LMFDB - Embedded newform 3450.2.a.bt.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("3450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$27.5483886973$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 690)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	3450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.96239 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.96239 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.35026 q^{11} -1.00000 q^{12} +4.96239 q^{13} -2.96239 q^{14} +1.00000 q^{16} -1.35026 q^{17} +1.00000 q^{18} -4.96239 q^{19} +2.96239 q^{21} -3.35026 q^{22} +1.00000 q^{23} -1.00000 q^{24} +4.96239 q^{26} -1.00000 q^{27} -2.96239 q^{28} +7.73813 q^{29} -4.00000 q^{31} +1.00000 q^{32} +3.35026 q^{33} -1.35026 q^{34} +1.00000 q^{36} +7.61213 q^{37} -4.96239 q^{38} -4.96239 q^{39} +4.70052 q^{41} +2.96239 q^{42} +10.3127 q^{43} -3.35026 q^{44} +1.00000 q^{46} -3.22425 q^{47} -1.00000 q^{48} +1.77575 q^{49} +1.35026 q^{51} +4.96239 q^{52} +6.96239 q^{53} -1.00000 q^{54} -2.96239 q^{56} +4.96239 q^{57} +7.73813 q^{58} +1.22425 q^{59} -11.0884 q^{61} -4.00000 q^{62} -2.96239 q^{63} +1.00000 q^{64} +3.35026 q^{66} +7.61213 q^{67} -1.35026 q^{68} -1.00000 q^{69} -2.18664 q^{71} +1.00000 q^{72} +9.92478 q^{73} +7.61213 q^{74} -4.96239 q^{76} +9.92478 q^{77} -4.96239 q^{78} -4.12601 q^{79} +1.00000 q^{81} +4.70052 q^{82} +6.38787 q^{83} +2.96239 q^{84} +10.3127 q^{86} -7.73813 q^{87} -3.35026 q^{88} +9.92478 q^{89} -14.7005 q^{91} +1.00000 q^{92} +4.00000 q^{93} -3.22425 q^{94} -1.00000 q^{96} -12.8872 q^{97} +1.77575 q^{98} -3.35026 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{12} + 4 q^{13} + 2 q^{14} + 3 q^{16} + 6 q^{17} + 3 q^{18} - 4 q^{19} - 2 q^{21} + 3 q^{23} - 3 q^{24} + 4 q^{26} - 3 q^{27} + 2 q^{28} + 14 q^{29} - 12 q^{31} + 3 q^{32} + 6 q^{34} + 3 q^{36} + 22 q^{37} - 4 q^{38} - 4 q^{39} - 6 q^{41} - 2 q^{42} + 10 q^{43} + 3 q^{46} - 8 q^{47} - 3 q^{48} + 7 q^{49} - 6 q^{51} + 4 q^{52} + 10 q^{53} - 3 q^{54} + 2 q^{56} + 4 q^{57} + 14 q^{58} + 2 q^{59} - 14 q^{61} - 12 q^{62} + 2 q^{63} + 3 q^{64} + 22 q^{67} + 6 q^{68} - 3 q^{69} + 6 q^{71} + 3 q^{72} + 8 q^{73} + 22 q^{74} - 4 q^{76} + 8 q^{77} - 4 q^{78} - 4 q^{79} + 3 q^{81} - 6 q^{82} + 20 q^{83} - 2 q^{84} + 10 q^{86} - 14 q^{87} + 8 q^{89} - 24 q^{91} + 3 q^{92} + 12 q^{93} - 8 q^{94} - 3 q^{96} - 6 q^{97} + 7 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^12 + 4 * q^13 + 2 * q^14 + 3 * q^16 + 6 * q^17 + 3 * q^18 - 4 * q^19 - 2 * q^21 + 3 * q^23 - 3 * q^24 + 4 * q^26 - 3 * q^27 + 2 * q^28 + 14 * q^29 - 12 * q^31 + 3 * q^32 + 6 * q^34 + 3 * q^36 + 22 * q^37 - 4 * q^38 - 4 * q^39 - 6 * q^41 - 2 * q^42 + 10 * q^43 + 3 * q^46 - 8 * q^47 - 3 * q^48 + 7 * q^49 - 6 * q^51 + 4 * q^52 + 10 * q^53 - 3 * q^54 + 2 * q^56 + 4 * q^57 + 14 * q^58 + 2 * q^59 - 14 * q^61 - 12 * q^62 + 2 * q^63 + 3 * q^64 + 22 * q^67 + 6 * q^68 - 3 * q^69 + 6 * q^71 + 3 * q^72 + 8 * q^73 + 22 * q^74 - 4 * q^76 + 8 * q^77 - 4 * q^78 - 4 * q^79 + 3 * q^81 - 6 * q^82 + 20 * q^83 - 2 * q^84 + 10 * q^86 - 14 * q^87 + 8 * q^89 - 24 * q^91 + 3 * q^92 + 12 * q^93 - 8 * q^94 - 3 * q^96 - 6 * q^97 + 7 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	−2.96239	−1.11968	−0.559839	−	0.828602i	$-0.689137\pi$
$7$	−2.96239	−1.11968	−0.559839	+	0.828602i	$0.689137\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.35026	−1.01014	−0.505071	−	0.863078i	$-0.668534\pi$
$11$	−3.35026	−1.01014	−0.505071	+	0.863078i	$0.668534\pi$
$12$	−1.00000	−0.288675
$13$	4.96239	1.37632	0.688159	−	0.725559i	$-0.258419\pi$
$13$	4.96239	1.37632	0.688159	+	0.725559i	$0.258419\pi$
$14$	−2.96239	−0.791732
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.35026	−0.327487	−0.163743	−	0.986503i	$-0.552357\pi$
$17$	−1.35026	−0.327487	−0.163743	+	0.986503i	$0.552357\pi$
$18$	1.00000	0.235702
$19$	−4.96239	−1.13845	−0.569225	−	0.822182i	$-0.692757\pi$
$19$	−4.96239	−1.13845	−0.569225	+	0.822182i	$0.692757\pi$
$20$	0	0
$21$	2.96239	0.646446
$22$	−3.35026	−0.714278
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	0	0
$26$	4.96239	0.973204
$27$	−1.00000	−0.192450
$28$	−2.96239	−0.559839
$29$	7.73813	1.43694	0.718468	−	0.695560i	$-0.244844\pi$
$29$	7.73813	1.43694	0.718468	+	0.695560i	$0.244844\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000	0.176777
$33$	3.35026	0.583206
$34$	−1.35026	−0.231568
$35$	0	0
$36$	1.00000	0.166667
$37$	7.61213	1.25143	0.625713	−	0.780053i	$-0.284808\pi$
$37$	7.61213	1.25143	0.625713	+	0.780053i	$0.284808\pi$
$38$	−4.96239	−0.805006
$39$	−4.96239	−0.794618
$40$	0	0
$41$	4.70052	0.734098	0.367049	−	0.930202i	$-0.380368\pi$
$41$	4.70052	0.734098	0.367049	+	0.930202i	$0.380368\pi$
$42$	2.96239	0.457106
$43$	10.3127	1.57266	0.786332	−	0.617804i	$-0.211977\pi$
$43$	10.3127	1.57266	0.786332	+	0.617804i	$0.211977\pi$
$44$	−3.35026	−0.505071
$45$	0	0
$46$	1.00000	0.147442
$47$	−3.22425	−0.470306	−0.235153	−	0.971958i	$-0.575559\pi$
$47$	−3.22425	−0.470306	−0.235153	+	0.971958i	$0.575559\pi$
$48$	−1.00000	−0.144338
$49$	1.77575	0.253678
$50$	0	0
$51$	1.35026	0.189074
$52$	4.96239	0.688159
$53$	6.96239	0.956358	0.478179	−	0.878263i	$-0.341297\pi$
$53$	6.96239	0.956358	0.478179	+	0.878263i	$0.341297\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−2.96239	−0.395866
$57$	4.96239	0.657284
$58$	7.73813	1.01607
$59$	1.22425	0.159384	0.0796921	−	0.996820i	$-0.474606\pi$
$59$	1.22425	0.159384	0.0796921	+	0.996820i	$0.474606\pi$
