LMFDB - Embedded newform 850.2.b.c.101.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [850,2,Mod(101,850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("850.101");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 850 = 2 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	850.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$6.78728417181$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 170)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			101.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	850.101
Dual form			850.2.b.c.101.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} +2.00000i q^{7} -1.00000 q^{8} +2.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} +2.00000i q^{7} -1.00000 q^{8} +2.00000 q^{9} -1.00000i q^{12} +1.00000 q^{13} -2.00000i q^{14} +1.00000 q^{16} +(-4.00000 + 1.00000i) q^{17} -2.00000 q^{18} +5.00000 q^{19} +2.00000 q^{21} +4.00000i q^{23} +1.00000i q^{24} -1.00000 q^{26} -5.00000i q^{27} +2.00000i q^{28} +9.00000i q^{29} +5.00000i q^{31} -1.00000 q^{32} +(4.00000 - 1.00000i) q^{34} +2.00000 q^{36} +2.00000i q^{37} -5.00000 q^{38} -1.00000i q^{39} -10.0000i q^{41} -2.00000 q^{42} +6.00000 q^{43} -4.00000i q^{46} +7.00000 q^{47} -1.00000i q^{48} +3.00000 q^{49} +(1.00000 + 4.00000i) q^{51} +1.00000 q^{52} +1.00000 q^{53} +5.00000i q^{54} -2.00000i q^{56} -5.00000i q^{57} -9.00000i q^{58} +5.00000 q^{59} +5.00000i q^{61} -5.00000i q^{62} +4.00000i q^{63} +1.00000 q^{64} +2.00000 q^{67} +(-4.00000 + 1.00000i) q^{68} +4.00000 q^{69} +5.00000i q^{71} -2.00000 q^{72} -11.0000i q^{73} -2.00000i q^{74} +5.00000 q^{76} +1.00000i q^{78} -16.0000i q^{79} +1.00000 q^{81} +10.0000i q^{82} +6.00000 q^{83} +2.00000 q^{84} -6.00000 q^{86} +9.00000 q^{87} -5.00000 q^{89} +2.00000i q^{91} +4.00000i q^{92} +5.00000 q^{93} -7.00000 q^{94} +1.00000i q^{96} +7.00000i q^{97} -3.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 4 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 4 q^{9} + 2 q^{13} + 2 q^{16} - 8 q^{17} - 4 q^{18} + 10 q^{19} + 4 q^{21} - 2 q^{26} - 2 q^{32} + 8 q^{34} + 4 q^{36} - 10 q^{38} - 4 q^{42} + 12 q^{43} + 14 q^{47} + 6 q^{49} + 2 q^{51} + 2 q^{52} + 2 q^{53} + 10 q^{59} + 2 q^{64} + 4 q^{67} - 8 q^{68} + 8 q^{69} - 4 q^{72} + 10 q^{76} + 2 q^{81} + 12 q^{83} + 4 q^{84} - 12 q^{86} + 18 q^{87} - 10 q^{89} + 10 q^{93} - 14 q^{94} - 6 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 4 * q^9 + 2 * q^13 + 2 * q^16 - 8 * q^17 - 4 * q^18 + 10 * q^19 + 4 * q^21 - 2 * q^26 - 2 * q^32 + 8 * q^34 + 4 * q^36 - 10 * q^38 - 4 * q^42 + 12 * q^43 + 14 * q^47 + 6 * q^49 + 2 * q^51 + 2 * q^52 + 2 * q^53 + 10 * q^59 + 2 * q^64 + 4 * q^67 - 8 * q^68 + 8 * q^69 - 4 * q^72 + 10 * q^76 + 2 * q^81 + 12 * q^83 + 4 * q^84 - 12 * q^86 + 18 * q^87 - 10 * q^89 + 10 * q^93 - 14 * q^94 - 6 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/850\mathbb{Z}\right)^\times$.

$n$	$477$	$751$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		−	1.00000i		−	0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$		−	1.00000i		−	0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	1.00000			0.500000
$5$		0			0
$5$		0			0
$6$			1.00000i			0.408248i
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	−1.00000			−0.353553
$9$	2.00000			0.666667
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		−	1.00000i		−	0.288675i
$13$	1.00000			0.277350			0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000			0.277350			0.138675	+	0.990338i	$0.455716\pi$
$14$		−	2.00000i		−	0.534522i
$15$		0			0
$16$	1.00000			0.250000
$17$	−4.00000	+	1.00000i	−0.970143	+	0.242536i
$17$	−4.00000	+	1.00000i	−0.970143	+	0.242536i
$18$	−2.00000			−0.471405
$19$	5.00000			1.14708			0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000			1.14708			0.573539	+	0.819178i	$0.305570\pi$
$20$		0			0
$21$	2.00000			0.436436
$22$		0			0
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$			1.00000i			0.204124i
$25$		0			0
$26$	−1.00000			−0.196116
$27$		−	5.00000i		−	0.962250i
$28$			2.00000i			0.377964i
$29$			9.00000i			1.67126i	0.549294	+	0.835629i	$0.314897\pi$
$29$			9.00000i			1.67126i	−0.549294	+	0.835629i	$0.685103\pi$
$30$		0			0
$31$			5.00000i			0.898027i	0.893525	+	0.449013i	$0.148224\pi$
$31$			5.00000i			0.898027i	−0.893525	+	0.449013i	$0.851776\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$	4.00000	−	1.00000i	0.685994	−	0.171499i
$35$		0			0
$36$	2.00000			0.333333
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	−5.00000			−0.811107
$39$		−	1.00000i		−	0.160128i
$40$		0			0
$41$		−	10.0000i		−	1.56174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$		−	10.0000i		−	1.56174i	0.624695	−	0.780869i	$-0.285223\pi$
$42$	−2.00000			−0.308607
$43$	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
$45$		0			0
$46$		−	4.00000i		−	0.589768i
$47$	7.00000			1.02105			0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000			1.02105			0.510527	+	0.859861i	$0.329450\pi$
$48$		−	1.00000i		−	0.144338i
$49$	3.00000			0.428571
$50$		0			0
$51$	1.00000	+	4.00000i	0.140028	+	0.560112i
$52$	1.00000			0.138675
$53$	1.00000			0.137361			0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000			0.137361			0.0686803	+	0.997639i	$0.478121\pi$
$54$			5.00000i			0.680414i
$55$		0			0
$56$		−	2.00000i		−	0.267261i
$57$		−	5.00000i		−	0.662266i
$58$		−	9.00000i		−	1.18176i
$59$	5.00000			0.650945			0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000			0.650945			0.325472	+	0.945552i	$0.394477\pi$
$60$		0			0
$61$			5.00000i			0.640184i	0.947386	+	0.320092i	$0.103714\pi$
$61$			5.00000i			0.640184i	−0.947386	+	0.320092i	$0.896286\pi$
$62$		−	5.00000i		−	0.635001i
$63$			4.00000i			0.503953i
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$	2.00000			0.244339			0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000			0.244339			0.122169	+	0.992509i	$0.461015\pi$
$68$	−4.00000	+	1.00000i	−0.485071	+	0.121268i
