LMFDB - Embedded newform 525.3.e.c.349.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,3,Mod(349,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("525.349");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	525.e (of order $2$, degree $1$, 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:	$14.3052138789$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.1
Character	$\chi$	$=$	525.349
Dual form			525.3.e.c.349.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.71214i q^{2} -1.73205 q^{3} +1.06857 q^{4} +2.96552i q^{6} +(-6.15534 - 3.33344i) q^{7} -8.67811i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q-1.71214i q^{2} -1.73205 q^{3} +1.06857 q^{4} +2.96552i q^{6} +(-6.15534 - 3.33344i) q^{7} -8.67811i q^{8} +3.00000 q^{9} +17.0001 q^{11} -1.85082 q^{12} +16.3319 q^{13} +(-5.70733 + 10.5388i) q^{14} -10.5839 q^{16} -13.4266 q^{17} -5.13643i q^{18} -13.7499i q^{19} +(10.6614 + 5.77369i) q^{21} -29.1066i q^{22} +16.6179i q^{23} +15.0309i q^{24} -27.9626i q^{26} -5.19615 q^{27} +(-6.57741 - 3.56202i) q^{28} -32.1793 q^{29} -6.74366i q^{31} -16.5913i q^{32} -29.4450 q^{33} +22.9883i q^{34} +3.20571 q^{36} -69.2141i q^{37} -23.5418 q^{38} -28.2878 q^{39} -39.7391i q^{41} +(9.88538 - 18.2538i) q^{42} -43.2210i q^{43} +18.1658 q^{44} +28.4522 q^{46} -40.1384 q^{47} +18.3318 q^{48} +(26.7763 + 41.0369i) q^{49} +23.2556 q^{51} +17.4518 q^{52} -22.5002i q^{53} +8.89655i q^{54} +(-28.9280 + 53.4167i) q^{56} +23.8155i q^{57} +55.0956i q^{58} -81.6005i q^{59} +14.9859i q^{61} -11.5461 q^{62} +(-18.4660 - 10.0003i) q^{63} -70.7422 q^{64} +50.4141i q^{66} +72.0872i q^{67} -14.3473 q^{68} -28.7831i q^{69} -25.7338 q^{71} -26.0343i q^{72} +75.0647 q^{73} -118.504 q^{74} -14.6927i q^{76} +(-104.641 - 56.6689i) q^{77} +48.4327i q^{78} -80.0480 q^{79} +9.00000 q^{81} -68.0389 q^{82} -102.112 q^{83} +(11.3924 + 6.16959i) q^{84} -74.0005 q^{86} +55.7362 q^{87} -147.529i q^{88} +128.381i q^{89} +(-100.529 - 54.4416i) q^{91} +17.7574i q^{92} +11.6804i q^{93} +68.7227i q^{94} +28.7371i q^{96} +159.448 q^{97} +(70.2610 - 45.8449i) q^{98} +51.0003 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 88 q^{4} + 72 q^{9}+O(q^{10}) $ 24 * q - 88 * q^4 + 72 * q^9	$ 24 q - 88 q^{4} + 72 q^{9} - 32 q^{11} + 80 q^{14} + 184 q^{16} + 72 q^{21} - 208 q^{29} - 264 q^{36} + 48 q^{39} - 384 q^{44} + 400 q^{46} - 120 q^{49} + 48 q^{51} - 736 q^{56} + 40 q^{64} + 64 q^{71} - 368 q^{74} - 240 q^{79} + 216 q^{81} - 216 q^{84} + 800 q^{86} + 48 q^{91} - 96 q^{99}+O(q^{100}) $ 24 * q - 88 * q^4 + 72 * q^9 - 32 * q^11 + 80 * q^14 + 184 * q^16 + 72 * q^21 - 208 * q^29 - 264 * q^36 + 48 * q^39 - 384 * q^44 + 400 * q^46 - 120 * q^49 + 48 * q^51 - 736 * q^56 + 40 * q^64 + 64 * q^71 - 368 * q^74 - 240 * q^79 + 216 * q^81 - 216 * q^84 + 800 * q^86 + 48 * q^91 - 96 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		−	1.71214i		−	0.856071i	−0.903762	−	0.428035i	$-0.859206\pi$
$2$		−	1.71214i		−	0.856071i	0.903762	−	0.428035i	$-0.140794\pi$
$3$	−1.73205			−0.577350
$3$	−1.73205			−0.577350
$4$	1.06857			0.267143
$5$		0			0
$5$		0			0
$6$			2.96552i			0.494253i
$7$	−6.15534	−	3.33344i	−0.879334	−	0.476206i
$7$	−6.15534	−	3.33344i	−0.879334	−	0.476206i
$8$		−	8.67811i		−	1.08476i
$9$	3.00000			0.333333
$10$		0			0
$11$	17.0001			1.54546			0.772732	−	0.634732i	$-0.218890\pi$
$11$	17.0001			1.54546			0.772732	+	0.634732i	$0.218890\pi$
$12$	−1.85082			−0.154235
$13$	16.3319			1.25630			0.628152	−	0.778091i	$-0.283812\pi$
$13$	16.3319			1.25630			0.628152	+	0.778091i	$0.283812\pi$
$14$	−5.70733	+	10.5388i	−0.407666	+	0.752772i
$15$		0			0
$16$	−10.5839			−0.661492
$17$	−13.4266			−0.789801			−0.394900	−	0.918724i	$-0.629221\pi$
$17$	−13.4266			−0.789801			−0.394900	+	0.918724i	$0.629221\pi$
$18$		−	5.13643i		−	0.285357i
$19$		−	13.7499i		−	0.723679i	−0.932240	−	0.361839i	$-0.882149\pi$
$19$		−	13.7499i		−	0.723679i	0.932240	−	0.361839i	$-0.117851\pi$
$20$		0			0
$21$	10.6614	+	5.77369i	0.507684	+	0.274938i
$22$		−	29.1066i		−	1.32303i
$23$			16.6179i			0.722518i	0.932466	+	0.361259i	$0.117653\pi$
$23$			16.6179i			0.722518i	−0.932466	+	0.361259i	$0.882347\pi$
$24$			15.0309i			0.626289i
$25$		0			0
$26$		−	27.9626i		−	1.07548i
$27$	−5.19615			−0.192450
$28$	−6.57741	−	3.56202i	−0.234907	−	0.127215i
$29$	−32.1793			−1.10963			−0.554816	−	0.831973i	$-0.687211\pi$
$29$	−32.1793			−1.10963			−0.554816	+	0.831973i	$0.687211\pi$
$30$		0			0
$31$		−	6.74366i		−	0.217538i	−0.994067	−	0.108769i	$-0.965309\pi$
$31$		−	6.74366i		−	0.217538i	0.994067	−	0.108769i	$-0.0346908\pi$
$32$		−	16.5913i		−	0.518480i
$33$	−29.4450			−0.892274
$34$			22.9883i			0.676125i
$35$		0			0
$36$	3.20571			0.0890475
$37$		−	69.2141i		−	1.87065i	−0.353789	−	0.935325i	$-0.615107\pi$
$37$		−	69.2141i		−	1.87065i	0.353789	−	0.935325i	$-0.384893\pi$
$38$	−23.5418			−0.619520
$39$	−28.2878			−0.725327
$40$		0			0
$41$		−	39.7391i		−	0.969246i	−0.874723	−	0.484623i	$-0.838957\pi$
$41$		−	39.7391i		−	0.969246i	0.874723	−	0.484623i	$-0.161043\pi$
$42$	9.88538	−	18.2538i	0.235366	−	0.434613i
$43$		−	43.2210i		−	1.00514i	−0.864537	−	0.502570i	$-0.832388\pi$
$43$		−	43.2210i		−	1.00514i	0.864537	−	0.502570i	$-0.167612\pi$
$44$	18.1658			0.412859
$45$		0			0
$46$	28.4522			0.618526
$47$	−40.1384			−0.854009			−0.427005	−	0.904249i	$-0.640431\pi$
$47$	−40.1384			−0.854009			−0.427005	+	0.904249i	$0.640431\pi$
$48$	18.3318			0.381913
$49$	26.7763	+	41.0369i	0.546456	+	0.837488i
$50$		0			0
$51$	23.2556			0.455992
$52$	17.4518			0.335612
$53$		−	22.5002i		−	0.424533i	−0.977212	−	0.212266i	$-0.931916\pi$
$53$		−	22.5002i		−	0.424533i	0.977212	−	0.212266i	$-0.0680845\pi$
$54$			8.89655i			0.164751i
$55$		0			0
$56$	−28.9280	+	53.4167i	−0.516571	+	0.953869i
$57$			23.8155i			0.417816i
$58$			55.0956i			0.949924i
$59$		−	81.6005i		−	1.38306i	−0.722348	−	0.691529i	$-0.756937\pi$
$59$		−	81.6005i		−	1.38306i	0.722348	−	0.691529i	$-0.243063\pi$
$60$		0			0
$61$			14.9859i			0.245671i	0.992427	+	0.122836i	$0.0391988\pi$
$61$			14.9859i			0.245671i	−0.992427	+	0.122836i	$0.960801\pi$
$62$	−11.5461			−0.186228
$63$	−18.4660	−	10.0003i	−0.293111	−	0.158735i
$64$	−70.7422			−1.10535
$65$		0			0
$66$			50.4141i			0.763850i
$67$			72.0872i			1.07593i	0.842968	+	0.537964i	$0.180806\pi$
$67$			72.0872i			1.07593i	−0.842968	+	0.537964i	$0.819194\pi$
