LMFDB - Embedded newform 345.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [345,2,Mod(1,345)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(345, 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("345.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 345 = 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	345.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:	$2.75483886973$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	345.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} -1.00000 q^{3} -1.00000 q^{5} -1.41421 q^{6} -3.82843 q^{7} -2.82843 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} -1.00000 q^{3} -1.00000 q^{5} -1.41421 q^{6} -3.82843 q^{7} -2.82843 q^{8} +1.00000 q^{9} -1.41421 q^{10} -5.41421 q^{11} +0.585786 q^{13} -5.41421 q^{14} +1.00000 q^{15} -4.00000 q^{16} +8.07107 q^{17} +1.41421 q^{18} +2.24264 q^{19} +3.82843 q^{21} -7.65685 q^{22} -1.00000 q^{23} +2.82843 q^{24} +1.00000 q^{25} +0.828427 q^{26} -1.00000 q^{27} -6.41421 q^{29} +1.41421 q^{30} -9.82843 q^{31} +5.41421 q^{33} +11.4142 q^{34} +3.82843 q^{35} +3.00000 q^{37} +3.17157 q^{38} -0.585786 q^{39} +2.82843 q^{40} -7.58579 q^{41} +5.41421 q^{42} +6.00000 q^{43} -1.00000 q^{45} -1.41421 q^{46} -11.4142 q^{47} +4.00000 q^{48} +7.65685 q^{49} +1.41421 q^{50} -8.07107 q^{51} -1.24264 q^{53} -1.41421 q^{54} +5.41421 q^{55} +10.8284 q^{56} -2.24264 q^{57} -9.07107 q^{58} +12.8995 q^{59} +2.58579 q^{61} -13.8995 q^{62} -3.82843 q^{63} +8.00000 q^{64} -0.585786 q^{65} +7.65685 q^{66} +5.48528 q^{67} +1.00000 q^{69} +5.41421 q^{70} -10.8995 q^{71} -2.82843 q^{72} -1.75736 q^{73} +4.24264 q^{74} -1.00000 q^{75} +20.7279 q^{77} -0.828427 q^{78} +7.31371 q^{79} +4.00000 q^{80} +1.00000 q^{81} -10.7279 q^{82} -0.0710678 q^{83} -8.07107 q^{85} +8.48528 q^{86} +6.41421 q^{87} +15.3137 q^{88} -18.1421 q^{89} -1.41421 q^{90} -2.24264 q^{91} +9.82843 q^{93} -16.1421 q^{94} -2.24264 q^{95} -6.82843 q^{97} +10.8284 q^{98} -5.41421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 8 q^{11} + 4 q^{13} - 8 q^{14} + 2 q^{15} - 8 q^{16} + 2 q^{17} - 4 q^{19} + 2 q^{21} - 4 q^{22} - 2 q^{23} + 2 q^{25} - 4 q^{26} - 2 q^{27} - 10 q^{29} - 14 q^{31} + 8 q^{33} + 20 q^{34} + 2 q^{35} + 6 q^{37} + 12 q^{38} - 4 q^{39} - 18 q^{41} + 8 q^{42} + 12 q^{43} - 2 q^{45} - 20 q^{47} + 8 q^{48} + 4 q^{49} - 2 q^{51} + 6 q^{53} + 8 q^{55} + 16 q^{56} + 4 q^{57} - 4 q^{58} + 6 q^{59} + 8 q^{61} - 8 q^{62} - 2 q^{63} + 16 q^{64} - 4 q^{65} + 4 q^{66} - 6 q^{67} + 2 q^{69} + 8 q^{70} - 2 q^{71} - 12 q^{73} - 2 q^{75} + 16 q^{77} + 4 q^{78} - 8 q^{79} + 8 q^{80} + 2 q^{81} + 4 q^{82} + 14 q^{83} - 2 q^{85} + 10 q^{87} + 8 q^{88} - 8 q^{89} + 4 q^{91} + 14 q^{93} - 4 q^{94} + 4 q^{95} - 8 q^{97} + 16 q^{98} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 - 8 * q^11 + 4 * q^13 - 8 * q^14 + 2 * q^15 - 8 * q^16 + 2 * q^17 - 4 * q^19 + 2 * q^21 - 4 * q^22 - 2 * q^23 + 2 * q^25 - 4 * q^26 - 2 * q^27 - 10 * q^29 - 14 * q^31 + 8 * q^33 + 20 * q^34 + 2 * q^35 + 6 * q^37 + 12 * q^38 - 4 * q^39 - 18 * q^41 + 8 * q^42 + 12 * q^43 - 2 * q^45 - 20 * q^47 + 8 * q^48 + 4 * q^49 - 2 * q^51 + 6 * q^53 + 8 * q^55 + 16 * q^56 + 4 * q^57 - 4 * q^58 + 6 * q^59 + 8 * q^61 - 8 * q^62 - 2 * q^63 + 16 * q^64 - 4 * q^65 + 4 * q^66 - 6 * q^67 + 2 * q^69 + 8 * q^70 - 2 * q^71 - 12 * q^73 - 2 * q^75 + 16 * q^77 + 4 * q^78 - 8 * q^79 + 8 * q^80 + 2 * q^81 + 4 * q^82 + 14 * q^83 - 2 * q^85 + 10 * q^87 + 8 * q^88 - 8 * q^89 + 4 * q^91 + 14 * q^93 - 4 * q^94 + 4 * q^95 - 8 * q^97 + 16 * q^98 - 8 * 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 $

$2$	1.41421	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$2$	1.41421	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.41421	−0.577350
$7$	−3.82843	−1.44701	−0.723505	−	0.690319i	$-0.757470\pi$
$7$	−3.82843	−1.44701	−0.723505	+	0.690319i	$0.757470\pi$
$8$	−2.82843	−1.00000
$9$	1.00000	0.333333
$10$	−1.41421	−0.447214
$11$	−5.41421	−1.63245	−0.816223	−	0.577736i	$-0.803936\pi$
$11$	−5.41421	−1.63245	−0.816223	+	0.577736i	$0.803936\pi$
$12$	0	0
$13$	0.585786	0.162468	0.0812340	−	0.996695i	$-0.474114\pi$
$13$	0.585786	0.162468	0.0812340	+	0.996695i	$0.474114\pi$
$14$	−5.41421	−1.44701
$15$	1.00000	0.258199
$16$	−4.00000	−1.00000
$17$	8.07107	1.95752	0.978761	−	0.205006i	$-0.0657214\pi$
$17$	8.07107	1.95752	0.978761	+	0.205006i	$0.0657214\pi$
$18$	1.41421	0.333333
$19$	2.24264	0.514497	0.257249	−	0.966345i	$-0.417184\pi$
$19$	2.24264	0.514497	0.257249	+	0.966345i	$0.417184\pi$
$20$	0	0
$21$	3.82843	0.835431
$22$	−7.65685	−1.63245
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	2.82843	0.577350
$25$	1.00000	0.200000
$26$	0.828427	0.162468
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.41421	−1.19109	−0.595545	−	0.803322i	$-0.703064\pi$
$29$	−6.41421	−1.19109	−0.595545	+	0.803322i	$0.703064\pi$
$30$	1.41421	0.258199
$31$	−9.82843	−1.76524	−0.882619	−	0.470089i	$-0.844222\pi$
$31$	−9.82843	−1.76524	−0.882619	+	0.470089i	$0.844222\pi$
$32$	0	0
$33$	5.41421	0.942494
$34$	11.4142	1.95752
$35$	3.82843	0.647122
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	3.17157	0.514497
$39$	−0.585786	−0.0938009
$40$	2.82843	0.447214
$41$	−7.58579	−1.18470	−0.592350	−	0.805680i	$-0.701800\pi$
$41$	−7.58579	−1.18470	−0.592350	+	0.805680i	$0.701800\pi$
$42$	5.41421	0.835431
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	−1.41421	−0.208514
$47$	−11.4142	−1.66493	−0.832467	−	0.554075i	$-0.813072\pi$
$47$	−11.4142	−1.66493	−0.832467	+	0.554075i	$0.813072\pi$
$48$	4.00000	0.577350
$49$	7.65685	1.09384
$50$	1.41421	0.200000
$51$	−8.07107	−1.13018
$52$	0	0
