LMFDB - Embedded newform 625.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("625.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$195.240640928$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-21.3970 q^{2} -45.8462 q^{3} +329.831 q^{4} +980.970 q^{6} +104.481 q^{7} -4318.57 q^{8} -85.1275 q^{9} +O(q^{10})$	\(q-21.3970 q^{2} -45.8462 q^{3} +329.831 q^{4} +980.970 q^{6} +104.481 q^{7} -4318.57 q^{8} -85.1275 q^{9} -6239.65 q^{11} -15121.5 q^{12} -7882.79 q^{13} -2235.58 q^{14} +50186.1 q^{16} -34462.9 q^{17} +1821.47 q^{18} +19754.9 q^{19} -4790.05 q^{21} +133510. q^{22} -89723.7 q^{23} +197990. q^{24} +168668. q^{26} +104168. q^{27} +34461.0 q^{28} -108162. q^{29} +266899. q^{31} -521054. q^{32} +286064. q^{33} +737401. q^{34} -28077.7 q^{36} +13673.2 q^{37} -422695. q^{38} +361396. q^{39} +295066. q^{41} +102493. q^{42} +804355. q^{43} -2.05803e6 q^{44} +1.91982e6 q^{46} +98532.0 q^{47} -2.30084e6 q^{48} -812627. q^{49} +1.57999e6 q^{51} -2.59999e6 q^{52} -1.09050e6 q^{53} -2.22889e6 q^{54} -451208. q^{56} -905687. q^{57} +2.31433e6 q^{58} -14776.3 q^{59} -579512. q^{61} -5.71084e6 q^{62} -8894.19 q^{63} +4.72516e6 q^{64} -6.12091e6 q^{66} +21392.3 q^{67} -1.13669e7 q^{68} +4.11349e6 q^{69} +4.15929e6 q^{71} +367629. q^{72} -5.15400e6 q^{73} -292564. q^{74} +6.51578e6 q^{76} -651924. q^{77} -7.73279e6 q^{78} +8.11379e6 q^{79} -4.58955e6 q^{81} -6.31353e6 q^{82} +2.17337e6 q^{83} -1.57991e6 q^{84} -1.72108e7 q^{86} +4.95880e6 q^{87} +2.69464e7 q^{88} +10155.1 q^{89} -823601. q^{91} -2.95937e7 q^{92} -1.22363e7 q^{93} -2.10829e6 q^{94} +2.38883e7 q^{96} -1.29273e6 q^{97} +1.73878e7 q^{98} +531166. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 23 q^{2} - 121 q^{3} + 1061 q^{4} - 431 q^{6} - 843 q^{7} - 4980 q^{8} + 14799 q^{9}+O(q^{10}) $ 22 * q - 23 * q^2 - 121 * q^3 + 1061 * q^4 - 431 * q^6 - 843 * q^7 - 4980 * q^8 + 14799 * q^9	\( 22 q - 23 q^{2} - 121 q^{3} + 1061 q^{4} - 431 q^{6} - 843 q^{7} - 4980 q^{8} + 14799 q^{9} - 781 q^{11} - 12608 q^{12} - 14686 q^{13} + 20762 q^{14} + 16117 q^{16} - 45648 q^{17} - 47171 q^{18} + 6185 q^{19} + 14149 q^{21} + 71124 q^{22} - 126921 q^{23} + 271570 q^{24} + 304129 q^{26} - 546520 q^{27} - 2019 q^{28} + 59330 q^{29} + 394804 q^{31} + 74397 q^{32} - 49067 q^{33} - 286938 q^{34} - 287278 q^{36} + 792122 q^{37} - 1338860 q^{38} + 635223 q^{39} - 160466 q^{41} - 3420191 q^{42} - 1527256 q^{43} - 1154853 q^{44} + 2653604 q^{46} - 1300863 q^{47} - 1885241 q^{48} + 1652981 q^{49} + 3408539 q^{51} - 1423303 q^{52} - 755656 q^{53} + 3117755 q^{54} - 2132625 q^{56} - 3026890 q^{57} + 5941470 q^{58} - 1548370 q^{59} - 6029951 q^{61} - 79936 q^{62} + 6962459 q^{63} - 5858224 q^{64} - 5380407 q^{66} - 7608838 q^{67} - 10737124 q^{68} + 13519553 q^{69} + 9483549 q^{71} - 6806340 q^{72} - 13548801 q^{73} - 15016023 q^{74} + 19635315 q^{76} + 2145019 q^{77} - 17222402 q^{78} + 10769160 q^{79} + 2757382 q^{81} - 12087571 q^{82} + 9632744 q^{83} + 19168542 q^{84} - 8511651 q^{86} - 298330 q^{87} + 21641425 q^{88} - 10850545 q^{89} - 6648131 q^{91} - 52978503 q^{92} - 51294822 q^{93} + 5863777 q^{94} + 2611654 q^{96} - 11579993 q^{97} - 7468074 q^{98} - 13552997 q^{99}+O(q^{100}) \) 22 * q - 23 * q^2 - 121 * q^3 + 1061 * q^4 - 431 * q^6 - 843 * q^7 - 4980 * q^8 + 14799 * q^9 - 781 * q^11 - 12608 * q^12 - 14686 * q^13 + 20762 * q^14 + 16117 * q^16 - 45648 * q^17 - 47171 * q^18 + 6185 * q^19 + 14149 * q^21 + 71124 * q^22 - 126921 * q^23 + 271570 * q^24 + 304129 * q^26 - 546520 * q^27 - 2019 * q^28 + 59330 * q^29 + 394804 * q^31 + 74397 * q^32 - 49067 * q^33 - 286938 * q^34 - 287278 * q^36 + 792122 * q^37 - 1338860 * q^38 + 635223 * q^39 - 160466 * q^41 - 3420191 * q^42 - 1527256 * q^43 - 1154853 * q^44 + 2653604 * q^46 - 1300863 * q^47 - 1885241 * q^48 + 1652981 * q^49 + 3408539 * q^51 - 1423303 * q^52 - 755656 * q^53 + 3117755 * q^54 - 2132625 * q^56 - 3026890 * q^57 + 5941470 * q^58 - 1548370 * q^59 - 6029951 * q^61 - 79936 * q^62 + 6962459 * q^63 - 5858224 * q^64 - 5380407 * q^66 - 7608838 * q^67 - 10737124 * q^68 + 13519553 * q^69 + 9483549 * q^71 - 6806340 * q^72 - 13548801 * q^73 - 15016023 * q^74 + 19635315 * q^76 + 2145019 * q^77 - 17222402 * q^78 + 10769160 * q^79 + 2757382 * q^81 - 12087571 * q^82 + 9632744 * q^83 + 19168542 * q^84 - 8511651 * q^86 - 298330 * q^87 + 21641425 * q^88 - 10850545 * q^89 - 6648131 * q^91 - 52978503 * q^92 - 51294822 * q^93 + 5863777 * q^94 + 2611654 * q^96 - 11579993 * q^97 - 7468074 * q^98 - 13552997 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−21.3970	−1.89124	−0.945622	−	0.325267i	$-0.894546\pi$
$2$	−21.3970	−1.89124	−0.945622	+	0.325267i	$0.894546\pi$
$3$	−45.8462	−0.980345	−0.490172	−	0.871626i	$-0.663066\pi$
$3$	−45.8462	−0.980345	−0.490172	+	0.871626i	$0.663066\pi$
$4$	329.831	2.57680
$5$	0	0
$5$	0	0
$6$	980.970	1.85407
$7$	104.481	0.115131	0.0575657	−	0.998342i	$-0.481666\pi$
$7$	104.481	0.115131	0.0575657	+	0.998342i	$0.481666\pi$
$8$	−4318.57	−2.98212
$9$	−85.1275	−0.0389243
$10$	0	0
$11$	−6239.65	−1.41347	−0.706734	−	0.707479i	$-0.749832\pi$
$11$	−6239.65	−1.41347	−0.706734	+	0.707479i	$0.749832\pi$
$12$	−15121.5	−2.52616
$13$	−7882.79	−0.995127	−0.497563	−	0.867428i	$-0.665772\pi$
$13$	−7882.79	−0.995127	−0.497563	+	0.867428i	$0.665772\pi$
$14$	−2235.58	−0.217742
$15$	0	0
$16$	50186.1	3.06312
$17$	−34462.9	−1.70130	−0.850649	−	0.525735i	$-0.823790\pi$
$17$	−34462.9	−1.70130	−0.850649	+	0.525735i	$0.823790\pi$
$18$	1821.47	0.0736154
$19$	19754.9	0.660750	0.330375	−	0.943850i	$-0.392825\pi$
$19$	19754.9	0.660750	0.330375	+	0.943850i	$0.392825\pi$
$20$	0	0
$21$	−4790.05	−0.112868
$22$	133510.	2.67321
$23$	−89723.7	−1.53766	−0.768829	−	0.639454i	$-0.779160\pi$
$23$	−89723.7	−1.53766	−0.768829	+	0.639454i	$0.779160\pi$
$24$	197990.	2.92351
$25$	0	0
$26$	168668.	1.88203
$27$	104168.	1.01850
$28$	34461.0	0.296671
$29$	−108162.	−0.823531	−0.411766	−	0.911290i	$-0.635088\pi$
$29$	−108162.	−0.823531	−0.411766	+	0.911290i	$0.635088\pi$
$30$	0	0
$31$	266899.	1.60909	0.804547	−	0.593889i	$-0.202408\pi$