$68$	−14.3473			−0.210989
$69$		−	28.7831i		−	0.417146i
$70$		0			0
$71$	−25.7338			−0.362448			−0.181224	−	0.983442i	$-0.558006\pi$
$71$	−25.7338			−0.362448			−0.181224	+	0.983442i	$0.558006\pi$
$72$		−	26.0343i		−	0.361588i
$73$	75.0647			1.02828			0.514142	−	0.857705i	$-0.328110\pi$
$73$	75.0647			1.02828			0.514142	+	0.857705i	$0.328110\pi$
$74$	−118.504			−1.60141
$75$		0			0
$76$		−	14.6927i		−	0.193325i
$77$	−104.641	−	56.6689i	−1.35898	−	0.735959i
$78$			48.4327i			0.620931i
$79$	−80.0480			−1.01327			−0.506633	−	0.862162i	$-0.669110\pi$
$79$	−80.0480			−1.01327			−0.506633	+	0.862162i	$0.669110\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$	−68.0389			−0.829743
$83$	−102.112			−1.23027			−0.615135	−	0.788421i	$-0.710899\pi$
$83$	−102.112			−1.23027			−0.615135	+	0.788421i	$0.710899\pi$
$84$	11.3924	+	6.16959i	0.135624	+	0.0734476i
$85$		0			0
$86$	−74.0005			−0.860471
$87$	55.7362			0.640646
$88$		−	147.529i		−	1.67646i
$89$			128.381i			1.44248i	0.692683	+	0.721242i	$0.256429\pi$
$89$			128.381i			1.44248i	−0.692683	+	0.721242i	$0.743571\pi$
$90$		0			0
$91$	−100.529	−	54.4416i	−1.10471	−	0.598259i
$92$			17.7574i			0.193015i
$93$			11.6804i			0.125595i
$94$			68.7227i			0.731092i
$95$		0			0
$96$			28.7371i			0.299344i
$97$	159.448			1.64379			0.821897	−	0.569636i	$-0.192916\pi$
$97$	159.448			1.64379			0.821897	+	0.569636i	$0.192916\pi$
$98$	70.2610	−	45.8449i	0.716949	−	0.467805i
$99$	51.0003			0.515155
$100$		0			0
$101$		−	24.1380i		−	0.238990i	−0.992835	−	0.119495i	$-0.961872\pi$
$101$		−	24.1380i		−	0.238990i	0.992835	−	0.119495i	$-0.0381276\pi$
$102$		−	39.8168i		−	0.390361i
$103$	87.3469			0.848028			0.424014	−	0.905656i	$-0.360621\pi$
$103$	87.3469			0.848028			0.424014	+	0.905656i	$0.360621\pi$
$104$		−	141.730i		−	1.36279i
$105$		0			0
$106$	−38.5236			−0.363430
$107$			168.359i			1.57344i	0.617307	+	0.786722i	$0.288224\pi$
$107$			168.359i			1.57344i	−0.617307	+	0.786722i	$0.711776\pi$
$108$	−5.55245			−0.0514116
$109$	155.570			1.42725			0.713624	−	0.700529i	$-0.247053\pi$
$109$	155.570			1.42725			0.713624	+	0.700529i	$0.247053\pi$
$110$		0			0
$111$			119.882i			1.08002i
$112$	65.1473	+	35.2807i	0.581672	+	0.315007i
$113$			20.9965i			0.185810i	0.995675	+	0.0929050i	$0.0296153\pi$
$113$			20.9965i			0.185810i	−0.995675	+	0.0929050i	$0.970385\pi$
$114$	40.7756			0.357680
$115$		0			0
$116$	−34.3859			−0.296430
$117$	48.9958			0.418768
$118$	−139.712			−1.18400
$119$	82.6453	+	44.7568i	0.694498	+	0.376108i
$120$		0			0
$121$	168.004			1.38846
$122$	25.6581			0.210312
$123$			68.8301i			0.559594i
$124$		−	7.20608i		−	0.0581135i
$125$		0			0
$126$	−17.1220	+	31.6164i	−0.135889	+	0.250924i
$127$		−	59.8712i		−	0.471427i	−0.971823	−	0.235713i	$-0.924257\pi$
$127$		−	59.8712i		−	0.471427i	0.971823	−	0.235713i	$-0.0757427\pi$
$128$			54.7554i			0.427776i
$129$			74.8609i			0.580317i
$130$		0			0
$131$		−	166.868i		−	1.27380i	−0.770947	−	0.636899i	$-0.780217\pi$
$131$		−	166.868i		−	1.27380i	0.770947	−	0.636899i	$-0.219783\pi$
$132$	−31.4641			−0.238364
$133$	−45.8345	+	84.6352i	−0.344620	+	0.636355i
$134$	123.423			0.921071
$135$		0			0
$136$			116.518i			0.856747i
$137$			126.139i			0.920726i	0.887731	+	0.460363i	$0.152281\pi$
$137$			126.139i			0.920726i	−0.887731	+	0.460363i	$0.847719\pi$
$138$	−49.2807			−0.357106
$139$		−	211.650i		−	1.52266i	−0.648365	−	0.761330i	$-0.724547\pi$
$139$		−	211.650i		−	1.52266i	0.648365	−	0.761330i	$-0.275453\pi$
$140$		0			0
$141$	69.5218			0.493062
$142$			44.0599i			0.310281i
$143$	277.645			1.94157
$144$	−31.7516			−0.220497
$145$		0			0
$146$		−	128.521i		−	0.880284i
$147$	−46.3780	−	71.0780i	−0.315496	−	0.483524i
$148$		−	73.9601i		−	0.499730i
$149$	64.1825			0.430755			0.215377	−	0.976531i	$-0.430902\pi$
$149$	64.1825			0.430755			0.215377	+	0.976531i	$0.430902\pi$
$150$		0			0
$151$	110.915			0.734538			0.367269	−	0.930115i	$-0.380293\pi$
$151$	110.915			0.734538			0.367269	+	0.930115i	$0.380293\pi$
$152$	−119.323			−0.785021
$153$	−40.2798			−0.263267
$154$	−97.0252	+	179.161i	−0.630033	+	1.16338i
$155$		0			0
$156$	−30.2275			−0.193766
$157$	290.451			1.85001			0.925004	−	0.379956i	$-0.124061\pi$
$157$	290.451			1.85001			0.925004	+	0.379956i	$0.124061\pi$
$158$			137.053i			0.867427i
$159$			38.9716i			0.245104i
$160$		0			0
$161$	55.3948	−	102.289i	0.344067	−	0.635334i
$162$		−	15.4093i		−	0.0951190i
$163$			53.8559i			0.330404i	0.986260	+	0.165202i	$0.0528276\pi$
$163$			53.8559i			0.330404i	−0.986260	+	0.165202i	$0.947172\pi$
$164$		−	42.4640i		−	0.258927i
$165$		0			0
$166$			174.831i			1.05320i
$167$	−41.7927			−0.250255			−0.125128	−	0.992141i	$-0.539934\pi$
$167$	−41.7927			−0.250255			−0.125128	+	0.992141i	$0.539934\pi$
$168$	50.1047	−	92.5204i	0.298242	−	0.550717i
$169$	97.7324			0.578298
$170$		0			0
$171$		−	41.2497i		−	0.241226i
$172$		−	46.1847i		−	0.268515i
$173$	−130.344			−0.753434			−0.376717	−	0.926328i	$-0.622947\pi$
$173$	−130.344			−0.753434			−0.376717	+	0.926328i	$0.622947\pi$
$174$		−	95.4283i		−	0.548439i
$175$		0			0
$176$	−179.927			−1.02231
$177$			141.336i			0.798509i
$178$	219.807			1.23487
$179$	−44.9934			−0.251360			−0.125680	−	0.992071i	$-0.540111\pi$
$179$	−44.9934			−0.251360			−0.125680	+	0.992071i	$0.540111\pi$
$180$		0			0
$181$			17.8944i			0.0988640i	0.998777	+	0.0494320i	$0.0157411\pi$
$181$			17.8944i			0.0988640i	−0.998777	+	0.0494320i	$0.984259\pi$
$182$	−93.2117	+	172.119i	−0.512152	+	0.945710i
$183$		−	25.9564i		−	0.141838i
$184$	144.212			0.783761
$185$		0			0
$186$	19.9985			0.107519
$187$	−228.254			−1.22061
$188$	−42.8907			−0.228142
$189$	31.9841	+	17.3211i	0.169228	+	0.0916459i
$190$		0			0
$191$	178.314			0.933583			0.466791	−	0.884367i	$-0.345410\pi$
$191$	178.314			0.933583			0.466791	+	0.884367i	$0.345410\pi$
$192$	122.529			0.638173
$193$			336.283i			1.74240i	0.490928	+	0.871200i	$0.336658\pi$
$193$			336.283i			1.74240i	−0.490928	+	0.871200i	$0.663342\pi$
$194$		−	272.998i		−	1.40720i
$195$		0			0
$196$	28.6124	+	43.8508i	0.145982	+	0.223729i
$197$			49.2082i			0.249788i	0.992170	+	0.124894i	$0.0398590\pi$
$197$			49.2082i			0.249788i	−0.992170	+	0.124894i	$0.960141\pi$
$198$		−	87.3198i		−	0.441009i
$199$		−	171.789i		−	0.863262i	−0.902050	−	0.431631i	$-0.857938\pi$
