LMFDB - Embedded newform 1620.3.t.c.269.4

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			269.4
Root			$3.54921 + 2.04914i$ of defining polynomial
Character	$\chi$	$=$	1620.269
Dual form			1620.3.t.c.1349.4

$q$-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 $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

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

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	0.856957	−	0.515388i	$-0.172352\pi$
$7$	5.87367	+	3.39116i	0.839096	+	0.484452i	−0.0178610	+	0.999840i	$0.505686\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	8.57321	+	4.94975i	0.779383	+	0.449977i	0.836212	−	0.548407i	$-0.184765\pi$
$11$	8.57321	+	4.94975i	0.779383	+	0.449977i	−0.0568285	+	0.998384i	$0.518099\pi$
$12$		0			0
$13$	−17.6210	+	10.1735i	−1.35546	+	0.782577i	−0.989008	−	0.147860i	$-0.952762\pi$
$13$	−17.6210	+	10.1735i	−1.35546	+	0.782577i	−0.366454	+	0.930436i	$0.619428\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−19.1833			−1.12843			−0.564215	−	0.825628i	$-0.690821\pi$
$17$	−19.1833			−1.12843			−0.564215	+	0.825628i	$0.690821\pi$
$18$		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	$-0.0998038\pi$
$23$	4.79583	+	8.30662i	0.208514	+	0.361158i	−0.742732	+	0.669588i	$0.766471\pi$
$24$		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	$-0.380069\pi$
$29$	−7.34847	−	4.24264i	−0.253395	−	0.146298i	−0.621318	+	0.783558i	$0.713402\pi$
$30$		0			0
$31$	19.0000	+	32.9090i	0.612903	+	1.06158i	0.990748	+	0.135711i	$0.0433318\pi$
$31$	19.0000	+	32.9090i	0.612903	+	1.06158i	−0.377845	+	0.925869i	$0.623335\pi$
$32$		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	$-0.970785\pi$
$37$		−	6.78233i		−	0.183306i	0.995791	−	0.0916531i	$-0.0292151\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−60.0125	+	34.6482i	−1.46372	+	0.845079i	−0.999181	−	0.0404739i	$-0.987113\pi$
$41$	−60.0125	+	34.6482i	−1.46372	+	0.845079i	−0.464539	+	0.885553i	$0.653780\pi$
$42$		0			0
$43$	−58.7367	−	33.9116i	−1.36597	−	0.788643i	−0.375559	−	0.926798i	$-0.622549\pi$
$43$	−58.7367	−	33.9116i	−1.36597	−	0.788643i	−0.990411	+	0.138155i	$0.955883\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−38.3667	+	66.4530i	−0.816312	+	1.41389i	0.0920704	+	0.995752i	$0.470652\pi$
$47$	−38.3667	+	66.4530i	−0.816312	+	1.41389i	−0.908382	+	0.418141i	$0.862682\pi$
$48$		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	$-0.416667\pi$
$59$	72.2599	−	41.7193i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$60$		0			0
$61$	35.0000	−	60.6218i	0.573770	−	0.993800i	−0.422404	−	0.906408i	$-0.638813\pi$
$61$	35.0000	−	60.6218i	0.573770	−	0.993800i	0.996174	−	0.0873918i	$-0.0278532\pi$
$62$		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	0.408002	−	0.912981i	$-0.366226\pi$
$67$	93.9787	−	54.2586i	1.40267	−	0.809830i	0.994666	+	0.103151i	$0.0328924\pi$
$68$		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	$-0.970384\pi$
$73$		−	13.5647i		−	0.185817i	0.995675	−	0.0929086i	$-0.0296164\pi$
$74$		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	−0.945208	−	0.326469i	$-0.894141\pi$
$79$	−15.0000	+	25.9808i	−0.189873	+	0.328870i	0.755334	+	0.655339i	$0.227474\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−67.1416	+	116.293i	−0.808935	+	1.40112i	0.104666	+	0.994507i	$0.466623\pi$
