LMFDB - Embedded newform 1620.3.t.c.1349.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(269,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 1, 3]))

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

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

chi := DirichletCharacter("1620.269");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.t (of order $6$, degree $2$, not 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:	$44.1418028264$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.1485512441856.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 24x^{6} + 455x^{4} - 2904x^{2} + 14641 $ x^8 - 24x^6 + 455x^4 - 2904*x^2 + 14641
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1349.1
Root			$2.32446 - 1.34203i$ of defining polynomial
Character	$\chi$	$=$	1620.1349
Dual form			1620.3.t.c.269.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.62266 + 3.44621i) q^{5} +(5.87367 - 3.39116i) q^{7} +O(q^{10})$	$q+(-3.62266 + 3.44621i) q^{5} +(5.87367 - 3.39116i) q^{7} +(-8.57321 + 4.94975i) q^{11} +(-17.6210 - 10.1735i) q^{13} +19.1833 q^{17} +12.0000 q^{19} +(-4.79583 + 8.30662i) q^{23} +(1.24734 - 24.9689i) q^{25} +(7.34847 - 4.24264i) q^{29} +(19.0000 - 32.9090i) q^{31} +(-9.59166 + 32.5269i) q^{35} +6.78233i q^{37} +(60.0125 + 34.6482i) q^{41} +(-58.7367 + 33.9116i) q^{43} +(38.3667 + 66.4530i) q^{47} +(-1.50000 + 2.59808i) q^{49} +(14.0000 - 47.4763i) q^{55} +(-72.2599 - 41.7193i) q^{59} +(35.0000 + 60.6218i) q^{61} +(98.8949 - 23.8705i) q^{65} +(93.9787 + 54.2586i) q^{67} -118.794i q^{71} +13.5647i q^{73} +(-33.5708 + 58.1464i) q^{77} +(-15.0000 - 25.9808i) q^{79} +(67.1416 + 116.293i) q^{83} +(-69.4947 + 66.1097i) q^{85} -32.5269i q^{89} -138.000 q^{91} +(-43.4719 + 41.3545i) q^{95} +(82.2314 - 47.4763i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 96 q^{19} - 84 q^{25} + 152 q^{31} - 12 q^{49} + 112 q^{55} + 280 q^{61} - 120 q^{79} - 368 q^{85} - 1104 q^{91}+O(q^{100}) $ 8 * q + 96 * q^19 - 84 * q^25 + 152 * q^31 - 12 * q^49 + 112 * q^55 + 280 * q^61 - 120 * q^79 - 368 * q^85 - 1104 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$e\left(\frac{5}{6}\right)$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−3.62266	+	3.44621i	−0.724532	+	0.689241i
$5$	−3.62266	+	3.44621i	−0.724532	+	0.689241i
$6$		0			0
$7$	5.87367	−	3.39116i	0.839096	−	0.484452i	−0.0178610	−	0.999840i	$-0.505686\pi$
$7$	5.87367	−	3.39116i	0.839096	−	0.484452i	0.856957	+	0.515388i	$0.172352\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−8.57321	+	4.94975i	−0.779383	+	0.449977i	−0.836212	−	0.548407i	$-0.815235\pi$
$11$	−8.57321	+	4.94975i	−0.779383	+	0.449977i	0.0568285	+	0.998384i	$0.481901\pi$
$12$		0			0
$13$	−17.6210	−	10.1735i	−1.35546	−	0.782577i	−0.366454	−	0.930436i	$-0.619428\pi$
$13$	−17.6210	−	10.1735i	−1.35546	−	0.782577i	−0.989008	+	0.147860i	$0.952762\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	19.1833			1.12843			0.564215	−	0.825628i	$-0.309179\pi$
$17$	19.1833			1.12843			0.564215	+	0.825628i	$0.309179\pi$
$18$		0			0
$19$	12.0000			0.631579			0.315789	−	0.948829i	$-0.397731\pi$
$19$	12.0000			0.631579			0.315789	+	0.948829i	$0.397731\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.79583	+	8.30662i	−0.208514	+	0.361158i	−0.951247	−	0.308431i	$-0.900196\pi$
$23$	−4.79583	+	8.30662i	−0.208514	+	0.361158i	0.742732	+	0.669588i	$0.233529\pi$
$24$		0			0
$25$	1.24734	−	24.9689i	0.0498936	−	0.998755i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.34847	−	4.24264i	0.253395	−	0.146298i	−0.367923	−	0.929856i	$-0.619931\pi$
$29$	7.34847	−	4.24264i	0.253395	−	0.146298i	0.621318	+	0.783558i	$0.286598\pi$
$30$		0			0
$31$	19.0000	−	32.9090i	0.612903	−	1.06158i	−0.377845	−	0.925869i	$-0.623335\pi$
$31$	19.0000	−	32.9090i	0.612903	−	1.06158i	0.990748	−	0.135711i	$-0.0433318\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−9.59166	+	32.5269i	−0.274048	+	0.929340i
$36$		0			0
$37$			6.78233i			0.183306i	0.995791	+	0.0916531i	$0.0292151\pi$
$37$			6.78233i			0.183306i	−0.995791	+	0.0916531i	$0.970785\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	60.0125	+	34.6482i	1.46372	+	0.845079i	0.999181	−	0.0404739i	$-0.0128868\pi$
$41$	60.0125	+	34.6482i	1.46372	+	0.845079i	0.464539	+	0.885553i	$0.346220\pi$
$42$		0			0
$43$	−58.7367	+	33.9116i	−1.36597	+	0.788643i	−0.990411	−	0.138155i	$-0.955883\pi$
$43$	−58.7367	+	33.9116i	−1.36597	+	0.788643i	−0.375559	+	0.926798i	$0.622549\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	38.3667	+	66.4530i	0.816312	+	1.41389i	0.908382	+	0.418141i	$0.137318\pi$
$47$	38.3667	+	66.4530i	0.816312	+	1.41389i	−0.0920704	+	0.995752i	$0.529348\pi$
$48$		0			0
$49$	−1.50000	+	2.59808i	−0.0306122	+	0.0530220i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	14.0000	−	47.4763i	0.254545	−	0.863206i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−72.2599	−	41.7193i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$59$	−72.2599	−	41.7193i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$60$		0			0
$61$	35.0000	+	60.6218i	0.573770	+	0.993800i	0.996174	+	0.0873918i	$0.0278532\pi$
$61$	35.0000	+	60.6218i	0.573770	+	0.993800i	−0.422404	+	0.906408i	$0.638813\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	98.8949	−	23.8705i	1.52146	−	0.367238i
$66$		0			0
$67$	93.9787	+	54.2586i	1.40267	+	0.809830i	0.994666	−	0.103151i	$-0.0328924\pi$
$67$	93.9787	+	54.2586i	1.40267	+	0.809830i	0.408002	+	0.912981i	$0.366226\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	118.794i		−	1.67315i	−0.547849	−	0.836577i	$-0.684553\pi$
$71$		−	118.794i		−	1.67315i	0.547849	−	0.836577i	$-0.315447\pi$
$72$		0			0
$73$			13.5647i			0.185817i	0.995675	+	0.0929086i	$0.0296164\pi$
$73$			13.5647i			0.185817i	−0.995675	+	0.0929086i	$0.970384\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−33.5708	+	58.1464i	−0.435985	+	0.755148i
$78$		0			0
$79$	−15.0000	−	25.9808i	−0.189873	−	0.328870i	0.755334	−	0.655339i	$-0.227474\pi$