$199$		−	171.789i		−	0.863262i	0.902050	−	0.431631i	$-0.142062\pi$
$200$		0			0
$201$		−	124.859i		−	0.621187i
$202$	−41.3277			−0.204592
$203$	198.075	+	107.268i	0.975737	+	0.528414i
$204$	24.8502			0.121815
$205$		0			0
$206$		−	149.550i		−	0.725972i
$207$			49.8537i			0.240839i
$208$	−172.855			−0.831035
$209$		−	233.750i		−	1.11842i
$210$		0			0
$211$	−5.09458			−0.0241449			−0.0120725	−	0.999927i	$-0.503843\pi$
$211$	−5.09458			−0.0241449			−0.0120725	+	0.999927i	$0.503843\pi$
$212$		−	24.0431i		−	0.113411i
$213$	44.5722			0.209259
$214$	288.254			1.34698
$215$		0			0
$216$			45.0928i			0.208763i
$217$	−22.4796	+	41.5095i	−0.103593	+	0.191288i
$218$		−	266.358i		−	1.22183i
$219$	−130.016			−0.593680
$220$		0			0
$221$	−219.283			−0.992229
$222$	205.256			0.924574
$223$	−310.066			−1.39043			−0.695215	−	0.718802i	$-0.744691\pi$
$223$	−310.066			−1.39043			−0.695215	+	0.718802i	$0.744691\pi$
$224$	−55.3063	+	102.125i	−0.246903	+	0.455917i
$225$		0			0
$226$	35.9490			0.159067
$227$	108.558			0.478228			0.239114	−	0.970991i	$-0.423143\pi$
$227$	108.558			0.478228			0.239114	+	0.970991i	$0.423143\pi$
$228$			25.4486i			0.111616i
$229$		−	236.483i		−	1.03268i	−0.856384	−	0.516339i	$-0.827294\pi$
$229$		−	236.483i		−	1.03268i	0.856384	−	0.516339i	$-0.172706\pi$
$230$		0			0
$231$	181.244	+	98.1534i	0.784607	+	0.424906i
$232$			279.256i			1.20369i
$233$		−	151.290i		−	0.649312i	−0.945832	−	0.324656i	$-0.894751\pi$
$233$		−	151.290i		−	0.649312i	0.945832	−	0.324656i	$-0.105249\pi$
$234$		−	83.8878i		−	0.358495i
$235$		0			0
$236$		−	87.1958i		−	0.369474i
$237$	138.647			0.585009
$238$	76.6300	−	141.500i	0.321975	−	0.594540i
$239$	−48.2956			−0.202074			−0.101037	−	0.994883i	$-0.532216\pi$
$239$	−48.2956			−0.202074			−0.101037	+	0.994883i	$0.532216\pi$
$240$		0			0
$241$			230.735i			0.957406i	0.877977	+	0.478703i	$0.158893\pi$
$241$			230.735i			0.957406i	−0.877977	+	0.478703i	$0.841107\pi$
$242$		−	287.646i		−	1.18862i
$243$	−15.5885			−0.0641500
$244$			16.0135i			0.0656292i
$245$		0			0
$246$	117.847			0.479053
$247$		−	224.563i		−	0.909160i
$248$	−58.5223			−0.235977
$249$	176.864			0.710297
$250$		0			0
$251$		−	86.6812i		−	0.345343i	−0.984979	−	0.172672i	$-0.944760\pi$
$251$		−	86.6812i		−	0.345343i	0.984979	−	0.172672i	$-0.0552399\pi$
$252$	−19.7322	−	10.6861i	−0.0783025	−	0.0424050i
$253$			282.506i			1.11663i
$254$	−102.508			−0.403575
$255$		0			0
$256$	−189.220			−0.739141
$257$	141.110			0.549065			0.274532	−	0.961578i	$-0.411477\pi$
$257$	141.110			0.549065			0.274532	+	0.961578i	$0.411477\pi$
$258$	128.173			0.496793
$259$	−230.721	+	426.036i	−0.890815	+	1.64493i
$260$		0			0
$261$	−96.5380			−0.369877
$262$	−285.701			−1.09046
$263$		−	38.0901i		−	0.144829i	−0.997375	−	0.0724147i	$-0.976930\pi$
$263$		−	38.0901i		−	0.144829i	0.997375	−	0.0724147i	$-0.0230705\pi$
$264$			255.527i			0.967907i
$265$		0			0
$266$	144.908	+	78.4752i	0.544765	+	0.295019i
$267$		−	222.363i		−	0.832819i
$268$			77.0302i			0.287426i
$269$			454.220i			1.68855i	0.535909	+	0.844275i	$0.319969\pi$
$269$			454.220i			1.68855i	−0.535909	+	0.844275i	$0.680031\pi$
$270$		0			0
$271$			78.7098i			0.290442i	0.989399	+	0.145221i	$0.0463893\pi$
$271$			78.7098i			0.290442i	−0.989399	+	0.145221i	$0.953611\pi$
$272$	142.106			0.522447
$273$	174.121	+	94.2956i	0.637805	+	0.345405i
$274$	215.969			0.788207
$275$		0			0
$276$		−	30.7567i		−	0.111437i
$277$		−	85.3396i		−	0.308085i	−0.988064	−	0.154043i	$-0.950771\pi$
$277$		−	85.3396i		−	0.308085i	0.988064	−	0.154043i	$-0.0492293\pi$
$278$	−362.374			−1.30350
$279$		−	20.2310i		−	0.0725125i
$280$		0			0
$281$	167.376			0.595643			0.297821	−	0.954622i	$-0.403740\pi$
$281$	167.376			0.595643			0.297821	+	0.954622i	$0.403740\pi$
$282$		−	119.031i		−	0.422096i
$283$	−149.591			−0.528588			−0.264294	−	0.964442i	$-0.585139\pi$
$283$	−149.591			−0.528588			−0.264294	+	0.964442i	$0.585139\pi$
$284$	−27.4984			−0.0968252
$285$		0			0
$286$		−	475.367i		−	1.66212i
$287$	−132.468	+	244.607i	−0.461561	+	0.852291i
$288$		−	49.7740i		−	0.172827i
$289$	−108.726			−0.376215
$290$		0			0
$291$	−276.172			−0.949045
$292$	80.2119			0.274698
$293$	145.805			0.497627			0.248813	−	0.968551i	$-0.419959\pi$
$293$	145.805			0.497627			0.248813	+	0.968551i	$0.419959\pi$
$294$	−121.696	+	79.4056i	−0.413931	+	0.270087i
$295$		0			0
$296$	−600.648			−2.02921
$297$	−88.3351			−0.297425
$298$		−	109.889i		−	0.368757i
$299$			271.403i			0.907701i
$300$		0			0
$301$	−144.075	+	266.040i	−0.478653	+	0.883853i
$302$		−	189.903i		−	0.628817i
$303$			41.8082i			0.137981i
$304$			145.527i			0.478708i
$305$		0			0
$306$			68.9648i			0.225375i
$307$	205.594			0.669686			0.334843	−	0.942274i	$-0.391317\pi$
$307$	205.594			0.669686			0.334843	+	0.942274i	$0.391317\pi$
$308$	−111.817	−	60.5547i	−0.363041	−	0.196606i
$309$	−151.289			−0.489609
$310$		0			0
$311$			424.383i			1.36458i	0.731084	+	0.682288i	$0.239015\pi$
$311$			424.383i			1.36458i	−0.731084	+	0.682288i	$0.760985\pi$
$312$			245.484i			0.786809i
$313$	363.503			1.16135			0.580676	−	0.814135i	$-0.302788\pi$
$313$	363.503			1.16135			0.580676	+	0.814135i	$0.302788\pi$
$314$		−	497.294i		−	1.58374i
$315$		0			0
$316$	−85.5369			−0.270686
$317$			441.407i			1.39245i	0.717822	+	0.696226i	$0.245139\pi$
$317$			441.407i			1.39245i	−0.717822	+	0.696226i	$0.754861\pi$
$318$	66.7248			0.209827
$319$	−547.052			−1.71490
$320$		0			0
$321$		−	291.606i		−	0.908429i
$322$	−175.133	−	94.8438i	−0.543891	−	0.294546i
$323$			184.615i			0.571562i
$324$	9.61713			0.0296825
$325$		0			0
$326$	92.2089			0.282849
$327$	−269.455			−0.824022
$328$	−344.860			−1.05140
$329$	247.066	+	133.799i	0.750959	+	0.406684i
$330$		0			0
$331$	509.327			1.53875			0.769376	−	0.638796i	$-0.220567\pi$
$331$	509.327			1.53875			0.769376	+	0.638796i	$0.220567\pi$
$332$	−109.114			−0.328658
$333$		−	207.642i		−	0.623550i
$334$			71.5550i			0.214236i
$335$		0			0
$336$	−112.838	−	61.1080i	−0.335829	−	0.181869i
$337$			442.557i			1.31323i	0.754228	+	0.656613i	$0.228012\pi$
$337$			442.557i			1.31323i	−0.754228	+	0.656613i	$0.771988\pi$
$338$		−	167.332i		−	0.495064i
$339$		−	36.3671i		−	0.107277i
$340$		0			0
$341$		−	114.643i		−	0.336197i
$342$	−70.6253			−0.206507
$343$	−28.0231	−	341.853i	−0.0816999	−	0.996657i
$344$	−375.077			−1.09034
$345$		0			0
