LMFDB - Embedded newform 625.8.a.e.1.7

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:	$48$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	$q-18.8926 q^{2} -78.3784 q^{3} +228.931 q^{4} +1480.77 q^{6} -1117.51 q^{7} -1906.86 q^{8} +3956.18 q^{9} +O(q^{10})$	\(q-18.8926 q^{2} -78.3784 q^{3} +228.931 q^{4} +1480.77 q^{6} -1117.51 q^{7} -1906.86 q^{8} +3956.18 q^{9} +1397.79 q^{11} -17943.3 q^{12} +14895.0 q^{13} +21112.7 q^{14} +6722.36 q^{16} -31479.8 q^{17} -74742.6 q^{18} +15261.7 q^{19} +87588.7 q^{21} -26407.9 q^{22} +46608.3 q^{23} +149457. q^{24} -281405. q^{26} -138666. q^{27} -255833. q^{28} +229981. q^{29} -242.785 q^{31} +117075. q^{32} -109556. q^{33} +594736. q^{34} +905694. q^{36} -438499. q^{37} -288334. q^{38} -1.16744e6 q^{39} -416899. q^{41} -1.65478e6 q^{42} +120210. q^{43} +319997. q^{44} -880554. q^{46} -522344. q^{47} -526888. q^{48} +425285. q^{49} +2.46734e6 q^{51} +3.40993e6 q^{52} -689500. q^{53} +2.61976e6 q^{54} +2.13093e6 q^{56} -1.19619e6 q^{57} -4.34495e6 q^{58} -1.62490e6 q^{59} +2.98861e6 q^{61} +4586.85 q^{62} -4.42107e6 q^{63} -3.07231e6 q^{64} +2.06981e6 q^{66} -3.10014e6 q^{67} -7.20671e6 q^{68} -3.65309e6 q^{69} -2.39052e6 q^{71} -7.54388e6 q^{72} +422147. q^{73} +8.28439e6 q^{74} +3.49388e6 q^{76} -1.56204e6 q^{77} +2.20561e7 q^{78} -5.57226e6 q^{79} +2.21622e6 q^{81} +7.87631e6 q^{82} +1.23102e6 q^{83} +2.00518e7 q^{84} -2.27109e6 q^{86} -1.80256e7 q^{87} -2.66538e6 q^{88} -108629. q^{89} -1.66453e7 q^{91} +1.06701e7 q^{92} +19029.1 q^{93} +9.86845e6 q^{94} -9.17615e6 q^{96} +1.29133e7 q^{97} -8.03474e6 q^{98} +5.52990e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 25 q^{2} - 95 q^{3} + 3419 q^{4} + 431 q^{6} - 4030 q^{7} - 3375 q^{8} + 36231 q^{9}+O(q^{10}) $ 48 * q - 25 * q^2 - 95 * q^3 + 3419 * q^4 + 431 * q^6 - 4030 * q^7 - 3375 * q^8 + 36231 * q^9	\( 48 q - 25 q^{2} - 95 q^{3} + 3419 q^{4} + 431 q^{6} - 4030 q^{7} - 3375 q^{8} + 36231 q^{9} + 781 q^{11} - 3925 q^{12} - 4290 q^{13} - 20762 q^{14} + 270603 q^{16} - 75075 q^{17} - 89950 q^{18} + 17750 q^{19} - 48034 q^{21} - 331305 q^{22} - 343890 q^{23} - 271570 q^{24} - 304129 q^{26} - 474740 q^{27} - 1146535 q^{28} - 59330 q^{29} - 385989 q^{31} - 1887300 q^{32} - 879805 q^{33} + 286938 q^{34} + 3553198 q^{36} - 935610 q^{37} - 984745 q^{38} - 294888 q^{39} + 160466 q^{41} + 783725 q^{42} + 146400 q^{43} + 2261658 q^{44} - 2639009 q^{46} - 4446810 q^{47} - 3994240 q^{48} + 7532484 q^{49} - 2294894 q^{51} - 4582065 q^{52} - 3977030 q^{53} - 3979475 q^{54} - 743430 q^{56} - 2455430 q^{57} - 14413560 q^{58} - 1614425 q^{59} + 7720866 q^{61} - 20362850 q^{62} - 26297840 q^{63} + 21801809 q^{64} + 945327 q^{66} - 3017910 q^{67} - 17494265 q^{68} - 13519553 q^{69} - 9483549 q^{71} - 21929370 q^{72} + 388070 q^{73} + 16144878 q^{74} - 13507955 q^{76} - 25473115 q^{77} + 3108110 q^{78} - 10950620 q^{79} + 34443488 q^{81} + 354040 q^{82} - 47217920 q^{83} - 27843102 q^{84} + 20021766 q^{86} - 56120960 q^{87} - 54397660 q^{88} + 10850545 q^{89} - 8553794 q^{91} + 20734425 q^{92} + 11206260 q^{93} - 52545997 q^{94} - 28125034 q^{96} - 32784020 q^{97} - 14131170 q^{98} + 27513602 q^{99}+O(q^{100}) \) 48 * q - 25 * q^2 - 95 * q^3 + 3419 * q^4 + 431 * q^6 - 4030 * q^7 - 3375 * q^8 + 36231 * q^9 + 781 * q^11 - 3925 * q^12 - 4290 * q^13 - 20762 * q^14 + 270603 * q^16 - 75075 * q^17 - 89950 * q^18 + 17750 * q^19 - 48034 * q^21 - 331305 * q^22 - 343890 * q^23 - 271570 * q^24 - 304129 * q^26 - 474740 * q^27 - 1146535 * q^28 - 59330 * q^29 - 385989 * q^31 - 1887300 * q^32 - 879805 * q^33 + 286938 * q^34 + 3553198 * q^36 - 935610 * q^37 - 984745 * q^38 - 294888 * q^39 + 160466 * q^41 + 783725 * q^42 + 146400 * q^43 + 2261658 * q^44 - 2639009 * q^46 - 4446810 * q^47 - 3994240 * q^48 + 7532484 * q^49 - 2294894 * q^51 - 4582065 * q^52 - 3977030 * q^53 - 3979475 * q^54 - 743430 * q^56 - 2455430 * q^57 - 14413560 * q^58 - 1614425 * q^59 + 7720866 * q^61 - 20362850 * q^62 - 26297840 * q^63 + 21801809 * q^64 + 945327 * q^66 - 3017910 * q^67 - 17494265 * q^68 - 13519553 * q^69 - 9483549 * q^71 - 21929370 * q^72 + 388070 * q^73 + 16144878 * q^74 - 13507955 * q^76 - 25473115 * q^77 + 3108110 * q^78 - 10950620 * q^79 + 34443488 * q^81 + 354040 * q^82 - 47217920 * q^83 - 27843102 * q^84 + 20021766 * q^86 - 56120960 * q^87 - 54397660 * q^88 + 10850545 * q^89 - 8553794 * q^91 + 20734425 * q^92 + 11206260 * q^93 - 52545997 * q^94 - 28125034 * q^96 - 32784020 * q^97 - 14131170 * q^98 + 27513602 * 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 $

$2$	−18.8926	−1.66989	−0.834944	−	0.550335i	$-0.814500\pi$
$2$	−18.8926	−1.66989	−0.834944	+	0.550335i	$0.814500\pi$
$3$	−78.3784	−1.67599	−0.837997	−	0.545676i	$-0.816273\pi$
$3$	−78.3784	−1.67599	−0.837997	+	0.545676i	$0.816273\pi$
$4$	228.931	1.78853
$5$	0	0
$5$	0	0
$6$	1480.77	2.79872
$7$	−1117.51	−1.23143	−0.615713	−	0.787971i	$-0.711132\pi$
$7$	−1117.51	−1.23143	−0.615713	+	0.787971i	$0.711132\pi$
$8$	−1906.86	−1.31675
$9$	3956.18	1.80895
$10$	0	0
$11$	1397.79	0.316641	0.158320	−	0.987388i	$-0.449392\pi$
$11$	1397.79	0.316641	0.158320	+	0.987388i	$0.449392\pi$
$12$	−17943.3	−2.99756
$13$	14895.0	1.88035	0.940174	−	0.340696i	$-0.110663\pi$
$13$	14895.0	1.88035	0.940174	+	0.340696i	$0.110663\pi$
$14$	21112.7	2.05634
$15$	0	0
$16$	6722.36	0.410300
$17$	−31479.8	−1.55403	−0.777017	−	0.629479i	$-0.783268\pi$
$17$	−31479.8	−1.55403	−0.777017	+	0.629479i	$0.783268\pi$
$18$	−74742.6	−3.02075
$19$	15261.7	0.510464	0.255232	−	0.966880i	$-0.417848\pi$
$19$	15261.7	0.510464	0.255232	+	0.966880i	$0.417848\pi$
$20$	0	0
$21$	87588.7	2.06386
$22$	−26407.9	−0.528754
$23$	46608.3	0.798760	0.399380	−	0.916785i	$-0.369225\pi$
$23$	46608.3	0.798760	0.399380	+	0.916785i	$0.369225\pi$
$24$	149457.	2.20687
$25$	0	0
$26$	−281405.	−3.13997
$27$	−138666.	−1.35580
$28$	−255833.	−2.20244
$29$	229981.	1.75105	0.875527	−	0.483169i	$-0.160514\pi$
$29$	229981.	1.75105	0.875527	+	0.483169i	$0.160514\pi$
$30$	0	0
$31$	−242.785	−0.00146371	−0.000731857	−	1.00000i	$-0.500233\pi$