$31$	266899.	1.60909	0.804547	+	0.593889i	$0.202408\pi$
$32$	−521054.	−2.81098
$33$	286064.	1.38569
$34$	737401.	3.21757
$35$	0	0
$36$	−28077.7	−0.100300
$37$	13673.2	0.0443775	0.0221888	−	0.999754i	$-0.492937\pi$
$37$	13673.2	0.0443775	0.0221888	+	0.999754i	$0.492937\pi$
$38$	−422695.	−1.24964
$39$	361396.	0.975567
$40$	0	0
$41$	295066.	0.668615	0.334308	−	0.942464i	$-0.391498\pi$
$41$	295066.	0.668615	0.334308	+	0.942464i	$0.391498\pi$
$42$	102493.	0.213462
$43$	804355.	1.54280	0.771398	−	0.636353i	$-0.219558\pi$
$43$	804355.	1.54280	0.771398	+	0.636353i	$0.219558\pi$
$44$	−2.05803e6	−3.64223
$45$	0	0
$46$	1.91982e6	2.90809
$47$	98532.0	0.138431	0.0692157	−	0.997602i	$-0.477950\pi$
$47$	98532.0	0.138431	0.0692157	+	0.997602i	$0.477950\pi$
$48$	−2.30084e6	−3.00291
$49$	−812627.	−0.986745
$50$	0	0
$51$	1.57999e6	1.66786
$52$	−2.59999e6	−2.56425
$53$	−1.09050e6	−1.00615	−0.503073	−	0.864244i	$-0.667797\pi$
$53$	−1.09050e6	−1.00615	−0.503073	+	0.864244i	$0.667797\pi$
$54$	−2.22889e6	−1.92624
$55$	0	0
$56$	−451208.	−0.343336
$57$	−905687.	−0.647763
$58$	2.31433e6	1.55750
$59$	−14776.3	−0.00936662	−0.00468331	−	0.999989i	$-0.501491\pi$
$59$	−14776.3	−0.00936662	−0.00468331	+	0.999989i	$0.501491\pi$
$60$	0	0
$61$	−579512.	−0.326895	−0.163447	−	0.986552i	$-0.552261\pi$
$61$	−579512.	−0.326895	−0.163447	+	0.986552i	$0.552261\pi$
$62$	−5.71084e6	−3.04319
$63$	−8894.19	−0.00448141
$64$	4.72516e6	2.25313
$65$	0	0
$66$	−6.12091e6	−2.62067
$67$	21392.3	0.00868950	0.00434475	−	0.999991i	$-0.498617\pi$
$67$	21392.3	0.00868950	0.00434475	+	0.999991i	$0.498617\pi$
$68$	−1.13669e7	−4.38391
$69$	4.11349e6	1.50744
$70$	0	0
$71$	4.15929e6	1.37916	0.689581	−	0.724208i	$-0.257795\pi$
$71$	4.15929e6	1.37916	0.689581	+	0.724208i	$0.257795\pi$
$72$	367629.	0.116077
$73$	−5.15400e6	−1.55065	−0.775326	−	0.631561i	$-0.782414\pi$
$73$	−5.15400e6	−1.55065	−0.775326	+	0.631561i	$0.782414\pi$
$74$	−292564.	−0.0839287
$75$	0	0
$76$	6.51578e6	1.70262
$77$	−651924.	−0.162734
$78$	−7.73279e6	−1.84504
$79$	8.11379e6	1.85152	0.925761	−	0.378109i	$-0.123426\pi$
$79$	8.11379e6	1.85152	0.925761	+	0.378109i	$0.123426\pi$
$80$	0	0
$81$	−4.58955e6	−0.959561
$82$	−6.31353e6	−1.26451
$83$	2.17337e6	0.417216	0.208608	−	0.977999i	$-0.433107\pi$
$83$	2.17337e6	0.417216	0.208608	+	0.977999i	$0.433107\pi$
$84$	−1.57991e6	−0.290840
$85$	0	0
$86$	−1.72108e7	−2.91780
$87$	4.95880e6	0.807345
$88$	2.69464e7	4.21513
$89$	10155.1	0.00152693	0.000763465	−	1.00000i	$-0.499757\pi$
$89$	10155.1	0.00152693	0.000763465		1.00000i	$0.499757\pi$
$90$	0	0
$91$	−823601.	−0.114570
$92$	−2.95937e7	−3.96225
$93$	−1.22363e7	−1.57747
$94$	−2.10829e6	−0.261808
$95$	0	0
$96$	2.38883e7	2.75573
$97$	−1.29273e6	−0.143816	−0.0719081	−	0.997411i	$-0.522909\pi$
$97$	−1.29273e6	−0.143816	−0.0719081	+	0.997411i	$0.522909\pi$
$98$	1.73878e7	1.86618
$99$	531166.	0.0550183
$100$	0	0
$101$	−3.42028e6	−0.330321	−0.165161	−	0.986267i	$-0.552814\pi$
$101$	−3.42028e6	−0.330321	−0.165161	+	0.986267i	$0.552814\pi$
$102$	−3.38070e7	−3.15433
$103$	1.38750e7	1.25113	0.625566	−	0.780171i	$-0.284868\pi$
$103$	1.38750e7	1.25113	0.625566	+	0.780171i	$0.284868\pi$
$104$	3.40424e7	2.96759
$105$	0	0
$106$	2.33335e7	1.90287
$107$	4.55629e6	0.359557	0.179779	−	0.983707i	$-0.442462\pi$
$107$	4.55629e6	0.359557	0.179779	+	0.983707i	$0.442462\pi$
$108$	3.43580e7	2.62449
$109$	−2.75061e6	−0.203440	−0.101720	−	0.994813i	$-0.532435\pi$
$109$	−2.75061e6	−0.203440	−0.101720	+	0.994813i	$0.532435\pi$
$110$	0	0
$111$	−626862.	−0.0435052
$112$	5.24349e6	0.352661
$113$	1.32896e7	0.866436	0.433218	−	0.901289i	$-0.357378\pi$
$113$	1.32896e7	0.866436	0.433218	+	0.901289i	$0.357378\pi$
$114$	1.93790e7	1.22508
$115$	0	0
$116$	−3.56751e7	−2.12208
$117$	671042.	0.0387346
$118$	316168.	0.0177146
$119$	−3.60071e6	−0.195873
$120$	0	0
$121$	1.94461e7	0.997892
$122$	1.23998e7	0.618238
$123$	−1.35277e7	−0.655474
$124$	8.80316e7	4.14632
$125$	0	0
$126$	190309.	0.00847544
$127$	1.94627e7	0.843122	0.421561	−	0.906800i	$-0.361482\pi$
$127$	1.94627e7	0.843122	0.421561	+	0.906800i	$0.361482\pi$
$128$	−3.44093e7	−1.45024
$129$	−3.68766e7	−1.51247
$130$	0	0
$131$	−4.15921e7	−1.61645	−0.808223	−	0.588876i	$-0.799570\pi$
$131$	−4.15921e7	−1.61645	−0.808223	+	0.588876i	$0.799570\pi$
$132$	9.43528e7	3.57064
$133$	2.06401e6	0.0760731
$134$	−457730.	−0.0164340
$135$	0	0
$136$	1.48830e8	5.07348
$137$	−2.16745e7	−0.720156	−0.360078	−	0.932922i	$-0.617250\pi$
$137$	−2.16745e7	−0.720156	−0.360078	+	0.932922i	$0.617250\pi$
$138$	−8.80163e7	−2.85093
$139$	1.18834e7	0.375308	0.187654	−	0.982235i	$-0.439912\pi$
$139$	1.18834e7	0.375308	0.187654	+	0.982235i	$0.439912\pi$
$140$	0	0
$141$	−4.51732e6	−0.135711
$142$	−8.89964e7	−2.60833
$143$	4.91859e7	1.40658
$144$	−4.27222e6	−0.119230
$145$	0	0
$146$	1.10280e8	2.93266
$147$	3.72558e7	0.967350
$148$	4.50983e6	0.114352
$149$	4.17848e7	1.03482	0.517412	−	0.855737i	$-0.326896\pi$
$149$	4.17848e7	1.03482	0.517412	+	0.855737i	$0.326896\pi$
$150$	0	0
$151$	−3.87555e7	−0.916039	−0.458019	−	0.888942i	$-0.651441\pi$
$151$	−3.87555e7	−0.916039	−0.458019	+	0.888942i	$0.651441\pi$
$152$	−8.53130e7	−1.97044
$153$	2.93374e6	0.0662218
$154$	1.39492e7	0.307771
$155$	0	0
$156$	1.19200e8	2.51385
$157$	3.70565e7	0.764216	0.382108	−	0.924118i	$-0.375198\pi$
$157$	3.70565e7	0.764216	0.382108	+	0.924118i	$0.375198\pi$
$158$	−1.73611e8	−3.50168
$159$	4.99954e7	0.986371
$160$	0	0
$161$	−9.37441e6	−0.177033
$162$	9.82025e7	1.81476
$163$	−5.92230e7	−1.07111	−0.535555	−	0.844500i	$-0.679898\pi$
$163$	−5.92230e7	−1.07111	−0.535555	+	0.844500i	$0.679898\pi$
$164$	9.73221e7	1.72289
$165$	0	0
$166$	−4.65036e7	−0.789058
$167$	7.21370e7	1.19853	0.599267	−	0.800549i	$-0.295459\pi$
$167$	7.21370e7	1.19853	0.599267	+	0.800549i	$0.295459\pi$
$168$	2.06862e7	0.336587
$169$	−610074.	−0.00972252
$170$	0	0
$171$	−1.68168e6	−0.0257192
$172$	2.65301e8	3.97548
$173$	1.27686e8	1.87491	0.937456	−	0.348103i	$-0.113174\pi$
$173$	1.27686e8	1.87491	0.937456	+	0.348103i	$0.113174\pi$
$174$	−1.06103e8	−1.52689
$175$	0	0
$176$	−3.13144e8	−4.32962
