LMFDB - Embedded newform 3450.2.a.bd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3450,2,Mod(1,3450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3450, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3450.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:	$27.5483886973$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.27492$ of defining polynomial
Character	$\chi$	$=$	3450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.27492 q^{11} -1.00000 q^{12} -3.27492 q^{13} +3.00000 q^{14} +1.00000 q^{16} +2.27492 q^{17} -1.00000 q^{18} +3.27492 q^{19} +3.00000 q^{21} +1.27492 q^{22} +1.00000 q^{23} +1.00000 q^{24} +3.27492 q^{26} -1.00000 q^{27} -3.00000 q^{28} -9.54983 q^{29} +6.00000 q^{31} -1.00000 q^{32} +1.27492 q^{33} -2.27492 q^{34} +1.00000 q^{36} -10.8248 q^{37} -3.27492 q^{38} +3.27492 q^{39} -2.72508 q^{41} -3.00000 q^{42} -7.27492 q^{43} -1.27492 q^{44} -1.00000 q^{46} -8.27492 q^{47} -1.00000 q^{48} +2.00000 q^{49} -2.27492 q^{51} -3.27492 q^{52} +10.5498 q^{53} +1.00000 q^{54} +3.00000 q^{56} -3.27492 q^{57} +9.54983 q^{58} +12.5498 q^{59} +0.549834 q^{61} -6.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} -1.27492 q^{66} +4.54983 q^{67} +2.27492 q^{68} -1.00000 q^{69} -8.82475 q^{71} -1.00000 q^{72} +15.5498 q^{73} +10.8248 q^{74} +3.27492 q^{76} +3.82475 q^{77} -3.27492 q^{78} -11.8248 q^{79} +1.00000 q^{81} +2.72508 q^{82} +2.45017 q^{83} +3.00000 q^{84} +7.27492 q^{86} +9.54983 q^{87} +1.27492 q^{88} +8.27492 q^{89} +9.82475 q^{91} +1.00000 q^{92} -6.00000 q^{93} +8.27492 q^{94} +1.00000 q^{96} -14.0000 q^{97} -2.00000 q^{98} -1.27492 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{7} - 2 q^{8} + 2 q^{9} + 5 q^{11} - 2 q^{12} + q^{13} + 6 q^{14} + 2 q^{16} - 3 q^{17} - 2 q^{18} - q^{19} + 6 q^{21} - 5 q^{22} + 2 q^{23} + 2 q^{24} - q^{26} - 2 q^{27} - 6 q^{28} - 4 q^{29} + 12 q^{31} - 2 q^{32} - 5 q^{33} + 3 q^{34} + 2 q^{36} + q^{37} + q^{38} - q^{39} - 13 q^{41} - 6 q^{42} - 7 q^{43} + 5 q^{44} - 2 q^{46} - 9 q^{47} - 2 q^{48} + 4 q^{49} + 3 q^{51} + q^{52} + 6 q^{53} + 2 q^{54} + 6 q^{56} + q^{57} + 4 q^{58} + 10 q^{59} - 14 q^{61} - 12 q^{62} - 6 q^{63} + 2 q^{64} + 5 q^{66} - 6 q^{67} - 3 q^{68} - 2 q^{69} + 5 q^{71} - 2 q^{72} + 16 q^{73} - q^{74} - q^{76} - 15 q^{77} + q^{78} - q^{79} + 2 q^{81} + 13 q^{82} + 20 q^{83} + 6 q^{84} + 7 q^{86} + 4 q^{87} - 5 q^{88} + 9 q^{89} - 3 q^{91} + 2 q^{92} - 12 q^{93} + 9 q^{94} + 2 q^{96} - 28 q^{97} - 4 q^{98} + 5 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^7 - 2 * q^8 + 2 * q^9 + 5 * q^11 - 2 * q^12 + q^13 + 6 * q^14 + 2 * q^16 - 3 * q^17 - 2 * q^18 - q^19 + 6 * q^21 - 5 * q^22 + 2 * q^23 + 2 * q^24 - q^26 - 2 * q^27 - 6 * q^28 - 4 * q^29 + 12 * q^31 - 2 * q^32 - 5 * q^33 + 3 * q^34 + 2 * q^36 + q^37 + q^38 - q^39 - 13 * q^41 - 6 * q^42 - 7 * q^43 + 5 * q^44 - 2 * q^46 - 9 * q^47 - 2 * q^48 + 4 * q^49 + 3 * q^51 + q^52 + 6 * q^53 + 2 * q^54 + 6 * q^56 + q^57 + 4 * q^58 + 10 * q^59 - 14 * q^61 - 12 * q^62 - 6 * q^63 + 2 * q^64 + 5 * q^66 - 6 * q^67 - 3 * q^68 - 2 * q^69 + 5 * q^71 - 2 * q^72 + 16 * q^73 - q^74 - q^76 - 15 * q^77 + q^78 - q^79 + 2 * q^81 + 13 * q^82 + 20 * q^83 + 6 * q^84 + 7 * q^86 + 4 * q^87 - 5 * q^88 + 9 * q^89 - 3 * q^91 + 2 * q^92 - 12 * q^93 + 9 * q^94 + 2 * q^96 - 28 * q^97 - 4 * q^98 + 5 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.27492	−0.384402	−0.192201	−	0.981356i	$-0.561563\pi$
$11$	−1.27492	−0.384402	−0.192201	+	0.981356i	$0.561563\pi$
$12$	−1.00000	−0.288675
$13$	−3.27492	−0.908299	−0.454149	−	0.890926i	$-0.650057\pi$
$13$	−3.27492	−0.908299	−0.454149	+	0.890926i	$0.650057\pi$
$14$	3.00000	0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	2.27492	0.551748	0.275874	−	0.961194i	$-0.411033\pi$
$17$	2.27492	0.551748	0.275874	+	0.961194i	$0.411033\pi$
$18$	−1.00000	−0.235702
$19$	3.27492	0.751318	0.375659	−	0.926758i	$-0.377416\pi$
$19$	3.27492	0.751318	0.375659	+	0.926758i	$0.377416\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	1.27492	0.271813
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	0	0
$26$	3.27492	0.642264
$27$	−1.00000	−0.192450
$28$	−3.00000	−0.566947
$29$	−9.54983	−1.77336	−0.886680	−	0.462384i	$-0.846994\pi$
$29$	−9.54983	−1.77336	−0.886680	+	0.462384i	$0.846994\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−1.00000	−0.176777
$33$	1.27492	0.221935
$34$	−2.27492	−0.390145
$35$	0	0
$36$	1.00000	0.166667
$37$	−10.8248	−1.77958	−0.889789	−	0.456372i	$-0.849149\pi$
$37$	−10.8248	−1.77958	−0.889789	+	0.456372i	$0.849149\pi$
$38$	−3.27492	−0.531262
$39$	3.27492	0.524406
$40$	0	0
$41$	−2.72508	−0.425586	−0.212793	−	0.977097i	$-0.568256\pi$
$41$	−2.72508	−0.425586	−0.212793	+	0.977097i	$0.568256\pi$
$42$	−3.00000	−0.462910
$43$	−7.27492	−1.10941	−0.554707	−	0.832046i	$-0.687170\pi$
$43$	−7.27492	−1.10941	−0.554707	+	0.832046i	$0.687170\pi$
$44$	−1.27492	−0.192201
$45$	0	0
$46$	−1.00000	−0.147442
$47$	−8.27492	−1.20702	−0.603510	−	0.797355i	$-0.706232\pi$
$47$	−8.27492	−1.20702	−0.603510	+	0.797355i	$0.706232\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	0	0
$51$	−2.27492	−0.318552
$52$	−3.27492	−0.454149
$53$	10.5498	1.44913	0.724566	−	0.689206i	$-0.242040\pi$
$53$	10.5498	1.44913	0.724566	+	0.689206i	$0.242040\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	3.00000	0.400892
