LMFDB - Embedded newform 425.4.c.a.424.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [425,4,Mod(424,425)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("425.424"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	425.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-16,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$25.0758117524$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			424.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	425.424
Dual form			425.4.c.a.424.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -8.00000 q^{3} +7.00000 q^{4} -8.00000i q^{6} -14.0000 q^{7} +15.0000i q^{8} +37.0000 q^{9} -20.0000i q^{11} -56.0000 q^{12} +58.0000i q^{13} -14.0000i q^{14} +41.0000 q^{16} +(68.0000 - 17.0000i) q^{17} +37.0000i q^{18} -80.0000 q^{19} +112.000 q^{21} +20.0000 q^{22} -118.000 q^{23} -120.000i q^{24} -58.0000 q^{26} -80.0000 q^{27} -98.0000 q^{28} +126.000i q^{29} -70.0000i q^{31} +161.000i q^{32} +160.000i q^{33} +(17.0000 + 68.0000i) q^{34} +259.000 q^{36} -134.000 q^{37} -80.0000i q^{38} -464.000i q^{39} +100.000i q^{41} +112.000i q^{42} -272.000i q^{43} -140.000i q^{44} -118.000i q^{46} -464.000i q^{47} -328.000 q^{48} -147.000 q^{49} +(-544.000 + 136.000i) q^{51} +406.000i q^{52} -642.000i q^{53} -80.0000i q^{54} -210.000i q^{56} +640.000 q^{57} -126.000 q^{58} +180.000 q^{59} -110.000i q^{61} +70.0000 q^{62} -518.000 q^{63} +167.000 q^{64} -160.000 q^{66} -924.000i q^{67} +(476.000 - 119.000i) q^{68} +944.000 q^{69} -90.0000i q^{71} +555.000i q^{72} -828.000 q^{73} -134.000i q^{74} -560.000 q^{76} +280.000i q^{77} +464.000 q^{78} -1334.00i q^{79} -359.000 q^{81} -100.000 q^{82} -552.000i q^{83} +784.000 q^{84} +272.000 q^{86} -1008.00i q^{87} +300.000 q^{88} -1490.00 q^{89} -812.000i q^{91} -826.000 q^{92} +560.000i q^{93} +464.000 q^{94} -1288.00i q^{96} +1376.00 q^{97} -147.000i q^{98} -740.000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 16 q^{3} + 14 q^{4} - 28 q^{7} + 74 q^{9} - 112 q^{12} + 82 q^{16} + 136 q^{17} - 160 q^{19} + 224 q^{21} + 40 q^{22} - 236 q^{23} - 116 q^{26} - 160 q^{27} - 196 q^{28} + 34 q^{34} + 518 q^{36} - 268 q^{37}+ \cdots + 2752 q^{97}+O(q^{100}) $ 2 * q - 16 * q^3 + 14 * q^4 - 28 * q^7 + 74 * q^9 - 112 * q^12 + 82 * q^16 + 136 * q^17 - 160 * q^19 + 224 * q^21 + 40 * q^22 - 236 * q^23 - 116 * q^26 - 160 * q^27 - 196 * q^28 + 34 * q^34 + 518 * q^36 - 268 * q^37 - 656 * q^48 - 294 * q^49 - 1088 * q^51 + 1280 * q^57 - 252 * q^58 + 360 * q^59 + 140 * q^62 - 1036 * q^63 + 334 * q^64 - 320 * q^66 + 952 * q^68 + 1888 * q^69 - 1656 * q^73 - 1120 * q^76 + 928 * q^78 - 718 * q^81 - 200 * q^82 + 1568 * q^84 + 544 * q^86 + 600 * q^88 - 2980 * q^89 - 1652 * q^92 + 928 * q^94 + 2752 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$.

$n$	$52$	$326$
$\chi(n)$	$-1$	$-1$

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 $

$2$			1.00000i			0.353553i	0.984251	+	0.176777i	$0.0565670\pi$
$2$			1.00000i			0.353553i	−0.984251	+	0.176777i	$0.943433\pi$
$3$	−8.00000			−1.53960			−0.769800	−	0.638285i	$-0.779644\pi$
$3$	−8.00000			−1.53960			−0.769800	+	0.638285i	$0.779644\pi$
$4$	7.00000			0.875000
$5$		0			0
$5$		0			0
$6$		−	8.00000i		−	0.544331i
$7$	−14.0000			−0.755929			−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−14.0000			−0.755929			−0.377964	+	0.925820i	$0.623376\pi$
$8$			15.0000i			0.662913i
$9$	37.0000			1.37037
$10$		0			0
$11$		−	20.0000i		−	0.548202i	−0.961701	−	0.274101i	$-0.911620\pi$
$11$		−	20.0000i		−	0.548202i	0.961701	−	0.274101i	$-0.0883803\pi$
$12$	−56.0000			−1.34715
$13$			58.0000i			1.23741i	0.785624	+	0.618704i	$0.212342\pi$
$13$			58.0000i			1.23741i	−0.785624	+	0.618704i	$0.787658\pi$
$14$		−	14.0000i		−	0.267261i
$15$		0			0
$16$	41.0000			0.640625
$17$	68.0000	−	17.0000i	0.970143	−	0.242536i
$17$	68.0000	−	17.0000i	0.970143	−	0.242536i
$18$			37.0000i			0.484499i
$19$	−80.0000			−0.965961			−0.482980	−	0.875631i	$-0.660446\pi$
$19$	−80.0000			−0.965961			−0.482980	+	0.875631i	$0.660446\pi$
$20$		0			0
$21$	112.000			1.16383
$22$	20.0000			0.193819
$23$	−118.000			−1.06977			−0.534885	−	0.844925i	$-0.679645\pi$
$23$	−118.000			−1.06977			−0.534885	+	0.844925i	$0.679645\pi$
$24$		−	120.000i		−	1.02062i
$25$		0			0
$26$	−58.0000			−0.437490
$27$	−80.0000			−0.570222
$28$	−98.0000			−0.661438
$29$			126.000i			0.806814i	0.915021	+	0.403407i	$0.132174\pi$
$29$			126.000i			0.806814i	−0.915021	+	0.403407i	$0.867826\pi$
$30$		0			0
$31$		−	70.0000i		−	0.405560i	−0.979224	−	0.202780i	$-0.935002\pi$
$31$		−	70.0000i		−	0.405560i	0.979224	−	0.202780i	$-0.0649977\pi$
$32$			161.000i			0.889408i
$33$			160.000i			0.844013i
$34$	17.0000	+	68.0000i	0.0857493	+	0.342997i
$35$		0			0
$36$	259.000			1.19907
$37$	−134.000			−0.595391			−0.297695	−	0.954661i	$-0.596218\pi$
$37$	−134.000			−0.595391			−0.297695	+	0.954661i	$0.596218\pi$
$38$		−	80.0000i		−	0.341519i
$39$		−	464.000i		−	1.90511i
$40$		0			0
$41$			100.000i			0.380912i	0.981696	+	0.190456i	$0.0609966\pi$
$41$			100.000i			0.380912i	−0.981696	+	0.190456i	$0.939003\pi$
$42$			112.000i			0.411476i
$43$		−	272.000i		−	0.964642i	−0.875995	−	0.482321i	$-0.839794\pi$
$43$		−	272.000i		−	0.964642i	0.875995	−	0.482321i	$-0.160206\pi$
$44$		−	140.000i		−	0.479677i
$45$		0			0
$46$		−	118.000i		−	0.378221i
$47$		−	464.000i		−	1.44003i	−0.693959	−	0.720014i	$-0.744135\pi$
$47$		−	464.000i		−	1.44003i	0.693959	−	0.720014i	$-0.255865\pi$
$48$	−328.000			−0.986307
$49$	−147.000			−0.428571
$50$		0			0
$51$	−544.000	+	136.000i	−1.49363	+	0.373408i
$52$			406.000i			1.08273i
$53$		−	642.000i		−	1.66388i	−0.554868	−	0.831939i	$-0.687231\pi$
$53$		−	642.000i		−	1.66388i	0.554868	−	0.831939i	$-0.312769\pi$
$54$		−	80.0000i		−	0.201604i
$55$		0			0
$56$		−	210.000i		−	0.501115i
$57$	640.000			1.48719
$58$	−126.000			−0.285252
$59$	180.000			0.397187			0.198593	−	0.980082i	$-0.436363\pi$
$59$	180.000			0.397187			0.198593	+	0.980082i	$0.436363\pi$
$60$		0			0
$61$		−	110.000i		−	0.230886i	−0.993314	−	0.115443i	$-0.963171\pi$
$61$		−	110.000i		−	0.230886i	0.993314	−	0.115443i	$-0.0368288\pi$
$62$	70.0000			0.143387
$63$	−518.000			−1.03590