$177$	677436.	0.00918252
$178$	−217288.	−0.00288780
$179$	1.04238e8	1.35844	0.679221	−	0.733934i	$-0.262318\pi$
$179$	1.04238e8	1.35844	0.679221	+	0.733934i	$0.262318\pi$
$180$	0	0
$181$	2.88149e7	0.361196	0.180598	−	0.983557i	$-0.442197\pi$
$181$	2.88149e7	0.361196	0.180598	+	0.983557i	$0.442197\pi$
$182$	1.76226e7	0.216680
$183$	2.65684e7	0.320470
$184$	3.87479e8	4.58549
$185$	0	0
$186$	2.61820e8	2.98337
$187$	2.15036e8	2.40473
$188$	3.24989e7	0.356711
$189$	1.08836e7	0.117262
$190$	0	0
$191$	2.27929e6	0.0236691	0.0118346	−	0.999930i	$-0.496233\pi$
$191$	2.27929e6	0.0236691	0.0118346	+	0.999930i	$0.496233\pi$
$192$	−2.16631e8	−2.20885
$193$	−1.57652e8	−1.57851	−0.789257	−	0.614062i	$-0.789534\pi$
$193$	−1.57652e8	−1.57851	−0.789257	+	0.614062i	$0.789534\pi$
$194$	2.76606e7	0.271992
$195$	0	0
$196$	−2.68029e8	−2.54265
$197$	1.51215e8	1.40916	0.704582	−	0.709622i	$-0.251134\pi$
$197$	1.51215e8	1.40916	0.704582	+	0.709622i	$0.251134\pi$
$198$	−1.13653e7	−0.104053
$199$	−6.63927e7	−0.597220	−0.298610	−	0.954375i	$-0.596523\pi$
$199$	−6.63927e7	−0.597220	−0.298610	+	0.954375i	$0.596523\pi$
$200$	0	0
$201$	−980753.	−0.00851870
$202$	7.31837e7	0.624719
$203$	−1.13008e7	−0.0948143
$204$	5.21130e8	4.29774
$205$	0	0
$206$	−2.96883e8	−2.36620
$207$	7.63795e6	0.0598523
$208$	−3.95607e8	−3.04819
$209$	−1.23264e8	−0.933949
$210$	0	0
$211$	−2.22339e8	−1.62940	−0.814699	−	0.579884i	$-0.803098\pi$
$211$	−2.22339e8	−1.62940	−0.814699	+	0.579884i	$0.803098\pi$
$212$	−3.59682e8	−2.59264
$213$	−1.90688e8	−1.35205
$214$	−9.74909e7	−0.680011
$215$	0	0
$216$	−4.49859e8	−3.03730
$217$	2.78859e7	0.185257
$218$	5.88547e7	0.384754
$219$	2.36291e8	1.52017
$220$	0	0
$221$	2.71664e8	1.69301
$222$	1.34130e7	0.0822790
$223$	9.10646e7	0.549899	0.274949	−	0.961459i	$-0.411339\pi$
$223$	9.10646e7	0.549899	0.274949	+	0.961459i	$0.411339\pi$
$224$	−5.44402e7	−0.323632
$225$	0	0
$226$	−2.84357e8	−1.63864
$227$	−6.61605e7	−0.375412	−0.187706	−	0.982225i	$-0.560105\pi$
$227$	−6.61605e7	−0.375412	−0.187706	+	0.982225i	$0.560105\pi$
$228$	−2.98724e8	−1.66916
$229$	−6.54623e7	−0.360219	−0.180110	−	0.983647i	$-0.557645\pi$
$229$	−6.54623e7	−0.360219	−0.180110	+	0.983647i	$0.557645\pi$
$230$	0	0
$231$	2.98882e7	0.159536
$232$	4.67104e8	2.45587
$233$	−2.22480e7	−0.115225	−0.0576124	−	0.998339i	$-0.518349\pi$
$233$	−2.22480e7	−0.115225	−0.0576124	+	0.998339i	$0.518349\pi$
$234$	−1.43583e7	−0.0732566
$235$	0	0
$236$	−4.87367e6	−0.0241360
$237$	−3.71986e8	−1.81513
$238$	7.70443e7	0.370443
$239$	3.39366e7	0.160796	0.0803981	−	0.996763i	$-0.474381\pi$
$239$	3.39366e7	0.160796	0.0803981	+	0.996763i	$0.474381\pi$
$240$	0	0
$241$	3.09435e7	0.142400	0.0711999	−	0.997462i	$-0.477317\pi$
$241$	3.09435e7	0.142400	0.0711999	+	0.997462i	$0.477317\pi$
$242$	−4.16088e8	−1.88726
$243$	−1.74029e7	−0.0778038
$244$	−1.91141e8	−0.842344
$245$	0	0
$246$	2.89451e8	1.23966
$247$	−1.55724e8	−0.657530
$248$	−1.15262e9	−4.79852
$249$	−9.96409e7	−0.409016
$250$	0	0
$251$	1.43260e8	0.571831	0.285915	−	0.958255i	$-0.407702\pi$
$251$	1.43260e8	0.571831	0.285915	+	0.958255i	$0.407702\pi$
$252$	−2.93358e6	−0.0115477
$253$	5.59845e8	2.17343
$254$	−4.16444e8	−1.59455
$255$	0	0
$256$	1.31435e8	0.489634
$257$	2.09379e8	0.769427	0.384714	−	0.923036i	$-0.374300\pi$
$257$	2.09379e8	0.769427	0.384714	+	0.923036i	$0.374300\pi$
$258$	7.89049e8	2.86045
$259$	1.42858e6	0.00510924
$260$	0	0
$261$	9.20752e6	0.0320554
$262$	8.89945e8	3.05709
$263$	−1.47033e8	−0.498391	−0.249196	−	0.968453i	$-0.580166\pi$
$263$	−1.47033e8	−0.498391	−0.249196	+	0.968453i	$0.580166\pi$
$264$	−1.23539e9	−4.13229
$265$	0	0
$266$	−4.41636e7	−0.143873
$267$	−465572.	−0.00149692
$268$	7.05583e6	0.0223911
$269$	−5.38366e8	−1.68634	−0.843169	−	0.537648i	$-0.819313\pi$
$269$	−5.38366e8	−1.68634	−0.843169	+	0.537648i	$0.819313\pi$
$270$	0	0
$271$	6.78663e7	0.207139	0.103570	−	0.994622i	$-0.466974\pi$
$271$	6.78663e7	0.207139	0.103570	+	0.994622i	$0.466974\pi$
$272$	−1.72956e9	−5.21127
$273$	3.77590e7	0.112318
$274$	4.63769e8	1.36199
$275$	0	0
$276$	1.35676e9	3.88437
$277$	4.62312e7	0.130694	0.0653470	−	0.997863i	$-0.479185\pi$
$277$	4.62312e7	0.130694	0.0653470	+	0.997863i	$0.479185\pi$
$278$	−2.54268e8	−0.709799
$279$	−2.27205e7	−0.0626329
$280$	0	0
$281$	−3.44319e7	−0.0925739	−0.0462870	−	0.998928i	$-0.514739\pi$
$281$	−3.44319e7	−0.0925739	−0.0462870	+	0.998928i	$0.514739\pi$
$282$	9.66569e7	0.256662
$283$	3.31394e8	0.869145	0.434572	−	0.900637i	$-0.356900\pi$
$283$	3.31394e8	0.869145	0.434572	+	0.900637i	$0.356900\pi$
$284$	1.37186e9	3.55383
$285$	0	0
$286$	−1.05243e9	−2.66019
$287$	3.08288e7	0.0769786
$288$	4.43560e7	0.109415
$289$	7.77350e8	1.89441
$290$	0	0
$291$	5.92669e7	0.140990
$292$	−1.69995e9	−3.99573
$293$	2.58787e8	0.601044	0.300522	−	0.953775i	$-0.402839\pi$
$293$	2.58787e8	0.601044	0.300522	+	0.953775i	$0.402839\pi$
$294$	−7.97163e8	−1.82950
$295$	0	0
$296$	−5.90486e7	−0.132339
$297$	−6.49974e8	−1.43962
$298$	−8.94069e8	−1.95710
$299$	7.07274e8	1.53017
$300$	0	0
$301$	8.40397e7	0.177624
$302$	8.29251e8	1.73245
$303$	1.56807e8	0.323829
$304$	9.91422e8	2.02396
$305$	0	0
$306$	−6.27731e7	−0.125242
$307$	−8.16485e8	−1.61051	−0.805256	−	0.592928i	$-0.797972\pi$
$307$	−8.16485e8	−1.61051	−0.805256	+	0.592928i	$0.797972\pi$
$308$	−2.15025e8	−0.419335
$309$	−6.36116e8	−1.22654
$310$	0	0
$311$	−6.90602e8	−1.30187	−0.650933	−	0.759135i	$-0.725622\pi$
$311$	−6.90602e8	−1.30187	−0.650933	+	0.759135i	$0.725622\pi$
$312$	−1.56072e9	−2.90926
$313$	2.33454e8	0.430324	0.215162	−	0.976578i	$-0.430972\pi$
$313$	2.33454e8	0.430324	0.215162	+	0.976578i	$0.430972\pi$
$314$	−7.92898e8	−1.44532
$315$	0	0
$316$	2.67618e9	4.77101
$317$	9.01852e8	1.59011	0.795056	−	0.606536i	$-0.207442\pi$
$317$	9.01852e8	1.59011	0.795056	+	0.606536i	$0.207442\pi$
$318$	−1.06975e9	−1.86547
$319$	6.74891e8	1.16404
$320$	0	0
$321$	−2.08889e8	−0.352490
$322$	2.00584e8	0.334812
$323$	−6.80811e8	−1.12413
$324$	−1.51378e9	−2.47260
$325$	0	0
$326$	1.26719e9	2.02573
$327$	1.26105e8	0.199441
$328$	−1.27427e9	−1.99389
$329$	1.02947e7	0.0159378
