LMFDB - Embedded newform 810.3.b.a.809.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,3,Mod(809,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("810.809");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	810.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:	$22.0709014132$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			809.4
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	810.809
Dual form			810.3.b.a.809.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} +(2.04989 + 4.56048i) q^{5} -11.8990i q^{7} -2.82843 q^{8} +O(q^{10})$	\(q-1.41421 q^{2} +2.00000 q^{4} +(2.04989 + 4.56048i) q^{5} -11.8990i q^{7} -2.82843 q^{8} +(-2.89898 - 6.44949i) q^{10} +6.43539i q^{11} +11.1464i q^{13} +16.8277i q^{14} +4.00000 q^{16} +20.5775 q^{17} -20.4495 q^{19} +(4.09978 + 9.12096i) q^{20} -9.10102i q^{22} +9.29593 q^{23} +(-16.5959 + 18.6969i) q^{25} -15.7634i q^{26} -23.7980i q^{28} -13.6814i q^{29} +53.8434 q^{31} -5.65685 q^{32} -29.1010 q^{34} +(54.2650 - 24.3916i) q^{35} +18.6515i q^{37} +28.9199 q^{38} +(-5.79796 - 12.8990i) q^{40} +0.460702i q^{41} -35.1918i q^{43} +12.8708i q^{44} -13.1464 q^{46} +64.3716 q^{47} -92.5857 q^{49} +(23.4702 - 26.4415i) q^{50} +22.2929i q^{52} +2.25697 q^{53} +(-29.3485 + 13.1918i) q^{55} +33.6554i q^{56} +19.3485i q^{58} +45.8263i q^{59} +9.89898 q^{61} -76.1460 q^{62} +8.00000 q^{64} +(-50.8330 + 22.8489i) q^{65} -41.9898i q^{67} +41.1551 q^{68} +(-76.7423 + 34.4949i) q^{70} +121.672i q^{71} +11.1010i q^{73} -26.3772i q^{74} -40.8990 q^{76} +76.5746 q^{77} +145.485 q^{79} +(8.19955 + 18.2419i) q^{80} -0.651531i q^{82} +21.2453 q^{83} +(42.1816 + 93.8434i) q^{85} +49.7688i q^{86} -18.2020i q^{88} +73.2999i q^{89} +132.631 q^{91} +18.5919 q^{92} -91.0352 q^{94} +(-41.9192 - 93.2594i) q^{95} -78.7878i q^{97} +130.936 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4}+O(q^{10}) $ 8 * q + 16 * q^4	$ 8 q + 16 q^{4} + 16 q^{10} + 32 q^{16} - 144 q^{19} + 24 q^{25} + 176 q^{31} - 272 q^{34} + 32 q^{40} + 32 q^{46} - 192 q^{49} - 176 q^{55} + 40 q^{61} + 64 q^{64} - 320 q^{70} - 288 q^{76} + 576 q^{79} - 368 q^{85} + 336 q^{91} - 160 q^{94}+O(q^{100}) $ 8 * q + 16 * q^4 + 16 * q^10 + 32 * q^16 - 144 * q^19 + 24 * q^25 + 176 * q^31 - 272 * q^34 + 32 * q^40 + 32 * q^46 - 192 * q^49 - 176 * q^55 + 40 * q^61 + 64 * q^64 - 320 * q^70 - 288 * q^76 + 576 * q^79 - 368 * q^85 + 336 * q^91 - 160 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$	2.04989	+	4.56048i	0.409978	+	0.912096i
$5$	2.04989	+	4.56048i	0.409978	+	0.912096i
$6$		0			0
$7$		−	11.8990i		−	1.69985i	−0.526900	−	0.849927i	$-0.676646\pi$
$7$		−	11.8990i		−	1.69985i	0.526900	−	0.849927i	$-0.323354\pi$
$8$	−2.82843			−0.353553
$9$		0			0
$10$	−2.89898	−	6.44949i	−0.289898	−	0.644949i
$11$			6.43539i			0.585036i	0.956260	+	0.292518i	$0.0944931\pi$
$11$			6.43539i			0.585036i	−0.956260	+	0.292518i	$0.905507\pi$
$12$		0			0
$13$			11.1464i			0.857418i	0.903443	+	0.428709i	$0.141031\pi$
$13$			11.1464i			0.857418i	−0.903443	+	0.428709i	$0.858969\pi$
$14$			16.8277i			1.20198i
$15$		0			0
$16$	4.00000			0.250000
$17$	20.5775			1.21044			0.605221	−	0.796057i	$-0.293085\pi$
$17$	20.5775			1.21044			0.605221	+	0.796057i	$0.293085\pi$
$18$		0			0
$19$	−20.4495			−1.07629			−0.538144	−	0.842853i	$-0.680875\pi$
$19$	−20.4495			−1.07629			−0.538144	+	0.842853i	$0.680875\pi$
$20$	4.09978	+	9.12096i	0.204989	+	0.456048i
$21$		0			0
$22$		−	9.10102i		−	0.413683i
$23$	9.29593			0.404171			0.202085	−	0.979368i	$-0.435228\pi$
$23$	9.29593			0.404171			0.202085	+	0.979368i	$0.435228\pi$
$24$		0			0
$25$	−16.5959	+	18.6969i	−0.663837	+	0.747878i
$26$		−	15.7634i		−	0.606286i
$27$		0			0
$28$		−	23.7980i		−	0.849927i
$29$		−	13.6814i		−	0.471774i	−0.971781	−	0.235887i	$-0.924201\pi$
$29$		−	13.6814i		−	0.471774i	0.971781	−	0.235887i	$-0.0757995\pi$
$30$		0			0
$31$	53.8434			1.73688			0.868441	−	0.495792i	$-0.165122\pi$
$31$	53.8434			1.73688			0.868441	+	0.495792i	$0.165122\pi$
$32$	−5.65685			−0.176777
$33$		0			0
$34$	−29.1010			−0.855912
$35$	54.2650	−	24.3916i	1.55043	−	0.696902i
$36$		0			0
$37$			18.6515i			0.504095i	0.967715	+	0.252048i	$0.0811040\pi$
$37$			18.6515i			0.504095i	−0.967715	+	0.252048i	$0.918896\pi$
$38$	28.9199			0.761051
$39$		0			0
$40$	−5.79796	−	12.8990i	−0.144949	−	0.322474i
$41$			0.460702i			0.0112366i	0.999984	+	0.00561831i	$0.00178837\pi$
$41$			0.460702i			0.0112366i	−0.999984	+	0.00561831i	$0.998212\pi$
$42$		0			0
$43$		−	35.1918i		−	0.818415i	−0.912441	−	0.409207i	$-0.865805\pi$
$43$		−	35.1918i		−	0.818415i	0.912441	−	0.409207i	$-0.134195\pi$
$44$			12.8708i			0.292518i
$45$		0			0
$46$	−13.1464			−0.285792
$47$	64.3716			1.36961			0.684804	−	0.728727i	$-0.259888\pi$
$47$	64.3716			1.36961			0.684804	+	0.728727i	$0.259888\pi$
$48$		0			0
$49$	−92.5857			−1.88950
$50$	23.4702	−	26.4415i	0.469403	−	0.528829i
$51$		0			0
$52$			22.2929i			0.428709i
$53$	2.25697			0.0425843			0.0212922	−	0.999773i	$-0.493222\pi$
$53$	2.25697			0.0425843			0.0212922	+	0.999773i	$0.493222\pi$
$54$		0			0
$55$	−29.3485	+	13.1918i	−0.533609	+	0.239852i
$56$			33.6554i			0.600989i
$57$		0			0
$58$			19.3485i			0.333594i
$59$			45.8263i			0.776717i	0.921508	+	0.388358i	$0.126958\pi$
$59$			45.8263i			0.776717i	−0.921508	+	0.388358i	$0.873042\pi$
$60$		0			0
$61$	9.89898			0.162278			0.0811392	−	0.996703i	$-0.474144\pi$
$61$	9.89898			0.162278			0.0811392	+	0.996703i	$0.474144\pi$
$62$	−76.1460			−1.22816
$63$		0			0
$64$	8.00000			0.125000
$65$	−50.8330	+	22.8489i	−0.782047	+	0.351522i
$66$		0			0
$67$		−	41.9898i		−	0.626713i	−0.949636	−	0.313357i	$-0.898546\pi$
$67$		−	41.9898i		−	0.626713i	0.949636	−	0.313357i	$-0.101454\pi$
$68$	41.1551			0.605221
$69$		0			0
$70$	−76.7423	+	34.4949i	−1.09632	+	0.492784i
$71$			121.672i			1.71369i	0.515572	+	0.856846i	$0.327579\pi$
$71$			121.672i			1.71369i	−0.515572	+	0.856846i	$0.672421\pi$
$72$		0			0
$73$			11.1010i			0.152069i	0.997105	+	0.0760344i	$0.0242259\pi$
$73$			11.1010i			0.152069i	−0.997105	+	0.0760344i	$0.975774\pi$
$74$		−	26.3772i		−	0.356449i