$57$	−3.27492	−0.433773
$58$	9.54983	1.25395
$59$	12.5498	1.63385	0.816925	−	0.576744i	$-0.195677\pi$
$59$	12.5498	1.63385	0.816925	+	0.576744i	$0.195677\pi$
$60$	0	0
$61$	0.549834	0.0703991	0.0351995	−	0.999380i	$-0.488793\pi$
$61$	0.549834	0.0703991	0.0351995	+	0.999380i	$0.488793\pi$
$62$	−6.00000	−0.762001
$63$	−3.00000	−0.377964
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.27492	−0.156931
$67$	4.54983	0.555851	0.277925	−	0.960603i	$-0.410353\pi$
$67$	4.54983	0.555851	0.277925	+	0.960603i	$0.410353\pi$
$68$	2.27492	0.275874
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−8.82475	−1.04731	−0.523653	−	0.851932i	$-0.675431\pi$
$71$	−8.82475	−1.04731	−0.523653	+	0.851932i	$0.675431\pi$
$72$	−1.00000	−0.117851
$73$	15.5498	1.81997	0.909985	−	0.414641i	$-0.136093\pi$
$73$	15.5498	1.81997	0.909985	+	0.414641i	$0.136093\pi$
$74$	10.8248	1.25835
$75$	0	0
$76$	3.27492	0.375659
$77$	3.82475	0.435871
$78$	−3.27492	−0.370811
$79$	−11.8248	−1.33039	−0.665194	−	0.746670i	$-0.731651\pi$
$79$	−11.8248	−1.33039	−0.665194	+	0.746670i	$0.731651\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.72508	0.300935
$83$	2.45017	0.268941	0.134470	−	0.990918i	$-0.457067\pi$
$83$	2.45017	0.268941	0.134470	+	0.990918i	$0.457067\pi$
$84$	3.00000	0.327327
$85$	0	0
$86$	7.27492	0.784474
$87$	9.54983	1.02385
$88$	1.27492	0.135907
$89$	8.27492	0.877139	0.438570	−	0.898697i	$-0.355485\pi$
$89$	8.27492	0.877139	0.438570	+	0.898697i	$0.355485\pi$
$90$	0	0
$91$	9.82475	1.02991
$92$	1.00000	0.104257
$93$	−6.00000	−0.622171
$94$	8.27492	0.853493
$95$	0	0
$96$	1.00000	0.102062
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−2.00000	−0.202031
$99$	−1.27492	−0.128134
$100$	0	0
$101$	8.27492	0.823385	0.411693	−	0.911323i	$-0.364938\pi$
$101$	8.27492	0.823385	0.411693	+	0.911323i	$0.364938\pi$
$102$	2.27492	0.225250
$103$	3.00000	0.295599	0.147799	−	0.989017i	$-0.452781\pi$
$103$	3.00000	0.295599	0.147799	+	0.989017i	$0.452781\pi$
$104$	3.27492	0.321132
$105$	0	0
$106$	−10.5498	−1.02469
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−1.00000	−0.0962250
$109$	2.82475	0.270562	0.135281	−	0.990807i	$-0.456806\pi$
$109$	2.82475	0.270562	0.135281	+	0.990807i	$0.456806\pi$
$110$	0	0
$111$	10.8248	1.02744
$112$	−3.00000	−0.283473
$113$	3.72508	0.350426	0.175213	−	0.984531i	$-0.443939\pi$
$113$	3.72508	0.350426	0.175213	+	0.984531i	$0.443939\pi$
$114$	3.27492	0.306724
$115$	0	0
$116$	−9.54983	−0.886680
$117$	−3.27492	−0.302766
$118$	−12.5498	−1.15531
$119$	−6.82475	−0.625624
$120$	0	0
$121$	−9.37459	−0.852235
$122$	−0.549834	−0.0497797
$123$	2.72508	0.245712
$124$	6.00000	0.538816
$125$	0	0
$126$	3.00000	0.267261
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	7.27492	0.640521
$130$	0	0
$131$	6.54983	0.572262	0.286131	−	0.958191i	$-0.407631\pi$
$131$	6.54983	0.572262	0.286131	+	0.958191i	$0.407631\pi$
$132$	1.27492	0.110967
$133$	−9.82475	−0.851914
$134$	−4.54983	−0.393046
$135$	0	0
$136$	−2.27492	−0.195073
$137$	−8.82475	−0.753949	−0.376975	−	0.926224i	$-0.623036\pi$
$137$	−8.82475	−0.753949	−0.376975	+	0.926224i	$0.623036\pi$
$138$	1.00000	0.0851257
$139$	6.27492	0.532232	0.266116	−	0.963941i	$-0.414260\pi$
$139$	6.27492	0.532232	0.266116	+	0.963941i	$0.414260\pi$
$140$	0	0
$141$	8.27492	0.696874
$142$	8.82475	0.740557
$143$	4.17525	0.349152
$144$	1.00000	0.0833333
$145$	0	0
$146$	−15.5498	−1.28691
$147$	−2.00000	−0.164957
$148$	−10.8248	−0.889789
$149$	−2.54983	−0.208891	−0.104445	−	0.994531i	$-0.533307\pi$
$149$	−2.54983	−0.208891	−0.104445	+	0.994531i	$0.533307\pi$
$150$	0	0
$151$	−2.54983	−0.207503	−0.103751	−	0.994603i	$-0.533085\pi$
$151$	−2.54983	−0.207503	−0.103751	+	0.994603i	$0.533085\pi$
$152$	−3.27492	−0.265631
$153$	2.27492	0.183916
$154$	−3.82475	−0.308207
$155$	0	0
$156$	3.27492	0.262203
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	11.8248	0.940727
$159$	−10.5498	−0.836656
$160$	0	0
$161$	−3.00000	−0.236433
$162$	−1.00000	−0.0785674
$163$	23.6495	1.85237	0.926186	−	0.377067i	$-0.123067\pi$
$163$	23.6495	1.85237	0.926186	+	0.377067i	$0.123067\pi$
$164$	−2.72508	−0.212793
$165$	0	0
$166$	−2.45017	−0.190170
$167$	12.2749	0.949862	0.474931	−	0.880023i	$-0.342473\pi$
$167$	12.2749	0.949862	0.474931	+	0.880023i	$0.342473\pi$
$168$	−3.00000	−0.231455
$169$	−2.27492	−0.174994
$170$	0	0
$171$	3.27492	0.250439
$172$	−7.27492	−0.554707
$173$	15.2749	1.16133	0.580665	−	0.814142i	$-0.302793\pi$
$173$	15.2749	1.16133	0.580665	+	0.814142i	$0.302793\pi$
$174$	−9.54983	−0.723971
$175$	0	0
$176$	−1.27492	−0.0961005
$177$	−12.5498	−0.943303
$178$	−8.27492	−0.620231
$179$	26.5498	1.98443	0.992214	−	0.124545i	$-0.0397473\pi$
$179$	26.5498	1.98443	0.992214	+	0.124545i	$0.0397473\pi$
$180$	0	0
$181$	1.72508	0.128224	0.0641122	−	0.997943i	$-0.479578\pi$
$181$	1.72508	0.128224	0.0641122	+	0.997943i	$0.479578\pi$
$182$	−9.82475	−0.728259
$183$	−0.549834	−0.0406449
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	6.00000	0.439941
$187$	−2.90033	−0.212093
$188$	−8.27492	−0.603510
$189$	3.00000	0.218218
$190$	0	0
$191$	23.2749	1.68411	0.842057	−	0.539389i	$-0.181345\pi$
$191$	23.2749	1.68411	0.842057	+	0.539389i	$0.181345\pi$
$192$	−1.00000	−0.0721688
$193$	7.72508	0.556064	0.278032	−	0.960572i	$-0.410318\pi$
$193$	7.72508	0.556064	0.278032	+	0.960572i	$0.410318\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	2.00000	0.142857
