LMFDB - Embedded newform 625.8.a.e.1.15

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			1.15
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)

$f(q)$	$=$	$q-9.78625 q^{2} -79.0302 q^{3} -32.2293 q^{4} +773.410 q^{6} +741.503 q^{7} +1568.04 q^{8} +4058.78 q^{9} +O(q^{10})$	\(q-9.78625 q^{2} -79.0302 q^{3} -32.2293 q^{4} +773.410 q^{6} +741.503 q^{7} +1568.04 q^{8} +4058.78 q^{9} +2548.10 q^{11} +2547.09 q^{12} +5811.51 q^{13} -7256.53 q^{14} -11219.9 q^{16} +26664.7 q^{17} -39720.2 q^{18} +21907.4 q^{19} -58601.1 q^{21} -24936.3 q^{22} +7844.69 q^{23} -123923. q^{24} -56872.8 q^{26} -147927. q^{27} -23898.1 q^{28} +152238. q^{29} -75660.0 q^{31} -90908.8 q^{32} -201377. q^{33} -260948. q^{34} -130812. q^{36} -609130. q^{37} -214391. q^{38} -459285. q^{39} +852763. q^{41} +573485. q^{42} -868704. q^{43} -82123.6 q^{44} -76770.1 q^{46} -933312. q^{47} +886713. q^{48} -273717. q^{49} -2.10732e6 q^{51} -187301. q^{52} +292116. q^{53} +1.44765e6 q^{54} +1.16271e6 q^{56} -1.73135e6 q^{57} -1.48984e6 q^{58} +1.98804e6 q^{59} -310096. q^{61} +740428. q^{62} +3.00960e6 q^{63} +2.32580e6 q^{64} +1.97072e6 q^{66} -3.09162e6 q^{67} -859387. q^{68} -619968. q^{69} -4.29481e6 q^{71} +6.36435e6 q^{72} -1.11256e6 q^{73} +5.96109e6 q^{74} -706061. q^{76} +1.88942e6 q^{77} +4.49467e6 q^{78} +1.71892e6 q^{79} +2.81417e6 q^{81} -8.34535e6 q^{82} +8.65834e6 q^{83} +1.88868e6 q^{84} +8.50136e6 q^{86} -1.20314e7 q^{87} +3.99553e6 q^{88} +194502. q^{89} +4.30925e6 q^{91} -252829. q^{92} +5.97943e6 q^{93} +9.13363e6 q^{94} +7.18454e6 q^{96} +1.56909e7 q^{97} +2.67866e6 q^{98} +1.03422e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−9.78625	−0.864990	−0.432495	−	0.901636i	$-0.642367\pi$
$2$	−9.78625	−0.864990	−0.432495	+	0.901636i	$0.642367\pi$
$3$	−79.0302	−1.68993	−0.844965	−	0.534821i	$-0.820379\pi$
$3$	−79.0302	−1.68993	−0.844965	+	0.534821i	$0.820379\pi$
$4$	−32.2293	−0.251792
$5$	0	0
$5$	0	0
$6$	773.410	1.46177
$7$	741.503	0.817089	0.408545	−	0.912738i	$-0.366036\pi$
$7$	741.503	0.817089	0.408545	+	0.912738i	$0.366036\pi$
$8$	1568.04	1.08279
$9$	4058.78	1.85587
$10$	0	0
$11$	2548.10	0.577221	0.288610	−	0.957447i	$-0.406807\pi$
$11$	2548.10	0.577221	0.288610	+	0.957447i	$0.406807\pi$
$12$	2547.09	0.425511
$13$	5811.51	0.733647	0.366823	−	0.930291i	$-0.380445\pi$
$13$	5811.51	0.733647	0.366823	+	0.930291i	$0.380445\pi$
$14$	−7256.53	−0.706774
$15$	0	0
$16$	−11219.9	−0.684809
$17$	26664.7	1.31633	0.658167	−	0.752872i	$-0.271332\pi$
$17$	26664.7	1.31633	0.658167	+	0.752872i	$0.271332\pi$
$18$	−39720.2	−1.60531
$19$	21907.4	0.732745	0.366373	−	0.930468i	$-0.380600\pi$
$19$	21907.4	0.732745	0.366373	+	0.930468i	$0.380600\pi$
$20$	0	0
$21$	−58601.1	−1.38082
$22$	−24936.3	−0.499290
$23$	7844.69	0.134440	0.0672200	−	0.997738i	$-0.478587\pi$
$23$	7844.69	0.134440	0.0672200	+	0.997738i	$0.478587\pi$
$24$	−123923.	−1.82984
$25$	0	0
$26$	−56872.8	−0.634597
$27$	−147927.	−1.44635
$28$	−23898.1	−0.205736
$29$	152238.	1.15913	0.579563	−	0.814928i	$-0.303223\pi$
$29$	152238.	1.15913	0.579563	+	0.814928i	$0.303223\pi$
$30$	0	0
$31$	−75660.0	−0.456142	−0.228071	−	0.973644i	$-0.573242\pi$
$31$	−75660.0	−0.456142	−0.228071	+	0.973644i	$0.573242\pi$
$32$	−90908.8	−0.490434
$33$	−201377.	−0.975463
$34$	−260948.	−1.13862
$35$	0	0
$36$	−130812.	−0.467292
$37$	−609130.	−1.97699	−0.988493	−	0.151265i	$-0.951665\pi$
$37$	−609130.	−1.97699	−0.988493	+	0.151265i	$0.951665\pi$
$38$	−214391.	−0.633817
$39$	−459285.	−1.23981
$40$	0	0
$41$	852763.	1.93235	0.966173	−	0.257894i	$-0.0830284\pi$
$41$	852763.	1.93235	0.966173	+	0.257894i	$0.0830284\pi$
$42$	573485.	1.19440
$43$	−868704.	−1.66622	−0.833110	−	0.553107i	$-0.813442\pi$
$43$	−868704.	−1.66622	−0.833110	+	0.553107i	$0.813442\pi$
$44$	−82123.6	−0.145339
$45$	0	0
$46$	−76770.1	−0.116289
$47$	−933312.	−1.31125	−0.655623	−	0.755088i	$-0.727594\pi$
$47$	−933312.	−1.31125	−0.655623	+	0.755088i	$0.727594\pi$
$48$	886713.	1.15728
$49$	−273717.	−0.332365
$50$	0	0
$51$	−2.10732e6	−2.22451
$52$	−187301.	−0.184726
$53$	292116.	0.269519	0.134759	−	0.990878i	$-0.456974\pi$
$53$	292116.	0.269519	0.134759	+	0.990878i	$0.456974\pi$
$54$	1.44765e6	1.25108
$55$	0	0
$56$	1.16271e6	0.884734
$57$	−1.73135e6	−1.23829
$58$	−1.48984e6	−1.00263
$59$	1.98804e6	1.26021	0.630105	−	0.776510i	$-0.283012\pi$
$59$	1.98804e6	1.26021	0.630105	+	0.776510i	$0.283012\pi$
$60$	0	0
$61$	−310096.	−0.174921	−0.0874604	−	0.996168i	$-0.527875\pi$
$61$	−310096.	−0.174921	−0.0874604	+	0.996168i	$0.527875\pi$
$62$	740428.	0.394559
$63$	3.00960e6	1.51641
$64$	2.32580e6	1.10903
$65$	0	0
$66$	1.97072e6	0.843766
$67$	−3.09162e6	−1.25581	−0.627906	−	0.778289i	$-0.716088\pi$
$67$	−3.09162e6	−1.25581	−0.627906	+	0.778289i	$0.716088\pi$
$68$	−859387.	−0.331442
$69$	−619968.	−0.227194
$70$	0	0
$71$	−4.29481e6	−1.42410	−0.712049	−	0.702130i	$-0.752233\pi$
$71$	−4.29481e6	−1.42410	−0.712049	+	0.702130i	$0.752233\pi$
$72$	6.36435e6	2.00951
$73$	−1.11256e6	−0.334730	−0.167365	−	0.985895i	$-0.553526\pi$
$73$	−1.11256e6	−0.334730	−0.167365	+	0.985895i	$0.553526\pi$
$74$	5.96109e6	1.71007
$75$	0	0
$76$	−706061.	−0.184499
$77$	1.88942e6	0.471641
$78$	4.49467e6	1.07243
$79$	1.71892e6	0.392249	0.196125	−	0.980579i	$-0.437164\pi$
$79$	1.71892e6	0.392249	0.196125	+	0.980579i	$0.437164\pi$
$80$	0	0
$81$	2.81417e6	0.588373
$82$	−8.34535e6	−1.67146