$75$		0			0
$76$	−40.8990			−0.538144
$77$	76.5746			0.994475
$78$		0			0
$79$	145.485			1.84158			0.920789	−	0.390061i	$-0.127546\pi$
$79$	145.485			1.84158			0.920789	+	0.390061i	$0.127546\pi$
$80$	8.19955	+	18.2419i	0.102494	+	0.228024i
$81$		0			0
$82$		−	0.651531i		−	0.00794550i
$83$	21.2453			0.255968			0.127984	−	0.991776i	$-0.459149\pi$
$83$	21.2453			0.255968			0.127984	+	0.991776i	$0.459149\pi$
$84$		0			0
$85$	42.1816	+	93.8434i	0.496254	+	1.10404i
$86$			49.7688i			0.578707i
$87$		0			0
$88$		−	18.2020i		−	0.206841i
$89$			73.2999i			0.823595i	0.911275	+	0.411797i	$0.135099\pi$
$89$			73.2999i			0.823595i	−0.911275	+	0.411797i	$0.864901\pi$
$90$		0			0
$91$	132.631			1.45748
$92$	18.5919			0.202085
$93$		0			0
$94$	−91.0352			−0.968460
$95$	−41.9192	−	93.2594i	−0.441254	−	0.981678i
$96$		0			0
$97$		−	78.7878i		−	0.812245i	−0.913819	−	0.406122i	$-0.866881\pi$
$97$		−	78.7878i		−	0.812245i	0.913819	−	0.406122i	$-0.133119\pi$
$98$	130.936			1.33608
$99$		0			0
$100$	−33.1918	+	37.3939i	−0.331918	+	0.373939i
$101$			144.314i			1.42885i	0.699711	+	0.714426i	$0.253312\pi$
$101$			144.314i			1.42885i	−0.699711	+	0.714426i	$0.746688\pi$
$102$		0			0
$103$		−	99.7775i		−	0.968714i	−0.874870	−	0.484357i	$-0.839054\pi$
$103$		−	99.7775i		−	0.968714i	0.874870	−	0.484357i	$-0.160946\pi$
$104$		−	31.5269i		−	0.303143i
$105$		0			0
$106$	−3.19184			−0.0301117
$107$	119.512			1.11693			0.558465	−	0.829528i	$-0.311391\pi$
$107$	119.512			1.11693			0.558465	+	0.829528i	$0.311391\pi$
$108$		0			0
$109$	151.091			1.38615			0.693077	−	0.720863i	$-0.256255\pi$
$109$	151.091			1.38615			0.693077	+	0.720863i	$0.256255\pi$
$110$	41.5050	−	18.6561i	0.377318	−	0.169601i
$111$		0			0
$112$		−	47.5959i		−	0.424964i
$113$	113.965			1.00854			0.504272	−	0.863545i	$-0.331761\pi$
$113$	113.965			1.00854			0.504272	+	0.863545i	$0.331761\pi$
$114$		0			0
$115$	19.0556	+	42.3939i	0.165701	+	0.368642i
$116$		−	27.3629i		−	0.235887i
$117$		0			0
$118$		−	64.8082i		−	0.549222i
$119$		−	244.852i		−	2.05758i
$120$		0			0
$121$	79.5857			0.657733
$122$	−13.9993			−0.114748
$123$		0			0
$124$	107.687			0.868441
$125$	−119.287	−	37.3587i	−0.954294	−	0.298869i
$126$		0			0
$127$		−	4.68673i		−	0.0369034i	−0.999830	−	0.0184517i	$-0.994126\pi$
$127$		−	4.68673i		−	0.0369034i	0.999830	−	0.0184517i	$-0.00587369\pi$
$128$	−11.3137			−0.0883883
$129$		0			0
$130$	71.8888	−	32.3133i	0.552991	−	0.248564i
$131$		−	5.37113i		−	0.0410010i	−0.999790	−	0.0205005i	$-0.993474\pi$
$131$		−	5.37113i		−	0.0410010i	0.999790	−	0.0205005i	$-0.00652596\pi$
$132$		0			0
$133$			243.328i			1.82953i
$134$			59.3825i			0.443153i
$135$		0			0
$136$	−58.2020			−0.427956
$137$	83.9956			0.613107			0.306553	−	0.951853i	$-0.400824\pi$
$137$	83.9956			0.613107			0.306553	+	0.951853i	$0.400824\pi$
$138$		0			0
$139$	68.7173			0.494369			0.247185	−	0.968968i	$-0.420495\pi$
$139$	68.7173			0.494369			0.247185	+	0.968968i	$0.420495\pi$
$140$	108.530	−	48.7832i	0.775215	−	0.348451i
$141$		0			0
$142$		−	172.070i		−	1.21176i
$143$	−71.7316			−0.501620
$144$		0			0
$145$	62.3939	−	28.0454i	0.430303	−	0.193417i
$146$		−	15.6992i		−	0.107529i
$147$		0			0
$148$			37.3031i			0.252048i
$149$			134.668i			0.903813i	0.892065	+	0.451906i	$0.149256\pi$
$149$			134.668i			0.903813i	−0.892065	+	0.451906i	$0.850744\pi$
$150$		0			0
$151$	−83.5505			−0.553315			−0.276657	−	0.960969i	$-0.589227\pi$
$151$	−83.5505			−0.553315			−0.276657	+	0.960969i	$0.589227\pi$
$152$	57.8399			0.380526
$153$		0			0
$154$	−108.293			−0.703200
$155$	110.373	+	245.551i	0.712083	+	1.58420i
$156$		0			0
$157$			133.303i			0.849064i	0.905413	+	0.424532i	$0.139561\pi$
$157$			133.303i			0.849064i	−0.905413	+	0.424532i	$0.860439\pi$
$158$	−205.746			−1.30219
$159$		0			0
$160$	−11.5959	−	25.7980i	−0.0724745	−	0.161237i
$161$		−	110.612i		−	0.687031i
$162$		0			0
$163$			196.808i			1.20741i	0.797207	+	0.603706i	$0.206310\pi$
$163$			196.808i			1.20741i	−0.797207	+	0.603706i	$0.793690\pi$
$164$			0.921404i			0.00561831i
$165$		0			0
$166$	−30.0454			−0.180996
$167$	−15.7458			−0.0942860			−0.0471430	−	0.998888i	$-0.515012\pi$
$167$	−15.7458			−0.0942860			−0.0471430	+	0.998888i	$0.515012\pi$
$168$		0			0
$169$	44.7571			0.264835
$170$	−59.6538	−	132.715i	−0.350905	−	0.780674i
$171$		0			0
$172$		−	70.3837i		−	0.409207i
$173$	−37.3410			−0.215844			−0.107922	−	0.994159i	$-0.534420\pi$
$173$	−37.3410			−0.215844			−0.107922	+	0.994159i	$0.534420\pi$
$174$		0			0
$175$	222.474	+	197.474i	1.27128	+	1.12843i
$176$			25.7416i			0.146259i
$177$		0			0
$178$		−	103.662i		−	0.582369i
$179$		−	236.209i		−	1.31960i	−0.751440	−	0.659802i	$-0.770640\pi$
$179$		−	236.209i		−	1.31960i	0.751440	−	0.659802i	$-0.229360\pi$
$180$		0			0
$181$	−265.050			−1.46436			−0.732182	−	0.681109i	$-0.761498\pi$
$181$	−265.050			−1.46436			−0.732182	+	0.681109i	$0.761498\pi$
$182$	−187.569			−1.03060
$183$		0			0
$184$	−26.2929			−0.142896
$185$	−85.0599	+	38.2336i	−0.459783	+	0.206668i
$186$		0			0
$187$			132.424i			0.708152i
$188$	128.743			0.684804
$189$		0			0
$190$	59.2827	+	131.889i	0.312014	+	0.694151i
$191$			94.5019i			0.494775i	0.968917	+	0.247387i	$0.0795720\pi$
$191$			94.5019i			0.494775i	−0.968917	+	0.247387i	$0.920428\pi$
$192$		0			0
$193$		−	42.3791i		−	0.219581i	−0.993955	−	0.109790i	$-0.964982\pi$
$193$		−	42.3791i		−	0.219581i	0.993955	−	0.109790i	$-0.0350180\pi$
$194$			111.423i			0.574344i
$195$		0			0
$196$	−185.171			−0.944752
$197$	−120.765			−0.613021			−0.306511	−	0.951867i	$-0.599161\pi$
$197$	−120.765			−0.613021			−0.306511	+	0.951867i	$0.599161\pi$
$198$		0			0
$199$	−29.4801			−0.148141			−0.0740706	−	0.997253i	$-0.523599\pi$
$199$	−29.4801			−0.148141			−0.0740706	+	0.997253i	$0.523599\pi$
$200$	46.9403	−	52.8829i	0.234702	−	0.264415i
$201$		0			0
$202$		−	204.091i		−	1.01035i
$203$	−162.795			−0.801946
$204$		0			0
$205$	−2.10102	+	0.944387i	−0.0102489	+	0.00460677i
$206$			141.107i			0.684984i
$207$		0			0
$208$			44.5857i			0.214354i
$209$		−	131.601i		−	0.629668i
$210$		0			0