$197$	4.45017	0.317061	0.158531	−	0.987354i	$-0.449324\pi$
$197$	4.45017	0.317061	0.158531	+	0.987354i	$0.449324\pi$
$198$	1.27492	0.0906044
$199$	11.0000	0.779769	0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000	0.779769	0.389885	+	0.920864i	$0.372515\pi$
$200$	0	0
$201$	−4.54983	−0.320921
$202$	−8.27492	−0.582221
$203$	28.6495	2.01080
$204$	−2.27492	−0.159276
$205$	0	0
$206$	−3.00000	−0.209020
$207$	1.00000	0.0695048
$208$	−3.27492	−0.227075
$209$	−4.17525	−0.288808
$210$	0	0
$211$	−20.8248	−1.43364	−0.716818	−	0.697261i	$-0.754402\pi$
$211$	−20.8248	−1.43364	−0.716818	+	0.697261i	$0.754402\pi$
$212$	10.5498	0.724566
$213$	8.82475	0.604662
$214$	−12.0000	−0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	−18.0000	−1.22192
$218$	−2.82475	−0.191316
$219$	−15.5498	−1.05076
$220$	0	0
$221$	−7.45017	−0.501152
$222$	−10.8248	−0.726510
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	3.00000	0.200446
$225$	0	0
$226$	−3.72508	−0.247789
$227$	10.2749	0.681970	0.340985	−	0.940069i	$-0.389239\pi$
$227$	10.2749	0.681970	0.340985	+	0.940069i	$0.389239\pi$
$228$	−3.27492	−0.216887
$229$	−19.0997	−1.26214	−0.631071	−	0.775725i	$-0.717384\pi$
$229$	−19.0997	−1.26214	−0.631071	+	0.775725i	$0.717384\pi$
$230$	0	0
$231$	−3.82475	−0.251650
$232$	9.54983	0.626977
$233$	−14.3746	−0.941710	−0.470855	−	0.882210i	$-0.656055\pi$
$233$	−14.3746	−0.941710	−0.470855	+	0.882210i	$0.656055\pi$
$234$	3.27492	0.214088
$235$	0	0
$236$	12.5498	0.816925
$237$	11.8248	0.768100
$238$	6.82475	0.442383
$239$	12.2749	0.793998	0.396999	−	0.917819i	$-0.370052\pi$
$239$	12.2749	0.793998	0.396999	+	0.917819i	$0.370052\pi$
$240$	0	0
$241$	10.5498	0.679575	0.339787	−	0.940502i	$-0.389645\pi$
$241$	10.5498	0.679575	0.339787	+	0.940502i	$0.389645\pi$
$242$	9.37459	0.602621
$243$	−1.00000	−0.0641500
$244$	0.549834	0.0351995
$245$	0	0
$246$	−2.72508	−0.173745
$247$	−10.7251	−0.682421
$248$	−6.00000	−0.381000
$249$	−2.45017	−0.155273
$250$	0	0
$251$	26.8248	1.69316	0.846582	−	0.532259i	$-0.178657\pi$
$251$	26.8248	1.69316	0.846582	+	0.532259i	$0.178657\pi$
$252$	−3.00000	−0.188982
$253$	−1.27492	−0.0801534
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	−7.27492	−0.452917
$259$	32.4743	2.01785
$260$	0	0
$261$	−9.54983	−0.591120
$262$	−6.54983	−0.404650
$263$	4.54983	0.280555	0.140277	−	0.990112i	$-0.455201\pi$
$263$	4.54983	0.280555	0.140277	+	0.990112i	$0.455201\pi$
$264$	−1.27492	−0.0784657
$265$	0	0
$266$	9.82475	0.602394
$267$	−8.27492	−0.506417
$268$	4.54983	0.277925
$269$	−3.27492	−0.199675	−0.0998376	−	0.995004i	$-0.531832\pi$
$269$	−3.27492	−0.199675	−0.0998376	+	0.995004i	$0.531832\pi$
$270$	0	0
$271$	1.45017	0.0880913	0.0440456	−	0.999030i	$-0.485975\pi$
$271$	1.45017	0.0880913	0.0440456	+	0.999030i	$0.485975\pi$
$272$	2.27492	0.137937
$273$	−9.82475	−0.594621
$274$	8.82475	0.533123
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	−5.82475	−0.349975	−0.174988	−	0.984571i	$-0.555989\pi$
$277$	−5.82475	−0.349975	−0.174988	+	0.984571i	$0.555989\pi$
$278$	−6.27492	−0.376345
$279$	6.00000	0.359211
$280$	0	0
$281$	5.37459	0.320621	0.160310	−	0.987067i	$-0.448750\pi$
$281$	5.37459	0.320621	0.160310	+	0.987067i	$0.448750\pi$
$282$	−8.27492	−0.492764
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	−8.82475	−0.523653
$285$	0	0
$286$	−4.17525	−0.246888
$287$	8.17525	0.482570
$288$	−1.00000	−0.0589256
$289$	−11.8248	−0.695574
$290$	0	0
$291$	14.0000	0.820695
$292$	15.5498	0.909985
$293$	21.6495	1.26478	0.632389	−	0.774651i	$-0.282075\pi$
$293$	21.6495	1.26478	0.632389	+	0.774651i	$0.282075\pi$
$294$	2.00000	0.116642
$295$	0	0
$296$	10.8248	0.629176
$297$	1.27492	0.0739782
$298$	2.54983	0.147708
$299$	−3.27492	−0.189393
$300$	0	0
$301$	21.8248	1.25796
$302$	2.54983	0.146726
$303$	−8.27492	−0.475382
$304$	3.27492	0.187829
$305$	0	0
$306$	−2.27492	−0.130048
$307$	−6.27492	−0.358128	−0.179064	−	0.983837i	$-0.557307\pi$
$307$	−6.27492	−0.358128	−0.179064	+	0.983837i	$0.557307\pi$
$308$	3.82475	0.217935
$309$	−3.00000	−0.170664
$310$	0	0
$311$	−10.8248	−0.613815	−0.306908	−	0.951739i	$-0.599294\pi$
$311$	−10.8248	−0.613815	−0.306908	+	0.951739i	$0.599294\pi$
$312$	−3.27492	−0.185406
$313$	32.0000	1.80875	0.904373	−	0.426742i	$-0.140339\pi$
$313$	32.0000	1.80875	0.904373	+	0.426742i	$0.140339\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	−11.8248	−0.665194
$317$	13.5498	0.761035	0.380517	−	0.924774i	$-0.375746\pi$
$317$	13.5498	0.761035	0.380517	+	0.924774i	$0.375746\pi$
$318$	10.5498	0.591605
$319$	12.1752	0.681683
$320$	0	0
$321$	−12.0000	−0.669775
$322$	3.00000	0.167183
$323$	7.45017	0.414538
$324$	1.00000	0.0555556
$325$	0	0
$326$	−23.6495	−1.30982
$327$	−2.82475	−0.156209
$328$	2.72508	0.150468
$329$	24.8248	1.36863
$330$	0	0
$331$	16.8248	0.924772	0.462386	−	0.886679i	$-0.346993\pi$
$331$	16.8248	0.924772	0.462386	+	0.886679i	$0.346993\pi$
$332$	2.45017	0.134470
$333$	−10.8248	−0.593193
$334$	−12.2749	−0.671654
$335$	0	0
$336$	3.00000	0.163663
$337$	−9.45017	−0.514783	−0.257392	−	0.966307i	$-0.582863\pi$
$337$	−9.45017	−0.514783	−0.257392	+	0.966307i	$0.582863\pi$
$338$	2.27492	0.123739
$339$	−3.72508	−0.202319
$340$	0	0
$341$	−7.64950	−0.414244
$342$	−3.27492	−0.177087
$343$	15.0000	0.809924
$344$	7.27492	0.392237
$345$	0	0