$69$	4.00000			0.481543
$70$		0			0
$71$			5.00000i			0.593391i	0.954972	+	0.296695i	$0.0958846\pi$
$71$			5.00000i			0.593391i	−0.954972	+	0.296695i	$0.904115\pi$
$72$	−2.00000			−0.235702
$73$		−	11.0000i		−	1.28745i	−0.765256	−	0.643726i	$-0.777388\pi$
$73$		−	11.0000i		−	1.28745i	0.765256	−	0.643726i	$-0.222612\pi$
$74$		−	2.00000i		−	0.232495i
$75$		0			0
$76$	5.00000			0.573539
$77$		0			0
$78$			1.00000i			0.113228i
$79$		−	16.0000i		−	1.80014i	−0.435745	−	0.900070i	$-0.643515\pi$
$79$		−	16.0000i		−	1.80014i	0.435745	−	0.900070i	$-0.356485\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$			10.0000i			1.10432i
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$	2.00000			0.218218
$85$		0			0
$86$	−6.00000			−0.646997
$87$	9.00000			0.964901
$88$		0			0
$89$	−5.00000			−0.529999			−0.264999	−	0.964249i	$-0.585372\pi$
$89$	−5.00000			−0.529999			−0.264999	+	0.964249i	$0.585372\pi$
$90$		0			0
$91$			2.00000i			0.209657i
$92$			4.00000i			0.417029i
$93$	5.00000			0.518476
$94$	−7.00000			−0.721995
$95$		0			0
$96$			1.00000i			0.102062i
$97$			7.00000i			0.710742i	0.934725	+	0.355371i	$0.115646\pi$
$97$			7.00000i			0.710742i	−0.934725	+	0.355371i	$0.884354\pi$
$98$	−3.00000			−0.303046
$99$		0			0
$100$		0			0
$101$	−8.00000			−0.796030			−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000			−0.796030			−0.398015	+	0.917379i	$0.630301\pi$
$102$	−1.00000	−	4.00000i	−0.0990148	−	0.396059i
$103$	16.0000			1.57653			0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000			1.57653			0.788263	+	0.615338i	$0.210980\pi$
$104$	−1.00000			−0.0980581
$105$		0			0
$106$	−1.00000			−0.0971286
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$		−	5.00000i		−	0.481125i
$109$			9.00000i			0.862044i	0.902342	+	0.431022i	$0.141847\pi$
$109$			9.00000i			0.862044i	−0.902342	+	0.431022i	$0.858153\pi$
$110$		0			0
$111$	2.00000			0.189832
$112$			2.00000i			0.188982i
$113$			9.00000i			0.846649i	0.905978	+	0.423324i	$0.139137\pi$
$113$			9.00000i			0.846649i	−0.905978	+	0.423324i	$0.860863\pi$
$114$			5.00000i			0.468293i
$115$		0			0
$116$			9.00000i			0.835629i
$117$	2.00000			0.184900
$118$	−5.00000			−0.460287
$119$	−2.00000	−	8.00000i	−0.183340	−	0.733359i
$120$		0			0
$121$	11.0000			1.00000
$122$		−	5.00000i		−	0.452679i
$123$	−10.0000			−0.901670
$124$			5.00000i			0.449013i
$125$		0			0
$126$		−	4.00000i		−	0.356348i
$127$	7.00000			0.621150			0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000			0.621150			0.310575	+	0.950549i	$0.399478\pi$
$128$	−1.00000			−0.0883883
$129$		−	6.00000i		−	0.528271i
$130$		0			0
$131$		−	20.0000i		−	1.74741i	−0.486458	−	0.873704i	$-0.661711\pi$
$131$		−	20.0000i		−	1.74741i	0.486458	−	0.873704i	$-0.338289\pi$
$132$		0			0
$133$			10.0000i			0.867110i
$134$	−2.00000			−0.172774
$135$		0			0
$136$	4.00000	−	1.00000i	0.342997	−	0.0857493i
$137$	2.00000			0.170872			0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000			0.170872			0.0854358	+	0.996344i	$0.472772\pi$
$138$	−4.00000			−0.340503
$139$			14.0000i			1.18746i	0.804663	+	0.593732i	$0.202346\pi$
$139$			14.0000i			1.18746i	−0.804663	+	0.593732i	$0.797654\pi$
$140$		0			0
$141$		−	7.00000i		−	0.589506i
$142$		−	5.00000i		−	0.419591i
$143$		0			0
$144$	2.00000			0.166667
$145$		0			0
$146$			11.0000i			0.910366i
$147$		−	3.00000i		−	0.247436i
$148$			2.00000i			0.164399i
$149$	−20.0000			−1.63846			−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−20.0000			−1.63846			−0.819232	+	0.573462i	$0.805600\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$	−5.00000			−0.405554
$153$	−8.00000	+	2.00000i	−0.646762	+	0.161690i
$154$		0			0
$155$		0			0
$156$		−	1.00000i		−	0.0800641i
$157$	−18.0000			−1.43656			−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000			−1.43656			−0.718278	+	0.695756i	$0.755069\pi$
$158$			16.0000i			1.27289i
$159$		−	1.00000i		−	0.0793052i
$160$		0			0
$161$	−8.00000			−0.630488
$162$	−1.00000			−0.0785674
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$		−	10.0000i		−	0.780869i
$165$		0			0
$166$	−6.00000			−0.465690
$167$		−	18.0000i		−	1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$		−	18.0000i		−	1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$	−2.00000			−0.154303
$169$	−12.0000			−0.923077
$170$		0			0
$171$	10.0000			0.764719
$172$	6.00000			0.457496
$173$		−	16.0000i		−	1.21646i	−0.793762	−	0.608229i	$-0.791880\pi$
$173$		−	16.0000i		−	1.21646i	0.793762	−	0.608229i	$-0.208120\pi$
$174$	−9.00000			−0.682288
$175$		0			0
$176$		0			0
$177$		−	5.00000i		−	0.375823i
$178$	5.00000			0.374766
$179$	−20.0000			−1.49487			−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000			−1.49487			−0.747435	+	0.664335i	$0.768715\pi$
$180$		0			0
$181$			10.0000i			0.743294i	0.928374	+	0.371647i	$0.121207\pi$
$181$			10.0000i			0.743294i	−0.928374	+	0.371647i	$0.878793\pi$
$182$		−	2.00000i		−	0.148250i
$183$	5.00000			0.369611
$184$		−	4.00000i		−	0.294884i
$185$		0			0
$186$	−5.00000			−0.366618
$187$		0			0
$188$	7.00000			0.510527
$189$	10.0000			0.727393
$190$		0			0
$191$	−18.0000			−1.30243			−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000			−1.30243			−0.651217	+	0.758891i	$0.725741\pi$
$192$		−	1.00000i		−	0.0721688i
$193$			14.0000i			1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$			14.0000i			1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$		−	7.00000i		−	0.502571i
$195$		0			0
$196$	3.00000			0.214286
$197$		−	8.00000i		−	0.569976i	−0.958531	−	0.284988i	$-0.908010\pi$
$197$		−	8.00000i		−	0.569976i	0.958531	−	0.284988i	$-0.0919897\pi$
$198$		0			0
$199$		−	1.00000i		−	0.0708881i	−0.999372	−	0.0354441i	$-0.988715\pi$
$199$		−	1.00000i		−	0.0708881i	0.999372	−	0.0354441i	$-0.0112846\pi$
$200$		0			0
$201$		−	2.00000i		−	0.141069i
$202$	8.00000			0.562878
$203$	−18.0000			−1.26335
$204$	1.00000	+	4.00000i	0.0700140	+	0.280056i
$205$		0			0
$206$	−16.0000			−1.11477