$211$	30.2474			0.143353			0.0716764	−	0.997428i	$-0.477165\pi$
$211$	30.2474			0.143353			0.0716764	+	0.997428i	$0.477165\pi$
$212$	4.51394			0.0212922
$213$		0			0
$214$	−169.015			−0.789789
$215$	160.492	−	72.1393i	0.746473	−	0.335532i
$216$		0			0
$217$		−	640.681i		−	2.95245i
$218$	−213.675			−0.980159
$219$		0			0
$220$	−58.6969	+	26.3837i	−0.266804	+	0.119926i
$221$			229.366i			1.03785i
$222$		0			0
$223$			14.1010i			0.0632333i	0.999500	+	0.0316166i	$0.0100656\pi$
$223$			14.1010i			0.0632333i	−0.999500	+	0.0316166i	$0.989934\pi$
$224$			67.3108i			0.300495i
$225$		0			0
$226$	−161.171			−0.713148
$227$	−258.186			−1.13738			−0.568692	−	0.822550i	$-0.692550\pi$
$227$	−258.186			−1.13738			−0.568692	+	0.822550i	$0.692550\pi$
$228$		0			0
$229$	−92.5255			−0.404042			−0.202021	−	0.979381i	$-0.564751\pi$
$229$	−92.5255			−0.404042			−0.202021	+	0.979381i	$0.564751\pi$
$230$	−26.9487	−	59.9540i	−0.117168	−	0.260670i
$231$		0			0
$232$			38.6969i			0.166797i
$233$	−320.619			−1.37605			−0.688023	−	0.725689i	$-0.741521\pi$
$233$	−320.619			−1.37605			−0.688023	+	0.725689i	$0.741521\pi$
$234$		0			0
$235$	131.955	+	293.565i	0.561509	+	1.24921i
$236$			91.6526i			0.388358i
$237$		0			0
$238$			346.272i			1.45493i
$239$		−	325.526i		−	1.36203i	−0.732268	−	0.681017i	$-0.761538\pi$
$239$		−	325.526i		−	1.36203i	0.732268	−	0.681017i	$-0.238462\pi$
$240$		0			0
$241$	−84.7571			−0.351689			−0.175845	−	0.984418i	$-0.556266\pi$
$241$	−84.7571			−0.351689			−0.175845	+	0.984418i	$0.556266\pi$
$242$	−112.551			−0.465088
$243$		0			0
$244$	19.7980			0.0811392
$245$	−189.790	−	422.235i	−0.774654	−	1.72341i
$246$		0			0
$247$		−	227.939i		−	0.922829i
$248$	−152.292			−0.614081
$249$		0			0
$250$	168.697	+	52.8332i	0.674788	+	0.211333i
$251$			77.5314i			0.308890i	0.988001	+	0.154445i	$0.0493589\pi$
$251$			77.5314i			0.308890i	−0.988001	+	0.154445i	$0.950641\pi$
$252$		0			0
$253$			59.8230i			0.236454i
$254$			6.62804i			0.0260947i
$255$		0			0
$256$	16.0000			0.0625000
$257$	−273.857			−1.06559			−0.532795	−	0.846244i	$-0.678858\pi$
$257$	−273.857			−1.06559			−0.532795	+	0.846244i	$0.678858\pi$
$258$		0			0
$259$	221.934			0.856889
$260$	−101.666	+	45.6979i	−0.391023	+	0.175761i
$261$		0			0
$262$			7.59592i			0.0289921i
$263$	37.8483			0.143910			0.0719549	−	0.997408i	$-0.477076\pi$
$263$	37.8483			0.143910			0.0719549	+	0.997408i	$0.477076\pi$
$264$		0			0
$265$	4.62653	+	10.2929i	0.0174586	+	0.0388410i
$266$		−	344.118i		−	1.29368i
$267$		0			0
$268$		−	83.9796i		−	0.313357i
$269$		−	135.240i		−	0.502749i	−0.967890	−	0.251375i	$-0.919117\pi$
$269$		−	135.240i		−	0.502749i	0.967890	−	0.251375i	$-0.0808826\pi$
$270$		0			0
$271$	147.914			0.545807			0.272904	−	0.962041i	$-0.412016\pi$
$271$	147.914			0.545807			0.272904	+	0.962041i	$0.412016\pi$
$272$	82.3101			0.302611
$273$		0			0
$274$	−118.788			−0.433532
$275$	−120.322	−	106.801i	−0.437535	−	0.388368i
$276$		0			0
$277$		−	464.813i		−	1.67802i	−0.544113	−	0.839012i	$-0.683134\pi$
$277$		−	464.813i		−	1.67802i	0.544113	−	0.839012i	$-0.316866\pi$
$278$	−97.1810			−0.349572
$279$		0			0
$280$	−153.485	+	68.9898i	−0.548160	+	0.246392i
$281$		−	8.62490i		−	0.0306936i	−0.999882	−	0.0153468i	$-0.995115\pi$
$281$		−	8.62490i		−	0.0306936i	0.999882	−	0.0153468i	$-0.00488523\pi$
$282$		0			0
$283$		−	81.4245i		−	0.287719i	−0.989598	−	0.143860i	$-0.954049\pi$
$283$		−	81.4245i		−	0.287719i	0.989598	−	0.143860i	$-0.0459513\pi$
$284$			243.344i			0.856846i
$285$		0			0
$286$	101.444			0.354699
$287$	5.48188			0.0191006
$288$		0			0
$289$	134.435			0.465172
$290$	−88.2383	+	39.6622i	−0.304270	+	0.136766i
$291$		0			0
$292$			22.2020i			0.0760344i
$293$	517.752			1.76707			0.883535	−	0.468365i	$-0.155157\pi$
$293$	517.752			1.76707			0.883535	+	0.468365i	$0.155157\pi$
$294$		0			0
$295$	−208.990	+	93.9388i	−0.708440	+	0.318437i
$296$		−	52.7545i		−	0.178225i
$297$		0			0
$298$		−	190.449i		−	0.639092i
$299$			103.616i			0.346543i
$300$		0			0
$301$	−418.747			−1.39119
$302$	118.158			0.391253
$303$		0			0
$304$	−81.7980			−0.269072
$305$	20.2918	+	45.1441i	0.0665305	+	0.148013i
$306$		0			0
$307$		−	4.59592i		−	0.0149704i	−0.999972	−	0.00748521i	$-0.997617\pi$
$307$		−	4.59592i		−	0.0149704i	0.999972	−	0.00748521i	$-0.00238264\pi$
$308$	153.149			0.497238
$309$		0			0
$310$	−156.091	−	347.262i	−0.503519	−	1.12020i
$311$		−	33.8625i		−	0.108883i	−0.998517	−	0.0544413i	$-0.982662\pi$
$311$		−	33.8625i		−	0.108883i	0.998517	−	0.0544413i	$-0.0173378\pi$
$312$		0			0
$313$		−	117.076i		−	0.374045i	−0.982356	−	0.187022i	$-0.940116\pi$
$313$		−	117.076i		−	0.374045i	0.982356	−	0.187022i	$-0.0598837\pi$
$314$		−	188.519i		−	0.600379i
$315$		0			0
$316$	290.969			0.920789
$317$	−446.341			−1.40802			−0.704008	−	0.710192i	$-0.748608\pi$
$317$	−446.341			−1.40802			−0.704008	+	0.710192i	$0.748608\pi$
$318$		0			0
$319$	88.0454			0.276004
$320$	16.3991	+	36.4838i	0.0512472	+	0.114012i
$321$		0			0
$322$			156.429i			0.485805i
$323$	−420.800			−1.30279
$324$		0			0
$325$	−208.404	−	184.985i	−0.641243	−	0.569185i
$326$		−	278.329i		−	0.853769i
$327$		0			0
$328$		−	1.30306i		−	0.00397275i
$329$		−	765.956i		−	2.32814i
$330$		0			0
$331$	519.333			1.56898			0.784490	−	0.620141i	$-0.212925\pi$
$331$	519.333			1.56898			0.784490	+	0.620141i	$0.212925\pi$
$332$	42.4906			0.127984
$333$		0			0
$334$	22.2679			0.0666702
$335$	191.494	−	86.0744i	0.571622	−	0.256938i
$336$		0			0
$337$		−	580.595i		−	1.72283i	−0.507899	−	0.861417i	$-0.669578\pi$
$337$		−	580.595i		−	1.72283i	0.507899	−	0.861417i	$-0.330422\pi$
$338$	−63.2962			−0.187267
$339$		0			0
$340$	84.3633	+	187.687i	0.248127	+	0.552020i
$341$			346.503i			1.01614i
$342$		0			0
$343$			518.626i			1.51203i
$344$			99.5375i			0.289353i
$345$		0			0
$346$	52.8082			0.152625
$347$	−260.437			−0.750538			−0.375269	−	0.926916i	$-0.622450\pi$
$347$	−260.437			−0.750538			−0.375269	+	0.926916i	$0.622450\pi$
$348$		0			0
$349$	−641.170			−1.83716			−0.918582	−	0.395230i	$-0.870665\pi$
$349$	−641.170			−1.83716			−0.918582	+	0.395230i	$0.870665\pi$