$53$	−1.24264	−0.170690	−0.0853449	−	0.996351i	$-0.527199\pi$
$53$	−1.24264	−0.170690	−0.0853449	+	0.996351i	$0.527199\pi$
$54$	−1.41421	−0.192450
$55$	5.41421	0.730052
$56$	10.8284	1.44701
$57$	−2.24264	−0.297045
$58$	−9.07107	−1.19109
$59$	12.8995	1.67937	0.839686	−	0.543073i	$-0.182739\pi$
$59$	12.8995	1.67937	0.839686	+	0.543073i	$0.182739\pi$
$60$	0	0
$61$	2.58579	0.331076	0.165538	−	0.986203i	$-0.447064\pi$
$61$	2.58579	0.331076	0.165538	+	0.986203i	$0.447064\pi$
$62$	−13.8995	−1.76524
$63$	−3.82843	−0.482336
$64$	8.00000	1.00000
$65$	−0.585786	−0.0726579
$66$	7.65685	0.942494
$67$	5.48528	0.670134	0.335067	−	0.942194i	$-0.391241\pi$
$67$	5.48528	0.670134	0.335067	+	0.942194i	$0.391241\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	5.41421	0.647122
$71$	−10.8995	−1.29353	−0.646766	−	0.762688i	$-0.723879\pi$
$71$	−10.8995	−1.29353	−0.646766	+	0.762688i	$0.723879\pi$
$72$	−2.82843	−0.333333
$73$	−1.75736	−0.205683	−0.102842	−	0.994698i	$-0.532794\pi$
$73$	−1.75736	−0.205683	−0.102842	+	0.994698i	$0.532794\pi$
$74$	4.24264	0.493197
$75$	−1.00000	−0.115470
$76$	0	0
$77$	20.7279	2.36217
$78$	−0.828427	−0.0938009
$79$	7.31371	0.822856	0.411428	−	0.911442i	$-0.365030\pi$
$79$	7.31371	0.822856	0.411428	+	0.911442i	$0.365030\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	−10.7279	−1.18470
$83$	−0.0710678	−0.00780071	−0.00390035	−	0.999992i	$-0.501242\pi$
$83$	−0.0710678	−0.00780071	−0.00390035	+	0.999992i	$0.501242\pi$
$84$	0	0
$85$	−8.07107	−0.875430
$86$	8.48528	0.914991
$87$	6.41421	0.687676
$88$	15.3137	1.63245
$89$	−18.1421	−1.92306	−0.961531	−	0.274696i	$-0.911423\pi$
$89$	−18.1421	−1.92306	−0.961531	+	0.274696i	$0.911423\pi$
$90$	−1.41421	−0.149071
$91$	−2.24264	−0.235093
$92$	0	0
$93$	9.82843	1.01916
$94$	−16.1421	−1.66493
$95$	−2.24264	−0.230090
$96$	0	0
$97$	−6.82843	−0.693322	−0.346661	−	0.937991i	$-0.612685\pi$
$97$	−6.82843	−0.693322	−0.346661	+	0.937991i	$0.612685\pi$
$98$	10.8284	1.09384
$99$	−5.41421	−0.544149
$100$	0	0
$101$	−18.5563	−1.84643	−0.923213	−	0.384289i	$-0.874447\pi$
$101$	−18.5563	−1.84643	−0.923213	+	0.384289i	$0.874447\pi$
$102$	−11.4142	−1.13018
$103$	−4.82843	−0.475759	−0.237880	−	0.971295i	$-0.576452\pi$
$103$	−4.82843	−0.475759	−0.237880	+	0.971295i	$0.576452\pi$
$104$	−1.65685	−0.162468
$105$	−3.82843	−0.373616
$106$	−1.75736	−0.170690
$107$	−1.58579	−0.153304	−0.0766519	−	0.997058i	$-0.524423\pi$
$107$	−1.58579	−0.153304	−0.0766519	+	0.997058i	$0.524423\pi$
$108$	0	0
$109$	5.75736	0.551455	0.275728	−	0.961236i	$-0.411081\pi$
$109$	5.75736	0.551455	0.275728	+	0.961236i	$0.411081\pi$
$110$	7.65685	0.730052
$111$	−3.00000	−0.284747
$112$	15.3137	1.44701
$113$	−7.58579	−0.713611	−0.356805	−	0.934179i	$-0.616134\pi$
$113$	−7.58579	−0.713611	−0.356805	+	0.934179i	$0.616134\pi$
$114$	−3.17157	−0.297045
$115$	1.00000	0.0932505
$116$	0	0
$117$	0.585786	0.0541560
$118$	18.2426	1.67937
$119$	−30.8995	−2.83255
$120$	−2.82843	−0.258199
$121$	18.3137	1.66488
$122$	3.65685	0.331076
$123$	7.58579	0.683987
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	−5.41421	−0.482336
$127$	8.24264	0.731416	0.365708	−	0.930730i	$-0.380827\pi$
$127$	8.24264	0.731416	0.365708	+	0.930730i	$0.380827\pi$
$128$	11.3137	1.00000
$129$	−6.00000	−0.528271
$130$	−0.828427	−0.0726579
$131$	−2.34315	−0.204722	−0.102361	−	0.994747i	$-0.532640\pi$
$131$	−2.34315	−0.204722	−0.102361	+	0.994747i	$0.532640\pi$
$132$	0	0
$133$	−8.58579	−0.744482
$134$	7.75736	0.670134
$135$	1.00000	0.0860663
$136$	−22.8284	−1.95752
$137$	−2.82843	−0.241649	−0.120824	−	0.992674i	$-0.538554\pi$
$137$	−2.82843	−0.241649	−0.120824	+	0.992674i	$0.538554\pi$
$138$	1.41421	0.120386
$139$	−11.0000	−0.933008	−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000	−0.933008	−0.466504	+	0.884519i	$0.654487\pi$
$140$	0	0
$141$	11.4142	0.961250
$142$	−15.4142	−1.29353
$143$	−3.17157	−0.265220
$144$	−4.00000	−0.333333
$145$	6.41421	0.532671
$146$	−2.48528	−0.205683
$147$	−7.65685	−0.631527
$148$	0	0
$149$	−5.07107	−0.415438	−0.207719	−	0.978189i	$-0.566604\pi$
$149$	−5.07107	−0.415438	−0.207719	+	0.978189i	$0.566604\pi$
$150$	−1.41421	−0.115470
$151$	13.3137	1.08345	0.541727	−	0.840554i	$-0.317771\pi$
$151$	13.3137	1.08345	0.541727	+	0.840554i	$0.317771\pi$
$152$	−6.34315	−0.514497
$153$	8.07107	0.652507
$154$	29.3137	2.36217
$155$	9.82843	0.789438
$156$	0	0
$157$	15.4853	1.23586	0.617930	−	0.786233i	$-0.287972\pi$
$157$	15.4853	1.23586	0.617930	+	0.786233i	$0.287972\pi$
$158$	10.3431	0.822856
$159$	1.24264	0.0985478
$160$	0	0
$161$	3.82843	0.301722
$162$	1.41421	0.111111
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	0	0
$165$	−5.41421	−0.421496
$166$	−0.100505	−0.00780071
$167$	18.7279	1.44921	0.724605	−	0.689164i	$-0.242022\pi$
$167$	18.7279	1.44921	0.724605	+	0.689164i	$0.242022\pi$
$168$	−10.8284	−0.835431
$169$	−12.6569	−0.973604
$170$	−11.4142	−0.875430
$171$	2.24264	0.171499
$172$	0	0
$173$	2.34315	0.178146	0.0890730	−	0.996025i	$-0.471610\pi$
$173$	2.34315	0.178146	0.0890730	+	0.996025i	$0.471610\pi$
$174$	9.07107	0.687676
$175$	−3.82843	−0.289402
$176$	21.6569	1.63245
$177$	−12.8995	−0.969585
$178$	−25.6569	−1.92306
$179$	3.65685	0.273326	0.136663	−	0.990618i	$-0.456362\pi$
$179$	3.65685	0.273326	0.136663	+	0.990618i	$0.456362\pi$
$180$	0	0
$181$	−1.17157	−0.0870823	−0.0435412	−	0.999052i	$-0.513864\pi$
$181$	−1.17157	−0.0870823	−0.0435412	+	0.999052i	$0.513864\pi$
$182$	−3.17157	−0.235093
$183$	−2.58579	−0.191147
$184$	2.82843	0.208514
$185$	−3.00000	−0.220564
$186$	13.8995	1.01916
$187$	−43.6985	−3.19555
$188$	0	0
$189$	3.82843	0.278477
$190$	−3.17157	−0.230090
$191$	1.75736	0.127158	0.0635790	−	0.997977i	$-0.479749\pi$