$60$	0	0
$61$	−11.0884	−1.41972	−0.709862	−	0.704341i	$-0.751243\pi$
$61$	−11.0884	−1.41972	−0.709862	+	0.704341i	$0.751243\pi$
$62$	−4.00000	−0.508001
$63$	−2.96239	−0.373226
$64$	1.00000	0.125000
$65$	0	0
$66$	3.35026	0.412389
$67$	7.61213	0.929969	0.464985	−	0.885319i	$-0.346060\pi$
$67$	7.61213	0.929969	0.464985	+	0.885319i	$0.346060\pi$
$68$	−1.35026	−0.163743
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−2.18664	−0.259507	−0.129753	−	0.991546i	$-0.541419\pi$
$71$	−2.18664	−0.259507	−0.129753	+	0.991546i	$0.541419\pi$
$72$	1.00000	0.117851
$73$	9.92478	1.16161	0.580804	−	0.814044i	$-0.302738\pi$
$73$	9.92478	1.16161	0.580804	+	0.814044i	$0.302738\pi$
$74$	7.61213	0.884892
$75$	0	0
$76$	−4.96239	−0.569225
$77$	9.92478	1.13103
$78$	−4.96239	−0.561880
$79$	−4.12601	−0.464212	−0.232106	−	0.972690i	$-0.574562\pi$
$79$	−4.12601	−0.464212	−0.232106	+	0.972690i	$0.574562\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	4.70052	0.519086
$83$	6.38787	0.701160	0.350580	−	0.936533i	$-0.385984\pi$
$83$	6.38787	0.701160	0.350580	+	0.936533i	$0.385984\pi$
$84$	2.96239	0.323223
$85$	0	0
$86$	10.3127	1.11204
$87$	−7.73813	−0.829615
$88$	−3.35026	−0.357139
$89$	9.92478	1.05202	0.526012	−	0.850477i	$-0.323687\pi$
$89$	9.92478	1.05202	0.526012	+	0.850477i	$0.323687\pi$
$90$	0	0
$91$	−14.7005	−1.54103
$92$	1.00000	0.104257
$93$	4.00000	0.414781
$94$	−3.22425	−0.332556
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−12.8872	−1.30849	−0.654247	−	0.756281i	$-0.727014\pi$
$97$	−12.8872	−1.30849	−0.654247	+	0.756281i	$0.727014\pi$
$98$	1.77575	0.179377
$99$	−3.35026	−0.336714
$100$	0	0
$101$	4.26187	0.424071	0.212036	−	0.977262i	$-0.431991\pi$
$101$	4.26187	0.424071	0.212036	+	0.977262i	$0.431991\pi$
$102$	1.35026	0.133696
$103$	0.261865	0.0258023	0.0129012	−	0.999917i	$-0.495893\pi$
$103$	0.261865	0.0258023	0.0129012	+	0.999917i	$0.495893\pi$
$104$	4.96239	0.486602
$105$	0	0
$106$	6.96239	0.676247
$107$	9.08840	0.878608	0.439304	−	0.898338i	$-0.355225\pi$
$107$	9.08840	0.878608	0.439304	+	0.898338i	$0.355225\pi$
$108$	−1.00000	−0.0962250
$109$	18.9380	1.81393	0.906963	−	0.421210i	$-0.138394\pi$
$109$	18.9380	1.81393	0.906963	+	0.421210i	$0.138394\pi$
$110$	0	0
$111$	−7.61213	−0.722511
$112$	−2.96239	−0.279919
$113$	6.64974	0.625555	0.312777	−	0.949826i	$-0.398741\pi$
$113$	6.64974	0.625555	0.312777	+	0.949826i	$0.398741\pi$
$114$	4.96239	0.464770
$115$	0	0
$116$	7.73813	0.718468
$117$	4.96239	0.458773
$118$	1.22425	0.112702
$119$	4.00000	0.366679
$120$	0	0
$121$	0.224254	0.0203867
$122$	−11.0884	−1.00390
$123$	−4.70052	−0.423832
$124$	−4.00000	−0.359211
$125$	0	0
$126$	−2.96239	−0.263911
$127$	−0.186642	−0.0165618	−0.00828091	−	0.999966i	$-0.502636\pi$
$127$	−0.186642	−0.0165618	−0.00828091	+	0.999966i	$0.502636\pi$
$128$	1.00000	0.0883883
$129$	−10.3127	−0.907978
$130$	0	0
$131$	7.92478	0.692391	0.346196	−	0.938162i	$-0.387473\pi$
$131$	7.92478	0.692391	0.346196	+	0.938162i	$0.387473\pi$
$132$	3.35026	0.291603
$133$	14.7005	1.27470
$134$	7.61213	0.657588
$135$	0	0
$136$	−1.35026	−0.115784
$137$	−9.19982	−0.785993	−0.392997	−	0.919540i	$-0.628562\pi$
$137$	−9.19982	−0.785993	−0.392997	+	0.919540i	$0.628562\pi$
$138$	−1.00000	−0.0851257
$139$	8.62530	0.731588	0.365794	−	0.930696i	$-0.380797\pi$
$139$	8.62530	0.731588	0.365794	+	0.930696i	$0.380797\pi$
$140$	0	0
$141$	3.22425	0.271531
$142$	−2.18664	−0.183499
$143$	−16.6253	−1.39028
$144$	1.00000	0.0833333
$145$	0	0
$146$	9.92478	0.821380
$147$	−1.77575	−0.146461
$148$	7.61213	0.625713
$149$	4.64974	0.380921	0.190461	−	0.981695i	$-0.439002\pi$
$149$	4.64974	0.380921	0.190461	+	0.981695i	$0.439002\pi$
$150$	0	0
$151$	10.7005	0.870796	0.435398	−	0.900238i	$-0.356608\pi$
$151$	10.7005	0.870796	0.435398	+	0.900238i	$0.356608\pi$
$152$	−4.96239	−0.402503
$153$	−1.35026	−0.109162
$154$	9.92478	0.799761
$155$	0	0
$156$	−4.96239	−0.397309
$157$	17.0132	1.35780	0.678900	−	0.734231i	$-0.262457\pi$
$157$	17.0132	1.35780	0.678900	+	0.734231i	$0.262457\pi$
$158$	−4.12601	−0.328248
$159$	−6.96239	−0.552153
$160$	0	0
$161$	−2.96239	−0.233469
$162$	1.00000	0.0785674
$163$	−12.6253	−0.988890	−0.494445	−	0.869209i	$-0.664629\pi$
$163$	−12.6253	−0.988890	−0.494445	+	0.869209i	$0.664629\pi$
$164$	4.70052	0.367049
$165$	0	0
$166$	6.38787	0.495795
$167$	24.6253	1.90556	0.952781	−	0.303657i	$-0.0982076\pi$
$167$	24.6253	1.90556	0.952781	+	0.303657i	$0.0982076\pi$
$168$	2.96239	0.228553
$169$	11.6253	0.894254
$170$	0	0
$171$	−4.96239	−0.379483
$172$	10.3127	0.786332
$173$	4.44851	0.338214	0.169107	−	0.985598i	$-0.445912\pi$
$173$	4.44851	0.338214	0.169107	+	0.985598i	$0.445912\pi$
$174$	−7.73813	−0.586626
$175$	0	0
$176$	−3.35026	−0.252535
$177$	−1.22425	−0.0920205
$178$	9.92478	0.743894
$179$	−13.8496	−1.03516	−0.517582	−	0.855634i	$-0.673168\pi$
$179$	−13.8496	−1.03516	−0.517582	+	0.855634i	$0.673168\pi$
$180$	0	0
$181$	−22.6859	−1.68623	−0.843116	−	0.537732i	$-0.819281\pi$
$181$	−22.6859	−1.68623	−0.843116	+	0.537732i	$0.819281\pi$
$182$	−14.7005	−1.08968
$183$	11.0884	0.819678
$184$	1.00000	0.0737210
$185$	0	0
$186$	4.00000	0.293294
$187$	4.52373	0.330808
$188$	−3.22425	−0.235153
$189$	2.96239	0.215482
$190$	0	0
$191$	−19.3258	−1.39837	−0.699184	−	0.714942i	$-0.746453\pi$
$191$	−19.3258	−1.39837	−0.699184	+	0.714942i	$0.746453\pi$
$192$	−1.00000	−0.0721688
$193$	5.92478	0.426475	0.213237	−	0.977000i	$-0.431599\pi$
$193$	5.92478	0.426475	0.213237	+	0.977000i	$0.431599\pi$