$64$	167.000			0.326172
$65$		0			0
$66$	−160.000			−0.298404
$67$		−	924.000i		−	1.68484i	−0.538818	−	0.842422i	$-0.681129\pi$
$67$		−	924.000i		−	1.68484i	0.538818	−	0.842422i	$-0.318871\pi$
$68$	476.000	−	119.000i	0.848875	−	0.212219i
$69$	944.000			1.64702
$70$		0			0
$71$		−	90.0000i		−	0.150437i	−0.997167	−	0.0752186i	$-0.976035\pi$
$71$		−	90.0000i		−	0.150437i	0.997167	−	0.0752186i	$-0.0239654\pi$
$72$			555.000i			0.908436i
$73$	−828.000			−1.32754			−0.663768	−	0.747939i	$-0.731044\pi$
$73$	−828.000			−1.32754			−0.663768	+	0.747939i	$0.731044\pi$
$74$		−	134.000i		−	0.210502i
$75$		0			0
$76$	−560.000			−0.845216
$77$			280.000i			0.414402i
$78$	464.000			0.673560
$79$		−	1334.00i		−	1.89983i	−0.312505	−	0.949916i	$-0.601168\pi$
$79$		−	1334.00i		−	1.89983i	0.312505	−	0.949916i	$-0.398832\pi$
$80$		0			0
$81$	−359.000			−0.492455
$82$	−100.000			−0.134673
$83$		−	552.000i		−	0.729998i	−0.931008	−	0.364999i	$-0.881069\pi$
$83$		−	552.000i		−	0.729998i	0.931008	−	0.364999i	$-0.118931\pi$
$84$	784.000			1.01835
$85$		0			0
$86$	272.000			0.341052
$87$		−	1008.00i		−	1.24217i
$88$	300.000			0.363410
$89$	−1490.00			−1.77460			−0.887302	−	0.461190i	$-0.847423\pi$
$89$	−1490.00			−1.77460			−0.887302	+	0.461190i	$0.847423\pi$
$90$		0			0
$91$		−	812.000i		−	0.935393i
$92$	−826.000			−0.936048
$93$			560.000i			0.624401i
$94$	464.000			0.509127
$95$		0			0
$96$		−	1288.00i		−	1.36933i
$97$	1376.00			1.44033			0.720163	−	0.693805i	$-0.244067\pi$
$97$	1376.00			1.44033			0.720163	+	0.693805i	$0.244067\pi$
$98$		−	147.000i		−	0.151523i
$99$		−	740.000i		−	0.751240i
$100$		0			0
$101$	642.000			0.632489			0.316244	−	0.948678i	$-0.397578\pi$
$101$	642.000			0.632489			0.316244	+	0.948678i	$0.397578\pi$
$102$	−136.000	−	544.000i	−0.132020	−	0.528079i
$103$			1248.00i			1.19387i	0.802288	+	0.596937i	$0.203616\pi$
$103$			1248.00i			1.19387i	−0.802288	+	0.596937i	$0.796384\pi$
$104$	−870.000			−0.820293
$105$		0			0
$106$	642.000			0.588269
$107$	−564.000			−0.509570			−0.254785	−	0.966998i	$-0.582005\pi$
$107$	−564.000			−0.509570			−0.254785	+	0.966998i	$0.582005\pi$
$108$	−560.000			−0.498945
$109$		−	1214.00i		−	1.06679i	−0.845866	−	0.533395i	$-0.820916\pi$
$109$		−	1214.00i		−	1.06679i	0.845866	−	0.533395i	$-0.179084\pi$
$110$		0			0
$111$	1072.00			0.916664
$112$	−574.000			−0.484267
$113$	−1388.00			−1.15550			−0.577752	−	0.816212i	$-0.696070\pi$
$113$	−1388.00			−1.15550			−0.577752	+	0.816212i	$0.696070\pi$
$114$			640.000i			0.525803i
$115$		0			0
$116$			882.000i			0.705962i
$117$			2146.00i			1.69571i
$118$			180.000i			0.140427i
$119$	−952.000	+	238.000i	−0.733359	+	0.183340i
$120$		0			0
$121$	931.000			0.699474
$122$	110.000			0.0816306
$123$		−	800.000i		−	0.586452i
$124$		−	490.000i		−	0.354865i
$125$		0			0
$126$		−	518.000i		−	0.366247i
$127$			1136.00i			0.793730i	0.917877	+	0.396865i	$0.129902\pi$
$127$			1136.00i			0.793730i	−0.917877	+	0.396865i	$0.870098\pi$
$128$			1455.00i			1.00473i
$129$			2176.00i			1.48516i
$130$		0			0
$131$			2620.00i			1.74741i	0.486458	+	0.873704i	$0.338289\pi$
$131$			2620.00i			1.74741i	−0.486458	+	0.873704i	$0.661711\pi$
$132$			1120.00i			0.738511i
$133$	1120.00			0.730198
$134$	924.000			0.595682
$135$		0			0
$136$	255.000	+	1020.00i	0.160780	+	0.643120i
$137$			1886.00i			1.17614i	0.808808	+	0.588072i	$0.200113\pi$
$137$			1886.00i			1.17614i	−0.808808	+	0.588072i	$0.799887\pi$
$138$			944.000i			0.582309i
$139$		−	4.00000i		−	0.00244083i	−0.999999	−	0.00122042i	$-0.999612\pi$
$139$		−	4.00000i		−	0.00244083i	0.999999	−	0.00122042i	$-0.000388470\pi$
$140$		0			0
$141$			3712.00i			2.21707i
$142$	90.0000			0.0531876
$143$	1160.00			0.678350
$144$	1517.00			0.877894
$145$		0			0
$146$		−	828.000i		−	0.469355i
$147$	1176.00			0.659829
$148$	−938.000			−0.520967
$149$	−870.000			−0.478343			−0.239172	−	0.970977i	$-0.576876\pi$
$149$	−870.000			−0.478343			−0.239172	+	0.970977i	$0.576876\pi$
$150$		0			0
$151$	−1368.00			−0.737260			−0.368630	−	0.929576i	$-0.620173\pi$
$151$	−1368.00			−0.737260			−0.368630	+	0.929576i	$0.620173\pi$
$152$		−	1200.00i		−	0.640348i
$153$	2516.00	−	629.000i	1.32945	−	0.332364i
$154$	−280.000			−0.146513
$155$		0			0
$156$		−	3248.00i		−	1.66698i
$157$		−	3374.00i		−	1.71512i	−0.514380	−	0.857562i	$-0.671978\pi$
$157$		−	3374.00i		−	1.71512i	0.514380	−	0.857562i	$-0.328022\pi$
$158$	1334.00			0.671692
$159$			5136.00i			2.56171i
$160$		0			0
$161$	1652.00			0.808670
$162$		−	359.000i		−	0.174109i
$163$	−1048.00			−0.503593			−0.251797	−	0.967780i	$-0.581021\pi$
$163$	−1048.00			−0.503593			−0.251797	+	0.967780i	$0.581021\pi$
$164$			700.000i			0.333298i
$165$		0			0
$166$	552.000			0.258093
$167$	1266.00			0.586623			0.293311	−	0.956017i	$-0.405243\pi$
$167$	1266.00			0.586623			0.293311	+	0.956017i	$0.405243\pi$
$168$			1680.00i			0.771517i
$169$	−1167.00			−0.531179
$170$		0			0
$171$	−2960.00			−1.32372
$172$		−	1904.00i		−	0.844062i
$173$	362.000			0.159089			0.0795444	−	0.996831i	$-0.474653\pi$
$173$	362.000			0.159089			0.0795444	+	0.996831i	$0.474653\pi$
$174$	1008.00			0.439174
$175$		0			0
$176$		−	820.000i		−	0.351192i
$177$	−1440.00			−0.611509
$178$		−	1490.00i		−	0.627417i
$179$	−1720.00			−0.718206			−0.359103	−	0.933298i	$-0.616917\pi$
$179$	−1720.00			−0.718206			−0.359103	+	0.933298i	$0.616917\pi$
$180$		0			0
$181$		−	1910.00i		−	0.784360i	−0.919888	−	0.392180i	$-0.871721\pi$
$181$		−	1910.00i		−	0.784360i	0.919888	−	0.392180i	$-0.128279\pi$
$182$	812.000			0.330711
$183$			880.000i			0.355473i
$184$		−	1770.00i		−	0.709164i
$185$		0			0
$186$	−560.000			−0.220759
$187$	−340.000	−	1360.00i	−0.132959	−	0.531834i
$188$		−	3248.00i		−	1.26003i
$189$	1120.00			0.431048
$190$		0			0
$191$	−4208.00			−1.59414			−0.797069	−	0.603889i	$-0.793617\pi$
$191$	−4208.00			−1.59414			−0.797069	+	0.603889i	$0.793617\pi$
$192$	−1336.00			−0.502174
$193$	2112.00			0.787695			0.393847	−	0.919176i	$-0.371144\pi$
$193$	2112.00			0.787695			0.393847	+	0.919176i	$0.371144\pi$
$194$			1376.00i			0.509232i
$195$		0			0
$196$	−1029.00			−0.375000
$197$	−2074.00			−0.750083			−0.375042	−	0.927008i	$-0.622372\pi$