$330$	0	0
$331$	−1.44345e7	−0.0218778	−0.0109389	−	0.999940i	$-0.503482\pi$
$331$	−1.44345e7	−0.0218778	−0.0109389	+	0.999940i	$0.503482\pi$
$332$	7.16846e8	1.07508
$333$	−1.16396e6	−0.00172736
$334$	−1.54351e9	−2.26672
$335$	0	0
$336$	−2.40394e8	−0.345729
$337$	−4.56976e7	−0.0650412	−0.0325206	−	0.999471i	$-0.510353\pi$
$337$	−4.56976e7	−0.0650412	−0.0325206	+	0.999471i	$0.510353\pi$
$338$	1.30537e7	0.0183877
$339$	−6.09276e8	−0.849406
$340$	0	0
$341$	−1.66536e9	−2.27440
$342$	3.59830e7	0.0486414
$343$	−1.70948e8	−0.228737
$344$	−3.47367e9	−4.60081
$345$	0	0
$346$	−2.73209e9	−3.54592
$347$	9.60442e8	1.23401	0.617004	−	0.786960i	$-0.288346\pi$
$347$	9.60442e8	1.23401	0.617004	+	0.786960i	$0.288346\pi$
$348$	1.63557e9	2.08037
$349$	−3.83365e8	−0.482752	−0.241376	−	0.970432i	$-0.577599\pi$
$349$	−3.83365e8	−0.482752	−0.241376	+	0.970432i	$0.577599\pi$
$350$	0	0
$351$	−8.21138e8	−1.01354
$352$	3.25120e9	3.97323
$353$	−8.99045e8	−1.08785	−0.543926	−	0.839133i	$-0.683063\pi$
$353$	−8.99045e8	−1.08785	−0.543926	+	0.839133i	$0.683063\pi$
$354$	−1.44951e7	−0.0173664
$355$	0	0
$356$	3.34947e6	0.00393460
$357$	1.65079e8	0.192023
$358$	−2.23038e9	−2.56914
$359$	−1.05710e8	−0.120583	−0.0602915	−	0.998181i	$-0.519203\pi$
$359$	−1.05710e8	−0.120583	−0.0602915	+	0.998181i	$0.519203\pi$
$360$	0	0
$361$	−5.03616e8	−0.563409
$362$	−6.16553e8	−0.683109
$363$	−8.91529e8	−0.978278
$364$	−2.71649e8	−0.295225
$365$	0	0
$366$	−5.68484e8	−0.606086
$367$	7.65818e8	0.808713	0.404357	−	0.914601i	$-0.367495\pi$
$367$	7.65818e8	0.808713	0.404357	+	0.914601i	$0.367495\pi$
$368$	−4.50289e9	−4.71003
$369$	−2.51183e7	−0.0260254
$370$	0	0
$371$	−1.13937e8	−0.115839
$372$	−4.03591e9	−4.06482
$373$	1.17236e9	1.16971	0.584856	−	0.811137i	$-0.301151\pi$
$373$	1.17236e9	1.16971	0.584856	+	0.811137i	$0.301151\pi$
$374$	−4.60113e9	−4.54793
$375$	0	0
$376$	−4.25518e8	−0.412819
$377$	8.52616e8	0.819518
$378$	−2.32876e8	−0.221771
$379$	1.80814e9	1.70606	0.853032	−	0.521859i	$-0.174761\pi$
$379$	1.80814e9	1.70606	0.853032	+	0.521859i	$0.174761\pi$
$380$	0	0
$381$	−8.92291e8	−0.826550
$382$	−4.87699e7	−0.0447641
$383$	6.00743e8	0.546378	0.273189	−	0.961960i	$-0.411922\pi$
$383$	6.00743e8	0.546378	0.273189	+	0.961960i	$0.411922\pi$
$384$	1.57754e9	1.42174
$385$	0	0
$386$	3.37328e9	2.98536
$387$	−6.84727e7	−0.0600522
$388$	−4.26384e8	−0.370587
$389$	1.34241e9	1.15628	0.578140	−	0.815938i	$-0.303779\pi$
$389$	1.34241e9	1.15628	0.578140	+	0.815938i	$0.303779\pi$
$390$	0	0
$391$	3.09214e9	2.61601
$392$	3.50939e9	2.94259
$393$	1.90684e9	1.58467
$394$	−3.23554e9	−2.66507
$395$	0	0
$396$	1.75195e8	0.141771
$397$	−1.31008e9	−1.05082	−0.525412	−	0.850848i	$-0.676089\pi$
$397$	−1.31008e9	−1.05082	−0.525412	+	0.850848i	$0.676089\pi$
$398$	1.42060e9	1.12949
$399$	−9.46270e7	−0.0745778
$400$	0	0
$401$	−2.16307e9	−1.67519	−0.837596	−	0.546289i	$-0.816040\pi$
$401$	−2.16307e9	−1.67519	−0.837596	+	0.546289i	$0.816040\pi$
$402$	2.09852e7	0.0161109
$403$	−2.10391e9	−1.60125
$404$	−1.12811e9	−0.851174
$405$	0	0
$406$	2.41803e8	0.179317
$407$	−8.53158e7	−0.0627262
$408$	−6.82331e9	−4.97376
$409$	1.51216e9	1.09286	0.546432	−	0.837503i	$-0.315986\pi$
$409$	1.51216e9	1.09286	0.546432	+	0.837503i	$0.315986\pi$
$410$	0	0
$411$	9.93693e8	0.706002
$412$	4.57641e9	3.22392
$413$	−1.54384e6	−0.00107839
$414$	−1.63429e8	−0.113195
$415$	0	0
$416$	4.10736e9	2.79728
$417$	−5.44807e8	−0.367931
$418$	2.63747e9	1.76633
$419$	5.88214e8	0.390649	0.195324	−	0.980739i	$-0.437424\pi$
$419$	5.88214e8	0.390649	0.195324	+	0.980739i	$0.437424\pi$
$420$	0	0
$421$	−2.25966e9	−1.47590	−0.737948	−	0.674858i	$-0.764205\pi$
$421$	−2.25966e9	−1.47590	−0.737948	+	0.674858i	$0.764205\pi$
$422$	4.75739e9	3.08159
$423$	−8.38778e6	−0.00538835
$424$	4.70942e9	3.00045
$425$	0	0
$426$	4.08014e9	2.55707
$427$	−6.05479e7	−0.0376358
$428$	1.50281e9	0.926509
$429$	−2.25499e9	−1.37893
$430$	0	0
$431$	2.66643e9	1.60421	0.802103	−	0.597185i	$-0.203714\pi$
$431$	2.66643e9	1.60421	0.802103	+	0.597185i	$0.203714\pi$
$432$	5.22781e9	3.11980
$433$	3.24035e9	1.91816	0.959080	−	0.283136i	$-0.0913749\pi$
$433$	3.24035e9	1.91816	0.959080	+	0.283136i	$0.0913749\pi$
$434$	−5.96673e8	−0.350367
$435$	0	0
$436$	−9.07235e8	−0.524224
$437$	−1.77248e9	−1.01601
$438$	−5.05592e9	−2.87502
$439$	1.01291e9	0.571404	0.285702	−	0.958319i	$-0.407773\pi$
$439$	1.01291e9	0.571404	0.285702	+	0.958319i	$0.407773\pi$
$440$	0	0
$441$	6.91769e7	0.0384084
$442$	−5.81278e9	−3.20189
$443$	8.18624e8	0.447375	0.223687	−	0.974661i	$-0.428191\pi$
$443$	8.18624e8	0.447375	0.223687	+	0.974661i	$0.428191\pi$
$444$	−2.06759e8	−0.112105
$445$	0	0
$446$	−1.94851e9	−1.03999
$447$	−1.91567e9	−1.01448
$448$	4.93689e8	0.259406
$449$	3.72052e8	0.193973	0.0969866	−	0.995286i	$-0.469080\pi$
$449$	3.72052e8	0.193973	0.0969866	+	0.995286i	$0.469080\pi$
$450$	0	0
$451$	−1.84111e9	−0.945066
$452$	4.38331e9	2.23264
$453$	1.77679e9	0.898034
$454$	1.41563e9	0.709996
$455$	0	0
$456$	3.91128e9	1.93171
$457$	−2.44939e8	−0.120047	−0.0600235	−	0.998197i	$-0.519118\pi$
$457$	−2.44939e8	−0.120047	−0.0600235	+	0.998197i	$0.519118\pi$
$458$	1.40070e9	0.681263
$459$	−3.58994e9	−1.73278
$460$	0	0
$461$	1.23419e9	0.586719	0.293359	−	0.956002i	$-0.405227\pi$
$461$	1.23419e9	0.586719	0.293359	+	0.956002i	$0.405227\pi$
$462$	−6.39518e8	−0.301721
$463$	−9.97984e8	−0.467294	−0.233647	−	0.972321i	$-0.575066\pi$
$463$	−9.97984e8	−0.467294	−0.233647	+	0.972321i	$0.575066\pi$
$464$	−5.42821e9	−2.52257
$465$	0	0
$466$	4.76041e8	0.217918
$467$	1.51176e9	0.686871	0.343435	−	0.939176i	$-0.388409\pi$
$467$	1.51176e9	0.686871	0.343435	+	0.939176i	$0.388409\pi$
$468$	2.21331e8	0.0998116
$469$	2.23508e6	0.00100043
$470$	0	0
$471$	−1.69890e9	−0.749195
$472$	6.38125e7	0.0279324
$473$	−5.01890e9	−2.18069
$474$	7.95938e9	3.43285
$475$	0	0
$476$	−1.18763e9	−0.504726
$477$	9.28318e7	0.0391636
$478$	−7.26141e8	−0.304105
$479$	2.16371e9	0.899548	0.449774	−	0.893142i	$-0.351505\pi$
$479$	2.16371e9	0.899548	0.449774	+	0.893142i	$0.351505\pi$
$480$	0	0
$481$	−1.07783e8	−0.0441612
