LMFDB - Embedded newform 625.2.a.e.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [625,2,Mod(1,625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 625 = 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	625.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$4.99065012633$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.6152203125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 $ x^8 - 3x^7 - 8x^6 + 20x^5 + 26x^4 - 35x^3 - 27x^2 + 16*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.68341$ of defining polynomial
Character	$\chi$	$=$	625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.68341 q^{2} -0.710340 q^{3} +0.833870 q^{4} -1.19579 q^{6} -4.59110 q^{7} -1.96307 q^{8} -2.49542 q^{9} +O(q^{10})$	\(q+1.68341 q^{2} -0.710340 q^{3} +0.833870 q^{4} -1.19579 q^{6} -4.59110 q^{7} -1.96307 q^{8} -2.49542 q^{9} +3.91479 q^{11} -0.592332 q^{12} -0.572939 q^{13} -7.72871 q^{14} -4.97240 q^{16} -0.232611 q^{17} -4.20081 q^{18} -5.55010 q^{19} +3.26125 q^{21} +6.59020 q^{22} -4.93267 q^{23} +1.39445 q^{24} -0.964492 q^{26} +3.90362 q^{27} -3.82839 q^{28} +4.13062 q^{29} -3.49531 q^{31} -4.44444 q^{32} -2.78083 q^{33} -0.391580 q^{34} -2.08085 q^{36} -5.41648 q^{37} -9.34309 q^{38} +0.406982 q^{39} +10.4227 q^{41} +5.49002 q^{42} +1.38833 q^{43} +3.26443 q^{44} -8.30370 q^{46} +0.920418 q^{47} +3.53210 q^{48} +14.0782 q^{49} +0.165233 q^{51} -0.477757 q^{52} +1.23118 q^{53} +6.57139 q^{54} +9.01268 q^{56} +3.94246 q^{57} +6.95353 q^{58} +4.50780 q^{59} -11.6588 q^{61} -5.88405 q^{62} +11.4567 q^{63} +2.46298 q^{64} -4.68128 q^{66} -2.95447 q^{67} -0.193968 q^{68} +3.50387 q^{69} +3.20551 q^{71} +4.89869 q^{72} -10.2922 q^{73} -9.11816 q^{74} -4.62806 q^{76} -17.9732 q^{77} +0.685118 q^{78} -9.61509 q^{79} +4.71335 q^{81} +17.5457 q^{82} -10.4834 q^{83} +2.71946 q^{84} +2.33712 q^{86} -2.93415 q^{87} -7.68502 q^{88} +7.25828 q^{89} +2.63042 q^{91} -4.11320 q^{92} +2.48286 q^{93} +1.54944 q^{94} +3.15707 q^{96} -8.31971 q^{97} +23.6994 q^{98} -9.76903 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{2} - 5 q^{3} + 11 q^{4} - 4 q^{6} - 10 q^{7} - 15 q^{8} + 9 q^{9}+O(q^{10}) $ 8 * q - 5 * q^2 - 5 * q^3 + 11 * q^4 - 4 * q^6 - 10 * q^7 - 15 * q^8 + 9 * q^9	$ 8 q - 5 q^{2} - 5 q^{3} + 11 q^{4} - 4 q^{6} - 10 q^{7} - 15 q^{8} + 9 q^{9} + q^{11} - 10 q^{12} - 10 q^{13} - 8 q^{14} + 13 q^{16} - 15 q^{17} + 5 q^{18} - 10 q^{19} - 14 q^{21} + 5 q^{22} - 30 q^{23} + 5 q^{24} + 11 q^{26} - 20 q^{27} + 5 q^{28} + 10 q^{29} - 9 q^{31} - 30 q^{32} - 5 q^{33} + 7 q^{34} + 3 q^{36} + 10 q^{37} - 20 q^{38} + 8 q^{39} - 4 q^{41} + 35 q^{42} - 18 q^{44} - 9 q^{46} - 30 q^{47} - 5 q^{48} - 4 q^{49} - 14 q^{51} - 5 q^{52} - 10 q^{53} - 20 q^{54} + 10 q^{57} + 30 q^{58} - 5 q^{59} + 6 q^{61} - 10 q^{62} - 9 q^{64} - 18 q^{66} - 10 q^{67} - 40 q^{68} + 3 q^{69} - 9 q^{71} + 15 q^{72} - 18 q^{74} - 10 q^{76} - 5 q^{77} + 30 q^{78} - 20 q^{79} + 8 q^{81} + 45 q^{82} - 40 q^{83} - 28 q^{84} - 24 q^{86} - 40 q^{87} + 40 q^{88} - 5 q^{89} + 6 q^{91} - 15 q^{92} + 40 q^{93} + 47 q^{94} + 71 q^{96} + 30 q^{98} - 22 q^{99}+O(q^{100}) $ 8 * q - 5 * q^2 - 5 * q^3 + 11 * q^4 - 4 * q^6 - 10 * q^7 - 15 * q^8 + 9 * q^9 + q^11 - 10 * q^12 - 10 * q^13 - 8 * q^14 + 13 * q^16 - 15 * q^17 + 5 * q^18 - 10 * q^19 - 14 * q^21 + 5 * q^22 - 30 * q^23 + 5 * q^24 + 11 * q^26 - 20 * q^27 + 5 * q^28 + 10 * q^29 - 9 * q^31 - 30 * q^32 - 5 * q^33 + 7 * q^34 + 3 * q^36 + 10 * q^37 - 20 * q^38 + 8 * q^39 - 4 * q^41 + 35 * q^42 - 18 * q^44 - 9 * q^46 - 30 * q^47 - 5 * q^48 - 4 * q^49 - 14 * q^51 - 5 * q^52 - 10 * q^53 - 20 * q^54 + 10 * q^57 + 30 * q^58 - 5 * q^59 + 6 * q^61 - 10 * q^62 - 9 * q^64 - 18 * q^66 - 10 * q^67 - 40 * q^68 + 3 * q^69 - 9 * q^71 + 15 * q^72 - 18 * q^74 - 10 * q^76 - 5 * q^77 + 30 * q^78 - 20 * q^79 + 8 * q^81 + 45 * q^82 - 40 * q^83 - 28 * q^84 - 24 * q^86 - 40 * q^87 + 40 * q^88 - 5 * q^89 + 6 * q^91 - 15 * q^92 + 40 * q^93 + 47 * q^94 + 71 * q^96 + 30 * q^98 - 22 * 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 $

$2$	1.68341	1.19035	0.595175	−	0.803596i	$-0.297083\pi$
$2$	1.68341	1.19035	0.595175	+	0.803596i	$0.297083\pi$
$3$	−0.710340	−0.410115	−0.205058	−	0.978750i	$-0.565738\pi$
$3$	−0.710340	−0.410115	−0.205058	+	0.978750i	$0.565738\pi$
$4$	0.833870	0.416935
$5$	0	0
$5$	0	0
$6$	−1.19579	−0.488181
$7$	−4.59110	−1.73527	−0.867637	−	0.497198i	$-0.834362\pi$
$7$	−4.59110	−1.73527	−0.867637	+	0.497198i	$0.834362\pi$
$8$	−1.96307	−0.694052
$9$	−2.49542	−0.831806
$10$	0	0
$11$	3.91479	1.18035	0.590177	−	0.807274i	$-0.299058\pi$
$11$	3.91479	1.18035	0.590177	+	0.807274i	$0.299058\pi$
$12$	−0.592332	−0.170991
$13$	−0.572939	−0.158905	−0.0794524	−	0.996839i	$-0.525317\pi$
$13$	−0.572939	−0.158905	−0.0794524	+	0.996839i	$0.525317\pi$
$14$	−7.72871	−2.06559
$15$	0	0
$16$	−4.97240	−1.24310
$17$	−0.232611	−0.0564165	−0.0282082	−	0.999602i	$-0.508980\pi$
$17$	−0.232611	−0.0564165	−0.0282082	+	0.999602i	$0.508980\pi$
$18$	−4.20081	−0.990140
$19$	−5.55010	−1.27328	−0.636640	−	0.771161i	$-0.719676\pi$
$19$	−5.55010	−1.27328	−0.636640	+	0.771161i	$0.719676\pi$
$20$	0	0
$21$	3.26125	0.711662
$22$	6.59020	1.40503
$23$	−4.93267	−1.02853	−0.514266	−	0.857631i	$-0.671936\pi$
$23$	−4.93267	−1.02853	−0.514266	+	0.857631i	$0.671936\pi$
$24$	1.39445	0.284641
$25$	0	0
$26$	−0.964492	−0.189152
$27$	3.90362	0.751251
$28$	−3.82839	−0.723497
$29$	4.13062	0.767037	0.383519	−	0.923533i	$-0.374712\pi$
$29$	4.13062	0.767037	0.383519	+	0.923533i	$0.374712\pi$
$30$	0	0
$31$	−3.49531	−0.627777	−0.313888	−	0.949460i	$-0.601632\pi$
$31$	−3.49531	−0.627777	−0.313888	+	0.949460i	$0.601632\pi$
$32$	−4.44444	−0.785674
$33$	−2.78083	−0.484081
$34$	−0.391580	−0.0671554
$35$	0	0
$36$	−2.08085	−0.346809
$37$	−5.41648	−0.890464	−0.445232	−	0.895415i	$-0.646879\pi$
$37$	−5.41648	−0.890464	−0.445232	+	0.895415i	$0.646879\pi$