$79$	−15.0000	−	25.9808i	−0.189873	−	0.328870i	−0.945208	+	0.326469i	$0.894141\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	67.1416	+	116.293i	0.808935	+	1.40112i	0.913602	+	0.406610i	$0.133289\pi$
$83$	67.1416	+	116.293i	0.808935	+	1.40112i	−0.104666	+	0.994507i	$0.533377\pi$
$84$		0			0
$85$	−69.4947	+	66.1097i	−0.817584	+	0.777761i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	32.5269i		−	0.365471i	−0.983162	−	0.182735i	$-0.941505\pi$
$89$		−	32.5269i		−	0.365471i	0.983162	−	0.182735i	$-0.0584952\pi$
$90$		0			0
$91$	−138.000			−1.51648
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−43.4719	+	41.3545i	−0.457599	+	0.435310i
$96$		0			0
$97$	82.2314	−	47.4763i	0.847746	−	0.489446i	−0.0121436	−	0.999926i	$-0.503866\pi$
$97$	82.2314	−	47.4763i	0.847746	−	0.489446i	0.859890	+	0.510480i	$0.170532\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−68.5857	+	39.5980i	−0.679066	+	0.392059i	−0.799503	−	0.600662i	$-0.794904\pi$
$101$	−68.5857	+	39.5980i	−0.679066	+	0.392059i	0.120437	+	0.992721i	$0.461570\pi$
$102$		0			0
$103$	−76.3577	−	44.0851i	−0.741337	−	0.428011i	0.0812182	−	0.996696i	$-0.474119\pi$
$103$	−76.3577	−	44.0851i	−0.741337	−	0.428011i	−0.822555	+	0.568685i	$0.807452\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	57.5500			0.537850			0.268925	−	0.963161i	$-0.413332\pi$
$107$	57.5500			0.537850			0.268925	+	0.963161i	$0.413332\pi$
$108$		0			0
$109$	74.0000			0.678899			0.339450	−	0.940624i	$-0.389759\pi$
$109$	74.0000			0.678899			0.339450	+	0.940624i	$0.389759\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−86.3250	+	149.519i	−0.763938	+	1.32318i	0.176869	+	0.984234i	$0.443403\pi$
$113$	−86.3250	+	149.519i	−0.763938	+	1.32318i	−0.940806	+	0.338945i	$0.889930\pi$
$114$		0			0
$115$	−11.2527	−	46.6195i	−0.0978492	−	0.405387i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	112.677	−	65.0538i	0.946862	−	0.546671i
$120$		0			0
$121$	−11.5000	+	19.9186i	−0.0950413	+	0.164616i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	81.5291	+	94.7523i	0.652233	+	0.758018i
$126$		0			0
$127$			169.558i			1.33510i	0.744563	+	0.667552i	$0.232658\pi$
$127$			169.558i			1.33510i	−0.744563	+	0.667552i	$0.767342\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	143.295	+	82.7315i	1.09386	+	0.631538i	0.934600	−	0.355699i	$-0.115757\pi$
$131$	143.295	+	82.7315i	1.09386	+	0.631538i	0.159256	+	0.987237i	$0.449091\pi$
$132$		0			0
$133$	70.4840	−	40.6940i	0.529955	−	0.305970i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	57.5500	+	99.6795i	0.420073	+	0.727587i	0.995946	−	0.0899515i	$-0.0286712\pi$
$137$	57.5500	+	99.6795i	0.420073	+	0.727587i	−0.575873	+	0.817539i	$0.695338\pi$
$138$		0			0
$139$	−31.0000	+	53.6936i	−0.223022	+	0.386285i	−0.955724	−	0.294264i	$-0.904925\pi$
$139$	−31.0000	+	53.6936i	−0.223022	+	0.386285i	0.732702	+	0.680549i	$0.238259\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	201.425			1.40857
$144$		0			0
$145$	−12.0000	+	40.6940i	−0.0827586	+	0.280648i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	137.171	+	79.1960i	0.920614	+	0.531517i	0.883831	−	0.467807i	$-0.154956\pi$
$149$	137.171	+	79.1960i	0.920614	+	0.531517i	0.0367828	+	0.999323i	$0.488289\pi$
$150$		0			0
$151$	−35.0000	−	60.6218i	−0.231788	−	0.401469i	0.726546	−	0.687118i	$-0.241124\pi$
$151$	−35.0000	−	60.6218i	−0.231788	−	0.401469i	−0.958334	+	0.285649i	$0.907791\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	44.5805	+	184.696i	0.287616	+	1.19159i
$156$		0			0
$157$	41.1157	+	23.7382i	0.261883	+	0.151198i	0.625193	−	0.780470i	$-0.285020\pi$
$157$	41.1157	+	23.7382i	0.261883	+	0.151198i	−0.363310	+	0.931668i	$0.618353\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			65.0538i			0.404061i
$162$		0			0
$163$		−	94.9526i		−	0.582531i	−0.956642	−	0.291266i	$-0.905924\pi$
$163$		−	94.9526i		−	0.582531i	0.956642	−	0.291266i	$-0.0940764\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−100.712	+	174.439i	−0.603069	+	1.04455i	0.389285	+	0.921117i	$0.372722\pi$
$167$	−100.712	+	174.439i	−0.603069	+	1.04455i	−0.992354	+	0.123428i	$0.960611\pi$
$168$		0			0
$169$	122.500	+	212.176i	0.724852	+	1.25548i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	134.283	+	232.585i	0.776204	+	1.34442i	0.934115	+	0.356971i	$0.116191\pi$
$173$	134.283	+	232.585i	0.776204	+	1.34442i	−0.157911	+	0.987453i	$0.550476\pi$
$174$		0			0
$175$	−77.3471	−	150.889i	−0.441983	−	0.862222i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			134.350i			0.750560i	0.926911	+	0.375280i	$0.122453\pi$
$179$			134.350i			0.750560i	−0.926911	+	0.375280i	$0.877547\pi$
$180$		0			0
$181$	−22.0000			−0.121547			−0.0607735	−	0.998152i	$-0.519357\pi$
$181$	−22.0000			−0.121547			−0.0607735	+	0.998152i	$0.519357\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−23.3733	−	24.5701i	−0.126342	−	0.132811i
$186$		0			0
$187$	−164.463	+	94.9526i	−0.879480	+	0.507768i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	169.015	−	97.5807i	0.884894	−	0.510894i	0.0126252	−	0.999920i	$-0.495981\pi$
$191$	169.015	−	97.5807i	0.884894	−	0.510894i	0.872269	+	0.489026i	$0.162648\pi$
$192$		0			0
$193$	234.947	+	135.647i	1.21734	+	0.702832i	0.964348	−	0.264636i	$-0.0852518\pi$
$193$	234.947	+	135.647i	1.21734	+	0.702832i	0.252993	+	0.967468i	$0.418585\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−163.058			−0.827707			−0.413853	−	0.910344i	$-0.635817\pi$
$197$	−163.058			−0.827707			−0.413853	+	0.910344i	$0.635817\pi$
$198$		0			0
$199$	294.000			1.47739			0.738693	−	0.674042i	$-0.235443\pi$
$199$	294.000			1.47739			0.738693	+	0.674042i	$0.235443\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	28.7750	−	49.8397i	0.141749	−	0.245516i
$204$		0			0
$205$	−336.810	+	81.2966i	−1.64297	+	0.396569i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−102.879	+	59.3970i	−0.492242	+	0.284196i
$210$		0			0
$211$	21.0000	−	36.3731i	0.0995261	−	0.172384i	−0.811963	−	0.583710i	$-0.801601\pi$