$83$	−67.1416	+	116.293i	−0.808935	+	1.40112i	−0.913602	+	0.406610i	$0.866711\pi$
$84$		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	0.859890	−	0.510480i	$-0.170532\pi$
$97$	82.2314	+	47.4763i	0.847746	+	0.489446i	−0.0121436	+	0.999926i	$0.503866\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	68.5857	+	39.5980i	0.679066	+	0.392059i	0.799503	−	0.600662i	$-0.205096\pi$
$101$	68.5857	+	39.5980i	0.679066	+	0.392059i	−0.120437	+	0.992721i	$0.538430\pi$
$102$		0			0
$103$	−76.3577	+	44.0851i	−0.741337	+	0.428011i	−0.822555	−	0.568685i	$-0.807452\pi$
$103$	−76.3577	+	44.0851i	−0.741337	+	0.428011i	0.0812182	+	0.996696i	$0.474119\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−57.5500			−0.537850			−0.268925	−	0.963161i	$-0.586668\pi$
$107$	−57.5500			−0.537850			−0.268925	+	0.963161i	$0.586668\pi$
$108$		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	0.940806	+	0.338945i	$0.110070\pi$
$113$	86.3250	+	149.519i	0.763938	+	1.32318i	−0.176869	+	0.984234i	$0.556597\pi$
$114$		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	$-0.767342\pi$
$127$		−	169.558i		−	1.33510i	0.744563	−	0.667552i	$-0.232658\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−143.295	+	82.7315i	−1.09386	+	0.631538i	−0.934600	−	0.355699i	$-0.884243\pi$
$131$	−143.295	+	82.7315i	−1.09386	+	0.631538i	−0.159256	+	0.987237i	$0.550909\pi$
$132$		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	$-0.971329\pi$
$137$	−57.5500	+	99.6795i	−0.420073	+	0.727587i	0.575873	+	0.817539i	$0.304662\pi$
$138$		0			0
$139$	−31.0000	−	53.6936i	−0.223022	−	0.386285i	0.732702	−	0.680549i	$-0.238259\pi$
$139$	−31.0000	−	53.6936i	−0.223022	−	0.386285i	−0.955724	+	0.294264i	$0.904925\pi$
$140$		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	$-0.845044\pi$
$149$	−137.171	+	79.1960i	−0.920614	+	0.531517i	−0.0367828	+	0.999323i	$0.511711\pi$
$150$		0			0
$151$	−35.0000	+	60.6218i	−0.231788	+	0.401469i	−0.958334	−	0.285649i	$-0.907791\pi$
$151$	−35.0000	+	60.6218i	−0.231788	+	0.401469i	0.726546	+	0.687118i	$0.241124\pi$
$152$		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	−0.363310	−	0.931668i	$-0.618353\pi$
$157$	41.1157	−	23.7382i	0.261883	−	0.151198i	0.625193	+	0.780470i	$0.285020\pi$
$158$		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	$0.0940764\pi$
$163$			94.9526i			0.582531i	−0.956642	+	0.291266i	$0.905924\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	100.712	+	174.439i	0.603069	+	1.04455i	0.992354	+	0.123428i	$0.0393888\pi$
$167$	100.712	+	174.439i	0.603069	+	1.04455i	−0.389285	+	0.921117i	$0.627278\pi$
$168$		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	0.157911	+	0.987453i	$0.449524\pi$
$173$	−134.283	+	232.585i	−0.776204	+	1.34442i	−0.934115	+	0.356971i	$0.883809\pi$
$174$		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	$-0.504019\pi$
$191$	−169.015	−	97.5807i	−0.884894	−	0.510894i	−0.872269	+	0.489026i	$0.837352\pi$
$192$		0			0
$193$	234.947	−	135.647i	1.21734	−	0.702832i	0.252993	−	0.967468i	$-0.418585\pi$
$193$	234.947	−	135.647i	1.21734	−	0.702832i	0.964348	+	0.264636i	$0.0852518\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	163.058			0.827707			0.413853	−	0.910344i	$-0.364183\pi$
$197$	163.058			0.827707			0.413853	+	0.910344i	$0.364183\pi$
$198$		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	0.911489	−	0.411325i	$-0.134934\pi$