$191$	1.75736	0.127158	0.0635790	+	0.997977i	$0.479749\pi$
$192$	−8.00000	−0.577350
$193$	8.48528	0.610784	0.305392	−	0.952227i	$-0.401213\pi$
$193$	8.48528	0.610784	0.305392	+	0.952227i	$0.401213\pi$
$194$	−9.65685	−0.693322
$195$	0.585786	0.0419490
$196$	0	0
$197$	11.1716	0.795942	0.397971	−	0.917398i	$-0.369715\pi$
$197$	11.1716	0.795942	0.397971	+	0.917398i	$0.369715\pi$
$198$	−7.65685	−0.544149
$199$	−6.48528	−0.459729	−0.229865	−	0.973223i	$-0.573828\pi$
$199$	−6.48528	−0.459729	−0.229865	+	0.973223i	$0.573828\pi$
$200$	−2.82843	−0.200000
$201$	−5.48528	−0.386902
$202$	−26.2426	−1.84643
$203$	24.5563	1.72352
$204$	0	0
$205$	7.58579	0.529814
$206$	−6.82843	−0.475759
$207$	−1.00000	−0.0695048
$208$	−2.34315	−0.162468
$209$	−12.1421	−0.839889
$210$	−5.41421	−0.373616
$211$	−10.6569	−0.733648	−0.366824	−	0.930290i	$-0.619555\pi$
$211$	−10.6569	−0.733648	−0.366824	+	0.930290i	$0.619555\pi$
$212$	0	0
$213$	10.8995	0.746821
$214$	−2.24264	−0.153304
$215$	−6.00000	−0.409197
$216$	2.82843	0.192450
$217$	37.6274	2.55432
$218$	8.14214	0.551455
$219$	1.75736	0.118751
$220$	0	0
$221$	4.72792	0.318034
$222$	−4.24264	−0.284747
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	−10.7279	−0.713611
$227$	6.34315	0.421009	0.210505	−	0.977593i	$-0.432489\pi$
$227$	6.34315	0.421009	0.210505	+	0.977593i	$0.432489\pi$
$228$	0	0
$229$	−15.7990	−1.04403	−0.522013	−	0.852937i	$-0.674819\pi$
$229$	−15.7990	−1.04403	−0.522013	+	0.852937i	$0.674819\pi$
$230$	1.41421	0.0932505
$231$	−20.7279	−1.36380
$232$	18.1421	1.19109
$233$	−1.65685	−0.108544	−0.0542721	−	0.998526i	$-0.517284\pi$
$233$	−1.65685	−0.108544	−0.0542721	+	0.998526i	$0.517284\pi$
$234$	0.828427	0.0541560
$235$	11.4142	0.744581
$236$	0	0
$237$	−7.31371	−0.475076
$238$	−43.6985	−2.83255
$239$	−4.41421	−0.285532	−0.142766	−	0.989756i	$-0.545600\pi$
$239$	−4.41421	−0.285532	−0.142766	+	0.989756i	$0.545600\pi$
$240$	−4.00000	−0.258199
$241$	20.5858	1.32605	0.663024	−	0.748599i	$-0.269273\pi$
$241$	20.5858	1.32605	0.663024	+	0.748599i	$0.269273\pi$
$242$	25.8995	1.66488
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−7.65685	−0.489178
$246$	10.7279	0.683987
$247$	1.31371	0.0835893
$248$	27.7990	1.76524
$249$	0.0710678	0.00450374
$250$	−1.41421	−0.0894427
$251$	24.6274	1.55447	0.777234	−	0.629211i	$-0.216622\pi$
$251$	24.6274	1.55447	0.777234	+	0.629211i	$0.216622\pi$
$252$	0	0
$253$	5.41421	0.340389
$254$	11.6569	0.731416
$255$	8.07107	0.505430
$256$	0	0
$257$	−18.0416	−1.12541	−0.562703	−	0.826659i	$-0.690239\pi$
$257$	−18.0416	−1.12541	−0.562703	+	0.826659i	$0.690239\pi$
$258$	−8.48528	−0.528271
$259$	−11.4853	−0.713661
$260$	0	0
$261$	−6.41421	−0.397030
$262$	−3.31371	−0.204722
$263$	−21.7279	−1.33980	−0.669901	−	0.742451i	$-0.733663\pi$
$263$	−21.7279	−1.33980	−0.669901	+	0.742451i	$0.733663\pi$
$264$	−15.3137	−0.942494
$265$	1.24264	0.0763348
$266$	−12.1421	−0.744482
$267$	18.1421	1.11028
$268$	0	0
$269$	4.55635	0.277806	0.138903	−	0.990306i	$-0.455642\pi$
$269$	4.55635	0.277806	0.138903	+	0.990306i	$0.455642\pi$
$270$	1.41421	0.0860663
$271$	−23.4853	−1.42663	−0.713315	−	0.700844i	$-0.752807\pi$
$271$	−23.4853	−1.42663	−0.713315	+	0.700844i	$0.752807\pi$
$272$	−32.2843	−1.95752
$273$	2.24264	0.135731
$274$	−4.00000	−0.241649
$275$	−5.41421	−0.326489
$276$	0	0
$277$	2.68629	0.161404	0.0807018	−	0.996738i	$-0.474284\pi$
$277$	2.68629	0.161404	0.0807018	+	0.996738i	$0.474284\pi$
$278$	−15.5563	−0.933008
$279$	−9.82843	−0.588413
$280$	−10.8284	−0.647122
$281$	14.5858	0.870115	0.435058	−	0.900403i	$-0.356728\pi$
$281$	14.5858	0.870115	0.435058	+	0.900403i	$0.356728\pi$
$282$	16.1421	0.961250
$283$	−17.3431	−1.03094	−0.515472	−	0.856907i	$-0.672383\pi$
$283$	−17.3431	−1.03094	−0.515472	+	0.856907i	$0.672383\pi$
$284$	0	0
$285$	2.24264	0.132843
$286$	−4.48528	−0.265220
$287$	29.0416	1.71427
$288$	0	0
$289$	48.1421	2.83189
$290$	9.07107	0.532671
$291$	6.82843	0.400289
$292$	0	0
$293$	−22.8995	−1.33780	−0.668901	−	0.743351i	$-0.733235\pi$
$293$	−22.8995	−1.33780	−0.668901	+	0.743351i	$0.733235\pi$
$294$	−10.8284	−0.631527
$295$	−12.8995	−0.751038
$296$	−8.48528	−0.493197
$297$	5.41421	0.314165
$298$	−7.17157	−0.415438
$299$	−0.585786	−0.0338769
$300$	0	0
$301$	−22.9706	−1.32400
$302$	18.8284	1.08345
$303$	18.5563	1.06603
$304$	−8.97056	−0.514497
$305$	−2.58579	−0.148062
$306$	11.4142	0.652507
$307$	−15.2132	−0.868263	−0.434132	−	0.900849i	$-0.642945\pi$
$307$	−15.2132	−0.868263	−0.434132	+	0.900849i	$0.642945\pi$
$308$	0	0
$309$	4.82843	0.274680
$310$	13.8995	0.789438
$311$	6.34315	0.359687	0.179843	−	0.983695i	$-0.442441\pi$
$311$	6.34315	0.359687	0.179843	+	0.983695i	$0.442441\pi$
$312$	1.65685	0.0938009
$313$	−7.97056	−0.450523	−0.225261	−	0.974298i	$-0.572324\pi$
$313$	−7.97056	−0.450523	−0.225261	+	0.974298i	$0.572324\pi$
$314$	21.8995	1.23586
$315$	3.82843	0.215707
$316$	0	0
$317$	1.55635	0.0874133	0.0437066	−	0.999044i	$-0.486083\pi$
$317$	1.55635	0.0874133	0.0437066	+	0.999044i	$0.486083\pi$
$318$	1.75736	0.0985478
$319$	34.7279	1.94439
$320$	−8.00000	−0.447214
$321$	1.58579	0.0885100
$322$	5.41421	0.301722
$323$	18.1005	1.00714
$324$	0	0
$325$	0.585786	0.0324936
$326$	−2.82843	−0.156652
$327$	−5.75736	−0.318383
$328$	21.4558	1.18470
$329$	43.6985	2.40918
$330$	−7.65685	−0.421496
$331$	1.00000	0.0549650	0.0274825	−	0.999622i	$-0.491251\pi$
$331$	1.00000	0.0549650	0.0274825	+	0.999622i	$0.491251\pi$
$332$	0	0
$333$	3.00000	0.164399
$334$	26.4853	1.44921
$335$	−5.48528	−0.299693
$336$	−15.3137	−0.835431
$337$	−2.97056	−0.161817	−0.0809084	−	0.996722i	$-0.525782\pi$
$337$	−2.97056	−0.161817	−0.0809084	+	0.996722i	$0.525782\pi$
$338$	−17.8995	−0.973604