$211$	21.0000	−	36.3731i	0.0995261	−	0.172384i	0.911489	+	0.411325i	$0.134934\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	95.9166	−	325.269i	0.446124	−	1.51288i
$216$		0			0
$217$		−	257.729i		−	1.18769i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−338.030	−	195.161i	−1.52955	−	0.883084i
$222$		0			0
$223$	170.336	−	98.3438i	0.763841	−	0.441004i	−0.0668324	−	0.997764i	$-0.521289\pi$
$223$	170.336	−	98.3438i	0.763841	−	0.441004i	0.830673	+	0.556761i	$0.187956\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	134.283	+	232.585i	0.591556	+	1.02461i	0.994023	+	0.109171i	$0.0348196\pi$
$227$	134.283	+	232.585i	0.591556	+	1.02461i	−0.402467	+	0.915435i	$0.631847\pi$
$228$		0			0
$229$	211.000	−	365.463i	0.921397	−	1.59591i	0.124142	−	0.992264i	$-0.460382\pi$
$229$	211.000	−	365.463i	0.921397	−	1.59591i	0.797255	−	0.603643i	$-0.206285\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−211.017			−0.905651			−0.452825	−	0.891599i	$-0.649584\pi$
$233$	−211.017			−0.905651			−0.452825	+	0.891599i	$0.649584\pi$
$234$		0			0
$235$	−368.000	−	108.517i	−1.56596	−	0.461776i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	61.2372	+	35.3553i	0.256223	+	0.147930i	0.622610	−	0.782532i	$-0.286072\pi$
$239$	61.2372	+	35.3553i	0.256223	+	0.147930i	−0.366388	+	0.930462i	$0.619406\pi$
$240$		0			0
$241$	−140.000	−	242.487i	−0.580913	−	1.00617i	−0.995371	−	0.0961024i	$-0.969362\pi$
$241$	−140.000	−	242.487i	−0.580913	−	1.00617i	0.414459	−	0.910068i	$-0.363971\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.51951	−	14.5813i	−0.0143654	−	0.0595153i
$246$		0			0
$247$	−211.452	−	122.082i	−0.856081	−	0.494259i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			80.6102i			0.321156i	0.987023	+	0.160578i	$0.0513358\pi$
$251$			80.6102i			0.321156i	−0.987023	+	0.160578i	$0.948664\pi$
$252$		0			0
$253$		−	94.9526i		−	0.375307i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$	23.0000	+	39.8372i	0.0888031	+	0.153811i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−33.5708	−	58.1464i	−0.127646	−	0.221089i	0.795118	−	0.606454i	$-0.207409\pi$
$263$	−33.5708	−	58.1464i	−0.127646	−	0.221089i	−0.922764	+	0.385365i	$0.874075\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	263.044i		−	0.977858i	−0.872324	−	0.488929i	$-0.837388\pi$
$269$		−	263.044i		−	0.977858i	0.872324	−	0.488929i	$-0.162612\pi$
$270$		0			0
$271$	322.000			1.18819			0.594096	−	0.804394i	$-0.297510\pi$
$271$	322.000			1.18819			0.594096	+	0.804394i	$0.297510\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	112.896	+	220.237i	0.410530	+	0.800863i
$276$		0			0
$277$	−29.3684	+	16.9558i	−0.106023	+	0.0612124i	−0.552074	−	0.833795i	$-0.686163\pi$
$277$	−29.3684	+	16.9558i	−0.106023	+	0.0612124i	0.446051	+	0.895008i	$0.352830\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	282.916	−	163.342i	1.00682	−	0.581287i	0.0965601	−	0.995327i	$-0.469216\pi$
$281$	282.916	−	163.342i	1.00682	−	0.581287i	0.910259	+	0.414040i	$0.135883\pi$
$282$		0			0
$283$	−93.9787	−	54.2586i	−0.332080	−	0.191727i	0.324684	−	0.945823i	$-0.394742\pi$
$283$	−93.9787	−	54.2586i	−0.332080	−	0.191727i	−0.656764	+	0.754096i	$0.728075\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	469.991			1.63760
$288$		0			0
$289$	79.0000			0.273356
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−100.712	+	174.439i	−0.343729	+	0.595355i	−0.985122	−	0.171857i	$-0.945023\pi$
$293$	−100.712	+	174.439i	−0.343729	+	0.595355i	0.641393	+	0.767212i	$0.278357\pi$
$294$		0			0
$295$	405.547	−	97.8877i	1.37473	−	0.331823i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	169.015	−	97.5807i	0.565267	−	0.326357i
$300$		0			0
$301$	−230.000	+	398.372i	−0.764120	+	1.32349i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−335.708	−	98.9949i	−1.10068	−	0.324574i
$306$		0			0
$307$			162.776i			0.530215i	0.964219	+	0.265107i	$0.0854074\pi$
$307$			162.776i			0.530215i	−0.964219	+	0.265107i	$0.914593\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−154.318	−	89.0955i	−0.496199	−	0.286481i	0.230944	−	0.972967i	$-0.425819\pi$
$311$	−154.318	−	89.0955i	−0.496199	−	0.286481i	−0.727142	+	0.686487i	$0.759152\pi$
$312$		0			0
$313$	46.9894	−	27.1293i	0.150126	−	0.0866751i	−0.423055	−	0.906104i	$-0.639042\pi$
$313$	46.9894	−	27.1293i	0.150126	−	0.0866751i	0.573181	+	0.819429i	$0.305709\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−139.079	−	240.892i	−0.438735	−	0.759912i	0.558857	−	0.829264i	$-0.311240\pi$
$317$	−139.079	−	240.892i	−0.438735	−	0.759912i	−0.997592	+	0.0693522i	$0.977907\pi$
$318$		0			0
$319$	−42.0000	+	72.7461i	−0.131661	+	0.228044i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	230.200			0.712693
$324$		0			0
$325$	−276.000	+	427.287i	−0.849231	+	1.31473i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	450.706	+	260.215i	1.36993	+	0.790928i
$330$		0			0
$331$	−20.0000	−	34.6410i	−0.0604230	−	0.104656i	0.834232	−	0.551414i	$-0.185912\pi$
$331$	−20.0000	−	34.6410i	−0.0604230	−	0.104656i	−0.894655	+	0.446759i	$0.852578\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−527.439	+	127.309i	−1.57445	+	0.380028i
$336$		0			0
$337$	−270.189	−	155.994i	−0.801747	−	0.462889i	0.0423345	−	0.999103i	$-0.486520\pi$
$337$	−270.189	−	155.994i	−0.801747	−	0.462889i	−0.844082	+	0.536215i	$0.819854\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			376.181i			1.10317i
$342$		0			0
$343$			352.681i			1.02822i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	201.425	−	348.878i	0.580475	−	1.00541i	−0.414948	−	0.909845i	$-0.636200\pi$
$347$	201.425	−	348.878i	0.580475	−	1.00541i	0.995423	−	0.0955674i	$-0.0304665\pi$
$348$		0			0
$349$	203.000	+	351.606i	0.581662	+	1.00747i	0.995283	+	0.0970186i	$0.0309306\pi$
$349$	203.000	+	351.606i	0.581662	+	1.00747i	−0.413621	+	0.910449i	$0.635736\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−268.567	−	465.171i	−0.760812	−	1.31776i	−0.942433	−	0.334396i	$-0.891468\pi$