$346$			223.167i			0.644993i
$347$		−	493.763i		−	1.42295i	−0.702712	−	0.711475i	$-0.748028\pi$
$347$		−	493.763i		−	1.42295i	0.702712	−	0.711475i	$-0.251972\pi$
$348$	59.5581			0.171144
$349$		−	324.460i		−	0.929686i	−0.885393	−	0.464843i	$-0.846111\pi$
$349$		−	324.460i		−	0.929686i	0.885393	−	0.464843i	$-0.153889\pi$
$350$		0			0
$351$	−84.8633			−0.241776
$352$		−	282.055i		−	0.801292i
$353$	529.424			1.49979			0.749893	−	0.661559i	$-0.230105\pi$
$353$	529.424			1.49979			0.749893	+	0.661559i	$0.230105\pi$
$354$	241.988			0.683581
$355$		0			0
$356$			137.184i			0.385349i
$357$	−143.146	−	77.5211i	−0.400969	−	0.217146i
$358$			77.0351i			0.215182i
$359$	−64.2261			−0.178903			−0.0894514	−	0.995991i	$-0.528511\pi$
$359$	−64.2261			−0.178903			−0.0894514	+	0.995991i	$0.528511\pi$
$360$		0			0
$361$	171.940			0.476289
$362$	30.6377			0.0846346
$363$	−290.991			−0.801627
$364$	−107.422	−	58.1747i	−0.295115	−	0.159821i
$365$		0			0
$366$	−44.4411			−0.121424
$367$	10.8172			0.0294747			0.0147373	−	0.999891i	$-0.495309\pi$
$367$	10.8172			0.0294747			0.0147373	+	0.999891i	$0.495309\pi$
$368$		−	175.882i		−	0.477940i
$369$		−	119.217i		−	0.323082i
$370$		0			0
$371$	−75.0033	+	138.497i	−0.202165	+	0.373306i
$372$			12.4813i			0.0335519i
$373$		−	532.850i		−	1.42855i	−0.699864	−	0.714276i	$-0.746756\pi$
$373$		−	532.850i		−	1.42855i	0.699864	−	0.714276i	$-0.253244\pi$
$374$			390.803i			1.04493i
$375$		0			0
$376$			348.326i			0.926398i
$377$	−525.551			−1.39403
$378$	29.6561	−	54.7613i	0.0784554	−	0.144871i
$379$	−516.003			−1.36149			−0.680743	−	0.732523i	$-0.738343\pi$
$379$	−516.003			−1.36149			−0.680743	+	0.732523i	$0.738343\pi$
$380$		0			0
$381$			103.700i			0.272178i
$382$		−	305.299i		−	0.799213i
$383$	415.069			1.08373			0.541866	−	0.840465i	$-0.317718\pi$
$383$	415.069			1.08373			0.541866	+	0.840465i	$0.317718\pi$
$384$		−	94.8391i		−	0.246977i
$385$		0			0
$386$	575.765			1.49162
$387$		−	129.663i		−	0.335046i
$388$	170.381			0.439127
$389$	−54.8032			−0.140882			−0.0704411	−	0.997516i	$-0.522441\pi$
$389$	−54.8032			−0.140882			−0.0704411	+	0.997516i	$0.522441\pi$
$390$		0			0
$391$		−	223.122i		−	0.570645i
$392$	356.123	−	232.368i	0.908477	−	0.592775i
$393$			289.023i			0.735428i
$394$	84.2514			0.213836
$395$		0			0
$396$	54.4974			0.137620
$397$	−19.9434			−0.0502352			−0.0251176	−	0.999685i	$-0.507996\pi$
$397$	−19.9434			−0.0502352			−0.0251176	+	0.999685i	$0.507996\pi$
$398$	−294.127			−0.739013
$399$	79.3877	−	146.593i	0.198967	−	0.367400i
$400$		0			0
$401$	−239.505			−0.597269			−0.298634	−	0.954368i	$-0.596531\pi$
$401$	−239.505			−0.597269			−0.298634	+	0.954368i	$0.596531\pi$
$402$	−213.776			−0.531781
$403$		−	110.137i		−	0.273293i
$404$		−	25.7931i		−	0.0638444i
$405$		0			0
$406$	183.658	−	339.132i	0.452359	−	0.835300i
$407$		−	1176.65i		−	2.89102i
$408$		−	201.814i		−	0.494643i
$409$			663.541i			1.62235i	0.584804	+	0.811175i	$0.301171\pi$
$409$			663.541i			1.62235i	−0.584804	+	0.811175i	$0.698829\pi$
$410$		0			0
$411$		−	218.480i		−	0.531581i
$412$	93.3363			0.226544
$413$	−272.010	+	502.278i	−0.658621	+	1.21617i
$414$	85.3566			0.206175
$415$		0			0
$416$		−	270.969i		−	0.651368i
$417$			366.588i			0.879108i
$418$	−400.213			−0.957447
$419$		−	110.648i		−	0.264077i	−0.991245	−	0.132039i	$-0.957848\pi$
$419$		−	110.648i		−	0.264077i	0.991245	−	0.132039i	$-0.0421523\pi$
$420$		0			0
$421$	−521.325			−1.23830			−0.619151	−	0.785272i	$-0.712523\pi$
$421$	−521.325			−1.23830			−0.619151	+	0.785272i	$0.712523\pi$
$422$			8.72264i			0.0206698i
$423$	−120.415			−0.284670
$424$	−195.260			−0.460518
$425$		0			0
$426$		−	76.3140i		−	0.179141i
$427$	49.9548	−	92.2435i	0.116990	−	0.216027i
$428$			179.903i			0.420334i
$429$	−480.895			−1.12097
$430$		0			0
$431$	573.019			1.32951			0.664755	−	0.747062i	$-0.268536\pi$
$431$	573.019			1.32951			0.664755	+	0.747062i	$0.268536\pi$
$432$	54.9954			0.127304
$433$	−429.740			−0.992472			−0.496236	−	0.868188i	$-0.665285\pi$
$433$	−429.740			−0.992472			−0.496236	+	0.868188i	$0.665285\pi$
$434$	71.0702	+	38.4883i	0.163756	+	0.0886827i
$435$		0			0
$436$	166.237			0.381279
$437$	228.495			0.522871
$438$			222.606i			0.508232i
$439$		−	83.0494i		−	0.189179i	−0.995516	−	0.0945893i	$-0.969846\pi$
$439$		−	83.0494i		−	0.189179i	0.995516	−	0.0945893i	$-0.0301538\pi$
$440$		0			0
$441$	80.3290	+	123.111i	0.182152	+	0.279163i
$442$			375.443i			0.849419i
$443$			410.010i			0.925530i	0.886481	+	0.462765i	$0.153143\pi$
$443$			410.010i			0.925530i	−0.886481	+	0.462765i	$0.846857\pi$
$444$			128.103i			0.288520i
$445$		0			0
$446$			530.876i			1.19031i
$447$	−111.167			−0.248696
$448$	435.442	+	235.815i	0.971969	+	0.526373i
$449$	205.948			0.458682			0.229341	−	0.973346i	$-0.426343\pi$
$449$	205.948			0.458682			0.229341	+	0.973346i	$0.426343\pi$
$450$		0			0
$451$		−	675.569i		−	1.49793i
$452$			22.4363i			0.0496378i
$453$	−192.111			−0.424086
$454$		−	185.866i		−	0.409397i
$455$		0			0
$456$	206.674			0.453232
$457$		−	640.868i		−	1.40234i	−0.712995	−	0.701169i	$-0.752662\pi$
$457$		−	640.868i		−	1.40234i	0.712995	−	0.701169i	$-0.247338\pi$
$458$	−404.893			−0.884046
$459$	69.7667			0.151997
$460$		0			0
$461$			166.085i			0.360272i	0.983642	+	0.180136i	$0.0576538\pi$
$461$			166.085i			0.360272i	−0.983642	+	0.180136i	$0.942346\pi$
$462$	168.052	−	310.316i	0.363750	−	0.671679i
$463$			10.4409i			0.0225505i	0.999936	+	0.0112753i	$0.00358911\pi$
$463$			10.4409i			0.0225505i	−0.999936	+	0.0112753i	$0.996411\pi$
$464$	340.582			0.734013
$465$		0			0
$466$	−259.029			−0.555857
$467$	269.736			0.577592			0.288796	−	0.957391i	$-0.406745\pi$
$467$	269.736			0.577592			0.288796	+	0.957391i	$0.406745\pi$
$468$	52.3555			0.111871
$469$	240.298	−	443.721i	0.512364	−	0.946100i
$470$		0			0
$471$	−503.077			−1.06810
$472$	−708.138			−1.50029
$473$		−	734.761i		−	1.55341i
$474$		−	237.384i		−	0.500809i
$475$		0			0
$476$	88.3123	+	47.8258i	0.185530	+	0.100474i
$477$		−	67.5007i		−	0.141511i
$478$			82.6889i			0.172989i
$479$			649.820i			1.35662i	0.734777	+	0.678309i	$0.237287\pi$
$479$			649.820i			1.35662i	−0.734777	+	0.678309i	$0.762713\pi$
$480$		0			0
$481$		−	1130.40i		−	2.35011i
$482$	395.051			0.819608
$483$	−95.9467	+	177.169i	−0.198647	+	0.366810i
$484$	179.524			0.370917
$485$		0			0