$194$	−12.8872	−0.925245
$195$	0	0
$196$	1.77575	0.126839
$197$	−8.07522	−0.575336	−0.287668	−	0.957730i	$-0.592880\pi$
$197$	−8.07522	−0.575336	−0.287668	+	0.957730i	$0.592880\pi$
$198$	−3.35026	−0.238093
$199$	15.9756	1.13248	0.566239	−	0.824241i	$-0.308398\pi$
$199$	15.9756	1.13248	0.566239	+	0.824241i	$0.308398\pi$
$200$	0	0
$201$	−7.61213	−0.536918
$202$	4.26187	0.299864
$203$	−22.9234	−1.60890
$204$	1.35026	0.0945372
$205$	0	0
$206$	0.261865	0.0182450
$207$	1.00000	0.0695048
$208$	4.96239	0.344080
$209$	16.6253	1.15000
$210$	0	0
$211$	21.7743	1.49901	0.749503	−	0.662000i	$-0.230292\pi$
$211$	21.7743	1.49901	0.749503	+	0.662000i	$0.230292\pi$
$212$	6.96239	0.478179
$213$	2.18664	0.149826
$214$	9.08840	0.621270
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	11.8496	0.804400
$218$	18.9380	1.28264
$219$	−9.92478	−0.670654
$220$	0	0
$221$	−6.70052	−0.450726
$222$	−7.61213	−0.510893
$223$	3.81336	0.255361	0.127681	−	0.991815i	$-0.459247\pi$
$223$	3.81336	0.255361	0.127681	+	0.991815i	$0.459247\pi$
$224$	−2.96239	−0.197933
$225$	0	0
$226$	6.64974	0.442334
$227$	16.9380	1.12421	0.562106	−	0.827065i	$-0.309991\pi$
$227$	16.9380	1.12421	0.562106	+	0.827065i	$0.309991\pi$
$228$	4.96239	0.328642
$229$	−26.9380	−1.78011	−0.890055	−	0.455853i	$-0.849334\pi$
$229$	−26.9380	−1.78011	−0.890055	+	0.455853i	$0.849334\pi$
$230$	0	0
$231$	−9.92478	−0.653002
$232$	7.73813	0.508033
$233$	27.4010	1.79510	0.897551	−	0.440910i	$-0.145344\pi$
$233$	27.4010	1.79510	0.897551	+	0.440910i	$0.145344\pi$
$234$	4.96239	0.324401
$235$	0	0
$236$	1.22425	0.0796921
$237$	4.12601	0.268013
$238$	4.00000	0.259281
$239$	7.48612	0.484237	0.242118	−	0.970247i	$-0.422158\pi$
$239$	7.48612	0.484237	0.242118	+	0.970247i	$0.422158\pi$
$240$	0	0
$241$	−13.0738	−0.842158	−0.421079	−	0.907024i	$-0.638348\pi$
$241$	−13.0738	−0.842158	−0.421079	+	0.907024i	$0.638348\pi$
$242$	0.224254	0.0144156
$243$	−1.00000	−0.0641500
$244$	−11.0884	−0.709862
$245$	0	0
$246$	−4.70052	−0.299694
$247$	−24.6253	−1.56687
$248$	−4.00000	−0.254000
$249$	−6.38787	−0.404815
$250$	0	0
$251$	−5.02302	−0.317050	−0.158525	−	0.987355i	$-0.550674\pi$
$251$	−5.02302	−0.317050	−0.158525	+	0.987355i	$0.550674\pi$
$252$	−2.96239	−0.186613
$253$	−3.35026	−0.210629
$254$	−0.186642	−0.0117110
$255$	0	0
$256$	1.00000	0.0625000
$257$	−19.4010	−1.21020	−0.605102	−	0.796148i	$-0.706868\pi$
$257$	−19.4010	−1.21020	−0.605102	+	0.796148i	$0.706868\pi$
$258$	−10.3127	−0.642038
$259$	−22.5501	−1.40119
$260$	0	0
$261$	7.73813	0.478979
$262$	7.92478	0.489594
$263$	−15.4763	−0.954308	−0.477154	−	0.878820i	$-0.658332\pi$
$263$	−15.4763	−0.954308	−0.477154	+	0.878820i	$0.658332\pi$
$264$	3.35026	0.206194
$265$	0	0
$266$	14.7005	0.901347
$267$	−9.92478	−0.607387
$268$	7.61213	0.464985
$269$	−20.2130	−1.23241	−0.616204	−	0.787587i	$-0.711330\pi$
$269$	−20.2130	−1.23241	−0.616204	+	0.787587i	$0.711330\pi$
$270$	0	0
$271$	30.3996	1.84665	0.923323	−	0.384024i	$-0.125462\pi$
$271$	30.3996	1.84665	0.923323	+	0.384024i	$0.125462\pi$
$272$	−1.35026	−0.0818716
$273$	14.7005	0.889716
$274$	−9.19982	−0.555781
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	−20.8119	−1.25047	−0.625234	−	0.780437i	$-0.714997\pi$
$277$	−20.8119	−1.25047	−0.625234	+	0.780437i	$0.714997\pi$
$278$	8.62530	0.517311
$279$	−4.00000	−0.239474
$280$	0	0
$281$	19.8496	1.18413	0.592063	−	0.805892i	$-0.298314\pi$
$281$	19.8496	1.18413	0.592063	+	0.805892i	$0.298314\pi$
$282$	3.22425	0.192002
$283$	−12.3879	−0.736383	−0.368191	−	0.929750i	$-0.620023\pi$
$283$	−12.3879	−0.736383	−0.368191	+	0.929750i	$0.620023\pi$
$284$	−2.18664	−0.129753
$285$	0	0
$286$	−16.6253	−0.983075
$287$	−13.9248	−0.821954
$288$	1.00000	0.0589256
$289$	−15.1768	−0.892753
$290$	0	0
$291$	12.8872	0.755459
$292$	9.92478	0.580804
$293$	−9.03761	−0.527983	−0.263991	−	0.964525i	$-0.585039\pi$
$293$	−9.03761	−0.527983	−0.263991	+	0.964525i	$0.585039\pi$
$294$	−1.77575	−0.103564
$295$	0	0
$296$	7.61213	0.442446
$297$	3.35026	0.194402
$298$	4.64974	0.269352
$299$	4.96239	0.286982
$300$	0	0
$301$	−30.5501	−1.76088
$302$	10.7005	0.615746
$303$	−4.26187	−0.244838
$304$	−4.96239	−0.284613
$305$	0	0
$306$	−1.35026	−0.0771893
$307$	30.5501	1.74359	0.871793	−	0.489875i	$-0.162958\pi$
$307$	30.5501	1.74359	0.871793	+	0.489875i	$0.162958\pi$
$308$	9.92478	0.565517
$309$	−0.261865	−0.0148970
$310$	0	0
$311$	14.4387	0.818741	0.409371	−	0.912368i	$-0.365748\pi$
$311$	14.4387	0.818741	0.409371	+	0.912368i	$0.365748\pi$
$312$	−4.96239	−0.280940
$313$	−27.9610	−1.58045	−0.790224	−	0.612818i	$-0.790036\pi$
$313$	−27.9610	−1.58045	−0.790224	+	0.612818i	$0.790036\pi$
$314$	17.0132	0.960109
$315$	0	0
$316$	−4.12601	−0.232106
$317$	−1.47627	−0.0829156	−0.0414578	−	0.999140i	$-0.513200\pi$
$317$	−1.47627	−0.0829156	−0.0414578	+	0.999140i	$0.513200\pi$
$318$	−6.96239	−0.390431
$319$	−25.9248	−1.45151
$320$	0	0
$321$	−9.08840	−0.507265
$322$	−2.96239	−0.165087
$323$	6.70052	0.372827
$324$	1.00000	0.0555556
$325$	0	0
$326$	−12.6253	−0.699251
$327$	−18.9380	−1.04727
$328$	4.70052	0.259543
$329$	9.55149	0.526591
$330$	0	0
$331$	23.1754	1.27383	0.636917	−	0.770932i	$-0.280209\pi$
$331$	23.1754	1.27383	0.636917	+	0.770932i	$0.280209\pi$
$332$	6.38787	0.350580
$333$	7.61213	0.417142
$334$	24.6253	1.34744
$335$	0	0
$336$	2.96239	0.161612
$337$	21.0376	1.14599	0.572996	−	0.819558i	$-0.305781\pi$
$337$	21.0376	1.14599	0.572996	+	0.819558i	$0.305781\pi$
$338$	11.6253	0.632333
$339$	−6.64974	−0.361164