$211$	21.0000	+	36.3731i	0.0995261	+	0.172384i	−0.811963	+	0.583710i	$0.801601\pi$
$212$		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	0.830673	−	0.556761i	$-0.187956\pi$
$223$	170.336	+	98.3438i	0.763841	+	0.441004i	−0.0668324	+	0.997764i	$0.521289\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−134.283	+	232.585i	−0.591556	+	1.02461i	0.402467	+	0.915435i	$0.368153\pi$
$227$	−134.283	+	232.585i	−0.591556	+	1.02461i	−0.994023	+	0.109171i	$0.965180\pi$
$228$		0			0
$229$	211.000	+	365.463i	0.921397	+	1.59591i	0.797255	+	0.603643i	$0.206285\pi$
$229$	211.000	+	365.463i	0.921397	+	1.59591i	0.124142	+	0.992264i	$0.460382\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	211.017			0.905651			0.452825	−	0.891599i	$-0.350416\pi$
$233$	211.017			0.905651			0.452825	+	0.891599i	$0.350416\pi$
$234$		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	$-0.713928\pi$
$239$	−61.2372	+	35.3553i	−0.256223	+	0.147930i	0.366388	+	0.930462i	$0.380594\pi$
$240$		0			0
$241$	−140.000	+	242.487i	−0.580913	+	1.00617i	0.414459	+	0.910068i	$0.363971\pi$
$241$	−140.000	+	242.487i	−0.580913	+	1.00617i	−0.995371	+	0.0961024i	$0.969362\pi$
$242$		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	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		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	$-0.792591\pi$
$263$	33.5708	−	58.1464i	0.127646	−	0.221089i	0.922764	+	0.385365i	$0.125925\pi$
$264$		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	0.446051	−	0.895008i	$-0.352830\pi$
$277$	−29.3684	−	16.9558i	−0.106023	−	0.0612124i	−0.552074	+	0.833795i	$0.686163\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−282.916	−	163.342i	−1.00682	−	0.581287i	−0.0965601	−	0.995327i	$-0.530784\pi$
$281$	−282.916	−	163.342i	−1.00682	−	0.581287i	−0.910259	+	0.414040i	$0.864117\pi$
$282$		0			0
$283$	−93.9787	+	54.2586i	−0.332080	+	0.191727i	−0.656764	−	0.754096i	$-0.728075\pi$
$283$	−93.9787	+	54.2586i	−0.332080	+	0.191727i	0.324684	+	0.945823i	$0.394742\pi$
$284$		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	$-0.0549767\pi$
$293$	100.712	+	174.439i	0.343729	+	0.595355i	−0.641393	+	0.767212i	$0.721643\pi$
$294$		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	$-0.914593\pi$
$307$		−	162.776i		−	0.530215i	0.964219	−	0.265107i	$-0.0854074\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	154.318	−	89.0955i	0.496199	−	0.286481i	−0.230944	−	0.972967i	$-0.574181\pi$
$311$	154.318	−	89.0955i	0.496199	−	0.286481i	0.727142	+	0.686487i	$0.240848\pi$
$312$		0			0
$313$	46.9894	+	27.1293i	0.150126	+	0.0866751i	0.573181	−	0.819429i	$-0.305709\pi$
$313$	46.9894	+	27.1293i	0.150126	+	0.0866751i	−0.423055	+	0.906104i	$0.639042\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	139.079	−	240.892i	0.438735	−	0.759912i	−0.558857	−	0.829264i	$-0.688760\pi$
$317$	139.079	−	240.892i	0.438735	−	0.759912i	0.997592	+	0.0693522i	$0.0220932\pi$
$318$		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	−0.894655	−	0.446759i	$-0.852578\pi$
$331$	−20.0000	+	34.6410i	−0.0604230	+	0.104656i	0.834232	+	0.551414i	$0.185912\pi$
$332$		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	−0.844082	−	0.536215i	$-0.819854\pi$
$337$	−270.189	+	155.994i	−0.801747	+	0.462889i	0.0423345	+	0.999103i	$0.486520\pi$
$338$		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	−0.995423	−	0.0955674i	$-0.969533\pi$