$353$	−268.567	−	465.171i	−0.760812	−	1.31776i	0.181621	−	0.983369i	$-0.441866\pi$
$354$		0			0
$355$	409.388	+	430.350i	1.15321	+	1.21225i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			161.220i			0.449082i	0.974465	+	0.224541i	$0.0720882\pi$
$359$			161.220i			0.449082i	−0.974465	+	0.224541i	$0.927912\pi$
$360$		0			0
$361$	−217.000			−0.601108
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−46.7466	−	49.1402i	−0.128073	−	0.134631i
$366$		0			0
$367$	217.326	−	125.473i	0.592168	−	0.341889i	−0.173786	−	0.984783i	$-0.555600\pi$
$367$	217.326	−	125.473i	0.592168	−	0.341889i	0.765954	+	0.642895i	$0.222267\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−88.1051	−	50.8675i	−0.236207	−	0.136374i	0.377225	−	0.926121i	$-0.376878\pi$
$373$	−88.1051	−	50.8675i	−0.236207	−	0.136374i	−0.613432	+	0.789748i	$0.710212\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−172.650			−0.457957
$378$		0			0
$379$	538.000			1.41953			0.709763	−	0.704441i	$-0.248802\pi$
$379$	538.000			1.41953			0.709763	+	0.704441i	$0.248802\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	134.283	−	232.585i	0.350609	−	0.607273i	−0.635747	−	0.771897i	$-0.719308\pi$
$383$	134.283	−	232.585i	0.350609	−	0.607273i	0.986356	+	0.164625i	$0.0526413\pi$
$384$		0			0
$385$	−78.7686	−	326.336i	−0.204594	−	0.847627i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−428.661	+	247.487i	−1.10196	+	0.636214i	−0.936734	−	0.350042i	$-0.886167\pi$
$389$	−428.661	+	247.487i	−1.10196	+	0.636214i	−0.165222	+	0.986256i	$0.552834\pi$
$390$		0			0
$391$	−92.0000	+	159.349i	−0.235294	+	0.407541i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	143.875	+	42.4264i	0.364240	+	0.107409i
$396$		0			0
$397$		−	278.076i		−	0.700442i	−0.936667	−	0.350221i	$-0.886106\pi$
$397$		−	278.076i		−	0.700442i	0.936667	−	0.350221i	$-0.113894\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−214.330	−	123.744i	−0.534490	−	0.308588i	0.208353	−	0.978054i	$-0.433190\pi$
$401$	−214.330	−	123.744i	−0.534490	−	0.308588i	−0.742843	+	0.669466i	$0.766523\pi$
$402$		0			0
$403$	−669.598	+	386.593i	−1.66153	+	0.959287i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−33.5708	−	58.1464i	−0.0824836	−	0.142866i
$408$		0			0
$409$	121.000	−	209.578i	0.295844	−	0.512416i	−0.679337	−	0.733826i	$-0.737733\pi$
$409$	121.000	−	209.578i	0.295844	−	0.512416i	0.975181	+	0.221410i	$0.0710660\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−565.908			−1.37024
$414$		0			0
$415$	−644.000	−	189.905i	−1.55181	−	0.457603i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−684.632	−	395.273i	−1.63397	−	0.943372i	−0.982853	−	0.184392i	$-0.940968\pi$
$419$	−684.632	−	395.273i	−1.63397	−	0.943372i	−0.651115	−	0.758979i	$-0.725698\pi$
$420$		0			0
$421$	−179.000	−	310.037i	−0.425178	−	0.736430i	0.571259	−	0.820770i	$-0.306455\pi$
$421$	−179.000	−	310.037i	−0.425178	−	0.736430i	−0.996437	+	0.0843398i	$0.973122\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	23.9281	−	478.986i	0.0563015	−	1.12703i
$426$		0			0
$427$	411.157	+	237.382i	0.962897	+	0.555929i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	14.1421i		−	0.0328124i	−0.999865	−	0.0164062i	$-0.994778\pi$
$431$		−	14.1421i		−	0.0328124i	0.999865	−	0.0164062i	$-0.00522249\pi$
$432$		0			0
$433$			257.729i			0.595216i	0.954688	+	0.297608i	$0.0961888\pi$
$433$			257.729i			0.595216i	−0.954688	+	0.297608i	$0.903811\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−57.5500	+	99.6795i	−0.131693	+	0.228100i
$438$		0			0
$439$	345.000	+	597.558i	0.785877	+	1.36118i	0.928474	+	0.371398i	$0.121121\pi$
$439$	345.000	+	597.558i	0.785877	+	1.36118i	−0.142597	+	0.989781i	$0.545545\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	201.425	+	348.878i	0.454684	+	0.787535i	0.998670	−	0.0515587i	$-0.0164189\pi$
$443$	201.425	+	348.878i	0.454684	+	0.787535i	−0.543986	+	0.839094i	$0.683086\pi$
$444$		0			0
$445$	112.094	+	117.834i	0.251898	+	0.264795i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	80.6102i		−	0.179533i	−0.995963	−	0.0897663i	$-0.971388\pi$
$449$		−	80.6102i		−	0.179533i	0.995963	−	0.0897663i	$-0.0286120\pi$
$450$		0			0
$451$	−686.000			−1.52106
$452$		0			0
$453$		0			0
$454$		0			0
$455$	499.927	−	475.576i	1.09874	−	1.04522i
$456$		0			0
$457$	−293.684	+	169.558i	−0.642633	+	0.371025i	−0.785628	−	0.618699i	$-0.787660\pi$
$457$	−293.684	+	169.558i	−0.642633	+	0.371025i	0.142995	+	0.989723i	$0.454327\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−558.484	+	322.441i	−1.21146	+	0.699438i	−0.963078	−	0.269224i	$-0.913233\pi$
$461$	−558.484	+	322.441i	−1.21146	+	0.699438i	−0.248384	+	0.968662i	$0.579899\pi$
$462$		0			0
$463$	229.073	+	132.255i	0.494758	+	0.285649i	0.726546	−	0.687117i	$-0.241124\pi$
$463$	229.073	+	132.255i	0.494758	+	0.285649i	−0.231788	+	0.972766i	$0.574458\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−460.400			−0.985867			−0.492933	−	0.870067i	$-0.664075\pi$
$467$	−460.400			−0.985867			−0.492933	+	0.870067i	$0.664075\pi$
$468$		0			0
$469$	736.000			1.56930
$470$		0			0
$471$		0			0
$472$		0			0
$473$	335.708	−	581.464i	0.709743	−	1.22931i
$474$		0			0
$475$	14.9681	−	299.626i	0.0315118	−	0.630792i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−68.5857	+	39.5980i	−0.143185	+	0.0826680i	−0.569881	−	0.821727i	$-0.693011\pi$
$479$	−68.5857	+	39.5980i	−0.143185	+	0.0826680i	0.426696	+	0.904395i	$0.359677\pi$
$480$		0			0
$481$	69.0000	−	119.512i	0.143451	−	0.248465i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−134.283	+	455.377i	−0.276873	+	0.938921i
$486$		0			0
$487$		−	793.533i		−	1.62943i	−0.579861	−	0.814715i	$-0.696893\pi$
$487$		−	793.533i		−	1.62943i	0.579861	−	0.814715i	$-0.303107\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	488.673	+	282.136i	0.995261	+	0.574614i	0.906843	−	0.421469i	$-0.138485\pi$
$491$	488.673	+	282.136i	0.995261	+	0.574614i	0.0884184	+	0.996083i	$0.471819\pi$
$492$		0			0