$350$	−314.626	−	279.271i	−0.898933	−	0.797917i
$351$		0			0
$352$		−	36.4041i		−	0.103421i
$353$	354.224			1.00347			0.501734	−	0.865022i	$-0.332696\pi$
$353$	354.224			1.00347			0.501734	+	0.865022i	$0.332696\pi$
$354$		0			0
$355$	−554.883	+	249.414i	−1.56305	+	0.702575i
$356$			146.600i			0.411797i
$357$		0			0
$358$			334.050i			0.933101i
$359$			388.215i			1.08138i	0.841222	+	0.540690i	$0.181837\pi$
$359$			388.215i			1.08138i	−0.841222	+	0.540690i	$0.818163\pi$
$360$		0			0
$361$	57.1816			0.158398
$362$	374.837			1.03546
$363$		0			0
$364$	265.262			0.728742
$365$	−50.6260	+	22.7558i	−0.138701	+	0.0623448i
$366$		0			0
$367$		−	475.292i		−	1.29507i	−0.762034	−	0.647537i	$-0.775799\pi$
$367$		−	475.292i		−	1.29507i	0.762034	−	0.647537i	$-0.224201\pi$
$368$	37.1837			0.101043
$369$		0			0
$370$	120.293	−	54.0704i	0.325116	−	0.146136i
$371$		−	26.8556i		−	0.0723871i
$372$		0			0
$373$			373.312i			1.00084i	0.865784	+	0.500419i	$0.166821\pi$
$373$			373.312i			1.00084i	−0.865784	+	0.500419i	$0.833179\pi$
$374$		−	187.277i		−	0.500739i
$375$		0			0
$376$	−182.070			−0.484230
$377$	152.499			0.404507
$378$		0			0
$379$	−348.899			−0.920578			−0.460289	−	0.887769i	$-0.652254\pi$
$379$	−348.899			−0.920578			−0.460289	+	0.887769i	$0.652254\pi$
$380$	−83.8383	−	186.519i	−0.220627	−	0.490839i
$381$		0			0
$382$		−	133.646i		−	0.349858i
$383$	−345.103			−0.901053			−0.450527	−	0.892763i	$-0.648764\pi$
$383$	−345.103			−0.901053			−0.450527	+	0.892763i	$0.648764\pi$
$384$		0			0
$385$	156.969	+	349.217i	0.407713	+	0.907057i
$386$			59.9331i			0.155267i
$387$		0			0
$388$		−	157.576i		−	0.406122i
$389$			320.388i			0.823618i	0.911270	+	0.411809i	$0.135103\pi$
$389$			320.388i			0.823618i	−0.911270	+	0.411809i	$0.864897\pi$
$390$		0			0
$391$	191.287			0.489226
$392$	261.872			0.668041
$393$		0			0
$394$	170.788			0.433471
$395$	298.227	+	663.480i	0.755006	+	1.67970i
$396$		0			0
$397$			205.344i			0.517239i	0.965979	+	0.258619i	$0.0832676\pi$
$397$			205.344i			0.517239i	−0.965979	+	0.258619i	$0.916732\pi$
$398$	41.6912			0.104752
$399$		0			0
$400$	−66.3837	+	74.7878i	−0.165959	+	0.186969i
$401$			411.329i			1.02576i	0.858461	+	0.512879i	$0.171421\pi$
$401$			411.329i			1.02576i	−0.858461	+	0.512879i	$0.828579\pi$
$402$		0			0
$403$			600.161i			1.48923i
$404$			288.628i			0.714426i
$405$		0			0
$406$	230.227			0.567062
$407$	−120.030			−0.294914
$408$		0			0
$409$	331.959			0.811636			0.405818	−	0.913954i	$-0.366987\pi$
$409$	331.959			0.811636			0.405818	+	0.913954i	$0.366987\pi$
$410$	2.97129	−	1.33557i	0.00724705	−	0.00325748i
$411$		0			0
$412$		−	199.555i		−	0.484357i
$413$	545.286			1.32031
$414$		0			0
$415$	43.5505	+	96.8888i	0.104941	+	0.233467i
$416$		−	63.0537i		−	0.151571i
$417$		0			0
$418$			186.111i			0.445242i
$419$			727.726i			1.73682i	0.495850	+	0.868408i	$0.334857\pi$
$419$			727.726i			1.73682i	−0.495850	+	0.868408i	$0.665143\pi$
$420$		0			0
$421$	141.171			0.335324			0.167662	−	0.985845i	$-0.446378\pi$
$421$	141.171			0.335324			0.167662	+	0.985845i	$0.446378\pi$
$422$	−42.7764			−0.101366
$423$		0			0
$424$	−6.38367			−0.0150558
$425$	−341.503	+	384.737i	−0.803536	+	0.905263i
$426$		0			0
$427$		−	117.788i		−	0.275850i
$428$	239.023			0.558465
$429$		0			0
$430$	−226.969	+	102.020i	−0.527836	+	0.237257i
$431$		−	650.423i		−	1.50910i	−0.656242	−	0.754551i	$-0.727855\pi$
$431$		−	650.423i		−	1.50910i	0.656242	−	0.754551i	$-0.272145\pi$
$432$		0			0
$433$			68.3179i			0.157778i	0.996883	+	0.0788890i	$0.0251373\pi$
$433$			68.3179i			0.157778i	−0.996883	+	0.0788890i	$0.974863\pi$
$434$			906.060i			2.08770i
$435$		0			0
$436$	302.182			0.693077
$437$	−190.097			−0.435005
$438$		0			0
$439$	−440.908			−1.00435			−0.502173	−	0.864767i	$-0.667466\pi$
$439$	−440.908			−1.00435			−0.502173	+	0.864767i	$0.667466\pi$
$440$	83.0100	−	37.3121i	0.188659	−	0.0848003i
$441$		0			0
$442$		−	324.372i		−	0.733874i
$443$	−749.837			−1.69263			−0.846317	−	0.532680i	$-0.821185\pi$
$443$	−749.837			−1.69263			−0.846317	+	0.532680i	$0.821185\pi$
$444$		0			0
$445$	−334.283	+	150.257i	−0.751197	+	0.337655i
$446$		−	19.9419i		−	0.0447127i
$447$		0			0
$448$		−	95.1918i		−	0.212482i
$449$		−	802.901i		−	1.78820i	−0.447869	−	0.894099i	$-0.647817\pi$
$449$		−	802.901i		−	1.78820i	0.447869	−	0.894099i	$-0.352183\pi$
$450$		0			0
$451$	−2.96480			−0.00657383
$452$	227.931			0.504272
$453$		0			0
$454$	365.131			0.804252
$455$	271.879	+	604.861i	0.597536	+	1.32937i
$456$		0			0
$457$		−	454.025i		−	0.993490i	−0.867896	−	0.496745i	$-0.834528\pi$
$457$		−	454.025i		−	0.993490i	0.867896	−	0.496745i	$-0.165472\pi$
$458$	130.851			0.285701
$459$		0			0
$460$	38.1112	+	84.7878i	0.0828505	+	0.184321i
$461$		−	506.248i		−	1.09815i	−0.835772	−	0.549076i	$-0.814980\pi$
$461$		−	506.248i		−	1.09815i	0.835772	−	0.549076i	$-0.185020\pi$
$462$		0			0
$463$			494.363i			1.06774i	0.845567	+	0.533870i	$0.179263\pi$
$463$			494.363i			1.06774i	−0.845567	+	0.533870i	$0.820737\pi$
$464$		−	54.7257i		−	0.117943i
$465$		0			0
$466$	453.423			0.973012
$467$	−390.044			−0.835211			−0.417606	−	0.908628i	$-0.637131\pi$
$467$	−390.044			−0.835211			−0.417606	+	0.908628i	$0.637131\pi$
$468$		0			0
$469$	−499.636			−1.06532
$470$	−186.612	−	415.164i	−0.397047	−	0.883328i
$471$		0			0
$472$		−	129.616i		−	0.274611i
$473$	226.473			0.478802
$474$		0			0
$475$	339.378	−	382.343i	0.714480	−	0.804932i
$476$		−	489.703i		−	1.02879i
$477$		0			0
$478$			460.363i			0.963103i
$479$		−	185.861i		−	0.388018i	−0.981000	−	0.194009i	$-0.937851\pi$
$479$		−	185.861i		−	0.388018i	0.981000	−	0.194009i	$-0.0621492\pi$
$480$		0			0
$481$	−207.898			−0.432220
$482$	119.865			0.248682
$483$		0			0
$484$	159.171			0.328867
$485$	359.310	−	161.506i	0.740845	−	0.333002i
$486$		0			0
$487$		−	828.261i		−	1.70074i	−0.526184	−	0.850371i	$-0.676378\pi$
$487$		−	828.261i		−	1.70074i	0.526184	−	0.850371i	$-0.323622\pi$
$488$	−27.9985			−0.0573741
$489$		0			0
$490$	268.404	+	597.131i	0.547763	+	1.21863i
$491$		−	253.072i		−	0.515422i	−0.966222	−	0.257711i	$-0.917032\pi$