$486$			26.6897i			0.0549170i
$487$			597.640i			1.22719i	0.789622	+	0.613593i	$0.210277\pi$
$487$			597.640i			1.22719i	−0.789622	+	0.613593i	$0.789723\pi$
$488$	130.050			0.266495
$489$		−	93.2811i		−	0.190759i
$490$		0			0
$491$	−107.625			−0.219195			−0.109598	−	0.993976i	$-0.534956\pi$
$491$	−107.625			−0.219195			−0.109598	+	0.993976i	$0.534956\pi$
$492$			73.5498i			0.149491i
$493$	432.059			0.876388
$494$	−384.483			−0.778306
$495$		0			0
$496$			71.3741i			0.143899i
$497$	158.400	+	85.7821i	0.318713	+	0.172600i
$498$		−	302.816i		−	0.608065i
$499$	420.611			0.842908			0.421454	−	0.906850i	$-0.361520\pi$
$499$	420.611			0.842908			0.421454	+	0.906850i	$0.361520\pi$
$500$		0			0
$501$	72.3870			0.144485
$502$	−148.410			−0.295638
$503$	−837.716			−1.66544			−0.832720	−	0.553694i	$-0.813218\pi$
$503$	−837.716			−1.66544			−0.832720	+	0.553694i	$0.813218\pi$
$504$	−86.7840	+	160.250i	−0.172190	+	0.317956i
$505$		0			0
$506$	483.691			0.955910
$507$	−169.278			−0.333881
$508$		−	63.9766i		−	0.125938i
$509$		−	511.304i		−	1.00453i	−0.864715	−	0.502264i	$-0.832501\pi$
$509$		−	511.304i		−	1.00453i	0.864715	−	0.502264i	$-0.167499\pi$
$510$		0			0
$511$	−462.049	−	250.224i	−0.904205	−	0.489675i
$512$			542.993i			1.06053i
$513$			71.4466i			0.139272i
$514$		−	241.600i		−	0.470038i
$515$		0			0
$516$			79.9942i			0.155027i
$517$	−682.358			−1.31984
$518$	729.434	+	395.027i	1.40817	+	0.762601i
$519$	225.762			0.434995
$520$		0			0
$521$			958.401i			1.83954i	0.392455	+	0.919771i	$0.371626\pi$
$521$			958.401i			1.83954i	−0.392455	+	0.919771i	$0.628374\pi$
$522$			165.287i			0.316641i
$523$	−152.860			−0.292275			−0.146138	−	0.989264i	$-0.546684\pi$
$523$	−152.860			−0.292275			−0.146138	+	0.989264i	$0.546684\pi$
$524$		−	178.310i		−	0.340286i
$525$		0			0
$526$	−65.2157			−0.123984
$527$			90.5445i			0.171811i
$528$	311.643			0.590232
$529$	252.845			0.477968
$530$		0			0
$531$		−	244.801i		−	0.461020i
$532$	−48.9774	+	90.4387i	−0.0920627	+	0.169998i
$533$		−	649.017i		−	1.21767i
$534$	−380.716			−0.712952
$535$		0			0
$536$	625.581			1.16713
$537$	77.9309			0.145123
$538$	777.689			1.44552
$539$	455.200	+	697.632i	0.844527	+	1.29431i
$540$		0			0
$541$	−285.016			−0.526832			−0.263416	−	0.964682i	$-0.584849\pi$
$541$	−285.016			−0.526832			−0.263416	+	0.964682i	$0.584849\pi$
$542$	134.762			0.248639
$543$		−	30.9940i		−	0.0570791i
$544$			222.766i			0.409495i
$545$		0			0
$546$	161.447	−	298.119i	0.295691	−	0.546006i
$547$		−	195.330i		−	0.357092i	−0.983932	−	0.178546i	$-0.942861\pi$
$547$		−	195.330i		−	0.357092i	0.983932	−	0.178546i	$-0.0571394\pi$
$548$			134.789i			0.245965i
$549$			44.9578i			0.0818904i
$550$		0			0
$551$			442.463i			0.803017i
$552$	−249.783			−0.452505
$553$	492.722	+	266.835i	0.890999	+	0.482523i
$554$	−146.113			−0.263743
$555$		0			0
$556$		−	226.163i		−	0.406767i
$557$		−	27.9390i		−	0.0501598i	−0.999685	−	0.0250799i	$-0.992016\pi$
$557$		−	27.9390i		−	0.0501598i	0.999685	−	0.0250799i	$-0.00798402\pi$
$558$	−34.6383			−0.0620759
$559$		−	705.883i		−	1.26276i
$560$		0			0
$561$	395.347			0.704719
$562$		−	286.571i		−	0.509912i
$563$	−459.585			−0.816314			−0.408157	−	0.912912i	$-0.633828\pi$
$563$	−459.585			−0.816314			−0.408157	+	0.912912i	$0.633828\pi$
$564$	74.2889			0.131718
$565$		0			0
$566$			256.120i			0.452509i
$567$	−55.3980	−	30.0010i	−0.0977037	−	0.0529118i
$568$			223.321i			0.393170i
$569$	635.353			1.11661			0.558306	−	0.829635i	$-0.311451\pi$
$569$	635.353			1.11661			0.558306	+	0.829635i	$0.311451\pi$
$570$		0			0
$571$	205.282			0.359513			0.179757	−	0.983711i	$-0.442469\pi$
$571$	205.282			0.359513			0.179757	+	0.983711i	$0.442469\pi$
$572$	296.683			0.518676
$573$	−308.849			−0.539004
$574$	418.803	+	226.804i	0.729621	+	0.395129i
$575$		0			0
$576$	−212.227			−0.368449
$577$	185.464			0.321429			0.160714	−	0.987001i	$-0.448620\pi$
$577$	185.464			0.321429			0.160714	+	0.987001i	$0.448620\pi$
$578$			186.155i			0.322067i
$579$		−	582.460i		−	1.00598i
$580$		0			0
$581$	628.537	+	340.386i	1.08182	+	0.585862i
$582$			472.846i			0.812450i
$583$		−	382.506i		−	0.656100i
$584$		−	651.420i		−	1.11545i
$585$		0			0
$586$		−	249.638i		−	0.426004i
$587$	673.958			1.14814			0.574070	−	0.818806i	$-0.305364\pi$
$587$	673.958			1.14814			0.574070	+	0.818806i	$0.305364\pi$
$588$	−49.5581	−	75.9519i	−0.0842825	−	0.129170i
$589$	−92.7247			−0.157427
$590$		0			0
$591$		−	85.2311i		−	0.144215i
$592$			732.553i			1.23742i
$593$	−0.486694			−0.000820731			−0.000410366	−	1.00000i	$-0.500131\pi$
$593$	−0.486694			−0.000820731			−0.000410366		1.00000i	$0.500131\pi$
$594$			151.242i			0.254617i
$595$		0			0
$596$	68.5835			0.115073
$597$			297.547i			0.498404i
$598$	464.680			0.777057
$599$	−580.285			−0.968756			−0.484378	−	0.874859i	$-0.660954\pi$
$599$	−580.285			−0.968756			−0.484378	+	0.874859i	$0.660954\pi$
$600$		0			0
$601$			781.851i			1.30092i	0.759542	+	0.650459i	$0.225423\pi$
$601$			781.851i			1.30092i	−0.759542	+	0.650459i	$0.774577\pi$
$602$	455.498	+	246.676i	0.756641	+	0.409761i
$603$			216.262i			0.358643i
$604$	118.521			0.196226
$605$		0			0
$606$	71.5816			0.118121
$607$	907.211			1.49458			0.747290	−	0.664498i	$-0.231354\pi$
$607$	907.211			1.49458			0.747290	+	0.664498i	$0.231354\pi$
$608$	−228.129			−0.375213
$609$	−343.075	−	185.794i	−0.563342	−	0.305080i
$610$		0			0
$611$	−655.539			−1.07289
$612$	−43.0418			−0.0703298
$613$			911.642i			1.48718i	0.668636	+	0.743590i	$0.266879\pi$
$613$			911.642i			1.48718i	−0.668636	+	0.743590i	$0.733121\pi$
$614$		−	352.006i		−	0.573299i
$615$		0			0
$616$	−491.779	+	908.089i	−0.798342	+	1.47417i
$617$			637.918i			1.03390i	0.856015	+	0.516951i	$0.172933\pi$
$617$			637.918i			1.03390i	−0.856015	+	0.516951i	$0.827067\pi$
$618$			259.029i			0.419140i
$619$			862.604i			1.39354i	0.717292	+	0.696772i	$0.245381\pi$
$619$			862.604i			1.39354i	−0.717292	+	0.696772i	$0.754619\pi$
$620$		0			0
$621$		−	86.3492i		−	0.139049i
$622$	726.604			1.16817
$623$	427.951	−	790.229i	0.686920	−	1.26843i
$624$	299.394			0.479798
$625$		0			0
$626$		−	622.369i		−	0.994200i
$627$			404.866i			0.645720i
$628$	310.368			0.494216
$629$			929.310i			1.47744i
$630$		0			0
$631$	−601.057			−0.952547			−0.476273	−	0.879297i	$-0.658013\pi$
$631$	−601.057			−0.952547			−0.476273	+	0.879297i	$0.658013\pi$