$38$	−9.34309	−1.51565
$39$	0.406982	0.0651693
$40$	0	0
$41$	10.4227	1.62775	0.813876	−	0.581039i	$-0.197354\pi$
$41$	10.4227	1.62775	0.813876	+	0.581039i	$0.197354\pi$
$42$	5.49002	0.847128
$43$	1.38833	0.211718	0.105859	−	0.994381i	$-0.466241\pi$
$43$	1.38833	0.211718	0.105859	+	0.994381i	$0.466241\pi$
$44$	3.26443	0.492131
$45$	0	0
$46$	−8.30370	−1.22431
$47$	0.920418	0.134257	0.0671284	−	0.997744i	$-0.478616\pi$
$47$	0.920418	0.134257	0.0671284	+	0.997744i	$0.478616\pi$
$48$	3.53210	0.509814
$49$	14.0782	2.01118
$50$	0	0
$51$	0.165233	0.0231373
$52$	−0.477757	−0.0662530
$53$	1.23118	0.169115	0.0845576	−	0.996419i	$-0.473052\pi$
$53$	1.23118	0.169115	0.0845576	+	0.996419i	$0.473052\pi$
$54$	6.57139	0.894253
$55$	0	0
$56$	9.01268	1.20437
$57$	3.94246	0.522191
$58$	6.95353	0.913043
$59$	4.50780	0.586866	0.293433	−	0.955980i	$-0.405202\pi$
$59$	4.50780	0.586866	0.293433	+	0.955980i	$0.405202\pi$
$60$	0	0
$61$	−11.6588	−1.49275	−0.746376	−	0.665525i	$-0.768208\pi$
$61$	−11.6588	−1.49275	−0.746376	+	0.665525i	$0.768208\pi$
$62$	−5.88405	−0.747275
$63$	11.4567	1.44341
$64$	2.46298	0.307873
$65$	0	0
$66$	−4.68128	−0.576226
$67$	−2.95447	−0.360946	−0.180473	−	0.983580i	$-0.557763\pi$
$67$	−2.95447	−0.360946	−0.180473	+	0.983580i	$0.557763\pi$
$68$	−0.193968	−0.0235220
$69$	3.50387	0.421817
$70$	0	0
$71$	3.20551	0.380424	0.190212	−	0.981743i	$-0.439082\pi$
$71$	3.20551	0.380424	0.190212	+	0.981743i	$0.439082\pi$
$72$	4.89869	0.577316
$73$	−10.2922	−1.20461	−0.602306	−	0.798266i	$-0.705751\pi$
$73$	−10.2922	−1.20461	−0.602306	+	0.798266i	$0.705751\pi$
$74$	−9.11816	−1.05996
$75$	0	0
$76$	−4.62806	−0.530875
$77$	−17.9732	−2.04824
$78$	0.685118	0.0775743
$79$	−9.61509	−1.08178	−0.540891	−	0.841093i	$-0.681913\pi$
$79$	−9.61509	−1.08178	−0.540891	+	0.841093i	$0.681913\pi$
$80$	0	0
$81$	4.71335	0.523706
$82$	17.5457	1.93759
$83$	−10.4834	−1.15070	−0.575351	−	0.817906i	$-0.695135\pi$
$83$	−10.4834	−1.15070	−0.575351	+	0.817906i	$0.695135\pi$
$84$	2.71946	0.296717
$85$	0	0
$86$	2.33712	0.252018
$87$	−2.93415	−0.314574
$88$	−7.68502	−0.819226
$89$	7.25828	0.769376	0.384688	−	0.923047i	$-0.374309\pi$
$89$	7.25828	0.769376	0.384688	+	0.923047i	$0.374309\pi$
$90$	0	0
$91$	2.63042	0.275743
$92$	−4.11320	−0.428831
$93$	2.48286	0.257461
$94$	1.54944	0.159813
$95$	0	0
$96$	3.15707	0.322217
$97$	−8.31971	−0.844739	−0.422369	−	0.906424i	$-0.638801\pi$
$97$	−8.31971	−0.844739	−0.422369	+	0.906424i	$0.638801\pi$
$98$	23.6994	2.39401
$99$	−9.76903	−0.981824
$100$	0	0
$101$	−3.56513	−0.354744	−0.177372	−	0.984144i	$-0.556760\pi$
$101$	−3.56513	−0.354744	−0.177372	+	0.984144i	$0.556760\pi$
$102$	0.278155	0.0275415
$103$	−0.399323	−0.0393465	−0.0196732	−	0.999806i	$-0.506263\pi$
$103$	−0.399323	−0.0393465	−0.0196732	+	0.999806i	$0.506263\pi$
$104$	1.12472	0.110288
$105$	0	0
$106$	2.07258	0.201306
$107$	−1.64372	−0.158904	−0.0794522	−	0.996839i	$-0.525317\pi$
$107$	−1.64372	−0.158904	−0.0794522	+	0.996839i	$0.525317\pi$
$108$	3.25511	0.313223
$109$	0.0749154	0.00717560	0.00358780	−	0.999994i	$-0.498858\pi$
$109$	0.0749154	0.00717560	0.00358780	+	0.999994i	$0.498858\pi$
$110$	0	0
$111$	3.84755	0.365193
$112$	22.8288	2.15712
$113$	14.1328	1.32951	0.664753	−	0.747063i	$-0.268537\pi$
$113$	14.1328	1.32951	0.664753	+	0.747063i	$0.268537\pi$
$114$	6.63677	0.621591
$115$	0	0
$116$	3.44440	0.319805
$117$	1.42972	0.132178
$118$	7.58848	0.698576
$119$	1.06794	0.0978981
$120$	0	0
$121$	4.32557	0.393234
$122$	−19.6265	−1.77690
$123$	−7.40366	−0.667565
$124$	−2.91464	−0.261742
$125$	0	0
$126$	19.2864	1.71817
$127$	11.8124	1.04819	0.524093	−	0.851661i	$-0.324405\pi$
$127$	11.8124	1.04819	0.524093	+	0.851661i	$0.324405\pi$
$128$	13.0351	1.15215
$129$	−0.986184	−0.0868287
$130$	0	0
$131$	−16.7373	−1.46234	−0.731170	−	0.682195i	$-0.761026\pi$
$131$	−16.7373	−1.46234	−0.731170	+	0.682195i	$0.761026\pi$
$132$	−2.31885	−0.201830
$133$	25.4811	2.20949
$134$	−4.97359	−0.429653
$135$	0	0
$136$	0.456633	0.0391560
$137$	10.4221	0.890423	0.445211	−	0.895425i	$-0.353128\pi$
$137$	10.4221	0.890423	0.445211	+	0.895425i	$0.353128\pi$
$138$	5.89845	0.502110
$139$	6.65993	0.564888	0.282444	−	0.959284i	$-0.408855\pi$
$139$	6.65993	0.564888	0.282444	+	0.959284i	$0.408855\pi$
$140$	0	0
$141$	−0.653810	−0.0550608
$142$	5.39619	0.452838
$143$	−2.24294	−0.187564
$144$	12.4082	1.03402
$145$	0	0
$146$	−17.3260	−1.43391
$147$	−10.0003	−0.824814
$148$	−4.51664	−0.371266
$149$	−12.0316	−0.985667	−0.492834	−	0.870124i	$-0.664039\pi$
$149$	−12.0316	−0.985667	−0.492834	+	0.870124i	$0.664039\pi$
$150$	0	0
$151$	−1.54218	−0.125501	−0.0627505	−	0.998029i	$-0.519987\pi$
$151$	−1.54218	−0.125501	−0.0627505	+	0.998029i	$0.519987\pi$
$152$	10.8953	0.883722
$153$	0.580462	0.0469276
$154$	−30.2563	−2.43812
$155$	0	0
$156$	0.339370	0.0271714
$157$	−9.82482	−0.784106	−0.392053	−	0.919943i	$-0.628235\pi$
$157$	−9.82482	−0.784106	−0.392053	+	0.919943i	$0.628235\pi$
$158$	−16.1861	−1.28770
$159$	−0.874555	−0.0693567
$160$	0	0
$161$	22.6464	1.78478
$162$	7.93451	0.623394
$163$	−5.58107	−0.437143	−0.218572	−	0.975821i	$-0.570140\pi$
$163$	−5.58107	−0.437143	−0.218572	+	0.975821i	$0.570140\pi$
$164$	8.69117	0.678667
$165$	0	0
$166$	−17.6479	−1.36974
$167$	−21.8607	−1.69163	−0.845817	−	0.533473i	$-0.820887\pi$
$167$	−21.8607	−1.69163	−0.845817	+	0.533473i	$0.820887\pi$
$168$	−6.40207	−0.493930
$169$	−12.6717	−0.974749
$170$	0	0
$171$	13.8498	1.05912
$172$	1.15768	0.0882726
$173$	23.4385	1.78200	0.891000	−	0.454004i	$-0.150005\pi$
$173$	23.4385	1.78200	0.891000	+	0.454004i	$0.150005\pi$
$174$	−4.93937	−0.374453
$175$	0	0
$176$	−19.4659	−1.46730
$177$	−3.20207	−0.240683
$178$	12.2187	0.915827
$179$	−6.31873	−0.472284	−0.236142	−	0.971719i	$-0.575883\pi$
$179$	−6.31873	−0.472284	−0.236142	+	0.971719i	$0.575883\pi$
$180$	0	0
$181$	−13.3377	−0.991385	−0.495693	−	0.868498i	$-0.665086\pi$