$347$	−201.425	−	348.878i	−0.580475	−	1.00541i	0.414948	−	0.909845i	$-0.363800\pi$
$348$		0			0
$349$	203.000	−	351.606i	0.581662	−	1.00747i	−0.413621	−	0.910449i	$-0.635736\pi$
$349$	203.000	−	351.606i	0.581662	−	1.00747i	0.995283	−	0.0970186i	$-0.0309306\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	268.567	−	465.171i	0.760812	−	1.31776i	−0.181621	−	0.983369i	$-0.558134\pi$
$353$	268.567	−	465.171i	0.760812	−	1.31776i	0.942433	−	0.334396i	$-0.108532\pi$
$354$		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	0.765954	−	0.642895i	$-0.222267\pi$
$367$	217.326	+	125.473i	0.592168	+	0.341889i	−0.173786	+	0.984783i	$0.555600\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−88.1051	+	50.8675i	−0.236207	+	0.136374i	−0.613432	−	0.789748i	$-0.710212\pi$
$373$	−88.1051	+	50.8675i	−0.236207	+	0.136374i	0.377225	+	0.926121i	$0.376878\pi$
$374$		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	$-0.280692\pi$
$383$	−134.283	−	232.585i	−0.350609	−	0.607273i	−0.986356	+	0.164625i	$0.947359\pi$
$384$		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	$-0.113833\pi$
$389$	428.661	+	247.487i	1.10196	+	0.636214i	0.165222	+	0.986256i	$0.447166\pi$
$390$		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	$0.113894\pi$
$397$			278.076i			0.700442i	−0.936667	+	0.350221i	$0.886106\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	214.330	−	123.744i	0.534490	−	0.308588i	−0.208353	−	0.978054i	$-0.566810\pi$
$401$	214.330	−	123.744i	0.534490	−	0.308588i	0.742843	+	0.669466i	$0.233477\pi$
$402$		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	0.975181	−	0.221410i	$-0.0710660\pi$
$409$	121.000	+	209.578i	0.295844	+	0.512416i	−0.679337	+	0.733826i	$0.737733\pi$
$410$		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	0.651115	−	0.758979i	$-0.274302\pi$
$419$	684.632	−	395.273i	1.63397	−	0.943372i	0.982853	−	0.184392i	$-0.0590317\pi$
$420$		0			0
$421$	−179.000	+	310.037i	−0.425178	+	0.736430i	−0.996437	−	0.0843398i	$-0.973122\pi$
$421$	−179.000	+	310.037i	−0.425178	+	0.736430i	0.571259	+	0.820770i	$0.306455\pi$
$422$		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	$-0.903811\pi$
$433$		−	257.729i		−	0.595216i	0.954688	−	0.297608i	$-0.0961888\pi$
$434$		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	−0.142597	−	0.989781i	$-0.545545\pi$
$439$	345.000	−	597.558i	0.785877	−	1.36118i	0.928474	−	0.371398i	$-0.121121\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−201.425	+	348.878i	−0.454684	+	0.787535i	−0.998670	−	0.0515587i	$-0.983581\pi$
$443$	−201.425	+	348.878i	−0.454684	+	0.787535i	0.543986	+	0.839094i	$0.316914\pi$
$444$		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	0.142995	−	0.989723i	$-0.454327\pi$
$457$	−293.684	−	169.558i	−0.642633	−	0.371025i	−0.785628	+	0.618699i	$0.787660\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	558.484	+	322.441i	1.21146	+	0.699438i	0.963078	−	0.269224i	$-0.0867673\pi$
$461$	558.484	+	322.441i	1.21146	+	0.699438i	0.248384	+	0.968662i	$0.420101\pi$
$462$		0			0
$463$	229.073	−	132.255i	0.494758	−	0.285649i	−0.231788	−	0.972766i	$-0.574458\pi$
$463$	229.073	−	132.255i	0.494758	−	0.285649i	0.726546	+	0.687117i	$0.241124\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	460.400			0.985867			0.492933	−	0.870067i	$-0.335925\pi$