$197$	−2074.00			−0.750083			−0.375042	+	0.927008i	$0.622372\pi$
$198$	740.000			0.265604
$199$			4466.00i			1.59089i	0.606028	+	0.795443i	$0.292762\pi$
$199$			4466.00i			1.59089i	−0.606028	+	0.795443i	$0.707238\pi$
$200$		0			0
$201$			7392.00i			2.59399i
$202$			642.000i			0.223619i
$203$		−	1764.00i		−	0.609894i
$204$	−3808.00	+	952.000i	−1.30693	+	0.326732i
$205$		0			0
$206$	−1248.00			−0.422098
$207$	−4366.00			−1.46598
$208$			2378.00i			0.792715i
$209$			1600.00i			0.529542i
$210$		0			0
$211$		−	2540.00i		−	0.828724i	−0.910112	−	0.414362i	$-0.864005\pi$
$211$		−	2540.00i		−	0.828724i	0.910112	−	0.414362i	$-0.135995\pi$
$212$		−	4494.00i		−	1.45589i
$213$			720.000i			0.231613i
$214$		−	564.000i		−	0.180160i
$215$		0			0
$216$		−	1200.00i		−	0.378008i
$217$			980.000i			0.306575i
$218$	1214.00			0.377167
$219$	6624.00			2.04387
$220$		0			0
$221$	986.000	+	3944.00i	0.300116	+	1.20046i
$222$			1072.00i			0.324090i
$223$		−	3832.00i		−	1.15072i	−0.817902	−	0.575358i	$-0.804863\pi$
$223$		−	3832.00i		−	1.15072i	0.817902	−	0.575358i	$-0.195137\pi$
$224$		−	2254.00i		−	0.672329i
$225$		0			0
$226$		−	1388.00i		−	0.408533i
$227$	6096.00			1.78240			0.891202	−	0.453607i	$-0.149863\pi$
$227$	6096.00			1.78240			0.891202	+	0.453607i	$0.149863\pi$
$228$	4480.00			1.30129
$229$	−2650.00			−0.764703			−0.382351	−	0.924017i	$-0.624886\pi$
$229$	−2650.00			−0.764703			−0.382351	+	0.924017i	$0.624886\pi$
$230$		0			0
$231$		−	2240.00i		−	0.638014i
$232$	−1890.00			−0.534847
$233$	−4228.00			−1.18878			−0.594389	−	0.804177i	$-0.702606\pi$
$233$	−4228.00			−1.18878			−0.594389	+	0.804177i	$0.702606\pi$
$234$	−2146.00			−0.599523
$235$		0			0
$236$	1260.00			0.347538
$237$			10672.0i			2.92498i
$238$	−238.000	−	952.000i	−0.0648204	−	0.259281i
$239$	80.0000			0.0216518			0.0108259	−	0.999941i	$-0.496554\pi$
$239$	80.0000			0.0216518			0.0108259	+	0.999941i	$0.496554\pi$
$240$		0			0
$241$		−	5980.00i		−	1.59836i	−0.601089	−	0.799182i	$-0.705266\pi$
$241$		−	5980.00i		−	1.59836i	0.601089	−	0.799182i	$-0.294734\pi$
$242$			931.000i			0.247301i
$243$	5032.00			1.32841
$244$		−	770.000i		−	0.202025i
$245$		0			0
$246$	800.000			0.207342
$247$		−	4640.00i		−	1.19529i
$248$	1050.00			0.268851
$249$			4416.00i			1.12391i
$250$		0			0
$251$	312.000			0.0784592			0.0392296	−	0.999230i	$-0.487510\pi$
$251$	312.000			0.0784592			0.0392296	+	0.999230i	$0.487510\pi$
$252$	−3626.00			−0.906415
$253$			2360.00i			0.586450i
$254$	−1136.00			−0.280626
$255$		0			0
$256$	−119.000			−0.0290527
$257$		−	434.000i		−	0.105339i	−0.998612	−	0.0526696i	$-0.983227\pi$
$257$		−	434.000i		−	0.105339i	0.998612	−	0.0526696i	$-0.0167730\pi$
$258$	−2176.00			−0.525085
$259$	1876.00			0.450073
$260$		0			0
$261$			4662.00i			1.10563i
$262$	−2620.00			−0.617802
$263$		−	1752.00i		−	0.410772i	−0.978681	−	0.205386i	$-0.934155\pi$
$263$		−	1752.00i		−	0.410772i	0.978681	−	0.205386i	$-0.0658449\pi$
$264$	−2400.00			−0.559507
$265$		0			0
$266$			1120.00i			0.258164i
$267$	11920.0			2.73218
$268$		−	6468.00i		−	1.47424i
$269$			1206.00i			0.273350i	0.990616	+	0.136675i	$0.0436416\pi$
$269$			1206.00i			0.273350i	−0.990616	+	0.136675i	$0.956358\pi$
$270$		0			0
$271$	5112.00			1.14587			0.572937	−	0.819599i	$-0.305804\pi$
$271$	5112.00			1.14587			0.572937	+	0.819599i	$0.305804\pi$
$272$	2788.00	−	697.000i	0.621498	−	0.155374i
$273$			6496.00i			1.44013i
$274$	−1886.00			−0.415830
$275$		0			0
$276$	6608.00			1.44114
$277$	−3154.00			−0.684135			−0.342068	−	0.939675i	$-0.611127\pi$
$277$	−3154.00			−0.684135			−0.342068	+	0.939675i	$0.611127\pi$
$278$	4.00000			0.000862964
$279$		−	2590.00i		−	0.555768i
$280$		0			0
$281$	4702.00			0.998213			0.499106	−	0.866541i	$-0.333662\pi$
$281$	4702.00			0.998213			0.499106	+	0.866541i	$0.333662\pi$
$282$	−3712.00			−0.783852
$283$	−1388.00			−0.291548			−0.145774	−	0.989318i	$-0.546567\pi$
$283$	−1388.00			−0.291548			−0.145774	+	0.989318i	$0.546567\pi$
$284$		−	630.000i		−	0.131632i
$285$		0			0
$286$			1160.00i			0.239833i
$287$		−	1400.00i		−	0.287942i
$288$			5957.00i			1.21882i
$289$	4335.00	−	2312.00i	0.882353	−	0.470588i
$290$		0			0
$291$	−11008.0			−2.21753
$292$	−5796.00			−1.16159
$293$			7318.00i			1.45912i	0.683917	+	0.729560i	$0.260275\pi$
$293$			7318.00i			1.45912i	−0.683917	+	0.729560i	$0.739725\pi$
$294$			1176.00i			0.233285i
$295$		0			0
$296$		−	2010.00i		−	0.394692i
$297$			1600.00i			0.312597i
$298$		−	870.000i		−	0.169120i
$299$		−	6844.00i		−	1.32374i
$300$		0			0
$301$			3808.00i			0.729201i
$302$		−	1368.00i		−	0.260661i
$303$	−5136.00			−0.973780
$304$	−3280.00			−0.618819
$305$		0			0
$306$	629.000	+	2516.00i	0.117508	+	0.470033i
$307$			216.000i			0.0401556i	0.999798	+	0.0200778i	$0.00639139\pi$
$307$			216.000i			0.0401556i	−0.999798	+	0.0200778i	$0.993609\pi$
$308$			1960.00i			0.362602i
$309$		−	9984.00i		−	1.83809i
$310$		0			0
$311$			7170.00i			1.30731i	0.756793	+	0.653655i	$0.226765\pi$
$311$			7170.00i			1.30731i	−0.756793	+	0.653655i	$0.773235\pi$
$312$	6960.00			1.26292
$313$	1472.00			0.265822			0.132911	−	0.991128i	$-0.457568\pi$
$313$	1472.00			0.265822			0.132911	+	0.991128i	$0.457568\pi$
$314$	3374.00			0.606388
$315$		0			0
$316$		−	9338.00i		−	1.66235i
$317$	1526.00			0.270374			0.135187	−	0.990820i	$-0.456836\pi$
$317$	1526.00			0.270374			0.135187	+	0.990820i	$0.456836\pi$
$318$	−5136.00			−0.905700
$319$	2520.00			0.442298
$320$		0			0
$321$	4512.00			0.784534
$322$			1652.00i			0.285908i
$323$	−5440.00	+	1360.00i	−0.937120	+	0.234280i
$324$	−2513.00			−0.430898
$325$		0			0
$326$		−	1048.00i		−	0.178047i
$327$			9712.00i			1.64243i
$328$	−1500.00			−0.252511
$329$			6496.00i			1.08856i
$330$		0			0
$331$	−2708.00			−0.449683			−0.224842	−	0.974395i	$-0.572186\pi$
$331$	−2708.00			−0.449683			−0.224842	+	0.974395i	$0.572186\pi$
$332$		−	3864.00i		−	0.638749i
$333$	−4958.00			−0.815906
$334$			1266.00i			0.207403i
$335$		0			0
$336$	4592.00			0.745578
$337$	10516.0			1.69983			0.849915	−	0.526919i	$-0.176653\pi$
$337$	10516.0			1.69983			0.849915	+	0.526919i	$0.176653\pi$
$338$		−	1167.00i		−	0.187800i