$181$	−13.3377	−0.991385	−0.495693	+	0.868498i	$0.665086\pi$
$182$	4.42808	0.328231
$183$	8.28169	0.612200
$184$	9.68319	0.713854
$185$	0	0
$186$	4.17968	0.306469
$187$	−0.910624	−0.0665914
$188$	0.767510	0.0559764
$189$	−17.9219	−1.30363
$190$	0	0
$191$	−2.78083	−0.201214	−0.100607	−	0.994926i	$-0.532078\pi$
$191$	−2.78083	−0.201214	−0.100607	+	0.994926i	$0.532078\pi$
$192$	−1.74956	−0.126263
$193$	22.5667	1.62438	0.812192	−	0.583391i	$-0.198274\pi$
$193$	22.5667	1.62438	0.812192	+	0.583391i	$0.198274\pi$
$194$	−14.0055	−1.00554
$195$	0	0
$196$	11.7394	0.838530
$197$	1.27182	0.0906135	0.0453068	−	0.998973i	$-0.485573\pi$
$197$	1.27182	0.0906135	0.0453068	+	0.998973i	$0.485573\pi$
$198$	−16.4453	−1.16872
$199$	−8.62648	−0.611515	−0.305757	−	0.952109i	$-0.598910\pi$
$199$	−8.62648	−0.611515	−0.305757	+	0.952109i	$0.598910\pi$
$200$	0	0
$201$	2.09868	0.148030
$202$	−6.00158	−0.422270
$203$	−18.9641	−1.33102
$204$	0.137783	0.00964674
$205$	0	0
$206$	−0.672225	−0.0468361
$207$	12.3091	0.855538
$208$	2.84888	0.197535
$209$	−21.7275	−1.50292
$210$	0	0
$211$	−21.0316	−1.44787	−0.723937	−	0.689866i	$-0.757670\pi$
$211$	−21.0316	−1.44787	−0.723937	+	0.689866i	$0.757670\pi$
$212$	1.02664	0.0705101
$213$	−2.27700	−0.156018
$214$	−2.76705	−0.189152
$215$	0	0
$216$	−7.66309	−0.521407
$217$	16.0473	1.08936
$218$	0.126113	0.00854148
$219$	7.31097	0.494029
$220$	0	0
$221$	0.133272	0.00896485
$222$	6.47700	0.434708
$223$	−6.33537	−0.424248	−0.212124	−	0.977243i	$-0.568038\pi$
$223$	−6.33537	−0.424248	−0.212124	+	0.977243i	$0.568038\pi$
$224$	20.4049	1.36336
$225$	0	0
$226$	23.7914	1.58258
$227$	−11.2624	−0.747512	−0.373756	−	0.927527i	$-0.621930\pi$
$227$	−11.2624	−0.747512	−0.373756	+	0.927527i	$0.621930\pi$
$228$	3.28750	0.217720
$229$	15.0408	0.993927	0.496964	−	0.867771i	$-0.334448\pi$
$229$	15.0408	0.993927	0.496964	+	0.867771i	$0.334448\pi$
$230$	0	0
$231$	12.7671	0.840013
$232$	−8.10872	−0.532363
$233$	−7.82749	−0.512796	−0.256398	−	0.966571i	$-0.582536\pi$
$233$	−7.82749	−0.512796	−0.256398	+	0.966571i	$0.582536\pi$
$234$	2.40681	0.157338
$235$	0	0
$236$	3.75892	0.244685
$237$	6.82999	0.443655
$238$	1.79778	0.116533
$239$	8.73886	0.565270	0.282635	−	0.959228i	$-0.408792\pi$
$239$	8.73886	0.565270	0.282635	+	0.959228i	$0.408792\pi$
$240$	0	0
$241$	−0.600892	−0.0387068	−0.0193534	−	0.999813i	$-0.506161\pi$
$241$	−0.600892	−0.0387068	−0.0193534	+	0.999813i	$0.506161\pi$
$242$	7.28171	0.468086
$243$	−15.0589	−0.966031
$244$	−9.72190	−0.622381
$245$	0	0
$246$	−12.4634	−0.794637
$247$	3.17987	0.202330
$248$	6.86156	0.435710
$249$	7.44678	0.471921
$250$	0	0
$251$	−14.1908	−0.895712	−0.447856	−	0.894106i	$-0.647812\pi$
$251$	−14.1908	−0.895712	−0.447856	+	0.894106i	$0.647812\pi$
$252$	9.55342	0.601809
$253$	−19.3103	−1.21403
$254$	19.8852	1.24771
$255$	0	0
$256$	17.0174	1.06359
$257$	−17.6859	−1.10322	−0.551609	−	0.834103i	$-0.685986\pi$
$257$	−17.6859	−1.10322	−0.551609	+	0.834103i	$0.685986\pi$
$258$	−1.66015	−0.103357
$259$	24.8676	1.54520
$260$	0	0
$261$	−10.3076	−0.638026
$262$	−28.1757	−1.74070
$263$	−24.3348	−1.50055	−0.750274	−	0.661127i	$-0.770078\pi$
$263$	−24.3348	−1.50055	−0.750274	+	0.661127i	$0.770078\pi$
$264$	5.45898	0.335977
$265$	0	0
$266$	42.8951	2.63007
$267$	−5.15585	−0.315533
$268$	−2.46365	−0.150491
$269$	29.9819	1.82803	0.914013	−	0.405685i	$-0.132967\pi$
$269$	29.9819	1.82803	0.914013	+	0.405685i	$0.132967\pi$
$270$	0	0
$271$	−27.6981	−1.68254	−0.841271	−	0.540613i	$-0.818192\pi$
$271$	−27.6981	−1.68254	−0.841271	+	0.540613i	$0.818192\pi$
$272$	1.15664	0.0701314
$273$	−1.86850	−0.113087
$274$	17.5447	1.05992
$275$	0	0
$276$	2.92177	0.175870
$277$	2.29407	0.137838	0.0689188	−	0.997622i	$-0.478045\pi$
$277$	2.29407	0.137838	0.0689188	+	0.997622i	$0.478045\pi$
$278$	11.2114	0.672415
$279$	8.72226	0.522188
$280$	0	0
$281$	−1.61829	−0.0965388	−0.0482694	−	0.998834i	$-0.515371\pi$
$281$	−1.61829	−0.0965388	−0.0482694	+	0.998834i	$0.515371\pi$
$282$	−1.10063	−0.0655416
$283$	−12.9922	−0.772306	−0.386153	−	0.922435i	$-0.626196\pi$
$283$	−12.9922	−0.772306	−0.386153	+	0.922435i	$0.626196\pi$
$284$	2.67298	0.158612
$285$	0	0
$286$	−3.77578	−0.223267
$287$	−47.8517	−2.82459
$288$	11.0907	0.653528
$289$	−16.9459	−0.996817
$290$	0	0
$291$	5.90983	0.346440
$292$	−8.58236	−0.502245
$293$	−11.7009	−0.683572	−0.341786	−	0.939778i	$-0.611032\pi$
$293$	−11.7009	−0.683572	−0.341786	+	0.939778i	$0.611032\pi$
$294$	−16.8347	−0.981818
$295$	0	0
$296$	10.6330	0.618028
$297$	15.2818	0.886742
$298$	−20.2541	−1.17329
$299$	2.82612	0.163439
$300$	0	0
$301$	−6.37395	−0.367388
$302$	−2.59613	−0.149390
$303$	2.53246	0.145486
$304$	27.5973	1.58281
$305$	0	0
$306$	0.977155	0.0558603
$307$	26.5673	1.51628	0.758138	−	0.652094i	$-0.226109\pi$
$307$	26.5673	1.51628	0.758138	+	0.652094i	$0.226109\pi$
$308$	−14.9873	−0.853982
$309$	0.283655	0.0161366
$310$	0	0
$311$	−13.4910	−0.765005	−0.382502	−	0.923955i	$-0.624938\pi$
$311$	−13.4910	−0.765005	−0.382502	+	0.923955i	$0.624938\pi$
$312$	−0.798936	−0.0452308
$313$	16.9944	0.960578	0.480289	−	0.877110i	$-0.340532\pi$
$313$	16.9944	0.960578	0.480289	+	0.877110i	$0.340532\pi$
$314$	−16.5392	−0.933361
$315$	0	0
$316$	−8.01774	−0.451033
$317$	16.7959	0.943351	0.471675	−	0.881772i	$-0.343649\pi$
$317$	16.7959	0.943351	0.471675	+	0.881772i	$0.343649\pi$
$318$	−1.47224	−0.0825588
$319$	16.1705	0.905375
$320$	0	0
$321$	1.16760	0.0651691
$322$	38.1231	2.12452
$323$	1.29101	0.0718340
$324$	3.93033	0.218351
$325$	0	0
$326$	−9.39524	−0.520354
$327$	−0.0532154	−0.00294282
$328$	−20.4605	−1.12974
$329$	−4.22574	−0.232972
$330$	0	0
$331$	12.8344	0.705442	0.352721	−	0.935729i	$-0.385257\pi$
$331$	12.8344	0.705442	0.352721	+	0.935729i	$0.385257\pi$
$332$	−8.74180	−0.479768
$333$	13.5164	0.740693
$334$	−36.8006	−2.01364
$335$	0	0
$336$	−16.2162	−0.884668