$31$	−242.785	−0.00146371	−0.000731857		1.00000i	$0.500233\pi$
$32$	117075.	0.631595
$33$	−109556.	−0.530687
$34$	594736.	2.59506
$35$	0	0
$36$	905694.	3.23536
$37$	−438499.	−1.42319	−0.711594	−	0.702591i	$-0.752026\pi$
$37$	−438499.	−1.42319	−0.711594	+	0.702591i	$0.752026\pi$
$38$	−288334.	−0.852418
$39$	−1.16744e6	−3.15145
$40$	0	0
$41$	−416899.	−0.944685	−0.472343	−	0.881415i	$-0.656591\pi$
$41$	−416899.	−0.944685	−0.472343	+	0.881415i	$0.656591\pi$
$42$	−1.65478e6	−3.44642
$43$	120210.	0.230570	0.115285	−	0.993332i	$-0.463222\pi$
$43$	120210.	0.230570	0.115285	+	0.993332i	$0.463222\pi$
$44$	319997.	0.566320
$45$	0	0
$46$	−880554.	−1.33384
$47$	−522344.	−0.733861	−0.366931	−	0.930248i	$-0.619591\pi$
$47$	−522344.	−0.733861	−0.366931	+	0.930248i	$0.619591\pi$
$48$	−526888.	−0.687661
$49$	425285.	0.516408
$50$	0	0
$51$	2.46734e6	2.60455
$52$	3.40993e6	3.36305
$53$	−689500.	−0.636163	−0.318082	−	0.948063i	$-0.603039\pi$
$53$	−689500.	−0.636163	−0.318082	+	0.948063i	$0.603039\pi$
$54$	2.61976e6	2.26403
$55$	0	0
$56$	2.13093e6	1.62148
$57$	−1.19619e6	−0.855534
$58$	−4.34495e6	−2.92406
$59$	−1.62490e6	−1.03002	−0.515010	−	0.857184i	$-0.672212\pi$
$59$	−1.62490e6	−1.03002	−0.515010	+	0.857184i	$0.672212\pi$
$60$	0	0
$61$	2.98861e6	1.68584	0.842918	−	0.538042i	$-0.180836\pi$
$61$	2.98861e6	1.68584	0.842918	+	0.538042i	$0.180836\pi$
$62$	4586.85	0.00244424
$63$	−4.42107e6	−2.22759
$64$	−3.07231e6	−1.46499
$65$	0	0
$66$	2.06981e6	0.886189
$67$	−3.10014e6	−1.25927	−0.629636	−	0.776890i	$-0.716796\pi$
$67$	−3.10014e6	−1.25927	−0.629636	+	0.776890i	$0.716796\pi$
$68$	−7.20671e6	−2.77943
$69$	−3.65309e6	−1.33872
$70$	0	0
$71$	−2.39052e6	−0.792662	−0.396331	−	0.918108i	$-0.629717\pi$
$71$	−2.39052e6	−0.792662	−0.396331	+	0.918108i	$0.629717\pi$
$72$	−7.54388e6	−2.38194
$73$	422147.	0.127009	0.0635043	−	0.997982i	$-0.479772\pi$
$73$	422147.	0.127009	0.0635043	+	0.997982i	$0.479772\pi$
$74$	8.28439e6	2.37657
$75$	0	0
$76$	3.49388e6	0.912978
$77$	−1.56204e6	−0.389919
$78$	2.20561e7	5.26257
$79$	−5.57226e6	−1.27156	−0.635779	−	0.771871i	$-0.719321\pi$
$79$	−5.57226e6	−1.27156	−0.635779	+	0.771871i	$0.719321\pi$
$80$	0	0
$81$	2.21622e6	0.463357
$82$	7.87631e6	1.57752
$83$	1.23102e6	0.236314	0.118157	−	0.992995i	$-0.462301\pi$
$83$	1.23102e6	0.236314	0.118157	+	0.992995i	$0.462301\pi$
$84$	2.00518e7	3.69127
$85$	0	0
$86$	−2.27109e6	−0.385026
$87$	−1.80256e7	−2.93475
$88$	−2.66538e6	−0.416937
$89$	−108629.	−0.0163335	−0.00816677	−	0.999967i	$-0.502600\pi$
$89$	−108629.	−0.0163335	−0.00816677	+	0.999967i	$0.502600\pi$
$90$	0	0
$91$	−1.66453e7	−2.31551
$92$	1.06701e7	1.42860
$93$	19029.1	0.00245317
$94$	9.86845e6	1.22547
$95$	0	0
$96$	−9.17615e6	−1.05855
$97$	1.29133e7	1.43660	0.718302	−	0.695732i	$-0.244920\pi$
$97$	1.29133e7	1.43660	0.718302	+	0.695732i	$0.244920\pi$
$98$	−8.03474e6	−0.862344
$99$	5.52990e6	0.572788
$100$	0	0
$101$	553287.	0.0534350	0.0267175	−	0.999643i	$-0.491495\pi$
$101$	553287.	0.0534350	0.0267175	+	0.999643i	$0.491495\pi$
$102$	−4.66145e7	−4.34931
$103$	8.87788e6	0.800533	0.400266	−	0.916399i	$-0.368918\pi$
$103$	8.87788e6	0.800533	0.400266	+	0.916399i	$0.368918\pi$
$104$	−2.84026e7	−2.47595
$105$	0	0
$106$	1.30265e7	1.06232
$107$	−6.19515e6	−0.488887	−0.244444	−	0.969663i	$-0.578605\pi$
$107$	−6.19515e6	−0.488887	−0.244444	+	0.969663i	$0.578605\pi$
$108$	−3.17449e7	−2.42488
$109$	−4.70867e6	−0.348262	−0.174131	−	0.984723i	$-0.555712\pi$
$109$	−4.70867e6	−0.348262	−0.174131	+	0.984723i	$0.555712\pi$
$110$	0	0
$111$	3.43688e7	2.38525
$112$	−7.51230e6	−0.505254
$113$	1.75418e7	1.14367	0.571833	−	0.820370i	$-0.306232\pi$
$113$	1.75418e7	1.14367	0.571833	+	0.820370i	$0.306232\pi$
$114$	2.25991e7	1.42865
$115$	0	0
$116$	5.26499e7	3.13181
$117$	5.89272e7	3.40146
$118$	3.06987e7	1.72002
$119$	3.51790e7	1.91368
$120$	0	0
$121$	−1.75334e7	−0.899739
$122$	−5.64628e7	−2.81516
$123$	3.26759e7	1.58329
$124$	−55581.1	−0.00261789
$125$	0	0
$126$	8.35256e7	3.71983
$127$	3.55473e7	1.53990	0.769952	−	0.638102i	$-0.220280\pi$
$127$	3.55473e7	1.53990	0.769952	+	0.638102i	$0.220280\pi$
$128$	4.30585e7	1.81478
$129$	−9.42190e6	−0.386433
$130$	0	0
$131$	1.43959e7	0.559485	0.279743	−	0.960075i	$-0.409751\pi$
$131$	1.43959e7	0.559485	0.279743	+	0.960075i	$0.409751\pi$
$132$	−2.50809e7	−0.949148
$133$	−1.70551e7	−0.628598
$134$	5.85698e7	2.10284
$135$	0	0
$136$	6.00275e7	2.04628
$137$	9.12485e6	0.303182	0.151591	−	0.988443i	$-0.451560\pi$
$137$	9.12485e6	0.303182	0.151591	+	0.988443i	$0.451560\pi$
$138$	6.90165e7	2.23551
$139$	−1.72294e6	−0.0544150	−0.0272075	−	0.999630i	$-0.508661\pi$
$139$	−1.72294e6	−0.0544150	−0.0272075	+	0.999630i	$0.508661\pi$
$140$	0	0
$141$	4.09405e7	1.22995
$142$	4.51632e7	1.32366
$143$	2.08200e7	0.595394
$144$	2.65949e7	0.742214
$145$	0	0
$146$	−7.97546e6	−0.212090
$147$	−3.33331e7	−0.865497
$148$	−1.00386e8	−2.54541
$149$	−4.56720e6	−0.113109	−0.0565546	−	0.998400i	$-0.518011\pi$
$149$	−4.56720e6	−0.113109	−0.0565546	+	0.998400i	$0.518011\pi$
$150$	0	0
$151$	4.86134e7	1.14904	0.574522	−	0.818489i	$-0.305188\pi$
$151$	4.86134e7	1.14904	0.574522	+	0.818489i	$0.305188\pi$
$152$	−2.91019e7	−0.672154
$153$	−1.24540e8	−2.81117
$154$	2.95110e7	0.651121
$155$	0	0
$156$	−2.67265e8	−5.63645
$157$	4.18202e7	0.862457	0.431229	−	0.902243i	$-0.358080\pi$
$157$	4.18202e7	0.862457	0.431229	+	0.902243i	$0.358080\pi$
$158$	1.05275e8	2.12336
$159$	5.40419e7	1.06621
$160$	0	0
$161$	−5.20853e7	−0.983613
$162$	−4.18703e7	−0.773755
$163$	−4.48654e7	−0.811438	−0.405719	−	0.913998i	$-0.632979\pi$
$163$	−4.48654e7	−0.811438	−0.405719	+	0.913998i	$0.632979\pi$
$164$	−9.54412e7	−1.68959
$165$	0	0
$166$	−2.32571e7	−0.394619
$167$	724330.	0.0120345	0.00601726	−	0.999982i	$-0.498085\pi$
$167$	724330.	0.0120345	0.00601726	+	0.999982i	$0.498085\pi$
$168$	−1.67019e8	−2.71759
$169$	1.59112e8	2.53571
$170$	0	0
$171$	6.03780e7	0.923405
$172$	2.75199e7	0.412380
$173$	−1.50092e7	−0.220392	−0.110196	−	0.993910i	$-0.535148\pi$
$173$	−1.50092e7	−0.220392	−0.110196	+	0.993910i	$0.535148\pi$
$174$	3.40551e8	4.90071
$175$	0	0