$340$	0	0
$341$	13.4010	0.725707
$342$	−4.96239	−0.268335
$343$	15.4763	0.835640
$344$	10.3127	0.556021
$345$	0	0
$346$	4.44851	0.239153
$347$	−3.37470	−0.181163	−0.0905817	−	0.995889i	$-0.528873\pi$
$347$	−3.37470	−0.181163	−0.0905817	+	0.995889i	$0.528873\pi$
$348$	−7.73813	−0.414808
$349$	−4.44851	−0.238123	−0.119062	−	0.992887i	$-0.537989\pi$
$349$	−4.44851	−0.238123	−0.119062	+	0.992887i	$0.537989\pi$
$350$	0	0
$351$	−4.96239	−0.264873
$352$	−3.35026	−0.178570
$353$	35.4010	1.88421	0.942104	−	0.335321i	$-0.108845\pi$
$353$	35.4010	1.88421	0.942104	+	0.335321i	$0.108845\pi$
$354$	−1.22425	−0.0650684
$355$	0	0
$356$	9.92478	0.526012
$357$	−4.00000	−0.211702
$358$	−13.8496	−0.731972
$359$	−34.5501	−1.82348	−0.911742	−	0.410764i	$-0.865262\pi$
$359$	−34.5501	−1.82348	−0.911742	+	0.410764i	$0.865262\pi$
$360$	0	0
$361$	5.62530	0.296068
$362$	−22.6859	−1.19235
$363$	−0.224254	−0.0117703
$364$	−14.7005	−0.770517
$365$	0	0
$366$	11.0884	0.579600
$367$	−22.8119	−1.19077	−0.595387	−	0.803439i	$-0.703001\pi$
$367$	−22.8119	−1.19077	−0.595387	+	0.803439i	$0.703001\pi$
$368$	1.00000	0.0521286
$369$	4.70052	0.244699
$370$	0	0
$371$	−20.6253	−1.07081
$372$	4.00000	0.207390
$373$	−10.8364	−0.561087	−0.280543	−	0.959841i	$-0.590515\pi$
$373$	−10.8364	−0.561087	−0.280543	+	0.959841i	$0.590515\pi$
$374$	4.52373	0.233917
$375$	0	0
$376$	−3.22425	−0.166278
$377$	38.3996	1.97768
$378$	2.96239	0.152369
$379$	−32.4387	−1.66626	−0.833131	−	0.553076i	$-0.813454\pi$
$379$	−32.4387	−1.66626	−0.833131	+	0.553076i	$0.813454\pi$
$380$	0	0
$381$	0.186642	0.00956198
$382$	−19.3258	−0.988795
$383$	32.9986	1.68615	0.843074	−	0.537797i	$-0.180743\pi$
$383$	32.9986	1.68615	0.843074	+	0.537797i	$0.180743\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	5.92478	0.301563
$387$	10.3127	0.524221
$388$	−12.8872	−0.654247
$389$	29.1246	1.47668	0.738338	−	0.674431i	$-0.235611\pi$
$389$	29.1246	1.47668	0.738338	+	0.674431i	$0.235611\pi$
$390$	0	0
$391$	−1.35026	−0.0682857
$392$	1.77575	0.0896887
$393$	−7.92478	−0.399752
$394$	−8.07522	−0.406824
$395$	0	0
$396$	−3.35026	−0.168357
$397$	−13.7381	−0.689497	−0.344749	−	0.938695i	$-0.612036\pi$
$397$	−13.7381	−0.689497	−0.344749	+	0.938695i	$0.612036\pi$
$398$	15.9756	0.800783
$399$	−14.7005	−0.735947
$400$	0	0
$401$	−1.44992	−0.0724057	−0.0362028	−	0.999344i	$-0.511526\pi$
$401$	−1.44992	−0.0724057	−0.0362028	+	0.999344i	$0.511526\pi$
$402$	−7.61213	−0.379658
$403$	−19.8496	−0.988777
$404$	4.26187	0.212036
$405$	0	0
$406$	−22.9234	−1.13767
$407$	−25.5026	−1.26412
$408$	1.35026	0.0668479
$409$	−17.8496	−0.882604	−0.441302	−	0.897359i	$-0.645483\pi$
$409$	−17.8496	−0.882604	−0.441302	+	0.897359i	$0.645483\pi$
$410$	0	0
$411$	9.19982	0.453793
$412$	0.261865	0.0129012
$413$	−3.62672	−0.178459
$414$	1.00000	0.0491473
$415$	0	0
$416$	4.96239	0.243301
$417$	−8.62530	−0.422383
$418$	16.6253	0.813170
$419$	17.9003	0.874489	0.437244	−	0.899343i	$-0.355954\pi$
$419$	17.9003	0.874489	0.437244	+	0.899343i	$0.355954\pi$
$420$	0	0
$421$	7.61213	0.370992	0.185496	−	0.982645i	$-0.440611\pi$
$421$	7.61213	0.370992	0.185496	+	0.982645i	$0.440611\pi$
$422$	21.7743	1.05996
$423$	−3.22425	−0.156769
$424$	6.96239	0.338123
$425$	0	0
$426$	2.18664	0.105943
$427$	32.8481	1.58963
$428$	9.08840	0.439304
$429$	16.6253	0.802677
$430$	0	0
$431$	−22.9525	−1.10558	−0.552792	−	0.833319i	$-0.686438\pi$
$431$	−22.9525	−1.10558	−0.552792	+	0.833319i	$0.686438\pi$
$432$	−1.00000	−0.0481125
$433$	−33.7645	−1.62262	−0.811309	−	0.584618i	$-0.801244\pi$
$433$	−33.7645	−1.62262	−0.811309	+	0.584618i	$0.801244\pi$
$434$	11.8496	0.568797
$435$	0	0
$436$	18.9380	0.906963
$437$	−4.96239	−0.237383
$438$	−9.92478	−0.474224
$439$	−23.4763	−1.12046	−0.560231	−	0.828337i	$-0.689287\pi$
$439$	−23.4763	−1.12046	−0.560231	+	0.828337i	$0.689287\pi$
$440$	0	0
$441$	1.77575	0.0845593
$442$	−6.70052	−0.318711
$443$	30.5501	1.45148	0.725739	−	0.687970i	$-0.241498\pi$
$443$	30.5501	1.45148	0.725739	+	0.687970i	$0.241498\pi$
$444$	−7.61213	−0.361256
$445$	0	0
$446$	3.81336	0.180568
$447$	−4.64974	−0.219925
$448$	−2.96239	−0.139960
$449$	32.7005	1.54323	0.771617	−	0.636088i	$-0.219448\pi$
$449$	32.7005	1.54323	0.771617	+	0.636088i	$0.219448\pi$
$450$	0	0
$451$	−15.7480	−0.741544
$452$	6.64974	0.312777
$453$	−10.7005	−0.502754
$454$	16.9380	0.794937
$455$	0	0
$456$	4.96239	0.232385
$457$	−5.81336	−0.271937	−0.135969	−	0.990713i	$-0.543415\pi$
$457$	−5.81336	−0.271937	−0.135969	+	0.990713i	$0.543415\pi$
$458$	−26.9380	−1.25873
$459$	1.35026	0.0630248
$460$	0	0
$461$	−23.9902	−1.11733	−0.558666	−	0.829392i	$-0.688687\pi$
$461$	−23.9902	−1.11733	−0.558666	+	0.829392i	$0.688687\pi$
$462$	−9.92478	−0.461742
$463$	35.3620	1.64341	0.821706	−	0.569911i	$-0.193022\pi$
$463$	35.3620	1.64341	0.821706	+	0.569911i	$0.193022\pi$
$464$	7.73813	0.359234
$465$	0	0
$466$	27.4010	1.26933
$467$	1.98541	0.0918739	0.0459369	−	0.998944i	$-0.485373\pi$
$467$	1.98541	0.0918739	0.0459369	+	0.998944i	$0.485373\pi$
$468$	4.96239	0.229386
$469$	−22.5501	−1.04127
$470$	0	0
$471$	−17.0132	−0.783926
$472$	1.22425	0.0563508
$473$	−34.5501	−1.58861
$474$	4.12601	0.189514
$475$	0	0
$476$	4.00000	0.183340
$477$	6.96239	0.318786
$478$	7.48612	0.342407
$479$	11.0738	0.505975	0.252988	−	0.967470i	$-0.418587\pi$
$479$	11.0738	0.505975	0.252988	+	0.967470i	$0.418587\pi$
$480$	0	0
$481$	37.7743	1.72236
$482$	−13.0738	−0.595496
$483$	2.96239	0.134793
$484$	0.224254	0.0101934