$493$	140.968	−	81.3880i	0.285939	−	0.165087i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−402.850	−	697.756i	−0.810563	−	1.40394i
$498$		0			0
$499$	36.0000	−	62.3538i	0.0721443	−	0.124958i	−0.827697	−	0.561176i	$-0.810349\pi$
$499$	36.0000	−	62.3538i	0.0721443	−	0.124958i	0.899841	+	0.436218i	$0.143682\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	230.200			0.457654			0.228827	−	0.973467i	$-0.426511\pi$
$503$	230.200			0.457654			0.228827	+	0.973467i	$0.426511\pi$
$504$		0			0
$505$	112.000	−	379.810i	0.221782	−	0.752100i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−426.211	−	246.073i	−0.837350	−	0.483444i	0.0190125	−	0.999819i	$-0.493948\pi$
$509$	−426.211	−	246.073i	−0.837350	−	0.483444i	−0.856363	+	0.516375i	$0.827281\pi$
$510$		0			0
$511$	46.0000	+	79.6743i	0.0900196	+	0.155918i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	428.545	−	103.439i	0.832125	−	0.200852i
$516$		0			0
$517$	−657.851	−	379.810i	−1.27244	−	0.734643i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			69.2965i			0.133007i	0.997786	+	0.0665033i	$0.0211843\pi$
$521$			69.2965i			0.133007i	−0.997786	+	0.0665033i	$0.978816\pi$
$522$		0			0
$523$		−	339.116i		−	0.648406i	−0.945987	−	0.324203i	$-0.894904\pi$
$523$		−	339.116i		−	0.648406i	0.945987	−	0.324203i	$-0.105096\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	364.483	−	631.303i	0.691619	−	1.19792i
$528$		0			0
$529$	218.500	+	378.453i	0.413043	+	0.715412i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−704.987	−	1221.07i	−1.32268	−	2.29095i
$534$		0			0
$535$	−208.484	+	198.329i	−0.389690	+	0.370708i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	29.6985i		−	0.0550992i
$540$		0			0
$541$	−590.000			−1.09057			−0.545287	−	0.838250i	$-0.683579\pi$
$541$	−590.000			−1.09057			−0.545287	+	0.838250i	$0.683579\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−268.077	+	255.019i	−0.491884	+	0.467925i
$546$		0			0
$547$	199.705	−	115.300i	0.365091	−	0.210785i	−0.306221	−	0.951961i	$-0.599065\pi$
$547$	199.705	−	115.300i	0.365091	−	0.210785i	0.671312	+	0.741175i	$0.265731\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	88.1816	−	50.9117i	0.160039	−	0.0923987i
$552$		0			0
$553$	−176.210	−	101.735i	−0.318644	−	0.183969i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−690.600			−1.23986			−0.619928	−	0.784659i	$-0.712838\pi$
$557$	−690.600			−1.23986			−0.619928	+	0.784659i	$0.712838\pi$
$558$		0			0
$559$	1380.00			2.46869
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−9.59166	+	16.6132i	−0.0170367	+	0.0295084i	−0.874418	−	0.485173i	$-0.838757\pi$
$563$	−9.59166	+	16.6132i	−0.0170367	+	0.0295084i	0.857381	+	0.514682i	$0.172090\pi$
$564$		0			0
$565$	−202.548	−	839.151i	−0.358492	−	1.48522i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−94.3054	+	54.4472i	−0.165739	+	0.0956893i	−0.580575	−	0.814207i	$-0.697172\pi$
$569$	−94.3054	+	54.4472i	−0.165739	+	0.0956893i	0.414836	+	0.909896i	$0.363839\pi$
$570$		0			0
$571$	352.000	−	609.682i	0.616462	−	1.06774i	−0.373664	−	0.927564i	$-0.621899\pi$
$571$	352.000	−	609.682i	0.616462	−	1.06774i	0.990126	−	0.140180i	$-0.0447681\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	201.425	+	130.108i	0.350304	+	0.226274i
$576$		0			0
$577$			1125.87i			1.95124i	0.219462	+	0.975621i	$0.429570\pi$
$577$			1125.87i			1.95124i	−0.219462	+	0.975621i	$0.570430\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	788.736	+	455.377i	1.35755	+	0.783781i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−489.175	−	847.276i	−0.833347	−	1.44340i	−0.895369	−	0.445325i	$-0.853088\pi$
$587$	−489.175	−	847.276i	−0.833347	−	1.44340i	0.0620218	−	0.998075i	$-0.480245\pi$
$588$		0			0
$589$	228.000	−	394.908i	0.387097	−	0.670471i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$	−184.000	+	623.974i	−0.309244	+	1.04870i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−257.196	−	148.492i	−0.429376	−	0.247901i	0.269705	−	0.962943i	$-0.413074\pi$
$599$	−257.196	−	148.492i	−0.429376	−	0.247901i	−0.699081	+	0.715043i	$0.746407\pi$
$600$		0			0
$601$	−10.0000	−	17.3205i	−0.0166389	−	0.0288195i	0.857586	−	0.514341i	$-0.171963\pi$
$601$	−10.0000	−	17.3205i	−0.0166389	−	0.0288195i	−0.874225	+	0.485521i	$0.838630\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−26.9829	−	111.790i	−0.0445999	−	0.184776i
$606$		0			0
$607$	5.87367	+	3.39116i	0.00967656	+	0.00558676i	0.504830	−	0.863219i	$-0.331555\pi$
$607$	5.87367	+	3.39116i	0.00967656	+	0.00558676i	−0.495154	+	0.868805i	$0.664888\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	1561.29i		−	2.55531i
$612$		0			0
$613$		−	47.4763i		−	0.0774491i	−0.999250	−	0.0387246i	$-0.987671\pi$
$613$		−	47.4763i		−	0.0774491i	0.999250	−	0.0387246i	$-0.0123295\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	19.1833	−	33.2265i	0.0310913	−	0.0538517i	−0.850061	−	0.526684i	$-0.823435\pi$
$617$	19.1833	−	33.2265i	0.0310913	−	0.0538517i	0.881152	+	0.472832i	$0.156768\pi$
$618$		0			0
$619$	−89.0000	−	154.153i	−0.143780	−	0.249035i	0.785137	−	0.619322i	$-0.212593\pi$
$619$	−89.0000	−	154.153i	−0.143780	−	0.249035i	−0.928917	+	0.370287i	$0.879259\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−110.304	−	191.052i	−0.177053	−	0.306665i
$624$		0			0
$625$	−621.888	−	62.2893i	−0.995021	−	0.0996629i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			130.108i			0.206848i
$630$		0			0
$631$	−154.000			−0.244057			−0.122029	−	0.992527i	$-0.538940\pi$
$631$	−154.000			−0.244057			−0.122029	+	0.992527i	$0.538940\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−584.333	−	614.252i	−0.920209	−	0.967326i
$636$		0			0
$637$	52.8630	−	30.5205i	0.0829875	−	0.0479128i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−699.329	+	403.758i	−1.09100	+	0.629888i	−0.933842	−	0.357686i	$-0.883566\pi$
$641$	−699.329	+	403.758i	−1.09100	+	0.629888i	−0.157156	+	0.987574i	$0.550232\pi$
$642$		0			0
$643$	−881.051	−	508.675i	−1.37022	−	0.791096i	−0.379263	−	0.925289i	$-0.623823\pi$