$176$	9.39643e6	0.129918
$177$	1.27357e8	1.72631
$178$	2.05228e6	0.0272752
$179$	1.00592e8	1.31093	0.655463	−	0.755227i	$-0.272473\pi$
$179$	1.00592e8	1.31093	0.655463	+	0.755227i	$0.272473\pi$
$180$	0	0
$181$	−3.06560e7	−0.384273	−0.192137	−	0.981368i	$-0.561542\pi$
$181$	−3.06560e7	−0.384273	−0.192137	+	0.981368i	$0.561542\pi$
$182$	3.14473e8	3.86664
$183$	−2.34243e8	−2.82545
$184$	−8.88756e7	−1.05177
$185$	0	0
$186$	−359510.	−0.00409653
$187$	−4.40020e7	−0.492070
$188$	−1.19581e8	−1.31253
$189$	1.54960e8	1.66957
$190$	0	0
$191$	−1.76074e8	−1.82843	−0.914214	−	0.405233i	$-0.867190\pi$
$191$	−1.76074e8	−1.82843	−0.914214	+	0.405233i	$0.867190\pi$
$192$	2.40803e8	2.45532
$193$	3.47547e7	0.347987	0.173994	−	0.984747i	$-0.444333\pi$
$193$	3.47547e7	0.347987	0.173994	+	0.984747i	$0.444333\pi$
$194$	−2.43967e8	−2.39897
$195$	0	0
$196$	9.73610e7	0.923610
$197$	1.07729e8	1.00393	0.501964	−	0.864888i	$-0.332611\pi$
$197$	1.07729e8	1.00393	0.501964	+	0.864888i	$0.332611\pi$
$198$	−1.04474e8	−0.956492
$199$	4.14665e7	0.373002	0.186501	−	0.982455i	$-0.440285\pi$
$199$	4.14665e7	0.373002	0.186501	+	0.982455i	$0.440285\pi$
$200$	0	0
$201$	2.42984e8	2.11053
$202$	−1.04530e7	−0.0892305
$203$	−2.57006e8	−2.15629
$204$	5.64851e8	4.65831
$205$	0	0
$206$	−1.67727e8	−1.33680
$207$	1.84391e8	1.44492
$208$	1.00129e8	0.771507
$209$	2.13326e7	0.161634
$210$	0	0
$211$	−7.84090e7	−0.574616	−0.287308	−	0.957838i	$-0.592760\pi$
$211$	−7.84090e7	−0.574616	−0.287308	+	0.957838i	$0.592760\pi$
$212$	−1.57848e8	−1.13779
$213$	1.87365e8	1.32850
$214$	1.17043e8	0.816387
$215$	0	0
$216$	2.64416e8	1.78525
$217$	271315.	0.00180245
$218$	8.89592e7	0.581558
$219$	−3.30872e7	−0.212866
$220$	0	0
$221$	−4.68891e8	−2.92212
$222$	−6.49318e8	−3.98311
$223$	−9.18459e7	−0.554616	−0.277308	−	0.960781i	$-0.589442\pi$
$223$	−9.18459e7	−0.554616	−0.277308	+	0.960781i	$0.589442\pi$
$224$	−1.30832e8	−0.777762
$225$	0	0
$226$	−3.31411e8	−1.90980
$227$	2.39383e7	0.135832	0.0679160	−	0.997691i	$-0.478365\pi$
$227$	2.39383e7	0.135832	0.0679160	+	0.997691i	$0.478365\pi$
$228$	−2.73845e8	−1.53015
$229$	−1.36945e8	−0.753565	−0.376783	−	0.926302i	$-0.622970\pi$
$229$	−1.36945e8	−0.753565	−0.376783	+	0.926302i	$0.622970\pi$
$230$	0	0
$231$	1.22430e8	0.653502
$232$	−4.38542e8	−2.30570
$233$	1.57209e8	0.814201	0.407101	−	0.913383i	$-0.366540\pi$
$233$	1.57209e8	0.814201	0.407101	+	0.913383i	$0.366540\pi$
$234$	−1.11329e9	−5.68006
$235$	0	0
$236$	−3.71992e8	−1.84222
$237$	4.36745e8	2.13112
$238$	−6.64623e8	−3.19563
$239$	−3.12103e8	−1.47878	−0.739392	−	0.673275i	$-0.764887\pi$
$239$	−3.12103e8	−1.47878	−0.739392	+	0.673275i	$0.764887\pi$
$240$	0	0
$241$	−1.24769e8	−0.574179	−0.287089	−	0.957904i	$-0.592688\pi$
$241$	−1.24769e8	−0.574179	−0.287089	+	0.957904i	$0.592688\pi$
$242$	3.31251e8	1.50246
$243$	1.29557e8	0.579216
$244$	6.84188e8	3.01516
$245$	0	0
$246$	−6.17333e8	−2.64391
$247$	2.27323e8	0.959850
$248$	462957.	0.00192735
$249$	−9.64850e7	−0.396061
$250$	0	0
$251$	2.12203e8	0.847020	0.423510	−	0.905891i	$-0.360798\pi$
$251$	2.12203e8	0.847020	0.423510	+	0.905891i	$0.360798\pi$
$252$	−1.01212e9	−3.98410
$253$	6.51485e7	0.252920
$254$	−6.71582e8	−2.57147
$255$	0	0
$256$	−4.20232e8	−1.56549
$257$	−2.55627e8	−0.939381	−0.469690	−	0.882831i	$-0.655634\pi$
$257$	−2.55627e8	−0.939381	−0.469690	+	0.882831i	$0.655634\pi$
$258$	1.78004e8	0.645300
$259$	4.90027e8	1.75255
$260$	0	0
$261$	9.09847e8	3.16757
$262$	−2.71976e8	−0.934278
$263$	−2.17389e8	−0.736871	−0.368436	−	0.929653i	$-0.620106\pi$
$263$	−2.17389e8	−0.736871	−0.368436	+	0.929653i	$0.620106\pi$
$264$	2.08909e8	0.698783
$265$	0	0
$266$	3.22215e8	1.04969
$267$	8.51416e6	0.0273749
$268$	−7.09719e8	−2.25224
$269$	−2.47358e8	−0.774806	−0.387403	−	0.921910i	$-0.626628\pi$
$269$	−2.47358e8	−0.774806	−0.387403	+	0.921910i	$0.626628\pi$
$270$	0	0
$271$	1.64905e8	0.503316	0.251658	−	0.967816i	$-0.419024\pi$
$271$	1.64905e8	0.503316	0.251658	+	0.967816i	$0.419024\pi$
$272$	−2.11619e8	−0.637621
$273$	1.30463e9	3.88077
$274$	−1.72392e8	−0.506280
$275$	0	0
$276$	−8.36307e8	−2.39433
$277$	−5.55785e8	−1.57119	−0.785593	−	0.618744i	$-0.787642\pi$
$277$	−5.55785e8	−1.57119	−0.785593	+	0.618744i	$0.787642\pi$
$278$	3.25509e7	0.0908669
$279$	−960501.	−0.00264779
$280$	0	0
$281$	4.69679e8	1.26278	0.631392	−	0.775464i	$-0.282484\pi$
$281$	4.69679e8	1.26278	0.631392	+	0.775464i	$0.282484\pi$
$282$	−7.73473e8	−2.05387
$283$	3.02302e8	0.792844	0.396422	−	0.918068i	$-0.370252\pi$
$283$	3.02302e8	0.792844	0.396422	+	0.918068i	$0.370252\pi$
$284$	−5.47265e8	−1.41770
$285$	0	0
$286$	−3.93344e8	−0.994242
$287$	4.65888e8	1.16331
$288$	4.63169e8	1.14253
$289$	5.80639e8	1.41502
$290$	0	0
$291$	−1.01213e9	−2.40774
$292$	9.66426e7	0.227158
$293$	4.11144e7	0.0954900	0.0477450	−	0.998860i	$-0.484797\pi$
$293$	4.11144e7	0.0954900	0.0477450	+	0.998860i	$0.484797\pi$
$294$	6.29751e8	1.44528
$295$	0	0
$296$	8.36155e8	1.87398
$297$	−1.93825e8	−0.429301
$298$	8.62864e7	0.188880
$299$	6.94230e8	1.50195
$300$	0	0
$301$	−1.34336e8	−0.283929
$302$	−9.18435e8	−1.91877
$303$	−4.33658e7	−0.0895567
$304$	1.02595e8	0.209444
$305$	0	0
$306$	2.35288e9	4.69435
$307$	1.72818e8	0.340882	0.170441	−	0.985368i	$-0.445481\pi$
$307$	1.72818e8	0.340882	0.170441	+	0.985368i	$0.445481\pi$
$308$	−3.57600e8	−0.697381
$309$	−6.95834e8	−1.34169
$310$	0	0
$311$	−3.85476e8	−0.726668	−0.363334	−	0.931659i	$-0.618362\pi$
$311$	−3.85476e8	−0.726668	−0.363334	+	0.931659i	$0.618362\pi$
$312$	2.22615e9	4.14967
$313$	−3.08958e8	−0.569501	−0.284751	−	0.958602i	$-0.591911\pi$
$313$	−3.08958e8	−0.569501	−0.284751	+	0.958602i	$0.591911\pi$
$314$	−7.90094e8	−1.44021
$315$	0	0
$316$	−1.27566e9	−2.27422
$317$	1.43703e8	0.253372	0.126686	−	0.991943i	$-0.459566\pi$
$317$	1.43703e8	0.253372	0.126686	+	0.991943i	$0.459566\pi$
$318$	−1.02099e9	−1.78044
$319$	3.21465e8	0.554455
$320$	0	0
$321$	4.85566e8	0.819372
$322$	9.84028e8	1.64252
$323$	−4.80435e8	−0.793279
$324$	5.07363e8	0.828727
$325$	0	0
$326$	8.47626e8	1.35501
$327$	3.69058e8	0.583684
$328$	7.94967e8	1.24392
$329$	5.83724e8	0.903695