$207$			8.00000i			0.556038i
$208$	1.00000			0.0693375
$209$		0			0
$210$		0			0
$211$		−	20.0000i		−	1.37686i	−0.725304	−	0.688428i	$-0.758301\pi$
$211$		−	20.0000i		−	1.37686i	0.725304	−	0.688428i	$-0.241699\pi$
$212$	1.00000			0.0686803
$213$	5.00000			0.342594
$214$		−	12.0000i		−	0.820303i
$215$		0			0
$216$			5.00000i			0.340207i
$217$	−10.0000			−0.678844
$218$		−	9.00000i		−	0.609557i
$219$	−11.0000			−0.743311
$220$		0			0
$221$	−4.00000	+	1.00000i	−0.269069	+	0.0672673i
$222$	−2.00000			−0.134231
$223$	−9.00000			−0.602685			−0.301342	−	0.953516i	$-0.597435\pi$
$223$	−9.00000			−0.602685			−0.301342	+	0.953516i	$0.597435\pi$
$224$		−	2.00000i		−	0.133631i
$225$		0			0
$226$		−	9.00000i		−	0.598671i
$227$			27.0000i			1.79205i	0.444001	+	0.896026i	$0.353559\pi$
$227$			27.0000i			1.79205i	−0.444001	+	0.896026i	$0.646441\pi$
$228$		−	5.00000i		−	0.331133i
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		−	9.00000i		−	0.590879i
$233$		−	21.0000i		−	1.37576i	−0.725826	−	0.687878i	$-0.758542\pi$
$233$		−	21.0000i		−	1.37576i	0.725826	−	0.687878i	$-0.241458\pi$
$234$	−2.00000			−0.130744
$235$		0			0
$236$	5.00000			0.325472
$237$	−16.0000			−1.03931
$238$	2.00000	+	8.00000i	0.129641	+	0.518563i
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$			10.0000i			0.644157i	0.946713	+	0.322078i	$0.104381\pi$
$241$			10.0000i			0.644157i	−0.946713	+	0.322078i	$0.895619\pi$
$242$	−11.0000			−0.707107
$243$		−	16.0000i		−	1.02640i
$244$			5.00000i			0.320092i
$245$		0			0
$246$	10.0000			0.637577
$247$	5.00000			0.318142
$248$		−	5.00000i		−	0.317500i
$249$		−	6.00000i		−	0.380235i
$250$		0			0
$251$	−8.00000			−0.504956			−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000			−0.504956			−0.252478	+	0.967603i	$0.581245\pi$
$252$			4.00000i			0.251976i
$253$		0			0
$254$	−7.00000			−0.439219
$255$		0			0
$256$	1.00000			0.0625000
$257$	−18.0000			−1.12281			−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000			−1.12281			−0.561405	+	0.827541i	$0.689739\pi$
$258$			6.00000i			0.373544i
$259$	−4.00000			−0.248548
$260$		0			0
$261$			18.0000i			1.11417i
$262$			20.0000i			1.23560i
$263$	21.0000			1.29492			0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000			1.29492			0.647458	+	0.762101i	$0.275832\pi$
$264$		0			0
$265$		0			0
$266$		−	10.0000i		−	0.613139i
$267$			5.00000i			0.305995i
$268$	2.00000			0.122169
$269$		−	1.00000i		−	0.0609711i	−0.999535	−	0.0304855i	$-0.990295\pi$
$269$		−	1.00000i		−	0.0609711i	0.999535	−	0.0304855i	$-0.00970535\pi$
$270$		0			0
$271$	22.0000			1.33640			0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000			1.33640			0.668202	+	0.743980i	$0.267064\pi$
$272$	−4.00000	+	1.00000i	−0.242536	+	0.0606339i
$273$	2.00000			0.121046
$274$	−2.00000			−0.120824
$275$		0			0
$276$	4.00000			0.240772
$277$		−	18.0000i		−	1.08152i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$		−	18.0000i		−	1.08152i	0.841178	−	0.540758i	$-0.181862\pi$
$278$		−	14.0000i		−	0.839664i
$279$			10.0000i			0.598684i
$280$		0			0
$281$	−23.0000			−1.37206			−0.686032	−	0.727571i	$-0.740649\pi$
$281$	−23.0000			−1.37206			−0.686032	+	0.727571i	$0.740649\pi$
$282$			7.00000i			0.416844i
$283$		−	21.0000i		−	1.24832i	−0.781296	−	0.624160i	$-0.785441\pi$
$283$		−	21.0000i		−	1.24832i	0.781296	−	0.624160i	$-0.214559\pi$
$284$			5.00000i			0.296695i
$285$		0			0
$286$		0			0
$287$	20.0000			1.18056
$288$	−2.00000			−0.117851
$289$	15.0000	−	8.00000i	0.882353	−	0.470588i
$290$		0			0
$291$	7.00000			0.410347
$292$		−	11.0000i		−	0.643726i
$293$	−29.0000			−1.69420			−0.847099	−	0.531435i	$-0.821653\pi$
$293$	−29.0000			−1.69420			−0.847099	+	0.531435i	$0.821653\pi$
$294$			3.00000i			0.174964i
$295$		0			0
$296$		−	2.00000i		−	0.116248i
$297$		0			0
$298$	20.0000			1.15857
$299$			4.00000i			0.231326i
$300$		0			0
$301$			12.0000i			0.691669i
$302$	8.00000			0.460348
$303$			8.00000i			0.459588i
$304$	5.00000			0.286770
$305$		0			0
$306$	8.00000	−	2.00000i	0.457330	−	0.114332i
$307$	2.00000			0.114146			0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000			0.114146			0.0570730	+	0.998370i	$0.481823\pi$
$308$		0			0
$309$		−	16.0000i		−	0.910208i
$310$		0			0
$311$			20.0000i			1.13410i	0.823685	+	0.567048i	$0.191915\pi$
$311$			20.0000i			1.13410i	−0.823685	+	0.567048i	$0.808085\pi$
$312$			1.00000i			0.0566139i
$313$			14.0000i			0.791327i	0.918396	+	0.395663i	$0.129485\pi$
$313$			14.0000i			0.791327i	−0.918396	+	0.395663i	$0.870515\pi$
$314$	18.0000			1.01580
$315$		0			0
$316$		−	16.0000i		−	0.900070i
$317$			12.0000i			0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$			12.0000i			0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$			1.00000i			0.0560772i
$319$		0			0
$320$		0			0
$321$	12.0000			0.669775
$322$	8.00000			0.445823
$323$	−20.0000	+	5.00000i	−1.11283	+	0.278207i
$324$	1.00000			0.0555556
$325$		0			0
$326$		−	4.00000i		−	0.221540i
$327$	9.00000			0.497701
$328$			10.0000i			0.552158i
$329$			14.0000i			0.771845i
$330$		0			0
$331$	7.00000			0.384755			0.192377	−	0.981321i	$-0.438380\pi$
$331$	7.00000			0.384755			0.192377	+	0.981321i	$0.438380\pi$
$332$	6.00000			0.329293
$333$			4.00000i			0.219199i
$334$			18.0000i			0.984916i
$335$		0			0
$336$	2.00000			0.109109
$337$		−	23.0000i		−	1.25289i	−0.779466	−	0.626445i	$-0.784509\pi$
$337$		−	23.0000i		−	1.25289i	0.779466	−	0.626445i	$-0.215491\pi$
$338$	12.0000			0.652714
$339$	9.00000			0.488813
$340$		0			0
$341$		0			0
$342$	−10.0000			−0.540738
$343$			20.0000i			1.07990i
$344$	−6.00000			−0.323498
$345$		0			0
$346$			16.0000i			0.860165i
$347$		−	23.0000i		−	1.23470i	−0.786687	−	0.617352i	$-0.788205\pi$
$347$		−	23.0000i		−	1.23470i	0.786687	−	0.617352i	$-0.211795\pi$
$348$	9.00000			0.482451
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$		−	5.00000i		−	0.266880i
$352$		0			0
$353$	−4.00000			−0.212899			−0.106449	−	0.994318i	$-0.533948\pi$