$491$		−	253.072i		−	0.515422i	0.966222	−	0.257711i	$-0.0829682\pi$
$492$		0			0
$493$		−	281.530i		−	0.571055i
$494$			322.354i			0.652539i
$495$		0			0
$496$	215.373			0.434221
$497$	1447.77			2.91303
$498$		0			0
$499$	975.614			1.95514			0.977569	−	0.210614i	$-0.0675462\pi$
$499$	975.614			1.95514			0.977569	+	0.210614i	$0.0675462\pi$
$500$	−238.573	−	74.7174i	−0.477147	−	0.149435i
$501$		0			0
$502$		−	109.646i		−	0.218418i
$503$	440.360			0.875467			0.437733	−	0.899105i	$-0.355781\pi$
$503$	440.360			0.875467			0.437733	+	0.899105i	$0.355781\pi$
$504$		0			0
$505$	−658.141	+	295.828i	−1.30325	+	0.585797i
$506$		−	84.6024i		−	0.167198i
$507$		0			0
$508$		−	9.37347i		−	0.0184517i
$509$		−	714.409i		−	1.40355i	−0.712397	−	0.701777i	$-0.752390\pi$
$509$		−	714.409i		−	1.40355i	0.712397	−	0.701777i	$-0.247610\pi$
$510$		0			0
$511$	132.091			0.258495
$512$	−22.6274			−0.0441942
$513$		0			0
$514$	387.292			0.753486
$515$	455.033	−	204.533i	0.883560	−	0.397151i
$516$		0			0
$517$			414.257i			0.801270i
$518$	−313.862			−0.605912
$519$		0			0
$520$	143.778	−	64.6265i	0.276495	−	0.124282i
$521$			328.944i			0.631370i	0.948864	+	0.315685i	$0.102234\pi$
$521$			328.944i			0.631370i	−0.948864	+	0.315685i	$0.897766\pi$
$522$		0			0
$523$		−	416.535i		−	0.796433i	−0.917291	−	0.398217i	$-0.869629\pi$
$523$		−	416.535i		−	0.796433i	0.917291	−	0.398217i	$-0.130371\pi$
$524$		−	10.7423i		−	0.0205005i
$525$		0			0
$526$	−53.5255			−0.101760
$527$	1107.96			2.10240
$528$		0			0
$529$	−442.586			−0.836646
$530$	−6.54291	−	14.5563i	−0.0123451	−	0.0274647i
$531$		0			0
$532$			486.656i			0.914767i
$533$	−5.13518			−0.00963448
$534$		0			0
$535$	244.985	+	545.030i	0.457916	+	1.01875i
$536$			118.765i			0.221577i
$537$		0			0
$538$			191.258i			0.355497i
$539$		−	595.825i		−	1.10543i
$540$		0			0
$541$	161.070			0.297727			0.148864	−	0.988858i	$-0.452438\pi$
$541$	161.070			0.297727			0.148864	+	0.988858i	$0.452438\pi$
$542$	−209.182			−0.385944
$543$		0			0
$544$	−116.404			−0.213978
$545$	309.719	+	689.046i	0.568292	+	1.26431i
$546$		0			0
$547$			474.373i			0.867228i	0.901099	+	0.433614i	$0.142762\pi$
$547$			474.373i			0.867228i	−0.901099	+	0.433614i	$0.857238\pi$
$548$	167.991			0.306553
$549$		0			0
$550$	170.161	+	151.040i	0.309384	+	0.274618i
$551$			279.778i			0.507765i
$552$		0			0
$553$		−	1731.12i		−	3.13041i
$554$			657.344i			1.18654i
$555$		0			0
$556$	137.435			0.247185
$557$	4.98583			0.00895122			0.00447561	−	0.999990i	$-0.498575\pi$
$557$	4.98583			0.00895122			0.00447561	+	0.999990i	$0.498575\pi$
$558$		0			0
$559$	392.263			0.701723
$560$	217.060	−	97.5663i	0.387607	−	0.174226i
$561$		0			0
$562$			12.1975i			0.0217037i
$563$	−281.753			−0.500449			−0.250225	−	0.968188i	$-0.580504\pi$
$563$	−281.753			−0.500449			−0.250225	+	0.968188i	$0.580504\pi$
$564$		0			0
$565$	233.616	+	519.737i	0.413480	+	0.919888i
$566$			115.152i			0.203448i
$567$		0			0
$568$		−	344.141i		−	0.605882i
$569$			471.654i			0.828917i	0.910068	+	0.414459i	$0.136029\pi$
$569$			471.654i			0.828917i	−0.910068	+	0.414459i	$0.863971\pi$
$570$		0			0
$571$	464.404			0.813317			0.406659	−	0.913580i	$-0.366694\pi$
$571$	464.404			0.813317			0.406659	+	0.913580i	$0.366694\pi$
$572$	−143.463			−0.250810
$573$		0			0
$574$	−7.75255			−0.0135062
$575$	−154.274	+	173.805i	−0.268303	+	0.302270i
$576$		0			0
$577$			393.090i			0.681265i	0.940197	+	0.340632i	$0.110641\pi$
$577$			393.090i			0.681265i	−0.940197	+	0.340632i	$0.889359\pi$
$578$	−190.119			−0.328926
$579$		0			0
$580$	124.788	−	56.0908i	0.215151	−	0.0967083i
$581$		−	252.798i		−	0.435108i
$582$		0			0
$583$			14.5245i			0.0249133i
$584$		−	31.3984i		−	0.0537644i
$585$		0			0
$586$	−732.211			−1.24951
$587$	48.4444			0.0825288			0.0412644	−	0.999148i	$-0.486861\pi$
$587$	48.4444			0.0825288			0.0412644	+	0.999148i	$0.486861\pi$
$588$		0			0
$589$	−1101.07			−1.86939
$590$	295.556	−	132.849i	0.500943	−	0.225169i
$591$		0			0
$592$			74.6061i			0.126024i
$593$	474.853			0.800764			0.400382	−	0.916348i	$-0.368877\pi$
$593$	474.853			0.800764			0.400382	+	0.916348i	$0.368877\pi$
$594$		0			0
$595$	1116.64	−	501.918i	1.87671	−	0.843560i
$596$			269.336i			0.451906i
$597$		0			0
$598$		−	146.536i		−	0.245043i
$599$		−	274.329i		−	0.457978i	−0.973429	−	0.228989i	$-0.926458\pi$
$599$		−	274.329i		−	0.457978i	0.973429	−	0.228989i	$-0.0735419\pi$
$600$		0			0
$601$	−1165.13			−1.93865			−0.969327	−	0.245776i	$-0.920957\pi$
$601$	−1165.13			−1.93865			−0.969327	+	0.245776i	$0.920957\pi$
$602$	592.198			0.983717
$603$		0			0
$604$	−167.101			−0.276657
$605$	163.142	+	362.949i	0.269656	+	0.599916i
$606$		0			0
$607$		−	843.615i		−	1.38981i	−0.719101	−	0.694906i	$-0.755446\pi$
$607$		−	843.615i		−	1.38981i	0.719101	−	0.694906i	$-0.244554\pi$
$608$	115.680			0.190263
$609$		0			0
$610$	−28.6969	−	63.8434i	−0.0470442	−	0.104661i
$611$			717.514i			1.17433i
$612$		0			0
$613$		−	443.551i		−	0.723573i	−0.932261	−	0.361787i	$-0.882167\pi$
$613$		−	443.551i		−	0.723573i	0.932261	−	0.361787i	$-0.117833\pi$
$614$			6.49961i			0.0105857i
$615$		0			0
$616$	−216.586			−0.351600
$617$	−826.928			−1.34024			−0.670120	−	0.742253i	$-0.733757\pi$
$617$	−826.928			−1.34024			−0.670120	+	0.742253i	$0.733757\pi$
$618$		0			0
$619$	213.753			0.345319			0.172660	−	0.984982i	$-0.444764\pi$
$619$	213.753			0.345319			0.172660	+	0.984982i	$0.444764\pi$
$620$	220.746	+	491.103i	0.356042	+	0.792102i
$621$		0			0
$622$			47.8888i			0.0769916i
$623$	872.194			1.39999
$624$		0			0
$625$	−74.1510	−	620.586i	−0.118642	−	0.992937i
$626$			165.570i			0.264490i
$627$		0			0
$628$			266.606i			0.424532i
$629$			383.802i			0.610179i
$630$		0			0
$631$	−292.150			−0.462995			−0.231498	−	0.972835i	$-0.574363\pi$
$631$	−292.150			−0.462995			−0.231498	+	0.972835i	$0.574363\pi$
$632$	−411.493			−0.651096
$633$		0			0
$634$	631.221			0.995617
$635$	21.3737	−	9.60728i	0.0336594	−	0.0151296i
$636$		0			0
$637$		−	1032.00i		−	1.62009i
$638$	−124.515			−0.195165
$639$		0			0
$640$	−23.1918	−	51.5959i	−0.0362372	−	0.0806186i