$339$	7.58579	0.412003
$340$	0	0
$341$	53.2132	2.88166
$342$	3.17157	0.171499
$343$	−2.51472	−0.135782
$344$	−16.9706	−0.914991
$345$	−1.00000	−0.0538382
$346$	3.31371	0.178146
$347$	18.8284	1.01076	0.505381	−	0.862896i	$-0.331352\pi$
$347$	18.8284	1.01076	0.505381	+	0.862896i	$0.331352\pi$
$348$	0	0
$349$	30.1127	1.61190	0.805948	−	0.591986i	$-0.201656\pi$
$349$	30.1127	1.61190	0.805948	+	0.591986i	$0.201656\pi$
$350$	−5.41421	−0.289402
$351$	−0.585786	−0.0312670
$352$	0	0
$353$	30.5269	1.62478	0.812392	−	0.583112i	$-0.198165\pi$
$353$	30.5269	1.62478	0.812392	+	0.583112i	$0.198165\pi$
$354$	−18.2426	−0.969585
$355$	10.8995	0.578485
$356$	0	0
$357$	30.8995	1.63537
$358$	5.17157	0.273326
$359$	−25.0711	−1.32320	−0.661600	−	0.749857i	$-0.730122\pi$
$359$	−25.0711	−1.32320	−0.661600	+	0.749857i	$0.730122\pi$
$360$	2.82843	0.149071
$361$	−13.9706	−0.735293
$362$	−1.65685	−0.0870823
$363$	−18.3137	−0.961220
$364$	0	0
$365$	1.75736	0.0919844
$366$	−3.65685	−0.191147
$367$	19.9706	1.04245	0.521227	−	0.853418i	$-0.325474\pi$
$367$	19.9706	1.04245	0.521227	+	0.853418i	$0.325474\pi$
$368$	4.00000	0.208514
$369$	−7.58579	−0.394900
$370$	−4.24264	−0.220564
$371$	4.75736	0.246990
$372$	0	0
$373$	20.9706	1.08581	0.542907	−	0.839793i	$-0.317323\pi$
$373$	20.9706	1.08581	0.542907	+	0.839793i	$0.317323\pi$
$374$	−61.7990	−3.19555
$375$	1.00000	0.0516398
$376$	32.2843	1.66493
$377$	−3.75736	−0.193514
$378$	5.41421	0.278477
$379$	−4.97056	−0.255321	−0.127660	−	0.991818i	$-0.540747\pi$
$379$	−4.97056	−0.255321	−0.127660	+	0.991818i	$0.540747\pi$
$380$	0	0
$381$	−8.24264	−0.422283
$382$	2.48528	0.127158
$383$	11.5858	0.592006	0.296003	−	0.955187i	$-0.404346\pi$
$383$	11.5858	0.592006	0.296003	+	0.955187i	$0.404346\pi$
$384$	−11.3137	−0.577350
$385$	−20.7279	−1.05639
$386$	12.0000	0.610784
$387$	6.00000	0.304997
$388$	0	0
$389$	−2.48528	−0.126009	−0.0630044	−	0.998013i	$-0.520068\pi$
$389$	−2.48528	−0.126009	−0.0630044	+	0.998013i	$0.520068\pi$
$390$	0.828427	0.0419490
$391$	−8.07107	−0.408171
$392$	−21.6569	−1.09384
$393$	2.34315	0.118196
$394$	15.7990	0.795942
$395$	−7.31371	−0.367993
$396$	0	0
$397$	7.31371	0.367065	0.183532	−	0.983014i	$-0.441247\pi$
$397$	7.31371	0.367065	0.183532	+	0.983014i	$0.441247\pi$
$398$	−9.17157	−0.459729
$399$	8.58579	0.429827
$400$	−4.00000	−0.200000
$401$	−3.85786	−0.192653	−0.0963263	−	0.995350i	$-0.530709\pi$
$401$	−3.85786	−0.192653	−0.0963263	+	0.995350i	$0.530709\pi$
$402$	−7.75736	−0.386902
$403$	−5.75736	−0.286794
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	34.7279	1.72352
$407$	−16.2426	−0.805118
$408$	22.8284	1.13018
$409$	−15.4853	−0.765698	−0.382849	−	0.923811i	$-0.625057\pi$
$409$	−15.4853	−0.765698	−0.382849	+	0.923811i	$0.625057\pi$
$410$	10.7279	0.529814
$411$	2.82843	0.139516
$412$	0	0
$413$	−49.3848	−2.43007
$414$	−1.41421	−0.0695048
$415$	0.0710678	0.00348858
$416$	0	0
$417$	11.0000	0.538672
$418$	−17.1716	−0.839889
$419$	24.0416	1.17451	0.587255	−	0.809402i	$-0.300208\pi$
$419$	24.0416	1.17451	0.587255	+	0.809402i	$0.300208\pi$
$420$	0	0
$421$	−10.2426	−0.499196	−0.249598	−	0.968350i	$-0.580298\pi$
$421$	−10.2426	−0.499196	−0.249598	+	0.968350i	$0.580298\pi$
$422$	−15.0711	−0.733648
$423$	−11.4142	−0.554978
$424$	3.51472	0.170690
$425$	8.07107	0.391504
$426$	15.4142	0.746821
$427$	−9.89949	−0.479070
$428$	0	0
$429$	3.17157	0.153125
$430$	−8.48528	−0.409197
$431$	11.7990	0.568337	0.284169	−	0.958774i	$-0.408282\pi$
$431$	11.7990	0.568337	0.284169	+	0.958774i	$0.408282\pi$
$432$	4.00000	0.192450
$433$	3.82843	0.183982	0.0919912	−	0.995760i	$-0.470677\pi$
$433$	3.82843	0.183982	0.0919912	+	0.995760i	$0.470677\pi$
$434$	53.2132	2.55432
$435$	−6.41421	−0.307538
$436$	0	0
$437$	−2.24264	−0.107280
$438$	2.48528	0.118751
$439$	−1.51472	−0.0722936	−0.0361468	−	0.999346i	$-0.511508\pi$
$439$	−1.51472	−0.0722936	−0.0361468	+	0.999346i	$0.511508\pi$
$440$	−15.3137	−0.730052
$441$	7.65685	0.364612
$442$	6.68629	0.318034
$443$	−34.5269	−1.64042	−0.820212	−	0.572060i	$-0.806144\pi$
$443$	−34.5269	−1.64042	−0.820212	+	0.572060i	$0.806144\pi$
$444$	0	0
$445$	18.1421	0.860020
$446$	−19.7990	−0.937509
$447$	5.07107	0.239853
$448$	−30.6274	−1.44701
$449$	−4.75736	−0.224514	−0.112257	−	0.993679i	$-0.535808\pi$
$449$	−4.75736	−0.224514	−0.112257	+	0.993679i	$0.535808\pi$
$450$	1.41421	0.0666667
$451$	41.0711	1.93396
$452$	0	0
$453$	−13.3137	−0.625533
$454$	8.97056	0.421009
$455$	2.24264	0.105137
$456$	6.34315	0.297045
$457$	−31.0000	−1.45012	−0.725059	−	0.688686i	$-0.758188\pi$
$457$	−31.0000	−1.45012	−0.725059	+	0.688686i	$0.758188\pi$
$458$	−22.3431	−1.04403
$459$	−8.07107	−0.376725
$460$	0	0
$461$	−26.4853	−1.23354	−0.616771	−	0.787142i	$-0.711560\pi$
$461$	−26.4853	−1.23354	−0.616771	+	0.787142i	$0.711560\pi$
$462$	−29.3137	−1.36380
$463$	2.92893	0.136119	0.0680595	−	0.997681i	$-0.478319\pi$
$463$	2.92893	0.136119	0.0680595	+	0.997681i	$0.478319\pi$
$464$	25.6569	1.19109
$465$	−9.82843	−0.455782
$466$	−2.34315	−0.108544
$467$	6.41421	0.296814	0.148407	−	0.988926i	$-0.452585\pi$
$467$	6.41421	0.296814	0.148407	+	0.988926i	$0.452585\pi$
$468$	0	0
$469$	−21.0000	−0.969690
$470$	16.1421	0.744581
$471$	−15.4853	−0.713524
$472$	−36.4853	−1.67937
$473$	−32.4853	−1.49367
$474$	−10.3431	−0.475076
$475$	2.24264	0.102899
$476$	0	0
$477$	−1.24264	−0.0568966
$478$	−6.24264	−0.285532
$479$	6.72792	0.307407	0.153703	−	0.988117i	$-0.450880\pi$
$479$	6.72792	0.307407	0.153703	+	0.988117i	$0.450880\pi$
$480$	0	0
$481$	1.75736	0.0801287
$482$	29.1127	1.32605
$483$	−3.82843	−0.174199
$484$	0	0
$485$	6.82843	0.310063
$486$	−1.41421	−0.0641500
$487$	−26.3848	−1.19561	−0.597804	−	0.801642i	$-0.703960\pi$