$632$			694.665i			1.09915i
$633$	8.82406			0.0139401
$634$	755.752			1.19204
$635$		0			0
$636$			41.6438i			0.0654777i
$637$	437.309	+	670.213i	0.686514	+	1.05214i
$638$			936.631i			1.46807i
$639$	−77.2014			−0.120816
$640$		0			0
$641$	−884.432			−1.37977			−0.689885	−	0.723919i	$-0.742339\pi$
$641$	−884.432			−1.37977			−0.689885	+	0.723919i	$0.742339\pi$
$642$	−499.270			−0.777680
$643$	385.448			0.599453			0.299726	−	0.954025i	$-0.403105\pi$
$643$	385.448			0.599453			0.299726	+	0.954025i	$0.403105\pi$
$644$	59.1933	−	109.303i	0.0919150	−	0.169725i
$645$		0			0
$646$	316.086			0.489298
$647$	−1096.93			−1.69540			−0.847702	−	0.530473i	$-0.822014\pi$
$647$	−1096.93			−1.69540			−0.847702	+	0.530473i	$0.822014\pi$
$648$		−	78.1030i		−	0.120529i
$649$		−	1387.22i		−	2.13747i
$650$		0			0
$651$	38.9358	−	71.8966i	0.0598093	−	0.110440i
$652$			57.5488i			0.0882650i
$653$			556.888i			0.852814i	0.904531	+	0.426407i	$0.140221\pi$
$653$			556.888i			0.852814i	−0.904531	+	0.426407i	$0.859779\pi$
$654$			461.345i			0.705421i
$655$		0			0
$656$			420.594i			0.641149i
$657$	225.194			0.342761
$658$	229.083	−	423.011i	0.348151	−	0.642874i
$659$	715.578			1.08585			0.542927	−	0.839780i	$-0.317316\pi$
$659$	715.578			1.08585			0.542927	+	0.839780i	$0.317316\pi$
$660$		0			0
$661$			280.365i			0.424152i	0.977253	+	0.212076i	$0.0680225\pi$
$661$			280.365i			0.424152i	−0.977253	+	0.212076i	$0.931978\pi$
$662$		−	872.040i		−	1.31728i
$663$	379.809			0.572864
$664$			886.143i			1.33455i
$665$		0			0
$666$	−355.513			−0.533803
$667$		−	534.753i		−	0.801729i
$668$	−44.6584			−0.0668539
$669$	537.050			0.802765
$670$		0			0
$671$			254.763i			0.379676i
$672$	95.7933	−	176.886i	0.142550	−	0.263224i
$673$		−	1067.08i		−	1.58556i	−0.609506	−	0.792781i	$-0.708632\pi$
$673$		−	1067.08i		−	1.58556i	0.609506	−	0.792781i	$-0.291368\pi$
$674$	757.721			1.12421
$675$		0			0
$676$	104.434			0.154488
$677$	313.071			0.462439			0.231219	−	0.972902i	$-0.425728\pi$
$677$	313.071			0.462439			0.231219	+	0.972902i	$0.425728\pi$
$678$	−62.2656			−0.0918371
$679$	−981.456	−	531.511i	−1.44544	−	0.782785i
$680$		0			0
$681$	−188.028			−0.276105
$682$	−196.285			−0.287808
$683$		−	505.514i		−	0.740138i	−0.929004	−	0.370069i	$-0.879334\pi$
$683$		−	505.514i		−	0.740138i	0.929004	−	0.370069i	$-0.120666\pi$
$684$		−	44.0782i		−	0.0644418i
$685$		0			0
$686$	−585.301	+	47.9795i	−0.853209	+	0.0699409i
$687$			409.601i			0.596217i
$688$			457.446i			0.664892i
$689$		−	367.473i		−	0.533342i
$690$		0			0
$691$		−	276.726i		−	0.400472i	−0.979748	−	0.200236i	$-0.935829\pi$
$691$		−	276.726i		−	0.400472i	0.979748	−	0.200236i	$-0.0641709\pi$
$692$	−139.282			−0.201274
$693$	−313.924	−	170.007i	−0.452993	−	0.245320i
$694$	−845.393			−1.21815
$695$		0			0
$696$		−	483.685i		−	0.694950i
$697$			533.561i			0.765511i
$698$	−555.522			−0.795877
$699$			262.041i			0.374880i
$700$		0			0
$701$	854.178			1.21851			0.609256	−	0.792973i	$-0.291468\pi$
$701$	854.178			1.21851			0.609256	+	0.792973i	$0.291468\pi$
$702$			145.298i			0.206977i
$703$	−951.687			−1.35375
$704$	−1202.63			−1.70828
$705$		0			0
$706$		−	906.450i		−	1.28392i
$707$	−80.4626	+	148.577i	−0.113809	+	0.210152i
$708$			151.028i			0.213316i
$709$	452.996			0.638922			0.319461	−	0.947599i	$-0.396498\pi$
$709$	452.996			0.638922			0.319461	+	0.947599i	$0.396498\pi$
$710$		0			0
$711$	−240.144			−0.337755
$712$	1114.11			1.56476
$713$	112.066			0.157175
$714$	−132.727	+	245.086i	−0.185892	+	0.343258i
$715$		0			0
$716$	−48.0786			−0.0671489
$717$	83.6504			0.116667
$718$			109.964i			0.153153i
$719$			840.685i			1.16924i	0.811306	+	0.584621i	$0.198757\pi$
$719$			840.685i			1.16924i	−0.811306	+	0.584621i	$0.801243\pi$
$720$		0			0
$721$	−537.650	−	291.166i	−0.745700	−	0.403836i
$722$		−	294.386i		−	0.407737i
$723$		−	399.645i		−	0.552759i
$724$			19.1214i			0.0264108i
$725$		0			0
$726$			498.217i			0.686250i
$727$	−1339.58			−1.84262			−0.921310	−	0.388829i	$-0.872880\pi$
$727$	−1339.58			−1.84262			−0.921310	+	0.388829i	$0.872880\pi$
$728$	−472.450	+	872.398i	−0.648970	+	1.19835i
$729$	27.0000			0.0370370
$730$		0			0
$731$			580.311i			0.793860i
$732$		−	27.7363i		−	0.0378911i
$733$	−536.275			−0.731616			−0.365808	−	0.930690i	$-0.619207\pi$
$733$	−536.275			−0.731616			−0.365808	+	0.930690i	$0.619207\pi$
$734$		−	18.5206i		−	0.0252324i
$735$		0			0
$736$	275.713			0.374611
$737$			1225.49i			1.66281i
$738$	−204.117			−0.276581
$739$	898.552			1.21590			0.607951	−	0.793974i	$-0.291992\pi$
$739$	898.552			1.21590			0.607951	+	0.793974i	$0.291992\pi$
$740$		0			0
$741$			388.954i			0.524904i
$742$	237.126	+	128.416i	0.319576	+	0.173068i
$743$		−	532.720i		−	0.716985i	−0.933533	−	0.358493i	$-0.883291\pi$
$743$		−	532.720i		−	0.716985i	0.933533	−	0.358493i	$-0.116709\pi$
$744$	101.364			0.136241
$745$		0			0
$746$	−912.314			−1.22294
$747$	−306.337			−0.410090
$748$	−243.905			−0.326076
$749$	561.214	−	1036.30i	0.749284	−	1.38358i
$750$		0			0
$751$	342.085			0.455506			0.227753	−	0.973719i	$-0.426862\pi$
$751$	342.085			0.455506			0.227753	+	0.973719i	$0.426862\pi$
$752$	424.820			0.564921
$753$			150.136i			0.199384i
$754$			899.818i			1.19339i
$755$		0			0
$756$	34.1772	+	18.5088i	0.0452080	+	0.0244825i
$757$			405.961i			0.536276i	0.963381	+	0.268138i	$0.0864083\pi$
$757$			405.961i			0.536276i	−0.963381	+	0.268138i	$0.913592\pi$
$758$			883.470i			1.16553i
$759$		−	489.315i		−	0.644684i
$760$		0			0
$761$		−	577.340i		−	0.758660i	−0.925261	−	0.379330i	$-0.876155\pi$
$761$		−	577.340i		−	0.758660i	0.925261	−	0.379330i	$-0.123845\pi$
$762$	177.549			0.233004
$763$	−957.586	−	518.584i	−1.25503	−	0.679664i
$764$	190.541			0.249400
$765$		0			0
$766$		−	710.657i		−	0.927751i
$767$		−	1332.69i		−	1.73754i
$768$	327.739			0.426743
$769$		−	828.522i		−	1.07740i	−0.842497	−	0.538701i	$-0.818915\pi$
$769$		−	828.522i		−	1.07740i	0.842497	−	0.538701i	$-0.181085\pi$
$770$		0			0
$771$	−244.409			−0.317003
$772$			359.342i			0.465469i
$773$	438.991			0.567906			0.283953	−	0.958838i	$-0.408354\pi$
$773$	438.991			0.567906			0.283953	+	0.958838i	$0.408354\pi$
$774$	−222.001			−0.286824
$775$		0			0
$776$		−	1383.71i		−	1.78313i
$777$	399.621	−	737.916i	0.514312	−	0.949699i
$778$			93.8308i			0.120605i
$779$	−546.408			−0.701423
$780$		0			0