$346$	−15.2749	−0.821185
$347$	19.4502	1.04414	0.522070	−	0.852903i	$-0.325160\pi$
$347$	19.4502	1.04414	0.522070	+	0.852903i	$0.325160\pi$
$348$	9.54983	0.511925
$349$	−27.2749	−1.45999	−0.729996	−	0.683451i	$-0.760478\pi$
$349$	−27.2749	−1.45999	−0.729996	+	0.683451i	$0.760478\pi$
$350$	0	0
$351$	3.27492	0.174802
$352$	1.27492	0.0679533
$353$	−24.3746	−1.29733	−0.648664	−	0.761075i	$-0.724672\pi$
$353$	−24.3746	−1.29733	−0.648664	+	0.761075i	$0.724672\pi$
$354$	12.5498	0.667016
$355$	0	0
$356$	8.27492	0.438570
$357$	6.82475	0.361204
$358$	−26.5498	−1.40320
$359$	−28.9244	−1.52657	−0.763286	−	0.646060i	$-0.776415\pi$
$359$	−28.9244	−1.52657	−0.763286	+	0.646060i	$0.776415\pi$
$360$	0	0
$361$	−8.27492	−0.435522
$362$	−1.72508	−0.0906683
$363$	9.37459	0.492038
$364$	9.82475	0.514957
$365$	0	0
$366$	0.549834	0.0287403
$367$	−9.27492	−0.484147	−0.242073	−	0.970258i	$-0.577827\pi$
$367$	−9.27492	−0.484147	−0.242073	+	0.970258i	$0.577827\pi$
$368$	1.00000	0.0521286
$369$	−2.72508	−0.141862
$370$	0	0
$371$	−31.6495	−1.64316
$372$	−6.00000	−0.311086
$373$	23.3746	1.21029	0.605145	−	0.796115i	$-0.293115\pi$
$373$	23.3746	1.21029	0.605145	+	0.796115i	$0.293115\pi$
$374$	2.90033	0.149973
$375$	0	0
$376$	8.27492	0.426746
$377$	31.2749	1.61074
$378$	−3.00000	−0.154303
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	−23.2749	−1.19085
$383$	−0.175248	−0.00895477	−0.00447739	−	0.999990i	$-0.501425\pi$
$383$	−0.175248	−0.00895477	−0.00447739	+	0.999990i	$0.501425\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−7.72508	−0.393196
$387$	−7.27492	−0.369805
$388$	−14.0000	−0.710742
$389$	23.0997	1.17120	0.585600	−	0.810600i	$-0.300859\pi$
$389$	23.0997	1.17120	0.585600	+	0.810600i	$0.300859\pi$
$390$	0	0
$391$	2.27492	0.115048
$392$	−2.00000	−0.101015
$393$	−6.54983	−0.330395
$394$	−4.45017	−0.224196
$395$	0	0
$396$	−1.27492	−0.0640670
$397$	21.4502	1.07655	0.538276	−	0.842768i	$-0.319076\pi$
$397$	21.4502	1.07655	0.538276	+	0.842768i	$0.319076\pi$
$398$	−11.0000	−0.551380
$399$	9.82475	0.491853
$400$	0	0
$401$	−9.64950	−0.481873	−0.240937	−	0.970541i	$-0.577455\pi$
$401$	−9.64950	−0.481873	−0.240937	+	0.970541i	$0.577455\pi$
$402$	4.54983	0.226925
$403$	−19.6495	−0.978811
$404$	8.27492	0.411693
$405$	0	0
$406$	−28.6495	−1.42185
$407$	13.8007	0.684073
$408$	2.27492	0.112625
$409$	−7.00000	−0.346128	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000	−0.346128	−0.173064	+	0.984911i	$0.555367\pi$
$410$	0	0
$411$	8.82475	0.435293
$412$	3.00000	0.147799
$413$	−37.6495	−1.85261
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	3.27492	0.160566
$417$	−6.27492	−0.307284
$418$	4.17525	0.204218
$419$	3.00000	0.146560	0.0732798	−	0.997311i	$-0.476653\pi$
$419$	3.00000	0.146560	0.0732798	+	0.997311i	$0.476653\pi$
$420$	0	0
$421$	31.0997	1.51571	0.757853	−	0.652426i	$-0.226249\pi$
$421$	31.0997	1.51571	0.757853	+	0.652426i	$0.226249\pi$
$422$	20.8248	1.01373
$423$	−8.27492	−0.402340
$424$	−10.5498	−0.512345
$425$	0	0
$426$	−8.82475	−0.427561
$427$	−1.64950	−0.0798251
$428$	12.0000	0.580042
$429$	−4.17525	−0.201583
$430$	0	0
$431$	−12.5498	−0.604504	−0.302252	−	0.953228i	$-0.597738\pi$
$431$	−12.5498	−0.604504	−0.302252	+	0.953228i	$0.597738\pi$
$432$	−1.00000	−0.0481125
$433$	−17.6495	−0.848181	−0.424091	−	0.905620i	$-0.639406\pi$
$433$	−17.6495	−0.848181	−0.424091	+	0.905620i	$0.639406\pi$
$434$	18.0000	0.864028
$435$	0	0
$436$	2.82475	0.135281
$437$	3.27492	0.156661
$438$	15.5498	0.743000
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	7.45017	0.354368
$443$	−15.6495	−0.743530	−0.371765	−	0.928327i	$-0.621247\pi$
$443$	−15.6495	−0.743530	−0.371765	+	0.928327i	$0.621247\pi$
$444$	10.8248	0.513720
$445$	0	0
$446$	−14.0000	−0.662919
$447$	2.54983	0.120603
$448$	−3.00000	−0.141737
$449$	15.0997	0.712597	0.356299	−	0.934372i	$-0.384039\pi$
$449$	15.0997	0.712597	0.356299	+	0.934372i	$0.384039\pi$
$450$	0	0
$451$	3.47425	0.163596
$452$	3.72508	0.175213
$453$	2.54983	0.119802
$454$	−10.2749	−0.482226
$455$	0	0
$456$	3.27492	0.153362
$457$	39.6495	1.85473	0.927363	−	0.374163i	$-0.122070\pi$
$457$	39.6495	1.85473	0.927363	+	0.374163i	$0.122070\pi$
$458$	19.0997	0.892469
$459$	−2.27492	−0.106184
$460$	0	0
$461$	0.0996689	0.00464204	0.00232102	−	0.999997i	$-0.499261\pi$
$461$	0.0996689	0.00464204	0.00232102	+	0.999997i	$0.499261\pi$
$462$	3.82475	0.177944
$463$	29.0997	1.35238	0.676188	−	0.736729i	$-0.263631\pi$
$463$	29.0997	1.35238	0.676188	+	0.736729i	$0.263631\pi$
$464$	−9.54983	−0.443340
$465$	0	0
$466$	14.3746	0.665890
$467$	21.0000	0.971764	0.485882	−	0.874024i	$-0.338498\pi$
$467$	21.0000	0.971764	0.485882	+	0.874024i	$0.338498\pi$
$468$	−3.27492	−0.151383
$469$	−13.6495	−0.630276
$470$	0	0
$471$	14.0000	0.645086
$472$	−12.5498	−0.577653
$473$	9.27492	0.426461
$474$	−11.8248	−0.543129
$475$	0	0
$476$	−6.82475	−0.312812
$477$	10.5498	0.483044
$478$	−12.2749	−0.561442
$479$	7.27492	0.332399	0.166200	−	0.986092i	$-0.446850\pi$
$479$	7.27492	0.332399	0.166200	+	0.986092i	$0.446850\pi$
$480$	0	0
$481$	35.4502	1.61639
$482$	−10.5498	−0.480532
$483$	3.00000	0.136505
$484$	−9.37459	−0.426118
$485$	0	0
$486$	1.00000	0.0453609
$487$	6.54983	0.296801	0.148401	−	0.988927i	$-0.452587\pi$
$487$	6.54983	0.296801	0.148401	+	0.988927i	$0.452587\pi$
$488$	−0.549834	−0.0248898
$489$	−23.6495	−1.06947
$490$	0	0