$353$	−4.00000			−0.212899			−0.106449	+	0.994318i	$0.533948\pi$
$354$			5.00000i			0.265747i
$355$		0			0
$356$	−5.00000			−0.264999
$357$	−8.00000	+	2.00000i	−0.423405	+	0.105851i
$358$	20.0000			1.05703
$359$	30.0000			1.58334			0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000			1.58334			0.791670	+	0.610949i	$0.209212\pi$
$360$		0			0
$361$	6.00000			0.315789
$362$		−	10.0000i		−	0.525588i
$363$		−	11.0000i		−	0.577350i
$364$			2.00000i			0.104828i
$365$		0			0
$366$	−5.00000			−0.261354
$367$			2.00000i			0.104399i	0.998637	+	0.0521996i	$0.0166232\pi$
$367$			2.00000i			0.104399i	−0.998637	+	0.0521996i	$0.983377\pi$
$368$			4.00000i			0.208514i
$369$		−	20.0000i		−	1.04116i
$370$		0			0
$371$			2.00000i			0.103835i
$372$	5.00000			0.259238
$373$	−34.0000			−1.76045			−0.880227	−	0.474554i	$-0.842610\pi$
$373$	−34.0000			−1.76045			−0.880227	+	0.474554i	$0.842610\pi$
$374$		0			0
$375$		0			0
$376$	−7.00000			−0.360997
$377$			9.00000i			0.463524i
$378$	−10.0000			−0.514344
$379$			4.00000i			0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$			4.00000i			0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$		0			0
$381$		−	7.00000i		−	0.358621i
$382$	18.0000			0.920960
$383$	31.0000			1.58403			0.792013	−	0.610504i	$-0.209033\pi$
$383$	31.0000			1.58403			0.792013	+	0.610504i	$0.209033\pi$
$384$			1.00000i			0.0510310i
$385$		0			0
$386$		−	14.0000i		−	0.712581i
$387$	12.0000			0.609994
$388$			7.00000i			0.355371i
$389$	10.0000			0.507020			0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000			0.507020			0.253510	+	0.967333i	$0.418415\pi$
$390$		0			0
$391$	−4.00000	−	16.0000i	−0.202289	−	0.809155i
$392$	−3.00000			−0.151523
$393$	−20.0000			−1.00887
$394$			8.00000i			0.403034i
$395$		0			0
$396$		0			0
$397$		−	18.0000i		−	0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$		−	18.0000i		−	0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$			1.00000i			0.0501255i
$399$	10.0000			0.500626
$400$		0			0
$401$		−	10.0000i		−	0.499376i	−0.968326	−	0.249688i	$-0.919672\pi$
$401$		−	10.0000i		−	0.499376i	0.968326	−	0.249688i	$-0.0803281\pi$
$402$			2.00000i			0.0997509i
$403$			5.00000i			0.249068i
$404$	−8.00000			−0.398015
$405$		0			0
$406$	18.0000			0.893325
$407$		0			0
$408$	−1.00000	−	4.00000i	−0.0495074	−	0.198030i
$409$	25.0000			1.23617			0.618085	−	0.786111i	$-0.287909\pi$
$409$	25.0000			1.23617			0.618085	+	0.786111i	$0.287909\pi$
$410$		0			0
$411$		−	2.00000i		−	0.0986527i
$412$	16.0000			0.788263
$413$			10.0000i			0.492068i
$414$		−	8.00000i		−	0.393179i
$415$		0			0
$416$	−1.00000			−0.0490290
$417$	14.0000			0.685583
$418$		0			0
$419$			24.0000i			1.17248i	0.810139	+	0.586238i	$0.199392\pi$
$419$			24.0000i			1.17248i	−0.810139	+	0.586238i	$0.800608\pi$
$420$		0			0
$421$	2.00000			0.0974740			0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000			0.0974740			0.0487370	+	0.998812i	$0.484480\pi$
$422$			20.0000i			0.973585i
$423$	14.0000			0.680703
$424$	−1.00000			−0.0485643
$425$		0			0
$426$	−5.00000			−0.242251
$427$	−10.0000			−0.483934
$428$			12.0000i			0.580042i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		−	5.00000i		−	0.240563i
$433$	6.00000			0.288342			0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000			0.288342			0.144171	+	0.989553i	$0.453949\pi$
$434$	10.0000			0.480015
$435$		0			0
$436$			9.00000i			0.431022i
$437$			20.0000i			0.956730i
$438$	11.0000			0.525600
$439$			24.0000i			1.14546i	0.819745	+	0.572729i	$0.194115\pi$
$439$			24.0000i			1.14546i	−0.819745	+	0.572729i	$0.805885\pi$
$440$		0			0
$441$	6.00000			0.285714
$442$	4.00000	−	1.00000i	0.190261	−	0.0475651i
$443$	26.0000			1.23530			0.617649	−	0.786454i	$-0.288085\pi$
$443$	26.0000			1.23530			0.617649	+	0.786454i	$0.288085\pi$
$444$	2.00000			0.0949158
$445$		0			0
$446$	9.00000			0.426162
$447$			20.0000i			0.945968i
$448$			2.00000i			0.0944911i
$449$		−	16.0000i		−	0.755087i	−0.925992	−	0.377543i	$-0.876769\pi$
$449$		−	16.0000i		−	0.755087i	0.925992	−	0.377543i	$-0.123231\pi$
$450$		0			0
$451$		0			0
$452$			9.00000i			0.423324i
$453$			8.00000i			0.375873i
$454$		−	27.0000i		−	1.26717i
$455$		0			0
$456$			5.00000i			0.234146i
$457$	−18.0000			−0.842004			−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000			−0.842004			−0.421002	+	0.907060i	$0.638322\pi$
$458$		0			0
$459$	5.00000	+	20.0000i	0.233380	+	0.933520i
$460$		0			0
$461$	32.0000			1.49039			0.745194	−	0.666847i	$-0.232357\pi$
$461$	32.0000			1.49039			0.745194	+	0.666847i	$0.232357\pi$
$462$		0			0
$463$	−29.0000			−1.34774			−0.673872	−	0.738848i	$-0.735370\pi$
$463$	−29.0000			−1.34774			−0.673872	+	0.738848i	$0.735370\pi$
$464$			9.00000i			0.417815i
$465$		0			0
$466$			21.0000i			0.972806i
$467$	12.0000			0.555294			0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000			0.555294			0.277647	+	0.960683i	$0.410445\pi$
$468$	2.00000			0.0924500
$469$			4.00000i			0.184703i
$470$		0			0
$471$			18.0000i			0.829396i
$472$	−5.00000			−0.230144
$473$		0			0
$474$	16.0000			0.734904
$475$		0			0
$476$	−2.00000	−	8.00000i	−0.0916698	−	0.366679i
$477$	2.00000			0.0915737
$478$		0			0
$479$		−	21.0000i		−	0.959514i	−0.877401	−	0.479757i	$-0.840725\pi$
$479$		−	21.0000i		−	0.959514i	0.877401	−	0.479757i	$-0.159275\pi$
$480$		0			0
$481$			2.00000i			0.0911922i
$482$		−	10.0000i		−	0.455488i
$483$			8.00000i			0.364013i
$484$	11.0000			0.500000
$485$		0			0
$486$			16.0000i			0.725775i
$487$			32.0000i			1.45006i	0.688718	+	0.725029i	$0.258174\pi$
$487$			32.0000i			1.45006i	−0.688718	+	0.725029i	$0.741826\pi$
$488$		−	5.00000i		−	0.226339i
$489$	4.00000			0.180886
$490$		0			0
$491$	−33.0000			−1.48927			−0.744635	−	0.667472i	$-0.767376\pi$
$491$	−33.0000			−1.48927			−0.744635	+	0.667472i	$0.767376\pi$
$492$	−10.0000			−0.450835
$493$	−9.00000	−	36.0000i	−0.405340	−	1.62136i
$494$	−5.00000			−0.224961
$495$		0			0