$643$	−881.051	−	508.675i	−1.37022	−	0.791096i	−0.990955	+	0.134193i	$0.957156\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	805.700			1.24529			0.622643	−	0.782506i	$-0.286059\pi$
$647$	805.700			1.24529			0.622643	+	0.782506i	$0.286059\pi$
$648$		0			0
$649$	826.000			1.27273
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−33.5708	+	58.1464i	−0.0514101	+	0.0890450i	−0.890585	−	0.454816i	$-0.849705\pi$
$653$	−33.5708	+	58.1464i	−0.0514101	+	0.0890450i	0.839175	+	0.543861i	$0.183038\pi$
$654$		0			0
$655$	−804.219	+	194.116i	−1.22782	+	0.296361i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	454.380	−	262.337i	0.689500	−	0.398083i	−0.113925	−	0.993489i	$-0.536342\pi$
$659$	454.380	−	262.337i	0.689500	−	0.398083i	0.803425	+	0.595406i	$0.203009\pi$
$660$		0			0
$661$	−37.0000	+	64.0859i	−0.0559758	+	0.0969529i	−0.892655	−	0.450740i	$-0.851160\pi$
$661$	−37.0000	+	64.0859i	−0.0559758	+	0.0969529i	0.836680	+	0.547693i	$0.184494\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−115.100	+	390.323i	−0.173083	+	0.586952i
$666$		0			0
$667$			81.3880i			0.122021i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−600.125	−	346.482i	−0.894374	−	0.516367i
$672$		0			0
$673$	599.114	−	345.899i	0.890214	−	0.513966i	0.0162018	−	0.999869i	$-0.494843\pi$
$673$	599.114	−	345.899i	0.890214	−	0.513966i	0.874013	+	0.485903i	$0.161509\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−139.079	−	240.892i	−0.205434	−	0.355823i	0.744837	−	0.667247i	$-0.232527\pi$
$677$	−139.079	−	240.892i	−0.205434	−	0.355823i	−0.950271	+	0.311424i	$0.899194\pi$
$678$		0			0
$679$	322.000	−	557.720i	0.474227	−	0.821385i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−422.033			−0.617911			−0.308955	−	0.951077i	$-0.599979\pi$
$683$	−422.033			−0.617911			−0.308955	+	0.951077i	$0.599979\pi$
$684$		0			0
$685$	−552.000	−	162.776i	−0.805839	−	0.237629i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	34.0000	+	58.8897i	0.0492041	+	0.0852239i	0.889578	−	0.456782i	$-0.150998\pi$
$691$	34.0000	+	58.8897i	0.0492041	+	0.0852239i	−0.840374	+	0.542006i	$0.817665\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−72.7366	−	301.346i	−0.104657	−	0.433591i
$696$		0			0
$697$	1151.24	+	664.668i	1.65171	+	0.953613i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			130.108i			0.185603i	0.995685	+	0.0928015i	$0.0295822\pi$
$701$			130.108i			0.185603i	−0.995685	+	0.0928015i	$0.970418\pi$
$702$		0			0
$703$			81.3880i			0.115772i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−268.567	+	465.171i	−0.379868	+	0.657950i
$708$		0			0
$709$	105.000	+	181.865i	0.148096	+	0.256510i	0.930524	−	0.366232i	$-0.119352\pi$
$709$	105.000	+	181.865i	0.148096	+	0.256510i	−0.782428	+	0.622741i	$0.786019\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	182.242	+	315.652i	0.255598	+	0.442709i
$714$		0			0
$715$	−729.694	+	694.152i	−1.02055	+	0.970841i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	121.622i		−	0.169155i	−0.996417	−	0.0845774i	$-0.973046\pi$
$719$		−	121.622i		−	0.169155i	0.996417	−	0.0845774i	$-0.0269540\pi$
$720$		0			0
$721$	−598.000			−0.829404
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−96.7679	−	188.775i	−0.133473	−	0.260379i
$726$		0			0
$727$	−17.6210	+	10.1735i	−0.0242380	+	0.0139938i	−0.512070	−	0.858944i	$-0.671121\pi$
$727$	−17.6210	+	10.1735i	−0.0242380	+	0.0139938i	0.487832	+	0.872938i	$0.337788\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1126.77	+	650.538i	−1.54140	+	0.889929i
$732$		0			0
$733$	1227.60	+	708.753i	1.67476	+	0.966922i	0.964916	+	0.262560i	$0.0845668\pi$
$733$	1227.60	+	708.753i	1.67476	+	0.966922i	0.709842	+	0.704361i	$0.248767\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1074.27			−1.45762
$738$		0			0
$739$	416.000			0.562923			0.281461	−	0.959573i	$-0.409181\pi$
$739$	416.000			0.562923			0.281461	+	0.959573i	$0.409181\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	402.850	−	697.756i	0.542194	−	0.939107i	−0.456584	−	0.889680i	$-0.650927\pi$
$743$	402.850	−	697.756i	0.542194	−	0.939107i	0.998778	−	0.0494266i	$-0.0157394\pi$
$744$		0			0
$745$	−769.851	+	185.821i	−1.03336	+	0.249424i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	338.030	−	195.161i	0.451308	−	0.260563i
$750$		0			0
$751$	−43.0000	+	74.4782i	−0.0572570	+	0.0991720i	−0.893233	−	0.449594i	$-0.851569\pi$
$751$	−43.0000	+	74.4782i	−0.0572570	+	0.0991720i	0.835976	+	0.548766i	$0.184902\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	335.708	+	98.9949i	0.444647	+	0.131119i
$756$		0			0
$757$			47.4763i			0.0627164i	0.999508	+	0.0313582i	$0.00998326\pi$
$757$			47.4763i			0.0627164i	−0.999508	+	0.0313582i	$0.990017\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	319.658	+	184.555i	0.420050	+	0.242516i	0.695099	−	0.718914i	$-0.255361\pi$
$761$	319.658	+	184.555i	0.420050	+	0.242516i	−0.275048	+	0.961430i	$0.588694\pi$
$762$		0			0
$763$	434.652	−	250.946i	0.569661	−	0.328894i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	848.862	+	1470.27i	1.10673	+	1.91691i
$768$		0			0
$769$	−192.000	+	332.554i	−0.249675	+	0.432450i	−0.963436	−	0.267940i	$-0.913657\pi$
$769$	−192.000	+	332.554i	−0.249675	+	0.432450i	0.713761	+	0.700390i	$0.246990\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−201.425			−0.260576			−0.130288	−	0.991476i	$-0.541590\pi$
$773$	−201.425			−0.260576			−0.130288	+	0.991476i	$0.541590\pi$
$774$		0			0
$775$	−798.000	−	515.457i	−1.02968	−	0.665106i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	720.150	+	415.779i	0.924454	+	0.533734i
$780$		0			0
$781$	588.000	+	1018.45i	0.752881	+	1.30403i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−230.755	+	55.6978i	−0.293955	+	0.0709526i
$786$		0			0
$787$	−411.157	−	237.382i	−0.522436	−	0.301628i	0.215495	−	0.976505i	$-0.430864\pi$
$787$	−411.157	−	237.382i	−0.522436	−	0.301628i	−0.737931	+	0.674877i	$0.764197\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1170.97i			1.48037i