$339$	11104.0			1.77902
$340$		0			0
$341$	−1400.00			−0.222329
$342$		−	2960.00i		−	0.468007i
$343$	6860.00			1.07990
$344$	4080.00			0.639473
$345$		0			0
$346$			362.000i			0.0562464i
$347$	−2704.00			−0.418324			−0.209162	−	0.977881i	$-0.567074\pi$
$347$	−2704.00			−0.418324			−0.209162	+	0.977881i	$0.567074\pi$
$348$		−	7056.00i		−	1.08690i
$349$	−9670.00			−1.48316			−0.741581	−	0.670864i	$-0.765923\pi$
$349$	−9670.00			−1.48316			−0.741581	+	0.670864i	$0.765923\pi$
$350$		0			0
$351$		−	4640.00i		−	0.705598i
$352$	3220.00			0.487576
$353$			7938.00i			1.19688i	0.801169	+	0.598438i	$0.204212\pi$
$353$			7938.00i			1.19688i	−0.801169	+	0.598438i	$0.795788\pi$
$354$		−	1440.00i		−	0.216201i
$355$		0			0
$356$	−10430.0			−1.55278
$357$	7616.00	−	1904.00i	1.12908	−	0.282270i
$358$		−	1720.00i		−	0.253924i
$359$	−2240.00			−0.329311			−0.164656	−	0.986351i	$-0.552651\pi$
$359$	−2240.00			−0.329311			−0.164656	+	0.986351i	$0.552651\pi$
$360$		0			0
$361$	−459.000			−0.0669194
$362$	1910.00			0.277313
$363$	−7448.00			−1.07691
$364$		−	5684.00i		−	0.818469i
$365$		0			0
$366$	−880.000			−0.125679
$367$	−6594.00			−0.937886			−0.468943	−	0.883229i	$-0.655365\pi$
$367$	−6594.00			−0.937886			−0.468943	+	0.883229i	$0.655365\pi$
$368$	−4838.00			−0.685321
$369$			3700.00i			0.521990i
$370$		0			0
$371$			8988.00i			1.25777i
$372$			3920.00i			0.546351i
$373$			4258.00i			0.591075i	0.955331	+	0.295537i	$0.0954987\pi$
$373$			4258.00i			0.591075i	−0.955331	+	0.295537i	$0.904501\pi$
$374$	1360.00	−	340.000i	0.188032	−	0.0470080i
$375$		0			0
$376$	6960.00			0.954613
$377$	−7308.00			−0.998358
$378$			1120.00i			0.152398i
$379$		−	8984.00i		−	1.21762i	−0.793317	−	0.608809i	$-0.791648\pi$
$379$		−	8984.00i		−	1.21762i	0.793317	−	0.608809i	$-0.208352\pi$
$380$		0			0
$381$		−	9088.00i		−	1.22203i
$382$		−	4208.00i		−	0.563613i
$383$		−	2912.00i		−	0.388502i	−0.980952	−	0.194251i	$-0.937772\pi$
$383$		−	2912.00i		−	0.388502i	0.980952	−	0.194251i	$-0.0622276\pi$
$384$		−	11640.0i		−	1.54688i
$385$		0			0
$386$			2112.00i			0.278492i
$387$		−	10064.0i		−	1.32192i
$388$	9632.00			1.26029
$389$	7950.00			1.03620			0.518099	−	0.855321i	$-0.326640\pi$
$389$	7950.00			1.03620			0.518099	+	0.855321i	$0.326640\pi$
$390$		0			0
$391$	−8024.00	+	2006.00i	−1.03783	+	0.259457i
$392$		−	2205.00i		−	0.284105i
$393$		−	20960.0i		−	2.69031i
$394$		−	2074.00i		−	0.265194i
$395$		0			0
$396$		−	5180.00i		−	0.657335i
$397$	−12634.0			−1.59718			−0.798592	−	0.601872i	$-0.794422\pi$
$397$	−12634.0			−1.59718			−0.798592	+	0.601872i	$0.794422\pi$
$398$	−4466.00			−0.562463
$399$	−8960.00			−1.12421
$400$		0			0
$401$		−	10740.0i		−	1.33748i	−0.743496	−	0.668741i	$-0.766834\pi$
$401$		−	10740.0i		−	1.33748i	0.743496	−	0.668741i	$-0.233166\pi$
$402$	−7392.00			−0.917113
$403$	4060.00			0.501844
$404$	4494.00			0.553428
$405$		0			0
$406$	1764.00			0.215630
$407$			2680.00i			0.326395i
$408$	−2040.00	−	8160.00i	−0.247537	−	0.990148i
$409$	−2950.00			−0.356646			−0.178323	−	0.983972i	$-0.557067\pi$
$409$	−2950.00			−0.356646			−0.178323	+	0.983972i	$0.557067\pi$
$410$		0			0
$411$		−	15088.0i		−	1.81079i
$412$			8736.00i			1.04464i
$413$	−2520.00			−0.300245
$414$		−	4366.00i		−	0.518302i
$415$		0			0
$416$	−9338.00			−1.10056
$417$			32.0000i			0.00375791i
$418$	−1600.00			−0.187221
$419$		−	5944.00i		−	0.693039i	−0.938043	−	0.346520i	$-0.887363\pi$
$419$		−	5944.00i		−	0.693039i	0.938043	−	0.346520i	$-0.112637\pi$
$420$		0			0
$421$	13682.0			1.58390			0.791948	−	0.610589i	$-0.209067\pi$
$421$	13682.0			1.58390			0.791948	+	0.610589i	$0.209067\pi$
$422$	2540.00			0.292998
$423$		−	17168.0i		−	1.97337i
$424$	9630.00			1.10301
$425$		0			0
$426$	−720.000			−0.0818876
$427$			1540.00i			0.174534i
$428$	−3948.00			−0.445873
$429$	−9280.00			−1.04439
$430$		0			0
$431$		−	410.000i		−	0.0458214i	−0.999738	−	0.0229107i	$-0.992707\pi$
$431$		−	410.000i		−	0.0458214i	0.999738	−	0.0229107i	$-0.00729333\pi$
$432$	−3280.00			−0.365299
$433$			2558.00i			0.283902i	0.989874	+	0.141951i	$0.0453376\pi$
$433$			2558.00i			0.283902i	−0.989874	+	0.141951i	$0.954662\pi$
$434$	−980.000			−0.108391
$435$		0			0
$436$		−	8498.00i		−	0.933441i
$437$	9440.00			1.03336
$438$			6624.00i			0.722619i
$439$			5946.00i			0.646440i	0.946324	+	0.323220i	$0.104765\pi$
$439$			5946.00i			0.646440i	−0.946324	+	0.323220i	$0.895235\pi$
$440$		0			0
$441$	−5439.00			−0.587302
$442$	−3944.00	+	986.000i	−0.424427	+	0.106107i
$443$		−	10852.0i		−	1.16387i	−0.813236	−	0.581935i	$-0.802296\pi$
$443$		−	10852.0i		−	1.16387i	0.813236	−	0.581935i	$-0.197704\pi$
$444$	7504.00			0.802081
$445$		0			0
$446$	3832.00			0.406840
$447$	6960.00			0.736458
$448$	−2338.00			−0.246563
$449$		−	3284.00i		−	0.345170i	−0.984995	−	0.172585i	$-0.944788\pi$
$449$		−	3284.00i		−	0.345170i	0.984995	−	0.172585i	$-0.0552120\pi$
$450$		0			0
$451$	2000.00			0.208817
$452$	−9716.00			−1.01107
$453$	10944.0			1.13509
$454$			6096.00i			0.630175i
$455$		0			0
$456$			9600.00i			0.985880i
$457$			4106.00i			0.420286i	0.977671	+	0.210143i	$0.0673929\pi$
$457$			4106.00i			0.420286i	−0.977671	+	0.210143i	$0.932607\pi$
$458$		−	2650.00i		−	0.270363i
$459$	−5440.00	+	1360.00i	−0.553197	+	0.138299i
$460$		0			0
$461$	8462.00			0.854912			0.427456	−	0.904036i	$-0.359410\pi$
$461$	8462.00			0.854912			0.427456	+	0.904036i	$0.359410\pi$
$462$	2240.00			0.225572
$463$		−	3992.00i		−	0.400700i	−0.979724	−	0.200350i	$-0.935792\pi$
$463$		−	3992.00i		−	0.400700i	0.979724	−	0.200350i	$-0.0642079\pi$
$464$			5166.00i			0.516865i
$465$		0			0
$466$		−	4228.00i		−	0.420297i
$467$		−	3164.00i		−	0.313517i	−0.987637	−	0.156759i	$-0.949896\pi$
$467$		−	3164.00i		−	0.313517i	0.987637	−	0.156759i	$-0.0501044\pi$
$468$			15022.0i			1.48374i
$469$			12936.0i			1.27362i
$470$		0			0
$471$			26992.0i			2.64061i
$472$			2700.00i			0.263300i
$473$	−5440.00			−0.528819
$474$	−10672.0			−1.03414
$475$		0			0
$476$	−6664.00	+	1666.00i	−0.641689	+	0.160422i
$477$		−	23754.0i		−	2.28013i
$478$			80.0000i			0.00765505i
$479$			4866.00i			0.464161i	0.972697	+	0.232081i	$0.0745533\pi$