$496$			5.00000i			0.224507i
$497$	−10.0000			−0.448561
$498$			6.00000i			0.268866i
$499$		−	6.00000i		−	0.268597i	−0.990941	−	0.134298i	$-0.957122\pi$
$499$		−	6.00000i		−	0.268597i	0.990941	−	0.134298i	$-0.0428781\pi$
$500$		0			0
$501$	−18.0000			−0.804181
$502$	8.00000			0.357057
$503$		−	6.00000i		−	0.267527i	−0.991013	−	0.133763i	$-0.957294\pi$
$503$		−	6.00000i		−	0.267527i	0.991013	−	0.133763i	$-0.0427062\pi$
$504$		−	4.00000i		−	0.178174i
$505$		0			0
$506$		0			0
$507$			12.0000i			0.532939i
$508$	7.00000			0.310575
$509$	20.0000			0.886484			0.443242	−	0.896402i	$-0.353828\pi$
$509$	20.0000			0.886484			0.443242	+	0.896402i	$0.353828\pi$
$510$		0			0
$511$	22.0000			0.973223
$512$	−1.00000			−0.0441942
$513$		−	25.0000i		−	1.10378i
$514$	18.0000			0.793946
$515$		0			0
$516$		−	6.00000i		−	0.264135i
$517$		0			0
$518$	4.00000			0.175750
$519$	−16.0000			−0.702322
$520$		0			0
$521$		−	40.0000i		−	1.75243i	−0.481919	−	0.876216i	$-0.660060\pi$
$521$		−	40.0000i		−	1.75243i	0.481919	−	0.876216i	$-0.339940\pi$
$522$		−	18.0000i		−	0.787839i
$523$	−34.0000			−1.48672			−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000			−1.48672			−0.743358	+	0.668894i	$0.766768\pi$
$524$		−	20.0000i		−	0.873704i
$525$		0			0
$526$	−21.0000			−0.915644
$527$	−5.00000	−	20.0000i	−0.217803	−	0.871214i
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$	10.0000			0.433963
$532$			10.0000i			0.433555i
$533$		−	10.0000i		−	0.433148i
$534$		−	5.00000i		−	0.216371i
$535$		0			0
$536$	−2.00000			−0.0863868
$537$			20.0000i			0.863064i
$538$			1.00000i			0.0431131i
$539$		0			0
$540$		0			0
$541$		−	10.0000i		−	0.429934i	−0.976621	−	0.214967i	$-0.931036\pi$
$541$		−	10.0000i		−	0.429934i	0.976621	−	0.214967i	$-0.0689643\pi$
$542$	−22.0000			−0.944981
$543$	10.0000			0.429141
$544$	4.00000	−	1.00000i	0.171499	−	0.0428746i
$545$		0			0
$546$	−2.00000			−0.0855921
$547$		−	13.0000i		−	0.555840i	−0.960604	−	0.277920i	$-0.910355\pi$
$547$		−	13.0000i		−	0.555840i	0.960604	−	0.277920i	$-0.0896450\pi$
$548$	2.00000			0.0854358
$549$			10.0000i			0.426790i
$550$		0			0
$551$			45.0000i			1.91706i
$552$	−4.00000			−0.170251
$553$	32.0000			1.36078
$554$			18.0000i			0.764747i
$555$		0			0
$556$			14.0000i			0.593732i
$557$	−3.00000			−0.127114			−0.0635570	−	0.997978i	$-0.520244\pi$
$557$	−3.00000			−0.127114			−0.0635570	+	0.997978i	$0.520244\pi$
$558$		−	10.0000i		−	0.423334i
$559$	6.00000			0.253773
$560$		0			0
$561$		0			0
$562$	23.0000			0.970196
$563$	−24.0000			−1.01148			−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000			−1.01148			−0.505740	+	0.862686i	$0.668780\pi$
$564$		−	7.00000i		−	0.294753i
$565$		0			0
$566$			21.0000i			0.882696i
$567$			2.00000i			0.0839921i
$568$		−	5.00000i		−	0.209795i
$569$	−15.0000			−0.628833			−0.314416	−	0.949285i	$-0.601809\pi$
$569$	−15.0000			−0.628833			−0.314416	+	0.949285i	$0.601809\pi$
$570$		0			0
$571$		−	20.0000i		−	0.836974i	−0.908223	−	0.418487i	$-0.862561\pi$
$571$		−	20.0000i		−	0.836974i	0.908223	−	0.418487i	$-0.137439\pi$
$572$		0			0
$573$			18.0000i			0.751961i
$574$	−20.0000			−0.834784
$575$		0			0
$576$	2.00000			0.0833333
$577$	−28.0000			−1.16566			−0.582828	−	0.812596i	$-0.698054\pi$
$577$	−28.0000			−1.16566			−0.582828	+	0.812596i	$0.698054\pi$
$578$	−15.0000	+	8.00000i	−0.623918	+	0.332756i
$579$	14.0000			0.581820
$580$		0			0
$581$			12.0000i			0.497844i
$582$	−7.00000			−0.290159
$583$		0			0
$584$			11.0000i			0.455183i
$585$		0			0
$586$	29.0000			1.19798
$587$	12.0000			0.495293			0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000			0.495293			0.247647	+	0.968850i	$0.420343\pi$
$588$		−	3.00000i		−	0.123718i
$589$			25.0000i			1.03011i
$590$		0			0
$591$	−8.00000			−0.329076
$592$			2.00000i			0.0821995i
$593$	6.00000			0.246390			0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000			0.246390			0.123195	+	0.992382i	$0.460686\pi$
$594$		0			0
$595$		0			0
$596$	−20.0000			−0.819232
$597$	−1.00000			−0.0409273
$598$		−	4.00000i		−	0.163572i
$599$	−20.0000			−0.817178			−0.408589	−	0.912719i	$-0.633979\pi$
$599$	−20.0000			−0.817178			−0.408589	+	0.912719i	$0.633979\pi$
$600$		0			0
$601$		−	30.0000i		−	1.22373i	−0.790964	−	0.611863i	$-0.790420\pi$
$601$		−	30.0000i		−	1.22373i	0.790964	−	0.611863i	$-0.209580\pi$
$602$		−	12.0000i		−	0.489083i
$603$	4.00000			0.162893
$604$	−8.00000			−0.325515
$605$		0			0
$606$		−	8.00000i		−	0.324978i
$607$		−	38.0000i		−	1.54237i	−0.636610	−	0.771186i	$-0.719664\pi$
$607$		−	38.0000i		−	1.54237i	0.636610	−	0.771186i	$-0.280336\pi$
$608$	−5.00000			−0.202777
$609$			18.0000i			0.729397i
$610$		0			0
$611$	7.00000			0.283190
$612$	−8.00000	+	2.00000i	−0.323381	+	0.0808452i
$613$	41.0000			1.65597			0.827987	−	0.560747i	$-0.189486\pi$
$613$	41.0000			1.65597			0.827987	+	0.560747i	$0.189486\pi$
$614$	−2.00000			−0.0807134
$615$		0			0
$616$		0			0
$617$		−	33.0000i		−	1.32853i	−0.747497	−	0.664265i	$-0.768745\pi$
$617$		−	33.0000i		−	1.32853i	0.747497	−	0.664265i	$-0.231255\pi$
$618$			16.0000i			0.643614i
$619$		−	26.0000i		−	1.04503i	−0.852631	−	0.522514i	$-0.824994\pi$
$619$		−	26.0000i		−	1.04503i	0.852631	−	0.522514i	$-0.175006\pi$
$620$		0			0
$621$	20.0000			0.802572
$622$		−	20.0000i		−	0.801927i
$623$		−	10.0000i		−	0.400642i
$624$		−	1.00000i		−	0.0400320i
$625$		0			0
$626$		−	14.0000i		−	0.559553i
$627$		0			0
$628$	−18.0000			−0.718278
$629$	−2.00000	−	8.00000i	−0.0797452	−	0.318981i
$630$		0			0
$631$	2.00000			0.0796187			0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000			0.0796187			0.0398094	+	0.999207i	$0.487325\pi$
$632$			16.0000i			0.636446i
$633$	−20.0000			−0.794929
$634$		−	12.0000i		−	0.476581i
$635$		0			0
$636$		−	1.00000i		−	0.0396526i
$637$	3.00000			0.118864
$638$		0			0
$639$			10.0000i			0.395594i
$640$		0			0
$641$		−	10.0000i		−	0.394976i	−0.980305	−	0.197488i	$-0.936722\pi$