$792$		0			0
$793$		−	1424.29i		−	1.79608i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	249.383	−	431.944i	0.312902	−	0.541963i	−0.666087	−	0.745874i	$-0.732032\pi$
$797$	249.383	−	431.944i	0.312902	−	0.541963i	0.978989	+	0.203911i	$0.0653654\pi$
$798$		0			0
$799$	736.000	+	1274.79i	0.921151	+	1.59548i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−67.1416	−	116.293i	−0.0836135	−	0.144823i
$804$		0			0
$805$	−224.189	−	235.668i	−0.278495	−	0.292755i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1480.68i			1.83026i	0.403157	+	0.915131i	$0.367913\pi$
$809$			1480.68i			1.83026i	−0.403157	+	0.915131i	$0.632087\pi$
$810$		0			0
$811$	−602.000			−0.742293			−0.371147	−	0.928574i	$-0.621035\pi$
$811$	−602.000			−0.742293			−0.371147	+	0.928574i	$0.621035\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	327.226	+	343.981i	0.401505	+	0.422063i
$816$		0			0
$817$	−704.840	+	406.940i	−0.862718	+	0.498090i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1217.40	−	702.864i	1.48282	−	0.856107i	0.483012	−	0.875614i	$-0.339543\pi$
$821$	1217.40	−	702.864i	1.48282	−	0.856107i	0.999810	+	0.0195065i	$0.00620952\pi$
$822$		0			0
$823$	1110.12	+	640.930i	1.34887	+	0.778773i	0.988090	−	0.153876i	$-0.0491757\pi$
$823$	1110.12	+	640.930i	1.34887	+	0.778773i	0.360784	+	0.932649i	$0.382509\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1170.18			1.41497			0.707487	−	0.706727i	$-0.249829\pi$
$827$	1170.18			1.41497			0.707487	+	0.706727i	$0.249829\pi$
$828$		0			0
$829$	1074.00			1.29554			0.647768	−	0.761837i	$-0.275702\pi$
$829$	1074.00			1.29554			0.647768	+	0.761837i	$0.275702\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−28.7750	+	49.8397i	−0.0345438	+	0.0598316i
$834$		0			0
$835$	−236.306	−	979.009i	−0.283001	−	1.17247i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−644.216	+	371.938i	−0.767838	+	0.443311i	−0.832103	−	0.554622i	$-0.812863\pi$
$839$	−644.216	+	371.938i	−0.767838	+	0.443311i	0.0642650	+	0.997933i	$0.479530\pi$
$840$		0			0
$841$	−384.500	+	665.974i	−0.457194	+	0.791883i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1174.98	−	346.482i	−1.39051	−	0.410038i
$846$		0			0
$847$			155.994i			0.184172i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−56.3383	−	32.5269i	−0.0662024	−	0.0382220i
$852$		0			0
$853$	146.842	−	84.7791i	0.172147	−	0.0993894i	−0.411451	−	0.911432i	$-0.634978\pi$
$853$	146.842	−	84.7791i	0.172147	−	0.0993894i	0.583598	+	0.812043i	$0.301644\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−57.5500	−	99.6795i	−0.0671528	−	0.116312i	0.830494	−	0.557027i	$-0.188058\pi$
$857$	−57.5500	−	99.6795i	−0.0671528	−	0.116312i	−0.897647	+	0.440715i	$0.854725\pi$
$858$		0			0
$859$	−77.0000	+	133.368i	−0.0896391	+	0.155260i	−0.907359	−	0.420357i	$-0.861905\pi$
$859$	−77.0000	+	133.368i	−0.0896391	+	0.155260i	0.817719	+	0.575617i	$0.195238\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1064.67			1.23369			0.616845	−	0.787085i	$-0.288411\pi$
$863$	1064.67			1.23369			0.616845	+	0.787085i	$0.288411\pi$
$864$		0			0
$865$	−1288.00	−	379.810i	−1.48902	−	0.439087i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	257.196	+	148.492i	0.295968	+	0.170877i
$870$		0			0
$871$	−1104.00	−	1912.18i	−1.26751	−	2.19539i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	800.196	+	280.065i	0.914510	+	0.320074i
$876$		0			0
$877$	1204.10	+	695.189i	1.37298	+	0.792690i	0.991302	−	0.131606i	$-0.0420135\pi$
$877$	1204.10	+	695.189i	1.37298	+	0.792690i	0.381677	+	0.924296i	$0.375347\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	861.256i		−	0.977589i	−0.872399	−	0.488795i	$-0.837437\pi$
$881$		−	861.256i		−	0.977589i	0.872399	−	0.488795i	$-0.162563\pi$
$882$		0			0
$883$			1017.35i			1.15215i	0.817396	+	0.576076i	$0.195417\pi$
$883$			1017.35i			1.15215i	−0.817396	+	0.576076i	$0.804583\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	302.137	−	523.317i	0.340628	−	0.589986i	−0.643921	−	0.765092i	$-0.722694\pi$
$887$	302.137	−	523.317i	0.340628	−	0.589986i	0.984550	+	0.175106i	$0.0560269\pi$
$888$		0			0
$889$	575.000	+	995.929i	0.646794	+	1.12028i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	460.400	+	797.436i	0.515565	+	0.892985i
$894$		0			0
$895$	−462.999	−	486.706i	−0.517317	−	0.543805i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	322.441i		−	0.358666i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	79.6985	−	75.8165i	0.0880647	−	0.0837752i
$906$		0			0
$907$	810.566	−	467.981i	0.893679	−	0.515966i	0.0185346	−	0.999828i	$-0.494100\pi$
$907$	810.566	−	467.981i	0.893679	−	0.515966i	0.875144	+	0.483863i	$0.160767\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	661.362	−	381.838i	0.725974	−	0.419141i	−0.0909736	−	0.995853i	$-0.528998\pi$
$911$	661.362	−	381.838i	0.725974	−	0.419141i	0.816947	+	0.576712i	$0.195665\pi$
$912$		0			0
$913$	−1151.24	−	664.668i	−1.26094	−	0.728005i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1122.22			1.22380
$918$		0			0
$919$	−54.0000			−0.0587595			−0.0293798	−	0.999568i	$-0.509353\pi$
$919$	−54.0000			−0.0587595			−0.0293798	+	0.999568i	$0.509353\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1208.55	+	2093.27i	−1.30937	+	2.26790i
$924$		0			0
$925$	169.347	+	8.45987i	0.183078	+	0.00914581i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−128.598	+	74.2462i	−0.138426	+	0.0799206i	−0.567614	−	0.823295i	$-0.692133\pi$
$929$	−128.598	+	74.2462i	−0.138426	+	0.0799206i	0.429187	+	0.903215i	$0.358800\pi$
$930$		0			0
$931$	−18.0000	+	31.1769i	−0.0193340	+	0.0334876i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	268.567	−	910.754i	0.287237	−	0.974068i
$936$		0			0
$937$			786.750i			0.839648i	0.907606	+	0.419824i	$0.137908\pi$
$937$			786.750i			0.839648i	−0.907606	+	0.419824i	$0.862092\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−296.388	−	171.120i	−0.314972	−	0.181849i	0.334177	−	0.942510i	$-0.391542\pi$