$487$	−26.3848	−1.19561	−0.597804	+	0.801642i	$0.703960\pi$
$488$	−7.31371	−0.331076
$489$	2.00000	0.0904431
$490$	−10.8284	−0.489178
$491$	−12.0711	−0.544760	−0.272380	−	0.962190i	$-0.587811\pi$
$491$	−12.0711	−0.544760	−0.272380	+	0.962190i	$0.587811\pi$
$492$	0	0
$493$	−51.7696	−2.33158
$494$	1.85786	0.0835893
$495$	5.41421	0.243351
$496$	39.3137	1.76524
$497$	41.7279	1.87175
$498$	0.100505	0.00450374
$499$	−13.0000	−0.581960	−0.290980	−	0.956729i	$-0.593981\pi$
$499$	−13.0000	−0.581960	−0.290980	+	0.956729i	$0.593981\pi$
$500$	0	0
$501$	−18.7279	−0.836702
$502$	34.8284	1.55447
$503$	−39.7279	−1.77138	−0.885690	−	0.464277i	$-0.846314\pi$
$503$	−39.7279	−1.77138	−0.885690	+	0.464277i	$0.846314\pi$
$504$	10.8284	0.482336
$505$	18.5563	0.825747
$506$	7.65685	0.340389
$507$	12.6569	0.562111
$508$	0	0
$509$	7.79899	0.345684	0.172842	−	0.984950i	$-0.444705\pi$
$509$	7.79899	0.345684	0.172842	+	0.984950i	$0.444705\pi$
$510$	11.4142	0.505430
$511$	6.72792	0.297626
$512$	−22.6274	−1.00000
$513$	−2.24264	−0.0990150
$514$	−25.5147	−1.12541
$515$	4.82843	0.212766
$516$	0	0
$517$	61.7990	2.71792
$518$	−16.2426	−0.713661
$519$	−2.34315	−0.102853
$520$	1.65685	0.0726579
$521$	−31.5563	−1.38251	−0.691254	−	0.722612i	$-0.742942\pi$
$521$	−31.5563	−1.38251	−0.691254	+	0.722612i	$0.742942\pi$
$522$	−9.07107	−0.397030
$523$	12.0000	0.524723	0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000	0.524723	0.262362	+	0.964970i	$0.415499\pi$
$524$	0	0
$525$	3.82843	0.167086
$526$	−30.7279	−1.33980
$527$	−79.3259	−3.45549
$528$	−21.6569	−0.942494
$529$	1.00000	0.0434783
$530$	1.75736	0.0763348
$531$	12.8995	0.559790
$532$	0	0
$533$	−4.44365	−0.192476
$534$	25.6569	1.11028
$535$	1.58579	0.0685595
$536$	−15.5147	−0.670134
$537$	−3.65685	−0.157805
$538$	6.44365	0.277806
$539$	−41.4558	−1.78563
$540$	0	0
$541$	5.85786	0.251849	0.125925	−	0.992040i	$-0.459810\pi$
$541$	5.85786	0.251849	0.125925	+	0.992040i	$0.459810\pi$
$542$	−33.2132	−1.42663
$543$	1.17157	0.0502770
$544$	0	0
$545$	−5.75736	−0.246618
$546$	3.17157	0.135731
$547$	12.9706	0.554581	0.277291	−	0.960786i	$-0.410564\pi$
$547$	12.9706	0.554581	0.277291	+	0.960786i	$0.410564\pi$
$548$	0	0
$549$	2.58579	0.110359
$550$	−7.65685	−0.326489
$551$	−14.3848	−0.612812
$552$	−2.82843	−0.120386
$553$	−28.0000	−1.19068
$554$	3.79899	0.161404
$555$	3.00000	0.127343
$556$	0	0
$557$	−6.89949	−0.292341	−0.146170	−	0.989259i	$-0.546695\pi$
$557$	−6.89949	−0.292341	−0.146170	+	0.989259i	$0.546695\pi$
$558$	−13.8995	−0.588413
$559$	3.51472	0.148657
$560$	−15.3137	−0.647122
$561$	43.6985	1.84495
$562$	20.6274	0.870115
$563$	20.8995	0.880809	0.440404	−	0.897800i	$-0.354835\pi$
$563$	20.8995	0.880809	0.440404	+	0.897800i	$0.354835\pi$
$564$	0	0
$565$	7.58579	0.319136
$566$	−24.5269	−1.03094
$567$	−3.82843	−0.160779
$568$	30.8284	1.29353
$569$	12.1421	0.509025	0.254512	−	0.967070i	$-0.418085\pi$
$569$	12.1421	0.509025	0.254512	+	0.967070i	$0.418085\pi$
$570$	3.17157	0.132843
$571$	28.7279	1.20223	0.601113	−	0.799164i	$-0.294724\pi$
$571$	28.7279	1.20223	0.601113	+	0.799164i	$0.294724\pi$
$572$	0	0
$573$	−1.75736	−0.0734147
$574$	41.0711	1.71427
$575$	−1.00000	−0.0417029
$576$	8.00000	0.333333
$577$	−3.02944	−0.126117	−0.0630586	−	0.998010i	$-0.520085\pi$
$577$	−3.02944	−0.126117	−0.0630586	+	0.998010i	$0.520085\pi$
$578$	68.0833	2.83189
$579$	−8.48528	−0.352636
$580$	0	0
$581$	0.272078	0.0112877
$582$	9.65685	0.400289
$583$	6.72792	0.278642
$584$	4.97056	0.205683
$585$	−0.585786	−0.0242193
$586$	−32.3848	−1.33780
$587$	−4.62742	−0.190994	−0.0954970	−	0.995430i	$-0.530444\pi$
$587$	−4.62742	−0.190994	−0.0954970	+	0.995430i	$0.530444\pi$
$588$	0	0
$589$	−22.0416	−0.908210
$590$	−18.2426	−0.751038
$591$	−11.1716	−0.459537
$592$	−12.0000	−0.493197
$593$	−33.5563	−1.37799	−0.688997	−	0.724764i	$-0.741949\pi$
$593$	−33.5563	−1.37799	−0.688997	+	0.724764i	$0.741949\pi$
$594$	7.65685	0.314165
$595$	30.8995	1.26676
$596$	0	0
$597$	6.48528	0.265425
$598$	−0.828427	−0.0338769
$599$	38.8284	1.58649	0.793243	−	0.608905i	$-0.208391\pi$
$599$	38.8284	1.58649	0.793243	+	0.608905i	$0.208391\pi$
$600$	2.82843	0.115470
$601$	4.31371	0.175960	0.0879799	−	0.996122i	$-0.471959\pi$
$601$	4.31371	0.175960	0.0879799	+	0.996122i	$0.471959\pi$
$602$	−32.4853	−1.32400
$603$	5.48528	0.223378
$604$	0	0
$605$	−18.3137	−0.744558
$606$	26.2426	1.06603
$607$	−1.61522	−0.0655599	−0.0327800	−	0.999463i	$-0.510436\pi$
$607$	−1.61522	−0.0655599	−0.0327800	+	0.999463i	$0.510436\pi$
$608$	0	0
$609$	−24.5563	−0.995073
$610$	−3.65685	−0.148062
$611$	−6.68629	−0.270498
$612$	0	0
$613$	12.6274	0.510017	0.255008	−	0.966939i	$-0.417922\pi$
$613$	12.6274	0.510017	0.255008	+	0.966939i	$0.417922\pi$
$614$	−21.5147	−0.868263
$615$	−7.58579	−0.305888
$616$	−58.6274	−2.36217
$617$	−14.8995	−0.599831	−0.299916	−	0.953966i	$-0.596959\pi$
$617$	−14.8995	−0.599831	−0.299916	+	0.953966i	$0.596959\pi$
$618$	6.82843	0.274680
$619$	8.28427	0.332973	0.166486	−	0.986044i	$-0.446758\pi$
$619$	8.28427	0.332973	0.166486	+	0.986044i	$0.446758\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	8.97056	0.359687
$623$	69.4558	2.78269
$624$	2.34315	0.0938009
$625$	1.00000	0.0400000
$626$	−11.2721	−0.450523
$627$	12.1421	0.484910
$628$	0	0
$629$	24.2132	0.965444
$630$	5.41421	0.215707
$631$	−9.21320	−0.366772	−0.183386	−	0.983041i	$-0.558706\pi$
$631$	−9.21320	−0.366772	−0.183386	+	0.983041i	$0.558706\pi$
$632$	−20.6863	−0.822856
$633$	10.6569	0.423572
$634$	2.20101	0.0874133
$635$	−8.24264	−0.327099
$636$	0	0
$637$	4.48528	0.177713
$638$	49.1127	1.94439
$639$	−10.8995	−0.431177
$640$	−11.3137	−0.447214