$491$	−8.54983	−0.385849	−0.192924	−	0.981214i	$-0.561797\pi$
$491$	−8.54983	−0.385849	−0.192924	+	0.981214i	$0.561797\pi$
$492$	2.72508	0.122856
$493$	−21.7251	−0.978449
$494$	10.7251	0.482544
$495$	0	0
$496$	6.00000	0.269408
$497$	26.4743	1.18753
$498$	2.45017	0.109795
$499$	11.7251	0.524887	0.262443	−	0.964947i	$-0.415472\pi$
$499$	11.7251	0.524887	0.262443	+	0.964947i	$0.415472\pi$
$500$	0	0
$501$	−12.2749	−0.548403
$502$	−26.8248	−1.19725
$503$	18.3746	0.819282	0.409641	−	0.912247i	$-0.365654\pi$
$503$	18.3746	0.819282	0.409641	+	0.912247i	$0.365654\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	1.27492	0.0566770
$507$	2.27492	0.101033
$508$	−8.00000	−0.354943
$509$	−30.4743	−1.35075	−0.675374	−	0.737476i	$-0.736018\pi$
$509$	−30.4743	−1.35075	−0.675374	+	0.737476i	$0.736018\pi$
$510$	0	0
$511$	−46.6495	−2.06365
$512$	−1.00000	−0.0441942
$513$	−3.27492	−0.144591
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	7.27492	0.320260
$517$	10.5498	0.463981
$518$	−32.4743	−1.42684
$519$	−15.2749	−0.670494
$520$	0	0
$521$	3.37459	0.147843	0.0739217	−	0.997264i	$-0.476449\pi$
$521$	3.37459	0.147843	0.0739217	+	0.997264i	$0.476449\pi$
$522$	9.54983	0.417985
$523$	−26.3746	−1.15328	−0.576640	−	0.816998i	$-0.695636\pi$
$523$	−26.3746	−1.15328	−0.576640	+	0.816998i	$0.695636\pi$
$524$	6.54983	0.286131
$525$	0	0
$526$	−4.54983	−0.198382
$527$	13.6495	0.594582
$528$	1.27492	0.0554837
$529$	1.00000	0.0434783
$530$	0	0
$531$	12.5498	0.544616
$532$	−9.82475	−0.425957
$533$	8.92442	0.386560
$534$	8.27492	0.358091
$535$	0	0
$536$	−4.54983	−0.196523
$537$	−26.5498	−1.14571
$538$	3.27492	0.141192
$539$	−2.54983	−0.109829
$540$	0	0
$541$	−31.8248	−1.36825	−0.684126	−	0.729363i	$-0.739816\pi$
$541$	−31.8248	−1.36825	−0.684126	+	0.729363i	$0.739816\pi$
$542$	−1.45017	−0.0622899
$543$	−1.72508	−0.0740304
$544$	−2.27492	−0.0975363
$545$	0	0
$546$	9.82475	0.420461
$547$	44.8248	1.91657	0.958284	−	0.285818i	$-0.0922652\pi$
$547$	44.8248	1.91657	0.958284	+	0.285818i	$0.0922652\pi$
$548$	−8.82475	−0.376975
$549$	0.549834	0.0234664
$550$	0	0
$551$	−31.2749	−1.33236
$552$	1.00000	0.0425628
$553$	35.4743	1.50852
$554$	5.82475	0.247470
$555$	0	0
$556$	6.27492	0.266116
$557$	20.5498	0.870724	0.435362	−	0.900255i	$-0.356620\pi$
$557$	20.5498	0.870724	0.435362	+	0.900255i	$0.356620\pi$
$558$	−6.00000	−0.254000
$559$	23.8248	1.00768
$560$	0	0
$561$	2.90033	0.122452
$562$	−5.37459	−0.226713
$563$	−6.72508	−0.283428	−0.141714	−	0.989908i	$-0.545261\pi$
$563$	−6.72508	−0.283428	−0.141714	+	0.989908i	$0.545261\pi$
$564$	8.27492	0.348437
$565$	0	0
$566$	−16.0000	−0.672530
$567$	−3.00000	−0.125988
$568$	8.82475	0.370278
$569$	−7.45017	−0.312327	−0.156164	−	0.987731i	$-0.549913\pi$
$569$	−7.45017	−0.312327	−0.156164	+	0.987731i	$0.549913\pi$
$570$	0	0
$571$	−32.5498	−1.36217	−0.681084	−	0.732205i	$-0.738491\pi$
$571$	−32.5498	−1.36217	−0.681084	+	0.732205i	$0.738491\pi$
$572$	4.17525	0.174576
$573$	−23.2749	−0.972324
$574$	−8.17525	−0.341228
$575$	0	0
$576$	1.00000	0.0416667
$577$	1.82475	0.0759654	0.0379827	−	0.999278i	$-0.487907\pi$
$577$	1.82475	0.0759654	0.0379827	+	0.999278i	$0.487907\pi$
$578$	11.8248	0.491845
$579$	−7.72508	−0.321043
$580$	0	0
$581$	−7.35050	−0.304950
$582$	−14.0000	−0.580319
$583$	−13.4502	−0.557049
$584$	−15.5498	−0.643457
$585$	0	0
$586$	−21.6495	−0.894333
$587$	41.0997	1.69636	0.848182	−	0.529704i	$-0.177697\pi$
$587$	41.0997	1.69636	0.848182	+	0.529704i	$0.177697\pi$
$588$	−2.00000	−0.0824786
$589$	19.6495	0.809644
$590$	0	0
$591$	−4.45017	−0.183055
$592$	−10.8248	−0.444895
$593$	−8.17525	−0.335717	−0.167859	−	0.985811i	$-0.553685\pi$
$593$	−8.17525	−0.335717	−0.167859	+	0.985811i	$0.553685\pi$
$594$	−1.27492	−0.0523105
$595$	0	0
$596$	−2.54983	−0.104445
$597$	−11.0000	−0.450200
$598$	3.27492	0.133921
$599$	36.7492	1.50153	0.750765	−	0.660569i	$-0.229685\pi$
$599$	36.7492	1.50153	0.750765	+	0.660569i	$0.229685\pi$
$600$	0	0
$601$	19.7251	0.804603	0.402301	−	0.915507i	$-0.368210\pi$
$601$	19.7251	0.804603	0.402301	+	0.915507i	$0.368210\pi$
$602$	−21.8248	−0.889510
$603$	4.54983	0.185284
$604$	−2.54983	−0.103751
$605$	0	0
$606$	8.27492	0.336146
$607$	−35.0997	−1.42465	−0.712326	−	0.701849i	$-0.752358\pi$
$607$	−35.0997	−1.42465	−0.712326	+	0.701849i	$0.752358\pi$
$608$	−3.27492	−0.132815
$609$	−28.6495	−1.16094
$610$	0	0
$611$	27.0997	1.09634
$612$	2.27492	0.0919581
$613$	−26.2749	−1.06123	−0.530617	−	0.847612i	$-0.678040\pi$
$613$	−26.2749	−1.06123	−0.530617	+	0.847612i	$0.678040\pi$
$614$	6.27492	0.253235
$615$	0	0
$616$	−3.82475	−0.154104
$617$	−37.6495	−1.51571	−0.757856	−	0.652422i	$-0.773753\pi$
$617$	−37.6495	−1.51571	−0.757856	+	0.652422i	$0.773753\pi$
$618$	3.00000	0.120678
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	10.8248	0.434033
$623$	−24.8248	−0.994583
$624$	3.27492	0.131102
$625$	0	0
$626$	−32.0000	−1.27898
$627$	4.17525	0.166743
$628$	−14.0000	−0.558661
$629$	−24.6254	−0.981880
$630$	0	0
$631$	−1.90033	−0.0756510	−0.0378255	−	0.999284i	$-0.512043\pi$
$631$	−1.90033	−0.0756510	−0.0378255	+	0.999284i	$0.512043\pi$
$632$	11.8248	0.470363
$633$	20.8248	0.827710
$634$	−13.5498	−0.538133
$635$	0	0
$636$	−10.5498	−0.418328
$637$	−6.54983	−0.259514
$638$	−12.1752	−0.482023
$639$	−8.82475	−0.349102
$640$	0	0
$641$	8.62541	0.340683	0.170342	−	0.985385i	$-0.445513\pi$