$641$		−	10.0000i		−	0.394976i	0.980305	−	0.197488i	$-0.0632784\pi$
$642$	−12.0000			−0.473602
$643$		−	36.0000i		−	1.41970i	−0.704352	−	0.709851i	$-0.748762\pi$
$643$		−	36.0000i		−	1.41970i	0.704352	−	0.709851i	$-0.251238\pi$
$644$	−8.00000			−0.315244
$645$		0			0
$646$	20.0000	−	5.00000i	0.786889	−	0.196722i
$647$	37.0000			1.45462			0.727310	−	0.686309i	$-0.240770\pi$
$647$	37.0000			1.45462			0.727310	+	0.686309i	$0.240770\pi$
$648$	−1.00000			−0.0392837
$649$		0			0
$650$		0			0
$651$			10.0000i			0.391931i
$652$			4.00000i			0.156652i
$653$			24.0000i			0.939193i	0.882881	+	0.469596i	$0.155601\pi$
$653$			24.0000i			0.939193i	−0.882881	+	0.469596i	$0.844399\pi$
$654$	−9.00000			−0.351928
$655$		0			0
$656$		−	10.0000i		−	0.390434i
$657$		−	22.0000i		−	0.858302i
$658$		−	14.0000i		−	0.545777i
$659$	5.00000			0.194772			0.0973862	−	0.995247i	$-0.468952\pi$
$659$	5.00000			0.194772			0.0973862	+	0.995247i	$0.468952\pi$
$660$		0			0
$661$	−8.00000			−0.311164			−0.155582	−	0.987823i	$-0.549725\pi$
$661$	−8.00000			−0.311164			−0.155582	+	0.987823i	$0.549725\pi$
$662$	−7.00000			−0.272063
$663$	1.00000	+	4.00000i	0.0388368	+	0.155347i
$664$	−6.00000			−0.232845
$665$		0			0
$666$		−	4.00000i		−	0.154997i
$667$	−36.0000			−1.39393
$668$		−	18.0000i		−	0.696441i
$669$			9.00000i			0.347960i
$670$		0			0
$671$		0			0
$672$	−2.00000			−0.0771517
$673$		−	1.00000i		−	0.0385472i	−0.999814	−	0.0192736i	$-0.993865\pi$
$673$		−	1.00000i		−	0.0385472i	0.999814	−	0.0192736i	$-0.00613535\pi$
$674$			23.0000i			0.885927i
$675$		0			0
$676$	−12.0000			−0.461538
$677$			2.00000i			0.0768662i	0.999261	+	0.0384331i	$0.0122367\pi$
$677$			2.00000i			0.0768662i	−0.999261	+	0.0384331i	$0.987763\pi$
$678$	−9.00000			−0.345643
$679$	−14.0000			−0.537271
$680$		0			0
$681$	27.0000			1.03464
$682$		0			0
$683$			39.0000i			1.49229i	0.665782	+	0.746147i	$0.268098\pi$
$683$			39.0000i			1.49229i	−0.665782	+	0.746147i	$0.731902\pi$
$684$	10.0000			0.382360
$685$		0			0
$686$		−	20.0000i		−	0.763604i
$687$		0			0
$688$	6.00000			0.228748
$689$	1.00000			0.0380970
$690$		0			0
$691$			30.0000i			1.14125i	0.821209	+	0.570627i	$0.193300\pi$
$691$			30.0000i			1.14125i	−0.821209	+	0.570627i	$0.806700\pi$
$692$		−	16.0000i		−	0.608229i
$693$		0			0
$694$			23.0000i			0.873068i
$695$		0			0
$696$	−9.00000			−0.341144
$697$	10.0000	+	40.0000i	0.378777	+	1.51511i
$698$	10.0000			0.378506
$699$	−21.0000			−0.794293
$700$		0			0
$701$	−48.0000			−1.81293			−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000			−1.81293			−0.906467	+	0.422276i	$0.861231\pi$
$702$			5.00000i			0.188713i
$703$			10.0000i			0.377157i
$704$		0			0
$705$		0			0
$706$	4.00000			0.150542
$707$		−	16.0000i		−	0.601742i
$708$		−	5.00000i		−	0.187912i
$709$		−	31.0000i		−	1.16423i	−0.813107	−	0.582115i	$-0.802225\pi$
$709$		−	31.0000i		−	1.16423i	0.813107	−	0.582115i	$-0.197775\pi$
$710$		0			0
$711$		−	32.0000i		−	1.20009i
$712$	5.00000			0.187383
$713$	−20.0000			−0.749006
$714$	8.00000	−	2.00000i	0.299392	−	0.0748481i
$715$		0			0
$716$	−20.0000			−0.747435
$717$		0			0
$718$	−30.0000			−1.11959
$719$		−	51.0000i		−	1.90198i	−0.309223	−	0.950990i	$-0.600069\pi$
$719$		−	51.0000i		−	1.90198i	0.309223	−	0.950990i	$-0.399931\pi$
$720$		0			0
$721$			32.0000i			1.19174i
$722$	−6.00000			−0.223297
$723$	10.0000			0.371904
$724$			10.0000i			0.371647i
$725$		0			0
$726$			11.0000i			0.408248i
$727$	−33.0000			−1.22390			−0.611951	−	0.790896i	$-0.709615\pi$
$727$	−33.0000			−1.22390			−0.611951	+	0.790896i	$0.709615\pi$
$728$		−	2.00000i		−	0.0741249i
$729$	−13.0000			−0.481481
$730$		0			0
$731$	−24.0000	+	6.00000i	−0.887672	+	0.221918i
$732$	5.00000			0.184805
$733$	6.00000			0.221615			0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000			0.221615			0.110808	+	0.993842i	$0.464656\pi$
$734$		−	2.00000i		−	0.0738213i
$735$		0			0
$736$		−	4.00000i		−	0.147442i
$737$		0			0
$738$			20.0000i			0.736210i
$739$	15.0000			0.551784			0.275892	−	0.961189i	$-0.411027\pi$
$739$	15.0000			0.551784			0.275892	+	0.961189i	$0.411027\pi$
$740$		0			0
$741$		−	5.00000i		−	0.183680i
$742$		−	2.00000i		−	0.0734223i
$743$			34.0000i			1.24734i	0.781688	+	0.623670i	$0.214359\pi$
$743$			34.0000i			1.24734i	−0.781688	+	0.623670i	$0.785641\pi$
$744$	−5.00000			−0.183309
$745$		0			0
$746$	34.0000			1.24483
$747$	12.0000			0.439057
$748$		0			0
$749$	−24.0000			−0.876941
$750$		0			0
$751$		−	25.0000i		−	0.912263i	−0.889912	−	0.456131i	$-0.849235\pi$
$751$		−	25.0000i		−	0.912263i	0.889912	−	0.456131i	$-0.150765\pi$
$752$	7.00000			0.255264
$753$			8.00000i			0.291536i
$754$		−	9.00000i		−	0.327761i
$755$		0			0
$756$	10.0000			0.363696
$757$	−13.0000			−0.472493			−0.236247	−	0.971693i	$-0.575917\pi$
$757$	−13.0000			−0.472493			−0.236247	+	0.971693i	$0.575917\pi$
$758$		−	4.00000i		−	0.145287i
$759$		0			0
$760$		0			0
$761$	22.0000			0.797499			0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000			0.797499			0.398750	+	0.917060i	$0.369444\pi$
$762$			7.00000i			0.253583i
$763$	−18.0000			−0.651644
$764$	−18.0000			−0.651217
$765$		0			0
$766$	−31.0000			−1.12008
$767$	5.00000			0.180540
$768$		−	1.00000i		−	0.0360844i
$769$	−35.0000			−1.26213			−0.631066	−	0.775729i	$-0.717382\pi$
$769$	−35.0000			−1.26213			−0.631066	+	0.775729i	$0.717382\pi$
$770$		0			0
$771$			18.0000i			0.648254i
$772$			14.0000i			0.503871i
$773$	26.0000			0.935155			0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000			0.935155			0.467578	+	0.883952i	$0.345127\pi$
$774$	−12.0000			−0.431331
$775$		0			0
$776$		−	7.00000i		−	0.251285i
$777$			4.00000i			0.143499i
$778$	−10.0000			−0.358517
$779$		−	50.0000i		−	1.79144i
$780$		0			0
$781$		0			0
$782$	4.00000	+	16.0000i	0.143040	+	0.572159i
$783$	45.0000			1.60817
$784$	3.00000			0.107143
$785$		0			0
$786$	20.0000			0.713376
$787$			7.00000i			0.249523i	0.992187	+	0.124762i	$0.0398166\pi$