$641$	19.0711	0.753262	0.376631	−	0.926363i	$-0.377082\pi$
$641$	19.0711	0.753262	0.376631	+	0.926363i	$0.377082\pi$
$642$	2.24264	0.0885100
$643$	33.0000	1.30139	0.650696	−	0.759338i	$-0.274477\pi$
$643$	33.0000	1.30139	0.650696	+	0.759338i	$0.274477\pi$
$644$	0	0
$645$	6.00000	0.236250
$646$	25.5980	1.00714
$647$	−21.2132	−0.833977	−0.416989	−	0.908912i	$-0.636914\pi$
$647$	−21.2132	−0.833977	−0.416989	+	0.908912i	$0.636914\pi$
$648$	−2.82843	−0.111111
$649$	−69.8406	−2.74148
$650$	0.828427	0.0324936
$651$	−37.6274	−1.47473
$652$	0	0
$653$	−5.41421	−0.211875	−0.105937	−	0.994373i	$-0.533784\pi$
$653$	−5.41421	−0.211875	−0.105937	+	0.994373i	$0.533784\pi$
$654$	−8.14214	−0.318383
$655$	2.34315	0.0915543
$656$	30.3431	1.18470
$657$	−1.75736	−0.0685611
$658$	61.7990	2.40918
$659$	−34.3848	−1.33944	−0.669720	−	0.742613i	$-0.733586\pi$
$659$	−34.3848	−1.33944	−0.669720	+	0.742613i	$0.733586\pi$
$660$	0	0
$661$	−39.4558	−1.53465	−0.767327	−	0.641256i	$-0.778414\pi$
$661$	−39.4558	−1.53465	−0.767327	+	0.641256i	$0.778414\pi$
$662$	1.41421	0.0549650
$663$	−4.72792	−0.183617
$664$	0.201010	0.00780071
$665$	8.58579	0.332943
$666$	4.24264	0.164399
$667$	6.41421	0.248359
$668$	0	0
$669$	14.0000	0.541271
$670$	−7.75736	−0.299693
$671$	−14.0000	−0.540464
$672$	0	0
$673$	−17.6985	−0.682226	−0.341113	−	0.940022i	$-0.610804\pi$
$673$	−17.6985	−0.682226	−0.341113	+	0.940022i	$0.610804\pi$
$674$	−4.20101	−0.161817
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−10.7574	−0.413439	−0.206719	−	0.978400i	$-0.566279\pi$
$677$	−10.7574	−0.413439	−0.206719	+	0.978400i	$0.566279\pi$
$678$	10.7279	0.412003
$679$	26.1421	1.00324
$680$	22.8284	0.875430
$681$	−6.34315	−0.243070
$682$	75.2548	2.88166
$683$	44.1838	1.69064	0.845322	−	0.534257i	$-0.179408\pi$
$683$	44.1838	1.69064	0.845322	+	0.534257i	$0.179408\pi$
$684$	0	0
$685$	2.82843	0.108069
$686$	−3.55635	−0.135782
$687$	15.7990	0.602769
$688$	−24.0000	−0.914991
$689$	−0.727922	−0.0277316
$690$	−1.41421	−0.0538382
$691$	−4.34315	−0.165221	−0.0826105	−	0.996582i	$-0.526326\pi$
$691$	−4.34315	−0.165221	−0.0826105	+	0.996582i	$0.526326\pi$
$692$	0	0
$693$	20.7279	0.787389
$694$	26.6274	1.01076
$695$	11.0000	0.417254
$696$	−18.1421	−0.687676
$697$	−61.2254	−2.31908
$698$	42.5858	1.61190
$699$	1.65685	0.0626680
$700$	0	0
$701$	0.384776	0.0145328	0.00726640	−	0.999974i	$-0.497687\pi$
$701$	0.384776	0.0145328	0.00726640	+	0.999974i	$0.497687\pi$
$702$	−0.828427	−0.0312670
$703$	6.72792	0.253748
$704$	−43.3137	−1.63245
$705$	−11.4142	−0.429884
$706$	43.1716	1.62478
$707$	71.0416	2.67180
$708$	0	0
$709$	−8.72792	−0.327784	−0.163892	−	0.986478i	$-0.552405\pi$
$709$	−8.72792	−0.327784	−0.163892	+	0.986478i	$0.552405\pi$
$710$	15.4142	0.578485
$711$	7.31371	0.274285
$712$	51.3137	1.92306
$713$	9.82843	0.368077
$714$	43.6985	1.63537
$715$	3.17157	0.118610
$716$	0	0
$717$	4.41421	0.164852
$718$	−35.4558	−1.32320
$719$	−6.21320	−0.231713	−0.115857	−	0.993266i	$-0.536961\pi$
$719$	−6.21320	−0.231713	−0.115857	+	0.993266i	$0.536961\pi$
$720$	4.00000	0.149071
$721$	18.4853	0.688428
$722$	−19.7574	−0.735293
$723$	−20.5858	−0.765594
$724$	0	0
$725$	−6.41421	−0.238218
$726$	−25.8995	−0.961220
$727$	−23.0000	−0.853023	−0.426511	−	0.904482i	$-0.640258\pi$
$727$	−23.0000	−0.853023	−0.426511	+	0.904482i	$0.640258\pi$
$728$	6.34315	0.235093
$729$	1.00000	0.0370370
$730$	2.48528	0.0919844
$731$	48.4264	1.79112
$732$	0	0
$733$	−0.514719	−0.0190116	−0.00950578	−	0.999955i	$-0.503026\pi$
$733$	−0.514719	−0.0190116	−0.00950578	+	0.999955i	$0.503026\pi$
$734$	28.2426	1.04245
$735$	7.65685	0.282427
$736$	0	0
$737$	−29.6985	−1.09396
$738$	−10.7279	−0.394900
$739$	−8.37258	−0.307990	−0.153995	−	0.988072i	$-0.549214\pi$
$739$	−8.37258	−0.307990	−0.153995	+	0.988072i	$0.549214\pi$
$740$	0	0
$741$	−1.31371	−0.0482603
$742$	6.72792	0.246990
$743$	−14.3431	−0.526199	−0.263099	−	0.964769i	$-0.584745\pi$
$743$	−14.3431	−0.526199	−0.263099	+	0.964769i	$0.584745\pi$
$744$	−27.7990	−1.01916
$745$	5.07107	0.185790
$746$	29.6569	1.08581
$747$	−0.0710678	−0.00260024
$748$	0	0
$749$	6.07107	0.221832
$750$	1.41421	0.0516398
$751$	−34.7279	−1.26724	−0.633620	−	0.773644i	$-0.718432\pi$
$751$	−34.7279	−1.26724	−0.633620	+	0.773644i	$0.718432\pi$
$752$	45.6569	1.66493
$753$	−24.6274	−0.897473
$754$	−5.31371	−0.193514
$755$	−13.3137	−0.484535
$756$	0	0
$757$	27.0000	0.981332	0.490666	−	0.871348i	$-0.336754\pi$
$757$	27.0000	0.981332	0.490666	+	0.871348i	$0.336754\pi$
$758$	−7.02944	−0.255321
$759$	−5.41421	−0.196524
$760$	6.34315	0.230090
$761$	−34.0711	−1.23508	−0.617538	−	0.786541i	$-0.711870\pi$
$761$	−34.0711	−1.23508	−0.617538	+	0.786541i	$0.711870\pi$
$762$	−11.6569	−0.422283
$763$	−22.0416	−0.797961
$764$	0	0
$765$	−8.07107	−0.291810
$766$	16.3848	0.592006
$767$	7.55635	0.272844
$768$	0	0
$769$	−28.0416	−1.01121	−0.505604	−	0.862766i	$-0.668730\pi$
$769$	−28.0416	−1.01121	−0.505604	+	0.862766i	$0.668730\pi$
$770$	−29.3137	−1.05639
$771$	18.0416	0.649753
$772$	0	0
$773$	−8.82843	−0.317536	−0.158768	−	0.987316i	$-0.550752\pi$
$773$	−8.82843	−0.317536	−0.158768	+	0.987316i	$0.550752\pi$
$774$	8.48528	0.304997
$775$	−9.82843	−0.353048
$776$	19.3137	0.693322
$777$	11.4853	0.412032
$778$	−3.51472	−0.126009
$779$	−17.0122	−0.609525
$780$	0	0
$781$	59.0122	2.11162
$782$	−11.4142	−0.408171
$783$	6.41421	0.229225
$784$	−30.6274	−1.09384
$785$	−15.4853	−0.552693
$786$	3.31371	0.118196
$787$	−41.6274	−1.48386	−0.741929	−	0.670479i	$-0.766089\pi$
$787$	−41.6274	−1.48386	−0.741929	+	0.670479i	$0.766089\pi$
$788$	0	0
$789$	21.7279	0.773535
$790$	−10.3431	−0.367993
$791$	29.0416	1.03260
$792$	15.3137	0.544149
$793$	1.51472	0.0537892