$467$	460.400			0.985867			0.492933	+	0.870067i	$0.335925\pi$
$468$		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	$-0.306989\pi$
$479$	68.5857	+	39.5980i	0.143185	+	0.0826680i	−0.426696	+	0.904395i	$0.640323\pi$
$480$		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	$0.303107\pi$
$487$			793.533i			1.62943i	−0.579861	+	0.814715i	$0.696893\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−488.673	+	282.136i	−0.995261	+	0.574614i	−0.906843	−	0.421469i	$-0.861515\pi$
$491$	−488.673	+	282.136i	−0.995261	+	0.574614i	−0.0884184	+	0.996083i	$0.528181\pi$
$492$		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	0.899841	−	0.436218i	$-0.143682\pi$
$499$	36.0000	+	62.3538i	0.0721443	+	0.124958i	−0.827697	+	0.561176i	$0.810349\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−230.200			−0.457654			−0.228827	−	0.973467i	$-0.573489\pi$
$503$	−230.200			−0.457654			−0.228827	+	0.973467i	$0.573489\pi$
$504$		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	$-0.506052\pi$
$509$	426.211	−	246.073i	0.837350	−	0.483444i	0.856363	+	0.516375i	$0.172719\pi$
$510$		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	$0.105096\pi$
$523$			339.116i			0.648406i	−0.945987	+	0.324203i	$0.894904\pi$
$524$		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	0.671312	−	0.741175i	$-0.265731\pi$
$547$	199.705	+	115.300i	0.365091	+	0.210785i	−0.306221	+	0.951961i	$0.599065\pi$
$548$		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	$-0.287162\pi$
$557$	690.600			1.23986			0.619928	+	0.784659i	$0.287162\pi$
$558$		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	$-0.161243\pi$
$563$	9.59166	+	16.6132i	0.0170367	+	0.0295084i	−0.857381	+	0.514682i	$0.827910\pi$
$564$		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	$-0.302828\pi$
$569$	94.3054	+	54.4472i	0.165739	+	0.0956893i	−0.414836	+	0.909896i	$0.636161\pi$
$570$		0			0
$571$	352.000	+	609.682i	0.616462	+	1.06774i	0.990126	+	0.140180i	$0.0447681\pi$
$571$	352.000	+	609.682i	0.616462	+	1.06774i	−0.373664	+	0.927564i	$0.621899\pi$
$572$		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	$-0.570430\pi$
$577$		−	1125.87i		−	1.95124i	0.219462	−	0.975621i	$-0.429570\pi$
$578$		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	−0.0620218	−	0.998075i	$-0.519755\pi$
$587$	489.175	−	847.276i	0.833347	−	1.44340i	0.895369	−	0.445325i	$-0.146912\pi$
$588$		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	$-0.586926\pi$
$599$	257.196	−	148.492i	0.429376	−	0.247901i	0.699081	+	0.715043i	$0.253593\pi$
$600$		0			0
$601$	−10.0000	+	17.3205i	−0.0166389	+	0.0288195i	−0.874225	−	0.485521i	$-0.838630\pi$
$601$	−10.0000	+	17.3205i	−0.0166389	+	0.0288195i	0.857586	+	0.514341i	$0.171963\pi$
$602$		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	−0.495154	−	0.868805i	$-0.664888\pi$
$607$	5.87367	−	3.39116i	0.00967656	−	0.00558676i	0.504830	+	0.863219i	$0.331555\pi$
$608$		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	$0.0123295\pi$
$613$			47.4763i			0.0774491i	−0.999250	+	0.0387246i	$0.987671\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−19.1833	−	33.2265i	−0.0310913	−	0.0538517i	0.850061	−	0.526684i	$-0.176565\pi$
$617$	−19.1833	−	33.2265i	−0.0310913	−	0.0538517i	−0.881152	+	0.472832i	$0.843232\pi$
$618$		0			0
$619$	−89.0000	+	154.153i	−0.143780	+	0.249035i	−0.928917	−	0.370287i	$-0.879259\pi$