$330$	0	0
$331$	−1.50008e8	−0.227362	−0.113681	−	0.993517i	$-0.536264\pi$
$331$	−1.50008e8	−0.227362	−0.113681	+	0.993517i	$0.536264\pi$
$332$	2.81818e8	0.422654
$333$	−1.73478e9	−2.57448
$334$	−1.36845e7	−0.0200963
$335$	0	0
$336$	5.88803e8	0.846803
$337$	6.85903e8	0.976243	0.488121	−	0.872776i	$-0.337682\pi$
$337$	6.85903e8	0.976243	0.488121	+	0.872776i	$0.337682\pi$
$338$	−3.00604e9	−4.23434
$339$	−1.37490e9	−1.91678
$340$	0	0
$341$	−339362.	−0.000463471 0
$342$	−1.14070e9	−1.54198
$343$	4.45058e8	0.595507
$344$	−2.29224e8	−0.303603
$345$	0	0
$346$	2.83563e8	0.368030
$347$	−2.12686e8	−0.273266	−0.136633	−	0.990622i	$-0.543628\pi$
$347$	−2.12686e8	−0.273266	−0.136633	+	0.990622i	$0.543628\pi$
$348$	−4.12662e9	−5.24889
$349$	5.11417e8	0.644001	0.322000	−	0.946740i	$-0.395645\pi$
$349$	5.11417e8	0.644001	0.322000	+	0.946740i	$0.395645\pi$
$350$	0	0
$351$	−2.06542e9	−2.54937
$352$	1.63646e8	0.199989
$353$	1.01828e9	1.23213	0.616064	−	0.787696i	$-0.288726\pi$
$353$	1.01828e9	1.23213	0.616064	+	0.787696i	$0.288726\pi$
$354$	−2.40612e9	−2.88274
$355$	0	0
$356$	−2.48686e7	−0.0292130
$357$	−2.75727e9	−3.20731
$358$	−1.90045e9	−2.18910
$359$	1.82932e8	0.208669	0.104335	−	0.994542i	$-0.466729\pi$
$359$	1.82932e8	0.208669	0.104335	+	0.994542i	$0.466729\pi$
$360$	0	0
$361$	−6.60952e8	−0.739426
$362$	5.79172e8	0.641693
$363$	1.37424e9	1.50796
$364$	−3.81063e9	−4.14135
$365$	0	0
$366$	4.42546e9	4.71819
$367$	−1.73284e9	−1.82990	−0.914950	−	0.403566i	$-0.867770\pi$
$367$	−1.73284e9	−1.82990	−0.914950	+	0.403566i	$0.867770\pi$
$368$	3.13318e8	0.327732
$369$	−1.64933e9	−1.70889
$370$	0	0
$371$	7.70523e8	0.783388
$372$	4.35636e6	0.00438757
$373$	−3.35837e7	−0.0335080	−0.0167540	−	0.999860i	$-0.505333\pi$
$373$	−3.35837e7	−0.0335080	−0.0167540	+	0.999860i	$0.505333\pi$
$374$	8.31314e8	0.821702
$375$	0	0
$376$	9.96036e8	0.966312
$377$	3.42557e9	3.29259
$378$	−2.92760e9	−2.78799
$379$	−8.94524e7	−0.0844024	−0.0422012	−	0.999109i	$-0.513437\pi$
$379$	−8.94524e7	−0.0844024	−0.0422012	+	0.999109i	$0.513437\pi$
$380$	0	0
$381$	−2.78614e9	−2.58087
$382$	3.32649e9	3.05327
$383$	1.75400e9	1.59527	0.797635	−	0.603141i	$-0.206084\pi$
$383$	1.75400e9	1.59527	0.797635	+	0.603141i	$0.206084\pi$
$384$	−3.37486e9	−3.04156
$385$	0	0
$386$	−6.56608e8	−0.581100
$387$	4.75574e8	0.417090
$388$	2.95626e9	2.56940
$389$	−8.42113e7	−0.0725348	−0.0362674	−	0.999342i	$-0.511547\pi$
$389$	−8.42113e7	−0.0725348	−0.0362674	+	0.999342i	$0.511547\pi$
$390$	0	0
$391$	−1.46722e9	−1.24130
$392$	−8.10958e8	−0.679981
$393$	−1.12833e9	−0.937693
$394$	−2.03529e9	−1.67645
$395$	0	0
$396$	1.26597e9	1.02445
$397$	−1.25794e9	−1.00900	−0.504501	−	0.863411i	$-0.668324\pi$
$397$	−1.25794e9	−1.00900	−0.504501	+	0.863411i	$0.668324\pi$
$398$	−7.83410e8	−0.622872
$399$	1.33675e9	1.05353
$400$	0	0
$401$	−4.17786e8	−0.323555	−0.161777	−	0.986827i	$-0.551723\pi$
$401$	−4.17786e8	−0.323555	−0.161777	+	0.986827i	$0.551723\pi$
$402$	−4.59061e9	−3.52435
$403$	−3.61628e6	−0.00275229
$404$	1.26665e8	0.0955699
$405$	0	0
$406$	4.85553e9	3.60077
$407$	−6.12928e8	−0.450639
$408$	−4.70486e9	−3.42954
$409$	1.38413e9	1.00034	0.500169	−	0.865928i	$-0.333271\pi$
$409$	1.38413e9	1.00034	0.500169	+	0.865928i	$0.333271\pi$
$410$	0	0
$411$	−7.15191e8	−0.508131
$412$	2.03243e9	1.43177
$413$	1.81585e9	1.26839
$414$	−3.48363e9	−2.41285
$415$	0	0
$416$	1.74383e9	1.18762
$417$	1.35041e8	0.0911991
$418$	−4.03029e8	−0.269910
$419$	−8.15832e7	−0.0541816	−0.0270908	−	0.999633i	$-0.508624\pi$
$419$	−8.15832e7	−0.0541816	−0.0270908	+	0.999633i	$0.508624\pi$
$420$	0	0
$421$	2.47598e9	1.61718	0.808592	−	0.588370i	$-0.200230\pi$
$421$	2.47598e9	1.61718	0.808592	+	0.588370i	$0.200230\pi$
$422$	1.48135e9	0.959544
$423$	−2.06649e9	−1.32752
$424$	1.31478e9	0.837669
$425$	0	0
$426$	−3.53982e9	−2.21844
$427$	−3.33981e9	−2.07598
$428$	−1.41827e9	−0.874388
$429$	−1.63184e9	−0.997876
$430$	0	0
$431$	−6.43784e8	−0.387320	−0.193660	−	0.981069i	$-0.562036\pi$
$431$	−6.43784e8	−0.387320	−0.193660	+	0.981069i	$0.562036\pi$
$432$	−9.32160e8	−0.556285
$433$	3.26706e7	0.0193397	0.00966986	−	0.999953i	$-0.496922\pi$
$433$	3.26706e7	0.0193397	0.00966986	+	0.999953i	$0.496922\pi$
$434$	−5.12585e6	−0.00300990
$435$	0	0
$436$	−1.07796e9	−0.622875
$437$	7.11322e8	0.407738
$438$	6.25104e8	0.355462
$439$	3.43012e9	1.93501	0.967506	−	0.252849i	$-0.0813676\pi$
$439$	3.43012e9	1.93501	0.967506	+	0.252849i	$0.0813676\pi$
$440$	0	0
$441$	1.68250e9	0.934158
$442$	8.85858e9	4.87962
$443$	−2.96159e9	−1.61850	−0.809249	−	0.587466i	$-0.800126\pi$
$443$	−2.96159e9	−1.61850	−0.809249	+	0.587466i	$0.800126\pi$
$444$	7.86811e9	4.26609
$445$	0	0
$446$	1.73521e9	0.926147
$447$	3.57970e8	0.189570
$448$	3.43334e9	1.80403
$449$	1.52875e9	0.797030	0.398515	−	0.917162i	$-0.369526\pi$
$449$	1.52875e9	0.797030	0.398515	+	0.917162i	$0.369526\pi$
$450$	0	0
$451$	−5.82736e8	−0.299126
$452$	4.01587e9	2.04548
$453$	−3.81024e9	−1.92579
$454$	−4.52257e8	−0.226824
$455$	0	0
$456$	2.28096e9	1.12653
$457$	3.39139e9	1.66216	0.831078	−	0.556156i	$-0.187724\pi$
$457$	3.39139e9	1.66216	0.831078	+	0.556156i	$0.187724\pi$
$458$	2.58724e9	1.25837
$459$	4.36516e9	2.10696
$460$	0	0
$461$	3.27030e9	1.55466	0.777329	−	0.629094i	$-0.216574\pi$
$461$	3.27030e9	1.55466	0.777329	+	0.629094i	$0.216574\pi$
$462$	−2.31303e9	−1.09128
$463$	−1.01485e9	−0.475193	−0.237597	−	0.971364i	$-0.576360\pi$
$463$	−1.01485e9	−0.475193	−0.237597	+	0.971364i	$0.576360\pi$
$464$	1.54602e9	0.718458
$465$	0	0
$466$	−2.97009e9	−1.35962
$467$	−3.80824e9	−1.73028	−0.865138	−	0.501533i	$-0.832770\pi$
$467$	−3.80824e9	−1.73028	−0.865138	+	0.501533i	$0.832770\pi$
$468$	1.34903e10	6.08360
$469$	3.46444e9	1.55070
$470$	0	0
$471$	−3.27780e9	−1.44547
$472$	3.09846e9	1.35628
$473$	1.68028e8	0.0730077
$474$	−8.25125e9	−3.55874
$475$	0	0
$476$	8.05357e9	3.42266
$477$	−2.72778e9	−1.15079
$478$	5.89644e9	2.46940
$479$	−3.15342e9	−1.31101	−0.655507	−	0.755189i	$-0.727545\pi$
$479$	−3.15342e9	−1.31101	−0.655507	+	0.755189i	$0.727545\pi$
$480$	0	0
$481$	−6.53143e9	−2.67609