$337$	21.2375	1.15688	0.578440	−	0.815725i	$-0.303662\pi$
$337$	21.2375	1.15688	0.578440	+	0.815725i	$0.303662\pi$
$338$	−21.3317	−1.16029
$339$	−10.0391	−0.545251
$340$	0	0
$341$	−13.6834	−0.740998
$342$	23.3149	1.26073
$343$	−32.4969	−1.75467
$344$	−2.72539	−0.146943
$345$	0	0
$346$	39.4567	2.12120
$347$	28.6281	1.53684	0.768419	−	0.639947i	$-0.221044\pi$
$347$	28.6281	1.53684	0.768419	+	0.639947i	$0.221044\pi$
$348$	−2.44670	−0.131157
$349$	19.9124	1.06588	0.532942	−	0.846152i	$-0.321086\pi$
$349$	19.9124	1.06588	0.532942	+	0.846152i	$0.321086\pi$
$350$	0	0
$351$	−2.23654	−0.119377
$352$	−17.3990	−0.927372
$353$	24.6916	1.31420	0.657099	−	0.753804i	$-0.271783\pi$
$353$	24.6916	1.31420	0.657099	+	0.753804i	$0.271783\pi$
$354$	−5.39041	−0.286497
$355$	0	0
$356$	6.05246	0.320780
$357$	−0.758602	−0.0401495
$358$	−10.6370	−0.562184
$359$	21.5011	1.13478	0.567391	−	0.823448i	$-0.307953\pi$
$359$	21.5011	1.13478	0.567391	+	0.823448i	$0.307953\pi$
$360$	0	0
$361$	11.8036	0.621241
$362$	−22.4529	−1.18010
$363$	−3.07263	−0.161271
$364$	2.19343	0.114967
$365$	0	0
$366$	13.9415	0.728733
$367$	−1.14383	−0.0597075	−0.0298538	−	0.999554i	$-0.509504\pi$
$367$	−1.14383	−0.0597075	−0.0298538	+	0.999554i	$0.509504\pi$
$368$	24.5272	1.27857
$369$	−26.0090	−1.35397
$370$	0	0
$371$	−5.65246	−0.293461
$372$	2.07038	0.107344
$373$	11.5395	0.597493	0.298747	−	0.954332i	$-0.403431\pi$
$373$	11.5395	0.597493	0.298747	+	0.954332i	$0.403431\pi$
$374$	−1.53295	−0.0792671
$375$	0	0
$376$	−1.80685	−0.0931812
$377$	−2.36660	−0.121886
$378$	−30.1699	−1.55177
$379$	21.1053	1.08410	0.542052	−	0.840345i	$-0.317648\pi$
$379$	21.1053	1.08410	0.542052	+	0.840345i	$0.317648\pi$
$380$	0	0
$381$	−8.39086	−0.429877
$382$	−4.68127	−0.239515
$383$	−0.858129	−0.0438483	−0.0219242	−	0.999760i	$-0.506979\pi$
$383$	−0.858129	−0.0438483	−0.0219242	+	0.999760i	$0.506979\pi$
$384$	−9.25935	−0.472514
$385$	0	0
$386$	37.9889	1.93359
$387$	−3.46445	−0.176108
$388$	−6.93756	−0.352201
$389$	−33.9346	−1.72055	−0.860277	−	0.509827i	$-0.829710\pi$
$389$	−33.9346	−1.72055	−0.860277	+	0.509827i	$0.829710\pi$
$390$	0	0
$391$	1.14739	0.0580262
$392$	−27.6366	−1.39586
$393$	11.8891	0.599728
$394$	2.14100	0.107862
$395$	0	0
$396$	−8.14610	−0.409357
$397$	−26.8286	−1.34649	−0.673245	−	0.739420i	$-0.735100\pi$
$397$	−26.8286	−1.34649	−0.673245	+	0.739420i	$0.735100\pi$
$398$	−14.5219	−0.727917
$399$	−18.1002	−0.906145
$400$	0	0
$401$	3.79757	0.189642	0.0948208	−	0.995494i	$-0.469772\pi$
$401$	3.79757	0.189642	0.0948208	+	0.995494i	$0.469772\pi$
$402$	3.53294	0.176207
$403$	2.00260	0.0997567
$404$	−2.97286	−0.147905
$405$	0	0
$406$	−31.9244	−1.58438
$407$	−21.2044	−1.05106
$408$	−0.324365	−0.0160585
$409$	−14.1754	−0.700927	−0.350463	−	0.936576i	$-0.613976\pi$
$409$	−14.1754	−0.700927	−0.350463	+	0.936576i	$0.613976\pi$
$410$	0	0
$411$	−7.40326	−0.365176
$412$	−0.332984	−0.0164049
$413$	−20.6958	−1.01837
$414$	20.7212	1.01839
$415$	0	0
$416$	2.54640	0.124847
$417$	−4.73082	−0.231669
$418$	−36.5762	−1.78900
$419$	−19.5969	−0.957369	−0.478685	−	0.877987i	$-0.658886\pi$
$419$	−19.5969	−0.957369	−0.478685	+	0.877987i	$0.658886\pi$
$420$	0	0
$421$	−25.7840	−1.25664	−0.628318	−	0.777957i	$-0.716256\pi$
$421$	−25.7840	−1.25664	−0.628318	+	0.777957i	$0.716256\pi$
$422$	−35.4048	−1.72348
$423$	−2.29683	−0.111676
$424$	−2.41689	−0.117375
$425$	0	0
$426$	−3.83313	−0.185716
$427$	53.5266	2.59033
$428$	−1.37065	−0.0662528
$429$	1.59325	0.0769227
$430$	0	0
$431$	9.42533	0.454002	0.227001	−	0.973894i	$-0.427108\pi$
$431$	9.42533	0.454002	0.227001	+	0.973894i	$0.427108\pi$
$432$	−19.4103	−0.933881
$433$	−1.75161	−0.0841770	−0.0420885	−	0.999114i	$-0.513401\pi$
$433$	−1.75161	−0.0841770	−0.0420885	+	0.999114i	$0.513401\pi$
$434$	27.0143	1.29673
$435$	0	0
$436$	0.0624697	0.00299176
$437$	27.3768	1.30961
$438$	12.3074	0.588068
$439$	28.0830	1.34033	0.670165	−	0.742212i	$-0.266223\pi$
$439$	28.0830	1.34033	0.670165	+	0.742212i	$0.266223\pi$
$440$	0	0
$441$	−35.1311	−1.67291
$442$	0.224352	0.0106713
$443$	29.9110	1.42111	0.710557	−	0.703640i	$-0.248443\pi$
$443$	29.9110	1.42111	0.710557	+	0.703640i	$0.248443\pi$
$444$	3.20835	0.152262
$445$	0	0
$446$	−10.6650	−0.505004
$447$	8.54653	0.404237
$448$	−11.3078	−0.534244
$449$	6.29974	0.297303	0.148652	−	0.988890i	$-0.452507\pi$
$449$	6.29974	0.297303	0.148652	+	0.988890i	$0.452507\pi$
$450$	0	0
$451$	40.8026	1.92132
$452$	11.7850	0.554318
$453$	1.09547	0.0514699
$454$	−18.9593	−0.889802
$455$	0	0
$456$	−7.73934	−0.362428
$457$	−19.1809	−0.897243	−0.448622	−	0.893722i	$-0.648085\pi$
$457$	−19.1809	−0.897243	−0.448622	+	0.893722i	$0.648085\pi$
$458$	25.3199	1.18312
$459$	−0.908025	−0.0423830
$460$	0	0
$461$	7.07110	0.329334	0.164667	−	0.986349i	$-0.447345\pi$
$461$	7.07110	0.329334	0.164667	+	0.986349i	$0.447345\pi$
$462$	21.4923	0.999910
$463$	9.61842	0.447006	0.223503	−	0.974703i	$-0.428251\pi$
$463$	9.61842	0.447006	0.223503	+	0.974703i	$0.428251\pi$
$464$	−20.5391	−0.953504
$465$	0	0
$466$	−13.1769	−0.610407
$467$	19.2220	0.889488	0.444744	−	0.895658i	$-0.353295\pi$
$467$	19.2220	0.889488	0.444744	+	0.895658i	$0.353295\pi$
$468$	1.19220	0.0551096
$469$	13.5643	0.626341
$470$	0	0
$471$	6.97896	0.321574
$472$	−8.84915	−0.407315
$473$	5.43500	0.249902
$474$	11.4977	0.528105
$475$	0	0
$476$	0.890525	0.0408172
$477$	−3.07230	−0.140671
$478$	14.7111	0.672869
$479$	−37.9996	−1.73625	−0.868123	−	0.496350i	$-0.834673\pi$
$479$	−37.9996	−1.73625	−0.868123	+	0.496350i	$0.834673\pi$
$480$	0	0
$481$	3.10332	0.141499
$482$	−1.01155	−0.0460747
$483$	−16.0866	−0.731967
$484$	3.60696	0.163953
$485$	0	0
$486$	−25.3504	−1.14992
$487$	16.7490	0.758969	0.379485	−	0.925198i	$-0.376101\pi$
$487$	16.7490	0.758969	0.379485	+	0.925198i	$0.376101\pi$
$488$	22.8870	1.03605
$489$	3.96446	0.179279
$490$	0	0
$491$	−8.95055	−0.403933	−0.201966	−	0.979392i	$-0.564733\pi$