$83$	8.65834e6	1.66212	0.831058	−	0.556186i	$-0.187736\pi$
$83$	8.65834e6	1.66212	0.831058	+	0.556186i	$0.187736\pi$
$84$	1.88868e6	0.347680
$85$	0	0
$86$	8.50136e6	1.44126
$87$	−1.20314e7	−1.95884
$88$	3.99553e6	0.625008
$89$	194502.	0.0292455	0.0146227	−	0.999893i	$-0.495345\pi$
$89$	194502.	0.0292455	0.0146227	+	0.999893i	$0.495345\pi$
$90$	0	0
$91$	4.30925e6	0.599455
$92$	−252829.	−0.0338509
$93$	5.97943e6	0.770849
$94$	9.13363e6	1.13422
$95$	0	0
$96$	7.18454e6	0.828800
$97$	1.56909e7	1.74560	0.872801	−	0.488075i	$-0.162301\pi$
$97$	1.56909e7	1.74560	0.872801	+	0.488075i	$0.162301\pi$
$98$	2.67866e6	0.287493
$99$	1.03422e7	1.07124
$100$	0	0
$101$	−3.67061e6	−0.354498	−0.177249	−	0.984166i	$-0.556720\pi$
$101$	−3.67061e6	−0.354498	−0.177249	+	0.984166i	$0.556720\pi$
$102$	2.06228e7	1.92418
$103$	−4.80891e6	−0.433627	−0.216814	−	0.976213i	$-0.569566\pi$
$103$	−4.80891e6	−0.433627	−0.216814	+	0.976213i	$0.569566\pi$
$104$	9.11270e6	0.794384
$105$	0	0
$106$	−2.85872e6	−0.233131
$107$	−1.00876e7	−0.796058	−0.398029	−	0.917373i	$-0.630306\pi$
$107$	−1.00876e7	−0.796058	−0.398029	+	0.917373i	$0.630306\pi$
$108$	4.76759e6	0.364180
$109$	95153.0	0.00703768	0.00351884	−	0.999994i	$-0.498880\pi$
$109$	95153.0	0.00703768	0.00351884	+	0.999994i	$0.498880\pi$
$110$	0	0
$111$	4.81397e7	3.34097
$112$	−8.31960e6	−0.559550
$113$	−1.68625e7	−1.09938	−0.549689	−	0.835369i	$-0.685254\pi$
$113$	−1.68625e7	−1.09938	−0.549689	+	0.835369i	$0.685254\pi$
$114$	1.69434e7	1.07111
$115$	0	0
$116$	−4.90653e6	−0.291858
$117$	2.35876e7	1.36155
$118$	−1.94555e7	−1.09007
$119$	1.97720e7	1.07556
$120$	0	0
$121$	−1.29944e7	−0.666816
$122$	3.03467e6	0.151305
$123$	−6.73941e7	−3.26553
$124$	2.43847e6	0.114853
$125$	0	0
$126$	−2.94526e7	−1.31168
$127$	−2.32131e7	−1.00559	−0.502794	−	0.864406i	$-0.667695\pi$
$127$	−2.32131e7	−1.00559	−0.502794	+	0.864406i	$0.667695\pi$
$128$	−1.11246e7	−0.468866
$129$	6.86539e7	2.81580
$130$	0	0
$131$	−1.92020e7	−0.746272	−0.373136	−	0.927777i	$-0.621718\pi$
$131$	−1.92020e7	−0.746272	−0.373136	+	0.927777i	$0.621718\pi$
$132$	6.49024e6	0.245614
$133$	1.62444e7	0.598718
$134$	3.02554e7	1.08627
$135$	0	0
$136$	4.18115e7	1.42531
$137$	−2.50112e7	−0.831023	−0.415512	−	0.909588i	$-0.636397\pi$
$137$	−2.50112e7	−0.831023	−0.415512	+	0.909588i	$0.636397\pi$
$138$	6.06716e6	0.196521
$139$	−5.40368e7	−1.70662	−0.853312	−	0.521400i	$-0.825410\pi$
$139$	−5.40368e7	−1.70662	−0.853312	+	0.521400i	$0.825410\pi$
$140$	0	0
$141$	7.37599e7	2.21592
$142$	4.20301e7	1.23183
$143$	1.48083e7	0.423476
$144$	−4.55392e7	−1.27091
$145$	0	0
$146$	1.08878e7	0.289538
$147$	2.16319e7	0.561674
$148$	1.96318e7	0.497789
$149$	1.69364e6	0.0419439	0.0209719	−	0.999780i	$-0.493324\pi$
$149$	1.69364e6	0.0419439	0.0209719	+	0.999780i	$0.493324\pi$
$150$	0	0
$151$	−5.30598e6	−0.125414	−0.0627070	−	0.998032i	$-0.519973\pi$
$151$	−5.30598e6	−0.125414	−0.0627070	+	0.998032i	$0.519973\pi$
$152$	3.43517e7	0.793407
$153$	1.08226e8	2.44294
$154$	−1.84904e7	−0.407965
$155$	0	0
$156$	1.48024e7	0.312174
$157$	−4.71279e7	−0.971916	−0.485958	−	0.873982i	$-0.661529\pi$
$157$	−4.71279e7	−0.971916	−0.485958	+	0.873982i	$0.661529\pi$
$158$	−1.68218e7	−0.339292
$159$	−2.30860e7	−0.455468
$160$	0	0
$161$	5.81686e6	0.109849
$162$	−2.75402e7	−0.508937
$163$	4.39007e7	0.793989	0.396995	−	0.917821i	$-0.370053\pi$
$163$	4.39007e7	0.793989	0.396995	+	0.917821i	$0.370053\pi$
$164$	−2.74840e7	−0.486549
$165$	0	0
$166$	−8.47326e7	−1.43771
$167$	−9.32969e7	−1.55010	−0.775050	−	0.631900i	$-0.782275\pi$
$167$	−9.32969e7	−1.55010	−0.775050	+	0.631900i	$0.782275\pi$
$168$	−9.18892e7	−1.49514
$169$	−2.89749e7	−0.461763
$170$	0	0
$171$	8.89172e7	1.35988
$172$	2.79978e7	0.419540
$173$	2.04364e7	0.300083	0.150042	−	0.988680i	$-0.452059\pi$
$173$	2.04364e7	0.300083	0.150042	+	0.988680i	$0.452059\pi$
$174$	1.17742e8	1.69438
$175$	0	0
$176$	−2.85895e7	−0.395286
$177$	−1.57115e8	−2.12967
$178$	−1.90344e6	−0.0252970
$179$	2.74971e6	0.0358345	0.0179173	−	0.999839i	$-0.494296\pi$
$179$	2.74971e6	0.0358345	0.0179173	+	0.999839i	$0.494296\pi$
$180$	0	0
$181$	1.00559e8	1.26050	0.630251	−	0.776391i	$-0.282952\pi$
$181$	1.00559e8	1.26050	0.630251	+	0.776391i	$0.282952\pi$
$182$	−4.21714e7	−0.518523
$183$	2.45069e7	0.295604
$184$	1.23008e7	0.145570
$185$	0	0
$186$	−5.85162e7	−0.666777
$187$	6.79444e7	0.759816
$188$	3.00800e7	0.330161
$189$	−1.09688e8	−1.18180
$190$	0	0
$191$	1.81077e7	0.188038	0.0940191	−	0.995570i	$-0.470029\pi$
$191$	1.81077e7	0.188038	0.0940191	+	0.995570i	$0.470029\pi$
$192$	−1.83809e8	−1.87418
$193$	−1.22225e8	−1.22380	−0.611899	−	0.790936i	$-0.709594\pi$
$193$	−1.22225e8	−1.22380	−0.611899	+	0.790936i	$0.709594\pi$
$194$	−1.53555e8	−1.50993
$195$	0	0
$196$	8.82172e6	0.0836868
$197$	−1.76319e8	−1.64312	−0.821558	−	0.570125i	$-0.806895\pi$
$197$	−1.76319e8	−1.64312	−0.821558	+	0.570125i	$0.806895\pi$
$198$	−1.01211e8	−0.926616
$199$	4.03749e7	0.363183	0.181592	−	0.983374i	$-0.441875\pi$
$199$	4.03749e7	0.363183	0.181592	+	0.983374i	$0.441875\pi$
$200$	0	0
$201$	2.44332e8	2.12224
$202$	3.59215e7	0.306637
$203$	1.12885e8	0.947109
$204$	6.79176e7	0.560114
$205$	0	0
$206$	4.70612e7	0.375084
$207$	3.18399e7	0.249503
$208$	−6.52046e7	−0.502408
$209$	5.58222e7	0.422956
$210$	0	0
$211$	−5.38765e7	−0.394830	−0.197415	−	0.980320i	$-0.563255\pi$
$211$	−5.38765e7	−0.394830	−0.197415	+	0.980320i	$0.563255\pi$