$641$			162.259i			0.253134i	0.991958	+	0.126567i	$0.0403959\pi$
$641$			162.259i			0.253134i	−0.991958	+	0.126567i	$0.959604\pi$
$642$		0			0
$643$			733.020i			1.14000i	0.821644	+	0.570000i	$0.193057\pi$
$643$			733.020i			1.14000i	−0.821644	+	0.570000i	$0.806943\pi$
$644$		−	221.224i		−	0.343516i
$645$		0			0
$646$	595.101			0.921209
$647$	196.964			0.304427			0.152213	−	0.988348i	$-0.451360\pi$
$647$	196.964			0.304427			0.152213	+	0.988348i	$0.451360\pi$
$648$		0			0
$649$	−294.910			−0.454407
$650$	294.728	+	261.609i	0.453428	+	0.402475i
$651$		0			0
$652$			393.616i			0.603706i
$653$	444.084			0.680067			0.340034	−	0.940413i	$-0.389562\pi$
$653$	444.084			0.680067			0.340034	+	0.940413i	$0.389562\pi$
$654$		0			0
$655$	24.4949	−	11.0102i	0.0373968	−	0.0168095i
$656$			1.84281i			0.00280916i
$657$		0			0
$658$			1083.23i			1.64624i
$659$			333.854i			0.506607i	0.967387	+	0.253303i	$0.0815171\pi$
$659$			333.854i			0.506607i	−0.967387	+	0.253303i	$0.918483\pi$
$660$		0			0
$661$	−492.041			−0.744389			−0.372194	−	0.928155i	$-0.621394\pi$
$661$	−492.041			−0.744389			−0.372194	+	0.928155i	$0.621394\pi$
$662$	−734.447			−1.10944
$663$		0			0
$664$	−60.0908			−0.0904982
$665$	−1109.69	+	498.795i	−1.66871	+	0.750068i
$666$		0			0
$667$		−	127.182i		−	0.190677i
$668$	−31.4915			−0.0471430
$669$		0			0
$670$	−270.813	+	121.728i	−0.404198	+	0.181683i
$671$			63.7038i			0.0949386i
$672$		0			0
$673$			757.448i			1.12548i	0.826634	+	0.562740i	$0.190253\pi$
$673$			757.448i			1.12548i	−0.826634	+	0.562740i	$0.809747\pi$
$674$			821.085i			1.21823i
$675$		0			0
$676$	89.5143			0.132418
$677$	−586.863			−0.866859			−0.433429	−	0.901188i	$-0.642697\pi$
$677$	−586.863			−0.866859			−0.433429	+	0.901188i	$0.642697\pi$
$678$		0			0
$679$	−937.494			−1.38070
$680$	−119.308	−	265.429i	−0.175452	−	0.390337i
$681$		0			0
$682$		−	490.030i		−	0.718518i
$683$	−759.768			−1.11240			−0.556199	−	0.831049i	$-0.687741\pi$
$683$	−759.768			−1.11240			−0.556199	+	0.831049i	$0.687741\pi$
$684$		0			0
$685$	172.182	+	383.060i	0.251360	+	0.559212i
$686$		−	733.447i		−	1.06917i
$687$		0			0
$688$		−	140.767i		−	0.204604i
$689$			25.1571i			0.0365125i
$690$		0			0
$691$	−457.839			−0.662574			−0.331287	−	0.943530i	$-0.607483\pi$
$691$	−457.839			−0.662574			−0.331287	+	0.943530i	$0.607483\pi$
$692$	−74.6820			−0.107922
$693$		0			0
$694$	368.313			0.530711
$695$	140.863	+	313.384i	0.202680	+	0.450912i
$696$		0			0
$697$			9.48011i			0.0136013i
$698$	906.752			1.29907
$699$		0			0
$700$	444.949	+	394.949i	0.635641	+	0.564213i
$701$			196.778i			0.280711i	0.990101	+	0.140355i	$0.0448245\pi$
$701$			196.778i			0.280711i	−0.990101	+	0.140355i	$0.955176\pi$
$702$		0			0
$703$		−	381.414i		−	0.542552i
$704$			51.4831i			0.0731295i
$705$		0			0
$706$	−500.949			−0.709559
$707$	1717.19			2.42884
$708$		0			0
$709$	480.848			0.678206			0.339103	−	0.940749i	$-0.389877\pi$
$709$	480.848			0.678206			0.339103	+	0.940749i	$0.389877\pi$
$710$	784.723	−	352.725i	1.10524	−	0.496796i
$711$		0			0
$712$		−	207.323i		−	0.291185i
$713$	500.524			0.701997
$714$		0			0
$715$	−147.042	−	327.131i	−0.205653	−	0.457525i
$716$		−	472.418i		−	0.659802i
$717$		0			0
$718$		−	549.019i		−	0.764651i
$719$			156.106i			0.217116i	0.994090	+	0.108558i	$0.0346232\pi$
$719$			156.106i			0.217116i	−0.994090	+	0.108558i	$0.965377\pi$
$720$		0			0
$721$	−1187.25			−1.64667
$722$	−80.8670			−0.112004
$723$		0			0
$724$	−530.100			−0.732182
$725$	255.801	+	227.056i	0.352829	+	0.313181i
$726$		0			0
$727$			1396.48i			1.92089i	0.278475	+	0.960443i	$0.410171\pi$
$727$			1396.48i			1.92089i	−0.278475	+	0.960443i	$0.589829\pi$
$728$	−375.137			−0.515299
$729$		0			0
$730$	71.5959	−	32.1816i	0.0980766	−	0.0440844i
$731$		−	724.161i		−	0.990644i
$732$		0			0
$733$			1068.01i			1.45704i	0.685025	+	0.728519i	$0.259791\pi$
$733$			1068.01i			1.45704i	−0.685025	+	0.728519i	$0.740209\pi$
$734$			672.164i			0.915755i
$735$		0			0
$736$	−52.5857			−0.0714480
$737$	270.221			0.366650
$738$		0			0
$739$	719.464			0.973565			0.486782	−	0.873523i	$-0.338170\pi$
$739$	719.464			0.973565			0.486782	+	0.873523i	$0.338170\pi$
$740$	−170.120	+	76.4671i	−0.229892	+	0.103334i
$741$		0			0
$742$			37.9796i			0.0511854i
$743$	624.551			0.840580			0.420290	−	0.907390i	$-0.361928\pi$
$743$	624.551			0.840580			0.420290	+	0.907390i	$0.361928\pi$
$744$		0			0
$745$	−614.151	+	276.055i	−0.824364	+	0.370543i
$746$		−	527.943i		−	0.707699i
$747$		0			0
$748$			264.849i			0.354076i
$749$		−	1422.06i		−	1.89862i
$750$		0			0
$751$	−1348.51			−1.79562			−0.897809	−	0.440386i	$-0.854842\pi$
$751$	−1348.51			−1.79562			−0.897809	+	0.440386i	$0.854842\pi$
$752$	257.486			0.342402
$753$		0			0
$754$	−215.666			−0.286030
$755$	−171.269	−	381.030i	−0.226847	−	0.504676i
$756$		0			0
$757$			19.4689i			0.0257185i	0.999917	+	0.0128592i	$0.00409333\pi$
$757$			19.4689i			0.0257185i	−0.999917	+	0.0128592i	$0.995907\pi$
$758$	493.418			0.650947
$759$		0			0
$760$	118.565	+	263.778i	0.156007	+	0.347076i
$761$		−	860.094i		−	1.13022i	−0.825017	−	0.565108i	$-0.808835\pi$
$761$		−	860.094i		−	1.13022i	0.825017	−	0.565108i	$-0.191165\pi$
$762$		0			0
$763$		−	1797.83i		−	2.35626i
$764$			189.004i			0.247387i
$765$		0			0
$766$	488.050			0.637141
$767$	−510.799			−0.665971
$768$		0			0
$769$	393.284			0.511422			0.255711	−	0.966753i	$-0.417690\pi$
$769$	393.284			0.511422			0.255711	+	0.966753i	$0.417690\pi$
$770$	−221.988	−	493.867i	−0.288296	−	0.641386i
$771$		0			0
$772$		−	84.7582i		−	0.109790i
$773$	626.967			0.811083			0.405541	−	0.914077i	$-0.367083\pi$
$773$	626.967			0.811083			0.405541	+	0.914077i	$0.367083\pi$
$774$		0			0
$775$	−893.580	+	1006.71i	−1.15301	+	1.29898i
$776$			222.845i			0.287172i
$777$		0			0
$778$		−	453.096i		−	0.582386i
$779$		−	9.42112i		−	0.0120939i
$780$		0			0
$781$	−783.008			−1.00257
$782$	−270.521			−0.345935
$783$		0			0
$784$	−370.343			−0.472376
$785$	−607.926	+	273.256i	−0.774428	+	0.348097i
$786$		0			0
$787$			69.7571i			0.0886368i	0.999017	+	0.0443184i	$0.0141116\pi$
