928$	−1.40037e9	−0.0575207
$929$	2.58465e10	1.05766	0.528831	−	0.848727i	$-0.322631\pi$
$929$	2.58465e10	1.05766	0.528831	+	0.848727i	$0.322631\pi$
$930$	0	0
$931$	−1.88021e10	−0.763629
$932$	−6.22787e9	−0.251990
$933$	−9.36303e7	−0.00377425
$934$	−4.51703e9	−0.181401
$935$	0	0
$936$	−2.25041e10	−0.897009
$937$	2.11341e10	0.839259	0.419630	−	0.907695i	$-0.362160\pi$
$937$	2.11341e10	0.839259	0.419630	+	0.907695i	$0.362160\pi$
$938$	6.12560e8	0.0242348
$939$	7.16517e7	0.00282421
$940$	0	0
$941$	1.80375e10	0.705687	0.352843	−	0.935682i	$-0.385215\pi$
$941$	1.80375e10	0.705687	0.352843	+	0.935682i	$0.385215\pi$
$942$	8.14931e6	0.000317645 0
$943$	4.55055e9	0.176715
$944$	1.70632e10	0.660173
$945$	0	0
$946$	−2.68548e10	−1.03134
$947$	−5.05378e10	−1.93371	−0.966855	−	0.255325i	$-0.917818\pi$
$947$	−5.05378e10	−1.93371	−0.966855	+	0.255325i	$0.917818\pi$
$948$	1.88937e7	0.000720257 0
$949$	7.84209e9	0.297852
$950$	0	0
$951$	−1.04335e8	−0.00393366
$952$	−4.31051e10	−1.61920
$953$	−3.60618e10	−1.34965	−0.674826	−	0.737977i	$-0.735781\pi$
$953$	−3.60618e10	−1.34965	−0.674826	+	0.737977i	$0.735781\pi$
$954$	9.68025e9	0.360967
$955$	0	0
$956$	1.15316e10	0.426861
$957$	1.25000e7	0.000461017 0
$958$	1.12705e10	0.414156
$959$	3.95682e9	0.144871
$960$	0	0
$961$	1.69003e9	0.0614276
$962$	−2.06927e10	−0.749384
$963$	9.86259e9	0.355876
$964$	−9.44104e9	−0.339430
$965$	0	0
$966$	7.52354e7	0.00268536
$967$	5.55248e10	1.97467	0.987334	−	0.158656i	$-0.0507161\pi$
$967$	5.55248e10	1.97467	0.987334	+	0.158656i	$0.0507161\pi$
$968$	9.42774e9	0.334075
$969$	4.87187e8	0.0172013
$970$	0	0
$971$	−2.82128e10	−0.988961	−0.494481	−	0.869189i	$-0.664642\pi$
$971$	−2.82128e10	−0.988961	−0.494481	+	0.869189i	$0.664642\pi$
$972$	−1.00723e8	−0.00351800
$973$	−1.92247e10	−0.669058
$974$	−3.02950e9	−0.105055
$975$	0	0
$976$	2.92922e10	1.00850
$977$	2.82394e9	0.0968779	0.0484389	−	0.998826i	$-0.484575\pi$
$977$	2.82394e9	0.0968779	0.0484389	+	0.998826i	$0.484575\pi$
$978$	2.13373e8	0.00729380
$979$	3.87248e10	1.31901
$980$	0	0
$981$	3.96357e10	1.34043
$982$	−2.73143e10	−0.920449
$983$	−3.19009e10	−1.07119	−0.535594	−	0.844476i	$-0.679912\pi$
$983$	−3.19009e10	−1.07119	−0.535594	+	0.844476i	$0.679912\pi$
$984$	−3.13689e7	−0.00104958
$985$	0	0
$986$	−5.96468e9	−0.198161
$987$	−1.53814e8	−0.00509196
$988$	1.18785e10	0.391843
$989$	3.72668e10	1.22500
$990$	0	0
$991$	4.95244e10	1.61645	0.808224	−	0.588875i	$-0.200429\pi$
$991$	4.95244e10	1.61645	0.808224	+	0.588875i	$0.200429\pi$
$992$	−1.54233e10	−0.501633