$212$	−9.41469e6	−0.0678626
$213$	3.39420e8	2.40663
$214$	9.87198e7	0.688583
$215$	0	0
$216$	−2.31956e8	−1.56610
$217$	−5.61021e7	−0.372709
$218$	−931191.	−0.00608753
$219$	8.79261e7	0.565670
$220$	0	0
$221$	1.54962e8	0.965724
$222$	−4.71107e8	−2.88991
$223$	−2.11569e8	−1.27757	−0.638786	−	0.769384i	$-0.720563\pi$
$223$	−2.11569e8	−1.27757	−0.638786	+	0.769384i	$0.720563\pi$
$224$	−6.74091e7	−0.400729
$225$	0	0
$226$	1.65020e8	0.950951
$227$	−9.36372e7	−0.531322	−0.265661	−	0.964066i	$-0.585590\pi$
$227$	−9.36372e7	−0.531322	−0.265661	+	0.964066i	$0.585590\pi$
$228$	5.58001e7	0.311791
$229$	−2.28720e8	−1.25858	−0.629288	−	0.777172i	$-0.716654\pi$
$229$	−2.28720e8	−1.25858	−0.629288	+	0.777172i	$0.716654\pi$
$230$	0	0
$231$	−1.49322e8	−0.797041
$232$	2.38716e8	1.25509
$233$	−1.30362e6	−0.00675156	−0.00337578	−	0.999994i	$-0.501075\pi$
$233$	−1.30362e6	−0.00675156	−0.00337578	+	0.999994i	$0.501075\pi$
$234$	−2.30834e8	−1.17773
$235$	0	0
$236$	−6.40732e7	−0.317310
$237$	−1.35847e8	−0.662874
$238$	−1.93493e8	−0.930351
$239$	−2.24137e8	−1.06199	−0.530997	−	0.847374i	$-0.678182\pi$
$239$	−2.24137e8	−1.06199	−0.530997	+	0.847374i	$0.678182\pi$
$240$	0	0
$241$	−2.56542e8	−1.18059	−0.590296	−	0.807187i	$-0.700989\pi$
$241$	−2.56542e8	−1.18059	−0.590296	+	0.807187i	$0.700989\pi$
$242$	1.27166e8	0.576789
$243$	1.01112e8	0.452045
$244$	9.99418e6	0.0440436
$245$	0	0
$246$	6.59535e8	2.82465
$247$	1.27315e8	0.537576
$248$	−1.18638e8	−0.493905
$249$	−6.84271e8	−2.80886
$250$	0	0
$251$	4.15348e7	0.165788	0.0828941	−	0.996558i	$-0.473584\pi$
$251$	4.15348e7	0.165788	0.0828941	+	0.996558i	$0.473584\pi$
$252$	−9.69973e7	−0.381819
$253$	1.99891e7	0.0776016
$254$	2.27169e8	0.869824
$255$	0	0
$256$	−1.88835e8	−0.703466
$257$	3.28442e8	1.20696	0.603480	−	0.797378i	$-0.293780\pi$
$257$	3.28442e8	1.20696	0.603480	+	0.797378i	$0.293780\pi$
$258$	−6.71864e8	−2.43564
$259$	−4.51671e8	−1.61537
$260$	0	0
$261$	6.17901e8	2.15118
$262$	1.87916e8	0.645518
$263$	5.53287e7	0.187545	0.0937724	−	0.995594i	$-0.470107\pi$
$263$	5.53287e7	0.187545	0.0937724	+	0.995594i	$0.470107\pi$
$264$	−3.15768e8	−1.05622
$265$	0	0
$266$	−1.58972e8	−0.517885
$267$	−1.53715e7	−0.0494228
$268$	9.96409e7	0.316203
$269$	−2.09311e7	−0.0655631	−0.0327816	−	0.999463i	$-0.510437\pi$
$269$	−2.09311e7	−0.0655631	−0.0327816	+	0.999463i	$0.510437\pi$
$270$	0	0
$271$	3.95523e8	1.20720	0.603601	−	0.797287i	$-0.293732\pi$
$271$	3.95523e8	1.20720	0.603601	+	0.797287i	$0.293732\pi$
$272$	−2.99176e8	−0.901438
$273$	−3.40561e8	−1.01304
$274$	2.44766e8	0.718827
$275$	0	0
$276$	1.99812e7	0.0572056
$277$	−4.12175e8	−1.16521	−0.582603	−	0.812757i	$-0.697966\pi$
$277$	−4.12175e8	−1.16521	−0.582603	+	0.812757i	$0.697966\pi$
$278$	5.28818e8	1.47621
$279$	−3.07087e8	−0.846539
$280$	0	0
$281$	6.81162e7	0.183138	0.0915689	−	0.995799i	$-0.470812\pi$
$281$	6.81162e7	0.183138	0.0915689	+	0.995799i	$0.470812\pi$
$282$	−7.21833e8	−1.91675
$283$	2.46310e8	0.645996	0.322998	−	0.946400i	$-0.395309\pi$
$283$	2.46310e8	0.645996	0.322998	+	0.946400i	$0.395309\pi$
$284$	1.38419e8	0.358576
$285$	0	0
$286$	−1.44918e8	−0.366303
$287$	6.32326e8	1.57890
$288$	−3.68979e8	−0.910180
$289$	3.00670e8	0.732736
$290$	0	0
$291$	−1.24005e9	−2.94995
$292$	3.58572e7	0.0842822
$293$	6.77732e8	1.57406	0.787030	−	0.616914i	$-0.211617\pi$
$293$	6.77732e8	1.57406	0.787030	+	0.616914i	$0.211617\pi$
$294$	−2.11695e8	−0.485843
$295$	0	0
$296$	−9.55142e8	−2.14066
$297$	−3.76933e8	−0.834866
$298$	−1.65743e7	−0.0362810
$299$	4.55895e7	0.0986315
$300$	0	0
$301$	−6.44147e8	−1.36145
$302$	5.19256e7	0.108482
$303$	2.90089e8	0.599077
$304$	−2.45799e8	−0.501791
$305$	0	0
$306$	−1.05913e9	−2.11312
$307$	−3.82363e8	−0.754209	−0.377105	−	0.926171i	$-0.623080\pi$
$307$	−3.82363e8	−0.754209	−0.377105	+	0.926171i	$0.623080\pi$
$308$	−6.08948e7	−0.118755
$309$	3.80050e8	0.732800
$310$	0	0
$311$	6.17563e8	1.16418	0.582090	−	0.813125i	$-0.302235\pi$
$311$	6.17563e8	1.16418	0.582090	+	0.813125i	$0.302235\pi$
$312$	−7.20179e8	−1.34245
$313$	7.93723e8	1.46307	0.731533	−	0.681806i	$-0.238805\pi$
$313$	7.93723e8	1.46307	0.731533	+	0.681806i	$0.238805\pi$
$314$	4.61205e8	0.840698
$315$	0	0
$316$	−5.53998e7	−0.0987651
$317$	−7.16446e8	−1.26321	−0.631606	−	0.775290i	$-0.717604\pi$
$317$	−7.16446e8	−1.26321	−0.631606	+	0.775290i	$0.717604\pi$
$318$	2.25925e8	0.393976
$319$	3.87918e8	0.669071
$320$	0	0
$321$	7.97226e8	1.34528
$322$	−5.69252e7	−0.0950188
$323$	5.84155e8	0.964537
$324$	−9.06988e7	−0.148147
$325$	0	0
$326$	−4.29623e8	−0.686793
$327$	−7.51996e6	−0.0118932
$328$	1.33717e9	2.09232
$329$	−6.92053e8	−1.07141
$330$	0	0
$331$	−3.29588e8	−0.499544	−0.249772	−	0.968305i	$-0.580356\pi$
$331$	−3.29588e8	−0.499544	−0.249772	+	0.968305i	$0.580356\pi$
$332$	−2.79052e8	−0.418507
$333$	−2.47232e9	−3.66902
$334$	9.13027e8	1.34082
$335$	0	0
$336$	6.57500e8	0.945601
$337$	−8.69084e8	−1.23696	−0.618482	−	0.785799i	$-0.712252\pi$
$337$	−8.69084e8	−1.23696	−0.618482	+	0.785799i	$0.712252\pi$
$338$	2.83556e8	0.399420
$339$	1.33265e9	1.85787
$340$	0	0
$341$	−1.92789e8	−0.263295
$342$	−8.70166e8	−1.17628
$343$	−8.13621e8	−1.08866
$344$	−1.36217e9	−1.80416
$345$	0	0
$346$	−1.99995e8	−0.259569
$347$	1.48475e9	1.90765	0.953826	−	0.300360i	$-0.0971068\pi$
$347$	1.48475e9	1.90765	0.953826	+	0.300360i	$0.0971068\pi$
$348$	3.87765e8	0.493220