$641$	8.62541	0.340683	0.170342	+	0.985385i	$0.445513\pi$
$642$	12.0000	0.473602
$643$	−7.27492	−0.286895	−0.143447	−	0.989658i	$-0.545819\pi$
$643$	−7.27492	−0.286895	−0.143447	+	0.989658i	$0.545819\pi$
$644$	−3.00000	−0.118217
$645$	0	0
$646$	−7.45017	−0.293123
$647$	15.1752	0.596601	0.298300	−	0.954472i	$-0.403580\pi$
$647$	15.1752	0.596601	0.298300	+	0.954472i	$0.403580\pi$
$648$	−1.00000	−0.0392837
$649$	−16.0000	−0.628055
$650$	0	0
$651$	18.0000	0.705476
$652$	23.6495	0.926186
$653$	−38.0997	−1.49096	−0.745478	−	0.666531i	$-0.767779\pi$
$653$	−38.0997	−1.49096	−0.745478	+	0.666531i	$0.767779\pi$
$654$	2.82475	0.110457
$655$	0	0
$656$	−2.72508	−0.106397
$657$	15.5498	0.606657
$658$	−24.8248	−0.967770
$659$	−26.0997	−1.01670	−0.508349	−	0.861151i	$-0.669744\pi$
$659$	−26.0997	−1.01670	−0.508349	+	0.861151i	$0.669744\pi$
$660$	0	0
$661$	40.8248	1.58790	0.793949	−	0.607984i	$-0.208021\pi$
$661$	40.8248	1.58790	0.793949	+	0.607984i	$0.208021\pi$
$662$	−16.8248	−0.653913
$663$	7.45017	0.289340
$664$	−2.45017	−0.0950849
$665$	0	0
$666$	10.8248	0.419451
$667$	−9.54983	−0.369771
$668$	12.2749	0.474931
$669$	−14.0000	−0.541271
$670$	0	0
$671$	−0.700993	−0.0270615
$672$	−3.00000	−0.115728
$673$	−6.45017	−0.248636	−0.124318	−	0.992242i	$-0.539674\pi$
$673$	−6.45017	−0.248636	−0.124318	+	0.992242i	$0.539674\pi$
$674$	9.45017	0.364007
$675$	0	0
$676$	−2.27492	−0.0874968
$677$	−17.0997	−0.657194	−0.328597	−	0.944470i	$-0.606576\pi$
$677$	−17.0997	−0.657194	−0.328597	+	0.944470i	$0.606576\pi$
$678$	3.72508	0.143061
$679$	42.0000	1.61181
$680$	0	0
$681$	−10.2749	−0.393736
$682$	7.64950	0.292915
$683$	22.5498	0.862845	0.431423	−	0.902150i	$-0.358012\pi$
$683$	22.5498	0.862845	0.431423	+	0.902150i	$0.358012\pi$
$684$	3.27492	0.125220
$685$	0	0
$686$	−15.0000	−0.572703
$687$	19.0997	0.728698
$688$	−7.27492	−0.277354
$689$	−34.5498	−1.31624
$690$	0	0
$691$	17.7251	0.674294	0.337147	−	0.941452i	$-0.390538\pi$
$691$	17.7251	0.674294	0.337147	+	0.941452i	$0.390538\pi$
$692$	15.2749	0.580665
$693$	3.82475	0.145290
$694$	−19.4502	−0.738318
$695$	0	0
$696$	−9.54983	−0.361986
$697$	−6.19934	−0.234817
$698$	27.2749	1.03237
$699$	14.3746	0.543697
$700$	0	0
$701$	17.6495	0.666613	0.333306	−	0.942819i	$-0.391836\pi$
$701$	17.6495	0.666613	0.333306	+	0.942819i	$0.391836\pi$
$702$	−3.27492	−0.123604
$703$	−35.4502	−1.33703
$704$	−1.27492	−0.0480503
$705$	0	0
$706$	24.3746	0.917350
$707$	−24.8248	−0.933631
$708$	−12.5498	−0.471652
$709$	−27.1752	−1.02059	−0.510294	−	0.860000i	$-0.670463\pi$
$709$	−27.1752	−1.02059	−0.510294	+	0.860000i	$0.670463\pi$
$710$	0	0
$711$	−11.8248	−0.443463
$712$	−8.27492	−0.310116
$713$	6.00000	0.224702
$714$	−6.82475	−0.255410
$715$	0	0
$716$	26.5498	0.992214
$717$	−12.2749	−0.458415
$718$	28.9244	1.07945
$719$	15.3746	0.573375	0.286688	−	0.958024i	$-0.407446\pi$
$719$	15.3746	0.573375	0.286688	+	0.958024i	$0.407446\pi$
$720$	0	0
$721$	−9.00000	−0.335178
$722$	8.27492	0.307961
$723$	−10.5498	−0.392353
$724$	1.72508	0.0641122
$725$	0	0
$726$	−9.37459	−0.347924
$727$	−34.1993	−1.26838	−0.634192	−	0.773176i	$-0.718667\pi$
$727$	−34.1993	−1.26838	−0.634192	+	0.773176i	$0.718667\pi$
$728$	−9.82475	−0.364130
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.5498	−0.612118
$732$	−0.549834	−0.0203225
$733$	−33.9244	−1.25303	−0.626514	−	0.779411i	$-0.715519\pi$
$733$	−33.9244	−1.25303	−0.626514	+	0.779411i	$0.715519\pi$
$734$	9.27492	0.342343
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−5.80066	−0.213670
$738$	2.72508	0.100312
$739$	−33.9244	−1.24793	−0.623965	−	0.781452i	$-0.714479\pi$
$739$	−33.9244	−1.24793	−0.623965	+	0.781452i	$0.714479\pi$
$740$	0	0
$741$	10.7251	0.393996
$742$	31.6495	1.16189
$743$	−21.8248	−0.800672	−0.400336	−	0.916368i	$-0.631107\pi$
$743$	−21.8248	−0.800672	−0.400336	+	0.916368i	$0.631107\pi$
$744$	6.00000	0.219971
$745$	0	0
$746$	−23.3746	−0.855804
$747$	2.45017	0.0896469
$748$	−2.90033	−0.106047
$749$	−36.0000	−1.31541
$750$	0	0
$751$	−8.64950	−0.315625	−0.157812	−	0.987469i	$-0.550444\pi$
$751$	−8.64950	−0.315625	−0.157812	+	0.987469i	$0.550444\pi$
$752$	−8.27492	−0.301755
$753$	−26.8248	−0.977548
$754$	−31.2749	−1.13897
$755$	0	0
$756$	3.00000	0.109109
$757$	−21.7251	−0.789612	−0.394806	−	0.918765i	$-0.629188\pi$
$757$	−21.7251	−0.789612	−0.394806	+	0.918765i	$0.629188\pi$
$758$	−12.0000	−0.435860
$759$	1.27492	0.0462766
$760$	0	0
$761$	−13.8248	−0.501147	−0.250573	−	0.968098i	$-0.580619\pi$
$761$	−13.8248	−0.501147	−0.250573	+	0.968098i	$0.580619\pi$
$762$	−8.00000	−0.289809
$763$	−8.47425	−0.306789
$764$	23.2749	0.842057
$765$	0	0
$766$	0.175248	0.00633198
$767$	−41.0997	−1.48402
$768$	−1.00000	−0.0360844
$769$	8.54983	0.308315	0.154157	−	0.988046i	$-0.450734\pi$
$769$	8.54983	0.308315	0.154157	+	0.988046i	$0.450734\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	7.72508	0.278032
$773$	46.7492	1.68145	0.840725	−	0.541462i	$-0.182129\pi$
$773$	46.7492	1.68145	0.840725	+	0.541462i	$0.182129\pi$
$774$	7.27492	0.261491
$775$	0	0
$776$	14.0000	0.502571
$777$	−32.4743	−1.16501
$778$	−23.0997	−0.828163
$779$	−8.92442	−0.319751
$780$	0	0
$781$	11.2508	0.402586
$782$	−2.27492	−0.0813509
$783$	9.54983	0.341283
$784$	2.00000	0.0714286
$785$	0	0
$786$	6.54983	0.233625
$787$	23.8248	0.849261	0.424630	−	0.905367i	$-0.360404\pi$