$482$	2.35721e9	0.958814
$483$	4.08236e9	1.64853
$484$	−4.01394e9	−1.60921
$485$	0	0
$486$	−2.44768e9	−0.967225
$487$	3.35676e9	1.31695	0.658475	−	0.752602i	$-0.271202\pi$
$487$	3.35676e9	1.31695	0.658475	+	0.752602i	$0.271202\pi$
$488$	−5.69887e9	−2.21983
$489$	3.51648e9	1.35996
$490$	0	0
$491$	−4.00657e9	−1.52752	−0.763760	−	0.645500i	$-0.776649\pi$
$491$	−4.00657e9	−1.52752	−0.763760	+	0.645500i	$0.776649\pi$
$492$	7.48053e9	2.83175
$493$	−7.23976e9	−2.72120
$494$	−4.29472e9	−1.60284
$495$	0	0
$496$	−1.63209e6	−0.000600562 0
$497$	2.67143e9	0.976105
$498$	1.82286e9	0.661378
$499$	−2.27845e9	−0.820893	−0.410447	−	0.911885i	$-0.634627\pi$
$499$	−2.27845e9	−0.820893	−0.410447	+	0.911885i	$0.634627\pi$
$500$	0	0
$501$	−5.67718e7	−0.0201698
$502$	−4.00907e9	−1.41443
$503$	2.42121e9	0.848292	0.424146	−	0.905594i	$-0.360574\pi$
$503$	2.42121e9	0.848292	0.424146	+	0.905594i	$0.360574\pi$
$504$	8.43035e9	2.93318
$505$	0	0
$506$	−1.23083e9	−0.422348
$507$	−1.24709e10	−4.24982
$508$	8.13789e9	2.75416
$509$	−4.03796e9	−1.35722	−0.678609	−	0.734500i	$-0.737417\pi$
$509$	−4.03796e9	−1.35722	−0.678609	+	0.734500i	$0.737417\pi$
$510$	0	0
$511$	−4.71753e8	−0.156402
$512$	2.42780e9	0.799407
$513$	−2.11627e9	−0.692087
$514$	4.82947e9	1.56866
$515$	0	0
$516$	−2.15697e9	−0.691146
$517$	−7.30125e8	−0.232370
$518$	−9.25789e9	−2.92656
$519$	1.17640e9	0.369376
$520$	0	0
$521$	1.63604e8	0.0506830	0.0253415	−	0.999679i	$-0.491933\pi$
$521$	1.63604e8	0.0506830	0.0253415	+	0.999679i	$0.491933\pi$
$522$	−1.71894e10	−5.28949
$523$	2.19768e9	0.671751	0.335876	−	0.941906i	$-0.390968\pi$
$523$	2.19768e9	0.671751	0.335876	+	0.941906i	$0.390968\pi$
$524$	3.29567e9	1.00065
$525$	0	0
$526$	4.10704e9	1.23049
$527$	7.64282e6	0.00227466
$528$	−7.36478e8	−0.217741
$529$	−1.23249e9	−0.361983
$530$	0	0
$531$	−6.42841e9	−1.86326
$532$	−3.90445e9	−1.12426
$533$	−6.20970e9	−1.77634
$534$	−1.60855e8	−0.0457130
$535$	0	0
$536$	5.91153e9	1.65815
$537$	−7.88425e9	−2.19710
$538$	4.67324e9	1.29384
$539$	5.94457e8	0.163516
$540$	0	0
$541$	6.56541e9	1.78267	0.891336	−	0.453343i	$-0.149769\pi$
$541$	6.56541e9	1.78267	0.891336	+	0.453343i	$0.149769\pi$
$542$	−3.11549e9	−0.840482
$543$	2.40277e9	0.644039
$544$	−3.68549e9	−0.981521
$545$	0	0
$546$	−2.46479e10	−6.48046
$547$	−1.54831e9	−0.404486	−0.202243	−	0.979335i	$-0.564823\pi$
$547$	−1.54831e9	−0.404486	−0.202243	+	0.979335i	$0.564823\pi$
$548$	2.08896e9	0.542249
$549$	1.18235e10	3.04960
$550$	0	0
$551$	3.50991e9	0.893850
$552$	6.96593e9	1.76276
$553$	6.22705e9	1.56583
$554$	1.05002e10	2.62370
$555$	0	0
$556$	−3.94435e8	−0.0973226
$557$	5.41329e9	1.32730	0.663649	−	0.748044i	$-0.269007\pi$
$557$	5.41329e9	1.32730	0.663649	+	0.748044i	$0.269007\pi$
$558$	1.81464e7	0.00442151
$559$	1.79053e9	0.433551
$560$	0	0
$561$	3.44881e9	0.824706
$562$	−8.87347e9	−2.10871
$563$	−4.25534e9	−1.00497	−0.502487	−	0.864585i	$-0.667581\pi$
$563$	−4.25534e9	−1.00497	−0.502487	+	0.864585i	$0.667581\pi$
$564$	9.37256e9	2.19979
$565$	0	0
$566$	−5.71127e9	−1.32396
$567$	−2.47665e9	−0.570590
$568$	4.55839e9	1.04374
$569$	−3.36106e9	−0.764864	−0.382432	−	0.923984i	$-0.624913\pi$
$569$	−3.36106e9	−0.764864	−0.382432	+	0.923984i	$0.624913\pi$
$570$	0	0
$571$	1.32397e9	0.297612	0.148806	−	0.988866i	$-0.452457\pi$
$571$	1.32397e9	0.297612	0.148806	+	0.988866i	$0.452457\pi$
$572$	4.76635e9	1.06488
$573$	1.38004e10	3.06443
$574$	−8.80186e9	−1.94260
$575$	0	0
$576$	−1.21546e10	−2.65010
$577$	7.23419e9	1.56774	0.783871	−	0.620923i	$-0.213242\pi$
$577$	7.23419e9	1.56774	0.783871	+	0.620923i	$0.213242\pi$
$578$	−1.09698e10	−2.36293
$579$	−2.72402e9	−0.583224
$580$	0	0
$581$	−1.37567e9	−0.291003
$582$	1.91217e10	4.02065
$583$	−9.63774e8	−0.201435
$584$	−8.04974e8	−0.167239
$585$	0	0
$586$	−7.76760e8	−0.159458
$587$	−7.57716e9	−1.54623	−0.773114	−	0.634267i	$-0.781302\pi$
$587$	−7.57716e9	−1.54623	−0.773114	+	0.634267i	$0.781302\pi$
$588$	−7.63100e9	−1.54796
$589$	−3.70531e6	−0.000747173 0
$590$	0	0
$591$	−8.44367e9	−1.68258
$592$	−2.94775e9	−0.583935
$593$	−4.10654e9	−0.808694	−0.404347	−	0.914606i	$-0.632501\pi$
$593$	−4.10654e9	−0.808694	−0.404347	+	0.914606i	$0.632501\pi$
$594$	3.66186e9	0.716885
$595$	0	0
$596$	−1.04558e9	−0.202299
$597$	−3.25008e9	−0.625149
$598$	−1.31158e10	−2.50808
$599$	8.51936e9	1.61962	0.809810	−	0.586692i	$-0.199570\pi$
$599$	8.51936e9	1.61962	0.809810	+	0.586692i	$0.199570\pi$
$600$	0	0
$601$	−1.75651e9	−0.330057	−0.165029	−	0.986289i	$-0.552772\pi$
$601$	−1.75651e9	−0.330057	−0.165029	+	0.986289i	$0.552772\pi$
$602$	2.53796e9	0.474130
$603$	−1.22647e10	−2.27796
$604$	1.11291e10	2.05510
$605$	0	0
$606$	8.19294e8	0.149550
$607$	−1.52080e9	−0.276002	−0.138001	−	0.990432i	$-0.544068\pi$
$607$	−1.52080e9	−0.276002	−0.138001	+	0.990432i	$0.544068\pi$
$608$	1.78676e9	0.322407
$609$	2.01438e10	3.61393
$610$	0	0
$611$	−7.78030e9	−1.37991
$612$	−2.85110e10	−5.02786
$613$	−4.34872e9	−0.762518	−0.381259	−	0.924468i	$-0.624509\pi$
$613$	−4.34872e9	−0.762518	−0.381259	+	0.924468i	$0.624509\pi$
$614$	−3.26498e9	−0.569234
$615$	0	0
$616$	2.97859e9	0.513427
$617$	−5.11666e8	−0.0876978	−0.0438489	−	0.999038i	$-0.513962\pi$
$617$	−5.11666e8	−0.0876978	−0.0438489	+	0.999038i	$0.513962\pi$
$618$	1.31461e10	2.24047
$619$	8.59569e8	0.145668	0.0728339	−	0.997344i	$-0.476796\pi$
$619$	8.59569e8	0.145668	0.0728339	+	0.997344i	$0.476796\pi$
$620$	0	0
$621$	−6.46297e9	−1.08296
$622$	7.28266e9	1.21345
$623$	1.21394e8	0.0201135
$624$	−7.84799e9	−1.29304
$625$	0	0
$626$	5.83704e9	0.951004
$627$	−1.67202e9	−0.270897
$628$	9.57396e9	1.54253
$629$	1.38038e10	2.21168
$630$	0	0
$631$	7.79789e9	1.23559	0.617795	−	0.786340i	$-0.288026\pi$
$631$	7.79789e9	1.23559	0.617795	+	0.786340i	$0.288026\pi$
$632$	1.06255e10	1.67433
$633$	6.14558e9	0.963052
$634$	−2.71493e9	−0.423102
$635$	0	0
$636$	1.23719e10	1.90694
$637$	6.33460e9	0.971027
$638$	−6.07332e9	−0.925877
$639$	−9.45733e9	−1.43389
$640$	0	0
$641$	−9.31139e9	−1.39640	−0.698202	−	0.715901i	$-0.746016\pi$
$641$	−9.31139e9	−1.39640	−0.698202	+	0.715901i	$0.746016\pi$
$642$	−9.17363e9	−1.36826