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−17.4372	−0.790157	−0.395078	−	0.918647i	$-0.629283\pi$
$487$	−17.4372	−0.790157	−0.395078	+	0.918647i	$0.629283\pi$
$488$	−11.0884	−0.501948
$489$	12.6253	0.570936
$490$	0	0
$491$	14.8773	0.671404	0.335702	−	0.941968i	$-0.391027\pi$
$491$	14.8773	0.671404	0.335702	+	0.941968i	$0.391027\pi$
$492$	−4.70052	−0.211916
$493$	−10.4485	−0.470577
$494$	−24.6253	−1.10794
$495$	0	0
$496$	−4.00000	−0.179605
$497$	6.47768	0.290564
$498$	−6.38787	−0.286247
$499$	−16.1016	−0.720805	−0.360403	−	0.932797i	$-0.617361\pi$
$499$	−16.1016	−0.720805	−0.360403	+	0.932797i	$0.617361\pi$
$500$	0	0
$501$	−24.6253	−1.10018
$502$	−5.02302	−0.224188
$503$	−38.8021	−1.73010	−0.865050	−	0.501686i	$-0.832713\pi$
$503$	−38.8021	−1.73010	−0.865050	+	0.501686i	$0.832713\pi$
$504$	−2.96239	−0.131955
$505$	0	0
$506$	−3.35026	−0.148937
$507$	−11.6253	−0.516298
$508$	−0.186642	−0.00828091
$509$	23.4372	1.03884	0.519419	−	0.854520i	$-0.326148\pi$
$509$	23.4372	1.03884	0.519419	+	0.854520i	$0.326148\pi$
$510$	0	0
$511$	−29.4010	−1.30063
$512$	1.00000	0.0441942
$513$	4.96239	0.219095
$514$	−19.4010	−0.855743
$515$	0	0
$516$	−10.3127	−0.453989
$517$	10.8021	0.475076
$518$	−22.5501	−0.990794
$519$	−4.44851	−0.195268
$520$	0	0
$521$	21.1490	0.926556	0.463278	−	0.886213i	$-0.346673\pi$
$521$	21.1490	0.926556	0.463278	+	0.886213i	$0.346673\pi$
$522$	7.73813	0.338689
$523$	−11.7626	−0.514341	−0.257171	−	0.966366i	$-0.582790\pi$
$523$	−11.7626	−0.514341	−0.257171	+	0.966366i	$0.582790\pi$
$524$	7.92478	0.346196
$525$	0	0
$526$	−15.4763	−0.674797
$527$	5.40105	0.235273
$528$	3.35026	0.145801
$529$	1.00000	0.0434783
$530$	0	0
$531$	1.22425	0.0531281
$532$	14.7005	0.637349
$533$	23.3258	1.01035
$534$	−9.92478	−0.429487
$535$	0	0
$536$	7.61213	0.328794
$537$	13.8496	0.597652
$538$	−20.2130	−0.871444
$539$	−5.94921	−0.256251
$540$	0	0
$541$	−21.1754	−0.910401	−0.455200	−	0.890389i	$-0.650432\pi$
$541$	−21.1754	−0.910401	−0.455200	+	0.890389i	$0.650432\pi$
$542$	30.3996	1.30578
$543$	22.6859	0.973547
$544$	−1.35026	−0.0578920
$545$	0	0
$546$	14.7005	0.629124
$547$	−5.29948	−0.226589	−0.113295	−	0.993561i	$-0.536140\pi$
$547$	−5.29948	−0.226589	−0.113295	+	0.993561i	$0.536140\pi$
$548$	−9.19982	−0.392997
$549$	−11.0884	−0.473241
$550$	0	0
$551$	−38.3996	−1.63588
$552$	−1.00000	−0.0425628
$553$	12.2228	0.519768
$554$	−20.8119	−0.884215
$555$	0	0
$556$	8.62530	0.365794
$557$	7.99015	0.338554	0.169277	−	0.985569i	$-0.445857\pi$
$557$	7.99015	0.338554	0.169277	+	0.985569i	$0.445857\pi$
$558$	−4.00000	−0.169334
$559$	51.1754	2.16449
$560$	0	0
$561$	−4.52373	−0.190992
$562$	19.8496	0.837303
$563$	−15.1636	−0.639070	−0.319535	−	0.947574i	$-0.603527\pi$
$563$	−15.1636	−0.639070	−0.319535	+	0.947574i	$0.603527\pi$
$564$	3.22425	0.135766
$565$	0	0
$566$	−12.3879	−0.520701
$567$	−2.96239	−0.124409
$568$	−2.18664	−0.0917495
$569$	−25.5223	−1.06995	−0.534976	−	0.844868i	$-0.679679\pi$
$569$	−25.5223	−1.06995	−0.534976	+	0.844868i	$0.679679\pi$
$570$	0	0
$571$	6.76590	0.283144	0.141572	−	0.989928i	$-0.454784\pi$
$571$	6.76590	0.283144	0.141572	+	0.989928i	$0.454784\pi$
$572$	−16.6253	−0.695139
$573$	19.3258	0.807348
$574$	−13.9248	−0.581209
$575$	0	0
$576$	1.00000	0.0416667
$577$	−4.00000	−0.166522	−0.0832611	−	0.996528i	$-0.526534\pi$
$577$	−4.00000	−0.166522	−0.0832611	+	0.996528i	$0.526534\pi$
$578$	−15.1768	−0.631271
$579$	−5.92478	−0.246225
$580$	0	0
$581$	−18.9234	−0.785073
$582$	12.8872	0.534190
$583$	−23.3258	−0.966057
$584$	9.92478	0.410690
$585$	0	0
$586$	−9.03761	−0.373340
$587$	12.8773	0.531504	0.265752	−	0.964041i	$-0.414380\pi$
$587$	12.8773	0.531504	0.265752	+	0.964041i	$0.414380\pi$
$588$	−1.77575	−0.0732305
$589$	19.8496	0.817887
$590$	0	0
$591$	8.07522	0.332170
$592$	7.61213	0.312856
$593$	−16.2981	−0.669281	−0.334641	−	0.942346i	$-0.608615\pi$
$593$	−16.2981	−0.669281	−0.334641	+	0.942346i	$0.608615\pi$
$594$	3.35026	0.137463
$595$	0	0
$596$	4.64974	0.190461
$597$	−15.9756	−0.653836
$598$	4.96239	0.202927
$599$	9.91493	0.405113	0.202556	−	0.979271i	$-0.435075\pi$
$599$	9.91493	0.405113	0.202556	+	0.979271i	$0.435075\pi$
$600$	0	0
$601$	−7.40105	−0.301895	−0.150948	−	0.988542i	$-0.548232\pi$
$601$	−7.40105	−0.301895	−0.150948	+	0.988542i	$0.548232\pi$
$602$	−30.5501	−1.24513
$603$	7.61213	0.309990
$604$	10.7005	0.435398
$605$	0	0
$606$	−4.26187	−0.173126
$607$	13.9610	0.566658	0.283329	−	0.959023i	$-0.408561\pi$
$607$	13.9610	0.566658	0.283329	+	0.959023i	$0.408561\pi$
$608$	−4.96239	−0.201251
$609$	22.9234	0.928902
$610$	0	0
$611$	−16.0000	−0.647291
$612$	−1.35026	−0.0545811
$613$	41.3865	1.67158	0.835792	−	0.549047i	$-0.185009\pi$
$613$	41.3865	1.67158	0.835792	+	0.549047i	$0.185009\pi$
$614$	30.5501	1.23290
$615$	0	0
$616$	9.92478	0.399881
$617$	−29.3014	−1.17963	−0.589815	−	0.807539i	$-0.700799\pi$
$617$	−29.3014	−1.17963	−0.589815	+	0.807539i	$0.700799\pi$
$618$	−0.261865	−0.0105338
$619$	−19.0376	−0.765186	−0.382593	−	0.923917i	$-0.624969\pi$
$619$	−19.0376	−0.765186	−0.382593	+	0.923917i	$0.624969\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	14.4387	0.578937
$623$	−29.4010	−1.17793
$624$	−4.96239	−0.198655
$625$	0	0
$626$	−27.9610	−1.11755
$627$	−16.6253	−0.663951
$628$	17.0132	0.678900
$629$	−10.2784	−0.409825
$630$	0	0
$631$	12.4993	0.497589	0.248794	−	0.968556i	$-0.419966\pi$
$631$	12.4993	0.497589	0.248794	+	0.968556i	$0.419966\pi$
$632$	−4.12601	−0.164124
$633$	−21.7743	−0.865452