$787$			7.00000i			0.249523i	−0.992187	+	0.124762i	$0.960183\pi$
$788$		−	8.00000i		−	0.284988i
$789$		−	21.0000i		−	0.747620i
$790$		0			0
$791$	−18.0000			−0.640006
$792$		0			0
$793$			5.00000i			0.177555i
$794$			18.0000i			0.638796i
$795$		0			0
$796$		−	1.00000i		−	0.0354441i
$797$	2.00000			0.0708436			0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000			0.0708436			0.0354218	+	0.999372i	$0.488723\pi$
$798$	−10.0000			−0.353996
$799$	−28.0000	+	7.00000i	−0.990569	+	0.247642i
$800$		0			0
$801$	−10.0000			−0.353333
$802$			10.0000i			0.353112i
$803$		0			0
$804$		−	2.00000i		−	0.0705346i
$805$		0			0
$806$		−	5.00000i		−	0.176117i
$807$	−1.00000			−0.0352017
$808$	8.00000			0.281439
$809$			24.0000i			0.843795i	0.906644	+	0.421898i	$0.138636\pi$
$809$			24.0000i			0.843795i	−0.906644	+	0.421898i	$0.861364\pi$
$810$		0			0
$811$			30.0000i			1.05344i	0.850038	+	0.526721i	$0.176579\pi$
$811$			30.0000i			1.05344i	−0.850038	+	0.526721i	$0.823421\pi$
$812$	−18.0000			−0.631676
$813$		−	22.0000i		−	0.771574i
$814$		0			0
$815$		0			0
$816$	1.00000	+	4.00000i	0.0350070	+	0.140028i
$817$	30.0000			1.04957
$818$	−25.0000			−0.874105
$819$			4.00000i			0.139771i
$820$		0			0
$821$			15.0000i			0.523504i	0.965135	+	0.261752i	$0.0843002\pi$
$821$			15.0000i			0.523504i	−0.965135	+	0.261752i	$0.915700\pi$
$822$			2.00000i			0.0697580i
$823$		−	46.0000i		−	1.60346i	−0.597687	−	0.801730i	$-0.703913\pi$
$823$		−	46.0000i		−	1.60346i	0.597687	−	0.801730i	$-0.296087\pi$
$824$	−16.0000			−0.557386
$825$		0			0
$826$		−	10.0000i		−	0.347945i
$827$			12.0000i			0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$			12.0000i			0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$			8.00000i			0.278019i
$829$	−10.0000			−0.347314			−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000			−0.347314			−0.173657	+	0.984806i	$0.555558\pi$
$830$		0			0
$831$	−18.0000			−0.624413
$832$	1.00000			0.0346688
$833$	−12.0000	+	3.00000i	−0.415775	+	0.103944i
$834$	−14.0000			−0.484780
$835$		0			0
$836$		0			0
$837$	25.0000			0.864126
$838$		−	24.0000i		−	0.829066i
$839$			9.00000i			0.310715i	0.987858	+	0.155357i	$0.0496529\pi$
$839$			9.00000i			0.310715i	−0.987858	+	0.155357i	$0.950347\pi$
$840$		0			0
$841$	−52.0000			−1.79310
$842$	−2.00000			−0.0689246
$843$			23.0000i			0.792162i
$844$		−	20.0000i		−	0.688428i
$845$		0			0
$846$	−14.0000			−0.481330
$847$			22.0000i			0.755929i
$848$	1.00000			0.0343401
$849$	−21.0000			−0.720718
$850$		0			0
$851$	−8.00000			−0.274236
$852$	5.00000			0.171297
$853$		−	16.0000i		−	0.547830i	−0.961754	−	0.273915i	$-0.911681\pi$
$853$		−	16.0000i		−	0.547830i	0.961754	−	0.273915i	$-0.0883186\pi$
$854$	10.0000			0.342193
$855$		0			0
$856$		−	12.0000i		−	0.410152i
$857$			17.0000i			0.580709i	0.956919	+	0.290354i	$0.0937732\pi$
$857$			17.0000i			0.580709i	−0.956919	+	0.290354i	$0.906227\pi$
$858$		0			0
$859$	25.0000			0.852989			0.426494	−	0.904490i	$-0.359748\pi$
$859$	25.0000			0.852989			0.426494	+	0.904490i	$0.359748\pi$
$860$		0			0
$861$		−	20.0000i		−	0.681598i
$862$		0			0
$863$	36.0000			1.22545			0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000			1.22545			0.612727	+	0.790295i	$0.290072\pi$
$864$			5.00000i			0.170103i
$865$		0			0
$866$	−6.00000			−0.203888
$867$	−8.00000	−	15.0000i	−0.271694	−	0.509427i
$868$	−10.0000			−0.339422
$869$		0			0
$870$		0			0
$871$	2.00000			0.0677674
$872$		−	9.00000i		−	0.304778i
$873$			14.0000i			0.473828i
$874$		−	20.0000i		−	0.676510i
$875$		0			0
$876$	−11.0000			−0.371656
$877$			32.0000i			1.08056i	0.841484	+	0.540282i	$0.181682\pi$
$877$			32.0000i			1.08056i	−0.841484	+	0.540282i	$0.818318\pi$
$878$		−	24.0000i		−	0.809961i
$879$			29.0000i			0.978146i
$880$		0			0
$881$			30.0000i			1.01073i	0.862907	+	0.505363i	$0.168641\pi$
$881$			30.0000i			1.01073i	−0.862907	+	0.505363i	$0.831359\pi$
$882$	−6.00000			−0.202031
$883$	−44.0000			−1.48072			−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000			−1.48072			−0.740359	+	0.672212i	$0.765344\pi$
$884$	−4.00000	+	1.00000i	−0.134535	+	0.0336336i
$885$		0			0
$886$	−26.0000			−0.873487
$887$		−	8.00000i		−	0.268614i	−0.990940	−	0.134307i	$-0.957119\pi$
$887$		−	8.00000i		−	0.268614i	0.990940	−	0.134307i	$-0.0428808\pi$
$888$	−2.00000			−0.0671156
$889$			14.0000i			0.469545i
$890$		0			0
$891$		0			0
$892$	−9.00000			−0.301342
$893$	35.0000			1.17123
$894$		−	20.0000i		−	0.668900i
$895$		0			0
$896$		−	2.00000i		−	0.0668153i
$897$	4.00000			0.133556
$898$			16.0000i			0.533927i
$899$	−45.0000			−1.50083
$900$		0			0
$901$	−4.00000	+	1.00000i	−0.133259	+	0.0333148i
$902$		0			0
$903$	12.0000			0.399335
$904$		−	9.00000i		−	0.299336i
$905$		0			0
$906$		−	8.00000i		−	0.265782i
$907$			7.00000i			0.232431i	0.993224	+	0.116216i	$0.0370764\pi$
$907$			7.00000i			0.232431i	−0.993224	+	0.116216i	$0.962924\pi$
$908$			27.0000i			0.896026i
$909$	−16.0000			−0.530687
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		−	5.00000i		−	0.165567i
$913$		0			0
$914$	18.0000			0.595387
$915$		0			0
$916$		0			0
$917$	40.0000			1.32092
$918$	−5.00000	−	20.0000i	−0.165025	−	0.660098i
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		−	2.00000i		−	0.0659022i
$922$	−32.0000			−1.05386
$923$			5.00000i			0.164577i
$924$		0			0
$925$		0			0
$926$	29.0000			0.952999
$927$	32.0000			1.05102
$928$		−	9.00000i		−	0.295439i
$929$			24.0000i			0.787414i	0.919236	+	0.393707i	$0.128808\pi$
$929$			24.0000i			0.787414i	−0.919236	+	0.393707i	$0.871192\pi$
$930$		0			0
$931$	15.0000			0.491605
$932$		−	21.0000i		−	0.687878i
$933$	20.0000			0.654771
$934$	−12.0000			−0.392652
$935$		0			0
$936$	−2.00000			−0.0653720
$937$	22.0000			0.718709			0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000			0.718709			0.359354	+	0.933201i	$0.382997\pi$
$938$		−	4.00000i		−	0.130605i
$939$	14.0000			0.456873
$940$		0			0
$941$		−	45.0000i		−	1.46696i	−0.679712	−	0.733479i	$-0.737895\pi$