$643$	−3.63024e9	−0.538514	−0.269257	−	0.963068i	$-0.586778\pi$
$643$	−3.63024e9	−0.538514	−0.269257	+	0.963068i	$0.586778\pi$
$644$	−1.19240e10	−1.75922
$645$	0	0
$646$	9.07668e9	1.32469
$647$	−1.25235e10	−1.81787	−0.908934	−	0.416940i	$-0.863102\pi$
$647$	−1.25235e10	−1.81787	−0.908934	+	0.416940i	$0.863102\pi$
$648$	−4.22603e9	−0.610126
$649$	−2.27127e9	−0.326146
$650$	0	0
$651$	−2.12652e7	−0.00302090
$652$	−1.02711e10	−1.45128
$653$	−5.25902e9	−0.739109	−0.369555	−	0.929209i	$-0.620490\pi$
$653$	−5.25902e9	−0.739109	−0.369555	+	0.929209i	$0.620490\pi$
$654$	−6.97248e9	−0.974688
$655$	0	0
$656$	−2.80255e9	−0.387605
$657$	1.67009e9	0.229753
$658$	−1.10281e10	−1.50907
$659$	−8.28334e9	−1.12747	−0.563737	−	0.825954i	$-0.690637\pi$
$659$	−8.28334e9	−1.12747	−0.563737	+	0.825954i	$0.690637\pi$
$660$	0	0
$661$	4.93416e9	0.664521	0.332260	−	0.943188i	$-0.392189\pi$
$661$	4.93416e9	0.664521	0.332260	+	0.943188i	$0.392189\pi$
$662$	2.83405e9	0.379669
$663$	3.67509e10	4.89746
$664$	−2.34737e9	−0.311167
$665$	0	0
$666$	3.27746e10	4.29909
$667$	1.07190e10	1.39867
$668$	1.65822e8	0.0215241
$669$	7.19873e9	0.929533
$670$	0	0
$671$	4.17745e9	0.533804
$672$	1.02544e10	1.30352
$673$	1.34683e10	1.70317	0.851586	−	0.524214i	$-0.175641\pi$
$673$	1.34683e10	1.70317	0.851586	+	0.524214i	$0.175641\pi$
$674$	−1.29585e10	−1.63022
$675$	0	0
$676$	3.64257e10	4.53518
$677$	9.00397e9	1.11525	0.557627	−	0.830092i	$-0.311712\pi$
$677$	9.00397e9	1.11525	0.557627	+	0.830092i	$0.311712\pi$
$678$	2.59754e10	3.20080
$679$	−1.44308e10	−1.76907
$680$	0	0
$681$	−1.87624e9	−0.227654
$682$	6.41144e6	0.000773945 0
$683$	1.77817e9	0.213550	0.106775	−	0.994283i	$-0.465947\pi$
$683$	1.77817e9	0.213550	0.106775	+	0.994283i	$0.465947\pi$
$684$	1.38224e10	1.65153
$685$	0	0
$686$	−8.40831e9	−0.994430
$687$	1.07335e10	1.26297
$688$	8.08098e8	0.0946029
$689$	−1.02701e10	−1.19621
$690$	0	0
$691$	4.37894e9	0.504889	0.252444	−	0.967611i	$-0.418766\pi$
$691$	4.37894e9	0.504889	0.252444	+	0.967611i	$0.418766\pi$
$692$	−3.43608e9	−0.394177
$693$	−6.17971e9	−0.705345
$694$	4.01820e9	0.456324
$695$	0	0
$696$	3.43722e10	3.86434
$697$	1.31239e10	1.46807
$698$	−9.66202e9	−1.07541
$699$	−1.23218e10	−1.36460
$700$	0	0
$701$	−4.15077e9	−0.455109	−0.227554	−	0.973765i	$-0.573073\pi$
$701$	−4.15077e9	−0.455109	−0.227554	+	0.973765i	$0.573073\pi$
$702$	3.90212e10	4.25717
$703$	−6.69223e9	−0.726486
$704$	−4.29444e9	−0.463876
$705$	0	0
$706$	−1.92380e10	−2.05751
$707$	−6.18304e8	−0.0658012
$708$	2.91561e10	3.08755
$709$	8.41776e9	0.887022	0.443511	−	0.896269i	$-0.353733\pi$
$709$	8.41776e9	0.887022	0.443511	+	0.896269i	$0.353733\pi$
$710$	0	0
$711$	−2.20448e10	−2.30019
$712$	2.07140e8	0.0215072
$713$	−1.13158e7	−0.00116916
$714$	5.20921e10	5.35585
$715$	0	0
$716$	2.30287e10	2.34463
$717$	2.44621e10	2.47843
$718$	−3.45606e9	−0.348454
$719$	−1.86618e10	−1.87242	−0.936211	−	0.351439i	$-0.885692\pi$
$719$	−1.86618e10	−1.87242	−0.936211	+	0.351439i	$0.885692\pi$
$720$	0	0
$721$	−9.92112e9	−0.985796
$722$	1.24871e10	1.23476
$723$	9.77920e9	0.962319
$724$	−7.01811e9	−0.687283
$725$	0	0
$726$	−2.59630e10	−2.51812
$727$	8.35324e9	0.806278	0.403139	−	0.915139i	$-0.367919\pi$
$727$	8.35324e9	0.806278	0.403139	+	0.915139i	$0.367919\pi$
$728$	3.17402e10	3.04895
$729$	−1.50014e10	−1.43412
$730$	0	0
$731$	−3.78420e9	−0.358313
$732$	−5.36256e10	−5.05339
$733$	−8.75589e9	−0.821177	−0.410588	−	0.911821i	$-0.634677\pi$
$733$	−8.75589e9	−0.821177	−0.410588	+	0.911821i	$0.634677\pi$
$734$	3.27379e10	3.05573
$735$	0	0
$736$	5.45667e9	0.504493
$737$	−4.33333e9	−0.398737
$738$	3.11601e10	2.85366
$739$	−6.34876e9	−0.578673	−0.289337	−	0.957227i	$-0.593435\pi$
$739$	−6.34876e9	−0.578673	−0.289337	+	0.957227i	$0.593435\pi$
$740$	0	0
$741$	−1.78172e10	−1.60870
$742$	−1.45572e10	−1.30817
$743$	1.42610e10	1.27553	0.637764	−	0.770232i	$-0.279860\pi$
$743$	1.42610e10	1.27553	0.637764	+	0.770232i	$0.279860\pi$
$744$	−3.62858e7	−0.00323022
$745$	0	0
$746$	6.34485e8	0.0559546
$747$	4.87012e9	0.427481
$748$	−1.00734e10	−0.880081
$749$	6.92314e9	0.602028
$750$	0	0
$751$	8.45970e9	0.728812	0.364406	−	0.931240i	$-0.381272\pi$
$751$	8.45970e9	0.728812	0.364406	+	0.931240i	$0.381272\pi$
$752$	−3.51138e9	−0.301104
$753$	−1.66321e10	−1.41960
$754$	−6.47179e10	−5.49826
$755$	0	0
$756$	3.54752e10	2.98606
$757$	3.65322e9	0.306084	0.153042	−	0.988220i	$-0.451093\pi$
$757$	3.65322e9	0.306084	0.153042	+	0.988220i	$0.451093\pi$
$758$	1.68999e9	0.140943
$759$	−5.10624e9	−0.423892
$760$	0	0
$761$	−7.89814e9	−0.649649	−0.324824	−	0.945774i	$-0.605305\pi$
$761$	−7.89814e9	−0.649649	−0.324824	+	0.945774i	$0.605305\pi$
$762$	5.26375e10	4.30976
$763$	5.26199e9	0.428858
$764$	−4.03088e10	−3.27019
$765$	0	0
$766$	−3.31377e10	−2.66392
$767$	−2.42029e10	−1.93680
$768$	3.29371e10	2.62374
$769$	1.30884e10	1.03787	0.518937	−	0.854812i	$-0.326328\pi$
$769$	1.30884e10	1.03787	0.518937	+	0.854812i	$0.326328\pi$
$770$	0	0
$771$	2.00357e10	1.57440
$772$	7.95645e9	0.622384
$773$	1.82493e9	0.142108	0.0710538	−	0.997472i	$-0.477364\pi$
$773$	1.82493e9	0.142108	0.0710538	+	0.997472i	$0.477364\pi$
$774$	−8.98484e9	−0.696493
$775$	0	0
$776$	−2.46239e10	−1.89165
$777$	−3.84075e10	−2.93726
$778$	1.59097e9	0.121125
$779$	−6.36258e9	−0.482228
$780$	0	0
$781$	−3.34144e9	−0.250989
$782$	2.77197e10	2.07283
$783$	−3.18905e10	−2.37408
$784$	2.85892e9	0.211883
$785$	0	0
$786$	2.13170e10	1.56584
$787$	1.58146e10	1.15650	0.578250	−	0.815860i	$-0.303736\pi$
$787$	1.58146e10	1.15650	0.578250	+	0.815860i	$0.303736\pi$
$788$	2.46627e10	1.79555
$789$	1.70386e10	1.23499
$790$	0	0
$791$	−1.96031e10	−1.40834
$792$	−1.05447e10	−0.754219
$793$	4.45153e10	3.16996
$794$	2.37657e10	1.68492
$795$	0	0
$796$	9.49298e9	0.667124
$797$	2.04440e9	0.143041	0.0715206	−	0.997439i	$-0.477215\pi$
$797$	2.04440e9	0.143041	0.0715206	+	0.997439i	$0.477215\pi$
$798$	−2.52547e10	−1.75927
$799$	1.64433e10	1.14045
$800$	0	0
$801$	−4.29755e8	−0.0295466
$802$	7.89307e9	0.540301
$803$	5.90071e8	0.0402161
$804$	5.56267e10	3.77474
$805$	0	0
$806$	6.83210e7	0.00459602