$349$	1.28449e9	1.61749	0.808745	−	0.588159i	$-0.200147\pi$
$349$	1.28449e9	1.61749	0.808745	+	0.588159i	$0.200147\pi$
$350$	0	0
$351$	−8.59680e8	−1.06111
$352$	−2.31645e8	−0.283089
$353$	−9.10271e8	−1.10144	−0.550718	−	0.834691i	$-0.685646\pi$
$353$	−9.10271e8	−1.10144	−0.550718	+	0.834691i	$0.685646\pi$
$354$	1.53757e9	1.84214
$355$	0	0
$356$	−6.26866e6	−0.00736376
$357$	−1.56258e9	−1.81763
$358$	−2.69094e7	−0.0309965
$359$	−7.02082e8	−0.800861	−0.400430	−	0.916327i	$-0.631139\pi$
$359$	−7.02082e8	−0.800861	−0.400430	+	0.916327i	$0.631139\pi$
$360$	0	0
$361$	−4.13938e8	−0.463085
$362$	−9.84091e8	−1.09032
$363$	1.02695e9	1.12687
$364$	−1.38884e8	−0.150938
$365$	0	0
$366$	−2.39831e8	−0.255695
$367$	−4.34446e8	−0.458780	−0.229390	−	0.973335i	$-0.573673\pi$
$367$	−4.34446e8	−0.458780	−0.229390	+	0.973335i	$0.573673\pi$
$368$	−8.80168e7	−0.0920658
$369$	3.46118e9	3.58618
$370$	0	0
$371$	2.16604e8	0.220221
$372$	−1.92713e8	−0.194093
$373$	−9.25995e7	−0.0923906	−0.0461953	−	0.998932i	$-0.514710\pi$
$373$	−9.25995e7	−0.0923906	−0.0461953	+	0.998932i	$0.514710\pi$
$374$	−6.64921e8	−0.657233
$375$	0	0
$376$	−1.46347e9	−1.41980
$377$	8.84733e8	0.850388
$378$	1.07344e9	1.02225
$379$	−5.72575e8	−0.540251	−0.270125	−	0.962825i	$-0.587065\pi$
$379$	−5.72575e8	−0.540251	−0.270125	+	0.962825i	$0.587065\pi$
$380$	0	0
$381$	1.83454e9	1.69937
$382$	−1.77206e8	−0.162651
$383$	−1.60707e9	−1.46164	−0.730818	−	0.682572i	$-0.760861\pi$
$383$	−1.60707e9	−1.46164	−0.730818	+	0.682572i	$0.760861\pi$
$384$	8.79179e8	0.792351
$385$	0	0
$386$	1.19613e9	1.05857
$387$	−3.52588e9	−3.09228
$388$	−5.05706e8	−0.439528
$389$	8.89912e8	0.766520	0.383260	−	0.923641i	$-0.374801\pi$
$389$	8.89912e8	0.766520	0.383260	+	0.923641i	$0.374801\pi$
$390$	0	0
$391$	2.09177e8	0.176968
$392$	−4.29200e8	−0.359881
$393$	1.51754e9	1.26115
$394$	1.72551e9	1.42128
$395$	0	0
$396$	−3.33321e8	−0.269731
$397$	−1.09740e9	−0.880235	−0.440117	−	0.897940i	$-0.645063\pi$
$397$	−1.09740e9	−0.880235	−0.440117	+	0.897940i	$0.645063\pi$
$398$	−3.95119e8	−0.314150
$399$	−1.28380e9	−1.01179
$400$	0	0
$401$	1.82602e8	0.141416	0.0707081	−	0.997497i	$-0.477474\pi$
$401$	1.82602e8	0.141416	0.0707081	+	0.997497i	$0.477474\pi$
$402$	−2.39109e9	−1.83571
$403$	−4.39699e8	−0.334647
$404$	1.18301e8	0.0892596
$405$	0	0
$406$	−1.10472e9	−0.819240
$407$	−1.55212e9	−1.14116
$408$	−3.30437e9	−2.40868
$409$	5.88413e8	0.425256	0.212628	−	0.977133i	$-0.431798\pi$
$409$	5.88413e8	0.425256	0.212628	+	0.977133i	$0.431798\pi$
$410$	0	0
$411$	1.97664e9	1.40437
$412$	1.54988e8	0.109184
$413$	1.47414e9	1.02970
$414$	−3.11593e8	−0.215817
$415$	0	0
$416$	−5.28317e8	−0.359805
$417$	4.27054e9	2.88408
$418$	−5.46290e8	−0.365853
$419$	−3.15815e8	−0.209741	−0.104871	−	0.994486i	$-0.533443\pi$
$419$	−3.15815e8	−0.209741	−0.104871	+	0.994486i	$0.533443\pi$
$420$	0	0
$421$	−1.67004e9	−1.09079	−0.545393	−	0.838180i	$-0.683620\pi$
$421$	−1.67004e9	−1.09079	−0.545393	+	0.838180i	$0.683620\pi$
$422$	5.27249e8	0.341525
$423$	−3.78811e9	−2.43350
$424$	4.58050e8	0.291832
$425$	0	0
$426$	−3.32165e9	−2.08171
$427$	−2.29937e8	−0.142926
$428$	3.25117e8	0.200441
$429$	−1.17030e9	−0.715645
$430$	0	0
$431$	1.96107e8	0.117984	0.0589920	−	0.998258i	$-0.481211\pi$
$431$	1.96107e8	0.117984	0.0589920	+	0.998258i	$0.481211\pi$
$432$	1.65973e9	0.990477
$433$	2.05254e9	1.21502	0.607510	−	0.794312i	$-0.292169\pi$
$433$	2.05254e9	1.21502	0.607510	+	0.794312i	$0.292169\pi$
$434$	5.49029e8	0.322390
$435$	0	0
$436$	−3.06672e6	−0.00177203
$437$	1.71857e8	0.0985103
$438$	−8.60467e8	−0.489299
$439$	2.50472e9	1.41297	0.706485	−	0.707729i	$-0.250280\pi$
$439$	2.50472e9	1.41297	0.706485	+	0.707729i	$0.250280\pi$
$440$	0	0
$441$	−1.11096e9	−0.616825
$442$	−1.51650e9	−0.835342
$443$	−9.55897e8	−0.522394	−0.261197	−	0.965286i	$-0.584117\pi$
$443$	−9.55897e8	−0.522394	−0.261197	+	0.965286i	$0.584117\pi$
$444$	−1.55151e9	−0.841229
$445$	0	0
$446$	2.07047e9	1.10509
$447$	−1.33849e8	−0.0708822
$448$	1.72459e9	0.906177
$449$	6.95750e7	0.0362736	0.0181368	−	0.999836i	$-0.494227\pi$
$449$	6.95750e7	0.0362736	0.0181368	+	0.999836i	$0.494227\pi$
$450$	0	0
$451$	2.17293e9	1.11539
$452$	5.43467e8	0.276814
$453$	4.19333e8	0.211941
$454$	9.16357e8	0.459589
$455$	0	0
$456$	−2.71483e9	−1.34080
$457$	−2.26929e9	−1.11220	−0.556101	−	0.831115i	$-0.687703\pi$
$457$	−2.26929e9	−1.11220	−0.556101	+	0.831115i	$0.687703\pi$
$458$	2.23831e9	1.08866
$459$	−3.94444e9	−1.90389
$460$	0	0
$461$	3.10799e9	1.47750	0.738748	−	0.673982i	$-0.235417\pi$
$461$	3.10799e9	1.47750	0.738748	+	0.673982i	$0.235417\pi$
$462$	1.46130e9	0.689432
$463$	1.67731e8	0.0785381	0.0392690	−	0.999229i	$-0.487497\pi$
$463$	1.67731e8	0.0785381	0.0392690	+	0.999229i	$0.487497\pi$
$464$	−1.70810e9	−0.793780
$465$	0	0
$466$	1.27575e7	0.00584003
$467$	−3.23090e9	−1.46796	−0.733980	−	0.679171i	$-0.762339\pi$
$467$	−3.23090e9	−1.46796	−0.733980	+	0.679171i	$0.762339\pi$
$468$	−7.60213e8	−0.342827
$469$	−2.29245e9	−1.02611
$470$	0	0
$471$	3.72453e9	1.64247
$472$	3.11733e9	1.36454
$473$	−2.21355e9	−0.961777
$474$	1.32943e9	0.573380
$475$	0	0
$476$	−6.37238e8	−0.270818
$477$	1.18563e9	0.500191
$478$	2.19346e9	0.918614
$479$	−3.56881e9	−1.48371	−0.741856	−	0.670559i	$-0.766054\pi$
$479$	−3.56881e9	−1.48371	−0.741856	+	0.670559i	$0.766054\pi$
$480$	0	0
$481$	−3.53996e9	−1.45041
$482$	2.51059e9	1.02120
$483$	−4.59708e8	−0.185638
$484$	4.18800e8	0.167899