$634$	−1.47627	−0.0586302
$635$	0	0
$636$	−6.96239	−0.276077
$637$	8.81194	0.349142
$638$	−25.9248	−1.02637
$639$	−2.18664	−0.0865022
$640$	0	0
$641$	−34.0263	−1.34396	−0.671980	−	0.740569i	$-0.734556\pi$
$641$	−34.0263	−1.34396	−0.671980	+	0.740569i	$0.734556\pi$
$642$	−9.08840	−0.358690
$643$	7.34041	0.289478	0.144739	−	0.989470i	$-0.453766\pi$
$643$	7.34041	0.289478	0.144739	+	0.989470i	$0.453766\pi$
$644$	−2.96239	−0.116734
$645$	0	0
$646$	6.70052	0.263629
$647$	14.9525	0.587845	0.293922	−	0.955829i	$-0.405039\pi$
$647$	14.9525	0.587845	0.293922	+	0.955829i	$0.405039\pi$
$648$	1.00000	0.0392837
$649$	−4.10157	−0.161001
$650$	0	0
$651$	−11.8496	−0.464421
$652$	−12.6253	−0.494445
$653$	−20.4485	−0.800212	−0.400106	−	0.916469i	$-0.631027\pi$
$653$	−20.4485	−0.800212	−0.400106	+	0.916469i	$0.631027\pi$
$654$	−18.9380	−0.740532
$655$	0	0
$656$	4.70052	0.183525
$657$	9.92478	0.387202
$658$	9.55149	0.372356
$659$	38.4241	1.49679	0.748395	−	0.663254i	$-0.230825\pi$
$659$	38.4241	1.49679	0.748395	+	0.663254i	$0.230825\pi$
$660$	0	0
$661$	−39.3112	−1.52903	−0.764515	−	0.644606i	$-0.777021\pi$
$661$	−39.3112	−1.52903	−0.764515	+	0.644606i	$0.777021\pi$
$662$	23.1754	0.900737
$663$	6.70052	0.260227
$664$	6.38787	0.247898
$665$	0	0
$666$	7.61213	0.294964
$667$	7.73813	0.299622
$668$	24.6253	0.952781
$669$	−3.81336	−0.147433
$670$	0	0
$671$	37.1490	1.43412
$672$	2.96239	0.114277
$673$	−25.5515	−0.984938	−0.492469	−	0.870330i	$-0.663905\pi$
$673$	−25.5515	−0.984938	−0.492469	+	0.870330i	$0.663905\pi$
$674$	21.0376	0.810339
$675$	0	0
$676$	11.6253	0.447127
$677$	−36.2130	−1.39178	−0.695889	−	0.718149i	$-0.744990\pi$
$677$	−36.2130	−1.39178	−0.695889	+	0.718149i	$0.744990\pi$
$678$	−6.64974	−0.255382
$679$	38.1768	1.46509
$680$	0	0
$681$	−16.9380	−0.649064
$682$	13.4010	0.513153
$683$	15.4763	0.592183	0.296092	−	0.955160i	$-0.404317\pi$
$683$	15.4763	0.592183	0.296092	+	0.955160i	$0.404317\pi$
$684$	−4.96239	−0.189742
$685$	0	0
$686$	15.4763	0.590887
$687$	26.9380	1.02775
$688$	10.3127	0.393166
$689$	34.5501	1.31625
$690$	0	0
$691$	−37.6531	−1.43239	−0.716195	−	0.697900i	$-0.754118\pi$
$691$	−37.6531	−1.43239	−0.716195	+	0.697900i	$0.754118\pi$
$692$	4.44851	0.169107
$693$	9.92478	0.377011
$694$	−3.37470	−0.128102
$695$	0	0
$696$	−7.73813	−0.293313
$697$	−6.34694	−0.240407
$698$	−4.44851	−0.168378
$699$	−27.4010	−1.03640
$700$	0	0
$701$	28.3488	1.07072	0.535361	−	0.844624i	$-0.320176\pi$
$701$	28.3488	1.07072	0.535361	+	0.844624i	$0.320176\pi$
$702$	−4.96239	−0.187293
$703$	−37.7743	−1.42469
$704$	−3.35026	−0.126268
$705$	0	0
$706$	35.4010	1.33234
$707$	−12.6253	−0.474823
$708$	−1.22425	−0.0460103
$709$	−43.5633	−1.63605	−0.818026	−	0.575181i	$-0.804932\pi$
$709$	−43.5633	−1.63605	−0.818026	+	0.575181i	$0.804932\pi$
$710$	0	0
$711$	−4.12601	−0.154737
$712$	9.92478	0.371947
$713$	−4.00000	−0.149801
$714$	−4.00000	−0.149696
$715$	0	0
$716$	−13.8496	−0.517582
$717$	−7.48612	−0.279574
$718$	−34.5501	−1.28940
$719$	32.3634	1.20695	0.603476	−	0.797381i	$-0.293782\pi$
$719$	32.3634	1.20695	0.603476	+	0.797381i	$0.293782\pi$
$720$	0	0
$721$	−0.775746	−0.0288903
$722$	5.62530	0.209352
$723$	13.0738	0.486220
$724$	−22.6859	−0.843116
$725$	0	0
$726$	−0.224254	−0.00832284
$727$	6.96239	0.258221	0.129110	−	0.991630i	$-0.458788\pi$
$727$	6.96239	0.258221	0.129110	+	0.991630i	$0.458788\pi$
$728$	−14.7005	−0.544838
$729$	1.00000	0.0370370
$730$	0	0
$731$	−13.9248	−0.515026
$732$	11.0884	0.409839
$733$	−9.68735	−0.357810	−0.178905	−	0.983866i	$-0.557256\pi$
$733$	−9.68735	−0.357810	−0.178905	+	0.983866i	$0.557256\pi$
$734$	−22.8119	−0.842004
$735$	0	0
$736$	1.00000	0.0368605
$737$	−25.5026	−0.939401
$738$	4.70052	0.173029
$739$	−42.9234	−1.57896	−0.789481	−	0.613775i	$-0.789650\pi$
$739$	−42.9234	−1.57896	−0.789481	+	0.613775i	$0.789650\pi$
$740$	0	0
$741$	24.6253	0.904633
$742$	−20.6253	−0.757179
$743$	16.9986	0.623618	0.311809	−	0.950145i	$-0.399065\pi$
$743$	16.9986	0.623618	0.311809	+	0.950145i	$0.399065\pi$
$744$	4.00000	0.146647
$745$	0	0
$746$	−10.8364	−0.396748
$747$	6.38787	0.233720
$748$	4.52373	0.165404
$749$	−26.9234	−0.983758
$750$	0	0
$751$	−17.9003	−0.653193	−0.326596	−	0.945164i	$-0.605902\pi$
$751$	−17.9003	−0.653193	−0.326596	+	0.945164i	$0.605902\pi$
$752$	−3.22425	−0.117576
$753$	5.02302	0.183049
$754$	38.3996	1.39843
$755$	0	0
$756$	2.96239	0.107741
$757$	−12.3879	−0.450245	−0.225122	−	0.974330i	$-0.572278\pi$
$757$	−12.3879	−0.450245	−0.225122	+	0.974330i	$0.572278\pi$
$758$	−32.4387	−1.17823
$759$	3.35026	0.121607
$760$	0	0
$761$	38.5764	1.39839	0.699197	−	0.714929i	$-0.253541\pi$
$761$	38.5764	1.39839	0.699197	+	0.714929i	$0.253541\pi$
$762$	0.186642	0.00676134
$763$	−56.1016	−2.03101
$764$	−19.3258	−0.699184
$765$	0	0
$766$	32.9986	1.19229
$767$	6.07522	0.219364
$768$	−1.00000	−0.0360844
$769$	34.2228	1.23411	0.617054	−	0.786921i	$-0.288326\pi$
$769$	34.2228	1.23411	0.617054	+	0.786921i	$0.288326\pi$
$770$	0	0
$771$	19.4010	0.698712
$772$	5.92478	0.213237
$773$	13.5613	0.487768	0.243884	−	0.969804i	$-0.421578\pi$
$773$	13.5613	0.487768	0.243884	+	0.969804i	$0.421578\pi$
$774$	10.3127	0.370681
$775$	0	0
$776$	−12.8872	−0.462622
$777$	22.5501	0.808980
$778$	29.1246	1.04417
$779$	−23.3258	−0.835734
$780$	0	0
$781$	7.32582	0.262139
$782$	−1.35026	−0.0482853
$783$	−7.73813	−0.276538
$784$	1.77575	0.0634195
$785$	0	0
$786$	−7.92478	−0.282667
$787$	−37.0132	−1.31938	−0.659689	−	0.751539i	$-0.729312\pi$