$482$	−6.62097e8	−0.269313
$483$	4.29781e8	0.173553
$484$	6.41392e9	2.57137
$485$	0	0
$486$	3.72371e8	0.147146
$487$	7.57260e8	0.297094	0.148547	−	0.988905i	$-0.452540\pi$
$487$	7.57260e8	0.297094	0.148547	+	0.988905i	$0.452540\pi$
$488$	2.50267e9	0.974840
$489$	2.71515e9	1.05006
$490$	0	0
$491$	−4.19786e9	−1.60045	−0.800226	−	0.599699i	$-0.795287\pi$
$491$	−4.19786e9	−1.60045	−0.800226	+	0.599699i	$0.795287\pi$
$492$	−4.46185e9	−1.68903
$493$	3.72756e9	1.40107
$494$	3.33202e9	1.24355
$495$	0	0
$496$	1.33946e10	4.92884
$497$	4.34567e8	0.158785
$498$	2.13201e9	0.773548
$499$	1.44053e9	0.519003	0.259502	−	0.965743i	$-0.416442\pi$
$499$	1.44053e9	0.519003	0.259502	+	0.965743i	$0.416442\pi$
$500$	0	0
$501$	−3.30720e9	−1.17498
$502$	−3.06534e9	−1.08147
$503$	−4.24069e9	−1.48576	−0.742880	−	0.669425i	$-0.766541\pi$
$503$	−4.24069e9	−1.48576	−0.742880	+	0.669425i	$0.766541\pi$
$504$	3.84102e7	0.0133641
$505$	0	0
$506$	−1.19790e10	−4.11049
$507$	2.79696e7	0.00953142
$508$	6.41941e9	2.17256
$509$	−2.15640e9	−0.724799	−0.362400	−	0.932023i	$-0.618042\pi$
$509$	−2.15640e9	−0.724799	−0.362400	+	0.932023i	$0.618042\pi$
$510$	0	0
$511$	−5.38494e8	−0.178529
$512$	1.59208e9	0.524228
$513$	2.05784e9	0.672977
$514$	−4.48008e9	−1.45518
$515$	0	0
$516$	−1.21631e10	−3.89734
$517$	−6.14805e8	−0.195668
$518$	−3.05674e7	−0.00966282
$519$	−5.85390e9	−1.83806
$520$	0	0
$521$	2.70359e9	0.837545	0.418773	−	0.908091i	$-0.362460\pi$
$521$	2.70359e9	0.837545	0.418773	+	0.908091i	$0.362460\pi$
$522$	−1.97013e8	−0.0606246
$523$	7.95053e8	0.243019	0.121510	−	0.992590i	$-0.461227\pi$
$523$	7.95053e8	0.243019	0.121510	+	0.992590i	$0.461227\pi$
$524$	−1.37184e10	−4.16527
$525$	0	0
$526$	3.14607e9	0.942579
$527$	−9.19811e9	−2.73755
$528$	1.43565e10	4.24452
$529$	4.64552e9	1.36439
$530$	0	0
$531$	1.25787e6	0.000364589 0
$532$	6.80774e8	0.196025
$533$	−2.32595e9	−0.665357
$534$	9.96185e6	0.00283104
$535$	0	0
$536$	−9.23840e7	−0.0259132
$537$	−4.77892e9	−1.33174
$538$	1.15194e10	3.18928
$539$	5.07051e9	1.39473
$540$	0	0
$541$	−1.89746e9	−0.515208	−0.257604	−	0.966251i	$-0.582933\pi$
$541$	−1.89746e9	−0.515208	−0.257604	+	0.966251i	$0.582933\pi$
$542$	−1.45214e9	−0.391751
$543$	−1.32105e9	−0.354096
$544$	1.79570e10	4.78231
$545$	0	0
$546$	−8.07928e8	−0.212422
$547$	−3.82406e9	−0.999008	−0.499504	−	0.866312i	$-0.666484\pi$
$547$	−3.82406e9	−0.999008	−0.499504	+	0.866312i	$0.666484\pi$
$548$	−7.14892e9	−1.85570
$549$	4.93324e7	0.0127242
$550$	0	0
$551$	−2.13672e9	−0.544149
$552$	−1.77644e10	−4.49536
$553$	8.47736e8	0.213168
$554$	−9.89207e8	−0.247174
$555$	0	0
$556$	3.91951e9	0.967096
$557$	2.92064e8	0.0716118	0.0358059	−	0.999359i	$-0.488600\pi$
$557$	2.92064e8	0.0716118	0.0358059	+	0.999359i	$0.488600\pi$
$558$	4.86149e8	0.118454
$559$	−6.34057e9	−1.53528
$560$	0	0
$561$	−9.85859e9	−2.35746
$562$	7.36739e8	0.175080
$563$	−5.30895e9	−1.25380	−0.626901	−	0.779099i	$-0.715677\pi$
$563$	−5.30895e9	−1.25380	−0.626901	+	0.779099i	$0.715677\pi$
$564$	−1.48995e9	−0.349699
$565$	0	0
$566$	−7.09083e9	−1.64377
$567$	−4.79520e8	−0.110476
$568$	−1.79622e10	−4.11283
$569$	7.91391e8	0.180094	0.0900468	−	0.995938i	$-0.471298\pi$
$569$	7.91391e8	0.180094	0.0900468	+	0.995938i	$0.471298\pi$
$570$	0	0
$571$	−2.08855e8	−0.0469482	−0.0234741	−	0.999724i	$-0.507473\pi$
$571$	−2.08855e8	−0.0469482	−0.0234741	+	0.999724i	$0.507473\pi$
$572$	1.62230e10	3.62448
$573$	−1.04497e8	−0.0232039
$574$	−6.59643e8	−0.145585
$575$	0	0
$576$	−4.02241e8	−0.0877017
$577$	5.53822e9	1.20020	0.600102	−	0.799923i	$-0.295127\pi$
$577$	5.53822e9	1.20020	0.600102	+	0.799923i	$0.295127\pi$
$578$	−1.66330e10	−3.58280
$579$	7.22774e9	1.54749
$580$	0	0
$581$	2.27076e8	0.0480347
$582$	−1.26813e9	−0.266646
$583$	6.80436e9	1.42216
$584$	2.22579e10	4.62423
$585$	0	0
$586$	−5.53727e9	−1.13672
$587$	−4.46179e9	−0.910491	−0.455245	−	0.890366i	$-0.650448\pi$
$587$	−4.46179e9	−0.910491	−0.455245	+	0.890366i	$0.650448\pi$
$588$	1.22881e10	2.49267
$589$	5.27257e9	1.06321
$590$	0	0
$591$	−6.93261e9	−1.38147
$592$	6.86203e8	0.135934
$593$	−1.64304e9	−0.323561	−0.161781	−	0.986827i	$-0.551724\pi$
$593$	−1.64304e9	−0.323561	−0.161781	+	0.986827i	$0.551724\pi$
$594$	1.39075e10	2.72268
$595$	0	0
$596$	1.37819e10	2.66654
$597$	3.04385e9	0.585482
$598$	−1.51335e10	−2.89392
$599$	−1.46040e9	−0.277638	−0.138819	−	0.990318i	$-0.544331\pi$
$599$	−1.46040e9	−0.277638	−0.138819	+	0.990318i	$0.544331\pi$
$600$	0	0
$601$	7.95128e8	0.149409	0.0747044	−	0.997206i	$-0.476199\pi$
$601$	7.95128e8	0.149409	0.0747044	+	0.997206i	$0.476199\pi$
$602$	−1.79820e9	−0.335931
$603$	−1.82107e6	−0.000338233 0
$604$	−1.27828e10	−2.36045
$605$	0	0
$606$	−3.35519e9	−0.612440
$607$	−2.29465e9	−0.416444	−0.208222	−	0.978082i	$-0.566768\pi$
$607$	−2.29465e9	−0.416444	−0.208222	+	0.978082i	$0.566768\pi$
$608$	−1.02934e10	−1.85736
$609$	5.18099e8	0.0929507
$610$	0	0
$611$	−7.76707e8	−0.137757
$612$	9.67637e8	0.170641
$613$	−7.96146e9	−1.39599	−0.697993	−	0.716105i	$-0.745923\pi$
$613$	−7.96146e9	−1.39599	−0.697993	+	0.716105i	$0.745923\pi$
$614$	1.74703e10	3.04587
$615$	0	0
$616$	2.81538e9	0.485294
$617$	−1.12966e10	−1.93620	−0.968099	−	0.250568i	$-0.919383\pi$
$617$	−1.12966e10	−1.93620	−0.968099	+	0.250568i	$0.919383\pi$
$618$	1.36110e10	2.31969
$619$	1.04664e9	0.177369	0.0886847	−	0.996060i	$-0.471734\pi$
$619$	1.04664e9	0.177369	0.0886847	+	0.996060i	$0.471734\pi$
$620$	0	0
$621$	−9.34638e9	−1.56611
$622$	1.47768e10	2.46215
$623$	1.06101e6	0.000175798 0
$624$	1.81371e10	2.98828
$625$	0	0
$626$	−4.99520e9	−0.813848
$627$	5.65117e9	0.915592
$628$	1.22224e10	1.96923
$629$	−4.71216e8	−0.0754993
$630$	0	0
$631$	−4.48346e9	−0.710412	−0.355206	−	0.934788i	$-0.615589\pi$
$631$	−4.48346e9	−0.710412	−0.355206	+	0.934788i	$0.615589\pi$
$632$	−3.50400e10	−5.52147
$633$	1.01934e10	1.59737
$634$	−1.92969e10	−3.00729
$635$	0	0
$636$	1.64900e10	2.54168
$637$	6.40577e9	0.981936
$638$	−1.44406e10	−2.20148
$639$	−3.54070e8	−0.0536829
$640$	0	0
$641$	1.00030e10	1.50012	0.750062	−	0.661367i	$-0.230024\pi$
$641$	1.00030e10	1.50012	0.750062	+	0.661367i	$0.230024\pi$