$941$	−296.388	−	171.120i	−0.314972	−	0.181849i	−0.649149	+	0.760661i	$0.724875\pi$
$942$		0			0
$943$	−575.620	+	332.334i	−0.610413	+	0.352422i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	201.425	+	348.878i	0.212698	+	0.368404i	0.952558	−	0.304357i	$-0.0984416\pi$
$947$	201.425	+	348.878i	0.212698	+	0.368404i	−0.739860	+	0.672761i	$0.765108\pi$
$948$		0			0
$949$	138.000	−	239.023i	0.145416	−	0.251868i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1074.27			−1.12725			−0.563623	−	0.826032i	$-0.690593\pi$
$953$	−1074.27			−1.12725			−0.563623	+	0.826032i	$0.690593\pi$
$954$		0			0
$955$	−276.000	+	935.962i	−0.289005	+	0.980064i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	676.059	+	390.323i	0.704963	+	0.407010i
$960$		0			0
$961$	−241.500	−	418.290i	−0.251301	−	0.435266i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1318.60	+	318.273i	−1.36642	+	0.329817i
$966$		0			0
$967$	−346.547	−	200.079i	−0.358373	−	0.206907i	0.309994	−	0.950739i	$-0.399673\pi$
$967$	−346.547	−	200.079i	−0.358373	−	0.206907i	−0.668367	+	0.743832i	$0.733006\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			683.065i			0.703466i	0.936100	+	0.351733i	$0.114407\pi$
$971$			683.065i			0.703466i	−0.936100	+	0.351733i	$0.885593\pi$
$972$		0			0
$973$			420.504i			0.432173i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−805.700	+	1395.51i	−0.824667	+	1.42837i	0.0775065	+	0.996992i	$0.475304\pi$
$977$	−805.700	+	1395.51i	−0.824667	+	1.42837i	−0.902174	+	0.431373i	$0.858029\pi$
$978$		0			0
$979$	161.000	+	278.860i	0.164454	+	0.284842i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	479.583	+	830.662i	0.487877	+	0.845028i	0.999903	−	0.0139422i	$-0.00443810\pi$
$983$	479.583	+	830.662i	0.487877	+	0.845028i	−0.512026	+	0.858970i	$0.671105\pi$
$984$		0			0
$985$	590.705	−	561.932i	0.599700	−	0.570490i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	650.538i		−	0.657774i
$990$		0			0
$991$	1054.00			1.06357			0.531786	−	0.846879i	$-0.321521\pi$
$991$	1054.00			1.06357			0.531786	+	0.846879i	$0.321521\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1065.06	+	1013.18i	−1.07041	+	1.01828i
$996$		0			0
$997$	1039.64	−	600.236i	1.04277	−	0.602042i	0.122152	−	0.992511i	$-0.461021\pi$
$997$	1039.64	−	600.236i	1.04277	−	0.602042i	0.920616	+	0.390469i	$0.127687\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.t.c.1349.1		8
3.2	odd	2	inner	1620.3.t.c.1349.4		8
5.4	even	2	inner	1620.3.t.c.1349.3		8
9.2	odd	6	inner	1620.3.t.c.269.3		8
9.4	even	3		180.3.b.a.89.4	yes	4
9.5	odd	6		180.3.b.a.89.1	✓	4
9.7	even	3	inner	1620.3.t.c.269.2		8
15.14	odd	2	inner	1620.3.t.c.1349.2		8
36.23	even	6		720.3.c.b.449.1		4
36.31	odd	6		720.3.c.b.449.4		4
45.4	even	6		180.3.b.a.89.2	yes	4
45.13	odd	12		900.3.g.c.701.3		4
45.14	odd	6		180.3.b.a.89.3	yes	4
45.22	odd	12		900.3.g.c.701.1		4
45.23	even	12		900.3.g.c.701.4		4
45.29	odd	6	inner	1620.3.t.c.269.1		8
45.32	even	12		900.3.g.c.701.2		4
45.34	even	6	inner	1620.3.t.c.269.4		8
72.5	odd	6		2880.3.c.c.449.4		4
72.13	even	6		2880.3.c.c.449.1		4
72.59	even	6		2880.3.c.f.449.4		4
72.67	odd	6		2880.3.c.f.449.1		4
180.23	odd	12		3600.3.l.r.1601.1		4
180.59	even	6		720.3.c.b.449.3		4
180.67	even	12		3600.3.l.r.1601.4		4
180.103	even	12		3600.3.l.r.1601.2		4
180.139	odd	6		720.3.c.b.449.2		4
180.167	odd	12		3600.3.l.r.1601.3		4
360.59	even	6		2880.3.c.f.449.2		4
360.139	odd	6		2880.3.c.f.449.3		4
360.149	odd	6		2880.3.c.c.449.2		4
360.229	even	6		2880.3.c.c.449.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.b.a.89.1	✓	4	9.5	odd	6
180.3.b.a.89.2	yes	4	45.4	even	6
180.3.b.a.89.3	yes	4	45.14	odd	6
180.3.b.a.89.4	yes	4	9.4	even	3
720.3.c.b.449.1		4	36.23	even	6
720.3.c.b.449.2		4	180.139	odd	6
720.3.c.b.449.3		4	180.59	even	6
720.3.c.b.449.4		4	36.31	odd	6
900.3.g.c.701.1		4	45.22	odd	12
900.3.g.c.701.2		4	45.32	even	12
900.3.g.c.701.3		4	45.13	odd	12
900.3.g.c.701.4		4	45.23	even	12
1620.3.t.c.269.1		8	45.29	odd	6	inner
1620.3.t.c.269.2		8	9.7	even	3	inner
1620.3.t.c.269.3		8	9.2	odd	6	inner
1620.3.t.c.269.4		8	45.34	even	6	inner
1620.3.t.c.1349.1		8	1.1	even	1	trivial
1620.3.t.c.1349.2		8	15.14	odd	2	inner
1620.3.t.c.1349.3		8	5.4	even	2	inner
1620.3.t.c.1349.4		8	3.2	odd	2	inner
2880.3.c.c.449.1		4	72.13	even	6
2880.3.c.c.449.2		4	360.149	odd	6
2880.3.c.c.449.3		4	360.229	even	6
2880.3.c.c.449.4		4	72.5	odd	6
2880.3.c.f.449.1		4	72.67	odd	6
2880.3.c.f.449.2		4	360.59	even	6
2880.3.c.f.449.3		4	360.139	odd	6
2880.3.c.f.449.4		4	72.59	even	6
3600.3.l.r.1601.1		4	180.23	odd	12
3600.3.l.r.1601.2		4	180.103	even	12
3600.3.l.r.1601.3		4	180.167	odd	12
3600.3.l.r.1601.4		4	180.67	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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.62266 + 3.44621i) q^{5} +(5.87367 - 3.39116i) q^{7} +O(q^{10})\)	\(q+(-3.62266 + 3.44621i) q^{5} +(5.87367 - 3.39116i) q^{7} +(-8.57321 + 4.94975i) q^{11} +(-17.6210 - 10.1735i) q^{13} +19.1833 q^{17} +12.0000 q^{19} +(-4.79583 + 8.30662i) q^{23} +(1.24734 - 24.9689i) q^{25} +(7.34847 - 4.24264i) q^{29} +(19.0000 - 32.9090i) q^{31} +(-9.59166 + 32.5269i) q^{35} +6.78233i q^{37} +(60.0125 + 34.6482i) q^{41} +(-58.7367 + 33.9116i) q^{43} +(38.3667 + 66.4530i) q^{47} +(-1.50000 + 2.59808i) q^{49} +(14.0000 - 47.4763i) q^{55} +(-72.2599 - 41.7193i) q^{59} +(35.0000 + 60.6218i) q^{61} +(98.8949 - 23.8705i) q^{65} +(93.9787 + 54.2586i) q^{67} -118.794i q^{71} +13.5647i q^{73} +(-33.5708 + 58.1464i) q^{77} +(-15.0000 - 25.9808i) q^{79} +(67.1416 + 116.293i) q^{83} +(-69.4947 + 66.1097i) q^{85} -32.5269i q^{89} -138.000 q^{91} +(-43.4719 + 41.3545i) q^{95} +(82.2314 - 47.4763i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 96 q^{19} - 84 q^{25} + 152 q^{31} - 12 q^{49} + 112 q^{55} + 280 q^{61} - 120 q^{79} - 368 q^{85} - 1104 q^{91}+O(q^{100}) \) 8 * q + 96 * q^19 - 84 * q^25 + 152 * q^31 - 12 * q^49 + 112 * q^55 + 280 * q^61 - 120 * q^79 - 368 * q^85 - 1104 * q^91

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