$491$	−8.95055	−0.403933	−0.201966	+	0.979392i	$0.564733\pi$
$492$	−6.17369	−0.278332
$493$	−0.960829	−0.0432736
$494$	5.35302	0.240844
$495$	0	0
$496$	17.3801	0.780389
$497$	−14.7168	−0.660140
$498$	12.5360	0.561751
$499$	−36.3310	−1.62640	−0.813200	−	0.581985i	$-0.802276\pi$
$499$	−36.3310	−1.62640	−0.813200	+	0.581985i	$0.802276\pi$
$500$	0	0
$501$	15.5286	0.693765
$502$	−23.8889	−1.06621
$503$	−12.2945	−0.548184	−0.274092	−	0.961703i	$-0.588377\pi$
$503$	−12.2945	−0.548184	−0.274092	+	0.961703i	$0.588377\pi$
$504$	−22.4904	−1.00180
$505$	0	0
$506$	−32.5072	−1.44512
$507$	9.00125	0.399759
$508$	9.85005	0.437025
$509$	−31.1292	−1.37978	−0.689889	−	0.723915i	$-0.742341\pi$
$509$	−31.1292	−1.37978	−0.689889	+	0.723915i	$0.742341\pi$
$510$	0	0
$511$	47.2526	2.09033
$512$	2.57716	0.113895
$513$	−21.6654	−0.956553
$514$	−29.7727	−1.31322
$515$	0	0
$516$	−0.822350	−0.0362019
$517$	3.60324	0.158470
$518$	41.8624	1.83933
$519$	−16.6493	−0.730825
$520$	0	0
$521$	16.2169	0.710476	0.355238	−	0.934776i	$-0.384400\pi$
$521$	16.2169	0.710476	0.355238	+	0.934776i	$0.384400\pi$
$522$	−17.3520	−0.759475
$523$	30.5932	1.33775	0.668873	−	0.743377i	$-0.266777\pi$
$523$	30.5932	1.33775	0.668873	+	0.743377i	$0.266777\pi$
$524$	−13.9567	−0.609701
$525$	0	0
$526$	−40.9655	−1.78618
$527$	0.813049	0.0354170
$528$	13.8274	0.601761
$529$	1.33119	0.0578776
$530$	0	0
$531$	−11.2488	−0.488158
$532$	21.2479	0.921214
$533$	−5.97157	−0.258657
$534$	−8.67941	−0.375595
$535$	0	0
$536$	5.79985	0.250515
$537$	4.48845	0.193691
$538$	50.4718	2.17599
$539$	55.1133	2.37390
$540$	0	0
$541$	18.3014	0.786840	0.393420	−	0.919359i	$-0.371292\pi$
$541$	18.3014	0.786840	0.393420	+	0.919359i	$0.371292\pi$
$542$	−46.6273	−2.00282
$543$	9.47432	0.406582
$544$	1.03383	0.0443250
$545$	0	0
$546$	−3.14545	−0.134613
$547$	6.55603	0.280316	0.140158	−	0.990129i	$-0.455239\pi$
$547$	6.55603	0.280316	0.140158	+	0.990129i	$0.455239\pi$
$548$	8.69071	0.371249
$549$	29.0935	1.24168
$550$	0	0
$551$	−22.9254	−0.976653
$552$	−6.87836	−0.292762
$553$	44.1439	1.87719
$554$	3.86187	0.164075
$555$	0	0
$556$	5.55352	0.235522
$557$	35.5383	1.50581	0.752904	−	0.658131i	$-0.228653\pi$
$557$	35.5383	1.50581	0.752904	+	0.658131i	$0.228653\pi$
$558$	14.6831	0.621587
$559$	−0.795427	−0.0336430
$560$	0	0
$561$	0.646853	0.0273101
$562$	−2.72424	−0.114915
$563$	−38.2479	−1.61196	−0.805979	−	0.591944i	$-0.798361\pi$
$563$	−38.2479	−1.61196	−0.805979	+	0.591944i	$0.798361\pi$
$564$	−0.545193	−0.0229568
$565$	0	0
$566$	−21.8712	−0.919315
$567$	−21.6395	−0.908773
$568$	−6.29266	−0.264034
$569$	7.55897	0.316889	0.158444	−	0.987368i	$-0.449352\pi$
$569$	7.55897	0.316889	0.158444	+	0.987368i	$0.449352\pi$
$570$	0	0
$571$	23.0262	0.963617	0.481809	−	0.876277i	$-0.339980\pi$
$571$	23.0262	0.963617	0.481809	+	0.876277i	$0.339980\pi$
$572$	−1.87032	−0.0782019
$573$	1.97533	0.0825207
$574$	−80.5540	−3.36226
$575$	0	0
$576$	−6.14617	−0.256090
$577$	−22.3168	−0.929059	−0.464529	−	0.885558i	$-0.653777\pi$
$577$	−22.3168	−0.929059	−0.464529	+	0.885558i	$0.653777\pi$
$578$	−28.5269	−1.18656
$579$	−16.0300	−0.666184
$580$	0	0
$581$	48.1304	1.99678
$582$	9.94866	0.412385
$583$	4.81980	0.199616
$584$	20.2044	0.836062
$585$	0	0
$586$	−19.6974	−0.813691
$587$	21.0786	0.870007	0.435004	−	0.900429i	$-0.356747\pi$
$587$	21.0786	0.870007	0.435004	+	0.900429i	$0.356747\pi$
$588$	−8.33899	−0.343894
$589$	19.3993	0.799335
$590$	0	0
$591$	−0.903426	−0.0371620
$592$	26.9329	1.10694
$593$	−34.3547	−1.41078	−0.705390	−	0.708819i	$-0.749228\pi$
$593$	−34.3547	−1.41078	−0.705390	+	0.708819i	$0.749228\pi$
$594$	25.7256	1.05553
$595$	0	0
$596$	−10.0328	−0.410959
$597$	6.12774	0.250792
$598$	4.75752	0.194549
$599$	0.498231	0.0203572	0.0101786	−	0.999948i	$-0.496760\pi$
$599$	0.498231	0.0203572	0.0101786	+	0.999948i	$0.496760\pi$
$600$	0	0
$601$	27.8635	1.13657	0.568287	−	0.822830i	$-0.307606\pi$
$601$	27.8635	1.13657	0.568287	+	0.822830i	$0.307606\pi$
$602$	−10.7300	−0.437321
$603$	7.37264	0.300237
$604$	−1.28598	−0.0523258
$605$	0	0
$606$	4.26316	0.173179
$607$	14.1000	0.572303	0.286152	−	0.958184i	$-0.407624\pi$
$607$	14.1000	0.572303	0.286152	+	0.958184i	$0.407624\pi$
$608$	24.6671	1.00038
$609$	13.4710	0.545871
$610$	0	0
$611$	−0.527344	−0.0213340
$612$	0.484030	0.0195657
$613$	31.8257	1.28543	0.642714	−	0.766106i	$-0.277808\pi$
$613$	31.8257	1.28543	0.642714	+	0.766106i	$0.277808\pi$
$614$	44.7237	1.80490
$615$	0	0
$616$	35.2827	1.42158
$617$	33.9662	1.36743	0.683715	−	0.729749i	$-0.260363\pi$
$617$	33.9662	1.36743	0.683715	+	0.729749i	$0.260363\pi$
$618$	0.477508	0.0192082
$619$	1.12842	0.0453550	0.0226775	−	0.999743i	$-0.492781\pi$
$619$	1.12842	0.0453550	0.0226775	+	0.999743i	$0.492781\pi$
$620$	0	0
$621$	−19.2552	−0.772686
$622$	−22.7109	−0.910624
$623$	−33.3235	−1.33508
$624$	−2.02368	−0.0810119
$625$	0	0
$626$	28.6085	1.14343
$627$	15.4339	0.616370
$628$	−8.19262	−0.326921
$629$	1.25993	0.0502369
$630$	0	0
$631$	32.7801	1.30496	0.652478	−	0.757807i	$-0.273729\pi$
$631$	32.7801	1.30496	0.652478	+	0.757807i	$0.273729\pi$
$632$	18.8751	0.750813
$633$	14.9396	0.593795
$634$	28.2744	1.12292
$635$	0	0
$636$	−0.729266	−0.0289173
$637$	−8.06597	−0.319586
$638$	27.2216	1.07771
$639$	−7.99909	−0.316439
$640$	0	0
$641$	30.4126	1.20123	0.600613	−	0.799540i	$-0.294923\pi$
$641$	30.4126	1.20123	0.600613	+	0.799540i	$0.294923\pi$
$642$	1.96555	0.0775741
$643$	−1.06932	−0.0421700	−0.0210850	−	0.999778i	$-0.506712\pi$
$643$	−1.06932	−0.0421700	−0.0210850	+	0.999778i	$0.506712\pi$
$644$	18.8841	0.744140
$645$	0	0
$646$	2.17331	0.0855076
$647$	−30.5680	−1.20175	−0.600876	−	0.799343i	$-0.705181\pi$
$647$	−30.5680	−1.20175	−0.600876	+	0.799343i	$0.705181\pi$
$648$	−9.25267	−0.363479
$649$	17.6471	0.692709
$650$	0	0
$651$	−11.3991	−0.446765
$652$	−4.65389	−0.182260