$479$			4866.00i			0.464161i	−0.972697	+	0.232081i	$0.925447\pi$
$480$		0			0
$481$		−	7772.00i		−	0.736742i
$482$	5980.00			0.565107
$483$	−13216.0			−1.24503
$484$	6517.00			0.612040
$485$		0			0
$486$			5032.00i			0.469663i
$487$	2926.00			0.272258			0.136129	−	0.990691i	$-0.456534\pi$
$487$	2926.00			0.272258			0.136129	+	0.990691i	$0.456534\pi$
$488$	1650.00			0.153057
$489$	8384.00			0.775332
$490$		0			0
$491$	−15128.0			−1.39046			−0.695231	−	0.718786i	$-0.744698\pi$
$491$	−15128.0			−1.39046			−0.695231	+	0.718786i	$0.744698\pi$
$492$		−	5600.00i		−	0.513145i
$493$	2142.00	+	8568.00i	0.195681	+	0.782725i
$494$	4640.00			0.422598
$495$		0			0
$496$		−	2870.00i		−	0.259812i
$497$			1260.00i			0.113720i
$498$	−4416.00			−0.397361
$499$		−	18684.0i		−	1.67617i	−0.545537	−	0.838087i	$-0.683674\pi$
$499$		−	18684.0i		−	1.67617i	0.545537	−	0.838087i	$-0.316326\pi$
$500$		0			0
$501$	−10128.0			−0.903165
$502$			312.000i			0.0277395i
$503$	8562.00			0.758968			0.379484	−	0.925198i	$-0.376102\pi$
$503$	8562.00			0.758968			0.379484	+	0.925198i	$0.376102\pi$
$504$		−	7770.00i		−	0.686713i
$505$		0			0
$506$	−2360.00			−0.207341
$507$	9336.00			0.817803
$508$			7952.00i			0.694514i
$509$	−16950.0			−1.47602			−0.738011	−	0.674788i	$-0.764235\pi$
$509$	−16950.0			−1.47602			−0.738011	+	0.674788i	$0.764235\pi$
$510$		0			0
$511$	11592.0			1.00352
$512$			11521.0i			0.994455i
$513$	6400.00			0.550813
$514$	434.000			0.0372430
$515$		0			0
$516$			15232.0i			1.29952i
$517$	−9280.00			−0.789427
$518$			1876.00i			0.159125i
$519$	−2896.00			−0.244933
$520$		0			0
$521$			17820.0i			1.49848i	0.662298	+	0.749240i	$0.269581\pi$
$521$			17820.0i			1.49848i	−0.662298	+	0.749240i	$0.730419\pi$
$522$	−4662.00			−0.390901
$523$			13548.0i			1.13272i	0.824158	+	0.566360i	$0.191649\pi$
$523$			13548.0i			1.13272i	−0.824158	+	0.566360i	$0.808351\pi$
$524$			18340.0i			1.52898i
$525$		0			0
$526$	1752.00			0.145230
$527$	−1190.00	−	4760.00i	−0.0983628	−	0.393451i
$528$			6560.00i			0.540696i
$529$	1757.00			0.144407
$530$		0			0
$531$	6660.00			0.544293
$532$	7840.00			0.638923
$533$	−5800.00			−0.471343
$534$			11920.0i			0.965972i
$535$		0			0
$536$	13860.0			1.11690
$537$	13760.0			1.10575
$538$	−1206.00			−0.0966438
$539$			2940.00i			0.234944i
$540$		0			0
$541$		−	7970.00i		−	0.633377i	−0.948530	−	0.316689i	$-0.897429\pi$
$541$		−	7970.00i		−	0.633377i	0.948530	−	0.316689i	$-0.102571\pi$
$542$			5112.00i			0.405128i
$543$			15280.0i			1.20760i
$544$	2737.00	+	10948.0i	0.215713	+	0.862852i
$545$		0			0
$546$	−6496.00			−0.509163
$547$	−22044.0			−1.72310			−0.861548	−	0.507676i	$-0.830505\pi$
$547$	−22044.0			−1.72310			−0.861548	+	0.507676i	$0.830505\pi$
$548$			13202.0i			1.02913i
$549$		−	4070.00i		−	0.316400i
$550$		0			0
$551$		−	10080.0i		−	0.779351i
$552$			14160.0i			1.09183i
$553$			18676.0i			1.43614i
$554$		−	3154.00i		−	0.241878i
$555$		0			0
$556$		−	28.0000i		−	0.00213573i
$557$			20166.0i			1.53404i	0.641623	+	0.767021i	$0.278262\pi$
$557$			20166.0i			1.53404i	−0.641623	+	0.767021i	$0.721738\pi$
$558$	2590.00			0.196494
$559$	15776.0			1.19366
$560$		0			0
$561$	2720.00	+	10880.0i	0.204703	+	0.818813i
$562$			4702.00i			0.352922i
$563$		−	1712.00i		−	0.128157i	−0.997945	−	0.0640783i	$-0.979589\pi$
$563$		−	1712.00i		−	0.128157i	0.997945	−	0.0640783i	$-0.0204107\pi$
$564$			25984.0i			1.93994i
$565$		0			0
$566$		−	1388.00i		−	0.103078i
$567$	5026.00			0.372261
$568$	1350.00			0.0997267
$569$	7270.00			0.535631			0.267816	−	0.963470i	$-0.413698\pi$
$569$	7270.00			0.535631			0.267816	+	0.963470i	$0.413698\pi$
$570$		0			0
$571$		−	26200.0i		−	1.92020i	−0.279652	−	0.960101i	$-0.590219\pi$
$571$		−	26200.0i		−	1.92020i	0.279652	−	0.960101i	$-0.409781\pi$
$572$	8120.00			0.593556
$573$	33664.0			2.45433
$574$	1400.00			0.101803
$575$		0			0
$576$	6179.00			0.446976
$577$		−	1794.00i		−	0.129437i	−0.997904	−	0.0647185i	$-0.979385\pi$
$577$		−	1794.00i		−	0.129437i	0.997904	−	0.0647185i	$-0.0206150\pi$
$578$	2312.00	+	4335.00i	0.166378	+	0.311959i
$579$	−16896.0			−1.21274
$580$		0			0
$581$			7728.00i			0.551827i
$582$		−	11008.0i		−	0.784014i
$583$	−12840.0			−0.912141
$584$		−	12420.0i		−	0.880040i
$585$		0			0
$586$	−7318.00			−0.515877
$587$			27876.0i			1.96008i	0.198806	+	0.980039i	$0.436294\pi$
$587$			27876.0i			1.96008i	−0.198806	+	0.980039i	$0.563706\pi$
$588$	8232.00			0.577350
$589$			5600.00i			0.391755i
$590$		0			0
$591$	16592.0			1.15483
$592$	−5494.00			−0.381422
$593$		−	9362.00i		−	0.648316i	−0.946003	−	0.324158i	$-0.894919\pi$
$593$		−	9362.00i		−	0.648316i	0.946003	−	0.324158i	$-0.105081\pi$
$594$	−1600.00			−0.110520
$595$		0			0
$596$	−6090.00			−0.418551
$597$		−	35728.0i		−	2.44933i
$598$	6844.00			0.468013
$599$	−13600.0			−0.927681			−0.463841	−	0.885919i	$-0.653529\pi$
$599$	−13600.0			−0.927681			−0.463841	+	0.885919i	$0.653529\pi$
$600$		0			0
$601$			5200.00i			0.352932i	0.984307	+	0.176466i	$0.0564666\pi$
$601$			5200.00i			0.352932i	−0.984307	+	0.176466i	$0.943533\pi$
$602$	−3808.00			−0.257811
$603$		−	34188.0i		−	2.30886i
$604$	−9576.00			−0.645103
$605$		0			0
$606$		−	5136.00i		−	0.344283i
$607$	−21854.0			−1.46133			−0.730664	−	0.682737i	$-0.760790\pi$
$607$	−21854.0			−1.46133			−0.730664	+	0.682737i	$0.760790\pi$
$608$		−	12880.0i		−	0.859133i
$609$			14112.0i			0.938994i
$610$		0			0
$611$	26912.0			1.78190
$612$	17612.0	−	4403.00i	1.16327	−	0.290818i
$613$			3558.00i			0.234431i	0.993106	+	0.117216i	$0.0373968\pi$
$613$			3558.00i			0.234431i	−0.993106	+	0.117216i	$0.962603\pi$
$614$	−216.000			−0.0141972
$615$		0			0
$616$	−4200.00			−0.274712
$617$	−804.000			−0.0524600			−0.0262300	−	0.999656i	$-0.508350\pi$
$617$	−804.000			−0.0524600			−0.0262300	+	0.999656i	$0.508350\pi$
$618$	9984.00			0.649863
$619$		−	4344.00i		−	0.282068i	−0.990005	−	0.141034i	$-0.954957\pi$
$619$		−	4344.00i		−	0.282068i	0.990005	−	0.141034i	$-0.0450427\pi$
$620$		0			0
$621$	9440.00			0.610007
$622$	−7170.00			−0.462204
$623$	20860.0			1.34147
$624$		−	19024.0i		−	1.22046i
$625$		0			0
$626$			1472.00i			0.0939824i
$627$		−	12800.0i		−	0.815284i
$628$		−	23618.0i		−	1.50073i