$485$	0	0
$486$	−9.89509e8	−0.391014
$487$	4.40604e9	1.72861	0.864305	−	0.502968i	$-0.167759\pi$
$487$	4.40604e9	1.72861	0.864305	+	0.502968i	$0.167759\pi$
$488$	−4.86244e8	−0.189402
$489$	−3.46948e9	−1.34179
$490$	0	0
$491$	−1.48886e8	−0.0567633	−0.0283817	−	0.999597i	$-0.509035\pi$
$491$	−1.48886e8	−0.0567633	−0.0283817	+	0.999597i	$0.509035\pi$
$492$	2.17207e9	0.822234
$493$	4.05939e9	1.52580
$494$	−1.24594e9	−0.464998
$495$	0	0
$496$	8.48899e8	0.312371
$497$	−3.18461e9	−1.16361
$498$	6.69644e9	2.42964
$499$	1.04117e9	0.375120	0.187560	−	0.982253i	$-0.439942\pi$
$499$	1.04117e9	0.375120	0.187560	+	0.982253i	$0.439942\pi$
$500$	0	0
$501$	7.37328e9	2.61956
$502$	−4.06470e8	−0.143405
$503$	−1.12650e9	−0.394678	−0.197339	−	0.980335i	$-0.563230\pi$
$503$	−1.12650e9	−0.394678	−0.197339	+	0.980335i	$0.563230\pi$
$504$	4.71918e9	1.64195
$505$	0	0
$506$	−1.95618e8	−0.0671246
$507$	2.28989e9	0.780347
$508$	7.48143e8	0.253199
$509$	3.61338e9	1.21451	0.607256	−	0.794506i	$-0.292270\pi$
$509$	3.61338e9	1.21451	0.607256	+	0.794506i	$0.292270\pi$
$510$	0	0
$511$	−8.24968e8	−0.273504
$512$	3.27193e9	1.07736
$513$	−3.24070e9	−1.05981
$514$	−3.21422e9	−1.04401
$515$	0	0
$516$	−2.21267e9	−0.708994
$517$	−2.37817e9	−0.756879
$518$	4.42017e9	1.39728
$519$	−1.61509e9	−0.507120
$520$	0	0
$521$	−3.03308e9	−0.939619	−0.469810	−	0.882768i	$-0.655677\pi$
$521$	−3.03308e9	−0.939619	−0.469810	+	0.882768i	$0.655677\pi$
$522$	−6.04693e9	−1.86075
$523$	−5.88407e9	−1.79855	−0.899273	−	0.437387i	$-0.855904\pi$
$523$	−5.88407e9	−1.79855	−0.899273	+	0.437387i	$0.855904\pi$
$524$	6.18868e8	0.187905
$525$	0	0
$526$	−5.41460e8	−0.162225
$527$	−2.01745e9	−0.600436
$528$	2.25943e9	0.668006
$529$	−3.34329e9	−0.981926
$530$	0	0
$531$	8.06901e9	2.33878
$532$	−5.23546e8	−0.150752
$533$	4.95584e9	1.41766
$534$	1.50430e8	0.0427502
$535$	0	0
$536$	−4.84780e9	−1.35978
$537$	−2.17310e8	−0.0605579
$538$	2.04837e8	0.0567115
$539$	−6.97458e8	−0.191848
$540$	0	0
$541$	−2.76954e9	−0.751999	−0.376000	−	0.926620i	$-0.622701\pi$
$541$	−2.76954e9	−0.751999	−0.376000	+	0.926620i	$0.622701\pi$
$542$	−3.87069e9	−1.04422
$543$	−7.94716e9	−2.13016
$544$	−2.42406e9	−0.645575
$545$	0	0
$546$	3.33281e9	0.876267
$547$	1.38231e9	0.361118	0.180559	−	0.983564i	$-0.442209\pi$
$547$	1.38231e9	0.361118	0.180559	+	0.983564i	$0.442209\pi$
$548$	8.06096e8	0.209245
$549$	−1.25861e9	−0.324629
$550$	0	0
$551$	3.33514e9	0.849344
$552$	−9.72137e8	−0.246003
$553$	1.27459e9	0.320503
$554$	4.03365e9	1.00789
$555$	0	0
$556$	1.74157e9	0.429714
$557$	2.29609e9	0.562984	0.281492	−	0.959564i	$-0.409171\pi$
$557$	2.29609e9	0.562984	0.281492	+	0.959564i	$0.409171\pi$
$558$	3.00523e9	0.732248
$559$	−5.04848e9	−1.22242
$560$	0	0
$561$	−5.36966e9	−1.28404
$562$	−6.66602e8	−0.158413
$563$	3.83669e9	0.906103	0.453052	−	0.891484i	$-0.350335\pi$
$563$	3.83669e9	0.906103	0.453052	+	0.891484i	$0.350335\pi$
$564$	−2.37723e9	−0.557949
$565$	0	0
$566$	−2.41045e9	−0.558780
$567$	2.08671e9	0.480753
$568$	−6.73445e9	−1.54200
$569$	−2.59591e9	−0.590740	−0.295370	−	0.955383i	$-0.595443\pi$
$569$	−2.59591e9	−0.590740	−0.295370	+	0.955383i	$0.595443\pi$
$570$	0	0
$571$	6.61910e8	0.148790	0.0743948	−	0.997229i	$-0.476298\pi$
$571$	6.61910e8	0.148790	0.0743948	+	0.997229i	$0.476298\pi$
$572$	−4.77262e8	−0.106628
$573$	−1.43105e9	−0.317771
$574$	−6.18810e9	−1.36573
$575$	0	0
$576$	9.43993e9	2.05821
$577$	3.46374e9	0.750638	0.375319	−	0.926896i	$-0.377533\pi$
$577$	3.46374e9	0.750638	0.375319	+	0.926896i	$0.377533\pi$
$578$	−2.94243e9	−0.633809
$579$	9.65949e9	2.06814
$580$	0	0
$581$	6.42018e9	1.35810
$582$	1.21355e10	2.55168
$583$	7.44340e8	0.155572
$584$	−1.74455e9	−0.362441
$585$	0	0
$586$	−6.63246e9	−1.36155
$587$	2.65724e9	0.542247	0.271124	−	0.962545i	$-0.412605\pi$
$587$	2.65724e9	0.542247	0.271124	+	0.962545i	$0.412605\pi$
$588$	−6.97182e8	−0.141425
$589$	−1.65751e9	−0.334236
$590$	0	0
$591$	1.39346e10	2.77675
$592$	6.83438e9	1.35386
$593$	−3.49842e9	−0.688938	−0.344469	−	0.938798i	$-0.611941\pi$
$593$	−3.49842e9	−0.688938	−0.344469	+	0.938798i	$0.611941\pi$
$594$	3.68876e9	0.722151
$595$	0	0
$596$	−5.45848e7	−0.0105611
$597$	−3.19084e9	−0.613755
$598$	−4.46150e8	−0.0853153
$599$	−3.46628e9	−0.658976	−0.329488	−	0.944160i	$-0.606876\pi$
$599$	−3.46628e9	−0.658976	−0.329488	+	0.944160i	$0.606876\pi$
$600$	0	0
$601$	−1.73033e8	−0.0325138	−0.0162569	−	0.999868i	$-0.505175\pi$
$601$	−1.73033e8	−0.0325138	−0.0162569	+	0.999868i	$0.505175\pi$
$602$	6.30378e9	1.17764
$603$	−1.25482e10	−2.33062
$604$	1.71008e8	0.0315782
$605$	0	0
$606$	−2.83889e9	−0.518196
$607$	−7.99736e9	−1.45140	−0.725698	−	0.688013i	$-0.758483\pi$
$607$	−7.99736e9	−1.45140	−0.725698	+	0.688013i	$0.758483\pi$
$608$	−1.99157e9	−0.359363
$609$	−8.92133e9	−1.60055
$610$	0	0
$611$	−5.42395e9	−0.961992
$612$	−3.48806e9	−0.615112
$613$	5.93751e9	1.04110	0.520550	−	0.853831i	$-0.325727\pi$
$613$	5.93751e9	1.04110	0.520550	+	0.853831i	$0.325727\pi$
$614$	3.74190e9	0.652384
$615$	0	0
$616$	2.96270e9	0.510687
$617$	5.97459e9	1.02402	0.512012	−	0.858978i	$-0.328900\pi$
$617$	5.97459e9	1.02402	0.512012	+	0.858978i	$0.328900\pi$
$618$	−3.71926e9	−0.633865
$619$	−1.71663e9	−0.290910	−0.145455	−	0.989365i	$-0.546465\pi$
$619$	−1.71663e9	−0.290910	−0.145455	+	0.989365i	$0.546465\pi$
$620$	0	0
$621$	−1.16044e9	−0.194448
$622$	−6.04363e9	−1.00700
$623$	1.44224e8	0.0238962