$781$	−437.477			−0.560150
$782$	−382.017			−0.488512
$783$	167.209			0.213549
$784$	−283.397	−	434.330i	−0.361476	−	0.553992i
$785$		0			0
$786$	494.849			0.629579
$787$	285.209			0.362400			0.181200	−	0.983446i	$-0.442002\pi$
$787$	285.209			0.362400			0.181200	+	0.983446i	$0.442002\pi$
$788$			52.5824i			0.0667289i
$789$			65.9740i			0.0836172i
$790$		0			0
$791$	69.9907	−	129.241i	0.0884838	−	0.163389i
$792$		−	442.586i		−	0.558821i
$793$			244.750i			0.308638i
$794$			34.1459i			0.0430049i
$795$		0			0
$796$		−	183.569i		−	0.230614i
$797$	1243.03			1.55964			0.779818	−	0.626006i	$-0.215311\pi$
$797$	1243.03			1.55964			0.779818	+	0.626006i	$0.215311\pi$
$798$	−250.987	−	135.923i	−0.314520	−	0.170330i
$799$	538.923			0.674497
$800$		0			0
$801$			385.143i			0.480828i
$802$			410.066i			0.511304i
$803$	1276.11			1.58918
$804$		−	133.420i		−	0.165946i
$805$		0			0
$806$	−188.570			−0.233958
$807$		−	786.732i		−	0.974885i
$808$	−209.472			−0.259248
$809$	630.338			0.779157			0.389578	−	0.920993i	$-0.372621\pi$
$809$	630.338			0.779157			0.389578	+	0.920993i	$0.372621\pi$
$810$		0			0
$811$			1121.08i			1.38234i	0.722692	+	0.691170i	$0.242905\pi$
$811$			1121.08i			1.38234i	−0.722692	+	0.691170i	$0.757095\pi$
$812$	211.657	+	114.623i	0.260661	+	0.141162i
$813$		−	136.329i		−	0.167687i
$814$	−2014.59			−2.47492
$815$		0			0
$816$	−246.134			−0.301635
$817$	−594.284			−0.727398
$818$	1136.08			1.38885
$819$	−301.586	−	163.325i	−0.368237	−	0.199420i
$820$		0			0
$821$	544.285			0.662954			0.331477	−	0.943463i	$-0.392453\pi$
$821$	544.285			0.662954			0.331477	+	0.943463i	$0.392453\pi$
$822$	−374.069			−0.455071
$823$		−	83.7611i		−	0.101775i	−0.998704	−	0.0508877i	$-0.983795\pi$
$823$		−	83.7611i		−	0.101775i	0.998704	−	0.0508877i	$-0.0162051\pi$
$824$		−	758.006i		−	0.919911i
$825$		0			0
$826$	859.972	+	465.721i	1.04113	+	0.563826i
$827$		−	957.378i		−	1.15765i	−0.815451	−	0.578826i	$-0.803511\pi$
$827$		−	957.378i		−	1.15765i	0.815451	−	0.578826i	$-0.196489\pi$
$828$			53.2722i			0.0643384i
$829$			9.08184i			0.0109552i	0.999985	+	0.00547759i	$0.00174358\pi$
$829$			9.08184i			0.0109552i	−0.999985	+	0.00547759i	$0.998256\pi$
$830$		0			0
$831$			147.812i			0.177873i
$832$	−1155.36			−1.38865
$833$	−359.515	−	550.987i	−0.431591	−	0.661449i
$834$	627.651			0.752579
$835$		0			0
$836$		−	249.778i		−	0.298778i
$837$			35.0411i			0.0418651i
$838$	−189.446			−0.226069
$839$			492.860i			0.587437i	0.955892	+	0.293719i	$0.0948929\pi$
$839$			492.860i			0.587437i	−0.955892	+	0.293719i	$0.905107\pi$
$840$		0			0
$841$	194.509			0.231283
$842$			892.583i			1.06007i
$843$	−289.903			−0.343894
$844$	−5.44391			−0.00645013
$845$		0			0
$846$			206.168i			0.243697i
$847$	−1034.12	−	560.030i	−1.22092	−	0.661193i
$848$			238.140i			0.280825i
$849$	259.098			0.305181
$850$		0			0
$851$	1150.19			1.35158
$852$	47.6286			0.0559021
$853$	519.198			0.608672			0.304336	−	0.952565i	$-0.401565\pi$
$853$	519.198			0.608672			0.304336	+	0.952565i	$0.401565\pi$
$854$	−157.934	−	85.5297i	−0.184934	−	0.100152i
$855$		0			0
$856$	1461.03			1.70682
$857$	−479.468			−0.559473			−0.279736	−	0.960077i	$-0.590247\pi$
$857$	−479.468			−0.559473			−0.279736	+	0.960077i	$0.590247\pi$
$858$			823.360i			0.959627i
$859$			1667.63i			1.94136i	0.240378	+	0.970679i	$0.422729\pi$
$859$			1667.63i			1.94136i	−0.240378	+	0.970679i	$0.577271\pi$
$860$		0			0
$861$	229.441	−	423.672i	0.266482	−	0.492070i
$862$		−	981.089i		−	1.13815i
$863$		−	1217.16i		−	1.41039i	−0.709016	−	0.705193i	$-0.750861\pi$
$863$		−	1217.16i		−	1.41039i	0.709016	−	0.705193i	$-0.249139\pi$
$864$			86.2112i			0.0997814i
$865$		0			0
$866$			735.776i			0.849626i
$867$	188.319			0.217208
$868$	−24.0211	+	44.3558i	−0.0276740	+	0.0511012i
$869$	−1360.82			−1.56597
$870$		0			0
$871$			1177.32i			1.35169i
$872$		−	1350.05i		−	1.54823i
$873$	478.344			0.547931
$874$		−	391.215i		−	0.447615i
$875$		0			0
$876$	−138.931			−0.158597
$877$		−	1027.18i		−	1.17125i	−0.810583	−	0.585623i	$-0.800850\pi$
$877$		−	1027.18i		−	1.17125i	0.810583	−	0.585623i	$-0.199150\pi$
$878$	−142.192			−0.161950
$879$	−252.541			−0.287305
$880$		0			0
$881$		−	149.054i		−	0.169187i	−0.996416	−	0.0845937i	$-0.973041\pi$
$881$		−	149.054i		−	0.169187i	0.996416	−	0.0845937i	$-0.0269592\pi$
$882$	210.783	−	137.535i	0.238983	−	0.155935i
$883$		−	201.243i		−	0.227908i	−0.993486	−	0.113954i	$-0.963648\pi$
$883$		−	201.243i		−	0.227908i	0.993486	−	0.113954i	$-0.0363516\pi$
$884$	−234.319			−0.265067
$885$		0			0
$886$	701.995			0.792319
$887$	−951.252			−1.07244			−0.536219	−	0.844079i	$-0.680148\pi$
$887$	−951.252			−1.07244			−0.536219	+	0.844079i	$0.680148\pi$
$888$	1040.35			1.17157
$889$	−199.577	+	368.527i	−0.224496	+	0.414542i
$890$		0			0
$891$	153.001			0.171718
$892$	−331.327			−0.371443
$893$			551.899i			0.618028i
$894$			190.334i			0.212902i
$895$		0			0
$896$	182.524	−	337.038i	0.203710	−	0.376158i
$897$		−	470.083i		−	0.524062i
$898$		−	352.613i		−	0.392664i
$899$			217.007i			0.241387i
$900$		0			0
$901$			302.102i			0.335296i
$902$	−1156.67			−1.28234
$903$	249.545	−	460.794i	0.276351	−	0.510293i
$904$	182.210			0.201560
$905$		0			0
$906$			328.921i			0.363047i
$907$		−	559.990i		−	0.617409i	−0.951158	−	0.308705i	$-0.900105\pi$
$907$		−	559.990i		−	0.617409i	0.951158	−	0.308705i	$-0.0998955\pi$
$908$	116.002			0.127755
$909$		−	72.4140i		−	0.0796633i
$910$		0			0
$911$	−674.618			−0.740525			−0.370262	−	0.928927i	$-0.620732\pi$
$911$	−674.618			−0.740525			−0.370262	+	0.928927i	$0.620732\pi$
$912$		−	252.061i		−	0.276382i
$913$	−1735.92			−1.90134
$914$	−1097.26			−1.20050
$915$		0			0
$916$		−	252.699i		−	0.275872i
$917$	−556.244	+	1027.13i	−0.606591	+	1.12009i
$918$		−	119.450i		−	0.130120i
$919$	−982.395			−1.06898			−0.534491	−	0.845174i	$-0.679497\pi$
$919$	−982.395			−1.06898			−0.534491	+	0.845174i	$0.679497\pi$
$920$		0			0
$921$	−356.099			−0.386644
$922$	284.362			0.308419
$923$	−420.283			−0.455345
$924$	193.672	+	104.884i	0.209602	+	0.113511i
$925$		0			0
$926$	17.8763			0.0193049
$927$	262.041			0.282676
$928$			533.898i			0.575322i
$929$		−	344.774i		−	0.371124i	−0.982633	−	0.185562i	$-0.940589\pi$
$929$		−	344.774i		−	0.371124i	0.982633	−	0.185562i	$-0.0594105\pi$
$930$		0			0
$931$	564.253	−	368.172i	0.606072	−	0.395458i