$629$	−9112.00	+	2278.00i	−0.577614	+	0.144404i
$630$		0			0
$631$	−24488.0			−1.54493			−0.772466	−	0.635056i	$-0.780977\pi$
$631$	−24488.0			−1.54493			−0.772466	+	0.635056i	$0.780977\pi$
$632$	20010.0			1.25942
$633$			20320.0i			1.27590i
$634$			1526.00i			0.0955918i
$635$		0			0
$636$			35952.0i			2.24149i
$637$		−	8526.00i		−	0.530318i
$638$			2520.00i			0.156376i
$639$		−	3330.00i		−	0.206155i
$640$		0			0
$641$			15500.0i			0.955091i	0.878607	+	0.477545i	$0.158473\pi$
$641$			15500.0i			0.955091i	−0.878607	+	0.477545i	$0.841527\pi$
$642$			4512.00i			0.277375i
$643$	−10928.0			−0.670231			−0.335115	−	0.942177i	$-0.608775\pi$
$643$	−10928.0			−0.670231			−0.335115	+	0.942177i	$0.608775\pi$
$644$	11564.0			0.707586
$645$		0			0
$646$	−1360.00	−	5440.00i	−0.0828305	−	0.331322i
$647$			12416.0i			0.754441i	0.926123	+	0.377221i	$0.123120\pi$
$647$			12416.0i			0.754441i	−0.926123	+	0.377221i	$0.876880\pi$
$648$		−	5385.00i		−	0.326455i
$649$		−	3600.00i		−	0.217739i
$650$		0			0
$651$		−	7840.00i		−	0.472003i
$652$	−7336.00			−0.440644
$653$	8782.00			0.526288			0.263144	−	0.964757i	$-0.415241\pi$
$653$	8782.00			0.526288			0.263144	+	0.964757i	$0.415241\pi$
$654$	−9712.00			−0.580687
$655$		0			0
$656$			4100.00i			0.244022i
$657$	−30636.0			−1.81921
$658$	−6496.00			−0.384864
$659$	−6020.00			−0.355851			−0.177926	−	0.984044i	$-0.556939\pi$
$659$	−6020.00			−0.355851			−0.177926	+	0.984044i	$0.556939\pi$
$660$		0			0
$661$	6342.00			0.373185			0.186592	−	0.982437i	$-0.440256\pi$
$661$	6342.00			0.373185			0.186592	+	0.982437i	$0.440256\pi$
$662$		−	2708.00i		−	0.158987i
$663$	−7888.00	−	31552.0i	−0.462058	−	1.84823i
$664$	8280.00			0.483925
$665$		0			0
$666$		−	4958.00i		−	0.288466i
$667$		−	14868.0i		−	0.863105i
$668$	8862.00			0.513295
$669$			30656.0i			1.77164i
$670$		0			0
$671$	−2200.00			−0.126572
$672$			18032.0i			1.03512i
$673$	18632.0			1.06718			0.533589	−	0.845744i	$-0.320843\pi$
$673$	18632.0			1.06718			0.533589	+	0.845744i	$0.320843\pi$
$674$			10516.0i			0.600981i
$675$		0			0
$676$	−8169.00			−0.464782
$677$	21186.0			1.20272			0.601362	−	0.798977i	$-0.294625\pi$
$677$	21186.0			1.20272			0.601362	+	0.798977i	$0.294625\pi$
$678$			11104.0i			0.628977i
$679$	−19264.0			−1.08878
$680$		0			0
$681$	−48768.0			−2.74419
$682$		−	1400.00i		−	0.0786052i
$683$	−15408.0			−0.863208			−0.431604	−	0.902063i	$-0.642052\pi$
$683$	−15408.0			−0.863208			−0.431604	+	0.902063i	$0.642052\pi$
$684$	−20720.0			−1.15826
$685$		0			0
$686$			6860.00i			0.381802i
$687$	21200.0			1.17734
$688$		−	11152.0i		−	0.617974i
$689$	37236.0			2.05889
$690$		0			0
$691$		−	3940.00i		−	0.216910i	−0.994101	−	0.108455i	$-0.965410\pi$
$691$		−	3940.00i		−	0.216910i	0.994101	−	0.108455i	$-0.0345903\pi$
$692$	2534.00			0.139203
$693$			10360.0i			0.567884i
$694$		−	2704.00i		−	0.147900i
$695$		0			0
$696$	15120.0			0.823451
$697$	1700.00	+	6800.00i	0.0923846	+	0.369539i
$698$		−	9670.00i		−	0.524377i
$699$	33824.0			1.83024
$700$		0			0
$701$	−14798.0			−0.797308			−0.398654	−	0.917101i	$-0.630523\pi$
$701$	−14798.0			−0.797308			−0.398654	+	0.917101i	$0.630523\pi$
$702$	4640.00			0.249467
$703$	10720.0			0.575124
$704$		−	3340.00i		−	0.178808i
$705$		0			0
$706$	−7938.00			−0.423159
$707$	−8988.00			−0.478117
$708$	−10080.0			−0.535070
$709$			14926.0i			0.790631i	0.918545	+	0.395316i	$0.129365\pi$
$709$			14926.0i			0.790631i	−0.918545	+	0.395316i	$0.870635\pi$
$710$		0			0
$711$		−	49358.0i		−	2.60347i
$712$		−	22350.0i		−	1.17641i
$713$			8260.00i			0.433856i
$714$	1904.00	+	7616.00i	0.0997975	+	0.399190i
$715$		0			0
$716$	−12040.0			−0.628430
$717$	−640.000			−0.0333351
$718$		−	2240.00i		−	0.116429i
$719$			7146.00i			0.370655i	0.982677	+	0.185327i	$0.0593346\pi$
$719$			7146.00i			0.370655i	−0.982677	+	0.185327i	$0.940665\pi$
$720$		0			0
$721$		−	17472.0i		−	0.902484i
$722$		−	459.000i		−	0.0236596i
$723$			47840.0i			2.46084i
$724$		−	13370.0i		−	0.686315i
$725$		0			0
$726$		−	7448.00i		−	0.380745i
$727$			19136.0i			0.976224i	0.872781	+	0.488112i	$0.162314\pi$
$727$			19136.0i			0.976224i	−0.872781	+	0.488112i	$0.837686\pi$
$728$	12180.0			0.620084
$729$	−30563.0			−1.55276
$730$		0			0
$731$	−4624.00	−	18496.0i	−0.233960	−	0.935840i
$732$			6160.00i			0.311038i
$733$		−	37922.0i		−	1.91089i	−0.295172	−	0.955444i	$-0.595377\pi$
$733$		−	37922.0i		−	1.91089i	0.295172	−	0.955444i	$-0.404623\pi$
$734$		−	6594.00i		−	0.331593i
$735$		0			0
$736$		−	18998.0i		−	0.951461i
$737$	−18480.0			−0.923636
$738$	−3700.00			−0.184551
$739$	34480.0			1.71633			0.858165	−	0.513375i	$-0.171605\pi$
$739$	34480.0			1.71633			0.858165	+	0.513375i	$0.171605\pi$
$740$		0			0
$741$			37120.0i			1.84027i
$742$	−8988.00			−0.444690
$743$	7062.00			0.348694			0.174347	−	0.984684i	$-0.444219\pi$
$743$	7062.00			0.348694			0.174347	+	0.984684i	$0.444219\pi$
$744$	−8400.00			−0.413923
$745$		0			0
$746$	−4258.00			−0.208976
$747$		−	20424.0i		−	1.00037i
$748$	−2380.00	−	9520.00i	−0.116339	−	0.465355i
$749$	7896.00			0.385198
$750$		0			0
$751$			16630.0i			0.808039i	0.914750	+	0.404020i	$0.132387\pi$
$751$			16630.0i			0.808039i	−0.914750	+	0.404020i	$0.867613\pi$
$752$		−	19024.0i		−	0.922518i
$753$	−2496.00			−0.120796
$754$		−	7308.00i		−	0.352973i
$755$		0			0
$756$	7840.00			0.377167
$757$		−	3314.00i		−	0.159114i	−0.996830	−	0.0795571i	$-0.974649\pi$
$757$		−	3314.00i		−	0.159114i	0.996830	−	0.0795571i	$-0.0253506\pi$
$758$	8984.00			0.430493
$759$		−	18880.0i		−	0.902899i
$760$		0			0
$761$	22902.0			1.09093			0.545464	−	0.838134i	$-0.316353\pi$
$761$	22902.0			1.09093			0.545464	+	0.838134i	$0.316353\pi$
$762$	9088.00			0.432052
$763$			16996.0i			0.806417i
$764$	−29456.0			−1.39487
$765$		0			0
$766$	2912.00			0.137356
$767$			10440.0i			0.491482i
$768$	952.000			0.0447296
$769$	22690.0			1.06401			0.532004	−	0.846742i	$-0.321439\pi$
$769$	22690.0			1.06401			0.532004	+	0.846742i	$0.321439\pi$
$770$		0			0
$771$			3472.00i			0.162180i
$772$	14784.0			0.689233
$773$			2418.00i			0.112509i	0.998416	+	0.0562545i	$0.0179158\pi$
$773$			2418.00i			0.112509i	−0.998416	+	0.0562545i	$0.982084\pi$