$624$	5.15314e9	0.849035
$625$	0	0
$626$	−7.76757e9	−1.26554
$627$	−4.41164e9	−0.714766
$628$	1.51890e9	0.244720
$629$	−1.62423e10	−2.60237
$630$	0	0
$631$	7.12744e9	1.12936	0.564678	−	0.825311i	$-0.309000\pi$
$631$	7.12744e9	1.12936	0.564678	+	0.825311i	$0.309000\pi$
$632$	2.69535e9	0.424723
$633$	4.25787e9	0.667236
$634$	7.01132e9	1.09267
$635$	0	0
$636$	7.44045e8	0.114683
$637$	−1.59071e9	−0.243839
$638$	−3.79626e9	−0.578740
$639$	−1.74317e10	−2.64293
$640$	0	0
$641$	4.94439e9	0.741497	0.370748	−	0.928733i	$-0.379101\pi$
$641$	4.94439e9	0.741497	0.370748	+	0.928733i	$0.379101\pi$
$642$	−7.80185e9	−1.16366
$643$	1.37344e9	0.203738	0.101869	−	0.994798i	$-0.467518\pi$
$643$	1.37344e9	0.203738	0.101869	+	0.994798i	$0.467518\pi$
$644$	−1.87474e8	−0.0276592
$645$	0	0
$646$	−5.71668e9	−0.834316
$647$	6.04509e9	0.877481	0.438741	−	0.898614i	$-0.355425\pi$
$647$	6.04509e9	0.877481	0.438741	+	0.898614i	$0.355425\pi$
$648$	4.41274e9	0.637083
$649$	5.06572e9	0.727420
$650$	0	0
$651$	4.43376e9	0.629852
$652$	−1.41489e9	−0.199920
$653$	5.13788e9	0.722084	0.361042	−	0.932549i	$-0.382421\pi$
$653$	5.13788e9	0.722084	0.361042	+	0.932549i	$0.382421\pi$
$654$	7.35922e7	0.0102875
$655$	0	0
$656$	−9.56793e9	−1.32329
$657$	−4.51565e9	−0.621214
$658$	6.77261e9	0.926756
$659$	2.14544e9	0.292024	0.146012	−	0.989283i	$-0.453356\pi$
$659$	2.14544e9	0.292024	0.146012	+	0.989283i	$0.453356\pi$
$660$	0	0
$661$	4.41672e9	0.594832	0.297416	−	0.954748i	$-0.403875\pi$
$661$	4.41672e9	0.594832	0.297416	+	0.954748i	$0.403875\pi$
$662$	3.22543e9	0.432101
$663$	−1.22467e10	−1.63201
$664$	1.35767e10	1.79972
$665$	0	0
$666$	2.41948e10	3.17367
$667$	1.19426e9	0.155833
$668$	3.00690e9	0.390302
$669$	1.67204e10	2.15901
$670$	0	0
$671$	−7.90155e8	−0.100968
$672$	5.32736e9	0.677204
$673$	1.18441e10	1.49779	0.748893	−	0.662691i	$-0.230586\pi$
$673$	1.18441e10	1.49779	0.748893	+	0.662691i	$0.230586\pi$
$674$	8.50508e9	1.06996
$675$	0	0
$676$	9.33842e8	0.116268
$677$	−4.44078e9	−0.550046	−0.275023	−	0.961438i	$-0.588685\pi$
$677$	−4.44078e9	−0.550046	−0.275023	+	0.961438i	$0.588685\pi$
$678$	−1.30416e10	−1.60704
$679$	1.16348e10	1.42631
$680$	0	0
$681$	7.40017e9	0.897898
$682$	1.88668e9	0.227748
$683$	1.48012e10	1.77756	0.888779	−	0.458335i	$-0.151554\pi$
$683$	1.48012e10	1.77756	0.888779	+	0.458335i	$0.151554\pi$
$684$	−2.86574e9	−0.342406
$685$	0	0
$686$	7.96230e9	0.941681
$687$	1.80758e10	2.12691
$688$	9.74679e9	1.14104
$689$	1.69763e9	0.197732
$690$	0	0
$691$	−4.30757e9	−0.496660	−0.248330	−	0.968675i	$-0.579882\pi$
$691$	−4.30757e9	−0.496660	−0.248330	+	0.968675i	$0.579882\pi$
$692$	−6.58650e8	−0.0755585
$693$	7.66875e9	0.875303
$694$	−1.45301e10	−1.65010
$695$	0	0
$696$	−1.88658e10	−2.12101
$697$	2.27387e10	2.54361
$698$	−1.25703e10	−1.39911
$699$	1.03025e8	0.0114097
$700$	0	0
$701$	−8.73667e9	−0.957928	−0.478964	−	0.877834i	$-0.658988\pi$
$701$	−8.73667e9	−0.957928	−0.478964	+	0.877834i	$0.658988\pi$
$702$	8.41304e9	0.917853
$703$	−1.33444e10	−1.44863
$704$	5.92638e9	0.640155
$705$	0	0
$706$	8.90814e9	0.952731
$707$	−2.72177e9	−0.289656
$708$	5.06372e9	0.536233
$709$	1.81555e10	1.91314	0.956570	−	0.291503i	$-0.0941552\pi$
$709$	1.81555e10	1.91314	0.956570	+	0.291503i	$0.0941552\pi$
$710$	0	0
$711$	6.97674e9	0.727962
$712$	3.04987e8	0.0316666
$713$	−5.93530e8	−0.0613238
$714$	1.52918e10	1.57223
$715$	0	0
$716$	−8.86214e7	−0.00902284
$717$	1.77136e10	1.79469
$718$	6.87074e9	0.692737
$719$	−1.35993e10	−1.36448	−0.682238	−	0.731130i	$-0.738993\pi$
$719$	−1.35993e10	−1.36448	−0.682238	+	0.731130i	$0.738993\pi$
$720$	0	0
$721$	−3.56582e9	−0.354312
$722$	4.05090e9	0.400564
$723$	2.02746e10	1.99512
$724$	−3.24093e9	−0.317384
$725$	0	0
$726$	−1.00500e10	−0.974734
$727$	5.25958e9	0.507669	0.253834	−	0.967248i	$-0.418308\pi$
$727$	5.25958e9	0.507669	0.253834	+	0.967248i	$0.418308\pi$
$728$	6.75709e9	0.649082
$729$	−1.41455e10	−1.35230
$730$	0	0
$731$	−2.31638e10	−2.19330
$732$	−7.89842e8	−0.0744306
$733$	1.96732e9	0.184506	0.0922530	−	0.995736i	$-0.470593\pi$
$733$	1.96732e9	0.184506	0.0922530	+	0.995736i	$0.470593\pi$
$734$	4.25160e9	0.396841
$735$	0	0
$736$	−7.13151e8	−0.0659340
$737$	−7.87776e9	−0.724881
$738$	−3.38720e10	−3.10201
$739$	3.16476e9	0.288460	0.144230	−	0.989544i	$-0.453930\pi$
$739$	3.16476e9	0.288460	0.144230	+	0.989544i	$0.453930\pi$
$740$	0	0
$741$	−1.00617e10	−0.908466
$742$	−2.11974e9	−0.190489
$743$	−9.63138e8	−0.0861445	−0.0430722	−	0.999072i	$-0.513715\pi$
$743$	−9.63138e8	−0.0861445	−0.0430722	+	0.999072i	$0.513715\pi$
$744$	9.37601e9	0.834666
$745$	0	0
$746$	9.06201e8	0.0799169
$747$	3.51423e10	3.08466
$748$	−2.18980e9	−0.191315
$749$	−7.47998e9	−0.650451
$750$	0	0
$751$	−2.02999e10	−1.74885	−0.874426	−	0.485159i	$-0.838762\pi$
$751$	−2.02999e10	−1.74885	−0.874426	+	0.485159i	$0.838762\pi$
$752$	1.04717e10	0.897954
$753$	−3.28250e9	−0.280171
$754$	−8.65821e9	−0.735578
$755$	0	0
$756$	3.53518e9	0.297568
$757$	4.83899e9	0.405433	0.202717	−	0.979237i	$-0.435023\pi$
$757$	4.83899e9	0.405433	0.202717	+	0.979237i	$0.435023\pi$
$758$	5.60336e9	0.467312
$759$	−1.57974e9	−0.131141
$760$	0	0
$761$	−1.27158e10	−1.04592	−0.522959	−	0.852358i	$-0.675172\pi$
$761$	−1.27158e10	−1.04592	−0.522959	+	0.852358i	$0.675172\pi$
$762$	−1.79532e10	−1.46994
$763$	7.05562e7	0.00575042
$764$	−5.83598e8	−0.0473464
$765$	0	0
$766$	1.57272e10	1.26430