$653$	33.3502	1.30509	0.652546	−	0.757749i	$-0.273701\pi$
$653$	33.3502	1.30509	0.652546	+	0.757749i	$0.273701\pi$
$654$	−0.0895834	−0.00350299
$655$	0	0
$656$	−51.8258	−2.02346
$657$	25.6833	1.00200
$658$	−7.11365	−0.277319
$659$	−10.1150	−0.394026	−0.197013	−	0.980401i	$-0.563124\pi$
$659$	−10.1150	−0.394026	−0.197013	+	0.980401i	$0.563124\pi$
$660$	0	0
$661$	−30.8383	−1.19947	−0.599735	−	0.800199i	$-0.704727\pi$
$661$	−30.8383	−1.19947	−0.599735	+	0.800199i	$0.704727\pi$
$662$	21.6055	0.839723
$663$	−0.0946685	−0.00367662
$664$	20.5797	0.798647
$665$	0	0
$666$	22.7536	0.881684
$667$	−20.3750	−0.788922
$668$	−18.2290	−0.705302
$669$	4.50027	0.173990
$670$	0	0
$671$	−45.6416	−1.76197
$672$	−14.4944	−0.559134
$673$	30.8250	1.18821	0.594107	−	0.804386i	$-0.297505\pi$
$673$	30.8250	1.18821	0.594107	+	0.804386i	$0.297505\pi$
$674$	35.7514	1.37709
$675$	0	0
$676$	−10.5666	−0.406407
$677$	−41.2521	−1.58545	−0.792723	−	0.609582i	$-0.791337\pi$
$677$	−41.2521	−1.58545	−0.792723	+	0.609582i	$0.791337\pi$
$678$	−16.9000	−0.649040
$679$	38.1967	1.46585
$680$	0	0
$681$	8.00014	0.306566
$682$	−23.0348	−0.882048
$683$	−22.3514	−0.855254	−0.427627	−	0.903955i	$-0.640650\pi$
$683$	−22.3514	−0.855254	−0.427627	+	0.903955i	$0.640650\pi$
$684$	11.5489	0.441585
$685$	0	0
$686$	−54.7056	−2.08867
$687$	−10.6841	−0.407625
$688$	−6.90332	−0.263186
$689$	−0.705390	−0.0268732
$690$	0	0
$691$	9.18901	0.349566	0.174783	−	0.984607i	$-0.444078\pi$
$691$	9.18901	0.349566	0.174783	+	0.984607i	$0.444078\pi$
$692$	19.5447	0.742978
$693$	44.8506	1.70373
$694$	48.1929	1.82938
$695$	0	0
$696$	5.75995	0.218330
$697$	−2.42443	−0.0918320
$698$	33.5207	1.26878
$699$	5.56018	0.210305
$700$	0	0
$701$	−35.9929	−1.35943	−0.679717	−	0.733475i	$-0.737897\pi$
$701$	−35.9929	−1.35943	−0.679717	+	0.733475i	$0.737897\pi$
$702$	−3.76501	−0.142101
$703$	30.0620	1.13381
$704$	9.64206	0.363399
$705$	0	0
$706$	41.5660	1.56436
$707$	16.3679	0.615578
$708$	−2.67012	−0.100349
$709$	20.2461	0.760359	0.380180	−	0.924913i	$-0.375862\pi$
$709$	20.2461	0.760359	0.380180	+	0.924913i	$0.375862\pi$
$710$	0	0
$711$	23.9937	0.899832
$712$	−14.2485	−0.533987
$713$	17.2412	0.645688
$714$	−1.27704	−0.0477920
$715$	0	0
$716$	−5.26900	−0.196912
$717$	−6.20756	−0.231826
$718$	36.1951	1.35079
$719$	−37.0265	−1.38086	−0.690428	−	0.723401i	$-0.742578\pi$
$719$	−37.0265	−1.38086	−0.690428	+	0.723401i	$0.742578\pi$
$720$	0	0
$721$	1.83333	0.0682769
$722$	19.8703	0.739495
$723$	0.426838	0.0158743
$724$	−11.1219	−0.413343
$725$	0	0
$726$	−5.17249	−0.191969
$727$	12.0790	0.447985	0.223993	−	0.974591i	$-0.428091\pi$
$727$	12.0790	0.447985	0.223993	+	0.974591i	$0.428091\pi$
$728$	−5.16372	−0.191380
$729$	−3.44309	−0.127522
$730$	0	0
$731$	−0.322940	−0.0119444
$732$	6.90586	0.255248
$733$	−44.6116	−1.64777	−0.823884	−	0.566759i	$-0.808197\pi$
$733$	−44.6116	−1.64777	−0.823884	+	0.566759i	$0.808197\pi$
$734$	−1.92554	−0.0710729
$735$	0	0
$736$	21.9229	0.808090
$737$	−11.5661	−0.426044
$738$	−43.7838	−1.61170
$739$	−14.6463	−0.538774	−0.269387	−	0.963032i	$-0.586821\pi$
$739$	−14.6463	−0.538774	−0.269387	+	0.963032i	$0.586821\pi$
$740$	0	0
$741$	−2.25879	−0.0829787
$742$	−9.51542	−0.349322
$743$	−24.4397	−0.896604	−0.448302	−	0.893882i	$-0.647971\pi$
$743$	−24.4397	−0.896604	−0.448302	+	0.893882i	$0.647971\pi$
$744$	−4.87404	−0.178691
$745$	0	0
$746$	19.4257	0.711227
$747$	26.1604	0.957161
$748$	−0.759342	−0.0277643
$749$	7.54648	0.275743
$750$	0	0
$751$	31.2863	1.14165	0.570826	−	0.821071i	$-0.306623\pi$
$751$	31.2863	1.14165	0.570826	+	0.821071i	$0.306623\pi$
$752$	−4.57669	−0.166895
$753$	10.0803	0.367345
$754$	−3.98395	−0.145087
$755$	0	0
$756$	−14.9445	−0.543528
$757$	11.3251	0.411617	0.205808	−	0.978592i	$-0.434018\pi$
$757$	11.3251	0.411617	0.205808	+	0.978592i	$0.434018\pi$
$758$	35.5288	1.29046
$759$	13.7169	0.497892
$760$	0	0
$761$	−41.3338	−1.49835	−0.749174	−	0.662373i	$-0.769549\pi$
$761$	−41.3338	−1.49835	−0.749174	+	0.662373i	$0.769549\pi$
$762$	−14.1253	−0.511704
$763$	−0.343944	−0.0124516
$764$	−2.31885	−0.0838930
$765$	0	0
$766$	−1.44458	−0.0521949
$767$	−2.58270	−0.0932558
$768$	−12.0882	−0.436195
$769$	22.6491	0.816747	0.408374	−	0.912815i	$-0.366096\pi$
$769$	22.6491	0.816747	0.408374	+	0.912815i	$0.366096\pi$
$770$	0	0
$771$	12.5630	0.452446
$772$	18.8177	0.677263
$773$	−16.8541	−0.606200	−0.303100	−	0.952959i	$-0.598022\pi$
$773$	−16.8541	−0.606200	−0.303100	+	0.952959i	$0.598022\pi$
$774$	−5.83210	−0.209630
$775$	0	0
$776$	16.3322	0.586292
$777$	−17.6645	−0.633710
$778$	−57.1259	−2.04806
$779$	−57.8470	−2.07258
$780$	0	0
$781$	12.5489	0.449035
$782$	1.93153	0.0690715
$783$	16.1244	0.576238
$784$	−70.0026	−2.50009
$785$	0	0
$786$	20.0143	0.713887
$787$	−20.1087	−0.716798	−0.358399	−	0.933568i	$-0.616677\pi$
$787$	−20.1087	−0.716798	−0.358399	+	0.933568i	$0.616677\pi$
$788$	1.06053	0.0377800
$789$	17.2860	0.615398
$790$	0	0
$791$	−64.8854	−2.30706
$792$	19.1773	0.681437
$793$	6.67976	0.237205
$794$	−45.1635	−1.60279
$795$	0	0
$796$	−7.19336	−0.254962
$797$	54.1727	1.91890	0.959448	−	0.281887i	$-0.0909602\pi$
$797$	54.1727	1.91890	0.959448	+	0.281887i	$0.0909602\pi$
$798$	−30.4701	−1.07863
$799$	−0.214100	−0.00757430
$800$	0	0
$801$	−18.1124	−0.639971
$802$	6.39287	0.225740
$803$	−40.2918	−1.42187
$804$	1.75003	0.0617187
$805$	0	0
$806$	3.37120	0.118745
$807$	−21.2973	−0.749701
$808$	6.99862	0.246211
$809$	15.3366	0.539206	0.269603	−	0.962972i	$-0.413108\pi$
$809$	15.3366	0.539206	0.269603	+	0.962972i	$0.413108\pi$
$810$	0	0
$811$	11.5666	0.406158	0.203079	−	0.979162i	$-0.434905\pi$
$811$	11.5666	0.406158	0.203079	+	0.979162i	$0.434905\pi$
$812$	−15.8136	−0.554949
$813$	19.6751	0.690036
$814$	−35.6957	−1.25113
$815$	0	0
$816$	−0.821605	−0.0287619
$817$	−7.70535	−0.269576
$818$	−23.8630	−0.834349
$819$	−6.56400	−0.229365
$820$	0	0