$794$	10.3431	0.367065
$795$	−1.24264	−0.0440719
$796$	0	0
$797$	25.2426	0.894140	0.447070	−	0.894499i	$-0.352467\pi$
$797$	25.2426	0.894140	0.447070	+	0.894499i	$0.352467\pi$
$798$	12.1421	0.429827
$799$	−92.1249	−3.25914
$800$	0	0
$801$	−18.1421	−0.641021
$802$	−5.45584	−0.192653
$803$	9.51472	0.335767
$804$	0	0
$805$	−3.82843	−0.134934
$806$	−8.14214	−0.286794
$807$	−4.55635	−0.160391
$808$	52.4853	1.84643
$809$	47.8701	1.68302	0.841511	−	0.540240i	$-0.181667\pi$
$809$	47.8701	1.68302	0.841511	+	0.540240i	$0.181667\pi$
$810$	−1.41421	−0.0496904
$811$	−17.2843	−0.606933	−0.303466	−	0.952842i	$-0.598144\pi$
$811$	−17.2843	−0.606933	−0.303466	+	0.952842i	$0.598144\pi$
$812$	0	0
$813$	23.4853	0.823665
$814$	−22.9706	−0.805118
$815$	2.00000	0.0700569
$816$	32.2843	1.13018
$817$	13.4558	0.470760
$818$	−21.8995	−0.765698
$819$	−2.24264	−0.0783642
$820$	0	0
$821$	−4.20101	−0.146616	−0.0733081	−	0.997309i	$-0.523356\pi$
$821$	−4.20101	−0.146616	−0.0733081	+	0.997309i	$0.523356\pi$
$822$	4.00000	0.139516
$823$	34.3431	1.19713	0.598563	−	0.801075i	$-0.295738\pi$
$823$	34.3431	1.19713	0.598563	+	0.801075i	$0.295738\pi$
$824$	13.6569	0.475759
$825$	5.41421	0.188499
$826$	−69.8406	−2.43007
$827$	32.0122	1.11317	0.556587	−	0.830790i	$-0.312111\pi$
$827$	32.0122	1.11317	0.556587	+	0.830790i	$0.312111\pi$
$828$	0	0
$829$	6.51472	0.226266	0.113133	−	0.993580i	$-0.463911\pi$
$829$	6.51472	0.226266	0.113133	+	0.993580i	$0.463911\pi$
$830$	0.100505	0.00348858
$831$	−2.68629	−0.0931864
$832$	4.68629	0.162468
$833$	61.7990	2.14121
$834$	15.5563	0.538672
$835$	−18.7279	−0.648106
$836$	0	0
$837$	9.82843	0.339720
$838$	34.0000	1.17451
$839$	40.6274	1.40261	0.701307	−	0.712859i	$-0.252600\pi$
$839$	40.6274	1.40261	0.701307	+	0.712859i	$0.252600\pi$
$840$	10.8284	0.373616
$841$	12.1421	0.418694
$842$	−14.4853	−0.499196
$843$	−14.5858	−0.502361
$844$	0	0
$845$	12.6569	0.435409
$846$	−16.1421	−0.554978
$847$	−70.1127	−2.40910
$848$	4.97056	0.170690
$849$	17.3431	0.595215
$850$	11.4142	0.391504
$851$	−3.00000	−0.102839
$852$	0	0
$853$	19.4558	0.666155	0.333078	−	0.942899i	$-0.391913\pi$
$853$	19.4558	0.666155	0.333078	+	0.942899i	$0.391913\pi$
$854$	−14.0000	−0.479070
$855$	−2.24264	−0.0766967
$856$	4.48528	0.153304
$857$	−57.5980	−1.96751	−0.983755	−	0.179518i	$-0.942546\pi$
$857$	−57.5980	−1.96751	−0.983755	+	0.179518i	$0.942546\pi$
$858$	4.48528	0.153125
$859$	−57.0000	−1.94481	−0.972407	−	0.233289i	$-0.925051\pi$
$859$	−57.0000	−1.94481	−0.972407	+	0.233289i	$0.925051\pi$
$860$	0	0
$861$	−29.0416	−0.989736
$862$	16.6863	0.568337
$863$	16.8284	0.572846	0.286423	−	0.958103i	$-0.407534\pi$
$863$	16.8284	0.572846	0.286423	+	0.958103i	$0.407534\pi$
$864$	0	0
$865$	−2.34315	−0.0796693
$866$	5.41421	0.183982
$867$	−48.1421	−1.63499
$868$	0	0
$869$	−39.5980	−1.34327
$870$	−9.07107	−0.307538
$871$	3.21320	0.108875
$872$	−16.2843	−0.551455
$873$	−6.82843	−0.231107
$874$	−3.17157	−0.107280
$875$	3.82843	0.129424
$876$	0	0
$877$	−18.4853	−0.624204	−0.312102	−	0.950049i	$-0.601033\pi$
$877$	−18.4853	−0.624204	−0.312102	+	0.950049i	$0.601033\pi$
$878$	−2.14214	−0.0722936
$879$	22.8995	0.772381
$880$	−21.6569	−0.730052
$881$	16.5858	0.558789	0.279395	−	0.960176i	$-0.409866\pi$
$881$	16.5858	0.558789	0.279395	+	0.960176i	$0.409866\pi$
$882$	10.8284	0.364612
$883$	17.8995	0.602366	0.301183	−	0.953566i	$-0.402619\pi$
$883$	17.8995	0.602366	0.301183	+	0.953566i	$0.402619\pi$
$884$	0	0
$885$	12.8995	0.433612
$886$	−48.8284	−1.64042
$887$	−17.7990	−0.597632	−0.298816	−	0.954311i	$-0.596592\pi$
$887$	−17.7990	−0.597632	−0.298816	+	0.954311i	$0.596592\pi$
$888$	8.48528	0.284747
$889$	−31.5563	−1.05837
$890$	25.6569	0.860020
$891$	−5.41421	−0.181383
$892$	0	0
$893$	−25.5980	−0.856604
$894$	7.17157	0.239853
$895$	−3.65685	−0.122235
$896$	−43.3137	−1.44701
$897$	0.585786	0.0195588
$898$	−6.72792	−0.224514
$899$	63.0416	2.10256
$900$	0	0
$901$	−10.0294	−0.334129
$902$	58.0833	1.93396
$903$	22.9706	0.764412
$904$	21.4558	0.713611
$905$	1.17157	0.0389444
$906$	−18.8284	−0.625533
$907$	2.85786	0.0948938	0.0474469	−	0.998874i	$-0.484892\pi$
$907$	2.85786	0.0948938	0.0474469	+	0.998874i	$0.484892\pi$
$908$	0	0
$909$	−18.5563	−0.615475
$910$	3.17157	0.105137
$911$	5.65685	0.187420	0.0937100	−	0.995600i	$-0.470127\pi$
$911$	5.65685	0.187420	0.0937100	+	0.995600i	$0.470127\pi$
$912$	8.97056	0.297045
$913$	0.384776	0.0127342
$914$	−43.8406	−1.45012
$915$	2.58579	0.0854835
$916$	0	0
$917$	8.97056	0.296234
$918$	−11.4142	−0.376725
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−2.82843	−0.0932505
$921$	15.2132	0.501292
$922$	−37.4558	−1.23354
$923$	−6.38478	−0.210157
$924$	0	0
$925$	3.00000	0.0986394
$926$	4.14214	0.136119
$927$	−4.82843	−0.158586
$928$	0	0
$929$	−11.8701	−0.389444	−0.194722	−	0.980858i	$-0.562380\pi$
$929$	−11.8701	−0.389444	−0.194722	+	0.980858i	$0.562380\pi$
$930$	−13.8995	−0.455782
$931$	17.1716	0.562776
$932$	0	0
$933$	−6.34315	−0.207665
$934$	9.07107	0.296814
$935$	43.6985	1.42909
$936$	−1.65685	−0.0541560
$937$	−20.2010	−0.659938	−0.329969	−	0.943992i	$-0.607038\pi$
$937$	−20.2010	−0.659938	−0.329969	+	0.943992i	$0.607038\pi$
$938$	−29.6985	−0.969690
$939$	7.97056	0.260109
$940$	0	0
$941$	−4.58579	−0.149492	−0.0747462	−	0.997203i	$-0.523815\pi$
$941$	−4.58579	−0.149492	−0.0747462	+	0.997203i	$0.523815\pi$
$942$	−21.8995	−0.713524
$943$	7.58579	0.247027
$944$	−51.5980	−1.67937
$945$	−3.82843	−0.124539
$946$	−45.9411	−1.49367
$947$	35.7990	1.16331	0.581655	−	0.813435i	$-0.302405\pi$
$947$	35.7990	1.16331	0.581655	+	0.813435i	$0.302405\pi$
$948$	0	0
$949$	−1.02944	−0.0334169
$950$	3.17157	0.102899
$951$	−1.55635	−0.0504681