$941$		−	45.0000i		−	1.46696i	0.679712	−	0.733479i	$-0.262105\pi$
$942$		−	18.0000i		−	0.586472i
$943$	40.0000			1.30258
$944$	5.00000			0.162736
$945$		0			0
$946$		0			0
$947$		−	3.00000i		−	0.0974869i	−0.998811	−	0.0487435i	$-0.984478\pi$
$947$		−	3.00000i		−	0.0974869i	0.998811	−	0.0487435i	$-0.0155217\pi$
$948$	−16.0000			−0.519656
$949$		−	11.0000i		−	0.357075i
$950$		0			0
$951$	12.0000			0.389127
$952$	2.00000	+	8.00000i	0.0648204	+	0.259281i
$953$	46.0000			1.49009			0.745043	−	0.667016i	$-0.232429\pi$
$953$	46.0000			1.49009			0.745043	+	0.667016i	$0.232429\pi$
$954$	−2.00000			−0.0647524
$955$		0			0
$956$		0			0
$957$		0			0
$958$			21.0000i			0.678479i
$959$			4.00000i			0.129167i
$960$		0			0
$961$	6.00000			0.193548
$962$		−	2.00000i		−	0.0644826i
$963$			24.0000i			0.773389i
$964$			10.0000i			0.322078i
$965$		0			0
$966$		−	8.00000i		−	0.257396i
$967$	−8.00000			−0.257263			−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000			−0.257263			−0.128631	+	0.991692i	$0.541058\pi$
$968$	−11.0000			−0.353553
$969$	5.00000	+	20.0000i	0.160623	+	0.642493i
$970$		0			0
$971$	57.0000			1.82922			0.914609	−	0.404341i	$-0.132499\pi$
$971$	57.0000			1.82922			0.914609	+	0.404341i	$0.132499\pi$
$972$		−	16.0000i		−	0.513200i
$973$	−28.0000			−0.897639
$974$		−	32.0000i		−	1.02535i
$975$		0			0
$976$			5.00000i			0.160046i
$977$	−8.00000			−0.255943			−0.127971	−	0.991778i	$-0.540847\pi$
$977$	−8.00000			−0.255943			−0.127971	+	0.991778i	$0.540847\pi$
$978$	−4.00000			−0.127906
$979$		0			0
$980$		0			0
$981$			18.0000i			0.574696i
$982$	33.0000			1.05307
$983$			24.0000i			0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$			24.0000i			0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	10.0000			0.318788
$985$		0			0
$986$	9.00000	+	36.0000i	0.286618	+	1.14647i
$987$	14.0000			0.445625
$988$	5.00000			0.159071
$989$			24.0000i			0.763156i
$990$		0			0
$991$		−	35.0000i		−	1.11181i	−0.831245	−	0.555906i	$-0.812372\pi$
$991$		−	35.0000i		−	1.11181i	0.831245	−	0.555906i	$-0.187628\pi$
$992$		−	5.00000i		−	0.158750i
$993$		−	7.00000i		−	0.222138i
$994$	10.0000			0.317181
$995$		0			0
$996$		−	6.00000i		−	0.190117i
$997$		−	18.0000i		−	0.570066i	−0.958518	−	0.285033i	$-0.907995\pi$
$997$		−	18.0000i		−	0.570066i	0.958518	−	0.285033i	$-0.0920045\pi$
$998$			6.00000i			0.189927i
$999$	10.0000			0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	850.2.b.c.101.1		2
5.2	odd	4		170.2.d.a.169.1	✓	2
5.3	odd	4		170.2.d.b.169.2	yes	2
5.4	even	2		850.2.b.i.101.2		2
15.2	even	4		1530.2.f.e.1189.2		2
15.8	even	4		1530.2.f.b.1189.1		2
17.16	even	2	inner	850.2.b.c.101.2		2
20.3	even	4		1360.2.o.a.849.1		2
20.7	even	4		1360.2.o.b.849.1		2
85.33	odd	4		170.2.d.a.169.2	yes	2
85.67	odd	4		170.2.d.b.169.1	yes	2
85.84	even	2		850.2.b.i.101.1		2
255.152	even	4		1530.2.f.b.1189.2		2
255.203	even	4		1530.2.f.e.1189.1		2
340.67	even	4		1360.2.o.a.849.2		2
340.203	even	4		1360.2.o.b.849.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.2.d.a.169.1	✓	2	5.2	odd	4
170.2.d.a.169.2	yes	2	85.33	odd	4
170.2.d.b.169.1	yes	2	85.67	odd	4
170.2.d.b.169.2	yes	2	5.3	odd	4
850.2.b.c.101.1		2	1.1	even	1	trivial
850.2.b.c.101.2		2	17.16	even	2	inner
850.2.b.i.101.1		2	85.84	even	2
850.2.b.i.101.2		2	5.4	even	2
1360.2.o.a.849.1		2	20.3	even	4
1360.2.o.a.849.2		2	340.67	even	4
1360.2.o.b.849.1		2	20.7	even	4
1360.2.o.b.849.2		2	340.203	even	4
1530.2.f.b.1189.1		2	15.8	even	4
1530.2.f.b.1189.2		2	255.152	even	4
1530.2.f.e.1189.1		2	255.203	even	4
1530.2.f.e.1189.2		2	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	850.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} +2.00000i q^{7} -1.00000 q^{8} +2.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} +2.00000i q^{7} -1.00000 q^{8} +2.00000 q^{9} -1.00000i q^{12} +1.00000 q^{13} -2.00000i q^{14} +1.00000 q^{16} +(-4.00000 + 1.00000i) q^{17} -2.00000 q^{18} +5.00000 q^{19} +2.00000 q^{21} +4.00000i q^{23} +1.00000i q^{24} -1.00000 q^{26} -5.00000i q^{27} +2.00000i q^{28} +9.00000i q^{29} +5.00000i q^{31} -1.00000 q^{32} +(4.00000 - 1.00000i) q^{34} +2.00000 q^{36} +2.00000i q^{37} -5.00000 q^{38} -1.00000i q^{39} -10.0000i q^{41} -2.00000 q^{42} +6.00000 q^{43} -4.00000i q^{46} +7.00000 q^{47} -1.00000i q^{48} +3.00000 q^{49} +(1.00000 + 4.00000i) q^{51} +1.00000 q^{52} +1.00000 q^{53} +5.00000i q^{54} -2.00000i q^{56} -5.00000i q^{57} -9.00000i q^{58} +5.00000 q^{59} +5.00000i q^{61} -5.00000i q^{62} +4.00000i q^{63} +1.00000 q^{64} +2.00000 q^{67} +(-4.00000 + 1.00000i) q^{68} +4.00000 q^{69} +5.00000i q^{71} -2.00000 q^{72} -11.0000i q^{73} -2.00000i q^{74} +5.00000 q^{76} +1.00000i q^{78} -16.0000i q^{79} +1.00000 q^{81} +10.0000i q^{82} +6.00000 q^{83} +2.00000 q^{84} -6.00000 q^{86} +9.00000 q^{87} -5.00000 q^{89} +2.00000i q^{91} +4.00000i q^{92} +5.00000 q^{93} -7.00000 q^{94} +1.00000i q^{96} +7.00000i q^{97} -3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 4 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 4 q^{9} + 2 q^{13} + 2 q^{16} - 8 q^{17} - 4 q^{18} + 10 q^{19} + 4 q^{21} - 2 q^{26} - 2 q^{32} + 8 q^{34} + 4 q^{36} - 10 q^{38} - 4 q^{42} + 12 q^{43} + 14 q^{47} + 6 q^{49} + 2 q^{51} + 2 q^{52} + 2 q^{53} + 10 q^{59} + 2 q^{64} + 4 q^{67} - 8 q^{68} + 8 q^{69} - 4 q^{72} + 10 q^{76} + 2 q^{81} + 12 q^{83} + 4 q^{84} - 12 q^{86} + 18 q^{87} - 10 q^{89} + 10 q^{93} - 14 q^{94} - 6 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 4 * q^9 + 2 * q^13 + 2 * q^16 - 8 * q^17 - 4 * q^18 + 10 * q^19 + 4 * q^21 - 2 * q^26 - 2 * q^32 + 8 * q^34 + 4 * q^36 - 10 * q^38 - 4 * q^42 + 12 * q^43 + 14 * q^47 + 6 * q^49 + 2 * q^51 + 2 * q^52 + 2 * q^53 + 10 * q^59 + 2 * q^64 + 4 * q^67 - 8 * q^68 + 8 * q^69 - 4 * q^72 + 10 * q^76 + 2 * q^81 + 12 * q^83 + 4 * q^84 - 12 * q^86 + 18 * q^87 - 10 * q^89 + 10 * q^93 - 14 * q^94 - 6 * q^98

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