$767$	1.15535e10	0.924549
$768$	1.49237e10	1.18881
$769$	1.67928e10	1.33162	0.665811	−	0.746121i	$-0.268086\pi$
$769$	1.67928e10	1.33162	0.665811	+	0.746121i	$0.268086\pi$
$770$	0	0
$771$	−2.59569e10	−2.03968
$772$	3.93924e9	0.308142
$773$	4.70349e9	0.366262	0.183131	−	0.983089i	$-0.441377\pi$
$773$	4.70349e9	0.366262	0.183131	+	0.983089i	$0.441377\pi$
$774$	3.45051e10	2.67479
$775$	0	0
$776$	2.46039e10	1.89012
$777$	3.56957e10	2.72987
$778$	−8.70890e9	−0.663032
$779$	1.86818e10	1.41592
$780$	0	0
$781$	−1.09436e10	−0.822019
$782$	−2.04706e9	−0.153076
$783$	−2.25202e10	−1.67651
$784$	3.07108e9	0.227607
$785$	0	0
$786$	−1.48510e10	−1.09088
$787$	−9.66196e9	−0.706568	−0.353284	−	0.935516i	$-0.614935\pi$
$787$	−9.66196e9	−0.706568	−0.353284	+	0.935516i	$0.614935\pi$
$788$	5.68266e9	0.413723
$789$	−4.37264e9	−0.316938
$790$	0	0
$791$	−1.25036e10	−0.898290
$792$	1.62170e10	1.15993
$793$	−1.80212e9	−0.128330
$794$	1.07394e10	0.761394
$795$	0	0
$796$	−1.30126e9	−0.0914466
$797$	−8.97258e9	−0.627788	−0.313894	−	0.949458i	$-0.601634\pi$
$797$	−8.97258e9	−0.627788	−0.313894	+	0.949458i	$0.601634\pi$
$798$	1.25636e10	0.875191
$799$	−2.48865e10	−1.72604
$800$	0	0
$801$	7.89440e8	0.0542757
$802$	−1.78698e9	−0.122324
$803$	−2.83492e9	−0.193213
$804$	−7.87465e9	−0.534361
$805$	0	0
$806$	4.30300e9	0.289467
$807$	1.65419e9	0.110797
$808$	−5.75568e9	−0.383846
$809$	1.83006e10	1.21519	0.607595	−	0.794247i	$-0.292134\pi$
$809$	1.83006e10	1.21519	0.607595	+	0.794247i	$0.292134\pi$
$810$	0	0
$811$	2.02719e10	1.33451	0.667255	−	0.744829i	$-0.267469\pi$
$811$	2.02719e10	1.33451	0.667255	+	0.744829i	$0.267469\pi$
$812$	−3.63821e9	−0.238474
$813$	−3.12583e10	−2.04009
$814$	1.51895e10	0.987091
$815$	0	0
$816$	2.36440e10	1.52337
$817$	−1.90310e10	−1.22091
$818$	−5.75835e9	−0.367842
$819$	1.74903e10	1.11251
$820$	0	0
$821$	−2.61445e10	−1.64884	−0.824420	−	0.565979i	$-0.808499\pi$
$821$	−2.61445e10	−1.64884	−0.824420	+	0.565979i	$0.808499\pi$
$822$	−1.93439e10	−1.21477
$823$	−1.65818e10	−1.03689	−0.518445	−	0.855111i	$-0.673489\pi$
$823$	−1.65818e10	−1.03689	−0.518445	+	0.855111i	$0.673489\pi$
$824$	−7.54059e9	−0.469526
$825$	0	0
$826$	−1.44263e10	−0.890684
$827$	−1.48018e10	−0.910011	−0.455005	−	0.890489i	$-0.650363\pi$
$827$	−1.48018e10	−0.910011	−0.455005	+	0.890489i	$0.650363\pi$
$828$	−1.02618e9	−0.0628227
$829$	2.77171e10	1.68969	0.844846	−	0.535010i	$-0.179692\pi$
$829$	2.77171e10	1.68969	0.844846	+	0.535010i	$0.179692\pi$
$830$	0	0
$831$	3.25743e10	1.96912
$832$	1.35164e10	0.813636
$833$	−7.29859e9	−0.437504
$834$	−4.17926e10	−2.49470
$835$	0	0
$836$	−1.79911e9	−0.106497
$837$	1.11922e10	0.659744
$838$	3.09064e9	0.181424
$839$	1.21267e10	0.708886	0.354443	−	0.935078i	$-0.384670\pi$
$839$	1.21267e10	0.708886	0.354443	+	0.935078i	$0.384670\pi$
$840$	0	0
$841$	5.92657e9	0.343572
$842$	1.63434e10	0.943519
$843$	−5.38324e9	−0.309490
$844$	1.73640e9	0.0994150
$845$	0	0
$846$	3.70714e10	2.10495
$847$	−9.63535e9	−0.544848
$848$	−3.27751e9	−0.184569
$849$	−1.94659e10	−1.09169
$850$	0	0
$851$	−4.77844e9	−0.265786
$852$	−1.09393e10	−0.605969
$853$	−2.63716e10	−1.45484	−0.727420	−	0.686193i	$-0.759281\pi$
$853$	−2.63716e10	−1.45484	−0.727420	+	0.686193i	$0.759281\pi$
$854$	2.25022e9	0.123629
$855$	0	0
$856$	−1.58178e10	−0.861962
$857$	−9.02680e9	−0.489892	−0.244946	−	0.969537i	$-0.578770\pi$
$857$	−9.02680e9	−0.489892	−0.244946	+	0.969537i	$0.578770\pi$
$858$	1.14529e10	0.619026
$859$	1.12502e10	0.605595	0.302798	−	0.953055i	$-0.402079\pi$
$859$	1.12502e10	0.605595	0.302798	+	0.953055i	$0.402079\pi$
$860$	0	0
$861$	−4.99729e10	−2.66823
$862$	−1.91915e9	−0.102055
$863$	−1.12255e10	−0.594522	−0.297261	−	0.954796i	$-0.596073\pi$
$863$	−1.12255e10	−0.594522	−0.297261	+	0.954796i	$0.596073\pi$
$864$	1.34479e10	0.709342
$865$	0	0
$866$	−2.00866e10	−1.05098
$867$	−2.37620e10	−1.23827
$868$	1.80813e9	0.0938450
$869$	4.37999e9	0.226414
$870$	0	0
$871$	−1.79670e10	−0.921322
$872$	1.49204e8	0.00762032
$873$	6.36857e10	3.23961
$874$	−1.68183e9	−0.0852104
$875$	0	0
$876$	−2.83380e9	−0.142431
$877$	−1.41855e10	−0.710142	−0.355071	−	0.934839i	$-0.615543\pi$
$877$	−1.41855e10	−0.710142	−0.355071	+	0.934839i	$0.615543\pi$
$878$	−2.45118e10	−1.22220
$879$	−5.35613e10	−2.66005
$880$	0	0
$881$	1.56693e8	0.00772031	0.00386015	−	0.999993i	$-0.498771\pi$
$881$	1.56693e8	0.00772031	0.00386015	+	0.999993i	$0.498771\pi$
$882$	1.08721e10	0.533548
$883$	4.49785e8	0.0219858	0.0109929	−	0.999940i	$-0.496501\pi$
$883$	4.49785e8	0.0219858	0.0109929	+	0.999940i	$0.496501\pi$
$884$	−4.99433e9	−0.243161
$885$	0	0
$886$	9.35464e9	0.451866
$887$	2.22799e10	1.07197	0.535983	−	0.844229i	$-0.319941\pi$
$887$	2.22799e10	1.07197	0.535983	+	0.844229i	$0.319941\pi$
$888$	7.54851e10	3.61756
$889$	−1.72126e10	−0.821656
$890$	0	0
$891$	7.17079e9	0.339621
$892$	6.81873e9	0.321682
$893$	−2.04464e10	−0.960810
$894$	1.30987e9	0.0613124
$895$	0	0
$896$	−8.24891e9	−0.383105
$897$	−3.60295e9	−0.166680
$898$	−6.80878e8	−0.0313763
$899$	−1.15183e10	−0.528726
$900$	0	0
$901$	7.78919e9	0.354777
$902$	−2.12648e10	−0.964802
$903$	5.09071e10	2.30076
$904$	−2.64411e10	−1.19039
$905$	0	0
$906$	−4.10369e9	−0.183327
$907$	−5.97324e9	−0.265818	−0.132909	−	0.991128i	$-0.542432\pi$
$907$	−5.97324e9	−0.265818	−0.132909	+	0.991128i	$0.542432\pi$
$908$	3.01787e9	0.133783
$909$	−1.48982e10	−0.657901
$910$	0	0