$821$	33.3675	1.16453	0.582267	−	0.812997i	$-0.302166\pi$
$821$	33.3675	1.16453	0.582267	+	0.812997i	$0.302166\pi$
$822$	−12.4627	−0.434688
$823$	−29.9821	−1.04511	−0.522554	−	0.852606i	$-0.675021\pi$
$823$	−29.9821	−1.04511	−0.522554	+	0.852606i	$0.675021\pi$
$824$	0.783901	0.0273085
$825$	0	0
$826$	−34.8395	−1.21222
$827$	24.7230	0.859702	0.429851	−	0.902900i	$-0.358566\pi$
$827$	24.7230	0.859702	0.429851	+	0.902900i	$0.358566\pi$
$828$	10.2642	0.356704
$829$	−0.211406	−0.00734242	−0.00367121	−	0.999993i	$-0.501169\pi$
$829$	−0.211406	−0.00734242	−0.00367121	+	0.999993i	$0.501169\pi$
$830$	0	0
$831$	−1.62957	−0.0565293
$832$	−1.41114	−0.0489225
$833$	−3.27475	−0.113464
$834$	−7.96391	−0.275768
$835$	0	0
$836$	−18.1179	−0.626620
$837$	−13.6444	−0.471618
$838$	−32.9895	−1.13961
$839$	−5.51714	−0.190473	−0.0952365	−	0.995455i	$-0.530361\pi$
$839$	−5.51714	−0.190473	−0.0952365	+	0.995455i	$0.530361\pi$
$840$	0	0
$841$	−11.9380	−0.411654
$842$	−43.4051	−1.49584
$843$	1.14953	0.0395920
$844$	−17.5376	−0.603670
$845$	0	0
$846$	−3.86650	−0.132933
$847$	−19.8591	−0.682368
$848$	−6.12191	−0.210227
$849$	9.22888	0.316734
$850$	0	0
$851$	26.7177	0.915871
$852$	−1.89873	−0.0650493
$853$	−1.53946	−0.0527100	−0.0263550	−	0.999653i	$-0.508390\pi$
$853$	−1.53946	−0.0527100	−0.0263550	+	0.999653i	$0.508390\pi$
$854$	90.1072	3.08341
$855$	0	0
$856$	3.22674	0.110288
$857$	−45.3407	−1.54881	−0.774404	−	0.632691i	$-0.781950\pi$
$857$	−45.3407	−1.54881	−0.774404	+	0.632691i	$0.781950\pi$
$858$	2.68209	0.0915651
$859$	−21.7964	−0.743685	−0.371842	−	0.928296i	$-0.621274\pi$
$859$	−21.7964	−0.743685	−0.371842	+	0.928296i	$0.621274\pi$
$860$	0	0
$861$	33.9910	1.15841
$862$	15.8667	0.540422
$863$	−12.7882	−0.435314	−0.217657	−	0.976025i	$-0.569841\pi$
$863$	−12.7882	−0.435314	−0.217657	+	0.976025i	$0.569841\pi$
$864$	−17.3494	−0.590238
$865$	0	0
$866$	−2.94868	−0.100200
$867$	12.0374	0.408810
$868$	13.3814	0.454194
$869$	−37.6410	−1.27688
$870$	0	0
$871$	1.69273	0.0573561
$872$	−0.147065	−0.00498023
$873$	20.7611	0.702658
$874$	46.0863	1.55889
$875$	0	0
$876$	6.09640	0.205978
$877$	9.55340	0.322595	0.161298	−	0.986906i	$-0.448432\pi$
$877$	9.55340	0.322595	0.161298	+	0.986906i	$0.448432\pi$
$878$	47.2753	1.59546
$879$	8.31160	0.280343
$880$	0	0
$881$	−39.1333	−1.31843	−0.659217	−	0.751953i	$-0.729112\pi$
$881$	−39.1333	−1.31843	−0.659217	+	0.751953i	$0.729112\pi$
$882$	−59.1400	−1.99135
$883$	−35.0254	−1.17870	−0.589349	−	0.807879i	$-0.700616\pi$
$883$	−35.0254	−1.17870	−0.589349	+	0.807879i	$0.700616\pi$
$884$	0.111132	0.00373776
$885$	0	0
$886$	50.3525	1.69162
$887$	36.2438	1.21695	0.608474	−	0.793574i	$-0.291782\pi$
$887$	36.2438	1.21695	0.608474	+	0.793574i	$0.291782\pi$
$888$	−7.55302	−0.253463
$889$	−54.2322	−1.81889
$890$	0	0
$891$	18.4518	0.618158
$892$	−5.28288	−0.176884
$893$	−5.10841	−0.170946
$894$	14.3873	0.481184
$895$	0	0
$896$	−59.8455	−1.99930
$897$	−2.00751	−0.0670287
$898$	10.6051	0.353895
$899$	−14.4378	−0.481528
$900$	0	0
$901$	−0.286386	−0.00954089
$902$	68.6876	2.28705
$903$	4.52768	0.150672
$904$	−27.7438	−0.922746
$905$	0	0
$906$	1.84413	0.0612672
$907$	−40.8532	−1.35651	−0.678255	−	0.734827i	$-0.737263\pi$
$907$	−40.8532	−1.35651	−0.678255	+	0.734827i	$0.737263\pi$
$908$	−9.39139	−0.311664
$909$	8.89649	0.295078
$910$	0	0
$911$	12.7468	0.422320	0.211160	−	0.977452i	$-0.432276\pi$
$911$	12.7468	0.422320	0.211160	+	0.977452i	$0.432276\pi$
$912$	−19.6035	−0.649136
$913$	−41.0403	−1.35824
$914$	−32.2893	−1.06803
$915$	0	0
$916$	12.5421	0.414403
$917$	76.8425	2.53756
$918$	−1.52858	−0.0504506
$919$	9.67206	0.319052	0.159526	−	0.987194i	$-0.449003\pi$
$919$	9.67206	0.319052	0.159526	+	0.987194i	$0.449003\pi$
$920$	0	0
$921$	−18.8718	−0.621848
$922$	11.9036	0.392023
$923$	−1.83656	−0.0604512
$924$	10.6461	0.350231
$925$	0	0
$926$	16.1917	0.532094
$927$	0.996477	0.0327286
$928$	−18.3583	−0.602641
$929$	10.9195	0.358257	0.179128	−	0.983826i	$-0.442672\pi$
$929$	10.9195	0.358257	0.179128	+	0.983826i	$0.442672\pi$
$930$	0	0
$931$	−78.1356	−2.56079
$932$	−6.52711	−0.213803
$933$	9.58320	0.313740
$934$	32.3585	1.05880
$935$	0	0
$936$	−2.80665	−0.0917383
$937$	15.9466	0.520954	0.260477	−	0.965480i	$-0.416120\pi$
$937$	15.9466	0.520954	0.260477	+	0.965480i	$0.416120\pi$
$938$	22.8343	0.745565
$939$	−12.0718	−0.393948
$940$	0	0
$941$	5.92701	0.193215	0.0966074	−	0.995323i	$-0.469201\pi$
$941$	5.92701	0.193215	0.0966074	+	0.995323i	$0.469201\pi$
$942$	11.7485	0.382785
$943$	−51.4117	−1.67419
$944$	−22.4146	−0.729533
$945$	0	0
$946$	9.14934	0.297471
$947$	−0.990382	−0.0321831	−0.0160916	−	0.999871i	$-0.505122\pi$
$947$	−0.990382	−0.0321831	−0.0160916	+	0.999871i	$0.505122\pi$
$948$	5.69532	0.184975
$949$	5.89681	0.191418
$950$	0	0
$951$	−11.9308	−0.386882
$952$	−2.09645	−0.0679463
$953$	−53.5993	−1.73625	−0.868125	−	0.496345i	$-0.834675\pi$
$953$	−53.5993	−1.73625	−0.868125	+	0.496345i	$0.834675\pi$
$954$	−5.17194	−0.167448
$955$	0	0
$956$	7.28708	0.235681
$957$	−11.4866	−0.371308
$958$	−63.9689	−2.06674
$959$	−47.8491	−1.54513
$960$	0	0
$961$	−18.7828	−0.605896
$962$	5.22415	0.168433
$963$	4.10176	0.132177
$964$	−0.501066	−0.0161382
$965$	0	0
$966$	−27.0804	−0.871298
$967$	−29.7350	−0.956212	−0.478106	−	0.878302i	$-0.658677\pi$
$967$	−29.7350	−0.956212	−0.478106	+	0.878302i	$0.658677\pi$
$968$	−8.49141	−0.272924
$969$	−0.917060	−0.0294602
$970$	0	0
$971$	60.6166	1.94528	0.972640	−	0.232317i	$-0.0746308\pi$
$971$	60.6166	1.94528	0.972640	+	0.232317i	$0.0746308\pi$
$972$	−12.5572	−0.402772
$973$	−30.5764	−0.980236
$974$	28.1954	0.903440
$975$	0	0
$976$	57.9720	1.85564
$977$	44.5130	1.42410	0.712048	−	0.702131i	$-0.247768\pi$
$977$	44.5130	1.42410	0.712048	+	0.702131i	$0.247768\pi$
$978$	6.67382	0.213405
$979$	28.4146	0.908135
$980$	0	0
$981$	−0.186945	−0.00596870
$982$	−15.0675	−0.480822