$787$			69.7571i			0.0886368i	−0.999017	+	0.0443184i	$0.985888\pi$
$788$	−241.530			−0.306511
$789$		0			0
$790$	−421.757	−	938.302i	−0.533870	−	1.18772i
$791$		−	1356.07i		−	1.71438i
$792$		0			0
$793$			110.338i			0.139140i
$794$		−	290.400i		−	0.365743i
$795$		0			0
$796$	−58.9602			−0.0740706
$797$	−1123.45			−1.40960			−0.704799	−	0.709407i	$-0.748963\pi$
$797$	−1123.45			−1.40960			−0.704799	+	0.709407i	$0.748963\pi$
$798$		0			0
$799$	1324.61			1.65783
$800$	93.8807	−	105.766i	0.117351	−	0.132207i
$801$		0			0
$802$		−	581.707i		−	0.725321i
$803$	−71.4394			−0.0889657
$804$		0			0
$805$	504.444	−	226.742i	0.626638	−	0.281668i
$806$		−	848.756i		−	1.05305i
$807$		0			0
$808$		−	408.182i		−	0.505175i
$809$			182.198i			0.225213i	0.993640	+	0.112607i	$0.0359200\pi$
$809$			182.198i			0.225213i	−0.993640	+	0.112607i	$0.964080\pi$
$810$		0			0
$811$	831.160			1.02486			0.512429	−	0.858729i	$-0.328746\pi$
$811$	831.160			1.02486			0.512429	+	0.858729i	$0.328746\pi$
$812$	−325.590			−0.400973
$813$		0			0
$814$	169.748			0.208536
$815$	−897.539	+	403.435i	−1.10128	+	0.495012i
$816$		0			0
$817$			719.655i			0.880851i
$818$	−469.461			−0.573913
$819$		0			0
$820$	−4.20204	+	1.88877i	−0.00512444	+	0.00230338i
$821$			767.607i			0.934965i	0.884002	+	0.467483i	$0.154839\pi$
$821$			767.607i			0.934965i	−0.884002	+	0.467483i	$0.845161\pi$
$822$		0			0
$823$			420.948i			0.511480i	0.966746	+	0.255740i	$0.0823191\pi$
$823$			420.948i			0.511480i	−0.966746	+	0.255740i	$0.917681\pi$
$824$			282.214i			0.342492i
$825$		0			0
$826$	−771.151			−0.933597
$827$	214.208			0.259018			0.129509	−	0.991578i	$-0.458660\pi$
$827$	214.208			0.259018			0.129509	+	0.991578i	$0.458660\pi$
$828$		0			0
$829$	−669.311			−0.807372			−0.403686	−	0.914898i	$-0.632271\pi$
$829$	−669.311			−0.807372			−0.403686	+	0.914898i	$0.632271\pi$
$830$	−61.5897	−	137.021i	−0.0742045	−	0.165086i
$831$		0			0
$832$			89.1714i			0.107177i
$833$	−1905.19			−2.28714
$834$		0			0
$835$	−32.2770	−	71.8082i	−0.0386551	−	0.0859978i
$836$		−	263.201i		−	0.314834i
$837$		0			0
$838$		−	1029.16i		−	1.22811i
$839$		−	1031.18i		−	1.22906i	−0.788893	−	0.614531i	$-0.789345\pi$
$839$		−	1031.18i		−	1.22906i	0.788893	−	0.614531i	$-0.210655\pi$
$840$		0			0
$841$	653.818			0.777430
$842$	−199.647			−0.237110
$843$		0			0
$844$	60.4949			0.0716764
$845$	91.7471	+	204.114i	0.108576	+	0.241555i
$846$		0			0
$847$		−	946.989i		−	1.11805i
$848$	9.02788			0.0106461
$849$		0			0
$850$	482.958	−	544.100i	0.568186	−	0.640118i
$851$			173.383i			0.203741i
$852$		0			0
$853$		−	511.033i		−	0.599101i	−0.954080	−	0.299550i	$-0.903163\pi$
$853$		−	511.033i		−	0.599101i	0.954080	−	0.299550i	$-0.0968367\pi$
$854$			166.577i			0.195055i
$855$		0			0
$856$	−338.030			−0.394894
$857$	200.281			0.233700			0.116850	−	0.993150i	$-0.462720\pi$
$857$	200.281			0.233700			0.116850	+	0.993150i	$0.462720\pi$
$858$		0			0
$859$	194.924			0.226920			0.113460	−	0.993543i	$-0.463807\pi$
$859$	194.924			0.226920			0.113460	+	0.993543i	$0.463807\pi$
$860$	320.983	−	144.279i	0.373236	−	0.167766i
$861$		0			0
$862$			919.837i			1.06710i
$863$	−400.540			−0.464125			−0.232063	−	0.972701i	$-0.574547\pi$
$863$	−400.540			−0.464125			−0.232063	+	0.972701i	$0.574547\pi$
$864$		0			0
$865$	−76.5449	−	170.293i	−0.0884912	−	0.196870i
$866$		−	96.6160i		−	0.111566i
$867$		0			0
$868$		−	1281.36i		−	1.47622i
$869$			936.251i			1.07739i
$870$		0			0
$871$	468.036			0.537355
$872$	−427.349			−0.490080
$873$		0			0
$874$	268.838			0.307595
$875$	−444.530	+	1419.39i	−0.508035	+	1.62216i
$876$		0			0
$877$		−	146.025i		−	0.166505i	−0.996528	−	0.0832526i	$-0.973469\pi$
$877$		−	146.025i		−	0.166505i	0.996528	−	0.0832526i	$-0.0265308\pi$
$878$	623.538			0.710180
$879$		0			0
$880$	−117.394	+	52.7673i	−0.133402	+	0.0599629i
$881$		−	350.906i		−	0.398305i	−0.979969	−	0.199152i	$-0.936181\pi$
$881$		−	350.906i		−	0.398305i	0.979969	−	0.199152i	$-0.0638189\pi$
$882$		0			0
$883$			215.102i			0.243604i	0.992554	+	0.121802i	$0.0388672\pi$
$883$			215.102i			0.243604i	−0.992554	+	0.121802i	$0.961133\pi$
$884$			458.732i			0.518927i
$885$		0			0
$886$	1060.43			1.19687
$887$	137.585			0.155113			0.0775563	−	0.996988i	$-0.475288\pi$
$887$	137.585			0.155113			0.0775563	+	0.996988i	$0.475288\pi$
$888$		0			0
$889$	−55.7673			−0.0627304
$890$	472.747	−	212.495i	0.531176	−	0.238758i
$891$		0			0
$892$			28.2020i			0.0316166i
$893$	−1316.37			−1.47409
$894$		0			0
$895$	1077.23	−	484.202i	1.20360	−	0.541008i
$896$			134.622i			0.150247i
$897$		0			0
$898$			1135.47i			1.26445i
$899$		−	736.654i		−	0.819415i
$900$		0			0
$901$	46.4428			0.0515459
$902$	4.19286			0.00464840
$903$		0			0
$904$	−322.343			−0.356574
$905$	−543.323	−	1208.75i	−0.600357	−	1.33564i
$906$		0			0
$907$			645.000i			0.711136i	0.934650	+	0.355568i	$0.115712\pi$
$907$			645.000i			0.711136i	−0.934650	+	0.355568i	$0.884288\pi$
$908$	−516.373			−0.568692
$909$		0			0
$910$	−384.495	−	855.403i	−0.422522	−	0.940003i
$911$		−	32.6279i		−	0.0358155i	−0.999840	−	0.0179078i	$-0.994299\pi$
$911$		−	32.6279i		−	0.0358155i	0.999840	−	0.0179078i	$-0.00570052\pi$
$912$		0			0
$913$			136.722i			0.149750i
$914$			642.088i			0.702504i
$915$		0			0
$916$	−185.051			−0.202021
$917$	−63.9109			−0.0696956
$918$		0			0
$919$	551.978			0.600628			0.300314	−	0.953840i	$-0.402908\pi$
$919$	551.978			0.600628			0.300314	+	0.953840i	$0.402908\pi$
$920$	−53.8974	−	119.908i	−0.0585841	−	0.130335i
$921$		0			0
$922$			715.943i			0.776511i
$923$	−1356.21			−1.46935
$924$		0			0
$925$	−348.727	−	309.539i	−0.377002	−	0.334637i
$926$		−	699.135i		−	0.755006i
$927$		0			0
$928$			77.3939i			0.0833986i
$929$		−	754.683i		−	0.812360i	−0.913793	−	0.406180i	$-0.866861\pi$
$929$		−	754.683i		−	0.812360i	0.913793	−	0.406180i	$-0.133139\pi$
$930$		0			0
$931$	1893.33			2.03365
$932$	−641.238			−0.688023
$933$		0			0
$934$	551.605			0.590584
$935$	−603.919	+	271.455i	−0.645903	+	0.290327i
$936$		0			0
$937$			1241.31i			1.32477i	0.749162	+	0.662386i	$0.230456\pi$
$937$			1241.31i			1.32477i	−0.749162	+	0.662386i	$0.769544\pi$