$807$	1.93875e10	1.29857
$808$	−1.05504e9	−0.0703606
$809$	7.20035e9	0.478117	0.239058	−	0.971005i	$-0.423161\pi$
$809$	7.20035e9	0.478117	0.239058	+	0.971005i	$0.423161\pi$
$810$	0	0
$811$	7.19747e9	0.473813	0.236906	−	0.971532i	$-0.423867\pi$
$811$	7.19747e9	0.473813	0.236906	+	0.971532i	$0.423867\pi$
$812$	−5.88368e10	−3.85659
$813$	−1.29250e10	−0.843554
$814$	1.15798e10	0.752517
$815$	0	0
$816$	1.65863e10	1.06865
$817$	1.83461e9	0.117698
$818$	−2.61499e10	−1.67045
$819$	−6.58517e10	−4.18864
$820$	0	0
$821$	6.01914e9	0.379606	0.189803	−	0.981822i	$-0.439215\pi$
$821$	6.01914e9	0.379606	0.189803	+	0.981822i	$0.439215\pi$
$822$	1.35118e10	0.848522
$823$	1.49550e10	0.935164	0.467582	−	0.883950i	$-0.345125\pi$
$823$	1.49550e10	0.935164	0.467582	+	0.883950i	$0.345125\pi$
$824$	−1.69289e10	−1.05410
$825$	0	0
$826$	−3.43061e10	−2.11807
$827$	−1.67090e10	−1.02726	−0.513631	−	0.858011i	$-0.671700\pi$
$827$	−1.67090e10	−1.02726	−0.513631	+	0.858011i	$0.671700\pi$
$828$	4.22129e10	2.58428
$829$	−2.34996e10	−1.43258	−0.716291	−	0.697801i	$-0.754162\pi$
$829$	−2.34996e10	−1.43258	−0.716291	+	0.697801i	$0.754162\pi$
$830$	0	0
$831$	4.35615e10	2.63330
$832$	−4.57620e10	−2.75470
$833$	−1.33879e10	−0.802516
$834$	−2.55129e9	−0.152292
$835$	0	0
$836$	4.88370e9	0.289086
$837$	3.36659e7	0.00198450
$838$	1.54132e9	0.0904772
$839$	4.13995e9	0.242007	0.121004	−	0.992652i	$-0.461389\pi$
$839$	4.13995e9	0.242007	0.121004	+	0.992652i	$0.461389\pi$
$840$	0	0
$841$	3.56415e10	2.06619
$842$	−4.67777e10	−2.70051
$843$	−3.68127e10	−2.11642
$844$	−1.79503e10	−1.02772
$845$	0	0
$846$	3.90413e10	2.21681
$847$	1.95937e10	1.10796
$848$	−4.63507e9	−0.261018
$849$	−2.36939e10	−1.32880
$850$	0	0
$851$	−2.04377e10	−1.13679
$852$	4.28938e10	2.37605
$853$	−4.71218e9	−0.259956	−0.129978	−	0.991517i	$-0.541491\pi$
$853$	−4.71218e9	−0.259956	−0.129978	+	0.991517i	$0.541491\pi$
$854$	6.30977e10	3.46666
$855$	0	0
$856$	1.18133e10	0.643743
$857$	3.05860e10	1.65993	0.829966	−	0.557814i	$-0.188360\pi$
$857$	3.05860e10	1.65993	0.829966	+	0.557814i	$0.188360\pi$
$858$	3.08297e10	1.66634
$859$	−2.82478e9	−0.152058	−0.0760288	−	0.997106i	$-0.524224\pi$
$859$	−2.82478e9	−0.152058	−0.0760288	+	0.997106i	$0.524224\pi$
$860$	0	0
$861$	−3.65156e10	−1.94970
$862$	1.21628e10	0.646780
$863$	−2.78503e10	−1.47500	−0.737500	−	0.675347i	$-0.763994\pi$
$863$	−2.78503e10	−1.47500	−0.737500	+	0.675347i	$0.763994\pi$
$864$	−1.62342e10	−0.856316
$865$	0	0
$866$	−6.17234e8	−0.0322952
$867$	−4.55095e10	−2.37157
$868$	6.21124e7	0.00322374
$869$	−7.78882e9	−0.402627
$870$	0	0
$871$	−4.61765e10	−2.36787
$872$	8.97878e9	0.458574
$873$	5.10874e10	2.59875
$874$	−1.34387e10	−0.680877
$875$	0	0
$876$	−7.57469e9	−0.380716
$877$	7.95069e9	0.398021	0.199011	−	0.979997i	$-0.436227\pi$
$877$	7.95069e9	0.398021	0.199011	+	0.979997i	$0.436227\pi$
$878$	−6.48040e10	−3.23125
$879$	−3.22249e9	−0.160041
$880$	0	0
$881$	−1.63779e10	−0.806942	−0.403471	−	0.914993i	$-0.632196\pi$
$881$	−1.63779e10	−0.806942	−0.403471	+	0.914993i	$0.632196\pi$
$882$	−3.17869e10	−1.55994
$883$	−1.10363e10	−0.539462	−0.269731	−	0.962936i	$-0.586935\pi$
$883$	−1.10363e10	−0.539462	−0.269731	+	0.962936i	$0.586935\pi$
$884$	−1.07344e11	−5.22630
$885$	0	0
$886$	5.59523e10	2.70271
$887$	−2.28262e10	−1.09825	−0.549124	−	0.835741i	$-0.685039\pi$
$887$	−2.28262e10	−1.09825	−0.549124	+	0.835741i	$0.685039\pi$
$888$	−6.55366e10	−3.14079
$889$	−3.97245e10	−1.89628
$890$	0	0
$891$	3.09781e9	0.146718
$892$	−2.10264e10	−0.991946
$893$	−7.97185e9	−0.374610
$894$	−6.76299e9	−0.316561
$895$	0	0
$896$	−4.81183e10	−2.23477
$897$	−5.44127e10	−2.51725
$898$	−2.88821e10	−1.33095
$899$	−5.58360e7	−0.00256304
$900$	0	0
$901$	2.17053e10	0.988619
$902$	1.10094e10	0.499506
$903$	1.05291e10	0.475864
$904$	−3.34497e10	−1.50592
$905$	0	0
$906$	7.19855e10	3.21585
$907$	−2.59492e10	−1.15478	−0.577389	−	0.816469i	$-0.695928\pi$
$907$	−2.59492e10	−1.15478	−0.577389	+	0.816469i	$0.695928\pi$
$908$	5.48022e9	0.242939
$909$	2.18890e9	0.0966614
$910$	0	0
$911$	−2.72877e10	−1.19578	−0.597892	−	0.801576i	$-0.703995\pi$
$911$	−2.72877e10	−1.19578	−0.597892	+	0.801576i	$0.703995\pi$
$912$	−8.04121e9	−0.351026
$913$	1.72070e9	0.0748267
$914$	−6.40723e10	−2.77561
$915$	0	0
$916$	−3.13509e10	−1.34777
$917$	−1.60875e10	−0.688964
$918$	−8.24694e10	−3.51838
$919$	1.96011e10	0.833059	0.416530	−	0.909122i	$-0.363246\pi$
$919$	1.96011e10	0.833059	0.416530	+	0.909122i	$0.363246\pi$
$920$	0	0
$921$	−1.35452e10	−0.571316
$922$	−6.17846e10	−2.59610
$923$	−3.56067e10	−1.49048
$924$	2.80281e10	1.16881
$925$	0	0
$926$	1.91733e10	0.793519
$927$	3.51225e10	1.44813
$928$	2.69250e10	1.10596
$929$	−5.18528e9	−0.212186	−0.106093	−	0.994356i	$-0.533834\pi$
$929$	−5.18528e9	−0.212186	−0.106093	+	0.994356i	$0.533834\pi$
$930$	0	0
$931$	6.49056e9	0.263608
$932$	3.59901e10	1.45622
$933$	3.02130e10	1.21789
$934$	7.19477e10	2.88937
$935$	0	0
$936$	−1.12366e11	−4.47887
$937$	−2.23328e10	−0.886858	−0.443429	−	0.896309i	$-0.646238\pi$
$937$	−2.23328e10	−0.886858	−0.443429	+	0.896309i	$0.646238\pi$
$938$	−6.54523e10	−2.58949
$939$	2.42157e10	0.954480
$940$	0	0
$941$	1.63132e9	0.0638226	0.0319113	−	0.999491i	$-0.489841\pi$
$941$	1.63132e9	0.0638226	0.0319113	+	0.999491i	$0.489841\pi$
$942$	6.19263e10	2.41378
$943$	−1.94310e10	−0.754577
$944$	−1.09232e10	−0.422618
$945$	0	0
$946$	−3.17450e9	−0.121915
$947$	3.08264e10	1.17950	0.589751	−	0.807585i	$-0.299226\pi$
$947$	3.08264e10	1.17950	0.589751	+	0.807585i	$0.299226\pi$
$948$	9.99846e10	3.81157
$949$	6.28786e9	0.238820
$950$	0	0
$951$	−1.12632e10	−0.424649
$952$	−6.70813e10	−2.51984
$953$	−2.48683e10	−0.930724	−0.465362	−	0.885120i	$-0.654076\pi$
$953$	−2.48683e10	−0.930724	−0.465362	+	0.885120i	$0.654076\pi$
$954$	5.15350e10	1.92169
$955$	0	0
$956$	−7.14501e10	−2.64484
$957$	−2.51959e10	−0.929262
$958$	5.95764e10	2.18925
$959$	−1.01971e10	−0.373346
$960$	0	0
$961$	−2.75126e10	−0.999998
$962$	1.23396e11	4.46877
$963$	−2.45091e10	−0.884374
$964$	−2.85635e10	−1.02693
$965$	0	0
$966$	−7.71265e10	−2.75286