$642$	4.46958e9	0.666645
$643$	7.33714e9	1.08840	0.544200	−	0.838956i	$-0.316833\pi$
$643$	7.33714e9	1.08840	0.544200	+	0.838956i	$0.316833\pi$
$644$	−3.09197e9	−0.456179
$645$	0	0
$646$	1.45673e10	2.12601
$647$	8.16612e8	0.118536	0.0592681	−	0.998242i	$-0.481123\pi$
$647$	8.16612e8	0.118536	0.0592681	+	0.998242i	$0.481123\pi$
$648$	1.98203e10	2.86153
$649$	9.21988e7	0.0132394
$650$	0	0
$651$	−1.27846e9	−0.181616
$652$	−1.95336e10	−2.76004
$653$	−1.06710e9	−0.149972	−0.0749860	−	0.997185i	$-0.523891\pi$
$653$	−1.06710e9	−0.149972	−0.0749860	+	0.997185i	$0.523891\pi$
$654$	−2.69826e9	−0.377192
$655$	0	0
$656$	1.48082e10	2.04805
$657$	4.38747e8	0.0603581
$658$	−2.20276e8	−0.0301423
$659$	1.09782e10	1.49428	0.747140	−	0.664667i	$-0.231426\pi$
$659$	1.09782e10	1.49428	0.747140	+	0.664667i	$0.231426\pi$
$660$	0	0
$661$	9.19554e9	1.23843	0.619216	−	0.785221i	$-0.287451\pi$
$661$	9.19554e9	1.23843	0.619216	+	0.785221i	$0.287451\pi$
$662$	3.08854e8	0.0413762
$663$	−1.24547e10	−1.65973
$664$	−9.38588e9	−1.24419
$665$	0	0
$666$	2.49053e7	0.00326687
$667$	9.70467e9	1.26631
$668$	2.37930e10	3.08839
$669$	−4.17496e9	−0.539090
$670$	0	0
$671$	3.61595e9	0.462055
$672$	2.49587e9	0.317271
$673$	6.69964e9	0.847225	0.423613	−	0.905843i	$-0.360762\pi$
$673$	6.69964e9	0.847225	0.423613	+	0.905843i	$0.360762\pi$
$674$	9.77791e8	0.123009
$675$	0	0
$676$	−2.01221e8	−0.0250530
$677$	−1.21341e9	−0.150296	−0.0751479	−	0.997172i	$-0.523943\pi$
$677$	−1.21341e9	−0.150296	−0.0751479	+	0.997172i	$0.523943\pi$
$678$	1.30367e10	1.60643
$679$	−1.35066e8	−0.0165578
$680$	0	0
$681$	3.03320e9	0.368033
$682$	3.56336e10	4.30145
$683$	5.70657e9	0.685334	0.342667	−	0.939457i	$-0.388670\pi$
$683$	5.70657e9	0.685334	0.342667	+	0.939457i	$0.388670\pi$
$684$	−5.54672e8	−0.0662735
$685$	0	0
$686$	3.65778e9	0.432597
$687$	3.00120e9	0.353139
$688$	4.03675e10	4.72576
$689$	8.59621e9	1.00124
$690$	0	0
$691$	−1.02351e10	−1.18010	−0.590050	−	0.807367i	$-0.700892\pi$
$691$	−1.02351e10	−1.18010	−0.590050	+	0.807367i	$0.700892\pi$
$692$	4.21147e10	4.83128
$693$	5.54967e7	0.00633433
$694$	−2.05506e10	−2.33381
$695$	0	0
$696$	−2.14149e10	−2.40760
$697$	−1.01688e10	−1.13751
$698$	8.20286e9	0.913002
$699$	1.01999e9	0.112960
$700$	0	0
$701$	−1.28337e10	−1.40714	−0.703571	−	0.710625i	$-0.748412\pi$
$701$	−1.28337e10	−1.40714	−0.703571	+	0.710625i	$0.748412\pi$
$702$	1.75699e10	1.91685
$703$	2.70112e8	0.0293224
$704$	−2.94834e10	−3.18473
$705$	0	0
$706$	1.92368e10	2.05739
$707$	−3.57354e8	−0.0380304
$708$	2.23439e8	0.0236616
$709$	−1.53819e10	−1.62087	−0.810436	−	0.585827i	$-0.800770\pi$
$709$	−1.53819e10	−1.62087	−0.810436	+	0.585827i	$0.800770\pi$
$710$	0	0
$711$	−6.90706e8	−0.0720692
$712$	−4.38556e7	−0.00455349
$713$	−2.39472e10	−2.47424
$714$	−3.53219e9	−0.363162
$715$	0	0
$716$	3.43810e10	3.50044
$717$	−1.55586e9	−0.157636
$718$	2.26188e9	0.228052
$719$	−8.84550e9	−0.887506	−0.443753	−	0.896149i	$-0.646353\pi$
$719$	−8.84550e9	−0.887506	−0.443753	+	0.896149i	$0.646353\pi$
$720$	0	0
$721$	1.44967e9	0.144045
$722$	1.07759e10	1.06554
$723$	−1.41864e9	−0.139601
$724$	9.50406e9	0.930731
$725$	0	0
$726$	1.90760e10	1.85016
$727$	4.05716e8	0.0391608	0.0195804	−	0.999808i	$-0.493767\pi$
$727$	4.05716e8	0.0391608	0.0195804	+	0.999808i	$0.493767\pi$
$728$	3.55678e9	0.341663
$729$	1.08352e10	1.03584
$730$	0	0
$731$	−2.77204e10	−2.62475
$732$	8.76308e9	0.825787
$733$	4.77550e9	0.447873	0.223937	−	0.974604i	$-0.428109\pi$
$733$	4.77550e9	0.447873	0.223937	+	0.974604i	$0.428109\pi$
$734$	−1.63862e10	−1.52947
$735$	0	0
$736$	4.67509e10	4.32233
$737$	−1.33480e8	−0.0122823
$738$	5.37455e8	0.0492204
$739$	−7.99975e9	−0.729157	−0.364578	−	0.931173i	$-0.618787\pi$
$739$	−7.99975e9	−0.729157	−0.364578	+	0.931173i	$0.618787\pi$
$740$	0	0
$741$	7.13934e9	0.644606
$742$	2.43790e9	0.219080
$743$	−1.02857e10	−0.919968	−0.459984	−	0.887927i	$-0.652145\pi$
$743$	−1.02857e10	−0.919968	−0.459984	+	0.887927i	$0.652145\pi$
$744$	5.28434e10	4.70420
$745$	0	0
$746$	−2.50849e10	−2.21221
$747$	−1.85014e8	−0.0162398
$748$	7.09256e10	6.19652
$749$	4.76045e8	0.0413963
$750$	0	0
$751$	−9.36299e9	−0.806631	−0.403315	−	0.915061i	$-0.632142\pi$
$751$	−9.36299e9	−0.806631	−0.403315	+	0.915061i	$0.632142\pi$
$752$	4.94494e9	0.424032
$753$	−6.56793e9	−0.560591
$754$	−1.82434e10	−1.54991
$755$	0	0
$756$	3.58975e9	0.302161
$757$	4.20100e9	0.351979	0.175990	−	0.984392i	$-0.443688\pi$
$757$	4.20100e9	0.351979	0.175990	+	0.984392i	$0.443688\pi$
$758$	−3.86888e10	−3.22658
$759$	−2.56668e10	−2.13071
$760$	0	0
$761$	−1.56158e10	−1.28445	−0.642225	−	0.766516i	$-0.721988\pi$
$761$	−1.56158e10	−1.28445	−0.642225	+	0.766516i	$0.721988\pi$
$762$	1.90923e10	1.56321
$763$	−2.87386e8	−0.0234223
$764$	7.51779e8	0.0609907
$765$	0	0
$766$	−1.28541e10	−1.03333
$767$	1.16478e8	0.00932098
$768$	−6.02579e9	−0.480010
$769$	−2.76858e9	−0.219540	−0.109770	−	0.993957i	$-0.535011\pi$
$769$	−2.76858e9	−0.219540	−0.109770	+	0.993957i	$0.535011\pi$
$770$	0	0
$771$	−9.59924e9	−0.754304
$772$	−5.19985e10	−4.06752
$773$	−5.69647e9	−0.443586	−0.221793	−	0.975094i	$-0.571191\pi$
$773$	−5.69647e9	−0.443586	−0.221793	+	0.975094i	$0.571191\pi$
$774$	1.46511e9	0.113573
$775$	0	0
$776$	5.58277e9	0.428878
$777$	−6.54951e7	−0.00500882
$778$	−2.87236e10	−2.18681
$779$	5.82901e9	0.441788
$780$	0	0
$781$	−2.59525e10	−1.94940
$782$	−6.61624e10	−4.94752
$783$	−1.12670e10	−0.838770
$784$	−4.07826e10	−3.02252
$785$	0	0
$786$	−4.08006e10	−2.99701
$787$	−2.08074e9	−0.152162	−0.0760810	−	0.997102i	$-0.524241\pi$
$787$	−2.08074e9	−0.152162	−0.0760810	+	0.997102i	$0.524241\pi$
$788$	4.98752e10	3.63114
$789$	6.74091e9	0.488595
$790$	0	0
$791$	1.38851e9	0.0997540
$792$	−2.29388e9	−0.164071
$793$	4.56817e9	0.325302
$794$	2.80317e10	1.98736
$795$	0	0
$796$	−2.18984e10	−1.53892
$797$	2.42407e10	1.69606	0.848031	−	0.529947i	$-0.177788\pi$
$797$	2.42407e10	1.69606	0.848031	+	0.529947i	$0.177788\pi$
$798$	2.02473e9	0.141045
$799$	−3.39569e9	−0.235513
$800$	0	0
$801$	−864478.	−5.94347e−5 0
$802$	4.62831e10	3.16820
$803$	3.21592e10	2.19180