$938$	706.592			0.753296
$939$		0			0
$940$	263.909	+	587.131i	0.280754	+	0.624607i
$941$			1468.28i			1.56034i	0.625570	+	0.780168i	$0.284867\pi$
$941$			1468.28i			1.56034i	−0.625570	+	0.780168i	$0.715133\pi$
$942$		0			0
$943$			4.28265i			0.00454152i
$944$			183.305i			0.194179i
$945$		0			0
$946$	−320.282			−0.338564
$947$	−259.055			−0.273553			−0.136777	−	0.990602i	$-0.543674\pi$
$947$	−259.055			−0.273553			−0.136777	+	0.990602i	$0.543674\pi$
$948$		0			0
$949$	−123.737			−0.130386
$950$	−479.953	+	540.714i	−0.505214	+	0.569173i
$951$		0			0
$952$			692.545i			0.727463i
$953$	−766.123			−0.803907			−0.401954	−	0.915660i	$-0.631669\pi$
$953$	−766.123			−0.803907			−0.401954	+	0.915660i	$0.631669\pi$
$954$		0			0
$955$	−430.974	+	193.718i	−0.451282	+	0.202846i
$956$		−	651.052i		−	0.681017i
$957$		0			0
$958$			262.847i			0.274370i
$959$		−	999.462i		−	1.04219i
$960$		0			0
$961$	1938.11			2.01676
$962$	294.012			0.305626
$963$		0			0
$964$	−169.514			−0.175845
$965$	193.269	−	86.8724i	0.200279	−	0.0900232i
$966$		0			0
$967$			676.725i			0.699820i	0.936783	+	0.349910i	$0.113788\pi$
$967$			676.725i			0.699820i	−0.936783	+	0.349910i	$0.886212\pi$
$968$	−225.102			−0.232544
$969$		0			0
$970$	−508.141	+	228.404i	−0.523857	+	0.235468i
$971$			332.231i			0.342154i	0.985258	+	0.171077i	$0.0547246\pi$
$971$			332.231i			0.342154i	−0.985258	+	0.171077i	$0.945275\pi$
$972$		0			0
$973$		−	817.666i		−	0.840356i
$974$			1171.34i			1.20261i
$975$		0			0
$976$	39.5959			0.0405696
$977$	−85.4034			−0.0874139			−0.0437069	−	0.999044i	$-0.513917\pi$
$977$	−85.4034			−0.0874139			−0.0437069	+	0.999044i	$0.513917\pi$
$978$		0			0
$979$	−471.714			−0.481832
$980$	−379.581	−	844.470i	−0.387327	−	0.861704i
$981$		0			0
$982$			357.898i			0.364458i
$983$	−1646.62			−1.67510			−0.837551	−	0.546360i	$-0.816013\pi$
$983$	−1646.62			−1.67510			−0.837551	+	0.546360i	$0.816013\pi$
$984$		0			0
$985$	−247.555	−	550.747i	−0.251325	−	0.559134i
$986$			398.144i			0.403797i
$987$		0			0
$988$		−	455.878i		−	0.461415i
$989$		−	327.141i		−	0.330779i
$990$		0			0
$991$	821.757			0.829220			0.414610	−	0.909999i	$-0.363918\pi$
$991$	821.757			0.829220			0.414610	+	0.909999i	$0.363918\pi$
$992$	−304.584			−0.307040
$993$		0			0
$994$	−2047.46			−2.05982
$995$	−60.4309	−	134.443i	−0.0607346	−	0.135119i
$996$		0			0
$997$		−	1874.35i		−	1.87999i	−0.341185	−	0.939996i	$-0.610828\pi$
$997$		−	1874.35i		−	1.87999i	0.341185	−	0.939996i	$-0.389172\pi$
$998$	−1379.73			−1.38249
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.3.b.a.809.4		8
3.2	odd	2	inner	810.3.b.a.809.5		8
5.4	even	2	inner	810.3.b.a.809.6		8
9.2	odd	6		270.3.j.a.179.2		8
9.4	even	3		270.3.j.a.89.4		8
9.5	odd	6		90.3.j.a.29.2	✓	8
9.7	even	3		90.3.j.a.59.3	yes	8
15.14	odd	2	inner	810.3.b.a.809.3		8
45.2	even	12		1350.3.i.a.1151.2		4
45.4	even	6		270.3.j.a.89.2		8
45.7	odd	12		450.3.i.a.401.1		4
45.13	odd	12		1350.3.i.c.251.1		4
45.14	odd	6		90.3.j.a.29.3	yes	8
45.22	odd	12		1350.3.i.a.251.2		4
45.23	even	12		450.3.i.c.101.2		4
45.29	odd	6		270.3.j.a.179.4		8
45.32	even	12		450.3.i.a.101.1		4
45.34	even	6		90.3.j.a.59.2	yes	8
45.38	even	12		1350.3.i.c.1151.1		4
45.43	odd	12		450.3.i.c.401.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.3.j.a.29.2	✓	8	9.5	odd	6
90.3.j.a.29.3	yes	8	45.14	odd	6
90.3.j.a.59.2	yes	8	45.34	even	6
90.3.j.a.59.3	yes	8	9.7	even	3
270.3.j.a.89.2		8	45.4	even	6
270.3.j.a.89.4		8	9.4	even	3
270.3.j.a.179.2		8	9.2	odd	6
270.3.j.a.179.4		8	45.29	odd	6
450.3.i.a.101.1		4	45.32	even	12
450.3.i.a.401.1		4	45.7	odd	12
450.3.i.c.101.2		4	45.23	even	12
450.3.i.c.401.2		4	45.43	odd	12
810.3.b.a.809.3		8	15.14	odd	2	inner
810.3.b.a.809.4		8	1.1	even	1	trivial
810.3.b.a.809.5		8	3.2	odd	2	inner
810.3.b.a.809.6		8	5.4	even	2	inner
1350.3.i.a.251.2		4	45.22	odd	12
1350.3.i.a.1151.2		4	45.2	even	12
1350.3.i.c.251.1		4	45.13	odd	12
1350.3.i.c.1151.1		4	45.38	even	12

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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	810.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +(2.04989 + 4.56048i) q^{5} -11.8990i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} +(2.04989 + 4.56048i) q^{5} -11.8990i q^{7} -2.82843 q^{8} +(-2.89898 - 6.44949i) q^{10} +6.43539i q^{11} +11.1464i q^{13} +16.8277i q^{14} +4.00000 q^{16} +20.5775 q^{17} -20.4495 q^{19} +(4.09978 + 9.12096i) q^{20} -9.10102i q^{22} +9.29593 q^{23} +(-16.5959 + 18.6969i) q^{25} -15.7634i q^{26} -23.7980i q^{28} -13.6814i q^{29} +53.8434 q^{31} -5.65685 q^{32} -29.1010 q^{34} +(54.2650 - 24.3916i) q^{35} +18.6515i q^{37} +28.9199 q^{38} +(-5.79796 - 12.8990i) q^{40} +0.460702i q^{41} -35.1918i q^{43} +12.8708i q^{44} -13.1464 q^{46} +64.3716 q^{47} -92.5857 q^{49} +(23.4702 - 26.4415i) q^{50} +22.2929i q^{52} +2.25697 q^{53} +(-29.3485 + 13.1918i) q^{55} +33.6554i q^{56} +19.3485i q^{58} +45.8263i q^{59} +9.89898 q^{61} -76.1460 q^{62} +8.00000 q^{64} +(-50.8330 + 22.8489i) q^{65} -41.9898i q^{67} +41.1551 q^{68} +(-76.7423 + 34.4949i) q^{70} +121.672i q^{71} +11.1010i q^{73} -26.3772i q^{74} -40.8990 q^{76} +76.5746 q^{77} +145.485 q^{79} +(8.19955 + 18.2419i) q^{80} -0.651531i q^{82} +21.2453 q^{83} +(42.1816 + 93.8434i) q^{85} +49.7688i q^{86} -18.2020i q^{88} +73.2999i q^{89} +132.631 q^{91} +18.5919 q^{92} -91.0352 q^{94} +(-41.9192 - 93.2594i) q^{95} -78.7878i q^{97} +130.936 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4}+O(q^{10}) \) 8 * q + 16 * q^4	\( 8 q + 16 q^{4} + 16 q^{10} + 32 q^{16} - 144 q^{19} + 24 q^{25} + 176 q^{31} - 272 q^{34} + 32 q^{40} + 32 q^{46} - 192 q^{49} - 176 q^{55} + 40 q^{61} + 64 q^{64} - 320 q^{70} - 288 q^{76} + 576 q^{79} - 368 q^{85} + 336 q^{91} - 160 q^{94}+O(q^{100}) \) 8 * q + 16 * q^4 + 16 * q^10 + 32 * q^16 - 144 * q^19 + 24 * q^25 + 176 * q^31 - 272 * q^34 + 32 * q^40 + 32 * q^46 - 192 * q^49 - 176 * q^55 + 40 * q^61 + 64 * q^64 - 320 * q^70 - 288 * q^76 + 576 * q^79 - 368 * q^85 + 336 * q^91 - 160 * q^94

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