$787$	23.8248	0.849261	0.424630	+	0.905367i	$0.360404\pi$
$788$	4.45017	0.158531
$789$	−4.54983	−0.161978
$790$	0	0
$791$	−11.1752	−0.397346
$792$	1.27492	0.0453022
$793$	−1.80066	−0.0639434
$794$	−21.4502	−0.761238
$795$	0	0
$796$	11.0000	0.389885
$797$	−12.5498	−0.444538	−0.222269	−	0.974985i	$-0.571346\pi$
$797$	−12.5498	−0.444538	−0.222269	+	0.974985i	$0.571346\pi$
$798$	−9.82475	−0.347792
$799$	−18.8248	−0.665972
$800$	0	0
$801$	8.27492	0.292380
$802$	9.64950	0.340736
$803$	−19.8248	−0.699600
$804$	−4.54983	−0.160460
$805$	0	0
$806$	19.6495	0.692124
$807$	3.27492	0.115283
$808$	−8.27492	−0.291111
$809$	13.4743	0.473730	0.236865	−	0.971543i	$-0.423880\pi$
$809$	13.4743	0.473730	0.236865	+	0.971543i	$0.423880\pi$
$810$	0	0
$811$	−41.0997	−1.44320	−0.721602	−	0.692308i	$-0.756594\pi$
$811$	−41.0997	−1.44320	−0.721602	+	0.692308i	$0.756594\pi$
$812$	28.6495	1.00540
$813$	−1.45017	−0.0508595
$814$	−13.8007	−0.483713
$815$	0	0
$816$	−2.27492	−0.0796380
$817$	−23.8248	−0.833523
$818$	7.00000	0.244749
$819$	9.82475	0.343305
$820$	0	0
$821$	−52.0241	−1.81565	−0.907827	−	0.419346i	$-0.862260\pi$
$821$	−52.0241	−1.81565	−0.907827	+	0.419346i	$0.862260\pi$
$822$	−8.82475	−0.307799
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	−3.00000	−0.104510
$825$	0	0
$826$	37.6495	1.30999
$827$	−48.0997	−1.67259	−0.836295	−	0.548280i	$-0.815283\pi$
$827$	−48.0997	−1.67259	−0.836295	+	0.548280i	$0.815283\pi$
$828$	1.00000	0.0347524
$829$	−4.37459	−0.151936	−0.0759678	−	0.997110i	$-0.524205\pi$
$829$	−4.37459	−0.151936	−0.0759678	+	0.997110i	$0.524205\pi$
$830$	0	0
$831$	5.82475	0.202058
$832$	−3.27492	−0.113537
$833$	4.54983	0.157642
$834$	6.27492	0.217283
$835$	0	0
$836$	−4.17525	−0.144404
$837$	−6.00000	−0.207390
$838$	−3.00000	−0.103633
$839$	3.82475	0.132045	0.0660225	−	0.997818i	$-0.478969\pi$
$839$	3.82475	0.132045	0.0660225	+	0.997818i	$0.478969\pi$
$840$	0	0
$841$	62.1993	2.14480
$842$	−31.0997	−1.07177
$843$	−5.37459	−0.185111
$844$	−20.8248	−0.716818
$845$	0	0
$846$	8.27492	0.284498
$847$	28.1238	0.966344
$848$	10.5498	0.362283
$849$	−16.0000	−0.549119
$850$	0	0
$851$	−10.8248	−0.371068
$852$	8.82475	0.302331
$853$	−18.7251	−0.641135	−0.320567	−	0.947226i	$-0.603874\pi$
$853$	−18.7251	−0.641135	−0.320567	+	0.947226i	$0.603874\pi$
$854$	1.64950	0.0564448
$855$	0	0
$856$	−12.0000	−0.410152
$857$	3.09967	0.105883	0.0529413	−	0.998598i	$-0.483140\pi$
$857$	3.09967	0.105883	0.0529413	+	0.998598i	$0.483140\pi$
$858$	4.17525	0.142541
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	−8.17525	−0.278612
$862$	12.5498	0.427449
$863$	53.9244	1.83561	0.917804	−	0.397033i	$-0.129960\pi$
$863$	53.9244	1.83561	0.917804	+	0.397033i	$0.129960\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	17.6495	0.599755
$867$	11.8248	0.401590
$868$	−18.0000	−0.610960
$869$	15.0756	0.511404
$870$	0	0
$871$	−14.9003	−0.504878
$872$	−2.82475	−0.0956582
$873$	−14.0000	−0.473828
$874$	−3.27492	−0.110776
$875$	0	0
$876$	−15.5498	−0.525380
$877$	−0.900331	−0.0304020	−0.0152010	−	0.999884i	$-0.504839\pi$
$877$	−0.900331	−0.0304020	−0.0152010	+	0.999884i	$0.504839\pi$
$878$	20.0000	0.674967
$879$	−21.6495	−0.730220
$880$	0	0
$881$	−30.7492	−1.03597	−0.517983	−	0.855391i	$-0.673317\pi$
$881$	−30.7492	−1.03597	−0.517983	+	0.855391i	$0.673317\pi$
$882$	−2.00000	−0.0673435
$883$	30.2749	1.01883	0.509416	−	0.860520i	$-0.329861\pi$
$883$	30.2749	1.01883	0.509416	+	0.860520i	$0.329861\pi$
$884$	−7.45017	−0.250576
$885$	0	0
$886$	15.6495	0.525755
$887$	47.3746	1.59068	0.795341	−	0.606162i	$-0.207292\pi$
$887$	47.3746	1.59068	0.795341	+	0.606162i	$0.207292\pi$
$888$	−10.8248	−0.363255
$889$	24.0000	0.804934
$890$	0	0
$891$	−1.27492	−0.0427113
$892$	14.0000	0.468755
$893$	−27.0997	−0.906856
$894$	−2.54983	−0.0852792
$895$	0	0
$896$	3.00000	0.100223
$897$	3.27492	0.109346
$898$	−15.0997	−0.503882
$899$	−57.2990	−1.91103
$900$	0	0
$901$	24.0000	0.799556
$902$	−3.47425	−0.115680
$903$	−21.8248	−0.726282
$904$	−3.72508	−0.123894
$905$	0	0
$906$	−2.54983	−0.0847126
$907$	0.374586	0.0124379	0.00621896	−	0.999981i	$-0.498020\pi$
$907$	0.374586	0.0124379	0.00621896	+	0.999981i	$0.498020\pi$
$908$	10.2749	0.340985
$909$	8.27492	0.274462
$910$	0	0
$911$	−43.8248	−1.45198	−0.725989	−	0.687706i	$-0.758618\pi$
$911$	−43.8248	−1.45198	−0.725989	+	0.687706i	$0.758618\pi$
$912$	−3.27492	−0.108443
$913$	−3.12376	−0.103381
$914$	−39.6495	−1.31149
$915$	0	0
$916$	−19.0997	−0.631071
$917$	−19.6495	−0.648884
$918$	2.27492	0.0750835
$919$	48.4743	1.59902	0.799509	−	0.600654i	$-0.205093\pi$
$919$	48.4743	1.59902	0.799509	+	0.600654i	$0.205093\pi$
$920$	0	0
$921$	6.27492	0.206766
$922$	−0.0996689	−0.00328242
$923$	28.9003	0.951266
$924$	−3.82475	−0.125825
$925$	0	0
$926$	−29.0997	−0.956274
$927$	3.00000	0.0985329
$928$	9.54983	0.313489
$929$	−8.37459	−0.274761	−0.137381	−	0.990518i	$-0.543868\pi$
$929$	−8.37459	−0.274761	−0.137381	+	0.990518i	$0.543868\pi$
$930$	0	0
$931$	6.54983	0.214662
$932$	−14.3746	−0.470855
$933$	10.8248	0.354386
$934$	−21.0000	−0.687141
$935$	0	0
$936$	3.27492	0.107044
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	13.6495	0.445672
$939$	−32.0000	−1.04428
$940$	0	0
$941$	−16.1993	−0.528083	−0.264042	−	0.964511i	$-0.585056\pi$
$941$	−16.1993	−0.528083	−0.264042	+	0.964511i	$0.585056\pi$
$942$	−14.0000	−0.456145