$911$	−3.95164e10	−1.73166	−0.865831	−	0.500337i	$-0.833209\pi$
$911$	−3.95164e10	−1.73166	−0.865831	+	0.500337i	$0.833209\pi$
$912$	1.94256e10	0.847992
$913$	2.20623e10	0.959408
$914$	2.22078e10	0.962044
$915$	0	0
$916$	7.37149e9	0.316899
$917$	−1.42383e10	−0.609771
$918$	3.86013e10	1.64684
$919$	−4.12588e10	−1.75353	−0.876763	−	0.480923i	$-0.840302\pi$
$919$	−4.12588e10	−1.75353	−0.876763	+	0.480923i	$0.840302\pi$
$920$	0	0
$921$	3.02183e10	1.27456
$922$	−3.04156e10	−1.27802
$923$	−2.49593e10	−1.04478
$924$	4.81253e9	0.200688
$925$	0	0
$926$	−1.64146e9	−0.0679347
$927$	−1.95183e10	−0.804754
$928$	−1.38398e10	−0.568475
$929$	−1.40970e10	−0.576860	−0.288430	−	0.957501i	$-0.593133\pi$
$929$	−1.40970e10	−0.576860	−0.288430	+	0.957501i	$0.593133\pi$
$930$	0	0
$931$	−5.99642e9	−0.243539
$932$	4.20147e7	0.00169999
$933$	−4.88062e10	−1.96738
$934$	3.16184e10	1.26977
$935$	0	0
$936$	3.69864e10	1.47427
$937$	−1.89332e10	−0.751857	−0.375928	−	0.926649i	$-0.622676\pi$
$937$	−1.89332e10	−0.751857	−0.375928	+	0.926649i	$0.622676\pi$
$938$	2.24344e10	0.887576
$939$	−6.27281e10	−2.47248
$940$	0	0
$941$	−2.47815e10	−0.969535	−0.484768	−	0.874643i	$-0.661096\pi$
$941$	−2.47815e10	−0.969535	−0.484768	+	0.874643i	$0.661096\pi$
$942$	−3.64491e10	−1.42072
$943$	6.68967e9	0.259785
$944$	−2.23056e10	−0.863004
$945$	0	0
$946$	2.16623e10	0.831928
$947$	−2.69114e10	−1.02970	−0.514850	−	0.857280i	$-0.672153\pi$
$947$	−2.69114e10	−1.02970	−0.514850	+	0.857280i	$0.672153\pi$
$948$	4.37826e9	0.166906
$949$	−6.46566e9	−0.245573
$950$	0	0
$951$	5.66209e10	2.13474
$952$	3.10033e10	1.16461
$953$	4.52288e9	0.169274	0.0846370	−	0.996412i	$-0.473027\pi$
$953$	4.52288e9	0.169274	0.0846370	+	0.996412i	$0.473027\pi$
$954$	−1.16029e10	−0.432660
$955$	0	0
$956$	7.22380e9	0.267401
$957$	−3.06572e10	−1.13068
$958$	3.49253e10	1.28340
$959$	−1.85459e10	−0.679020
$960$	0	0
$961$	−2.17882e10	−0.791934
$962$	3.46429e10	1.25459
$963$	−4.09434e10	−1.47738
$964$	8.26819e9	0.297263
$965$	0	0
$966$	4.49882e9	0.160575
$967$	2.20599e10	0.784532	0.392266	−	0.919852i	$-0.371691\pi$
$967$	2.20599e10	0.784532	0.392266	+	0.919852i	$0.371691\pi$
$968$	−2.03757e10	−0.722020
$969$	−4.61659e10	−1.63000
$970$	0	0
$971$	2.52789e10	0.886116	0.443058	−	0.896493i	$-0.353894\pi$
$971$	2.52789e10	0.886116	0.443058	+	0.896493i	$0.353894\pi$
$972$	−3.25878e9	−0.113821
$973$	−4.00684e10	−1.39446
$974$	−4.31186e10	−1.49523
$975$	0	0
$976$	3.47925e9	0.119787
$977$	2.42839e10	0.833080	0.416540	−	0.909117i	$-0.363243\pi$
$977$	2.42839e10	0.833080	0.416540	+	0.909117i	$0.363243\pi$
$978$	3.39532e10	1.16063
$979$	4.95610e8	0.0168811
$980$	0	0
$981$	3.86205e8	0.0130610
$982$	1.45703e9	0.0490997
$983$	1.48304e10	0.497984	0.248992	−	0.968506i	$-0.419901\pi$
$983$	1.48304e10	0.497984	0.248992	+	0.968506i	$0.419901\pi$
$984$	−1.05677e11	−3.53588
$985$	0	0
$986$	−3.97262e10	−1.31980
$987$	5.46931e10	1.81060
$988$	−4.10327e9	−0.135357
$989$	−6.81472e9	−0.224007
$990$	0	0
$991$	−1.36268e10	−0.444769	−0.222384	−	0.974959i	$-0.571384\pi$
$991$	−1.36268e10	−0.444769	−0.222384	+	0.974959i	$0.571384\pi$
$992$	6.87816e9	0.223708
$993$	2.60474e10	0.844195
$994$	3.11654e10	1.00652
$995$	0	0
$996$	2.20536e10	0.707248
$997$	9.39609e9	0.300271	0.150136	−	0.988665i	$-0.452029\pi$
$997$	9.39609e9	0.300271	0.150136	+	0.988665i	$0.452029\pi$
$998$	−1.01892e10	−0.324476
$999$	9.01068e10	2.85942

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.8.a.e.1.15	✓	48
5.4	even	2		625.8.a.f.1.34	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.8.a.e.1.15	✓	48	1.1	even	1	trivial
625.8.a.f.1.34	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)

\(f(q)\)	\(=\)	\(q-9.78625 q^{2} -79.0302 q^{3} -32.2293 q^{4} +773.410 q^{6} +741.503 q^{7} +1568.04 q^{8} +4058.78 q^{9} +O(q^{10})\)	\(q-9.78625 q^{2} -79.0302 q^{3} -32.2293 q^{4} +773.410 q^{6} +741.503 q^{7} +1568.04 q^{8} +4058.78 q^{9} +2548.10 q^{11} +2547.09 q^{12} +5811.51 q^{13} -7256.53 q^{14} -11219.9 q^{16} +26664.7 q^{17} -39720.2 q^{18} +21907.4 q^{19} -58601.1 q^{21} -24936.3 q^{22} +7844.69 q^{23} -123923. q^{24} -56872.8 q^{26} -147927. q^{27} -23898.1 q^{28} +152238. q^{29} -75660.0 q^{31} -90908.8 q^{32} -201377. q^{33} -260948. q^{34} -130812. q^{36} -609130. q^{37} -214391. q^{38} -459285. q^{39} +852763. q^{41} +573485. q^{42} -868704. q^{43} -82123.6 q^{44} -76770.1 q^{46} -933312. q^{47} +886713. q^{48} -273717. q^{49} -2.10732e6 q^{51} -187301. q^{52} +292116. q^{53} +1.44765e6 q^{54} +1.16271e6 q^{56} -1.73135e6 q^{57} -1.48984e6 q^{58} +1.98804e6 q^{59} -310096. q^{61} +740428. q^{62} +3.00960e6 q^{63} +2.32580e6 q^{64} +1.97072e6 q^{66} -3.09162e6 q^{67} -859387. q^{68} -619968. q^{69} -4.29481e6 q^{71} +6.36435e6 q^{72} -1.11256e6 q^{73} +5.96109e6 q^{74} -706061. q^{76} +1.88942e6 q^{77} +4.49467e6 q^{78} +1.71892e6 q^{79} +2.81417e6 q^{81} -8.34535e6 q^{82} +8.65834e6 q^{83} +1.88868e6 q^{84} +8.50136e6 q^{86} -1.20314e7 q^{87} +3.99553e6 q^{88} +194502. q^{89} +4.30925e6 q^{91} -252829. q^{92} +5.97943e6 q^{93} +9.13363e6 q^{94} +7.18454e6 q^{96} +1.56909e7 q^{97} +2.67866e6 q^{98} +1.03422e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 25 q^{2} - 95 q^{3} + 3419 q^{4} + 431 q^{6} - 4030 q^{7} - 3375 q^{8} + 36231 q^{9}+O(q^{10}) \) 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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