$952$	87.3970	2.83255
$953$	52.6274	1.70477	0.852385	−	0.522915i	$-0.175156\pi$
$953$	52.6274	1.70477	0.852385	+	0.522915i	$0.175156\pi$
$954$	−1.75736	−0.0568966
$955$	−1.75736	−0.0568668
$956$	0	0
$957$	−34.7279	−1.12259
$958$	9.51472	0.307407
$959$	10.8284	0.349668
$960$	8.00000	0.258199
$961$	65.5980	2.11606
$962$	2.48528	0.0801287
$963$	−1.58579	−0.0511013
$964$	0	0
$965$	−8.48528	−0.273151
$966$	−5.41421	−0.174199
$967$	31.5563	1.01478	0.507392	−	0.861715i	$-0.330610\pi$
$967$	31.5563	1.01478	0.507392	+	0.861715i	$0.330610\pi$
$968$	−51.7990	−1.66488
$969$	−18.1005	−0.581472
$970$	9.65685	0.310063
$971$	−54.4264	−1.74663	−0.873313	−	0.487159i	$-0.838033\pi$
$971$	−54.4264	−1.74663	−0.873313	+	0.487159i	$0.838033\pi$
$972$	0	0
$973$	42.1127	1.35007
$974$	−37.3137	−1.19561
$975$	−0.585786	−0.0187602
$976$	−10.3431	−0.331076
$977$	−18.4142	−0.589123	−0.294561	−	0.955633i	$-0.595174\pi$
$977$	−18.4142	−0.589123	−0.294561	+	0.955633i	$0.595174\pi$
$978$	2.82843	0.0904431
$979$	98.2254	3.13930
$980$	0	0
$981$	5.75736	0.183818
$982$	−17.0711	−0.544760
$983$	−58.0122	−1.85030	−0.925151	−	0.379600i	$-0.876062\pi$
$983$	−58.0122	−1.85030	−0.925151	+	0.379600i	$0.876062\pi$
$984$	−21.4558	−0.683987
$985$	−11.1716	−0.355956
$986$	−73.2132	−2.33158
$987$	−43.6985	−1.39094
$988$	0	0
$989$	−6.00000	−0.190789
$990$	7.65685	0.243351
$991$	12.6569	0.402058	0.201029	−	0.979585i	$-0.435571\pi$
$991$	12.6569	0.402058	0.201029	+	0.979585i	$0.435571\pi$
$992$	0	0
$993$	−1.00000	−0.0317340
$994$	59.0122	1.87175
$995$	6.48528	0.205597
$996$	0	0
$997$	−5.85786	−0.185520	−0.0927602	−	0.995688i	$-0.529569\pi$
$997$	−5.85786	−0.185520	−0.0927602	+	0.995688i	$0.529569\pi$
$998$	−18.3848	−0.581960
$999$	−3.00000	−0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	345.2.a.h.1.2	✓	2
3.2	odd	2		1035.2.a.j.1.1		2
4.3	odd	2		5520.2.a.bm.1.2		2
5.2	odd	4		1725.2.b.s.1174.4		4
5.3	odd	4		1725.2.b.s.1174.1		4
5.4	even	2		1725.2.a.z.1.1		2
15.14	odd	2		5175.2.a.bj.1.2		2
23.22	odd	2		7935.2.a.q.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.a.h.1.2	✓	2	1.1	even	1	trivial
1035.2.a.j.1.1		2	3.2	odd	2
1725.2.a.z.1.1		2	5.4	even	2
1725.2.b.s.1174.1		4	5.3	odd	4
1725.2.b.s.1174.4		4	5.2	odd	4
5175.2.a.bj.1.2		2	15.14	odd	2
5520.2.a.bm.1.2		2	4.3	odd	2
7935.2.a.q.1.2		2	23.22	odd	2

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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:	\( N \)	\(=\)	\( 345 = 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	345.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -1.00000 q^{3} -1.00000 q^{5} -1.41421 q^{6} -3.82843 q^{7} -2.82843 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} -1.00000 q^{3} -1.00000 q^{5} -1.41421 q^{6} -3.82843 q^{7} -2.82843 q^{8} +1.00000 q^{9} -1.41421 q^{10} -5.41421 q^{11} +0.585786 q^{13} -5.41421 q^{14} +1.00000 q^{15} -4.00000 q^{16} +8.07107 q^{17} +1.41421 q^{18} +2.24264 q^{19} +3.82843 q^{21} -7.65685 q^{22} -1.00000 q^{23} +2.82843 q^{24} +1.00000 q^{25} +0.828427 q^{26} -1.00000 q^{27} -6.41421 q^{29} +1.41421 q^{30} -9.82843 q^{31} +5.41421 q^{33} +11.4142 q^{34} +3.82843 q^{35} +3.00000 q^{37} +3.17157 q^{38} -0.585786 q^{39} +2.82843 q^{40} -7.58579 q^{41} +5.41421 q^{42} +6.00000 q^{43} -1.00000 q^{45} -1.41421 q^{46} -11.4142 q^{47} +4.00000 q^{48} +7.65685 q^{49} +1.41421 q^{50} -8.07107 q^{51} -1.24264 q^{53} -1.41421 q^{54} +5.41421 q^{55} +10.8284 q^{56} -2.24264 q^{57} -9.07107 q^{58} +12.8995 q^{59} +2.58579 q^{61} -13.8995 q^{62} -3.82843 q^{63} +8.00000 q^{64} -0.585786 q^{65} +7.65685 q^{66} +5.48528 q^{67} +1.00000 q^{69} +5.41421 q^{70} -10.8995 q^{71} -2.82843 q^{72} -1.75736 q^{73} +4.24264 q^{74} -1.00000 q^{75} +20.7279 q^{77} -0.828427 q^{78} +7.31371 q^{79} +4.00000 q^{80} +1.00000 q^{81} -10.7279 q^{82} -0.0710678 q^{83} -8.07107 q^{85} +8.48528 q^{86} +6.41421 q^{87} +15.3137 q^{88} -18.1421 q^{89} -1.41421 q^{90} -2.24264 q^{91} +9.82843 q^{93} -16.1421 q^{94} -2.24264 q^{95} -6.82843 q^{97} +10.8284 q^{98} -5.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 8 q^{11} + 4 q^{13} - 8 q^{14} + 2 q^{15} - 8 q^{16} + 2 q^{17} - 4 q^{19} + 2 q^{21} - 4 q^{22} - 2 q^{23} + 2 q^{25} - 4 q^{26} - 2 q^{27} - 10 q^{29} - 14 q^{31} + 8 q^{33} + 20 q^{34} + 2 q^{35} + 6 q^{37} + 12 q^{38} - 4 q^{39} - 18 q^{41} + 8 q^{42} + 12 q^{43} - 2 q^{45} - 20 q^{47} + 8 q^{48} + 4 q^{49} - 2 q^{51} + 6 q^{53} + 8 q^{55} + 16 q^{56} + 4 q^{57} - 4 q^{58} + 6 q^{59} + 8 q^{61} - 8 q^{62} - 2 q^{63} + 16 q^{64} - 4 q^{65} + 4 q^{66} - 6 q^{67} + 2 q^{69} + 8 q^{70} - 2 q^{71} - 12 q^{73} - 2 q^{75} + 16 q^{77} + 4 q^{78} - 8 q^{79} + 8 q^{80} + 2 q^{81} + 4 q^{82} + 14 q^{83} - 2 q^{85} + 10 q^{87} + 8 q^{88} - 8 q^{89} + 4 q^{91} + 14 q^{93} - 4 q^{94} + 4 q^{95} - 8 q^{97} + 16 q^{98} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 - 8 * q^11 + 4 * q^13 - 8 * q^14 + 2 * q^15 - 8 * q^16 + 2 * q^17 - 4 * q^19 + 2 * q^21 - 4 * q^22 - 2 * q^23 + 2 * q^25 - 4 * q^26 - 2 * q^27 - 10 * q^29 - 14 * q^31 + 8 * q^33 + 20 * q^34 + 2 * q^35 + 6 * q^37 + 12 * q^38 - 4 * q^39 - 18 * q^41 + 8 * q^42 + 12 * q^43 - 2 * q^45 - 20 * q^47 + 8 * q^48 + 4 * q^49 - 2 * q^51 + 6 * q^53 + 8 * q^55 + 16 * q^56 + 4 * q^57 - 4 * q^58 + 6 * q^59 + 8 * q^61 - 8 * q^62 - 2 * q^63 + 16 * q^64 - 4 * q^65 + 4 * q^66 - 6 * q^67 + 2 * q^69 + 8 * q^70 - 2 * q^71 - 12 * q^73 - 2 * q^75 + 16 * q^77 + 4 * q^78 - 8 * q^79 + 8 * q^80 + 2 * q^81 + 4 * q^82 + 14 * q^83 - 2 * q^85 + 10 * q^87 + 8 * q^88 - 8 * q^89 + 4 * q^91 + 14 * q^93 - 4 * q^94 + 4 * q^95 - 8 * q^97 + 16 * q^98 - 8 * q^99

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