$619$	−89.0000	+	154.153i	−0.143780	+	0.249035i	0.785137	+	0.619322i	$0.212593\pi$
$620$		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	$-0.116434\pi$
$641$	699.329	+	403.758i	1.09100	+	0.629888i	0.157156	+	0.987574i	$0.449768\pi$
$642$		0			0
$643$	−881.051	+	508.675i	−1.37022	+	0.791096i	−0.990955	−	0.134193i	$-0.957156\pi$
$643$	−881.051	+	508.675i	−1.37022	+	0.791096i	−0.379263	+	0.925289i	$0.623823\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−805.700			−1.24529			−0.622643	−	0.782506i	$-0.713941\pi$
$647$	−805.700			−1.24529			−0.622643	+	0.782506i	$0.713941\pi$
$648$		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	$-0.150295\pi$
$653$	33.5708	+	58.1464i	0.0514101	+	0.0890450i	−0.839175	+	0.543861i	$0.816962\pi$
$654$		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	$-0.463658\pi$
$659$	−454.380	−	262.337i	−0.689500	−	0.398083i	−0.803425	+	0.595406i	$0.796991\pi$
$660$		0			0
$661$	−37.0000	−	64.0859i	−0.0559758	−	0.0969529i	0.836680	−	0.547693i	$-0.184494\pi$
$661$	−37.0000	−	64.0859i	−0.0559758	−	0.0969529i	−0.892655	+	0.450740i	$0.851160\pi$
$662$		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	0.874013	−	0.485903i	$-0.161509\pi$
$673$	599.114	+	345.899i	0.890214	+	0.513966i	0.0162018	+	0.999869i	$0.494843\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	139.079	−	240.892i	0.205434	−	0.355823i	−0.744837	−	0.667247i	$-0.767473\pi$
$677$	139.079	−	240.892i	0.205434	−	0.355823i	0.950271	+	0.311424i	$0.100806\pi$
$678$		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	$-0.400021\pi$
$683$	422.033			0.617911			0.308955	+	0.951077i	$0.400021\pi$
$684$		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	−0.840374	−	0.542006i	$-0.817665\pi$
$691$	34.0000	−	58.8897i	0.0492041	−	0.0852239i	0.889578	+	0.456782i	$0.150998\pi$
$692$		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	−0.782428	−	0.622741i	$-0.786019\pi$
$709$	105.000	−	181.865i	0.148096	−	0.256510i	0.930524	+	0.366232i	$0.119352\pi$
$710$		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	0.487832	−	0.872938i	$-0.337788\pi$
$727$	−17.6210	−	10.1735i	−0.0242380	−	0.0139938i	−0.512070	+	0.858944i	$0.671121\pi$
$728$		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	0.709842	−	0.704361i	$-0.248767\pi$
$733$	1227.60	−	708.753i	1.67476	−	0.966922i	0.964916	−	0.262560i	$-0.0845668\pi$
$734$		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	−0.998778	−	0.0494266i	$-0.984261\pi$
$743$	−402.850	−	697.756i	−0.542194	−	0.939107i	0.456584	−	0.889680i	$-0.349073\pi$
$744$		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	0.835976	−	0.548766i	$-0.184902\pi$
$751$	−43.0000	−	74.4782i	−0.0572570	−	0.0991720i	−0.893233	+	0.449594i	$0.851569\pi$
$752$		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	$-0.990017\pi$
$757$		−	47.4763i		−	0.0627164i	0.999508	−	0.0313582i	$-0.00998326\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−319.658	+	184.555i	−0.420050	+	0.242516i	−0.695099	−	0.718914i	$-0.744639\pi$
$761$	−319.658	+	184.555i	−0.420050	+	0.242516i	0.275048	+	0.961430i	$0.411306\pi$
$762$		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	0.713761	−	0.700390i	$-0.246990\pi$
$769$	−192.000	−	332.554i	−0.249675	−	0.432450i	−0.963436	+	0.267940i	$0.913657\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	201.425			0.260576			0.130288	−	0.991476i	$-0.458410\pi$