$774$	10064.0			0.467368
$775$		0			0
$776$			20640.0i			0.954810i
$777$	−15008.0			−0.692933
$778$			7950.00i			0.366351i
$779$		−	8000.00i		−	0.367946i
$780$		0			0
$781$	−1800.00			−0.0824700
$782$	−2006.00	−	8024.00i	−0.0917320	−	0.366928i
$783$		−	10080.0i		−	0.460064i
$784$	−6027.00			−0.274554
$785$		0			0
$786$	20960.0			0.951169
$787$	−8484.00			−0.384272			−0.192136	−	0.981368i	$-0.561541\pi$
$787$	−8484.00			−0.384272			−0.192136	+	0.981368i	$0.561541\pi$
$788$	−14518.0			−0.656323
$789$			14016.0i			0.632424i
$790$		0			0
$791$	19432.0			0.873480
$792$	11100.0			0.498007
$793$	6380.00			0.285700
$794$		−	12634.0i		−	0.564690i
$795$		0			0
$796$			31262.0i			1.39203i
$797$			5346.00i			0.237597i	0.992918	+	0.118799i	$0.0379043\pi$
$797$			5346.00i			0.237597i	−0.992918	+	0.118799i	$0.962096\pi$
$798$		−	8960.00i		−	0.397469i
$799$	−7888.00	−	31552.0i	−0.349258	−	1.39703i
$800$		0			0
$801$	−55130.0			−2.43186
$802$	10740.0			0.472871
$803$			16560.0i			0.727758i
$804$			51744.0i			2.26974i
$805$		0			0
$806$			4060.00i			0.177429i
$807$		−	9648.00i		−	0.420850i
$808$			9630.00i			0.419285i
$809$			11236.0i			0.488303i	0.969737	+	0.244151i	$0.0785093\pi$
$809$			11236.0i			0.488303i	−0.969737	+	0.244151i	$0.921491\pi$
$810$		0			0
$811$		−	28880.0i		−	1.25045i	−0.780445	−	0.625224i	$-0.785007\pi$
$811$		−	28880.0i		−	1.25045i	0.780445	−	0.625224i	$-0.214993\pi$
$812$		−	12348.0i		−	0.533657i
$813$	−40896.0			−1.76419
$814$	−2680.00			−0.115398
$815$		0			0
$816$	−22304.0	+	5576.00i	−0.956858	+	0.239215i
$817$			21760.0i			0.931807i
$818$		−	2950.00i		−	0.126093i
$819$		−	30044.0i		−	1.28183i
$820$		0			0
$821$			21110.0i			0.897374i	0.893689	+	0.448687i	$0.148108\pi$
$821$			21110.0i			0.897374i	−0.893689	+	0.448687i	$0.851892\pi$
$822$	15088.0			0.640212
$823$	−33958.0			−1.43828			−0.719138	−	0.694867i	$-0.755463\pi$
$823$	−33958.0			−1.43828			−0.719138	+	0.694867i	$0.755463\pi$
$824$	−18720.0			−0.791435
$825$		0			0
$826$		−	2520.00i		−	0.106153i
$827$	8436.00			0.354714			0.177357	−	0.984147i	$-0.443245\pi$
$827$	8436.00			0.354714			0.177357	+	0.984147i	$0.443245\pi$
$828$	−30562.0			−1.28273
$829$	8210.00			0.343963			0.171981	−	0.985100i	$-0.444983\pi$
$829$	8210.00			0.343963			0.171981	+	0.985100i	$0.444983\pi$
$830$		0			0
$831$	25232.0			1.05330
$832$			9686.00i			0.403608i
$833$	−9996.00	+	2499.00i	−0.415775	+	0.103944i
$834$	−32.0000			−0.00132862
$835$		0			0
$836$			11200.0i			0.463349i
$837$			5600.00i			0.231260i
$838$	5944.00			0.245026
$839$			4126.00i			0.169780i	0.996390	+	0.0848900i	$0.0270539\pi$
$839$			4126.00i			0.169780i	−0.996390	+	0.0848900i	$0.972946\pi$
$840$		0			0
$841$	8513.00			0.349051
$842$			13682.0i			0.559992i
$843$	−37616.0			−1.53685
$844$		−	17780.0i		−	0.725134i
$845$		0			0
$846$	17168.0			0.697693
$847$	−13034.0			−0.528753
$848$		−	26322.0i		−	1.06592i
$849$	11104.0			0.448867
$850$		0			0
$851$	15812.0			0.636931
$852$			5040.00i			0.202661i
$853$	−20838.0			−0.836436			−0.418218	−	0.908347i	$-0.637345\pi$
$853$	−20838.0			−0.836436			−0.418218	+	0.908347i	$0.637345\pi$
$854$	−1540.00			−0.0617069
$855$		0			0
$856$		−	8460.00i		−	0.337800i
$857$	−2864.00			−0.114157			−0.0570784	−	0.998370i	$-0.518178\pi$
$857$	−2864.00			−0.114157			−0.0570784	+	0.998370i	$0.518178\pi$
$858$		−	9280.00i		−	0.369247i
$859$	11980.0			0.475847			0.237923	−	0.971284i	$-0.423533\pi$
$859$	11980.0			0.475847			0.237923	+	0.971284i	$0.423533\pi$
$860$		0			0
$861$			11200.0i			0.443316i
$862$	410.000			0.0162003
$863$			13728.0i			0.541491i	0.962651	+	0.270745i	$0.0872702\pi$
$863$			13728.0i			0.541491i	−0.962651	+	0.270745i	$0.912730\pi$
$864$		−	12880.0i		−	0.507160i
$865$		0			0
$866$	−2558.00			−0.100375
$867$	−34680.0	+	18496.0i	−1.35847	+	0.724518i
$868$			6860.00i			0.268253i
$869$	−26680.0			−1.04149
$870$		0			0
$871$	53592.0			2.08484
$872$	18210.0			0.707189
$873$	50912.0			1.97378
$874$			9440.00i			0.365346i
$875$		0			0
$876$	46368.0			1.78839
$877$	8506.00			0.327511			0.163756	−	0.986501i	$-0.447639\pi$
$877$	8506.00			0.327511			0.163756	+	0.986501i	$0.447639\pi$
$878$	−5946.00			−0.228551
$879$		−	58544.0i		−	2.24646i
$880$		0			0
$881$		−	48600.0i		−	1.85854i	−0.369399	−	0.929271i	$-0.620436\pi$
$881$		−	48600.0i		−	1.85854i	0.369399	−	0.929271i	$-0.379564\pi$
$882$		−	5439.00i		−	0.207642i
$883$			24668.0i			0.940141i	0.882629	+	0.470070i	$0.155771\pi$
$883$			24668.0i			0.940141i	−0.882629	+	0.470070i	$0.844229\pi$
$884$	6902.00	+	27608.0i	0.262601	+	1.05040i
$885$		0			0
$886$	10852.0			0.411490
$887$	14086.0			0.533215			0.266607	−	0.963805i	$-0.414097\pi$
$887$	14086.0			0.533215			0.266607	+	0.963805i	$0.414097\pi$
$888$			16080.0i			0.607668i
$889$		−	15904.0i		−	0.600003i
$890$		0			0
$891$			7180.00i			0.269965i
$892$		−	26824.0i		−	1.00688i
$893$			37120.0i			1.39101i
$894$			6960.00i			0.260377i
$895$		0			0
$896$		−	20370.0i		−	0.759502i
$897$			54752.0i			2.03803i
$898$	3284.00			0.122036
$899$	8820.00			0.327212
$900$		0			0
$901$	−10914.0	−	43656.0i	−0.403549	−	1.61420i
$902$			2000.00i			0.0738278i
$903$		−	30464.0i		−	1.12268i
$904$		−	20820.0i		−	0.765999i
$905$		0			0
$906$			10944.0i			0.401314i
$907$	−52364.0			−1.91700			−0.958499	−	0.285094i	$-0.907975\pi$
$907$	−52364.0			−1.91700			−0.958499	+	0.285094i	$0.907975\pi$
$908$	42672.0			1.55960
$909$	23754.0			0.866744
$910$		0			0
$911$			34570.0i			1.25725i	0.777708	+	0.628625i	$0.216382\pi$
$911$			34570.0i			1.25725i	−0.777708	+	0.628625i	$0.783618\pi$
$912$	26240.0			0.952734
$913$	−11040.0			−0.400187
$914$	−4106.00			−0.148593
$915$		0			0
$916$	−18550.0			−0.669115
$917$		−	36680.0i		−	1.32092i
$918$	−1360.00	−	5440.00i	−0.0488962	−	0.195585i
$919$	−45480.0			−1.63248			−0.816239	−	0.577715i	$-0.803945\pi$
$919$	−45480.0			−1.63248			−0.816239	+	0.577715i	$0.803945\pi$
$920$		0			0
$921$		−	1728.00i		−	0.0618236i
$922$			8462.00i			0.302257i
$923$	5220.00			0.186152
$924$		−	15680.0i		−	0.558262i
$925$		0			0
$926$	3992.00			0.141669
$927$			46176.0i			1.63605i
$928$	−20286.0			−0.717587
$929$		−	1284.00i		−	0.0453463i	−0.999743	−	0.0226731i	$-0.992782\pi$