$993$	5.46334e7	0.00177066
$994$	−2.49397e10	−0.805450
$995$	0	0
$996$	−2.19607e7	−0.000704270 0
$997$	2.93484e10	0.937889	0.468945	−	0.883228i	$-0.344634\pi$
$997$	2.93484e10	0.937889	0.468945	+	0.883228i	$0.344634\pi$
$998$	−2.36809e10	−0.754123
$999$	−3.08845e8	−0.00980078

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.8.a.a.1.17	✓	22
5.4	even	2		625.8.a.b.1.6	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.8.a.a.1.17	✓	22	1.1	even	1	trivial
625.8.a.b.1.6	yes	22	5.4	even	2

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:	\( N \)	\(=\)	\( 625 = 5^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	625.a (trivial)

\(f(q)\)	\(=\)	\(q+9.79880 q^{2} +0.219481 q^{3} -31.9834 q^{4} +2.15065 q^{6} +700.879 q^{7} -1567.65 q^{8} -2186.95 q^{9} +O(q^{10})\)	\(q+9.79880 q^{2} +0.219481 q^{3} -31.9834 q^{4} +2.15065 q^{6} +700.879 q^{7} -1567.65 q^{8} -2186.95 q^{9} -3670.59 q^{11} -7.01975 q^{12} -6564.10 q^{13} +6867.78 q^{14} -11267.2 q^{16} +39231.7 q^{17} -21429.5 q^{18} +56579.9 q^{19} +153.830 q^{21} -35967.4 q^{22} +49912.5 q^{23} -344.068 q^{24} -64320.3 q^{26} -959.998 q^{27} -22416.5 q^{28} -15515.9 q^{29} -170888. q^{31} +90253.8 q^{32} -805.624 q^{33} +384424. q^{34} +69946.2 q^{36} +321714. q^{37} +554415. q^{38} -1440.69 q^{39} +91170.6 q^{41} +1507.35 q^{42} +746642. q^{43} +117398. q^{44} +489083. q^{46} -999896. q^{47} -2472.93 q^{48} -332311. q^{49} +8610.61 q^{51} +209942. q^{52} -451725. q^{53} -9406.84 q^{54} -1.09873e6 q^{56} +12418.2 q^{57} -152037. q^{58} -1.51441e6 q^{59} -2.59978e6 q^{61} -1.67450e6 q^{62} -1.53279e6 q^{63} +2.32658e6 q^{64} -7894.15 q^{66} +89193.3 q^{67} -1.25477e6 q^{68} +10954.8 q^{69} -3.63140e6 q^{71} +3.42837e6 q^{72} -1.19469e6 q^{73} +3.15241e6 q^{74} -1.80962e6 q^{76} -2.57264e6 q^{77} -14117.1 q^{78} -2.69150e6 q^{79} +4.78265e6 q^{81} +893363. q^{82} +3.12842e6 q^{83} -4920.00 q^{84} +7.31620e6 q^{86} -3405.44 q^{87} +5.75419e6 q^{88} -1.05500e7 q^{89} -4.60064e6 q^{91} -1.59637e6 q^{92} -37506.6 q^{93} -9.79779e6 q^{94} +19809.0 q^{96} +2.81105e6 q^{97} -3.25625e6 q^{98} +8.02740e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 23 q^{2} - 121 q^{3} + 1061 q^{4} - 431 q^{6} - 843 q^{7} - 4980 q^{8} + 14799 q^{9}+O(q^{10}) \) 22 * q - 23 * q^2 - 121 * q^3 + 1061 * q^4 - 431 * q^6 - 843 * q^7 - 4980 * q^8 + 14799 * q^9	\( 22 q - 23 q^{2} - 121 q^{3} + 1061 q^{4} - 431 q^{6} - 843 q^{7} - 4980 q^{8} + 14799 q^{9} - 781 q^{11} - 12608 q^{12} - 14686 q^{13} + 20762 