$983$	38.2401	1.21967	0.609834	−	0.792529i	$-0.291236\pi$
$983$	38.2401	1.21967	0.609834	+	0.792529i	$0.291236\pi$
$984$	14.5339	0.463325
$985$	0	0
$986$	−1.61747	−0.0515107
$987$	3.00171	0.0955455
$988$	2.65160	0.0843586
$989$	−6.84815	−0.217759
$990$	0	0
$991$	22.7082	0.721349	0.360675	−	0.932692i	$-0.382547\pi$
$991$	22.7082	0.721349	0.360675	+	0.932692i	$0.382547\pi$
$992$	15.5347	0.493228
$993$	−9.11678	−0.289312
$994$	−24.7745	−0.785799
$995$	0	0
$996$	6.20965	0.196760
$997$	20.6284	0.653309	0.326654	−	0.945144i	$-0.394079\pi$
$997$	20.6284	0.653309	0.326654	+	0.945144i	$0.394079\pi$
$998$	−61.1600	−1.93599
$999$	−21.1439	−0.668962

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.2.a.e.1.7	✓	8
3.2	odd	2		5625.2.a.be.1.2		8
4.3	odd	2		10000.2.a.bn.1.4		8
5.2	odd	4		625.2.b.d.624.12		16
5.3	odd	4		625.2.b.d.624.5		16
5.4	even	2		625.2.a.g.1.2	yes	8
15.14	odd	2		5625.2.a.s.1.7		8
20.19	odd	2		10000.2.a.be.1.5		8
25.2	odd	20		625.2.e.k.124.6		32
25.3	odd	20		625.2.e.j.249.3		32
25.4	even	10		625.2.d.m.376.4		16
25.6	even	5		625.2.d.q.251.1		16
25.8	odd	20		625.2.e.j.374.6		32
25.9	even	10		625.2.d.n.126.1		16
25.11	even	5		625.2.d.p.501.4		16
25.12	odd	20		625.2.e.k.499.3		32
25.13	odd	20		625.2.e.k.499.6		32
25.14	even	10		625.2.d.n.501.1		16
25.16	even	5		625.2.d.p.126.4		16
25.17	odd	20		625.2.e.j.374.3		32
25.19	even	10		625.2.d.m.251.4		16
25.21	even	5		625.2.d.q.376.1		16
25.22	odd	20		625.2.e.j.249.6		32
25.23	odd	20		625.2.e.k.124.3		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.e.1.7	✓	8	1.1	even	1	trivial
625.2.a.g.1.2	yes	8	5.4	even	2
625.2.b.d.624.5		16	5.3	odd	4
625.2.b.d.624.12		16	5.2	odd	4
625.2.d.m.251.4		16	25.19	even	10
625.2.d.m.376.4		16	25.4	even	10
625.2.d.n.126.1		16	25.9	even	10
625.2.d.n.501.1		16	25.14	even	10
625.2.d.p.126.4		16	25.16	even	5
625.2.d.p.501.4		16	25.11	even	5
625.2.d.q.251.1		16	25.6	even	5
625.2.d.q.376.1		16	25.21	even	5
625.2.e.j.249.3		32	25.3	odd	20
625.2.e.j.249.6		32	25.22	odd	20
625.2.e.j.374.3		32	25.17	odd	20
625.2.e.j.374.6		32	25.8	odd	20
625.2.e.k.124.3		32	25.23	odd	20
625.2.e.k.124.6		32	25.2	odd	20
625.2.e.k.499.3		32	25.12	odd	20
625.2.e.k.499.6		32	25.13	odd	20
5625.2.a.s.1.7		8	15.14	odd	2
5625.2.a.be.1.2		8	3.2	odd	2
10000.2.a.be.1.5		8	20.19	odd	2
10000.2.a.bn.1.4		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 625 = 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	625.a (trivial)

\(f(q)\)	\(=\)	\(q+1.68341 q^{2} -0.710340 q^{3} +0.833870 q^{4} -1.19579 q^{6} -4.59110 q^{7} -1.96307 q^{8} -2.49542 q^{9} +O(q^{10})\)	\(q+1.68341 q^{2} -0.710340 q^{3} +0.833870 q^{4} -1.19579 q^{6} -4.59110 q^{7} -1.96307 q^{8} -2.49542 q^{9} +3.91479 q^{11} -0.592332 q^{12} -0.572939 q^{13} -7.72871 q^{14} -4.97240 q^{16} -0.232611 q^{17} -4.20081 q^{18} -5.55010 q^{19} +3.26125 q^{21} +6.59020 q^{22} -4.93267 q^{23} +1.39445 q^{24} -0.964492 q^{26} +3.90362 q^{27} -3.82839 q^{28} +4.13062 q^{29} -3.49531 q^{31} -4.44444 q^{32} -2.78083 q^{33} -0.391580 q^{34} -2.08085 q^{36} -5.41648 q^{37} -9.34309 q^{38} +0.406982 q^{39} +10.4227 q^{41} +5.49002 q^{42} +1.38833 q^{43} +3.26443 q^{44} -8.30370 q^{46} +0.920418 q^{47} +3.53210 q^{48} +14.0782 q^{49} +0.165233 q^{51} -0.477757 q^{52} +1.23118 q^{53} +6.57139 q^{54} +9.01268 q^{56} +3.94246 q^{57} +6.95353 q^{58} +4.50780 q^{59} -11.6588 q^{61} -5.88405 q^{62} +11.4567 q^{63} +2.46298 q^{64} -4.68128 q^{66} -2.95447 q^{67} -0.193968 q^{68} +3.50387 q^{69} +3.20551 q^{71} +4.89869 q^{72} -10.2922 q^{73} -9.11816 q^{74} -4.62806 q^{76} -17.9732 q^{77} +0.685118 q^{78} -9.61509 q^{79} +4.71335 q^{81} +17.5457 q^{82} -10.4834 q^{83} +2.71946 q^{84} +2.33712 q^{86} -2.93415 q^{87} -7.68502 q^{88} +7.25828 q^{89} +2.63042 q^{91} -4.11320 q^{92} +2.48286 q^{93} +1.54944 q^{94} +3.15707 q^{96} -8.31971 q^{97} +23.6994 q^{98} -9.76903 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{2} - 5 q^{3} + 11 q^{4} - 4 q^{6} - 10 q^{7} - 15 q^{8} + 9 q^{9}+O(q^{10}) \) 8 * q - 5 * q^2 - 5 * q^3 + 11 * q^4 - 4 * q^6 - 10 * q^7 - 15 * q^8 + 9 * q^9	\( 8 q - 5 q^{2} - 5 q^{3} + 11 q^{4} - 4 q^{6} - 10 q^{7} - 15 q^{8} + 9 q^{9} + q^{11} - 10 q^{12} - 10 q^{13} - 8 q^{14} + 13 q^{16} - 15 q^{17} + 5 q^{18} - 10 q^{19} - 14 q^{21} + 5 q^{22} - 30 q^{23} + 5 q^{24} + 11 q^{26} - 20 q^{27} + 5 q^{28} + 10 q^{29} - 9 q^{31} - 30 q^{32} - 5 q^{33} + 7 q^{34} + 3 q^{36} + 10 q^{37} - 20 q^{38} + 8 q^{39} - 4 q^{41} + 35 q^{42} - 18 q^{44} - 9 q^{46} - 30 q^{47} - 5 q^{48} - 4 q^{49} - 14 q^{51} - 5 q^{52} - 10 q^{53} - 20 q^{54} + 10 q^{57} + 30 q^{58} - 5 q^{59} + 6 q^{61} - 10 q^{62} - 9 q^{64} - 18 q^{66} - 10 q^{67} - 40 q^{68} + 3 q^{69} - 9 q^{71} + 15 q^{72} - 18 q^{74} - 10 q^{76} - 5 q^{77} + 30 q^{78} - 20 q^{79} + 8 q^{81} + 45 q^{82} - 40 q^{83} - 28 q^{84} - 24 q^{86} - 40 q^{87} + 40 q^{88} - 5 q^{89} + 6 q^{91} - 15 q^{92} + 40 q^{93} + 47 q^{94} + 71 q^{96} + 30 q^{98} - 22 q^{99}+O(q^{100}) \) 8 * q - 5 * q^2 - 5 * q^3 + 11 * q^4 - 4 * q^6 - 10 * q^7 - 15 * q^8 + 9 * q^9 + q^11 - 10 * q^12 - 10 * q^13 - 8 * q^14 + 13 * q^16 - 15 * q^17 + 5 * q^18 - 10 * q^19 - 14 * q^21 + 5 * q^22 - 30 * q^23 + 5 * q^24 + 11 * q^26 - 20 * q^27 + 5 * q^28 + 10 * q^29 - 9 * q^31 - 30 * q^32 - 5 * q^33 + 7 * q^34 + 3 * q^36 + 10 * q^37 - 20 * q^38 + 8 * q^39 - 4 * q^41 + 35 * q^42 - 18 * q^44 - 9 * q^46 - 30 * q^47 - 5 * q^48 - 4 * q^49 - 14 * q^51 - 5 * q^52 - 10 * q^53 - 20 * q^54 + 10 * q^57 + 30 * q^58 - 5 * q^59 + 6 * q^61 - 10 * q^62 - 9 * q^64 - 18 * q^66 - 10 * q^67 - 40 * q^68 + 3 * q^69 - 9 * q^71 + 15 * q^72 - 18 * q^74 - 10 * q^76 - 5 * q^77 + 30 * q^78 - 20 * q^79 + 8 * q^81 + 45 * q^82 - 40 * q^83 - 28 * q^84 - 24 * q^86 - 40 * q^87 + 40 * q^88 - 5 * q^89 + 6 * q^91 - 15 * q^92 + 40 * q^93 + 47 * q^94 + 71 * q^96 + 30 * q^98 - 22 * q^99

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