$943$	−2.72508	−0.0887409
$944$	12.5498	0.408462
$945$	0	0
$946$	−9.27492	−0.301554
$947$	51.6495	1.67838	0.839192	−	0.543836i	$-0.183029\pi$
$947$	51.6495	1.67838	0.839192	+	0.543836i	$0.183029\pi$
$948$	11.8248	0.384050
$949$	−50.9244	−1.65308
$950$	0	0
$951$	−13.5498	−0.439383
$952$	6.82475	0.221191
$953$	−35.9244	−1.16371	−0.581853	−	0.813294i	$-0.697672\pi$
$953$	−35.9244	−1.16371	−0.581853	+	0.813294i	$0.697672\pi$
$954$	−10.5498	−0.341564
$955$	0	0
$956$	12.2749	0.396999
$957$	−12.1752	−0.393570
$958$	−7.27492	−0.235042
$959$	26.4743	0.854898
$960$	0	0
$961$	5.00000	0.161290
$962$	−35.4502	−1.14296
$963$	12.0000	0.386695
$964$	10.5498	0.339787
$965$	0	0
$966$	−3.00000	−0.0965234
$967$	−25.6495	−0.824832	−0.412416	−	0.910996i	$-0.635315\pi$
$967$	−25.6495	−0.824832	−0.412416	+	0.910996i	$0.635315\pi$
$968$	9.37459	0.301311
$969$	−7.45017	−0.239334
$970$	0	0
$971$	4.09967	0.131565	0.0657823	−	0.997834i	$-0.479046\pi$
$971$	4.09967	0.131565	0.0657823	+	0.997834i	$0.479046\pi$
$972$	−1.00000	−0.0320750
$973$	−18.8248	−0.603494
$974$	−6.54983	−0.209870
$975$	0	0
$976$	0.549834	0.0175998
$977$	−33.9244	−1.08534	−0.542669	−	0.839947i	$-0.682586\pi$
$977$	−33.9244	−1.08534	−0.542669	+	0.839947i	$0.682586\pi$
$978$	23.6495	0.756228
$979$	−10.5498	−0.337174
$980$	0	0
$981$	2.82475	0.0901874
$982$	8.54983	0.272836
$983$	−36.9244	−1.17771	−0.588853	−	0.808240i	$-0.700420\pi$
$983$	−36.9244	−1.17771	−0.588853	+	0.808240i	$0.700420\pi$
$984$	−2.72508	−0.0868725
$985$	0	0
$986$	21.7251	0.691868
$987$	−24.8248	−0.790181
$988$	−10.7251	−0.341210
$989$	−7.27492	−0.231329
$990$	0	0
$991$	10.5498	0.335127	0.167563	−	0.985861i	$-0.446410\pi$
$991$	10.5498	0.335127	0.167563	+	0.985861i	$0.446410\pi$
$992$	−6.00000	−0.190500
$993$	−16.8248	−0.533917
$994$	−26.4743	−0.839712
$995$	0	0
$996$	−2.45017	−0.0776365
$997$	−8.72508	−0.276326	−0.138163	−	0.990409i	$-0.544120\pi$
$997$	−8.72508	−0.276326	−0.138163	+	0.990409i	$0.544120\pi$
$998$	−11.7251	−0.371151
$999$	10.8248	0.342480

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.a.bd.1.1	✓	2
5.2	odd	4		3450.2.d.z.2899.1		4
5.3	odd	4		3450.2.d.z.2899.3		4
5.4	even	2		3450.2.a.bm.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3450.2.a.bd.1.1	✓	2	1.1	even	1	trivial
3450.2.a.bm.1.1	yes	2	5.4	even	2
3450.2.d.z.2899.1		4	5.2	odd	4
3450.2.d.z.2899.3		4	5.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.27492 q^{11} -1.00000 q^{12} -3.27492 q^{13} +3.00000 q^{14} +1.00000 q^{16} +2.27492 q^{17} -1.00000 q^{18} +3.27492 q^{19} +3.00000 q^{21} +1.27492 q^{22} +1.00000 q^{23} +1.00000 q^{24} +3.27492 q^{26} -1.00000 q^{27} -3.00000 q^{28} -9.54983 q^{29} +6.00000 q^{31} -1.00000 q^{32} +1.27492 q^{33} -2.27492 q^{34} +1.00000 q^{36} -10.8248 q^{37} -3.27492 q^{38} +3.27492 q^{39} -2.72508 q^{41} -3.00000 q^{42} -7.27492 q^{43} -1.27492 q^{44} -1.00000 q^{46} -8.27492 q^{47} -1.00000 q^{48} +2.00000 q^{49} -2.27492 q^{51} -3.27492 q^{52} +10.5498 q^{53} +1.00000 q^{54} +3.00000 q^{56} -3.27492 q^{57} +9.54983 q^{58} +12.5498 q^{59} +0.549834 q^{61} -6.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} -1.27492 q^{66} +4.54983 q^{67} +2.27492 q^{68} -1.00000 q^{69} -8.82475 q^{71} -1.00000 q^{72} +15.5498 q^{73} +10.8248 q^{74} +3.27492 q^{76} +3.82475 q^{77} -3.27492 q^{78} -11.8248 q^{79} +1.00000 q^{81} +2.72508 q^{82} +2.45017 q^{83} +3.00000 q^{84} +7.27492 q^{86} +9.54983 q^{87} +1.27492 q^{88} +8.27492 q^{89} +9.82475 q^{91} +1.00000 q^{92} -6.00000 q^{93} +8.27492 q^{94} +1.00000 q^{96} -14.0000 q^{97} -2.00000 q^{98} -1.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{7} - 2 q^{8} + 2 q^{9} + 5 q^{11} - 2 q^{12} + q^{13} + 6 q^{14} + 2 q^{16} - 3 q^{17} - 2 q^{18} - q^{19} + 6 q^{21} - 5 q^{22} + 2 q^{23} + 2 q^{24} - q^{26} - 2 q^{27} - 6 q^{28} - 4 q^{29} + 12 q^{31} - 2 q^{32} - 5 q^{33} + 3 q^{34} + 2 q^{36} + q^{37} + q^{38} - q^{39} - 13 q^{41} - 6 q^{42} - 7 q^{43} + 5 q^{44} - 2 q^{46} - 9 q^{47} - 2 q^{48} + 4 q^{49} + 3 q^{51} + q^{52} + 6 q^{53} + 2 q^{54} + 6 q^{56} + q^{57} + 4 q^{58} + 10 q^{59} - 14 q^{61} - 12 q^{62} - 6 q^{63} + 2 q^{64} + 5 q^{66} - 6 q^{67} - 3 q^{68} - 2 q^{69} + 5 q^{71} - 2 q^{72} + 16 q^{73} - q^{74} - q^{76} - 15 q^{77} + q^{78} - q^{79} + 2 q^{81} + 13 q^{82} + 20 q^{83} + 6 q^{84} + 7 q^{86} + 4 q^{87} - 5 q^{88} + 9 q^{89} - 3 q^{91} + 2 q^{92} - 12 q^{93} + 9 q^{94} + 2 q^{96} - 28 q^{97} - 4 q^{98} + 5 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^7 - 2 * q^8 + 2 * q^9 + 5 * q^11 - 2 * q^12 + q^13 + 6 * q^14 + 2 * q^16 - 3 * q^17 - 2 * q^18 - q^19 + 6 * q^21 - 5 * q^22 + 2 * q^23 + 2 * q^24 - q^26 - 2 * q^27 - 6 * q^28 - 4 * q^29 + 12 * q^31 - 2 * q^32 - 5 * q^33 + 3 * q^34 + 2 * q^36 + q^37 + q^38 - q^39 - 13 * q^41 - 6 * q^42 - 7 * q^43 + 5 * q^44 - 2 * q^46 - 9 * q^47 - 2 * q^48 + 4 * q^49 + 3 * q^51 + q^52 + 6 * q^53 + 2 * q^54 + 6 * q^56 + q^57 + 4 * q^58 + 10 * q^59 - 14 * q^61 - 12 * q^62 - 6 * q^63 + 2 * q^64 + 5 * q^66 - 6 * q^67 - 3 * q^68 - 2 * q^69 + 5 * q^71 - 2 * q^72 + 16 * q^73 - q^74 - q^76 - 15 * q^77 + q^78 - q^79 + 2 * q^81 + 13 * q^82 + 20 * q^83 + 6 * q^84 + 7 * q^86 + 4 * q^87 - 5 * q^88 + 9 * q^89 - 3 * q^91 + 2 * q^92 - 12 * q^93 + 9 * q^94 + 2 * q^96 - 28 * q^97 - 4 * q^98 + 5 * q^99

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