$787$	−37.0132	−1.31938	−0.659689	+	0.751539i	$0.729312\pi$
$788$	−8.07522	−0.287668
$789$	15.4763	0.550970
$790$	0	0
$791$	−19.6991	−0.700420
$792$	−3.35026	−0.119046
$793$	−55.0249	−1.95399
$794$	−13.7381	−0.487548
$795$	0	0
$796$	15.9756	0.566239
$797$	−47.6893	−1.68924	−0.844620	−	0.535366i	$-0.820174\pi$
$797$	−47.6893	−1.68924	−0.844620	+	0.535366i	$0.820174\pi$
$798$	−14.7005	−0.520393
$799$	4.35359	0.154019
$800$	0	0
$801$	9.92478	0.350675
$802$	−1.44992	−0.0511985
$803$	−33.2506	−1.17339
$804$	−7.61213	−0.268459
$805$	0	0
$806$	−19.8496	−0.699171
$807$	20.2130	0.711531
$808$	4.26187	0.149932
$809$	−9.17538	−0.322589	−0.161295	−	0.986906i	$-0.551567\pi$
$809$	−9.17538	−0.322589	−0.161295	+	0.986906i	$0.551567\pi$
$810$	0	0
$811$	−38.5501	−1.35368	−0.676838	−	0.736132i	$-0.736650\pi$
$811$	−38.5501	−1.35368	−0.676838	+	0.736132i	$0.736650\pi$
$812$	−22.9234	−0.804452
$813$	−30.3996	−1.06616
$814$	−25.5026	−0.893866
$815$	0	0
$816$	1.35026	0.0472686
$817$	−51.1754	−1.79040
$818$	−17.8496	−0.624095
$819$	−14.7005	−0.513678
$820$	0	0
$821$	43.4372	1.51597	0.757985	−	0.652272i	$-0.226184\pi$
$821$	43.4372	1.51597	0.757985	+	0.652272i	$0.226184\pi$
$822$	9.19982	0.320880
$823$	3.51247	0.122437	0.0612184	−	0.998124i	$-0.480501\pi$
$823$	3.51247	0.122437	0.0612184	+	0.998124i	$0.480501\pi$
$824$	0.261865	0.00912250
$825$	0	0
$826$	−3.62672	−0.126190
$827$	9.56325	0.332547	0.166273	−	0.986080i	$-0.446827\pi$
$827$	9.56325	0.332547	0.166273	+	0.986080i	$0.446827\pi$
$828$	1.00000	0.0347524
$829$	30.5764	1.06196	0.530982	−	0.847383i	$-0.321823\pi$
$829$	30.5764	1.06196	0.530982	+	0.847383i	$0.321823\pi$
$830$	0	0
$831$	20.8119	0.721958
$832$	4.96239	0.172040
$833$	−2.39772	−0.0830762
$834$	−8.62530	−0.298670
$835$	0	0
$836$	16.6253	0.574998
$837$	4.00000	0.138260
$838$	17.9003	0.618357
$839$	22.1768	0.765628	0.382814	−	0.923825i	$-0.374955\pi$
$839$	22.1768	0.765628	0.382814	+	0.923825i	$0.374955\pi$
$840$	0	0
$841$	30.8787	1.06478
$842$	7.61213	0.262331
$843$	−19.8496	−0.683655
$844$	21.7743	0.749503
$845$	0	0
$846$	−3.22425	−0.110852
$847$	−0.664327	−0.0228265
$848$	6.96239	0.239089
$849$	12.3879	0.425151
$850$	0	0
$851$	7.61213	0.260940
$852$	2.18664	0.0749131
$853$	−29.7381	−1.01821	−0.509107	−	0.860703i	$-0.670024\pi$
$853$	−29.7381	−1.01821	−0.509107	+	0.860703i	$0.670024\pi$
$854$	32.8481	1.12404
$855$	0	0
$856$	9.08840	0.310635
$857$	22.2520	0.760114	0.380057	−	0.924963i	$-0.375904\pi$
$857$	22.2520	0.760114	0.380057	+	0.924963i	$0.375904\pi$
$858$	16.6253	0.567578
$859$	11.0738	0.377833	0.188917	−	0.981993i	$-0.439502\pi$
$859$	11.0738	0.377833	0.188917	+	0.981993i	$0.439502\pi$
$860$	0	0
$861$	13.9248	0.474555
$862$	−22.9525	−0.781767
$863$	−22.8218	−0.776863	−0.388431	−	0.921478i	$-0.626983\pi$
$863$	−22.8218	−0.776863	−0.388431	+	0.921478i	$0.626983\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−33.7645	−1.14736
$867$	15.1768	0.515431
$868$	11.8496	0.402200
$869$	13.8232	0.468920
$870$	0	0
$871$	37.7743	1.27993
$872$	18.9380	0.641320
$873$	−12.8872	−0.436164
$874$	−4.96239	−0.167855
$875$	0	0
$876$	−9.92478	−0.335327
$877$	−8.81194	−0.297558	−0.148779	−	0.988870i	$-0.547534\pi$
$877$	−8.81194	−0.297558	−0.148779	+	0.988870i	$0.547534\pi$
$878$	−23.4763	−0.792286
$879$	9.03761	0.304831
$880$	0	0
$881$	21.4010	0.721020	0.360510	−	0.932755i	$-0.382603\pi$
$881$	21.4010	0.721020	0.360510	+	0.932755i	$0.382603\pi$
$882$	1.77575	0.0597925
$883$	6.17679	0.207866	0.103933	−	0.994584i	$-0.466857\pi$
$883$	6.17679	0.207866	0.103933	+	0.994584i	$0.466857\pi$
$884$	−6.70052	−0.225363
$885$	0	0
$886$	30.5501	1.02635
$887$	−35.5975	−1.19525	−0.597624	−	0.801776i	$-0.703889\pi$
$887$	−35.5975	−1.19525	−0.597624	+	0.801776i	$0.703889\pi$
$888$	−7.61213	−0.255446
$889$	0.552907	0.0185439
$890$	0	0
$891$	−3.35026	−0.112238
$892$	3.81336	0.127681
$893$	16.0000	0.535420
$894$	−4.64974	−0.155511
$895$	0	0
$896$	−2.96239	−0.0989665
$897$	−4.96239	−0.165689
$898$	32.7005	1.09123
$899$	−30.9525	−1.03232
$900$	0	0
$901$	−9.40105	−0.313194
$902$	−15.7480	−0.524351
$903$	30.5501	1.01664
$904$	6.64974	0.221167
$905$	0	0
$906$	−10.7005	−0.355501
$907$	5.53690	0.183850	0.0919249	−	0.995766i	$-0.470698\pi$
$907$	5.53690	0.183850	0.0919249	+	0.995766i	$0.470698\pi$
$908$	16.9380	0.562106
$909$	4.26187	0.141357
$910$	0	0
$911$	−15.4763	−0.512752	−0.256376	−	0.966577i	$-0.582528\pi$
$911$	−15.4763	−0.512752	−0.256376	+	0.966577i	$0.582528\pi$
$912$	4.96239	0.164321
$913$	−21.4010	−0.708271
$914$	−5.81336	−0.192289
$915$	0	0
$916$	−26.9380	−0.890055
$917$	−23.4763	−0.775255
$918$	1.35026	0.0445653
$919$	−5.17347	−0.170657	−0.0853285	−	0.996353i	$-0.527194\pi$
$919$	−5.17347	−0.170657	−0.0853285	+	0.996353i	$0.527194\pi$
$920$	0	0
$921$	−30.5501	−1.00666
$922$	−23.9902	−0.790074
$923$	−10.8510	−0.357164
$924$	−9.92478	−0.326501
$925$	0	0
$926$	35.3620	1.16207
$927$	0.261865	0.00860078
$928$	7.73813	0.254017
$929$	−44.3996	−1.45670	−0.728352	−	0.685203i	$-0.759714\pi$
$929$	−44.3996	−1.45670	−0.728352	+	0.685203i	$0.759714\pi$
$930$	0	0
$931$	−8.81194	−0.288800
$932$	27.4010	0.897551
$933$	−14.4387	−0.472700
$934$	1.98541	0.0649647
$935$	0	0
$936$	4.96239	0.162201
$937$	3.99015	0.130353	0.0651763	−	0.997874i	$-0.479239\pi$
$937$	3.99015	0.130353	0.0651763	+	0.997874i	$0.479239\pi$
$938$	−22.5501	−0.736286
$939$	27.9610	0.912472
$940$	0	0
$941$	−2.30280	−0.0750692	−0.0375346	−	0.999295i	$-0.511950\pi$
$941$	−2.30280	−0.0750692	−0.0375346	+	0.999295i	$0.511950\pi$