q^{14} + 16117 q^{16} - 45648 q^{17} - 47171 q^{18} + 6185 q^{19} + 14149 q^{21} + 71124 q^{22} - 126921 q^{23} + 271570 q^{24} + 304129 q^{26} - 546520 q^{27} - 2019 q^{28} + 59330 q^{29} + 394804 q^{31} + 74397 q^{32} - 49067 q^{33} - 286938 q^{34} - 287278 q^{36} + 792122 q^{37} - 1338860 q^{38} + 635223 q^{39} - 160466 q^{41} - 3420191 q^{42} - 1527256 q^{43} - 1154853 q^{44} + 2653604 q^{46} - 1300863 q^{47} - 1885241 q^{48} + 1652981 q^{49} + 3408539 q^{51} - 1423303 q^{52} - 755656 q^{53} + 3117755 q^{54} - 2132625 q^{56} - 3026890 q^{57} + 5941470 q^{58} - 1548370 q^{59} - 6029951 q^{61} - 79936 q^{62} + 6962459 q^{63} - 5858224 q^{64} - 5380407 q^{66} - 7608838 q^{67} - 10737124 q^{68} + 13519553 q^{69} + 9483549 q^{71} - 6806340 q^{72} - 13548801 q^{73} - 15016023 q^{74} + 19635315 q^{76} + 2145019 q^{77} - 17222402 q^{78} + 10769160 q^{79} + 2757382 q^{81} - 12087571 q^{82} + 9632744 q^{83} + 19168542 q^{84} - 8511651 q^{86} - 298330 q^{87} + 21641425 q^{88} - 10850545 q^{89} - 6648131 q^{91} - 52978503 q^{92} - 51294822 q^{93} + 5863777 q^{94} + 2611654 q^{96} - 11579993 q^{97} - 7468074 q^{98} - 13552997 q^{99}+O(q^{100}) \) 22 * q - 23 * q^2 - 121 * q^3 + 1061 * q^4 - 431 * q^6 - 843 * q^7 - 4980 * q^8 + 14799 * q^9 - 781 * q^11 - 12608 * q^12 - 14686 * q^13 + 20762 * q^14 + 16117 * q^16 - 45648 * q^17 - 47171 * q^18 + 6185 * q^19 + 14149 * q^21 + 71124 * q^22 - 126921 * q^23 + 271570 * q^24 + 304129 * q^26 - 546520 * q^27 - 2019 * q^28 + 59330 * q^29 + 394804 * q^31 + 74397 * q^32 - 49067 * q^33 - 286938 * q^34 - 287278 * q^36 + 792122 * q^37 - 1338860 * q^38 + 635223 * q^39 - 160466 * q^41 - 3420191 * q^42 - 1527256 * q^43 - 1154853 * q^44 + 2653604 * q^46 - 1300863 * q^47 - 1885241 * q^48 + 1652981 * q^49 + 3408539 * q^51 - 1423303 * q^52 - 755656 * q^53 + 3117755 * q^54 - 2132625 * q^56 - 3026890 * q^57 + 5941470 * q^58 - 1548370 * q^59 - 6029951 * q^61 - 79936 * q^62 + 6962459 * q^63 - 5858224 * q^64 - 5380407 * q^66 - 7608838 * q^67 - 10737124 * q^68 + 13519553 * q^69 + 9483549 * q^71 - 6806340 * q^72 - 13548801 * q^73 - 15016023 * q^74 + 19635315 * q^76 + 2145019 * q^77 - 17222402 * q^78 + 10769160 * q^79 + 2757382 * q^81 - 12087571 * q^82 + 9632744 * q^83 + 19168542 * q^84 - 8511651 * q^86 - 298330 * q^87 + 21641425 * q^88 - 10850545 * q^89 - 6648131 * q^91 - 52978503 * q^92 - 51294822 * q^93 + 5863777 * q^94 + 2611654 * q^96 - 11579993 * q^97 - 7468074 * q^98 - 13552997 * q^99

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