$773$	201.425			0.260576			0.130288	+	0.991476i	$0.458410\pi$
$774$		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	−0.737931	−	0.674877i	$-0.764197\pi$
$787$	−411.157	+	237.382i	−0.522436	+	0.301628i	0.215495	+	0.976505i	$0.430864\pi$
$788$		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	$-0.267968\pi$
$797$	−249.383	−	431.944i	−0.312902	−	0.541963i	−0.978989	+	0.203911i	$0.934635\pi$
$798$		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	$-0.660457\pi$
$821$	−1217.40	−	702.864i	−1.48282	−	0.856107i	−0.999810	+	0.0195065i	$0.993790\pi$
$822$		0			0
$823$	1110.12	−	640.930i	1.34887	−	0.778773i	0.360784	−	0.932649i	$-0.382509\pi$
$823$	1110.12	−	640.930i	1.34887	−	0.778773i	0.988090	+	0.153876i	$0.0491757\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1170.18			−1.41497			−0.707487	−	0.706727i	$-0.750171\pi$
$827$	−1170.18			−1.41497			−0.707487	+	0.706727i	$0.750171\pi$
$828$		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	$-0.187137\pi$
$839$	644.216	+	371.938i	0.767838	+	0.443311i	−0.0642650	+	0.997933i	$0.520470\pi$
$840$		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	0.583598	−	0.812043i	$-0.301644\pi$
$853$	146.842	+	84.7791i	0.172147	+	0.0993894i	−0.411451	+	0.911432i	$0.634978\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	57.5500	−	99.6795i	0.0671528	−	0.116312i	−0.830494	−	0.557027i	$-0.811942\pi$
$857$	57.5500	−	99.6795i	0.0671528	−	0.116312i	0.897647	+	0.440715i	$0.145275\pi$
$858$		0			0
$859$	−77.0000	−	133.368i	−0.0896391	−	0.155260i	0.817719	−	0.575617i	$-0.195238\pi$
$859$	−77.0000	−	133.368i	−0.0896391	−	0.155260i	−0.907359	+	0.420357i	$0.861905\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1064.67			−1.23369			−0.616845	−	0.787085i	$-0.711589\pi$
$863$	−1064.67			−1.23369			−0.616845	+	0.787085i	$0.711589\pi$
$864$		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	0.381677	−	0.924296i	$-0.375347\pi$
$877$	1204.10	−	695.189i	1.37298	−	0.792690i	0.991302	+	0.131606i	$0.0420135\pi$
$878$		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	$-0.804583\pi$
$883$		−	1017.35i		−	1.15215i	0.817396	−	0.576076i	$-0.195417\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−302.137	−	523.317i	−0.340628	−	0.589986i	0.643921	−	0.765092i	$-0.277306\pi$
$887$	−302.137	−	523.317i	−0.340628	−	0.589986i	−0.984550	+	0.175106i	$0.943973\pi$
$888$		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	0.875144	−	0.483863i	$-0.160767\pi$
$907$	810.566	+	467.981i	0.893679	+	0.515966i	0.0185346	+	0.999828i	$0.494100\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−661.362	−	381.838i	−0.725974	−	0.419141i	0.0909736	−	0.995853i	$-0.471002\pi$
$911$	−661.362	−	381.838i	−0.725974	−	0.419141i	−0.816947	+	0.576712i	$0.804335\pi$
$912$		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	$-0.307867\pi$
$929$	128.598	+	74.2462i	0.138426	+	0.0799206i	−0.429187	+	0.903215i	$0.641200\pi$
$930$		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	$-0.862092\pi$
$937$		−	786.750i		−	0.839648i	0.907606	−	0.419824i	$-0.137908\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	296.388	−	171.120i	0.314972	−	0.181849i	−0.334177	−	0.942510i	$-0.608458\pi$
$941$	296.388	−	171.120i	0.314972	−	0.181849i	0.649149	+	0.760661i	$0.275125\pi$
$942$		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	$-0.901558\pi$