$929$		−	1284.00i		−	0.0453463i	0.999743	−	0.0226731i	$-0.00721770\pi$
$930$		0			0
$931$	11760.0			0.413983
$932$	−29596.0			−1.04018
$933$		−	57360.0i		−	2.01274i
$934$	3164.00			0.110845
$935$		0			0
$936$	−32190.0			−1.12411
$937$			10166.0i			0.354438i	0.984171	+	0.177219i	$0.0567101\pi$
$937$			10166.0i			0.354438i	−0.984171	+	0.177219i	$0.943290\pi$
$938$	−12936.0			−0.450294
$939$	−11776.0			−0.409260
$940$		0			0
$941$		−	47310.0i		−	1.63896i	−0.573107	−	0.819480i	$-0.694262\pi$
$941$		−	47310.0i		−	1.63896i	0.573107	−	0.819480i	$-0.305738\pi$
$942$	−26992.0			−0.933595
$943$		−	11800.0i		−	0.407488i
$944$	7380.00			0.254448
$945$		0			0
$946$		−	5440.00i		−	0.186966i
$947$	−1284.00			−0.0440595			−0.0220298	−	0.999757i	$-0.507013\pi$
$947$	−1284.00			−0.0440595			−0.0220298	+	0.999757i	$0.507013\pi$
$948$			74704.0i			2.55936i
$949$		−	48024.0i		−	1.64270i
$950$		0			0
$951$	−12208.0			−0.416269
$952$	−3570.00	−	14280.0i	−0.121538	−	0.486153i
$953$		−	40602.0i		−	1.38009i	−0.723765	−	0.690046i	$-0.757590\pi$
$953$		−	40602.0i		−	1.38009i	0.723765	−	0.690046i	$-0.242410\pi$
$954$	23754.0			0.806147
$955$		0			0
$956$	560.000			0.0189453
$957$	−20160.0			−0.680962
$958$	−4866.00			−0.164106
$959$		−	26404.0i		−	0.889082i
$960$		0			0
$961$	24891.0			0.835521
$962$	7772.00			0.260477
$963$	−20868.0			−0.698299
$964$		−	41860.0i		−	1.39857i
$965$		0			0
$966$		−	13216.0i		−	0.440184i
$967$		−	39624.0i		−	1.31771i	−0.752272	−	0.658853i	$-0.771042\pi$
$967$		−	39624.0i		−	1.31771i	0.752272	−	0.658853i	$-0.228958\pi$
$968$			13965.0i			0.463690i
$969$	43520.0	−	10880.0i	1.44279	−	0.360698i
$970$		0			0
$971$	8772.00			0.289914			0.144957	−	0.989438i	$-0.453696\pi$
$971$	8772.00			0.289914			0.144957	+	0.989438i	$0.453696\pi$
$972$	35224.0			1.16236
$973$			56.0000i			0.00184510i
$974$			2926.00i			0.0962578i
$975$		0			0
$976$		−	4510.00i		−	0.147911i
$977$		−	17974.0i		−	0.588576i	−0.955717	−	0.294288i	$-0.904917\pi$
$977$		−	17974.0i		−	0.588576i	0.955717	−	0.294288i	$-0.0950826\pi$
$978$			8384.00i			0.274121i
$979$			29800.0i			0.972842i
$980$		0			0
$981$		−	44918.0i		−	1.46190i
$982$		−	15128.0i		−	0.491603i
$983$	−47558.0			−1.54310			−0.771549	−	0.636170i	$-0.780518\pi$
$983$	−47558.0			−1.54310			−0.771549	+	0.636170i	$0.780518\pi$
$984$	12000.0			0.388766
$985$		0			0
$986$	−8568.00	+	2142.00i	−0.276735	+	0.0691837i
$987$		−	51968.0i		−	1.67595i
$988$		−	32480.0i		−	1.04588i
$989$			32096.0i			1.03194i
$990$		0			0
$991$		−	5490.00i		−	0.175979i	−0.996121	−	0.0879897i	$-0.971956\pi$
$991$		−	5490.00i		−	0.175979i	0.996121	−	0.0879897i	$-0.0280443\pi$
$992$	11270.0			0.360709
$993$	21664.0			0.692333
$994$	−1260.00			−0.0402060
$995$		0			0
$996$			30912.0i			0.983418i
$997$	45286.0			1.43854			0.719269	−	0.694732i	$-0.244477\pi$
$997$	45286.0			1.43854			0.719269	+	0.694732i	$0.244477\pi$
$998$	18684.0			0.592617
$999$	10720.0			0.339505

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.4.c.a.424.2		2
5.2	odd	4		425.4.d.a.101.2		2
5.3	odd	4		85.4.d.a.16.1	✓	2
5.4	even	2		425.4.c.b.424.1		2
15.8	even	4		765.4.g.a.271.2		2
17.16	even	2		425.4.c.b.424.2		2
85.13	odd	4		1445.4.a.e.1.1		1
85.33	odd	4		85.4.d.a.16.2	yes	2
85.38	odd	4		1445.4.a.d.1.1		1
85.67	odd	4		425.4.d.a.101.1		2
85.84	even	2	inner	425.4.c.a.424.1		2
255.203	even	4		765.4.g.a.271.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.4.d.a.16.1	✓	2	5.3	odd	4
85.4.d.a.16.2	yes	2	85.33	odd	4
425.4.c.a.424.1		2	85.84	even	2	inner
425.4.c.a.424.2		2	1.1	even	1	trivial
425.4.c.b.424.1		2	5.4	even	2
425.4.c.b.424.2		2	17.16	even	2
425.4.d.a.101.1		2	85.67	odd	4
425.4.d.a.101.2		2	5.2	odd	4
765.4.g.a.271.1		2	255.203	even	4
765.4.g.a.271.2		2	15.8	even	4
1445.4.a.d.1.1		1	85.38	odd	4
1445.4.a.e.1.1		1	85.13	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	425.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -8.00000 q^{3} +7.00000 q^{4} -8.00000i q^{6} -14.0000 q^{7} +15.0000i q^{8} +37.0000 q^{9} -20.0000i q^{11} -56.0000 q^{12} +58.0000i q^{13} -14.0000i q^{14} +41.0000 q^{16} +(68.0000 - 17.0000i) q^{17} +37.0000i q^{18} -80.0000 q^{19} +112.000 q^{21} +20.0000 q^{22} -118.000 q^{23} -120.000i q^{24} -58.0000 q^{26} -80.0000 q^{27} -98.0000 q^{28} +126.000i q^{29} -70.0000i q^{31} +161.000i q^{32} +160.000i q^{33} +(17.0000 + 68.0000i) q^{34} +259.000 q^{36} -134.000 q^{37} -80.0000i q^{38} -464.000i q^{39} +100.000i q^{41} +112.000i q^{42} -272.000i q^{43} -140.000i q^{44} -118.000i q^{46} -464.000i q^{47} -328.000 q^{48} -147.000 q^{49} +(-544.000 + 136.000i) q^{51} +406.000i q^{52} -642.000i q^{53} -80.0000i q^{54} -210.000i q^{56} +640.000 q^{57} -126.000 q^{58} +180.000 q^{59} -110.000i q^{61} +70.0000 q^{62} -518.000 q^{63} +167.000 q^{64} -160.000 q^{66} -924.000i q^{67} +(476.000 - 119.000i) q^{68} +944.000 q^{69} -90.0000i q^{71} +555.000i q^{72} -828.000 q^{73} -134.000i q^{74} -560.000 q^{76} +280.000i q^{77} +464.000 q^{78} -1334.00i q^{79} -359.000 q^{81} -100.000 q^{82} -552.000i q^{83} +784.000 q^{84} +272.000 q^{86} -1008.00i q^{87} +300.000 q^{88} -1490.00 q^{89} -812.000i q^{91} -826.000 q^{92} +560.000i q^{93} +464.000 q^{94} -1288.00i q^{96} +1376.00 q^{97} -147.000i q^{98} -740.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 16 q^{3} + 14 q^{4} - 28 q^{7} + 74 q^{9} - 112 q^{12} + 82 q^{16} + 136 q^{17} - 160 q^{19} + 224 q^{21} + 40 q^{22} - 236 q^{23} - 116 q^{26} - 160 q^{27} - 196 q^{28} + 34 q^{34} + 518 q^{36} - 268 q^{37}+ \cdots + 2752 q^{97}+O(q^{100}) \) 2 * q - 16 * q^3 + 14 * q^4 - 28 * q^7 + 74 * q^9 - 112 * q^12 + 82 * q^16 + 136 * q^17 - 160 * q^19 + 224 * q^21 + 40 * q^22 - 236 * q^23 - 116 * q^26 - 160 * q^27 - 196 * q^28 + 34 * q^34 + 518 * q^36 - 268 * q^37 - 656 * q^48 - 294 * q^49 - 1088 * q^51 + 1280 * q^57 - 252 * q^58 + 360 * q^59 + 140 * q^62 - 1036 * q^63 + 334 * q^64 - 320 * q^66 + 952 * q^68 + 1888 * q^69 - 1656 * q^73 - 1120 * q^76 + 928 * q^78 - 718 * q^81 - 200 * q^82 + 1568 * q^84 + 544 * q^86 + 600 * q^88 - 2980 * q^89 - 1652 * q^92 + 928 * q^94 + 2752 * q^97

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