$932$		−	161.664i		−	0.173459i
$933$		−	735.053i		−	0.787838i
$934$		−	461.826i		−	0.494460i
$935$		0			0
$936$		−	425.191i		−	0.454264i
$937$	635.256			0.677967			0.338984	−	0.940792i	$-0.389917\pi$
$937$	635.256			0.677967			0.338984	+	0.940792i	$0.389917\pi$
$938$	−759.713	−	411.425i	−0.809929	−	0.438620i
$939$	−629.606			−0.670507
$940$		0			0
$941$		−	1207.78i		−	1.28351i	−0.766909	−	0.641756i	$-0.778206\pi$
$941$		−	1207.78i		−	1.28351i	0.766909	−	0.641756i	$-0.221794\pi$
$942$			861.338i			0.914372i
$943$	660.380			0.700297
$944$			863.649i			0.914883i
$945$		0			0
$946$	−1258.02			−1.32983
$947$		−	254.133i		−	0.268356i	−0.990957	−	0.134178i	$-0.957161\pi$
$947$		−	254.133i		−	0.268356i	0.990957	−	0.134178i	$-0.0428394\pi$
$948$	148.154			0.156281
$949$	1225.95			1.29184
$950$		0			0
$951$		−	764.540i		−	0.803933i
$952$	388.405	−	717.205i	0.407988	−	0.753367i
$953$		−	424.523i		−	0.445460i	−0.974880	−	0.222730i	$-0.928503\pi$
$953$		−	424.523i		−	0.445460i	0.974880	−	0.222730i	$-0.0714968\pi$
$954$	−115.571			−0.121143
$955$		0			0
$956$	−51.6072			−0.0539824
$957$	947.522			0.990096
$958$	1112.58			1.16136
$959$	420.479	−	776.431i	0.438455	−	0.809625i
$960$		0			0
$961$	915.523			0.952677
$962$	−1935.41			−2.01186
$963$			505.076i			0.524482i
$964$			246.557i			0.255764i
$965$		0			0
$966$	303.339	+	164.274i	0.314016	+	0.170056i
$967$		−	46.7338i		−	0.0483286i	−0.999708	−	0.0241643i	$-0.992308\pi$
$967$		−	46.7338i		−	0.0483286i	0.999708	−	0.0241643i	$-0.00769249\pi$
$968$		−	1457.95i		−	1.50615i
$969$		−	319.762i		−	0.329991i
$970$		0			0
$971$		−	1724.03i		−	1.77552i	−0.460303	−	0.887762i	$-0.652259\pi$
$971$		−	1724.03i		−	1.77552i	0.460303	−	0.887762i	$-0.347741\pi$
$972$	−16.6574			−0.0171372
$973$	−705.522	+	1302.77i	−0.725100	+	1.33893i
$974$	1023.24			1.05056
$975$		0			0
$976$		−	158.609i		−	0.162510i
$977$		−	1613.20i		−	1.65118i	−0.564271	−	0.825589i	$-0.690843\pi$
$977$		−	1613.20i		−	1.65118i	0.564271	−	0.825589i	$-0.309157\pi$
$978$	−159.711			−0.163303
$979$			2182.49i			2.22931i
$980$		0			0
$981$	466.710			0.475749
$982$			184.269i			0.187647i
$983$	−31.6909			−0.0322389			−0.0161195	−	0.999870i	$-0.505131\pi$
$983$	−31.6909			−0.0322389			−0.0161195	+	0.999870i	$0.505131\pi$
$984$	597.315			0.607028
$985$		0			0
$986$		−	739.747i		−	0.750250i
$987$	−427.930	−	231.747i	−0.433566	−	0.234799i
$988$		−	239.961i		−	0.242875i
$989$	718.242			0.726231
$990$		0			0
$991$	−236.475			−0.238623			−0.119311	−	0.992857i	$-0.538069\pi$
$991$	−236.475			−0.238623			−0.119311	+	0.992857i	$0.538069\pi$
$992$	−111.886			−0.112789
$993$	−882.180			−0.888399
$994$	146.871	−	271.204i	0.147758	−	0.272841i
$995$		0			0
$996$	188.992			0.189751
$997$	948.441			0.951295			0.475648	−	0.879636i	$-0.342214\pi$
$997$	948.441			0.951295			0.475648	+	0.879636i	$0.342214\pi$
$998$		−	720.146i		−	0.721589i
$999$			359.647i			0.360007i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.3.e.c.349.1		24
5.2	odd	4		525.3.h.d.76.8		12
5.3	odd	4		105.3.h.a.76.5	✓	12
5.4	even	2	inner	525.3.e.c.349.16		24
7.6	odd	2	inner	525.3.e.c.349.15		24
15.8	even	4		315.3.h.d.181.7		12
20.3	even	4		1680.3.s.c.1441.11		12
35.13	even	4		105.3.h.a.76.6	yes	12
35.27	even	4		525.3.h.d.76.7		12
35.34	odd	2	inner	525.3.e.c.349.2		24
105.83	odd	4		315.3.h.d.181.8		12
140.83	odd	4		1680.3.s.c.1441.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.h.a.76.5	✓	12	5.3	odd	4
105.3.h.a.76.6	yes	12	35.13	even	4
315.3.h.d.181.7		12	15.8	even	4
315.3.h.d.181.8		12	105.83	odd	4
525.3.e.c.349.1		24	1.1	even	1	trivial
525.3.e.c.349.2		24	35.34	odd	2	inner
525.3.e.c.349.15		24	7.6	odd	2	inner
525.3.e.c.349.16		24	5.4	even	2	inner
525.3.h.d.76.7		12	35.27	even	4
525.3.h.d.76.8		12	5.2	odd	4
1680.3.s.c.1441.2		12	140.83	odd	4
1680.3.s.c.1441.11		12	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	525.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.71214i q^{2} -1.73205 q^{3} +1.06857 q^{4} +2.96552i q^{6} +(-6.15534 - 3.33344i) q^{7} -8.67811i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.71214i q^{2} -1.73205 q^{3} +1.06857 q^{4} +2.96552i q^{6} +(-6.15534 - 3.33344i) q^{7} -8.67811i q^{8} +3.00000 q^{9} +17.0001 q^{11} -1.85082 q^{12} +16.3319 q^{13} +(-5.70733 + 10.5388i) q^{14} -10.5839 q^{16} -13.4266 q^{17} -5.13643i q^{18} -13.7499i q^{19} +(10.6614 + 5.77369i) q^{21} -29.1066i q^{22} +16.6179i q^{23} +15.0309i q^{24} -27.9626i q^{26} -5.19615 q^{27} +(-6.57741 - 3.56202i) q^{28} -32.1793 q^{29} -6.74366i q^{31} -16.5913i q^{32} -29.4450 q^{33} +22.9883i q^{34} +3.20571 q^{36} -69.2141i q^{37} -23.5418 q^{38} -28.2878 q^{39} -39.7391i q^{41} +(9.88538 - 18.2538i) q^{42} -43.2210i q^{43} +18.1658 q^{44} +28.4522 q^{46} -40.1384 q^{47} +18.3318 q^{48} +(26.7763 + 41.0369i) q^{49} +23.2556 q^{51} +17.4518 q^{52} -22.5002i q^{53} +8.89655i q^{54} +(-28.9280 + 53.4167i) q^{56} +23.8155i q^{57} +55.0956i q^{58} -81.6005i q^{59} +14.9859i q^{61} -11.5461 q^{62} +(-18.4660 - 10.0003i) q^{63} -70.7422 q^{64} +50.4141i q^{66} +72.0872i q^{67} -14.3473 q^{68} -28.7831i q^{69} -25.7338 q^{71} -26.0343i q^{72} +75.0647 q^{73} -118.504 q^{74} -14.6927i q^{76} +(-104.641 - 56.6689i) q^{77} +48.4327i q^{78} -80.0480 q^{79} +9.00000 q^{81} -68.0389 q^{82} -102.112 q^{83} +(11.3924 + 6.16959i) q^{84} -74.0005 q^{86} +55.7362 q^{87} -147.529i q^{88} +128.381i q^{89} +(-100.529 - 54.4416i) q^{91} +17.7574i q^{92} +11.6804i q^{93} +68.7227i q^{94} +28.7371i q^{96} +159.448 q^{97} +(70.2610 - 45.8449i) q^{98} +51.0003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 88 q^{4} + 72 q^{9}+O(q^{10}) \) 24 * q - 88 * q^4 + 72 * q^9	\( 24 q - 88 q^{4} + 72 q^{9} - 32 q^{11} + 80 q^{14} + 184 q^{16} + 72 q^{21} - 208 q^{29} - 264 q^{36} + 48 q^{39} - 384 q^{44} + 400 q^{46} - 120 q^{49} + 48 q^{51} - 736 q^{56} + 40 q^{64} + 64 q^{71} - 368 q^{74} - 240 q^{79} + 216 q^{81} - 216 q^{84} + 800 q^{86} + 48 q^{91} - 96 q^{99}+O(q^{100}) \) 24 * q - 88 * q^4 + 72 * q^9 - 32 * q^11 + 80 * q^14 + 184 * q^16 + 72 * q^21 - 208 * q^29 - 264 * q^36 + 48 * q^39 - 384 * q^44 + 400 * q^46 - 120 * q^49 + 48 * q^51 - 736 * q^56 + 40 * q^64 + 64 * q^71 - 368 * q^74 - 240 * q^79 + 216 * q^81 - 216 * q^84 + 800 * q^86 + 48 * q^91 - 96 * q^99

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