$942$	−17.0132	−0.554319
$943$	4.70052	0.153070
$944$	1.22425	0.0398461
$945$	0	0
$946$	−34.5501	−1.12332
$947$	−24.7269	−0.803515	−0.401758	−	0.915746i	$-0.631601\pi$
$947$	−24.7269	−0.803515	−0.401758	+	0.915746i	$0.631601\pi$
$948$	4.12601	0.134007
$949$	49.2506	1.59874
$950$	0	0
$951$	1.47627	0.0478713
$952$	4.00000	0.129641
$953$	11.6775	0.378271	0.189136	−	0.981951i	$-0.439431\pi$
$953$	11.6775	0.378271	0.189136	+	0.981951i	$0.439431\pi$
$954$	6.96239	0.225416
$955$	0	0
$956$	7.48612	0.242118
$957$	25.9248	0.838029
$958$	11.0738	0.357779
$959$	27.2534	0.880059
$960$	0	0
$961$	−15.0000	−0.483871
$962$	37.7743	1.21789
$963$	9.08840	0.292869
$964$	−13.0738	−0.421079
$965$	0	0
$966$	2.96239	0.0953133
$967$	−51.2116	−1.64685	−0.823427	−	0.567423i	$-0.807941\pi$
$967$	−51.2116	−1.64685	−0.823427	+	0.567423i	$0.807941\pi$
$968$	0.224254	0.00720779
$969$	−6.70052	−0.215252
$970$	0	0
$971$	14.8265	0.475806	0.237903	−	0.971289i	$-0.423540\pi$
$971$	14.8265	0.475806	0.237903	+	0.971289i	$0.423540\pi$
$972$	−1.00000	−0.0320750
$973$	−25.5515	−0.819143
$974$	−17.4372	−0.558725
$975$	0	0
$976$	−11.0884	−0.354931
$977$	16.6958	0.534145	0.267073	−	0.963676i	$-0.413944\pi$
$977$	16.6958	0.534145	0.267073	+	0.963676i	$0.413944\pi$
$978$	12.6253	0.403713
$979$	−33.2506	−1.06269
$980$	0	0
$981$	18.9380	0.604642
$982$	14.8773	0.474754
$983$	30.0263	0.957692	0.478846	−	0.877899i	$-0.341055\pi$
$983$	30.0263	0.957692	0.478846	+	0.877899i	$0.341055\pi$
$984$	−4.70052	−0.149847
$985$	0	0
$986$	−10.4485	−0.332748
$987$	−9.55149	−0.304027
$988$	−24.6253	−0.783435
$989$	10.3127	0.327923
$990$	0	0
$991$	−1.40105	−0.0445057	−0.0222529	−	0.999752i	$-0.507084\pi$
$991$	−1.40105	−0.0445057	−0.0222529	+	0.999752i	$0.507084\pi$
$992$	−4.00000	−0.127000
$993$	−23.1754	−0.735448
$994$	6.47768	0.205460
$995$	0	0
$996$	−6.38787	−0.202408
$997$	−13.2144	−0.418504	−0.209252	−	0.977862i	$-0.567103\pi$
$997$	−13.2144	−0.418504	−0.209252	+	0.977862i	$0.567103\pi$
$998$	−16.1016	−0.509686
$999$	−7.61213	−0.240837

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.a.bt.1.1		3
5.2	odd	4		690.2.d.c.139.4	yes	6
5.3	odd	4		690.2.d.c.139.1	✓	6
5.4	even	2		3450.2.a.bo.1.3		3
15.2	even	4		2070.2.d.e.829.3		6
15.8	even	4		2070.2.d.e.829.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.d.c.139.1	✓	6	5.3	odd	4
690.2.d.c.139.4	yes	6	5.2	odd	4
2070.2.d.e.829.3		6	15.2	even	4
2070.2.d.e.829.6		6	15.8	even	4
3450.2.a.bo.1.3		3	5.4	even	2
3450.2.a.bt.1.1		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.96239 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.96239 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.35026 q^{11} -1.00000 q^{12} +4.96239 q^{13} -2.96239 q^{14} +1.00000 q^{16} -1.35026 q^{17} +1.00000 q^{18} -4.96239 q^{19} +2.96239 q^{21} -3.35026 q^{22} +1.00000 q^{23} -1.00000 q^{24} +4.96239 q^{26} -1.00000 q^{27} -2.96239 q^{28} +7.73813 q^{29} -4.00000 q^{31} +1.00000 q^{32} +3.35026 q^{33} -1.35026 q^{34} +1.00000 q^{36} +7.61213 q^{37} -4.96239 q^{38} -4.96239 q^{39} +4.70052 q^{41} +2.96239 q^{42} +10.3127 q^{43} -3.35026 q^{44} +1.00000 q^{46} -3.22425 q^{47} -1.00000 q^{48} +1.77575 q^{49} +1.35026 q^{51} +4.96239 q^{52} +6.96239 q^{53} -1.00000 q^{54} -2.96239 q^{56} +4.96239 q^{57} +7.73813 q^{58} +1.22425 q^{59} -11.0884 q^{61} -4.00000 q^{62} -2.96239 q^{63} +1.00000 q^{64} +3.35026 q^{66} +7.61213 q^{67} -1.35026 q^{68} -1.00000 q^{69} -2.18664 q^{71} +1.00000 q^{72} +9.92478 q^{73} +7.61213 q^{74} -4.96239 q^{76} +9.92478 q^{77} -4.96239 q^{78} -4.12601 q^{79} +1.00000 q^{81} +4.70052 q^{82} +6.38787 q^{83} +2.96239 q^{84} +10.3127 q^{86} -7.73813 q^{87} -3.35026 q^{88} +9.92478 q^{89} -14.7005 q^{91} +1.00000 q^{92} +4.00000 q^{93} -3.22425 q^{94} -1.00000 q^{96} -12.8872 q^{97} +1.77575 q^{98} -3.35026 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{12} + 4 q^{13} + 2 q^{14} + 3 q^{16} + 6 q^{17} + 3 q^{18} - 4 q^{19} - 2 q^{21} + 3 q^{23} - 3 q^{24} + 4 q^{26} - 3 q^{27} + 2 q^{28} + 14 q^{29} - 12 q^{31} + 3 q^{32} + 6 q^{34} + 3 q^{36} + 22 q^{37} - 4 q^{38} - 4 q^{39} - 6 q^{41} - 2 q^{42} + 10 q^{43} + 3 q^{46} - 8 q^{47} - 3 q^{48} + 7 q^{49} - 6 q^{51} + 4 q^{52} + 10 q^{53} - 3 q^{54} + 2 q^{56} + 4 q^{57} + 14 q^{58} + 2 q^{59} - 14 q^{61} - 12 q^{62} + 2 q^{63} + 3 q^{64} + 22 q^{67} + 6 q^{68} - 3 q^{69} + 6 q^{71} + 3 q^{72} + 8 q^{73} + 22 q^{74} - 4 q^{76} + 8 q^{77} - 4 q^{78} - 4 q^{79} + 3 q^{81} - 6 q^{82} + 20 q^{83} - 2 q^{84} + 10 q^{86} - 14 q^{87} + 8 q^{89} - 24 q^{91} + 3 q^{92} + 12 q^{93} - 8 q^{94} - 3 q^{96} - 6 q^{97} + 7 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^12 + 4 * q^13 + 2 * q^14 + 3 * q^16 + 6 * q^17 + 3 * q^18 - 4 * q^19 - 2 * q^21 + 3 * q^23 - 3 * q^24 + 4 * q^26 - 3 * q^27 + 2 * q^28 + 14 * q^29 - 12 * q^31 + 3 * q^32 + 6 * q^34 + 3 * q^36 + 22 * q^37 - 4 * q^38 - 4 * q^39 - 6 * q^41 - 2 * q^42 + 10 * q^43 + 3 * q^46 - 8 * q^47 - 3 * q^48 + 7 * q^49 - 6 * q^51 + 4 * q^52 + 10 * q^53 - 3 * q^54 + 2 * q^56 + 4 * q^57 + 14 * q^58 + 2 * q^59 - 14 * q^61 - 12 * q^62 + 2 * q^63 + 3 * q^64 + 22 * q^67 + 6 * q^68 - 3 * q^69 + 6 * q^71 + 3 * q^72 + 8 * q^73 + 22 * q^74 - 4 * q^76 + 8 * q^77 - 4 * q^78 - 4 * q^79 + 3 * q^81 - 6 * q^82 + 20 * q^83 - 2 * q^84 + 10 * q^86 - 14 * q^87 + 8 * q^89 - 24 * q^91 + 3 * q^92 + 12 * q^93 - 8 * q^94 - 3 * q^96 - 6 * q^97 + 7 * q^98

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