$804$	−3.23483e8	−0.0219510
$805$	0	0
$806$	4.50174e10	3.02836
$807$	2.46820e10	1.65319
$808$	1.47707e10	0.985059
$809$	2.54306e9	0.168864	0.0844319	−	0.996429i	$-0.473092\pi$
$809$	2.54306e9	0.168864	0.0844319	+	0.996429i	$0.473092\pi$
$810$	0	0
$811$	−8.54054e9	−0.562228	−0.281114	−	0.959674i	$-0.590704\pi$
$811$	−8.54054e9	−0.562228	−0.281114	+	0.959674i	$0.590704\pi$
$812$	−3.72736e9	−0.244318
$813$	−3.11141e9	−0.203068
$814$	1.82550e9	0.118631
$815$	0	0
$816$	7.92936e10	5.10884
$817$	1.58900e10	1.01940
$818$	−3.23557e10	−2.06687
$819$	7.01111e7	0.00445957
$820$	0	0
$821$	−8.52173e9	−0.537436	−0.268718	−	0.963219i	$-0.586600\pi$
$821$	−8.52173e9	−0.537436	−0.268718	+	0.963219i	$0.586600\pi$
$822$	−2.12620e10	−1.33522
$823$	−7.97383e9	−0.498618	−0.249309	−	0.968424i	$-0.580203\pi$
$823$	−7.97383e9	−0.498618	−0.249309	+	0.968424i	$0.580203\pi$
$824$	−5.99203e10	−3.73103
$825$	0	0
$826$	3.30335e7	0.00203950
$827$	1.30555e10	0.802645	0.401323	−	0.915937i	$-0.368551\pi$
$827$	1.30555e10	0.802645	0.401323	+	0.915937i	$0.368551\pi$
$828$	2.51923e9	0.154228
$829$	1.25527e9	0.0765236	0.0382618	−	0.999268i	$-0.487818\pi$
$829$	1.25527e9	0.0765236	0.0382618	+	0.999268i	$0.487818\pi$
$830$	0	0
$831$	−2.11952e9	−0.128125
$832$	−3.72475e10	−2.24215
$833$	2.80054e10	1.67875
$834$	1.16572e10	0.695848
$835$	0	0
$836$	−4.06562e10	−2.40660
$837$	2.78025e10	1.63887
$838$	−1.25860e10	−0.738812
$839$	2.57204e10	1.50353	0.751763	−	0.659434i	$-0.229204\pi$
$839$	2.57204e10	1.50353	0.751763	+	0.659434i	$0.229204\pi$
$840$	0	0
$841$	−5.55094e9	−0.321796
$842$	4.83499e10	2.79128
$843$	1.57857e9	0.0907544
$844$	−7.33343e10	−4.19864
$845$	0	0
$846$	1.79473e8	0.0101907
$847$	2.03174e9	0.114889
$848$	−5.47281e10	−3.08195
$849$	−1.51932e10	−0.852061
$850$	0	0
$851$	−1.22681e9	−0.0682374
$852$	−6.28947e10	−3.48398
$853$	3.97562e8	0.0219323	0.0109661	−	0.999940i	$-0.496509\pi$
$853$	3.97562e8	0.0219323	0.0109661	+	0.999940i	$0.496509\pi$
$854$	1.29554e9	0.0711786
$855$	0	0
$856$	−1.96767e10	−1.07224
$857$	−1.36235e10	−0.739361	−0.369680	−	0.929159i	$-0.620533\pi$
$857$	−1.36235e10	−0.739361	−0.369680	+	0.929159i	$0.620533\pi$
$858$	4.82499e10	2.60790
$859$	−1.19878e10	−0.645302	−0.322651	−	0.946518i	$-0.604574\pi$
$859$	−1.19878e10	−0.645302	−0.322651	+	0.946518i	$0.604574\pi$
$860$	0	0
$861$	−1.41338e9	−0.0754656
$862$	−5.70536e10	−3.03395
$863$	−3.00996e9	−0.159413	−0.0797064	−	0.996818i	$-0.525398\pi$
$863$	−3.00996e9	−0.159413	−0.0797064	+	0.996818i	$0.525398\pi$
$864$	−5.42774e10	−2.86299
$865$	0	0
$866$	−6.93338e10	−3.62771
$867$	−3.56386e10	−1.85718
$868$	9.19762e9	0.477372
$869$	−5.06272e10	−2.61707
$870$	0	0
$871$	−1.68631e8	−0.00864715
$872$	1.18787e10	0.606682
$873$	1.10047e8	0.00559795
$874$	3.79258e10	1.92152
$875$	0	0
$876$	7.79362e10	3.91719
$877$	−1.20934e10	−0.605412	−0.302706	−	0.953084i	$-0.597890\pi$
$877$	−1.20934e10	−0.605412	−0.302706	+	0.953084i	$0.597890\pi$
$878$	−2.16731e10	−1.08066
$879$	−1.18644e10	−0.589231
$880$	0	0
$881$	1.00531e10	0.495317	0.247658	−	0.968847i	$-0.420339\pi$
$881$	1.00531e10	0.495317	0.247658	+	0.968847i	$0.420339\pi$
$882$	−1.48018e9	−0.0726396
$883$	3.45116e9	0.168695	0.0843476	−	0.996436i	$-0.473119\pi$
$883$	3.45116e9	0.168695	0.0843476	+	0.996436i	$0.473119\pi$
$884$	8.96031e10	4.36255
$885$	0	0
$886$	−1.75161e10	−0.846095
$887$	−1.30756e10	−0.629115	−0.314557	−	0.949238i	$-0.601856\pi$
$887$	−1.30756e10	−0.629115	−0.314557	+	0.949238i	$0.601856\pi$
$888$	2.70715e9	0.129738
$889$	2.03348e9	0.0970698
$890$	0	0
$891$	2.86372e10	1.35631
$892$	3.00359e10	1.41698
$893$	1.94649e9	0.0914686
$894$	4.09896e10	1.91864
$895$	0	0
$896$	−3.59512e9	−0.166969
$897$	−3.24258e10	−1.50009
$898$	−7.96080e9	−0.366851
$899$	−2.88683e10	−1.32514
$900$	0	0
$901$	3.75819e10	1.71175
$902$	3.93942e10	1.78735
$903$	−3.85290e9	−0.174133
$904$	−5.73920e10	−2.58382
$905$	0	0
$906$	−3.80180e10	−1.69840
$907$	−2.30295e10	−1.02485	−0.512423	−	0.858733i	$-0.671252\pi$
$907$	−2.30295e10	−1.02485	−0.512423	+	0.858733i	$0.671252\pi$
$908$	−2.18218e10	−0.967363
$909$	2.91160e8	0.0128575
$910$	0	0
$911$	5.27930e9	0.231346	0.115673	−	0.993287i	$-0.463098\pi$
$911$	5.27930e9	0.231346	0.115673	+	0.993287i	$0.463098\pi$
$912$	−4.54529e10	−1.98417
$913$	−1.35611e10	−0.589722
$914$	5.24095e9	0.227038
$915$	0	0
$916$	−2.15915e10	−0.928215
$917$	−4.34558e9	−0.186104
$918$	7.68139e10	3.27711
$919$	6.34113e9	0.269502	0.134751	−	0.990879i	$-0.456977\pi$
$919$	6.34113e9	0.269502	0.134751	+	0.990879i	$0.456977\pi$
$920$	0	0
$921$	3.74327e10	1.57886
$922$	−2.64080e10	−1.10963
$923$	−3.27869e10	−1.37244
$924$	9.85807e9	0.411093
$925$	0	0
$926$	2.13539e10	0.883767
$927$	−1.18114e9	−0.0486995
$928$	5.63581e10	2.31493
$929$	1.44774e9	0.0592429	0.0296215	−	0.999561i	$-0.490570\pi$
$929$	1.44774e9	0.0592429	0.0296215	+	0.999561i	$0.490570\pi$
$930$	0	0
$931$	−1.60534e10	−0.651992
$932$	−7.33809e9	−0.296912
$933$	3.16615e10	1.27628
$934$	−3.23472e10	−1.29904
$935$	0	0
$936$	−2.89795e9	−0.115511
$937$	−4.10761e10	−1.63117	−0.815587	−	0.578634i	$-0.803586\pi$
$937$	−4.10761e10	−1.63117	−0.815587	+	0.578634i	$0.803586\pi$
$938$	−4.78240e7	−0.00189206
$939$	−1.07030e10	−0.421866
$940$	0	0
$941$	2.11198e10	0.826278	0.413139	−	0.910668i	$-0.364432\pi$
$941$	2.11198e10	0.826278	0.413139	+	0.910668i	$0.364432\pi$
$942$	3.63514e10	1.41691
$943$	−2.64745e10	−1.02810
$944$	−7.41564e8	−0.0286911
$945$	0	0
$946$	1.07389e11	4.12422
$947$	6.57627e9	0.251626	0.125813	−	0.992054i	$-0.459846\pi$
$947$	6.57627e9	0.251626	0.125813	+	0.992054i	$0.459846\pi$
$948$	−1.22693e11	−4.67723
$949$	4.06279e10	1.54310
$950$	0	0
$951$	−4.13464e10	−1.55886
$952$	1.55499e10	0.584116
$953$	−1.17416e10	−0.439441	−0.219720	−	0.975563i	$-0.570515\pi$
$953$	−1.17416e10	−0.439441	−0.219720	+	0.975563i	$0.570515\pi$
$954$	−1.98632e9	−0.0740679
$955$	0	0
$956$	1.11933e10	0.414340
$957$	−3.09412e10	−1.14116
$958$	−4.62968e10	−1.70126
$959$	−2.26457e9	−0.0829126
$960$	0	0
$961$	4.37226e10	1.58918
$962$	2.30622e9	0.0835197
$963$	−3.87865e8	−0.0139955
$964$	1.02061e10	0.366936