$967$	3.19433e10	1.13602	0.568012	−	0.823020i	$-0.307713\pi$
$967$	3.19433e10	1.13602	0.568012	+	0.823020i	$0.307713\pi$
$968$	3.34337e10	1.18473
$969$	3.76557e10	1.32953
$970$	0	0
$971$	−2.26783e10	−0.794958	−0.397479	−	0.917611i	$-0.630115\pi$
$971$	−2.26783e10	−0.794958	−0.397479	+	0.917611i	$0.630115\pi$
$972$	2.96598e10	1.03594
$973$	1.92540e9	0.0670080
$974$	−6.34181e10	−2.19916
$975$	0	0
$976$	2.00906e10	0.691700
$977$	−1.18844e10	−0.407707	−0.203853	−	0.979001i	$-0.565347\pi$
$977$	−1.18844e10	−0.407707	−0.203853	+	0.979001i	$0.565347\pi$
$978$	−6.64356e10	−2.27099
$979$	−1.51840e8	−0.00517186
$980$	0	0
$981$	−1.86284e10	−0.629989
$982$	7.56946e10	2.55079
$983$	−4.01250e10	−1.34734	−0.673670	−	0.739032i	$-0.735283\pi$
$983$	−4.01250e10	−1.34734	−0.673670	+	0.739032i	$0.735283\pi$
$984$	−6.23083e10	−2.08479
$985$	0	0
$986$	1.36778e11	4.54410
$987$	−4.57514e10	−1.51459
$988$	5.20413e10	1.71672
$989$	5.60281e9	0.184170
$990$	0	0
$991$	2.24992e10	0.734362	0.367181	−	0.930150i	$-0.380323\pi$
$991$	2.24992e10	0.734362	0.367181	+	0.930150i	$0.380323\pi$
$992$	−2.84240e7	−0.000924474 0
$993$	1.17574e10	0.381057
$994$	−5.04703e10	−1.62999
$995$	0	0
$996$	−2.20885e10	−0.708366
$997$	1.91411e10	0.611693	0.305846	−	0.952081i	$-0.401061\pi$
$997$	1.91411e10	0.611693	0.305846	+	0.952081i	$0.401061\pi$
$998$	4.30458e10	1.37080
$999$	6.08047e10	1.92956

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.8.a.e.1.7	✓	48
5.4	even	2		625.8.a.f.1.42	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.8.a.e.1.7	✓	48	1.1	even	1	trivial
625.8.a.f.1.42	yes	48	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)

\(f(q)\)	\(=\)	\(q-18.8926 q^{2} -78.3784 q^{3} +228.931 q^{4} +1480.77 q^{6} -1117.51 q^{7} -1906.86 q^{8} +3956.18 q^{9} +O(q^{10})\)	\(q-18.8926 q^{2} -78.3784 q^{3} +228.931 q^{4} +1480.77 q^{6} -1117.51 q^{7} -1906.86 q^{8} +3956.18 q^{9} +1397.79 q^{11} -17943.3 q^{12} +14895.0 q^{13} +21112.7 q^{14} +6722.36 q^{16} -31479.8 q^{17} -74742.6 q^{18} +15261.7 q^{19} +87588.7 q^{21} -26407.9 q^{22} +46608.3 q^{23} +149457. q^{24} -281405. q^{26} -138666. q^{27} -255833. q^{28} +229981. q^{29} -242.785 q^{31} +117075. q^{32} -109556. q^{33} +594736. q^{34} +905694. q^{36} -438499. q^{37} -288334. q^{38} -1.16744e6 q^{39} -416899. q^{41} -1.65478e6 q^{42} +120210. q^{43} +319997. q^{44} -880554. q^{46} -522344. q^{47} -526888. q^{48} +425285. q^{49} +2.46734e6 q^{51} +3.40993e6 q^{52} -689500. q^{53} +2.61976e6 q^{54} +2.13093e6 q^{56} -1.19619e6 q^{57} -4.34495e6 q^{58} -1.62490e6 q^{59} +2.98861e6 q^{61} +4586.85 q^{62} -4.42107e6 q^{63} -3.07231e6 q^{64} +2.06981e6 q^{66} -3.10014e6 q^{67} -7.20671e6 q^{68} -3.65309e6 q^{69} -2.39052e6 q^{71} -7.54388e6 q^{72} +422147. q^{73} +8.28439e6 q^{74} +3.49388e6 q^{76} -1.56204e6 q^{77} +2.20561e7 q^{78} -5.57226e6 q^{79} +2.21622e6 q^{81} +7.87631e6 q^{82} +1.23102e6 q^{83} +2.00518e7 q^{84} -2.27109e6 q^{86} -1.80256e7 q^{87} -2.66538e6 q^{88} -108629. q^{89} -1.66453e7 q^{91} +1.06701e7 q^{92} +19029.1 q^{93} +9.86845e6 q^{94} -9.17615e6 q^{96} +1.29133e7 q^{97} -8.03474e6 q^{98} +5.52990e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 25 q^{2} - 95 q^{3} + 3419 q^{4} + 431 q^{6} - 4030 q^{7} - 3375 q^{8} + 36231 q^{9}+O(q^{10}) \) 48 * q - 25 * q^2 - 95 * q^3 + 3419 * q^4 + 431 * q^6 - 4030 * q^7 - 3375 * q^8 + 36231 * q^9	\( 48 q - 25 q^{2} - 95 q^{3} + 3419 q^{4} + 431 q^{6} - 4030 q^{7} - 3375 q^{8} + 36231 q^{9} + 781 q^{11} - 3925 q^{12} - 4290 q^{13} - 20762 q^{14} + 270603 q^{16} - 75075 q^{17} - 89950 q^{18} + 17750 q^{19} - 48034 q^{21} - 331305 q^{22} - 343890 q^{23} - 271570 q^{24} - 304129 q^{26} - 474740 q^{27} - 1146535 q^{28} - 59330 q^{29} - 385989 q^{31} - 1887300 q^{32} - 879805 q^{33} + 286938 q^{34} + 3553198 q^{36} - 935610 q^{37} - 984745 q^{38} - 294888 q^{39} + 160466 q^{41} + 783725 q^{42} + 146400 q^{43} + 2261658 q^{44} - 2639009 q^{46} - 4446810 q^{47} - 3994240 q^{48} + 7532484 q^{49} - 2294894 q^{51} - 4582065 q^{52} - 3977030 q^{53} - 3979475 q^{54} - 743430 q^{56} - 2455430 q^{57} - 14413560 q^{58} - 1614425 q^{59} + 7720866 q^{61} - 20362850 q^{62} - 26297840 q^{63} + 21801809 q^{64} + 945327 q^{66} - 3017910 q^{67} - 17494265 q^{68} - 13519553 q^{69} - 9483549 q^{71} - 21929370 q^{72} + 388070 q^{73} + 16144878 q^{74} - 13507955 q^{76} - 25473115 q^{77} + 3108110 q^{78} - 10950620 q^{79} + 34443488 q^{81} + 354040 q^{82} - 47217920 q^{83} - 27843102 q^{84} + 20021766 q^{86} - 56120960 q^{87} - 54397660 q^{88} + 10850545 q^{89} - 8553794 q^{91} + 20734425 q^{92} + 11206260 q^{93} - 52545997 q^{94} - 28125034 q^{96} - 32784020 q^{97} - 14131170 q^{98} + 27513602 q^{99}+O(q^{100}) \) 48 * q - 25 * q^2 - 95 * q^3 + 3419 * q^4 + 431 * q^6 - 4030 * q^7 - 3375 * q^8 + 36231 * q^9 + 781 * q^11 - 3925 * q^12 - 4290 * q^13 - 20762 * q^14 + 270603 * q^16 - 75075 * q^17 - 89950 * q^18 + 17750 * q^19 - 48034 * q^21 - 331305 * q^22 - 343890 * q^23 - 271570 * q^24 - 304129 * q^26 - 474740 * q^27 - 1146535 * q^28 - 59330 * q^29 - 385989 * q^31 - 1887300 * q^32 - 879805 * q^33 + 286938 * q^34 + 3553198 * q^36 - 935610 * q^37 - 984745 * q^38 - 294888 * q^39 + 160466 * q^41 + 783725 * q^42 + 146400 * q^43 + 2261658 * q^44 - 2639009 * q^46 - 4446810 * q^47 - 3994240 * q^48 + 7532484 * q^49 - 2294894 * q^51 - 4582065 * q^52 - 3977030 * q^53 - 3979475 * q^54 - 743430 * q^56 - 2455430 * q^57 - 14413560 * q^58 - 1614425 * q^59 + 7720866 * q^61 - 20362850 * q^62 - 26297840 * q^63 + 21801809 * q^64 + 945327 * q^66 - 3017910 * q^67 - 17494265 * q^68 - 13519553 * q^69 - 9483549 * q^71 - 21929370 * q^72 + 388070 * q^73 + 16144878 * q^74 - 13507955 * q^76 - 25473115 * q^77 + 3108110 * q^78 - 10950620 * q^79 + 34443488 * q^81 + 354040 * q^82 - 47217920 * q^83 - 27843102 * q^84 + 20021766 * q^86 - 56120960 * q^87 - 54397660 * q^88 + 10850545 * q^89 - 8553794 * q^91 + 20734425 * q^92 + 11206260 * q^93 - 52545997 * q^94 - 28125034 * q^96 - 32784020 * q^97 - 14131170 * q^98 + 27513602 * q^99

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