$947$	−201.425	+	348.878i	−0.212698	+	0.368404i	0.739860	+	0.672761i	$0.234892\pi$
$948$		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	$-0.309407\pi$
$953$	1074.27			1.12725			0.563623	+	0.826032i	$0.309407\pi$
$954$		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	−0.668367	−	0.743832i	$-0.733006\pi$
$967$	−346.547	+	200.079i	−0.358373	+	0.206907i	0.309994	+	0.950739i	$0.399673\pi$
$968$		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	0.902174	+	0.431373i	$0.141971\pi$
$977$	805.700	+	1395.51i	0.824667	+	1.42837i	−0.0775065	+	0.996992i	$0.524696\pi$
$978$		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	$-0.995562\pi$
$983$	−479.583	+	830.662i	−0.487877	+	0.845028i	0.512026	+	0.858970i	$0.328895\pi$
$984$		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	0.920616	−	0.390469i	$-0.127687\pi$
$997$	1039.64	+	600.236i	1.04277	+	0.602042i	0.122152	+	0.992511i	$0.461021\pi$
$998$		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.269.4		8
3.2	odd	2	inner	1620.3.t.c.269.1		8
5.4	even	2	inner	1620.3.t.c.269.2		8
9.2	odd	6		180.3.b.a.89.3	yes	4
9.4	even	3	inner	1620.3.t.c.1349.3		8
9.5	odd	6	inner	1620.3.t.c.1349.2		8
9.7	even	3		180.3.b.a.89.2	yes	4
15.14	odd	2	inner	1620.3.t.c.269.3		8
36.7	odd	6		720.3.c.b.449.2		4
36.11	even	6		720.3.c.b.449.3		4
45.2	even	12		900.3.g.c.701.4		4
45.4	even	6	inner	1620.3.t.c.1349.1		8
45.7	odd	12		900.3.g.c.701.3		4
45.14	odd	6	inner	1620.3.t.c.1349.4		8
45.29	odd	6		180.3.b.a.89.1	✓	4
45.34	even	6		180.3.b.a.89.4	yes	4
45.38	even	12		900.3.g.c.701.2		4
45.43	odd	12		900.3.g.c.701.1		4
72.11	even	6		2880.3.c.f.449.2		4
72.29	odd	6		2880.3.c.c.449.2		4
72.43	odd	6		2880.3.c.f.449.3		4
72.61	even	6		2880.3.c.c.449.3		4
180.7	even	12		3600.3.l.r.1601.2		4
180.43	even	12		3600.3.l.r.1601.4		4
180.47	odd	12		3600.3.l.r.1601.1		4
180.79	odd	6		720.3.c.b.449.4		4
180.83	odd	12		3600.3.l.r.1601.3		4
180.119	even	6		720.3.c.b.449.1		4
360.29	odd	6		2880.3.c.c.449.4		4
360.259	odd	6		2880.3.c.f.449.1		4
360.299	even	6		2880.3.c.f.449.4		4
360.349	even	6		2880.3.c.c.449.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.b.a.89.1	✓	4	45.29	odd	6
180.3.b.a.89.2	yes	4	9.7	even	3
180.3.b.a.89.3	yes	4	9.2	odd	6
180.3.b.a.89.4	yes	4	45.34	even	6
720.3.c.b.449.1		4	180.119	even	6
720.3.c.b.449.2		4	36.7	odd	6
720.3.c.b.449.3		4	36.11	even	6
720.3.c.b.449.4		4	180.79	odd	6
900.3.g.c.701.1		4	45.43	odd	12
900.3.g.c.701.2		4	45.38	even	12
900.3.g.c.701.3		4	45.7	odd	12
900.3.g.c.701.4		4	45.2	even	12
1620.3.t.c.269.1		8	3.2	odd	2	inner
1620.3.t.c.269.2		8	5.4	even	2	inner
1620.3.t.c.269.3		8	15.14	odd	2	inner
1620.3.t.c.269.4		8	1.1	even	1	trivial
1620.3.t.c.1349.1		8	45.4	even	6	inner
1620.3.t.c.1349.2		8	9.5	odd	6	inner
1620.3.t.c.1349.3		8	9.4	even	3	inner
1620.3.t.c.1349.4		8	45.14	odd	6	inner
2880.3.c.c.449.1		4	360.349	even	6
2880.3.c.c.449.2		4	72.29	odd	6
2880.3.c.c.449.3		4	72.61	even	6
2880.3.c.c.449.4		4	360.29	odd	6
2880.3.c.f.449.1		4	360.259	odd	6
2880.3.c.f.449.2		4	72.11	even	6
2880.3.c.f.449.3		4	72.43	odd	6
2880.3.c.f.449.4		4	360.299	even	6
3600.3.l.r.1601.1		4	180.47	odd	12
3600.3.l.r.1601.2		4	180.7	even	12
3600.3.l.r.1601.3		4	180.83	odd	12
3600.3.l.r.1601.4		4	180.43	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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