$965$	0	0
$966$	−9.19602e9	−0.328231
$967$	−4.62138e10	−1.64354	−0.821768	−	0.569823i	$-0.807012\pi$
$967$	−4.62138e10	−1.64354	−0.821768	+	0.569823i	$0.807012\pi$
$968$	−8.39794e10	−2.97584
$969$	3.12126e10	1.10204
$970$	0	0
$971$	−5.52265e10	−1.93589	−0.967944	−	0.251166i	$-0.919186\pi$
$971$	−5.52265e10	−1.93589	−0.967944	+	0.251166i	$0.919186\pi$
$972$	−5.74003e9	−0.200485
$973$	1.24159e9	0.0432097
$974$	−1.62031e10	−0.561877
$975$	0	0
$976$	−2.90835e10	−1.00132
$977$	−4.63906e10	−1.59147	−0.795736	−	0.605644i	$-0.792915\pi$
$977$	−4.63906e10	−1.59147	−0.795736	+	0.605644i	$0.792915\pi$
$978$	−5.80960e10	−1.98591
$979$	−6.33643e7	−0.00215827
$980$	0	0
$981$	2.34152e8	0.00791875
$982$	8.98215e10	3.02684
$983$	3.07681e10	1.03315	0.516575	−	0.856242i	$-0.327207\pi$
$983$	3.07681e10	1.03315	0.516575	+	0.856242i	$0.327207\pi$
$984$	5.84203e10	1.95470
$985$	0	0
$986$	−7.97585e10	−2.64977
$987$	−4.71973e8	−0.0156245
$988$	−5.13625e10	−1.69433
$989$	−7.21698e10	−2.37229
$990$	0	0
$991$	1.40198e10	0.457599	0.228800	−	0.973474i	$-0.426520\pi$
$991$	1.40198e10	0.457599	0.228800	+	0.973474i	$0.426520\pi$
$992$	−1.39069e11	−4.52313
$993$	6.61766e8	0.0214478
$994$	−9.29842e9	−0.300301
$995$	0	0
$996$	−3.28647e10	−1.05395
$997$	1.75155e10	0.559745	0.279872	−	0.960037i	$-0.409708\pi$
$997$	1.75155e10	0.559745	0.279872	+	0.960037i	$0.409708\pi$
$998$	−3.08230e10	−0.981562
$999$	1.42431e9	0.0451987

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.8.a.a.1.1	✓	22
5.4	even	2		625.8.a.b.1.22	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.8.a.a.1.1	✓	22	1.1	even	1	trivial
625.8.a.b.1.22	yes	22	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-21.3970 q^{2} -45.8462 q^{3} +329.831 q^{4} +980.970 q^{6} +104.481 q^{7} -4318.57 q^{8} -85.1275 q^{9} +O(q^{10})\)	\(q-21.3970 q^{2} -45.8462 q^{3} +329.831 q^{4} +980.970 q^{6} +104.481 q^{7} -4318.57 q^{8} -85.1275 q^{9} -6239.65 q^{11} -15121.5 q^{12} -7882.79 q^{13} -2235.58 q^{14} +50186.1 q^{16} -34462.9 q^{17} +1821.47 q^{18} +19754.9 q^{19} -4790.05 q^{21} +133510. q^{22} -89723.7 q^{23} +197990. q^{24} +168668. q^{26} +104168. q^{27} +34461.0 q^{28} -108162. q^{29} +266899. q^{31} -521054. q^{32} +286064. q^{33} +737401. q^{34} -28077.7 q^{36} +13673.2 q^{37} -422695. q^{38} +361396. q^{39} +295066. q^{41} +102493. q^{42} +804355. q^{43} -2.05803e6 q^{44} +1.91982e6 q^{46} +98532.0 q^{47} -2.30084e6 q^{48} -812627. q^{49} +1.57999e6 q^{51} -2.59999e6 q^{52} -1.09050e6 q^{53} -2.22889e6 q^{54} -451208. q^{56} -905687. q^{57} +2.31433e6 q^{58} -14776.3 q^{59} -579512. q^{61} -5.71084e6 q^{62} -8894.19 q^{63} +4.72516e6 q^{64} -6.12091e6 q^{66} +21392.3 q^{67} -1.13669e7 q^{68} +4.11349e6 q^{69} +4.15929e6 q^{71} +367629. q^{72} -5.15400e6 q^{73} -292564. q^{74} +6.51578e6 q^{76} -651924. q^{77} -7.73279e6 q^{78} +8.11379e6 q^{79} -4.58955e6 q^{81} -6.31353e6 q^{82} +2.17337e6 q^{83} -1.57991e6 q^{84} -1.72108e7 q^{86} +4.95880e6 q^{87} +2.69464e7 q^{88} +10155.1 q^{89} -823601. q^{91} -2.95937e7 q^{92} -1.22363e7 q^{93} -2.10829e6 q^{94} +2.38883e7 q^{96} -1.29273e6 q^{97} +1.73878e7 q^{98} +531166. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 23 q^{2} - 121 q^{3} + 1061 q^{4} - 431 q^{6} - 843 q^{7} - 4980 q^{8} + 14799 q^{9}+O(q^{10}) \) 22 * q - 23 * q^2 - 121 * q^3 + 1061 * q^4 - 431 * q^6 - 843 * q^7 - 4980 * q^8 + 14799 * q^9	\( 22 q - 23 q^{2} - 121 q^{3} + 1061 q^{4} - 431 q^{6} - 843 q^{7} - 4980 q^{8} + 14799 q^{9} - 781 q^{11} - 12608 q^{12} - 14686 q^{13} + 20762 q^{14} + 16117 q^{16} - 45648 q^{17} - 47171 q^{18} + 6185 q^{19} + 14149 q^{21} + 71124 q^{22} - 126921 q^{23} + 271570 q^{24} + 304129 q^{26} - 546520 q^{27} - 2019 q^{28} + 59330 q^{29} + 394804 q^{31} + 74397 q^{32} - 49067 q^{33} - 286938 q^{34} - 287278 q^{36} + 792122 q^{37} - 1338860 q^{38} + 635223 q^{39} - 160466 q^{41} - 3420191 q^{42} - 1527256 q^{43} - 1154853 q^{44} + 2653604 q^{46} - 1300863 q^{47} - 1885241 q^{48} + 1652981 q^{49} + 3408539 q^{51} - 1423303 q^{52} - 755656 q^{53} + 3117755 q^{54} - 2132625 q^{56} - 3026890 q^{57} + 5941470 q^{58} - 1548370 q^{59} - 6029951 q^{61} - 79936 q^{62} + 6962459 q^{63} - 5858224 q^{64} - 5380407 q^{66} - 7608838 q^{67} - 10737124 q^{68} + 13519553 q^{69} + 9483549 q^{71} - 6806340 q^{72} - 13548801 q^{73} - 15016023 q^{74} + 19635315 q^{76} + 2145019 q^{77} - 17222402 q^{78} + 10769160 q^{79} + 2757382 q^{81} - 12087571 q^{82} + 9632744 q^{83} + 19168542 q^{84} - 8511651 q^{86} - 298330 q^{87} + 21641425 q^{88} - 10850545 q^{89} - 6648131 q^{91} - 52978503 q^{92} - 51294822 q^{93} + 5863777 q^{94} + 2611654 q^{96} - 11579993 q^{97} - 7468074 q^{98} - 13552997 q^{99}+O(q^{100}) \) 22 * q - 23 * q^2 - 121 * q^3 + 1061 * q^4 - 431 * q^6 - 843 * q^7 - 4980 * q^8 + 14799 * q^9 - 781 * q^11 - 12608 * q^12 - 14686 * q^13 + 20762 * q^14 + 16117 * q^16 - 45648 * q^17 - 47171 * q^18 + 6185 * q^19 + 14149 * q^21 + 71124 * q^22 - 126921 * q^23 + 271570 * q^24 + 304129 * q^26 - 546520 * q^27 - 2019 * q^28 + 59330 * q^29 + 394804 * q^31 + 74397 * q^32 - 49067 * q^33 - 286938 * q^34 - 287278 * q^36 + 792122 * q^37 - 1338860 * q^38 + 635223 * q^39 - 160466 * q^41 - 3420191 * q^42 - 1527256 * q^43 - 1154853 * q^44 + 2653604 * q^46 - 1300863 * q^47 - 1885241 * q^48 + 1652981 * q^49 + 3408539 * q^51 - 1423303 * q^52 - 755656 * q^53 + 3117755 * q^54 - 2132625 * q^56 - 3026890 * q^57 + 5941470 * q^58 - 1548370 * q^59 - 6029951 * q^61 - 79936 * q^62 + 6962459 * q^63 - 5858224 * q^64 - 5380407 * q^66 - 7608838 * q^67 - 10737124 * q^68 + 13519553 * q^69 + 9483549 * q^71 - 6806340 * q^72 - 13548801 * q^73 - 15016023 * q^74 + 19635315 * q^76 + 2145019 * q^77 - 17222402 * q^78 + 10769160 * q^79 + 2757382 * q^81 - 12087571 * q^82 + 9632744 * q^83 + 19168542 * q^84 - 8511651 * q^86 - 298330 * q^87 + 21641425 * q^88 - 10850545 * q^89 - 6648131 * q^91 - 52978503 * q^92 - 51294822 * q^93 + 5863777 * q^94 + 2611654 * q^96 - 11579993 * q^97 - 7468074 * q^98 - 13552997 * q^99

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